LMFDB - Embedded newform 960.4.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(1,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("960.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.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:	$56.6418336055$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +5.00000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +5.00000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +4.00000 q^{11} +6.00000 q^{13} +15.0000 q^{15} -2.00000 q^{17} -16.0000 q^{19} +24.0000 q^{21} +60.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +142.000 q^{29} +176.000 q^{31} +12.0000 q^{33} +40.0000 q^{35} +214.000 q^{37} +18.0000 q^{39} -278.000 q^{41} -68.0000 q^{43} +45.0000 q^{45} -116.000 q^{47} -279.000 q^{49} -6.00000 q^{51} +350.000 q^{53} +20.0000 q^{55} -48.0000 q^{57} +684.000 q^{59} +394.000 q^{61} +72.0000 q^{63} +30.0000 q^{65} +108.000 q^{67} +180.000 q^{69} +96.0000 q^{71} -398.000 q^{73} +75.0000 q^{75} +32.0000 q^{77} -136.000 q^{79} +81.0000 q^{81} +436.000 q^{83} -10.0000 q^{85} +426.000 q^{87} -750.000 q^{89} +48.0000 q^{91} +528.000 q^{93} -80.0000 q^{95} +82.0000 q^{97} +36.0000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	4.00000	0.109640	0.0548202	−	0.998496i	$-0.482541\pi$
$11$	4.00000	0.109640	0.0548202	+	0.998496i	$0.482541\pi$
$12$	0	0
$13$	6.00000	0.128008	0.0640039	−	0.997950i	$-0.479613\pi$
$13$	6.00000	0.128008	0.0640039	+	0.997950i	$0.479613\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−2.00000	−0.0285336	−0.0142668	−	0.999898i	$-0.504541\pi$
$17$	−2.00000	−0.0285336	−0.0142668	+	0.999898i	$0.504541\pi$
$18$	0	0
$19$	−16.0000	−0.193192	−0.0965961	−	0.995324i	$-0.530796\pi$
$19$	−16.0000	−0.193192	−0.0965961	+	0.995324i	$0.530796\pi$
$20$	0	0
$21$	24.0000	0.249392
$22$	0	0
$23$	60.0000	0.543951	0.271975	−	0.962304i	$-0.412323\pi$
$23$	60.0000	0.543951	0.271975	+	0.962304i	$0.412323\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	142.000	0.909267	0.454633	−	0.890679i	$-0.349770\pi$
$29$	142.000	0.909267	0.454633	+	0.890679i	$0.349770\pi$
$30$	0	0
$31$	176.000	1.01969	0.509847	−	0.860265i	$-0.329702\pi$
$31$	176.000	1.01969	0.509847	+	0.860265i	$0.329702\pi$
$32$	0	0
$33$	12.0000	0.0633010
$34$	0	0
$35$	40.0000	0.193178
$36$	0	0
$37$	214.000	0.950848	0.475424	−	0.879757i	$-0.342295\pi$
$37$	214.000	0.950848	0.475424	+	0.879757i	$0.342295\pi$
$38$	0	0
$39$	18.0000	0.0739053
$40$	0	0
$41$	−278.000	−1.05893	−0.529467	−	0.848330i	$-0.677608\pi$
$41$	−278.000	−1.05893	−0.529467	+	0.848330i	$0.677608\pi$
$42$	0	0
$43$	−68.0000	−0.241161	−0.120580	−	0.992704i	$-0.538476\pi$
$43$	−68.0000	−0.241161	−0.120580	+	0.992704i	$0.538476\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−116.000	−0.360007	−0.180004	−	0.983666i	$-0.557611\pi$
$47$	−116.000	−0.360007	−0.180004	+	0.983666i	$0.557611\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	−6.00000	−0.0164739
$52$	0	0
$53$	350.000	0.907098	0.453549	−	0.891231i	$-0.350158\pi$
$53$	350.000	0.907098	0.453549	+	0.891231i	$0.350158\pi$
$54$	0	0
$55$	20.0000	0.0490327
$56$	0	0
$57$	−48.0000	−0.111540
$58$	0	0
$59$	684.000	1.50931	0.754654	−	0.656123i	$-0.227805\pi$
$59$	684.000	1.50931	0.754654	+	0.656123i	$0.227805\pi$
$60$	0	0
$61$	394.000	0.826992	0.413496	−	0.910506i	$-0.364308\pi$
$61$	394.000	0.826992	0.413496	+	0.910506i	$0.364308\pi$
$62$	0	0
$63$	72.0000	0.143986
$64$	0	0
$65$	30.0000	0.0572468
$66$	0	0
$67$	108.000	0.196930	0.0984649	−	0.995141i	$-0.468607\pi$
$67$	108.000	0.196930	0.0984649	+	0.995141i	$0.468607\pi$
$68$	0	0
$69$	180.000	0.314050
$70$	0	0
$71$	96.0000	0.160466	0.0802331	−	0.996776i	$-0.474434\pi$
$71$	96.0000	0.160466	0.0802331	+	0.996776i	$0.474434\pi$
$72$	0	0
$73$	−398.000	−0.638115	−0.319057	−	0.947735i	$-0.603366\pi$
$73$	−398.000	−0.638115	−0.319057	+	0.947735i	$0.603366\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	32.0000	0.0473602
$78$	0	0
$79$	−136.000	−0.193686	−0.0968430	−	0.995300i	$-0.530874\pi$
$79$	−136.000	−0.193686	−0.0968430	+	0.995300i	$0.530874\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	436.000	0.576593	0.288296	−	0.957541i	$-0.406911\pi$
$83$	436.000	0.576593	0.288296	+	0.957541i	$0.406911\pi$
$84$	0	0
$85$	−10.0000	−0.0127606
$86$	0	0
$87$	426.000	0.524965
$88$	0	0
$89$	−750.000	−0.893257	−0.446628	−	0.894720i	$-0.647375\pi$
