LMFDB - Embedded newform 432.8.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,8,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, 8, names="a")

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

chi := DirichletCharacter("432.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 8 $
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:	$134.950331009$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 257x^{2} - 702x - 63 $ x^4 - 2x^3 - 257x^2 - 702*x - 63
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 216)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$18.2252$ 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)

$f(q)$	$=$	$q-13.8479 q^{5} -58.0683 q^{7} -7294.19 q^{11} +13723.1 q^{13} -34022.0 q^{17} +42942.8 q^{19} -20040.8 q^{23} -77933.2 q^{25} +138701. q^{29} -245242. q^{31} +804.123 q^{35} +210298. q^{37} -220731. q^{41} +364301. q^{43} +331125. q^{47} -820171. q^{49} +1.55208e6 q^{53} +101009. q^{55} -2.51316e6 q^{59} +2.37709e6 q^{61} -190036. q^{65} +224931. q^{67} +673820. q^{71} -780367. q^{73} +423561. q^{77} +1.02182e6 q^{79} -9.98352e6 q^{83} +471133. q^{85} +8.64638e6 q^{89} -796877. q^{91} -594667. q^{95} -6.34072e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 104 q^{5} + 492 q^{7} + 2104 q^{11} + 2356 q^{13} - 4136 q^{17} - 5516 q^{19} + 17848 q^{23} + 66476 q^{25} + 150720 q^{29} + 78256 q^{31} - 195432 q^{35} - 42324 q^{37} + 280704 q^{41} + 51200 q^{43}+ \cdots + 9596660 q^{97}+O(q^{100}) $ 4 * q + 104 * q^5 + 492 * q^7 + 2104 * q^11 + 2356 * q^13 - 4136 * q^17 - 5516 * q^19 + 17848 * q^23 + 66476 * q^25 + 150720 * q^29 + 78256 * q^31 - 195432 * q^35 - 42324 * q^37 + 280704 * q^41 + 51200 * q^43 - 829608 * q^47 + 13256 * q^49 + 1763120 * q^53 + 519968 * q^55 - 1881992 * q^59 + 869116 * q^61 + 5534696 * q^65 + 249700 * q^67 - 9344848 * q^71 - 461660 * q^73 + 11536584 * q^77 + 936116 * q^79 - 15826384 * q^83 + 5871776 * q^85 + 22823736 * q^89 + 2375484 * q^91 - 32094136 * q^95 + 9596660 * 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
$5$	−13.8479	−0.0495438	−0.0247719	−	0.999693i	$-0.507886\pi$
$5$	−13.8479	−0.0495438	−0.0247719	+	0.999693i	$0.507886\pi$
$6$	0	0
$7$	−58.0683	−0.0639876	−0.0319938	−	0.999488i	$-0.510186\pi$
$7$	−58.0683	−0.0639876	−0.0319938	+	0.999488i	$0.510186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−7294.19	−1.65235	−0.826176	−	0.563412i	$-0.809488\pi$
$11$	−7294.19	−1.65235	−0.826176	+	0.563412i	$0.809488\pi$
$12$	0	0
$13$	13723.1	1.73241	0.866205	−	0.499689i	$-0.166552\pi$
$13$	13723.1	1.73241	0.866205	+	0.499689i	$0.166552\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−34022.0	−1.67953	−0.839766	−	0.542949i	$-0.817308\pi$
$17$	−34022.0	−1.67953	−0.839766	+	0.542949i	$0.817308\pi$
$18$	0	0
$19$	42942.8	1.43632	0.718162	−	0.695876i	$-0.244984\pi$
$19$	42942.8	1.43632	0.718162	+	0.695876i	$0.244984\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−20040.8	−0.343454	−0.171727	−	0.985145i	$-0.554935\pi$
$23$	−20040.8	−0.343454	−0.171727	+	0.985145i	$0.554935\pi$
$24$	0	0
$25$	−77933.2	−0.997545
$26$	0	0
$27$	0	0
$28$	0	0
$29$	138701.	1.05606	0.528028	−	0.849227i	$-0.322932\pi$
$29$	138701.	1.05606	0.528028	+	0.849227i	$0.322932\pi$
$30$	0	0
$31$	−245242.	−1.47852	−0.739262	−	0.673418i	$-0.764825\pi$
$31$	−245242.	−1.47852	−0.739262	+	0.673418i	$0.764825\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	804.123	0.00317018
$36$	0	0
$37$	210298.	0.682542	0.341271	−	0.939965i	$-0.389143\pi$
$37$	210298.	0.682542	0.341271	+	0.939965i	$0.389143\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−220731.	−0.500172	−0.250086	−	0.968224i	$-0.580459\pi$
$41$	−220731.	−0.500172	−0.250086	+	0.968224i	$0.580459\pi$
$42$	0	0
$43$	364301.	0.698749	0.349375	−	0.936983i	$-0.386394\pi$
$43$	364301.	0.698749	0.349375	+	0.936983i	$0.386394\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	331125.	0.465211	0.232605	−	0.972571i	$-0.425275\pi$
$47$	331125.	0.465211	0.232605	+	0.972571i	$0.425275\pi$
$48$	0	0
$49$	−820171.	−0.995906
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.55208e6	1.43202	0.716010	−	0.698090i	$-0.245966\pi$
$53$	1.55208e6	1.43202	0.716010	+	0.698090i	$0.245966\pi$
$54$	0	0
$55$	101009.	0.0818637
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.51316e6	−1.59308	−0.796540	−	0.604586i	$-0.793339\pi$
$59$	−2.51316e6	−1.59308	−0.796540	+	0.604586i	$0.793339\pi$
$60$	0	0
$61$	2.37709e6	1.34088	0.670442	−	0.741962i	$-0.266104\pi$
$61$	2.37709e6	1.34088	0.670442	+	0.741962i	$0.266104\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−190036.	−0.0858301
$66$	0	0
$67$	224931.	0.0913666	0.0456833	−	0.998956i	$-0.485453\pi$
