LMFDB - Embedded newform 784.4.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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([0, 0, 0]))

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

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

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.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:	$46.2574974445$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{3} +12.0000 q^{5} -11.0000 q^{9} +O(q^{10})$

$q+4.00000 q^{3} +12.0000 q^{5} -11.0000 q^{9} -12.0000 q^{11} -76.0000 q^{13} +48.0000 q^{15} +8.00000 q^{17} -100.000 q^{19} +56.0000 q^{23} +19.0000 q^{25} -152.000 q^{27} -166.000 q^{29} -232.000 q^{31} -48.0000 q^{33} -414.000 q^{37} -304.000 q^{39} -72.0000 q^{41} +452.000 q^{43} -132.000 q^{45} +424.000 q^{47} +32.0000 q^{51} -18.0000 q^{53} -144.000 q^{55} -400.000 q^{57} +444.000 q^{59} +284.000 q^{61} -912.000 q^{65} -524.000 q^{67} +224.000 q^{69} +1008.00 q^{71} -896.000 q^{73} +76.0000 q^{75} +40.0000 q^{79} -311.000 q^{81} +1388.00 q^{83} +96.0000 q^{85} -664.000 q^{87} -448.000 q^{89} -928.000 q^{93} -1200.00 q^{95} +824.000 q^{97} +132.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	4.00000	0.769800	0.384900	−	0.922958i	$-0.374236\pi$
$3$	4.00000	0.769800	0.384900	+	0.922958i	$0.374236\pi$
$4$	0	0
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	−76.0000	−1.62143	−0.810716	−	0.585440i	$-0.800922\pi$
$13$	−76.0000	−1.62143	−0.810716	+	0.585440i	$0.800922\pi$
$14$	0	0
$15$	48.0000	0.826236
$16$	0	0
$17$	8.00000	0.114134	0.0570672	−	0.998370i	$-0.481825\pi$
$17$	8.00000	0.114134	0.0570672	+	0.998370i	$0.481825\pi$
$18$	0	0
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000	−1.20745	−0.603726	+	0.797192i	$0.706318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	56.0000	0.507687	0.253844	−	0.967245i	$-0.418305\pi$
$23$	56.0000	0.507687	0.253844	+	0.967245i	$0.418305\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	0	0
$27$	−152.000	−1.08342
$28$	0	0
$29$	−166.000	−1.06295	−0.531473	−	0.847075i	$-0.678361\pi$
$29$	−166.000	−1.06295	−0.531473	+	0.847075i	$0.678361\pi$
$30$	0	0
$31$	−232.000	−1.34414	−0.672071	−	0.740486i	$-0.734595\pi$
$31$	−232.000	−1.34414	−0.672071	+	0.740486i	$0.734595\pi$
$32$	0	0
$33$	−48.0000	−0.253204
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−414.000	−1.83949	−0.919746	−	0.392515i	$-0.871605\pi$
$37$	−414.000	−1.83949	−0.919746	+	0.392515i	$0.871605\pi$
$38$	0	0
$39$	−304.000	−1.24818
$40$	0	0
$41$	−72.0000	−0.274256	−0.137128	−	0.990553i	$-0.543787\pi$
$41$	−72.0000	−0.274256	−0.137128	+	0.990553i	$0.543787\pi$
$42$	0	0
$43$	452.000	1.60301	0.801504	−	0.597989i	$-0.204033\pi$
$43$	452.000	1.60301	0.801504	+	0.597989i	$0.204033\pi$
$44$	0	0
$45$	−132.000	−0.437276
$46$	0	0
$47$	424.000	1.31589	0.657944	−	0.753067i	$-0.271426\pi$
$47$	424.000	1.31589	0.657944	+	0.753067i	$0.271426\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	32.0000	0.0878607
$52$	0	0
$53$	−18.0000	−0.0466508	−0.0233254	−	0.999728i	$-0.507425\pi$
$53$	−18.0000	−0.0466508	−0.0233254	+	0.999728i	$0.507425\pi$
$54$	0	0
$55$	−144.000	−0.353036
$56$	0	0
$57$	−400.000	−0.929496
$58$	0	0
$59$	444.000	0.979727	0.489863	−	0.871799i	$-0.337047\pi$
$59$	444.000	0.979727	0.489863	+	0.871799i	$0.337047\pi$
$60$	0	0
$61$	284.000	0.596106	0.298053	−	0.954549i	$-0.403663\pi$
$61$	284.000	0.596106	0.298053	+	0.954549i	$0.403663\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−912.000	−1.74030
$66$	0	0
$67$	−524.000	−0.955474	−0.477737	−	0.878503i	$-0.658543\pi$
$67$	−524.000	−0.955474	−0.477737	+	0.878503i	$0.658543\pi$
$68$	0	0
$69$	224.000	0.390818
$70$	0	0
$71$	1008.00	1.68490	0.842448	−	0.538778i	$-0.181114\pi$
$71$	1008.00	1.68490	0.842448	+	0.538778i	$0.181114\pi$
$72$	0	0
$73$	−896.000	−1.43656	−0.718280	−	0.695754i	$-0.755070\pi$
$73$	−896.000	−1.43656	−0.718280	+	0.695754i	$0.755070\pi$
$74$	0	0
$75$	76.0000	0.117010
$76$	0	0
$77$	0	0
$78$	0	0
$79$	40.0000	0.0569665	0.0284832	−	0.999594i	$-0.490932\pi$
$79$	40.0000	0.0569665	0.0284832	+	0.999594i	$0.490932\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	1388.00	1.83558	0.917788	−	0.397071i	$-0.129973\pi$
$83$	1388.00	1.83558	0.917788	+	0.397071i	$0.129973\pi$
$84$	0	0
$85$	96.0000	0.122502
$86$	0	0
$87$	−664.000	−0.818256
