LMFDB - Embedded newform 5070.2.b.ba.1351.8

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:	$8$
Coefficient field:	8.0.17284886784.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704 $ x^8 - 2x^7 + 2x^6 + 30x^5 + 185x^4 + 36x^3 + 8x^2 + 208*x + 2704
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.8
Root			$1.33404 - 1.33404i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.ba.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +3.64466i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +3.64466i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.66808i q^{11} -1.00000 q^{12} -3.64466 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} -6.31274i q^{19} +1.00000i q^{20} +3.64466i q^{21} +1.66808 q^{22} -1.24453 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.64466i q^{28} +10.0448 q^{29} +1.00000 q^{30} +4.21957i q^{31} +1.00000i q^{32} -1.66808i q^{33} -4.00000i q^{34} +3.64466 q^{35} -1.00000 q^{36} +9.86423i q^{37} +6.31274 q^{38} -1.00000 q^{40} +9.28932i q^{41} -3.64466 q^{42} +7.57286 q^{43} +1.66808i q^{44} -1.00000i q^{45} -1.24453i q^{46} -6.82522i q^{47} +1.00000 q^{48} -6.28354 q^{49} -1.00000i q^{50} -4.00000 q^{51} -0.848634 q^{53} +1.00000i q^{54} -1.66808 q^{55} +3.64466 q^{56} -6.31274i q^{57} +10.0448i q^{58} -6.10876i q^{59} +1.00000i q^{60} +7.46410 q^{61} -4.21957 q^{62} +3.64466i q^{63} -1.00000 q^{64} +1.66808 q^{66} +14.7534i q^{67} +4.00000 q^{68} -1.24453 q^{69} +3.64466i q^{70} +3.51093i q^{71} -1.00000i q^{72} +12.2175i q^{73} -9.86423 q^{74} -1.00000 q^{75} +6.31274i q^{76} +6.07957 q^{77} +9.93398 q^{79} -1.00000i q^{80} +1.00000 q^{81} -9.28932 q^{82} +7.95317i q^{83} -3.64466i q^{84} +4.00000i q^{85} +7.57286i q^{86} +10.0448 q^{87} -1.66808 q^{88} +5.95162i q^{89} +1.00000 q^{90} +1.24453 q^{92} +4.21957i q^{93} +6.82522 q^{94} -6.31274 q^{95} +1.00000i q^{96} +2.75342i q^{97} -6.28354i q^{98} -1.66808i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^9	$ 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9} + 8 q^{10} - 8 q^{12} + 4 q^{14} + 8 q^{16} - 32 q^{17} - 4 q^{22} - 8 q^{23} - 8 q^{25} + 8 q^{27} + 16 q^{29} + 8 q^{30} - 4 q^{35} - 8 q^{36} - 8 q^{40} + 4 q^{42} - 28 q^{43} + 8 q^{48} - 28 q^{49} - 32 q^{51} + 16 q^{53} + 4 q^{55} - 4 q^{56} + 32 q^{61} - 8 q^{62} - 8 q^{64} - 4 q^{66} + 32 q^{68} - 8 q^{69} - 20 q^{74} - 8 q^{75} + 16 q^{77} - 20 q^{79} + 8 q^{81} - 8 q^{82} + 16 q^{87} + 4 q^{88} + 8 q^{90} + 8 q^{92} + 16 q^{94}+O(q^{100}) $ 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^9 + 8 * q^10 - 8 * q^12 + 4 * q^14 + 8 * q^16 - 32 * q^17 - 4 * q^22 - 8 * q^23 - 8 * q^25 + 8 * q^27 + 16 * q^29 + 8 * q^30 - 4 * q^35 - 8 * q^36 - 8 * q^40 + 4 * q^42 - 28 * q^43 + 8 * q^48 - 28 * q^49 - 32 * q^51 + 16 * q^53 + 4 * q^55 - 4 * q^56 + 32 * q^61 - 8 * q^62 - 8 * q^64 - 4 * q^66 + 32 * q^68 - 8 * q^69 - 20 * q^74 - 8 * q^75 + 16 * q^77 - 20 * q^79 + 8 * q^81 - 8 * q^82 + 16 * q^87 + 4 * q^88 + 8 * q^90 + 8 * q^92 + 16 * q^94

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
$7$	3.64466i	1.37755i	0.724974	+	0.688776i	$0.241852\pi$
$7$	3.64466i	1.37755i	−0.724974	+	0.688776i	$0.758148\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	− 1.66808i	− 0.502944i	−0.967865	−	0.251472i	$-0.919085\pi$
$11$	− 1.66808i	− 0.502944i	0.967865	−	0.251472i	$-0.0809146\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−3.64466	−0.974076
$15$	− 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
$19$	− 6.31274i	− 1.44824i	−0.689674	−	0.724120i	$-0.742246\pi$
$19$	− 6.31274i	− 1.44824i	0.689674	−	0.724120i	$-0.257754\pi$
$20$	1.00000i	0.223607i
$21$	3.64466i	0.795330i
$22$	1.66808	0.355635
$23$	−1.24453	−0.259503	−0.129752	−	0.991547i	$-0.541418\pi$
$23$	−1.24453	−0.259503	−0.129752	+	0.991547i	$0.541418\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 3.64466i	− 0.688776i
$29$	10.0448	1.86527	0.932635	−	0.360821i	$-0.117504\pi$
$29$	10.0448	1.86527	0.932635	+	0.360821i	$0.117504\pi$
$30$	1.00000	0.182574
$31$	4.21957i	0.757857i	0.925426	+	0.378928i	$0.123707\pi$
$31$	4.21957i	0.757857i	−0.925426	+	0.378928i	$0.876293\pi$
$32$	1.00000i	0.176777i
$33$	− 1.66808i	− 0.290375i
$34$	− 4.00000i	− 0.685994i
$35$	3.64466	0.616060
$36$	−1.00000	−0.166667
$37$	9.86423i	1.62167i	0.585275	+	0.810835i	$0.300986\pi$
$37$	9.86423i	1.62167i	−0.585275	+	0.810835i	$0.699014\pi$
$38$	6.31274	1.02406
$39$	0	0
$40$	−1.00000	−0.158114
$41$	9.28932i	1.45075i	0.688355	+	0.725374i	$0.258333\pi$
$41$	9.28932i	1.45075i	−0.688355	+	0.725374i	$0.741667\pi$
$42$	−3.64466	−0.562383
$43$	7.57286	1.15485	0.577425	−	0.816443i	$-0.304057\pi$
$43$	7.57286	1.15485	0.577425	+	0.816443i	$0.304057\pi$
$44$	1.66808i	0.251472i
$45$	− 1.00000i	− 0.149071i
$46$	− 1.24453i	− 0.183496i
$47$	− 6.82522i	− 0.995560i	−0.867303	−	0.497780i	$-0.834149\pi$
$47$	− 6.82522i	− 0.995560i	0.867303	−	0.497780i	$-0.165851\pi$
$48$	1.00000	0.144338
$49$	−6.28354	−0.897649
$50$	− 1.00000i	− 0.141421i
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−0.848634	−0.116569	−0.0582844	−	0.998300i	$-0.518563\pi$
$53$	−0.848634	−0.116569	−0.0582844	+	0.998300i	$0.518563\pi$
$54$	1.00000i	0.136083i
$55$	−1.66808	−0.224923
$56$	3.64466	0.487038
$57$	− 6.31274i	− 0.836142i
$58$	10.0448i	1.31895i
$59$	− 6.10876i	− 0.795293i	−0.917539	−	0.397646i	$-0.869827\pi$
$59$	− 6.10876i	− 0.795293i	0.917539	−	0.397646i	$-0.130173\pi$
$60$	1.00000i	0.129099i
$61$	7.46410	0.955680	0.477840	−	0.878447i	$-0.341420\pi$
$61$	7.46410	0.955680	0.477840	+	0.878447i	$0.341420\pi$
$62$	−4.21957	−0.535886
$63$	3.64466i	0.459184i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.66808	0.205326
$67$	14.7534i	1.80242i	0.433386	+	0.901209i	$0.357319\pi$
$67$	14.7534i	1.80242i	−0.433386	+	0.901209i	$0.642681\pi$
$68$	4.00000	0.485071
$69$	−1.24453	−0.149824
$70$	3.64466i	0.435620i
$71$	3.51093i	0.416671i	0.978057	+	0.208336i	$0.0668046\pi$
