LMFDB - Embedded newform 378.4.g.c.163.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(109,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.g (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 18x^{6} + 9x^{5} + 283x^{4} - 48x^{3} + 186x^{2} + 40x + 100 $ x^8 - x^7 + 18x^6 + 9x^5 + 283x^4 - 48x^3 + 186x^2 + 40x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.3
Root			$-1.84763 - 3.20018i$ of defining polynomial
Character	$\chi$	$=$	378.163
Dual form			378.4.g.c.109.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.04834 - 8.74398i) q^{5} +(-5.48778 - 17.6885i) q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.04834 - 8.74398i) q^{5} +(-5.48778 - 17.6885i) q^{7} +8.00000 q^{8} +(10.0967 + 17.4880i) q^{10} +(-18.0748 - 31.3065i) q^{11} -4.01927 q^{13} +(36.1252 + 8.18341i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(-18.9802 - 32.8747i) q^{17} +(-28.5163 + 49.3917i) q^{19} -40.3867 q^{20} +72.2993 q^{22} +(-39.2198 + 67.9308i) q^{23} +(11.5285 + 19.9680i) q^{25} +(4.01927 - 6.96157i) q^{26} +(-50.2993 + 54.3873i) q^{28} -215.332 q^{29} +(91.1642 + 157.901i) q^{31} +(-16.0000 - 27.7128i) q^{32} +75.9208 q^{34} +(-182.372 - 41.3127i) q^{35} +(-156.279 + 270.683i) q^{37} +(-57.0327 - 98.7835i) q^{38} +(40.3867 - 69.9518i) q^{40} -186.557 q^{41} +262.266 q^{43} +(-72.2993 + 125.226i) q^{44} +(-78.4397 - 135.862i) q^{46} +(68.9606 - 119.443i) q^{47} +(-282.768 + 194.142i) q^{49} -46.1142 q^{50} +(8.03853 + 13.9231i) q^{52} +(-273.767 - 474.179i) q^{53} -364.991 q^{55} +(-43.9023 - 141.508i) q^{56} +(215.332 - 372.967i) q^{58} +(-1.86703 - 3.23378i) q^{59} +(57.5008 - 99.5943i) q^{61} -364.657 q^{62} +64.0000 q^{64} +(-20.2906 + 35.1444i) q^{65} +(-313.900 - 543.690i) q^{67} +(-75.9208 + 131.499i) q^{68} +(253.928 - 274.566i) q^{70} -533.107 q^{71} +(149.197 + 258.417i) q^{73} +(-312.557 - 541.365i) q^{74} +228.131 q^{76} +(-454.576 + 491.521i) q^{77} +(433.508 - 750.857i) q^{79} +(80.7734 + 139.904i) q^{80} +(186.557 - 323.126i) q^{82} +244.224 q^{83} -383.274 q^{85} +(-262.266 + 454.258i) q^{86} +(-144.599 - 250.452i) q^{88} +(223.508 - 387.128i) q^{89} +(22.0569 + 71.0949i) q^{91} +313.759 q^{92} +(137.921 + 238.887i) q^{94} +(287.920 + 498.692i) q^{95} -1803.99 q^{97} +(-53.4948 - 683.911i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8	$ 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8} + 4 q^{10} - 32 q^{11} - 4 q^{13} - 36 q^{14} - 64 q^{16} + 58 q^{17} + 70 q^{19} - 16 q^{20} + 128 q^{22} - 86 q^{23} - 156 q^{25} + 4 q^{26} + 48 q^{28} + 212 q^{29} - 64 q^{31} - 128 q^{32} - 232 q^{34} - 8 q^{35} - 146 q^{37} + 140 q^{38} + 16 q^{40} + 780 q^{41} + 880 q^{43} - 128 q^{44} - 172 q^{46} + 306 q^{47} + 50 q^{49} + 624 q^{50} + 8 q^{52} + 90 q^{53} - 64 q^{55} + 48 q^{56} - 212 q^{58} - 148 q^{59} - 364 q^{61} + 256 q^{62} + 512 q^{64} - 1296 q^{65} - 954 q^{67} + 232 q^{68} + 20 q^{70} + 1360 q^{71} - 54 q^{73} - 292 q^{74} - 560 q^{76} - 2224 q^{77} - 226 q^{79} + 32 q^{80} - 780 q^{82} + 3136 q^{83} + 3920 q^{85} - 880 q^{86} - 256 q^{88} + 1458 q^{89} + 3836 q^{91} + 688 q^{92} + 612 q^{94} - 1310 q^{95} - 4344 q^{97} - 776 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8 + 4 * q^10 - 32 * q^11 - 4 * q^13 - 36 * q^14 - 64 * q^16 + 58 * q^17 + 70 * q^19 - 16 * q^20 + 128 * q^22 - 86 * q^23 - 156 * q^25 + 4 * q^26 + 48 * q^28 + 212 * q^29 - 64 * q^31 - 128 * q^32 - 232 * q^34 - 8 * q^35 - 146 * q^37 + 140 * q^38 + 16 * q^40 + 780 * q^41 + 880 * q^43 - 128 * q^44 - 172 * q^46 + 306 * q^47 + 50 * q^49 + 624 * q^50 + 8 * q^52 + 90 * q^53 - 64 * q^55 + 48 * q^56 - 212 * q^58 - 148 * q^59 - 364 * q^61 + 256 * q^62 + 512 * q^64 - 1296 * q^65 - 954 * q^67 + 232 * q^68 + 20 * q^70 + 1360 * q^71 - 54 * q^73 - 292 * q^74 - 560 * q^76 - 2224 * q^77 - 226 * q^79 + 32 * q^80 - 780 * q^82 + 3136 * q^83 + 3920 * q^85 - 880 * q^86 - 256 * q^88 + 1458 * q^89 + 3836 * q^91 + 688 * q^92 + 612 * q^94 - 1310 * q^95 - 4344 * q^97 - 776 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	5.04834	−	8.74398i	0.451537	−	0.782085i	−0.546945	−	0.837169i	$-0.684209\pi$
$5$	5.04834	−	8.74398i	0.451537	−	0.782085i	0.998482	+	0.0550835i	$0.0175425\pi$
$6$		0			0
$7$	−5.48778	−	17.6885i	−0.296312	−	0.955091i
$7$	−5.48778	−	17.6885i	−0.296312	−	0.955091i
$8$	8.00000			0.353553
$9$		0			0
$10$	10.0967	+	17.4880i	0.319285	+	0.553018i
$11$	−18.0748	−	31.3065i	−0.495433	−	0.858116i	0.504553	−	0.863381i	$-0.331657\pi$
$11$	−18.0748	−	31.3065i	−0.495433	−	0.858116i	−0.999986	+	0.00526523i	$0.998324\pi$
$12$		0			0
$13$	−4.01927			−0.0857495			−0.0428748	−	0.999080i	$-0.513652\pi$
$13$	−4.01927			−0.0857495			−0.0428748	+	0.999080i	$0.513652\pi$
$14$	36.1252	+	8.18341i	0.689634	+	0.156222i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	−18.9802	−	32.8747i	−0.270787	−	0.469017i	0.698277	−	0.715828i	$-0.253951\pi$
$17$	−18.9802	−	32.8747i	−0.270787	−	0.469017i	−0.969064	+	0.246811i	$0.920617\pi$
$18$		0			0
$19$	−28.5163	+	49.3917i	−0.344321	+	0.596381i	−0.985230	−	0.171236i	$-0.945224\pi$
$19$	−28.5163	+	49.3917i	−0.344321	+	0.596381i	0.640909	+	0.767617i	$0.278557\pi$
$20$	−40.3867			−0.451537
$21$		0			0
$22$	72.2993			0.700648
$23$	−39.2198	+	67.9308i	−0.355561	+	0.615850i	−0.987214	−	0.159402i	$-0.949044\pi$
$23$	−39.2198	+	67.9308i	−0.355561	+	0.615850i	0.631653	+	0.775251i	$0.282377\pi$
$24$		0			0
$25$	11.5285	+	19.9680i	0.0922284	+	0.159744i
$26$	4.01927	−	6.96157i	0.0303170	−	0.0525107i
$27$		0			0
$28$	−50.2993	+	54.3873i	−0.339488	+	0.367080i
$29$	−215.332			−1.37884			−0.689418	−	0.724364i	$-0.742134\pi$
$29$	−215.332			−1.37884			−0.689418	+	0.724364i	$0.742134\pi$
$30$		0			0
$31$	91.1642	+	157.901i	0.528180	+	0.914834i	0.999460	+	0.0328507i	$0.0104586\pi$
$31$	91.1642	+	157.901i	0.528180	+	0.914834i	−0.471281	+	0.881983i	$0.656208\pi$
$32$	−16.0000	−	27.7128i	−0.0883883	−	0.153093i
$33$		0			0
$34$	75.9208			0.382950
$35$	−182.372	−	41.3127i	−0.880759	−	0.199518i
$36$		0			0
$37$	−156.279	+	270.683i	−0.694380	+	1.20270i	0.276010	+	0.961155i	$0.410988\pi$
$37$	−156.279	+	270.683i	−0.694380	+	1.20270i	−0.970389	+	0.241546i	$0.922346\pi$
$38$	−57.0327	−	98.7835i	−0.243472	−	0.421705i
$39$		0			0
$40$	40.3867	−	69.9518i	0.159642	−	0.276509i
$41$	−186.557			−0.710617			−0.355308	−	0.934749i	$-0.615624\pi$
$41$	−186.557			−0.710617			−0.355308	+	0.934749i	$0.615624\pi$
$42$		0			0
$43$	262.266			0.930121			0.465061	−	0.885279i	$-0.346033\pi$
$43$	262.266			0.930121			0.465061	+	0.885279i	$0.346033\pi$
$44$	−72.2993	+	125.226i	−0.247717	+	0.429058i
$45$		0			0
$46$	−78.4397	−	135.862i	−0.251420	−	0.435472i
$47$	68.9606	−	119.443i	0.214020	−	0.370693i	−0.738949	−	0.673761i	$-0.764678\pi$
$47$	68.9606	−	119.443i	0.214020	−	0.370693i	0.952969	+	0.303068i	$0.0980109\pi$
$48$		0			0
$49$	−282.768	+	194.142i	−0.824398	+	0.566011i
$50$	−46.1142			−0.130431
$51$		0			0
$52$	8.03853	+	13.9231i	0.0214374	+	0.0371306i
$53$	−273.767	−	474.179i	−0.709526	−	1.22893i	−0.965033	−	0.262127i	$-0.915576\pi$
$53$	−273.767	−	474.179i	−0.709526	−	1.22893i	0.255508	−	0.966807i	$-0.417757\pi$
$54$		0			0
$55$	−364.991			−0.894826
$56$	−43.9023	−	141.508i	−0.104762	−	0.337676i
$57$		0			0
$58$	215.332	−	372.967i	0.487492	−	0.844361i
$59$	−1.86703	−	3.23378i	−0.00411976	−	0.00713564i	0.863958	−	0.503564i	$-0.167978\pi$
