LMFDB - Embedded newform 2205.4.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2205,4,Mod(1,2205)] 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("2205.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2205, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,0,-7,-5,0,0,15,0,5,-12,0,78] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$130.099211563$
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	$\chi$	$=$	2205.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -7.00000 q^{4} -5.00000 q^{5} +15.0000 q^{8} +5.00000 q^{10} -12.0000 q^{11} +78.0000 q^{13} +41.0000 q^{16} -94.0000 q^{17} -40.0000 q^{19} +35.0000 q^{20} +12.0000 q^{22} -32.0000 q^{23} +25.0000 q^{25} -78.0000 q^{26} +50.0000 q^{29} +248.000 q^{31} -161.000 q^{32} +94.0000 q^{34} -434.000 q^{37} +40.0000 q^{38} -75.0000 q^{40} +402.000 q^{41} -68.0000 q^{43} +84.0000 q^{44} +32.0000 q^{46} +536.000 q^{47} -25.0000 q^{50} -546.000 q^{52} -22.0000 q^{53} +60.0000 q^{55} -50.0000 q^{58} -560.000 q^{59} +278.000 q^{61} -248.000 q^{62} -167.000 q^{64} -390.000 q^{65} -164.000 q^{67} +658.000 q^{68} -672.000 q^{71} -82.0000 q^{73} +434.000 q^{74} +280.000 q^{76} -1000.00 q^{79} -205.000 q^{80} -402.000 q^{82} -448.000 q^{83} +470.000 q^{85} +68.0000 q^{86} -180.000 q^{88} -870.000 q^{89} +224.000 q^{92} -536.000 q^{94} +200.000 q^{95} -1026.00 q^{97} +O(q^{100})$

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 $

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	15.0000	0.662913
$9$	0	0
$10$	5.00000	0.158114
$11$	−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
$15$	0	0
$16$	41.0000	0.640625
$17$	−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	$0.577636\pi$
$20$	35.0000	0.391312
$21$	0	0
$22$	12.0000	0.116291
$23$	−32.0000	−0.290107	−0.145054	−	0.989424i	$-0.546335\pi$
$23$	−32.0000	−0.290107	−0.145054	+	0.989424i	$0.546335\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−78.0000	−0.588348
$27$	0	0
$28$	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	$-0.244865\pi$
$31$	248.000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−161.000	−0.889408
$33$	0	0
$34$	94.0000	0.474143
$35$	0	0
$36$	0	0
$37$	−434.000	−1.92836	−0.964178	−	0.265257i	$-0.914543\pi$
$37$	−434.000	−1.92836	−0.964178	+	0.265257i	$0.914543\pi$
$38$	40.0000	0.170759
$39$	0	0
$40$	−75.0000	−0.296464
$41$	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	$-0.538476\pi$
$43$	−68.0000	−0.241161	−0.120580	+	0.992704i	$0.538476\pi$
$44$	84.0000	0.287806
$45$	0	0
$46$	32.0000	0.102568
$47$	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
$49$	0	0
$50$	−25.0000	−0.0707107
$51$	0	0
$52$	−546.000	−1.45609
$53$	−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
$57$	0	0
$58$	−50.0000	−0.113195
$59$	−560.000	−1.23569	−0.617846	−	0.786299i	$-0.711994\pi$
$59$	−560.000	−1.23569	−0.617846	+	0.786299i	$0.711994\pi$
$60$	0	0
$61$	278.000	0.583512	0.291756	−	0.956493i	$-0.405760\pi$
$61$	278.000	0.583512	0.291756	+	0.956493i	$0.405760\pi$
$62$	−248.000	−0.508001
$63$	0	0
$64$	−167.000	−0.326172
$65$	−390.000	−0.744208
$66$	0	0
$67$	−164.000	−0.299042	−0.149521	−	0.988759i	$-0.547773\pi$
$67$	−164.000	−0.299042	−0.149521	+	0.988759i	$0.547773\pi$
$68$	658.000	1.17344
$69$	0	0
$70$	0	0
$71$	−672.000	−1.12326	−0.561632	−	0.827387i	$-0.689826\pi$
$71$	−672.000	−1.12326	−0.561632	+	0.827387i	$0.689826\pi$
$72$	0	0
$73$	−82.0000	−0.131471	−0.0657354	−	0.997837i	$-0.520939\pi$
$73$	−82.0000	−0.131471	−0.0657354	+	0.997837i	$0.520939\pi$
$74$	434.000	0.681777
$75$	0	0
$76$	280.000	0.422608
$77$	0	0
$78$	0	0
$79$	−1000.00	−1.42416	−0.712081	−	0.702097i	$-0.752247\pi$
$79$	−1000.00	−1.42416	−0.712081	+	0.702097i	$0.752247\pi$
$80$	−205.000	−0.286496
$81$	0	0
$82$	−402.000	−0.541384
