LMFDB - Embedded newform 800.6.f.c.49.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(49,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$128.307055850$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 $ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{93}\cdot 3^{4}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.14
Root			$3.72553 + 1.45618i$ of defining polynomial
Character	$\chi$	$=$	800.49
Dual form			800.6.f.c.49.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.8240 q^{3} +163.706i q^{7} -125.841 q^{9} -321.520i q^{11} -128.246 q^{13} +2110.72i q^{17} -1454.37i q^{19} +1771.96i q^{21} +1231.18i q^{23} -3992.34 q^{27} -4073.19i q^{29} +3956.03 q^{31} -3480.14i q^{33} +10656.6 q^{37} -1388.13 q^{39} -5907.19 q^{41} -16439.6 q^{43} +23238.8i q^{47} -9992.68 q^{49} +22846.5i q^{51} -30634.0 q^{53} -15742.1i q^{57} -25262.4i q^{59} -39115.5i q^{61} -20600.9i q^{63} -20894.5 q^{67} +13326.3i q^{69} -13889.1 q^{71} -43451.2i q^{73} +52634.8 q^{77} -12546.4 q^{79} -12633.8 q^{81} -6680.84 q^{83} -44088.2i q^{87} +90400.9 q^{89} -20994.6i q^{91} +42820.2 q^{93} -149616. i q^{97} +40460.4i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 36 q^{3} + 1620 q^{9} + 11664 q^{27} - 7160 q^{31} + 3608 q^{37} - 44904 q^{39} + 11608 q^{41} - 51772 q^{43} - 18756 q^{49} - 928 q^{53} - 161604 q^{67} + 200312 q^{71} - 26008 q^{77} + 282080 q^{79}+ \cdots - 293472 q^{93}+O(q^{100}) $ 20 * q + 36 * q^3 + 1620 * q^9 + 11664 * q^27 - 7160 * q^31 + 3608 * q^37 - 44904 * q^39 + 11608 * q^41 - 51772 * q^43 - 18756 * q^49 - 928 * q^53 - 161604 * q^67 + 200312 * q^71 - 26008 * q^77 + 282080 * q^79 + 65172 * q^81 - 99092 * q^83 + 3160 * q^89 - 293472 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	10.8240	0.694361	0.347180	−	0.937798i	$-0.387139\pi$
$3$	10.8240	0.694361	0.347180	+	0.937798i	$0.387139\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	163.706i	1.26276i	0.775475	+	0.631378i	$0.217510\pi$
$7$	163.706i	1.26276i	−0.775475	+	0.631378i	$0.782490\pi$
$8$	0	0
$9$	−125.841	−0.517863
$10$	0	0
$11$	− 321.520i	− 0.801174i	−0.916259	−	0.400587i	$-0.868806\pi$
$11$	− 321.520i	− 0.801174i	0.916259	−	0.400587i	$-0.131194\pi$
$12$	0	0
$13$	−128.246	−0.210467	−0.105233	−	0.994448i	$-0.533559\pi$
$13$	−128.246	−0.210467	−0.105233	+	0.994448i	$0.533559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2110.72i	1.77137i	0.464290	+	0.885683i	$0.346310\pi$
$17$	2110.72i	1.77137i	−0.464290	+	0.885683i	$0.653690\pi$
$18$	0	0
$19$	− 1454.37i	− 0.924252i	−0.886814	−	0.462126i	$-0.847087\pi$
$19$	− 1454.37i	− 0.924252i	0.886814	−	0.462126i	$-0.152913\pi$
$20$	0	0
$21$	1771.96i	0.876809i
$22$	0	0
$23$	1231.18i	0.485289i	0.970115	+	0.242645i	$0.0780149\pi$
$23$	1231.18i	0.485289i	−0.970115	+	0.242645i	$0.921985\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3992.34	−1.05394
$28$	0	0
$29$	− 4073.19i	− 0.899372i	−0.893187	−	0.449686i	$-0.851536\pi$
$29$	− 4073.19i	− 0.899372i	0.893187	−	0.449686i	$-0.148464\pi$
$30$	0	0
$31$	3956.03	0.739360	0.369680	−	0.929159i	$-0.379467\pi$
$31$	3956.03	0.739360	0.369680	+	0.929159i	$0.379467\pi$
$32$	0	0
$33$	− 3480.14i	− 0.556304i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10656.6	1.27972	0.639861	−	0.768490i	$-0.278992\pi$
$37$	10656.6	1.27972	0.639861	+	0.768490i	$0.278992\pi$
$38$	0	0
$39$	−1388.13	−0.146140
$40$	0	0
$41$	−5907.19	−0.548809	−0.274404	−	0.961614i	$-0.588481\pi$
$41$	−5907.19	−0.548809	−0.274404	+	0.961614i	$0.588481\pi$
$42$	0	0
$43$	−16439.6	−1.35588	−0.677938	−	0.735119i	$-0.737126\pi$
$43$	−16439.6	−1.35588	−0.677938	+	0.735119i	$0.737126\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	23238.8i	1.53450i	0.641345	+	0.767252i	$0.278377\pi$
$47$	23238.8i	1.53450i	−0.641345	+	0.767252i	$0.721623\pi$
$48$	0	0
$49$	−9992.68	−0.594555
$50$	0	0
$51$	22846.5i	1.22997i
$52$	0	0
$53$	−30634.0	−1.49801	−0.749003	−	0.662567i	$-0.769467\pi$
$53$	−30634.0	−1.49801	−0.749003	+	0.662567i	$0.769467\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 15742.1i	− 0.641764i
$58$	0	0
$59$	− 25262.4i	− 0.944810i	−0.881382	−	0.472405i	$-0.843386\pi$
$59$	− 25262.4i	− 0.944810i	0.881382	−	0.472405i	$-0.156614\pi$
$60$	0	0
$61$	− 39115.5i	− 1.34594i	−0.739672	−	0.672968i	$-0.765019\pi$
$61$	− 39115.5i	− 1.34594i	0.739672	−	0.672968i	$-0.234981\pi$
$62$	0	0
$63$	− 20600.9i	− 0.653935i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−20894.5	−0.568651	−0.284325	−	0.958728i	$-0.591770\pi$
$67$	−20894.5	−0.568651	−0.284325	+	0.958728i	$0.591770\pi$
$68$	0	0
$69$	13326.3i	0.336966i
$70$	0	0
$71$	−13889.1	−0.326984	−0.163492	−	0.986545i	$-0.552276\pi$
$71$	−13889.1	−0.326984	−0.163492	+	0.986545i	$0.552276\pi$
$72$	0	0
$73$	− 43451.2i	− 0.954321i	−0.878816	−	0.477161i	$-0.841666\pi$
$73$	− 43451.2i	− 0.954321i	0.878816	−	0.477161i	$-0.158334\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	52634.8	1.01169
