LMFDB - Embedded newform 8100.2.d.q.649.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8100,2,Mod(649,8100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [8,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4057180416.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 22x^{4} + 12x^{2} + 1 $ x^8 + 9x^6 + 22x^4 + 12*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-0.318459i$ of defining polynomial
Character	$\chi$	$=$	8100.649
Dual form			8100.2.d.q.649.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.40179i q^{7} -4.40179 q^{11} -2.06320i q^{13} -1.40179i q^{17} +6.35717 q^{19} -0.107826i q^{23} -9.08178 q^{29} +3.06320 q^{31} +1.95538i q^{37} -8.69575 q^{41} +7.12641i q^{43} -7.48357i q^{47} -4.57217 q^{49} -13.0975i q^{53} +13.1793 q^{59} -1.72462 q^{61} -12.3757i q^{67} -7.50961 q^{71} +5.42037i q^{73} +14.9740i q^{77} -9.43613 q^{79} -8.97396i q^{83} -4.01576 q^{89} -7.01858 q^{91} +2.17103i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{11} + 8 q^{19} - 18 q^{29} + 4 q^{31} - 18 q^{41} - 18 q^{49} - 30 q^{59} - 2 q^{61} - 24 q^{71} + 14 q^{79} - 6 q^{89} - 22 q^{91}+O(q^{100}) $ 8 * q - 6 * q^11 + 8 * q^19 - 18 * q^29 + 4 * q^31 - 18 * q^41 - 18 * q^49 - 30 * q^59 - 2 * q^61 - 24 * q^71 + 14 * q^79 - 6 * q^89 - 22 * q^91

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
$7$	− 3.40179i	− 1.28576i	−0.765969	−	0.642878i	$-0.777740\pi$
$7$	− 3.40179i	− 1.28576i	0.765969	−	0.642878i	$-0.222260\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.40179	−1.32719	−0.663595	−	0.748092i	$-0.730970\pi$
$11$	−4.40179	−1.32719	−0.663595	+	0.748092i	$0.730970\pi$
$12$	0	0
$13$	− 2.06320i	− 0.572229i	−0.958195	−	0.286115i	$-0.907636\pi$
$13$	− 2.06320i	− 0.572229i	0.958195	−	0.286115i	$-0.0923638\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.40179i	− 0.339984i	−0.985445	−	0.169992i	$-0.945626\pi$
$17$	− 1.40179i	− 0.339984i	0.985445	−	0.169992i	$-0.0543741\pi$
$18$	0	0
$19$	6.35717	1.45843	0.729217	−	0.684283i	$-0.239885\pi$
$19$	6.35717	1.45843	0.729217	+	0.684283i	$0.239885\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 0.107826i	− 0.0224832i	−0.999937	−	0.0112416i	$-0.996422\pi$
$23$	− 0.107826i	− 0.0224832i	0.999937	−	0.0112416i	$-0.00357839\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.08178	−1.68644	−0.843222	−	0.537565i	$-0.819344\pi$
$29$	−9.08178	−1.68644	−0.843222	+	0.537565i	$0.819344\pi$
$30$	0	0
$31$	3.06320	0.550167	0.275084	−	0.961420i	$-0.411294\pi$
$31$	3.06320	0.550167	0.275084	+	0.961420i	$0.411294\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.95538i	0.321462i	0.986998	+	0.160731i	$0.0513852\pi$
$37$	1.95538i	0.321462i	−0.986998	+	0.160731i	$0.948615\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.69575	−1.35805	−0.679024	−	0.734116i	$-0.737597\pi$
$41$	−8.69575	−1.35805	−0.679024	+	0.734116i	$0.737597\pi$
$42$	0	0
$43$	7.12641i	1.08677i	0.839485	+	0.543383i	$0.182857\pi$
$43$	7.12641i	1.08677i	−0.839485	+	0.543383i	$0.817143\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.48357i	− 1.09159i	−0.837918	−	0.545795i	$-0.816228\pi$
$47$	− 7.48357i	− 1.09159i	0.837918	−	0.545795i	$-0.183772\pi$
$48$	0	0
$49$	−4.57217	−0.653167
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 13.0975i	− 1.79909i	−0.436833	−	0.899543i	$-0.643900\pi$
$53$	− 13.0975i	− 1.79909i	0.436833	−	0.899543i	$-0.356100\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.1793	1.71580	0.857901	−	0.513815i	$-0.171768\pi$
$59$	13.1793	1.71580	0.857901	+	0.513815i	$0.171768\pi$
$60$	0	0
$61$	−1.72462	−0.220814	−0.110407	−	0.993886i	$-0.535216\pi$
$61$	−1.72462	−0.220814	−0.110407	+	0.993886i	$0.535216\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.3757i	− 1.51194i	−0.654608	−	0.755969i	$-0.727166\pi$
$67$	− 12.3757i	− 1.51194i	0.654608	−	0.755969i	$-0.272834\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.50961	−0.891227	−0.445614	−	0.895225i	$-0.647014\pi$
$71$	−7.50961	−0.891227	−0.445614	+	0.895225i	$0.647014\pi$
$72$	0	0
$73$	5.42037i	0.634406i	0.948358	+	0.317203i	$0.102744\pi$
$73$	5.42037i	0.634406i	−0.948358	+	0.317203i	$0.897256\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.9740i	1.70644i
$78$	0	0
$79$	−9.43613	−1.06165	−0.530824	−	0.847482i	$-0.678117\pi$
$79$	−9.43613	−1.06165	−0.530824	+	0.847482i	$0.678117\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.97396i	− 0.985020i	−0.870307	−	0.492510i	$-0.836080\pi$