$88$	0	0
$89$	−448.000	−0.533572	−0.266786	−	0.963756i	$-0.585962\pi$
$89$	−448.000	−0.533572	−0.266786	+	0.963756i	$0.585962\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−928.000	−1.03472
$94$	0	0
$95$	−1200.00	−1.29597
$96$	0	0
$97$	824.000	0.862521	0.431260	−	0.902227i	$-0.358069\pi$
$97$	824.000	0.862521	0.431260	+	0.902227i	$0.358069\pi$
$98$	0	0
$99$	132.000	0.134005
$100$	0	0
$101$	1916.00	1.88762	0.943808	−	0.330496i	$-0.107216\pi$
$101$	1916.00	1.88762	0.943808	+	0.330496i	$0.107216\pi$
$102$	0	0
$103$	−824.000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−824.000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1796.00	−1.62267	−0.811336	−	0.584580i	$-0.801259\pi$
$107$	−1796.00	−1.62267	−0.811336	+	0.584580i	$0.801259\pi$
$108$	0	0
$109$	−1238.00	−1.08788	−0.543940	−	0.839124i	$-0.683068\pi$
$109$	−1238.00	−1.08788	−0.543940	+	0.839124i	$0.683068\pi$
$110$	0	0
$111$	−1656.00	−1.41604
$112$	0	0
$113$	626.000	0.521143	0.260571	−	0.965455i	$-0.416089\pi$
$113$	626.000	0.521143	0.260571	+	0.965455i	$0.416089\pi$
$114$	0	0
$115$	672.000	0.544907
$116$	0	0
$117$	836.000	0.660583
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	−288.000	−0.211123
$124$	0	0
$125$	−1272.00	−0.910169
$126$	0	0
$127$	160.000	0.111793	0.0558965	−	0.998437i	$-0.482198\pi$
$127$	160.000	0.111793	0.0558965	+	0.998437i	$0.482198\pi$
$128$	0	0
$129$	1808.00	1.23400
$130$	0	0
$131$	−724.000	−0.482872	−0.241436	−	0.970417i	$-0.577618\pi$
$131$	−724.000	−0.482872	−0.241436	+	0.970417i	$0.577618\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1824.00	−1.16285
$136$	0	0
$137$	−1370.00	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−1370.00	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	852.000	0.519897	0.259949	−	0.965622i	$-0.416294\pi$
$139$	852.000	0.519897	0.259949	+	0.965622i	$0.416294\pi$
$140$	0	0
$141$	1696.00	1.01297
$142$	0	0
$143$	912.000	0.533324
$144$	0	0
$145$	−1992.00	−1.14087
$146$	0	0
$147$	0	0
$148$	0	0
$149$	750.000	0.412365	0.206183	−	0.978514i	$-0.433896\pi$
$149$	750.000	0.412365	0.206183	+	0.978514i	$0.433896\pi$
$150$	0	0
$151$	440.000	0.237130	0.118565	−	0.992946i	$-0.462171\pi$
$151$	440.000	0.237130	0.118565	+	0.992946i	$0.462171\pi$
$152$	0	0
$153$	−88.0000	−0.0464992
$154$	0	0
$155$	−2784.00	−1.44269
$156$	0	0
$157$	−1660.00	−0.843837	−0.421919	−	0.906634i	$-0.638643\pi$
$157$	−1660.00	−0.843837	−0.421919	+	0.906634i	$0.638643\pi$
$158$	0	0
$159$	−72.0000	−0.0359118
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2532.00	−1.21670	−0.608348	−	0.793670i	$-0.708168\pi$
$163$	−2532.00	−1.21670	−0.608348	+	0.793670i	$0.708168\pi$
$164$	0	0
$165$	−576.000	−0.271767
$166$	0	0
$167$	1048.00	0.485609	0.242804	−	0.970075i	$-0.421933\pi$
$167$	1048.00	0.485609	0.242804	+	0.970075i	$0.421933\pi$
$168$	0	0
$169$	3579.00	1.62904
$170$	0	0
$171$	1100.00	0.491925
$172$	0	0
$173$	884.000	0.388493	0.194246	−	0.980953i	$-0.437774\pi$
$173$	884.000	0.388493	0.194246	+	0.980953i	$0.437774\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1776.00	0.754194
$178$	0	0
$179$	−396.000	−0.165354	−0.0826772	−	0.996576i	$-0.526347\pi$
$179$	−396.000	−0.165354	−0.0826772	+	0.996576i	$0.526347\pi$
$180$	0	0
$181$	228.000	0.0936304	0.0468152	−	0.998904i	$-0.485093\pi$
$181$	228.000	0.0936304	0.0468152	+	0.998904i	$0.485093\pi$
$182$	0	0
$183$	1136.00	0.458883
$184$	0	0
$185$	−4968.00	−1.97435
$186$	0	0
$187$	−96.0000	−0.0375413
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−888.000	−0.336405	−0.168203	−	0.985752i	$-0.553796\pi$
$191$	−888.000	−0.336405	−0.168203	+	0.985752i	$0.553796\pi$
$192$	0	0
$193$	−2942.00	−1.09725	−0.548626	−	0.836068i	$-0.684849\pi$
$193$	−2942.00	−1.09725	−0.548626	+	0.836068i	$0.684849\pi$
$194$	0	0
$195$	−3648.00	−1.33969
$196$	0	0
$197$	1086.00	0.392763	0.196381	−	0.980528i	$-0.437081\pi$
$197$	1086.00	0.392763	0.196381	+	0.980528i	$0.437081\pi$
$198$	0	0
$199$	−1544.00	−0.550006	−0.275003	−	0.961443i	$-0.588679\pi$
$199$	−1544.00	−0.550006	−0.275003	+	0.961443i	$0.588679\pi$
$200$	0	0
$201$	−2096.00	−0.735525
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−864.000	−0.294363
$206$	0	0
$207$	−616.000	−0.206836
$208$	0	0
$209$	1200.00	0.397157
$210$	0	0
$211$	−4700.00	−1.53347	−0.766733	−	0.641966i	$-0.778119\pi$
