LMFDB - Embedded newform 8100.2.a.x.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8100,2,Mod(1,8100)] 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("8100.1"); 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, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-1,0,0,0,-3,0,2,0,0,0,-9,0,-4,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.6788256372$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3981.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 900)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.318459$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.40179 q^{7} -4.40179 q^{11} -2.06320 q^{13} +1.40179 q^{17} -6.35717 q^{19} -0.107826 q^{23} +9.08178 q^{29} +3.06320 q^{31} -1.95538 q^{37} -8.69575 q^{41} +7.12641 q^{43} +7.48357 q^{47} +4.57217 q^{49} -13.0975 q^{53} -13.1793 q^{59} -1.72462 q^{61} +12.3757 q^{67} -7.50961 q^{71} +5.42037 q^{73} -14.9740 q^{77} +9.43613 q^{79} -8.97396 q^{83} +4.01576 q^{89} -7.01858 q^{91} -2.17103 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{7} - 3 q^{11} + 2 q^{13} - 9 q^{17} - 4 q^{19} + 3 q^{23} + 9 q^{29} + 2 q^{31} - q^{37} - 9 q^{41} + 8 q^{43} - 12 q^{47} + 9 q^{49} - 12 q^{53} + 15 q^{59} - q^{61} + 11 q^{67} - 12 q^{71}+ \cdots + 5 q^{97}+O(q^{100}) $ 4 * q - q^7 - 3 * q^11 + 2 * q^13 - 9 * q^17 - 4 * q^19 + 3 * q^23 + 9 * q^29 + 2 * q^31 - q^37 - 9 * q^41 + 8 * q^43 - 12 * q^47 + 9 * q^49 - 12 * q^53 + 15 * q^59 - q^61 + 11 * q^67 - 12 * q^71 - 10 * q^73 - 36 * q^77 - 7 * q^79 - 12 * q^83 + 3 * q^89 - 11 * q^91 + 5 * q^97

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
$7$	3.40179	1.28576	0.642878	−	0.765969i	$-0.277740\pi$
$7$	3.40179	1.28576	0.642878	+	0.765969i	$0.277740\pi$
$8$	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
$13$	−2.06320	−0.572229	−0.286115	−	0.958195i	$-0.592364\pi$
$13$	−2.06320	−0.572229	−0.286115	+	0.958195i	$0.592364\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.40179	0.339984	0.169992	−	0.985445i	$-0.445626\pi$
$17$	1.40179	0.339984	0.169992	+	0.985445i	$0.445626\pi$
$18$	0	0
$19$	−6.35717	−1.45843	−0.729217	−	0.684283i	$-0.760115\pi$
$19$	−6.35717	−1.45843	−0.729217	+	0.684283i	$0.760115\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.107826	−0.0224832	−0.0112416	−	0.999937i	$-0.503578\pi$
$23$	−0.107826	−0.0224832	−0.0112416	+	0.999937i	$0.503578\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.08178	1.68644	0.843222	−	0.537565i	$-0.180656\pi$
$29$	9.08178	1.68644	0.843222	+	0.537565i	$0.180656\pi$
$30$	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
$37$	−1.95538	−0.321462	−0.160731	−	0.986998i	$-0.551385\pi$
$37$	−1.95538	−0.321462	−0.160731	+	0.986998i	$0.551385\pi$
$38$	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
$43$	7.12641	1.08677	0.543383	−	0.839485i	$-0.317143\pi$
$43$	7.12641	1.08677	0.543383	+	0.839485i	$0.317143\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.48357	1.09159	0.545795	−	0.837918i	$-0.316228\pi$
$47$	7.48357	1.09159	0.545795	+	0.837918i	$0.316228\pi$
$48$	0	0
$49$	4.57217	0.653167
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.0975	−1.79909	−0.899543	−	0.436833i	$-0.856100\pi$
$53$	−13.0975	−1.79909	−0.899543	+	0.436833i	$0.856100\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.1793	−1.71580	−0.857901	−	0.513815i	$-0.828232\pi$
$59$	−13.1793	−1.71580	−0.857901	+	0.513815i	$0.828232\pi$
$60$	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
