LMFDB - Embedded newform 1848.2.v.e.881.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1848,2,Mod(881,1848)] 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("1848.881"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1848, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1848.v (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$14.7563542935$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.16
Character	$\chi$	$=$	1848.881
Dual form			1848.2.v.e.881.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.228186 + 1.71695i) q^{3} +1.84204 q^{5} +(0.286690 + 2.63017i) q^{7} +(-2.89586 - 0.783571i) q^{9} +1.00000i q^{11} +6.65872i q^{13} +(-0.420328 + 3.16270i) q^{15} -6.91093 q^{17} +0.932936i q^{19} +(-4.58130 - 0.107936i) q^{21} -6.36510i q^{23} -1.60689 q^{25} +(2.00615 - 4.79326i) q^{27} +1.98122i q^{29} -5.52322i q^{31} +(-1.71695 - 0.228186i) q^{33} +(0.528094 + 4.84488i) q^{35} +0.344178 q^{37} +(-11.4327 - 1.51943i) q^{39} -1.92582 q^{41} -1.37134 q^{43} +(-5.33429 - 1.44337i) q^{45} +10.2056 q^{47} +(-6.83562 + 1.50809i) q^{49} +(1.57698 - 11.8657i) q^{51} +6.02011i q^{53} +1.84204i q^{55} +(-1.60181 - 0.212883i) q^{57} +10.1230 q^{59} +13.4033i q^{61} +(1.23071 - 7.84126i) q^{63} +12.2656i q^{65} -7.35794 q^{67} +(10.9286 + 1.45243i) q^{69} +7.58725i q^{71} -7.92560i q^{73} +(0.366670 - 2.75895i) q^{75} +(-2.63017 + 0.286690i) q^{77} -0.993655 q^{79} +(7.77203 + 4.53823i) q^{81} -6.57988 q^{83} -12.7302 q^{85} +(-3.40166 - 0.452087i) q^{87} -9.35319 q^{89} +(-17.5136 + 1.90899i) q^{91} +(9.48311 + 1.26032i) q^{93} +1.71851i q^{95} -11.9745i q^{97} +(0.783571 - 2.89586i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 4 q^{7} + 4 q^{9} - 16 q^{15} + 6 q^{21} + 68 q^{25} - 68 q^{37} - 32 q^{39} + 120 q^{43} - 64 q^{49} + 56 q^{51} - 44 q^{57} + 42 q^{63} + 4 q^{67} - 56 q^{79} - 28 q^{81} - 44 q^{91} - 28 q^{93}+O(q^{100}) $ 32 * q + 4 * q^7 + 4 * q^9 - 16 * q^15 + 6 * q^21 + 68 * q^25 - 68 * q^37 - 32 * q^39 + 120 * q^43 - 64 * q^49 + 56 * q^51 - 44 * q^57 + 42 * q^63 + 4 * q^67 - 56 * q^79 - 28 * q^81 - 44 * q^91 - 28 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1848\mathbb{Z}\right)^\times$.

$n$	$463$	$617$	$673$	$925$	$1585$
$\chi(n)$	$1$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.228186	+	1.71695i	−0.131743	+	0.991284i
$3$	−0.228186	+	1.71695i	−0.131743	+	0.991284i
$4$		0			0
$5$	1.84204			0.823785			0.411893	−	0.911232i	$-0.364868\pi$
$5$	1.84204			0.823785			0.411893	+	0.911232i	$0.364868\pi$
$6$		0			0
$7$	0.286690	+	2.63017i	0.108359	+	0.994112i
$7$	0.286690	+	2.63017i	0.108359	+	0.994112i
$8$		0			0
$9$	−2.89586	−	0.783571i	−0.965287	−	0.261190i
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$			6.65872i			1.84680i	0.383845	+	0.923398i	$0.374600\pi$
$13$			6.65872i			1.84680i	−0.383845	+	0.923398i	$0.625400\pi$
$14$		0			0
$15$	−0.420328	+	3.16270i	−0.108528	+	0.816605i
$16$		0			0
$17$	−6.91093			−1.67615			−0.838073	−	0.545558i	$-0.816318\pi$
$17$	−6.91093			−1.67615			−0.838073	+	0.545558i	$0.816318\pi$
$18$		0			0
$19$			0.932936i			0.214030i	0.994257	+	0.107015i	$0.0341293\pi$
$19$			0.932936i			0.214030i	−0.994257	+	0.107015i	$0.965871\pi$
$20$		0			0
$21$	−4.58130	−	0.107936i	−0.999723	−	0.0235536i
$22$		0			0
$23$		−	6.36510i		−	1.32722i	−0.748081	−	0.663608i	$-0.769024\pi$
$23$		−	6.36510i		−	1.32722i	0.748081	−	0.663608i	$-0.230976\pi$
$24$		0			0
$25$	−1.60689			−0.321378
$26$		0			0
$27$	2.00615	−	4.79326i	0.386084	−	0.922464i
$28$		0			0
$29$			1.98122i			0.367903i	0.982935	+	0.183951i	$0.0588889\pi$
$29$			1.98122i			0.367903i	−0.982935	+	0.183951i	$0.941111\pi$
$30$		0			0
$31$		−	5.52322i		−	0.992000i	−0.868323	−	0.496000i	$-0.834802\pi$
$31$		−	5.52322i		−	0.992000i	0.868323	−	0.496000i	$-0.165198\pi$
$32$		0			0
$33$	−1.71695	−	0.228186i	−0.298883	−	0.0397221i
$34$		0			0
$35$	0.528094	+	4.84488i	0.0892642	+	0.818935i
$36$		0			0
$37$	0.344178			0.0565826			0.0282913	−	0.999600i	$-0.490993\pi$
$37$	0.344178			0.0565826			0.0282913	+	0.999600i	$0.490993\pi$
$38$		0			0
$39$	−11.4327	−	1.51943i	−1.83070	−	0.243303i
$40$		0			0
$41$	−1.92582			−0.300763			−0.150381	−	0.988628i	$-0.548050\pi$
$41$	−1.92582			−0.300763			−0.150381	+	0.988628i	$0.548050\pi$
$42$		0			0
$43$	−1.37134			−0.209128			−0.104564	−	0.994518i	$-0.533345\pi$
$43$	−1.37134			−0.209128			−0.104564	+	0.994518i	$0.533345\pi$
$44$		0			0
$45$	−5.33429	−	1.44337i	−0.795189	−	0.215165i
$46$		0			0
$47$	10.2056			1.48864			0.744319	−	0.667825i	$-0.232774\pi$
$47$	10.2056			1.48864			0.744319	+	0.667825i	$0.232774\pi$
$48$		0			0
$49$	−6.83562	+	1.50809i	−0.976517	+	0.215441i
$50$		0			0
$51$	1.57698	−	11.8657i	0.220821	−	1.66154i
$52$		0			0
$53$			6.02011i			0.826926i	0.910521	+	0.413463i	$0.135681\pi$
$53$			6.02011i			0.826926i	−0.910521	+	0.413463i	$0.864319\pi$
$54$		0			0
$55$			1.84204i			0.248381i
$56$		0			0
$57$	−1.60181	−	0.212883i	−0.212165	−	0.0281971i
$58$		0			0
$59$	10.1230			1.31791			0.658953	−	0.752184i	$-0.270999\pi$
$59$	10.1230			1.31791			0.658953	+	0.752184i	$0.270999\pi$
$60$		0			0
$61$			13.4033i			1.71612i	0.513550	+	0.858059i	$0.328330\pi$
$61$			13.4033i			1.71612i	−0.513550	+	0.858059i	$0.671670\pi$
$62$		0			0
$63$	1.23071	−	7.84126i	0.155055	−	0.987906i
$64$		0			0
$65$			12.2656i			1.52136i
$66$		0			0
$67$	−7.35794			−0.898916			−0.449458	−	0.893301i	$-0.648383\pi$
$67$	−7.35794			−0.898916			−0.449458	+	0.893301i	$0.648383\pi$
$68$		0			0
$69$	10.9286	+	1.45243i	1.31565	+	0.174852i
$70$		0			0
$71$			7.58725i			0.900441i	0.892917	+	0.450221i	$0.148655\pi$
$71$			7.58725i			0.900441i	−0.892917	+	0.450221i	$0.851345\pi$
$72$		0			0
$73$		−	7.92560i		−	0.927622i	−0.885934	−	0.463811i	$-0.846482\pi$
$73$		−	7.92560i		−	0.927622i	0.885934	−	0.463811i	$-0.153518\pi$
