LMFDB - Embedded newform 5376.2.c.u.2689.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5376,2,Mod(2689,5376)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5376, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5376.2689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5376 = 2^{8} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5376.c (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:	$42.9275761266$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2689.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5376.2689
Dual form			5376.2.c.u.2689.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} -2.00000 q^{15} +2.00000 q^{17} +4.00000i q^{19} -1.00000i q^{21} +1.00000 q^{25} +1.00000i q^{27} +6.00000i q^{29} -2.00000i q^{35} -6.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} -8.00000i q^{43} +2.00000i q^{45} +8.00000 q^{47} +1.00000 q^{49} -2.00000i q^{51} -6.00000i q^{53} +4.00000 q^{57} +12.0000i q^{59} +10.0000i q^{61} -1.00000 q^{63} +4.00000 q^{65} +16.0000i q^{67} +8.00000 q^{71} +6.00000 q^{73} -1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -4.00000i q^{85} +6.00000 q^{87} +14.0000 q^{89} +2.00000i q^{91} +8.00000 q^{95} -6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^7 - 2 * q^9	$ 2 q + 2 q^{7} - 2 q^{9} - 4 q^{15} + 4 q^{17} + 2 q^{25} + 4 q^{39} + 12 q^{41} + 16 q^{47} + 2 q^{49} + 8 q^{57} - 2 q^{63} + 8 q^{65} + 16 q^{71} + 12 q^{73} + 16 q^{79} + 2 q^{81} + 12 q^{87} + 28 q^{89} + 16 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 2 * q^9 - 4 * q^15 + 4 * q^17 + 2 * q^25 + 4 * q^39 + 12 * q^41 + 16 * q^47 + 2 * q^49 + 8 * q^57 - 2 * q^63 + 8 * q^65 + 16 * q^71 + 12 * q^73 + 16 * q^79 + 2 * q^81 + 12 * q^87 + 28 * q^89 + 16 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times$.

$n$	$1793$	$2815$	$4609$	$5125$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 2.00000i	− 0.280056i
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	12.0000i	1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$	12.0000i	1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$	0	0
$61$	10.0000i	1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$	10.0000i	1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	16.0000i	1.95471i	0.211604	+	0.977356i	$0.432131\pi$
$67$	16.0000i	1.95471i	−0.211604	+	0.977356i	$0.567869\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	− 4.00000i	− 0.433861i
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	2.00000i	0.209657i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.00000i	− 0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	− 2.00000i	− 0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	− 18.0000i	− 1.72409i	−0.506834	−	0.862044i	$-0.669184\pi$
$109$	− 18.0000i	− 1.72409i	0.506834	−	0.862044i	$-0.330816\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	− 12.0000i	− 1.07331i
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	− 12.0000i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 12.0000i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	4.00000i	0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$	4.00000i	0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$	0	0
$141$	− 8.00000i	− 0.673722i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	10.0000i	0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$	10.0000i	0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.00000i	0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$	8.00000i	0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	24.0000i	1.79384i	0.442189	+	0.896922i	$0.354202\pi$
$179$	24.0000i	1.79384i	−0.442189	+	0.896922i	$0.645798\pi$
$180$	0	0
$181$	− 10.0000i	− 0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$	− 10.0000i	− 0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.00000i	0.0727393i
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	− 4.00000i	− 0.286446i
$196$	0	0
$197$	− 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
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	− 12.0000i	− 0.838116i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 6.00000i	− 0.405442i
$220$	0	0
$221$	4.00000i	0.269069i
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	− 10.0000i	− 0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$	− 10.0000i	− 0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	− 16.0000i	− 1.04372i
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 2.00000i	− 0.127775i
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	20.0000i	1.26239i	0.775625	+	0.631194i	$0.217435\pi$
$251$	20.0000i	1.26239i	−0.775625	+	0.631194i	$0.782565\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	− 6.00000i	− 0.372822i
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	− 14.0000i	− 0.856786i
$268$	0	0
$269$	− 14.0000i	− 0.853595i	−0.904347	−	0.426798i	$-0.859642\pi$
$269$	− 14.0000i	− 0.853595i	0.904347	−	0.426798i	$-0.140358\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	6.00000i	0.351726i
