LMFDB - Embedded newform 960.4.a.s.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,4,Mod(1,960)] 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("960.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,-5,0,-32,0,9,0,-60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.6418336055$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
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)

$f(q)$

$=$

$q+3.00000 q^{3} -5.00000 q^{5} -32.0000 q^{7} +9.00000 q^{9} -60.0000 q^{11} +34.0000 q^{13} -15.0000 q^{15} +42.0000 q^{17} -76.0000 q^{19} -96.0000 q^{21} +25.0000 q^{25} +27.0000 q^{27} -6.00000 q^{29} +232.000 q^{31} -180.000 q^{33} +160.000 q^{35} -134.000 q^{37} +102.000 q^{39} +234.000 q^{41} -412.000 q^{43} -45.0000 q^{45} +360.000 q^{47} +681.000 q^{49} +126.000 q^{51} -222.000 q^{53} +300.000 q^{55} -228.000 q^{57} +660.000 q^{59} +490.000 q^{61} -288.000 q^{63} -170.000 q^{65} +812.000 q^{67} -120.000 q^{71} +746.000 q^{73} +75.0000 q^{75} +1920.00 q^{77} -152.000 q^{79} +81.0000 q^{81} -804.000 q^{83} -210.000 q^{85} -18.0000 q^{87} -678.000 q^{89} -1088.00 q^{91} +696.000 q^{93} +380.000 q^{95} +194.000 q^{97} -540.000 q^{99} +O(q^{100})$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−32.0000	−1.72784	−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000	−1.72784	−0.863919	+	0.503631i	$0.831997\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−60.0000	−1.64461	−0.822304	−	0.569049i	$-0.807311\pi$
$11$	−60.0000	−1.64461	−0.822304	+	0.569049i	$0.807311\pi$
$12$	0	0
$13$	34.0000	0.725377	0.362689	−	0.931910i	$-0.381859\pi$
$13$	34.0000	0.725377	0.362689	+	0.931910i	$0.381859\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	42.0000	0.599206	0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000	0.599206	0.299603	+	0.954064i	$0.403146\pi$
$18$	0	0
$19$	−76.0000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−76.0000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−96.0000	−0.997567
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−6.00000	−0.0384197	−0.0192099	−	0.999815i	$-0.506115\pi$
$29$	−6.00000	−0.0384197	−0.0192099	+	0.999815i	$0.506115\pi$
$30$	0	0
$31$	232.000	1.34414	0.672071	−	0.740486i	$-0.265405\pi$
$31$	232.000	1.34414	0.672071	+	0.740486i	$0.265405\pi$
$32$	0	0
$33$	−180.000	−0.949514
$34$	0	0
$35$	160.000	0.772712
$36$	0	0
$37$	−134.000	−0.595391	−0.297695	−	0.954661i	$-0.596218\pi$
$37$	−134.000	−0.595391	−0.297695	+	0.954661i	$0.596218\pi$
$38$	0	0
$39$	102.000	0.418797
$40$	0	0
$41$	234.000	0.891333	0.445667	−	0.895199i	$-0.352967\pi$
$41$	234.000	0.891333	0.445667	+	0.895199i	$0.352967\pi$
$42$	0	0
$43$	−412.000	−1.46115	−0.730575	−	0.682833i	$-0.760748\pi$
$43$	−412.000	−1.46115	−0.730575	+	0.682833i	$0.760748\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	360.000	1.11726	0.558632	−	0.829416i	$-0.311326\pi$
$47$	360.000	1.11726	0.558632	+	0.829416i	$0.311326\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	126.000	0.345952
$52$	0	0
$53$	−222.000	−0.575359	−0.287680	−	0.957727i	$-0.592884\pi$
$53$	−222.000	−0.575359	−0.287680	+	0.957727i	$0.592884\pi$
$54$	0	0
$55$	300.000	0.735491
$56$	0	0
$57$	−228.000	−0.529813
$58$	0	0
$59$	660.000	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$59$	660.000	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$60$	0	0
$61$	490.000	1.02849	0.514246	−	0.857642i	$-0.328072\pi$
$61$	490.000	1.02849	0.514246	+	0.857642i	$0.328072\pi$
$62$	0	0
$63$	−288.000	−0.575946
$64$	0	0
$65$	−170.000	−0.324399
$66$	0	0
$67$	812.000	1.48062	0.740310	−	0.672265i	$-0.234679\pi$
$67$	812.000	1.48062	0.740310	+	0.672265i	$0.234679\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−120.000	−0.200583	−0.100291	−	0.994958i	$-0.531978\pi$
$71$	−120.000	−0.200583	−0.100291	+	0.994958i	$0.531978\pi$
$72$	0	0
$73$	746.000	1.19606	0.598032	−	0.801472i	$-0.295949\pi$
$73$	746.000	1.19606	0.598032	+	0.801472i	$0.295949\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	1920.00	2.84161
$78$	0	0
$79$	−152.000	−0.216473	−0.108236	−	0.994125i	$-0.534520\pi$
$79$	−152.000	−0.216473	−0.108236	+	0.994125i	$0.534520\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−804.000	−1.06326	−0.531629	−	0.846977i	$-0.678420\pi$
$83$	−804.000	−1.06326	−0.531629	+	0.846977i	$0.678420\pi$
$84$	0	0
$85$	−210.000	−0.267973
$86$	0	0
$87$	−18.0000	−0.0221816
$88$	0	0
$89$	−678.000	−0.807504	−0.403752	−	0.914868i	$-0.632294\pi$