$83$	− 8.97396i	− 0.985020i	0.870307	−	0.492510i	$-0.163920\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.01576	−0.425670	−0.212835	−	0.977088i	$-0.568270\pi$
$89$	−4.01576	−0.425670	−0.212835	+	0.977088i	$0.568270\pi$
$90$	0	0
$91$	−7.01858	−0.735747
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.17103i	0.220435i	0.993908	+	0.110217i	$0.0351547\pi$
$97$	2.17103i	0.220435i	−0.993908	+	0.110217i	$0.964845\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.97396	0.295920	0.147960	−	0.988993i	$-0.452729\pi$
$101$	2.97396	0.295920	0.147960	+	0.988993i	$0.452729\pi$
$102$	0	0
$103$	− 6.63537i	− 0.653802i	−0.945059	−	0.326901i	$-0.893996\pi$
$103$	− 6.63537i	− 0.653802i	0.945059	−	0.326901i	$-0.106004\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.7878i	1.33292i	0.745541	+	0.666459i	$0.232191\pi$
$107$	13.7878i	1.33292i	−0.745541	+	0.666459i	$0.767809\pi$
$108$	0	0
$109$	18.1347	1.73699	0.868495	−	0.495699i	$-0.165088\pi$
$109$	18.1347	1.73699	0.868495	+	0.495699i	$0.165088\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.0817817i	0.00769338i	0.999993	+	0.00384669i	$0.00122444\pi$
$113$	0.0817817i	0.00769338i	−0.999993	+	0.00384669i	$0.998776\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.76859	−0.437136
$120$	0	0
$121$	8.37574	0.761431
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	21.5811i	1.91501i	0.288410	+	0.957507i	$0.406873\pi$
$127$	21.5811i	1.91501i	−0.288410	+	0.957507i	$0.593127\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.7878	−0.942536	−0.471268	−	0.881990i	$-0.656204\pi$
$131$	−10.7878	−0.942536	−0.471268	+	0.881990i	$0.656204\pi$
$132$	0	0
$133$	− 21.6257i	− 1.87519i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.5096i	1.15420i	0.816672	+	0.577102i	$0.195817\pi$
$137$	13.5096i	1.15420i	−0.816672	+	0.577102i	$0.804183\pi$
$138$	0	0
$139$	−14.6100	−1.23920	−0.619601	−	0.784917i	$-0.712706\pi$
$139$	−14.6100	−1.23920	−0.619601	+	0.784917i	$0.712706\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.08178i	0.759457i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.133870	−0.0109671	−0.00548353	−	0.999985i	$-0.501745\pi$
$149$	−0.133870	−0.0109671	−0.00548353	+	0.999985i	$0.501745\pi$
$150$	0	0
$151$	6.14498	0.500072	0.250036	−	0.968237i	$-0.419558\pi$
$151$	6.14498	0.500072	0.250036	+	0.968237i	$0.419558\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.7747i	1.17915i	0.807713	+	0.589575i	$0.200705\pi$
$157$	14.7747i	1.17915i	−0.807713	+	0.589575i	$0.799295\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.366801	−0.0289079
$162$	0	0
$163$	15.9451i	1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$163$	15.9451i	1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.4993i	1.58629i	0.609036	+	0.793143i	$0.291557\pi$
$167$	20.4993i	1.58629i	−0.609036	+	0.793143i	$0.708443\pi$
$168$	0	0
$169$	8.74320	0.672553
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 22.5914i	− 1.71759i	−0.512318	−	0.858796i	$-0.671213\pi$
$173$	− 22.5914i	− 1.71759i	0.512318	−	0.858796i	$-0.328787\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.5879	−1.53881	−0.769407	−	0.638759i	$-0.779448\pi$
$179$	−20.5879	−1.53881	−0.769407	+	0.638759i	$0.779448\pi$
$180$	0	0
$181$	−8.32830	−0.619038	−0.309519	−	0.950893i	$-0.600168\pi$
$181$	−8.32830	−0.619038	−0.309519	+	0.950893i	$0.600168\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.17038i	0.451223i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.78435	−0.635613	−0.317807	−	0.948156i	$-0.602946\pi$
$191$	−8.78435	−0.635613	−0.317807	+	0.948156i	$0.602946\pi$
$192$	0	0
$193$	− 7.23076i	− 0.520482i	−0.965544	−	0.260241i	$-0.916198\pi$
$193$	− 7.23076i	− 0.520482i	0.965544	−	0.260241i	$-0.0838019\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.17932i	− 0.297764i	−0.988855	−	0.148882i	$-0.952432\pi$
$197$	− 4.17932i	− 0.297764i	0.988855	−	0.148882i	$-0.0475675\pi$
$198$	0	0
$199$	−15.6518	−1.10953	−0.554763	−	0.832009i	$-0.687191\pi$
$199$	−15.6518	−1.10953	−0.554763	+	0.832009i	$0.687191\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	30.8943i	2.16836i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−27.9829	−1.93562
$210$	0	0
$211$	−24.1539	−1.66282	−0.831412	−	0.555656i	$-0.812467\pi$
$211$	−24.1539	−1.66282	−0.831412	+	0.555656i	$0.812467\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 10.4204i	− 0.707381i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.89217	−0.194549
