LMFDB - Embedded newform 432.6.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,6,Mod(1,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("432.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	432.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$69.2858101592$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	432.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+58.7878 q^{5} -167.000 q^{7} +764.241 q^{11} -235.000 q^{13} +176.363 q^{17} -1361.00 q^{19} +2410.30 q^{23} +331.000 q^{25} +470.302 q^{29} -3500.00 q^{31} -9817.55 q^{35} +13115.0 q^{37} +9406.04 q^{41} -104.000 q^{43} +20516.9 q^{47} +11082.0 q^{49} -1058.18 q^{53} +44928.0 q^{55} -30746.0 q^{59} -7393.00 q^{61} -13815.1 q^{65} -38861.0 q^{67} +2469.09 q^{71} +5465.00 q^{73} -127628. q^{77} +82903.0 q^{79} +13286.0 q^{83} +10368.0 q^{85} +89768.9 q^{89} +39245.0 q^{91} -80010.1 q^{95} -49603.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 334 q^{7} - 470 q^{13} - 2722 q^{19} + 662 q^{25} - 7000 q^{31} + 26230 q^{37} - 208 q^{43} + 22164 q^{49} + 89856 q^{55} - 14786 q^{61} - 77722 q^{67} + 10930 q^{73} + 165806 q^{79} + 20736 q^{85}+ \cdots - 99206 q^{97}+O(q^{100}) $ 2 * q - 334 * q^7 - 470 * q^13 - 2722 * q^19 + 662 * q^25 - 7000 * q^31 + 26230 * q^37 - 208 * q^43 + 22164 * q^49 + 89856 * q^55 - 14786 * q^61 - 77722 * q^67 + 10930 * q^73 + 165806 * q^79 + 20736 * q^85 + 78490 * q^91 - 99206 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	58.7878	1.05163	0.525814	−	0.850600i	$-0.323761\pi$
$5$	58.7878	1.05163	0.525814	+	0.850600i	$0.323761\pi$
$6$	0	0
$7$	−167.000	−1.28816	−0.644082	−	0.764956i	$-0.722761\pi$
$7$	−167.000	−1.28816	−0.644082	+	0.764956i	$0.722761\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	764.241	1.90436	0.952179	−	0.305541i	$-0.0988374\pi$
$11$	764.241	1.90436	0.952179	+	0.305541i	$0.0988374\pi$
$12$	0	0
$13$	−235.000	−0.385664	−0.192832	−	0.981232i	$-0.561767\pi$
$13$	−235.000	−0.385664	−0.192832	+	0.981232i	$0.561767\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	176.363	0.148008	0.0740041	−	0.997258i	$-0.476422\pi$
$17$	176.363	0.148008	0.0740041	+	0.997258i	$0.476422\pi$
$18$	0	0
$19$	−1361.00	−0.864916	−0.432458	−	0.901654i	$-0.642354\pi$
$19$	−1361.00	−0.864916	−0.432458	+	0.901654i	$0.642354\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2410.30	0.950060	0.475030	−	0.879970i	$-0.342437\pi$
$23$	2410.30	0.950060	0.475030	+	0.879970i	$0.342437\pi$
$24$	0	0
$25$	331.000	0.105920
$26$	0	0
$27$	0	0
$28$	0	0
$29$	470.302	0.103844	0.0519221	−	0.998651i	$-0.483465\pi$
$29$	470.302	0.103844	0.0519221	+	0.998651i	$0.483465\pi$
$30$	0	0
$31$	−3500.00	−0.654130	−0.327065	−	0.945002i	$-0.606060\pi$
$31$	−3500.00	−0.654130	−0.327065	+	0.945002i	$0.606060\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−9817.55	−1.35467
$36$	0	0
$37$	13115.0	1.57494	0.787470	−	0.616353i	$-0.211391\pi$
$37$	13115.0	1.57494	0.787470	+	0.616353i	$0.211391\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9406.04	0.873871	0.436935	−	0.899493i	$-0.356064\pi$
$41$	9406.04	0.873871	0.436935	+	0.899493i	$0.356064\pi$
$42$	0	0
$43$	−104.000	−0.00857753	−0.00428876	−	0.999991i	$-0.501365\pi$
$43$	−104.000	−0.00857753	−0.00428876	+	0.999991i	$0.501365\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	20516.9	1.35478	0.677388	−	0.735626i	$-0.263112\pi$
$47$	20516.9	1.35478	0.677388	+	0.735626i	$0.263112\pi$
$48$	0	0
$49$	11082.0	0.659368
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1058.18	−0.0517452	−0.0258726	−	0.999665i	$-0.508236\pi$
$53$	−1058.18	−0.0517452	−0.0258726	+	0.999665i	$0.508236\pi$
$54$	0	0
$55$	44928.0	2.00267
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−30746.0	−1.14990	−0.574948	−	0.818190i	$-0.694978\pi$
$59$	−30746.0	−1.14990	−0.574948	+	0.818190i	$0.694978\pi$
$60$	0	0
$61$	−7393.00	−0.254388	−0.127194	−	0.991878i	$-0.540597\pi$
$61$	−7393.00	−0.254388	−0.127194	+	0.991878i	$0.540597\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13815.1	−0.405575
$66$	0	0
$67$	−38861.0	−1.05761	−0.528807	−	0.848742i	$-0.677360\pi$
$67$	−38861.0	−1.05761	−0.528807	+	0.848742i	$0.677360\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2469.09	0.0581287	0.0290643	−	0.999578i	$-0.490747\pi$
$71$	2469.09	0.0581287	0.0290643	+	0.999578i	$0.490747\pi$
$72$	0	0
$73$	5465.00	0.120028	0.0600141	−	0.998198i	$-0.480885\pi$
$73$	5465.00	0.120028	0.0600141	+	0.998198i	$0.480885\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−127628.	−2.45313
$78$	0	0
$79$	82903.0	1.49452	0.747261	−	0.664530i	$-0.231368\pi$