$74$		0			0
$75$	0.366670	−	2.75895i	0.0423394	−	0.318577i
$76$		0			0
$77$	−2.63017	+	0.286690i	−0.299736	+	0.0326713i
$78$		0			0
$79$	−0.993655			−0.111795			−0.0558974	−	0.998437i	$-0.517802\pi$
$79$	−0.993655			−0.111795			−0.0558974	+	0.998437i	$0.517802\pi$
$80$		0			0
$81$	7.77203	+	4.53823i	0.863559	+	0.504247i
$82$		0			0
$83$	−6.57988			−0.722235			−0.361118	−	0.932520i	$-0.617605\pi$
$83$	−6.57988			−0.722235			−0.361118	+	0.932520i	$0.617605\pi$
$84$		0			0
$85$	−12.7302			−1.38078
$86$		0			0
$87$	−3.40166	−	0.452087i	−0.364696	−	0.0484688i
$88$		0			0
$89$	−9.35319			−0.991436			−0.495718	−	0.868484i	$-0.665095\pi$
$89$	−9.35319			−0.991436			−0.495718	+	0.868484i	$0.665095\pi$
$90$		0			0
$91$	−17.5136	+	1.90899i	−1.83592	+	0.200116i
$92$		0			0
$93$	9.48311	+	1.26032i	0.983353	+	0.130689i
$94$		0			0
$95$			1.71851i			0.176315i
$96$		0			0
$97$		−	11.9745i		−	1.21582i	−0.794004	−	0.607912i	$-0.792007\pi$
$97$		−	11.9745i		−	1.21582i	0.794004	−	0.607912i	$-0.207993\pi$
$98$		0			0
$99$	0.783571	−	2.89586i	0.0787518	−	0.291045i
$100$		0			0
$101$	9.97020			0.992072			0.496036	−	0.868302i	$-0.334788\pi$
$101$	9.97020			0.992072			0.496036	+	0.868302i	$0.334788\pi$
$102$		0			0
$103$			11.1682i			1.10044i	0.835021	+	0.550218i	$0.185455\pi$
$103$			11.1682i			1.10044i	−0.835021	+	0.550218i	$0.814545\pi$
$104$		0			0
$105$	−8.43895	−	0.198822i	−0.823557	−	0.0194031i
$106$		0			0
$107$		−	8.60700i		−	0.832070i	−0.909349	−	0.416035i	$-0.863419\pi$
$107$		−	8.60700i		−	0.832070i	0.909349	−	0.416035i	$-0.136581\pi$
$108$		0			0
$109$	−3.61223			−0.345989			−0.172995	−	0.984923i	$-0.555344\pi$
$109$	−3.61223			−0.345989			−0.172995	+	0.984923i	$0.555344\pi$
$110$		0			0
$111$	−0.0785368	+	0.590939i	−0.00745438	+	0.0560894i
$112$		0			0
$113$		−	11.5392i		−	1.08551i	−0.839890	−	0.542757i	$-0.817380\pi$
$113$		−	11.5392i		−	1.08551i	0.839890	−	0.542757i	$-0.182620\pi$
$114$		0			0
$115$		−	11.7248i		−	1.09334i
$116$		0			0
$117$	5.21757	−	19.2827i	0.482365	−	1.78269i
$118$		0			0
$119$	−1.98129	−	18.1769i	−0.181625	−	1.66628i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	0.439446	−	3.30654i	0.0396235	−	0.298141i
$124$		0			0
$125$	−12.1702			−1.08853
$126$		0			0
$127$	5.31196			0.471360			0.235680	−	0.971831i	$-0.424268\pi$
$127$	5.31196			0.471360			0.235680	+	0.971831i	$0.424268\pi$
$128$		0			0
$129$	0.312922	−	2.35453i	0.0275512	−	0.207305i
$130$		0			0
$131$	3.98885			0.348508			0.174254	−	0.984701i	$-0.444249\pi$
$131$	3.98885			0.348508			0.174254	+	0.984701i	$0.444249\pi$
$132$		0			0
$133$	−2.45378	+	0.267463i	−0.212770	+	0.0231920i
$134$		0			0
$135$	3.69541	−	8.82938i	0.318050	−	0.759912i
$136$		0			0
$137$			17.9141i			1.53051i	0.643729	+	0.765253i	$0.277386\pi$
$137$			17.9141i			1.53051i	−0.643729	+	0.765253i	$0.722614\pi$
$138$		0			0
$139$			22.2150i			1.88425i	0.335263	+	0.942124i	$0.391175\pi$
$139$			22.2150i			1.88425i	−0.335263	+	0.942124i	$0.608825\pi$
$140$		0			0
$141$	−2.32877	+	17.5225i	−0.196118	+	1.47566i
$142$		0			0
$143$	−6.65872			−0.556830
$144$		0			0
$145$			3.64948i			0.303073i
$146$		0			0
$147$	−1.02952	−	12.0806i	−0.0849137	−	0.996388i
$148$		0			0
$149$			3.26433i			0.267424i	0.991020	+	0.133712i	$0.0426898\pi$
$149$			3.26433i			0.267424i	−0.991020	+	0.133712i	$0.957310\pi$
$150$		0			0
$151$	7.13598			0.580718			0.290359	−	0.956918i	$-0.406225\pi$
$151$	7.13598			0.580718			0.290359	+	0.956918i	$0.406225\pi$
$152$		0			0
$153$	20.0131	+	5.41520i	1.61796	+	0.437793i
$154$		0			0
$155$		−	10.1740i		−	0.817195i
$156$		0			0
$157$		−	18.7298i		−	1.49480i	−0.664374	−	0.747400i	$-0.731302\pi$
$157$		−	18.7298i		−	1.49480i	0.664374	−	0.747400i	$-0.268698\pi$
$158$		0			0
$159$	−10.3363	−	1.37371i	−0.819718	−	0.108942i
$160$		0			0
$161$	16.7413	−	1.82481i	1.31940	−	0.143815i
$162$		0			0
$163$	5.35976			0.419809			0.209904	−	0.977722i	$-0.432685\pi$
$163$	5.35976			0.419809			0.209904	+	0.977722i	$0.432685\pi$
$164$		0			0
$165$	−3.16270	−	0.420328i	−0.246216	−	0.0327225i
$166$		0			0
$167$	−10.7726			−0.833610			−0.416805	−	0.908996i	$-0.636850\pi$
$167$	−10.7726			−0.833610			−0.416805	+	0.908996i	$0.636850\pi$
$168$		0			0
$169$	−31.3385			−2.41065
$170$		0			0
$171$	0.731022	−	2.70166i	0.0559026	−	0.206601i
$172$		0			0
$173$	−9.06474			−0.689180			−0.344590	−	0.938753i	$-0.611982\pi$
$173$	−9.06474			−0.689180			−0.344590	+	0.938753i	$0.611982\pi$
$174$		0			0
$175$	−0.460679	−	4.22640i	−0.0348241	−	0.319486i
$176$		0			0
$177$	−2.30994	+	17.3808i	−0.173626	+	1.30642i
$178$		0			0
$179$			19.1310i			1.42992i	0.699167	+	0.714959i	$0.253555\pi$
$179$			19.1310i			1.42992i	−0.699167	+	0.714959i	$0.746445\pi$
$180$		0			0
$181$			14.1149i			1.04915i	0.851365	+	0.524574i	$0.175776\pi$
$181$			14.1149i			1.04915i	−0.851365	+	0.524574i	$0.824224\pi$
$182$		0			0
$183$	−23.0129	−	3.05845i	−1.70116	−	0.226087i
$184$		0			0
$185$	0.633990			0.0466119
$186$		0			0
$187$		−	6.91093i		−	0.505377i
$188$		0			0
$189$	13.1823	+	3.90234i	0.958868	+	0.283854i
$190$		0			0
$191$		−	13.9988i		−	1.01292i	−0.862265	−	0.506458i	$-0.830955\pi$
$191$		−	13.9988i		−	1.01292i	0.862265	−	0.506458i	$-0.169045\pi$
$192$		0			0
$193$	14.0720			1.01292			0.506462	−	0.862262i	$-0.330953\pi$
$193$	14.0720			1.01292			0.506462	+	0.862262i	$0.330953\pi$
$194$		0			0
$195$	−21.0595	−	2.79885i	−1.50810	−	0.200430i
$196$		0			0
$197$		−	0.145471i		−	0.0103644i	−0.999987	−	0.00518219i	$-0.998350\pi$
$197$		−	0.145471i		−	0.0103644i	0.999987	−	0.00518219i	$-0.00164955\pi$
$198$		0			0
$199$			14.2886i			1.01289i	0.862273	+	0.506444i	$0.169040\pi$
$199$			14.2886i			1.01289i	−0.862273	+	0.506444i	$0.830960\pi$
$200$		0			0
$201$	1.67898	−	12.6333i	0.118426	−	0.891081i
$202$		0			0
