LMFDB - Embedded newform 324.3.k.a.125.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,3,Mod(17,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 11]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("324.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	324.k (of order $18$, degree $6$, 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:	$8.82836056527$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$6$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			125.3
Character	$\chi$	$=$	324.125
Dual form			324.3.k.a.197.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.740753 - 2.03520i) q^{5} +(1.08248 + 6.13906i) q^{7} +O(q^{10})$	$q+(-0.740753 - 2.03520i) q^{5} +(1.08248 + 6.13906i) q^{7} +(5.10694 - 14.0312i) q^{11} +(9.95988 + 8.35733i) q^{13} +(3.36308 + 1.94167i) q^{17} +(6.39866 + 11.0828i) q^{19} +(35.3663 + 6.23604i) q^{23} +(15.5578 - 13.0545i) q^{25} +(-7.13197 - 8.49955i) q^{29} +(6.25720 - 35.4864i) q^{31} +(11.6924 - 6.75059i) q^{35} +(-16.3431 + 28.3071i) q^{37} +(14.5164 - 17.3000i) q^{41} +(58.2594 + 21.2047i) q^{43} +(3.36255 - 0.592908i) q^{47} +(9.52866 - 3.46815i) q^{49} +79.3144i q^{53} -32.3393 q^{55} +(-7.36318 - 20.2302i) q^{59} +(13.7025 + 77.7109i) q^{61} +(9.63104 - 26.4611i) q^{65} +(-84.5379 - 70.9357i) q^{67} +(-71.2183 - 41.1179i) q^{71} +(-3.09644 - 5.36320i) q^{73} +(91.6666 + 16.1633i) q^{77} +(49.2237 - 41.3036i) q^{79} +(-97.6080 - 116.325i) q^{83} +(1.46049 - 8.28284i) q^{85} +(-97.5489 + 56.3199i) q^{89} +(-40.5248 + 70.1909i) q^{91} +(17.8159 - 21.2322i) q^{95} +(-52.0684 - 18.9513i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 9 q^{5}+O(q^{10}) $ 36 * q + 9 * q^5	$ 36 q + 9 q^{5} - 36 q^{11} + 18 q^{23} - 9 q^{25} + 18 q^{29} + 45 q^{31} + 243 q^{35} + 198 q^{41} + 90 q^{43} + 243 q^{47} + 72 q^{49} - 252 q^{59} - 144 q^{61} - 747 q^{65} + 108 q^{67} - 324 q^{71} - 63 q^{73} - 495 q^{77} + 36 q^{79} + 27 q^{83} - 180 q^{85} + 567 q^{89} + 99 q^{91} + 1044 q^{95} - 216 q^{97}+O(q^{100}) $ 36 * q + 9 * q^5 - 36 * q^11 + 18 * q^23 - 9 * q^25 + 18 * q^29 + 45 * q^31 + 243 * q^35 + 198 * q^41 + 90 * q^43 + 243 * q^47 + 72 * q^49 - 252 * q^59 - 144 * q^61 - 747 * q^65 + 108 * q^67 - 324 * q^71 - 63 * q^73 - 495 * q^77 + 36 * q^79 + 27 * q^83 - 180 * q^85 + 567 * q^89 + 99 * q^91 + 1044 * q^95 - 216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{5}{18}\right)$

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
$5$	−0.740753	−	2.03520i	−0.148151	−	0.407040i	0.843313	−	0.537422i	$-0.180602\pi$
$5$	−0.740753	−	2.03520i	−0.148151	−	0.407040i	−0.991464	+	0.130382i	$0.958380\pi$
$6$		0			0
$7$	1.08248	+	6.13906i	0.154640	+	0.877008i	0.959114	+	0.283020i	$0.0913363\pi$
$7$	1.08248	+	6.13906i	0.154640	+	0.877008i	−0.804474	+	0.593988i	$0.797553\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.10694	−	14.0312i	0.464267	−	1.27556i	−0.457979	−	0.888963i	$-0.651427\pi$
$11$	5.10694	−	14.0312i	0.464267	−	1.27556i	0.922247	−	0.386601i	$-0.126351\pi$
$12$		0			0
$13$	9.95988	+	8.35733i	0.766144	+	0.642871i	0.939718	−	0.341949i	$-0.111087\pi$
$13$	9.95988	+	8.35733i	0.766144	+	0.642871i	−0.173574	+	0.984821i	$0.555532\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.36308	+	1.94167i	0.197828	+	0.114216i	0.595642	−	0.803250i	$-0.296898\pi$
$17$	3.36308	+	1.94167i	0.197828	+	0.114216i	−0.397814	+	0.917466i	$0.630231\pi$
$18$		0			0
$19$	6.39866	+	11.0828i	0.336772	+	0.583306i	0.983824	−	0.179140i	$-0.0573316\pi$
$19$	6.39866	+	11.0828i	0.336772	+	0.583306i	−0.647052	+	0.762446i	$0.723998\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	35.3663	+	6.23604i	1.53767	+	0.271132i	0.877348	−	0.479854i	$-0.159310\pi$
$23$	35.3663	+	6.23604i	1.53767	+	0.271132i	0.660318	+	0.750986i	$0.270422\pi$
$24$		0			0
$25$	15.5578	−	13.0545i	0.622311	−	0.522181i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.13197	−	8.49955i	−0.245930	−	0.293088i	0.628932	−	0.777460i	$-0.283492\pi$
$29$	−7.13197	−	8.49955i	−0.245930	−	0.293088i	−0.874862	+	0.484372i	$0.839048\pi$
$30$		0			0
$31$	6.25720	−	35.4864i	0.201845	−	1.14472i	−0.700482	−	0.713670i	$-0.747032\pi$
$31$	6.25720	−	35.4864i	0.201845	−	1.14472i	0.902327	−	0.431052i	$-0.141857\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	11.6924	−	6.75059i	0.334068	−	0.192874i
$36$		0			0
$37$	−16.3431	+	28.3071i	−0.441705	+	0.765056i	−0.997816	−	0.0660523i	$-0.978960\pi$
$37$	−16.3431	+	28.3071i	−0.441705	+	0.765056i	0.556111	+	0.831108i	$0.312293\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	14.5164	−	17.3000i	0.354059	−	0.421951i	−0.559390	−	0.828905i	$-0.688964\pi$
$41$	14.5164	−	17.3000i	0.354059	−	0.421951i	0.913449	+	0.406954i	$0.133409\pi$
$42$		0			0
$43$	58.2594	+	21.2047i	1.35487	+	0.493133i	0.914465	−	0.404666i	$-0.132612\pi$
$43$	58.2594	+	21.2047i	1.35487	+	0.493133i	0.440406	+	0.897799i	$0.354834\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.36255	−	0.592908i	0.0715435	−	0.0126151i	−0.137762	−	0.990465i	$-0.543991\pi$
$47$	3.36255	−	0.592908i	0.0715435	−	0.0126151i	0.209305	+	0.977850i	$0.432880\pi$
$48$		0			0
$49$	9.52866	−	3.46815i	0.194462	−	0.0707785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			79.3144i			1.49650i	0.663418	+	0.748249i	$0.269105\pi$
$53$			79.3144i			1.49650i	−0.663418	+	0.748249i	$0.730895\pi$
$54$		0			0
$55$	−32.3393			−0.587987
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.36318	−	20.2302i	−0.124800	−	0.342884i	0.861521	−	0.507722i	$-0.169512\pi$
$59$	−7.36318	−	20.2302i	−0.124800	−	0.342884i	−0.986321	+	0.164838i	$0.947290\pi$
$60$		0			0
$61$	13.7025	+	77.7109i	0.224632	+	1.27395i	0.863388	+	0.504540i	$0.168338\pi$
$61$	13.7025	+	77.7109i	0.224632	+	1.27395i	−0.638757	+	0.769409i	$0.720551\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	9.63104	−	26.4611i	0.148170	−	0.407093i
$66$		0			0
$67$	−84.5379	−	70.9357i	−1.26176	−	1.05874i	−0.995492	−	0.0948429i	$-0.969765\pi$
$67$	−84.5379	−	70.9357i	−1.26176	−	1.05874i	−0.266267	−	0.963899i	$-0.585790\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−71.2183	−	41.1179i	−1.00307	−	0.579125i	−0.0939180	−	0.995580i	$-0.529939\pi$
