LMFDB - Embedded newform 1521.2.b.c.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.1452461474$
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 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.c.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{7} -3.00000i q^{8} +1.00000 q^{10} -2.00000i q^{11} -2.00000 q^{14} -1.00000 q^{16} -7.00000 q^{17} -6.00000i q^{19} +1.00000i q^{20} -2.00000 q^{22} -6.00000 q^{23} +4.00000 q^{25} -2.00000i q^{28} +1.00000 q^{29} +4.00000i q^{31} -5.00000i q^{32} +7.00000i q^{34} +2.00000 q^{35} -1.00000i q^{37} -6.00000 q^{38} +3.00000 q^{40} -9.00000i q^{41} -6.00000 q^{43} -2.00000i q^{44} +6.00000i q^{46} +6.00000i q^{47} +3.00000 q^{49} -4.00000i q^{50} +9.00000 q^{53} +2.00000 q^{55} -6.00000 q^{56} -1.00000i q^{58} +1.00000 q^{61} +4.00000 q^{62} -7.00000 q^{64} -2.00000i q^{67} -7.00000 q^{68} -2.00000i q^{70} -6.00000i q^{71} -11.0000i q^{73} -1.00000 q^{74} -6.00000i q^{76} -4.00000 q^{77} -4.00000 q^{79} -1.00000i q^{80} -9.00000 q^{82} +14.0000i q^{83} -7.00000i q^{85} +6.00000i q^{86} -6.00000 q^{88} -14.0000i q^{89} -6.00000 q^{92} +6.00000 q^{94} +6.00000 q^{95} -2.00000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{10} - 4 q^{14} - 2 q^{16} - 14 q^{17} - 4 q^{22} - 12 q^{23} + 8 q^{25} + 2 q^{29} + 4 q^{35} - 12 q^{38} + 6 q^{40} - 12 q^{43} + 6 q^{49} + 18 q^{53} + 4 q^{55} - 12 q^{56} + 2 q^{61}+ \cdots + 12 q^{95}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^10 - 4 * q^14 - 2 * q^16 - 14 * q^17 - 4 * q^22 - 12 * q^23 + 8 * q^25 + 2 * q^29 + 4 * q^35 - 12 * q^38 + 6 * q^40 - 12 * q^43 + 6 * q^49 + 18 * q^53 + 4 * q^55 - 12 * q^56 + 2 * q^61 + 8 * q^62 - 14 * q^64 - 14 * q^68 - 2 * q^74 - 8 * q^77 - 8 * q^79 - 18 * q^82 - 12 * q^88 - 12 * q^92 + 12 * q^94 + 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000i	0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$	1.00000i	0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 3.00000i	− 1.06066i
$9$	0	0
$10$	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	$-0.0974911\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	− 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
$21$	0	0
$22$	−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	$0.715123\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	− 5.00000i	− 0.883883i
$33$	0	0
$34$	7.00000i	1.20049i
$35$	2.00000	0.338062
$36$	0	0
$37$	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	− 1.00000i	− 0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	3.00000	0.474342
$41$	− 9.00000i	− 1.40556i	−0.711405	−	0.702782i	$-0.751941\pi$
$41$	− 9.00000i	− 1.40556i	0.711405	−	0.702782i	$-0.248059\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	− 2.00000i	− 0.301511i
$45$	0	0
$46$	6.00000i	0.884652i
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	− 4.00000i	− 0.565685i
$51$	0	0
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	−6.00000	−0.801784
$57$	0	0
$58$	− 1.00000i	− 0.131306i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	−7.00000	−0.848875
$69$	0	0
$70$	− 2.00000i	− 0.239046i
$71$	− 6.00000i	− 0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$	− 6.00000i	− 0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$	0	0
$73$	− 11.0000i	− 1.28745i	−0.765256	−	0.643726i	$-0.777388\pi$
$73$	− 11.0000i	− 1.28745i	0.765256	−	0.643726i	$-0.222612\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	− 6.00000i	− 0.688247i
$77$	−4.00000	−0.455842
$78$	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
$81$	0	0
$82$	−9.00000	−0.993884
$83$	14.0000i	1.53670i	0.640030	+	0.768350i	$0.278922\pi$
$83$	14.0000i	1.53670i	−0.640030	+	0.768350i	$0.721078\pi$
$84$	0	0
$85$	− 7.00000i	− 0.759257i
$86$	6.00000i	0.646997i
$87$	0	0
$88$	−6.00000	−0.639602
$89$	− 14.0000i	− 1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$	− 14.0000i	− 1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	6.00000	0.618853
$95$	6.00000	0.615587
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	4.00000	0.400000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	0	0
