LMFDB - Embedded newform 5070.2.b.ba.1351.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5070,2,Mod(1351,5070)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [8,0,8,-8,0,0,0,0,8,8,0,-8,0,4,0,8,-32,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == traces)

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			1351.4
Root			$-1.70006 - 1.70006i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.ba.1351.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +4.64466i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.40013i q^{11} -1.00000 q^{12} +4.64466 q^{14} +1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000i q^{18} -8.04479i q^{19} -1.00000i q^{20} +4.64466i q^{21} -4.40013 q^{22} +0.976584 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -4.64466i q^{28} -4.31274 q^{29} +1.00000 q^{30} -6.44069i q^{31} -1.00000i q^{32} -4.40013i q^{33} +4.00000i q^{34} -4.64466 q^{35} -1.00000 q^{36} -3.79603i q^{37} -8.04479 q^{38} -1.00000 q^{40} +7.28932i q^{41} +4.64466 q^{42} -0.716456 q^{43} +4.40013i q^{44} +1.00000i q^{45} -0.976584i q^{46} -9.75342i q^{47} +1.00000 q^{48} -14.5729 q^{49} +1.00000i q^{50} -4.00000 q^{51} +13.5089 q^{53} -1.00000i q^{54} +4.40013 q^{55} -4.64466 q^{56} -8.04479i q^{57} +4.31274i q^{58} -2.18056i q^{59} -1.00000i q^{60} +7.46410 q^{61} -6.44069 q^{62} +4.64466i q^{63} -1.00000 q^{64} -4.40013 q^{66} +1.82522i q^{67} +4.00000 q^{68} +0.976584 q^{69} +4.64466i q^{70} -7.95317i q^{71} +1.00000i q^{72} +4.36112i q^{73} -3.79603 q^{74} -1.00000 q^{75} +8.04479i q^{76} +20.4371 q^{77} -14.9340 q^{79} +1.00000i q^{80} +1.00000 q^{81} +7.28932 q^{82} -3.51093i q^{83} -4.64466i q^{84} -4.00000i q^{85} +0.716456i q^{86} -4.31274 q^{87} +4.40013 q^{88} -8.17274i q^{89} +1.00000 q^{90} -0.976584 q^{92} -6.44069i q^{93} -9.75342 q^{94} +8.04479 q^{95} -1.00000i q^{96} +13.8252i q^{97} +14.5729i q^{98} -4.40013i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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
$7$	4.64466i	1.75552i	0.479104	+	0.877758i	$0.340962\pi$
$7$	4.64466i	1.75552i	−0.479104	+	0.877758i	$0.659038\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	− 4.40013i	− 1.32669i	−0.748315	−	0.663344i	$-0.769137\pi$
$11$	− 4.40013i	− 1.32669i	0.748315	−	0.663344i	$-0.230863\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	4.64466	1.24134
$15$	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
$19$	− 8.04479i	− 1.84560i	−0.385279	−	0.922800i	$-0.625895\pi$
$19$	− 8.04479i	− 1.84560i	0.385279	−	0.922800i	$-0.374105\pi$
$20$	− 1.00000i	− 0.223607i
$21$	4.64466i	1.01355i
$22$	−4.40013	−0.938110
$23$	0.976584	0.203632	0.101816	−	0.994803i	$-0.467535\pi$
$23$	0.976584	0.203632	0.101816	+	0.994803i	$0.467535\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 4.64466i	− 0.877758i
$29$	−4.31274	−0.800855	−0.400427	−	0.916328i	$-0.631138\pi$
$29$	−4.31274	−0.800855	−0.400427	+	0.916328i	$0.631138\pi$
$30$	1.00000	0.182574
$31$	− 6.44069i	− 1.15678i	−0.815760	−	0.578391i	$-0.803681\pi$
$31$	− 6.44069i	− 1.15678i	0.815760	−	0.578391i	$-0.196319\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.40013i	− 0.765964i
$34$	4.00000i	0.685994i
$35$	−4.64466	−0.785091
$36$	−1.00000	−0.166667
$37$	− 3.79603i	− 0.624063i	−0.950072	−	0.312031i	$-0.898991\pi$
$37$	− 3.79603i	− 0.624063i	0.950072	−	0.312031i	$-0.101009\pi$
$38$	−8.04479	−1.30504
$39$	0	0
$40$	−1.00000	−0.158114
$41$	7.28932i	1.13840i	0.822199	+	0.569200i	$0.192747\pi$
$41$	7.28932i	1.13840i	−0.822199	+	0.569200i	$0.807253\pi$
$42$	4.64466	0.716687
$43$	−0.716456	−0.109259	−0.0546293	−	0.998507i	$-0.517398\pi$
$43$	−0.716456	−0.109259	−0.0546293	+	0.998507i	$0.517398\pi$
$44$	4.40013i	0.663344i
$45$	1.00000i	0.149071i
$46$	− 0.976584i	− 0.143989i
$47$	− 9.75342i	− 1.42268i	−0.702847	−	0.711341i	$-0.748088\pi$
$47$	− 9.75342i	− 1.42268i	0.702847	−	0.711341i	$-0.251912\pi$
$48$	1.00000	0.144338
$49$	−14.5729	−2.08184
$50$	1.00000i	0.141421i
$51$	−4.00000	−0.560112
$52$	0	0
$53$	13.5089	1.85559	0.927794	−	0.373092i	$-0.121703\pi$
$53$	13.5089	1.85559	0.927794	+	0.373092i	$0.121703\pi$
$54$	− 1.00000i	− 0.136083i
$55$	4.40013	0.593313
$56$	−4.64466	−0.620669
$57$	− 8.04479i	− 1.06556i
$58$	4.31274i	0.566290i
$59$	− 2.18056i	− 0.283884i	−0.989875	−	0.141942i	$-0.954665\pi$
$59$	− 2.18056i	− 0.283884i	0.989875	−	0.141942i	$-0.0453347\pi$
$60$	− 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	$0.341420\pi$
$62$	−6.44069	−0.817968
$63$	4.64466i	0.585172i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.40013	−0.541618
$67$	1.82522i	0.222986i	0.993765	+	0.111493i	$0.0355632\pi$
$67$	1.82522i	0.222986i	−0.993765	+	0.111493i	$0.964437\pi$
$68$	4.00000	0.485071
$69$	0.976584	0.117567
$70$	4.64466i	0.555143i
$71$	− 7.95317i	− 0.943867i	−0.881634	−	0.471934i	$-0.843556\pi$
$71$	− 7.95317i	− 0.943867i	0.881634	−	0.471934i	$-0.156444\pi$
$72$	1.00000i	0.117851i
$73$	4.36112i	0.510430i	0.966884	+	0.255215i	$0.0821463\pi$
$73$	4.36112i	0.510430i	−0.966884	+	0.255215i	$0.917854\pi$
$74$	−3.79603	−0.441279
$75$	−1.00000	−0.115470
$76$	8.04479i	0.922800i
$77$	20.4371	2.32902
$78$	0	0
$79$	−14.9340	−1.68020	−0.840102	−	0.542429i	$-0.817505\pi$
$79$	−14.9340	−1.68020	−0.840102	+	0.542429i	$0.817505\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	7.28932	0.804971
$83$	− 3.51093i	− 0.385375i	−0.981260	−	0.192688i	$-0.938280\pi$
$83$	− 3.51093i	− 0.385375i	0.981260	−	0.192688i	$-0.0617204\pi$
$84$	− 4.64466i	− 0.506774i
$85$	− 4.00000i	− 0.433861i
$86$	0.716456i	0.0772575i
$87$	−4.31274	−0.462374
$88$	4.40013	0.469055
$89$	− 8.17274i	− 0.866308i	−0.901320	−	0.433154i	$-0.857401\pi$
