LMFDB - Embedded newform 1045.2.b.b.419.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(419,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1045.419");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.b (of order $2$, degree $1$, 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:	$8.34436701122$
Analytic rank:	$0$
Dimension:	$16$
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} + 19x^{14} + 144x^{12} + 552x^{10} + 1119x^{8} + 1146x^{6} + 524x^{4} + 83x^{2} + 4 $ x^16 + 19x^14 + 144x^12 + 552x^10 + 1119x^8 + 1146x^6 + 524x^4 + 83*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			419.3
Root			$-2.05535i$ of defining polynomial
Character	$\chi$	$=$	1045.419
Dual form			1045.2.b.b.419.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.05535i q^{2} -2.80376i q^{3} -2.22445 q^{4} +(-2.02523 + 0.947852i) q^{5} -5.76269 q^{6} -1.02917i q^{7} +0.461316i q^{8} -4.86106 q^{9} +O(q^{10})$	\(q-2.05535i q^{2} -2.80376i q^{3} -2.22445 q^{4} +(-2.02523 + 0.947852i) q^{5} -5.76269 q^{6} -1.02917i q^{7} +0.461316i q^{8} -4.86106 q^{9} +(1.94816 + 4.16256i) q^{10} -1.00000 q^{11} +6.23681i q^{12} +1.06173i q^{13} -2.11531 q^{14} +(2.65755 + 5.67827i) q^{15} -3.50073 q^{16} +1.41848i q^{17} +9.99115i q^{18} -1.00000 q^{19} +(4.50503 - 2.10845i) q^{20} -2.88555 q^{21} +2.05535i q^{22} +3.94773i q^{23} +1.29342 q^{24} +(3.20315 - 3.83925i) q^{25} +2.18222 q^{26} +5.21795i q^{27} +2.28934i q^{28} +1.85764 q^{29} +(11.6708 - 5.46218i) q^{30} -7.16673 q^{31} +8.11784i q^{32} +2.80376i q^{33} +2.91547 q^{34} +(0.975504 + 2.08432i) q^{35} +10.8132 q^{36} -1.81613i q^{37} +2.05535i q^{38} +2.97682 q^{39} +(-0.437259 - 0.934273i) q^{40} +1.21675 q^{41} +5.93081i q^{42} -7.57572i q^{43} +2.22445 q^{44} +(9.84478 - 4.60756i) q^{45} +8.11394 q^{46} -9.29494i q^{47} +9.81520i q^{48} +5.94080 q^{49} +(-7.89098 - 6.58359i) q^{50} +3.97708 q^{51} -2.36175i q^{52} +9.70613i q^{53} +10.7247 q^{54} +(2.02523 - 0.947852i) q^{55} +0.474774 q^{56} +2.80376i q^{57} -3.81810i q^{58} -7.10538 q^{59} +(-5.91157 - 12.6310i) q^{60} -2.23668 q^{61} +14.7301i q^{62} +5.00287i q^{63} +9.68351 q^{64} +(-1.00636 - 2.15025i) q^{65} +5.76269 q^{66} +4.13469i q^{67} -3.15534i q^{68} +11.0685 q^{69} +(4.28399 - 2.00500i) q^{70} -15.3770 q^{71} -2.24248i q^{72} +6.22562i q^{73} -3.73277 q^{74} +(-10.7643 - 8.98087i) q^{75} +2.22445 q^{76} +1.02917i q^{77} -6.11840i q^{78} +2.22307 q^{79} +(7.08980 - 3.31817i) q^{80} +0.0467005 q^{81} -2.50085i q^{82} -10.1783i q^{83} +6.41876 q^{84} +(-1.34451 - 2.87276i) q^{85} -15.5707 q^{86} -5.20838i q^{87} -0.461316i q^{88} +4.69482 q^{89} +(-9.47013 - 20.2344i) q^{90} +1.09270 q^{91} -8.78150i q^{92} +20.0938i q^{93} -19.1043 q^{94} +(2.02523 - 0.947852i) q^{95} +22.7605 q^{96} -6.64102i q^{97} -12.2104i q^{98} +4.86106 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) $ 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9	$ 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + 10 q^{10} - 16 q^{11} + 4 q^{14} + 3 q^{15} - 18 q^{16} - 16 q^{19} - 2 q^{20} - 10 q^{21} + 10 q^{24} - 7 q^{25} - 24 q^{26} + 2 q^{29} + 4 q^{30} - 32 q^{31} - 16 q^{34} - 18 q^{35} + 18 q^{36} + 40 q^{39} - 28 q^{40} + 6 q^{41} + 6 q^{44} + 16 q^{45} + 38 q^{49} - 30 q^{50} - 16 q^{51} + 18 q^{54} - 3 q^{55} + 12 q^{56} + 24 q^{59} - 20 q^{60} - 42 q^{61} + 62 q^{64} - 20 q^{65} + 8 q^{66} + 30 q^{69} - 18 q^{70} - 46 q^{71} - 2 q^{74} - 25 q^{75} + 6 q^{76} + 74 q^{79} - 22 q^{80} - 56 q^{81} + 34 q^{84} - 18 q^{85} + 8 q^{86} + 14 q^{89} - 4 q^{90} - 24 q^{91} + 64 q^{94} - 3 q^{95} + 54 q^{96} + 8 q^{99}+O(q^{100}) $ 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9 + 10 * q^10 - 16 * q^11 + 4 * q^14 + 3 * q^15 - 18 * q^16 - 16 * q^19 - 2 * q^20 - 10 * q^21 + 10 * q^24 - 7 * q^25 - 24 * q^26 + 2 * q^29 + 4 * q^30 - 32 * q^31 - 16 * q^34 - 18 * q^35 + 18 * q^36 + 40 * q^39 - 28 * q^40 + 6 * q^41 + 6 * q^44 + 16 * q^45 + 38 * q^49 - 30 * q^50 - 16 * q^51 + 18 * q^54 - 3 * q^55 + 12 * q^56 + 24 * q^59 - 20 * q^60 - 42 * q^61 + 62 * q^64 - 20 * q^65 + 8 * q^66 + 30 * q^69 - 18 * q^70 - 46 * q^71 - 2 * q^74 - 25 * q^75 + 6 * q^76 + 74 * q^79 - 22 * q^80 - 56 * q^81 + 34 * q^84 - 18 * q^85 + 8 * q^86 + 14 * q^89 - 4 * q^90 - 24 * q^91 + 64 * q^94 - 3 * q^95 + 54 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$.

$n$	$496$	$761$	$837$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	2.05535i		−	1.45335i	−0.686982	−	0.726675i	$-0.741065\pi$
$2$		−	2.05535i		−	1.45335i	0.686982	−	0.726675i	$-0.258935\pi$
$3$		−	2.80376i		−	1.61875i	−0.587292	−	0.809375i	$-0.699806\pi$
$3$		−	2.80376i		−	1.61875i	0.587292	−	0.809375i	$-0.300194\pi$
$4$	−2.22445			−1.11222
$5$	−2.02523	+	0.947852i	−0.905713	+	0.423892i
$5$	−2.02523	+	0.947852i	−0.905713	+	0.423892i
$6$	−5.76269			−2.35261
$7$		−	1.02917i		−	0.388991i	−0.980903	−	0.194495i	$-0.937693\pi$
$7$		−	1.02917i		−	0.388991i	0.980903	−	0.194495i	$-0.0623070\pi$
$8$			0.461316i			0.163100i
$9$	−4.86106			−1.62035
$10$	1.94816	+	4.16256i	0.616063	+	1.31632i
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$			6.23681i			1.80041i
$13$			1.06173i			0.294470i	0.989102	+	0.147235i	$0.0470374\pi$
$13$			1.06173i			0.294470i	−0.989102	+	0.147235i	$0.952963\pi$
$14$	−2.11531			−0.565340
$15$	2.65755	+	5.67827i	0.686176	+	1.46612i
$16$	−3.50073			−0.875182
$17$			1.41848i			0.344033i	0.985094	+	0.172016i	$0.0550282\pi$
$17$			1.41848i			0.344033i	−0.985094	+	0.172016i	$0.944972\pi$
$18$			9.99115i			2.35494i
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$	4.50503	−	2.10845i	1.00735	−	0.471463i
$21$	−2.88555			−0.629679
$22$			2.05535i			0.438201i
$23$			3.94773i			0.823158i	0.911374	+	0.411579i	$0.135023\pi$
$23$			3.94773i			0.823158i	−0.911374	+	0.411579i	$0.864977\pi$
$24$	1.29342			0.264018
$25$	3.20315	−	3.83925i	0.640631	−	0.767849i
$26$	2.18222			0.427968
$27$			5.21795i			1.00420i
$28$			2.28934i			0.432645i
$29$	1.85764			0.344956			0.172478	−	0.985013i	$-0.444823\pi$
$29$	1.85764			0.344956			0.172478	+	0.985013i	$0.444823\pi$
$30$	11.6708	−	5.46218i	2.13079	−	0.997253i
$31$	−7.16673			−1.28718			−0.643592	−	0.765369i	$-0.722556\pi$
$31$	−7.16673			−1.28718			−0.643592	+	0.765369i	$0.722556\pi$
$32$			8.11784i			1.43505i
$33$			2.80376i			0.488072i
$34$	2.91547			0.500000
$35$	0.975504	+	2.08432i	0.164890	+	0.352314i
$36$	10.8132			1.80219
$37$		−	1.81613i		−	0.298569i	−0.988794	−	0.149285i	$-0.952303\pi$
$37$		−	1.81613i		−	0.298569i	0.988794	−	0.149285i	$-0.0476971\pi$
$38$			2.05535i			0.333421i
$39$	2.97682			0.476673
$40$	−0.437259	−	0.934273i	−0.0691367	−	0.147722i
$41$	1.21675			0.190025			0.0950125	−	0.995476i	$-0.469711\pi$
$41$	1.21675			0.190025			0.0950125	+	0.995476i	$0.469711\pi$
$42$			5.93081i			0.915144i
$43$		−	7.57572i		−	1.15529i	−0.816290	−	0.577643i	$-0.803973\pi$
$43$		−	7.57572i		−	1.15529i	0.816290	−	0.577643i	$-0.196027\pi$
$44$	2.22445			0.335348
$45$	9.84478	−	4.60756i	1.46757	−	0.686855i
$46$	8.11394			1.19634
$47$		−	9.29494i		−	1.35581i	−0.735151	−	0.677903i	$-0.762889\pi$
$47$		−	9.29494i		−	1.35581i	0.735151	−	0.677903i	$-0.237111\pi$
$48$			9.81520i			1.41670i
$49$	5.94080			0.848686