$67$	224931.	0.0913666	0.0456833	+	0.998956i	$0.485453\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	673820.	0.223429	0.111715	−	0.993740i	$-0.464366\pi$
$71$	673820.	0.223429	0.111715	+	0.993740i	$0.464366\pi$
$72$	0	0
$73$	−780367.	−0.234784	−0.117392	−	0.993086i	$-0.537453\pi$
$73$	−780367.	−0.234784	−0.117392	+	0.993086i	$0.537453\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	423561.	0.105730
$78$	0	0
$79$	1.02182e6	0.233174	0.116587	−	0.993180i	$-0.462805\pi$
$79$	1.02182e6	0.233174	0.116587	+	0.993180i	$0.462805\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.98352e6	−1.91651	−0.958253	−	0.285921i	$-0.907701\pi$
$83$	−9.98352e6	−1.91651	−0.958253	+	0.285921i	$0.907701\pi$
$84$	0	0
$85$	471133.	0.0832103
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.64638e6	1.30008	0.650039	−	0.759901i	$-0.274753\pi$
$89$	8.64638e6	1.30008	0.650039	+	0.759901i	$0.274753\pi$
$90$	0	0
$91$	−796877.	−0.110853
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−594667.	−0.0711609
$96$	0	0
$97$	−6.34072e6	−0.705403	−0.352701	−	0.935736i	$-0.614737\pi$
$97$	−6.34072e6	−0.705403	−0.352701	+	0.935736i	$0.614737\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.13461e7	1.09577	0.547887	−	0.836552i	$-0.315432\pi$
$101$	1.13461e7	1.09577	0.547887	+	0.836552i	$0.315432\pi$
$102$	0	0
$103$	2.99883e6	0.270409	0.135205	−	0.990818i	$-0.456831\pi$
$103$	2.99883e6	0.270409	0.135205	+	0.990818i	$0.456831\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.20558e7	−0.951378	−0.475689	−	0.879614i	$-0.657801\pi$
$107$	−1.20558e7	−0.951378	−0.475689	+	0.879614i	$0.657801\pi$
$108$	0	0
$109$	9.98019e6	0.738152	0.369076	−	0.929399i	$-0.379674\pi$
$109$	9.98019e6	0.738152	0.369076	+	0.929399i	$0.379674\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.39434e7	1.56103	0.780515	−	0.625137i	$-0.214957\pi$
$113$	2.39434e7	1.56103	0.780515	+	0.625137i	$0.214957\pi$
$114$	0	0
$115$	277524.	0.0170160
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.97560e6	0.107469
$120$	0	0
$121$	3.37180e7	1.73027
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.16108e6	0.0989659
$126$	0	0
$127$	2.58136e7	1.11824	0.559120	−	0.829087i	$-0.311139\pi$
$127$	2.58136e7	1.11824	0.559120	+	0.829087i	$0.311139\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.37342e7	−0.533771	−0.266885	−	0.963728i	$-0.585995\pi$
$131$	−1.37342e7	−0.533771	−0.266885	+	0.963728i	$0.585995\pi$
$132$	0	0
$133$	−2.49361e6	−0.0919069
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.00927e7	0.335341	0.167671	−	0.985843i	$-0.446375\pi$
$137$	1.00927e7	0.335341	0.167671	+	0.985843i	$0.446375\pi$
$138$	0	0
$139$	3.50551e7	1.10713	0.553566	−	0.832805i	$-0.313267\pi$
$139$	3.50551e7	1.10713	0.553566	+	0.832805i	$0.313267\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.00099e8	−2.86255
$144$	0	0
$145$	−1.92072e6	−0.0523209
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.17120e7	−0.290053	−0.145027	−	0.989428i	$-0.546327\pi$
$149$	−1.17120e7	−0.290053	−0.145027	+	0.989428i	$0.546327\pi$
$150$	0	0
$151$	−1.77937e7	−0.420578	−0.210289	−	0.977639i	$-0.567441\pi$
$151$	−1.77937e7	−0.420578	−0.210289	+	0.977639i	$0.567441\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.39608e6	0.0732516
$156$	0	0
$157$	−4.52444e7	−0.933074	−0.466537	−	0.884502i	$-0.654499\pi$
$157$	−4.52444e7	−0.933074	−0.466537	+	0.884502i	$0.654499\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.16374e6	0.0219768
$162$	0	0
$163$	−2.30806e7	−0.417437	−0.208719	−	0.977976i	$-0.566929\pi$
$163$	−2.30806e7	−0.417437	−0.208719	+	0.977976i	$0.566929\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.45870e7	1.07309	0.536547	−	0.843871i	$-0.319729\pi$
$167$	6.45870e7	1.07309	0.536547	+	0.843871i	$0.319729\pi$
$168$	0	0
$169$	1.25575e8	2.00124
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.75061e7	0.697571	0.348785	−	0.937203i	$-0.386594\pi$
$173$	4.75061e7	0.697571	0.348785	+	0.937203i	$0.386594\pi$
$174$	0	0
$175$	4.52545e6	0.0638305
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.29893e8	1.69278	0.846390	−	0.532564i	$-0.178771\pi$
$179$	1.29893e8	1.69278	0.846390	+	0.532564i	$0.178771\pi$
$180$	0	0
$181$	1.06317e8	1.33269	0.666345	−	0.745643i	$-0.267858\pi$
$181$	1.06317e8	1.33269	0.666345	+	0.745643i	$0.267858\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.91219e6	−0.0338157
$186$	0	0
$187$	2.48163e8	2.77518
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.66823e7	−0.588614	−0.294307	−	0.955711i	$-0.595089\pi$
$191$	−5.66823e7	−0.588614	−0.294307	+	0.955711i	$0.595089\pi$
$192$	0	0
$193$	4.86693e7	0.487309	0.243655	−	0.969862i	$-0.421654\pi$
$193$	4.86693e7	0.487309	0.243655	+	0.969862i	$0.421654\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.55409e8	1.44825	0.724127	−	0.689667i	$-0.242243\pi$