$89$	− 8.17274i	− 0.866308i	0.901320	−	0.433154i	$-0.142599\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−0.976584	−0.101816
$93$	− 6.44069i	− 0.667868i
$94$	−9.75342	−1.00599
$95$	8.04479	0.825378
$96$	− 1.00000i	− 0.102062i
$97$	13.8252i	1.40374i	0.712306	+	0.701869i	$0.247651\pi$
$97$	13.8252i	1.40374i	−0.712306	+	0.701869i	$0.752349\pi$
$98$	14.5729i	1.47208i
$99$	− 4.40013i	− 0.442229i
$100$	1.00000	0.100000
$101$	6.80025	0.676651	0.338325	−	0.941029i	$-0.390140\pi$
$101$	6.80025	0.676651	0.338325	+	0.941029i	$0.390140\pi$
$102$	4.00000i	0.396059i
$103$	−5.29137	−0.521374	−0.260687	−	0.965423i	$-0.583949\pi$
$103$	−5.29137	−0.521374	−0.260687	+	0.965423i	$0.583949\pi$
$104$	0	0
$105$	−4.64466	−0.453272
$106$	− 13.5089i	− 1.31210i
$107$	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
$109$	− 12.8003i	− 1.22604i	−0.790067	−	0.613021i	$-0.789954\pi$
$109$	− 12.8003i	− 1.22604i	0.790067	−	0.613021i	$-0.210046\pi$
$110$	− 4.40013i	− 0.419536i
$111$	− 3.79603i	− 0.360303i
$112$	4.64466i	0.438879i
$113$	13.0662	1.22916	0.614580	−	0.788854i	$-0.289325\pi$
$113$	13.0662	1.22916	0.614580	+	0.788854i	$0.289325\pi$
$114$	−8.04479	−0.753463
$115$	0.976584i	0.0910669i
$116$	4.31274	0.400427
$117$	0	0
$118$	−2.18056	−0.200737
$119$	− 18.5786i	− 1.70310i
$120$	−1.00000	−0.0912871
$121$	−8.36112	−0.760101
$122$	− 7.46410i	− 0.675768i
$123$	7.28932i	0.657256i
$124$	6.44069i	0.578391i
$125$	− 1.00000i	− 0.0894427i
$126$	4.64466	0.413779
$127$	−12.2196	−1.08431	−0.542156	−	0.840278i	$-0.682392\pi$
$127$	−12.2196	−1.08431	−0.542156	+	0.840278i	$0.682392\pi$
$128$	1.00000i	0.0883883i
$129$	−0.716456	−0.0630805
$130$	0	0
$131$	−0.378757	−0.0330921	−0.0165461	−	0.999863i	$-0.505267\pi$
$131$	−0.378757	−0.0330921	−0.0165461	+	0.999863i	$0.505267\pi$
$132$	4.40013i	0.382982i
$133$	37.3653	3.23998
$134$	1.82522	0.157675
$135$	1.00000i	0.0860663i
$136$	− 4.00000i	− 0.342997i
$137$	− 1.25235i	− 0.106996i	−0.998568	−	0.0534979i	$-0.982963\pi$
$137$	− 1.25235i	− 0.106996i	0.998568	−	0.0534979i	$-0.0170371\pi$
$138$	− 0.976584i	− 0.0831323i
$139$	3.35534	0.284596	0.142298	−	0.989824i	$-0.454551\pi$
$139$	3.35534	0.284596	0.142298	+	0.989824i	$0.454551\pi$
$140$	4.64466	0.392545
$141$	− 9.75342i	− 0.821386i
$142$	−7.95317	−0.667415
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 4.31274i	− 0.358153i
$146$	4.36112	0.360929
$147$	−14.5729	−1.20195
$148$	3.79603i	0.312031i
$149$	− 4.61970i	− 0.378460i	−0.981933	−	0.189230i	$-0.939401\pi$
$149$	− 4.61970i	− 0.378460i	0.981933	−	0.189230i	$-0.0605992\pi$
$150$	1.00000i	0.0816497i
$151$	− 11.2425i	− 0.914901i	−0.889235	−	0.457450i	$-0.848763\pi$
$151$	− 11.2425i	− 0.914901i	0.889235	−	0.457450i	$-0.151237\pi$
$152$	8.04479	0.652518
$153$	−4.00000	−0.323381
$154$	− 20.4371i	− 1.64687i
$155$	6.44069	0.517328
$156$	0	0
$157$	−1.50311	−0.119961	−0.0599807	−	0.998200i	$-0.519104\pi$
$157$	−1.50311	−0.119961	−0.0599807	+	0.998200i	$0.519104\pi$
$158$	14.9340i	1.18808i
$159$	13.5089	1.07132
$160$	1.00000	0.0790569
$161$	4.53590i	0.357479i
$162$	− 1.00000i	− 0.0785674i
$163$	− 0.176330i	− 0.0138112i	−0.999976	−	0.00690561i	$-0.997802\pi$
$163$	− 0.176330i	− 0.0138112i	0.999976	−	0.00690561i	$-0.00219814\pi$
$164$	− 7.28932i	− 0.569200i
$165$	4.40013	0.342549
$166$	−3.51093	−0.272501
$167$	− 8.68162i	− 0.671804i	−0.941897	−	0.335902i	$-0.890959\pi$
$167$	− 8.68162i	− 0.671804i	0.941897	−	0.335902i	$-0.109041\pi$
$168$	−4.64466	−0.358343
$169$	0	0
$170$	−4.00000	−0.306786
$171$	− 8.04479i	− 0.615200i
$172$	0.716456	0.0546293
$173$	−1.78043	−0.135364	−0.0676818	−	0.997707i	$-0.521560\pi$
$173$	−1.78043	−0.135364	−0.0676818	+	0.997707i	$0.521560\pi$
$174$	4.31274i	0.326948i
$175$	− 4.64466i	− 0.351103i
$176$	− 4.40013i	− 0.331672i
$177$	− 2.18056i	− 0.163901i
$178$	−8.17274	−0.612572
$179$	17.4157	1.30171	0.650856	−	0.759201i	$-0.274410\pi$
$179$	17.4157	1.30171	0.650856	+	0.759201i	$0.274410\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−10.3611	−0.770136	−0.385068	−	0.922888i	$-0.625822\pi$
$181$	−10.3611	−0.770136	−0.385068	+	0.922888i	$0.625822\pi$
$182$	0	0
$183$	7.46410	0.551762
$184$	0.976584i	0.0719947i
$185$	3.79603	0.279089
$186$	−6.44069	−0.472254
$187$	17.6005i	1.28708i
$188$	9.75342i	0.711341i
$189$	4.64466i	0.337849i
$190$	− 8.04479i	− 0.583630i
$191$	−0.897014	−0.0649057	−0.0324528	−	0.999473i	$-0.510332\pi$
$191$	−0.897014	−0.0649057	−0.0324528	+	0.999473i	$0.510332\pi$
$192$	−1.00000	−0.0721688
$193$	− 20.2644i	− 1.45866i	−0.684162	−	0.729330i	$-0.739832\pi$
$193$	− 20.2644i	− 1.45866i	0.684162	−	0.729330i	$-0.260168\pi$
$194$	13.8252	0.992593
$195$	0	0
$196$	14.5729	1.04092
$197$	− 9.18056i	− 0.654088i	−0.945009	−	0.327044i	$-0.893948\pi$
$197$	− 9.18056i	− 0.654088i	0.945009	−	0.327044i	$-0.106052\pi$
$198$	−4.40013	−0.312703
$199$	16.2175	1.14963	0.574815	−	0.818284i	$-0.305074\pi$
$199$	16.2175	1.14963	0.574815	+	0.818284i	$0.305074\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	1.82522i	0.128741i
$202$	− 6.80025i	− 0.478464i
$203$	− 20.0312i	− 1.40591i
$204$	4.00000	0.280056
$205$	−7.28932	−0.509108
$206$	5.29137i	0.368667i
$207$	0.976584	0.0678773
$208$	0	0
$209$	−35.3981	−2.44854
$210$	4.64466i	0.320512i
$211$	−1.61970	−0.111504	−0.0557522	−	0.998445i	$-0.517756\pi$
$211$	−1.61970	−0.111504	−0.0557522	+	0.998445i	$0.517756\pi$
$212$	−13.5089	−0.927794
$213$	− 7.95317i	− 0.544942i
$214$	− 16.9282i	− 1.15719i
$215$	− 0.716456i	− 0.0488619i
$216$	1.00000i	0.0680414i
$217$	29.9148	2.03075
$218$	−12.8003	−0.866943
$219$	4.36112i	0.294697i
$220$	−4.40013	−0.296656
$221$	0	0
$222$	−3.79603	−0.254773
$223$	− 2.23671i	− 0.149781i	−0.997192	−	0.0748906i	$-0.976139\pi$