$59$	−1.86703	−	3.23378i	−0.00411976	−	0.00713564i	−0.868078	+	0.496428i	$0.834645\pi$
$60$		0			0
$61$	57.5008	−	99.5943i	0.120692	−	0.209045i	−0.799349	−	0.600868i	$-0.794822\pi$
$61$	57.5008	−	99.5943i	0.120692	−	0.209045i	0.920041	+	0.391822i	$0.128155\pi$
$62$	−364.657			−0.746959
$63$		0			0
$64$	64.0000			0.125000
$65$	−20.2906	+	35.1444i	−0.0387191	+	0.0670635i
$66$		0			0
$67$	−313.900	−	543.690i	−0.572373	−	0.991378i	−0.996322	−	0.0856924i	$-0.972690\pi$
$67$	−313.900	−	543.690i	−0.572373	−	0.991378i	0.423949	−	0.905686i	$-0.360644\pi$
$68$	−75.9208	+	131.499i	−0.135393	+	0.234508i
$69$		0			0
$70$	253.928	−	274.566i	0.433574	−	0.468812i
$71$	−533.107			−0.891100			−0.445550	−	0.895257i	$-0.646992\pi$
$71$	−533.107			−0.891100			−0.445550	+	0.895257i	$0.646992\pi$
$72$		0			0
$73$	149.197	+	258.417i	0.239208	+	0.414320i	0.960487	−	0.278324i	$-0.0897789\pi$
$73$	149.197	+	258.417i	0.239208	+	0.414320i	−0.721279	+	0.692644i	$0.756446\pi$
$74$	−312.557	−	541.365i	−0.491001	−	0.850438i
$75$		0			0
$76$	228.131			0.344321
$77$	−454.576	+	491.521i	−0.672775	+	0.727454i
$78$		0			0
$79$	433.508	−	750.857i	0.617385	−	1.06934i	−0.372576	−	0.928002i	$-0.621525\pi$
$79$	433.508	−	750.857i	0.617385	−	1.06934i	0.989961	−	0.141340i	$-0.0451412\pi$
$80$	80.7734	+	139.904i	0.112884	+	0.195521i
$81$		0			0
$82$	186.557	−	323.126i	0.251241	−	0.435162i
$83$	244.224			0.322977			0.161489	−	0.986875i	$-0.448370\pi$
$83$	244.224			0.322977			0.161489	+	0.986875i	$0.448370\pi$
$84$		0			0
$85$	−383.274			−0.489081
$86$	−262.266	+	454.258i	−0.328847	+	0.569580i
$87$		0			0
$88$	−144.599	−	250.452i	−0.175162	−	0.303390i
$89$	223.508	−	387.128i	0.266200	−	0.461073i	−0.701677	−	0.712495i	$-0.747565\pi$
$89$	223.508	−	387.128i	0.266200	−	0.461073i	0.967877	+	0.251423i	$0.0808984\pi$
$90$		0			0
$91$	22.0569	+	71.0949i	0.0254087	+	0.0818986i
$92$	313.759			0.355561
$93$		0			0
$94$	137.921	+	238.887i	0.151335	+	0.262120i
$95$	287.920	+	498.692i	0.310947	+	0.538576i
$96$		0			0
$97$	−1803.99			−1.88832			−0.944162	−	0.329481i	$-0.893126\pi$
$97$	−1803.99			−1.88832			−0.944162	+	0.329481i	$0.893126\pi$
$98$	−53.4948	−	683.911i	−0.0551407	−	0.704954i
$99$		0			0
$100$	46.1142	−	79.8721i	0.0461142	−	0.0798721i
$101$	−842.134	−	1458.62i	−0.829658	−	1.43701i	−0.898307	−	0.439369i	$-0.855202\pi$
$101$	−842.134	−	1458.62i	−0.829658	−	1.43701i	0.0686484	−	0.997641i	$-0.478131\pi$
$102$		0			0
$103$	−820.532	+	1421.20i	−0.784946	+	1.35957i	0.144085	+	0.989565i	$0.453976\pi$
$103$	−820.532	+	1421.20i	−0.784946	+	1.35957i	−0.929031	+	0.370001i	$0.879357\pi$
$104$	−32.1541			−0.0303170
$105$		0			0
$106$	1095.07			1.00342
$107$	−160.547	+	278.076i	−0.145053	+	0.251239i	−0.929393	−	0.369092i	$-0.879669\pi$
$107$	−160.547	+	278.076i	−0.145053	+	0.251239i	0.784340	+	0.620332i	$0.213002\pi$
$108$		0			0
$109$	−280.005	−	484.982i	−0.246051	−	0.426173i	0.716376	−	0.697715i	$-0.245800\pi$
$109$	−280.005	−	484.982i	−0.246051	−	0.426173i	−0.962427	+	0.271542i	$0.912466\pi$
$110$	364.991	−	632.184i	0.316369	−	0.547967i
$111$		0			0
$112$	289.002	+	65.4673i	0.243822	+	0.0552329i
$113$	1870.65			1.55731			0.778655	−	0.627452i	$-0.215902\pi$
$113$	1870.65			1.55731			0.778655	+	0.627452i	$0.215902\pi$
$114$		0			0
$115$	395.990	+	685.875i	0.321098	+	0.556158i
$116$	430.665	+	745.934i	0.344709	+	0.597053i
$117$		0			0
$118$	7.46811			0.00582623
$119$	−477.346	+	516.141i	−0.367716	+	0.397602i
$120$		0			0
$121$	12.1012	−	20.9598i	0.00909179	−	0.0157474i
$122$	115.002	+	199.189i	0.0853423	+	0.147817i
$123$		0			0
$124$	364.657	−	631.604i	0.264090	−	0.457417i
$125$	1494.88			1.06965
$126$		0			0
$127$	790.048			0.552011			0.276006	−	0.961156i	$-0.410989\pi$
$127$	790.048			0.552011			0.276006	+	0.961156i	$0.410989\pi$
$128$	−64.0000	+	110.851i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−40.5812	−	70.2888i	−0.0273785	−	0.0474210i
$131$	−819.041	+	1418.62i	−0.546259	+	0.946148i	0.452267	+	0.891882i	$0.350615\pi$
$131$	−819.041	+	1418.62i	−0.546259	+	0.946148i	−0.998527	+	0.0542661i	$0.982718\pi$
$132$		0			0
$133$	1030.16	+	233.361i	0.671625	+	0.152143i
$134$	1255.60			0.809457
$135$		0			0
$136$	−151.842	−	262.997i	−0.0957376	−	0.165822i
$137$	−994.660	−	1722.80i	−0.620289	−	1.07437i	−0.989432	−	0.144999i	$-0.953682\pi$
$137$	−994.660	−	1722.80i	−0.620289	−	1.07437i	0.369143	−	0.929373i	$-0.379651\pi$
$138$		0			0
$139$	−655.628			−0.400069			−0.200035	−	0.979789i	$-0.564105\pi$
$139$	−655.628			−0.400069			−0.200035	+	0.979789i	$0.564105\pi$
$140$	221.634	+	714.382i	0.133796	+	0.431259i
$141$		0			0
$142$	533.107	−	923.368i	0.315052	−	0.545685i
$143$	72.6475	+	125.829i	0.0424832	+	0.0735830i
$144$		0			0
$145$	−1087.07	+	1882.86i	−0.622596	+	1.07837i
$146$	−596.788			−0.338291
$147$		0			0
$148$	1250.23			0.694380
$149$	−1494.77	+	2589.01i	−0.821853	+	1.42349i	0.0824482	+	0.996595i	$0.473726\pi$
$149$	−1494.77	+	2589.01i	−0.821853	+	1.42349i	−0.904301	+	0.426895i	$0.859607\pi$
$150$		0			0
$151$	547.979	+	949.128i	0.295324	+	0.511516i	0.975060	−	0.221940i	$-0.0712391\pi$
$151$	547.979	+	949.128i	0.295324	+	0.511516i	−0.679736	+	0.733457i	$0.737906\pi$
$152$	−228.131	+	395.134i	−0.121736	+	0.210853i
$153$		0			0
$154$	−396.763	−	1278.87i	−0.207611	−	0.669183i
$155$	1840.91			0.953971
$156$		0			0
$157$	−250.513	−	433.901i	−0.127345	−	0.220567i	0.795302	−	0.606213i	$-0.207312\pi$
$157$	−250.513	−	433.901i	−0.127345	−	0.220567i	−0.922647	+	0.385646i	$0.873979\pi$
$158$	867.015	+	1501.71i	0.436557	+	0.756139i
$159$		0			0
$160$	−323.094			−0.159642
$161$	1416.83	+	320.952i	0.693550	+	0.157109i
$162$		0			0
$163$	−1134.21	+	1964.50i	−0.545018	+	0.943998i	0.453588	+	0.891211i	$0.350144\pi$
$163$	−1134.21	+	1964.50i	−0.545018	+	0.943998i	−0.998606	+	0.0527870i	$0.983190\pi$
$164$	373.114	+	646.252i	0.177654	+	0.307706i
$165$		0			0
$166$	−244.224	+	423.009i	−0.114190	+	0.197782i
$167$	1369.97			0.634799			0.317400	−	0.948292i	$-0.397190\pi$
$167$	1369.97			0.634799			0.317400	+	0.948292i	$0.397190\pi$
$168$		0			0
$169$	−2180.85			−0.992647
$170$	383.274	−	663.850i	0.172916	−	0.299500i
$171$		0			0
$172$	−524.532	−	908.516i	−0.232530	−	0.402754i
$173$	921.885	−	1596.75i	0.405142	−	0.701727i	−0.589196	−	0.807990i	$-0.700555\pi$
$173$	921.885	−	1596.75i	0.405142	−	0.701727i	0.994338	+	0.106264i	$0.0338887\pi$
$174$		0			0
$175$	289.939	−	313.503i	0.125242	−	0.135421i
$176$	578.395			0.247717
$177$		0			0
$178$	447.017	+	774.255i	0.188232	+	0.326028i
$179$	530.785	+	919.346i	0.221635	+	0.383884i	0.955305	−	0.295623i	$-0.0955272\pi$
$179$	530.785	+	919.346i	0.221635	+	0.383884i	−0.733669	+	0.679507i	$0.762194\pi$
$180$		0			0
$181$	2441.54			1.00264			0.501320	−	0.865262i	$-0.332848\pi$
$181$	2441.54			1.00264			0.501320	+	0.865262i	$0.332848\pi$
$182$	−145.197	−	32.8913i	−0.0591358	−	0.0133960i
$183$		0			0
$184$	−313.759	+	543.446i	−0.125710	+	0.217736i
$185$	1577.90	+	2732.99i	0.627077	+	1.08613i
$186$		0			0
$187$	−686.128	+	1188.41i	−0.268314	+	0.464733i
$188$	−551.685			−0.214020
$189$		0			0
$190$	−1151.68			−0.439746
$191$	1846.08	−	3197.50i	0.699359	−	1.21132i	−0.269331	−	0.963048i	$-0.586802\pi$
$191$	1846.08	−	3197.50i	0.699359	−	1.21132i	0.968689	−	0.248277i	$-0.0798642\pi$
$192$		0			0
$193$	−1571.16	−	2721.33i	−0.585982	−	1.01495i	−0.994752	−	0.102313i	$-0.967376\pi$