$222$	0	0
$223$	− 1.44641i	− 0.0968589i	−0.998827	−	0.0484295i	$-0.984578\pi$
$223$	− 1.44641i	− 0.0968589i	0.998827	−	0.0484295i	$-0.0154216\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.5254i	1.36232i	0.732136	+	0.681158i	$0.238524\pi$
$227$	20.5254i	1.36232i	−0.732136	+	0.681158i	$0.761476\pi$
$228$	0	0
$229$	−20.9486	−1.38432	−0.692160	−	0.721744i	$-0.743341\pi$
$229$	−20.9486	−1.38432	−0.692160	+	0.721744i	$0.743341\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.90112i	0.648644i	0.945947	+	0.324322i	$0.105136\pi$
$233$	9.90112i	0.648644i	−0.945947	+	0.324322i	$0.894864\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.10783	−0.395082	−0.197541	−	0.980295i	$-0.563296\pi$
$239$	−6.10783	−0.395082	−0.197541	+	0.980295i	$0.563296\pi$
$240$	0	0
$241$	−0.296783	−0.0191175	−0.00955875	−	0.999954i	$-0.503043\pi$
$241$	−0.296783	−0.0191175	−0.00955875	+	0.999954i	$0.503043\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 13.1161i	− 0.834559i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.51643	−0.285075	−0.142537	−	0.989789i	$-0.545526\pi$
$251$	−4.51643	−0.285075	−0.142537	+	0.989789i	$0.545526\pi$
$252$	0	0
$253$	0.474626i	0.0298395i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.5982i	0.848233i	0.905608	+	0.424117i	$0.139415\pi$
$257$	13.5982i	0.848233i	−0.905608	+	0.424117i	$0.860585\pi$
$258$	0	0
$259$	6.65178	0.413321
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.12358i	0.439259i	0.975583	+	0.219630i	$0.0704849\pi$
$263$	7.12358i	0.439259i	−0.975583	+	0.219630i	$0.929515\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−8.80358	−0.536764	−0.268382	−	0.963313i	$-0.586489\pi$
$269$	−8.80358	−0.536764	−0.268382	+	0.963313i	$0.586489\pi$
$270$	0	0
$271$	−9.95885	−0.604957	−0.302478	−	0.953156i	$-0.597814\pi$
$271$	−9.95885	−0.604957	−0.302478	+	0.953156i	$0.597814\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 8.92933i	− 0.536512i	−0.963348	−	0.268256i	$-0.913553\pi$
$277$	− 8.92933i	− 0.536512i	0.963348	−	0.268256i	$-0.0864472\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	32.0804	1.91376	0.956879	−	0.290486i	$-0.0938169\pi$
$281$	32.0804	1.91376	0.956879	+	0.290486i	$0.0938169\pi$
$282$	0	0
$283$	6.36745i	0.378506i	0.981928	+	0.189253i	$0.0606066\pi$
$283$	6.36745i	0.378506i	−0.981928	+	0.189253i	$0.939393\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	29.5811i	1.74612i
$288$	0	0
$289$	15.0350	0.884411
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 27.9011i	− 1.63000i	−0.579460	−	0.815000i	$-0.696737\pi$
$293$	− 27.9011i	− 1.63000i	0.579460	−	0.815000i	$-0.303263\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.222466	−0.0128656
$300$	0	0
$301$	24.2425	1.39732
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.2054i	− 0.696597i	−0.937384	−	0.348299i	$-0.886760\pi$
$307$	− 12.2054i	− 0.696597i	0.937384	−	0.348299i	$-0.113240\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.90246	−0.277993	−0.138996	−	0.990293i	$-0.544388\pi$
$311$	−4.90246	−0.277993	−0.138996	+	0.990293i	$0.544388\pi$
$312$	0	0
$313$	17.5886i	0.994165i	0.867703	+	0.497083i	$0.165595\pi$
$313$	17.5886i	0.994165i	−0.867703	+	0.497083i	$0.834405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.5722i	1.09928i	0.835401	+	0.549641i	$0.185236\pi$
$317$	19.5722i	1.09928i	−0.835401	+	0.549641i	$0.814764\pi$
$318$	0	0
$319$	39.9761	2.23823
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8.91140i	− 0.495844i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−25.4575	−1.40352
$330$	0	0
$331$	14.5907	0.801980	0.400990	−	0.916082i	$-0.368666\pi$
$331$	14.5907	0.801980	0.400990	+	0.916082i	$0.368666\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 9.90858i	− 0.539755i	−0.962895	−	0.269877i	$-0.913017\pi$
$337$	− 9.90858i	− 0.539755i	0.962895	−	0.269877i	$-0.0869832\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−13.4836	−0.730176
$342$	0	0
$343$	− 8.25897i	− 0.445943i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.3200i	0.554007i	0.960869	+	0.277004i	$0.0893414\pi$
$347$	10.3200i	0.554007i	−0.960869	+	0.277004i	$0.910659\pi$
$348$	0	0
$349$	−17.1353	−0.917234	−0.458617	−	0.888634i	$-0.651655\pi$
$349$	−17.1353	−0.917234	−0.458617	+	0.888634i	$0.651655\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.2611i	0.705817i	0.935658	+	0.352909i	$0.114807\pi$