$78$	0	0
$79$	−12546.4	−0.226179	−0.113089	−	0.993585i	$-0.536075\pi$
$79$	−12546.4	−0.226179	−0.113089	+	0.993585i	$0.536075\pi$
$80$	0	0
$81$	−12633.8	−0.213955
$82$	0	0
$83$	−6680.84	−0.106448	−0.0532238	−	0.998583i	$-0.516950\pi$
$83$	−6680.84	−0.106448	−0.0532238	+	0.998583i	$0.516950\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 44088.2i	− 0.624488i
$88$	0	0
$89$	90400.9	1.20976	0.604878	−	0.796318i	$-0.293222\pi$
$89$	90400.9	1.20976	0.604878	+	0.796318i	$0.293222\pi$
$90$	0	0
$91$	− 20994.6i	− 0.265769i
$92$	0	0
$93$	42820.2	0.513382
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 149616.i	− 1.61454i	−0.590184	−	0.807269i	$-0.700945\pi$
$97$	− 149616.i	− 1.61454i	0.590184	−	0.807269i	$-0.299055\pi$
$98$	0	0
$99$	40460.4i	0.414898i
$100$	0	0
$101$	− 114822.i	− 1.12001i	−0.828491	−	0.560003i	$-0.810800\pi$
$101$	− 114822.i	− 1.12001i	0.828491	−	0.560003i	$-0.189200\pi$
$102$	0	0
$103$	− 38586.9i	− 0.358382i	−0.983814	−	0.179191i	$-0.942652\pi$
$103$	− 38586.9i	− 0.358382i	0.983814	−	0.179191i	$-0.0573481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−189459.	−1.59976	−0.799881	−	0.600158i	$-0.795104\pi$
$107$	−189459.	−1.59976	−0.799881	+	0.600158i	$0.795104\pi$
$108$	0	0
$109$	24392.6i	0.196649i	0.995154	+	0.0983245i	$0.0313483\pi$
$109$	24392.6i	0.196649i	−0.995154	+	0.0983245i	$0.968652\pi$
$110$	0	0
$111$	115348.	0.888589
$112$	0	0
$113$	− 52918.6i	− 0.389863i	−0.980817	−	0.194932i	$-0.937552\pi$
$113$	− 52918.6i	− 0.389863i	0.980817	−	0.194932i	$-0.0624485\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16138.5	0.108993
$118$	0	0
$119$	−345538.	−2.23681
$120$	0	0
$121$	57675.7	0.358120
$122$	0	0
$123$	−63939.5	−0.381071
$124$	0	0
$125$	0	0
$126$	0	0
$127$	293650.i	1.61555i	0.589489	+	0.807777i	$0.299329\pi$
$127$	293650.i	1.61555i	−0.589489	+	0.807777i	$0.700671\pi$
$128$	0	0
$129$	−177942.	−0.941467
$130$	0	0
$131$	− 317738.i	− 1.61767i	−0.588032	−	0.808837i	$-0.700097\pi$
$131$	− 317738.i	− 1.61767i	0.588032	−	0.808837i	$-0.299903\pi$
$132$	0	0
$133$	238089.	1.16711
$134$	0	0
$135$	0	0
$136$	0	0
$137$	162285.i	0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$137$	162285.i	0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$138$	0	0
$139$	− 306759.i	− 1.34667i	−0.739338	−	0.673334i	$-0.764861\pi$
$139$	− 306759.i	− 1.34667i	0.739338	−	0.673334i	$-0.235139\pi$
$140$	0	0
$141$	251537.i	1.06550i
$142$	0	0
$143$	41233.5i	0.168621i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−108161.	−0.412835
$148$	0	0
$149$	368107.i	1.35834i	0.733981	+	0.679170i	$0.237660\pi$
$149$	368107.i	1.35834i	−0.733981	+	0.679170i	$0.762340\pi$
$150$	0	0
$151$	−336822.	−1.20215	−0.601075	−	0.799193i	$-0.705261\pi$
$151$	−336822.	−1.20215	−0.601075	+	0.799193i	$0.705261\pi$
$152$	0	0
$153$	− 265615.i	− 0.917326i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−271253.	−0.878264	−0.439132	−	0.898423i	$-0.644714\pi$
$157$	−271253.	−0.878264	−0.439132	+	0.898423i	$0.644714\pi$
$158$	0	0
$159$	−331582.	−1.04016
$160$	0	0
$161$	−201551.	−0.612802
$162$	0	0
$163$	121354.	0.357756	0.178878	−	0.983871i	$-0.442753\pi$
$163$	121354.	0.357756	0.178878	+	0.983871i	$0.442753\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	56095.4i	0.155645i	0.996967	+	0.0778227i	$0.0247968\pi$
$167$	56095.4i	0.155645i	−0.996967	+	0.0778227i	$0.975203\pi$
$168$	0	0
$169$	−354846.	−0.955704
$170$	0	0
$171$	183019.i	0.478636i
$172$	0	0
$173$	167666.	0.425921	0.212960	−	0.977061i	$-0.431689\pi$
$173$	167666.	0.425921	0.212960	+	0.977061i	$0.431689\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 273440.i	− 0.656039i
$178$	0	0
$179$	− 535921.i	− 1.25017i	−0.780558	−	0.625083i	$-0.785065\pi$
$179$	− 535921.i	− 1.25017i	0.780558	−	0.625083i	$-0.214935\pi$
$180$	0	0
$181$	113446.i	0.257391i	0.991684	+	0.128696i	$0.0410790\pi$
$181$	113446.i	0.257391i	−0.991684	+	0.128696i	$0.958921\pi$
$182$	0	0
$183$	− 423387.i	− 0.934565i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	678640.	1.41917
$188$	0	0
$189$	− 653570.i	− 1.33088i
$190$	0	0
$191$	−391808.	−0.777124	−0.388562	−	0.921423i	$-0.627028\pi$
$191$	−391808.	−0.777124	−0.388562	+	0.921423i	$0.627028\pi$
$192$	0	0
$193$	− 261513.i	− 0.505359i	−0.967550	−	0.252680i	$-0.918688\pi$
$193$	− 261513.i	− 0.505359i	0.967550	−	0.252680i	$-0.0813118\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	898947.	1.65032	0.825161	−	0.564898i	$-0.191084\pi$
$197$	898947.	1.65032	0.825161	+	0.564898i	$0.191084\pi$
$198$	0	0
$199$	−83719.3	−0.149862	−0.0749312	−	0.997189i	$-0.523874\pi$
$199$	−83719.3	−0.149862	−0.0749312	+	0.997189i	$0.523874\pi$
$200$	0	0
$201$	−226163.	−0.394849
$202$	0	0
$203$	666805.	1.13569
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 154932.i	− 0.251313i
$208$	0	0
$209$	−467609.	−0.740487
$210$	0	0
