LMFDB - Embedded newform 360.4.f.d.289.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,4,Mod(289,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("360.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	360.f (of order $2$, degree $1$, 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:	$21.2406876021$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{129})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 65x^{2} + 1024 $ x^4 + 65*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.3
Root			$6.17891i$ of defining polynomial
Character	$\chi$	$=$	360.289
Dual form			360.4.f.d.289.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.178908 - 11.1789i) q^{5} -35.0735i q^{7} -25.6422 q^{11} -37.6422i q^{13} +95.7891i q^{17} -50.8625 q^{19} +110.863i q^{23} +(-124.936 - 4.00000i) q^{25} -54.5047 q^{29} +198.441 q^{31} +(-392.083 - 6.27493i) q^{35} +266.945i q^{37} -103.853 q^{41} -108.000i q^{43} -597.009i q^{47} -887.147 q^{49} -305.642i q^{53} +(-4.58760 + 286.652i) q^{55} -223.533 q^{59} +485.450 q^{61} +(-420.799 - 6.73450i) q^{65} -876.166i q^{67} -585.597 q^{71} +1137.60i q^{73} +899.360i q^{77} -685.009 q^{79} +305.725i q^{83} +(1070.82 + 17.1375i) q^{85} +887.175 q^{89} -1320.24 q^{91} +(-9.09973 + 568.588i) q^{95} -556.550i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 22 q^{5} - 148 q^{11} + 160 q^{19} + 100 q^{29} - 24 q^{31} - 796 q^{35} - 688 q^{41} - 3276 q^{49} + 1072 q^{55} + 1332 q^{59} + 488 q^{61} - 820 q^{65} - 616 q^{71} - 2104 q^{79} + 2148 q^{85} + 1368 q^{89}+ \cdots - 2944 q^{95}+O(q^{100}) $ 4 * q - 22 * q^5 - 148 * q^11 + 160 * q^19 + 100 * q^29 - 24 * q^31 - 796 * q^35 - 688 * q^41 - 3276 * q^49 + 1072 * q^55 + 1332 * q^59 + 488 * q^61 - 820 * q^65 - 616 * q^71 - 2104 * q^79 + 2148 * q^85 + 1368 * q^89 + 1352 * q^91 - 2944 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$.

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.178908	−	11.1789i	0.0160020	−	0.999872i
$5$	0.178908	−	11.1789i	0.0160020	−	0.999872i
$6$		0			0
$7$		−	35.0735i		−	1.89379i	−0.321545	−	0.946894i	$-0.604202\pi$
$7$		−	35.0735i		−	1.89379i	0.321545	−	0.946894i	$-0.395798\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−25.6422			−0.702855			−0.351428	−	0.936215i	$-0.614304\pi$
$11$	−25.6422			−0.702855			−0.351428	+	0.936215i	$0.614304\pi$
$12$		0			0
$13$		−	37.6422i		−	0.803082i	−0.915841	−	0.401541i	$-0.868475\pi$
$13$		−	37.6422i		−	0.803082i	0.915841	−	0.401541i	$-0.131525\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			95.7891i			1.36660i	0.730136	+	0.683302i	$0.239457\pi$
$17$			95.7891i			1.36660i	−0.730136	+	0.683302i	$0.760543\pi$
$18$		0			0
$19$	−50.8625			−0.614140			−0.307070	−	0.951687i	$-0.599349\pi$
$19$	−50.8625			−0.614140			−0.307070	+	0.951687i	$0.599349\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			110.863i			1.00506i	0.864559	+	0.502531i	$0.167598\pi$
$23$			110.863i			1.00506i	−0.864559	+	0.502531i	$0.832402\pi$
$24$		0			0
$25$	−124.936	−	4.00000i	−0.999488	−	0.0320000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−54.5047			−0.349009			−0.174505	−	0.984656i	$-0.555832\pi$
$29$	−54.5047			−0.349009			−0.174505	+	0.984656i	$0.555832\pi$
$30$		0			0
$31$	198.441			1.14971			0.574855	−	0.818255i	$-0.305059\pi$
$31$	198.441			1.14971			0.574855	+	0.818255i	$0.305059\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−392.083	−	6.27493i	−1.89355	−	0.0303045i
$36$		0			0
$37$			266.945i			1.18610i	0.805167	+	0.593048i	$0.202076\pi$
$37$			266.945i			1.18610i	−0.805167	+	0.593048i	$0.797924\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−103.853			−0.395589			−0.197794	−	0.980244i	$-0.563378\pi$
$41$	−103.853			−0.395589			−0.197794	+	0.980244i	$0.563378\pi$
$42$		0			0
$43$		−	108.000i		−	0.383020i	−0.981491	−	0.191510i	$-0.938662\pi$
$43$		−	108.000i		−	0.383020i	0.981491	−	0.191510i	$-0.0613384\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	597.009i		−	1.85283i	−0.376510	−	0.926413i	$-0.622876\pi$
$47$		−	597.009i		−	1.85283i	0.376510	−	0.926413i	$-0.377124\pi$
$48$		0			0
$49$	−887.147			−2.58643
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	305.642i		−	0.792136i	−0.918221	−	0.396068i	$-0.870375\pi$
