LMFDB - Embedded newform 8100.2.d.f.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(649,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8100.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$64.6788256372$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	8100.649
Dual form			8100.2.d.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)

$f(q)$	$=$	$q-1.00000i q^{7} +O(q^{10})$	$q-1.00000i q^{7} +4.00000i q^{13} +6.00000i q^{17} -2.00000 q^{19} -3.00000i q^{23} +3.00000 q^{29} -10.0000 q^{31} -10.0000i q^{37} -9.00000 q^{41} +4.00000i q^{43} -9.00000i q^{47} +6.00000 q^{49} -6.00000i q^{53} -6.00000 q^{59} -1.00000 q^{61} +11.0000i q^{67} -12.0000 q^{71} +4.00000i q^{73} +10.0000 q^{79} -9.00000i q^{83} +9.00000 q^{89} +4.00000 q^{91} -10.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 4 q^{19} + 6 q^{29} - 20 q^{31} - 18 q^{41} + 12 q^{49} - 12 q^{59} - 2 q^{61} - 24 q^{71} + 20 q^{79} + 18 q^{89} + 8 q^{91}+O(q^{100}) $ 2 * q - 4 * q^19 + 6 * q^29 - 20 * q^31 - 18 * q^41 + 12 * q^49 - 12 * q^59 - 2 * q^61 - 24 * q^71 + 20 * q^79 + 18 * q^89 + 8 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$.

$n$	$4051$	$6401$	$7777$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	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
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	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
$57$	0	0
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.0000i	1.34386i	0.740613	+	0.671932i	$0.234535\pi$
$67$	11.0000i	1.34386i	−0.740613	+	0.671932i	$0.765465\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9.00000i	− 0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$	− 9.00000i	− 0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$	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
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 7.00000i	− 0.621150i	−0.950549	−	0.310575i	$-0.899478\pi$
$127$	− 7.00000i	− 0.621150i	0.950549	−	0.310575i	$-0.100522\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	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
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.0000i	1.16073i	0.814355	+	0.580367i	$0.197091\pi$
$167$	15.0000i	1.16073i	−0.814355	+	0.580367i	$0.802909\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.00000i	− 0.210559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	10.0000i	0.678844i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	− 11.0000i	− 0.736614i	−0.929704	−	0.368307i	$-0.879937\pi$
$223$	− 11.0000i	− 0.736614i	0.929704	−	0.368307i	$-0.120063\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 30.0000i	− 1.96537i	−0.185296	−	0.982683i	$-0.559325\pi$
$233$	− 30.0000i	− 1.96537i	0.185296	−	0.982683i	$-0.440675\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	0	0
$274$	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
$281$	−33.0000	−1.96861	−0.984307	−	0.176462i	$-0.943535\pi$
$281$	−33.0000	−1.96861	−0.984307	+	0.176462i	$0.943535\pi$
$282$	0	0
$283$	1.00000i	0.0594438i	0.999558	+	0.0297219i	$0.00946217\pi$
$283$	1.00000i	0.0594438i	−0.999558	+	0.0297219i	$0.990538\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.00000i	0.531253i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.0000i	1.65512i	0.561379	+	0.827559i	$0.310271\pi$
$307$	29.0000i	1.65512i	−0.561379	+	0.827559i	$0.689729\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	16.0000i	0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$	16.0000i	0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 12.0000i	− 0.667698i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.00000	−0.496186
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.00000i	0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$	8.00000i	0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	0	0
$349$	−23.0000	−1.23116	−0.615581	−	0.788074i	$-0.711079\pi$
