LMFDB - Embedded newform 6300.2.ch.f.4301.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(1601,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 0, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6300.1601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.ch (of order $6$, degree $2$, 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:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4301.4
Character	$\chi$	$=$	6300.4301
Dual form			6300.2.ch.f.1601.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.43620 + 2.22201i) q^{7} +O(q^{10})$	$q+(-1.43620 + 2.22201i) q^{7} +(-0.601378 - 0.347206i) q^{11} -3.76549i q^{13} +(-0.0200491 + 0.0347261i) q^{17} +(0.107408 - 0.0620123i) q^{19} +(3.35023 - 1.93426i) q^{23} +4.39545i q^{29} +(-2.60805 - 1.50576i) q^{31} +(5.56393 + 9.63702i) q^{37} -7.91118 q^{41} +3.73746 q^{43} +(-3.29812 - 5.71251i) q^{47} +(-2.87464 - 6.38251i) q^{49} +(7.75992 + 4.48019i) q^{53} +(2.90185 - 5.02615i) q^{59} +(2.65084 - 1.53047i) q^{61} +(-2.93799 + 5.08874i) q^{67} -7.19849i q^{71} +(-7.38874 - 4.26589i) q^{73} +(1.63520 - 0.837609i) q^{77} +(3.98269 + 6.89822i) q^{79} +6.32489 q^{83} +(-5.00494 - 8.66880i) q^{89} +(8.36695 + 5.40801i) q^{91} +11.5960i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 24 q^{19} + 24 q^{31} - 40 q^{49} - 24 q^{61} - 32 q^{79} - 56 q^{91}+O(q^{100}) $ 32 * q + 24 * q^19 + 24 * q^31 - 40 * q^49 - 24 * q^61 - 32 * q^79 - 56 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.43620	+	2.22201i	−0.542834	+	0.839840i
$7$	−1.43620	+	2.22201i	−0.542834	+	0.839840i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.601378	−	0.347206i	−0.181322	−	0.104687i	0.406591	−	0.913610i	$-0.366717\pi$
$11$	−0.601378	−	0.347206i	−0.181322	−	0.104687i	−0.587914	+	0.808924i	$0.700051\pi$
$12$		0			0
$13$		−	3.76549i		−	1.04436i	−0.852836	−	0.522179i	$-0.825119\pi$
$13$		−	3.76549i		−	1.04436i	0.852836	−	0.522179i	$-0.174881\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.0200491	+	0.0347261i	−0.00486263	+	0.00842231i	−0.868446	−	0.495783i	$-0.834881\pi$
$17$	−0.0200491	+	0.0347261i	−0.00486263	+	0.00842231i	0.863584	+	0.504205i	$0.168214\pi$
$18$		0			0
$19$	0.107408	−	0.0620123i	0.0246412	−	0.0142266i	−0.487629	−	0.873051i	$-0.662138\pi$
$19$	0.107408	−	0.0620123i	0.0246412	−	0.0142266i	0.512270	+	0.858824i	$0.328805\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.35023	−	1.93426i	0.698571	−	0.403320i	−0.108244	−	0.994124i	$-0.534523\pi$
$23$	3.35023	−	1.93426i	0.698571	−	0.403320i	0.806815	+	0.590804i	$0.201189\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.39545i			0.816214i	0.912934	+	0.408107i	$0.133811\pi$
$29$			4.39545i			0.816214i	−0.912934	+	0.408107i	$0.866189\pi$
$30$		0			0
$31$	−2.60805	−	1.50576i	−0.468420	−	0.270442i	0.247158	−	0.968975i	$-0.420503\pi$
$31$	−2.60805	−	1.50576i	−0.468420	−	0.270442i	−0.715578	+	0.698533i	$0.753837\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.56393	+	9.63702i	0.914705	+	1.58432i	0.807332	+	0.590097i	$0.200911\pi$
$37$	5.56393	+	9.63702i	0.914705	+	1.58432i	0.107373	+	0.994219i	$0.465756\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.91118			−1.23552			−0.617759	−	0.786367i	$-0.711959\pi$
$41$	−7.91118			−1.23552			−0.617759	+	0.786367i	$0.711959\pi$
$42$		0			0
$43$	3.73746			0.569957			0.284979	−	0.958534i	$-0.408014\pi$
$43$	3.73746			0.569957			0.284979	+	0.958534i	$0.408014\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.29812	−	5.71251i	−0.481080	−	0.833255i	0.518684	−	0.854966i	$-0.326422\pi$
$47$	−3.29812	−	5.71251i	−0.481080	−	0.833255i	−0.999764	+	0.0217106i	$0.993089\pi$
$48$		0			0
$49$	−2.87464	−	6.38251i	−0.410663	−	0.911787i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.75992	+	4.48019i	1.06591	+	0.615402i	0.927061	−	0.374912i	$-0.122327\pi$
$53$	7.75992	+	4.48019i	1.06591	+	0.615402i	0.138847	+	0.990314i	$0.455660\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.90185	−	5.02615i	0.377789	−	0.654349i	−0.612952	−	0.790121i	$-0.710018\pi$
$59$	2.90185	−	5.02615i	0.377789	−	0.654349i	0.990740	+	0.135771i	$0.0433513\pi$
$60$		0			0
$61$	2.65084	−	1.53047i	0.339406	−	0.195956i	−0.320603	−	0.947214i	$-0.603886\pi$
$61$	2.65084	−	1.53047i	0.339406	−	0.195956i	0.660009	+	0.751257i	$0.270552\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.93799	+	5.08874i	−0.358932	+	0.621689i	−0.987783	−	0.155838i	$-0.950192\pi$
$67$	−2.93799	+	5.08874i	−0.358932	+	0.621689i	0.628851	+	0.777526i	$0.283526\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.19849i		−	0.854303i	−0.904180	−	0.427152i	$-0.859517\pi$
$71$		−	7.19849i		−	0.854303i	0.904180	−	0.427152i	$-0.140483\pi$
$72$		0			0
$73$	−7.38874	−	4.26589i	−0.864787	−	0.499285i	0.000825643	−	1.00000i	$-0.499737\pi$
$73$	−7.38874	−	4.26589i	−0.864787	−	0.499285i	−0.865612	+	0.500715i	$0.833071\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.63520	−	0.837609i	0.186348	−	0.0954544i
$78$		0			0
$79$	3.98269	+	6.89822i	0.448088	+	0.776110i	0.998262	−	0.0589396i	$-0.0187719\pi$
$79$	3.98269	+	6.89822i	0.448088	+	0.776110i	−0.550174	+	0.835050i	$0.685439\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.32489			0.694246			0.347123	−	0.937820i	$-0.387159\pi$
$83$	6.32489			0.694246			0.347123	+	0.937820i	$0.387159\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.00494	−	8.66880i	−0.530522	−	0.918891i	−0.999366	−	0.0356101i	$-0.988663\pi$
