LMFDB - Embedded newform 5070.2.b.p.1351.2

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			1351.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.p.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -0.267949i q^{11} +1.00000 q^{12} +3.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000i q^{18} +5.73205i q^{19} -1.00000i q^{20} -3.00000i q^{21} -0.267949 q^{22} +3.46410 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -3.00000i q^{28} +1.46410 q^{29} -1.00000 q^{30} +4.92820i q^{31} -1.00000i q^{32} +0.267949i q^{33} -4.00000i q^{34} -3.00000 q^{35} -1.00000 q^{36} +5.92820i q^{37} +5.73205 q^{38} -1.00000 q^{40} +4.00000i q^{41} -3.00000 q^{42} +6.00000 q^{43} +0.267949i q^{44} +1.00000i q^{45} -3.46410i q^{46} -6.46410i q^{47} -1.00000 q^{48} -2.00000 q^{49} +1.00000i q^{50} -4.00000 q^{51} -0.267949 q^{53} +1.00000i q^{54} +0.267949 q^{55} -3.00000 q^{56} -5.73205i q^{57} -1.46410i q^{58} -11.4641i q^{59} +1.00000i q^{60} -0.535898 q^{61} +4.92820 q^{62} +3.00000i q^{63} -1.00000 q^{64} +0.267949 q^{66} -1.46410i q^{67} -4.00000 q^{68} -3.46410 q^{69} +3.00000i q^{70} -12.9282i q^{71} +1.00000i q^{72} -6.92820i q^{73} +5.92820 q^{74} +1.00000 q^{75} -5.73205i q^{76} +0.803848 q^{77} +3.07180 q^{79} +1.00000i q^{80} +1.00000 q^{81} +4.00000 q^{82} +9.46410i q^{83} +3.00000i q^{84} +4.00000i q^{85} -6.00000i q^{86} -1.46410 q^{87} +0.267949 q^{88} +14.1244i q^{89} +1.00000 q^{90} -3.46410 q^{92} -4.92820i q^{93} -6.46410 q^{94} -5.73205 q^{95} +1.00000i q^{96} +8.39230i q^{97} +2.00000i q^{98} -0.267949i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	− 0.267949i	− 0.0807897i	−0.999184	−	0.0403949i	$-0.987138\pi$
$11$	− 0.267949i	− 0.0807897i	0.999184	−	0.0403949i	$-0.0128616\pi$
$12$	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
$19$	5.73205i	1.31502i	0.753445	+	0.657511i	$0.228391\pi$
$19$	5.73205i	1.31502i	−0.753445	+	0.657511i	$0.771609\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 3.00000i	− 0.654654i
$22$	−0.267949	−0.0571270
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 3.00000i	− 0.566947i
$29$	1.46410	0.271877	0.135938	−	0.990717i	$-0.456595\pi$
$29$	1.46410	0.271877	0.135938	+	0.990717i	$0.456595\pi$
$30$	−1.00000	−0.182574
$31$	4.92820i	0.885131i	0.896736	+	0.442566i	$0.145932\pi$
$31$	4.92820i	0.885131i	−0.896736	+	0.442566i	$0.854068\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0.267949i	0.0466440i
$34$	− 4.00000i	− 0.685994i
$35$	−3.00000	−0.507093
$36$	−1.00000	−0.166667
$37$	5.92820i	0.974591i	0.873237	+	0.487295i	$0.162016\pi$
$37$	5.92820i	0.974591i	−0.873237	+	0.487295i	$0.837984\pi$
$38$	5.73205	0.929861
$39$	0	0
$40$	−1.00000	−0.158114
$41$	4.00000i	0.624695i	0.949968	+	0.312348i	$0.101115\pi$
$41$	4.00000i	0.624695i	−0.949968	+	0.312348i	$0.898885\pi$
$42$	−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	$0.348747\pi$
$44$	0.267949i	0.0403949i
$45$	1.00000i	0.149071i
$46$	− 3.46410i	− 0.510754i
$47$	− 6.46410i	− 0.942886i	−0.881897	−	0.471443i	$-0.843733\pi$
$47$	− 6.46410i	− 0.942886i	0.881897	−	0.471443i	$-0.156267\pi$
$48$	−1.00000	−0.144338
$49$	−2.00000	−0.285714
$50$	1.00000i	0.141421i
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−0.267949	−0.0368057	−0.0184028	−	0.999831i	$-0.505858\pi$
$53$	−0.267949	−0.0368057	−0.0184028	+	0.999831i	$0.505858\pi$
$54$	1.00000i	0.136083i
$55$	0.267949	0.0361303
$56$	−3.00000	−0.400892
$57$	− 5.73205i	− 0.759229i
$58$	− 1.46410i	− 0.192246i
$59$	− 11.4641i	− 1.49250i	−0.665666	−	0.746249i	$-0.731853\pi$
$59$	− 11.4641i	− 1.49250i	0.665666	−	0.746249i	$-0.268147\pi$
$60$	1.00000i	0.129099i
$61$	−0.535898	−0.0686148	−0.0343074	−	0.999411i	$-0.510923\pi$
$61$	−0.535898	−0.0686148	−0.0343074	+	0.999411i	$0.510923\pi$
$62$	4.92820	0.625882
$63$	3.00000i	0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0.267949	0.0329823
$67$	− 1.46410i	− 0.178868i	−0.995993	−	0.0894342i	$-0.971494\pi$
$67$	− 1.46410i	− 0.178868i	0.995993	−	0.0894342i	$-0.0285059\pi$
$68$	−4.00000	−0.485071
$69$	−3.46410	−0.417029
$70$	3.00000i	0.358569i
$71$	− 12.9282i	− 1.53430i	−0.641470	−	0.767148i	$-0.721675\pi$
$71$	− 12.9282i	− 1.53430i	0.641470	−	0.767148i	$-0.278325\pi$
$72$	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	$-0.132883\pi$
$74$	5.92820	0.689140
$75$	1.00000	0.115470
$76$	− 5.73205i	− 0.657511i
$77$	0.803848	0.0916069
$78$	0	0
$79$	3.07180	0.345604	0.172802	−	0.984957i	$-0.444718\pi$
$79$	3.07180	0.345604	0.172802	+	0.984957i	$0.444718\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	9.46410i	1.03882i	0.854525	+	0.519410i	$0.173848\pi$
$83$	9.46410i	1.03882i	−0.854525	+	0.519410i	$0.826152\pi$
$84$	3.00000i	0.327327i
$85$	4.00000i	0.433861i
$86$	− 6.00000i	− 0.646997i
$87$	−1.46410	−0.156968
$88$	0.267949	0.0285635
$89$	14.1244i	1.49718i	0.663034	+	0.748589i	$0.269269\pi$
$89$	14.1244i	1.49718i	−0.663034	+	0.748589i	$0.730731\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−3.46410	−0.361158
$93$	− 4.92820i	− 0.511031i
$94$	−6.46410	−0.666721
$95$	−5.73205	−0.588096
$96$	1.00000i	0.102062i
$97$	8.39230i	0.852109i	0.904697	+	0.426055i	$0.140097\pi$
$97$	8.39230i	0.852109i	−0.904697	+	0.426055i	$0.859903\pi$
$98$	2.00000i	0.202031i
$99$	− 0.267949i	− 0.0269299i
$100$	1.00000	0.100000
$101$	12.3923	1.23308	0.616540	−	0.787323i	$-0.288534\pi$
$101$	12.3923	1.23308	0.616540	+	0.787323i	$0.288534\pi$
$102$	4.00000i	0.396059i
$103$	−11.5885	−1.14184	−0.570922	−	0.821004i	$-0.693414\pi$