$211$	−4700.00	−1.53347	−0.766733	+	0.641966i	$0.778119\pi$
$212$	0	0
$213$	4032.00	1.29703
$214$	0	0
$215$	5424.00	1.72053
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3584.00	−1.10586
$220$	0	0
$221$	−608.000	−0.185061
$222$	0	0
$223$	3440.00	1.03300	0.516501	−	0.856287i	$-0.327234\pi$
$223$	3440.00	1.03300	0.516501	+	0.856287i	$0.327234\pi$
$224$	0	0
$225$	−209.000	−0.0619259
$226$	0	0
$227$	−4244.00	−1.24090	−0.620450	−	0.784246i	$-0.713050\pi$
$227$	−4244.00	−1.24090	−0.620450	+	0.784246i	$0.713050\pi$
$228$	0	0
$229$	3700.00	1.06770	0.533849	−	0.845580i	$-0.320745\pi$
$229$	3700.00	1.06770	0.533849	+	0.845580i	$0.320745\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3270.00	0.919420	0.459710	−	0.888069i	$-0.347953\pi$
$233$	3270.00	0.919420	0.459710	+	0.888069i	$0.347953\pi$
$234$	0	0
$235$	5088.00	1.41236
$236$	0	0
$237$	160.000	0.0438528
$238$	0	0
$239$	1856.00	0.502321	0.251160	−	0.967945i	$-0.419188\pi$
$239$	1856.00	0.502321	0.251160	+	0.967945i	$0.419188\pi$
$240$	0	0
$241$	−936.000	−0.250179	−0.125089	−	0.992145i	$-0.539922\pi$
$241$	−936.000	−0.250179	−0.125089	+	0.992145i	$0.539922\pi$
$242$	0	0
$243$	2860.00	0.755017
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7600.00	1.95780
$248$	0	0
$249$	5552.00	1.41303
$250$	0	0
$251$	−3852.00	−0.968670	−0.484335	−	0.874883i	$-0.660938\pi$
$251$	−3852.00	−0.968670	−0.484335	+	0.874883i	$0.660938\pi$
$252$	0	0
$253$	−672.000	−0.166989
$254$	0	0
$255$	384.000	0.0943020
$256$	0	0
$257$	6720.00	1.63106	0.815529	−	0.578716i	$-0.196446\pi$
$257$	6720.00	1.63106	0.815529	+	0.578716i	$0.196446\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1826.00	0.433052
$262$	0	0
$263$	−3072.00	−0.720257	−0.360129	−	0.932903i	$-0.617267\pi$
$263$	−3072.00	−0.720257	−0.360129	+	0.932903i	$0.617267\pi$
$264$	0	0
$265$	−216.000	−0.0500708
$266$	0	0
$267$	−1792.00	−0.410744
$268$	0	0
$269$	−2556.00	−0.579339	−0.289669	−	0.957127i	$-0.593545\pi$
$269$	−2556.00	−0.579339	−0.289669	+	0.957127i	$0.593545\pi$
$270$	0	0
$271$	−7616.00	−1.70716	−0.853578	−	0.520966i	$-0.825572\pi$
$271$	−7616.00	−1.70716	−0.853578	+	0.520966i	$0.825572\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−228.000	−0.0499961
$276$	0	0
$277$	1934.00	0.419505	0.209752	−	0.977755i	$-0.432734\pi$
$277$	1934.00	0.419505	0.209752	+	0.977755i	$0.432734\pi$
$278$	0	0
$279$	2552.00	0.547614
$280$	0	0
$281$	−6602.00	−1.40157	−0.700787	−	0.713371i	$-0.747168\pi$
$281$	−6602.00	−1.40157	−0.700787	+	0.713371i	$0.747168\pi$
$282$	0	0
$283$	−4708.00	−0.988910	−0.494455	−	0.869203i	$-0.664632\pi$
$283$	−4708.00	−0.988910	−0.494455	+	0.869203i	$0.664632\pi$
$284$	0	0
$285$	−4800.00	−0.997640
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4849.00	−0.986973
$290$	0	0
$291$	3296.00	0.663969
$292$	0	0
$293$	−3732.00	−0.744115	−0.372058	−	0.928210i	$-0.621348\pi$
$293$	−3732.00	−0.744115	−0.372058	+	0.928210i	$0.621348\pi$
$294$	0	0
$295$	5328.00	1.05155
$296$	0	0
$297$	1824.00	0.356361
$298$	0	0
$299$	−4256.00	−0.823180
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7664.00	1.45309
$304$	0	0
$305$	3408.00	0.639808
$306$	0	0
$307$	4812.00	0.894578	0.447289	−	0.894390i	$-0.352390\pi$
$307$	4812.00	0.894578	0.447289	+	0.894390i	$0.352390\pi$
$308$	0	0
$309$	−3296.00	−0.606805
$310$	0	0
$311$	8480.00	1.54616	0.773081	−	0.634307i	$-0.218714\pi$
$311$	8480.00	1.54616	0.773081	+	0.634307i	$0.218714\pi$
$312$	0	0
$313$	−6568.00	−1.18609	−0.593044	−	0.805170i	$-0.702074\pi$
$313$	−6568.00	−1.18609	−0.593044	+	0.805170i	$0.702074\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4710.00	0.834511	0.417255	−	0.908789i	$-0.362992\pi$
$317$	4710.00	0.834511	0.417255	+	0.908789i	$0.362992\pi$
$318$	0	0
$319$	1992.00	0.349626
$320$	0	0
$321$	−7184.00	−1.24913
$322$	0	0
$323$	−800.000	−0.137812
$324$	0	0
$325$	−1444.00	−0.246458
$326$	0	0
$327$	−4952.00	−0.837450
$328$	0	0
$329$	0	0
$330$	0	0
$331$	7796.00	1.29458	0.647291	−	0.762243i	$-0.275902\pi$
$331$	7796.00	1.29458	0.647291	+	0.762243i	$0.275902\pi$
$332$	0	0
$333$	4554.00	0.749422
$334$	0	0
$335$	−6288.00	−1.02552
$336$	0	0
$337$	−4434.00	−0.716722	−0.358361	−	0.933583i	$-0.616664\pi$
$337$	−4434.00	−0.716722	−0.358361	+	0.933583i	$0.616664\pi$
$338$	0	0
$339$	2504.00	0.401176
$340$	0	0
$341$	2784.00	0.442117
