LMFDB - Embedded newform 324.4.i.a.37.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(37,324)]

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, 14]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("324.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.i (of order $9$, 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:	$19.1166188419$
Analytic rank:	$0$
Dimension:	$54$
Relative dimension:	$9$ over $\Q(\zeta_{9})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			37.7
Character	$\chi$	$=$	324.37
Dual form			324.4.i.a.289.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.50317 - 2.36696i) q^{5} +(1.34685 + 7.63837i) q^{7} +O(q^{10})$	$q+(6.50317 - 2.36696i) q^{5} +(1.34685 + 7.63837i) q^{7} +(-13.7454 - 5.00292i) q^{11} +(40.8838 + 34.3056i) q^{13} +(29.6006 - 51.2697i) q^{17} +(46.8884 + 81.2131i) q^{19} +(-4.51998 + 25.6341i) q^{23} +(-59.0669 + 49.5630i) q^{25} +(172.381 - 144.645i) q^{29} +(4.27764 - 24.2597i) q^{31} +(26.8385 + 46.4856i) q^{35} +(-38.3554 + 66.4335i) q^{37} +(305.098 + 256.008i) q^{41} +(282.678 + 102.886i) q^{43} +(3.08562 + 17.4994i) q^{47} +(265.784 - 96.7374i) q^{49} +242.682 q^{53} -101.230 q^{55} +(211.368 - 76.9317i) q^{59} +(81.1159 + 460.031i) q^{61} +(347.074 + 126.325i) q^{65} +(333.309 + 279.680i) q^{67} +(-277.317 + 480.327i) q^{71} +(-226.923 - 393.043i) q^{73} +(19.7011 - 111.731i) q^{77} +(-537.292 + 450.842i) q^{79} +(427.701 - 358.883i) q^{83} +(71.1442 - 403.479i) q^{85} +(-578.094 - 1001.29i) q^{89} +(-206.974 + 358.490i) q^{91} +(497.151 + 417.160i) q^{95} +(1388.22 + 505.270i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q - 12 q^{5}+O(q^{10}) $ 54 * q - 12 * q^5	$ 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) $ 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{7}{9}\right)$

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
$5$	6.50317	−	2.36696i	0.581661	−	0.211707i	−0.0343968	−	0.999408i	$-0.510951\pi$
$5$	6.50317	−	2.36696i	0.581661	−	0.211707i	0.616058	+	0.787701i	$0.288729\pi$
$6$		0			0
$7$	1.34685	+	7.63837i	0.0727231	+	0.412433i	0.999337	+	0.0364180i	$0.0115948\pi$
$7$	1.34685	+	7.63837i	0.0727231	+	0.412433i	−0.926614	+	0.376015i	$0.877294\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.7454	−	5.00292i	−0.376763	−	0.137131i	0.146695	−	0.989182i	$-0.453136\pi$
$11$	−13.7454	−	5.00292i	−0.376763	−	0.137131i	−0.523459	+	0.852051i	$0.675359\pi$
$12$		0			0
$13$	40.8838	+	34.3056i	0.872241	+	0.731897i	0.964569	−	0.263832i	$-0.0849864\pi$
$13$	40.8838	+	34.3056i	0.872241	+	0.731897i	−0.0923280	+	0.995729i	$0.529431\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	29.6006	−	51.2697i	0.422306	−	0.731455i	−0.573859	−	0.818954i	$-0.694554\pi$
$17$	29.6006	−	51.2697i	0.422306	−	0.731455i	0.996165	+	0.0874994i	$0.0278876\pi$
$18$		0			0
$19$	46.8884	+	81.2131i	0.566155	+	0.980609i	0.996941	+	0.0781550i	$0.0249029\pi$
$19$	46.8884	+	81.2131i	0.566155	+	0.980609i	−0.430786	+	0.902454i	$0.641764\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.51998	+	25.6341i	−0.0409774	+	0.232395i	−0.998417	−	0.0562379i	$-0.982089\pi$
$23$	−4.51998	+	25.6341i	−0.0409774	+	0.232395i	0.957440	+	0.288633i	$0.0932006\pi$
$24$		0			0
$25$	−59.0669	+	49.5630i	−0.472535	+	0.396504i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	172.381	−	144.645i	1.10380	−	0.926201i	0.106129	−	0.994352i	$-0.466154\pi$
$29$	172.381	−	144.645i	1.10380	−	0.926201i	0.997675	+	0.0681509i	$0.0217099\pi$
$30$		0			0
$31$	4.27764	−	24.2597i	0.0247834	−	0.140554i	−0.969905	−	0.243482i	$-0.921710\pi$
$31$	4.27764	−	24.2597i	0.0247834	−	0.140554i	0.994689	+	0.102929i	$0.0328213\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	26.8385	+	46.4856i	0.129615	+	0.224500i
$36$		0			0
$37$	−38.3554	+	66.4335i	−0.170421	+	0.295178i	−0.938567	−	0.345096i	$-0.887846\pi$
$37$	−38.3554	+	66.4335i	−0.170421	+	0.295178i	0.768146	+	0.640275i	$0.221180\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	305.098	+	256.008i	1.16215	+	0.975164i	0.999933	−	0.0115905i	$-0.00368946\pi$
$41$	305.098	+	256.008i	1.16215	+	0.975164i	0.162222	+	0.986754i	$0.448134\pi$
$42$		0			0
$43$	282.678	+	102.886i	1.00251	+	0.364885i	0.790553	−	0.612394i	$-0.209793\pi$
$43$	282.678	+	102.886i	1.00251	+	0.364885i	0.211960	+	0.977278i	$0.432016\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.08562	+	17.4994i	0.00957624	+	0.0543096i	0.989221	−	0.146427i	$-0.0467775\pi$
$47$	3.08562	+	17.4994i	0.00957624	+	0.0543096i	−0.979645	+	0.200737i	$0.935666\pi$
$48$		0			0
$49$	265.784	−	96.7374i	0.774880	−	0.282033i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	242.682			0.628960			0.314480	−	0.949264i	$-0.398170\pi$
$53$	242.682			0.628960			0.314480	+	0.949264i	$0.398170\pi$
$54$		0			0
$55$	−101.230			−0.248180
$56$		0			0
$57$		0			0
$58$		0			0
$59$	211.368	−	76.9317i	0.466403	−	0.169757i	−0.0981193	−	0.995175i	$-0.531283\pi$
$59$	211.368	−	76.9317i	0.466403	−	0.169757i	0.564522	+	0.825418i	$0.309060\pi$
$60$		0			0
$61$	81.1159	+	460.031i	0.170259	+	0.965589i	0.943475	+	0.331445i	$0.107536\pi$
$61$	81.1159	+	460.031i	0.170259	+	0.965589i	−0.773215	+	0.634144i	$0.781353\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	347.074	+	126.325i	0.662296	+	0.241056i
$66$		0			0
$67$	333.309	+	279.680i	0.607764	+	0.509975i	0.893931	−	0.448205i	$-0.147937\pi$
$67$	333.309	+	279.680i	0.607764	+	0.509975i	−0.286166	+	0.958180i	$0.592381\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−277.317	+	480.327i	−0.463541	+	0.802877i	−0.999134	−	0.0415991i	$-0.986755\pi$
$71$	−277.317	+	480.327i	−0.463541	+	0.802877i	0.535593	+	0.844476i	$0.320088\pi$
$72$		0			0
$73$	−226.923	−	393.043i	−0.363827	−	0.630166i	0.624760	−	0.780817i	$-0.285197\pi$
$73$	−226.923	−	393.043i	−0.363827	−	0.630166i	−0.988587	+	0.150650i	$0.951863\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	19.7011	−	111.731i	0.0291578	−	0.165362i
$78$		0			0
$79$	−537.292	+	450.842i	−0.765191	+	0.642072i	−0.939473	−	0.342624i	$-0.888684\pi$