$71$	−71.2183	−	41.1179i	−1.00307	−	0.579125i	−0.909157	+	0.416455i	$0.863272\pi$
$72$		0			0
$73$	−3.09644	−	5.36320i	−0.0424170	−	0.0734685i	0.844037	−	0.536284i	$-0.180172\pi$
$73$	−3.09644	−	5.36320i	−0.0424170	−	0.0734685i	−0.886454	+	0.462816i	$0.846839\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	91.6666	+	16.1633i	1.19048	+	0.209913i
$78$		0			0
$79$	49.2237	−	41.3036i	0.623085	−	0.522830i	−0.275687	−	0.961248i	$-0.588905\pi$
$79$	49.2237	−	41.3036i	0.623085	−	0.522830i	0.898772	+	0.438417i	$0.144461\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−97.6080	−	116.325i	−1.17600	−	1.40150i	−0.897470	−	0.441075i	$-0.854597\pi$
$83$	−97.6080	−	116.325i	−1.17600	−	1.40150i	−0.278530	−	0.960428i	$-0.589847\pi$
$84$		0			0
$85$	1.46049	−	8.28284i	0.0171822	−	0.0974451i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−97.5489	+	56.3199i	−1.09606	+	0.632808i	−0.935182	−	0.354167i	$-0.884764\pi$
$89$	−97.5489	+	56.3199i	−1.09606	+	0.632808i	−0.160873	+	0.986975i	$0.551431\pi$
$90$		0			0
$91$	−40.5248	+	70.1909i	−0.445327	+	0.771329i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	17.8159	−	21.2322i	0.187536	−	0.223497i
$96$		0			0
$97$	−52.0684	−	18.9513i	−0.536787	−	0.195375i	0.0593790	−	0.998236i	$-0.481088\pi$
$97$	−52.0684	−	18.9513i	−0.536787	−	0.195375i	−0.596166	+	0.802861i	$0.703310\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−77.5927	+	13.6817i	−0.768245	+	0.135462i	−0.544017	−	0.839074i	$-0.683097\pi$
$101$	−77.5927	+	13.6817i	−0.768245	+	0.135462i	−0.224228	+	0.974537i	$0.571986\pi$
$102$		0			0
$103$	49.2174	−	17.9137i	0.477839	−	0.173919i	−0.0918613	−	0.995772i	$-0.529282\pi$
$103$	49.2174	−	17.9137i	0.477839	−	0.173919i	0.569700	+	0.821853i	$0.307059\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			189.920i			1.77495i	0.460854	+	0.887476i	$0.347543\pi$
$107$			189.920i			1.77495i	−0.460854	+	0.887476i	$0.652457\pi$
$108$		0			0
$109$	−161.394			−1.48068			−0.740338	−	0.672234i	$-0.765335\pi$
$109$	−161.394			−1.48068			−0.740338	+	0.672234i	$0.765335\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	71.3701	+	196.088i	0.631594	+	1.73529i	0.676649	+	0.736306i	$0.263432\pi$
$113$	71.3701	+	196.088i	0.631594	+	1.73529i	−0.0450545	+	0.998985i	$0.514346\pi$
$114$		0			0
$115$	−13.5061	−	76.5969i	−0.117444	−	0.666060i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.27958	+	22.7480i	−0.0695763	+	0.191159i
$120$		0			0
$121$	−78.1025	−	65.5358i	−0.645475	−	0.541618i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−84.9844	−	49.0657i	−0.679875	−	0.392526i
$126$		0			0
$127$	−28.4088	−	49.2055i	−0.223691	−	0.387445i	0.732235	−	0.681052i	$-0.238477\pi$
$127$	−28.4088	−	49.2055i	−0.223691	−	0.387445i	−0.955926	+	0.293608i	$0.905144\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−136.396	−	24.0503i	−1.04119	−	0.183590i	−0.373194	−	0.927753i	$-0.621737\pi$
$131$	−136.396	−	24.0503i	−1.04119	−	0.183590i	−0.667998	+	0.744163i	$0.732849\pi$
$132$		0			0
$133$	−61.1116	+	51.2787i	−0.459486	+	0.385554i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	34.3399	+	40.9247i	0.250656	+	0.298720i	0.876671	−	0.481091i	$-0.159759\pi$
$137$	34.3399	+	40.9247i	0.250656	+	0.298720i	−0.626014	+	0.779811i	$0.715315\pi$
$138$		0			0
$139$	34.7093	−	196.846i	0.249707	−	1.41616i	−0.559595	−	0.828767i	$-0.689043\pi$
$139$	34.7093	−	196.846i	0.249707	−	1.41616i	0.809302	−	0.587393i	$-0.199846\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	168.128	−	97.0687i	1.17572	−	0.678802i
$144$		0			0
$145$	−12.0153	+	20.8111i	−0.0828639	+	0.143525i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	43.0873	−	51.3494i	0.289176	−	0.344627i	−0.601825	−	0.798628i	$-0.705559\pi$
$149$	43.0873	−	51.3494i	0.289176	−	0.344627i	0.891001	+	0.454001i	$0.150004\pi$
$150$		0			0
$151$	209.460	+	76.2371i	1.38715	+	0.504881i	0.924338	−	0.381575i	$-0.124618\pi$
$151$	209.460	+	76.2371i	1.38715	+	0.504881i	0.462812	+	0.886456i	$0.346840\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−76.8569	+	13.5520i	−0.495851	+	0.0874320i
$156$		0			0
$157$	−113.832	+	41.4314i	−0.725043	+	0.263894i	−0.678065	−	0.735002i	$-0.737181\pi$
$157$	−113.832	+	41.4314i	−0.725043	+	0.263894i	−0.0469779	+	0.998896i	$0.514959\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			223.866i			1.39047i
$162$		0			0
$163$	70.9962			0.435560			0.217780	−	0.975998i	$-0.430119\pi$
$163$	70.9962			0.435560			0.217780	+	0.975998i	$0.430119\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−57.0403	−	156.717i	−0.341559	−	0.938425i	−0.984943	−	0.172882i	$-0.944692\pi$
$167$	−57.0403	−	156.717i	−0.341559	−	0.938425i	0.643384	−	0.765544i	$-0.277530\pi$
$168$		0			0
$169$	0.00765212	+	0.0433973i	4.52788e−5	+	0.000256789i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	78.3577	−	215.286i	0.452935	−	1.24443i	−0.477714	−	0.878515i	$-0.658535\pi$
$173$	78.3577	−	215.286i	0.452935	−	1.24443i	0.930649	−	0.365913i	$-0.119243\pi$
$174$		0			0
$175$	96.9835	+	81.3789i	0.554192	+	0.465022i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−129.909	−	75.0033i	−0.725751	−	0.419013i	0.0911147	−	0.995840i	$-0.470957\pi$
$179$	−129.909	−	75.0033i	−0.725751	−	0.419013i	−0.816866	+	0.576828i	$0.804290\pi$
$180$		0			0
$181$	74.7777	+	129.519i	0.413137	+	0.715574i	0.995231	−	0.0975476i	$-0.0310998\pi$
$181$	74.7777	+	129.519i	0.413137	+	0.715574i	−0.582094	+	0.813121i	$0.697767\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	69.7168	+	12.2929i	0.376847	+	0.0664483i
$186$		0			0
$187$	44.4191	−	37.2720i	0.237535	−	0.199316i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	48.7526	+	58.1011i	0.255249	+	0.304194i	0.878418	−	0.477893i	$-0.158599\pi$
$191$	48.7526	+	58.1011i	0.255249	+	0.304194i	−0.623169	+	0.782087i	$0.714155\pi$
$192$		0			0
$193$	−23.6459	+	134.103i	−0.122518	+	0.694833i	0.860234	+	0.509900i	$0.170318\pi$
$193$	−23.6459	+	134.103i	−0.122518	+	0.694833i	−0.982751	+	0.184932i	$0.940793\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	285.910	−	165.070i	1.45132	−	0.837919i	0.452762	−	0.891631i	$-0.350439\pi$
$197$	285.910	−	165.070i	1.45132	−	0.837919i	0.998556	+	0.0537122i	$0.0171054\pi$
$198$		0			0
