LMFDB - Embedded newform 324.5.k.a.17.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,5,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, 5, names="a")

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

chi := DirichletCharacter("324.17");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
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:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			17.8
Character	$\chi$	$=$	324.17
Dual form			324.5.k.a.305.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(15.6508 + 2.75966i) q^{5} +(-42.2367 + 35.4408i) q^{7} +O(q^{10})$	$q+(15.6508 + 2.75966i) q^{5} +(-42.2367 + 35.4408i) q^{7} +(90.3346 - 15.9284i) q^{11} +(-153.162 + 55.7463i) q^{13} +(-148.420 - 85.6904i) q^{17} +(41.1095 + 71.2038i) q^{19} +(-8.90402 + 10.6114i) q^{23} +(-349.975 - 127.381i) q^{25} +(209.718 - 576.196i) q^{29} +(-832.166 - 698.270i) q^{31} +(-758.845 + 438.119i) q^{35} +(-446.904 + 774.060i) q^{37} +(474.238 + 1302.96i) q^{41} +(-392.197 - 2224.26i) q^{43} +(-662.180 - 789.156i) q^{47} +(110.960 - 629.288i) q^{49} +4085.25i q^{53} +1457.77 q^{55} +(-6022.47 - 1061.92i) q^{59} +(-1479.50 + 1241.44i) q^{61} +(-2550.95 + 449.801i) q^{65} +(1639.89 - 596.872i) q^{67} +(-8290.96 - 4786.79i) q^{71} +(-4369.09 - 7567.48i) q^{73} +(-3250.92 + 3874.30i) q^{77} +(-8849.56 - 3220.98i) q^{79} +(-2050.61 + 5634.01i) q^{83} +(-2086.42 - 1750.72i) q^{85} +(2113.06 - 1219.97i) q^{89} +(4493.36 - 7782.72i) q^{91} +(446.900 + 1227.85i) q^{95} +(-7.43876 - 42.1873i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{5}+O(q^{10}) $ 72 * q - 9 * q^5	$ 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) $ 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{11}{18}\right)$

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
$5$	15.6508	+	2.75966i	0.626033	+	0.110387i	0.477660	−	0.878545i	$-0.341485\pi$
$5$	15.6508	+	2.75966i	0.626033	+	0.110387i	0.148373	+	0.988931i	$0.452596\pi$
$6$		0			0
$7$	−42.2367	+	35.4408i	−0.861974	+	0.723282i	−0.962392	−	0.271663i	$-0.912426\pi$
$7$	−42.2367	+	35.4408i	−0.861974	+	0.723282i	0.100418	+	0.994945i	$0.467982\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	90.3346	−	15.9284i	0.746567	−	0.131640i	0.212593	−	0.977141i	$-0.431809\pi$
$11$	90.3346	−	15.9284i	0.746567	−	0.131640i	0.533974	+	0.845501i	$0.320698\pi$
$12$		0			0
$13$	−153.162	+	55.7463i	−0.906283	+	0.329860i	−0.752768	−	0.658286i	$-0.771282\pi$
$13$	−153.162	+	55.7463i	−0.906283	+	0.329860i	−0.153515	+	0.988146i	$0.549059\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−148.420	−	85.6904i	−0.513565	−	0.296507i	0.220733	−	0.975334i	$-0.429155\pi$
$17$	−148.420	−	85.6904i	−0.513565	−	0.296507i	−0.734298	+	0.678828i	$0.762488\pi$
$18$		0			0
$19$	41.1095	+	71.2038i	0.113877	+	0.197240i	0.917330	−	0.398127i	$-0.130340\pi$
$19$	41.1095	+	71.2038i	0.113877	+	0.197240i	−0.803453	+	0.595368i	$0.797006\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.90402	+	10.6114i	−0.0168318	+	0.0200593i	−0.774395	−	0.632703i	$-0.781945\pi$
$23$	−8.90402	+	10.6114i	−0.0168318	+	0.0200593i	0.757563	+	0.652762i	$0.226390\pi$
$24$		0			0
$25$	−349.975	−	127.381i	−0.559960	−	0.203809i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	209.718	−	576.196i	0.249368	−	0.685132i	−0.750342	−	0.661050i	$-0.770111\pi$
$29$	209.718	−	576.196i	0.249368	−	0.685132i	0.999710	−	0.0240827i	$-0.00766651\pi$
$30$		0			0
$31$	−832.166	−	698.270i	−0.865937	−	0.726608i	0.0973014	−	0.995255i	$-0.468979\pi$
$31$	−832.166	−	698.270i	−0.865937	−	0.726608i	−0.963239	+	0.268647i	$0.913423\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−758.845	+	438.119i	−0.619465	+	0.357648i
$36$		0			0
$37$	−446.904	+	774.060i	−0.326445	+	0.565420i	−0.981804	−	0.189898i	$-0.939184\pi$
$37$	−446.904	+	774.060i	−0.326445	+	0.565420i	0.655359	+	0.755318i	$0.272518\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	474.238	+	1302.96i	0.282117	+	0.775109i	0.997110	+	0.0759772i	$0.0242076\pi$
$41$	474.238	+	1302.96i	0.282117	+	0.775109i	−0.714993	+	0.699132i	$0.753570\pi$
$42$		0			0
$43$	−392.197	−	2224.26i	−0.212113	−	1.20295i	−0.885847	−	0.463978i	$-0.846422\pi$
$43$	−392.197	−	2224.26i	−0.212113	−	1.20295i	0.673734	−	0.738974i	$-0.264689\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−662.180	−	789.156i	−0.299765	−	0.357246i	0.595046	−	0.803692i	$-0.297134\pi$
$47$	−662.180	−	789.156i	−0.299765	−	0.357246i	−0.894810	+	0.446446i	$0.852689\pi$
$48$		0			0
$49$	110.960	−	629.288i	0.0462142	−	0.262094i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4085.25i			1.45434i	0.686455	+	0.727172i	$0.259166\pi$
$53$			4085.25i			1.45434i	−0.686455	+	0.727172i	$0.740834\pi$
$54$		0			0
$55$	1457.77			0.481907
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6022.47	−	1061.92i	−1.73010	−	0.305063i	−0.782057	−	0.623207i	$-0.785829\pi$
$59$	−6022.47	−	1061.92i	−1.73010	−	0.305063i	−0.948042	+	0.318144i	$0.896940\pi$
$60$		0			0
$61$	−1479.50	+	1241.44i	−0.397607	+	0.333632i	−0.819568	−	0.572982i	$-0.805786\pi$
$61$	−1479.50	+	1241.44i	−0.397607	+	0.333632i	0.421961	+	0.906614i	$0.361342\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2550.95	+	449.801i	−0.603775	+	0.106462i
$66$		0			0
$67$	1639.89	−	596.872i	0.365313	−	0.132963i	−0.152840	−	0.988251i	$-0.548842\pi$
$67$	1639.89	−	596.872i	0.365313	−	0.132963i	0.518153	+	0.855288i	$0.326620\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8290.96	−	4786.79i	−1.64471	−	0.949571i	−0.979129	−	0.203242i	$-0.934852\pi$
$71$	−8290.96	−	4786.79i	−1.64471	−	0.949571i	−0.665577	−	0.746329i	$-0.731814\pi$
$72$		0			0
$73$	−4369.09	−	7567.48i	−0.819870	−	1.42006i	−0.905778	−	0.423753i	$-0.860712\pi$
$73$	−4369.09	−	7567.48i	−0.819870	−	1.42006i	0.0859076	−	0.996303i	$-0.472621\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3250.92	+	3874.30i	−0.548309	+	0.653449i
$78$		0			0
$79$	−8849.56	−	3220.98i	−1.41797	−	0.516099i	−0.484513	−	0.874784i	$-0.661003\pi$
