LMFDB - Embedded newform 784.4.f.i.783.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(783,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("784.783");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, 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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.16928550682624.32
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 62x^{6} - 152x^{5} + 1187x^{4} - 1424x^{3} + 7038x^{2} + 1452x + 5287 $ x^8 - 4x^7 + 62x^6 - 152x^5 + 1187x^4 - 1424x^3 + 7038x^2 + 1452*x + 5287
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.6
Root			$1.20711 - 4.91262i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.4.f.i.783.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.34835 q^{3} +6.94269i q^{5} +13.3015 q^{9} +O(q^{10})$	$q+6.34835 q^{3} +6.94269i q^{5} +13.3015 q^{9} -2.01259i q^{11} -11.3810i q^{13} +44.0746i q^{15} +71.5824i q^{17} -81.8905 q^{19} +192.197i q^{23} +76.7990 q^{25} -86.9627 q^{27} -92.2010 q^{29} +252.658 q^{31} -12.7766i q^{33} +277.186 q^{37} -72.2509i q^{39} +276.725i q^{41} +418.608i q^{43} +92.3484i q^{45} -416.439 q^{47} +454.430i q^{51} -310.583 q^{53} +13.9728 q^{55} -519.869 q^{57} +184.740 q^{59} -148.839i q^{61} +79.0152 q^{65} +180.324i q^{67} +1220.13i q^{69} -456.443i q^{71} +874.619i q^{73} +487.547 q^{75} -428.266i q^{79} -911.211 q^{81} -337.100 q^{83} -496.975 q^{85} -585.324 q^{87} -35.0305i q^{89} +1603.96 q^{93} -568.541i q^{95} +1604.99i q^{97} -26.7705i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 344 q^{9}+O(q^{10}) $ 8 * q + 344 * q^9	$ 8 q + 344 q^{9} + 456 q^{25} - 896 q^{29} + 208 q^{53} + 672 q^{57} + 3008 q^{65} + 6728 q^{81} - 16 q^{85} + 6496 q^{93}+O(q^{100}) $ 8 * q + 344 * q^9 + 456 * q^25 - 896 * q^29 + 208 * q^53 + 672 * q^57 + 3008 * q^65 + 6728 * q^81 - 16 * q^85 + 6496 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	6.34835	1.22174	0.610870	−	0.791731i	$-0.290820\pi$
$3$	6.34835	1.22174	0.610870	+	0.791731i	$0.290820\pi$
$4$	0	0
$5$	6.94269i	0.620973i	0.950578	+	0.310487i	$0.100492\pi$
$5$	6.94269i	0.620973i	−0.950578	+	0.310487i	$0.899508\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	13.3015	0.492649
$10$	0	0
$11$	− 2.01259i	− 0.0551653i	−0.999620	−	0.0275826i	$-0.991219\pi$
$11$	− 2.01259i	− 0.0551653i	0.999620	−	0.0275826i	$-0.00878094\pi$
$12$	0	0
$13$	− 11.3810i	− 0.242810i	−0.992603	−	0.121405i	$-0.961260\pi$
$13$	− 11.3810i	− 0.242810i	0.992603	−	0.121405i	$-0.0387400\pi$
$14$	0	0
$15$	44.0746i	0.758668i
$16$	0	0
$17$	71.5824i	1.02125i	0.859803	+	0.510626i	$0.170586\pi$
$17$	71.5824i	1.02125i	−0.859803	+	0.510626i	$0.829414\pi$
$18$	0	0
$19$	−81.8905	−0.988788	−0.494394	−	0.869238i	$-0.664610\pi$
$19$	−81.8905	−0.988788	−0.494394	+	0.869238i	$0.664610\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	192.197i	1.74243i	0.490904	+	0.871214i	$0.336667\pi$
$23$	192.197i	1.74243i	−0.490904	+	0.871214i	$0.663333\pi$
$24$	0	0
$25$	76.7990	0.614392
$26$	0	0
$27$	−86.9627	−0.619851
$28$	0	0
$29$	−92.2010	−0.590390	−0.295195	−	0.955437i	$-0.595385\pi$
$29$	−92.2010	−0.590390	−0.295195	+	0.955437i	$0.595385\pi$
$30$	0	0
$31$	252.658	1.46383	0.731914	−	0.681397i	$-0.238627\pi$
$31$	252.658	1.46383	0.731914	+	0.681397i	$0.238627\pi$
$32$	0	0
$33$	− 12.7766i	− 0.0673976i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	277.186	1.23160	0.615798	−	0.787904i	$-0.288834\pi$
$37$	277.186	1.23160	0.615798	+	0.787904i	$0.288834\pi$
$38$	0	0
$39$	− 72.2509i	− 0.296651i
$40$	0	0
$41$	276.725i	1.05408i	0.849841	+	0.527039i	$0.176698\pi$
$41$	276.725i	1.05408i	−0.849841	+	0.527039i	$0.823302\pi$
$42$	0	0
$43$	418.608i	1.48458i	0.670077	+	0.742292i	$0.266261\pi$
$43$	418.608i	1.48458i	−0.670077	+	0.742292i	$0.733739\pi$
$44$	0	0
$45$	92.3484i	0.305922i
$46$	0	0
$47$	−416.439	−1.29242	−0.646211	−	0.763159i	$-0.723647\pi$
$47$	−416.439	−1.29242	−0.646211	+	0.763159i	$0.723647\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	454.430i	1.24770i
$52$	0	0
$53$	−310.583	−0.804940	−0.402470	−	0.915433i	$-0.631848\pi$
$53$	−310.583	−0.804940	−0.402470	+	0.915433i	$0.631848\pi$
$54$	0	0
$55$	13.9728	0.0342562
$56$	0	0
$57$	−519.869	−1.20804
$58$	0	0
$59$	184.740	0.407646	0.203823	−	0.979008i	$-0.434663\pi$
$59$	184.740	0.407646	0.203823	+	0.979008i	$0.434663\pi$
$60$	0	0
$61$	− 148.839i	− 0.312409i	−0.987725	−	0.156204i	$-0.950074\pi$
$61$	− 148.839i	− 0.312409i	0.987725	−	0.156204i	$-0.0499258\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	79.0152	0.150779
$66$	0	0
$67$	180.324i	0.328807i	0.986393	+	0.164403i	$0.0525699\pi$
$67$	180.324i	0.328807i	−0.986393	+	0.164403i	$0.947430\pi$
$68$	0	0
$69$	1220.13i	2.12879i