$89$	−750.000	−0.893257	−0.446628	+	0.894720i	$0.647375\pi$
$90$	0	0
$91$	48.0000	0.0552941
$92$	0	0
$93$	528.000	0.588721
$94$	0	0
$95$	−80.0000	−0.0863982
$96$	0	0
$97$	82.0000	0.0858334	0.0429167	−	0.999079i	$-0.486335\pi$
$97$	82.0000	0.0858334	0.0429167	+	0.999079i	$0.486335\pi$
$98$	0	0
$99$	36.0000	0.0365468
$100$	0	0
$101$	−154.000	−0.151719	−0.0758593	−	0.997119i	$-0.524170\pi$
$101$	−154.000	−0.151719	−0.0758593	+	0.997119i	$0.524170\pi$
$102$	0	0
$103$	1216.00	1.16326	0.581631	−	0.813453i	$-0.302415\pi$
$103$	1216.00	1.16326	0.581631	+	0.813453i	$0.302415\pi$
$104$	0	0
$105$	120.000	0.111531
$106$	0	0
$107$	732.000	0.661356	0.330678	−	0.943744i	$-0.392723\pi$
$107$	732.000	0.661356	0.330678	+	0.943744i	$0.392723\pi$
$108$	0	0
$109$	554.000	0.486822	0.243411	−	0.969923i	$-0.421734\pi$
$109$	554.000	0.486822	0.243411	+	0.969923i	$0.421734\pi$
$110$	0	0
$111$	642.000	0.548972
$112$	0	0
$113$	−474.000	−0.394603	−0.197302	−	0.980343i	$-0.563218\pi$
$113$	−474.000	−0.394603	−0.197302	+	0.980343i	$0.563218\pi$
$114$	0	0
$115$	300.000	0.243262
$116$	0	0
$117$	54.0000	0.0426692
$118$	0	0
$119$	−16.0000	−0.0123254
$120$	0	0
$121$	−1315.00	−0.987979
$122$	0	0
$123$	−834.000	−0.611376
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	840.000	0.586913	0.293456	−	0.955972i	$-0.405194\pi$
$127$	840.000	0.586913	0.293456	+	0.955972i	$0.405194\pi$
$128$	0	0
$129$	−204.000	−0.139234
$130$	0	0
$131$	1484.00	0.989753	0.494877	−	0.868963i	$-0.335213\pi$
$131$	1484.00	0.989753	0.494877	+	0.868963i	$0.335213\pi$
$132$	0	0
$133$	−128.000	−0.0834512
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	1006.00	0.627360	0.313680	−	0.949529i	$-0.398438\pi$
$137$	1006.00	0.627360	0.313680	+	0.949529i	$0.398438\pi$
$138$	0	0
$139$	1400.00	0.854291	0.427146	−	0.904183i	$-0.359519\pi$
$139$	1400.00	0.854291	0.427146	+	0.904183i	$0.359519\pi$
$140$	0	0
$141$	−348.000	−0.207850
$142$	0	0
$143$	24.0000	0.0140348
$144$	0	0
$145$	710.000	0.406636
$146$	0	0
$147$	−837.000	−0.469623
$148$	0	0
$149$	−2.00000	−0.00109964	−0.000549820	−	1.00000i	$-0.500175\pi$
$149$	−2.00000	−0.00109964	−0.000549820		1.00000i	$0.500175\pi$
$150$	0	0
$151$	2248.00	1.21152	0.605760	−	0.795647i	$-0.292869\pi$
$151$	2248.00	1.21152	0.605760	+	0.795647i	$0.292869\pi$
$152$	0	0
$153$	−18.0000	−0.00951120
$154$	0	0
$155$	880.000	0.456021
$156$	0	0
$157$	−2890.00	−1.46909	−0.734545	−	0.678560i	$-0.762604\pi$
$157$	−2890.00	−1.46909	−0.734545	+	0.678560i	$0.762604\pi$
$158$	0	0
$159$	1050.00	0.523713
$160$	0	0
$161$	480.000	0.234965
$162$	0	0
$163$	−252.000	−0.121093	−0.0605465	−	0.998165i	$-0.519284\pi$
$163$	−252.000	−0.121093	−0.0605465	+	0.998165i	$0.519284\pi$
$164$	0	0
$165$	60.0000	0.0283091
$166$	0	0
$167$	884.000	0.409617	0.204808	−	0.978802i	$-0.434343\pi$
$167$	884.000	0.409617	0.204808	+	0.978802i	$0.434343\pi$
$168$	0	0
$169$	−2161.00	−0.983614
$170$	0	0
$171$	−144.000	−0.0643974
$172$	0	0
$173$	1398.00	0.614381	0.307191	−	0.951648i	$-0.400611\pi$
$173$	1398.00	0.614381	0.307191	+	0.951648i	$0.400611\pi$
$174$	0	0
$175$	200.000	0.0863919
$176$	0	0
$177$	2052.00	0.871400
$178$	0	0
$179$	3612.00	1.50823	0.754116	−	0.656741i	$-0.228066\pi$
$179$	3612.00	1.50823	0.754116	+	0.656741i	$0.228066\pi$
$180$	0	0
$181$	−3150.00	−1.29358	−0.646789	−	0.762669i	$-0.723889\pi$
$181$	−3150.00	−1.29358	−0.646789	+	0.762669i	$0.723889\pi$
$182$	0	0
$183$	1182.00	0.477464
$184$	0	0
$185$	1070.00	0.425232
$186$	0	0
$187$	−8.00000	−0.00312844
$188$	0	0
$189$	216.000	0.0831306
$190$	0	0
$191$	−3040.00	−1.15166	−0.575829	−	0.817570i	$-0.695321\pi$
$191$	−3040.00	−1.15166	−0.575829	+	0.817570i	$0.695321\pi$
$192$	0	0
$193$	1994.00	0.743685	0.371843	−	0.928296i	$-0.378726\pi$
$193$	1994.00	0.743685	0.371843	+	0.928296i	$0.378726\pi$
$194$	0	0
$195$	90.0000	0.0330515
$196$	0	0
$197$	−4066.00	−1.47051	−0.735255	−	0.677790i	$-0.762938\pi$
$197$	−4066.00	−1.47051	−0.735255	+	0.677790i	$0.762938\pi$
$198$	0	0
$199$	3904.00	1.39069	0.695345	−	0.718676i	$-0.255252\pi$
$199$	3904.00	1.39069	0.695345	+	0.718676i	$0.255252\pi$
$200$	0	0
$201$	324.000	0.113698
$202$	0	0
$203$	1136.00	0.392766
$204$	0	0
$205$	−1390.00	−0.473570
$206$	0	0
$207$	540.000	0.181317
$208$	0	0
$209$	−64.0000	−0.0211817
$210$	0	0
$211$	−560.000	−0.182711	−0.0913554	−	0.995818i	$-0.529120\pi$