$79$	−537.292	+	450.842i	−0.765191	+	0.642072i	0.174281	+	0.984696i	$0.444240\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	427.701	−	358.883i	0.565617	−	0.474609i	−0.314571	−	0.949234i	$-0.601861\pi$
$83$	427.701	−	358.883i	0.565617	−	0.474609i	0.880188	+	0.474625i	$0.157416\pi$
$84$		0			0
$85$	71.1442	−	403.479i	0.0907844	−	0.514864i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−578.094	−	1001.29i	−0.688515	−	1.19254i	−0.972318	−	0.233660i	$-0.924930\pi$
$89$	−578.094	−	1001.29i	−0.688515	−	1.19254i	0.283804	−	0.958882i	$-0.408404\pi$
$90$		0			0
$91$	−206.974	+	358.490i	−0.238426	+	0.412967i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	497.151	+	417.160i	0.536912	+	0.450523i
$96$		0			0
$97$	1388.22	+	505.270i	1.45312	+	0.528891i	0.943460	−	0.331488i	$-0.107551\pi$
$97$	1388.22	+	505.270i	1.45312	+	0.528891i	0.509656	+	0.860378i	$0.329773\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−285.927	−	1621.57i	−0.281691	−	1.59755i	−0.716870	−	0.697207i	$-0.754426\pi$
$101$	−285.927	−	1621.57i	−0.281691	−	1.59755i	0.435178	−	0.900344i	$-0.356685\pi$
$102$		0			0
$103$	495.107	−	180.204i	0.473634	−	0.172389i	−0.0941639	−	0.995557i	$-0.530018\pi$
$103$	495.107	−	180.204i	0.473634	−	0.172389i	0.567798	+	0.823168i	$0.307796\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1997.93			−1.80512			−0.902559	−	0.430566i	$-0.858314\pi$
$107$	−1997.93			−1.80512			−0.902559	+	0.430566i	$0.858314\pi$
$108$		0			0
$109$	−1268.88			−1.11502			−0.557509	−	0.830171i	$-0.688243\pi$
$109$	−1268.88			−1.11502			−0.557509	+	0.830171i	$0.688243\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−216.017	+	78.6237i	−0.179833	+	0.0654539i	−0.430367	−	0.902654i	$-0.641616\pi$
$113$	−216.017	+	78.6237i	−0.179833	+	0.0654539i	0.250534	+	0.968108i	$0.419394\pi$
$114$		0			0
$115$	31.2806	+	177.401i	0.0253647	+	0.143850i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	431.484	+	157.047i	0.332387	+	0.120979i
$120$		0			0
$121$	−855.698	−	718.016i	−0.642899	−	0.539456i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−699.341	+	1211.29i	−0.500408	+	0.866732i
$126$		0			0
$127$	−923.845	−	1600.15i	−0.645496	−	1.11803i	−0.984187	−	0.177134i	$-0.943317\pi$
$127$	−923.845	−	1600.15i	−0.645496	−	1.11803i	0.338691	−	0.940898i	$-0.390016\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−462.901	+	2625.24i	−0.308732	+	1.75091i	0.296665	+	0.954981i	$0.404125\pi$
$131$	−462.901	+	2625.24i	−0.308732	+	1.75091i	−0.605397	+	0.795924i	$0.706986\pi$
$132$		0			0
$133$	−557.184	+	467.533i	−0.363263	+	0.304814i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	150.629	−	126.392i	0.0939348	−	0.0788206i	−0.594611	−	0.804014i	$-0.702694\pi$
$137$	150.629	−	126.392i	0.0939348	−	0.0788206i	0.688546	+	0.725193i	$0.258249\pi$
$138$		0			0
$139$	236.445	−	1340.95i	0.144281	−	0.818257i	−0.823661	−	0.567082i	$-0.808072\pi$
$139$	236.445	−	1340.95i	0.144281	−	0.818257i	0.967942	−	0.251174i	$-0.0808168\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−390.337	−	676.083i	−0.228263	−	0.395363i
$144$		0			0
$145$	778.653	−	1348.67i	0.445956	−	0.772419i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	582.798	+	489.026i	0.320434	+	0.268876i	0.788789	−	0.614664i	$-0.210709\pi$
$149$	582.798	+	489.026i	0.320434	+	0.268876i	−0.468355	+	0.883541i	$0.655153\pi$
$150$		0			0
$151$	−2128.86	−	774.842i	−1.14731	−	0.417588i	−0.302764	−	0.953066i	$-0.597909\pi$
$151$	−2128.86	−	774.842i	−1.14731	−	0.417588i	−0.844549	+	0.535478i	$0.820132\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−29.6035	−	167.890i	−0.0153407	−	0.0870016i
$156$		0			0
$157$	3067.14	−	1116.35i	1.55914	−	0.567480i	0.588599	−	0.808425i	$-0.299680\pi$
$157$	3067.14	−	1116.35i	1.55914	−	0.567480i	0.970539	+	0.240946i	$0.0774576\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−201.890			−0.0988272
$162$		0			0
$163$	−3101.93			−1.49057			−0.745283	−	0.666749i	$-0.767685\pi$
$163$	−3101.93			−1.49057			−0.745283	+	0.666749i	$0.767685\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−875.969	+	318.826i	−0.405895	+	0.147734i	−0.536896	−	0.843649i	$-0.680403\pi$
$167$	−875.969	+	318.826i	−0.405895	+	0.147734i	0.131001	+	0.991382i	$0.458181\pi$
$168$		0			0
$169$	113.107	+	641.464i	0.0514827	+	0.291973i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2953.58	−	1075.01i	−1.29801	−	0.472439i	−0.401665	−	0.915787i	$-0.631568\pi$
$173$	−2953.58	−	1075.01i	−1.29801	−	0.472439i	−0.896349	+	0.443348i	$0.853790\pi$
$174$		0			0
$175$	−458.134	−	384.420i	−0.197895	−	0.166054i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1491.37	+	2583.12i	−0.622737	+	1.07861i	0.366237	+	0.930522i	$0.380646\pi$
$179$	−1491.37	+	2583.12i	−0.622737	+	1.07861i	−0.988974	+	0.148090i	$0.952687\pi$
$180$		0			0
$181$	−475.603	−	823.768i	−0.195311	−	0.338289i	0.751691	−	0.659515i	$-0.229238\pi$
$181$	−475.603	−	823.768i	−0.195311	−	0.338289i	−0.947002	+	0.321226i	$0.895905\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−92.1862	+	522.814i	−0.0366360	+	0.207773i
$186$		0			0
$187$	−663.370	+	556.634i	−0.259414	+	0.217674i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2073.86	−	1740.18i	0.785651	−	0.659240i	−0.159014	−	0.987276i	$-0.550831\pi$
$191$	2073.86	−	1740.18i	0.785651	−	0.659240i	0.944665	+	0.328037i	$0.106387\pi$
$192$		0			0
$193$	742.402	−	4210.37i	0.276887	−	1.57031i	−0.456015	−	0.889972i	$-0.650724\pi$
$193$	742.402	−	4210.37i	0.276887	−	1.57031i	0.732903	−	0.680334i	$-0.238165\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2280.07	−	3949.20i	−0.824610	−	1.42827i	−0.902217	−	0.431282i	$-0.858061\pi$
$197$	−2280.07	−	3949.20i	−0.824610	−	1.42827i	0.0776069	−	0.996984i	$-0.475272\pi$
$198$		0			0
$199$	−1556.41	+	2695.78i	−0.554428	+	0.960297i	0.443520	+	0.896264i	$0.353730\pi$
$199$	−1556.41	+	2695.78i	−0.554428	+	0.960297i	−0.997948	+	0.0640327i	$0.979604\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1337.02	+	1121.89i	0.462268	+	0.387889i
