LMFDB - Embedded newform 1690.2.d.e.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1351,1690)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1690, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1690.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.d (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:	$13.4947179416$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1690.1351
Dual form			1690.2.d.e.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +2.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +2.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000i q^{11} -2.00000 q^{12} -4.00000 q^{14} +2.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000i q^{18} -6.00000i q^{19} -1.00000i q^{20} -8.00000i q^{21} -2.00000 q^{22} -6.00000 q^{23} +2.00000i q^{24} -1.00000 q^{25} -4.00000 q^{27} +4.00000i q^{28} +2.00000 q^{29} +2.00000 q^{30} +6.00000i q^{31} -1.00000i q^{32} -4.00000i q^{33} +2.00000i q^{34} +4.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} -6.00000 q^{38} -1.00000 q^{40} -10.0000i q^{41} -8.00000 q^{42} +10.0000 q^{43} +2.00000i q^{44} +1.00000i q^{45} +6.00000i q^{46} -12.0000i q^{47} +2.00000 q^{48} -9.00000 q^{49} +1.00000i q^{50} -4.00000 q^{51} +2.00000 q^{53} +4.00000i q^{54} +2.00000 q^{55} +4.00000 q^{56} -12.0000i q^{57} -2.00000i q^{58} +10.0000i q^{59} -2.00000i q^{60} +2.00000 q^{61} +6.00000 q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} +2.00000 q^{68} -12.0000 q^{69} -4.00000i q^{70} -10.0000i q^{71} +1.00000i q^{72} +10.0000i q^{73} -2.00000 q^{74} -2.00000 q^{75} +6.00000i q^{76} -8.00000 q^{77} -4.00000 q^{79} +1.00000i q^{80} -11.0000 q^{81} -10.0000 q^{82} +8.00000i q^{84} -2.00000i q^{85} -10.0000i q^{86} +4.00000 q^{87} +2.00000 q^{88} -14.0000i q^{89} +1.00000 q^{90} +6.00000 q^{92} +12.0000i q^{93} -12.0000 q^{94} +6.00000 q^{95} -2.00000i q^{96} -14.0000i q^{97} +9.00000i q^{98} -2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q + 4 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} - 4 q^{12} - 8 q^{14} + 2 q^{16} - 4 q^{17} - 4 q^{22} - 12 q^{23} - 2 q^{25} - 8 q^{27} + 4 q^{29} + 4 q^{30} + 8 q^{35} - 2 q^{36} - 12 q^{38} - 2 q^{40} - 16 q^{42} + 20 q^{43} + 4 q^{48} - 18 q^{49} - 8 q^{51} + 4 q^{53} + 4 q^{55} + 8 q^{56} + 4 q^{61} + 12 q^{62} - 2 q^{64} - 8 q^{66} + 4 q^{68} - 24 q^{69} - 4 q^{74} - 4 q^{75} - 16 q^{77} - 8 q^{79} - 22 q^{81} - 20 q^{82} + 8 q^{87} + 4 q^{88} + 2 q^{90} + 12 q^{92} - 24 q^{94} + 12 q^{95}+O(q^{100}) $ 2 * q + 4 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 - 4 * q^12 - 8 * q^14 + 2 * q^16 - 4 * q^17 - 4 * q^22 - 12 * q^23 - 2 * q^25 - 8 * q^27 + 4 * q^29 + 4 * q^30 + 8 * q^35 - 2 * q^36 - 12 * q^38 - 2 * q^40 - 16 * q^42 + 20 * q^43 + 4 * q^48 - 18 * q^49 - 8 * q^51 + 4 * q^53 + 4 * q^55 + 8 * q^56 + 4 * q^61 + 12 * q^62 - 2 * q^64 - 8 * q^66 + 4 * q^68 - 24 * q^69 - 4 * q^74 - 4 * q^75 - 16 * q^77 - 8 * q^79 - 22 * q^81 - 20 * q^82 + 8 * q^87 + 4 * q^88 + 2 * q^90 + 12 * q^92 - 24 * q^94 + 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-1$	$1$

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 2.00000i	− 0.816497i
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	−2.00000	−0.577350
$13$	0	0
$13$	0	0
$14$	−4.00000	−1.06904
$15$	2.00000i	0.516398i
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	− 1.00000i	− 0.235702i
$19$	− 6.00000i	− 1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	− 6.00000i	− 1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 8.00000i	− 1.74574i
$22$	−2.00000	−0.426401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	2.00000i	0.408248i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	4.00000i	0.755929i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	6.00000i	1.07763i	0.842424	+	0.538816i	$0.181128\pi$
$31$	6.00000i	1.07763i	−0.842424	+	0.538816i	$0.818872\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	2.00000i	0.342997i
$35$	4.00000	0.676123
$36$	−1.00000	−0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 10.0000i	− 1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	− 10.0000i	− 1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$	−8.00000	−1.23443
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	2.00000i	0.301511i
$45$	1.00000i	0.149071i
$46$	6.00000i	0.884652i
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	2.00000	0.288675
$49$	−9.00000	−1.28571
$50$	1.00000i	0.141421i
$51$	−4.00000	−0.560112
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	4.00000i	0.544331i
$55$	2.00000	0.269680
$56$	4.00000	0.534522
$57$	− 12.0000i	− 1.58944i
$58$	− 2.00000i	− 0.262613i
$59$	10.0000i	1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$	10.0000i	1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$	− 2.00000i	− 0.258199i