$211$	−560.000	−0.182711	−0.0913554	+	0.995818i	$0.529120\pi$
$212$	0	0
$213$	288.000	0.0926452
$214$	0	0
$215$	−340.000	−0.107850
$216$	0	0
$217$	1408.00	0.440467
$218$	0	0
$219$	−1194.00	−0.368416
$220$	0	0
$221$	−12.0000	−0.00365252
$222$	0	0
$223$	−608.000	−0.182577	−0.0912885	−	0.995824i	$-0.529099\pi$
$223$	−608.000	−0.182577	−0.0912885	+	0.995824i	$0.529099\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	5124.00	1.49820	0.749101	−	0.662456i	$-0.230486\pi$
$227$	5124.00	1.49820	0.749101	+	0.662456i	$0.230486\pi$
$228$	0	0
$229$	−1190.00	−0.343395	−0.171697	−	0.985150i	$-0.554925\pi$
$229$	−1190.00	−0.343395	−0.171697	+	0.985150i	$0.554925\pi$
$230$	0	0
$231$	96.0000	0.0273434
$232$	0	0
$233$	3862.00	1.08587	0.542936	−	0.839774i	$-0.317313\pi$
$233$	3862.00	1.08587	0.542936	+	0.839774i	$0.317313\pi$
$234$	0	0
$235$	−580.000	−0.161000
$236$	0	0
$237$	−408.000	−0.111825
$238$	0	0
$239$	−1640.00	−0.443861	−0.221931	−	0.975062i	$-0.571236\pi$
$239$	−1640.00	−0.443861	−0.221931	+	0.975062i	$0.571236\pi$
$240$	0	0
$241$	−2334.00	−0.623843	−0.311921	−	0.950108i	$-0.600973\pi$
$241$	−2334.00	−0.623843	−0.311921	+	0.950108i	$0.600973\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1395.00	−0.363768
$246$	0	0
$247$	−96.0000	−0.0247301
$248$	0	0
$249$	1308.00	0.332896
$250$	0	0
$251$	−772.000	−0.194136	−0.0970681	−	0.995278i	$-0.530946\pi$
$251$	−772.000	−0.194136	−0.0970681	+	0.995278i	$0.530946\pi$
$252$	0	0
$253$	240.000	0.0596390
$254$	0	0
$255$	−30.0000	−0.00736734
$256$	0	0
$257$	6214.00	1.50824	0.754122	−	0.656734i	$-0.228063\pi$
$257$	6214.00	1.50824	0.754122	+	0.656734i	$0.228063\pi$
$258$	0	0
$259$	1712.00	0.410728
$260$	0	0
$261$	1278.00	0.303089
$262$	0	0
$263$	−4948.00	−1.16010	−0.580051	−	0.814580i	$-0.696967\pi$
$263$	−4948.00	−1.16010	−0.580051	+	0.814580i	$0.696967\pi$
$264$	0	0
$265$	1750.00	0.405667
$266$	0	0
$267$	−2250.00	−0.515722
$268$	0	0
$269$	7038.00	1.59522	0.797610	−	0.603173i	$-0.206097\pi$
$269$	7038.00	1.59522	0.797610	+	0.603173i	$0.206097\pi$
$270$	0	0
$271$	728.000	0.163184	0.0815920	−	0.996666i	$-0.474000\pi$
$271$	728.000	0.163184	0.0815920	+	0.996666i	$0.474000\pi$
$272$	0	0
$273$	144.000	0.0319241
$274$	0	0
$275$	100.000	0.0219281
$276$	0	0
$277$	−7274.00	−1.57781	−0.788903	−	0.614518i	$-0.789351\pi$
$277$	−7274.00	−1.57781	−0.788903	+	0.614518i	$0.789351\pi$
$278$	0	0
$279$	1584.00	0.339898
$280$	0	0
$281$	1714.00	0.363874	0.181937	−	0.983310i	$-0.441763\pi$
$281$	1714.00	0.363874	0.181937	+	0.983310i	$0.441763\pi$
$282$	0	0
$283$	−4316.00	−0.906571	−0.453285	−	0.891365i	$-0.649748\pi$
$283$	−4316.00	−0.906571	−0.453285	+	0.891365i	$0.649748\pi$
$284$	0	0
$285$	−240.000	−0.0498820
$286$	0	0
$287$	−2224.00	−0.457417
$288$	0	0
$289$	−4909.00	−0.999186
$290$	0	0
$291$	246.000	0.0495559
$292$	0	0
$293$	6078.00	1.21188	0.605940	−	0.795511i	$-0.292797\pi$
$293$	6078.00	1.21188	0.605940	+	0.795511i	$0.292797\pi$
$294$	0	0
$295$	3420.00	0.674983
$296$	0	0
$297$	108.000	0.0211003
$298$	0	0
$299$	360.000	0.0696299
$300$	0	0
$301$	−544.000	−0.104172
$302$	0	0
$303$	−462.000	−0.0875947
$304$	0	0
$305$	1970.00	0.369842
$306$	0	0
$307$	−1564.00	−0.290756	−0.145378	−	0.989376i	$-0.546440\pi$
$307$	−1564.00	−0.290756	−0.145378	+	0.989376i	$0.546440\pi$
$308$	0	0
$309$	3648.00	0.671610
$310$	0	0
$311$	−5216.00	−0.951036	−0.475518	−	0.879706i	$-0.657739\pi$
$311$	−5216.00	−0.951036	−0.475518	+	0.879706i	$0.657739\pi$
$312$	0	0
$313$	−5790.00	−1.04559	−0.522796	−	0.852458i	$-0.675111\pi$
$313$	−5790.00	−1.04559	−0.522796	+	0.852458i	$0.675111\pi$
$314$	0	0
$315$	360.000	0.0643927
$316$	0	0
$317$	−3946.00	−0.699146	−0.349573	−	0.936909i	$-0.613673\pi$
$317$	−3946.00	−0.699146	−0.349573	+	0.936909i	$0.613673\pi$
$318$	0	0
$319$	568.000	0.0996925
$320$	0	0
$321$	2196.00	0.381834
$322$	0	0
$323$	32.0000	0.00551247
$324$	0	0
$325$	150.000	0.0256015
$326$	0	0
$327$	1662.00	0.281067
$328$	0	0
$329$	−928.000	−0.155508
$330$	0	0
$331$	−9992.00	−1.65924	−0.829622	−	0.558325i	$-0.811444\pi$
$331$	−9992.00	−1.65924	−0.829622	+	0.558325i	$0.811444\pi$
$332$	0	0
$333$	1926.00	0.316949
$334$	0	0
$335$	540.000	0.0880697
$336$	0	0
$337$	3274.00	0.529217	0.264609	−	0.964356i	$-0.414757\pi$
$337$	3274.00	0.529217	0.264609	+	0.964356i	$0.414757\pi$
$338$	0	0
$339$	−1422.00	−0.227824
$340$	0	0
$341$	704.000	0.111800
