LMFDB - Embedded newform 1900.2.l.d.493.7

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			493.7
Character	$\chi$	$=$	1900.493
Dual form			1900.2.l.d.1557.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.814717 - 0.814717i) q^{3} +(-1.28987 + 1.28987i) q^{7} +1.67247i q^{9} -0.814579 q^{11} +(2.02624 - 2.02624i) q^{13} +(-1.28987 + 1.28987i) q^{17} +(-4.09740 + 1.48705i) q^{19} +2.10176i q^{21} +(1.75613 + 1.75613i) q^{23} +(3.80674 + 3.80674i) q^{27} +1.12978 q^{29} +4.96326i q^{31} +(-0.663651 + 0.663651i) q^{33} +(3.80674 + 3.80674i) q^{37} -3.30163i q^{39} +5.22718i q^{41} +(-1.22474 - 1.22474i) q^{43} +(-4.40100 + 4.40100i) q^{47} +3.67247i q^{49} +2.10176i q^{51} +(5.41508 - 5.41508i) q^{53} +(-2.12670 + 4.54975i) q^{57} -3.99128 q^{59} -6.46115 q^{61} +(-2.15727 - 2.15727i) q^{63} +(6.22979 + 6.22979i) q^{67} +2.86150 q^{69} +13.0519i q^{71} +(-0.985576 - 0.985576i) q^{73} +(1.05070 - 1.05070i) q^{77} -6.19916 q^{79} +1.18542 q^{81} +(5.56062 + 5.56062i) q^{83} +(0.920451 - 0.920451i) q^{87} +2.10176 q^{89} +5.22718i q^{91} +(4.04365 + 4.04365i) q^{93} +(3.65568 + 3.65568i) q^{97} -1.36236i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.814717	−	0.814717i	0.470377	−	0.470377i	−0.431659	−	0.902037i	$-0.642072\pi$
$3$	0.814717	−	0.814717i	0.470377	−	0.470377i	0.902037	+	0.431659i	$0.142072\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.28987	+	1.28987i	−0.487525	+	0.487525i	−0.907524	−	0.419999i	$-0.862030\pi$
$7$	−1.28987	+	1.28987i	−0.487525	+	0.487525i	0.419999	+	0.907524i	$0.362030\pi$
$8$		0			0
$9$			1.67247i			0.557491i
$10$		0			0
$11$	−0.814579			−0.245605			−0.122802	−	0.992431i	$-0.539188\pi$
$11$	−0.814579			−0.245605			−0.122802	+	0.992431i	$0.539188\pi$
$12$		0			0
$13$	2.02624	−	2.02624i	0.561979	−	0.561979i	−0.367890	−	0.929869i	$-0.619920\pi$
$13$	2.02624	−	2.02624i	0.561979	−	0.561979i	0.929869	+	0.367890i	$0.119920\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.28987	+	1.28987i	−0.312839	+	0.312839i	−0.846009	−	0.533169i	$-0.821001\pi$
$17$	−1.28987	+	1.28987i	−0.312839	+	0.312839i	0.533169	+	0.846009i	$0.321001\pi$
$18$		0			0
$19$	−4.09740	+	1.48705i	−0.940008	+	0.341153i
$19$	−4.09740	+	1.48705i	−0.940008	+	0.341153i
$20$		0			0
$21$			2.10176i			0.458641i
$22$		0			0
$23$	1.75613	+	1.75613i	0.366179	+	0.366179i	0.866082	−	0.499903i	$-0.166631\pi$
$23$	1.75613	+	1.75613i	0.366179	+	0.366179i	−0.499903	+	0.866082i	$0.666631\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.80674	+	3.80674i	0.732608	+	0.732608i
$28$		0			0
$29$	1.12978			0.209795			0.104897	−	0.994483i	$-0.466549\pi$
$29$	1.12978			0.209795			0.104897	+	0.994483i	$0.466549\pi$
$30$		0			0
$31$			4.96326i			0.891428i	0.895175	+	0.445714i	$0.147050\pi$
$31$			4.96326i			0.891428i	−0.895175	+	0.445714i	$0.852950\pi$
$32$		0			0
$33$	−0.663651	+	0.663651i	−0.115527	+	0.115527i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.80674	+	3.80674i	0.625825	+	0.625825i	0.947015	−	0.321190i	$-0.104083\pi$
$37$	3.80674	+	3.80674i	0.625825	+	0.625825i	−0.321190	+	0.947015i	$0.604083\pi$
$38$		0			0
$39$		−	3.30163i		−	0.528684i
$40$		0			0
$41$			5.22718i			0.816348i	0.912904	+	0.408174i	$0.133834\pi$
$41$			5.22718i			0.816348i	−0.912904	+	0.408174i	$0.866166\pi$
$42$		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
$47$	−4.40100	+	4.40100i	−0.641951	+	0.641951i	−0.951035	−	0.309083i	$-0.899978\pi$
$47$	−4.40100	+	4.40100i	−0.641951	+	0.641951i	0.309083	+	0.951035i	$0.399978\pi$
$48$		0			0
$49$			3.67247i			0.524639i
$50$		0			0
$51$			2.10176i			0.294305i
$52$		0			0
$53$	5.41508	−	5.41508i	0.743818	−	0.743818i	−0.229492	−	0.973310i	$-0.573707\pi$
$53$	5.41508	−	5.41508i	0.743818	−	0.743818i	0.973310	+	0.229492i	$0.0737066\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.12670	+	4.54975i	−0.281688	+	0.602629i
$58$		0			0
$59$	−3.99128			−0.519621			−0.259810	−	0.965660i	$-0.583660\pi$
$59$	−3.99128			−0.519621			−0.259810	+	0.965660i	$0.583660\pi$
$60$		0			0
$61$	−6.46115			−0.827266			−0.413633	−	0.910444i	$-0.635740\pi$
$61$	−6.46115			−0.827266			−0.413633	+	0.910444i	$0.635740\pi$
$62$		0			0
$63$	−2.15727	−	2.15727i	−0.271791	−	0.271791i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.22979	+	6.22979i	0.761091	+	0.761091i	0.976520	−	0.215429i	$-0.0691150\pi$
$67$	6.22979	+	6.22979i	0.761091	+	0.761091i	−0.215429	+	0.976520i	$0.569115\pi$
$68$		0			0
$69$	2.86150			0.344484
$70$		0			0
$71$			13.0519i			1.54898i	0.632586	+	0.774490i	$0.281994\pi$
$71$			13.0519i			1.54898i	−0.632586	+	0.774490i	$0.718006\pi$
$72$		0			0
$73$	−0.985576	−	0.985576i	−0.115353	−	0.115353i	0.647074	−	0.762427i	$-0.275992\pi$
$73$	−0.985576	−	0.985576i	−0.115353	−	0.115353i	−0.762427	+	0.647074i	$0.775992\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.05070	−	1.05070i	0.119738	−	0.119738i
$78$		0			0
$79$	−6.19916			−0.697460			−0.348730	−	0.937223i	$-0.613387\pi$
$79$	−6.19916			−0.697460			−0.348730	+	0.937223i	$0.613387\pi$
$80$		0			0
$81$	1.18542			0.131713
$82$		0			0
$83$	5.56062	+	5.56062i	0.610357	+	0.610357i	0.943039	−	0.332682i	$-0.107954\pi$
$83$	5.56062	+	5.56062i	0.610357	+	0.610357i	−0.332682	+	0.943039i	$0.607954\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	0.920451	−	0.920451i	0.0986827	−	0.0986827i
$88$		0			0
$89$	2.10176			0.222786			0.111393	−	0.993776i	$-0.464469\pi$
$89$	2.10176			0.222786			0.111393	+	0.993776i	$0.464469\pi$
$90$		0			0
$91$			5.22718i			0.547957i
$92$		0			0
$93$	4.04365	+	4.04365i	0.419307	+	0.419307i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.65568	+	3.65568i	0.371178	+	0.371178i	0.867906	−	0.496728i	$-0.165465\pi$