$204$		0			0
$205$	2590.07	+	942.707i	0.882429	+	0.321178i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−238.198	−	1350.89i	−0.0788349	−	0.447095i
$210$		0			0
$211$	−1845.57	+	671.733i	−0.602153	+	0.219166i	−0.625066	−	0.780572i	$-0.714928\pi$
$211$	−1845.57	+	671.733i	−0.602153	+	0.219166i	0.0229132	+	0.999737i	$0.492706\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2081.83			0.660371
$216$		0			0
$217$	191.066			0.0597714
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2969.02	−	1080.64i	0.903702	−	0.328920i
$222$		0			0
$223$	737.919	+	4184.95i	0.221591	+	1.25670i	0.869096	+	0.494643i	$0.164701\pi$
$223$	737.919	+	4184.95i	0.221591	+	1.25670i	−0.647506	+	0.762061i	$0.724188\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1470.13	+	535.082i	0.429849	+	0.156452i	0.547879	−	0.836558i	$-0.315436\pi$
$227$	1470.13	+	535.082i	0.429849	+	0.156452i	−0.118030	+	0.993010i	$0.537658\pi$
$228$		0			0
$229$	3409.84	+	2861.19i	0.983967	+	0.825647i	0.984683	−	0.174353i	$-0.0557834\pi$
$229$	3409.84	+	2861.19i	0.983967	+	0.825647i	−0.000715825		1.00000i	$0.500228\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1513.22	+	2620.97i	−0.425469	+	0.736934i	−0.996464	−	0.0840197i	$-0.973224\pi$
$233$	−1513.22	+	2620.97i	−0.425469	+	0.736934i	0.570995	+	0.820953i	$0.306557\pi$
$234$		0			0
$235$	61.4866	+	106.498i	0.0170679	+	0.0295624i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−352.245	+	1997.68i	−0.0953340	+	0.540666i	0.899311	+	0.437311i	$0.144069\pi$
$239$	−352.245	+	1997.68i	−0.0953340	+	0.540666i	−0.994645	+	0.103355i	$0.967042\pi$
$240$		0			0
$241$	1684.85	−	1413.75i	0.450334	−	0.377875i	−0.389226	−	0.921142i	$-0.627257\pi$
$241$	1684.85	−	1413.75i	0.450334	−	0.377875i	0.839560	+	0.543267i	$0.182813\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1499.46	−	1258.20i	0.391009	−	0.328096i
$246$		0			0
$247$	−869.087	+	4928.84i	−0.223881	+	1.26969i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1848.46	−	3201.63i	−0.464836	−	0.805119i	0.534359	−	0.845258i	$-0.320553\pi$
$251$	−1848.46	−	3201.63i	−0.464836	−	0.805119i	−0.999194	+	0.0401391i	$0.987220\pi$
$252$		0			0
$253$	190.374	−	329.738i	0.0473072	−	0.0819385i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2233.86	−	1874.44i	−0.542197	−	0.454957i	0.330091	−	0.943949i	$-0.392920\pi$
$257$	−2233.86	−	1874.44i	−0.542197	−	0.454957i	−0.872289	+	0.488992i	$0.837365\pi$
$258$		0			0
$259$	−559.102	−	203.497i	−0.134135	−	0.0488211i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	270.803	+	1535.80i	0.0634920	+	0.360081i	0.999957	+	0.00931899i	$0.00296637\pi$
$263$	270.803	+	1535.80i	0.0634920	+	0.360081i	−0.936465	+	0.350762i	$0.885923\pi$
$264$		0			0
$265$	1578.20	−	574.417i	0.365841	−	0.133155i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7131.45			1.61640			0.808201	−	0.588906i	$-0.200441\pi$
$269$	7131.45			1.61640			0.808201	+	0.588906i	$0.200441\pi$
$270$		0			0
$271$	5551.33			1.24435			0.622175	−	0.782878i	$-0.286249\pi$
$271$	5551.33			1.24435			0.622175	+	0.782878i	$0.286249\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1059.86	−	385.757i	0.232407	−	0.0845891i
$276$		0			0
$277$	680.783	+	3860.91i	0.147669	+	0.837471i	0.965185	+	0.261567i	$0.0842391\pi$
$277$	680.783	+	3860.91i	0.147669	+	0.837471i	−0.817517	+	0.575905i	$0.804650\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5176.79	+	1884.20i	1.09901	+	0.400006i	0.826950	−	0.562275i	$-0.190074\pi$
$281$	5176.79	+	1884.20i	1.09901	+	0.400006i	0.272057	+	0.962281i	$0.412296\pi$
$282$		0			0
$283$	−962.891	−	807.961i	−0.202254	−	0.169711i	0.536035	−	0.844196i	$-0.319922\pi$
$283$	−962.891	−	807.961i	−0.202254	−	0.169711i	−0.738289	+	0.674484i	$0.764366\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1544.56	+	2675.26i	−0.317674	+	0.550228i
$288$		0			0
$289$	704.111	+	1219.56i	0.143316	+	0.248230i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1207.87	−	6850.17i	0.240834	−	1.36584i	−0.589137	−	0.808033i	$-0.700532\pi$
$293$	1207.87	−	6850.17i	0.240834	−	1.36584i	0.829971	−	0.557806i	$-0.188357\pi$
$294$		0			0
$295$	1192.47	−	1000.60i	0.235350	−	0.197482i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1064.19	+	892.959i	−0.205831	+	0.172713i
$300$		0			0
$301$	−405.159	+	2297.77i	−0.0775847	+	0.440005i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1616.38	+	2799.66i	0.303455	+	0.525600i
$306$		0			0
$307$	1319.29	−	2285.08i	0.245263	−	0.424808i	−0.716942	−	0.697132i	$-0.754459\pi$
$307$	1319.29	−	2285.08i	0.245263	−	0.424808i	0.962206	+	0.272324i	$0.0877923\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4431.20	+	3718.22i	0.807943	+	0.677945i	0.950116	−	0.311897i	$-0.100964\pi$
$311$	4431.20	+	3718.22i	0.807943	+	0.677945i	−0.142173	+	0.989842i	$0.545409\pi$
$312$		0			0
$313$	−1731.49	−	630.210i	−0.312682	−	0.113807i	0.180912	−	0.983499i	$-0.442095\pi$
$313$	−1731.49	−	630.210i	−0.312682	−	0.113807i	−0.493594	+	0.869692i	$0.664317\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	71.8687	+	407.588i	0.0127336	+	0.0722158i	0.990512	−	0.137424i	$-0.0438821\pi$
$317$	71.8687	+	407.588i	0.0127336	+	0.0722158i	−0.977779	+	0.209639i	$0.932771\pi$
$318$		0			0
$319$	−3093.09	+	1125.79i	−0.542883	+	0.197593i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5551.70			0.956361
$324$		0			0
$325$	−4115.17			−0.702364
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−129.511	+	47.1381i	−0.0217026	+	0.00789912i
$330$		0			0
$331$	−1439.06	−	8161.32i	−0.238967	−	1.35525i	−0.834097	−	0.551618i	$-0.814011\pi$
$331$	−1439.06	−	8161.32i	−0.238967	−	1.35525i	0.595131	−	0.803629i	$-0.297100\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2829.56	+	1029.87i	0.461478	+	0.167964i
$336$		0			0
$337$	−992.442	−	832.758i	−0.160421	−	0.134609i	0.559044	−	0.829138i	$-0.311168\pi$
$337$	−992.442	−	832.758i	−0.160421	−	0.134609i	−0.719465	+	0.694529i	$0.755613\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−180.167	+	312.059i	−0.0286117	+	0.0495570i
