LMFDB - Embedded newform 1900.2.l.d.493.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1900,2,Mod(493,1900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1900.493"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.l (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			493.4
Character	$\chi$	$=$	1900.493
Dual form			1900.2.l.d.1557.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.05614 + 1.05614i) q^{3} +(1.45445 - 1.45445i) q^{7} +0.769150i q^{9} +4.10096 q^{11} +(3.51885 - 3.51885i) q^{13} +(1.45445 - 1.45445i) q^{17} +(-0.485200 - 4.33181i) q^{19} +3.07220i q^{21} +(-2.62617 - 2.62617i) q^{23} +(-3.98074 - 3.98074i) q^{27} -10.7212 q^{29} +8.61938i q^{31} +(-4.33118 + 4.33118i) q^{33} +(-3.98074 - 3.98074i) q^{37} +7.43277i q^{39} -10.2360i q^{41} +(-1.22474 - 1.22474i) q^{43} +(8.21426 - 8.21426i) q^{47} +2.76915i q^{49} +3.07220i q^{51} +(6.22537 - 6.22537i) q^{53} +(5.08742 + 4.06255i) q^{57} +5.17399 q^{59} +10.9954 q^{61} +(1.11869 + 1.11869i) q^{63} +(5.16923 + 5.16923i) q^{67} +5.54718 q^{69} +3.93059i q^{71} +(-8.64384 - 8.64384i) q^{73} +(5.96464 - 5.96464i) q^{77} -3.55740 q^{79} +6.10096 q^{81} +(-4.31032 - 4.31032i) q^{83} +(11.3230 - 11.3230i) q^{87} +3.07220 q^{89} -10.2360i q^{91} +(-9.10325 - 9.10325i) q^{93} +(1.40657 + 1.40657i) q^{97} +3.15425i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{11} + 24 q^{61} + 24 q^{81}+O(q^{100}) $ 24 * q - 24 * q^11 + 24 * q^61 + 24 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$1$

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
$3$	−1.05614	+	1.05614i	−0.609761	+	0.609761i	−0.942884	−	0.333123i	$-0.891898\pi$
$3$	−1.05614	+	1.05614i	−0.609761	+	0.609761i	0.333123	+	0.942884i	$0.391898\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.45445	−	1.45445i	0.549730	−	0.549730i	−0.376632	−	0.926363i	$-0.622918\pi$
$7$	1.45445	−	1.45445i	0.549730	−	0.549730i	0.926363	+	0.376632i	$0.122918\pi$
$8$		0			0
$9$			0.769150i			0.256383i
$10$		0			0
$11$	4.10096			1.23649			0.618243	−	0.785987i	$-0.287845\pi$
$11$	4.10096			1.23649			0.618243	+	0.785987i	$0.287845\pi$
$12$		0			0
$13$	3.51885	−	3.51885i	0.975953	−	0.975953i	−0.0237648	−	0.999718i	$-0.507565\pi$
$13$	3.51885	−	3.51885i	0.975953	−	0.975953i	0.999718	+	0.0237648i	$0.00756528\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.45445	−	1.45445i	0.352756	−	0.352756i	−0.508378	−	0.861134i	$-0.669755\pi$
$17$	1.45445	−	1.45445i	0.352756	−	0.352756i	0.861134	+	0.508378i	$0.169755\pi$
$18$		0			0
$19$	−0.485200	−	4.33181i	−0.111313	−	0.993785i
$19$	−0.485200	−	4.33181i	−0.111313	−	0.993785i
$20$		0			0
$21$			3.07220i			0.670408i
$22$		0			0
$23$	−2.62617	−	2.62617i	−0.547594	−	0.547594i	0.378150	−	0.925744i	$-0.376560\pi$
$23$	−2.62617	−	2.62617i	−0.547594	−	0.547594i	−0.925744	+	0.378150i	$0.876560\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.98074	−	3.98074i	−0.766093	−	0.766093i
$28$		0			0
$29$	−10.7212			−1.99087			−0.995436	−	0.0954276i	$-0.969578\pi$
$29$	−10.7212			−1.99087			−0.995436	+	0.0954276i	$0.969578\pi$
$30$		0			0
$31$			8.61938i			1.54809i	0.633133	+	0.774043i	$0.281769\pi$
$31$			8.61938i			1.54809i	−0.633133	+	0.774043i	$0.718231\pi$
$32$		0			0
$33$	−4.33118	+	4.33118i	−0.753961	+	0.753961i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.98074	−	3.98074i	−0.654429	−	0.654429i	0.299627	−	0.954056i	$-0.403138\pi$
$37$	−3.98074	−	3.98074i	−0.654429	−	0.654429i	−0.954056	+	0.299627i	$0.903138\pi$
$38$		0			0
$39$			7.43277i			1.19020i
$40$		0			0
$41$		−	10.2360i		−	1.59859i	−0.600938	−	0.799296i	$-0.705206\pi$
$41$		−	10.2360i		−	1.59859i	0.600938	−	0.799296i	$-0.294794\pi$
$42$		0			0
$43$	−1.22474	−	1.22474i	−0.186772	−	0.186772i	0.607527	−	0.794299i	$-0.292162\pi$
$43$	−1.22474	−	1.22474i	−0.186772	−	0.186772i	−0.794299	+	0.607527i	$0.792162\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.21426	−	8.21426i	1.19817	−	1.19817i	0.223460	−	0.974713i	$-0.428265\pi$
$47$	8.21426	−	8.21426i	1.19817	−	1.19817i	0.974713	−	0.223460i	$-0.0717353\pi$
$48$		0			0
$49$			2.76915i			0.395593i
$50$		0			0
$51$			3.07220i			0.430194i
$52$		0			0
$53$	6.22537	−	6.22537i	0.855120	−	0.855120i	−0.135638	−	0.990758i	$-0.543309\pi$
$53$	6.22537	−	6.22537i	0.855120	−	0.855120i	0.990758	+	0.135638i	$0.0433085\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	5.08742	+	4.06255i	0.673846	+	0.538097i
$58$		0			0
$59$	5.17399			0.673597			0.336798	−	0.941577i	$-0.390656\pi$
$59$	5.17399			0.673597			0.336798	+	0.941577i	$0.390656\pi$
$60$		0			0
$61$	10.9954			1.40782			0.703910	−	0.710289i	$-0.251436\pi$
$61$	10.9954			1.40782			0.703910	+	0.710289i	$0.251436\pi$
$62$		0			0
$63$	1.11869	+	1.11869i	0.140942	+	0.140942i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.16923	+	5.16923i	0.631522	+	0.631522i	0.948450	−	0.316928i	$-0.102651\pi$
$67$	5.16923	+	5.16923i	0.631522	+	0.631522i	−0.316928	+	0.948450i	$0.602651\pi$
$68$		0			0
$69$	5.54718			0.667803
$70$		0			0
$71$			3.93059i			0.466475i	0.972420	+	0.233238i	$0.0749320\pi$
$71$			3.93059i			0.466475i	−0.972420	+	0.233238i	$0.925068\pi$
$72$		0			0
$73$	−8.64384	−	8.64384i	−1.01168	−	1.01168i	−0.999931	−	0.0117536i	$-0.996259\pi$
$73$	−8.64384	−	8.64384i	−1.01168	−	1.01168i	−0.0117536	−	0.999931i	$-0.503741\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.96464	−	5.96464i	0.679734	−	0.679734i
$78$		0			0
$79$	−3.55740			−0.400238			−0.200119	−	0.979772i	$-0.564133\pi$
$79$	−3.55740			−0.400238			−0.200119	+	0.979772i	$0.564133\pi$
$80$		0			0
$81$	6.10096			0.677884
$82$		0			0
$83$	−4.31032	−	4.31032i	−0.473119	−	0.473119i	0.429803	−	0.902923i	$-0.358583\pi$