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	6.00000	0.762001
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	2.00000	0.242536
$69$	−12.0000	−1.44463
$70$	− 4.00000i	− 0.478091i
$71$	− 10.0000i	− 1.18678i	−0.804914	−	0.593391i	$-0.797789\pi$
$71$	− 10.0000i	− 1.18678i	0.804914	−	0.593391i	$-0.202211\pi$
$72$	1.00000i	0.117851i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	−2.00000	−0.232495
$75$	−2.00000	−0.230940
$76$	6.00000i	0.688247i
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000i	0.111803i
$81$	−11.0000	−1.22222
$82$	−10.0000	−1.10432
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	8.00000i	0.872872i
$85$	− 2.00000i	− 0.216930i
$86$	− 10.0000i	− 1.07833i
$87$	4.00000	0.428845
$88$	2.00000	0.213201
$89$	− 14.0000i	− 1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$	− 14.0000i	− 1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	6.00000	0.625543
$93$	12.0000i	1.24434i
$94$	−12.0000	−1.23771
$95$	6.00000	0.615587
$96$	− 2.00000i	− 0.204124i
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	9.00000i	0.909137i
$99$	− 2.00000i	− 0.201008i
$100$	1.00000	0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	4.00000i	0.396059i
$103$	18.0000	1.77359	0.886796	−	0.462160i	$-0.152926\pi$
$103$	18.0000	1.77359	0.886796	+	0.462160i	$0.152926\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	− 2.00000i	− 0.194257i
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	4.00000	0.384900
$109$	6.00000i	0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	6.00000i	0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$	− 2.00000i	− 0.190693i
$111$	− 4.00000i	− 0.379663i
$112$	− 4.00000i	− 0.377964i
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−12.0000	−1.12390
$115$	− 6.00000i	− 0.559503i
$116$	−2.00000	−0.185695
$117$	0	0
$118$	10.0000	0.920575
$119$	8.00000i	0.733359i
$120$	−2.00000	−0.182574
$121$	7.00000	0.636364
$122$	− 2.00000i	− 0.181071i
$123$	− 20.0000i	− 1.80334i
$124$	− 6.00000i	− 0.538816i
$125$	− 1.00000i	− 0.0894427i
$126$	−4.00000	−0.356348
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	1.00000i	0.0883883i
$129$	20.0000	1.76090
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	4.00000i	0.348155i
$133$	−24.0000	−2.08106
$134$	12.0000	1.03664
$135$	− 4.00000i	− 0.344265i
$136$	− 2.00000i	− 0.171499i
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	12.0000i	1.02151i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	−4.00000	−0.338062
$141$	− 24.0000i	− 2.02116i
$142$	−10.0000	−0.839181
$143$	0	0
$144$	1.00000	0.0833333
$145$	2.00000i	0.166091i
$146$	10.0000	0.827606
$147$	−18.0000	−1.48461
$148$	2.00000i	0.164399i
$149$	− 2.00000i	− 0.163846i	−0.996639	−	0.0819232i	$-0.973894\pi$
$149$	− 2.00000i	− 0.163846i	0.996639	−	0.0819232i	$-0.0261062\pi$
$150$	2.00000i	0.163299i
$151$	6.00000i	0.488273i	0.969741	+	0.244137i	$0.0785045\pi$
$151$	6.00000i	0.488273i	−0.969741	+	0.244137i	$0.921495\pi$
$152$	6.00000	0.486664
$153$	−2.00000	−0.161690
$154$	8.00000i	0.644658i
$155$	−6.00000	−0.481932
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	4.00000i	0.318223i
$159$	4.00000	0.317221
$160$	1.00000	0.0790569
$161$	24.0000i	1.89146i
$162$	11.0000i	0.864242i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	10.0000i	0.780869i
$165$	4.00000	0.311400
$166$	0	0
$167$	20.0000i	1.54765i	0.633402	+	0.773823i	$0.281658\pi$
$167$	20.0000i	1.54765i	−0.633402	+	0.773823i	$0.718342\pi$
$168$	8.00000	0.617213
$169$	0	0
$170$	−2.00000	−0.153393
$171$	− 6.00000i	− 0.458831i
$172$	−10.0000	−0.762493
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	− 4.00000i	− 0.303239i
$175$	4.00000i	0.302372i
$176$	− 2.00000i	− 0.150756i
$177$	20.0000i	1.50329i
$178$	−14.0000	−1.04934
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	− 6.00000i	− 0.442326i
$185$	2.00000	0.147043
$186$	12.0000	0.879883
$187$	4.00000i	0.292509i
$188$	12.0000i	0.875190i
$189$	16.0000i	1.16383i
$190$	− 6.00000i	− 0.435286i
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−2.00000	−0.144338
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	−2.00000	−0.142134
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	24.0000i	1.69283i
$202$	14.0000i	0.985037i
$203$	− 8.00000i	− 0.561490i
$204$	4.00000	0.280056
$205$	10.0000	0.698430
$206$	− 18.0000i	− 1.25412i
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−12.0000	−0.830057
$210$	− 8.00000i	− 0.552052i
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	−2.00000	−0.137361
$213$	− 20.0000i	− 1.37038i
$214$	− 6.00000i	− 0.410152i
$215$	10.0000i	0.681994i
$216$	− 4.00000i	− 0.272166i
$217$	24.0000	1.62923
$218$	6.00000	0.406371
$219$	20.0000i	1.35147i
$220$	−2.00000	−0.134840
$221$	0	0
$222$	−4.00000	−0.268462
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	−4.00000	−0.267261
$225$	−1.00000	−0.0666667
