LMFDB - Embedded newform 324.4.i.a.73.3

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			73.3
Character	$\chi$	$=$	324.73
Dual form			324.4.i.a.253.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.52940 + 6.31792i) q^{5} +(24.9076 - 9.06562i) q^{7} +O(q^{10})$	$q+(-7.52940 + 6.31792i) q^{5} +(24.9076 - 9.06562i) q^{7} +(-17.0456 - 14.3030i) q^{11} +(13.5975 + 77.1153i) q^{13} +(-32.2518 - 55.8618i) q^{17} +(35.2151 - 60.9943i) q^{19} +(112.980 + 41.1215i) q^{23} +(-4.93023 + 27.9607i) q^{25} +(-30.3150 + 171.925i) q^{29} +(139.932 + 50.9312i) q^{31} +(-130.263 + 225.623i) q^{35} +(144.919 + 251.008i) q^{37} +(76.1146 + 431.667i) q^{41} +(-76.4782 - 64.1728i) q^{43} +(290.224 - 105.633i) q^{47} +(275.449 - 231.129i) q^{49} +533.670 q^{53} +218.708 q^{55} +(-320.392 + 268.841i) q^{59} +(-87.3117 + 31.7789i) q^{61} +(-589.589 - 494.724i) q^{65} +(-80.8767 - 458.675i) q^{67} +(451.994 + 782.877i) q^{71} +(-86.0801 + 149.095i) q^{73} +(-554.231 - 201.723i) q^{77} +(-27.8206 + 157.779i) q^{79} +(-103.082 + 584.607i) q^{83} +(595.767 + 216.841i) q^{85} +(818.257 - 1417.26i) q^{89} +(1037.78 + 1797.48i) q^{91} +(120.209 + 681.737i) q^{95} +(-323.876 - 271.765i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{5}{9}\right)$

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
$5$	−7.52940	+	6.31792i	−0.673450	+	0.565092i	−0.914084	−	0.405524i	$-0.867089\pi$
$5$	−7.52940	+	6.31792i	−0.673450	+	0.565092i	0.240634	+	0.970616i	$0.422645\pi$
$6$		0			0
$7$	24.9076	−	9.06562i	1.34488	−	0.489497i	0.433536	−	0.901136i	$-0.357266\pi$
$7$	24.9076	−	9.06562i	1.34488	−	0.489497i	0.911347	+	0.411639i	$0.135044\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−17.0456	−	14.3030i	−0.467223	−	0.392046i	0.378558	−	0.925578i	$-0.376420\pi$
$11$	−17.0456	−	14.3030i	−0.467223	−	0.392046i	−0.845780	+	0.533531i	$0.820865\pi$
$12$		0			0
$13$	13.5975	+	77.1153i	0.290098	+	1.64522i	0.686487	+	0.727142i	$0.259152\pi$
$13$	13.5975	+	77.1153i	0.290098	+	1.64522i	−0.396389	+	0.918083i	$0.629737\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−32.2518	−	55.8618i	−0.460130	−	0.796969i	0.538837	−	0.842410i	$-0.318864\pi$
$17$	−32.2518	−	55.8618i	−0.460130	−	0.796969i	−0.998967	+	0.0454411i	$0.985531\pi$
$18$		0			0
$19$	35.2151	−	60.9943i	0.425205	−	0.736477i	−0.571234	−	0.820787i	$-0.693535\pi$
$19$	35.2151	−	60.9943i	0.425205	−	0.736477i	0.996440	+	0.0843100i	$0.0268686\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	112.980	+	41.1215i	1.02426	+	0.372801i	0.798894	−	0.601472i	$-0.205419\pi$
$23$	112.980	+	41.1215i	1.02426	+	0.372801i	0.225369	+	0.974274i	$0.427641\pi$
$24$		0			0
$25$	−4.93023	+	27.9607i	−0.0394419	+	0.223686i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−30.3150	+	171.925i	−0.194116	+	1.10089i	0.719556	+	0.694434i	$0.244345\pi$
$29$	−30.3150	+	171.925i	−0.194116	+	1.10089i	−0.913672	+	0.406452i	$0.866766\pi$
$30$		0			0
$31$	139.932	+	50.9312i	0.810728	+	0.295081i	0.713925	−	0.700223i	$-0.246916\pi$
$31$	139.932	+	50.9312i	0.810728	+	0.295081i	0.0968036	+	0.995304i	$0.469138\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−130.263	+	225.623i	−0.629100	+	1.08963i
$36$		0			0
$37$	144.919	+	251.008i	0.643908	+	1.11528i	0.984553	+	0.175090i	$0.0560215\pi$
$37$	144.919	+	251.008i	0.643908	+	1.11528i	−0.340644	+	0.940192i	$0.610645\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	76.1146	+	431.667i	0.289929	+	1.64427i	0.687126	+	0.726538i	$0.258872\pi$
$41$	76.1146	+	431.667i	0.289929	+	1.64427i	−0.397197	+	0.917733i	$0.630017\pi$
$42$		0			0
$43$	−76.4782	−	64.1728i	−0.271228	−	0.227588i	0.497021	−	0.867739i	$-0.334427\pi$
$43$	−76.4782	−	64.1728i	−0.271228	−	0.227588i	−0.768249	+	0.640151i	$0.778872\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	290.224	−	105.633i	0.900712	−	0.327832i	0.150174	−	0.988660i	$-0.452017\pi$
$47$	290.224	−	105.633i	0.900712	−	0.327832i	0.750538	+	0.660827i	$0.229794\pi$
$48$		0			0
$49$	275.449	−	231.129i	0.803057	−	0.673845i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	533.670			1.38312			0.691558	−	0.722321i	$-0.256925\pi$
$53$	533.670			1.38312			0.691558	+	0.722321i	$0.256925\pi$
$54$		0			0
$55$	218.708			0.536193
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−320.392	+	268.841i	−0.706975	+	0.593222i	−0.923748	−	0.383000i	$-0.874891\pi$
$59$	−320.392	+	268.841i	−0.706975	+	0.593222i	0.216774	+	0.976222i	$0.430447\pi$
$60$		0			0
$61$	−87.3117	+	31.7789i	−0.183264	+	0.0667027i	−0.432022	−	0.901863i	$-0.642200\pi$
$61$	−87.3117	+	31.7789i	−0.183264	+	0.0667027i	0.248758	+	0.968566i	$0.419978\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−589.589	−	494.724i	−1.12507	−	0.944045i
$66$		0			0
$67$	−80.8767	−	458.675i	−0.147473	−	0.836359i	−0.965349	−	0.260964i	$-0.915960\pi$
$67$	−80.8767	−	458.675i	−0.147473	−	0.836359i	0.817876	−	0.575395i	$-0.195152\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	451.994	+	782.877i	0.755519	+	1.30860i	0.945116	+	0.326735i	$0.105949\pi$
$71$	451.994	+	782.877i	0.755519	+	1.30860i	−0.189597	+	0.981862i	$0.560718\pi$
$72$		0			0
$73$	−86.0801	+	149.095i	−0.138012	+	0.239045i	−0.926744	−	0.375693i	$-0.877405\pi$
$73$	−86.0801	+	149.095i	−0.138012	+	0.239045i	0.788732	+	0.614738i	$0.210738\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−554.231	−	201.723i	−0.820265	−	0.298552i
$78$		0			0
$79$	−27.8206	+	157.779i	−0.0396211	+	0.224702i	−0.998189	−	0.0601633i	$-0.980838\pi$
$79$	−27.8206	+	157.779i	−0.0396211	+	0.224702i	0.958567	+	0.284866i	$0.0919490\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−103.082	+	584.607i	−0.136322	+	0.773120i	0.837608	+	0.546272i	$0.183953\pi$
$83$	−103.082	+	584.607i	−0.136322	+	0.773120i	−0.973930	+	0.226849i	$0.927158\pi$
$84$		0			0
$85$	595.767	+	216.841i	0.760235	+	0.276703i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	818.257	−	1417.26i	0.974551	−	1.68797i	0.293143	−	0.956068i	$-0.405299\pi$