$79$	82903.0	1.49452	0.747261	+	0.664530i	$0.231368\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13286.0	0.211690	0.105845	−	0.994383i	$-0.466245\pi$
$83$	13286.0	0.211690	0.105845	+	0.994383i	$0.466245\pi$
$84$	0	0
$85$	10368.0	0.155649
$86$	0	0
$87$	0	0
$88$	0	0
$89$	89768.9	1.20130	0.600649	−	0.799513i	$-0.294909\pi$
$89$	89768.9	1.20130	0.600649	+	0.799513i	$0.294909\pi$
$90$	0	0
$91$	39245.0	0.496799
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−80010.1	−0.909570
$96$	0	0
$97$	−49603.0	−0.535277	−0.267639	−	0.963519i	$-0.586243\pi$
$97$	−49603.0	−0.535277	−0.267639	+	0.963519i	$0.586243\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	138739.	1.35330	0.676652	−	0.736303i	$-0.263430\pi$
$101$	138739.	1.35330	0.676652	+	0.736303i	$0.263430\pi$
$102$	0	0
$103$	94933.0	0.881707	0.440853	−	0.897579i	$-0.354676\pi$
$103$	94933.0	0.881707	0.440853	+	0.897579i	$0.354676\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	168780.	1.42515	0.712575	−	0.701596i	$-0.247529\pi$
$107$	168780.	1.42515	0.712575	+	0.701596i	$0.247529\pi$
$108$	0	0
$109$	124850.	1.00652	0.503260	−	0.864135i	$-0.332134\pi$
$109$	124850.	1.00652	0.503260	+	0.864135i	$0.332134\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	172542.	1.27116	0.635578	−	0.772037i	$-0.280762\pi$
$113$	172542.	1.27116	0.635578	+	0.772037i	$0.280762\pi$
$114$	0	0
$115$	141696.	0.999109
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−29452.7	−0.190659
$120$	0	0
$121$	423013.	2.62658
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−164253.	−0.940239
$126$	0	0
$127$	70108.0	0.385708	0.192854	−	0.981227i	$-0.438226\pi$
$127$	70108.0	0.385708	0.192854	+	0.981227i	$0.438226\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	139445.	0.709943	0.354971	−	0.934877i	$-0.384491\pi$
$131$	139445.	0.709943	0.354971	+	0.934877i	$0.384491\pi$
$132$	0	0
$133$	227287.	1.11415
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9464.83	0.0430835	0.0215418	−	0.999768i	$-0.493143\pi$
$137$	9464.83	0.0430835	0.0215418	+	0.999768i	$0.493143\pi$
$138$	0	0
$139$	150913.	0.662506	0.331253	−	0.943542i	$-0.392529\pi$
$139$	150913.	0.662506	0.331253	+	0.943542i	$0.392529\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−179597.	−0.734443
$144$	0	0
$145$	27648.0	0.109205
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−230801.	−0.851670	−0.425835	−	0.904801i	$-0.640020\pi$
$149$	−230801.	−0.851670	−0.425835	+	0.904801i	$0.640020\pi$
$150$	0	0
$151$	−129461.	−0.462058	−0.231029	−	0.972947i	$-0.574209\pi$
$151$	−129461.	−0.462058	−0.231029	+	0.972947i	$0.574209\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−205757.	−0.687901
$156$	0	0
$157$	257822.	0.834778	0.417389	−	0.908728i	$-0.362945\pi$
$157$	257822.	0.834778	0.417389	+	0.908728i	$0.362945\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−402520.	−1.22383
$162$	0	0
$163$	131569.	0.387869	0.193934	−	0.981015i	$-0.437875\pi$
$163$	131569.	0.387869	0.193934	+	0.981015i	$0.437875\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	257079.	0.713305	0.356652	−	0.934237i	$-0.383918\pi$
$167$	257079.	0.713305	0.356652	+	0.934237i	$0.383918\pi$
$168$	0	0
$169$	−316068.	−0.851263
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−167428.	−0.425316	−0.212658	−	0.977127i	$-0.568212\pi$
$173$	−167428.	−0.425316	−0.212658	+	0.977127i	$0.568212\pi$
$174$	0	0
$175$	−55277.0	−0.136442
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−389057.	−0.907572	−0.453786	−	0.891111i	$-0.649927\pi$
$179$	−389057.	−0.907572	−0.453786	+	0.891111i	$0.649927\pi$
$180$	0	0
$181$	165305.	0.375050	0.187525	−	0.982260i	$-0.439953\pi$
$181$	165305.	0.375050	0.187525	+	0.982260i	$0.439953\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	771001.	1.65625
$186$	0	0
$187$	134784.	0.281861
$188$	0	0
$189$	0	0
$190$	0	0
$191$	343967.	0.682234	0.341117	−	0.940021i	$-0.389195\pi$
$191$	343967.	0.682234	0.341117	+	0.940021i	$0.389195\pi$
$192$	0	0
$193$	−251785.	−0.486560	−0.243280	−	0.969956i	$-0.578223\pi$
$193$	−251785.	−0.486560	−0.243280	+	0.969956i	$0.578223\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	820971.	1.50717	0.753585	−	0.657350i	$-0.228323\pi$
$197$	820971.	1.50717	0.753585	+	0.657350i	$0.228323\pi$
$198$	0	0
$199$	336157.	0.601741	0.300870	−	0.953665i	$-0.402723\pi$
$199$	336157.	0.601741	0.300870	+	0.953665i	$0.402723\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−78540.4	−0.133768
$204$	0	0
$205$	552960.	0.918986
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.04013e6	−1.64711
$210$	0	0
$211$	821557.	1.27037	0.635187	−	0.772358i	$-0.280923\pi$