$71$	3.51093i	0.416671i	−0.978057	+	0.208336i	$0.933195\pi$
$72$	− 1.00000i	− 0.117851i
$73$	12.2175i	1.42995i	0.699149	+	0.714976i	$0.253563\pi$
$73$	12.2175i	1.42995i	−0.699149	+	0.714976i	$0.746437\pi$
$74$	−9.86423	−1.14669
$75$	−1.00000	−0.115470
$76$	6.31274i	0.724120i
$77$	6.07957	0.692831
$78$	0	0
$79$	9.93398	1.11766	0.558830	−	0.829282i	$-0.311250\pi$
$79$	9.93398	1.11766	0.558830	+	0.829282i	$0.311250\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−9.28932	−1.02583
$83$	7.95317i	0.872974i	0.899711	+	0.436487i	$0.143777\pi$
$83$	7.95317i	0.872974i	−0.899711	+	0.436487i	$0.856223\pi$
$84$	− 3.64466i	− 0.397665i
$85$	4.00000i	0.433861i
$86$	7.57286i	0.816603i
$87$	10.0448	1.07691
$88$	−1.66808	−0.177817
$89$	5.95162i	0.630870i	0.948947	+	0.315435i	$0.102150\pi$
$89$	5.95162i	0.630870i	−0.948947	+	0.315435i	$0.897850\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	1.24453	0.129752
$93$	4.21957i	0.437549i
$94$	6.82522	0.703967
$95$	−6.31274	−0.647673
$96$	1.00000i	0.102062i
$97$	2.75342i	0.279568i	0.990182	+	0.139784i	$0.0446407\pi$
$97$	2.75342i	0.279568i	−0.990182	+	0.139784i	$0.955359\pi$
$98$	− 6.28354i	− 0.634734i
$99$	− 1.66808i	− 0.167648i
$100$	1.00000	0.100000
$101$	−5.33615	−0.530967	−0.265483	−	0.964115i	$-0.585532\pi$
$101$	−5.33615	−0.530967	−0.265483	+	0.964115i	$0.585532\pi$
$102$	− 4.00000i	− 0.396059i
$103$	−7.51248	−0.740227	−0.370113	−	0.928987i	$-0.620681\pi$
$103$	−7.51248	−0.740227	−0.370113	+	0.928987i	$0.620681\pi$
$104$	0	0
$105$	3.64466	0.355682
$106$	− 0.848634i	− 0.0824266i
$107$	16.9282	1.63651	0.818256	−	0.574855i	$-0.194941\pi$
$107$	16.9282	1.63651	0.818256	+	0.574855i	$0.194941\pi$
$108$	−1.00000	−0.0962250
$109$	0.663848i	0.0635851i	0.999494	+	0.0317926i	$0.0101216\pi$
$109$	0.663848i	0.0635851i	−0.999494	+	0.0317926i	$0.989878\pi$
$110$	− 1.66808i	− 0.159045i
$111$	9.86423i	0.936271i
$112$	3.64466i	0.344388i
$113$	−17.8700	−1.68107	−0.840534	−	0.541758i	$-0.817759\pi$
$113$	−17.8700	−1.68107	−0.840534	+	0.541758i	$0.817759\pi$
$114$	6.31274	0.591242
$115$	1.24453i	0.116053i
$116$	−10.0448	−0.932635
$117$	0	0
$118$	6.10876	0.562357
$119$	− 14.5786i	− 1.33642i
$120$	−1.00000	−0.0912871
$121$	8.21752	0.747047
$122$	7.46410i	0.675768i
$123$	9.28932i	0.837590i
$124$	− 4.21957i	− 0.378928i
$125$	1.00000i	0.0894427i
$126$	−3.64466	−0.324692
$127$	−14.4407	−1.28140	−0.640702	−	0.767790i	$-0.721357\pi$
$127$	−14.4407	−1.28140	−0.640702	+	0.767790i	$0.721357\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	7.57286	0.666753
$130$	0	0
$131$	−10.8892	−0.951393	−0.475697	−	0.879609i	$-0.657804\pi$
$131$	−10.8892	−0.951393	−0.475697	+	0.879609i	$0.657804\pi$
$132$	1.66808i	0.145187i
$133$	23.0078	1.99503
$134$	−14.7534	−1.27450
$135$	− 1.00000i	− 0.0860663i
$136$	4.00000i	0.342997i
$137$	− 7.03696i	− 0.601208i	−0.953749	−	0.300604i	$-0.902812\pi$
$137$	− 7.03696i	− 0.601208i	0.953749	−	0.300604i	$-0.0971883\pi$
$138$	− 1.24453i	− 0.105942i
$139$	11.6447	0.987687	0.493844	−	0.869551i	$-0.335592\pi$
$139$	11.6447	0.987687	0.493844	+	0.869551i	$0.335592\pi$
$140$	−3.64466	−0.308030
$141$	− 6.82522i	− 0.574787i
$142$	−3.51093	−0.294631
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 10.0448i	− 0.834174i
$146$	−12.2175	−1.01113
$147$	−6.28354	−0.518258
$148$	− 9.86423i	− 0.810835i
$149$	0.772609i	0.0632946i	0.999499	+	0.0316473i	$0.0100753\pi$
$149$	0.772609i	0.0632946i	−0.999499	+	0.0316473i	$0.989925\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	− 9.77838i	− 0.795754i	−0.917439	−	0.397877i	$-0.869747\pi$
$151$	− 9.77838i	− 0.795754i	0.917439	−	0.397877i	$-0.130253\pi$
$152$	−6.31274	−0.512030
$153$	−4.00000	−0.323381
$154$	6.07957i	0.489906i
$155$	4.21957	0.338924
$156$	0	0
$157$	−12.0135	−0.958786	−0.479393	−	0.877600i	$-0.659143\pi$
$157$	−12.0135	−0.958786	−0.479393	+	0.877600i	$0.659143\pi$
$158$	9.93398i	0.790305i
$159$	−0.848634	−0.0673011
$160$	1.00000	0.0790569
$161$	− 4.53590i	− 0.357479i
$162$	1.00000i	0.0785674i
$163$	10.0916i	0.790437i	0.918587	+	0.395218i	$0.129331\pi$
$163$	10.0916i	0.790437i	−0.918587	+	0.395218i	$0.870669\pi$
$164$	− 9.28932i	− 0.725374i
$165$	−1.66808	−0.129860
$166$	−7.95317	−0.617285
$167$	− 7.89701i	− 0.611089i	−0.952178	−	0.305545i	$-0.901161\pi$
$167$	− 7.89701i	− 0.611089i	0.952178	−	0.305545i	$-0.0988385\pi$
$168$	3.64466	0.281192
$169$	0	0
$170$	−4.00000	−0.306786
$171$	− 6.31274i	− 0.482747i
$172$	−7.57286	−0.577425
$173$	0.440685	0.0335047	0.0167523	−	0.999860i	$-0.494667\pi$
$173$	0.440685	0.0335047	0.0167523	+	0.999860i	$0.494667\pi$
$174$	10.0448i	0.761493i
$175$	− 3.64466i	− 0.275510i
$176$	− 1.66808i	− 0.125736i
$177$	− 6.10876i	− 0.459163i
$178$	−5.95162	−0.446093
$179$	19.6368	1.46773	0.733863	−	0.679297i	$-0.237715\pi$
$179$	19.6368	1.46773	0.733863	+	0.679297i	$0.237715\pi$
$180$	1.00000i	0.0745356i
$181$	6.21752	0.462145	0.231072	−	0.972937i	$-0.425777\pi$
$181$	6.21752	0.462145	0.231072	+	0.972937i	$0.425777\pi$
$182$	0	0
$183$	7.46410	0.551762
$184$	1.24453i	0.0917482i
$185$	9.86423	0.725232
$186$	−4.21957	−0.309394
$187$	6.67230i	0.487927i
$188$	6.82522i	0.497780i
$189$	3.64466i	0.265110i
$190$	− 6.31274i	− 0.457974i
$191$	15.6816	1.13468	0.567341	−	0.823483i	$-0.307972\pi$
$191$	15.6816	1.13468	0.567341	+	0.823483i	$0.307972\pi$
$192$	−1.00000	−0.0721688
$193$	8.12795i	0.585063i	0.956256	+	0.292531i	$0.0944976\pi$
$193$	8.12795i	0.585063i	−0.956256	+	0.292531i	$0.905502\pi$
$194$	−2.75342	−0.197684
$195$	0	0
$196$	6.28354	0.448825
$197$	0.891239i	0.0634981i	0.999496	+	0.0317491i	$0.0101077\pi$
$197$	0.891239i	0.0634981i	−0.999496	+	0.0317491i	$0.989892\pi$
$198$	1.66808	0.118545
$199$	−0.361116	−0.0255988	−0.0127994	−	0.999918i	$-0.504074\pi$
$199$	−0.361116	−0.0255988	−0.0127994	+	0.999918i	$0.504074\pi$
$200$	1.00000i	0.0707107i
$201$	14.7534i	1.04063i
$202$	− 5.33615i	− 0.375450i
$203$	36.6098i	2.56951i
$204$	4.00000	0.280056
