LMFDB - Embedded newform 840.4.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [840,4,Mod(1,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("840.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,5,0,7,0,9,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.5616044048$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} -16.0000 q^{11} -62.0000 q^{13} +15.0000 q^{15} -14.0000 q^{17} -56.0000 q^{19} +21.0000 q^{21} -136.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -154.000 q^{29} -116.000 q^{31} -48.0000 q^{33} +35.0000 q^{35} +6.00000 q^{37} -186.000 q^{39} -150.000 q^{41} -20.0000 q^{43} +45.0000 q^{45} +152.000 q^{47} +49.0000 q^{49} -42.0000 q^{51} -78.0000 q^{53} -80.0000 q^{55} -168.000 q^{57} +124.000 q^{59} +166.000 q^{61} +63.0000 q^{63} -310.000 q^{65} +140.000 q^{67} -408.000 q^{69} +204.000 q^{71} -210.000 q^{73} +75.0000 q^{75} -112.000 q^{77} -984.000 q^{79} +81.0000 q^{81} +628.000 q^{83} -70.0000 q^{85} -462.000 q^{87} +138.000 q^{89} -434.000 q^{91} -348.000 q^{93} -280.000 q^{95} -1202.00 q^{97} -144.000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−16.0000	−0.438562	−0.219281	−	0.975662i	$-0.570371\pi$
$11$	−16.0000	−0.438562	−0.219281	+	0.975662i	$0.570371\pi$
$12$	0	0
$13$	−62.0000	−1.32275	−0.661373	−	0.750057i	$-0.730026\pi$
$13$	−62.0000	−1.32275	−0.661373	+	0.750057i	$0.730026\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−14.0000	−0.199735	−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000	−0.199735	−0.0998676	+	0.995001i	$0.531842\pi$
$18$	0	0
$19$	−56.0000	−0.676173	−0.338086	−	0.941115i	$-0.609780\pi$
$19$	−56.0000	−0.676173	−0.338086	+	0.941115i	$0.609780\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	−136.000	−1.23295	−0.616477	−	0.787373i	$-0.711441\pi$
$23$	−136.000	−1.23295	−0.616477	+	0.787373i	$0.711441\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−154.000	−0.986106	−0.493053	−	0.869999i	$-0.664119\pi$
$29$	−154.000	−0.986106	−0.493053	+	0.869999i	$0.664119\pi$
$30$	0	0
$31$	−116.000	−0.672071	−0.336036	−	0.941849i	$-0.609086\pi$
$31$	−116.000	−0.672071	−0.336036	+	0.941849i	$0.609086\pi$
$32$	0	0
$33$	−48.0000	−0.253204
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	6.00000	0.0266593	0.0133296	−	0.999911i	$-0.495757\pi$
$37$	6.00000	0.0266593	0.0133296	+	0.999911i	$0.495757\pi$
$38$	0	0
$39$	−186.000	−0.763688
$40$	0	0
$41$	−150.000	−0.571367	−0.285684	−	0.958324i	$-0.592221\pi$
$41$	−150.000	−0.571367	−0.285684	+	0.958324i	$0.592221\pi$
$42$	0	0
$43$	−20.0000	−0.0709296	−0.0354648	−	0.999371i	$-0.511291\pi$
$43$	−20.0000	−0.0709296	−0.0354648	+	0.999371i	$0.511291\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	152.000	0.471734	0.235867	−	0.971785i	$-0.424207\pi$
$47$	152.000	0.471734	0.235867	+	0.971785i	$0.424207\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−42.0000	−0.115317
$52$	0	0
$53$	−78.0000	−0.202153	−0.101077	−	0.994879i	$-0.532229\pi$
$53$	−78.0000	−0.202153	−0.101077	+	0.994879i	$0.532229\pi$
$54$	0	0
$55$	−80.0000	−0.196131
$56$	0	0
$57$	−168.000	−0.390388
$58$	0	0
$59$	124.000	0.273617	0.136809	−	0.990597i	$-0.456315\pi$
$59$	124.000	0.273617	0.136809	+	0.990597i	$0.456315\pi$
$60$	0	0
$61$	166.000	0.348428	0.174214	−	0.984708i	$-0.444262\pi$
$61$	166.000	0.348428	0.174214	+	0.984708i	$0.444262\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−310.000	−0.591550
$66$	0	0
$67$	140.000	0.255279	0.127640	−	0.991821i	$-0.459260\pi$
$67$	140.000	0.255279	0.127640	+	0.991821i	$0.459260\pi$
$68$	0	0
$69$	−408.000	−0.711847
$70$	0	0
$71$	204.000	0.340991	0.170495	−	0.985358i	$-0.445463\pi$
$71$	204.000	0.340991	0.170495	+	0.985358i	$0.445463\pi$
$72$	0	0
$73$	−210.000	−0.336694	−0.168347	−	0.985728i	$-0.553843\pi$
$73$	−210.000	−0.336694	−0.168347	+	0.985728i	$0.553843\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−112.000	−0.165761
$78$	0	0
$79$	−984.000	−1.40138	−0.700688	−	0.713468i	$-0.747123\pi$
$79$	−984.000	−1.40138	−0.700688	+	0.713468i	$0.747123\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	628.000	0.830505	0.415253	−	0.909706i	$-0.363693\pi$
$83$	628.000	0.830505	0.415253	+	0.909706i	$0.363693\pi$
$84$	0	0
$85$	−70.0000	−0.0893243
$86$	0	0
$87$	−462.000	−0.569329
$88$	0	0
$89$	138.000	0.164359	0.0821796	−	0.996618i	$-0.473812\pi$