$292$	0	0
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 8.00000i	− 0.461112i
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	20.0000	1.14520
$306$	0	0
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	16.0000i	0.910208i
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	0	0
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	0	0
$315$	2.00000i	0.112687i
$316$	0	0
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	2.00000i	0.110940i
$326$	0	0
$327$	−18.0000	−0.995402
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	8.00000i	0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$	8.00000i	0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	32.0000	1.74835
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	0	0
$339$	− 2.00000i	− 0.108625i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.0000i	0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$	16.0000i	0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$	0	0
$349$	2.00000i	0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$	2.00000i	0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	− 16.0000i	− 0.849192i
$356$	0	0
$357$	− 2.00000i	− 0.105851i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 11.0000i	− 0.577350i
$364$	0	0
$365$	− 12.0000i	− 0.628109i
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	− 6.00000i	− 0.311504i
$372$	0	0
$373$	2.00000i	0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$	2.00000i	0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	− 24.0000i	− 1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$	− 24.0000i	− 1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$	0	0
$381$	16.0000i	0.819705i
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.00000i	0.406663i
$388$	0	0
$389$	− 14.0000i	− 0.709828i	−0.934899	−	0.354914i	$-0.884510\pi$
$389$	− 14.0000i	− 0.709828i	0.934899	−	0.354914i	$-0.115490\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	− 16.0000i	− 0.805047i
$396$	0	0
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 2.00000i	− 0.0993808i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	− 6.00000i	− 0.295958i
$412$	0	0
$413$	12.0000i	0.590481i
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	− 20.0000i	− 0.977064i	−0.872546	−	0.488532i	$-0.837533\pi$
$419$	− 20.0000i	− 0.977064i	0.872546	−	0.488532i	$-0.162467\pi$
$420$	0	0
$421$	− 14.0000i	− 0.682318i	−0.940006	−	0.341159i	$-0.889181\pi$
$421$	− 14.0000i	− 0.682318i	0.940006	−	0.341159i	$-0.110819\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	10.0000i	0.483934i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	− 12.0000i	− 0.575356i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	8.00000i	0.380091i	0.981775	+	0.190046i	$0.0608636\pi$
$443$	8.00000i	0.380091i	−0.981775	+	0.190046i	$0.939136\pi$
$444$	0	0
$445$	− 28.0000i	− 1.32733i
$446$	0	0
$447$	10.0000	0.472984
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	16.0000i	0.751746i
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	0	0
$459$	2.00000i	0.0933520i
$460$	0	0
$461$	2.00000i	0.0931493i	0.998915	+	0.0465746i	$0.0148305\pi$
$461$	2.00000i	0.0931493i	−0.998915	+	0.0465746i	$0.985169\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.0000i	0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$	20.0000i	0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	0	0
$469$	16.0000i	0.738811i
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000i	0.183533i
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.0000i	0.544892i
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	− 8.00000i	− 0.361035i	−0.983572	−	0.180517i	$-0.942223\pi$
$491$	− 8.00000i	− 0.361035i	0.983572	−	0.180517i	$-0.0577772\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	− 16.0000i	− 0.716258i	−0.933672	−	0.358129i	$-0.883415\pi$
$499$	− 16.0000i	− 0.716258i	0.933672	−	0.358129i	$-0.116585\pi$
$500$	0	0
$501$	− 8.00000i	− 0.357414i
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	− 30.0000i	− 1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$509$	− 30.0000i	− 1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	32.0000i	1.41009i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	0	0
$525$	− 1.00000i	− 0.0436436i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 12.0000i	− 0.520756i
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	16.0000	0.691740
$536$	0	0
$537$	24.0000	1.03568
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.0000i	0.945854i	0.881102	+	0.472927i	$0.156803\pi$
$541$	22.0000i	0.945854i	−0.881102	+	0.472927i	$0.843197\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−36.0000	−1.54207
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	0	0
$549$	− 10.0000i	− 0.426790i
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	12.0000i	0.509372i
$556$	0	0
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	− 4.00000i	− 0.168281i