$353$	13.2611i	0.705817i	−0.935658	+	0.352909i	$0.885193\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.2817	−0.753758	−0.376879	−	0.926263i	$-0.623003\pi$
$359$	−14.2817	−0.753758	−0.376879	+	0.926263i	$0.623003\pi$
$360$	0	0
$361$	21.4136	1.12703
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.58111i	0.500130i	0.968229	+	0.250065i	$0.0804520\pi$
$367$	9.58111i	0.500130i	−0.968229	+	0.250065i	$0.919548\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−44.5551	−2.31318
$372$	0	0
$373$	24.1058i	1.24815i	0.781363	+	0.624076i	$0.214525\pi$
$373$	24.1058i	1.24815i	−0.781363	+	0.624076i	$0.785475\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.7376i	0.965033i
$378$	0	0
$379$	−2.73973	−0.140730	−0.0703651	−	0.997521i	$-0.522416\pi$
$379$	−2.73973	−0.140730	−0.0703651	+	0.997521i	$0.522416\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 21.3532i	− 1.09110i	−0.838080	−	0.545548i	$-0.816322\pi$
$383$	− 21.3532i	− 1.09110i	0.838080	−	0.545548i	$-0.183678\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.6697	0.743784	0.371892	−	0.928276i	$-0.378709\pi$
$389$	14.6697	0.743784	0.371892	+	0.928276i	$0.378709\pi$
$390$	0	0
$391$	−0.151149	−0.00764393
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.43065i	0.322745i	0.986894	+	0.161373i	$0.0515921\pi$
$397$	6.43065i	0.322745i	−0.986894	+	0.161373i	$0.948408\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0750	0.902621	0.451310	−	0.892367i	$-0.350957\pi$
$401$	18.0750	0.902621	0.451310	+	0.892367i	$0.350957\pi$
$402$	0	0
$403$	− 6.32001i	− 0.314822i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.60716i	− 0.426641i
$408$	0	0
$409$	18.7906	0.929137	0.464569	−	0.885537i	$-0.346209\pi$
$409$	18.7906	0.929137	0.464569	+	0.885537i	$0.346209\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 44.8333i	− 2.20610i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−39.6594	−1.93749	−0.968745	−	0.248060i	$-0.920207\pi$
$419$	−39.6594	−1.93749	−0.968745	+	0.248060i	$0.920207\pi$
$420$	0	0
$421$	−37.5365	−1.82942	−0.914708	−	0.404115i	$-0.867580\pi$
$421$	−37.5365	−1.82942	−0.914708	+	0.404115i	$0.867580\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.86678i	0.283913i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.7357	0.806132	0.403066	−	0.915171i	$-0.367945\pi$
$431$	16.7357	0.806132	0.403066	+	0.915171i	$0.367945\pi$
$432$	0	0
$433$	− 21.0008i	− 1.00924i	−0.863343	−	0.504618i	$-0.831633\pi$
$433$	− 21.0008i	− 1.00924i	0.863343	−	0.504618i	$-0.168367\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 0.685467i	− 0.0327903i
$438$	0	0
$439$	6.03152	0.287869	0.143934	−	0.989587i	$-0.454025\pi$
$439$	6.03152	0.287869	0.143934	+	0.989587i	$0.454025\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6.32548i	− 0.300533i	−0.988646	−	0.150266i	$-0.951987\pi$
$443$	− 6.32548i	− 0.300533i	0.988646	−	0.150266i	$-0.0480131\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.6594	0.880593	0.440296	−	0.897853i	$-0.354873\pi$
$449$	18.6594	0.880593	0.440296	+	0.897853i	$0.354873\pi$
$450$	0	0
$451$	38.2769	1.80239
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 27.5846i	− 1.29035i	−0.764034	−	0.645176i	$-0.776784\pi$
$457$	− 27.5846i	− 1.29035i	0.764034	−	0.645176i	$-0.223216\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.5288	−0.630101	−0.315051	−	0.949075i	$-0.602021\pi$
$461$	−13.5288	−0.630101	−0.315051	+	0.949075i	$0.602021\pi$
$462$	0	0
$463$	− 27.4842i	− 1.27730i	−0.769497	−	0.638650i	$-0.779493\pi$
$463$	− 27.4842i	− 1.27730i	0.769497	−	0.638650i	$-0.220507\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.7515i	1.51556i	0.652511	+	0.757779i	$0.273716\pi$
$467$	32.7515i	1.51556i	−0.652511	+	0.757779i	$0.726284\pi$
$468$	0	0
$469$	−42.0997	−1.94398
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 31.3689i	− 1.44234i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.85567	−0.267552	−0.133776	−	0.991012i	$-0.542710\pi$
$479$	−5.85567	−0.267552	−0.133776	+	0.991012i	$0.542710\pi$
$480$	0	0
$481$	4.03434	0.183950
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 16.4864i	− 0.747070i	−0.927616	−	0.373535i	$-0.878146\pi$
$487$	− 16.4864i	− 0.747070i	0.927616	−	0.373535i	$-0.121854\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13.7060	−0.618545	−0.309272	−	0.950973i	$-0.600085\pi$
$491$	−13.7060	−0.618545	−0.309272	+	0.950973i	$0.600085\pi$
$492$	0	0