$203$	−5.21094	+	0.567995i	−0.365736	+	0.0398654i
$204$		0			0
$205$	−3.54744			−0.247764
$206$		0			0
$207$	−4.98751	+	18.4325i	−0.346656	+	1.28114i
$208$		0			0
$209$	−0.932936			−0.0645326
$210$		0			0
$211$	−16.2575			−1.11921			−0.559607	−	0.828758i	$-0.689048\pi$
$211$	−16.2575			−1.11921			−0.559607	+	0.828758i	$0.689048\pi$
$212$		0			0
$213$	−13.0270	−	1.73131i	−0.892593	−	0.118627i
$214$		0			0
$215$	−2.52607			−0.172276
$216$		0			0
$217$	14.5270	−	1.58345i	0.986159	−	0.107492i
$218$		0			0
$219$	13.6079	+	1.80851i	0.919536	+	0.122208i
$220$		0			0
$221$		−	46.0179i		−	3.09550i
$222$		0			0
$223$			11.0700i			0.741300i	0.928773	+	0.370650i	$0.120865\pi$
$223$			11.0700i			0.741300i	−0.928773	+	0.370650i	$0.879135\pi$
$224$		0			0
$225$	4.65333	+	1.25911i	0.310222	+	0.0839408i
$226$		0			0
$227$	13.1716			0.874233			0.437116	−	0.899405i	$-0.356000\pi$
$227$	13.1716			0.874233			0.437116	+	0.899405i	$0.356000\pi$
$228$		0			0
$229$			16.6082i			1.09750i	0.835985	+	0.548752i	$0.184897\pi$
$229$			16.6082i			1.09750i	−0.835985	+	0.548752i	$0.815103\pi$
$230$		0			0
$231$	0.107936	−	4.58130i	0.00710166	−	0.301428i
$232$		0			0
$233$			13.9908i			0.916568i	0.888806	+	0.458284i	$0.151536\pi$
$233$			13.9908i			0.916568i	−0.888806	+	0.458284i	$0.848464\pi$
$234$		0			0
$235$	18.7991			1.22632
$236$		0			0
$237$	0.226738	−	1.70606i	0.0147282	−	0.110820i
$238$		0			0
$239$			8.11661i			0.525020i	0.964929	+	0.262510i	$0.0845503\pi$
$239$			8.11661i			0.525020i	−0.964929	+	0.262510i	$0.915450\pi$
$240$		0			0
$241$			23.2933i			1.50046i	0.661179	+	0.750228i	$0.270056\pi$
$241$			23.2933i			1.50046i	−0.661179	+	0.750228i	$0.729944\pi$
$242$		0			0
$243$	−9.56540	+	12.3087i	−0.613620	+	0.789601i
$244$		0			0
$245$	−12.5915	+	2.77796i	−0.804440	+	0.177477i
$246$		0			0
$247$	−6.21216			−0.395270
$248$		0			0
$249$	1.50144	−	11.2973i	0.0951498	−	0.715940i
$250$		0			0
$251$	28.7735			1.81617			0.908083	−	0.418790i	$-0.137546\pi$
$251$	28.7735			1.81617			0.908083	+	0.418790i	$0.137546\pi$
$252$		0			0
$253$	6.36510			0.400171
$254$		0			0
$255$	2.90486	−	21.8572i	0.181909	−	1.36875i
$256$		0			0
$257$	−12.8504			−0.801586			−0.400793	−	0.916169i	$-0.631265\pi$
$257$	−12.8504			−0.801586			−0.400793	+	0.916169i	$0.631265\pi$
$258$		0			0
$259$	0.0986725	+	0.905249i	0.00613121	+	0.0562494i
$260$		0			0
$261$	1.55242	−	5.73733i	0.0960926	−	0.355132i
$262$		0			0
$263$		−	20.6328i		−	1.27227i	−0.771576	−	0.636137i	$-0.780531\pi$
$263$		−	20.6328i		−	1.27227i	0.771576	−	0.636137i	$-0.219469\pi$
$264$		0			0
$265$			11.0893i			0.681209i
$266$		0			0
$267$	2.13427	−	16.0590i	0.130615	−	0.982794i
$268$		0			0
$269$	−7.96994			−0.485936			−0.242968	−	0.970034i	$-0.578121\pi$
$269$	−7.96994			−0.485936			−0.242968	+	0.970034i	$0.578121\pi$
$270$		0			0
$271$		−	6.32592i		−	0.384272i	−0.981368	−	0.192136i	$-0.938458\pi$
$271$		−	6.32592i		−	0.384272i	0.981368	−	0.192136i	$-0.0615415\pi$
$272$		0			0
$273$	0.718715	−	30.5056i	0.0434986	−	1.84628i
$274$		0			0
$275$		−	1.60689i		−	0.0968991i
$276$		0			0
$277$	11.3204			0.680178			0.340089	−	0.940393i	$-0.389543\pi$
$277$	11.3204			0.680178			0.340089	+	0.940393i	$0.389543\pi$
$278$		0			0
$279$	−4.32783	+	15.9945i	−0.259101	+	0.957565i
$280$		0			0
$281$			29.1605i			1.73957i	0.493431	+	0.869785i	$0.335743\pi$
$281$			29.1605i			1.73957i	−0.493431	+	0.869785i	$0.664257\pi$
$282$		0			0
$283$		−	10.7006i		−	0.636082i	−0.948077	−	0.318041i	$-0.896975\pi$
$283$		−	10.7006i		−	0.636082i	0.948077	−	0.318041i	$-0.103025\pi$
$284$		0			0
$285$	−2.95060	−	0.392140i	−0.174778	−	0.0232283i
$286$		0			0
$287$	−0.552113	−	5.06524i	−0.0325902	−	0.298992i
$288$		0			0
$289$	30.7609			1.80947
$290$		0			0
$291$	20.5596	+	2.73241i	1.20523	+	0.160177i
$292$		0			0
$293$	15.4708			0.903812			0.451906	−	0.892065i	$-0.350744\pi$
$293$	15.4708			0.903812			0.451906	+	0.892065i	$0.350744\pi$
$294$		0			0
$295$	18.6470			1.08567
$296$		0			0
$297$	4.79326	+	2.00615i	0.278133	+	0.116409i
$298$		0			0
$299$	42.3834			2.45110
$300$		0			0
$301$	−0.393150	−	3.60687i	−0.0226608	−	0.207896i
$302$		0			0
$303$	−2.27506	+	17.1184i	−0.130699	+	0.983425i
$304$		0			0
$305$			24.6894i			1.41371i
$306$		0			0
$307$			15.8705i			0.905777i	0.891567	+	0.452889i	$0.149606\pi$
$307$			15.8705i			0.905777i	−0.891567	+	0.452889i	$0.850394\pi$
$308$		0			0
$309$	−19.1753	−	2.54843i	−1.09084	−	0.144975i
$310$		0			0
$311$	−10.8099			−0.612974			−0.306487	−	0.951875i	$-0.599154\pi$
$311$	−10.8099			−0.612974			−0.306487	+	0.951875i	$0.599154\pi$
$312$		0			0
$313$			7.74653i			0.437859i	0.975741	+	0.218930i	$0.0702566\pi$
$313$			7.74653i			0.437859i	−0.975741	+	0.218930i	$0.929743\pi$
$314$		0			0
$315$	2.26702	−	14.4439i	0.127732	−	0.813822i
$316$		0			0
$317$		−	32.4347i		−	1.82171i	−0.412722	−	0.910857i	$-0.635422\pi$
$317$		−	32.4347i		−	1.82171i	0.412722	−	0.910857i	$-0.364578\pi$
$318$		0			0
$319$	−1.98122			−0.110927
$320$		0			0
$321$	14.7778	+	1.96400i	0.824817	+	0.109620i
$322$		0			0
$323$		−	6.44746i		−	0.358746i
$324$		0			0
$325$		−	10.6998i		−	0.593519i
$326$		0			0
$327$	0.824262	−	6.20204i	0.0455818	−	0.342974i
$328$		0			0
$329$	2.92584	+	26.8424i	0.161307	+	1.47987i
$330$		0			0
$331$	4.54109			0.249601			0.124800	−	0.992182i	$-0.460171\pi$
$331$	4.54109			0.249601			0.124800	+	0.992182i	$0.460171\pi$
$332$		0			0
$333$	−0.996693	−	0.269688i	−0.0546185	−	0.0147788i
$334$		0			0
$335$	−13.5536			−0.740514
$336$		0			0
$337$	−0.520860			−0.0283731			−0.0141865	−	0.999899i	$-0.504516\pi$
$337$	−0.520860			−0.0283731			−0.0141865	+	0.999899i	$0.504516\pi$
$338$		0			0
$339$	19.8122	+	2.63308i	1.07605	+	0.143009i
$340$		0			0
$341$	5.52322			0.299099
$342$		0			0
$343$	−5.92623	−	17.5465i	−0.319987	−	0.947422i