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2688.00	0.419470
$346$	0	0
$347$	−1068.00	−0.165225	−0.0826127	−	0.996582i	$-0.526326\pi$
$347$	−1068.00	−0.165225	−0.0826127	+	0.996582i	$0.526326\pi$
$348$	0	0
$349$	6708.00	1.02886	0.514428	−	0.857533i	$-0.328004\pi$
$349$	6708.00	1.02886	0.514428	+	0.857533i	$0.328004\pi$
$350$	0	0
$351$	11552.0	1.75670
$352$	0	0
$353$	9392.00	1.41611	0.708053	−	0.706159i	$-0.249574\pi$
$353$	9392.00	1.41611	0.708053	+	0.706159i	$0.249574\pi$
$354$	0	0
$355$	12096.0	1.80842
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13496.0	1.98410	0.992050	−	0.125847i	$-0.0401649\pi$
$359$	13496.0	1.98410	0.992050	+	0.125847i	$0.0401649\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	0	0
$363$	−4748.00	−0.686516
$364$	0	0
$365$	−10752.0	−1.54188
$366$	0	0
$367$	4976.00	0.707752	0.353876	−	0.935292i	$-0.384863\pi$
$367$	4976.00	0.707752	0.353876	+	0.935292i	$0.384863\pi$
$368$	0	0
$369$	792.000	0.111734
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5074.00	−0.704348	−0.352174	−	0.935935i	$-0.614557\pi$
$373$	−5074.00	−0.704348	−0.352174	+	0.935935i	$0.614557\pi$
$374$	0	0
$375$	−5088.00	−0.700649
$376$	0	0
$377$	12616.0	1.72349
$378$	0	0
$379$	−9852.00	−1.33526	−0.667630	−	0.744494i	$-0.732691\pi$
$379$	−9852.00	−1.33526	−0.667630	+	0.744494i	$0.732691\pi$
$380$	0	0
$381$	640.000	0.0860583
$382$	0	0
$383$	2568.00	0.342607	0.171304	−	0.985218i	$-0.445202\pi$
$383$	2568.00	0.342607	0.171304	+	0.985218i	$0.445202\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4972.00	−0.653077
$388$	0	0
$389$	10370.0	1.35162	0.675810	−	0.737076i	$-0.263794\pi$
$389$	10370.0	1.35162	0.675810	+	0.737076i	$0.263794\pi$
$390$	0	0
$391$	448.000	0.0579446
$392$	0	0
$393$	−2896.00	−0.371715
$394$	0	0
$395$	480.000	0.0611428
$396$	0	0
$397$	−11836.0	−1.49630	−0.748151	−	0.663529i	$-0.769058\pi$
$397$	−11836.0	−1.49630	−0.748151	+	0.663529i	$0.769058\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7950.00	0.990035	0.495018	−	0.868883i	$-0.335162\pi$
$401$	7950.00	0.990035	0.495018	+	0.868883i	$0.335162\pi$
$402$	0	0
$403$	17632.0	2.17944
$404$	0	0
$405$	−3732.00	−0.457888
$406$	0	0
$407$	4968.00	0.605048
$408$	0	0
$409$	1816.00	0.219549	0.109774	−	0.993957i	$-0.464987\pi$
$409$	1816.00	0.219549	0.109774	+	0.993957i	$0.464987\pi$
$410$	0	0
$411$	−5480.00	−0.657685
$412$	0	0
$413$	0	0
$414$	0	0
$415$	16656.0	1.97015
$416$	0	0
$417$	3408.00	0.400217
$418$	0	0
$419$	11012.0	1.28394	0.641971	−	0.766729i	$-0.278117\pi$
$419$	11012.0	1.28394	0.641971	+	0.766729i	$0.278117\pi$
$420$	0	0
$421$	2526.00	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	2526.00	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	−4664.00	−0.536103
$424$	0	0
$425$	152.000	0.0173484
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3648.00	0.410553
$430$	0	0
$431$	−5136.00	−0.573996	−0.286998	−	0.957931i	$-0.592657\pi$
$431$	−5136.00	−0.573996	−0.286998	+	0.957931i	$0.592657\pi$
$432$	0	0
$433$	−3656.00	−0.405765	−0.202882	−	0.979203i	$-0.565031\pi$
$433$	−3656.00	−0.405765	−0.202882	+	0.979203i	$0.565031\pi$
$434$	0	0
$435$	−7968.00	−0.878245
$436$	0	0
$437$	−5600.00	−0.613008
$438$	0	0
$439$	5376.00	0.584470	0.292235	−	0.956346i	$-0.405601\pi$
$439$	5376.00	0.584470	0.292235	+	0.956346i	$0.405601\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11236.0	−1.20505	−0.602526	−	0.798099i	$-0.705839\pi$
$443$	−11236.0	−1.20505	−0.602526	+	0.798099i	$0.705839\pi$
$444$	0	0
$445$	−5376.00	−0.572690
$446$	0	0
$447$	3000.00	0.317439
$448$	0	0
$449$	−1790.00	−0.188141	−0.0940705	−	0.995566i	$-0.529988\pi$
$449$	−1790.00	−0.188141	−0.0940705	+	0.995566i	$0.529988\pi$
$450$	0	0
$451$	864.000	0.0902088
$452$	0	0
$453$	1760.00	0.182543
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8966.00	−0.917750	−0.458875	−	0.888501i	$-0.651747\pi$
$457$	−8966.00	−0.917750	−0.458875	+	0.888501i	$0.651747\pi$
$458$	0	0
$459$	−1216.00	−0.123656
$460$	0	0
$461$	13732.0	1.38734	0.693669	−	0.720294i	$-0.255993\pi$
$461$	13732.0	1.38734	0.693669	+	0.720294i	$0.255993\pi$
$462$	0	0
$463$	9368.00	0.940319	0.470160	−	0.882581i	$-0.344196\pi$
$463$	9368.00	0.940319	0.470160	+	0.882581i	$0.344196\pi$
$464$	0	0
$465$	−11136.0	−1.11058
$466$	0	0
$467$	−10004.0	−0.991285	−0.495642	−	0.868527i	$-0.665067\pi$