$103$	−11.5885	−1.14184	−0.570922	+	0.821004i	$0.693414\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0.267949i	0.0260255i
$107$	−12.9282	−1.24982	−0.624908	−	0.780698i	$-0.714864\pi$
$107$	−12.9282	−1.24982	−0.624908	+	0.780698i	$0.714864\pi$
$108$	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	$-0.165822\pi$
$110$	− 0.267949i	− 0.0255480i
$111$	− 5.92820i	− 0.562680i
$112$	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	$0.690905\pi$
$114$	−5.73205	−0.536856
$115$	3.46410i	0.323029i
$116$	−1.46410	−0.135938
$117$	0	0
$118$	−11.4641	−1.05536
$119$	12.0000i	1.10004i
$120$	1.00000	0.0912871
$121$	10.9282	0.993473
$122$	0.535898i	0.0485180i
$123$	− 4.00000i	− 0.360668i
$124$	− 4.92820i	− 0.442566i
$125$	− 1.00000i	− 0.0894427i
$126$	3.00000	0.267261
$127$	−12.6603	−1.12342	−0.561708	−	0.827336i	$-0.689856\pi$
$127$	−12.6603	−1.12342	−0.561708	+	0.827336i	$0.689856\pi$
$128$	1.00000i	0.0883883i
$129$	−6.00000	−0.528271
$130$	0	0
$131$	14.3205	1.25119	0.625594	−	0.780149i	$-0.284857\pi$
$131$	14.3205	1.25119	0.625594	+	0.780149i	$0.284857\pi$
$132$	− 0.267949i	− 0.0233220i
$133$	−17.1962	−1.49110
$134$	−1.46410	−0.126479
$135$	− 1.00000i	− 0.0860663i
$136$	4.00000i	0.342997i
$137$	13.4641i	1.15032i	0.818042	+	0.575158i	$0.195059\pi$
$137$	13.4641i	1.15032i	−0.818042	+	0.575158i	$0.804941\pi$
$138$	3.46410i	0.294884i
$139$	−19.7846	−1.67811	−0.839054	−	0.544048i	$-0.816891\pi$
$139$	−19.7846	−1.67811	−0.839054	+	0.544048i	$0.816891\pi$
$140$	3.00000	0.253546
$141$	6.46410i	0.544376i
$142$	−12.9282	−1.08491
$143$	0	0
$144$	1.00000	0.0833333
$145$	1.46410i	0.121587i
$146$	−6.92820	−0.573382
$147$	2.00000	0.164957
$148$	− 5.92820i	− 0.487295i
$149$	18.7846i	1.53890i	0.638710	+	0.769448i	$0.279468\pi$
$149$	18.7846i	1.53890i	−0.638710	+	0.769448i	$0.720532\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	22.7846i	1.85419i	0.374832	+	0.927093i	$0.377700\pi$
$151$	22.7846i	1.85419i	−0.374832	+	0.927093i	$0.622300\pi$
$152$	−5.73205	−0.464931
$153$	4.00000	0.323381
$154$	− 0.803848i	− 0.0647759i
$155$	−4.92820	−0.395843
$156$	0	0
$157$	21.1962	1.69164	0.845819	−	0.533471i	$-0.179113\pi$
$157$	21.1962	1.69164	0.845819	+	0.533471i	$0.179113\pi$
$158$	− 3.07180i	− 0.244379i
$159$	0.267949	0.0212498
$160$	1.00000	0.0790569
$161$	10.3923i	0.819028i
$162$	− 1.00000i	− 0.0785674i
$163$	− 2.92820i	− 0.229355i	−0.993403	−	0.114677i	$-0.963417\pi$
$163$	− 2.92820i	− 0.229355i	0.993403	−	0.114677i	$-0.0365834\pi$
$164$	− 4.00000i	− 0.312348i
$165$	−0.267949	−0.0208598
$166$	9.46410	0.734557
$167$	0.464102i	0.0359133i	0.999839	+	0.0179566i	$0.00571608\pi$
$167$	0.464102i	0.0359133i	−0.999839	+	0.0179566i	$0.994284\pi$
$168$	3.00000	0.231455
$169$	0	0
$170$	4.00000	0.306786
$171$	5.73205i	0.438341i
$172$	−6.00000	−0.457496
$173$	−2.12436	−0.161512	−0.0807559	−	0.996734i	$-0.525733\pi$
$173$	−2.12436	−0.161512	−0.0807559	+	0.996734i	$0.525733\pi$
$174$	1.46410i	0.110993i
$175$	− 3.00000i	− 0.226779i
$176$	− 0.267949i	− 0.0201974i
$177$	11.4641i	0.861695i
$178$	14.1244	1.05867
$179$	9.07180	0.678058	0.339029	−	0.940776i	$-0.389902\pi$
$179$	9.07180	0.678058	0.339029	+	0.940776i	$0.389902\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	16.9282	1.25826	0.629132	−	0.777299i	$-0.283411\pi$
$181$	16.9282	1.25826	0.629132	+	0.777299i	$0.283411\pi$
$182$	0	0
$183$	0.535898	0.0396147
$184$	3.46410i	0.255377i
$185$	−5.92820	−0.435850
$186$	−4.92820	−0.361353
$187$	− 1.07180i	− 0.0783775i
$188$	6.46410i	0.471443i
$189$	− 3.00000i	− 0.218218i
$190$	5.73205i	0.415847i
$191$	−17.3205	−1.25327	−0.626634	−	0.779314i	$-0.715568\pi$
$191$	−17.3205	−1.25327	−0.626634	+	0.779314i	$0.715568\pi$
$192$	1.00000	0.0721688
$193$	9.85641i	0.709480i	0.934965	+	0.354740i	$0.115431\pi$
$193$	9.85641i	0.709480i	−0.934965	+	0.354740i	$0.884569\pi$
$194$	8.39230	0.602532
$195$	0	0
$196$	2.00000	0.142857
$197$	11.3923i	0.811668i	0.913947	+	0.405834i	$0.133019\pi$
$197$	11.3923i	0.811668i	−0.913947	+	0.405834i	$0.866981\pi$
$198$	−0.267949	−0.0190423
$199$	−24.9282	−1.76711	−0.883557	−	0.468324i	$-0.844858\pi$
$199$	−24.9282	−1.76711	−0.883557	+	0.468324i	$0.844858\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	1.46410i	0.103270i
$202$	− 12.3923i	− 0.871920i
$203$	4.39230i	0.308279i
$204$	4.00000	0.280056
$205$	−4.00000	−0.279372
$206$	11.5885i	0.807406i
$207$	3.46410	0.240772
$208$	0	0
$209$	1.53590	0.106240
$210$	− 3.00000i	− 0.207020i
$211$	−8.07180	−0.555685	−0.277843	−	0.960627i	$-0.589619\pi$
$211$	−8.07180	−0.555685	−0.277843	+	0.960627i	$0.589619\pi$
$212$	0.267949	0.0184028
$213$	12.9282i	0.885826i
$214$	12.9282i	0.883754i
$215$	6.00000i	0.409197i
$216$	− 1.00000i	− 0.0680414i
$217$	−14.7846	−1.00364
$218$	−10.3923	−0.703856
$219$	6.92820i	0.468165i
$220$	−0.267949	−0.0180651
$221$	0	0
$222$	−5.92820	−0.397875
$223$	18.8564i	1.26272i	0.775491	+	0.631359i	$0.217503\pi$
$223$	18.8564i	1.26272i	−0.775491	+	0.631359i	$0.782497\pi$
$224$	3.00000	0.200446
$225$	−1.00000	−0.0666667
$226$	12.0000i	0.798228i
$227$	6.53590i	0.433803i	0.976194	+	0.216901i	$0.0695950\pi$
$227$	6.53590i	0.433803i	−0.976194	+	0.216901i	$0.930405\pi$
$228$	5.73205i	0.379614i
$229$	4.53590i	0.299741i	0.988706	+	0.149870i	$0.0478856\pi$
$229$	4.53590i	0.299741i	−0.988706	+	0.149870i	$0.952114\pi$
$230$	3.46410	0.228416
$231$	−0.803848	−0.0528893
$232$	1.46410i	0.0961230i
$233$	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
$235$	6.46410	0.421671