$83$	−448.000	−0.592463	−0.296231	−	0.955116i	$-0.595730\pi$
$83$	−448.000	−0.592463	−0.296231	+	0.955116i	$0.595730\pi$
$84$	0	0
$85$	470.000	0.599749
$86$	68.0000	0.0852631
$87$	0	0
$88$	−180.000	−0.218046
$89$	−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
$91$	0	0
$92$	224.000	0.253844
$93$	0	0
$94$	−536.000	−0.588130
$95$	200.000	0.215995
$96$	0	0
$97$	−1026.00	−1.07396	−0.536982	−	0.843594i	$-0.680436\pi$
$97$	−1026.00	−1.07396	−0.536982	+	0.843594i	$0.680436\pi$
$98$	0	0
$99$	0	0
$100$	−175.000	−0.175000
$101$	482.000	0.474859	0.237430	−	0.971405i	$-0.423695\pi$
$101$	482.000	0.474859	0.237430	+	0.971405i	$0.423695\pi$
$102$	0	0
$103$	−272.000	−0.260203	−0.130102	−	0.991501i	$-0.541530\pi$
$103$	−272.000	−0.260203	−0.130102	+	0.991501i	$0.541530\pi$
$104$	1170.00	1.10315
$105$	0	0
$106$	22.0000	0.0201588
$107$	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	$-0.671860\pi$
$109$	−1170.00	−1.02813	−0.514063	+	0.857753i	$0.671860\pi$
$110$	−60.0000	−0.0520071
$111$	0	0
$112$	0	0
$113$	798.000	0.664332	0.332166	−	0.943221i	$-0.392221\pi$
$113$	798.000	0.664332	0.332166	+	0.943221i	$0.392221\pi$
$114$	0	0
$115$	160.000	0.129740
$116$	−350.000	−0.280144
$117$	0	0
$118$	560.000	0.436883
$119$	0	0
$120$	0	0
$121$	−1187.00	−0.891811
$122$	−278.000	−0.206303
$123$	0	0
$124$	−1736.00	−1.25724
$125$	−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	$0.412613\pi$
$128$	1455.00	1.00473
$129$	0	0
$130$	390.000	0.263117
$131$	1112.00	0.741648	0.370824	−	0.928703i	$-0.379075\pi$
$131$	1112.00	0.741648	0.370824	+	0.928703i	$0.379075\pi$
$132$	0	0
$133$	0	0
$134$	164.000	0.105727
$135$	0	0
$136$	−1410.00	−0.889018
$137$	694.000	0.432791	0.216396	−	0.976306i	$-0.430570\pi$
$137$	694.000	0.432791	0.216396	+	0.976306i	$0.430570\pi$
$138$	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
$141$	0	0
$142$	672.000	0.397134
$143$	−936.000	−0.547358
$144$	0	0
$145$	−250.000	−0.143182
$146$	82.0000	0.0464820
$147$	0	0
$148$	3038.00	1.68731
$149$	−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	$0.445525\pi$
$152$	−600.000	−0.320174
$153$	0	0
$154$	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	$0.440266\pi$
$158$	1000.00	0.503517
$159$	0	0
$160$	805.000	0.397755
$161$	0	0
$162$	0	0
$163$	2532.00	1.21670	0.608348	−	0.793670i	$-0.291832\pi$
$163$	2532.00	1.21670	0.608348	+	0.793670i	$0.291832\pi$
$164$	−2814.00	−1.33986
$165$	0	0
$166$	448.000	0.209467
$167$	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
$170$	−470.000	−0.212043
$171$	0	0
$172$	476.000	0.211015
$173$	3042.00	1.33687	0.668436	−	0.743769i	$-0.266964\pi$
$173$	3042.00	1.33687	0.668436	+	0.743769i	$0.266964\pi$
$174$	0	0
$175$	0	0
$176$	−492.000	−0.210715
$177$	0	0
$178$	870.000	0.366344
$179$	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
$183$	0	0
$184$	−480.000	−0.192316
$185$	2170.00	0.862387
$186$	0	0
$187$	1128.00	0.441110
$188$	−3752.00	−1.45555
$189$	0	0
$190$	−200.000	−0.0763659
$191$	2888.00	1.09408	0.547038	−	0.837108i	$-0.315755\pi$
$191$	2888.00	1.09408	0.547038	+	0.837108i	$0.315755\pi$
$192$	0	0
$193$	1602.00	0.597484	0.298742	−	0.954334i	$-0.403433\pi$
$193$	1602.00	0.597484	0.298742	+	0.954334i	$0.403433\pi$
$194$	1026.00	0.379704
$195$	0	0
$196$	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	$-0.573213\pi$
$199$	−1280.00	−0.455964	−0.227982	+	0.973665i	$0.573213\pi$
$200$	375.000	0.132583
$201$	0	0
$202$	−482.000	−0.167888
$203$	0	0
$204$	0	0
$205$	−2010.00	−0.684802
$206$	272.000	0.0919958
$207$	0	0
$208$	3198.00	1.06606
$209$	480.000	0.158863
$210$	0	0
$211$	−68.0000	−0.0221863	−0.0110932	−	0.999938i	$-0.503531\pi$
$211$	−68.0000	−0.0221863	−0.0110932	+	0.999938i	$0.503531\pi$
$212$	154.000	0.0498904
$213$	0	0
$214$	−444.000	−0.141828
$215$	340.000	0.107850