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	900.000	0.140447
$346$	0	0
$347$	−5580.00	−0.863257	−0.431628	−	0.902052i	$-0.642061\pi$
$347$	−5580.00	−0.863257	−0.431628	+	0.902052i	$0.642061\pi$
$348$	0	0
$349$	4154.00	0.637130	0.318565	−	0.947901i	$-0.396799\pi$
$349$	4154.00	0.637130	0.318565	+	0.947901i	$0.396799\pi$
$350$	0	0
$351$	162.000	0.0246351
$352$	0	0
$353$	−2802.00	−0.422480	−0.211240	−	0.977434i	$-0.567750\pi$
$353$	−2802.00	−0.422480	−0.211240	+	0.977434i	$0.567750\pi$
$354$	0	0
$355$	480.000	0.0717627
$356$	0	0
$357$	−48.0000	−0.00711605
$358$	0	0
$359$	11296.0	1.66067	0.830334	−	0.557265i	$-0.188149\pi$
$359$	11296.0	1.66067	0.830334	+	0.557265i	$0.188149\pi$
$360$	0	0
$361$	−6603.00	−0.962677
$362$	0	0
$363$	−3945.00	−0.570410
$364$	0	0
$365$	−1990.00	−0.285374
$366$	0	0
$367$	−10536.0	−1.49857	−0.749284	−	0.662248i	$-0.769602\pi$
$367$	−10536.0	−1.49857	−0.749284	+	0.662248i	$0.769602\pi$
$368$	0	0
$369$	−2502.00	−0.352978
$370$	0	0
$371$	2800.00	0.391830
$372$	0	0
$373$	2222.00	0.308447	0.154224	−	0.988036i	$-0.450712\pi$
$373$	2222.00	0.308447	0.154224	+	0.988036i	$0.450712\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	852.000	0.116393
$378$	0	0
$379$	−1512.00	−0.204924	−0.102462	−	0.994737i	$-0.532672\pi$
$379$	−1512.00	−0.204924	−0.102462	+	0.994737i	$0.532672\pi$
$380$	0	0
$381$	2520.00	0.338854
$382$	0	0
$383$	−7916.00	−1.05611	−0.528053	−	0.849211i	$-0.677078\pi$
$383$	−7916.00	−1.05611	−0.528053	+	0.849211i	$0.677078\pi$
$384$	0	0
$385$	160.000	0.0211801
$386$	0	0
$387$	−612.000	−0.0803868
$388$	0	0
$389$	9950.00	1.29688	0.648438	−	0.761267i	$-0.275422\pi$
$389$	9950.00	1.29688	0.648438	+	0.761267i	$0.275422\pi$
$390$	0	0
$391$	−120.000	−0.0155209
$392$	0	0
$393$	4452.00	0.571434
$394$	0	0
$395$	−680.000	−0.0866190
$396$	0	0
$397$	−9554.00	−1.20781	−0.603906	−	0.797055i	$-0.706390\pi$
$397$	−9554.00	−1.20781	−0.603906	+	0.797055i	$0.706390\pi$
$398$	0	0
$399$	−384.000	−0.0481806
$400$	0	0
$401$	−6390.00	−0.795764	−0.397882	−	0.917437i	$-0.630255\pi$
$401$	−6390.00	−0.795764	−0.397882	+	0.917437i	$0.630255\pi$
$402$	0	0
$403$	1056.00	0.130529
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	856.000	0.104251
$408$	0	0
$409$	−13030.0	−1.57529	−0.787643	−	0.616132i	$-0.788699\pi$
$409$	−13030.0	−1.57529	−0.787643	+	0.616132i	$0.788699\pi$
$410$	0	0
$411$	3018.00	0.362207
$412$	0	0
$413$	5472.00	0.651960
$414$	0	0
$415$	2180.00	0.257860
$416$	0	0
$417$	4200.00	0.493225
$418$	0	0
$419$	2100.00	0.244849	0.122424	−	0.992478i	$-0.460933\pi$
$419$	2100.00	0.244849	0.122424	+	0.992478i	$0.460933\pi$
$420$	0	0
$421$	−3478.00	−0.402630	−0.201315	−	0.979527i	$-0.564522\pi$
$421$	−3478.00	−0.402630	−0.201315	+	0.979527i	$0.564522\pi$
$422$	0	0
$423$	−1044.00	−0.120002
$424$	0	0
$425$	−50.0000	−0.00570672
$426$	0	0
$427$	3152.00	0.357227
$428$	0	0
$429$	72.0000	0.00810301
$430$	0	0
$431$	−8504.00	−0.950402	−0.475201	−	0.879877i	$-0.657625\pi$
$431$	−8504.00	−0.950402	−0.475201	+	0.879877i	$0.657625\pi$
$432$	0	0
$433$	−16102.0	−1.78710	−0.893548	−	0.448967i	$-0.851792\pi$
$433$	−16102.0	−1.78710	−0.893548	+	0.448967i	$0.851792\pi$
$434$	0	0
$435$	2130.00	0.234772
$436$	0	0
$437$	−960.000	−0.105087
$438$	0	0
$439$	1072.00	0.116546	0.0582731	−	0.998301i	$-0.481441\pi$
$439$	1072.00	0.116546	0.0582731	+	0.998301i	$0.481441\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	0	0
$443$	−11076.0	−1.18789	−0.593947	−	0.804505i	$-0.702431\pi$
$443$	−11076.0	−1.18789	−0.593947	+	0.804505i	$0.702431\pi$
$444$	0	0
$445$	−3750.00	−0.399477
$446$	0	0
$447$	−6.00000	−0.000634878 0
$448$	0	0
$449$	−9910.00	−1.04161	−0.520804	−	0.853676i	$-0.674368\pi$
$449$	−9910.00	−1.04161	−0.520804	+	0.853676i	$0.674368\pi$
$450$	0	0
$451$	−1112.00	−0.116102
$452$	0	0
$453$	6744.00	0.699472
$454$	0	0
$455$	240.000	0.0247283
$456$	0	0
$457$	−3798.00	−0.388759	−0.194380	−	0.980926i	$-0.562269\pi$
$457$	−3798.00	−0.388759	−0.194380	+	0.980926i	$0.562269\pi$
$458$	0	0
$459$	−54.0000	−0.00549129
$460$	0	0
$461$	8406.00	0.849255	0.424627	−	0.905368i	$-0.360405\pi$
$461$	8406.00	0.849255	0.424627	+	0.905368i	$0.360405\pi$
$462$	0	0
$463$	−18040.0	−1.81078	−0.905389	−	0.424584i	$-0.860420\pi$
$463$	−18040.0	−1.81078	−0.905389	+	0.424584i	$0.860420\pi$
$464$	0	0
$465$	2640.00	0.263284
$466$	0	0
$467$	−8484.00	−0.840670	−0.420335	−	0.907369i	$-0.638087\pi$