$89$	−5.00494	−	8.66880i	−0.530522	−	0.918891i	0.468844	−	0.883281i	$-0.344671\pi$
$90$		0			0
$91$	8.36695	+	5.40801i	0.877094	+	0.566913i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			11.5960i			1.17740i	0.808353	+	0.588698i	$0.200359\pi$
$97$			11.5960i			1.17740i	−0.808353	+	0.588698i	$0.799641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.67710	−	2.90483i	0.166878	−	0.289041i	−0.770442	−	0.637509i	$-0.779965\pi$
$101$	1.67710	−	2.90483i	0.166878	−	0.289041i	0.937321	+	0.348468i	$0.113298\pi$
$102$		0			0
$103$	9.53305	−	5.50391i	0.939320	−	0.542316i	0.0495727	−	0.998771i	$-0.484214\pi$
$103$	9.53305	−	5.50391i	0.939320	−	0.542316i	0.889747	+	0.456454i	$0.150881\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.7697	−	6.79524i	1.13782	−	0.656920i	0.191929	−	0.981409i	$-0.438526\pi$
$107$	11.7697	−	6.79524i	1.13782	−	0.656920i	0.945890	+	0.324489i	$0.105192\pi$
$108$		0			0
$109$	3.47371	−	6.01665i	0.332721	−	0.576290i	−0.650323	−	0.759658i	$-0.725366\pi$
$109$	3.47371	−	6.01665i	0.332721	−	0.576290i	0.983044	+	0.183367i	$0.0586998\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	13.5489i		−	1.27457i	−0.770628	−	0.637285i	$-0.780057\pi$
$113$		−	13.5489i		−	1.27457i	0.770628	−	0.637285i	$-0.219943\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.0483670	−	0.0944230i	−0.00443380	−	0.00865575i
$120$		0			0
$121$	−5.25890	−	9.10868i	−0.478081	−	0.828061i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	4.96761			0.440804			0.220402	−	0.975409i	$-0.429263\pi$
$127$	4.96761			0.440804			0.220402	+	0.975409i	$0.429263\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.74775	+	11.6874i	0.589554	+	1.02114i	0.994291	+	0.106704i	$0.0340297\pi$
$131$	6.74775	+	11.6874i	0.589554	+	1.02114i	−0.404737	+	0.914433i	$0.632637\pi$
$132$		0			0
$133$	−0.0164686	+	0.327725i	−0.00142801	+	0.0284173i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.3748	+	5.98987i	0.886375	+	0.511749i	0.872755	−	0.488158i	$-0.162331\pi$
$137$	10.3748	+	5.98987i	0.886375	+	0.511749i	0.0136198	+	0.999907i	$0.495665\pi$
$138$		0			0
$139$		−	1.63912i		−	0.139029i	−0.997581	−	0.0695143i	$-0.977855\pi$
$139$		−	1.63912i		−	0.139029i	0.997581	−	0.0695143i	$-0.0221449\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.30740	+	2.26448i	−0.109330	+	0.189366i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.96150	+	5.75128i	−0.816078	+	0.471163i	−0.849062	−	0.528293i	$-0.822832\pi$
$149$	−9.96150	+	5.75128i	−0.816078	+	0.471163i	0.0329840	+	0.999456i	$0.489499\pi$
$150$		0			0
$151$	−7.24992	+	12.5572i	−0.589990	+	1.02189i	0.404243	+	0.914652i	$0.367535\pi$
$151$	−7.24992	+	12.5572i	−0.589990	+	1.02189i	−0.994233	+	0.107241i	$0.965798\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	9.25395	+	5.34277i	0.738546	+	0.426400i	0.821540	−	0.570150i	$-0.193115\pi$
$157$	9.25395	+	5.34277i	0.738546	+	0.426400i	−0.0829945	+	0.996550i	$0.526448\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.513680	+	10.2222i	−0.0404836	+	0.805624i
$162$		0			0
$163$	6.01293	+	10.4147i	0.470969	+	0.815742i	0.999449	−	0.0332039i	$-0.0105711\pi$
$163$	6.01293	+	10.4147i	0.470969	+	0.815742i	−0.528480	+	0.848946i	$0.677238\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	25.1151			1.94347			0.971733	−	0.236081i	$-0.0758629\pi$
$167$	25.1151			1.94347			0.971733	+	0.236081i	$0.0758629\pi$
$168$		0			0
$169$	−1.17891			−0.0906854
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.48559	+	7.76926i	0.341033	+	0.590686i	0.984625	−	0.174683i	$-0.0558900\pi$
$173$	4.48559	+	7.76926i	0.341033	+	0.590686i	−0.643592	+	0.765369i	$0.722557\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21.7179	+	12.5388i	1.62327	+	0.937198i	0.986036	+	0.166530i	$0.0532562\pi$
$179$	21.7179	+	12.5388i	1.62327	+	0.937198i	0.637237	+	0.770668i	$0.280077\pi$
$180$		0			0
$181$			4.30930i			0.320308i	0.987092	+	0.160154i	$0.0511991\pi$
$181$			4.30930i			0.320308i	−0.987092	+	0.160154i	$0.948801\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.0241142	−	0.0139223i	0.00176341	−	0.00101810i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.9583	−	6.32679i	0.792916	−	0.457790i	−0.0480723	−	0.998844i	$-0.515308\pi$
$191$	10.9583	−	6.32679i	0.792916	−	0.457790i	0.840988	+	0.541054i	$0.181974\pi$
$192$		0			0
$193$	−8.48545	+	14.6972i	−0.610796	+	1.05793i	0.380310	+	0.924859i	$0.375817\pi$
$193$	−8.48545	+	14.6972i	−0.610796	+	1.05793i	−0.991106	+	0.133071i	$0.957516\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	15.0304i		−	1.07087i	−0.844576	−	0.535436i	$-0.820147\pi$
$197$		−	15.0304i		−	1.07087i	0.844576	−	0.535436i	$-0.179853\pi$
$198$		0			0
$199$	14.4468	+	8.34085i	1.02411	+	0.591267i	0.915290	−	0.402795i	$-0.131961\pi$
$199$	14.4468	+	8.34085i	1.02411	+	0.591267i	0.108815	+	0.994062i	$0.465294\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.76672	−	6.31276i	−0.685489	−	0.443069i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.0861242			−0.00595733
$210$		0			0
$211$	−15.4819			−1.06582			−0.532908	−	0.846173i	$-0.678901\pi$
$211$	−15.4819			−1.06582			−0.532908	+	0.846173i	$0.678901\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	7.09150	−	3.63253i	0.481402	−	0.246592i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.130761	+	0.0754947i	0.00879592	+	0.00507833i
$222$		0			0
$223$			25.1597i			1.68482i	0.538841	+	0.842408i	$0.318862\pi$