$53$		−	305.642i		−	0.792136i	0.918221	−	0.396068i	$-0.129625\pi$
$54$		0			0
$55$	−4.58760	+	286.652i	−0.0112471	+	0.702765i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−223.533			−0.493246			−0.246623	−	0.969111i	$-0.579321\pi$
$59$	−223.533			−0.493246			−0.246623	+	0.969111i	$0.579321\pi$
$60$		0			0
$61$	485.450			1.01894			0.509471	−	0.860488i	$-0.329841\pi$
$61$	485.450			1.01894			0.509471	+	0.860488i	$0.329841\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−420.799	−	6.73450i	−0.802979	−	0.0128510i
$66$		0			0
$67$		−	876.166i		−	1.59762i	−0.601582	−	0.798811i	$-0.705463\pi$
$67$		−	876.166i		−	1.59762i	0.601582	−	0.798811i	$-0.294537\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−585.597			−0.978839			−0.489420	−	0.872048i	$-0.662791\pi$
$71$	−585.597			−0.978839			−0.489420	+	0.872048i	$0.662791\pi$
$72$		0			0
$73$			1137.60i			1.82391i	0.410287	+	0.911957i	$0.365429\pi$
$73$			1137.60i			1.82391i	−0.410287	+	0.911957i	$0.634571\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			899.360i			1.33106i
$78$		0			0
$79$	−685.009			−0.975564			−0.487782	−	0.872965i	$-0.662194\pi$
$79$	−685.009			−0.975564			−0.487782	+	0.872965i	$0.662194\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			305.725i			0.404309i	0.979354	+	0.202155i	$0.0647944\pi$
$83$			305.725i			0.404309i	−0.979354	+	0.202155i	$0.935206\pi$
$84$		0			0
$85$	1070.82	+	17.1375i	1.36643	+	0.0218685i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	887.175			1.05663			0.528317	−	0.849047i	$-0.322823\pi$
$89$	887.175			1.05663			0.528317	+	0.849047i	$0.322823\pi$
$90$		0			0
$91$	−1320.24			−1.52087
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.09973	+	568.588i	−0.00982750	+	0.614062i
$96$		0			0
$97$		−	556.550i		−	0.582568i	−0.956637	−	0.291284i	$-0.905918\pi$
$97$		−	556.550i		−	0.582568i	0.956637	−	0.291284i	$-0.0940824\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1591.09			−1.56752			−0.783760	−	0.621063i	$-0.786701\pi$
$101$	−1591.09			−1.56752			−0.783760	+	0.621063i	$0.786701\pi$
$102$		0			0
$103$		−	1350.95i		−	1.29236i	−0.763187	−	0.646178i	$-0.776366\pi$
$103$		−	1350.95i		−	1.29236i	0.763187	−	0.646178i	$-0.223634\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1333.51i		−	1.20481i	−0.798190	−	0.602406i	$-0.794209\pi$
$107$		−	1333.51i		−	1.20481i	0.798190	−	0.602406i	$-0.205791\pi$
$108$		0			0
$109$	609.910			0.535952			0.267976	−	0.963426i	$-0.413645\pi$
$109$	609.910			0.535952			0.267976	+	0.963426i	$0.413645\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			241.808i			0.201304i	0.994922	+	0.100652i	$0.0320929\pi$
$113$			241.808i			0.201304i	−0.994922	+	0.100652i	$0.967907\pi$
$114$		0			0
$115$	1239.32	+	19.8342i	1.00493	+	0.0160831i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3359.65			2.58806
$120$		0			0
$121$	−673.478			−0.505994
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−67.0677	+	1395.93i	−0.0479898	+	0.998848i
$126$		0			0
$127$		−	1045.94i		−	0.730802i	−0.930850	−	0.365401i	$-0.880932\pi$
$127$		−	1045.94i		−	0.730802i	0.930850	−	0.365401i	$-0.119068\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	886.524			0.591267			0.295633	−	0.955301i	$-0.404469\pi$
$131$	886.524			0.591267			0.295633	+	0.955301i	$0.404469\pi$
$132$		0			0
$133$			1783.92i			1.16305i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			160.723i			0.100230i	0.998743	+	0.0501150i	$0.0159588\pi$
$137$			160.723i			0.100230i	−0.998743	+	0.0501150i	$0.984041\pi$
$138$		0			0
$139$	57.2655			0.0349439			0.0174719	−	0.999847i	$-0.494438\pi$
$139$	57.2655			0.0349439			0.0174719	+	0.999847i	$0.494438\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			965.228i			0.564450i
$144$		0			0
$145$	−9.75135	+	609.303i	−0.00558487	+	0.348965i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1105.15			−0.607633			−0.303817	−	0.952731i	$-0.598261\pi$
$149$	−1105.15			−0.607633			−0.303817	+	0.952731i	$0.598261\pi$
$150$		0			0
$151$	2289.63			1.23396			0.616980	−	0.786979i	$-0.288356\pi$
$151$	2289.63			1.23396			0.616980	+	0.786979i	$0.288356\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	35.5027	−	2218.35i	0.0183977	−	1.14956i