$89$	818.257	−	1417.26i	0.974551	−	1.68797i	0.681408	−	0.731904i	$-0.261368\pi$
$90$		0			0
$91$	1037.78	+	1797.48i	1.19548	+	2.07063i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	120.209	+	681.737i	0.129823	+	0.736260i
$96$		0			0
$97$	−323.876	−	271.765i	−0.339017	−	0.284469i	0.457345	−	0.889289i	$-0.348800\pi$
$97$	−323.876	−	271.765i	−0.339017	−	0.284469i	−0.796362	+	0.604820i	$0.793245\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−743.112	+	270.471i	−0.732103	+	0.266464i	−0.681055	−	0.732233i	$-0.738478\pi$
$101$	−743.112	+	270.471i	−0.732103	+	0.266464i	−0.0510481	+	0.998696i	$0.516256\pi$
$102$		0			0
$103$	696.301	−	584.266i	0.666103	−	0.558927i	−0.245806	−	0.969319i	$-0.579053\pi$
$103$	696.301	−	584.266i	0.666103	−	0.558927i	0.911909	+	0.410392i	$0.134608\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−225.538			−0.203772			−0.101886	−	0.994796i	$-0.532488\pi$
$107$	−225.538			−0.203772			−0.101886	+	0.994796i	$0.532488\pi$
$108$		0			0
$109$	−1916.33			−1.68396			−0.841979	−	0.539510i	$-0.818610\pi$
$109$	−1916.33			−1.68396			−0.841979	+	0.539510i	$0.818610\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	496.602	−	416.698i	0.413419	−	0.346900i	−0.412234	−	0.911078i	$-0.635251\pi$
$113$	496.602	−	416.698i	0.413419	−	0.346900i	0.825653	+	0.564178i	$0.190807\pi$
$114$		0			0
$115$	−1110.48	+	404.180i	−0.900456	+	0.327739i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1309.74	−	1099.00i	−1.00894	−	0.846597i
$120$		0			0
$121$	−145.148	−	823.172i	−0.109051	−	0.618462i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−753.840	−	1305.69i	−0.539404	−	0.934275i
$126$		0			0
$127$	−180.673	+	312.934i	−0.126237	+	0.218649i	−0.922216	−	0.386676i	$-0.873623\pi$
$127$	−180.673	+	312.934i	−0.126237	+	0.218649i	0.795979	+	0.605324i	$0.206957\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	659.456	+	240.022i	0.439824	+	0.160083i	0.552434	−	0.833556i	$-0.313699\pi$
$131$	659.456	+	240.022i	0.439824	+	0.160083i	−0.112611	+	0.993639i	$0.535921\pi$
$132$		0			0
$133$	324.171	−	1838.47i	0.211348	−	1.19861i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	490.451	−	2781.48i	0.305854	−	1.73459i	−0.313602	−	0.949555i	$-0.601536\pi$
$137$	490.451	−	2781.48i	0.305854	−	1.73459i	0.619456	−	0.785031i	$-0.287353\pi$
$138$		0			0
$139$	−928.816	−	338.061i	−0.566771	−	0.206288i	0.0427113	−	0.999087i	$-0.486400\pi$
$139$	−928.816	−	338.061i	−0.566771	−	0.206288i	−0.609482	+	0.792800i	$0.708623\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	871.200	−	1508.96i	0.509464	−	0.882418i
$144$		0			0
$145$	−857.954	−	1486.02i	−0.491374	−	0.851085i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	339.478	+	1925.27i	0.186652	+	1.05855i	0.923815	+	0.382840i	$0.125054\pi$
$149$	339.478	+	1925.27i	0.186652	+	1.05855i	−0.737163	+	0.675715i	$0.763835\pi$
$150$		0			0
$151$	−2176.29	−	1826.12i	−1.17287	−	0.984156i	−0.172872	−	0.984944i	$-0.555305\pi$
$151$	−2176.29	−	1826.12i	−1.17287	−	0.984156i	−1.00000		0.000788384i	$0.999749\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1375.38	+	500.599i	−0.712733	+	0.259413i
$156$		0			0
$157$	544.235	−	456.668i	0.276654	−	0.232140i	−0.493894	−	0.869522i	$-0.664427\pi$
$157$	544.235	−	456.668i	0.276654	−	0.232140i	0.770548	+	0.637382i	$0.219983\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3186.86			1.56000
$162$		0			0
$163$	1340.84			0.644311			0.322156	−	0.946687i	$-0.395593\pi$
$163$	1340.84			0.644311			0.322156	+	0.946687i	$0.395593\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1287.33	+	1080.20i	−0.596508	+	0.500530i	−0.890321	−	0.455333i	$-0.849520\pi$
$167$	−1287.33	+	1080.20i	−0.596508	+	0.500530i	0.293813	+	0.955863i	$0.405076\pi$
$168$		0			0
$169$	−3697.37	+	1345.73i	−1.68292	+	0.612531i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	118.475	+	99.4124i	0.0520664	+	0.0436889i	0.668450	−	0.743758i	$-0.266958\pi$
$173$	118.475	+	99.4124i	0.0520664	+	0.0436889i	−0.616383	+	0.787446i	$0.711403\pi$
$174$		0			0
$175$	130.681	+	741.130i	0.0564490	+	0.320138i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	426.813	+	739.262i	0.178221	+	0.308687i	0.941271	−	0.337652i	$-0.109633\pi$
$179$	426.813	+	739.262i	0.178221	+	0.308687i	−0.763050	+	0.646339i	$0.776299\pi$
$180$		0			0
$181$	−985.862	+	1707.56i	−0.404854	+	0.701228i	−0.994304	−	0.106577i	$-0.966011\pi$
$181$	−985.862	+	1707.56i	−0.404854	+	0.701228i	0.589451	+	0.807804i	$0.299344\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2677.00	−	974.349i	−1.06388	−	0.387219i
$186$		0			0
$187$	−249.238	+	1413.50i	−0.0974656	+	0.552755i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	887.225	−	5031.70i	0.336112	−	1.90618i	−0.0798581	−	0.996806i	$-0.525447\pi$
$191$	887.225	−	5031.70i	0.336112	−	1.90618i	0.415970	−	0.909378i	$-0.363442\pi$
$192$		0			0
$193$	456.904	+	166.299i	0.170408	+	0.0620233i	0.425815	−	0.904810i	$-0.359987\pi$
$193$	456.904	+	166.299i	0.170408	+	0.0620233i	−0.255408	+	0.966833i	$0.582210\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1019.37	+	1765.60i	−0.368665	+	0.638546i	−0.989357	−	0.145508i	$-0.953518\pi$
$197$	−1019.37	+	1765.60i	−0.368665	+	0.638546i	0.620692	+	0.784054i	$0.286852\pi$
$198$		0			0
$199$	2237.59	+	3875.62i	0.797079	+	1.38058i	0.921511	+	0.388353i	$0.126956\pi$
$199$	2237.59	+	3875.62i	0.797079	+	1.38058i	−0.124432	+	0.992228i	$0.539711\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	803.533	+	4557.06i	0.277817	+	1.57558i
$204$		0			0
$205$	−3300.34	−	2769.31i	−1.12442	−	0.943498i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1472.66	+	536.006i	−0.487399	+	0.177399i
$210$		0			0
$211$	3003.60	−	2520.32i	0.979983	−	0.822304i	−0.00410365	−	0.999992i	$-0.501306\pi$
$211$	3003.60	−	2520.32i	0.979983	−	0.822304i	0.984087	+	0.177688i	$0.0568618\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	981.273			0.311266
