LMFDB - Embedded newform 225.6.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,6,Mod(1,225)] 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("225.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,7,0,17,0,0,-12] 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:	yes
Analytic conductor:	$36.0863594579$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+7.00000 q^{2} +17.0000 q^{4} -12.0000 q^{7} -105.000 q^{8} -112.000 q^{11} +974.000 q^{13} -84.0000 q^{14} -1279.00 q^{16} +2182.00 q^{17} +1420.00 q^{19} -784.000 q^{22} +3216.00 q^{23} +6818.00 q^{26} -204.000 q^{28} +4150.00 q^{29} -5688.00 q^{31} -5593.00 q^{32} +15274.0 q^{34} -6482.00 q^{37} +9940.00 q^{38} -5402.00 q^{41} +21764.0 q^{43} -1904.00 q^{44} +22512.0 q^{46} -368.000 q^{47} -16663.0 q^{49} +16558.0 q^{52} +12586.0 q^{53} +1260.00 q^{56} +29050.0 q^{58} +25520.0 q^{59} +11782.0 q^{61} -39816.0 q^{62} +1777.00 q^{64} +13188.0 q^{67} +37094.0 q^{68} +35968.0 q^{71} -73186.0 q^{73} -45374.0 q^{74} +24140.0 q^{76} +1344.00 q^{77} -52440.0 q^{79} -37814.0 q^{82} +69036.0 q^{83} +152348. q^{86} +11760.0 q^{88} +33870.0 q^{89} -11688.0 q^{91} +54672.0 q^{92} -2576.00 q^{94} -143042. q^{97} -116641. q^{98} +O(q^{100})$

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 $

$2$	7.00000	1.23744	0.618718	−	0.785613i	$-0.287652\pi$
$2$	7.00000	1.23744	0.618718	+	0.785613i	$0.287652\pi$
$3$	0	0
$3$	0	0
$4$	17.0000	0.531250
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−12.0000	−0.0925627	−0.0462814	−	0.998928i	$-0.514737\pi$
$7$	−12.0000	−0.0925627	−0.0462814	+	0.998928i	$0.514737\pi$
$8$	−105.000	−0.580049
$9$	0	0
$10$	0	0
$11$	−112.000	−0.279085	−0.139542	−	0.990216i	$-0.544563\pi$
$11$	−112.000	−0.279085	−0.139542	+	0.990216i	$0.544563\pi$
$12$	0	0
$13$	974.000	1.59846	0.799228	−	0.601028i	$-0.205242\pi$
$13$	974.000	1.59846	0.799228	+	0.601028i	$0.205242\pi$
$14$	−84.0000	−0.114541
$15$	0	0
$16$	−1279.00	−1.24902
$17$	2182.00	1.83119	0.915593	−	0.402106i	$-0.131722\pi$
$17$	2182.00	1.83119	0.915593	+	0.402106i	$0.131722\pi$
$18$	0	0
$19$	1420.00	0.902411	0.451205	−	0.892420i	$-0.350994\pi$
$19$	1420.00	0.902411	0.451205	+	0.892420i	$0.350994\pi$
$20$	0	0
$21$	0	0
$22$	−784.000	−0.345350
$23$	3216.00	1.26764	0.633821	−	0.773480i	$-0.281486\pi$
$23$	3216.00	1.26764	0.633821	+	0.773480i	$0.281486\pi$
$24$	0	0
$25$	0	0
$26$	6818.00	1.97799
$27$	0	0
$28$	−204.000	−0.0491739
$29$	4150.00	0.916333	0.458166	−	0.888867i	$-0.348506\pi$
$29$	4150.00	0.916333	0.458166	+	0.888867i	$0.348506\pi$
$30$	0	0
$31$	−5688.00	−1.06305	−0.531527	−	0.847041i	$-0.678382\pi$
$31$	−5688.00	−1.06305	−0.531527	+	0.847041i	$0.678382\pi$
$32$	−5593.00	−0.965539
$33$	0	0
$34$	15274.0	2.26598
$35$	0	0
$36$	0	0
$37$	−6482.00	−0.778403	−0.389202	−	0.921153i	$-0.627249\pi$
$37$	−6482.00	−0.778403	−0.389202	+	0.921153i	$0.627249\pi$
$38$	9940.00	1.11668
$39$	0	0
$40$	0	0
$41$	−5402.00	−0.501874	−0.250937	−	0.968003i	$-0.580739\pi$
$41$	−5402.00	−0.501874	−0.250937	+	0.968003i	$0.580739\pi$
$42$	0	0
$43$	21764.0	1.79501	0.897506	−	0.441001i	$-0.145377\pi$
$43$	21764.0	1.79501	0.897506	+	0.441001i	$0.145377\pi$
$44$	−1904.00	−0.148264
$45$	0	0
$46$	22512.0	1.56863
$47$	−368.000	−0.0242998	−0.0121499	−	0.999926i	$-0.503868\pi$
$47$	−368.000	−0.0242998	−0.0121499	+	0.999926i	$0.503868\pi$
$48$	0	0
$49$	−16663.0	−0.991432
$50$	0	0
$51$	0	0
$52$	16558.0	0.849180
$53$	12586.0	0.615457	0.307729	−	0.951474i	$-0.400431\pi$
$53$	12586.0	0.615457	0.307729	+	0.951474i	$0.400431\pi$
$54$	0	0
$55$	0	0
$56$	1260.00	0.0536909
$57$	0	0
$58$	29050.0	1.13390
$59$	25520.0	0.954444	0.477222	−	0.878783i	$-0.341644\pi$
$59$	25520.0	0.954444	0.477222	+	0.878783i	$0.341644\pi$
$60$	0	0
$61$	11782.0	0.405410	0.202705	−	0.979240i	$-0.435027\pi$
$61$	11782.0	0.405410	0.202705	+	0.979240i	$0.435027\pi$
$62$	−39816.0	−1.31546
$63$	0	0
$64$	1777.00	0.0542297
$65$	0	0
$66$	0	0
$67$	13188.0	0.358915	0.179458	−	0.983766i	$-0.442566\pi$
$67$	13188.0	0.358915	0.179458	+	0.983766i	$0.442566\pi$
$68$	37094.0	0.972818
$69$	0	0
$70$	0	0
$71$	35968.0	0.846780	0.423390	−	0.905948i	$-0.360840\pi$
$71$	35968.0	0.846780	0.423390	+	0.905948i	$0.360840\pi$
$72$	0	0
$73$	−73186.0	−1.60739	−0.803694	−	0.595042i	$-0.797135\pi$
$73$	−73186.0	−1.60739	−0.803694	+	0.595042i	$0.797135\pi$
$74$	−45374.0	−0.963225
$75$	0	0
$76$	24140.0	0.479406
$77$	1344.00	0.0258329
$78$	0	0
$79$	−52440.0	−0.945355	−0.472678	−	0.881235i	$-0.656712\pi$
$79$	−52440.0	−0.945355	−0.472678	+	0.881235i	$0.656712\pi$
$80$	0	0
$81$	0	0
$82$	−37814.0	−0.621038