$156$		0			0
$157$		−	161.514i		−	0.0821034i	−0.999157	−	0.0410517i	$-0.986929\pi$
$157$		−	161.514i		−	0.0821034i	0.999157	−	0.0410517i	$-0.0130708\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3888.33			1.90338
$162$		0			0
$163$		−	1594.15i		−	0.766032i	−0.923742	−	0.383016i	$-0.874885\pi$
$163$		−	1594.15i		−	0.766032i	0.923742	−	0.383016i	$-0.125115\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1017.39i		−	0.471427i	−0.971823	−	0.235713i	$-0.924257\pi$
$167$		−	1017.39i		−	0.471427i	0.971823	−	0.235713i	$-0.0757427\pi$
$168$		0			0
$169$	780.066			0.355060
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2444.49i		−	1.07428i	−0.843492	−	0.537142i	$-0.819504\pi$
$173$		−	2444.49i		−	1.07428i	0.843492	−	0.537142i	$-0.180496\pi$
$174$		0			0
$175$	−140.294	+	4381.94i	−0.0606012	+	1.89282i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	222.780			0.0930242			0.0465121	−	0.998918i	$-0.485189\pi$
$179$	222.780			0.0930242			0.0465121	+	0.998918i	$0.485189\pi$
$180$		0			0
$181$	−100.034			−0.0410798			−0.0205399	−	0.999789i	$-0.506539\pi$
$181$	−100.034			−0.0410798			−0.0205399	+	0.999789i	$0.506539\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2984.16	+	47.7588i	1.18594	+	0.0189800i
$186$		0			0
$187$		−	2456.24i		−	0.960525i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−702.403			−0.266095			−0.133047	−	0.991110i	$-0.542476\pi$
$191$	−702.403			−0.266095			−0.133047	+	0.991110i	$0.542476\pi$
$192$		0			0
$193$			4126.63i			1.53907i	0.638602	+	0.769537i	$0.279513\pi$
$193$			4126.63i			1.53907i	−0.638602	+	0.769537i	$0.720487\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3104.45i		−	1.12276i	−0.827559	−	0.561378i	$-0.810271\pi$
$197$		−	3104.45i		−	1.12276i	0.827559	−	0.561378i	$-0.189729\pi$
$198$		0			0
$199$	367.616			0.130953			0.0654764	−	0.997854i	$-0.479143\pi$
$199$	367.616			0.130953			0.0654764	+	0.997854i	$0.479143\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1911.67i			0.660950i
$204$		0			0
$205$	−18.5802	+	1160.96i	−0.00633023	+	0.395538i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1304.23			0.431652
$210$		0			0
$211$	2594.85			0.846619			0.423310	−	0.905985i	$-0.360868\pi$
$211$	2594.85			0.846619			0.423310	+	0.905985i	$0.360868\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1207.32	−	19.3221i	−0.382971	−	0.00612910i
$216$		0			0
$217$		−	6960.00i		−	2.17731i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3605.71			1.09749
$222$		0			0
$223$			1834.41i			0.550857i	0.961321	+	0.275429i	$0.0888198\pi$
$223$			1834.41i			0.550857i	−0.961321	+	0.275429i	$0.911180\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	175.811i		−	0.0514053i	−0.999670	−	0.0257027i	$-0.991818\pi$
$227$		−	175.811i		−	0.0514053i	0.999670	−	0.0257027i	$-0.00818231\pi$
$228$		0			0
$229$	−1622.35			−0.468158			−0.234079	−	0.972218i	$-0.575207\pi$
$229$	−1622.35			−0.468158			−0.234079	+	0.972218i	$0.575207\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3965.65i			1.11501i	0.830172	+	0.557507i	$0.188242\pi$
$233$			3965.65i			1.11501i	−0.830172	+	0.557507i	$0.811758\pi$
$234$		0			0
$235$	−6673.91	−	106.810i	−1.85259	−	0.0296490i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6323.25			−1.71137			−0.855685	−	0.517497i	$-0.826864\pi$
$239$	−6323.25			−1.71137			−0.855685	+	0.517497i	$0.826864\pi$
$240$		0			0
$241$	3407.51			0.910775			0.455388	−	0.890293i	$-0.349501\pi$
$241$	3407.51			0.910775			0.455388	+	0.890293i	$0.349501\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−158.718	+	9917.33i	−0.0413882	+	2.58610i
$246$		0			0
$247$			1914.58i			0.493205i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1345.81			−0.338433			−0.169216	−	0.985579i	$-0.554124\pi$
$251$	−1345.81			−0.338433			−0.169216	+	0.985579i	$0.554124\pi$
$252$		0			0
$253$		−	2842.76i		−	0.706414i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	4697.19i		−	1.14009i	−0.821614	−	0.570044i	$-0.806926\pi$
$257$		−	4697.19i		−	1.14009i	0.821614	−	0.570044i	$-0.193074\pi$
$258$		0			0
$259$	9362.70			2.24621