$205$	9.28932	0.648794
$206$	− 7.51248i	− 0.523419i
$207$	−1.24453	−0.0865010
$208$	0	0
$209$	−10.5301	−0.728384
$210$	3.64466i	0.251505i
$211$	2.22739	0.153340	0.0766700	−	0.997057i	$-0.475571\pi$
$211$	2.22739	0.153340	0.0766700	+	0.997057i	$0.475571\pi$
$212$	0.848634	0.0582844
$213$	3.51093i	0.240565i
$214$	16.9282i	1.15719i
$215$	− 7.57286i	− 0.516465i
$216$	− 1.00000i	− 0.0680414i
$217$	−15.3789	−1.04399
$218$	−0.663848	−0.0449615
$219$	12.2175i	0.825584i
$220$	1.66808	0.112462
$221$	0	0
$222$	−9.86423	−0.662044
$223$	6.08380i	0.407401i	0.979033	+	0.203701i	$0.0652969\pi$
$223$	6.08380i	0.407401i	−0.979033	+	0.203701i	$0.934703\pi$
$224$	−3.64466	−0.243519
$225$	−1.00000	−0.0666667
$226$	− 17.8700i	− 1.18869i
$227$	15.3205i	1.01686i	0.861104	+	0.508429i	$0.169774\pi$
$227$	15.3205i	1.01686i	−0.861104	+	0.508429i	$0.830226\pi$
$228$	6.31274i	0.418071i
$229$	− 22.2644i	− 1.47127i	−0.677378	−	0.735635i	$-0.736884\pi$
$229$	− 22.2644i	− 1.47127i	0.677378	−	0.735635i	$-0.263116\pi$
$230$	−1.24453	−0.0820621
$231$	6.07957	0.400006
$232$	− 10.0448i	− 0.659473i
$233$	10.8366	0.709928	0.354964	−	0.934880i	$-0.384493\pi$
$233$	10.8366	0.709928	0.354964	+	0.934880i	$0.384493\pi$
$234$	0	0
$235$	−6.82522	−0.445228
$236$	6.10876i	0.397646i
$237$	9.93398	0.645281
$238$	14.5786	0.944993
$239$	16.4975i	1.06714i	0.845757	+	0.533568i	$0.179149\pi$
$239$	16.4975i	1.06714i	−0.845757	+	0.533568i	$0.820851\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 4.40435i	− 0.283709i	−0.989887	−	0.141855i	$-0.954693\pi$
$241$	− 4.40435i	− 0.283709i	0.989887	−	0.141855i	$-0.0453066\pi$
$242$	8.21752i	0.528242i
$243$	1.00000	0.0641500
$244$	−7.46410	−0.477840
$245$	6.28354i	0.401441i
$246$	−9.28932	−0.592265
$247$	0	0
$248$	4.21957	0.267943
$249$	7.95317i	0.504011i
$250$	−1.00000	−0.0632456
$251$	−11.9453	−0.753984	−0.376992	−	0.926217i	$-0.623041\pi$
$251$	−11.9453	−0.753984	−0.376992	+	0.926217i	$0.623041\pi$
$252$	− 3.64466i	− 0.229592i
$253$	2.07598i	0.130515i
$254$	− 14.4407i	− 0.906089i
$255$	4.00000i	0.250490i
$256$	1.00000	0.0625000
$257$	−19.4621	−1.21401	−0.607005	−	0.794698i	$-0.707629\pi$
$257$	−19.4621	−1.21401	−0.607005	+	0.794698i	$0.707629\pi$
$258$	7.57286i	0.471466i
$259$	−35.9518	−2.23393
$260$	0	0
$261$	10.0448	0.621757
$262$	− 10.8892i	− 0.672737i
$263$	2.03478	0.125470	0.0627350	−	0.998030i	$-0.480018\pi$
$263$	2.03478	0.125470	0.0627350	+	0.998030i	$0.480018\pi$
$264$	−1.66808	−0.102663
$265$	0.848634i	0.0521312i
$266$	23.0078i	1.41070i
$267$	5.95162i	0.364233i
$268$	− 14.7534i	− 0.901209i
$269$	20.5287	1.25166	0.625829	−	0.779960i	$-0.284761\pi$
$269$	20.5287	1.25166	0.625829	+	0.779960i	$0.284761\pi$
$270$	1.00000	0.0608581
$271$	25.5401i	1.55145i	0.631072	+	0.775725i	$0.282615\pi$
$271$	25.5401i	1.55145i	−0.631072	+	0.775725i	$0.717385\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	7.03696	0.425119
$275$	1.66808i	0.100589i
$276$	1.24453	0.0749121
$277$	−18.0604	−1.08514	−0.542572	−	0.840010i	$-0.682549\pi$
$277$	−18.0604	−1.08514	−0.542572	+	0.840010i	$0.682549\pi$
$278$	11.6447i	0.698400i
$279$	4.21957i	0.252619i
$280$	− 3.64466i	− 0.217810i
$281$	20.2175i	1.20608i	0.797712	+	0.603038i	$0.206043\pi$
$281$	20.2175i	1.20608i	−0.797712	+	0.603038i	$0.793957\pi$
$282$	6.82522	0.406436
$283$	−8.69149	−0.516656	−0.258328	−	0.966057i	$-0.583171\pi$
$283$	−8.69149	−0.516656	−0.258328	+	0.966057i	$0.583171\pi$
$284$	− 3.51093i	− 0.208336i
$285$	−6.31274	−0.373934
$286$	0	0
$287$	−33.8564	−1.99848
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	10.0448	0.589850
$291$	2.75342i	0.161408i
$292$	− 12.2175i	− 0.714976i
$293$	− 7.54790i	− 0.440953i	−0.975392	−	0.220476i	$-0.929239\pi$
$293$	− 7.54790i	− 0.440953i	0.975392	−	0.220476i	$-0.0707612\pi$
$294$	− 6.28354i	− 0.366464i
$295$	−6.10876	−0.355666
$296$	9.86423	0.573347
$297$	− 1.66808i	− 0.0967916i
$298$	−0.772609	−0.0447560
$299$	0	0
$300$	1.00000	0.0577350
$301$	27.6005i	1.59087i
$302$	9.77838	0.562683
$303$	−5.33615	−0.306554
$304$	− 6.31274i	− 0.362060i
$305$	− 7.46410i	− 0.427393i
$306$	− 4.00000i	− 0.228665i
$307$	26.0427i	1.48634i	0.669104	+	0.743169i	$0.266678\pi$
$307$	26.0427i	1.48634i	−0.669104	+	0.743169i	$0.733322\pi$
$308$	−6.07957	−0.346416
$309$	−7.51248	−0.427370
$310$	4.21957i	0.239655i
$311$	25.3789	1.43910	0.719552	−	0.694438i	$-0.244347\pi$
$311$	25.3789	1.43910	0.719552	+	0.694438i	$0.244347\pi$
$312$	0	0
$313$	−31.4600	−1.77822	−0.889112	−	0.457689i	$-0.848677\pi$
$313$	−31.4600	−1.77822	−0.889112	+	0.457689i	$0.848677\pi$
$314$	− 12.0135i	− 0.677964i
$315$	3.64466	0.205353
$316$	−9.93398	−0.558830
$317$	− 24.7093i	− 1.38781i	−0.720066	−	0.693905i	$-0.755889\pi$
$317$	− 24.7093i	− 1.38781i	0.720066	−	0.693905i	$-0.244111\pi$
$318$	− 0.848634i	− 0.0475890i
$319$	− 16.7555i	− 0.938126i
$320$	1.00000i	0.0559017i
$321$	16.9282	0.944840
$322$	4.53590	0.252776
$323$	25.2509i	1.40500i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−10.0916	−0.558923
$327$	0.663848i	0.0367109i
$328$	9.28932	0.512917
$329$	24.8756	1.37144
$330$	− 1.66808i	− 0.0918246i
$331$	5.60360i	0.308002i	0.988071	+	0.154001i	$0.0492159\pi$
$331$	5.60360i	0.308002i	−0.988071	+	0.154001i	$0.950784\pi$
$332$	− 7.95317i	− 0.436487i
$333$	9.86423i	0.540556i
$334$	7.89701	0.432105
$335$	14.7534	0.806065
$336$	3.64466i	0.198832i
$337$	21.7868	1.18680	0.593402	−	0.804906i	$-0.297784\pi$
$337$	21.7868	1.18680	0.593402	+	0.804906i	$0.297784\pi$
$338$	0	0
$339$	−17.8700	−0.970565
$340$	− 4.00000i	− 0.216930i
$341$	7.03856	0.381159
$342$	6.31274	0.341354
$343$	2.61124i	0.140994i
$344$	− 7.57286i	− 0.408301i
$345$	1.24453i	0.0670034i
$346$	0.440685i	0.0236914i
$347$	9.68162	0.519737	0.259868	−	0.965644i	$-0.416321\pi$
$347$	9.68162	0.519737	0.259868	+	0.965644i	$0.416321\pi$
$348$	−10.0448	−0.538457
$349$	19.3205i	1.03420i	0.855924	+	0.517102i	$0.172989\pi$