$50$	−7.89098	−	6.58359i	−1.11595	−	0.931060i
$51$	3.97708			0.556903
$52$		−	2.36175i		−	0.327516i
$53$			9.70613i			1.33324i	0.745398	+	0.666620i	$0.232259\pi$
$53$			9.70613i			1.33324i	−0.745398	+	0.666620i	$0.767741\pi$
$54$	10.7247			1.45945
$55$	2.02523	−	0.947852i	0.273083	−	0.127808i
$56$	0.474774			0.0634443
$57$			2.80376i			0.371367i
$58$		−	3.81810i		−	0.501341i
$59$	−7.10538			−0.925042			−0.462521	−	0.886608i	$-0.653055\pi$
$59$	−7.10538			−0.925042			−0.462521	+	0.886608i	$0.653055\pi$
$60$	−5.91157	−	12.6310i	−0.763181	−	1.63066i
$61$	−2.23668			−0.286378			−0.143189	−	0.989695i	$-0.545736\pi$
$61$	−2.23668			−0.286378			−0.143189	+	0.989695i	$0.545736\pi$
$62$			14.7301i			1.87073i
$63$			5.00287i			0.630302i
$64$	9.68351			1.21044
$65$	−1.00636	−	2.15025i	−0.124824	−	0.266705i
$66$	5.76269			0.709338
$67$			4.13469i			0.505133i	0.967580	+	0.252566i	$0.0812746\pi$
$67$			4.13469i			0.505133i	−0.967580	+	0.252566i	$0.918725\pi$
$68$		−	3.15534i		−	0.382641i
$69$	11.0685			1.33249
$70$	4.28399	−	2.00500i	0.512035	−	0.239643i
$71$	−15.3770			−1.82492			−0.912459	−	0.409168i	$-0.865819\pi$
$71$	−15.3770			−1.82492			−0.912459	+	0.409168i	$0.865819\pi$
$72$		−	2.24248i		−	0.264279i
$73$			6.22562i			0.728654i	0.931271	+	0.364327i	$0.118701\pi$
$73$			6.22562i			0.728654i	−0.931271	+	0.364327i	$0.881299\pi$
$74$	−3.73277			−0.433926
$75$	−10.7643	−	8.98087i	−1.24296	−	1.03702i
$76$	2.22445			0.255162
$77$			1.02917i			0.117285i
$78$		−	6.11840i		−	0.692773i
$79$	2.22307			0.250115			0.125057	−	0.992150i	$-0.460089\pi$
$79$	2.22307			0.250115			0.125057	+	0.992150i	$0.460089\pi$
$80$	7.08980	−	3.31817i	0.792664	−	0.370983i
$81$	0.0467005			0.00518894
$82$		−	2.50085i		−	0.276173i
$83$		−	10.1783i		−	1.11721i	−0.829433	−	0.558606i	$-0.811336\pi$
$83$		−	10.1783i		−	1.11721i	0.829433	−	0.558606i	$-0.188664\pi$
$84$	6.41876			0.700344
$85$	−1.34451	−	2.87276i	−0.145833	−	0.311595i
$86$	−15.5707			−1.67903
$87$		−	5.20838i		−	0.558397i
$88$		−	0.461316i		−	0.0491764i
$89$	4.69482			0.497650			0.248825	−	0.968548i	$-0.419956\pi$
$89$	4.69482			0.497650			0.248825	+	0.968548i	$0.419956\pi$
$90$	−9.47013	−	20.2344i	−0.998240	−	2.13290i
$91$	1.09270			0.114546
$92$		−	8.78150i		−	0.915535i
$93$			20.0938i			2.08363i
$94$	−19.1043			−1.97046
$95$	2.02523	−	0.947852i	0.207785	−	0.0972476i
$96$	22.7605			2.32298
$97$		−	6.64102i		−	0.674294i	−0.941452	−	0.337147i	$-0.890538\pi$
$97$		−	6.64102i		−	0.674294i	0.941452	−	0.337147i	$-0.109462\pi$
$98$		−	12.2104i		−	1.23344i
$99$	4.86106			0.488555
$100$	−7.12524	+	8.54020i	−0.712524	+	0.854020i
$101$	−15.8963			−1.58174			−0.790869	−	0.611985i	$-0.790371\pi$
$101$	−15.8963			−1.58174			−0.790869	+	0.611985i	$0.790371\pi$
$102$		−	8.17428i		−	0.809375i
$103$		−	12.6170i		−	1.24319i	−0.783338	−	0.621596i	$-0.786485\pi$
$103$		−	12.6170i		−	1.24319i	0.783338	−	0.621596i	$-0.213515\pi$
$104$	−0.489791			−0.0480280
$105$	5.84392	−	2.73508i	0.570308	−	0.266916i
$106$	19.9494			1.93766
$107$			13.8474i			1.33868i	0.742957	+	0.669339i	$0.233423\pi$
$107$			13.8474i			1.33868i	−0.742957	+	0.669339i	$0.766577\pi$
$108$		−	11.6071i		−	1.11689i
$109$	−11.6709			−1.11787			−0.558936	−	0.829211i	$-0.688790\pi$
$109$	−11.6709			−1.11787			−0.558936	+	0.829211i	$0.688790\pi$
$110$	−1.94816	−	4.16256i	−0.185750	−	0.396884i
$111$	−5.09198			−0.483309
$112$			3.60286i			0.340438i
$113$		−	6.60763i		−	0.621593i	−0.950476	−	0.310797i	$-0.899404\pi$
$113$		−	6.60763i		−	0.621593i	0.950476	−	0.310797i	$-0.100596\pi$
$114$	5.76269			0.539726
$115$	−3.74186	−	7.99507i	−0.348930	−	0.745544i
$116$	−4.13223			−0.383668
$117$		−	5.16111i		−	0.477145i
$118$			14.6040i			1.34441i
$119$	1.45987			0.133826
$120$	−2.61947	+	1.22597i	−0.239124	+	0.111915i
$121$	1.00000			0.0909091
$122$			4.59716i			0.416207i
$123$		−	3.41148i		−	0.307603i
$124$	15.9420			1.43164
$125$	−2.84810	+	10.8115i	−0.254742	+	0.967009i
$126$	10.2826			0.916049
$127$		−	20.1320i		−	1.78642i	−0.449636	−	0.893212i	$-0.648446\pi$
$127$		−	20.1320i		−	1.78642i	0.449636	−	0.893212i	$-0.351554\pi$
$128$		−	3.66729i		−	0.324145i
$129$	−21.2405			−1.87012
$130$	−4.41950	+	2.06842i	−0.387616	+	0.181412i
$131$	−2.02571			−0.176987			−0.0884937	−	0.996077i	$-0.528205\pi$
$131$	−2.02571			−0.176987			−0.0884937	+	0.996077i	$0.528205\pi$
$132$		−	6.23681i		−	0.542845i
$133$			1.02917i			0.0892406i
$134$	8.49822			0.734134
$135$	−4.94585	−	10.5676i	−0.425671	−	0.909512i
$136$	−0.654369			−0.0561117
$137$			14.2319i			1.21591i	0.793971	+	0.607956i	$0.208010\pi$
$137$			14.2319i			1.21591i	−0.793971	+	0.607956i	$0.791990\pi$
$138$		−	22.7495i		−	1.93657i
$139$	7.59104			0.643863			0.321932	−	0.946763i	$-0.395668\pi$
$139$	7.59104			0.643863			0.321932	+	0.946763i	$0.395668\pi$
$140$	−2.16996	−	4.63645i	−0.183395	−	0.391852i
$141$	−26.0608			−2.19471
$142$			31.6051i			2.65224i
$143$		−	1.06173i		−	0.0887861i
$144$	17.0172			1.41810
$145$	−3.76216	+	1.76077i	−0.312431	+	0.146224i
$146$	12.7958			1.05899
$147$		−	16.6566i		−	1.37381i
$148$			4.03988i			0.332076i
$149$	7.41690			0.607616			0.303808	−	0.952733i	$-0.401742\pi$
$149$	7.41690			0.607616			0.303808	+	0.952733i	$0.401742\pi$
$150$	−18.4588	+	22.1244i	−1.50715	+	1.80645i
$151$	−4.73096			−0.385000			−0.192500	−	0.981297i	$-0.561660\pi$
$151$	−4.73096			−0.385000			−0.192500	+	0.981297i	$0.561660\pi$
$152$		−	0.461316i		−	0.0374177i
$153$		−	6.89533i		−	0.557454i
$154$	2.11531			0.170456
$155$	14.5143	−	6.79300i	1.16582	−	0.545627i
$156$	−6.62179			−0.530167
$157$		−	2.65936i		−	0.212240i	−0.994353	−	0.106120i	$-0.966157\pi$
$157$		−	2.65936i		−	0.212240i	0.994353	−	0.106120i	$-0.0338427\pi$
$158$		−	4.56917i		−	0.363504i
$159$	27.2136			2.15818
$160$	−7.69451	−	16.4405i	−0.608305	−	1.29974i
$161$	4.06289			0.320201
$162$		−	0.0959856i		−	0.00754134i
$163$			0.842202i			0.0659664i	0.999456	+	0.0329832i	$0.0105008\pi$
$163$			0.842202i			0.0659664i	−0.999456	+	0.0329832i	$0.989499\pi$
$164$	−2.70660			−0.211350
$165$	−2.65755	−	5.67827i	−0.206890	−	0.442053i
$166$	−20.9199			−1.62370
$167$			19.6260i			1.51871i	0.650679	+	0.759353i	$0.274484\pi$
$167$			19.6260i			1.51871i	−0.650679	+	0.759353i	$0.725516\pi$
$168$		−	1.33115i		−	0.102701i
$169$	11.8727			0.913287
$170$	−5.90452	+	2.76344i	−0.452856	+	0.211946i
$171$	4.86106			0.371734
$172$			16.8518i			1.28494i
$173$			14.9622i			1.13756i	0.822491	+	0.568778i	$0.192584\pi$
$173$			14.9622i			1.13756i	−0.822491	+	0.568778i	$0.807416\pi$
$174$	−10.7050			−0.811546
$175$	−3.95125	−	3.29660i	−0.298686	−	0.249200i
$176$	3.50073			0.263877
$177$			19.9218i			1.49741i
$178$		−	9.64948i		−	0.723259i
$179$	−5.19344			−0.388176			−0.194088	−	0.980984i	$-0.562175\pi$
$179$	−5.19344			−0.388176			−0.194088	+	0.980984i	$0.562175\pi$
$180$	−21.8992	+	10.2493i	−1.63227	+	0.763936i
$181$	−5.63160			−0.418594			−0.209297	−	0.977852i	$-0.567117\pi$
$181$	−5.63160			−0.418594			−0.209297	+	0.977852i	$0.567117\pi$
$182$		−	2.24588i		−	0.166476i
$183$			6.27112i			0.463574i
$184$	−1.82115			−0.134257
$185$	1.72142	+	3.67808i	0.126561	+	0.270418i
$186$	41.2997			3.02824
$187$		−	1.41848i		−	0.103730i