$467$	−10004.0	−0.991285	−0.495642	+	0.868527i	$0.665067\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6640.00	−0.649586
$472$	0	0
$473$	−5424.00	−0.527264
$474$	0	0
$475$	−1900.00	−0.183533
$476$	0	0
$477$	198.000	0.0190059
$478$	0	0
$479$	−10568.0	−1.00807	−0.504034	−	0.863684i	$-0.668151\pi$
$479$	−10568.0	−1.00807	−0.504034	+	0.863684i	$0.668151\pi$
$480$	0	0
$481$	31464.0	2.98261
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9888.00	0.925755
$486$	0	0
$487$	9688.00	0.901448	0.450724	−	0.892663i	$-0.351166\pi$
$487$	9688.00	0.901448	0.450724	+	0.892663i	$0.351166\pi$
$488$	0	0
$489$	−10128.0	−0.936613
$490$	0	0
$491$	−19044.0	−1.75039	−0.875197	−	0.483766i	$-0.839268\pi$
$491$	−19044.0	−1.75039	−0.875197	+	0.483766i	$0.839268\pi$
$492$	0	0
$493$	−1328.00	−0.121319
$494$	0	0
$495$	1584.00	0.143829
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1932.00	−0.173323	−0.0866615	−	0.996238i	$-0.527620\pi$
$499$	−1932.00	−0.173323	−0.0866615	+	0.996238i	$0.527620\pi$
$500$	0	0
$501$	4192.00	0.373822
$502$	0	0
$503$	15824.0	1.40270	0.701349	−	0.712818i	$-0.252581\pi$
$503$	15824.0	1.40270	0.701349	+	0.712818i	$0.252581\pi$
$504$	0	0
$505$	22992.0	2.02600
$506$	0	0
$507$	14316.0	1.25404
$508$	0	0
$509$	−8348.00	−0.726952	−0.363476	−	0.931604i	$-0.618410\pi$
$509$	−8348.00	−0.726952	−0.363476	+	0.931604i	$0.618410\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	15200.0	1.30818
$514$	0	0
$515$	−9888.00	−0.846053
$516$	0	0
$517$	−5088.00	−0.432824
$518$	0	0
$519$	3536.00	0.299062
$520$	0	0
$521$	11208.0	0.942479	0.471239	−	0.882005i	$-0.343807\pi$
$521$	11208.0	0.942479	0.471239	+	0.882005i	$0.343807\pi$
$522$	0	0
$523$	1372.00	0.114710	0.0573550	−	0.998354i	$-0.481733\pi$
$523$	1372.00	0.114710	0.0573550	+	0.998354i	$0.481733\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1856.00	−0.153413
$528$	0	0
$529$	−9031.00	−0.742254
$530$	0	0
$531$	−4884.00	−0.399148
$532$	0	0
$533$	5472.00	0.444688
$534$	0	0
$535$	−21552.0	−1.74163
$536$	0	0
$537$	−1584.00	−0.127290
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3834.00	−0.304689	−0.152344	−	0.988327i	$-0.548682\pi$
$541$	−3834.00	−0.304689	−0.152344	+	0.988327i	$0.548682\pi$
$542$	0	0
$543$	912.000	0.0720767
$544$	0	0
$545$	−14856.0	−1.16764
$546$	0	0
$547$	−20516.0	−1.60366	−0.801829	−	0.597554i	$-0.796139\pi$
$547$	−20516.0	−1.60366	−0.801829	+	0.597554i	$0.796139\pi$
$548$	0	0
$549$	−3124.00	−0.242858
$550$	0	0
$551$	16600.0	1.28346
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−19872.0	−1.51985
$556$	0	0
$557$	−18346.0	−1.39559	−0.697796	−	0.716296i	$-0.745836\pi$
$557$	−18346.0	−1.39559	−0.697796	+	0.716296i	$0.745836\pi$
$558$	0	0
$559$	−34352.0	−2.59917
$560$	0	0
$561$	−384.000	−0.0288993
$562$	0	0
$563$	−6108.00	−0.457232	−0.228616	−	0.973517i	$-0.573420\pi$
$563$	−6108.00	−0.457232	−0.228616	+	0.973517i	$0.573420\pi$
$564$	0	0
$565$	7512.00	0.559349
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2678.00	0.197307	0.0986534	−	0.995122i	$-0.468546\pi$
$569$	2678.00	0.197307	0.0986534	+	0.995122i	$0.468546\pi$
$570$	0	0
$571$	17748.0	1.30075	0.650377	−	0.759611i	$-0.274611\pi$
$571$	17748.0	1.30075	0.650377	+	0.759611i	$0.274611\pi$
$572$	0	0
$573$	−3552.00	−0.258965
$574$	0	0
$575$	1064.00	0.0771685
$576$	0	0
$577$	13840.0	0.998556	0.499278	−	0.866442i	$-0.333599\pi$
$577$	13840.0	0.998556	0.499278	+	0.866442i	$0.333599\pi$
$578$	0	0
$579$	−11768.0	−0.844666
$580$	0	0
$581$	0	0
$582$	0	0
$583$	216.000	0.0153444
$584$	0	0
$585$	10032.0	0.709012
$586$	0	0
$587$	−11964.0	−0.841239	−0.420619	−	0.907237i	$-0.638187\pi$
$587$	−11964.0	−0.841239	−0.420619	+	0.907237i	$0.638187\pi$
$588$	0	0
$589$	23200.0	1.62299
$590$	0	0
$591$	4344.00	0.302349
$592$	0	0
$593$	−26032.0	−1.80271	−0.901354	−	0.433083i	$-0.857426\pi$
$593$	−26032.0	−1.80271	−0.901354	+	0.433083i	$0.857426\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6176.00	−0.423395
$598$	0	0
$599$	−25440.0	−1.73531	−0.867655	−	0.497167i	$-0.834373\pi$
$599$	−25440.0	−1.73531	−0.867655	+	0.497167i	$0.834373\pi$
$600$	0	0
$601$	13280.0	0.901335	0.450668	−	0.892692i	$-0.351186\pi$
$601$	13280.0	0.901335	0.450668	+	0.892692i	$0.351186\pi$
$602$	0	0
$603$	5764.00	0.389267
$604$	0	0
$605$	−14244.0	−0.957192
$606$	0	0
$607$	−9504.00	−0.635511	−0.317756	−	0.948173i	$-0.602929\pi$
