LMFDB - Embedded newform 2240.4.a.bk.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2240,4,Mod(1,2240)] 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("2240.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,8,0,5,0,7,0,37,0,-12] 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:	$132.164278413$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2240.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +37.0000 q^{9} -12.0000 q^{11} +78.0000 q^{13} +40.0000 q^{15} -94.0000 q^{17} -40.0000 q^{19} +56.0000 q^{21} +32.0000 q^{23} +25.0000 q^{25} +80.0000 q^{27} +50.0000 q^{29} -248.000 q^{31} -96.0000 q^{33} +35.0000 q^{35} +434.000 q^{37} +624.000 q^{39} +402.000 q^{41} +68.0000 q^{43} +185.000 q^{45} +536.000 q^{47} +49.0000 q^{49} -752.000 q^{51} -22.0000 q^{53} -60.0000 q^{55} -320.000 q^{57} +560.000 q^{59} +278.000 q^{61} +259.000 q^{63} +390.000 q^{65} +164.000 q^{67} +256.000 q^{69} +672.000 q^{71} +82.0000 q^{73} +200.000 q^{75} -84.0000 q^{77} -1000.00 q^{79} -359.000 q^{81} +448.000 q^{83} -470.000 q^{85} +400.000 q^{87} -870.000 q^{89} +546.000 q^{91} -1984.00 q^{93} -200.000 q^{95} +1026.00 q^{97} -444.000 q^{99} +O(q^{100})$

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
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	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
$9$	37.0000	1.37037
$10$	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
$13$	78.0000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	78.0000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	40.0000	0.688530
$16$	0	0
$17$	−94.0000	−1.34108	−0.670540	−	0.741874i	$-0.733937\pi$
$17$	−94.0000	−1.34108	−0.670540	+	0.741874i	$0.733937\pi$
$18$	0	0
$19$	−40.0000	−0.482980	−0.241490	−	0.970403i	$-0.577636\pi$
$19$	−40.0000	−0.482980	−0.241490	+	0.970403i	$0.577636\pi$
$20$	0	0
$21$	56.0000	0.581914
$22$	0	0
$23$	32.0000	0.290107	0.145054	−	0.989424i	$-0.453665\pi$
$23$	32.0000	0.290107	0.145054	+	0.989424i	$0.453665\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	80.0000	0.570222
$28$	0	0
$29$	50.0000	0.320164	0.160082	−	0.987104i	$-0.448824\pi$
$29$	50.0000	0.320164	0.160082	+	0.987104i	$0.448824\pi$
$30$	0	0
$31$	−248.000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−248.000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−96.0000	−0.506408
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	434.000	1.92836	0.964178	−	0.265257i	$-0.0854567\pi$
$37$	434.000	1.92836	0.964178	+	0.265257i	$0.0854567\pi$
$38$	0	0
$39$	624.000	2.56205
$40$	0	0
$41$	402.000	1.53126	0.765632	−	0.643278i	$-0.222426\pi$
$41$	402.000	1.53126	0.765632	+	0.643278i	$0.222426\pi$
$42$	0	0
$43$	68.0000	0.241161	0.120580	−	0.992704i	$-0.461524\pi$
$43$	68.0000	0.241161	0.120580	+	0.992704i	$0.461524\pi$
$44$	0	0
$45$	185.000	0.612848
$46$	0	0
$47$	536.000	1.66348	0.831741	−	0.555164i	$-0.187345\pi$
$47$	536.000	1.66348	0.831741	+	0.555164i	$0.187345\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−752.000	−2.06473
$52$	0	0
$53$	−22.0000	−0.0570176	−0.0285088	−	0.999594i	$-0.509076\pi$
$53$	−22.0000	−0.0570176	−0.0285088	+	0.999594i	$0.509076\pi$
$54$	0	0
$55$	−60.0000	−0.147098
$56$	0	0
$57$	−320.000	−0.743597
$58$	0	0
$59$	560.000	1.23569	0.617846	−	0.786299i	$-0.288006\pi$
$59$	560.000	1.23569	0.617846	+	0.786299i	$0.288006\pi$
$60$	0	0
$61$	278.000	0.583512	0.291756	−	0.956493i	$-0.405760\pi$
$61$	278.000	0.583512	0.291756	+	0.956493i	$0.405760\pi$
$62$	0	0
$63$	259.000	0.517951
$64$	0	0
$65$	390.000	0.744208
$66$	0	0
$67$	164.000	0.299042	0.149521	−	0.988759i	$-0.452227\pi$
$67$	164.000	0.299042	0.149521	+	0.988759i	$0.452227\pi$
$68$	0	0
$69$	256.000	0.446649
$70$	0	0
$71$	672.000	1.12326	0.561632	−	0.827387i	$-0.310174\pi$
$71$	672.000	1.12326	0.561632	+	0.827387i	$0.310174\pi$
$72$	0	0
$73$	82.0000	0.131471	0.0657354	−	0.997837i	$-0.479061\pi$
$73$	82.0000	0.131471	0.0657354	+	0.997837i	$0.479061\pi$
$74$	0	0
$75$	200.000	0.307920
$76$	0	0
$77$	−84.0000	−0.124321
$78$	0	0
$79$	−1000.00	−1.42416	−0.712081	−	0.702097i	$-0.752247\pi$
$79$	−1000.00	−1.42416	−0.712081	+	0.702097i	$0.752247\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	448.000	0.592463	0.296231	−	0.955116i	$-0.404270\pi$
$83$	448.000	0.592463	0.296231	+	0.955116i	$0.404270\pi$
$84$	0	0
$85$	−470.000	−0.599749
$86$	0	0
$87$	400.000	0.492925
$88$	0	0