$83$	−4.31032	−	4.31032i	−0.473119	−	0.473119i	−0.902923	+	0.429803i	$0.858583\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	11.3230	−	11.3230i	1.21396	−	1.21396i
$88$		0			0
$89$	3.07220			0.325652			0.162826	−	0.986655i	$-0.447939\pi$
$89$	3.07220			0.325652			0.162826	+	0.986655i	$0.447939\pi$
$90$		0			0
$91$		−	10.2360i		−	1.07302i
$92$		0			0
$93$	−9.10325	−	9.10325i	−0.943963	−	0.943963i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.40657	+	1.40657i	0.142816	+	0.142816i	0.774900	−	0.632084i	$-0.217800\pi$
$97$	1.40657	+	1.40657i	0.142816	+	0.142816i	−0.632084	+	0.774900i	$0.717800\pi$
$98$		0			0
$99$			3.15425i			0.317014i
$100$		0			0
$101$	10.6880			1.06349			0.531747	−	0.846903i	$-0.321536\pi$
$101$	10.6880			1.06349			0.531747	+	0.846903i	$0.321536\pi$
$102$		0			0
$103$	−12.3184	+	12.3184i	−1.21377	+	1.21377i	−0.243988	+	0.969778i	$0.578456\pi$
$103$	−12.3184	+	12.3184i	−1.21377	+	1.21377i	−0.969778	+	0.243988i	$0.921544\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.76266	+	3.76266i	0.363750	+	0.363750i	0.865191	−	0.501442i	$-0.167197\pi$
$107$	3.76266	+	3.76266i	0.363750	+	0.363750i	−0.501442	+	0.865191i	$0.667197\pi$
$108$		0			0
$109$	6.03239			0.577798			0.288899	−	0.957360i	$-0.406711\pi$
$109$	6.03239			0.577798			0.288899	+	0.957360i	$0.406711\pi$
$110$		0			0
$111$	8.40841			0.798091
$112$		0			0
$113$	5.51967	−	5.51967i	0.519247	−	0.519247i	−0.398097	−	0.917343i	$-0.630329\pi$
$113$	5.51967	−	5.51967i	0.519247	−	0.519247i	0.917343	+	0.398097i	$0.130329\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.70652	+	2.70652i	0.250218	+	0.250218i
$118$		0			0
$119$		−	4.23085i		−	0.387841i
$120$		0			0
$121$	5.81787			0.528897
$122$		0			0
$123$	10.8106	+	10.8106i	0.974758	+	0.974758i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.461890	−	0.461890i	−0.0409861	−	0.0409861i	0.686317	−	0.727303i	$-0.259226\pi$
$127$	−0.461890	−	0.461890i	−0.0409861	−	0.0409861i	−0.727303	+	0.686317i	$0.759226\pi$
$128$		0			0
$129$	2.58700			0.227772
$130$		0			0
$131$	1.23085			0.107540			0.0537699	−	0.998553i	$-0.482876\pi$
$131$	1.23085			0.107540			0.0537699	+	0.998553i	$0.482876\pi$
$132$		0			0
$133$	−7.00610	−	5.59470i	−0.607506	−	0.485122i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.76973	+	4.76973i	−0.407506	+	0.407506i	−0.880868	−	0.473362i	$-0.843040\pi$
$137$	−4.76973	+	4.76973i	−0.407506	+	0.407506i	0.473362	+	0.880868i	$0.343040\pi$
$138$		0			0
$139$			15.4526i			1.31067i	0.755339	+	0.655335i	$0.227472\pi$
$139$			15.4526i			1.31067i	−0.755339	+	0.655335i	$0.772528\pi$
$140$		0			0
$141$			17.3508i			1.46120i
$142$		0			0
$143$	14.4307	−	14.4307i	1.20675	−	1.20675i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.92460	−	2.92460i	−0.241217	−	0.241217i
$148$		0			0
$149$		−	20.0964i		−	1.64636i	−0.567780	−	0.823180i	$-0.692198\pi$
$149$		−	20.0964i		−	1.64636i	0.567780	−	0.823180i	$-0.307802\pi$
$150$		0			0
$151$		−	11.5796i		−	0.942332i	−0.882045	−	0.471166i	$-0.843833\pi$
$151$		−	11.5796i		−	0.942332i	0.882045	−	0.471166i	$-0.156167\pi$
$152$		0			0
$153$	1.11869	+	1.11869i	0.0904407	+	0.0904407i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.87850	−	7.87850i	0.628773	−	0.628773i	−0.318986	−	0.947759i	$-0.603342\pi$
$157$	7.87850	−	7.87850i	0.628773	−	0.628773i	0.947759	+	0.318986i	$0.103342\pi$
$158$		0			0
$159$			13.1497i			1.04284i
$160$		0			0
$161$	−7.63926			−0.602058
$162$		0			0
$163$	13.1132	+	13.1132i	1.02711	+	1.02711i	0.999622	+	0.0274862i	$0.00875024\pi$
$163$	13.1132	+	13.1132i	1.02711	+	1.02711i	0.0274862	+	0.999622i	$0.491250\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.46271	+	2.46271i	0.190570	+	0.190570i	0.795943	−	0.605372i	$-0.206976\pi$
$167$	2.46271	+	2.46271i	0.190570	+	0.190570i	−0.605372	+	0.795943i	$0.706976\pi$
$168$		0			0
$169$		−	11.7646i		−	0.904968i
$170$		0			0
$171$	3.33181	−	0.373192i	0.254790	−	0.0285387i
$172$		0			0
$173$	−13.2422	+	13.2422i	−1.00678	+	1.00678i	−0.00680600	+	0.999977i	$0.502166\pi$
$173$	−13.2422	+	13.2422i	−1.00678	+	1.00678i	−0.999977	+	0.00680600i	$0.997834\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−5.46445	+	5.46445i	−0.410733	+	0.410733i
$178$		0			0
$179$	13.3082			0.994700			0.497350	−	0.867550i	$-0.334307\pi$
$179$	13.3082			0.994700			0.497350	+	0.867550i	$0.334307\pi$
$180$		0			0
$181$		−	23.6562i		−	1.75835i	−0.476500	−	0.879174i	$-0.658095\pi$
$181$		−	23.6562i		−	1.75835i	0.476500	−	0.879174i	$-0.341905\pi$
$182$		0			0
$183$	−11.6127	+	11.6127i	−0.858434	+	0.858434i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	5.96464	−	5.96464i	0.436178	−	0.436178i
$188$		0			0
$189$	−11.5796			−0.842290
$190$		0			0
$191$	4.10096			0.296735			0.148368	−	0.988932i	$-0.452598\pi$
$191$	4.10096			0.296735			0.148368	+	0.988932i	$0.452598\pi$
$192$		0			0
$193$	13.0241	−	13.0241i	0.937494	−	0.937494i	−0.0606644	−	0.998158i	$-0.519322\pi$
$193$	13.0241	−	13.0241i	0.937494	−	0.937494i	0.998158	+	0.0606644i	$0.0193219\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.1839	+	13.1839i	−0.939311	+	0.939311i	−0.998261	−	0.0589495i	$-0.981225\pi$
$197$	−13.1839	+	13.1839i	−0.939311	+	0.939311i	0.0589495	+	0.998261i	$0.481225\pi$
$198$		0			0
$199$			13.1776i			0.934132i	0.884222	+	0.467066i	$0.154689\pi$
$199$			13.1776i			0.934132i	−0.884222	+	0.467066i	$0.845311\pi$
$200$		0			0
$201$	−10.9188			−0.770155
$202$		0			0
$203$	−15.5934	+	15.5934i	−1.09444	+	1.09444i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.01992	−	2.01992i	0.140394	−	0.140394i
$208$		0			0
$209$	−1.98979	−	17.7646i	−0.137636	−	1.22880i
$210$		0			0
$211$			18.4822i			1.27236i	0.771539	+	0.636182i	$0.219487\pi$
$211$			18.4822i			1.27236i	−0.771539	+	0.636182i	$0.780513\pi$
