LMFDB - Embedded newform 1575.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{2} +1.00000 q^{4} -7.00000 q^{7} +21.0000 q^{8} +O(q^{10})$

$q-3.00000 q^{2} +1.00000 q^{4} -7.00000 q^{7} +21.0000 q^{8} +6.00000 q^{11} -41.0000 q^{13} +21.0000 q^{14} -71.0000 q^{16} +27.0000 q^{17} -4.00000 q^{19} -18.0000 q^{22} +75.0000 q^{23} +123.000 q^{26} -7.00000 q^{28} +123.000 q^{29} -205.000 q^{31} +45.0000 q^{32} -81.0000 q^{34} +262.000 q^{37} +12.0000 q^{38} -57.0000 q^{41} -407.000 q^{43} +6.00000 q^{44} -225.000 q^{46} -60.0000 q^{47} +49.0000 q^{49} -41.0000 q^{52} +327.000 q^{53} -147.000 q^{56} -369.000 q^{58} -33.0000 q^{59} -427.000 q^{61} +615.000 q^{62} +433.000 q^{64} +628.000 q^{67} +27.0000 q^{68} -300.000 q^{71} -98.0000 q^{73} -786.000 q^{74} -4.00000 q^{76} -42.0000 q^{77} +686.000 q^{79} +171.000 q^{82} +1401.00 q^{83} +1221.00 q^{86} +126.000 q^{88} -714.000 q^{89} +287.000 q^{91} +75.0000 q^{92} +180.000 q^{94} -494.000 q^{97} -147.000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	21.0000	0.928078
$9$	0	0
$10$	0	0
$11$	6.00000	0.164461	0.0822304	−	0.996613i	$-0.473796\pi$
$11$	6.00000	0.164461	0.0822304	+	0.996613i	$0.473796\pi$
$12$	0	0
$13$	−41.0000	−0.874720	−0.437360	−	0.899287i	$-0.644086\pi$
$13$	−41.0000	−0.874720	−0.437360	+	0.899287i	$0.644086\pi$
$14$	21.0000	0.400892
$15$	0	0
$16$	−71.0000	−1.10938
$17$	27.0000	0.385204	0.192602	−	0.981277i	$-0.438307\pi$
$17$	27.0000	0.385204	0.192602	+	0.981277i	$0.438307\pi$
$18$	0	0
$19$	−4.00000	−0.0482980	−0.0241490	−	0.999708i	$-0.507688\pi$
$19$	−4.00000	−0.0482980	−0.0241490	+	0.999708i	$0.507688\pi$
$20$	0	0
$21$	0	0
$22$	−18.0000	−0.174437
$23$	75.0000	0.679938	0.339969	−	0.940437i	$-0.389583\pi$
$23$	75.0000	0.679938	0.339969	+	0.940437i	$0.389583\pi$
$24$	0	0
$25$	0	0
$26$	123.000	0.927780
$27$	0	0
$28$	−7.00000	−0.0472456
$29$	123.000	0.787604	0.393802	−	0.919195i	$-0.371159\pi$
$29$	123.000	0.787604	0.393802	+	0.919195i	$0.371159\pi$
$30$	0	0
$31$	−205.000	−1.18771	−0.593856	−	0.804571i	$-0.702395\pi$
$31$	−205.000	−1.18771	−0.593856	+	0.804571i	$0.702395\pi$
$32$	45.0000	0.248592
$33$	0	0
$34$	−81.0000	−0.408570
$35$	0	0
$36$	0	0
$37$	262.000	1.16412	0.582061	−	0.813145i	$-0.302246\pi$
$37$	262.000	1.16412	0.582061	+	0.813145i	$0.302246\pi$
$38$	12.0000	0.0512278
$39$	0	0
$40$	0	0
$41$	−57.0000	−0.217120	−0.108560	−	0.994090i	$-0.534624\pi$
$41$	−57.0000	−0.217120	−0.108560	+	0.994090i	$0.534624\pi$
$42$	0	0
$43$	−407.000	−1.44342	−0.721708	−	0.692197i	$-0.756643\pi$
$43$	−407.000	−1.44342	−0.721708	+	0.692197i	$0.756643\pi$
$44$	6.00000	0.0205576
$45$	0	0
$46$	−225.000	−0.721183
$47$	−60.0000	−0.186211	−0.0931053	−	0.995656i	$-0.529679\pi$
$47$	−60.0000	−0.186211	−0.0931053	+	0.995656i	$0.529679\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−41.0000	−0.109340
$53$	327.000	0.847489	0.423744	−	0.905782i	$-0.360715\pi$
$53$	327.000	0.847489	0.423744	+	0.905782i	$0.360715\pi$
$54$	0	0
$55$	0	0
$56$	−147.000	−0.350780
$57$	0	0
$58$	−369.000	−0.835381
$59$	−33.0000	−0.0728175	−0.0364088	−	0.999337i	$-0.511592\pi$
$59$	−33.0000	−0.0728175	−0.0364088	+	0.999337i	$0.511592\pi$
$60$	0	0
$61$	−427.000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−427.000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	615.000	1.25976
$63$	0	0
$64$	433.000	0.845703
$65$	0	0
$66$	0	0
$67$	628.000	1.14511	0.572555	−	0.819866i	$-0.305952\pi$
$67$	628.000	1.14511	0.572555	+	0.819866i	$0.305952\pi$
$68$	27.0000	0.0481505
$69$	0	0
$70$	0	0
$71$	−300.000	−0.501457	−0.250729	−	0.968057i	$-0.580670\pi$
$71$	−300.000	−0.501457	−0.250729	+	0.968057i	$0.580670\pi$
$72$	0	0
$73$	−98.0000	−0.157124	−0.0785619	−	0.996909i	$-0.525033\pi$
$73$	−98.0000	−0.157124	−0.0785619	+	0.996909i	$0.525033\pi$
$74$	−786.000	−1.23474
$75$	0	0
$76$	−4.00000	−0.00603726
$77$	−42.0000	−0.0621603
$78$	0	0
$79$	686.000	0.976975	0.488488	−	0.872571i	$-0.337549\pi$
$79$	686.000	0.976975	0.488488	+	0.872571i	$0.337549\pi$
$80$	0	0
$81$	0	0
$82$	171.000	0.230290
$83$	1401.00	1.85277	0.926384	−	0.376580i	$-0.122900\pi$
$83$	1401.00	1.85277	0.926384	+	0.376580i	$0.122900\pi$
$84$	0	0
$85$	0	0
$86$	1221.00	1.53097
$87$	0	0
$88$	126.000	0.152632
$89$	−714.000	−0.850380	−0.425190	−	0.905104i	$-0.639793\pi$
$89$	−714.000	−0.850380	−0.425190	+	0.905104i	$0.639793\pi$
$90$	0	0
$91$	287.000	0.330613
$92$	75.0000	0.0849923
$93$	0	0