$211$	821557.	1.27037	0.635187	+	0.772358i	$0.280923\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6113.93	−0.00902036
$216$	0	0
$217$	584500.	0.842627
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−41445.4	−0.0570815
$222$	0	0
$223$	−670388.	−0.902743	−0.451371	−	0.892336i	$-0.649065\pi$
$223$	−670388.	−0.902743	−0.451371	+	0.892336i	$0.649065\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−860065.	−1.10781	−0.553907	−	0.832579i	$-0.686864\pi$
$227$	−860065.	−1.10781	−0.553907	+	0.832579i	$0.686864\pi$
$228$	0	0
$229$	−1.43277e6	−1.80546	−0.902732	−	0.430203i	$-0.858442\pi$
$229$	−1.43277e6	−1.80546	−0.902732	+	0.430203i	$0.858442\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	407046.	0.491195	0.245598	−	0.969372i	$-0.421016\pi$
$233$	407046.	0.491195	0.245598	+	0.969372i	$0.421016\pi$
$234$	0	0
$235$	1.20614e6	1.42472
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−115577.	−0.130881	−0.0654404	−	0.997856i	$-0.520845\pi$
$239$	−115577.	−0.130881	−0.0654404	+	0.997856i	$0.520845\pi$
$240$	0	0
$241$	−1.55192e6	−1.72118	−0.860590	−	0.509298i	$-0.829905\pi$
$241$	−1.55192e6	−1.72118	−0.860590	+	0.509298i	$0.829905\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	651486.	0.693410
$246$	0	0
$247$	319835.	0.333567
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.70226e6	1.70546	0.852729	−	0.522353i	$-0.174946\pi$
$251$	1.70226e6	1.70546	0.852729	+	0.522353i	$0.174946\pi$
$252$	0	0
$253$	1.84205e6	1.80925
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.38210e6	−1.30529	−0.652645	−	0.757664i	$-0.726340\pi$
$257$	−1.38210e6	−1.30529	−0.652645	+	0.757664i	$0.726340\pi$
$258$	0	0
$259$	−2.19020e6	−2.02878
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.61678e6	1.44132	0.720662	−	0.693286i	$-0.243838\pi$
$263$	1.61678e6	1.44132	0.720662	+	0.693286i	$0.243838\pi$
$264$	0	0
$265$	−62208.0	−0.0544166
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−406870.	−0.342827	−0.171413	−	0.985199i	$-0.554833\pi$
$269$	−406870.	−0.342827	−0.171413	+	0.985199i	$0.554833\pi$
$270$	0	0
$271$	−246053.	−0.203519	−0.101760	−	0.994809i	$-0.532447\pi$
$271$	−246053.	−0.203519	−0.101760	+	0.994809i	$0.532447\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	252964.	0.201710
$276$	0	0
$277$	−347350.	−0.271999	−0.136000	−	0.990709i	$-0.543425\pi$
$277$	−347350.	−0.271999	−0.136000	+	0.990709i	$0.543425\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	611393.	0.461907	0.230953	−	0.972965i	$-0.425815\pi$
$281$	611393.	0.461907	0.230953	+	0.972965i	$0.425815\pi$
$282$	0	0
$283$	−2.05827e6	−1.52770	−0.763848	−	0.645397i	$-0.776692\pi$
$283$	−2.05827e6	−1.52770	−0.763848	+	0.645397i	$0.776692\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.57081e6	−1.12569
$288$	0	0
$289$	−1.38875e6	−0.978094
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.26844e6	1.54369	0.771843	−	0.635813i	$-0.219335\pi$
$293$	2.26844e6	1.54369	0.771843	+	0.635813i	$0.219335\pi$
$294$	0	0
$295$	−1.80749e6	−1.20926
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−566420.	−0.366404
$300$	0	0
$301$	17368.0	0.0110493
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−434618.	−0.267521
$306$	0	0
$307$	−902576.	−0.546560	−0.273280	−	0.961935i	$-0.588109\pi$
$307$	−902576.	−0.546560	−0.273280	+	0.961935i	$0.588109\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.45964e6	−0.855746	−0.427873	−	0.903839i	$-0.640737\pi$
$311$	−1.45964e6	−0.855746	−0.427873	+	0.903839i	$0.640737\pi$
$312$	0	0
$313$	−2.75992e6	−1.59234	−0.796169	−	0.605075i	$-0.793143\pi$
$313$	−2.75992e6	−1.59234	−0.796169	+	0.605075i	$0.793143\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17048.4	0.00952877	0.00476438	−	0.999989i	$-0.498483\pi$
$317$	17048.4	0.00952877	0.00476438	+	0.999989i	$0.498483\pi$
$318$	0	0
$319$	359424.	0.197756
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−240030.	−0.128015
$324$	0	0
$325$	−77785.0	−0.0408496
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.42633e6	−1.74518
$330$	0	0
$331$	−3.37236e6	−1.69186	−0.845929	−	0.533296i	$-0.820953\pi$
$331$	−3.37236e6	−1.69186	−0.845929	+	0.533296i	$0.820953\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.28455e6	−1.11222
$336$	0	0
$337$	−360523.	−0.172925	−0.0864626	−	0.996255i	$-0.527556\pi$
$337$	−360523.	−0.172925	−0.0864626	+	0.996255i	$0.527556\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.67484e6	−1.24570
$342$	0	0
$343$	956075.	0.438790
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.44206e6	0.642926	0.321463	−	0.946922i	$-0.395825\pi$
$347$	1.44206e6	0.642926	0.321463	+	0.946922i	$0.395825\pi$
$348$	0	0