$199$	−29.8782	+	51.7505i	−0.150141	+	0.260053i	−0.931279	−	0.364306i	$-0.881306\pi$
$199$	−29.8782	+	51.7505i	−0.150141	+	0.260053i	0.781138	+	0.624359i	$0.214640\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	44.4590	−	52.9842i	0.219010	−	0.261006i
$204$		0			0
$205$	−45.9621	−	16.7288i	−0.224205	−	0.0816040i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	188.183	−	33.1817i	0.900396	−	0.158764i
$210$		0			0
$211$	10.6114	−	3.86224i	0.0502911	−	0.0183045i	−0.316752	−	0.948508i	$-0.602592\pi$
$211$	10.6114	−	3.86224i	0.0502911	−	0.0183045i	0.367043	+	0.930204i	$0.380370\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	134.277i		−	0.624545i
$216$		0			0
$217$	224.626			1.03514
$218$		0			0
$219$		0			0
$220$		0			0
$221$	17.2686	+	47.4452i	0.0781386	+	0.214684i
$222$		0			0
$223$	19.2101	+	108.946i	0.0861441	+	0.488547i	0.997104	+	0.0760521i	$0.0242315\pi$
$223$	19.2101	+	108.946i	0.0861441	+	0.488547i	−0.910960	+	0.412495i	$0.864657\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−112.933	+	310.280i	−0.497500	+	1.36687i	0.396182	+	0.918172i	$0.370335\pi$
$227$	−112.933	+	310.280i	−0.497500	+	1.36687i	−0.893683	+	0.448699i	$0.851887\pi$
$228$		0			0
$229$	346.931	+	291.110i	1.51498	+	1.27122i	0.853273	+	0.521464i	$0.174614\pi$
$229$	346.931	+	291.110i	1.51498	+	1.27122i	0.661711	+	0.749759i	$0.269831\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−312.892	−	180.648i	−1.34289	−	0.775315i	−0.355655	−	0.934617i	$-0.615742\pi$
$233$	−312.892	−	180.648i	−1.34289	−	0.775315i	−0.987230	+	0.159302i	$0.949076\pi$
$234$		0			0
$235$	−3.69750	−	6.40426i	−0.0157340	−	0.0272522i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	125.627	+	22.1514i	0.525636	+	0.0926838i	0.430168	−	0.902749i	$-0.358454\pi$
$239$	125.627	+	22.1514i	0.525636	+	0.0926838i	0.0954677	+	0.995433i	$0.469565\pi$
$240$		0			0
$241$	−100.014	+	83.9215i	−0.414995	+	0.348222i	−0.826255	−	0.563296i	$-0.809533\pi$
$241$	−100.014	+	83.9215i	−0.414995	+	0.348222i	0.411260	+	0.911518i	$0.365089\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14.1168	−	16.8237i	−0.0576194	−	0.0686681i
$246$		0			0
$247$	−28.8928	+	163.859i	−0.116975	+	0.663397i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	376.247	−	217.226i	1.49899	−	0.865444i	0.498994	−	0.866606i	$-0.333703\pi$
$251$	376.247	−	217.226i	1.49899	−	0.865444i	0.999999	+	0.00116146i	$0.000369704\pi$
$252$		0			0
$253$	268.113	−	464.385i	1.05973	−	1.83551i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−104.428	+	124.452i	−0.406334	+	0.484250i	−0.929941	−	0.367710i	$-0.880142\pi$
$257$	−104.428	+	124.452i	−0.406334	+	0.484250i	0.523606	+	0.851960i	$0.324586\pi$
$258$		0			0
$259$	−191.470	−	69.6893i	−0.739266	−	0.269071i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	297.767	−	52.5043i	1.13219	−	0.199636i	0.424005	−	0.905660i	$-0.360624\pi$
$263$	297.767	−	52.5043i	1.13219	−	0.199636i	0.708188	+	0.706024i	$0.249513\pi$
$264$		0			0
$265$	161.421	−	58.7524i	0.609135	−	0.221707i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			252.041i			0.936956i	0.883475	+	0.468478i	$0.155197\pi$
$269$			252.041i			0.936956i	−0.883475	+	0.468478i	$0.844803\pi$
$270$		0			0
$271$	107.217			0.395635			0.197818	−	0.980239i	$-0.436615\pi$
$271$	107.217			0.395635			0.197818	+	0.980239i	$0.436615\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−103.718	−	284.963i	−0.377157	−	1.03623i
$276$		0			0
$277$	−46.8272	−	265.570i	−0.169051	−	0.958738i	−0.944788	−	0.327681i	$-0.893733\pi$
$277$	−46.8272	−	265.570i	−0.169051	−	0.958738i	0.775737	−	0.631056i	$-0.217378\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.79982	−	15.9349i	0.0206399	−	0.0567077i	−0.928945	−	0.370218i	$-0.879283\pi$
$281$	5.79982	−	15.9349i	0.0206399	−	0.0567077i	0.949585	+	0.313511i	$0.101505\pi$
$282$		0			0
$283$	−250.525	−	210.215i	−0.885247	−	0.742810i	0.0820041	−	0.996632i	$-0.473868\pi$
$283$	−250.525	−	210.215i	−0.885247	−	0.742810i	−0.967251	+	0.253822i	$0.918312\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	121.919	+	70.3903i	0.424807	+	0.245262i
$288$		0			0
$289$	−136.960	−	237.221i	−0.473909	−	0.820835i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−264.414	−	46.6234i	−0.902437	−	0.159124i	−0.296870	−	0.954918i	$-0.595943\pi$
$293$	−264.414	−	46.6234i	−0.902437	−	0.159124i	−0.605568	+	0.795794i	$0.707054\pi$
$294$		0			0
$295$	−35.7182	+	29.9711i	−0.121078	+	0.101597i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	300.128	+	357.678i	1.00377	+	1.19625i
$300$		0			0
$301$	−67.1121	+	380.612i	−0.222964	+	1.26449i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	148.007	−	85.4519i	0.485269	−	0.280170i
$306$		0			0
$307$	−51.1411	+	88.5790i	−0.166583	+	0.288531i	−0.937216	−	0.348748i	$-0.886607\pi$
$307$	−51.1411	+	88.5790i	−0.166583	+	0.288531i	0.770633	+	0.637279i	$0.219940\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−300.925	+	358.629i	−0.967605	+	1.15315i	0.0205661	+	0.999788i	$0.493453\pi$
$311$	−300.925	+	358.629i	−0.967605	+	1.15315i	−0.988171	+	0.153358i	$0.950991\pi$
$312$		0			0
$313$	−173.348	−	63.0934i	−0.553827	−	0.201576i	0.0499192	−	0.998753i	$-0.484104\pi$
$313$	−173.348	−	63.0934i	−0.553827	−	0.201576i	−0.603746	+	0.797177i	$0.706326\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−96.3583	+	16.9906i	−0.303969	+	0.0535980i	−0.323552	−	0.946210i	$-0.604877\pi$
$317$	−96.3583	+	16.9906i	−0.303969	+	0.0535980i	0.0195831	+	0.999808i	$0.493766\pi$
$318$		0			0
$319$	−155.682	+	56.6634i	−0.488030	+	0.177628i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			49.6965i			0.153859i
$324$		0			0
$325$	264.055			0.812476
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.27979	+	20.0011i	0.0221270	+	0.0607935i
$330$		0			0
$331$	−18.3892	−	104.290i	−0.0555564	−	0.315076i	0.944347	−	0.328950i	$-0.106695\pi$
$331$	−18.3892	−	104.290i	−0.0555564	−	0.315076i	−0.999904	+	0.0138738i	$0.995584\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−81.7468	+	224.597i	−0.244020	+	0.670440i
$336$		0			0
$337$	−3.17508	−	2.66420i	−0.00942159	−	0.00790565i	0.638065	−	0.769983i	$-0.279735\pi$
