LMFDB - Embedded newform 378.4.k.c.269.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(215,378)]

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([3, 5]))

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

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, 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:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 178 x^{14} + 23185 x^{12} - 1395488 x^{10} + 61706754 x^{8} - 468877357 x^{6} + \cdots + 6975757441 $ x^16 - 178x^14 + 23185x^12 - 1395488x^10 + 61706754x^8 - 468877357x^6 + 2731971910x^4 - 4899926507*x^2 + 6975757441
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.6
Root			$-2.11897 + 1.22339i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.c.215.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.89811 - 6.75173i) q^{5} +(15.4219 - 10.2550i) q^{7} -8.00000i q^{8} +O(q^{10})$	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.89811 - 6.75173i) q^{5} +(15.4219 - 10.2550i) q^{7} -8.00000i q^{8} +(-13.5035 - 7.79622i) q^{10} +(44.0180 + 25.4138i) q^{11} -50.9038i q^{13} +(16.4565 - 33.1841i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(-4.37255 + 7.57348i) q^{17} +(-87.9687 + 50.7888i) q^{19} -31.1849 q^{20} +101.655 q^{22} +(36.9867 - 21.3543i) q^{23} +(32.1094 - 55.6152i) q^{25} +(-50.9038 - 88.1679i) q^{26} +(-4.68064 - 73.9330i) q^{28} -218.184i q^{29} +(-87.6221 - 50.5886i) q^{31} +(-27.7128 - 16.0000i) q^{32} +17.4902i q^{34} +(-129.355 - 64.1492i) q^{35} +(-137.147 - 237.545i) q^{37} +(-101.578 + 175.937i) q^{38} +(-54.0138 + 31.1849i) q^{40} +6.40338 q^{41} -262.501 q^{43} +(176.072 - 101.655i) q^{44} +(42.7085 - 73.9733i) q^{46} +(53.2835 + 92.2898i) q^{47} +(132.669 - 316.303i) q^{49} -128.438i q^{50} +(-176.336 - 101.808i) q^{52} +(555.867 + 320.930i) q^{53} -396.263i q^{55} +(-82.0401 - 123.375i) q^{56} +(-218.184 - 377.906i) q^{58} +(52.5654 - 91.0459i) q^{59} +(471.522 - 272.233i) q^{61} -202.354 q^{62} -64.0000 q^{64} +(-343.688 + 198.429i) q^{65} +(109.449 - 189.572i) q^{67} +(17.4902 + 30.2939i) q^{68} +(-288.199 + 18.2457i) q^{70} +542.115i q^{71} +(-778.156 - 449.268i) q^{73} +(-475.090 - 274.293i) q^{74} +406.310i q^{76} +(939.460 - 59.4764i) q^{77} +(561.113 + 971.876i) q^{79} +(-62.3698 + 108.028i) q^{80} +(11.0910 - 6.40338i) q^{82} -855.676 q^{83} +68.1788 q^{85} +(-454.665 + 262.501i) q^{86} +(203.310 - 352.144i) q^{88} +(658.192 + 1140.02i) q^{89} +(-522.019 - 785.032i) q^{91} -170.834i q^{92} +(184.580 + 106.567i) q^{94} +(685.824 + 395.961i) q^{95} -1240.73i q^{97} +(-86.5135 - 680.523i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} + 50 q^{7}+O(q^{10}) $ 16 * q + 32 * q^4 + 50 * q^7	$ 16 q + 32 q^{4} + 50 q^{7} + 60 q^{10} - 128 q^{16} - 498 q^{19} - 240 q^{22} - 470 q^{25} + 160 q^{28} - 582 q^{31} + 40 q^{37} + 240 q^{40} + 1900 q^{43} - 456 q^{46} - 1634 q^{49} - 720 q^{52} + 1200 q^{58} - 1302 q^{61} - 1024 q^{64} - 100 q^{67} - 1620 q^{70} - 5280 q^{73} + 590 q^{79} - 480 q^{82} + 2580 q^{85} - 480 q^{88} - 2382 q^{91} - 3396 q^{94}+O(q^{100}) $ 16 * q + 32 * q^4 + 50 * q^7 + 60 * q^10 - 128 * q^16 - 498 * q^19 - 240 * q^22 - 470 * q^25 + 160 * q^28 - 582 * q^31 + 40 * q^37 + 240 * q^40 + 1900 * q^43 - 456 * q^46 - 1634 * q^49 - 720 * q^52 + 1200 * q^58 - 1302 * q^61 - 1024 * q^64 - 100 * q^67 - 1620 * q^70 - 5280 * q^73 + 590 * q^79 - 480 * q^82 + 2580 * q^85 - 480 * q^88 - 2382 * q^91 - 3396 * q^94

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	1.73205	−	1.00000i	0.612372	−	0.353553i
$2$	1.73205	−	1.00000i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	−3.89811	−	6.75173i	−0.348658	−	0.603893i	0.637354	−	0.770572i	$-0.280029\pi$
$5$	−3.89811	−	6.75173i	−0.348658	−	0.603893i	−0.986011	+	0.166679i	$0.946696\pi$
$6$		0			0
$7$	15.4219	−	10.2550i	0.832704	−	0.553719i
$7$	15.4219	−	10.2550i	0.832704	−	0.553719i
$8$		−	8.00000i		−	0.353553i
$9$		0			0
$10$	−13.5035	−	7.79622i	−0.427017	−	0.246538i
$11$	44.0180	+	25.4138i	1.20654	+	0.696595i	0.962001	−	0.273045i	$-0.0880306\pi$
$11$	44.0180	+	25.4138i	1.20654	+	0.696595i	0.244537	+	0.969640i	$0.421364\pi$
$12$		0			0
$13$		−	50.9038i		−	1.08601i	−0.839729	−	0.543006i	$-0.817286\pi$
$13$		−	50.9038i		−	1.08601i	0.839729	−	0.543006i	$-0.182714\pi$
$14$	16.4565	−	33.1841i	0.314156	−	0.633487i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	−4.37255	+	7.57348i	−0.0623823	+	0.108049i	−0.895530	−	0.445001i	$-0.853203\pi$
$17$	−4.37255	+	7.57348i	−0.0623823	+	0.108049i	0.833148	+	0.553051i	$0.186536\pi$
$18$		0			0
$19$	−87.9687	+	50.7888i	−1.06218	+	0.613250i	−0.926034	−	0.377439i	$-0.876805\pi$
$19$	−87.9687	+	50.7888i	−1.06218	+	0.613250i	−0.136145	+	0.990689i	$0.543471\pi$
$20$	−31.1849			−0.348658
$21$		0			0
$22$	101.655			0.985135
$23$	36.9867	−	21.3543i	0.335315	−	0.193594i	−0.322883	−	0.946439i	$-0.604652\pi$
$23$	36.9867	−	21.3543i	0.335315	−	0.193594i	0.658198	+	0.752844i	$0.271319\pi$
$24$		0			0
$25$	32.1094	−	55.6152i	0.256876	−	0.444921i
$26$	−50.9038	−	88.1679i	−0.383964	−	0.665044i
$27$		0			0
$28$	−4.68064	−	73.9330i	−0.0315914	−	0.499001i
$29$		−	218.184i		−	1.39710i	−0.715563	−	0.698548i	$-0.753830\pi$
$29$		−	218.184i		−	1.39710i	0.715563	−	0.698548i	$-0.246170\pi$
$30$		0			0
$31$	−87.6221	−	50.5886i	−0.507658	−	0.293096i	0.224213	−	0.974540i	$-0.428019\pi$
$31$	−87.6221	−	50.5886i	−0.507658	−	0.293096i	−0.731870	+	0.681444i	$0.761352\pi$
$32$	−27.7128	−	16.0000i	−0.153093	−	0.0883883i
$33$		0			0
$34$			17.4902i			0.0882219i
$35$	−129.355	−	64.1492i	−0.624715	−	0.309806i
$36$		0			0
$37$	−137.147	−	237.545i	−0.609373	−	1.05546i	−0.991344	−	0.131290i	$-0.958088\pi$
$37$	−137.147	−	237.545i	−0.609373	−	1.05546i	0.381971	−	0.924174i	$-0.375245\pi$
$38$	−101.578	+	175.937i	−0.433633	+	0.751074i
$39$		0			0
$40$	−54.0138	+	31.1849i	−0.213508	+	0.123269i
$41$	6.40338			0.0243912			0.0121956	−	0.999926i	$-0.496118\pi$
$41$	6.40338			0.0243912			0.0121956	+	0.999926i	$0.496118\pi$
$42$		0			0
$43$	−262.501			−0.930955			−0.465477	−	0.885060i	$-0.654117\pi$
$43$	−262.501			−0.930955			−0.465477	+	0.885060i	$0.654117\pi$
$44$	176.072	−	101.655i	0.603269	−	0.348298i
$45$		0			0
$46$	42.7085	−	73.9733i	0.136892	−	0.237104i
$47$	53.2835	+	92.2898i	0.165366	+	0.286422i	0.936785	−	0.349905i	$-0.113786\pi$
$47$	53.2835	+	92.2898i	0.165366	+	0.286422i	−0.771419	+	0.636327i	$0.780453\pi$
$48$		0			0
$49$	132.669	−	316.303i	0.386791	−	0.922167i
$50$		−	128.438i		−	0.363277i
$51$		0			0
$52$	−176.336	−	101.808i	−0.470257	−	0.271503i
$53$	555.867	+	320.930i	1.44065	+	0.831758i	0.997893	−	0.0648858i	$-0.0206683\pi$
$53$	555.867	+	320.930i	1.44065	+	0.831758i	0.442754	+	0.896643i	$0.354002\pi$
$54$		0			0
$55$		−	396.263i		−	0.971494i
$56$	−82.0401	−	123.375i	−0.195769	−	0.294405i
$57$		0			0
$58$	−218.184	−	377.906i	−0.493948	−	0.855543i
$59$	52.5654	−	91.0459i	0.115990	−	0.200901i	−0.802185	−	0.597076i	$-0.796329\pi$
$59$	52.5654	−	91.0459i	0.115990	−	0.200901i	0.918175	+	0.396175i	$0.129662\pi$
$60$		0			0
$61$	471.522	−	272.233i	0.989707	−	0.571408i	0.0845205	−	0.996422i	$-0.473064\pi$
$61$	471.522	−	272.233i	0.989707	−	0.571408i	0.905187	+	0.425014i	$0.139731\pi$
$62$	−202.354			−0.414501
$63$		0			0
$64$	−64.0000			−0.125000
