LMFDB - Embedded newform 5070.2.b.y.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, not 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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$1.24698i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.y.1351.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -5.04892i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -5.04892i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.74094i q^{11} -1.00000 q^{12} -5.04892 q^{14} +1.00000i q^{15} +1.00000 q^{16} +1.08815 q^{17} -1.00000i q^{18} +4.44504i q^{19} -1.00000i q^{20} -5.04892i q^{21} +4.74094 q^{22} -2.69202 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +5.04892i q^{28} -7.93900 q^{29} +1.00000 q^{30} +7.85086i q^{31} -1.00000i q^{32} +4.74094i q^{33} -1.08815i q^{34} +5.04892 q^{35} -1.00000 q^{36} +7.14675i q^{37} +4.44504 q^{38} -1.00000 q^{40} -0.664874i q^{41} -5.04892 q^{42} -4.35690 q^{43} -4.74094i q^{44} +1.00000i q^{45} +2.69202i q^{46} -7.00000i q^{47} +1.00000 q^{48} -18.4916 q^{49} +1.00000i q^{50} +1.08815 q^{51} -10.8605 q^{53} -1.00000i q^{54} -4.74094 q^{55} +5.04892 q^{56} +4.44504i q^{57} +7.93900i q^{58} -12.2784i q^{59} -1.00000i q^{60} -11.0707 q^{61} +7.85086 q^{62} -5.04892i q^{63} -1.00000 q^{64} +4.74094 q^{66} +9.58211i q^{67} -1.08815 q^{68} -2.69202 q^{69} -5.04892i q^{70} +4.47219i q^{71} +1.00000i q^{72} +8.17390i q^{73} +7.14675 q^{74} -1.00000 q^{75} -4.44504i q^{76} +23.9366 q^{77} +14.1468 q^{79} +1.00000i q^{80} +1.00000 q^{81} -0.664874 q^{82} +4.86294i q^{83} +5.04892i q^{84} +1.08815i q^{85} +4.35690i q^{86} -7.93900 q^{87} -4.74094 q^{88} -14.8877i q^{89} +1.00000 q^{90} +2.69202 q^{92} +7.85086i q^{93} -7.00000 q^{94} -4.44504 q^{95} -1.00000i q^{96} +14.9269i q^{97} +18.4916i q^{98} +4.74094i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} - 6 q^{12} - 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} - 28 q^{29} + 6 q^{30} + 12 q^{35} - 6 q^{36} + 26 q^{38} - 6 q^{40} - 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} + 12 q^{56} - 42 q^{61} + 20 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} - 12 q^{74} - 6 q^{75} + 42 q^{77} + 30 q^{79} + 6 q^{81} - 6 q^{82} - 28 q^{87} + 6 q^{90} + 6 q^{92} - 42 q^{94} - 26 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 - 6 * q^12 - 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 - 28 * q^29 + 6 * q^30 + 12 * q^35 - 6 * q^36 + 26 * q^38 - 6 * q^40 - 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 + 12 * q^56 - 42 * q^61 + 20 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 - 12 * q^74 - 6 * q^75 + 42 * q^77 + 30 * q^79 + 6 * q^81 - 6 * q^82 - 28 * q^87 + 6 * q^90 + 6 * q^92 - 42 * q^94 - 26 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 1.00000i	− 0.408248i
$7$	− 5.04892i	− 1.90831i	−0.299311	−	0.954156i	$-0.596757\pi$
$7$	− 5.04892i	− 1.90831i	0.299311	−	0.954156i	$-0.403243\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	4.74094i	1.42945i	0.699407	+	0.714723i	$0.253447\pi$
$11$	4.74094i	1.42945i	−0.699407	+	0.714723i	$0.746553\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−5.04892	−1.34938
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	1.08815	0.263914	0.131957	−	0.991255i	$-0.457874\pi$
$17$	1.08815	0.263914	0.131957	+	0.991255i	$0.457874\pi$
$18$	− 1.00000i	− 0.235702i
$19$	4.44504i	1.01976i	0.860245	+	0.509881i	$0.170311\pi$
$19$	4.44504i	1.01976i	−0.860245	+	0.509881i	$0.829689\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 5.04892i	− 1.10176i
$22$	4.74094	1.01077
$23$	−2.69202	−0.561325	−0.280663	−	0.959806i	$-0.590554\pi$
$23$	−2.69202	−0.561325	−0.280663	+	0.959806i	$0.590554\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	5.04892i	0.954156i
$29$	−7.93900	−1.47424	−0.737118	−	0.675764i	$-0.763814\pi$
$29$	−7.93900	−1.47424	−0.737118	+	0.675764i	$0.763814\pi$
$30$	1.00000	0.182574
$31$	7.85086i	1.41006i	0.709180	+	0.705028i	$0.249065\pi$
$31$	7.85086i	1.41006i	−0.709180	+	0.705028i	$0.750935\pi$
$32$	− 1.00000i	− 0.176777i
$33$	4.74094i	0.825292i
$34$	− 1.08815i	− 0.186615i
$35$	5.04892	0.853423
$36$	−1.00000	−0.166667
$37$	7.14675i	1.17492i	0.809254	+	0.587459i	$0.199872\pi$
$37$	7.14675i	1.17492i	−0.809254	+	0.587459i	$0.800128\pi$
$38$	4.44504	0.721081
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 0.664874i	− 0.103836i	−0.998651	−	0.0519180i	$-0.983467\pi$
$41$	− 0.664874i	− 0.103836i	0.998651	−	0.0519180i	$-0.0165334\pi$
$42$	−5.04892	−0.779065
$43$	−4.35690	−0.664420	−0.332210	−	0.943205i	$-0.607794\pi$
$43$	−4.35690	−0.664420	−0.332210	+	0.943205i	$0.607794\pi$
$44$	− 4.74094i	− 0.714723i
$45$	1.00000i	0.149071i
$46$	2.69202i	0.396917i
$47$	− 7.00000i	− 1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$	− 7.00000i	− 1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	1.00000	0.144338
$49$	−18.4916	−2.64165
$50$	1.00000i	0.141421i
$51$	1.08815	0.152371
$52$	0	0
$53$	−10.8605	−1.49181	−0.745905	−	0.666052i	$-0.767983\pi$
$53$	−10.8605	−1.49181	−0.745905	+	0.666052i	$0.767983\pi$
$54$	− 1.00000i	− 0.136083i
$55$	−4.74094	−0.639268
$56$	5.04892	0.674690
$57$	4.44504i	0.588760i
$58$	7.93900i	1.04244i
$59$	− 12.2784i	− 1.59852i	−0.600988	−	0.799258i	$-0.705226\pi$
$59$	− 12.2784i	− 1.59852i	0.600988	−	0.799258i	$-0.294774\pi$
$60$	− 1.00000i	− 0.129099i
$61$	−11.0707	−1.41746	−0.708728	−	0.705482i	$-0.750731\pi$
$61$	−11.0707	−1.41746	−0.708728	+	0.705482i	$0.750731\pi$
$62$	7.85086	0.997060
$63$	− 5.04892i	− 0.636104i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.74094	0.583569
$67$	9.58211i	1.17064i	0.810802	+	0.585320i	$0.199031\pi$
$67$	9.58211i	1.17064i	−0.810802	+	0.585320i	$0.800969\pi$
$68$	−1.08815	−0.131957
$69$	−2.69202	−0.324081
$70$	− 5.04892i	− 0.603461i
