LMFDB - Embedded newform 5400.2.f.f.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5400,2,Mod(649,5400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5400.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5400.f (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:	$43.1192170915$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5400.649
Dual form			5400.2.f.f.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{7} -4.00000 q^{11} -2.00000i q^{13} +5.00000i q^{17} +5.00000 q^{19} -1.00000i q^{23} -2.00000 q^{29} +7.00000 q^{31} +6.00000i q^{37} +4.00000i q^{43} +4.00000i q^{47} +3.00000 q^{49} -9.00000i q^{53} +14.0000 q^{59} -11.0000 q^{61} -14.0000i q^{67} -12.0000i q^{73} +8.00000i q^{77} +3.00000 q^{79} +1.00000i q^{83} -4.00000 q^{91} -16.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{11} + 10 q^{19} - 4 q^{29} + 14 q^{31} + 6 q^{49} + 28 q^{59} - 22 q^{61} + 6 q^{79} - 8 q^{91}+O(q^{100}) $ 2 * q - 8 * q^11 + 10 * q^19 - 4 * q^29 + 14 * q^31 + 6 * q^49 + 28 * q^59 - 22 * q^61 + 6 * q^79 - 8 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$	$2701$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.00000i	1.21268i	0.795206	+	0.606339i	$0.207363\pi$
$17$	5.00000i	1.21268i	−0.795206	+	0.606339i	$0.792637\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.00000i	− 0.208514i	−0.994550	−	0.104257i	$-0.966753\pi$
$23$	− 1.00000i	− 0.208514i	0.994550	−	0.104257i	$-0.0332465\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 14.0000i	− 1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	− 14.0000i	− 1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 12.0000i	− 1.40449i	−0.711934	−	0.702247i	$-0.752180\pi$
$73$	− 12.0000i	− 1.40449i	0.711934	−	0.702247i	$-0.247820\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.00000i	0.911685i
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.00000i	0.109764i	0.998493	+	0.0548821i	$0.0174783\pi$
$83$	1.00000i	0.109764i	−0.998493	+	0.0548821i	$0.982522\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 16.0000i	− 1.62455i	−0.583272	−	0.812277i	$-0.698228\pi$
$97$	− 16.0000i	− 1.62455i	0.583272	−	0.812277i	$-0.301772\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	19.0000	1.81987	0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000	1.81987	0.909935	+	0.414751i	$0.136131\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.0000	0.916698
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	− 10.0000i	− 0.867110i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.0000i	− 1.45241i	−0.687479	−	0.726204i	$-0.741283\pi$
$137$	− 17.0000i	− 1.45241i	0.687479	−	0.726204i	$-0.258717\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00000i	0.668994i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 16.0000i	− 1.27694i	−0.769647	−	0.638470i	$-0.779568\pi$
$157$	− 16.0000i	− 1.27694i	0.769647	−	0.638470i	$-0.220432\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	− 14.0000i	− 1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$	− 14.0000i	− 1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.0000i	0.988372i	0.869356	+	0.494186i	$0.164534\pi$
$173$	13.0000i	0.988372i	−0.869356	+	0.494186i	$0.835466\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 20.0000i	− 1.46254i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.00000i	− 0.356235i	−0.984009	−	0.178118i	$-0.942999\pi$
$197$	− 5.00000i	− 0.356235i	0.984009	−	0.178118i	$-0.0570008\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	4.00000i	0.280745i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−20.0000	−1.38343
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 14.0000i	− 0.950382i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.0000	0.672673
$222$	0	0
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	0	0
$229$	29.0000	1.91637	0.958187	−	0.286143i	$-0.0923732\pi$