$106$	− 9.00000i	− 0.874157i
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	− 2.00000i	− 0.190693i
$111$	0	0
$112$	2.00000i	0.188982i
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	− 6.00000i	− 0.559503i
$116$	1.00000	0.0928477
$117$	0	0
$118$	0	0
$119$	14.0000i	1.28338i
$120$	0	0
$121$	7.00000	0.636364
$122$	− 1.00000i	− 0.0905357i
$123$	0	0
$124$	4.00000i	0.359211i
$125$	9.00000i	0.804984i
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	− 3.00000i	− 0.265165i
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	−2.00000	−0.172774
$135$	0	0
$136$	21.0000i	1.80074i
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−6.00000	−0.503509
$143$	0	0
$144$	0	0
$145$	1.00000i	0.0830455i
$146$	−11.0000	−0.910366
$147$	0	0
$148$	− 1.00000i	− 0.0821995i
$149$	− 3.00000i	− 0.245770i	−0.992421	−	0.122885i	$-0.960785\pi$
$149$	− 3.00000i	− 0.245770i	0.992421	−	0.122885i	$-0.0392146\pi$
$150$	0	0
$151$	2.00000i	0.162758i	0.996683	+	0.0813788i	$0.0259324\pi$
$151$	2.00000i	0.162758i	−0.996683	+	0.0813788i	$0.974068\pi$
$152$	−18.0000	−1.45999
$153$	0	0
$154$	4.00000i	0.322329i
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	4.00000i	0.318223i
$159$	0	0
$160$	5.00000	0.395285
$161$	12.0000i	0.945732i
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	− 9.00000i	− 0.702782i
$165$	0	0
$166$	14.0000	1.08661
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	0	0
$170$	−7.00000	−0.536875
$171$	0	0
$172$	−6.00000	−0.457496
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	− 8.00000i	− 0.604743i
$176$	2.00000i	0.150756i
$177$	0	0
$178$	−14.0000	−1.04934
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	0	0
$184$	18.0000i	1.32698i
$185$	1.00000	0.0735215
$186$	0	0
$187$	14.0000i	1.02378i
$188$	6.00000i	0.437595i
$189$	0	0
$190$	− 6.00000i	− 0.435286i
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	9.00000i	0.647834i	0.946085	+	0.323917i	$0.105000\pi$
$193$	9.00000i	0.647834i	−0.946085	+	0.323917i	$0.895000\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	3.00000	0.214286
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	− 12.0000i	− 0.848528i
$201$	0	0
$202$	− 3.00000i	− 0.211079i
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	9.00000	0.628587
$206$	6.00000i	0.418040i
$207$	0	0
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	− 6.00000i	− 0.410152i
$215$	− 6.00000i	− 0.409197i
$216$	0	0
$217$	8.00000	0.543075
$218$	−2.00000	−0.135457
$219$	0	0
$220$	2.00000	0.134840
$221$	0	0
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	− 15.0000i	− 0.997785i
$227$	14.0000i	0.929213i	0.885517	+	0.464606i	$0.153804\pi$
$227$	14.0000i	0.929213i	−0.885517	+	0.464606i	$0.846196\pi$
$228$	0	0
$229$	22.0000i	1.45380i	0.686743	+	0.726900i	$0.259040\pi$
$229$	22.0000i	1.45380i	−0.686743	+	0.726900i	$0.740960\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	− 3.00000i	− 0.196960i
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	0	0
$238$	14.0000	0.907485
$239$	− 30.0000i	− 1.94054i	−0.242028	−	0.970269i	$-0.577812\pi$
$239$	− 30.0000i	− 1.94054i	0.242028	−	0.970269i	$-0.422188\pi$
$240$	0	0
$241$	− 7.00000i	− 0.450910i	−0.974254	−	0.225455i	$-0.927613\pi$
$241$	− 7.00000i	− 0.450910i	0.974254	−	0.225455i	$-0.0723868\pi$
$242$	− 7.00000i	− 0.449977i
$243$	0	0
$244$	1.00000	0.0640184
$245$	3.00000i	0.191663i
$246$	0	0
$247$	0	0
$248$	12.0000	0.762001
$249$	0	0
$250$	9.00000	0.569210
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	20.0000i	1.25491i
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	− 8.00000i	− 0.494242i
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	9.00000i	0.552866i
$266$	12.0000i	0.735767i
$267$	0	0
$268$	− 2.00000i	− 0.122169i
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	7.00000	0.424437
$273$	0	0
$274$	−3.00000	−0.181237
$275$	− 8.00000i	− 0.482418i
$276$	0	0
$277$	31.0000	1.86261	0.931305	−	0.364241i	$-0.118672\pi$
$277$	31.0000	1.86261	0.931305	+	0.364241i	$0.118672\pi$
$278$	− 12.0000i	− 0.719712i
$279$	0	0
$280$	− 6.00000i	− 0.358569i