$71$	4.47219i	0.530751i	0.964145	+	0.265376i	$0.0854960\pi$
$71$	4.47219i	0.530751i	−0.964145	+	0.265376i	$0.914504\pi$
$72$	1.00000i	0.117851i
$73$	8.17390i	0.956683i	0.878174	+	0.478341i	$0.158762\pi$
$73$	8.17390i	0.956683i	−0.878174	+	0.478341i	$0.841238\pi$
$74$	7.14675	0.830793
$75$	−1.00000	−0.115470
$76$	− 4.44504i	− 0.509881i
$77$	23.9366	2.72783
$78$	0	0
$79$	14.1468	1.59163	0.795817	−	0.605537i	$-0.207042\pi$
$79$	14.1468	1.59163	0.795817	+	0.605537i	$0.207042\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	−0.664874	−0.0734231
$83$	4.86294i	0.533777i	0.963727	+	0.266888i	$0.0859955\pi$
$83$	4.86294i	0.533777i	−0.963727	+	0.266888i	$0.914005\pi$
$84$	5.04892i	0.550882i
$85$	1.08815i	0.118026i
$86$	4.35690i	0.469816i
$87$	−7.93900	−0.851150
$88$	−4.74094	−0.505386
$89$	− 14.8877i	− 1.57809i	−0.614334	−	0.789046i	$-0.710575\pi$
$89$	− 14.8877i	− 1.57809i	0.614334	−	0.789046i	$-0.289425\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	2.69202	0.280663
$93$	7.85086i	0.814096i
$94$	−7.00000	−0.721995
$95$	−4.44504	−0.456052
$96$	− 1.00000i	− 0.102062i
$97$	14.9269i	1.51560i	0.652487	+	0.757800i	$0.273726\pi$
$97$	14.9269i	1.51560i	−0.652487	+	0.757800i	$0.726274\pi$
$98$	18.4916i	1.86793i
$99$	4.74094i	0.476482i
$100$	1.00000	0.100000
$101$	−11.4819	−1.14249	−0.571245	−	0.820780i	$-0.693539\pi$
$101$	−11.4819	−1.14249	−0.571245	+	0.820780i	$0.693539\pi$
$102$	− 1.08815i	− 0.107743i
$103$	−4.16421	−0.410312	−0.205156	−	0.978729i	$-0.565770\pi$
$103$	−4.16421	−0.410312	−0.205156	+	0.978729i	$0.565770\pi$
$104$	0	0
$105$	5.04892	0.492724
$106$	10.8605i	1.05487i
$107$	−0.796561	−0.0770065	−0.0385032	−	0.999258i	$-0.512259\pi$
$107$	−0.796561	−0.0770065	−0.0385032	+	0.999258i	$0.512259\pi$
$108$	−1.00000	−0.0962250
$109$	0.704103i	0.0674408i	0.999431	+	0.0337204i	$0.0107356\pi$
$109$	0.704103i	0.0674408i	−0.999431	+	0.0337204i	$0.989264\pi$
$110$	4.74094i	0.452031i
$111$	7.14675i	0.678340i
$112$	− 5.04892i	− 0.477078i
$113$	14.4983	1.36388	0.681941	−	0.731407i	$-0.261136\pi$
$113$	14.4983	1.36388	0.681941	+	0.731407i	$0.261136\pi$
$114$	4.44504	0.416316
$115$	− 2.69202i	− 0.251032i
$116$	7.93900	0.737118
$117$	0	0
$118$	−12.2784	−1.13032
$119$	− 5.49396i	− 0.503630i
$120$	−1.00000	−0.0912871
$121$	−11.4765	−1.04332
$122$	11.0707i	1.00229i
$123$	− 0.664874i	− 0.0599497i
$124$	− 7.85086i	− 0.705028i
$125$	− 1.00000i	− 0.0894427i
$126$	−5.04892	−0.449793
$127$	2.93900	0.260794	0.130397	−	0.991462i	$-0.458375\pi$
$127$	2.93900	0.260794	0.130397	+	0.991462i	$0.458375\pi$
$128$	1.00000i	0.0883883i
$129$	−4.35690	−0.383603
$130$	0	0
$131$	−3.56465	−0.311445	−0.155722	−	0.987801i	$-0.549771\pi$
$131$	−3.56465	−0.311445	−0.155722	+	0.987801i	$0.549771\pi$
$132$	− 4.74094i	− 0.412646i
$133$	22.4426	1.94602
$134$	9.58211	0.827768
$135$	1.00000i	0.0860663i
$136$	1.08815i	0.0933077i
$137$	9.47650i	0.809632i	0.914398	+	0.404816i	$0.132664\pi$
$137$	9.47650i	0.809632i	−0.914398	+	0.404816i	$0.867336\pi$
$138$	2.69202i	0.229160i
$139$	−14.2078	−1.20509	−0.602543	−	0.798087i	$-0.705846\pi$
$139$	−14.2078	−1.20509	−0.602543	+	0.798087i	$0.705846\pi$
$140$	−5.04892	−0.426711
$141$	− 7.00000i	− 0.589506i
$142$	4.47219	0.375298
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 7.93900i	− 0.659298i
$146$	8.17390	0.676477
$147$	−18.4916	−1.52516
$148$	− 7.14675i	− 0.587459i
$149$	10.2010i	0.835702i	0.908516	+	0.417851i	$0.137217\pi$
$149$	10.2010i	0.835702i	−0.908516	+	0.417851i	$0.862783\pi$
$150$	1.00000i	0.0816497i
$151$	− 3.86294i	− 0.314361i	−0.987570	−	0.157181i	$-0.949760\pi$
$151$	− 3.86294i	− 0.314361i	0.987570	−	0.157181i	$-0.0502405\pi$
$152$	−4.44504	−0.360541
$153$	1.08815	0.0879714
$154$	− 23.9366i	− 1.92887i
$155$	−7.85086	−0.630596
$156$	0	0
$157$	−11.2174	−0.895249	−0.447625	−	0.894222i	$-0.647730\pi$
$157$	−11.2174	−0.895249	−0.447625	+	0.894222i	$0.647730\pi$
$158$	− 14.1468i	− 1.12546i
$159$	−10.8605	−0.861297
$160$	1.00000	0.0790569
$161$	13.5918i	1.07118i
$162$	− 1.00000i	− 0.0785674i
$163$	6.50902i	0.509826i	0.966964	+	0.254913i	$0.0820469\pi$
$163$	6.50902i	0.509826i	−0.966964	+	0.254913i	$0.917953\pi$
$164$	0.664874i	0.0519180i
$165$	−4.74094	−0.369082
$166$	4.86294	0.377437
$167$	− 9.67456i	− 0.748640i	−0.927300	−	0.374320i	$-0.877876\pi$
$167$	− 9.67456i	− 0.748640i	0.927300	−	0.374320i	$-0.122124\pi$
$168$	5.04892	0.389532
$169$	0	0
$170$	1.08815	0.0834570
$171$	4.44504i	0.339921i
$172$	4.35690	0.332210
$173$	13.8388	1.05214	0.526071	−	0.850441i	$-0.323665\pi$
$173$	13.8388	1.05214	0.526071	+	0.850441i	$0.323665\pi$
$174$	7.93900i	0.601854i
$175$	5.04892i	0.381662i
$176$	4.74094i	0.357362i
$177$	− 12.2784i	− 0.922904i
$178$	−14.8877	−1.11588
$179$	14.2078	1.06194	0.530969	−	0.847392i	$-0.321828\pi$
$179$	14.2078	1.06194	0.530969	+	0.847392i	$0.321828\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	24.9028	1.85101	0.925504	−	0.378739i	$-0.123642\pi$
$181$	24.9028	1.85101	0.925504	+	0.378739i	$0.123642\pi$
$182$	0	0
$183$	−11.0707	−0.818369
$184$	− 2.69202i	− 0.198458i
$185$	−7.14675	−0.525440
$186$	7.85086	0.575653
$187$	5.15883i	0.377251i
$188$	7.00000i	0.510527i
$189$	− 5.04892i	− 0.367255i
$190$	4.44504i	0.322477i
$191$	2.44504	0.176917	0.0884585	−	0.996080i	$-0.471806\pi$
$191$	2.44504	0.176917	0.0884585	+	0.996080i	$0.471806\pi$
$192$	−1.00000	−0.0721688
$193$	15.2959i	1.10102i	0.834828	+	0.550511i	$0.185567\pi$
$193$	15.2959i	1.10102i	−0.834828	+	0.550511i	$0.814433\pi$
$194$	14.9269	1.07169
$195$	0	0
$196$	18.4916	1.32083
$197$	3.44935i	0.245756i	0.992422	+	0.122878i	$0.0392124\pi$
$197$	3.44935i	0.245756i	−0.992422	+	0.122878i	$0.960788\pi$
$198$	4.74094	0.336924
$199$	9.18598	0.651177	0.325588	−	0.945512i	$-0.394438\pi$
$199$	9.18598	0.651177	0.325588	+	0.945512i	$0.394438\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	9.58211i	0.675870i
