LMFDB - Embedded newform 5070.2.b.o.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:	$\Z[a_1, \ldots, a_{23}]$
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.o.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.46410i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000i q^{18} -7.46410i q^{19} +1.00000i q^{20} -2.00000i q^{21} +6.46410 q^{22} +3.73205 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} +0.267949 q^{29} +1.00000 q^{30} +1.73205i q^{31} -1.00000i q^{32} -6.46410i q^{33} +4.00000i q^{34} +2.00000 q^{35} -1.00000 q^{36} +9.19615i q^{37} -7.46410 q^{38} +1.00000 q^{40} +2.00000i q^{41} -2.00000 q^{42} +11.9282 q^{43} -6.46410i q^{44} -1.00000i q^{45} -3.73205i q^{46} +3.53590i q^{47} -1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} +4.00000 q^{51} +0.928203 q^{53} +1.00000i q^{54} +6.46410 q^{55} -2.00000 q^{56} +7.46410i q^{57} -0.267949i q^{58} -8.46410i q^{59} -1.00000i q^{60} -10.3923 q^{61} +1.73205 q^{62} +2.00000i q^{63} -1.00000 q^{64} -6.46410 q^{66} -11.4641i q^{67} +4.00000 q^{68} -3.73205 q^{69} -2.00000i q^{70} +12.3923i q^{71} +1.00000i q^{72} -2.00000i q^{73} +9.19615 q^{74} +1.00000 q^{75} +7.46410i q^{76} -12.9282 q^{77} -13.9282 q^{79} -1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} -8.92820i q^{83} +2.00000i q^{84} +4.00000i q^{85} -11.9282i q^{86} -0.267949 q^{87} -6.46410 q^{88} -0.535898i q^{89} -1.00000 q^{90} -3.73205 q^{92} -1.73205i q^{93} +3.53590 q^{94} -7.46410 q^{95} +1.00000i q^{96} -0.535898i q^{97} -3.00000i q^{98} +6.46410i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{14} + 4 q^{16} - 16 q^{17} + 12 q^{22} + 8 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} + 4 q^{30} + 8 q^{35} - 4 q^{36} - 16 q^{38} + 4 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{48} + 12 q^{49} + 16 q^{51} - 24 q^{53} + 12 q^{55} - 8 q^{56} - 4 q^{64} - 12 q^{66} + 16 q^{68} - 8 q^{69} + 16 q^{74} + 4 q^{75} - 24 q^{77} - 28 q^{79} + 4 q^{81} + 8 q^{82} - 8 q^{87} - 12 q^{88} - 4 q^{90} - 8 q^{92} + 28 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 + 8 * q^14 + 4 * q^16 - 16 * q^17 + 12 * q^22 + 8 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 + 4 * q^30 + 8 * q^35 - 4 * q^36 - 16 * q^38 + 4 * q^40 - 8 * q^42 + 20 * q^43 - 4 * q^48 + 12 * q^49 + 16 * q^51 - 24 * q^53 + 12 * q^55 - 8 * q^56 - 4 * q^64 - 12 * q^66 + 16 * q^68 - 8 * q^69 + 16 * q^74 + 4 * q^75 - 24 * q^77 - 28 * q^79 + 4 * q^81 + 8 * q^82 - 8 * q^87 - 12 * q^88 - 4 * q^90 - 8 * q^92 + 28 * 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
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	6.46410i	1.94900i	0.224388	+	0.974500i	$0.427962\pi$
$11$	6.46410i	1.94900i	−0.224388	+	0.974500i	$0.572038\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	− 1.00000i	− 0.235702i
$19$	− 7.46410i	− 1.71238i	−0.516659	−	0.856191i	$-0.672825\pi$
$19$	− 7.46410i	− 1.71238i	0.516659	−	0.856191i	$-0.327175\pi$
$20$	1.00000i	0.223607i
$21$	− 2.00000i	− 0.436436i
$22$	6.46410	1.37815
$23$	3.73205	0.778186	0.389093	−	0.921198i	$-0.372788\pi$
$23$	3.73205	0.778186	0.389093	+	0.921198i	$0.372788\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 2.00000i	− 0.377964i
$29$	0.267949	0.0497569	0.0248785	−	0.999690i	$-0.492080\pi$
$29$	0.267949	0.0497569	0.0248785	+	0.999690i	$0.492080\pi$
$30$	1.00000	0.182574
$31$	1.73205i	0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$	1.73205i	0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 6.46410i	− 1.12526i
$34$	4.00000i	0.685994i
$35$	2.00000	0.338062
$36$	−1.00000	−0.166667
$37$	9.19615i	1.51184i	0.654665	+	0.755919i	$0.272810\pi$
$37$	9.19615i	1.51184i	−0.654665	+	0.755919i	$0.727190\pi$
$38$	−7.46410	−1.21084
$39$	0	0
$40$	1.00000	0.158114
$41$	2.00000i	0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$	2.00000i	0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$	−2.00000	−0.308607
$43$	11.9282	1.81903	0.909517	−	0.415667i	$-0.136452\pi$
$43$	11.9282	1.81903	0.909517	+	0.415667i	$0.136452\pi$
$44$	− 6.46410i	− 0.974500i
$45$	− 1.00000i	− 0.149071i
$46$	− 3.73205i	− 0.550261i
$47$	3.53590i	0.515764i	0.966176	+	0.257882i	$0.0830245\pi$
$47$	3.53590i	0.515764i	−0.966176	+	0.257882i	$0.916975\pi$
$48$	−1.00000	−0.144338
$49$	3.00000	0.428571
$50$	1.00000i	0.141421i
$51$	4.00000	0.560112
$52$	0	0
$53$	0.928203	0.127499	0.0637493	−	0.997966i	$-0.479694\pi$
$53$	0.928203	0.127499	0.0637493	+	0.997966i	$0.479694\pi$
$54$	1.00000i	0.136083i
$55$	6.46410	0.871619
$56$	−2.00000	−0.267261
$57$	7.46410i	0.988644i
$58$	− 0.267949i	− 0.0351835i
$59$	− 8.46410i	− 1.10193i	−0.834528	−	0.550966i	$-0.814259\pi$
$59$	− 8.46410i	− 1.10193i	0.834528	−	0.550966i	$-0.185741\pi$
$60$	− 1.00000i	− 0.129099i
$61$	−10.3923	−1.33060	−0.665299	−	0.746577i	$-0.731696\pi$
$61$	−10.3923	−1.33060	−0.665299	+	0.746577i	$0.731696\pi$
$62$	1.73205	0.219971
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−6.46410	−0.795676
$67$	− 11.4641i	− 1.40056i	−0.713867	−	0.700281i	$-0.753058\pi$
$67$	− 11.4641i	− 1.40056i	0.713867	−	0.700281i	$-0.246942\pi$
$68$	4.00000	0.485071
$69$	−3.73205	−0.449286
$70$	− 2.00000i	− 0.239046i
$71$	12.3923i	1.47070i	0.677690	+	0.735348i	$0.262981\pi$
$71$	12.3923i	1.47070i	−0.677690	+	0.735348i	$0.737019\pi$
$72$	1.00000i	0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	9.19615	1.06903
$75$	1.00000	0.115470
$76$	7.46410i	0.856191i
$77$	−12.9282	−1.47331
$78$	0	0
$79$	−13.9282	−1.56705	−0.783523	−	0.621363i	$-0.786579\pi$
$79$	−13.9282	−1.56705	−0.783523	+	0.621363i	$0.786579\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	− 8.92820i	− 0.979998i	−0.871723	−	0.489999i	$-0.836997\pi$
$83$	− 8.92820i	− 0.979998i	0.871723	−	0.489999i	$-0.163003\pi$
$84$	2.00000i	0.218218i