$65$	−343.688	+	198.429i	−0.655835	+	0.378647i
$66$		0			0
$67$	109.449	−	189.572i	0.199572	−	0.345670i	−0.748817	−	0.662776i	$-0.769378\pi$
$67$	109.449	−	189.572i	0.199572	−	0.345670i	0.948390	+	0.317107i	$0.102711\pi$
$68$	17.4902	+	30.2939i	0.0311912	+	0.0540247i
$69$		0			0
$70$	−288.199	+	18.2457i	−0.492091	+	0.0311539i
$71$			542.115i			0.906158i	0.891470	+	0.453079i	$0.149674\pi$
$71$			542.115i			0.906158i	−0.891470	+	0.453079i	$0.850326\pi$
$72$		0			0
$73$	−778.156	−	449.268i	−1.24762	−	0.720314i	−0.276986	−	0.960874i	$-0.589335\pi$
$73$	−778.156	−	449.268i	−1.24762	−	0.720314i	−0.970634	+	0.240561i	$0.922669\pi$
$74$	−475.090	−	274.293i	−0.746326	−	0.430891i
$75$		0			0
$76$			406.310i			0.613250i
$77$	939.460	−	59.4764i	1.39041	−	0.0880256i
$78$		0			0
$79$	561.113	+	971.876i	0.799115	+	1.38411i	0.920193	+	0.391465i	$0.128032\pi$
$79$	561.113	+	971.876i	0.799115	+	1.38411i	−0.121078	+	0.992643i	$0.538635\pi$
$80$	−62.3698	+	108.028i	−0.0871644	+	0.150973i
$81$		0			0
$82$	11.0910	−	6.40338i	0.0149365	−	0.00862360i
$83$	−855.676			−1.13160			−0.565799	−	0.824543i	$-0.691432\pi$
$83$	−855.676			−1.13160			−0.565799	+	0.824543i	$0.691432\pi$
$84$		0			0
$85$	68.1788			0.0870003
$86$	−454.665	+	262.501i	−0.570091	+	0.329142i
$87$		0			0
$88$	203.310	−	352.144i	0.246284	−	0.426576i
$89$	658.192	+	1140.02i	0.783913	+	1.35778i	0.929646	+	0.368453i	$0.120112\pi$
$89$	658.192	+	1140.02i	0.783913	+	1.35778i	−0.145733	+	0.989324i	$0.546554\pi$
$90$		0			0
$91$	−522.019	−	785.032i	−0.601346	−	0.904327i
$92$		−	170.834i		−	0.193594i
$93$		0			0
$94$	184.580	+	106.567i	0.202531	+	0.116931i
$95$	685.824	+	395.961i	0.740674	+	0.427628i
$96$		0			0
$97$		−	1240.73i		−	1.29873i	−0.760475	−	0.649367i	$-0.775034\pi$
$97$		−	1240.73i		−	1.29873i	0.760475	−	0.649367i	$-0.224966\pi$
$98$	−86.5135	−	680.523i	−0.0891753	−	0.701461i
$99$		0			0
$100$	−128.438	−	222.461i	−0.128438	−	0.222461i
$101$	−250.942	+	434.645i	−0.247225	+	0.428206i	−0.962755	−	0.270376i	$-0.912852\pi$
$101$	−250.942	+	434.645i	−0.247225	+	0.428206i	0.715530	+	0.698582i	$0.246185\pi$
$102$		0			0
$103$	−305.111	+	176.156i	−0.291878	+	0.168516i	−0.638789	−	0.769382i	$-0.720564\pi$
$103$	−305.111	+	176.156i	−0.291878	+	0.168516i	0.346910	+	0.937898i	$0.387231\pi$
$104$	−407.230			−0.383964
$105$		0			0
$106$	1283.72			1.17628
$107$	935.207	−	539.942i	0.844952	−	0.487833i	−0.0139923	−	0.999902i	$-0.504454\pi$
$107$	935.207	−	539.942i	0.844952	−	0.487833i	0.858944	+	0.512069i	$0.171121\pi$
$108$		0			0
$109$	−242.598	+	420.192i	−0.213180	+	0.369239i	−0.952708	−	0.303887i	$-0.901716\pi$
$109$	−242.598	+	420.192i	−0.213180	+	0.369239i	0.739528	+	0.673126i	$0.235049\pi$
$110$	−396.263	−	686.348i	−0.343475	−	0.594916i
$111$		0			0
$112$	−265.473	−	131.652i	−0.223972	−	0.111071i
$113$			1686.83i			1.40428i	0.712038	+	0.702140i	$0.247772\pi$
$113$			1686.83i			1.40428i	−0.712038	+	0.702140i	$0.752228\pi$
$114$		0			0
$115$	−288.356	−	166.483i	−0.233821	−	0.134996i
$116$	−755.812	−	436.368i	−0.604960	−	0.349274i
$117$		0			0
$118$		−	210.262i		−	0.164035i
$119$	10.2332	+	161.638i	0.00788297	+	0.124515i
$120$		0			0
$121$	626.222	+	1084.65i	0.470490	+	0.814913i
$122$	544.466	−	943.043i	0.404046	−	0.699829i
$123$		0			0
$124$	−350.488	+	202.354i	−0.253829	+	0.146548i
$125$	−1475.19			−1.05556
$126$		0			0
$127$	1425.59			0.996070			0.498035	−	0.867157i	$-0.334055\pi$
$127$	1425.59			0.996070			0.498035	+	0.867157i	$0.334055\pi$
$128$	−110.851	+	64.0000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	−396.857	+	687.377i	−0.267744	+	0.463746i
$131$	576.837	+	999.111i	0.384721	+	0.666357i	0.991730	−	0.128338i	$-0.0409643\pi$
$131$	576.837	+	999.111i	0.384721	+	0.666357i	−0.607009	+	0.794695i	$0.707631\pi$
$132$		0			0
$133$	−835.804	+	1685.38i	−0.544913	+	1.09880i
$134$		−	437.797i		−	0.282238i
$135$		0			0
$136$	60.5878	+	34.9804i	0.0382012	+	0.0220555i
$137$	−1704.44	−	984.057i	−1.06292	−	0.613676i	−0.136681	−	0.990615i	$-0.543643\pi$
$137$	−1704.44	−	984.057i	−1.06292	−	0.613676i	−0.926238	+	0.376939i	$0.876977\pi$
$138$		0			0
$139$			821.420i			0.501237i	0.968086	+	0.250619i	$0.0806340\pi$
$139$			821.420i			0.501237i	−0.968086	+	0.250619i	$0.919366\pi$
$140$	−480.930	+	319.802i	−0.290329	+	0.193058i
$141$		0			0
$142$	542.115	+	938.971i	0.320375	+	0.554906i
$143$	1293.66	−	2240.68i	0.756511	−	1.31032i
$144$		0			0
$145$	−1473.12	+	850.506i	−0.843696	+	0.487108i
$146$	−1797.07			−1.01868
$147$		0			0
$148$	−1097.17			−0.609373
$149$	2337.37	−	1349.48i	1.28513	−	0.741971i	0.307350	−	0.951597i	$-0.400558\pi$
$149$	2337.37	−	1349.48i	1.28513	−	0.741971i	0.977782	+	0.209626i	$0.0672245\pi$
$150$		0			0
$151$	−636.455	+	1102.37i	−0.343007	+	0.594105i	−0.984990	−	0.172614i	$-0.944779\pi$
$151$	−636.455	+	1102.37i	−0.343007	+	0.594105i	0.641983	+	0.766719i	$0.278112\pi$
$152$	406.310	+	703.750i	0.216816	+	0.375537i
$153$		0			0
$154$	1567.72	−	1042.48i	0.820325	−	0.545488i
$155$			788.800i			0.408761i
$156$		0			0
$157$	2684.88	+	1550.11i	1.36482	+	0.787979i	0.990261	−	0.139225i	$-0.0444611\pi$
$157$	2684.88	+	1550.11i	1.36482	+	0.787979i	0.374558	+	0.927203i	$0.377794\pi$
$158$	1943.75	+	1122.23i	0.978712	+	0.565060i
$159$		0			0
$160$			249.479i			0.123269i
$161$	351.416	−	708.622i	0.172021	−	0.346877i
$162$		0			0
$163$	1415.80	+	2452.24i	0.680333	+	1.17837i	0.974879	+	0.222735i	$0.0714983\pi$
$163$	1415.80	+	2452.24i	0.680333	+	1.17837i	−0.294546	+	0.955637i	$0.595168\pi$
$164$	12.8068	−	22.1820i	0.00609780	−	0.0105617i
$165$		0			0
$166$	−1482.07	+	855.676i	−0.692959	+	0.400080i
$167$	3845.26			1.78177			0.890883	−	0.454234i	$-0.150087\pi$
$167$	3845.26			1.78177			0.890883	+	0.454234i	$0.150087\pi$
$168$		0			0
$169$	−394.194			−0.179424
$170$	118.089	−	68.1788i	0.0532766	−	0.0307593i
$171$		0			0
$172$	−525.002	+	909.331i	−0.232739	+	0.403115i
$173$	−1098.40	−	1902.48i	−0.482714	−	0.836084i	0.517090	−	0.855931i	$-0.327015\pi$
$173$	−1098.40	−	1902.48i	−0.482714	−	0.836084i	−0.999803	+	0.0198470i	$0.993682\pi$
$174$		0			0
$175$	−75.1464	−	1186.97i	−0.0324602	−	0.512725i
$176$		−	813.242i		−	0.348298i
$177$		0			0
$178$	2280.05	+	1316.38i	0.960093	+	0.554310i
$179$	2000.24	+	1154.84i	0.835222	+	0.482216i	0.855637	−	0.517576i	$-0.173165\pi$
$179$	2000.24	+	1154.84i	0.835222	+	0.482216i	−0.0204150	+	0.999792i	$0.506499\pi$
$180$		0			0
$181$			3240.23i			1.33063i	0.746561	+	0.665317i	$0.231704\pi$
$181$			3240.23i			1.33063i	−0.746561	+	0.665317i	$0.768296\pi$
$182$	−1689.20	−	837.697i	−0.687975	−	0.341177i
$183$		0			0
$184$	−170.834	−	295.893i	−0.0684459	−	0.118552i
$185$	−1069.23	+	1851.95i	−0.424925	+	0.735992i
$186$		0			0
$187$	−384.942	+	222.246i	−0.150533	+	0.0869105i
$188$	426.268			0.165366
$189$		0			0
$190$	1583.84			0.604758
$191$	95.2419	−	54.9879i	0.0360809	−	0.0208313i	−0.481851	−	0.876253i	$-0.660035\pi$
$191$	95.2419	−	54.9879i	0.0360809	−	0.0208313i	0.517932	+	0.855422i	$0.326702\pi$
$192$		0			0
$193$	2499.43	−	4329.13i	0.932190	−	1.61460i	0.152621	−	0.988285i	$-0.451229\pi$
$193$	2499.43	−	4329.13i	0.932190	−	1.61460i	0.779569	−	0.626316i	$-0.215438\pi$
$194$	−1240.73	−	2149.01i	−0.459172	−	0.795308i
$195$		0			0
$196$	−830.369	−	1092.19i	−0.302612	−	0.398027i