$202$	11.4819i	0.807862i
$203$	40.0834i	2.81330i
$204$	−1.08815	−0.0761855
$205$	0.664874	0.0464368
$206$	4.16421i	0.290134i
$207$	−2.69202	−0.187108
$208$	0	0
$209$	−21.0737	−1.45770
$210$	− 5.04892i	− 0.348408i
$211$	−20.7875	−1.43107	−0.715534	−	0.698578i	$-0.753817\pi$
$211$	−20.7875	−1.43107	−0.715534	+	0.698578i	$0.753817\pi$
$212$	10.8605	0.745905
$213$	4.47219i	0.306429i
$214$	0.796561i	0.0544518i
$215$	− 4.35690i	− 0.297138i
$216$	1.00000i	0.0680414i
$217$	39.6383	2.69082
$218$	0.704103	0.0476879
$219$	8.17390i	0.552341i
$220$	4.74094	0.319634
$221$	0	0
$222$	7.14675	0.479659
$223$	17.3502i	1.16185i	0.813955	+	0.580927i	$0.197310\pi$
$223$	17.3502i	1.16185i	−0.813955	+	0.580927i	$0.802690\pi$
$224$	−5.04892	−0.337345
$225$	−1.00000	−0.0666667
$226$	− 14.4983i	− 0.964411i
$227$	− 13.7573i	− 0.913106i	−0.889696	−	0.456553i	$-0.849084\pi$
$227$	− 13.7573i	− 0.913106i	0.889696	−	0.456553i	$-0.150916\pi$
$228$	− 4.44504i	− 0.294380i
$229$	− 1.75541i	− 0.116001i	−0.998317	−	0.0580005i	$-0.981528\pi$
$229$	− 1.75541i	− 0.116001i	0.998317	−	0.0580005i	$-0.0184725\pi$
$230$	−2.69202	−0.177507
$231$	23.9366	1.57491
$232$	− 7.93900i	− 0.521221i
$233$	5.49396	0.359921	0.179961	−	0.983674i	$-0.442403\pi$
$233$	5.49396	0.359921	0.179961	+	0.983674i	$0.442403\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	12.2784i	0.799258i
$237$	14.1468	0.918930
$238$	−5.49396	−0.356120
$239$	− 17.4004i	− 1.12554i	−0.826613	−	0.562770i	$-0.809736\pi$
$239$	− 17.4004i	− 1.12554i	0.826613	−	0.562770i	$-0.190264\pi$
$240$	1.00000i	0.0645497i
$241$	3.15452i	0.203201i	0.994825	+	0.101600i	$0.0323963\pi$
$241$	3.15452i	0.203201i	−0.994825	+	0.101600i	$0.967604\pi$
$242$	11.4765i	0.737737i
$243$	1.00000	0.0641500
$244$	11.0707	0.708728
$245$	− 18.4916i	− 1.18138i
$246$	−0.664874	−0.0423908
$247$	0	0
$248$	−7.85086	−0.498530
$249$	4.86294i	0.308176i
$250$	−1.00000	−0.0632456
$251$	−23.9433	−1.51129	−0.755644	−	0.654982i	$-0.772676\pi$
$251$	−23.9433	−1.51129	−0.755644	+	0.654982i	$0.772676\pi$
$252$	5.04892i	0.318052i
$253$	− 12.7627i	− 0.802385i
$254$	− 2.93900i	− 0.184409i
$255$	1.08815i	0.0681423i
$256$	1.00000	0.0625000
$257$	−12.3381	−0.769630	−0.384815	−	0.922994i	$-0.625735\pi$
$257$	−12.3381	−0.769630	−0.384815	+	0.922994i	$0.625735\pi$
$258$	4.35690i	0.271248i
$259$	36.0834	2.24211
$260$	0	0
$261$	−7.93900	−0.491412
$262$	3.56465i	0.220225i
$263$	2.70709	0.166926	0.0834631	−	0.996511i	$-0.473402\pi$
$263$	2.70709	0.166926	0.0834631	+	0.996511i	$0.473402\pi$
$264$	−4.74094	−0.291785
$265$	− 10.8605i	− 0.667158i
$266$	− 22.4426i	− 1.37605i
$267$	− 14.8877i	− 0.911112i
$268$	− 9.58211i	− 0.585320i
$269$	20.9681	1.27845	0.639223	−	0.769022i	$-0.279256\pi$
$269$	20.9681	1.27845	0.639223	+	0.769022i	$0.279256\pi$
$270$	1.00000	0.0608581
$271$	7.99761i	0.485820i	0.970049	+	0.242910i	$0.0781020\pi$
$271$	7.99761i	0.485820i	−0.970049	+	0.242910i	$0.921898\pi$
$272$	1.08815	0.0659785
$273$	0	0
$274$	9.47650	0.572496
$275$	− 4.74094i	− 0.285889i
$276$	2.69202	0.162041
$277$	5.78554	0.347620	0.173810	−	0.984779i	$-0.444392\pi$
$277$	5.78554	0.347620	0.173810	+	0.984779i	$0.444392\pi$
$278$	14.2078i	0.852124i
$279$	7.85086i	0.470018i
$280$	5.04892i	0.301731i
$281$	− 19.5579i	− 1.16673i	−0.812210	−	0.583365i	$-0.801736\pi$
$281$	− 19.5579i	− 1.16673i	0.812210	−	0.583365i	$-0.198264\pi$
$282$	−7.00000	−0.416844
$283$	−31.0726	−1.84707	−0.923537	−	0.383508i	$-0.874716\pi$
$283$	−31.0726	−1.84707	−0.923537	+	0.383508i	$0.874716\pi$
$284$	− 4.47219i	− 0.265376i
$285$	−4.44504	−0.263302
$286$	0	0
$287$	−3.35690	−0.198151
$288$	− 1.00000i	− 0.0589256i
$289$	−15.8159	−0.930349
$290$	−7.93900	−0.466194
$291$	14.9269i	0.875032i
$292$	− 8.17390i	− 0.478341i
$293$	10.0543i	0.587378i	0.955901	+	0.293689i	$0.0948830\pi$
$293$	10.0543i	0.587378i	−0.955901	+	0.293689i	$0.905117\pi$
$294$	18.4916i	1.07845i
$295$	12.2784	0.714878
$296$	−7.14675	−0.415397
$297$	4.74094i	0.275097i
$298$	10.2010	0.590931
$299$	0	0
$300$	1.00000	0.0577350
$301$	21.9976i	1.26792i
$302$	−3.86294	−0.222287
$303$	−11.4819	−0.659617
$304$	4.44504i	0.254941i
$305$	− 11.0707i	− 0.633906i
$306$	− 1.08815i	− 0.0622052i
$307$	23.6872i	1.35190i	0.736947	+	0.675951i	$0.236267\pi$
$307$	23.6872i	1.35190i	−0.736947	+	0.675951i	$0.763733\pi$
$308$	−23.9366	−1.36391
$309$	−4.16421	−0.236894
$310$	7.85086i	0.445899i
$311$	−6.77240	−0.384027	−0.192014	−	0.981392i	$-0.561502\pi$
$311$	−6.77240	−0.384027	−0.192014	+	0.981392i	$0.561502\pi$
$312$	0	0
$313$	7.16123	0.404776	0.202388	−	0.979305i	$-0.435130\pi$
$313$	7.16123	0.404776	0.202388	+	0.979305i	$0.435130\pi$
$314$	11.2174i	0.633037i
$315$	5.04892	0.284474
$316$	−14.1468	−0.795817
$317$	− 1.20344i	− 0.0675919i	−0.999429	−	0.0337959i	$-0.989240\pi$
$317$	− 1.20344i	− 0.0675919i	0.999429	−	0.0337959i	$-0.0107596\pi$
$318$	10.8605i	0.609029i
$319$	− 37.6383i	− 2.10734i
$320$	− 1.00000i	− 0.0559017i
$321$	−0.796561	−0.0444597
$322$	13.5918	0.757441
$323$	4.83685i	0.269130i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	6.50902	0.360502
$327$	0.704103i	0.0389370i
$328$	0.664874	0.0367115
$329$	−35.3424	−1.94849
$330$	4.74094i	0.260980i
$331$	20.9933i	1.15390i	0.816781	+	0.576948i	$0.195756\pi$
$331$	20.9933i	1.15390i	−0.816781	+	0.576948i	$0.804244\pi$
$332$	− 4.86294i	− 0.266888i
$333$	7.14675i	0.391640i
$334$	−9.67456	−0.529369
$335$	−9.58211	−0.523526
$336$	− 5.04892i	− 0.275441i
$337$	3.01938	0.164476	0.0822380	−	0.996613i	$-0.473793\pi$
$337$	3.01938	0.164476	0.0822380	+	0.996613i	$0.473793\pi$
$338$	0	0
$339$	14.4983	0.787438
$340$	− 1.08815i	− 0.0590130i
$341$	−37.2204	−2.01560
$342$	4.44504	0.240360
$343$	58.0200i	3.13278i
$344$	− 4.35690i	− 0.234908i
$345$	− 2.69202i	− 0.144934i
$346$	− 13.8388i	− 0.743977i