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4700.66i			1.10211i	0.834468	+	0.551056i	$0.185775\pi$
$263$			4700.66i			1.10211i	−0.834468	+	0.551056i	$0.814225\pi$
$264$		0			0
$265$	−3416.75	−	54.6819i	−0.792034	−	0.0126758i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7962.67			−1.80481			−0.902403	−	0.430894i	$-0.858198\pi$
$269$	−7962.67			−1.80481			−0.902403	+	0.430894i	$0.858198\pi$
$270$		0			0
$271$	−6122.73			−1.37243			−0.686217	−	0.727397i	$-0.740730\pi$
$271$	−6122.73			−1.37243			−0.686217	+	0.727397i	$0.740730\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3203.63	+	102.569i	0.702495	+	0.0224914i
$276$		0			0
$277$		−	8417.57i		−	1.82586i	−0.408117	−	0.912930i	$-0.633814\pi$
$277$		−	8417.57i		−	1.82586i	0.408117	−	0.912930i	$-0.366186\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3030.99			−0.643466			−0.321733	−	0.946830i	$-0.604265\pi$
$281$	−3030.99			−0.643466			−0.321733	+	0.946830i	$0.604265\pi$
$282$		0			0
$283$		−	2890.81i		−	0.607211i	−0.952798	−	0.303606i	$-0.901809\pi$
$283$		−	2890.81i		−	0.607211i	0.952798	−	0.303606i	$-0.0981905\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			3642.49i			0.749161i
$288$		0			0
$289$	−4262.55			−0.867606
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	8966.75i		−	1.78786i	−0.448206	−	0.893930i	$-0.647937\pi$
$293$		−	8966.75i		−	1.78786i	0.448206	−	0.893930i	$-0.352063\pi$
$294$		0			0
$295$	−39.9919	+	2498.86i	−0.00789295	+	0.493183i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4173.11			0.807147
$300$		0			0
$301$	−3787.93			−0.725358
$302$		0			0
$303$		0			0
$304$		0			0
$305$	86.8511	−	5426.80i	0.0163052	−	1.01881i
$306$		0			0
$307$		−	3298.42i		−	0.613194i	−0.951839	−	0.306597i	$-0.900810\pi$
$307$		−	3298.42i		−	0.613194i	0.951839	−	0.306597i	$-0.0991904\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3394.92			0.618998			0.309499	−	0.950900i	$-0.399839\pi$
$311$	3394.92			0.618998			0.309499	+	0.950900i	$0.399839\pi$
$312$		0			0
$313$			5946.95i			1.07393i	0.843603	+	0.536967i	$0.180430\pi$
$313$			5946.95i			1.07393i	−0.843603	+	0.536967i	$0.819570\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2520.83i		−	0.446637i	−0.974746	−	0.223318i	$-0.928311\pi$
$317$		−	2520.83i		−	0.446637i	0.974746	−	0.223318i	$-0.0716889\pi$
$318$		0			0
$319$	1397.62			0.245303
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4872.08i		−	0.839286i
$324$		0			0
$325$	−150.569	+	4702.86i	−0.0256986	+	0.802671i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−20939.2			−3.50886
$330$		0			0
$331$	−4586.86			−0.761682			−0.380841	−	0.924641i	$-0.624365\pi$
$331$	−4586.86			−0.761682			−0.380841	+	0.924641i	$0.624365\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9794.58	−	156.753i	−1.59742	−	0.0255652i
$336$		0			0
$337$		−	8582.12i		−	1.38723i	−0.720344	−	0.693617i	$-0.756016\pi$
$337$		−	8582.12i		−	1.38723i	0.720344	−	0.693617i	$-0.243984\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5088.45			−0.808080
$342$		0			0
$343$			19085.1i			3.00437i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	2539.38i		−	0.392857i	−0.980518	−	0.196428i	$-0.937066\pi$
$347$		−	2539.38i		−	0.392857i	0.980518	−	0.196428i	$-0.0629343\pi$
$348$		0			0
$349$	9002.82			1.38083			0.690415	−	0.723413i	$-0.257428\pi$
$349$	9002.82			1.38083			0.690415	+	0.723413i	$0.257428\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			3928.08i			0.592267i	0.955146	+	0.296134i	$0.0956974\pi$
$353$			3928.08i			0.592267i	−0.955146	+	0.296134i	$0.904303\pi$
$354$		0			0
$355$	−104.768	+	6546.34i	−0.0156634	+	0.978714i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10001.7			1.47039			0.735197	−	0.677854i	$-0.237090\pi$
$359$	10001.7			1.47039			0.735197	+	0.677854i	$0.237090\pi$
$360$		0			0
$361$	−4272.00			−0.622832
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12717.1	+	203.526i	1.82368	+	0.0291863i
$366$		0			0
$367$			5967.79i			0.848818i	0.905471	+	0.424409i	$0.139518\pi$
$367$			5967.79i			0.848818i	−0.905471	+	0.424409i	$0.860482\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−10719.9			−1.50014
$372$		0			0
$373$		−	6931.55i		−	0.962204i	−0.876665	−	0.481102i	$-0.840237\pi$