$67$	12.3757	1.51194	0.755969	−	0.654608i	$-0.227166\pi$
$67$	12.3757	1.51194	0.755969	+	0.654608i	$0.227166\pi$
$68$	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
$73$	5.42037	0.634406	0.317203	−	0.948358i	$-0.397256\pi$
$73$	5.42037	0.634406	0.317203	+	0.948358i	$0.397256\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.9740	−1.70644
$78$	0	0
$79$	9.43613	1.06165	0.530824	−	0.847482i	$-0.321883\pi$
$79$	9.43613	1.06165	0.530824	+	0.847482i	$0.321883\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.97396	−0.985020	−0.492510	−	0.870307i	$-0.663920\pi$
$83$	−8.97396	−0.985020	−0.492510	+	0.870307i	$0.663920\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.01576	0.425670	0.212835	−	0.977088i	$-0.431730\pi$
$89$	4.01576	0.425670	0.212835	+	0.977088i	$0.431730\pi$
$90$	0	0
$91$	−7.01858	−0.735747
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.17103	−0.220435	−0.110217	−	0.993908i	$-0.535155\pi$
$97$	−2.17103	−0.220435	−0.110217	+	0.993908i	$0.535155\pi$
$98$	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
$103$	−6.63537	−0.653802	−0.326901	−	0.945059i	$-0.606004\pi$
$103$	−6.63537	−0.653802	−0.326901	+	0.945059i	$0.606004\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.7878	−1.33292	−0.666459	−	0.745541i	$-0.732191\pi$
$107$	−13.7878	−1.33292	−0.666459	+	0.745541i	$0.732191\pi$
$108$	0	0
$109$	−18.1347	−1.73699	−0.868495	−	0.495699i	$-0.834912\pi$
$109$	−18.1347	−1.73699	−0.868495	+	0.495699i	$0.834912\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.0817817	0.00769338	0.00384669	−	0.999993i	$-0.498776\pi$
$113$	0.0817817	0.00769338	0.00384669	+	0.999993i	$0.498776\pi$
$114$	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
$127$	−21.5811	−1.91501	−0.957507	−	0.288410i	$-0.906873\pi$
$127$	−21.5811	−1.91501	−0.957507	+	0.288410i	$0.906873\pi$
$128$	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
$133$	−21.6257	−1.87519
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.5096	−1.15420	−0.577102	−	0.816672i	$-0.695817\pi$
$137$	−13.5096	−1.15420	−0.577102	+	0.816672i	$0.695817\pi$
$138$	0	0
$139$	14.6100	1.23920	0.619601	−	0.784917i	$-0.287294\pi$
$139$	14.6100	1.23920	0.619601	+	0.784917i	$0.287294\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.08178	0.759457
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.133870	0.0109671	0.00548353	−	0.999985i	$-0.498255\pi$
$149$	0.133870	0.0109671	0.00548353	+	0.999985i	$0.498255\pi$
$150$	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
$157$	−14.7747	−1.17915	−0.589575	−	0.807713i	$-0.700705\pi$
$157$	−14.7747	−1.17915	−0.589575	+	0.807713i	$0.700705\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.366801	−0.0289079
$162$	0	0
$163$	15.9451	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$163$	15.9451	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.4993	−1.58629	−0.793143	−	0.609036i	$-0.791557\pi$
$167$	−20.4993	−1.58629	−0.793143	+	0.609036i	$0.791557\pi$
$168$	0	0
$169$	−8.74320	−0.672553
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.5914	−1.71759	−0.858796	−	0.512318i	$-0.828787\pi$
$173$	−22.5914	−1.71759	−0.858796	+	0.512318i	$0.828787\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.5879	1.53881	0.769407	−	0.638759i	$-0.220552\pi$
$179$	20.5879	1.53881	0.769407	+	0.638759i	$0.220552\pi$
$180$	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
$187$	−6.17038	−0.451223
$188$	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