$197$	1.55409e8	1.44825	0.724127	+	0.689667i	$0.242243\pi$
$198$	0	0
$199$	1.94608e8	1.75055	0.875275	−	0.483626i	$-0.160680\pi$
$199$	1.94608e8	1.75055	0.875275	+	0.483626i	$0.160680\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.05413e6	−0.0675744
$204$	0	0
$205$	3.05666e6	0.0247804
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.13233e8	−2.37331
$210$	0	0
$211$	−8.38100e7	−0.614197	−0.307098	−	0.951678i	$-0.599358\pi$
$211$	−8.38100e7	−0.614197	−0.307098	+	0.951678i	$0.599358\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.04481e6	−0.0346186
$216$	0	0
$217$	1.42408e7	0.0946071
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.66887e8	−2.90964
$222$	0	0
$223$	−1.71955e8	−1.03836	−0.519180	−	0.854665i	$-0.673763\pi$
$223$	−1.71955e8	−1.03836	−0.519180	+	0.854665i	$0.673763\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.93562e8	1.09832	0.549160	−	0.835717i	$-0.314948\pi$
$227$	1.93562e8	1.09832	0.549160	+	0.835717i	$0.314948\pi$
$228$	0	0
$229$	3.43678e8	1.89116	0.945580	−	0.325391i	$-0.105496\pi$
$229$	3.43678e8	1.89116	0.945580	+	0.325391i	$0.105496\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.27531e8	0.660497	0.330249	−	0.943894i	$-0.392867\pi$
$233$	1.27531e8	0.660497	0.330249	+	0.943894i	$0.392867\pi$
$234$	0	0
$235$	−4.58539e6	−0.0230483
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.73575e8	1.29623	0.648116	−	0.761541i	$-0.275557\pi$
$239$	2.73575e8	1.29623	0.648116	+	0.761541i	$0.275557\pi$
$240$	0	0
$241$	−1.44202e8	−0.663609	−0.331805	−	0.943348i	$-0.607658\pi$
$241$	−1.44202e8	−0.663609	−0.331805	+	0.943348i	$0.607658\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.13576e7	0.0493409
$246$	0	0
$247$	5.89308e8	2.48830
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.23278e8	−1.29038	−0.645190	−	0.764022i	$-0.723222\pi$
$251$	−3.23278e8	−1.29038	−0.645190	+	0.764022i	$0.723222\pi$
$252$	0	0
$253$	1.46182e8	0.567507
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.17666e6	−0.00432401	−0.00216200	−	0.999998i	$-0.500688\pi$
$257$	−1.17666e6	−0.00432401	−0.00216200	+	0.999998i	$0.500688\pi$
$258$	0	0
$259$	−1.22117e7	−0.0436742
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.23924e8	1.43695	0.718477	−	0.695551i	$-0.244840\pi$
$263$	4.23924e8	1.43695	0.718477	+	0.695551i	$0.244840\pi$
$264$	0	0
$265$	−2.14931e7	−0.0709477
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.33873e8	−0.419334	−0.209667	−	0.977773i	$-0.567238\pi$
$269$	−1.33873e8	−0.419334	−0.209667	+	0.977773i	$0.567238\pi$
$270$	0	0
$271$	2.05415e8	0.626961	0.313480	−	0.949595i	$-0.398505\pi$
$271$	2.05415e8	0.626961	0.313480	+	0.949595i	$0.398505\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.68460e8	1.64830
$276$	0	0
$277$	3.12347e8	0.882996	0.441498	−	0.897262i	$-0.354447\pi$
$277$	3.12347e8	0.882996	0.441498	+	0.897262i	$0.354447\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.66094e8	0.446561	0.223281	−	0.974754i	$-0.428323\pi$
$281$	1.66094e8	0.446561	0.223281	+	0.974754i	$0.428323\pi$
$282$	0	0
$283$	3.28103e8	0.860513	0.430256	−	0.902707i	$-0.358423\pi$
$283$	3.28103e8	0.860513	0.430256	+	0.902707i	$0.358423\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.28175e7	0.0320048
$288$	0	0
$289$	7.47155e8	1.82083
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.99707e8	−1.16059	−0.580295	−	0.814406i	$-0.697063\pi$
$293$	−4.99707e8	−1.16059	−0.580295	+	0.814406i	$0.697063\pi$
$294$	0	0
$295$	3.48019e7	0.0789271
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.75023e8	−0.595003
$300$	0	0
$301$	−2.11543e7	−0.0447113
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.29177e7	−0.0664325
$306$	0	0
$307$	3.32792e8	0.656431	0.328215	−	0.944603i	$-0.393553\pi$
$307$	3.32792e8	0.656431	0.328215	+	0.944603i	$0.393553\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.86382e8	0.351353	0.175676	−	0.984448i	$-0.443789\pi$
$311$	1.86382e8	0.351353	0.175676	+	0.984448i	$0.443789\pi$
$312$	0	0
$313$	5.99210e7	0.110452	0.0552261	−	0.998474i	$-0.482412\pi$
$313$	5.99210e7	0.110452	0.0552261	+	0.998474i	$0.482412\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.25382e7	−0.0221070	−0.0110535	−	0.999939i	$-0.503519\pi$
$317$	−1.25382e7	−0.0221070	−0.0110535	+	0.999939i	$0.503519\pi$
$318$	0	0
$319$	−1.01171e9	−1.74497
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.46100e9	−2.41235
$324$	0	0
$325$	−1.06949e9	−1.72816
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.92279e7	−0.0297677
$330$	0	0
$331$	2.89103e8	0.438183	0.219091	−	0.975704i	$-0.429691\pi$
$331$	2.89103e8	0.438183	0.219091	+	0.975704i	$0.429691\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.11482e6	−0.00452665