$89$	138.000	0.164359	0.0821796	+	0.996618i	$0.473812\pi$
$90$	0	0
$91$	−434.000	−0.499951
$92$	0	0
$93$	−348.000	−0.388021
$94$	0	0
$95$	−280.000	−0.302394
$96$	0	0
$97$	−1202.00	−1.25819	−0.629096	−	0.777328i	$-0.716575\pi$
$97$	−1202.00	−1.25819	−0.629096	+	0.777328i	$0.716575\pi$
$98$	0	0
$99$	−144.000	−0.146187
$100$	0	0
$101$	270.000	0.266000	0.133000	−	0.991116i	$-0.457539\pi$
$101$	270.000	0.266000	0.133000	+	0.991116i	$0.457539\pi$
$102$	0	0
$103$	−488.000	−0.466836	−0.233418	−	0.972377i	$-0.574991\pi$
$103$	−488.000	−0.466836	−0.233418	+	0.972377i	$0.574991\pi$
$104$	0	0
$105$	105.000	0.0975900
$106$	0	0
$107$	−1092.00	−0.986613	−0.493307	−	0.869855i	$-0.664212\pi$
$107$	−1092.00	−0.986613	−0.493307	+	0.869855i	$0.664212\pi$
$108$	0	0
$109$	1910.00	1.67839	0.839196	−	0.543828i	$-0.183026\pi$
$109$	1910.00	1.67839	0.839196	+	0.543828i	$0.183026\pi$
$110$	0	0
$111$	18.0000	0.0153918
$112$	0	0
$113$	−1002.00	−0.834161	−0.417081	−	0.908869i	$-0.636947\pi$
$113$	−1002.00	−0.834161	−0.417081	+	0.908869i	$0.636947\pi$
$114$	0	0
$115$	−680.000	−0.551394
$116$	0	0
$117$	−558.000	−0.440916
$118$	0	0
$119$	−98.0000	−0.0754928
$120$	0	0
$121$	−1075.00	−0.807663
$122$	0	0
$123$	−450.000	−0.329879
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	984.000	0.687527	0.343763	−	0.939056i	$-0.388298\pi$
$127$	984.000	0.687527	0.343763	+	0.939056i	$0.388298\pi$
$128$	0	0
$129$	−60.0000	−0.0409512
$130$	0	0
$131$	−84.0000	−0.0560238	−0.0280119	−	0.999608i	$-0.508918\pi$
$131$	−84.0000	−0.0560238	−0.0280119	+	0.999608i	$0.508918\pi$
$132$	0	0
$133$	−392.000	−0.255569
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	566.000	0.352968	0.176484	−	0.984303i	$-0.443528\pi$
$137$	566.000	0.352968	0.176484	+	0.984303i	$0.443528\pi$
$138$	0	0
$139$	864.000	0.527220	0.263610	−	0.964629i	$-0.415087\pi$
$139$	864.000	0.527220	0.263610	+	0.964629i	$0.415087\pi$
$140$	0	0
$141$	456.000	0.272356
$142$	0	0
$143$	992.000	0.580106
$144$	0	0
$145$	−770.000	−0.441000
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−1274.00	−0.700471	−0.350235	−	0.936662i	$-0.613898\pi$
$149$	−1274.00	−0.700471	−0.350235	+	0.936662i	$0.613898\pi$
$150$	0	0
$151$	−952.000	−0.513064	−0.256532	−	0.966536i	$-0.582580\pi$
$151$	−952.000	−0.513064	−0.256532	+	0.966536i	$0.582580\pi$
$152$	0	0
$153$	−126.000	−0.0665784
$154$	0	0
$155$	−580.000	−0.300559
$156$	0	0
$157$	−2342.00	−1.19052	−0.595261	−	0.803532i	$-0.702951\pi$
$157$	−2342.00	−1.19052	−0.595261	+	0.803532i	$0.702951\pi$
$158$	0	0
$159$	−234.000	−0.116713
$160$	0	0
$161$	−952.000	−0.466013
$162$	0	0
$163$	3460.00	1.66263	0.831313	−	0.555804i	$-0.187590\pi$
$163$	3460.00	1.66263	0.831313	+	0.555804i	$0.187590\pi$
$164$	0	0
$165$	−240.000	−0.113236
$166$	0	0
$167$	600.000	0.278020	0.139010	−	0.990291i	$-0.455608\pi$
$167$	600.000	0.278020	0.139010	+	0.990291i	$0.455608\pi$
$168$	0	0
$169$	1647.00	0.749659
$170$	0	0
$171$	−504.000	−0.225391
$172$	0	0
$173$	−1970.00	−0.865759	−0.432880	−	0.901452i	$-0.642502\pi$
$173$	−1970.00	−0.865759	−0.432880	+	0.901452i	$0.642502\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	372.000	0.157973
$178$	0	0
$179$	1480.00	0.617991	0.308996	−	0.951063i	$-0.400007\pi$
$179$	1480.00	0.617991	0.308996	+	0.951063i	$0.400007\pi$
$180$	0	0
$181$	−4522.00	−1.85700	−0.928502	−	0.371328i	$-0.878902\pi$
$181$	−4522.00	−1.85700	−0.928502	+	0.371328i	$0.878902\pi$
$182$	0	0
$183$	498.000	0.201165
$184$	0	0
$185$	30.0000	0.0119224
$186$	0	0
$187$	224.000	0.0875963
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	1468.00	0.556130	0.278065	−	0.960562i	$-0.410307\pi$
$191$	1468.00	0.556130	0.278065	+	0.960562i	$0.410307\pi$
$192$	0	0
$193$	−3662.00	−1.36579	−0.682893	−	0.730519i	$-0.739278\pi$
$193$	−3662.00	−1.36579	−0.682893	+	0.730519i	$0.739278\pi$
$194$	0	0
$195$	−930.000	−0.341532
$196$	0	0
$197$	−1798.00	−0.650265	−0.325132	−	0.945668i	$-0.605409\pi$
$197$	−1798.00	−0.650265	−0.325132	+	0.945668i	$0.605409\pi$
$198$	0	0
$199$	−460.000	−0.163862	−0.0819310	−	0.996638i	$-0.526109\pi$
$199$	−460.000	−0.163862	−0.0819310	+	0.996638i	$0.526109\pi$
$200$	0	0
$201$	420.000	0.147386
$202$	0	0
$203$	−1078.00	−0.372713
$204$	0	0
$205$	−750.000	−0.255523
$206$	0	0
$207$	−1224.00	−0.410985
$208$	0	0
$209$	896.000	0.296544