$281$	− 19.0000i	− 1.13344i	−0.823909	−	0.566722i	$-0.808211\pi$
$281$	− 19.0000i	− 1.13344i	0.823909	−	0.566722i	$-0.191789\pi$
$282$	0	0
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	− 6.00000i	− 0.356034i
$285$	0	0
$286$	0	0
$287$	−18.0000	−1.06251
$288$	0	0
$289$	32.0000	1.88235
$290$	1.00000	0.0587220
$291$	0	0
$292$	− 11.0000i	− 0.643726i
$293$	− 9.00000i	− 0.525786i	−0.964825	−	0.262893i	$-0.915323\pi$
$293$	− 9.00000i	− 0.525786i	0.964825	−	0.262893i	$-0.0846766\pi$
$294$	0	0
$295$	0	0
$296$	−3.00000	−0.174371
$297$	0	0
$298$	−3.00000	−0.173785
$299$	0	0
$300$	0	0
$301$	12.0000i	0.691669i
$302$	2.00000	0.115087
$303$	0	0
$304$	6.00000i	0.344124i
$305$	1.00000i	0.0572598i
$306$	0	0
$307$	− 14.0000i	− 0.799022i	−0.916728	−	0.399511i	$-0.869180\pi$
$307$	− 14.0000i	− 0.799022i	0.916728	−	0.399511i	$-0.130820\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	4.00000i	0.227185i
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	3.00000i	0.169300i
$315$	0	0
$316$	−4.00000	−0.225018
$317$	25.0000i	1.40414i	0.712108	+	0.702070i	$0.247741\pi$
$317$	25.0000i	1.40414i	−0.712108	+	0.702070i	$0.752259\pi$
$318$	0	0
$319$	− 2.00000i	− 0.111979i
$320$	− 7.00000i	− 0.391312i
$321$	0	0
$322$	12.0000	0.668734
$323$	42.0000i	2.33694i
$324$	0	0
$325$	0	0
$326$	4.00000	0.221540
$327$	0	0
$328$	−27.0000	−1.49083
$329$	12.0000	0.661581
$330$	0	0
$331$	− 4.00000i	− 0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$	− 4.00000i	− 0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$	14.0000i	0.768350i
$333$	0	0
$334$	16.0000	0.875481
$335$	2.00000	0.109272
$336$	0	0
$337$	33.0000	1.79762	0.898812	−	0.438334i	$-0.144431\pi$
$337$	33.0000	1.79762	0.898812	+	0.438334i	$0.144431\pi$
$338$	0	0
$339$	0	0
$340$	− 7.00000i	− 0.379628i
$341$	8.00000	0.433224
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	18.0000i	0.970495i
$345$	0	0
$346$	6.00000i	0.322562i
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	26.0000i	1.39175i	0.718164	+	0.695874i	$0.244983\pi$
$349$	26.0000i	1.39175i	−0.718164	+	0.695874i	$0.755017\pi$
$350$	−8.00000	−0.427618
$351$	0	0
$352$	−10.0000	−0.533002
$353$	11.0000i	0.585471i	0.956193	+	0.292735i	$0.0945655\pi$
$353$	11.0000i	0.585471i	−0.956193	+	0.292735i	$0.905434\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	− 14.0000i	− 0.741999i
$357$	0	0
$358$	2.00000i	0.105703i
$359$	− 18.0000i	− 0.950004i	−0.879985	−	0.475002i	$-0.842447\pi$
$359$	− 18.0000i	− 0.950004i	0.879985	−	0.475002i	$-0.157553\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	− 7.00000i	− 0.367912i
$363$	0	0
$364$	0	0
$365$	11.0000	0.575766
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	− 1.00000i	− 0.0519875i
$371$	− 18.0000i	− 0.934513i
$372$	0	0
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	14.0000	0.723923
$375$	0	0
$376$	18.0000	0.928279
$377$	0	0
$378$	0	0
$379$	− 36.0000i	− 1.84920i	−0.380945	−	0.924598i	$-0.624401\pi$
$379$	− 36.0000i	− 1.84920i	0.380945	−	0.924598i	$-0.375599\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	− 4.00000i	− 0.204658i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	0	0
$385$	− 4.00000i	− 0.203859i
$386$	9.00000	0.458088
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	19.0000	0.963338	0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000	0.963338	0.481669	+	0.876353i	$0.340031\pi$
$390$	0	0
$391$	42.0000	2.12403
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	− 4.00000i	− 0.201262i
$396$	0	0
$397$	34.0000i	1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$	34.0000i	1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	14.0000i	0.701757i
$399$	0	0
$400$	−4.00000	−0.200000
$401$	1.00000i	0.0499376i	0.999688	+	0.0249688i	$0.00794864\pi$
$401$	1.00000i	0.0499376i	−0.999688	+	0.0249688i	$0.992051\pi$
$402$	0	0
$403$	0	0
$404$	3.00000	0.149256
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	7.00000i	0.346128i	0.984911	+	0.173064i	$0.0553667\pi$
$409$	7.00000i	0.346128i	−0.984911	+	0.173064i	$0.944633\pi$
$410$	− 9.00000i	− 0.444478i