$193$	−1571.16	−	2721.33i	−0.585982	−	1.01495i	0.408770	−	0.912637i	$-0.365958\pi$
$194$	1803.99	−	3124.60i	0.667623	−	1.15636i
$195$		0			0
$196$	1238.06	+	591.255i	0.451189	+	0.215472i
$197$	2273.51			0.822236			0.411118	−	0.911582i	$-0.365138\pi$
$197$	2273.51			0.822236			0.411118	+	0.911582i	$0.365138\pi$
$198$		0			0
$199$	1368.26	+	2369.89i	0.487403	+	0.844206i	0.999895	−	0.0144854i	$-0.00461100\pi$
$199$	1368.26	+	2369.89i	0.487403	+	0.844206i	−0.512492	+	0.858692i	$0.671278\pi$
$200$	92.2284	+	159.744i	0.0326076	+	0.0564781i
$201$		0			0
$202$	3368.54			1.17331
$203$	1181.70	+	3808.92i	0.408566	+	1.31691i
$204$		0			0
$205$	−941.803	+	1631.25i	−0.320870	+	0.555763i
$206$	−1641.06	−	2842.41i	−0.555041	−	0.961359i
$207$		0			0
$208$	32.1541	−	55.6926i	0.0107187	−	0.0185653i
$209$	2061.71			0.682352
$210$		0			0
$211$	2915.84			0.951349			0.475675	−	0.879621i	$-0.342204\pi$
$211$	2915.84			0.951349			0.475675	+	0.879621i	$0.342204\pi$
$212$	−1095.07	+	1896.72i	−0.354763	+	0.614467i
$213$		0			0
$214$	−321.094	−	556.151i	−0.102568	−	0.177653i
$215$	1324.01	−	2293.25i	0.419984	−	0.727434i
$216$		0			0
$217$	2292.75	−	2479.09i	0.717244	−	0.775536i
$218$	1120.02			0.347969
$219$		0			0
$220$	729.983	+	1264.37i	0.223707	+	0.387471i
$221$	76.2865	+	132.132i	0.0232198	+	0.0402179i
$222$		0			0
$223$	−1875.11			−0.563078			−0.281539	−	0.959550i	$-0.590845\pi$
$223$	−1875.11			−0.563078			−0.281539	+	0.959550i	$0.590845\pi$
$224$	−402.395	+	435.098i	−0.120027	+	0.129782i
$225$		0			0
$226$	−1870.65	+	3240.06i	−0.550592	+	0.953654i
$227$	−2019.15	−	3497.27i	−0.590377	−	1.02256i	−0.994182	−	0.107717i	$-0.965646\pi$
$227$	−2019.15	−	3497.27i	−0.590377	−	1.02256i	0.403805	−	0.914845i	$-0.367688\pi$
$228$		0			0
$229$	−686.456	+	1188.98i	−0.198089	+	0.343099i	−0.947909	−	0.318542i	$-0.896807\pi$
$229$	−686.456	+	1188.98i	−0.198089	+	0.343099i	0.749820	+	0.661642i	$0.230140\pi$
$230$	−1583.96			−0.454101
$231$		0			0
$232$	−1722.66			−0.487492
$233$	−2087.15	+	3615.06i	−0.586841	+	1.01644i	0.407802	+	0.913070i	$0.366295\pi$
$233$	−2087.15	+	3615.06i	−0.586841	+	1.01644i	−0.994643	+	0.103368i	$0.967038\pi$
$234$		0			0
$235$	−696.273	−	1205.98i	−0.193276	−	0.334764i
$236$	−7.46811	+	12.9351i	−0.00205988	+	0.00356782i
$237$		0			0
$238$	−416.637	−	1342.93i	−0.113473	−	0.365753i
$239$	5544.70			1.50066			0.750329	−	0.661065i	$-0.229895\pi$
$239$	5544.70			1.50066			0.750329	+	0.661065i	$0.229895\pi$
$240$		0			0
$241$	−2445.07	−	4234.98i	−0.653530	−	1.13195i	−0.982260	−	0.187524i	$-0.939954\pi$
$241$	−2445.07	−	4234.98i	−0.653530	−	1.13195i	0.328730	−	0.944424i	$-0.393379\pi$
$242$	24.2023	+	41.9197i	0.00642887	+	0.0111351i
$243$		0			0
$244$	−460.007			−0.120692
$245$	270.060	+	3452.61i	0.0704224	+	0.900324i
$246$		0			0
$247$	114.615	−	198.519i	0.0295253	−	0.0511394i
$248$	729.313	+	1263.21i	0.186740	+	0.323443i
$249$		0			0
$250$	−1494.88	+	2589.22i	−0.378179	+	0.655026i
$251$	−7560.15			−1.90117			−0.950583	−	0.310470i	$-0.899513\pi$
$251$	−7560.15			−1.90117			−0.950583	+	0.310470i	$0.899513\pi$
$252$		0			0
$253$	2835.57			0.704627
$254$	−790.048	+	1368.40i	−0.195165	+	0.338037i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	−1616.43	+	2799.74i	−0.392335	+	0.679545i	−0.992757	−	0.120139i	$-0.961666\pi$
$257$	−1616.43	+	2799.74i	−0.392335	+	0.679545i	0.600422	+	0.799683i	$0.294999\pi$
$258$		0			0
$259$	5645.60	+	1278.89i	1.35444	+	0.306821i
$260$	162.325			0.0387191
$261$		0			0
$262$	−1638.08	−	2837.24i	−0.386263	−	0.669028i
$263$	−2142.26	−	3710.51i	−0.502272	−	0.869961i	−0.999997	−	0.00262575i	$-0.999164\pi$
$263$	−2142.26	−	3710.51i	−0.502272	−	0.869961i	0.497724	−	0.867335i	$-0.334169\pi$
$264$		0			0
$265$	−5528.28			−1.28151
$266$	−1434.35	+	1550.93i	−0.330623	+	0.357494i
$267$		0			0
$268$	−1255.60	+	2174.76i	−0.286186	+	0.495689i
$269$	−3348.26	−	5799.36i	−0.758912	−	1.31447i	−0.943406	−	0.331640i	$-0.892398\pi$
$269$	−3348.26	−	5799.36i	−0.758912	−	1.31447i	0.184495	−	0.982834i	$-0.440935\pi$
$270$		0			0
$271$	3157.89	−	5469.63i	0.707854	−	1.22604i	−0.257798	−	0.966199i	$-0.582997\pi$
$271$	3157.89	−	5469.63i	0.707854	−	1.22604i	0.965652	−	0.259840i	$-0.0836698\pi$
$272$	607.367			0.135393
$273$		0			0
$274$	3978.64			0.877221
$275$	416.753	−	721.837i	0.0913860	−	0.158285i
$276$		0			0
$277$	−1377.62	−	2386.12i	−0.298821	−	0.517573i	0.677045	−	0.735941i	$-0.263260\pi$
$277$	−1377.62	−	2386.12i	−0.298821	−	0.517573i	−0.975866	+	0.218368i	$0.929927\pi$
$278$	655.628	−	1135.58i	0.141446	−	0.244991i
$279$		0			0
$280$	−1458.98	−	330.501i	−0.311395	−	0.0705401i
$281$	−1453.00			−0.308464			−0.154232	−	0.988035i	$-0.549290\pi$
$281$	−1453.00			−0.308464			−0.154232	+	0.988035i	$0.549290\pi$
$282$		0			0
$283$	−2715.92	−	4704.11i	−0.570476	−	0.988093i	−0.996517	−	0.0833890i	$-0.973426\pi$
$283$	−2715.92	−	4704.11i	−0.570476	−	0.988093i	0.426042	−	0.904704i	$-0.359908\pi$
$284$	1066.21	+	1846.74i	0.222775	+	0.385858i
$285$		0			0
$286$	−290.590			−0.0600803
$287$	1023.78	+	3299.92i	0.210565	+	0.678704i
$288$		0			0
$289$	1736.00	−	3006.85i	0.353349	−	0.612018i
$290$	−2174.14	−	3765.73i	−0.440242	−	0.762521i
$291$		0			0
$292$	596.788	−	1033.67i	0.119604	−	0.207160i
$293$	−6227.21			−1.24163			−0.620815	−	0.783957i	$-0.713198\pi$
$293$	−6227.21			−1.24163			−0.620815	+	0.783957i	$0.713198\pi$
$294$		0			0
$295$	−37.7015			−0.00744091
$296$	−1250.23	+	2165.46i	−0.245500	+	0.425219i
$297$		0			0
$298$	−2989.53	−	5178.02i	−0.581138	−	1.00656i
$299$	157.635	−	273.032i	0.0304892	−	0.0528088i
$300$		0			0
$301$	−1439.26	−	4639.10i	−0.275606	−	0.888350i
$302$	−2191.92			−0.417651
$303$		0			0
$304$	−456.261	−	790.268i	−0.0860802	−	0.149095i
$305$	−580.567	−	1005.57i	−0.108994	−	0.188783i
$306$		0			0
$307$	8621.61			1.60281			0.801403	−	0.598125i	$-0.204088\pi$
$307$	8621.61			1.60281			0.801403	+	0.598125i	$0.204088\pi$
$308$	2611.83	+	591.655i	0.483191	+	0.109457i
$309$		0			0
$310$	−1840.91	+	3188.55i	−0.337280	+	0.584186i
$311$	−3712.11	−	6429.56i	−0.676831	−	1.17230i	−0.975930	−	0.218083i	$-0.930020\pi$
$311$	−3712.11	−	6429.56i	−0.676831	−	1.17230i	0.299100	−	0.954222i	$-0.403314\pi$
$312$		0			0
$313$	−1633.30	+	2828.96i	−0.294951	+	0.510870i	−0.974974	−	0.222321i	$-0.928637\pi$
$313$	−1633.30	+	2828.96i	−0.294951	+	0.510870i	0.680022	+	0.733191i	$0.261970\pi$
$314$	1002.05			0.180092
$315$		0			0
$316$	−3468.06			−0.617385
$317$	−2440.25	+	4226.63i	−0.432359	+	0.748868i	−0.997076	−	0.0764167i	$-0.975652\pi$
$317$	−2440.25	+	4226.63i	−0.432359	+	0.748868i	0.564717	+	0.825285i	$0.308985\pi$
$318$		0			0
$319$	3892.10	+	6741.31i	0.683121	+	1.18320i
$320$	323.094	−	559.615i	0.0564421	−	0.0977607i
$321$		0			0
$322$	−1972.73	+	2133.06i	−0.341416	+	0.369164i
$323$	2164.98			0.372950
$324$		0			0
$325$	−46.3363	−	80.2568i	−0.00790854	−	0.0136980i
$326$	−2268.41	−	3929.01i	−0.385386	−	0.667508i
$327$		0			0
$328$	−1492.46			−0.251241
$329$	−2491.22	−	564.333i	−0.417463	−	0.0945675i
$330$		0			0
$331$	4799.58	−	8313.11i	0.797005	−	1.38045i	−0.124554	−	0.992213i	$-0.539750\pi$
$331$	4799.58	−	8313.11i	0.797005	−	1.38045i	0.921558	−	0.388240i	$-0.126917\pi$
$332$	−488.449	−	846.018i	−0.0807443	−	0.139853i
$333$		0			0
$334$	−1369.97	+	2372.86i	−0.224436	+	0.388734i
$335$	−6338.69			−1.03379
$336$		0			0
$337$	−2895.79			−0.468082			−0.234041	−	0.972227i	$-0.575195\pi$