$467$	−8484.00	−0.840670	−0.420335	+	0.907369i	$0.638087\pi$
$468$	0	0
$469$	864.000	0.0850657
$470$	0	0
$471$	−8670.00	−0.848179
$472$	0	0
$473$	−272.000	−0.0264410
$474$	0	0
$475$	−400.000	−0.0386384
$476$	0	0
$477$	3150.00	0.302366
$478$	0	0
$479$	2136.00	0.203750	0.101875	−	0.994797i	$-0.467516\pi$
$479$	2136.00	0.203750	0.101875	+	0.994797i	$0.467516\pi$
$480$	0	0
$481$	1284.00	0.121716
$482$	0	0
$483$	1440.00	0.135657
$484$	0	0
$485$	410.000	0.0383859
$486$	0	0
$487$	−3152.00	−0.293287	−0.146643	−	0.989189i	$-0.546847\pi$
$487$	−3152.00	−0.293287	−0.146643	+	0.989189i	$0.546847\pi$
$488$	0	0
$489$	−756.000	−0.0699131
$490$	0	0
$491$	−11812.0	−1.08568	−0.542839	−	0.839837i	$-0.682651\pi$
$491$	−11812.0	−1.08568	−0.542839	+	0.839837i	$0.682651\pi$
$492$	0	0
$493$	−284.000	−0.0259447
$494$	0	0
$495$	180.000	0.0163442
$496$	0	0
$497$	768.000	0.0693149
$498$	0	0
$499$	10352.0	0.928696	0.464348	−	0.885653i	$-0.346289\pi$
$499$	10352.0	0.928696	0.464348	+	0.885653i	$0.346289\pi$
$500$	0	0
$501$	2652.00	0.236492
$502$	0	0
$503$	1300.00	0.115237	0.0576184	−	0.998339i	$-0.481649\pi$
$503$	1300.00	0.115237	0.0576184	+	0.998339i	$0.481649\pi$
$504$	0	0
$505$	−770.000	−0.0678506
$506$	0	0
$507$	−6483.00	−0.567890
$508$	0	0
$509$	4398.00	0.382982	0.191491	−	0.981494i	$-0.438668\pi$
$509$	4398.00	0.382982	0.191491	+	0.981494i	$0.438668\pi$
$510$	0	0
$511$	−3184.00	−0.275640
$512$	0	0
$513$	−432.000	−0.0371799
$514$	0	0
$515$	6080.00	0.520227
$516$	0	0
$517$	−464.000	−0.0394714
$518$	0	0
$519$	4194.00	0.354713
$520$	0	0
$521$	1650.00	0.138748	0.0693741	−	0.997591i	$-0.477900\pi$
$521$	1650.00	0.138748	0.0693741	+	0.997591i	$0.477900\pi$
$522$	0	0
$523$	−17276.0	−1.44441	−0.722205	−	0.691679i	$-0.756871\pi$
$523$	−17276.0	−1.44441	−0.722205	+	0.691679i	$0.756871\pi$
$524$	0	0
$525$	600.000	0.0498784
$526$	0	0
$527$	−352.000	−0.0290956
$528$	0	0
$529$	−8567.00	−0.704118
$530$	0	0
$531$	6156.00	0.503103
$532$	0	0
$533$	−1668.00	−0.135552
$534$	0	0
$535$	3660.00	0.295767
$536$	0	0
$537$	10836.0	0.870778
$538$	0	0
$539$	−1116.00	−0.0891828
$540$	0	0
$541$	−3878.00	−0.308185	−0.154093	−	0.988056i	$-0.549245\pi$
$541$	−3878.00	−0.308185	−0.154093	+	0.988056i	$0.549245\pi$
$542$	0	0
$543$	−9450.00	−0.746848
$544$	0	0
$545$	2770.00	0.217713
$546$	0	0
$547$	9604.00	0.750708	0.375354	−	0.926881i	$-0.377521\pi$
$547$	9604.00	0.750708	0.375354	+	0.926881i	$0.377521\pi$
$548$	0	0
$549$	3546.00	0.275664
$550$	0	0
$551$	−2272.00	−0.175663
$552$	0	0
$553$	−1088.00	−0.0836645
$554$	0	0
$555$	3210.00	0.245508
$556$	0	0
$557$	−13546.0	−1.03045	−0.515227	−	0.857054i	$-0.672292\pi$
$557$	−13546.0	−1.03045	−0.515227	+	0.857054i	$0.672292\pi$
$558$	0	0
$559$	−408.000	−0.0308704
$560$	0	0
$561$	−24.0000	−0.00180620
$562$	0	0
$563$	−16884.0	−1.26390	−0.631950	−	0.775009i	$-0.717745\pi$
$563$	−16884.0	−1.26390	−0.631950	+	0.775009i	$0.717745\pi$
$564$	0	0
$565$	−2370.00	−0.176472
$566$	0	0
$567$	648.000	0.0479955
$568$	0	0
$569$	13746.0	1.01276	0.506382	−	0.862309i	$-0.330983\pi$
$569$	13746.0	1.01276	0.506382	+	0.862309i	$0.330983\pi$
$570$	0	0
$571$	−15176.0	−1.11225	−0.556126	−	0.831098i	$-0.687713\pi$
$571$	−15176.0	−1.11225	−0.556126	+	0.831098i	$0.687713\pi$
$572$	0	0
$573$	−9120.00	−0.664910
$574$	0	0
$575$	1500.00	0.108790
$576$	0	0
$577$	15106.0	1.08990	0.544949	−	0.838469i	$-0.316549\pi$
$577$	15106.0	1.08990	0.544949	+	0.838469i	$0.316549\pi$
$578$	0	0
$579$	5982.00	0.429367
$580$	0	0
$581$	3488.00	0.249065
$582$	0	0
$583$	1400.00	0.0994547
$584$	0	0
$585$	270.000	0.0190823
$586$	0	0
$587$	−17332.0	−1.21869	−0.609343	−	0.792907i	$-0.708567\pi$
$587$	−17332.0	−1.21869	−0.609343	+	0.792907i	$0.708567\pi$
$588$	0	0
$589$	−2816.00	−0.196997
$590$	0	0
$591$	−12198.0	−0.848999
$592$	0	0
$593$	−1290.00	−0.0893321	−0.0446661	−	0.999002i	$-0.514222\pi$
$593$	−1290.00	−0.0893321	−0.0446661	+	0.999002i	$0.514222\pi$
$594$	0	0
$595$	−80.0000	−0.00551207
$596$	0	0
$597$	11712.0	0.802915
$598$	0	0
$599$	20544.0	1.40134	0.700672	−	0.713484i	$-0.252884\pi$
$599$	20544.0	1.40134	0.700672	+	0.713484i	$0.252884\pi$
$600$	0	0
$601$	−2630.00	−0.178502	−0.0892512	−	0.996009i	$-0.528447\pi$
$601$	−2630.00	−0.178502	−0.0892512	+	0.996009i	$0.528447\pi$
$602$	0	0
$603$	972.000	0.0656433
$604$	0	0
$605$	−6575.00	−0.441838
$606$	0	0
$607$	−20320.0	−1.35875	−0.679377	−	0.733790i	$-0.737750\pi$