$236$	11.4641i	0.746249i
$237$	−3.07180	−0.199535
$238$	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
$241$	− 25.1962i	− 1.62303i	−0.584334	−	0.811513i	$-0.698644\pi$
$241$	− 25.1962i	− 1.62303i	0.584334	−	0.811513i	$-0.301356\pi$
$242$	− 10.9282i	− 0.702492i
$243$	−1.00000	−0.0641500
$244$	0.535898	0.0343074
$245$	− 2.00000i	− 0.127775i
$246$	−4.00000	−0.255031
$247$	0	0
$248$	−4.92820	−0.312941
$249$	− 9.46410i	− 0.599763i
$250$	−1.00000	−0.0632456
$251$	19.5359	1.23309	0.616547	−	0.787318i	$-0.288531\pi$
$251$	19.5359	1.23309	0.616547	+	0.787318i	$0.288531\pi$
$252$	− 3.00000i	− 0.188982i
$253$	− 0.928203i	− 0.0583556i
$254$	12.6603i	0.794375i
$255$	− 4.00000i	− 0.250490i
$256$	1.00000	0.0625000
$257$	14.5359	0.906724	0.453362	−	0.891326i	$-0.350224\pi$
$257$	14.5359	0.906724	0.453362	+	0.891326i	$0.350224\pi$
$258$	6.00000i	0.373544i
$259$	−17.7846	−1.10508
$260$	0	0
$261$	1.46410	0.0906256
$262$	− 14.3205i	− 0.884724i
$263$	−24.5167	−1.51176	−0.755881	−	0.654709i	$-0.772791\pi$
$263$	−24.5167	−1.51176	−0.755881	+	0.654709i	$0.772791\pi$
$264$	−0.267949	−0.0164911
$265$	− 0.267949i	− 0.0164600i
$266$	17.1962i	1.05436i
$267$	− 14.1244i	− 0.864397i
$268$	1.46410i	0.0894342i
$269$	−17.0718	−1.04089	−0.520443	−	0.853896i	$-0.674233\pi$
$269$	−17.0718	−1.04089	−0.520443	+	0.853896i	$0.674233\pi$
$270$	−1.00000	−0.0608581
$271$	− 24.3923i	− 1.48173i	−0.671656	−	0.740863i	$-0.734417\pi$
$271$	− 24.3923i	− 1.48173i	0.671656	−	0.740863i	$-0.265583\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	13.4641	0.813396
$275$	0.267949i	0.0161579i
$276$	3.46410	0.208514
$277$	11.5885	0.696283	0.348141	−	0.937442i	$-0.386813\pi$
$277$	11.5885	0.696283	0.348141	+	0.937442i	$0.386813\pi$
$278$	19.7846i	1.18660i
$279$	4.92820i	0.295044i
$280$	− 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	$-0.0662565\pi$
$282$	6.46410	0.384932
$283$	−6.39230	−0.379983	−0.189992	−	0.981786i	$-0.560846\pi$
$283$	−6.39230	−0.379983	−0.189992	+	0.981786i	$0.560846\pi$
$284$	12.9282i	0.767148i
$285$	5.73205	0.339537
$286$	0	0
$287$	−12.0000	−0.708338
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	1.46410	0.0859750
$291$	− 8.39230i	− 0.491966i
$292$	6.92820i	0.405442i
$293$	29.2487i	1.70873i	0.519675	+	0.854364i	$0.326053\pi$
$293$	29.2487i	1.70873i	−0.519675	+	0.854364i	$0.673947\pi$
$294$	− 2.00000i	− 0.116642i
$295$	11.4641	0.667466
$296$	−5.92820	−0.344570
$297$	0.267949i	0.0155480i
$298$	18.7846	1.08816
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	18.0000i	1.03750i
$302$	22.7846	1.31111
$303$	−12.3923	−0.711919
$304$	5.73205i	0.328756i
$305$	− 0.535898i	− 0.0306855i
$306$	− 4.00000i	− 0.228665i
$307$	28.2487i	1.61224i	0.591753	+	0.806120i	$0.298436\pi$
$307$	28.2487i	1.61224i	−0.591753	+	0.806120i	$0.701564\pi$
$308$	−0.803848	−0.0458035
$309$	11.5885	0.659244
$310$	4.92820i	0.279903i
$311$	18.9282	1.07332	0.536660	−	0.843799i	$-0.319686\pi$
$311$	18.9282	1.07332	0.536660	+	0.843799i	$0.319686\pi$
$312$	0	0
$313$	−33.3205	−1.88339	−0.941693	−	0.336473i	$-0.890766\pi$
$313$	−33.3205	−1.88339	−0.941693	+	0.336473i	$0.890766\pi$
$314$	− 21.1962i	− 1.19617i
$315$	−3.00000	−0.169031
$316$	−3.07180	−0.172802
$317$	− 30.4641i	− 1.71103i	−0.517774	−	0.855517i	$-0.673239\pi$
$317$	− 30.4641i	− 1.71103i	0.517774	−	0.855517i	$-0.326761\pi$
$318$	− 0.267949i	− 0.0150258i
$319$	− 0.392305i	− 0.0219649i
$320$	− 1.00000i	− 0.0559017i
$321$	12.9282	0.721582
$322$	10.3923	0.579141
$323$	22.9282i	1.27576i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−2.92820	−0.162178
$327$	10.3923i	0.574696i
$328$	−4.00000	−0.220863
$329$	19.3923	1.06913
$330$	0.267949i	0.0147501i
$331$	14.3923i	0.791073i	0.918450	+	0.395536i	$0.129441\pi$
$331$	14.3923i	0.791073i	−0.918450	+	0.395536i	$0.870559\pi$
$332$	− 9.46410i	− 0.519410i
$333$	5.92820i	0.324864i
$334$	0.464102	0.0253945
$335$	1.46410	0.0799924
$336$	− 3.00000i	− 0.163663i
$337$	−26.3923	−1.43768	−0.718840	−	0.695175i	$-0.755327\pi$
$337$	−26.3923	−1.43768	−0.718840	+	0.695175i	$0.755327\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	− 4.00000i	− 0.216930i
$341$	1.32051	0.0715095
$342$	5.73205	0.309954
$343$	15.0000i	0.809924i
$344$	6.00000i	0.323498i
$345$	− 3.46410i	− 0.186501i
$346$	2.12436i	0.114206i
$347$	4.39230	0.235791	0.117896	−	0.993026i	$-0.462385\pi$
$347$	4.39230	0.235791	0.117896	+	0.993026i	$0.462385\pi$
$348$	1.46410	0.0784841
$349$	10.5359i	0.563974i	0.959418	+	0.281987i	$0.0909935\pi$
$349$	10.5359i	0.563974i	−0.959418	+	0.281987i	$0.909007\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	−0.267949	−0.0142817
$353$	− 7.60770i	− 0.404917i	−0.979291	−	0.202458i	$-0.935107\pi$
$353$	− 7.60770i	− 0.404917i	0.979291	−	0.202458i	$-0.0648931\pi$
$354$	11.4641	0.609310
$355$	12.9282	0.686158
$356$	− 14.1244i	− 0.748589i
$357$	− 12.0000i	− 0.635107i
$358$	− 9.07180i	− 0.479459i
$359$	− 0.928203i	− 0.0489887i	−0.999700	−	0.0244943i	$-0.992202\pi$
$359$	− 0.928203i	− 0.0489887i	0.999700	−	0.0244943i	$-0.00779757\pi$
$360$	−1.00000	−0.0527046
$361$	−13.8564	−0.729285
$362$	− 16.9282i	− 0.889727i
$363$	−10.9282	−0.573582
$364$	0	0
$365$	6.92820	0.362639
$366$	− 0.535898i	− 0.0280119i
$367$	21.3205	1.11292	0.556461	−	0.830874i	$-0.312159\pi$
$367$	21.3205	1.11292	0.556461	+	0.830874i	$0.312159\pi$
$368$	3.46410	0.180579
$369$	4.00000i	0.208232i
$370$	5.92820i	0.308193i
$371$	− 0.803848i	− 0.0417337i
$372$	4.92820i	0.255515i
$373$	9.07180	0.469720	0.234860	−	0.972029i	$-0.424537\pi$
