LMFDB - Embedded newform 5070.2.b.p.1351.4

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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.p.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -3.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -3.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.73205i q^{11} +1.00000 q^{12} +3.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000i q^{18} -2.26795i q^{19} +1.00000i q^{20} +3.00000i q^{21} -3.73205 q^{22} -3.46410 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +3.00000i q^{28} -5.46410 q^{29} -1.00000 q^{30} +8.92820i q^{31} +1.00000i q^{32} -3.73205i q^{33} +4.00000i q^{34} -3.00000 q^{35} -1.00000 q^{36} +7.92820i q^{37} +2.26795 q^{38} -1.00000 q^{40} -4.00000i q^{41} -3.00000 q^{42} +6.00000 q^{43} -3.73205i q^{44} -1.00000i q^{45} -3.46410i q^{46} -0.464102i q^{47} -1.00000 q^{48} -2.00000 q^{49} -1.00000i q^{50} -4.00000 q^{51} -3.73205 q^{53} -1.00000i q^{54} +3.73205 q^{55} -3.00000 q^{56} +2.26795i q^{57} -5.46410i q^{58} +4.53590i q^{59} -1.00000i q^{60} -7.46410 q^{61} -8.92820 q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.73205 q^{66} -5.46410i q^{67} -4.00000 q^{68} +3.46410 q^{69} -3.00000i q^{70} -0.928203i q^{71} -1.00000i q^{72} -6.92820i q^{73} -7.92820 q^{74} +1.00000 q^{75} +2.26795i q^{76} +11.1962 q^{77} +16.9282 q^{79} -1.00000i q^{80} +1.00000 q^{81} +4.00000 q^{82} -2.53590i q^{83} -3.00000i q^{84} -4.00000i q^{85} +6.00000i q^{86} +5.46410 q^{87} +3.73205 q^{88} +10.1244i q^{89} +1.00000 q^{90} +3.46410 q^{92} -8.92820i q^{93} +0.464102 q^{94} -2.26795 q^{95} -1.00000i q^{96} +12.3923i q^{97} -2.00000i q^{98} +3.73205i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9	$ 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} + 4 q^{12} + 12 q^{14} + 4 q^{16} + 16 q^{17} - 8 q^{22} - 4 q^{25} - 4 q^{27} - 8 q^{29} - 4 q^{30} - 12 q^{35} - 4 q^{36} + 16 q^{38} - 4 q^{40} - 12 q^{42} + 24 q^{43} - 4 q^{48} - 8 q^{49} - 16 q^{51} - 8 q^{53} + 8 q^{55} - 12 q^{56} - 16 q^{61} - 8 q^{62} - 4 q^{64} + 8 q^{66} - 16 q^{68} - 4 q^{74} + 4 q^{75} + 24 q^{77} + 40 q^{79} + 4 q^{81} + 16 q^{82} + 8 q^{87} + 8 q^{88} + 4 q^{90} - 12 q^{94} - 16 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 + 4 * q^10 + 4 * q^12 + 12 * q^14 + 4 * q^16 + 16 * q^17 - 8 * q^22 - 4 * q^25 - 4 * q^27 - 8 * q^29 - 4 * q^30 - 12 * q^35 - 4 * q^36 + 16 * q^38 - 4 * q^40 - 12 * q^42 + 24 * q^43 - 4 * q^48 - 8 * q^49 - 16 * q^51 - 8 * q^53 + 8 * q^55 - 12 * q^56 - 16 * q^61 - 8 * q^62 - 4 * q^64 + 8 * q^66 - 16 * q^68 - 4 * q^74 + 4 * q^75 + 24 * q^77 + 40 * q^79 + 4 * q^81 + 16 * q^82 + 8 * q^87 + 8 * q^88 + 4 * q^90 - 12 * q^94 - 16 * 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
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	3.73205i	1.12526i	0.826710	+	0.562628i	$0.190210\pi$
$11$	3.73205i	1.12526i	−0.826710	+	0.562628i	$0.809790\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	3.00000	0.801784
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	1.00000i	0.235702i
$19$	− 2.26795i	− 0.520303i	−0.965568	−	0.260152i	$-0.916227\pi$
$19$	− 2.26795i	− 0.520303i	0.965568	−	0.260152i	$-0.0837725\pi$
$20$	1.00000i	0.223607i
$21$	3.00000i	0.654654i
$22$	−3.73205	−0.795676
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	3.00000i	0.566947i
$29$	−5.46410	−1.01466	−0.507329	−	0.861752i	$-0.669367\pi$
$29$	−5.46410	−1.01466	−0.507329	+	0.861752i	$0.669367\pi$
$30$	−1.00000	−0.182574
$31$	8.92820i	1.60355i	0.597624	+	0.801776i	$0.296111\pi$
$31$	8.92820i	1.60355i	−0.597624	+	0.801776i	$0.703889\pi$
$32$	1.00000i	0.176777i
$33$	− 3.73205i	− 0.649667i
$34$	4.00000i	0.685994i
$35$	−3.00000	−0.507093
$36$	−1.00000	−0.166667
$37$	7.92820i	1.30339i	0.758482	+	0.651694i	$0.225941\pi$
$37$	7.92820i	1.30339i	−0.758482	+	0.651694i	$0.774059\pi$
$38$	2.26795	0.367910
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 4.00000i	− 0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$	− 4.00000i	− 0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$	−3.00000	−0.462910
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	− 3.73205i	− 0.562628i
$45$	− 1.00000i	− 0.149071i
$46$	− 3.46410i	− 0.510754i
$47$	− 0.464102i	− 0.0676962i	−0.999427	−	0.0338481i	$-0.989224\pi$
$47$	− 0.464102i	− 0.0676962i	0.999427	−	0.0338481i	$-0.0107762\pi$
$48$	−1.00000	−0.144338
$49$	−2.00000	−0.285714
$50$	− 1.00000i	− 0.141421i
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−3.73205	−0.512637	−0.256318	−	0.966592i	$-0.582510\pi$
$53$	−3.73205	−0.512637	−0.256318	+	0.966592i	$0.582510\pi$
$54$	− 1.00000i	− 0.136083i
$55$	3.73205	0.503230
$56$	−3.00000	−0.400892
$57$	2.26795i	0.300397i
$58$	− 5.46410i	− 0.717472i
$59$	4.53590i	0.590524i	0.955416	+	0.295262i	$0.0954069\pi$
$59$	4.53590i	0.590524i	−0.955416	+	0.295262i	$0.904593\pi$
$60$	− 1.00000i	− 0.129099i
$61$	−7.46410	−0.955680	−0.477840	−	0.878447i	$-0.658580\pi$
$61$	−7.46410	−0.955680	−0.477840	+	0.878447i	$0.658580\pi$
$62$	−8.92820	−1.13388
$63$	− 3.00000i	− 0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.73205	0.459384
$67$	− 5.46410i	− 0.667546i	−0.942653	−	0.333773i	$-0.891678\pi$
$67$	− 5.46410i	− 0.667546i	0.942653	−	0.333773i	$-0.108322\pi$
$68$	−4.00000	−0.485071
$69$	3.46410	0.417029
$70$	− 3.00000i	− 0.358569i
$71$	− 0.928203i	− 0.110157i	−0.998482	−	0.0550787i	$-0.982459\pi$
$71$	− 0.928203i	− 0.110157i	0.998482	−	0.0550787i	$-0.0175410\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 6.92820i	− 0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$	− 6.92820i	− 0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$	−7.92820	−0.921635
$75$	1.00000	0.115470
$76$	2.26795i	0.260152i
$77$	11.1962	1.27592
$78$	0	0
$79$	16.9282	1.90457	0.952286	−	0.305208i	$-0.0987259\pi$