$70$	0	0
$71$	− 456.443i	− 0.762954i	−0.924378	−	0.381477i	$-0.875415\pi$
$71$	− 456.443i	− 0.762954i	0.924378	−	0.381477i	$-0.124585\pi$
$72$	0	0
$73$	874.619i	1.40228i	0.713024	+	0.701139i	$0.247325\pi$
$73$	874.619i	1.40228i	−0.713024	+	0.701139i	$0.752675\pi$
$74$	0	0
$75$	487.547	0.750627
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 428.266i	− 0.609921i	−0.952365	−	0.304960i	$-0.901357\pi$
$79$	− 428.266i	− 0.609921i	0.952365	−	0.304960i	$-0.0986432\pi$
$80$	0	0
$81$	−911.211	−1.24995
$82$	0	0
$83$	−337.100	−0.445802	−0.222901	−	0.974841i	$-0.571553\pi$
$83$	−337.100	−0.445802	−0.222901	+	0.974841i	$0.571553\pi$
$84$	0	0
$85$	−496.975	−0.634170
$86$	0	0
$87$	−585.324	−0.721303
$88$	0	0
$89$	− 35.0305i	− 0.0417216i	−0.999782	−	0.0208608i	$-0.993359\pi$
$89$	− 35.0305i	− 0.0417216i	0.999782	−	0.0208608i	$-0.00664069\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1603.96	1.78842
$94$	0	0
$95$	− 568.541i	− 0.614011i
$96$	0	0
$97$	1604.99i	1.68003i	0.542567	+	0.840013i	$0.317452\pi$
$97$	1604.99i	1.68003i	−0.542567	+	0.840013i	$0.682548\pi$
$98$	0	0
$99$	− 26.7705i	− 0.0271771i
$100$	0	0
$101$	486.846i	0.479633i	0.970818	+	0.239817i	$0.0770873\pi$
$101$	486.846i	0.479633i	−0.970818	+	0.239817i	$0.922913\pi$
$102$	0	0
$103$	373.244	0.357057	0.178528	−	0.983935i	$-0.442866\pi$
$103$	373.244	0.357057	0.178528	+	0.983935i	$0.442866\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1898.43i	− 1.71521i	−0.514307	−	0.857606i	$-0.671951\pi$
$107$	− 1898.43i	− 1.71521i	0.514307	−	0.857606i	$-0.328049\pi$
$108$	0	0
$109$	426.794	0.375041	0.187520	−	0.982261i	$-0.439955\pi$
$109$	426.794	0.375041	0.187520	+	0.982261i	$0.439955\pi$
$110$	0	0
$111$	1759.67	1.50469
$112$	0	0
$113$	−1763.07	−1.46775	−0.733873	−	0.679287i	$-0.762289\pi$
$113$	−1763.07	−1.46775	−0.733873	+	0.679287i	$0.762289\pi$
$114$	0	0
$115$	−1334.36	−1.08200
$116$	0	0
$117$	− 151.385i	− 0.119620i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1326.95	0.996957
$122$	0	0
$123$	1756.75i	1.28781i
$124$	0	0
$125$	1401.03i	1.00249i
$126$	0	0
$127$	− 2502.99i	− 1.74885i	−0.485157	−	0.874427i	$-0.661238\pi$
$127$	− 2502.99i	− 1.74885i	0.485157	−	0.874427i	$-0.338762\pi$
$128$	0	0
$129$	2657.47i	1.81378i
$130$	0	0
$131$	2303.11	1.53606	0.768029	−	0.640415i	$-0.221238\pi$
$131$	2303.11	1.53606	0.768029	+	0.640415i	$0.221238\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 603.756i	− 0.384911i
$136$	0	0
$137$	34.5025	0.0215164	0.0107582	−	0.999942i	$-0.496575\pi$
$137$	34.5025	0.0215164	0.0107582	+	0.999942i	$0.496575\pi$
$138$	0	0
$139$	−1413.13	−0.862303	−0.431151	−	0.902280i	$-0.641892\pi$
$139$	−1413.13	−0.862303	−0.431151	+	0.902280i	$0.641892\pi$
$140$	0	0
$141$	−2643.70	−1.57900
$142$	0	0
$143$	−22.9054	−0.0133947
$144$	0	0
$145$	− 640.123i	− 0.366616i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2030.16	1.11622	0.558112	−	0.829766i	$-0.311526\pi$
$149$	2030.16	1.11622	0.558112	+	0.829766i	$0.311526\pi$
$150$	0	0
$151$	− 2094.45i	− 1.12877i	−0.825513	−	0.564383i	$-0.809114\pi$
$151$	− 2094.45i	− 1.12877i	0.825513	−	0.564383i	$-0.190886\pi$
$152$	0	0
$153$	952.154i	0.503118i
$154$	0	0
$155$	1754.13i	0.908999i
$156$	0	0
$157$	453.932i	0.230750i	0.993322	+	0.115375i	$0.0368070\pi$
$157$	453.932i	0.230750i	−0.993322	+	0.115375i	$0.963193\pi$
$158$	0	0
$159$	−1971.69	−0.983428
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 214.942i	− 0.103286i	−0.998666	−	0.0516429i	$-0.983554\pi$
$163$	− 214.942i	− 0.103286i	0.998666	−	0.0516429i	$-0.0164458\pi$
$164$	0	0
$165$	88.7040	0.0418521
$166$	0	0
$167$	264.078	0.122365	0.0611826	−	0.998127i	$-0.480513\pi$
$167$	264.078	0.122365	0.0611826	+	0.998127i	$0.480513\pi$
$168$	0	0
$169$	2067.47	0.941043
$170$	0	0
$171$	−1089.27	−0.487125
$172$	0	0
$173$	1005.10i	0.441715i	0.975306	+	0.220857i	$0.0708855\pi$
$173$	1005.10i	0.441715i	−0.975306	+	0.220857i	$0.929114\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1172.79	0.498037
$178$	0	0
$179$	368.496i	0.153870i	0.997036	+	0.0769348i	$0.0245133\pi$
$179$	368.496i	0.153870i	−0.997036	+	0.0769348i	$0.975487\pi$
$180$	0	0
$181$	2476.12i	1.01684i	0.861108	+	0.508422i	$0.169771\pi$
$181$	2476.12i	1.01684i	−0.861108	+	0.508422i	$0.830229\pi$
$182$	0	0
$183$	− 944.884i	− 0.381682i
$184$	0	0
$185$	1924.42i	0.764789i
$186$	0	0
$187$	144.066	0.0563376
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2146.17i	0.813042i	0.913641	+	0.406521i	$0.133258\pi$
$191$	2146.17i	0.813042i	−0.913641	+	0.406521i	$0.866742\pi$
$192$	0	0
$193$	−2751.95	−1.02637	−0.513186	−	0.858278i	$-0.671535\pi$
$193$	−2751.95	−1.02637	−0.513186	+	0.858278i	$0.671535\pi$
$194$	0	0
$195$	501.616	0.184213