$97$	3.65568	+	3.65568i	0.371178	+	0.371178i	−0.496728	+	0.867906i	$0.665465\pi$
$98$		0			0
$99$		−	1.36236i		−	0.136922i
$100$		0			0
$101$	−9.47857			−0.943153			−0.471576	−	0.881825i	$-0.656315\pi$
$101$	−9.47857			−0.943153			−0.471576	+	0.881825i	$0.656315\pi$
$102$		0			0
$103$	0.0211000	−	0.0211000i	0.00207904	−	0.00207904i	−0.706066	−	0.708146i	$-0.749532\pi$
$103$	0.0211000	−	0.0211000i	0.00207904	−	0.00207904i	0.708146	+	0.706066i	$0.249532\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.57412	+	2.57412i	0.248849	+	0.248849i	0.820498	−	0.571649i	$-0.193696\pi$
$107$	2.57412	+	2.57412i	0.248849	+	0.248849i	−0.571649	+	0.820498i	$0.693696\pi$
$108$		0			0
$109$	6.95890			0.666542			0.333271	−	0.942831i	$-0.391848\pi$
$109$	6.95890			0.666542			0.333271	+	0.942831i	$0.391848\pi$
$110$		0			0
$111$	6.20284			0.588747
$112$		0			0
$113$	10.7002	−	10.7002i	1.00659	−	1.00659i	0.00661092	−	0.999978i	$-0.497896\pi$
$113$	10.7002	−	10.7002i	1.00659	−	1.00659i	0.999978	−	0.00661092i	$-0.00210434\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.38883	+	3.38883i	0.313298	+	0.313298i
$118$		0			0
$119$		−	3.32753i		−	0.305034i
$120$		0			0
$121$	−10.3365			−0.939678
$122$		0			0
$123$	4.25867	+	4.25867i	0.383992	+	0.383992i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	5.83299	+	5.83299i	0.517594	+	0.517594i	0.916843	−	0.399249i	$-0.130729\pi$
$127$	5.83299	+	5.83299i	0.517594	+	0.517594i	−0.399249	+	0.916843i	$0.630729\pi$
$128$		0			0
$129$	−1.99564			−0.175706
$130$		0			0
$131$	0.327528			0.0286163			0.0143081	−	0.999898i	$-0.495445\pi$
$131$	0.327528			0.0286163			0.0143081	+	0.999898i	$0.495445\pi$
$132$		0			0
$133$	3.36701	−	7.20321i	0.291957	−	0.624598i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.5898	−	10.5898i	0.904752	−	0.904752i	−0.0910909	−	0.995843i	$-0.529035\pi$
$137$	10.5898	−	10.5898i	0.904752	−	0.904752i	0.995843	+	0.0910909i	$0.0290354\pi$
$138$		0			0
$139$		−	21.2672i		−	1.80386i	−0.431878	−	0.901932i	$-0.642149\pi$
$139$		−	21.2672i		−	1.80386i	0.431878	−	0.901932i	$-0.357851\pi$
$140$		0			0
$141$			7.17114i			0.603919i
$142$		0			0
$143$	−1.65053	+	1.65053i	−0.138025	+	0.138025i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.99203	+	2.99203i	0.246778	+	0.246778i
$148$		0			0
$149$			2.27573i			0.186435i	0.995646	+	0.0932176i	$0.0297152\pi$
$149$			2.27573i			0.186435i	−0.995646	+	0.0932176i	$0.970285\pi$
$150$		0			0
$151$		−	9.82041i		−	0.799173i	−0.916695	−	0.399587i	$-0.869154\pi$
$151$		−	9.82041i		−	0.799173i	0.916695	−	0.399587i	$-0.130846\pi$
$152$		0			0
$153$	−2.15727	−	2.15727i	−0.174405	−	0.174405i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−5.26840	+	5.26840i	−0.420464	+	0.420464i	−0.885363	−	0.464899i	$-0.846090\pi$
$157$	−5.26840	+	5.26840i	−0.420464	+	0.420464i	0.464899	+	0.885363i	$0.346090\pi$
$158$		0			0
$159$		−	8.82351i		−	0.699750i
$160$		0			0
$161$	−4.53037			−0.357043
$162$		0			0
$163$	0.497982	+	0.497982i	0.0390050	+	0.0390050i	0.726340	−	0.687335i	$-0.241220\pi$
$163$	0.497982	+	0.497982i	0.0390050	+	0.0390050i	−0.687335	+	0.726340i	$0.741220\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.84096	+	2.84096i	0.219840	+	0.219840i	0.808431	−	0.588591i	$-0.200317\pi$
$167$	2.84096	+	2.84096i	0.219840	+	0.219840i	−0.588591	+	0.808431i	$0.700317\pi$
$168$		0			0
$169$			4.78868i			0.368360i
$170$		0			0
$171$	−2.48705	−	6.85279i	−0.190190	−	0.524046i
$172$		0			0
$173$	11.6871	−	11.6871i	0.888552	−	0.888552i	−0.105832	−	0.994384i	$-0.533751\pi$
$173$	11.6871	−	11.6871i	0.888552	−	0.888552i	0.994384	+	0.105832i	$0.0337507\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−3.25177	+	3.25177i	−0.244418	+	0.244418i
$178$		0			0
$179$	−3.12542			−0.233605			−0.116802	−	0.993155i	$-0.537264\pi$
$179$	−3.12542			−0.233605			−0.116802	+	0.993155i	$0.537264\pi$
$180$		0			0
$181$			11.1080i			0.825650i	0.910810	+	0.412825i	$0.135458\pi$
$181$			11.1080i			0.825650i	−0.910810	+	0.412825i	$0.864542\pi$
$182$		0			0
$183$	−5.26401	+	5.26401i	−0.389127	+	0.389127i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.05070	−	1.05070i	0.0768348	−	0.0768348i
$188$		0			0
$189$	−9.82041			−0.714329
$190$		0			0
$191$	−0.814579			−0.0589408			−0.0294704	−	0.999566i	$-0.509382\pi$
$191$	−0.814579			−0.0589408			−0.0294704	+	0.999566i	$0.509382\pi$
$192$		0			0
$193$	−5.30621	+	5.30621i	−0.381949	+	0.381949i	−0.871804	−	0.489855i	$-0.837050\pi$
$193$	−5.30621	+	5.30621i	−0.381949	+	0.381949i	0.489855	+	0.871804i	$0.337050\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.08966	−	7.08966i	0.505117	−	0.505117i	−0.407907	−	0.913024i	$-0.633741\pi$
$197$	7.08966	−	7.08966i	0.505117	−	0.505117i	0.913024	+	0.407907i	$0.133741\pi$
$198$		0			0
$199$			11.8753i			0.841818i	0.907103	+	0.420909i	$0.138289\pi$
$199$			11.8753i			0.841818i	−0.907103	+	0.420909i	$0.861711\pi$
$200$		0			0
$201$	10.1510			0.715999
$202$		0			0
$203$	−1.45727	+	1.45727i	−0.102280	+	0.102280i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.93708	+	2.93708i	−0.204141	+	0.204141i
$208$		0			0
$209$	3.33765	−	1.21132i	0.230870	−	0.0837887i
$210$		0			0
$211$		−	7.11670i		−	0.489934i	−0.969531	−	0.244967i	$-0.921223\pi$
$211$		−	7.11670i		−	0.489934i	0.969531	−	0.244967i	$-0.0787771\pi$
$212$		0			0
$213$	10.6336	+	10.6336i	0.728605	+	0.728605i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−6.40196	−	6.40196i	−0.434593	−	0.434593i
$218$		0			0
$219$	−1.60593			−0.108519
$220$		0			0
$221$			5.22718i			0.351618i
$222$		0			0
$223$	−15.4938	+	15.4938i	−1.03754	+	1.03754i	−0.0382753	+	0.999267i	$0.512186\pi$