$349$	−23.0000	−1.23116	−0.615581	+	0.788074i	$0.711079\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.0000i	1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$	32.0000i	1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	− 40.0000i	− 1.99254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.00000i	0.295241i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.00000i	0.0483934i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	0	0
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.00000i	0.287019i
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 27.0000i	− 1.28281i	−0.767203	−	0.641404i	$-0.778352\pi$
$443$	− 27.0000i	− 1.28281i	0.767203	−	0.641404i	$-0.221648\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	0	0
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	11.0000	0.507933
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	18.0000i	0.810679i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.0000i	0.538274i
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.00000i	0.133763i	0.997761	+	0.0668817i	$0.0213050\pi$
$503$	3.00000i	0.133763i	−0.997761	+	0.0668817i	$0.978695\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.00000	−0.398918	−0.199459	−	0.979906i	$-0.563918\pi$
$509$	−9.00000	−0.398918	−0.199459	+	0.979906i	$0.563918\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−27.0000	−1.18289	−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000	−1.18289	−0.591446	+	0.806345i	$0.701443\pi$
$522$	0	0
$523$	7.00000i	0.306089i	0.988219	+	0.153044i	$0.0489077\pi$
$523$	7.00000i	0.306089i	−0.988219	+	0.153044i	$0.951092\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 60.0000i	− 2.61364i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 36.0000i	− 1.55933i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 7.00000i	− 0.299298i	−0.988739	−	0.149649i	$-0.952186\pi$
$547$	− 7.00000i	− 0.299298i	0.988739	−	0.149649i	$-0.0478144\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	− 10.0000i	− 0.425243i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 39.0000i	− 1.64365i	−0.569737	−	0.821827i	$-0.692955\pi$
$563$	− 39.0000i	− 1.64365i	0.569737	−	0.821827i	$-0.307045\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	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
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	38.0000i	1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$	38.0000i	1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.00000	−0.373383
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.0000i	1.11441i	0.830375	+	0.557205i	$0.188126\pi$
$587$	27.0000i	1.11441i	−0.830375	+	0.557205i	$0.811874\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7.00000i	− 0.284121i	−0.989858	−	0.142061i	$-0.954627\pi$
$607$	− 7.00000i	− 0.284121i	0.989858	−	0.142061i	$-0.0453728\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	36.0000	1.45640
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 9.00000i	− 0.360577i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	60.0000	2.39236
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	24.0000i	0.950915i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−21.0000	−0.829450	−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000	−0.829450	−0.414725	+	0.909947i	$0.636122\pi$
$642$	0	0
$643$	31.0000i	1.22252i	0.791430	+	0.611260i	$0.209337\pi$
$643$	31.0000i	1.22252i	−0.791430	+	0.611260i	$0.790663\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.00000i	− 0.353827i	−0.984226	−	0.176913i	$-0.943389\pi$
$647$	− 9.00000i	− 0.353827i	0.984226	−	0.176913i	$-0.0566112\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.00000i	− 0.348481i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.0000i	1.07932i	0.841883	+	0.539660i	$0.181447\pi$
$673$	28.0000i	1.07932i	−0.841883	+	0.539660i	$0.818553\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 54.0000i	− 2.04540i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.0000	0.793159	0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000	0.793159	0.396580	+	0.918000i	$0.370197\pi$