$349$	677579.	0.297781	0.148890	−	0.988854i	$-0.452430\pi$
$349$	677579.	0.297781	0.148890	+	0.988854i	$0.452430\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.84484e6	1.64226	0.821129	−	0.570743i	$-0.193345\pi$
$353$	3.84484e6	1.64226	0.821129	+	0.570743i	$0.193345\pi$
$354$	0	0
$355$	145152.	0.0611297
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.36699e6	−0.559796	−0.279898	−	0.960030i	$-0.590301\pi$
$359$	−1.36699e6	−0.559796	−0.279898	+	0.960030i	$0.590301\pi$
$360$	0	0
$361$	−623778.	−0.251920
$362$	0	0
$363$	0	0
$364$	0	0
$365$	321275.	0.126225
$366$	0	0
$367$	1.16951e6	0.453251	0.226625	−	0.973982i	$-0.427231\pi$
$367$	1.16951e6	0.453251	0.226625	+	0.973982i	$0.427231\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	176716.	0.0666563
$372$	0	0
$373$	2.52666e6	0.940318	0.470159	−	0.882582i	$-0.344197\pi$
$373$	2.52666e6	0.940318	0.470159	+	0.882582i	$0.344197\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−110521.	−0.0400490
$378$	0	0
$379$	−219269.	−0.0784114	−0.0392057	−	0.999231i	$-0.512483\pi$
$379$	−219269.	−0.0784114	−0.0392057	+	0.999231i	$0.512483\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	59728.4	0.0208058	0.0104029	−	0.999946i	$-0.496689\pi$
$383$	59728.4	0.0208058	0.0104029	+	0.999946i	$0.496689\pi$
$384$	0	0
$385$	−7.50298e6	−2.57977
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.70653e6	−0.906857	−0.453428	−	0.891293i	$-0.649799\pi$
$389$	−2.70653e6	−0.906857	−0.453428	+	0.891293i	$0.649799\pi$
$390$	0	0
$391$	425088.	0.140617
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.87368e6	1.57168
$396$	0	0
$397$	4.43128e6	1.41108	0.705542	−	0.708668i	$-0.250704\pi$
$397$	4.43128e6	1.41108	0.705542	+	0.708668i	$0.250704\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.37585e6	0.737832	0.368916	−	0.929463i	$-0.379729\pi$
$401$	2.37585e6	0.737832	0.368916	+	0.929463i	$0.379729\pi$
$402$	0	0
$403$	822500.	0.252274
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.00230e7	2.99925
$408$	0	0
$409$	−1.71805e6	−0.507840	−0.253920	−	0.967225i	$-0.581720\pi$
$409$	−1.71805e6	−0.507840	−0.253920	+	0.967225i	$0.581720\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.13458e6	1.48126
$414$	0	0
$415$	781056.	0.222619
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5.34633e6	−1.48772	−0.743860	−	0.668336i	$-0.767007\pi$
$419$	−5.34633e6	−1.48772	−0.743860	+	0.668336i	$0.767007\pi$
$420$	0	0
$421$	−5.68202e6	−1.56242	−0.781209	−	0.624269i	$-0.785397\pi$
$421$	−5.68202e6	−1.56242	−0.781209	+	0.624269i	$0.785397\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	58376.2	0.0156770
$426$	0	0
$427$	1.23463e6	0.327693
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.29890e6	0.596109	0.298055	−	0.954549i	$-0.403662\pi$
$431$	2.29890e6	0.596109	0.298055	+	0.954549i	$0.403662\pi$
$432$	0	0
$433$	−4.72608e6	−1.21138	−0.605691	−	0.795700i	$-0.707103\pi$
$433$	−4.72608e6	−1.21138	−0.605691	+	0.795700i	$0.707103\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.28042e6	−0.821723
$438$	0	0
$439$	−6.12223e6	−1.51617	−0.758086	−	0.652155i	$-0.773865\pi$
$439$	−6.12223e6	−1.51617	−0.758086	+	0.652155i	$0.773865\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.68429e6	−1.61825	−0.809125	−	0.587636i	$-0.800059\pi$
$443$	−6.68429e6	−1.61825	−0.809125	+	0.587636i	$0.800059\pi$
$444$	0	0
$445$	5.27731e6	1.26332
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.01701e6	1.17443	0.587217	−	0.809429i	$-0.300223\pi$
$449$	5.01701e6	1.17443	0.587217	+	0.809429i	$0.300223\pi$
$450$	0	0
$451$	7.18848e6	1.66416
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.30713e6	0.522448
$456$	0	0
$457$	683222.	0.153028	0.0765141	−	0.997069i	$-0.475621\pi$
$457$	683222.	0.153028	0.0765141	+	0.997069i	$0.475621\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.04123e6	−1.10480	−0.552400	−	0.833579i	$-0.686288\pi$
$461$	−5.04123e6	−1.10480	−0.552400	+	0.833579i	$0.686288\pi$
$462$	0	0
$463$	−5.04086e6	−1.09283	−0.546415	−	0.837515i	$-0.684008\pi$
$463$	−5.04086e6	−1.09283	−0.546415	+	0.837515i	$0.684008\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.12347e6	−1.72365	−0.861825	−	0.507205i	$-0.830679\pi$
$467$	−8.12347e6	−1.72365	−0.861825	+	0.507205i	$0.830679\pi$
$468$	0	0
$469$	6.48979e6	1.36238
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−79481.0	−0.0163347
$474$	0	0
$475$	−450491.	−0.0916119
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.60739e6	−1.31580	−0.657902	−	0.753104i	$-0.728556\pi$
$479$	−6.60739e6	−1.31580	−0.657902	+	0.753104i	$0.728556\pi$
$480$	0	0
$481$	−3.08202e6	−0.607398