$223$			25.1597i			1.68482i	−0.538841	+	0.842408i	$0.681138\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.7885	−	23.8824i	0.915175	−	1.58513i	0.108530	−	0.994093i	$-0.465386\pi$
$227$	13.7885	−	23.8824i	0.915175	−	1.58513i	0.806645	−	0.591036i	$-0.201281\pi$
$228$		0			0
$229$	−19.8150	+	11.4402i	−1.30941	+	0.755990i	−0.981998	−	0.188891i	$-0.939511\pi$
$229$	−19.8150	+	11.4402i	−1.30941	+	0.755990i	−0.327415	+	0.944881i	$0.606178\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.37912	−	4.83769i	0.548935	−	0.316928i	−0.199758	−	0.979845i	$-0.564015\pi$
$233$	8.37912	−	4.83769i	0.548935	−	0.316928i	0.748692	+	0.662918i	$0.230682\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4.30863i		−	0.278702i	−0.990243	−	0.139351i	$-0.955498\pi$
$239$		−	4.30863i		−	0.278702i	0.990243	−	0.139351i	$-0.0445016\pi$
$240$		0			0
$241$	5.56526	+	3.21310i	0.358490	+	0.206974i	0.668418	−	0.743786i	$-0.266972\pi$
$241$	5.56526	+	3.21310i	0.358490	+	0.206974i	−0.309928	+	0.950760i	$0.600305\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.233507	−	0.404445i	−0.0148577	−	0.0257342i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.89601			0.624631			0.312315	−	0.949978i	$-0.398895\pi$
$251$	9.89601			0.624631			0.312315	+	0.949978i	$0.398895\pi$
$252$		0			0
$253$	−2.68634			−0.168889
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.7617	+	22.1039i	0.796051	+	1.37880i	0.922170	+	0.386786i	$0.126415\pi$
$257$	12.7617	+	22.1039i	0.796051	+	1.37880i	−0.126119	+	0.992015i	$0.540252\pi$
$258$		0			0
$259$	−29.4045	−	1.47761i	−1.82710	−	0.0918144i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.24202	−	1.29443i	−0.138249	−	0.0798182i	0.429280	−	0.903171i	$-0.358767\pi$
$263$	−2.24202	−	1.29443i	−0.138249	−	0.0798182i	−0.567529	+	0.823353i	$0.692101\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.89054	+	6.73861i	−0.237210	+	0.410860i	−0.959913	−	0.280299i	$-0.909566\pi$
$269$	−3.89054	+	6.73861i	−0.237210	+	0.410860i	0.722702	+	0.691159i	$0.242900\pi$
$270$		0			0
$271$	25.5035	−	14.7245i	1.54923	−	0.894448i	0.551029	−	0.834486i	$-0.314235\pi$
$271$	25.5035	−	14.7245i	1.54923	−	0.894448i	0.998201	−	0.0599626i	$-0.0190981\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.739081	−	1.28012i	0.0444070	−	0.0769153i	−0.842968	−	0.537964i	$-0.819193\pi$
$277$	0.739081	−	1.28012i	0.0444070	−	0.0769153i	0.887375	+	0.461049i	$0.152527\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			4.00483i			0.238908i	0.992840	+	0.119454i	$0.0381145\pi$
$281$			4.00483i			0.238908i	−0.992840	+	0.119454i	$0.961886\pi$
$282$		0			0
$283$	16.8194	+	9.71071i	0.999812	+	0.577242i	0.908193	−	0.418552i	$-0.137462\pi$
$283$	16.8194	+	9.71071i	0.999812	+	0.577242i	0.0916195	+	0.995794i	$0.470796\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	11.3621	−	17.5787i	0.670681	−	1.03764i
$288$		0			0
$289$	8.49920	+	14.7210i	0.499953	+	0.865943i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.39101			−0.139684			−0.0698421	−	0.997558i	$-0.522250\pi$
$293$	−2.39101			−0.139684			−0.0698421	+	0.997558i	$0.522250\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.28342	−	12.6153i	−0.421211	−	0.729559i
$300$		0			0
$301$	−5.36775	+	8.30466i	−0.309392	+	0.478673i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	1.67320i		−	0.0954947i	−0.998859	−	0.0477474i	$-0.984796\pi$
$307$		−	1.67320i		−	0.0954947i	0.998859	−	0.0477474i	$-0.0152042\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.00471	−	1.74021i	0.0569719	−	0.0986783i	−0.836133	−	0.548527i	$-0.815189\pi$
$311$	1.00471	−	1.74021i	0.0569719	−	0.0986783i	0.893105	+	0.449849i	$0.148522\pi$
$312$		0			0
$313$	−18.8564	+	10.8867i	−1.06583	+	0.615355i	−0.927038	−	0.374967i	$-0.877654\pi$
$313$	−18.8564	+	10.8867i	−1.06583	+	0.615355i	−0.138788	+	0.990322i	$0.544321\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.6685	−	8.46884i	0.823863	−	0.475657i	−0.0278839	−	0.999611i	$-0.508877\pi$
$317$	14.6685	−	8.46884i	0.823863	−	0.475657i	0.851747	+	0.523954i	$0.175544\pi$
$318$		0			0
$319$	1.52613	−	2.64333i	0.0854466	−	0.147998i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			0.00497317i			0.000276715i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	17.4300	+	0.875881i	0.960948	+	0.0482889i
$330$		0			0
$331$	−13.4474	−	23.2916i	−0.739137	−	1.28022i	−0.952884	−	0.303334i	$-0.901900\pi$
$331$	−13.4474	−	23.2916i	−0.739137	−	1.28022i	0.213747	−	0.976889i	$-0.431433\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	16.5625			0.902216			0.451108	−	0.892469i	$-0.351029\pi$
$337$	16.5625			0.902216			0.451108	+	0.892469i	$0.351029\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.04562	+	1.81106i	0.0566233	+	0.0980745i
$342$		0			0
$343$	18.3106	+	2.77912i	0.988677	+	0.150058i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	27.5258	+	15.8920i	1.47766	+	0.853130i	0.999681	−	0.0252399i	$-0.00803498\pi$
$347$	27.5258	+	15.8920i	1.47766	+	0.853130i	0.477982	+	0.878369i	$0.341368\pi$
$348$		0			0
$349$		−	27.5288i		−	1.47358i	−0.676119	−	0.736792i	$-0.736340\pi$
$349$		−	27.5288i		−	1.47358i	0.676119	−	0.736792i	$-0.263660\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.531900	+	0.921277i	−0.0283102	+	0.0490347i	−0.879833	−	0.475282i	$-0.842346\pi$
$353$	−0.531900	+	0.921277i	−0.0283102	+	0.0490347i	0.851523	+	0.524317i	$0.175679\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.1737	+	12.2246i	−1.11750	+	0.645192i	−0.940763	−	0.339066i	$-0.889889\pi$