$210$	0	0
$211$	−3484.00	−1.13672	−0.568361	−	0.822779i	$-0.692422\pi$
$211$	−3484.00	−1.13672	−0.568361	+	0.822779i	$0.692422\pi$
$212$	0	0
$213$	612.000	0.196871
$214$	0	0
$215$	−100.000	−0.0317207
$216$	0	0
$217$	−812.000	−0.254019
$218$	0	0
$219$	−630.000	−0.194390
$220$	0	0
$221$	868.000	0.264199
$222$	0	0
$223$	−1480.00	−0.444431	−0.222216	−	0.974998i	$-0.571329\pi$
$223$	−1480.00	−0.444431	−0.222216	+	0.974998i	$0.571329\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	2004.00	0.585948	0.292974	−	0.956120i	$-0.405355\pi$
$227$	2004.00	0.585948	0.292974	+	0.956120i	$0.405355\pi$
$228$	0	0
$229$	142.000	0.0409765	0.0204883	−	0.999790i	$-0.493478\pi$
$229$	142.000	0.0409765	0.0204883	+	0.999790i	$0.493478\pi$
$230$	0	0
$231$	−336.000	−0.0957021
$232$	0	0
$233$	4590.00	1.29056	0.645281	−	0.763945i	$-0.276740\pi$
$233$	4590.00	1.29056	0.645281	+	0.763945i	$0.276740\pi$
$234$	0	0
$235$	760.000	0.210966
$236$	0	0
$237$	−2952.00	−0.809084
$238$	0	0
$239$	4548.00	1.23090	0.615451	−	0.788175i	$-0.288974\pi$
$239$	4548.00	1.23090	0.615451	+	0.788175i	$0.288974\pi$
$240$	0	0
$241$	−6118.00	−1.63525	−0.817625	−	0.575752i	$-0.804709\pi$
$241$	−6118.00	−1.63525	−0.817625	+	0.575752i	$0.804709\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	3472.00	0.894405
$248$	0	0
$249$	1884.00	0.479493
$250$	0	0
$251$	1740.00	0.437561	0.218781	−	0.975774i	$-0.429792\pi$
$251$	1740.00	0.437561	0.218781	+	0.975774i	$0.429792\pi$
$252$	0	0
$253$	2176.00	0.540727
$254$	0	0
$255$	−210.000	−0.0515714
$256$	0	0
$257$	4554.00	1.10533	0.552667	−	0.833402i	$-0.313610\pi$
$257$	4554.00	1.10533	0.552667	+	0.833402i	$0.313610\pi$
$258$	0	0
$259$	42.0000	0.0100763
$260$	0	0
$261$	−1386.00	−0.328702
$262$	0	0
$263$	2448.00	0.573955	0.286977	−	0.957937i	$-0.407350\pi$
$263$	2448.00	0.573955	0.286977	+	0.957937i	$0.407350\pi$
$264$	0	0
$265$	−390.000	−0.0904057
$266$	0	0
$267$	414.000	0.0948928
$268$	0	0
$269$	−1994.00	−0.451957	−0.225978	−	0.974132i	$-0.572558\pi$
$269$	−1994.00	−0.451957	−0.225978	+	0.974132i	$0.572558\pi$
$270$	0	0
$271$	3396.00	0.761226	0.380613	−	0.924734i	$-0.375713\pi$
$271$	3396.00	0.761226	0.380613	+	0.924734i	$0.375713\pi$
$272$	0	0
$273$	−1302.00	−0.288647
$274$	0	0
$275$	−400.000	−0.0877124
$276$	0	0
$277$	−3754.00	−0.814282	−0.407141	−	0.913365i	$-0.633474\pi$
$277$	−3754.00	−0.814282	−0.407141	+	0.913365i	$0.633474\pi$
$278$	0	0
$279$	−1044.00	−0.224024
$280$	0	0
$281$	1250.00	0.265369	0.132685	−	0.991158i	$-0.457640\pi$
$281$	1250.00	0.265369	0.132685	+	0.991158i	$0.457640\pi$
$282$	0	0
$283$	308.000	0.0646951	0.0323475	−	0.999477i	$-0.489702\pi$
$283$	308.000	0.0646951	0.0323475	+	0.999477i	$0.489702\pi$
$284$	0	0
$285$	−840.000	−0.174587
$286$	0	0
$287$	−1050.00	−0.215957
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	−3606.00	−0.726417
$292$	0	0
$293$	38.0000	0.00757674	0.00378837	−	0.999993i	$-0.498794\pi$
$293$	38.0000	0.00757674	0.00378837	+	0.999993i	$0.498794\pi$
$294$	0	0
$295$	620.000	0.122365
$296$	0	0
$297$	−432.000	−0.0844013
$298$	0	0
$299$	8432.00	1.63089
$300$	0	0
$301$	−140.000	−0.0268089
$302$	0	0
$303$	810.000	0.153575
$304$	0	0
$305$	830.000	0.155822
$306$	0	0
$307$	5860.00	1.08941	0.544703	−	0.838629i	$-0.316642\pi$
$307$	5860.00	1.08941	0.544703	+	0.838629i	$0.316642\pi$
$308$	0	0
$309$	−1464.00	−0.269528
$310$	0	0
$311$	3400.00	0.619924	0.309962	−	0.950749i	$-0.399684\pi$
$311$	3400.00	0.619924	0.309962	+	0.950749i	$0.399684\pi$
$312$	0	0
$313$	2054.00	0.370923	0.185462	−	0.982652i	$-0.440622\pi$
$313$	2054.00	0.370923	0.185462	+	0.982652i	$0.440622\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	−1254.00	−0.222182	−0.111091	−	0.993810i	$-0.535434\pi$
$317$	−1254.00	−0.222182	−0.111091	+	0.993810i	$0.535434\pi$
$318$	0	0
$319$	2464.00	0.432469
$320$	0	0
$321$	−3276.00	−0.569621
$322$	0	0
$323$	784.000	0.135056
$324$	0	0
$325$	−1550.00	−0.264549
$326$	0	0
$327$	5730.00	0.969021
$328$	0	0
$329$	1064.00	0.178299
$330$	0	0
$331$	−132.000	−0.0219196	−0.0109598	−	0.999940i	$-0.503489\pi$
$331$	−132.000	−0.0219196	−0.0109598	+	0.999940i	$0.503489\pi$
$332$	0	0
$333$	54.0000	0.00888643
$334$	0	0
$335$	700.000	0.114164
$336$	0	0
$337$	−398.000	−0.0643337	−0.0321668	−	0.999483i	$-0.510241\pi$
$337$	−398.000	−0.0643337	−0.0321668	+	0.999483i	$0.510241\pi$