$94$	180.000	0.197506
$95$	0	0
$96$	0	0
$97$	−494.000	−0.517094	−0.258547	−	0.965999i	$-0.583244\pi$
$97$	−494.000	−0.517094	−0.258547	+	0.965999i	$0.583244\pi$
$98$	−147.000	−0.151523
$99$	0	0
$100$	0	0
$101$	−624.000	−0.614756	−0.307378	−	0.951588i	$-0.599452\pi$
$101$	−624.000	−0.614756	−0.307378	+	0.951588i	$0.599452\pi$
$102$	0	0
$103$	769.000	0.735649	0.367824	−	0.929895i	$-0.380103\pi$
$103$	769.000	0.735649	0.367824	+	0.929895i	$0.380103\pi$
$104$	−861.000	−0.811808
$105$	0	0
$106$	−981.000	−0.898898
$107$	1662.00	1.50160	0.750802	−	0.660527i	$-0.229667\pi$
$107$	1662.00	1.50160	0.750802	+	0.660527i	$0.229667\pi$
$108$	0	0
$109$	188.000	0.165203	0.0826015	−	0.996583i	$-0.473677\pi$
$109$	188.000	0.165203	0.0826015	+	0.996583i	$0.473677\pi$
$110$	0	0
$111$	0	0
$112$	497.000	0.419304
$113$	−18.0000	−0.0149849	−0.00749247	−	0.999972i	$-0.502385\pi$
$113$	−18.0000	−0.0149849	−0.00749247	+	0.999972i	$0.502385\pi$
$114$	0	0
$115$	0	0
$116$	123.000	0.0984505
$117$	0	0
$118$	99.0000	0.0772347
$119$	−189.000	−0.145593
$120$	0	0
$121$	−1295.00	−0.972953
$122$	1281.00	0.950625
$123$	0	0
$124$	−205.000	−0.148464
$125$	0	0
$126$	0	0
$127$	1450.00	1.01312	0.506562	−	0.862204i	$-0.330916\pi$
$127$	1450.00	1.01312	0.506562	+	0.862204i	$0.330916\pi$
$128$	−1659.00	−1.14560
$129$	0	0
$130$	0	0
$131$	−2664.00	−1.77675	−0.888377	−	0.459115i	$-0.848167\pi$
$131$	−2664.00	−1.77675	−0.888377	+	0.459115i	$0.848167\pi$
$132$	0	0
$133$	28.0000	0.0182549
$134$	−1884.00	−1.21457
$135$	0	0
$136$	567.000	0.357499
$137$	1692.00	1.05516	0.527581	−	0.849504i	$-0.323099\pi$
$137$	1692.00	1.05516	0.527581	+	0.849504i	$0.323099\pi$
$138$	0	0
$139$	1268.00	0.773744	0.386872	−	0.922134i	$-0.373556\pi$
$139$	1268.00	0.773744	0.386872	+	0.922134i	$0.373556\pi$
$140$	0	0
$141$	0	0
$142$	900.000	0.531876
$143$	−246.000	−0.143857
$144$	0	0
$145$	0	0
$146$	294.000	0.166655
$147$	0	0
$148$	262.000	0.145515
$149$	−2169.00	−1.19256	−0.596280	−	0.802777i	$-0.703355\pi$
$149$	−2169.00	−1.19256	−0.596280	+	0.802777i	$0.703355\pi$
$150$	0	0
$151$	518.000	0.279167	0.139584	−	0.990210i	$-0.455424\pi$
$151$	518.000	0.279167	0.139584	+	0.990210i	$0.455424\pi$
$152$	−84.0000	−0.0448243
$153$	0	0
$154$	126.000	0.0659310
$155$	0	0
$156$	0	0
$157$	886.000	0.450385	0.225193	−	0.974314i	$-0.427699\pi$
$157$	886.000	0.450385	0.225193	+	0.974314i	$0.427699\pi$
$158$	−2058.00	−1.03624
$159$	0	0
$160$	0	0
$161$	−525.000	−0.256993
$162$	0	0
$163$	−3893.00	−1.87070	−0.935348	−	0.353730i	$-0.884913\pi$
$163$	−3893.00	−1.87070	−0.935348	+	0.353730i	$0.884913\pi$
$164$	−57.0000	−0.0271400
$165$	0	0
$166$	−4203.00	−1.96516
$167$	2046.00	0.948049	0.474025	−	0.880512i	$-0.342801\pi$
$167$	2046.00	0.948049	0.474025	+	0.880512i	$0.342801\pi$
$168$	0	0
$169$	−516.000	−0.234866
$170$	0	0
$171$	0	0
$172$	−407.000	−0.180427
$173$	−3540.00	−1.55573	−0.777865	−	0.628432i	$-0.783697\pi$
$173$	−3540.00	−1.55573	−0.777865	+	0.628432i	$0.783697\pi$
$174$	0	0
$175$	0	0
$176$	−426.000	−0.182449
$177$	0	0
$178$	2142.00	0.901965
$179$	3246.00	1.35540	0.677702	−	0.735336i	$-0.262976\pi$
$179$	3246.00	1.35540	0.677702	+	0.735336i	$0.262976\pi$
$180$	0	0
$181$	1334.00	0.547820	0.273910	−	0.961755i	$-0.411683\pi$
$181$	1334.00	0.547820	0.273910	+	0.961755i	$0.411683\pi$
$182$	−861.000	−0.350668
$183$	0	0
$184$	1575.00	0.631036
$185$	0	0
$186$	0	0
$187$	162.000	0.0633509
$188$	−60.0000	−0.0232763
$189$	0	0
$190$	0	0
$191$	873.000	0.330723	0.165361	−	0.986233i	$-0.447121\pi$
$191$	873.000	0.330723	0.165361	+	0.986233i	$0.447121\pi$
$192$	0	0
$193$	1006.00	0.375199	0.187600	−	0.982246i	$-0.439929\pi$
$193$	1006.00	0.375199	0.187600	+	0.982246i	$0.439929\pi$
$194$	1482.00	0.548461
$195$	0	0
$196$	49.0000	0.0178571
$197$	591.000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	591.000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	0	0
$199$	−2584.00	−0.920477	−0.460238	−	0.887795i	$-0.652236\pi$
$199$	−2584.00	−0.920477	−0.460238	+	0.887795i	$0.652236\pi$
$200$	0	0
$201$	0	0
$202$	1872.00	0.652047
$203$	−861.000	−0.297686
$204$	0	0
$205$	0	0
$206$	−2307.00	−0.780273
$207$	0	0
$208$	2911.00	0.970392
$209$	−24.0000	−0.00794313
$210$	0	0
$211$	−1441.00	−0.470154	−0.235077	−	0.971977i	$-0.575534\pi$
$211$	−1441.00	−0.470154	−0.235077	+	0.971977i	$0.575534\pi$
$212$	327.000	0.105936
$213$	0	0
$214$	−4986.00	−1.59269
$215$	0	0
$216$	0	0
$217$	1435.00	0.448913