$411$	0	0
$412$	−6.00000	−0.295599
$413$	0	0
$414$	0	0
$415$	−14.0000	−0.687233
$416$	0	0
$417$	0	0
$418$	12.0000i	0.586939i
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	− 19.0000i	− 0.926003i	−0.886357	−	0.463002i	$-0.846772\pi$
$421$	− 19.0000i	− 0.926003i	0.886357	−	0.463002i	$-0.153228\pi$
$422$	8.00000i	0.389434i
$423$	0	0
$424$	− 27.0000i	− 1.31124i
$425$	−28.0000	−1.35820
$426$	0	0
$427$	− 2.00000i	− 0.0967868i
$428$	6.00000	0.290021
$429$	0	0
$430$	−6.00000	−0.289346
$431$	30.0000i	1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$	30.0000i	1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	− 8.00000i	− 0.384012i
$435$	0	0
$436$	− 2.00000i	− 0.0957826i
$437$	36.0000i	1.72211i
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	− 6.00000i	− 0.286039i
$441$	0	0
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	14.0000	0.663664
$446$	16.0000	0.757622
$447$	0	0
$448$	14.0000i	0.661438i
$449$	34.0000i	1.60456i	0.596948	+	0.802280i	$0.296380\pi$
$449$	34.0000i	1.60456i	−0.596948	+	0.802280i	$0.703620\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	15.0000	0.705541
$453$	0	0
$454$	14.0000	0.657053
$455$	0	0
$456$	0	0
$457$	− 13.0000i	− 0.608114i	−0.952654	−	0.304057i	$-0.901659\pi$
$457$	− 13.0000i	− 0.608114i	0.952654	−	0.304057i	$-0.0983414\pi$
$458$	22.0000	1.02799
$459$	0	0
$460$	− 6.00000i	− 0.279751i
$461$	− 19.0000i	− 0.884918i	−0.896789	−	0.442459i	$-0.854106\pi$
$461$	− 19.0000i	− 0.884918i	0.896789	−	0.442459i	$-0.145894\pi$
$462$	0	0
$463$	26.0000i	1.20832i	0.796862	+	0.604161i	$0.206492\pi$
$463$	26.0000i	1.20832i	−0.796862	+	0.604161i	$0.793508\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	− 10.0000i	− 0.463241i
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	6.00000i	0.276759i
$471$	0	0
$472$	0	0
$473$	12.0000i	0.551761i
$474$	0	0
$475$	− 24.0000i	− 1.10120i
$476$	14.0000i	0.641689i
$477$	0	0
$478$	−30.0000	−1.37217
$479$	24.0000i	1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$	24.0000i	1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$	0	0
$481$	0	0
$482$	−7.00000	−0.318841
$483$	0	0
$484$	7.00000	0.318182
$485$	2.00000	0.0908153
$486$	0	0
$487$	18.0000i	0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$	18.0000i	0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$	− 3.00000i	− 0.135804i
$489$	0	0
$490$	3.00000	0.135526
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	−7.00000	−0.315264
$494$	0	0
$495$	0	0
$496$	− 4.00000i	− 0.179605i
$497$	−12.0000	−0.538274
$498$	0	0
$499$	24.0000i	1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$	24.0000i	1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$	9.00000i	0.402492i
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	3.00000i	0.133498i
$506$	12.0000	0.533465
$507$	0	0
$508$	−20.0000	−0.887357
$509$	− 7.00000i	− 0.310270i	−0.987893	−	0.155135i	$-0.950419\pi$
$509$	− 7.00000i	− 0.310270i	0.987893	−	0.155135i	$-0.0495812\pi$
$510$	0	0
$511$	−22.0000	−0.973223
$512$	11.0000i	0.486136i
$513$	0	0
$514$	7.00000i	0.308757i
$515$	− 6.00000i	− 0.264392i
$516$	0	0
$517$	12.0000	0.527759
$518$	2.00000i	0.0878750i
$519$	0	0
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	− 30.0000i	− 1.30806i
$527$	− 28.0000i	− 1.21970i
$528$	0	0
$529$	13.0000	0.565217
$530$	9.00000	0.390935
$531$	0	0
$532$	−12.0000	−0.520266
$533$	0	0
$534$	0	0
$535$	6.00000i	0.259403i
$536$	−6.00000	−0.259161
$537$	0	0
$538$	− 14.0000i	− 0.603583i
$539$	− 6.00000i	− 0.258438i
$540$	0	0
$541$	− 45.0000i	− 1.93470i	−0.253442	−	0.967351i	$-0.581563\pi$
$541$	− 45.0000i	− 1.93470i	0.253442	−	0.967351i	$-0.418437\pi$
$542$	0	0
$543$	0	0
$544$	35.0000i	1.50061i
$545$	2.00000	0.0856706
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	− 3.00000i	− 0.128154i
$549$	0	0
$550$	−8.00000	−0.341121
$551$	− 6.00000i	− 0.255609i
$552$	0	0
$553$	8.00000i	0.340195i
$554$	− 31.0000i	− 1.31706i
$555$	0	0
$556$	12.0000	0.508913
$557$	− 9.00000i	− 0.381342i	−0.981654	−	0.190671i	$-0.938934\pi$
$557$	− 9.00000i	− 0.381342i	0.981654	−	0.190671i	$-0.0610664\pi$