$347$	11.2610	0.604521	0.302261	−	0.953225i	$-0.402259\pi$
$347$	11.2610	0.604521	0.302261	+	0.953225i	$0.402259\pi$
$348$	7.93900	0.425575
$349$	5.84010i	0.312613i	0.987709	+	0.156307i	$0.0499588\pi$
$349$	5.84010i	0.312613i	−0.987709	+	0.156307i	$0.950041\pi$
$350$	5.04892	0.269876
$351$	0	0
$352$	4.74094	0.252693
$353$	0.675628i	0.0359601i	0.999838	+	0.0179800i	$0.00572353\pi$
$353$	0.675628i	0.0359601i	−0.999838	+	0.0179800i	$0.994276\pi$
$354$	−12.2784	−0.652592
$355$	−4.47219	−0.237359
$356$	14.8877i	0.789046i
$357$	− 5.49396i	− 0.290771i
$358$	− 14.2078i	− 0.750903i
$359$	8.52350i	0.449853i	0.974376	+	0.224927i	$0.0722142\pi$
$359$	8.52350i	0.449853i	−0.974376	+	0.224927i	$0.927786\pi$
$360$	−1.00000	−0.0527046
$361$	−0.758397	−0.0399156
$362$	− 24.9028i	− 1.30886i
$363$	−11.4765	−0.602360
$364$	0	0
$365$	−8.17390	−0.427841
$366$	11.0707i	0.578674i
$367$	−9.40044	−0.490699	−0.245349	−	0.969435i	$-0.578903\pi$
$367$	−9.40044	−0.490699	−0.245349	+	0.969435i	$0.578903\pi$
$368$	−2.69202	−0.140331
$369$	− 0.664874i	− 0.0346120i
$370$	7.14675i	0.371542i
$371$	54.8340i	2.84684i
$372$	− 7.85086i	− 0.407048i
$373$	23.6582	1.22497	0.612487	−	0.790481i	$-0.290169\pi$
$373$	23.6582	1.22497	0.612487	+	0.790481i	$0.290169\pi$
$374$	5.15883	0.266757
$375$	− 1.00000i	− 0.0516398i
$376$	7.00000	0.360997
$377$	0	0
$378$	−5.04892	−0.259688
$379$	16.3437i	0.839522i	0.907635	+	0.419761i	$0.137886\pi$
$379$	16.3437i	0.839522i	−0.907635	+	0.419761i	$0.862114\pi$
$380$	4.44504	0.228026
$381$	2.93900	0.150570
$382$	− 2.44504i	− 0.125099i
$383$	− 4.79656i	− 0.245093i	−0.992463	−	0.122546i	$-0.960894\pi$
$383$	− 4.79656i	− 0.245093i	0.992463	−	0.122546i	$-0.0391060\pi$
$384$	1.00000i	0.0510310i
$385$	23.9366i	1.21992i
$386$	15.2959	0.778541
$387$	−4.35690	−0.221473
$388$	− 14.9269i	− 0.757800i
$389$	−21.4534	−1.08773	−0.543865	−	0.839173i	$-0.683040\pi$
$389$	−21.4534	−1.08773	−0.543865	+	0.839173i	$0.683040\pi$
$390$	0	0
$391$	−2.92931	−0.148142
$392$	− 18.4916i	− 0.933965i
$393$	−3.56465	−0.179813
$394$	3.44935	0.173776
$395$	14.1468i	0.711800i
$396$	− 4.74094i	− 0.238241i
$397$	− 1.36765i	− 0.0686404i	−0.999411	−	0.0343202i	$-0.989073\pi$
$397$	− 1.36765i	− 0.0686404i	0.999411	−	0.0343202i	$-0.0109266\pi$
$398$	− 9.18598i	− 0.460452i
$399$	22.4426	1.12354
$400$	−1.00000	−0.0500000
$401$	18.8931i	0.943475i	0.881739	+	0.471737i	$0.156373\pi$
$401$	18.8931i	0.943475i	−0.881739	+	0.471737i	$0.843627\pi$
$402$	9.58211	0.477912
$403$	0	0
$404$	11.4819	0.571245
$405$	1.00000i	0.0496904i
$406$	40.0834	1.98930
$407$	−33.8823	−1.67948
$408$	1.08815i	0.0538713i
$409$	− 25.9638i	− 1.28383i	−0.766778	−	0.641913i	$-0.778141\pi$
$409$	− 25.9638i	− 1.28383i	0.766778	−	0.641913i	$-0.221859\pi$
$410$	− 0.664874i	− 0.0328358i
$411$	9.47650i	0.467441i
$412$	4.16421	0.205156
$413$	−61.9928	−3.05047
$414$	2.69202i	0.132306i
$415$	−4.86294	−0.238712
$416$	0	0
$417$	−14.2078	−0.695757
$418$	21.0737i	1.03075i
$419$	−32.4282	−1.58422	−0.792110	−	0.610378i	$-0.791017\pi$
$419$	−32.4282	−1.58422	−0.792110	+	0.610378i	$0.791017\pi$
$420$	−5.04892	−0.246362
$421$	− 33.2868i	− 1.62230i	−0.584839	−	0.811150i	$-0.698842\pi$
$421$	− 33.2868i	− 1.62230i	0.584839	−	0.811150i	$-0.301158\pi$
$422$	20.7875i	1.01192i
$423$	− 7.00000i	− 0.340352i
$424$	− 10.8605i	− 0.527435i
$425$	−1.08815	−0.0527828
$426$	4.47219	0.216678
$427$	55.8950i	2.70495i
$428$	0.796561	0.0385032
$429$	0	0
$430$	−4.35690	−0.210108
$431$	11.1578i	0.537451i	0.963217	+	0.268725i	$0.0866024\pi$
$431$	11.1578i	0.537451i	−0.963217	+	0.268725i	$0.913398\pi$
$432$	1.00000	0.0481125
$433$	−14.0828	−0.676775	−0.338387	−	0.941007i	$-0.609881\pi$
$433$	−14.0828	−0.676775	−0.338387	+	0.941007i	$0.609881\pi$
$434$	− 39.6383i	− 1.90270i
$435$	− 7.93900i	− 0.380646i
$436$	− 0.704103i	− 0.0337204i
$437$	− 11.9661i	− 0.572418i
$438$	8.17390	0.390564
$439$	28.9933	1.38377	0.691887	−	0.722006i	$-0.256780\pi$
$439$	28.9933	1.38377	0.691887	+	0.722006i	$0.256780\pi$
$440$	− 4.74094i	− 0.226015i
$441$	−18.4916	−0.880551
$442$	0	0
$443$	33.3080	1.58251	0.791255	−	0.611486i	$-0.209428\pi$
$443$	33.3080	1.58251	0.791255	+	0.611486i	$0.209428\pi$
$444$	− 7.14675i	− 0.339170i
$445$	14.8877	0.705744
$446$	17.3502	0.821555
$447$	10.2010i	0.482493i
$448$	5.04892i	0.238539i
$449$	37.7899i	1.78341i	0.452614	+	0.891707i	$0.350492\pi$
$449$	37.7899i	1.78341i	−0.452614	+	0.891707i	$0.649508\pi$
$450$	1.00000i	0.0471405i
$451$	3.15213	0.148428
$452$	−14.4983	−0.681941
$453$	− 3.86294i	− 0.181497i
$454$	−13.7573	−0.645664
$455$	0	0
$456$	−4.44504	−0.208158
$457$	− 10.5700i	− 0.494445i	−0.968959	−	0.247222i	$-0.920482\pi$
$457$	− 10.5700i	− 0.494445i	0.968959	−	0.247222i	$-0.0795178\pi$
$458$	−1.75541	−0.0820251
$459$	1.08815	0.0507903
$460$	2.69202i	0.125516i
$461$	− 20.1420i	− 0.938105i	−0.883170	−	0.469052i	$-0.844596\pi$
$461$	− 20.1420i	− 0.938105i	0.883170	−	0.469052i	$-0.155404\pi$
$462$	− 23.9366i	− 1.11363i
$463$	− 15.9065i	− 0.739237i	−0.929184	−	0.369618i	$-0.879488\pi$
$463$	− 15.9065i	− 0.739237i	0.929184	−	0.369618i	$-0.120512\pi$
$464$	−7.93900	−0.368559
$465$	−7.85086	−0.364075
$466$	− 5.49396i	− 0.254503i
$467$	25.6136	1.18525	0.592627	−	0.805477i	$-0.298091\pi$
$467$	25.6136	1.18525	0.592627	+	0.805477i	$0.298091\pi$
$468$	0	0
$469$	48.3793	2.23395
$470$	− 7.00000i	− 0.322886i
$471$	−11.2174	−0.516872
$472$	12.2784	0.565161
$473$	− 20.6558i	− 0.949754i
$474$	− 14.1468i	− 0.649782i
$475$	− 4.44504i	− 0.203953i
$476$	5.49396i	0.251815i
$477$	−10.8605	−0.497270
$478$	−17.4004	−0.795877
$479$	7.59658i	0.347097i	0.984825	+	0.173548i	$0.0555233\pi$
$479$	7.59658i	0.347097i	−0.984825	+	0.173548i	$0.944477\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	3.15452	0.143685
$483$	13.5918i	0.618448i
$484$	11.4765	0.521659