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	40.0000i	1.67395i	0.547243	+	0.836974i	$0.315677\pi$
$571$	40.0000i	1.67395i	−0.547243	+	0.836974i	$0.684323\pi$
$572$	0	0
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−46.0000	−1.91501	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−46.0000	−1.91501	−0.957503	+	0.288425i	$0.906868\pi$
$578$	0	0
$579$	− 2.00000i	− 0.0831172i
$580$	0	0
$581$	− 12.0000i	− 0.497844i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	10.0000	0.410651	0.205325	−	0.978694i	$-0.434175\pi$
$593$	10.0000	0.410651	0.205325	+	0.978694i	$0.434175\pi$
$594$	0	0
$595$	− 4.00000i	− 0.163984i
$596$	0	0
$597$	− 16.0000i	− 0.654836i
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	− 16.0000i	− 0.651570i
$604$	0	0
$605$	− 22.0000i	− 0.894427i
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	16.0000i	0.647291i
$612$	0	0
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.0000	0.560898
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	32.0000i	1.26988i
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−46.0000	−1.81689	−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000	−1.81689	−0.908445	+	0.418004i	$0.862730\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	16.0000i	0.629999i
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.0000i	0.860927i	0.902608	+	0.430463i	$0.141650\pi$
$653$	22.0000i	0.860927i	−0.902608	+	0.430463i	$0.858350\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	48.0000i	1.86981i	0.354892	+	0.934907i	$0.384518\pi$
$659$	48.0000i	1.86981i	−0.354892	+	0.934907i	$0.615482\pi$
$660$	0	0
$661$	− 18.0000i	− 0.700119i	−0.936727	−	0.350059i	$-0.886161\pi$
$661$	− 18.0000i	− 0.700119i	0.936727	−	0.350059i	$-0.113839\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	0	0
$668$	0	0
$669$	8.00000i	0.309298i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	− 12.0000i	− 0.458496i
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	− 28.0000i	− 1.06517i	−0.846376	−	0.532585i	$-0.821221\pi$
$691$	− 28.0000i	− 1.06517i	0.846376	−	0.532585i	$-0.178779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	10.0000i	0.378235i
$700$	0	0
$701$	22.0000i	0.830929i	0.909610	+	0.415464i	$0.136381\pi$
$701$	22.0000i	0.830929i	−0.909610	+	0.415464i	$0.863619\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	− 2.00000i	− 0.0752177i
$708$	0	0
$709$	42.0000i	1.57734i	0.614815	+	0.788672i	$0.289231\pi$
$709$	42.0000i	1.57734i	−0.614815	+	0.788672i	$0.710769\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.0000i	0.597531i
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	− 10.0000i	− 0.371904i
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 16.0000i	− 0.591781i
$732$	0	0
$733$	42.0000i	1.55131i	0.631160	+	0.775653i	$0.282579\pi$
$733$	42.0000i	1.55131i	−0.631160	+	0.775653i	$0.717421\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	0	0
$738$	0	0
$739$	24.0000i	0.882854i	0.897297	+	0.441427i	$0.145528\pi$
$739$	24.0000i	0.882854i	−0.897297	+	0.441427i	$0.854472\pi$
$740$	0	0
$741$	8.00000i	0.293887i
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	20.0000	0.732743
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	8.00000i	0.292314i
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	32.0000i	1.16460i
$756$	0	0
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	0	0
$763$	− 18.0000i	− 0.651644i
$764$	0	0
$765$	4.00000i	0.144620i
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	6.00000i	0.216085i
$772$	0	0
$773$	46.0000i	1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$	46.0000i	1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6.00000	−0.215249
$778$	0	0
$779$	24.0000i	0.859889i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	−28.0000	−0.999363
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	0	0
$789$	24.0000i	0.854423i
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	12.0000i	0.425596i
$796$	0	0
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.0000	−0.492823
$808$	0	0
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	− 52.0000i	− 1.82597i	−0.407997	−	0.912983i	$-0.633772\pi$
$811$	− 52.0000i	− 1.82597i	0.407997	−	0.912983i	$-0.366228\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	0	0
$815$	16.0000	0.560456
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	− 2.00000i	− 0.0698857i
$820$	0	0
$821$	− 22.0000i	− 0.767805i	−0.923374	−	0.383903i	$-0.874580\pi$
$821$	− 22.0000i	− 0.767805i	0.923374	−	0.383903i	$-0.125420\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 24.0000i	− 0.834562i	−0.908778	−	0.417281i	$-0.862983\pi$
$827$	− 24.0000i	− 0.834562i	0.908778	−	0.417281i	$-0.137017\pi$
$828$	0	0
$829$	− 22.0000i	− 0.764092i	−0.924143	−	0.382046i	$-0.875220\pi$
$829$	− 22.0000i	− 0.764092i	0.924143	−	0.382046i	$-0.124780\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	− 16.0000i	− 0.553703i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	− 6.00000i	− 0.206651i
$844$	0	0
$845$	− 18.0000i	− 0.619219i