$493$	12.7307i	0.573364i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.5461i	1.14590i
$498$	0	0
$499$	−4.46964	−0.200088	−0.100044	−	0.994983i	$-0.531898\pi$
$499$	−4.46964	−0.200088	−0.100044	+	0.994983i	$0.531898\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.1010i	0.807084i	0.914961	+	0.403542i	$0.132221\pi$
$503$	18.1010i	0.807084i	−0.914961	+	0.403542i	$0.867779\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.2088	0.940065	0.470033	−	0.882649i	$-0.344242\pi$
$509$	21.2088	0.940065	0.470033	+	0.882649i	$0.344242\pi$
$510$	0	0
$511$	18.4389	0.815691
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	32.9411i	1.44875i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.4507	−1.24645	−0.623224	−	0.782043i	$-0.714178\pi$
$521$	−28.4507	−1.24645	−0.623224	+	0.782043i	$0.714178\pi$
$522$	0	0
$523$	− 26.5111i	− 1.15925i	−0.814884	−	0.579625i	$-0.803199\pi$
$523$	− 26.5111i	− 1.15925i	0.814884	−	0.579625i	$-0.196801\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.29396i	− 0.187048i
$528$	0	0
$529$	22.9884	0.999495
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.9411i	0.777115i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.1257	0.866876
$540$	0	0
$541$	−14.3132	−0.615372	−0.307686	−	0.951488i	$-0.599555\pi$
$541$	−14.3132	−0.615372	−0.307686	+	0.951488i	$0.599555\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13.6511i	0.583680i	0.956467	+	0.291840i	$0.0942675\pi$
$547$	13.6511i	0.583680i	−0.956467	+	0.291840i	$0.905733\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−57.7344	−2.45957
$552$	0	0
$553$	32.0997i	1.36502i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23.4665i	− 0.994306i	−0.867663	−	0.497153i	$-0.834379\pi$
$557$	− 23.4665i	− 0.994306i	0.867663	−	0.497153i	$-0.165621\pi$
$558$	0	0
$559$	14.7032	0.621880
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.582452i	0.0245474i	0.999925	+	0.0122737i	$0.00390694\pi$
$563$	0.582452i	0.0245474i	−0.999925	+	0.0122737i	$0.996093\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.55988	0.233082	0.116541	−	0.993186i	$-0.462819\pi$
$569$	5.55988	0.233082	0.116541	+	0.993186i	$0.462819\pi$
$570$	0	0
$571$	−22.7412	−0.951690	−0.475845	−	0.879529i	$-0.657858\pi$
$571$	−22.7412	−0.951690	−0.475845	+	0.879529i	$0.657858\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 30.5522i	− 1.27191i	−0.771728	−	0.635953i	$-0.780607\pi$
$577$	− 30.5522i	− 1.27191i	0.771728	−	0.635953i	$-0.219393\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−30.5275	−1.26649
$582$	0	0
$583$	57.6526i	2.38773i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.3635i	0.840490i	0.907411	+	0.420245i	$0.138056\pi$
$587$	20.3635i	0.840490i	−0.907411	+	0.420245i	$0.861944\pi$
$588$	0	0
$589$	19.4733	0.802383
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 15.5199i	− 0.637326i	−0.947868	−	0.318663i	$-0.896766\pi$
$593$	− 15.5199i	− 0.637326i	0.947868	−	0.318663i	$-0.103234\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−29.2576	−1.19543	−0.597717	−	0.801707i	$-0.703925\pi$
$599$	−29.2576	−1.19543	−0.597717	+	0.801707i	$0.703925\pi$
$600$	0	0
$601$	−18.3051	−0.746680	−0.373340	−	0.927695i	$-0.621787\pi$
$601$	−18.3051	−0.746680	−0.373340	+	0.927695i	$0.621787\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	16.8407i	0.683544i	0.939783	+	0.341772i	$0.111027\pi$
$607$	16.8407i	0.683544i	−0.939783	+	0.341772i	$0.888973\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.4401	−0.624640
$612$	0	0
$613$	− 13.5954i	− 0.549113i	−0.961571	−	0.274556i	$-0.911469\pi$
$613$	− 13.5954i	− 0.549113i	0.961571	−	0.274556i	$-0.0885310\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 29.3134i	− 1.18011i	−0.807362	−	0.590056i	$-0.799106\pi$
$617$	− 29.3134i	− 1.18011i	0.807362	−	0.590056i	$-0.200894\pi$
$618$	0	0
$619$	−12.1642	−0.488921	−0.244461	−	0.969659i	$-0.578611\pi$
$619$	−12.1642	−0.488921	−0.244461	+	0.969659i	$0.578611\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.6608i	0.547307i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.74103	0.109292
$630$	0	0
$631$	5.45471	0.217148	0.108574	−	0.994088i	$-0.465371\pi$
$631$	5.45471	0.217148	0.108574	+	0.994088i	$0.465371\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.43331i	0.373761i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.67999	−0.0663557	−0.0331779	−	0.999449i	$-0.510563\pi$