$193$	−7.23076	−0.520482	−0.260241	−	0.965544i	$-0.583802\pi$
$193$	−7.23076	−0.520482	−0.260241	+	0.965544i	$0.583802\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.17932	0.297764	0.148882	−	0.988855i	$-0.452432\pi$
$197$	4.17932	0.297764	0.148882	+	0.988855i	$0.452432\pi$
$198$	0	0
$199$	15.6518	1.10953	0.554763	−	0.832009i	$-0.312809\pi$
$199$	15.6518	1.10953	0.554763	+	0.832009i	$0.312809\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	30.8943	2.16836
$204$	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
$217$	10.4204	0.707381
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.89217	−0.194549
$222$	0	0
$223$	−1.44641	−0.0968589	−0.0484295	−	0.998827i	$-0.515422\pi$
$223$	−1.44641	−0.0968589	−0.0484295	+	0.998827i	$0.515422\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.5254	−1.36232	−0.681158	−	0.732136i	$-0.738524\pi$
$227$	−20.5254	−1.36232	−0.681158	+	0.732136i	$0.738524\pi$
$228$	0	0
$229$	20.9486	1.38432	0.692160	−	0.721744i	$-0.256659\pi$
$229$	20.9486	1.38432	0.692160	+	0.721744i	$0.256659\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.90112	0.648644	0.324322	−	0.945947i	$-0.394864\pi$
$233$	9.90112	0.648644	0.324322	+	0.945947i	$0.394864\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.10783	0.395082	0.197541	−	0.980295i	$-0.436704\pi$
$239$	6.10783	0.395082	0.197541	+	0.980295i	$0.436704\pi$
$240$	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
$247$	13.1161	0.834559
$248$	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
$253$	0.474626	0.0298395
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.5982	−0.848233	−0.424117	−	0.905608i	$-0.639415\pi$
$257$	−13.5982	−0.848233	−0.424117	+	0.905608i	$0.639415\pi$
$258$	0	0
$259$	−6.65178	−0.413321
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.12358	0.439259	0.219630	−	0.975583i	$-0.429515\pi$
$263$	7.12358	0.439259	0.219630	+	0.975583i	$0.429515\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.80358	0.536764	0.268382	−	0.963313i	$-0.413511\pi$
$269$	8.80358	0.536764	0.268382	+	0.963313i	$0.413511\pi$
$270$	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
$277$	8.92933	0.536512	0.268256	−	0.963348i	$-0.413553\pi$
$277$	8.92933	0.536512	0.268256	+	0.963348i	$0.413553\pi$
$278$	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
$283$	6.36745	0.378506	0.189253	−	0.981928i	$-0.439393\pi$
$283$	6.36745	0.378506	0.189253	+	0.981928i	$0.439393\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−29.5811	−1.74612
$288$	0	0
$289$	−15.0350	−0.884411
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−27.9011	−1.63000	−0.815000	−	0.579460i	$-0.803263\pi$
$293$	−27.9011	−1.63000	−0.815000	+	0.579460i	$0.803263\pi$
$294$	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
$307$	12.2054	0.696597	0.348299	−	0.937384i	$-0.386760\pi$
$307$	12.2054	0.696597	0.348299	+	0.937384i	$0.386760\pi$
$308$	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
$313$	17.5886	0.994165	0.497083	−	0.867703i	$-0.334405\pi$
$313$	17.5886	0.994165	0.497083	+	0.867703i	$0.334405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.5722	−1.09928	−0.549641	−	0.835401i	$-0.685236\pi$
$317$	−19.5722	−1.09928	−0.549641	+	0.835401i	$0.685236\pi$
$318$	0	0
$319$	−39.9761	−2.23823
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.91140	−0.495844
$324$	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
$337$	9.90858	0.539755	0.269877	−	0.962895i	$-0.413017\pi$