$212$		0			0
$213$	−4.15124	−	4.15124i	−0.284438	−	0.284438i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	12.5365	+	12.5365i	0.851030	+	0.851030i
$218$		0			0
$219$	18.2582			1.23377
$220$		0			0
$221$		−	10.2360i		−	0.688546i
$222$		0			0
$223$	17.2229	−	17.2229i	1.15333	−	1.15333i	0.167450	−	0.985881i	$-0.446447\pi$
$223$	17.2229	−	17.2229i	1.15333	−	1.15333i	0.985881	−	0.167450i	$-0.0535533\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.4619	+	17.4619i	1.15899	+	1.15899i	0.984694	+	0.174292i	$0.0557636\pi$
$227$	17.4619	+	17.4619i	1.15899	+	1.15899i	0.174292	+	0.984694i	$0.444236\pi$
$228$		0			0
$229$			9.53373i			0.630007i	0.949091	+	0.315003i	$0.102006\pi$
$229$			9.53373i			0.630007i	−0.949091	+	0.315003i	$0.897994\pi$
$230$		0			0
$231$			12.5990i			0.828950i
$232$		0			0
$233$	−11.9117	−	11.9117i	−0.780361	−	0.780361i	0.199531	−	0.979892i	$-0.436058\pi$
$233$	−11.9117	−	11.9117i	−0.780361	−	0.780361i	−0.979892	+	0.199531i	$0.936058\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	3.75710	−	3.75710i	0.244050	−	0.244050i
$238$		0			0
$239$			10.7448i			0.695023i	0.937676	+	0.347511i	$0.112973\pi$
$239$			10.7448i			0.695023i	−0.937676	+	0.347511i	$0.887027\pi$
$240$		0			0
$241$			5.06198i			0.326071i	0.986620	+	0.163036i	$0.0521285\pi$
$241$			5.06198i			0.326071i	−0.986620	+	0.163036i	$0.947871\pi$
$242$		0			0
$243$	5.49877	−	5.49877i	0.352746	−	0.352746i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−16.9503	−	13.5356i	−1.07852	−	0.861252i
$248$		0			0
$249$	9.10458			0.576979
$250$		0			0
$251$	−13.5337			−0.854241			−0.427121	−	0.904195i	$-0.640472\pi$
$251$	−13.5337			−0.854241			−0.427121	+	0.904195i	$0.640472\pi$
$252$		0			0
$253$	−10.7698	−	10.7698i	−0.677092	−	0.677092i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.2077	−	14.2077i	−0.886255	−	0.886255i	0.107906	−	0.994161i	$-0.465585\pi$
$257$	−14.2077	−	14.2077i	−0.886255	−	0.886255i	−0.994161	+	0.107906i	$0.965585\pi$
$258$		0			0
$259$	−11.5796			−0.719519
$260$		0			0
$261$		−	8.24619i		−	0.510426i
$262$		0			0
$263$	−9.17947	−	9.17947i	−0.566030	−	0.566030i	0.364984	−	0.931014i	$-0.381075\pi$
$263$	−9.17947	−	9.17947i	−0.566030	−	0.566030i	−0.931014	+	0.364984i	$0.881075\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−3.24466	+	3.24466i	−0.198570	+	0.198570i
$268$		0			0
$269$	−7.92198			−0.483012			−0.241506	−	0.970399i	$-0.577641\pi$
$269$	−7.92198			−0.483012			−0.241506	+	0.970399i	$0.577641\pi$
$270$		0			0
$271$	−11.6880			−0.709994			−0.354997	−	0.934867i	$-0.615518\pi$
$271$	−11.6880			−0.709994			−0.354997	+	0.934867i	$0.615518\pi$
$272$		0			0
$273$	10.8106	+	10.8106i	0.654287	+	0.654287i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.03751	+	2.03751i	−0.122422	+	0.122422i	−0.765663	−	0.643241i	$-0.777589\pi$
$277$	−2.03751	+	2.03751i	−0.122422	+	0.122422i	0.643241	+	0.765663i	$0.277589\pi$
$278$		0			0
$279$	−6.62959			−0.396903
$280$		0			0
$281$		−	7.11479i		−	0.424433i	−0.977223	−	0.212217i	$-0.931932\pi$
$281$		−	7.11479i		−	0.424433i	0.977223	−	0.212217i	$-0.0680682\pi$
$282$		0			0
$283$	19.7140	+	19.7140i	1.17187	+	1.17187i	0.981763	+	0.190112i	$0.0608850\pi$
$283$	19.7140	+	19.7140i	1.17187	+	1.17187i	0.190112	+	0.981763i	$0.439115\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−14.8877	−	14.8877i	−0.878794	−	0.878794i
$288$		0			0
$289$			12.7691i			0.751126i
$290$		0			0
$291$	−2.97107			−0.174167
$292$		0			0
$293$	2.35126	−	2.35126i	0.137362	−	0.137362i	−0.635082	−	0.772444i	$-0.719034\pi$
$293$	2.35126	−	2.35126i	0.137362	−	0.137362i	0.772444	+	0.635082i	$0.219034\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−16.3248	−	16.3248i	−0.947264	−	0.947264i
$298$		0			0
$299$	−18.4822			−1.06885
$300$		0			0
$301$	−3.56266			−0.205348
$302$		0			0
$303$	−11.2880	+	11.2880i	−0.648477	+	0.648477i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.65038	−	1.65038i	−0.0941924	−	0.0941924i	0.658440	−	0.752633i	$-0.271216\pi$
$307$	−1.65038	−	1.65038i	−0.0941924	−	0.0941924i	−0.752633	+	0.658440i	$0.771216\pi$
$308$		0			0
$309$		−	26.0198i		−	1.48021i
$310$		0			0
$311$	−3.66819			−0.208004			−0.104002	−	0.994577i	$-0.533165\pi$
$311$	−3.66819			−0.208004			−0.104002	+	0.994577i	$0.533165\pi$
$312$		0			0
$313$	15.8455	+	15.8455i	0.895639	+	0.895639i	0.995047	−	0.0994081i	$-0.0316949\pi$
$313$	15.8455	+	15.8455i	0.895639	+	0.895639i	−0.0994081	+	0.995047i	$0.531695\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.65120	−	3.65120i	−0.205072	−	0.205072i	0.597097	−	0.802169i	$-0.296321\pi$
$317$	−3.65120	−	3.65120i	−0.205072	−	0.205072i	−0.802169	+	0.597097i	$0.796321\pi$
$318$		0			0
$319$	−43.9671			−2.46169
$320$		0			0
$321$	−7.94776			−0.443601
$322$		0			0
$323$	−7.00610	−	5.59470i	−0.389830	−	0.311298i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−6.37102	+	6.37102i	−0.352318	+	0.352318i
$328$		0			0
$329$		−	23.8945i		−	1.31734i
$330$		0			0
$331$			13.0960i			0.719820i	0.932987	+	0.359910i	$0.117193\pi$
$331$			13.0960i			0.719820i	−0.932987	+	0.359910i	$0.882807\pi$
$332$		0			0
$333$	3.06178	−	3.06178i	0.167785	−	0.167785i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	8.66718	+	8.66718i	0.472131	+	0.472131i	0.902604	−	0.430473i	$-0.141653\pi$
$337$	8.66718	+	8.66718i	0.472131	+	0.472131i	−0.430473	+	0.902604i	$0.641653\pi$
$338$		0			0
$339$			11.6591i			0.633233i
$340$		0			0
$341$			35.3477i			1.91419i
$342$		0			0
$343$	14.2087	+	14.2087i	0.767200	+	0.767200i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.1640	+	17.1640i	−0.921413	+	0.921413i	−0.997129	−	0.0757163i	$-0.975876\pi$
$347$	−17.1640	+	17.1640i	−0.921413	+	0.921413i	0.0757163	+	0.997129i	$0.475876\pi$