$223$	− 2.23671i	− 0.149781i	0.997192	−	0.0748906i	$-0.0238608\pi$
$224$	4.64466	0.310334
$225$	−1.00000	−0.0666667
$226$	− 13.0662i	− 0.869148i
$227$	− 15.3205i	− 1.01686i	−0.861104	−	0.508429i	$-0.830226\pi$
$227$	− 15.3205i	− 1.01686i	0.861104	−	0.508429i	$-0.169774\pi$
$228$	8.04479i	0.532779i
$229$	10.1279i	0.669274i	0.942347	+	0.334637i	$0.108614\pi$
$229$	10.1279i	0.669274i	−0.942347	+	0.334637i	$0.891386\pi$
$230$	0.976584	0.0643940
$231$	20.4371	1.34466
$232$	− 4.31274i	− 0.283145i
$233$	20.7519	1.35950	0.679750	−	0.733444i	$-0.262088\pi$
$233$	20.7519	1.35950	0.679750	+	0.733444i	$0.262088\pi$
$234$	0	0
$235$	9.75342	0.636243
$236$	2.18056i	0.141942i
$237$	−14.9340	−0.970066
$238$	−18.5786	−1.20427
$239$	24.3539i	1.57532i	0.616107	+	0.787662i	$0.288709\pi$
$239$	24.3539i	1.57532i	−0.616107	+	0.787662i	$0.711291\pi$
$240$	1.00000i	0.0645497i
$241$	− 19.8685i	− 1.27984i	−0.768442	−	0.639920i	$-0.778968\pi$
$241$	− 19.8685i	− 1.27984i	0.768442	−	0.639920i	$-0.221032\pi$
$242$	8.36112i	0.537473i
$243$	1.00000	0.0641500
$244$	−7.46410	−0.477840
$245$	− 14.5729i	− 0.931026i
$246$	7.28932	0.464750
$247$	0	0
$248$	6.44069	0.408984
$249$	− 3.51093i	− 0.222496i
$250$	−1.00000	−0.0632456
$251$	−13.5713	−0.856614	−0.428307	−	0.903633i	$-0.640890\pi$
$251$	−13.5713	−0.856614	−0.428307	+	0.903633i	$0.640890\pi$
$252$	− 4.64466i	− 0.292586i
$253$	− 4.29709i	− 0.270156i
$254$	12.2196i	0.766724i
$255$	− 4.00000i	− 0.250490i
$256$	1.00000	0.0625000
$257$	−0.662300	−0.0413132	−0.0206566	−	0.999787i	$-0.506576\pi$
$257$	−0.662300	−0.0413132	−0.0206566	+	0.999787i	$0.506576\pi$
$258$	0.716456i	0.0446046i
$259$	17.6312	1.09555
$260$	0	0
$261$	−4.31274	−0.266952
$262$	0.378757i	0.0233997i
$263$	30.7498	1.89612	0.948058	−	0.318098i	$-0.103044\pi$
$263$	30.7498	1.89612	0.948058	+	0.318098i	$0.103044\pi$
$264$	4.40013	0.270809
$265$	13.5089i	0.829844i
$266$	− 37.3653i	− 2.29101i
$267$	− 8.17274i	− 0.500163i
$268$	− 1.82522i	− 0.111493i
$269$	−3.74410	−0.228282	−0.114141	−	0.993465i	$-0.536412\pi$
$269$	−3.74410	−0.228282	−0.114141	+	0.993465i	$0.536412\pi$
$270$	1.00000	0.0608581
$271$	− 27.7612i	− 1.68637i	−0.537622	−	0.843186i	$-0.680677\pi$
$271$	− 27.7612i	− 1.68637i	0.537622	−	0.843186i	$-0.319323\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−1.25235	−0.0756575
$275$	4.40013i	0.265338i
$276$	−0.976584	−0.0587834
$277$	−11.9922	−0.720540	−0.360270	−	0.932848i	$-0.617315\pi$
$277$	−11.9922	−0.720540	−0.360270	+	0.932848i	$0.617315\pi$
$278$	− 3.35534i	− 0.201240i
$279$	− 6.44069i	− 0.385594i
$280$	− 4.64466i	− 0.277572i
$281$	− 3.63888i	− 0.217078i	−0.994092	−	0.108539i	$-0.965383\pi$
$281$	− 3.63888i	− 0.217078i	0.994092	−	0.108539i	$-0.0346172\pi$
$282$	−9.75342	−0.580808
$283$	−4.84441	−0.287970	−0.143985	−	0.989580i	$-0.545992\pi$
$283$	−4.84441	−0.287970	−0.143985	+	0.989580i	$0.545992\pi$
$284$	7.95317i	0.471934i
$285$	8.04479	0.476532
$286$	0	0
$287$	−33.8564	−1.99848
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	−4.31274	−0.253253
$291$	13.8252i	0.810449i
$292$	− 4.36112i	− 0.255215i
$293$	3.70081i	0.216204i	0.994140	+	0.108102i	$0.0344773\pi$
$293$	3.70081i	0.216204i	−0.994140	+	0.108102i	$0.965523\pi$
$294$	14.5729i	0.849907i
$295$	2.18056	0.126957
$296$	3.79603	0.220640
$297$	− 4.40013i	− 0.255321i
$298$	−4.61970	−0.267612
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 3.32770i	− 0.191805i
$302$	−11.2425	−0.646932
$303$	6.80025	0.390664
$304$	− 8.04479i	− 0.461400i
$305$	7.46410i	0.427393i
$306$	4.00000i	0.228665i
$307$	7.11454i	0.406048i	0.979174	+	0.203024i	$0.0650770\pi$
$307$	7.11454i	0.406048i	−0.979174	+	0.203024i	$0.934923\pi$
$308$	−20.4371	−1.16451
$309$	−5.29137	−0.301015
$310$	− 6.44069i	− 0.365806i
$311$	−19.9148	−1.12926	−0.564632	−	0.825343i	$-0.690982\pi$
$311$	−19.9148	−1.12926	−0.564632	+	0.825343i	$0.690982\pi$
$312$	0	0
$313$	6.13950	0.347025	0.173513	−	0.984832i	$-0.444488\pi$
$313$	6.13950	0.347025	0.173513	+	0.984832i	$0.444488\pi$
$314$	1.50311i	0.0848255i
$315$	−4.64466	−0.261697
$316$	14.9340	0.840102
$317$	− 7.85286i	− 0.441061i	−0.975380	−	0.220530i	$-0.929221\pi$
$317$	− 7.85286i	− 0.441061i	0.975380	−	0.220530i	$-0.0707788\pi$
$318$	− 13.5089i	− 0.757541i
$319$	18.9766i	1.06248i
$320$	− 1.00000i	− 0.0559017i
$321$	16.9282	0.944840
$322$	4.53590	0.252776
$323$	32.1791i	1.79050i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−0.176330	−0.00976601
$327$	− 12.8003i	− 0.707856i
$328$	−7.28932	−0.402485
$329$	45.3013	2.49754
$330$	− 4.40013i	− 0.242219i
$331$	31.9959i	1.75865i	0.476218	+	0.879327i	$0.342007\pi$
$331$	31.9959i	1.75865i	−0.476218	+	0.879327i	$0.657993\pi$
$332$	3.51093i	0.192688i
$333$	− 3.79603i	− 0.208021i
$334$	−8.68162	−0.475037
$335$	−1.82522	−0.0997223
$336$	4.64466i	0.253387i
$337$	−35.6432	−1.94161	−0.970806	−	0.239867i	$-0.922896\pi$
$337$	−35.6432	−1.94161	−0.970806	+	0.239867i	$0.922896\pi$
$338$	0	0
$339$	13.0662	0.709656
$340$	4.00000i	0.216930i
$341$	−28.3398	−1.53469
$342$	−8.04479	−0.435012
$343$	− 35.1734i	− 1.89918i
$344$	− 0.716456i	− 0.0386287i
$345$	0.976584i	0.0525775i
$346$	1.78043i	0.0957166i
$347$	−6.89701	−0.370251	−0.185126	−	0.982715i	$-0.559269\pi$
$347$	−6.89701	−0.370251	−0.185126	+	0.982715i	$0.559269\pi$
$348$	4.31274	0.231187
$349$	− 19.3205i	− 1.03420i	−0.855924	−	0.517102i	$-0.827011\pi$
$349$	− 19.3205i	− 1.03420i	0.855924	−	0.517102i	$-0.172989\pi$
$350$	−4.64466	−0.248267
$351$	0	0
$352$	−4.40013	−0.234528
$353$	27.4173i	1.45927i	0.683835	+	0.729637i	$0.260311\pi$
$353$	27.4173i	1.45927i	−0.683835	+	0.729637i	$0.739689\pi$
$354$	−2.18056	−0.115895
$355$	7.95317	0.422110
$356$	8.17274i	0.433154i
$357$	− 18.5786i	− 0.983286i