$85$	4.00000i	0.433861i
$86$	− 11.9282i	− 1.28625i
$87$	−0.267949	−0.0287272
$88$	−6.46410	−0.689076
$89$	− 0.535898i	− 0.0568051i	−0.999597	−	0.0284026i	$-0.990958\pi$
$89$	− 0.535898i	− 0.0568051i	0.999597	−	0.0284026i	$-0.00904203\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−3.73205	−0.389093
$93$	− 1.73205i	− 0.179605i
$94$	3.53590	0.364700
$95$	−7.46410	−0.765801
$96$	1.00000i	0.102062i
$97$	− 0.535898i	− 0.0544122i	−0.999630	−	0.0272061i	$-0.991339\pi$
$97$	− 0.535898i	− 0.0544122i	0.999630	−	0.0272061i	$-0.00866105\pi$
$98$	− 3.00000i	− 0.303046i
$99$	6.46410i	0.649667i
$100$	1.00000	0.100000
$101$	2.92820	0.291367	0.145684	−	0.989331i	$-0.453462\pi$
$101$	2.92820	0.291367	0.145684	+	0.989331i	$0.453462\pi$
$102$	− 4.00000i	− 0.396059i
$103$	−11.8564	−1.16825	−0.584123	−	0.811665i	$-0.698562\pi$
$103$	−11.8564	−1.16825	−0.584123	+	0.811665i	$0.698562\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	− 0.928203i	− 0.0901551i
$107$	−7.85641	−0.759507	−0.379754	−	0.925088i	$-0.623991\pi$
$107$	−7.85641	−0.759507	−0.379754	+	0.925088i	$0.623991\pi$
$108$	1.00000	0.0962250
$109$	15.8564i	1.51877i	0.650643	+	0.759384i	$0.274500\pi$
$109$	15.8564i	1.51877i	−0.650643	+	0.759384i	$0.725500\pi$
$110$	− 6.46410i	− 0.616328i
$111$	− 9.19615i	− 0.872860i
$112$	2.00000i	0.188982i
$113$	0.803848	0.0756196	0.0378098	−	0.999285i	$-0.487962\pi$
$113$	0.803848	0.0756196	0.0378098	+	0.999285i	$0.487962\pi$
$114$	7.46410	0.699077
$115$	− 3.73205i	− 0.348016i
$116$	−0.267949	−0.0248785
$117$	0	0
$118$	−8.46410	−0.779184
$119$	− 8.00000i	− 0.733359i
$120$	−1.00000	−0.0912871
$121$	−30.7846	−2.79860
$122$	10.3923i	0.940875i
$123$	− 2.00000i	− 0.180334i
$124$	− 1.73205i	− 0.155543i
$125$	1.00000i	0.0894427i
$126$	2.00000	0.178174
$127$	−4.92820	−0.437307	−0.218654	−	0.975803i	$-0.570166\pi$
$127$	−4.92820	−0.437307	−0.218654	+	0.975803i	$0.570166\pi$
$128$	1.00000i	0.0883883i
$129$	−11.9282	−1.05022
$130$	0	0
$131$	−18.6603	−1.63035	−0.815177	−	0.579212i	$-0.803360\pi$
$131$	−18.6603	−1.63035	−0.815177	+	0.579212i	$0.803360\pi$
$132$	6.46410i	0.562628i
$133$	14.9282	1.29444
$134$	−11.4641	−0.990348
$135$	1.00000i	0.0860663i
$136$	− 4.00000i	− 0.342997i
$137$	2.46410i	0.210522i	0.994445	+	0.105261i	$0.0335679\pi$
$137$	2.46410i	0.210522i	−0.994445	+	0.105261i	$0.966432\pi$
$138$	3.73205i	0.317693i
$139$	−12.9282	−1.09656	−0.548278	−	0.836296i	$-0.684716\pi$
$139$	−12.9282	−1.09656	−0.548278	+	0.836296i	$0.684716\pi$
$140$	−2.00000	−0.169031
$141$	− 3.53590i	− 0.297776i
$142$	12.3923	1.03994
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 0.267949i	− 0.0222520i
$146$	−2.00000	−0.165521
$147$	−3.00000	−0.247436
$148$	− 9.19615i	− 0.755919i
$149$	13.5359i	1.10890i	0.832216	+	0.554452i	$0.187072\pi$
$149$	13.5359i	1.10890i	−0.832216	+	0.554452i	$0.812928\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	10.3923i	0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$	10.3923i	0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$	7.46410	0.605419
$153$	−4.00000	−0.323381
$154$	12.9282i	1.04178i
$155$	1.73205	0.139122
$156$	0	0
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	13.9282i	1.10807i
$159$	−0.928203	−0.0736113
$160$	−1.00000	−0.0790569
$161$	7.46410i	0.588254i
$162$	− 1.00000i	− 0.0785674i
$163$	− 15.0526i	− 1.17901i	−0.807766	−	0.589504i	$-0.799323\pi$
$163$	− 15.0526i	− 1.17901i	0.807766	−	0.589504i	$-0.200677\pi$
$164$	− 2.00000i	− 0.156174i
$165$	−6.46410	−0.503230
$166$	−8.92820	−0.692963
$167$	16.3205i	1.26292i	0.775409	+	0.631459i	$0.217544\pi$
$167$	16.3205i	1.26292i	−0.775409	+	0.631459i	$0.782456\pi$
$168$	2.00000	0.154303
$169$	0	0
$170$	4.00000	0.306786
$171$	− 7.46410i	− 0.570794i
$172$	−11.9282	−0.909517
$173$	10.9282	0.830856	0.415428	−	0.909626i	$-0.363632\pi$
$173$	10.9282	0.830856	0.415428	+	0.909626i	$0.363632\pi$
$174$	0.267949i	0.0203132i
$175$	− 2.00000i	− 0.151186i
$176$	6.46410i	0.487250i
$177$	8.46410i	0.636201i
$178$	−0.535898	−0.0401673
$179$	−19.7321	−1.47484	−0.737421	−	0.675433i	$-0.763957\pi$
$179$	−19.7321	−1.47484	−0.737421	+	0.675433i	$0.763957\pi$
$180$	1.00000i	0.0745356i
$181$	−2.92820	−0.217652	−0.108826	−	0.994061i	$-0.534709\pi$
$181$	−2.92820	−0.217652	−0.108826	+	0.994061i	$0.534709\pi$
$182$	0	0
$183$	10.3923	0.768221
$184$	3.73205i	0.275130i
$185$	9.19615	0.676115
$186$	−1.73205	−0.127000
$187$	− 25.8564i	− 1.89081i
$188$	− 3.53590i	− 0.257882i
$189$	− 2.00000i	− 0.145479i
$190$	7.46410i	0.541503i
$191$	21.4641	1.55309	0.776544	−	0.630063i	$-0.216971\pi$
$191$	21.4641	1.55309	0.776544	+	0.630063i	$0.216971\pi$
$192$	1.00000	0.0721688
$193$	11.3205i	0.814868i	0.913235	+	0.407434i	$0.133576\pi$
$193$	11.3205i	0.814868i	−0.913235	+	0.407434i	$0.866424\pi$
$194$	−0.535898	−0.0384753
$195$	0	0
$196$	−3.00000	−0.214286
$197$	4.39230i	0.312939i	0.987683	+	0.156469i	$0.0500113\pi$
$197$	4.39230i	0.312939i	−0.987683	+	0.156469i	$0.949989\pi$
$198$	6.46410	0.459384
$199$	−5.07180	−0.359530	−0.179765	−	0.983710i	$-0.557534\pi$
$199$	−5.07180	−0.359530	−0.179765	+	0.983710i	$0.557534\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	11.4641i	0.808615i
$202$	− 2.92820i	− 0.206028i
$203$	0.535898i	0.0376127i
$204$	−4.00000	−0.280056
$205$	2.00000	0.139686
$206$	11.8564i	0.826075i
$207$	3.73205	0.259395
$208$	0	0
$209$	48.2487	3.33743
$210$	2.00000i	0.138013i
$211$	−11.3205	−0.779336	−0.389668	−	0.920955i	$-0.627410\pi$
$211$	−11.3205	−0.779336	−0.389668	+	0.920955i	$0.627410\pi$
$212$	−0.928203	−0.0637493
$213$	− 12.3923i	− 0.849107i
$214$	7.85641i	0.537053i
$215$	− 11.9282i	− 0.813497i
$216$	− 1.00000i	− 0.0680414i
$217$	−3.46410	−0.235159
$218$	15.8564	1.07393
$219$	2.00000i	0.135147i
$220$	−6.46410	−0.435810
$221$	0	0