$373$		−	6931.55i		−	0.962204i	0.876665	−	0.481102i	$-0.159763\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2051.68i			0.280283i
$378$		0			0
$379$	−9711.70			−1.31624			−0.658122	−	0.752911i	$-0.728649\pi$
$379$	−9711.70			−1.31624			−0.658122	+	0.752911i	$0.728649\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			5664.84i			0.755769i	0.925853	+	0.377885i	$0.123348\pi$
$383$			5664.84i			0.755769i	−0.925853	+	0.377885i	$0.876652\pi$
$384$		0			0
$385$	10053.9	+	160.903i	1.33089	+	0.0212997i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8918.17			1.16239			0.581195	−	0.813765i	$-0.302586\pi$
$389$	8918.17			1.16239			0.581195	+	0.813765i	$0.302586\pi$
$390$		0			0
$391$	−10619.4			−1.37352
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−122.554	+	7657.66i	−0.0156110	+	0.975439i
$396$		0			0
$397$			12849.9i			1.62448i	0.583321	+	0.812242i	$0.301753\pi$
$397$			12849.9i			1.62448i	−0.583321	+	0.812242i	$0.698247\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3563.08			−0.443721			−0.221860	−	0.975078i	$-0.571213\pi$
$401$	−3563.08			−0.443721			−0.221860	+	0.975078i	$0.571213\pi$
$402$		0			0
$403$		−	7469.74i		−	0.923311i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	6845.06i		−	0.833654i
$408$		0			0
$409$	2026.69			0.245020			0.122510	−	0.992467i	$-0.460906\pi$
$409$	2026.69			0.245020			0.122510	+	0.992467i	$0.460906\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			7840.07i			0.934104i
$414$		0			0
$415$	3417.67	+	54.6968i	0.404258	+	0.00646978i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1670.39			−0.194759			−0.0973793	−	0.995247i	$-0.531046\pi$
$419$	−1670.39			−0.194759			−0.0973793	+	0.995247i	$0.531046\pi$
$420$		0			0
$421$	6079.66			0.703812			0.351906	−	0.936035i	$-0.385534\pi$
$421$	6079.66			0.703812			0.351906	+	0.936035i	$0.385534\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	383.156	−	11967.5i	0.0437313	−	1.36590i
$426$		0			0
$427$		−	17026.4i		−	1.92966i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1719.37			−0.192155			−0.0960777	−	0.995374i	$-0.530630\pi$
$431$	−1719.37			−0.192155			−0.0960777	+	0.995374i	$0.530630\pi$
$432$		0			0
$433$		−	12024.9i		−	1.33459i	−0.744792	−	0.667296i	$-0.767451\pi$
$433$		−	12024.9i		−	1.33459i	0.744792	−	0.667296i	$-0.232549\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	5638.75i		−	0.617249i
$438$		0			0
$439$	7542.97			0.820060			0.410030	−	0.912072i	$-0.365518\pi$
$439$	7542.97			0.820060			0.410030	+	0.912072i	$0.365518\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	4578.13i		−	0.491001i	−0.969396	−	0.245501i	$-0.921048\pi$
$443$		−	4578.13i		−	0.491001i	0.969396	−	0.245501i	$-0.0789524\pi$
$444$		0			0
$445$	158.723	−	9917.65i	0.0169083	−	1.05650i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6875.21			0.722630			0.361315	−	0.932444i	$-0.382328\pi$
$449$	6875.21			0.722630			0.361315	+	0.932444i	$0.382328\pi$
$450$		0			0
$451$	2663.02			0.278042
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−236.202	+	14758.9i	−0.0243370	+	1.52067i
$456$		0			0
$457$		−	7351.97i		−	0.752540i	−0.926510	−	0.376270i	$-0.877207\pi$
$457$		−	7351.97i		−	0.752540i	0.926510	−	0.376270i	$-0.122793\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15614.5			1.57752			0.788762	−	0.614698i	$-0.210722\pi$
$461$	15614.5			1.57752			0.788762	+	0.614698i	$0.210722\pi$
$462$		0			0
$463$			1684.73i			0.169106i	0.996419	+	0.0845530i	$0.0269462\pi$
$463$			1684.73i			0.169106i	−0.996419	+	0.0845530i	$0.973054\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	10235.2i		−	1.01420i	−0.861888	−	0.507099i	$-0.830718\pi$
$467$		−	10235.2i		−	1.01420i	0.861888	−	0.507099i	$-0.169282\pi$
$468$		0			0
$469$	−30730.2			−3.02556
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2769.36i			0.269207i
$474$		0			0
$475$	6354.56	+	203.450i	0.613826	+	0.0196525i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12247.2			1.16825			0.584123	−	0.811665i	$-0.301439\pi$
$479$	12247.2			1.16825			0.584123	+	0.811665i	$0.301439\pi$
$480$		0			0
$481$	10048.4			0.952532
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6221.62	−	99.5714i	−0.582493	−	0.00932228i