$702$	0	0
$703$	20.0000i	0.754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18.0000i	− 0.676960i
$708$	0	0
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	30.0000i	1.12351i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 19.0000i	− 0.704671i	−0.935874	−	0.352335i	$-0.885388\pi$
$727$	− 19.0000i	− 0.704671i	0.935874	−	0.352335i	$-0.114612\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.0000i	1.21065i	0.795977	+	0.605326i	$0.206957\pi$
$743$	33.0000i	1.21065i	−0.795977	+	0.605326i	$0.793043\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.00000	−0.328853
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.0000i	0.726912i	0.931611	+	0.363456i	$0.118403\pi$
$757$	20.0000i	0.726912i	−0.931611	+	0.363456i	$0.881597\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	0	0
$763$	− 19.0000i	− 0.687846i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 24.0000i	− 0.866590i
$768$	0	0
$769$	31.0000	1.11789	0.558944	−	0.829205i	$-0.311207\pi$
$769$	31.0000	1.11789	0.558944	+	0.829205i	$0.311207\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 36.0000i	− 1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$	− 36.0000i	− 1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.0000	0.644917
$780$	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	$-0.747371\pi$
$787$	− 40.0000i	− 1.42585i	0.701242	−	0.712923i	$-0.252629\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.0000i	0.850124i	0.905164	+	0.425062i	$0.139748\pi$
$797$	24.0000i	0.850124i	−0.905164	+	0.425062i	$0.860252\pi$
$798$	0	0
$799$	54.0000	1.91038
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.00000i	− 0.279885i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	0	0
$823$	49.0000i	1.70803i	0.520246	+	0.854016i	$0.325840\pi$
$823$	49.0000i	1.70803i	−0.520246	+	0.854016i	$0.674160\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 27.0000i	− 0.938882i	−0.882964	−	0.469441i	$-0.844455\pi$
$827$	− 27.0000i	− 0.938882i	0.882964	−	0.469441i	$-0.155545\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	36.0000i	1.24733i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.0000i	0.377964i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−30.0000	−1.02839
$852$	0	0
$853$	4.00000i	0.136957i	0.997653	+	0.0684787i	$0.0218145\pi$
$853$	4.00000i	0.136957i	−0.997653	+	0.0684787i	$0.978185\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.0000i	1.63965i	0.572615	+	0.819824i	$0.305929\pi$
$857$	48.0000i	1.63965i	−0.572615	+	0.819824i	$0.694071\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.0000i	0.510606i	0.966861	+	0.255303i	$0.0821752\pi$
$863$	15.0000i	0.510606i	−0.966861	+	0.255303i	$0.917825\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−44.0000	−1.49088
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 4.00000i	− 0.135070i	−0.997717	−	0.0675352i	$-0.978487\pi$
$877$	− 4.00000i	− 0.135070i	0.997717	−	0.0675352i	$-0.0215135\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	0	0
$883$	− 29.0000i	− 0.975928i	−0.872864	−	0.487964i	$-0.837740\pi$
$883$	− 29.0000i	− 0.975928i	0.872864	−	0.487964i	$-0.162260\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18.0000i	0.602347i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−30.0000	−1.00056
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.00000i	− 0.0332045i	−0.999862	−	0.0166022i	$-0.994715\pi$
$907$	− 1.00000i	− 0.0332045i	0.999862	−	0.0166022i	$-0.00528490\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.00000i	0.198137i
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 48.0000i	− 1.57994i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 16.0000i	− 0.522697i	−0.965244	−	0.261349i	$-0.915833\pi$
$937$	− 16.0000i	− 0.522697i	0.965244	−	0.261349i	$-0.0841672\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−45.0000	−1.46696	−0.733479	−	0.679712i	$-0.762105\pi$