$226$	− 2.00000i	− 0.133038i
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	12.0000i	0.794719i
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	−6.00000	−0.395628
$231$	−16.0000	−1.05272
$232$	2.00000i	0.131306i
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	− 10.0000i	− 0.650945i
$237$	−8.00000	−0.519656
$238$	8.00000	0.518563
$239$	26.0000i	1.68180i	0.541190	+	0.840900i	$0.317974\pi$
$239$	26.0000i	1.68180i	−0.541190	+	0.840900i	$0.682026\pi$
$240$	2.00000i	0.129099i
$241$	− 22.0000i	− 1.41714i	−0.705638	−	0.708572i	$-0.749340\pi$
$241$	− 22.0000i	− 1.41714i	0.705638	−	0.708572i	$-0.250660\pi$
$242$	− 7.00000i	− 0.449977i
$243$	−10.0000	−0.641500
$244$	−2.00000	−0.128037
$245$	− 9.00000i	− 0.574989i
$246$	−20.0000	−1.27515
$247$	0	0
$248$	−6.00000	−0.381000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	4.00000i	0.251976i
$253$	12.0000i	0.754434i
$254$	− 14.0000i	− 0.878438i
$255$	− 4.00000i	− 0.250490i
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	− 20.0000i	− 1.24515i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	2.00000	0.123797
$262$	− 4.00000i	− 0.247121i
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	4.00000	0.246183
$265$	2.00000i	0.122859i
$266$	24.0000i	1.47153i
$267$	− 28.0000i	− 1.71357i
$268$	− 12.0000i	− 0.733017i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−4.00000	−0.243432
$271$	− 2.00000i	− 0.121491i	−0.998153	−	0.0607457i	$-0.980652\pi$
$271$	− 2.00000i	− 0.121491i	0.998153	−	0.0607457i	$-0.0193479\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−18.0000	−1.08742
$275$	2.00000i	0.120605i
$276$	12.0000	0.722315
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	8.00000i	0.479808i
$279$	6.00000i	0.359211i
$280$	4.00000i	0.239046i
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	−24.0000	−1.42918
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	10.0000i	0.593391i
$285$	12.0000	0.710819
$286$	0	0
$287$	−40.0000	−2.36113
$288$	− 1.00000i	− 0.0589256i
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	− 28.0000i	− 1.64139i
$292$	− 10.0000i	− 0.585206i
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	18.0000i	1.04978i
$295$	−10.0000	−0.582223
$296$	2.00000	0.116248
$297$	8.00000i	0.464207i
$298$	−2.00000	−0.115857
$299$	0	0
$300$	2.00000	0.115470
$301$	− 40.0000i	− 2.30556i
$302$	6.00000	0.345261
$303$	−28.0000	−1.60856
$304$	− 6.00000i	− 0.344124i
$305$	2.00000i	0.114520i
$306$	2.00000i	0.114332i
$307$	24.0000i	1.36975i	0.728659	+	0.684876i	$0.240144\pi$
$307$	24.0000i	1.36975i	−0.728659	+	0.684876i	$0.759856\pi$
$308$	8.00000	0.455842
$309$	36.0000	2.04797
$310$	6.00000i	0.340777i
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	− 10.0000i	− 0.564333i
$315$	4.00000	0.225374
$316$	4.00000	0.225018
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	− 4.00000i	− 0.224309i
$319$	− 4.00000i	− 0.223957i
$320$	− 1.00000i	− 0.0559017i
$321$	12.0000	0.669775
$322$	24.0000	1.33747
$323$	12.0000i	0.667698i
$324$	11.0000	0.611111
$325$	0	0
$326$	−4.00000	−0.221540
$327$	12.0000i	0.663602i
$328$	10.0000	0.552158
$329$	−48.0000	−2.64633
$330$	− 4.00000i	− 0.220193i
$331$	14.0000i	0.769510i	0.923019	+	0.384755i	$0.125714\pi$
$331$	14.0000i	0.769510i	−0.923019	+	0.384755i	$0.874286\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	20.0000	1.09435
$335$	−12.0000	−0.655630
$336$	− 8.00000i	− 0.436436i
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	4.00000	0.217250
$340$	2.00000i	0.108465i
$341$	12.0000	0.649836
$342$	−6.00000	−0.324443
$343$	8.00000i	0.431959i
$344$	10.0000i	0.539164i
$345$	− 12.0000i	− 0.646058i
$346$	10.0000i	0.537603i
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	−4.00000	−0.214423
$349$	2.00000i	0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$	2.00000i	0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	−2.00000	−0.106600
$353$	− 34.0000i	− 1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	− 34.0000i	− 1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$	20.0000	1.06299
$355$	10.0000	0.530745
$356$	14.0000i	0.741999i
$357$	16.0000i	0.846810i
$358$	− 4.00000i	− 0.211407i
$359$	− 6.00000i	− 0.316668i	−0.987386	−	0.158334i	$-0.949388\pi$
$359$	− 6.00000i	− 0.316668i	0.987386	−	0.158334i	$-0.0506123\pi$
$360$	−1.00000	−0.0527046
$361$	−17.0000	−0.894737
$362$	10.0000i	0.525588i
$363$	14.0000	0.734809
$364$	0	0
$365$	−10.0000	−0.523424
$366$	− 4.00000i	− 0.209083i
$367$	30.0000	1.56599	0.782994	−	0.622030i	$-0.213692\pi$
$367$	30.0000	1.56599	0.782994	+	0.622030i	$0.213692\pi$
$368$	−6.00000	−0.312772
$369$	− 10.0000i	− 0.520579i
$370$	− 2.00000i	− 0.103975i
$371$	− 8.00000i	− 0.415339i
$372$	− 12.0000i	− 0.622171i
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	4.00000	0.206835
$375$	− 2.00000i	− 0.103280i
$376$	12.0000	0.618853
$377$	0	0