$222$	−9.19615	−0.617205
$223$	− 20.5359i	− 1.37519i	−0.726097	−	0.687593i	$-0.758667\pi$
$223$	− 20.5359i	− 1.37519i	0.726097	−	0.687593i	$-0.241333\pi$
$224$	2.00000	0.133631
$225$	−1.00000	−0.0666667
$226$	− 0.803848i	− 0.0534711i
$227$	16.3923i	1.08800i	0.839087	+	0.543998i	$0.183090\pi$
$227$	16.3923i	1.08800i	−0.839087	+	0.543998i	$0.816910\pi$
$228$	− 7.46410i	− 0.494322i
$229$	7.85641i	0.519166i	0.965721	+	0.259583i	$0.0835851\pi$
$229$	7.85641i	0.519166i	−0.965721	+	0.259583i	$0.916415\pi$
$230$	−3.73205	−0.246084
$231$	12.9282	0.850613
$232$	0.267949i	0.0175917i
$233$	6.12436	0.401220	0.200610	−	0.979671i	$-0.435708\pi$
$233$	6.12436	0.401220	0.200610	+	0.979671i	$0.435708\pi$
$234$	0	0
$235$	3.53590	0.230657
$236$	8.46410i	0.550966i
$237$	13.9282	0.904734
$238$	−8.00000	−0.518563
$239$	16.3923i	1.06033i	0.847894	+	0.530165i	$0.177870\pi$
$239$	16.3923i	1.06033i	−0.847894	+	0.530165i	$0.822130\pi$
$240$	1.00000i	0.0645497i
$241$	− 17.7321i	− 1.14222i	−0.820873	−	0.571111i	$-0.806513\pi$
$241$	− 17.7321i	− 1.14222i	0.820873	−	0.571111i	$-0.193487\pi$
$242$	30.7846i	1.97891i
$243$	−1.00000	−0.0641500
$244$	10.3923	0.665299
$245$	− 3.00000i	− 0.191663i
$246$	−2.00000	−0.127515
$247$	0	0
$248$	−1.73205	−0.109985
$249$	8.92820i	0.565802i
$250$	1.00000	0.0632456
$251$	15.7321	0.992998	0.496499	−	0.868037i	$-0.334619\pi$
$251$	15.7321	0.992998	0.496499	+	0.868037i	$0.334619\pi$
$252$	− 2.00000i	− 0.125988i
$253$	24.1244i	1.51669i
$254$	4.92820i	0.309223i
$255$	− 4.00000i	− 0.250490i
$256$	1.00000	0.0625000
$257$	−5.33975	−0.333084	−0.166542	−	0.986034i	$-0.553260\pi$
$257$	−5.33975	−0.333084	−0.166542	+	0.986034i	$0.553260\pi$
$258$	11.9282i	0.742617i
$259$	−18.3923	−1.14284
$260$	0	0
$261$	0.267949	0.0165856
$262$	18.6603i	1.15283i
$263$	6.12436	0.377644	0.188822	−	0.982011i	$-0.439533\pi$
$263$	6.12436	0.377644	0.188822	+	0.982011i	$0.439533\pi$
$264$	6.46410	0.397838
$265$	− 0.928203i	− 0.0570191i
$266$	− 14.9282i	− 0.915307i
$267$	0.535898i	0.0327964i
$268$	11.4641i	0.700281i
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	1.00000	0.0608581
$271$	1.19615i	0.0726611i	0.999340	+	0.0363305i	$0.0115669\pi$
$271$	1.19615i	0.0726611i	−0.999340	+	0.0363305i	$0.988433\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	2.46410	0.148862
$275$	− 6.46410i	− 0.389800i
$276$	3.73205	0.224643
$277$	−3.92820	−0.236023	−0.118011	−	0.993012i	$-0.537652\pi$
$277$	−3.92820	−0.236023	−0.118011	+	0.993012i	$0.537652\pi$
$278$	12.9282i	0.775382i
$279$	1.73205i	0.103695i
$280$	2.00000i	0.119523i
$281$	8.92820i	0.532612i	0.963889	+	0.266306i	$0.0858032\pi$
$281$	8.92820i	0.532612i	−0.963889	+	0.266306i	$0.914197\pi$
$282$	−3.53590	−0.210560
$283$	−9.92820	−0.590170	−0.295085	−	0.955471i	$-0.595348\pi$
$283$	−9.92820	−0.590170	−0.295085	+	0.955471i	$0.595348\pi$
$284$	− 12.3923i	− 0.735348i
$285$	7.46410	0.442135
$286$	0	0
$287$	−4.00000	−0.236113
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	−0.267949	−0.0157345
$291$	0.535898i	0.0314149i
$292$	2.00000i	0.117041i
$293$	31.8564i	1.86107i	0.366202	+	0.930536i	$0.380658\pi$
$293$	31.8564i	1.86107i	−0.366202	+	0.930536i	$0.619342\pi$
$294$	3.00000i	0.174964i
$295$	−8.46410	−0.492799
$296$	−9.19615	−0.534516
$297$	− 6.46410i	− 0.375085i
$298$	13.5359	0.784114
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	23.8564i	1.37506i
$302$	10.3923	0.598010
$303$	−2.92820	−0.168221
$304$	− 7.46410i	− 0.428096i
$305$	10.3923i	0.595062i
$306$	4.00000i	0.228665i
$307$	− 19.4641i	− 1.11087i	−0.831558	−	0.555437i	$-0.812551\pi$
$307$	− 19.4641i	− 1.11087i	0.831558	−	0.555437i	$-0.187449\pi$
$308$	12.9282	0.736653
$309$	11.8564	0.674487
$310$	− 1.73205i	− 0.0983739i
$311$	−28.3923	−1.60998	−0.804990	−	0.593288i	$-0.797829\pi$
$311$	−28.3923	−1.60998	−0.804990	+	0.593288i	$0.797829\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	5.00000i	0.282166i
$315$	2.00000	0.112687
$316$	13.9282	0.783523
$317$	− 14.5359i	− 0.816417i	−0.912889	−	0.408209i	$-0.866154\pi$
$317$	− 14.5359i	− 0.816417i	0.912889	−	0.408209i	$-0.133846\pi$
$318$	0.928203i	0.0520511i
$319$	1.73205i	0.0969762i
$320$	1.00000i	0.0559017i
$321$	7.85641	0.438502
$322$	7.46410	0.415958
$323$	29.8564i	1.66125i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−15.0526	−0.833684
$327$	− 15.8564i	− 0.876861i
$328$	−2.00000	−0.110432
$329$	−7.07180	−0.389881
$330$	6.46410i	0.355837i
$331$	16.7846i	0.922566i	0.887253	+	0.461283i	$0.152611\pi$
$331$	16.7846i	0.922566i	−0.887253	+	0.461283i	$0.847389\pi$
$332$	8.92820i	0.489999i
$333$	9.19615i	0.503946i
$334$	16.3205	0.893018
$335$	−11.4641	−0.626351
$336$	− 2.00000i	− 0.109109i
$337$	9.32051	0.507720	0.253860	−	0.967241i	$-0.418300\pi$
$337$	9.32051	0.507720	0.253860	+	0.967241i	$0.418300\pi$
$338$	0	0
$339$	−0.803848	−0.0436590
$340$	− 4.00000i	− 0.216930i
$341$	−11.1962	−0.606306
$342$	−7.46410	−0.403612
$343$	20.0000i	1.07990i
$344$	11.9282i	0.643126i
$345$	3.73205i	0.200927i
$346$	− 10.9282i	− 0.587504i
$347$	1.60770	0.0863056	0.0431528	−	0.999068i	$-0.486260\pi$
$347$	1.60770	0.0863056	0.0431528	+	0.999068i	$0.486260\pi$
$348$	0.267949	0.0143636
$349$	21.4641i	1.14895i	0.818523	+	0.574474i	$0.194793\pi$
$349$	21.4641i	1.14895i	−0.818523	+	0.574474i	$0.805207\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	6.46410	0.344538
$353$	2.00000i	0.106449i	0.998583	+	0.0532246i	$0.0169499\pi$
$353$	2.00000i	0.106449i	−0.998583	+	0.0532246i	$0.983050\pi$
$354$	8.46410	0.449862
$355$	12.3923	0.657715
$356$	0.535898i	0.0284026i
$357$	8.00000i	0.423405i
$358$	19.7321i	1.04287i
$359$	− 5.07180i	− 0.267679i	−0.991003	−	0.133840i	$-0.957269\pi$
$359$	− 5.07180i	− 0.267679i	0.991003	−	0.133840i	$-0.0427307\pi$
$360$	1.00000	0.0527046