$83$	69036.0	1.09997	0.549984	−	0.835175i	$-0.314634\pi$
$83$	69036.0	1.09997	0.549984	+	0.835175i	$0.314634\pi$
$84$	0	0
$85$	0	0
$86$	152348.	2.22122
$87$	0	0
$88$	11760.0	0.161883
$89$	33870.0	0.453252	0.226626	−	0.973982i	$-0.427230\pi$
$89$	33870.0	0.453252	0.226626	+	0.973982i	$0.427230\pi$
$90$	0	0
$91$	−11688.0	−0.147957
$92$	54672.0	0.673435
$93$	0	0
$94$	−2576.00	−0.0300695
$95$	0	0
$96$	0	0
$97$	−143042.	−1.54360	−0.771799	−	0.635867i	$-0.780643\pi$
$97$	−143042.	−1.54360	−0.771799	+	0.635867i	$0.780643\pi$
$98$	−116641.	−1.22683
$99$	0	0
$100$	0	0
$101$	−63042.0	−0.614931	−0.307466	−	0.951559i	$-0.599481\pi$
$101$	−63042.0	−0.614931	−0.307466	+	0.951559i	$0.599481\pi$
$102$	0	0
$103$	−38636.0	−0.358839	−0.179419	−	0.983773i	$-0.557422\pi$
$103$	−38636.0	−0.358839	−0.179419	+	0.983773i	$0.557422\pi$
$104$	−102270.	−0.927182
$105$	0	0
$106$	88102.0	0.761590
$107$	−69228.0	−0.584551	−0.292275	−	0.956334i	$-0.594412\pi$
$107$	−69228.0	−0.584551	−0.292275	+	0.956334i	$0.594412\pi$
$108$	0	0
$109$	51590.0	0.415910	0.207955	−	0.978138i	$-0.433319\pi$
$109$	51590.0	0.415910	0.207955	+	0.978138i	$0.433319\pi$
$110$	0	0
$111$	0	0
$112$	15348.0	0.115613
$113$	20206.0	0.148862	0.0744311	−	0.997226i	$-0.476286\pi$
$113$	20206.0	0.148862	0.0744311	+	0.997226i	$0.476286\pi$
$114$	0	0
$115$	0	0
$116$	70550.0	0.486802
$117$	0	0
$118$	178640.	1.18106
$119$	−26184.0	−0.169500
$120$	0	0
$121$	−148507.	−0.922112
$122$	82474.0	0.501669
$123$	0	0
$124$	−96696.0	−0.564747
$125$	0	0
$126$	0	0
$127$	198908.	1.09432	0.547158	−	0.837029i	$-0.315710\pi$
$127$	198908.	1.09432	0.547158	+	0.837029i	$0.315710\pi$
$128$	191415.	1.03264
$129$	0	0
$130$	0	0
$131$	−150672.	−0.767104	−0.383552	−	0.923519i	$-0.625299\pi$
$131$	−150672.	−0.767104	−0.383552	+	0.923519i	$0.625299\pi$
$132$	0	0
$133$	−17040.0	−0.0835296
$134$	92316.0	0.444135
$135$	0	0
$136$	−229110.	−1.06218
$137$	−19098.0	−0.0869334	−0.0434667	−	0.999055i	$-0.513840\pi$
$137$	−19098.0	−0.0869334	−0.0434667	+	0.999055i	$0.513840\pi$
$138$	0	0
$139$	196460.	0.862456	0.431228	−	0.902243i	$-0.358080\pi$
$139$	196460.	0.862456	0.431228	+	0.902243i	$0.358080\pi$
$140$	0	0
$141$	0	0
$142$	251776.	1.04784
$143$	−109088.	−0.446105
$144$	0	0
$145$	0	0
$146$	−512302.	−1.98904
$147$	0	0
$148$	−110194.	−0.413527
$149$	362710.	1.33842	0.669212	−	0.743071i	$-0.266632\pi$
$149$	362710.	1.33842	0.669212	+	0.743071i	$0.266632\pi$
$150$	0	0
$151$	76072.0	0.271508	0.135754	−	0.990743i	$-0.456654\pi$
$151$	76072.0	0.271508	0.135754	+	0.990743i	$0.456654\pi$
$152$	−149100.	−0.523442
$153$	0	0
$154$	9408.00	0.0319665
$155$	0	0
$156$	0	0
$157$	−252722.	−0.818265	−0.409132	−	0.912475i	$-0.634169\pi$
$157$	−252722.	−0.818265	−0.409132	+	0.912475i	$0.634169\pi$
$158$	−367080.	−1.16982
$159$	0	0
$160$	0	0
$161$	−38592.0	−0.117336
$162$	0	0
$163$	−85916.0	−0.253282	−0.126641	−	0.991949i	$-0.540420\pi$
$163$	−85916.0	−0.253282	−0.126641	+	0.991949i	$0.540420\pi$
$164$	−91834.0	−0.266621
$165$	0	0
$166$	483252.	1.36114
$167$	316272.	0.877545	0.438773	−	0.898598i	$-0.355413\pi$
$167$	316272.	0.877545	0.438773	+	0.898598i	$0.355413\pi$
$168$	0	0
$169$	577383.	1.55506
$170$	0	0
$171$	0	0
$172$	369988.	0.953601
$173$	−597534.	−1.51791	−0.758957	−	0.651140i	$-0.774291\pi$
$173$	−597534.	−1.51791	−0.758957	+	0.651140i	$0.774291\pi$
$174$	0	0
$175$	0	0
$176$	143248.	0.348584
$177$	0	0
$178$	237090.	0.560871
$179$	282680.	0.659421	0.329710	−	0.944082i	$-0.393049\pi$
$179$	282680.	0.659421	0.329710	+	0.944082i	$0.393049\pi$
$180$	0	0
$181$	−294658.	−0.668531	−0.334266	−	0.942479i	$-0.608488\pi$
$181$	−294658.	−0.668531	−0.334266	+	0.942479i	$0.608488\pi$
$182$	−81816.0	−0.183088
$183$	0	0
$184$	−337680.	−0.735294
$185$	0	0
$186$	0	0
$187$	−244384.	−0.511056
$188$	−6256.00	−0.0129093
$189$	0	0
$190$	0	0
$191$	−723272.	−1.43456	−0.717279	−	0.696786i	$-0.754613\pi$
$191$	−723272.	−1.43456	−0.717279	+	0.696786i	$0.754613\pi$
$192$	0	0
$193$	−80426.0	−0.155419	−0.0777093	−	0.996976i	$-0.524761\pi$
$193$	−80426.0	−0.155419	−0.0777093	+	0.996976i	$0.524761\pi$
$194$	−1.00129e6	−1.91011
$195$	0	0
$196$	−283271.	−0.526698
$197$	509802.	0.935914	0.467957	−	0.883751i	$-0.344990\pi$
$197$	509802.	0.935914	0.467957	+	0.883751i	$0.344990\pi$
$198$	0	0
$199$	−435320.	−0.779248	−0.389624	−	0.920974i	$-0.627395\pi$
$199$	−435320.	−0.779248	−0.389624	+	0.920974i	$0.627395\pi$
$200$	0	0
$201$	0	0
$202$	−441294.	−0.760939
$203$	−49800.0	−0.0848182
$204$	0	0
$205$	0	0
$206$	−270452.	−0.444040
$207$	0	0
$208$	−1.24575e6	−1.99651
$209$	−159040.	−0.251849
$210$	0	0
$211$	−1.12275e6	−1.73611	−0.868053	−	0.496472i	$-0.834629\pi$