$218$	−564.000	−0.175224
$219$	0	0
$220$	0	0
$221$	−1107.00	−0.336945
$222$	0	0
$223$	−3827.00	−1.14921	−0.574607	−	0.818429i	$-0.694845\pi$
$223$	−3827.00	−1.14921	−0.574607	+	0.818429i	$0.694845\pi$
$224$	−315.000	−0.0939590
$225$	0	0
$226$	54.0000	0.0158939
$227$	−5421.00	−1.58504	−0.792521	−	0.609845i	$-0.791232\pi$
$227$	−5421.00	−1.58504	−0.792521	+	0.609845i	$0.791232\pi$
$228$	0	0
$229$	−5290.00	−1.52652	−0.763260	−	0.646092i	$-0.776402\pi$
$229$	−5290.00	−1.52652	−0.763260	+	0.646092i	$0.776402\pi$
$230$	0	0
$231$	0	0
$232$	2583.00	0.730958
$233$	4908.00	1.37997	0.689987	−	0.723822i	$-0.257616\pi$
$233$	4908.00	1.37997	0.689987	+	0.723822i	$0.257616\pi$
$234$	0	0
$235$	0	0
$236$	−33.0000	−0.00910219
$237$	0	0
$238$	567.000	0.154425
$239$	−1056.00	−0.285803	−0.142902	−	0.989737i	$-0.545643\pi$
$239$	−1056.00	−0.285803	−0.142902	+	0.989737i	$0.545643\pi$
$240$	0	0
$241$	5342.00	1.42784	0.713918	−	0.700229i	$-0.246919\pi$
$241$	5342.00	1.42784	0.713918	+	0.700229i	$0.246919\pi$
$242$	3885.00	1.03197
$243$	0	0
$244$	−427.000	−0.112032
$245$	0	0
$246$	0	0
$247$	164.000	0.0422472
$248$	−4305.00	−1.10229
$249$	0	0
$250$	0	0
$251$	−5805.00	−1.45979	−0.729897	−	0.683557i	$-0.760432\pi$
$251$	−5805.00	−1.45979	−0.729897	+	0.683557i	$0.760432\pi$
$252$	0	0
$253$	450.000	0.111823
$254$	−4350.00	−1.07458
$255$	0	0
$256$	1513.00	0.369385
$257$	543.000	0.131795	0.0658977	−	0.997826i	$-0.479009\pi$
$257$	543.000	0.131795	0.0658977	+	0.997826i	$0.479009\pi$
$258$	0	0
$259$	−1834.00	−0.439997
$260$	0	0
$261$	0	0
$262$	7992.00	1.88453
$263$	−5193.00	−1.21754	−0.608772	−	0.793345i	$-0.708338\pi$
$263$	−5193.00	−1.21754	−0.608772	+	0.793345i	$0.708338\pi$
$264$	0	0
$265$	0	0
$266$	−84.0000	−0.0193623
$267$	0	0
$268$	628.000	0.143139
$269$	−2154.00	−0.488222	−0.244111	−	0.969747i	$-0.578496\pi$
$269$	−2154.00	−0.488222	−0.244111	+	0.969747i	$0.578496\pi$
$270$	0	0
$271$	2396.00	0.537072	0.268536	−	0.963270i	$-0.413460\pi$
$271$	2396.00	0.537072	0.268536	+	0.963270i	$0.413460\pi$
$272$	−1917.00	−0.427335
$273$	0	0
$274$	−5076.00	−1.11917
$275$	0	0
$276$	0	0
$277$	−4286.00	−0.929678	−0.464839	−	0.885395i	$-0.653888\pi$
$277$	−4286.00	−0.929678	−0.464839	+	0.885395i	$0.653888\pi$
$278$	−3804.00	−0.820679
$279$	0	0
$280$	0	0
$281$	−2208.00	−0.468748	−0.234374	−	0.972146i	$-0.575304\pi$
$281$	−2208.00	−0.468748	−0.234374	+	0.972146i	$0.575304\pi$
$282$	0	0
$283$	−3620.00	−0.760377	−0.380188	−	0.924909i	$-0.624141\pi$
$283$	−3620.00	−0.760377	−0.380188	+	0.924909i	$0.624141\pi$
$284$	−300.000	−0.0626821
$285$	0	0
$286$	738.000	0.152583
$287$	399.000	0.0820635
$288$	0	0
$289$	−4184.00	−0.851618
$290$	0	0
$291$	0	0
$292$	−98.0000	−0.0196405
$293$	−1392.00	−0.277548	−0.138774	−	0.990324i	$-0.544316\pi$
$293$	−1392.00	−0.277548	−0.138774	+	0.990324i	$0.544316\pi$
$294$	0	0
$295$	0	0
$296$	5502.00	1.08040
$297$	0	0
$298$	6507.00	1.26490
$299$	−3075.00	−0.594755
$300$	0	0
$301$	2849.00	0.545560
$302$	−1554.00	−0.296101
$303$	0	0
$304$	284.000	0.0535806
$305$	0	0
$306$	0	0
$307$	−9002.00	−1.67352	−0.836761	−	0.547568i	$-0.815554\pi$
$307$	−9002.00	−1.67352	−0.836761	+	0.547568i	$0.815554\pi$
$308$	−42.0000	−0.00777004
$309$	0	0
$310$	0	0
$311$	−6666.00	−1.21542	−0.607708	−	0.794161i	$-0.707911\pi$
$311$	−6666.00	−1.21542	−0.607708	+	0.794161i	$0.707911\pi$
$312$	0	0
$313$	2878.00	0.519726	0.259863	−	0.965646i	$-0.416323\pi$
$313$	2878.00	0.519726	0.259863	+	0.965646i	$0.416323\pi$
$314$	−2658.00	−0.477706
$315$	0	0
$316$	686.000	0.122122
$317$	4611.00	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$317$	4611.00	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$318$	0	0
$319$	738.000	0.129530
$320$	0	0
$321$	0	0
$322$	1575.00	0.272582
$323$	−108.000	−0.0186046
$324$	0	0
$325$	0	0
$326$	11679.0	1.98417
$327$	0	0
$328$	−1197.00	−0.201504
$329$	420.000	0.0703810
$330$	0	0
$331$	−7459.00	−1.23862	−0.619311	−	0.785146i	$-0.712588\pi$
$331$	−7459.00	−1.23862	−0.619311	+	0.785146i	$0.712588\pi$
$332$	1401.00	0.231596
$333$	0	0
$334$	−6138.00	−1.00556
$335$	0	0
$336$	0	0
$337$	−5843.00	−0.944476	−0.472238	−	0.881471i	$-0.656554\pi$
$337$	−5843.00	−0.944476	−0.472238	+	0.881471i	$0.656554\pi$
$338$	1548.00	0.249113
$339$	0	0
$340$	0	0
$341$	−1230.00	−0.195332
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−8547.00	−1.33960
$345$	0	0
$346$	10620.0	1.65010