$197$			508.689i			0.183973i	0.995760	+	0.0919863i	$0.0293216\pi$
$197$			508.689i			0.183973i	−0.995760	+	0.0919863i	$0.970678\pi$
$198$		0			0
$199$	−1043.93	−	602.713i	−0.371871	−	0.214700i	0.302405	−	0.953180i	$-0.402211\pi$
$199$	−1043.93	−	602.713i	−0.371871	−	0.214700i	−0.674275	+	0.738480i	$0.735544\pi$
$200$	−444.921	−	256.876i	−0.157304	−	0.0908192i
$201$		0			0
$202$			1003.77i			0.349629i
$203$	−2237.48	−	3364.81i	−0.773598	−	1.16337i
$204$		0			0
$205$	−24.9611	−	43.2339i	−0.00850419	−	0.0147297i
$206$	−352.311	+	610.221i	−0.119159	+	0.206389i
$207$		0			0
$208$	−705.343	+	407.230i	−0.235129	+	0.135752i
$209$	−5162.94			−1.70875
$210$		0			0
$211$	3127.48			1.02040			0.510200	−	0.860056i	$-0.329571\pi$
$211$	3127.48			1.02040			0.510200	+	0.860056i	$0.329571\pi$
$212$	2223.47	−	1283.72i	0.720323	−	0.415879i
$213$		0			0
$214$	1079.88	−	1870.41i	0.344950	−	0.597471i
$215$	1023.26	+	1772.34i	0.324585	+	0.562197i
$216$		0			0
$217$	−1870.08	+	118.394i	−0.585021	+	0.0370372i
$218$			970.391i			0.301483i
$219$		0			0
$220$	−1372.70	−	792.527i	−0.420669	−	0.242873i
$221$	385.519	+	222.579i	0.117343	+	0.0677480i
$222$		0			0
$223$			1206.04i			0.362164i	0.983468	+	0.181082i	$0.0579599\pi$
$223$			1206.04i			0.362164i	−0.983468	+	0.181082i	$0.942040\pi$
$224$	−591.464	+	37.4451i	−0.176423	+	0.0111692i
$225$		0			0
$226$	1686.83	+	2921.68i	0.496488	+	0.859943i
$227$	236.529	−	409.681i	0.0691586	−	0.119786i	−0.829373	−	0.558696i	$-0.811302\pi$
$227$	236.529	−	409.681i	0.0691586	−	0.119786i	0.898531	+	0.438910i	$0.144635\pi$
$228$		0			0
$229$	−1774.94	+	1024.76i	−0.512188	+	0.295712i	−0.733733	−	0.679438i	$-0.762223\pi$
$229$	−1774.94	+	1024.76i	−0.512188	+	0.295712i	0.221545	+	0.975150i	$0.428890\pi$
$230$	−665.930			−0.190914
$231$		0			0
$232$	−1745.47			−0.493948
$233$	−1548.92	+	894.267i	−0.435506	+	0.251439i	−0.701689	−	0.712483i	$-0.747571\pi$
$233$	−1548.92	+	894.267i	−0.435506	+	0.251439i	0.266184	+	0.963922i	$0.414237\pi$
$234$		0			0
$235$	415.410	−	719.512i	0.115312	−	0.199727i
$236$	−210.262	−	364.184i	−0.0579952	−	0.100451i
$237$		0			0
$238$	179.362	+	269.732i	0.0488501	+	0.0734627i
$239$			4369.08i			1.18248i	0.806496	+	0.591239i	$0.201361\pi$
$239$			4369.08i			1.18248i	−0.806496	+	0.591239i	$0.798639\pi$
$240$		0			0
$241$	2425.16	+	1400.17i	0.648208	+	0.374243i	0.787769	−	0.615970i	$-0.211236\pi$
$241$	2425.16	+	1400.17i	0.648208	+	0.374243i	−0.139561	+	0.990213i	$0.544569\pi$
$242$	2169.30	+	1252.44i	0.576230	+	0.332687i
$243$		0			0
$244$		−	2177.86i		−	0.571408i
$245$	−2652.75	+	337.239i	−0.691748	+	0.0879405i
$246$		0			0
$247$	2585.34	+	4477.94i	0.665997	+	1.15354i
$248$	−404.709	+	700.976i	−0.103625	+	0.179484i
$249$		0			0
$250$	−2555.11	+	1475.19i	−0.646397	+	0.373198i
$251$	3078.51			0.774159			0.387080	−	0.922046i	$-0.373484\pi$
$251$	3078.51			0.774159			0.387080	+	0.922046i	$0.373484\pi$
$252$		0			0
$253$	2170.77			0.539428
$254$	2469.20	−	1425.59i	0.609966	−	0.352164i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	4.62749	+	8.01505i	0.00112317	+	0.00194539i	0.866586	−	0.499027i	$-0.166309\pi$
$257$	4.62749	+	8.01505i	0.00112317	+	0.00194539i	−0.865463	+	0.500972i	$0.832976\pi$
$258$		0			0
$259$	−4551.09	−	2256.95i	−1.09186	−	0.541468i
$260$			1587.43i			0.378647i
$261$		0			0
$262$	1998.22	+	1153.67i	0.471185	+	0.272039i
$263$	−650.240	−	375.416i	−0.152454	−	0.0880196i	0.421832	−	0.906674i	$-0.361387\pi$
$263$	−650.240	−	375.416i	−0.152454	−	0.0880196i	−0.574287	+	0.818654i	$0.694720\pi$
$264$		0			0
$265$		−	5004.09i		−	1.15999i
$266$	237.724	+	3754.97i	0.0547962	+	0.865533i
$267$		0			0
$268$	−437.797	−	758.287i	−0.0997862	−	0.172835i
$269$	332.781	−	576.394i	0.0754276	−	0.130644i	−0.825845	−	0.563898i	$-0.809301\pi$
$269$	332.781	−	576.394i	0.0754276	−	0.130644i	0.901272	+	0.433253i	$0.142635\pi$
$270$		0			0
$271$	−2885.66	+	1666.04i	−0.646832	+	0.373449i	−0.787242	−	0.616645i	$-0.788491\pi$
$271$	−2885.66	+	1666.04i	−0.646832	+	0.373449i	0.140409	+	0.990094i	$0.455158\pi$
$272$	139.922			0.0311912
$273$		0			0
$274$	−3936.23			−0.867870
$275$	2826.79	−	1632.05i	0.619861	−	0.357877i
$276$		0			0
$277$	2828.26	−	4898.68i	0.613478	−	1.06258i	−0.377171	−	0.926144i	$-0.623103\pi$
$277$	2828.26	−	4898.68i	0.613478	−	1.06258i	0.990649	−	0.136432i	$-0.0435635\pi$
$278$	821.420	+	1422.74i	0.177214	+	0.306944i
$279$		0			0
$280$	−513.194	+	1034.84i	−0.109533	+	0.220870i
$281$			4156.44i			0.882392i	0.897411	+	0.441196i	$0.145446\pi$
$281$			4156.44i			0.882392i	−0.897411	+	0.441196i	$0.854554\pi$
$282$		0			0
$283$	−5249.78	−	3030.96i	−1.10271	−	0.636650i	−0.165779	−	0.986163i	$-0.553014\pi$
$283$	−5249.78	−	3030.96i	−1.10271	−	0.636650i	−0.936932	+	0.349512i	$0.886347\pi$
$284$	1877.94	+	1084.23i	0.392378	+	0.226540i
$285$		0			0
$286$		−	5174.63i		−	1.06987i
$287$	98.7522	−	65.6668i	0.0203107	−	0.0135059i
$288$		0			0
$289$	2418.26	+	4188.55i	0.492217	+	0.852545i
$290$	−1701.01	+	2946.24i	−0.344438	+	0.596583i
$291$		0			0
$292$	−3112.62	+	1797.07i	−0.623810	+	0.360157i
$293$	−5099.64			−1.01681			−0.508403	−	0.861119i	$-0.669764\pi$
$293$	−5099.64			−1.01681			−0.508403	+	0.861119i	$0.669764\pi$
$294$		0			0
$295$	−819.623			−0.161764
$296$	−1900.36	+	1097.17i	−0.373163	+	0.215446i
$297$		0			0
$298$	2698.96	−	4674.73i	0.524653	−	0.908725i
$299$	−1087.01	−	1882.76i	−0.210246	−	0.364157i
$300$		0			0
$301$	−4048.26	+	2691.95i	−0.775210	+	0.515487i
$302$			2545.82i			0.485085i
$303$		0			0
$304$	1407.50	+	812.620i	0.265545	+	0.153312i
$305$	−3676.09	−	2122.39i	−0.690138	−	0.398451i
$306$		0			0
$307$		−	4775.43i		−	0.887778i	−0.896082	−	0.443889i	$-0.853598\pi$
$307$		−	4775.43i		−	0.887778i	0.896082	−	0.443889i	$-0.146402\pi$
$308$	1672.89	−	3373.34i	0.309486	−	0.624070i
$309$		0			0
$310$	788.800	+	1366.24i	0.144519	+	0.250314i
$311$	5031.96	−	8715.62i	0.917481	−	1.58912i	0.114253	−	0.993452i	$-0.463553\pi$
$311$	5031.96	−	8715.62i	0.917481	−	1.58912i	0.803228	−	0.595672i	$-0.203114\pi$
$312$		0			0
$313$	−8486.84	+	4899.88i	−1.53260	+	0.884848i	−0.533361	+	0.845888i	$0.679071\pi$
$313$	−8486.84	+	4899.88i	−1.53260	+	0.884848i	−0.999241	+	0.0389605i	$0.987595\pi$
$314$	6200.46			1.11437
$315$		0			0
$316$	4488.90			0.799115
$317$	−3218.25	+	1858.06i	−0.570205	+	0.329208i	−0.757231	−	0.653147i	$-0.773448\pi$
$317$	−3218.25	+	1858.06i	−0.570205	+	0.329208i	0.187026	+	0.982355i	$0.440115\pi$
$318$		0			0
$319$	5544.89	−	9604.03i	0.973211	−	1.68565i
$320$	249.479	+	432.111i	0.0435822	+	0.0754866i
$321$		0			0
$322$	−99.9516	−	1578.78i	−0.0172984	−	0.273237i
$323$		−	888.306i		−	0.153024i
$324$		0			0
$325$	−2831.02	−	1634.49i	−0.483190	−	0.278970i
$326$	4904.49	+	2831.61i	0.833235	+	0.481068i
$327$		0			0
$328$		−	51.2270i		−	0.00862360i
$329$	1768.17	+	876.859i	0.296298	+	0.146939i
$330$		0			0
$331$	−3354.20	−	5809.64i	−0.556989	−	0.964734i	−0.997746	−	0.0671070i	$-0.978623\pi$
$331$	−3354.20	−	5809.64i	−0.556989	−	0.964734i	0.440757	−	0.897627i	$-0.354710\pi$
$332$	−1711.35	+	2964.15i	−0.282899	+	0.489996i
$333$		0			0
$334$	6660.18	−	3845.26i	1.09110	−	0.629949i
$335$	−1706.58			−0.278330
$336$		0			0
$337$	−735.278			−0.118852			−0.0594260	−	0.998233i	$-0.518927\pi$