$188$			20.6761i			1.50796i
$189$	5.37018			0.390623
$190$	−1.94816	−	4.16256i	−0.141335	−	0.301984i
$191$	−21.7700			−1.57522			−0.787611	−	0.616173i	$-0.788682\pi$
$191$	−21.7700			−1.57522			−0.787611	+	0.616173i	$0.788682\pi$
$192$		−	27.1502i		−	1.95940i
$193$		−	5.93030i		−	0.426872i	−0.976957	−	0.213436i	$-0.931534\pi$
$193$		−	5.93030i		−	0.426872i	0.976957	−	0.213436i	$-0.0684655\pi$
$194$	−13.6496			−0.979984
$195$	−6.02877	+	2.82159i	−0.431729	+	0.202058i
$196$	−13.2150			−0.943928
$197$			24.8072i			1.76744i	0.468014	+	0.883721i	$0.344970\pi$
$197$			24.8072i			1.76744i	−0.468014	+	0.883721i	$0.655030\pi$
$198$		−	9.99115i		−	0.710040i
$199$	8.66573			0.614297			0.307149	−	0.951662i	$-0.400625\pi$
$199$	8.66573			0.614297			0.307149	+	0.951662i	$0.400625\pi$
$200$	1.77110	+	1.47767i	0.125236	+	0.104487i
$201$	11.5927			0.817684
$202$			32.6723i			2.29882i
$203$		−	1.91184i		−	0.134185i
$204$	−8.84681			−0.619401
$205$	−2.46421	+	1.15330i	−0.172108	+	0.0805501i
$206$	−25.9323			−1.80679
$207$		−	19.1901i		−	1.33381i
$208$		−	3.71682i		−	0.257715i
$209$	1.00000			0.0691714
$210$	−5.62153	−	12.0113i	−0.387922	−	0.828857i
$211$	−20.2880			−1.39668			−0.698342	−	0.715764i	$-0.746079\pi$
$211$	−20.2880			−1.39668			−0.698342	+	0.715764i	$0.746079\pi$
$212$		−	21.5908i		−	1.48286i
$213$			43.1135i			2.95409i
$214$	28.4612			1.94557
$215$	7.18066	+	15.3426i	0.489717	+	1.04636i
$216$	−2.40712			−0.163784
$217$			7.37581i			0.500703i
$218$			23.9878i			1.62466i
$219$	17.4551			1.17951
$220$	−4.50503	+	2.10845i	−0.303729	+	0.142151i
$221$	−1.50604			−0.101307
$222$			10.4658i			0.702417i
$223$			2.35772i			0.157885i	0.996879	+	0.0789425i	$0.0251543\pi$
$223$			2.35772i			0.157885i	−0.996879	+	0.0789425i	$0.974846\pi$
$224$	8.35467			0.558220
$225$	−15.5707	+	18.6628i	−1.03805	+	1.24419i
$226$	−13.5810			−0.903392
$227$		−	19.1924i		−	1.27384i	−0.770928	−	0.636922i	$-0.780207\pi$
$227$		−	19.1924i		−	1.27384i	0.770928	−	0.636922i	$-0.219793\pi$
$228$		−	6.23681i		−	0.413043i
$229$	3.74063			0.247188			0.123594	−	0.992333i	$-0.460558\pi$
$229$	3.74063			0.247188			0.123594	+	0.992333i	$0.460558\pi$
$230$	−16.4326	+	7.69081i	−1.08354	+	0.507117i
$231$	2.88555			0.189855
$232$			0.856960i			0.0562622i
$233$		−	14.2206i		−	0.931621i	−0.884884	−	0.465811i	$-0.845763\pi$
$233$		−	14.2206i		−	0.931621i	0.884884	−	0.465811i	$-0.154237\pi$
$234$	−10.6079			−0.693458
$235$	8.81023	+	18.8244i	0.574716	+	1.22797i
$236$	15.8055			1.02885
$237$		−	6.23294i		−	0.404873i
$238$		−	3.00053i		−	0.194495i
$239$	3.45458			0.223458			0.111729	−	0.993739i	$-0.464361\pi$
$239$	3.45458			0.223458			0.111729	+	0.993739i	$0.464361\pi$
$240$	−9.30335	−	19.8781i	−0.600529	−	1.28312i
$241$	−19.8133			−1.27629			−0.638143	−	0.769918i	$-0.720297\pi$
$241$	−19.8133			−1.27629			−0.638143	+	0.769918i	$0.720297\pi$
$242$		−	2.05535i		−	0.132123i
$243$			15.5229i			0.995796i
$244$	4.97538			0.318516
$245$	−12.0315	+	5.63100i	−0.768666	+	0.359751i
$246$	−7.01178			−0.447055
$247$		−	1.06173i		−	0.0675561i
$248$		−	3.30613i		−	0.209939i
$249$	−28.5375			−1.80849
$250$	22.2214	+	5.85383i	1.40540	+	0.370229i
$251$	11.8530			0.748154			0.374077	−	0.927398i	$-0.377960\pi$
$251$	11.8530			0.748154			0.374077	+	0.927398i	$0.377960\pi$
$252$		−	11.1286i		−	0.701037i
$253$		−	3.94773i		−	0.248191i
$254$	−41.3782			−2.59630
$255$	−8.05453	+	3.76969i	−0.504394	+	0.236067i
$256$	11.8295			0.739343
$257$		−	4.32353i		−	0.269694i	−0.990866	−	0.134847i	$-0.956946\pi$
$257$		−	4.32353i		−	0.269694i	0.990866	−	0.134847i	$-0.0430543\pi$
$258$			43.6565i			2.71794i
$259$	−1.86911			−0.116141
$260$	2.23859	+	4.78311i	0.138832	+	0.296636i
$261$	−9.03011			−0.558950
$262$			4.16354i			0.257225i
$263$		−	16.7851i		−	1.03501i	−0.855679	−	0.517507i	$-0.826860\pi$
$263$		−	16.7851i		−	1.03501i	0.855679	−	0.517507i	$-0.173140\pi$
$264$	−1.29342			−0.0796044
$265$	−9.19997	−	19.6572i	−0.565150	−	1.20753i
$266$	2.11531			0.129698
$267$		−	13.1631i		−	0.805571i
$268$		−	9.19740i		−	0.561820i
$269$	1.62791			0.0992557			0.0496278	−	0.998768i	$-0.484196\pi$
$269$	1.62791			0.0992557			0.0496278	+	0.998768i	$0.484196\pi$
$270$	−21.7200	+	10.1654i	−1.32184	+	0.618648i
$271$	6.83153			0.414986			0.207493	−	0.978237i	$-0.433470\pi$
$271$	6.83153			0.414986			0.207493	+	0.978237i	$0.433470\pi$
$272$		−	4.96573i		−	0.301091i
$273$		−	3.06367i		−	0.185422i
$274$	29.2514			1.76714
$275$	−3.20315	+	3.83925i	−0.193157	+	0.231515i
$276$	−24.6212			−1.48202
$277$		−	27.0022i		−	1.62240i	−0.584767	−	0.811201i	$-0.698814\pi$
$277$		−	27.0022i		−	1.62240i	0.584767	−	0.811201i	$-0.301186\pi$
$278$		−	15.6022i		−	0.935758i
$279$	34.8379			2.08569
$280$	−0.961529	+	0.450015i	−0.0574623	+	0.0268936i
$281$	2.51531			0.150051			0.0750255	−	0.997182i	$-0.476096\pi$
$281$	2.51531			0.150051			0.0750255	+	0.997182i	$0.476096\pi$
$282$			53.5639i			3.18968i
$283$		−	9.44200i		−	0.561269i	−0.959815	−	0.280634i	$-0.909455\pi$
$283$		−	9.44200i		−	0.561269i	0.959815	−	0.280634i	$-0.0905448\pi$
$284$	34.2054			2.02972
$285$	−2.65755	−	5.67827i	−0.157420	−	0.336352i
$286$	−2.18222			−0.129037
$287$		−	1.25225i		−	0.0739180i
$288$		−	39.4613i		−	2.32528i
$289$	14.9879			0.881641
$290$	3.61899	+	7.73255i	0.212515	+	0.454071i
$291$	−18.6198			−1.09151
$292$		−	13.8486i		−	0.810426i
$293$		−	9.07011i		−	0.529882i	−0.964265	−	0.264941i	$-0.914648\pi$
$293$		−	9.07011i		−	0.529882i	0.964265	−	0.264941i	$-0.0853524\pi$
$294$	−34.2350			−1.99663
$295$	14.3901	−	6.73485i	0.837822	−	0.392118i
$296$	0.837808			0.0486966
$297$		−	5.21795i		−	0.302776i
$298$		−	15.2443i		−	0.883078i
$299$	−4.19141			−0.242395
$300$	23.9446	+	19.9775i	1.38244	+	1.15340i
$301$	−7.79673			−0.449396
$302$			9.72376i			0.559540i
$303$			44.5693i			2.56044i
$304$	3.50073			0.200781
$305$	4.52981	−	2.12005i	0.259376	−	0.121393i
$306$	−14.1723			−0.810176
$307$		−	14.4451i		−	0.824425i	−0.911088	−	0.412213i	$-0.864756\pi$
$307$		−	14.4451i		−	0.824425i	0.911088	−	0.412213i	$-0.135244\pi$
$308$		−	2.28934i		−	0.130447i
$309$	−35.3751			−2.01242
$310$	−13.9620	−	29.8319i	−0.792987	−	1.69434i
$311$	7.96488			0.451647			0.225824	−	0.974168i	$-0.427493\pi$
$311$	7.96488			0.451647			0.225824	+	0.974168i	$0.427493\pi$
$312$			1.37326i			0.0777453i
$313$		−	14.1783i		−	0.801407i	−0.916208	−	0.400704i	$-0.868766\pi$
$313$		−	14.1783i		−	0.801407i	0.916208	−	0.400704i	$-0.131234\pi$
$314$	−5.46590			−0.308458
$315$	−4.74198	−	10.1320i	−0.267180	−	0.570873i
$316$	−4.94510			−0.278183
$317$		−	3.67505i		−	0.206411i	−0.994660	−	0.103206i	$-0.967090\pi$
$317$		−	3.67505i		−	0.206411i	0.994660	−	0.103206i	$-0.0329100\pi$
$318$		−	55.9334i		−	3.13659i
$319$	−1.85764			−0.104008
$320$	−19.6114	+	9.17854i	−1.09631	+	0.513096i
$321$	38.8247			2.16699
$322$		−	8.35065i		−	0.465364i
$323$		−	1.41848i		−	0.0789265i
$324$	−0.103883			−0.00577126
$325$	4.07623	+	3.40087i	0.226109	+	0.188647i
$326$	1.73102			0.0958722
$327$			32.7225i			1.80956i
$328$			0.561308i			0.0309930i
$329$	−9.56611			−0.527397
$330$	−11.6708	+	5.46218i	−0.642457	+	0.300683i
$331$	−18.1033			−0.995046			−0.497523	−	0.867451i	$-0.665757\pi$