$378$	16.0000	0.822951
$379$	− 6.00000i	− 0.308199i	−0.988055	−	0.154100i	$-0.950752\pi$
$379$	− 6.00000i	− 0.308199i	0.988055	−	0.154100i	$-0.0492477\pi$
$380$	−6.00000	−0.307794
$381$	28.0000	1.43448
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	2.00000i	0.102062i
$385$	− 8.00000i	− 0.407718i
$386$	14.0000	0.712581
$387$	10.0000	0.508329
$388$	14.0000i	0.710742i
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	− 9.00000i	− 0.454569i
$393$	8.00000	0.403547
$394$	6.00000	0.302276
$395$	− 4.00000i	− 0.201262i
$396$	2.00000i	0.100504i
$397$	38.0000i	1.90717i	0.301131	+	0.953583i	$0.402636\pi$
$397$	38.0000i	1.90717i	−0.301131	+	0.953583i	$0.597364\pi$
$398$	0	0
$399$	−48.0000	−2.40301
$400$	−1.00000	−0.0500000
$401$	− 6.00000i	− 0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$	− 6.00000i	− 0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$	24.0000	1.19701
$403$	0	0
$404$	14.0000	0.696526
$405$	− 11.0000i	− 0.546594i
$406$	−8.00000	−0.397033
$407$	−4.00000	−0.198273
$408$	− 4.00000i	− 0.198030i
$409$	− 34.0000i	− 1.68119i	−0.541663	−	0.840596i	$-0.682205\pi$
$409$	− 34.0000i	− 1.68119i	0.541663	−	0.840596i	$-0.317795\pi$
$410$	− 10.0000i	− 0.493865i
$411$	− 36.0000i	− 1.77575i
$412$	−18.0000	−0.886796
$413$	40.0000	1.96827
$414$	6.00000i	0.294884i
$415$	0	0
$416$	0	0
$417$	−16.0000	−0.783523
$418$	12.0000i	0.586939i
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	−8.00000	−0.390360
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	− 28.0000i	− 1.36302i
$423$	− 12.0000i	− 0.583460i
$424$	2.00000i	0.0971286i
$425$	2.00000	0.0970143
$426$	−20.0000	−0.969003
$427$	− 8.00000i	− 0.387147i
$428$	−6.00000	−0.290021
$429$	0	0
$430$	10.0000	0.482243
$431$	− 18.0000i	− 0.867029i	−0.901146	−	0.433515i	$-0.857273\pi$
$431$	− 18.0000i	− 0.867029i	0.901146	−	0.433515i	$-0.142727\pi$
$432$	−4.00000	−0.192450
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	− 24.0000i	− 1.15204i
$435$	4.00000i	0.191785i
$436$	− 6.00000i	− 0.287348i
$437$	36.0000i	1.72211i
$438$	20.0000	0.955637
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	2.00000i	0.0953463i
$441$	−9.00000	−0.428571
$442$	0	0
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	4.00000i	0.189832i
$445$	14.0000	0.663664
$446$	−4.00000	−0.189405
$447$	− 4.00000i	− 0.189194i
$448$	4.00000i	0.188982i
$449$	− 6.00000i	− 0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$	− 6.00000i	− 0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	1.00000i	0.0471405i
$451$	−20.0000	−0.941763
$452$	−2.00000	−0.0940721
$453$	12.0000i	0.563809i
$454$	4.00000	0.187729
$455$	0	0
$456$	12.0000	0.561951
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	10.0000	0.467269
$459$	8.00000	0.373408
$460$	6.00000i	0.279751i
$461$	6.00000i	0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$	6.00000i	0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$	16.0000i	0.744387i
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	2.00000	0.0928477
$465$	−12.0000	−0.556487
$466$	− 6.00000i	− 0.277945i
$467$	−10.0000	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$468$	0	0
$469$	48.0000	2.21643
$470$	− 12.0000i	− 0.553519i
$471$	20.0000	0.921551
$472$	−10.0000	−0.460287
$473$	− 20.0000i	− 0.919601i
$474$	8.00000i	0.367452i
$475$	6.00000i	0.275299i
$476$	− 8.00000i	− 0.366679i
$477$	2.00000	0.0915737
$478$	26.0000	1.18921
$479$	2.00000i	0.0913823i	0.998956	+	0.0456912i	$0.0145490\pi$
$479$	2.00000i	0.0913823i	−0.998956	+	0.0456912i	$0.985451\pi$
$480$	2.00000	0.0912871
$481$	0	0
$482$	−22.0000	−1.00207
$483$	48.0000i	2.18408i
$484$	−7.00000	−0.318182
$485$	14.0000	0.635707
$486$	10.0000i	0.453609i
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	2.00000i	0.0905357i
$489$	− 8.00000i	− 0.361773i
$490$	−9.00000	−0.406579
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	20.0000i	0.901670i
$493$	−4.00000	−0.180151
$494$	0	0
$495$	2.00000	0.0898933
$496$	6.00000i	0.269408i
$497$	−40.0000	−1.79425
$498$	0	0
$499$	− 38.0000i	− 1.70111i	−0.525883	−	0.850557i	$-0.676265\pi$
$499$	− 38.0000i	− 1.70111i	0.525883	−	0.850557i	$-0.323735\pi$
$500$	1.00000i	0.0447214i
$501$	40.0000i	1.78707i
$502$	0	0
$503$	−14.0000	−0.624229	−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000	−0.624229	−0.312115	+	0.950044i	$0.601037\pi$
$504$	4.00000	0.178174
$505$	− 14.0000i	− 0.622992i
$506$	12.0000	0.533465
$507$	0	0
$508$	−14.0000	−0.621150
$509$	6.00000i	0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$	6.00000i	0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$	−4.00000	−0.177123
$511$	40.0000	1.76950
$512$	− 1.00000i	− 0.0441942i
$513$	24.0000i	1.05963i
$514$	− 30.0000i	− 1.32324i
$515$	18.0000i	0.793175i
$516$	−20.0000	−0.880451
$517$	−24.0000	−1.05552
$518$	8.00000i	0.351500i
$519$	−20.0000	−0.877903