$337$	−2895.79			−0.468082			−0.234041	+	0.972227i	$0.575195\pi$
$338$	2180.85	−	3777.34i	0.350954	−	0.607870i
$339$		0			0
$340$	766.548	+	1327.70i	0.122270	+	0.211778i
$341$	3295.55	−	5708.07i	0.523356	−	0.906478i
$342$		0			0
$343$	4985.85	+	3936.35i	0.784871	+	0.619659i
$344$	2098.13			0.328847
$345$		0			0
$346$	1843.77	+	3193.50i	0.286479	+	0.496196i
$347$	−2678.81	−	4639.83i	−0.414426	−	0.717807i	0.580942	−	0.813945i	$-0.302684\pi$
$347$	−2678.81	−	4639.83i	−0.414426	−	0.717807i	−0.995368	+	0.0961381i	$0.969351\pi$
$348$		0			0
$349$	−6049.21			−0.927813			−0.463906	−	0.885884i	$-0.653553\pi$
$349$	−6049.21			−0.927813			−0.463906	+	0.885884i	$0.653553\pi$
$350$	253.065	+	815.692i	0.0386482	+	0.124573i
$351$		0			0
$352$	−578.395	+	1001.81i	−0.0875811	+	0.151695i
$353$	−3440.28	−	5958.74i	−0.518718	−	0.898446i	−0.999763	−	0.0217505i	$-0.993076\pi$
$353$	−3440.28	−	5958.74i	−0.518718	−	0.898446i	0.481045	−	0.876696i	$-0.340257\pi$
$354$		0			0
$355$	−2691.30	+	4661.47i	−0.402365	+	0.696916i
$356$	−1788.07			−0.266200
$357$		0			0
$358$	−2123.14			−0.313440
$359$	−2995.95	+	5189.15i	−0.440447	+	0.762876i	−0.997723	−	0.0674511i	$-0.978513\pi$
$359$	−2995.95	+	5189.15i	−0.440447	+	0.762876i	0.557276	+	0.830328i	$0.311847\pi$
$360$		0			0
$361$	1803.14	+	3123.13i	0.262886	+	0.455333i
$362$	−2441.54	+	4228.86i	−0.354487	+	0.613989i
$363$		0			0
$364$	202.166	−	218.597i	0.0291110	−	0.0314769i
$365$	3012.79			0.432045
$366$		0			0
$367$	3285.52	+	5690.70i	0.467311	+	0.809406i	0.999302	−	0.0373437i	$-0.0118896\pi$
$367$	3285.52	+	5690.70i	0.467311	+	0.809406i	−0.531992	+	0.846749i	$0.678556\pi$
$368$	−627.518	−	1086.89i	−0.0888903	−	0.153962i
$369$		0			0
$370$	−6311.58			−0.886820
$371$	−6885.16	+	7444.74i	−0.963503	+	1.04181i
$372$		0			0
$373$	1866.81	−	3233.41i	0.259141	−	0.448846i	−0.706871	−	0.707343i	$-0.749894\pi$
$373$	1866.81	−	3233.41i	0.259141	−	0.448846i	0.966012	+	0.258497i	$0.0832272\pi$
$374$	−1372.26	−	2376.82i	−0.189726	−	0.328616i
$375$		0			0
$376$	551.685	−	955.546i	0.0756675	−	0.131060i
$377$	865.479			0.118235
$378$		0			0
$379$	8419.14			1.14106			0.570531	−	0.821276i	$-0.306738\pi$
$379$	8419.14			1.14106			0.570531	+	0.821276i	$0.306738\pi$
$380$	1151.68	−	1994.77i	0.155474	−	0.269288i
$381$		0			0
$382$	3692.15	+	6395.00i	0.494521	+	0.856536i
$383$	2665.21	−	4616.28i	0.355577	−	0.615877i	−0.631640	−	0.775262i	$-0.717618\pi$
$383$	2665.21	−	4616.28i	0.355577	−	0.615877i	0.987217	+	0.159385i	$0.0509511\pi$
$384$		0			0
$385$	2002.99	+	6456.16i	0.265148	+	0.854640i
$386$	6284.64			0.828704
$387$		0			0
$388$	3607.98	+	6249.21i	0.472081	+	0.817668i
$389$	3905.58	+	6764.67i	0.509051	+	0.881703i	0.999945	+	0.0104832i	$0.00333696\pi$
$389$	3905.58	+	6764.67i	0.509051	+	0.881703i	−0.490894	+	0.871219i	$0.663330\pi$
$390$		0			0
$391$	2977.60			0.385125
$392$	−2262.15	+	1553.13i	−0.291469	+	0.200115i
$393$		0			0
$394$	−2273.51	+	3937.83i	−0.290704	+	0.503515i
$395$	−4376.99	−	7581.16i	−0.557544	−	0.965695i
$396$		0			0
$397$	−455.847	+	789.550i	−0.0576279	+	0.0998145i	−0.893400	−	0.449262i	$-0.851687\pi$
$397$	−455.847	+	789.550i	−0.0576279	+	0.0998145i	0.835772	+	0.549076i	$0.185020\pi$
$398$	−5473.03			−0.689292
$399$		0			0
$400$	−368.913			−0.0461142
$401$	834.008	−	1444.54i	0.103861	−	0.179893i	−0.809411	−	0.587242i	$-0.800214\pi$
$401$	834.008	−	1444.54i	0.103861	−	0.179893i	0.913272	+	0.407349i	$0.133547\pi$
$402$		0			0
$403$	−366.413	−	634.646i	−0.0452912	−	0.0784466i
$404$	−3368.54	+	5834.48i	−0.414829	+	0.718505i
$405$		0			0
$406$	−7778.93	−	1762.15i	−0.950892	−	0.215405i
$407$	11298.8			1.37608
$408$		0			0
$409$	−4555.24	−	7889.90i	−0.550714	−	0.953864i	−0.998223	−	0.0595852i	$-0.981022\pi$
$409$	−4555.24	−	7889.90i	−0.550714	−	0.953864i	0.447509	−	0.894279i	$-0.352311\pi$
$410$	−1883.61	−	3262.50i	−0.226889	−	0.392984i
$411$		0			0
$412$	6564.26			0.784946
$413$	−46.9551	+	50.7713i	−0.00559445	+	0.00604913i
$414$		0			0
$415$	1232.93	−	2135.49i	0.145836	−	0.252596i
$416$	64.3083	+	111.385i	0.00757926	+	0.0131277i
$417$		0			0
$418$	−2061.71	+	3570.99i	−0.241248	+	0.417853i
$419$	−12626.4			−1.47217			−0.736085	−	0.676889i	$-0.763328\pi$
$419$	−12626.4			−1.47217			−0.736085	+	0.676889i	$0.763328\pi$
$420$		0			0
$421$	−4805.61			−0.556320			−0.278160	−	0.960535i	$-0.589725\pi$
$421$	−4805.61			−0.556320			−0.278160	+	0.960535i	$0.589725\pi$
$422$	−2915.84	+	5050.38i	−0.336353	+	0.582580i
$423$		0			0
$424$	−2190.14	−	3793.43i	−0.250855	−	0.434494i
$425$	437.628	−	757.994i	0.0499484	−	0.0865132i
$426$		0			0
$427$	−2077.23	−	470.553i	−0.235420	−	0.0533294i
$428$	1284.38			0.145053
$429$		0			0
$430$	2648.02	+	4586.50i	0.296974	+	0.514374i
$431$	3341.06	+	5786.89i	0.373395	+	0.646739i	0.990085	−	0.140467i	$-0.0448603\pi$
$431$	3341.06	+	5786.89i	0.373395	+	0.646739i	−0.616690	+	0.787206i	$0.711527\pi$
$432$		0			0
$433$	−8502.27			−0.943632			−0.471816	−	0.881697i	$-0.656401\pi$
$433$	−8502.27			−0.943632			−0.471816	+	0.881697i	$0.656401\pi$
$434$	2001.16	+	6450.24i	0.221333	+	0.713414i
$435$		0			0
$436$	−1120.02	+	1939.93i	−0.123026	+	0.213087i
$437$	−2236.81	−	3874.27i	−0.244854	−	0.424100i
$438$		0			0
$439$	6814.77	−	11803.5i	0.740891	−	1.28326i	−0.211199	−	0.977443i	$-0.567737\pi$
$439$	6814.77	−	11803.5i	0.740891	−	1.28326i	0.952090	−	0.305818i	$-0.0989299\pi$
$440$	−2919.93			−0.316369
$441$		0			0
$442$	−305.146			−0.0328378
$443$	−3427.16	+	5936.02i	−0.367561	+	0.636634i	−0.989184	−	0.146683i	$-0.953140\pi$
$443$	−3427.16	+	5936.02i	−0.367561	+	0.636634i	0.621623	+	0.783317i	$0.286474\pi$
$444$		0			0
$445$	−2256.69	−	3908.70i	−0.240399	−	0.416383i
$446$	1875.11	−	3247.78i	0.199078	−	0.344813i
$447$		0			0
$448$	−351.218	−	1132.07i	−0.0370391	−	0.119386i
$449$	10372.3			1.09019			0.545097	−	0.838373i	$-0.316493\pi$
$449$	10372.3			1.09019			0.545097	+	0.838373i	$0.316493\pi$
$450$		0			0
$451$	3371.98	+	5840.45i	0.352063	+	0.609791i
$452$	−3741.30	−	6480.13i	−0.389328	−	0.674335i
$453$		0			0
$454$	8076.59			0.834919
$455$	733.003	+	166.047i	0.0755247	+	0.0171085i
$456$		0			0
$457$	3950.40	−	6842.30i	0.404359	−	0.700370i	−0.589888	−	0.807485i	$-0.700828\pi$
$457$	3950.40	−	6842.30i	0.404359	−	0.700370i	0.994247	+	0.107115i	$0.0341613\pi$
$458$	−1372.91	−	2377.95i	−0.140070	−	0.242608i
$459$		0			0
$460$	1583.96	−	2743.50i	0.160549	−	0.278079i
$461$	11359.7			1.14767			0.573834	−	0.818972i	$-0.305456\pi$
$461$	11359.7			1.14767			0.573834	+	0.818972i	$0.305456\pi$
$462$		0			0
$463$	7346.85			0.737445			0.368723	−	0.929539i	$-0.379795\pi$
$463$	7346.85			0.737445			0.368723	+	0.929539i	$0.379795\pi$
$464$	1722.66	−	2983.73i	0.172354	−	0.298527i
$465$		0			0
$466$	−4174.31	−	7230.11i	−0.414959	−	0.718731i
$467$	473.289	−	819.760i	0.0468976	−	0.0812291i	−0.841624	−	0.540064i	$-0.818400\pi$
$467$	473.289	−	819.760i	0.0468976	−	0.0812291i	0.888521	+	0.458835i	$0.151733\pi$
$468$		0			0
$469$	−7894.47	+	8536.08i	−0.777256	+	0.840426i
$470$	2785.09			0.273333
$471$		0			0
$472$	−14.9362	−	25.8703i	−0.00145656	−	0.00252283i
$473$	−4740.41	−	8210.64i	−0.460813	−	0.798151i
$474$		0			0
$475$	−1315.01			−0.127025
$476$	2742.66	+	621.292i	0.264096	+	0.0598253i
$477$		0			0
$478$	−5544.70	+	9603.71i	−0.530562	+	0.918961i
$479$	7657.75	+	13263.6i	0.730462	+	1.26520i	0.956686	+	0.291122i	$0.0940287\pi$