$607$	−20320.0	−1.35875	−0.679377	+	0.733790i	$0.737750\pi$
$608$	0	0
$609$	3408.00	0.226764
$610$	0	0
$611$	−696.000	−0.0460837
$612$	0	0
$613$	−15386.0	−1.01376	−0.506880	−	0.862017i	$-0.669201\pi$
$613$	−15386.0	−1.01376	−0.506880	+	0.862017i	$0.669201\pi$
$614$	0	0
$615$	−4170.00	−0.273416
$616$	0	0
$617$	8934.00	0.582932	0.291466	−	0.956581i	$-0.405857\pi$
$617$	8934.00	0.582932	0.291466	+	0.956581i	$0.405857\pi$
$618$	0	0
$619$	20408.0	1.32515	0.662574	−	0.748996i	$-0.269464\pi$
$619$	20408.0	1.32515	0.662574	+	0.748996i	$0.269464\pi$
$620$	0	0
$621$	1620.00	0.104683
$622$	0	0
$623$	−6000.00	−0.385851
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−192.000	−0.0122293
$628$	0	0
$629$	−428.000	−0.0271311
$630$	0	0
$631$	24000.0	1.51414	0.757072	−	0.653331i	$-0.226629\pi$
$631$	24000.0	1.51414	0.757072	+	0.653331i	$0.226629\pi$
$632$	0	0
$633$	−1680.00	−0.105488
$634$	0	0
$635$	4200.00	0.262475
$636$	0	0
$637$	−1674.00	−0.104123
$638$	0	0
$639$	864.000	0.0534888
$640$	0	0
$641$	15810.0	0.974193	0.487096	−	0.873348i	$-0.338056\pi$
$641$	15810.0	0.974193	0.487096	+	0.873348i	$0.338056\pi$
$642$	0	0
$643$	7716.00	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	7716.00	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	−1020.00	−0.0622674
$646$	0	0
$647$	27324.0	1.66030	0.830152	−	0.557536i	$-0.188253\pi$
$647$	27324.0	1.66030	0.830152	+	0.557536i	$0.188253\pi$
$648$	0	0
$649$	2736.00	0.165481
$650$	0	0
$651$	4224.00	0.254304
$652$	0	0
$653$	−9450.00	−0.566320	−0.283160	−	0.959073i	$-0.591383\pi$
$653$	−9450.00	−0.566320	−0.283160	+	0.959073i	$0.591383\pi$
$654$	0	0
$655$	7420.00	0.442631
$656$	0	0
$657$	−3582.00	−0.212705
$658$	0	0
$659$	10804.0	0.638640	0.319320	−	0.947647i	$-0.396545\pi$
$659$	10804.0	0.638640	0.319320	+	0.947647i	$0.396545\pi$
$660$	0	0
$661$	−1534.00	−0.0902658	−0.0451329	−	0.998981i	$-0.514371\pi$
$661$	−1534.00	−0.0902658	−0.0451329	+	0.998981i	$0.514371\pi$
$662$	0	0
$663$	−36.0000	−0.00210878
$664$	0	0
$665$	−640.000	−0.0373205
$666$	0	0
$667$	8520.00	0.494596
$668$	0	0
$669$	−1824.00	−0.105411
$670$	0	0
$671$	1576.00	0.0906718
$672$	0	0
$673$	1306.00	0.0748033	0.0374016	−	0.999300i	$-0.488092\pi$
$673$	1306.00	0.0748033	0.0374016	+	0.999300i	$0.488092\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	6398.00	0.363213	0.181606	−	0.983371i	$-0.441870\pi$
$677$	6398.00	0.363213	0.181606	+	0.983371i	$0.441870\pi$
$678$	0	0
$679$	656.000	0.0370765
$680$	0	0
$681$	15372.0	0.864987
$682$	0	0
$683$	−20396.0	−1.14265	−0.571326	−	0.820723i	$-0.693571\pi$
$683$	−20396.0	−1.14265	−0.571326	+	0.820723i	$0.693571\pi$
$684$	0	0
$685$	5030.00	0.280564
$686$	0	0
$687$	−3570.00	−0.198259
$688$	0	0
$689$	2100.00	0.116116
$690$	0	0
$691$	−11592.0	−0.638177	−0.319089	−	0.947725i	$-0.603377\pi$
$691$	−11592.0	−0.638177	−0.319089	+	0.947725i	$0.603377\pi$
$692$	0	0
$693$	288.000	0.0157867
$694$	0	0
$695$	7000.00	0.382051
$696$	0	0
$697$	556.000	0.0302152
$698$	0	0
$699$	11586.0	0.626928
$700$	0	0
$701$	−3618.00	−0.194936	−0.0974679	−	0.995239i	$-0.531074\pi$
$701$	−3618.00	−0.194936	−0.0974679	+	0.995239i	$0.531074\pi$
$702$	0	0
$703$	−3424.00	−0.183696
$704$	0	0
$705$	−1740.00	−0.0929535
$706$	0	0
$707$	−1232.00	−0.0655362
$708$	0	0
$709$	−30518.0	−1.61654	−0.808270	−	0.588811i	$-0.799596\pi$
$709$	−30518.0	−1.61654	−0.808270	+	0.588811i	$0.799596\pi$
$710$	0	0
$711$	−1224.00	−0.0645620
$712$	0	0
$713$	10560.0	0.554664
$714$	0	0
$715$	120.000	0.00627657
$716$	0	0
$717$	−4920.00	−0.256263
$718$	0	0
$719$	13088.0	0.678860	0.339430	−	0.940631i	$-0.389766\pi$
$719$	13088.0	0.678860	0.339430	+	0.940631i	$0.389766\pi$
$720$	0	0
$721$	9728.00	0.502482
$722$	0	0
$723$	−7002.00	−0.360176
$724$	0	0
$725$	3550.00	0.181853
$726$	0	0
$727$	−6064.00	−0.309355	−0.154678	−	0.987965i	$-0.549434\pi$
$727$	−6064.00	−0.309355	−0.154678	+	0.987965i	$0.549434\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	136.000	0.00688118
$732$	0	0
$733$	−3522.00	−0.177473	−0.0887367	−	0.996055i	$-0.528283\pi$
$733$	−3522.00	−0.177473	−0.0887367	+	0.996055i	$0.528283\pi$
$734$	0	0
$735$	−4185.00	−0.210022
$736$	0	0
$737$	432.000	0.0215915
$738$	0	0
$739$	39040.0	1.94331	0.971657	−	0.236394i	$-0.0759655\pi$
$739$	39040.0	1.94331	0.971657	+	0.236394i	$0.0759655\pi$
$740$	0	0
$741$	−288.000	−0.0142779
$742$	0	0