$336$	0	0
$337$	4.21523e8	0.599952	0.299976	−	0.953947i	$-0.403021\pi$
$337$	4.21523e8	0.599952	0.299976	+	0.953947i	$0.403021\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.78884e9	2.44304
$342$	0	0
$343$	9.54476e7	0.127713
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.50682e8	1.09299	0.546493	−	0.837464i	$-0.315963\pi$
$347$	8.50682e8	1.09299	0.546493	+	0.837464i	$0.315963\pi$
$348$	0	0
$349$	−6.98241e8	−0.879259	−0.439629	−	0.898179i	$-0.644890\pi$
$349$	−6.98241e8	−0.879259	−0.439629	+	0.898179i	$0.644890\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.28309e9	−1.55256	−0.776278	−	0.630391i	$-0.782895\pi$
$353$	−1.28309e9	−1.55256	−0.776278	+	0.630391i	$0.782895\pi$
$354$	0	0
$355$	−9.33099e6	−0.0110695
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.05640e9	−1.20503	−0.602515	−	0.798108i	$-0.705835\pi$
$359$	−1.05640e9	−1.20503	−0.602515	+	0.798108i	$0.705835\pi$
$360$	0	0
$361$	9.50210e8	1.06303
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.08064e7	0.0116321
$366$	0	0
$367$	1.97860e8	0.208942	0.104471	−	0.994528i	$-0.466685\pi$
$367$	1.97860e8	0.208942	0.104471	+	0.994528i	$0.466685\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9.01267e7	−0.0916315
$372$	0	0
$373$	−5.90029e8	−0.588698	−0.294349	−	0.955698i	$-0.595103\pi$
$373$	−5.90029e8	−0.588698	−0.294349	+	0.955698i	$0.595103\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.90341e9	1.82952
$378$	0	0
$379$	8.71294e7	0.0822105	0.0411053	−	0.999155i	$-0.486912\pi$
$379$	8.71294e7	0.0822105	0.0411053	+	0.999155i	$0.486912\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.07952e8	−0.825786	−0.412893	−	0.910780i	$-0.635482\pi$
$383$	−9.07952e8	−0.825786	−0.412893	+	0.910780i	$0.635482\pi$
$384$	0	0
$385$	−5.86543e6	−0.00523826
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.58207e9	1.36271	0.681353	−	0.731955i	$-0.261392\pi$
$389$	1.58207e9	1.36271	0.681353	+	0.731955i	$0.261392\pi$
$390$	0	0
$391$	6.81829e8	0.576842
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.41501e7	−0.0115523
$396$	0	0
$397$	1.47758e9	1.18518	0.592590	−	0.805505i	$-0.298106\pi$
$397$	1.47758e9	1.18518	0.592590	+	0.805505i	$0.298106\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.76301e8	−0.601208	−0.300604	−	0.953749i	$-0.597188\pi$
$401$	−7.76301e8	−0.601208	−0.300604	+	0.953749i	$0.597188\pi$
$402$	0	0
$403$	−3.36548e9	−2.56141
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.53395e9	−1.12780
$408$	0	0
$409$	−1.16457e9	−0.841655	−0.420828	−	0.907141i	$-0.638260\pi$
$409$	−1.16457e9	−0.841655	−0.420828	+	0.907141i	$0.638260\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.45935e8	0.101937
$414$	0	0
$415$	1.38251e8	0.0949509
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.11979e8	0.140781	0.0703906	−	0.997520i	$-0.477575\pi$
$419$	2.11979e8	0.140781	0.0703906	+	0.997520i	$0.477575\pi$
$420$	0	0
$421$	−2.32841e9	−1.52080	−0.760401	−	0.649454i	$-0.774997\pi$
$421$	−2.32841e9	−1.52080	−0.760401	+	0.649454i	$0.774997\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.65144e9	1.67541
$426$	0	0
$427$	−1.38034e8	−0.0858000
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.70305e8	−0.463439	−0.231719	−	0.972783i	$-0.574435\pi$
$431$	−7.70305e8	−0.463439	−0.231719	+	0.972783i	$0.574435\pi$
$432$	0	0
$433$	−1.95227e9	−1.15567	−0.577833	−	0.816155i	$-0.696101\pi$
$433$	−1.95227e9	−1.15567	−0.577833	+	0.816155i	$0.696101\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.60610e8	−0.493311
$438$	0	0
$439$	−7.79733e8	−0.439865	−0.219933	−	0.975515i	$-0.570584\pi$
$439$	−7.79733e8	−0.439865	−0.219933	+	0.975515i	$0.570584\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.34329e9	0.734104	0.367052	−	0.930200i	$-0.380367\pi$
$443$	1.34329e9	0.734104	0.367052	+	0.930200i	$0.380367\pi$
$444$	0	0
$445$	−1.19734e8	−0.0644107
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.07166e9	1.08008	0.540040	−	0.841639i	$-0.318409\pi$
$449$	2.07166e9	1.08008	0.540040	+	0.841639i	$0.318409\pi$
$450$	0	0
$451$	1.61005e9	0.826460
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.10351e7	0.00549206
$456$	0	0
$457$	−1.29492e9	−0.634653	−0.317326	−	0.948316i	$-0.602785\pi$
$457$	−1.29492e9	−0.634653	−0.317326	+	0.948316i	$0.602785\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.55949e9	−1.69213	−0.846067	−	0.533076i	$-0.821036\pi$
$461$	−3.55949e9	−1.69213	−0.846067	+	0.533076i	$0.821036\pi$
$462$	0	0
$463$	−1.80217e9	−0.843843	−0.421922	−	0.906632i	$-0.638644\pi$
$463$	−1.80217e9	−0.843843	−0.421922	+	0.906632i	$0.638644\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.81494e9	−0.824620	−0.412310	−	0.911044i	$-0.635278\pi$
$467$	−1.81494e9	−0.824620	−0.412310	+	0.911044i	$0.635278\pi$