$348$		0			0
$349$			4.66819i			0.249882i	0.992164	+	0.124941i	$0.0398742\pi$
$349$			4.66819i			0.249882i	−0.992164	+	0.124941i	$0.960126\pi$
$350$		0			0
$351$	−28.0152			−1.49534
$352$		0			0
$353$	0.00559594	+	0.00559594i	0.000297842	+	0.000297842i	0.707256	−	0.706958i	$-0.249933\pi$
$353$	0.00559594	+	0.00559594i	0.000297842	+	0.000297842i	−0.706958	+	0.707256i	$0.749933\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	4.46836	+	4.46836i	0.236491	+	0.236491i
$358$		0			0
$359$		−	13.0442i		−	0.688444i	−0.938888	−	0.344222i	$-0.888143\pi$
$359$		−	13.0442i		−	0.688444i	0.938888	−	0.344222i	$-0.111857\pi$
$360$		0			0
$361$	−18.5292	+	4.20359i	−0.975219	+	0.221242i
$362$		0			0
$363$	−6.14447	+	6.14447i	−0.322501	+	0.322501i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−18.7952	+	18.7952i	−0.981099	+	0.981099i	−0.999825	−	0.0187258i	$-0.994039\pi$
$367$	−18.7952	+	18.7952i	−0.981099	+	0.981099i	0.0187258	+	0.999825i	$0.494039\pi$
$368$		0			0
$369$	7.87300			0.409852
$370$		0			0
$371$		−	18.1090i		−	0.940171i
$372$		0			0
$373$	7.74340	−	7.74340i	0.400938	−	0.400938i	−0.477626	−	0.878563i	$-0.658502\pi$
$373$	7.74340	−	7.74340i	0.400938	−	0.400938i	0.878563	+	0.477626i	$0.158502\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−37.7262	+	37.7262i	−1.94300	+	1.94300i
$378$		0			0
$379$	34.2653			1.76009			0.880046	−	0.474889i	$-0.157512\pi$
$379$	34.2653			1.76009			0.880046	+	0.474889i	$0.157512\pi$
$380$		0			0
$381$	0.975639			0.0499835
$382$		0			0
$383$	7.04252	−	7.04252i	0.359856	−	0.359856i	−0.503904	−	0.863760i	$-0.668103\pi$
$383$	7.04252	−	7.04252i	0.359856	−	0.359856i	0.863760	+	0.503904i	$0.168103\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.942012	−	0.942012i	0.0478852	−	0.0478852i
$388$		0			0
$389$		−	21.6545i		−	1.09793i	−0.835847	−	0.548963i	$-0.815023\pi$
$389$		−	21.6545i		−	1.09793i	0.835847	−	0.548963i	$-0.184977\pi$
$390$		0			0
$391$	−7.63926			−0.386334
$392$		0			0
$393$	−1.29995	+	1.29995i	−0.0655736	+	0.0655736i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.252896	+	0.252896i	−0.0126925	+	0.0126925i	−0.713425	−	0.700732i	$-0.752857\pi$
$397$	−0.252896	+	0.252896i	−0.0126925	+	0.0126925i	0.700732	+	0.713425i	$0.252857\pi$
$398$		0			0
$399$	13.3082	−	1.49063i	0.666242	−	0.0746249i
$400$		0			0
$401$			21.3422i			1.06578i	0.846185	+	0.532888i	$0.178893\pi$
$401$			21.3422i			1.06578i	−0.846185	+	0.532888i	$0.821107\pi$
$402$		0			0
$403$	30.3303	+	30.3303i	1.51086	+	1.51086i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−16.3248	−	16.3248i	−0.809193	−	0.809193i
$408$		0			0
$409$	−27.7477			−1.37204			−0.686019	−	0.727584i	$-0.740643\pi$
$409$	−27.7477			−1.37204			−0.686019	+	0.727584i	$0.740643\pi$
$410$		0			0
$411$		−	10.0750i		−	0.496962i
$412$		0			0
$413$	7.52531	−	7.52531i	0.370297	−	0.370297i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−16.3200	−	16.3200i	−0.799195	−	0.799195i
$418$		0			0
$419$		−	17.0720i		−	0.834023i	−0.908901	−	0.417012i	$-0.863077\pi$
$419$		−	17.0720i		−	0.834023i	0.908901	−	0.417012i	$-0.136923\pi$
$420$		0			0
$421$		−	12.8230i		−	0.624954i	−0.949925	−	0.312477i	$-0.898841\pi$
$421$		−	12.8230i		−	0.624954i	0.949925	−	0.312477i	$-0.101159\pi$
$422$		0			0
$423$	6.31800	+	6.31800i	0.307192	+	0.307192i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	15.9923	−	15.9923i	0.773922	−	0.773922i
$428$		0			0
$429$			30.4815i			1.47166i
$430$		0			0
$431$		−	18.5824i		−	0.895081i	−0.894264	−	0.447540i	$-0.852300\pi$
$431$		−	18.5824i		−	0.895081i	0.894264	−	0.447540i	$-0.147700\pi$
$432$		0			0
$433$	−15.7258	+	15.7258i	−0.755733	+	0.755733i	−0.975543	−	0.219810i	$-0.929456\pi$
$433$	−15.7258	+	15.7258i	−0.755733	+	0.755733i	0.219810	+	0.975543i	$0.429456\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.1018	+	12.6503i	−0.483237	+	0.605145i
$438$		0			0
$439$	8.03399			0.383442			0.191721	−	0.981450i	$-0.438593\pi$
$439$	8.03399			0.383442			0.191721	+	0.981450i	$0.438593\pi$
$440$		0			0
$441$	−2.12989			−0.101423
$442$		0			0
$443$	−14.1547	−	14.1547i	−0.672508	−	0.672508i	0.285785	−	0.958294i	$-0.407746\pi$
$443$	−14.1547	−	14.1547i	−0.672508	−	0.672508i	−0.958294	+	0.285785i	$0.907746\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	21.2245	+	21.2245i	1.00389	+	1.00389i
$448$		0			0
$449$	11.5306			0.544162			0.272081	−	0.962274i	$-0.412288\pi$
$449$	11.5306			0.544162			0.272081	+	0.962274i	$0.412288\pi$
$450$		0			0
$451$		−	41.9773i		−	1.97664i
$452$		0			0
$453$	12.2296	+	12.2296i	0.574597	+	0.574597i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−23.5119	+	23.5119i	−1.09984	+	1.09984i	−0.105409	+	0.994429i	$0.533615\pi$
$457$	−23.5119	+	23.5119i	−1.09984	+	1.09984i	−0.994429	+	0.105409i	$0.966385\pi$
$458$		0			0
$459$	−11.5796			−0.540488
$460$		0			0
$461$	2.00457			0.0933621			0.0466810	−	0.998910i	$-0.485136\pi$
$461$	2.00457			0.0933621			0.0466810	+	0.998910i	$0.485136\pi$
$462$		0			0
$463$	14.9035	+	14.9035i	0.692622	+	0.692622i	0.962808	−	0.270186i	$-0.0870852\pi$
$463$	14.9035	+	14.9035i	0.692622	+	0.692622i	−0.270186	+	0.962808i	$0.587085\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.72502	−	7.72502i	0.357471	−	0.357471i	−0.505409	−	0.862880i	$-0.668658\pi$
$467$	7.72502	−	7.72502i	0.357471	−	0.357471i	0.862880	+	0.505409i	$0.168658\pi$
$468$		0			0
$469$	15.0368			0.694334
$470$		0			0
$471$			16.6416i			0.766802i
$472$		0			0
$473$	−5.02263	−	5.02263i	−0.230941	−	0.230941i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.78824	+	4.78824i	0.219238	+	0.219238i
$478$		0			0
$479$			26.2019i			1.19720i	0.801049	+	0.598598i	$0.204275\pi$
$479$			26.2019i			1.19720i	−0.801049	+	0.598598i	$0.795725\pi$