$89$	−678.000	−0.807504	−0.403752	+	0.914868i	$0.632294\pi$
$90$	0	0
$91$	−1088.00	−1.25333
$92$	0	0
$93$	696.000	0.776041
$94$	0	0
$95$	380.000	0.410391
$96$	0	0
$97$	194.000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	194.000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−540.000	−0.548202
$100$	0	0
$101$	−798.000	−0.786178	−0.393089	−	0.919500i	$-0.628594\pi$
$101$	−798.000	−0.786178	−0.393089	+	0.919500i	$0.628594\pi$
$102$	0	0
$103$	−1088.00	−1.04081	−0.520407	−	0.853918i	$-0.674220\pi$
$103$	−1088.00	−1.04081	−0.520407	+	0.853918i	$0.674220\pi$
$104$	0	0
$105$	480.000	0.446126
$106$	0	0
$107$	1716.00	1.55039	0.775196	−	0.631721i	$-0.217651\pi$
$107$	1716.00	1.55039	0.775196	+	0.631721i	$0.217651\pi$
$108$	0	0
$109$	970.000	0.852378	0.426189	−	0.904634i	$-0.359856\pi$
$109$	970.000	0.852378	0.426189	+	0.904634i	$0.359856\pi$
$110$	0	0
$111$	−402.000	−0.343749
$112$	0	0
$113$	426.000	0.354643	0.177322	−	0.984153i	$-0.443257\pi$
$113$	426.000	0.354643	0.177322	+	0.984153i	$0.443257\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	306.000	0.241792
$118$	0	0
$119$	−1344.00	−1.03533
$120$	0	0
$121$	2269.00	1.70473
$122$	0	0
$123$	702.000	0.514611
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−200.000	−0.139741	−0.0698706	−	0.997556i	$-0.522259\pi$
$127$	−200.000	−0.139741	−0.0698706	+	0.997556i	$0.522259\pi$
$128$	0	0
$129$	−1236.00	−0.843595
$130$	0	0
$131$	60.0000	0.0400170	0.0200085	−	0.999800i	$-0.493631\pi$
$131$	60.0000	0.0400170	0.0200085	+	0.999800i	$0.493631\pi$
$132$	0	0
$133$	2432.00	1.58557
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	642.000	0.400363	0.200182	−	0.979759i	$-0.435847\pi$
$137$	642.000	0.400363	0.200182	+	0.979759i	$0.435847\pi$
$138$	0	0
$139$	−2836.00	−1.73055	−0.865275	−	0.501298i	$-0.832856\pi$
$139$	−2836.00	−1.73055	−0.865275	+	0.501298i	$0.832856\pi$
$140$	0	0
$141$	1080.00	0.645053
$142$	0	0
$143$	−2040.00	−1.19296
$144$	0	0
$145$	30.0000	0.0171818
$146$	0	0
$147$	2043.00	1.14628
$148$	0	0
$149$	1554.00	0.854420	0.427210	−	0.904152i	$-0.359496\pi$
$149$	1554.00	0.854420	0.427210	+	0.904152i	$0.359496\pi$
$150$	0	0
$151$	2272.00	1.22446	0.612228	−	0.790682i	$-0.290274\pi$
$151$	2272.00	1.22446	0.612228	+	0.790682i	$0.290274\pi$
$152$	0	0
$153$	378.000	0.199735
$154$	0	0
$155$	−1160.00	−0.601119
$156$	0	0
$157$	−1694.00	−0.861120	−0.430560	−	0.902562i	$-0.641684\pi$
$157$	−1694.00	−0.861120	−0.430560	+	0.902562i	$0.641684\pi$
$158$	0	0
$159$	−666.000	−0.332184
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−52.0000	−0.0249874	−0.0124937	−	0.999922i	$-0.503977\pi$
$163$	−52.0000	−0.0249874	−0.0124937	+	0.999922i	$0.503977\pi$
$164$	0	0
$165$	900.000	0.424636
$166$	0	0
$167$	1200.00	0.556041	0.278020	−	0.960575i	$-0.410322\pi$
$167$	1200.00	0.556041	0.278020	+	0.960575i	$0.410322\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	−684.000	−0.305888
$172$	0	0
$173$	−54.0000	−0.0237315	−0.0118657	−	0.999930i	$-0.503777\pi$
$173$	−54.0000	−0.0237315	−0.0118657	+	0.999930i	$0.503777\pi$
$174$	0	0
$175$	−800.000	−0.345568
$176$	0	0
$177$	1980.00	0.840824
$178$	0	0
$179$	876.000	0.365784	0.182892	−	0.983133i	$-0.441454\pi$
$179$	876.000	0.365784	0.182892	+	0.983133i	$0.441454\pi$
$180$	0	0
$181$	−3854.00	−1.58268	−0.791341	−	0.611375i	$-0.790617\pi$
$181$	−3854.00	−1.58268	−0.791341	+	0.611375i	$0.790617\pi$
$182$	0	0
$183$	1470.00	0.593801
$184$	0	0
$185$	670.000	0.266267
$186$	0	0
$187$	−2520.00	−0.985458
$188$	0	0
$189$	−864.000	−0.332522
$190$	0	0
$191$	2784.00	1.05468	0.527338	−	0.849656i	$-0.323190\pi$
$191$	2784.00	1.05468	0.527338	+	0.849656i	$0.323190\pi$
$192$	0	0
$193$	914.000	0.340887	0.170443	−	0.985367i	$-0.445480\pi$
$193$	914.000	0.340887	0.170443	+	0.985367i	$0.445480\pi$
$194$	0	0
$195$	−510.000	−0.187292
$196$	0	0
$197$	5202.00	1.88136	0.940678	−	0.339300i	$-0.110190\pi$
$197$	5202.00	1.88136	0.940678	+	0.339300i	$0.110190\pi$
$198$	0	0
$199$	−3152.00	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$199$	−3152.00	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$200$	0	0
$201$	2436.00	0.854837
$202$	0	0
$203$	192.000	0.0663830
$204$	0	0
$205$	−1170.00	−0.398616
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4560.00	1.50920
$210$	0	0
$211$	740.000	0.241439	0.120720	−	0.992687i	$-0.461480\pi$
$211$	740.000	0.241439	0.120720	+	0.992687i	$0.461480\pi$
$212$	0	0
$213$	−360.000	−0.115807
$214$	0	0
$215$	2060.00	0.653446
$216$	0	0