$337$	9.90858	0.539755	0.269877	+	0.962895i	$0.413017\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−13.4836	−0.730176
$342$	0	0
$343$	−8.25897	−0.445943
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.3200	−0.554007	−0.277004	−	0.960869i	$-0.589341\pi$
$347$	−10.3200	−0.554007	−0.277004	+	0.960869i	$0.589341\pi$
$348$	0	0
$349$	17.1353	0.917234	0.458617	−	0.888634i	$-0.348345\pi$
$349$	17.1353	0.917234	0.458617	+	0.888634i	$0.348345\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.2611	0.705817	0.352909	−	0.935658i	$-0.385193\pi$
$353$	13.2611	0.705817	0.352909	+	0.935658i	$0.385193\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.2817	0.753758	0.376879	−	0.926263i	$-0.376997\pi$
$359$	14.2817	0.753758	0.376879	+	0.926263i	$0.376997\pi$
$360$	0	0
$361$	21.4136	1.12703
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−9.58111	−0.500130	−0.250065	−	0.968229i	$-0.580452\pi$
$367$	−9.58111	−0.500130	−0.250065	+	0.968229i	$0.580452\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−44.5551	−2.31318
$372$	0	0
$373$	24.1058	1.24815	0.624076	−	0.781363i	$-0.285475\pi$
$373$	24.1058	1.24815	0.624076	+	0.781363i	$0.285475\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.7376	−0.965033
$378$	0	0
$379$	2.73973	0.140730	0.0703651	−	0.997521i	$-0.477584\pi$
$379$	2.73973	0.140730	0.0703651	+	0.997521i	$0.477584\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.3532	−1.09110	−0.545548	−	0.838080i	$-0.683678\pi$
$383$	−21.3532	−1.09110	−0.545548	+	0.838080i	$0.683678\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.6697	−0.743784	−0.371892	−	0.928276i	$-0.621291\pi$
$389$	−14.6697	−0.743784	−0.371892	+	0.928276i	$0.621291\pi$
$390$	0	0
$391$	−0.151149	−0.00764393
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.43065	−0.322745	−0.161373	−	0.986894i	$-0.551592\pi$
$397$	−6.43065	−0.322745	−0.161373	+	0.986894i	$0.551592\pi$
$398$	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
$403$	−6.32001	−0.314822
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.60716	0.426641
$408$	0	0
$409$	−18.7906	−0.929137	−0.464569	−	0.885537i	$-0.653791\pi$
$409$	−18.7906	−0.929137	−0.464569	+	0.885537i	$0.653791\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−44.8333	−2.20610
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	39.6594	1.93749	0.968745	−	0.248060i	$-0.0797929\pi$
$419$	39.6594	1.93749	0.968745	+	0.248060i	$0.0797929\pi$
$420$	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
$427$	−5.86678	−0.283913
$428$	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
$433$	−21.0008	−1.00924	−0.504618	−	0.863343i	$-0.668367\pi$
$433$	−21.0008	−1.00924	−0.504618	+	0.863343i	$0.668367\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.685467	0.0327903
$438$	0	0
$439$	−6.03152	−0.287869	−0.143934	−	0.989587i	$-0.545975\pi$
$439$	−6.03152	−0.287869	−0.143934	+	0.989587i	$0.545975\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.32548	−0.300533	−0.150266	−	0.988646i	$-0.548013\pi$
$443$	−6.32548	−0.300533	−0.150266	+	0.988646i	$0.548013\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.6594	−0.880593	−0.440296	−	0.897853i	$-0.645127\pi$
$449$	−18.6594	−0.880593	−0.440296	+	0.897853i	$0.645127\pi$
$450$	0	0
$451$	38.2769	1.80239
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.5846	1.29035	0.645176	−	0.764034i	$-0.276784\pi$
$457$	27.5846	1.29035	0.645176	+	0.764034i	$0.276784\pi$