$89$	−870.000	−1.03618	−0.518089	−	0.855327i	$-0.673356\pi$
$89$	−870.000	−1.03618	−0.518089	+	0.855327i	$0.673356\pi$
$90$	0	0
$91$	546.000	0.628971
$92$	0	0
$93$	−1984.00	−2.21216
$94$	0	0
$95$	−200.000	−0.215995
$96$	0	0
$97$	1026.00	1.07396	0.536982	−	0.843594i	$-0.319564\pi$
$97$	1026.00	1.07396	0.536982	+	0.843594i	$0.319564\pi$
$98$	0	0
$99$	−444.000	−0.450744
$100$	0	0
$101$	−482.000	−0.474859	−0.237430	−	0.971405i	$-0.576305\pi$
$101$	−482.000	−0.474859	−0.237430	+	0.971405i	$0.576305\pi$
$102$	0	0
$103$	272.000	0.260203	0.130102	−	0.991501i	$-0.458470\pi$
$103$	272.000	0.260203	0.130102	+	0.991501i	$0.458470\pi$
$104$	0	0
$105$	280.000	0.260240
$106$	0	0
$107$	444.000	0.401150	0.200575	−	0.979678i	$-0.435719\pi$
$107$	444.000	0.401150	0.200575	+	0.979678i	$0.435719\pi$
$108$	0	0
$109$	1170.00	1.02813	0.514063	−	0.857753i	$-0.328140\pi$
$109$	1170.00	1.02813	0.514063	+	0.857753i	$0.328140\pi$
$110$	0	0
$111$	3472.00	2.96890
$112$	0	0
$113$	−798.000	−0.664332	−0.332166	−	0.943221i	$-0.607779\pi$
$113$	−798.000	−0.664332	−0.332166	+	0.943221i	$0.607779\pi$
$114$	0	0
$115$	160.000	0.129740
$116$	0	0
$117$	2886.00	2.28043
$118$	0	0
$119$	−658.000	−0.506880
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	3216.00	2.35754
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	776.000	0.542196	0.271098	−	0.962552i	$-0.412613\pi$
$127$	776.000	0.542196	0.271098	+	0.962552i	$0.412613\pi$
$128$	0	0
$129$	544.000	0.371291
$130$	0	0
$131$	−1112.00	−0.741648	−0.370824	−	0.928703i	$-0.620925\pi$
$131$	−1112.00	−0.741648	−0.370824	+	0.928703i	$0.620925\pi$
$132$	0	0
$133$	−280.000	−0.182549
$134$	0	0
$135$	400.000	0.255011
$136$	0	0
$137$	−694.000	−0.432791	−0.216396	−	0.976306i	$-0.569430\pi$
$137$	−694.000	−0.432791	−0.216396	+	0.976306i	$0.569430\pi$
$138$	0	0
$139$	−360.000	−0.219675	−0.109837	−	0.993950i	$-0.535033\pi$
$139$	−360.000	−0.219675	−0.109837	+	0.993950i	$0.535033\pi$
$140$	0	0
$141$	4288.00	2.56110
$142$	0	0
$143$	−936.000	−0.547358
$144$	0	0
$145$	250.000	0.143182
$146$	0	0
$147$	392.000	0.219943
$148$	0	0
$149$	−2270.00	−1.24809	−0.624046	−	0.781388i	$-0.714512\pi$
$149$	−2270.00	−1.24809	−0.624046	+	0.781388i	$0.714512\pi$
$150$	0	0
$151$	632.000	0.340606	0.170303	−	0.985392i	$-0.445525\pi$
$151$	632.000	0.340606	0.170303	+	0.985392i	$0.445525\pi$
$152$	0	0
$153$	−3478.00	−1.83778
$154$	0	0
$155$	−1240.00	−0.642575
$156$	0	0
$157$	734.000	0.373118	0.186559	−	0.982444i	$-0.440266\pi$
$157$	734.000	0.373118	0.186559	+	0.982444i	$0.440266\pi$
$158$	0	0
$159$	−176.000	−0.0877843
$160$	0	0
$161$	224.000	0.109650
$162$	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
$165$	−480.000	−0.226472
$166$	0	0
$167$	416.000	0.192761	0.0963804	−	0.995345i	$-0.469273\pi$
$167$	416.000	0.192761	0.0963804	+	0.995345i	$0.469273\pi$
$168$	0	0
$169$	3887.00	1.76923
$170$	0	0
$171$	−1480.00	−0.661862
$172$	0	0
$173$	−3042.00	−1.33687	−0.668436	−	0.743769i	$-0.733036\pi$
$173$	−3042.00	−1.33687	−0.668436	+	0.743769i	$0.733036\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	4480.00	1.90247
$178$	0	0
$179$	180.000	0.0751611	0.0375805	−	0.999294i	$-0.488035\pi$
$179$	180.000	0.0751611	0.0375805	+	0.999294i	$0.488035\pi$
$180$	0	0
$181$	1958.00	0.804072	0.402036	−	0.915624i	$-0.368303\pi$
$181$	1958.00	0.804072	0.402036	+	0.915624i	$0.368303\pi$
$182$	0	0
$183$	2224.00	0.898376
$184$	0	0
$185$	2170.00	0.862387
$186$	0	0
$187$	1128.00	0.441110
$188$	0	0
$189$	560.000	0.215524
$190$	0	0
$191$	−2888.00	−1.09408	−0.547038	−	0.837108i	$-0.684245\pi$
$191$	−2888.00	−1.09408	−0.547038	+	0.837108i	$0.684245\pi$
$192$	0	0
$193$	1602.00	0.597484	0.298742	−	0.954334i	$-0.403433\pi$
$193$	1602.00	0.597484	0.298742	+	0.954334i	$0.403433\pi$
$194$	0	0
$195$	3120.00	1.14578
$196$	0	0
$197$	4794.00	1.73380	0.866899	−	0.498483i	$-0.166109\pi$
$197$	4794.00	1.73380	0.866899	+	0.498483i	$0.166109\pi$
$198$	0	0
$199$	1280.00	0.455964	0.227982	−	0.973665i	$-0.426787\pi$
$199$	1280.00	0.455964	0.227982	+	0.973665i	$0.426787\pi$
$200$	0	0
$201$	1312.00	0.460405
$202$	0	0
$203$	350.000	0.121011
$204$	0	0
$205$	2010.00	0.684802
$206$	0	0
$207$	1184.00	0.397554
$208$	0	0
$209$	480.000	0.158863
$210$	0	0
$211$	68.0000	0.0221863	0.0110932	−	0.999938i	$-0.496469\pi$
$211$	68.0000	0.0221863	0.0110932	+	0.999938i	$0.496469\pi$
$212$	0	0
$213$	5376.00	1.72938
$214$	0	0
$215$	340.000	0.107850
$216$	0	0
$217$	−1736.00	−0.543075