$607$	−9504.00	−0.635511	−0.317756	+	0.948173i	$0.602929\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32224.0	−2.13362
$612$	0	0
$613$	−20254.0	−1.33450	−0.667252	−	0.744832i	$-0.732530\pi$
$613$	−20254.0	−1.33450	−0.667252	+	0.744832i	$0.732530\pi$
$614$	0	0
$615$	−3456.00	−0.226601
$616$	0	0
$617$	−3386.00	−0.220932	−0.110466	−	0.993880i	$-0.535234\pi$
$617$	−3386.00	−0.220932	−0.110466	+	0.993880i	$0.535234\pi$
$618$	0	0
$619$	13196.0	0.856853	0.428427	−	0.903577i	$-0.359068\pi$
$619$	13196.0	0.856853	0.428427	+	0.903577i	$0.359068\pi$
$620$	0	0
$621$	−8512.00	−0.550040
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	4800.00	0.305731
$628$	0	0
$629$	−3312.00	−0.209949
$630$	0	0
$631$	8976.00	0.566290	0.283145	−	0.959077i	$-0.408622\pi$
$631$	8976.00	0.566290	0.283145	+	0.959077i	$0.408622\pi$
$632$	0	0
$633$	−18800.0	−1.18046
$634$	0	0
$635$	1920.00	0.119989
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−11088.0	−0.686439
$640$	0	0
$641$	−20802.0	−1.28179	−0.640897	−	0.767627i	$-0.721437\pi$
$641$	−20802.0	−1.28179	−0.640897	+	0.767627i	$0.721437\pi$
$642$	0	0
$643$	24964.0	1.53108	0.765540	−	0.643389i	$-0.222472\pi$
$643$	24964.0	1.53108	0.765540	+	0.643389i	$0.222472\pi$
$644$	0	0
$645$	21696.0	1.32446
$646$	0	0
$647$	−6920.00	−0.420484	−0.210242	−	0.977649i	$-0.567425\pi$
$647$	−6920.00	−0.420484	−0.210242	+	0.977649i	$0.567425\pi$
$648$	0	0
$649$	−5328.00	−0.322253
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9002.00	0.539472	0.269736	−	0.962934i	$-0.413063\pi$
$653$	9002.00	0.539472	0.269736	+	0.962934i	$0.413063\pi$
$654$	0	0
$655$	−8688.00	−0.518272
$656$	0	0
$657$	9856.00	0.585265
$658$	0	0
$659$	−20260.0	−1.19760	−0.598799	−	0.800899i	$-0.704355\pi$
$659$	−20260.0	−1.19760	−0.598799	+	0.800899i	$0.704355\pi$
$660$	0	0
$661$	2780.00	0.163585	0.0817923	−	0.996649i	$-0.473936\pi$
$661$	2780.00	0.163585	0.0817923	+	0.996649i	$0.473936\pi$
$662$	0	0
$663$	−2432.00	−0.142460
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9296.00	−0.539644
$668$	0	0
$669$	13760.0	0.795205
$670$	0	0
$671$	−3408.00	−0.196072
$672$	0	0
$673$	−3106.00	−0.177901	−0.0889506	−	0.996036i	$-0.528351\pi$
$673$	−3106.00	−0.177901	−0.0889506	+	0.996036i	$0.528351\pi$
$674$	0	0
$675$	−2888.00	−0.164680
$676$	0	0
$677$	−5788.00	−0.328583	−0.164292	−	0.986412i	$-0.552534\pi$
$677$	−5788.00	−0.328583	−0.164292	+	0.986412i	$0.552534\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−16976.0	−0.955245
$682$	0	0
$683$	−12708.0	−0.711945	−0.355972	−	0.934497i	$-0.615850\pi$
$683$	−12708.0	−0.711945	−0.355972	+	0.934497i	$0.615850\pi$
$684$	0	0
$685$	−16440.0	−0.916993
$686$	0	0
$687$	14800.0	0.821914
$688$	0	0
$689$	1368.00	0.0756410
$690$	0	0
$691$	31084.0	1.71128	0.855638	−	0.517575i	$-0.173165\pi$
$691$	31084.0	1.71128	0.855638	+	0.517575i	$0.173165\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10224.0	0.558012
$696$	0	0
$697$	−576.000	−0.0313021
$698$	0	0
$699$	13080.0	0.707770
$700$	0	0
$701$	−31270.0	−1.68481	−0.842405	−	0.538845i	$-0.818861\pi$
$701$	−31270.0	−1.68481	−0.842405	+	0.538845i	$0.818861\pi$
$702$	0	0
$703$	41400.0	2.22110
$704$	0	0
$705$	20352.0	1.08723
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36766.0	−1.94750	−0.973749	−	0.227624i	$-0.926904\pi$
$709$	−36766.0	−1.94750	−0.973749	+	0.227624i	$0.926904\pi$
$710$	0	0
$711$	−440.000	−0.0232086
$712$	0	0
$713$	−12992.0	−0.682404
$714$	0	0
$715$	10944.0	0.572423
$716$	0	0
$717$	7424.00	0.386687
$718$	0	0
$719$	−14216.0	−0.737368	−0.368684	−	0.929555i	$-0.620191\pi$
$719$	−14216.0	−0.737368	−0.368684	+	0.929555i	$0.620191\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3744.00	−0.192588
$724$	0	0
$725$	−3154.00	−0.161568
$726$	0	0
$727$	−1656.00	−0.0844809	−0.0422405	−	0.999107i	$-0.513450\pi$
$727$	−1656.00	−0.0844809	−0.0422405	+	0.999107i	$0.513450\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	3616.00	0.182958
$732$	0	0
$733$	18700.0	0.942292	0.471146	−	0.882055i	$-0.343840\pi$
$733$	18700.0	0.942292	0.471146	+	0.882055i	$0.343840\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6288.00	0.314276
$738$	0	0
$739$	−11012.0	−0.548150	−0.274075	−	0.961708i	$-0.588372\pi$
$739$	−11012.0	−0.548150	−0.274075	+	0.961708i	$0.588372\pi$
$740$	0	0
$741$	30400.0	1.50711
$742$	0	0