$211$	−1.12275e6	−1.73611	−0.868053	+	0.496472i	$0.834629\pi$
$212$	213962.	0.326962
$213$	0	0
$214$	−484596.	−0.723345
$215$	0	0
$216$	0	0
$217$	68256.0	0.0983992
$218$	361130.	0.514662
$219$	0	0
$220$	0	0
$221$	2.12527e6	2.92707
$222$	0	0
$223$	325084.	0.437757	0.218879	−	0.975752i	$-0.429760\pi$
$223$	325084.	0.437757	0.218879	+	0.975752i	$0.429760\pi$
$224$	67116.0	0.0893729
$225$	0	0
$226$	141442.	0.184207
$227$	−311308.	−0.400983	−0.200491	−	0.979695i	$-0.564254\pi$
$227$	−311308.	−0.400983	−0.200491	+	0.979695i	$0.564254\pi$
$228$	0	0
$229$	−615450.	−0.775540	−0.387770	−	0.921756i	$-0.626754\pi$
$229$	−615450.	−0.775540	−0.387770	+	0.921756i	$0.626754\pi$
$230$	0	0
$231$	0	0
$232$	−435750.	−0.531517
$233$	−304434.	−0.367370	−0.183685	−	0.982985i	$-0.558803\pi$
$233$	−304434.	−0.367370	−0.183685	+	0.982985i	$0.558803\pi$
$234$	0	0
$235$	0	0
$236$	433840.	0.507049
$237$	0	0
$238$	−183288.	−0.209745
$239$	−780760.	−0.884144	−0.442072	−	0.896980i	$-0.645756\pi$
$239$	−780760.	−0.884144	−0.442072	+	0.896980i	$0.645756\pi$
$240$	0	0
$241$	635842.	0.705191	0.352595	−	0.935776i	$-0.385299\pi$
$241$	635842.	0.705191	0.352595	+	0.935776i	$0.385299\pi$
$242$	−1.03955e6	−1.14105
$243$	0	0
$244$	200294.	0.215374
$245$	0	0
$246$	0	0
$247$	1.38308e6	1.44246
$248$	597240.	0.616623
$249$	0	0
$250$	0	0
$251$	−1.20559e6	−1.20786	−0.603929	−	0.797038i	$-0.706399\pi$
$251$	−1.20559e6	−1.20786	−0.603929	+	0.797038i	$0.706399\pi$
$252$	0	0
$253$	−360192.	−0.353780
$254$	1.39236e6	1.35415
$255$	0	0
$256$	1.28304e6	1.22360
$257$	1.96702e6	1.85770	0.928852	−	0.370452i	$-0.120797\pi$
$257$	1.96702e6	1.85770	0.928852	+	0.370452i	$0.120797\pi$
$258$	0	0
$259$	77784.0	0.0720511
$260$	0	0
$261$	0	0
$262$	−1.05470e6	−0.949243
$263$	−625264.	−0.557409	−0.278705	−	0.960377i	$-0.589905\pi$
$263$	−625264.	−0.557409	−0.278705	+	0.960377i	$0.589905\pi$
$264$	0	0
$265$	0	0
$266$	−119280.	−0.103363
$267$	0	0
$268$	224196.	0.190674
$269$	−527050.	−0.444090	−0.222045	−	0.975036i	$-0.571273\pi$
$269$	−527050.	−0.444090	−0.222045	+	0.975036i	$0.571273\pi$
$270$	0	0
$271$	2.10923e6	1.74462	0.872311	−	0.488952i	$-0.162621\pi$
$271$	2.10923e6	1.74462	0.872311	+	0.488952i	$0.162621\pi$
$272$	−2.79078e6	−2.28719
$273$	0	0
$274$	−133686.	−0.107575
$275$	0	0
$276$	0	0
$277$	267438.	0.209423	0.104711	−	0.994503i	$-0.466608\pi$
$277$	267438.	0.209423	0.104711	+	0.994503i	$0.466608\pi$
$278$	1.37522e6	1.06724
$279$	0	0
$280$	0	0
$281$	−838002.	−0.633110	−0.316555	−	0.948574i	$-0.602526\pi$
$281$	−838002.	−0.633110	−0.316555	+	0.948574i	$0.602526\pi$
$282$	0	0
$283$	2.23772e6	1.66089	0.830444	−	0.557102i	$-0.188087\pi$
$283$	2.23772e6	1.66089	0.830444	+	0.557102i	$0.188087\pi$
$284$	611456.	0.449852
$285$	0	0
$286$	−763616.	−0.552027
$287$	64824.0	0.0464549
$288$	0	0
$289$	3.34127e6	2.35324
$290$	0	0
$291$	0	0
$292$	−1.24416e6	−0.853925
$293$	785706.	0.534676	0.267338	−	0.963603i	$-0.413856\pi$
$293$	785706.	0.534676	0.267338	+	0.963603i	$0.413856\pi$
$294$	0	0
$295$	0	0
$296$	680610.	0.451512
$297$	0	0
$298$	2.53897e6	1.65622
$299$	3.13238e6	2.02627
$300$	0	0
$301$	−261168.	−0.166151
$302$	532504.	0.335974
$303$	0	0
$304$	−1.81618e6	−1.12713
$305$	0	0
$306$	0	0
$307$	2.94955e6	1.78612	0.893058	−	0.449942i	$-0.148555\pi$
$307$	2.94955e6	1.78612	0.893058	+	0.449942i	$0.148555\pi$
$308$	22848.0	0.0137237
$309$	0	0
$310$	0	0
$311$	3.07757e6	1.80429	0.902146	−	0.431431i	$-0.141991\pi$
$311$	3.07757e6	1.80429	0.902146	+	0.431431i	$0.141991\pi$
$312$	0	0
$313$	−1.61367e6	−0.931007	−0.465503	−	0.885046i	$-0.654127\pi$
$313$	−1.61367e6	−0.931007	−0.465503	+	0.885046i	$0.654127\pi$
$314$	−1.76905e6	−1.01255
$315$	0	0
$316$	−891480.	−0.502220
$317$	−2.00496e6	−1.12062	−0.560308	−	0.828284i	$-0.689317\pi$
$317$	−2.00496e6	−1.12062	−0.560308	+	0.828284i	$0.689317\pi$
$318$	0	0
$319$	−464800.	−0.255735
$320$	0	0
$321$	0	0
$322$	−270144.	−0.145196
$323$	3.09844e6	1.65248
$324$	0	0
$325$	0	0
$326$	−601412.	−0.313421
$327$	0	0
$328$	567210.	0.291111
$329$	4416.00	0.00224926
$330$	0	0
$331$	470772.	0.236179	0.118089	−	0.993003i	$-0.462323\pi$
$331$	470772.	0.236179	0.118089	+	0.993003i	$0.462323\pi$
$332$	1.17361e6	0.584358
$333$	0	0
$334$	2.21390e6	1.08591
$335$	0	0
$336$	0	0
$337$	−2.31548e6	−1.11062	−0.555311	−	0.831642i	$-0.687401\pi$
$337$	−2.31548e6	−1.11062	−0.555311	+	0.831642i	$0.687401\pi$
$338$	4.04168e6	1.92429
$339$	0	0
$340$	0	0
$341$	637056.	0.296682
$342$	0	0
$343$	401640.	0.184332
$344$	−2.28522e6	−1.04119
$345$	0	0
$346$	−4.18274e6	−1.87832
$347$	1.25393e6	0.559050	0.279525	−	0.960138i	$-0.409823\pi$