$216$	0	0
$217$	0	0
$218$	1170.00	0.363497
$219$	0	0
$220$	−420.000	−0.128711
$221$	−7332.00	−2.23169
$222$	0	0
$223$	1728.00	0.518903	0.259452	−	0.965756i	$-0.416458\pi$
$223$	1728.00	0.518903	0.259452	+	0.965756i	$0.416458\pi$
$224$	0	0
$225$	0	0
$226$	−798.000	−0.234877
$227$	−4864.00	−1.42218	−0.711090	−	0.703101i	$-0.751798\pi$
$227$	−4864.00	−1.42218	−0.711090	+	0.703101i	$0.751798\pi$
$228$	0	0
$229$	5510.00	1.59000	0.795002	−	0.606606i	$-0.207470\pi$
$229$	5510.00	1.59000	0.795002	+	0.606606i	$0.207470\pi$
$230$	−160.000	−0.0458699
$231$	0	0
$232$	750.000	0.212241
$233$	−5322.00	−1.49638	−0.748188	−	0.663486i	$-0.769076\pi$
$233$	−5322.00	−1.49638	−0.748188	+	0.663486i	$0.769076\pi$
$234$	0	0
$235$	−2680.00	−0.743932
$236$	3920.00	1.08123
$237$	0	0
$238$	0	0
$239$	1840.00	0.497990	0.248995	−	0.968505i	$-0.419900\pi$
$239$	1840.00	0.497990	0.248995	+	0.968505i	$0.419900\pi$
$240$	0	0
$241$	438.000	0.117071	0.0585354	−	0.998285i	$-0.481357\pi$
$241$	438.000	0.117071	0.0585354	+	0.998285i	$0.481357\pi$
$242$	1187.00	0.315303
$243$	0	0
$244$	−1946.00	−0.510573
$245$	0	0
$246$	0	0
$247$	−3120.00	−0.803728
$248$	3720.00	0.952501
$249$	0	0
$250$	125.000	0.0316228
$251$	5592.00	1.40623	0.703115	−	0.711076i	$-0.251792\pi$
$251$	5592.00	1.40623	0.703115	+	0.711076i	$0.251792\pi$
$252$	0	0
$253$	384.000	0.0954224
$254$	−776.000	−0.191695
$255$	0	0
$256$	−119.000	−0.0290527
$257$	−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
$259$	0	0
$260$	2730.00	0.651182
$261$	0	0
$262$	−1112.00	−0.262212
$263$	728.000	0.170686	0.0853430	−	0.996352i	$-0.472801\pi$
$263$	728.000	0.170686	0.0853430	+	0.996352i	$0.472801\pi$
$264$	0	0
$265$	110.000	0.0254990
$266$	0	0
$267$	0	0
$268$	1148.00	0.261661
$269$	5810.00	1.31688	0.658442	−	0.752631i	$-0.271216\pi$
$269$	5810.00	1.31688	0.658442	+	0.752631i	$0.271216\pi$
$270$	0	0
$271$	6528.00	1.46328	0.731638	−	0.681693i	$-0.238756\pi$
$271$	6528.00	1.46328	0.731638	+	0.681693i	$0.238756\pi$
$272$	−3854.00	−0.859129
$273$	0	0
$274$	−694.000	−0.153015
$275$	−300.000	−0.0657843
$276$	0	0
$277$	5126.00	1.11188	0.555941	−	0.831222i	$-0.312358\pi$
$277$	5126.00	1.11188	0.555941	+	0.831222i	$0.312358\pi$
$278$	360.000	0.0776668
$279$	0	0
$280$	0	0
$281$	2358.00	0.500592	0.250296	−	0.968169i	$-0.419472\pi$
$281$	2358.00	0.500592	0.250296	+	0.968169i	$0.419472\pi$
$282$	0	0
$283$	−392.000	−0.0823392	−0.0411696	−	0.999152i	$-0.513108\pi$
$283$	−392.000	−0.0823392	−0.0411696	+	0.999152i	$0.513108\pi$
$284$	4704.00	0.982856
$285$	0	0
$286$	936.000	0.193520
$287$	0	0
$288$	0	0
$289$	3923.00	0.798494
$290$	250.000	0.0506224
$291$	0	0
$292$	574.000	0.115037
$293$	1202.00	0.239664	0.119832	−	0.992794i	$-0.461764\pi$
$293$	1202.00	0.239664	0.119832	+	0.992794i	$0.461764\pi$
$294$	0	0
$295$	2800.00	0.552618
$296$	−6510.00	−1.27833
$297$	0	0
$298$	2270.00	0.441267
$299$	−2496.00	−0.482767
$300$	0	0
$301$	0	0
$302$	−632.000	−0.120422
$303$	0	0
$304$	−1640.00	−0.309409
$305$	−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
$309$	0	0
$310$	1240.00	0.227185
$311$	−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	$-0.419889\pi$
$313$	2758.00	0.498056	0.249028	+	0.968496i	$0.419889\pi$
$314$	−734.000	−0.131917
$315$	0	0
$316$	7000.00	1.24614
$317$	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
$320$	835.000	0.145868
$321$	0	0
$322$	0	0
$323$	3760.00	0.647715
$324$	0	0
$325$	1950.00	0.332820
$326$	−2532.00	−0.430167
$327$	0	0
$328$	6030.00	1.01509
$329$	0	0
$330$	0	0
$331$	1932.00	0.320823	0.160411	−	0.987050i	$-0.448718\pi$
$331$	1932.00	0.320823	0.160411	+	0.987050i	$0.448718\pi$
$332$	3136.00	0.518405
$333$	0	0
$334$	−416.000	−0.0681512
$335$	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	$0.438230\pi$
$338$	−3887.00	−0.625518