$338$	0	0
$339$	−3006.00	−0.481603
$340$	0	0
$341$	1856.00	0.294745
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−2040.00	−0.318348
$346$	0	0
$347$	10084.0	1.56005	0.780025	−	0.625748i	$-0.215206\pi$
$347$	10084.0	1.56005	0.780025	+	0.625748i	$0.215206\pi$
$348$	0	0
$349$	822.000	0.126076	0.0630382	−	0.998011i	$-0.479921\pi$
$349$	822.000	0.126076	0.0630382	+	0.998011i	$0.479921\pi$
$350$	0	0
$351$	−1674.00	−0.254563
$352$	0	0
$353$	5698.00	0.859133	0.429566	−	0.903035i	$-0.358666\pi$
$353$	5698.00	0.859133	0.429566	+	0.903035i	$0.358666\pi$
$354$	0	0
$355$	1020.00	0.152496
$356$	0	0
$357$	−294.000	−0.0435858
$358$	0	0
$359$	−2364.00	−0.347541	−0.173770	−	0.984786i	$-0.555595\pi$
$359$	−2364.00	−0.347541	−0.173770	+	0.984786i	$0.555595\pi$
$360$	0	0
$361$	−3723.00	−0.542790
$362$	0	0
$363$	−3225.00	−0.466305
$364$	0	0
$365$	−1050.00	−0.150574
$366$	0	0
$367$	776.000	0.110373	0.0551865	−	0.998476i	$-0.482425\pi$
$367$	776.000	0.110373	0.0551865	+	0.998476i	$0.482425\pi$
$368$	0	0
$369$	−1350.00	−0.190456
$370$	0	0
$371$	−546.000	−0.0764068
$372$	0	0
$373$	−8930.00	−1.23962	−0.619809	−	0.784752i	$-0.712790\pi$
$373$	−8930.00	−1.23962	−0.619809	+	0.784752i	$0.712790\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	9548.00	1.30437
$378$	0	0
$379$	6868.00	0.930832	0.465416	−	0.885092i	$-0.345905\pi$
$379$	6868.00	0.930832	0.465416	+	0.885092i	$0.345905\pi$
$380$	0	0
$381$	2952.00	0.396944
$382$	0	0
$383$	10848.0	1.44728	0.723638	−	0.690179i	$-0.242468\pi$
$383$	10848.0	1.44728	0.723638	+	0.690179i	$0.242468\pi$
$384$	0	0
$385$	−560.000	−0.0741305
$386$	0	0
$387$	−180.000	−0.0236432
$388$	0	0
$389$	5046.00	0.657692	0.328846	−	0.944383i	$-0.393340\pi$
$389$	5046.00	0.657692	0.328846	+	0.944383i	$0.393340\pi$
$390$	0	0
$391$	1904.00	0.246264
$392$	0	0
$393$	−252.000	−0.0323453
$394$	0	0
$395$	−4920.00	−0.626714
$396$	0	0
$397$	7090.00	0.896315	0.448157	−	0.893955i	$-0.352080\pi$
$397$	7090.00	0.896315	0.448157	+	0.893955i	$0.352080\pi$
$398$	0	0
$399$	−1176.00	−0.147553
$400$	0	0
$401$	−2382.00	−0.296637	−0.148318	−	0.988940i	$-0.547386\pi$
$401$	−2382.00	−0.296637	−0.148318	+	0.988940i	$0.547386\pi$
$402$	0	0
$403$	7192.00	0.888980
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−96.0000	−0.0116918
$408$	0	0
$409$	1498.00	0.181104	0.0905518	−	0.995892i	$-0.471137\pi$
$409$	1498.00	0.181104	0.0905518	+	0.995892i	$0.471137\pi$
$410$	0	0
$411$	1698.00	0.203786
$412$	0	0
$413$	868.000	0.103418
$414$	0	0
$415$	3140.00	0.371413
$416$	0	0
$417$	2592.00	0.304390
$418$	0	0
$419$	8940.00	1.04236	0.521178	−	0.853448i	$-0.325493\pi$
$419$	8940.00	1.04236	0.521178	+	0.853448i	$0.325493\pi$
$420$	0	0
$421$	3966.00	0.459124	0.229562	−	0.973294i	$-0.426271\pi$
$421$	3966.00	0.459124	0.229562	+	0.973294i	$0.426271\pi$
$422$	0	0
$423$	1368.00	0.157245
$424$	0	0
$425$	−350.000	−0.0399470
$426$	0	0
$427$	1162.00	0.131693
$428$	0	0
$429$	2976.00	0.334925
$430$	0	0
$431$	−8924.00	−0.997341	−0.498670	−	0.866792i	$-0.666178\pi$
$431$	−8924.00	−0.997341	−0.498670	+	0.866792i	$0.666178\pi$
$432$	0	0
$433$	5342.00	0.592887	0.296444	−	0.955050i	$-0.404199\pi$
$433$	5342.00	0.592887	0.296444	+	0.955050i	$0.404199\pi$
$434$	0	0
$435$	−2310.00	−0.254612
$436$	0	0
$437$	7616.00	0.833690
$438$	0	0
$439$	−4460.00	−0.484884	−0.242442	−	0.970166i	$-0.577948\pi$
$439$	−4460.00	−0.484884	−0.242442	+	0.970166i	$0.577948\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	14340.0	1.53795	0.768977	−	0.639276i	$-0.220766\pi$
$443$	14340.0	1.53795	0.768977	+	0.639276i	$0.220766\pi$
$444$	0	0
$445$	690.000	0.0735037
$446$	0	0
$447$	−3822.00	−0.404417
$448$	0	0
$449$	18018.0	1.89381	0.946906	−	0.321509i	$-0.104190\pi$
$449$	18018.0	1.89381	0.946906	+	0.321509i	$0.104190\pi$
$450$	0	0
$451$	2400.00	0.250580
$452$	0	0
$453$	−2856.00	−0.296218
$454$	0	0
$455$	−2170.00	−0.223585
$456$	0	0
$457$	34.0000	0.00348020	0.00174010	−	0.999998i	$-0.499446\pi$
$457$	34.0000	0.00348020	0.00174010	+	0.999998i	$0.499446\pi$
$458$	0	0
$459$	−378.000	−0.0384391
$460$	0	0
$461$	−9498.00	−0.959579	−0.479790	−	0.877384i	$-0.659287\pi$
$461$	−9498.00	−0.959579	−0.479790	+	0.877384i	$0.659287\pi$
$462$	0	0
$463$	−3432.00	−0.344489	−0.172245	−	0.985054i	$-0.555102\pi$
$463$	−3432.00	−0.344489	−0.172245	+	0.985054i	$0.555102\pi$