$218$	0	0
$219$	656.000	0.202413
$220$	0	0
$221$	−7332.00	−2.23169
$222$	0	0
$223$	−1728.00	−0.518903	−0.259452	−	0.965756i	$-0.583542\pi$
$223$	−1728.00	−0.518903	−0.259452	+	0.965756i	$0.583542\pi$
$224$	0	0
$225$	925.000	0.274074
$226$	0	0
$227$	4864.00	1.42218	0.711090	−	0.703101i	$-0.248202\pi$
$227$	4864.00	1.42218	0.711090	+	0.703101i	$0.248202\pi$
$228$	0	0
$229$	5510.00	1.59000	0.795002	−	0.606606i	$-0.207470\pi$
$229$	5510.00	1.59000	0.795002	+	0.606606i	$0.207470\pi$
$230$	0	0
$231$	−672.000	−0.191404
$232$	0	0
$233$	5322.00	1.49638	0.748188	−	0.663486i	$-0.230924\pi$
$233$	5322.00	1.49638	0.748188	+	0.663486i	$0.230924\pi$
$234$	0	0
$235$	2680.00	0.743932
$236$	0	0
$237$	−8000.00	−2.19264
$238$	0	0
$239$	−1840.00	−0.497990	−0.248995	−	0.968505i	$-0.580100\pi$
$239$	−1840.00	−0.497990	−0.248995	+	0.968505i	$0.580100\pi$
$240$	0	0
$241$	−438.000	−0.117071	−0.0585354	−	0.998285i	$-0.518643\pi$
$241$	−438.000	−0.117071	−0.0585354	+	0.998285i	$0.518643\pi$
$242$	0	0
$243$	−5032.00	−1.32841
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	−3120.00	−0.803728
$248$	0	0
$249$	3584.00	0.912156
$250$	0	0
$251$	−5592.00	−1.40623	−0.703115	−	0.711076i	$-0.748208\pi$
$251$	−5592.00	−1.40623	−0.703115	+	0.711076i	$0.748208\pi$
$252$	0	0
$253$	−384.000	−0.0954224
$254$	0	0
$255$	−3760.00	−0.923374
$256$	0	0
$257$	−1974.00	−0.479123	−0.239562	−	0.970881i	$-0.577004\pi$
$257$	−1974.00	−0.479123	−0.239562	+	0.970881i	$0.577004\pi$
$258$	0	0
$259$	3038.00	0.728850
$260$	0	0
$261$	1850.00	0.438744
$262$	0	0
$263$	−728.000	−0.170686	−0.0853430	−	0.996352i	$-0.527199\pi$
$263$	−728.000	−0.170686	−0.0853430	+	0.996352i	$0.527199\pi$
$264$	0	0
$265$	−110.000	−0.0254990
$266$	0	0
$267$	−6960.00	−1.59530
$268$	0	0
$269$	−5810.00	−1.31688	−0.658442	−	0.752631i	$-0.728784\pi$
$269$	−5810.00	−1.31688	−0.658442	+	0.752631i	$0.728784\pi$
$270$	0	0
$271$	−6528.00	−1.46328	−0.731638	−	0.681693i	$-0.761244\pi$
$271$	−6528.00	−1.46328	−0.731638	+	0.681693i	$0.761244\pi$
$272$	0	0
$273$	4368.00	0.968364
$274$	0	0
$275$	−300.000	−0.0657843
$276$	0	0
$277$	−5126.00	−1.11188	−0.555941	−	0.831222i	$-0.687642\pi$
$277$	−5126.00	−1.11188	−0.555941	+	0.831222i	$0.687642\pi$
$278$	0	0
$279$	−9176.00	−1.96901
$280$	0	0
$281$	−2358.00	−0.500592	−0.250296	−	0.968169i	$-0.580528\pi$
$281$	−2358.00	−0.500592	−0.250296	+	0.968169i	$0.580528\pi$
$282$	0	0
$283$	−392.000	−0.0823392	−0.0411696	−	0.999152i	$-0.513108\pi$
$283$	−392.000	−0.0823392	−0.0411696	+	0.999152i	$0.513108\pi$
$284$	0	0
$285$	−1600.00	−0.332547
$286$	0	0
$287$	2814.00	0.578764
$288$	0	0
$289$	3923.00	0.798494
$290$	0	0
$291$	8208.00	1.65348
$292$	0	0
$293$	−1202.00	−0.239664	−0.119832	−	0.992794i	$-0.538236\pi$
$293$	−1202.00	−0.239664	−0.119832	+	0.992794i	$0.538236\pi$
$294$	0	0
$295$	2800.00	0.552618
$296$	0	0
$297$	−960.000	−0.187558
$298$	0	0
$299$	2496.00	0.482767
$300$	0	0
$301$	476.000	0.0911501
$302$	0	0
$303$	−3856.00	−0.731094
$304$	0	0
$305$	1390.00	0.260955
$306$	0	0
$307$	6384.00	1.18682	0.593411	−	0.804900i	$-0.297781\pi$
$307$	6384.00	1.18682	0.593411	+	0.804900i	$0.297781\pi$
$308$	0	0
$309$	2176.00	0.400609
$310$	0	0
$311$	−4968.00	−0.905818	−0.452909	−	0.891557i	$-0.649614\pi$
$311$	−4968.00	−0.905818	−0.452909	+	0.891557i	$0.649614\pi$
$312$	0	0
$313$	−2758.00	−0.498056	−0.249028	−	0.968496i	$-0.580111\pi$
$313$	−2758.00	−0.498056	−0.249028	+	0.968496i	$0.580111\pi$
$314$	0	0
$315$	1295.00	0.231635
$316$	0	0
$317$	6274.00	1.11162	0.555809	−	0.831310i	$-0.312409\pi$
$317$	6274.00	1.11162	0.555809	+	0.831310i	$0.312409\pi$
$318$	0	0
$319$	−600.000	−0.105309
$320$	0	0
$321$	3552.00	0.617612
$322$	0	0
$323$	3760.00	0.647715
$324$	0	0
$325$	1950.00	0.332820
$326$	0	0
$327$	9360.00	1.58290
$328$	0	0
$329$	3752.00	0.628737
$330$	0	0
$331$	−1932.00	−0.320823	−0.160411	−	0.987050i	$-0.551282\pi$
$331$	−1932.00	−0.320823	−0.160411	+	0.987050i	$0.551282\pi$
$332$	0	0
$333$	16058.0	2.64256
$334$	0	0
$335$	820.000	0.133735
$336$	0	0
$337$	2386.00	0.385679	0.192839	−	0.981230i	$-0.438230\pi$
$337$	2386.00	0.385679	0.192839	+	0.981230i	$0.438230\pi$
$338$	0	0
$339$	−6384.00	−1.02281
$340$	0	0
$341$	2976.00	0.472608
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	1280.00	0.199747
$346$	0	0
$347$	−6076.00	−0.939991	−0.469995	−	0.882669i	$-0.655744\pi$
$347$	−6076.00	−0.939991	−0.469995	+	0.882669i	$0.655744\pi$