$331$	−18.1033			−0.995046			−0.497523	+	0.867451i	$0.665757\pi$
$332$			22.6411i			1.24259i
$333$			8.82830i			0.483788i
$334$	40.3382			2.20721
$335$	−3.91907	−	8.37372i	−0.214122	−	0.457505i
$336$	10.1015			0.551084
$337$		−	9.60959i		−	0.523468i	−0.965140	−	0.261734i	$-0.915706\pi$
$337$		−	9.60959i		−	0.523468i	0.965140	−	0.261734i	$-0.0842943\pi$
$338$		−	24.4026i		−	1.32733i
$339$	−18.5262			−1.00620
$340$	2.99080	+	6.39031i	0.162199	+	0.346563i
$341$	7.16673			0.388100
$342$		−	9.99115i		−	0.540260i
$343$		−	13.3183i		−	0.719122i
$344$	3.49480			0.188427
$345$	−22.4162	+	10.4913i	−1.20685	+	0.564831i
$346$	30.7525			1.65327
$347$			1.39188i			0.0747199i	0.999302	+	0.0373599i	$0.0118948\pi$
$347$			1.39188i			0.0747199i	−0.999302	+	0.0373599i	$0.988105\pi$
$348$			11.5858i			0.621062i
$349$	−32.7429			−1.75269			−0.876345	−	0.481685i	$-0.840025\pi$
$349$	−32.7429			−1.75269			−0.876345	+	0.481685i	$0.840025\pi$
$350$	−6.77565	+	8.12118i	−0.362174	+	0.434096i
$351$	−5.54004			−0.295705
$352$		−	8.11784i		−	0.432682i
$353$			0.426322i			0.0226908i	0.999936	+	0.0113454i	$0.00361144\pi$
$353$			0.426322i			0.0226908i	−0.999936	+	0.0113454i	$0.996389\pi$
$354$	40.9461			2.17626
$355$	31.1421	−	14.5751i	1.65285	−	0.773569i
$356$	−10.4434			−0.553498
$357$		−	4.09311i		−	0.216630i
$358$			10.6743i			0.564156i
$359$	−31.9618			−1.68688			−0.843440	−	0.537223i	$-0.819473\pi$
$359$	−31.9618			−1.68688			−0.843440	+	0.537223i	$0.819473\pi$
$360$	2.12554	+	4.54155i	0.112026	+	0.239361i
$361$	1.00000			0.0526316
$362$			11.5749i			0.608363i
$363$		−	2.80376i		−	0.147159i
$364$	−2.43066			−0.127401
$365$	−5.90097	−	12.6083i	−0.308871	−	0.659951i
$366$	12.8893			0.673735
$367$		−	6.03820i		−	0.315192i	−0.987504	−	0.157596i	$-0.949626\pi$
$367$		−	6.03820i		−	0.315192i	0.987504	−	0.157596i	$-0.0503743\pi$
$368$		−	13.8199i		−	0.720413i
$369$	−5.91471			−0.307907
$370$	7.55974	−	3.53811i	0.393012	−	0.183938i
$371$	9.98929			0.518618
$372$		−	44.6976i		−	2.31746i
$373$			23.6955i			1.22691i	0.789730	+	0.613454i	$0.210221\pi$
$373$			23.6955i			1.22691i	−0.789730	+	0.613454i	$0.789779\pi$
$374$	−2.91547			−0.150756
$375$	30.3128	+	7.98539i	1.56535	+	0.412364i
$376$	4.28791			0.221132
$377$			1.97231i			0.101579i
$378$		−	11.0376i		−	0.567711i
$379$	13.6890			0.703156			0.351578	−	0.936159i	$-0.385645\pi$
$379$	13.6890			0.703156			0.351578	+	0.936159i	$0.385645\pi$
$380$	−4.50503	+	2.10845i	−0.231103	+	0.108161i
$381$	−56.4452			−2.89177
$382$			44.7449i			2.28935i
$383$		−	14.0596i		−	0.718411i	−0.933258	−	0.359206i	$-0.883048\pi$
$383$		−	14.0596i		−	0.718411i	0.933258	−	0.359206i	$-0.116952\pi$
$384$	−10.2822			−0.524711
$385$	−0.975504	−	2.08432i	−0.0497163	−	0.106227i
$386$	−12.1888			−0.620394
$387$			36.8260i			1.87197i
$388$			14.7726i			0.749965i
$389$	1.99458			0.101129			0.0505647	−	0.998721i	$-0.483898\pi$
$389$	1.99458			0.101129			0.0505647	+	0.998721i	$0.483898\pi$
$390$	5.79934	+	12.3912i	0.293661	+	0.627453i
$391$	−5.59978			−0.283193
$392$			2.74059i			0.138421i
$393$			5.67961i			0.286499i
$394$	50.9874			2.56871
$395$	−4.50223	+	2.10714i	−0.226532	+	0.106022i
$396$	−10.8132			−0.543382
$397$		−	27.3650i		−	1.37341i	−0.726936	−	0.686705i	$-0.759057\pi$
$397$		−	27.3650i		−	1.37341i	0.726936	−	0.686705i	$-0.240943\pi$
$398$		−	17.8111i		−	0.892788i
$399$	2.88555			0.144458
$400$	−11.2134	+	13.4402i	−0.560669	+	0.672008i
$401$	16.9851			0.848193			0.424096	−	0.905617i	$-0.360592\pi$
$401$	16.9851			0.848193			0.424096	+	0.905617i	$0.360592\pi$
$402$		−	23.8269i		−	1.18838i
$403$		−	7.60911i		−	0.379037i
$404$	35.3604			1.75925
$405$	−0.0945794	+	0.0442651i	−0.00469969	+	0.00219955i
$406$	−3.92949			−0.195017
$407$			1.81613i			0.0900221i
$408$			1.83469i			0.0908308i
$409$	−14.5878			−0.721321			−0.360660	−	0.932697i	$-0.617449\pi$
$409$	−14.5878			−0.721321			−0.360660	+	0.932697i	$0.617449\pi$
$410$	2.37044	+	5.06481i	0.117067	+	0.250133i
$411$	39.9027			1.96826
$412$			28.0659i			1.38271i
$413$			7.31267i			0.359833i
$414$	−39.4423			−1.93848
$415$	9.64752	+	20.6134i	0.473578	+	1.01187i
$416$	−8.61893			−0.422578
$417$		−	21.2834i		−	1.04225i
$418$		−	2.05535i		−	0.100530i
$419$	6.52166			0.318604			0.159302	−	0.987230i	$-0.449076\pi$
$419$	6.52166			0.318604			0.159302	+	0.987230i	$0.449076\pi$
$420$	−12.9995	+	6.08403i	−0.634310	+	0.296870i
$421$	35.6247			1.73624			0.868120	−	0.496355i	$-0.165328\pi$
$421$	35.6247			1.73624			0.868120	+	0.496355i	$0.165328\pi$
$422$			41.6989i			2.02987i
$423$			45.1832i			2.19688i
$424$	−4.47759			−0.217451
$425$	5.44591	+	4.54362i	0.264165	+	0.220398i
$426$	88.6131			4.29332
$427$			2.30194i			0.111398i
$428$		−	30.8028i		−	1.48891i
$429$	−2.97682			−0.143722
$430$	31.5344	−	14.7587i	1.52072	−	0.711729i
$431$	−20.7424			−0.999126			−0.499563	−	0.866278i	$-0.666506\pi$
$431$	−20.7424			−0.999126			−0.499563	+	0.866278i	$0.666506\pi$
$432$		−	18.2666i		−	0.878854i
$433$		−	35.5039i		−	1.70621i	−0.521742	−	0.853103i	$-0.674718\pi$
$433$		−	35.5039i		−	1.70621i	0.521742	−	0.853103i	$-0.325282\pi$
$434$	15.1598			0.727696
$435$	4.93677	+	10.5482i	0.236700	+	0.505747i
$436$	25.9614			1.24332
$437$		−	3.94773i		−	0.188845i
$438$		−	35.8763i		−	1.71424i
$439$	2.09314			0.0999001			0.0499500	−	0.998752i	$-0.484094\pi$
$439$	2.09314			0.0999001			0.0499500	+	0.998752i	$0.484094\pi$
$440$	0.437259	+	0.934273i	0.0208455	+	0.0445397i
$441$	−28.8786			−1.37517
$442$			3.09544i			0.147235i
$443$		−	17.6113i		−	0.836740i	−0.908277	−	0.418370i	$-0.862602\pi$
$443$		−	17.6113i		−	0.836740i	0.908277	−	0.418370i	$-0.137398\pi$
$444$	11.3268			0.537548
$445$	−9.50811	+	4.44999i	−0.450728	+	0.210950i
$446$	4.84594			0.229462
$447$		−	20.7952i		−	0.983578i
$448$		−	9.96602i		−	0.470850i
$449$	−6.54425			−0.308842			−0.154421	−	0.988005i	$-0.549351\pi$
$449$	−6.54425			−0.308842			−0.154421	+	0.988005i	$0.549351\pi$
$450$	38.3585	+	32.0032i	1.80824	+	1.50864i
$451$	−1.21675			−0.0572947
$452$			14.6983i			0.691351i
$453$			13.2645i			0.623219i
$454$	−39.4470			−1.85134
$455$	−2.21298	+	1.03572i	−0.103746	+	0.0485552i
$456$	−1.29342			−0.0605698
$457$			14.0648i			0.657924i	0.944343	+	0.328962i	$0.106699\pi$
$457$			14.0648i			0.657924i	−0.944343	+	0.328962i	$0.893301\pi$
$458$		−	7.68829i		−	0.359250i
$459$	−7.40158			−0.345476
$460$	8.32357	+	17.7846i	0.388088	+	0.829212i
$461$	−5.88635			−0.274155			−0.137077	−	0.990560i	$-0.543771\pi$
$461$	−5.88635			−0.274155			−0.137077	+	0.990560i	$0.543771\pi$
$462$		−	5.93081i		−	0.275926i
$463$		−	29.7862i		−	1.38428i	−0.721762	−	0.692141i	$-0.756668\pi$
$463$		−	29.7862i		−	1.38428i	0.721762	−	0.692141i	$-0.243332\pi$
$464$	−6.50311			−0.301899
$465$	−19.0459	−	40.6946i	−0.883234	−	1.88717i
$466$	−29.2282			−1.35397
$467$		−	8.73185i		−	0.404062i	−0.979379	−	0.202031i	$-0.935246\pi$
$467$		−	8.73185i		−	0.404062i	0.979379	−	0.202031i	$-0.0647541\pi$
$468$			11.4806i			0.530692i
$469$	4.25531			0.196492
$470$	38.6907	−	18.1081i	1.78467	−	0.835263i
$471$	−7.45619			−0.343563
$472$		−	3.27783i		−	0.150874i
$473$			7.57572i			0.348332i
$474$	−12.8109			−0.588422
$475$	−3.20315	+	3.83925i	−0.146971	+	0.176157i