$464$	0	0
$465$	−1740.00	−0.173528
$466$	0	0
$467$	13396.0	1.32739	0.663697	−	0.748002i	$-0.268986\pi$
$467$	13396.0	1.32739	0.663697	+	0.748002i	$0.268986\pi$
$468$	0	0
$469$	980.000	0.0964866
$470$	0	0
$471$	−7026.00	−0.687348
$472$	0	0
$473$	320.000	0.0311070
$474$	0	0
$475$	−1400.00	−0.135235
$476$	0	0
$477$	−702.000	−0.0673844
$478$	0	0
$479$	−14752.0	−1.40717	−0.703587	−	0.710609i	$-0.748419\pi$
$479$	−14752.0	−1.40717	−0.703587	+	0.710609i	$0.748419\pi$
$480$	0	0
$481$	−372.000	−0.0352635
$482$	0	0
$483$	−2856.00	−0.269053
$484$	0	0
$485$	−6010.00	−0.562680
$486$	0	0
$487$	−6224.00	−0.579130	−0.289565	−	0.957158i	$-0.593511\pi$
$487$	−6224.00	−0.579130	−0.289565	+	0.957158i	$0.593511\pi$
$488$	0	0
$489$	10380.0	0.959918
$490$	0	0
$491$	−8400.00	−0.772071	−0.386035	−	0.922484i	$-0.626156\pi$
$491$	−8400.00	−0.772071	−0.386035	+	0.922484i	$0.626156\pi$
$492$	0	0
$493$	2156.00	0.196960
$494$	0	0
$495$	−720.000	−0.0653770
$496$	0	0
$497$	1428.00	0.128882
$498$	0	0
$499$	−5188.00	−0.465424	−0.232712	−	0.972546i	$-0.574760\pi$
$499$	−5188.00	−0.465424	−0.232712	+	0.972546i	$0.574760\pi$
$500$	0	0
$501$	1800.00	0.160515
$502$	0	0
$503$	−18648.0	−1.65303	−0.826514	−	0.562916i	$-0.809679\pi$
$503$	−18648.0	−1.65303	−0.826514	+	0.562916i	$0.809679\pi$
$504$	0	0
$505$	1350.00	0.118959
$506$	0	0
$507$	4941.00	0.432816
$508$	0	0
$509$	18950.0	1.65018	0.825092	−	0.564998i	$-0.191123\pi$
$509$	18950.0	1.65018	0.825092	+	0.564998i	$0.191123\pi$
$510$	0	0
$511$	−1470.00	−0.127258
$512$	0	0
$513$	−1512.00	−0.130129
$514$	0	0
$515$	−2440.00	−0.208775
$516$	0	0
$517$	−2432.00	−0.206884
$518$	0	0
$519$	−5910.00	−0.499846
$520$	0	0
$521$	−5158.00	−0.433735	−0.216868	−	0.976201i	$-0.569584\pi$
$521$	−5158.00	−0.433735	−0.216868	+	0.976201i	$0.569584\pi$
$522$	0	0
$523$	15004.0	1.25445	0.627227	−	0.778837i	$-0.284190\pi$
$523$	15004.0	1.25445	0.627227	+	0.778837i	$0.284190\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	1624.00	0.134236
$528$	0	0
$529$	6329.00	0.520178
$530$	0	0
$531$	1116.00	0.0912058
$532$	0	0
$533$	9300.00	0.755774
$534$	0	0
$535$	−5460.00	−0.441227
$536$	0	0
$537$	4440.00	0.356797
$538$	0	0
$539$	−784.000	−0.0626517
$540$	0	0
$541$	−20906.0	−1.66140	−0.830702	−	0.556718i	$-0.812060\pi$
$541$	−20906.0	−1.66140	−0.830702	+	0.556718i	$0.812060\pi$
$542$	0	0
$543$	−13566.0	−1.07214
$544$	0	0
$545$	9550.00	0.750600
$546$	0	0
$547$	8732.00	0.682547	0.341274	−	0.939964i	$-0.389142\pi$
$547$	8732.00	0.682547	0.341274	+	0.939964i	$0.389142\pi$
$548$	0	0
$549$	1494.00	0.116143
$550$	0	0
$551$	8624.00	0.666778
$552$	0	0
$553$	−6888.00	−0.529670
$554$	0	0
$555$	90.0000	0.00688340
$556$	0	0
$557$	8602.00	0.654360	0.327180	−	0.944962i	$-0.393902\pi$
$557$	8602.00	0.654360	0.327180	+	0.944962i	$0.393902\pi$
$558$	0	0
$559$	1240.00	0.0938218
$560$	0	0
$561$	672.000	0.0505737
$562$	0	0
$563$	−12052.0	−0.902187	−0.451093	−	0.892477i	$-0.648966\pi$
$563$	−12052.0	−0.902187	−0.451093	+	0.892477i	$0.648966\pi$
$564$	0	0
$565$	−5010.00	−0.373048
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	14746.0	1.08644	0.543220	−	0.839590i	$-0.317205\pi$
$569$	14746.0	1.08644	0.543220	+	0.839590i	$0.317205\pi$
$570$	0	0
$571$	−524.000	−0.0384041	−0.0192020	−	0.999816i	$-0.506113\pi$
$571$	−524.000	−0.0384041	−0.0192020	+	0.999816i	$0.506113\pi$
$572$	0	0
$573$	4404.00	0.321082
$574$	0	0
$575$	−3400.00	−0.246591
$576$	0	0
$577$	−16162.0	−1.16609	−0.583044	−	0.812441i	$-0.698139\pi$
$577$	−16162.0	−1.16609	−0.583044	+	0.812441i	$0.698139\pi$
$578$	0	0
$579$	−10986.0	−0.788536
$580$	0	0
$581$	4396.00	0.313902
$582$	0	0
$583$	1248.00	0.0886567
$584$	0	0
$585$	−2790.00	−0.197183
$586$	0	0
$587$	23460.0	1.64957	0.824785	−	0.565446i	$-0.191296\pi$
$587$	23460.0	1.64957	0.824785	+	0.565446i	$0.191296\pi$
$588$	0	0
$589$	6496.00	0.454436
$590$	0	0
$591$	−5394.00	−0.375431
$592$	0	0
$593$	4818.00	0.333645	0.166823	−	0.985987i	$-0.446649\pi$
$593$	4818.00	0.333645	0.166823	+	0.985987i	$0.446649\pi$
$594$	0	0
$595$	−490.000	−0.0337614
$596$	0	0
$597$	−1380.00	−0.0946058
$598$	0	0
$599$	−27612.0	−1.88347	−0.941733	−	0.336362i	$-0.890803\pi$
$599$	−27612.0	−1.88347	−0.941733	+	0.336362i	$0.890803\pi$
$600$	0	0
$601$	−7550.00	−0.512431	−0.256215	−	0.966620i	$-0.582476\pi$