$337$	−3.17508	−	2.66420i	−0.00942159	−	0.00790565i	−0.647486	+	0.762077i	$0.724180\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−465.961	−	269.023i	−1.36646	−	0.788924i
$342$		0			0
$343$	184.333	+	319.274i	0.537414	+	0.930828i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−387.103	−	68.2568i	−1.11557	−	0.196705i	−0.414676	−	0.909969i	$-0.636105\pi$
$347$	−387.103	−	68.2568i	−1.11557	−	0.196705i	−0.700896	+	0.713264i	$0.747216\pi$
$348$		0			0
$349$	−246.286	+	206.658i	−0.705690	+	0.592144i	−0.923386	−	0.383872i	$-0.874590\pi$
$349$	−246.286	+	206.658i	−0.705690	+	0.592144i	0.217696	+	0.976017i	$0.430146\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−112.600	−	134.192i	−0.318982	−	0.380147i	0.582598	−	0.812760i	$-0.302036\pi$
$353$	−112.600	−	134.192i	−0.318982	−	0.380147i	−0.901580	+	0.432613i	$0.857592\pi$
$354$		0			0
$355$	−30.9281	+	175.402i	−0.0871213	+	0.494089i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	455.354	−	262.899i	1.26840	−	0.732309i	0.293711	−	0.955894i	$-0.405110\pi$
$359$	455.354	−	262.899i	1.26840	−	0.732309i	0.974684	+	0.223586i	$0.0717762\pi$
$360$		0			0
$361$	98.6142	−	170.805i	0.273170	−	0.473143i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.62149	+	10.2747i	−0.0236205	+	0.0281498i
$366$		0			0
$367$	−304.854	−	110.958i	−0.830664	−	0.302337i	−0.108533	−	0.994093i	$-0.534615\pi$
$367$	−304.854	−	110.958i	−0.830664	−	0.302337i	−0.722131	+	0.691756i	$0.756837\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−486.916	+	85.8564i	−1.31244	+	0.231419i
$372$		0			0
$373$	227.629	−	82.8503i	0.610267	−	0.222119i	−0.0183533	−	0.999832i	$-0.505842\pi$
$373$	227.629	−	82.8503i	0.610267	−	0.222119i	0.628620	+	0.777713i	$0.283620\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	144.259i		−	0.382649i
$378$		0			0
$379$	−605.266			−1.59701			−0.798504	−	0.601989i	$-0.794375\pi$
$379$	−605.266			−1.59701			−0.798504	+	0.601989i	$0.794375\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−92.4520	−	254.010i	−0.241389	−	0.663211i	−0.999933	−	0.0115889i	$-0.996311\pi$
$383$	−92.4520	−	254.010i	−0.241389	−	0.663211i	0.758544	−	0.651622i	$-0.225911\pi$
$384$		0			0
$385$	−35.0067	−	198.533i	−0.0909265	−	0.515670i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−56.4976	+	155.226i	−0.145238	+	0.399038i	−0.990886	−	0.134703i	$-0.956992\pi$
$389$	−56.4976	+	155.226i	−0.145238	+	0.399038i	0.845648	+	0.533741i	$0.179214\pi$
$390$		0			0
$391$	106.831	+	89.6421i	0.273226	+	0.229264i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−120.524	−	69.5844i	−0.305123	−	0.176163i
$396$		0			0
$397$	−192.854	−	334.032i	−0.485778	−	0.841392i	0.514089	−	0.857737i	$-0.328130\pi$
$397$	−192.854	−	334.032i	−0.485778	−	0.841392i	−0.999866	+	0.0163455i	$0.994797\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−238.103	−	41.9840i	−0.593773	−	0.104698i	−0.131317	−	0.991340i	$-0.541921\pi$
$401$	−238.103	−	41.9840i	−0.593773	−	0.104698i	−0.462456	+	0.886642i	$0.653032\pi$
$402$		0			0
$403$	358.892	−	301.146i	0.890551	−	0.747261i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	313.719	+	373.876i	0.770808	+	0.918614i
$408$		0			0
$409$	26.9195	−	152.668i	0.0658180	−	0.373272i	−0.934052	−	0.357137i	$-0.883753\pi$
$409$	26.9195	−	152.668i	0.0658180	−	0.373272i	0.999870	−	0.0161348i	$-0.00513609\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	116.224	−	67.1018i	0.281413	−	0.162474i
$414$		0			0
$415$	−164.441	+	284.820i	−0.396243	+	0.686313i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−308.155	+	367.245i	−0.735453	+	0.876479i	−0.996034	−	0.0889728i	$-0.971642\pi$
$419$	−308.155	+	367.245i	−0.735453	+	0.876479i	0.260581	+	0.965452i	$0.416086\pi$
$420$		0			0
$421$	−647.463	−	235.657i	−1.53792	−	0.559756i	−0.572372	−	0.819994i	$-0.693977\pi$
$421$	−647.463	−	235.657i	−1.53792	−	0.559756i	−0.965545	+	0.260238i	$0.916199\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	77.6696	−	13.6953i	0.182752	−	0.0322241i
$426$		0			0
$427$	−462.239	+	168.241i	−1.08253	+	0.394008i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			506.298i			1.17471i	0.809331	+	0.587353i	$0.199830\pi$
$431$			506.298i			1.17471i	−0.809331	+	0.587353i	$0.800170\pi$
$432$		0			0
$433$	425.824			0.983427			0.491714	−	0.870757i	$-0.336371\pi$
$433$	425.824			0.983427			0.491714	+	0.870757i	$0.336371\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	157.184	+	431.861i	0.359690	+	0.988239i
$438$		0			0
$439$	79.7889	+	452.505i	0.181751	+	1.03076i	0.930059	+	0.367411i	$0.119756\pi$
$439$	79.7889	+	452.505i	0.181751	+	1.03076i	−0.748307	+	0.663353i	$0.769133\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.6549	+	51.2541i	−0.0421105	+	0.115698i	−0.958966	−	0.283523i	$-0.908497\pi$
$443$	−18.6549	+	51.2541i	−0.0421105	+	0.115698i	0.916855	+	0.399220i	$0.130719\pi$
$444$		0			0
$445$	186.882	+	156.813i	0.419959	+	0.352388i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−182.250	−	105.222i	−0.405902	−	0.234348i	0.283125	−	0.959083i	$-0.408629\pi$
$449$	−182.250	−	105.222i	−0.405902	−	0.234348i	−0.689027	+	0.724735i	$0.741962\pi$
$450$		0			0
$451$	−168.605	−	292.033i	−0.373848	−	0.647523i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	172.871	+	30.4819i	0.379937	+	0.0669932i
$456$		0			0
$457$	−150.910	+	126.628i	−0.330218	+	0.277086i	−0.792789	−	0.609497i	$-0.791372\pi$
$457$	−150.910	+	126.628i	−0.330218	+	0.277086i	0.462571	+	0.886582i	$0.346927\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−128.532	−	153.179i	−0.278812	−	0.332275i	0.608406	−	0.793626i	$-0.291809\pi$
$461$	−128.532	−	153.179i	−0.278812	−	0.332275i	−0.887218	+	0.461351i	$0.847365\pi$
$462$		0			0
$463$	67.1681	−	380.929i	0.145071	−	0.822741i	−0.822238	−	0.569143i	$-0.807275\pi$
$463$	67.1681	−	380.929i	0.145071	−	0.822741i	0.967310	−	0.253598i	$-0.0816139\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−300.701	+	173.610i	−0.643900	+	0.371756i	−0.786115	−	0.618080i	$-0.787911\pi$
$467$	−300.701	+	173.610i	−0.643900	+	0.371756i	0.142215	+	0.989836i	$0.454577\pi$
$468$		0			0
$469$	343.968	−	595.770i	0.733407	−	1.27030i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	595.055	−	709.159i	1.25804	−	1.49928i
$474$		0			0