$373$	9.07180	0.469720	0.234860	+	0.972029i	$0.424537\pi$
$374$	−1.07180	−0.0554213
$375$	1.00000i	0.0516398i
$376$	6.46410	0.333361
$377$	0	0
$378$	−3.00000	−0.154303
$379$	9.73205i	0.499902i	0.968259	+	0.249951i	$0.0804145\pi$
$379$	9.73205i	0.499902i	−0.968259	+	0.249951i	$0.919586\pi$
$380$	5.73205	0.294048
$381$	12.6603	0.648604
$382$	17.3205i	0.886194i
$383$	4.78461i	0.244482i	0.992500	+	0.122241i	$0.0390081\pi$
$383$	4.78461i	0.244482i	−0.992500	+	0.122241i	$0.960992\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	0.803848i	0.0409679i
$386$	9.85641	0.501678
$387$	6.00000	0.304997
$388$	− 8.39230i	− 0.426055i
$389$	17.8564	0.905356	0.452678	−	0.891674i	$-0.350469\pi$
$389$	17.8564	0.905356	0.452678	+	0.891674i	$0.350469\pi$
$390$	0	0
$391$	13.8564	0.700749
$392$	− 2.00000i	− 0.101015i
$393$	−14.3205	−0.722374
$394$	11.3923	0.573936
$395$	3.07180i	0.154559i
$396$	0.267949i	0.0134650i
$397$	− 5.92820i	− 0.297528i	−0.988873	−	0.148764i	$-0.952471\pi$
$397$	− 5.92820i	− 0.297528i	0.988873	−	0.148764i	$-0.0475295\pi$
$398$	24.9282i	1.24954i
$399$	17.1962	0.860884
$400$	−1.00000	−0.0500000
$401$	22.1244i	1.10484i	0.833567	+	0.552419i	$0.186295\pi$
$401$	22.1244i	1.10484i	−0.833567	+	0.552419i	$0.813705\pi$
$402$	1.46410	0.0730228
$403$	0	0
$404$	−12.3923	−0.616540
$405$	1.00000i	0.0496904i
$406$	4.39230	0.217986
$407$	1.58846	0.0787369
$408$	− 4.00000i	− 0.198030i
$409$	− 39.0526i	− 1.93102i	−0.260357	−	0.965512i	$-0.583840\pi$
$409$	− 39.0526i	− 1.93102i	0.260357	−	0.965512i	$-0.416160\pi$
$410$	4.00000i	0.197546i
$411$	− 13.4641i	− 0.664135i
$412$	11.5885	0.570922
$413$	34.3923	1.69233
$414$	− 3.46410i	− 0.170251i
$415$	−9.46410	−0.464574
$416$	0	0
$417$	19.7846	0.968857
$418$	− 1.53590i	− 0.0751232i
$419$	9.85641	0.481517	0.240758	−	0.970585i	$-0.422604\pi$
$419$	9.85641	0.481517	0.240758	+	0.970585i	$0.422604\pi$
$420$	−3.00000	−0.146385
$421$	− 21.8564i	− 1.06522i	−0.846362	−	0.532608i	$-0.821212\pi$
$421$	− 21.8564i	− 1.06522i	0.846362	−	0.532608i	$-0.178788\pi$
$422$	8.07180i	0.392929i
$423$	− 6.46410i	− 0.314295i
$424$	− 0.267949i	− 0.0130128i
$425$	−4.00000	−0.194029
$426$	12.9282	0.626373
$427$	− 1.60770i	− 0.0778018i
$428$	12.9282	0.624908
$429$	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
$433$	−8.78461	−0.422161	−0.211081	−	0.977469i	$-0.567698\pi$
$433$	−8.78461	−0.422161	−0.211081	+	0.977469i	$0.567698\pi$
$434$	14.7846i	0.709684i
$435$	− 1.46410i	− 0.0701983i
$436$	10.3923i	0.497701i
$437$	19.8564i	0.949861i
$438$	6.92820	0.331042
$439$	−13.3205	−0.635753	−0.317877	−	0.948132i	$-0.602970\pi$
$439$	−13.3205	−0.635753	−0.317877	+	0.948132i	$0.602970\pi$
$440$	0.267949i	0.0127740i
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	−19.8564	−0.943406	−0.471703	−	0.881757i	$-0.656361\pi$
$443$	−19.8564	−0.943406	−0.471703	+	0.881757i	$0.656361\pi$
$444$	5.92820i	0.281340i
$445$	−14.1244	−0.669559
$446$	18.8564	0.892877
$447$	− 18.7846i	− 0.888482i
$448$	− 3.00000i	− 0.141737i
$449$	− 6.12436i	− 0.289026i	−0.989503	−	0.144513i	$-0.953838\pi$
$449$	− 6.12436i	− 0.289026i	0.989503	−	0.144513i	$-0.0461616\pi$
$450$	1.00000i	0.0471405i
$451$	1.07180	0.0504689
$452$	12.0000	0.564433
$453$	− 22.7846i	− 1.07051i
$454$	6.53590	0.306745
$455$	0	0
$456$	5.73205	0.268428
$457$	7.46410i	0.349156i	0.984643	+	0.174578i	$0.0558561\pi$
$457$	7.46410i	0.349156i	−0.984643	+	0.174578i	$0.944144\pi$
$458$	4.53590	0.211949
$459$	−4.00000	−0.186704
$460$	− 3.46410i	− 0.161515i
$461$	− 11.6077i	− 0.540624i	−0.962773	−	0.270312i	$-0.912873\pi$
$461$	− 11.6077i	− 0.540624i	0.962773	−	0.270312i	$-0.0871269\pi$
$462$	0.803848i	0.0373984i
$463$	40.7846i	1.89542i	0.319131	+	0.947711i	$0.396609\pi$
$463$	40.7846i	1.89542i	−0.319131	+	0.947711i	$0.603391\pi$
$464$	1.46410	0.0679692
$465$	4.92820	0.228540
$466$	− 18.0000i	− 0.833834i
$467$	24.3923	1.12874	0.564371	−	0.825522i	$-0.309119\pi$
$467$	24.3923	1.12874	0.564371	+	0.825522i	$0.309119\pi$
$468$	0	0
$469$	4.39230	0.202818
$470$	− 6.46410i	− 0.298167i
$471$	−21.1962	−0.976667
$472$	11.4641	0.527678
$473$	− 1.60770i	− 0.0739219i
$474$	3.07180i	0.141092i
$475$	− 5.73205i	− 0.263005i
$476$	− 12.0000i	− 0.550019i
$477$	−0.267949	−0.0122686
$478$	−3.46410	−0.158444
$479$	22.2487i	1.01657i	0.861189	+	0.508285i	$0.169720\pi$
$479$	22.2487i	1.01657i	−0.861189	+	0.508285i	$0.830280\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−25.1962	−1.14765
$483$	− 10.3923i	− 0.472866i
$484$	−10.9282	−0.496737
$485$	−8.39230	−0.381075
$486$	1.00000i	0.0453609i
$487$	21.0000i	0.951601i	0.879553	+	0.475800i	$0.157842\pi$
$487$	21.0000i	0.951601i	−0.879553	+	0.475800i	$0.842158\pi$
$488$	− 0.535898i	− 0.0242590i
$489$	2.92820i	0.132418i
$490$	−2.00000	−0.0903508
$491$	5.39230	0.243351	0.121676	−	0.992570i	$-0.461173\pi$
$491$	5.39230	0.243351	0.121676	+	0.992570i	$0.461173\pi$
$492$	4.00000i	0.180334i
$493$	5.85641	0.263759
$494$	0	0
$495$	0.267949	0.0120434
$496$	4.92820i	0.221283i
$497$	38.7846	1.73973
$498$	−9.46410	−0.424097
$499$	− 33.3205i	− 1.49163i	−0.666153	−	0.745815i	$-0.732060\pi$
$499$	− 33.3205i	− 1.49163i	0.666153	−	0.745815i	$-0.267940\pi$
$500$	1.00000i	0.0447214i
$501$	− 0.464102i	− 0.0207345i
$502$	− 19.5359i	− 0.871930i
$503$	3.73205	0.166404	0.0832020	−	0.996533i	$-0.473485\pi$
$503$	3.73205	0.166404	0.0832020	+	0.996533i	$0.473485\pi$
$504$	−3.00000	−0.133631
$505$	12.3923i	0.551450i
$506$	−0.928203	−0.0412637
$507$	0	0
$508$	12.6603	0.561708