$217$	−7424.00	−2.32246
$218$	0	0
$219$	2238.00	0.690548
$220$	0	0
$221$	1428.00	0.434650
$222$	0	0
$223$	520.000	0.156151	0.0780757	−	0.996947i	$-0.475122\pi$
$223$	520.000	0.156151	0.0780757	+	0.996947i	$0.475122\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	396.000	0.115786	0.0578930	−	0.998323i	$-0.481562\pi$
$227$	396.000	0.115786	0.0578930	+	0.998323i	$0.481562\pi$
$228$	0	0
$229$	1330.00	0.383794	0.191897	−	0.981415i	$-0.438536\pi$
$229$	1330.00	0.383794	0.191897	+	0.981415i	$0.438536\pi$
$230$	0	0
$231$	5760.00	1.64061
$232$	0	0
$233$	4866.00	1.36816	0.684082	−	0.729405i	$-0.260203\pi$
$233$	4866.00	1.36816	0.684082	+	0.729405i	$0.260203\pi$
$234$	0	0
$235$	−1800.00	−0.499656
$236$	0	0
$237$	−456.000	−0.124981
$238$	0	0
$239$	1824.00	0.493660	0.246830	−	0.969059i	$-0.420611\pi$
$239$	1824.00	0.493660	0.246830	+	0.969059i	$0.420611\pi$
$240$	0	0
$241$	6482.00	1.73254	0.866270	−	0.499575i	$-0.166511\pi$
$241$	6482.00	1.73254	0.866270	+	0.499575i	$0.166511\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−3405.00	−0.887908
$246$	0	0
$247$	−2584.00	−0.665652
$248$	0	0
$249$	−2412.00	−0.613873
$250$	0	0
$251$	1476.00	0.371172	0.185586	−	0.982628i	$-0.440582\pi$
$251$	1476.00	0.371172	0.185586	+	0.982628i	$0.440582\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−630.000	−0.154714
$256$	0	0
$257$	4314.00	1.04708	0.523541	−	0.852001i	$-0.324611\pi$
$257$	4314.00	1.04708	0.523541	+	0.852001i	$0.324611\pi$
$258$	0	0
$259$	4288.00	1.02874
$260$	0	0
$261$	−54.0000	−0.0128066
$262$	0	0
$263$	5280.00	1.23794	0.618971	−	0.785414i	$-0.287550\pi$
$263$	5280.00	1.23794	0.618971	+	0.785414i	$0.287550\pi$
$264$	0	0
$265$	1110.00	0.257309
$266$	0	0
$267$	−2034.00	−0.466213
$268$	0	0
$269$	−5526.00	−1.25251	−0.626257	−	0.779617i	$-0.715414\pi$
$269$	−5526.00	−1.25251	−0.626257	+	0.779617i	$0.715414\pi$
$270$	0	0
$271$	−2024.00	−0.453687	−0.226844	−	0.973931i	$-0.572841\pi$
$271$	−2024.00	−0.453687	−0.226844	+	0.973931i	$0.572841\pi$
$272$	0	0
$273$	−3264.00	−0.723613
$274$	0	0
$275$	−1500.00	−0.328921
$276$	0	0
$277$	−2054.00	−0.445534	−0.222767	−	0.974872i	$-0.571509\pi$
$277$	−2054.00	−0.445534	−0.222767	+	0.974872i	$0.571509\pi$
$278$	0	0
$279$	2088.00	0.448048
$280$	0	0
$281$	−7302.00	−1.55018	−0.775090	−	0.631850i	$-0.782296\pi$
$281$	−7302.00	−1.55018	−0.775090	+	0.631850i	$0.782296\pi$
$282$	0	0
$283$	−3724.00	−0.782222	−0.391111	−	0.920344i	$-0.627909\pi$
$283$	−3724.00	−0.782222	−0.391111	+	0.920344i	$0.627909\pi$
$284$	0	0
$285$	1140.00	0.236940
$286$	0	0
$287$	−7488.00	−1.54008
$288$	0	0
$289$	−3149.00	−0.640953
$290$	0	0
$291$	582.000	0.117242
$292$	0	0
$293$	7218.00	1.43918	0.719591	−	0.694399i	$-0.244330\pi$
$293$	7218.00	1.43918	0.719591	+	0.694399i	$0.244330\pi$
$294$	0	0
$295$	−3300.00	−0.651300
$296$	0	0
$297$	−1620.00	−0.316505
$298$	0	0
$299$	0	0
$300$	0	0
$301$	13184.0	2.52463
$302$	0	0
$303$	−2394.00	−0.453900
$304$	0	0
$305$	−2450.00	−0.459956
$306$	0	0
$307$	2540.00	0.472200	0.236100	−	0.971729i	$-0.424131\pi$
$307$	2540.00	0.472200	0.236100	+	0.971729i	$0.424131\pi$
$308$	0	0
$309$	−3264.00	−0.600914
$310$	0	0
$311$	−1560.00	−0.284436	−0.142218	−	0.989835i	$-0.545423\pi$
$311$	−1560.00	−0.284436	−0.142218	+	0.989835i	$0.545423\pi$
$312$	0	0
$313$	−934.000	−0.168667	−0.0843335	−	0.996438i	$-0.526876\pi$
$313$	−934.000	−0.168667	−0.0843335	+	0.996438i	$0.526876\pi$
$314$	0	0
$315$	1440.00	0.257571
$316$	0	0
$317$	1674.00	0.296597	0.148298	−	0.988943i	$-0.452620\pi$
$317$	1674.00	0.296597	0.148298	+	0.988943i	$0.452620\pi$
$318$	0	0
$319$	360.000	0.0631854
$320$	0	0
$321$	5148.00	0.895119
$322$	0	0
$323$	−3192.00	−0.549869
$324$	0	0
$325$	850.000	0.145075
$326$	0	0
$327$	2910.00	0.492120
$328$	0	0
$329$	−11520.0	−1.93045
$330$	0	0
$331$	−3988.00	−0.662237	−0.331118	−	0.943589i	$-0.607426\pi$
$331$	−3988.00	−0.662237	−0.331118	+	0.943589i	$0.607426\pi$
$332$	0	0
$333$	−1206.00	−0.198464
$334$	0	0
$335$	−4060.00	−0.662154
$336$	0	0
$337$	2.00000	0.000323285 0	0.000161642	−	1.00000i	$-0.499949\pi$
$337$	2.00000	0.000323285 0	0.000161642		1.00000i	$0.499949\pi$
$338$	0	0
$339$	1278.00	0.204753
$340$	0	0
$341$	−13920.0	−2.21059
$342$	0	0
$343$	−10816.0	−1.70265
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1764.00	0.272901	0.136450	−	0.990647i	$-0.456431\pi$
$347$	1764.00	0.272901	0.136450	+	0.990647i	$0.456431\pi$
$348$	0	0
$349$	−4310.00	−0.661057	−0.330529	−	0.943796i	$-0.607227\pi$