$486$		0			0
$487$		−	7895.91i		−	0.734698i	−0.930083	−	0.367349i	$-0.880265\pi$
$487$		−	7895.91i		−	0.734698i	0.930083	−	0.367349i	$-0.119735\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7625.48			0.700882			0.350441	−	0.936585i	$-0.386032\pi$
$491$	7625.48			0.700882			0.350441	+	0.936585i	$0.386032\pi$
$492$		0			0
$493$		−	5220.96i		−	0.476958i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			20538.9i			1.85371i
$498$		0			0
$499$	8655.23			0.776476			0.388238	−	0.921559i	$-0.373084\pi$
$499$	8655.23			0.776476			0.388238	+	0.921559i	$0.373084\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			118.441i			0.0104990i	0.999986	+	0.00524951i	$0.00167098\pi$
$503$			118.441i			0.0104990i	−0.999986	+	0.00524951i	$0.998329\pi$
$504$		0			0
$505$	−284.660	+	17786.7i	−0.0250835	+	1.56732i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5359.43			0.466704			0.233352	−	0.972392i	$-0.425030\pi$
$509$	5359.43			0.466704			0.233352	+	0.972392i	$0.425030\pi$
$510$		0			0
$511$	39899.5			3.45411
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−15102.1	−	241.695i	−1.29219	−	0.0206803i
$516$		0			0
$517$			15308.6i			1.30227i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10862.4			0.913414			0.456707	−	0.889617i	$-0.349029\pi$
$521$	10862.4			0.913414			0.456707	+	0.889617i	$0.349029\pi$
$522$		0			0
$523$			9553.39i			0.798740i	0.916790	+	0.399370i	$0.130771\pi$
$523$			9553.39i			0.798740i	−0.916790	+	0.399370i	$0.869229\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			19008.5i			1.57120i
$528$		0			0
$529$	−123.501			−0.0101505
$530$		0			0
$531$		0			0
$532$		0			0
$533$			3909.26i			0.317690i
$534$		0			0
$535$	−14907.1	−	238.575i	−1.20466	−	0.0192795i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	22748.4			1.81789
$540$		0			0
$541$	−6132.47			−0.487348			−0.243674	−	0.969857i	$-0.578353\pi$
$541$	−6132.47			−0.487348			−0.243674	+	0.969857i	$0.578353\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	109.118	−	6818.12i	0.00857633	−	0.535883i
$546$		0			0
$547$		−	2853.85i		−	0.223075i	−0.993760	−	0.111537i	$-0.964422\pi$
$547$		−	2853.85i		−	0.223075i	0.993760	−	0.111537i	$-0.0355775\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2772.25			0.214341
$552$		0			0
$553$			24025.6i			1.84751i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	5192.60i		−	0.395005i	−0.980302	−	0.197502i	$-0.936717\pi$
$557$		−	5192.60i		−	0.395005i	0.980302	−	0.197502i	$-0.0632830\pi$
$558$		0			0
$559$	−4065.36			−0.307596
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	10907.2i		−	0.816492i	−0.912872	−	0.408246i	$-0.866141\pi$
$563$		−	10907.2i		−	0.816492i	0.912872	−	0.408246i	$-0.133859\pi$
$564$		0			0
$565$	2703.15	+	43.2615i	0.201278	+	0.00322128i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−155.257			−0.0114389			−0.00571945	−	0.999984i	$-0.501821\pi$
$569$	−155.257			−0.0114389			−0.00571945	+	0.999984i	$0.501821\pi$
$570$		0			0
$571$	−4925.15			−0.360965			−0.180483	−	0.983578i	$-0.557766\pi$
$571$	−4925.15			−0.360965			−0.180483	+	0.983578i	$0.557766\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	443.450	−	13850.7i	0.0321620	−	1.00455i
$576$		0			0
$577$			5292.05i			0.381822i	0.981607	+	0.190911i	$0.0611441\pi$
$577$			5292.05i			0.381822i	−0.981607	+	0.190911i	$0.938856\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10722.8			0.765677
$582$		0			0
$583$			7837.33i			0.556757i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	19658.5i		−	1.38227i	−0.722727	−	0.691134i	$-0.757112\pi$
$587$		−	19658.5i		−	1.38227i	0.722727	−	0.691134i	$-0.242888\pi$
$588$		0			0
$589$	−10093.2			−0.706083
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6578.08i		−	0.455530i	−0.973716	−	0.227765i	$-0.926858\pi$
$593$		−	6578.08i		−	0.455530i	0.973716	−	0.227765i	$-0.0731418\pi$
$594$		0			0
$595$	601.070	−	37557.3i	0.0414142	−	2.58773i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16915.9			1.15386			0.576931	−	0.816793i	$-0.304250\pi$
$599$	16915.9			1.15386			0.576931	+	0.816793i	$0.304250\pi$
$600$		0			0
$601$	19801.4			1.34395			0.671977	−	0.740572i	$-0.265445\pi$