$743$	14868.0	0.734124	0.367062	−	0.930197i	$-0.380364\pi$
$743$	14868.0	0.734124	0.367062	+	0.930197i	$0.380364\pi$
$744$	0	0
$745$	−10.0000	−0.000491774 0
$746$	0	0
$747$	3924.00	0.192198
$748$	0	0
$749$	5856.00	0.285679
$750$	0	0
$751$	4016.00	0.195134	0.0975672	−	0.995229i	$-0.468894\pi$
$751$	4016.00	0.195134	0.0975672	+	0.995229i	$0.468894\pi$
$752$	0	0
$753$	−2316.00	−0.112085
$754$	0	0
$755$	11240.0	0.541809
$756$	0	0
$757$	7958.00	0.382085	0.191043	−	0.981582i	$-0.438813\pi$
$757$	7958.00	0.382085	0.191043	+	0.981582i	$0.438813\pi$
$758$	0	0
$759$	720.000	0.0344326
$760$	0	0
$761$	15890.0	0.756915	0.378457	−	0.925619i	$-0.376455\pi$
$761$	15890.0	0.756915	0.378457	+	0.925619i	$0.376455\pi$
$762$	0	0
$763$	4432.00	0.210287
$764$	0	0
$765$	−90.0000	−0.00425354
$766$	0	0
$767$	4104.00	0.193203
$768$	0	0
$769$	6834.00	0.320469	0.160234	−	0.987079i	$-0.448775\pi$
$769$	6834.00	0.320469	0.160234	+	0.987079i	$0.448775\pi$
$770$	0	0
$771$	18642.0	0.870785
$772$	0	0
$773$	3694.00	0.171881	0.0859405	−	0.996300i	$-0.472611\pi$
$773$	3694.00	0.171881	0.0859405	+	0.996300i	$0.472611\pi$
$774$	0	0
$775$	4400.00	0.203939
$776$	0	0
$777$	5136.00	0.237134
$778$	0	0
$779$	4448.00	0.204578
$780$	0	0
$781$	384.000	0.0175936
$782$	0	0
$783$	3834.00	0.174988
$784$	0	0
$785$	−14450.0	−0.656997
$786$	0	0
$787$	38284.0	1.73402	0.867012	−	0.498287i	$-0.166037\pi$
$787$	38284.0	1.73402	0.867012	+	0.498287i	$0.166037\pi$
$788$	0	0
$789$	−14844.0	−0.669785
$790$	0	0
$791$	−3792.00	−0.170453
$792$	0	0
$793$	2364.00	0.105861
$794$	0	0
$795$	5250.00	0.234212
$796$	0	0
$797$	−3258.00	−0.144798	−0.0723992	−	0.997376i	$-0.523066\pi$
$797$	−3258.00	−0.144798	−0.0723992	+	0.997376i	$0.523066\pi$
$798$	0	0
$799$	232.000	0.0102723
$800$	0	0
$801$	−6750.00	−0.297752
$802$	0	0
$803$	−1592.00	−0.0699632
$804$	0	0
$805$	2400.00	0.105079
$806$	0	0
$807$	21114.0	0.921001
$808$	0	0
$809$	29754.0	1.29307	0.646536	−	0.762884i	$-0.276217\pi$
$809$	29754.0	1.29307	0.646536	+	0.762884i	$0.276217\pi$
$810$	0	0
$811$	−42848.0	−1.85524	−0.927618	−	0.373530i	$-0.878147\pi$
$811$	−42848.0	−1.85524	−0.927618	+	0.373530i	$0.878147\pi$
$812$	0	0
$813$	2184.00	0.0942143
$814$	0	0
$815$	−1260.00	−0.0541544
$816$	0	0
$817$	1088.00	0.0465903
$818$	0	0
$819$	432.000	0.0184314
$820$	0	0
$821$	6454.00	0.274356	0.137178	−	0.990546i	$-0.456197\pi$
$821$	6454.00	0.274356	0.137178	+	0.990546i	$0.456197\pi$
$822$	0	0
$823$	−19128.0	−0.810158	−0.405079	−	0.914282i	$-0.632756\pi$
$823$	−19128.0	−0.810158	−0.405079	+	0.914282i	$0.632756\pi$
$824$	0	0
$825$	300.000	0.0126602
$826$	0	0
$827$	7060.00	0.296856	0.148428	−	0.988923i	$-0.452579\pi$
$827$	7060.00	0.296856	0.148428	+	0.988923i	$0.452579\pi$
$828$	0	0
$829$	4666.00	0.195485	0.0977424	−	0.995212i	$-0.468838\pi$
$829$	4666.00	0.195485	0.0977424	+	0.995212i	$0.468838\pi$
$830$	0	0
$831$	−21822.0	−0.910947
$832$	0	0
$833$	558.000	0.0232095
$834$	0	0
$835$	4420.00	0.183186
$836$	0	0
$837$	4752.00	0.196240
$838$	0	0
$839$	33792.0	1.39050	0.695250	−	0.718768i	$-0.255294\pi$
$839$	33792.0	1.39050	0.695250	+	0.718768i	$0.255294\pi$
$840$	0	0
$841$	−4225.00	−0.173234
$842$	0	0
$843$	5142.00	0.210083
$844$	0	0
$845$	−10805.0	−0.439886
$846$	0	0
$847$	−10520.0	−0.426767
$848$	0	0
$849$	−12948.0	−0.523409
$850$	0	0
$851$	12840.0	0.517214
$852$	0	0
$853$	−23146.0	−0.929078	−0.464539	−	0.885553i	$-0.653780\pi$
$853$	−23146.0	−0.929078	−0.464539	+	0.885553i	$0.653780\pi$
$854$	0	0
$855$	−720.000	−0.0287994
$856$	0	0
$857$	−40698.0	−1.62219	−0.811095	−	0.584914i	$-0.801128\pi$
$857$	−40698.0	−1.62219	−0.811095	+	0.584914i	$0.801128\pi$
$858$	0	0
$859$	40136.0	1.59421	0.797103	−	0.603844i	$-0.206365\pi$
$859$	40136.0	1.59421	0.797103	+	0.603844i	$0.206365\pi$
$860$	0	0
$861$	−6672.00	−0.264090
$862$	0	0
$863$	−1396.00	−0.0550642	−0.0275321	−	0.999621i	$-0.508765\pi$
$863$	−1396.00	−0.0550642	−0.0275321	+	0.999621i	$0.508765\pi$
$864$	0	0
$865$	6990.00	0.274760
$866$	0	0
$867$	−14727.0	−0.576880
$868$	0	0
$869$	−544.000	−0.0212358
$870$	0	0
$871$	648.000	0.0252085
$872$	0	0
$873$	738.000	0.0286111
$874$	0	0
$875$	1000.00	0.0386356
$876$	0	0
$877$	34646.0	1.33399	0.666997	−	0.745061i	$-0.267579\pi$
$877$	34646.0	1.33399	0.666997	+	0.745061i	$0.267579\pi$
$878$	0	0
$879$	18234.0	0.699679
$880$	0	0
$881$	−15302.0	−0.585173	−0.292587	−	0.956239i	$-0.594516\pi$