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	6.00000i	0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$	6.00000i	0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	20.0000i	0.682391i	0.939992	+	0.341196i	$0.110832\pi$
$859$	20.0000i	0.682391i	−0.939992	+	0.341196i	$0.889168\pi$
$860$	0	0
$861$	− 6.00000i	− 0.204479i
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−28.0000	−0.952029
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−32.0000	−1.08428
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	− 12.0000i	− 0.405674i
$876$	0	0
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	0	0
$885$	− 24.0000i	− 0.806751i
$886$	0	0
$887$	56.0000	1.88030	0.940148	−	0.340766i	$-0.110687\pi$
$887$	56.0000	1.88030	0.940148	+	0.340766i	$0.110687\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	48.0000	1.60446
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	32.0000i	1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$	32.0000i	1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$	0	0
$909$	2.00000i	0.0663358i
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 20.0000i	− 0.661180i
$916$	0	0
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	0	0
$923$	16.0000i	0.526646i
$924$	0	0
$925$	− 6.00000i	− 0.197279i
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	4.00000i	0.131095i
$932$	0	0
$933$	− 32.0000i	− 1.04763i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	34.0000i	1.10955i
$940$	0	0
$941$	42.0000i	1.36916i	0.728937	+	0.684580i	$0.240015\pi$
$941$	42.0000i	1.36916i	−0.728937	+	0.684580i	$0.759985\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	48.0000i	1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$	48.0000i	1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	− 48.0000i	− 1.55324i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 8.00000i	− 0.257796i
$964$	0	0
$965$	− 4.00000i	− 0.128765i
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	44.0000i	1.41203i	0.708198	+	0.706014i	$0.249508\pi$
$971$	44.0000i	1.41203i	−0.708198	+	0.706014i	$0.750492\pi$
$972$	0	0
$973$	4.00000i	0.128234i
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	18.0000i	0.574696i
$982$	0	0
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	8.00000	0.253872
$994$	0	0
$995$	− 32.0000i	− 1.01447i
$996$	0	0
$997$	− 34.0000i	− 1.07679i	−0.842692	−	0.538395i	$-0.819031\pi$
$997$	− 34.0000i	− 1.07679i	0.842692	−	0.538395i	$-0.180969\pi$
$998$	0	0
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.c.u.2689.1		2
4.3	odd	2		5376.2.c.o.2689.2		2
8.3	odd	2		5376.2.c.o.2689.1		2
8.5	even	2	inner	5376.2.c.u.2689.2		2
16.3	odd	4		1344.2.a.d.1.1		1
16.5	even	4		672.2.a.d.1.1	✓	1
16.11	odd	4		672.2.a.h.1.1	yes	1
16.13	even	4		1344.2.a.l.1.1		1
48.5	odd	4		2016.2.a.a.1.1		1
48.11	even	4		2016.2.a.b.1.1		1
48.29	odd	4		4032.2.a.bd.1.1		1
48.35	even	4		4032.2.a.bi.1.1		1
112.13	odd	4		9408.2.a.bd.1.1		1
112.27	even	4		4704.2.a.c.1.1		1
112.69	odd	4		4704.2.a.v.1.1		1
112.83	even	4		9408.2.a.cz.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.a.d.1.1	✓	1	16.5	even	4
672.2.a.h.1.1	yes	1	16.11	odd	4
1344.2.a.d.1.1		1	16.3	odd	4
1344.2.a.l.1.1		1	16.13	even	4
2016.2.a.a.1.1		1	48.5	odd	4
2016.2.a.b.1.1		1	48.11	even	4
4032.2.a.bd.1.1		1	48.29	odd	4
4032.2.a.bi.1.1		1	48.35	even	4
4704.2.a.c.1.1		1	112.27	even	4
4704.2.a.v.1.1		1	112.69	odd	4
5376.2.c.o.2689.1		2	8.3	odd	2
5376.2.c.o.2689.2		2	4.3	odd	2
5376.2.c.u.2689.1		2	1.1	even	1	trivial
5376.2.c.u.2689.2		2	8.5	even	2	inner
9408.2.a.bd.1.1		1	112.13	odd	4
9408.2.a.cz.1.1		1	112.83	even	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:	\( N \)	\(=\)	\( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5376.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} -2.00000 q^{15} +2.00000 q^{17} +4.00000i q^{19} -1.00000i q^{21} +1.00000 q^{25} +1.00000i q^{27} +6.00000i q^{29} -2.00000i q^{35} -6.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} -8.00000i q^{43} +2.00000i q^{45} +8.00000 q^{47} +1.00000 q^{49} -2.00000i q^{51} -6.00000i q^{53} +4.00000 q^{57} +12.0000i q^{59} +10.0000i q^{61} -1.00000 q^{63} +4.00000 q^{65} +16.0000i q^{67} +8.00000 q^{71} +6.00000 q^{73} -1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -4.00000i q^{85} +6.00000 q^{87} +14.0000 q^{89} +2.00000i q^{91} +8.00000 q^{95} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^7 - 2 * q^9	\( 2 q + 2 q^{7} - 2 q^{9} - 4 q^{15} + 4 q^{17} + 2 q^{25} + 4 q^{39} + 12 q^{41} + 16 q^{47} + 2 q^{49} + 8 q^{57} - 2 q^{63} + 8 q^{65} + 16 q^{71} + 12 q^{73} + 16 q^{79} + 2 q^{81} + 12 q^{87} + 28 q^{89} + 16 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 2 * q^9 - 4 * q^15 + 4 * q^17 + 2 * q^25 + 4 * q^39 + 12 * q^41 + 16 * q^47 + 2 * q^49 + 8 * q^57 - 2 * q^63 + 8 * q^65 + 16 * q^71 + 12 * q^73 + 16 * q^79 + 2 * q^81 + 12 * q^87 + 28 * q^89 + 16 * q^95 - 12 * q^97

Introduction

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