$79$	−8849.56	−	3220.98i	−1.41797	−	0.516099i	−0.933459	+	0.358685i	$0.883225\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2050.61	+	5634.01i	−0.297665	+	0.817827i	0.697225	+	0.716853i	$0.254418\pi$
$83$	−2050.61	+	5634.01i	−0.297665	+	0.817827i	−0.994889	+	0.100974i	$0.967804\pi$
$84$		0			0
$85$	−2086.42	−	1750.72i	−0.288778	−	0.242314i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2113.06	−	1219.97i	0.266767	−	0.154018i	−0.360651	−	0.932701i	$-0.617445\pi$
$89$	2113.06	−	1219.97i	0.266767	−	0.154018i	0.627417	+	0.778683i	$0.284112\pi$
$90$		0			0
$91$	4493.36	−	7782.72i	0.542611	−	0.939829i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	446.900	+	1227.85i	0.0495180	+	0.136050i
$96$		0			0
$97$	−7.43876	−	42.1873i	−0.000790600	−	0.00448372i	0.984410	−	0.175890i	$-0.0562802\pi$
$97$	−7.43876	−	42.1873i	−0.000790600	−	0.00448372i	−0.985201	+	0.171406i	$0.945169\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1258.51	−	1499.84i	−0.123371	−	0.147028i	0.700823	−	0.713335i	$-0.252816\pi$
$101$	−1258.51	−	1499.84i	−0.123371	−	0.147028i	−0.824195	+	0.566307i	$0.808372\pi$
$102$		0			0
$103$	−2632.38	+	14929.0i	−0.248127	+	1.40720i	0.564988	+	0.825099i	$0.308881\pi$
$103$	−2632.38	+	14929.0i	−0.248127	+	1.40720i	−0.813116	+	0.582102i	$0.802230\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1094.68i			0.0956134i	0.998857	+	0.0478067i	$0.0152232\pi$
$107$			1094.68i			0.0956134i	−0.998857	+	0.0478067i	$0.984777\pi$
$108$		0			0
$109$	−3612.18			−0.304030			−0.152015	−	0.988378i	$-0.548576\pi$
$109$	−3612.18			−0.304030			−0.152015	+	0.988378i	$0.548576\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	21824.1	+	3848.18i	1.70915	+	0.301369i	0.940875	−	0.338754i	$-0.110006\pi$
$113$	21824.1	+	3848.18i	1.70915	+	0.301369i	0.768272	+	0.640123i	$0.221117\pi$
$114$		0			0
$115$	−168.639	+	141.505i	−0.0127515	+	0.0106998i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9305.72	−	1640.85i	0.657137	−	0.115871i
$120$		0			0
$121$	−5851.41	+	2129.74i	−0.399659	+	0.145464i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−13727.8	−	7925.76i	−0.878580	−	0.507249i
$126$		0			0
$127$	4525.22	+	7837.91i	0.280564	+	0.485951i	0.971524	−	0.236942i	$-0.0761451\pi$
$127$	4525.22	+	7837.91i	0.280564	+	0.485951i	−0.690960	+	0.722893i	$0.742812\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3507.15	+	4179.65i	−0.204367	+	0.243555i	−0.858487	−	0.512836i	$-0.828595\pi$
$131$	−3507.15	+	4179.65i	−0.204367	+	0.243555i	0.654119	+	0.756391i	$0.273039\pi$
$132$		0			0
$133$	−4259.85	−	1550.46i	−0.240819	−	0.0876511i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10373.9	−	28501.9i	0.552712	−	1.51856i	−0.277280	−	0.960789i	$-0.589433\pi$
$137$	10373.9	−	28501.9i	0.552712	−	1.51856i	0.829992	−	0.557775i	$-0.188345\pi$
$138$		0			0
$139$	−1417.90	−	1189.76i	−0.0733866	−	0.0615787i	0.605356	−	0.795954i	$-0.293031\pi$
$139$	−1417.90	−	1189.76i	−0.0733866	−	0.0615787i	−0.678743	+	0.734376i	$0.737475\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12947.9	+	7475.45i	−0.633178	+	0.365566i
$144$		0			0
$145$	4872.37	−	8439.20i	0.231742	−	0.401389i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9454.94	+	25977.2i	0.425879	+	1.17009i	0.948292	+	0.317400i	$0.102810\pi$
$149$	9454.94	+	25977.2i	0.425879	+	1.17009i	−0.522413	+	0.852693i	$0.674968\pi$
$150$		0			0
$151$	−4418.24	−	25057.1i	−0.193774	−	1.09895i	−0.914154	−	0.405367i	$-0.867144\pi$
$151$	−4418.24	−	25057.1i	−0.193774	−	1.09895i	0.720380	−	0.693579i	$-0.243967\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−11097.1	−	13225.0i	−0.461898	−	0.550468i
$156$		0			0
$157$	−7015.15	+	39784.9i	−0.284602	+	1.61406i	0.422102	+	0.906548i	$0.361293\pi$
$157$	−7015.15	+	39784.9i	−0.284602	+	1.61406i	−0.706704	+	0.707510i	$0.749819\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	763.757i		−	0.0294648i
$162$		0			0
$163$	38592.0			1.45252			0.726260	−	0.687420i	$-0.241257\pi$
$163$	38592.0			1.45252			0.726260	+	0.687420i	$0.241257\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13099.4	+	2309.78i	0.469699	+	0.0828206i	0.403488	−	0.914985i	$-0.367798\pi$
$167$	13099.4	+	2309.78i	0.469699	+	0.0828206i	0.0662110	+	0.997806i	$0.478909\pi$
$168$		0			0
$169$	−1528.12	+	1282.25i	−0.0535038	+	0.0448950i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	50954.6	−	8984.67i	1.70252	−	0.300199i	0.763944	−	0.645283i	$-0.223260\pi$
$173$	50954.6	−	8984.67i	1.70252	−	0.300199i	0.938572	+	0.345083i	$0.112149\pi$
$174$		0			0
$175$	19296.3	−	7023.27i	0.630083	−	0.229331i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−21079.6	−	12170.3i	−0.657895	−	0.379836i	0.133579	−	0.991038i	$-0.457353\pi$
$179$	−21079.6	−	12170.3i	−0.657895	−	0.379836i	−0.791475	+	0.611202i	$0.790686\pi$
$180$		0			0
$181$	17380.2	+	30103.4i	0.530516	+	0.918880i	0.999366	+	0.0356028i	$0.0113351\pi$
$181$	17380.2	+	30103.4i	0.530516	+	0.918880i	−0.468850	+	0.883278i	$0.655332\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9130.56	+	10881.4i	−0.266780	+	0.317936i
$186$		0			0
$187$	−14772.4	−	5376.71i	−0.422442	−	0.153756i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7974.79	−	21910.6i	0.218601	−	0.600602i	−0.781116	−	0.624386i	$-0.785349\pi$
$191$	7974.79	−	21910.6i	0.218601	−	0.600602i	0.999717	+	0.0237840i	$0.00757140\pi$
$192$		0			0
$193$	55178.3	+	46300.1i	1.48134	+	1.24299i	0.904744	+	0.425956i	$0.140062\pi$
$193$	55178.3	+	46300.1i	1.48134	+	1.24299i	0.576592	+	0.817032i	$0.304382\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14378.1	+	8301.21i	−0.370484	+	0.213899i	−0.673670	−	0.739032i	$-0.735283\pi$
$197$	−14378.1	+	8301.21i	−0.370484	+	0.213899i	0.303186	+	0.952931i	$0.401950\pi$
$198$		0			0
$199$	29153.3	−	50495.1i	0.736177	−	1.27510i	−0.218028	−	0.975942i	$-0.569963\pi$
$199$	29153.3	−	50495.1i	0.736177	−	1.27510i	0.954205	−	0.299153i	$-0.0967041\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	11563.1	+	31769.2i	0.280596	+	0.770930i
$204$		0			0