$348$	0	0
$349$	−2210.00	−0.338964	−0.169482	−	0.985533i	$-0.554210\pi$
$349$	−2210.00	−0.338964	−0.169482	+	0.985533i	$0.554210\pi$
$350$	0	0
$351$	6240.00	0.948908
$352$	0	0
$353$	−2598.00	−0.391721	−0.195861	−	0.980632i	$-0.562750\pi$
$353$	−2598.00	−0.391721	−0.195861	+	0.980632i	$0.562750\pi$
$354$	0	0
$355$	3360.00	0.502339
$356$	0	0
$357$	−5264.00	−0.780393
$358$	0	0
$359$	−13320.0	−1.95822	−0.979112	−	0.203320i	$-0.934827\pi$
$359$	−13320.0	−1.95822	−0.979112	+	0.203320i	$0.934827\pi$
$360$	0	0
$361$	−5259.00	−0.766730
$362$	0	0
$363$	−9496.00	−1.37303
$364$	0	0
$365$	410.000	0.0587956
$366$	0	0
$367$	10816.0	1.53839	0.769197	−	0.639012i	$-0.220656\pi$
$367$	10816.0	1.53839	0.769197	+	0.639012i	$0.220656\pi$
$368$	0	0
$369$	14874.0	2.09840
$370$	0	0
$371$	−154.000	−0.0215506
$372$	0	0
$373$	11098.0	1.54057	0.770285	−	0.637700i	$-0.220114\pi$
$373$	11098.0	1.54057	0.770285	+	0.637700i	$0.220114\pi$
$374$	0	0
$375$	1000.00	0.137706
$376$	0	0
$377$	3900.00	0.532786
$378$	0	0
$379$	−7100.00	−0.962276	−0.481138	−	0.876645i	$-0.659776\pi$
$379$	−7100.00	−0.962276	−0.481138	+	0.876645i	$0.659776\pi$
$380$	0	0
$381$	6208.00	0.834765
$382$	0	0
$383$	−728.000	−0.0971255	−0.0485627	−	0.998820i	$-0.515464\pi$
$383$	−728.000	−0.0971255	−0.0485627	+	0.998820i	$0.515464\pi$
$384$	0	0
$385$	−420.000	−0.0555979
$386$	0	0
$387$	2516.00	0.330479
$388$	0	0
$389$	6810.00	0.887611	0.443806	−	0.896123i	$-0.353628\pi$
$389$	6810.00	0.887611	0.443806	+	0.896123i	$0.353628\pi$
$390$	0	0
$391$	−3008.00	−0.389057
$392$	0	0
$393$	−8896.00	−1.14184
$394$	0	0
$395$	−5000.00	−0.636905
$396$	0	0
$397$	574.000	0.0725648	0.0362824	−	0.999342i	$-0.488448\pi$
$397$	574.000	0.0725648	0.0362824	+	0.999342i	$0.488448\pi$
$398$	0	0
$399$	−2240.00	−0.281053
$400$	0	0
$401$	6162.00	0.767371	0.383685	−	0.923464i	$-0.374655\pi$
$401$	6162.00	0.767371	0.383685	+	0.923464i	$0.374655\pi$
$402$	0	0
$403$	−19344.0	−2.39105
$404$	0	0
$405$	−1795.00	−0.220233
$406$	0	0
$407$	−5208.00	−0.634278
$408$	0	0
$409$	8210.00	0.992563	0.496282	−	0.868162i	$-0.334698\pi$
$409$	8210.00	0.992563	0.496282	+	0.868162i	$0.334698\pi$
$410$	0	0
$411$	−5552.00	−0.666326
$412$	0	0
$413$	3920.00	0.467047
$414$	0	0
$415$	2240.00	0.264957
$416$	0	0
$417$	−2880.00	−0.338212
$418$	0	0
$419$	−4800.00	−0.559655	−0.279827	−	0.960050i	$-0.590277\pi$
$419$	−4800.00	−0.559655	−0.279827	+	0.960050i	$0.590277\pi$
$420$	0	0
$421$	9938.00	1.15047	0.575236	−	0.817988i	$-0.304910\pi$
$421$	9938.00	1.15047	0.575236	+	0.817988i	$0.304910\pi$
$422$	0	0
$423$	19832.0	2.27959
$424$	0	0
$425$	−2350.00	−0.268216
$426$	0	0
$427$	1946.00	0.220547
$428$	0	0
$429$	−7488.00	−0.842713
$430$	0	0
$431$	−9248.00	−1.03355	−0.516776	−	0.856121i	$-0.672868\pi$
$431$	−9248.00	−1.03355	−0.516776	+	0.856121i	$0.672868\pi$
$432$	0	0
$433$	−1118.00	−0.124082	−0.0620412	−	0.998074i	$-0.519761\pi$
$433$	−1118.00	−0.124082	−0.0620412	+	0.998074i	$0.519761\pi$
$434$	0	0
$435$	2000.00	0.220443
$436$	0	0
$437$	−1280.00	−0.140116
$438$	0	0
$439$	−11960.0	−1.30027	−0.650136	−	0.759818i	$-0.725288\pi$
$439$	−11960.0	−1.30027	−0.650136	+	0.759818i	$0.725288\pi$
$440$	0	0
$441$	1813.00	0.195767
$442$	0	0
$443$	−7332.00	−0.786352	−0.393176	−	0.919463i	$-0.628624\pi$
$443$	−7332.00	−0.786352	−0.393176	+	0.919463i	$0.628624\pi$
$444$	0	0
$445$	−4350.00	−0.463393
$446$	0	0
$447$	−18160.0	−1.92156
$448$	0	0
$449$	1890.00	0.198652	0.0993259	−	0.995055i	$-0.468331\pi$
$449$	1890.00	0.198652	0.0993259	+	0.995055i	$0.468331\pi$
$450$	0	0
$451$	−4824.00	−0.503666
$452$	0	0
$453$	5056.00	0.524396
$454$	0	0
$455$	2730.00	0.281284
$456$	0	0
$457$	−7014.00	−0.717945	−0.358973	−	0.933348i	$-0.616873\pi$
$457$	−7014.00	−0.717945	−0.358973	+	0.933348i	$0.616873\pi$
$458$	0	0
$459$	−7520.00	−0.764714
$460$	0	0
$461$	8318.00	0.840364	0.420182	−	0.907440i	$-0.361966\pi$
$461$	8318.00	0.840364	0.420182	+	0.907440i	$0.361966\pi$
$462$	0	0
$463$	6432.00	0.645616	0.322808	−	0.946464i	$-0.395373\pi$
$463$	6432.00	0.645616	0.322808	+	0.946464i	$0.395373\pi$
$464$	0	0
$465$	−9920.00	−0.989310
$466$	0	0
$467$	10064.0	0.997230	0.498615	−	0.866824i	$-0.333842\pi$
$467$	10064.0	0.997230	0.498615	+	0.866824i	$0.333842\pi$
$468$	0	0
$469$	1148.00	0.113027
$470$	0	0
$471$	5872.00	0.574453
$472$	0	0
$473$	−816.000	−0.0793229
$474$	0	0
$475$	−1000.00	−0.0965961
$476$	0	0
$477$	−814.000	−0.0781352
$478$	0	0