$468$	0	0
$469$	−1.30614e7	−0.00584633
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.65728e9	−1.15458
$474$	0	0
$475$	−3.34667e9	−1.43280
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.94674e8	0.0809346	0.0404673	−	0.999181i	$-0.487115\pi$
$479$	1.94674e8	0.0809346	0.0404673	+	0.999181i	$0.487115\pi$
$480$	0	0
$481$	2.88595e9	1.18244
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.78056e7	0.0349483
$486$	0	0
$487$	−4.65390e9	−1.82585	−0.912925	−	0.408127i	$-0.866182\pi$
$487$	−4.65390e9	−1.82585	−0.912925	+	0.408127i	$0.866182\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.19925e8	−0.198224	−0.0991118	−	0.995076i	$-0.531600\pi$
$491$	−5.19925e8	−0.198224	−0.0991118	+	0.995076i	$0.531600\pi$
$492$	0	0
$493$	−4.71888e9	−1.77368
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.91276e7	−0.0142967
$498$	0	0
$499$	1.15867e9	0.417454	0.208727	−	0.977974i	$-0.433068\pi$
$499$	1.15867e9	0.417454	0.208727	+	0.977974i	$0.433068\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.43771e9	−0.503715	−0.251857	−	0.967764i	$-0.581041\pi$
$503$	−1.43771e9	−0.503715	−0.251857	+	0.967764i	$0.581041\pi$
$504$	0	0
$505$	−1.57119e8	−0.0542888
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.21519e9	1.41679	0.708394	−	0.705817i	$-0.249420\pi$
$509$	4.21519e9	1.41679	0.708394	+	0.705817i	$0.249420\pi$
$510$	0	0
$511$	4.53145e7	0.0150233
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.15275e7	−0.0133971
$516$	0	0
$517$	−2.41529e9	−0.768692
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.02371e9	−1.24651	−0.623254	−	0.782020i	$-0.714190\pi$
$521$	−4.02371e9	−1.24651	−0.623254	+	0.782020i	$0.714190\pi$
$522$	0	0
$523$	2.70580e9	0.827066	0.413533	−	0.910489i	$-0.364295\pi$
$523$	2.70580e9	0.827066	0.413533	+	0.910489i	$0.364295\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.34360e9	2.48323
$528$	0	0
$529$	−3.00319e9	−0.882039
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.02911e9	−0.866503
$534$	0	0
$535$	1.66948e8	0.0471348
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.98248e9	1.64559
$540$	0	0
$541$	1.52701e9	0.414621	0.207311	−	0.978275i	$-0.433529\pi$
$541$	1.52701e9	0.414621	0.207311	+	0.978275i	$0.433529\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.38205e8	−0.0365708
$546$	0	0
$547$	−1.50698e9	−0.393687	−0.196843	−	0.980435i	$-0.563069\pi$
$547$	−1.50698e9	−0.393687	−0.196843	+	0.980435i	$0.563069\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.95621e9	1.51684
$552$	0	0
$553$	−5.93353e7	−0.0149202
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.28476e9	−0.805398	−0.402699	−	0.915332i	$-0.631928\pi$
$557$	−3.28476e9	−0.805398	−0.402699	+	0.915332i	$0.631928\pi$
$558$	0	0
$559$	4.99935e9	1.21052
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3.39840e9	−0.802593	−0.401296	−	0.915948i	$-0.631440\pi$
$563$	−3.39840e9	−0.802593	−0.401296	+	0.915948i	$0.631440\pi$
$564$	0	0
$565$	−3.31566e8	−0.0773393
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.44799e8	0.215004	0.107502	−	0.994205i	$-0.465715\pi$
$569$	9.44799e8	0.215004	0.107502	+	0.994205i	$0.465715\pi$
$570$	0	0
$571$	−7.79275e9	−1.75172	−0.875860	−	0.482566i	$-0.839705\pi$
$571$	−7.79275e9	−1.75172	−0.875860	+	0.482566i	$0.839705\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.56185e9	0.342611
$576$	0	0
$577$	7.06901e9	1.53194	0.765972	−	0.642874i	$-0.222258\pi$
$577$	7.06901e9	1.53194	0.765972	+	0.642874i	$0.222258\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.79725e8	0.122633
$582$	0	0
$583$	−1.13212e10	−2.36620
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.39133e9	−1.30424	−0.652121	−	0.758115i	$-0.726121\pi$
$587$	−6.39133e9	−1.30424	−0.652121	+	0.758115i	$0.726121\pi$
$588$	0	0
$589$	−1.05314e10	−2.12364
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.79150e9	−0.746654	−0.373327	−	0.927700i	$-0.621783\pi$
$593$	−3.79150e9	−0.746654	−0.373327	+	0.927700i	$0.621783\pi$
$594$	0	0
$595$	−2.73579e7	−0.00532442
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.10616e9	−0.210293	−0.105147	−	0.994457i	$-0.533531\pi$
$599$	−1.10616e9	−0.210293	−0.105147	+	0.994457i	$0.533531\pi$
$600$	0	0
$601$	6.73569e8	0.126567	0.0632836	−	0.997996i	$-0.479843\pi$
$601$	6.73569e8	0.126567	0.0632836	+	0.997996i	$0.479843\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.66923e8	−0.0857239
$606$	0	0
$607$	2.79627e9	0.507480	0.253740	−	0.967272i	$-0.418339\pi$
$607$	2.79627e9	0.507480	0.253740	+	0.967272i	$0.418339\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.54407e9	0.805936
$612$	0	0
$613$	−6.59459e9	−1.15631	−0.578157	−	0.815925i	$-0.696228\pi$