$223$	−15.4938	+	15.4938i	−1.03754	+	1.03754i	−0.999267	+	0.0382753i	$0.987814\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.720040	−	0.720040i	−0.0477908	−	0.0477908i	0.682808	−	0.730598i	$-0.260759\pi$
$227$	−0.720040	−	0.720040i	−0.0477908	−	0.0477908i	−0.730598	+	0.682808i	$0.760759\pi$
$228$		0			0
$229$		−	6.11621i		−	0.404170i	−0.979368	−	0.202085i	$-0.935228\pi$
$229$		−	6.11621i		−	0.404170i	0.979368	−	0.202085i	$-0.0647718\pi$
$230$		0			0
$231$		−	1.71205i		−	0.112644i
$232$		0			0
$233$	−11.3800	−	11.3800i	−0.745532	−	0.745532i	0.228105	−	0.973637i	$-0.426747\pi$
$233$	−11.3800	−	11.3800i	−0.745532	−	0.745532i	−0.973637	+	0.228105i	$0.926747\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−5.05056	+	5.05056i	−0.328069	+	0.328069i
$238$		0			0
$239$			20.1769i			1.30514i	0.757729	+	0.652569i	$0.226309\pi$
$239$			20.1769i			1.30514i	−0.757729	+	0.652569i	$0.773691\pi$
$240$		0			0
$241$		−	1.23590i		−	0.0796111i	−0.999207	−	0.0398055i	$-0.987326\pi$
$241$		−	1.23590i		−	0.0796111i	0.999207	−	0.0398055i	$-0.0126738\pi$
$242$		0			0
$243$	−10.4544	+	10.4544i	−0.670653	+	0.670653i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5.28920	+	11.3155i	−0.336544	+	0.719985i
$248$		0			0
$249$	9.06066			0.574196
$250$		0			0
$251$	2.11621			0.133574			0.0667869	−	0.997767i	$-0.478725\pi$
$251$	2.11621			0.133574			0.0667869	+	0.997767i	$0.478725\pi$
$252$		0			0
$253$	−1.43051	−	1.43051i	−0.0899353	−	0.0899353i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.9562	−	18.9562i	−1.18246	−	1.18246i	−0.979106	−	0.203351i	$-0.934817\pi$
$257$	−18.9562	−	18.9562i	−1.18246	−	1.18246i	−0.203351	−	0.979106i	$-0.565183\pi$
$258$		0			0
$259$	−9.82041			−0.610210
$260$		0			0
$261$			1.88952i			0.116959i
$262$		0			0
$263$	−9.75416	−	9.75416i	−0.601468	−	0.601468i	0.339234	−	0.940702i	$-0.389832\pi$
$263$	−9.75416	−	9.75416i	−0.601468	−	0.601468i	−0.940702	+	0.339234i	$0.889832\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.71234	−	1.71234i	0.104793	−	0.104793i
$268$		0			0
$269$	28.4696			1.73582			0.867910	−	0.496722i	$-0.165463\pi$
$269$	28.4696			1.73582			0.867910	+	0.496722i	$0.165463\pi$
$270$		0			0
$271$	8.47857			0.515036			0.257518	−	0.966273i	$-0.417095\pi$
$271$	8.47857			0.515036			0.257518	+	0.966273i	$0.417095\pi$
$272$		0			0
$273$	4.25867	+	4.25867i	0.257747	+	0.257747i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	12.2157	−	12.2157i	0.733972	−	0.733972i	−0.237432	−	0.971404i	$-0.576306\pi$
$277$	12.2157	−	12.2157i	0.733972	−	0.733972i	0.971404	+	0.237432i	$0.0763057\pi$
$278$		0			0
$279$	−8.30092			−0.496963
$280$		0			0
$281$		−	12.3983i		−	0.739621i	−0.929107	−	0.369811i	$-0.879423\pi$
$281$		−	12.3983i		−	0.739621i	0.929107	−	0.369811i	$-0.120577\pi$
$282$		0			0
$283$	−7.68617	−	7.68617i	−0.456895	−	0.456895i	0.440740	−	0.897635i	$-0.354716\pi$
$283$	−7.68617	−	7.68617i	−0.456895	−	0.456895i	−0.897635	+	0.440740i	$0.854716\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.74238	−	6.74238i	−0.397990	−	0.397990i
$288$		0			0
$289$			13.6725i			0.804263i
$290$		0			0
$291$	5.95669			0.349187
$292$		0			0
$293$	13.1443	−	13.1443i	0.767901	−	0.767901i	−0.209836	−	0.977737i	$-0.567293\pi$
$293$	13.1443	−	13.1443i	0.767901	−	0.767901i	0.977737	+	0.209836i	$0.0672930\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−3.10089	−	3.10089i	−0.179932	−	0.179932i
$298$		0			0
$299$	7.11670			0.411570
$300$		0			0
$301$	3.15952			0.182112
$302$		0			0
$303$	−7.72235	+	7.72235i	−0.443638	+	0.443638i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−4.20355	−	4.20355i	−0.239909	−	0.239909i	0.576903	−	0.816813i	$-0.304261\pi$
$307$	−4.20355	−	4.20355i	−0.239909	−	0.239909i	−0.816813	+	0.576903i	$0.804261\pi$
$308$		0			0
$309$		−	0.0343811i		−	0.00195587i
$310$		0			0
$311$	−9.48705			−0.537961			−0.268981	−	0.963146i	$-0.586687\pi$
$311$	−9.48705			−0.537961			−0.268981	+	0.963146i	$0.586687\pi$
$312$		0			0
$313$	2.12387	+	2.12387i	0.120048	+	0.120048i	0.764579	−	0.644531i	$-0.222947\pi$
$313$	2.12387	+	2.12387i	0.120048	+	0.120048i	−0.644531	+	0.764579i	$0.722947\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.8775	−	12.8775i	−0.723272	−	0.723272i	0.245998	−	0.969270i	$-0.420884\pi$
$317$	−12.8775	−	12.8775i	−0.723272	−	0.723272i	−0.969270	+	0.245998i	$0.920884\pi$
$318$		0			0
$319$	−0.920294			−0.0515266
$320$		0			0
$321$	4.19435			0.234106
$322$		0			0
$323$	3.36701	−	7.20321i	0.187345	−	0.400797i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	5.66954	−	5.66954i	0.313526	−	0.313526i
$328$		0			0
$329$		−	11.3534i		−	0.625935i
$330$		0			0
$331$		−	32.4608i		−	1.78421i	−0.451829	−	0.892105i	$-0.649228\pi$
$331$		−	32.4608i		−	1.78421i	0.451829	−	0.892105i	$-0.350772\pi$
$332$		0			0
$333$	−6.36667	+	6.36667i	−0.348891	+	0.348891i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−12.8986	−	12.8986i	−0.702631	−	0.702631i	0.262343	−	0.964975i	$-0.415505\pi$
$337$	−12.8986	−	12.8986i	−0.702631	−	0.702631i	−0.964975	+	0.262343i	$0.915505\pi$
$338$		0			0
$339$		−	17.4353i		−	0.946953i
$340$		0			0
$341$		−	4.04297i		−	0.218939i
$342$		0			0
$343$	−13.7661	−	13.7661i	−0.743299	−	0.743299i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.86778	+	7.86778i	−0.422365	+	0.422365i	−0.886017	−	0.463652i	$-0.846539\pi$
$347$	−7.86778	+	7.86778i	−0.422365	+	0.422365i	0.463652	+	0.886017i	$0.346539\pi$
$348$		0			0
$349$			10.4871i			0.561359i	0.959802	+	0.280679i	$0.0905598\pi$
$349$			10.4871i			0.561359i	−0.959802	+	0.280679i	$0.909440\pi$
$350$		0			0
$351$	15.4268			0.823420
$352$		0			0
$353$	21.3855	+	21.3855i	1.13823	+	1.13823i	0.988767	+	0.149466i	$0.0477555\pi$