$349$	−4310.00	−0.661057	−0.330529	+	0.943796i	$0.607227\pi$
$350$	0	0
$351$	918.000	0.139599
$352$	0	0
$353$	138.000	0.0208074	0.0104037	−	0.999946i	$-0.496688\pi$
$353$	138.000	0.0208074	0.0104037	+	0.999946i	$0.496688\pi$
$354$	0	0
$355$	600.000	0.0897034
$356$	0	0
$357$	−4032.00	−0.597748
$358$	0	0
$359$	11976.0	1.76064	0.880319	−	0.474382i	$-0.157328\pi$
$359$	11976.0	1.76064	0.880319	+	0.474382i	$0.157328\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	0	0
$363$	6807.00	0.984228
$364$	0	0
$365$	−3730.00	−0.534896
$366$	0	0
$367$	−9704.00	−1.38023	−0.690115	−	0.723699i	$-0.742440\pi$
$367$	−9704.00	−1.38023	−0.690115	+	0.723699i	$0.742440\pi$
$368$	0	0
$369$	2106.00	0.297111
$370$	0	0
$371$	7104.00	0.994128
$372$	0	0
$373$	8122.00	1.12746	0.563728	−	0.825960i	$-0.309367\pi$
$373$	8122.00	1.12746	0.563728	+	0.825960i	$0.309367\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−204.000	−0.0278688
$378$	0	0
$379$	3404.00	0.461350	0.230675	−	0.973031i	$-0.425907\pi$
$379$	3404.00	0.461350	0.230675	+	0.973031i	$0.425907\pi$
$380$	0	0
$381$	−600.000	−0.0806796
$382$	0	0
$383$	2520.00	0.336204	0.168102	−	0.985770i	$-0.446236\pi$
$383$	2520.00	0.336204	0.168102	+	0.985770i	$0.446236\pi$
$384$	0	0
$385$	−9600.00	−1.27081
$386$	0	0
$387$	−3708.00	−0.487050
$388$	0	0
$389$	−1566.00	−0.204111	−0.102056	−	0.994779i	$-0.532542\pi$
$389$	−1566.00	−0.204111	−0.102056	+	0.994779i	$0.532542\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	180.000	0.0231038
$394$	0	0
$395$	760.000	0.0968095
$396$	0	0
$397$	4354.00	0.550431	0.275215	−	0.961383i	$-0.411251\pi$
$397$	4354.00	0.550431	0.275215	+	0.961383i	$0.411251\pi$
$398$	0	0
$399$	7296.00	0.915431
$400$	0	0
$401$	−8046.00	−1.00199	−0.500995	−	0.865450i	$-0.667033\pi$
$401$	−8046.00	−1.00199	−0.500995	+	0.865450i	$0.667033\pi$
$402$	0	0
$403$	7888.00	0.975011
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	8040.00	0.979184
$408$	0	0
$409$	−2806.00	−0.339237	−0.169618	−	0.985510i	$-0.554253\pi$
$409$	−2806.00	−0.339237	−0.169618	+	0.985510i	$0.554253\pi$
$410$	0	0
$411$	1926.00	0.231150
$412$	0	0
$413$	−21120.0	−2.51634
$414$	0	0
$415$	4020.00	0.475504
$416$	0	0
$417$	−8508.00	−0.999133
$418$	0	0
$419$	11580.0	1.35017	0.675084	−	0.737741i	$-0.264108\pi$
$419$	11580.0	1.35017	0.675084	+	0.737741i	$0.264108\pi$
$420$	0	0
$421$	370.000	0.0428330	0.0214165	−	0.999771i	$-0.493182\pi$
$421$	370.000	0.0428330	0.0214165	+	0.999771i	$0.493182\pi$
$422$	0	0
$423$	3240.00	0.372421
$424$	0	0
$425$	1050.00	0.119841
$426$	0	0
$427$	−15680.0	−1.77707
$428$	0	0
$429$	−6120.00	−0.688756
$430$	0	0
$431$	−5040.00	−0.563267	−0.281634	−	0.959522i	$-0.590876\pi$
$431$	−5040.00	−0.563267	−0.281634	+	0.959522i	$0.590876\pi$
$432$	0	0
$433$	−3742.00	−0.415310	−0.207655	−	0.978202i	$-0.566583\pi$
$433$	−3742.00	−0.415310	−0.207655	+	0.978202i	$0.566583\pi$
$434$	0	0
$435$	90.0000	0.00991993
$436$	0	0
$437$	0	0
$438$	0	0
$439$	6208.00	0.674924	0.337462	−	0.941339i	$-0.390432\pi$
$439$	6208.00	0.674924	0.337462	+	0.941339i	$0.390432\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	0	0
$443$	−15564.0	−1.66923	−0.834614	−	0.550835i	$-0.814309\pi$
$443$	−15564.0	−1.66923	−0.834614	+	0.550835i	$0.814309\pi$
$444$	0	0
$445$	3390.00	0.361127
$446$	0	0
$447$	4662.00	0.493300
$448$	0	0
$449$	−15774.0	−1.65795	−0.828977	−	0.559283i	$-0.811076\pi$
$449$	−15774.0	−1.65795	−0.828977	+	0.559283i	$0.811076\pi$
$450$	0	0
$451$	−14040.0	−1.46589
$452$	0	0
$453$	6816.00	0.706940
$454$	0	0
$455$	5440.00	0.560508
$456$	0	0
$457$	9722.00	0.995133	0.497567	−	0.867426i	$-0.334227\pi$
$457$	9722.00	0.995133	0.497567	+	0.867426i	$0.334227\pi$
$458$	0	0
$459$	1134.00	0.115317
$460$	0	0
$461$	10890.0	1.10021	0.550106	−	0.835095i	$-0.314587\pi$
$461$	10890.0	1.10021	0.550106	+	0.835095i	$0.314587\pi$
$462$	0	0
$463$	−15128.0	−1.51848	−0.759242	−	0.650809i	$-0.774430\pi$
$463$	−15128.0	−1.51848	−0.759242	+	0.650809i	$0.774430\pi$
$464$	0	0
$465$	−3480.00	−0.347056
$466$	0	0
$467$	10668.0	1.05708	0.528540	−	0.848909i	$-0.322740\pi$
$467$	10668.0	1.05708	0.528540	+	0.848909i	$0.322740\pi$
$468$	0	0
$469$	−25984.0	−2.55827
$470$	0	0
$471$	−5082.00	−0.497168
$472$	0	0
$473$	24720.0	2.40302
$474$	0	0
$475$	−1900.00	−0.183533
$476$	0	0
$477$	−1998.00	−0.191786
$478$	0	0
$479$	−15264.0	−1.45601	−0.728006	−	0.685571i	$-0.759553\pi$
$479$	−15264.0	−1.45601	−0.728006	+	0.685571i	$0.759553\pi$
$480$	0	0
$481$	−4556.00	−0.431883