$342$		0			0
$343$	2427.08	+	4203.82i	0.382069	+	0.661763i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−229.552	+	1301.85i	−0.0355129	+	0.201404i	−0.997402	−	0.0720360i	$-0.977050\pi$
$347$	−229.552	+	1301.85i	−0.0355129	+	0.201404i	0.961889	+	0.273440i	$0.0881615\pi$
$348$		0			0
$349$	1971.21	−	1654.04i	0.302339	−	0.253693i	−0.478978	−	0.877827i	$-0.658993\pi$
$349$	1971.21	−	1654.04i	0.302339	−	0.253693i	0.781317	+	0.624134i	$0.214548\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7466.29	+	6264.96i	−1.12575	+	0.944618i	−0.998881	−	0.0473007i	$-0.984938\pi$
$353$	−7466.29	+	6264.96i	−1.12575	+	0.944618i	−0.126872	+	0.991919i	$0.540494\pi$
$354$		0			0
$355$	−666.523	+	3780.04i	−0.0996490	+	0.565137i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5633.32	−	9757.19i	−0.828176	−	1.43444i	−0.899468	−	0.436987i	$-0.856046\pi$
$359$	−5633.32	−	9757.19i	−0.828176	−	1.43444i	0.0712921	−	0.997455i	$-0.477288\pi$
$360$		0			0
$361$	−967.548	+	1675.84i	−0.141063	+	0.244328i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2406.04	−	2018.90i	−0.345035	−	0.289518i
$366$		0			0
$367$	−721.883	−	262.744i	−0.102676	−	0.0373709i	0.290171	−	0.956975i	$-0.406288\pi$
$367$	−721.883	−	262.744i	−0.102676	−	0.0373709i	−0.392847	+	0.919604i	$0.628510\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	326.856	+	1853.69i	0.0457399	+	0.259404i
$372$		0			0
$373$	−3384.75	+	1231.95i	−0.469854	+	0.171013i	−0.566086	−	0.824346i	$-0.691543\pi$
$373$	−3384.75	+	1231.95i	−0.469854	+	0.171013i	0.0962323	+	0.995359i	$0.469321\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12009.7			1.64067
$378$		0			0
$379$	11322.2			1.53451			0.767257	−	0.641340i	$-0.221621\pi$
$379$	11322.2			1.53451			0.767257	+	0.641340i	$0.221621\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6956.61	+	2532.00i	−0.928110	+	0.337805i	−0.761460	−	0.648212i	$-0.775517\pi$
$383$	−6956.61	+	2532.00i	−0.928110	+	0.337805i	−0.166650	+	0.986016i	$0.553295\pi$
$384$		0			0
$385$	−136.342	−	773.235i	−0.0180484	−	0.102358i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6863.75	−	2498.20i	−0.894617	−	0.325614i	−0.146523	−	0.989207i	$-0.546808\pi$
$389$	−6863.75	−	2498.20i	−0.894617	−	0.325614i	−0.748094	+	0.663593i	$0.769031\pi$
$390$		0			0
$391$	1180.46	+	990.522i	0.152681	+	0.128115i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2426.98	+	4203.65i	−0.309151	+	0.535465i
$396$		0			0
$397$	4660.31	+	8071.90i	0.589155	+	1.02045i	0.994343	+	0.106213i	$0.0338725\pi$
$397$	4660.31	+	8071.90i	0.589155	+	1.02045i	−0.405189	+	0.914233i	$0.632794\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1648.21	−	9347.46i	0.205256	−	1.16406i	−0.691781	−	0.722108i	$-0.743173\pi$
$401$	1648.21	−	9347.46i	0.205256	−	1.16406i	0.897037	−	0.441956i	$-0.145715\pi$
$402$		0			0
$403$	1007.13	−	845.082i	0.124488	−	0.104458i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	859.572	−	721.267i	0.104687	−	0.0878424i
$408$		0			0
$409$	727.417	−	4125.39i	0.0879424	−	0.498746i	−0.908740	−	0.417362i	$-0.862955\pi$
$409$	727.417	−	4125.39i	0.0879424	−	0.498746i	0.996683	−	0.0813844i	$-0.0259341\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	872.313	+	1510.89i	0.103932	+	0.180015i
$414$		0			0
$415$	1931.95	−	3346.23i	0.228519	−	0.395807i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12673.7	−	10634.5i	−1.47769	−	1.23993i	−0.908611	−	0.417643i	$-0.862856\pi$
$419$	−12673.7	−	10634.5i	−1.47769	−	1.23993i	−0.569077	−	0.822284i	$-0.692699\pi$
$420$		0			0
$421$	−6440.45	−	2344.13i	−0.745578	−	0.271368i	−0.0588342	−	0.998268i	$-0.518738\pi$
$421$	−6440.45	−	2344.13i	−0.745578	−	0.271368i	−0.686744	+	0.726900i	$0.740961\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	792.666	+	4495.43i	0.0904705	+	0.513084i
$426$		0			0
$427$	−3404.63	+	1239.19i	−0.385859	+	0.140441i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16248.1			1.81588			0.907938	−	0.419104i	$-0.137656\pi$
$431$	16248.1			1.81588			0.907938	+	0.419104i	$0.137656\pi$
$432$		0			0
$433$	−6251.26			−0.693803			−0.346901	−	0.937902i	$-0.612766\pi$
$433$	−6251.26			−0.693803			−0.346901	+	0.937902i	$0.612766\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2293.76	+	834.860i	−0.251088	+	0.0913885i
$438$		0			0
$439$	−2321.71	−	13167.1i	−0.252412	−	1.43150i	−0.802629	−	0.596479i	$-0.796566\pi$
$439$	−2321.71	−	13167.1i	−0.252412	−	1.43150i	0.550216	−	0.835022i	$-0.314545\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13255.0	−	4824.43i	−1.42159	−	0.517417i	−0.487081	−	0.873357i	$-0.661939\pi$
$443$	−13255.0	−	4824.43i	−1.42159	−	0.517417i	−0.934509	+	0.355940i	$0.884161\pi$
$444$		0			0
$445$	−6129.45	−	5143.22i	−0.652952	−	0.547892i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6125.14	+	10609.0i	−0.643793	+	1.11508i	0.340786	+	0.940141i	$0.389307\pi$
$449$	−6125.14	+	10609.0i	−0.643793	+	1.11508i	−0.984579	+	0.174941i	$0.944026\pi$
$450$		0			0
$451$	−2912.91	−	5045.32i	−0.304133	−	0.526773i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−497.457	+	2821.22i	−0.0512553	+	0.290683i
$456$		0			0
$457$	−3161.89	+	2653.14i	−0.323648	+	0.271573i	−0.790106	−	0.612971i	$-0.789974\pi$
$457$	−3161.89	+	2653.14i	−0.323648	+	0.271573i	0.466458	+	0.884544i	$0.345530\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11312.5	+	9492.29i	−1.14289	+	0.959002i	−0.999530	−	0.0306683i	$-0.990236\pi$
$461$	−11312.5	+	9492.29i	−1.14289	+	0.959002i	−0.143364	+	0.989670i	$0.545792\pi$
$462$		0			0
$463$	−1803.16	+	10226.2i	−0.180993	+	1.02646i	0.750003	+	0.661434i	$0.230052\pi$
$463$	−1803.16	+	10226.2i	−0.180993	+	1.02646i	−0.930996	+	0.365029i	$0.881059\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9089.22	−	15743.0i	−0.900640	−	1.55995i	−0.826665	−	0.562694i	$-0.809765\pi$
$467$	−9089.22	−	15743.0i	−0.900640	−	1.55995i	−0.0739745	−	0.997260i	$-0.523568\pi$
$468$		0			0