$485$	−14.9269	−0.677796
$486$	− 1.00000i	− 0.0453609i
$487$	0.779103i	0.0353045i	0.999844	+	0.0176523i	$0.00561918\pi$
$487$	0.779103i	0.0353045i	−0.999844	+	0.0176523i	$0.994381\pi$
$488$	− 11.0707i	− 0.501146i
$489$	6.50902i	0.294348i
$490$	−18.4916	−0.835364
$491$	−11.3948	−0.514240	−0.257120	−	0.966379i	$-0.582774\pi$
$491$	−11.3948	−0.514240	−0.257120	+	0.966379i	$0.582774\pi$
$492$	0.664874i	0.0299749i
$493$	−8.63879	−0.389072
$494$	0	0
$495$	−4.74094	−0.213089
$496$	7.85086i	0.352514i
$497$	22.5797	1.01284
$498$	4.86294	0.217913
$499$	23.5555i	1.05449i	0.849713	+	0.527246i	$0.176775\pi$
$499$	23.5555i	1.05449i	−0.849713	+	0.527246i	$0.823225\pi$
$500$	1.00000i	0.0447214i
$501$	− 9.67456i	− 0.432228i
$502$	23.9433i	1.06864i
$503$	−12.6165	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$503$	−12.6165	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$504$	5.04892	0.224897
$505$	− 11.4819i	− 0.510937i
$506$	−12.7627	−0.567372
$507$	0	0
$508$	−2.93900	−0.130397
$509$	− 8.45473i	− 0.374749i	−0.982289	−	0.187375i	$-0.940002\pi$
$509$	− 8.45473i	− 0.374749i	0.982289	−	0.187375i	$-0.0599978\pi$
$510$	1.08815	0.0481839
$511$	41.2693	1.82565
$512$	− 1.00000i	− 0.0441942i
$513$	4.44504i	0.196253i
$514$	12.3381i	0.544211i
$515$	− 4.16421i	− 0.183497i
$516$	4.35690	0.191802
$517$	33.1866	1.45954
$518$	− 36.0834i	− 1.58541i
$519$	13.8388	0.607455
$520$	0	0
$521$	34.7254	1.52135	0.760674	−	0.649134i	$-0.224869\pi$
$521$	34.7254	1.52135	0.760674	+	0.649134i	$0.224869\pi$
$522$	7.93900i	0.347481i
$523$	29.3163	1.28191	0.640957	−	0.767577i	$-0.278538\pi$
$523$	29.3163	1.28191	0.640957	+	0.767577i	$0.278538\pi$
$524$	3.56465	0.155722
$525$	5.04892i	0.220353i
$526$	− 2.70709i	− 0.118035i
$527$	8.54288i	0.372134i
$528$	4.74094i	0.206323i
$529$	−15.7530	−0.684914
$530$	−10.8605	−0.471752
$531$	− 12.2784i	− 0.532839i
$532$	−22.4426	−0.973012
$533$	0	0
$534$	−14.8877	−0.644253
$535$	− 0.796561i	− 0.0344383i
$536$	−9.58211	−0.413884
$537$	14.2078	0.613110
$538$	− 20.9681i	− 0.903998i
$539$	− 87.6674i	− 3.77610i
$540$	− 1.00000i	− 0.0430331i
$541$	− 29.1032i	− 1.25124i	−0.780126	−	0.625622i	$-0.784845\pi$
$541$	− 29.1032i	− 1.25124i	0.780126	−	0.625622i	$-0.215155\pi$
$542$	7.99761	0.343527
$543$	24.9028	1.06868
$544$	− 1.08815i	− 0.0466539i
$545$	−0.704103	−0.0301605
$546$	0	0
$547$	−2.37004	−0.101336	−0.0506678	−	0.998716i	$-0.516135\pi$
$547$	−2.37004	−0.101336	−0.0506678	+	0.998716i	$0.516135\pi$
$548$	− 9.47650i	− 0.404816i
$549$	−11.0707	−0.472485
$550$	−4.74094	−0.202154
$551$	− 35.2892i	− 1.50337i
$552$	− 2.69202i	− 0.114580i
$553$	− 71.4258i	− 3.03733i
$554$	− 5.78554i	− 0.245804i
$555$	−7.14675	−0.303363
$556$	14.2078	0.602543
$557$	− 28.8364i	− 1.22184i	−0.791694	−	0.610918i	$-0.790800\pi$
$557$	− 28.8364i	− 1.22184i	0.791694	−	0.610918i	$-0.209200\pi$
$558$	7.85086	0.332353
$559$	0	0
$560$	5.04892	0.213356
$561$	5.15883i	0.217806i
$562$	−19.5579	−0.825002
$563$	35.7133	1.50514	0.752568	−	0.658514i	$-0.228815\pi$
$563$	35.7133	1.50514	0.752568	+	0.658514i	$0.228815\pi$
$564$	7.00000i	0.294753i
$565$	14.4983i	0.609947i
$566$	31.0726i	1.30608i
$567$	− 5.04892i	− 0.212035i
$568$	−4.47219	−0.187649
$569$	−42.1618	−1.76752	−0.883758	−	0.467945i	$-0.844995\pi$
$569$	−42.1618	−1.76752	−0.883758	+	0.467945i	$0.844995\pi$
$570$	4.44504i	0.186182i
$571$	−23.1497	−0.968786	−0.484393	−	0.874850i	$-0.660960\pi$
$571$	−23.1497	−0.968786	−0.484393	+	0.874850i	$0.660960\pi$
$572$	0	0
$573$	2.44504	0.102143
$574$	3.35690i	0.140114i
$575$	2.69202	0.112265
$576$	−1.00000	−0.0416667
$577$	− 41.7318i	− 1.73732i	−0.495409	−	0.868660i	$-0.664982\pi$
$577$	− 41.7318i	− 1.73732i	0.495409	−	0.868660i	$-0.335018\pi$
$578$	15.8159i	0.657856i
$579$	15.2959i	0.635676i
$580$	7.93900i	0.329649i
$581$	24.5526	1.01861
$582$	14.9269	0.618741
$583$	− 51.4892i	− 2.13246i
$584$	−8.17390	−0.338238
$585$	0	0
$586$	10.0543	0.415339
$587$	− 3.52648i	− 0.145554i	−0.997348	−	0.0727768i	$-0.976814\pi$
$587$	− 3.52648i	− 0.145554i	0.997348	−	0.0727768i	$-0.0231861\pi$
$588$	18.4916	0.762579
$589$	−34.8974	−1.43792
$590$	− 12.2784i	− 0.505495i
$591$	3.44935i	0.141887i
$592$	7.14675i	0.293730i
$593$	− 7.53617i	− 0.309473i	−0.987956	−	0.154737i	$-0.950547\pi$
$593$	− 7.53617i	− 0.309473i	0.987956	−	0.154737i	$-0.0494529\pi$
$594$	4.74094	0.194523
$595$	5.49396	0.225230
$596$	− 10.2010i	− 0.417851i
$597$	9.18598	0.375957
$598$	0	0
$599$	−44.2161	−1.80662	−0.903311	−	0.428987i	$-0.858871\pi$
$599$	−44.2161	−1.80662	−0.903311	+	0.428987i	$0.858871\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−40.7251	−1.66121	−0.830606	−	0.556860i	$-0.812006\pi$
$601$	−40.7251	−1.66121	−0.830606	+	0.556860i	$0.812006\pi$
$602$	21.9976	0.896556
$603$	9.58211i	0.390213i
$604$	3.86294i	0.157181i
$605$	− 11.4765i	− 0.466586i
$606$	11.4819i	0.466419i
$607$	40.9788	1.66328	0.831640	−	0.555316i	$-0.187403\pi$
$607$	40.9788	1.66328	0.831640	+	0.555316i	$0.187403\pi$
$608$	4.44504	0.180270
$609$	40.0834i	1.62426i
$610$	−11.0707	−0.448239
$611$	0	0
$612$	−1.08815	−0.0439857
$613$	− 40.5491i	− 1.63776i	−0.573963	−	0.818882i	$-0.694595\pi$
$613$	− 40.5491i	− 1.63776i	0.573963	−	0.818882i	$-0.305405\pi$
$614$	23.6872	0.955939
$615$	0.664874	0.0268103
$616$	23.9366i	0.964433i
$617$	− 3.06829i	− 0.123525i	−0.998091	−	0.0617624i	$-0.980328\pi$
$617$	− 3.06829i	− 0.123525i	0.998091	−	0.0617624i	$-0.0196721\pi$
$618$	4.16421i	0.167509i
$619$	31.0847i	1.24940i	0.780865	+	0.624700i	$0.214779\pi$
$619$	31.0847i	1.24940i	−0.780865	+	0.624700i	$0.785221\pi$
$620$	7.85086	0.315298
$621$	−2.69202	−0.108027
$622$	6.77240i	0.271548i
$623$	−75.1667	−3.01149
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 7.16123i	− 0.286220i
$627$	−21.0737	−0.841601
$628$	11.2174	0.447625
$629$	7.77671i	0.310078i