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−970.000	−0.0908153
$486$	0	0
$487$	5776.00	0.537445	0.268722	−	0.963218i	$-0.413399\pi$
$487$	5776.00	0.537445	0.268722	+	0.963218i	$0.413399\pi$
$488$	0	0
$489$	−156.000	−0.0144265
$490$	0	0
$491$	14244.0	1.30921	0.654606	−	0.755971i	$-0.272835\pi$
$491$	14244.0	1.30921	0.654606	+	0.755971i	$0.272835\pi$
$492$	0	0
$493$	−252.000	−0.0230213
$494$	0	0
$495$	2700.00	0.245164
$496$	0	0
$497$	3840.00	0.346575
$498$	0	0
$499$	−17116.0	−1.53551	−0.767753	−	0.640746i	$-0.778625\pi$
$499$	−17116.0	−1.53551	−0.767753	+	0.640746i	$0.778625\pi$
$500$	0	0
$501$	3600.00	0.321030
$502$	0	0
$503$	16848.0	1.49347	0.746735	−	0.665122i	$-0.231620\pi$
$503$	16848.0	1.49347	0.746735	+	0.665122i	$0.231620\pi$
$504$	0	0
$505$	3990.00	0.351589
$506$	0	0
$507$	−3123.00	−0.273565
$508$	0	0
$509$	3834.00	0.333868	0.166934	−	0.985968i	$-0.446613\pi$
$509$	3834.00	0.333868	0.166934	+	0.985968i	$0.446613\pi$
$510$	0	0
$511$	−23872.0	−2.06660
$512$	0	0
$513$	−2052.00	−0.176604
$514$	0	0
$515$	5440.00	0.465466
$516$	0	0
$517$	−21600.0	−1.83746
$518$	0	0
$519$	−162.000	−0.0137014
$520$	0	0
$521$	−18822.0	−1.58274	−0.791369	−	0.611338i	$-0.790631\pi$
$521$	−18822.0	−1.58274	−0.791369	+	0.611338i	$0.790631\pi$
$522$	0	0
$523$	−15340.0	−1.28255	−0.641273	−	0.767313i	$-0.721593\pi$
$523$	−15340.0	−1.28255	−0.641273	+	0.767313i	$0.721593\pi$
$524$	0	0
$525$	−2400.00	−0.199513
$526$	0	0
$527$	9744.00	0.805418
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	5940.00	0.485450
$532$	0	0
$533$	7956.00	0.646553
$534$	0	0
$535$	−8580.00	−0.693357
$536$	0	0
$537$	2628.00	0.211185
$538$	0	0
$539$	−40860.0	−3.26524
$540$	0	0
$541$	−18950.0	−1.50596	−0.752980	−	0.658044i	$-0.771384\pi$
$541$	−18950.0	−1.50596	−0.752980	+	0.658044i	$0.771384\pi$
$542$	0	0
$543$	−11562.0	−0.913762
$544$	0	0
$545$	−4850.00	−0.381195
$546$	0	0
$547$	−10036.0	−0.784476	−0.392238	−	0.919864i	$-0.628299\pi$
$547$	−10036.0	−0.784476	−0.392238	+	0.919864i	$0.628299\pi$
$548$	0	0
$549$	4410.00	0.342831
$550$	0	0
$551$	456.000	0.0352564
$552$	0	0
$553$	4864.00	0.374030
$554$	0	0
$555$	2010.00	0.153729
$556$	0	0
$557$	−10326.0	−0.785506	−0.392753	−	0.919644i	$-0.628477\pi$
$557$	−10326.0	−0.785506	−0.392753	+	0.919644i	$0.628477\pi$
$558$	0	0
$559$	−14008.0	−1.05988
$560$	0	0
$561$	−7560.00	−0.568954
$562$	0	0
$563$	4524.00	0.338657	0.169328	−	0.985560i	$-0.445840\pi$
$563$	4524.00	0.338657	0.169328	+	0.985560i	$0.445840\pi$
$564$	0	0
$565$	−2130.00	−0.158601
$566$	0	0
$567$	−2592.00	−0.191982
$568$	0	0
$569$	16362.0	1.20550	0.602751	−	0.797929i	$-0.294071\pi$
$569$	16362.0	1.20550	0.602751	+	0.797929i	$0.294071\pi$
$570$	0	0
$571$	6620.00	0.485181	0.242591	−	0.970129i	$-0.422003\pi$
$571$	6620.00	0.485181	0.242591	+	0.970129i	$0.422003\pi$
$572$	0	0
$573$	8352.00	0.608918
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8834.00	0.637373	0.318687	−	0.947860i	$-0.396758\pi$
$577$	8834.00	0.637373	0.318687	+	0.947860i	$0.396758\pi$
$578$	0	0
$579$	2742.00	0.196811
$580$	0	0
$581$	25728.0	1.83714
$582$	0	0
$583$	13320.0	0.946240
$584$	0	0
$585$	−1530.00	−0.108133
$586$	0	0
$587$	3636.00	0.255662	0.127831	−	0.991796i	$-0.459198\pi$
$587$	3636.00	0.255662	0.127831	+	0.991796i	$0.459198\pi$
$588$	0	0
$589$	−17632.0	−1.23347
$590$	0	0
$591$	15606.0	1.08620
$592$	0	0
$593$	6570.00	0.454971	0.227485	−	0.973782i	$-0.426950\pi$
$593$	6570.00	0.454971	0.227485	+	0.973782i	$0.426950\pi$
$594$	0	0
$595$	6720.00	0.463014
$596$	0	0
$597$	−9456.00	−0.648255
$598$	0	0
$599$	−16584.0	−1.13123	−0.565613	−	0.824671i	$-0.691360\pi$
$599$	−16584.0	−1.13123	−0.565613	+	0.824671i	$0.691360\pi$
$600$	0	0
$601$	−502.000	−0.0340716	−0.0170358	−	0.999855i	$-0.505423\pi$
$601$	−502.000	−0.0340716	−0.0170358	+	0.999855i	$0.505423\pi$
$602$	0	0
$603$	7308.00	0.493540
$604$	0	0
$605$	−11345.0	−0.762380
$606$	0	0
$607$	18568.0	1.24160	0.620801	−	0.783969i	$-0.286808\pi$
$607$	18568.0	1.24160	0.620801	+	0.783969i	$0.286808\pi$
$608$	0	0
$609$	576.000	0.0383263
$610$	0	0
$611$	12240.0	0.810438
$612$	0	0
$613$	13114.0	0.864061	0.432031	−	0.901859i	$-0.357797\pi$
$613$	13114.0	0.864061	0.432031	+	0.901859i	$0.357797\pi$
$614$	0	0
$615$	−3510.00	−0.230141
$616$	0	0
$617$	5250.00	0.342556	0.171278	−	0.985223i	$-0.445210\pi$
$617$	5250.00	0.342556	0.171278	+	0.985223i	$0.445210\pi$
$618$	0	0
$619$	−10804.0	−0.701534	−0.350767	−	0.936463i	$-0.614079\pi$