$79$	16.9282	1.90457	0.952286	+	0.305208i	$0.0987259\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	− 2.53590i	− 0.278351i	−0.990268	−	0.139176i	$-0.955555\pi$
$83$	− 2.53590i	− 0.278351i	0.990268	−	0.139176i	$-0.0444452\pi$
$84$	− 3.00000i	− 0.327327i
$85$	− 4.00000i	− 0.433861i
$86$	6.00000i	0.646997i
$87$	5.46410	0.585813
$88$	3.73205	0.397838
$89$	10.1244i	1.07318i	0.843843	+	0.536590i	$0.180288\pi$
$89$	10.1244i	1.07318i	−0.843843	+	0.536590i	$0.819712\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	3.46410	0.361158
$93$	− 8.92820i	− 0.925812i
$94$	0.464102	0.0478684
$95$	−2.26795	−0.232687
$96$	− 1.00000i	− 0.102062i
$97$	12.3923i	1.25825i	0.777305	+	0.629124i	$0.216586\pi$
$97$	12.3923i	1.25825i	−0.777305	+	0.629124i	$0.783414\pi$
$98$	− 2.00000i	− 0.202031i
$99$	3.73205i	0.375085i
$100$	1.00000	0.100000
$101$	−8.39230	−0.835066	−0.417533	−	0.908662i	$-0.637105\pi$
$101$	−8.39230	−0.835066	−0.417533	+	0.908662i	$0.637105\pi$
$102$	− 4.00000i	− 0.396059i
$103$	19.5885	1.93011	0.965054	−	0.262051i	$-0.0843989\pi$
$103$	19.5885	1.93011	0.965054	+	0.262051i	$0.0843989\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	− 3.73205i	− 0.362489i
$107$	0.928203	0.0897328	0.0448664	−	0.998993i	$-0.485714\pi$
$107$	0.928203	0.0897328	0.0448664	+	0.998993i	$0.485714\pi$
$108$	1.00000	0.0962250
$109$	− 10.3923i	− 0.995402i	−0.867349	−	0.497701i	$-0.834178\pi$
$109$	− 10.3923i	− 0.995402i	0.867349	−	0.497701i	$-0.165822\pi$
$110$	3.73205i	0.355837i
$111$	− 7.92820i	− 0.752512i
$112$	− 3.00000i	− 0.283473i
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	−2.26795	−0.212413
$115$	3.46410i	0.323029i
$116$	5.46410	0.507329
$117$	0	0
$118$	−4.53590	−0.417563
$119$	− 12.0000i	− 1.10004i
$120$	1.00000	0.0912871
$121$	−2.92820	−0.266200
$122$	− 7.46410i	− 0.675768i
$123$	4.00000i	0.360668i
$124$	− 8.92820i	− 0.801776i
$125$	1.00000i	0.0894427i
$126$	3.00000	0.267261
$127$	4.66025	0.413531	0.206765	−	0.978391i	$-0.433706\pi$
$127$	4.66025	0.413531	0.206765	+	0.978391i	$0.433706\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−20.3205	−1.77541	−0.887706	−	0.460412i	$-0.847702\pi$
$131$	−20.3205	−1.77541	−0.887706	+	0.460412i	$0.847702\pi$
$132$	3.73205i	0.324833i
$133$	−6.80385	−0.589968
$134$	5.46410	0.472026
$135$	1.00000i	0.0860663i
$136$	− 4.00000i	− 0.342997i
$137$	− 6.53590i	− 0.558399i	−0.960233	−	0.279200i	$-0.909931\pi$
$137$	− 6.53590i	− 0.558399i	0.960233	−	0.279200i	$-0.0900692\pi$
$138$	3.46410i	0.294884i
$139$	21.7846	1.84775	0.923873	−	0.382699i	$-0.125005\pi$
$139$	21.7846	1.84775	0.923873	+	0.382699i	$0.125005\pi$
$140$	3.00000	0.253546
$141$	0.464102i	0.0390844i
$142$	0.928203	0.0778931
$143$	0	0
$144$	1.00000	0.0833333
$145$	5.46410i	0.453769i
$146$	6.92820	0.573382
$147$	2.00000	0.164957
$148$	− 7.92820i	− 0.651694i
$149$	22.7846i	1.86659i	0.359113	+	0.933294i	$0.383079\pi$
$149$	22.7846i	1.86659i	−0.359113	+	0.933294i	$0.616921\pi$
$150$	1.00000i	0.0816497i
$151$	18.7846i	1.52867i	0.644819	+	0.764335i	$0.276933\pi$
$151$	18.7846i	1.52867i	−0.644819	+	0.764335i	$0.723067\pi$
$152$	−2.26795	−0.183955
$153$	4.00000	0.323381
$154$	11.1962i	0.902212i
$155$	8.92820	0.717131
$156$	0	0
$157$	10.8038	0.862241	0.431120	−	0.902294i	$-0.358118\pi$
$157$	10.8038	0.862241	0.431120	+	0.902294i	$0.358118\pi$
$158$	16.9282i	1.34674i
$159$	3.73205	0.295971
$160$	1.00000	0.0790569
$161$	10.3923i	0.819028i
$162$	1.00000i	0.0785674i
$163$	− 10.9282i	− 0.855963i	−0.903788	−	0.427981i	$-0.859225\pi$
$163$	− 10.9282i	− 0.855963i	0.903788	−	0.427981i	$-0.140775\pi$
$164$	4.00000i	0.312348i
$165$	−3.73205	−0.290540
$166$	2.53590	0.196824
$167$	6.46410i	0.500207i	0.968219	+	0.250104i	$0.0804647\pi$
$167$	6.46410i	0.500207i	−0.968219	+	0.250104i	$0.919535\pi$
$168$	3.00000	0.231455
$169$	0	0
$170$	4.00000	0.306786
$171$	− 2.26795i	− 0.173434i
$172$	−6.00000	−0.457496
$173$	22.1244	1.68208	0.841042	−	0.540970i	$-0.181943\pi$
$173$	22.1244	1.68208	0.841042	+	0.540970i	$0.181943\pi$
$174$	5.46410i	0.414232i
$175$	3.00000i	0.226779i
$176$	3.73205i	0.281314i
$177$	− 4.53590i	− 0.340939i
$178$	−10.1244	−0.758853
$179$	22.9282	1.71373	0.856867	−	0.515537i	$-0.172408\pi$
$179$	22.9282	1.71373	0.856867	+	0.515537i	$0.172408\pi$
$180$	1.00000i	0.0745356i
$181$	3.07180	0.228325	0.114162	−	0.993462i	$-0.463582\pi$
$181$	3.07180	0.228325	0.114162	+	0.993462i	$0.463582\pi$
$182$	0	0
$183$	7.46410	0.551762
$184$	3.46410i	0.255377i
$185$	7.92820	0.582893
$186$	8.92820	0.654648
$187$	14.9282i	1.09166i
$188$	0.464102i	0.0338481i
$189$	3.00000i	0.218218i
$190$	− 2.26795i	− 0.164534i
$191$	17.3205	1.25327	0.626634	−	0.779314i	$-0.284432\pi$
$191$	17.3205	1.25327	0.626634	+	0.779314i	$0.284432\pi$
$192$	1.00000	0.0721688
$193$	17.8564i	1.28533i	0.766146	+	0.642666i	$0.222172\pi$
$193$	17.8564i	1.28533i	−0.766146	+	0.642666i	$0.777828\pi$
$194$	−12.3923	−0.889716
$195$	0	0
$196$	2.00000	0.142857
$197$	9.39230i	0.669174i	0.942365	+	0.334587i	$0.108597\pi$
$197$	9.39230i	0.669174i	−0.942365	+	0.334587i	$0.891403\pi$
$198$	−3.73205	−0.265225
$199$	−11.0718	−0.784859	−0.392429	−	0.919782i	$-0.628365\pi$
$199$	−11.0718	−0.784859	−0.392429	+	0.919782i	$0.628365\pi$
$200$	1.00000i	0.0707107i
$201$	5.46410i	0.385408i
$202$	− 8.39230i	− 0.590481i
$203$	16.3923i	1.15051i
$204$	4.00000	0.280056
$205$	−4.00000	−0.279372
$206$	19.5885i	1.36479i
$207$	−3.46410	−0.240772
$208$	0	0
$209$	8.46410	0.585474
$210$	3.00000i	0.207020i
$211$	−21.9282	−1.50960	−0.754800	−	0.655955i	$-0.772266\pi$
$211$	−21.9282	−1.50960	−0.754800	+	0.655955i	$0.772266\pi$
$212$	3.73205	0.256318
$213$	0.928203i	0.0635994i
$214$	0.928203i	0.0634507i
$215$	− 6.00000i	− 0.409197i
$216$	1.00000i	0.0680414i
$217$	26.7846	1.81826
$218$	10.3923	0.703856
$219$	6.92820i	0.468165i
$220$	−3.73205	−0.251615