$476$	−3.24739			−0.148844
$477$		−	47.1820i		−	2.16032i
$478$		−	7.10036i		−	0.324763i
$479$	−28.0305			−1.28075			−0.640374	−	0.768063i	$-0.721221\pi$
$479$	−28.0305			−1.28075			−0.640374	+	0.768063i	$0.721221\pi$
$480$	−46.0953	+	21.5735i	−2.10395	+	0.984693i
$481$	1.92823			0.0879198
$482$			40.7231i			1.85489i
$483$		−	11.3914i		−	0.518325i
$484$	−2.22445			−0.101111
$485$	6.29471	+	13.4496i	0.285828	+	0.610716i
$486$	31.9050			1.44724
$487$			26.5126i			1.20140i	0.799474	+	0.600701i	$0.205112\pi$
$487$			26.5126i			1.20140i	−0.799474	+	0.600701i	$0.794888\pi$
$488$		−	1.03182i		−	0.0467082i
$489$	2.36133			0.106783
$490$	11.5737	+	24.7289i	0.522844	+	1.11714i
$491$	−35.5006			−1.60212			−0.801059	−	0.598586i	$-0.795730\pi$
$491$	−35.5006			−1.60212			−0.801059	+	0.598586i	$0.795730\pi$
$492$			7.58866i			0.342123i
$493$			2.63504i			0.118676i
$494$	−2.18222			−0.0981825
$495$	−9.84478	+	4.60756i	−0.442490	+	0.207095i
$496$	25.0888			1.12652
$497$			15.8256i			0.709877i
$498$			58.6544i			2.62836i
$499$	35.0740			1.57013			0.785065	−	0.619414i	$-0.212630\pi$
$499$	35.0740			1.57013			0.785065	+	0.619414i	$0.212630\pi$
$500$	6.33545	−	24.0496i	0.283330	−	1.07553i
$501$	55.0265			2.45840
$502$		−	24.3620i		−	1.08733i
$503$		−	6.72014i		−	0.299636i	−0.988714	−	0.149818i	$-0.952131\pi$
$503$		−	6.72014i		−	0.299636i	0.988714	−	0.149818i	$-0.0478688\pi$
$504$	−2.30790			−0.102802
$505$	32.1937	−	15.0673i	1.43260	−	0.670487i
$506$	−8.11394			−0.360709
$507$		−	33.2883i		−	1.47838i
$508$			44.7825i			1.98690i
$509$	−43.6239			−1.93359			−0.966797	−	0.255546i	$-0.917745\pi$
$509$	−43.6239			−1.93359			−0.966797	+	0.255546i	$0.917745\pi$
$510$	7.74801	+	16.5548i	0.343088	+	0.733061i
$511$	6.40725			0.283440
$512$		−	31.6483i		−	1.39867i
$513$		−	5.21795i		−	0.230378i
$514$	−8.88634			−0.391960
$515$	11.9591	+	25.5524i	0.526979	+	1.12597i
$516$	47.2483			2.07999
$517$			9.29494i			0.408791i
$518$			3.84167i			0.168793i
$519$	41.9504			1.84142
$520$	0.991943	−	0.464250i	0.0434996	−	0.0203587i
$521$	18.1966			0.797206			0.398603	−	0.917124i	$-0.369495\pi$
$521$	18.1966			0.797206			0.398603	+	0.917124i	$0.369495\pi$
$522$			18.5600i			0.812349i
$523$			30.0628i			1.31455i	0.753649	+	0.657277i	$0.228292\pi$
$523$			30.0628i			1.31455i	−0.753649	+	0.657277i	$0.771708\pi$
$524$	4.50609			0.196850
$525$	−9.24287	+	11.0783i	−0.403392	+	0.483499i
$526$	−34.4992			−1.50424
$527$		−	10.1659i		−	0.442833i
$528$		−	9.81520i		−	0.427152i
$529$	7.41547			0.322412
$530$	−40.4023	+	18.9091i	−1.75496	+	0.821360i
$531$	34.5397			1.49889
$532$		−	2.28934i		−	0.0992555i
$533$			1.29186i			0.0559567i
$534$	−27.0548			−1.17078
$535$	−13.1253	−	28.0442i	−0.567455	−	1.21246i
$536$	−1.90740			−0.0823870
$537$			14.5612i			0.628360i
$538$		−	3.34593i		−	0.144253i
$539$	−5.94080			−0.255888
$540$	11.0018	+	23.5070i	0.473441	+	1.01158i
$541$	−25.4250			−1.09311			−0.546553	−	0.837425i	$-0.684060\pi$
$541$	−25.4250			−1.09311			−0.546553	+	0.837425i	$0.684060\pi$
$542$		−	14.0412i		−	0.603119i
$543$			15.7896i			0.677598i
$544$	−11.5150			−0.493703
$545$	23.6364	−	11.0623i	1.01247	−	0.473858i
$546$	−6.29690			−0.269482
$547$		−	21.7689i		−	0.930771i	−0.885108	−	0.465385i	$-0.845916\pi$
$547$		−	21.7689i		−	0.930771i	0.885108	−	0.465385i	$-0.154084\pi$
$548$		−	31.6581i		−	1.35237i
$549$	10.8726			0.464033
$550$	7.89098	+	6.58359i	0.336472	+	0.280725i
$551$	−1.85764			−0.0791383
$552$			5.10606i			0.217328i
$553$		−	2.28792i		−	0.0972923i
$554$	−55.4988			−2.35792
$555$	10.3125	−	4.82644i	0.437739	−	0.204871i
$556$	−16.8859			−0.716120
$557$		−	1.68626i		−	0.0714492i	−0.999362	−	0.0357246i	$-0.988626\pi$
$557$		−	1.68626i		−	0.0714492i	0.999362	−	0.0357246i	$-0.0113739\pi$
$558$		−	71.6039i		−	3.03124i
$559$	8.04334			0.340197
$560$	−3.41498	−	7.29663i	−0.144309	−	0.308339i
$561$	−3.97708			−0.167913
$562$		−	5.16984i		−	0.218076i
$563$			29.5858i			1.24689i	0.781866	+	0.623446i	$0.214268\pi$
$563$			29.5858i			1.24689i	−0.781866	+	0.623446i	$0.785732\pi$
$564$	57.9708			2.44101
$565$	6.26305	+	13.3820i	0.263489	+	0.562985i
$566$	−19.4066			−0.815719
$567$		−	0.0480629i		−	0.00201845i
$568$		−	7.09367i		−	0.297644i
$569$	7.48022			0.313587			0.156794	−	0.987631i	$-0.449884\pi$
$569$	7.48022			0.313587			0.156794	+	0.987631i	$0.449884\pi$
$570$	−11.6708	+	5.46218i	−0.488836	+	0.228785i
$571$	−20.4271			−0.854850			−0.427425	−	0.904051i	$-0.640579\pi$
$571$	−20.4271			−0.854850			−0.427425	+	0.904051i	$0.640579\pi$
$572$			2.36175i			0.0987499i
$573$			61.0378i			2.54989i
$574$	−2.57381			−0.107429
$575$	15.1563	+	12.6452i	0.632061	+	0.527340i
$576$	−47.0721			−1.96134
$577$		−	37.2768i		−	1.55185i	−0.630824	−	0.775926i	$-0.717283\pi$
$577$		−	37.2768i		−	1.55185i	0.630824	−	0.775926i	$-0.282717\pi$
$578$		−	30.8053i		−	1.28133i
$579$	−16.6271			−0.690999
$580$	8.36873	−	3.91674i	0.347493	−	0.162634i
$581$	−10.4752			−0.434586
$582$			38.2702i			1.58635i
$583$		−	9.70613i		−	0.401987i
$584$	−2.87198			−0.118843
$585$	4.89197	+	10.4525i	0.202258	+	0.432156i
$586$	−18.6422			−0.770103
$587$		−	44.6132i		−	1.84139i	−0.390288	−	0.920693i	$-0.627625\pi$
$587$		−	44.6132i		−	1.84139i	0.390288	−	0.920693i	$-0.372375\pi$
$588$			37.0517i			1.52798i
$589$	7.16673			0.295300
$590$	−13.8424	−	29.5766i	−0.569885	−	1.21765i
$591$	69.5535			2.86105
$592$			6.35777i			0.261303i
$593$		−	0.147572i		−	0.00606004i	−0.999995	−	0.00303002i	$-0.999036\pi$
$593$		−	0.147572i		−	0.00606004i	0.999995	−	0.00303002i	$-0.000964487\pi$
$594$	−10.7247			−0.440040
$595$	−2.95657	+	1.38374i	−0.121208	+	0.0567277i
$596$	−16.4985			−0.675805
$597$		−	24.2966i		−	0.994394i
$598$			8.61479i			0.352285i
$599$	3.92910			0.160539			0.0802694	−	0.996773i	$-0.474422\pi$
$599$	3.92910			0.160539			0.0802694	+	0.996773i	$0.474422\pi$
$600$	4.14302	−	4.96575i	0.169138	−	0.202726i
$601$	22.2592			0.907972			0.453986	−	0.891009i	$-0.350002\pi$
$601$	22.2592			0.907972			0.453986	+	0.891009i	$0.350002\pi$
$602$			16.0250i			0.653129i
$603$		−	20.0990i		−	0.818493i
$604$	10.5238			0.428206
$605$	−2.02523	+	0.947852i	−0.0823375	+	0.0385357i
$606$	91.6053			3.72121
$607$		−	10.0862i		−	0.409386i	−0.978826	−	0.204693i	$-0.934380\pi$
$607$		−	10.0862i		−	0.409386i	0.978826	−	0.204693i	$-0.0656196\pi$
$608$		−	8.11784i		−	0.329222i
$609$	−5.36033			−0.217211
$610$	−4.35743	−	9.31033i	−0.176427	−	0.376964i
$611$	9.86869			0.399245
$612$			15.3383i			0.620014i
$613$			46.7707i			1.88905i	0.328437	+	0.944526i	$0.393478\pi$
$613$			46.7707i			1.88905i	−0.328437	+	0.944526i	$0.606522\pi$
$614$	−29.6897			−1.19818
$615$	3.23358	+	6.90905i	0.130391	+	0.278600i
$616$	−0.474774			−0.0191292
$617$		−	40.4764i		−	1.62952i	−0.579799	−	0.814760i	$-0.696869\pi$
$617$		−	40.4764i		−	1.62952i	0.579799	−	0.814760i	$-0.303131\pi$
$618$			72.7080i			2.92474i
$619$	37.4998			1.50725			0.753623	−	0.657307i	$-0.228305\pi$
$619$	37.4998			1.50725			0.753623	+	0.657307i	$0.228305\pi$
$620$	−32.2863	+	15.1107i	−1.29665	+	0.606859i
$621$	−20.5990			−0.826611
$622$		−	16.3706i		−	0.656401i
$623$		−	4.83178i		−	0.193581i
$624$	−10.4211			−0.417176
$625$	−4.47962	−	24.5954i	−0.179185	−	0.983815i