$619$	−10804.0	−0.701534	−0.350767	+	0.936463i	$0.614079\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	21696.0	1.39524
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	13680.0	0.871334
$628$	0	0
$629$	−5628.00	−0.356762
$630$	0	0
$631$	27088.0	1.70896	0.854482	−	0.519481i	$-0.173875\pi$
$631$	27088.0	1.70896	0.854482	+	0.519481i	$0.173875\pi$
$632$	0	0
$633$	2220.00	0.139395
$634$	0	0
$635$	1000.00	0.0624942
$636$	0	0
$637$	23154.0	1.44018
$638$	0	0
$639$	−1080.00	−0.0668609
$640$	0	0
$641$	18930.0	1.16644	0.583222	−	0.812313i	$-0.301792\pi$
$641$	18930.0	1.16644	0.583222	+	0.812313i	$0.301792\pi$
$642$	0	0
$643$	20108.0	1.23325	0.616627	−	0.787256i	$-0.288499\pi$
$643$	20108.0	1.23325	0.616627	+	0.787256i	$0.288499\pi$
$644$	0	0
$645$	6180.00	0.377267
$646$	0	0
$647$	7152.00	0.434581	0.217291	−	0.976107i	$-0.430278\pi$
$647$	7152.00	0.434581	0.217291	+	0.976107i	$0.430278\pi$
$648$	0	0
$649$	−39600.0	−2.39512
$650$	0	0
$651$	−22272.0	−1.34087
$652$	0	0
$653$	31626.0	1.89528	0.947642	−	0.319333i	$-0.103459\pi$
$653$	31626.0	1.89528	0.947642	+	0.319333i	$0.103459\pi$
$654$	0	0
$655$	−300.000	−0.0178961
$656$	0	0
$657$	6714.00	0.398688
$658$	0	0
$659$	28092.0	1.66056	0.830280	−	0.557347i	$-0.188181\pi$
$659$	28092.0	1.66056	0.830280	+	0.557347i	$0.188181\pi$
$660$	0	0
$661$	13186.0	0.775909	0.387955	−	0.921678i	$-0.373182\pi$
$661$	13186.0	0.775909	0.387955	+	0.921678i	$0.373182\pi$
$662$	0	0
$663$	4284.00	0.250945
$664$	0	0
$665$	−12160.0	−0.709090
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1560.00	0.0901541
$670$	0	0
$671$	−29400.0	−1.69147
$672$	0	0
$673$	5138.00	0.294287	0.147144	−	0.989115i	$-0.452992\pi$
$673$	5138.00	0.294287	0.147144	+	0.989115i	$0.452992\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−6078.00	−0.345047	−0.172523	−	0.985005i	$-0.555192\pi$
$677$	−6078.00	−0.345047	−0.172523	+	0.985005i	$0.555192\pi$
$678$	0	0
$679$	−6208.00	−0.350871
$680$	0	0
$681$	1188.00	0.0668491
$682$	0	0
$683$	32244.0	1.80642	0.903208	−	0.429203i	$-0.141205\pi$
$683$	32244.0	1.80642	0.903208	+	0.429203i	$0.141205\pi$
$684$	0	0
$685$	−3210.00	−0.179048
$686$	0	0
$687$	3990.00	0.221584
$688$	0	0
$689$	−7548.00	−0.417353
$690$	0	0
$691$	4484.00	0.246859	0.123429	−	0.992353i	$-0.460611\pi$
$691$	4484.00	0.246859	0.123429	+	0.992353i	$0.460611\pi$
$692$	0	0
$693$	17280.0	0.947205
$694$	0	0
$695$	14180.0	0.773925
$696$	0	0
$697$	9828.00	0.534092
$698$	0	0
$699$	14598.0	0.789910
$700$	0	0
$701$	30426.0	1.63934	0.819668	−	0.572839i	$-0.194158\pi$
$701$	30426.0	1.63934	0.819668	+	0.572839i	$0.194158\pi$
$702$	0	0
$703$	10184.0	0.546368
$704$	0	0
$705$	−5400.00	−0.288476
$706$	0	0
$707$	25536.0	1.35839
$708$	0	0
$709$	−13262.0	−0.702489	−0.351245	−	0.936284i	$-0.614241\pi$
$709$	−13262.0	−0.702489	−0.351245	+	0.936284i	$0.614241\pi$
$710$	0	0
$711$	−1368.00	−0.0721575
$712$	0	0
$713$	0	0
$714$	0	0
$715$	10200.0	0.533508
$716$	0	0
$717$	5472.00	0.285015
$718$	0	0
$719$	−13920.0	−0.722014	−0.361007	−	0.932563i	$-0.617567\pi$
$719$	−13920.0	−0.722014	−0.361007	+	0.932563i	$0.617567\pi$
$720$	0	0
$721$	34816.0	1.79836
$722$	0	0
$723$	19446.0	1.00028
$724$	0	0
$725$	−150.000	−0.00768395
$726$	0	0
$727$	9376.00	0.478317	0.239159	−	0.970981i	$-0.423128\pi$
$727$	9376.00	0.478317	0.239159	+	0.970981i	$0.423128\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−17304.0	−0.875529
$732$	0	0
$733$	−6014.00	−0.303045	−0.151523	−	0.988454i	$-0.548418\pi$
$733$	−6014.00	−0.303045	−0.151523	+	0.988454i	$0.548418\pi$
$734$	0	0
$735$	−10215.0	−0.512634
$736$	0	0
$737$	−48720.0	−2.43504
$738$	0	0
$739$	−7468.00	−0.371739	−0.185869	−	0.982574i	$-0.559510\pi$
$739$	−7468.00	−0.371739	−0.185869	+	0.982574i	$0.559510\pi$
$740$	0	0
$741$	−7752.00	−0.384314
$742$	0	0
$743$	−31248.0	−1.54290	−0.771452	−	0.636287i	$-0.780469\pi$
$743$	−31248.0	−1.54290	−0.771452	+	0.636287i	$0.780469\pi$
$744$	0	0
$745$	−7770.00	−0.382108
$746$	0	0
$747$	−7236.00	−0.354420
$748$	0	0
$749$	−54912.0	−2.67883
$750$	0	0
$751$	−32840.0	−1.59567	−0.797835	−	0.602875i	$-0.794022\pi$
$751$	−32840.0	−1.59567	−0.797835	+	0.602875i	$0.794022\pi$
$752$	0	0
$753$	4428.00	0.214297
$754$	0	0
$755$	−11360.0	−0.547593
$756$	0	0
$757$	19066.0	0.915410	0.457705	−	0.889104i	$-0.348672\pi$
$757$	19066.0	0.915410	0.457705	+	0.889104i	$0.348672\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6858.00	0.326678	0.163339	−	0.986570i	$-0.447773\pi$
$761$	6858.00	0.326678	0.163339	+	0.986570i	$0.447773\pi$
$762$	0	0