$339$	0	0
$340$	−3290.00	−0.524780
$341$	−2976.00	−0.472608
$342$	0	0
$343$	0	0
$344$	−1020.00	−0.159868
$345$	0	0
$346$	−3042.00	−0.472656
$347$	−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
$351$	0	0
$352$	1932.00	0.292545
$353$	−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
$356$	6090.00	0.906655
$357$	0	0
$358$	−180.000	−0.0265735
$359$	13320.0	1.95822	0.979112	−	0.203320i	$-0.0651731\pi$
$359$	13320.0	1.95822	0.979112	+	0.203320i	$0.0651731\pi$
$360$	0	0
$361$	−5259.00	−0.766730
$362$	−1958.00	−0.284282
$363$	0	0
$364$	0	0
$365$	410.000	0.0587956
$366$	0	0
$367$	−10816.0	−1.53839	−0.769197	−	0.639012i	$-0.779344\pi$
$367$	−10816.0	−1.53839	−0.769197	+	0.639012i	$0.779344\pi$
$368$	−1312.00	−0.185850
$369$	0	0
$370$	−2170.00	−0.304900
$371$	0	0
$372$	0	0
$373$	−11098.0	−1.54057	−0.770285	−	0.637700i	$-0.779886\pi$
$373$	−11098.0	−1.54057	−0.770285	+	0.637700i	$0.779886\pi$
$374$	−1128.00	−0.155956
$375$	0	0
$376$	8040.00	1.10274
$377$	3900.00	0.532786
$378$	0	0
$379$	7100.00	0.962276	0.481138	−	0.876645i	$-0.340224\pi$
$379$	7100.00	0.962276	0.481138	+	0.876645i	$0.340224\pi$
$380$	−1400.00	−0.188996
$381$	0	0
$382$	−2888.00	−0.386814
$383$	−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
$385$	0	0
$386$	−1602.00	−0.211243
$387$	0	0
$388$	7182.00	0.939719
$389$	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
$393$	0	0
$394$	−4794.00	−0.612990
$395$	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	$0.488448\pi$
$398$	1280.00	0.161208
$399$	0	0
$400$	1025.00	0.128125
$401$	−6162.00	−0.767371	−0.383685	−	0.923464i	$-0.625345\pi$
$401$	−6162.00	−0.767371	−0.383685	+	0.923464i	$0.625345\pi$
$402$	0	0
$403$	19344.0	2.39105
$404$	−3374.00	−0.415502
$405$	0	0
$406$	0	0
$407$	5208.00	0.634278
$408$	0	0
$409$	−8210.00	−0.992563	−0.496282	−	0.868162i	$-0.665302\pi$
$409$	−8210.00	−0.992563	−0.496282	+	0.868162i	$0.665302\pi$
$410$	2010.00	0.242114
$411$	0	0
$412$	1904.00	0.227678
$413$	0	0
$414$	0	0
$415$	2240.00	0.264957
$416$	−12558.0	−1.48006
$417$	0	0
$418$	−480.000	−0.0561664
$419$	4800.00	0.559655	0.279827	−	0.960050i	$-0.409723\pi$
$419$	4800.00	0.559655	0.279827	+	0.960050i	$0.409723\pi$
$420$	0	0
$421$	−9938.00	−1.15047	−0.575236	−	0.817988i	$-0.695090\pi$
$421$	−9938.00	−1.15047	−0.575236	+	0.817988i	$0.695090\pi$
$422$	68.0000	0.00784405
$423$	0	0
$424$	−330.000	−0.0377977
$425$	−2350.00	−0.268216
$426$	0	0
$427$	0	0
$428$	−3108.00	−0.351007
$429$	0	0
$430$	−340.000	−0.0381308
$431$	9248.00	1.03355	0.516776	−	0.856121i	$-0.327132\pi$
$431$	9248.00	1.03355	0.516776	+	0.856121i	$0.327132\pi$
$432$	0	0
$433$	1118.00	0.124082	0.0620412	−	0.998074i	$-0.480239\pi$
$433$	1118.00	0.124082	0.0620412	+	0.998074i	$0.480239\pi$
$434$	0	0
$435$	0	0
$436$	8190.00	0.899610
$437$	1280.00	0.140116
$438$	0	0
$439$	11960.0	1.30027	0.650136	−	0.759818i	$-0.274712\pi$
$439$	11960.0	1.30027	0.650136	+	0.759818i	$0.274712\pi$
$440$	900.000	0.0975132
$441$	0	0
$442$	7332.00	0.789022
$443$	−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
$446$	−1728.00	−0.183460
$447$	0	0
$448$	0	0
$449$	−1890.00	−0.198652	−0.0993259	−	0.995055i	$-0.531669\pi$
$449$	−1890.00	−0.198652	−0.0993259	+	0.995055i	$0.531669\pi$
$450$	0	0
$451$	−4824.00	−0.503666
$452$	−5586.00	−0.581291
$453$	0	0
$454$	4864.00	0.502817
$455$	0	0
$456$	0	0
$457$	−7014.00	−0.717945	−0.358973	−	0.933348i	$-0.616873\pi$
$457$	−7014.00	−0.717945	−0.358973	+	0.933348i	$0.616873\pi$
$458$	−5510.00	−0.562152
$459$	0	0
$460$	−1120.00	−0.113522
$461$	−8318.00	−0.840364	−0.420182	−	0.907440i	$-0.638034\pi$
$461$	−8318.00	−0.840364	−0.420182	+	0.907440i	$0.638034\pi$
$462$	0	0
$463$	6432.00	0.645616	0.322808	−	0.946464i	$-0.395373\pi$
$463$	6432.00	0.645616	0.322808	+	0.946464i	$0.395373\pi$