$349$	19.3205i	1.03420i	−0.855924	+	0.517102i	$0.827011\pi$
$350$	3.64466	0.194815
$351$	0	0
$352$	1.66808	0.0889087
$353$	− 22.9750i	− 1.22284i	−0.791307	−	0.611419i	$-0.790599\pi$
$353$	− 22.9750i	− 1.22284i	0.791307	−	0.611419i	$-0.209401\pi$
$354$	6.10876	0.324677
$355$	3.51093	0.186341
$356$	− 5.95162i	− 0.315435i
$357$	− 14.5786i	− 0.771583i
$358$	19.6368i	1.03784i
$359$	− 2.21752i	− 0.117036i	−0.998286	−	0.0585182i	$-0.981362\pi$
$359$	− 2.21752i	− 0.117036i	0.998286	−	0.0585182i	$-0.0186376\pi$
$360$	−1.00000	−0.0527046
$361$	−20.8506	−1.09740
$362$	6.21752i	0.326786i
$363$	8.21752	0.431308
$364$	0	0
$365$	12.2175	0.639494
$366$	7.46410i	0.390155i
$367$	−6.08112	−0.317432	−0.158716	−	0.987324i	$-0.550735\pi$
$367$	−6.08112	−0.317432	−0.158716	+	0.987324i	$0.550735\pi$
$368$	−1.24453	−0.0648758
$369$	9.28932i	0.483583i
$370$	9.86423i	0.512817i
$371$	− 3.09298i	− 0.160580i
$372$	− 4.21957i	− 0.218774i
$373$	15.6781	0.811780	0.405890	−	0.913922i	$-0.366962\pi$
$373$	15.6781	0.811780	0.405890	+	0.913922i	$0.366962\pi$
$374$	−6.67230	−0.345017
$375$	1.00000i	0.0516398i
$376$	−6.82522	−0.351984
$377$	0	0
$378$	−3.64466	−0.187461
$379$	30.7166i	1.57781i	0.614518	+	0.788903i	$0.289350\pi$
$379$	30.7166i	1.57781i	−0.614518	+	0.788903i	$0.710650\pi$
$380$	6.31274	0.323837
$381$	−14.4407	−0.739819
$382$	15.6816i	0.802342i
$383$	− 20.0619i	− 1.02512i	−0.858652	−	0.512558i	$-0.828698\pi$
$383$	− 20.0619i	− 1.02512i	0.858652	−	0.512558i	$-0.171302\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	− 6.07957i	− 0.309844i
$386$	−8.12795	−0.413702
$387$	7.57286	0.384950
$388$	− 2.75342i	− 0.139784i
$389$	27.0314	1.37055	0.685273	−	0.728287i	$-0.259683\pi$
$389$	27.0314	1.37055	0.685273	+	0.728287i	$0.259683\pi$
$390$	0	0
$391$	4.97813	0.251755
$392$	6.28354i	0.317367i
$393$	−10.8892	−0.549287
$394$	−0.891239	−0.0449000
$395$	− 9.93398i	− 0.499833i
$396$	1.66808i	0.0838240i
$397$	3.73628i	0.187518i	0.995595	+	0.0937592i	$0.0298884\pi$
$397$	3.73628i	0.187518i	−0.995595	+	0.0937592i	$0.970112\pi$
$398$	− 0.361116i	− 0.0181011i
$399$	23.0078	1.15183
$400$	−1.00000	−0.0500000
$401$	28.0911i	1.40280i	0.712766	+	0.701402i	$0.247442\pi$
$401$	28.0911i	1.40280i	−0.712766	+	0.701402i	$0.752558\pi$
$402$	−14.7534	−0.735834
$403$	0	0
$404$	5.33615	0.265483
$405$	− 1.00000i	− 0.0496904i
$406$	−36.6098	−1.81692
$407$	16.4543	0.815609
$408$	4.00000i	0.198030i
$409$	− 27.3804i	− 1.35388i	−0.736040	−	0.676938i	$-0.763307\pi$
$409$	− 27.3804i	− 1.35388i	0.736040	−	0.676938i	$-0.236693\pi$
$410$	9.28932i	0.458767i
$411$	− 7.03696i	− 0.347108i
$412$	7.51248	0.370113
$413$	22.2644	1.09556
$414$	− 1.24453i	− 0.0611654i
$415$	7.95317	0.390406
$416$	0	0
$417$	11.6447	0.570241
$418$	− 10.5301i	− 0.515045i
$419$	−13.1614	−0.642975	−0.321487	−	0.946914i	$-0.604183\pi$
$419$	−13.1614	−0.642975	−0.321487	+	0.946914i	$0.604183\pi$
$420$	−3.64466	−0.177841
$421$	− 1.29341i	− 0.0630370i	−0.999503	−	0.0315185i	$-0.989966\pi$
$421$	− 1.29341i	− 0.0630370i	0.999503	−	0.0315185i	$-0.0100343\pi$
$422$	2.22739i	0.108428i
$423$	− 6.82522i	− 0.331853i
$424$	0.848634i	0.0412133i
$425$	4.00000	0.194029
$426$	−3.51093	−0.170105
$427$	27.2041i	1.31650i
$428$	−16.9282	−0.818256
$429$	0	0
$430$	7.57286	0.365196
$431$	12.1279i	0.584183i	0.956390	+	0.292091i	$0.0943511\pi$
$431$	12.1279i	0.584183i	−0.956390	+	0.292091i	$0.905649\pi$
$432$	1.00000	0.0481125
$433$	2.07802	0.0998633	0.0499317	−	0.998753i	$-0.484100\pi$
$433$	2.07802	0.0998633	0.0499317	+	0.998753i	$0.484100\pi$
$434$	− 15.3789i	− 0.738210i
$435$	− 10.0448i	− 0.481611i
$436$	− 0.663848i	− 0.0317926i
$437$	7.85641i	0.375823i
$438$	−12.2175	−0.583776
$439$	2.39640	0.114374	0.0571869	−	0.998363i	$-0.481787\pi$
$439$	2.39640	0.114374	0.0571869	+	0.998363i	$0.481787\pi$
$440$	1.66808i	0.0795224i
$441$	−6.28354	−0.299216
$442$	0	0
$443$	21.9959	1.04506	0.522529	−	0.852622i	$-0.324989\pi$
$443$	21.9959	1.04506	0.522529	+	0.852622i	$0.324989\pi$
$444$	− 9.86423i	− 0.468136i
$445$	5.95162	0.282134
$446$	−6.08380	−0.288076
$447$	0.772609i	0.0365432i
$448$	− 3.64466i	− 0.172194i
$449$	− 29.2253i	− 1.37923i	−0.724178	−	0.689613i	$-0.757780\pi$
$449$	− 29.2253i	− 1.37923i	0.724178	−	0.689613i	$-0.242220\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	15.4953	0.729645
$452$	17.8700	0.840534
$453$	− 9.77838i	− 0.459429i
$454$	−15.3205	−0.719027
$455$	0	0
$456$	−6.31274	−0.295621
$457$	− 39.6432i	− 1.85443i	−0.374526	−	0.927216i	$-0.622195\pi$
$457$	− 39.6432i	− 1.85443i	0.374526	−	0.927216i	$-0.377805\pi$
$458$	22.2644	1.04034
$459$	−4.00000	−0.186704
$460$	− 1.24453i	− 0.0580266i
$461$	− 16.6758i	− 0.776672i	−0.921518	−	0.388336i	$-0.873050\pi$
$461$	− 16.6758i	− 0.776672i	0.921518	−	0.388336i	$-0.126950\pi$
$462$	6.07957i	0.282847i
$463$	32.2175i	1.49728i	0.662979	+	0.748638i	$0.269292\pi$
$463$	32.2175i	1.49728i	−0.662979	+	0.748638i	$0.730708\pi$
$464$	10.0448	0.466318
$465$	4.21957	0.195678
$466$	10.8366i	0.501995i
$467$	−6.88137	−0.318432	−0.159216	−	0.987244i	$-0.550897\pi$
$467$	−6.88137	−0.318432	−0.159216	+	0.987244i	$0.550897\pi$
$468$	0	0
$469$	−53.7712	−2.48292
$470$	− 6.82522i	− 0.314824i
$471$	−12.0135	−0.553555
$472$	−6.10876	−0.281179
$473$	− 12.6321i	− 0.580825i
$474$	9.93398i	0.456283i
$475$	6.31274i	0.289648i
$476$	14.5786i	0.668211i
$477$	−0.848634	−0.0388563
$478$	−16.4975	−0.754579
$479$	− 18.9709i	− 0.866805i	−0.901201	−	0.433402i	$-0.857313\pi$
$479$	− 18.9709i	− 0.866805i	0.901201	−	0.433402i	$-0.142687\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	4.40435	0.200613
$483$	− 4.53590i	− 0.206391i
$484$	−8.21752	−0.373524
$485$	2.75342	0.125026
$486$	1.00000i	0.0453609i
$487$	− 1.91620i	− 0.0868314i	−0.999057	−	0.0434157i	$-0.986176\pi$
$487$	− 1.91620i	− 0.0868314i	0.999057	−	0.0434157i	$-0.0138240\pi$
$488$	− 7.46410i	− 0.337884i
$489$	10.0916i	0.456359i
$490$	−6.28354	−0.283862
$491$	33.6375	1.51804	0.759019	−	0.651069i	$-0.225679\pi$