$337$	−735.278			−0.118852			−0.0594260	+	0.998233i	$0.518927\pi$
$338$	−682.764	+	394.194i	−0.109874	+	0.0634359i
$339$		0			0
$340$	136.358	−	236.178i	0.0217501	−	0.0376722i
$341$	−2571.30	−	4453.62i	−0.408339	−	0.707264i
$342$		0			0
$343$	−1197.69	−	6238.52i	−0.188539	−	0.982066i
$344$			2100.01i			0.329142i
$345$		0			0
$346$	−3804.95	−	2196.79i	−0.591201	−	0.341330i
$347$	−9227.87	−	5327.71i	−1.42760	−	0.824226i	−0.430671	−	0.902509i	$-0.641723\pi$
$347$	−9227.87	−	5327.71i	−1.42760	−	0.824226i	−0.996931	+	0.0782826i	$0.975056\pi$
$348$		0			0
$349$		−	10961.2i		−	1.68120i	−0.541653	−	0.840602i	$-0.682201\pi$
$349$		−	10961.2i		−	1.68120i	0.541653	−	0.840602i	$-0.317799\pi$
$350$	−1317.13	−	1980.75i	−0.201153	−	0.302502i
$351$		0			0
$352$	−813.242	−	1408.58i	−0.123142	−	0.213288i
$353$	1529.90	−	2649.87i	0.230676	−	0.399542i	−0.727331	−	0.686286i	$-0.759240\pi$
$353$	1529.90	−	2649.87i	0.230676	−	0.399542i	0.958007	+	0.286744i	$0.0925730\pi$
$354$		0			0
$355$	3660.21	−	2113.23i	0.547223	−	0.315939i
$356$	5265.54			0.783913
$357$		0			0
$358$	4619.35			0.681956
$359$	−5412.09	+	3124.67i	−0.795653	+	0.459370i	−0.841949	−	0.539557i	$-0.818592\pi$
$359$	−5412.09	+	3124.67i	−0.795653	+	0.459370i	0.0462959	+	0.998928i	$0.485258\pi$
$360$		0			0
$361$	1729.50	−	2995.58i	0.252150	−	0.436736i
$362$	3240.23	+	5612.25i	0.470450	+	0.814844i
$363$		0			0
$364$	−3763.47	+	238.262i	−0.541921	+	0.0343086i
$365$			7005.19i			1.00457i
$366$		0			0
$367$	10383.2	+	5994.76i	1.47684	+	0.852654i	0.999658	−	0.0261512i	$-0.00832512\pi$
$367$	10383.2	+	5994.76i	1.47684	+	0.852654i	0.477181	+	0.878805i	$0.341658\pi$
$368$	−591.787	−	341.668i	−0.0838288	−	0.0483986i
$369$		0			0
$370$			4276.91i			0.600935i
$371$	11863.7	−	751.079i	1.66019	−	0.105105i
$372$		0			0
$373$	1431.68	+	2479.73i	0.198738	+	0.344225i	0.948120	−	0.317914i	$-0.102982\pi$
$373$	1431.68	+	2479.73i	0.198738	+	0.344225i	−0.749381	+	0.662139i	$0.769649\pi$
$374$	−444.493	+	769.884i	−0.0614550	+	0.106443i
$375$		0			0
$376$	738.318	−	426.268i	0.101266	−	0.0584657i
$377$	−11106.4			−1.51726
$378$		0			0
$379$	−7545.36			−1.02264			−0.511318	−	0.859392i	$-0.670843\pi$
$379$	−7545.36			−1.02264			−0.511318	+	0.859392i	$0.670843\pi$
$380$	2743.30	−	1583.84i	0.370337	−	0.213814i
$381$		0			0
$382$	109.976	−	190.484i	0.0147300	−	0.0255131i
$383$	−1640.98	−	2842.25i	−0.218929	−	0.379197i	0.735552	−	0.677469i	$-0.236923\pi$
$383$	−1640.98	−	2842.25i	−0.218929	−	0.379197i	−0.954481	+	0.298272i	$0.903590\pi$
$384$		0			0
$385$	−4063.69	−	6111.13i	−0.537934	−	0.808966i
$386$		−	9997.71i		−	1.31832i
$387$		0			0
$388$	−4298.02	−	2481.46i	−0.562368	−	0.324683i
$389$	−1010.94	−	583.664i	−0.131765	−	0.0760744i	0.432669	−	0.901553i	$-0.357572\pi$
$389$	−1010.94	−	583.664i	−0.131765	−	0.0760744i	−0.564433	+	0.825479i	$0.690905\pi$
$390$		0			0
$391$			373.490i			0.0483075i
$392$	−2530.43	−	1061.35i	−0.326035	−	0.136751i
$393$		0			0
$394$	508.689	+	881.076i	0.0650441	+	0.112660i
$395$	4374.56	−	7576.96i	0.557235	−	0.965160i
$396$		0			0
$397$	−11853.7	+	6843.75i	−1.49854	+	0.865183i	−0.999999	−	0.00168199i	$-0.999465\pi$
$397$	−11853.7	+	6843.75i	−1.49854	+	0.865183i	−0.498543	+	0.866865i	$0.666131\pi$
$398$	−2410.85			−0.303631
$399$		0			0
$400$	−1027.50			−0.128438
$401$	−3691.74	+	2131.42i	−0.459742	+	0.265432i	−0.711936	−	0.702245i	$-0.752181\pi$
$401$	−3691.74	+	2131.42i	−0.459742	+	0.265432i	0.252194	+	0.967677i	$0.418848\pi$
$402$		0			0
$403$	−2575.15	+	4460.29i	−0.318306	+	0.551323i
$404$	1003.77	+	1738.58i	0.123612	+	0.214103i
$405$		0			0
$406$	−7240.25	−	3590.54i	−0.885043	−	0.438906i
$407$		−	13941.7i		−	1.69794i
$408$		0			0
$409$	9296.72	+	5367.46i	1.12394	+	0.648909i	0.942405	−	0.334474i	$-0.108559\pi$
$409$	9296.72	+	5367.46i	1.12394	+	0.648909i	0.181539	+	0.983384i	$0.441892\pi$
$410$	−86.4678	−	49.9222i	−0.0104155	−	0.00601337i
$411$		0			0
$412$			1409.25i			0.168516i
$413$	−123.020	−	1943.16i	−0.0146572	−	0.231517i
$414$		0			0
$415$	3335.52	+	5777.29i	0.394540	+	0.683364i
$416$	−814.460	+	1410.69i	−0.0959909	+	0.166261i
$417$		0			0
$418$	−8942.48	+	5162.94i	−1.04639	+	0.604133i
$419$	−894.104			−0.104248			−0.0521239	−	0.998641i	$-0.516599\pi$
$419$	−894.104			−0.104248			−0.0521239	+	0.998641i	$0.516599\pi$
$420$		0			0
$421$	4275.53			0.494956			0.247478	−	0.968893i	$-0.420398\pi$
$421$	4275.53			0.494956			0.247478	+	0.968893i	$0.420398\pi$
$422$	5416.95	−	3127.48i	0.624865	−	0.360766i
$423$		0			0
$424$	2567.44	−	4446.94i	0.294071	−	0.509345i
$425$	280.800	+	486.360i	0.0320490	+	0.0555105i
$426$		0			0
$427$	4480.00	−	9033.81i	0.507734	−	1.02383i
$428$		−	4319.54i		−	0.487833i
$429$		0			0
$430$	3544.67	+	2046.52i	0.397533	+	0.229516i
$431$	4324.96	+	2497.01i	0.483355	+	0.279065i	0.721813	−	0.692088i	$-0.243309\pi$
$431$	4324.96	+	2497.01i	0.483355	+	0.279065i	−0.238459	+	0.971153i	$0.576642\pi$
$432$		0			0
$433$		−	2469.93i		−	0.274128i	−0.990562	−	0.137064i	$-0.956233\pi$
$433$		−	2469.93i		−	0.274128i	0.990562	−	0.137064i	$-0.0437666\pi$
$434$	−3120.69	+	2075.15i	−0.345156	+	0.229517i
$435$		0			0
$436$	970.391	+	1680.77i	0.106590	+	0.184620i
$437$	−2169.11	+	3757.01i	−0.237443	+	0.411264i
$438$		0			0
$439$	−7186.67	+	4149.22i	−0.781323	+	0.451097i	−0.836899	−	0.547357i	$-0.815634\pi$
$439$	−7186.67	+	4149.22i	−0.781323	+	0.451097i	0.0555758	+	0.998454i	$0.482301\pi$
$440$	−3170.11			−0.343475
$441$		0			0
$442$	890.317			0.0958101
$443$	10784.7	−	6226.57i	1.15666	−	0.667795i	0.206155	−	0.978519i	$-0.433905\pi$
$443$	10784.7	−	6226.57i	1.15666	−	0.667795i	0.950500	+	0.310724i	$0.100572\pi$
$444$		0			0
$445$	5131.42	−	8887.87i	0.546635	−	0.946799i
$446$	1206.04	+	2088.93i	0.128044	+	0.221779i
$447$		0			0
$448$	−987.001	+	656.321i	−0.104088	+	0.0692149i
$449$		−	7411.31i		−	0.778979i	−0.921031	−	0.389489i	$-0.872652\pi$
$449$		−	7411.31i		−	0.778979i	0.921031	−	0.389489i	$-0.127348\pi$
$450$		0			0
$451$	281.864	+	162.734i	0.0294289	+	0.0169908i
$452$	5843.36	+	3373.66i	0.608072	+	0.351070i
$453$		0			0
$454$		−	946.118i		−	0.0978051i
$455$	−3265.44	+	6584.67i	−0.336453	+	0.678449i
$456$		0			0
$457$	6156.75	+	10663.8i	0.630198	+	1.09154i	0.987511	+	0.157551i	$0.0503597\pi$
$457$	6156.75	+	10663.8i	0.630198	+	1.09154i	−0.357313	+	0.933985i	$0.616307\pi$
$458$	−2049.52	+	3549.87i	−0.209100	+	0.362171i
$459$		0			0
$460$	−1153.43	+	665.930i	−0.116910	+	0.0674982i
$461$	16338.5			1.65067			0.825334	−	0.564645i	$-0.190987\pi$
$461$	16338.5			1.65067			0.825334	+	0.564645i	$0.190987\pi$
$462$		0			0
$463$	13862.3			1.39144			0.695720	−	0.718313i	$-0.255086\pi$
$463$	13862.3			1.39144			0.695720	+	0.718313i	$0.255086\pi$
$464$	−3023.25	+	1745.47i	−0.302480	+	0.174637i
$465$		0			0
$466$	−1788.53	+	3097.83i	−0.177794	+	0.307949i
$467$	8822.10	+	15280.3i	0.874172	+	1.51411i	0.857643	+	0.514246i	$0.171928\pi$
$467$	8822.10	+	15280.3i	0.874172	+	1.51411i	0.0165291	+	0.999863i	$0.494738\pi$
$468$		0			0
$469$	−256.146	−	4045.96i	−0.0252191	−	0.398347i
$470$		−	1661.64i		−	0.163076i
$471$		0			0
$472$	−728.368	−	420.523i	−0.0710293	−	0.0410088i
$473$	−11554.8	−	6671.15i	−1.12323	−	0.648499i
$474$		0			0
$475$			6523.20i			0.630115i
$476$	580.397	+	287.827i	0.0558875	+	0.0277154i
$477$		0			0