$221$	0	0
$222$	7.92820	0.532106
$223$	8.85641i	0.593069i	0.955022	+	0.296534i	$0.0958309\pi$
$223$	8.85641i	0.593069i	−0.955022	+	0.296534i	$0.904169\pi$
$224$	3.00000	0.200446
$225$	−1.00000	−0.0666667
$226$	− 12.0000i	− 0.798228i
$227$	− 13.4641i	− 0.893644i	−0.894623	−	0.446822i	$-0.852556\pi$
$227$	− 13.4641i	− 0.893644i	0.894623	−	0.446822i	$-0.147444\pi$
$228$	− 2.26795i	− 0.150199i
$229$	− 11.4641i	− 0.757569i	−0.925485	−	0.378785i	$-0.876342\pi$
$229$	− 11.4641i	− 0.757569i	0.925485	−	0.378785i	$-0.123658\pi$
$230$	−3.46410	−0.228416
$231$	−11.1962	−0.736653
$232$	5.46410i	0.358736i
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−0.464102	−0.0302747
$236$	− 4.53590i	− 0.295262i
$237$	−16.9282	−1.09960
$238$	12.0000	0.777844
$239$	− 3.46410i	− 0.224074i	−0.993704	−	0.112037i	$-0.964262\pi$
$239$	− 3.46410i	− 0.224074i	0.993704	−	0.112037i	$-0.0357375\pi$
$240$	1.00000i	0.0645497i
$241$	14.8038i	0.953600i	0.879012	+	0.476800i	$0.158203\pi$
$241$	14.8038i	0.953600i	−0.879012	+	0.476800i	$0.841797\pi$
$242$	− 2.92820i	− 0.188232i
$243$	−1.00000	−0.0641500
$244$	7.46410	0.477840
$245$	2.00000i	0.127775i
$246$	−4.00000	−0.255031
$247$	0	0
$248$	8.92820	0.566941
$249$	2.53590i	0.160706i
$250$	−1.00000	−0.0632456
$251$	26.4641	1.67040	0.835200	−	0.549947i	$-0.185352\pi$
$251$	26.4641	1.67040	0.835200	+	0.549947i	$0.185352\pi$
$252$	3.00000i	0.188982i
$253$	− 12.9282i	− 0.812789i
$254$	4.66025i	0.292410i
$255$	4.00000i	0.250490i
$256$	1.00000	0.0625000
$257$	21.4641	1.33889	0.669447	−	0.742860i	$-0.266531\pi$
$257$	21.4641	1.33889	0.669447	+	0.742860i	$0.266531\pi$
$258$	− 6.00000i	− 0.373544i
$259$	23.7846	1.47790
$260$	0	0
$261$	−5.46410	−0.338219
$262$	− 20.3205i	− 1.25541i
$263$	20.5167	1.26511	0.632556	−	0.774515i	$-0.282006\pi$
$263$	20.5167	1.26511	0.632556	+	0.774515i	$0.282006\pi$
$264$	−3.73205	−0.229692
$265$	3.73205i	0.229258i
$266$	− 6.80385i	− 0.417171i
$267$	− 10.1244i	− 0.619601i
$268$	5.46410i	0.333773i
$269$	−30.9282	−1.88573	−0.942863	−	0.333181i	$-0.891878\pi$
$269$	−30.9282	−1.88573	−0.942863	+	0.333181i	$0.891878\pi$
$270$	−1.00000	−0.0608581
$271$	3.60770i	0.219152i	0.993978	+	0.109576i	$0.0349493\pi$
$271$	3.60770i	0.219152i	−0.993978	+	0.109576i	$0.965051\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	6.53590	0.394848
$275$	− 3.73205i	− 0.225051i
$276$	−3.46410	−0.208514
$277$	−19.5885	−1.17696	−0.588478	−	0.808513i	$-0.700273\pi$
$277$	−19.5885	−1.17696	−0.588478	+	0.808513i	$0.700273\pi$
$278$	21.7846i	1.30655i
$279$	8.92820i	0.534518i
$280$	3.00000i	0.179284i
$281$	− 6.92820i	− 0.413302i	−0.978415	−	0.206651i	$-0.933744\pi$
$281$	− 6.92820i	− 0.413302i	0.978415	−	0.206651i	$-0.0662565\pi$
$282$	−0.464102	−0.0276368
$283$	14.3923	0.855534	0.427767	−	0.903889i	$-0.359300\pi$
$283$	14.3923	0.855534	0.427767	+	0.903889i	$0.359300\pi$
$284$	0.928203i	0.0550787i
$285$	2.26795	0.134342
$286$	0	0
$287$	−12.0000	−0.708338
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	−5.46410	−0.320863
$291$	− 12.3923i	− 0.726450i
$292$	6.92820i	0.405442i
$293$	19.2487i	1.12452i	0.826960	+	0.562261i	$0.190068\pi$
$293$	19.2487i	1.12452i	−0.826960	+	0.562261i	$0.809932\pi$
$294$	2.00000i	0.116642i
$295$	4.53590	0.264090
$296$	7.92820	0.460817
$297$	− 3.73205i	− 0.216556i
$298$	−22.7846	−1.31988
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 18.0000i	− 1.03750i
$302$	−18.7846	−1.08093
$303$	8.39230	0.482125
$304$	− 2.26795i	− 0.130076i
$305$	7.46410i	0.427393i
$306$	4.00000i	0.228665i
$307$	20.2487i	1.15565i	0.816159	+	0.577827i	$0.196099\pi$
$307$	20.2487i	1.15565i	−0.816159	+	0.577827i	$0.803901\pi$
$308$	−11.1962	−0.637960
$309$	−19.5885	−1.11435
$310$	8.92820i	0.507088i
$311$	5.07180	0.287595	0.143798	−	0.989607i	$-0.454069\pi$
$311$	5.07180	0.287595	0.143798	+	0.989607i	$0.454069\pi$
$312$	0	0
$313$	1.32051	0.0746395	0.0373198	−	0.999303i	$-0.488118\pi$
$313$	1.32051	0.0746395	0.0373198	+	0.999303i	$0.488118\pi$
$314$	10.8038i	0.609696i
$315$	−3.00000	−0.169031
$316$	−16.9282	−0.952286
$317$	23.5359i	1.32191i	0.750427	+	0.660954i	$0.229848\pi$
$317$	23.5359i	1.32191i	−0.750427	+	0.660954i	$0.770152\pi$
$318$	3.73205i	0.209283i
$319$	− 20.3923i	− 1.14175i
$320$	1.00000i	0.0559017i
$321$	−0.928203	−0.0518073
$322$	−10.3923	−0.579141
$323$	− 9.07180i	− 0.504768i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	10.9282	0.605257
$327$	10.3923i	0.574696i
$328$	−4.00000	−0.220863
$329$	−1.39230	−0.0767603
$330$	− 3.73205i	− 0.205443i
$331$	6.39230i	0.351353i	0.984448	+	0.175676i	$0.0562112\pi$
$331$	6.39230i	0.351353i	−0.984448	+	0.175676i	$0.943789\pi$
$332$	2.53590i	0.139176i
$333$	7.92820i	0.434463i
$334$	−6.46410	−0.353700
$335$	−5.46410	−0.298536
$336$	3.00000i	0.163663i
$337$	−5.60770	−0.305471	−0.152735	−	0.988267i	$-0.548808\pi$
$337$	−5.60770	−0.305471	−0.152735	+	0.988267i	$0.548808\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	4.00000i	0.216930i
$341$	−33.3205	−1.80441
$342$	2.26795	0.122637
$343$	− 15.0000i	− 0.809924i
$344$	− 6.00000i	− 0.323498i
$345$	− 3.46410i	− 0.186501i
$346$	22.1244i	1.18941i
$347$	−16.3923	−0.879985	−0.439993	−	0.898001i	$-0.645019\pi$
$347$	−16.3923	−0.879985	−0.439993	+	0.898001i	$0.645019\pi$
$348$	−5.46410	−0.292907
$349$	− 17.4641i	− 0.934832i	−0.884038	−	0.467416i	$-0.845185\pi$
$349$	− 17.4641i	− 0.934832i	0.884038	−	0.467416i	$-0.154815\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	−3.73205	−0.198919
$353$	28.3923i	1.51117i	0.655051	+	0.755585i	$0.272647\pi$
$353$	28.3923i	1.51117i	−0.655051	+	0.755585i	$0.727353\pi$
$354$	4.53590	0.241080
$355$	−0.928203	−0.0492639
$356$	− 10.1244i	− 0.536590i
$357$	12.0000i	0.635107i
$358$	22.9282i	1.21179i
$359$	− 12.9282i	− 0.682324i	−0.940004	−	0.341162i	$-0.889179\pi$