$479$	1400.00	0.133544	0.0667721	−	0.997768i	$-0.478730\pi$
$479$	1400.00	0.133544	0.0667721	+	0.997768i	$0.478730\pi$
$480$	0	0
$481$	33852.0	3.20898
$482$	0	0
$483$	1792.00	0.168817
$484$	0	0
$485$	5130.00	0.480291
$486$	0	0
$487$	13376.0	1.24461	0.622304	−	0.782775i	$-0.286197\pi$
$487$	13376.0	1.24461	0.622304	+	0.782775i	$0.286197\pi$
$488$	0	0
$489$	−20256.0	−1.87323
$490$	0	0
$491$	−7092.00	−0.651848	−0.325924	−	0.945396i	$-0.605675\pi$
$491$	−7092.00	−0.651848	−0.325924	+	0.945396i	$0.605675\pi$
$492$	0	0
$493$	−4700.00	−0.429366
$494$	0	0
$495$	−2220.00	−0.201579
$496$	0	0
$497$	4704.00	0.424554
$498$	0	0
$499$	820.000	0.0735636	0.0367818	−	0.999323i	$-0.488289\pi$
$499$	820.000	0.0735636	0.0367818	+	0.999323i	$0.488289\pi$
$500$	0	0
$501$	3328.00	0.296775
$502$	0	0
$503$	−4568.00	−0.404925	−0.202462	−	0.979290i	$-0.564894\pi$
$503$	−4568.00	−0.404925	−0.202462	+	0.979290i	$0.564894\pi$
$504$	0	0
$505$	−2410.00	−0.212364
$506$	0	0
$507$	31096.0	2.72391
$508$	0	0
$509$	−19810.0	−1.72507	−0.862537	−	0.505994i	$-0.831126\pi$
$509$	−19810.0	−1.72507	−0.862537	+	0.505994i	$0.831126\pi$
$510$	0	0
$511$	574.000	0.0496913
$512$	0	0
$513$	−3200.00	−0.275406
$514$	0	0
$515$	1360.00	0.116367
$516$	0	0
$517$	−6432.00	−0.547155
$518$	0	0
$519$	−24336.0	−2.05825
$520$	0	0
$521$	−1838.00	−0.154557	−0.0772785	−	0.997010i	$-0.524623\pi$
$521$	−1838.00	−0.154557	−0.0772785	+	0.997010i	$0.524623\pi$
$522$	0	0
$523$	−2072.00	−0.173236	−0.0866178	−	0.996242i	$-0.527606\pi$
$523$	−2072.00	−0.173236	−0.0866178	+	0.996242i	$0.527606\pi$
$524$	0	0
$525$	1400.00	0.116383
$526$	0	0
$527$	23312.0	1.92692
$528$	0	0
$529$	−11143.0	−0.915838
$530$	0	0
$531$	20720.0	1.69335
$532$	0	0
$533$	31356.0	2.54818
$534$	0	0
$535$	2220.00	0.179400
$536$	0	0
$537$	1440.00	0.115718
$538$	0	0
$539$	−588.000	−0.0469888
$540$	0	0
$541$	3498.00	0.277987	0.138993	−	0.990293i	$-0.455613\pi$
$541$	3498.00	0.277987	0.138993	+	0.990293i	$0.455613\pi$
$542$	0	0
$543$	15664.0	1.23795
$544$	0	0
$545$	5850.00	0.459792
$546$	0	0
$547$	−5076.00	−0.396772	−0.198386	−	0.980124i	$-0.563570\pi$
$547$	−5076.00	−0.396772	−0.198386	+	0.980124i	$0.563570\pi$
$548$	0	0
$549$	10286.0	0.799628
$550$	0	0
$551$	−2000.00	−0.154633
$552$	0	0
$553$	−7000.00	−0.538283
$554$	0	0
$555$	17360.0	1.32773
$556$	0	0
$557$	8674.00	0.659837	0.329918	−	0.944009i	$-0.392979\pi$
$557$	8674.00	0.659837	0.329918	+	0.944009i	$0.392979\pi$
$558$	0	0
$559$	5304.00	0.401315
$560$	0	0
$561$	9024.00	0.679133
$562$	0	0
$563$	−16072.0	−1.20312	−0.601558	−	0.798829i	$-0.705453\pi$
$563$	−16072.0	−1.20312	−0.601558	+	0.798829i	$0.705453\pi$
$564$	0	0
$565$	−3990.00	−0.297098
$566$	0	0
$567$	−2513.00	−0.186131
$568$	0	0
$569$	2730.00	0.201138	0.100569	−	0.994930i	$-0.467934\pi$
$569$	2730.00	0.201138	0.100569	+	0.994930i	$0.467934\pi$
$570$	0	0
$571$	−19932.0	−1.46082	−0.730410	−	0.683009i	$-0.760671\pi$
$571$	−19932.0	−1.46082	−0.730410	+	0.683009i	$0.760671\pi$
$572$	0	0
$573$	−23104.0	−1.68444
$574$	0	0
$575$	800.000	0.0580214
$576$	0	0
$577$	−20054.0	−1.44690	−0.723448	−	0.690379i	$-0.757444\pi$
$577$	−20054.0	−1.44690	−0.723448	+	0.690379i	$0.757444\pi$
$578$	0	0
$579$	12816.0	0.919887
$580$	0	0
$581$	3136.00	0.223930
$582$	0	0
$583$	264.000	0.0187543
$584$	0	0
$585$	14430.0	1.01984
$586$	0	0
$587$	2544.00	0.178879	0.0894396	−	0.995992i	$-0.471492\pi$
$587$	2544.00	0.178879	0.0894396	+	0.995992i	$0.471492\pi$
$588$	0	0
$589$	9920.00	0.693967
$590$	0	0
$591$	38352.0	2.66936
$592$	0	0
$593$	14202.0	0.983484	0.491742	−	0.870741i	$-0.336360\pi$
$593$	14202.0	0.983484	0.491742	+	0.870741i	$0.336360\pi$
$594$	0	0
$595$	−3290.00	−0.226684
$596$	0	0
$597$	10240.0	0.702002
$598$	0	0
$599$	−19600.0	−1.33695	−0.668476	−	0.743734i	$-0.733053\pi$
$599$	−19600.0	−1.33695	−0.668476	+	0.743734i	$0.733053\pi$
$600$	0	0
$601$	−27078.0	−1.83783	−0.918914	−	0.394458i	$-0.870932\pi$
$601$	−27078.0	−1.83783	−0.918914	+	0.394458i	$0.870932\pi$
$602$	0	0
$603$	6068.00	0.409798
$604$	0	0
$605$	−5935.00	−0.398830
$606$	0	0
$607$	−2704.00	−0.180811	−0.0904053	−	0.995905i	$-0.528816\pi$
$607$	−2704.00	−0.180811	−0.0904053	+	0.995905i	$0.528816\pi$
$608$	0	0
$609$	2800.00	0.186308
$610$	0	0
$611$	41808.0	2.76820
$612$	0	0
$613$	−12702.0	−0.836915	−0.418458	−	0.908236i	$-0.637429\pi$
$613$	−12702.0	−0.836915	−0.418458	+	0.908236i	$0.637429\pi$
$614$	0	0
$615$	16080.0	1.05432
$616$	0	0