$479$	7657.75	+	13263.6i	0.730462	+	1.26520i	−0.226224	+	0.974075i	$0.572638\pi$
$480$		0			0
$481$	628.125	−	1087.95i	0.0595427	−	0.103131i
$482$	9780.28			0.924231
$483$		0			0
$484$	−96.8094			−0.00909179
$485$	−9107.15	+	15774.1i	−0.852649	+	1.47683i
$486$		0			0
$487$	4320.19	+	7482.78i	0.401984	+	0.696257i	0.993965	−	0.109695i	$-0.0349873\pi$
$487$	4320.19	+	7482.78i	0.401984	+	0.696257i	−0.591981	+	0.805952i	$0.701654\pi$
$488$	460.007	−	796.755i	0.0426711	−	0.0739086i
$489$		0			0
$490$	−6250.16	−	2984.86i	−0.576232	−	0.275188i
$491$	13425.6			1.23399			0.616996	−	0.786966i	$-0.288349\pi$
$491$	13425.6			1.23399			0.616996	+	0.786966i	$0.288349\pi$
$492$		0			0
$493$	4087.05	+	7078.99i	0.373371	+	0.646697i
$494$	229.229	+	397.037i	0.0208776	+	0.0361610i
$495$		0			0
$496$	−2917.25			−0.264090
$497$	2925.57	+	9429.88i	0.264044	+	0.851082i
$498$		0			0
$499$	5489.01	−	9507.24i	0.492429	−	0.852911i	−0.507533	−	0.861632i	$-0.669443\pi$
$499$	5489.01	−	9507.24i	0.492429	−	0.852911i	0.999962	+	0.00872085i	$0.00277597\pi$
$500$	−2989.77	−	5178.43i	−0.267413	−	0.463173i
$501$		0			0
$502$	7560.15	−	13094.6i	0.672164	−	1.16422i
$503$	7829.41			0.694028			0.347014	−	0.937860i	$-0.387196\pi$
$503$	7829.41			0.694028			0.347014	+	0.937860i	$0.387196\pi$
$504$		0			0
$505$	−17005.5			−1.49849
$506$	−2835.57	+	4911.35i	−0.249123	+	0.431494i
$507$		0			0
$508$	−1580.10	−	2736.81i	−0.138003	−	0.239028i
$509$	9091.90	−	15747.6i	0.791732	−	1.37132i	−0.133162	−	0.991094i	$-0.542513\pi$
$509$	9091.90	−	15747.6i	0.791732	−	1.37132i	0.924894	−	0.380225i	$-0.124153\pi$
$510$		0			0
$511$	3752.25	−	4057.21i	0.324833	−	0.351234i
$512$	512.000			0.0441942
$513$		0			0
$514$	−3232.86	−	5599.48i	−0.277423	−	0.480511i
$515$	8284.65	+	14349.4i	0.708865	+	1.22779i
$516$		0			0
$517$	−4985.80			−0.424130
$518$	−7860.71	+	8499.57i	−0.666756	+	0.720946i
$519$		0			0
$520$	−162.325	+	281.155i	−0.0136893	+	0.0237105i
$521$	−11202.3	−	19402.9i	−0.941997	−	1.63159i	−0.761656	−	0.647982i	$-0.775613\pi$
$521$	−11202.3	−	19402.9i	−0.941997	−	1.63159i	−0.180341	−	0.983604i	$-0.557720\pi$
$522$		0			0
$523$	−2843.67	+	4925.38i	−0.237753	+	0.411801i	−0.960069	−	0.279763i	$-0.909744\pi$
$523$	−2843.67	+	4925.38i	−0.237753	+	0.411801i	0.722316	+	0.691563i	$0.243078\pi$
$524$	6552.33			0.546259
$525$		0			0
$526$	8569.05			0.710320
$527$	3460.63	−	5993.99i	0.286048	−	0.495450i
$528$		0			0
$529$	3007.11	+	5208.46i	0.247153	+	0.428081i
$530$	5528.28	−	9575.27i	0.453082	−	0.784761i
$531$		0			0
$532$	−1251.93	−	4035.30i	−0.102027	−	0.328858i
$533$	749.822			0.0609351
$534$		0			0
$535$	1620.99	+	2807.64i	0.130994	+	0.226888i
$536$	−2511.20	−	4349.52i	−0.202364	−	0.350505i
$537$		0			0
$538$	13393.1			1.07326
$539$	11188.9	+	5343.42i	0.894137	+	0.427008i
$540$		0			0
$541$	6508.78	−	11273.5i	0.517254	−	0.895910i	−0.482545	−	0.875871i	$-0.660288\pi$
$541$	6508.78	−	11273.5i	0.517254	−	0.895910i	0.999799	−	0.0200392i	$-0.00637910\pi$
$542$	6315.79	+	10939.3i	0.500528	+	0.866940i
$543$		0			0
$544$	−607.367	+	1051.99i	−0.0478688	+	0.0829112i
$545$	−5654.23			−0.444405
$546$		0			0
$547$	−6353.31			−0.496614			−0.248307	−	0.968681i	$-0.579874\pi$
$547$	−6353.31			−0.496614			−0.248307	+	0.968681i	$0.579874\pi$
$548$	−3978.64	+	6891.21i	−0.310144	+	0.537186i
$549$		0			0
$550$	833.506	+	1443.67i	0.0646197	+	0.111925i
$551$	6140.49	−	10635.6i	0.474762	−	0.822311i
$552$		0			0
$553$	−15660.6	−	3547.57i	−1.20426	−	0.272799i
$554$	5510.50			0.422597
$555$		0			0
$556$	1311.26	+	2271.16i	0.100017	+	0.173235i
$557$	7759.69	+	13440.2i	0.590285	+	1.02240i	0.994194	+	0.107604i	$0.0343179\pi$
$557$	7759.69	+	13440.2i	0.590285	+	1.02240i	−0.403909	+	0.914799i	$0.632349\pi$
$558$		0			0
$559$	−1054.12			−0.0797574
$560$	2031.42	−	2196.52i	0.153292	−	0.165750i
$561$		0			0
$562$	1453.00	−	2516.66i	0.109059	−	0.188895i
$563$	9095.49	+	15753.9i	0.680869	+	1.17930i	0.974716	+	0.223448i	$0.0717311\pi$
$563$	9095.49	+	15753.9i	0.680869	+	1.17930i	−0.293847	+	0.955853i	$0.594936\pi$
$564$		0			0
$565$	9443.68	−	16356.9i	0.703184	−	1.21795i
$566$	10863.7			0.806774
$567$		0			0
$568$	−4264.85			−0.315052
$569$	−6159.38	+	10668.4i	−0.453804	+	0.786012i	−0.998619	−	0.0525449i	$-0.983267\pi$
$569$	−6159.38	+	10668.4i	−0.453804	+	0.786012i	0.544814	+	0.838557i	$0.316600\pi$
$570$		0			0
$571$	8694.98	+	15060.2i	0.637257	+	1.10376i	0.986032	+	0.166555i	$0.0532644\pi$
$571$	8694.98	+	15060.2i	0.637257	+	1.10376i	−0.348775	+	0.937206i	$0.613402\pi$
$572$	290.590	−	503.317i	0.0212416	−	0.0367915i
$573$		0			0
$574$	−6739.41	−	1526.67i	−0.490065	−	0.111014i
$575$	−1808.59			−0.131171
$576$		0			0
$577$	−1625.45	−	2815.37i	−0.117276	−	0.203129i	0.801411	−	0.598114i	$-0.204083\pi$
$577$	−1625.45	−	2815.37i	−0.117276	−	0.203129i	−0.918687	+	0.394985i	$0.870750\pi$
$578$	3472.01	+	6013.69i	0.249855	+	0.432762i
$579$		0			0
$580$	8696.57			0.622596
$581$	−1340.25	−	4319.97i	−0.0957022	−	0.308473i
$582$		0			0
$583$	−9896.60	+	17141.4i	−0.703045	+	1.21771i
$584$	1193.58	+	2067.33i	0.0845728	+	0.146484i
$585$		0			0
$586$	6227.21	−	10785.8i	0.438982	−	0.760339i
$587$	−6993.54			−0.491745			−0.245872	−	0.969302i	$-0.579074\pi$
$587$	−6993.54			−0.491745			−0.245872	+	0.969302i	$0.579074\pi$
$588$		0			0
$589$	−10398.7			−0.727453
$590$	37.7015	−	65.3010i	0.00263076	−	0.00455661i
$591$		0			0
$592$	−2500.46	−	4330.92i	−0.173595	−	0.300675i
$593$	−5001.10	+	8662.15i	−0.346324	+	0.599851i	−0.985593	−	0.169132i	$-0.945904\pi$
$593$	−5001.10	+	8662.15i	−0.346324	+	0.599851i	0.639269	+	0.768983i	$0.279237\pi$
$594$		0			0
$595$	2103.32	+	6779.56i	0.144921	+	0.467117i
$596$	11958.1			0.821853
$597$		0			0
$598$	315.270	+	546.064i	0.0215591	+	0.0373415i
$599$	−12362.1	−	21411.8i	−0.843241	−	1.46054i	−0.887140	−	0.461500i	$-0.847312\pi$
$599$	−12362.1	−	21411.8i	−0.843241	−	1.46054i	0.0438997	−	0.999036i	$-0.486022\pi$
$600$		0			0
$601$	−4825.50			−0.327515			−0.163757	−	0.986501i	$-0.552361\pi$
$601$	−4825.50			−0.327515			−0.163757	+	0.986501i	$0.552361\pi$
$602$	9474.42	+	2146.23i	0.641443	+	0.145305i
$603$		0			0
$604$	2191.92	−	3796.51i	0.147662	−	0.255758i
$605$	−122.182	−	211.625i	−0.00821056	−	0.0142211i
$606$		0			0
$607$	−12365.1	+	21417.0i	−0.826826	+	1.43210i	0.0736896	+	0.997281i	$0.476523\pi$
$607$	−12365.1	+	21417.0i	−0.826826	+	1.43210i	−0.900516	+	0.434824i	$0.856811\pi$
$608$	1825.04			0.121736
$609$		0			0
$610$	2322.27			0.154141
$611$	−277.171	+	480.074i	−0.0183521	+	0.0317868i
$612$		0			0
$613$	3804.29	+	6589.22i	0.250659	+	0.434154i	0.963707	−	0.266961i	$-0.0860195\pi$
$613$	3804.29	+	6589.22i	0.250659	+	0.434154i	−0.713049	+	0.701115i	$0.752686\pi$
$614$	−8621.61	+	14933.1i	−0.566677	+	0.981514i
$615$		0			0
$616$	−3636.61	+	3932.16i	−0.237862	+	0.257194i
$617$	25089.8			1.63708			0.818538	−	0.574452i	$-0.194785\pi$
$617$	25089.8			1.63708			0.818538	+	0.574452i	$0.194785\pi$
$618$		0			0
$619$	−11452.5	−	19836.3i	−0.743642	−	1.28803i	−0.950827	−	0.309724i	$-0.899763\pi$
$619$	−11452.5	−	19836.3i	−0.743642	−	1.28803i	0.207185	−	0.978302i	$-0.433570\pi$
$620$	−3681.82	−	6377.10i	−0.238493	−	0.413082i
$621$		0			0
$622$	14848.4			0.957183
$623$	−8074.29	−	1829.06i	−0.519245	−	0.117624i
$624$		0			0
$625$	6105.62	−	10575.2i	0.390760	−	0.676815i
$626$	−3266.61	−	5657.93i	−0.208562	−	0.361240i