$353$	21.3855	+	21.3855i	1.13823	+	1.13823i	0.149466	+	0.988767i	$0.452244\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−2.71099	−	2.71099i	−0.143481	−	0.143481i
$358$		0			0
$359$			21.4701i			1.13315i	0.824011	+	0.566574i	$0.191731\pi$
$359$			21.4701i			1.13315i	−0.824011	+	0.566574i	$0.808269\pi$
$360$		0			0
$361$	14.5774	−	12.1861i	0.767229	−	0.641373i
$362$		0			0
$363$	−8.42129	+	8.42129i	−0.442003	+	0.442003i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2.37229	+	2.37229i	−0.123832	+	0.123832i	−0.766307	−	0.642475i	$-0.777908\pi$
$367$	−2.37229	+	2.37229i	−0.123832	+	0.123832i	0.642475	+	0.766307i	$0.277908\pi$
$368$		0			0
$369$	−8.74231			−0.455106
$370$		0			0
$371$			13.9695i			0.725260i
$372$		0			0
$373$	−1.23263	+	1.23263i	−0.0638229	+	0.0638229i	−0.738298	−	0.674475i	$-0.764370\pi$
$373$	−1.23263	+	1.23263i	−0.0638229	+	0.0638229i	0.674475	+	0.738298i	$0.264370\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.28921	−	2.28921i	0.117900	−	0.117900i
$378$		0			0
$379$	−9.48238			−0.487077			−0.243539	−	0.969891i	$-0.578308\pi$
$379$	−9.48238			−0.487077			−0.243539	+	0.969891i	$0.578308\pi$
$380$		0			0
$381$	9.50447			0.486929
$382$		0			0
$383$	−10.1734	+	10.1734i	−0.519837	+	0.519837i	−0.917522	−	0.397685i	$-0.869814\pi$
$383$	−10.1734	+	10.1734i	−0.519837	+	0.519837i	0.397685	+	0.917522i	$0.369814\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.04835	−	2.04835i	0.104124	−	0.104124i
$388$		0			0
$389$			24.8964i			1.26230i	0.775662	+	0.631149i	$0.217416\pi$
$389$			24.8964i			1.26230i	−0.775662	+	0.631149i	$0.782584\pi$
$390$		0			0
$391$	−4.53037			−0.229110
$392$		0			0
$393$	0.266843	−	0.266843i	0.0134604	−	0.0134604i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−9.59220	+	9.59220i	−0.481418	+	0.481418i	−0.905584	−	0.424166i	$-0.860567\pi$
$397$	−9.59220	+	9.59220i	−0.481418	+	0.481418i	0.424166	+	0.905584i	$0.360567\pi$
$398$		0			0
$399$	−3.12542	−	8.61174i	−0.156467	−	0.431126i
$400$		0			0
$401$		−	34.3504i		−	1.71538i	−0.514171	−	0.857688i	$-0.671900\pi$
$401$		−	34.3504i		−	1.71538i	0.514171	−	0.857688i	$-0.328100\pi$
$402$		0			0
$403$	10.0568	+	10.0568i	0.500963	+	0.500963i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.10089	−	3.10089i	−0.153705	−	0.153705i
$408$		0			0
$409$	20.5387			1.01557			0.507786	−	0.861483i	$-0.330464\pi$
$409$	20.5387			1.01557			0.507786	+	0.861483i	$0.330464\pi$
$410$		0			0
$411$		−	17.2555i		−	0.851149i
$412$		0			0
$413$	5.14823	−	5.14823i	0.253328	−	0.253328i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−17.3268	−	17.3268i	−0.848497	−	0.848497i
$418$		0			0
$419$		−	3.22874i		−	0.157734i	−0.996885	−	0.0788670i	$-0.974870\pi$
$419$		−	3.22874i		−	0.157734i	0.996885	−	0.0788670i	$-0.0251303\pi$
$420$		0			0
$421$			7.22282i			0.352019i	0.984388	+	0.176009i	$0.0563189\pi$
$421$			7.22282i			0.352019i	−0.984388	+	0.176009i	$0.943681\pi$
$422$		0			0
$423$	−7.36054	−	7.36054i	−0.357882	−	0.357882i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8.33404	−	8.33404i	0.403313	−	0.403313i
$428$		0			0
$429$			2.68944i			0.129847i
$430$		0			0
$431$		−	24.9741i		−	1.20296i	−0.798888	−	0.601480i	$-0.794578\pi$
$431$		−	24.9741i		−	1.20296i	0.798888	−	0.601480i	$-0.205422\pi$
$432$		0			0
$433$	−12.3085	+	12.3085i	−0.591510	+	0.591510i	−0.938039	−	0.346529i	$-0.887360\pi$
$433$	−12.3085	+	12.3085i	−0.591510	+	0.591510i	0.346529	+	0.938039i	$0.387360\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.80704	−	4.58412i	−0.469134	−	0.219288i
$438$		0			0
$439$	−31.2249			−1.49029			−0.745143	−	0.666905i	$-0.767619\pi$
$439$	−31.2249			−1.49029			−0.745143	+	0.666905i	$0.767619\pi$
$440$		0			0
$441$	−6.14211			−0.292481
$442$		0			0
$443$	−28.4510	−	28.4510i	−1.35175	−	1.35175i	−0.883709	−	0.468037i	$-0.844961\pi$
$443$	−28.4510	−	28.4510i	−1.35175	−	1.35175i	−0.468037	−	0.883709i	$-0.655039\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1.85408	+	1.85408i	0.0876948	+	0.0876948i
$448$		0			0
$449$	29.5477			1.39444			0.697220	−	0.716857i	$-0.254420\pi$
$449$	29.5477			1.39444			0.697220	+	0.716857i	$0.254420\pi$
$450$		0			0
$451$		−	4.25795i		−	0.200499i
$452$		0			0
$453$	−8.00085	−	8.00085i	−0.375913	−	0.375913i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	9.90857	−	9.90857i	0.463503	−	0.463503i	−0.436299	−	0.899802i	$-0.643711\pi$
$457$	9.90857	−	9.90857i	0.463503	−	0.463503i	0.899802	+	0.436299i	$0.143711\pi$
$458$		0			0
$459$	−9.82041			−0.458377
$460$		0			0
$461$	19.4612			0.906396			0.453198	−	0.891410i	$-0.350283\pi$
$461$	19.4612			0.906396			0.453198	+	0.891410i	$0.350283\pi$
$462$		0			0
$463$	0.0755138	+	0.0755138i	0.00350942	+	0.00350942i	0.708859	−	0.705350i	$-0.249210\pi$
$463$	0.0755138	+	0.0755138i	0.00350942	+	0.00350942i	−0.705350	+	0.708859i	$0.749210\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.0440	−	11.0440i	0.511057	−	0.511057i	−0.403793	−	0.914850i	$-0.632309\pi$
$467$	11.0440	−	11.0440i	0.511057	−	0.511057i	0.914850	+	0.403793i	$0.132309\pi$
$468$		0			0
$469$	−16.0712			−0.742101
$470$		0			0
$471$			8.58451i			0.395553i
$472$		0			0
$473$	0.997651	+	0.997651i	0.0458720	+	0.0458720i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	9.05657	+	9.05657i	0.414672	+	0.414672i
$478$		0			0
$479$			16.3708i			0.748003i	0.927428	+	0.374001i	$0.122014\pi$
$479$			16.3708i			0.748003i	−0.927428	+	0.374001i	$0.877986\pi$
$480$		0			0
$481$	15.4268			0.703400
$482$		0			0
$483$	−3.69097	+	3.69097i	−0.167945	+	0.167945i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−10.9881	−	10.9881i	−0.497920	−	0.497920i	0.412870	−	0.910790i	$-0.364526\pi$