$617$	12666.0	0.826441	0.413220	−	0.910631i	$-0.364404\pi$
$617$	12666.0	0.826441	0.413220	+	0.910631i	$0.364404\pi$
$618$	0	0
$619$	−960.000	−0.0623355	−0.0311677	−	0.999514i	$-0.509923\pi$
$619$	−960.000	−0.0623355	−0.0311677	+	0.999514i	$0.509923\pi$
$620$	0	0
$621$	2560.00	0.165426
$622$	0	0
$623$	−6090.00	−0.391638
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	3840.00	0.244585
$628$	0	0
$629$	−40796.0	−2.58608
$630$	0	0
$631$	23232.0	1.46569	0.732846	−	0.680395i	$-0.238192\pi$
$631$	23232.0	1.46569	0.732846	+	0.680395i	$0.238192\pi$
$632$	0	0
$633$	544.000	0.0341581
$634$	0	0
$635$	3880.00	0.242477
$636$	0	0
$637$	3822.00	0.237729
$638$	0	0
$639$	24864.0	1.53929
$640$	0	0
$641$	12162.0	0.749407	0.374704	−	0.927145i	$-0.377744\pi$
$641$	12162.0	0.749407	0.374704	+	0.927145i	$0.377744\pi$
$642$	0	0
$643$	488.000	0.0299298	0.0149649	−	0.999888i	$-0.495236\pi$
$643$	488.000	0.0299298	0.0149649	+	0.999888i	$0.495236\pi$
$644$	0	0
$645$	2720.00	0.166046
$646$	0	0
$647$	−3984.00	−0.242082	−0.121041	−	0.992647i	$-0.538623\pi$
$647$	−3984.00	−0.242082	−0.121041	+	0.992647i	$0.538623\pi$
$648$	0	0
$649$	−6720.00	−0.406445
$650$	0	0
$651$	−13888.0	−0.836119
$652$	0	0
$653$	30538.0	1.83008	0.915042	−	0.403360i	$-0.132158\pi$
$653$	30538.0	1.83008	0.915042	+	0.403360i	$0.132158\pi$
$654$	0	0
$655$	−5560.00	−0.331675
$656$	0	0
$657$	3034.00	0.180164
$658$	0	0
$659$	−22740.0	−1.34420	−0.672098	−	0.740463i	$-0.734606\pi$
$659$	−22740.0	−1.34420	−0.672098	+	0.740463i	$0.734606\pi$
$660$	0	0
$661$	18718.0	1.10143	0.550715	−	0.834693i	$-0.314355\pi$
$661$	18718.0	1.10143	0.550715	+	0.834693i	$0.314355\pi$
$662$	0	0
$663$	−58656.0	−3.43591
$664$	0	0
$665$	−1400.00	−0.0816386
$666$	0	0
$667$	1600.00	0.0928819
$668$	0	0
$669$	−13824.0	−0.798904
$670$	0	0
$671$	−3336.00	−0.191930
$672$	0	0
$673$	10802.0	0.618702	0.309351	−	0.950948i	$-0.399888\pi$
$673$	10802.0	0.618702	0.309351	+	0.950948i	$0.399888\pi$
$674$	0	0
$675$	2000.00	0.114044
$676$	0	0
$677$	−346.000	−0.0196423	−0.00982117	−	0.999952i	$-0.503126\pi$
$677$	−346.000	−0.0196423	−0.00982117	+	0.999952i	$0.503126\pi$
$678$	0	0
$679$	7182.00	0.405920
$680$	0	0
$681$	38912.0	2.18959
$682$	0	0
$683$	11628.0	0.651439	0.325720	−	0.945466i	$-0.394393\pi$
$683$	11628.0	0.651439	0.325720	+	0.945466i	$0.394393\pi$
$684$	0	0
$685$	−3470.00	−0.193550
$686$	0	0
$687$	44080.0	2.44797
$688$	0	0
$689$	−1716.00	−0.0948830
$690$	0	0
$691$	−2472.00	−0.136092	−0.0680458	−	0.997682i	$-0.521676\pi$
$691$	−2472.00	−0.136092	−0.0680458	+	0.997682i	$0.521676\pi$
$692$	0	0
$693$	−3108.00	−0.170365
$694$	0	0
$695$	−1800.00	−0.0982416
$696$	0	0
$697$	−37788.0	−2.05355
$698$	0	0
$699$	42576.0	2.30382
$700$	0	0
$701$	2018.00	0.108729	0.0543643	−	0.998521i	$-0.482687\pi$
$701$	2018.00	0.108729	0.0543643	+	0.998521i	$0.482687\pi$
$702$	0	0
$703$	−17360.0	−0.931358
$704$	0	0
$705$	21440.0	1.14536
$706$	0	0
$707$	−3374.00	−0.179480
$708$	0	0
$709$	−790.000	−0.0418464	−0.0209232	−	0.999781i	$-0.506661\pi$
$709$	−790.000	−0.0418464	−0.0209232	+	0.999781i	$0.506661\pi$
$710$	0	0
$711$	−37000.0	−1.95163
$712$	0	0
$713$	−7936.00	−0.416838
$714$	0	0
$715$	−4680.00	−0.244786
$716$	0	0
$717$	−14720.0	−0.766706
$718$	0	0
$719$	18200.0	0.944013	0.472007	−	0.881595i	$-0.343530\pi$
$719$	18200.0	0.944013	0.472007	+	0.881595i	$0.343530\pi$
$720$	0	0
$721$	1904.00	0.0983477
$722$	0	0
$723$	−3504.00	−0.180242
$724$	0	0
$725$	1250.00	0.0640329
$726$	0	0
$727$	29056.0	1.48229	0.741147	−	0.671343i	$-0.234282\pi$
$727$	29056.0	1.48229	0.741147	+	0.671343i	$0.234282\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	−6392.00	−0.323415
$732$	0	0
$733$	−7082.00	−0.356862	−0.178431	−	0.983952i	$-0.557102\pi$
$733$	−7082.00	−0.356862	−0.178431	+	0.983952i	$0.557102\pi$
$734$	0	0
$735$	1960.00	0.0983615
$736$	0	0
$737$	−1968.00	−0.0983612
$738$	0	0
$739$	−11060.0	−0.550539	−0.275270	−	0.961367i	$-0.588767\pi$
$739$	−11060.0	−0.550539	−0.275270	+	0.961367i	$0.588767\pi$
$740$	0	0
$741$	−24960.0	−1.23742
$742$	0	0
$743$	33072.0	1.63297	0.816483	−	0.577369i	$-0.195921\pi$
$743$	33072.0	1.63297	0.816483	+	0.577369i	$0.195921\pi$
$744$	0	0
$745$	−11350.0	−0.558164
$746$	0	0
$747$	16576.0	0.811893
$748$	0	0
$749$	3108.00	0.151621
$750$	0	0
$751$	29072.0	1.41259	0.706293	−	0.707919i	$-0.250366\pi$
$751$	29072.0	1.41259	0.706293	+	0.707919i	$0.250366\pi$
$752$	0	0
$753$	−44736.0	−2.16503