$601$	−7550.00	−0.512431	−0.256215	+	0.966620i	$0.582476\pi$
$602$	0	0
$603$	1260.00	0.0850931
$604$	0	0
$605$	−5375.00	−0.361198
$606$	0	0
$607$	3192.00	0.213442	0.106721	−	0.994289i	$-0.465965\pi$
$607$	3192.00	0.213442	0.106721	+	0.994289i	$0.465965\pi$
$608$	0	0
$609$	−3234.00	−0.215186
$610$	0	0
$611$	−9424.00	−0.623984
$612$	0	0
$613$	−9738.00	−0.641622	−0.320811	−	0.947143i	$-0.603955\pi$
$613$	−9738.00	−0.641622	−0.320811	+	0.947143i	$0.603955\pi$
$614$	0	0
$615$	−2250.00	−0.147526
$616$	0	0
$617$	7326.00	0.478012	0.239006	−	0.971018i	$-0.423178\pi$
$617$	7326.00	0.478012	0.239006	+	0.971018i	$0.423178\pi$
$618$	0	0
$619$	−3232.00	−0.209863	−0.104931	−	0.994479i	$-0.533462\pi$
$619$	−3232.00	−0.209863	−0.104931	+	0.994479i	$0.533462\pi$
$620$	0	0
$621$	−3672.00	−0.237282
$622$	0	0
$623$	966.000	0.0621219
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	2688.00	0.171210
$628$	0	0
$629$	−84.0000	−0.00532480
$630$	0	0
$631$	−20048.0	−1.26482	−0.632408	−	0.774636i	$-0.717933\pi$
$631$	−20048.0	−1.26482	−0.632408	+	0.774636i	$0.717933\pi$
$632$	0	0
$633$	−10452.0	−0.656287
$634$	0	0
$635$	4920.00	0.307471
$636$	0	0
$637$	−3038.00	−0.188964
$638$	0	0
$639$	1836.00	0.113664
$640$	0	0
$641$	−28518.0	−1.75724	−0.878622	−	0.477518i	$-0.841536\pi$
$641$	−28518.0	−1.75724	−0.878622	+	0.477518i	$0.841536\pi$
$642$	0	0
$643$	−300.000	−0.0183994	−0.00919972	−	0.999958i	$-0.502928\pi$
$643$	−300.000	−0.0183994	−0.00919972	+	0.999958i	$0.502928\pi$
$644$	0	0
$645$	−300.000	−0.0183139
$646$	0	0
$647$	6024.00	0.366040	0.183020	−	0.983109i	$-0.441413\pi$
$647$	6024.00	0.366040	0.183020	+	0.983109i	$0.441413\pi$
$648$	0	0
$649$	−1984.00	−0.119998
$650$	0	0
$651$	−2436.00	−0.146658
$652$	0	0
$653$	−13598.0	−0.814902	−0.407451	−	0.913227i	$-0.633582\pi$
$653$	−13598.0	−0.814902	−0.407451	+	0.913227i	$0.633582\pi$
$654$	0	0
$655$	−420.000	−0.0250546
$656$	0	0
$657$	−1890.00	−0.112231
$658$	0	0
$659$	−22224.0	−1.31369	−0.656847	−	0.754024i	$-0.728110\pi$
$659$	−22224.0	−1.31369	−0.656847	+	0.754024i	$0.728110\pi$
$660$	0	0
$661$	−29058.0	−1.70987	−0.854936	−	0.518734i	$-0.826404\pi$
$661$	−29058.0	−1.70987	−0.854936	+	0.518734i	$0.826404\pi$
$662$	0	0
$663$	2604.00	0.152535
$664$	0	0
$665$	−1960.00	−0.114294
$666$	0	0
$667$	20944.0	1.21582
$668$	0	0
$669$	−4440.00	−0.256592
$670$	0	0
$671$	−2656.00	−0.152807
$672$	0	0
$673$	−4230.00	−0.242280	−0.121140	−	0.992635i	$-0.538655\pi$
$673$	−4230.00	−0.242280	−0.121140	+	0.992635i	$0.538655\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	15974.0	0.906840	0.453420	−	0.891297i	$-0.350204\pi$
$677$	15974.0	0.906840	0.453420	+	0.891297i	$0.350204\pi$
$678$	0	0
$679$	−8414.00	−0.475552
$680$	0	0
$681$	6012.00	0.338297
$682$	0	0
$683$	−31068.0	−1.74053	−0.870266	−	0.492581i	$-0.836053\pi$
$683$	−31068.0	−1.74053	−0.870266	+	0.492581i	$0.836053\pi$
$684$	0	0
$685$	2830.00	0.157852
$686$	0	0
$687$	426.000	0.0236578
$688$	0	0
$689$	4836.00	0.267398
$690$	0	0
$691$	−7648.00	−0.421047	−0.210524	−	0.977589i	$-0.567517\pi$
$691$	−7648.00	−0.421047	−0.210524	+	0.977589i	$0.567517\pi$
$692$	0	0
$693$	−1008.00	−0.0552536
$694$	0	0
$695$	4320.00	0.235780
$696$	0	0
$697$	2100.00	0.114122
$698$	0	0
$699$	13770.0	0.745106
$700$	0	0
$701$	12454.0	0.671014	0.335507	−	0.942038i	$-0.391092\pi$
$701$	12454.0	0.671014	0.335507	+	0.942038i	$0.391092\pi$
$702$	0	0
$703$	−336.000	−0.0180263
$704$	0	0
$705$	2280.00	0.121801
$706$	0	0
$707$	1890.00	0.100539
$708$	0	0
$709$	−31618.0	−1.67481	−0.837404	−	0.546584i	$-0.815928\pi$
$709$	−31618.0	−1.67481	−0.837404	+	0.546584i	$0.815928\pi$
$710$	0	0
$711$	−8856.00	−0.467125
$712$	0	0
$713$	15776.0	0.828634
$714$	0	0
$715$	4960.00	0.259431
$716$	0	0
$717$	13644.0	0.710662
$718$	0	0
$719$	−18280.0	−0.948163	−0.474081	−	0.880481i	$-0.657220\pi$
$719$	−18280.0	−0.948163	−0.474081	+	0.880481i	$0.657220\pi$
$720$	0	0
$721$	−3416.00	−0.176447
$722$	0	0
$723$	−18354.0	−0.944111
$724$	0	0
$725$	−3850.00	−0.197221
$726$	0	0
$727$	23312.0	1.18926	0.594632	−	0.803998i	$-0.297298\pi$
$727$	23312.0	1.18926	0.594632	+	0.803998i	$0.297298\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	280.000	0.0141671
$732$	0	0
$733$	28962.0	1.45939	0.729697	−	0.683771i	$-0.239661\pi$
$733$	28962.0	1.45939	0.729697	+	0.683771i	$0.239661\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	−2240.00	−0.111956