$358$	− 17.4157i	− 0.920449i
$359$	− 14.3611i	− 0.757951i	−0.925407	−	0.378975i	$-0.876277\pi$
$359$	− 14.3611i	− 0.757951i	0.925407	−	0.378975i	$-0.123723\pi$
$360$	−1.00000	−0.0527046
$361$	−45.7186	−2.40624
$362$	10.3611i	0.544568i
$363$	−8.36112	−0.438845
$364$	0	0
$365$	−4.36112	−0.228271
$366$	− 7.46410i	− 0.390155i
$367$	−13.7753	−0.719064	−0.359532	−	0.933133i	$-0.617064\pi$
$367$	−13.7753	−0.719064	−0.359532	+	0.933133i	$0.617064\pi$
$368$	0.976584	0.0509079
$369$	7.28932i	0.379467i
$370$	− 3.79603i	− 0.197346i
$371$	62.7442i	3.25752i
$372$	6.44069i	0.333934i
$373$	−33.4627	−1.73263	−0.866316	−	0.499496i	$-0.833519\pi$
$373$	−33.4627	−1.73263	−0.866316	+	0.499496i	$0.833519\pi$
$374$	17.6005	0.910101
$375$	− 1.00000i	− 0.0516398i
$376$	9.75342	0.502994
$377$	0	0
$378$	4.64466	0.238896
$379$	33.3768i	1.71445i	0.514939	+	0.857227i	$0.327814\pi$
$379$	33.3768i	1.71445i	−0.514939	+	0.857227i	$0.672186\pi$
$380$	−8.04479	−0.412689
$381$	−12.2196	−0.626027
$382$	0.897014i	0.0458952i
$383$	7.33038i	0.374565i	0.982306	+	0.187282i	$0.0599679\pi$
$383$	7.33038i	0.374565i	−0.982306	+	0.187282i	$0.940032\pi$
$384$	1.00000i	0.0510310i
$385$	20.4371i	1.04157i
$386$	−20.2644	−1.03143
$387$	−0.716456	−0.0364195
$388$	− 13.8252i	− 0.701869i
$389$	−32.6198	−1.65389	−0.826946	−	0.562282i	$-0.809924\pi$
$389$	−32.6198	−1.65389	−0.826946	+	0.562282i	$0.809924\pi$
$390$	0	0
$391$	−3.90633	−0.197552
$392$	− 14.5729i	− 0.736041i
$393$	−0.378757	−0.0191058
$394$	−9.18056	−0.462510
$395$	− 14.9340i	− 0.751410i
$396$	4.40013i	0.221115i
$397$	14.4683i	0.726145i	0.931761	+	0.363072i	$0.118272\pi$
$397$	14.4683i	0.726145i	−0.931761	+	0.363072i	$0.881728\pi$
$398$	− 16.2175i	− 0.812911i
$399$	37.3653	1.87060
$400$	−1.00000	−0.0500000
$401$	7.28727i	0.363909i	0.983307	+	0.181955i	$0.0582424\pi$
$401$	7.28727i	0.363909i	−0.983307	+	0.181955i	$0.941758\pi$
$402$	1.82522	0.0910336
$403$	0	0
$404$	−6.80025	−0.338325
$405$	1.00000i	0.0496904i
$406$	−20.0312	−0.994131
$407$	−16.7030	−0.827937
$408$	− 4.00000i	− 0.198030i
$409$	− 24.5766i	− 1.21523i	−0.794230	−	0.607617i	$-0.792126\pi$
$409$	− 24.5766i	− 1.21523i	0.794230	−	0.607617i	$-0.207874\pi$
$410$	7.28932i	0.359994i
$411$	− 1.25235i	− 0.0617741i
$412$	5.29137	0.260687
$413$	10.1279	0.498364
$414$	− 0.976584i	− 0.0479965i
$415$	3.51093	0.172345
$416$	0	0
$417$	3.35534	0.164312
$418$	35.3981i	1.73138i
$419$	15.5537	0.759847	0.379923	−	0.925018i	$-0.375950\pi$
$419$	15.5537	0.759847	0.379923	+	0.925018i	$0.375950\pi$
$420$	4.64466	0.226636
$421$	22.3143i	1.08753i	0.839237	+	0.543766i	$0.183002\pi$
$421$	22.3143i	1.08753i	−0.839237	+	0.543766i	$0.816998\pi$
$422$	1.61970i	0.0788456i
$423$	− 9.75342i	− 0.474228i
$424$	13.5089i	0.656050i
$425$	4.00000	0.194029
$426$	−7.95317	−0.385332
$427$	34.6682i	1.67771i
$428$	−16.9282	−0.818256
$429$	0	0
$430$	−0.716456	−0.0345506
$431$	− 24.2644i	− 1.16877i	−0.811476	−	0.584386i	$-0.801335\pi$
$431$	− 24.2644i	− 1.16877i	0.811476	−	0.584386i	$-0.198665\pi$
$432$	1.00000	0.0481125
$433$	23.0989	1.11006	0.555031	−	0.831830i	$-0.312706\pi$
$433$	23.0989	1.11006	0.555031	+	0.831830i	$0.312706\pi$
$434$	− 29.9148i	− 1.43596i
$435$	− 4.31274i	− 0.206780i
$436$	12.8003i	0.613021i
$437$	− 7.85641i	− 0.375823i
$438$	4.36112	0.208382
$439$	39.9959	1.90890	0.954450	−	0.298370i	$-0.0964430\pi$
$439$	39.9959	1.90890	0.954450	+	0.298370i	$0.0964430\pi$
$440$	4.40013i	0.209768i
$441$	−14.5729	−0.693946
$442$	0	0
$443$	−15.6036	−0.741350	−0.370675	−	0.928763i	$-0.620874\pi$
$443$	−15.6036	−0.741350	−0.370675	+	0.928763i	$0.620874\pi$
$444$	3.79603i	0.180151i
$445$	8.17274	0.387425
$446$	−2.23671	−0.105911
$447$	− 4.61970i	− 0.218504i
$448$	− 4.64466i	− 0.219440i
$449$	27.0042i	1.27441i	0.770696	+	0.637203i	$0.219909\pi$
$449$	27.0042i	1.27441i	−0.770696	+	0.637203i	$0.780091\pi$
$450$	1.00000i	0.0471405i
$451$	32.0739	1.51030
$452$	−13.0662	−0.614580
$453$	− 11.2425i	− 0.528218i
$454$	−15.3205	−0.719027
$455$	0	0
$456$	8.04479	0.376732
$457$	− 17.7868i	− 0.832033i	−0.909357	−	0.416017i	$-0.863426\pi$
$457$	− 17.7868i	− 0.832033i	0.909357	−	0.416017i	$-0.136574\pi$
$458$	10.1279	0.473248
$459$	−4.00000	−0.186704
$460$	− 0.976584i	− 0.0455334i
$461$	24.9652i	1.16274i	0.813638	+	0.581372i	$0.197484\pi$
$461$	24.9652i	1.16274i	−0.813638	+	0.581372i	$0.802516\pi$
$462$	− 20.4371i	− 0.950820i
$463$	− 15.6389i	− 0.726801i	−0.931633	−	0.363400i	$-0.881616\pi$
$463$	− 15.6389i	− 0.726801i	0.931633	−	0.363400i	$-0.118384\pi$
$464$	−4.31274	−0.200214
$465$	6.44069	0.298680
$466$	− 20.7519i	− 0.961312i
$467$	−2.43914	−0.112870	−0.0564349	−	0.998406i	$-0.517973\pi$
$467$	−2.43914	−0.112870	−0.0564349	+	0.998406i	$0.517973\pi$
$468$	0	0
$469$	−8.47751	−0.391455
$470$	− 9.75342i	− 0.449892i
$471$	−1.50311	−0.0692598
$472$	2.18056	0.100368
$473$	3.15250i	0.144952i
$474$	14.9340i	0.685940i
$475$	8.04479i	0.369120i
$476$	18.5786i	0.851550i
$477$	13.5089	0.618529
$478$	24.3539	1.11392
$479$	− 14.1863i	− 0.648190i	−0.946025	−	0.324095i	$-0.894940\pi$
$479$	− 14.1863i	− 0.648190i	0.946025	−	0.324095i	$-0.105060\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−19.8685	−0.904983
$483$	4.53590i	0.206391i
$484$	8.36112	0.380051
$485$	−13.8252	−0.627771
$486$	− 1.00000i	− 0.0453609i
$487$	5.76329i	0.261160i	0.991438	+	0.130580i	$0.0416839\pi$
$487$	5.76329i	0.261160i	−0.991438	+	0.130580i	$0.958316\pi$
$488$	7.46410i	0.337884i
$489$	− 0.176330i	− 0.00797392i
$490$	−14.5729	−0.658335
$491$	1.07534	0.0485295	0.0242647	−	0.999706i	$-0.492276\pi$
$491$	1.07534	0.0485295	0.0242647	+	0.999706i	$0.492276\pi$
$492$	− 7.28932i	− 0.328628i
$493$	17.2509	0.776943
$494$	0	0