$458$	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
$463$	−27.4842	−1.27730	−0.638650	−	0.769497i	$-0.720507\pi$
$463$	−27.4842	−1.27730	−0.638650	+	0.769497i	$0.720507\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.7515	−1.51556	−0.757779	−	0.652511i	$-0.773716\pi$
$467$	−32.7515	−1.51556	−0.757779	+	0.652511i	$0.773716\pi$
$468$	0	0
$469$	42.0997	1.94398
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−31.3689	−1.44234
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.85567	0.267552	0.133776	−	0.991012i	$-0.457290\pi$
$479$	5.85567	0.267552	0.133776	+	0.991012i	$0.457290\pi$
$480$	0	0
$481$	4.03434	0.183950
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.4864	0.747070	0.373535	−	0.927616i	$-0.378146\pi$
$487$	16.4864	0.747070	0.373535	+	0.927616i	$0.378146\pi$
$488$	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
$493$	12.7307	0.573364
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−25.5461	−1.14590
$498$	0	0
$499$	4.46964	0.200088	0.100044	−	0.994983i	$-0.468102\pi$
$499$	4.46964	0.200088	0.100044	+	0.994983i	$0.468102\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.1010	0.807084	0.403542	−	0.914961i	$-0.367779\pi$
$503$	18.1010	0.807084	0.403542	+	0.914961i	$0.367779\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.2088	−0.940065	−0.470033	−	0.882649i	$-0.655758\pi$
$509$	−21.2088	−0.940065	−0.470033	+	0.882649i	$0.655758\pi$
$510$	0	0
$511$	18.4389	0.815691
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−32.9411	−1.44875
$518$	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
$523$	−26.5111	−1.15925	−0.579625	−	0.814884i	$-0.696801\pi$
$523$	−26.5111	−1.15925	−0.579625	+	0.814884i	$0.696801\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.29396	0.187048
$528$	0	0
$529$	−22.9884	−0.999495
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.9411	0.777115
$534$	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
$547$	−13.6511	−0.583680	−0.291840	−	0.956467i	$-0.594267\pi$
$547$	−13.6511	−0.583680	−0.291840	+	0.956467i	$0.594267\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−57.7344	−2.45957
$552$	0	0
$553$	32.0997	1.36502
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.4665	0.994306	0.497153	−	0.867663i	$-0.334379\pi$
$557$	23.4665	0.994306	0.497153	+	0.867663i	$0.334379\pi$
$558$	0	0
$559$	−14.7032	−0.621880
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.582452	0.0245474	0.0122737	−	0.999925i	$-0.496093\pi$
$563$	0.582452	0.0245474	0.0122737	+	0.999925i	$0.496093\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.55988	−0.233082	−0.116541	−	0.993186i	$-0.537181\pi$
$569$	−5.55988	−0.233082	−0.116541	+	0.993186i	$0.537181\pi$
$570$	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
$577$	30.5522	1.27191	0.635953	−	0.771728i	$-0.280607\pi$
$577$	30.5522	1.27191	0.635953	+	0.771728i	$0.280607\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−30.5275	−1.26649
$582$	0	0
$583$	57.6526	2.38773
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.3635	−0.840490	−0.420245	−	0.907411i	$-0.638056\pi$
$587$	−20.3635	−0.840490	−0.420245	+	0.907411i	$0.638056\pi$
$588$	0	0
$589$	−19.4733	−0.802383
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−15.5199	−0.637326	−0.318663	−	0.947868i	$-0.603234\pi$
$593$	−15.5199	−0.637326	−0.318663	+	0.947868i	$0.603234\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	29.2576	1.19543	0.597717	−	0.801707i	$-0.296075\pi$