$205$	3826.49	+	21701.1i	0.0910527	+	0.516386i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4847.78	+	5777.36i	0.110981	+	0.132262i
$210$		0			0
$211$	−3487.31	+	19777.5i	−0.0783296	+	0.444229i	0.920268	+	0.391289i	$0.127971\pi$
$211$	−3487.31	+	19777.5i	−0.0783296	+	0.444229i	−0.998598	+	0.0529407i	$0.983141\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	35893.8i		−	0.776502i
$216$		0			0
$217$	59895.2			1.27196
$218$		0			0
$219$		0			0
$220$		0			0
$221$	27509.2	+	4850.62i	0.563240	+	0.0993145i
$222$		0			0
$223$	−40313.8	+	33827.3i	−0.810669	+	0.680232i	−0.950767	−	0.309905i	$-0.899703\pi$
$223$	−40313.8	+	33827.3i	−0.810669	+	0.680232i	0.140098	+	0.990138i	$0.455258\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	71977.3	−	12691.5i	1.39683	−	0.246299i	0.575991	−	0.817456i	$-0.304616\pi$
$227$	71977.3	−	12691.5i	1.39683	−	0.246299i	0.820841	+	0.571157i	$0.193505\pi$
$228$		0			0
$229$	9448.24	−	3438.88i	0.180169	−	0.0655761i	−0.250360	−	0.968153i	$-0.580549\pi$
$229$	9448.24	−	3438.88i	0.180169	−	0.0655761i	0.430529	+	0.902577i	$0.358327\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	56992.0	+	32904.4i	1.04979	+	0.606096i	0.922590	−	0.385781i	$-0.126068\pi$
$233$	56992.0	+	32904.4i	1.04979	+	0.606096i	0.127199	+	0.991877i	$0.459401\pi$
$234$		0			0
$235$	−8185.87	−	14178.3i	−0.148228	−	0.256738i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	29590.7	−	35264.8i	0.518036	−	0.617371i	−0.442079	−	0.896976i	$-0.645759\pi$
$239$	29590.7	−	35264.8i	0.518036	−	0.617371i	0.960115	+	0.279605i	$0.0902035\pi$
$240$		0			0
$241$	−44245.1	−	16103.9i	−0.761782	−	0.277266i	−0.0682271	−	0.997670i	$-0.521734\pi$
$241$	−44245.1	−	16103.9i	−0.761782	−	0.277266i	−0.693555	+	0.720404i	$0.743956\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3473.25	−	9542.66i	0.0578633	−	0.158978i
$246$		0			0
$247$	−10265.8	−	8613.99i	−0.168266	−	0.141192i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−64264.4	+	37103.1i	−1.02005	+	0.588929i	−0.914120	−	0.405444i	$-0.867117\pi$
$251$	−64264.4	+	37103.1i	−1.02005	+	0.588929i	−0.105935	+	0.994373i	$0.533783\pi$
$252$		0			0
$253$	−635.318	+	1100.40i	−0.00992545	+	0.0171914i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4345.92	+	11940.3i	0.0657984	+	0.180780i	0.968234	−	0.250044i	$-0.0804452\pi$
$257$	4345.92	+	11940.3i	0.0657984	+	0.180780i	−0.902436	+	0.430824i	$0.858223\pi$
$258$		0			0
$259$	−8557.57	−	48532.4i	−0.127571	−	0.723490i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−52260.8	−	62282.0i	−0.755552	−	0.900432i	0.242006	−	0.970275i	$-0.422195\pi$
$263$	−52260.8	−	62282.0i	−0.755552	−	0.900432i	−0.997558	+	0.0698426i	$0.977750\pi$
$264$		0			0
$265$	−11273.9	+	63937.6i	−0.160540	+	0.910468i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	46817.2i		−	0.646995i	−0.946229	−	0.323497i	$-0.895141\pi$
$269$		−	46817.2i		−	0.646995i	0.946229	−	0.323497i	$-0.104859\pi$
$270$		0			0
$271$	−11920.6			−0.162315			−0.0811574	−	0.996701i	$-0.525862\pi$
$271$	−11920.6			−0.162315			−0.0811574	+	0.996701i	$0.525862\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−33643.8	−	5932.32i	−0.444877	−	0.0784438i
$276$		0			0
$277$	−19940.7	+	16732.2i	−0.259884	+	0.218069i	−0.763415	−	0.645909i	$-0.776479\pi$
$277$	−19940.7	+	16732.2i	−0.259884	+	0.218069i	0.503530	+	0.863978i	$0.332034\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	48575.1	−	8565.11i	0.615179	−	0.108473i	0.142628	−	0.989776i	$-0.454445\pi$
$281$	48575.1	−	8565.11i	0.615179	−	0.108473i	0.472550	+	0.881304i	$0.343334\pi$
$282$		0			0
$283$	−82298.0	+	29954.0i	−1.02758	+	0.374009i	−0.800160	−	0.599787i	$-0.795252\pi$
$283$	−82298.0	+	29954.0i	−1.02758	+	0.374009i	−0.227422	+	0.973796i	$0.573030\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−66208.2	−	38225.3i	−0.803800	−	0.464074i
$288$		0			0
$289$	−27074.8	−	46894.9i	−0.324168	−	0.561475i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−41178.7	+	49074.9i	−0.479664	+	0.571642i	−0.950558	−	0.310548i	$-0.899487\pi$
$293$	−41178.7	+	49074.9i	−0.479664	+	0.571642i	0.470893	+	0.882190i	$0.343932\pi$
$294$		0			0
$295$	−91326.2	−	33240.0i	−1.04942	−	0.381959i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	772.209	−	2121.63i	0.00863759	−	0.0237316i
$300$		0			0
$301$	95394.7	+	80045.6i	1.05291	+	0.883496i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−26581.3	+	15346.7i	−0.285744	+	0.164974i
$306$		0			0
$307$	−8879.41	+	15379.6i	−0.0942123	+	0.163180i	−0.909280	−	0.416186i	$-0.863367\pi$
$307$	−8879.41	+	15379.6i	−0.0942123	+	0.163180i	0.815067	+	0.579366i	$0.196700\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	58887.5	+	161792.i	0.608838	+	1.67277i	0.732768	+	0.680479i	$0.238228\pi$
$311$	58887.5	+	161792.i	0.608838	+	1.67277i	−0.123930	+	0.992291i	$0.539550\pi$
$312$		0			0
$313$	−2027.01	−	11495.8i	−0.0206903	−	0.117341i	0.972713	−	0.232010i	$-0.0745302\pi$
$313$	−2027.01	−	11495.8i	−0.0206903	−	0.117341i	−0.993404	+	0.114669i	$0.963419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−31005.0	−	36950.4i	−0.308542	−	0.367706i	0.589384	−	0.807853i	$-0.299371\pi$
$317$	−31005.0	−	36950.4i	−0.308542	−	0.367706i	−0.897925	+	0.440147i	$0.854926\pi$
$318$		0			0
$319$	9766.92	−	55391.0i	0.0959790	−	0.544324i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	14090.8i		−	0.135061i
$324$		0			0
$325$	60703.8			0.574710
$326$		0			0
$327$		0			0
$328$		0			0
$329$	55936.7	+	9863.14i	0.516779	+	0.0911221i
$330$		0			0
$331$	109654.	−	92010.6i	1.00085	−	0.839812i	0.0137467	−	0.999906i	$-0.495624\pi$
$331$	109654.	−	92010.6i	1.00085	−	0.839812i	0.987102	+	0.160094i	$0.0511797\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	27312.8	−	4815.99i	0.243376	−	0.0429137i
$336$		0			0
$337$	−99403.1	+	36179.8i	−0.875266	+	0.318571i	−0.740298	−	0.672279i	$-0.765316\pi$
$337$	−99403.1	+	36179.8i	−0.875266	+	0.318571i	−0.134968	+	0.990850i	$0.543093\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−86295.7	−	49822.9i	−0.742131	−	0.428469i