$211$	553641.i	0.856095i	0.903756	+	0.428047i	$0.140798\pi$
$211$	553641.i	0.856095i	−0.903756	+	0.428047i	$0.859202\pi$
$212$	0	0
$213$	−150335.	−0.227045
$214$	0	0
$215$	0	0
$216$	0	0
$217$	647627.i	0.933632i
$218$	0	0
$219$	− 470316.i	− 0.662643i
$220$	0	0
$221$	− 270691.i	− 0.372814i
$222$	0	0
$223$	− 279400.i	− 0.376239i	−0.982146	−	0.188120i	$-0.939761\pi$
$223$	− 279400.i	− 0.376239i	0.982146	−	0.188120i	$-0.0602392\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−593068.	−0.763905	−0.381953	−	0.924182i	$-0.624748\pi$
$227$	−593068.	−0.763905	−0.381953	+	0.924182i	$0.624748\pi$
$228$	0	0
$229$	− 927873.i	− 1.16923i	−0.811311	−	0.584615i	$-0.801246\pi$
$229$	− 927873.i	− 1.16923i	0.811311	−	0.584615i	$-0.198754\pi$
$230$	0	0
$231$	569720.	0.702476
$232$	0	0
$233$	− 1.09279e6i	− 1.31871i	−0.751833	−	0.659353i	$-0.770830\pi$
$233$	− 1.09279e6i	− 1.31871i	0.751833	−	0.659353i	$-0.229170\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−135803.	−0.157050
$238$	0	0
$239$	−797967.	−0.903630	−0.451815	−	0.892112i	$-0.649223\pi$
$239$	−797967.	−0.903630	−0.451815	+	0.892112i	$0.649223\pi$
$240$	0	0
$241$	−1.61861e6	−1.79515	−0.897573	−	0.440865i	$-0.854672\pi$
$241$	−1.61861e6	−1.79515	−0.897573	+	0.440865i	$0.854672\pi$
$242$	0	0
$243$	833389.	0.905383
$244$	0	0
$245$	0	0
$246$	0	0
$247$	186516.i	0.194525i
$248$	0	0
$249$	−72313.5	−0.0739130
$250$	0	0
$251$	449688.i	0.450533i	0.974297	+	0.225266i	$0.0723253\pi$
$251$	449688.i	0.450533i	−0.974297	+	0.225266i	$0.927675\pi$
$252$	0	0
$253$	395848.	0.388801
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 396434.i	− 0.374402i	−0.982322	−	0.187201i	$-0.940058\pi$
$257$	− 396434.i	− 0.374402i	0.982322	−	0.187201i	$-0.0599416\pi$
$258$	0	0
$259$	1.74456e6i	1.61598i
$260$	0	0
$261$	512573.i	0.465752i
$262$	0	0
$263$	− 1.93423e6i	− 1.72432i	−0.506635	−	0.862160i	$-0.669111\pi$
$263$	− 1.93423e6i	− 1.72432i	0.506635	−	0.862160i	$-0.330889\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	978500.	0.840007
$268$	0	0
$269$	− 670016.i	− 0.564553i	−0.959333	−	0.282276i	$-0.908910\pi$
$269$	− 670016.i	− 0.564553i	0.959333	−	0.282276i	$-0.0910895\pi$
$270$	0	0
$271$	−2.08940e6	−1.72822	−0.864109	−	0.503305i	$-0.832117\pi$
$271$	−2.08940e6	−1.72822	−0.864109	+	0.503305i	$0.832117\pi$
$272$	0	0
$273$	− 227246.i	− 0.184539i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−828834.	−0.649035	−0.324517	−	0.945880i	$-0.605202\pi$
$277$	−828834.	−0.649035	−0.324517	+	0.945880i	$0.605202\pi$
$278$	0	0
$279$	−497830.	−0.382887
$280$	0	0
$281$	2.39932e6	1.81268	0.906342	−	0.422545i	$-0.138863\pi$
$281$	2.39932e6	1.81268	0.906342	+	0.422545i	$0.138863\pi$
$282$	0	0
$283$	−2.00868e6	−1.49089	−0.745444	−	0.666568i	$-0.767763\pi$
$283$	−2.00868e6	−1.49089	−0.745444	+	0.666568i	$0.767763\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 967042.i	− 0.693012i
$288$	0	0
$289$	−3.03529e6	−2.13774
$290$	0	0
$291$	− 1.61944e6i	− 1.12107i
$292$	0	0
$293$	1.74203e6	1.18546	0.592729	−	0.805402i	$-0.298050\pi$
$293$	1.74203e6	1.18546	0.592729	+	0.805402i	$0.298050\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.28362e6i	0.844393i
$298$	0	0
$299$	− 157893.i	− 0.102137i
$300$	0	0
$301$	− 2.69126e6i	− 1.71214i
$302$	0	0
$303$	− 1.24283e6i	− 0.777688i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−978690.	−0.592651	−0.296326	−	0.955087i	$-0.595761\pi$
$307$	−978690.	−0.592651	−0.296326	+	0.955087i	$0.595761\pi$
$308$	0	0
$309$	− 417665.i	− 0.248847i
$310$	0	0
$311$	1.57652e6	0.924268	0.462134	−	0.886810i	$-0.347084\pi$
$311$	1.57652e6	0.924268	0.462134	+	0.886810i	$0.347084\pi$
$312$	0	0
$313$	1.60962e6i	0.928670i	0.885660	+	0.464335i	$0.153707\pi$
$313$	1.60962e6i	0.928670i	−0.885660	+	0.464335i	$0.846293\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.50670e6	−0.842130	−0.421065	−	0.907030i	$-0.638344\pi$
$317$	−1.50670e6	−0.842130	−0.421065	+	0.907030i	$0.638344\pi$
$318$	0	0
$319$	−1.30961e6	−0.720553
$320$	0	0
$321$	−2.05070e6	−1.11081
$322$	0	0
$323$	3.06977e6	1.63719
$324$	0	0
$325$	0	0
$326$	0	0
$327$	264026.i	0.136545i
$328$	0	0
$329$	−3.80433e6	−1.93771
$330$	0	0
$331$	394453.i	0.197891i	0.995093	+	0.0989454i	$0.0315469\pi$
$331$	394453.i	0.197891i	−0.995093	+	0.0989454i	$0.968453\pi$
$332$	0	0
$333$	−1.34104e6	−0.662721
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.36879e6i	0.656543i	0.944583	+	0.328272i	$0.106466\pi$
$337$	1.36879e6i	0.656543i	−0.944583	+	0.328272i	$0.893534\pi$
$338$	0	0
$339$	− 572791.i	− 0.270706i
$340$	0	0
$341$	− 1.27195e6i	− 0.592356i
$342$	0	0
$343$	1.11555e6i	0.511979i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−569376.	−0.253849	−0.126925	−	0.991912i	$-0.540511\pi$
$347$	−569376.	−0.253849	−0.126925	+	0.991912i	$0.540511\pi$
$348$	0	0
$349$	− 2.20053e6i	− 0.967081i	−0.875322	−	0.483541i	$-0.839351\pi$
$349$	− 2.20053e6i	− 0.967081i	0.875322	−	0.483541i	$-0.160649\pi$