$480$		0			0
$481$	−28.0152			−1.27738
$482$		0			0
$483$	8.06810	−	8.06810i	0.367111	−	0.367111i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	8.09866	+	8.09866i	0.366985	+	0.366985i	0.866377	−	0.499391i	$-0.166443\pi$
$487$	8.09866	+	8.09866i	0.366985	+	0.366985i	−0.499391	+	0.866377i	$0.666443\pi$
$488$		0			0
$489$	−27.6988			−1.25258
$490$		0			0
$491$	8.35617			0.377109			0.188554	−	0.982063i	$-0.439620\pi$
$491$	8.35617			0.377109			0.188554	+	0.982063i	$0.439620\pi$
$492$		0			0
$493$	−15.5934	+	15.5934i	−0.702292	+	0.702292i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.71685	+	5.71685i	0.256436	+	0.256436i
$498$		0			0
$499$		−	8.23542i		−	0.368668i	−0.982864	−	0.184334i	$-0.940987\pi$
$499$		−	8.23542i		−	0.368668i	0.982864	−	0.184334i	$-0.0590128\pi$
$500$		0			0
$501$	−5.20192			−0.232405
$502$		0			0
$503$	−2.72663	−	2.72663i	−0.121574	−	0.121574i	0.643702	−	0.765276i	$-0.277398\pi$
$503$	−2.72663	−	2.72663i	−0.121574	−	0.121574i	−0.765276	+	0.643702i	$0.777398\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	12.4250	+	12.4250i	0.551814	+	0.551814i
$508$		0			0
$509$	41.4803			1.83858			0.919291	−	0.393578i	$-0.128763\pi$
$509$	41.4803			1.83858			0.919291	+	0.393578i	$0.128763\pi$
$510$		0			0
$511$	−25.1441			−1.11231
$512$		0			0
$513$	−15.3123	+	19.1753i	−0.676057	+	0.846608i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	33.6864	−	33.6864i	1.48152	−	1.48152i
$518$		0			0
$519$		−	27.9711i		−	1.22779i
$520$		0			0
$521$		−	6.89078i		−	0.301890i	−0.988542	−	0.150945i	$-0.951768\pi$
$521$		−	6.89078i		−	0.301890i	0.988542	−	0.150945i	$-0.0482317\pi$
$522$		0			0
$523$	−3.64638	+	3.64638i	−0.159445	+	0.159445i	−0.782321	−	0.622876i	$-0.785964\pi$
$523$	−3.64638	+	3.64638i	−0.159445	+	0.159445i	0.622876	+	0.782321i	$0.285964\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.5365	+	12.5365i	0.546097	+	0.546097i
$528$		0			0
$529$		−	9.20649i		−	0.400282i
$530$		0			0
$531$			3.97957i			0.172699i
$532$		0			0
$533$	−36.0188	−	36.0188i	−1.56015	−	1.56015i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−14.0553	+	14.0553i	−0.606529	+	0.606529i
$538$		0			0
$539$			11.3562i			0.489145i
$540$		0			0
$541$	28.2937			1.21644			0.608221	−	0.793767i	$-0.291883\pi$
$541$	28.2937			1.21644			0.608221	+	0.793767i	$0.291883\pi$
$542$		0			0
$543$	24.9841	+	24.9841i	1.07217	+	1.07217i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−18.8942	−	18.8942i	−0.807857	−	0.807857i	0.176452	−	0.984309i	$-0.443538\pi$
$547$	−18.8942	−	18.8942i	−0.807857	−	0.807857i	−0.984309	+	0.176452i	$0.943538\pi$
$548$		0			0
$549$			8.45713i			0.360942i
$550$		0			0
$551$	5.20192	+	46.4421i	0.221609	+	1.97850i
$552$		0			0
$553$	−5.17406	+	5.17406i	−0.220023	+	0.220023i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.9692	+	21.9692i	−0.930864	+	0.930864i	−0.997760	−	0.0668956i	$-0.978691\pi$
$557$	−21.9692	+	21.9692i	−0.930864	+	0.930864i	0.0668956	+	0.997760i	$0.478691\pi$
$558$		0			0
$559$	−8.61938			−0.364561
$560$		0			0
$561$			12.5990i			0.531928i
$562$		0			0
$563$	30.5069	−	30.5069i	1.28571	−	1.28571i	0.348345	−	0.937367i	$-0.386744\pi$
$563$	30.5069	−	30.5069i	1.28571	−	1.28571i	0.937367	−	0.348345i	$-0.113256\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	8.87354	−	8.87354i	0.372654	−	0.372654i
$568$		0			0
$569$	7.86118			0.329558			0.164779	−	0.986331i	$-0.447309\pi$
$569$	7.86118			0.329558			0.164779	+	0.986331i	$0.447309\pi$
$570$		0			0
$571$	−2.72043			−0.113846			−0.0569232	−	0.998379i	$-0.518129\pi$
$571$	−2.72043			−0.113846			−0.0569232	+	0.998379i	$0.518129\pi$
$572$		0			0
$573$	−4.33118	+	4.33118i	−0.180937	+	0.180937i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.50083	−	1.50083i	0.0624805	−	0.0624805i	−0.675176	−	0.737657i	$-0.735932\pi$
$577$	1.50083	−	1.50083i	0.0624805	−	0.0624805i	0.737657	+	0.675176i	$0.235932\pi$
$578$		0			0
$579$			27.5104i			1.14329i
$580$		0			0
$581$	−12.5383			−0.520176
$582$		0			0
$583$	25.5300	−	25.5300i	1.05734	−	1.05734i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.18499	−	3.18499i	0.131459	−	0.131459i	−0.638316	−	0.769774i	$-0.720369\pi$
$587$	3.18499	−	3.18499i	0.131459	−	0.131459i	0.769774	+	0.638316i	$0.220369\pi$
$588$		0			0
$589$	37.3375	−	4.18213i	1.53847	−	0.172322i
$590$		0			0
$591$		−	27.8479i		−	1.14551i
$592$		0			0
$593$	18.8714	+	18.8714i	0.774954	+	0.774954i	0.978968	−	0.204014i	$-0.0653987\pi$
$593$	18.8714	+	18.8714i	0.774954	+	0.774954i	−0.204014	+	0.978968i	$0.565399\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−13.9173	−	13.9173i	−0.569597	−	0.569597i
$598$		0			0
$599$	−35.3967			−1.44627			−0.723135	−	0.690706i	$-0.757300\pi$
$599$	−35.3967			−1.44627			−0.723135	+	0.690706i	$0.757300\pi$
$600$		0			0
$601$			8.29518i			0.338367i	0.985585	+	0.169184i	$0.0541131\pi$
$601$			8.29518i			0.338367i	−0.985585	+	0.169184i	$0.945887\pi$
$602$		0			0
$603$	−3.97591	+	3.97591i	−0.161912	+	0.161912i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−18.3857	−	18.3857i	−0.746251	−	0.746251i	0.227522	−	0.973773i	$-0.426938\pi$
$607$	−18.3857	−	18.3857i	−0.746251	−	0.746251i	−0.973773	+	0.227522i	$0.926938\pi$
$608$		0			0
$609$		−	32.9376i		−	1.33470i
$610$		0			0
$611$		−	57.8095i		−	2.33872i
$612$		0			0
$613$	−22.0862	−	22.0862i	−0.892053	−	0.892053i	0.102663	−	0.994716i	$-0.467264\pi$
$613$	−22.0862	−	22.0862i	−0.892053	−	0.892053i	−0.994716	+	0.102663i	$0.967264\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−27.3926	+	27.3926i	−1.10278	+	1.10278i	−0.108711	+	0.994073i	$0.534672\pi$
$617$	−27.3926	+	27.3926i	−1.10278	+	1.10278i	−0.994073	+	0.108711i	$0.965328\pi$
$618$		0			0