$626$	−29.1414			−1.16472
$627$		−	2.80376i		−	0.111971i
$628$			5.91559i			0.236058i
$629$	2.57615			0.102718
$630$	−20.8247	+	9.74641i	−0.829677	+	0.388306i
$631$	37.9809			1.51200			0.755999	−	0.654573i	$-0.227152\pi$
$631$	37.9809			1.51200			0.755999	+	0.654573i	$0.227152\pi$
$632$			1.02554i			0.0407936i
$633$			56.8827i			2.26088i
$634$	−7.55350			−0.299988
$635$	19.0821	+	40.7720i	0.757251	+	1.61799i
$636$	−60.5353			−2.40038
$637$			6.30751i			0.249913i
$638$			3.81810i			0.151160i
$639$	74.7486			2.95701
$640$	3.47605	+	7.42712i	0.137403	+	0.293583i
$641$	41.8147			1.65158			0.825791	−	0.563977i	$-0.190729\pi$
$641$	41.8147			1.65158			0.825791	+	0.563977i	$0.190729\pi$
$642$		−	79.7983i		−	3.14939i
$643$		−	39.5687i		−	1.56044i	−0.625508	−	0.780218i	$-0.715108\pi$
$643$		−	39.5687i		−	1.56044i	0.625508	−	0.780218i	$-0.284892\pi$
$644$	−9.03769			−0.356135
$645$	43.0169	−	20.1328i	1.69379	−	0.792729i
$646$	−2.91547			−0.114708
$647$		−	11.2149i		−	0.440905i	−0.975398	−	0.220453i	$-0.929247\pi$
$647$		−	11.2149i		−	0.440905i	0.975398	−	0.220453i	$-0.0707534\pi$
$648$			0.0215437i			0.000846315i
$649$	7.10538			0.278911
$650$	6.98997	−	8.37806i	0.274169	−	0.328615i
$651$	20.6800			0.810513
$652$		−	1.87343i		−	0.0733693i
$653$		−	8.45436i		−	0.330845i	−0.986223	−	0.165422i	$-0.947101\pi$
$653$		−	8.45436i		−	0.330845i	0.986223	−	0.165422i	$-0.0528987\pi$
$654$	67.2560			2.62992
$655$	4.10255	−	1.92008i	0.160300	−	0.0750236i
$656$	−4.25953			−0.166307
$657$		−	30.2631i		−	1.18068i
$658$			19.6617i			0.766491i
$659$	4.19404			0.163376			0.0816882	−	0.996658i	$-0.473969\pi$
$659$	4.19404			0.163376			0.0816882	+	0.996658i	$0.473969\pi$
$660$	5.91157	+	12.6310i	0.230108	+	0.491661i
$661$	−46.6704			−1.81527			−0.907634	−	0.419763i	$-0.862113\pi$
$661$	−46.6704			−1.81527			−0.907634	+	0.419763i	$0.862113\pi$
$662$			37.2085i			1.44615i
$663$			4.22258i			0.163991i
$664$	4.69541			0.182217
$665$	−0.975504	−	2.08432i	−0.0378284	−	0.0808264i
$666$	18.1452			0.703112
$667$			7.33346i			0.283953i
$668$		−	43.6570i		−	1.68914i
$669$	6.61049			0.255576
$670$	−17.2109	+	8.05505i	−0.664914	+	0.311194i
$671$	2.23668			0.0863462
$672$		−	23.4245i		−	0.903618i
$673$			21.7354i			0.837837i	0.908024	+	0.418918i	$0.137591\pi$
$673$			21.7354i			0.837837i	−0.908024	+	0.418918i	$0.862409\pi$
$674$	−19.7510			−0.760782
$675$	20.0330	+	16.7139i	0.771070	+	0.643318i
$676$	−26.4103			−1.01578
$677$		−	30.6531i		−	1.17809i	−0.808099	−	0.589047i	$-0.799503\pi$
$677$		−	30.6531i		−	1.17809i	0.808099	−	0.589047i	$-0.200497\pi$
$678$			38.0777i			1.46237i
$679$	−6.83476			−0.262294
$680$	1.32525	−	0.620245i	0.0508211	−	0.0237853i
$681$	−53.8108			−2.06203
$682$		−	14.7301i		−	0.564045i
$683$			27.9782i			1.07056i	0.844676	+	0.535278i	$0.179793\pi$
$683$			27.9782i			1.07056i	−0.844676	+	0.535278i	$0.820207\pi$
$684$	−10.8132			−0.413452
$685$	−13.4897	−	28.8229i	−0.515415	−	1.10127i
$686$	−27.3738			−1.04514
$687$		−	10.4878i		−	0.400136i
$688$			26.5205i			1.01109i
$689$	−10.3053			−0.392599
$690$	21.5632	+	46.0731i	0.820896	+	1.75397i
$691$	23.1023			0.878854			0.439427	−	0.898278i	$-0.355181\pi$
$691$	23.1023			0.878854			0.439427	+	0.898278i	$0.355181\pi$
$692$		−	33.2827i		−	1.26522i
$693$		−	5.00287i		−	0.190043i
$694$	2.86079			0.108594
$695$	−15.3736	+	7.19518i	−0.583155	+	0.272929i
$696$	2.40271			0.0910744
$697$			1.72595i			0.0653748i
$698$			67.2981i			2.54727i
$699$	−39.8711			−1.50806
$700$	8.78934	+	7.33311i	0.332206	+	0.277166i
$701$	43.2476			1.63344			0.816719	−	0.577036i	$-0.195791\pi$
$701$	43.2476			1.63344			0.816719	+	0.577036i	$0.195791\pi$
$702$			11.3867i			0.429763i
$703$			1.81613i			0.0684965i
$704$	−9.68351			−0.364961
$705$	52.7792	−	24.7018i	1.98778	−	0.930322i
$706$	0.876240			0.0329777
$707$			16.3600i			0.615282i
$708$		−	44.3149i		−	1.66546i
$709$	24.7373			0.929029			0.464515	−	0.885565i	$-0.346229\pi$
$709$	24.7373			0.929029			0.464515	+	0.885565i	$0.346229\pi$
$710$	−29.9570	−	64.0078i	−1.12427	−	2.40217i
$711$	−10.8065			−0.405274
$712$			2.16579i			0.0811666i
$713$		−	28.2923i		−	1.05955i
$714$	−8.41275			−0.314839
$715$	1.00636	+	2.15025i	0.0376357	+	0.0804147i
$716$	11.5525			0.431739
$717$		−	9.68580i		−	0.361723i
$718$			65.6926i			2.45163i
$719$	39.2250			1.46285			0.731424	−	0.681923i	$-0.238856\pi$
$719$	39.2250			1.46285			0.731424	+	0.681923i	$0.238856\pi$
$720$	−34.4639	+	16.1298i	−1.28439	+	0.601123i
$721$	−12.9851			−0.483590
$722$		−	2.05535i		−	0.0764921i
$723$			55.5516i			2.06599i
$724$	12.5272			0.465570
$725$	5.95032	−	7.13195i	0.220989	−	0.264874i
$726$	−5.76269			−0.213874
$727$			2.91909i			0.108263i	0.998534	+	0.0541315i	$0.0172390\pi$
$727$			2.91909i			0.108263i	−0.998534	+	0.0541315i	$0.982761\pi$
$728$			0.504080i			0.0186825i
$729$	43.6626			1.61713
$730$	−25.9145	+	12.1285i	−0.959139	+	0.448897i
$731$	10.7460			0.397456
$732$		−	13.9498i		−	0.515598i
$733$		−	28.1294i		−	1.03898i	−0.854475	−	0.519492i	$-0.826121\pi$
$733$		−	28.1294i		−	1.03898i	0.854475	−	0.519492i	$-0.173879\pi$
$734$	−12.4106			−0.458083
$735$	15.7880	+	33.7335i	0.582348	+	1.24428i
$736$	−32.0470			−1.18127
$737$		−	4.13469i		−	0.152303i
$738$			12.1568i			0.447497i
$739$	43.2020			1.58921			0.794605	−	0.607127i	$-0.207678\pi$
$739$	43.2020			1.58921			0.794605	+	0.607127i	$0.207678\pi$
$740$	−3.82921	−	8.18170i	−0.140764	−	0.300765i
$741$	−2.97682			−0.109356
$742$		−	20.5314i		−	0.753733i
$743$		−	13.6537i		−	0.500907i	−0.968129	−	0.250454i	$-0.919420\pi$
$743$		−	13.6537i		−	0.500907i	0.968129	−	0.250454i	$-0.0805798\pi$
$744$	−9.26958			−0.339839
$745$	−15.0210	+	7.03012i	−0.550325	+	0.257564i
$746$	48.7025			1.78313
$747$			49.4773i			1.81028i
$748$			3.15534i			0.115371i
$749$	14.2514			0.520734
$750$	16.4127	−	62.3033i	0.599308	−	2.27499i
$751$	−39.1791			−1.42966			−0.714832	−	0.699296i	$-0.753497\pi$
$751$	−39.1791			−1.42966			−0.714832	+	0.699296i	$0.753497\pi$
$752$			32.5391i			1.18658i
$753$		−	33.2329i		−	1.21107i
$754$	4.05378			0.147630
$755$	9.58131	−	4.48425i	0.348699	−	0.163199i
$756$	−11.9457			−0.434460
$757$			29.1901i			1.06093i	0.847706	+	0.530466i	$0.177983\pi$
$757$			29.1901i			1.06093i	−0.847706	+	0.530466i	$0.822017\pi$
$758$		−	28.1356i		−	1.02193i
$759$	−11.0685			−0.401760
$760$	0.437259	+	0.934273i	0.0158611	+	0.0338896i
$761$	12.8923			0.467344			0.233672	−	0.972315i	$-0.424926\pi$
$761$	12.8923			0.467344			0.233672	+	0.972315i	$0.424926\pi$
$762$			116.014i			4.20276i
$763$			12.0114i			0.434842i
$764$	48.4262			1.75200
$765$	6.53575	+	13.9647i	0.236301	+	0.504893i
$766$	−28.8973			−1.04410
$767$		−	7.54398i		−	0.272397i
$768$		−	33.1670i		−	1.19681i
$769$	−33.6067			−1.21189			−0.605944	−	0.795507i	$-0.707205\pi$
$769$	−33.6067			−1.21189			−0.605944	+	0.795507i	$0.707205\pi$
$770$	−4.28399	+	2.00500i	−0.154384	+	0.0722551i
$771$	−12.1221			−0.436568
$772$			13.1916i			0.474777i
$773$		−	34.8153i		−	1.25222i	−0.779735	−	0.626109i	$-0.784646\pi$
$773$		−	34.8153i		−	1.25222i	0.779735	−	0.626109i	$-0.215354\pi$
$774$	75.6901			2.72063
$775$	−22.9561	+	27.5149i	−0.824609	+	0.988363i
$776$	3.06361			0.109977