$359$	− 12.9282i	− 0.682324i	0.940004	−	0.341162i	$-0.110821\pi$
$360$	−1.00000	−0.0527046
$361$	13.8564	0.729285
$362$	3.07180i	0.161450i
$363$	2.92820	0.153691
$364$	0	0
$365$	−6.92820	−0.362639
$366$	7.46410i	0.390155i
$367$	−13.3205	−0.695325	−0.347662	−	0.937620i	$-0.613024\pi$
$367$	−13.3205	−0.695325	−0.347662	+	0.937620i	$0.613024\pi$
$368$	−3.46410	−0.180579
$369$	− 4.00000i	− 0.208232i
$370$	7.92820i	0.412168i
$371$	11.1962i	0.581275i
$372$	8.92820i	0.462906i
$373$	22.9282	1.18718	0.593589	−	0.804769i	$-0.297711\pi$
$373$	22.9282	1.18718	0.593589	+	0.804769i	$0.297711\pi$
$374$	−14.9282	−0.771919
$375$	− 1.00000i	− 0.0516398i
$376$	−0.464102	−0.0239342
$377$	0	0
$378$	−3.00000	−0.154303
$379$	− 6.26795i	− 0.321963i	−0.986957	−	0.160981i	$-0.948534\pi$
$379$	− 6.26795i	− 0.321963i	0.986957	−	0.160981i	$-0.0514659\pi$
$380$	2.26795	0.116343
$381$	−4.66025	−0.238752
$382$	17.3205i	0.886194i
$383$	36.7846i	1.87961i	0.341717	+	0.939803i	$0.388992\pi$
$383$	36.7846i	1.87961i	−0.341717	+	0.939803i	$0.611008\pi$
$384$	1.00000i	0.0510310i
$385$	− 11.1962i	− 0.570609i
$386$	−17.8564	−0.908867
$387$	6.00000	0.304997
$388$	− 12.3923i	− 0.629124i
$389$	−9.85641	−0.499740	−0.249870	−	0.968279i	$-0.580388\pi$
$389$	−9.85641	−0.499740	−0.249870	+	0.968279i	$0.580388\pi$
$390$	0	0
$391$	−13.8564	−0.700749
$392$	2.00000i	0.101015i
$393$	20.3205	1.02503
$394$	−9.39230	−0.473177
$395$	− 16.9282i	− 0.851750i
$396$	− 3.73205i	− 0.187543i
$397$	− 7.92820i	− 0.397905i	−0.980009	−	0.198953i	$-0.936246\pi$
$397$	− 7.92820i	− 0.397905i	0.980009	−	0.198953i	$-0.0637540\pi$
$398$	− 11.0718i	− 0.554979i
$399$	6.80385	0.340618
$400$	−1.00000	−0.0500000
$401$	2.12436i	0.106085i	0.998592	+	0.0530426i	$0.0168919\pi$
$401$	2.12436i	0.106085i	−0.998592	+	0.0530426i	$0.983108\pi$
$402$	−5.46410	−0.272525
$403$	0	0
$404$	8.39230	0.417533
$405$	− 1.00000i	− 0.0496904i
$406$	−16.3923	−0.813536
$407$	−29.5885	−1.46665
$408$	4.00000i	0.198030i
$409$	0.947441i	0.0468479i	0.999726	+	0.0234240i	$0.00745676\pi$
$409$	0.947441i	0.0468479i	−0.999726	+	0.0234240i	$0.992543\pi$
$410$	− 4.00000i	− 0.197546i
$411$	6.53590i	0.322392i
$412$	−19.5885	−0.965054
$413$	13.6077	0.669591
$414$	− 3.46410i	− 0.170251i
$415$	−2.53590	−0.124482
$416$	0	0
$417$	−21.7846	−1.06680
$418$	8.46410i	0.413993i
$419$	−17.8564	−0.872343	−0.436171	−	0.899864i	$-0.643666\pi$
$419$	−17.8564	−0.872343	−0.436171	+	0.899864i	$0.643666\pi$
$420$	−3.00000	−0.146385
$421$	− 5.85641i	− 0.285424i	−0.989764	−	0.142712i	$-0.954418\pi$
$421$	− 5.85641i	− 0.285424i	0.989764	−	0.142712i	$-0.0455822\pi$
$422$	− 21.9282i	− 1.06745i
$423$	− 0.464102i	− 0.0225654i
$424$	3.73205i	0.181244i
$425$	−4.00000	−0.194029
$426$	−0.928203	−0.0449716
$427$	22.3923i	1.08364i
$428$	−0.928203	−0.0448664
$429$	0	0
$430$	6.00000	0.289346
$431$	13.8564i	0.667440i	0.942672	+	0.333720i	$0.108304\pi$
$431$	13.8564i	0.667440i	−0.942672	+	0.333720i	$0.891696\pi$
$432$	−1.00000	−0.0481125
$433$	32.7846	1.57553	0.787764	−	0.615977i	$-0.211239\pi$
$433$	32.7846	1.57553	0.787764	+	0.615977i	$0.211239\pi$
$434$	26.7846i	1.28570i
$435$	− 5.46410i	− 0.261984i
$436$	10.3923i	0.497701i
$437$	7.85641i	0.375823i
$438$	−6.92820	−0.331042
$439$	21.3205	1.01757	0.508786	−	0.860893i	$-0.330094\pi$
$439$	21.3205	1.01757	0.508786	+	0.860893i	$0.330094\pi$
$440$	− 3.73205i	− 0.177919i
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	7.85641	0.373269	0.186635	−	0.982429i	$-0.440242\pi$
$443$	7.85641	0.373269	0.186635	+	0.982429i	$0.440242\pi$
$444$	7.92820i	0.376256i
$445$	10.1244	0.479940
$446$	−8.85641	−0.419363
$447$	− 22.7846i	− 1.07768i
$448$	3.00000i	0.141737i
$449$	− 18.1244i	− 0.855341i	−0.903935	−	0.427671i	$-0.859334\pi$
$449$	− 18.1244i	− 0.855341i	0.903935	−	0.427671i	$-0.140666\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	14.9282	0.702942
$452$	12.0000	0.564433
$453$	− 18.7846i	− 0.882578i
$454$	13.4641	0.631902
$455$	0	0
$456$	2.26795	0.106206
$457$	− 0.535898i	− 0.0250683i	−0.999921	−	0.0125341i	$-0.996010\pi$
$457$	− 0.535898i	− 0.0250683i	0.999921	−	0.0125341i	$-0.00398984\pi$
$458$	11.4641	0.535682
$459$	−4.00000	−0.186704
$460$	− 3.46410i	− 0.161515i
$461$	32.3923i	1.50866i	0.656495	+	0.754330i	$0.272038\pi$
$461$	32.3923i	1.50866i	−0.656495	+	0.754330i	$0.727962\pi$
$462$	− 11.1962i	− 0.520892i
$463$	0.784610i	0.0364639i	0.999834	+	0.0182320i	$0.00580373\pi$
$463$	0.784610i	0.0364639i	−0.999834	+	0.0182320i	$0.994196\pi$
$464$	−5.46410	−0.253665
$465$	−8.92820	−0.414036
$466$	18.0000i	0.833834i
$467$	3.60770	0.166944	0.0834721	−	0.996510i	$-0.473399\pi$
$467$	3.60770	0.166944	0.0834721	+	0.996510i	$0.473399\pi$
$468$	0	0
$469$	−16.3923	−0.756926
$470$	− 0.464102i	− 0.0214074i
$471$	−10.8038	−0.497815
$472$	4.53590	0.208782
$473$	22.3923i	1.02960i
$474$	− 16.9282i	− 0.777538i
$475$	2.26795i	0.104061i
$476$	12.0000i	0.550019i
$477$	−3.73205	−0.170879
$478$	3.46410	0.158444
$479$	26.2487i	1.19933i	0.800250	+	0.599667i	$0.204700\pi$
$479$	26.2487i	1.19933i	−0.800250	+	0.599667i	$0.795300\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−14.8038	−0.674297
$483$	− 10.3923i	− 0.472866i
$484$	2.92820	0.133100
$485$	12.3923	0.562706
$486$	− 1.00000i	− 0.0453609i
$487$	− 21.0000i	− 0.951601i	−0.879553	−	0.475800i	$-0.842158\pi$
$487$	− 21.0000i	− 0.951601i	0.879553	−	0.475800i	$-0.157842\pi$
$488$	7.46410i	0.337884i
$489$	10.9282i	0.494190i
$490$	−2.00000	−0.0903508
$491$	−15.3923	−0.694645	−0.347322	−	0.937746i	$-0.612909\pi$
$491$	−15.3923	−0.694645	−0.347322	+	0.937746i	$0.612909\pi$
$492$	− 4.00000i	− 0.180334i
$493$	−21.8564	−0.984363
$494$	0	0
$495$	3.73205	0.167743
$496$	8.92820i	0.400888i
$497$	−2.78461	−0.124907
$498$	−2.53590	−0.113636
$499$	− 1.32051i	− 0.0591141i	−0.999563	−	0.0295570i	$-0.990590\pi$