$491$	33.6375	1.51804	0.759019	+	0.651069i	$0.225679\pi$
$492$	− 9.28932i	− 0.418795i
$493$	−40.1791	−1.80958
$494$	0	0
$495$	−1.66808	−0.0749744
$496$	4.21957i	0.189464i
$497$	−12.7962	−0.573986
$498$	−7.95317	−0.356390
$499$	− 1.82522i	− 0.0817080i	−0.999165	−	0.0408540i	$-0.986992\pi$
$499$	− 1.82522i	− 0.0817080i	0.999165	−	0.0408540i	$-0.0130078\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 7.89701i	− 0.352813i
$502$	− 11.9453i	− 0.533147i
$503$	−19.9985	−0.891687	−0.445843	−	0.895111i	$-0.647096\pi$
$503$	−19.9985	−0.891687	−0.445843	+	0.895111i	$0.647096\pi$
$504$	3.64466	0.162346
$505$	5.33615i	0.237456i
$506$	−2.07598	−0.0922884
$507$	0	0
$508$	14.4407	0.640702
$509$	5.89124i	0.261125i	0.991440	+	0.130562i	$0.0416783\pi$
$509$	5.89124i	0.261125i	−0.991440	+	0.130562i	$0.958322\pi$
$510$	−4.00000	−0.177123
$511$	−44.5287	−1.96983
$512$	1.00000i	0.0441942i
$513$	− 6.31274i	− 0.278714i
$514$	− 19.4621i	− 0.858434i
$515$	7.51248i	0.331040i
$516$	−7.57286	−0.333377
$517$	−11.3850	−0.500711
$518$	− 35.9518i	− 1.57963i
$519$	0.440685	0.0193439
$520$	0	0
$521$	32.0370	1.40356	0.701782	−	0.712391i	$-0.252388\pi$
$521$	32.0370	1.40356	0.701782	+	0.712391i	$0.252388\pi$
$522$	10.0448i	0.439648i
$523$	38.7186	1.69305	0.846523	−	0.532352i	$-0.178692\pi$
$523$	38.7186	1.69305	0.846523	+	0.532352i	$0.178692\pi$
$524$	10.8892	0.475697
$525$	− 3.64466i	− 0.159066i
$526$	2.03478i	0.0887207i
$527$	− 16.8783i	− 0.735229i
$528$	− 1.66808i	− 0.0725937i
$529$	−21.4511	−0.932658
$530$	−0.848634	−0.0368623
$531$	− 6.10876i	− 0.265098i
$532$	−23.0078	−0.997513
$533$	0	0
$534$	−5.95162	−0.257552
$535$	− 16.9282i	− 0.731870i
$536$	14.7534	0.637251
$537$	19.6368	0.847392
$538$	20.5287i	0.885056i
$539$	10.4814i	0.451467i
$540$	1.00000i	0.0430331i
$541$	− 25.9616i	− 1.11618i	−0.829781	−	0.558089i	$-0.811535\pi$
$541$	− 25.9616i	− 1.11618i	0.829781	−	0.558089i	$-0.188465\pi$
$542$	−25.5401	−1.09704
$543$	6.21752	0.266819
$544$	− 4.00000i	− 0.171499i
$545$	0.663848	0.0284361
$546$	0	0
$547$	−17.7596	−0.759348	−0.379674	−	0.925120i	$-0.623964\pi$
$547$	−17.7596	−0.759348	−0.379674	+	0.925120i	$0.623964\pi$
$548$	7.03696i	0.300604i
$549$	7.46410	0.318560
$550$	−1.66808	−0.0711270
$551$	− 63.4101i	− 2.70136i
$552$	1.24453i	0.0529708i
$553$	36.2060i	1.53963i
$554$	− 18.0604i	− 0.767312i
$555$	9.86423	0.418713
$556$	−11.6447	−0.493844
$557$	− 26.2202i	− 1.11099i	−0.831521	−	0.555493i	$-0.812530\pi$
$557$	− 26.2202i	− 1.11099i	0.831521	−	0.555493i	$-0.187470\pi$
$558$	−4.21957	−0.178629
$559$	0	0
$560$	3.64466	0.154015
$561$	6.67230i	0.281705i
$562$	−20.2175	−0.852825
$563$	25.9928	1.09547	0.547733	−	0.836653i	$-0.315491\pi$
$563$	25.9928	1.09547	0.547733	+	0.836653i	$0.315491\pi$
$564$	6.82522i	0.287394i
$565$	17.8700i	0.751797i
$566$	− 8.69149i	− 0.365331i
$567$	3.64466i	0.153061i
$568$	3.51093	0.147316
$569$	25.4699	1.06775	0.533876	−	0.845563i	$-0.320735\pi$
$569$	25.4699	1.06775	0.533876	+	0.845563i	$0.320735\pi$
$570$	− 6.31274i	− 0.264411i
$571$	−8.44491	−0.353409	−0.176704	−	0.984264i	$-0.556544\pi$
$571$	−8.44491	−0.353409	−0.176704	+	0.984264i	$0.556544\pi$
$572$	0	0
$573$	15.6816	0.655109
$574$	− 33.8564i	− 1.41314i
$575$	1.24453	0.0519006
$576$	−1.00000	−0.0416667
$577$	35.4216i	1.47462i	0.675554	+	0.737311i	$0.263905\pi$
$577$	35.4216i	1.47462i	−0.675554	+	0.737311i	$0.736095\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	8.12795i	0.337786i
$580$	10.0448i	0.417087i
$581$	−28.9866	−1.20257
$582$	−2.75342	−0.114133
$583$	1.41559i	0.0586276i
$584$	12.2175	0.505565
$585$	0	0
$586$	7.54790	0.311801
$587$	27.3789i	1.13005i	0.825075	+	0.565024i	$0.191133\pi$
$587$	27.3789i	1.13005i	−0.825075	+	0.565024i	$0.808867\pi$
$588$	6.28354	0.259129
$589$	26.6370	1.09756
$590$	− 6.10876i	− 0.251494i
$591$	0.891239i	0.0366607i
$592$	9.86423i	0.405417i
$593$	− 12.0619i	− 0.495324i	−0.968846	−	0.247662i	$-0.920338\pi$
$593$	− 12.0619i	− 0.495324i	0.968846	−	0.247662i	$-0.0796623\pi$
$594$	1.66808	0.0684420
$595$	−14.5786	−0.597666
$596$	− 0.772609i	− 0.0316473i
$597$	−0.361116	−0.0147795
$598$	0	0
$599$	−28.6129	−1.16909	−0.584546	−	0.811360i	$-0.698727\pi$
$599$	−28.6129	−1.16909	−0.584546	+	0.811360i	$0.698727\pi$
$600$	1.00000i	0.0408248i
$601$	−19.1676	−0.781862	−0.390931	−	0.920420i	$-0.627847\pi$
$601$	−19.1676	−0.781862	−0.390931	+	0.920420i	$0.627847\pi$
$602$	−27.6005	−1.12491
$603$	14.7534i	0.600806i
$604$	9.77838i	0.397877i
$605$	− 8.21752i	− 0.334090i
$606$	− 5.33615i	− 0.216766i
$607$	18.9382	0.768678	0.384339	−	0.923192i	$-0.374429\pi$
$607$	18.9382	0.768678	0.384339	+	0.923192i	$0.374429\pi$
$608$	6.31274	0.256015
$609$	36.6098i	1.48351i
$610$	7.46410	0.302213
$611$	0	0
$612$	4.00000	0.161690
$613$	− 17.9694i	− 0.725779i	−0.931832	−	0.362890i	$-0.881790\pi$
$613$	− 17.9694i	− 0.725779i	0.931832	−	0.362890i	$-0.118210\pi$
$614$	−26.0427	−1.05100
$615$	9.28932	0.374582
$616$	− 6.07957i	− 0.244953i
$617$	18.0151i	0.725260i	0.931933	+	0.362630i	$0.118121\pi$
$617$	18.0151i	0.725260i	−0.931933	+	0.362630i	$0.881879\pi$
$618$	− 7.51248i	− 0.302196i
$619$	− 25.0505i	− 1.00687i	−0.864035	−	0.503433i	$-0.832070\pi$
$619$	− 25.0505i	− 1.00687i	0.864035	−	0.503433i	$-0.167930\pi$
$620$	−4.21957	−0.169462
$621$	−1.24453	−0.0499414
$622$	25.3789i	1.01760i
$623$	−21.6916	−0.869057
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 31.4600i	− 1.25739i
$627$	−10.5301	−0.420533
$628$	12.0135	0.479393
$629$	− 39.4569i	− 1.57325i
$630$	3.64466i	0.145207i
$631$	1.41727i	0.0564206i	0.999602	+	0.0282103i	$0.00898081\pi$
$631$	1.41727i	0.0564206i	−0.999602	+	0.0282103i	$0.991019\pi$
$632$	− 9.93398i	− 0.395152i
$633$	2.22739	0.0885308
$634$	24.7093	0.981330
$635$	14.4407i	0.573061i
$636$	0.848634	0.0336505
$637$	0	0
$638$	16.7555	0.663355
$639$	3.51093i	0.138890i
$640$	−1.00000	−0.0395285
$641$	−4.61970	−0.182467	−0.0912335	−	0.995830i	$-0.529081\pi$