$601$	19801.4			1.34395			0.671977	+	0.740572i	$0.265445\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−120.491	+	7528.75i	−0.00809695	+	0.505930i
$606$		0			0
$607$			22498.4i			1.50442i	0.658926	+	0.752208i	$0.271011\pi$
$607$			22498.4i			1.50442i	−0.658926	+	0.752208i	$0.728989\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−22472.7			−1.48797
$612$		0			0
$613$		−	23829.3i		−	1.57008i	−0.619446	−	0.785039i	$-0.712643\pi$
$613$		−	23829.3i		−	1.57008i	0.619446	−	0.785039i	$-0.287357\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	4277.52i		−	0.279103i	−0.990215	−	0.139551i	$-0.955434\pi$
$617$		−	4277.52i		−	0.279103i	0.990215	−	0.139551i	$-0.0445660\pi$
$618$		0			0
$619$	−4995.99			−0.324404			−0.162202	−	0.986758i	$-0.551860\pi$
$619$	−4995.99			−0.324404			−0.162202	+	0.986758i	$0.551860\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	31116.3i		−	2.00104i
$624$		0			0
$625$	15593.0	+	999.488i	0.997952	+	0.0639672i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−25570.5			−1.62092
$630$		0			0
$631$	−11328.0			−0.714675			−0.357337	−	0.933975i	$-0.616315\pi$
$631$	−11328.0			−0.714675			−0.357337	+	0.933975i	$0.616315\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11692.4	−	187.127i	−0.730708	−	0.0116943i
$636$		0			0
$637$			33394.1i			2.07712i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5955.35			−0.366961			−0.183481	−	0.983023i	$-0.558736\pi$
$641$	−5955.35			−0.366961			−0.183481	+	0.983023i	$0.558736\pi$
$642$		0			0
$643$		−	15727.7i		−	0.964605i	−0.876005	−	0.482302i	$-0.839801\pi$
$643$		−	15727.7i		−	0.964605i	0.876005	−	0.482302i	$-0.160199\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			9582.24i			0.582252i	0.956685	+	0.291126i	$0.0940298\pi$
$647$			9582.24i			0.582252i	−0.956685	+	0.291126i	$0.905970\pi$
$648$		0			0
$649$	5731.87			0.346681
$650$		0			0
$651$		0			0
$652$		0			0
$653$			8313.70i			0.498224i	0.968475	+	0.249112i	$0.0801388\pi$
$653$			8313.70i			0.498224i	−0.968475	+	0.249112i	$0.919861\pi$
$654$		0			0
$655$	158.606	−	9910.37i	0.00946148	−	0.591191i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15095.5			0.892317			0.446158	−	0.894954i	$-0.352792\pi$
$659$	15095.5			0.892317			0.446158	+	0.894954i	$0.352792\pi$
$660$		0			0
$661$	8266.98			0.486457			0.243229	−	0.969969i	$-0.421793\pi$
$661$	8266.98			0.486457			0.243229	+	0.969969i	$0.421793\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	19942.3	+	319.159i	1.16290	+	0.0186112i
$666$		0			0
$667$		−	6042.53i		−	0.350776i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12448.0			−0.716170
$672$		0			0
$673$		−	11186.7i		−	0.640735i	−0.947293	−	0.320367i	$-0.896194\pi$
$673$		−	11186.7i		−	0.640735i	0.947293	−	0.320367i	$-0.103806\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	7675.84i		−	0.435756i	−0.975976	−	0.217878i	$-0.930087\pi$
$677$		−	7675.84i		−	0.435756i	0.975976	−	0.217878i	$-0.0699134\pi$
$678$		0			0
$679$	−19520.1			−1.10326
$680$		0			0
$681$		0			0
$682$		0			0
$683$			10550.9i			0.591099i	0.955327	+	0.295549i	$0.0955027\pi$
$683$			10550.9i			0.591099i	−0.955327	+	0.295549i	$0.904497\pi$
$684$		0			0
$685$	1796.71	+	28.7547i	0.100217	+	0.00160388i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−11505.0			−0.636150
$690$		0			0
$691$	−26950.9			−1.48373			−0.741867	−	0.670547i	$-0.766060\pi$
$691$	−26950.9			−1.48373			−0.741867	+	0.670547i	$0.766060\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.2453	−	640.166i	0.000559173	−	0.0349394i
$696$		0			0
$697$		−	9947.99i		−	0.540613i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−12791.6			−0.689204			−0.344602	−	0.938749i	$-0.611986\pi$
$701$	−12791.6			−0.689204			−0.344602	+	0.938749i	$0.611986\pi$
$702$		0			0
$703$		−	13577.5i		−	0.728429i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			55805.1i			2.96855i
$708$		0			0
$709$	16238.3			0.860142			0.430071	−	0.902795i	$-0.358489\pi$
$709$	16238.3			0.860142			0.430071	+	0.902795i	$0.358489\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			21999.6i			1.15553i
$714$		0			0
$715$	10790.2	+	172.687i	0.564378	+	0.00903236i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24285.8			−1.25968			−0.629839	−	0.776726i	$-0.716879\pi$
