LMFDB - Embedded newform 960.4.a.b.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,-24,0,9,0,-52] 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 15)
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)

$f(q)$

$=$

$q-3.00000 q^{3} -5.00000 q^{5} -24.0000 q^{7} +9.00000 q^{9} -52.0000 q^{11} -22.0000 q^{13} +15.0000 q^{15} -14.0000 q^{17} +20.0000 q^{19} +72.0000 q^{21} -168.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -230.000 q^{29} -288.000 q^{31} +156.000 q^{33} +120.000 q^{35} +34.0000 q^{37} +66.0000 q^{39} +122.000 q^{41} +188.000 q^{43} -45.0000 q^{45} +256.000 q^{47} +233.000 q^{49} +42.0000 q^{51} +338.000 q^{53} +260.000 q^{55} -60.0000 q^{57} -100.000 q^{59} -742.000 q^{61} -216.000 q^{63} +110.000 q^{65} +84.0000 q^{67} +504.000 q^{69} -328.000 q^{71} -38.0000 q^{73} -75.0000 q^{75} +1248.00 q^{77} -240.000 q^{79} +81.0000 q^{81} -1212.00 q^{83} +70.0000 q^{85} +690.000 q^{87} +330.000 q^{89} +528.000 q^{91} +864.000 q^{93} -100.000 q^{95} +866.000 q^{97} -468.000 q^{99} +O(q^{100})$

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
$7$	−24.0000	−1.29588	−0.647939	−	0.761692i	$-0.724369\pi$
$7$	−24.0000	−1.29588	−0.647939	+	0.761692i	$0.724369\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−52.0000	−1.42533	−0.712663	−	0.701506i	$-0.752511\pi$
$11$	−52.0000	−1.42533	−0.712663	+	0.701506i	$0.752511\pi$
$12$	0	0
$13$	−22.0000	−0.469362	−0.234681	−	0.972072i	$-0.575405\pi$
$13$	−22.0000	−0.469362	−0.234681	+	0.972072i	$0.575405\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−14.0000	−0.199735	−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000	−0.199735	−0.0998676	+	0.995001i	$0.531842\pi$
$18$	0	0
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	72.0000	0.748176
$22$	0	0
$23$	−168.000	−1.52306	−0.761531	−	0.648129i	$-0.775552\pi$
$23$	−168.000	−1.52306	−0.761531	+	0.648129i	$0.775552\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−230.000	−1.47276	−0.736378	−	0.676570i	$-0.763465\pi$
$29$	−230.000	−1.47276	−0.736378	+	0.676570i	$0.763465\pi$
$30$	0	0
$31$	−288.000	−1.66859	−0.834296	−	0.551317i	$-0.814125\pi$
$31$	−288.000	−1.66859	−0.834296	+	0.551317i	$0.814125\pi$
$32$	0	0
$33$	156.000	0.822913
$34$	0	0
$35$	120.000	0.579534
$36$	0	0
$37$	34.0000	0.151069	0.0755347	−	0.997143i	$-0.475934\pi$
$37$	34.0000	0.151069	0.0755347	+	0.997143i	$0.475934\pi$
$38$	0	0
$39$	66.0000	0.270986
$40$	0	0
$41$	122.000	0.464712	0.232356	−	0.972631i	$-0.425357\pi$
$41$	122.000	0.464712	0.232356	+	0.972631i	$0.425357\pi$
$42$	0	0
$43$	188.000	0.666738	0.333369	−	0.942796i	$-0.391815\pi$
$43$	188.000	0.666738	0.333369	+	0.942796i	$0.391815\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	256.000	0.794499	0.397249	−	0.917711i	$-0.369965\pi$
$47$	256.000	0.794499	0.397249	+	0.917711i	$0.369965\pi$
$48$	0	0
$49$	233.000	0.679300
$50$	0	0
$51$	42.0000	0.115317
$52$	0	0
$53$	338.000	0.875998	0.437999	−	0.898976i	$-0.355687\pi$
$53$	338.000	0.875998	0.437999	+	0.898976i	$0.355687\pi$