$777$			5.24053i			0.188003i
$778$		−	4.09956i		−	0.146976i
$779$	−1.21675			−0.0435947
$780$	13.4107	−	6.27647i	0.480179	−	0.224734i
$781$	15.3770			0.550233
$782$			11.5095i			0.411579i
$783$			9.69309i			0.346403i
$784$	−20.7971			−0.742755
$785$	2.52068	+	5.38582i	0.0899668	+	0.192228i
$786$	11.6736			0.416382
$787$			13.8735i			0.494536i	0.968947	+	0.247268i	$0.0795328\pi$
$787$			13.8735i			0.494536i	−0.968947	+	0.247268i	$0.920467\pi$
$788$		−	55.1824i		−	1.96579i
$789$	−47.0614			−1.67543
$790$	4.33090	+	9.25365i	0.154086	+	0.329230i
$791$	−6.80039			−0.241794
$792$			2.24248i			0.0796831i
$793$		−	2.37475i		−	0.0843297i
$794$	−56.2445			−1.99604
$795$	−55.1140	+	25.7945i	−1.95469	+	0.914836i
$796$	−19.2765			−0.683236
$797$			1.91740i			0.0679178i	0.999423	+	0.0339589i	$0.0108115\pi$
$797$			1.91740i			0.0679178i	−0.999423	+	0.0339589i	$0.989188\pi$
$798$		−	5.93081i		−	0.209948i
$799$	13.1847			0.466442
$800$	31.1664	+	26.0027i	1.10190	+	0.919334i
$801$	−22.8218			−0.806368
$802$		−	34.9102i		−	1.23272i
$803$		−	6.22562i		−	0.219697i
$804$	−25.7873			−0.909447
$805$	−8.22831	+	3.85102i	−0.290010	+	0.135731i
$806$	−15.6394			−0.550873
$807$		−	4.56428i		−	0.160670i
$808$		−	7.33320i		−	0.257981i
$809$	−47.2032			−1.65958			−0.829788	−	0.558079i	$-0.811539\pi$
$809$	−47.2032			−1.65958			−0.829788	+	0.558079i	$0.811539\pi$
$810$	0.0909802	+	0.194393i	0.00319672	+	0.00683029i
$811$	−32.9669			−1.15762			−0.578812	−	0.815461i	$-0.696484\pi$
$811$	−32.9669			−1.15762			−0.578812	+	0.815461i	$0.696484\pi$
$812$			4.25278i			0.149243i
$813$		−	19.1539i		−	0.671758i
$814$	3.73277			0.130834
$815$	−0.798283	−	1.70566i	−0.0279626	−	0.0597466i
$816$	−13.9227			−0.487392
$817$			7.57572i			0.265041i
$818$			29.9830i			1.04833i
$819$	−5.31168			−0.185605
$820$	5.48151	−	2.56546i	0.191423	−	0.0895897i
$821$	44.8045			1.56369			0.781845	−	0.623473i	$-0.214279\pi$
$821$	44.8045			1.56369			0.781845	+	0.623473i	$0.214279\pi$
$822$		−	82.0139i		−	2.86056i
$823$			35.7590i			1.24648i	0.782031	+	0.623240i	$0.214184\pi$
$823$			35.7590i			1.24648i	−0.782031	+	0.623240i	$0.785816\pi$
$824$	5.82043			0.202764
$825$	10.7643	+	8.98087i	0.374765	+	0.312674i
$826$	15.0301			0.522963
$827$		−	34.4851i		−	1.19916i	−0.800314	−	0.599582i	$-0.795334\pi$
$827$		−	34.4851i		−	1.19916i	0.800314	−	0.599582i	$-0.204666\pi$
$828$			42.6874i			1.48349i
$829$	−14.7995			−0.514008			−0.257004	−	0.966410i	$-0.582735\pi$
$829$	−14.7995			−0.514008			−0.257004	+	0.966410i	$0.582735\pi$
$830$	42.3677	−	19.8290i	1.47061	−	0.688274i
$831$	−75.7075			−2.62626
$832$			10.2812i			0.356438i
$833$			8.42693i			0.291976i
$834$	−43.7448			−1.51476
$835$	−18.6025	−	39.7473i	−0.643767	−	1.37551i
$836$	−2.22445			−0.0769341
$837$		−	37.3957i		−	1.29258i
$838$		−	13.4043i		−	0.463043i
$839$	18.2520			0.630130			0.315065	−	0.949070i	$-0.397974\pi$
$839$	18.2520			0.630130			0.315065	+	0.949070i	$0.397974\pi$
$840$	1.26173	+	2.69589i	0.0435340	+	0.0930172i
$841$	−25.5492			−0.881006
$842$		−	73.2210i		−	2.52336i
$843$		−	7.05232i		−	0.242895i
$844$	45.1296			1.55343
$845$	−24.0451	+	11.2536i	−0.827176	+	0.387135i
$846$	92.8672			3.19284
$847$		−	1.02917i		−	0.0353628i
$848$		−	33.9785i		−	1.16683i
$849$	−26.4731			−0.908554
$850$	9.33871	−	11.1932i	0.320315	−	0.383924i
$851$	7.16957			0.245770
$852$		−	95.9036i		−	3.28560i
$853$			21.7411i			0.744401i	0.928152	+	0.372201i	$0.121397\pi$
$853$			21.7411i			0.744401i	−0.928152	+	0.372201i	$0.878603\pi$
$854$	4.73127			0.161901
$855$	−9.84478	+	4.60756i	−0.336684	+	0.157575i
$856$	−6.38802			−0.218338
$857$			15.2552i			0.521109i	0.965459	+	0.260554i	$0.0839054\pi$
$857$			15.2552i			0.521109i	−0.965459	+	0.260554i	$0.916095\pi$
$858$			6.11840i			0.208879i
$859$	−41.9667			−1.43189			−0.715943	−	0.698159i	$-0.754003\pi$
$859$	−41.9667			−1.43189			−0.715943	+	0.698159i	$0.754003\pi$
$860$	−15.9730	−	34.1288i	−0.544674	−	1.16378i
$861$	−3.51101			−0.119655
$862$			42.6328i			1.45208i
$863$			19.8494i			0.675681i	0.941203	+	0.337840i	$0.109696\pi$
$863$			19.8494i			0.675681i	−0.941203	+	0.337840i	$0.890304\pi$
$864$	−42.3585			−1.44107
$865$	−14.1820	−	30.3020i	−0.482201	−	1.03030i
$866$	−72.9727			−2.47971
$867$		−	42.0225i		−	1.42716i
$868$		−	16.4071i		−	0.556893i
$869$	−2.22307			−0.0754124
$870$	21.6802	−	10.1468i	0.735027	−	0.344008i
$871$	−4.38991			−0.148746
$872$		−	5.38399i		−	0.182325i
$873$			32.2824i			1.09259i
$874$	−8.11394			−0.274458
$875$	11.1269	+	2.93119i	0.376158	+	0.0990923i
$876$	−38.8280			−1.31188
$877$			8.72116i			0.294493i	0.989100	+	0.147246i	$0.0470410\pi$
$877$			8.72116i			0.294493i	−0.989100	+	0.147246i	$0.952959\pi$
$878$		−	4.30212i		−	0.145190i
$879$	−25.4304			−0.857746
$880$	−7.08980	+	3.31817i	−0.238997	+	0.111856i
$881$	−36.0915			−1.21595			−0.607977	−	0.793955i	$-0.708019\pi$
$881$	−36.0915			−1.21595			−0.607977	+	0.793955i	$0.708019\pi$
$882$			59.3555i			1.99860i
$883$			6.08363i			0.204730i	0.994747	+	0.102365i	$0.0326410\pi$
$883$			6.08363i			0.204730i	−0.994747	+	0.102365i	$0.967359\pi$
$884$	3.35011			0.112676
$885$	−18.8829	−	40.3463i	−0.634741	−	1.35622i
$886$	−36.1974			−1.21607
$887$		−	14.7548i		−	0.495418i	−0.968835	−	0.247709i	$-0.920322\pi$
$887$		−	14.7548i		−	0.495418i	0.968835	−	0.247709i	$-0.0796776\pi$
$888$		−	2.34901i		−	0.0788277i
$889$	−20.7193			−0.694903
$890$	9.14627	+	19.5425i	0.306584	+	0.655065i
$891$	−0.0467005			−0.00156452
$892$		−	5.24463i		−	0.175603i
$893$			9.29494i			0.311043i
$894$	−42.7413			−1.42948
$895$	10.5179	−	4.92262i	0.351576	−	0.164545i
$896$	−3.77427			−0.126090
$897$			11.7517i			0.392377i
$898$			13.4507i			0.448856i
$899$	−13.3132			−0.444021
$900$	34.6362	−	41.5144i	1.15454	−	1.38381i
$901$	−13.7680			−0.458678
$902$			2.50085i			0.0832692i
$903$			21.8601i			0.727459i
$904$	3.04820			0.101382
$905$	11.4053	−	5.33792i	0.379125	−	0.177439i
$906$	27.2631			0.905755
$907$			32.9622i			1.09449i	0.836972	+	0.547245i	$0.184323\pi$
$907$			32.9622i			1.09449i	−0.836972	+	0.547245i	$0.815677\pi$
$908$			42.6925i			1.41680i
$909$	77.2727			2.56297
$910$	2.12876	+	4.54843i	0.0705677	+	0.150779i
$911$	33.2950			1.10311			0.551555	−	0.834138i	$-0.314035\pi$
$911$	33.2950			1.10311			0.551555	+	0.834138i	$0.314035\pi$
$912$		−	9.81520i		−	0.325014i
$913$			10.1783i			0.336852i
$914$	28.9081			0.956194
$915$	−5.94409	−	12.7005i	−0.196506	−	0.419865i
$916$	−8.32084			−0.274928
$917$			2.08481i			0.0688465i
$918$			15.2128i			0.502097i
$919$	25.1476			0.829541			0.414771	−	0.909926i	$-0.363862\pi$
$919$	25.1476			0.829541			0.414771	+	0.909926i	$0.363862\pi$
$920$	3.68825	−	1.72618i	0.121598	−	0.0569104i
$921$	−40.5006			−1.33454
$922$			12.0985i			0.398442i
$923$		−	16.3262i		−	0.537384i
$924$	−6.41876			−0.211162
$925$	−6.97256	−	5.81733i	−0.229256	−	0.191273i
$926$	−61.2209			−2.01185
$927$			61.3320i			2.01441i
$928$			15.0801i			0.495027i
$929$	−2.19388			−0.0719788			−0.0359894	−	0.999352i	$-0.511458\pi$
$929$	−2.19388			−0.0719788			−0.0359894	+	0.999352i	$0.511458\pi$
$930$	−83.6415	+	39.1460i	−2.74271	+	1.28365i
$931$	−5.94080			−0.194702
$932$			31.6329i			1.03617i