$487$	−10.9881	−	10.9881i	−0.497920	−	0.497920i	−0.910790	+	0.412870i	$0.864526\pi$
$488$		0			0
$489$	0.811429			0.0366941
$490$		0			0
$491$	−5.99152			−0.270393			−0.135197	−	0.990819i	$-0.543167\pi$
$491$	−5.99152			−0.270393			−0.135197	+	0.990819i	$0.543167\pi$
$492$		0			0
$493$	−1.45727	+	1.45727i	−0.0656321	+	0.0656321i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.8353	−	16.8353i	−0.755167	−	0.755167i
$498$		0			0
$499$		−	24.7887i		−	1.10969i	−0.831952	−	0.554847i	$-0.812777\pi$
$499$		−	24.7887i		−	1.10969i	0.831952	−	0.554847i	$-0.187223\pi$
$500$		0			0
$501$	4.62916			0.206816
$502$		0			0
$503$	19.7596	+	19.7596i	0.881036	+	0.881036i	0.993640	−	0.112604i	$-0.0359192\pi$
$503$	19.7596	+	19.7596i	0.881036	+	0.881036i	−0.112604	+	0.993640i	$0.535919\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	3.90142	+	3.90142i	0.173268	+	0.173268i
$508$		0			0
$509$	35.0067			1.55165			0.775823	−	0.630950i	$-0.217335\pi$
$509$	35.0067			1.55165			0.775823	+	0.630950i	$0.217335\pi$
$510$		0			0
$511$	2.54253			0.112475
$512$		0			0
$513$	−21.2586	−	9.93693i	−0.938589	−	0.438726i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	3.58496	−	3.58496i	0.157666	−	0.157666i
$518$		0			0
$519$		−	19.0433i		−	0.835909i
$520$		0			0
$521$		−	17.9091i		−	0.784611i	−0.919835	−	0.392306i	$-0.871678\pi$
$521$		−	17.9091i		−	0.784611i	0.919835	−	0.392306i	$-0.128322\pi$
$522$		0			0
$523$	−27.1034	+	27.1034i	−1.18515	+	1.18515i	−0.206756	+	0.978392i	$0.566291\pi$
$523$	−27.1034	+	27.1034i	−1.18515	+	1.18515i	−0.978392	+	0.206756i	$0.933709\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.40196	−	6.40196i	−0.278874	−	0.278874i
$528$		0			0
$529$		−	16.8320i		−	0.731826i
$530$		0			0
$531$		−	6.67531i		−	0.289684i
$532$		0			0
$533$	10.5915	+	10.5915i	0.458770	+	0.458770i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.54633	+	2.54633i	−0.109882	+	0.109882i
$538$		0			0
$539$		−	2.99152i		−	0.128854i
$540$		0			0
$541$	−21.3660			−0.918598			−0.459299	−	0.888282i	$-0.651899\pi$
$541$	−21.3660			−0.918598			−0.459299	+	0.888282i	$0.651899\pi$
$542$		0			0
$543$	9.04987	+	9.04987i	0.388367	+	0.388367i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−9.86437	−	9.86437i	−0.421770	−	0.421770i	0.464043	−	0.885813i	$-0.346398\pi$
$547$	−9.86437	−	9.86437i	−0.421770	−	0.421770i	−0.885813	+	0.464043i	$0.846398\pi$
$548$		0			0
$549$		−	10.8061i		−	0.461193i
$550$		0			0
$551$	−4.62916	+	1.68004i	−0.197209	+	0.0715721i
$552$		0			0
$553$	7.99611	−	7.99611i	0.340029	−	0.340029i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.24759	+	8.24759i	−0.349462	+	0.349462i	−0.859909	−	0.510447i	$-0.829480\pi$
$557$	−8.24759	+	8.24759i	−0.349462	+	0.349462i	0.510447	+	0.859909i	$0.329480\pi$
$558$		0			0
$559$	−4.96326			−0.209924
$560$		0			0
$561$		−	1.71205i		−	0.0722827i
$562$		0			0
$563$	15.1284	−	15.1284i	0.637585	−	0.637585i	−0.312374	−	0.949959i	$-0.601124\pi$
$563$	15.1284	−	15.1284i	0.637585	−	0.637585i	0.949959	+	0.312374i	$0.101124\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.52904	+	1.52904i	−0.0642136	+	0.0642136i
$568$		0			0
$569$	26.1039			1.09433			0.547166	−	0.837024i	$-0.315707\pi$
$569$	26.1039			1.09433			0.547166	+	0.837024i	$0.315707\pi$
$570$		0			0
$571$	−20.6814			−0.865490			−0.432745	−	0.901516i	$-0.642455\pi$
$571$	−20.6814			−0.865490			−0.432745	+	0.901516i	$0.642455\pi$
$572$		0			0
$573$	−0.663651	+	0.663651i	−0.0277244	+	0.0277244i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	22.9238	−	22.9238i	0.954328	−	0.954328i	−0.0446734	−	0.999002i	$-0.514225\pi$
$577$	22.9238	−	22.9238i	0.954328	−	0.954328i	0.999002	+	0.0446734i	$0.0142247\pi$
$578$		0			0
$579$			8.64612i			0.359321i
$580$		0			0
$581$	−14.3449			−0.595129
$582$		0			0
$583$	−4.41101	+	4.41101i	−0.182685	+	0.182685i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.1193	−	19.1193i	0.789137	−	0.789137i	−0.192216	−	0.981353i	$-0.561567\pi$
$587$	19.1193	−	19.1193i	0.789137	−	0.789137i	0.981353	+	0.192216i	$0.0615674\pi$
$588$		0			0
$589$	−7.38062	−	20.3365i	−0.304113	−	0.837949i
$590$		0			0
$591$		−	11.5521i		−	0.475191i
$592$		0			0
$593$	16.1701	+	16.1701i	0.664027	+	0.664027i	0.956327	−	0.292300i	$-0.0944206\pi$
$593$	16.1701	+	16.1701i	0.664027	+	0.664027i	−0.292300	+	0.956327i	$0.594421\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	9.67502	+	9.67502i	0.395972	+	0.395972i
$598$		0			0
$599$	23.7702			0.971225			0.485612	−	0.874174i	$-0.338597\pi$
$599$	23.7702			0.971225			0.485612	+	0.874174i	$0.338597\pi$
$600$		0			0
$601$		−	21.6168i		−	0.881767i	−0.897564	−	0.440883i	$-0.854665\pi$
$601$		−	21.6168i		−	0.881767i	0.897564	−	0.440883i	$-0.145335\pi$
$602$		0			0
$603$	−10.4192	+	10.4192i	−0.424301	+	0.424301i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	12.3860	+	12.3860i	0.502733	+	0.502733i	0.912286	−	0.409553i	$-0.134315\pi$
$607$	12.3860	+	12.3860i	0.502733	+	0.502733i	−0.409553	+	0.912286i	$0.634315\pi$
$608$		0			0
$609$			2.37452i			0.0962205i
$610$		0			0
$611$			17.8350i			0.721526i
$612$		0			0
$613$	−24.8736	−	24.8736i	−1.00463	−	1.00463i	−0.999989	−	0.00464487i	$-0.998521\pi$
$613$	−24.8736	−	24.8736i	−1.00463	−	1.00463i	−0.00464487	−	0.999989i	$-0.501479\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	20.8558	−	20.8558i	0.839621	−	0.839621i	−0.149188	−	0.988809i	$-0.547666\pi$
$617$	20.8558	−	20.8558i	0.839621	−	0.839621i	0.988809	+	0.149188i	$0.0476659\pi$
$618$		0			0
$619$			29.3016i			1.17773i	0.808231	+	0.588866i	$0.200425\pi$
$619$			29.3016i			1.17773i	−0.808231	+	0.588866i	$0.799575\pi$
$620$		0			0
$621$			13.3703i			0.536531i
$622$		0			0