$54$	0	0
$55$	260.000	0.637425
$56$	0	0
$57$	−60.0000	−0.139424
$58$	0	0
$59$	−100.000	−0.220659	−0.110330	−	0.993895i	$-0.535191\pi$
$59$	−100.000	−0.220659	−0.110330	+	0.993895i	$0.535191\pi$
$60$	0	0
$61$	−742.000	−1.55743	−0.778716	−	0.627376i	$-0.784129\pi$
$61$	−742.000	−1.55743	−0.778716	+	0.627376i	$0.784129\pi$
$62$	0	0
$63$	−216.000	−0.431959
$64$	0	0
$65$	110.000	0.209905
$66$	0	0
$67$	84.0000	0.153168	0.0765838	−	0.997063i	$-0.475599\pi$
$67$	84.0000	0.153168	0.0765838	+	0.997063i	$0.475599\pi$
$68$	0	0
$69$	504.000	0.879340
$70$	0	0
$71$	−328.000	−0.548260	−0.274130	−	0.961693i	$-0.588390\pi$
$71$	−328.000	−0.548260	−0.274130	+	0.961693i	$0.588390\pi$
$72$	0	0
$73$	−38.0000	−0.0609255	−0.0304628	−	0.999536i	$-0.509698\pi$
$73$	−38.0000	−0.0609255	−0.0304628	+	0.999536i	$0.509698\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	1248.00	1.84705
$78$	0	0
$79$	−240.000	−0.341799	−0.170899	−	0.985288i	$-0.554667\pi$
$79$	−240.000	−0.341799	−0.170899	+	0.985288i	$0.554667\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1212.00	−1.60282	−0.801411	−	0.598114i	$-0.795917\pi$
$83$	−1212.00	−1.60282	−0.801411	+	0.598114i	$0.795917\pi$
$84$	0	0
$85$	70.0000	0.0893243
$86$	0	0
$87$	690.000	0.850296
$88$	0	0
$89$	330.000	0.393033	0.196516	−	0.980501i	$-0.437037\pi$
$89$	330.000	0.393033	0.196516	+	0.980501i	$0.437037\pi$
$90$	0	0
$91$	528.000	0.608236
$92$	0	0
$93$	864.000	0.963362
$94$	0	0
$95$	−100.000	−0.107998
$96$	0	0
$97$	866.000	0.906484	0.453242	−	0.891387i	$-0.350267\pi$
$97$	866.000	0.906484	0.453242	+	0.891387i	$0.350267\pi$
$98$	0	0
$99$	−468.000	−0.475109
$100$	0	0
$101$	1218.00	1.19996	0.599978	−	0.800017i	$-0.295176\pi$
$101$	1218.00	1.19996	0.599978	+	0.800017i	$0.295176\pi$
$102$	0	0
$103$	−88.0000	−0.0841835	−0.0420917	−	0.999114i	$-0.513402\pi$
$103$	−88.0000	−0.0841835	−0.0420917	+	0.999114i	$0.513402\pi$
$104$	0	0
$105$	−360.000	−0.334594
$106$	0	0
$107$	−36.0000	−0.0325257	−0.0162629	−	0.999868i	$-0.505177\pi$
$107$	−36.0000	−0.0325257	−0.0162629	+	0.999868i	$0.505177\pi$
$108$	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
$111$	−102.000	−0.0872199
$112$	0	0
$113$	1042.00	0.867461	0.433731	−	0.901043i	$-0.357197\pi$
$113$	1042.00	0.867461	0.433731	+	0.901043i	$0.357197\pi$
$114$	0	0
$115$	840.000	0.681134
$116$	0	0
$117$	−198.000	−0.156454
$118$	0	0
$119$	336.000	0.258833
$120$	0	0
$121$	1373.00	1.03156
$122$	0	0
$123$	−366.000	−0.268302
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1936.00	1.35269	0.676347	−	0.736583i	$-0.263562\pi$
$127$	1936.00	1.35269	0.676347	+	0.736583i	$0.263562\pi$
$128$	0	0
$129$	−564.000	−0.384941
$130$	0	0
$131$	−732.000	−0.488207	−0.244104	−	0.969749i	$-0.578494\pi$
$131$	−732.000	−0.488207	−0.244104	+	0.969749i	$0.578494\pi$
$132$	0	0
$133$	−480.000	−0.312942
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−2214.00	−1.38069	−0.690346	−	0.723479i	$-0.742542\pi$
$137$	−2214.00	−1.38069	−0.690346	+	0.723479i	$0.742542\pi$
$138$	0	0