$478$	4369.08	+	7567.47i	0.418069	+	0.724117i
$479$	−2801.64	+	4852.59i	−0.267245	+	0.462882i	−0.968149	−	0.250373i	$-0.919447\pi$
$479$	−2801.64	+	4852.59i	−0.267245	+	0.462882i	0.700904	+	0.713255i	$0.252780\pi$
$480$		0			0
$481$	−12091.9	+	6981.29i	−1.14625	+	0.661786i
$482$	5600.66			0.529260
$483$		0			0
$484$	5009.78			0.470490
$485$	−8377.08	+	4836.51i	−0.784296	+	0.452813i
$486$		0			0
$487$	733.532	−	1270.51i	0.0682536	−	0.118219i	−0.829879	−	0.557943i	$-0.811591\pi$
$487$	733.532	−	1270.51i	0.0682536	−	0.118219i	0.898133	+	0.439725i	$0.144924\pi$
$488$	−2177.86	−	3772.17i	−0.202023	−	0.349914i
$489$		0			0
$490$	−4257.47	+	3236.87i	−0.392516	+	0.298422i
$491$		−	379.554i		−	0.0348860i	−0.999848	−	0.0174430i	$-0.994447\pi$
$491$		−	379.554i		−	0.0348860i	0.999848	−	0.0174430i	$-0.00555256\pi$
$492$		0			0
$493$	1652.41	+	954.021i	0.150955	+	0.0871541i
$494$	8955.88	+	5170.68i	0.815676	+	0.470931i
$495$		0			0
$496$			1618.84i			0.146548i
$497$	5559.40	+	8360.44i	0.501757	+	0.754561i
$498$		0			0
$499$	6499.43	+	11257.3i	0.583075	+	1.00992i	0.995112	+	0.0987491i	$0.0314841\pi$
$499$	6499.43	+	11257.3i	0.583075	+	1.00992i	−0.412037	+	0.911167i	$0.635183\pi$
$500$	−2950.39	+	5110.22i	−0.263891	+	0.457072i
$501$		0			0
$502$	5332.14	−	3078.51i	0.474074	−	0.273707i
$503$	−10899.0			−0.966129			−0.483064	−	0.875585i	$-0.660476\pi$
$503$	−10899.0			−0.966129			−0.483064	+	0.875585i	$0.660476\pi$
$504$		0			0
$505$	3912.81			0.344787
$506$	3759.89	−	2170.77i	0.330331	−	0.190717i
$507$		0			0
$508$	2851.19	−	4938.40i	0.249018	−	0.431311i
$509$	−8142.34	−	14103.0i	−0.709043	−	1.22810i	−0.965212	−	0.261467i	$-0.915794\pi$
$509$	−8142.34	−	14103.0i	−0.709043	−	1.22810i	0.256169	−	0.966632i	$-0.417540\pi$
$510$		0			0
$511$	−16607.9	+	1051.43i	−1.43775	+	0.0910227i
$512$			512.000i			0.0441942i
$513$		0			0
$514$	16.0301	+	9.25498i	0.00137560	+	0.000794201i
$515$	2378.71	+	1373.35i	0.203531	+	0.117509i
$516$		0			0
$517$			5416.55i			0.460773i
$518$	−10139.7	+	641.935i	−0.860061	+	0.0544498i
$519$		0			0
$520$	1587.43	+	2749.51i	0.133872	+	0.231873i
$521$	−7475.64	+	12948.2i	−0.628626	+	1.08881i	0.359202	+	0.933260i	$0.383049\pi$
$521$	−7475.64	+	12948.2i	−0.628626	+	1.08881i	−0.987828	+	0.155552i	$0.950284\pi$
$522$		0			0
$523$	−11840.7	+	6836.22i	−0.989975	+	0.571562i	−0.905267	−	0.424844i	$-0.860329\pi$
$523$	−11840.7	+	6836.22i	−0.989975	+	0.571562i	−0.0847081	+	0.996406i	$0.526996\pi$
$524$	4614.70			0.384721
$525$		0			0
$526$	−1501.66			−0.124478
$527$	766.264	−	442.403i	0.0633377	−	0.0365680i
$528$		0			0
$529$	−5171.49	+	8957.29i	−0.425042	+	0.736195i
$530$	−5004.09	−	8667.33i	−0.410120	−	0.710349i
$531$		0			0
$532$	4166.72	+	6266.07i	0.339568	+	0.510655i
$533$		−	325.956i		−	0.0264892i
$534$		0			0
$535$	−7291.08	−	4209.51i	−0.589198	−	0.340174i
$536$	−1516.57	−	875.594i	−0.122213	−	0.0705595i
$537$		0			0
$538$		−	1331.12i		−	0.106671i
$539$	13878.3	−	10551.4i	1.10906	−	0.843194i
$540$		0			0
$541$	−3105.32	−	5378.57i	−0.246780	−	0.427436i	0.715850	−	0.698254i	$-0.246039\pi$
$541$	−3105.32	−	5378.57i	−0.246780	−	0.427436i	−0.962631	+	0.270818i	$0.912706\pi$
$542$	−3332.08	+	5771.33i	−0.264068	+	0.457379i
$543$		0			0
$544$	242.351	−	139.922i	0.0191006	−	0.0110277i
$545$	3782.69			0.297308
$546$		0			0
$547$	10396.4			0.812649			0.406324	−	0.913729i	$-0.366810\pi$
$547$	10396.4			0.812649			0.406324	+	0.913729i	$0.366810\pi$
$548$	−6817.75	+	3936.23i	−0.531459	+	0.306838i
$549$		0			0
$550$	3264.09	−	5653.57i	0.253057	−	0.438308i
$551$	11081.3	+	19193.4i	0.856768	+	1.48397i
$552$		0			0
$553$	18620.0	+	9233.94i	1.43183	+	0.710067i
$554$		−	11313.0i		−	0.867589i
$555$		0			0
$556$	2845.48	+	1642.84i	0.217042	+	0.125309i
$557$	−11076.9	−	6395.26i	−0.842628	−	0.486492i	0.0155285	−	0.999879i	$-0.495057\pi$
$557$	−11076.9	−	6395.26i	−0.842628	−	0.486492i	−0.858157	+	0.513388i	$0.828390\pi$
$558$		0			0
$559$			13362.3i			1.01103i
$560$	145.965	+	2305.59i	0.0110146	+	0.173981i
$561$		0			0
$562$	4156.44	+	7199.16i	0.311973	+	0.540353i
$563$	3100.58	−	5370.36i	0.232103	−	0.402014i	−0.726324	−	0.687353i	$-0.758773\pi$
$563$	3100.58	−	5370.36i	0.232103	−	0.402014i	0.958427	+	0.285339i	$0.0921061\pi$
$564$		0			0
$565$	11389.0	−	6575.46i	0.848035	−	0.489613i
$566$	−12123.9			−0.900360
$567$		0			0
$568$	4336.92			0.320375
$569$	−22006.0	+	12705.2i	−1.62134	+	0.936079i	−0.634774	+	0.772698i	$0.718907\pi$
$569$	−22006.0	+	12705.2i	−1.62134	+	0.936079i	−0.986563	+	0.163381i	$0.947760\pi$
$570$		0			0
$571$	−4142.12	+	7174.37i	−0.303577	+	0.525811i	−0.976944	−	0.213498i	$-0.931514\pi$
$571$	−4142.12	+	7174.37i	−0.303577	+	0.525811i	0.673366	+	0.739309i	$0.264848\pi$
$572$	−5174.63	−	8962.73i	−0.378256	−	0.655158i
$573$		0			0
$574$	105.377	−	212.490i	0.00766264	−	0.0154515i
$575$		−	2742.69i		−	0.198919i
$576$		0			0
$577$	−1424.66	−	822.525i	−0.102789	−	0.0593452i	0.447724	−	0.894172i	$-0.352235\pi$
$577$	−1424.66	−	822.525i	−0.102789	−	0.0593452i	−0.550513	+	0.834826i	$0.685568\pi$
$578$	8377.10	+	4836.52i	0.602840	+	0.348050i
$579$		0			0
$580$			6804.05i			0.487108i
$581$	−13196.1	+	8774.97i	−0.942286	+	0.626587i
$582$		0			0
$583$	16312.1	+	28253.4i	1.15880	+	2.00710i
$584$	−3594.15	+	6225.25i	−0.254669	+	0.441100i
$585$		0			0
$586$	−8832.83	+	5099.64i	−0.622664	+	0.359495i
$587$	21115.2			1.48470			0.742350	−	0.670013i	$-0.233711\pi$
$587$	21115.2			1.48470			0.742350	+	0.670013i	$0.233711\pi$
$588$		0			0
$589$	10277.3			0.718964
$590$	−1419.63	+	819.623i	−0.0990597	+	0.0571921i
$591$		0			0
$592$	−2194.35	+	3800.72i	−0.152343	+	0.263866i
$593$	6622.57	+	11470.6i	0.458611	+	0.794338i	0.998888	−	0.0471494i	$-0.0150137\pi$
$593$	6622.57	+	11470.6i	0.458611	+	0.794338i	−0.540277	+	0.841488i	$0.681680\pi$
$594$		0			0
$595$	1051.45	−	699.174i	0.0724455	−	0.0481737i
$596$		−	10795.8i		−	0.741971i
$597$		0			0
$598$	−3765.52	−	2174.02i	−0.257498	−	0.148666i
$599$	11800.0	+	6812.73i	0.804900	+	0.464709i	0.845182	−	0.534479i	$-0.179492\pi$
$599$	11800.0	+	6812.73i	0.804900	+	0.464709i	−0.0402819	+	0.999188i	$0.512826\pi$
$600$		0			0
$601$		−	15150.8i		−	1.02831i	−0.857697	−	0.514156i	$-0.828105\pi$
$601$		−	15150.8i		−	1.02831i	0.857697	−	0.514156i	$-0.171895\pi$
$602$	−4319.85	+	8710.87i	−0.292465	+	0.589748i
$603$		0			0
$604$	2545.82	+	4409.49i	0.171503	+	0.297052i
$605$	4882.17	−	8456.17i	0.328080	−	0.568251i
$606$		0			0
$607$	−3988.38	+	2302.69i	−0.266694	+	0.153976i	−0.627384	−	0.778710i	$-0.715875\pi$
$607$	−3988.38	+	2302.69i	−0.266694	+	0.153976i	0.360690	+	0.932686i	$0.382541\pi$
$608$	3250.48			0.216816
$609$		0			0
$610$	−8489.56			−0.563495
$611$	4697.90	−	2712.33i	0.311058	−	0.179590i
$612$		0			0
$613$	2333.08	−	4041.01i	0.153723	−	0.266256i	−0.778870	−	0.627185i	$-0.784207\pi$
$613$	2333.08	−	4041.01i	0.153723	−	0.266256i	0.932593	+	0.360929i	$0.117540\pi$
$614$	−4775.43	−	8271.28i	−0.313877	−	0.543651i
$615$		0			0
$616$	−475.811	−	7515.68i	−0.0311217	−	0.491583i
$617$			16316.4i			1.06463i	0.846548	+	0.532313i	$0.178677\pi$
$617$			16316.4i			1.06463i	−0.846548	+	0.532313i	$0.821323\pi$
$618$		0			0
$619$	4523.82	+	2611.83i	0.293744	+	0.169593i	0.639629	−	0.768684i	$-0.279088\pi$
$619$	4523.82	+	2611.83i	0.293744	+	0.169593i	−0.345885	+	0.938277i	$0.612421\pi$