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	−4.00000	−0.174741
$525$	8.00000i	0.349149i
$526$	− 2.00000i	− 0.0872041i
$527$	− 12.0000i	− 0.522728i
$528$	− 4.00000i	− 0.174078i
$529$	13.0000	0.565217
$530$	2.00000	0.0868744
$531$	10.0000i	0.433963i
$532$	24.0000	1.04053
$533$	0	0
$534$	−28.0000	−1.21168
$535$	6.00000i	0.259403i
$536$	−12.0000	−0.518321
$537$	8.00000	0.345225
$538$	− 6.00000i	− 0.258678i
$539$	18.0000i	0.775315i
$540$	4.00000i	0.172133i
$541$	− 38.0000i	− 1.63375i	−0.576816	−	0.816874i	$-0.695705\pi$
$541$	− 38.0000i	− 1.63375i	0.576816	−	0.816874i	$-0.304295\pi$
$542$	−2.00000	−0.0859074
$543$	−20.0000	−0.858282
$544$	2.00000i	0.0857493i
$545$	−6.00000	−0.257012
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	18.0000i	0.768922i
$549$	2.00000	0.0853579
$550$	2.00000	0.0852803
$551$	− 12.0000i	− 0.511217i
$552$	− 12.0000i	− 0.510754i
$553$	16.0000i	0.680389i
$554$	2.00000i	0.0849719i
$555$	4.00000	0.169791
$556$	8.00000	0.339276
$557$	38.0000i	1.61011i	0.593199	+	0.805056i	$0.297865\pi$
$557$	38.0000i	1.61011i	−0.593199	+	0.805056i	$0.702135\pi$
$558$	6.00000	0.254000
$559$	0	0
$560$	4.00000	0.169031
$561$	8.00000i	0.337760i
$562$	−6.00000	−0.253095
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	24.0000i	1.01058i
$565$	2.00000i	0.0841406i
$566$	14.0000i	0.588464i
$567$	44.0000i	1.84783i
$568$	10.0000	0.419591
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	− 12.0000i	− 0.502625i
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	0	0
$574$	40.0000i	1.66957i
$575$	6.00000	0.250217
$576$	−1.00000	−0.0416667
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	13.0000i	0.540729i
$579$	28.0000i	1.16364i
$580$	− 2.00000i	− 0.0830455i
$581$	0	0
$582$	−28.0000	−1.16064
$583$	− 4.00000i	− 0.165663i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	22.0000	0.908812
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	18.0000	0.742307
$589$	36.0000	1.48335
$590$	10.0000i	0.411693i
$591$	12.0000i	0.493614i
$592$	− 2.00000i	− 0.0821995i
$593$	− 2.00000i	− 0.0821302i	−0.999156	−	0.0410651i	$-0.986925\pi$
$593$	− 2.00000i	− 0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$	8.00000	0.328244
$595$	−8.00000	−0.327968
$596$	2.00000i	0.0819232i
$597$	0	0
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	− 2.00000i	− 0.0816497i
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−40.0000	−1.63028
$603$	12.0000i	0.488678i
$604$	− 6.00000i	− 0.244137i
$605$	7.00000i	0.284590i
$606$	28.0000i	1.13742i
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	−6.00000	−0.243332
$609$	− 16.0000i	− 0.648353i
$610$	2.00000	0.0809776
$611$	0	0
$612$	2.00000	0.0808452
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	24.0000	0.968561
$615$	20.0000	0.806478
$616$	− 8.00000i	− 0.322329i
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	− 36.0000i	− 1.44813i
$619$	− 46.0000i	− 1.84890i	−0.381308	−	0.924448i	$-0.624526\pi$
$619$	− 46.0000i	− 1.84890i	0.381308	−	0.924448i	$-0.375474\pi$
$620$	6.00000	0.240966
$621$	24.0000	0.963087
$622$	− 12.0000i	− 0.481156i
$623$	−56.0000	−2.24359
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000i	0.239808i
$627$	−24.0000	−0.958468
$628$	−10.0000	−0.399043
$629$	4.00000i	0.159490i
$630$	− 4.00000i	− 0.159364i
$631$	30.0000i	1.19428i	0.802137	+	0.597141i	$0.203697\pi$
$631$	30.0000i	1.19428i	−0.802137	+	0.597141i	$0.796303\pi$
$632$	− 4.00000i	− 0.159111i
$633$	56.0000	2.22580
$634$	18.0000	0.714871
$635$	14.0000i	0.555573i
$636$	−4.00000	−0.158610
$637$	0	0
$638$	−4.00000	−0.158362
$639$	− 10.0000i	− 0.395594i
$640$	−1.00000	−0.0395285
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	− 12.0000i	− 0.473602i
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	− 24.0000i	− 0.945732i
$645$	20.0000i	0.787499i
$646$	12.0000	0.472134
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	− 11.0000i	− 0.432121i
$649$	20.0000	0.785069
$650$	0	0
$651$	48.0000	1.88127
$652$	4.00000i	0.156652i
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	12.0000	0.469237
$655$	4.00000i	0.156293i
$656$	− 10.0000i	− 0.390434i
$657$	10.0000i	0.390137i
$658$	48.0000i	1.87123i
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	−4.00000	−0.155700
$661$	− 30.0000i	− 1.16686i	−0.812162	−	0.583432i	$-0.801709\pi$
$661$	− 30.0000i	− 1.16686i	0.812162	−	0.583432i	$-0.198291\pi$
$662$	14.0000	0.544125
$663$	0	0
$664$	0	0
$665$	− 24.0000i	− 0.930680i
$666$	−2.00000	−0.0774984
$667$	−12.0000	−0.464642
$668$	− 20.0000i	− 0.773823i
$669$	− 8.00000i	− 0.309298i
$670$	12.0000i	0.463600i
$671$	− 4.00000i	− 0.154418i
$672$	−8.00000	−0.308607
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	− 22.0000i	− 0.847408i
$675$	4.00000	0.153960
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	− 4.00000i	− 0.153619i
$679$	−56.0000	−2.14908
$680$	2.00000	0.0766965