$361$	−36.7128	−1.93225
$362$	2.92820i	0.153903i
$363$	30.7846	1.61577
$364$	0	0
$365$	−2.00000	−0.104685
$366$	− 10.3923i	− 0.543214i
$367$	15.6077	0.814715	0.407358	−	0.913269i	$-0.366450\pi$
$367$	15.6077	0.814715	0.407358	+	0.913269i	$0.366450\pi$
$368$	3.73205	0.194547
$369$	2.00000i	0.104116i
$370$	− 9.19615i	− 0.478085i
$371$	1.85641i	0.0963798i
$372$	1.73205i	0.0898027i
$373$	−15.7846	−0.817296	−0.408648	−	0.912692i	$-0.634000\pi$
$373$	−15.7846	−0.817296	−0.408648	+	0.912692i	$0.634000\pi$
$374$	−25.8564	−1.33700
$375$	− 1.00000i	− 0.0516398i
$376$	−3.53590	−0.182350
$377$	0	0
$378$	−2.00000	−0.102869
$379$	27.8564i	1.43089i	0.698670	+	0.715444i	$0.253775\pi$
$379$	27.8564i	1.43089i	−0.698670	+	0.715444i	$0.746225\pi$
$380$	7.46410	0.382900
$381$	4.92820	0.252479
$382$	− 21.4641i	− 1.09820i
$383$	− 25.3923i	− 1.29749i	−0.761007	−	0.648743i	$-0.775295\pi$
$383$	− 25.3923i	− 1.29749i	0.761007	−	0.648743i	$-0.224705\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	12.9282i	0.658882i
$386$	11.3205	0.576199
$387$	11.9282	0.606345
$388$	0.535898i	0.0272061i
$389$	−23.7321	−1.20326	−0.601631	−	0.798774i	$-0.705482\pi$
$389$	−23.7321	−1.20326	−0.601631	+	0.798774i	$0.705482\pi$
$390$	0	0
$391$	−14.9282	−0.754952
$392$	3.00000i	0.151523i
$393$	18.6603	0.941285
$394$	4.39230	0.221281
$395$	13.9282i	0.700804i
$396$	− 6.46410i	− 0.324833i
$397$	− 12.1244i	− 0.608504i	−0.952592	−	0.304252i	$-0.901594\pi$
$397$	− 12.1244i	− 0.608504i	0.952592	−	0.304252i	$-0.0984065\pi$
$398$	5.07180i	0.254226i
$399$	−14.9282	−0.747345
$400$	−1.00000	−0.0500000
$401$	− 32.0000i	− 1.59800i	−0.601329	−	0.799002i	$-0.705362\pi$
$401$	− 32.0000i	− 1.59800i	0.601329	−	0.799002i	$-0.294638\pi$
$402$	11.4641	0.571777
$403$	0	0
$404$	−2.92820	−0.145684
$405$	− 1.00000i	− 0.0496904i
$406$	0.535898	0.0265962
$407$	−59.4449	−2.94657
$408$	4.00000i	0.198030i
$409$	− 4.00000i	− 0.197787i	−0.995098	−	0.0988936i	$-0.968470\pi$
$409$	− 4.00000i	− 0.197787i	0.995098	−	0.0988936i	$-0.0315304\pi$
$410$	− 2.00000i	− 0.0987730i
$411$	− 2.46410i	− 0.121545i
$412$	11.8564	0.584123
$413$	16.9282	0.832982
$414$	− 3.73205i	− 0.183420i
$415$	−8.92820	−0.438268
$416$	0	0
$417$	12.9282	0.633097
$418$	− 48.2487i	− 2.35992i
$419$	22.3923	1.09394	0.546968	−	0.837154i	$-0.315782\pi$
$419$	22.3923	1.09394	0.546968	+	0.837154i	$0.315782\pi$
$420$	2.00000	0.0975900
$421$	− 4.39230i	− 0.214068i	−0.994255	−	0.107034i	$-0.965865\pi$
$421$	− 4.39230i	− 0.214068i	0.994255	−	0.107034i	$-0.0341353\pi$
$422$	11.3205i	0.551074i
$423$	3.53590i	0.171921i
$424$	0.928203i	0.0450775i
$425$	4.00000	0.194029
$426$	−12.3923	−0.600409
$427$	− 20.7846i	− 1.00584i
$428$	7.85641	0.379754
$429$	0	0
$430$	−11.9282	−0.575229
$431$	28.3923i	1.36761i	0.729665	+	0.683805i	$0.239676\pi$
$431$	28.3923i	1.36761i	−0.729665	+	0.683805i	$0.760324\pi$
$432$	−1.00000	−0.0481125
$433$	19.3205	0.928484	0.464242	−	0.885708i	$-0.346327\pi$
$433$	19.3205	0.928484	0.464242	+	0.885708i	$0.346327\pi$
$434$	3.46410i	0.166282i
$435$	0.267949i	0.0128472i
$436$	− 15.8564i	− 0.759384i
$437$	− 27.8564i	− 1.33255i
$438$	2.00000	0.0955637
$439$	−17.8564	−0.852240	−0.426120	−	0.904667i	$-0.640120\pi$
$439$	−17.8564	−0.852240	−0.426120	+	0.904667i	$0.640120\pi$
$440$	6.46410i	0.308164i
$441$	3.00000	0.142857
$442$	0	0
$443$	16.3923	0.778822	0.389411	−	0.921064i	$-0.372679\pi$
$443$	16.3923	0.778822	0.389411	+	0.921064i	$0.372679\pi$
$444$	9.19615i	0.436430i
$445$	−0.535898	−0.0254040
$446$	−20.5359	−0.972403
$447$	− 13.5359i	− 0.640226i
$448$	− 2.00000i	− 0.0944911i
$449$	15.7128i	0.741533i	0.928726	+	0.370767i	$0.120905\pi$
$449$	15.7128i	0.741533i	−0.928726	+	0.370767i	$0.879095\pi$
$450$	1.00000i	0.0471405i
$451$	−12.9282	−0.608765
$452$	−0.803848	−0.0378098
$453$	− 10.3923i	− 0.488273i
$454$	16.3923	0.769329
$455$	0	0
$456$	−7.46410	−0.349539
$457$	24.5359i	1.14774i	0.818946	+	0.573870i	$0.194559\pi$
$457$	24.5359i	1.14774i	−0.818946	+	0.573870i	$0.805441\pi$
$458$	7.85641	0.367106
$459$	4.00000	0.186704
$460$	3.73205i	0.174008i
$461$	− 0.464102i	− 0.0216154i	−0.999942	−	0.0108077i	$-0.996560\pi$
$461$	− 0.464102i	− 0.0216154i	0.999942	−	0.0108077i	$-0.00344026\pi$
$462$	− 12.9282i	− 0.601474i
$463$	− 7.07180i	− 0.328654i	−0.986406	−	0.164327i	$-0.947455\pi$
$463$	− 7.07180i	− 0.328654i	0.986406	−	0.164327i	$-0.0525453\pi$
$464$	0.267949	0.0124392
$465$	−1.73205	−0.0803219
$466$	− 6.12436i	− 0.283705i
$467$	−15.8564	−0.733747	−0.366873	−	0.930271i	$-0.619572\pi$
$467$	−15.8564	−0.733747	−0.366873	+	0.930271i	$0.619572\pi$
$468$	0	0
$469$	22.9282	1.05873
$470$	− 3.53590i	− 0.163099i
$471$	5.00000	0.230388
$472$	8.46410	0.389592
$473$	77.1051i	3.54530i
$474$	− 13.9282i	− 0.639744i
$475$	7.46410i	0.342476i
$476$	8.00000i	0.366679i
$477$	0.928203	0.0424995
$478$	16.3923	0.749767
$479$	5.46410i	0.249661i	0.992178	+	0.124831i	$0.0398387\pi$
$479$	5.46410i	0.249661i	−0.992178	+	0.124831i	$0.960161\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−17.7321	−0.807673
$483$	− 7.46410i	− 0.339628i
$484$	30.7846	1.39930
$485$	−0.535898	−0.0243339
$486$	1.00000i	0.0453609i
$487$	− 23.1769i	− 1.05025i	−0.851026	−	0.525123i	$-0.824019\pi$
$487$	− 23.1769i	− 1.05025i	0.851026	−	0.525123i	$-0.175981\pi$
$488$	− 10.3923i	− 0.470438i
$489$	15.0526i	0.680700i
$490$	−3.00000	−0.135526
$491$	−17.3205	−0.781664	−0.390832	−	0.920462i	$-0.627813\pi$
$491$	−17.3205	−0.781664	−0.390832	+	0.920462i	$0.627813\pi$
$492$	2.00000i	0.0901670i
$493$	−1.07180	−0.0482713
$494$	0	0
$495$	6.46410	0.290540
$496$	1.73205i	0.0777714i
$497$	−24.7846	−1.11174
$498$	8.92820	0.400082
$499$	− 6.53590i	− 0.292587i	−0.989241	−	0.146293i	$-0.953266\pi$
$499$	− 6.53590i	− 0.292587i	0.989241	−	0.146293i	$-0.0467344\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 16.3205i	− 0.729147i