$763$	−31040.0	−1.47277
$764$	0	0
$765$	−1890.00	−0.0893243
$766$	0	0
$767$	22440.0	1.05640
$768$	0	0
$769$	22178.0	1.04000	0.519999	−	0.854167i	$-0.325932\pi$
$769$	22178.0	1.04000	0.519999	+	0.854167i	$0.325932\pi$
$770$	0	0
$771$	12942.0	0.604533
$772$	0	0
$773$	−14286.0	−0.664724	−0.332362	−	0.943152i	$-0.607846\pi$
$773$	−14286.0	−0.664724	−0.332362	+	0.943152i	$0.607846\pi$
$774$	0	0
$775$	5800.00	0.268829
$776$	0	0
$777$	12864.0	0.593943
$778$	0	0
$779$	−17784.0	−0.817943
$780$	0	0
$781$	7200.00	0.329880
$782$	0	0
$783$	−162.000	−0.00739388
$784$	0	0
$785$	8470.00	0.385105
$786$	0	0
$787$	−18868.0	−0.854602	−0.427301	−	0.904109i	$-0.640535\pi$
$787$	−18868.0	−0.854602	−0.427301	+	0.904109i	$0.640535\pi$
$788$	0	0
$789$	15840.0	0.714726
$790$	0	0
$791$	−13632.0	−0.612766
$792$	0	0
$793$	16660.0	0.746045
$794$	0	0
$795$	3330.00	0.148557
$796$	0	0
$797$	21690.0	0.963989	0.481994	−	0.876174i	$-0.339913\pi$
$797$	21690.0	0.963989	0.481994	+	0.876174i	$0.339913\pi$
$798$	0	0
$799$	15120.0	0.669471
$800$	0	0
$801$	−6102.00	−0.269168
$802$	0	0
$803$	−44760.0	−1.96706
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16578.0	−0.723139
$808$	0	0
$809$	−24726.0	−1.07456	−0.537281	−	0.843404i	$-0.680548\pi$
$809$	−24726.0	−1.07456	−0.537281	+	0.843404i	$0.680548\pi$
$810$	0	0
$811$	−2644.00	−0.114480	−0.0572401	−	0.998360i	$-0.518230\pi$
$811$	−2644.00	−0.114480	−0.0572401	+	0.998360i	$0.518230\pi$
$812$	0	0
$813$	−6072.00	−0.261936
$814$	0	0
$815$	260.000	0.0111747
$816$	0	0
$817$	31312.0	1.34084
$818$	0	0
$819$	−9792.00	−0.417778
$820$	0	0
$821$	37842.0	1.60864	0.804321	−	0.594195i	$-0.202529\pi$
$821$	37842.0	1.60864	0.804321	+	0.594195i	$0.202529\pi$
$822$	0	0
$823$	880.000	0.0372720	0.0186360	−	0.999826i	$-0.494068\pi$
$823$	880.000	0.0372720	0.0186360	+	0.999826i	$0.494068\pi$
$824$	0	0
$825$	−4500.00	−0.189903
$826$	0	0
$827$	−12876.0	−0.541406	−0.270703	−	0.962663i	$-0.587256\pi$
$827$	−12876.0	−0.541406	−0.270703	+	0.962663i	$0.587256\pi$
$828$	0	0
$829$	25498.0	1.06825	0.534127	−	0.845404i	$-0.320641\pi$
$829$	25498.0	1.06825	0.534127	+	0.845404i	$0.320641\pi$
$830$	0	0
$831$	−6162.00	−0.257229
$832$	0	0
$833$	28602.0	1.18968
$834$	0	0
$835$	−6000.00	−0.248669
$836$	0	0
$837$	6264.00	0.258680
$838$	0	0
$839$	40584.0	1.66998	0.834991	−	0.550263i	$-0.185473\pi$
$839$	40584.0	1.66998	0.834991	+	0.550263i	$0.185473\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	−21906.0	−0.894997
$844$	0	0
$845$	5205.00	0.211902
$846$	0	0
$847$	−72608.0	−2.94550
$848$	0	0
$849$	−11172.0	−0.451616
$850$	0	0
$851$	0	0
$852$	0	0
$853$	25738.0	1.03312	0.516561	−	0.856251i	$-0.327212\pi$
$853$	25738.0	1.03312	0.516561	+	0.856251i	$0.327212\pi$
$854$	0	0
$855$	3420.00	0.136797
$856$	0	0
$857$	13314.0	0.530686	0.265343	−	0.964154i	$-0.414515\pi$
$857$	13314.0	0.530686	0.265343	+	0.964154i	$0.414515\pi$
$858$	0	0
$859$	24524.0	0.974096	0.487048	−	0.873375i	$-0.338074\pi$
$859$	24524.0	0.974096	0.487048	+	0.873375i	$0.338074\pi$
$860$	0	0
$861$	−22464.0	−0.889165
$862$	0	0
$863$	−5592.00	−0.220572	−0.110286	−	0.993900i	$-0.535177\pi$
$863$	−5592.00	−0.220572	−0.110286	+	0.993900i	$0.535177\pi$
$864$	0	0
$865$	270.000	0.0106130
$866$	0	0
$867$	−9447.00	−0.370054
$868$	0	0
$869$	9120.00	0.356012
$870$	0	0
$871$	27608.0	1.07401
$872$	0	0
$873$	1746.00	0.0676897
$874$	0	0
$875$	4000.00	0.154542
$876$	0	0
$877$	14386.0	0.553912	0.276956	−	0.960883i	$-0.410674\pi$
$877$	14386.0	0.553912	0.276956	+	0.960883i	$0.410674\pi$
$878$	0	0
$879$	21654.0	0.830912
$880$	0	0
$881$	47106.0	1.80141	0.900705	−	0.434432i	$-0.143051\pi$
$881$	47106.0	1.80141	0.900705	+	0.434432i	$0.143051\pi$
$882$	0	0
$883$	51548.0	1.96458	0.982292	−	0.187354i	$-0.0599913\pi$
$883$	51548.0	1.96458	0.982292	+	0.187354i	$0.0599913\pi$
$884$	0	0
$885$	−9900.00	−0.376028
$886$	0	0
$887$	−34080.0	−1.29007	−0.645036	−	0.764152i	$-0.723158\pi$
$887$	−34080.0	−1.29007	−0.645036	+	0.764152i	$0.723158\pi$
$888$	0	0
$889$	6400.00	0.241450
$890$	0	0
$891$	−4860.00	−0.182734
$892$	0	0
$893$	−27360.0	−1.02527
$894$	0	0
$895$	−4380.00	−0.163584
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1392.00	−0.0516416
$900$	0	0
$901$	−9324.00	−0.344759
$902$	0	0
$903$	39552.0	1.45759
$904$	0	0
$905$	19270.0	0.707797
$906$	0	0
$907$	25748.0	0.942611	0.471306	−	0.881970i	$-0.343783\pi$
$907$	25748.0	0.942611	0.471306	+	0.881970i	$0.343783\pi$
$908$	0	0
$909$	−7182.00	−0.262059
$910$	0	0