$359$	−21.1737	+	12.2246i	−1.11750	+	0.645192i	−0.176742	+	0.984257i	$0.556556\pi$
$360$		0			0
$361$	−9.49231	+	16.4412i	−0.499595	+	0.865324i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	5.94383	+	3.43167i	0.310266	+	0.179132i	0.647045	−	0.762452i	$-0.276004\pi$
$367$	5.94383	+	3.43167i	0.310266	+	0.179132i	−0.336780	+	0.941583i	$0.609338\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−21.0999	+	10.8081i	−1.09545	+	0.561131i
$372$		0			0
$373$	−7.13877	−	12.3647i	−0.369631	−	0.640220i	0.619877	−	0.784699i	$-0.287183\pi$
$373$	−7.13877	−	12.3647i	−0.369631	−	0.640220i	−0.989508	+	0.144479i	$0.953849\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	16.5510			0.852421
$378$		0			0
$379$	35.6634			1.83190			0.915952	−	0.401288i	$-0.131437\pi$
$379$	35.6634			1.83190			0.915952	+	0.401288i	$0.131437\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.9004	−	24.0762i	−0.710277	−	1.23024i	−0.964753	−	0.263157i	$-0.915236\pi$
$383$	−13.9004	−	24.0762i	−0.710277	−	1.23024i	0.254476	−	0.967079i	$-0.418097\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.3179	+	7.68908i	0.675243	+	0.389852i	0.798060	−	0.602578i	$-0.205860\pi$
$389$	13.3179	+	7.68908i	0.675243	+	0.389852i	−0.122817	+	0.992429i	$0.539193\pi$
$390$		0			0
$391$			0.155120i			0.00784478i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−17.1518	+	9.90259i	−0.860823	+	0.496997i	−0.864288	−	0.502997i	$-0.832231\pi$
$397$	−17.1518	+	9.90259i	−0.860823	+	0.496997i	0.00346457	+	0.999994i	$0.498897\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	17.5175	−	10.1137i	0.874782	−	0.505056i	0.00584759	−	0.999983i	$-0.498139\pi$
$401$	17.5175	−	10.1137i	0.874782	−	0.505056i	0.868934	+	0.494927i	$0.164805\pi$
$402$		0			0
$403$	−5.66992	+	9.82059i	−0.282439	+	0.489198i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	7.72733i		−	0.383029i
$408$		0			0
$409$	14.1784	+	8.18591i	0.701078	+	0.404767i	0.807749	−	0.589527i	$-0.200686\pi$
$409$	14.1784	+	8.18591i	0.701078	+	0.404767i	−0.106671	+	0.994294i	$0.534019\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.00050	+	13.6665i	0.344472	+	0.672485i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−32.6180			−1.59349			−0.796746	−	0.604314i	$-0.793447\pi$
$419$	−32.6180			−1.59349			−0.796746	+	0.604314i	$0.793447\pi$
$420$		0			0
$421$	1.78518			0.0870045			0.0435022	−	0.999053i	$-0.486148\pi$
$421$	1.78518			0.0870045			0.0435022	+	0.999053i	$0.486148\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.406445	+	8.08826i	−0.0196693	+	0.391418i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−27.0011	−	15.5891i	−1.30060	−	0.750901i	−0.320092	−	0.947386i	$-0.603714\pi$
$431$	−27.0011	−	15.5891i	−1.30060	−	0.750901i	−0.980507	+	0.196485i	$0.937047\pi$
$432$		0			0
$433$			1.35671i			0.0651994i	0.999468	+	0.0325997i	$0.0103786\pi$
$433$			1.35671i			0.0651994i	−0.999468	+	0.0325997i	$0.989621\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.239895	−	0.415511i	0.0114757	−	0.0198766i
$438$		0			0
$439$	−7.38345	+	4.26284i	−0.352393	+	0.203454i	−0.665739	−	0.746185i	$-0.731883\pi$
$439$	−7.38345	+	4.26284i	−0.352393	+	0.203454i	0.313346	+	0.949639i	$0.398550\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.6655	−	7.31244i	0.601757	−	0.347424i	−0.167976	−	0.985791i	$-0.553723\pi$
$443$	12.6655	−	7.31244i	0.601757	−	0.347424i	0.769732	+	0.638367i	$0.220390\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	18.0320i		−	0.850984i	−0.904962	−	0.425492i	$-0.860101\pi$
$449$		−	18.0320i		−	0.850984i	0.904962	−	0.425492i	$-0.139899\pi$
$450$		0			0
$451$	4.75761	+	2.74681i	0.224027	+	0.129342i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.0717	+	24.3728i	0.658244	+	1.14011i	0.981070	+	0.193654i	$0.0620339\pi$
$457$	14.0717	+	24.3728i	0.658244	+	1.14011i	−0.322826	+	0.946458i	$0.604633\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−21.9940			−1.02436			−0.512181	−	0.858878i	$-0.671162\pi$
$461$	−21.9940			−1.02436			−0.512181	+	0.858878i	$0.671162\pi$
$462$		0			0
$463$	15.7014			0.729705			0.364852	−	0.931065i	$-0.381119\pi$
$463$	15.7014			0.729705			0.364852	+	0.931065i	$0.381119\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.8548	−	29.1933i	−0.779945	−	1.35091i	−0.931973	−	0.362529i	$-0.881913\pi$
$467$	−16.8548	−	29.1933i	−0.779945	−	1.35091i	0.152027	−	0.988376i	$-0.451420\pi$
$468$		0			0
$469$	−7.08768	−	13.8367i	−0.327278	−	0.638919i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.24763	−	1.29767i	−0.103346	−	0.0596668i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.7047	−	30.6655i	0.808950	−	1.40114i	−0.104641	−	0.994510i	$-0.533369\pi$
$479$	17.7047	−	30.6655i	0.808950	−	1.40114i	0.913592	−	0.406633i	$-0.133297\pi$
$480$		0			0
$481$	36.2881	−	20.9509i	1.65459	−	0.955280i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−2.52026	+	4.36522i	−0.114204	+	0.197807i	−0.917461	−	0.397825i	$-0.869765\pi$
$487$	−2.52026	+	4.36522i	−0.114204	+	0.197807i	0.803257	+	0.595632i	$0.203098\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	12.6130i		−	0.569215i	−0.958644	−	0.284608i	$-0.908137\pi$
$491$		−	12.6130i		−	0.569215i	0.958644	−	0.284608i	$-0.0918633\pi$
$492$		0			0
$493$	−0.152637	−	0.0881249i	−0.00687441	−	0.00396894i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.9951	+	10.3385i	0.717478	+	0.463745i