$613$	−6.59459e9	−1.15631	−0.578157	+	0.815925i	$0.696228\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.93189e9	−1.53089	−0.765447	−	0.643499i	$-0.777482\pi$
$617$	−8.93189e9	−1.53089	−0.765447	+	0.643499i	$0.777482\pi$
$618$	0	0
$619$	−3.24802e8	−0.0550430	−0.0275215	−	0.999621i	$-0.508761\pi$
$619$	−3.24802e8	−0.0550430	−0.0275215	+	0.999621i	$0.508761\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.02080e8	−0.0831888
$624$	0	0
$625$	6.05861e9	0.992642
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.15476e9	−1.14635
$630$	0	0
$631$	9.69591e9	1.53634	0.768168	−	0.640249i	$-0.221169\pi$
$631$	9.69591e9	1.53634	0.768168	+	0.640249i	$0.221169\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.57464e8	−0.0554018
$636$	0	0
$637$	−1.12553e10	−1.72532
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.94358e8	0.0291473	0.0145737	−	0.999894i	$-0.495361\pi$
$641$	1.94358e8	0.0291473	0.0145737	+	0.999894i	$0.495361\pi$
$642$	0	0
$643$	1.03639e10	1.53739	0.768697	−	0.639613i	$-0.220905\pi$
$643$	1.03639e10	1.53739	0.768697	+	0.639613i	$0.220905\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.70895e9	0.538376	0.269188	−	0.963088i	$-0.413245\pi$
$647$	3.70895e9	0.538376	0.269188	+	0.963088i	$0.413245\pi$
$648$	0	0
$649$	1.83314e10	2.63233
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.39530e9	1.03935	0.519673	−	0.854365i	$-0.326054\pi$
$653$	7.39530e9	1.03935	0.519673	+	0.854365i	$0.326054\pi$
$654$	0	0
$655$	1.90190e8	0.0264450
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.00786e10	1.37183	0.685914	−	0.727682i	$-0.259402\pi$
$659$	1.00786e10	1.37183	0.685914	+	0.727682i	$0.259402\pi$
$660$	0	0
$661$	1.08007e10	1.45461	0.727305	−	0.686314i	$-0.240773\pi$
$661$	1.08007e10	1.45461	0.727305	+	0.686314i	$0.240773\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.45313e7	0.00455341
$666$	0	0
$667$	−2.77969e9	−0.362706
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.73389e10	−2.21561
$672$	0	0
$673$	8.93825e8	0.113032	0.0565158	−	0.998402i	$-0.482001\pi$
$673$	8.93825e8	0.113032	0.0565158	+	0.998402i	$0.482001\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.11217e9	0.509344	0.254672	−	0.967028i	$-0.418033\pi$
$677$	4.11217e9	0.509344	0.254672	+	0.967028i	$0.418033\pi$
$678$	0	0
$679$	3.68194e8	0.0451370
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.74856e9	0.330091	0.165045	−	0.986286i	$-0.447223\pi$
$683$	2.74856e9	0.330091	0.165045	+	0.986286i	$0.447223\pi$
$684$	0	0
$685$	−1.39763e8	−0.0166141
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.12994e10	2.48085
$690$	0	0
$691$	−6.46154e8	−0.0745012	−0.0372506	−	0.999306i	$-0.511860\pi$
$691$	−6.46154e8	−0.0745012	−0.0372506	+	0.999306i	$0.511860\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.85440e8	−0.0548515
$696$	0	0
$697$	7.50969e9	0.840055
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.37473e9	0.698954	0.349477	−	0.936945i	$-0.386359\pi$
$701$	6.37473e9	0.698954	0.349477	+	0.936945i	$0.386359\pi$
$702$	0	0
$703$	9.03079e9	0.980352
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.58847e8	−0.0701159
$708$	0	0
$709$	−1.02707e10	−1.08228	−0.541140	−	0.840933i	$-0.682007\pi$
$709$	−1.02707e10	−1.08228	−0.541140	+	0.840933i	$0.682007\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.91485e9	0.507805
$714$	0	0
$715$	1.38616e9	0.141821
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.25037e10	1.25455	0.627273	−	0.778800i	$-0.284171\pi$
$719$	1.25037e10	1.25455	0.627273	+	0.778800i	$0.284171\pi$
$720$	0	0
$721$	−1.74137e8	−0.0173028
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.08094e10	−1.05346
$726$	0	0
$727$	−8.62049e9	−0.832073	−0.416037	−	0.909348i	$-0.636581\pi$
$727$	−8.62049e9	−0.832073	−0.416037	+	0.909348i	$0.636581\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.23942e10	−1.17357
$732$	0	0
$733$	3.69857e9	0.346872	0.173436	−	0.984845i	$-0.444513\pi$
$733$	3.69857e9	0.346872	0.173436	+	0.984845i	$0.444513\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.64069e9	−0.150970
$738$	0	0
$739$	3.37949e9	0.308032	0.154016	−	0.988068i	$-0.450779\pi$
$739$	3.37949e9	0.308032	0.154016	+	0.988068i	$0.450779\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.43356e10	−1.28219	−0.641097	−	0.767460i	$-0.721520\pi$
$743$	−1.43356e10	−1.28219	−0.641097	+	0.767460i	$0.721520\pi$
$744$	0	0
$745$	1.62186e8	0.0143703
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.00060e8	0.0608764
$750$	0	0
$751$	2.21735e9	0.191026	0.0955132	−	0.995428i	$-0.469551\pi$
$751$	2.21735e9	0.191026	0.0955132	+	0.995428i	$0.469551\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.46405e8	0.0208370
$756$	0	0
$757$	2.62317e9	0.219781	0.109891	−	0.993944i	$-0.464950\pi$