$475$	244.230	+	88.8924i	0.514168	+	0.187142i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	239.163	−	42.1710i	0.499297	−	0.0880396i	0.0816727	−	0.996659i	$-0.473974\pi$
$479$	239.163	−	42.1710i	0.499297	−	0.0880396i	0.417625	+	0.908620i	$0.362863\pi$
$480$		0			0
$481$	−399.347	+	145.350i	−0.830242	+	0.302183i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			120.008i			0.247439i
$486$		0			0
$487$	294.321			0.604355			0.302177	−	0.953252i	$-0.402287\pi$
$487$	294.321			0.604355			0.302177	+	0.953252i	$0.402287\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	81.7251	+	224.538i	0.166446	+	0.457307i	0.994672	−	0.103087i	$-0.0328719\pi$
$491$	81.7251	+	224.538i	0.166446	+	0.457307i	−0.828226	+	0.560394i	$0.810650\pi$
$492$		0			0
$493$	−7.48201	−	42.4326i	−0.0151765	−	0.0860702i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	175.333	−	481.723i	0.352782	−	0.969261i
$498$		0			0
$499$	−184.698	−	154.980i	−0.370136	−	0.310581i	0.438679	−	0.898644i	$-0.355446\pi$
$499$	−184.698	−	154.980i	−0.370136	−	0.310581i	−0.808815	+	0.588063i	$0.799891\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	138.798	+	80.1352i	0.275941	+	0.159314i	0.631584	−	0.775307i	$-0.282405\pi$
$503$	138.798	+	80.1352i	0.275941	+	0.159314i	−0.355644	+	0.934622i	$0.615738\pi$
$504$		0			0
$505$	85.3220	+	147.782i	0.168954	+	0.292638i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−78.7326	−	13.8827i	−0.154681	−	0.0272744i	0.0957713	−	0.995403i	$-0.469468\pi$
$509$	−78.7326	−	13.8827i	−0.154681	−	0.0272744i	−0.250452	+	0.968129i	$0.580579\pi$
$510$		0			0
$511$	29.5731	−	24.8148i	0.0578731	−	0.0485613i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−72.9158	−	86.8977i	−0.141584	−	0.168733i
$516$		0			0
$517$	8.85312	−	50.2085i	0.0171240	−	0.0971151i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−599.564	+	346.158i	−1.15079	+	0.664411i	−0.949081	−	0.315033i	$-0.897984\pi$
$521$	−599.564	+	346.158i	−1.15079	+	0.664411i	−0.201713	+	0.979445i	$0.564651\pi$
$522$		0			0
$523$	−33.5131	+	58.0464i	−0.0640786	+	0.110987i	−0.896285	−	0.443479i	$-0.853744\pi$
$523$	−33.5131	+	58.0464i	−0.0640786	+	0.110987i	0.832206	+	0.554466i	$0.187077\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	89.9464	−	107.194i	0.170676	−	0.203404i
$528$		0			0
$529$	714.791	+	260.163i	1.35121	+	0.491801i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	289.164	−	50.9873i	0.542521	−	0.0956611i
$534$		0			0
$535$	386.525	−	140.684i	0.722477	−	0.262960i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	151.410i		−	0.280909i
$540$		0			0
$541$	−584.288			−1.08002			−0.540008	−	0.841660i	$-0.681579\pi$
$541$	−584.288			−1.08002			−0.540008	+	0.841660i	$0.681579\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	119.553	+	328.469i	0.219363	+	0.602695i
$546$		0			0
$547$	13.0135	+	73.8033i	0.0237907	+	0.134924i	0.994390	−	0.105778i	$-0.0337334\pi$
$547$	13.0135	+	73.8033i	0.0237907	+	0.134924i	−0.970599	+	0.240702i	$0.922622\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	48.5638	−	133.428i	0.0881377	−	0.242156i
$552$		0			0
$553$	306.849	+	257.477i	0.554881	+	0.465600i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	460.828	+	266.059i	0.827340	+	0.477665i	0.852941	−	0.522007i	$-0.174817\pi$
$557$	460.828	+	266.059i	0.827340	+	0.477665i	−0.0256012	+	0.999672i	$0.508150\pi$
$558$		0			0
$559$	403.042	+	698.090i	0.721006	+	1.24882i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	457.883	+	80.7371i	0.813291	+	0.143405i	0.564799	−	0.825229i	$-0.308954\pi$
$563$	457.883	+	80.7371i	0.813291	+	0.143405i	0.248492	+	0.968634i	$0.420065\pi$
$564$		0			0
$565$	346.211	−	290.505i	0.612762	−	0.514168i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−324.973	−	387.288i	−0.571130	−	0.680647i	0.400732	−	0.916195i	$-0.368756\pi$
$569$	−324.973	−	387.288i	−0.571130	−	0.680647i	−0.971863	+	0.235549i	$0.924311\pi$
$570$		0			0
$571$	4.63998	−	26.3146i	0.00812605	−	0.0460851i	−0.980475	−	0.196643i	$-0.936996\pi$
$571$	4.63998	−	26.3146i	0.00812605	−	0.0460851i	0.988601	+	0.150558i	$0.0481070\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	631.630	−	364.672i	1.09849	−	0.634212i
$576$		0			0
$577$	−568.853	+	985.283i	−0.985881	+	1.70760i	−0.347926	+	0.937522i	$0.613114\pi$
$577$	−568.853	+	985.283i	−0.985881	+	1.70760i	−0.637955	+	0.770074i	$0.720219\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	608.465	−	725.141i	1.04727	−	1.24809i
$582$		0			0
$583$	1112.88	+	405.054i	1.90888	+	0.694775i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−413.543	+	72.9187i	−0.704502	+	0.124223i	−0.514411	−	0.857544i	$-0.671989\pi$
$587$	−413.543	+	72.9187i	−0.704502	+	0.124223i	−0.190091	+	0.981767i	$0.560878\pi$
$588$		0			0
$589$	433.327	−	157.718i	0.735699	−	0.267772i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	608.635i		−	1.02637i	−0.858279	−	0.513183i	$-0.828466\pi$
$593$		−	608.635i		−	1.02637i	0.858279	−	0.513183i	$-0.171534\pi$
$594$		0			0
$595$	52.4298			0.0881173
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−222.904	−	612.422i	−0.372126	−	1.02241i	−0.974538	−	0.224223i	$-0.928016\pi$
$599$	−222.904	−	612.422i	−0.372126	−	1.02241i	0.602412	−	0.798185i	$-0.294206\pi$
$600$		0			0
$601$	29.5147	+	167.386i	0.0491092	+	0.278512i	0.999467	−	0.0326472i	$-0.0103938\pi$
$601$	29.5147	+	167.386i	0.0491092	+	0.278512i	−0.950358	+	0.311160i	$0.899283\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−75.5239	+	207.500i	−0.124833	+	0.342976i
$606$		0			0
$607$	−275.061	−	230.804i	−0.453149	−	0.380237i	0.387454	−	0.921889i	$-0.373355\pi$
$607$	−275.061	−	230.804i	−0.453149	−	0.380237i	−0.840603	+	0.541652i	$0.817799\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	38.4457	+	22.1966i	0.0629225	+	0.0363283i
$612$		0			0
$613$	112.374	+	194.637i	0.183317	+	0.317515i	0.943008	−	0.332769i	$-0.107983\pi$
$613$	112.374	+	194.637i	0.183317	+	0.317515i	−0.759691	+	0.650284i	$0.774650\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	369.095	+	65.0815i	0.598210	+	0.105481i	0.464550	−	0.885547i	$-0.346216\pi$
$617$	369.095	+	65.0815i	0.598210	+	0.105481i	0.133660	+	0.991027i	$0.457327\pi$
$618$		0			0