$499$	− 1.32051i	− 0.0591141i	0.999563	−	0.0295570i	$-0.00940967\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 6.46410i	− 0.288795i
$502$	26.4641i	1.18115i
$503$	0.267949	0.0119473	0.00597363	−	0.999982i	$-0.498099\pi$
$503$	0.267949	0.0119473	0.00597363	+	0.999982i	$0.498099\pi$
$504$	−3.00000	−0.133631
$505$	8.39230i	0.373453i
$506$	12.9282	0.574729
$507$	0	0
$508$	−4.66025	−0.206765
$509$	− 5.32051i	− 0.235827i	−0.993024	−	0.117914i	$-0.962379\pi$
$509$	− 5.32051i	− 0.235827i	0.993024	−	0.117914i	$-0.0376206\pi$
$510$	−4.00000	−0.177123
$511$	−20.7846	−0.919457
$512$	1.00000i	0.0441942i
$513$	2.26795i	0.100132i
$514$	21.4641i	0.946741i
$515$	− 19.5885i	− 0.863171i
$516$	6.00000	0.264135
$517$	1.73205	0.0761755
$518$	23.7846i	1.04504i
$519$	−22.1244	−0.971151
$520$	0	0
$521$	−27.3923	−1.20008	−0.600039	−	0.799970i	$-0.704848\pi$
$521$	−27.3923	−1.20008	−0.600039	+	0.799970i	$0.704848\pi$
$522$	− 5.46410i	− 0.239157i
$523$	13.7128	0.599619	0.299810	−	0.953999i	$-0.403077\pi$
$523$	13.7128	0.599619	0.299810	+	0.953999i	$0.403077\pi$
$524$	20.3205	0.887706
$525$	− 3.00000i	− 0.130931i
$526$	20.5167i	0.894569i
$527$	35.7128i	1.55567i
$528$	− 3.73205i	− 0.162417i
$529$	−11.0000	−0.478261
$530$	−3.73205	−0.162110
$531$	4.53590i	0.196841i
$532$	6.80385	0.294984
$533$	0	0
$534$	10.1244	0.438124
$535$	− 0.928203i	− 0.0401297i
$536$	−5.46410	−0.236013
$537$	−22.9282	−0.989425
$538$	− 30.9282i	− 1.33341i
$539$	− 7.46410i	− 0.321502i
$540$	− 1.00000i	− 0.0430331i
$541$	− 26.7846i	− 1.15156i	−0.817605	−	0.575780i	$-0.804698\pi$
$541$	− 26.7846i	− 1.15156i	0.817605	−	0.575780i	$-0.195302\pi$
$542$	−3.60770	−0.154964
$543$	−3.07180	−0.131823
$544$	4.00000i	0.171499i
$545$	−10.3923	−0.445157
$546$	0	0
$547$	29.3205	1.25365	0.626827	−	0.779158i	$-0.284353\pi$
$547$	29.3205	1.25365	0.626827	+	0.779158i	$0.284353\pi$
$548$	6.53590i	0.279200i
$549$	−7.46410	−0.318560
$550$	3.73205	0.159135
$551$	12.3923i	0.527930i
$552$	− 3.46410i	− 0.147442i
$553$	− 50.7846i	− 2.15958i
$554$	− 19.5885i	− 0.832234i
$555$	−7.92820	−0.336533
$556$	−21.7846	−0.923873
$557$	− 43.3923i	− 1.83859i	−0.393568	−	0.919295i	$-0.628759\pi$
$557$	− 43.3923i	− 1.83859i	0.393568	−	0.919295i	$-0.371241\pi$
$558$	−8.92820	−0.377961
$559$	0	0
$560$	−3.00000	−0.126773
$561$	− 14.9282i	− 0.630269i
$562$	6.92820	0.292249
$563$	19.3205	0.814262	0.407131	−	0.913370i	$-0.366529\pi$
$563$	19.3205	0.814262	0.407131	+	0.913370i	$0.366529\pi$
$564$	− 0.464102i	− 0.0195422i
$565$	12.0000i	0.504844i
$566$	14.3923i	0.604954i
$567$	− 3.00000i	− 0.125988i
$568$	−0.928203	−0.0389465
$569$	18.3205	0.768036	0.384018	−	0.923326i	$-0.374540\pi$
$569$	18.3205	0.768036	0.384018	+	0.923326i	$0.374540\pi$
$570$	2.26795i	0.0949939i
$571$	−16.8564	−0.705419	−0.352709	−	0.935733i	$-0.614740\pi$
$571$	−16.8564	−0.705419	−0.352709	+	0.935733i	$0.614740\pi$
$572$	0	0
$573$	−17.3205	−0.723575
$574$	− 12.0000i	− 0.500870i
$575$	3.46410	0.144463
$576$	−1.00000	−0.0416667
$577$	25.3205i	1.05411i	0.849832	+	0.527053i	$0.176703\pi$
$577$	25.3205i	1.05411i	−0.849832	+	0.527053i	$0.823297\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	− 17.8564i	− 0.742087i
$580$	− 5.46410i	− 0.226884i
$581$	−7.60770	−0.315620
$582$	12.3923	0.513678
$583$	− 13.9282i	− 0.576847i
$584$	−6.92820	−0.286691
$585$	0	0
$586$	−19.2487	−0.795157
$587$	4.92820i	0.203409i	0.994815	+	0.101704i	$0.0324296\pi$
$587$	4.92820i	0.203409i	−0.994815	+	0.101704i	$0.967570\pi$
$588$	−2.00000	−0.0824786
$589$	20.2487	0.834334
$590$	4.53590i	0.186740i
$591$	− 9.39230i	− 0.386348i
$592$	7.92820i	0.325847i
$593$	− 3.21539i	− 0.132040i	−0.997818	−	0.0660201i	$-0.978970\pi$
$593$	− 3.21539i	− 0.132040i	0.997818	−	0.0660201i	$-0.0210302\pi$
$594$	3.73205	0.153128
$595$	−12.0000	−0.491952
$596$	− 22.7846i	− 0.933294i
$597$	11.0718	0.453138
$598$	0	0
$599$	15.0718	0.615817	0.307908	−	0.951416i	$-0.400371\pi$
$599$	15.0718	0.615817	0.307908	+	0.951416i	$0.400371\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−24.7128	−1.00806	−0.504028	−	0.863687i	$-0.668149\pi$
$601$	−24.7128	−1.00806	−0.504028	+	0.863687i	$0.668149\pi$
$602$	18.0000	0.733625
$603$	− 5.46410i	− 0.222515i
$604$	− 18.7846i	− 0.764335i
$605$	2.92820i	0.119048i
$606$	8.39230i	0.340914i
$607$	37.4449	1.51984	0.759920	−	0.650017i	$-0.225238\pi$
$607$	37.4449	1.51984	0.759920	+	0.650017i	$0.225238\pi$
$608$	2.26795	0.0919775
$609$	− 16.3923i	− 0.664250i
$610$	−7.46410	−0.302213
$611$	0	0
$612$	−4.00000	−0.161690
$613$	45.0000i	1.81753i	0.417305	+	0.908766i	$0.362975\pi$
$613$	45.0000i	1.81753i	−0.417305	+	0.908766i	$0.637025\pi$
$614$	−20.2487	−0.817171
$615$	4.00000	0.161296
$616$	− 11.1962i	− 0.451106i
$617$	28.5359i	1.14881i	0.818571	+	0.574406i	$0.194767\pi$
$617$	28.5359i	1.14881i	−0.818571	+	0.574406i	$0.805233\pi$
$618$	− 19.5885i	− 0.787963i
$619$	− 42.5167i	− 1.70889i	−0.519543	−	0.854444i	$-0.673898\pi$
$619$	− 42.5167i	− 1.70889i	0.519543	−	0.854444i	$-0.326102\pi$
$620$	−8.92820	−0.358565
$621$	3.46410	0.139010
$622$	5.07180i	0.203361i
$623$	30.3731	1.21687
$624$	0	0
$625$	1.00000	0.0400000
$626$	1.32051i	0.0527781i
$627$	−8.46410	−0.338024
$628$	−10.8038	−0.431120
$629$	31.7128i	1.26447i
$630$	− 3.00000i	− 0.119523i
$631$	8.00000i	0.318475i	0.987240	+	0.159237i	$0.0509036\pi$
$631$	8.00000i	0.318475i	−0.987240	+	0.159237i	$0.949096\pi$
$632$	− 16.9282i	− 0.673368i
$633$	21.9282	0.871568
$634$	−23.5359	−0.934730
$635$	− 4.66025i	− 0.184937i
$636$	−3.73205	−0.147985
$637$	0	0
$638$	20.3923	0.807339
$639$	− 0.928203i	− 0.0367192i
$640$	−1.00000	−0.0395285
$641$	7.53590	0.297650	0.148825	−	0.988864i	$-0.452451\pi$
$641$	7.53590	0.297650	0.148825	+	0.988864i	$0.452451\pi$
$642$	− 0.928203i	− 0.0366333i
$643$	28.7846i	1.13515i	0.823320	+	0.567577i	$0.192119\pi$
$643$	28.7846i	1.13515i	−0.823320	+	0.567577i	$0.807881\pi$