$342$		0			0
$343$	−48575.1	−	84134.5i	−0.412881	−	0.715131i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−67627.8	+	80595.6i	−0.561651	+	0.669349i	−0.969895	−	0.243524i	$-0.921697\pi$
$347$	−67627.8	+	80595.6i	−0.561651	+	0.669349i	0.408244	+	0.912873i	$0.366141\pi$
$348$		0			0
$349$	−7016.58	−	2553.83i	−0.0576069	−	0.0209672i	0.313056	−	0.949735i	$-0.398647\pi$
$349$	−7016.58	−	2553.83i	−0.0576069	−	0.0209672i	−0.370663	+	0.928767i	$0.620869\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−79381.1	+	218098.i	−0.637042	+	1.75026i	0.0237832	+	0.999717i	$0.492429\pi$
$353$	−79381.1	+	218098.i	−0.637042	+	1.75026i	−0.660825	+	0.750540i	$0.729793\pi$
$354$		0			0
$355$	−116550.	−	97797.5i	−0.924820	−	0.776016i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−202651.	+	117000.i	−1.57239	+	0.907817i	−0.576509	+	0.817091i	$0.695586\pi$
$359$	−202651.	+	117000.i	−1.57239	+	0.907817i	−0.995876	+	0.0907263i	$0.971081\pi$
$360$		0			0
$361$	61780.5	−	107007.i	0.474064	−	0.821103i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−47496.1	−	130495.i	−0.356511	−	0.979505i
$366$		0			0
$367$	−29927.4	−	169727.i	−0.222196	−	1.26014i	−0.867973	−	0.496611i	$-0.834577\pi$
$367$	−29927.4	−	169727.i	−0.222196	−	1.26014i	0.645777	−	0.763526i	$-0.276534\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−144785.	−	172548.i	−1.05190	−	1.25361i
$372$		0			0
$373$	11618.6	−	65892.6i	0.0835098	−	0.473608i	−0.914158	−	0.405357i	$-0.867147\pi$
$373$	11618.6	−	65892.6i	0.0835098	−	0.473608i	0.997668	−	0.0682506i	$-0.0217417\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			99942.3i			0.703180i
$378$		0			0
$379$	−21007.8			−0.146252			−0.0731261	−	0.997323i	$-0.523298\pi$
$379$	−21007.8			−0.146252			−0.0731261	+	0.997323i	$0.523298\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6817.31	−	1202.08i	−0.0464746	−	0.00819473i	0.150363	−	0.988631i	$-0.451956\pi$
$383$	−6817.31	−	1202.08i	−0.0464746	−	0.00819473i	−0.196837	+	0.980436i	$0.563067\pi$
$384$		0			0
$385$	−61571.4	+	51664.6i	−0.415392	+	0.348555i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22440.6	+	3956.88i	−0.148298	+	0.0261489i	−0.247304	−	0.968938i	$-0.579545\pi$
$389$	−22440.6	+	3956.88i	−0.148298	+	0.0261489i	0.0990064	+	0.995087i	$0.468434\pi$
$390$		0			0
$391$	2230.83	−	811.956i	0.0145919	−	0.00531103i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−129614.	−	74832.8i	−0.830727	−	0.479620i
$396$		0			0
$397$	31654.4	+	54827.1i	0.200841	+	0.347868i	0.948800	−	0.315878i	$-0.102299\pi$
$397$	31654.4	+	54827.1i	0.200841	+	0.347868i	−0.747958	+	0.663746i	$0.768966\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−39783.3	+	47411.9i	−0.247407	+	0.294848i	−0.875428	−	0.483348i	$-0.839421\pi$
$401$	−39783.3	+	47411.9i	−0.247407	+	0.294848i	0.628021	+	0.778196i	$0.283865\pi$
$402$		0			0
$403$	166382.	+	60558.1i	1.02446	+	0.372874i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−28041.3	+	77042.9i	−0.169281	+	0.465097i
$408$		0			0
$409$	−48448.0	−	40652.7i	−0.289621	−	0.243021i	0.486388	−	0.873743i	$-0.338314\pi$
$409$	−48448.0	−	40652.7i	−0.289621	−	0.243021i	−0.776008	+	0.630722i	$0.782759\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	292005.	−	168589.i	1.71195	−	0.988393i
$414$		0			0
$415$	−47641.7	+	82517.9i	−0.276625	+	0.479129i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−52225.0	−	143487.i	−0.297475	−	0.817306i	−0.994920	−	0.100668i	$-0.967902\pi$
$419$	−52225.0	−	143487.i	−0.297475	−	0.817306i	0.697445	−	0.716638i	$-0.254320\pi$
$420$		0			0
$421$	39245.9	+	222575.i	0.221427	+	1.25577i	0.869399	+	0.494110i	$0.164506\pi$
$421$	39245.9	+	222575.i	0.221427	+	1.25577i	−0.647972	+	0.761664i	$0.724383\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	41028.1	+	48895.3i	0.227145	+	0.270701i
$426$		0			0
$427$	18491.3	−	104869.i	0.101417	−	0.575164i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			180330.i			0.970764i	0.874302	+	0.485382i	$0.161319\pi$
$431$			180330.i			0.970764i	−0.874302	+	0.485382i	$0.838681\pi$
$432$		0			0
$433$	−323998.			−1.72809			−0.864045	−	0.503414i	$-0.832077\pi$
$433$	−323998.			−1.72809			−0.864045	+	0.503414i	$0.832077\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1121.61	−	197.770i	−0.00587326	−	0.00103562i
$438$		0			0
$439$	−72842.5	+	61122.1i	−0.377969	+	0.317154i	−0.811904	−	0.583790i	$-0.801569\pi$
$439$	−72842.5	+	61122.1i	−0.377969	+	0.317154i	0.433936	+	0.900944i	$0.357125\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	101340.	−	17869.0i	0.516387	−	0.0910529i	0.0906194	−	0.995886i	$-0.471115\pi$
$443$	101340.	−	17869.0i	0.516387	−	0.0910529i	0.425767	+	0.904833i	$0.360004\pi$
$444$		0			0
$445$	36437.8	−	13262.3i	0.184006	−	0.0669728i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	225220.	+	130031.i	1.11716	+	0.644992i	0.940674	−	0.339312i	$-0.110194\pi$
$449$	225220.	+	130031.i	1.11716	+	0.644992i	0.176484	+	0.984303i	$0.443528\pi$
$450$		0			0
$451$	63594.2	+	110148.i	0.312654	+	0.541533i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	91802.5	−	109406.i	0.443437	−	0.528467i
$456$		0			0
$457$	−224130.	−	81576.6i	−1.07317	−	0.390601i	−0.255807	−	0.966728i	$-0.582341\pi$
$457$	−224130.	−	81576.6i	−1.07317	−	0.390601i	−0.817360	+	0.576127i	$0.804563\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−102315.	+	281109.i	−0.481437	+	1.32274i	0.426825	+	0.904334i	$0.359632\pi$
$461$	−102315.	+	281109.i	−0.481437	+	1.32274i	−0.908262	+	0.418403i	$0.862590\pi$
$462$		0			0
$463$	173161.	+	145299.i	0.807769	+	0.677798i	0.950074	−	0.312025i	$-0.101007\pi$
$463$	173161.	+	145299.i	0.807769	+	0.677798i	−0.142306	+	0.989823i	$0.545452\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	342708.	−	197863.i	1.57141	−	0.907256i	0.575417	−	0.817860i	$-0.304840\pi$
$467$	342708.	−	197863.i	1.57141	−	0.907256i	0.995996	−	0.0893959i	$-0.0284936\pi$
$468$		0			0
$469$	−48110.1	+	83329.1i	−0.218721	+	0.378836i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−70857.9	−	194680.i	−0.316713	−	0.870162i