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.91605e6	−0.562912
$486$	0	0
$487$	2.49806e6	0.477289	0.238644	−	0.971107i	$-0.423297\pi$
$487$	2.49806e6	0.477289	0.238644	+	0.971107i	$0.423297\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.70141e6	1.06728	0.533640	−	0.845711i	$-0.320824\pi$
$491$	5.70141e6	1.06728	0.533640	+	0.845711i	$0.320824\pi$
$492$	0	0
$493$	82944.0	0.0153698
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−412337.	−0.0748793
$498$	0	0
$499$	−4.19754e6	−0.754646	−0.377323	−	0.926082i	$-0.623155\pi$
$499$	−4.19754e6	−0.754646	−0.377323	+	0.926082i	$0.623155\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.37784e6	−0.947738	−0.473869	−	0.880595i	$-0.657143\pi$
$503$	−5.37784e6	−0.947738	−0.473869	+	0.880595i	$0.657143\pi$
$504$	0	0
$505$	8.15616e6	1.42317
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.35623e6	0.745275	0.372637	−	0.927977i	$-0.378454\pi$
$509$	4.35623e6	0.745275	0.372637	+	0.927977i	$0.378454\pi$
$510$	0	0
$511$	−912655.	−0.154616
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.58090e6	0.927227
$516$	0	0
$517$	1.56799e7	2.57998
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.29639e6	−0.693440	−0.346720	−	0.937969i	$-0.612705\pi$
$521$	−4.29639e6	−0.693440	−0.346720	+	0.937969i	$0.612705\pi$
$522$	0	0
$523$	1.07625e7	1.72052	0.860261	−	0.509854i	$-0.170301\pi$
$523$	1.07625e7	1.72052	0.860261	+	0.509854i	$0.170301\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−617271.	−0.0968166
$528$	0	0
$529$	−626807.	−0.0973856
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.21042e6	−0.337021
$534$	0	0
$535$	9.92218e6	1.49873
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.46932e6	1.25567
$540$	0	0
$541$	7.49825e6	1.10146	0.550728	−	0.834685i	$-0.314350\pi$
$541$	7.49825e6	1.10146	0.550728	+	0.834685i	$0.314350\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.33965e6	1.05848
$546$	0	0
$547$	3.63295e6	0.519148	0.259574	−	0.965723i	$-0.416418\pi$
$547$	3.63295e6	0.519148	0.259574	+	0.965723i	$0.416418\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−640081.	−0.0898165
$552$	0	0
$553$	−1.38448e7	−1.92519
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.12137e6	−0.972581	−0.486290	−	0.873797i	$-0.661650\pi$
$557$	−7.12137e6	−0.972581	−0.486290	+	0.873797i	$0.661650\pi$
$558$	0	0
$559$	24440.0	0.00330805
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.41768e6	0.454424	0.227212	−	0.973845i	$-0.427039\pi$
$563$	3.41768e6	0.454424	0.227212	+	0.973845i	$0.427039\pi$
$564$	0	0
$565$	1.01434e7	1.33678
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.43699e6	−0.704008	−0.352004	−	0.935999i	$-0.614500\pi$
$569$	−5.43699e6	−0.704008	−0.352004	+	0.935999i	$0.614500\pi$
$570$	0	0
$571$	3.92190e6	0.503391	0.251696	−	0.967806i	$-0.419012\pi$
$571$	3.92190e6	0.503391	0.251696	+	0.967806i	$0.419012\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	797809.	0.100630
$576$	0	0
$577$	1.07034e7	1.33839	0.669193	−	0.743088i	$-0.266640\pi$
$577$	1.07034e7	1.33839	0.669193	+	0.743088i	$0.266640\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.21877e6	−0.272691
$582$	0	0
$583$	−808704.	−0.0985413
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.99424e6	−0.358667	−0.179333	−	0.983788i	$-0.557394\pi$
$587$	−2.99424e6	−0.358667	−0.179333	+	0.983788i	$0.557394\pi$
$588$	0	0
$589$	4.76350e6	0.565767
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5.85279e6	−0.683481	−0.341740	−	0.939794i	$-0.611016\pi$
$593$	−5.85279e6	−0.683481	−0.341740	+	0.939794i	$0.611016\pi$
$594$	0	0
$595$	−1.73146e6	−0.200502
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.93059e6	0.219848	0.109924	−	0.993940i	$-0.464939\pi$
$599$	1.93059e6	0.219848	0.109924	+	0.993940i	$0.464939\pi$
$600$	0	0
$601$	−6.64461e6	−0.750384	−0.375192	−	0.926947i	$-0.622423\pi$
$601$	−6.64461e6	−0.750384	−0.375192	+	0.926947i	$0.622423\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.48680e7	2.76218
$606$	0	0
$607$	6.72545e6	0.740883	0.370441	−	0.928856i	$-0.379206\pi$
$607$	6.72545e6	0.740883	0.370441	+	0.928856i	$0.379206\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.82148e6	−0.522489
$612$	0	0
$613$	3.03643e6	0.326372	0.163186	−	0.986595i	$-0.447823\pi$
$613$	3.03643e6	0.326372	0.163186	+	0.986595i	$0.447823\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.93504e6	0.627640	0.313820	−	0.949483i	$-0.398391\pi$
$617$	5.93504e6	0.627640	0.313820	+	0.949483i	$0.398391\pi$
$618$	0	0
$619$	−1.21004e7	−1.26932	−0.634661	−	0.772791i	$-0.718860\pi$
$619$	−1.21004e7	−1.26932	−0.634661	+	0.772791i	$0.718860\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.49914e7	−1.54747