$630$	− 5.04892i	− 0.201154i
$631$	− 23.7101i	− 0.943883i	−0.881630	−	0.471942i	$-0.843553\pi$
$631$	− 23.7101i	− 0.943883i	0.881630	−	0.471942i	$-0.156447\pi$
$632$	14.1468i	0.562728i
$633$	−20.7875	−0.826227
$634$	−1.20344	−0.0477947
$635$	2.93900i	0.116631i
$636$	10.8605	0.430649
$637$	0	0
$638$	−37.6383	−1.49012
$639$	4.47219i	0.176917i
$640$	−1.00000	−0.0395285
$641$	48.3866	1.91115	0.955577	−	0.294742i	$-0.0952336\pi$
$641$	48.3866	1.91115	0.955577	+	0.294742i	$0.0952336\pi$
$642$	0.796561i	0.0314378i
$643$	20.4209i	0.805321i	0.915349	+	0.402660i	$0.131914\pi$
$643$	20.4209i	0.805321i	−0.915349	+	0.402660i	$0.868086\pi$
$644$	− 13.5918i	− 0.535592i
$645$	− 4.35690i	− 0.171553i
$646$	4.83685	0.190303
$647$	0.292913	0.0115156	0.00575780	−	0.999983i	$-0.498167\pi$
$647$	0.292913	0.0115156	0.00575780	+	0.999983i	$0.498167\pi$
$648$	1.00000i	0.0392837i
$649$	58.2113	2.28499
$650$	0	0
$651$	39.6383	1.55355
$652$	− 6.50902i	− 0.254913i
$653$	21.8323	0.854365	0.427183	−	0.904165i	$-0.359506\pi$
$653$	21.8323	0.854365	0.427183	+	0.904165i	$0.359506\pi$
$654$	0.704103	0.0275326
$655$	− 3.56465i	− 0.139282i
$656$	− 0.664874i	− 0.0259590i
$657$	8.17390i	0.318894i
$658$	35.3424i	1.37779i
$659$	−17.0519	−0.664248	−0.332124	−	0.943236i	$-0.607765\pi$
$659$	−17.0519	−0.664248	−0.332124	+	0.943236i	$0.607765\pi$
$660$	4.74094	0.184541
$661$	31.8098i	1.23726i	0.785683	+	0.618629i	$0.212312\pi$
$661$	31.8098i	1.23726i	−0.785683	+	0.618629i	$0.787688\pi$
$662$	20.9933	0.815928
$663$	0	0
$664$	−4.86294	−0.188719
$665$	22.4426i	0.870289i
$666$	7.14675	0.276931
$667$	21.3720	0.827526
$668$	9.67456i	0.374320i
$669$	17.3502i	0.670797i
$670$	9.58211i	0.370189i
$671$	− 52.4855i	− 2.02618i
$672$	−5.04892	−0.194766
$673$	4.10513	0.158241	0.0791206	−	0.996865i	$-0.474789\pi$
$673$	4.10513	0.158241	0.0791206	+	0.996865i	$0.474789\pi$
$674$	− 3.01938i	− 0.116302i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	11.0543	0.424851	0.212426	−	0.977177i	$-0.431864\pi$
$677$	11.0543	0.424851	0.212426	+	0.977177i	$0.431864\pi$
$678$	− 14.4983i	− 0.556803i
$679$	75.3648	2.89223
$680$	−1.08815	−0.0417285
$681$	− 13.7573i	− 0.527182i
$682$	37.2204i	1.42524i
$683$	9.06590i	0.346897i	0.984843	+	0.173449i	$0.0554910\pi$
$683$	9.06590i	0.346897i	−0.984843	+	0.173449i	$0.944509\pi$
$684$	− 4.44504i	− 0.169960i
$685$	−9.47650	−0.362078
$686$	58.0200	2.21521
$687$	− 1.75541i	− 0.0669732i
$688$	−4.35690	−0.166105
$689$	0	0
$690$	−2.69202	−0.102484
$691$	− 12.4282i	− 0.472790i	−0.971657	−	0.236395i	$-0.924034\pi$
$691$	− 12.4282i	− 0.472790i	0.971657	−	0.236395i	$-0.0759659\pi$
$692$	−13.8388	−0.526071
$693$	23.9366	0.909277
$694$	− 11.2610i	− 0.427461i
$695$	− 14.2078i	− 0.538931i
$696$	− 7.93900i	− 0.300927i
$697$	− 0.723480i	− 0.0274038i
$698$	5.84010	0.221051
$699$	5.49396	0.207801
$700$	− 5.04892i	− 0.190831i
$701$	38.5418	1.45570	0.727852	−	0.685734i	$-0.240519\pi$
$701$	38.5418	1.45570	0.727852	+	0.685734i	$0.240519\pi$
$702$	0	0
$703$	−31.7676	−1.19814
$704$	− 4.74094i	− 0.178681i
$705$	7.00000	0.263635
$706$	0.675628	0.0254276
$707$	57.9711i	2.18023i
$708$	12.2784i	0.461452i
$709$	− 38.7851i	− 1.45660i	−0.685256	−	0.728302i	$-0.740310\pi$
$709$	− 38.7851i	− 1.45660i	0.685256	−	0.728302i	$-0.259690\pi$
$710$	4.47219i	0.167838i
$711$	14.1468	0.530545
$712$	14.8877	0.557940
$713$	− 21.1347i	− 0.791500i
$714$	−5.49396	−0.205606
$715$	0	0
$716$	−14.2078	−0.530969
$717$	− 17.4004i	− 0.649831i
$718$	8.52350	0.318094
$719$	12.1021	0.451334	0.225667	−	0.974205i	$-0.427544\pi$
$719$	12.1021	0.451334	0.225667	+	0.974205i	$0.427544\pi$
$720$	1.00000i	0.0372678i
$721$	21.0248i	0.783003i
$722$	0.758397i	0.0282246i
$723$	3.15452i	0.117318i
$724$	−24.9028	−0.925504
$725$	7.93900	0.294847
$726$	11.4765i	0.425933i
$727$	−40.6088	−1.50610	−0.753048	−	0.657965i	$-0.771417\pi$
$727$	−40.6088	−1.50610	−0.753048	+	0.657965i	$0.771417\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	8.17390i	0.302530i
$731$	−4.74094	−0.175350
$732$	11.0707	0.409184
$733$	32.6601i	1.20633i	0.797617	+	0.603164i	$0.206094\pi$
$733$	32.6601i	1.20633i	−0.797617	+	0.603164i	$0.793906\pi$
$734$	9.40044i	0.346976i
$735$	− 18.4916i	− 0.682072i
$736$	2.69202i	0.0992292i
$737$	−45.4282	−1.67337
$738$	−0.664874	−0.0244744
$739$	− 3.72348i	− 0.136970i	−0.997652	−	0.0684852i	$-0.978183\pi$
$739$	− 3.72348i	− 0.136970i	0.997652	−	0.0684852i	$-0.0218166\pi$
$740$	7.14675	0.262720
$741$	0	0
$742$	54.8340	2.01302
$743$	− 31.3099i	− 1.14865i	−0.818628	−	0.574324i	$-0.805265\pi$
$743$	− 31.3099i	− 1.14865i	0.818628	−	0.574324i	$-0.194735\pi$
$744$	−7.85086	−0.287826
$745$	−10.2010	−0.373737
$746$	− 23.6582i	− 0.866187i
$747$	4.86294i	0.177926i
$748$	− 5.15883i	− 0.188626i
$749$	4.02177i	0.146952i
$750$	−1.00000	−0.0365148
$751$	−2.66296	−0.0971726	−0.0485863	−	0.998819i	$-0.515472\pi$
$751$	−2.66296	−0.0971726	−0.0485863	+	0.998819i	$0.515472\pi$
$752$	− 7.00000i	− 0.255264i
$753$	−23.9433	−0.872543
$754$	0	0
$755$	3.86294	0.140587
$756$	5.04892i	0.183627i
$757$	32.2553	1.17234	0.586170	−	0.810188i	$-0.300635\pi$
$757$	32.2553	1.17234	0.586170	+	0.810188i	$0.300635\pi$
$758$	16.3437	0.593632
$759$	− 12.7627i	− 0.463257i
$760$	− 4.44504i	− 0.161239i
$761$	− 27.2228i	− 0.986826i	−0.869795	−	0.493413i	$-0.835749\pi$
$761$	− 27.2228i	− 0.986826i	0.869795	−	0.493413i	$-0.164251\pi$
$762$	− 2.93900i	− 0.106469i
$763$	3.55496	0.128698
$764$	−2.44504	−0.0884585
$765$	1.08815i	0.0393420i
$766$	−4.79656	−0.173307
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 5.62133i	− 0.202710i	−0.994850	−	0.101355i	$-0.967682\pi$
$769$	− 5.62133i	− 0.202710i	0.994850	−	0.101355i	$-0.0323179\pi$
$770$	23.9366	0.862615
$771$	−12.3381	−0.444346
$772$	− 15.2959i	− 0.550511i
$773$	35.7918i	1.28734i	0.765303	+	0.643670i	$0.222589\pi$