$941$	−45.0000	−1.46696	−0.733479	+	0.679712i	$0.762105\pi$
$942$	0	0
$943$	27.0000i	0.879241i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 15.0000i	− 0.487435i	−0.969846	−	0.243717i	$-0.921633\pi$
$947$	− 15.0000i	− 0.487435i	0.969846	−	0.243717i	$-0.0783669\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.0000i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$	24.0000i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.00000i	− 0.0321578i	−0.999871	−	0.0160789i	$-0.994882\pi$
$967$	− 1.00000i	− 0.0321578i	0.999871	−	0.0160789i	$-0.00511830\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	0	0
$973$	20.0000i	0.641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24.0000i	− 0.767828i	−0.923369	−	0.383914i	$-0.874576\pi$
$977$	− 24.0000i	− 0.767828i	0.923369	−	0.383914i	$-0.125424\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9.00000i	0.287055i	0.989646	+	0.143528i	$0.0458446\pi$
$983$	9.00000i	0.287055i	−0.989646	+	0.143528i	$0.954155\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.0000	0.381578
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.d.f.649.1		2
3.2	odd	2		8100.2.d.e.649.1		2
5.2	odd	4		8100.2.a.h.1.1		1
5.3	odd	4		1620.2.a.b.1.1		1
5.4	even	2	inner	8100.2.d.f.649.2		2
9.2	odd	6		900.2.s.a.49.2		4
9.4	even	3		2700.2.s.a.2449.2		4
9.5	odd	6		900.2.s.a.349.1		4
9.7	even	3		2700.2.s.a.1549.1		4
15.2	even	4		8100.2.a.i.1.1		1
15.8	even	4		1620.2.a.e.1.1		1
15.14	odd	2		8100.2.d.e.649.2		2
20.3	even	4		6480.2.a.h.1.1		1
45.2	even	12		900.2.i.a.301.1		2
45.4	even	6		2700.2.s.a.2449.1		4
45.7	odd	12		2700.2.i.a.901.1		2
45.13	odd	12		540.2.i.a.181.1		2
45.14	odd	6		900.2.s.a.349.2		4
45.22	odd	12		2700.2.i.a.1801.1		2
45.23	even	12		180.2.i.a.61.1	✓	2
45.29	odd	6		900.2.s.a.49.1		4
45.32	even	12		900.2.i.a.601.1		2
45.34	even	6		2700.2.s.a.1549.2		4
45.38	even	12		180.2.i.a.121.1	yes	2
45.43	odd	12		540.2.i.a.361.1		2
60.23	odd	4		6480.2.a.t.1.1		1
180.23	odd	12		720.2.q.a.241.1		2
180.43	even	12		2160.2.q.e.1441.1		2
180.83	odd	12		720.2.q.a.481.1		2
180.103	even	12		2160.2.q.e.721.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.i.a.61.1	✓	2	45.23	even	12
180.2.i.a.121.1	yes	2	45.38	even	12
540.2.i.a.181.1		2	45.13	odd	12
540.2.i.a.361.1		2	45.43	odd	12
720.2.q.a.241.1		2	180.23	odd	12
720.2.q.a.481.1		2	180.83	odd	12
900.2.i.a.301.1		2	45.2	even	12
900.2.i.a.601.1		2	45.32	even	12
900.2.s.a.49.1		4	45.29	odd	6
900.2.s.a.49.2		4	9.2	odd	6
900.2.s.a.349.1		4	9.5	odd	6
900.2.s.a.349.2		4	45.14	odd	6
1620.2.a.b.1.1		1	5.3	odd	4
1620.2.a.e.1.1		1	15.8	even	4
2160.2.q.e.721.1		2	180.103	even	12
2160.2.q.e.1441.1		2	180.43	even	12
2700.2.i.a.901.1		2	45.7	odd	12
2700.2.i.a.1801.1		2	45.22	odd	12
2700.2.s.a.1549.1		4	9.7	even	3
2700.2.s.a.1549.2		4	45.34	even	6
2700.2.s.a.2449.1		4	45.4	even	6
2700.2.s.a.2449.2		4	9.4	even	3
6480.2.a.h.1.1		1	20.3	even	4
6480.2.a.t.1.1		1	60.23	odd	4
8100.2.a.h.1.1		1	5.2	odd	4
8100.2.a.i.1.1		1	15.2	even	4
8100.2.d.e.649.1		2	3.2	odd	2
8100.2.d.e.649.2		2	15.14	odd	2
8100.2.d.f.649.1		2	1.1	even	1	trivial
8100.2.d.f.649.2		2	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{7} +O(q^{10})\)	\(q-1.00000i q^{7} +4.00000i q^{13} +6.00000i q^{17} -2.00000 q^{19} -3.00000i q^{23} +3.00000 q^{29} -10.0000 q^{31} -10.0000i q^{37} -9.00000 q^{41} +4.00000i q^{43} -9.00000i q^{47} +6.00000 q^{49} -6.00000i q^{53} -6.00000 q^{59} -1.00000 q^{61} +11.0000i q^{67} -12.0000 q^{71} +4.00000i q^{73} +10.0000 q^{79} -9.00000i q^{83} +9.00000 q^{89} +4.00000 q^{91} -10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 4 q^{19} + 6 q^{29} - 20 q^{31} - 18 q^{41} + 12 q^{49} - 12 q^{59} - 2 q^{61} - 24 q^{71} + 20 q^{79} + 18 q^{89} + 8 q^{91}+O(q^{100}) \) 2 * q - 4 * q^19 + 6 * q^29 - 20 * q^31 - 18 * q^41 + 12 * q^49 - 12 * q^59 - 2 * q^61 - 24 * q^71 + 20 * q^79 + 18 * q^89 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more