$624$	0	0
$625$	−1.06904e7	−1.09470
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.31300e6	0.233104
$630$	0	0
$631$	6.17037e6	0.616933	0.308466	−	0.951235i	$-0.400184\pi$
$631$	6.17037e6	0.616933	0.308466	+	0.951235i	$0.400184\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.12149e6	0.405621
$636$	0	0
$637$	−2.60427e6	−0.254295
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.13478e7	−1.09085	−0.545427	−	0.838158i	$-0.683632\pi$
$641$	−1.13478e7	−1.09085	−0.545427	+	0.838158i	$0.683632\pi$
$642$	0	0
$643$	1.00778e7	0.961257	0.480629	−	0.876924i	$-0.340408\pi$
$643$	1.00778e7	0.961257	0.480629	+	0.876924i	$0.340408\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.07287e6	0.664255	0.332128	−	0.943234i	$-0.392233\pi$
$647$	7.07287e6	0.664255	0.332128	+	0.943234i	$0.392233\pi$
$648$	0	0
$649$	−2.34973e7	−2.18981
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.84773e7	−1.69573	−0.847865	−	0.530213i	$-0.822112\pi$
$653$	−1.84773e7	−1.69573	−0.847865	+	0.530213i	$0.822112\pi$
$654$	0	0
$655$	8.19763e6	0.746595
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.53893e7	1.38040	0.690202	−	0.723616i	$-0.257521\pi$
$659$	1.53893e7	1.38040	0.690202	+	0.723616i	$0.257521\pi$
$660$	0	0
$661$	5.21093e6	0.463886	0.231943	−	0.972729i	$-0.425492\pi$
$661$	5.21093e6	0.463886	0.231943	+	0.972729i	$0.425492\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.33617e7	1.17168
$666$	0	0
$667$	1.13357e6	0.0986582
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.65003e6	−0.484445
$672$	0	0
$673$	1.60404e7	1.36514	0.682569	−	0.730821i	$-0.260863\pi$
$673$	1.60404e7	1.36514	0.682569	+	0.730821i	$0.260863\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.48811e7	1.24785	0.623927	−	0.781482i	$-0.285536\pi$
$677$	1.48811e7	1.24785	0.623927	+	0.781482i	$0.285536\pi$
$678$	0	0
$679$	8.28370e6	0.689525
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.62881e6	−0.461705	−0.230853	−	0.972989i	$-0.574152\pi$
$683$	−5.62881e6	−0.461705	−0.230853	+	0.972989i	$0.574152\pi$
$684$	0	0
$685$	556416.	0.0453078
$686$	0	0
$687$	0	0
$688$	0	0
$689$	248672.	0.0199563
$690$	0	0
$691$	1.01916e7	0.811987	0.405993	−	0.913876i	$-0.366926\pi$
$691$	1.01916e7	0.811987	0.405993	+	0.913876i	$0.366926\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.87184e6	0.696709
$696$	0	0
$697$	1.65888e6	0.129340
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.39524e7	1.84100	0.920501	−	0.390739i	$-0.127781\pi$
$701$	2.39524e7	1.84100	0.920501	+	0.390739i	$0.127781\pi$
$702$	0	0
$703$	−1.78495e7	−1.36219
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.31694e7	−1.74328
$708$	0	0
$709$	1.49472e7	1.11672	0.558360	−	0.829599i	$-0.311431\pi$
$709$	1.49472e7	1.11672	0.558360	+	0.829599i	$0.311431\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.43604e6	−0.621463
$714$	0	0
$715$	−1.05581e7	−0.772360
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.61670e7	1.88770	0.943848	−	0.330380i	$-0.107177\pi$
$719$	2.61670e7	1.88770	0.943848	+	0.330380i	$0.107177\pi$
$720$	0	0
$721$	−1.58538e7	−1.13578
$722$	0	0
$723$	0	0
$724$	0	0
$725$	155670.	0.0109992
$726$	0	0
$727$	2.91140e6	0.204299	0.102149	−	0.994769i	$-0.467428\pi$
$727$	2.91140e6	0.204299	0.102149	+	0.994769i	$0.467428\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−18341.8	−0.00126954
$732$	0	0
$733$	6.78250e6	0.466262	0.233131	−	0.972445i	$-0.425103\pi$
$733$	6.78250e6	0.466262	0.233131	+	0.972445i	$0.425103\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.96992e7	−2.01407
$738$	0	0
$739$	−1.52588e7	−1.02780	−0.513901	−	0.857850i	$-0.671800\pi$
$739$	−1.52588e7	−1.02780	−0.513901	+	0.857850i	$0.671800\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.96863e6	−0.130825	−0.0654125	−	0.997858i	$-0.520836\pi$
$743$	−1.96863e6	−0.130825	−0.0654125	+	0.997858i	$0.520836\pi$
$744$	0	0
$745$	−1.35683e7	−0.895640
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.81862e7	−1.83583
$750$	0	0
$751$	1.62379e7	1.05058	0.525290	−	0.850924i	$-0.323957\pi$
$751$	1.62379e7	1.05058	0.525290	+	0.850924i	$0.323957\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.61072e6	−0.485913
$756$	0	0
$757$	9.17096e6	0.581668	0.290834	−	0.956774i	$-0.406067\pi$
$757$	9.17096e6	0.581668	0.290834	+	0.956774i	$0.406067\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.31222e7	−1.44733	−0.723666	−	0.690151i	$-0.757544\pi$
$761$	−2.31222e7	−1.44733	−0.723666	+	0.690151i	$0.757544\pi$
$762$	0	0
$763$	−2.08500e7	−1.29656
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.22531e6	0.443474
$768$	0	0