$623$	−2.71099	+	2.71099i	−0.108614	+	0.108614i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1.73236	−	3.70613i	0.0691838	−	0.148008i
$628$		0			0
$629$	−9.82041			−0.391565
$630$		0			0
$631$	1.24983			0.0497550			0.0248775	−	0.999691i	$-0.492080\pi$
$631$	1.24983			0.0497550			0.0248775	+	0.999691i	$0.492080\pi$
$632$		0			0
$633$	−5.79810	−	5.79810i	−0.230454	−	0.230454i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	7.44132	+	7.44132i	0.294836	+	0.294836i
$638$		0			0
$639$	−21.8290			−0.863542
$640$		0			0
$641$			19.0906i			0.754031i	0.926207	+	0.377016i	$0.123050\pi$
$641$			19.0906i			0.754031i	−0.926207	+	0.377016i	$0.876950\pi$
$642$		0			0
$643$	−10.6002	−	10.6002i	−0.418032	−	0.418032i	0.466493	−	0.884525i	$-0.345517\pi$
$643$	−10.6002	−	10.6002i	−0.418032	−	0.418032i	−0.884525	+	0.466493i	$0.845517\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.8150	+	18.8150i	−0.739693	+	0.739693i	−0.972518	−	0.232826i	$-0.925203\pi$
$647$	−18.8150	+	18.8150i	−0.739693	+	0.739693i	0.232826	+	0.972518i	$0.425203\pi$
$648$		0			0
$649$	3.25121			0.127621
$650$		0			0
$651$	−10.4316			−0.408846
$652$		0			0
$653$	19.4236	+	19.4236i	0.760103	+	0.760103i	0.976341	−	0.216238i	$-0.0693787\pi$
$653$	19.4236	+	19.4236i	0.760103	+	0.760103i	−0.216238	+	0.976341i	$0.569379\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.64835	−	1.64835i	0.0643082	−	0.0643082i
$658$		0			0
$659$	42.5996			1.65944			0.829722	−	0.558176i	$-0.188499\pi$
$659$	42.5996			1.65944			0.829722	+	0.558176i	$0.188499\pi$
$660$		0			0
$661$			32.4608i			1.26258i	0.775547	+	0.631290i	$0.217474\pi$
$661$			32.4608i			1.26258i	−0.775547	+	0.631290i	$0.782526\pi$
$662$		0			0
$663$	4.25867	+	4.25867i	0.165393	+	0.165393i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.98404	+	1.98404i	0.0768224	+	0.0768224i
$668$		0			0
$669$			25.2462i			0.976073i
$670$		0			0
$671$	5.26312			0.203180
$672$		0			0
$673$	12.3929	−	12.3929i	0.477712	−	0.477712i	−0.426687	−	0.904399i	$-0.640319\pi$
$673$	12.3929	−	12.3929i	0.477712	−	0.477712i	0.904399	+	0.426687i	$0.140319\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.09185	+	5.09185i	0.195696	+	0.195696i	0.798152	−	0.602456i	$-0.205811\pi$
$677$	5.09185	+	5.09185i	0.195696	+	0.195696i	−0.602456	+	0.798152i	$0.705811\pi$
$678$		0			0
$679$	−9.43069			−0.361917
$680$		0			0
$681$	−1.17326			−0.0449594
$682$		0			0
$683$	10.3598	−	10.3598i	0.396406	−	0.396406i	−0.480558	−	0.876963i	$-0.659566\pi$
$683$	10.3598	−	10.3598i	0.396406	−	0.396406i	0.876963	+	0.480558i	$0.159566\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−4.98298	−	4.98298i	−0.190113	−	0.190113i
$688$		0			0
$689$		−	21.9445i		−	0.836020i
$690$		0			0
$691$	6.40567			0.243683			0.121842	−	0.992550i	$-0.461120\pi$
$691$	6.40567			0.243683			0.121842	+	0.992550i	$0.461120\pi$
$692$		0			0
$693$	1.75727	+	1.75727i	0.0667531	+	0.0667531i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−6.74238	−	6.74238i	−0.255386	−	0.255386i
$698$		0			0
$699$	−18.5430			−0.701362
$700$		0			0
$701$	1.59958			0.0604152			0.0302076	−	0.999544i	$-0.490383\pi$
$701$	1.59958			0.0604152			0.0302076	+	0.999544i	$0.490383\pi$
$702$		0			0
$703$	−21.2586	−	9.93693i	−0.801782	−	0.374778i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.2261	−	12.2261i	0.459811	−	0.459811i
$708$		0			0
$709$		−	27.8579i		−	1.04623i	−0.852264	−	0.523113i	$-0.824771\pi$
$709$		−	27.8579i		−	1.04623i	0.852264	−	0.523113i	$-0.175229\pi$
$710$		0			0
$711$		−	10.3679i		−	0.388827i
$712$		0			0
$713$	−8.71615	+	8.71615i	−0.326422	+	0.326422i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	16.4385	+	16.4385i	0.613907	+	0.613907i
$718$		0			0
$719$		−	16.7972i		−	0.626428i	−0.949682	−	0.313214i	$-0.898594\pi$
$719$		−	16.7972i		−	0.626428i	0.949682	−	0.313214i	$-0.101406\pi$
$720$		0			0
$721$			0.0544325i			0.00202717i
$722$		0			0
$723$	−1.00691	−	1.00691i	−0.0374472	−	0.0374472i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−23.0113	+	23.0113i	−0.853443	+	0.853443i	−0.990556	−	0.137112i	$-0.956218\pi$
$727$	−23.0113	+	23.0113i	−0.853443	+	0.853443i	0.137112	+	0.990556i	$0.456218\pi$
$728$		0			0
$729$			20.5911i			0.762633i
$730$		0			0
$731$	3.15952			0.116859
$732$		0			0
$733$	32.3085	+	32.3085i	1.19334	+	1.19334i	0.976123	+	0.217219i	$0.0696984\pi$
$733$	32.3085	+	32.3085i	1.19334	+	1.19334i	0.217219	+	0.976123i	$0.430302\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.07466	−	5.07466i	−0.186927	−	0.186927i
$738$		0			0
$739$		−	2.71947i		−	0.100037i	−0.998748	−	0.0500186i	$-0.984072\pi$
$739$		−	2.71947i		−	0.100037i	0.998748	−	0.0500186i	$-0.0159281\pi$
$740$		0			0
$741$	4.90969	+	13.5281i	0.180362	+	0.496967i
$742$		0			0
$743$	36.7784	−	36.7784i	1.34927	−	1.34927i	0.462814	−	0.886456i	$-0.346840\pi$
$743$	36.7784	−	36.7784i	1.34927	−	1.34927i	0.886456	−	0.462814i	$-0.153160\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−9.29998	+	9.29998i	−0.340268	+	0.340268i
$748$		0			0
$749$	−6.64055			−0.242640
$750$		0			0
$751$			10.0843i			0.367982i	0.982928	+	0.183991i	$0.0589018\pi$
$751$			10.0843i			0.367982i	−0.982928	+	0.183991i	$0.941098\pi$
$752$		0			0
$753$	1.72411	−	1.72411i	0.0628301	−	0.0628301i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−34.0571	+	34.0571i	−1.23782	+	1.23782i	−0.276937	+	0.960888i	$0.589319\pi$
$757$	−34.0571	+	34.0571i	−1.23782	+	1.23782i	−0.960888	+	0.276937i	$0.910681\pi$
$758$		0			0
$759$	−2.33092			−0.0846070
$760$		0			0
$761$	25.9397			0.940314			0.470157	−	0.882583i	$-0.344197\pi$
$761$	25.9397			0.940314			0.470157	+	0.882583i	$0.344197\pi$
$762$		0			0