$498$		0			0
$499$	−0.115742	−	0.200472i	−0.00518134	−	0.00897434i	0.863423	−	0.504480i	$-0.168316\pi$
$499$	−0.115742	−	0.200472i	−0.00518134	−	0.00897434i	−0.868604	+	0.495506i	$0.834983\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17.9750			0.801465			0.400732	−	0.916195i	$-0.368756\pi$
$503$	17.9750			0.801465			0.400732	+	0.916195i	$0.368756\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.49089	−	16.4387i	−0.420676	−	0.728632i	0.575330	−	0.817922i	$-0.304874\pi$
$509$	−9.49089	−	16.4387i	−0.420676	−	0.728632i	−0.996006	+	0.0892892i	$0.971540\pi$
$510$		0			0
$511$	20.0906	−	10.2912i	0.888755	−	0.455254i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			4.58051i			0.201450i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	14.6325	−	25.3443i	0.641064	−	1.11035i	−0.344132	−	0.938921i	$-0.611827\pi$
$521$	14.6325	−	25.3443i	0.641064	−	1.11035i	0.985196	−	0.171433i	$-0.0548399\pi$
$522$		0			0
$523$	13.5626	−	7.83035i	0.593050	−	0.342397i	−0.173253	−	0.984877i	$-0.555428\pi$
$523$	13.5626	−	7.83035i	0.593050	−	0.342397i	0.766302	+	0.642480i	$0.222094\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.104578	−	0.0603783i	0.00455550	−	0.00263012i
$528$		0			0
$529$	−4.01731	+	6.95819i	−0.174666	+	0.302530i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			29.7895i			1.29032i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.487301	+	4.83640i	−0.0209895	+	0.208318i
$540$		0			0
$541$	−11.3407	−	19.6426i	−0.487573	−	0.844501i	0.512325	−	0.858792i	$-0.328784\pi$
$541$	−11.3407	−	19.6426i	−0.487573	−	0.844501i	−0.999898	+	0.0142904i	$0.995451\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.67849			0.285552			0.142776	−	0.989755i	$-0.454397\pi$
$547$	6.67849			0.285552			0.142776	+	0.989755i	$0.454397\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.272572	+	0.472108i	0.0116120	+	0.0201125i
$552$		0			0
$553$	−21.0479	−	1.05768i	−0.895046	−	0.0449772i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.3470	−	14.6341i	−1.07399	−	0.620066i	−0.144718	−	0.989473i	$-0.546227\pi$
$557$	−25.3470	−	14.6341i	−1.07399	−	0.620066i	−0.929268	+	0.369407i	$0.879561\pi$
$558$		0			0
$559$		−	14.0734i		−	0.595240i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.1420	−	33.1550i	0.806741	−	1.39732i	−0.108369	−	0.994111i	$-0.534563\pi$
$563$	19.1420	−	33.1550i	0.806741	−	1.39732i	0.915110	−	0.403205i	$-0.132104\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17.1572	+	9.90573i	−0.719268	+	0.415270i	−0.814483	−	0.580187i	$-0.802979\pi$
$569$	−17.1572	+	9.90573i	−0.719268	+	0.415270i	0.0952150	+	0.995457i	$0.469646\pi$
$570$		0			0
$571$	−10.4378	+	18.0788i	−0.436809	+	0.756575i	−0.997441	−	0.0714899i	$-0.977225\pi$
$571$	−10.4378	+	18.0788i	−0.436809	+	0.756575i	0.560633	+	0.828065i	$0.310558\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	10.8306	+	6.25307i	0.450885	+	0.260319i	0.708204	−	0.706008i	$-0.249506\pi$
$577$	10.8306	+	6.25307i	0.450885	+	0.260319i	−0.257319	+	0.966327i	$0.582839\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.08382	+	14.0539i	−0.376861	+	0.583056i
$582$		0			0
$583$	−3.11110	−	5.38858i	−0.128849	−	0.223172i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.4930			−0.845838			−0.422919	−	0.906168i	$-0.638994\pi$
$587$	−20.4930			−0.845838			−0.422919	+	0.906168i	$0.638994\pi$
$588$		0			0
$589$	−0.373502			−0.0153899
$590$		0			0
$591$		0			0
$592$		0			0
$593$	16.6389	+	28.8194i	0.683277	+	1.18347i	0.973975	+	0.226656i	$0.0727793\pi$
$593$	16.6389	+	28.8194i	0.683277	+	1.18347i	−0.290698	+	0.956815i	$0.593887\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	33.8034	+	19.5164i	1.38117	+	0.797419i	0.992298	−	0.123873i	$-0.0395317\pi$
$599$	33.8034	+	19.5164i	1.38117	+	0.797419i	0.388871	+	0.921292i	$0.372865\pi$
$600$		0			0
$601$			14.7657i			0.602304i	0.953576	+	0.301152i	$0.0973712\pi$
$601$			14.7657i			0.602304i	−0.953576	+	0.301152i	$0.902629\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	3.43776	−	1.98479i	0.139534	−	0.0805602i	−0.428608	−	0.903491i	$-0.640996\pi$
$607$	3.43776	−	1.98479i	0.139534	−	0.0805602i	0.568142	+	0.822930i	$0.307662\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21.5104	+	12.4190i	−0.870218	+	0.502420i
$612$		0			0
$613$	17.5733	−	30.4379i	0.709780	−	1.22937i	−0.255159	−	0.966899i	$-0.582128\pi$
$613$	17.5733	−	30.4379i	0.709780	−	1.22937i	0.964939	−	0.262476i	$-0.0845389\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			24.3102i			0.978692i	0.872090	+	0.489346i	$0.162764\pi$
$617$			24.3102i			0.978692i	−0.872090	+	0.489346i	$0.837236\pi$
$618$		0			0
$619$	−6.92178	−	3.99629i	−0.278210	−	0.160625i	0.354403	−	0.935093i	$-0.384684\pi$
$619$	−6.92178	−	3.99629i	−0.278210	−	0.160625i	−0.632613	+	0.774468i	$0.718017\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	26.4503	+	1.32916i	1.05971	+	0.0532516i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.446208			−0.0177915
$630$		0			0
$631$	−14.1828			−0.564607			−0.282304	−	0.959325i	$-0.591099\pi$
$631$	−14.1828			−0.564607			−0.282304	+	0.959325i	$0.591099\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−24.0333	+	10.8244i	−0.952233	+	0.428879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20.1549	+	11.6364i	0.796070	+	0.459612i	0.842095	−	0.539329i	$-0.181322\pi$
$641$	20.1549	+	11.6364i	0.796070	+	0.459612i	−0.0460248	+	0.998940i	$0.514655\pi$