$509$	− 29.3205i	− 1.29961i	−0.760102	−	0.649804i	$-0.774851\pi$
$509$	− 29.3205i	− 1.29961i	0.760102	−	0.649804i	$-0.225149\pi$
$510$	−4.00000	−0.177123
$511$	20.7846	0.919457
$512$	− 1.00000i	− 0.0441942i
$513$	− 5.73205i	− 0.253076i
$514$	− 14.5359i	− 0.641151i
$515$	− 11.5885i	− 0.510648i
$516$	6.00000	0.264135
$517$	−1.73205	−0.0761755
$518$	17.7846i	0.781411i
$519$	2.12436	0.0932489
$520$	0	0
$521$	−6.60770	−0.289488	−0.144744	−	0.989469i	$-0.546236\pi$
$521$	−6.60770	−0.289488	−0.144744	+	0.989469i	$0.546236\pi$
$522$	− 1.46410i	− 0.0640820i
$523$	−41.7128	−1.82397	−0.911987	−	0.410219i	$-0.865452\pi$
$523$	−41.7128	−1.82397	−0.911987	+	0.410219i	$0.865452\pi$
$524$	−14.3205	−0.625594
$525$	3.00000i	0.130931i
$526$	24.5167i	1.06898i
$527$	19.7128i	0.858704i
$528$	0.267949i	0.0116610i
$529$	−11.0000	−0.478261
$530$	−0.267949	−0.0116390
$531$	− 11.4641i	− 0.497500i
$532$	17.1962	0.745548
$533$	0	0
$534$	−14.1244	−0.611221
$535$	− 12.9282i	− 0.558935i
$536$	1.46410	0.0632396
$537$	−9.07180	−0.391477
$538$	17.0718i	0.736017i
$539$	0.535898i	0.0230828i
$540$	1.00000i	0.0430331i
$541$	− 14.7846i	− 0.635640i	−0.948151	−	0.317820i	$-0.897049\pi$
$541$	− 14.7846i	− 0.635640i	0.948151	−	0.317820i	$-0.102951\pi$
$542$	−24.3923	−1.04774
$543$	−16.9282	−0.726459
$544$	− 4.00000i	− 0.171499i
$545$	10.3923	0.445157
$546$	0	0
$547$	−5.32051	−0.227488	−0.113744	−	0.993510i	$-0.536284\pi$
$547$	−5.32051	−0.227488	−0.113744	+	0.993510i	$0.536284\pi$
$548$	− 13.4641i	− 0.575158i
$549$	−0.535898	−0.0228716
$550$	0.267949	0.0114254
$551$	8.39230i	0.357524i
$552$	− 3.46410i	− 0.147442i
$553$	9.21539i	0.391878i
$554$	− 11.5885i	− 0.492346i
$555$	5.92820	0.251638
$556$	19.7846	0.839054
$557$	22.6077i	0.957919i	0.877837	+	0.478959i	$0.158986\pi$
$557$	22.6077i	0.957919i	−0.877837	+	0.478959i	$0.841014\pi$
$558$	4.92820	0.208627
$559$	0	0
$560$	−3.00000	−0.126773
$561$	1.07180i	0.0452513i
$562$	−6.92820	−0.292249
$563$	−15.3205	−0.645682	−0.322841	−	0.946453i	$-0.604638\pi$
$563$	−15.3205	−0.645682	−0.322841	+	0.946453i	$0.604638\pi$
$564$	− 6.46410i	− 0.272188i
$565$	− 12.0000i	− 0.504844i
$566$	6.39230i	0.268689i
$567$	3.00000i	0.125988i
$568$	12.9282	0.542455
$569$	−16.3205	−0.684191	−0.342096	−	0.939665i	$-0.611137\pi$
$569$	−16.3205	−0.684191	−0.342096	+	0.939665i	$0.611137\pi$
$570$	− 5.73205i	− 0.240089i
$571$	10.8564	0.454326	0.227163	−	0.973857i	$-0.427055\pi$
$571$	10.8564	0.454326	0.227163	+	0.973857i	$0.427055\pi$
$572$	0	0
$573$	17.3205	0.723575
$574$	12.0000i	0.500870i
$575$	−3.46410	−0.144463
$576$	−1.00000	−0.0416667
$577$	9.32051i	0.388018i	0.981000	+	0.194009i	$0.0621491\pi$
$577$	9.32051i	0.388018i	−0.981000	+	0.194009i	$0.937851\pi$
$578$	1.00000i	0.0415945i
$579$	− 9.85641i	− 0.409618i
$580$	− 1.46410i	− 0.0607935i
$581$	−28.3923	−1.17791
$582$	−8.39230	−0.347872
$583$	0.0717968i	0.00297352i
$584$	6.92820	0.286691
$585$	0	0
$586$	29.2487	1.20825
$587$	8.92820i	0.368506i	0.982879	+	0.184253i	$0.0589866\pi$
$587$	8.92820i	0.368506i	−0.982879	+	0.184253i	$0.941013\pi$
$588$	−2.00000	−0.0824786
$589$	−28.2487	−1.16397
$590$	− 11.4641i	− 0.471970i
$591$	− 11.3923i	− 0.468617i
$592$	5.92820i	0.243648i
$593$	44.7846i	1.83908i	0.392992	+	0.919542i	$0.371440\pi$
$593$	44.7846i	1.83908i	−0.392992	+	0.919542i	$0.628560\pi$
$594$	0.267949	0.0109941
$595$	−12.0000	−0.491952
$596$	− 18.7846i	− 0.769448i
$597$	24.9282	1.02024
$598$	0	0
$599$	28.9282	1.18197	0.590987	−	0.806681i	$-0.298738\pi$
$599$	28.9282	1.18197	0.590987	+	0.806681i	$0.298738\pi$
$600$	1.00000i	0.0408248i
$601$	30.7128	1.25280	0.626401	−	0.779501i	$-0.284527\pi$
$601$	30.7128	1.25280	0.626401	+	0.779501i	$0.284527\pi$
$602$	18.0000	0.733625
$603$	− 1.46410i	− 0.0596228i
$604$	− 22.7846i	− 0.927093i
$605$	10.9282i	0.444295i
$606$	12.3923i	0.503403i
$607$	−21.4449	−0.870420	−0.435210	−	0.900329i	$-0.643326\pi$
$607$	−21.4449	−0.870420	−0.435210	+	0.900329i	$0.643326\pi$
$608$	5.73205	0.232465
$609$	− 4.39230i	− 0.177985i
$610$	−0.535898	−0.0216979
$611$	0	0
$612$	−4.00000	−0.161690
$613$	− 45.0000i	− 1.81753i	−0.417305	−	0.908766i	$-0.637025\pi$
$613$	− 45.0000i	− 1.81753i	0.417305	−	0.908766i	$-0.362975\pi$
$614$	28.2487	1.14003
$615$	4.00000	0.161296
$616$	0.803848i	0.0323879i
$617$	− 35.4641i	− 1.42773i	−0.700283	−	0.713865i	$-0.746943\pi$
$617$	− 35.4641i	− 1.42773i	0.700283	−	0.713865i	$-0.253057\pi$
$618$	− 11.5885i	− 0.466156i
$619$	− 2.51666i	− 0.101153i	−0.998720	−	0.0505766i	$-0.983894\pi$
$619$	− 2.51666i	− 0.101153i	0.998720	−	0.0505766i	$-0.0161059\pi$
$620$	4.92820	0.197921
$621$	−3.46410	−0.139010
$622$	− 18.9282i	− 0.758952i
$623$	−42.3731	−1.69764
$624$	0	0
$625$	1.00000	0.0400000
$626$	33.3205i	1.33176i
$627$	−1.53590	−0.0613379
$628$	−21.1962	−0.845819
$629$	23.7128i	0.945492i
$630$	3.00000i	0.119523i
$631$	− 8.00000i	− 0.318475i	−0.987240	−	0.159237i	$-0.949096\pi$
$631$	− 8.00000i	− 0.318475i	0.987240	−	0.159237i	$-0.0509036\pi$
$632$	3.07180i	0.122190i
$633$	8.07180	0.320825
$634$	−30.4641	−1.20988
$635$	− 12.6603i	− 0.502407i
$636$	−0.267949	−0.0106249
$637$	0	0
$638$	−0.392305	−0.0155315
$639$	− 12.9282i	− 0.511432i
$640$	−1.00000	−0.0395285
$641$	14.4641	0.571298	0.285649	−	0.958334i	$-0.407791\pi$
$641$	14.4641	0.571298	0.285649	+	0.958334i	$0.407791\pi$
$642$	− 12.9282i	− 0.510235i
$643$	12.7846i	0.504176i	0.967704	+	0.252088i	$0.0811172\pi$
$643$	12.7846i	0.504176i	−0.967704	+	0.252088i	$0.918883\pi$
$644$	− 10.3923i	− 0.409514i