$216$		0			0
$217$	3947.10			1.23478
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3869.25	−	3246.69i	1.17771	−	0.988217i
$222$		0			0
$223$	1464.87	−	533.169i	0.439888	−	0.160106i	−0.112576	−	0.993643i	$-0.535910\pi$
$223$	1464.87	−	533.169i	0.439888	−	0.160106i	0.552464	+	0.833537i	$0.313688\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	195.605	+	164.132i	0.0571927	+	0.0479904i	0.670936	−	0.741515i	$-0.265893\pi$
$227$	195.605	+	164.132i	0.0571927	+	0.0479904i	−0.613743	+	0.789506i	$0.710337\pi$
$228$		0			0
$229$	−897.549	−	5090.25i	−0.259003	−	1.46888i	−0.785584	−	0.618755i	$-0.787637\pi$
$229$	−897.549	−	5090.25i	−0.259003	−	1.46888i	0.526581	−	0.850125i	$-0.323474\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2015.27	−	3490.54i	−0.566629	−	0.981430i	−0.996896	−	0.0787281i	$-0.974914\pi$
$233$	−2015.27	−	3490.54i	−0.566629	−	0.981430i	0.430268	−	0.902701i	$-0.358419\pi$
$234$		0			0
$235$	−1517.83	+	2628.96i	−0.421329	+	0.729764i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5365.88	−	1953.02i	−1.45226	−	0.528579i	−0.509037	−	0.860745i	$-0.669998\pi$
$239$	−5365.88	−	1953.02i	−1.45226	−	0.528579i	−0.943221	+	0.332166i	$0.892221\pi$
$240$		0			0
$241$	−438.520	+	2486.97i	−0.117210	+	0.664730i	0.868422	+	0.495825i	$0.165134\pi$
$241$	−438.520	+	2486.97i	−0.117210	+	0.664730i	−0.985632	+	0.168905i	$0.945977\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−613.710	+	3480.52i	−0.160035	+	0.907602i
$246$		0			0
$247$	5182.43	+	1886.25i	1.33502	+	0.485908i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	289.705	−	501.784i	0.0728526	−	0.126184i	−0.827298	−	0.561764i	$-0.810123\pi$
$251$	289.705	−	501.784i	0.0728526	−	0.126184i	0.900150	+	0.435579i	$0.143456\pi$
$252$		0			0
$253$	−1337.66	−	2316.90i	−0.332403	−	0.575740i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	614.801	+	3486.71i	0.149223	+	0.846284i	0.963879	+	0.266340i	$0.0858144\pi$
$257$	614.801	+	3486.71i	0.149223	+	0.846284i	−0.814657	+	0.579944i	$0.803075\pi$
$258$		0			0
$259$	5885.13	+	4938.21i	1.41191	+	1.18473i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2173.81	+	791.204i	−0.509670	+	0.185505i	−0.584038	−	0.811726i	$-0.698528\pi$
$263$	−2173.81	+	791.204i	−0.509670	+	0.185505i	0.0743683	+	0.997231i	$0.476306\pi$
$264$		0			0
$265$	−4018.21	+	3371.68i	−0.931460	+	0.781587i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	595.561			0.134989			0.0674944	−	0.997720i	$-0.478500\pi$
$269$	595.561			0.134989			0.0674944	+	0.997720i	$0.478500\pi$
$270$		0			0
$271$	−498.234			−0.111681			−0.0558406	−	0.998440i	$-0.517784\pi$
$271$	−498.234			−0.111681			−0.0558406	+	0.998440i	$0.517784\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	483.961	−	406.091i	0.106123	−	0.0890481i
$276$		0			0
$277$	8126.82	−	2957.92i	1.76279	−	0.641604i	0.762804	−	0.646630i	$-0.223822\pi$
$277$	8126.82	−	2957.92i	1.76279	−	0.641604i	0.999987	+	0.00502614i	$0.00159988\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1183.18	+	992.805i	0.251183	+	0.210768i	0.759682	−	0.650295i	$-0.225355\pi$
$281$	1183.18	+	992.805i	0.251183	+	0.210768i	−0.508498	+	0.861063i	$0.669799\pi$
$282$		0			0
$283$	1007.68	+	5714.81i	0.211661	+	1.20039i	0.886607	+	0.462523i	$0.153056\pi$
$283$	1007.68	+	5714.81i	0.211661	+	1.20039i	−0.674946	+	0.737867i	$0.735833\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5809.16	+	10061.8i	1.19479	+	2.06943i
$288$		0			0
$289$	376.139	−	651.493i	0.0765600	−	0.132606i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3902.90	+	1420.54i	0.778191	+	0.283238i	0.700418	−	0.713733i	$-0.252997\pi$
$293$	3902.90	+	1420.54i	0.778191	+	0.283238i	0.0777730	+	0.996971i	$0.475219\pi$
$294$		0			0
$295$	713.846	−	4048.42i	0.140887	−	0.799011i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1634.84	+	9271.66i	−0.316206	+	1.79329i
$300$		0			0
$301$	−2486.65	−	905.067i	−0.476174	−	0.173313i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	456.629	−	790.904i	0.0857261	−	0.148482i
$306$		0			0
$307$	−3125.08	−	5412.80i	−0.580970	−	1.00627i	−0.995365	−	0.0961718i	$-0.969340\pi$
$307$	−3125.08	−	5412.80i	−0.580970	−	1.00627i	0.414395	−	0.910097i	$-0.363993\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	893.683	+	5068.33i	0.162946	+	0.924111i	0.951157	+	0.308707i	$0.0998961\pi$
$311$	893.683	+	5068.33i	0.162946	+	0.924111i	−0.788212	+	0.615404i	$0.788993\pi$
$312$		0			0
$313$	4309.20	+	3615.85i	0.778180	+	0.652971i	0.942790	−	0.333388i	$-0.108192\pi$
$313$	4309.20	+	3615.85i	0.778180	+	0.652971i	−0.164610	+	0.986359i	$0.552636\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2826.93	−	1028.92i	0.500872	−	0.182302i	−0.0792143	−	0.996858i	$-0.525241\pi$
$317$	2826.93	−	1028.92i	0.500872	−	0.182302i	0.580086	+	0.814555i	$0.303019\pi$
$318$		0			0
$319$	2975.78	−	2496.98i	0.522294	−	0.438257i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4543.01			−0.782599
$324$		0			0
$325$	−2223.24			−0.379456
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6271.14	−	5262.11i	1.05088	−	0.881792i
$330$		0			0
$331$	−4837.44	+	1760.69i	−0.803293	+	0.292375i	−0.710850	−	0.703343i	$-0.751690\pi$
$331$	−4837.44	+	1760.69i	−0.803293	+	0.292375i	−0.0924426	+	0.995718i	$0.529467\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3506.82	+	2942.57i	0.571935	+	0.479910i
$336$		0			0
$337$	935.450	+	5305.20i	0.151208	+	0.857545i	0.962171	+	0.272447i	$0.0878328\pi$
$337$	935.450	+	5305.20i	0.151208	+	0.857545i	−0.810963	+	0.585098i	$0.801056\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1656.77	−	2869.60i	−0.263105	−	0.455712i
$342$		0			0
$343$	219.637	−	380.422i	0.0345751	−	0.0598859i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9458.09	−	3442.46i	−1.46322	−	0.532568i	−0.516969	−	0.856004i	$-0.672940\pi$
$347$	−9458.09	−	3442.46i	−1.46322	−	0.532568i	−0.946250	+	0.323436i	$0.895162\pi$