$344$		0			0
$345$	20.1309	+	2.67543i	1.08381	+	0.144040i
$346$		0			0
$347$			5.36414i			0.287962i	0.989580	+	0.143981i	$0.0459905\pi$
$347$			5.36414i			0.287962i	−0.989580	+	0.143981i	$0.954010\pi$
$348$		0			0
$349$			15.3892i			0.823765i	0.911237	+	0.411883i	$0.135129\pi$
$349$			15.3892i			0.823765i	−0.911237	+	0.411883i	$0.864871\pi$
$350$		0			0
$351$	31.9170	+	13.3584i	1.70360	+	0.713018i
$352$		0			0
$353$	−2.83659			−0.150977			−0.0754883	−	0.997147i	$-0.524052\pi$
$353$	−2.83659			−0.150977			−0.0754883	+	0.997147i	$0.524052\pi$
$354$		0			0
$355$			13.9760i			0.741770i
$356$		0			0
$357$	31.6611	+	0.745938i	1.67568	+	0.0394792i
$358$		0			0
$359$		−	5.10231i		−	0.269290i	−0.990894	−	0.134645i	$-0.957011\pi$
$359$		−	5.10231i		−	0.269290i	0.990894	−	0.134645i	$-0.0429894\pi$
$360$		0			0
$361$	18.1296			0.954191
$362$		0			0
$363$	0.228186	−	1.71695i	0.0119767	−	0.0901167i
$364$		0			0
$365$		−	14.5993i		−	0.764161i
$366$		0			0
$367$			22.3203i			1.16511i	0.812791	+	0.582555i	$0.197947\pi$
$367$			22.3203i			1.16511i	−0.812791	+	0.582555i	$0.802053\pi$
$368$		0			0
$369$	5.57691	+	1.50902i	0.290322	+	0.0785562i
$370$		0			0
$371$	−15.8339	+	1.72591i	−0.822057	+	0.0896045i
$372$		0			0
$373$	5.28115			0.273448			0.136724	−	0.990609i	$-0.456343\pi$
$373$	5.28115			0.273448			0.136724	+	0.990609i	$0.456343\pi$
$374$		0			0
$375$	2.77706	−	20.8956i	0.143407	−	1.07904i
$376$		0			0
$377$	−13.1924			−0.679441
$378$		0			0
$379$	−12.7099			−0.652866			−0.326433	−	0.945220i	$-0.605847\pi$
$379$	−12.7099			−0.652866			−0.326433	+	0.945220i	$0.605847\pi$
$380$		0			0
$381$	−1.21212	+	9.12039i	−0.0620986	+	0.467252i
$382$		0			0
$383$	14.7446			0.753414			0.376707	−	0.926332i	$-0.377056\pi$
$383$	14.7446			0.753414			0.376707	+	0.926332i	$0.377056\pi$
$384$		0			0
$385$	−4.84488	+	0.528094i	−0.246918	+	0.0269142i
$386$		0			0
$387$	3.97122	+	1.07454i	0.201868	+	0.0546221i
$388$		0			0
$389$		−	4.52999i		−	0.229679i	−0.993384	−	0.114840i	$-0.963365\pi$
$389$		−	4.52999i		−	0.229679i	0.993384	−	0.114840i	$-0.0366355\pi$
$390$		0			0
$391$			43.9888i			2.22461i
$392$		0			0
$393$	−0.910201	+	6.84867i	−0.0459136	+	0.345470i
$394$		0			0
$395$	−1.83035			−0.0920950
$396$		0			0
$397$			23.5710i			1.18300i	0.806306	+	0.591498i	$0.201463\pi$
$397$			23.5710i			1.18300i	−0.806306	+	0.591498i	$0.798537\pi$
$398$		0			0
$399$	0.100697	−	4.27407i	0.00504117	−	0.213971i
$400$		0			0
$401$			0.373777i			0.0186655i	0.999956	+	0.00933275i	$0.00297075\pi$
$401$			0.373777i			0.0186655i	−0.999956	+	0.00933275i	$0.997029\pi$
$402$		0			0
$403$	36.7776			1.83202
$404$		0			0
$405$	14.3164	+	8.35959i	0.711387	+	0.415391i
$406$		0			0
$407$			0.344178i			0.0170603i
$408$		0			0
$409$			16.5090i			0.816315i	0.912911	+	0.408158i	$0.133829\pi$
$409$			16.5090i			0.816315i	−0.912911	+	0.408158i	$0.866171\pi$
$410$		0			0
$411$	−30.7577	−	4.08776i	−1.51717	−	0.201634i
$412$		0			0
$413$	2.90217	+	26.6253i	0.142807	+	1.31015i
$414$		0			0
$415$	−12.1204			−0.594967
$416$		0			0
$417$	−38.1421	−	5.06915i	−1.86783	−	0.248237i
$418$		0			0
$419$	6.25550			0.305601			0.152801	−	0.988257i	$-0.451171\pi$
$419$	6.25550			0.305601			0.152801	+	0.988257i	$0.451171\pi$
$420$		0			0
$421$	32.9673			1.60673			0.803365	−	0.595487i	$-0.203041\pi$
$421$	32.9673			1.60673			0.803365	+	0.595487i	$0.203041\pi$
$422$		0			0
$423$	−29.5540	−	7.99680i	−1.43696	−	0.388818i
$424$		0			0
$425$	11.1051			0.538676
$426$		0			0
$427$	−35.2530	+	3.84260i	−1.70601	+	0.185956i
$428$		0			0
$429$	1.51943	−	11.4327i	0.0733587	−	0.551976i
$430$		0			0
$431$		−	17.8039i		−	0.857585i	−0.903403	−	0.428793i	$-0.858939\pi$
$431$		−	17.8039i		−	0.857585i	0.903403	−	0.428793i	$-0.141061\pi$
$432$		0			0
$433$			8.07028i			0.387833i	0.981018	+	0.193916i	$0.0621190\pi$
$433$			8.07028i			0.387833i	−0.981018	+	0.193916i	$0.937881\pi$
$434$		0			0
$435$	−6.26599	−	0.832761i	−0.300431	−	0.0399279i
$436$		0			0
$437$	5.93824			0.284064
$438$		0			0
$439$		−	9.24798i		−	0.441382i	−0.975344	−	0.220691i	$-0.929169\pi$
$439$		−	9.24798i		−	0.441382i	0.975344	−	0.220691i	$-0.0708313\pi$
$440$		0			0
$441$	20.9767	+	0.988975i	0.998890	+	0.0470940i
$442$		0			0
$443$		−	41.4320i		−	1.96849i	−0.176805	−	0.984246i	$-0.556576\pi$
$443$		−	41.4320i		−	1.96849i	0.176805	−	0.984246i	$-0.443424\pi$
$444$		0			0
$445$	−17.2289			−0.816730
$446$		0			0
$447$	−5.60471	−	0.744875i	−0.265093	−	0.0352314i
$448$		0			0
$449$			23.0269i			1.08671i	0.839504	+	0.543354i	$0.182846\pi$
$449$			23.0269i			1.08671i	−0.839504	+	0.543354i	$0.817154\pi$
$450$		0			0
$451$		−	1.92582i		−	0.0906833i
$452$		0			0
$453$	−1.62833	+	12.2522i	−0.0765058	+	0.575656i
$454$		0			0
$455$	−32.2607	+	3.51643i	−1.51240	+	0.164853i
$456$		0			0
$457$	18.3371			0.857772			0.428886	−	0.903359i	$-0.358906\pi$
$457$	18.3371			0.857772			0.428886	+	0.903359i	$0.358906\pi$
$458$		0			0
$459$	−13.8644	+	33.1259i	−0.647133	+	1.54618i
$460$		0			0
$461$	3.22224			0.150075			0.0750374	−	0.997181i	$-0.476092\pi$
$461$	3.22224			0.150075			0.0750374	+	0.997181i	$0.476092\pi$
$462$		0			0
$463$	41.8139			1.94326			0.971629	−	0.236510i	$-0.0760036\pi$
$463$	41.8139			1.94326			0.971629	+	0.236510i	$0.0760036\pi$
$464$		0			0
$465$	17.4683	+	2.32157i	0.810072	+	0.107660i
$466$		0			0
$467$	17.9760			0.831829			0.415915	−	0.909404i	$-0.363462\pi$
$467$	17.9760			0.831829			0.415915	+	0.909404i	$0.363462\pi$
$468$		0			0
$469$	−2.10945	−	19.3527i	−0.0974053	−	0.893623i
$470$		0			0
$471$	32.1582	+	4.27388i	1.48177	+	0.196930i
$472$		0			0
$473$		−	1.37134i		−	0.0630544i
$474$		0			0
$475$		−	1.49913i		−	0.0687846i
$476$		0			0
$477$	4.71718	−	17.4334i	0.215985	−	0.798221i
$478$		0			0
$479$	−18.9634			−0.866458			−0.433229	−	0.901284i	$-0.642626\pi$