$495$	4.40013	0.197771
$496$	− 6.44069i	− 0.289195i
$497$	36.9398	1.65697
$498$	−3.51093	−0.157329
$499$	− 14.7534i	− 0.660454i	−0.943902	−	0.330227i	$-0.892875\pi$
$499$	− 14.7534i	− 0.660454i	0.943902	−	0.330227i	$-0.107125\pi$
$500$	1.00000i	0.0447214i
$501$	− 8.68162i	− 0.387866i
$502$	13.5713i	0.605717i
$503$	−26.6618	−1.18879	−0.594395	−	0.804173i	$-0.702609\pi$
$503$	−26.6618	−1.18879	−0.594395	+	0.804173i	$0.702609\pi$
$504$	−4.64466	−0.206890
$505$	6.80025i	0.302607i
$506$	−4.29709	−0.191029
$507$	0	0
$508$	12.2196	0.542156
$509$	− 14.1806i	− 0.628542i	−0.949333	−	0.314271i	$-0.898240\pi$
$509$	− 14.1806i	− 0.628542i	0.949333	−	0.314271i	$-0.101760\pi$
$510$	−4.00000	−0.177123
$511$	−20.2559	−0.896068
$512$	− 1.00000i	− 0.0441942i
$513$	− 8.04479i	− 0.355186i
$514$	0.662300i	0.0292128i
$515$	− 5.29137i	− 0.233165i
$516$	0.716456	0.0315402
$517$	−42.9163	−1.88746
$518$	− 17.6312i	− 0.774673i
$519$	−1.78043	−0.0781523
$520$	0	0
$521$	23.7476	1.04040	0.520202	−	0.854043i	$-0.325857\pi$
$521$	23.7476	1.04040	0.520202	+	0.854043i	$0.325857\pi$
$522$	4.31274i	0.188763i
$523$	13.8506	0.605646	0.302823	−	0.953047i	$-0.402071\pi$
$523$	13.8506	0.605646	0.302823	+	0.953047i	$0.402071\pi$
$524$	0.378757	0.0165461
$525$	− 4.64466i	− 0.202710i
$526$	− 30.7498i	− 1.34076i
$527$	25.7627i	1.12224i
$528$	− 4.40013i	− 0.191491i
$529$	−22.0463	−0.958534
$530$	13.5089	0.586789
$531$	− 2.18056i	− 0.0946282i
$532$	−37.3653	−1.61999
$533$	0	0
$534$	−8.17274	−0.353669
$535$	16.9282i	0.731870i
$536$	−1.82522	−0.0788374
$537$	17.4157	0.751544
$538$	3.74410i	0.161420i
$539$	64.1224i	2.76195i
$540$	− 1.00000i	− 0.0430331i
$541$	− 14.8898i	− 0.640164i	−0.947390	−	0.320082i	$-0.896290\pi$
$541$	− 14.8898i	− 0.640164i	0.947390	−	0.320082i	$-0.103710\pi$
$542$	−27.7612	−1.19245
$543$	−10.3611	−0.444638
$544$	4.00000i	0.171499i
$545$	12.8003	0.548303
$546$	0	0
$547$	−22.2019	−0.949284	−0.474642	−	0.880179i	$-0.657422\pi$
$547$	−22.2019	−0.949284	−0.474642	+	0.880179i	$0.657422\pi$
$548$	1.25235i	0.0534979i
$549$	7.46410	0.318560
$550$	4.40013	0.187622
$551$	34.6950i	1.47806i
$552$	0.976584i	0.0415662i
$553$	− 69.3632i	− 2.94963i
$554$	11.9922i	0.509499i
$555$	3.79603	0.161132
$556$	−3.35534	−0.142298
$557$	− 1.89969i	− 0.0804927i	−0.999190	−	0.0402463i	$-0.987186\pi$
$557$	− 1.89969i	− 0.0804927i	0.999190	−	0.0402463i	$-0.0128143\pi$
$558$	−6.44069	−0.272656
$559$	0	0
$560$	−4.64466	−0.196273
$561$	17.6005i	0.743094i
$562$	−3.63888	−0.153497
$563$	1.72000	0.0724894	0.0362447	−	0.999343i	$-0.488460\pi$
$563$	1.72000	0.0724894	0.0362447	+	0.999343i	$0.488460\pi$
$564$	9.75342i	0.410693i
$565$	13.0662i	0.549697i
$566$	4.84441i	0.203626i
$567$	4.64466i	0.195057i
$568$	7.95317	0.333707
$569$	0.601920	0.0252338	0.0126169	−	0.999920i	$-0.495984\pi$
$569$	0.601920	0.0252338	0.0126169	+	0.999920i	$0.495984\pi$
$570$	− 8.04479i	− 0.336959i
$571$	11.9808	0.501381	0.250691	−	0.968067i	$-0.419342\pi$
$571$	11.9808	0.501381	0.250691	+	0.968067i	$0.419342\pi$
$572$	0	0
$573$	−0.897014	−0.0374733
$574$	33.8564i	1.41314i
$575$	−0.976584	−0.0407264
$576$	−1.00000	−0.0416667
$577$	43.0293i	1.79133i	0.444725	+	0.895667i	$0.353301\pi$
$577$	43.0293i	1.79133i	−0.444725	+	0.895667i	$0.646699\pi$
$578$	1.00000i	0.0415945i
$579$	− 20.2644i	− 0.842158i
$580$	4.31274i	0.179077i
$581$	16.3071	0.676532
$582$	13.8252	0.573074
$583$	− 59.4408i	− 2.46179i
$584$	−4.36112	−0.180464
$585$	0	0
$586$	3.70081	0.152879
$587$	17.9148i	0.739423i	0.929147	+	0.369711i	$0.120543\pi$
$587$	17.9148i	0.739423i	−0.929147	+	0.369711i	$0.879457\pi$
$588$	14.5729	0.600975
$589$	−51.8139	−2.13496
$590$	− 2.18056i	− 0.0897722i
$591$	− 9.18056i	− 0.377638i
$592$	− 3.79603i	− 0.156016i
$593$	− 0.669624i	− 0.0274981i	−0.999905	−	0.0137491i	$-0.995623\pi$
$593$	− 0.669624i	− 0.0274981i	0.999905	−	0.0137491i	$-0.00437660\pi$
$594$	−4.40013	−0.180539
$595$	18.5786	0.761650
$596$	4.61970i	0.189230i
$597$	16.2175	0.663739
$598$	0	0
$599$	1.29241	0.0528066	0.0264033	−	0.999651i	$-0.491595\pi$
$599$	1.29241	0.0528066	0.0264033	+	0.999651i	$0.491595\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−11.4734	−0.468011	−0.234005	−	0.972235i	$-0.575183\pi$
$601$	−11.4734	−0.468011	−0.234005	+	0.972235i	$0.575183\pi$
$602$	−3.32770	−0.135627
$603$	1.82522i	0.0743286i
$604$	11.2425i	0.457450i
$605$	− 8.36112i	− 0.339928i
$606$	− 6.80025i	− 0.276241i
$607$	−24.1344	−0.979583	−0.489792	−	0.871839i	$-0.662927\pi$
$607$	−24.1344	−0.979583	−0.489792	+	0.871839i	$0.662927\pi$
$608$	−8.04479	−0.326259
$609$	− 20.0312i	− 0.811705i
$610$	7.46410	0.302213
$611$	0	0
$612$	4.00000	0.161690
$613$	− 28.9502i	− 1.16929i	−0.811290	−	0.584644i	$-0.801234\pi$
$613$	− 28.9502i	− 1.16929i	0.811290	−	0.584644i	$-0.198766\pi$
$614$	7.11454	0.287119
$615$	−7.28932	−0.293934
$616$	20.4371i	0.823434i
$617$	− 0.841311i	− 0.0338699i	−0.999857	−	0.0169349i	$-0.994609\pi$
$617$	− 0.841311i	− 0.0338699i	0.999857	−	0.0169349i	$-0.00539082\pi$
$618$	5.29137i	0.212850i
$619$	6.25076i	0.251239i	0.992078	+	0.125620i	$0.0400919\pi$
$619$	6.25076i	0.251239i	−0.992078	+	0.125620i	$0.959908\pi$
$620$	−6.44069	−0.258664
$621$	0.976584	0.0391889
$622$	19.9148i	0.798510i
$623$	37.9596	1.52082
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 6.13950i	− 0.245384i
$627$	−35.3981	−1.41366
$628$	1.50311	0.0599807
$629$	15.1841i	0.605430i
$630$	4.64466i	0.185048i
$631$	3.02496i	0.120422i	0.998186	+	0.0602110i	$0.0191773\pi$
$631$	3.02496i	0.120422i	−0.998186	+	0.0602110i	$0.980823\pi$
$632$	− 14.9340i	− 0.594042i
$633$	−1.61970	−0.0643771
$634$	−7.85286	−0.311877
$635$	− 12.2196i	− 0.484919i
$636$	−13.5089	−0.535662
$637$	0	0
$638$	18.9766	0.751290