$757$	2.62317e9	0.219781	0.109891	+	0.993944i	$0.464950\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.36033e9	0.523158	0.261579	−	0.965182i	$-0.415757\pi$
$761$	6.36033e9	0.523158	0.261579	+	0.965182i	$0.415757\pi$
$762$	0	0
$763$	−5.79532e8	−0.0472326
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.44883e10	−2.75987
$768$	0	0
$769$	3.79748e9	0.301129	0.150565	−	0.988600i	$-0.451891\pi$
$769$	3.79748e9	0.301129	0.150565	+	0.988600i	$0.451891\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.16854e10	−0.909948	−0.454974	−	0.890505i	$-0.650351\pi$
$773$	−1.16854e10	−0.909948	−0.454974	+	0.890505i	$0.650351\pi$
$774$	0	0
$775$	1.91125e10	1.47489
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.47879e9	−0.718409
$780$	0	0
$781$	−4.91497e9	−0.369183
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.26540e8	0.0462280
$786$	0	0
$787$	1.91531e10	1.40064	0.700320	−	0.713829i	$-0.253040\pi$
$787$	1.91531e10	1.40064	0.700320	+	0.713829i	$0.253040\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.39035e9	−0.0998866
$792$	0	0
$793$	3.26211e10	2.32296
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.18466e10	1.52855	0.764275	−	0.644891i	$-0.223097\pi$
$797$	2.18466e10	1.52855	0.764275	+	0.644891i	$0.223097\pi$
$798$	0	0
$799$	−1.12655e10	−0.781336
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.69214e9	0.387946
$804$	0	0
$805$	−1.61153e7	−0.00108881
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.24627e9	−0.481166	−0.240583	−	0.970629i	$-0.577339\pi$
$809$	−7.24627e9	−0.481166	−0.240583	+	0.970629i	$0.577339\pi$
$810$	0	0
$811$	2.41465e10	1.58958	0.794789	−	0.606886i	$-0.207582\pi$
$811$	2.41465e10	1.58958	0.794789	+	0.606886i	$0.207582\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.19618e8	0.0206814
$816$	0	0
$817$	1.56441e10	1.00363
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.40199e10	1.51485	0.757424	−	0.652923i	$-0.226457\pi$
$821$	2.40199e10	1.51485	0.757424	+	0.652923i	$0.226457\pi$
$822$	0	0
$823$	8.52444e9	0.533048	0.266524	−	0.963828i	$-0.414125\pi$
$823$	8.52444e9	0.533048	0.266524	+	0.963828i	$0.414125\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.13365e9	−0.0696965	−0.0348483	−	0.999393i	$-0.511095\pi$
$827$	−1.13365e9	−0.0696965	−0.0348483	+	0.999393i	$0.511095\pi$
$828$	0	0
$829$	2.87884e10	1.75500	0.877498	−	0.479580i	$-0.159211\pi$
$829$	2.87884e10	1.75500	0.877498	+	0.479580i	$0.159211\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.79038e10	1.67265
$834$	0	0
$835$	−8.94394e8	−0.0531651
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.77071e10	1.61966	0.809830	−	0.586664i	$-0.199559\pi$
$839$	2.77071e10	1.61966	0.809830	+	0.586664i	$0.199559\pi$
$840$	0	0
$841$	1.98810e9	0.115253
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.73895e9	−0.0991491
$846$	0	0
$847$	−1.95794e9	−0.110716
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.21456e9	−0.234422
$852$	0	0
$853$	−1.19666e10	−0.660159	−0.330080	−	0.943953i	$-0.607076\pi$
$853$	−1.19666e10	−0.660159	−0.330080	+	0.943953i	$0.607076\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.93427e10	−1.04975	−0.524873	−	0.851180i	$-0.675887\pi$
$857$	−1.93427e10	−1.04975	−0.524873	+	0.851180i	$0.675887\pi$
$858$	0	0
$859$	−1.03032e10	−0.554621	−0.277310	−	0.960780i	$-0.589443\pi$
$859$	−1.03032e10	−0.554621	−0.277310	+	0.960780i	$0.589443\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.18771e9	0.433635	0.216818	−	0.976212i	$-0.430432\pi$
$863$	8.18771e9	0.433635	0.216818	+	0.976212i	$0.430432\pi$
$864$	0	0
$865$	−6.57860e8	−0.0345603
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.45335e9	−0.385285
$870$	0	0
$871$	3.08675e9	0.158284
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.25490e8	−0.00633259
$876$	0	0
$877$	−3.72988e10	−1.86723	−0.933613	−	0.358284i	$-0.883362\pi$
$877$	−3.72988e10	−1.86723	−0.933613	+	0.358284i	$0.883362\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.03129e10	−1.00082	−0.500410	−	0.865789i	$-0.666817\pi$
$881$	−2.03129e10	−1.00082	−0.500410	+	0.865789i	$0.666817\pi$
$882$	0	0
$883$	−1.32308e10	−0.646729	−0.323365	−	0.946274i	$-0.604814\pi$
$883$	−1.32308e10	−0.646729	−0.323365	+	0.946274i	$0.604814\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.66742e10	0.802256	0.401128	−	0.916022i	$-0.368618\pi$
$887$	1.66742e10	0.802256	0.401128	+	0.916022i	$0.368618\pi$
$888$	0	0
$889$	−1.49895e9	−0.0715534
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.42194e10	0.668194
$894$	0	0
$895$	−1.79875e9	−0.0838667
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.40153e10	−1.56140
$900$	0	0
$901$	−5.28049e10	−2.40512
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.47227e9	−0.0660265
$906$	0	0
$907$	2.61012e10	1.16154	0.580772	−	0.814066i	$-0.302751\pi$