$620$	2732.48	+	1577.60i	0.176999	+	0.102190i
$621$		0			0
$622$		−	20127.9i		−	1.29751i
$623$	21841.5	+	10831.5i	1.40459	+	0.696559i
$624$		0			0
$625$	1736.79	+	3008.20i	0.111154	+	0.192525i
$626$	−9799.76	+	16973.7i	−0.625682	+	1.08371i
$627$		0			0
$628$	10739.5	−	6200.46i	0.682409	−	0.393989i
$629$	2398.72			0.152056
$630$		0			0
$631$	−12467.8			−0.786584			−0.393292	−	0.919414i	$-0.628664\pi$
$631$	−12467.8			−0.786584			−0.393292	+	0.919414i	$0.628664\pi$
$632$	7775.00	−	4488.90i	0.489356	−	0.282530i
$633$		0			0
$634$	−3716.11	+	6436.50i	−0.232785	+	0.403196i
$635$	−5557.12	−	9625.22i	−0.347288	−	0.601520i
$636$		0			0
$637$	−16101.0	−	6753.37i	−1.00149	−	0.420060i
$638$		−	22179.6i		−	1.37633i
$639$		0			0
$640$	864.221	+	498.958i	0.0533771	+	0.0308173i
$641$	−5896.71	−	3404.47i	−0.363348	−	0.209779i	0.307200	−	0.951645i	$-0.400608\pi$
$641$	−5896.71	−	3404.47i	−0.363348	−	0.209779i	−0.670548	+	0.741866i	$0.733941\pi$
$642$		0			0
$643$			8746.71i			0.536449i	0.963356	+	0.268224i	$0.0864369\pi$
$643$			8746.71i			0.536449i	−0.963356	+	0.268224i	$0.913563\pi$
$644$	−1751.91	−	2634.58i	−0.107197	−	0.161207i
$645$		0			0
$646$	−888.306	−	1538.59i	−0.0541020	−	0.0937075i
$647$	5728.79	−	9922.55i	0.348102	−	0.602930i	−0.637810	−	0.770193i	$-0.720160\pi$
$647$	5728.79	−	9922.55i	0.348102	−	0.602930i	0.985912	+	0.167263i	$0.0534930\pi$
$648$		0			0
$649$	4627.65	−	2671.77i	0.279894	−	0.161597i
$650$	−6537.97			−0.394523
$651$		0			0
$652$	11326.4			0.680333
$653$	−19349.2	+	11171.3i	−1.15956	+	0.669472i	−0.951198	−	0.308580i	$-0.900146\pi$
$653$	−19349.2	+	11171.3i	−1.15956	+	0.669472i	−0.208361	+	0.978052i	$0.566813\pi$
$654$		0			0
$655$	4497.15	−	7789.29i	0.268272	−	0.464661i
$656$	−51.2270	−	88.7278i	−0.00304890	−	0.00528085i
$657$		0			0
$658$	3939.41	−	249.401i	0.233395	−	0.0147761i
$659$			6944.96i			0.410527i	0.978707	+	0.205263i	$0.0658051\pi$
$659$			6944.96i			0.410527i	−0.978707	+	0.205263i	$0.934195\pi$
$660$		0			0
$661$	−16338.1	−	9432.82i	−0.961391	−	0.555059i	−0.0647898	−	0.997899i	$-0.520638\pi$
$661$	−16338.1	−	9432.82i	−0.961391	−	0.555059i	−0.896601	+	0.442840i	$0.853971\pi$
$662$	−11619.3	−	6708.40i	−0.682170	−	0.393851i
$663$		0			0
$664$			6845.41i			0.400080i
$665$	14637.3	−	926.675i	0.853548	−	0.0540374i
$666$		0			0
$667$	−4659.16	−	8069.90i	−0.270470	−	0.468468i
$668$	7690.51	−	13320.4i	0.445441	−	0.771527i
$669$		0			0
$670$	−2955.89	+	1706.58i	−0.170442	+	0.0984045i
$671$	27673.9			1.59216
$672$		0			0
$673$	5622.15			0.322018			0.161009	−	0.986953i	$-0.448525\pi$
$673$	5622.15			0.322018			0.161009	+	0.986953i	$0.448525\pi$
$674$	−1273.54	+	735.278i	−0.0727817	+	0.0420206i
$675$		0			0
$676$	−788.388	+	1365.53i	−0.0448560	+	0.0776928i
$677$	13723.5	+	23769.8i	0.779081	+	1.34941i	0.932472	+	0.361243i	$0.117647\pi$
$677$	13723.5	+	23769.8i	0.779081	+	1.34941i	−0.153391	+	0.988166i	$0.549019\pi$
$678$		0			0
$679$	−12723.7	−	19134.4i	−0.719133	−	1.08146i
$680$		−	545.430i		−	0.0307593i
$681$		0			0
$682$	−8907.24	−	5142.60i	−0.500111	−	0.288739i
$683$	−27472.6	−	15861.3i	−1.53911	−	0.888604i	−0.998891	−	0.0470784i	$-0.985009\pi$
$683$	−27472.6	−	15861.3i	−1.53911	−	0.888604i	−0.540217	−	0.841526i	$-0.681658\pi$
$684$		0			0
$685$			15343.9i			0.855852i
$686$	−8312.97	−	9607.75i	−0.462669	−	0.534731i
$687$		0			0
$688$	2100.01	+	3637.32i	0.116369	+	0.201558i
$689$	16336.6	−	28295.7i	0.903299	−	1.56456i
$690$		0			0
$691$	6896.96	−	3981.96i	0.379700	−	0.219220i	−0.297988	−	0.954570i	$-0.596315\pi$
$691$	6896.96	−	3981.96i	0.379700	−	0.219220i	0.677688	+	0.735350i	$0.262982\pi$
$692$	−8787.16			−0.482714
$693$		0			0
$694$	−21310.8			−1.16563
$695$	5546.01	−	3201.99i	0.302694	−	0.174760i
$696$		0			0
$697$	−27.9991	+	48.4959i	−0.00152158	+	0.00263545i
$698$	−10961.2	−	18985.4i	−0.594395	−	1.02952i
$699$		0			0
$700$	−4262.09	−	2113.63i	−0.230131	−	0.114125i
$701$			21374.8i			1.15166i	0.817568	+	0.575831i	$0.195322\pi$
$701$			21374.8i			1.15166i	−0.817568	+	0.575831i	$0.804678\pi$
$702$		0			0
$703$	24129.2	+	13931.0i	1.29453	+	0.747395i
$704$	−2817.15	−	1626.48i	−0.150817	−	0.0870744i
$705$		0			0
$706$		−	6119.61i		−	0.326225i
$707$	587.286	+	9276.47i	0.0312407	+	0.493462i
$708$		0			0
$709$	−3514.63	−	6087.51i	−0.186170	−	0.322456i	0.757800	−	0.652487i	$-0.226274\pi$
$709$	−3514.63	−	6087.51i	−0.186170	−	0.322456i	−0.943970	+	0.330031i	$0.892941\pi$
$710$	4226.45	−	7320.43i	0.223403	−	0.386945i
$711$		0			0
$712$	9120.18	−	5265.54i	0.480047	−	0.277155i
$713$	−4321.13			−0.226967
$714$		0			0
$715$	−20171.3			−1.05505
$716$	8000.95	−	4619.35i	0.417611	−	0.241108i
$717$		0			0
$718$	−6249.35	+	10824.2i	−0.324824	+	0.562612i
$719$	−5369.20	−	9299.72i	−0.278494	−	0.482366i	0.692517	−	0.721402i	$-0.256502\pi$
$719$	−5369.20	−	9299.72i	−0.278494	−	0.482366i	−0.971011	+	0.239036i	$0.923169\pi$
$720$		0			0
$721$	−2898.90	+	5845.57i	−0.149738	+	0.301942i
$722$		−	6917.98i		−	0.356594i
$723$		0			0
$724$	11224.5	+	6480.47i	0.576181	+	0.332658i
$725$	−12134.4	−	7005.77i	−0.621598	−	0.358880i
$726$		0			0
$727$		−	15722.6i		−	0.802092i	−0.916058	−	0.401046i	$-0.868647\pi$
$727$		−	15722.6i		−	0.802092i	0.916058	−	0.401046i	$-0.131353\pi$
$728$	−6280.26	+	4176.15i	−0.319728	+	0.212608i
$729$		0			0
$730$	7005.19	+	12133.4i	0.355170	+	0.615172i
$731$	1147.80	−	1988.05i	0.0580751	−	0.100589i
$732$		0			0
$733$	−14966.9	+	8641.17i	−0.754184	+	0.435428i	−0.827204	−	0.561902i	$-0.810070\pi$
$733$	−14966.9	+	8641.17i	−0.754184	+	0.435428i	0.0730197	+	0.997331i	$0.476736\pi$
$734$	23979.0			1.20583
$735$		0			0
$736$	−1366.67			−0.0684459
$737$	9635.47	−	5563.04i	0.481584	−	0.278043i
$738$		0			0
$739$	17235.3	−	29852.5i	0.857932	−	1.48598i	−0.0159660	−	0.999873i	$-0.505082\pi$
$739$	17235.3	−	29852.5i	0.857932	−	1.48598i	0.873898	−	0.486109i	$-0.161584\pi$
$740$	4276.91	+	7407.82i	0.212462	+	0.367996i
$741$		0			0
$742$	19797.4	−	13164.6i	0.979495	−	0.651330i
$743$			33600.2i			1.65904i	0.558474	+	0.829522i	$0.311387\pi$
$743$			33600.2i			1.65904i	−0.558474	+	0.829522i	$0.688613\pi$
$744$		0			0
$745$	−18222.6	−	10520.8i	−0.896142	−	0.517388i
$746$	4959.47	+	2863.35i	0.243404	+	0.140529i
$747$		0			0
$748$			1777.97i			0.0869105i
$749$	8885.55	−	17917.5i	0.433472	−	0.874087i
$750$		0			0
$751$	−17479.2	−	30274.9i	−0.849303	−	1.47104i	−0.881831	−	0.471565i	$-0.843689\pi$
$751$	−17479.2	−	30274.9i	−0.849303	−	1.47104i	0.0325279	−	0.999471i	$-0.489644\pi$
$752$	852.536	−	1476.64i	0.0413415	−	0.0716056i
$753$		0			0
$754$	−19236.8	+	11106.4i	−0.929131	+	0.536434i
$755$	9923.90			0.478368
$756$		0			0
$757$	38297.1			1.83875			0.919374	−	0.393386i	$-0.128696\pi$
$757$	38297.1			1.83875			0.919374	+	0.393386i	$0.128696\pi$
$758$	−13068.9	+	7545.36i	−0.626234	+	0.361556i
$759$		0			0
$760$	3167.68	−	5486.59i	0.151189	−	0.261868i
$761$	−9539.02	−	16522.1i	−0.454388	−	0.787023i	0.544265	−	0.838913i	$-0.316809\pi$
$761$	−9539.02	−	16522.1i	−0.454388	−	0.787023i	−0.998653	+	0.0518907i	$0.983475\pi$
$762$		0			0
$763$	567.757	+	8967.99i	0.0269386	+	0.425509i
$764$		−	439.903i		−	0.0208313i
$765$		0			0
$766$	−5684.51	−	3281.95i	−0.268133	−	0.154806i
$767$	−4634.58	−	2675.78i	−0.218181	−	0.125967i