$933$		−	22.3316i		−	0.731104i
$934$	−17.9470			−0.587243
$935$	1.34451	+	2.87276i	0.0439703	+	0.0939494i
$936$	2.38090			0.0778223
$937$		−	19.9424i		−	0.651490i	−0.945458	−	0.325745i	$-0.894385\pi$
$937$		−	19.9424i		−	0.651490i	0.945458	−	0.325745i	$-0.105615\pi$
$938$		−	8.74614i		−	0.285571i
$939$	−39.7526			−1.29728
$940$	−19.5979	−	41.8740i	−0.639213	−	1.36578i
$941$	−29.2455			−0.953377			−0.476689	−	0.879072i	$-0.658163\pi$
$941$	−29.2455			−0.953377			−0.476689	+	0.879072i	$0.658163\pi$
$942$			15.3250i			0.499317i
$943$			4.80341i			0.156421i
$944$	24.8740			0.809581
$945$	−10.8759	+	5.09013i	−0.353792	+	0.165582i
$946$	15.5707			0.506248
$947$			22.0101i			0.715233i	0.933868	+	0.357617i	$0.116411\pi$
$947$			22.0101i			0.715233i	−0.933868	+	0.357617i	$0.883589\pi$
$948$			13.8648i			0.450309i
$949$	−6.60991			−0.214567
$950$	7.89098	+	6.58359i	0.256017	+	0.213600i
$951$	−10.3040			−0.334129
$952$			0.673459i			0.0218269i
$953$		−	4.17843i		−	0.135353i	−0.997707	−	0.0676764i	$-0.978441\pi$
$953$		−	4.17843i		−	0.135353i	0.997707	−	0.0676764i	$-0.0215585\pi$
$954$	−96.9754			−3.13969
$955$	44.0894	−	20.6347i	1.42670	−	0.667725i
$956$	−7.68453			−0.248535
$957$			5.20838i			0.168363i
$958$			57.6125i			1.86137i
$959$	14.6471			0.472979
$960$	25.7344	+	54.9856i	0.830574	+	1.77465i
$961$	20.3621			0.656841
$962$		−	3.96318i		−	0.127778i
$963$		−	67.3130i		−	2.16913i
$964$	44.0736			1.41951
$965$	5.62104	+	12.0102i	0.180948	+	0.386624i
$966$	−23.4132			−0.753307
$967$			39.8457i			1.28135i	0.767811	+	0.640676i	$0.221346\pi$
$967$			39.8457i			1.28135i	−0.767811	+	0.640676i	$0.778654\pi$
$968$			0.461316i			0.0148273i
$969$	−3.97708			−0.127762
$970$	27.6436	−	12.9378i	0.887584	−	0.415408i
$971$	−5.23210			−0.167906			−0.0839531	−	0.996470i	$-0.526755\pi$
$971$	−5.23210			−0.167906			−0.0839531	+	0.996470i	$0.526755\pi$
$972$		−	34.5299i		−	1.10755i
$973$		−	7.81249i		−	0.250457i
$974$	54.4926			1.74606
$975$	9.53523	−	11.4288i	0.305372	−	0.366013i
$976$	7.83003			0.250633
$977$			48.6872i			1.55764i	0.627247	+	0.778821i	$0.284182\pi$
$977$			48.6872i			1.55764i	−0.627247	+	0.778821i	$0.715818\pi$
$978$		−	4.85335i		−	0.155193i
$979$	−4.69482			−0.150047
$980$	26.7635	−	12.5259i	0.854928	−	0.400124i
$981$	56.7331			1.81135
$982$			72.9659i			2.32844i
$983$			5.87633i			0.187426i	0.995599	+	0.0937129i	$0.0298736\pi$
$983$			5.87633i			0.187426i	−0.995599	+	0.0937129i	$0.970126\pi$
$984$	1.57377			0.0501700
$985$	−23.5136	−	50.2405i	−0.749205	−	1.60079i
$986$	5.41591			0.172478
$987$			26.8210i			0.853723i
$988$			2.36175i			0.0751374i
$989$	29.9068			0.950982
$990$	9.47013	+	20.2344i	0.300981	+	0.643092i
$991$	49.6851			1.57830			0.789149	−	0.614201i	$-0.210522\pi$
$991$	49.6851			1.57830			0.789149	+	0.614201i	$0.210522\pi$
$992$		−	58.1784i		−	1.84717i
$993$			50.7572i			1.61073i
$994$	32.5271			1.03170
$995$	−17.5501	+	8.21383i	−0.556377	+	0.260396i
$996$	63.4801			2.01144
$997$			5.26243i			0.166663i	0.996522	+	0.0833314i	$0.0265560\pi$
$997$			5.26243i			0.166663i	−0.996522	+	0.0833314i	$0.973444\pi$
$998$		−	72.0893i		−	2.28195i
$999$	9.47646			0.299822

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.b.b.419.3	✓	16
5.2	odd	4		5225.2.a.z.1.14		16
5.3	odd	4		5225.2.a.z.1.3		16
5.4	even	2	inner	1045.2.b.b.419.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.b.419.3	✓	16	1.1	even	1	trivial
1045.2.b.b.419.14	yes	16	5.4	even	2	inner
5225.2.a.z.1.3		16	5.3	odd	4
5225.2.a.z.1.14		16	5.2	odd	4

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.05535i q^{2} -2.80376i q^{3} -2.22445 q^{4} +(-2.02523 + 0.947852i) q^{5} -5.76269 q^{6} -1.02917i q^{7} +0.461316i q^{8} -4.86106 q^{9} +O(q^{10})\)	\(q-2.05535i q^{2} -2.80376i q^{3} -2.22445 q^{4} +(-2.02523 + 0.947852i) q^{5} -5.76269 q^{6} -1.02917i q^{7} +0.461316i q^{8} -4.86106 q^{9} +(1.94816 + 4.16256i) q^{10} -1.00000 q^{11} +6.23681i q^{12} +1.06173i q^{13} -2.11531 q^{14} +(2.65755 + 5.67827i) q^{15} -3.50073 q^{16} +1.41848i q^{17} +9.99115i q^{18} -1.00000 q^{19} +(4.50503 - 2.10845i) q^{20} -2.88555 q^{21} +2.05535i q^{22} +3.94773i q^{23} +1.29342 q^{24} +(3.20315 - 3.83925i) q^{25} +2.18222 q^{26} +5.21795i q^{27} +2.28934i q^{28} +1.85764 q^{29} +(11.6708 - 5.46218i) q^{30} -7.16673 q^{31} +8.11784i q^{32} +2.80376i q^{33} +2.91547 q^{34} +(0.975504 + 2.08432i) q^{35} +10.8132 q^{36} -1.81613i q^{37} +2.05535i q^{38} +2.97682 q^{39} +(-0.437259 - 0.934273i) q^{40} +1.21675 q^{41} +5.93081i q^{42} -7.57572i q^{43} +2.22445 q^{44} +(9.84478 - 4.60756i) q^{45} +8.11394 q^{46} -9.29494i q^{47} +9.81520i q^{48} +5.94080 q^{49} +(-7.89098 - 6.58359i) q^{50} +3.97708 q^{51} -2.36175i q^{52} +9.70613i q^{53} +10.7247 q^{54} +(2.02523 - 0.947852i) q^{55} +0.474774 q^{56} +2.80376i q^{57} -3.81810i q^{58} -7.10538 q^{59} +(-5.91157 - 12.6310i) q^{60} -2.23668 q^{61} +14.7301i q^{62} +5.00287i q^{63} +9.68351 q^{64} +(-1.00636 - 2.15025i) q^{65} +5.76269 q^{66} +4.13469i q^{67} -3.15534i q^{68} +11.0685 q^{69} +(4.28399 - 2.00500i) q^{70} -15.3770 q^{71} -2.24248i q^{72} +6.22562i q^{73} -3.73277 q^{74} +(-10.7643 - 8.98087i) q^{75} +2.22445 q^{76} +1.02917i q^{77} -6.11840i q^{78} +2.22307 q^{79} +(7.08980 - 3.31817i) q^{80} +0.0467005 q^{81} -2.50085i q^{82} -10.1783i q^{83} +6.41876 q^{84} +(-1.34451 - 2.87276i) q^{85} -15.5707 q^{86} -5.20838i q^{87} -0.461316i q^{88} +4.69482 q^{89} +(-9.47013 - 20.2344i) q^{90} +1.09270 q^{91} -8.78150i q^{92} +20.0938i q^{93} -19.1043 q^{94} +(2.02523 - 0.947852i) q^{95} +22.7605 q^{96} -6.64102i q^{97} -12.2104i q^{98} +4.86106 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) \) 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9	\( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + 10 q^{10} - 16 q^{11} + 4 q^{14} + 3 q^{15} - 18 q^{16} - 16 q^{19} - 2 q^{20} - 10 q^{21} + 10 q^{24} - 7 q^{25} - 24 q^{26} + 2 q^{29} + 4 q^{30} - 32 q^{31} - 16 q^{34} - 18 q^{35} + 18 q^{36} + 40 q^{39} - 28 q^{40} + 6 q^{41} + 6 q^{44} + 16 q^{45} + 38 q^{49} - 30 q^{50} - 16 q^{51} + 18 q^{54} - 3 q^{55} + 12 q^{56} + 24 q^{59} - 20 q^{60} - 42 q^{61} + 62 q^{64} - 20 q^{65} + 8 q^{66} + 30 q^{69} - 18 q^{70} - 46 q^{71} - 2 q^{74} - 25 q^{75} + 6 q^{76} + 74 q^{79} - 22 q^{80} - 56 q^{81} + 34 q^{84} - 18 q^{85} + 8 q^{86} + 14 q^{89} - 4 q^{90} - 24 q^{91} + 64 q^{94} - 3 q^{95} + 54 q^{96} + 8 q^{99}+O(q^{100}) \) 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9 + 10 * q^10 - 16 * q^11 + 4 * q^14 + 3 * q^15 - 18 * q^16 - 16 * q^19 - 2 * q^20 - 10 * q^21 + 10 * q^24 - 7 * q^25 - 24 * q^26 + 2 * q^29 + 4 * q^30 - 32 * q^31 - 16 * q^34 - 18 * q^35 + 18 * q^36 + 40 * q^39 - 28 * q^40 + 6 * q^41 + 6 * q^44 + 16 * q^45 + 38 * q^49 - 30 * q^50 - 16 * q^51 + 18 * q^54 - 3 * q^55 + 12 * q^56 + 24 * q^59 - 20 * q^60 - 42 * q^61 + 62 * q^64 - 20 * q^65 + 8 * q^66 + 30 * q^69 - 18 * q^70 - 46 * q^71 - 2 * q^74 - 25 * q^75 + 6 * q^76 + 74 * q^79 - 22 * q^80 - 56 * q^81 + 34 * q^84 - 18 * q^85 + 8 * q^86 + 14 * q^89 - 4 * q^90 - 24 * q^91 + 64 * q^94 - 3 * q^95 + 54 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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