$641$	−4.61970	−0.182467	−0.0912335	+	0.995830i	$0.529081\pi$
$642$	16.9282i	0.668103i
$643$	− 48.0896i	− 1.89647i	−0.317573	−	0.948234i	$-0.602868\pi$
$643$	− 48.0896i	− 1.89647i	0.317573	−	0.948234i	$-0.397132\pi$
$644$	4.53590i	0.178739i
$645$	− 7.57286i	− 0.298181i
$646$	−25.2509	−0.993485
$647$	−3.74565	−0.147257	−0.0736283	−	0.997286i	$-0.523458\pi$
$647$	−3.74565	−0.147257	−0.0736283	+	0.997286i	$0.523458\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−10.1899	−0.399988
$650$	0	0
$651$	−15.3789	−0.602746
$652$	− 10.0916i	− 0.395218i
$653$	42.2899	1.65493	0.827466	−	0.561516i	$-0.189782\pi$
$653$	42.2899	1.65493	0.827466	+	0.561516i	$0.189782\pi$
$654$	−0.663848	−0.0259585
$655$	10.8892i	0.425476i
$656$	9.28932i	0.362687i
$657$	12.2175i	0.476651i
$658$	24.8756i	0.969752i
$659$	−29.1750	−1.13650	−0.568248	−	0.822858i	$-0.692378\pi$
$659$	−29.1750	−1.13650	−0.568248	+	0.822858i	$0.692378\pi$
$660$	1.66808	0.0649298
$661$	− 44.4662i	− 1.72954i	−0.502172	−	0.864768i	$-0.667465\pi$
$661$	− 44.4662i	− 1.72954i	0.502172	−	0.864768i	$-0.332535\pi$
$662$	−5.60360	−0.217790
$663$	0	0
$664$	7.95317	0.308643
$665$	− 23.0078i	− 0.892203i
$666$	−9.86423	−0.382231
$667$	−12.5011	−0.484043
$668$	7.89701i	0.305545i
$669$	6.08380i	0.235213i
$670$	14.7534i	0.569974i
$671$	− 12.4507i	− 0.480654i
$672$	−3.64466	−0.140596
$673$	−7.90633	−0.304767	−0.152383	−	0.988321i	$-0.548695\pi$
$673$	−7.90633	−0.304767	−0.152383	+	0.988321i	$0.548695\pi$
$674$	21.7868i	0.839198i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−7.05615	−0.271190	−0.135595	−	0.990764i	$-0.543295\pi$
$677$	−7.05615	−0.271190	−0.135595	+	0.990764i	$0.543295\pi$
$678$	− 17.8700i	− 0.686293i
$679$	−10.0353	−0.385119
$680$	4.00000	0.153393
$681$	15.3205i	0.587083i
$682$	7.03856i	0.269520i
$683$	5.95317i	0.227792i	0.993493	+	0.113896i	$0.0363330\pi$
$683$	5.95317i	0.227792i	−0.993493	+	0.113896i	$0.963667\pi$
$684$	6.31274i	0.241373i
$685$	−7.03696	−0.268869
$686$	−2.61124	−0.0996976
$687$	− 22.2644i	− 0.849438i
$688$	7.57286	0.288713
$689$	0	0
$690$	−1.24453	−0.0473786
$691$	18.7591i	0.713628i	0.934175	+	0.356814i	$0.116137\pi$
$691$	18.7591i	0.713628i	−0.934175	+	0.356814i	$0.883863\pi$
$692$	−0.440685	−0.0167523
$693$	6.07957	0.230944
$694$	9.68162i	0.367509i
$695$	− 11.6447i	− 0.441707i
$696$	− 10.0448i	− 0.380747i
$697$	− 37.1573i	− 1.40743i
$698$	−19.3205	−0.731292
$699$	10.8366	0.409877
$700$	3.64466i	0.137755i
$701$	−28.5298	−1.07755	−0.538777	−	0.842448i	$-0.681113\pi$
$701$	−28.5298	−1.07755	−0.538777	+	0.842448i	$0.681113\pi$
$702$	0	0
$703$	62.2703	2.34857
$704$	1.66808i	0.0628680i
$705$	−6.82522	−0.257053
$706$	22.9750	0.864677
$707$	− 19.4485i	− 0.731435i
$708$	6.10876i	0.229581i
$709$	23.1926i	0.871015i	0.900185	+	0.435507i	$0.143431\pi$
$709$	23.1926i	0.871015i	−0.900185	+	0.435507i	$0.856569\pi$
$710$	3.51093i	0.131763i
$711$	9.93398	0.372553
$712$	5.95162	0.223046
$713$	− 5.25139i	− 0.196666i
$714$	14.5786	0.545592
$715$	0	0
$716$	−19.6368	−0.733863
$717$	16.4975i	0.616111i
$718$	2.21752	0.0827572
$719$	−11.7128	−0.436814	−0.218407	−	0.975858i	$-0.570086\pi$
$719$	−11.7128	−0.436814	−0.218407	+	0.975858i	$0.570086\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 27.3804i	− 1.01970i
$722$	− 20.8506i	− 0.775980i
$723$	− 4.40435i	− 0.163800i
$724$	−6.21752	−0.231072
$725$	−10.0448	−0.373054
$726$	8.21752i	0.304981i
$727$	3.82677	0.141927	0.0709634	−	0.997479i	$-0.477393\pi$
$727$	3.82677	0.141927	0.0709634	+	0.997479i	$0.477393\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.2175i	0.452191i
$731$	−30.2915	−1.12037
$732$	−7.46410	−0.275881
$733$	− 12.9340i	− 0.477727i	−0.971053	−	0.238864i	$-0.923225\pi$
$733$	− 12.9340i	− 0.477727i	0.971053	−	0.238864i	$-0.0767749\pi$
$734$	− 6.08112i	− 0.224458i
$735$	6.28354i	0.231772i
$736$	− 1.24453i	− 0.0458741i
$737$	24.6098	0.906515
$738$	−9.28932	−0.341945
$739$	− 35.3462i	− 1.30023i	−0.759836	−	0.650115i	$-0.774721\pi$
$739$	− 35.3462i	− 1.30023i	0.759836	−	0.650115i	$-0.225279\pi$
$740$	−9.86423	−0.362616
$741$	0	0
$742$	3.09298	0.113547
$743$	− 37.0953i	− 1.36090i	−0.732796	−	0.680448i	$-0.761785\pi$
$743$	− 37.0953i	− 1.36090i	0.732796	−	0.680448i	$-0.238215\pi$
$744$	4.21957	0.154697
$745$	0.772609	0.0283062
$746$	15.6781i	0.574015i
$747$	7.95317i	0.290991i
$748$	− 6.67230i	− 0.243964i
$749$	61.6975i	2.25438i
$750$	−1.00000	−0.0365148
$751$	9.65807	0.352428	0.176214	−	0.984352i	$-0.443615\pi$
$751$	9.65807	0.352428	0.176214	+	0.984352i	$0.443615\pi$
$752$	− 6.82522i	− 0.248890i
$753$	−11.9453	−0.435313
$754$	0	0
$755$	−9.77838	−0.355872
$756$	− 3.64466i	− 0.132555i
$757$	−43.6885	−1.58789	−0.793943	−	0.607992i	$-0.791975\pi$
$757$	−43.6885	−1.58789	−0.793943	+	0.607992i	$0.791975\pi$
$758$	−30.7166	−1.11568
$759$	2.07598i	0.0753531i
$760$	6.31274i	0.228987i
$761$	− 39.9007i	− 1.44640i	−0.690639	−	0.723200i	$-0.742671\pi$
$761$	− 39.9007i	− 1.44640i	0.690639	−	0.723200i	$-0.257329\pi$
$762$	− 14.4407i	− 0.523131i
$763$	−2.41950	−0.0875918
$764$	−15.6816	−0.567341
$765$	4.00000i	0.144620i
$766$	20.0619	0.724867
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 17.5588i	− 0.633187i	−0.948561	−	0.316594i	$-0.897461\pi$
$769$	− 17.5588i	− 0.633187i	0.948561	−	0.316594i	$-0.102539\pi$
$770$	6.07957	0.219092
$771$	−19.4621	−0.700909
$772$	− 8.12795i	− 0.292531i
$773$	− 12.8372i	− 0.461723i	−0.972987	−	0.230861i	$-0.925846\pi$
$773$	− 12.8372i	− 0.461723i	0.972987	−	0.230861i	$-0.0741544\pi$
$774$	7.57286i	0.272201i
$775$	− 4.21957i	− 0.151571i
$776$	2.75342	0.0988420
$777$	−35.9518	−1.28976
$778$	27.0314i	0.969122i
$779$	58.6410	2.10103
$780$	0	0
$781$	5.85651	0.209562
$782$	4.97813i	0.178018i
$783$	10.0448	0.358971
$784$	−6.28354	−0.224412
$785$	12.0135i	0.428782i
$786$	− 10.8892i	− 0.388405i
$787$	6.95735i	0.248003i	0.992282	+	0.124001i	$0.0395727\pi$
$787$	6.95735i	0.248003i	−0.992282	+	0.124001i	$0.960427\pi$