$639$	− 7.95317i	− 0.314622i
$640$	−1.00000	−0.0395285
$641$	−0.772609	−0.0305162	−0.0152581	−	0.999884i	$-0.504857\pi$
$641$	−0.772609	−0.0305162	−0.0152581	+	0.999884i	$0.504857\pi$
$642$	− 16.9282i	− 0.668103i
$643$	19.3745i	0.764057i	0.924151	+	0.382028i	$0.124774\pi$
$643$	19.3745i	0.764057i	−0.924151	+	0.382028i	$0.875226\pi$
$644$	− 4.53590i	− 0.178739i
$645$	− 0.716456i	− 0.0282104i
$646$	32.1791	1.26607
$647$	27.1905	1.06897	0.534485	−	0.845178i	$-0.320506\pi$
$647$	27.1905	1.06897	0.534485	+	0.845178i	$0.320506\pi$
$648$	1.00000i	0.0392837i
$649$	−9.59473	−0.376626
$650$	0	0
$651$	29.9148	1.17245
$652$	0.176330i	0.00690561i
$653$	15.7960	0.618144	0.309072	−	0.951039i	$-0.399982\pi$
$653$	15.7960	0.618144	0.309072	+	0.951039i	$0.399982\pi$
$654$	−12.8003	−0.500530
$655$	− 0.378757i	− 0.0147993i
$656$	7.28932i	0.284600i
$657$	4.36112i	0.170143i
$658$	− 45.3013i	− 1.76603i
$659$	30.4762	1.18719	0.593593	−	0.804766i	$-0.297709\pi$
$659$	30.4762	1.18719	0.593593	+	0.804766i	$0.297709\pi$
$660$	−4.40013	−0.171275
$661$	27.8876i	1.08470i	0.840152	+	0.542351i	$0.182466\pi$
$661$	27.8876i	1.08470i	−0.840152	+	0.542351i	$0.817534\pi$
$662$	31.9959	1.24356
$663$	0	0
$664$	3.51093	0.136251
$665$	37.3653i	1.44896i
$666$	−3.79603	−0.147093
$667$	−4.21175	−0.163079
$668$	8.68162i	0.335902i
$669$	− 2.23671i	− 0.0864762i
$670$	1.82522i	0.0705143i
$671$	− 32.8430i	− 1.26789i
$672$	4.64466	0.179172
$673$	0.978131	0.0377042	0.0188521	−	0.999822i	$-0.493999\pi$
$673$	0.978131	0.0377042	0.0188521	+	0.999822i	$0.493999\pi$
$674$	35.6432i	1.37293i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−19.1926	−0.737630	−0.368815	−	0.929503i	$-0.620236\pi$
$677$	−19.1926	−0.737630	−0.368815	+	0.929503i	$0.620236\pi$
$678$	− 13.0662i	− 0.501803i
$679$	−64.2134	−2.46429
$680$	4.00000	0.153393
$681$	− 15.3205i	− 0.587083i
$682$	28.3398i	1.08519i
$683$	− 1.51093i	− 0.0578143i	−0.999582	−	0.0289071i	$-0.990797\pi$
$683$	− 1.51093i	− 0.0578143i	0.999582	−	0.0289071i	$-0.00920271\pi$
$684$	8.04479i	0.307600i
$685$	1.25235	0.0478500
$686$	−35.1734	−1.34293
$687$	10.1279i	0.386405i
$688$	−0.716456	−0.0273146
$689$	0	0
$690$	0.976584	0.0371779
$691$	− 33.1166i	− 1.25981i	−0.776670	−	0.629907i	$-0.783093\pi$
$691$	− 33.1166i	− 1.25981i	0.776670	−	0.629907i	$-0.216907\pi$
$692$	1.78043	0.0676818
$693$	20.4371	0.776341
$694$	6.89701i	0.261807i
$695$	3.35534i	0.127275i
$696$	− 4.31274i	− 0.163474i
$697$	− 29.1573i	− 1.10441i
$698$	−19.3205	−0.731292
$699$	20.7519	0.784908
$700$	4.64466i	0.175552i
$701$	27.8695	1.05262	0.526308	−	0.850294i	$-0.323576\pi$
$701$	27.8695	1.05262	0.526308	+	0.850294i	$0.323576\pi$
$702$	0	0
$703$	−30.5382	−1.15177
$704$	4.40013i	0.165836i
$705$	9.75342	0.367335
$706$	27.4173	1.03186
$707$	31.5849i	1.18787i
$708$	2.18056i	0.0819504i
$709$	− 11.0562i	− 0.415223i	−0.978211	−	0.207611i	$-0.933431\pi$
$709$	− 11.0562i	− 0.415223i	0.978211	−	0.207611i	$-0.0665689\pi$
$710$	− 7.95317i	− 0.298477i
$711$	−14.9340	−0.560068
$712$	8.17274	0.306286
$713$	− 6.28987i	− 0.235557i
$714$	−18.5786	−0.695288
$715$	0	0
$716$	−17.4157	−0.650856
$717$	24.3539i	0.909514i
$718$	−14.3611	−0.535952
$719$	−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
$721$	− 24.5766i	− 0.915280i
$722$	45.7186i	1.70147i
$723$	− 19.8685i	− 0.738916i
$724$	10.3611	0.385068
$725$	4.31274	0.160171
$726$	8.36112i	0.310310i
$727$	−19.4152	−0.720071	−0.360035	−	0.932939i	$-0.617235\pi$
$727$	−19.4152	−0.720071	−0.360035	+	0.932939i	$0.617235\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.36112i	0.161412i
$731$	2.86583	0.105996
$732$	−7.46410	−0.275881
$733$	− 11.9340i	− 0.440792i	−0.975411	−	0.220396i	$-0.929265\pi$
$733$	− 11.9340i	− 0.440792i	0.975411	−	0.220396i	$-0.0707349\pi$
$734$	13.7753i	0.508455i
$735$	− 14.5729i	− 0.537528i
$736$	− 0.976584i	− 0.0359973i
$737$	8.03119	0.295833
$738$	7.28932	0.268324
$739$	− 19.8628i	− 0.730666i	−0.930877	−	0.365333i	$-0.880955\pi$
$739$	− 19.8628i	− 0.730666i	0.930877	−	0.365333i	$-0.119045\pi$
$740$	−3.79603	−0.139545
$741$	0	0
$742$	62.7442	2.30341
$743$	− 16.4877i	− 0.604873i	−0.953169	−	0.302437i	$-0.902200\pi$
$743$	− 16.4877i	− 0.604873i	0.953169	−	0.302437i	$-0.0978001\pi$
$744$	6.44069	0.236127
$745$	4.61970	0.169253
$746$	33.4627i	1.22516i
$747$	− 3.51093i	− 0.128458i
$748$	− 17.6005i	− 0.643538i
$749$	78.6257i	2.87292i
$750$	−1.00000	−0.0365148
$751$	46.6624	1.70274	0.851368	−	0.524569i	$-0.175773\pi$
$751$	46.6624	1.70274	0.851368	+	0.524569i	$0.175773\pi$
$752$	− 9.75342i	− 0.355671i
$753$	−13.5713	−0.494566
$754$	0	0
$755$	11.2425	0.409156
$756$	− 4.64466i	− 0.168925i
$757$	2.63597	0.0958061	0.0479030	−	0.998852i	$-0.484746\pi$
$757$	2.63597	0.0958061	0.0479030	+	0.998852i	$0.484746\pi$
$758$	33.3768	1.21230
$759$	− 4.29709i	− 0.155975i
$760$	8.04479i	0.291815i
$761$	0.0800681i	0.00290246i	0.999999	+	0.00145123i	$0.000461942\pi$
$761$	0.0800681i	0.00290246i	−0.999999	+	0.00145123i	$0.999538\pi$
$762$	12.2196i	0.442668i
$763$	59.4528	2.15234
$764$	0.897014	0.0324528
$765$	− 4.00000i	− 0.144620i
$766$	7.33038	0.264857
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 5.68317i	− 0.204940i	−0.994736	−	0.102470i	$-0.967325\pi$
$769$	− 5.68317i	− 0.204940i	0.994736	−	0.102470i	$-0.0326746\pi$
$770$	20.4371	0.736502
$771$	−0.662300	−0.0238522
$772$	20.2644i	0.729330i
$773$	− 7.58851i	− 0.272940i	−0.990644	−	0.136470i	$-0.956424\pi$
$773$	− 7.58851i	− 0.272940i	0.990644	−	0.136470i	$-0.0435757\pi$
$774$	0.716456i	0.0257525i
$775$	6.44069i	0.231356i
$776$	−13.8252	−0.496296
$777$	17.6312	0.632517
$778$	32.6198i	1.16948i
$779$	58.6410	2.10103
$780$	0	0
$781$	−34.9949	−1.25222
$782$	3.90633i	0.139690i