$738$	0	0
$739$	−14340.0	−0.713810	−0.356905	−	0.934141i	$-0.616168\pi$
$739$	−14340.0	−0.713810	−0.356905	+	0.934141i	$0.616168\pi$
$740$	0	0
$741$	10416.0	0.516385
$742$	0	0
$743$	−8976.00	−0.443200	−0.221600	−	0.975138i	$-0.571128\pi$
$743$	−8976.00	−0.443200	−0.221600	+	0.975138i	$0.571128\pi$
$744$	0	0
$745$	−6370.00	−0.313260
$746$	0	0
$747$	5652.00	0.276835
$748$	0	0
$749$	−7644.00	−0.372905
$750$	0	0
$751$	−34008.0	−1.65242	−0.826211	−	0.563360i	$-0.809508\pi$
$751$	−34008.0	−1.65242	−0.826211	+	0.563360i	$0.809508\pi$
$752$	0	0
$753$	5220.00	0.252626
$754$	0	0
$755$	−4760.00	−0.229449
$756$	0	0
$757$	21854.0	1.04927	0.524635	−	0.851327i	$-0.324202\pi$
$757$	21854.0	1.04927	0.524635	+	0.851327i	$0.324202\pi$
$758$	0	0
$759$	6528.00	0.312189
$760$	0	0
$761$	−9798.00	−0.466724	−0.233362	−	0.972390i	$-0.574973\pi$
$761$	−9798.00	−0.466724	−0.233362	+	0.972390i	$0.574973\pi$
$762$	0	0
$763$	13370.0	0.634373
$764$	0	0
$765$	−630.000	−0.0297748
$766$	0	0
$767$	−7688.00	−0.361926
$768$	0	0
$769$	14938.0	0.700492	0.350246	−	0.936658i	$-0.386098\pi$
$769$	14938.0	0.700492	0.350246	+	0.936658i	$0.386098\pi$
$770$	0	0
$771$	13662.0	0.638165
$772$	0	0
$773$	−9546.00	−0.444173	−0.222087	−	0.975027i	$-0.571287\pi$
$773$	−9546.00	−0.444173	−0.222087	+	0.975027i	$0.571287\pi$
$774$	0	0
$775$	−2900.00	−0.134414
$776$	0	0
$777$	126.000	0.00581754
$778$	0	0
$779$	8400.00	0.386343
$780$	0	0
$781$	−3264.00	−0.149546
$782$	0	0
$783$	−4158.00	−0.189776
$784$	0	0
$785$	−11710.0	−0.532418
$786$	0	0
$787$	9812.00	0.444422	0.222211	−	0.974999i	$-0.428673\pi$
$787$	9812.00	0.444422	0.222211	+	0.974999i	$0.428673\pi$
$788$	0	0
$789$	7344.00	0.331373
$790$	0	0
$791$	−7014.00	−0.315283
$792$	0	0
$793$	−10292.0	−0.460882
$794$	0	0
$795$	−1170.00	−0.0521958
$796$	0	0
$797$	−24546.0	−1.09092	−0.545460	−	0.838137i	$-0.683645\pi$
$797$	−24546.0	−1.09092	−0.545460	+	0.838137i	$0.683645\pi$
$798$	0	0
$799$	−2128.00	−0.0942218
$800$	0	0
$801$	1242.00	0.0547864
$802$	0	0
$803$	3360.00	0.147661
$804$	0	0
$805$	−4760.00	−0.208407
$806$	0	0
$807$	−5982.00	−0.260937
$808$	0	0
$809$	−12414.0	−0.539497	−0.269748	−	0.962931i	$-0.586941\pi$
$809$	−12414.0	−0.539497	−0.269748	+	0.962931i	$0.586941\pi$
$810$	0	0
$811$	−30592.0	−1.32458	−0.662288	−	0.749250i	$-0.730414\pi$
$811$	−30592.0	−1.32458	−0.662288	+	0.749250i	$0.730414\pi$
$812$	0	0
$813$	10188.0	0.439494
$814$	0	0
$815$	17300.0	0.743549
$816$	0	0
$817$	1120.00	0.0479606
$818$	0	0
$819$	−3906.00	−0.166650
$820$	0	0
$821$	22190.0	0.943284	0.471642	−	0.881790i	$-0.343661\pi$
$821$	22190.0	0.943284	0.471642	+	0.881790i	$0.343661\pi$
$822$	0	0
$823$	−26960.0	−1.14188	−0.570940	−	0.820992i	$-0.693421\pi$
$823$	−26960.0	−1.14188	−0.570940	+	0.820992i	$0.693421\pi$
$824$	0	0
$825$	−1200.00	−0.0506408
$826$	0	0
$827$	−3876.00	−0.162977	−0.0814883	−	0.996674i	$-0.525967\pi$
$827$	−3876.00	−0.162977	−0.0814883	+	0.996674i	$0.525967\pi$
$828$	0	0
$829$	24190.0	1.01345	0.506727	−	0.862107i	$-0.330855\pi$
$829$	24190.0	1.01345	0.506727	+	0.862107i	$0.330855\pi$
$830$	0	0
$831$	−11262.0	−0.470126
$832$	0	0
$833$	−686.000	−0.0285336
$834$	0	0
$835$	3000.00	0.124334
$836$	0	0
$837$	−3132.00	−0.129340
$838$	0	0
$839$	6496.00	0.267303	0.133651	−	0.991028i	$-0.457330\pi$
$839$	6496.00	0.267303	0.133651	+	0.991028i	$0.457330\pi$
$840$	0	0
$841$	−673.000	−0.0275944
$842$	0	0
$843$	3750.00	0.153211
$844$	0	0
$845$	8235.00	0.335258
$846$	0	0
$847$	−7525.00	−0.305268
$848$	0	0
$849$	924.000	0.0373517
$850$	0	0
$851$	−816.000	−0.0328697
$852$	0	0
$853$	28058.0	1.12625	0.563123	−	0.826373i	$-0.309600\pi$
$853$	28058.0	1.12625	0.563123	+	0.826373i	$0.309600\pi$
$854$	0	0
$855$	−2520.00	−0.100798
$856$	0	0
$857$	−33390.0	−1.33090	−0.665450	−	0.746443i	$-0.731760\pi$
$857$	−33390.0	−1.33090	−0.665450	+	0.746443i	$0.731760\pi$
$858$	0	0
$859$	33608.0	1.33491	0.667456	−	0.744649i	$-0.267383\pi$
$859$	33608.0	1.33491	0.667456	+	0.744649i	$0.267383\pi$
$860$	0	0
$861$	−3150.00	−0.124683
$862$	0	0
$863$	−16608.0	−0.655090	−0.327545	−	0.944836i	$-0.606221\pi$
$863$	−16608.0	−0.655090	−0.327545	+	0.944836i	$0.606221\pi$
$864$	0	0
$865$	−9850.00	−0.387179
$866$	0	0
$867$	−14151.0	−0.554317
$868$	0	0
$869$	15744.0	0.614590
$870$	0	0