$881$	−15302.0	−0.585173	−0.292587	+	0.956239i	$0.594516\pi$
$882$	0	0
$883$	−26300.0	−1.00234	−0.501170	−	0.865349i	$-0.667097\pi$
$883$	−26300.0	−1.00234	−0.501170	+	0.865349i	$0.667097\pi$
$884$	0	0
$885$	10260.0	0.389702
$886$	0	0
$887$	12052.0	0.456219	0.228110	−	0.973635i	$-0.426746\pi$
$887$	12052.0	0.456219	0.228110	+	0.973635i	$0.426746\pi$
$888$	0	0
$889$	6720.00	0.253523
$890$	0	0
$891$	324.000	0.0121823
$892$	0	0
$893$	1856.00	0.0695506
$894$	0	0
$895$	18060.0	0.674502
$896$	0	0
$897$	1080.00	0.0402008
$898$	0	0
$899$	24992.0	0.927174
$900$	0	0
$901$	−700.000	−0.0258828
$902$	0	0
$903$	−1632.00	−0.0601435
$904$	0	0
$905$	−15750.0	−0.578506
$906$	0	0
$907$	18260.0	0.668482	0.334241	−	0.942488i	$-0.391520\pi$
$907$	18260.0	0.668482	0.334241	+	0.942488i	$0.391520\pi$
$908$	0	0
$909$	−1386.00	−0.0505728
$910$	0	0
$911$	10280.0	0.373866	0.186933	−	0.982373i	$-0.440145\pi$
$911$	10280.0	0.373866	0.186933	+	0.982373i	$0.440145\pi$
$912$	0	0
$913$	1744.00	0.0632179
$914$	0	0
$915$	5910.00	0.213528
$916$	0	0
$917$	11872.0	0.427533
$918$	0	0
$919$	34504.0	1.23850	0.619250	−	0.785194i	$-0.287437\pi$
$919$	34504.0	1.23850	0.619250	+	0.785194i	$0.287437\pi$
$920$	0	0
$921$	−4692.00	−0.167868
$922$	0	0
$923$	576.000	0.0205409
$924$	0	0
$925$	5350.00	0.190170
$926$	0	0
$927$	10944.0	0.387754
$928$	0	0
$929$	28266.0	0.998253	0.499127	−	0.866529i	$-0.333654\pi$
$929$	28266.0	0.998253	0.499127	+	0.866529i	$0.333654\pi$
$930$	0	0
$931$	4464.00	0.157145
$932$	0	0
$933$	−15648.0	−0.549081
$934$	0	0
$935$	−40.0000	−0.00139908
$936$	0	0
$937$	−51494.0	−1.79534	−0.897671	−	0.440666i	$-0.854742\pi$
$937$	−51494.0	−1.79534	−0.897671	+	0.440666i	$0.854742\pi$
$938$	0	0
$939$	−17370.0	−0.603673
$940$	0	0
$941$	−5338.00	−0.184924	−0.0924622	−	0.995716i	$-0.529474\pi$
$941$	−5338.00	−0.184924	−0.0924622	+	0.995716i	$0.529474\pi$
$942$	0	0
$943$	−16680.0	−0.576008
$944$	0	0
$945$	1080.00	0.0371771
$946$	0	0
$947$	−47644.0	−1.63487	−0.817435	−	0.576021i	$-0.804605\pi$
$947$	−47644.0	−1.63487	−0.817435	+	0.576021i	$0.804605\pi$
$948$	0	0
$949$	−2388.00	−0.0816836
$950$	0	0
$951$	−11838.0	−0.403652
$952$	0	0
$953$	−13906.0	−0.472675	−0.236338	−	0.971671i	$-0.575947\pi$
$953$	−13906.0	−0.472675	−0.236338	+	0.971671i	$0.575947\pi$
$954$	0	0
$955$	−15200.0	−0.515037
$956$	0	0
$957$	1704.00	0.0575575
$958$	0	0
$959$	8048.00	0.270994
$960$	0	0
$961$	1185.00	0.0397771
$962$	0	0
$963$	6588.00	0.220452
$964$	0	0
$965$	9970.00	0.332586
$966$	0	0
$967$	−8536.00	−0.283867	−0.141933	−	0.989876i	$-0.545332\pi$
$967$	−8536.00	−0.283867	−0.141933	+	0.989876i	$0.545332\pi$
$968$	0	0
$969$	96.0000	0.00318263
$970$	0	0
$971$	22572.0	0.746004	0.373002	−	0.927831i	$-0.378328\pi$
$971$	22572.0	0.746004	0.373002	+	0.927831i	$0.378328\pi$
$972$	0	0
$973$	11200.0	0.369019
$974$	0	0
$975$	450.000	0.0147811
$976$	0	0
$977$	−51474.0	−1.68557	−0.842784	−	0.538253i	$-0.819085\pi$
$977$	−51474.0	−1.68557	−0.842784	+	0.538253i	$0.819085\pi$
$978$	0	0
$979$	−3000.00	−0.0979371
$980$	0	0
$981$	4986.00	0.162274
$982$	0	0
$983$	−6412.00	−0.208048	−0.104024	−	0.994575i	$-0.533172\pi$
$983$	−6412.00	−0.208048	−0.104024	+	0.994575i	$0.533172\pi$
$984$	0	0
$985$	−20330.0	−0.657632
$986$	0	0
$987$	−2784.00	−0.0897829
$988$	0	0
$989$	−4080.00	−0.131179
$990$	0	0
$991$	54008.0	1.73120	0.865601	−	0.500735i	$-0.166937\pi$
$991$	54008.0	1.73120	0.865601	+	0.500735i	$0.166937\pi$
$992$	0	0
$993$	−29976.0	−0.957965
$994$	0	0
$995$	19520.0	0.621935
$996$	0	0
$997$	30846.0	0.979842	0.489921	−	0.871767i	$-0.337026\pi$
$997$	30846.0	0.979842	0.489921	+	0.871767i	$0.337026\pi$
$998$	0	0
$999$	5778.00	0.182991

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.a.bh.1.1		1
4.3	odd	2		960.4.a.m.1.1		1
8.3	odd	2		480.4.a.g.1.1	yes	1
8.5	even	2		480.4.a.b.1.1	✓	1
24.5	odd	2		1440.4.a.p.1.1		1
24.11	even	2		1440.4.a.m.1.1		1
40.19	odd	2		2400.4.a.f.1.1		1
40.29	even	2		2400.4.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.b.1.1	✓	1	8.5	even	2
480.4.a.g.1.1	yes	1	8.3	odd	2
960.4.a.m.1.1		1	4.3	odd	2
960.4.a.bh.1.1		1	1.1	even	1	trivial
1440.4.a.m.1.1		1	24.11	even	2
1440.4.a.p.1.1		1	24.5	odd	2
2400.4.a.f.1.1		1	40.19	odd	2
2400.4.a.q.1.1		1	40.29	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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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