$783$	−4.31274	−0.154125
$784$	−14.5729	−0.520459
$785$	− 1.50311i	− 0.0536484i
$786$	0.378757i	0.0135098i
$787$	− 4.73623i	− 0.168828i	−0.996431	−	0.0844142i	$-0.973098\pi$
$787$	− 4.73623i	− 0.168828i	0.996431	−	0.0844142i	$-0.0269019\pi$
$788$	9.18056i	0.327044i
$789$	30.7498	1.09472
$790$	−14.9340	−0.531327
$791$	60.6878i	2.15781i
$792$	4.40013	0.156352
$793$	0	0
$794$	14.4683	0.513462
$795$	13.5089i	0.479111i
$796$	−16.2175	−0.574815
$797$	41.0636	1.45455	0.727274	−	0.686347i	$-0.240787\pi$
$797$	41.0636	1.45455	0.727274	+	0.686347i	$0.240787\pi$
$798$	− 37.3653i	− 1.32272i
$799$	39.0137i	1.38020i
$800$	1.00000i	0.0353553i
$801$	− 8.17274i	− 0.288769i
$802$	7.28727	0.257323
$803$	19.1895	0.677181
$804$	− 1.82522i	− 0.0643705i
$805$	−4.53590	−0.159869
$806$	0	0
$807$	−3.74410	−0.131799
$808$	6.80025i	0.239232i
$809$	−27.6005	−0.970382	−0.485191	−	0.874408i	$-0.661250\pi$
$809$	−27.6005	−0.970382	−0.485191	+	0.874408i	$0.661250\pi$
$810$	1.00000	0.0351364
$811$	− 10.6930i	− 0.375482i	−0.982219	−	0.187741i	$-0.939883\pi$
$811$	− 10.6930i	− 0.375482i	0.982219	−	0.187741i	$-0.0601165\pi$
$812$	20.0312i	0.702957i
$813$	− 27.7612i	− 0.973628i
$814$	16.7030i	0.585440i
$815$	0.176330	0.00617657
$816$	−4.00000	−0.140028
$817$	5.76374i	0.201648i
$818$	−24.5766	−0.859300
$819$	0	0
$820$	7.28932	0.254554
$821$	− 17.5635i	− 0.612972i	−0.951875	−	0.306486i	$-0.900847\pi$
$821$	− 17.5635i	− 0.612972i	0.951875	−	0.306486i	$-0.0991532\pi$
$822$	−1.25235	−0.0436809
$823$	38.7130	1.34945	0.674725	−	0.738069i	$-0.264262\pi$
$823$	38.7130	1.34945	0.674725	+	0.738069i	$0.264262\pi$
$824$	− 5.29137i	− 0.184333i
$825$	4.40013i	0.153193i
$826$	− 10.1279i	− 0.352396i
$827$	− 28.4703i	− 0.990010i	−0.868890	−	0.495005i	$-0.835166\pi$
$827$	− 28.4703i	− 0.990010i	0.868890	−	0.495005i	$-0.164834\pi$
$828$	−0.976584	−0.0339386
$829$	−8.28709	−0.287822	−0.143911	−	0.989591i	$-0.545968\pi$
$829$	−8.28709	−0.287822	−0.143911	+	0.989591i	$0.545968\pi$
$830$	− 3.51093i	− 0.121866i
$831$	−11.9922	−0.416004
$832$	0	0
$833$	58.2915	2.01968
$834$	− 3.35534i	− 0.116186i
$835$	8.68162	0.300440
$836$	35.3981	1.22427
$837$	− 6.44069i	− 0.222623i
$838$	− 15.5537i	− 0.537293i
$839$	− 42.5630i	− 1.46944i	−0.678372	−	0.734719i	$-0.737314\pi$
$839$	− 42.5630i	− 1.46944i	0.678372	−	0.734719i	$-0.262686\pi$
$840$	− 4.64466i	− 0.160256i
$841$	−10.4003	−0.358631
$842$	22.3143	0.769001
$843$	− 3.63888i	− 0.125330i
$844$	1.61970	0.0557522
$845$	0	0
$846$	−9.75342	−0.335330
$847$	− 38.8345i	− 1.33437i
$848$	13.5089	0.463897
$849$	−4.84441	−0.166260
$850$	− 4.00000i	− 0.137199i
$851$	− 3.70714i	− 0.127079i
$852$	7.95317i	0.272471i
$853$	− 28.1380i	− 0.963425i	−0.876329	−	0.481713i	$-0.840015\pi$
$853$	− 28.1380i	− 0.963425i	0.876329	−	0.481713i	$-0.159985\pi$
$854$	34.6682	1.18632
$855$	8.04479	0.275126
$856$	16.9282i	0.578594i
$857$	−10.3355	−0.353053	−0.176526	−	0.984296i	$-0.556486\pi$
$857$	−10.3355	−0.353053	−0.176526	+	0.984296i	$0.556486\pi$
$858$	0	0
$859$	0.380304	0.0129758	0.00648791	−	0.999979i	$-0.497935\pi$
$859$	0.380304	0.0129758	0.00648791	+	0.999979i	$0.497935\pi$
$860$	0.716456i	0.0244310i
$861$	−33.8564	−1.15382
$862$	−24.2644	−0.826447
$863$	− 47.1484i	− 1.60495i	−0.596685	−	0.802475i	$-0.703516\pi$
$863$	− 47.1484i	− 1.60495i	0.596685	−	0.802475i	$-0.296484\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	− 1.78043i	− 0.0605365i
$866$	− 23.0989i	− 0.784932i
$867$	−1.00000	−0.0339618
$868$	−29.9148	−1.01537
$869$	65.7114i	2.22911i
$870$	−4.31274	−0.146215
$871$	0	0
$872$	12.8003	0.433471
$873$	13.8252i	0.467913i
$874$	−7.85641	−0.265747
$875$	4.64466	0.157018
$876$	− 4.36112i	− 0.147348i
$877$	40.0412i	1.35209i	0.736858	+	0.676047i	$0.236309\pi$
$877$	40.0412i	1.35209i	−0.736858	+	0.676047i	$0.763691\pi$
$878$	− 39.9959i	− 1.34980i
$879$	3.70081i	0.124825i
$880$	4.40013	0.148328
$881$	15.4449	0.520352	0.260176	−	0.965561i	$-0.416219\pi$
$881$	15.4449	0.520352	0.260176	+	0.965561i	$0.416219\pi$
$882$	14.5729i	0.490694i
$883$	−6.02142	−0.202637	−0.101318	−	0.994854i	$-0.532306\pi$
$883$	−6.02142	−0.202637	−0.101318	+	0.994854i	$0.532306\pi$
$884$	0	0
$885$	2.18056	0.0732987
$886$	15.6036i	0.524213i
$887$	−49.1380	−1.64989	−0.824947	−	0.565210i	$-0.808795\pi$
$887$	−49.1380	−1.64989	−0.824947	+	0.565210i	$0.808795\pi$
$888$	3.79603	0.127386
$889$	− 56.7557i	− 1.90353i
$890$	− 8.17274i	− 0.273951i
$891$	− 4.40013i	− 0.147410i
$892$	2.23671i	0.0748906i
$893$	−78.4642	−2.62570
$894$	−4.61970	−0.154506
$895$	17.4157i	0.582143i
$896$	−4.64466	−0.155167
$897$	0	0
$898$	27.0042	0.901141
$899$	27.7770i	0.926414i
$900$	1.00000	0.0333333
$901$	−54.0356	−1.80019
$902$	− 32.0739i	− 1.06795i
$903$	− 3.32770i	− 0.110739i
$904$	13.0662i	0.434574i
$905$	− 10.3611i	− 0.344415i
$906$	−11.2425	−0.373507
$907$	−16.3669	−0.543454	−0.271727	−	0.962374i	$-0.587595\pi$
$907$	−16.3669	−0.543454	−0.271727	+	0.962374i	$0.587595\pi$
$908$	15.3205i	0.508429i
$909$	6.80025	0.225550
$910$	0	0
$911$	−10.4819	−0.347280	−0.173640	−	0.984809i	$-0.555553\pi$
$911$	−10.4819	−0.347280	−0.173640	+	0.984809i	$0.555553\pi$
$912$	− 8.04479i	− 0.266389i
$913$	−15.4486	−0.511273
$914$	−17.7868	−0.588336
$915$	7.46410i	0.246756i
$916$	− 10.1279i	− 0.334637i
$917$	− 1.75920i	− 0.0580938i
$918$	4.00000i	0.132020i
$919$	3.21130	0.105931	0.0529655	−	0.998596i	$-0.483133\pi$
$919$	3.21130	0.105931	0.0529655	+	0.998596i	$0.483133\pi$
$920$	−0.976584	−0.0321970
$921$	7.11454i	0.234432i
$922$	24.9652	0.822184
$923$	0	0
$924$	−20.4371	−0.672331
$925$	3.79603i	0.124813i
$926$	−15.6389	−0.513926
$927$	−5.29137	−0.173791
$928$	4.31274i	0.141572i
$929$	46.4788i	1.52492i	0.647036	+	0.762460i	$0.276008\pi$