$743$	22024.0	1.08746	0.543730	−	0.839260i	$-0.317012\pi$
$743$	22024.0	1.08746	0.543730	+	0.839260i	$0.317012\pi$
$744$	0	0
$745$	9000.00	0.442597
$746$	0	0
$747$	−15268.0	−0.747827
$748$	0	0
$749$	0	0
$750$	0	0
$751$	8224.00	0.399598	0.199799	−	0.979837i	$-0.435971\pi$
$751$	8224.00	0.399598	0.199799	+	0.979837i	$0.435971\pi$
$752$	0	0
$753$	−15408.0	−0.745682
$754$	0	0
$755$	5280.00	0.254515
$756$	0	0
$757$	626.000	0.0300560	0.0150280	−	0.999887i	$-0.495216\pi$
$757$	626.000	0.0300560	0.0150280	+	0.999887i	$0.495216\pi$
$758$	0	0
$759$	−2688.00	−0.128548
$760$	0	0
$761$	19368.0	0.922588	0.461294	−	0.887247i	$-0.347385\pi$
$761$	19368.0	0.922588	0.461294	+	0.887247i	$0.347385\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1056.00	−0.0499082
$766$	0	0
$767$	−33744.0	−1.58856
$768$	0	0
$769$	26216.0	1.22935	0.614677	−	0.788779i	$-0.289286\pi$
$769$	26216.0	1.22935	0.614677	+	0.788779i	$0.289286\pi$
$770$	0	0
$771$	26880.0	1.25559
$772$	0	0
$773$	24156.0	1.12397	0.561986	−	0.827146i	$-0.310037\pi$
$773$	24156.0	1.12397	0.561986	+	0.827146i	$0.310037\pi$
$774$	0	0
$775$	−4408.00	−0.204310
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7200.00	0.331151
$780$	0	0
$781$	−12096.0	−0.554198
$782$	0	0
$783$	25232.0	1.15162
$784$	0	0
$785$	−19920.0	−0.905701
$786$	0	0
$787$	−27988.0	−1.26768	−0.633840	−	0.773464i	$-0.718522\pi$
$787$	−27988.0	−1.26768	−0.633840	+	0.773464i	$0.718522\pi$
$788$	0	0
$789$	−12288.0	−0.554454
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−21584.0	−0.966545
$794$	0	0
$795$	−864.000	−0.0385446
$796$	0	0
$797$	3812.00	0.169420	0.0847101	−	0.996406i	$-0.473004\pi$
$797$	3812.00	0.169420	0.0847101	+	0.996406i	$0.473004\pi$
$798$	0	0
$799$	3392.00	0.150188
$800$	0	0
$801$	4928.00	0.217381
$802$	0	0
$803$	10752.0	0.472515
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10224.0	−0.445975
$808$	0	0
$809$	11482.0	0.498993	0.249497	−	0.968376i	$-0.419735\pi$
$809$	11482.0	0.498993	0.249497	+	0.968376i	$0.419735\pi$
$810$	0	0
$811$	−3764.00	−0.162974	−0.0814870	−	0.996674i	$-0.525967\pi$
$811$	−3764.00	−0.162974	−0.0814870	+	0.996674i	$0.525967\pi$
$812$	0	0
$813$	−30464.0	−1.31417
$814$	0	0
$815$	−30384.0	−1.30590
$816$	0	0
$817$	−45200.0	−1.93555
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23762.0	−1.01011	−0.505055	−	0.863087i	$-0.668528\pi$
$821$	−23762.0	−1.01011	−0.505055	+	0.863087i	$0.668528\pi$
$822$	0	0
$823$	16640.0	0.704780	0.352390	−	0.935853i	$-0.385369\pi$
$823$	16640.0	0.704780	0.352390	+	0.935853i	$0.385369\pi$
$824$	0	0
$825$	−912.000	−0.0384870
$826$	0	0
$827$	10172.0	0.427709	0.213854	−	0.976866i	$-0.431398\pi$
$827$	10172.0	0.427709	0.213854	+	0.976866i	$0.431398\pi$
$828$	0	0
$829$	11804.0	0.494535	0.247268	−	0.968947i	$-0.420467\pi$
$829$	11804.0	0.494535	0.247268	+	0.968947i	$0.420467\pi$
$830$	0	0
$831$	7736.00	0.322935
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12576.0	0.521210
$836$	0	0
$837$	35264.0	1.45627
$838$	0	0
$839$	26568.0	1.09324	0.546621	−	0.837380i	$-0.315914\pi$
$839$	26568.0	1.09324	0.546621	+	0.837380i	$0.315914\pi$
$840$	0	0
$841$	3167.00	0.129854
$842$	0	0
$843$	−26408.0	−1.07893
$844$	0	0
$845$	42948.0	1.74847
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−18832.0	−0.761263
$850$	0	0
$851$	−23184.0	−0.933886
$852$	0	0
$853$	3604.00	0.144664	0.0723321	−	0.997381i	$-0.476956\pi$
$853$	3604.00	0.144664	0.0723321	+	0.997381i	$0.476956\pi$
$854$	0	0
$855$	13200.0	0.527989
$856$	0	0
$857$	−34536.0	−1.37658	−0.688289	−	0.725437i	$-0.741638\pi$
$857$	−34536.0	−1.37658	−0.688289	+	0.725437i	$0.741638\pi$
$858$	0	0
$859$	−24348.0	−0.967105	−0.483552	−	0.875315i	$-0.660654\pi$
$859$	−24348.0	−0.967105	−0.483552	+	0.875315i	$0.660654\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17032.0	0.671815	0.335907	−	0.941895i	$-0.390957\pi$
$863$	17032.0	0.671815	0.335907	+	0.941895i	$0.390957\pi$
$864$	0	0
$865$	10608.0	0.416974
$866$	0	0
$867$	−19396.0	−0.759772
$868$	0	0
$869$	−480.000	−0.0187375
$870$	0	0
$871$	39824.0	1.54924
$872$	0	0
$873$	−9064.00	−0.351397
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−11126.0	−0.428390	−0.214195	−	0.976791i	$-0.568713\pi$
$877$	−11126.0	−0.428390	−0.214195	+	0.976791i	$0.568713\pi$
$878$	0	0
$879$	−14928.0	−0.572820
$880$	0	0