$347$	2346.00	0.362939	0.181470	−	0.983397i	$-0.441915\pi$
$347$	2346.00	0.362939	0.181470	+	0.983397i	$0.441915\pi$
$348$	0	0
$349$	5807.00	0.890664	0.445332	−	0.895366i	$-0.353086\pi$
$349$	5807.00	0.890664	0.445332	+	0.895366i	$0.353086\pi$
$350$	0	0
$351$	0	0
$352$	270.000	0.0408837
$353$	−5190.00	−0.782538	−0.391269	−	0.920276i	$-0.627964\pi$
$353$	−5190.00	−0.782538	−0.391269	+	0.920276i	$0.627964\pi$
$354$	0	0
$355$	0	0
$356$	−714.000	−0.106298
$357$	0	0
$358$	−9738.00	−1.43762
$359$	−8883.00	−1.30592	−0.652962	−	0.757391i	$-0.726474\pi$
$359$	−8883.00	−1.30592	−0.652962	+	0.757391i	$0.726474\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	−4002.00	−0.581051
$363$	0	0
$364$	287.000	0.0413266
$365$	0	0
$366$	0	0
$367$	−9965.00	−1.41735	−0.708677	−	0.705533i	$-0.750708\pi$
$367$	−9965.00	−1.41735	−0.708677	+	0.705533i	$0.750708\pi$
$368$	−5325.00	−0.754307
$369$	0	0
$370$	0	0
$371$	−2289.00	−0.320321
$372$	0	0
$373$	−11660.0	−1.61858	−0.809292	−	0.587406i	$-0.800149\pi$
$373$	−11660.0	−1.61858	−0.809292	+	0.587406i	$0.800149\pi$
$374$	−486.000	−0.0671937
$375$	0	0
$376$	−1260.00	−0.172818
$377$	−5043.00	−0.688933
$378$	0	0
$379$	3203.00	0.434108	0.217054	−	0.976160i	$-0.430355\pi$
$379$	3203.00	0.434108	0.217054	+	0.976160i	$0.430355\pi$
$380$	0	0
$381$	0	0
$382$	−2619.00	−0.350785
$383$	−8220.00	−1.09666	−0.548332	−	0.836261i	$-0.684737\pi$
$383$	−8220.00	−1.09666	−0.548332	+	0.836261i	$0.684737\pi$
$384$	0	0
$385$	0	0
$386$	−3018.00	−0.397959
$387$	0	0
$388$	−494.000	−0.0646367
$389$	2226.00	0.290135	0.145068	−	0.989422i	$-0.453660\pi$
$389$	2226.00	0.290135	0.145068	+	0.989422i	$0.453660\pi$
$390$	0	0
$391$	2025.00	0.261915
$392$	1029.00	0.132583
$393$	0	0
$394$	−1773.00	−0.226707
$395$	0	0
$396$	0	0
$397$	−10451.0	−1.32121	−0.660605	−	0.750733i	$-0.729700\pi$
$397$	−10451.0	−1.32121	−0.660605	+	0.750733i	$0.729700\pi$
$398$	7752.00	0.976313
$399$	0	0
$400$	0	0
$401$	1320.00	0.164383	0.0821916	−	0.996617i	$-0.473808\pi$
$401$	1320.00	0.164383	0.0821916	+	0.996617i	$0.473808\pi$
$402$	0	0
$403$	8405.00	1.03892
$404$	−624.000	−0.0768445
$405$	0	0
$406$	2583.00	0.315744
$407$	1572.00	0.191452
$408$	0	0
$409$	2402.00	0.290394	0.145197	−	0.989403i	$-0.453618\pi$
$409$	2402.00	0.290394	0.145197	+	0.989403i	$0.453618\pi$
$410$	0	0
$411$	0	0
$412$	769.000	0.0919561
$413$	231.000	0.0275224
$414$	0	0
$415$	0	0
$416$	−1845.00	−0.217448
$417$	0	0
$418$	72.0000	0.00842496
$419$	3333.00	0.388610	0.194305	−	0.980941i	$-0.437755\pi$
$419$	3333.00	0.388610	0.194305	+	0.980941i	$0.437755\pi$
$420$	0	0
$421$	−1462.00	−0.169248	−0.0846241	−	0.996413i	$-0.526969\pi$
$421$	−1462.00	−0.169248	−0.0846241	+	0.996413i	$0.526969\pi$
$422$	4323.00	0.498674
$423$	0	0
$424$	6867.00	0.786535
$425$	0	0
$426$	0	0
$427$	2989.00	0.338754
$428$	1662.00	0.187700
$429$	0	0
$430$	0	0
$431$	10089.0	1.12754	0.563770	−	0.825932i	$-0.309350\pi$
$431$	10089.0	1.12754	0.563770	+	0.825932i	$0.309350\pi$
$432$	0	0
$433$	−3242.00	−0.359817	−0.179908	−	0.983683i	$-0.557580\pi$
$433$	−3242.00	−0.359817	−0.179908	+	0.983683i	$0.557580\pi$
$434$	−4305.00	−0.476144
$435$	0	0
$436$	188.000	0.0206504
$437$	−300.000	−0.0328397
$438$	0	0
$439$	1799.00	0.195584	0.0977922	−	0.995207i	$-0.468822\pi$
$439$	1799.00	0.195584	0.0977922	+	0.995207i	$0.468822\pi$
$440$	0	0
$441$	0	0
$442$	3321.00	0.357384
$443$	8772.00	0.940791	0.470395	−	0.882456i	$-0.344111\pi$
$443$	8772.00	0.940791	0.470395	+	0.882456i	$0.344111\pi$
$444$	0	0
$445$	0	0
$446$	11481.0	1.21893
$447$	0	0
$448$	−3031.00	−0.319646
$449$	−1560.00	−0.163966	−0.0819832	−	0.996634i	$-0.526125\pi$
$449$	−1560.00	−0.163966	−0.0819832	+	0.996634i	$0.526125\pi$
$450$	0	0
$451$	−342.000	−0.0357077
$452$	−18.0000	−0.00187312
$453$	0	0
$454$	16263.0	1.68119
$455$	0	0
$456$	0	0
$457$	−11615.0	−1.18890	−0.594449	−	0.804133i	$-0.702630\pi$
$457$	−11615.0	−1.18890	−0.594449	+	0.804133i	$0.702630\pi$
$458$	15870.0	1.61912
$459$	0	0
$460$	0	0
$461$	−6960.00	−0.703166	−0.351583	−	0.936157i	$-0.614356\pi$
$461$	−6960.00	−0.703166	−0.351583	+	0.936157i	$0.614356\pi$
$462$	0	0
$463$	−13052.0	−1.31010	−0.655052	−	0.755584i	$-0.727353\pi$
$463$	−13052.0	−1.31010	−0.655052	+	0.755584i	$0.727353\pi$
$464$	−8733.00	−0.873749
$465$	0	0
$466$	−14724.0	−1.46368
$467$	14013.0	1.38853	0.694266	−	0.719719i	$-0.255729\pi$
$467$	14013.0	1.38853	0.694266	+	0.719719i	$0.255729\pi$
$468$	0	0
$469$	−4396.00	−0.432811