$229$	29.0000	1.91637	0.958187	+	0.286143i	$0.0923732\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	11.0000	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 10.0000i	− 0.636285i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	4.00000i	0.251478i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.0000i	1.68421i	0.539311	+	0.842107i	$0.318685\pi$
$257$	27.0000i	1.68421i	−0.539311	+	0.842107i	$0.681315\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−9.00000	−0.546711	−0.273356	−	0.961913i	$-0.588134\pi$
$271$	−9.00000	−0.546711	−0.273356	+	0.961913i	$0.588134\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.00000i	0.240337i	0.992754	+	0.120168i	$0.0383434\pi$
$277$	4.00000i	0.240337i	−0.992754	+	0.120168i	$0.961657\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	0	0
$283$	8.00000i	0.475551i	0.971320	+	0.237775i	$0.0764182\pi$
$283$	8.00000i	0.475551i	−0.971320	+	0.237775i	$0.923582\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 7.00000i	− 0.408944i	−0.978872	−	0.204472i	$-0.934452\pi$
$293$	− 7.00000i	− 0.408944i	0.978872	−	0.204472i	$-0.0655478\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 22.0000i	− 1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$	− 22.0000i	− 1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.0000	1.24751	0.623753	−	0.781622i	$-0.285607\pi$
$311$	22.0000	1.24751	0.623753	+	0.781622i	$0.285607\pi$
$312$	0	0
$313$	8.00000i	0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$	8.00000i	0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 27.0000i	− 1.51647i	−0.651981	−	0.758236i	$-0.726062\pi$
$317$	− 27.0000i	− 1.51647i	0.651981	−	0.758236i	$-0.273938\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	25.0000i	1.39104i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−28.0000	−1.51629
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.0000i	1.07366i	0.843692	+	0.536828i	$0.180378\pi$
$347$	20.0000i	1.07366i	−0.843692	+	0.536828i	$0.819622\pi$
$348$	0	0
$349$	3.00000	0.160586	0.0802932	−	0.996771i	$-0.474414\pi$
$349$	3.00000	0.160586	0.0802932	+	0.996771i	$0.474414\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.0000i	1.14839i	0.818718	+	0.574195i	$0.194685\pi$
$367$	22.0000i	1.14839i	−0.818718	+	0.574195i	$0.805315\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−18.0000	−0.934513
$372$	0	0
$373$	− 32.0000i	− 1.65690i	−0.560065	−	0.828449i	$-0.689224\pi$
$373$	− 32.0000i	− 1.65690i	0.560065	−	0.828449i	$-0.310776\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000i	0.206010i
$378$	0	0
$379$	23.0000	1.18143	0.590715	−	0.806880i	$-0.298846\pi$
$379$	23.0000	1.18143	0.590715	+	0.806880i	$0.298846\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	27.0000i	1.37964i	0.723983	+	0.689818i	$0.242309\pi$
$383$	27.0000i	1.37964i	−0.723983	+	0.689818i	$0.757691\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	− 14.0000i	− 0.697390i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 24.0000i	− 1.18964i
$408$	0	0
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 28.0000i	− 1.37779i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	3.00000	0.146211	0.0731055	−	0.997324i	$-0.476709\pi$
$421$	3.00000	0.146211	0.0731055	+	0.997324i	$0.476709\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	22.0000i	1.06465i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	0	0
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.00000i	− 0.239182i
$438$	0	0
$439$	17.0000	0.811366	0.405683	−	0.914014i	$-0.367034\pi$
$439$	17.0000	0.811366	0.405683	+	0.914014i	$0.367034\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 39.0000i	− 1.85295i	−0.376361	−	0.926473i	$-0.622825\pi$
$443$	− 39.0000i	− 1.85295i	0.376361	−	0.926473i	$-0.377175\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.0000i	− 1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$	− 28.0000i	− 1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	− 18.0000i	− 0.836531i	−0.908325	−	0.418265i	$-0.862638\pi$