$479$	−18.9634			−0.866458			−0.433229	+	0.901284i	$0.642626\pi$
$480$		0			0
$481$			2.29179i			0.104496i
$482$		0			0
$483$	−0.687023	+	29.1605i	−0.0312606	+	1.32685i
$484$		0			0
$485$		−	22.0575i		−	1.00158i
$486$		0			0
$487$	−24.0057			−1.08780			−0.543901	−	0.839149i	$-0.683053\pi$
$487$	−24.0057			−1.08780			−0.543901	+	0.839149i	$0.683053\pi$
$488$		0			0
$489$	−1.22302	+	9.20246i	−0.0553070	+	0.416150i
$490$		0			0
$491$		−	0.887157i		−	0.0400368i	−0.999800	−	0.0200184i	$-0.993628\pi$
$491$		−	0.887157i		−	0.0400368i	0.999800	−	0.0200184i	$-0.00637248\pi$
$492$		0			0
$493$		−	13.6921i		−	0.616659i
$494$		0			0
$495$	1.44337	−	5.33429i	0.0648746	−	0.239759i
$496$		0			0
$497$	−19.9558	+	2.17519i	−0.895139	+	0.0975706i
$498$		0			0
$499$	−2.21216			−0.0990298			−0.0495149	−	0.998773i	$-0.515768\pi$
$499$	−2.21216			−0.0990298			−0.0495149	+	0.998773i	$0.515768\pi$
$500$		0			0
$501$	2.45816	−	18.4961i	0.109823	−	0.826345i
$502$		0			0
$503$	41.6130			1.85543			0.927716	−	0.373286i	$-0.121769\pi$
$503$	41.6130			1.85543			0.927716	+	0.373286i	$0.121769\pi$
$504$		0			0
$505$	18.3655			0.817255
$506$		0			0
$507$	7.15101	−	53.8068i	0.317588	−	2.38964i
$508$		0			0
$509$	−23.8214			−1.05586			−0.527932	−	0.849287i	$-0.677032\pi$
$509$	−23.8214			−1.05586			−0.527932	+	0.849287i	$0.677032\pi$
$510$		0			0
$511$	20.8457	−	2.27219i	0.922160	−	0.100516i
$512$		0			0
$513$	4.47181	+	1.87161i	0.197435	+	0.0826337i
$514$		0			0
$515$			20.5723i			0.906523i
$516$		0			0
$517$			10.2056i			0.448841i
$518$		0			0
$519$	2.06845	−	15.5637i	0.0907949	−	0.683173i
$520$		0			0
$521$	−14.9553			−0.655202			−0.327601	−	0.944816i	$-0.606240\pi$
$521$	−14.9553			−0.655202			−0.327601	+	0.944816i	$0.606240\pi$
$522$		0			0
$523$		−	31.1782i		−	1.36333i	−0.731666	−	0.681664i	$-0.761257\pi$
$523$		−	31.1782i		−	1.36333i	0.731666	−	0.681664i	$-0.238743\pi$
$524$		0			0
$525$	7.36165	+	0.173441i	0.321289	+	0.00756959i
$526$		0			0
$527$			38.1706i			1.66274i
$528$		0			0
$529$	−17.5145			−0.761502
$530$		0			0
$531$	−29.3149	−	7.93211i	−1.27216	−	0.344224i
$532$		0			0
$533$		−	12.8235i		−	0.555447i
$534$		0			0
$535$		−	15.8544i		−	0.685447i
$536$		0			0
$537$	−32.8470	−	4.36543i	−1.41745	−	0.188382i
$538$		0			0
$539$	−1.50809	−	6.83562i	−0.0649580	−	0.294431i
$540$		0			0
$541$	−13.0446			−0.560832			−0.280416	−	0.959879i	$-0.590472\pi$
$541$	−13.0446			−0.560832			−0.280416	+	0.959879i	$0.590472\pi$
$542$		0			0
$543$	−24.2346	−	3.22082i	−1.04000	−	0.138218i
$544$		0			0
$545$	−6.65388			−0.285021
$546$		0			0
$547$	30.1498			1.28911			0.644557	−	0.764556i	$-0.277042\pi$
$547$	30.1498			1.28911			0.644557	+	0.764556i	$0.277042\pi$
$548$		0			0
$549$	10.5024	−	38.8142i	0.448233	−	1.65655i
$550$		0			0
$551$	−1.84835			−0.0787423
$552$		0			0
$553$	−0.284871	−	2.61348i	−0.0121139	−	0.111137i
$554$		0			0
$555$	−0.144668	+	1.08853i	−0.00614081	+	0.0462056i
$556$		0			0
$557$		−	42.3853i		−	1.79592i	−0.440075	−	0.897961i	$-0.645048\pi$
$557$		−	42.3853i		−	1.79592i	0.440075	−	0.897961i	$-0.354952\pi$
$558$		0			0
$559$		−	9.13138i		−	0.386216i
$560$		0			0
$561$	11.8657	+	1.57698i	0.500972	+	0.0665801i
$562$		0			0
$563$	41.1298			1.73341			0.866707	−	0.498817i	$-0.166232\pi$
$563$	41.1298			1.73341			0.866707	+	0.498817i	$0.166232\pi$
$564$		0			0
$565$		−	21.2556i		−	0.894231i
$566$		0			0
$567$	−9.70815	+	21.7429i	−0.407704	+	0.913114i
$568$		0			0
$569$			41.3862i			1.73500i	0.497439	+	0.867499i	$0.334274\pi$
$569$			41.3862i			1.73500i	−0.497439	+	0.867499i	$0.665726\pi$
$570$		0			0
$571$	13.7792			0.576642			0.288321	−	0.957534i	$-0.406903\pi$
$571$	13.7792			0.576642			0.288321	+	0.957534i	$0.406903\pi$
$572$		0			0
$573$	24.0352	+	3.19433i	1.00409	+	0.133445i
$574$		0			0
$575$			10.2280i			0.426538i
$576$		0			0
$577$		−	42.7307i		−	1.77890i	−0.457031	−	0.889451i	$-0.651087\pi$
$577$		−	42.7307i		−	1.77890i	0.457031	−	0.889451i	$-0.348913\pi$
$578$		0			0
$579$	−3.21104	+	24.1610i	−0.133446	+	1.00410i
$580$		0			0
$581$	−1.88638	−	17.3062i	−0.0782604	−	0.717983i
$582$		0			0
$583$	−6.02011			−0.249328
$584$		0			0
$585$	9.61098	−	35.5195i	0.397365	−	1.46855i
$586$		0			0
$587$	2.56414			0.105833			0.0529166	−	0.998599i	$-0.483148\pi$
$587$	2.56414			0.105833			0.0529166	+	0.998599i	$0.483148\pi$
$588$		0			0
$589$	5.15281			0.212318
$590$		0			0
$591$	0.249767	+	0.0331945i	0.0102740	+	0.00136544i
$592$		0			0
$593$	−29.2689			−1.20193			−0.600965	−	0.799276i	$-0.705217\pi$
$593$	−29.2689			−1.20193			−0.600965	+	0.799276i	$0.705217\pi$
$594$		0			0
$595$	−3.64962	−	33.4826i	−0.149620	−	1.37265i
$596$		0			0
$597$	−24.5328	−	3.26045i	−1.00406	−	0.133441i
$598$		0			0
$599$			14.6814i			0.599867i	0.953960	+	0.299933i	$0.0969645\pi$
$599$			14.6814i			0.599867i	−0.953960	+	0.299933i	$0.903036\pi$
$600$		0			0
$601$			2.34778i			0.0957681i	0.998853	+	0.0478840i	$0.0152478\pi$
$601$			2.34778i			0.0957681i	−0.998853	+	0.0478840i	$0.984752\pi$
$602$		0			0
$603$	21.3076	+	5.76547i	0.867712	+	0.234788i
$604$		0			0
$605$	−1.84204			−0.0748896
$606$		0			0
$607$		−	8.22535i		−	0.333857i	−0.985969	−	0.166928i	$-0.946615\pi$
$607$		−	8.22535i		−	0.333857i	0.985969	−	0.166928i	$-0.0533848\pi$
$608$		0			0
$609$	0.213844	−	9.07656i	0.00866542	−	0.367801i
$610$		0			0
$611$			67.9561i			2.74921i
$612$		0			0
$613$	40.3157			1.62834			0.814168	−	0.580629i	$-0.197193\pi$
$613$	40.3157			1.62834			0.814168	+	0.580629i	$0.197193\pi$
$614$		0			0
$615$	0.809476	−	6.09079i	0.0326412	−	0.245604i
$616$		0			0
$617$		−	39.3539i		−	1.58433i	−0.610309	−	0.792164i	$-0.708955\pi$
$617$		−	39.3539i		−	1.58433i	0.610309	−	0.792164i	$-0.291045\pi$
$618$		0			0
$619$			9.53937i			0.383420i	0.981452	+	0.191710i	$0.0614033\pi$