$619$			18.5672i			0.746280i	0.927775	+	0.373140i	$0.121719\pi$
$619$			18.5672i			0.746280i	−0.927775	+	0.373140i	$0.878281\pi$
$620$		0			0
$621$			20.9082i			0.839016i
$622$		0			0
$623$	4.46836	−	4.46836i	0.179021	−	0.179021i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	20.8633	+	16.6603i	0.833201	+	0.665350i
$628$		0			0
$629$	−11.5796			−0.461708
$630$		0			0
$631$	−32.7600			−1.30416			−0.652078	−	0.758152i	$-0.726102\pi$
$631$	−32.7600			−1.30416			−0.652078	+	0.758152i	$0.726102\pi$
$632$		0			0
$633$	−19.5197	−	19.5197i	−0.775838	−	0.775838i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	9.74422	+	9.74422i	0.386080	+	0.386080i
$638$		0			0
$639$	−3.02321			−0.119596
$640$		0			0
$641$		−	34.0041i		−	1.34308i	−0.740967	−	0.671542i	$-0.765632\pi$
$641$		−	34.0041i		−	1.34308i	0.740967	−	0.671542i	$-0.234368\pi$
$642$		0			0
$643$	−12.8129	−	12.8129i	−0.505292	−	0.505292i	0.407786	−	0.913078i	$-0.366301\pi$
$643$	−12.8129	−	12.8129i	−0.505292	−	0.505292i	−0.913078	+	0.407786i	$0.866301\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−13.2833	+	13.2833i	−0.522219	+	0.522219i	−0.918241	−	0.396022i	$-0.870390\pi$
$647$	−13.2833	+	13.2833i	−0.522219	+	0.522219i	0.396022	+	0.918241i	$0.370390\pi$
$648$		0			0
$649$	21.2183			0.832893
$650$		0			0
$651$	−26.4804			−1.03785
$652$		0			0
$653$	−6.91330	−	6.91330i	−0.270538	−	0.270538i	0.558779	−	0.829317i	$-0.311270\pi$
$653$	−6.91330	−	6.91330i	−0.270538	−	0.270538i	−0.829317	+	0.558779i	$0.811270\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	6.64840	−	6.64840i	0.259379	−	0.259379i
$658$		0			0
$659$	15.4612			0.602282			0.301141	−	0.953580i	$-0.402633\pi$
$659$	15.4612			0.602282			0.301141	+	0.953580i	$0.402633\pi$
$660$		0			0
$661$		−	13.0960i		−	0.509374i	−0.967023	−	0.254687i	$-0.918027\pi$
$661$		−	13.0960i		−	0.509374i	0.967023	−	0.254687i	$-0.0819725\pi$
$662$		0			0
$663$	10.8106	+	10.8106i	0.419849	+	0.419849i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	28.1556	+	28.1556i	1.09019	+	1.09019i
$668$		0			0
$669$			36.3795i			1.40651i
$670$		0			0
$671$	45.0918			1.74075
$672$		0			0
$673$	−33.5477	+	33.5477i	−1.29317	+	1.29317i	−0.360356	+	0.932815i	$0.617345\pi$
$673$	−33.5477	+	33.5477i	−1.29317	+	1.29317i	−0.932815	+	0.360356i	$0.882655\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	29.3184	+	29.3184i	1.12680	+	1.12680i	0.990695	+	0.136102i	$0.0434574\pi$
$677$	29.3184	+	29.3184i	1.12680	+	1.12680i	0.136102	+	0.990695i	$0.456543\pi$
$678$		0			0
$679$	4.09158			0.157021
$680$		0			0
$681$	−36.8843			−1.41341
$682$		0			0
$683$	−21.9045	+	21.9045i	−0.838153	+	0.838153i	−0.988616	−	0.150463i	$-0.951924\pi$
$683$	−21.9045	+	21.9045i	−0.838153	+	0.838153i	0.150463	+	0.988616i	$0.451924\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−10.0689	−	10.0689i	−0.384153	−	0.384153i
$688$		0			0
$689$		−	43.8122i		−	1.66911i
$690$		0			0
$691$	10.8168			0.411491			0.205746	−	0.978606i	$-0.434038\pi$
$691$	10.8168			0.411491			0.205746	+	0.978606i	$0.434038\pi$
$692$		0			0
$693$	4.58770	+	4.58770i	0.174272	+	0.174272i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−14.8877	−	14.8877i	−0.563913	−	0.563913i
$698$		0			0
$699$	25.1607			0.951667
$700$		0			0
$701$	25.2739			0.954584			0.477292	−	0.878745i	$-0.341618\pi$
$701$	25.2739			0.954584			0.477292	+	0.878745i	$0.341618\pi$
$702$		0			0
$703$	−15.3123	+	19.1753i	−0.577516	+	0.723209i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	15.5451	−	15.5451i	0.584635	−	0.584635i
$708$		0			0
$709$		−	31.8701i		−	1.19691i	−0.801158	−	0.598454i	$-0.795782\pi$
$709$		−	31.8701i		−	1.19691i	0.801158	−	0.598454i	$-0.204218\pi$
$710$		0			0
$711$		−	2.73617i		−	0.102614i
$712$		0			0
$713$	22.6359	−	22.6359i	0.847723	−	0.847723i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−11.3480	−	11.3480i	−0.423798	−	0.423798i
$718$		0			0
$719$		−	14.5916i		−	0.544175i	−0.962273	−	0.272087i	$-0.912286\pi$
$719$		−	14.5916i		−	0.544175i	0.962273	−	0.272087i	$-0.0877139\pi$
$720$		0			0
$721$			35.8329i			1.33449i
$722$		0			0
$723$	−5.34615	−	5.34615i	−0.198825	−	0.198825i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−2.73782	+	2.73782i	−0.101540	+	0.101540i	−0.756052	−	0.654512i	$-0.772874\pi$
$727$	−2.73782	+	2.73782i	−0.101540	+	0.101540i	0.654512	+	0.756052i	$0.272874\pi$
$728$		0			0
$729$			29.9178i			1.10807i
$730$		0			0
$731$	−3.56266			−0.131770
$732$		0			0
$733$	23.4578	+	23.4578i	0.866433	+	0.866433i	0.992076	−	0.125643i	$-0.0400993\pi$
$733$	23.4578	+	23.4578i	0.866433	+	0.866433i	−0.125643	+	0.992076i	$0.540099\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	21.1988	+	21.1988i	0.780868	+	0.780868i
$738$		0			0
$739$			34.3993i			1.26540i	0.774398	+	0.632699i	$0.218053\pi$
$739$			34.3993i			1.26540i	−0.774398	+	0.632699i	$0.781947\pi$
$740$		0			0
$741$	32.1974	−	3.60638i	1.18280	−	0.132484i
$742$		0			0
$743$	−10.2709	+	10.2709i	−0.376804	+	0.376804i	−0.869948	−	0.493144i	$-0.835848\pi$
$743$	−10.2709	+	10.2709i	−0.376804	+	0.376804i	0.493144	+	0.869948i	$0.335848\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	3.31528	−	3.31528i	0.121300	−	0.121300i
$748$		0			0
$749$	10.9452			0.399929
$750$		0			0
$751$		−	7.27579i		−	0.265497i	−0.991150	−	0.132749i	$-0.957620\pi$
$751$		−	7.27579i		−	0.265497i	0.991150	−	0.132749i	$-0.0423803\pi$
$752$		0			0
$753$	14.2935	−	14.2935i	0.520883	−	0.520883i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	20.7863	−	20.7863i	0.755490	−	0.755490i	−0.220008	−	0.975498i	$-0.570608\pi$
$757$	20.7863	−	20.7863i	0.755490	−	0.755490i	0.975498	+	0.220008i	$0.0706083\pi$
$758$		0			0
$759$	22.7488			0.825728
$760$		0			0
$761$	−11.6834			−0.423523			−0.211762	−	0.977321i	$-0.567920\pi$