$645$	− 6.00000i	− 0.236250i
$646$	22.9282	0.902098
$647$	−35.7321	−1.40477	−0.702386	−	0.711796i	$-0.747882\pi$
$647$	−35.7321	−1.40477	−0.702386	+	0.711796i	$0.747882\pi$
$648$	1.00000i	0.0392837i
$649$	−3.07180	−0.120579
$650$	0	0
$651$	14.7846	0.579455
$652$	2.92820i	0.114677i
$653$	−1.58846	−0.0621611	−0.0310806	−	0.999517i	$-0.509895\pi$
$653$	−1.58846	−0.0621611	−0.0310806	+	0.999517i	$0.509895\pi$
$654$	10.3923	0.406371
$655$	14.3205i	0.559549i
$656$	4.00000i	0.156174i
$657$	− 6.92820i	− 0.270295i
$658$	− 19.3923i	− 0.755991i
$659$	−39.7128	−1.54699	−0.773496	−	0.633801i	$-0.781494\pi$
$659$	−39.7128	−1.54699	−0.773496	+	0.633801i	$0.781494\pi$
$660$	0.267949	0.0104299
$661$	21.6077i	0.840442i	0.907422	+	0.420221i	$0.138047\pi$
$661$	21.6077i	0.840442i	−0.907422	+	0.420221i	$0.861953\pi$
$662$	14.3923	0.559373
$663$	0	0
$664$	−9.46410	−0.367278
$665$	− 17.1962i	− 0.666838i
$666$	5.92820	0.229713
$667$	5.07180	0.196381
$668$	− 0.464102i	− 0.0179566i
$669$	− 18.8564i	− 0.729031i
$670$	− 1.46410i	− 0.0565632i
$671$	0.143594i	0.00554337i
$672$	−3.00000	−0.115728
$673$	−16.7846	−0.646999	−0.323500	−	0.946228i	$-0.604859\pi$
$673$	−16.7846	−0.646999	−0.323500	+	0.946228i	$0.604859\pi$
$674$	26.3923i	1.01659i
$675$	1.00000	0.0384900
$676$	0	0
$677$	10.9282	0.420005	0.210002	−	0.977701i	$-0.432653\pi$
$677$	10.9282	0.420005	0.210002	+	0.977701i	$0.432653\pi$
$678$	− 12.0000i	− 0.460857i
$679$	−25.1769	−0.966201
$680$	−4.00000	−0.153393
$681$	− 6.53590i	− 0.250456i
$682$	− 1.32051i	− 0.0505649i
$683$	41.3205i	1.58109i	0.612407	+	0.790543i	$0.290201\pi$
$683$	41.3205i	1.58109i	−0.612407	+	0.790543i	$0.709799\pi$
$684$	− 5.73205i	− 0.219170i
$685$	−13.4641	−0.514437
$686$	15.0000	0.572703
$687$	− 4.53590i	− 0.173055i
$688$	6.00000	0.228748
$689$	0	0
$690$	−3.46410	−0.131876
$691$	− 15.0526i	− 0.572626i	−0.958136	−	0.286313i	$-0.907570\pi$
$691$	− 15.0526i	− 0.572626i	0.958136	−	0.286313i	$-0.0924298\pi$
$692$	2.12436	0.0807559
$693$	0.803848	0.0305356
$694$	− 4.39230i	− 0.166730i
$695$	− 19.7846i	− 0.750473i
$696$	− 1.46410i	− 0.0554966i
$697$	16.0000i	0.606043i
$698$	10.5359	0.398790
$699$	−18.0000	−0.680823
$700$	3.00000i	0.113389i
$701$	−4.39230	−0.165895	−0.0829475	−	0.996554i	$-0.526433\pi$
$701$	−4.39230	−0.165895	−0.0829475	+	0.996554i	$0.526433\pi$
$702$	0	0
$703$	−33.9808	−1.28161
$704$	0.267949i	0.0100987i
$705$	−6.46410	−0.243452
$706$	−7.60770	−0.286319
$707$	37.1769i	1.39818i
$708$	− 11.4641i	− 0.430847i
$709$	− 27.3205i	− 1.02604i	−0.858376	−	0.513022i	$-0.828526\pi$
$709$	− 27.3205i	− 1.02604i	0.858376	−	0.513022i	$-0.171474\pi$
$710$	− 12.9282i	− 0.485187i
$711$	3.07180	0.115201
$712$	−14.1244	−0.529333
$713$	17.0718i	0.639344i
$714$	−12.0000	−0.449089
$715$	0	0
$716$	−9.07180	−0.339029
$717$	3.46410i	0.129369i
$718$	−0.928203	−0.0346402
$719$	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
$721$	− 34.7654i	− 1.29473i
$722$	13.8564i	0.515682i
$723$	25.1962i	0.937055i
$724$	−16.9282	−0.629132
$725$	−1.46410	−0.0543754
$726$	10.9282i	0.405584i
$727$	−4.66025	−0.172839	−0.0864196	−	0.996259i	$-0.527543\pi$
$727$	−4.66025	−0.172839	−0.0864196	+	0.996259i	$0.527543\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 6.92820i	− 0.256424i
$731$	24.0000	0.887672
$732$	−0.535898	−0.0198074
$733$	− 20.8564i	− 0.770349i	−0.922844	−	0.385174i	$-0.874141\pi$
$733$	− 20.8564i	− 0.770349i	0.922844	−	0.385174i	$-0.125859\pi$
$734$	− 21.3205i	− 0.786954i
$735$	2.00000i	0.0737711i
$736$	− 3.46410i	− 0.127688i
$737$	−0.392305	−0.0144507
$738$	4.00000	0.147242
$739$	− 35.8372i	− 1.31829i	−0.752015	−	0.659146i	$-0.770918\pi$
$739$	− 35.8372i	− 1.31829i	0.752015	−	0.659146i	$-0.229082\pi$
$740$	5.92820	0.217925
$741$	0	0
$742$	−0.803848	−0.0295102
$743$	10.9282i	0.400917i	0.979702	+	0.200458i	$0.0642432\pi$
$743$	10.9282i	0.400917i	−0.979702	+	0.200458i	$0.935757\pi$
$744$	4.92820	0.180677
$745$	−18.7846	−0.688215
$746$	− 9.07180i	− 0.332142i
$747$	9.46410i	0.346273i
$748$	1.07180i	0.0391888i
$749$	− 38.7846i	− 1.41716i
$750$	1.00000	0.0365148
$751$	−21.1769	−0.772757	−0.386378	−	0.922340i	$-0.626274\pi$
$751$	−21.1769	−0.772757	−0.386378	+	0.922340i	$0.626274\pi$
$752$	− 6.46410i	− 0.235722i
$753$	−19.5359	−0.711928
$754$	0	0
$755$	−22.7846	−0.829217
$756$	3.00000i	0.109109i
$757$	21.7321	0.789865	0.394932	−	0.918710i	$-0.370768\pi$
$757$	21.7321	0.789865	0.394932	+	0.918710i	$0.370768\pi$
$758$	9.73205	0.353484
$759$	0.928203i	0.0336916i
$760$	− 5.73205i	− 0.207923i
$761$	7.98076i	0.289302i	0.989483	+	0.144651i	$0.0462060\pi$
$761$	7.98076i	0.289302i	−0.989483	+	0.144651i	$0.953794\pi$
$762$	− 12.6603i	− 0.458633i
$763$	31.1769	1.12868
$764$	17.3205	0.626634
$765$	4.00000i	0.144620i
$766$	4.78461	0.172875
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 15.7128i	− 0.566619i	−0.959029	−	0.283309i	$-0.908568\pi$
$769$	− 15.7128i	− 0.566619i	0.959029	−	0.283309i	$-0.0914323\pi$
$770$	0.803848	0.0289687
$771$	−14.5359	−0.523498
$772$	− 9.85641i	− 0.354740i
$773$	− 32.6077i	− 1.17282i	−0.810015	−	0.586409i	$-0.800541\pi$
$773$	− 32.6077i	− 1.17282i	0.810015	−	0.586409i	$-0.199459\pi$
$774$	− 6.00000i	− 0.215666i
$775$	− 4.92820i	− 0.177026i
$776$	−8.39230	−0.301266
$777$	17.7846	0.638019
$778$	− 17.8564i	− 0.640183i
$779$	−22.9282	−0.821488
$780$	0	0
$781$	−3.46410	−0.123955
$782$	− 13.8564i	− 0.495504i
$783$	−1.46410	−0.0523227
$784$	−2.00000	−0.0714286
$785$	21.1962i	0.756523i
$786$	14.3205i	0.510796i