$470$	0	0
$471$	0	0
$472$	−693.000	−0.0675803
$473$	−2442.00	−0.237385
$474$	0	0
$475$	0	0
$476$	−189.000	−0.0181992
$477$	0	0
$478$	3168.00	0.303140
$479$	−1056.00	−0.100730	−0.0503652	−	0.998731i	$-0.516039\pi$
$479$	−1056.00	−0.100730	−0.0503652	+	0.998731i	$0.516039\pi$
$480$	0	0
$481$	−10742.0	−1.01828
$482$	−16026.0	−1.51445
$483$	0	0
$484$	−1295.00	−0.121619
$485$	0	0
$486$	0	0
$487$	−7886.00	−0.733776	−0.366888	−	0.930265i	$-0.619577\pi$
$487$	−7886.00	−0.733776	−0.366888	+	0.930265i	$0.619577\pi$
$488$	−8967.00	−0.831797
$489$	0	0
$490$	0	0
$491$	−10590.0	−0.973361	−0.486680	−	0.873580i	$-0.661792\pi$
$491$	−10590.0	−0.973361	−0.486680	+	0.873580i	$0.661792\pi$
$492$	0	0
$493$	3321.00	0.303388
$494$	−492.000	−0.0448100
$495$	0	0
$496$	14555.0	1.31762
$497$	2100.00	0.189533
$498$	0	0
$499$	−14899.0	−1.33661	−0.668307	−	0.743885i	$-0.732981\pi$
$499$	−14899.0	−1.33661	−0.668307	+	0.743885i	$0.732981\pi$
$500$	0	0
$501$	0	0
$502$	17415.0	1.54835
$503$	21558.0	1.91098	0.955491	−	0.295021i	$-0.0953267\pi$
$503$	21558.0	1.91098	0.955491	+	0.295021i	$0.0953267\pi$
$504$	0	0
$505$	0	0
$506$	−1350.00	−0.118606
$507$	0	0
$508$	1450.00	0.126640
$509$	15240.0	1.32711	0.663557	−	0.748126i	$-0.269046\pi$
$509$	15240.0	1.32711	0.663557	+	0.748126i	$0.269046\pi$
$510$	0	0
$511$	686.000	0.0593872
$512$	8733.00	0.753804
$513$	0	0
$514$	−1629.00	−0.139790
$515$	0	0
$516$	0	0
$517$	−360.000	−0.0306243
$518$	5502.00	0.466687
$519$	0	0
$520$	0	0
$521$	16797.0	1.41246	0.706228	−	0.707984i	$-0.250395\pi$
$521$	16797.0	1.41246	0.706228	+	0.707984i	$0.250395\pi$
$522$	0	0
$523$	−22520.0	−1.88285	−0.941425	−	0.337222i	$-0.890513\pi$
$523$	−22520.0	−1.88285	−0.941425	+	0.337222i	$0.890513\pi$
$524$	−2664.00	−0.222094
$525$	0	0
$526$	15579.0	1.29140
$527$	−5535.00	−0.457511
$528$	0	0
$529$	−6542.00	−0.537684
$530$	0	0
$531$	0	0
$532$	28.0000	0.00228187
$533$	2337.00	0.189919
$534$	0	0
$535$	0	0
$536$	13188.0	1.06275
$537$	0	0
$538$	6462.00	0.517838
$539$	294.000	0.0234944
$540$	0	0
$541$	−12004.0	−0.953960	−0.476980	−	0.878914i	$-0.658269\pi$
$541$	−12004.0	−0.953960	−0.476980	+	0.878914i	$0.658269\pi$
$542$	−7188.00	−0.569651
$543$	0	0
$544$	1215.00	0.0957586
$545$	0	0
$546$	0	0
$547$	16423.0	1.28372	0.641862	−	0.766820i	$-0.278162\pi$
$547$	16423.0	1.28372	0.641862	+	0.766820i	$0.278162\pi$
$548$	1692.00	0.131895
$549$	0	0
$550$	0	0
$551$	−492.000	−0.0380398
$552$	0	0
$553$	−4802.00	−0.369262
$554$	12858.0	0.986072
$555$	0	0
$556$	1268.00	0.0967179
$557$	714.000	0.0543145	0.0271572	−	0.999631i	$-0.491355\pi$
$557$	714.000	0.0543145	0.0271572	+	0.999631i	$0.491355\pi$
$558$	0	0
$559$	16687.0	1.26258
$560$	0	0
$561$	0	0
$562$	6624.00	0.497183
$563$	16293.0	1.21966	0.609830	−	0.792533i	$-0.291238\pi$
$563$	16293.0	1.21966	0.609830	+	0.792533i	$0.291238\pi$
$564$	0	0
$565$	0	0
$566$	10860.0	0.806501
$567$	0	0
$568$	−6300.00	−0.465391
$569$	17370.0	1.27977	0.639884	−	0.768471i	$-0.278982\pi$
$569$	17370.0	1.27977	0.639884	+	0.768471i	$0.278982\pi$
$570$	0	0
$571$	−12589.0	−0.922650	−0.461325	−	0.887231i	$-0.652626\pi$
$571$	−12589.0	−0.922650	−0.461325	+	0.887231i	$0.652626\pi$
$572$	−246.000	−0.0179821
$573$	0	0
$574$	−1197.00	−0.0870415
$575$	0	0
$576$	0	0
$577$	13318.0	0.960894	0.480447	−	0.877024i	$-0.340475\pi$
$577$	13318.0	0.960894	0.480447	+	0.877024i	$0.340475\pi$
$578$	12552.0	0.903277
$579$	0	0
$580$	0	0
$581$	−9807.00	−0.700280
$582$	0	0
$583$	1962.00	0.139379
$584$	−2058.00	−0.145823
$585$	0	0
$586$	4176.00	0.294384
$587$	−7551.00	−0.530942	−0.265471	−	0.964119i	$-0.585528\pi$
$587$	−7551.00	−0.530942	−0.265471	+	0.964119i	$0.585528\pi$
$588$	0	0
$589$	820.000	0.0573642
$590$	0	0
$591$	0	0
$592$	−18602.0	−1.29145
$593$	−6402.00	−0.443337	−0.221668	−	0.975122i	$-0.571150\pi$
$593$	−6402.00	−0.443337	−0.221668	+	0.975122i	$0.571150\pi$
$594$	0	0
$595$	0	0
$596$	−2169.00	−0.149070
$597$	0	0
$598$	9225.00	0.630833
$599$	15753.0	1.07454	0.537271	−	0.843410i	$-0.319455\pi$
$599$	15753.0	1.07454	0.537271	+	0.843410i	$0.319455\pi$
$600$	0	0
$601$	9764.00	0.662699	0.331349	−	0.943508i	$-0.392496\pi$
$601$	9764.00	0.662699	0.331349	+	0.943508i	$0.392496\pi$
$602$	−8547.00	−0.578654
$603$	0	0
$604$	518.000	0.0348959
$605$	0	0
$606$	0	0
$607$	−7772.00	−0.519696	−0.259848	−	0.965649i	$-0.583673\pi$
$607$	−7772.00	−0.519696	−0.259848	+	0.965649i	$0.583673\pi$
$608$	−180.000	−0.0120065