$642$		0			0
$643$		−	43.2624i		−	1.70610i	−0.521826	−	0.853052i	$-0.674749\pi$
$643$		−	43.2624i		−	1.70610i	0.521826	−	0.853052i	$-0.325251\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.00423	+	12.1317i	−0.275365	+	0.476946i	−0.970227	−	0.242197i	$-0.922132\pi$
$647$	−7.00423	+	12.1317i	−0.275365	+	0.476946i	0.694862	+	0.719143i	$0.255465\pi$
$648$		0			0
$649$	−3.49022	+	2.01508i	−0.137003	+	0.0790988i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.47903	+	1.43127i	−0.0970120	+	0.0560099i	−0.547721	−	0.836661i	$-0.684505\pi$
$653$	−2.47903	+	1.43127i	−0.0970120	+	0.0560099i	0.450709	+	0.892671i	$0.351171\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			41.1201i			1.60181i	0.598789	+	0.800907i	$0.295649\pi$
$659$			41.1201i			1.60181i	−0.598789	+	0.800907i	$0.704351\pi$
$660$		0			0
$661$	−11.9588	−	6.90443i	−0.465144	−	0.268551i	0.249061	−	0.968488i	$-0.419878\pi$
$661$	−11.9588	−	6.90443i	−0.465144	−	0.268551i	−0.714205	+	0.699937i	$0.753212\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8.50192	+	14.7258i	0.329196	+	0.570184i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.12555			−0.0820559
$672$		0			0
$673$	−36.5668			−1.40955			−0.704773	−	0.709433i	$-0.748951\pi$
$673$	−36.5668			−1.40955			−0.704773	+	0.709433i	$0.748951\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.20030	−	12.4713i	−0.276730	−	0.479310i	0.693840	−	0.720129i	$-0.255917\pi$
$677$	−7.20030	−	12.4713i	−0.276730	−	0.479310i	−0.970570	+	0.240819i	$0.922584\pi$
$678$		0			0
$679$	−25.7664	−	16.6542i	−0.988825	−	0.639131i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.1129	−	6.41606i	−0.425225	−	0.245504i	0.272085	−	0.962273i	$-0.412287\pi$
$683$	−11.1129	−	6.41606i	−0.425225	−	0.245504i	−0.697310	+	0.716769i	$0.745620\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.8701	−	29.2199i	0.642701	−	1.11319i
$690$		0			0
$691$	−28.9442	+	16.7110i	−1.10109	+	0.635715i	−0.936507	−	0.350648i	$-0.885961\pi$
$691$	−28.9442	+	16.7110i	−1.10109	+	0.635715i	−0.164584	+	0.986363i	$0.552628\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.158612	−	0.274724i	0.00600786	−	0.0104059i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			7.88834i			0.297938i	0.988842	+	0.148969i	$0.0475955\pi$
$701$			7.88834i			0.297938i	−0.988842	+	0.148969i	$0.952404\pi$
$702$		0			0
$703$	1.19523	+	0.690065i	0.0450789	+	0.0260263i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.04589	+	7.89847i	0.152161	+	0.297052i
$708$		0			0
$709$	17.1731	+	29.7448i	0.644951	+	1.11709i	0.984313	+	0.176432i	$0.0564556\pi$
$709$	17.1731	+	29.7448i	0.644951	+	1.11709i	−0.339362	+	0.940656i	$0.610211\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−11.6501			−0.436299
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10.5666	−	18.3019i	−0.394068	−	0.682546i	0.598914	−	0.800814i	$-0.295599\pi$
$719$	−10.5666	−	18.3019i	−0.394068	−	0.682546i	−0.992982	+	0.118268i	$0.962266\pi$
$720$		0			0
$721$	−1.46167	+	29.0873i	−0.0544355	+	1.08327i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			33.6575i			1.24829i	0.781310	+	0.624143i	$0.214552\pi$
$727$			33.6575i			1.24829i	−0.781310	+	0.624143i	$0.785448\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.0749327	+	0.129787i	−0.00277149	+	0.00480036i
$732$		0			0
$733$	27.4201	−	15.8310i	1.01278	−	0.584731i	0.100778	−	0.994909i	$-0.467867\pi$
$733$	27.4201	−	15.8310i	1.01278	−	0.584731i	0.912006	+	0.410178i	$0.134533\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.53368	−	2.04017i	0.130165	−	0.0751507i
$738$		0			0
$739$	−7.29352	+	12.6327i	−0.268296	+	0.464703i	−0.968422	−	0.249317i	$-0.919794\pi$
$739$	−7.29352	+	12.6327i	−0.268296	+	0.464703i	0.700126	+	0.714020i	$0.253127\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	36.3676i		−	1.33420i	−0.744969	−	0.667099i	$-0.767536\pi$
$743$		−	36.3676i		−	1.33420i	0.744969	−	0.667099i	$-0.232464\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.80461	+	35.9117i	−0.0659390	+	1.31218i
$750$		0			0
$751$	14.1725	+	24.5475i	0.517162	+	0.895751i	0.999801	+	0.0199317i	$0.00634489\pi$
$751$	14.1725	+	24.5475i	0.517162	+	0.895751i	−0.482639	+	0.875819i	$0.660322\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−45.7899			−1.66426			−0.832130	−	0.554580i	$-0.812879\pi$
$757$	−45.7899			−1.66426			−0.832130	+	0.554580i	$0.812879\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.45015	+	4.24378i	0.0888179	+	0.153837i	0.907012	−	0.421105i	$-0.138358\pi$
$761$	2.45015	+	4.24378i	0.0888179	+	0.153837i	−0.818194	+	0.574942i	$0.805024\pi$
$762$		0			0
$763$	8.38008	+	16.3597i	0.303379	+	0.592263i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.9259	−	10.9269i	−0.683375	−	0.394547i
$768$		0			0
$769$		−	24.7837i		−	0.893722i	−0.894603	−	0.446861i	$-0.852542\pi$
$769$		−	24.7837i		−	0.893722i	0.894603	−	0.446861i	$-0.147458\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7.53601	+	13.0528i	−0.271052	+	0.469475i	−0.969131	−	0.246545i	$-0.920705\pi$
$773$	−7.53601	+	13.0528i	−0.271052	+	0.469475i	0.698080	+	0.716020i	$0.254038\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.849727	+	0.490590i	−0.0304446	+	0.0175772i
$780$		0			0
$781$	−2.49936	+	4.32901i	−0.0894340	+	0.154904i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	3.96684	+	2.29025i	0.141402	+	0.0816387i	0.569032	−	0.822315i	$-0.307318\pi$