$348$		0			0
$349$	197.712	−	1121.28i	0.0303246	−	0.171979i	−0.965884	−	0.258975i	$-0.916615\pi$
$349$	197.712	−	1121.28i	0.0303246	−	0.171979i	0.996209	+	0.0869952i	$0.0277265\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−910.038	+	5161.08i	−0.137214	+	0.778177i	0.836079	+	0.548609i	$0.184842\pi$
$353$	−910.038	+	5161.08i	−0.137214	+	0.778177i	−0.973293	+	0.229568i	$0.926269\pi$
$354$		0			0
$355$	−8349.40	−	3038.93i	−1.24828	−	0.454337i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−324.335	+	561.764i	−0.0476817	+	0.0825871i	−0.888881	−	0.458138i	$-0.848517\pi$
$359$	−324.335	+	561.764i	−0.0476817	+	0.0825871i	0.841200	+	0.540725i	$0.181850\pi$
$360$		0			0
$361$	949.293	+	1644.22i	0.138401	+	0.239718i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−293.839	−	1666.44i	−0.0421376	−	0.238974i
$366$		0			0
$367$	−4084.64	−	3427.42i	−0.580971	−	0.487493i	0.304295	−	0.952578i	$-0.401579\pi$
$367$	−4084.64	−	3427.42i	−0.580971	−	0.487493i	−0.885266	+	0.465085i	$0.846024\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13292.4	−	4838.04i	1.86013	−	0.677032i
$372$		0			0
$373$	−5635.34	+	4728.61i	−0.782270	+	0.656403i	−0.943819	−	0.330462i	$-0.892795\pi$
$373$	−5635.34	+	4728.61i	−0.782270	+	0.656403i	0.161549	+	0.986865i	$0.448351\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13670.3			−1.86752
$378$		0			0
$379$	12285.0			1.66501			0.832507	−	0.554015i	$-0.186905\pi$
$379$	12285.0			1.66501			0.832507	+	0.554015i	$0.186905\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9861.87	−	8275.10i	1.31571	−	1.10401i	0.328518	−	0.944498i	$-0.393451\pi$
$383$	9861.87	−	8275.10i	1.31571	−	1.10401i	0.987195	−	0.159517i	$-0.0509937\pi$
$384$		0			0
$385$	5447.50	−	1982.73i	0.721117	−	0.262465i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3905.60	−	3277.18i	−0.509053	−	0.427146i	0.351743	−	0.936097i	$-0.385589\pi$
$389$	−3905.60	−	3277.18i	−0.509053	−	0.427146i	−0.860796	+	0.508951i	$0.830034\pi$
$390$		0			0
$391$	−1346.70	−	7637.53i	−0.174183	−	0.987843i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−787.359	−	1363.75i	−0.100295	−	0.173715i
$396$		0			0
$397$	3842.87	−	6656.05i	0.485814	−	0.841455i	−0.514053	−	0.857759i	$-0.671856\pi$
$397$	3842.87	−	6656.05i	0.485814	−	0.841455i	0.999867	+	0.0163035i	$0.00518981\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10580.7	−	3851.08i	−1.31765	−	0.479585i	−0.414944	−	0.909847i	$-0.636199\pi$
$401$	−10580.7	−	3851.08i	−1.31765	−	0.479585i	−0.902704	+	0.430262i	$0.858421\pi$
$402$		0			0
$403$	−2024.84	+	11483.4i	−0.250284	+	1.41943i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1119.92	−	6351.37i	0.136394	−	0.773527i
$408$		0			0
$409$	9890.09	+	3599.70i	1.19568	+	0.435192i	0.861715	−	0.507393i	$-0.169391\pi$
$409$	9890.09	+	3599.70i	1.19568	+	0.435192i	0.333966	+	0.942585i	$0.391613\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5542.98	+	9600.73i	−0.660417	+	1.14388i
$414$		0			0
$415$	−2917.35	−	5053.01i	−0.345078	−	0.597692i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	605.417	+	3433.49i	0.0705884	+	0.400327i	0.999546	+	0.0301399i	$0.00959528\pi$
$419$	605.417	+	3433.49i	0.0705884	+	0.400327i	−0.928957	+	0.370187i	$0.879294\pi$
$420$		0			0
$421$	6215.42	+	5215.36i	0.719527	+	0.603755i	0.927255	−	0.374432i	$-0.122162\pi$
$421$	6215.42	+	5215.36i	0.719527	+	0.603755i	−0.207727	+	0.978187i	$0.566607\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1720.95	−	626.373i	0.196419	−	0.0714907i
$426$		0			0
$427$	−1886.63	+	1583.07i	−0.213818	+	0.179415i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3793.19			0.423925			0.211962	−	0.977278i	$-0.432015\pi$
$431$	3793.19			0.423925			0.211962	+	0.977278i	$0.432015\pi$
$432$		0			0
$433$	−13262.0			−1.47190			−0.735948	−	0.677038i	$-0.763263\pi$
$433$	−13262.0			−1.47190			−0.735948	+	0.677038i	$0.763263\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6486.79	−	5443.07i	0.710081	−	0.595829i
$438$		0			0
$439$	−9674.73	+	3521.31i	−1.05182	+	0.382832i	−0.809350	−	0.587327i	$-0.800180\pi$
$439$	−9674.73	+	3521.31i	−1.05182	+	0.382832i	−0.242471	+	0.970159i	$0.577958\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6433.59	−	5398.43i	−0.689998	−	0.578977i	0.228910	−	0.973447i	$-0.426484\pi$
$443$	−6433.59	−	5398.43i	−0.689998	−	0.578977i	−0.918909	+	0.394470i	$0.870928\pi$
$444$		0			0
$445$	2793.16	+	15840.8i	0.297547	+	1.68748i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5426.60	−	9399.15i	−0.570373	−	0.987914i	−0.996528	−	0.0832641i	$-0.973466\pi$
$449$	−5426.60	−	9399.15i	−0.570373	−	0.987914i	0.426155	−	0.904650i	$-0.359868\pi$
$450$		0			0
$451$	4876.71	−	8446.71i	0.509169	−	0.881907i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−19170.2	−	6977.38i	−1.97519	−	0.718911i
$456$		0			0
$457$	2521.85	−	14302.1i	0.258134	−	1.46395i	−0.529765	−	0.848144i	$-0.677720\pi$
$457$	2521.85	−	14302.1i	0.258134	−	1.46395i	0.787899	−	0.615805i	$-0.211169\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1197.28	−	6790.14i	0.120961	−	0.686005i	−0.862664	−	0.505778i	$-0.831206\pi$
$461$	1197.28	−	6790.14i	0.120961	−	0.686005i	0.983625	−	0.180227i	$-0.0576833\pi$
$462$		0			0
$463$	−16976.9	−	6179.10i	−1.70407	−	0.620231i	−0.707792	−	0.706421i	$-0.750309\pi$
$463$	−16976.9	−	6179.10i	−1.70407	−	0.620231i	−0.996279	+	0.0861899i	$0.972531\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4432.93	+	7678.07i	−0.439254	+	0.760811i	−0.997632	−	0.0687761i	$-0.978091\pi$
$467$	−4432.93	+	7678.07i	−0.439254	+	0.760811i	0.558378	+	0.829587i	$0.311424\pi$
$468$		0			0
$469$	−6172.61	−	10691.3i	−0.607729	−	1.05262i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	385.756	+	2187.73i	0.0374991	+	0.212668i
$474$		0			0
$475$	1531.83	+	1285.36i	0.147969	+	0.124160i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13232.7	−	4816.31i	1.26225	−	0.459422i	0.377727	−	0.925917i	$-0.376706\pi$