$609$	0	0
$610$	0	0
$611$	2460.00	0.162882
$612$	0	0
$613$	24262.0	1.59859	0.799293	−	0.600942i	$-0.205208\pi$
$613$	24262.0	1.59859	0.799293	+	0.600942i	$0.205208\pi$
$614$	27006.0	1.77504
$615$	0	0
$616$	−882.000	−0.0576896
$617$	−25818.0	−1.68459	−0.842296	−	0.539015i	$-0.818797\pi$
$617$	−25818.0	−1.68459	−0.842296	+	0.539015i	$0.818797\pi$
$618$	0	0
$619$	22430.0	1.45644	0.728221	−	0.685342i	$-0.240347\pi$
$619$	22430.0	1.45644	0.728221	+	0.685342i	$0.240347\pi$
$620$	0	0
$621$	0	0
$622$	19998.0	1.28914
$623$	4998.00	0.321414
$624$	0	0
$625$	0	0
$626$	−8634.00	−0.551252
$627$	0	0
$628$	886.000	0.0562982
$629$	7074.00	0.448424
$630$	0	0
$631$	−2830.00	−0.178543	−0.0892714	−	0.996007i	$-0.528454\pi$
$631$	−2830.00	−0.178543	−0.0892714	+	0.996007i	$0.528454\pi$
$632$	14406.0	0.906709
$633$	0	0
$634$	−13833.0	−0.866528
$635$	0	0
$636$	0	0
$637$	−2009.00	−0.124960
$638$	−2214.00	−0.137387
$639$	0	0
$640$	0	0
$641$	5202.00	0.320541	0.160270	−	0.987073i	$-0.448763\pi$
$641$	5202.00	0.320541	0.160270	+	0.987073i	$0.448763\pi$
$642$	0	0
$643$	22030.0	1.35113	0.675566	−	0.737299i	$-0.263899\pi$
$643$	22030.0	1.35113	0.675566	+	0.737299i	$0.263899\pi$
$644$	−525.000	−0.0321241
$645$	0	0
$646$	324.000	0.0197331
$647$	−20370.0	−1.23775	−0.618877	−	0.785488i	$-0.712412\pi$
$647$	−20370.0	−1.23775	−0.618877	+	0.785488i	$0.712412\pi$
$648$	0	0
$649$	−198.000	−0.0119756
$650$	0	0
$651$	0	0
$652$	−3893.00	−0.233837
$653$	−31626.0	−1.89528	−0.947642	−	0.319333i	$-0.896541\pi$
$653$	−31626.0	−1.89528	−0.947642	+	0.319333i	$0.896541\pi$
$654$	0	0
$655$	0	0
$656$	4047.00	0.240867
$657$	0	0
$658$	−1260.00	−0.0746503
$659$	11142.0	0.658620	0.329310	−	0.944222i	$-0.393184\pi$
$659$	11142.0	0.658620	0.329310	+	0.944222i	$0.393184\pi$
$660$	0	0
$661$	−5518.00	−0.324698	−0.162349	−	0.986733i	$-0.551907\pi$
$661$	−5518.00	−0.324698	−0.162349	+	0.986733i	$0.551907\pi$
$662$	22377.0	1.31376
$663$	0	0
$664$	29421.0	1.71951
$665$	0	0
$666$	0	0
$667$	9225.00	0.535522
$668$	2046.00	0.118506
$669$	0	0
$670$	0	0
$671$	−2562.00	−0.147399
$672$	0	0
$673$	12517.0	0.716931	0.358466	−	0.933543i	$-0.383300\pi$
$673$	12517.0	0.716931	0.358466	+	0.933543i	$0.383300\pi$
$674$	17529.0	1.00177
$675$	0	0
$676$	−516.000	−0.0293582
$677$	−2604.00	−0.147828	−0.0739142	−	0.997265i	$-0.523549\pi$
$677$	−2604.00	−0.147828	−0.0739142	+	0.997265i	$0.523549\pi$
$678$	0	0
$679$	3458.00	0.195443
$680$	0	0
$681$	0	0
$682$	3690.00	0.207181
$683$	25986.0	1.45582	0.727911	−	0.685671i	$-0.240491\pi$
$683$	25986.0	1.45582	0.727911	+	0.685671i	$0.240491\pi$
$684$	0	0
$685$	0	0
$686$	1029.00	0.0572703
$687$	0	0
$688$	28897.0	1.60129
$689$	−13407.0	−0.741315
$690$	0	0
$691$	−24610.0	−1.35486	−0.677430	−	0.735587i	$-0.736906\pi$
$691$	−24610.0	−1.35486	−0.677430	+	0.735587i	$0.736906\pi$
$692$	−3540.00	−0.194466
$693$	0	0
$694$	−7038.00	−0.384955
$695$	0	0
$696$	0	0
$697$	−1539.00	−0.0836353
$698$	−17421.0	−0.944691
$699$	0	0
$700$	0	0
$701$	−10725.0	−0.577857	−0.288928	−	0.957351i	$-0.593299\pi$
$701$	−10725.0	−0.577857	−0.288928	+	0.957351i	$0.593299\pi$
$702$	0	0
$703$	−1048.00	−0.0562248
$704$	2598.00	0.139085
$705$	0	0
$706$	15570.0	0.830007
$707$	4368.00	0.232356
$708$	0	0
$709$	−34516.0	−1.82832	−0.914158	−	0.405359i	$-0.867147\pi$
$709$	−34516.0	−1.82832	−0.914158	+	0.405359i	$0.867147\pi$
$710$	0	0
$711$	0	0
$712$	−14994.0	−0.789219
$713$	−15375.0	−0.807571
$714$	0	0
$715$	0	0
$716$	3246.00	0.169426
$717$	0	0
$718$	26649.0	1.38514
$719$	−17958.0	−0.931461	−0.465730	−	0.884927i	$-0.654208\pi$
$719$	−17958.0	−0.931461	−0.465730	+	0.884927i	$0.654208\pi$
$720$	0	0
$721$	−5383.00	−0.278049
$722$	20529.0	1.05819
$723$	0	0
$724$	1334.00	0.0684775
$725$	0	0
$726$	0	0
$727$	−2255.00	−0.115039	−0.0575195	−	0.998344i	$-0.518319\pi$
$727$	−2255.00	−0.115039	−0.0575195	+	0.998344i	$0.518319\pi$
$728$	6027.00	0.306834
$729$	0	0
$730$	0	0
$731$	−10989.0	−0.556009
$732$	0	0
$733$	−9539.00	−0.480670	−0.240335	−	0.970690i	$-0.577257\pi$
$733$	−9539.00	−0.480670	−0.240335	+	0.970690i	$0.577257\pi$
$734$	29895.0	1.50333
$735$	0	0
$736$	3375.00	0.169027
$737$	3768.00	0.188326
$738$	0	0
$739$	37979.0	1.89050	0.945250	−	0.326346i	$-0.105817\pi$
$739$	37979.0	1.89050	0.945250	+	0.326346i	$0.105817\pi$
$740$	0	0
$741$	0	0
$742$	6867.00	0.339751
$743$	−20535.0	−1.01394	−0.506969	−	0.861964i	$-0.669234\pi$
$743$	−20535.0	−1.01394	−0.506969	+	0.861964i	$0.669234\pi$