$196$	0	0
$197$	−1963.83	−0.710238	−0.355119	−	0.934821i	$-0.615560\pi$
$197$	−1963.83	−0.710238	−0.355119	+	0.934821i	$0.615560\pi$
$198$	0	0
$199$	4509.69	1.60645	0.803224	−	0.595677i	$-0.203116\pi$
$199$	4509.69	1.60645	0.803224	+	0.595677i	$0.203116\pi$
$200$	0	0
$201$	1144.76i	0.401716i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1921.22	−0.654554
$206$	0	0
$207$	2556.51i	0.858405i
$208$	0	0
$209$	164.812i	0.0545467i
$210$	0	0
$211$	4946.41i	1.61386i	0.590645	+	0.806931i	$0.298873\pi$
$211$	4946.41i	1.61386i	−0.590645	+	0.806931i	$0.701127\pi$
$212$	0	0
$213$	− 2897.66i	− 0.932132i
$214$	0	0
$215$	−2906.27	−0.921887
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5552.38i	1.71322i
$220$	0	0
$221$	814.683	0.247971
$222$	0	0
$223$	−553.486	−0.166207	−0.0831035	−	0.996541i	$-0.526483\pi$
$223$	−553.486	−0.166207	−0.0831035	+	0.996541i	$0.526483\pi$
$224$	0	0
$225$	1021.54	0.302679
$226$	0	0
$227$	3155.06	0.922507	0.461253	−	0.887269i	$-0.347400\pi$
$227$	3155.06	0.922507	0.461253	+	0.887269i	$0.347400\pi$
$228$	0	0
$229$	− 6017.72i	− 1.73651i	−0.496114	−	0.868257i	$-0.665240\pi$
$229$	− 6017.72i	− 1.73651i	0.496114	−	0.868257i	$-0.334760\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3102.08	−0.872206	−0.436103	−	0.899897i	$-0.643642\pi$
$233$	−3102.08	−0.872206	−0.436103	+	0.899897i	$0.643642\pi$
$234$	0	0
$235$	− 2891.21i	− 0.802560i
$236$	0	0
$237$	− 2718.78i	− 0.745164i
$238$	0	0
$239$	822.531i	0.222616i	0.993786	+	0.111308i	$0.0355039\pi$
$239$	822.531i	0.222616i	−0.993786	+	0.111308i	$0.964496\pi$
$240$	0	0
$241$	− 889.011i	− 0.237619i	−0.992917	−	0.118810i	$-0.962092\pi$
$241$	− 889.011i	− 0.237619i	0.992917	−	0.118810i	$-0.0379078\pi$
$242$	0	0
$243$	−3436.69	−0.907258
$244$	0	0
$245$	0	0
$246$	0	0
$247$	932.000i	0.240088i
$248$	0	0
$249$	−2140.03	−0.544654
$250$	0	0
$251$	−5246.06	−1.31924	−0.659619	−	0.751600i	$-0.729282\pi$
$251$	−5246.06	−1.31924	−0.659619	+	0.751600i	$0.729282\pi$
$252$	0	0
$253$	386.813	0.0961215
$254$	0	0
$255$	−3154.97	−0.774791
$256$	0	0
$257$	6711.12i	1.62890i	0.580231	+	0.814452i	$0.302962\pi$
$257$	6711.12i	1.62890i	−0.580231	+	0.814452i	$0.697038\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1226.41	−0.290855
$262$	0	0
$263$	− 5274.05i	− 1.23655i	−0.785963	−	0.618273i	$-0.787832\pi$
$263$	− 5274.05i	− 1.23655i	0.785963	−	0.618273i	$-0.212168\pi$
$264$	0	0
$265$	− 2156.28i	− 0.499847i
$266$	0	0
$267$	− 222.386i	− 0.0509730i
$268$	0	0
$269$	− 7012.51i	− 1.58944i	−0.606974	−	0.794722i	$-0.707617\pi$
$269$	− 7012.51i	− 1.58944i	0.606974	−	0.794722i	$-0.292383\pi$
$270$	0	0
$271$	264.078	0.0591942	0.0295971	−	0.999562i	$-0.490578\pi$
$271$	264.078	0.0591942	0.0295971	+	0.999562i	$0.490578\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 154.565i	− 0.0338931i
$276$	0	0
$277$	6212.47	1.34755	0.673775	−	0.738937i	$-0.264672\pi$
$277$	6212.47	1.34755	0.673775	+	0.738937i	$0.264672\pi$
$278$	0	0
$279$	3360.73	0.721153
$280$	0	0
$281$	6480.47	1.37577	0.687887	−	0.725818i	$-0.258538\pi$
$281$	6480.47	1.37577	0.687887	+	0.725818i	$0.258538\pi$
$282$	0	0
$283$	3795.51	0.797243	0.398622	−	0.917115i	$-0.369489\pi$
$283$	3795.51	0.797243	0.398622	+	0.917115i	$0.369489\pi$
$284$	0	0
$285$	− 3609.29i	− 0.750162i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−211.040	−0.0429554
$290$	0	0
$291$	10189.1i	2.05255i
$292$	0	0
$293$	− 4841.65i	− 0.965365i	−0.875795	−	0.482682i	$-0.839663\pi$
$293$	− 4841.65i	− 0.965365i	0.875795	−	0.482682i	$-0.160337\pi$
$294$	0	0
$295$	1282.59i	0.253137i
$296$	0	0
$297$	175.020i	0.0341943i
$298$	0	0
$299$	2187.40	0.423080
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3090.67i	0.585987i
$304$	0	0
$305$	1033.35	0.193998
$306$	0	0
$307$	2539.11	0.472036	0.236018	−	0.971749i	$-0.424158\pi$
$307$	2539.11	0.472036	0.236018	+	0.971749i	$0.424158\pi$
$308$	0	0
$309$	2369.48	0.436231
$310$	0	0
$311$	947.212	0.172706	0.0863528	−	0.996265i	$-0.472479\pi$
$311$	947.212	0.172706	0.0863528	+	0.996265i	$0.472479\pi$
$312$	0	0
$313$	3887.25i	0.701982i	0.936379	+	0.350991i	$0.114155\pi$
$313$	3887.25i	0.701982i	−0.936379	+	0.350991i	$0.885845\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5985.84	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$317$	−5985.84	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$318$	0	0
$319$	185.563i	0.0325690i
$320$	0	0
$321$	− 12051.9i	− 2.09554i
$322$	0	0
$323$	− 5861.92i	− 1.00980i
$324$	0	0
$325$	− 874.053i	− 0.149181i
$326$	0	0
$327$	2709.44	0.458202
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 10860.5i	− 1.80346i	−0.432302	−	0.901729i	$-0.642299\pi$
$331$	− 10860.5i	− 1.80346i	0.432302	−	0.901729i	$-0.357701\pi$