$469$	−1687.38	+	2922.63i	−0.166132	+	0.287749i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3370.80	−	2828.43i	−0.327673	−	0.274950i
$474$		0			0
$475$	−6794.72	−	2473.07i	−0.656343	−	0.238889i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	878.825	+	4984.07i	0.0838300	+	0.475423i	0.997603	+	0.0691981i	$0.0220441\pi$
$479$	878.825	+	4984.07i	0.0838300	+	0.475423i	−0.913773	+	0.406225i	$0.866845\pi$
$480$		0			0
$481$	−3847.16	+	1400.25i	−0.364689	+	0.132736i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10223.8			0.957191
$486$		0			0
$487$	16549.8			1.53993			0.769963	−	0.638089i	$-0.220275\pi$
$487$	16549.8			1.53993			0.769963	+	0.638089i	$0.220275\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1518.20	+	552.579i	−0.139542	+	0.0507893i	−0.410847	−	0.911704i	$-0.634767\pi$
$491$	−1518.20	+	552.579i	−0.139542	+	0.0507893i	0.271305	+	0.962493i	$0.412545\pi$
$492$		0			0
$493$	−2313.32	−	13119.5i	−0.211332	−	1.19852i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4042.41	−	1471.32i	−0.364843	−	0.132792i
$498$		0			0
$499$	−4436.72	−	3722.85i	−0.398026	−	0.333983i	0.421704	−	0.906733i	$-0.361432\pi$
$499$	−4436.72	−	3722.85i	−0.398026	−	0.333983i	−0.819730	+	0.572750i	$0.805876\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2966.70	+	5138.47i	−0.262979	+	0.455493i	−0.967032	−	0.254654i	$-0.918038\pi$
$503$	−2966.70	+	5138.47i	−0.262979	+	0.455493i	0.704053	+	0.710147i	$0.251372\pi$
$504$		0			0
$505$	−5697.64	−	9868.59i	−0.502062	−	0.869597i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1327.68	−	7529.66i	0.115616	−	0.655690i	−0.870828	−	0.491589i	$-0.836416\pi$
$509$	1327.68	−	7529.66i	0.115616	−	0.655690i	0.986443	−	0.164102i	$-0.0524725\pi$
$510$		0			0
$511$	2696.57	−	2262.69i	0.233443	−	0.195882i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2793.23	−	2343.79i	0.238998	−	0.200544i
$516$		0			0
$517$	45.1350	−	255.974i	0.00383953	−	0.0217751i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−294.488	−	510.068i	−0.0247635	−	0.0428916i	0.853378	−	0.521292i	$-0.174550\pi$
$521$	−294.488	−	510.068i	−0.0247635	−	0.0428916i	−0.878142	+	0.478401i	$0.841217\pi$
$522$		0			0
$523$	4872.57	−	8439.54i	0.407386	−	0.705613i	−0.587210	−	0.809434i	$-0.699774\pi$
$523$	4872.57	−	8439.54i	0.407386	−	0.705613i	0.994596	+	0.103822i	$0.0331071\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1117.17	−	937.415i	−0.0923426	−	0.0774847i
$528$		0			0
$529$	10796.6	+	3929.63i	0.887365	+	0.322974i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3691.08	+	20933.2i	0.299960	+	1.70115i
$534$		0			0
$535$	−12992.9	+	4729.03i	−1.04997	+	0.382157i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4137.28			−0.330622
$540$		0			0
$541$	−4636.36			−0.368452			−0.184226	−	0.982884i	$-0.558978\pi$
$541$	−4636.36			−0.368452			−0.184226	+	0.982884i	$0.558978\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8251.77	+	3003.40i	−0.648563	+	0.236058i
$546$		0			0
$547$	−3576.84	−	20285.3i	−0.279588	−	1.58562i	−0.723999	−	0.689801i	$-0.757698\pi$
$547$	−3576.84	−	20285.3i	−0.279588	−	1.58562i	0.444411	−	0.895823i	$-0.353413\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	19829.7	+	7217.42i	1.53317	+	0.558027i
$552$		0			0
$553$	−4167.35	−	3496.82i	−0.320459	−	0.268897i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2376.10	+	4115.53i	−0.180752	+	0.313071i	−0.942137	−	0.335229i	$-0.891186\pi$
$557$	−2376.10	+	4115.53i	−0.180752	+	0.313071i	0.761385	+	0.648300i	$0.224520\pi$
$558$		0			0
$559$	8027.38	+	13903.8i	0.607374	+	1.05200i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1868.80	+	10598.5i	−0.139894	+	0.793379i	0.831432	+	0.555627i	$0.187522\pi$
$563$	−1868.80	+	10598.5i	−0.139894	+	0.793379i	−0.971326	+	0.237752i	$0.923589\pi$
$564$		0			0
$565$	−1218.69	+	1022.61i	−0.0907449	+	0.0761440i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2744.52	+	2302.93i	−0.202208	+	0.169673i	−0.738269	−	0.674507i	$-0.764356\pi$
$569$	−2744.52	+	2302.93i	−0.202208	+	0.169673i	0.536061	+	0.844180i	$0.319912\pi$
$570$		0			0
$571$	588.801	−	3339.25i	0.0431533	−	0.244735i	−0.955599	−	0.294670i	$-0.904790\pi$
$571$	588.801	−	3339.25i	0.0431533	−	0.244735i	0.998752	+	0.0499351i	$0.0159015\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1003.52	−	1738.15i	−0.0727821	−	0.126062i
$576$		0			0
$577$	−1802.55	+	3122.12i	−0.130054	+	0.225261i	−0.923697	−	0.383123i	$-0.874848\pi$
$577$	−1802.55	+	3122.12i	−0.130054	+	0.225261i	0.793643	+	0.608384i	$0.208182\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3317.33	+	2783.57i	0.236878	+	0.198764i
$582$		0			0
$583$	−3335.76	−	1214.12i	−0.236969	−	0.0862497i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3392.97	−	19242.5i	−0.238574	−	1.35302i	−0.834955	−	0.550318i	$-0.814506\pi$
$587$	−3392.97	−	19242.5i	−0.238574	−	1.35302i	0.596381	−	0.802702i	$-0.296605\pi$
$588$		0			0
$589$	2170.78	−	790.099i	0.151860	−	0.0552724i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9718.65			0.673013			0.336507	−	0.941681i	$-0.390755\pi$
$593$	9718.65			0.673013			0.336507	+	0.941681i	$0.390755\pi$
$594$		0			0
$595$	3177.74			0.218949
$596$		0			0
$597$		0			0
$598$		0			0
$599$	23413.0	−	8521.62i	1.59704	−	0.581275i	0.618222	−	0.786003i	$-0.287853\pi$
$599$	23413.0	−	8521.62i	1.59704	−	0.581275i	0.978819	+	0.204728i	$0.0656309\pi$
$600$		0			0
$601$	4759.65	+	26993.3i	0.323045	+	1.83208i	0.523069	+	0.852290i	$0.324787\pi$
$601$	4759.65	+	26993.3i	0.323045	+	1.83208i	−0.200024	+	0.979791i	$0.564102\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7264.26	−	2643.98i	−0.488156	−	0.177674i
$606$		0			0
$607$	19958.2	+	16746.9i	1.33456	+	1.11983i	0.982988	+	0.183671i	$0.0587979\pi$
$607$	19958.2	+	16746.9i	1.33456	+	1.11983i	0.351574	+	0.936160i	$0.385646\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−474.175	+	821.296i	−0.0313962	+	0.0543798i
$612$		0			0
$613$	−694.965	−	1203.72i	−0.0457902	−	0.0793109i	0.842222	−	0.539131i	$-0.181247\pi$