$641$	−1.67999	−0.0663557	−0.0331779	+	0.999449i	$0.510563\pi$
$642$	0	0
$643$	5.68346i	0.224134i	0.993701	+	0.112067i	$0.0357471\pi$
$643$	5.68346i	0.224134i	−0.993701	+	0.112067i	$0.964253\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 34.3064i	− 1.34872i	−0.738401	−	0.674361i	$-0.764419\pi$
$647$	− 34.3064i	− 1.34872i	0.738401	−	0.674361i	$-0.235581\pi$
$648$	0	0
$649$	−58.0126	−2.27719
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 12.6297i	− 0.494239i	−0.968985	−	0.247120i	$-0.920516\pi$
$653$	− 12.6297i	− 0.494239i	0.968985	−	0.247120i	$-0.0794841\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.33394	0.246735	0.123368	−	0.992361i	$-0.460631\pi$
$659$	6.33394	0.246735	0.123368	+	0.992361i	$0.460631\pi$
$660$	0	0
$661$	−37.6896	−1.46596	−0.732978	−	0.680253i	$-0.761870\pi$
$661$	−37.6896	−1.46596	−0.732978	+	0.680253i	$0.761870\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.979251i	0.0379167i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.59140	0.293063
$672$	0	0
$673$	− 30.6546i	− 1.18165i	−0.806800	−	0.590824i	$-0.798803\pi$
$673$	− 30.6546i	− 1.18165i	0.806800	−	0.590824i	$-0.201197\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.8716i	1.10963i	0.831975	+	0.554813i	$0.187210\pi$
$677$	28.8716i	1.10963i	−0.831975	+	0.554813i	$0.812790\pi$
$678$	0	0
$679$	7.38538	0.283425
$680$	0	0
$681$	0	0
$682$	0	0
$683$	47.0352i	1.79975i	0.436147	+	0.899875i	$0.356343\pi$
$683$	47.0352i	1.79975i	−0.436147	+	0.899875i	$0.643657\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−27.0229	−1.02949
$690$	0	0
$691$	0.850371	0.0323496	0.0161748	−	0.999869i	$-0.494851\pi$
$691$	0.850371	0.0323496	0.0161748	+	0.999869i	$0.494851\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.1896i	0.461714i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−35.6821	−1.34770	−0.673848	−	0.738870i	$-0.735360\pi$
$701$	−35.6821	−1.34770	−0.673848	+	0.738870i	$0.735360\pi$
$702$	0	0
$703$	12.4307i	0.468831i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10.1168i	− 0.380480i
$708$	0	0
$709$	−15.3909	−0.578016	−0.289008	−	0.957327i	$-0.593325\pi$
$709$	−15.3909	−0.578016	−0.289008	+	0.957327i	$0.593325\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 0.330292i	− 0.0123695i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−25.5846	−0.954144	−0.477072	−	0.878864i	$-0.658302\pi$
$719$	−25.5846	−0.954144	−0.477072	+	0.878864i	$0.658302\pi$
$720$	0	0
$721$	−22.5721	−0.840630
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.13734i	0.227621i	0.993502	+	0.113811i	$0.0363057\pi$
$727$	6.13734i	0.227621i	−0.993502	+	0.113811i	$0.963694\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.98971	0.369483
$732$	0	0
$733$	− 5.13040i	− 0.189496i	−0.995501	−	0.0947478i	$-0.969796\pi$
$733$	− 5.13040i	− 0.189496i	0.995501	−	0.0947478i	$-0.0302045\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	54.4754i	2.00663i
$738$	0	0
$739$	−18.8784	−0.694454	−0.347227	−	0.937781i	$-0.612877\pi$
$739$	−18.8784	−0.694454	−0.347227	+	0.937781i	$0.612877\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 21.9569i	− 0.805519i	−0.915306	−	0.402759i	$-0.868051\pi$
$743$	− 21.9569i	− 0.805519i	0.915306	−	0.402759i	$-0.131949\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	46.9032	1.71381
$750$	0	0
$751$	−3.16074	−0.115337	−0.0576686	−	0.998336i	$-0.518367\pi$
$751$	−3.16074	−0.115337	−0.0576686	+	0.998336i	$0.518367\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	39.1753i	1.42385i	0.702255	+	0.711926i	$0.252177\pi$
$757$	39.1753i	1.42385i	−0.702255	+	0.711926i	$0.747823\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.0783	0.365339	0.182669	−	0.983174i	$-0.441526\pi$
$761$	10.0783	0.365339	0.182669	+	0.983174i	$0.441526\pi$
$762$	0	0
$763$	− 61.6904i	− 2.23334i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 27.1916i	− 0.981832i
$768$	0	0
$769$	−21.1579	−0.762974	−0.381487	−	0.924374i	$-0.624588\pi$
$769$	−21.1579	−0.762974	−0.381487	+	0.924374i	$0.624588\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 43.2543i	− 1.55575i	−0.628420	−	0.777874i	$-0.716298\pi$
$773$	− 43.2543i	− 1.55575i	0.628420	−	0.777874i	$-0.283702\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−55.2803	−1.98062
$780$	0	0
$781$	33.0557	1.18283
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.17997i	− 0.0420615i	−0.999779	−	0.0210307i	$-0.993305\pi$