$599$	29.2576	1.19543	0.597717	+	0.801707i	$0.296075\pi$
$600$	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
$607$	−16.8407	−0.683544	−0.341772	−	0.939783i	$-0.611027\pi$
$607$	−16.8407	−0.683544	−0.341772	+	0.939783i	$0.611027\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.4401	−0.624640
$612$	0	0
$613$	−13.5954	−0.549113	−0.274556	−	0.961571i	$-0.588531\pi$
$613$	−13.5954	−0.549113	−0.274556	+	0.961571i	$0.588531\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.3134	1.18011	0.590056	−	0.807362i	$-0.299106\pi$
$617$	29.3134	1.18011	0.590056	+	0.807362i	$0.299106\pi$
$618$	0	0
$619$	12.1642	0.488921	0.244461	−	0.969659i	$-0.421389\pi$
$619$	12.1642	0.488921	0.244461	+	0.969659i	$0.421389\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.6608	0.547307
$624$	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
$637$	−9.43331	−0.373761
$638$	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
$643$	5.68346	0.224134	0.112067	−	0.993701i	$-0.464253\pi$
$643$	5.68346	0.224134	0.112067	+	0.993701i	$0.464253\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.3064	1.34872	0.674361	−	0.738401i	$-0.264419\pi$
$647$	34.3064	1.34872	0.674361	+	0.738401i	$0.264419\pi$
$648$	0	0
$649$	58.0126	2.27719
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.6297	−0.494239	−0.247120	−	0.968985i	$-0.579484\pi$
$653$	−12.6297	−0.494239	−0.247120	+	0.968985i	$0.579484\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−6.33394	−0.246735	−0.123368	−	0.992361i	$-0.539369\pi$
$659$	−6.33394	−0.246735	−0.123368	+	0.992361i	$0.539369\pi$
$660$	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
$667$	−0.979251	−0.0379167
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.59140	0.293063
$672$	0	0
$673$	−30.6546	−1.18165	−0.590824	−	0.806800i	$-0.701197\pi$
$673$	−30.6546	−1.18165	−0.590824	+	0.806800i	$0.701197\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.8716	−1.10963	−0.554813	−	0.831975i	$-0.687210\pi$
$677$	−28.8716	−1.10963	−0.554813	+	0.831975i	$0.687210\pi$
$678$	0	0
$679$	−7.38538	−0.283425
$680$	0	0
$681$	0	0
$682$	0	0
$683$	47.0352	1.79975	0.899875	−	0.436147i	$-0.143657\pi$
$683$	47.0352	1.79975	0.899875	+	0.436147i	$0.143657\pi$
$684$	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
$697$	−12.1896	−0.461714
$698$	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
$703$	12.4307	0.468831
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.1168	0.380480
$708$	0	0
$709$	15.3909	0.578016	0.289008	−	0.957327i	$-0.406675\pi$
$709$	15.3909	0.578016	0.289008	+	0.957327i	$0.406675\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.330292	−0.0123695
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.5846	0.954144	0.477072	−	0.878864i	$-0.341698\pi$
$719$	25.5846	0.954144	0.477072	+	0.878864i	$0.341698\pi$
$720$	0	0
$721$	−22.5721	−0.840630
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−6.13734	−0.227621	−0.113811	−	0.993502i	$-0.536306\pi$
$727$	−6.13734	−0.227621	−0.113811	+	0.993502i	$0.536306\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.98971	0.369483
$732$	0	0
$733$	−5.13040	−0.189496	−0.0947478	−	0.995501i	$-0.530204\pi$
$733$	−5.13040	−0.189496	−0.0947478	+	0.995501i	$0.530204\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−54.4754	−2.00663
$738$	0	0
$739$	18.8784	0.694454	0.347227	−	0.937781i	$-0.387123\pi$
$739$	18.8784	0.694454	0.347227	+	0.937781i	$0.387123\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.9569	−0.805519	−0.402759	−	0.915306i	$-0.631949\pi$