$502$	− 15.7321i	− 0.702156i
$503$	−31.1769	−1.39011	−0.695055	−	0.718957i	$-0.744620\pi$
$503$	−31.1769	−1.39011	−0.695055	+	0.718957i	$0.744620\pi$
$504$	−2.00000	−0.0890871
$505$	− 2.92820i	− 0.130303i
$506$	24.1244	1.07246
$507$	0	0
$508$	4.92820	0.218654
$509$	1.39230i	0.0617128i	0.999524	+	0.0308564i	$0.00982346\pi$
$509$	1.39230i	0.0617128i	−0.999524	+	0.0308564i	$0.990177\pi$
$510$	−4.00000	−0.177123
$511$	4.00000	0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	7.46410i	0.329548i
$514$	5.33975i	0.235526i
$515$	11.8564i	0.522456i
$516$	11.9282	0.525110
$517$	−22.8564	−1.00522
$518$	18.3923i	0.808111i
$519$	−10.9282	−0.479695
$520$	0	0
$521$	−17.3205	−0.758825	−0.379413	−	0.925228i	$-0.623874\pi$
$521$	−17.3205	−0.758825	−0.379413	+	0.925228i	$0.623874\pi$
$522$	− 0.267949i	− 0.0117278i
$523$	11.7846	0.515305	0.257653	−	0.966238i	$-0.417051\pi$
$523$	11.7846	0.515305	0.257653	+	0.966238i	$0.417051\pi$
$524$	18.6603	0.815177
$525$	2.00000i	0.0872872i
$526$	− 6.12436i	− 0.267035i
$527$	− 6.92820i	− 0.301797i
$528$	− 6.46410i	− 0.281314i
$529$	−9.07180	−0.394426
$530$	−0.928203	−0.0403186
$531$	− 8.46410i	− 0.367311i
$532$	−14.9282	−0.647220
$533$	0	0
$534$	0.535898	0.0231906
$535$	7.85641i	0.339662i
$536$	11.4641	0.495174
$537$	19.7321	0.851501
$538$	− 12.0000i	− 0.517357i
$539$	19.3923i	0.835286i
$540$	− 1.00000i	− 0.0430331i
$541$	− 26.9282i	− 1.15773i	−0.815422	−	0.578867i	$-0.803495\pi$
$541$	− 26.9282i	− 1.15773i	0.815422	−	0.578867i	$-0.196505\pi$
$542$	1.19615	0.0513791
$543$	2.92820	0.125661
$544$	4.00000i	0.171499i
$545$	15.8564	0.679214
$546$	0	0
$547$	22.9282	0.980339	0.490170	−	0.871627i	$-0.336935\pi$
$547$	22.9282	0.980339	0.490170	+	0.871627i	$0.336935\pi$
$548$	− 2.46410i	− 0.105261i
$549$	−10.3923	−0.443533
$550$	−6.46410	−0.275630
$551$	− 2.00000i	− 0.0852029i
$552$	− 3.73205i	− 0.158847i
$553$	− 27.8564i	− 1.18457i
$554$	3.92820i	0.166893i
$555$	−9.19615	−0.390355
$556$	12.9282	0.548278
$557$	17.7128i	0.750516i	0.926920	+	0.375258i	$0.122446\pi$
$557$	17.7128i	0.750516i	−0.926920	+	0.375258i	$0.877554\pi$
$558$	1.73205	0.0733236
$559$	0	0
$560$	2.00000	0.0845154
$561$	25.8564i	1.09166i
$562$	8.92820	0.376614
$563$	−4.67949	−0.197217	−0.0986085	−	0.995126i	$-0.531439\pi$
$563$	−4.67949	−0.197217	−0.0986085	+	0.995126i	$0.531439\pi$
$564$	3.53590i	0.148888i
$565$	− 0.803848i	− 0.0338181i
$566$	9.92820i	0.417314i
$567$	2.00000i	0.0839921i
$568$	−12.3923	−0.519970
$569$	29.3205	1.22918	0.614590	−	0.788847i	$-0.289322\pi$
$569$	29.3205	1.22918	0.614590	+	0.788847i	$0.289322\pi$
$570$	− 7.46410i	− 0.312637i
$571$	−17.1769	−0.718832	−0.359416	−	0.933178i	$-0.617024\pi$
$571$	−17.1769	−0.718832	−0.359416	+	0.933178i	$0.617024\pi$
$572$	0	0
$573$	−21.4641	−0.896676
$574$	4.00000i	0.166957i
$575$	−3.73205	−0.155637
$576$	−1.00000	−0.0416667
$577$	10.0000i	0.416305i	0.978096	+	0.208153i	$0.0667451\pi$
$577$	10.0000i	0.416305i	−0.978096	+	0.208153i	$0.933255\pi$
$578$	1.00000i	0.0415945i
$579$	− 11.3205i	− 0.470464i
$580$	0.267949i	0.0111260i
$581$	17.8564	0.740809
$582$	0.535898	0.0222137
$583$	6.00000i	0.248495i
$584$	2.00000	0.0827606
$585$	0	0
$586$	31.8564	1.31598
$587$	− 2.39230i	− 0.0987410i	−0.998781	−	0.0493705i	$-0.984278\pi$
$587$	− 2.39230i	− 0.0987410i	0.998781	−	0.0493705i	$-0.0157215\pi$
$588$	3.00000	0.123718
$589$	12.9282	0.532697
$590$	8.46410i	0.348462i
$591$	− 4.39230i	− 0.180675i
$592$	9.19615i	0.377960i
$593$	− 45.1051i	− 1.85225i	−0.377223	−	0.926123i	$-0.623121\pi$
$593$	− 45.1051i	− 1.85225i	0.377223	−	0.926123i	$-0.376879\pi$
$594$	−6.46410	−0.265225
$595$	−8.00000	−0.327968
$596$	− 13.5359i	− 0.554452i
$597$	5.07180	0.207575
$598$	0	0
$599$	−10.3923	−0.424618	−0.212309	−	0.977203i	$-0.568098\pi$
$599$	−10.3923	−0.424618	−0.212309	+	0.977203i	$0.568098\pi$
$600$	1.00000i	0.0408248i
$601$	−19.7846	−0.807031	−0.403516	−	0.914973i	$-0.632212\pi$
$601$	−19.7846	−0.807031	−0.403516	+	0.914973i	$0.632212\pi$
$602$	23.8564	0.972315
$603$	− 11.4641i	− 0.466854i
$604$	− 10.3923i	− 0.422857i
$605$	30.7846i	1.25157i
$606$	2.92820i	0.118950i
$607$	−19.1769	−0.778367	−0.389183	−	0.921160i	$-0.627243\pi$
$607$	−19.1769	−0.778367	−0.389183	+	0.921160i	$0.627243\pi$
$608$	−7.46410	−0.302709
$609$	− 0.535898i	− 0.0217157i
$610$	10.3923	0.420772
$611$	0	0
$612$	4.00000	0.161690
$613$	− 39.0526i	− 1.57732i	−0.614831	−	0.788659i	$-0.710776\pi$
$613$	− 39.0526i	− 1.57732i	0.614831	−	0.788659i	$-0.289224\pi$
$614$	−19.4641	−0.785507
$615$	−2.00000	−0.0806478
$616$	− 12.9282i	− 0.520892i
$617$	22.4641i	0.904371i	0.891924	+	0.452185i	$0.149355\pi$
$617$	22.4641i	0.904371i	−0.891924	+	0.452185i	$0.850645\pi$
$618$	− 11.8564i	− 0.476935i
$619$	24.2487i	0.974638i	0.873224	+	0.487319i	$0.162025\pi$
$619$	24.2487i	0.974638i	−0.873224	+	0.487319i	$0.837975\pi$
$620$	−1.73205	−0.0695608
$621$	−3.73205	−0.149762
$622$	28.3923i	1.13843i
$623$	1.07180	0.0429406
$624$	0	0
$625$	1.00000	0.0400000
$626$	28.0000i	1.11911i
$627$	−48.2487	−1.92687
$628$	5.00000	0.199522
$629$	− 36.7846i	− 1.46670i
$630$	− 2.00000i	− 0.0796819i
$631$	− 31.4641i	− 1.25257i	−0.779596	−	0.626283i	$-0.784575\pi$
$631$	− 31.4641i	− 1.25257i	0.779596	−	0.626283i	$-0.215425\pi$
$632$	− 13.9282i	− 0.554034i
$633$	11.3205	0.449950
$634$	−14.5359	−0.577294
$635$	4.92820i	0.195570i
$636$	0.928203	0.0368057
$637$	0	0
$638$	1.73205	0.0685725
$639$	12.3923i	0.490232i
$640$	1.00000	0.0395285
$641$	0.143594	0.00567160	0.00283580	−	0.999996i	$-0.499097\pi$
$641$	0.143594	0.00567160	0.00283580	+	0.999996i	$0.499097\pi$
$642$	− 7.85641i	− 0.310068i
$643$	20.5359i	0.809857i	0.914348	+	0.404928i	$0.132704\pi$
$643$	20.5359i	0.809857i	−0.914348	+	0.404928i	$0.867296\pi$