$613$	−694.965	−	1203.72i	−0.0457902	−	0.0793109i	−0.888012	+	0.459820i	$0.847914\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1604.15	−	9097.57i	0.104669	−	0.593605i	−0.886684	−	0.462377i	$-0.846997\pi$
$617$	1604.15	−	9097.57i	0.104669	−	0.593605i	0.991352	−	0.131228i	$-0.0418921\pi$
$618$		0			0
$619$	5940.36	−	4984.55i	0.385724	−	0.323661i	−0.429221	−	0.903200i	$-0.641212\pi$
$619$	5940.36	−	4984.55i	0.385724	−	0.323661i	0.814945	+	0.579539i	$0.196767\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6869.59	−	5764.27i	0.441773	−	0.370691i
$624$		0			0
$625$	−7.17646	+	40.6997i	−0.000459293	+	0.00260478i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2270.68	+	3932.94i	0.143940	+	0.249311i
$630$		0			0
$631$	−1569.14	+	2717.83i	−0.0989958	+	0.171466i	−0.911269	−	0.411811i	$-0.864896\pi$
$631$	−1569.14	+	2717.83i	−0.0989958	+	0.171466i	0.812273	+	0.583277i	$0.198230\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9795.40	−	8219.32i	−0.612155	−	0.513659i
$636$		0			0
$637$	14184.9	+	5162.88i	0.882301	+	0.321131i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1044.51	−	5923.70i	−0.0643613	−	0.365011i	−0.999930	−	0.0118653i	$-0.996223\pi$
$641$	−1044.51	−	5923.70i	−0.0643613	−	0.365011i	0.935568	−	0.353146i	$-0.114888\pi$
$642$		0			0
$643$	−3903.59	+	1420.79i	−0.239413	+	0.0871393i	−0.458940	−	0.888467i	$-0.651771\pi$
$643$	−3903.59	+	1420.79i	−0.239413	+	0.0871393i	0.219527	+	0.975606i	$0.429549\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3878.14			0.235650			0.117825	−	0.993034i	$-0.462408\pi$
$647$	3878.14			0.235650			0.117825	+	0.993034i	$0.462408\pi$
$648$		0			0
$649$	−3290.22			−0.199002
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3094.96	+	1126.48i	−0.185475	+	0.0675075i	−0.433088	−	0.901352i	$-0.642576\pi$
$653$	−3094.96	+	1126.48i	−0.185475	+	0.0675075i	0.247613	+	0.968859i	$0.420354\pi$
$654$		0			0
$655$	3203.52	+	18168.1i	0.191102	+	1.08379i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14236.8	+	5181.76i	0.841556	+	0.306301i	0.726593	−	0.687068i	$-0.241103\pi$
$659$	14236.8	+	5181.76i	0.841556	+	0.306301i	0.114963	+	0.993370i	$0.463325\pi$
$660$		0			0
$661$	3706.41	+	3110.04i	0.218098	+	0.183006i	0.745290	−	0.666740i	$-0.232311\pi$
$661$	3706.41	+	3110.04i	0.218098	+	0.183006i	−0.527193	+	0.849746i	$0.676755\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2516.83	+	4359.28i	−0.146765	+	0.254204i
$666$		0			0
$667$	2928.68	+	5072.62i	0.170013	+	0.294471i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1186.53	−	6729.13i	0.0682643	−	0.387146i
$672$		0			0
$673$	−15159.9	+	12720.6i	−0.868305	+	0.728595i	−0.963741	−	0.266841i	$-0.914020\pi$
$673$	−15159.9	+	12720.6i	−0.868305	+	0.728595i	0.0954352	+	0.995436i	$0.469576\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6219.26	+	5218.58i	−0.353066	+	0.296257i	−0.802020	−	0.597298i	$-0.796241\pi$
$677$	−6219.26	+	5218.58i	−0.353066	+	0.296257i	0.448954	+	0.893555i	$0.351797\pi$
$678$		0			0
$679$	−1989.72	+	11284.2i	−0.112457	+	0.637775i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9569.27	−	16574.5i	−0.536102	−	0.928557i	−0.999109	−	0.0422019i	$-0.986563\pi$
$683$	−9569.27	−	16574.5i	−0.536102	−	0.928557i	0.463007	−	0.886355i	$-0.346771\pi$
$684$		0			0
$685$	680.397	−	1178.48i	0.0379513	−	0.0657336i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9921.75	+	8325.33i	0.548604	+	0.460334i
$690$		0			0
$691$	101.637	+	36.9930i	0.00559547	+	0.00203658i	0.344816	−	0.938670i	$-0.387941\pi$
$691$	101.637	+	36.9930i	0.00559547	+	0.00203658i	−0.339221	+	0.940707i	$0.610163\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1636.33	−	9280.06i	−0.0893085	−	0.506493i
$696$		0			0
$697$	22156.5	−	8064.32i	1.20407	−	0.438247i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27394.8			1.47601			0.738007	−	0.674793i	$-0.235767\pi$
$701$	27394.8			1.47601			0.738007	+	0.674793i	$0.235767\pi$
$702$		0			0
$703$	−7193.70			−0.385939
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12001.1	−	4368.04i	0.638398	−	0.232358i
$708$		0			0
$709$	2265.35	+	12847.4i	0.119996	+	0.680529i	0.984155	+	0.177310i	$0.0567397\pi$
$709$	2265.35	+	12847.4i	0.119996	+	0.680529i	−0.864159	+	0.503218i	$0.832149\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	602.540	+	219.307i	0.0316484	+	0.0115191i
$714$		0			0
$715$	−4138.69	−	3472.77i	−0.216473	−	0.181642i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7365.93	−	12758.2i	0.382063	−	0.661752i	−0.609294	−	0.792944i	$-0.708547\pi$
$719$	7365.93	−	12758.2i	0.382063	−	0.661752i	0.991357	+	0.131192i	$0.0418805\pi$
$720$		0			0
$721$	2043.30	+	3539.10i	0.105543	+	0.182806i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3012.97	+	17087.4i	−0.154343	+	0.875325i
$726$		0			0
$727$	616.890	−	517.632i	0.0314707	−	0.0264070i	−0.626917	−	0.779086i	$-0.715683\pi$
$727$	616.890	−	517.632i	0.0314707	−	0.0264070i	0.658387	+	0.752679i	$0.271239\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	13642.4	−	11447.3i	0.690263	−	0.579200i
$732$		0			0
$733$	314.266	−	1782.29i	0.0158358	−	0.0898095i	−0.975866	−	0.218372i	$-0.929925\pi$
$733$	314.266	−	1782.29i	0.0158358	−	0.0898095i	0.991701	+	0.128563i	$0.0410364\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3182.26	−	5511.83i	−0.159050	−	0.275483i
$738$		0			0
$739$	−2005.53	+	3473.68i	−0.0998303	+	0.172911i	−0.911614	−	0.411047i	$-0.865163\pi$
$739$	−2005.53	+	3473.68i	−0.0998303	+	0.172911i	0.811784	+	0.583958i	$0.198497\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	20551.0	+	17244.3i	1.01473	+	0.851456i	0.988956	−	0.148211i	$-0.0473516\pi$
$743$	20551.0	+	17244.3i	1.01473	+	0.851456i	0.0257705	+	0.999668i	$0.491796\pi$
$744$		0			0
$745$	4947.54	+	1800.76i	0.243307	+	0.0885566i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2690.92	−	15261.0i	−0.131274	−	0.744490i
$750$		0			0
$751$	33872.2	−	12328.5i	1.64582	−	0.599031i	0.657780	−	0.753210i	$-0.271496\pi$