$871$	−8680.00	−0.337670
$872$	0	0
$873$	−10818.0	−0.419397
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−194.000	−0.00746969	−0.00373484	−	0.999993i	$-0.501189\pi$
$877$	−194.000	−0.00746969	−0.00373484	+	0.999993i	$0.501189\pi$
$878$	0	0
$879$	114.000	0.00437443
$880$	0	0
$881$	−3294.00	−0.125968	−0.0629839	−	0.998015i	$-0.520062\pi$
$881$	−3294.00	−0.125968	−0.0629839	+	0.998015i	$0.520062\pi$
$882$	0	0
$883$	49772.0	1.89690	0.948449	−	0.316929i	$-0.102652\pi$
$883$	49772.0	1.89690	0.948449	+	0.316929i	$0.102652\pi$
$884$	0	0
$885$	1860.00	0.0706477
$886$	0	0
$887$	19856.0	0.751634	0.375817	−	0.926694i	$-0.377362\pi$
$887$	19856.0	0.751634	0.375817	+	0.926694i	$0.377362\pi$
$888$	0	0
$889$	6888.00	0.259861
$890$	0	0
$891$	−1296.00	−0.0487291
$892$	0	0
$893$	−8512.00	−0.318973
$894$	0	0
$895$	7400.00	0.276374
$896$	0	0
$897$	25296.0	0.941593
$898$	0	0
$899$	17864.0	0.662734
$900$	0	0
$901$	1092.00	0.0403771
$902$	0	0
$903$	−420.000	−0.0154781
$904$	0	0
$905$	−22610.0	−0.830477
$906$	0	0
$907$	−3468.00	−0.126960	−0.0634802	−	0.997983i	$-0.520220\pi$
$907$	−3468.00	−0.126960	−0.0634802	+	0.997983i	$0.520220\pi$
$908$	0	0
$909$	2430.00	0.0886667
$910$	0	0
$911$	−32452.0	−1.18022	−0.590111	−	0.807322i	$-0.700916\pi$
$911$	−32452.0	−1.18022	−0.590111	+	0.807322i	$0.700916\pi$
$912$	0	0
$913$	−10048.0	−0.364228
$914$	0	0
$915$	2490.00	0.0899638
$916$	0	0
$917$	−588.000	−0.0211750
$918$	0	0
$919$	−44368.0	−1.59256	−0.796281	−	0.604926i	$-0.793203\pi$
$919$	−44368.0	−1.59256	−0.796281	+	0.604926i	$0.793203\pi$
$920$	0	0
$921$	17580.0	0.628969
$922$	0	0
$923$	−12648.0	−0.451044
$924$	0	0
$925$	150.000	0.00533186
$926$	0	0
$927$	−4392.00	−0.155612
$928$	0	0
$929$	36146.0	1.27655	0.638273	−	0.769810i	$-0.279649\pi$
$929$	36146.0	1.27655	0.638273	+	0.769810i	$0.279649\pi$
$930$	0	0
$931$	−2744.00	−0.0965961
$932$	0	0
$933$	10200.0	0.357913
$934$	0	0
$935$	1120.00	0.0391742
$936$	0	0
$937$	−29042.0	−1.01255	−0.506276	−	0.862372i	$-0.668978\pi$
$937$	−29042.0	−1.01255	−0.506276	+	0.862372i	$0.668978\pi$
$938$	0	0
$939$	6162.00	0.214153
$940$	0	0
$941$	1622.00	0.0561910	0.0280955	−	0.999605i	$-0.491056\pi$
$941$	1622.00	0.0561910	0.0280955	+	0.999605i	$0.491056\pi$
$942$	0	0
$943$	20400.0	0.704470
$944$	0	0
$945$	945.000	0.0325300
$946$	0	0
$947$	−39460.0	−1.35404	−0.677021	−	0.735964i	$-0.736729\pi$
$947$	−39460.0	−1.35404	−0.677021	+	0.735964i	$0.736729\pi$
$948$	0	0
$949$	13020.0	0.445360
$950$	0	0
$951$	−3762.00	−0.128277
$952$	0	0
$953$	48990.0	1.66521	0.832603	−	0.553870i	$-0.186850\pi$
$953$	48990.0	1.66521	0.832603	+	0.553870i	$0.186850\pi$
$954$	0	0
$955$	7340.00	0.248709
$956$	0	0
$957$	7392.00	0.249686
$958$	0	0
$959$	3962.00	0.133409
$960$	0	0
$961$	−16335.0	−0.548320
$962$	0	0
$963$	−9828.00	−0.328871
$964$	0	0
$965$	−18310.0	−0.610798
$966$	0	0
$967$	16104.0	0.535543	0.267771	−	0.963482i	$-0.413713\pi$
$967$	16104.0	0.535543	0.267771	+	0.963482i	$0.413713\pi$
$968$	0	0
$969$	2352.00	0.0779743
$970$	0	0
$971$	39532.0	1.30653	0.653266	−	0.757129i	$-0.273398\pi$
$971$	39532.0	1.30653	0.653266	+	0.757129i	$0.273398\pi$
$972$	0	0
$973$	6048.00	0.199270
$974$	0	0
$975$	−4650.00	−0.152738
$976$	0	0
$977$	−7610.00	−0.249197	−0.124599	−	0.992207i	$-0.539764\pi$
$977$	−7610.00	−0.249197	−0.124599	+	0.992207i	$0.539764\pi$
$978$	0	0
$979$	−2208.00	−0.0720817
$980$	0	0
$981$	17190.0	0.559464
$982$	0	0
$983$	−22512.0	−0.730439	−0.365219	−	0.930921i	$-0.619006\pi$
$983$	−22512.0	−0.730439	−0.365219	+	0.930921i	$0.619006\pi$
$984$	0	0
$985$	−8990.00	−0.290807
$986$	0	0
$987$	3192.00	0.102941
$988$	0	0
$989$	2720.00	0.0874530
$990$	0	0
$991$	−48280.0	−1.54759	−0.773797	−	0.633434i	$-0.781645\pi$
$991$	−48280.0	−1.54759	−0.773797	+	0.633434i	$0.781645\pi$
$992$	0	0
$993$	−396.000	−0.0126553
$994$	0	0
$995$	−2300.00	−0.0732813
$996$	0	0
$997$	−25926.0	−0.823555	−0.411778	−	0.911284i	$-0.635092\pi$
$997$	−25926.0	−0.823555	−0.411778	+	0.911284i	$0.635092\pi$
$998$	0	0
$999$	162.000	0.00513058

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.4.a.f.1.1	✓	1
4.3	odd	2		1680.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.4.a.f.1.1	✓	1	1.1	even	1	trivial
1680.4.a.f.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	840.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more