$619$	−511.061	+	428.831i	−0.825623	+	0.692780i	−0.954282	−	0.298909i	$-0.903377\pi$
$619$	−511.061	+	428.831i	−0.825623	+	0.692780i	0.128658	+	0.991689i	$0.458933\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−451.346	−	537.893i	−0.724472	−	0.863392i
$624$		0			0
$625$	51.2603	−	290.712i	0.0820165	−	0.465139i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−109.926	+	63.4659i	−0.174763	+	0.100900i
$630$		0			0
$631$	−288.004	+	498.837i	−0.456425	+	0.790551i	−0.998769	−	0.0496058i	$-0.984204\pi$
$631$	−288.004	+	498.837i	−0.456425	+	0.790551i	0.542344	+	0.840156i	$0.317537\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−79.0992	+	94.2667i	−0.124566	+	0.148452i
$636$		0			0
$637$	123.889	+	45.0918i	0.194488	+	0.0707877i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	732.193	−	129.105i	1.14227	−	0.201412i	0.429670	−	0.902986i	$-0.358630\pi$
$641$	732.193	−	129.105i	1.14227	−	0.201412i	0.712597	+	0.701574i	$0.247519\pi$
$642$		0			0
$643$	717.060	−	260.989i	1.11518	−	0.405892i	0.282289	−	0.959330i	$-0.408906\pi$
$643$	717.060	−	260.989i	1.11518	−	0.405892i	0.832891	+	0.553437i	$0.186684\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			128.575i			0.198725i	0.995051	+	0.0993624i	$0.0316803\pi$
$647$			128.575i			0.198725i	−0.995051	+	0.0993624i	$0.968320\pi$
$648$		0			0
$649$	−321.457			−0.495311
$650$		0			0
$651$		0			0
$652$		0			0
$653$	61.1625	+	168.042i	0.0936638	+	0.257339i	0.977674	−	0.210129i	$-0.0673884\pi$
$653$	61.1625	+	168.042i	0.0936638	+	0.257339i	−0.884010	+	0.467468i	$0.845166\pi$
$654$		0			0
$655$	52.0886	+	295.409i	0.0795245	+	0.451006i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	69.7138	−	191.537i	0.105787	−	0.290648i	−0.875494	−	0.483229i	$-0.839464\pi$
$659$	69.7138	−	191.537i	0.105787	−	0.290648i	0.981281	+	0.192581i	$0.0616859\pi$
$660$		0			0
$661$	−588.511	−	493.819i	−0.890334	−	0.747079i	0.0779433	−	0.996958i	$-0.475165\pi$
$661$	−588.511	−	493.819i	−0.890334	−	0.747079i	−0.968277	+	0.249879i	$0.919609\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	149.631	+	86.3896i	0.225009	+	0.129909i
$666$		0			0
$667$	−199.228	−	345.073i	−0.298693	−	0.517351i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1160.36	+	204.602i	1.72929	+	0.304921i
$672$		0			0
$673$	897.798	−	753.342i	1.33402	−	1.11938i	0.350905	−	0.936411i	$-0.385874\pi$
$673$	897.798	−	753.342i	1.33402	−	1.11938i	0.983119	−	0.182967i	$-0.0585703\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−192.459	−	229.364i	−0.284283	−	0.338795i	0.604939	−	0.796272i	$-0.293198\pi$
$677$	−192.459	−	229.364i	−0.284283	−	0.338795i	−0.889221	+	0.457477i	$0.848753\pi$
$678$		0			0
$679$	59.9803	−	340.165i	0.0883363	−	0.500980i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−812.585	+	469.146i	−1.18973	+	0.686890i	−0.958245	−	0.285949i	$-0.907691\pi$
$683$	−812.585	+	469.146i	−1.18973	+	0.686890i	−0.231484	+	0.972839i	$0.574358\pi$
$684$		0			0
$685$	57.8526	−	100.204i	0.0844564	−	0.146283i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−662.856	+	789.962i	−0.962056	+	1.14653i
$690$		0			0
$691$	−31.6500	−	11.5197i	−0.0458032	−	0.0166710i	0.319017	−	0.947749i	$-0.396647\pi$
$691$	−31.6500	−	11.5197i	−0.0458032	−	0.0166710i	−0.364820	+	0.931078i	$0.618870\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−426.333	+	75.1739i	−0.613428	+	0.108164i
$696$		0			0
$697$	82.4108	−	29.9951i	0.118236	−	0.0430345i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1259.81i			1.79716i	0.438814	+	0.898578i	$0.355399\pi$
$701$			1259.81i			1.79716i	−0.438814	+	0.898578i	$0.644601\pi$
$702$		0			0
$703$	−418.296			−0.595015
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−167.985	−	461.536i	−0.237603	−	0.652809i
$708$		0			0
$709$	51.2300	+	290.540i	0.0722567	+	0.409788i	0.999386	+	0.0350457i	$0.0111577\pi$
$709$	51.2300	+	290.540i	0.0722567	+	0.409788i	−0.927129	+	0.374742i	$0.877731\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	442.589	−	1216.00i	0.620741	−	1.70547i
$714$		0			0
$715$	−322.096	−	270.270i	−0.450483	−	0.378000i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	719.610	+	415.467i	1.00085	+	0.577840i	0.908499	−	0.417886i	$-0.137229\pi$
$719$	719.610	+	415.467i	1.00085	+	0.577840i	0.0923493	+	0.995727i	$0.470562\pi$
$720$		0			0
$721$	163.250	+	282.757i	0.226422	+	0.392174i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−221.915	−	39.1297i	−0.306090	−	0.0539719i
$726$		0			0
$727$	171.551	−	143.949i	0.235972	−	0.198004i	−0.517132	−	0.855906i	$-0.673000\pi$
$727$	171.551	−	143.949i	0.235972	−	0.198004i	0.753104	+	0.657902i	$0.228556\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	154.758	+	184.434i	0.211708	+	0.252303i
$732$		0			0
$733$	−47.7555	+	270.835i	−0.0651508	+	0.369488i	0.934749	+	0.355309i	$0.115625\pi$
$733$	−47.7555	+	270.835i	−0.0651508	+	0.369488i	−0.999899	+	0.0141788i	$0.995487\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1427.04	+	823.904i	−1.93629	+	1.11792i
$738$		0			0
$739$	235.936	−	408.653i	0.319264	−	0.552981i	−0.661071	−	0.750323i	$-0.729898\pi$
$739$	235.936	−	408.653i	0.319264	−	0.552981i	0.980335	+	0.197343i	$0.0632311\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	245.380	−	292.432i	0.330255	−	0.393583i	−0.575208	−	0.818007i	$-0.695079\pi$
$743$	245.380	−	292.432i	0.330255	−	0.393583i	0.905464	+	0.424424i	$0.139523\pi$
$744$		0			0
$745$	−136.423	−	49.6541i	−0.183119	−	0.0666497i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1165.93	+	205.585i	−1.55665	+	0.274479i
$750$		0			0
$751$	−1053.31	+	383.374i	−1.40255	+	0.510485i	−0.928933	−	0.370248i	$-0.879273\pi$
$751$	−1053.31	+	383.374i	−1.40255	+	0.510485i	−0.473613	+	0.880733i	$0.657051\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	482.765i		−	0.639424i
$756$		0			0
$757$	157.155			0.207602			0.103801	−	0.994598i	$-0.466899\pi$
$757$	157.155			0.207602			0.103801	+	0.994598i	$0.466899\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	440.078	+	1209.11i	0.578290	+	1.58884i	0.791062	+	0.611736i	$0.209528\pi$
$761$	440.078	+	1209.11i	0.578290	+	1.58884i	−0.212773	+	0.977102i	$0.568249\pi$
$762$		0			0