$463$	− 18.0000i	− 0.836531i	0.908325	−	0.418265i	$-0.137362\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 13.0000i	− 0.601568i	−0.953692	−	0.300784i	$-0.902752\pi$
$467$	− 13.0000i	− 0.601568i	0.953692	−	0.300784i	$-0.0972484\pi$
$468$	0	0
$469$	−28.0000	−1.29292
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 40.0000i	− 1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$	− 40.0000i	− 1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	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
$493$	− 10.0000i	− 0.450377i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	25.0000	1.11915	0.559577	−	0.828778i	$-0.310964\pi$
$499$	25.0000	1.11915	0.559577	+	0.828778i	$0.310964\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.0000i	0.668817i	0.942428	+	0.334408i	$0.108537\pi$
$503$	15.0000i	0.668817i	−0.942428	+	0.334408i	$0.891463\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−40.0000	−1.75243	−0.876216	−	0.481919i	$-0.839940\pi$
$521$	−40.0000	−1.75243	−0.876216	+	0.481919i	$0.839940\pi$
$522$	0	0
$523$	10.0000i	0.437269i	0.975807	+	0.218635i	$0.0701603\pi$
$523$	10.0000i	0.437269i	−0.975807	+	0.218635i	$0.929840\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	35.0000i	1.52462i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.0000	−0.426014
$552$	0	0
$553$	− 6.00000i	− 0.255146i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10.0000i	− 0.423714i	−0.977301	−	0.211857i	$-0.932049\pi$
$557$	− 10.0000i	− 0.423714i	0.977301	−	0.211857i	$-0.0679510\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−5.00000	−0.209243	−0.104622	−	0.994512i	$-0.533363\pi$
$571$	−5.00000	−0.209243	−0.104622	+	0.994512i	$0.533363\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.0000i	0.666089i	0.942911	+	0.333044i	$0.108076\pi$
$577$	16.0000i	0.666089i	−0.942911	+	0.333044i	$0.891924\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.00000	0.0829740
$582$	0	0
$583$	36.0000i	1.49097i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.0000i	− 0.619116i	−0.950881	−	0.309558i	$-0.899819\pi$
$587$	− 15.0000i	− 0.619116i	0.950881	−	0.309558i	$-0.100181\pi$
$588$	0	0
$589$	35.0000	1.44215
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.00000i	0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$	9.00000i	0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	0	0
$601$	−11.0000	−0.448699	−0.224350	−	0.974509i	$-0.572026\pi$
$601$	−11.0000	−0.448699	−0.224350	+	0.974509i	$0.572026\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.0000i	1.46119i	0.682808	+	0.730597i	$0.260758\pi$
$607$	36.0000i	1.46119i	−0.682808	+	0.730597i	$0.739242\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.0000i	1.32853i	0.747497	+	0.664265i	$0.231255\pi$
$617$	33.0000i	1.32853i	−0.747497	+	0.664265i	$0.768745\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−30.0000	−1.19618
$630$	0	0
$631$	13.0000	0.517522	0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000	0.517522	0.258761	+	0.965941i	$0.416686\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	6.00000i	0.236617i	0.992977	+	0.118308i	$0.0377472\pi$
$643$	6.00000i	0.236617i	−0.992977	+	0.118308i	$0.962253\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.0000i	1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$	33.0000i	1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$	0	0
$649$	−56.0000	−2.19819
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 41.0000i	− 1.60445i	−0.597019	−	0.802227i	$-0.703648\pi$
$653$	− 41.0000i	− 1.60445i	0.597019	−	0.802227i	$-0.296352\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.0000	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.00000i	0.0774403i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	44.0000	1.69860
$672$	0	0
$673$	30.0000i	1.15642i	0.815890	+	0.578208i	$0.196248\pi$