$350$	0	0
$351$	511999.	0.221820
$352$	0	0
$353$	− 2.51557e6i	− 1.07448i	−0.843428	−	0.537242i	$-0.819466\pi$
$353$	− 2.51557e6i	− 1.07448i	0.843428	−	0.537242i	$-0.180534\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.74011e6	−1.55315
$358$	0	0
$359$	−794808.	−0.325481	−0.162741	−	0.986669i	$-0.552033\pi$
$359$	−794808.	−0.325481	−0.162741	+	0.986669i	$0.552033\pi$
$360$	0	0
$361$	360911.	0.145758
$362$	0	0
$363$	624282.	0.248665
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 874302.i	− 0.338841i	−0.985544	−	0.169421i	$-0.945810\pi$
$367$	− 874302.i	− 0.338841i	0.985544	−	0.169421i	$-0.0541896\pi$
$368$	0	0
$369$	743365.	0.284208
$370$	0	0
$371$	− 5.01497e6i	− 1.89162i
$372$	0	0
$373$	−4.86205e6	−1.80945	−0.904726	−	0.425994i	$-0.859924\pi$
$373$	−4.86205e6	−1.80945	−0.904726	+	0.425994i	$0.859924\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	522368.i	0.189288i
$378$	0	0
$379$	− 1.24150e6i	− 0.443964i	−0.975051	−	0.221982i	$-0.928747\pi$
$379$	− 1.24150e6i	− 0.443964i	0.975051	−	0.221982i	$-0.0712527\pi$
$380$	0	0
$381$	3.17848e6i	1.12178i
$382$	0	0
$383$	30969.9i	0.0107880i	0.999985	+	0.00539402i	$0.00171698\pi$
$383$	30969.9i	0.0107880i	−0.999985	+	0.00539402i	$0.998283\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.06877e6	0.702158
$388$	0	0
$389$	4.27860e6i	1.43360i	0.697279	+	0.716800i	$0.254394\pi$
$389$	4.27860e6i	1.43360i	−0.697279	+	0.716800i	$0.745606\pi$
$390$	0	0
$391$	−2.59867e6	−0.859626
$392$	0	0
$393$	− 3.43920e6i	− 1.12325i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.31119e6	0.735968	0.367984	−	0.929832i	$-0.380048\pi$
$397$	2.31119e6	0.735968	0.367984	+	0.929832i	$0.380048\pi$
$398$	0	0
$399$	2.57708e6	0.810392
$400$	0	0
$401$	996347.	0.309421	0.154710	−	0.987960i	$-0.450556\pi$
$401$	996347.	0.309421	0.154710	+	0.987960i	$0.450556\pi$
$402$	0	0
$403$	−507344.	−0.155611
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.42633e6i	− 1.02528i
$408$	0	0
$409$	5.17948e6	1.53101	0.765505	−	0.643430i	$-0.222489\pi$
$409$	5.17948e6	1.53101	0.765505	+	0.643430i	$0.222489\pi$
$410$	0	0
$411$	1.75658e6i	0.512936i
$412$	0	0
$413$	4.13561e6	1.19306
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 3.32037e6i	− 0.935074i
$418$	0	0
$419$	3.57698e6i	0.995363i	0.867360	+	0.497682i	$0.165815\pi$
$419$	3.57698e6i	0.995363i	−0.867360	+	0.497682i	$0.834185\pi$
$420$	0	0
$421$	2.15848e6i	0.593531i	0.954950	+	0.296765i	$0.0959080\pi$
$421$	2.15848e6i	0.593531i	−0.954950	+	0.296765i	$0.904092\pi$
$422$	0	0
$423$	− 2.92438e6i	− 0.794664i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.40344e6	1.69959
$428$	0	0
$429$	446312.i	0.117084i
$430$	0	0
$431$	1.80890e6	0.469052	0.234526	−	0.972110i	$-0.424646\pi$
$431$	1.80890e6	0.469052	0.234526	+	0.972110i	$0.424646\pi$
$432$	0	0
$433$	1.28911e6i	0.330423i	0.986258	+	0.165211i	$0.0528306\pi$
$433$	1.28911e6i	0.330423i	−0.986258	+	0.165211i	$0.947169\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.79058e6	0.448530
$438$	0	0
$439$	1.89694e6	0.469777	0.234889	−	0.972022i	$-0.424527\pi$
$439$	1.89694e6	0.469777	0.234889	+	0.972022i	$0.424527\pi$
$440$	0	0
$441$	1.25749e6	0.307898
$442$	0	0
$443$	−6.01979e6	−1.45738	−0.728689	−	0.684845i	$-0.759870\pi$
$443$	−6.01979e6	−1.45738	−0.728689	+	0.684845i	$0.759870\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.98440e6i	0.943178i
$448$	0	0
$449$	2.40081e6	0.562007	0.281003	−	0.959707i	$-0.409333\pi$
$449$	2.40081e6	0.562007	0.281003	+	0.959707i	$0.409333\pi$
$450$	0	0
$451$	1.89928e6i	0.439691i
$452$	0	0
$453$	−3.64577e6	−0.834726
$454$	0	0
$455$	0	0
$456$	0	0
$457$	858952.i	0.192388i	0.995363	+	0.0961941i	$0.0306669\pi$
$457$	858952.i	0.192388i	−0.995363	+	0.0961941i	$0.969333\pi$
$458$	0	0
$459$	− 8.42671e6i	− 1.86692i
$460$	0	0
$461$	2.33481e6i	0.511680i	0.966719	+	0.255840i	$0.0823521\pi$
$461$	2.33481e6i	0.511680i	−0.966719	+	0.255840i	$0.917648\pi$
$462$	0	0
$463$	1.35195e6i	0.293096i	0.989204	+	0.146548i	$0.0468162\pi$
$463$	1.35195e6i	0.293096i	−0.989204	+	0.146548i	$0.953184\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.44013e6	−1.36648	−0.683239	−	0.730195i	$-0.739429\pi$
$467$	−6.44013e6	−1.36648	−0.683239	+	0.730195i	$0.739429\pi$
$468$	0	0
$469$	− 3.42056e6i	− 0.718068i
$470$	0	0
$471$	−2.93604e6	−0.609832
$472$	0	0
$473$	5.28566e6i	1.08629i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.85500e6	0.775762
$478$	0	0
$479$	−1.85358e6	−0.369124	−0.184562	−	0.982821i	$-0.559087\pi$
$479$	−1.85358e6	−0.369124	−0.184562	+	0.982821i	$0.559087\pi$
$480$	0	0
$481$	−1.36667e6	−0.269339
$482$	0	0
$483$	−2.18159e6	−0.425506
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.51668e6i	0.480846i	0.970668	+	0.240423i	$0.0772861\pi$
$487$	2.51668e6i	0.480846i	−0.970668	+	0.240423i	$0.922714\pi$
$488$	0	0
$489$	1.31354e6	0.248412
$490$	0	0