$681$	8.00000i	0.306561i
$682$	− 12.0000i	− 0.459504i
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	6.00000i	0.229416i
$685$	18.0000	0.687745
$686$	8.00000	0.305441
$687$	20.0000i	0.763048i
$688$	10.0000	0.381246
$689$	0	0
$690$	−12.0000	−0.456832
$691$	38.0000i	1.44559i	0.691063	+	0.722794i	$0.257142\pi$
$691$	38.0000i	1.44559i	−0.691063	+	0.722794i	$0.742858\pi$
$692$	10.0000	0.380143
$693$	−8.00000	−0.303895
$694$	− 6.00000i	− 0.227757i
$695$	− 8.00000i	− 0.303457i
$696$	4.00000i	0.151620i
$697$	20.0000i	0.757554i
$698$	2.00000	0.0757011
$699$	12.0000	0.453882
$700$	− 4.00000i	− 0.151186i
$701$	−38.0000	−1.43524	−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000	−1.43524	−0.717620	+	0.696435i	$0.754769\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	2.00000i	0.0753778i
$705$	24.0000	0.903892
$706$	−34.0000	−1.27961
$707$	56.0000i	2.10610i
$708$	− 20.0000i	− 0.751646i
$709$	− 22.0000i	− 0.826227i	−0.910679	−	0.413114i	$-0.864441\pi$
$709$	− 22.0000i	− 0.826227i	0.910679	−	0.413114i	$-0.135559\pi$
$710$	− 10.0000i	− 0.375293i
$711$	−4.00000	−0.150012
$712$	14.0000	0.524672
$713$	− 36.0000i	− 1.34821i
$714$	16.0000	0.598785
$715$	0	0
$716$	−4.00000	−0.149487
$717$	52.0000i	1.94198i
$718$	−6.00000	−0.223918
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	1.00000i	0.0372678i
$721$	− 72.0000i	− 2.68142i
$722$	17.0000i	0.632674i
$723$	− 44.0000i	− 1.63638i
$724$	10.0000	0.371647
$725$	−2.00000	−0.0742781
$726$	− 14.0000i	− 0.519589i
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	10.0000i	0.370117i
$731$	−20.0000	−0.739727
$732$	−4.00000	−0.147844
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	− 30.0000i	− 1.10732i
$735$	− 18.0000i	− 0.663940i
$736$	6.00000i	0.221163i
$737$	24.0000	0.884051
$738$	−10.0000	−0.368105
$739$	42.0000i	1.54499i	0.635018	+	0.772497i	$0.280993\pi$
$739$	42.0000i	1.54499i	−0.635018	+	0.772497i	$0.719007\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	−8.00000	−0.293689
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	−12.0000	−0.439941
$745$	2.00000	0.0732743
$746$	14.0000i	0.512576i
$747$	0	0
$748$	− 4.00000i	− 0.146254i
$749$	− 24.0000i	− 0.876941i
$750$	−2.00000	−0.0730297
$751$	−44.0000	−1.60558	−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000	−1.60558	−0.802791	+	0.596260i	$0.796653\pi$
$752$	− 12.0000i	− 0.437595i
$753$	0	0
$754$	0	0
$755$	−6.00000	−0.218362
$756$	− 16.0000i	− 0.581914i
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	−6.00000	−0.217930
$759$	24.0000i	0.871145i
$760$	6.00000i	0.217643i
$761$	− 14.0000i	− 0.507500i	−0.967270	−	0.253750i	$-0.918336\pi$
$761$	− 14.0000i	− 0.507500i	0.967270	−	0.253750i	$-0.0816640\pi$
$762$	− 28.0000i	− 1.01433i
$763$	24.0000	0.868858
$764$	0	0
$765$	− 2.00000i	− 0.0723102i
$766$	−12.0000	−0.433578
$767$	0	0
$768$	2.00000	0.0721688
$769$	− 10.0000i	− 0.360609i	−0.983611	−	0.180305i	$-0.942292\pi$
$769$	− 10.0000i	− 0.360609i	0.983611	−	0.180305i	$-0.0577084\pi$
$770$	−8.00000	−0.288300
$771$	60.0000	2.16085
$772$	− 14.0000i	− 0.503871i
$773$	34.0000i	1.22290i	0.791285	+	0.611448i	$0.209412\pi$
$773$	34.0000i	1.22290i	−0.791285	+	0.611448i	$0.790588\pi$
$774$	− 10.0000i	− 0.359443i
$775$	− 6.00000i	− 0.215526i
$776$	14.0000	0.502571
$777$	−16.0000	−0.573997
$778$	− 10.0000i	− 0.358517i
$779$	−60.0000	−2.14972
$780$	0	0
$781$	−20.0000	−0.715656
$782$	− 12.0000i	− 0.429119i
$783$	−8.00000	−0.285897
$784$	−9.00000	−0.321429
$785$	10.0000i	0.356915i
$786$	− 8.00000i	− 0.285351i
$787$	− 8.00000i	− 0.285169i	−0.989783	−	0.142585i	$-0.954459\pi$
$787$	− 8.00000i	− 0.285169i	0.989783	−	0.142585i	$-0.0455413\pi$
$788$	− 6.00000i	− 0.213741i
$789$	4.00000	0.142404
$790$	−4.00000	−0.142314
$791$	− 8.00000i	− 0.284447i
$792$	2.00000	0.0710669
$793$	0	0
$794$	38.0000	1.34857
$795$	4.00000i	0.141865i
$796$	0	0
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	48.0000i	1.69918i
$799$	24.0000i	0.849059i
$800$	1.00000i	0.0353553i
$801$	− 14.0000i	− 0.494666i
$802$	−6.00000	−0.211867
$803$	20.0000	0.705785
$804$	− 24.0000i	− 0.846415i
$805$	−24.0000	−0.845889
$806$	0	0
$807$	12.0000	0.422420
$808$	− 14.0000i	− 0.492518i
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	−11.0000	−0.386501
$811$	38.0000i	1.33436i	0.744896	+	0.667180i	$0.232499\pi$
$811$	38.0000i	1.33436i	−0.744896	+	0.667180i	$0.767501\pi$
$812$	8.00000i	0.280745i
$813$	− 4.00000i	− 0.140286i
$814$	4.00000i	0.140200i
$815$	4.00000	0.140114
$816$	−4.00000	−0.140028
$817$	− 60.0000i	− 2.09913i
$818$	−34.0000	−1.18878
$819$	0	0
$820$	−10.0000	−0.349215
$821$	− 50.0000i	− 1.74501i	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	− 50.0000i	− 1.74501i	0.488603	−	0.872506i	$-0.337507\pi$
$822$	−36.0000	−1.25564
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	18.0000i	0.627060i