$787$	1.46410i	0.0521896i	0.999659	+	0.0260948i	$0.00830717\pi$
$787$	1.46410i	0.0521896i	−0.999659	+	0.0260948i	$0.991693\pi$
$788$	− 11.3923i	− 0.405834i
$789$	24.5167	0.872816
$790$	3.07180	0.109290
$791$	− 36.0000i	− 1.28001i
$792$	0.267949	0.00952116
$793$	0	0
$794$	−5.92820	−0.210384
$795$	0.267949i	0.00950318i
$796$	24.9282	0.883557
$797$	−10.1436	−0.359305	−0.179652	−	0.983730i	$-0.557497\pi$
$797$	−10.1436	−0.359305	−0.179652	+	0.983730i	$0.557497\pi$
$798$	− 17.1962i	− 0.608737i
$799$	− 25.8564i	− 0.914734i
$800$	1.00000i	0.0353553i
$801$	14.1244i	0.499060i
$802$	22.1244	0.781238
$803$	−1.85641	−0.0655112
$804$	− 1.46410i	− 0.0516349i
$805$	−10.3923	−0.366281
$806$	0	0
$807$	17.0718	0.600956
$808$	12.3923i	0.435960i
$809$	−6.78461	−0.238534	−0.119267	−	0.992862i	$-0.538054\pi$
$809$	−6.78461	−0.238534	−0.119267	+	0.992862i	$0.538054\pi$
$810$	1.00000	0.0351364
$811$	− 23.5885i	− 0.828303i	−0.910208	−	0.414151i	$-0.864078\pi$
$811$	− 23.5885i	− 0.828303i	0.910208	−	0.414151i	$-0.135922\pi$
$812$	− 4.39230i	− 0.154140i
$813$	24.3923i	0.855475i
$814$	− 1.58846i	− 0.0556754i
$815$	2.92820	0.102570
$816$	−4.00000	−0.140028
$817$	34.3923i	1.20323i
$818$	−39.0526	−1.36544
$819$	0	0
$820$	4.00000	0.139686
$821$	14.6795i	0.512318i	0.966635	+	0.256159i	$0.0824570\pi$
$821$	14.6795i	0.512318i	−0.966635	+	0.256159i	$0.917543\pi$
$822$	−13.4641	−0.469614
$823$	−8.41154	−0.293208	−0.146604	−	0.989195i	$-0.546834\pi$
$823$	−8.41154	−0.293208	−0.146604	+	0.989195i	$0.546834\pi$
$824$	− 11.5885i	− 0.403703i
$825$	− 0.267949i	− 0.00932879i
$826$	− 34.3923i	− 1.19666i
$827$	− 46.4974i	− 1.61687i	−0.588583	−	0.808437i	$-0.700314\pi$
$827$	− 46.4974i	− 1.61687i	0.588583	−	0.808437i	$-0.299686\pi$
$828$	−3.46410	−0.120386
$829$	−4.67949	−0.162525	−0.0812627	−	0.996693i	$-0.525895\pi$
$829$	−4.67949	−0.162525	−0.0812627	+	0.996693i	$0.525895\pi$
$830$	9.46410i	0.328504i
$831$	−11.5885	−0.401999
$832$	0	0
$833$	−8.00000	−0.277184
$834$	− 19.7846i	− 0.685085i
$835$	−0.464102	−0.0160609
$836$	−1.53590	−0.0531202
$837$	− 4.92820i	− 0.170344i
$838$	− 9.85641i	− 0.340484i
$839$	19.1769i	0.662061i	0.943620	+	0.331030i	$0.107396\pi$
$839$	19.1769i	0.662061i	−0.943620	+	0.331030i	$0.892604\pi$
$840$	3.00000i	0.103510i
$841$	−26.8564	−0.926083
$842$	−21.8564	−0.753222
$843$	6.92820i	0.238620i
$844$	8.07180	0.277843
$845$	0	0
$846$	−6.46410	−0.222240
$847$	32.7846i	1.12649i
$848$	−0.267949	−0.00920141
$849$	6.39230	0.219383
$850$	4.00000i	0.137199i
$851$	20.5359i	0.703962i
$852$	− 12.9282i	− 0.442913i
$853$	24.6410i	0.843692i	0.906667	+	0.421846i	$0.138618\pi$
$853$	24.6410i	0.843692i	−0.906667	+	0.421846i	$0.861382\pi$
$854$	−1.60770	−0.0550142
$855$	−5.73205	−0.196032
$856$	− 12.9282i	− 0.441877i
$857$	−28.9282	−0.988169	−0.494084	−	0.869414i	$-0.664497\pi$
$857$	−28.9282	−0.988169	−0.494084	+	0.869414i	$0.664497\pi$
$858$	0	0
$859$	−6.07180	−0.207167	−0.103584	−	0.994621i	$-0.533031\pi$
$859$	−6.07180	−0.207167	−0.103584	+	0.994621i	$0.533031\pi$
$860$	− 6.00000i	− 0.204598i
$861$	12.0000	0.408959
$862$	13.8564	0.471951
$863$	− 2.92820i	− 0.0996772i	−0.998757	−	0.0498386i	$-0.984129\pi$
$863$	− 2.92820i	− 0.0996772i	0.998757	−	0.0498386i	$-0.0158707\pi$
$864$	1.00000i	0.0340207i
$865$	− 2.12436i	− 0.0722303i
$866$	8.78461i	0.298513i
$867$	1.00000	0.0339618
$868$	14.7846	0.501822
$869$	− 0.823085i	− 0.0279213i
$870$	−1.46410	−0.0496377
$871$	0	0
$872$	10.3923	0.351928
$873$	8.39230i	0.284036i
$874$	19.8564	0.671653
$875$	3.00000	0.101419
$876$	− 6.92820i	− 0.234082i
$877$	21.7128i	0.733190i	0.930381	+	0.366595i	$0.119476\pi$
$877$	21.7128i	0.733190i	−0.930381	+	0.366595i	$0.880524\pi$
$878$	13.3205i	0.449545i
$879$	− 29.2487i	− 0.986535i
$880$	0.267949	0.00903257
$881$	39.1051	1.31748	0.658742	−	0.752369i	$-0.271089\pi$
$881$	39.1051	1.31748	0.658742	+	0.752369i	$0.271089\pi$
$882$	2.00000i	0.0673435i
$883$	−13.3205	−0.448271	−0.224135	−	0.974558i	$-0.571956\pi$
$883$	−13.3205	−0.448271	−0.224135	+	0.974558i	$0.571956\pi$
$884$	0	0
$885$	−11.4641	−0.385362
$886$	19.8564i	0.667089i
$887$	55.7321	1.87130	0.935650	−	0.352930i	$-0.114815\pi$
$887$	55.7321	1.87130	0.935650	+	0.352930i	$0.114815\pi$
$888$	5.92820	0.198937
$889$	− 37.9808i	− 1.27383i
$890$	14.1244i	0.473449i
$891$	− 0.267949i	− 0.00897664i
$892$	− 18.8564i	− 0.631359i
$893$	37.0526	1.23992
$894$	−18.7846	−0.628251
$895$	9.07180i	0.303237i
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−6.12436	−0.204372
$899$	7.21539i	0.240647i
$900$	1.00000	0.0333333
$901$	−1.07180	−0.0357067
$902$	− 1.07180i	− 0.0356869i
$903$	− 18.0000i	− 0.599002i
$904$	− 12.0000i	− 0.399114i
$905$	16.9282i	0.562713i
$906$	−22.7846	−0.756968
$907$	20.9282	0.694910	0.347455	−	0.937697i	$-0.387046\pi$
$907$	20.9282	0.694910	0.347455	+	0.937697i	$0.387046\pi$
$908$	− 6.53590i	− 0.216901i
$909$	12.3923	0.411027
$910$	0	0
$911$	12.7846	0.423573	0.211787	−	0.977316i	$-0.432072\pi$
$911$	12.7846	0.423573	0.211787	+	0.977316i	$0.432072\pi$
$912$	− 5.73205i	− 0.189807i
$913$	2.53590	0.0839260
$914$	7.46410	0.246891
$915$	0.535898i	0.0177163i
$916$	− 4.53590i	− 0.149870i
$917$	42.9615i	1.41871i
$918$	4.00000i	0.132020i
$919$	−47.9615	−1.58210	−0.791052	−	0.611748i	$-0.790466\pi$
$919$	−47.9615	−1.58210	−0.791052	+	0.611748i	$0.790466\pi$
$920$	−3.46410	−0.114208
$921$	− 28.2487i	− 0.930827i
$922$	−11.6077	−0.382279
$923$	0	0
$924$	0.803848	0.0264446
$925$	− 5.92820i	− 0.194918i
$926$	40.7846	1.34027
$927$	−11.5885	−0.380615