$768$		0			0
$769$			12553.3i			0.588667i	0.955703	+	0.294334i	$0.0950977\pi$
$769$			12553.3i			0.588667i	−0.955703	+	0.294334i	$0.904902\pi$
$770$	−13149.6	−	6521.10i	−0.615429	−	0.305200i
$771$		0			0
$772$	−9997.71	−	17316.5i	−0.466095	−	0.807300i
$773$	−4533.66	+	7852.53i	−0.210950	+	0.365376i	−0.952012	−	0.306060i	$-0.900989\pi$
$773$	−4533.66	+	7852.53i	−0.210950	+	0.365376i	0.741062	+	0.671437i	$0.234322\pi$
$774$		0			0
$775$	−5626.99	+	3248.74i	−0.260810	+	0.150579i
$776$	−9925.85			−0.459172
$777$		0			0
$778$	−2334.66			−0.107585
$779$	−563.297	+	325.220i	−0.0259078	+	0.0149579i
$780$		0			0
$781$	−13777.2	+	23862.8i	−0.631226	+	1.09331i
$782$	373.490	+	646.904i	0.0170793	+	0.0295822i
$783$		0			0
$784$	−5444.18	+	692.108i	−0.248004	+	0.0315282i
$785$		−	24170.1i		−	1.09894i
$786$		0			0
$787$	1848.67	+	1067.33i	0.0837333	+	0.0483435i	0.541282	−	0.840841i	$-0.317939\pi$
$787$	1848.67	+	1067.33i	0.0837333	+	0.0483435i	−0.457549	+	0.889185i	$0.651272\pi$
$788$	1762.15	+	1017.38i	0.0796625	+	0.0459932i
$789$		0			0
$790$		−	17498.2i		−	0.788050i
$791$	17298.5	+	26014.1i	0.777577	+	1.16935i
$792$		0			0
$793$	−13857.7	−	24002.2i	−0.620556	−	1.07483i
$794$	−13687.5	+	23707.4i	−0.611777	+	1.05963i
$795$		0			0
$796$	−4175.72	+	2410.85i	−0.185935	+	0.107350i
$797$	16730.6			0.743574			0.371787	−	0.928318i	$-0.378745\pi$
$797$	16730.6			0.743574			0.371787	+	0.928318i	$0.378745\pi$
$798$		0			0
$799$	−931.939			−0.0412636
$800$	−1779.69	+	1027.50i	−0.0786518	+	0.0454096i
$801$		0			0
$802$	−4262.85	+	7383.47i	−0.187689	+	0.325087i
$803$	−22835.2	−	39551.8i	−1.00353	−	1.73817i
$804$		0			0
$805$	−6154.28	+	389.623i	−0.269453	+	0.0170589i
$806$			10300.6i			0.450153i
$807$		0			0
$808$	3477.16	+	2007.54i	0.151394	+	0.0874072i
$809$	21928.9	+	12660.6i	0.953001	+	0.550215i	0.894012	−	0.448043i	$-0.147879\pi$
$809$	21928.9	+	12660.6i	0.953001	+	0.550215i	0.0589890	+	0.998259i	$0.481212\pi$
$810$		0			0
$811$		−	23863.6i		−	1.03325i	−0.856212	−	0.516625i	$-0.827188\pi$
$811$		−	23863.6i		−	1.03325i	0.856212	−	0.516625i	$-0.172812\pi$
$812$	−16131.0	+	1021.24i	−0.697152	+	0.0441362i
$813$		0			0
$814$	−13941.7	−	24147.7i	−0.600314	−	1.03977i
$815$	11037.9	−	19118.3i	0.474407	−	0.821697i
$816$		0			0
$817$	23091.9	−	13332.1i	0.988841	−	0.570908i
$818$	21469.8			0.917696
$819$		0			0
$820$	−199.689			−0.00850419
$821$	−20798.2	+	12007.9i	−0.884121	+	0.510447i	−0.872015	−	0.489479i	$-0.837187\pi$
$821$	−20798.2	+	12007.9i	−0.884121	+	0.510447i	−0.0121059	+	0.999927i	$0.503854\pi$
$822$		0			0
$823$	16008.0	−	27726.6i	0.678011	−	1.17435i	−0.297569	−	0.954700i	$-0.596176\pi$
$823$	16008.0	−	27726.6i	0.678011	−	1.17435i	0.975579	−	0.219648i	$-0.0704910\pi$
$824$	1409.25	+	2440.89i	0.0595794	+	0.103195i
$825$		0			0
$826$	−2156.24	−	3242.63i	−0.0908294	−	0.136593i
$827$		−	33661.4i		−	1.41538i	−0.706522	−	0.707691i	$-0.749737\pi$
$827$		−	33661.4i		−	1.41538i	0.706522	−	0.707691i	$-0.250263\pi$
$828$		0			0
$829$	2750.33	+	1587.90i	0.115227	+	0.0665261i	0.556506	−	0.830844i	$-0.312142\pi$
$829$	2750.33	+	1587.90i	0.115227	+	0.0665261i	−0.441279	+	0.897370i	$0.645475\pi$
$830$	11554.6	+	6671.04i	0.483211	+	0.278982i
$831$		0			0
$832$			3257.84i			0.135752i
$833$	1815.41	+	2387.82i	0.0755107	+	0.0993194i
$834$		0			0
$835$	−14989.2	−	25962.1i	−0.621226	−	1.07600i
$836$	−10325.9	+	17885.0i	−0.427187	+	0.739909i
$837$		0			0
$838$	−1548.63	+	894.104i	−0.0638385	+	0.0368572i
$839$	−40349.4			−1.66033			−0.830165	−	0.557518i	$-0.811754\pi$
$839$	−40349.4			−1.66033			−0.830165	+	0.557518i	$0.811754\pi$
$840$		0			0
$841$	−23215.3			−0.951877
$842$	7405.43	−	4275.53i	0.303097	−	0.174993i
$843$		0			0
$844$	6254.96	−	10833.9i	0.255100	−	0.441846i
$845$	1536.61	+	2661.49i	0.0625575	+	0.108353i
$846$		0			0
$847$	20780.6	+	10305.4i	0.843012	+	0.418062i
$848$		−	10269.8i		−	0.415879i
$849$		0			0
$850$	972.721	+	561.601i	0.0392518	+	0.0226621i
$851$	−10145.2	−	5857.33i	−0.408664	−	0.235942i
$852$		0			0
$853$		−	46022.2i		−	1.84733i	−0.383204	−	0.923664i	$-0.625179\pi$
$853$		−	46022.2i		−	1.84733i	0.383204	−	0.923664i	$-0.374821\pi$
$854$	−1274.23	−	20127.0i	−0.0510575	−	0.806478i
$855$		0			0
$856$	−4319.54	−	7481.66i	−0.172475	−	0.298736i
$857$	7564.70	−	13102.4i	0.301523	−	0.522253i	−0.674958	−	0.737856i	$-0.735838\pi$
$857$	7564.70	−	13102.4i	0.301523	−	0.522253i	0.976481	+	0.215603i	$0.0691717\pi$
$858$		0			0
$859$	39374.0	−	22732.6i	1.56394	−	0.902941i	0.567087	−	0.823658i	$-0.308070\pi$
$859$	39374.0	−	22732.6i	1.56394	−	0.902941i	0.996852	−	0.0792832i	$-0.0252631\pi$
$860$	8186.07			0.324585
$861$		0			0
$862$	9988.06			0.394657
$863$	−5596.89	+	3231.36i	−0.220765	+	0.127459i	−0.606304	−	0.795233i	$-0.707349\pi$
$863$	−5596.89	+	3231.36i	−0.220765	+	0.127459i	0.385539	+	0.922691i	$0.374015\pi$
$864$		0			0
$865$	−8563.34	+	14832.1i	−0.336604	+	0.583015i
$866$	−2469.93	−	4278.05i	−0.0969189	−	0.167868i
$867$		0			0
$868$	−3330.04	+	6714.95i	−0.130218	+	0.262581i
$869$			57040.0i			2.22664i
$870$		0			0
$871$	−9649.91	−	5571.38i	−0.375402	−	0.216738i
$872$	3361.53	+	1940.78i	0.130546	+	0.0753706i
$873$		0			0
$874$			8676.45i			0.335796i
$875$	−22750.3	+	15128.1i	−0.878970	+	0.584485i
$876$		0			0
$877$	16030.7	+	27766.0i	0.617240	+	1.06909i	0.989987	+	0.141158i	$0.0450826\pi$
$877$	16030.7	+	27766.0i	0.617240	+	1.06909i	−0.372747	+	0.927933i	$0.621584\pi$
$878$	−8298.45	+	14373.3i	−0.318974	+	0.552479i
$879$		0			0
$880$	−5490.79	+	3170.11i	−0.210335	+	0.121437i
$881$	−17364.4			−0.664042			−0.332021	−	0.943272i	$-0.607730\pi$
$881$	−17364.4			−0.664042			−0.332021	+	0.943272i	$0.607730\pi$
$882$		0			0
$883$	−22921.5			−0.873578			−0.436789	−	0.899564i	$-0.643884\pi$
$883$	−22921.5			−0.873578			−0.436789	+	0.899564i	$0.643884\pi$
$884$	1542.07	−	890.317i	0.0586715	−	0.0338740i
$885$		0			0
$886$	12453.1	−	21569.5i	0.472202	−	0.817879i
$887$	19471.0	+	33724.7i	0.737058	+	1.27662i	0.953814	+	0.300397i	$0.0971191\pi$
$887$	19471.0	+	33724.7i	0.737058	+	1.27662i	−0.216756	+	0.976226i	$0.569548\pi$
$888$		0			0
$889$	21985.3	−	14619.5i	0.829432	−	0.551543i
$890$		−	20525.7i		−	0.773058i
$891$		0			0
$892$	4177.85	+	2412.08i	0.156822	+	0.0905410i
$893$	−9374.56	−	5412.41i	−0.351297	−	0.202821i
$894$		0			0
$895$		−	18006.8i		−	0.672513i
$896$	−1053.21	+	2123.78i	−0.0392695	+	0.0791859i
$897$		0			0
$898$	−7411.31	−	12836.8i	−0.275411	−	0.477025i
$899$	−11037.6	+	19117.7i	−0.409484	+	0.709246i
$900$		0			0
$901$	−4861.12	+	2806.57i	−0.179742	+	0.103774i
$902$	650.937			0.0240286
$903$		0			0
$904$	13494.7			0.496488
$905$	21877.2	−	12630.8i	0.803560	−	0.463936i
$906$		0			0
$907$	−19008.5	+	32923.6i	−0.695882	+	1.20530i	0.274000	+	0.961730i	$0.411653\pi$
$907$	−19008.5	+	32923.6i	−0.695882	+	1.20530i	−0.969882	+	0.243574i	$0.921680\pi$
$908$	−946.118	−	1638.72i	−0.0345793	−	0.0598931i
$909$		0			0
$910$	928.773	+	14670.4i	0.0338335	+	0.534418i
$911$			22877.1i			0.832000i	0.909365	+	0.416000i	$0.136568\pi$
$911$			22877.1i			0.832000i	−0.909365	+	0.416000i	$0.863432\pi$
$912$		0			0
$913$	−37665.1	−	21746.0i	−1.36532	−	0.788266i
$914$	21327.6	+	12313.5i	0.771832	+	0.445617i
$915$		0			0
$916$			8198.07i			0.295712i
$917$	19141.8	+	9492.70i	0.689333	+	0.341850i