$558$	0	0
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−19.0000	−0.801467
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	15.0000i	0.631055i
$566$	− 18.0000i	− 0.756596i
$567$	0	0
$568$	−18.0000	−0.755263
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	0	0
$573$	0	0
$574$	18.0000i	0.751305i
$575$	−24.0000	−1.00087
$576$	0	0
$577$	11.0000i	0.457936i	0.973434	+	0.228968i	$0.0735351\pi$
$577$	11.0000i	0.457936i	−0.973434	+	0.228968i	$0.926465\pi$
$578$	− 32.0000i	− 1.33102i
$579$	0	0
$580$	1.00000i	0.0415227i
$581$	28.0000	1.16164
$582$	0	0
$583$	− 18.0000i	− 0.745484i
$584$	−33.0000	−1.36555
$585$	0	0
$586$	−9.00000	−0.371787
$587$	− 16.0000i	− 0.660391i	−0.943913	−	0.330195i	$-0.892885\pi$
$587$	− 16.0000i	− 0.660391i	0.943913	−	0.330195i	$-0.107115\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	1.00000i	0.0410997i
$593$	13.0000i	0.533846i	0.963718	+	0.266923i	$0.0860069\pi$
$593$	13.0000i	0.533846i	−0.963718	+	0.266923i	$0.913993\pi$
$594$	0	0
$595$	−14.0000	−0.573944
$596$	− 3.00000i	− 0.122885i
$597$	0	0
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	12.0000	0.489083
$603$	0	0
$604$	2.00000i	0.0813788i
$605$	7.00000i	0.284590i
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−30.0000	−1.21666
$609$	0	0
$610$	1.00000	0.0404888
$611$	0	0
$612$	0	0
$613$	− 23.0000i	− 0.928961i	−0.885583	−	0.464481i	$-0.846241\pi$
$613$	− 23.0000i	− 0.928961i	0.885583	−	0.464481i	$-0.153759\pi$
$614$	−14.0000	−0.564994
$615$	0	0
$616$	12.0000i	0.483494i
$617$	− 13.0000i	− 0.523360i	−0.965155	−	0.261680i	$-0.915723\pi$
$617$	− 13.0000i	− 0.523360i	0.965155	−	0.261680i	$-0.0842766\pi$
$618$	0	0
$619$	24.0000i	0.964641i	0.875995	+	0.482321i	$0.160206\pi$
$619$	24.0000i	0.964641i	−0.875995	+	0.482321i	$0.839794\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	− 18.0000i	− 0.721734i
$623$	−28.0000	−1.12180
$624$	0	0
$625$	11.0000	0.440000
$626$	− 6.00000i	− 0.239808i
$627$	0	0
$628$	−3.00000	−0.119713
$629$	7.00000i	0.279108i
$630$	0	0
$631$	− 20.0000i	− 0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$	− 20.0000i	− 0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$	12.0000i	0.477334i
$633$	0	0
$634$	25.0000	0.992877
$635$	− 20.0000i	− 0.793676i
$636$	0	0
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	3.00000	0.118585
$641$	−31.0000	−1.22443	−0.612213	−	0.790693i	$-0.709721\pi$
$641$	−31.0000	−1.22443	−0.612213	+	0.790693i	$0.709721\pi$
$642$	0	0
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	12.0000i	0.472866i
$645$	0	0
$646$	42.0000	1.65247
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	8.00000i	0.312586i
$656$	9.00000i	0.351391i
$657$	0	0
$658$	− 12.0000i	− 0.467809i
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	0	0
$661$	− 45.0000i	− 1.75030i	−0.483854	−	0.875149i	$-0.660764\pi$
$661$	− 45.0000i	− 1.75030i	0.483854	−	0.875149i	$-0.339236\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	42.0000	1.62992
$665$	− 12.0000i	− 0.465340i
$666$	0	0
$667$	−6.00000	−0.232321
$668$	16.0000i	0.619059i
$669$	0	0
$670$	− 2.00000i	− 0.0772667i
$671$	− 2.00000i	− 0.0772091i
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	− 33.0000i	− 1.27111i
$675$	0	0
$676$	0	0
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	−21.0000	−0.805313
$681$	0	0
$682$	− 8.00000i	− 0.306336i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	3.00000	0.114624
$686$	−20.0000	−0.763604
$687$	0	0
$688$	6.00000	0.228748
$689$	0	0
$690$	0	0
$691$	42.0000i	1.59776i	0.601494	+	0.798878i	$0.294573\pi$
$691$	42.0000i	1.59776i	−0.601494	+	0.798878i	$0.705427\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	18.0000i	0.683271i
$695$	12.0000i	0.455186i
$696$	0	0
$697$	63.0000i	2.38630i
$698$	26.0000	0.984115
$699$	0	0
$700$	− 8.00000i	− 0.302372i
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	14.0000i	0.527645i
$705$	0	0
$706$	11.0000	0.413990
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	11.0000i	0.413114i	0.978435	+	0.206557i	$0.0662258\pi$
$709$	11.0000i	0.413114i	−0.978435	+	0.206557i	$0.933774\pi$