$719$	−24285.8			−1.25968			−0.629839	+	0.776726i	$0.716879\pi$
$720$		0			0
$721$	−47382.3			−2.44745
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6809.60	+	218.019i	0.348831	+	0.0111683i
$726$		0			0
$727$			4466.82i			0.227875i	0.993488	+	0.113937i	$0.0363464\pi$
$727$			4466.82i			0.227875i	−0.993488	+	0.113937i	$0.963654\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10345.2			0.523436
$732$		0			0
$733$			30802.1i			1.55212i	0.630661	+	0.776059i	$0.282784\pi$
$733$			30802.1i			1.55212i	−0.630661	+	0.776059i	$0.717216\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			22466.8i			1.12290i
$738$		0			0
$739$	12920.5			0.643151			0.321576	−	0.946884i	$-0.395788\pi$
$739$	12920.5			0.643151			0.321576	+	0.946884i	$0.395788\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			2571.29i			0.126960i	0.997983	+	0.0634802i	$0.0202200\pi$
$743$			2571.29i			0.126960i	−0.997983	+	0.0634802i	$0.979780\pi$
$744$		0			0
$745$	−197.720	+	12354.4i	−0.00972337	+	0.607555i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−46770.7			−2.28166
$750$		0			0
$751$	−13427.4			−0.652426			−0.326213	−	0.945296i	$-0.605773\pi$
$751$	−13427.4			−0.652426			−0.326213	+	0.945296i	$0.605773\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	409.635	−	25595.6i	0.0197459	−	1.23380i
$756$		0			0
$757$			7103.66i			0.341066i	0.985352	+	0.170533i	$0.0545489\pi$
$757$			7103.66i			0.341066i	−0.985352	+	0.170533i	$0.945451\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19205.2			−0.914831			−0.457416	−	0.889253i	$-0.651225\pi$
$761$	−19205.2			−0.914831			−0.457416	+	0.889253i	$0.651225\pi$
$762$		0			0
$763$		−	21391.6i		−	1.01498i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			8414.27i			0.396117i
$768$		0			0
$769$	−19508.1			−0.914799			−0.457399	−	0.889261i	$-0.651219\pi$
$769$	−19508.1			−0.914799			−0.457399	+	0.889261i	$0.651219\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			38852.5i			1.80780i	0.427746	+	0.903899i	$0.359308\pi$
$773$			38852.5i			1.80780i	−0.427746	+	0.903899i	$0.640692\pi$
$774$		0			0
$775$	−24792.4	−	793.763i	−1.14912	−	0.0367907i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5282.23			0.242947
$780$		0			0
$781$	15016.0			0.687982
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1805.55	−	28.8962i	−0.0820929	−	0.00131382i
$786$		0			0
$787$			24851.1i			1.12560i	0.826594	+	0.562799i	$0.190275\pi$
$787$			24851.1i			1.12560i	−0.826594	+	0.562799i	$0.809725\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	8481.04			0.381228
$792$		0			0
$793$		−	18273.4i		−	0.818295i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	33040.1i		−	1.46843i	−0.678916	−

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:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	360.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.178908 - 11.1789i) q^{5} -35.0735i q^{7} -25.6422 q^{11} -37.6422i q^{13} +95.7891i q^{17} -50.8625 q^{19} +110.863i q^{23} +(-124.936 - 4.00000i) q^{25} -54.5047 q^{29} +198.441 q^{31} +(-392.083 - 6.27493i) q^{35} +266.945i q^{37} -103.853 q^{41} -108.000i q^{43} -597.009i q^{47} -887.147 q^{49} -305.642i q^{53} +(-4.58760 + 286.652i) q^{55} -223.533 q^{59} +485.450 q^{61} +(-420.799 - 6.73450i) q^{65} -876.166i q^{67} -585.597 q^{71} +1137.60i q^{73} +899.360i q^{77} -685.009 q^{79} +305.725i q^{83} +(1070.82 + 17.1375i) q^{85} +887.175 q^{89} -1320.24 q^{91} +(-9.09973 + 568.588i) q^{95} -556.550i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 22 q^{5} - 148 q^{11} + 160 q^{19} + 100 q^{29} - 24 q^{31} - 796 q^{35} - 688 q^{41} - 3276 q^{49} + 1072 q^{55} + 1332 q^{59} + 488 q^{61} - 820 q^{65} - 616 q^{71} - 2104 q^{79} + 2148 q^{85} + 1368 q^{89}+ \cdots - 2944 q^{95}+O(q^{100}) \) 4 * q - 22 * q^5 - 148 * q^11 + 160 * q^19 + 100 * q^29 - 24 * q^31 - 796 * q^35 - 688 * q^41 - 3276 * q^49 + 1072 * q^55 + 1332 * q^59 + 488 * q^61 - 820 * q^65 - 616 * q^71 - 2104 * q^79 + 2148 * q^85 + 1368 * q^89 + 1352 * q^91 - 2944 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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