$763$	−8.97608	+	8.97608i	−0.324956	+	0.324956i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.08731	+	8.08731i	−0.292016	+	0.292016i
$768$		0			0
$769$			3.36191i			0.121234i	0.998161	+	0.0606168i	$0.0193068\pi$
$769$			3.36191i			0.121234i	−0.998161	+	0.0606168i	$0.980693\pi$
$770$		0			0
$771$	−30.8879			−1.11240
$772$		0			0
$773$	26.7419	−	26.7419i	0.961838	−	0.961838i	−0.0374597	−	0.999298i	$-0.511927\pi$
$773$	26.7419	−	26.7419i	0.961838	−	0.961838i	0.999298	+	0.0374597i	$0.0119266\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−8.00085	+	8.00085i	−0.287029	+	0.287029i
$778$		0			0
$779$	−7.77308	−	21.4178i	−0.278499	−	0.767374i
$780$		0			0
$781$		−	10.6318i		−	0.380437i
$782$		0			0
$783$	4.30078	+	4.30078i	0.153697	+	0.153697i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	20.7014	+	20.7014i	0.737927	+	0.737927i	0.972176	−	0.234250i	$-0.0752634\pi$
$787$	20.7014	+	20.7014i	0.737927	+	0.737927i	−0.234250	+	0.972176i	$0.575263\pi$
$788$		0			0
$789$	−15.8938			−0.565833
$790$		0			0
$791$			27.6037i			0.981475i
$792$		0			0
$793$	−13.0919	+	13.0919i	−0.464906	+	0.464906i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.6541	+	22.6541i	0.802449	+	0.802449i	0.983478	−	0.181028i	$-0.0579426\pi$
$797$	22.6541	+	22.6541i	0.802449	+	0.802449i	−0.181028	+	0.983478i	$0.557943\pi$
$798$		0			0
$799$		−	11.3534i		−	0.401655i
$800$		0			0
$801$			3.51513i			0.124201i
$802$		0			0
$803$	0.802829	+	0.802829i	0.0283312	+	0.0283312i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	23.1946	−	23.1946i	0.816490	−	0.816490i
$808$		0			0
$809$			45.6677i			1.60559i	0.596255	+	0.802795i	$0.296655\pi$
$809$			45.6677i			1.60559i	−0.596255	+	0.802795i	$0.703345\pi$
$810$		0			0
$811$		−	0.421719i		−	0.0148085i	−0.999973	−	0.00740427i	$-0.997643\pi$
$811$		−	0.421719i		−	0.0148085i	0.999973	−	0.00740427i	$-0.00235687\pi$
$812$		0			0
$813$	6.90763	−	6.90763i	0.242261	−	0.242261i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	6.83953	+	3.19701i	0.239285	+	0.111849i
$818$		0			0
$819$	−8.74231			−0.305481
$820$		0			0
$821$	−3.24983			−0.113420			−0.0567100	−	0.998391i	$-0.518061\pi$
$821$	−3.24983			−0.113420			−0.0567100	+	0.998391i	$0.518061\pi$
$822$		0			0
$823$	−28.5795	−	28.5795i	−0.996220	−	0.996220i	0.00377327	−	0.999993i	$-0.498799\pi$
$823$	−28.5795	−	28.5795i	−0.996220	−	0.996220i	−0.999993	+	0.00377327i	$0.998799\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.78457	−	2.78457i	−0.0968290	−	0.0968290i	0.657033	−	0.753862i	$-0.271811\pi$
$827$	−2.78457	−	2.78457i	−0.0968290	−	0.0968290i	−0.753862	+	0.657033i	$0.771811\pi$
$828$		0			0
$829$	−28.2601			−0.981513			−0.490757	−	0.871297i	$-0.663280\pi$
$829$	−28.2601			−0.981513			−0.490757	+	0.871297i	$0.663280\pi$
$830$		0			0
$831$		−	19.9047i		−	0.690487i
$832$		0			0
$833$	−4.73701	−	4.73701i	−0.164128	−	0.164128i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−18.8939	+	18.8939i	−0.653067	+	0.653067i
$838$		0			0
$839$	4.75103			0.164024			0.0820118	−	0.996631i	$-0.473865\pi$
$839$	4.75103			0.164024			0.0820118	+	0.996631i	$0.473865\pi$
$840$		0			0
$841$	−27.7236			−0.955986
$842$		0			0
$843$	−10.1011	−	10.1011i	−0.347901	−	0.347901i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	13.3327	−	13.3327i	0.458117	−	0.458117i
$848$		0			0
$849$	−12.5241			−0.429826
$850$		0			0
$851$			13.3703i			0.458328i
$852$		0			0
$853$	7.41359	+	7.41359i	0.253837	+	0.253837i	0.822542	−	0.568705i	$-0.192555\pi$
$853$	7.41359	+	7.41359i	0.253837	+	0.253837i	−0.568705	+	0.822542i	$0.692555\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.2874	−	12.2874i	−0.419730	−	0.419730i	0.465380	−	0.885111i	$-0.345918\pi$
$857$	−12.2874	−	12.2874i	−0.419730	−	0.419730i	−0.885111	+	0.465380i	$0.845918\pi$
$858$		0			0
$859$			33.0174i			1.12654i	0.826273	+	0.563270i	$0.190457\pi$
$859$			33.0174i			1.12654i	−0.826273	+	0.563270i	$0.809543\pi$
$860$		0			0
$861$	−10.9863			−0.374411
$862$		0			0
$863$	19.5463	−	19.5463i	0.665364	−	0.665364i	−0.291275	−	0.956639i	$-0.594080\pi$
$863$	19.5463	−	19.5463i	0.665364	−	0.665364i	0.956639	+	0.291275i	$0.0940796\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	11.1392	+	11.1392i	0.378307	+	0.378307i
$868$		0			0
$869$	5.04970			0.171299
$870$		0			0
$871$	25.2462			0.855433
$872$		0			0
$873$	−6.11402	+	6.11402i	−0.206928	+	0.206928i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	39.0504	+	39.0504i	1.31864	+	1.31864i	0.914854	+	0.403784i	$0.132305\pi$
$877$	39.0504	+	39.0504i	1.31864	+	1.31864i	0.403784	+	0.914854i	$0.367695\pi$
$878$		0			0
$879$		−	21.4178i		−	0.722406i
$880$		0			0
$881$	22.6037			0.761538			0.380769	−	0.924670i	$-0.375659\pi$
$881$	22.6037			0.761538			0.380769	+	0.924670i	$0.375659\pi$
$882$		0			0
$883$	36.9728	+	36.9728i	1.24423	+	1.24423i	0.958228	+	0.286007i	$0.0923279\pi$
$883$	36.9728	+	36.9728i	1.24423	+	1.24423i	0.286007	+	0.958228i	$0.407672\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.0509	+	23.0509i	0.773974	+	0.773974i	0.978799	−	0.204825i	$-0.0656625\pi$
$887$	23.0509	+	23.0509i	0.773974	+	0.773974i	−0.204825	+	0.978799i	$0.565662\pi$
$888$		0			0
$889$	−15.0476			−0.504680
$890$		0			0
$891$	−0.965619			−0.0323495
$892$		0			0
$893$	11.4881	−	24.5771i	0.384436	−	0.822443i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	5.79810	−	5.79810i	0.193593	−	0.193593i
$898$		0			0
$899$			5.60739i			0.187017i
$900$		0			0
$901$			13.9695i			0.465391i
$902$		0			0
$903$	2.57412	−	2.57412i	0.0856613	−	0.0856613i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	15.0759	+	15.0759i	0.500587	+	0.500587i	0.911620	−	0.411033i	$-0.134832\pi$
$907$	15.0759	+	15.0759i	0.500587	+	0.500587i	−0.411033	+	0.911620i	$0.634832\pi$