$347$	1.25393e6	0.559050	0.279525	+	0.960138i	$0.409823\pi$
$348$	0	0
$349$	616390.	0.270889	0.135445	−	0.990785i	$-0.456754\pi$
$349$	616390.	0.270889	0.135445	+	0.990785i	$0.456754\pi$
$350$	0	0
$351$	0	0
$352$	626416.	0.269467
$353$	−281274.	−0.120141	−0.0600707	−	0.998194i	$-0.519133\pi$
$353$	−281274.	−0.120141	−0.0600707	+	0.998194i	$0.519133\pi$
$354$	0	0
$355$	0	0
$356$	575790.	0.240790
$357$	0	0
$358$	1.97876e6	0.815991
$359$	1.19148e6	0.487922	0.243961	−	0.969785i	$-0.421553\pi$
$359$	1.19148e6	0.487922	0.243961	+	0.969785i	$0.421553\pi$
$360$	0	0
$361$	−459699.	−0.185655
$362$	−2.06261e6	−0.827265
$363$	0	0
$364$	−198696.	−0.0786024
$365$	0	0
$366$	0	0
$367$	793068.	0.307359	0.153679	−	0.988121i	$-0.450888\pi$
$367$	793068.	0.307359	0.153679	+	0.988121i	$0.450888\pi$
$368$	−4.11326e6	−1.58331
$369$	0	0
$370$	0	0
$371$	−151032.	−0.0569684
$372$	0	0
$373$	−635626.	−0.236554	−0.118277	−	0.992981i	$-0.537737\pi$
$373$	−635626.	−0.236554	−0.118277	+	0.992981i	$0.537737\pi$
$374$	−1.71069e6	−0.632400
$375$	0	0
$376$	38640.0	0.0140951
$377$	4.04210e6	1.46472
$378$	0	0
$379$	−2.12834e6	−0.761102	−0.380551	−	0.924760i	$-0.624266\pi$
$379$	−2.12834e6	−0.761102	−0.380551	+	0.924760i	$0.624266\pi$
$380$	0	0
$381$	0	0
$382$	−5.06290e6	−1.77518
$383$	−4.88174e6	−1.70051	−0.850253	−	0.526375i	$-0.823551\pi$
$383$	−4.88174e6	−1.70051	−0.850253	+	0.526375i	$0.823551\pi$
$384$	0	0
$385$	0	0
$386$	−562982.	−0.192321
$387$	0	0
$388$	−2.43171e6	−0.820036
$389$	2.30607e6	0.772678	0.386339	−	0.922357i	$-0.373740\pi$
$389$	2.30607e6	0.772678	0.386339	+	0.922357i	$0.373740\pi$
$390$	0	0
$391$	7.01731e6	2.32129
$392$	1.74962e6	0.575079
$393$	0	0
$394$	3.56861e6	1.15813
$395$	0	0
$396$	0	0
$397$	423398.	0.134826	0.0674128	−	0.997725i	$-0.478526\pi$
$397$	423398.	0.134826	0.0674128	+	0.997725i	$0.478526\pi$
$398$	−3.04724e6	−0.964271
$399$	0	0
$400$	0	0
$401$	2.60756e6	0.809791	0.404896	−	0.914363i	$-0.367308\pi$
$401$	2.60756e6	0.809791	0.404896	+	0.914363i	$0.367308\pi$
$402$	0	0
$403$	−5.54011e6	−1.69924
$404$	−1.07171e6	−0.326682
$405$	0	0
$406$	−348600.	−0.104957
$407$	725984.	0.217241
$408$	0	0
$409$	−4.80871e6	−1.42141	−0.710707	−	0.703489i	$-0.751625\pi$
$409$	−4.80871e6	−1.42141	−0.710707	+	0.703489i	$0.751625\pi$
$410$	0	0
$411$	0	0
$412$	−656812.	−0.190633
$413$	−306240.	−0.0883460
$414$	0	0
$415$	0	0
$416$	−5.44758e6	−1.54337
$417$	0	0
$418$	−1.11328e6	−0.311648
$419$	−139760.	−0.0388909	−0.0194454	−	0.999811i	$-0.506190\pi$
$419$	−139760.	−0.0388909	−0.0194454	+	0.999811i	$0.506190\pi$
$420$	0	0
$421$	−3.00310e6	−0.825780	−0.412890	−	0.910781i	$-0.635481\pi$
$421$	−3.00310e6	−0.825780	−0.412890	+	0.910781i	$0.635481\pi$
$422$	−7.85924e6	−2.14832
$423$	0	0
$424$	−1.32153e6	−0.356995
$425$	0	0
$426$	0	0
$427$	−141384.	−0.0375259
$428$	−1.17688e6	−0.310543
$429$	0	0
$430$	0	0
$431$	−5.97955e6	−1.55051	−0.775257	−	0.631646i	$-0.782379\pi$
$431$	−5.97955e6	−1.55051	−0.775257	+	0.631646i	$0.782379\pi$
$432$	0	0
$433$	−1.75235e6	−0.449159	−0.224580	−	0.974456i	$-0.572101\pi$
$433$	−1.75235e6	−0.449159	−0.224580	+	0.974456i	$0.572101\pi$
$434$	477792.	0.121763
$435$	0	0
$436$	877030.	0.220952
$437$	4.56672e6	1.14393
$438$	0	0
$439$	−4.72556e6	−1.17029	−0.585143	−	0.810930i	$-0.698962\pi$
$439$	−4.72556e6	−1.17029	−0.585143	+	0.810930i	$0.698962\pi$
$440$	0	0
$441$	0	0
$442$	1.48769e7	3.62206
$443$	2.48584e6	0.601815	0.300908	−	0.953653i	$-0.402710\pi$
$443$	2.48584e6	0.601815	0.300908	+	0.953653i	$0.402710\pi$
$444$	0	0
$445$	0	0
$446$	2.27559e6	0.541697
$447$	0	0
$448$	−21324.0	−0.00501965
$449$	−1.46233e6	−0.342318	−0.171159	−	0.985243i	$-0.554751\pi$
$449$	−1.46233e6	−0.342318	−0.171159	+	0.985243i	$0.554751\pi$
$450$	0	0
$451$	605024.	0.140066
$452$	343502.	0.0790830
$453$	0	0
$454$	−2.17916e6	−0.496191
$455$	0	0
$456$	0	0
$457$	1.45684e6	0.326303	0.163151	−	0.986601i	$-0.447834\pi$
$457$	1.45684e6	0.326303	0.163151	+	0.986601i	$0.447834\pi$
$458$	−4.30815e6	−0.959681
$459$	0	0
$460$	0	0
$461$	−4.32280e6	−0.947356	−0.473678	−	0.880698i	$-0.657074\pi$
$461$	−4.32280e6	−0.947356	−0.473678	+	0.880698i	$0.657074\pi$
$462$	0	0
$463$	−1.07848e6	−0.233807	−0.116904	−	0.993143i	$-0.537297\pi$
$463$	−1.07848e6	−0.233807	−0.116904	+	0.993143i	$0.537297\pi$
$464$	−5.30785e6	−1.14452
$465$	0	0
$466$	−2.13104e6	−0.454597
$467$	−3.40023e6	−0.721466	−0.360733	−	0.932669i	$-0.617473\pi$
$467$	−3.40023e6	−0.721466	−0.360733	+	0.932669i	$0.617473\pi$
$468$	0	0
$469$	−158256.	−0.0332222
$470$	0	0
$471$	0	0
$472$	−2.67960e6	−0.553624
$473$	−2.43757e6	−0.500961