$474$		0			0
$475$	−5317.33	−	30156.1i	−0.0235671	−	0.133656i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−148819.	−	177356.i	−0.648616	−	0.772990i	0.337089	−	0.941473i	$-0.390558\pi$
$479$	−148819.	−	177356.i	−0.648616	−	0.772990i	−0.985705	+	0.168483i	$0.946113\pi$
$480$		0			0
$481$	25297.6	−	143470.i	0.109342	−	0.620111i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	680.795i		−	0.00289423i
$486$		0			0
$487$	−57016.1			−0.240403			−0.120201	−	0.992750i	$-0.538354\pi$
$487$	−57016.1			−0.240403			−0.120201	+	0.992750i	$0.538354\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−346891.	−	61166.3i	−1.43890	−	0.253717i	−0.600872	−	0.799345i	$-0.705180\pi$
$491$	−346891.	−	61166.3i	−1.43890	−	0.253717i	−0.838027	+	0.545629i	$0.816291\pi$
$492$		0			0
$493$	−80500.9	+	67548.3i	−0.331213	+	0.277920i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	519831.	−	91660.2i	2.10450	−	0.371080i
$498$		0			0
$499$	158650.	−	57743.7i	0.637145	−	0.231902i	−0.00319343	−	0.999995i	$-0.501017\pi$
$499$	158650.	−	57743.7i	0.637145	−	0.231902i	0.640338	+	0.768093i	$0.278794\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	191717.	+	110688.i	0.757750	+	0.437487i	0.828487	−	0.560008i	$-0.189202\pi$
$503$	191717.	+	110688.i	0.757750	+	0.437487i	−0.0707376	+	0.997495i	$0.522535\pi$
$504$		0			0
$505$	−15557.7	−	26946.7i	−0.0610046	−	0.105663i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	162835.	−	194059.i	0.628509	−	0.749028i	−0.354000	−	0.935246i	$-0.615179\pi$
$509$	162835.	−	194059.i	0.628509	−	0.749028i	0.982508	+	0.186218i	$0.0596230\pi$
$510$		0			0
$511$	452734.	+	164782.i	1.73381	+	0.631055i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−82398.0	+	226387.i	−0.310672	+	0.853565i
$516$		0			0
$517$	−72387.8	−	60740.6i	−0.270822	−	0.227247i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	89226.4	−	51514.9i	0.328714	−	0.189783i	−0.326556	−	0.945178i	$-0.605888\pi$
$521$	89226.4	−	51514.9i	0.328714	−	0.189783i	0.655270	+	0.755395i	$0.272555\pi$
$522$		0			0
$523$	−34436.6	+	59645.9i	−0.125897	+	0.218061i	−0.922083	−	0.386991i	$-0.873514\pi$
$523$	−34436.6	+	59645.9i	−0.125897	+	0.218061i	0.796186	+	0.605052i	$0.206848\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	63675.1	+	174946.i	0.229271	+	0.629916i
$528$		0			0
$529$	48560.6	+	275401.i	0.173529	+	0.984132i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−145270.	−	173126.i	−0.511355	−	0.609409i
$534$		0			0
$535$	−3020.94	+	17132.6i	−0.0105544	+	0.0598572i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	58613.9i		−	0.201754i
$540$		0			0
$541$	172390.			0.589005			0.294502	−	0.955651i	$-0.404846\pi$
$541$	172390.			0.589005			0.294502	+	0.955651i	$0.404846\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−56533.6	−	9968.39i	−0.190333	−	0.0335608i
$546$		0			0
$547$	285530.	−	239588.i	0.954283	−	0.800738i	−0.0257309	−	0.999669i	$-0.508191\pi$
$547$	285530.	−	239588.i	0.954283	−	0.800738i	0.980014	+	0.198931i	$0.0637469\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	49648.8	−	8754.42i	0.163533	−	0.0288353i
$552$		0			0
$553$	487931.	−	177592.i	1.59554	−	0.580729i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	109992.	+	63503.9i	0.354528	+	0.204687i	0.666678	−	0.745346i	$-0.267716\pi$
$557$	109992.	+	63503.9i	0.354528	+	0.204687i	−0.312150	+	0.950033i	$0.601049\pi$
$558$		0			0
$559$	184064.	+	318808.i	0.589040	+	1.02025i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	203868.	−	242961.i	0.643181	−	0.766513i	−0.341688	−	0.939813i	$-0.610999\pi$
$563$	203868.	−	242961.i	0.643181	−	0.766513i	0.984869	+	0.173300i	$0.0554431\pi$
$564$		0			0
$565$	330946.	+	120454.i	1.03672	+	0.377334i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−198514.	+	545413.i	−0.613150	+	1.68462i	0.110006	+	0.993931i	$0.464913\pi$
$569$	−198514.	+	545413.i	−0.613150	+	1.68462i	−0.723156	+	0.690685i	$0.757309\pi$
$570$		0			0
$571$	−403041.	−	338192.i	−1.23617	−	1.03727i	−0.997814	−	0.0660883i	$-0.978948\pi$
$571$	−403041.	−	338192.i	−1.23617	−	1.03727i	−0.238353	−	0.971179i	$-0.576607\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4467.87	−	2579.53i	0.0135134	−	0.00780197i
$576$		0			0
$577$	−200068.	+	346528.i	−0.600933	+	1.04085i	0.391747	+	0.920073i	$0.371871\pi$
$577$	−200068.	+	346528.i	−0.600933	+	1.04085i	−0.992680	+	0.120774i	$0.961462\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−113063.	−	310638.i	−0.334940	−	0.920241i
$582$		0			0
$583$	65071.7	+	369040.i	0.191450	+	1.08577i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1859.82	+	2216.45i	0.00539752	+	0.00643252i	0.768736	−	0.639566i	$-0.220886\pi$
$587$	1859.82	+	2216.45i	0.00539752	+	0.00643252i	−0.763339	+	0.645998i	$0.776441\pi$
$588$		0			0
$589$	15509.5	−	87958.9i	0.0447062	−	0.253542i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			66977.9i			0.190468i	0.995455	+	0.0952340i	$0.0303599\pi$
$593$			66977.9i			0.190468i	−0.995455	+	0.0952340i	$0.969640\pi$
$594$		0			0
$595$	150171.			0.424181
$596$		0			0
$597$		0			0
$598$		0			0
$599$	475745.	+	83886.7i	1.32593	+	0.233797i	0.791372	−	0.611335i	$-0.209367\pi$
$599$	475745.	+	83886.7i	1.32593	+	0.233797i	0.534558	+	0.845132i	$0.320478\pi$
$600$		0			0
$601$	458187.	−	384464.i	1.26851	−	1.06441i	0.273788	−	0.961790i	$-0.411723\pi$
$601$	458187.	−	384464.i	1.26851	−	1.06441i	0.994721	−	0.102615i	$-0.0327210\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−97456.8	+	17184.3i	−0.266257	+	0.0469483i
$606$		0			0
$607$	−598800.	+	217945.i	−1.62519	+	0.591521i	−0.984361	−	0.176162i	$-0.943632\pi$
$607$	−598800.	+	217945.i	−1.62519	+	0.591521i	−0.640829	+	0.767683i	$0.721409\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	145413.	+	83954.3i	0.389512	+	0.224885i
$612$		0			0
$613$	−99379.8	−	172131.i	−0.264471	−	0.458076i	0.702954	−	0.711235i	$-0.251864\pi$