$491$	5.39115e6i	1.00920i	0.863353	+	0.504601i	$0.168360\pi$
$491$	5.39115e6i	1.00920i	−0.863353	+	0.504601i	$0.831640\pi$
$492$	0	0
$493$	8.59736e6	1.59312
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.27372e6i	− 0.412902i
$498$	0	0
$499$	3.18358e6i	0.572353i	0.958177	+	0.286177i	$0.0923844\pi$
$499$	3.18358e6i	0.572353i	−0.958177	+	0.286177i	$0.907616\pi$
$500$	0	0
$501$	607177.i	0.108074i
$502$	0	0
$503$	8.99291e6i	1.58482i	0.609989	+	0.792410i	$0.291174\pi$
$503$	8.99291e6i	1.58482i	−0.609989	+	0.792410i	$0.708826\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.84086e6	−0.663603
$508$	0	0
$509$	5.35388e6i	0.915956i	0.888964	+	0.457978i	$0.151426\pi$
$509$	5.35388e6i	0.915956i	−0.888964	+	0.457978i	$0.848574\pi$
$510$	0	0
$511$	7.11322e6	1.20508
$512$	0	0
$513$	5.80633e6i	0.974111i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.47173e6	1.22941
$518$	0	0
$519$	1.81481e6	0.295743
$520$	0	0
$521$	−2.62401e6	−0.423518	−0.211759	−	0.977322i	$-0.567919\pi$
$521$	−2.62401e6	−0.423518	−0.211759	+	0.977322i	$0.567919\pi$
$522$	0	0
$523$	−1.31228e6	−0.209784	−0.104892	−	0.994484i	$-0.533450\pi$
$523$	−1.31228e6	−0.209784	−0.104892	+	0.994484i	$0.533450\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.35008e6i	1.30968i
$528$	0	0
$529$	4.92055e6	0.764494
$530$	0	0
$531$	3.17904e6i	0.489282i
$532$	0	0
$533$	757570.	0.115506
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 5.80081e6i	− 0.868067i
$538$	0	0
$539$	3.21285e6i	0.476342i
$540$	0	0
$541$	− 6.84935e6i	− 1.00613i	−0.864247	−	0.503067i	$-0.832205\pi$
$541$	− 6.84935e6i	− 1.00613i	0.864247	−	0.503067i	$-0.167795\pi$
$542$	0	0
$543$	1.22794e6i	0.178722i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6.39084e6	−0.913251	−0.456625	−	0.889659i	$-0.650942\pi$
$547$	−6.39084e6	−0.913251	−0.456625	+	0.889659i	$0.650942\pi$
$548$	0	0
$549$	4.92232e6i	0.697010i
$550$	0	0
$551$	−5.92391e6	−0.831246
$552$	0	0
$553$	− 2.05392e6i	− 0.285609i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−463389.	−0.0632861	−0.0316430	−	0.999499i	$-0.510074\pi$
$557$	−463389.	−0.0632861	−0.0316430	+	0.999499i	$0.510074\pi$
$558$	0	0
$559$	2.10830e6	0.285367
$560$	0	0
$561$	7.34561e6	0.985418
$562$	0	0
$563$	1.07609e7	1.43080	0.715400	−	0.698715i	$-0.246244\pi$
$563$	1.07609e7	1.43080	0.715400	+	0.698715i	$0.246244\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.06823e6i	− 0.270173i
$568$	0	0
$569$	1.04253e7	1.34992	0.674961	−	0.737853i	$-0.264160\pi$
$569$	1.04253e7	1.34992	0.674961	+	0.737853i	$0.264160\pi$
$570$	0	0
$571$	1.58675e6i	0.203666i	0.994802	+	0.101833i	$0.0324707\pi$
$571$	1.58675e6i	0.203666i	−0.994802	+	0.101833i	$0.967529\pi$
$572$	0	0
$573$	−4.24094e6	−0.539605
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.79056e6i	0.348941i	0.984662	+	0.174471i	$0.0558214\pi$
$577$	2.79056e6i	0.348941i	−0.984662	+	0.174471i	$0.944179\pi$
$578$	0	0
$579$	− 2.83062e6i	− 0.350901i
$580$	0	0
$581$	− 1.09369e6i	− 0.134417i
$582$	0	0
$583$	9.84944e6i	1.20016i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.94272e6	−0.232710	−0.116355	−	0.993208i	$-0.537121\pi$
$587$	−1.94272e6	−0.232710	−0.116355	+	0.993208i	$0.537121\pi$
$588$	0	0
$589$	− 5.75353e6i	− 0.683355i
$590$	0	0
$591$	9.73022e6	1.14592
$592$	0	0
$593$	8.14862e6i	0.951584i	0.879558	+	0.475792i	$0.157839\pi$
$593$	8.14862e6i	0.951584i	−0.879558	+	0.475792i	$0.842161\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−906179.	−0.104059
$598$	0	0
$599$	1.49677e6	0.170447	0.0852234	−	0.996362i	$-0.472840\pi$
$599$	1.49677e6	0.170447	0.0852234	+	0.996362i	$0.472840\pi$
$600$	0	0
$601$	−9.04082e6	−1.02099	−0.510495	−	0.859881i	$-0.670538\pi$
$601$	−9.04082e6	−1.02099	−0.510495	+	0.859881i	$0.670538\pi$
$602$	0	0
$603$	2.62938e6	0.294483
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 3.63200e6i	− 0.400105i	−0.979785	−	0.200052i	$-0.935889\pi$
$607$	− 3.63200e6i	− 0.400105i	0.979785	−	0.200052i	$-0.0641112\pi$
$608$	0	0
$609$	7.21751e6	0.788577
$610$	0	0
$611$	− 2.98027e6i	− 0.322962i
$612$	0	0
$613$	−1.89937e6	−0.204154	−0.102077	−	0.994776i	$-0.532549\pi$
$613$	−1.89937e6	−0.204154	−0.102077	+	0.994776i	$0.532549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5.96746e6i	− 0.631069i	−0.948914	−	0.315534i	$-0.897816\pi$
$617$	− 5.96746e6i	− 0.631069i	0.948914	−	0.315534i	$-0.102184\pi$
$618$	0	0
$619$	1.46307e7i	1.53475i	0.641196	+	0.767377i	$0.278439\pi$
$619$	1.46307e7i	1.53475i	−0.641196	+	0.767377i	$0.721561\pi$
$620$	0	0
$621$	− 4.91527e6i	− 0.511468i
$622$	0	0
$623$	1.47992e7i	1.52763i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5.06141e6	−0.514165
$628$	0	0
$629$	2.24932e7i	2.26686i
$630$	0	0
$631$	1.47747e7	1.47722	0.738609	−	0.674134i	$-0.235483\pi$
$631$	1.47747e7	1.47722	0.738609	+	0.674134i	$0.235483\pi$
$632$	0	0
$633$	5.99261e6i	0.594438i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.28152e6	0.125134