$332$	0	0
$333$	3686.99	0.606744
$334$	0	0
$335$	−1251.93	−0.204180
$336$	0	0
$337$	9732.16	1.57313	0.786565	−	0.617508i	$-0.211858\pi$
$337$	9732.16	1.57313	0.786565	+	0.617508i	$0.211858\pi$
$338$	0	0
$339$	−11192.6	−1.79320
$340$	0	0
$341$	− 508.496i	− 0.0807525i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8471.01	−1.32192
$346$	0	0
$347$	− 11446.1i	− 1.77077i	−0.464856	−	0.885386i	$-0.653894\pi$
$347$	− 11446.1i	− 1.77077i	0.464856	−	0.885386i	$-0.346106\pi$
$348$	0	0
$349$	− 11071.4i	− 1.69811i	−0.528306	−	0.849054i	$-0.677173\pi$
$349$	− 11071.4i	− 1.69811i	0.528306	−	0.849054i	$-0.322827\pi$
$350$	0	0
$351$	989.727i	0.150506i
$352$	0	0
$353$	2564.12i	0.386613i	0.981138	+	0.193306i	$0.0619211\pi$
$353$	2564.12i	0.386613i	−0.981138	+	0.193306i	$0.938079\pi$
$354$	0	0
$355$	3168.94	0.473774
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 12352.3i	− 1.81597i	−0.419007	−	0.907983i	$-0.637622\pi$
$359$	− 12352.3i	− 1.81597i	0.419007	−	0.907983i	$-0.362378\pi$
$360$	0	0
$361$	−152.950	−0.0222992
$362$	0	0
$363$	8423.94	1.21802
$364$	0	0
$365$	−6072.21	−0.870778
$366$	0	0
$367$	−5129.40	−0.729571	−0.364785	−	0.931092i	$-0.618858\pi$
$367$	−5129.40	−0.729571	−0.364785	+	0.931092i	$0.618858\pi$
$368$	0	0
$369$	3680.86i	0.519290i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1885.56	0.261744	0.130872	−	0.991399i	$-0.458222\pi$
$373$	1885.56	0.261744	0.130872	+	0.991399i	$0.458222\pi$
$374$	0	0
$375$	8894.22i	1.22479i
$376$	0	0
$377$	1049.34i	0.143353i
$378$	0	0
$379$	− 10668.8i	− 1.44597i	−0.690865	−	0.722984i	$-0.742770\pi$
$379$	− 10668.8i	− 1.44597i	0.690865	−	0.722984i	$-0.257230\pi$
$380$	0	0
$381$	− 15889.9i	− 2.13665i
$382$	0	0
$383$	7776.57	1.03750	0.518752	−	0.854925i	$-0.326397\pi$
$383$	7776.57	1.03750	0.518752	+	0.854925i	$0.326397\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5568.12i	0.731378i
$388$	0	0
$389$	8944.72	1.16585	0.582925	−	0.812526i	$-0.301908\pi$
$389$	8944.72	1.16585	0.582925	+	0.812526i	$0.301908\pi$
$390$	0	0
$391$	−13757.9	−1.77946
$392$	0	0
$393$	14620.9	1.87666
$394$	0	0
$395$	2973.32	0.378745
$396$	0	0
$397$	− 2611.00i	− 0.330082i	−0.986287	−	0.165041i	$-0.947224\pi$
$397$	− 2611.00i	− 0.330082i	0.986287	−	0.165041i	$-0.0527756\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7779.71	−0.968828	−0.484414	−	0.874839i	$-0.660967\pi$
$401$	−7779.71	−0.968828	−0.484414	+	0.874839i	$0.660967\pi$
$402$	0	0
$403$	− 2875.51i	− 0.355433i
$404$	0	0
$405$	− 6326.26i	− 0.776183i
$406$	0	0
$407$	− 557.861i	− 0.0679413i
$408$	0	0
$409$	− 476.212i	− 0.0575725i	−0.999586	−	0.0287863i	$-0.990836\pi$
$409$	− 476.212i	− 0.0575725i	0.999586	−	0.0287863i	$-0.00916422\pi$
$410$	0	0
$411$	219.034	0.0262875
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 2340.39i	− 0.276831i
$416$	0	0
$417$	−8971.04	−1.05351
$418$	0	0
$419$	7231.02	0.843099	0.421550	−	0.906805i	$-0.361486\pi$
$419$	7231.02	0.843099	0.421550	+	0.906805i	$0.361486\pi$
$420$	0	0
$421$	112.110	0.0129784	0.00648921	−	0.999979i	$-0.497934\pi$
$421$	112.110	0.0129784	0.00648921	+	0.999979i	$0.497934\pi$
$422$	0	0
$423$	−5539.27	−0.636710
$424$	0	0
$425$	5497.46i	0.627449i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−145.411	−0.0163648
$430$	0	0
$431$	10980.2i	1.22714i	0.789640	+	0.613571i	$0.210267\pi$
$431$	10980.2i	1.22714i	−0.789640	+	0.613571i	$0.789733\pi$
$432$	0	0
$433$	− 7744.68i	− 0.859551i	−0.902936	−	0.429776i	$-0.858593\pi$
$433$	− 7744.68i	− 0.859551i	0.902936	−	0.429776i	$-0.141407\pi$
$434$	0	0
$435$	− 4063.73i	− 0.447910i
$436$	0	0
$437$	− 15739.1i	− 1.72289i
$438$	0	0
$439$	5183.76	0.563570	0.281785	−	0.959478i	$-0.409073\pi$
$439$	5183.76	0.563570	0.281785	+	0.959478i	$0.409073\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6956.94i	− 0.746126i	−0.927806	−	0.373063i	$-0.878307\pi$
$443$	− 6956.94i	− 0.746126i	0.927806	−	0.373063i	$-0.121693\pi$
$444$	0	0
$445$	243.206	0.0259080
$446$	0	0
$447$	12888.2	1.36373
$448$	0	0
$449$	1906.59	0.200396	0.100198	−	0.994968i	$-0.468052\pi$
$449$	1906.59	0.200396	0.100198	+	0.994968i	$0.468052\pi$
$450$	0	0
$451$	556.933	0.0581484
$452$	0	0
$453$	− 13296.3i	− 1.37906i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5974.66	−0.611560	−0.305780	−	0.952102i	$-0.598917\pi$
$457$	−5974.66	−0.611560	−0.305780	+	0.952102i	$0.598917\pi$
$458$	0	0
$459$	− 6225.00i	− 0.633024i
$460$	0	0
$461$	12381.0i	1.25085i	0.780283	+	0.625426i	$0.215075\pi$
$461$	12381.0i	1.25085i	−0.780283	+	0.625426i	$0.784925\pi$
$462$	0	0
$463$	9559.14i	0.959505i	0.877404	+	0.479753i	$0.159274\pi$
$463$	9559.14i	0.959505i	−0.877404	+	0.479753i	$0.840726\pi$
$464$	0	0
$465$	11135.8i	1.11056i
$466$	0	0