$788$	− 0.891239i	− 0.0317491i
$789$	2.03478	0.0724402
$790$	9.93398	0.353435
$791$	− 65.1301i	− 2.31576i
$792$	−1.66808	−0.0592725
$793$	0	0
$794$	−3.73628	−0.132596
$795$	0.848634i	0.0300979i
$796$	0.361116	0.0127994
$797$	−34.1354	−1.20914	−0.604569	−	0.796553i	$-0.706655\pi$
$797$	−34.1354	−1.20914	−0.604569	+	0.796553i	$0.706655\pi$
$798$	23.0078i	0.814466i
$799$	27.3009i	0.965835i
$800$	− 1.00000i	− 0.0353553i
$801$	5.95162i	0.210290i
$802$	−28.0911	−0.991932
$803$	20.3798	0.719186
$804$	− 14.7534i	− 0.520313i
$805$	−4.53590	−0.159869
$806$	0	0
$807$	20.5287	0.722645
$808$	5.33615i	0.187725i
$809$	−3.32770	−0.116996	−0.0584978	−	0.998288i	$-0.518631\pi$
$809$	−3.32770	−0.116996	−0.0584978	+	0.998288i	$0.518631\pi$
$810$	1.00000	0.0351364
$811$	20.6083i	0.723655i	0.932245	+	0.361827i	$0.117847\pi$
$811$	20.6083i	0.723655i	−0.932245	+	0.361827i	$0.882153\pi$
$812$	− 36.6098i	− 1.28475i
$813$	25.5401i	0.895730i
$814$	16.4543i	0.576722i
$815$	10.0916	0.353494
$816$	−4.00000	−0.140028
$817$	− 47.8055i	− 1.67250i
$818$	27.3804	0.957335
$819$	0	0
$820$	−9.28932	−0.324397
$821$	1.58005i	0.0551442i	0.999620	+	0.0275721i	$0.00877758\pi$
$821$	1.58005i	0.0551442i	−0.999620	+	0.0275721i	$0.991222\pi$
$822$	7.03696	0.245442
$823$	−37.5168	−1.30776	−0.653878	−	0.756600i	$-0.726859\pi$
$823$	−37.5168	−1.30776	−0.653878	+	0.756600i	$0.726859\pi$
$824$	7.51248i	0.261710i
$825$	1.66808i	0.0580750i
$826$	22.2644i	0.774676i
$827$	49.4912i	1.72098i	0.509469	+	0.860489i	$0.329842\pi$
$827$	49.4912i	1.72098i	−0.509469	+	0.860489i	$0.670158\pi$
$828$	1.24453	0.0432505
$829$	−49.1385	−1.70665	−0.853326	−	0.521378i	$-0.825418\pi$
$829$	−49.1385	−1.70665	−0.853326	+	0.521378i	$0.825418\pi$
$830$	7.95317i	0.276058i
$831$	−18.0604	−0.626508
$832$	0	0
$833$	25.1342	0.870848
$834$	11.6447i	0.403222i
$835$	−7.89701	−0.273287
$836$	10.5301	0.364192
$837$	4.21957i	0.145850i
$838$	− 13.1614i	− 0.454652i
$839$	21.5421i	0.743717i	0.928289	+	0.371858i	$0.121279\pi$
$839$	21.5421i	0.743717i	−0.928289	+	0.371858i	$0.878721\pi$
$840$	− 3.64466i	− 0.125753i
$841$	71.8977	2.47923
$842$	1.29341	0.0445739
$843$	20.2175i	0.696328i
$844$	−2.22739	−0.0766700
$845$	0	0
$846$	6.82522	0.234656
$847$	29.9501i	1.02910i
$848$	−0.848634	−0.0291422
$849$	−8.69149	−0.298291
$850$	4.00000i	0.137199i
$851$	− 12.2764i	− 0.420828i
$852$	− 3.51093i	− 0.120283i
$853$	− 2.79821i	− 0.0958088i	−0.998852	−	0.0479044i	$-0.984746\pi$
$853$	− 2.79821i	− 0.0958088i	0.998852	−	0.0479044i	$-0.0152543\pi$
$854$	−27.2041	−0.930905
$855$	−6.31274	−0.215891
$856$	− 16.9282i	− 0.578594i
$857$	−48.9658	−1.67264	−0.836320	−	0.548242i	$-0.815297\pi$
$857$	−48.9658	−1.67264	−0.836320	+	0.548242i	$0.815297\pi$
$858$	0	0
$859$	4.22739	0.144237	0.0721184	−	0.997396i	$-0.477024\pi$
$859$	4.22739	0.144237	0.0721184	+	0.997396i	$0.477024\pi$
$860$	7.57286i	0.258232i
$861$	−33.8564	−1.15382
$862$	−12.1279	−0.413080
$863$	19.0285i	0.647738i	0.946102	+	0.323869i	$0.104984\pi$
$863$	19.0285i	0.647738i	−0.946102	+	0.323869i	$0.895016\pi$
$864$	1.00000i	0.0340207i
$865$	− 0.440685i	− 0.0149837i
$866$	2.07802i	0.0706141i
$867$	−1.00000	−0.0339618
$868$	15.3789	0.521994
$869$	− 16.5706i	− 0.562120i
$870$	10.0448	0.340550
$871$	0	0
$872$	0.663848	0.0224807
$873$	2.75342i	0.0931892i
$874$	−7.85641	−0.265747
$875$	−3.64466	−0.123212
$876$	− 12.2175i	− 0.412792i
$877$	− 13.5473i	− 0.457459i	−0.973490	−	0.228729i	$-0.926543\pi$
$877$	− 13.5473i	− 0.457459i	0.973490	−	0.228729i	$-0.0734571\pi$
$878$	2.39640i	0.0808745i
$879$	− 7.54790i	− 0.254584i
$880$	−1.66808	−0.0562308
$881$	−4.98081	−0.167808	−0.0839039	−	0.996474i	$-0.526739\pi$
$881$	−4.98081	−0.167808	−0.0839039	+	0.996474i	$0.526739\pi$
$882$	− 6.28354i	− 0.211578i
$883$	30.9829	1.04266	0.521330	−	0.853355i	$-0.325436\pi$
$883$	30.9829	1.04266	0.521330	+	0.853355i	$0.325436\pi$
$884$	0	0
$885$	−6.10876	−0.205344
$886$	21.9959i	0.738967i
$887$	17.1765	0.576731	0.288365	−	0.957520i	$-0.406888\pi$
$887$	17.1765	0.576731	0.288365	+	0.957520i	$0.406888\pi$
$888$	9.86423	0.331022
$889$	− 52.6314i	− 1.76520i
$890$	5.95162i	0.199499i
$891$	− 1.66808i	− 0.0558826i
$892$	− 6.08380i	− 0.203701i
$893$	−43.0858	−1.44181
$894$	−0.772609	−0.0258399
$895$	− 19.6368i	− 0.656387i
$896$	3.64466	0.121760
$897$	0	0
$898$	29.2253	0.975261
$899$	42.3847i	1.41361i
$900$	1.00000	0.0333333
$901$	3.39454	0.113088
$902$	15.4953i	0.515937i
$903$	27.6005i	0.918487i
$904$	17.8700i	0.594347i
$905$	− 6.21752i	− 0.206677i
$906$	9.77838	0.324865
$907$	25.0797	0.832758	0.416379	−	0.909191i	$-0.363299\pi$
$907$	25.0797	0.832758	0.416379	+	0.909191i	$0.363299\pi$
$908$	− 15.3205i	− 0.508429i
$909$	−5.33615	−0.176989
$910$	0	0
$911$	18.2332	0.604092	0.302046	−	0.953293i	$-0.402330\pi$
$911$	18.2332	0.604092	0.302046	+	0.953293i	$0.402330\pi$
$912$	− 6.31274i	− 0.209036i
$913$	13.2665	0.439057
$914$	39.6432	1.31128
$915$	− 7.46410i	− 0.246756i
$916$	22.2644i	0.735635i
$917$	− 39.6874i	− 1.31059i
$918$	− 4.00000i	− 0.132020i
$919$	−34.3882	−1.13436	−0.567181	−	0.823593i	$-0.691966\pi$
$919$	−34.3882	−1.13436	−0.567181	+	0.823593i	$0.691966\pi$
$920$	1.24453	0.0410310
$921$	26.0427i	0.858137i
$922$	16.6758	0.549190
$923$	0	0
$924$	−6.07957	−0.200003
$925$	− 9.86423i	− 0.324334i
$926$	−32.2175	−1.05873
$927$	−7.51248	−0.246742
$928$	10.0448i	0.329736i
$929$	− 31.0904i	− 1.02004i	−0.860161	−	0.510022i	$-0.829637\pi$
$929$	− 31.0904i	− 1.02004i	0.860161	−	0.510022i	$-0.170363\pi$
$930$	4.21957i	0.138365i
$931$	39.6663i	1.30001i
$932$	−10.8366	−0.354964
$933$	25.3789	0.830868
$934$	− 6.88137i	− 0.225165i
$935$	6.67230	0.218208
$936$	0	0
$937$	−18.8783	−0.616726	−0.308363	−	0.951269i	$-0.599781\pi$
$937$	−18.8783	−0.616726	−0.308363	+	0.951269i	$0.599781\pi$
$938$	− 53.7712i	− 1.75569i
$939$	−31.4600	−1.02666
$940$	6.82522	0.222614
$941$	28.9398i	0.943409i	0.881757	+	0.471705i	$0.156361\pi$