$464$	2050.00	0.205105
$465$	0	0
$466$	5322.00	0.529049
$467$	−10064.0	−0.997230	−0.498615	−	0.866824i	$-0.666158\pi$
$467$	−10064.0	−0.997230	−0.498615	+	0.866824i	$0.666158\pi$
$468$	0	0
$469$	0	0
$470$	2680.00	0.263020
$471$	0	0
$472$	−8400.00	−0.819155
$473$	816.000	0.0793229
$474$	0	0
$475$	−1000.00	−0.0965961
$476$	0	0
$477$	0	0
$478$	−1840.00	−0.176066
$479$	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
$482$	−438.000	−0.0413908
$483$	0	0
$484$	8309.00	0.780334
$485$	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	$0.286197\pi$
$488$	4170.00	0.386818
$489$	0	0
$490$	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
$494$	3120.00	0.284161
$495$	0	0
$496$	10168.0	0.920477
$497$	0	0
$498$	0	0
$499$	−820.000	−0.0735636	−0.0367818	−	0.999323i	$-0.511711\pi$
$499$	−820.000	−0.0735636	−0.0367818	+	0.999323i	$0.511711\pi$
$500$	875.000	0.0782624
$501$	0	0
$502$	−5592.00	−0.497178
$503$	−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
$506$	−384.000	−0.0337369
$507$	0	0
$508$	−5432.00	−0.474421
$509$	19810.0	1.72507	0.862537	−	0.505994i	$-0.168874\pi$
$509$	19810.0	1.72507	0.862537	+	0.505994i	$0.168874\pi$
$510$	0	0
$511$	0	0
$512$	−11521.0	−0.994455
$513$	0	0
$514$	1974.00	0.169396
$515$	1360.00	0.116367
$516$	0	0
$517$	−6432.00	−0.547155
$518$	0	0
$519$	0	0
$520$	−5850.00	−0.493345
$521$	−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	$0.527606\pi$
$524$	−7784.00	−0.648942
$525$	0	0
$526$	−728.000	−0.0603466
$527$	−23312.0	−1.92692
$528$	0	0
$529$	−11143.0	−0.915838
$530$	−110.000	−0.00901527
$531$	0	0
$532$	0	0
$533$	31356.0	2.54818
$534$	0	0
$535$	−2220.00	−0.179400
$536$	−2460.00	−0.198238
$537$	0	0
$538$	−5810.00	−0.465589
$539$	0	0
$540$	0	0
$541$	−3498.00	−0.277987	−0.138993	−	0.990293i	$-0.544387\pi$
$541$	−3498.00	−0.277987	−0.138993	+	0.990293i	$0.544387\pi$
$542$	−6528.00	−0.517346
$543$	0	0
$544$	15134.0	1.19277
$545$	5850.00	0.459792
$546$	0	0
$547$	5076.00	0.396772	0.198386	−	0.980124i	$-0.436430\pi$
$547$	5076.00	0.396772	0.198386	+	0.980124i	$0.436430\pi$
$548$	−4858.00	−0.378692
$549$	0	0
$550$	300.000	0.0232583
$551$	−2000.00	−0.154633
$552$	0	0
$553$	0	0
$554$	−5126.00	−0.393110
$555$	0	0
$556$	2520.00	0.192215
$557$	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
$561$	0	0
$562$	−2358.00	−0.176986
$563$	16072.0	1.20312	0.601558	−	0.798829i	$-0.294547\pi$
$563$	16072.0	1.20312	0.601558	+	0.798829i	$0.294547\pi$
$564$	0	0
$565$	−3990.00	−0.297098
$566$	392.000	0.0291113
$567$	0	0
$568$	−10080.0	−0.744626
$569$	−2730.00	−0.201138	−0.100569	−	0.994930i	$-0.532066\pi$
$569$	−2730.00	−0.201138	−0.100569	+	0.994930i	$0.532066\pi$
$570$	0	0
$571$	19932.0	1.46082	0.730410	−	0.683009i	$-0.239329\pi$
$571$	19932.0	1.46082	0.730410	+	0.683009i	$0.239329\pi$
$572$	6552.00	0.478939
$573$	0	0
$574$	0	0
$575$	−800.000	−0.0580214
$576$	0	0
$577$	20054.0	1.44690	0.723448	−	0.690379i	$-0.242556\pi$
$577$	20054.0	1.44690	0.723448	+	0.690379i	$0.242556\pi$
$578$	−3923.00	−0.282310
$579$	0	0
$580$	1750.00	0.125284
$581$	0	0
$582$	0	0
$583$	264.000	0.0187543
$584$	−1230.00	−0.0871537
$585$	0	0
$586$	−1202.00	−0.0847341
$587$	−2544.00	−0.178879	−0.0894396	−	0.995992i	$-0.528508\pi$
$587$	−2544.00	−0.178879	−0.0894396	+	0.995992i	$0.528508\pi$
$588$	0	0
$589$	−9920.00	−0.693967
$590$	−2800.00	−0.195380
$591$	0	0
$592$	−17794.0	−1.23535
$593$	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
$595$	0	0
$596$	15890.0	1.09208
$597$	0	0
$598$	2496.00	0.170684
$599$	19600.0	1.33695	0.668476	−	0.743734i	$-0.266947\pi$
$599$	19600.0	1.33695	0.668476	+	0.743734i	$0.266947\pi$
$600$	0	0
$601$	27078.0	1.83783	0.918914	−	0.394458i	$-0.129068\pi$
$601$	27078.0	1.83783	0.918914	+	0.394458i	$0.129068\pi$
$602$	0	0
$603$	0	0
$604$	−4424.00	−0.298030