$928$	− 1.46410i	− 0.0480615i
$929$	57.8564i	1.89821i	0.314964	+	0.949104i	$0.398007\pi$
$929$	57.8564i	1.89821i	−0.314964	+	0.949104i	$0.601993\pi$
$930$	− 4.92820i	− 0.161602i
$931$	− 11.4641i	− 0.375721i
$932$	−18.0000	−0.589610
$933$	−18.9282	−0.619682
$934$	− 24.3923i	− 0.798141i
$935$	1.07180	0.0350515
$936$	0	0
$937$	−21.7128	−0.709327	−0.354663	−	0.934994i	$-0.615405\pi$
$937$	−21.7128	−0.709327	−0.354663	+	0.934994i	$0.615405\pi$
$938$	− 4.39230i	− 0.143414i
$939$	33.3205	1.08737
$940$	−6.46410	−0.210836
$941$	60.4974i	1.97216i	0.166273	+	0.986080i	$0.446827\pi$
$941$	60.4974i	1.97216i	−0.166273	+	0.986080i	$0.553173\pi$
$942$	21.1962i	0.690608i
$943$	13.8564i	0.451227i
$944$	− 11.4641i	− 0.373125i
$945$	3.00000	0.0975900
$946$	−1.60770	−0.0522707
$947$	− 34.3923i	− 1.11760i	−0.829303	−	0.558800i	$-0.811262\pi$
$947$	− 34.3923i	− 1.11760i	0.829303	−	0.558800i	$-0.188738\pi$
$948$	3.07180	0.0997673
$949$	0	0
$950$	−5.73205	−0.185972
$951$	30.4641i	0.987866i
$952$	−12.0000	−0.388922
$953$	−53.3205	−1.72722	−0.863610	−	0.504160i	$-0.831802\pi$
$953$	−53.3205	−1.72722	−0.863610	+	0.504160i	$0.831802\pi$
$954$	0.267949i	0.00867518i
$955$	− 17.3205i	− 0.560478i
$956$	3.46410i	0.112037i
$957$	0.392305i	0.0126814i
$958$	22.2487	0.718823
$959$	−40.3923	−1.30434
$960$	1.00000i	0.0322749i
$961$	6.71281	0.216542
$962$	0	0
$963$	−12.9282	−0.416606
$964$	25.1962i	0.811513i
$965$	−9.85641	−0.317289
$966$	−10.3923	−0.334367
$967$	1.14359i	0.0367755i	0.999831	+	0.0183877i	$0.00585333\pi$
$967$	1.14359i	0.0367755i	−0.999831	+	0.0183877i	$0.994147\pi$
$968$	10.9282i	0.351246i
$969$	− 22.9282i	− 0.736560i
$970$	8.39230i	0.269461i
$971$	−41.3923	−1.32834	−0.664171	−	0.747581i	$-0.731215\pi$
$971$	−41.3923	−1.32834	−0.664171	+	0.747581i	$0.731215\pi$
$972$	1.00000	0.0320750
$973$	− 59.3538i	− 1.90280i
$974$	21.0000	0.672883
$975$	0	0
$976$	−0.535898	−0.0171537
$977$	5.46410i	0.174812i	0.996173	+	0.0874060i	$0.0278578\pi$
$977$	5.46410i	0.174812i	−0.996173	+	0.0874060i	$0.972142\pi$
$978$	2.92820	0.0936336
$979$	3.78461	0.120957
$980$	2.00000i	0.0638877i
$981$	− 10.3923i	− 0.331801i
$982$	− 5.39230i	− 0.172075i
$983$	− 46.1769i	− 1.47281i	−0.676538	−	0.736407i	$-0.736521\pi$
$983$	− 46.1769i	− 1.47281i	0.676538	−	0.736407i	$-0.263479\pi$
$984$	4.00000	0.127515
$985$	−11.3923	−0.362989
$986$	− 5.85641i	− 0.186506i
$987$	−19.3923	−0.617264
$988$	0	0
$989$	20.7846	0.660912
$990$	− 0.267949i	− 0.00851598i
$991$	39.1769	1.24450	0.622248	−	0.782820i	$-0.286220\pi$
$991$	39.1769	1.24450	0.622248	+	0.782820i	$0.286220\pi$
$992$	4.92820	0.156471
$993$	− 14.3923i	− 0.456726i
$994$	− 38.7846i	− 1.23017i
$995$	− 24.9282i	− 0.790277i
$996$	9.46410i	0.299882i
$997$	61.9808	1.96295	0.981475	−	0.191589i	$-0.0613641\pi$
$997$	61.9808	1.96295	0.981475	+	0.191589i	$0.0613641\pi$
$998$	−33.3205	−1.05474
$999$	− 5.92820i	− 0.187560i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.p.1351.2		4
13.3	even	3		390.2.bb.a.121.2	✓	4
13.4	even	6		390.2.bb.a.361.2	yes	4
13.5	odd	4		5070.2.a.ba.1.1		2
13.8	odd	4		5070.2.a.be.1.2		2
13.12	even	2	inner	5070.2.b.p.1351.3		4
39.17	odd	6		1170.2.bs.d.361.1		4
39.29	odd	6		1170.2.bs.d.901.1		4
65.3	odd	12		1950.2.y.e.199.2		4
65.4	even	6		1950.2.bc.a.751.1		4
65.17	odd	12		1950.2.y.e.49.2		4
65.29	even	6		1950.2.bc.a.901.1		4
65.42	odd	12		1950.2.y.d.199.1		4
65.43	odd	12		1950.2.y.d.49.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.a.121.2	✓	4	13.3	even	3
390.2.bb.a.361.2	yes	4	13.4	even	6
1170.2.bs.d.361.1		4	39.17	odd	6
1170.2.bs.d.901.1		4	39.29	odd	6
1950.2.y.d.49.1		4	65.43	odd	12
1950.2.y.d.199.1		4	65.42	odd	12
1950.2.y.e.49.2		4	65.17	odd	12
1950.2.y.e.199.2		4	65.3	odd	12
1950.2.bc.a.751.1		4	65.4	even	6
1950.2.bc.a.901.1		4	65.29	even	6
5070.2.a.ba.1.1		2	13.5	odd	4
5070.2.a.be.1.2		2	13.8	odd	4
5070.2.b.p.1351.2		4	1.1	even	1	trivial
5070.2.b.p.1351.3		4	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -0.267949i q^{11} +1.00000 q^{12} +3.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000i q^{18} +5.73205i q^{19} -1.00000i q^{20} -3.00000i q^{21} -0.267949 q^{22} +3.46410 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -3.00000i q^{28} +1.46410 q^{29} -1.00000 q^{30} +4.92820i q^{31} -1.00000i q^{32} +0.267949i q^{33} -4.00000i q^{34} -3.00000 q^{35} -1.00000 q^{36} +5.92820i q^{37} +5.73205 q^{38} -1.00000 q^{40} +4.00000i q^{41} -3.00000 q^{42} +6.00000 q^{43} +0.267949i q^{44} +1.00000i q^{45} -3.46410i q^{46} -6.46410i q^{47} -1.00000 q^{48} -2.00000 q^{49} +1.00000i q^{50} -4.00000 q^{51} -0.267949 q^{53} +1.00000i q^{54} +0.267949 q^{55} -3.00000 q^{56} -5.73205i q^{57} -1.46410i q^{58} -11.4641i q^{59} +1.00000i q^{60} -0.535898 q^{61} +4.92820 q^{62} +3.00000i q^{63} -1.00000 q^{64} +0.267949 q^{66} -1.46410i q^{67} -4.00000 q^{68} -3.46410 q^{69} +3.00000i q^{70} -12.9282i q^{71} +1.00000i q^{72} -6.92820i q^{73} +5.92820 q^{74} +1.00000 q^{75} -5.73205i q^{76} +0.803848 q^{77} +3.07180 q^{79} +1.00000i q^{80} +1.00000 q^{81} +4.00000 q^{82} +9.46410i q^{83} +3.00000i q^{84} +4.00000i q^{85} -6.00000i q^{86} -1.46410 q^{87} +0.267949 q^{88} +14.1244i q^{89} +1.00000 q^{90} -3.46410 q^{92} -4.92820i q^{93} -6.46410 q^{94} -5.73205 q^{95} +1.00000i q^{96} +8.39230i q^{97} +2.00000i q^{98} -0.267949i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 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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