$773$	35.7918i	1.28734i	−0.765303	+	0.643670i	$0.777411\pi$
$774$	4.35690i	0.156605i
$775$	− 7.85086i	− 0.282011i
$776$	−14.9269	−0.535845
$777$	36.0834	1.29448
$778$	21.4534i	0.769142i
$779$	2.95539	0.105888
$780$	0	0
$781$	−21.2024	−0.758681
$782$	2.92931i	0.104752i
$783$	−7.93900	−0.283717
$784$	−18.4916	−0.660413
$785$	− 11.2174i	− 0.400368i
$786$	3.56465i	0.127147i
$787$	18.5442i	0.661030i	0.943801	+	0.330515i	$0.107222\pi$
$787$	18.5442i	0.661030i	−0.943801	+	0.330515i	$0.892778\pi$
$788$	− 3.44935i	− 0.122878i
$789$	2.70709	0.0963748
$790$	14.1468	0.503319
$791$	− 73.2006i	− 2.60271i
$792$	−4.74094	−0.168462
$793$	0	0
$794$	−1.36765	−0.0485361
$795$	− 10.8605i	− 0.385184i
$796$	−9.18598	−0.325588
$797$	33.0898	1.17210	0.586050	−	0.810275i	$-0.300682\pi$
$797$	33.0898	1.17210	0.586050	+	0.810275i	$0.300682\pi$
$798$	− 22.4426i	− 0.794461i
$799$	− 7.61702i	− 0.269471i
$800$	1.00000i	0.0353553i
$801$	− 14.8877i	− 0.526031i
$802$	18.8931	0.667137
$803$	−38.7520	−1.36753
$804$	− 9.58211i	− 0.337935i
$805$	−13.5918	−0.479048
$806$	0	0
$807$	20.9681	0.738111
$808$	− 11.4819i	− 0.403931i
$809$	11.4004	0.400818	0.200409	−	0.979712i	$-0.435773\pi$
$809$	11.4004	0.400818	0.200409	+	0.979712i	$0.435773\pi$
$810$	1.00000	0.0351364
$811$	38.9724i	1.36851i	0.729245	+	0.684253i	$0.239871\pi$
$811$	38.9724i	1.36851i	−0.729245	+	0.684253i	$0.760129\pi$
$812$	− 40.0834i	− 1.40665i
$813$	7.99761i	0.280488i
$814$	33.8823i	1.18757i
$815$	−6.50902	−0.228001
$816$	1.08815	0.0380927
$817$	− 19.3666i	− 0.677551i
$818$	−25.9638	−0.907801
$819$	0	0
$820$	−0.664874	−0.0232184
$821$	− 7.12498i	− 0.248664i	−0.992241	−	0.124332i	$-0.960321\pi$
$821$	− 7.12498i	− 0.248664i	0.992241	−	0.124332i	$-0.0396787\pi$
$822$	9.47650	0.330531
$823$	−19.1323	−0.666909	−0.333455	−	0.942766i	$-0.608214\pi$
$823$	−19.1323	−0.666909	−0.333455	+	0.942766i	$0.608214\pi$
$824$	− 4.16421i	− 0.145067i
$825$	− 4.74094i	− 0.165058i
$826$	61.9928i	2.15701i
$827$	− 42.0538i	− 1.46235i	−0.682187	−	0.731177i	$-0.738971\pi$
$827$	− 42.0538i	− 1.46235i	0.682187	−	0.731177i	$-0.261029\pi$
$828$	2.69202	0.0935542
$829$	17.6364	0.612537	0.306269	−	0.951945i	$-0.400919\pi$
$829$	17.6364	0.612537	0.306269	+	0.951945i	$0.400919\pi$
$830$	4.86294i	0.168795i
$831$	5.78554	0.200698
$832$	0	0
$833$	−20.1215	−0.697169
$834$	14.2078i	0.491974i
$835$	9.67456	0.334802
$836$	21.0737	0.728848
$837$	7.85086i	0.271365i
$838$	32.4282i	1.12021i
$839$	44.4905i	1.53598i	0.640460	+	0.767991i	$0.278744\pi$
$839$	44.4905i	1.53598i	−0.640460	+	0.767991i	$0.721256\pi$
$840$	5.04892i	0.174204i
$841$	34.0277	1.17337
$842$	−33.2868	−1.14714
$843$	− 19.5579i	− 0.673611i
$844$	20.7875	0.715534
$845$	0	0
$846$	−7.00000	−0.240665
$847$	57.9439i	1.99098i
$848$	−10.8605	−0.372953
$849$	−31.0726	−1.06641
$850$	1.08815i	0.0373231i
$851$	− 19.2392i	− 0.659512i
$852$	− 4.47219i	− 0.153215i
$853$	− 7.59743i	− 0.260131i	−0.991505	−	0.130066i	$-0.958481\pi$
$853$	− 7.59743i	− 0.260131i	0.991505	−	0.130066i	$-0.0415188\pi$
$854$	55.8950	1.91269
$855$	−4.44504	−0.152017
$856$	− 0.796561i	− 0.0272259i
$857$	34.2978	1.17159	0.585796	−	0.810459i	$-0.300782\pi$
$857$	34.2978	1.17159	0.585796	+	0.810459i	$0.300782\pi$
$858$	0	0
$859$	−30.8412	−1.05229	−0.526144	−	0.850396i	$-0.676363\pi$
$859$	−30.8412	−1.05229	−0.526144	+	0.850396i	$0.676363\pi$
$860$	4.35690i	0.148569i
$861$	−3.35690	−0.114403
$862$	11.1578	0.380035
$863$	− 16.1075i	− 0.548306i	−0.961686	−	0.274153i	$-0.911602\pi$
$863$	− 16.1075i	− 0.548306i	0.961686	−	0.274153i	$-0.0883975\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	13.8388i	0.470532i
$866$	14.0828i	0.478552i
$867$	−15.8159	−0.537137
$868$	−39.6383	−1.34541
$869$	67.0689i	2.27516i
$870$	−7.93900	−0.269157
$871$	0	0
$872$	−0.704103	−0.0238439
$873$	14.9269i	0.505200i
$874$	−11.9661	−0.404761
$875$	−5.04892	−0.170685
$876$	− 8.17390i	− 0.276170i
$877$	− 32.7342i	− 1.10536i	−0.833395	−	0.552678i	$-0.813606\pi$
$877$	− 32.7342i	− 1.10536i	0.833395	−	0.552678i	$-0.186394\pi$
$878$	− 28.9933i	− 0.978476i
$879$	10.0543i	0.339123i
$880$	−4.74094	−0.159817
$881$	9.32736	0.314247	0.157123	−	0.987579i	$-0.449778\pi$
$881$	9.32736	0.314247	0.157123	+	0.987579i	$0.449778\pi$
$882$	18.4916i	0.622643i
$883$	−45.1135	−1.51819	−0.759095	−	0.650980i	$-0.774358\pi$
$883$	−45.1135	−1.51819	−0.759095	+	0.650980i	$0.774358\pi$
$884$	0	0
$885$	12.2784	0.412735
$886$	− 33.3080i	− 1.11900i
$887$	−37.4383	−1.25706	−0.628528	−	0.777787i	$-0.716342\pi$
$887$	−37.4383	−1.25706	−0.628528	+	0.777787i	$0.716342\pi$
$888$	−7.14675	−0.239829
$889$	− 14.8388i	− 0.497676i
$890$	− 14.8877i	− 0.499037i
$891$	4.74094i	0.158827i
$892$	− 17.3502i	− 0.580927i
$893$	31.1153	1.04123
$894$	10.2010	0.341174
$895$	14.2078i	0.474913i
$896$	5.04892	0.168672
$897$	0	0
$898$	37.7899	1.26106
$899$	− 62.3279i	− 2.07875i
$900$	1.00000	0.0333333
$901$	−11.8179	−0.393710
$902$	− 3.15213i	− 0.104954i
$903$	21.9976i	0.732035i
$904$	14.4983i	0.482205i
$905$	24.9028i	0.827796i
$906$	−3.86294	−0.128337
$907$	35.9004	1.19205	0.596026	−	0.802965i	$-0.296746\pi$
$907$	35.9004	1.19205	0.596026	+	0.802965i	$0.296746\pi$
$908$	13.7573i	0.456553i
$909$	−11.4819	−0.380830
$910$	0	0
$911$	34.0025	1.12655	0.563277	−	0.826268i	$-0.309541\pi$
$911$	34.0025	1.12655	0.563277	+	0.826268i	$0.309541\pi$
$912$	4.44504i	0.147190i
$913$	−23.0549	−0.763005
$914$	−10.5700	−0.349625
$915$	− 11.0707i	− 0.365986i
$916$	1.75541i	0.0580005i
$917$	17.9976i	0.594333i
$918$	− 1.08815i	− 0.0359142i
$919$	−27.7778	−0.916304	−0.458152	−	0.888874i	$-0.651489\pi$
$919$	−27.7778	−0.916304	−0.458152	+	0.888874i	$0.651489\pi$
$920$	2.69202	0.0887533
$921$	23.6872i	0.780521i
$922$	−20.1420	−0.663340
$923$	0	0
$924$	−23.9366	−0.787457
$925$	− 7.14675i	− 0.234984i
$926$	−15.9065	−0.522719