$754$	0	0
$755$	3160.00	0.152323
$756$	0	0
$757$	13234.0	0.635400	0.317700	−	0.948191i	$-0.397089\pi$
$757$	13234.0	0.635400	0.317700	+	0.948191i	$0.397089\pi$
$758$	0	0
$759$	−3072.00	−0.146912
$760$	0	0
$761$	−22398.0	−1.06692	−0.533460	−	0.845825i	$-0.679109\pi$
$761$	−22398.0	−1.06692	−0.533460	+	0.845825i	$0.679109\pi$
$762$	0	0
$763$	8190.00	0.388595
$764$	0	0
$765$	−17390.0	−0.821878
$766$	0	0
$767$	43680.0	2.05631
$768$	0	0
$769$	6890.00	0.323095	0.161547	−	0.986865i	$-0.448352\pi$
$769$	6890.00	0.323095	0.161547	+	0.986865i	$0.448352\pi$
$770$	0	0
$771$	−15792.0	−0.737659
$772$	0	0
$773$	−16722.0	−0.778071	−0.389035	−	0.921223i	$-0.627192\pi$
$773$	−16722.0	−0.778071	−0.389035	+	0.921223i	$0.627192\pi$
$774$	0	0
$775$	−6200.00	−0.287368
$776$	0	0
$777$	24304.0	1.12214
$778$	0	0
$779$	−16080.0	−0.739571
$780$	0	0
$781$	−8064.00	−0.369466
$782$	0	0
$783$	4000.00	0.182565
$784$	0	0
$785$	3670.00	0.166864
$786$	0	0
$787$	32624.0	1.47766	0.738831	−	0.673891i	$-0.235378\pi$
$787$	32624.0	1.47766	0.738831	+	0.673891i	$0.235378\pi$
$788$	0	0
$789$	−5824.00	−0.262788
$790$	0	0
$791$	−5586.00	−0.251094
$792$	0	0
$793$	21684.0	0.971023
$794$	0	0
$795$	−880.000	−0.0392583
$796$	0	0
$797$	−11346.0	−0.504261	−0.252130	−	0.967693i	$-0.581131\pi$
$797$	−11346.0	−0.504261	−0.252130	+	0.967693i	$0.581131\pi$
$798$	0	0
$799$	−50384.0	−2.23086
$800$	0	0
$801$	−32190.0	−1.41995
$802$	0	0
$803$	−984.000	−0.0432436
$804$	0	0
$805$	1120.00	0.0490370
$806$	0	0
$807$	−46480.0	−2.02748
$808$	0	0
$809$	−35190.0	−1.52931	−0.764657	−	0.644438i	$-0.777091\pi$
$809$	−35190.0	−1.52931	−0.764657	+	0.644438i	$0.777091\pi$
$810$	0	0
$811$	−30432.0	−1.31765	−0.658824	−	0.752297i	$-0.728946\pi$
$811$	−30432.0	−1.31765	−0.658824	+	0.752297i	$0.728946\pi$
$812$	0	0
$813$	−52224.0	−2.25286
$814$	0	0
$815$	−12660.0	−0.544123
$816$	0	0
$817$	−2720.00	−0.116476
$818$	0	0
$819$	20202.0	0.861923
$820$	0	0
$821$	−12702.0	−0.539955	−0.269977	−	0.962867i	$-0.587016\pi$
$821$	−12702.0	−0.539955	−0.269977	+	0.962867i	$0.587016\pi$
$822$	0	0
$823$	16952.0	0.717995	0.358997	−	0.933339i	$-0.383119\pi$
$823$	16952.0	0.717995	0.358997	+	0.933339i	$0.383119\pi$
$824$	0	0
$825$	−2400.00	−0.101282
$826$	0	0
$827$	25404.0	1.06818	0.534089	−	0.845428i	$-0.320655\pi$
$827$	25404.0	1.06818	0.534089	+	0.845428i	$0.320655\pi$
$828$	0	0
$829$	−26250.0	−1.09976	−0.549879	−	0.835244i	$-0.685326\pi$
$829$	−26250.0	−1.09976	−0.549879	+	0.835244i	$0.685326\pi$
$830$	0	0
$831$	−41008.0	−1.71186
$832$	0	0
$833$	−4606.00	−0.191583
$834$	0	0
$835$	2080.00	0.0862052
$836$	0	0
$837$	−19840.0	−0.819320
$838$	0	0
$839$	−15360.0	−0.632045	−0.316023	−	0.948752i	$-0.602348\pi$
$839$	−15360.0	−0.632045	−0.316023	+	0.948752i	$0.602348\pi$
$840$	0	0
$841$	−21889.0	−0.897495
$842$	0	0
$843$	−18864.0	−0.770713
$844$	0	0
$845$	19435.0	0.791224
$846$	0	0
$847$	−8309.00	−0.337073
$848$	0	0
$849$	−3136.00	−0.126769
$850$	0	0
$851$	13888.0	0.559430
$852$	0	0
$853$	−10362.0	−0.415930	−0.207965	−	0.978136i	$-0.566684\pi$
$853$	−10362.0	−0.415930	−0.207965	+	0.978136i	$0.566684\pi$
$854$	0	0
$855$	−7400.00	−0.295994
$856$	0	0
$857$	4506.00	0.179606	0.0898028	−	0.995960i	$-0.471376\pi$
$857$	4506.00	0.179606	0.0898028	+	0.995960i	$0.471376\pi$
$858$	0	0
$859$	−24200.0	−0.961226	−0.480613	−	0.876933i	$-0.659586\pi$
$859$	−24200.0	−0.961226	−0.480613	+	0.876933i	$0.659586\pi$
$860$	0	0
$861$	22512.0	0.891065
$862$	0	0
$863$	−37008.0	−1.45975	−0.729877	−	0.683579i	$-0.760422\pi$
$863$	−37008.0	−1.45975	−0.729877	+	0.683579i	$0.760422\pi$
$864$	0	0
$865$	−15210.0	−0.597868
$866$	0	0
$867$	31384.0	1.22936
$868$	0	0
$869$	12000.0	0.468437
$870$	0	0
$871$	12792.0	0.497635
$872$	0	0
$873$	37962.0	1.47173
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−3446.00	−0.132683	−0.0663416	−	0.997797i	$-0.521133\pi$
$877$	−3446.00	−0.132683	−0.0663416	+	0.997797i	$0.521133\pi$
$878$	0	0
$879$	−9616.00	−0.368987
$880$	0	0
$881$	−16158.0	−0.617908	−0.308954	−	0.951077i	$-0.599979\pi$
$881$	−16158.0	−0.617908	−0.308954	+	0.951077i	$0.599979\pi$
$882$	0	0
$883$	44708.0	1.70390	0.851950	−	0.523623i	$-0.175420\pi$
$883$	44708.0	1.70390	0.851950	+	0.523623i	$0.175420\pi$
$884$	0	0
$885$	22400.0	0.850811
$886$	0	0
$887$	−23504.0	−0.889726	−0.444863	−	0.895599i	$-0.646748\pi$
$887$	−23504.0	−0.889726	−0.444863	+	0.895599i	$0.646748\pi$
$888$	0	0
$889$	5432.00	0.204931