$627$		0			0
$628$	−1002.05	+	1735.60i	−0.0636723	+	0.110284i
$629$	11864.8			0.752116
$630$		0			0
$631$	2253.93			0.142199			0.0710996	−	0.997469i	$-0.477349\pi$
$631$	2253.93			0.142199			0.0710996	+	0.997469i	$0.477349\pi$
$632$	3468.06	−	6006.86i	0.218279	−	0.378070i
$633$		0			0
$634$	−4880.49	−	8453.26i	−0.305724	−	0.529530i
$635$	3988.43	−	6908.17i	0.249254	−	0.431720i
$636$		0			0
$637$	1136.52	−	780.307i	0.0706917	−	0.0485352i
$638$	−15568.4			−0.966079
$639$		0			0
$640$	646.187	+	1119.23i	0.0399106	+	0.0691272i
$641$	−6014.77	−	10417.9i	−0.370623	−	0.641938i	0.619039	−	0.785361i	$-0.287522\pi$
$641$	−6014.77	−	10417.9i	−0.370623	−	0.641938i	−0.989661	+	0.143423i	$0.954189\pi$
$642$		0			0
$643$	−4738.59			−0.290625			−0.145312	−	0.989386i	$-0.546419\pi$
$643$	−4738.59			−0.290625			−0.145312	+	0.989386i	$0.546419\pi$
$644$	−1721.84	−	5549.93i	−0.105357	−	0.339593i
$645$		0			0
$646$	−2164.98	+	3749.86i	−0.131858	+	0.228384i
$647$	2869.84	+	4970.71i	0.174382	+	0.302038i	0.939947	−	0.341320i	$-0.110874\pi$
$647$	2869.84	+	4970.71i	0.174382	+	0.302038i	−0.765565	+	0.643358i	$0.777541\pi$
$648$		0			0
$649$	−67.4924	+	116.900i	−0.00408214	+	0.00707047i
$650$	185.345			0.0111844
$651$		0			0
$652$	9073.65			0.545018
$653$	863.084	−	1494.91i	0.0517230	−	0.0895868i	−0.839005	−	0.544124i	$-0.816862\pi$
$653$	863.084	−	1494.91i	0.0517230	−	0.0895868i	0.890728	+	0.454537i	$0.150195\pi$
$654$		0			0
$655$	8269.59	+	14323.4i	0.493313	+	0.854442i
$656$	1492.46	−	2585.01i	0.0888271	−	0.153853i
$657$		0			0
$658$	3468.67	−	3750.58i	0.205506	−	0.222208i
$659$	10987.7			0.649501			0.324751	−	0.945800i	$-0.394720\pi$
$659$	10987.7			0.649501			0.324751	+	0.945800i	$0.394720\pi$
$660$		0			0
$661$	−16020.4	−	27748.1i	−0.942695	−	1.63280i	−0.760302	−	0.649570i	$-0.774949\pi$
$661$	−16020.4	−	27748.1i	−0.942695	−	1.63280i	−0.182393	−	0.983226i	$-0.558384\pi$
$662$	9599.15	+	16626.2i	0.563567	+	0.976127i
$663$		0			0
$664$	1953.80			0.114190
$665$	7241.09	−	7829.60i	0.422252	−	0.456570i
$666$		0			0
$667$	8445.31	−	14627.7i	0.490260	−	0.849156i
$668$	−2739.94	−	4745.72i	−0.158700	−	0.274876i
$669$		0			0
$670$	6338.69	−	10978.9i	0.365500	−	0.633065i
$671$	−4157.27			−0.239180
$672$		0			0
$673$	−13026.7			−0.746124			−0.373062	−	0.927806i	$-0.621692\pi$
$673$	−13026.7			−0.746124			−0.373062	+	0.927806i	$0.621692\pi$
$674$	2895.79	−	5015.65i	0.165492	−	0.286641i
$675$		0			0
$676$	4361.69	+	7554.67i	0.248162	+	0.429829i
$677$	−2360.39	+	4088.32i	−0.133999	+	0.232093i	−0.925215	−	0.379444i	$-0.876115\pi$
$677$	−2360.39	+	4088.32i	−0.133999	+	0.232093i	0.791216	+	0.611537i	$0.209449\pi$
$678$		0			0
$679$	9899.91	+	31909.9i	0.559534	+	1.80352i
$680$	−3066.19			−0.172916
$681$		0			0
$682$	6591.11	+	11416.1i	0.370068	+	0.640977i
$683$	−5585.00	−	9673.51i	−0.312890	−	0.541942i	0.666096	−	0.745866i	$-0.267964\pi$
$683$	−5585.00	−	9673.51i	−0.312890	−	0.541942i	−0.978987	+	0.203924i	$0.934631\pi$
$684$		0			0
$685$	−20085.5			−1.12033
$686$	−11803.8	+	4699.40i	−0.656956	+	0.261551i
$687$		0			0
$688$	−2098.13	+	3634.07i	−0.116265	+	0.201377i
$689$	1100.34	+	1905.85i	0.0608415	+	0.105381i
$690$		0			0
$691$	6986.42	−	12100.8i	0.384625	−	0.666190i	−0.607092	−	0.794632i	$-0.707664\pi$
$691$	6986.42	−	12100.8i	0.384625	−	0.666190i	0.991717	+	0.128441i	$0.0409974\pi$
$692$	−7375.08			−0.405142
$693$		0			0
$694$	10715.2			0.586087
$695$	−3309.83	+	5732.79i	−0.180646	+	0.312888i
$696$		0			0
$697$	3540.89	+	6133.00i	0.192426	+	0.333291i
$698$	6049.21	−	10477.5i	0.328031	−	0.568167i
$699$		0			0
$700$	−1665.88	−	377.371i	−0.0899493	−	0.0203761i
$701$	−9694.83			−0.522352			−0.261176	−	0.965291i	$-0.584110\pi$
$701$	−9694.83			−0.522352			−0.261176	+	0.965291i	$0.584110\pi$
$702$		0			0
$703$	−8912.99	−	15437.7i	−0.478179	−	0.828230i
$704$	−1156.79	−	2003.62i	−0.0619292	−	0.107264i
$705$		0			0
$706$	13761.1			0.733578
$707$	−21179.4	+	22900.7i	−1.12664	+	1.21820i
$708$		0			0
$709$	−12712.8	+	22019.3i	−0.673399	+	1.16636i	0.303535	+	0.952820i	$0.401833\pi$
$709$	−12712.8	+	22019.3i	−0.673399	+	1.16636i	−0.976934	+	0.213541i	$0.931500\pi$
$710$	−5382.61	−	9322.95i	−0.284515	−	0.492794i
$711$		0			0
$712$	1788.07	−	3097.02i	0.0941160	−	0.163014i
$713$	−14301.8			−0.751200
$714$		0			0
$715$	1467.00			0.0767309
$716$	2123.14	−	3677.39i	0.110818	−	0.191942i
$717$		0			0
$718$	−5991.91	−	10378.3i	−0.311443	−	0.539435i
$719$	−8796.40	+	15235.8i	−0.456259	+	0.790264i	−0.998760	−	0.0497912i	$-0.984144\pi$
$719$	−8796.40	+	15235.8i	−0.456259	+	0.790264i	0.542500	+	0.840056i	$0.317478\pi$
$720$		0			0
$721$	29641.9	+	6714.76i	1.53110	+	0.346839i
$722$	−7212.55			−0.371778
$723$		0			0
$724$	−4883.07	−	8457.73i	−0.250660	−	0.434156i
$725$	−2482.47	−	4299.76i	−0.127168	−	0.220261i
$726$		0			0
$727$	−10087.1			−0.514596			−0.257298	−	0.966332i	$-0.582832\pi$
$727$	−10087.1			−0.514596			−0.257298	+	0.966332i	$0.582832\pi$
$728$	176.455	+	568.759i	0.00898332	+	0.0289555i
$729$		0			0
$730$	−3012.79	+	5218.30i	−0.152751	+	0.264572i
$731$	−4977.86	−	8621.91i	−0.251865	−	0.436242i
$732$		0			0
$733$	−15904.3	+	27547.1i	−0.801417	+	1.38810i	0.117266	+	0.993101i	$0.462587\pi$
$733$	−15904.3	+	27547.1i	−0.801417	+	1.38810i	−0.918683	+	0.394995i	$0.870746\pi$
$734$	−13142.1			−0.660877
$735$		0			0
$736$	2510.07			0.125710
$737$	−11347.4	+	19654.2i	−0.567145	+	0.982324i
$738$		0			0
$739$	8287.63	+	14354.6i	0.412538	+	0.714536i	0.995166	−	0.0982021i	$-0.0313092\pi$
$739$	8287.63	+	14354.6i	0.412538	+	0.714536i	−0.582629	+	0.812738i	$0.697976\pi$
$740$	6311.58	−	10932.0i	0.313538	−	0.543064i
$741$		0			0
$742$	−6009.51	−	19370.2i	−0.297326	−	0.958358i
$743$	−11053.4			−0.545773			−0.272886	−	0.962046i	$-0.587978\pi$
$743$	−11053.4			−0.545773			−0.272886	+	0.962046i	$0.587978\pi$
$744$		0			0
$745$	15092.2	+	26140.4i	0.742194	+	1.28552i
$746$	3733.61	+	6466.81i	0.183240	+	0.317382i
$747$		0			0
$748$	5489.02			0.268314
$749$	5799.80	+	1313.82i	0.282937	+	0.0640935i
$750$		0			0
$751$	−1589.84	+	2753.68i	−0.0772490	+	0.133799i	−0.902062	−	0.431606i	$-0.857947\pi$
$751$	−1589.84	+	2753.68i	−0.0772490	+	0.133799i	0.824813	+	0.565405i	$0.191280\pi$
$752$	1103.37	+	1911.09i	0.0535050	+	0.0926733i
$753$		0			0
$754$	−865.479	+	1499.05i	−0.0418022	+	0.0724036i
$755$	11065.5			0.533399
$756$		0			0
$757$	−31130.8			−1.49467			−0.747336	−	0.664446i	$-0.768667\pi$
$757$	−31130.8			−1.49467			−0.747336	+	0.664446i	$0.768667\pi$
$758$	−8419.14	+	14582.4i	−0.403426	+	0.698755i
$759$		0			0
$760$	2303.36	+	3989.54i	0.109936	+	0.190416i
$761$	−3915.15	+	6781.24i	−0.186497	+	0.323022i	−0.944080	−	0.329717i	$-0.893047\pi$
$761$	−3915.15	+	6781.24i	−0.186497	+	0.323022i	0.757583	+	0.652739i	$0.226380\pi$
$762$		0			0
$763$	−7042.02	+	7614.35i	−0.334126	+	0.361282i
$764$	−14768.6			−0.699359
$765$		0			0
$766$	5330.42	+	9232.56i	0.251431	+	0.435491i
$767$	7.50408	+	12.9974i	0.000353268	+	0.000611878i
$768$		0			0
$769$	9389.21			0.440291			0.220145	−	0.975467i	$-0.429347\pi$
$769$	9389.21			0.440291			0.220145	+	0.975467i	$0.429347\pi$
$770$	−13185.4	−	2986.88i	−0.617102	−	0.139792i
$771$		0			0
$772$	−6284.64	+	10885.3i	−0.292991	+	0.507475i
$773$	−13896.1	−	24068.8i	−0.646584	−	1.11992i	−0.983933	−	0.178537i	$-0.942864\pi$
$773$	−13896.1	−	24068.8i	−0.646584	−	1.11992i	0.337349	−	0.941380i	$-0.390470\pi$