$644$	− 7.46410i	− 0.294127i
$645$	11.9282i	0.469673i
$646$	29.8564	1.17468
$647$	−13.3205	−0.523683	−0.261842	−	0.965111i	$-0.584330\pi$
$647$	−13.3205	−0.523683	−0.261842	+	0.965111i	$0.584330\pi$
$648$	1.00000i	0.0392837i
$649$	54.7128	2.14767
$650$	0	0
$651$	3.46410	0.135769
$652$	15.0526i	0.589504i
$653$	4.24871	0.166265	0.0831325	−	0.996539i	$-0.473508\pi$
$653$	4.24871	0.166265	0.0831325	+	0.996539i	$0.473508\pi$
$654$	−15.8564	−0.620035
$655$	18.6603i	0.729116i
$656$	2.00000i	0.0780869i
$657$	− 2.00000i	− 0.0780274i
$658$	7.07180i	0.275687i
$659$	−0.267949	−0.0104378	−0.00521891	−	0.999986i	$-0.501661\pi$
$659$	−0.267949	−0.0104378	−0.00521891	+	0.999986i	$0.501661\pi$
$660$	6.46410	0.251615
$661$	− 8.67949i	− 0.337593i	−0.985651	−	0.168797i	$-0.946012\pi$
$661$	− 8.67949i	− 0.337593i	0.985651	−	0.168797i	$-0.0539881\pi$
$662$	16.7846	0.652352
$663$	0	0
$664$	8.92820	0.346481
$665$	− 14.9282i	− 0.578891i
$666$	9.19615	0.356344
$667$	1.00000	0.0387202
$668$	− 16.3205i	− 0.631459i
$669$	20.5359i	0.793964i
$670$	11.4641i	0.442897i
$671$	− 67.1769i	− 2.59334i
$672$	−2.00000	−0.0771517
$673$	32.0000	1.23351	0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000	1.23351	0.616755	+	0.787155i	$0.288447\pi$
$674$	− 9.32051i	− 0.359013i
$675$	1.00000	0.0384900
$676$	0	0
$677$	32.3923	1.24494	0.622469	−	0.782645i	$-0.286130\pi$
$677$	32.3923	1.24494	0.622469	+	0.782645i	$0.286130\pi$
$678$	0.803848i	0.0308716i
$679$	1.07180	0.0411318
$680$	−4.00000	−0.153393
$681$	− 16.3923i	− 0.628154i
$682$	11.1962i	0.428723i
$683$	40.7846i	1.56058i	0.625418	+	0.780290i	$0.284928\pi$
$683$	40.7846i	1.56058i	−0.625418	+	0.780290i	$0.715072\pi$
$684$	7.46410i	0.285397i
$685$	2.46410	0.0941485
$686$	20.0000	0.763604
$687$	− 7.85641i	− 0.299741i
$688$	11.9282	0.454758
$689$	0	0
$690$	3.73205	0.142077
$691$	− 27.1769i	− 1.03386i	−0.856028	−	0.516929i	$-0.827075\pi$
$691$	− 27.1769i	− 1.03386i	0.856028	−	0.516929i	$-0.172925\pi$
$692$	−10.9282	−0.415428
$693$	−12.9282	−0.491102
$694$	− 1.60770i	− 0.0610273i
$695$	12.9282i	0.490395i
$696$	− 0.267949i	− 0.0101566i
$697$	− 8.00000i	− 0.303022i
$698$	21.4641	0.812428
$699$	−6.12436	−0.231644
$700$	2.00000i	0.0755929i
$701$	−0.267949	−0.0101203	−0.00506015	−	0.999987i	$-0.501611\pi$
$701$	−0.267949	−0.0101203	−0.00506015	+	0.999987i	$0.501611\pi$
$702$	0	0
$703$	68.6410	2.58884
$704$	− 6.46410i	− 0.243625i
$705$	−3.53590	−0.133170
$706$	2.00000	0.0752710
$707$	5.85641i	0.220253i
$708$	− 8.46410i	− 0.318100i
$709$	22.9282i	0.861087i	0.902570	+	0.430543i	$0.141678\pi$
$709$	22.9282i	0.861087i	−0.902570	+	0.430543i	$0.858322\pi$
$710$	− 12.3923i	− 0.465075i
$711$	−13.9282	−0.522348
$712$	0.535898	0.0200836
$713$	6.46410i	0.242083i
$714$	8.00000	0.299392
$715$	0	0
$716$	19.7321	0.737421
$717$	− 16.3923i	− 0.612182i
$718$	−5.07180	−0.189278
$719$	−34.6410	−1.29189	−0.645946	−	0.763383i	$-0.723537\pi$
$719$	−34.6410	−1.29189	−0.645946	+	0.763383i	$0.723537\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 23.7128i	− 0.883111i
$722$	36.7128i	1.36631i
$723$	17.7321i	0.659462i
$724$	2.92820	0.108826
$725$	−0.267949	−0.00995138
$726$	− 30.7846i	− 1.14252i
$727$	−31.7128	−1.17616	−0.588082	−	0.808802i	$-0.700117\pi$
$727$	−31.7128	−1.17616	−0.588082	+	0.808802i	$0.700117\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	2.00000i	0.0740233i
$731$	−47.7128	−1.76472
$732$	−10.3923	−0.384111
$733$	50.9282i	1.88108i	0.339688	+	0.940538i	$0.389678\pi$
$733$	50.9282i	1.88108i	−0.339688	+	0.940538i	$0.610322\pi$
$734$	− 15.6077i	− 0.576091i
$735$	3.00000i	0.110657i
$736$	− 3.73205i	− 0.137565i
$737$	74.1051	2.72970
$738$	2.00000	0.0736210
$739$	− 19.3205i	− 0.710716i	−0.934730	−	0.355358i	$-0.884359\pi$
$739$	− 19.3205i	− 0.710716i	0.934730	−	0.355358i	$-0.115641\pi$
$740$	−9.19615	−0.338057
$741$	0	0
$742$	1.85641	0.0681508
$743$	− 40.4641i	− 1.48448i	−0.670132	−	0.742242i	$-0.733763\pi$
$743$	− 40.4641i	− 1.48448i	0.670132	−	0.742242i	$-0.266237\pi$
$744$	1.73205	0.0635001
$745$	13.5359	0.495917
$746$	15.7846i	0.577916i
$747$	− 8.92820i	− 0.326666i
$748$	25.8564i	0.945404i
$749$	− 15.7128i	− 0.574134i
$750$	−1.00000	−0.0365148
$751$	14.0718	0.513487	0.256744	−	0.966480i	$-0.417350\pi$
$751$	14.0718	0.513487	0.256744	+	0.966480i	$0.417350\pi$
$752$	3.53590i	0.128941i
$753$	−15.7321	−0.573308
$754$	0	0
$755$	10.3923	0.378215
$756$	2.00000i	0.0727393i
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	27.8564	1.01179
$759$	− 24.1244i	− 0.875659i
$760$	− 7.46410i	− 0.270751i
$761$	− 5.07180i	− 0.183852i	−0.995766	−	0.0919262i	$-0.970698\pi$
$761$	− 5.07180i	− 0.183852i	0.995766	−	0.0919262i	$-0.0293024\pi$
$762$	− 4.92820i	− 0.178530i
$763$	−31.7128	−1.14808
$764$	−21.4641	−0.776544
$765$	4.00000i	0.144620i
$766$	−25.3923	−0.917461
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 11.5885i	− 0.417890i	−0.977927	−	0.208945i	$-0.932997\pi$
$769$	− 11.5885i	− 0.417890i	0.977927	−	0.208945i	$-0.0670030\pi$
$770$	12.9282	0.465900
$771$	5.33975	0.192306
$772$	− 11.3205i	− 0.407434i
$773$	− 27.7128i	− 0.996761i	−0.866959	−	0.498380i	$-0.833928\pi$
$773$	− 27.7128i	− 0.996761i	0.866959	−	0.498380i	$-0.166072\pi$
$774$	− 11.9282i	− 0.428750i
$775$	− 1.73205i	− 0.0622171i
$776$	0.535898	0.0192376
$777$	18.3923	0.659820
$778$	23.7321i	0.850835i
$779$	14.9282	0.534858
$780$	0	0
$781$	−80.1051	−2.86639
$782$	14.9282i	0.533831i
$783$	−0.267949	−0.00957572
$784$	3.00000	0.107143
$785$	5.00000i	0.178458i
$786$	− 18.6603i	− 0.665589i
$787$	− 1.73205i	− 0.0617409i	−0.999523	−	0.0308705i	$-0.990172\pi$
$787$	− 1.73205i	− 0.0617409i	0.999523	−	0.0308705i	$-0.00982794\pi$
$788$	− 4.39230i	− 0.156469i
$789$	−6.12436	−0.218033
$790$	13.9282	0.495543
$791$	1.60770i	0.0571631i