$743$	−21.9569	−0.805519	−0.402759	+	0.915306i	$0.631949\pi$
$744$	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
$757$	−39.1753	−1.42385	−0.711926	−	0.702255i	$-0.752177\pi$
$757$	−39.1753	−1.42385	−0.711926	+	0.702255i	$0.752177\pi$
$758$	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
$763$	−61.6904	−2.23334
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.1916	0.981832
$768$	0	0
$769$	21.1579	0.762974	0.381487	−	0.924374i	$-0.375412\pi$
$769$	21.1579	0.762974	0.381487	+	0.924374i	$0.375412\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−43.2543	−1.55575	−0.777874	−	0.628420i	$-0.783702\pi$
$773$	−43.2543	−1.55575	−0.777874	+	0.628420i	$0.783702\pi$
$774$	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
$787$	1.17997	0.0420615	0.0210307	−	0.999779i	$-0.493305\pi$
$787$	1.17997	0.0420615	0.0210307	+	0.999779i	$0.493305\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.278204	0.00989180
$792$	0	0
$793$	3.55823	0.126357
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.14281	0.0404805	0.0202403	−	0.999795i	$-0.493557\pi$
$797$	1.14281	0.0404805	0.0202403	+	0.999795i	$0.493557\pi$
$798$	0	0
$799$	10.4904	0.371123
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−23.8593	−0.841977
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−11.3723	−0.399828	−0.199914	−	0.979813i	$-0.564066\pi$
$809$	−11.3723	−0.399828	−0.199914	+	0.979813i	$0.564066\pi$
$810$	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
$817$	−45.3037	−1.58498
$818$	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
$823$	−7.32084	−0.255188	−0.127594	−	0.991826i	$-0.540725\pi$
$823$	−7.32084	−0.255188	−0.127594	+	0.991826i	$0.540725\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.8401	0.516040	0.258020	−	0.966140i	$-0.416930\pi$
$827$	14.8401	0.516040	0.258020	+	0.966140i	$0.416930\pi$
$828$	0	0
$829$	−23.3346	−0.810444	−0.405222	−	0.914218i	$-0.632806\pi$
$829$	−23.3346	−0.810444	−0.405222	+	0.914218i	$0.632806\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.40921	0.222066
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.6765	−0.506690	−0.253345	−	0.967376i	$-0.581531\pi$
$839$	−14.6765	−0.506690	−0.253345	+	0.967376i	$0.581531\pi$
$840$	0	0
$841$	53.4788	1.84410
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	28.4925	0.979014
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.210840	0.00722751
$852$	0	0
$853$	−24.5222	−0.839624	−0.419812	−	0.907611i	$-0.637904\pi$
$853$	−24.5222	−0.839624	−0.419812	+	0.907611i	$0.637904\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.58793	−0.190880	−0.0954400	−	0.995435i	$-0.530426\pi$
$857$	−5.58793	−0.190880	−0.0954400	+	0.995435i	$0.530426\pi$
$858$	0	0
$859$	−40.0619	−1.36689	−0.683447	−	0.730001i	$-0.739520\pi$
$859$	−40.0619	−1.36689	−0.683447	+	0.730001i	$0.739520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.2157	0.926432	0.463216	−	0.886246i	$-0.346695\pi$
$863$	27.2157	0.926432	0.463216	+	0.886246i	$0.346695\pi$
$864$	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
$877$	23.8799	0.806366	0.403183	−	0.915119i	$-0.367904\pi$
$877$	23.8799	0.806366	0.403183	+	0.915119i	$0.367904\pi$
$878$	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
$883$	−28.3449	−0.953881	−0.476941	−	0.878936i	$-0.658254\pi$
$883$	−28.3449	−0.953881	−0.476941	+	0.878936i	$0.658254\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.515088	−0.0172950	−0.00864749	−	0.999963i	$-0.502753\pi$