$467$	−15638.8	−1.54963	−0.774814	−	0.632189i	$-0.782157\pi$
$467$	−15638.8	−1.54963	−0.774814	+	0.632189i	$0.782157\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2881.72i	0.281917i
$472$	0	0
$473$	842.485	0.0818975
$474$	0	0
$475$	−6289.11	−0.607503
$476$	0	0
$477$	−4131.22	−0.396553
$478$	0	0
$479$	3212.33	0.306420	0.153210	−	0.988194i	$-0.451039\pi$
$479$	3212.33	0.306420	0.153210	+	0.988194i	$0.451039\pi$
$480$	0	0
$481$	− 3154.67i	− 0.299044i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11143.0	−1.04325
$486$	0	0
$487$	− 12900.2i	− 1.20033i	−0.799875	−	0.600166i	$-0.795101\pi$
$487$	− 12900.2i	− 1.20033i	0.799875	−	0.600166i	$-0.204899\pi$
$488$	0	0
$489$	− 1364.53i	− 0.126188i
$490$	0	0
$491$	9459.34i	0.869438i	0.900566	+	0.434719i	$0.143152\pi$
$491$	9459.34i	0.869438i	−0.900566	+	0.434719i	$0.856848\pi$
$492$	0	0
$493$	− 6599.97i	− 0.602936i
$494$	0	0
$495$	185.859	0.0168763
$496$	0	0
$497$	0	0
$498$	0	0
$499$	11725.9i	1.05195i	0.850501	+	0.525973i	$0.176299\pi$
$499$	11725.9i	1.05195i	−0.850501	+	0.525973i	$0.823701\pi$
$500$	0	0
$501$	1676.46	0.149499
$502$	0	0
$503$	−18280.0	−1.62041	−0.810203	−	0.586149i	$-0.800643\pi$
$503$	−18280.0	−1.62041	−0.810203	+	0.586149i	$0.800643\pi$
$504$	0	0
$505$	−3380.02	−0.297840
$506$	0	0
$507$	13125.0	1.14971
$508$	0	0
$509$	15875.6i	1.38246i	0.722633	+	0.691232i	$0.242932\pi$
$509$	15875.6i	1.38246i	−0.722633	+	0.691232i	$0.757068\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7121.42	0.612901
$514$	0	0
$515$	2591.32i	0.221723i
$516$	0	0
$517$	838.119i	0.0712968i
$518$	0	0
$519$	6380.75i	0.539661i
$520$	0	0
$521$	16824.0i	1.41473i	0.706849	+	0.707365i	$0.250116\pi$
$521$	16824.0i	1.41473i	−0.706849	+	0.707365i	$0.749884\pi$
$522$	0	0
$523$	−8798.84	−0.735653	−0.367826	−	0.929894i	$-0.619898\pi$
$523$	−8798.84	−0.735653	−0.367826	+	0.929894i	$0.619898\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18085.9i	1.49494i
$528$	0	0
$529$	−24772.7	−2.03605
$530$	0	0
$531$	2457.32	0.200826
$532$	0	0
$533$	3149.42	0.255941
$534$	0	0
$535$	13180.2	1.06510
$536$	0	0
$537$	2339.34i	0.187989i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	20544.0	1.63264	0.816319	−	0.577602i	$-0.196011\pi$
$541$	20544.0	1.63264	0.816319	+	0.577602i	$0.196011\pi$
$542$	0	0
$543$	15719.3i	1.24232i
$544$	0	0
$545$	2963.10i	0.232890i
$546$	0	0
$547$	− 11300.4i	− 0.883307i	−0.897186	−	0.441654i	$-0.854392\pi$
$547$	− 11300.4i	− 0.883307i	0.897186	−	0.441654i	$-0.145608\pi$
$548$	0	0
$549$	− 1979.79i	− 0.153908i
$550$	0	0
$551$	7550.38	0.583770
$552$	0	0
$553$	0	0
$554$	0	0
$555$	12216.9i	0.934373i
$556$	0	0
$557$	22747.3	1.73040	0.865202	−	0.501423i	$-0.167190\pi$
$557$	22747.3	1.73040	0.865202	+	0.501423i	$0.167190\pi$
$558$	0	0
$559$	4764.20	0.360472
$560$	0	0
$561$	914.580	0.0688299
$562$	0	0
$563$	−269.343	−0.0201624	−0.0100812	−	0.999949i	$-0.503209\pi$
$563$	−269.343	−0.0201624	−0.0100812	+	0.999949i	$0.503209\pi$
$564$	0	0
$565$	− 12240.4i	− 0.911431i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1167.37	−0.0860086	−0.0430043	−	0.999075i	$-0.513693\pi$
$569$	−1167.37	−0.0860086	−0.0430043	+	0.999075i	$0.513693\pi$
$570$	0	0
$571$	− 4687.58i	− 0.343553i	−0.985136	−	0.171777i	$-0.945049\pi$
$571$	− 4687.58i	− 0.343553i	0.985136	−	0.171777i	$-0.0549508\pi$
$572$	0	0
$573$	13624.6i	0.993327i
$574$	0	0
$575$	14760.5i	1.07053i
$576$	0	0
$577$	10061.8i	0.725961i	0.931797	+	0.362981i	$0.118241\pi$
$577$	10061.8i	0.725961i	−0.931797	+	0.362981i	$0.881759\pi$
$578$	0	0
$579$	−17470.3	−1.25396
$580$	0	0
$581$	0	0
$582$	0	0
$583$	625.075i	0.0444047i
$584$	0	0
$585$	1051.02	0.0742810
$586$	0	0
$587$	−298.565	−0.0209933	−0.0104967	−	0.999945i	$-0.503341\pi$
$587$	−298.565	−0.0209933	−0.0104967	+	0.999945i	$0.503341\pi$
$588$	0	0
$589$	−20690.3	−1.44742
$590$	0	0
$591$	−12467.1	−0.867727
$592$	0	0
$593$	13929.2i	0.964591i	0.876008	+	0.482296i	$0.160197\pi$
$593$	13929.2i	0.964591i	−0.876008	+	0.482296i	$0.839803\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	28629.1	1.96266
$598$	0	0
$599$	− 15329.5i	− 1.04565i	−0.852439	−	0.522827i	$-0.824877\pi$
$599$	− 15329.5i	− 1.04565i	0.852439	−	0.522827i	$-0.175123\pi$
$600$	0	0
$601$	5792.35i	0.393136i	0.980490	+	0.196568i	$0.0629797\pi$
$601$	5792.35i	0.393136i	−0.980490	+	0.196568i	$0.937020\pi$
$602$	0	0
$603$	2398.58i	0.161986i
$604$	0	0
$605$	9212.61i	0.619084i
$606$	0	0
$607$	−20161.8	−1.34818	−0.674089	−	0.738650i	$-0.735463\pi$
$607$	−20161.8	−1.34818	−0.674089	+	0.738650i	$0.735463\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4739.51i	0.313814i
$612$	0	0
$613$	−8562.45	−0.564167	−0.282083	−	0.959390i	$-0.591025\pi$