$761$	−11.6834			−0.423523			−0.211762	+	0.977321i	$0.567920\pi$
$762$		0			0
$763$	8.77380	−	8.77380i	0.317633	−	0.317633i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18.2065	−	18.2065i	0.657398	−	0.657398i
$768$		0			0
$769$			30.2506i			1.09087i	0.838154	+	0.545433i	$0.183635\pi$
$769$			30.2506i			1.09087i	−0.838154	+	0.545433i	$0.816365\pi$
$770$		0			0
$771$	30.0107			1.08081
$772$		0			0
$773$	−11.4594	+	11.4594i	−0.412167	+	0.412167i	−0.882493	−	0.470326i	$-0.844136\pi$
$773$	−11.4594	+	11.4594i	−0.412167	+	0.412167i	0.470326	+	0.882493i	$0.344136\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	12.2296	−	12.2296i	0.438735	−	0.438735i
$778$		0			0
$779$	−44.3403	+	4.96650i	−1.58866	+	0.177943i
$780$		0			0
$781$			16.1192i			0.576790i
$782$		0			0
$783$	42.6782	+	42.6782i	1.52519	+	1.52519i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	16.1828	+	16.1828i	0.576856	+	0.576856i	0.934036	−	0.357180i	$-0.116262\pi$
$787$	16.1828	+	16.1828i	0.576856	+	0.576856i	−0.357180	+	0.934036i	$0.616262\pi$
$788$		0			0
$789$	19.3895			0.690286
$790$		0			0
$791$		−	16.0562i		−	0.570892i
$792$		0			0
$793$	38.6912	−	38.6912i	1.37397	−	1.37397i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.02244	−	9.02244i	−0.319591	−	0.319591i	0.529019	−	0.848610i	$-0.322560\pi$
$797$	−9.02244	−	9.02244i	−0.319591	−	0.319591i	−0.848610	+	0.529019i	$0.822560\pi$
$798$		0			0
$799$		−	23.8945i		−	0.845326i
$800$		0			0
$801$			2.36298i			0.0834918i
$802$		0			0
$803$	−35.4480	−	35.4480i	−1.25093	−	1.25093i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	8.36670	−	8.36670i	0.294522	−	0.294522i
$808$		0			0
$809$		−	14.7265i		−	0.517757i	−0.965910	−	0.258878i	$-0.916647\pi$
$809$		−	14.7265i		−	0.517757i	0.965910	−	0.258878i	$-0.0833529\pi$
$810$		0			0
$811$			43.3699i			1.52292i	0.648210	+	0.761462i	$0.275518\pi$
$811$			43.3699i			1.52292i	−0.648210	+	0.761462i	$0.724482\pi$
$812$		0			0
$813$	12.3441	−	12.3441i	0.432927	−	0.432927i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.71112	+	5.89961i	−0.164821	+	0.206401i
$818$		0			0
$819$	7.87300			0.275105
$820$		0			0
$821$	30.7600			1.07353			0.536766	−	0.843731i	$-0.319646\pi$
$821$	30.7600			1.07353			0.536766	+	0.843731i	$0.319646\pi$
$822$		0			0
$823$	−40.0454	−	40.0454i	−1.39589	−	1.39589i	−0.811392	−	0.584503i	$-0.801290\pi$
$823$	−40.0454	−	40.0454i	−1.39589	−	1.39589i	−0.584503	−	0.811392i	$-0.698710\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.2558	−	24.2558i	−0.843456	−	0.843456i	0.145851	−	0.989307i	$-0.453408\pi$
$827$	−24.2558	−	24.2558i	−0.843456	−	0.843456i	−0.989307	+	0.145851i	$0.953408\pi$
$828$		0			0
$829$	−46.7663			−1.62426			−0.812131	−	0.583476i	$-0.801692\pi$
$829$	−46.7663			−1.62426			−0.812131	+	0.583476i	$0.801692\pi$
$830$		0			0
$831$		−	4.30378i		−	0.149297i
$832$		0			0
$833$	4.02759	+	4.02759i	0.139548	+	0.139548i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	34.3115	−	34.3115i	1.18598	−	1.18598i
$838$		0			0
$839$	−2.69901			−0.0931800			−0.0465900	−	0.998914i	$-0.514835\pi$
$839$	−2.69901			−0.0931800			−0.0465900	+	0.998914i	$0.514835\pi$
$840$		0			0
$841$	85.9437			2.96357
$842$		0			0
$843$	7.51420	+	7.51420i	0.258803	+	0.258803i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	8.46181	−	8.46181i	0.290751	−	0.290751i
$848$		0			0
$849$	−41.6413			−1.42913
$850$		0			0
$851$			20.9082i			0.716723i
$852$		0			0
$853$	4.66927	+	4.66927i	0.159873	+	0.159873i	0.782510	−	0.622638i	$-0.213939\pi$
$853$	4.66927	+	4.66927i	0.159873	+	0.159873i	−0.622638	+	0.782510i	$0.713939\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−28.0442	−	28.0442i	−0.957970	−	0.957970i	0.0411816	−	0.999152i	$-0.486888\pi$
$857$	−28.0442	−	28.0442i	−0.957970	−	0.957970i	−0.999152	+	0.0411816i	$0.986888\pi$
$858$		0			0
$859$			30.3074i			1.03408i	0.855962	+	0.517038i	$0.172966\pi$
$859$			30.3074i			1.03408i	−0.855962	+	0.517038i	$0.827034\pi$
$860$		0			0
$861$	31.4469			1.07171
$862$		0			0
$863$	−10.1852	+	10.1852i	−0.346708	+	0.346708i	−0.858882	−	0.512174i	$-0.828840\pi$
$863$	−10.1852	+	10.1852i	−0.346708	+	0.346708i	0.512174	+	0.858882i	$0.328840\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−13.4860	−	13.4860i	−0.458008	−	0.458008i
$868$		0			0
$869$	−14.5887			−0.494889
$870$		0			0
$871$	36.3795			1.23267
$872$		0			0
$873$	−1.08187	+	1.08187i	−0.0366156	+	0.0366156i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	4.26635	+	4.26635i	0.144065	+	0.144065i	0.775461	−	0.631396i	$-0.217518\pi$
$877$	4.26635	+	4.26635i	0.144065	+	0.144065i	−0.631396	+	0.775461i	$0.717518\pi$
$878$		0			0
$879$			4.96650i			0.167516i
$880$		0			0
$881$	−30.2704			−1.01984			−0.509918	−	0.860223i	$-0.670324\pi$
$881$	−30.2704			−1.01984			−0.509918	+	0.860223i	$0.670324\pi$
$882$		0			0
$883$	−19.5085	−	19.5085i	−0.656514	−	0.656514i	0.298040	−	0.954553i	$-0.403667\pi$
$883$	−19.5085	−	19.5085i	−0.656514	−	0.656514i	−0.954553	+	0.298040i	$0.903667\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.39132	−	3.39132i	−0.113869	−	0.113869i	0.647876	−	0.761746i	$-0.275657\pi$
$887$	−3.39132	−	3.39132i	−0.113869	−	0.113869i	−0.761746	+	0.647876i	$0.775657\pi$
$888$		0			0
$889$	−1.34359			−0.0450626
$890$		0			0
$891$	25.0198			0.838195
$892$		0			0
$893$	−39.5682	−	31.5971i	−1.32410	−	1.05736i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	19.5197	−	19.5197i	0.651744	−	0.651744i
$898$		0			0
$899$		−	92.4099i		−	3.08204i
$900$		0			0
$901$		−	18.1090i		−	0.603297i
$902$		0			0
$903$	3.76266	−	3.76266i	0.125213	−	0.125213i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−10.5356	−	10.5356i	−0.349830	−	0.349830i	0.510216	−	0.860046i	$-0.329565\pi$