$744$	0	0
$745$	0	0
$746$	34980.0	1.71677
$747$	0	0
$748$	162.000	0.00791886
$749$	−11634.0	−0.567553
$750$	0	0
$751$	−2452.00	−0.119141	−0.0595704	−	0.998224i	$-0.518973\pi$
$751$	−2452.00	−0.119141	−0.0595704	+	0.998224i	$0.518973\pi$
$752$	4260.00	0.206577
$753$	0	0
$754$	15129.0	0.730724
$755$	0	0
$756$	0	0
$757$	36850.0	1.76927	0.884634	−	0.466286i	$-0.154408\pi$
$757$	36850.0	1.76927	0.884634	+	0.466286i	$0.154408\pi$
$758$	−9609.00	−0.460441
$759$	0	0
$760$	0	0
$761$	−30258.0	−1.44133	−0.720665	−	0.693284i	$-0.756163\pi$
$761$	−30258.0	−1.44133	−0.720665	+	0.693284i	$0.756163\pi$
$762$	0	0
$763$	−1316.00	−0.0624409
$764$	873.000	0.0413404
$765$	0	0
$766$	24660.0	1.16319
$767$	1353.00	0.0636949
$768$	0	0
$769$	29288.0	1.37341	0.686705	−	0.726936i	$-0.259056\pi$
$769$	29288.0	1.37341	0.686705	+	0.726936i	$0.259056\pi$
$770$	0	0
$771$	0	0
$772$	1006.00	0.0468999
$773$	40668.0	1.89227	0.946136	−	0.323769i	$-0.104950\pi$
$773$	40668.0	1.89227	0.946136	+	0.323769i	$0.104950\pi$
$774$	0	0
$775$	0	0
$776$	−10374.0	−0.479903
$777$	0	0
$778$	−6678.00	−0.307735
$779$	228.000	0.0104865
$780$	0	0
$781$	−1800.00	−0.0824700
$782$	−6075.00	−0.277803
$783$	0	0
$784$	−3479.00	−0.158482
$785$	0	0
$786$	0	0
$787$	−13118.0	−0.594163	−0.297081	−	0.954852i	$-0.596013\pi$
$787$	−13118.0	−0.594163	−0.297081	+	0.954852i	$0.596013\pi$
$788$	591.000	0.0267176
$789$	0	0
$790$	0	0
$791$	126.000	0.00566377
$792$	0	0
$793$	17507.0	0.783975
$794$	31353.0	1.40136
$795$	0	0
$796$	−2584.00	−0.115060
$797$	−37278.0	−1.65678	−0.828391	−	0.560151i	$-0.810743\pi$
$797$	−37278.0	−1.65678	−0.828391	+	0.560151i	$0.810743\pi$
$798$	0	0
$799$	−1620.00	−0.0717290
$800$	0	0
$801$	0	0
$802$	−3960.00	−0.174355
$803$	−588.000	−0.0258407
$804$	0	0
$805$	0	0
$806$	−25215.0	−1.10194
$807$	0	0
$808$	−13104.0	−0.570541
$809$	−5268.00	−0.228941	−0.114470	−	0.993427i	$-0.536517\pi$
$809$	−5268.00	−0.228941	−0.114470	+	0.993427i	$0.536517\pi$
$810$	0	0
$811$	34994.0	1.51517	0.757587	−	0.652735i	$-0.226378\pi$
$811$	34994.0	1.51517	0.757587	+	0.652735i	$0.226378\pi$
$812$	−861.000	−0.0372108
$813$	0	0
$814$	−4716.00	−0.203066
$815$	0	0
$816$	0	0
$817$	1628.00	0.0697142
$818$	−7206.00	−0.308010
$819$	0	0
$820$	0	0
$821$	−28818.0	−1.22504	−0.612518	−	0.790456i	$-0.709843\pi$
$821$	−28818.0	−1.22504	−0.612518	+	0.790456i	$0.709843\pi$
$822$	0	0
$823$	20962.0	0.887836	0.443918	−	0.896067i	$-0.353588\pi$
$823$	20962.0	0.887836	0.443918	+	0.896067i	$0.353588\pi$
$824$	16149.0	0.682739
$825$	0	0
$826$	−693.000	−0.0291920
$827$	−28266.0	−1.18852	−0.594259	−	0.804273i	$-0.702555\pi$
$827$	−28266.0	−1.18852	−0.594259	+	0.804273i	$0.702555\pi$
$828$	0	0
$829$	42491.0	1.78019	0.890093	−	0.455780i	$-0.150640\pi$
$829$	42491.0	1.78019	0.890093	+	0.455780i	$0.150640\pi$
$830$	0	0
$831$	0	0
$832$	−17753.0	−0.739753
$833$	1323.00	0.0550291
$834$	0	0
$835$	0	0
$836$	−24.0000	−0.000992892 0
$837$	0	0
$838$	−9999.00	−0.412183
$839$	40512.0	1.66702	0.833510	−	0.552505i	$-0.186328\pi$
$839$	40512.0	1.66702	0.833510	+	0.552505i	$0.186328\pi$
$840$	0	0
$841$	−9260.00	−0.379679
$842$	4386.00	0.179515
$843$	0	0
$844$	−1441.00	−0.0587693
$845$	0	0
$846$	0	0
$847$	9065.00	0.367742
$848$	−23217.0	−0.940183
$849$	0	0
$850$	0	0
$851$	19650.0	0.791532
$852$	0	0
$853$	40525.0	1.62667	0.813335	−	0.581796i	$-0.197650\pi$
$853$	40525.0	1.62667	0.813335	+	0.581796i	$0.197650\pi$
$854$	−8967.00	−0.359303
$855$	0	0
$856$	34902.0	1.39360
$857$	−7182.00	−0.286269	−0.143134	−	0.989703i	$-0.545718\pi$
$857$	−7182.00	−0.286269	−0.143134	+	0.989703i	$0.545718\pi$
$858$	0	0
$859$	−7864.00	−0.312359	−0.156179	−	0.987729i	$-0.549918\pi$
$859$	−7864.00	−0.312359	−0.156179	+	0.987729i	$0.549918\pi$
$860$	0	0
$861$	0	0
$862$	−30267.0	−1.19594
$863$	28872.0	1.13883	0.569417	−	0.822049i	$-0.307169\pi$
$863$	28872.0	1.13883	0.569417	+	0.822049i	$0.307169\pi$
$864$	0	0
$865$	0	0
$866$	9726.00	0.381643
$867$	0	0
$868$	1435.00	0.0561141
$869$	4116.00	0.160674
$870$	0	0
$871$	−25748.0	−1.00165
$872$	3948.00	0.153321
$873$	0	0
$874$	900.000	0.0348318
$875$	0	0
$876$	0	0
$877$	47392.0	1.82476	0.912380	−	0.409345i	$-0.134243\pi$
$877$	47392.0	1.82476	0.912380	+	0.409345i	$0.134243\pi$
$878$	−5397.00	−0.207449
$879$	0	0
$880$	0	0
$881$	−17769.0	−0.679515	−0.339758	−	0.940513i	$-0.610345\pi$
$881$	−17769.0	−0.679515	−0.339758	+	0.940513i	$0.610345\pi$