$908$		0			0
$909$		−	15.8526i		−	0.525799i
$910$		0			0
$911$			11.6386i			0.385603i	0.981238	+	0.192802i	$0.0617573\pi$
$911$			11.6386i			0.385603i	−0.981238	+	0.192802i	$0.938243\pi$
$912$		0			0
$913$	−4.52956	−	4.52956i	−0.149907	−	0.149907i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.422468	+	0.422468i	−0.0139511	+	0.0139511i
$918$		0			0
$919$			30.8409i			1.01735i	0.860959	+	0.508674i	$0.169864\pi$
$919$			30.8409i			1.01735i	−0.860959	+	0.508674i	$0.830136\pi$
$920$		0			0
$921$	−6.84941			−0.225696
$922$		0			0
$923$	26.4464	+	26.4464i	0.870494	+	0.870494i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	0.0352892	+	0.0352892i	0.00115905	+	0.00115905i
$928$		0			0
$929$		−	34.2070i		−	1.12229i	−0.827716	−	0.561147i	$-0.810360\pi$
$929$		−	34.2070i		−	1.12229i	0.827716	−	0.561147i	$-0.189640\pi$
$930$		0			0
$931$	−5.46115	−	15.0476i	−0.178982	−	0.493165i
$932$		0			0
$933$	−7.72926	+	7.72926i	−0.253045	+	0.253045i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−38.2293	+	38.2293i	−1.24890	+	1.24890i	−0.292688	+	0.956208i	$0.594550\pi$
$937$	−38.2293	+	38.2293i	−1.24890	+	1.24890i	−0.956208	+	0.292688i	$0.905450\pi$
$938$		0			0
$939$	3.46070			0.112936
$940$		0			0
$941$			53.6559i			1.74913i	0.484906	+	0.874566i	$0.338854\pi$
$941$			53.6559i			1.74913i	−0.484906	+	0.874566i	$0.661146\pi$
$942$		0			0
$943$	−9.17962	+	9.17962i	−0.298930	+	0.298930i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.6170	+	27.6170i	−0.897431	+	0.897431i	−0.995208	−	0.0977775i	$-0.968827\pi$
$947$	−27.6170	+	27.6170i	−0.897431	+	0.897431i	0.0977775	+	0.995208i	$0.468827\pi$
$948$		0			0
$949$	−3.99403			−0.129652
$950$		0			0
$951$	−20.9830			−0.680422
$952$		0			0
$953$	1.51366	−	1.51366i	0.0490322	−	0.0490322i	−0.682166	−	0.731198i	$-0.738962\pi$
$953$	1.51366	−	1.51366i	0.0490322	−	0.0490322i	0.731198	+	0.682166i	$0.238962\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−0.749780	+	0.749780i	−0.0242369	+	0.0242369i
$958$		0			0
$959$			27.3190i			0.882178i
$960$		0			0
$961$	6.36604			0.205356
$962$		0			0
$963$	−4.30514	+	4.30514i	−0.138731	+	0.138731i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	12.5200	−	12.5200i	0.402617	−	0.402617i	−0.476537	−	0.879154i	$-0.658108\pi$
$967$	12.5200	−	12.5200i	0.402617	−	0.402617i	0.879154	+	0.476537i	$0.158108\pi$
$968$		0			0
$969$	−3.12542	−	8.61174i	−0.100403	−	0.276649i
$970$		0			0
$971$			46.1692i			1.48164i	0.671704	+	0.740819i	$0.265563\pi$
$971$			46.1692i			1.48164i	−0.671704	+	0.740819i	$0.734437\pi$
$972$		0			0
$973$	27.4320	+	27.4320i	0.879429	+	0.879429i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.8919	−	25.8919i	−0.828354	−	0.828354i	0.158935	−	0.987289i	$-0.449194\pi$
$977$	−25.8919	−	25.8919i	−0.828354	−	0.828354i	−0.987289	+	0.158935i	$0.949194\pi$
$978$		0			0
$979$	−1.71205			−0.0547173
$980$		0			0
$981$			11.6386i			0.371591i
$982$		0			0
$983$	31.9637	−	31.9637i	1.01948	−	1.01948i	0.0196770	−	0.999806i	$-0.493736\pi$
$983$	31.9637	−	31.9637i	1.01948	−	1.01948i	0.999806	−	0.0196770i	$-0.00626377\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−9.24983	−	9.24983i	−0.294425	−	0.294425i
$988$		0			0
$989$		−	4.30163i		−	0.136784i
$990$		0			0
$991$			38.9980i			1.23881i	0.785070	+	0.619407i	$0.212627\pi$
$991$			38.9980i			1.23881i	−0.785070	+	0.619407i	$0.787373\pi$
$992$		0			0
$993$	−26.4464	−	26.4464i	−0.839251	−	0.839251i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.08045	−	9.08045i	0.287581	−	0.287581i	−0.548542	−	0.836123i	$-0.684817\pi$
$997$	9.08045	−	9.08045i	0.287581	−	0.287581i	0.836123	+	0.548542i	$0.184817\pi$
$998$		0			0
$999$			28.9826i			0.916968i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.l.d.493.7	yes	24
5.2	odd	4	inner	1900.2.l.d.1557.5	yes	24
5.3	odd	4	inner	1900.2.l.d.1557.8	yes	24
5.4	even	2	inner	1900.2.l.d.493.6	yes	24
19.18	odd	2	inner	1900.2.l.d.493.5	✓	24
95.18	even	4	inner	1900.2.l.d.1557.6	yes	24
95.37	even	4	inner	1900.2.l.d.1557.7	yes	24
95.94	odd	2	inner	1900.2.l.d.493.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.l.d.493.5	✓	24	19.18	odd	2	inner
1900.2.l.d.493.6	yes	24	5.4	even	2	inner
1900.2.l.d.493.7	yes	24	1.1	even	1	trivial
1900.2.l.d.493.8	yes	24	95.94	odd	2	inner
1900.2.l.d.1557.5	yes	24	5.2	odd	4	inner
1900.2.l.d.1557.6	yes	24	95.18	even	4	inner
1900.2.l.d.1557.7	yes	24	95.37	even	4	inner
1900.2.l.d.1557.8	yes	24	5.3	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)

\(f(q)\)	\(=\)	\(q+(0.814717 - 0.814717i) q^{3} +(-1.28987 + 1.28987i) q^{7} +1.67247i q^{9} -0.814579 q^{11} +(2.02624 - 2.02624i) q^{13} +(-1.28987 + 1.28987i) q^{17} +(-4.09740 + 1.48705i) q^{19} +2.10176i q^{21} +(1.75613 + 1.75613i) q^{23} +(3.80674 + 3.80674i) q^{27} +1.12978 q^{29} +4.96326i q^{31} +(-0.663651 + 0.663651i) q^{33} +(3.80674 + 3.80674i) q^{37} -3.30163i q^{39} +5.22718i q^{41} +(-1.22474 - 1.22474i) q^{43} +(-4.40100 + 4.40100i) q^{47} +3.67247i q^{49} +2.10176i q^{51} +(5.41508 - 5.41508i) q^{53} +(-2.12670 + 4.54975i) q^{57} -3.99128 q^{59} -6.46115 q^{61} +(-2.15727 - 2.15727i) q^{63} +(6.22979 + 6.22979i) q^{67} +2.86150 q^{69} +13.0519i q^{71} +(-0.985576 - 0.985576i) q^{73} +(1.05070 - 1.05070i) q^{77} -6.19916 q^{79} +1.18542 q^{81} +(5.56062 + 5.56062i) q^{83} +(0.920451 - 0.920451i) q^{87} +2.10176 q^{89} +5.22718i q^{91} +(4.04365 + 4.04365i) q^{93} +(3.65568 + 3.65568i) q^{97} -1.36236i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{11} + 24 q^{61} + 24 q^{81}+O(q^{100}) \) 24 * q - 24 * q^11 + 24 * q^61 + 24 * q^81

Introduction

L-functions

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