$774$		0			0
$775$	−2101.98	+	3640.74i	−0.0974263	+	0.168747i
$776$	−14431.9			−0.667623
$777$		0			0
$778$	−15622.3			−0.719907
$779$	5319.92	−	9214.37i	0.244680	−	0.423798i
$780$		0			0
$781$	9635.81	+	16689.7i	0.441481	+	0.764667i
$782$	−2977.60	+	5157.36i	−0.136162	+	0.235840i
$783$		0			0
$784$	−427.958	−	5471.29i	−0.0194952	−	0.249239i
$785$	−5058.69			−0.230003
$786$		0			0
$787$	−1413.31	−	2447.93i	−0.0640141	−	0.110876i	0.832242	−	0.554412i	$-0.187057\pi$
$787$	−1413.31	−	2447.93i	−0.0640141	−	0.110876i	−0.896256	+	0.443537i	$0.853724\pi$
$788$	−4547.01	−	7875.65i	−0.205559	−	0.356039i
$789$		0			0
$790$	17507.9			0.788487
$791$	−10265.7	−	33089.1i	−0.461450	−	1.48737i
$792$		0			0
$793$	−231.111	+	400.296i	−0.0103493	+	0.0179255i
$794$	−911.693	−	1579.10i	−0.0407491	−	0.0705795i
$795$		0			0
$796$	5473.03	−	9479.56i	0.243701	−	0.422103i
$797$	−29567.2			−1.31408			−0.657041	−	0.753855i	$-0.728192\pi$
$797$	−29567.2			−1.31408			−0.657041	+	0.753855i	$0.728192\pi$
$798$		0			0
$799$	−5235.54			−0.231815
$800$	368.913	−	638.977i	0.0163038	−	0.0282391i
$801$		0			0
$802$	1668.02	+	2889.09i	0.0734410	+	0.127204i
$803$	5393.42	−	9341.67i	0.237023	−	0.410536i
$804$		0			0
$805$	9959.02	−	10768.4i	0.436036	−	0.471474i
$806$	1465.65			0.0640514
$807$		0			0
$808$	−6737.07	−	11669.0i	−0.293328	−	0.508060i
$809$	4385.17	+	7595.33i	0.190574	+	0.330084i	0.945441	−	0.325795i	$-0.105632\pi$
$809$	4385.17	+	7595.33i	0.190574	+	0.330084i	−0.754867	+	0.655878i	$0.772298\pi$
$810$		0			0
$811$	−2017.43			−0.0873508			−0.0436754	−	0.999046i	$-0.513907\pi$
$811$	−2017.43			−0.0873508			−0.0436754	+	0.999046i	$0.513907\pi$
$812$	10831.1	−	11711.4i	0.468099	−	0.506143i
$813$		0			0
$814$	−11298.8	+	19570.2i	−0.486516	+	0.842671i
$815$	11451.7	+	19835.0i	0.492192	+	0.852501i
$816$		0			0
$817$	−7478.87	+	12953.8i	−0.320260	+	0.554707i
$818$	18220.9			0.778827
$819$		0			0
$820$	7534.42			0.320870
$821$	22549.5	−	39056.9i	0.958567	−	1.66029i	0.232582	−	0.972577i	$-0.425283\pi$
$821$	22549.5	−	39056.9i	0.958567	−	1.66029i	0.725985	−	0.687710i	$-0.241384\pi$
$822$		0			0
$823$	14738.3	+	25527.5i	0.624236	+	1.08121i	0.988688	+	0.149986i	$0.0479228\pi$
$823$	14738.3	+	25527.5i	0.624236	+	1.08121i	−0.364452	+	0.931222i	$0.618744\pi$
$824$	−6564.26	+	11369.6i	−0.277520	+	0.480679i
$825$		0			0
$826$	−40.9833	−	132.100i	−0.00172638	−	0.00556458i
$827$	−20417.9			−0.858526			−0.429263	−	0.903179i	$-0.641227\pi$
$827$	−20417.9			−0.858526			−0.429263	+	0.903179i	$0.641227\pi$
$828$		0			0
$829$	−9150.67	−	15849.4i	−0.383373	−	0.664021i	0.608169	−	0.793807i	$-0.291904\pi$
$829$	−9150.67	−	15849.4i	−0.383373	−	0.664021i	−0.991542	+	0.129786i	$0.958571\pi$
$830$	2465.86	+	4270.99i	0.103122	+	0.178612i
$831$		0			0
$832$	−257.233			−0.0107187
$833$	11749.3	+	5611.07i	0.488704	+	0.233388i
$834$		0			0
$835$	6916.08	−	11979.0i	0.286636	−	0.496467i
$836$	−4123.42	−	7141.98i	−0.170588	−	0.295467i
$837$		0			0
$838$	12626.4	−	21869.5i	0.520491	−	0.901516i
$839$	4380.29			0.180244			0.0901218	−	0.995931i	$-0.471274\pi$
$839$	4380.29			0.180244			0.0901218	+	0.995931i	$0.471274\pi$
$840$		0			0
$841$	21979.1			0.901188
$842$	4805.61	−	8323.55i	0.196689	−	0.340675i
$843$		0			0
$844$	−5831.68	−	10100.8i	−0.237837	−	0.411946i
$845$	−11009.6	+	19069.3i	−0.448217	+	0.776335i
$846$		0			0
$847$	−437.157	−	99.0289i	−0.0177342	−	0.00401732i
$848$	8760.56			0.354763
$849$		0			0
$850$	875.257	+	1515.99i	0.0353189	+	0.0611741i
$851$	−12258.4	−	21232.3i	−0.493789	−	0.855267i
$852$		0			0
$853$	−6855.49			−0.275179			−0.137589	−	0.990489i	$-0.543935\pi$
$853$	−6855.49			−0.275179			−0.137589	+	0.990489i	$0.543935\pi$
$854$	2892.25	−	3127.31i	0.115891	−	0.125310i
$855$		0			0
$856$	−1284.38	+	2224.61i	−0.0512840	+	0.0888265i
$857$	−8858.08	−	15342.7i	−0.353076	−	0.611546i	0.633710	−	0.773570i	$-0.281531\pi$
$857$	−8858.08	−	15342.7i	−0.353076	−	0.611546i	−0.986787	+	0.162024i	$0.948198\pi$
$858$		0			0
$859$	11521.0	−	19955.0i	0.457616	−	0.792615i	−0.541218	−	0.840882i	$-0.682037\pi$
$859$	11521.0	−	19955.0i	0.457616	−	0.792615i	0.998834	+	0.0482676i	$0.0153700\pi$
$860$	−10592.1			−0.419984
$861$		0			0
$862$	−13364.2			−0.528060
$863$	5983.70	−	10364.1i	0.236022	−	0.408803i	−0.723547	−	0.690275i	$-0.757489\pi$
$863$	5983.70	−	10364.1i	0.236022	−	0.408803i	0.959569	+	0.281472i	$0.0908228\pi$
$864$		0			0
$865$	−9307.97	−	16121.9i	−0.365873	−	0.633711i
$866$	8502.27	−	14726.4i	0.333624	−	0.577855i
$867$		0			0
$868$	−13173.3	−	2984.14i	−0.515128	−	0.116691i
$869$	−31342.3			−1.22349
$870$		0			0
$871$	1261.65	+	2185.24i	0.0490807	+	0.0850102i
$872$	−2240.04	−	3879.86i	−0.0869922	−	0.150675i
$873$		0			0
$874$	8947.25			0.346276
$875$	−8203.60	−	26442.3i	−0.316951	−	1.02162i
$876$		0			0
$877$	21877.6	−	37893.1i	0.842364	−	1.45902i	−0.0455264	−	0.998963i	$-0.514497\pi$
$877$	21877.6	−	37893.1i	0.842364	−	1.45902i	0.887891	−	0.460055i	$-0.152170\pi$
$878$	13629.5	+	23607.1i	0.523889	+	0.907403i
$879$		0			0
$880$	2919.93	−	5057.47i	0.111853	−	0.193736i
$881$	36896.0			1.41096			0.705482	−	0.708728i	$-0.250731\pi$
$881$	36896.0			1.41096			0.705482	+	0.708728i	$0.250731\pi$
$882$		0			0
$883$	−10988.6			−0.418793			−0.209397	−	0.977831i	$-0.567150\pi$
$883$	−10988.6			−0.418793			−0.209397	+	0.977831i	$0.567150\pi$
$884$	305.146	−	528.528i	0.0116099	−	0.0201090i
$885$		0			0
$886$	−6854.32	−	11872.0i	−0.259905	−	0.450168i
$887$	−16024.0	+	27754.4i	−0.606578	+	1.05062i	0.385222	+	0.922824i	$0.374125\pi$
$887$	−16024.0	+	27754.4i	−0.606578	+	1.05062i	−0.991800	+	0.127800i	$0.959209\pi$
$888$		0			0
$889$	−4335.61	−	13974.8i	−0.163568	−	0.527221i
$890$	9026.77			0.339975
$891$		0			0
$892$	3750.21	+	6495.56i	0.140770	+	0.243820i
$893$	3933.01	+	6812.17i	0.147383	+	0.255275i
$894$		0			0
$895$	10718.3			0.400306
$896$	2312.01	+	523.739i	0.0862042	+	0.0195278i
$897$		0			0
$898$	−10372.3	+	17965.3i	−0.385442	+	0.667605i
$899$	−19630.6	−	34001.2i	−0.728273	−	1.26141i
$900$		0			0
$901$	−10392.3	+	18000.0i	−0.384260	+	0.665558i
$902$	−13487.9			−0.497893
$903$		0			0
$904$	14965.2			0.550592
$905$	12325.7	−	21348.7i	0.452729	−	0.784150i
$906$		0			0
$907$	−26578.9	−	46035.9i	−0.973028	−	1.68533i	−0.686293	−	0.727325i	$-0.740763\pi$
$907$	−26578.9	−	46035.9i	−0.973028	−	1.68533i	−0.286735	−	0.958010i	$-0.592570\pi$
$908$	−8076.59	+	13989.1i	−0.295188	+	0.511281i
$909$		0			0
$910$	−1020.60	+	1103.55i	−0.0371788	+	0.0402004i
$911$	−42187.2			−1.53427			−0.767137	−	0.641484i	$-0.778319\pi$
$911$	−42187.2			−1.53427			−0.767137	+	0.641484i	$0.778319\pi$
$912$		0			0
$913$	−4414.31	−	7645.82i	−0.160014	−	0.277152i
$914$	7900.81	+	13684.6i	0.285925	+	0.495237i
$915$		0			0
$916$	5491.65			0.198089
$917$	29588.0	+	6702.55i	1.06552	+	0.241372i
$918$		0			0
$919$	6003.98	−	10399.2i	0.215509	−	0.373273i	−0.737921	−	0.674887i	$-0.764192\pi$
$919$	6003.98	−	10399.2i	0.215509	−	0.373273i	0.953430	+	0.301614i	$0.0975256\pi$
$920$	3167.92	+	5487.00i	0.113525	+	0.196632i
$921$		0			0
$922$	−11359.7	+	19675.6i	−0.405762	+	0.702800i
$923$	2142.70			0.0764114
$924$		0			0
$925$	−7206.66			−0.256166
$926$	−7346.85	+	12725.1i	−0.260726	+	0.451591i
$927$		0			0
$928$	3445.32	+	5967.47i	0.121873	+	0.211090i
$929$	−10558.0	+	18287.0i	−0.372871	+	0.645831i	−0.990006	−	0.141026i	$-0.954960\pi$