$619$			9.53937i			0.383420i	−0.981452	+	0.191710i	$0.938597\pi$
$620$		0			0
$621$	−30.5096	−	12.7694i	−1.22431	−	0.512417i
$622$		0			0
$623$	−2.68146	−	24.6005i	−0.107431	−	0.985598i
$624$		0			0
$625$	−14.3835			−0.575338
$626$		0			0
$627$	0.212883	−	1.60181i	0.00850174	−	0.0639701i
$628$		0			0
$629$	−2.37859			−0.0948407
$630$		0			0
$631$	0.282052			0.0112283			0.00561416	−	0.999984i	$-0.498213\pi$
$631$	0.282052			0.0112283			0.00561416	+	0.999984i	$0.498213\pi$
$632$		0			0
$633$	3.70974	−	27.9134i	0.147449	−	1.10946i
$634$		0			0
$635$	9.78484			0.388299
$636$		0			0
$637$	−10.0419	−	45.5164i	−0.397876	−	1.80343i
$638$		0			0
$639$	5.94515	−	21.9716i	0.235186	−	0.869185i
$640$		0			0
$641$		−	11.4777i		−	0.453341i	−0.973971	−	0.226671i	$-0.927216\pi$
$641$		−	11.4777i		−	0.453341i	0.973971	−	0.226671i	$-0.0727841\pi$
$642$		0			0
$643$			30.6971i			1.21058i	0.796007	+	0.605288i	$0.206942\pi$
$643$			30.6971i			1.21058i	−0.796007	+	0.605288i	$0.793058\pi$
$644$		0			0
$645$	0.576414	−	4.33714i	0.0226963	−	0.170775i
$646$		0			0
$647$	−28.6596			−1.12673			−0.563363	−	0.826209i	$-0.690493\pi$
$647$	−28.6596			−1.12673			−0.563363	+	0.826209i	$0.690493\pi$
$648$		0			0
$649$			10.1230i			0.397364i
$650$		0			0
$651$	−0.596154	+	25.3036i	−0.0233651	+	0.991724i
$652$		0			0
$653$			16.9380i			0.662837i	0.943484	+	0.331418i	$0.107527\pi$
$653$			16.9380i			0.662837i	−0.943484	+	0.331418i	$0.892473\pi$
$654$		0			0
$655$	7.34762			0.287095
$656$		0			0
$657$	−6.21027	+	22.9515i	−0.242286	+	0.895422i
$658$		0			0
$659$		−	16.7175i		−	0.651221i	−0.945504	−	0.325610i	$-0.894430\pi$
$659$		−	16.7175i		−	0.651221i	0.945504	−	0.325610i	$-0.105570\pi$
$660$		0			0
$661$		−	6.43346i		−	0.250233i	−0.992142	−	0.125116i	$-0.960070\pi$
$661$		−	6.43346i		−	0.250233i	0.992142	−	0.125116i	$-0.0399304\pi$
$662$		0			0
$663$	79.0106	+	10.5007i	3.06852	+	0.407812i
$664$		0			0
$665$	−4.51997	+	0.492678i	−0.175277	+	0.0191052i
$666$		0			0
$667$	12.6107			0.488286
$668$		0			0
$669$	−19.0066	−	2.52602i	−0.734839	−	0.0976614i
$670$		0			0
$671$	−13.4033			−0.517429
$672$		0			0
$673$	13.4728			0.519339			0.259669	−	0.965698i	$-0.416386\pi$
$673$	13.4728			0.519339			0.259669	+	0.965698i	$0.416386\pi$
$674$		0			0
$675$	−3.22366	+	7.70224i	−0.124079	+	0.296459i
$676$		0			0
$677$	−12.5143			−0.480965			−0.240482	−	0.970654i	$-0.577306\pi$
$677$	−12.5143			−0.480965			−0.240482	+	0.970654i	$0.577306\pi$
$678$		0			0
$679$	31.4949	−	3.43296i	1.20867	−	0.131745i
$680$		0			0
$681$	−3.00559	+	22.6151i	−0.115174	+	0.866613i
$682$		0			0
$683$		−	23.3443i		−	0.893243i	−0.894723	−	0.446622i	$-0.852627\pi$
$683$		−	23.3443i		−	0.893243i	0.894723	−	0.446622i	$-0.147373\pi$
$684$		0			0
$685$			32.9985i			1.26081i
$686$		0			0
$687$	−28.5156	−	3.78977i	−1.08794	−	0.144589i
$688$		0			0
$689$	−40.0862			−1.52716
$690$		0			0
$691$			40.0019i			1.52174i	0.648902	+	0.760872i	$0.275228\pi$
$691$			40.0019i			1.52174i	−0.648902	+	0.760872i	$0.724772\pi$
$692$		0			0
$693$	7.84126	+	1.23071i	0.297865	+	0.0467509i
$694$		0			0
$695$			40.9209i			1.55222i
$696$		0			0
$697$	13.3092			0.504122
$698$		0			0
$699$	−24.0216	−	3.19251i	−0.908579	−	0.120752i
$700$		0			0
$701$		−	11.8879i		−	0.449001i	−0.974474	−	0.224500i	$-0.927925\pi$
$701$		−	11.8879i		−	0.449001i	0.974474	−	0.224500i	$-0.0720750\pi$
$702$		0			0
$703$			0.321097i			0.0121104i
$704$		0			0
$705$	−4.28969	+	32.2772i	−0.161559	+	1.21563i
$706$		0			0
$707$	2.85836	+	26.2234i	0.107500	+	0.986231i
$708$		0			0
$709$	−16.7356			−0.628518			−0.314259	−	0.949337i	$-0.601756\pi$
$709$	−16.7356			−0.628518			−0.314259	+	0.949337i	$0.601756\pi$
$710$		0			0
$711$	2.87749	+	0.778599i	0.107914	+	0.0291997i
$712$		0			0
$713$	−35.1559			−1.31660
$714$		0			0
$715$	−12.2656			−0.458708
$716$		0			0
$717$	−13.9358	−	1.85210i	−0.520444	−	0.0691679i
$718$		0			0
$719$	37.4305			1.39592			0.697960	−	0.716137i	$-0.254091\pi$
$719$	37.4305			1.39592			0.697960	+	0.716137i	$0.254091\pi$
$720$		0			0
$721$	−29.3743	+	3.20181i	−1.09396	+	0.119242i
$722$		0			0
$723$	−39.9936	−	5.31522i	−1.48738	−	0.197675i
$724$		0			0
$725$		−	3.18360i		−	0.118236i
$726$		0			0
$727$		−	35.7945i		−	1.32754i	−0.747935	−	0.663772i	$-0.768954\pi$
$727$		−	35.7945i		−	1.32754i	0.747935	−	0.663772i	$-0.231046\pi$
$728$		0			0
$729$	−18.9507	−	19.2320i	−0.701878	−	0.712297i
$730$		0			0
$731$	9.47725			0.350529
$732$		0			0
$733$			13.7817i			0.509038i	0.967068	+	0.254519i	$0.0819171\pi$
$733$			13.7817i			0.509038i	−0.967068	+	0.254519i	$0.918083\pi$
$734$		0			0
$735$	−1.89642	−	22.2529i	−0.0699506	−	0.820810i
$736$		0			0
$737$		−	7.35794i		−	0.271033i
$738$		0			0
$739$	−40.2594			−1.48097			−0.740483	−	0.672076i	$-0.765403\pi$
$739$	−40.2594			−1.48097			−0.740483	+	0.672076i	$0.765403\pi$
$740$		0			0
$741$	1.41753	−	10.6660i	0.0520742	−	0.391825i
$742$		0			0
$743$			48.9106i			1.79435i	0.441670	+	0.897177i	$0.354386\pi$
$743$			48.9106i			1.79435i	−0.441670	+	0.897177i	$0.645614\pi$
$744$		0			0
$745$			6.01303i			0.220300i
$746$		0			0
$747$	19.0544	+	5.15580i	0.697165	+	0.188641i
$748$		0			0
$749$	22.6379	−	2.46754i	0.827170	−	0.0901619i
$750$		0			0
$751$	−14.2946			−0.521618			−0.260809	−	0.965390i	$-0.583989\pi$
$751$	−14.2946			−0.521618			−0.260809	+	0.965390i	$0.583989\pi$
$752$		0			0
$753$	−6.56571	+	49.4027i	−0.239268	+	1.80034i
$754$		0			0
$755$	13.1448			0.478387
$756$		0			0
$757$	−35.5185			−1.29094			−0.645472	−	0.763784i	$-0.723339\pi$
$757$	−35.5185			−1.29094			−0.645472	+	0.763784i	$0.723339\pi$
$758$		0			0
$759$	−1.45243	+	10.9286i	−0.0527198	+	0.396683i
$760$		0			0
$761$	33.5220			1.21517			0.607586	−	0.794254i	$-0.292138\pi$
$761$	33.5220			1.21517			0.607586	+	0.794254i	$0.292138\pi$
$762$		0			0