$792$	−6.46410	−0.229692
$793$	0	0
$794$	−12.1244	−0.430277
$795$	0.928203i	0.0329200i
$796$	5.07180	0.179765
$797$	−10.1436	−0.359305	−0.179652	−	0.983730i	$-0.557497\pi$
$797$	−10.1436	−0.359305	−0.179652	+	0.983730i	$0.557497\pi$
$798$	14.9282i	0.528453i
$799$	− 14.1436i	− 0.500364i
$800$	1.00000i	0.0353553i
$801$	− 0.535898i	− 0.0189350i
$802$	−32.0000	−1.12996
$803$	12.9282	0.456226
$804$	− 11.4641i	− 0.404308i
$805$	7.46410	0.263075
$806$	0	0
$807$	−12.0000	−0.422420
$808$	2.92820i	0.103014i
$809$	−42.7846	−1.50423	−0.752113	−	0.659034i	$-0.770965\pi$
$809$	−42.7846	−1.50423	−0.752113	+	0.659034i	$0.770965\pi$
$810$	−1.00000	−0.0351364
$811$	− 40.7846i	− 1.43214i	−0.698028	−	0.716071i	$-0.745939\pi$
$811$	− 40.7846i	− 1.43214i	0.698028	−	0.716071i	$-0.254061\pi$
$812$	− 0.535898i	− 0.0188063i
$813$	− 1.19615i	− 0.0419509i
$814$	59.4449i	2.08354i
$815$	−15.0526	−0.527268
$816$	4.00000	0.140028
$817$	− 89.0333i	− 3.11488i
$818$	−4.00000	−0.139857
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	14.6077i	0.509812i	0.966966	+	0.254906i	$0.0820445\pi$
$821$	14.6077i	0.509812i	−0.966966	+	0.254906i	$0.917955\pi$
$822$	−2.46410	−0.0859454
$823$	39.1769	1.36562	0.682811	−	0.730595i	$-0.260757\pi$
$823$	39.1769	1.36562	0.682811	+	0.730595i	$0.260757\pi$
$824$	− 11.8564i	− 0.413037i
$825$	6.46410i	0.225051i
$826$	− 16.9282i	− 0.589008i
$827$	− 17.3205i	− 0.602293i	−0.953578	−	0.301147i	$-0.902631\pi$
$827$	− 17.3205i	− 0.602293i	0.953578	−	0.301147i	$-0.0973693\pi$
$828$	−3.73205	−0.129698
$829$	16.5359	0.574315	0.287158	−	0.957883i	$-0.407290\pi$
$829$	16.5359	0.574315	0.287158	+	0.957883i	$0.407290\pi$
$830$	8.92820i	0.309902i
$831$	3.92820	0.136268
$832$	0	0
$833$	−12.0000	−0.415775
$834$	− 12.9282i	− 0.447667i
$835$	16.3205	0.564794
$836$	−48.2487	−1.66872
$837$	− 1.73205i	− 0.0598684i
$838$	− 22.3923i	− 0.773529i
$839$	43.5692i	1.50418i	0.659062	+	0.752088i	$0.270953\pi$
$839$	43.5692i	1.50418i	−0.659062	+	0.752088i	$0.729047\pi$
$840$	− 2.00000i	− 0.0690066i
$841$	−28.9282	−0.997524
$842$	−4.39230	−0.151369
$843$	− 8.92820i	− 0.307504i
$844$	11.3205	0.389668
$845$	0	0
$846$	3.53590	0.121567
$847$	− 61.5692i	− 2.11554i
$848$	0.928203	0.0318746
$849$	9.92820	0.340735
$850$	− 4.00000i	− 0.137199i
$851$	34.3205i	1.17649i
$852$	12.3923i	0.424553i
$853$	− 43.8372i	− 1.50096i	−0.660895	−	0.750478i	$-0.729823\pi$
$853$	− 43.8372i	− 1.50096i	0.660895	−	0.750478i	$-0.270177\pi$
$854$	−20.7846	−0.711235
$855$	−7.46410	−0.255267
$856$	− 7.85641i	− 0.268526i
$857$	24.5167	0.837473	0.418737	−	0.908108i	$-0.362473\pi$
$857$	24.5167	0.837473	0.418737	+	0.908108i	$0.362473\pi$
$858$	0	0
$859$	−35.1769	−1.20022	−0.600110	−	0.799917i	$-0.704877\pi$
$859$	−35.1769	−1.20022	−0.600110	+	0.799917i	$0.704877\pi$
$860$	11.9282i	0.406748i
$861$	4.00000	0.136320
$862$	28.3923	0.967046
$863$	− 35.5359i	− 1.20966i	−0.796356	−	0.604828i	$-0.793242\pi$
$863$	− 35.5359i	− 1.20966i	0.796356	−	0.604828i	$-0.206758\pi$
$864$	1.00000i	0.0340207i
$865$	− 10.9282i	− 0.371570i
$866$	− 19.3205i	− 0.656538i
$867$	1.00000	0.0339618
$868$	3.46410	0.117579
$869$	− 90.0333i	− 3.05417i
$870$	0.267949	0.00908433
$871$	0	0
$872$	−15.8564	−0.536966
$873$	− 0.535898i	− 0.0181374i
$874$	−27.8564	−0.942257
$875$	−2.00000	−0.0676123
$876$	− 2.00000i	− 0.0675737i
$877$	3.87564i	0.130871i	0.997857	+	0.0654356i	$0.0208437\pi$
$877$	3.87564i	0.130871i	−0.997857	+	0.0654356i	$0.979156\pi$
$878$	17.8564i	0.602625i
$879$	− 31.8564i	− 1.07449i
$880$	6.46410	0.217905
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	− 3.00000i	− 0.101015i
$883$	29.9282	1.00716	0.503582	−	0.863947i	$-0.332015\pi$
$883$	29.9282	1.00716	0.503582	+	0.863947i	$0.332015\pi$
$884$	0	0
$885$	8.46410	0.284518
$886$	− 16.3923i	− 0.550710i
$887$	14.1244	0.474249	0.237125	−	0.971479i	$-0.423795\pi$
$887$	14.1244	0.474249	0.237125	+	0.971479i	$0.423795\pi$
$888$	9.19615	0.308603
$889$	− 9.85641i	− 0.330573i
$890$	0.535898i	0.0179634i
$891$	6.46410i	0.216556i
$892$	20.5359i	0.687593i
$893$	26.3923	0.883185
$894$	−13.5359	−0.452708
$895$	19.7321i	0.659570i
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	15.7128	0.524343
$899$	0.464102i	0.0154787i
$900$	1.00000	0.0333333
$901$	−3.71281	−0.123692
$902$	12.9282i	0.430462i
$903$	− 23.8564i	− 0.793891i
$904$	0.803848i	0.0267356i
$905$	2.92820i	0.0973368i
$906$	−10.3923	−0.345261
$907$	−20.8564	−0.692526	−0.346263	−	0.938138i	$-0.612549\pi$
$907$	−20.8564	−0.692526	−0.346263	+	0.938138i	$0.612549\pi$
$908$	− 16.3923i	− 0.543998i
$909$	2.92820	0.0971224
$910$	0	0
$911$	24.2487	0.803396	0.401698	−	0.915772i	$-0.368420\pi$
$911$	24.2487	0.803396	0.401698	+	0.915772i	$0.368420\pi$
$912$	7.46410i	0.247161i
$913$	57.7128	1.91002
$914$	24.5359	0.811575
$915$	− 10.3923i	− 0.343559i
$916$	− 7.85641i	− 0.259583i
$917$	− 37.3205i	− 1.23243i
$918$	− 4.00000i	− 0.132020i
$919$	14.6410	0.482963	0.241481	−	0.970405i	$-0.422367\pi$
$919$	14.6410	0.482963	0.241481	+	0.970405i	$0.422367\pi$
$920$	3.73205	0.123042
$921$	19.4641i	0.641364i
$922$	−0.464102	−0.0152844
$923$	0	0
$924$	−12.9282	−0.425307
$925$	− 9.19615i	− 0.302368i
$926$	−7.07180	−0.232394
$927$	−11.8564	−0.389415
$928$	− 0.267949i	− 0.00879586i
$929$	6.14359i	0.201565i	0.994908	+	0.100782i	$0.0321346\pi$
$929$	6.14359i	0.201565i	−0.994908	+	0.100782i	$0.967865\pi$
$930$	1.73205i	0.0567962i
$931$	− 22.3923i	− 0.733878i
$932$	−6.12436	−0.200610
$933$	28.3923	0.929522
$934$	15.8564i	0.518837i
$935$	−25.8564	−0.845595
$936$	0	0
$937$	−24.6410	−0.804987	−0.402493	−	0.915423i	$-0.631856\pi$
$937$	−24.6410	−0.804987	−0.402493	+	0.915423i	$0.631856\pi$
$938$	− 22.9282i	− 0.748632i
$939$	28.0000	0.913745
$940$	−3.53590	−0.115328
$941$	14.7846i	0.481965i	0.970530	+	0.240982i	$0.0774696\pi$