$638$	0	0
$639$	1.74781e6	0.169333
$640$	0	0
$641$	1.81017e7	1.74010	0.870050	−	0.492964i	$-0.164087\pi$
$641$	1.81017e7	1.74010	0.870050	+	0.492964i	$0.164087\pi$
$642$	0	0
$643$	−7.21849e6	−0.688524	−0.344262	−	0.938874i	$-0.611871\pi$
$643$	−7.21849e6	−0.688524	−0.344262	+	0.938874i	$0.611871\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.44672e6i	0.229786i	0.993378	+	0.114893i	$0.0366526\pi$
$647$	2.44672e6i	0.229786i	−0.993378	+	0.114893i	$0.963347\pi$
$648$	0	0
$649$	−8.12237e6	−0.756957
$650$	0	0
$651$	7.00992e6i	0.648277i
$652$	0	0
$653$	−5.92231e6	−0.543511	−0.271755	−	0.962366i	$-0.587604\pi$
$653$	−5.92231e6	−0.543511	−0.271755	+	0.962366i	$0.587604\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.46793e6i	0.494208i
$658$	0	0
$659$	2.68146e6i	0.240523i	0.992742	+	0.120262i	$0.0383734\pi$
$659$	2.68146e6i	0.240523i	−0.992742	+	0.120262i	$0.961627\pi$
$660$	0	0
$661$	1.14098e7i	1.01572i	0.861440	+	0.507859i	$0.169563\pi$
$661$	1.14098e7i	1.01572i	−0.861440	+	0.507859i	$0.830437\pi$
$662$	0	0
$663$	− 2.92996e6i	− 0.258868i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.01481e6	0.436456
$668$	0	0
$669$	− 3.02423e6i	− 0.261246i
$670$	0	0
$671$	−1.25764e7	−1.07833
$672$	0	0
$673$	− 5.13646e6i	− 0.437146i	−0.975821	−	0.218573i	$-0.929860\pi$
$673$	− 5.13646e6i	− 0.437146i	0.975821	−	0.218573i	$-0.0701402\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.91407e6	−0.412068	−0.206034	−	0.978545i	$-0.566056\pi$
$677$	−4.91407e6	−0.412068	−0.206034	+	0.978545i	$0.566056\pi$
$678$	0	0
$679$	2.44930e7	2.03877
$680$	0	0
$681$	−6.41937e6	−0.530426
$682$	0	0
$683$	−4.30841e6	−0.353399	−0.176700	−	0.984265i	$-0.556542\pi$
$683$	−4.30841e6	−0.353399	−0.176700	+	0.984265i	$0.556542\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.00433e7i	− 0.811868i
$688$	0	0
$689$	3.92867e6	0.315281
$690$	0	0
$691$	2.17672e6i	0.173423i	0.996233	+	0.0867116i	$0.0276359\pi$
$691$	2.17672e6i	0.173423i	−0.996233	+	0.0867116i	$0.972364\pi$
$692$	0	0
$693$	−6.62361e6	−0.523916
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.24684e7i	− 0.972142i
$698$	0	0
$699$	− 1.18284e7i	− 0.915658i
$700$	0	0
$701$	1.31991e6i	0.101450i	0.998713	+	0.0507248i	$0.0161531\pi$
$701$	1.31991e6i	0.101450i	−0.998713	+	0.0507248i	$0.983847\pi$
$702$	0	0
$703$	− 1.54987e7i	− 1.18279i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.87970e7	1.41429
$708$	0	0
$709$	1.33410e7i	0.996721i	0.866970	+	0.498360i	$0.166064\pi$
$709$	1.33410e7i	0.996721i	−0.866970	+	0.498360i	$0.833936\pi$
$710$	0	0
$711$	1.57885e6	0.117130
$712$	0	0
$713$	4.87058e6i	0.358803i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.63721e6	−0.627445
$718$	0	0
$719$	2.25289e7	1.62524	0.812620	−	0.582794i	$-0.198040\pi$
$719$	2.25289e7	1.62524	0.812620	+	0.582794i	$0.198040\pi$
$720$	0	0
$721$	6.31691e6	0.452550
$722$	0	0
$723$	−1.75199e7	−1.24648
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.49088e7i	− 1.04618i	−0.852277	−	0.523091i	$-0.824779\pi$
$727$	− 1.49088e7i	− 1.04618i	0.852277	−	0.523091i	$-0.175221\pi$
$728$	0	0
$729$	1.20906e7	0.842617
$730$	0	0
$731$	− 3.46994e7i	− 2.40175i
$732$	0	0
$733$	7.98329e6	0.548810	0.274405	−	0.961614i	$-0.411519\pi$
$733$	7.98329e6	0.548810	0.274405	+	0.961614i	$0.411519\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.71802e6i	0.455588i
$738$	0	0
$739$	3.81018e6i	0.256646i	0.991732	+	0.128323i	$0.0409594\pi$
$739$	3.81018e6i	0.256646i	−0.991732	+	0.128323i	$0.959041\pi$
$740$	0	0
$741$	2.01885e6i	0.135070i
$742$	0	0
$743$	− 8.99457e6i	− 0.597734i	−0.954295	−	0.298867i	$-0.903391\pi$
$743$	− 8.99457e6i	− 0.597734i	0.954295	−	0.298867i	$-0.0966088\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	840722.	0.0551253
$748$	0	0
$749$	− 3.10156e7i	− 2.02011i
$750$	0	0
$751$	−1.18325e7	−0.765559	−0.382779	−	0.923840i	$-0.625033\pi$
$751$	−1.18325e7	−0.765559	−0.382779	+	0.923840i	$0.625033\pi$
$752$	0	0
$753$	4.86742e6i	0.312832i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.14086e6	0.326059	0.163030	−	0.986621i	$-0.447873\pi$
$757$	5.14086e6	0.326059	0.163030	+	0.986621i	$0.447873\pi$
$758$	0	0
$759$	4.28467e6	0.269968
$760$	0	0
$761$	8.21979e6	0.514516	0.257258	−	0.966343i	$-0.417181\pi$
$761$	8.21979e6	0.514516	0.257258	+	0.966343i	$0.417181\pi$
$762$	0	0
$763$	−3.99322e6	−0.248320
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.23979e6i	0.198851i
$768$	0	0
$769$	1.11390e7	0.679251	0.339626	−	0.940561i	$-0.389700\pi$
$769$	1.11390e7	0.679251	0.339626	+	0.940561i	$0.389700\pi$
$770$	0	0
$771$	− 4.29101e6i	− 0.259970i
$772$	0	0
$773$	−5.63671e6	−0.339294	−0.169647	−	0.985505i	$-0.554263\pi$
$773$	−5.63671e6	−0.339294	−0.169647	+	0.985505i	$0.554263\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.88831e7i	1.12207i
$778$	0	0
$779$	8.59123e6i	0.507238i
$780$	0	0
$781$	4.46561e6i	0.261971i