$763$	−1.03559	−	9.50080i	−0.0374909	−	0.343952i
$764$		0			0
$765$	36.8649	+	9.97502i	1.33285	+	0.360648i
$766$		0			0
$767$			67.4064i			2.43390i
$768$		0			0
$769$			27.6629i			0.997548i	0.866732	+	0.498774i	$0.166216\pi$
$769$			27.6629i			0.997548i	−0.866732	+	0.498774i	$0.833784\pi$
$770$		0			0
$771$	2.93228	−	22.0635i	0.105604	−	0.794599i
$772$		0			0
$773$	15.1488			0.544863			0.272431	−	0.962175i	$-0.412172\pi$
$773$	15.1488			0.544863			0.272431	+	0.962175i	$0.412172\pi$
$774$		0			0
$775$			8.87520i			0.318807i
$776$		0			0
$777$	−1.57679	−	0.0371492i	−0.0565669	−	0.00133272i
$778$		0			0
$779$		−	1.79667i		−	0.0643723i
$780$		0			0
$781$	−7.58725			−0.271493
$782$		0			0
$783$	9.49649	+	3.97462i	0.339377	+	0.142041i
$784$		0			0
$785$		−	34.5010i		−	1.23140i
$786$		0			0
$787$			2.56564i			0.0914552i	0.998954	+	0.0457276i	$0.0145606\pi$
$787$			2.56564i			0.0914552i	−0.998954	+	0.0457276i	$0.985439\pi$
$788$		0			0
$789$	35.4256	+	4.70813i	1.26118	+	0.167614i
$790$		0			0
$791$	30.3500	−	3.30816i	1.07912	−	0.117625i
$792$		0			0
$793$	−89.2489			−3.16932
$794$		0			0
$795$	−19.0398	−	2.53042i	−0.675272	−	0.0897448i
$796$		0			0
$797$	24.3030			0.860856			0.430428	−	0.902625i	$-0.358363\pi$
$797$	24.3030			0.860856			0.430428	+	0.902625i	$0.358363\pi$
$798$		0			0
$799$	−70.5301			−2.49517
$800$		0			0
$801$	27.0855	+	7.32888i	0.957020	+	0.258953i
$802$		0			0
$803$	7.92560			0.279688
$804$		0			0
$805$	30.8382	−	3.36137i	1.08690	−	0.118473i
$806$		0			0
$807$	1.81863	−	13.6840i	0.0640188	−	0.481700i
$808$		0			0
$809$		−	38.6623i		−	1.35929i	−0.733539	−	0.679647i	$-0.762133\pi$
$809$		−	38.6623i		−	1.35929i	0.733539	−	0.679647i	$-0.237867\pi$
$810$		0			0
$811$			5.11273i			0.179532i	0.995963	+	0.0897661i	$0.0286120\pi$
$811$			5.11273i			0.179532i	−0.995963	+	0.0897661i	$0.971388\pi$
$812$		0			0
$813$	10.8613	+	1.44349i	0.380923	+	0.0506253i
$814$		0			0
$815$	9.87289			0.345832
$816$		0			0
$817$		−	1.27938i		−	0.0447597i
$818$		0			0
$819$	52.2127	+	8.19496i	1.82446	+	0.286355i
$820$		0			0
$821$		−	38.7122i		−	1.35106i	−0.737330	−	0.675532i	$-0.763914\pi$
$821$		−	38.7122i		−	1.35106i	0.737330	−	0.675532i	$-0.236086\pi$
$822$		0			0
$823$	−8.97914			−0.312993			−0.156497	−	0.987679i	$-0.550020\pi$
$823$	−8.97914			−0.312993			−0.156497	+	0.987679i	$0.550020\pi$
$824$		0			0
$825$	2.75895	+	0.366670i	0.0960545	+	0.0127658i
$826$		0			0
$827$			35.9159i			1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$827$			35.9159i			1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$828$		0			0
$829$		−	44.0237i		−	1.52901i	−0.644620	−	0.764503i	$-0.722984\pi$
$829$		−	44.0237i		−	1.52901i	0.644620	−	0.764503i	$-0.277016\pi$
$830$		0			0
$831$	−2.58317	+	19.4366i	−0.0896090	+	0.674250i
$832$		0			0
$833$	47.2405	−	10.4223i	1.63679	−	0.361111i
$834$		0			0
$835$	−19.8436			−0.686716
$836$		0			0
$837$	−26.4742	−	11.0804i	−0.915084	−	0.382995i
$838$		0			0
$839$	−17.4186			−0.601356			−0.300678	−	0.953726i	$-0.597213\pi$
$839$	−17.4186			−0.601356			−0.300678	+	0.953726i	$0.597213\pi$
$840$		0			0
$841$	25.0748			0.864648
$842$		0			0
$843$	−50.0673	−	6.65403i	−1.72441	−	0.229177i
$844$		0			0
$845$	−57.7268			−1.98586
$846$		0			0
$847$	−0.286690	−	2.63017i	−0.00985078	−	0.0903738i
$848$		0			0
$849$	18.3724	+	2.44172i	0.630538	+	0.0837996i
$850$		0			0
$851$		−	2.19073i		−	0.0750973i
$852$		0			0
$853$		−	10.1747i		−	0.348376i	−0.984712	−	0.174188i	$-0.944270\pi$
$853$		−	10.1747i		−	0.348376i	0.984712	−	0.174188i	$-0.0557301\pi$
$854$		0			0
$855$	1.34657	−	4.97656i	0.0460518	−	0.170195i
$856$		0			0
$857$	−9.30990			−0.318020			−0.159010	−	0.987277i	$-0.550830\pi$
$857$	−9.30990			−0.318020			−0.159010	+	0.987277i	$0.550830\pi$
$858$		0			0
$859$		−	27.0188i		−	0.921870i	−0.887434	−	0.460935i	$-0.847514\pi$
$859$		−	27.0188i		−	0.921870i	0.887434	−	0.460935i	$-0.152486\pi$
$860$		0			0
$861$	8.82277	+	0.207865i	0.300679	+	0.00708403i
$862$		0			0
$863$		−	1.59975i		−	0.0544562i	−0.999629	−	0.0272281i	$-0.991332\pi$
$863$		−	1.59975i		−	0.0544562i	0.999629	−	0.0272281i	$-0.00866805\pi$
$864$		0			0
$865$	−16.6976			−0.567736
$866$		0			0
$867$	−7.01923	+	52.8151i	−0.238385	+	1.79370i
$868$		0			0
$869$		−	0.993655i		−	0.0337074i
$870$		0			0
$871$		−	48.9945i		−	1.66011i
$872$		0			0
$873$	−9.38285	+	34.6764i	−0.317561	+	1.17362i
$874$		0			0
$875$	−3.48906	−	32.0096i	−0.117952	−	1.08212i
$876$		0			0
$877$	−49.6065			−1.67509			−0.837546	−	0.546368i	$-0.816010\pi$
$877$	−49.6065			−1.67509			−0.837546	+	0.546368i	$0.816010\pi$
$878$		0			0
$879$	−3.53022	+	26.5626i	−0.119071	+	0.895935i
$880$		0			0
$881$	29.9882			1.01033			0.505164	−	0.863024i	$-0.331432\pi$
$881$	29.9882			1.01033			0.505164	+	0.863024i	$0.331432\pi$
$882$		0			0
$883$	−14.4088			−0.484896			−0.242448	−	0.970164i	$-0.577950\pi$
$883$	−14.4088			−0.484896			−0.242448	+	0.970164i	$0.577950\pi$
$884$		0			0
$885$	−4.25500	+	32.0161i	−0.143030	+	1.07621i
$886$		0			0
$887$	46.2449			1.55275			0.776377	−	0.630269i	$-0.217056\pi$
$887$	46.2449			1.55275			0.776377	+	0.630269i	$0.217056\pi$
$888$		0			0
$889$	1.52288	+	13.9714i	0.0510759	+	0.468585i
$890$		0			0
$891$	−4.53823	+	7.77203i	−0.152036	+	0.260373i
$892$		0			0
$893$			9.52116i			0.318614i
$894$		0			0
$895$			35.2400i			1.17794i
$896$		0			0
$897$	−9.67131	+	72.7704i	−0.322916	+	2.42973i
$898$		0			0
$899$	10.9427			0.364959
$900$		0			0
$901$		−	41.6046i		−	1.38605i
$902$		0			0
$903$	6.28254	+	0.148017i	0.209070	+	0.00492570i
$904$		0			0
$905$			26.0001i			0.864273i
$906$		0			0
$907$	46.2720			1.53644			0.768218	−	0.640188i	$-0.221144\pi$
$907$	46.2720			1.53644			0.768218	+	0.640188i	$0.221144\pi$
$908$		0			0
$909$	−28.8723	−	7.81236i	−0.957635	−	0.259120i
$910$		0			0