$887$	−0.515088	−0.0172950	−0.00864749	+	0.999963i	$0.502753\pi$
$888$	0	0
$889$	−73.4144	−2.46224
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−47.5743	−1.59201
$894$	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
$907$	−2.16421	−0.0718615	−0.0359308	−	0.999354i	$-0.511440\pi$
$907$	−2.16421	−0.0718615	−0.0359308	+	0.999354i	$0.511440\pi$
$908$	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
$913$	39.5015	1.30731
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−36.6979	−1.21187
$918$	0	0
$919$	18.6992	0.616830	0.308415	−	0.951252i	$-0.400201\pi$
$919$	18.6992	0.616830	0.308415	+	0.951252i	$0.400201\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	15.4939	0.509986
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.20671	−0.0723997	−0.0361999	−	0.999345i	$-0.511525\pi$
$929$	−2.20671	−0.0723997	−0.0361999	+	0.999345i	$0.511525\pi$
$930$	0	0
$931$	−29.0660	−0.952600
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.3429	0.697242	0.348621	−	0.937264i	$-0.386650\pi$
$937$	21.3429	0.697242	0.348621	+	0.937264i	$0.386650\pi$
$938$	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
$943$	0.937627	0.0305333
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.6889	0.769787	0.384894	−	0.922961i	$-0.374238\pi$
$947$	23.6889	0.769787	0.384894	+	0.922961i	$0.374238\pi$
$948$	0	0
$949$	−11.1833	−0.363026
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.8125	0.415038	0.207519	−	0.978231i	$-0.433461\pi$
$953$	12.8125	0.415038	0.207519	+	0.978231i	$0.433461\pi$
$954$	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
$967$	10.9265	0.351373	0.175686	−	0.984446i	$-0.443786\pi$
$967$	10.9265	0.351373	0.175686	+	0.984446i	$0.443786\pi$
$968$	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
$973$	49.7001	1.59331
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37.4980	−1.19967	−0.599833	−	0.800125i	$-0.704767\pi$
$977$	−37.4980	−1.19967	−0.599833	+	0.800125i	$0.704767\pi$
$978$	0	0
$979$	−17.6765	−0.564944
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23.5322	0.750560	0.375280	−	0.926911i	$-0.377547\pi$
$983$	23.5322	0.750560	0.375280	+	0.926911i	$0.377547\pi$
$984$	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
$997$	42.8580	1.35733	0.678663	−	0.734450i	$-0.262560\pi$
$997$	42.8580	1.35733	0.678663	+	0.734450i	$0.262560\pi$
$998$	0	0
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q+3.40179 q^{7} -4.40179 q^{11} -2.06320 q^{13} +1.40179 q^{17} -6.35717 q^{19} -0.107826 q^{23} +9.08178 q^{29} +3.06320 q^{31} -1.95538 q^{37} -8.69575 q^{41} +7.12641 q^{43} +7.48357 q^{47} +4.57217 q^{49} -13.0975 q^{53} -13.1793 q^{59} -1.72462 q^{61} +12.3757 q^{67} -7.50961 q^{71} +5.42037 q^{73} -14.9740 q^{77} +9.43613 q^{79} -8.97396 q^{83} +4.01576 q^{89} -7.01858 q^{91} -2.17103 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{7} - 3 q^{11} + 2 q^{13} - 9 q^{17} - 4 q^{19} + 3 q^{23} + 9 q^{29} + 2 q^{31} - q^{37} - 9 q^{41} + 8 q^{43} - 12 q^{47} + 9 q^{49} - 12 q^{53} + 15 q^{59} - q^{61} + 11 q^{67} - 12 q^{71}+ \cdots + 5 q^{97}+O(q^{100}) \) 4 * q - q^7 - 3 * q^11 + 2 * q^13 - 9 * q^17 - 4 * q^19 + 3 * q^23 + 9 * q^29 + 2 * q^31 - q^37 - 9 * q^41 + 8 * q^43 - 12 * q^47 + 9 * q^49 - 12 * q^53 + 15 * q^59 - q^61 + 11 * q^67 - 12 * q^71 - 10 * q^73 - 36 * q^77 - 7 * q^79 - 12 * q^83 + 3 * q^89 - 11 * q^91 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more