$825$	4.00000i	0.139262i
$826$	− 40.0000i	− 1.39178i
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	6.00000	0.208514
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	0	0
$833$	18.0000	0.623663
$834$	16.0000i	0.554035i
$835$	−20.0000	−0.692129
$836$	12.0000	0.415029
$837$	− 24.0000i	− 0.829561i
$838$	− 16.0000i	− 0.552711i
$839$	26.0000i	0.897620i	0.893627	+	0.448810i	$0.148152\pi$
$839$	26.0000i	0.897620i	−0.893627	+	0.448810i	$0.851848\pi$
$840$	8.00000i	0.276026i
$841$	−25.0000	−0.862069
$842$	−10.0000	−0.344623
$843$	− 12.0000i	− 0.413302i
$844$	−28.0000	−0.963800
$845$	0	0
$846$	−12.0000	−0.412568
$847$	− 28.0000i	− 0.962091i
$848$	2.00000	0.0686803
$849$	−28.0000	−0.960958
$850$	− 2.00000i	− 0.0685994i
$851$	12.0000i	0.411355i
$852$	20.0000i	0.685189i
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	−8.00000	−0.273754
$855$	6.00000	0.205196
$856$	6.00000i	0.205076i
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	− 10.0000i	− 0.340997i
$861$	−80.0000	−2.72639
$862$	−18.0000	−0.613082
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	4.00000i	0.136083i
$865$	− 10.0000i	− 0.340010i
$866$	− 38.0000i	− 1.29129i
$867$	−26.0000	−0.883006
$868$	−24.0000	−0.814613
$869$	8.00000i	0.271381i
$870$	4.00000	0.135613
$871$	0	0
$872$	−6.00000	−0.203186
$873$	− 14.0000i	− 0.473828i
$874$	36.0000	1.21772
$875$	−4.00000	−0.135225
$876$	− 20.0000i	− 0.675737i
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	− 32.0000i	− 1.07995i
$879$	44.0000i	1.48408i
$880$	2.00000	0.0674200
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	9.00000i	0.303046i
$883$	−38.0000	−1.27880	−0.639401	−	0.768874i	$-0.720818\pi$
$883$	−38.0000	−1.27880	−0.639401	+	0.768874i	$0.720818\pi$
$884$	0	0
$885$	−20.0000	−0.672293
$886$	14.0000i	0.470339i
$887$	22.0000	0.738688	0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000	0.738688	0.369344	+	0.929293i	$0.379582\pi$
$888$	4.00000	0.134231
$889$	− 56.0000i	− 1.87818i
$890$	− 14.0000i	− 0.469281i
$891$	22.0000i	0.737028i
$892$	4.00000i	0.133930i
$893$	−72.0000	−2.40939
$894$	−4.00000	−0.133780
$895$	4.00000i	0.133705i
$896$	4.00000	0.133631
$897$	0	0
$898$	−6.00000	−0.200223
$899$	12.0000i	0.400222i
$900$	1.00000	0.0333333
$901$	−4.00000	−0.133259
$902$	20.0000i	0.665927i
$903$	− 80.0000i	− 2.66223i
$904$	2.00000i	0.0665190i
$905$	− 10.0000i	− 0.332411i
$906$	12.0000	0.398673
$907$	14.0000	0.464862	0.232431	−	0.972613i	$-0.425332\pi$
$907$	14.0000	0.464862	0.232431	+	0.972613i	$0.425332\pi$
$908$	− 4.00000i	− 0.132745i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	− 12.0000i	− 0.397360i
$913$	0	0
$914$	10.0000	0.330771
$915$	4.00000i	0.132236i
$916$	− 10.0000i	− 0.330409i
$917$	− 16.0000i	− 0.528367i
$918$	− 8.00000i	− 0.264039i
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	6.00000	0.197814
$921$	48.0000i	1.58165i
$922$	6.00000	0.197599
$923$	0	0
$924$	16.0000	0.526361
$925$	2.00000i	0.0657596i
$926$	16.0000	0.525793
$927$	18.0000	0.591198
$928$	− 2.00000i	− 0.0656532i
$929$	6.00000i	0.196854i	0.995144	+	0.0984268i	$0.0313810\pi$
$929$	6.00000i	0.196854i	−0.995144	+	0.0984268i	$0.968619\pi$
$930$	12.0000i	0.393496i
$931$	54.0000i	1.76978i
$932$	−6.00000	−0.196537
$933$	24.0000	0.785725
$934$	10.0000i	0.327210i
$935$	−4.00000	−0.130814
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	− 48.0000i	− 1.56726i
$939$	−12.0000	−0.391605
$940$	−12.0000	−0.391397
$941$	− 50.0000i	− 1.62995i	−0.579494	−	0.814977i	$-0.696750\pi$
$941$	− 50.0000i	− 1.62995i	0.579494	−	0.814977i	$-0.303250\pi$
$942$	− 20.0000i	− 0.651635i
$943$	60.0000i	1.95387i
$944$	10.0000i	0.325472i
$945$	−16.0000	−0.520480
$946$	−20.0000	−0.650256
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	8.00000	0.259828
$949$	0	0
$950$	6.00000	0.194666
$951$	36.0000i	1.16738i
$952$	−8.00000	−0.259281
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	− 2.00000i	− 0.0647524i
$955$	0	0
$956$	− 26.0000i	− 0.840900i
$957$	− 8.00000i	− 0.258603i
$958$	2.00000	0.0646171
$959$	−72.0000	−2.32500
$960$	− 2.00000i	− 0.0645497i
$961$	−5.00000	−0.161290
$962$	0	0
$963$	6.00000	0.193347
$964$	22.0000i	0.708572i
$965$	−14.0000	−0.450676
$966$	48.0000	1.54437
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	7.00000i	0.224989i
$969$	24.0000i	0.770991i
$970$	− 14.0000i	− 0.449513i
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	10.0000	0.320750
$973$	32.0000i	1.02587i
$974$	16.0000	0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	−8.00000	−0.255812
$979$	−28.0000	−0.894884
$980$	9.00000i	0.287494i
$981$	6.00000i	0.191565i
$982$	0	0
$983$	− 16.0000i	− 0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$	− 16.0000i	− 0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$	20.0000	0.637577
$985$	−6.00000	−0.191176
$986$	4.00000i	0.127386i
$987$	−96.0000	−3.05571