$782$	0	0
$783$	1.62615e7i	0.947888i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.07197e7	1.76799	0.883996	−	0.467495i	$-0.154843\pi$
$787$	3.07197e7	1.76799	0.883996	+	0.467495i	$0.154843\pi$
$788$	0	0
$789$	− 2.09361e7i	− 1.19730i
$790$	0	0
$791$	8.66309e6	0.492302
$792$	0	0
$793$	5.01639e6i	0.283275i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.22339e7	−1.23985	−0.619926	−	0.784660i	$-0.712837\pi$
$797$	−2.22339e7	−1.23985	−0.619926	+	0.784660i	$0.712837\pi$
$798$	0	0
$799$	−4.90505e7	−2.71817
$800$	0	0
$801$	−1.13761e7	−0.626488
$802$	0	0
$803$	−1.39704e7	−0.764577
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 7.25226e6i	− 0.392003i
$808$	0	0
$809$	−1.51247e7	−0.812484	−0.406242	−	0.913765i	$-0.633161\pi$
$809$	−1.51247e7	−0.812484	−0.406242	+	0.913765i	$0.633161\pi$
$810$	0	0
$811$	1.35982e7i	0.725990i	0.931791	+	0.362995i	$0.118246\pi$
$811$	1.35982e7i	0.725990i	−0.931791	+	0.362995i	$0.881754\pi$
$812$	0	0
$813$	−2.26157e7	−1.20001
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.39092e7i	1.25317i
$818$	0	0
$819$	2.64197e6i	0.137632i
$820$	0	0
$821$	2.56951e7i	1.33043i	0.746651	+	0.665216i	$0.231660\pi$
$821$	2.56951e7i	1.33043i	−0.746651	+	0.665216i	$0.768340\pi$
$822$	0	0
$823$	− 8.73184e6i	− 0.449372i	−0.974431	−	0.224686i	$-0.927864\pi$
$823$	− 8.73184e6i	− 0.449372i	0.974431	−	0.224686i	$-0.0721357\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.46565e6	0.0745191	0.0372596	−	0.999306i	$-0.488137\pi$
$827$	1.46565e6	0.0745191	0.0372596	+	0.999306i	$0.488137\pi$
$828$	0	0
$829$	− 4.49887e6i	− 0.227362i	−0.993517	−	0.113681i	$-0.963736\pi$
$829$	− 4.49887e6i	− 0.227362i	0.993517	−	0.113681i	$-0.0362641\pi$
$830$	0	0
$831$	−8.97130e6	−0.450664
$832$	0	0
$833$	− 2.10918e7i	− 1.05317i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.57938e7	−0.779244
$838$	0	0
$839$	−1.43801e7	−0.705273	−0.352636	−	0.935760i	$-0.614715\pi$
$839$	−1.43801e7	−0.705273	−0.352636	+	0.935760i	$0.614715\pi$
$840$	0	0
$841$	3.92030e6	0.191130
$842$	0	0
$843$	2.59702e7	1.25866
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.44185e6i	0.452219i
$848$	0	0
$849$	−2.17420e7	−1.03521
$850$	0	0
$851$	1.31202e7i	0.621036i
$852$	0	0
$853$	−2.62365e7	−1.23462	−0.617310	−	0.786720i	$-0.711778\pi$
$853$	−2.62365e7	−1.23462	−0.617310	+	0.786720i	$0.711778\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.34066e6i	0.294905i	0.989069	+	0.147453i	$0.0471074\pi$
$857$	6.34066e6i	0.294905i	−0.989069	+	0.147453i	$0.952893\pi$
$858$	0	0
$859$	1.09488e7i	0.506273i	0.967431	+	0.253137i	$0.0814622\pi$
$859$	1.09488e7i	0.506273i	−0.967431	+	0.253137i	$0.918538\pi$
$860$	0	0
$861$	− 1.04673e7i	− 0.481200i
$862$	0	0
$863$	3.28604e6i	0.150192i	0.997176	+	0.0750958i	$0.0239263\pi$
$863$	3.28604e6i	0.150192i	−0.997176	+	0.0750958i	$0.976074\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.28540e7	−1.48436
$868$	0	0
$869$	4.03393e6i	0.181209i
$870$	0	0
$871$	2.67963e6	0.119682
$872$	0	0
$873$	1.88278e7i	0.836110i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.75326e6	−0.340397	−0.170198	−	0.985410i	$-0.554441\pi$
$877$	−7.75326e6	−0.340397	−0.170198	+	0.985410i	$0.554441\pi$
$878$	0	0
$879$	1.88557e7	0.823136
$880$	0	0
$881$	−449453.	−0.0195094	−0.00975471	−	0.999952i	$-0.503105\pi$
$881$	−449453.	−0.0195094	−0.00975471	+	0.999952i	$0.503105\pi$
$882$	0	0
$883$	2.42001e7	1.04452	0.522259	−	0.852787i	$-0.325089\pi$
$883$	2.42001e7	1.04452	0.522259	+	0.852787i	$0.325089\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.80204e6i	0.375642i	0.982203	+	0.187821i	$0.0601425\pi$
$887$	8.80204e6i	0.375642i	−0.982203	+	0.187821i	$0.939857\pi$
$888$	0	0
$889$	−4.80724e7	−2.04005
$890$	0	0
$891$	4.06203e6i	0.171415i
$892$	0	0
$893$	3.37977e7	1.41827
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.70903e6i	− 0.0709202i
$898$	0	0
$899$	− 1.61137e7i	− 0.664959i
$900$	0	0
$901$	− 6.46597e7i	− 2.65352i
$902$	0	0
$903$	− 2.91302e7i	− 1.18884i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.35035e7	−0.545040	−0.272520	−	0.962150i	$-0.587857\pi$
$907$	−1.35035e7	−0.545040	−0.272520	+	0.962150i	$0.587857\pi$
$908$	0	0
$909$	1.44492e7i	0.580010i
$910$	0	0
$911$	1.74802e7	0.697831	0.348915	−	0.937154i	$-0.386550\pi$
$911$	1.74802e7	0.697831	0.348915	+	0.937154i	$0.386550\pi$
$912$	0	0
$913$	2.14803e6i	0.0852830i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.20157e7	2.04273
$918$	0	0
$919$	3.84416e7	1.50146	0.750728	−	0.660612i	$-0.229703\pi$
$919$	3.84416e7	1.50146	0.750728	+	0.660612i	$0.229703\pi$
$920$	0	0
$921$	−1.05934e7	−0.411514
$922$	0	0
$923$	1.78121e6	0.0688194
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.85580e6i	0.185593i
$928$	0	0
$929$	694053.	0.0263848	0.0131924	−	0.999913i	$-0.495801\pi$
$929$	694053.	0.0263848	0.0131924	+	0.999913i	$0.495801\pi$
$930$	0	0
$931$	1.45330e7i	0.549518i