$941$	28.9398i	0.943409i	−0.881757	+	0.471705i	$0.843639\pi$
$942$	− 12.0135i	− 0.391423i
$943$	− 11.5609i	− 0.376473i
$944$	− 6.10876i	− 0.198823i
$945$	3.64466	0.118561
$946$	12.6321	0.410705
$947$	7.86896i	0.255707i	0.991793	+	0.127853i	$0.0408087\pi$
$947$	7.86896i	0.255707i	−0.991793	+	0.127853i	$0.959191\pi$
$948$	−9.93398	−0.322641
$949$	0	0
$950$	−6.31274	−0.204812
$951$	− 24.7093i	− 0.801253i
$952$	−14.5786	−0.472496
$953$	55.0968	1.78476	0.892381	−	0.451283i	$-0.149033\pi$
$953$	55.0968	1.78476	0.892381	+	0.451283i	$0.149033\pi$
$954$	− 0.848634i	− 0.0274755i
$955$	− 15.6816i	− 0.507445i
$956$	− 16.4975i	− 0.533568i
$957$	− 16.7555i	− 0.541627i
$958$	18.9709	0.612923
$959$	25.6473	0.828196
$960$	1.00000i	0.0322749i
$961$	13.1952	0.425653
$962$	0	0
$963$	16.9282	0.545504
$964$	4.40435i	0.141855i
$965$	8.12795	0.261648
$966$	4.53590	0.145940
$967$	− 31.5523i	− 1.01465i	−0.861754	−	0.507326i	$-0.830634\pi$
$967$	− 31.5523i	− 1.01465i	0.861754	−	0.507326i	$-0.169366\pi$
$968$	− 8.21752i	− 0.264121i
$969$	25.2509i	0.811177i
$970$	2.75342i	0.0884070i
$971$	3.67585	0.117964	0.0589818	−	0.998259i	$-0.481215\pi$
$971$	3.67585	0.117964	0.0589818	+	0.998259i	$0.481215\pi$
$972$	−1.00000	−0.0320750
$973$	42.4408i	1.36059i
$974$	1.91620	0.0613991
$975$	0	0
$976$	7.46410	0.238920
$977$	42.0205i	1.34435i	0.740391	+	0.672177i	$0.234641\pi$
$977$	42.0205i	1.34435i	−0.740391	+	0.672177i	$0.765359\pi$
$978$	−10.0916	−0.322694
$979$	9.92775	0.317292
$980$	− 6.28354i	− 0.200720i
$981$	0.663848i	0.0211950i
$982$	33.6375i	1.07341i
$983$	13.4307i	0.428372i	0.976793	+	0.214186i	$0.0687099\pi$
$983$	13.4307i	0.428372i	−0.976793	+	0.214186i	$0.931290\pi$
$984$	9.28932	0.296133
$985$	0.891239	0.0283972
$986$	− 40.1791i	− 1.27956i
$987$	24.8756	0.791799
$988$	0	0
$989$	−9.42468	−0.299687
$990$	− 1.66808i	− 0.0530149i
$991$	43.7988	1.39132	0.695658	−	0.718373i	$-0.255113\pi$
$991$	43.7988	1.39132	0.695658	+	0.718373i	$0.255113\pi$
$992$	−4.21957	−0.133971
$993$	5.60360i	0.177825i
$994$	− 12.7962i	− 0.405870i
$995$	0.361116i	0.0114481i
$996$	− 7.95317i	− 0.252006i
$997$	−23.5125	−0.744648	−0.372324	−	0.928103i	$-0.621439\pi$
$997$	−23.5125	−0.744648	−0.372324	+	0.928103i	$0.621439\pi$
$998$	1.82522	0.0577763
$999$	9.86423i	0.312090i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.ba.1351.8		8
13.5	odd	4		5070.2.a.ca.1.1		4
13.8	odd	4		5070.2.a.bz.1.4		4
13.9	even	3		390.2.bb.c.361.3	yes	8
13.10	even	6		390.2.bb.c.121.3	✓	8
13.12	even	2	inner	5070.2.b.ba.1351.1		8
39.23	odd	6		1170.2.bs.f.901.1		8
39.35	odd	6		1170.2.bs.f.361.1		8
65.9	even	6		1950.2.bc.g.751.2		8
65.22	odd	12		1950.2.y.k.49.2		8
65.23	odd	12		1950.2.y.k.199.2		8
65.48	odd	12		1950.2.y.j.49.3		8
65.49	even	6		1950.2.bc.g.901.2		8
65.62	odd	12		1950.2.y.j.199.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.c.121.3	✓	8	13.10	even	6
390.2.bb.c.361.3	yes	8	13.9	even	3
1170.2.bs.f.361.1		8	39.35	odd	6
1170.2.bs.f.901.1		8	39.23	odd	6
1950.2.y.j.49.3		8	65.48	odd	12
1950.2.y.j.199.3		8	65.62	odd	12
1950.2.y.k.49.2		8	65.22	odd	12
1950.2.y.k.199.2		8	65.23	odd	12
1950.2.bc.g.751.2		8	65.9	even	6
1950.2.bc.g.901.2		8	65.49	even	6
5070.2.a.bz.1.4		4	13.8	odd	4
5070.2.a.ca.1.1		4	13.5	odd	4
5070.2.b.ba.1351.1		8	13.12	even	2	inner
5070.2.b.ba.1351.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +3.64466i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +3.64466i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.66808i q^{11} -1.00000 q^{12} -3.64466 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} -6.31274i q^{19} +1.00000i q^{20} +3.64466i q^{21} +1.66808 q^{22} -1.24453 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.64466i q^{28} +10.0448 q^{29} +1.00000 q^{30} +4.21957i q^{31} +1.00000i q^{32} -1.66808i q^{33} -4.00000i q^{34} +3.64466 q^{35} -1.00000 q^{36} +9.86423i q^{37} +6.31274 q^{38} -1.00000 q^{40} +9.28932i q^{41} -3.64466 q^{42} +7.57286 q^{43} +1.66808i q^{44} -1.00000i q^{45} -1.24453i q^{46} -6.82522i q^{47} +1.00000 q^{48} -6.28354 q^{49} -1.00000i q^{50} -4.00000 q^{51} -0.848634 q^{53} +1.00000i q^{54} -1.66808 q^{55} +3.64466 q^{56} -6.31274i q^{57} +10.0448i q^{58} -6.10876i q^{59} +1.00000i q^{60} +7.46410 q^{61} -4.21957 q^{62} +3.64466i q^{63} -1.00000 q^{64} +1.66808 q^{66} +14.7534i q^{67} +4.00000 q^{68} -1.24453 q^{69} +3.64466i q^{70} +3.51093i q^{71} -1.00000i q^{72} +12.2175i q^{73} -9.86423 q^{74} -1.00000 q^{75} +6.31274i q^{76} +6.07957 q^{77} +9.93398 q^{79} -1.00000i q^{80} +1.00000 q^{81} -9.28932 q^{82} +7.95317i q^{83} -3.64466i q^{84} +4.00000i q^{85} +7.57286i q^{86} +10.0448 q^{87} -1.66808 q^{88} +5.95162i q^{89} +1.00000 q^{90} +1.24453 q^{92} +4.21957i q^{93} +6.82522 q^{94} -6.31274 q^{95} +1.00000i q^{96} +2.75342i q^{97} -6.28354i q^{98} -1.66808i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^9	\( 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9} + 8 q^{10} - 8 q^{12} + 4 q^{14} + 8 q^{16} - 32 q^{17} - 4 q^{22} - 8 q^{23} - 8 q^{25} + 8 q^{27} + 16 q^{29} + 8 q^{30} - 4 q^{35} - 8 q^{36} - 8 q^{40} + 4 q^{42} - 28 q^{43} + 8 q^{48} - 28 q^{49} - 32 q^{51} + 16 q^{53} + 4 q^{55} - 4 q^{56} + 32 q^{61} - 8 q^{62} - 8 q^{64} - 4 q^{66} + 32 q^{68} - 8 q^{69} - 20 q^{74} - 8 q^{75} + 16 q^{77} - 20 q^{79} + 8 q^{81} - 8 q^{82} + 16 q^{87} + 4 q^{88} + 8 q^{90} + 8 q^{92} + 16 q^{94}+O(q^{100}) \) 8 * q + 8 * q^3 - 8 * q^4 + 8 * q^9 + 8 * q^10 - 8 * q^12 + 4 * q^14 + 8 * q^16 - 32 * q^17 - 4 * q^22 - 8 * q^23 - 8 * q^25 + 8 * q^27 + 16 * q^29 + 8 * q^30 - 4 * q^35 - 8 * q^36 - 8 * q^40 + 4 * q^42 - 28 * q^43 + 8 * q^48 - 28 * q^49 - 32 * q^51 + 16 * q^53 + 4 * q^55 - 4 * q^56 + 32 * q^61 - 8 * q^62 - 8 * q^64 - 4 * q^66 + 32 * q^68 - 8 * q^69 - 20 * q^74 - 8 * q^75 + 16 * q^77 - 20 * q^79 + 8 * q^81 - 8 * q^82 + 16 * q^87 + 4 * q^88 + 8 * q^90 + 8 * q^92 + 16 * q^94

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