$479$	13232.7	−	4816.31i	1.26225	−	0.459422i	0.884524	+	0.466495i	$0.154484\pi$
$480$		0			0
$481$	−17386.0	+	14588.6i	−1.64809	+	1.38291i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4155.58			0.389062
$486$		0			0
$487$	−19389.4			−1.80414			−0.902072	−	0.431586i	$-0.857954\pi$
$487$	−19389.4			−1.80414			−0.902072	+	0.431586i	$0.857954\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13194.3	−	11071.3i	1.21273	−	1.01760i	0.213556	−	0.976931i	$-0.431495\pi$
$491$	13194.3	−	11071.3i	1.21273	−	1.01760i	0.999173	−	0.0406697i	$-0.0129491\pi$
$492$		0			0
$493$	10581.8	−	3851.45i	0.966691	−	0.351847i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18355.3	+	15402.0i	1.65664	+	1.39009i
$498$		0			0
$499$	−2323.05	−	13174.7i	−0.208405	−	1.18192i	−0.891991	−	0.452054i	$-0.850691\pi$
$499$	−2323.05	−	13174.7i	−0.208405	−	1.18192i	0.683585	−	0.729871i	$-0.260420\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6922.31	−	11989.8i	−0.613620	−	1.06282i	−0.990625	−	0.136609i	$-0.956379\pi$
$503$	−6922.31	−	11989.8i	−0.613620	−	1.06282i	0.377005	−	0.926211i	$-0.376954\pi$
$504$		0			0
$505$	3886.38	−	6731.40i	0.342458	−	0.593155i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14055.9	+	5115.94i	1.22400	+	0.445501i	0.871540	−	0.490324i	$-0.163122\pi$
$509$	14055.9	+	5115.94i	1.22400	+	0.445501i	0.352464	+	0.935825i	$0.385344\pi$
$510$		0			0
$511$	−792.407	+	4493.96i	−0.0685989	+	0.389044i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1551.39	+	8798.34i	−0.132742	+	0.752818i
$516$		0			0
$517$	−6457.91	−	2350.49i	−0.549359	−	0.199950i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3924.43	−	6797.31i	0.330004	−	0.571584i	−0.652508	−	0.757782i	$-0.726283\pi$
$521$	3924.43	−	6797.31i	0.330004	−	0.571584i	0.982512	+	0.186198i	$0.0596164\pi$
$522$		0			0
$523$	7321.97	+	12682.0i	0.612175	+	1.06032i	0.990873	+	0.134798i	$0.0430385\pi$
$523$	7321.97	+	12682.0i	0.612175	+	1.06032i	−0.378698	+	0.925520i	$0.623628\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1667.96	−	9459.49i	−0.137870	−	0.781901i
$528$		0			0
$529$	1753.12	+	1471.05i	0.144088	+	0.120905i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−32253.2	+	11739.2i	−2.62109	+	0.953998i
$534$		0			0
$535$	1698.17	−	1424.93i	0.137230	−	0.115150i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8001.03			−0.639385
$540$		0			0
$541$	−12240.4			−0.972745			−0.486372	−	0.873752i	$-0.661680\pi$
$541$	−12240.4			−0.972745			−0.486372	+	0.873752i	$0.661680\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14428.8	−	12107.2i	1.13406	−	0.951591i
$546$		0			0
$547$	13010.6	−	4735.49i	1.01699	−	0.370155i	0.220879	−	0.975301i	$-0.429108\pi$
$547$	13010.6	−	4735.49i	1.01699	−	0.370155i	0.796114	+	0.605146i	$0.206885\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	9418.91	+	7903.41i	0.728238	+	0.611064i
$552$		0			0
$553$	737.416	+	4182.09i	0.0567054	+	0.321593i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2049.36	+	3549.59i	0.155896	+	0.270020i	0.933385	−	0.358877i	$-0.116840\pi$
$557$	2049.36	+	3549.59i	0.155896	+	0.270020i	−0.777489	+	0.628897i	$0.783507\pi$
$558$		0			0
$559$	3908.79	−	6770.22i	0.295750	−	0.512254i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19726.1	−	7179.71i	−1.47665	−	0.537458i	−0.526755	−	0.850017i	$-0.676592\pi$
$563$	−19726.1	−	7179.71i	−1.47665	−	0.537458i	−0.949898	+	0.312559i	$0.898814\pi$
$564$		0			0
$565$	−1106.45	+	6274.97i	−0.0823869	+	0.467239i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3207.06	+	18188.1i	−0.236286	+	1.34005i	0.603602	+	0.797286i	$0.293732\pi$
$569$	−3207.06	+	18188.1i	−0.236286	+	1.34005i	−0.839888	+	0.542760i	$0.817379\pi$
$570$		0			0
$571$	8850.02	+	3221.15i	0.648620	+	0.236078i	0.645315	−	0.763916i	$-0.276726\pi$
$571$	8850.02	+	3221.15i	0.648620	+	0.236078i	0.00330448	+	0.999995i	$0.498948\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1706.81	+	2956.28i	−0.123789	+	0.214409i
$576$		0			0
$577$	−6534.09	−	11317.4i	−0.471435	−	0.816549i	0.528031	−	0.849225i	$-0.322930\pi$
$577$	−6534.09	−	11317.4i	−0.471435	−	0.816549i	−0.999466	+	0.0326762i	$0.989597\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2732.30	+	15495.7i	0.195103	+	1.10649i
$582$		0			0
$583$	−9096.74	−	7633.07i	−0.646223	−	0.542246i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2383.29	−	867.448i	0.167579	−	0.0609939i	−0.256868	−	0.966446i	$-0.582691\pi$
$587$	2383.29	−	867.448i	0.167579	−	0.0609939i	0.424447	+	0.905453i	$0.360468\pi$
$588$		0			0
$589$	8034.24	−	6741.53i	0.562046	−	0.471613i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8431.98			−0.583912			−0.291956	−	0.956432i	$-0.594306\pi$
$593$	−8431.98			−0.583912			−0.291956	+	0.956432i	$0.594306\pi$
$594$		0			0
$595$	16804.9			1.15787
$596$		0			0
$597$		0			0
$598$		0			0
$599$	285.369	−	239.453i	0.0194655	−	0.0163335i	−0.633003	−	0.774149i	$-0.718178\pi$
$599$	285.369	−	239.453i	0.0194655	−	0.0163335i	0.652468	+	0.757816i	$0.273733\pi$
$600$		0			0
$601$	3783.87	−	1377.22i	0.256818	−	0.0934740i	−0.210403	−	0.977615i	$-0.567478\pi$
$601$	3783.87	−	1377.22i	0.256818	−	0.0934740i	0.467221	+	0.884141i	$0.345255\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6293.61	+	5280.96i	0.422928	+	0.354879i
$606$		0			0
$607$	−240.592	−	1364.47i	−0.0160879	−	0.0912388i	0.975707	−	0.219081i	$-0.0703060\pi$
$607$	−240.592	−	1364.47i	−0.0160879	−	0.0912388i	−0.991795	+	0.127842i	$0.959195\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12092.2	+	20944.3i	0.800652	+	1.38677i
$612$		0			0
$613$	−6625.82	+	11476.3i	−0.436565	+	0.756153i	−0.997422	−	0.0717601i	$-0.977138\pi$
$613$	−6625.82	+	11476.3i	−0.436565	+	0.756153i	0.560857	+	0.827913i	$0.310472\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5742.14	−	2089.97i	−0.374667	−	0.136368i	0.147821	−	0.989014i	$-0.452774\pi$