$710$	− 6.00000i	− 0.225176i
$711$	0	0
$712$	−42.0000	−1.57402
$713$	− 24.0000i	− 0.898807i
$714$	0	0
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	0	0
$718$	−18.0000	−0.671754
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	12.0000i	0.446903i
$722$	17.0000i	0.632674i
$723$	0	0
$724$	7.00000	0.260153
$725$	4.00000	0.148556
$726$	0	0
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	0	0
$729$	0	0
$730$	− 11.0000i	− 0.407128i
$731$	42.0000	1.55343
$732$	0	0
$733$	− 15.0000i	− 0.554038i	−0.960864	−	0.277019i	$-0.910654\pi$
$733$	− 15.0000i	− 0.554038i	0.960864	−	0.277019i	$-0.0893464\pi$
$734$	− 10.0000i	− 0.369107i
$735$	0	0
$736$	30.0000i	1.10581i
$737$	−4.00000	−0.147342
$738$	0	0
$739$	16.0000i	0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$	16.0000i	0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	−18.0000	−0.660801
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	0	0
$745$	3.00000	0.109911
$746$	11.0000i	0.402739i
$747$	0	0
$748$	14.0000i	0.511891i
$749$	− 12.0000i	− 0.438470i
$750$	0	0
$751$	34.0000	1.24068	0.620339	−	0.784334i	$-0.286995\pi$
$751$	34.0000	1.24068	0.620339	+	0.784334i	$0.286995\pi$
$752$	− 6.00000i	− 0.218797i
$753$	0	0
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	−36.0000	−1.30758
$759$	0	0
$760$	− 18.0000i	− 0.652929i
$761$	50.0000i	1.81250i	0.422744	+	0.906249i	$0.361067\pi$
$761$	50.0000i	1.81250i	−0.422744	+	0.906249i	$0.638933\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	4.00000	0.144715
$765$	0	0
$766$	−8.00000	−0.289052
$767$	0	0
$768$	0	0
$769$	30.0000i	1.08183i	0.841078	+	0.540914i	$0.181921\pi$
$769$	30.0000i	1.08183i	−0.841078	+	0.540914i	$0.818079\pi$
$770$	−4.00000	−0.144150
$771$	0	0
$772$	9.00000i	0.323917i
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	0	0
$775$	16.0000i	0.574737i
$776$	−6.00000	−0.215387
$777$	0	0
$778$	− 19.0000i	− 0.681183i
$779$	−54.0000	−1.93475
$780$	0	0
$781$	−12.0000	−0.429394
$782$	− 42.0000i	− 1.50192i
$783$	0	0
$784$	−3.00000	−0.107143
$785$	− 3.00000i	− 0.107075i
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	−4.00000	−0.142314
$791$	− 30.0000i	− 1.06668i
$792$	0	0
$793$	0	0
$794$	34.0000	1.20661
$795$	0	0
$796$	−14.0000	−0.496217
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	− 42.0000i	− 1.48585i
$800$	− 20.0000i	− 0.707107i
$801$	0	0
$802$	1.00000	0.0353112
$803$	−22.0000	−0.776363
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	0	0
$808$	− 9.00000i	− 0.316619i
$809$	−33.0000	−1.16022	−0.580109	−	0.814539i	$-0.696990\pi$
$809$	−33.0000	−1.16022	−0.580109	+	0.814539i	$0.696990\pi$
$810$	0	0
$811$	− 28.0000i	− 0.983213i	−0.870817	−	0.491606i	$-0.836410\pi$
$811$	− 28.0000i	− 0.983213i	0.870817	−	0.491606i	$-0.163590\pi$
$812$	− 2.00000i	− 0.0701862i
$813$	0	0
$814$	2.00000i	0.0701000i
$815$	−4.00000	−0.140114
$816$	0	0
$817$	36.0000i	1.25948i
$818$	7.00000	0.244749
$819$	0	0
$820$	9.00000	0.314294
$821$	− 50.0000i	− 1.74501i	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	− 50.0000i	− 1.74501i	0.488603	−	0.872506i	$-0.337507\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	18.0000i	0.627060i
$825$	0	0
$826$	0	0
$827$	16.0000i	0.556375i	0.960527	+	0.278187i	$0.0897336\pi$
$827$	16.0000i	0.556375i	−0.960527	+	0.278187i	$0.910266\pi$
$828$	0	0
$829$	−17.0000	−0.590434	−0.295217	−	0.955430i	$-0.595392\pi$
$829$	−17.0000	−0.590434	−0.295217	+	0.955430i	$0.595392\pi$
$830$	14.0000i	0.485947i
$831$	0	0
$832$	0	0
$833$	−21.0000	−0.727607
$834$	0	0
$835$	−16.0000	−0.553703
$836$	−12.0000	−0.415029
$837$	0	0
$838$	16.0000i	0.552711i
$839$	12.0000i	0.414286i	0.978311	+	0.207143i	$0.0664165\pi$
$839$	12.0000i	0.414286i	−0.978311	+	0.207143i	$0.933583\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−19.0000	−0.654783
$843$	0	0
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	− 14.0000i	− 0.481046i
$848$	−9.00000	−0.309061
$849$	0	0
$850$	28.0000i	0.960392i
$851$	6.00000i	0.205677i
$852$	0	0
$853$	− 21.0000i	− 0.719026i	−0.933140	−	0.359513i	$-0.882943\pi$