$927$	−4.16421	−0.136771
$928$	7.93900i	0.260610i
$929$	− 44.9342i	− 1.47424i	−0.675760	−	0.737122i	$-0.736184\pi$
$929$	− 44.9342i	− 1.47424i	0.675760	−	0.737122i	$-0.263816\pi$
$930$	7.85086i	0.257440i
$931$	− 82.1958i	− 2.69386i
$932$	−5.49396	−0.179961
$933$	−6.77240	−0.221718
$934$	− 25.6136i	− 0.838101i
$935$	−5.15883	−0.168712
$936$	0	0
$937$	−4.50843	−0.147284	−0.0736421	−	0.997285i	$-0.523462\pi$
$937$	−4.50843	−0.147284	−0.0736421	+	0.997285i	$0.523462\pi$
$938$	− 48.3793i	− 1.57964i
$939$	7.16123	0.233698
$940$	−7.00000	−0.228315
$941$	− 18.4926i	− 0.602843i	−0.953491	−	0.301421i	$-0.902539\pi$
$941$	− 18.4926i	− 0.602843i	0.953491	−	0.301421i	$-0.0974610\pi$
$942$	11.2174i	0.365484i
$943$	1.78986i	0.0582857i
$944$	− 12.2784i	− 0.399629i
$945$	5.04892	0.164241
$946$	−20.6558	−0.671577
$947$	60.3062i	1.95969i	0.199766	+	0.979844i	$0.435982\pi$
$947$	60.3062i	1.95969i	−0.199766	+	0.979844i	$0.564018\pi$
$948$	−14.1468	−0.459465
$949$	0	0
$950$	−4.44504	−0.144216
$951$	− 1.20344i	− 0.0390242i
$952$	5.49396	0.178060
$953$	−23.6394	−0.765755	−0.382877	−	0.923799i	$-0.625067\pi$
$953$	−23.6394	−0.765755	−0.382877	+	0.923799i	$0.625067\pi$
$954$	10.8605i	0.351623i
$955$	2.44504i	0.0791197i
$956$	17.4004i	0.562770i
$957$	− 37.6383i	− 1.21667i
$958$	7.59658	0.245434
$959$	47.8461	1.54503
$960$	− 1.00000i	− 0.0322749i
$961$	−30.6359	−0.988256
$962$	0	0
$963$	−0.796561	−0.0256688
$964$	− 3.15452i	− 0.101600i
$965$	−15.2959	−0.492392
$966$	13.5918	0.437309
$967$	− 22.4910i	− 0.723261i	−0.932321	−	0.361631i	$-0.882220\pi$
$967$	− 22.4910i	− 0.723261i	0.932321	−	0.361631i	$-0.117780\pi$
$968$	− 11.4765i	− 0.368869i
$969$	4.83685i	0.155382i
$970$	14.9269i	0.479275i
$971$	−15.4257	−0.495033	−0.247517	−	0.968884i	$-0.579614\pi$
$971$	−15.4257	−0.495033	−0.247517	+	0.968884i	$0.579614\pi$
$972$	−1.00000	−0.0320750
$973$	71.7338i	2.29968i
$974$	0.779103	0.0249641
$975$	0	0
$976$	−11.0707	−0.354364
$977$	38.5018i	1.23178i	0.787831	+	0.615892i	$0.211204\pi$
$977$	38.5018i	1.23178i	−0.787831	+	0.615892i	$0.788796\pi$
$978$	6.50902	0.208136
$979$	70.5816	2.25580
$980$	18.4916i	0.590691i
$981$	0.704103i	0.0224803i
$982$	11.3948i	0.363623i
$983$	− 3.83340i	− 0.122266i	−0.998130	−	0.0611332i	$-0.980529\pi$
$983$	− 3.83340i	− 0.122266i	0.998130	−	0.0611332i	$-0.0194714\pi$
$984$	0.664874	0.0211954
$985$	−3.44935	−0.109906
$986$	8.63879i	0.275115i
$987$	−35.3424	−1.12496
$988$	0	0
$989$	11.7289	0.372956
$990$	4.74094i	0.150677i
$991$	−46.7670	−1.48560	−0.742802	−	0.669512i	$-0.766503\pi$
$991$	−46.7670	−1.48560	−0.742802	+	0.669512i	$0.766503\pi$
$992$	7.85086	0.249265
$993$	20.9933i	0.666202i
$994$	− 22.5797i	− 0.716185i
$995$	9.18598i	0.291215i
$996$	− 4.86294i	− 0.154088i
$997$	−42.8920	−1.35840	−0.679202	−	0.733952i	$-0.737674\pi$
$997$	−42.8920	−1.35840	−0.679202	+	0.733952i	$0.737674\pi$
$998$	23.5555	0.745638
$999$	7.14675i	0.226113i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.y.1351.1		6
13.5	odd	4		5070.2.a.bq.1.3	✓	3
13.8	odd	4		5070.2.a.bv.1.1	yes	3
13.12	even	2	inner	5070.2.b.y.1351.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bq.1.3	✓	3	13.5	odd	4
5070.2.a.bv.1.1	yes	3	13.8	odd	4
5070.2.b.y.1351.1		6	1.1	even	1	trivial
5070.2.b.y.1351.6		6	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -5.04892i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -5.04892i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.74094i q^{11} -1.00000 q^{12} -5.04892 q^{14} +1.00000i q^{15} +1.00000 q^{16} +1.08815 q^{17} -1.00000i q^{18} +4.44504i q^{19} -1.00000i q^{20} -5.04892i q^{21} +4.74094 q^{22} -2.69202 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +5.04892i q^{28} -7.93900 q^{29} +1.00000 q^{30} +7.85086i q^{31} -1.00000i q^{32} +4.74094i q^{33} -1.08815i q^{34} +5.04892 q^{35} -1.00000 q^{36} +7.14675i q^{37} +4.44504 q^{38} -1.00000 q^{40} -0.664874i q^{41} -5.04892 q^{42} -4.35690 q^{43} -4.74094i q^{44} +1.00000i q^{45} +2.69202i q^{46} -7.00000i q^{47} +1.00000 q^{48} -18.4916 q^{49} +1.00000i q^{50} +1.08815 q^{51} -10.8605 q^{53} -1.00000i q^{54} -4.74094 q^{55} +5.04892 q^{56} +4.44504i q^{57} +7.93900i q^{58} -12.2784i q^{59} -1.00000i q^{60} -11.0707 q^{61} +7.85086 q^{62} -5.04892i q^{63} -1.00000 q^{64} +4.74094 q^{66} +9.58211i q^{67} -1.08815 q^{68} -2.69202 q^{69} -5.04892i q^{70} +4.47219i q^{71} +1.00000i q^{72} +8.17390i q^{73} +7.14675 q^{74} -1.00000 q^{75} -4.44504i q^{76} +23.9366 q^{77} +14.1468 q^{79} +1.00000i q^{80} +1.00000 q^{81} -0.664874 q^{82} +4.86294i q^{83} +5.04892i q^{84} +1.08815i q^{85} +4.35690i q^{86} -7.93900 q^{87} -4.74094 q^{88} -14.8877i q^{89} +1.00000 q^{90} +2.69202 q^{92} +7.85086i q^{93} -7.00000 q^{94} -4.44504 q^{95} -1.00000i q^{96} +14.9269i q^{97} +18.4916i q^{98} +4.74094i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} - 6 q^{12} - 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} - 28 q^{29} + 6 q^{30} + 12 q^{35} - 6 q^{36} + 26 q^{38} - 6 q^{40} - 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} + 12 q^{56} - 42 q^{61} + 20 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} - 12 q^{74} - 6 q^{75} + 42 q^{77} + 30 q^{79} + 6 q^{81} - 6 q^{82} - 28 q^{87} + 6 q^{90} + 6 q^{92} - 42 q^{94} - 26 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 - 6 * q^12 - 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 - 28 * q^29 + 6 * q^30 + 12 * q^35 - 6 * q^36 + 26 * q^38 - 6 * q^40 - 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 + 12 * q^56 - 42 * q^61 + 20 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 - 12 * q^74 - 6 * q^75 + 42 * q^77 + 30 * q^79 + 6 * q^81 - 6 * q^82 - 28 * q^87 + 6 * q^90 + 6 * q^92 - 42 * q^94 - 26 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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