$617$	−5742.14	−	2089.97i	−0.374667	−	0.136368i	−0.522489	+	0.852646i	$0.674996\pi$
$618$		0			0
$619$	158.970	−	901.563i	0.0103224	−	0.0585410i	−0.979211	−	0.202842i	$-0.934982\pi$
$619$	158.970	−	901.563i	0.0103224	−	0.0585410i	0.989534	+	0.144301i	$0.0460933\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7532.44	−	42718.6i	0.484399	−	2.74716i
$624$		0			0
$625$	10590.2	+	3854.53i	0.677774	+	0.246690i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9347.83	−	16190.9i	0.592563	−	1.02635i
$630$		0			0
$631$	3125.11	+	5412.85i	0.197161	+	0.341493i	0.947607	−	0.319439i	$-0.103495\pi$
$631$	3125.11	+	5412.85i	0.197161	+	0.341493i	−0.750446	+	0.660932i	$0.770161\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−616.735	−	3497.68i	−0.0385423	−	0.218585i
$636$		0			0
$637$	21569.0	+	18098.5i	1.34159	+	1.12573i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7732.55	+	2814.42i	−0.476470	+	0.173421i	−0.569081	−	0.822281i	$-0.692701\pi$
$641$	−7732.55	+	2814.42i	−0.476470	+	0.173421i	0.0926108	+	0.995702i	$0.470479\pi$
$642$		0			0
$643$	−1394.39	+	1170.03i	−0.0855201	+	0.0717599i	−0.684545	−	0.728970i	$-0.739999\pi$
$643$	−1394.39	+	1170.03i	−0.0855201	+	0.0717599i	0.599025	+	0.800730i	$0.295555\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14438.2			0.877318			0.438659	−	0.898654i	$-0.355454\pi$
$647$	14438.2			0.877318			0.438659	+	0.898654i	$0.355454\pi$
$648$		0			0
$649$	9306.51			0.562885
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15367.0	+	12894.4i	−0.920914	+	0.772739i	−0.974164	−	0.225842i	$-0.927487\pi$
$653$	−15367.0	+	12894.4i	−0.920914	+	0.772739i	0.0532495	+	0.998581i	$0.483042\pi$
$654$		0			0
$655$	−6481.75	+	2359.16i	−0.386661	+	0.140733i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−14421.1	−	12100.7i	−0.852451	−	0.715291i	0.107877	−	0.994164i	$-0.465595\pi$
$659$	−14421.1	−	12100.7i	−0.852451	−	0.715291i	−0.960328	+	0.278873i	$0.910039\pi$
$660$		0			0
$661$	387.406	+	2197.09i	0.0227963	+	0.129284i	0.994082	−	0.108632i	$-0.0346469\pi$
$661$	387.406	+	2197.09i	0.0227963	+	0.129284i	−0.971286	+	0.237916i	$0.923536\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	9174.47	+	15890.6i	0.534993	+	0.926636i
$666$		0			0
$667$	−10494.8	+	18177.6i	−0.609237	+	1.05523i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1942.82	+	707.127i	0.111776	+	0.0406831i
$672$		0			0
$673$	−2561.47	+	14526.8i	−0.146712	+	0.832048i	0.819264	+	0.573417i	$0.194382\pi$
$673$	−2561.47	+	14526.8i	−0.146712	+	0.832048i	−0.965976	+	0.258631i	$0.916729\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−666.714	+	3781.13i	−0.0378492	+	0.214654i	−0.997867	−	0.0652871i	$-0.979204\pi$
$677$	−666.714	+	3781.13i	−0.0378492	+	0.214654i	0.960017	+	0.279941i	$0.0903148\pi$
$678$		0			0
$679$	−10530.7	−	3832.86i	−0.595185	−	0.216630i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6452.62	+	11176.3i	−0.361497	+	0.626132i	−0.988207	−	0.153121i	$-0.951068\pi$
$683$	−6452.62	+	11176.3i	−0.361497	+	0.626132i	0.626710	+	0.779252i	$0.284401\pi$
$684$		0			0
$685$	13880.4	+	24041.5i	0.774222	+	1.34099i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7256.57	+	41154.1i	0.401239	+	2.27554i
$690$		0			0
$691$	−5853.56	−	4911.72i	−0.322257	−	0.270406i	0.467279	−	0.884110i	$-0.345234\pi$
$691$	−5853.56	−	4911.72i	−0.322257	−	0.270406i	−0.789536	+	0.613704i	$0.789679\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9129.27	−	3322.78i	0.498263	−	0.181353i
$696$		0			0
$697$	21658.9	−	18174.0i	1.17703	−	0.987644i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−16562.4			−0.892373			−0.446187	−	0.894940i	$-0.647218\pi$
$701$	−16562.4			−0.892373			−0.446187	+	0.894940i	$0.647218\pi$
$702$		0			0
$703$	20413.4			1.09517
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−16057.1	+	13473.5i	−0.854159	+	0.716725i
$708$		0			0
$709$	866.193	−	315.268i	0.0458823	−	0.0166998i	−0.318977	−	0.947762i	$-0.603339\pi$
$709$	866.193	−	315.268i	0.0458823	−	0.0166998i	0.364859	+	0.931063i	$0.381117\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13715.2	+	11508.4i	0.720392	+	0.604481i
$714$		0			0
$715$	2973.89	+	16865.8i	0.155548	+	0.882159i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1108.55	−	1920.06i	−0.0574992	−	0.0995915i	0.835843	−	0.548969i	$-0.184979\pi$
$719$	−1108.55	−	1920.06i	−0.0574992	−	0.0995915i	−0.893342	+	0.449377i	$0.851646\pi$
$720$		0			0
$721$	12046.4	−	20865.0i	0.622237	−	1.07775i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4657.69	−	1695.26i	−0.238596	−	0.0868420i
$726$		0			0
$727$	2305.90	−	13077.4i	0.117636	−	0.667145i	−0.867776	−	0.496956i	$-0.834451\pi$
$727$	2305.90	−	13077.4i	0.117636	−	0.667145i	0.985411	−	0.170189i	$-0.0544378\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1118.25	+	6341.90i	−0.0565799	+	0.320880i
$732$		0			0
$733$	18468.5	+	6721.99i	0.930628	+	0.338721i	0.762458	−	0.647037i	$-0.223992\pi$
$733$	18468.5	+	6721.99i	0.930628	+	0.338721i	0.168169	+	0.985758i	$0.446214\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5181.82	+	8975.18i	−0.258989	+	0.448582i
$738$		0			0
$739$	6917.27	+	11981.1i	0.344325	+	0.596388i	0.985231	−	0.171231i	$-0.0547746\pi$
$739$	6917.27	+	11981.1i	0.344325	+	0.596388i	−0.640906	+	0.767619i	$0.721441\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3872.43	+	21961.6i	0.191206	+	1.08438i	0.917720	+	0.397229i	$0.130028\pi$
$743$	3872.43	+	21961.6i	0.191206	+	1.08438i	−0.726514	+	0.687152i	$0.758861\pi$
$744$		0			0
$745$	−14719.8	−	12351.4i	−0.723881	−	0.607408i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5617.60	+	2044.64i	−0.274049	+	0.0997457i
$750$		0			0
$751$	18445.2	−	15477.4i	0.896238	−	0.752033i	−0.0732136	−	0.997316i	$-0.523325\pi$
$751$	18445.2	−	15477.4i	0.896238	−	0.752033i	0.969451	+	0.245283i	$0.0788810\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	27923.4			1.34601