$911$	24768.0	0.900769	0.450384	−	0.892835i	$-0.351287\pi$
$911$	24768.0	0.900769	0.450384	+	0.892835i	$0.351287\pi$
$912$	0	0
$913$	48240.0	1.74864
$914$	0	0
$915$	−7350.00	−0.265556
$916$	0	0
$917$	−1920.00	−0.0691428
$918$	0	0
$919$	31264.0	1.12220	0.561101	−	0.827747i	$-0.310378\pi$
$919$	31264.0	1.12220	0.561101	+	0.827747i	$0.310378\pi$
$920$	0	0
$921$	7620.00	0.272625
$922$	0	0
$923$	−4080.00	−0.145498
$924$	0	0
$925$	−3350.00	−0.119078
$926$	0	0
$927$	−9792.00	−0.346938
$928$	0	0
$929$	−6174.00	−0.218043	−0.109022	−	0.994039i	$-0.534772\pi$
$929$	−6174.00	−0.218043	−0.109022	+	0.994039i	$0.534772\pi$
$930$	0	0
$931$	−51756.0	−1.82195
$932$	0	0
$933$	−4680.00	−0.164219
$934$	0	0
$935$	12600.0	0.440710
$936$	0	0
$937$	28922.0	1.00837	0.504184	−	0.863596i	$-0.331793\pi$
$937$	28922.0	1.00837	0.504184	+	0.863596i	$0.331793\pi$
$938$	0	0
$939$	−2802.00	−0.0973800
$940$	0	0
$941$	−29238.0	−1.01289	−0.506446	−	0.862272i	$-0.669041\pi$
$941$	−29238.0	−1.01289	−0.506446	+	0.862272i	$0.669041\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	4320.00	0.148709
$946$	0	0
$947$	−2868.00	−0.0984134	−0.0492067	−	0.998789i	$-0.515669\pi$
$947$	−2868.00	−0.0984134	−0.0492067	+	0.998789i	$0.515669\pi$
$948$	0	0
$949$	25364.0	0.867598
$950$	0	0
$951$	5022.00	0.171240
$952$	0	0
$953$	24018.0	0.816390	0.408195	−	0.912895i	$-0.366158\pi$
$953$	24018.0	0.816390	0.408195	+	0.912895i	$0.366158\pi$
$954$	0	0
$955$	−13920.0	−0.471666
$956$	0	0
$957$	1080.00	0.0364801
$958$	0	0
$959$	−20544.0	−0.691763
$960$	0	0
$961$	24033.0	0.806720
$962$	0	0
$963$	15444.0	0.516797
$964$	0	0
$965$	−4570.00	−0.152449
$966$	0	0
$967$	−25712.0	−0.855059	−0.427530	−	0.904001i	$-0.640616\pi$
$967$	−25712.0	−0.855059	−0.427530	+	0.904001i	$0.640616\pi$
$968$	0	0
$969$	−9576.00	−0.317467
$970$	0	0
$971$	−12396.0	−0.409688	−0.204844	−	0.978795i	$-0.565669\pi$
$971$	−12396.0	−0.409688	−0.204844	+	0.978795i	$0.565669\pi$
$972$	0	0
$973$	90752.0	2.99011
$974$	0	0
$975$	2550.00	0.0837593
$976$	0	0
$977$	−46614.0	−1.52642	−0.763211	−	0.646150i	$-0.776378\pi$
$977$	−46614.0	−1.52642	−0.763211	+	0.646150i	$0.776378\pi$
$978$	0	0
$979$	40680.0	1.32803
$980$	0	0
$981$	8730.00	0.284126
$982$	0	0
$983$	672.000	0.0218041	0.0109021	−	0.999941i	$-0.496530\pi$
$983$	672.000	0.0218041	0.0109021	+	0.999941i	$0.496530\pi$
$984$	0	0
$985$	−26010.0	−0.841368
$986$	0	0
$987$	−34560.0	−1.11455
$988$	0	0
$989$	0	0
$990$	0	0
$991$	38776.0	1.24295	0.621473	−	0.783435i	$-0.286534\pi$
$991$	38776.0	1.24295	0.621473	+	0.783435i	$0.286534\pi$
$992$	0	0
$993$	−11964.0	−0.382342
$994$	0	0
$995$	15760.0	0.502136
$996$	0	0
$997$	−30422.0	−0.966374	−0.483187	−	0.875517i	$-0.660521\pi$
$997$	−30422.0	−0.966374	−0.483187	+	0.875517i	$0.660521\pi$
$998$	0	0
$999$	−3618.00	−0.114583

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.a.s.1.1		1
4.3	odd	2		960.4.a.j.1.1		1
8.3	odd	2		30.4.a.a.1.1	✓	1
8.5	even	2		240.4.a.c.1.1		1
24.5	odd	2		720.4.a.b.1.1		1
24.11	even	2		90.4.a.d.1.1		1
40.3	even	4		150.4.c.a.49.2		2
40.13	odd	4		1200.4.f.u.49.1		2
40.19	odd	2		150.4.a.e.1.1		1
40.27	even	4		150.4.c.a.49.1		2
40.29	even	2		1200.4.a.bk.1.1		1
40.37	odd	4		1200.4.f.u.49.2		2
56.27	even	2		1470.4.a.a.1.1		1
72.11	even	6		810.4.e.e.271.1		2
72.43	odd	6		810.4.e.m.271.1		2
72.59	even	6		810.4.e.e.541.1		2
72.67	odd	6		810.4.e.m.541.1		2
120.59	even	2		450.4.a.b.1.1		1
120.83	odd	4		450.4.c.k.199.1		2
120.107	odd	4		450.4.c.k.199.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.a.1.1	✓	1	8.3	odd	2
90.4.a.d.1.1		1	24.11	even	2
150.4.a.e.1.1		1	40.19	odd	2
150.4.c.a.49.1		2	40.27	even	4
150.4.c.a.49.2		2	40.3	even	4
240.4.a.c.1.1		1	8.5	even	2
450.4.a.b.1.1		1	120.59	even	2
450.4.c.k.199.1		2	120.83	odd	4
450.4.c.k.199.2		2	120.107	odd	4
720.4.a.b.1.1		1	24.5	odd	2
810.4.e.e.271.1		2	72.11	even	6
810.4.e.e.541.1		2	72.59	even	6
810.4.e.m.271.1		2	72.43	odd	6
810.4.e.m.541.1		2	72.67	odd	6
960.4.a.j.1.1		1	4.3	odd	2
960.4.a.s.1.1		1	1.1	even	1	trivial
1200.4.a.bk.1.1		1	40.29	even	2
1200.4.f.u.49.1		2	40.13	odd	4
1200.4.f.u.49.2		2	40.37	odd	4
1470.4.a.a.1.1		1	56.27	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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.a (trivial)

Introduction

L-functions

Modular forms

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Fields

Representations

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Database

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