$918$		0			0
$919$	314.315	+	544.409i	0.0112821	+	0.0195412i	0.871611	−	0.490198i	$-0.163075\pi$
$919$	314.315	+	544.409i	0.0112821	+	0.0195412i	−0.860329	+	0.509739i	$0.829742\pi$
$920$	−1331.86	+	2306.85i	−0.0477284	+	0.0826680i
$921$		0			0
$922$	28299.0	−	16338.5i	1.01082	−	0.583599i
$923$	27595.7			0.984099
$924$		0			0
$925$	−17614.8			−0.626132
$926$	24010.2	−	13862.3i	0.852079	−	0.491948i
$927$		0			0
$928$	−3490.95	+	6046.50i	−0.123487	+	0.213886i
$929$	21843.2	+	37833.5i	0.771423	+	1.33614i	0.936783	+	0.349911i	$0.113788\pi$
$929$	21843.2	+	37833.5i	0.771423	+	1.33614i	−0.165359	+	0.986233i	$0.552878\pi$
$930$		0			0
$931$	4393.91	+	34562.9i	0.154677	+	1.21671i
$932$			7154.13i			0.251439i
$933$		0			0
$934$	30560.7	+	17644.2i	1.07064	+	0.618133i
$935$	3001.09	+	1732.68i	0.104969	+	0.0606040i
$936$		0			0
$937$			27163.5i			0.947059i	0.880778	+	0.473529i	$0.157020\pi$
$937$			27163.5i			0.947059i	−0.880778	+	0.473529i	$0.842980\pi$
$938$	−4489.62	−	6751.66i	−0.156281	−	0.235021i
$939$		0			0
$940$	−1661.64	−	2878.05i	−0.0576561	−	0.0998633i
$941$	13299.2	−	23034.9i	0.460725	−	0.797999i	−0.538272	−	0.842771i	$-0.680923\pi$
$941$	13299.2	−	23034.9i	0.460725	−	0.797999i	0.998997	+	0.0447721i	$0.0142562\pi$
$942$		0			0
$943$	236.840	−	136.739i	0.00817875	−	0.00472200i
$944$	−1682.09			−0.0579952
$945$		0			0
$946$	−26684.6			−0.917116
$947$	17614.5	−	10169.7i	0.604429	−	0.348967i	−0.166353	−	0.986066i	$-0.553199\pi$
$947$	17614.5	−	10169.7i	0.604429	−	0.348967i	0.770782	+	0.637099i	$0.219866\pi$
$948$		0			0
$949$	−22869.5	+	39611.1i	−0.782270	+	1.35493i
$950$	6523.20	+	11298.5i	0.222779	+	0.385865i
$951$		0			0
$952$	1293.10	−	81.8654i	0.0440228	−	0.00278705i
$953$		−	45566.5i		−	1.54884i	−0.632673	−	0.774419i	$-0.718042\pi$
$953$		−	45566.5i		−	1.54884i	0.632673	−	0.774419i	$-0.281958\pi$
$954$		0			0
$955$	−742.527	−	428.698i	−0.0251598	−	0.0145260i
$956$	15134.9	+	8738.16i	0.512028	+	0.295620i
$957$		0			0
$958$			11206.6i			0.377942i
$959$	−36377.2	+	2303.01i	−1.22490	+	0.0775475i
$960$		0			0
$961$	−9777.08	−	16934.4i	−0.328189	−	0.568440i
$962$	−13962.6	+	24183.9i	−0.467954	+	0.810519i
$963$		0			0
$964$	9700.63	−	5600.66i	0.324104	−	0.187121i
$965$	−38972.2			−1.30006
$966$		0			0
$967$	−34217.5			−1.13791			−0.568955	−	0.822368i	$-0.692652\pi$
$967$	−34217.5			−1.13791			−0.568955	+	0.822368i	$0.692652\pi$
$968$	8677.19	−	5009.78i	0.288115	−	0.166343i
$969$		0			0
$970$	−9673.02	+	16754.2i	−0.320187	+	0.554581i
$971$	16355.4	+	28328.5i	0.540547	+	0.936255i	0.998873	+	0.0474705i	$0.0151160\pi$
$971$	16355.4	+	28328.5i	0.540547	+	0.936255i	−0.458326	+	0.888784i	$0.651551\pi$
$972$		0			0
$973$	8423.68	+	12667.9i	0.277544	+	0.417382i
$974$		−	2934.13i		−	0.0965252i
$975$		0			0
$976$	−7544.34	−	4355.73i	−0.247427	−	0.142852i
$977$	14095.0	+	8137.73i	0.461554	+	0.266478i	0.712697	−	0.701472i	$-0.247473\pi$
$977$	14095.0	+	8137.73i	0.461554	+	0.266478i	−0.251144	+	0.967950i	$0.580807\pi$
$978$		0			0
$979$			66908.7i			2.18428i
$980$	−4137.28	+	9863.89i	−0.134858	+	0.321521i
$981$		0			0
$982$	−379.554	−	657.406i	−0.0123341	−	0.0213632i
$983$	−16406.9	+	28417.6i	−0.532349	+	0.922056i	0.466938	+	0.884290i	$0.345357\pi$
$983$	−16406.9	+	28417.6i	−0.532349	+	0.922056i	−0.999287	+	0.0377653i	$0.987976\pi$
$984$		0			0
$985$	3434.53	−	1982.93i	0.111100	−	0.0641435i
$986$	3816.09			0.123254
$987$		0			0
$988$	20682.7			0.665997
$989$	−9709.04	+	5605.52i	−0.312163	+	0.180228i
$990$		0			0
$991$	4299.83	−	7447.52i	0.137829	−	0.238727i	−0.788846	−	0.614591i	$-0.789321\pi$
$991$	4299.83	−	7447.52i	0.137829	−	0.238727i	0.926675	+	0.375865i	$0.122654\pi$
$992$	1618.84	+	2803.91i	0.0518126	+	0.0897420i
$993$		0			0
$994$	17989.6	+	8921.31i	0.574040	+	0.284675i
$995$			9397.78i			0.299427i
$996$		0			0
$997$	27178.0	+	15691.2i	0.863326	+	0.498441i	0.865125	−	0.501557i	$-0.167239\pi$
$997$	27178.0	+	15691.2i	0.863326	+	0.498441i	−0.00179867	+	0.999998i	$0.500573\pi$
$998$	22514.7	+	12998.9i	0.714119	+	0.412297i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.c.269.6	yes	16
3.2	odd	2	inner	378.4.k.c.269.3	yes	16
7.5	odd	6	inner	378.4.k.c.215.3	✓	16
21.5	even	6	inner	378.4.k.c.215.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.c.215.3	✓	16	7.5	odd	6	inner
378.4.k.c.215.6	yes	16	21.5	even	6	inner
378.4.k.c.269.3	yes	16	3.2	odd	2	inner
378.4.k.c.269.6	yes	16	1.1	even	1	trivial

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.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.89811 - 6.75173i) q^{5} +(15.4219 - 10.2550i) q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-3.89811 - 6.75173i) q^{5} +(15.4219 - 10.2550i) q^{7} -8.00000i q^{8} +(-13.5035 - 7.79622i) q^{10} +(44.0180 + 25.4138i) q^{11} -50.9038i q^{13} +(16.4565 - 33.1841i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(-4.37255 + 7.57348i) q^{17} +(-87.9687 + 50.7888i) q^{19} -31.1849 q^{20} +101.655 q^{22} +(36.9867 - 21.3543i) q^{23} +(32.1094 - 55.6152i) q^{25} +(-50.9038 - 88.1679i) q^{26} +(-4.68064 - 73.9330i) q^{28} -218.184i q^{29} +(-87.6221 - 50.5886i) q^{31} +(-27.7128 - 16.0000i) q^{32} +17.4902i q^{34} +(-129.355 - 64.1492i) q^{35} +(-137.147 - 237.545i) q^{37} +(-101.578 + 175.937i) q^{38} +(-54.0138 + 31.1849i) q^{40} +6.40338 q^{41} -262.501 q^{43} +(176.072 - 101.655i) q^{44} +(42.7085 - 73.9733i) q^{46} +(53.2835 + 92.2898i) q^{47} +(132.669 - 316.303i) q^{49} -128.438i q^{50} +(-176.336 - 101.808i) q^{52} +(555.867 + 320.930i) q^{53} -396.263i q^{55} +(-82.0401 - 123.375i) q^{56} +(-218.184 - 377.906i) q^{58} +(52.5654 - 91.0459i) q^{59} +(471.522 - 272.233i) q^{61} -202.354 q^{62} -64.0000 q^{64} +(-343.688 + 198.429i) q^{65} +(109.449 - 189.572i) q^{67} +(17.4902 + 30.2939i) q^{68} +(-288.199 + 18.2457i) q^{70} +542.115i q^{71} +(-778.156 - 449.268i) q^{73} +(-475.090 - 274.293i) q^{74} +406.310i q^{76} +(939.460 - 59.4764i) q^{77} +(561.113 + 971.876i) q^{79} +(-62.3698 + 108.028i) q^{80} +(11.0910 - 6.40338i) q^{82} -855.676 q^{83} +68.1788 q^{85} +(-454.665 + 262.501i) q^{86} +(203.310 - 352.144i) q^{88} +(658.192 + 1140.02i) q^{89} +(-522.019 - 785.032i) q^{91} -170.834i q^{92} +(184.580 + 106.567i) q^{94} +(685.824 + 395.961i) q^{95} -1240.73i q^{97} +(-86.5135 - 680.523i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} + 50 q^{7}+O(q^{10}) \) 16 * q + 32 * q^4 + 50 * q^7	\( 16 q + 32 q^{4} + 50 q^{7} + 60 q^{10} - 128 q^{16} - 498 q^{19} - 240 q^{22} - 470 q^{25} + 160 q^{28} - 582 q^{31} + 40 q^{37} + 240 q^{40} + 1900 q^{43} - 456 q^{46} - 1634 q^{49} - 720 q^{52} + 1200 q^{58} - 1302 q^{61} - 1024 q^{64} - 100 q^{67} - 1620 q^{70} - 5280 q^{73} + 590 q^{79} - 480 q^{82} + 2580 q^{85} - 480 q^{88} - 2382 q^{91} - 3396 q^{94}+O(q^{100}) \) 16 * q + 32 * q^4 + 50 * q^7 + 60 * q^10 - 128 * q^16 - 498 * q^19 - 240 * q^22 - 470 * q^25 + 160 * q^28 - 582 * q^31 + 40 * q^37 + 240 * q^40 + 1900 * q^43 - 456 * q^46 - 1634 * q^49 - 720 * q^52 + 1200 * q^58 - 1302 * q^61 - 1024 * q^64 - 100 * q^67 - 1620 * q^70 - 5280 * q^73 + 590 * q^79 - 480 * q^82 + 2580 * q^85 - 480 * q^88 - 2382 * q^91 - 3396 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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