$756$		0			0
$757$	15873.5			0.762132			0.381066	−	0.924548i	$-0.375557\pi$
$757$	15873.5			0.762132			0.381066	+	0.924548i	$0.375557\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	21981.7	−	18444.8i	1.04709	−	0.878613i	0.0543053	−	0.998524i	$-0.482706\pi$
$761$	21981.7	−	18444.8i	1.04709	−	0.878613i	0.992785	+	0.119912i	$0.0382611\pi$
$762$		0			0
$763$	−47731.2	+	17372.7i	−2.26473	+	0.824293i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−25088.3	−	21051.6i	−1.18108	−	0.991040i
$768$		0			0
$769$	−2217.08	−	12573.7i	−0.103966	−	0.589620i	−0.991628	−	0.129125i	$-0.958783\pi$
$769$	−2217.08	−	12573.7i	−0.103966	−	0.589620i	0.887662	−	0.460495i	$-0.152328\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3017.93	−	5227.21i	−0.140424	−	0.243221i	0.787233	−	0.616656i	$-0.211513\pi$
$773$	−3017.93	−	5227.21i	−0.140424	−	0.243221i	−0.927656	+	0.373435i	$0.878180\pi$
$774$		0			0
$775$	−2113.97	+	3661.51i	−0.0979821	+	0.169710i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	29009.7	+	10558.7i	1.33425	+	0.485626i
$780$		0			0
$781$	3492.95	−	19809.5i	0.160035	−	0.907605i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1212.58	+	6876.87i	−0.0551321	+	0.312670i
$786$		0			0
$787$	20849.4	+	7588.56i	0.944346	+	0.343714i	0.767881	−	0.640593i	$-0.221311\pi$
$787$	20849.4	+	7588.56i	0.944346	+	0.343714i	0.176465	+	0.984307i	$0.443534\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	8591.51	−	14880.9i	0.386194	−	0.668907i
$792$		0			0
$793$	−3637.86	−	6300.95i	−0.162905	−	0.282161i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2619.80	−	14857.6i	−0.116434	−	0.660332i	−0.986030	−	0.166567i	$-0.946732\pi$
$797$	−2619.80	−	14857.6i	−0.116434	−	0.660332i	0.869596	−	0.493764i	$-0.164379\pi$
$798$		0			0
$799$	−15261.1	−	12805.6i	−0.675717	−	0.566994i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3599.79	−	1310.22i	0.158199	−	0.0575798i
$804$		0			0
$805$	−23995.1	+	20134.3i	−1.05058	+	0.881542i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	4962.96			0.215684			0.107842	−	0.994168i	$-0.465606\pi$
$809$	4962.96			0.215684			0.107842	+	0.994168i	$0.465606\pi$
$810$		0			0
$811$	−18808.2			−0.814358			−0.407179	−	0.913348i	$-0.633488\pi$
$811$	−18808.2			−0.814358			−0.407179	+	0.913348i	$0.633488\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10095.7	+	8471.32i	−0.433912	+	0.364095i
$816$		0			0
$817$	−6607.37	+	2404.88i	−0.282941	+	0.102982i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10726.0	−	9000.16i	−0.455955	−	0.382592i	0.385685	−	0.922630i	$-0.373965\pi$
$821$	−10726.0	−	9000.16i	−0.455955	−	0.382592i	−0.841640	+	0.540039i	$0.818410\pi$
$822$		0			0
$823$	−6387.96	−	36227.9i	−0.270559	−	1.53442i	−0.752724	−	0.658336i	$-0.771260\pi$
$823$	−6387.96	−	36227.9i	−0.270559	−	1.53442i	0.482165	−	0.876081i	$-0.339851\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−10316.7	−	17869.0i	−0.433791	−	0.751349i	0.563405	−	0.826181i	$-0.309491\pi$
$827$	−10316.7	−	17869.0i	−0.433791	−	0.751349i	−0.997196	+	0.0748324i	$0.976158\pi$
$828$		0			0
$829$	−19800.1	+	34294.7i	−0.829535	+	1.43680i	0.0688680	+	0.997626i	$0.478061\pi$
$829$	−19800.1	+	34294.7i	−0.829535	+	1.43680i	−0.898403	+	0.439171i	$0.855272\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21795.0	−	7932.73i	−0.906545	−	0.329955i
$834$		0			0
$835$	2868.23	−	16266.5i	0.118873	−	0.674163i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3742.81	+	21226.5i	−0.154012	+	0.873446i	0.805671	+	0.592363i	$0.201805\pi$
$839$	−3742.81	+	21226.5i	−0.154012	+	0.873446i	−0.959683	+	0.281083i	$0.909306\pi$
$840$		0			0
$841$	−5721.07	−	2082.30i	−0.234576	−	0.0853786i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	19336.7	−	33492.2i	0.787223	−	1.36351i
$846$		0			0
$847$	−11077.8	−	19187.4i	−0.449397	−	0.778378i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6051.23	+	34318.3i	0.243753	+	1.38239i
$852$		0			0
$853$	−10060.5	−	8441.78i	−0.403828	−	0.338852i	0.418143	−	0.908381i	$-0.362681\pi$
$853$	−10060.5	−	8441.78i	−0.403828	−	0.338852i	−0.821971	+	0.569529i	$0.807126\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6420.15	−	2336.74i	0.255902	−	0.0931408i	−0.210884	−	0.977511i	$-0.567634\pi$
$857$	6420.15	−	2336.74i	0.255902	−	0.0931408i	0.466786	+	0.884370i	$0.345412\pi$
$858$		0			0
$859$	2844.30	−	2386.65i	0.112976	−	0.0947979i	−0.584550	−	0.811358i	$-0.698729\pi$
$859$	2844.30	−	2386.65i	0.112976	−	0.0947979i	0.697525	+	0.716560i	$0.254284\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	28016.1			1.10507			0.552537	−	0.833488i	$-0.313660\pi$
$863$	28016.1			1.10507			0.552537	+	0.833488i	$0.313660\pi$
$864$		0			0
$865$	−1520.13			−0.0597524
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2730.92	−	2291.52i	0.106606	−	0.0894527i
$870$		0			0
$871$	34271.1	−	12473.7i	1.33322	−	0.485251i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−30613.2	−	25687.5i	−1.18276	−	0.992454i
$876$		0			0
$877$	4156.37	+	23571.9i	0.160035	+	0.907602i	0.954037	+	0.299688i	$0.0968825\pi$
$877$	4156.37	+	23571.9i	0.160035	+	0.907602i	−0.794003	+	0.607914i	$0.792006\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−6076.95	−	10525.6i	−0.232392	−	0.402515i	0.726119	−	0.687569i	$-0.241322\pi$
$881$	−6076.95	−	10525.6i	−0.232392	−	0.402515i	−0.958512	+	0.285054i	$0.907989\pi$
$882$		0			0
$883$	−14612.5	+	25309.6i	−0.556909	+	0.964595i	0.440843	+	0.897584i	$0.354679\pi$
$883$	−14612.5	+	25309.6i	−0.556909	+	0.964595i	−0.997752	+	0.0670105i	$0.978654\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	31069.7	+	11308.4i	1.17612	+	0.428072i	0.854830	−	0.518909i	$-0.173662\pi$
$887$	31069.7	+	11308.4i	1.17612	+	0.428072i	0.321290	+	0.946981i	$0.395884\pi$
$888$		0			0
$889$	−1663.17	+	9432.33i	−0.0627459	+	0.355850i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3777.25	−	21421.9i	0.141546	−	0.802750i