$988$	0	0
$989$	−60.0000	−1.90789
$990$	− 2.00000i	− 0.0635642i
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	6.00000	0.190500
$993$	28.0000i	0.888553i
$994$	40.0000i	1.26872i
$995$	0	0
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	−38.0000	−1.20287
$999$	8.00000i	0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.d.e.1351.1		2
13.2	odd	12		1690.2.e.g.191.1		2
13.3	even	3		1690.2.l.a.1161.2		4
13.4	even	6		1690.2.l.a.361.2		4
13.5	odd	4		1690.2.a.e.1.1		1
13.6	odd	12		1690.2.e.g.991.1		2
13.7	odd	12		1690.2.e.a.991.1		2
13.8	odd	4		130.2.a.c.1.1	✓	1
13.9	even	3		1690.2.l.a.361.1		4
13.10	even	6		1690.2.l.a.1161.1		4
13.11	odd	12		1690.2.e.a.191.1		2
13.12	even	2	inner	1690.2.d.e.1351.2		2
39.8	even	4		1170.2.a.d.1.1		1
52.47	even	4		1040.2.a.b.1.1		1
65.8	even	4		650.2.b.g.599.1		2
65.34	odd	4		650.2.a.c.1.1		1
65.44	odd	4		8450.2.a.n.1.1		1
65.47	even	4		650.2.b.g.599.2		2
91.34	even	4		6370.2.a.l.1.1		1
104.21	odd	4		4160.2.a.c.1.1		1
104.99	even	4		4160.2.a.t.1.1		1
156.47	odd	4		9360.2.a.by.1.1		1
195.8	odd	4		5850.2.e.u.5149.2		2
195.47	odd	4		5850.2.e.u.5149.1		2
195.164	even	4		5850.2.a.cb.1.1		1
260.99	even	4		5200.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.a.c.1.1	✓	1	13.8	odd	4
650.2.a.c.1.1		1	65.34	odd	4
650.2.b.g.599.1		2	65.8	even	4
650.2.b.g.599.2		2	65.47	even	4
1040.2.a.b.1.1		1	52.47	even	4
1170.2.a.d.1.1		1	39.8	even	4
1690.2.a.e.1.1		1	13.5	odd	4
1690.2.d.e.1351.1		2	1.1	even	1	trivial
1690.2.d.e.1351.2		2	13.12	even	2	inner
1690.2.e.a.191.1		2	13.11	odd	12
1690.2.e.a.991.1		2	13.7	odd	12
1690.2.e.g.191.1		2	13.2	odd	12
1690.2.e.g.991.1		2	13.6	odd	12
1690.2.l.a.361.1		4	13.9	even	3
1690.2.l.a.361.2		4	13.4	even	6
1690.2.l.a.1161.1		4	13.10	even	6
1690.2.l.a.1161.2		4	13.3	even	3
4160.2.a.c.1.1		1	104.21	odd	4
4160.2.a.t.1.1		1	104.99	even	4
5200.2.a.bd.1.1		1	260.99	even	4
5850.2.a.cb.1.1		1	195.164	even	4
5850.2.e.u.5149.1		2	195.47	odd	4
5850.2.e.u.5149.2		2	195.8	odd	4
6370.2.a.l.1.1		1	91.34	even	4
8450.2.a.n.1.1		1	65.44	odd	4
9360.2.a.by.1.1		1	156.47	odd	4

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:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +2.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +2.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000i q^{11} -2.00000 q^{12} -4.00000 q^{14} +2.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000i q^{18} -6.00000i q^{19} -1.00000i q^{20} -8.00000i q^{21} -2.00000 q^{22} -6.00000 q^{23} +2.00000i q^{24} -1.00000 q^{25} -4.00000 q^{27} +4.00000i q^{28} +2.00000 q^{29} +2.00000 q^{30} +6.00000i q^{31} -1.00000i q^{32} -4.00000i q^{33} +2.00000i q^{34} +4.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} -6.00000 q^{38} -1.00000 q^{40} -10.0000i q^{41} -8.00000 q^{42} +10.0000 q^{43} +2.00000i q^{44} +1.00000i q^{45} +6.00000i q^{46} -12.0000i q^{47} +2.00000 q^{48} -9.00000 q^{49} +1.00000i q^{50} -4.00000 q^{51} +2.00000 q^{53} +4.00000i q^{54} +2.00000 q^{55} +4.00000 q^{56} -12.0000i q^{57} -2.00000i q^{58} +10.0000i q^{59} -2.00000i q^{60} +2.00000 q^{61} +6.00000 q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} +2.00000 q^{68} -12.0000 q^{69} -4.00000i q^{70} -10.0000i q^{71} +1.00000i q^{72} +10.0000i q^{73} -2.00000 q^{74} -2.00000 q^{75} +6.00000i q^{76} -8.00000 q^{77} -4.00000 q^{79} +1.00000i q^{80} -11.0000 q^{81} -10.0000 q^{82} +8.00000i q^{84} -2.00000i q^{85} -10.0000i q^{86} +4.00000 q^{87} +2.00000 q^{88} -14.0000i q^{89} +1.00000 q^{90} +6.00000 q^{92} +12.0000i q^{93} -12.0000 q^{94} +6.00000 q^{95} -2.00000i q^{96} -14.0000i q^{97} +9.00000i q^{98} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q + 4 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} - 4 q^{12} - 8 q^{14} + 2 q^{16} - 4 q^{17} - 4 q^{22} - 12 q^{23} - 2 q^{25} - 8 q^{27} + 4 q^{29} + 4 q^{30} + 8 q^{35} - 2 q^{36} - 12 q^{38} - 2 q^{40} - 16 q^{42} + 20 q^{43} + 4 q^{48} - 18 q^{49} - 8 q^{51} + 4 q^{53} + 4 q^{55} + 8 q^{56} + 4 q^{61} + 12 q^{62} - 2 q^{64} - 8 q^{66} + 4 q^{68} - 24 q^{69} - 4 q^{74} - 4 q^{75} - 16 q^{77} - 8 q^{79} - 22 q^{81} - 20 q^{82} + 8 q^{87} + 4 q^{88} + 2 q^{90} + 12 q^{92} - 24 q^{94} + 12 q^{95}+O(q^{100}) \) 2 * q + 4 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 - 4 * q^12 - 8 * q^14 + 2 * q^16 - 4 * q^17 - 4 * q^22 - 12 * q^23 - 2 * q^25 - 8 * q^27 + 4 * q^29 + 4 * q^30 + 8 * q^35 - 2 * q^36 - 12 * q^38 - 2 * q^40 - 16 * q^42 + 20 * q^43 + 4 * q^48 - 18 * q^49 - 8 * q^51 + 4 * q^53 + 4 * q^55 + 8 * q^56 + 4 * q^61 + 12 * q^62 - 2 * q^64 - 8 * q^66 + 4 * q^68 - 24 * q^69 - 4 * q^74 - 4 * q^75 - 16 * q^77 - 8 * q^79 - 22 * q^81 - 20 * q^82 + 8 * q^87 + 4 * q^88 + 2 * q^90 + 12 * q^92 - 24 * q^94 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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