$763$	−174.706	−	990.806i	−0.228972	−	1.29857i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	95.7338	−	263.026i	0.124816	−	0.342929i
$768$		0			0
$769$	−687.236	−	576.660i	−0.893675	−	0.749882i	0.0752688	−	0.997163i	$-0.476019\pi$
$769$	−687.236	−	576.660i	−0.893675	−	0.749882i	−0.968944	+	0.247281i	$0.920463\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	211.789	+	122.276i	0.273983	+	0.158184i	0.630696	−	0.776030i	$-0.282769\pi$
$773$	211.789	+	122.276i	0.273983	+	0.158184i	−0.356713	+	0.934214i	$0.616103\pi$
$774$		0			0
$775$	−365.910	−	633.774i	−0.472141	−	0.817773i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	284.618	+	50.1859i	0.365364	+	0.0644235i
$780$		0			0
$781$	−940.641	+	789.292i	−1.20441	+	1.01062i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	168.642	+	200.980i	0.214831	+	0.256026i
$786$		0			0
$787$	75.9828	−	430.920i	0.0965474	−	0.547547i	−0.897715	−	0.440577i	$-0.854774\pi$
$787$	75.9828	−	430.920i	0.0965474	−	0.547547i	0.994262	−	0.106970i	$-0.0341150\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1126.54	+	650.407i	−1.42419	+	0.822259i
$792$		0			0
$793$	−512.980	+	888.507i	−0.646885	+	1.12044i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	275.128	−	327.885i	0.345204	−	0.411399i	−0.565308	−	0.824880i	$-0.691243\pi$
$797$	275.128	−	327.885i	0.345204	−	0.411399i	0.910513	+	0.413481i	$0.135687\pi$
$798$		0			0
$799$	12.4597	+	4.53497i	0.0155942	+	0.00567581i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−91.0655	+	16.0573i	−0.113407	+	0.0199966i
$804$		0			0
$805$	455.613	−	165.830i	0.565979	−	0.205999i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1161.28i		−	1.43545i	−0.696326	−	0.717725i	$-0.745183\pi$
$809$		−	1161.28i		−	1.43545i	0.696326	−	0.717725i	$-0.254817\pi$
$810$		0			0
$811$	−611.140			−0.753563			−0.376782	−	0.926302i	$-0.622969\pi$
$811$	−611.140			−0.753563			−0.376782	+	0.926302i	$0.622969\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−52.5907	−	144.492i	−0.0645284	−	0.177290i
$816$		0			0
$817$	137.775	+	781.360i	0.168635	+	0.956377i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−531.207	+	1459.48i	−0.647024	+	1.77769i	−0.0185843	+	0.999827i	$0.505916\pi$
$821$	−531.207	+	1459.48i	−0.647024	+	1.77769i	−0.628440	+	0.777858i	$0.716306\pi$
$822$		0			0
$823$	463.298	+	388.753i	0.562938	+	0.472361i	0.879294	−	0.476279i	$-0.158015\pi$
$823$	463.298	+	388.753i	0.562938	+	0.472361i	−0.316356	+	0.948641i	$0.602459\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−79.8064	−	46.0762i	−0.0965010	−	0.0557149i	0.450973	−	0.892538i	$-0.351077\pi$
$827$	−79.8064	−	46.0762i	−0.0965010	−	0.0557149i	−0.547474	+	0.836823i	$0.684410\pi$
$828$		0			0
$829$	−418.250	−	724.429i	−0.504523	−	0.873859i	−0.999986	−	0.00523058i	$-0.998335\pi$
$829$	−418.250	−	724.429i	−0.504523	−	0.873859i	0.495463	−	0.868629i	$-0.334998\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	38.7796	+	6.83789i	0.0465541	+	0.00820875i
$834$		0			0
$835$	−276.698	+	232.177i	−0.331375	+	0.278056i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−278.941	−	332.429i	−0.332469	−	0.396221i	0.573750	−	0.819031i	$-0.305488\pi$
$839$	−278.941	−	332.429i	−0.332469	−	0.396221i	−0.906218	+	0.422810i	$0.861044\pi$
$840$		0			0
$841$	124.661	−	706.986i	0.148229	−	0.840649i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.0826539	−	0.0477203i	9.78153e−5	−	5.64737e-5i
$846$		0			0
$847$	317.784	−	550.417i	0.375187	−	0.649843i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−754.519	+	899.200i	−0.886626	+	1.05664i
$852$		0			0
$853$	375.612	+	136.712i	0.440342	+	0.160271i	0.552671	−	0.833400i	$-0.313609\pi$
$853$	375.612	+	136.712i	0.440342	+	0.160271i	−0.112329	+	0.993671i	$0.535831\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−314.017	+	55.3697i	−0.366414	+	0.0646087i	−0.353824	−	0.935312i	$-0.615119\pi$
$857$	−314.017	+	55.3697i	−0.366414	+	0.0646087i	−0.0125903	+	0.999921i	$0.504008\pi$
$858$		0			0
$859$	259.079	−	94.2970i	0.301605	−	0.109775i	−0.186785	−	0.982401i	$-0.559807\pi$
$859$	259.079	−	94.2970i	0.301605	−	0.109775i	0.488390	+	0.872626i	$0.337584\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	764.036i		−	0.885325i	−0.896688	−	0.442663i	$-0.854034\pi$
$863$		−	764.036i		−	0.885325i	0.896688	−	0.442663i	$-0.145966\pi$
$864$		0			0
$865$	−496.194			−0.573635
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−328.157	−	901.603i	−0.377626	−	1.03752i
$870$		0			0
$871$	−249.154	−	1413.02i	−0.286055	−	1.62230i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	209.223	−	574.837i	0.239113	−	0.656956i
$876$		0			0
$877$	611.740	+	513.311i	0.697537	+	0.585303i	0.921072	−	0.389393i	$-0.127315\pi$
$877$	611.740	+	513.311i	0.697537	+	0.585303i	−0.223535	+	0.974696i	$0.571760\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	659.646	+	380.847i	0.748747	+	0.432290i	0.825241	−	0.564781i	$-0.191039\pi$
$881$	659.646	+	380.847i	0.748747	+	0.432290i	−0.0764937	+	0.997070i	$0.524373\pi$
$882$		0			0
$883$	463.143	+	802.187i	0.524511	+	0.908479i	0.999593	+	0.0285375i	$0.00908500\pi$
$883$	463.143	+	802.187i	0.524511	+	0.908479i	−0.475082	+	0.879941i	$0.657582\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−169.111	−	29.8188i	−0.190655	−	0.0336176i	0.0775053	−	0.996992i	$-0.475305\pi$
$887$	−169.111	−	29.8188i	−0.190655	−	0.0336176i	−0.268160	+	0.963374i	$0.586416\pi$
$888$		0			0
$889$	271.323	−	227.667i	0.305201	−	0.256094i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	28.0869	+	33.4726i	0.0314523	+	0.0374834i
$894$		0			0
$895$	−56.4159	+	319.951i	−0.0630346	+	0.357487i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−346.244	+	199.904i	−0.385144	+	0.222363i
$900$		0			0
$901$	−154.003	+	266.740i	−0.170924	+	0.296049i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	208.205	−	248.129i	0.230061	−	0.274176i
$906$		0			0
$907$	−479.601	−	174.560i	−0.528777	−	0.192459i	0.0638151	−	0.997962i	$-0.479673\pi$
$907$	−479.601	−	174.560i	−0.528777	−	0.192459i	−0.592592	+	0.805503i	$0.701895\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	653.228	−	115.182i	0.717045	−	0.126434i	0.196791	−	0.980446i	$-0.436948\pi$