$605$	5935.00	0.398830
$606$	0	0
$607$	2704.00	0.180811	0.0904053	−	0.995905i	$-0.471184\pi$
$607$	2704.00	0.180811	0.0904053	+	0.995905i	$0.471184\pi$
$608$	6440.00	0.429567
$609$	0	0
$610$	1390.00	0.0922614
$611$	41808.0	2.76820
$612$	0	0
$613$	12702.0	0.836915	0.418458	−	0.908236i	$-0.362571\pi$
$613$	12702.0	0.836915	0.418458	+	0.908236i	$0.362571\pi$
$614$	−6384.00	−0.419605
$615$	0	0
$616$	0	0
$617$	−12666.0	−0.826441	−0.413220	−	0.910631i	$-0.635596\pi$
$617$	−12666.0	−0.826441	−0.413220	+	0.910631i	$0.635596\pi$
$618$	0	0
$619$	−960.000	−0.0623355	−0.0311677	−	0.999514i	$-0.509923\pi$
$619$	−960.000	−0.0623355	−0.0311677	+	0.999514i	$0.509923\pi$
$620$	8680.00	0.562254
$621$	0	0
$622$	4968.00	0.320255
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−2758.00	−0.176089
$627$	0	0
$628$	−5138.00	−0.326479
$629$	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	$0.238192\pi$
$632$	−15000.0	−0.944095
$633$	0	0
$634$	−6274.00	−0.393016
$635$	−3880.00	−0.242477
$636$	0	0
$637$	0	0
$638$	600.000	0.0372323
$639$	0	0
$640$	−7275.00	−0.449328
$641$	−12162.0	−0.749407	−0.374704	−	0.927145i	$-0.622256\pi$
$641$	−12162.0	−0.749407	−0.374704	+	0.927145i	$0.622256\pi$
$642$	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
$645$	0	0
$646$	−3760.00	−0.229002
$647$	−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
$650$	−1950.00	−0.117670
$651$	0	0
$652$	−17724.0	−1.06461
$653$	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
$656$	16482.0	0.980966
$657$	0	0
$658$	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	$0.314355\pi$
$662$	−1932.00	−0.113428
$663$	0	0
$664$	−6720.00	−0.392751
$665$	0	0
$666$	0	0
$667$	−1600.00	−0.0928819
$668$	−2912.00	−0.168666
$669$	0	0
$670$	−820.000	−0.0472826
$671$	−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	$0.399888\pi$
$674$	−2386.00	−0.136358
$675$	0	0
$676$	−27209.0	−1.54808
$677$	346.000	0.0196423	0.00982117	−	0.999952i	$-0.496874\pi$
$677$	346.000	0.0196423	0.00982117	+	0.999952i	$0.496874\pi$
$678$	0	0
$679$	0	0
$680$	7050.00	0.397581
$681$	0	0
$682$	2976.00	0.167092
$683$	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
$687$	0	0
$688$	−2788.00	−0.154493
$689$	−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	$0.521676\pi$
$692$	−21294.0	−1.16976
$693$	0	0
$694$	6076.00	0.332337
$695$	1800.00	0.0982416
$696$	0	0
$697$	−37788.0	−2.05355
$698$	2210.00	0.119842
$699$	0	0
$700$	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
$704$	2004.00	0.107285
$705$	0	0
$706$	2598.00	0.138494
$707$	0	0
$708$	0	0
$709$	790.000	0.0418464	0.0209232	−	0.999781i	$-0.493339\pi$
$709$	790.000	0.0418464	0.0209232	+	0.999781i	$0.493339\pi$
$710$	−3360.00	−0.177604
$711$	0	0
$712$	−13050.0	−0.686895
$713$	−7936.00	−0.416838
$714$	0	0
$715$	4680.00	0.244786
$716$	−1260.00	−0.0657659
$717$	0	0
$718$	−13320.0	−0.692337
$719$	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
$721$	0	0
$722$	5259.00	0.271080
$723$	0	0
$724$	−13706.0	−0.703563
$725$	1250.00	0.0640329
$726$	0	0
$727$	−29056.0	−1.48229	−0.741147	−	0.671343i	$-0.765718\pi$
$727$	−29056.0	−1.48229	−0.741147	+	0.671343i	$0.765718\pi$
$728$	0	0
$729$	0	0
$730$	−410.000	−0.0207874
$731$	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	$0.557102\pi$
$734$	10816.0	0.543904
$735$	0	0
$736$	5152.00	0.258023
$737$	1968.00	0.0983612
$738$	0	0
$739$	11060.0	0.550539	0.275270	−	0.961367i	$-0.411233\pi$
$739$	11060.0	0.550539	0.275270	+	0.961367i	$0.411233\pi$
$740$	−15190.0	−0.754589
$741$	0	0
$742$	0	0
$743$	−33072.0	−1.63297	−0.816483	−	0.577369i	$-0.804079\pi$
$743$	−33072.0	−1.63297	−0.816483	+	0.577369i	$0.804079\pi$
$744$	0	0
$745$	11350.0	0.558164
$746$	11098.0	0.544674
$747$	0	0
$748$	−7896.00	−0.385971
$749$	0	0
$750$	0	0