$673$	30.0000i	1.15642i	−0.815890	+	0.578208i	$0.803752\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2.00000i	− 0.0768662i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	− 2.00000i	− 0.0768662i	0.999261	−	0.0384331i	$-0.0122367\pi$
$678$	0	0
$679$	−32.0000	−1.22805
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−18.0000	−0.685745
$690$	0	0
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	30.0000i	1.13147i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.0000i	0.902613i
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 7.00000i	− 0.262152i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	46.0000	1.71551	0.857755	−	0.514058i	$-0.171858\pi$
$719$	46.0000	1.71551	0.857755	+	0.514058i	$0.171858\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−20.0000	−0.739727
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	56.0000i	2.06279i
$738$	0	0
$739$	15.0000	0.551784	0.275892	−	0.961189i	$-0.411027\pi$
$739$	15.0000	0.551784	0.275892	+	0.961189i	$0.411027\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−3.00000	−0.109472	−0.0547358	−	0.998501i	$-0.517432\pi$
$751$	−3.00000	−0.109472	−0.0547358	+	0.998501i	$0.517432\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	− 38.0000i	− 1.37569i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 28.0000i	− 1.01102i
$768$	0	0
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15.0000i	0.539513i	0.962929	+	0.269756i	$0.0869431\pi$
$773$	15.0000i	0.539513i	−0.962929	+	0.269756i	$0.913057\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	40.0000i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$	40.0000i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	22.0000i	0.781243i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.00000i	0.106265i	0.998587	+	0.0531327i	$0.0169206\pi$
$797$	3.00000i	0.106265i	−0.998587	+	0.0531327i	$0.983079\pi$
$798$	0	0
$799$	−20.0000	−0.707549
$800$	0	0
$801$	0	0
$802$	0	0
$803$	48.0000i	1.69388i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20.0000i	0.699711i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	0	0
$823$	− 20.0000i	− 0.697156i	−0.937280	−	0.348578i	$-0.886665\pi$
$823$	− 20.0000i	− 0.697156i	0.937280	−	0.348578i	$-0.113335\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 39.0000i	− 1.35616i	−0.734987	−	0.678081i	$-0.762812\pi$
$827$	− 39.0000i	− 1.35616i	0.734987	−	0.678081i	$-0.237188\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	15.0000i	0.519719i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	− 28.0000i	− 0.958702i	−0.877623	−	0.479351i	$-0.840872\pi$
$853$	− 28.0000i	− 0.958702i	0.877623	−	0.479351i	$-0.159128\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.0000i	0.717346i	0.933463	+	0.358673i	$0.116771\pi$
$857$	21.0000i	0.717346i	−0.933463	+	0.358673i	$0.883229\pi$
$858$	0	0
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 33.0000i	− 1.12333i	−0.827364	−	0.561667i	$-0.810160\pi$
$863$	− 33.0000i	− 1.12333i	0.827364	−	0.561667i	$-0.189840\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−28.0000	−0.948744
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.0000i	1.08056i	0.841484	+	0.540282i	$0.181682\pi$
$877$	32.0000i	1.08056i	−0.841484	+	0.540282i	$0.818318\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	− 22.0000i	− 0.740359i	−0.928960	−	0.370179i	$-0.879296\pi$
$883$	− 22.0000i	− 0.740359i	0.928960	−	0.370179i	$-0.120704\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 23.0000i	− 0.772264i	−0.922443	−	0.386132i	$-0.873811\pi$
$887$	− 23.0000i	− 0.772264i	0.922443	−	0.386132i	$-0.126189\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	20.0000i	0.669274i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.0000	−0.466926
$900$	0	0
$901$	45.0000	1.49917
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.00000	0.0662630	0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000	0.0662630	0.0331315	+	0.999451i	$0.489452\pi$
$912$	0	0