$751$	33872.2	−	12328.5i	1.64582	−	0.599031i	0.988043	+	0.154179i	$0.0492734\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−15678.4			−0.755754
$756$		0			0
$757$	−20176.1			−0.968708			−0.484354	−	0.874872i	$-0.660945\pi$
$757$	−20176.1			−0.968708			−0.484354	+	0.874872i	$0.660945\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3112.67	−	1132.92i	0.148271	−	0.0539662i	−0.266819	−	0.963747i	$-0.585972\pi$
$761$	3112.67	−	1132.92i	0.148271	−	0.0539662i	0.415090	+	0.909781i	$0.363750\pi$
$762$		0			0
$763$	−1709.00	−	9692.20i	−0.0810876	−	0.459871i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11280.7	+	4105.84i	0.531060	+	0.193290i
$768$		0			0
$769$	−872.027	−	731.718i	−0.0408922	−	0.0343126i	0.622113	−	0.782928i	$-0.286275\pi$
$769$	−872.027	−	731.718i	−0.0408922	−	0.0343126i	−0.663005	+	0.748615i	$0.730719\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20299.3	+	35159.4i	−0.944521	+	1.63596i	−0.187814	+	0.982205i	$0.560140\pi$
$773$	−20299.3	+	35159.4i	−0.944521	+	1.63596i	−0.756707	+	0.653754i	$0.773193\pi$
$774$		0			0
$775$	949.716	+	1644.96i	0.0440191	+	0.0762434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6485.62	+	36781.8i	−0.298295	+	1.69171i
$780$		0			0
$781$	6214.87	−	5214.89i	0.284744	−	0.238929i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	17303.8	−	14519.6i	0.786750	−	0.660161i
$786$		0			0
$787$	4660.94	−	26433.5i	0.211111	−	1.19727i	−0.676418	−	0.736518i	$-0.736469\pi$
$787$	4660.94	−	26433.5i	0.211111	−	1.19727i	0.887529	−	0.460752i	$-0.152420\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−891.499	−	1544.12i	−0.0400734	−	0.0694091i
$792$		0			0
$793$	−12465.3	+	21590.5i	−0.558204	+	0.966838i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	23016.9	+	19313.5i	1.02296	+	0.858368i	0.989997	−	0.141088i	$-0.0450602\pi$
$797$	23016.9	+	19313.5i	1.02296	+	0.858368i	0.0329662	+	0.999456i	$0.489505\pi$
$798$		0			0
$799$	988.525	+	359.794i	0.0437691	+	0.0159306i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1152.79	+	6537.81i	0.0506615	+	0.287315i
$804$		0			0
$805$	−1312.93	+	477.866i	−0.0574839	+	0.0209224i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−21235.4			−0.922865			−0.461432	−	0.887175i	$-0.652664\pi$
$809$	−21235.4			−0.922865			−0.461432	+	0.887175i	$0.652664\pi$
$810$		0			0
$811$	16153.8			0.699427			0.349714	−	0.936857i	$-0.386279\pi$
$811$	16153.8			0.699427			0.349714	+	0.936857i	$0.386279\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20172.4	+	7342.15i	−0.867004	+	0.315564i
$816$		0			0
$817$	4898.61	+	27781.4i	0.209768	+	1.18965i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−20735.6	−	7547.15i	−0.881459	−	0.320825i	−0.138661	−	0.990340i	$-0.544280\pi$
$821$	−20735.6	−	7547.15i	−0.881459	−	0.320825i	−0.742798	+	0.669515i	$0.766502\pi$
$822$		0			0
$823$	−12610.1	−	10581.1i	−0.534094	−	0.448158i	0.335418	−	0.942069i	$-0.391122\pi$
$823$	−12610.1	−	10581.1i	−0.534094	−	0.448158i	−0.869512	+	0.493911i	$0.835567\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3411.08	+	5908.16i	−0.143428	+	0.248424i	−0.928785	−	0.370618i	$-0.879146\pi$
$827$	−3411.08	+	5908.16i	−0.143428	+	0.248424i	0.785357	+	0.619043i	$0.212479\pi$
$828$		0			0
$829$	−16850.1	−	29185.1i	−0.705943	−	1.22273i	−0.966350	−	0.257230i	$-0.917190\pi$
$829$	−16850.1	−	29185.1i	−0.705943	−	1.22273i	0.260408	−	0.965499i	$-0.416143\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2907.66	−	16490.2i	0.120942	−	0.685894i
$834$		0			0
$835$	−4941.92	+	4146.76i	−0.204817	+	0.171862i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17763.1	−	14905.0i	0.730931	−	0.613324i	−0.199454	−	0.979907i	$-0.563917\pi$
$839$	17763.1	−	14905.0i	0.730931	−	0.613324i	0.930385	+	0.366583i	$0.119473\pi$
$840$		0			0
$841$	4557.96	−	25849.5i	0.186886	−	1.05988i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2253.88	+	3903.83i	0.0917582	+	0.158930i
$846$		0			0
$847$	4331.97	−	7503.20i	0.175736	−	0.304384i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1529.60	−	1283.48i	−0.0616144	−	0.0517007i
$852$		0			0
$853$	−27893.1	−	10152.3i	−1.11963	−	0.407511i	−0.285111	−	0.958495i	$-0.592030\pi$
$853$	−27893.1	−	10152.3i	−1.11963	−	0.407511i	−0.834516	+	0.550984i	$0.814253\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1789.08	+	10146.4i	0.0713114	+	0.404427i	0.999479	+	0.0322654i	$0.0102722\pi$
$857$	1789.08	+	10146.4i	0.0713114	+	0.404427i	−0.928168	+	0.372162i	$0.878617\pi$
$858$		0			0
$859$	−955.259	+	347.686i	−0.0379430	+	0.0138101i	−0.360922	−	0.932596i	$-0.617538\pi$
$859$	−955.259	+	347.686i	−0.0379430	+	0.0138101i	0.322979	+	0.946406i	$0.395316\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−43831.8			−1.72891			−0.864456	−	0.502709i	$-0.832337\pi$
$863$	−43831.8			−1.72891			−0.864456	+	0.502709i	$0.832337\pi$
$864$		0			0
$865$	−21752.1			−0.855023
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9640.83	−	3508.97i	0.376344	−	0.136978i
$870$		0			0
$871$	4032.38	+	22868.7i	0.156868	+	0.889642i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10194.2	−	3710.39i	−0.393860	−	0.143353i
$876$		0			0
$877$	6970.64	+	5849.06i	0.268394	+	0.225210i	0.767045	−	0.641594i	$-0.221726\pi$
$877$	6970.64	+	5849.06i	0.268394	+	0.225210i	−0.498650	+	0.866803i	$0.666171\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	17753.9	−	30750.6i	0.678937	−	1.17595i	−0.296365	−	0.955075i	$-0.595774\pi$
$881$	17753.9	−	30750.6i	0.678937	−	1.17595i	0.975301	−	0.220878i	$-0.0708923\pi$
$882$		0			0
$883$	−23165.8	−	40124.3i	−0.882889	−	1.52921i	−0.848114	−	0.529814i	$-0.822262\pi$
$883$	−23165.8	−	40124.3i	−0.882889	−	1.52921i	−0.0347750	−	0.999395i	$-0.511071\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3331.76	+	18895.3i	−0.126121	+	0.715268i	0.854515	+	0.519427i	$0.173855\pi$
$887$	−3331.76	+	18895.3i	−0.126121	+	0.715268i	−0.980636	+	0.195841i	$0.937256\pi$
$888$		0			0
$889$	10978.2	−	9211.82i	0.414171	−	0.347530i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1276.50	+	1071.11i	−0.0478348	+	0.0401382i