$882$	0	0
$883$	−32549.0	−1.24050	−0.620250	−	0.784404i	$-0.712969\pi$
$883$	−32549.0	−1.24050	−0.620250	+	0.784404i	$0.712969\pi$
$884$	−1107.00	−0.0421181
$885$	0	0
$886$	−26316.0	−0.997859
$887$	6714.00	0.254153	0.127077	−	0.991893i	$-0.459441\pi$
$887$	6714.00	0.254153	0.127077	+	0.991893i	$0.459441\pi$
$888$	0	0
$889$	−10150.0	−0.382925
$890$	0	0
$891$	0	0
$892$	−3827.00	−0.143652
$893$	240.000	0.00899361
$894$	0	0
$895$	0	0
$896$	11613.0	0.432995
$897$	0	0
$898$	4680.00	0.173913
$899$	−25215.0	−0.935448
$900$	0	0
$901$	8829.00	0.326456
$902$	1026.00	0.0378737
$903$	0	0
$904$	−378.000	−0.0139072
$905$	0	0
$906$	0	0
$907$	−29693.0	−1.08703	−0.543517	−	0.839398i	$-0.682908\pi$
$907$	−29693.0	−1.08703	−0.543517	+	0.839398i	$0.682908\pi$
$908$	−5421.00	−0.198130
$909$	0	0
$910$	0	0
$911$	8199.00	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	8199.00	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	8406.00	0.304708
$914$	34845.0	1.26102
$915$	0	0
$916$	−5290.00	−0.190815
$917$	18648.0	0.671550
$918$	0	0
$919$	−15388.0	−0.552343	−0.276172	−	0.961108i	$-0.589066\pi$
$919$	−15388.0	−0.552343	−0.276172	+	0.961108i	$0.589066\pi$
$920$	0	0
$921$	0	0
$922$	20880.0	0.745820
$923$	12300.0	0.438634
$924$	0	0
$925$	0	0
$926$	39156.0	1.38957
$927$	0	0
$928$	5535.00	0.195792
$929$	411.000	0.0145150	0.00725752	−	0.999974i	$-0.497690\pi$
$929$	411.000	0.0145150	0.00725752	+	0.999974i	$0.497690\pi$
$930$	0	0
$931$	−196.000	−0.00689972
$932$	4908.00	0.172497
$933$	0	0
$934$	−42039.0	−1.47276
$935$	0	0
$936$	0	0
$937$	−26066.0	−0.908793	−0.454397	−	0.890800i	$-0.650145\pi$
$937$	−26066.0	−0.908793	−0.454397	+	0.890800i	$0.650145\pi$
$938$	13188.0	0.459066
$939$	0	0
$940$	0	0
$941$	−24048.0	−0.833095	−0.416548	−	0.909114i	$-0.636760\pi$
$941$	−24048.0	−0.833095	−0.416548	+	0.909114i	$0.636760\pi$
$942$	0	0
$943$	−4275.00	−0.147628
$944$	2343.00	0.0807819
$945$	0	0
$946$	7326.00	0.251785
$947$	−22968.0	−0.788131	−0.394065	−	0.919082i	$-0.628932\pi$
$947$	−22968.0	−0.788131	−0.394065	+	0.919082i	$0.628932\pi$
$948$	0	0
$949$	4018.00	0.137439
$950$	0	0
$951$	0	0
$952$	−3969.00	−0.135122
$953$	−10248.0	−0.348337	−0.174169	−	0.984716i	$-0.555724\pi$
$953$	−10248.0	−0.348337	−0.174169	+	0.984716i	$0.555724\pi$
$954$	0	0
$955$	0	0
$956$	−1056.00	−0.0357254
$957$	0	0
$958$	3168.00	0.106841
$959$	−11844.0	−0.398814
$960$	0	0
$961$	12234.0	0.410661
$962$	32226.0	1.08005
$963$	0	0
$964$	5342.00	0.178479
$965$	0	0
$966$	0	0
$967$	43060.0	1.43197	0.715986	−	0.698115i	$-0.245978\pi$
$967$	43060.0	1.43197	0.715986	+	0.698115i	$0.245978\pi$
$968$	−27195.0	−0.902976
$969$	0	0
$970$	0	0
$971$	12348.0	0.408101	0.204051	−	0.978960i	$-0.434589\pi$
$971$	12348.0	0.408101	0.204051	+	0.978960i	$0.434589\pi$
$972$	0	0
$973$	−8876.00	−0.292448
$974$	23658.0	0.778287
$975$	0	0
$976$	30317.0	0.994286
$977$	−44190.0	−1.44705	−0.723523	−	0.690301i	$-0.757478\pi$
$977$	−44190.0	−1.44705	−0.723523	+	0.690301i	$0.757478\pi$
$978$	0	0
$979$	−4284.00	−0.139854
$980$	0	0
$981$	0	0
$982$	31770.0	1.03240
$983$	60168.0	1.95225	0.976125	−	0.217211i	$-0.0696959\pi$
$983$	60168.0	1.95225	0.976125	+	0.217211i	$0.0696959\pi$
$984$	0	0
$985$	0	0
$986$	−9963.00	−0.321792
$987$	0	0
$988$	164.000	0.00528091
$989$	−30525.0	−0.981434
$990$	0	0
$991$	8804.00	0.282208	0.141104	−	0.989995i	$-0.454935\pi$
$991$	8804.00	0.282208	0.141104	+	0.989995i	$0.454935\pi$
$992$	−9225.00	−0.295256
$993$	0	0
$994$	−6300.00	−0.201030
$995$	0	0
$996$	0	0
$997$	−15302.0	−0.486077	−0.243039	−	0.970017i	$-0.578144\pi$
$997$	−15302.0	−0.486077	−0.243039	+	0.970017i	$0.578144\pi$
$998$	44697.0	1.41769
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.a.1.1		1
3.2	odd	2		525.4.a.h.1.1	yes	1
5.4	even	2		1575.4.a.j.1.1		1
15.2	even	4		525.4.d.d.274.2		2
15.8	even	4		525.4.d.d.274.1		2
15.14	odd	2		525.4.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.c.1.1	✓	1	15.14	odd	2
525.4.a.h.1.1	yes	1	3.2	odd	2
525.4.d.d.274.1		2	15.8	even	4
525.4.d.d.274.2		2	15.2	even	4
1575.4.a.a.1.1		1	1.1	even	1	trivial
1575.4.a.j.1.1		1	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

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L-functions

Modular forms

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Representations

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Database

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