$474$	0	0
$475$	0	0
$476$	−445128.	−0.0900466
$477$	0	0
$478$	−5.46532e6	−1.09407
$479$	−2.75268e6	−0.548172	−0.274086	−	0.961705i	$-0.588375\pi$
$479$	−2.75268e6	−0.548172	−0.274086	+	0.961705i	$0.588375\pi$
$480$	0	0
$481$	−6.31347e6	−1.24424
$482$	4.45089e6	0.872629
$483$	0	0
$484$	−2.52462e6	−0.489872
$485$	0	0
$486$	0	0
$487$	−3.56997e6	−0.682091	−0.341046	−	0.940047i	$-0.610781\pi$
$487$	−3.56997e6	−0.682091	−0.341046	+	0.940047i	$0.610781\pi$
$488$	−1.23711e6	−0.235157
$489$	0	0
$490$	0	0
$491$	−1.96455e6	−0.367756	−0.183878	−	0.982949i	$-0.558865\pi$
$491$	−1.96455e6	−0.367756	−0.183878	+	0.982949i	$0.558865\pi$
$492$	0	0
$493$	9.05530e6	1.67798
$494$	9.68156e6	1.78496
$495$	0	0
$496$	7.27495e6	1.32778
$497$	−431616.	−0.0783802
$498$	0	0
$499$	5.14798e6	0.925519	0.462760	−	0.886484i	$-0.346859\pi$
$499$	5.14798e6	0.925519	0.462760	+	0.886484i	$0.346859\pi$
$500$	0	0
$501$	0	0
$502$	−8.43914e6	−1.49465
$503$	1.97502e6	0.348057	0.174029	−	0.984741i	$-0.444321\pi$
$503$	1.97502e6	0.348057	0.174029	+	0.984741i	$0.444321\pi$
$504$	0	0
$505$	0	0
$506$	−2.52134e6	−0.437780
$507$	0	0
$508$	3.38144e6	0.581356
$509$	3.32447e6	0.568759	0.284379	−	0.958712i	$-0.408213\pi$
$509$	3.32447e6	0.568759	0.284379	+	0.958712i	$0.408213\pi$
$510$	0	0
$511$	878232.	0.148784
$512$	2.85601e6	0.481487
$513$	0	0
$514$	1.37692e7	2.29879
$515$	0	0
$516$	0	0
$517$	41216.0	0.00678171
$518$	544488.	0.0891587
$519$	0	0
$520$	0	0
$521$	2.97960e6	0.480910	0.240455	−	0.970660i	$-0.422703\pi$
$521$	2.97960e6	0.480910	0.240455	+	0.970660i	$0.422703\pi$
$522$	0	0
$523$	6.19108e6	0.989720	0.494860	−	0.868973i	$-0.335219\pi$
$523$	6.19108e6	0.989720	0.494860	+	0.868973i	$0.335219\pi$
$524$	−2.56142e6	−0.407524
$525$	0	0
$526$	−4.37685e6	−0.689759
$527$	−1.24112e7	−1.94665
$528$	0	0
$529$	3.90631e6	0.606915
$530$	0	0
$531$	0	0
$532$	−289680.	−0.0443751
$533$	−5.26155e6	−0.802224
$534$	0	0
$535$	0	0
$536$	−1.38474e6	−0.208188
$537$	0	0
$538$	−3.68935e6	−0.549533
$539$	1.86626e6	0.276694
$540$	0	0
$541$	−8.55398e6	−1.25654	−0.628268	−	0.777997i	$-0.716236\pi$
$541$	−8.55398e6	−1.25654	−0.628268	+	0.777997i	$0.716236\pi$
$542$	1.47646e7	2.15886
$543$	0	0
$544$	−1.22039e7	−1.76808
$545$	0	0
$546$	0	0
$547$	2.54371e6	0.363495	0.181748	−	0.983345i	$-0.441825\pi$
$547$	2.54371e6	0.363495	0.181748	+	0.983345i	$0.441825\pi$
$548$	−324666.	−0.0461833
$549$	0	0
$550$	0	0
$551$	5.89300e6	0.826908
$552$	0	0
$553$	629280.	0.0875046
$554$	1.87207e6	0.259147
$555$	0	0
$556$	3.33982e6	0.458180
$557$	7.58704e6	1.03618	0.518089	−	0.855327i	$-0.326644\pi$
$557$	7.58704e6	1.03618	0.518089	+	0.855327i	$0.326644\pi$
$558$	0	0
$559$	2.11981e7	2.86925
$560$	0	0
$561$	0	0
$562$	−5.86601e6	−0.783434
$563$	7.50940e6	0.998468	0.499234	−	0.866467i	$-0.333615\pi$
$563$	7.50940e6	0.998468	0.499234	+	0.866467i	$0.333615\pi$
$564$	0	0
$565$	0	0
$566$	1.56641e7	2.05524
$567$	0	0
$568$	−3.77664e6	−0.491173
$569$	−1.38890e7	−1.79842	−0.899209	−	0.437519i	$-0.855857\pi$
$569$	−1.38890e7	−1.79842	−0.899209	+	0.437519i	$0.855857\pi$
$570$	0	0
$571$	−1.93539e6	−0.248415	−0.124207	−	0.992256i	$-0.539639\pi$
$571$	−1.93539e6	−0.248415	−0.124207	+	0.992256i	$0.539639\pi$
$572$	−1.85450e6	−0.236993
$573$	0	0
$574$	453768.	0.0574849
$575$	0	0
$576$	0	0
$577$	4.89408e6	0.611972	0.305986	−	0.952036i	$-0.401014\pi$
$577$	4.89408e6	0.611972	0.305986	+	0.952036i	$0.401014\pi$
$578$	2.33889e7	2.91199
$579$	0	0
$580$	0	0
$581$	−828432.	−0.101816
$582$	0	0
$583$	−1.40963e6	−0.171765
$584$	7.68453e6	0.932363
$585$	0	0
$586$	5.49994e6	0.661628
$587$	−6.43883e6	−0.771279	−0.385640	−	0.922650i	$-0.626019\pi$
$587$	−6.43883e6	−0.771279	−0.385640	+	0.922650i	$0.626019\pi$
$588$	0	0
$589$	−8.07696e6	−0.959312
$590$	0	0
$591$	0	0
$592$	8.29048e6	0.972244
$593$	4.30365e6	0.502574	0.251287	−	0.967913i	$-0.419146\pi$
$593$	4.30365e6	0.502574	0.251287	+	0.967913i	$0.419146\pi$
$594$	0	0
$595$	0	0
$596$	6.16607e6	0.711038
$597$	0	0
$598$	2.19267e7	2.50738
$599$	4.50988e6	0.513568	0.256784	−	0.966469i	$-0.417337\pi$
$599$	4.50988e6	0.513568	0.256784	+	0.966469i	$0.417337\pi$
$600$	0	0
$601$	−5.11596e6	−0.577751	−0.288876	−	0.957367i	$-0.593281\pi$
$601$	−5.11596e6	−0.577751	−0.288876	+	0.957367i	$0.593281\pi$
$602$	−1.82818e6	−0.205602
$603$	0	0
$604$	1.29322e6	0.144239
$605$	0	0
$606$	0	0
$607$	−1.61925e7	−1.78378	−0.891891	−	0.452250i	$-0.850622\pi$
$607$	−1.61925e7	−1.78378	−0.891891	+	0.452250i	$0.850622\pi$
$608$	−7.94206e6	−0.871313
$609$	0	0
$610$	0	0
$611$	−358432.	−0.0388422
$612$	0	0
$613$	−1.55525e7	−1.67166	−0.835830	−	0.548988i	$-0.815013\pi$