$644$	− 10.3923i	− 0.409514i
$645$	6.00000i	0.236250i
$646$	9.07180	0.356925
$647$	−32.2679	−1.26858	−0.634292	−	0.773094i	$-0.718708\pi$
$647$	−32.2679	−1.26858	−0.634292	+	0.773094i	$0.718708\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−16.9282	−0.664490
$650$	0	0
$651$	−26.7846	−1.04977
$652$	10.9282i	0.427981i
$653$	29.5885	1.15789	0.578943	−	0.815368i	$-0.303465\pi$
$653$	29.5885	1.15789	0.578943	+	0.815368i	$0.303465\pi$
$654$	−10.3923	−0.406371
$655$	20.3205i	0.793988i
$656$	− 4.00000i	− 0.156174i
$657$	− 6.92820i	− 0.270295i
$658$	− 1.39230i	− 0.0542777i
$659$	15.7128	0.612084	0.306042	−	0.952018i	$-0.400995\pi$
$659$	15.7128	0.612084	0.306042	+	0.952018i	$0.400995\pi$
$660$	3.73205	0.145270
$661$	− 42.3923i	− 1.64887i	−0.565957	−	0.824435i	$-0.691493\pi$
$661$	− 42.3923i	− 1.64887i	0.565957	−	0.824435i	$-0.308507\pi$
$662$	−6.39230	−0.248444
$663$	0	0
$664$	−2.53590	−0.0984119
$665$	6.80385i	0.263842i
$666$	−7.92820	−0.307212
$667$	18.9282	0.732903
$668$	− 6.46410i	− 0.250104i
$669$	− 8.85641i	− 0.342408i
$670$	− 5.46410i	− 0.211097i
$671$	− 27.8564i	− 1.07538i
$672$	−3.00000	−0.115728
$673$	24.7846	0.955376	0.477688	−	0.878529i	$-0.341475\pi$
$673$	24.7846	0.955376	0.477688	+	0.878529i	$0.341475\pi$
$674$	− 5.60770i	− 0.216000i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−2.92820	−0.112540	−0.0562700	−	0.998416i	$-0.517921\pi$
$677$	−2.92820	−0.112540	−0.0562700	+	0.998416i	$0.517921\pi$
$678$	12.0000i	0.460857i
$679$	37.1769	1.42672
$680$	−4.00000	−0.153393
$681$	13.4641i	0.515945i
$682$	− 33.3205i	− 1.27591i
$683$	− 6.67949i	− 0.255584i	−0.991801	−	0.127792i	$-0.959211\pi$
$683$	− 6.67949i	− 0.255584i	0.991801	−	0.127792i	$-0.0407889\pi$
$684$	2.26795i	0.0867172i
$685$	−6.53590	−0.249724
$686$	15.0000	0.572703
$687$	11.4641i	0.437383i
$688$	6.00000	0.228748
$689$	0	0
$690$	3.46410	0.131876
$691$	− 23.0526i	− 0.876961i	−0.898741	−	0.438480i	$-0.855517\pi$
$691$	− 23.0526i	− 0.876961i	0.898741	−	0.438480i	$-0.144483\pi$
$692$	−22.1244	−0.841042
$693$	11.1962	0.425307
$694$	− 16.3923i	− 0.622243i
$695$	− 21.7846i	− 0.826337i
$696$	− 5.46410i	− 0.207116i
$697$	− 16.0000i	− 0.606043i
$698$	17.4641	0.661026
$699$	−18.0000	−0.680823
$700$	− 3.00000i	− 0.113389i
$701$	16.3923	0.619129	0.309564	−	0.950878i	$-0.399817\pi$
$701$	16.3923	0.619129	0.309564	+	0.950878i	$0.399817\pi$
$702$	0	0
$703$	17.9808	0.678157
$704$	− 3.73205i	− 0.140657i
$705$	0.464102	0.0174791
$706$	−28.3923	−1.06856
$707$	25.1769i	0.946875i
$708$	4.53590i	0.170470i
$709$	− 7.32051i	− 0.274927i	−0.990507	−	0.137464i	$-0.956105\pi$
$709$	− 7.32051i	− 0.274927i	0.990507	−	0.137464i	$-0.0438951\pi$
$710$	− 0.928203i	− 0.0348348i
$711$	16.9282	0.634857
$712$	10.1244	0.379426
$713$	− 30.9282i	− 1.15827i
$714$	−12.0000	−0.449089
$715$	0	0
$716$	−22.9282	−0.856867
$717$	3.46410i	0.129369i
$718$	12.9282	0.482476
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 58.7654i	− 2.18854i
$722$	13.8564i	0.515682i
$723$	− 14.8038i	− 0.550561i
$724$	−3.07180	−0.114162
$725$	5.46410	0.202932
$726$	2.92820i	0.108676i
$727$	12.6603	0.469543	0.234771	−	0.972051i	$-0.424566\pi$
$727$	12.6603	0.469543	0.234771	+	0.972051i	$0.424566\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 6.92820i	− 0.256424i
$731$	24.0000	0.887672
$732$	−7.46410	−0.275881
$733$	− 6.85641i	− 0.253247i	−0.991951	−	0.126624i	$-0.959586\pi$
$733$	− 6.85641i	− 0.253247i	0.991951	−	0.126624i	$-0.0404140\pi$
$734$	− 13.3205i	− 0.491669i
$735$	− 2.00000i	− 0.0737711i
$736$	− 3.46410i	− 0.127688i
$737$	20.3923	0.751160
$738$	4.00000	0.147242
$739$	− 43.8372i	− 1.61258i	−0.591523	−	0.806288i	$-0.701473\pi$
$739$	− 43.8372i	− 1.61258i	0.591523	−	0.806288i	$-0.298527\pi$
$740$	−7.92820	−0.291447
$741$	0	0
$742$	−11.1962	−0.411024
$743$	2.92820i	0.107425i	0.998556	+	0.0537127i	$0.0171055\pi$
$743$	2.92820i	0.107425i	−0.998556	+	0.0537127i	$0.982894\pi$
$744$	−8.92820	−0.327324
$745$	22.7846	0.834764
$746$	22.9282i	0.839461i
$747$	− 2.53590i	− 0.0927837i
$748$	− 14.9282i	− 0.545829i
$749$	− 2.78461i	− 0.101747i
$750$	1.00000	0.0365148
$751$	41.1769	1.50257	0.751283	−	0.659980i	$-0.229435\pi$
$751$	41.1769	1.50257	0.751283	+	0.659980i	$0.229435\pi$
$752$	− 0.464102i	− 0.0169240i
$753$	−26.4641	−0.964405
$754$	0	0
$755$	18.7846	0.683642
$756$	− 3.00000i	− 0.109109i
$757$	18.2679	0.663960	0.331980	−	0.943286i	$-0.392283\pi$
$757$	18.2679	0.663960	0.331980	+	0.943286i	$0.392283\pi$
$758$	6.26795	0.227662
$759$	12.9282i	0.469264i
$760$	2.26795i	0.0822672i
$761$	43.9808i	1.59430i	0.603780	+	0.797151i	$0.293660\pi$
$761$	43.9808i	1.59430i	−0.603780	+	0.797151i	$0.706340\pi$
$762$	− 4.66025i	− 0.168823i
$763$	−31.1769	−1.12868
$764$	−17.3205	−0.626634
$765$	− 4.00000i	− 0.144620i
$766$	−36.7846	−1.32908
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 39.7128i	− 1.43208i	−0.698059	−	0.716040i	$-0.745953\pi$
$769$	− 39.7128i	− 1.43208i	0.698059	−	0.716040i	$-0.254047\pi$
$770$	11.1962	0.403481
$771$	−21.4641	−0.773011
$772$	− 17.8564i	− 0.642666i
$773$	53.3923i	1.92039i	0.279334	+	0.960194i	$0.409886\pi$
$773$	53.3923i	1.92039i	−0.279334	+	0.960194i	$0.590114\pi$
$774$	6.00000i	0.215666i
$775$	− 8.92820i	− 0.320711i
$776$	12.3923	0.444858
$777$	−23.7846	−0.853268
$778$	− 9.85641i	− 0.353369i
$779$	−9.07180	−0.325031
$780$	0	0
$781$	3.46410	0.123955
$782$	− 13.8564i	− 0.495504i
$783$	5.46410	0.195271
$784$	−2.00000	−0.0714286
$785$	− 10.8038i	− 0.385606i
$786$	20.3205i	0.724809i
$787$	5.46410i	0.194774i	0.995247	+	0.0973871i	$0.0310485\pi$
$787$	5.46410i	0.194774i	−0.995247	+	0.0973871i	$0.968952\pi$
$788$	− 9.39230i	− 0.334587i
$789$	−20.5167	−0.730412
$790$	16.9282	0.602278
$791$	36.0000i	1.28001i
$792$	3.73205	0.132613
$793$	0	0
$794$	7.92820	0.281361
$795$	− 3.73205i	− 0.132362i