$913$	− 4.00000i	− 0.132381i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 36.0000i	− 1.18882i
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8.00000	−0.262471	−0.131236	−	0.991351i	$-0.541894\pi$
$929$	−8.00000	−0.262471	−0.131236	+	0.991351i	$0.541894\pi$
$930$	0	0
$931$	15.0000	0.491605
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	56.0000i	1.82944i	0.404088	+	0.914720i	$0.367589\pi$
$937$	56.0000i	1.82944i	−0.404088	+	0.914720i	$0.632411\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 53.0000i	− 1.72227i	−0.508378	−	0.861134i	$-0.669755\pi$
$947$	− 53.0000i	− 1.72227i	0.508378	−	0.861134i	$-0.330245\pi$
$948$	0	0
$949$	−24.0000	−0.779073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34.0000	−1.09792
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	30.0000i	0.964735i	0.875969	+	0.482367i	$0.160223\pi$
$967$	30.0000i	0.964735i	−0.875969	+	0.482367i	$0.839777\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14.0000	−0.449281	−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000	−0.449281	−0.224641	+	0.974442i	$0.572121\pi$
$972$	0	0
$973$	24.0000i	0.769405i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.00000i	0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$	2.00000i	0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.00000i	0.0956851i	0.998855	+	0.0478426i	$0.0152346\pi$
$983$	3.00000i	0.0956851i	−0.998855	+	0.0478426i	$0.984765\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−55.0000	−1.74713	−0.873566	−	0.486705i	$-0.838199\pi$
$991$	−55.0000	−1.74713	−0.873566	+	0.486705i	$0.838199\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	48.0000i	1.52018i	0.649821	+	0.760088i	$0.274844\pi$
$997$	48.0000i	1.52018i	−0.649821	+	0.760088i	$0.725156\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5400.2.f.f.649.1		2
3.2	odd	2		5400.2.f.x.649.1		2
5.2	odd	4		1080.2.a.e.1.1	✓	1
5.3	odd	4		5400.2.a.j.1.1		1
5.4	even	2	inner	5400.2.f.f.649.2		2
15.2	even	4		1080.2.a.l.1.1	yes	1
15.8	even	4		5400.2.a.q.1.1		1
15.14	odd	2		5400.2.f.x.649.2		2
20.7	even	4		2160.2.a.e.1.1		1
40.27	even	4		8640.2.a.bi.1.1		1
40.37	odd	4		8640.2.a.cd.1.1		1
45.2	even	12		3240.2.q.b.1081.1		2
45.7	odd	12		3240.2.q.p.1081.1		2
45.22	odd	12		3240.2.q.p.2161.1		2
45.32	even	12		3240.2.q.b.2161.1		2
60.47	odd	4		2160.2.a.m.1.1		1
120.77	even	4		8640.2.a.t.1.1		1
120.107	odd	4		8640.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.e.1.1	✓	1	5.2	odd	4
1080.2.a.l.1.1	yes	1	15.2	even	4
2160.2.a.e.1.1		1	20.7	even	4
2160.2.a.m.1.1		1	60.47	odd	4
3240.2.q.b.1081.1		2	45.2	even	12
3240.2.q.b.2161.1		2	45.32	even	12
3240.2.q.p.1081.1		2	45.7	odd	12
3240.2.q.p.2161.1		2	45.22	odd	12
5400.2.a.j.1.1		1	5.3	odd	4
5400.2.a.q.1.1		1	15.8	even	4
5400.2.f.f.649.1		2	1.1	even	1	trivial
5400.2.f.f.649.2		2	5.4	even	2	inner
5400.2.f.x.649.1		2	3.2	odd	2
5400.2.f.x.649.2		2	15.14	odd	2
8640.2.a.k.1.1		1	120.107	odd	4
8640.2.a.t.1.1		1	120.77	even	4
8640.2.a.bi.1.1		1	40.27	even	4
8640.2.a.cd.1.1		1	40.37	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:	\( N \)	\(=\)	\( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5400.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{7} -4.00000 q^{11} -2.00000i q^{13} +5.00000i q^{17} +5.00000 q^{19} -1.00000i q^{23} -2.00000 q^{29} +7.00000 q^{31} +6.00000i q^{37} +4.00000i q^{43} +4.00000i q^{47} +3.00000 q^{49} -9.00000i q^{53} +14.0000 q^{59} -11.0000 q^{61} -14.0000i q^{67} -12.0000i q^{73} +8.00000i q^{77} +3.00000 q^{79} +1.00000i q^{83} -4.00000 q^{91} -16.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{11} + 10 q^{19} - 4 q^{29} + 14 q^{31} + 6 q^{49} + 28 q^{59} - 22 q^{61} + 6 q^{79} - 8 q^{91}+O(q^{100}) \) 2 * q - 8 * q^11 + 10 * q^19 - 4 * q^29 + 14 * q^31 + 6 * q^49 + 28 * q^59 - 22 * q^61 + 6 * q^79 - 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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