$613$	−1.55525e7	−1.67166	−0.835830	+	0.548988i	$0.815013\pi$
$614$	2.06468e7	2.21021
$615$	0	0
$616$	−141120.	−0.0149843
$617$	−1.46710e7	−1.55148	−0.775740	−	0.631053i	$-0.782623\pi$
$617$	−1.46710e7	−1.55148	−0.775740	+	0.631053i	$0.782623\pi$
$618$	0	0
$619$	−9.66826e6	−1.01420	−0.507098	−	0.861889i	$-0.669282\pi$
$619$	−9.66826e6	−1.01420	−0.507098	+	0.861889i	$0.669282\pi$
$620$	0	0
$621$	0	0
$622$	2.15430e7	2.23270
$623$	−406440.	−0.0419543
$624$	0	0
$625$	0	0
$626$	−1.12957e7	−1.15206
$627$	0	0
$628$	−4.29627e6	−0.434703
$629$	−1.41437e7	−1.42540
$630$	0	0
$631$	−1.16557e6	−0.116537	−0.0582686	−	0.998301i	$-0.518558\pi$
$631$	−1.16557e6	−0.116537	−0.0582686	+	0.998301i	$0.518558\pi$
$632$	5.50620e6	0.548352
$633$	0	0
$634$	−1.40347e7	−1.38669
$635$	0	0
$636$	0	0
$637$	−1.62298e7	−1.58476
$638$	−3.25360e6	−0.316455
$639$	0	0
$640$	0	0
$641$	−1.95088e6	−0.187537	−0.0937683	−	0.995594i	$-0.529891\pi$
$641$	−1.95088e6	−0.187537	−0.0937683	+	0.995594i	$0.529891\pi$
$642$	0	0
$643$	1.17387e7	1.11968	0.559839	−	0.828601i	$-0.310863\pi$
$643$	1.17387e7	1.11968	0.559839	+	0.828601i	$0.310863\pi$
$644$	−656064.	−0.0623349
$645$	0	0
$646$	2.16891e7	2.04484
$647$	−1.01369e7	−0.952015	−0.476008	−	0.879441i	$-0.657917\pi$
$647$	−1.01369e7	−0.952015	−0.476008	+	0.879441i	$0.657917\pi$
$648$	0	0
$649$	−2.85824e6	−0.266371
$650$	0	0
$651$	0	0
$652$	−1.46057e6	−0.134556
$653$	2.47095e6	0.226767	0.113384	−	0.993551i	$-0.463831\pi$
$653$	2.47095e6	0.226767	0.113384	+	0.993551i	$0.463831\pi$
$654$	0	0
$655$	0	0
$656$	6.90916e6	0.626853
$657$	0	0
$658$	30912.0	0.00278332
$659$	−1.62242e7	−1.45529	−0.727644	−	0.685955i	$-0.759384\pi$
$659$	−1.62242e7	−1.45529	−0.727644	+	0.685955i	$0.759384\pi$
$660$	0	0
$661$	1.54679e7	1.37698	0.688490	−	0.725246i	$-0.258274\pi$
$661$	1.54679e7	1.37698	0.688490	+	0.725246i	$0.258274\pi$
$662$	3.29540e6	0.292256
$663$	0	0
$664$	−7.24878e6	−0.638035
$665$	0	0
$666$	0	0
$667$	1.33464e7	1.16158
$668$	5.37662e6	0.466196
$669$	0	0
$670$	0	0
$671$	−1.31958e6	−0.113144
$672$	0	0
$673$	1.94441e7	1.65482	0.827410	−	0.561598i	$-0.189813\pi$
$673$	1.94441e7	1.65482	0.827410	+	0.561598i	$0.189813\pi$
$674$	−1.62084e7	−1.37433
$675$	0	0
$676$	9.81551e6	0.826126
$677$	643242.	0.0539390	0.0269695	−	0.999636i	$-0.491414\pi$
$677$	643242.	0.0539390	0.0269695	+	0.999636i	$0.491414\pi$
$678$	0	0
$679$	1.71650e6	0.142880
$680$	0	0
$681$	0	0
$682$	4.45939e6	0.367126
$683$	1.14412e6	0.0938465	0.0469233	−	0.998898i	$-0.485058\pi$
$683$	1.14412e6	0.0938465	0.0469233	+	0.998898i	$0.485058\pi$
$684$	0	0
$685$	0	0
$686$	2.81148e6	0.228100
$687$	0	0
$688$	−2.78362e7	−2.24201
$689$	1.22588e7	0.983781
$690$	0	0
$691$	1.63625e6	0.130363	0.0651816	−	0.997873i	$-0.479237\pi$
$691$	1.63625e6	0.130363	0.0651816	+	0.997873i	$0.479237\pi$
$692$	−1.01581e7	−0.806392
$693$	0	0
$694$	8.77752e6	0.691789
$695$	0	0
$696$	0	0
$697$	−1.17872e7	−0.919025
$698$	4.31473e6	0.335209
$699$	0	0
$700$	0	0
$701$	−1.58303e7	−1.21673	−0.608364	−	0.793658i	$-0.708174\pi$
$701$	−1.58303e7	−1.21673	−0.608364	+	0.793658i	$0.708174\pi$
$702$	0	0
$703$	−9.20444e6	−0.702440
$704$	−199024.	−0.0151347
$705$	0	0
$706$	−1.96892e6	−0.148667
$707$	756504.	0.0569197
$708$	0	0
$709$	910870.	0.0680520	0.0340260	−	0.999421i	$-0.489167\pi$
$709$	910870.	0.0680520	0.0340260	+	0.999421i	$0.489167\pi$
$710$	0	0
$711$	0	0
$712$	−3.55635e6	−0.262908
$713$	−1.82926e7	−1.34757
$714$	0	0
$715$	0	0
$716$	4.80556e6	0.350317
$717$	0	0
$718$	8.34036e6	0.603773
$719$	1.52246e7	1.09831	0.549155	−	0.835721i	$-0.314950\pi$
$719$	1.52246e7	1.09831	0.549155	+	0.835721i	$0.314950\pi$
$720$	0	0
$721$	463632.	0.0332151
$722$	−3.21789e6	−0.229736
$723$	0	0
$724$	−5.00919e6	−0.355157
$725$	0	0
$726$	0	0
$727$	−1.81793e7	−1.27567	−0.637837	−	0.770171i	$-0.720171\pi$
$727$	−1.81793e7	−1.27567	−0.637837	+	0.770171i	$0.720171\pi$
$728$	1.22724e6	0.0858225
$729$	0	0
$730$	0	0
$731$	4.74890e7	3.28700
$732$	0	0
$733$	−2.08512e7	−1.43341	−0.716707	−	0.697374i	$-0.754352\pi$
$733$	−2.08512e7	−1.43341	−0.716707	+	0.697374i	$0.754352\pi$
$734$	5.55148e6	0.380337
$735$	0	0
$736$	−1.79871e7	−1.22396
$737$	−1.47706e6	−0.100168
$738$	0	0
$739$	−2.12513e7	−1.43144	−0.715722	−	0.698385i	$-0.753902\pi$
$739$	−2.12513e7	−1.43144	−0.715722	+	0.698385i	$0.753902\pi$
$740$	0	0
$741$	0	0
$742$	−1.05722e6	−0.0704948
$743$	7.12262e6	0.473334	0.236667	−	0.971591i	$-0.423945\pi$
$743$	7.12262e6	0.473334	0.236667	+	0.971591i	$0.423945\pi$
$744$	0	0
$745$	0	0
$746$	−4.44938e6	−0.292720
$747$	0	0
$748$	−4.15453e6	−0.271499
$749$	830736.	0.0541076