$787$	3.96684	+	2.29025i	0.141402	+	0.0816387i	−0.427630	+	0.903954i	$0.640651\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	30.1057	+	19.4589i	1.07043	+	0.691880i
$792$		0			0
$793$	−5.76295	−	9.98173i	−0.204648	−	0.354462i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−23.9082			−0.846872			−0.423436	−	0.905926i	$-0.639176\pi$
$797$	−23.9082			−0.846872			−0.423436	+	0.905926i	$0.639176\pi$
$798$		0			0
$799$	0.264498			0.00935725
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.96229	+	5.13083i	0.104537	+	0.181063i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	44.4970	+	25.6904i	1.56443	+	0.903226i	0.996799	+	0.0799442i	$0.0254742\pi$
$809$	44.4970	+	25.6904i	1.56443	+	0.903226i	0.567633	+	0.823281i	$0.307859\pi$
$810$		0			0
$811$		−	33.9917i		−	1.19361i	−0.802387	−	0.596804i	$-0.796437\pi$
$811$		−	33.9917i		−	1.19361i	0.802387	−	0.596804i	$-0.203563\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.401435	−	0.231768i	0.0140444	−	0.00810855i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	30.1076	−	17.3826i	1.05076	−	0.606658i	0.127899	−	0.991787i	$-0.459177\pi$
$821$	30.1076	−	17.3826i	1.05076	−	0.606658i	0.922863	+	0.385129i	$0.125843\pi$
$822$		0			0
$823$	16.3784	−	28.3682i	0.570915	−	0.988853i	−0.425557	−	0.904931i	$-0.639922\pi$
$823$	16.3784	−	28.3682i	0.570915	−	0.988853i	0.996472	−	0.0839221i	$-0.0267447\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			5.43764i			0.189085i	0.995521	+	0.0945427i	$0.0301389\pi$
$827$			5.43764i			0.189085i	−0.995521	+	0.0945427i	$0.969861\pi$
$828$		0			0
$829$	34.4858	+	19.9104i	1.19774	+	0.691515i	0.960051	−	0.279826i	$-0.0902768\pi$
$829$	34.4858	+	19.9104i	1.19774	+	0.691515i	0.237689	+	0.971341i	$0.423610\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.279274	+	0.0281388i	0.00967626	+	0.000974951i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.3647			−0.530449			−0.265225	−	0.964187i	$-0.585446\pi$
$839$	−15.3647			−0.530449			−0.265225	+	0.964187i	$0.585446\pi$
$840$		0			0
$841$	9.68003			0.333794
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	27.7924	+	1.39660i	0.954958	+	0.0479879i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	37.2809	+	21.5241i	1.27797	+	0.737838i
$852$		0			0
$853$			32.0274i			1.09660i	0.836282	+	0.548299i	$0.184724\pi$
$853$			32.0274i			1.09660i	−0.836282	+	0.548299i	$0.815276\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.69864	+	4.67417i	−0.0921836	+	0.159667i	−0.908430	−	0.418038i	$-0.862718\pi$
$857$	−2.69864	+	4.67417i	−0.0921836	+	0.159667i	0.816246	+	0.577704i	$0.196051\pi$
$858$		0			0
$859$	5.91952	−	3.41764i	0.201971	−	0.116608i	−0.395603	−	0.918422i	$-0.629464\pi$
$859$	5.91952	−	3.41764i	0.201971	−	0.116608i	0.597575	+	0.801813i	$0.296131\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.0819	+	5.82079i	−0.343192	+	0.198142i	−0.661683	−	0.749784i	$-0.730157\pi$
$863$	−10.0819	+	5.82079i	−0.343192	+	0.198142i	0.318491	+	0.947926i	$0.396824\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	5.53125i		−	0.187635i
$870$		0			0
$871$	19.1616	+	11.0630i	0.649266	+	0.374854i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−16.5919	−	28.7380i	−0.560269	−	0.970415i	−0.997473	−	0.0710519i	$-0.977364\pi$
$877$	−16.5919	−	28.7380i	−0.560269	−	0.970415i	0.437204	−	0.899363i	$-0.355969\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−41.0831			−1.38412			−0.692062	−	0.721838i	$-0.743298\pi$
$881$	−41.0831			−1.38412			−0.692062	+	0.721838i	$0.743298\pi$
$882$		0			0
$883$	−11.7151			−0.394244			−0.197122	−	0.980379i	$-0.563159\pi$
$883$	−11.7151			−0.394244			−0.197122	+	0.980379i	$0.563159\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.99958	+	13.8557i	0.268600	+	0.465228i	0.968500	−	0.249012i	$-0.0801058\pi$
$887$	7.99958	+	13.8557i	0.268600	+	0.465228i	−0.699901	+	0.714240i	$0.746772\pi$
$888$		0			0
$889$	−7.13450	+	11.0381i	−0.239283	+	0.370205i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.708492	−	0.409048i	−0.0237088	−	0.0136883i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.61849	−	11.4636i	0.220739	−	0.382331i
$900$		0			0
$901$	−0.311159	+	0.179648i	−0.0103662	+	0.00598494i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.23355	−	5.60068i	0.107368	−	0.185968i	−0.807335	−	0.590093i	$-0.799091\pi$
$907$	3.23355	−	5.60068i	0.107368	−	0.185968i	0.914703	+	0.404126i	$0.132424\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	30.3458i		−	1.00540i	−0.864461	−	0.502700i	$-0.832340\pi$
$911$		−	30.3458i		−	1.00540i	0.864461	−	0.502700i	$-0.167660\pi$
$912$		0			0
$913$	−3.80365	−	2.19604i	−0.125882	−	0.0726783i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−35.6607	−	1.79200i	−1.17762	−	0.0591770i
$918$		0			0
$919$	−23.9218	−	41.4337i	−0.789107	−	1.36677i	−0.926515	−	0.376258i	$-0.877210\pi$
$919$	−23.9218	−	41.4337i	−0.789107	−	1.36677i	0.137408	−	0.990515i	$-0.456123\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−27.1058			−0.892199
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15.7384	−	27.2597i	−0.516360	−	0.894361i	−0.999820	−	0.0189947i	$-0.993953\pi$
$929$	−15.7384	−	27.2597i	−0.516360	−	0.894361i	0.483460	−	0.875366i	$-0.339380\pi$
$930$		0			0
$931$	−0.704555	−	0.507273i	−0.0230909	−	0.0166252i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	51.6376i		−	1.68693i	−0.537186	−	0.843464i	$-0.680513\pi$