$929$	−10558.0	+	18287.0i	−0.372871	+	0.645831i	0.617135	+	0.786857i	$0.288293\pi$
$930$		0			0
$931$	−1525.47	−	19502.6i	−0.0537008	−	0.686544i
$932$	16697.2			0.586841
$933$		0			0
$934$	946.578	+	1639.52i	0.0331616	+	0.0574376i
$935$	6927.61	+	11999.0i	0.242307	+	0.419688i
$936$		0			0
$937$	9506.74			0.331453			0.165727	−	0.986172i	$-0.447003\pi$
$937$	9506.74			0.331453			0.165727	+	0.986172i	$0.447003\pi$
$938$	−6890.46	−	22209.7i	−0.239852	−	0.773105i
$939$		0			0
$940$	−2785.09	+	4823.92i	−0.0966380	+	0.167382i
$941$	26695.9	+	46238.6i	0.924825	+	1.60184i	0.791843	+	0.610725i	$0.209122\pi$
$941$	26695.9	+	46238.6i	0.924825	+	1.60184i	0.132982	+	0.991119i	$0.457545\pi$
$942$		0			0
$943$	7316.73	−	12673.0i	0.252668	−	0.437633i
$944$	59.7448			0.00205988
$945$		0			0
$946$	18961.7			0.651688
$947$	2189.61	−	3792.52i	0.0751350	−	0.130138i	−0.826010	−	0.563655i	$-0.809395\pi$
$947$	2189.61	−	3792.52i	0.0751350	−	0.130138i	0.901145	+	0.433518i	$0.142728\pi$
$948$		0			0
$949$	−599.662	−	1038.65i	−0.0205120	−	0.0355278i
$950$	1315.01	−	2277.66i	0.0449100	−	0.0777863i
$951$		0			0
$952$	−3818.77	+	4129.13i	−0.130007	+	0.140573i
$953$	−57958.4			−1.97005			−0.985025	−	0.172411i	$-0.944844\pi$
$953$	−57958.4			−1.97005			−0.985025	+	0.172411i	$0.944844\pi$
$954$		0			0
$955$	−18639.2	−	32284.1i	−0.631573	−	1.09392i
$956$	−11089.4	−	19207.4i	−0.375164	−	0.649804i
$957$		0			0
$958$	−30631.0			−1.03303
$959$	−25015.4	+	27048.4i	−0.842323	+	0.910782i
$960$		0			0
$961$	−1726.31	+	2990.06i	−0.0579475	+	0.100368i
$962$	1256.25	+	2175.89i	0.0421031	+	0.0729247i
$963$		0			0
$964$	−9780.28	+	16939.9i	−0.326765	+	0.565974i
$965$	−31727.0			−1.05837
$966$		0			0
$967$	17466.6			0.580857			0.290429	−	0.956897i	$-0.406202\pi$
$967$	17466.6			0.580857			0.290429	+	0.956897i	$0.406202\pi$
$968$	96.8094	−	167.679i	0.00321443	−	0.00556756i
$969$		0			0
$970$	−18214.3	−	31548.1i	−0.602914	−	1.04428i
$971$	15895.2	−	27531.3i	0.525336	−	0.909908i	−0.474229	−	0.880402i	$-0.657273\pi$
$971$	15895.2	−	27531.3i	0.525336	−	0.909908i	0.999565	−	0.0295066i	$-0.00939360\pi$
$972$		0			0
$973$	3597.94	+	11597.1i	0.118545	+	0.382102i
$974$	−17280.7			−0.568491
$975$		0			0
$976$	920.013	+	1593.51i	0.0301731	+	0.0522613i
$977$	29302.2	+	50753.0i	0.959531	+	1.66196i	0.723641	+	0.690177i	$0.242467\pi$
$977$	29302.2	+	50753.0i	0.959531	+	1.66196i	0.235890	+	0.971780i	$0.424200\pi$
$978$		0			0
$979$	−16159.5			−0.527538
$980$	11420.1	−	7840.74i	0.372246	−	0.255575i
$981$		0			0
$982$	−13425.6	+	23253.9i	−0.436282	+	0.755663i
$983$	20073.7	+	34768.8i	0.651326	+	1.12813i	0.982801	+	0.184665i	$0.0591202\pi$
$983$	20073.7	+	34768.8i	0.651326	+	1.12813i	−0.331476	+	0.943464i	$0.607547\pi$
$984$		0			0
$985$	11477.4	−	19879.5i	0.371270	−	0.643059i
$986$	−16348.2			−0.528026
$987$		0			0
$988$	−916.918			−0.0295253
$989$	−10286.0	+	17815.9i	−0.330715	+	0.572815i
$990$		0			0
$991$	−21436.6	−	37129.4i	−0.687142	−	1.19016i	−0.972759	−	0.231821i	$-0.925532\pi$
$991$	−21436.6	−	37129.4i	−0.687142	−	1.19016i	0.285617	−	0.958344i	$-0.407802\pi$
$992$	2917.25	−	5052.83i	0.0933699	−	0.161721i
$993$		0			0
$994$	−19258.6	−	4362.63i	−0.614533	−	0.139210i
$995$	27629.7			0.880322
$996$		0			0
$997$	9752.05	+	16891.0i	0.309780	+	0.536554i	0.978314	−	0.207127i	$-0.0664113\pi$
$997$	9752.05	+	16891.0i	0.309780	+	0.536554i	−0.668534	+	0.743681i	$0.733078\pi$
$998$	10978.0	+	19014.5i	0.348200	+	0.603099i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.g.c.163.3	yes	8
3.2	odd	2		378.4.g.f.163.2	yes	8
7.4	even	3	inner	378.4.g.c.109.3	✓	8
21.11	odd	6		378.4.g.f.109.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.c.109.3	✓	8	7.4	even	3	inner
378.4.g.c.163.3	yes	8	1.1	even	1	trivial
378.4.g.f.109.2	yes	8	21.11	odd	6
378.4.g.f.163.2	yes	8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.04834 - 8.74398i) q^{5} +(-5.48778 - 17.6885i) q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.04834 - 8.74398i) q^{5} +(-5.48778 - 17.6885i) q^{7} +8.00000 q^{8} +(10.0967 + 17.4880i) q^{10} +(-18.0748 - 31.3065i) q^{11} -4.01927 q^{13} +(36.1252 + 8.18341i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(-18.9802 - 32.8747i) q^{17} +(-28.5163 + 49.3917i) q^{19} -40.3867 q^{20} +72.2993 q^{22} +(-39.2198 + 67.9308i) q^{23} +(11.5285 + 19.9680i) q^{25} +(4.01927 - 6.96157i) q^{26} +(-50.2993 + 54.3873i) q^{28} -215.332 q^{29} +(91.1642 + 157.901i) q^{31} +(-16.0000 - 27.7128i) q^{32} +75.9208 q^{34} +(-182.372 - 41.3127i) q^{35} +(-156.279 + 270.683i) q^{37} +(-57.0327 - 98.7835i) q^{38} +(40.3867 - 69.9518i) q^{40} -186.557 q^{41} +262.266 q^{43} +(-72.2993 + 125.226i) q^{44} +(-78.4397 - 135.862i) q^{46} +(68.9606 - 119.443i) q^{47} +(-282.768 + 194.142i) q^{49} -46.1142 q^{50} +(8.03853 + 13.9231i) q^{52} +(-273.767 - 474.179i) q^{53} -364.991 q^{55} +(-43.9023 - 141.508i) q^{56} +(215.332 - 372.967i) q^{58} +(-1.86703 - 3.23378i) q^{59} +(57.5008 - 99.5943i) q^{61} -364.657 q^{62} +64.0000 q^{64} +(-20.2906 + 35.1444i) q^{65} +(-313.900 - 543.690i) q^{67} +(-75.9208 + 131.499i) q^{68} +(253.928 - 274.566i) q^{70} -533.107 q^{71} +(149.197 + 258.417i) q^{73} +(-312.557 - 541.365i) q^{74} +228.131 q^{76} +(-454.576 + 491.521i) q^{77} +(433.508 - 750.857i) q^{79} +(80.7734 + 139.904i) q^{80} +(186.557 - 323.126i) q^{82} +244.224 q^{83} -383.274 q^{85} +(-262.266 + 454.258i) q^{86} +(-144.599 - 250.452i) q^{88} +(223.508 - 387.128i) q^{89} +(22.0569 + 71.0949i) q^{91} +313.759 q^{92} +(137.921 + 238.887i) q^{94} +(287.920 + 498.692i) q^{95} -1803.99 q^{97} +(-53.4948 - 683.911i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8	\( 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8} + 4 q^{10} - 32 q^{11} - 4 q^{13} - 36 q^{14} - 64 q^{16} + 58 q^{17} + 70 q^{19} - 16 q^{20} + 128 q^{22} - 86 q^{23} - 156 q^{25} + 4 q^{26} + 48 q^{28} + 212 q^{29} - 64 q^{31} - 128 q^{32} - 232 q^{34} - 8 q^{35} - 146 q^{37} + 140 q^{38} + 16 q^{40} + 780 q^{41} + 880 q^{43} - 128 q^{44} - 172 q^{46} + 306 q^{47} + 50 q^{49} + 624 q^{50} + 8 q^{52} + 90 q^{53} - 64 q^{55} + 48 q^{56} - 212 q^{58} - 148 q^{59} - 364 q^{61} + 256 q^{62} + 512 q^{64} - 1296 q^{65} - 954 q^{67} + 232 q^{68} + 20 q^{70} + 1360 q^{71} - 54 q^{73} - 292 q^{74} - 560 q^{76} - 2224 q^{77} - 226 q^{79} + 32 q^{80} - 780 q^{82} + 3136 q^{83} + 3920 q^{85} - 880 q^{86} - 256 q^{88} + 1458 q^{89} + 3836 q^{91} + 688 q^{92} + 612 q^{94} - 1310 q^{95} - 4344 q^{97} - 776 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8 + 4 * q^10 - 32 * q^11 - 4 * q^13 - 36 * q^14 - 64 * q^16 + 58 * q^17 + 70 * q^19 - 16 * q^20 + 128 * q^22 - 86 * q^23 - 156 * q^25 + 4 * q^26 + 48 * q^28 + 212 * q^29 - 64 * q^31 - 128 * q^32 - 232 * q^34 - 8 * q^35 - 146 * q^37 + 140 * q^38 + 16 * q^40 + 780 * q^41 + 880 * q^43 - 128 * q^44 - 172 * q^46 + 306 * q^47 + 50 * q^49 + 624 * q^50 + 8 * q^52 + 90 * q^53 - 64 * q^55 + 48 * q^56 - 212 * q^58 - 148 * q^59 - 364 * q^61 + 256 * q^62 + 512 * q^64 - 1296 * q^65 - 954 * q^67 + 232 * q^68 + 20 * q^70 + 1360 * q^71 - 54 * q^73 - 292 * q^74 - 560 * q^76 - 2224 * q^77 - 226 * q^79 + 32 * q^80 - 780 * q^82 + 3136 * q^83 + 3920 * q^85 - 880 * q^86 - 256 * q^88 + 1458 * q^89 + 3836 * q^91 + 688 * q^92 + 612 * q^94 - 1310 * q^95 - 4344 * q^97 - 776 * q^98

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