$787$	− 1.17997i	− 0.0420615i	0.999779	−	0.0210307i	$-0.00669478\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.278204	0.00989180
$792$	0	0
$793$	3.55823i	0.126357i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.14281i	− 0.0404805i	−0.999795	−	0.0202403i	$-0.993557\pi$
$797$	− 1.14281i	− 0.0404805i	0.999795	−	0.0202403i	$-0.00644312\pi$
$798$	0	0
$799$	−10.4904	−0.371123
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 23.8593i	− 0.841977i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	11.3723	0.399828	0.199914	−	0.979813i	$-0.435934\pi$
$809$	11.3723	0.399828	0.199914	+	0.979813i	$0.435934\pi$
$810$	0	0
$811$	2.08590	0.0732459	0.0366230	−	0.999329i	$-0.488340\pi$
$811$	2.08590	0.0732459	0.0366230	+	0.999329i	$0.488340\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	45.3037i	1.58498i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.7913	0.551120	0.275560	−	0.961284i	$-0.411137\pi$
$821$	15.7913	0.551120	0.275560	+	0.961284i	$0.411137\pi$
$822$	0	0
$823$	− 7.32084i	− 0.255188i	−0.991826	−	0.127594i	$-0.959275\pi$
$823$	− 7.32084i	− 0.255188i	0.991826	−	0.127594i	$-0.0407255\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 14.8401i	− 0.516040i	−0.966140	−	0.258020i	$-0.916930\pi$
$827$	− 14.8401i	− 0.516040i	0.966140	−	0.258020i	$-0.0830701\pi$
$828$	0	0
$829$	23.3346	0.810444	0.405222	−	0.914218i	$-0.367194\pi$
$829$	23.3346	0.810444	0.405222	+	0.914218i	$0.367194\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.40921i	0.222066i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	14.6765	0.506690	0.253345	−	0.967376i	$-0.418469\pi$
$839$	14.6765	0.506690	0.253345	+	0.967376i	$0.418469\pi$
$840$	0	0
$841$	53.4788	1.84410
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 28.4925i	− 0.979014i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.210840	0.00722751
$852$	0	0
$853$	− 24.5222i	− 0.839624i	−0.907611	−	0.419812i	$-0.862096\pi$
$853$	− 24.5222i	− 0.839624i	0.907611	−	0.419812i	$-0.137904\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.58793i	0.190880i	0.995435	+	0.0954400i	$0.0304258\pi$
$857$	5.58793i	0.190880i	−0.995435	+	0.0954400i	$0.969574\pi$
$858$	0	0
$859$	40.0619	1.36689	0.683447	−	0.730001i	$-0.260480\pi$
$859$	40.0619	1.36689	0.683447	+	0.730001i	$0.260480\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.2157i	0.926432i	0.886246	+	0.463216i	$0.153305\pi$
$863$	27.2157i	0.926432i	−0.886246	+	0.463216i	$0.846695\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	41.5358	1.40901
$870$	0	0
$871$	−25.5337	−0.865175
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 23.8799i	− 0.806366i	−0.915119	−	0.403183i	$-0.867904\pi$
$877$	− 23.8799i	− 0.806366i	0.915119	−	0.403183i	$-0.132096\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28.3585	0.955421	0.477710	−	0.878517i	$-0.341467\pi$
$881$	28.3585	0.955421	0.477710	+	0.878517i	$0.341467\pi$
$882$	0	0
$883$	− 28.3449i	− 0.953881i	−0.878936	−	0.476941i	$-0.841746\pi$
$883$	− 28.3449i	− 0.953881i	0.878936	−	0.476941i	$-0.158254\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.515088i	0.0172950i	0.999963	+	0.00864749i	$0.00275261\pi$
$887$	0.515088i	0.0172950i	−0.999963	+	0.00864749i	$0.997247\pi$
$888$	0	0
$889$	73.4144	2.46224
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 47.5743i	− 1.59201i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−27.8193	−0.927827
$900$	0	0
$901$	−18.3600	−0.611660
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.16421i	0.0718615i	0.999354	+	0.0359308i	$0.0114396\pi$
$907$	2.16421i	0.0718615i	−0.999354	+	0.0359308i	$0.988560\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38.4665	1.27445	0.637226	−	0.770677i	$-0.280082\pi$
$911$	38.4665	1.27445	0.637226	+	0.770677i	$0.280082\pi$
$912$	0	0
$913$	39.5015i	1.30731i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36.6979i	1.21187i
$918$	0	0
$919$	−18.6992	−0.616830	−0.308415	−	0.951252i	$-0.599799\pi$
$919$	−18.6992	−0.616830	−0.308415	+	0.951252i	$0.599799\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	15.4939i	0.509986i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.20671	0.0723997	0.0361999	−	0.999345i	$-0.488475\pi$
$929$	2.20671	0.0723997	0.0361999	+	0.999345i	$0.488475\pi$
$930$	0	0
$931$	−29.0660	−0.952600
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 21.3429i	− 0.697242i	−0.937264	−	0.348621i	$-0.886650\pi$