$937$		−	51.6376i		−	1.68693i	0.537186	−	0.843464i	$-0.319487\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.60381	+	4.50994i	−0.0848819	+	0.147020i	−0.905341	−	0.424686i	$-0.860385\pi$
$941$	−2.60381	+	4.50994i	−0.0848819	+	0.147020i	0.820459	+	0.571705i	$0.193718\pi$
$942$		0			0
$943$	−26.5043	+	15.3022i	−0.863097	+	0.498309i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−46.2239	+	26.6874i	−1.50207	+	0.867223i	−0.502076	+	0.864823i	$0.667430\pi$
$947$	−46.2239	+	26.6874i	−1.50207	+	0.867223i	−0.999997	+	0.00239939i	$0.999236\pi$
$948$		0			0
$949$	−16.0632	+	27.8222i	−0.521433	+	0.903148i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			58.6135i			1.89868i	0.314257	+	0.949338i	$0.398245\pi$
$953$			58.6135i			1.89868i	−0.314257	+	0.949338i	$0.601755\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−28.2098	+	14.4501i	−0.910941	+	0.466618i
$960$		0			0
$961$	−10.9654	−	18.9926i	−0.353722	−	0.612664i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	14.9345			0.480262			0.240131	−	0.970740i	$-0.422810\pi$
$967$	14.9345			0.480262			0.240131	+	0.970740i	$0.422810\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11.9098	+	20.6284i	0.382205	+	0.661998i	0.991377	−	0.131040i	$-0.0418316\pi$
$971$	11.9098	+	20.6284i	0.382205	+	0.661998i	−0.609173	+	0.793038i	$0.708498\pi$
$972$		0			0
$973$	3.64214	+	2.35411i	0.116762	+	0.0754694i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.56268	+	0.902213i	0.0499945	+	0.0288643i	0.524789	−	0.851232i	$-0.324144\pi$
$977$	1.56268	+	0.902213i	0.0499945	+	0.0288643i	−0.474794	+	0.880097i	$0.657478\pi$
$978$		0			0
$979$			6.95097i			0.222154i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.58383	−	14.8676i	0.273782	−	0.474204i	−0.696045	−	0.717998i	$-0.745059\pi$
$983$	8.58383	−	14.8676i	0.273782	−	0.474204i	0.969827	+	0.243794i	$0.0783921\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.5213	−	7.22920i	0.398155	−	0.229875i
$990$		0			0
$991$	−7.64187	+	13.2361i	−0.242752	+	0.420459i	−0.961497	−	0.274815i	$-0.911383\pi$
$991$	−7.64187	+	13.2361i	−0.242752	+	0.420459i	0.718745	+	0.695274i	$0.244717\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−43.8863	−	25.3378i	−1.38989	−	0.802456i	−0.396591	−	0.917995i	$-0.629807\pi$
$997$	−43.8863	−	25.3378i	−1.38989	−	0.802456i	−0.993303	+	0.115540i	$0.963140\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.ch.f.4301.4		32
3.2	odd	2	inner	6300.2.ch.f.4301.3		32
5.2	odd	4		1260.2.dc.a.269.11	yes	32
5.3	odd	4		1260.2.dc.a.269.12	yes	32
5.4	even	2	inner	6300.2.ch.f.4301.14		32
7.5	odd	6	inner	6300.2.ch.f.1601.3		32
15.2	even	4		1260.2.dc.a.269.6	yes	32
15.8	even	4		1260.2.dc.a.269.5	yes	32
15.14	odd	2	inner	6300.2.ch.f.4301.13		32
21.5	even	6	inner	6300.2.ch.f.1601.4		32
35.3	even	12		8820.2.f.a.4409.29		32
35.12	even	12		1260.2.dc.a.89.5	✓	32
35.17	even	12		8820.2.f.a.4409.32		32
35.18	odd	12		8820.2.f.a.4409.3		32
35.19	odd	6	inner	6300.2.ch.f.1601.13		32
35.32	odd	12		8820.2.f.a.4409.2		32
35.33	even	12		1260.2.dc.a.89.6	yes	32
105.17	odd	12		8820.2.f.a.4409.1		32
105.32	even	12		8820.2.f.a.4409.31		32
105.38	odd	12		8820.2.f.a.4409.4		32
105.47	odd	12		1260.2.dc.a.89.12	yes	32
105.53	even	12		8820.2.f.a.4409.30		32
105.68	odd	12		1260.2.dc.a.89.11	yes	32
105.89	even	6	inner	6300.2.ch.f.1601.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.dc.a.89.5	✓	32	35.12	even	12
1260.2.dc.a.89.6	yes	32	35.33	even	12
1260.2.dc.a.89.11	yes	32	105.68	odd	12
1260.2.dc.a.89.12	yes	32	105.47	odd	12
1260.2.dc.a.269.5	yes	32	15.8	even	4
1260.2.dc.a.269.6	yes	32	15.2	even	4
1260.2.dc.a.269.11	yes	32	5.2	odd	4
1260.2.dc.a.269.12	yes	32	5.3	odd	4
6300.2.ch.f.1601.3		32	7.5	odd	6	inner
6300.2.ch.f.1601.4		32	21.5	even	6	inner
6300.2.ch.f.1601.13		32	35.19	odd	6	inner
6300.2.ch.f.1601.14		32	105.89	even	6	inner
6300.2.ch.f.4301.3		32	3.2	odd	2	inner
6300.2.ch.f.4301.4		32	1.1	even	1	trivial
6300.2.ch.f.4301.13		32	15.14	odd	2	inner
6300.2.ch.f.4301.14		32	5.4	even	2	inner
8820.2.f.a.4409.1		32	105.17	odd	12
8820.2.f.a.4409.2		32	35.32	odd	12
8820.2.f.a.4409.3		32	35.18	odd	12
8820.2.f.a.4409.4		32	105.38	odd	12
8820.2.f.a.4409.29		32	35.3	even	12
8820.2.f.a.4409.30		32	105.53	even	12
8820.2.f.a.4409.31		32	105.32	even	12
8820.2.f.a.4409.32		32	35.17	even	12

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:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.ch (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.43620 + 2.22201i) q^{7} +O(q^{10})\)	\(q+(-1.43620 + 2.22201i) q^{7} +(-0.601378 - 0.347206i) q^{11} -3.76549i q^{13} +(-0.0200491 + 0.0347261i) q^{17} +(0.107408 - 0.0620123i) q^{19} +(3.35023 - 1.93426i) q^{23} +4.39545i q^{29} +(-2.60805 - 1.50576i) q^{31} +(5.56393 + 9.63702i) q^{37} -7.91118 q^{41} +3.73746 q^{43} +(-3.29812 - 5.71251i) q^{47} +(-2.87464 - 6.38251i) q^{49} +(7.75992 + 4.48019i) q^{53} +(2.90185 - 5.02615i) q^{59} +(2.65084 - 1.53047i) q^{61} +(-2.93799 + 5.08874i) q^{67} -7.19849i q^{71} +(-7.38874 - 4.26589i) q^{73} +(1.63520 - 0.837609i) q^{77} +(3.98269 + 6.89822i) q^{79} +6.32489 q^{83} +(-5.00494 - 8.66880i) q^{89} +(8.36695 + 5.40801i) q^{91} +11.5960i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 24 q^{19} + 24 q^{31} - 40 q^{49} - 24 q^{61} - 32 q^{79} - 56 q^{91}+O(q^{100}) \) 32 * q + 24 * q^19 + 24 * q^31 - 40 * q^49 - 24 * q^61 - 32 * q^79 - 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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