$769$	9.38318e6	0.572182	0.286091	−	0.958202i	$-0.407644\pi$
$769$	9.38318e6	0.572182	0.286091	+	0.958202i	$0.407644\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−89239.8	−0.00537168	−0.00268584	−	0.999996i	$-0.500855\pi$
$773$	−89239.8	−0.00537168	−0.00268584	+	0.999996i	$0.500855\pi$
$774$	0	0
$775$	−1.15850e6	−0.0692854
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.28016e7	−0.755825
$780$	0	0
$781$	1.88698e6	0.110698
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.51568e7	0.877875
$786$	0	0
$787$	−3.04100e7	−1.75017	−0.875084	−	0.483971i	$-0.839194\pi$
$787$	−3.04100e7	−1.75017	−0.875084	+	0.483971i	$0.839194\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.88145e7	−1.63746
$792$	0	0
$793$	1.73736e6	0.0981083
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.44534e6	0.136362	0.0681809	−	0.997673i	$-0.478281\pi$
$797$	2.44534e6	0.136362	0.0681809	+	0.997673i	$0.478281\pi$
$798$	0	0
$799$	3.61843e6	0.200518
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.17658e6	0.228576
$804$	0	0
$805$	−2.36632e7	−1.28702
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.11075e7	1.13388	0.566938	−	0.823760i	$-0.308128\pi$
$809$	2.11075e7	1.13388	0.566938	+	0.823760i	$0.308128\pi$
$810$	0	0
$811$	1.56012e7	0.832925	0.416462	−	0.909153i	$-0.363270\pi$
$811$	1.56012e7	0.832925	0.416462	+	0.909153i	$0.363270\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.73465e6	0.407893
$816$	0	0
$817$	141544.	0.00741885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.04677e6	0.209532	0.104766	−	0.994497i	$-0.466591\pi$
$821$	4.04677e6	0.209532	0.104766	+	0.994497i	$0.466591\pi$
$822$	0	0
$823$	−2.21529e7	−1.14007	−0.570035	−	0.821620i	$-0.693071\pi$
$823$	−2.21529e7	−1.14007	−0.570035	+	0.821620i	$0.693071\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.72252e7	−0.875792	−0.437896	−	0.899026i	$-0.644276\pi$
$827$	−1.72252e7	−0.875792	−0.437896	+	0.899026i	$0.644276\pi$
$828$	0	0
$829$	−2.31834e7	−1.17163	−0.585816	−	0.810444i	$-0.699226\pi$
$829$	−2.31834e7	−1.17163	−0.585816	+	0.810444i	$0.699226\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.95446e6	0.0975919
$834$	0	0
$835$	1.51131e7	0.750131
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.07272e7	−0.526118	−0.263059	−	0.964780i	$-0.584731\pi$
$839$	−1.07272e7	−0.526118	−0.263059	+	0.964780i	$0.584731\pi$
$840$	0	0
$841$	−2.02900e7	−0.989216
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.85809e7	−0.895211
$846$	0	0
$847$	−7.06432e7	−3.38346
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.16111e7	1.49629
$852$	0	0
$853$	−421069.	−0.0198144	−0.00990719	−	0.999951i	$-0.503154\pi$
$853$	−421069.	−0.0198144	−0.00990719	+	0.999951i	$0.503154\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.71932e7	1.26476	0.632381	−	0.774658i	$-0.282078\pi$
$857$	2.71932e7	1.26476	0.632381	+	0.774658i	$0.282078\pi$
$858$	0	0
$859$	−2.01529e7	−0.931869	−0.465934	−	0.884819i	$-0.654282\pi$
$859$	−2.01529e7	−0.931869	−0.465934	+	0.884819i	$0.654282\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.88086e7	0.859666	0.429833	−	0.902908i	$-0.358572\pi$
$863$	1.88086e7	0.859666	0.429833	+	0.902908i	$0.358572\pi$
$864$	0	0
$865$	−9.84269e6	−0.447274
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.33579e7	2.84611
$870$	0	0
$871$	9.13234e6	0.407884
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.74302e7	1.21118
$876$	0	0
$877$	−2.21143e7	−0.970898	−0.485449	−	0.874265i	$-0.661344\pi$
$877$	−2.21143e7	−0.970898	−0.485449	+	0.874265i	$0.661344\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.00370e7	0.435677	0.217838	−	0.975985i	$-0.430099\pi$
$881$	1.00370e7	0.435677	0.217838	+	0.975985i	$0.430099\pi$
$882$	0	0
$883$	2.12042e7	0.915207	0.457604	−	0.889156i	$-0.348708\pi$
$883$	2.12042e7	0.915207	0.457604	+	0.889156i	$0.348708\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.73507e7	0.740472	0.370236	−	0.928938i	$-0.379277\pi$
$887$	1.73507e7	0.740472	0.370236	+	0.928938i	$0.379277\pi$
$888$	0	0
$889$	−1.17080e7	−0.496855
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.79235e7	−1.17177
$894$	0	0
$895$	−2.28718e7	−0.954427
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.64606e6	−0.0679275
$900$	0	0
$901$	−186624.	−0.00765871
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.71791e6	0.394413
$906$	0	0
$907$	1.02757e7	0.414756	0.207378	−	0.978261i	$-0.433507\pi$
$907$	1.02757e7	0.414756	0.207378	+	0.978261i	$0.433507\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.94713e7	−1.57574	−0.787871	−	0.615840i	$-0.788817\pi$
$911$	−3.94713e7	−1.57574	−0.787871	+	0.615840i	$0.788817\pi$