$853$	− 21.0000i	− 0.719026i	0.933140	−	0.359513i	$-0.117057\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	− 18.0000i	− 0.615227i
$857$	−31.0000	−1.05894	−0.529470	−	0.848329i	$-0.677609\pi$
$857$	−31.0000	−1.05894	−0.529470	+	0.848329i	$0.677609\pi$
$858$	0	0
$859$	34.0000	1.16007	0.580033	−	0.814593i	$-0.303040\pi$
$859$	34.0000	1.16007	0.580033	+	0.814593i	$0.303040\pi$
$860$	− 6.00000i	− 0.204598i
$861$	0	0
$862$	30.0000	1.02180
$863$	− 10.0000i	− 0.340404i	−0.985409	−	0.170202i	$-0.945558\pi$
$863$	− 10.0000i	− 0.340404i	0.985409	−	0.170202i	$-0.0544420\pi$
$864$	0	0
$865$	− 6.00000i	− 0.204006i
$866$	19.0000i	0.645646i
$867$	0	0
$868$	8.00000	0.271538
$869$	8.00000i	0.271381i
$870$	0	0
$871$	0	0
$872$	−6.00000	−0.203186
$873$	0	0
$874$	36.0000	1.21772
$875$	18.0000	0.608511
$876$	0	0
$877$	17.0000i	0.574049i	0.957923	+	0.287025i	$0.0926662\pi$
$877$	17.0000i	0.574049i	−0.957923	+	0.287025i	$0.907334\pi$
$878$	14.0000i	0.472477i
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	37.0000	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$881$	37.0000	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	0	0
$886$	− 4.00000i	− 0.134383i
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	40.0000i	1.34156i
$890$	− 14.0000i	− 0.469281i
$891$	0	0
$892$	16.0000i	0.535720i
$893$	36.0000	1.20469
$894$	0	0
$895$	− 2.00000i	− 0.0668526i
$896$	−6.00000	−0.200446
$897$	0	0
$898$	34.0000	1.13459
$899$	4.00000i	0.133407i
$900$	0	0
$901$	−63.0000	−2.09883
$902$	18.0000i	0.599334i
$903$	0	0
$904$	− 45.0000i	− 1.49668i
$905$	7.00000i	0.232688i
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	14.0000i	0.464606i
$909$	0	0
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	28.0000	0.926665
$914$	−13.0000	−0.430002
$915$	0	0
$916$	22.0000i	0.726900i
$917$	− 16.0000i	− 0.528367i
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	−18.0000	−0.593442
$921$	0	0
$922$	−19.0000	−0.625732
$923$	0	0
$924$	0	0
$925$	− 4.00000i	− 0.131519i
$926$	26.0000	0.854413
$927$	0	0
$928$	− 5.00000i	− 0.164133i
$929$	27.0000i	0.885841i	0.896561	+	0.442921i	$0.146058\pi$
$929$	27.0000i	0.885841i	−0.896561	+	0.442921i	$0.853942\pi$
$930$	0	0
$931$	− 18.0000i	− 0.589926i
$932$	10.0000	0.327561
$933$	0	0
$934$	− 6.00000i	− 0.196326i
$935$	−14.0000	−0.457849
$936$	0	0
$937$	−49.0000	−1.60076	−0.800380	−	0.599493i	$-0.795369\pi$
$937$	−49.0000	−1.60076	−0.800380	+	0.599493i	$0.795369\pi$
$938$	4.00000i	0.130605i
$939$	0	0
$940$	−6.00000	−0.195698
$941$	38.0000i	1.23876i	0.785090	+	0.619382i	$0.212617\pi$
$941$	38.0000i	1.23876i	−0.785090	+	0.619382i	$0.787383\pi$
$942$	0	0
$943$	54.0000i	1.75848i
$944$	0	0
$945$	0	0
$946$	12.0000	0.390154
$947$	− 48.0000i	− 1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$	− 48.0000i	− 1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$	0	0
$949$	0	0
$950$	−24.0000	−0.778663
$951$	0	0
$952$	42.0000	1.36123
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	4.00000i	0.129437i
$956$	− 30.0000i	− 0.970269i
$957$	0	0
$958$	24.0000	0.775405
$959$	−6.00000	−0.193750
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	− 7.00000i	− 0.225455i
$965$	−9.00000	−0.289720
$966$	0	0
$967$	2.00000i	0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$	2.00000i	0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	− 21.0000i	− 0.674966i
$969$	0	0
$970$	− 2.00000i	− 0.0642161i
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	− 24.0000i	− 0.769405i
$974$	18.0000	0.576757
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	− 33.0000i	− 1.05576i	−0.849318	−	0.527882i	$-0.822986\pi$
$977$	− 33.0000i	− 1.05576i	0.849318	−	0.527882i	$-0.177014\pi$
$978$	0	0
$979$	−28.0000	−0.894884
$980$	3.00000i	0.0958315i
$981$	0	0
$982$	− 6.00000i	− 0.191468i
$983$	4.00000i	0.127580i	0.997963	+	0.0637901i	$0.0203188\pi$
$983$	4.00000i	0.127580i	−0.997963	+	0.0637901i	$0.979681\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	7.00000i	0.222925i
$987$	0	0
$988$	0	0
$989$	36.0000	1.14473
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	20.0000	0.635001