$929$	46.4788i	1.52492i	−0.647036	+	0.762460i	$0.723992\pi$
$930$	− 6.44069i	− 0.211198i
$931$	117.236i	3.84224i
$932$	−20.7519	−0.679750
$933$	−19.9148	−0.651981
$934$	2.43914i	0.0798110i
$935$	−17.6005	−0.575598
$936$	0	0
$937$	−27.7627	−0.906969	−0.453485	−	0.891264i	$-0.649819\pi$
$937$	−27.7627	−0.906969	−0.453485	+	0.891264i	$0.649819\pi$
$938$	8.47751i	0.276801i
$939$	6.13950	0.200355
$940$	−9.75342	−0.318122
$941$	20.7962i	0.677935i	0.940798	+	0.338968i	$0.110078\pi$
$941$	20.7962i	0.677935i	−0.940798	+	0.338968i	$0.889922\pi$
$942$	1.50311i	0.0489741i
$943$	7.11863i	0.231814i
$944$	− 2.18056i	− 0.0709711i
$945$	−4.64466	−0.151091
$946$	3.15250	0.102497
$947$	− 9.05925i	− 0.294386i	−0.989108	−	0.147193i	$-0.952976\pi$
$947$	− 9.05925i	− 0.294386i	0.989108	−	0.147193i	$-0.0470238\pi$
$948$	14.9340	0.485033
$949$	0	0
$950$	8.04479	0.261007
$951$	− 7.85286i	− 0.254646i
$952$	18.5786	0.602137
$953$	15.2762	0.494845	0.247423	−	0.968908i	$-0.420416\pi$
$953$	15.2762	0.494845	0.247423	+	0.968908i	$0.420416\pi$
$954$	− 13.5089i	− 0.437366i
$955$	− 0.897014i	− 0.0290267i
$956$	− 24.3539i	− 0.787662i
$957$	18.9766i	0.613426i
$958$	−14.1863	−0.458340
$959$	5.81676	0.187833
$960$	− 1.00000i	− 0.0322749i
$961$	−10.4824	−0.338143
$962$	0	0
$963$	16.9282	0.545504
$964$	19.8685i	0.639920i
$965$	20.2644	0.652333
$966$	4.53590	0.145940
$967$	− 46.3036i	− 1.48902i	−0.667610	−	0.744511i	$-0.732683\pi$
$967$	− 46.3036i	− 1.48902i	0.667610	−	0.744511i	$-0.267317\pi$
$968$	− 8.36112i	− 0.268736i
$969$	32.1791i	1.03374i
$970$	13.8252i	0.443901i
$971$	11.9652	0.383981	0.191990	−	0.981397i	$-0.438506\pi$
$971$	11.9652	0.383981	0.191990	+	0.981397i	$0.438506\pi$
$972$	−1.00000	−0.0320750
$973$	15.5844i	0.499613i
$974$	5.76329	0.184668
$975$	0	0
$976$	7.46410	0.238920
$977$	− 1.76415i	− 0.0564403i	−0.999602	−	0.0282201i	$-0.991016\pi$
$977$	− 1.76415i	− 0.0564403i	0.999602	−	0.0282201i	$-0.00898394\pi$
$978$	−0.176330	−0.00563841
$979$	−35.9611	−1.14932
$980$	14.5729i	0.465513i
$981$	− 12.8003i	− 0.408681i
$982$	− 1.07534i	− 0.0343155i
$983$	− 54.2821i	− 1.73133i	−0.500623	−	0.865666i	$-0.666896\pi$
$983$	− 54.2821i	− 1.73133i	0.500623	−	0.865666i	$-0.333104\pi$
$984$	−7.28932	−0.232375
$985$	9.18056	0.292517
$986$	− 17.2509i	− 0.549382i
$987$	45.3013	1.44196
$988$	0	0
$989$	−0.699680	−0.0222485
$990$	− 4.40013i	− 0.139845i
$991$	−17.4783	−0.555218	−0.277609	−	0.960694i	$-0.589542\pi$
$991$	−17.4783	−0.555218	−0.277609	+	0.960694i	$0.589542\pi$
$992$	−6.44069	−0.204492
$993$	31.9959i	1.01536i
$994$	− 36.9398i	− 1.17166i
$995$	16.2175i	0.514130i
$996$	3.51093i	0.111248i
$997$	−21.2914	−0.674304	−0.337152	−	0.941450i	$-0.609464\pi$
$997$	−21.2914	−0.674304	−0.337152	+	0.941450i	$0.609464\pi$
$998$	−14.7534	−0.467011
$999$	− 3.79603i	− 0.120101i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.ba.1351.4		8
13.3	even	3		390.2.bb.c.121.4	✓	8
13.4	even	6		390.2.bb.c.361.4	yes	8
13.5	odd	4		5070.2.a.bz.1.1		4
13.8	odd	4		5070.2.a.ca.1.4		4
13.12	even	2	inner	5070.2.b.ba.1351.5		8
39.17	odd	6		1170.2.bs.f.361.2		8
39.29	odd	6		1170.2.bs.f.901.2		8
65.3	odd	12		1950.2.y.k.199.1		8
65.4	even	6		1950.2.bc.g.751.1		8
65.17	odd	12		1950.2.y.k.49.1		8
65.29	even	6		1950.2.bc.g.901.1		8
65.42	odd	12		1950.2.y.j.199.4		8
65.43	odd	12		1950.2.y.j.49.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.c.121.4	✓	8	13.3	even	3
390.2.bb.c.361.4	yes	8	13.4	even	6
1170.2.bs.f.361.2		8	39.17	odd	6
1170.2.bs.f.901.2		8	39.29	odd	6
1950.2.y.j.49.4		8	65.43	odd	12
1950.2.y.j.199.4		8	65.42	odd	12
1950.2.y.k.49.1		8	65.17	odd	12
1950.2.y.k.199.1		8	65.3	odd	12
1950.2.bc.g.751.1		8	65.4	even	6
1950.2.bc.g.901.1		8	65.29	even	6
5070.2.a.bz.1.1		4	13.5	odd	4
5070.2.a.ca.1.4		4	13.8	odd	4
5070.2.b.ba.1351.4		8	1.1	even	1	trivial
5070.2.b.ba.1351.5		8	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +4.64466i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.40013i q^{11} -1.00000 q^{12} +4.64466 q^{14} +1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000i q^{18} -8.04479i q^{19} -1.00000i q^{20} +4.64466i q^{21} -4.40013 q^{22} +0.976584 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -4.64466i q^{28} -4.31274 q^{29} +1.00000 q^{30} -6.44069i q^{31} -1.00000i q^{32} -4.40013i q^{33} +4.00000i q^{34} -4.64466 q^{35} -1.00000 q^{36} -3.79603i q^{37} -8.04479 q^{38} -1.00000 q^{40} +7.28932i q^{41} +4.64466 q^{42} -0.716456 q^{43} +4.40013i q^{44} +1.00000i q^{45} -0.976584i q^{46} -9.75342i q^{47} +1.00000 q^{48} -14.5729 q^{49} +1.00000i q^{50} -4.00000 q^{51} +13.5089 q^{53} -1.00000i q^{54} +4.40013 q^{55} -4.64466 q^{56} -8.04479i q^{57} +4.31274i q^{58} -2.18056i q^{59} -1.00000i q^{60} +7.46410 q^{61} -6.44069 q^{62} +4.64466i q^{63} -1.00000 q^{64} -4.40013 q^{66} +1.82522i q^{67} +4.00000 q^{68} +0.976584 q^{69} +4.64466i q^{70} -7.95317i q^{71} +1.00000i q^{72} +4.36112i q^{73} -3.79603 q^{74} -1.00000 q^{75} +8.04479i q^{76} +20.4371 q^{77} -14.9340 q^{79} +1.00000i q^{80} +1.00000 q^{81} +7.28932 q^{82} -3.51093i q^{83} -4.64466i q^{84} -4.00000i q^{85} +0.716456i q^{86} -4.31274 q^{87} +4.40013 q^{88} -8.17274i q^{89} +1.00000 q^{90} -0.976584 q^{92} -6.44069i q^{93} -9.75342 q^{94} +8.04479 q^{95} -1.00000i q^{96} +13.8252i q^{97} +14.5729i q^{98} -4.40013i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9} + 8 q^{10} - 8 q^{12} + 4 q^{14} + 8 q^{16} - 32 q^{17} - 4 q^{22} - 8 q^{23} - 8 q^{25} + 8 q^{27} + 16 q^{29} + 8 q^{30} - 4 q^{35} - 8 q^{36} - 8 q^{40} + 4 q^{42} - 28 q^{43}+ \cdots + 16 q^{94}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

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