$890$	0	0
$891$	4308.00	0.161979
$892$	0	0
$893$	−21440.0	−0.803429
$894$	0	0
$895$	900.000	0.0336131
$896$	0	0
$897$	19968.0	0.743269
$898$	0	0
$899$	−12400.0	−0.460026
$900$	0	0
$901$	2068.00	0.0764651
$902$	0	0
$903$	3808.00	0.140335
$904$	0	0
$905$	9790.00	0.359592
$906$	0	0
$907$	−42436.0	−1.55354	−0.776772	−	0.629782i	$-0.783144\pi$
$907$	−42436.0	−1.55354	−0.776772	+	0.629782i	$0.783144\pi$
$908$	0	0
$909$	−17834.0	−0.650733
$910$	0	0
$911$	−7968.00	−0.289782	−0.144891	−	0.989448i	$-0.546283\pi$
$911$	−7968.00	−0.289782	−0.144891	+	0.989448i	$0.546283\pi$
$912$	0	0
$913$	−5376.00	−0.194874
$914$	0	0
$915$	11120.0	0.401766
$916$	0	0
$917$	−7784.00	−0.280317
$918$	0	0
$919$	14880.0	0.534109	0.267054	−	0.963681i	$-0.413950\pi$
$919$	14880.0	0.534109	0.267054	+	0.963681i	$0.413950\pi$
$920$	0	0
$921$	51072.0	1.82723
$922$	0	0
$923$	52416.0	1.86922
$924$	0	0
$925$	10850.0	0.385671
$926$	0	0
$927$	10064.0	0.356575
$928$	0	0
$929$	27610.0	0.975086	0.487543	−	0.873099i	$-0.337893\pi$
$929$	27610.0	0.975086	0.487543	+	0.873099i	$0.337893\pi$
$930$	0	0
$931$	−1960.00	−0.0689972
$932$	0	0
$933$	−39744.0	−1.39460
$934$	0	0
$935$	5640.00	0.197270
$936$	0	0
$937$	−28094.0	−0.979499	−0.489750	−	0.871863i	$-0.662912\pi$
$937$	−28094.0	−0.979499	−0.489750	+	0.871863i	$0.662912\pi$
$938$	0	0
$939$	−22064.0	−0.766807
$940$	0	0
$941$	12198.0	0.422575	0.211288	−	0.977424i	$-0.432234\pi$
$941$	12198.0	0.422575	0.211288	+	0.977424i	$0.432234\pi$
$942$	0	0
$943$	12864.0	0.444231
$944$	0	0
$945$	2800.00	0.0963852
$946$	0	0
$947$	−31316.0	−1.07459	−0.537293	−	0.843396i	$-0.680553\pi$
$947$	−31316.0	−1.07459	−0.537293	+	0.843396i	$0.680553\pi$
$948$	0	0
$949$	6396.00	0.218781
$950$	0	0
$951$	50192.0	1.71145
$952$	0	0
$953$	27322.0	0.928695	0.464348	−	0.885653i	$-0.346289\pi$
$953$	27322.0	0.928695	0.464348	+	0.885653i	$0.346289\pi$
$954$	0	0
$955$	−14440.0	−0.489285
$956$	0	0
$957$	−4800.00	−0.162134
$958$	0	0
$959$	−4858.00	−0.163580
$960$	0	0
$961$	31713.0	1.06452
$962$	0	0
$963$	16428.0	0.549725
$964$	0	0
$965$	8010.00	0.267203
$966$	0	0
$967$	5296.00	0.176120	0.0880599	−	0.996115i	$-0.471933\pi$
$967$	5296.00	0.176120	0.0880599	+	0.996115i	$0.471933\pi$
$968$	0	0
$969$	30080.0	0.997223
$970$	0	0
$971$	−512.000	−0.0169216	−0.00846079	−	0.999964i	$-0.502693\pi$
$971$	−512.000	−0.0169216	−0.00846079	+	0.999964i	$0.502693\pi$
$972$	0	0
$973$	−2520.00	−0.0830293
$974$	0	0
$975$	15600.0	0.512410
$976$	0	0
$977$	−20734.0	−0.678955	−0.339478	−	0.940614i	$-0.610250\pi$
$977$	−20734.0	−0.678955	−0.339478	+	0.940614i	$0.610250\pi$
$978$	0	0
$979$	10440.0	0.340821
$980$	0	0
$981$	43290.0	1.40891
$982$	0	0
$983$	−61168.0	−1.98470	−0.992348	−	0.123472i	$-0.960597\pi$
$983$	−61168.0	−1.98470	−0.992348	+	0.123472i	$0.960597\pi$
$984$	0	0
$985$	23970.0	0.775378
$986$	0	0
$987$	30016.0	0.968004
$988$	0	0
$989$	2176.00	0.0699624
$990$	0	0
$991$	−47928.0	−1.53631	−0.768155	−	0.640264i	$-0.778825\pi$
$991$	−47928.0	−1.53631	−0.768155	+	0.640264i	$0.778825\pi$
$992$	0	0
$993$	−15456.0	−0.493939
$994$	0	0
$995$	6400.00	0.203913
$996$	0	0
$997$	9454.00	0.300312	0.150156	−	0.988662i	$-0.452022\pi$
$997$	9454.00	0.300312	0.150156	+	0.988662i	$0.452022\pi$
$998$	0	0
$999$	34720.0	1.09959

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.bk.1.1		1
4.3	odd	2		2240.4.a.b.1.1		1
8.3	odd	2		560.4.a.p.1.1		1
8.5	even	2		35.4.a.a.1.1	✓	1
24.5	odd	2		315.4.a.c.1.1		1
40.13	odd	4		175.4.b.a.99.1		2
40.29	even	2		175.4.a.a.1.1		1
40.37	odd	4		175.4.b.a.99.2		2
56.5	odd	6		245.4.e.b.116.1		2
56.13	odd	2		245.4.a.d.1.1		1
56.37	even	6		245.4.e.e.116.1		2
56.45	odd	6		245.4.e.b.226.1		2
56.53	even	6		245.4.e.e.226.1		2
120.29	odd	2		1575.4.a.g.1.1		1
168.125	even	2		2205.4.a.i.1.1		1
280.69	odd	2		1225.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.a.1.1	✓	1	8.5	even	2
175.4.a.a.1.1		1	40.29	even	2
175.4.b.a.99.1		2	40.13	odd	4
175.4.b.a.99.2		2	40.37	odd	4
245.4.a.d.1.1		1	56.13	odd	2
245.4.e.b.116.1		2	56.5	odd	6
245.4.e.b.226.1		2	56.45	odd	6
245.4.e.e.116.1		2	56.37	even	6
245.4.e.e.226.1		2	56.53	even	6
315.4.a.c.1.1		1	24.5	odd	2
560.4.a.p.1.1		1	8.3	odd	2
1225.4.a.e.1.1		1	280.69	odd	2
1575.4.a.g.1.1		1	120.29	odd	2
2205.4.a.i.1.1		1	168.125	even	2
2240.4.a.b.1.1		1	4.3	odd	2
2240.4.a.bk.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

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