$941$	14.7846i	0.481965i	−0.970530	+	0.240982i	$0.922530\pi$
$942$	− 5.00000i	− 0.162909i
$943$	7.46410i	0.243065i
$944$	− 8.46410i	− 0.275483i
$945$	−2.00000	−0.0650600
$946$	77.1051	2.50690
$947$	9.46410i	0.307542i	0.988107	+	0.153771i	$0.0491418\pi$
$947$	9.46410i	0.307542i	−0.988107	+	0.153771i	$0.950858\pi$
$948$	−13.9282	−0.452367
$949$	0	0
$950$	7.46410	0.242167
$951$	14.5359i	0.471359i
$952$	8.00000	0.259281
$953$	−12.2679	−0.397398	−0.198699	−	0.980061i	$-0.563672\pi$
$953$	−12.2679	−0.397398	−0.198699	+	0.980061i	$0.563672\pi$
$954$	− 0.928203i	− 0.0300517i
$955$	− 21.4641i	− 0.694562i
$956$	− 16.3923i	− 0.530165i
$957$	− 1.73205i	− 0.0559893i
$958$	5.46410	0.176537
$959$	−4.92820	−0.159140
$960$	− 1.00000i	− 0.0322749i
$961$	28.0000	0.903226
$962$	0	0
$963$	−7.85641	−0.253169
$964$	17.7321i	0.571111i
$965$	11.3205	0.364420
$966$	−7.46410	−0.240154
$967$	41.4641i	1.33340i	0.745328	+	0.666698i	$0.232293\pi$
$967$	41.4641i	1.33340i	−0.745328	+	0.666698i	$0.767707\pi$
$968$	− 30.7846i	− 0.989455i
$969$	− 29.8564i	− 0.959126i
$970$	0.535898i	0.0172067i
$971$	4.53590	0.145564	0.0727820	−	0.997348i	$-0.476812\pi$
$971$	4.53590	0.145564	0.0727820	+	0.997348i	$0.476812\pi$
$972$	1.00000	0.0320750
$973$	− 25.8564i	− 0.828918i
$974$	−23.1769	−0.742636
$975$	0	0
$976$	−10.3923	−0.332650
$977$	12.6077i	0.403356i	0.979452	+	0.201678i	$0.0646394\pi$
$977$	12.6077i	0.403356i	−0.979452	+	0.201678i	$0.935361\pi$
$978$	15.0526	0.481328
$979$	3.46410	0.110713
$980$	3.00000i	0.0958315i
$981$	15.8564i	0.506256i
$982$	17.3205i	0.552720i
$983$	36.6077i	1.16760i	0.811896	+	0.583802i	$0.198436\pi$
$983$	36.6077i	1.16760i	−0.811896	+	0.583802i	$0.801564\pi$
$984$	2.00000	0.0637577
$985$	4.39230	0.139950
$986$	1.07180i	0.0341330i
$987$	7.07180	0.225098
$988$	0	0
$989$	44.5167	1.41555
$990$	− 6.46410i	− 0.205443i
$991$	−52.8564	−1.67904	−0.839520	−	0.543329i	$-0.817163\pi$
$991$	−52.8564	−1.67904	−0.839520	+	0.543329i	$0.817163\pi$
$992$	1.73205	0.0549927
$993$	− 16.7846i	− 0.532643i
$994$	24.7846i	0.786120i
$995$	5.07180i	0.160787i
$996$	− 8.92820i	− 0.282901i
$997$	35.5692	1.12649	0.563244	−	0.826290i	$-0.309553\pi$
$997$	35.5692	1.12649	0.563244	+	0.826290i	$0.309553\pi$
$998$	−6.53590	−0.206890
$999$	− 9.19615i	− 0.290953i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.o.1351.2		4
13.5	odd	4		5070.2.a.y.1.1		2
13.8	odd	4		5070.2.a.bg.1.2		2
13.9	even	3		390.2.bb.b.361.1	yes	4
13.10	even	6		390.2.bb.b.121.1	✓	4
13.12	even	2	inner	5070.2.b.o.1351.3		4
39.23	odd	6		1170.2.bs.e.901.2		4
39.35	odd	6		1170.2.bs.e.361.2		4
65.9	even	6		1950.2.bc.b.751.2		4
65.22	odd	12		1950.2.y.c.49.2		4
65.23	odd	12		1950.2.y.c.199.2		4
65.48	odd	12		1950.2.y.f.49.1		4
65.49	even	6		1950.2.bc.b.901.2		4
65.62	odd	12		1950.2.y.f.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.b.121.1	✓	4	13.10	even	6
390.2.bb.b.361.1	yes	4	13.9	even	3
1170.2.bs.e.361.2		4	39.35	odd	6
1170.2.bs.e.901.2		4	39.23	odd	6
1950.2.y.c.49.2		4	65.22	odd	12
1950.2.y.c.199.2		4	65.23	odd	12
1950.2.y.f.49.1		4	65.48	odd	12
1950.2.y.f.199.1		4	65.62	odd	12
1950.2.bc.b.751.2		4	65.9	even	6
1950.2.bc.b.901.2		4	65.49	even	6
5070.2.a.y.1.1		2	13.5	odd	4
5070.2.a.bg.1.2		2	13.8	odd	4
5070.2.b.o.1351.2		4	1.1	even	1	trivial
5070.2.b.o.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.46410i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000i q^{18} -7.46410i q^{19} +1.00000i q^{20} -2.00000i q^{21} +6.46410 q^{22} +3.73205 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} +0.267949 q^{29} +1.00000 q^{30} +1.73205i q^{31} -1.00000i q^{32} -6.46410i q^{33} +4.00000i q^{34} +2.00000 q^{35} -1.00000 q^{36} +9.19615i q^{37} -7.46410 q^{38} +1.00000 q^{40} +2.00000i q^{41} -2.00000 q^{42} +11.9282 q^{43} -6.46410i q^{44} -1.00000i q^{45} -3.73205i q^{46} +3.53590i q^{47} -1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} +4.00000 q^{51} +0.928203 q^{53} +1.00000i q^{54} +6.46410 q^{55} -2.00000 q^{56} +7.46410i q^{57} -0.267949i q^{58} -8.46410i q^{59} -1.00000i q^{60} -10.3923 q^{61} +1.73205 q^{62} +2.00000i q^{63} -1.00000 q^{64} -6.46410 q^{66} -11.4641i q^{67} +4.00000 q^{68} -3.73205 q^{69} -2.00000i q^{70} +12.3923i q^{71} +1.00000i q^{72} -2.00000i q^{73} +9.19615 q^{74} +1.00000 q^{75} +7.46410i q^{76} -12.9282 q^{77} -13.9282 q^{79} -1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} -8.92820i q^{83} +2.00000i q^{84} +4.00000i q^{85} -11.9282i q^{86} -0.267949 q^{87} -6.46410 q^{88} -0.535898i q^{89} -1.00000 q^{90} -3.73205 q^{92} -1.73205i q^{93} +3.53590 q^{94} -7.46410 q^{95} +1.00000i q^{96} -0.535898i q^{97} -3.00000i q^{98} +6.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} + 4 q^{12} + 8 q^{14} + 4 q^{16} - 16 q^{17} + 12 q^{22} + 8 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} + 4 q^{30} + 8 q^{35} - 4 q^{36} - 16 q^{38} + 4 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{48} + 12 q^{49} + 16 q^{51} - 24 q^{53} + 12 q^{55} - 8 q^{56} - 4 q^{64} - 12 q^{66} + 16 q^{68} - 8 q^{69} + 16 q^{74} + 4 q^{75} - 24 q^{77} - 28 q^{79} + 4 q^{81} + 8 q^{82} - 8 q^{87} - 12 q^{88} - 4 q^{90} - 8 q^{92} + 28 q^{94} - 16 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 + 4 * q^12 + 8 * q^14 + 4 * q^16 - 16 * q^17 + 12 * q^22 + 8 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 + 4 * q^30 + 8 * q^35 - 4 * q^36 - 16 * q^38 + 4 * q^40 - 8 * q^42 + 20 * q^43 - 4 * q^48 + 12 * q^49 + 16 * q^51 - 24 * q^53 + 12 * q^55 - 8 * q^56 - 4 * q^64 - 12 * q^66 + 16 * q^68 - 8 * q^69 + 16 * q^74 + 4 * q^75 - 24 * q^77 - 28 * q^79 + 4 * q^81 + 8 * q^82 - 8 * q^87 - 12 * q^88 - 4 * q^90 - 8 * q^92 + 28 * q^94 - 16 * q^95

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