$796$	11.0718	0.392429
$797$	−37.8564	−1.34094	−0.670471	−	0.741935i	$-0.733908\pi$
$797$	−37.8564	−1.34094	−0.670471	+	0.741935i	$0.733908\pi$
$798$	6.80385i	0.240854i
$799$	− 1.85641i	− 0.0656749i
$800$	− 1.00000i	− 0.0353553i
$801$	10.1244i	0.357727i
$802$	−2.12436	−0.0750136
$803$	25.8564	0.912453
$804$	− 5.46410i	− 0.192704i
$805$	10.3923	0.366281
$806$	0	0
$807$	30.9282	1.08872
$808$	8.39230i	0.295240i
$809$	34.7846	1.22296	0.611481	−	0.791259i	$-0.290574\pi$
$809$	34.7846	1.22296	0.611481	+	0.791259i	$0.290574\pi$
$810$	1.00000	0.0351364
$811$	− 7.58846i	− 0.266467i	−0.991085	−	0.133233i	$-0.957464\pi$
$811$	− 7.58846i	− 0.266467i	0.991085	−	0.133233i	$-0.0425360\pi$
$812$	− 16.3923i	− 0.575257i
$813$	− 3.60770i	− 0.126527i
$814$	− 29.5885i	− 1.03707i
$815$	−10.9282	−0.382798
$816$	−4.00000	−0.140028
$817$	− 13.6077i	− 0.476073i
$818$	−0.947441	−0.0331265
$819$	0	0
$820$	4.00000	0.139686
$821$	− 49.3205i	− 1.72130i	−0.509199	−	0.860649i	$-0.670058\pi$
$821$	− 49.3205i	− 1.72130i	0.509199	−	0.860649i	$-0.329942\pi$
$822$	−6.53590	−0.227966
$823$	−39.5885	−1.37997	−0.689983	−	0.723825i	$-0.742382\pi$
$823$	−39.5885	−1.37997	−0.689983	+	0.723825i	$0.742382\pi$
$824$	− 19.5885i	− 0.682396i
$825$	3.73205i	0.129933i
$826$	13.6077i	0.473472i
$827$	− 50.4974i	− 1.75597i	−0.478690	−	0.877984i	$-0.658888\pi$
$827$	− 50.4974i	− 1.75597i	0.478690	−	0.877984i	$-0.341112\pi$
$828$	3.46410	0.120386
$829$	−39.3205	−1.36566	−0.682829	−	0.730578i	$-0.739251\pi$
$829$	−39.3205	−1.36566	−0.682829	+	0.730578i	$0.739251\pi$
$830$	− 2.53590i	− 0.0880223i
$831$	19.5885	0.679516
$832$	0	0
$833$	−8.00000	−0.277184
$834$	− 21.7846i	− 0.754339i
$835$	6.46410	0.223699
$836$	−8.46410	−0.292737
$837$	− 8.92820i	− 0.308604i
$838$	− 17.8564i	− 0.616839i
$839$	43.1769i	1.49063i	0.666711	+	0.745316i	$0.267702\pi$
$839$	43.1769i	1.49063i	−0.666711	+	0.745316i	$0.732298\pi$
$840$	− 3.00000i	− 0.103510i
$841$	0.856406	0.0295313
$842$	5.85641	0.201825
$843$	6.92820i	0.238620i
$844$	21.9282	0.754800
$845$	0	0
$846$	0.464102	0.0159561
$847$	8.78461i	0.301843i
$848$	−3.73205	−0.128159
$849$	−14.3923	−0.493943
$850$	− 4.00000i	− 0.137199i
$851$	− 27.4641i	− 0.941457i
$852$	− 0.928203i	− 0.0317997i
$853$	44.6410i	1.52848i	0.644932	+	0.764240i	$0.276886\pi$
$853$	44.6410i	1.52848i	−0.644932	+	0.764240i	$0.723114\pi$
$854$	−22.3923	−0.766249
$855$	−2.26795	−0.0775622
$856$	− 0.928203i	− 0.0317253i
$857$	−15.0718	−0.514843	−0.257421	−	0.966299i	$-0.582873\pi$
$857$	−15.0718	−0.514843	−0.257421	+	0.966299i	$0.582873\pi$
$858$	0	0
$859$	−19.9282	−0.679942	−0.339971	−	0.940436i	$-0.610417\pi$
$859$	−19.9282	−0.679942	−0.339971	+	0.940436i	$0.610417\pi$
$860$	6.00000i	0.204598i
$861$	12.0000	0.408959
$862$	−13.8564	−0.471951
$863$	− 10.9282i	− 0.372000i	−0.982550	−	0.186000i	$-0.940447\pi$
$863$	− 10.9282i	− 0.372000i	0.982550	−	0.186000i	$-0.0595525\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	− 22.1244i	− 0.752251i
$866$	32.7846i	1.11407i
$867$	1.00000	0.0339618
$868$	−26.7846	−0.909129
$869$	63.1769i	2.14313i
$870$	5.46410	0.185250
$871$	0	0
$872$	−10.3923	−0.351928
$873$	12.3923i	0.419416i
$874$	−7.85641	−0.265747
$875$	3.00000	0.101419
$876$	− 6.92820i	− 0.234082i
$877$	33.7128i	1.13840i	0.822199	+	0.569200i	$0.192747\pi$
$877$	33.7128i	1.13840i	−0.822199	+	0.569200i	$0.807253\pi$
$878$	21.3205i	0.719532i
$879$	− 19.2487i	− 0.649243i
$880$	3.73205	0.125807
$881$	−37.1051	−1.25010	−0.625052	−	0.780583i	$-0.714922\pi$
$881$	−37.1051	−1.25010	−0.625052	+	0.780583i	$0.714922\pi$
$882$	− 2.00000i	− 0.0673435i
$883$	21.3205	0.717492	0.358746	−	0.933435i	$-0.383204\pi$
$883$	21.3205	0.717492	0.358746	+	0.933435i	$0.383204\pi$
$884$	0	0
$885$	−4.53590	−0.152473
$886$	7.85641i	0.263941i
$887$	52.2679	1.75499	0.877493	−	0.479589i	$-0.159214\pi$
$887$	52.2679	1.75499	0.877493	+	0.479589i	$0.159214\pi$
$888$	−7.92820	−0.266053
$889$	− 13.9808i	− 0.468900i
$890$	10.1244i	0.339369i
$891$	3.73205i	0.125028i
$892$	− 8.85641i	− 0.296534i
$893$	−1.05256	−0.0352225
$894$	22.7846	0.762031
$895$	− 22.9282i	− 0.766405i
$896$	−3.00000	−0.100223
$897$	0	0
$898$	18.1244	0.604818
$899$	− 48.7846i	− 1.62706i
$900$	1.00000	0.0333333
$901$	−14.9282	−0.497331
$902$	14.9282i	0.497055i
$903$	18.0000i	0.599002i
$904$	12.0000i	0.399114i
$905$	− 3.07180i	− 0.102110i
$906$	18.7846	0.624077
$907$	7.07180	0.234815	0.117408	−	0.993084i	$-0.462542\pi$
$907$	7.07180	0.234815	0.117408	+	0.993084i	$0.462542\pi$
$908$	13.4641i	0.446822i
$909$	−8.39230	−0.278355
$910$	0	0
$911$	−28.7846	−0.953677	−0.476838	−	0.878991i	$-0.658217\pi$
$911$	−28.7846	−0.953677	−0.476838	+	0.878991i	$0.658217\pi$
$912$	2.26795i	0.0750993i
$913$	9.46410	0.313216
$914$	0.535898	0.0177259
$915$	− 7.46410i	− 0.246756i
$916$	11.4641i	0.378785i
$917$	60.9615i	2.01313i
$918$	− 4.00000i	− 0.132020i
$919$	55.9615	1.84600	0.923000	−	0.384800i	$-0.125729\pi$
$919$	55.9615	1.84600	0.923000	+	0.384800i	$0.125729\pi$
$920$	3.46410	0.114208
$921$	− 20.2487i	− 0.667218i
$922$	−32.3923	−1.06678
$923$	0	0
$924$	11.1962	0.368326
$925$	− 7.92820i	− 0.260678i
$926$	−0.784610	−0.0257839
$927$	19.5885	0.643369
$928$	− 5.46410i	− 0.179368i
$929$	− 30.1436i	− 0.988979i	−0.869184	−	0.494490i	$-0.835355\pi$
$929$	− 30.1436i	− 0.988979i	0.869184	−	0.494490i	$-0.164645\pi$
$930$	− 8.92820i	− 0.292767i
$931$	4.53590i	0.148658i
$932$	−18.0000	−0.589610
$933$	−5.07180	−0.166043
$934$	3.60770i	0.118047i
$935$	14.9282	0.488204
$936$	0	0
$937$	33.7128	1.10135	0.550675	−	0.834720i	$-0.314370\pi$
$937$	33.7128	1.10135	0.550675	+	0.834720i	$0.314370\pi$
$938$	− 16.3923i	− 0.535228i
$939$	−1.32051	−0.0430932
$940$	0.464102	0.0151373
$941$	36.4974i	1.18978i	0.803806	+	0.594891i	$0.202805\pi$
$941$	36.4974i	1.18978i	−0.803806	+	0.594891i	$0.797195\pi$