$881$	1344.00	0.0513967	0.0256984	−	0.999670i	$-0.491819\pi$
$881$	1344.00	0.0513967	0.0256984	+	0.999670i	$0.491819\pi$
$882$	0	0
$883$	29252.0	1.11485	0.557423	−	0.830229i	$-0.311790\pi$
$883$	29252.0	1.11485	0.557423	+	0.830229i	$0.311790\pi$
$884$	0	0
$885$	21312.0	0.809486
$886$	0	0
$887$	−9080.00	−0.343717	−0.171858	−	0.985122i	$-0.554977\pi$
$887$	−9080.00	−0.343717	−0.171858	+	0.985122i	$0.554977\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3732.00	0.140322
$892$	0	0
$893$	−42400.0	−1.58887
$894$	0	0
$895$	−4752.00	−0.177477
$896$	0	0
$897$	−17024.0	−0.633684
$898$	0	0
$899$	38512.0	1.42875
$900$	0	0
$901$	−144.000	−0.00532446
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2736.00	0.100495
$906$	0	0
$907$	27052.0	0.990349	0.495175	−	0.868793i	$-0.335104\pi$
$907$	27052.0	0.990349	0.495175	+	0.868793i	$0.335104\pi$
$908$	0	0
$909$	−21076.0	−0.769028
$910$	0	0
$911$	−23424.0	−0.851890	−0.425945	−	0.904749i	$-0.640058\pi$
$911$	−23424.0	−0.851890	−0.425945	+	0.904749i	$0.640058\pi$
$912$	0	0
$913$	−16656.0	−0.603760
$914$	0	0
$915$	13632.0	0.492525
$916$	0	0
$917$	0	0
$918$	0	0
$919$	11376.0	0.408335	0.204167	−	0.978936i	$-0.434551\pi$
$919$	11376.0	0.408335	0.204167	+	0.978936i	$0.434551\pi$
$920$	0	0
$921$	19248.0	0.688646
$922$	0	0
$923$	−76608.0	−2.73194
$924$	0	0
$925$	−7866.00	−0.279603
$926$	0	0
$927$	9064.00	0.321144
$928$	0	0
$929$	−31864.0	−1.12532	−0.562661	−	0.826688i	$-0.690222\pi$
$929$	−31864.0	−1.12532	−0.562661	+	0.826688i	$0.690222\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	33920.0	1.19024
$934$	0	0
$935$	−1152.00	−0.0402935
$936$	0	0
$937$	−44064.0	−1.53629	−0.768147	−	0.640273i	$-0.778821\pi$
$937$	−44064.0	−1.53629	−0.768147	+	0.640273i	$0.778821\pi$
$938$	0	0
$939$	−26272.0	−0.913050
$940$	0	0
$941$	−16260.0	−0.563295	−0.281648	−	0.959518i	$-0.590881\pi$
$941$	−16260.0	−0.563295	−0.281648	+	0.959518i	$0.590881\pi$
$942$	0	0
$943$	−4032.00	−0.139236
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25756.0	0.883799	0.441899	−	0.897065i	$-0.354305\pi$
$947$	25756.0	0.883799	0.441899	+	0.897065i	$0.354305\pi$
$948$	0	0
$949$	68096.0	2.32928
$950$	0	0
$951$	18840.0	0.642407
$952$	0	0
$953$	−20502.0	−0.696878	−0.348439	−	0.937331i	$-0.613288\pi$
$953$	−20502.0	−0.696878	−0.348439	+	0.937331i	$0.613288\pi$
$954$	0	0
$955$	−10656.0	−0.361068
$956$	0	0
$957$	7968.00	0.269142
$958$	0	0
$959$	0	0
$960$	0	0
$961$	24033.0	0.806720
$962$	0	0
$963$	19756.0	0.661088
$964$	0	0
$965$	−35304.0	−1.17770
$966$	0	0
$967$	2696.00	0.0896562	0.0448281	−	0.998995i	$-0.485726\pi$
$967$	2696.00	0.0896562	0.0448281	+	0.998995i	$0.485726\pi$
$968$	0	0
$969$	−3200.00	−0.106088
$970$	0	0
$971$	4060.00	0.134183	0.0670915	−	0.997747i	$-0.478628\pi$
$971$	4060.00	0.134183	0.0670915	+	0.997747i	$0.478628\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−5776.00	−0.189723
$976$	0	0
$977$	−28786.0	−0.942626	−0.471313	−	0.881966i	$-0.656220\pi$
$977$	−28786.0	−0.942626	−0.471313	+	0.881966i	$0.656220\pi$
$978$	0	0
$979$	5376.00	0.175503
$980$	0	0
$981$	13618.0	0.443210
$982$	0	0
$983$	24808.0	0.804936	0.402468	−	0.915434i	$-0.368152\pi$
$983$	24808.0	0.804936	0.402468	+	0.915434i	$0.368152\pi$
$984$	0	0
$985$	13032.0	0.421557
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25312.0	0.813827
$990$	0	0
$991$	51656.0	1.65581	0.827905	−	0.560869i	$-0.189533\pi$
$991$	51656.0	1.65581	0.827905	+	0.560869i	$0.189533\pi$
$992$	0	0
$993$	31184.0	0.996570
$994$	0	0
$995$	−18528.0	−0.590329
$996$	0	0
$997$	−40484.0	−1.28600	−0.643000	−	0.765867i	$-0.722310\pi$
$997$	−40484.0	−1.28600	−0.643000	+	0.765867i	$0.722310\pi$
$998$	0	0
$999$	62928.0	1.99295

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.o.1.1		1
4.3	odd	2		392.4.a.b.1.1	✓	1
7.6	odd	2		784.4.a.d.1.1		1
28.3	even	6		392.4.i.c.177.1		2
28.11	odd	6		392.4.i.f.177.1		2
28.19	even	6		392.4.i.c.361.1		2
28.23	odd	6		392.4.i.f.361.1		2
28.27	even	2		392.4.a.d.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.4.a.b.1.1	✓	1	4.3	odd	2
392.4.a.d.1.1	yes	1	28.27	even	2
392.4.i.c.177.1		2	28.3	even	6
392.4.i.c.361.1		2	28.19	even	6
392.4.i.f.177.1		2	28.11	odd	6
392.4.i.f.361.1		2	28.23	odd	6
784.4.a.d.1.1		1	7.6	odd	2
784.4.a.o.1.1		1	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.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more