$894$		0			0
$895$	−3584.46	+	20328.5i	−0.133872	+	0.759224i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2771.65	−	4800.64i	−0.102825	−	0.178098i
$900$		0			0
$901$	7183.51	−	12442.2i	0.265613	−	0.460056i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5042.75	−	4231.37i	−0.185223	−	0.155420i
$906$		0			0
$907$	−2377.37	−	865.291i	−0.0870333	−	0.0316775i	0.298137	−	0.954523i	$-0.403635\pi$
$907$	−2377.37	−	865.291i	−0.0870333	−	0.0316775i	−0.385170	+	0.922846i	$0.625857\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2356.17	−	13362.5i	−0.0856898	−	0.485971i	−0.997206	−	0.0747059i	$-0.976198\pi$
$911$	−2356.17	−	13362.5i	−0.0856898	−	0.485971i	0.911516	−	0.411265i	$-0.134913\pi$
$912$		0			0
$913$	−7674.38	+	2793.25i	−0.278187	+	0.101252i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−20676.0			−0.744583
$918$		0			0
$919$	−34243.4			−1.22915			−0.614574	−	0.788859i	$-0.710672\pi$
$919$	−34243.4			−1.22915			−0.614574	+	0.788859i	$0.710672\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−27815.7	+	10124.1i	−0.991943	+	0.361038i
$924$		0			0
$925$	−1027.11	−	5825.03i	−0.0365093	−	0.207055i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6133.04	−	2232.25i	−0.216597	−	0.0788349i	0.231443	−	0.972849i	$-0.425655\pi$
$929$	−6133.04	−	2232.25i	−0.216597	−	0.0788349i	−0.448040	+	0.894014i	$0.647878\pi$
$930$		0			0
$931$	20318.5	+	17049.3i	0.715267	+	0.600180i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2996.48	+	5190.05i	−0.104808	+	0.181533i
$936$		0			0
$937$	−4402.77	−	7625.83i	−0.153503	−	0.265875i	0.779010	−	0.627012i	$-0.215722\pi$
$937$	−4402.77	−	7625.83i	−0.153503	−	0.265875i	−0.932513	+	0.361137i	$0.882389\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	7494.30	−	42502.3i	0.259625	−	1.47241i	−0.524291	−	0.851539i	$-0.675669\pi$
$941$	7494.30	−	42502.3i	0.259625	−	1.47241i	0.783916	−	0.620867i	$-0.213219\pi$
$942$		0			0
$943$	−7941.57	+	6663.77i	−0.274245	+	0.230119i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5003.21	+	4198.19i	−0.171682	+	0.144058i	−0.724580	−	0.689191i	$-0.757966\pi$
$947$	−5003.21	+	4198.19i	−0.171682	+	0.144058i	0.552898	+	0.833249i	$0.313522\pi$
$948$		0			0
$949$	4206.07	−	23853.8i	0.143872	−	0.815941i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−573.765	−	993.790i	−0.0195027	−	0.0337797i	0.856109	−	0.516795i	$-0.172875\pi$
$953$	−573.765	−	993.790i	−0.0195027	−	0.0337797i	−0.875612	+	0.483015i	$0.839542\pi$
$954$		0			0
$955$	9367.74	−	16225.4i	0.317417	−	0.549782i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1168.31	+	980.324i	0.0393395	+	0.0330097i
$960$		0			0
$961$	27424.1	+	9981.57i	0.920551	+	0.335053i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5137.81	−	29138.0i	−0.171391	−	0.972004i
$966$		0			0
$967$	−4869.23	+	1772.25i	−0.161927	+	0.0589368i	−0.421712	−	0.906730i	$-0.638571\pi$
$967$	−4869.23	+	1772.25i	−0.161927	+	0.0589368i	0.259785	+	0.965667i	$0.416348\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−59813.0			−1.97682			−0.988409	−	0.151812i	$-0.951489\pi$
$971$	−59813.0			−1.97682			−0.988409	+	0.151812i	$0.951489\pi$
$972$		0			0
$973$	10561.1			0.347969
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17492.6	+	6366.79i	−0.572813	+	0.208487i	−0.612154	−	0.790739i	$-0.709697\pi$
$977$	−17492.6	+	6366.79i	−0.572813	+	0.208487i	0.0393403	+	0.999226i	$0.487474\pi$
$978$		0			0
$979$	2936.77	+	16655.3i	0.0958730	+	0.543723i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2809.40	−	1022.54i	−0.0911555	−	0.0331779i	0.296040	−	0.955176i	$-0.404334\pi$
$983$	−2809.40	−	1022.54i	−0.0911555	−	0.0331779i	−0.387195	+	0.921998i	$0.626556\pi$
$984$		0			0
$985$	−24175.3	−	20285.4i	−0.782018	−	0.656191i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3915.10	+	6781.15i	−0.125878	+	0.218026i
$990$		0			0
$991$	11105.6	+	19235.4i	0.355984	+	0.616583i	0.987286	−	0.158954i	$-0.0508123\pi$
$991$	11105.6	+	19235.4i	0.355984	+	0.616583i	−0.631302	+	0.775537i	$0.717479\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3740.80	+	21215.1i	−0.119187	+	0.675944i
$996$		0			0
$997$	−17248.2	+	14473.0i	−0.547901	+	0.459744i	−0.874230	−	0.485513i	$-0.838633\pi$
$997$	−17248.2	+	14473.0i	−0.547901	+	0.459744i	0.326328	+	0.945256i	$0.394188\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.i.a.37.7		54
3.2	odd	2		108.4.i.a.49.3	✓	54
27.11	odd	18		108.4.i.a.97.3	yes	54
27.16	even	9	inner	324.4.i.a.289.7		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.3	✓	54	3.2	odd	2
108.4.i.a.97.3	yes	54	27.11	odd	18
324.4.i.a.37.7		54	1.1	even	1	trivial
324.4.i.a.289.7		54	27.16	even	9	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.i (of order \(9\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.50317 - 2.36696i) q^{5} +(1.34685 + 7.63837i) q^{7} +O(q^{10})\)	\(q+(6.50317 - 2.36696i) q^{5} +(1.34685 + 7.63837i) q^{7} +(-13.7454 - 5.00292i) q^{11} +(40.8838 + 34.3056i) q^{13} +(29.6006 - 51.2697i) q^{17} +(46.8884 + 81.2131i) q^{19} +(-4.51998 + 25.6341i) q^{23} +(-59.0669 + 49.5630i) q^{25} +(172.381 - 144.645i) q^{29} +(4.27764 - 24.2597i) q^{31} +(26.8385 + 46.4856i) q^{35} +(-38.3554 + 66.4335i) q^{37} +(305.098 + 256.008i) q^{41} +(282.678 + 102.886i) q^{43} +(3.08562 + 17.4994i) q^{47} +(265.784 - 96.7374i) q^{49} +242.682 q^{53} -101.230 q^{55} +(211.368 - 76.9317i) q^{59} +(81.1159 + 460.031i) q^{61} +(347.074 + 126.325i) q^{65} +(333.309 + 279.680i) q^{67} +(-277.317 + 480.327i) q^{71} +(-226.923 - 393.043i) q^{73} +(19.7011 - 111.731i) q^{77} +(-537.292 + 450.842i) q^{79} +(427.701 - 358.883i) q^{83} +(71.1442 - 403.479i) q^{85} +(-578.094 - 1001.29i) q^{89} +(-206.974 + 358.490i) q^{91} +(497.151 + 417.160i) q^{95} +(1388.22 + 505.270i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 54 * q - 12 * q^5	\( 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) \) 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more