$937$	− 21.3429i	− 0.697242i	0.937264	−	0.348621i	$-0.113350\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.17932	−0.0384448	−0.0192224	−	0.999815i	$-0.506119\pi$
$941$	−1.17932	−0.0384448	−0.0192224	+	0.999815i	$0.506119\pi$
$942$	0	0
$943$	0.937627i	0.0305333i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 23.6889i	− 0.769787i	−0.922961	−	0.384894i	$-0.874238\pi$
$947$	− 23.6889i	− 0.769787i	0.922961	−	0.384894i	$-0.125762\pi$
$948$	0	0
$949$	11.1833	0.363026
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.8125i	0.415038i	0.978231	+	0.207519i	$0.0665389\pi$
$953$	12.8125i	0.415038i	−0.978231	+	0.207519i	$0.933461\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	45.9569	1.48402
$960$	0	0
$961$	−21.6168	−0.697316
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 10.9265i	− 0.351373i	−0.984446	−	0.175686i	$-0.943786\pi$
$967$	− 10.9265i	− 0.351373i	0.984446	−	0.175686i	$-0.0562145\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15.7933	0.506831	0.253415	−	0.967358i	$-0.418446\pi$
$971$	15.7933	0.506831	0.253415	+	0.967358i	$0.418446\pi$
$972$	0	0
$973$	49.7001i	1.59331i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.4980i	1.19967i	0.800125	+	0.599833i	$0.204767\pi$
$977$	37.4980i	1.19967i	−0.800125	+	0.599833i	$0.795233\pi$
$978$	0	0
$979$	17.6765	0.564944
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23.5322i	0.750560i	0.926911	+	0.375280i	$0.122453\pi$
$983$	23.5322i	0.750560i	−0.926911	+	0.375280i	$0.877547\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.768410	0.0244340
$990$	0	0
$991$	46.7019	1.48353	0.741767	−	0.670658i	$-0.233988\pi$
$991$	46.7019	1.48353	0.741767	+	0.670658i	$0.233988\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 42.8580i	− 1.35733i	−0.734450	−	0.678663i	$-0.762560\pi$
$997$	− 42.8580i	− 1.35733i	0.734450	−	0.678663i	$-0.237440\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.d.q.649.2		8
3.2	odd	2		8100.2.d.s.649.2		8
5.2	odd	4		8100.2.a.x.1.4		4
5.3	odd	4		8100.2.a.z.1.1		4
5.4	even	2	inner	8100.2.d.q.649.7		8
9.2	odd	6		2700.2.s.d.1549.2		16
9.4	even	3		900.2.s.d.349.5		16
9.5	odd	6		2700.2.s.d.2449.7		16
9.7	even	3		900.2.s.d.49.4		16
15.2	even	4		8100.2.a.y.1.4		4
15.8	even	4		8100.2.a.ba.1.1		4
15.14	odd	2		8100.2.d.s.649.7		8
45.2	even	12		2700.2.i.e.901.1		8
45.4	even	6		900.2.s.d.349.4		16
45.7	odd	12		900.2.i.d.301.1	✓	8
45.13	odd	12		900.2.i.e.601.4	yes	8
45.14	odd	6		2700.2.s.d.2449.2		16
45.22	odd	12		900.2.i.d.601.1	yes	8
45.23	even	12		2700.2.i.d.1801.4		8
45.29	odd	6		2700.2.s.d.1549.7		16
45.32	even	12		2700.2.i.e.1801.1		8
45.34	even	6		900.2.s.d.49.5		16
45.38	even	12		2700.2.i.d.901.4		8
45.43	odd	12		900.2.i.e.301.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.i.d.301.1	✓	8	45.7	odd	12
900.2.i.d.601.1	yes	8	45.22	odd	12
900.2.i.e.301.4	yes	8	45.43	odd	12
900.2.i.e.601.4	yes	8	45.13	odd	12
900.2.s.d.49.4		16	9.7	even	3
900.2.s.d.49.5		16	45.34	even	6
900.2.s.d.349.4		16	45.4	even	6
900.2.s.d.349.5		16	9.4	even	3
2700.2.i.d.901.4		8	45.38	even	12
2700.2.i.d.1801.4		8	45.23	even	12
2700.2.i.e.901.1		8	45.2	even	12
2700.2.i.e.1801.1		8	45.32	even	12
2700.2.s.d.1549.2		16	9.2	odd	6
2700.2.s.d.1549.7		16	45.29	odd	6
2700.2.s.d.2449.2		16	45.14	odd	6
2700.2.s.d.2449.7		16	9.5	odd	6
8100.2.a.x.1.4		4	5.2	odd	4
8100.2.a.y.1.4		4	15.2	even	4
8100.2.a.z.1.1		4	5.3	odd	4
8100.2.a.ba.1.1		4	15.8	even	4
8100.2.d.q.649.2		8	1.1	even	1	trivial
8100.2.d.q.649.7		8	5.4	even	2	inner
8100.2.d.s.649.2		8	3.2	odd	2
8100.2.d.s.649.7		8	15.14	odd	2

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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q-3.40179i q^{7} -4.40179 q^{11} -2.06320i q^{13} -1.40179i q^{17} +6.35717 q^{19} -0.107826i q^{23} -9.08178 q^{29} +3.06320 q^{31} +1.95538i q^{37} -8.69575 q^{41} +7.12641i q^{43} -7.48357i q^{47} -4.57217 q^{49} -13.0975i q^{53} +13.1793 q^{59} -1.72462 q^{61} -12.3757i q^{67} -7.50961 q^{71} +5.42037i q^{73} +14.9740i q^{77} -9.43613 q^{79} -8.97396i q^{83} -4.01576 q^{89} -7.01858 q^{91} +2.17103i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{11} + 8 q^{19} - 18 q^{29} + 4 q^{31} - 18 q^{41} - 18 q^{49} - 30 q^{59} - 2 q^{61} - 24 q^{71} + 14 q^{79} - 6 q^{89} - 22 q^{91}+O(q^{100}) \) 8 * q - 6 * q^11 + 8 * q^19 - 18 * q^29 + 4 * q^31 - 18 * q^41 - 18 * q^49 - 30 * q^59 - 2 * q^61 - 24 * q^71 + 14 * q^79 - 6 * q^89 - 22 * q^91

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