$911$	653.228	−	115.182i	0.717045	−	0.126434i	0.520255	+	0.854011i	$0.325837\pi$
$912$		0			0
$913$	−2130.65	+	775.495i	−2.33369	+	0.849392i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	863.378i		−	0.941525i
$918$		0			0
$919$	63.4363			0.0690275			0.0345138	−	0.999404i	$-0.489012\pi$
$919$	63.4363			0.0690275			0.0345138	+	0.999404i	$0.489012\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−365.690	−	1004.72i	−0.396197	−	1.08854i
$924$		0			0
$925$	115.273	+	653.746i	0.124620	+	0.706753i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−444.979	+	1222.57i	−0.478987	+	1.31601i	0.431368	+	0.902176i	$0.358031\pi$
$929$	−444.979	+	1222.57i	−0.478987	+	1.31601i	−0.910354	+	0.413829i	$0.864191\pi$
$930$		0			0
$931$	99.4075	+	83.4128i	0.106775	+	0.0895948i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−108.760	−	62.7924i	−0.116320	−	0.0671576i
$936$		0			0
$937$	88.1210	+	152.630i	0.0940459	+	0.162892i	0.909210	−	0.416338i	$-0.136687\pi$
$937$	88.1210	+	152.630i	0.0940459	+	0.162892i	−0.815164	+	0.579230i	$0.803353\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1167.87	−	205.926i	−1.24109	−	0.218838i	−0.485708	−	0.874121i	$-0.661438\pi$
$941$	−1167.87	−	205.926i	−1.24109	−	0.218838i	−0.755383	+	0.655284i	$0.772549\pi$
$942$		0			0
$943$	621.276	−	521.312i	0.658829	−	0.552823i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−169.717	−	202.261i	−0.179216	−	0.213581i	0.668957	−	0.743302i	$-0.266741\pi$
$947$	−169.717	−	202.261i	−0.179216	−	0.213581i	−0.848172	+	0.529721i	$0.822297\pi$
$948$		0			0
$949$	13.9818	−	79.2948i	0.0147332	−	0.0835561i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	117.730	−	67.9713i	0.123536	−	0.0713235i	−0.436959	−	0.899482i	$-0.643944\pi$
$953$	117.730	−	67.9713i	0.123536	−	0.0713235i	0.560495	+	0.828158i	$0.310611\pi$
$954$		0			0
$955$	82.1337	−	142.260i	0.0860039	−	0.148963i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−214.067	+	255.115i	−0.223219	+	0.266022i
$960$		0			0
$961$	−317.085	−	115.410i	−0.329954	−	0.120093i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	290.442	−	51.2127i	0.300976	−	0.0530702i
$966$		0			0
$967$	434.941	−	158.306i	0.449784	−	0.163708i	−0.107189	−	0.994239i	$-0.534185\pi$
$967$	434.941	−	158.306i	0.449784	−	0.163708i	0.556973	+	0.830531i	$0.311963\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	241.390i		−	0.248599i	−0.992245	−	0.124299i	$-0.960332\pi$
$971$		−	241.390i		−	0.248599i	0.992245	−	0.124299i	$-0.0396683\pi$
$972$		0			0
$973$	1246.02			1.28060
$974$		0			0
$975$		0			0
$976$		0			0
$977$	543.473	+	1493.18i	0.556267	+	1.52833i	0.825009	+	0.565119i	$0.191170\pi$
$977$	543.473	+	1493.18i	0.556267	+	1.52833i	−0.268742	+	0.963212i	$0.586608\pi$
$978$		0			0
$979$	292.059	+	1656.35i	0.298324	+	1.69188i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.4309	−	31.4063i	0.0116286	−	0.0319494i	−0.933743	−	0.357945i	$-0.883477\pi$
$983$	11.4309	−	31.4063i	0.0116286	−	0.0319494i	0.945371	+	0.325996i	$0.105700\pi$
$984$		0			0
$985$	−547.739	−	459.608i	−0.556080	−	0.466607i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1928.19	+	1113.24i	1.94963	+	1.12562i
$990$		0			0
$991$	90.3973	+	156.573i	0.0912182	+	0.157995i	0.908024	−	0.418918i	$-0.137591\pi$
$991$	90.3973	+	156.573i	0.0912182	+	0.157995i	−0.816806	+	0.576913i	$0.804257\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	127.455	+	22.4737i	0.128095	+	0.0225867i
$996$		0			0
$997$	−371.647	+	311.849i	−0.372766	+	0.312787i	−0.809854	−	0.586631i	$-0.800454\pi$
$997$	−371.647	+	311.849i	−0.372766	+	0.312787i	0.437089	+	0.899418i	$0.356010\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.3.k.a.125.3		36
3.2	odd	2		108.3.k.a.5.1	✓	36
12.11	even	2		432.3.bc.b.113.6		36
27.4	even	9		2916.3.c.b.1457.22		36
27.11	odd	18	inner	324.3.k.a.197.3		36
27.16	even	9		108.3.k.a.65.1	yes	36
27.23	odd	18		2916.3.c.b.1457.15		36
108.43	odd	18		432.3.bc.b.65.6		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.k.a.5.1	✓	36	3.2	odd	2
108.3.k.a.65.1	yes	36	27.16	even	9
324.3.k.a.125.3		36	1.1	even	1	trivial
324.3.k.a.197.3		36	27.11	odd	18	inner
432.3.bc.b.65.6		36	108.43	odd	18
432.3.bc.b.113.6		36	12.11	even	2
2916.3.c.b.1457.15		36	27.23	odd	18
2916.3.c.b.1457.22		36	27.4	even	9

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.k (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.740753 - 2.03520i) q^{5} +(1.08248 + 6.13906i) q^{7} +O(q^{10})\)	\(q+(-0.740753 - 2.03520i) q^{5} +(1.08248 + 6.13906i) q^{7} +(5.10694 - 14.0312i) q^{11} +(9.95988 + 8.35733i) q^{13} +(3.36308 + 1.94167i) q^{17} +(6.39866 + 11.0828i) q^{19} +(35.3663 + 6.23604i) q^{23} +(15.5578 - 13.0545i) q^{25} +(-7.13197 - 8.49955i) q^{29} +(6.25720 - 35.4864i) q^{31} +(11.6924 - 6.75059i) q^{35} +(-16.3431 + 28.3071i) q^{37} +(14.5164 - 17.3000i) q^{41} +(58.2594 + 21.2047i) q^{43} +(3.36255 - 0.592908i) q^{47} +(9.52866 - 3.46815i) q^{49} +79.3144i q^{53} -32.3393 q^{55} +(-7.36318 - 20.2302i) q^{59} +(13.7025 + 77.7109i) q^{61} +(9.63104 - 26.4611i) q^{65} +(-84.5379 - 70.9357i) q^{67} +(-71.2183 - 41.1179i) q^{71} +(-3.09644 - 5.36320i) q^{73} +(91.6666 + 16.1633i) q^{77} +(49.2237 - 41.3036i) q^{79} +(-97.6080 - 116.325i) q^{83} +(1.46049 - 8.28284i) q^{85} +(-97.5489 + 56.3199i) q^{89} +(-40.5248 + 70.1909i) q^{91} +(17.8159 - 21.2322i) q^{95} +(-52.0684 - 18.9513i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 9 q^{5}+O(q^{10}) \) 36 * q + 9 * q^5	\( 36 q + 9 q^{5} - 36 q^{11} + 18 q^{23} - 9 q^{25} + 18 q^{29} + 45 q^{31} + 243 q^{35} + 198 q^{41} + 90 q^{43} + 243 q^{47} + 72 q^{49} - 252 q^{59} - 144 q^{61} - 747 q^{65} + 108 q^{67} - 324 q^{71} - 63 q^{73} - 495 q^{77} + 36 q^{79} + 27 q^{83} - 180 q^{85} + 567 q^{89} + 99 q^{91} + 1044 q^{95} - 216 q^{97}+O(q^{100}) \) 36 * q + 9 * q^5 - 36 * q^11 + 18 * q^23 - 9 * q^25 + 18 * q^29 + 45 * q^31 + 243 * q^35 + 198 * q^41 + 90 * q^43 + 243 * q^47 + 72 * q^49 - 252 * q^59 - 144 * q^61 - 747 * q^65 + 108 * q^67 - 324 * q^71 - 63 * q^73 - 495 * q^77 + 36 * q^79 + 27 * q^83 - 180 * q^85 + 567 * q^89 + 99 * q^91 + 1044 * q^95 - 216 * q^97

Introduction

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