$993$	0	0
$994$	12.0000i	0.380617i
$995$	− 14.0000i	− 0.443830i
$996$	0	0
$997$	−35.0000	−1.10846	−0.554231	−	0.832363i	$-0.686987\pi$
$997$	−35.0000	−1.10846	−0.554231	+	0.832363i	$0.686987\pi$
$998$	24.0000	0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.c.1351.1		2
3.2	odd	2		507.2.b.b.337.2		2
13.2	odd	12		117.2.g.b.100.1		2
13.5	odd	4		1521.2.a.a.1.1		1
13.6	odd	12		117.2.g.b.55.1		2
13.8	odd	4		1521.2.a.d.1.1		1
13.12	even	2	inner	1521.2.b.c.1351.2		2
39.2	even	12		39.2.e.a.22.1	yes	2
39.5	even	4		507.2.a.c.1.1		1
39.8	even	4		507.2.a.b.1.1		1
39.11	even	12		507.2.e.c.22.1		2
39.17	odd	6		507.2.j.d.361.1		4
39.20	even	12		507.2.e.c.484.1		2
39.23	odd	6		507.2.j.d.316.2		4
39.29	odd	6		507.2.j.d.316.1		4
39.32	even	12		39.2.e.a.16.1	✓	2
39.35	odd	6		507.2.j.d.361.2		4
39.38	odd	2		507.2.b.b.337.1		2
52.15	even	12		1872.2.t.j.1153.1		2
52.19	even	12		1872.2.t.j.289.1		2
156.47	odd	4		8112.2.a.bc.1.1		1
156.71	odd	12		624.2.q.c.289.1		2
156.83	odd	4		8112.2.a.w.1.1		1
156.119	odd	12		624.2.q.c.529.1		2
195.2	odd	12		975.2.bb.d.724.1		4
195.32	odd	12		975.2.bb.d.874.2		4
195.119	even	12		975.2.i.f.451.1		2
195.149	even	12		975.2.i.f.601.1		2
195.158	odd	12		975.2.bb.d.724.2		4
195.188	odd	12		975.2.bb.d.874.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.a.16.1	✓	2	39.32	even	12
39.2.e.a.22.1	yes	2	39.2	even	12
117.2.g.b.55.1		2	13.6	odd	12
117.2.g.b.100.1		2	13.2	odd	12
507.2.a.b.1.1		1	39.8	even	4
507.2.a.c.1.1		1	39.5	even	4
507.2.b.b.337.1		2	39.38	odd	2
507.2.b.b.337.2		2	3.2	odd	2
507.2.e.c.22.1		2	39.11	even	12
507.2.e.c.484.1		2	39.20	even	12
507.2.j.d.316.1		4	39.29	odd	6
507.2.j.d.316.2		4	39.23	odd	6
507.2.j.d.361.1		4	39.17	odd	6
507.2.j.d.361.2		4	39.35	odd	6
624.2.q.c.289.1		2	156.71	odd	12
624.2.q.c.529.1		2	156.119	odd	12
975.2.i.f.451.1		2	195.119	even	12
975.2.i.f.601.1		2	195.149	even	12
975.2.bb.d.724.1		4	195.2	odd	12
975.2.bb.d.724.2		4	195.158	odd	12
975.2.bb.d.874.1		4	195.188	odd	12
975.2.bb.d.874.2		4	195.32	odd	12
1521.2.a.a.1.1		1	13.5	odd	4
1521.2.a.d.1.1		1	13.8	odd	4
1521.2.b.c.1351.1		2	1.1	even	1	trivial
1521.2.b.c.1351.2		2	13.12	even	2	inner
1872.2.t.j.289.1		2	52.19	even	12
1872.2.t.j.1153.1		2	52.15	even	12
8112.2.a.w.1.1		1	156.83	odd	4
8112.2.a.bc.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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} +1.00000i q^{5} -2.00000i q^{7} -3.00000i q^{8} +1.00000 q^{10} -2.00000i q^{11} -2.00000 q^{14} -1.00000 q^{16} -7.00000 q^{17} -6.00000i q^{19} +1.00000i q^{20} -2.00000 q^{22} -6.00000 q^{23} +4.00000 q^{25} -2.00000i q^{28} +1.00000 q^{29} +4.00000i q^{31} -5.00000i q^{32} +7.00000i q^{34} +2.00000 q^{35} -1.00000i q^{37} -6.00000 q^{38} +3.00000 q^{40} -9.00000i q^{41} -6.00000 q^{43} -2.00000i q^{44} +6.00000i q^{46} +6.00000i q^{47} +3.00000 q^{49} -4.00000i q^{50} +9.00000 q^{53} +2.00000 q^{55} -6.00000 q^{56} -1.00000i q^{58} +1.00000 q^{61} +4.00000 q^{62} -7.00000 q^{64} -2.00000i q^{67} -7.00000 q^{68} -2.00000i q^{70} -6.00000i q^{71} -11.0000i q^{73} -1.00000 q^{74} -6.00000i q^{76} -4.00000 q^{77} -4.00000 q^{79} -1.00000i q^{80} -9.00000 q^{82} +14.0000i q^{83} -7.00000i q^{85} +6.00000i q^{86} -6.00000 q^{88} -14.0000i q^{89} -6.00000 q^{92} +6.00000 q^{94} +6.00000 q^{95} -2.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{10} - 4 q^{14} - 2 q^{16} - 14 q^{17} - 4 q^{22} - 12 q^{23} + 8 q^{25} + 2 q^{29} + 4 q^{35} - 12 q^{38} + 6 q^{40} - 12 q^{43} + 6 q^{49} + 18 q^{53} + 4 q^{55} - 12 q^{56} + 2 q^{61}+ \cdots + 12 q^{95}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^10 - 4 * q^14 - 2 * q^16 - 14 * q^17 - 4 * q^22 - 12 * q^23 + 8 * q^25 + 2 * q^29 + 4 * q^35 - 12 * q^38 + 6 * q^40 - 12 * q^43 + 6 * q^49 + 18 * q^53 + 4 * q^55 - 12 * q^56 + 2 * q^61 + 8 * q^62 - 14 * q^64 - 14 * q^68 - 2 * q^74 - 8 * q^77 - 8 * q^79 - 18 * q^82 - 12 * q^88 - 12 * q^92 + 12 * q^94 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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