$750$	0	0
$751$	−1.00277e7	−0.648785	−0.324393	−	0.945923i	$-0.605160\pi$
$751$	−1.00277e7	−0.648785	−0.324393	+	0.945923i	$0.605160\pi$
$752$	470672.	0.0303511
$753$	0	0
$754$	2.82947e7	1.81249
$755$	0	0
$756$	0	0
$757$	2.18303e7	1.38459	0.692294	−	0.721616i	$-0.256600\pi$
$757$	2.18303e7	1.38459	0.692294	+	0.721616i	$0.256600\pi$
$758$	−1.48984e7	−0.941816
$759$	0	0
$760$	0	0
$761$	−2.56780e7	−1.60731	−0.803655	−	0.595096i	$-0.797114\pi$
$761$	−2.56780e7	−1.60731	−0.803655	+	0.595096i	$0.797114\pi$
$762$	0	0
$763$	−619080.	−0.0384978
$764$	−1.22956e7	−0.762109
$765$	0	0
$766$	−3.41722e7	−2.10427
$767$	2.48565e7	1.52564
$768$	0	0
$769$	5.81453e6	0.354567	0.177284	−	0.984160i	$-0.443269\pi$
$769$	5.81453e6	0.354567	0.177284	+	0.984160i	$0.443269\pi$
$770$	0	0
$771$	0	0
$772$	−1.36724e6	−0.0825662
$773$	1.55507e7	0.936057	0.468029	−	0.883713i	$-0.344964\pi$
$773$	1.55507e7	0.936057	0.468029	+	0.883713i	$0.344964\pi$
$774$	0	0
$775$	0	0
$776$	1.50194e7	0.895362
$777$	0	0
$778$	1.61425e7	0.956140
$779$	−7.67084e6	−0.452897
$780$	0	0
$781$	−4.02842e6	−0.236323
$782$	4.91212e7	2.87245
$783$	0	0
$784$	2.13120e7	1.23832
$785$	0	0
$786$	0	0
$787$	−2.35987e7	−1.35816	−0.679079	−	0.734065i	$-0.737621\pi$
$787$	−2.35987e7	−1.35816	−0.679079	+	0.734065i	$0.737621\pi$
$788$	8.66663e6	0.497204
$789$	0	0
$790$	0	0
$791$	−242472.	−0.0137791
$792$	0	0
$793$	1.14757e7	0.648030
$794$	2.96379e6	0.166838
$795$	0	0
$796$	−7.40044e6	−0.413976
$797$	−2.01127e7	−1.12157	−0.560783	−	0.827963i	$-0.689500\pi$
$797$	−2.01127e7	−1.12157	−0.560783	+	0.827963i	$0.689500\pi$
$798$	0	0
$799$	−802976.	−0.0444975
$800$	0	0
$801$	0	0
$802$	1.82529e7	1.00207
$803$	8.19683e6	0.448598
$804$	0	0
$805$	0	0
$806$	−3.87808e7	−2.10271
$807$	0	0
$808$	6.61941e6	0.356690
$809$	4.24111e6	0.227829	0.113914	−	0.993491i	$-0.463661\pi$
$809$	4.24111e6	0.227829	0.113914	+	0.993491i	$0.463661\pi$
$810$	0	0
$811$	6.04321e6	0.322638	0.161319	−	0.986902i	$-0.448425\pi$
$811$	6.04321e6	0.322638	0.161319	+	0.986902i	$0.448425\pi$
$812$	−846600.	−0.0450597
$813$	0	0
$814$	5.08189e6	0.268822
$815$	0	0
$816$	0	0
$817$	3.09049e7	1.61984
$818$	−3.36610e7	−1.75891
$819$	0	0
$820$	0	0
$821$	−1.66230e7	−0.860702	−0.430351	−	0.902662i	$-0.641610\pi$
$821$	−1.66230e7	−0.860702	−0.430351	+	0.902662i	$0.641610\pi$
$822$	0	0
$823$	2.59172e7	1.33380	0.666898	−	0.745149i	$-0.267622\pi$
$823$	2.59172e7	1.33380	0.666898	+	0.745149i	$0.267622\pi$
$824$	4.05678e6	0.208144
$825$	0	0
$826$	−2.14368e6	−0.109323
$827$	1.67704e7	0.852668	0.426334	−	0.904566i	$-0.359805\pi$
$827$	1.67704e7	0.852668	0.426334	+	0.904566i	$0.359805\pi$
$828$	0	0
$829$	6.15999e6	0.311310	0.155655	−	0.987811i	$-0.450251\pi$
$829$	6.15999e6	0.311310	0.155655	+	0.987811i	$0.450251\pi$
$830$	0	0
$831$	0	0
$832$	1.73080e6	0.0866838
$833$	−3.63587e7	−1.81550
$834$	0	0
$835$	0	0
$836$	−2.70368e6	−0.133795
$837$	0	0
$838$	−978320.	−0.0481250
$839$	−2.80172e6	−0.137410	−0.0687052	−	0.997637i	$-0.521887\pi$
$839$	−2.80172e6	−0.137410	−0.0687052	+	0.997637i	$0.521887\pi$
$840$	0	0
$841$	−3.28865e6	−0.160335
$842$	−2.10217e7	−1.02185
$843$	0	0
$844$	−1.90867e7	−0.922306
$845$	0	0
$846$	0	0
$847$	1.78208e6	0.0853532
$848$	−1.60975e7	−0.768721
$849$	0	0
$850$	0	0
$851$	−2.08461e7	−0.986736
$852$	0	0
$853$	−2.54991e7	−1.19992	−0.599960	−	0.800030i	$-0.704817\pi$
$853$	−2.54991e7	−1.19992	−0.599960	+	0.800030i	$0.704817\pi$
$854$	−989688.	−0.0464359
$855$	0	0
$856$	7.26894e6	0.339068
$857$	−1.19499e7	−0.555794	−0.277897	−	0.960611i	$-0.589637\pi$
$857$	−1.19499e7	−0.555794	−0.277897	+	0.960611i	$0.589637\pi$
$858$	0	0
$859$	−1.01568e7	−0.469651	−0.234825	−	0.972038i	$-0.575452\pi$
$859$	−1.01568e7	−0.469651	−0.234825	+	0.972038i	$0.575452\pi$
$860$	0	0
$861$	0	0
$862$	−4.18569e7	−1.91866
$863$	3.66497e7	1.67511	0.837556	−	0.546351i	$-0.183984\pi$
$863$	3.66497e7	1.67511	0.837556	+	0.546351i	$0.183984\pi$
$864$	0	0
$865$	0	0
$866$	−1.22664e7	−0.555806
$867$	0	0
$868$	1.16035e6	0.0522746
$869$	5.87328e6	0.263834
$870$	0	0
$871$	1.28451e7	0.573710
$872$	−5.41695e6	−0.241248
$873$	0	0
$874$	3.19670e7	1.41555
$875$	0	0
$876$	0	0
$877$	8.08232e6	0.354844	0.177422	−	0.984135i	$-0.443224\pi$
$877$	8.08232e6	0.354844	0.177422	+	0.984135i	$0.443224\pi$
$878$	−3.30789e7	−1.44815
$879$	0	0
$880$	0	0
$881$	−288202.	−0.0125100	−0.00625500	−	0.999980i	$-0.501991\pi$
$881$	−288202.	−0.0125100	−0.00625500	+	0.999980i	$0.501991\pi$
$882$	0	0
$883$	−6.20688e6	−0.267899	−0.133950	−	0.990988i	$-0.542766\pi$
$883$	−6.20688e6	−0.267899	−0.133950	+	0.990988i	$0.542766\pi$
$884$	3.61296e7	1.55501