$613$	−8562.45	−0.564167	−0.282083	+	0.959390i	$0.591025\pi$
$614$	0	0
$615$	−12196.5	−0.799695
$616$	0	0
$617$	−12256.5	−0.799723	−0.399862	−	0.916576i	$-0.630942\pi$
$617$	−12256.5	−0.799723	−0.399862	+	0.916576i	$0.630942\pi$
$618$	0	0
$619$	12946.4	0.840645	0.420322	−	0.907375i	$-0.361917\pi$
$619$	12946.4	0.840645	0.420322	+	0.907375i	$0.361917\pi$
$620$	0	0
$621$	− 16714.0i	− 1.08005i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−127.041	−0.00813065
$626$	0	0
$627$	1046.28i	0.0666419i
$628$	0	0
$629$	19841.6i	1.25777i
$630$	0	0
$631$	24477.7i	1.54428i	0.635453	+	0.772140i	$0.280814\pi$
$631$	24477.7i	1.54428i	−0.635453	+	0.772140i	$0.719186\pi$
$632$	0	0
$633$	31401.5i	1.97172i
$634$	0	0
$635$	17377.5	1.08599
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 6071.38i	− 0.375868i
$640$	0	0
$641$	14547.6	0.896405	0.448202	−	0.893932i	$-0.352064\pi$
$641$	14547.6	0.896405	0.448202	+	0.893932i	$0.352064\pi$
$642$	0	0
$643$	17488.0	1.07256	0.536282	−	0.844039i	$-0.319828\pi$
$643$	17488.0	1.07256	0.536282	+	0.844039i	$0.319828\pi$
$644$	0	0
$645$	−18450.0	−1.12631
$646$	0	0
$647$	13545.4	0.823067	0.411533	−	0.911395i	$-0.364993\pi$
$647$	13545.4	0.823067	0.411533	+	0.911395i	$0.364993\pi$
$648$	0	0
$649$	− 371.806i	− 0.0224879i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2451.08	−0.146888	−0.0734442	−	0.997299i	$-0.523399\pi$
$653$	−2451.08	−0.146888	−0.0734442	+	0.997299i	$0.523399\pi$
$654$	0	0
$655$	15989.8i	0.953852i
$656$	0	0
$657$	11633.8i	0.690831i
$658$	0	0
$659$	− 14589.7i	− 0.862419i	−0.902252	−	0.431210i	$-0.858087\pi$
$659$	− 14589.7i	− 0.862419i	0.902252	−	0.431210i	$-0.141913\pi$
$660$	0	0
$661$	− 8036.29i	− 0.472883i	−0.971646	−	0.236441i	$-0.924019\pi$
$661$	− 8036.29i	− 0.472883i	0.971646	−	0.236441i	$-0.0759812\pi$
$662$	0	0
$663$	5171.89	0.302956
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 17720.8i	− 1.02871i
$668$	0	0
$669$	−3513.72	−0.203062
$670$	0	0
$671$	−299.552	−0.0172341
$672$	0	0
$673$	−290.299	−0.0166274	−0.00831369	−	0.999965i	$-0.502646\pi$
$673$	−290.299	−0.0166274	−0.00831369	+	0.999965i	$0.502646\pi$
$674$	0	0
$675$	−6678.65	−0.380832
$676$	0	0
$677$	2307.73i	0.131009i	0.997852	+	0.0655046i	$0.0208657\pi$
$677$	2307.73i	0.131009i	−0.997852	+	0.0655046i	$0.979134\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	20029.4	1.12706
$682$	0	0
$683$	− 6677.56i	− 0.374099i	−0.982350	−	0.187050i	$-0.940107\pi$
$683$	− 6677.56i	− 0.374099i	0.982350	−	0.187050i	$-0.0598925\pi$
$684$	0	0
$685$	239.541i	0.0133611i
$686$	0	0
$687$	− 38202.6i	− 2.12157i
$688$	0	0
$689$	3534.76i	0.195448i
$690$	0	0
$691$	32363.5	1.78171	0.890857	−	0.454284i	$-0.150105\pi$
$691$	32363.5	1.78171	0.890857	+	0.454284i	$0.150105\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 9810.93i	− 0.535467i
$696$	0	0
$697$	−19808.6	−1.07648
$698$	0	0
$699$	−19693.1	−1.06561
$700$	0	0
$701$	−19434.3	−1.04711	−0.523555	−	0.851992i	$-0.675394\pi$
$701$	−19434.3	−1.04711	−0.523555	+	0.851992i	$0.675394\pi$
$702$	0	0
$703$	−22698.9	−1.21779
$704$	0	0
$705$	− 18354.4i	− 0.980519i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	27781.0	1.47156	0.735781	−	0.677220i	$-0.236815\pi$
$709$	27781.0	1.47156	0.735781	+	0.677220i	$0.236815\pi$
$710$	0	0
$711$	− 5696.59i	− 0.300477i
$712$	0	0
$713$	48560.1i	2.55062i
$714$	0	0
$715$	− 159.025i	− 0.00831775i
$716$	0	0
$717$	5221.71i	0.271978i
$718$	0	0
$719$	−9786.09	−0.507593	−0.253797	−	0.967258i	$-0.581679\pi$
$719$	−9786.09	−0.507593	−0.253797	+	0.967258i	$0.581679\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 5643.75i	− 0.290309i
$724$	0	0
$725$	−7080.94	−0.362731
$726$	0	0
$727$	19674.1	1.00367	0.501837	−	0.864962i	$-0.332658\pi$
$727$	19674.1	1.00367	0.501837	+	0.864962i	$0.332658\pi$
$728$	0	0
$729$	2785.40	0.141513
$730$	0	0
$731$	−29965.0	−1.51613
$732$	0	0
$733$	11333.6i	0.571099i	0.958364	+	0.285549i	$0.0921761\pi$
$733$	11333.6i	0.571099i	−0.958364	+	0.285549i	$0.907824\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	362.917	0.0181387
$738$	0	0
$739$	16949.6i	0.843709i	0.906664	+	0.421854i	$0.138621\pi$
$739$	16949.6i	0.843709i	−0.906664	+	0.421854i	$0.861379\pi$
$740$	0	0
$741$	5916.66i	0.293325i
$742$	0	0
$743$	− 23375.0i	− 1.15417i	−0.816685	−	0.577083i	$-0.804191\pi$
$743$	− 23375.0i	− 1.15417i	0.816685	−	0.577083i	$-0.195809\pi$
$744$	0	0
$745$	14094.8i	0.693145i
$746$	0	0
$747$	−4483.95	−0.219624
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1318.01i	0.0640412i	0.999487	+	0.0320206i	$0.0101942\pi$
$751$	1318.01i	0.0640412i	−0.999487	+	0.0320206i	$0.989806\pi$
$752$	0	0
$753$	−33303.8	−1.61176
$754$	0	0
$755$	14541.1	0.700933
$756$	0	0
$757$	−7569.63	−0.363439	−0.181719	−	0.983350i	$-0.558166\pi$