$894$		0			0
$895$	−7884.24	−	2869.63i	−0.294459	−	0.107174i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12998.4	+	22513.9i	−0.482226	+	0.835239i
$900$		0			0
$901$	−17211.8	−	29811.7i	−0.636414	−	1.10230i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3365.29	−	19085.5i	−0.123609	−	0.701021i
$906$		0			0
$907$	5150.43	+	4321.72i	0.188553	+	0.158214i	0.732177	−	0.681114i	$-0.238504\pi$
$907$	5150.43	+	4321.72i	0.188553	+	0.158214i	−0.543625	+	0.839328i	$0.682949\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7244.57	+	2636.81i	−0.263472	+	0.0958961i	−0.470379	−	0.882465i	$-0.655883\pi$
$911$	−7244.57	+	2636.81i	−0.263472	+	0.0958961i	0.206906	+	0.978361i	$0.433660\pi$
$912$		0			0
$913$	10118.7	−	8490.62i	0.366792	−	0.307775i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18601.4			0.669871
$918$		0			0
$919$	22717.1			0.815418			0.407709	−	0.913112i	$-0.366328\pi$
$919$	22717.1			0.815418			0.407709	+	0.913112i	$0.366328\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−54225.8	+	45500.8i	−1.93376	+	1.62262i
$924$		0			0
$925$	−7732.85	+	2814.53i	−0.274870	+	0.100044i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−29601.6	−	24838.7i	−1.04542	−	0.877213i	−0.0528166	−	0.998604i	$-0.516820\pi$
$929$	−29601.6	−	24838.7i	−1.04542	−	0.877213i	−0.992605	+	0.121392i	$0.961264\pi$
$930$		0			0
$931$	−4397.60	−	24940.0i	−0.154807	−	0.877955i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−7053.75	−	12217.4i	−0.246719	−	0.427330i
$936$		0			0
$937$	−15175.5	+	26284.7i	−0.529095	+	0.916419i	0.470330	+	0.882491i	$0.344135\pi$
$937$	−15175.5	+	26284.7i	−0.529095	+	0.916419i	−0.999424	+	0.0339279i	$0.989198\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22898.9	+	8334.50i	0.793285	+	0.288732i	0.706701	−	0.707512i	$-0.250183\pi$
$941$	22898.9	+	8334.50i	0.793285	+	0.288732i	0.0865842	+	0.996245i	$0.472405\pi$
$942$		0			0
$943$	−9151.35	+	51899.9i	−0.316022	+	1.79225i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1088.11	+	6170.97i	−0.0373377	+	0.211752i	−0.997769	−	0.0667655i	$-0.978732\pi$
$947$	−1088.11	+	6170.97i	−0.0373377	+	0.211752i	0.960431	+	0.278518i	$0.0898432\pi$
$948$		0			0
$949$	−12668.0	−	4610.77i	−0.433319	−	0.157715i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3531.00	−	6115.87i	0.120021	−	0.207883i	−0.799755	−	0.600327i	$-0.795037\pi$
$953$	3531.00	−	6115.87i	0.120021	−	0.207883i	0.919776	+	0.392444i	$0.128370\pi$
$954$		0			0
$955$	25109.6	+	43491.1i	0.850814	+	1.47365i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12999.9	−	73726.3i	−0.437737	−	2.48253i
$960$		0			0
$961$	−5834.18	−	4895.46i	−0.195837	−	0.164327i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4490.88	+	1634.55i	−0.149810	+	0.0545263i
$966$		0			0
$967$	8819.45	−	7400.40i	0.293293	−	0.246102i	−0.484253	−	0.874928i	$-0.660909\pi$
$967$	8819.45	−	7400.40i	0.293293	−	0.246102i	0.777546	+	0.628826i	$0.216464\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	25765.5			0.851549			0.425774	−	0.904829i	$-0.360002\pi$
$971$	25765.5			0.851549			0.425774	+	0.904829i	$0.360002\pi$
$972$		0			0
$973$	−26199.3			−0.863218
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12494.6	+	10484.2i	−0.409149	+	0.343317i	−0.824017	−	0.566565i	$-0.808272\pi$
$977$	−12494.6	+	10484.2i	−0.409149	+	0.343317i	0.414868	+	0.909882i	$0.363828\pi$
$978$		0			0
$979$	−34218.8	+	12454.6i	−1.11710	+	0.406590i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−14280.0	−	11982.4i	−0.463338	−	0.388787i	0.381019	−	0.924567i	$-0.375573\pi$
$983$	−14280.0	−	11982.4i	−0.463338	−	0.388787i	−0.844358	+	0.535780i	$0.820018\pi$
$984$		0			0
$985$	−3479.67	−	19734.2i	−0.112560	−	0.638358i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6001.65	−	10395.2i	−0.192964	−	0.334223i
$990$		0			0
$991$	3160.07	−	5473.39i	0.101294	−	0.175447i	−0.810924	−	0.585152i	$-0.801035\pi$
$991$	3160.07	−	5473.39i	0.101294	−	0.175447i	0.912218	+	0.409705i	$0.134368\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−41333.6	−	15044.2i	−1.31695	−	0.479330i
$996$		0			0
$997$	5035.77	−	28559.3i	0.159964	−	0.907203i	−0.794141	−	0.607733i	$-0.792079\pi$
$997$	5035.77	−	28559.3i	0.159964	−	0.907203i	0.954106	−	0.299470i	$-0.0968099\pi$
$998$		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.73.3		54
3.2	odd	2		108.4.i.a.25.4	yes	54
27.13	even	9	inner	324.4.i.a.253.3		54
27.14	odd	18		108.4.i.a.13.4	✓	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.13.4	✓	54	27.14	odd	18
108.4.i.a.25.4	yes	54	3.2	odd	2
324.4.i.a.73.3		54	1.1	even	1	trivial
324.4.i.a.253.3		54	27.13	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)

\(f(q)\)	\(=\)	\(q+(-7.52940 + 6.31792i) q^{5} +(24.9076 - 9.06562i) q^{7} +O(q^{10})\)	\(q+(-7.52940 + 6.31792i) q^{5} +(24.9076 - 9.06562i) q^{7} +(-17.0456 - 14.3030i) q^{11} +(13.5975 + 77.1153i) q^{13} +(-32.2518 - 55.8618i) q^{17} +(35.2151 - 60.9943i) q^{19} +(112.980 + 41.1215i) q^{23} +(-4.93023 + 27.9607i) q^{25} +(-30.3150 + 171.925i) q^{29} +(139.932 + 50.9312i) q^{31} +(-130.263 + 225.623i) q^{35} +(144.919 + 251.008i) q^{37} +(76.1146 + 431.667i) q^{41} +(-76.4782 - 64.1728i) q^{43} +(290.224 - 105.633i) q^{47} +(275.449 - 231.129i) q^{49} +533.670 q^{53} +218.708 q^{55} +(-320.392 + 268.841i) q^{59} +(-87.3117 + 31.7789i) q^{61} +(-589.589 - 494.724i) q^{65} +(-80.8767 - 458.675i) q^{67} +(451.994 + 782.877i) q^{71} +(-86.0801 + 149.095i) q^{73} +(-554.231 - 201.723i) q^{77} +(-27.8206 + 157.779i) q^{79} +(-103.082 + 584.607i) q^{83} +(595.767 + 216.841i) q^{85} +(818.257 - 1417.26i) q^{89} +(1037.78 + 1797.48i) q^{91} +(120.209 + 681.737i) q^{95} +(-323.876 - 271.765i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 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

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Database

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