$932$	0	0
$933$	1.70643e7	0.641776
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.08849e7i	− 1.52130i	−0.649165	−	0.760648i	$-0.724881\pi$
$937$	− 4.08849e7i	− 1.52130i	0.649165	−	0.760648i	$-0.275119\pi$
$938$	0	0
$939$	1.74225e7i	0.644832i
$940$	0	0
$941$	1.21324e7i	0.446655i	0.974743	+	0.223328i	$0.0716920\pi$
$941$	1.21324e7i	0.446655i	−0.974743	+	0.223328i	$0.928308\pi$
$942$	0	0
$943$	− 7.27279e6i	− 0.266331i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.38948e7	1.22817	0.614084	−	0.789241i	$-0.289526\pi$
$947$	3.38948e7	1.22817	0.614084	+	0.789241i	$0.289526\pi$
$948$	0	0
$949$	5.57242e6i	0.200853i
$950$	0	0
$951$	−1.63086e7	−0.584742
$952$	0	0
$953$	− 5.18152e7i	− 1.84810i	−0.382274	−	0.924049i	$-0.624859\pi$
$953$	− 5.18152e7i	− 1.84810i	0.382274	−	0.924049i	$-0.375141\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.41753e7	−0.500324
$958$	0	0
$959$	−2.65671e7	−0.932819
$960$	0	0
$961$	−1.29789e7	−0.453347
$962$	0	0
$963$	2.38416e7	0.828458
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.41440e7i	− 1.51812i	−0.651023	−	0.759058i	$-0.725660\pi$
$967$	− 4.41440e7i	− 1.51812i	0.651023	−	0.759058i	$-0.274340\pi$
$968$	0	0
$969$	3.32272e7	1.13680
$970$	0	0
$971$	− 2.25369e7i	− 0.767091i	−0.923522	−	0.383546i	$-0.874703\pi$
$971$	− 2.25369e7i	− 0.767091i	0.923522	−	0.383546i	$-0.125297\pi$
$972$	0	0
$973$	5.02184e7	1.70052
$974$	0	0
$975$	0	0
$976$	0	0
$977$	508955.i	0.0170586i	0.999964	+	0.00852929i	$0.00271499\pi$
$977$	508955.i	0.0170586i	−0.999964	+	0.00852929i	$0.997285\pi$
$978$	0	0
$979$	− 2.90657e7i	− 0.969224i
$980$	0	0
$981$	− 3.06958e6i	− 0.101837i
$982$	0	0
$983$	1.38312e7i	0.456536i	0.973598	+	0.228268i	$0.0733063\pi$
$983$	1.38312e7i	0.456536i	−0.973598	+	0.228268i	$0.926694\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.11781e7	−1.34547
$988$	0	0
$989$	− 2.02400e7i	− 0.657992i
$990$	0	0
$991$	−1.13368e7	−0.366696	−0.183348	−	0.983048i	$-0.558693\pi$
$991$	−1.13368e7	−0.366696	−0.183348	+	0.983048i	$0.558693\pi$
$992$	0	0
$993$	4.26957e6i	0.137408i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.66390e7	−1.16736	−0.583681	−	0.811983i	$-0.698388\pi$
$997$	−3.66390e7	−1.16736	−0.583681	+	0.811983i	$0.698388\pi$
$998$	0	0
$999$	−4.25449e7	−1.34876

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.f.c.49.14		20
4.3	odd	2		200.6.f.b.149.7		20
5.2	odd	4		160.6.d.a.81.7		20
5.3	odd	4		800.6.d.c.401.14		20
5.4	even	2		800.6.f.b.49.7		20
8.3	odd	2		200.6.f.c.149.13		20
8.5	even	2		800.6.f.b.49.8		20
20.3	even	4		200.6.d.b.101.4		20
20.7	even	4		40.6.d.a.21.17	✓	20
20.19	odd	2		200.6.f.c.149.14		20
40.3	even	4		200.6.d.b.101.3		20
40.13	odd	4		800.6.d.c.401.7		20
40.19	odd	2		200.6.f.b.149.8		20
40.27	even	4		40.6.d.a.21.18	yes	20
40.29	even	2	inner	800.6.f.c.49.13		20
40.37	odd	4		160.6.d.a.81.14		20
60.47	odd	4		360.6.k.b.181.4		20
120.107	odd	4		360.6.k.b.181.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.17	✓	20	20.7	even	4
40.6.d.a.21.18	yes	20	40.27	even	4
160.6.d.a.81.7		20	5.2	odd	4
160.6.d.a.81.14		20	40.37	odd	4
200.6.d.b.101.3		20	40.3	even	4
200.6.d.b.101.4		20	20.3	even	4
200.6.f.b.149.7		20	4.3	odd	2
200.6.f.b.149.8		20	40.19	odd	2
200.6.f.c.149.13		20	8.3	odd	2
200.6.f.c.149.14		20	20.19	odd	2
360.6.k.b.181.3		20	120.107	odd	4
360.6.k.b.181.4		20	60.47	odd	4
800.6.d.c.401.7		20	40.13	odd	4
800.6.d.c.401.14		20	5.3	odd	4
800.6.f.b.49.7		20	5.4	even	2
800.6.f.b.49.8		20	8.5	even	2
800.6.f.c.49.13		20	40.29	even	2	inner
800.6.f.c.49.14		20	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.8240 q^{3} +163.706i q^{7} -125.841 q^{9} -321.520i q^{11} -128.246 q^{13} +2110.72i q^{17} -1454.37i q^{19} +1771.96i q^{21} +1231.18i q^{23} -3992.34 q^{27} -4073.19i q^{29} +3956.03 q^{31} -3480.14i q^{33} +10656.6 q^{37} -1388.13 q^{39} -5907.19 q^{41} -16439.6 q^{43} +23238.8i q^{47} -9992.68 q^{49} +22846.5i q^{51} -30634.0 q^{53} -15742.1i q^{57} -25262.4i q^{59} -39115.5i q^{61} -20600.9i q^{63} -20894.5 q^{67} +13326.3i q^{69} -13889.1 q^{71} -43451.2i q^{73} +52634.8 q^{77} -12546.4 q^{79} -12633.8 q^{81} -6680.84 q^{83} -44088.2i q^{87} +90400.9 q^{89} -20994.6i q^{91} +42820.2 q^{93} -149616. i q^{97} +40460.4i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 36 q^{3} + 1620 q^{9} + 11664 q^{27} - 7160 q^{31} + 3608 q^{37} - 44904 q^{39} + 11608 q^{41} - 51772 q^{43} - 18756 q^{49} - 928 q^{53} - 161604 q^{67} + 200312 q^{71} - 26008 q^{77} + 282080 q^{79}+ \cdots - 293472 q^{93}+O(q^{100}) \) 20 * q + 36 * q^3 + 1620 * q^9 + 11664 * q^27 - 7160 * q^31 + 3608 * q^37 - 44904 * q^39 + 11608 * q^41 - 51772 * q^43 - 18756 * q^49 - 928 * q^53 - 161604 * q^67 + 200312 * q^71 - 26008 * q^77 + 282080 * q^79 + 65172 * q^81 - 99092 * q^83 + 3160 * q^89 - 293472 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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