$751$	29072.0	1.41259	0.706293	−	0.707919i	$-0.250366\pi$
$751$	29072.0	1.41259	0.706293	+	0.707919i	$0.250366\pi$
$752$	21976.0	1.06567
$753$	0	0
$754$	−3900.00	−0.188368
$755$	−3160.00	−0.152323
$756$	0	0
$757$	−13234.0	−0.635400	−0.317700	−	0.948191i	$-0.602911\pi$
$757$	−13234.0	−0.635400	−0.317700	+	0.948191i	$0.602911\pi$
$758$	−7100.00	−0.340216
$759$	0	0
$760$	3000.00	0.143186
$761$	−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
$763$	0	0
$764$	−20216.0	−0.957316
$765$	0	0
$766$	728.000	0.0343390
$767$	−43680.0	−2.05631
$768$	0	0
$769$	−6890.00	−0.323095	−0.161547	−	0.986865i	$-0.551648\pi$
$769$	−6890.00	−0.323095	−0.161547	+	0.986865i	$0.551648\pi$
$770$	0	0
$771$	0	0
$772$	−11214.0	−0.522799
$773$	16722.0	0.778071	0.389035	−	0.921223i	$-0.372808\pi$
$773$	16722.0	0.778071	0.389035	+	0.921223i	$0.372808\pi$
$774$	0	0
$775$	6200.00	0.287368
$776$	−15390.0	−0.711944
$777$	0	0
$778$	−6810.00	−0.313818
$779$	−16080.0	−0.739571
$780$	0	0
$781$	8064.00	0.369466
$782$	−3008.00	−0.137552
$783$	0	0
$784$	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	$0.235378\pi$
$788$	−33558.0	−1.51707
$789$	0	0
$790$	−5000.00	−0.225180
$791$	0	0
$792$	0	0
$793$	21684.0	0.971023
$794$	−574.000	−0.0256555
$795$	0	0
$796$	8960.00	0.398968
$797$	11346.0	0.504261	0.252130	−	0.967693i	$-0.418869\pi$
$797$	11346.0	0.504261	0.252130	+	0.967693i	$0.418869\pi$
$798$	0	0
$799$	−50384.0	−2.23086
$800$	−4025.00	−0.177882
$801$	0	0
$802$	6162.00	0.271306
$803$	984.000	0.0432436
$804$	0	0
$805$	0	0
$806$	−19344.0	−0.845364
$807$	0	0
$808$	7230.00	0.314790
$809$	35190.0	1.52931	0.764657	−	0.644438i	$-0.222909\pi$
$809$	35190.0	1.52931	0.764657	+	0.644438i	$0.222909\pi$
$810$	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
$813$	0	0
$814$	−5208.00	−0.224251
$815$	−12660.0	−0.544123
$816$	0	0
$817$	2720.00	0.116476
$818$	8210.00	0.350924
$819$	0	0
$820$	14070.0	0.599202
$821$	−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	$0.383119\pi$
$824$	−4080.00	−0.172492
$825$	0	0
$826$	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	$0.685326\pi$
$830$	−2240.00	−0.0936765
$831$	0	0
$832$	−13026.0	−0.542783
$833$	0	0
$834$	0	0
$835$	−2080.00	−0.0862052
$836$	−3360.00	−0.139005
$837$	0	0
$838$	−4800.00	−0.197868
$839$	−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
$842$	9938.00	0.406753
$843$	0	0
$844$	476.000	0.0194130
$845$	−19435.0	−0.791224
$846$	0	0
$847$	0	0
$848$	−902.000	−0.0365269
$849$	0	0
$850$	2350.00	0.0948286
$851$	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
$855$	0	0
$856$	6660.00	0.265928
$857$	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	$0.659586\pi$
$860$	−2380.00	−0.0943690
$861$	0	0
$862$	−9248.00	−0.365415
$863$	37008.0	1.45975	0.729877	−	0.683579i	$-0.239578\pi$
$863$	37008.0	1.45975	0.729877	+	0.683579i	$0.239578\pi$
$864$	0	0
$865$	−15210.0	−0.597868
$866$	−1118.00	−0.0438697
$867$	0	0
$868$	0	0
$869$	12000.0	0.468437
$870$	0	0
$871$	−12792.0	−0.497635
$872$	−17550.0	−0.681557
$873$	0	0
$874$	−1280.00	−0.0495385
$875$	0	0
$876$	0	0
$877$	3446.00	0.132683	0.0663416	−	0.997797i	$-0.478867\pi$
$877$	3446.00	0.132683	0.0663416	+	0.997797i	$0.478867\pi$
$878$	−11960.0	−0.459716
$879$	0	0
$880$	2460.00	0.0942348
$881$	−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	$-0.824580\pi$
$883$	−44708.0	−1.70390	−0.851950	+	0.523623i	$0.824580\pi$
$884$	51324.0	1.95273
$885$	0	0
$886$	7332.00	0.278017
$887$	−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
$889$	0	0
$890$	−4350.00	−0.163834
$891$	0	0
$892$	−12096.0	−0.454040
$893$	−21440.0	−0.803429
$894$	0	0
$895$	−900.000	−0.0336131
$896$	0	0
$897$	0	0
$898$	1890.00	0.0702340
$899$	12400.0	0.460026
$900$	0	0
$901$	2068.00	0.0764651
$902$	4824.00	0.178073
$903$	0	0
$904$	11970.0	0.440394