$907$	2.61012e10	1.16154	0.580772	+	0.814066i	$0.302751\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.90651e9	0.390295	0.195148	−	0.980774i	$-0.437481\pi$
$911$	8.90651e9	0.390295	0.195148	+	0.980774i	$0.437481\pi$
$912$	0	0
$913$	7.28216e10	3.16674
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.97523e8	0.0341547
$918$	0	0
$919$	2.06931e10	0.879469	0.439734	−	0.898128i	$-0.355073\pi$
$919$	2.06931e10	0.879469	0.439734	+	0.898128i	$0.355073\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.24690e9	0.387071
$924$	0	0
$925$	−1.63892e10	−0.680867
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.34441e10	−0.959352	−0.479676	−	0.877446i	$-0.659246\pi$
$929$	−2.34441e10	−0.959352	−0.479676	+	0.877446i	$0.659246\pi$
$930$	0	0
$931$	−3.52204e10	−1.43044
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.43653e9	−0.137493
$936$	0	0
$937$	3.30493e10	1.31242	0.656212	−	0.754577i	$-0.272158\pi$
$937$	3.30493e10	1.31242	0.656212	+	0.754577i	$0.272158\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.64769e10	−1.42710	−0.713550	−	0.700604i	$-0.752914\pi$
$941$	−3.64769e10	−1.42710	−0.713550	+	0.700604i	$0.752914\pi$
$942$	0	0
$943$	4.42363e9	0.171786
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.25570e9	0.0480465	0.0240232	−	0.999711i	$-0.492352\pi$
$947$	1.25570e9	0.0480465	0.0240232	+	0.999711i	$0.492352\pi$
$948$	0	0
$949$	−1.07091e10	−0.406742
$950$	0	0
$951$	0	0
$952$	0	0
$953$	7.65511e9	0.286501	0.143250	−	0.989686i	$-0.454245\pi$
$953$	7.65511e9	0.286501	0.143250	+	0.989686i	$0.454245\pi$
$954$	0	0
$955$	7.84931e8	0.0291622
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.86068e8	−0.0214577
$960$	0	0
$961$	3.26309e10	1.18603
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.73968e8	−0.0241431
$966$	0	0
$967$	4.60711e9	0.163846	0.0819230	−	0.996639i	$-0.473894\pi$
$967$	4.60711e9	0.163846	0.0819230	+	0.996639i	$0.473894\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.40647e10	1.89516	0.947581	−	0.319516i	$-0.103520\pi$
$971$	5.40647e10	1.89516	0.947581	+	0.319516i	$0.103520\pi$
$972$	0	0
$973$	−2.03559e9	−0.0708427
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.27015e10	−1.12186	−0.560928	−	0.827865i	$-0.689556\pi$
$977$	−3.27015e10	−1.12186	−0.560928	+	0.827865i	$0.689556\pi$
$978$	0	0
$979$	−6.30683e10	−2.14819
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.71354e9	0.225431	0.112716	−	0.993627i	$-0.464045\pi$
$983$	6.71354e9	0.225431	0.112716	+	0.993627i	$0.464045\pi$
$984$	0	0
$985$	−2.15209e9	−0.0717519
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.30091e9	−0.239988
$990$	0	0
$991$	−1.63296e10	−0.532987	−0.266494	−	0.963837i	$-0.585865\pi$
$991$	−1.63296e10	−0.532987	−0.266494	+	0.963837i	$0.585865\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.69491e9	−0.0867288
$996$	0	0
$997$	−4.89868e10	−1.56548	−0.782738	−	0.622352i	$-0.786177\pi$
$997$	−4.89868e10	−1.56548	−0.782738	+	0.622352i	$0.786177\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.8.a.y.1.3		4
3.2	odd	2		432.8.a.x.1.2		4
4.3	odd	2		216.8.a.g.1.3	yes	4
12.11	even	2		216.8.a.f.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.8.a.f.1.2	✓	4	12.11	even	2
216.8.a.g.1.3	yes	4	4.3	odd	2
432.8.a.x.1.2		4	3.2	odd	2
432.8.a.y.1.3		4	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	432.a (trivial)

\(f(q)\)	\(=\)	\(q-13.8479 q^{5} -58.0683 q^{7} -7294.19 q^{11} +13723.1 q^{13} -34022.0 q^{17} +42942.8 q^{19} -20040.8 q^{23} -77933.2 q^{25} +138701. q^{29} -245242. q^{31} +804.123 q^{35} +210298. q^{37} -220731. q^{41} +364301. q^{43} +331125. q^{47} -820171. q^{49} +1.55208e6 q^{53} +101009. q^{55} -2.51316e6 q^{59} +2.37709e6 q^{61} -190036. q^{65} +224931. q^{67} +673820. q^{71} -780367. q^{73} +423561. q^{77} +1.02182e6 q^{79} -9.98352e6 q^{83} +471133. q^{85} +8.64638e6 q^{89} -796877. q^{91} -594667. q^{95} -6.34072e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 104 q^{5} + 492 q^{7} + 2104 q^{11} + 2356 q^{13} - 4136 q^{17} - 5516 q^{19} + 17848 q^{23} + 66476 q^{25} + 150720 q^{29} + 78256 q^{31} - 195432 q^{35} - 42324 q^{37} + 280704 q^{41} + 51200 q^{43}+ \cdots + 9596660 q^{97}+O(q^{100}) \) 4 * q + 104 * q^5 + 492 * q^7 + 2104 * q^11 + 2356 * q^13 - 4136 * q^17 - 5516 * q^19 + 17848 * q^23 + 66476 * q^25 + 150720 * q^29 + 78256 * q^31 - 195432 * q^35 - 42324 * q^37 + 280704 * q^41 + 51200 * q^43 - 829608 * q^47 + 13256 * q^49 + 1763120 * q^53 + 519968 * q^55 - 1881992 * q^59 + 869116 * q^61 + 5534696 * q^65 + 249700 * q^67 - 9344848 * q^71 - 461660 * q^73 + 11536584 * q^77 + 936116 * q^79 - 15826384 * q^83 + 5871776 * q^85 + 22823736 * q^89 + 2375484 * q^91 - 32094136 * q^95 + 9596660 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more