$757$	−7569.63	−0.363439	−0.181719	+	0.983350i	$0.558166\pi$
$758$	0	0
$759$	2455.62	0.117435
$760$	0	0
$761$	40270.0i	1.91825i	0.282985	+	0.959124i	$0.408675\pi$
$761$	40270.0i	1.91825i	−0.282985	+	0.959124i	$0.591325\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6610.52	−0.312423
$766$	0	0
$767$	− 2102.54i	− 0.0989807i
$768$	0	0
$769$	1523.73i	0.0714528i	0.999362	+	0.0357264i	$0.0113745\pi$
$769$	1523.73i	0.0714528i	−0.999362	+	0.0357264i	$0.988626\pi$
$770$	0	0
$771$	42604.5i	1.99010i
$772$	0	0
$773$	6015.36i	0.279893i	0.990159	+	0.139947i	$0.0446931\pi$
$773$	6015.36i	0.279893i	−0.990159	+	0.139947i	$0.955307\pi$
$774$	0	0
$775$	19403.9	0.899364
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 22661.1i	− 1.04226i
$780$	0	0
$781$	−918.630	−0.0420886
$782$	0	0
$783$	8018.05	0.365954
$784$	0	0
$785$	−3151.51	−0.143290
$786$	0	0
$787$	−22547.0	−1.02124	−0.510619	−	0.859807i	$-0.670584\pi$
$787$	−22547.0	−1.02124	−0.510619	+	0.859807i	$0.670584\pi$
$788$	0	0
$789$	− 33481.5i	− 1.51074i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1693.95	−0.0758561
$794$	0	0
$795$	− 13688.8i	− 0.610683i
$796$	0	0
$797$	32775.0i	1.45665i	0.685231	+	0.728326i	$0.259701\pi$
$797$	32775.0i	1.45665i	−0.685231	+	0.728326i	$0.740299\pi$
$798$	0	0
$799$	− 29809.7i	− 1.31989i
$800$	0	0
$801$	− 465.959i	− 0.0205541i
$802$	0	0
$803$	1760.25	0.0773571
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 44517.9i	− 1.94189i
$808$	0	0
$809$	26945.1	1.17100	0.585501	−	0.810672i	$-0.300898\pi$
$809$	26945.1	1.17100	0.585501	+	0.810672i	$0.300898\pi$
$810$	0	0
$811$	−34211.3	−1.48128	−0.740642	−	0.671900i	$-0.765479\pi$
$811$	−34211.3	−1.48128	−0.740642	+	0.671900i	$0.765479\pi$
$812$	0	0
$813$	1676.46	0.0723199
$814$	0	0
$815$	1492.28	0.0641377
$816$	0	0
$817$	− 34280.0i	− 1.46794i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22612.8	−0.961258	−0.480629	−	0.876924i	$-0.659592\pi$
$821$	−22612.8	−0.961258	−0.480629	+	0.876924i	$0.659592\pi$
$822$	0	0
$823$	− 12932.6i	− 0.547755i	−0.961765	−	0.273877i	$-0.911694\pi$
$823$	− 12932.6i	− 0.547755i	0.961765	−	0.273877i	$-0.0883062\pi$
$824$	0	0
$825$	− 981.230i	− 0.0414085i
$826$	0	0
$827$	− 18340.4i	− 0.771172i	−0.922672	−	0.385586i	$-0.873999\pi$
$827$	− 18340.4i	− 0.771172i	0.922672	−	0.385586i	$-0.126001\pi$
$828$	0	0
$829$	18232.3i	0.763851i	0.924193	+	0.381925i	$0.124739\pi$
$829$	18232.3i	0.763851i	−0.924193	+	0.381925i	$0.875261\pi$
$830$	0	0
$831$	39438.9	1.64636
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1833.42i	0.0759856i
$836$	0	0
$837$	−21971.8	−0.907356
$838$	0	0
$839$	31127.7	1.28087	0.640434	−	0.768014i	$-0.278755\pi$
$839$	31127.7	1.28087	0.640434	+	0.768014i	$0.278755\pi$
$840$	0	0
$841$	−15888.0	−0.651440
$842$	0	0
$843$	41140.3	1.68084
$844$	0	0
$845$	14353.8i	0.584363i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	24095.2	0.974024
$850$	0	0
$851$	53274.3i	2.14597i
$852$	0	0
$853$	− 29716.0i	− 1.19280i	−0.802688	−	0.596399i	$-0.796598\pi$
$853$	− 29716.0i	− 1.19280i	0.802688	−	0.596399i	$-0.203402\pi$
$854$	0	0
$855$	− 7562.45i	− 0.302492i
$856$	0	0
$857$	− 26268.3i	− 1.04704i	−0.852015	−	0.523518i	$-0.824619\pi$
$857$	− 26268.3i	− 1.04704i	0.852015	−	0.523518i	$-0.175381\pi$
$858$	0	0
$859$	−9898.86	−0.393184	−0.196592	−	0.980485i	$-0.562987\pi$
$859$	−9898.86	−0.393184	−0.196592	+	0.980485i	$0.562987\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24931.7i	0.983414i	0.870761	+	0.491707i	$0.163627\pi$
$863$	24931.7i	0.983414i	−0.870761	+	0.491707i	$0.836373\pi$
$864$	0	0
$865$	−6978.13	−0.274293
$866$	0	0
$867$	−1339.75	−0.0524803
$868$	0	0
$869$	−861.923	−0.0336464
$870$	0	0
$871$	2052.27	0.0798377
$872$	0	0
$873$	21348.9i	0.827662i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13437.8	0.517403	0.258701	−	0.965957i	$-0.416705\pi$
$877$	13437.8	0.517403	0.258701	+	0.965957i	$0.416705\pi$
$878$	0	0
$879$	− 30736.4i	− 1.17943i
$880$	0	0
$881$	1327.03i	0.0507477i	0.999678	+	0.0253738i	$0.00807761\pi$
$881$	1327.03i	0.0507477i	−0.999678	+	0.0253738i	$0.991922\pi$
$882$	0	0
$883$	10846.1i	0.413365i	0.978408	+	0.206682i	$0.0662667\pi$
$883$	10846.1i	0.413365i	−0.978408	+	0.206682i	$0.933733\pi$
$884$	0	0
$885$	8142.35i	0.309268i
$886$	0	0
$887$	−45321.9	−1.71563	−0.857813	−	0.513962i	$-0.828177\pi$
$887$	−45321.9	−1.71563	−0.857813	+	0.513962i	$0.828177\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1833.89i	0.0689536i
$892$	0	0
$893$	34102.4	1.27793
$894$	0	0
$895$	−2558.35	−0.0955489
$896$	0	0
$897$	13886.4	0.516893
$898$	0	0
$899$	−23295.3	−0.864229
$900$	0	0
$901$	− 22232.3i	− 0.822047i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17191.0	−0.631433