$905$	−9790.00	−0.359592
$906$	0	0
$907$	42436.0	1.55354	0.776772	−	0.629782i	$-0.216856\pi$
$907$	42436.0	1.55354	0.776772	+	0.629782i	$0.216856\pi$
$908$	34048.0	1.24441
$909$	0	0
$910$	0	0
$911$	7968.00	0.289782	0.144891	−	0.989448i	$-0.453717\pi$
$911$	7968.00	0.289782	0.144891	+	0.989448i	$0.453717\pi$
$912$	0	0
$913$	5376.00	0.194874
$914$	7014.00	0.253832
$915$	0	0
$916$	−38570.0	−1.39125
$917$	0	0
$918$	0	0
$919$	14880.0	0.534109	0.267054	−	0.963681i	$-0.413950\pi$
$919$	14880.0	0.534109	0.267054	+	0.963681i	$0.413950\pi$
$920$	2400.00	0.0860061
$921$	0	0
$922$	8318.00	0.297114
$923$	−52416.0	−1.86922
$924$	0	0
$925$	−10850.0	−0.385671
$926$	−6432.00	−0.228260
$927$	0	0
$928$	−8050.00	−0.284757
$929$	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
$931$	0	0
$932$	37254.0	1.30933
$933$	0	0
$934$	10064.0	0.352574
$935$	−5640.00	−0.197270
$936$	0	0
$937$	28094.0	0.979499	0.489750	−	0.871863i	$-0.337088\pi$
$937$	28094.0	0.979499	0.489750	+	0.871863i	$0.337088\pi$
$938$	0	0
$939$	0	0
$940$	18760.0	0.650940
$941$	−12198.0	−0.422575	−0.211288	−	0.977424i	$-0.567766\pi$
$941$	−12198.0	−0.422575	−0.211288	+	0.977424i	$0.567766\pi$
$942$	0	0
$943$	−12864.0	−0.444231
$944$	−22960.0	−0.791615
$945$	0	0
$946$	−816.000	−0.0280449
$947$	−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
$950$	1000.00	0.0341519
$951$	0	0
$952$	0	0
$953$	−27322.0	−0.928695	−0.464348	−	0.885653i	$-0.653711\pi$
$953$	−27322.0	−0.928695	−0.464348	+	0.885653i	$0.653711\pi$
$954$	0	0
$955$	−14440.0	−0.489285
$956$	−12880.0	−0.435742
$957$	0	0
$958$	−1400.00	−0.0472150
$959$	0	0
$960$	0	0
$961$	31713.0	1.06452
$962$	33852.0	1.13454
$963$	0	0
$964$	−3066.00	−0.102437
$965$	−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	$0.471933\pi$
$968$	−17805.0	−0.591193
$969$	0	0
$970$	−5130.00	−0.169809
$971$	512.000	0.0169216	0.00846079	−	0.999964i	$-0.497307\pi$
$971$	512.000	0.0169216	0.00846079	+	0.999964i	$0.497307\pi$
$972$	0	0
$973$	0	0
$974$	−13376.0	−0.440036
$975$	0	0
$976$	11398.0	0.373813
$977$	20734.0	0.678955	0.339478	−	0.940614i	$-0.389750\pi$
$977$	20734.0	0.678955	0.339478	+	0.940614i	$0.389750\pi$
$978$	0	0
$979$	10440.0	0.340821
$980$	0	0
$981$	0	0
$982$	7092.00	0.230463
$983$	−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
$986$	4700.00	0.151804
$987$	0	0
$988$	21840.0	0.703262
$989$	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	$0.778825\pi$
$992$	−39928.0	−1.27794
$993$	0	0
$994$	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	$0.452022\pi$
$998$	820.000	0.0260087
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.i.1.1		1
3.2	odd	2		245.4.a.d.1.1		1
7.6	odd	2		315.4.a.c.1.1		1
15.14	odd	2		1225.4.a.e.1.1		1
21.2	odd	6		245.4.e.b.116.1		2
21.5	even	6		245.4.e.e.116.1		2
21.11	odd	6		245.4.e.b.226.1		2
21.17	even	6		245.4.e.e.226.1		2
21.20	even	2		35.4.a.a.1.1	✓	1
35.34	odd	2		1575.4.a.g.1.1		1
84.83	odd	2		560.4.a.p.1.1		1
105.62	odd	4		175.4.b.a.99.2		2
105.83	odd	4		175.4.b.a.99.1		2
105.104	even	2		175.4.a.a.1.1		1
168.83	odd	2		2240.4.a.b.1.1		1
168.125	even	2		2240.4.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.a.1.1	✓	1	21.20	even	2
175.4.a.a.1.1		1	105.104	even	2
175.4.b.a.99.1		2	105.83	odd	4
175.4.b.a.99.2		2	105.62	odd	4
245.4.a.d.1.1		1	3.2	odd	2
245.4.e.b.116.1		2	21.2	odd	6
245.4.e.b.226.1		2	21.11	odd	6
245.4.e.e.116.1		2	21.5	even	6
245.4.e.e.226.1		2	21.17	even	6
315.4.a.c.1.1		1	7.6	odd	2
560.4.a.p.1.1		1	84.83	odd	2
1225.4.a.e.1.1		1	15.14	odd	2
1575.4.a.g.1.1		1	35.34	odd	2
2205.4.a.i.1.1		1	1.1	even	1	trivial
2240.4.a.b.1.1		1	168.83	odd	2
2240.4.a.bk.1.1		1	168.125	even	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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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