$613$	−99379.8	−	172131.i	−0.264471	−	0.458076i	−0.967425	+	0.253159i	$0.918531\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	52048.6	−	62029.1i	0.136722	−	0.162939i	−0.693339	−	0.720612i	$-0.743861\pi$
$617$	52048.6	−	62029.1i	0.136722	−	0.162939i	0.830061	+	0.557673i	$0.188306\pi$
$618$		0			0
$619$	195774.	+	71256.0i	0.510945	+	0.185969i	0.584610	−	0.811314i	$-0.301247\pi$
$619$	195774.	+	71256.0i	0.510945	+	0.185969i	−0.0736652	+	0.997283i	$0.523470\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−46011.8	+	126416.i	−0.118548	+	0.325707i
$624$		0			0
$625$	−14665.4	−	12305.8i	−0.0375435	−	0.0315027i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	132659.	−	76590.7i	0.335301	−	0.193586i
$630$		0			0
$631$	−233911.	+	405146.i	−0.587479	+	1.01754i	0.407082	+	0.913392i	$0.366546\pi$
$631$	−233911.	+	405146.i	−0.587479	+	1.01754i	−0.994561	+	0.104153i	$0.966787\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	49193.4	+	135158.i	0.122000	+	0.335192i
$636$		0			0
$637$	18085.6	+	102568.i	0.0445711	+	0.252775i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−448373.	−	534350.i	−1.09125	−	1.30050i	−0.950593	−	0.310440i	$-0.899524\pi$
$641$	−448373.	−	534350.i	−1.09125	−	1.30050i	−0.140655	−	0.990059i	$-0.544921\pi$
$642$		0			0
$643$	−18041.5	+	102318.i	−0.0436365	+	0.247475i	−0.998821	−	0.0485347i	$-0.984545\pi$
$643$	−18041.5	+	102318.i	−0.0436365	+	0.247475i	0.955185	+	0.296009i	$0.0956560\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	221844.i		−	0.529956i	−0.964254	−	0.264978i	$-0.914635\pi$
$647$		−	221844.i		−	0.529956i	0.964254	−	0.264978i	$-0.0853647\pi$
$648$		0			0
$649$	−560953.			−1.33179
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−280560.	−	49470.3i	−0.657960	−	0.116016i	−0.165307	−	0.986242i	$-0.552862\pi$
$653$	−280560.	−	49470.3i	−0.657960	−	0.116016i	−0.492653	+	0.870226i	$0.663973\pi$
$654$		0			0
$655$	−66424.2	+	55736.5i	−0.154826	+	0.129914i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	320732.	−	56553.8i	0.738537	−	0.130224i	0.208290	−	0.978067i	$-0.433210\pi$
$659$	320732.	−	56553.8i	0.738537	−	0.130224i	0.530247	+	0.847843i	$0.322099\pi$
$660$		0			0
$661$	631353.	−	229794.i	1.44500	−	0.525939i	0.503813	−	0.863813i	$-0.331930\pi$
$661$	631353.	−	229794.i	1.44500	−	0.525939i	0.941191	+	0.337874i	$0.109708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−62391.5	−	36021.8i	−0.141085	−	0.0814557i
$666$		0			0
$667$	4246.91	+	7355.87i	0.00954600	+	0.0165342i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−113875.	+	135711.i	−0.252921	+	0.301420i
$672$		0			0
$673$	377311.	+	137330.i	0.833046	+	0.303204i	0.723108	−	0.690734i	$-0.242713\pi$
$673$	377311.	+	137330.i	0.833046	+	0.303204i	0.109938	+	0.993938i	$0.464935\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−298505.	+	820137.i	−0.651290	+	1.78941i	−0.0383669	+	0.999264i	$0.512216\pi$
$677$	−298505.	+	820137.i	−0.651290	+	1.78941i	−0.612924	+	0.790142i	$0.710007\pi$
$678$		0			0
$679$	1809.34	+	1518.22i	0.00392447	+	0.00329302i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−67606.5	+	39032.6i	−0.144926	+	0.0836732i	−0.570710	−	0.821152i	$-0.693332\pi$
$683$	−67606.5	+	39032.6i	−0.144926	+	0.0836732i	0.425784	+	0.904825i	$0.359999\pi$
$684$		0			0
$685$	241015.	−	417451.i	0.513645	−	0.889660i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−227738.	−	625704.i	−0.479730	−	1.31805i
$690$		0			0
$691$	−120873.	−	685503.i	−0.253147	−	1.43567i	−0.800786	−	0.598951i	$-0.795584\pi$
$691$	−120873.	−	685503.i	−0.253147	−	1.43567i	0.547639	−	0.836715i	$-0.315527\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−18908.0	−	22533.7i	−0.0391450	−	0.0466512i
$696$		0			0
$697$	41264.5	−	234023.i	0.0849398	−	0.481718i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	296521.i		−	0.603419i	−0.953400	−	0.301710i	$-0.902443\pi$
$701$		−	296521.i		−	0.603419i	0.953400	−	0.301710i	$-0.0975573\pi$
$702$		0			0
$703$	−73488.0			−0.148698
$704$		0			0
$705$		0			0
$706$		0			0
$707$	106311.	+	18745.5i	0.212686	+	0.0375023i
$708$		0			0
$709$	−691450.	+	580195.i	−1.37552	+	1.15420i	−0.404689	+	0.914454i	$0.632620\pi$
$709$	−691450.	+	580195.i	−1.37552	+	1.15420i	−0.970835	+	0.239748i	$0.922935\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14819.2	−	2613.03i	0.0291506	−	0.00514003i
$714$		0			0
$715$	−223275.	+	81265.3i	−0.436744	+	0.158962i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	802547.	+	463351.i	1.55243	+	0.896297i	0.997943	+	0.0641058i	$0.0204195\pi$
$719$	802547.	+	463351.i	1.55243	+	0.896297i	0.554489	+	0.832191i	$0.312914\pi$
$720$		0			0
$721$	−417913.	−	723846.i	−0.803924	−	1.39244i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−146792.	+	174940.i	−0.279272	+	0.332823i
$726$		0			0
$727$	−327145.	−	119071.i	−0.618972	−	0.225287i	0.0134524	−	0.999910i	$-0.495718\pi$
$727$	−327145.	−	119071.i	−0.618972	−	0.225287i	−0.632424	+	0.774622i	$0.717940\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−132388.	+	363732.i	−0.247750	+	0.680686i
$732$		0			0
$733$	−319835.	−	268374.i	−0.595276	−	0.499496i	0.294647	−	0.955606i	$-0.404798\pi$
$733$	−319835.	−	268374.i	−0.595276	−	0.499496i	−0.889923	+	0.456110i	$0.849242\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	138632.	−	80039.1i	0.255228	−	0.147356i
$738$		0			0
$739$	−247544.	+	428759.i	−0.453277	+	0.785098i	−0.998587	−	0.0531359i	$-0.983078\pi$
$739$	−247544.	+	428759.i	−0.453277	+	0.785098i	0.545311	+	0.838234i	$0.316412\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−36587.0	−	100522.i	−0.0662750	−	0.182089i	0.902134	−	0.431456i	$-0.142000\pi$
$743$	−36587.0	−	100522.i	−0.0662750	−	0.182089i	−0.968409	+	0.249367i	$0.919778\pi$
$744$		0			0
$745$	76289.3	+	432658.i	0.137452	+	0.779529i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−38796.3	−	46235.6i	−0.0691555	−	0.0824163i
$750$		0			0
$751$	−21750.0	+	123351.i	−0.0385638	+	0.218706i	−0.998000	−	0.0632212i	$-0.979863\pi$
$751$	−21750.0	+	123351.i	−0.0385638	+	0.218706i	0.959436	+	0.281927i	$0.0909738\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	404357.i		−	0.709367i