$906$	0	0
$907$	− 5059.34i	− 0.185218i	−0.995703	−	0.0926089i	$-0.970479\pi$
$907$	− 5059.34i	− 0.185218i	0.995703	−	0.0926089i	$-0.0295206\pi$
$908$	0	0
$909$	6475.79i	0.236291i
$910$	0	0
$911$	38008.9i	1.38232i	0.722702	+	0.691159i	$0.242900\pi$
$911$	38008.9i	1.38232i	−0.722702	+	0.691159i	$0.757100\pi$
$912$	0	0
$913$	678.444i	0.0245928i
$914$	0	0
$915$	6560.04	0.237015
$916$	0	0
$917$	0	0
$918$	0	0
$919$	44547.9i	1.59902i	0.600653	+	0.799510i	$0.294907\pi$
$919$	44547.9i	1.59902i	−0.600653	+	0.799510i	$0.705093\pi$
$920$	0	0
$921$	16119.2	0.576705
$922$	0	0
$923$	−5194.79	−0.185253
$924$	0	0
$925$	21287.6	0.756683
$926$	0	0
$927$	4964.72	0.175904
$928$	0	0
$929$	29299.2i	1.03474i	0.855761	+	0.517371i	$0.173089\pi$
$929$	29299.2i	1.03474i	−0.855761	+	0.517371i	$0.826911\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	6013.23	0.211001
$934$	0	0
$935$	1000.20i	0.0349842i
$936$	0	0
$937$	− 35972.9i	− 1.25420i	−0.778939	−	0.627099i	$-0.784242\pi$
$937$	− 35972.9i	− 1.25420i	0.778939	−	0.627099i	$-0.215758\pi$
$938$	0	0
$939$	24677.6i	0.857639i
$940$	0	0
$941$	− 26797.7i	− 0.928353i	−0.885743	−	0.464177i	$-0.846350\pi$
$941$	− 26797.7i	− 0.928353i	0.885743	−	0.464177i	$-0.153650\pi$
$942$	0	0
$943$	−53185.7	−1.83665
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30497.6i	1.04650i	0.852178	+	0.523252i	$0.175281\pi$
$947$	30497.6i	1.04650i	−0.852178	+	0.523252i	$0.824719\pi$
$948$	0	0
$949$	9954.08	0.340488
$950$	0	0
$951$	−38000.2	−1.29573
$952$	0	0
$953$	−7122.53	−0.242100	−0.121050	−	0.992646i	$-0.538626\pi$
$953$	−7122.53	−0.242100	−0.121050	+	0.992646i	$0.538626\pi$
$954$	0	0
$955$	−14900.2	−0.504878
$956$	0	0
$957$	1178.02i	0.0397908i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	34045.0	1.14279
$962$	0	0
$963$	− 25251.9i	− 0.844997i
$964$	0	0
$965$	− 19105.9i	− 0.637349i
$966$	0	0
$967$	− 31747.1i	− 1.05576i	−0.849320	−	0.527879i	$-0.822988\pi$
$967$	− 31747.1i	− 1.05576i	0.849320	−	0.527879i	$-0.177012\pi$
$968$	0	0
$969$	− 37213.5i	− 1.23371i
$970$	0	0
$971$	40890.2	1.35142	0.675710	−	0.737167i	$-0.263837\pi$
$971$	40890.2	1.35142	0.675710	+	0.737167i	$0.263837\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 5548.79i	− 0.182260i
$976$	0	0
$977$	23215.4	0.760212	0.380106	−	0.924943i	$-0.375887\pi$
$977$	23215.4	0.760212	0.380106	+	0.924943i	$0.375887\pi$
$978$	0	0
$979$	−70.5019	−0.00230158
$980$	0	0
$981$	5677.01	0.184763
$982$	0	0
$983$	33405.6	1.08390	0.541949	−	0.840411i	$-0.317686\pi$
$983$	33405.6	1.08390	0.541949	+	0.840411i	$0.317686\pi$
$984$	0	0
$985$	− 13634.3i	− 0.441039i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−80455.2	−2.58678
$990$	0	0
$991$	38876.1i	1.24616i	0.782160	+	0.623078i	$0.214118\pi$
$991$	38876.1i	1.24616i	−0.782160	+	0.623078i	$0.785882\pi$
$992$	0	0
$993$	− 68945.9i	− 2.20336i
$994$	0	0
$995$	31309.4i	0.997562i
$996$	0	0
$997$	21309.0i	0.676892i	0.940986	+	0.338446i	$0.109901\pi$
$997$	21309.0i	0.676892i	−0.940986	+	0.338446i	$0.890099\pi$
$998$	0	0
$999$	−24104.8	−0.763407

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.i.783.6	yes	8
4.3	odd	2	inner	784.4.f.i.783.4	yes	8
7.6	odd	2	inner	784.4.f.i.783.3	✓	8
28.27	even	2	inner	784.4.f.i.783.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.4.f.i.783.3	✓	8	7.6	odd	2	inner
784.4.f.i.783.4	yes	8	4.3	odd	2	inner
784.4.f.i.783.5	yes	8	28.27	even	2	inner
784.4.f.i.783.6	yes	8	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.34835 q^{3} +6.94269i q^{5} +13.3015 q^{9} +O(q^{10})\)	\(q+6.34835 q^{3} +6.94269i q^{5} +13.3015 q^{9} -2.01259i q^{11} -11.3810i q^{13} +44.0746i q^{15} +71.5824i q^{17} -81.8905 q^{19} +192.197i q^{23} +76.7990 q^{25} -86.9627 q^{27} -92.2010 q^{29} +252.658 q^{31} -12.7766i q^{33} +277.186 q^{37} -72.2509i q^{39} +276.725i q^{41} +418.608i q^{43} +92.3484i q^{45} -416.439 q^{47} +454.430i q^{51} -310.583 q^{53} +13.9728 q^{55} -519.869 q^{57} +184.740 q^{59} -148.839i q^{61} +79.0152 q^{65} +180.324i q^{67} +1220.13i q^{69} -456.443i q^{71} +874.619i q^{73} +487.547 q^{75} -428.266i q^{79} -911.211 q^{81} -337.100 q^{83} -496.975 q^{85} -585.324 q^{87} -35.0305i q^{89} +1603.96 q^{93} -568.541i q^{95} +1604.99i q^{97} -26.7705i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 344 q^{9}+O(q^{10}) \) 8 * q + 344 * q^9	\( 8 q + 344 q^{9} + 456 q^{25} - 896 q^{29} + 208 q^{53} + 672 q^{57} + 3008 q^{65} + 6728 q^{81} - 16 q^{85} + 6496 q^{93}+O(q^{100}) \) 8 * q + 344 * q^9 + 456 * q^25 - 896 * q^29 + 208 * q^53 + 672 * q^57 + 3008 * q^65 + 6728 * q^81 - 16 * q^85 + 6496 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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