$885$	0	0
$886$	1.74009e7	0.744708
$887$	−1.49976e7	−0.640050	−0.320025	−	0.947409i	$-0.603691\pi$
$887$	−1.49976e7	−0.640050	−0.320025	+	0.947409i	$0.603691\pi$
$888$	0	0
$889$	−2.38690e6	−0.101293
$890$	0	0
$891$	0	0
$892$	5.52643e6	0.232559
$893$	−522560.	−0.0219284
$894$	0	0
$895$	0	0
$896$	−2.29698e6	−0.0955844
$897$	0	0
$898$	−1.02363e7	−0.423597
$899$	−2.36052e7	−0.974111
$900$	0	0
$901$	2.74627e7	1.12702
$902$	4.23517e6	0.173322
$903$	0	0
$904$	−2.12163e6	−0.0863473
$905$	0	0
$906$	0	0
$907$	3.92150e7	1.58283	0.791415	−	0.611279i	$-0.209345\pi$
$907$	3.92150e7	1.58283	0.791415	+	0.611279i	$0.209345\pi$
$908$	−5.29224e6	−0.213022
$909$	0	0
$910$	0	0
$911$	3.72997e7	1.48905	0.744526	−	0.667594i	$-0.232676\pi$
$911$	3.72997e7	1.48905	0.744526	+	0.667594i	$0.232676\pi$
$912$	0	0
$913$	−7.73203e6	−0.306985
$914$	1.01979e7	0.403779
$915$	0	0
$916$	−1.04626e7	−0.412005
$917$	1.80806e6	0.0710052
$918$	0	0
$919$	5.78776e6	0.226059	0.113029	−	0.993592i	$-0.463945\pi$
$919$	5.78776e6	0.226059	0.113029	+	0.993592i	$0.463945\pi$
$920$	0	0
$921$	0	0
$922$	−3.02596e7	−1.17229
$923$	3.50328e7	1.35354
$924$	0	0
$925$	0	0
$926$	−7.54933e6	−0.289322
$927$	0	0
$928$	−2.32110e7	−0.884755
$929$	−1.62700e7	−0.618513	−0.309256	−	0.950979i	$-0.600080\pi$
$929$	−1.62700e7	−0.618513	−0.309256	+	0.950979i	$0.600080\pi$
$930$	0	0
$931$	−2.36615e7	−0.894679
$932$	−5.17538e6	−0.195165
$933$	0	0
$934$	−2.38016e7	−0.892769
$935$	0	0
$936$	0	0
$937$	1.20396e7	0.447983	0.223992	−	0.974591i	$-0.428091\pi$
$937$	1.20396e7	0.447983	0.223992	+	0.974591i	$0.428091\pi$
$938$	−1.10779e6	−0.0411103
$939$	0	0
$940$	0	0
$941$	−3.10171e7	−1.14190	−0.570949	−	0.820985i	$-0.693425\pi$
$941$	−3.10171e7	−1.14190	−0.570949	+	0.820985i	$0.693425\pi$
$942$	0	0
$943$	−1.73728e7	−0.636197
$944$	−3.26401e7	−1.19212
$945$	0	0
$946$	−1.70630e7	−0.619908
$947$	3.27325e6	0.118605	0.0593027	−	0.998240i	$-0.481112\pi$
$947$	3.27325e6	0.118605	0.0593027	+	0.998240i	$0.481112\pi$
$948$	0	0
$949$	−7.12832e7	−2.56934
$950$	0	0
$951$	0	0
$952$	2.74932e6	0.0983180
$953$	−2.62021e7	−0.934552	−0.467276	−	0.884112i	$-0.654765\pi$
$953$	−2.62021e7	−0.934552	−0.467276	+	0.884112i	$0.654765\pi$
$954$	0	0
$955$	0	0
$956$	−1.32729e7	−0.469701
$957$	0	0
$958$	−1.92688e7	−0.678328
$959$	229176.	0.00804679
$960$	0	0
$961$	3.72419e6	0.130084
$962$	−4.41943e7	−1.53967
$963$	0	0
$964$	1.08093e7	0.374633
$965$	0	0
$966$	0	0
$967$	2.01481e7	0.692897	0.346449	−	0.938069i	$-0.387387\pi$
$967$	2.01481e7	0.692897	0.346449	+	0.938069i	$0.387387\pi$
$968$	1.55932e7	0.534869
$969$	0	0
$970$	0	0
$971$	1.57046e7	0.534537	0.267269	−	0.963622i	$-0.413879\pi$
$971$	1.57046e7	0.534537	0.267269	+	0.963622i	$0.413879\pi$
$972$	0	0
$973$	−2.35752e6	−0.0798313
$974$	−2.49898e7	−0.844045
$975$	0	0
$976$	−1.50692e7	−0.506367
$977$	2.84554e7	0.953736	0.476868	−	0.878975i	$-0.341772\pi$
$977$	2.84554e7	0.953736	0.476868	+	0.878975i	$0.341772\pi$
$978$	0	0
$979$	−3.79344e6	−0.126496
$980$	0	0
$981$	0	0
$982$	−1.37519e7	−0.455075
$983$	−7.50074e6	−0.247583	−0.123791	−	0.992308i	$-0.539505\pi$
$983$	−7.50074e6	−0.247583	−0.123791	+	0.992308i	$0.539505\pi$
$984$	0	0
$985$	0	0
$986$	6.33871e7	2.07639
$987$	0	0
$988$	2.35124e7	0.766309
$989$	6.99930e7	2.27543
$990$	0	0
$991$	3.22184e7	1.04212	0.521062	−	0.853519i	$-0.325536\pi$
$991$	3.22184e7	1.04212	0.521062	+	0.853519i	$0.325536\pi$
$992$	3.18130e7	1.02642
$993$	0	0
$994$	−3.02131e6	−0.0969906
$995$	0	0
$996$	0	0
$997$	−1.22112e7	−0.389065	−0.194532	−	0.980896i	$-0.562319\pi$
$997$	−1.22112e7	−0.389065	−0.194532	+	0.980896i	$0.562319\pi$
$998$	3.60359e7	1.14527
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.a.h.1.1		1
3.2	odd	2		75.6.a.a.1.1		1
5.2	odd	4		225.6.b.a.199.2		2
5.3	odd	4		225.6.b.a.199.1		2
5.4	even	2		45.6.a.a.1.1		1
15.2	even	4		75.6.b.a.49.1		2
15.8	even	4		75.6.b.a.49.2		2
15.14	odd	2		15.6.a.b.1.1	✓	1
20.19	odd	2		720.6.a.q.1.1		1
60.59	even	2		240.6.a.b.1.1		1
105.104	even	2		735.6.a.b.1.1		1
120.29	odd	2		960.6.a.k.1.1		1
120.59	even	2		960.6.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.6.a.b.1.1	✓	1	15.14	odd	2
45.6.a.a.1.1		1	5.4	even	2
75.6.a.a.1.1		1	3.2	odd	2
75.6.b.a.49.1		2	15.2	even	4
75.6.b.a.49.2		2	15.8	even	4
225.6.a.h.1.1		1	1.1	even	1	trivial
225.6.b.a.199.1		2	5.3	odd	4
225.6.b.a.199.2		2	5.2	odd	4
240.6.a.b.1.1		1	60.59	even	2
720.6.a.q.1.1		1	20.19	odd	2
735.6.a.b.1.1		1	105.104	even	2
960.6.a.k.1.1		1	120.29	odd	2
960.6.a.x.1.1		1	120.59	even	2

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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