$911$			32.5083i			1.07705i	0.842610	+	0.538524i	$0.181018\pi$
$911$			32.5083i			1.07705i	−0.842610	+	0.538524i	$0.818982\pi$
$912$		0			0
$913$		−	6.57988i		−	0.217762i
$914$		0			0
$915$	−42.3906	−	5.63379i	−1.40139	−	0.186247i
$916$		0			0
$917$	1.14356	+	10.4914i	0.0377638	+	0.346456i
$918$		0			0
$919$	−25.9753			−0.856847			−0.428424	−	0.903578i	$-0.640931\pi$
$919$	−25.9753			−0.856847			−0.428424	+	0.903578i	$0.640931\pi$
$920$		0			0
$921$	−27.2489	−	3.62143i	−0.897882	−	0.119330i
$922$		0			0
$923$	−50.5214			−1.66293
$924$		0			0
$925$	−0.553057			−0.0181844
$926$		0			0
$927$	8.75108	−	32.3416i	0.287423	−	1.06224i
$928$		0			0
$929$	30.5766			1.00319			0.501593	−	0.865104i	$-0.332748\pi$
$929$	30.5766			1.00319			0.501593	+	0.865104i	$0.332748\pi$
$930$		0			0
$931$	−1.40695	−	6.37720i	−0.0461109	−	0.209004i
$932$		0			0
$933$	2.46667	−	18.5601i	0.0807553	−	0.607631i
$934$		0			0
$935$		−	12.7302i		−	0.416322i
$936$		0			0
$937$			24.4170i			0.797670i	0.917023	+	0.398835i	$0.130585\pi$
$937$			24.4170i			0.797670i	−0.917023	+	0.398835i	$0.869415\pi$
$938$		0			0
$939$	−13.3004	−	1.76765i	−0.434043	−	0.0576851i
$940$		0			0
$941$	−46.0463			−1.50107			−0.750533	−	0.660833i	$-0.770203\pi$
$941$	−46.0463			−1.50107			−0.750533	+	0.660833i	$0.770203\pi$
$942$		0			0
$943$			12.2580i			0.399177i
$944$		0			0
$945$	24.2822	+	7.18827i	0.789901	+	0.233835i
$946$		0			0
$947$		−	3.11758i		−	0.101308i	−0.998716	−	0.0506538i	$-0.983869\pi$
$947$		−	3.11758i		−	0.101308i	0.998716	−	0.0506538i	$-0.0161305\pi$
$948$		0			0
$949$	52.7743			1.71313
$950$		0			0
$951$	55.6889	+	7.40115i	1.80584	+	0.239999i
$952$		0			0
$953$		−	34.9315i		−	1.13154i	−0.824563	−	0.565771i	$-0.808579\pi$
$953$		−	34.9315i		−	1.13154i	0.824563	−	0.565771i	$-0.191421\pi$
$954$		0			0
$955$		−	25.7863i		−	0.834425i
$956$		0			0
$957$	0.452087	−	3.40166i	0.0146139	−	0.109960i
$958$		0			0
$959$	−47.1172	+	5.13580i	−1.52149	+	0.165844i
$960$		0			0
$961$	0.494041			0.0159368
$962$		0			0
$963$	−6.74419	+	24.9247i	−0.217328	+	0.803186i
$964$		0			0
$965$	25.9212			0.834432
$966$		0			0
$967$	−42.9211			−1.38025			−0.690125	−	0.723690i	$-0.742445\pi$
$967$	−42.9211			−1.38025			−0.690125	+	0.723690i	$0.742445\pi$
$968$		0			0
$969$	11.0700	+	1.47122i	0.355619	+	0.0472624i
$970$		0			0
$971$	−41.2692			−1.32439			−0.662195	−	0.749331i	$-0.730375\pi$
$971$	−41.2692			−1.32439			−0.662195	+	0.749331i	$0.730375\pi$
$972$		0			0
$973$	−58.4292	+	6.36881i	−1.87315	+	0.204175i
$974$		0			0
$975$	18.3711	+	2.44155i	0.588346	+	0.0781922i
$976$		0			0
$977$			22.6863i			0.725798i	0.931828	+	0.362899i	$0.118213\pi$
$977$			22.6863i			0.725798i	−0.931828	+	0.362899i	$0.881787\pi$
$978$		0			0
$979$		−	9.35319i		−	0.298929i
$980$		0			0
$981$	10.4605	+	2.83044i	0.333979	+	0.0903690i
$982$		0			0
$983$	56.3951			1.79873			0.899363	−	0.437204i	$-0.144031\pi$
$983$	56.3951			1.79873			0.899363	+	0.437204i	$0.144031\pi$
$984$		0			0
$985$		−	0.267963i		−	0.00853802i
$986$		0			0
$987$	−46.7549	−	1.10155i	−1.48822	−	0.0350627i
$988$		0			0
$989$			8.72874i			0.277558i
$990$		0			0
$991$	27.2609			0.865971			0.432985	−	0.901401i	$-0.357460\pi$
$991$	27.2609			0.865971			0.432985	+	0.901401i	$0.357460\pi$
$992$		0			0
$993$	−1.03621	+	7.79684i	−0.0328833	+	0.247425i
$994$		0			0
$995$			26.3201i			0.834403i
$996$		0			0
$997$			22.3734i			0.708573i	0.935137	+	0.354286i	$0.115276\pi$
$997$			22.3734i			0.708573i	−0.935137	+	0.354286i	$0.884724\pi$
$998$		0			0
$999$	0.690474	−	1.64974i	0.0218456	−	0.0521954i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1848.2.v.e.881.16	yes	32
3.2	odd	2	inner	1848.2.v.e.881.18	yes	32
7.6	odd	2	inner	1848.2.v.e.881.17	yes	32
21.20	even	2	inner	1848.2.v.e.881.15	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.v.e.881.15	✓	32	21.20	even	2	inner
1848.2.v.e.881.16	yes	32	1.1	even	1	trivial
1848.2.v.e.881.17	yes	32	7.6	odd	2	inner
1848.2.v.e.881.18	yes	32	3.2	odd	2	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:	\( N \)	\(=\)	\( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1848.v (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.228186 + 1.71695i) q^{3} +1.84204 q^{5} +(0.286690 + 2.63017i) q^{7} +(-2.89586 - 0.783571i) q^{9} +1.00000i q^{11} +6.65872i q^{13} +(-0.420328 + 3.16270i) q^{15} -6.91093 q^{17} +0.932936i q^{19} +(-4.58130 - 0.107936i) q^{21} -6.36510i q^{23} -1.60689 q^{25} +(2.00615 - 4.79326i) q^{27} +1.98122i q^{29} -5.52322i q^{31} +(-1.71695 - 0.228186i) q^{33} +(0.528094 + 4.84488i) q^{35} +0.344178 q^{37} +(-11.4327 - 1.51943i) q^{39} -1.92582 q^{41} -1.37134 q^{43} +(-5.33429 - 1.44337i) q^{45} +10.2056 q^{47} +(-6.83562 + 1.50809i) q^{49} +(1.57698 - 11.8657i) q^{51} +6.02011i q^{53} +1.84204i q^{55} +(-1.60181 - 0.212883i) q^{57} +10.1230 q^{59} +13.4033i q^{61} +(1.23071 - 7.84126i) q^{63} +12.2656i q^{65} -7.35794 q^{67} +(10.9286 + 1.45243i) q^{69} +7.58725i q^{71} -7.92560i q^{73} +(0.366670 - 2.75895i) q^{75} +(-2.63017 + 0.286690i) q^{77} -0.993655 q^{79} +(7.77203 + 4.53823i) q^{81} -6.57988 q^{83} -12.7302 q^{85} +(-3.40166 - 0.452087i) q^{87} -9.35319 q^{89} +(-17.5136 + 1.90899i) q^{91} +(9.48311 + 1.26032i) q^{93} +1.71851i q^{95} -11.9745i q^{97} +(0.783571 - 2.89586i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{7} + 4 q^{9} - 16 q^{15} + 6 q^{21} + 68 q^{25} - 68 q^{37} - 32 q^{39} + 120 q^{43} - 64 q^{49} + 56 q^{51} - 44 q^{57} + 42 q^{63} + 4 q^{67} - 56 q^{79} - 28 q^{81} - 44 q^{91} - 28 q^{93}+O(q^{100}) \) 32 * q + 4 * q^7 + 4 * q^9 - 16 * q^15 + 6 * q^21 + 68 * q^25 - 68 * q^37 - 32 * q^39 + 120 * q^43 - 64 * q^49 + 56 * q^51 - 44 * q^57 + 42 * q^63 + 4 * q^67 - 56 * q^79 - 28 * q^81 - 44 * q^91 - 28 * q^93

Introduction

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