$942$	− 10.8038i	− 0.352008i
$943$	13.8564i	0.451227i
$944$	4.53590i	0.147631i
$945$	3.00000	0.0975900
$946$	−22.3923	−0.728037
$947$	13.6077i	0.442191i	0.975252	+	0.221095i	$0.0709632\pi$
$947$	13.6077i	0.442191i	−0.975252	+	0.221095i	$0.929037\pi$
$948$	16.9282	0.549802
$949$	0	0
$950$	−2.26795	−0.0735820
$951$	− 23.5359i	− 0.763204i
$952$	−12.0000	−0.388922
$953$	−18.6795	−0.605088	−0.302544	−	0.953135i	$-0.597836\pi$
$953$	−18.6795	−0.605088	−0.302544	+	0.953135i	$0.597836\pi$
$954$	− 3.73205i	− 0.120830i
$955$	− 17.3205i	− 0.560478i
$956$	3.46410i	0.112037i
$957$	20.3923i	0.659190i
$958$	−26.2487	−0.848057
$959$	−19.6077	−0.633165
$960$	− 1.00000i	− 0.0322749i
$961$	−48.7128	−1.57138
$962$	0	0
$963$	0.928203	0.0299109
$964$	− 14.8038i	− 0.476800i
$965$	17.8564	0.574818
$966$	10.3923	0.334367
$967$	− 28.8564i	− 0.927959i	−0.885846	−	0.463980i	$-0.846421\pi$
$967$	− 28.8564i	− 0.927959i	0.885846	−	0.463980i	$-0.153579\pi$
$968$	2.92820i	0.0941160i
$969$	9.07180i	0.291428i
$970$	12.3923i	0.397893i
$971$	−20.6077	−0.661332	−0.330666	−	0.943748i	$-0.607273\pi$
$971$	−20.6077	−0.661332	−0.330666	+	0.943748i	$0.607273\pi$
$972$	1.00000	0.0320750
$973$	− 65.3538i	− 2.09515i
$974$	21.0000	0.672883
$975$	0	0
$976$	−7.46410	−0.238920
$977$	1.46410i	0.0468408i	0.999726	+	0.0234204i	$0.00745562\pi$
$977$	1.46410i	0.0468408i	−0.999726	+	0.0234204i	$0.992544\pi$
$978$	−10.9282	−0.349445
$979$	−37.7846	−1.20760
$980$	− 2.00000i	− 0.0638877i
$981$	− 10.3923i	− 0.331801i
$982$	− 15.3923i	− 0.491188i
$983$	− 16.1769i	− 0.515963i	−0.966150	−	0.257982i	$-0.916943\pi$
$983$	− 16.1769i	− 0.515963i	0.966150	−	0.257982i	$-0.0830574\pi$
$984$	4.00000	0.127515
$985$	9.39230	0.299264
$986$	− 21.8564i	− 0.696050i
$987$	1.39230	0.0443176
$988$	0	0
$989$	−20.7846	−0.660912
$990$	3.73205i	0.118612i
$991$	−23.1769	−0.736239	−0.368119	−	0.929778i	$-0.619998\pi$
$991$	−23.1769	−0.736239	−0.368119	+	0.929778i	$0.619998\pi$
$992$	−8.92820	−0.283471
$993$	− 6.39230i	− 0.202854i
$994$	− 2.78461i	− 0.0883225i
$995$	11.0718i	0.351000i
$996$	− 2.53590i	− 0.0803530i
$997$	10.0192	0.317312	0.158656	−	0.987334i	$-0.449284\pi$
$997$	10.0192	0.317312	0.158656	+	0.987334i	$0.449284\pi$
$998$	1.32051	0.0418000
$999$	− 7.92820i	− 0.250837i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.p.1351.4		4
13.3	even	3		390.2.bb.a.121.1	✓	4
13.4	even	6		390.2.bb.a.361.1	yes	4
13.5	odd	4		5070.2.a.be.1.1		2
13.8	odd	4		5070.2.a.ba.1.2		2
13.12	even	2	inner	5070.2.b.p.1351.1		4
39.17	odd	6		1170.2.bs.d.361.2		4
39.29	odd	6		1170.2.bs.d.901.2		4
65.3	odd	12		1950.2.y.d.199.2		4
65.4	even	6		1950.2.bc.a.751.2		4
65.17	odd	12		1950.2.y.d.49.2		4
65.29	even	6		1950.2.bc.a.901.2		4
65.42	odd	12		1950.2.y.e.199.1		4
65.43	odd	12		1950.2.y.e.49.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.a.121.1	✓	4	13.3	even	3
390.2.bb.a.361.1	yes	4	13.4	even	6
1170.2.bs.d.361.2		4	39.17	odd	6
1170.2.bs.d.901.2		4	39.29	odd	6
1950.2.y.d.49.2		4	65.17	odd	12
1950.2.y.d.199.2		4	65.3	odd	12
1950.2.y.e.49.1		4	65.43	odd	12
1950.2.y.e.199.1		4	65.42	odd	12
1950.2.bc.a.751.2		4	65.4	even	6
1950.2.bc.a.901.2		4	65.29	even	6
5070.2.a.ba.1.2		2	13.8	odd	4
5070.2.a.be.1.1		2	13.5	odd	4
5070.2.b.p.1351.1		4	13.12	even	2	inner
5070.2.b.p.1351.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -3.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -3.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.73205i q^{11} +1.00000 q^{12} +3.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000i q^{18} -2.26795i q^{19} +1.00000i q^{20} +3.00000i q^{21} -3.73205 q^{22} -3.46410 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +3.00000i q^{28} -5.46410 q^{29} -1.00000 q^{30} +8.92820i q^{31} +1.00000i q^{32} -3.73205i q^{33} +4.00000i q^{34} -3.00000 q^{35} -1.00000 q^{36} +7.92820i q^{37} +2.26795 q^{38} -1.00000 q^{40} -4.00000i q^{41} -3.00000 q^{42} +6.00000 q^{43} -3.73205i q^{44} -1.00000i q^{45} -3.46410i q^{46} -0.464102i q^{47} -1.00000 q^{48} -2.00000 q^{49} -1.00000i q^{50} -4.00000 q^{51} -3.73205 q^{53} -1.00000i q^{54} +3.73205 q^{55} -3.00000 q^{56} +2.26795i q^{57} -5.46410i q^{58} +4.53590i q^{59} -1.00000i q^{60} -7.46410 q^{61} -8.92820 q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.73205 q^{66} -5.46410i q^{67} -4.00000 q^{68} +3.46410 q^{69} -3.00000i q^{70} -0.928203i q^{71} -1.00000i q^{72} -6.92820i q^{73} -7.92820 q^{74} +1.00000 q^{75} +2.26795i q^{76} +11.1962 q^{77} +16.9282 q^{79} -1.00000i q^{80} +1.00000 q^{81} +4.00000 q^{82} -2.53590i q^{83} -3.00000i q^{84} -4.00000i q^{85} +6.00000i q^{86} +5.46410 q^{87} +3.73205 q^{88} +10.1244i q^{89} +1.00000 q^{90} +3.46410 q^{92} -8.92820i q^{93} +0.464102 q^{94} -2.26795 q^{95} -1.00000i q^{96} +12.3923i q^{97} -2.00000i q^{98} +3.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} + 4 q^{12} + 12 q^{14} + 4 q^{16} + 16 q^{17} - 8 q^{22} - 4 q^{25} - 4 q^{27} - 8 q^{29} - 4 q^{30} - 12 q^{35} - 4 q^{36} + 16 q^{38} - 4 q^{40} - 12 q^{42} + 24 q^{43} - 4 q^{48} - 8 q^{49} - 16 q^{51} - 8 q^{53} + 8 q^{55} - 12 q^{56} - 16 q^{61} - 8 q^{62} - 4 q^{64} + 8 q^{66} - 16 q^{68} - 4 q^{74} + 4 q^{75} + 24 q^{77} + 40 q^{79} + 4 q^{81} + 16 q^{82} + 8 q^{87} + 8 q^{88} + 4 q^{90} - 12 q^{94} - 16 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 + 4 * q^10 + 4 * q^12 + 12 * q^14 + 4 * q^16 + 16 * q^17 - 8 * q^22 - 4 * q^25 - 4 * q^27 - 8 * q^29 - 4 * q^30 - 12 * q^35 - 4 * q^36 + 16 * q^38 - 4 * q^40 - 12 * q^42 + 24 * q^43 - 4 * q^48 - 8 * q^49 - 16 * q^51 - 8 * q^53 + 8 * q^55 - 12 * q^56 - 16 * q^61 - 8 * q^62 - 4 * q^64 + 8 * q^66 - 16 * q^68 - 4 * q^74 + 4 * q^75 + 24 * q^77 + 40 * q^79 + 4 * q^81 + 16 * q^82 + 8 * q^87 + 8 * q^88 + 4 * q^90 - 12 * q^94 - 16 * q^95

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