$912$	0	0
$913$	1.01537e7	0.403133
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.32872e7	−0.914523
$918$	0	0
$919$	1.65850e6	0.0647779	0.0323889	−	0.999475i	$-0.489688\pi$
$919$	1.65850e6	0.0647779	0.0323889	+	0.999475i	$0.489688\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−580235.	−0.0224181
$924$	0	0
$925$	4.34106e6	0.166818
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.65517e6	−0.176969	−0.0884843	−	0.996078i	$-0.528202\pi$
$929$	−4.65517e6	−0.176969	−0.0884843	+	0.996078i	$0.528202\pi$
$930$	0	0
$931$	−1.50826e7	−0.570298
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.92365e6	0.296412
$936$	0	0
$937$	−3.86106e7	−1.43667	−0.718337	−	0.695696i	$-0.755096\pi$
$937$	−3.86106e7	−1.43667	−0.718337	+	0.695696i	$0.755096\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.14452e7	−1.15766	−0.578828	−	0.815449i	$-0.696490\pi$
$941$	−3.14452e7	−1.15766	−0.578828	+	0.815449i	$0.696490\pi$
$942$	0	0
$943$	2.26714e7	0.830230
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.34273e6	0.338531	0.169266	−	0.985570i	$-0.445860\pi$
$947$	9.34273e6	0.338531	0.169266	+	0.985570i	$0.445860\pi$
$948$	0	0
$949$	−1.28428e6	−0.0462906
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.89849e6	−0.174715	−0.0873574	−	0.996177i	$-0.527842\pi$
$953$	−4.89849e6	−0.174715	−0.0873574	+	0.996177i	$0.527842\pi$
$954$	0	0
$955$	2.02211e7	0.717456
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.58063e6	−0.0554987
$960$	0	0
$961$	−1.63792e7	−0.572114
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.48019e7	−0.511680
$966$	0	0
$967$	1.13925e7	0.391791	0.195896	−	0.980625i	$-0.437239\pi$
$967$	1.13925e7	0.391791	0.195896	+	0.980625i	$0.437239\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.48394e7	0.505089	0.252544	−	0.967585i	$-0.418733\pi$
$971$	1.48394e7	0.505089	0.252544	+	0.967585i	$0.418733\pi$
$972$	0	0
$973$	−2.52025e7	−0.853416
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.53280e7	−0.848915	−0.424458	−	0.905448i	$-0.639535\pi$
$977$	−2.53280e7	−0.848915	−0.424458	+	0.905448i	$0.639535\pi$
$978$	0	0
$979$	6.86051e7	2.28770
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.48906e7	−1.15166	−0.575830	−	0.817569i	$-0.695321\pi$
$983$	−3.48906e7	−1.15166	−0.575830	+	0.817569i	$0.695321\pi$
$984$	0	0
$985$	4.82630e7	1.58498
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−250671.	−0.00814917
$990$	0	0
$991$	−2.38192e7	−0.770447	−0.385224	−	0.922823i	$-0.625876\pi$
$991$	−2.38192e7	−0.770447	−0.385224	+	0.922823i	$0.625876\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.97619e7	0.632807
$996$	0	0
$997$	3.78624e7	1.20634	0.603171	−	0.797612i	$-0.293904\pi$
$997$	3.78624e7	1.20634	0.603171	+	0.797612i	$0.293904\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.6.a.o.1.2		2
3.2	odd	2	inner	432.6.a.o.1.1		2
4.3	odd	2		27.6.a.c.1.2	yes	2
12.11	even	2		27.6.a.c.1.1	✓	2
20.19	odd	2		675.6.a.j.1.1		2
36.7	odd	6		81.6.c.f.28.1		4
36.11	even	6		81.6.c.f.28.2		4
36.23	even	6		81.6.c.f.55.2		4
36.31	odd	6		81.6.c.f.55.1		4
60.59	even	2		675.6.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.a.c.1.1	✓	2	12.11	even	2
27.6.a.c.1.2	yes	2	4.3	odd	2
81.6.c.f.28.1		4	36.7	odd	6
81.6.c.f.28.2		4	36.11	even	6
81.6.c.f.55.1		4	36.31	odd	6
81.6.c.f.55.2		4	36.23	even	6
432.6.a.o.1.1		2	3.2	odd	2	inner
432.6.a.o.1.2		2	1.1	even	1	trivial
675.6.a.j.1.1		2	20.19	odd	2
675.6.a.j.1.2		2	60.59	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	432.a (trivial)

\(f(q)\)	\(=\)	\(q+58.7878 q^{5} -167.000 q^{7} +764.241 q^{11} -235.000 q^{13} +176.363 q^{17} -1361.00 q^{19} +2410.30 q^{23} +331.000 q^{25} +470.302 q^{29} -3500.00 q^{31} -9817.55 q^{35} +13115.0 q^{37} +9406.04 q^{41} -104.000 q^{43} +20516.9 q^{47} +11082.0 q^{49} -1058.18 q^{53} +44928.0 q^{55} -30746.0 q^{59} -7393.00 q^{61} -13815.1 q^{65} -38861.0 q^{67} +2469.09 q^{71} +5465.00 q^{73} -127628. q^{77} +82903.0 q^{79} +13286.0 q^{83} +10368.0 q^{85} +89768.9 q^{89} +39245.0 q^{91} -80010.1 q^{95} -49603.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 334 q^{7} - 470 q^{13} - 2722 q^{19} + 662 q^{25} - 7000 q^{31} + 26230 q^{37} - 208 q^{43} + 22164 q^{49} + 89856 q^{55} - 14786 q^{61} - 77722 q^{67} + 10930 q^{73} + 165806 q^{79} + 20736 q^{85}+ \cdots - 99206 q^{97}+O(q^{100}) \) 2 * q - 334 * q^7 - 470 * q^13 - 2722 * q^19 + 662 * q^25 - 7000 * q^31 + 26230 * q^37 - 208 * q^43 + 22164 * q^49 + 89856 * q^55 - 14786 * q^61 - 77722 * q^67 + 10930 * q^73 + 165806 * q^79 + 20736 * q^85 + 78490 * q^91 - 99206 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more