$907$	−10.5356	−	10.5356i	−0.349830	−	0.349830i	−0.860046	+	0.510216i	$0.829565\pi$
$908$		0			0
$909$			8.22066i			0.272662i
$910$		0			0
$911$			4.63981i			0.153724i	0.997042	+	0.0768618i	$0.0244900\pi$
$911$			4.63981i			0.153724i	−0.997042	+	0.0768618i	$0.975510\pi$
$912$		0			0
$913$	−17.6765	−	17.6765i	−0.585005	−	0.585005i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.79021	−	1.79021i	0.0591180	−	0.0591180i
$918$		0			0
$919$			6.15777i			0.203126i	0.994829	+	0.101563i	$0.0323843\pi$
$919$			6.15777i			0.203126i	−0.994829	+	0.101563i	$0.967616\pi$
$920$		0			0
$921$	3.48606			0.114870
$922$		0			0
$923$	13.8311	+	13.8311i	0.455258	+	0.455258i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−9.47468	−	9.47468i	−0.311189	−	0.311189i
$928$		0			0
$929$			40.1360i			1.31682i	0.752660	+	0.658409i	$0.228770\pi$
$929$			40.1360i			1.31682i	−0.752660	+	0.658409i	$0.771230\pi$
$930$		0			0
$931$	11.9954	−	1.34359i	0.393134	−	0.0440345i
$932$		0			0
$933$	3.87411	−	3.87411i	0.126833	−	0.126833i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	3.99880	−	3.99880i	0.130635	−	0.130635i	−0.638766	−	0.769401i	$-0.720555\pi$
$937$	3.99880	−	3.99880i	0.130635	−	0.130635i	0.769401	+	0.638766i	$0.220555\pi$
$938$		0			0
$939$	−33.4700			−1.09225
$940$		0			0
$941$			21.9788i			0.716487i	0.933628	+	0.358244i	$0.116624\pi$
$941$			21.9788i			0.716487i	−0.933628	+	0.358244i	$0.883376\pi$
$942$		0			0
$943$	−26.8814	+	26.8814i	−0.875379	+	0.875379i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3.14525	−	3.14525i	0.102207	−	0.102207i	−0.654154	−	0.756361i	$-0.726975\pi$
$947$	3.14525	−	3.14525i	0.102207	−	0.102207i	0.756361	+	0.654154i	$0.226975\pi$
$948$		0			0
$949$	−60.8327			−1.97471
$950$		0			0
$951$	7.71234			0.250090
$952$		0			0
$953$	−6.19964	+	6.19964i	−0.200826	+	0.200826i	−0.800354	−	0.599528i	$-0.795355\pi$
$953$	−6.19964	+	6.19964i	−0.200826	+	0.200826i	0.599528	+	0.800354i	$0.295355\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	46.4353	−	46.4353i	1.50104	−	1.50104i
$958$		0			0
$959$			13.8747i			0.448037i
$960$		0			0
$961$	−43.2937			−1.39657
$962$		0			0
$963$	−2.89405	+	2.89405i	−0.0932593	+	0.0932593i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−12.1358	+	12.1358i	−0.390261	+	0.390261i	−0.874780	−	0.484520i	$-0.838994\pi$
$967$	−12.1358	+	12.1358i	−0.390261	+	0.390261i	0.484520	+	0.874780i	$0.338994\pi$
$968$		0			0
$969$	13.3082	−	1.49063i	0.427520	−	0.0478860i
$970$		0			0
$971$			53.6571i			1.72194i	0.508657	+	0.860969i	$0.330142\pi$
$971$			53.6571i			1.72194i	−0.508657	+	0.860969i	$0.669858\pi$
$972$		0			0
$973$	22.4750	+	22.4750i	0.720515	+	0.720515i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.928606	+	0.928606i	0.0297087	+	0.0297087i	0.721805	−	0.692096i	$-0.243313\pi$
$977$	0.928606	+	0.928606i	0.0297087	+	0.0297087i	−0.692096	+	0.721805i	$0.743313\pi$
$978$		0			0
$979$	12.5990			0.402664
$980$		0			0
$981$			4.63981i			0.148138i
$982$		0			0
$983$	24.7900	−	24.7900i	0.790679	−	0.790679i	−0.190926	−	0.981605i	$-0.561149\pi$
$983$	24.7900	−	24.7900i	0.790679	−	0.790679i	0.981605	+	0.190926i	$0.0611489\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	25.2358	+	25.2358i	0.803265	+	0.803265i
$988$		0			0
$989$			6.43277i			0.204550i
$990$		0			0
$991$			36.3063i			1.15331i	0.816988	+	0.576654i	$0.195642\pi$
$991$			36.3063i			1.15331i	−0.816988	+	0.576654i	$0.804358\pi$
$992$		0			0
$993$	−13.8311	−	13.8311i	−0.438918	−	0.438918i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	32.0552	−	32.0552i	1.01520	−	1.01520i	0.0153168	−	0.999883i	$-0.495124\pi$
$997$	32.0552	−	32.0552i	1.01520	−	1.01520i	0.999883	−	0.0153168i	$-0.00487567\pi$
$998$		0			0
$999$			31.6926i			1.00271i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.l.d.493.4	yes	24
5.2	odd	4	inner	1900.2.l.d.1557.10	yes	24
5.3	odd	4	inner	1900.2.l.d.1557.3	yes	24
5.4	even	2	inner	1900.2.l.d.493.9	yes	24
19.18	odd	2	inner	1900.2.l.d.493.10	yes	24
95.18	even	4	inner	1900.2.l.d.1557.9	yes	24
95.37	even	4	inner	1900.2.l.d.1557.4	yes	24
95.94	odd	2	inner	1900.2.l.d.493.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.l.d.493.3	✓	24	95.94	odd	2	inner
1900.2.l.d.493.4	yes	24	1.1	even	1	trivial
1900.2.l.d.493.9	yes	24	5.4	even	2	inner
1900.2.l.d.493.10	yes	24	19.18	odd	2	inner
1900.2.l.d.1557.3	yes	24	5.3	odd	4	inner
1900.2.l.d.1557.4	yes	24	95.37	even	4	inner
1900.2.l.d.1557.9	yes	24	95.18	even	4	inner
1900.2.l.d.1557.10	yes	24	5.2	odd	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.05614 + 1.05614i) q^{3} +(1.45445 - 1.45445i) q^{7} +0.769150i q^{9} +4.10096 q^{11} +(3.51885 - 3.51885i) q^{13} +(1.45445 - 1.45445i) q^{17} +(-0.485200 - 4.33181i) q^{19} +3.07220i q^{21} +(-2.62617 - 2.62617i) q^{23} +(-3.98074 - 3.98074i) q^{27} -10.7212 q^{29} +8.61938i q^{31} +(-4.33118 + 4.33118i) q^{33} +(-3.98074 - 3.98074i) q^{37} +7.43277i q^{39} -10.2360i q^{41} +(-1.22474 - 1.22474i) q^{43} +(8.21426 - 8.21426i) q^{47} +2.76915i q^{49} +3.07220i q^{51} +(6.22537 - 6.22537i) q^{53} +(5.08742 + 4.06255i) q^{57} +5.17399 q^{59} +10.9954 q^{61} +(1.11869 + 1.11869i) q^{63} +(5.16923 + 5.16923i) q^{67} +5.54718 q^{69} +3.93059i q^{71} +(-8.64384 - 8.64384i) q^{73} +(5.96464 - 5.96464i) q^{77} -3.55740 q^{79} +6.10096 q^{81} +(-4.31032 - 4.31032i) q^{83} +(11.3230 - 11.3230i) q^{87} +3.07220 q^{89} -10.2360i q^{91} +(-9.10325 - 9.10325i) q^{93} +(1.40657 + 1.40657i) q^{97} +3.15425i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{11} + 24 q^{61} + 24 q^{81}+O(q^{100}) \) 24 * q - 24 * q^11 + 24 * q^61 + 24 * q^81

Introduction

L-functions

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