$756$		0			0
$757$	−382031.			−0.666664			−0.333332	−	0.942809i	$-0.608173\pi$
$757$	−382031.			−0.666664			−0.333332	+	0.942809i	$0.608173\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−913614.	−	161095.i	−1.57759	−	0.278171i	−0.684828	−	0.728704i	$-0.740123\pi$
$761$	−913614.	−	161095.i	−1.57759	−	0.278171i	−0.892760	+	0.450533i	$0.851234\pi$
$762$		0			0
$763$	152567.	−	128019.i	0.262066	−	0.219899i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	981611.	−	173085.i	1.66859	−	0.294217i
$768$		0			0
$769$	274516.	−	99915.7i	0.464211	−	0.168959i	−0.0993172	−	0.995056i	$-0.531666\pi$
$769$	274516.	−	99915.7i	0.464211	−	0.168959i	0.563528	+	0.826097i	$0.309444\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	476347.	+	275019.i	0.797195	+	0.460261i	0.842489	−	0.538713i	$-0.181089\pi$
$773$	476347.	+	275019.i	0.797195	+	0.460261i	−0.0452945	+	0.998974i	$0.514423\pi$
$774$		0			0
$775$	202291.	+	350379.i	0.336801	+	0.583357i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−73279.8	+	87331.5i	−0.120756	+	0.143912i
$780$		0			0
$781$	−825207.	−	300351.i	−1.35288	−	0.492410i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−219586.	+	603308.i	−0.356341	+	0.979038i
$786$		0			0
$787$	−180456.	−	151420.i	−0.291354	−	0.244475i	0.485381	−	0.874303i	$-0.338681\pi$
$787$	−180456.	−	151420.i	−0.291354	−	0.244475i	−0.776735	+	0.629828i	$0.783125\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.05816e6	+	610930.i	−1.69122	+	0.976424i
$792$		0			0
$793$	157396.	−	272618.i	0.250293	−	0.433520i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	24808.0	+	68159.4i	0.0390548	+	0.107302i	0.957687	−	0.287811i	$-0.0929275\pi$
$797$	24808.0	+	68159.4i	0.0390548	+	0.107302i	−0.918632	+	0.395113i	$0.870705\pi$
$798$		0			0
$799$	30657.8	+	173869.i	0.0480228	+	0.272351i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−515218.	−	614013.i	−0.799024	−	0.952240i
$804$		0			0
$805$	2107.71	−	11953.4i	0.00325252	−	0.0184459i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			361473.i			0.552305i	0.961114	+	0.276153i	$0.0890596\pi$
$809$			361473.i			0.552305i	−0.961114	+	0.276153i	$0.910940\pi$
$810$		0			0
$811$	−40313.3			−0.0612925			−0.0306462	−	0.999530i	$-0.509757\pi$
$811$	−40313.3			−0.0612925			−0.0306462	+	0.999530i	$0.509757\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	603997.	+	106501.i	0.909326	+	0.160339i
$816$		0			0
$817$	142253.	−	119364.i	0.213116	−	0.178826i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.02982e6	+	181584.i	−1.52782	+	0.269397i	−0.873503	−	0.486819i	$-0.838157\pi$
$821$	−1.02982e6	+	181584.i	−1.52782	+	0.269397i	−0.654322	+	0.756216i	$0.727046\pi$
$822$		0			0
$823$	−155218.	+	56494.7i	−0.229162	+	0.0834081i	−0.454049	−	0.890977i	$-0.650021\pi$
$823$	−155218.	+	56494.7i	−0.229162	+	0.0834081i	0.224887	+	0.974385i	$0.427799\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.03033e6	−	594863.i	−1.50649	−	0.869774i	−0.999972	−	0.00754577i	$-0.997598\pi$
$827$	−1.03033e6	−	594863.i	−1.50649	−	0.869774i	−0.506521	−	0.862228i	$-0.669069\pi$
$828$		0			0
$829$	−347059.	−	601125.i	−0.505004	−	0.874693i	−0.999983	−	0.00578777i	$-0.998158\pi$
$829$	−347059.	−	601125.i	−0.505004	−	0.874693i	0.494979	−	0.868905i	$-0.335176\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−70392.7	+	83890.7i	−0.101447	+	0.120899i
$834$		0			0
$835$	198643.	+	72300.0i	0.284905	+	0.103697i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−63511.9	+	174498.i	−0.0902259	+	0.247894i	−0.976595	−	0.215085i	$-0.930997\pi$
$839$	−63511.9	+	174498.i	−0.0902259	+	0.247894i	0.886369	+	0.462979i	$0.153219\pi$
$840$		0			0
$841$	253788.	+	212954.i	0.358822	+	0.301088i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−27454.9	+	15851.1i	−0.0384510	+	0.0221997i
$846$		0			0
$847$	171665.	−	297332.i	0.239284	−	0.414453i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4234.62	−	11634.5i	−0.00584729	−	0.0160653i
$852$		0			0
$853$	−149347.	−	846990.i	−0.205257	−	1.16407i	−0.897034	−	0.441961i	$-0.854283\pi$
$853$	−149347.	−	846990.i	−0.205257	−	1.16407i	0.691777	−	0.722112i	$-0.256828\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−636576.	−	758642.i	−0.866740	−	1.03294i	−0.999128	−	0.0417421i	$-0.986709\pi$
$857$	−636576.	−	758642.i	−0.866740	−	1.03294i	0.132389	−	0.991198i	$-0.457735\pi$
$858$		0			0
$859$	188027.	−	1.06636e6i	0.254821	−	1.44516i	−0.541712	−	0.840564i	$-0.682223\pi$
$859$	188027.	−	1.06636e6i	0.254821	−	1.44516i	0.796532	−	0.604596i	$-0.206665\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			870848.i			1.16929i	0.811290	+	0.584643i	$0.198766\pi$
$863$			870848.i			1.16929i	−0.811290	+	0.584643i	$0.801234\pi$
$864$		0			0
$865$	822277.			1.09897
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−850727.	−	150006.i	−1.12655	−	0.198641i
$870$		0			0
$871$	−217895.	+	182836.i	−0.287218	+	0.241005i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	860714.	−	151767.i	1.12420	−	0.198226i
$876$		0			0
$877$	−763778.	+	277993.i	−0.993043	+	0.361438i	−0.786898	−	0.617083i	$-0.788314\pi$
$877$	−763778.	+	277993.i	−0.993043	+	0.361438i	−0.206145	+	0.978521i	$0.566092\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.00260e6	−	578853.i	−1.29175	−	0.745790i	−0.312782	−	0.949825i	$-0.601261\pi$
$881$	−1.00260e6	−	578853.i	−1.29175	−	0.745790i	−0.978964	+	0.204035i	$0.934594\pi$
$882$		0			0
$883$	758959.	+	1.31456e6i	0.973413	+	1.68600i	0.685077	+	0.728471i	$0.259769\pi$
$883$	758959.	+	1.31456e6i	0.973413	+	1.68600i	0.288336	+	0.957529i	$0.406898\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−672195.	+	801091.i	−0.854374	+	1.01820i	0.145211	+	0.989401i	$0.453614\pi$
$887$	−672195.	+	801091.i	−0.854374	+	1.01820i	−0.999585	+	0.0288025i	$0.990831\pi$
$888$		0			0
$889$	−468913.	−	170670.i	−0.593319	−	0.215951i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	28969.0	−	79591.5i	0.0363270	−	0.0998077i
$894$		0			0
$895$	−296328.	−	248648.i	−0.369936	−	0.310413i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−576861.	+	333051.i	−0.713759	+	0.412089i
$900$		0			0
$901$	350067.	−	606334.i	0.431223	−	0.746899i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	188940.	+	519108.i	0.230689	+	0.633812i
$906$		0			0
$907$	−105260.	−	596961.i	−0.127953	−	0.725657i	−0.979510	−	0.201395i	$-0.935452\pi$
$907$	−105260.	−	596961.i	−0.127953	−	0.725657i	0.851557	−	0.524262i	$-0.175659\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−137807.	−	164232.i	−0.166048	−	0.197888i	0.676604	−	0.736347i	$-0.263451\pi$
$911$	−137807.	−	164232.i	−0.166048	−	0.197888i	−0.842652	+	0.538459i	$0.819007\pi$
$912$		0			0
$913$	−95500.3	+	541609.i	−0.114568	+	0.649747i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	300831.i		−	0.357754i
$918$		0			0
$919$	−197124.			−0.233404			−0.116702	−	0.993167i	$-0.537232\pi$
$919$	−197124.			−0.233404			−0.116702	+	0.993167i	$0.537232\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.53670e6	+	270962.i	1.80379	+	0.318057i
$924$		0			0
$925$	255005.	−	213975.i	0.298034	−	0.250080i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.10768e6	−	195314.i	1.28346	−	0.226309i	0.510013	−	0.860166i	$-0.329640\pi$
$929$	1.10768e6	−	195314.i	1.28346	−	0.226309i	0.773450	+	0.633857i	$0.218529\pi$
$930$		0			0
$931$	49369.2	−	17968.9i	0.0569583	−	0.0207311i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−216362.	−	124917.i	−0.247490	−	0.142889i
$936$		0			0
$937$	−135548.	−	234776.i	−0.154388	−	0.267408i	0.778448	−	0.627709i	$-0.216007\pi$
$937$	−135548.	−	234776.i	−0.154388	−	0.267408i	−0.932836	+	0.360301i	$0.882674\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	252005.	−	300328.i	0.284596	−	0.339169i	−0.604739	−	0.796423i	$-0.706723\pi$
$941$	252005.	−	300328.i	0.284596	−	0.339169i	0.889336	+	0.457255i	$0.151167\pi$
$942$		0			0
$943$	−18048.8	−	6569.23i	−0.0202967	−	0.00738740i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	401266.	−	1.10247e6i	0.447437	−	1.22932i	−0.487065	−	0.873366i	$-0.661932\pi$
$947$	401266.	−	1.10247e6i	0.447437	−	1.22932i	0.934502	−	0.355958i	$-0.115845\pi$
$948$		0			0
$949$	1.09104e6	+	915488.i	1.21145	+	1.01653i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	885071.	−	510996.i	0.974524	−	0.562642i	0.0739114	−	0.997265i	$-0.476452\pi$
$953$	885071.	−	510996.i	0.974524	−	0.562642i	0.900612	+	0.434623i	$0.143118\pi$
$954$		0			0
$955$	185278.	−	320911.i	0.203150	−	0.351866i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	571974.	+	1.57149e6i	0.621927	+	1.70873i
$960$		0			0
$961$	44551.1	+	252662.i	0.0482405	+	0.273585i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	735813.	+	876908.i	0.790156	+	0.941672i
$966$		0			0
$967$	−276356.	+	1.56729e6i	−0.295540	+	1.67609i	0.369461	+	0.929246i	$0.379542\pi$
$967$	−276356.	+	1.56729e6i	−0.295540	+	1.67609i	−0.665000	+	0.746843i	$0.731569\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			625381.i			0.663294i	0.943404	+	0.331647i	$0.107604\pi$
$971$			625381.i			0.663294i	−0.943404	+	0.331647i	$0.892396\pi$
$972$		0			0
$973$	102054.			0.107796
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−812926.	−	143341.i	−0.851651	−	0.150169i	−0.269250	−	0.963070i	$-0.586776\pi$
$977$	−812926.	−	143341.i	−0.851651	−	0.150169i	−0.582402	+	0.812901i	$0.697887\pi$
$978$		0			0
$979$	171450.	−	143864.i	0.178884	−	0.150102i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.48167e6	−	261258.i	1.53336	−	0.270372i	0.657691	−	0.753287i	$-0.271533\pi$
$983$	1.48167e6	−	261258.i	1.53336	−	0.270372i	0.875667	+	0.482915i	$0.160422\pi$
$984$		0			0
$985$	−247938.	+	90242.1i	−0.255547	+	0.0930115i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	27094.6	+	15643.1i	0.0277007	+	0.0159930i
$990$		0			0
$991$	230605.	+	399420.i	0.234813	+	0.406708i	0.959218	−	0.282666i	$-0.0912189\pi$
$991$	230605.	+	399420.i	0.234813	+	0.406708i	−0.724405	+	0.689374i	$0.757886\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	595623.	−	709836.i	0.601625	−	0.716988i
$996$		0			0
$997$	−932582.	−	339432.i	−0.938203	−	0.341478i	−0.172747	−	0.984966i	$-0.555264\pi$
$997$	−932582.	−	339432.i	−0.938203	−	0.341478i	−0.765456	+	0.643488i	$0.777487\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.k.a.17.8		72
3.2	odd	2		108.5.k.a.77.9	✓	72
27.7	even	9		108.5.k.a.101.9	yes	72
27.20	odd	18	inner	324.5.k.a.305.8		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.77.9	✓	72	3.2	odd	2
108.5.k.a.101.9	yes	72	27.7	even	9
324.5.k.a.17.8		72	1.1	even	1	trivial
324.5.k.a.305.8		72	27.20	odd	18	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.k (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(15.6508 + 2.75966i) q^{5} +(-42.2367 + 35.4408i) q^{7} +O(q^{10})\)	\(q+(15.6508 + 2.75966i) q^{5} +(-42.2367 + 35.4408i) q^{7} +(90.3346 - 15.9284i) q^{11} +(-153.162 + 55.7463i) q^{13} +(-148.420 - 85.6904i) q^{17} +(41.1095 + 71.2038i) q^{19} +(-8.90402 + 10.6114i) q^{23} +(-349.975 - 127.381i) q^{25} +(209.718 - 576.196i) q^{29} +(-832.166 - 698.270i) q^{31} +(-758.845 + 438.119i) q^{35} +(-446.904 + 774.060i) q^{37} +(474.238 + 1302.96i) q^{41} +(-392.197 - 2224.26i) q^{43} +(-662.180 - 789.156i) q^{47} +(110.960 - 629.288i) q^{49} +4085.25i q^{53} +1457.77 q^{55} +(-6022.47 - 1061.92i) q^{59} +(-1479.50 + 1241.44i) q^{61} +(-2550.95 + 449.801i) q^{65} +(1639.89 - 596.872i) q^{67} +(-8290.96 - 4786.79i) q^{71} +(-4369.09 - 7567.48i) q^{73} +(-3250.92 + 3874.30i) q^{77} +(-8849.56 - 3220.98i) q^{79} +(-2050.61 + 5634.01i) q^{83} +(-2086.42 - 1750.72i) q^{85} +(2113.06 - 1219.97i) q^{89} +(4493.36 - 7782.72i) q^{91} +(446.900 + 1227.85i) q^{95} +(-7.43876 - 42.1873i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{5}+O(q^{10}) \) 72 * q - 9 * q^5	\( 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) \) 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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