$139$	−20.0000	−0.0122042	−0.00610208	−	0.999981i	$-0.501942\pi$
$139$	−20.0000	−0.0122042	−0.00610208	+	0.999981i	$0.501942\pi$
$140$	0	0
$141$	−768.000	−0.458704
$142$	0	0
$143$	1144.00	0.668994
$144$	0	0
$145$	1150.00	0.658637
$146$	0	0
$147$	−699.000	−0.392194
$148$	0	0
$149$	1330.00	0.731261	0.365630	−	0.930760i	$-0.380853\pi$
$149$	1330.00	0.731261	0.365630	+	0.930760i	$0.380853\pi$
$150$	0	0
$151$	−1208.00	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−1208.00	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	−126.000	−0.0665784
$154$	0	0
$155$	1440.00	0.746217
$156$	0	0
$157$	3514.00	1.78629	0.893146	−	0.449768i	$-0.148493\pi$
$157$	3514.00	1.78629	0.893146	+	0.449768i	$0.148493\pi$
$158$	0	0
$159$	−1014.00	−0.505757
$160$	0	0
$161$	4032.00	1.97370
$162$	0	0
$163$	2068.00	0.993732	0.496866	−	0.867827i	$-0.334484\pi$
$163$	2068.00	0.993732	0.496866	+	0.867827i	$0.334484\pi$
$164$	0	0
$165$	−780.000	−0.368018
$166$	0	0
$167$	−24.0000	−0.0111208	−0.00556041	−	0.999985i	$-0.501770\pi$
$167$	−24.0000	−0.0111208	−0.00556041	+	0.999985i	$0.501770\pi$
$168$	0	0
$169$	−1713.00	−0.779700
$170$	0	0
$171$	180.000	0.0804967
$172$	0	0
$173$	618.000	0.271593	0.135797	−	0.990737i	$-0.456641\pi$
$173$	618.000	0.271593	0.135797	+	0.990737i	$0.456641\pi$
$174$	0	0
$175$	−600.000	−0.259176
$176$	0	0
$177$	300.000	0.127398
$178$	0	0
$179$	−3340.00	−1.39466	−0.697328	−	0.716752i	$-0.745628\pi$
$179$	−3340.00	−1.39466	−0.697328	+	0.716752i	$0.745628\pi$
$180$	0	0
$181$	178.000	0.0730974	0.0365487	−	0.999332i	$-0.488364\pi$
$181$	178.000	0.0730974	0.0365487	+	0.999332i	$0.488364\pi$
$182$	0	0
$183$	2226.00	0.899184
$184$	0	0
$185$	−170.000	−0.0675603
$186$	0	0
$187$	728.000	0.284688
$188$	0	0
$189$	648.000	0.249392
$190$	0	0
$191$	−1888.00	−0.715240	−0.357620	−	0.933867i	$-0.616412\pi$
$191$	−1888.00	−0.715240	−0.357620	+	0.933867i	$0.616412\pi$
$192$	0	0
$193$	1922.00	0.716832	0.358416	−	0.933562i	$-0.383317\pi$
$193$	1922.00	0.716832	0.358416	+	0.933562i	$0.383317\pi$
$194$	0	0
$195$	−330.000	−0.121189
$196$	0	0
$197$	−2526.00	−0.913554	−0.456777	−	0.889581i	$-0.650996\pi$
$197$	−2526.00	−0.913554	−0.456777	+	0.889581i	$0.650996\pi$
$198$	0	0
$199$	−1160.00	−0.413217	−0.206609	−	0.978424i	$-0.566243\pi$
$199$	−1160.00	−0.413217	−0.206609	+	0.978424i	$0.566243\pi$
$200$	0	0
$201$	−252.000	−0.0884314
$202$	0	0
$203$	5520.00	1.90851
$204$	0	0
$205$	−610.000	−0.207826
$206$	0	0
$207$	−1512.00	−0.507687
$208$	0	0
$209$	−1040.00	−0.344202
$210$	0	0
$211$	4468.00	1.45777	0.728886	−	0.684635i	$-0.240039\pi$
$211$	4468.00	1.45777	0.728886	+	0.684635i	$0.240039\pi$
$212$	0	0
$213$	984.000	0.316538
$214$	0	0
$215$	−940.000	−0.298174
$216$	0	0
$217$	6912.00	2.16229
$218$	0	0
$219$	114.000	0.0351754
$220$	0	0
$221$	308.000	0.0937481
$222$	0	0
$223$	6032.00	1.81136	0.905678	−	0.423965i	$-0.139362\pi$
$223$	6032.00	1.81136	0.905678	+	0.423965i	$0.139362\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−2636.00	−0.770738	−0.385369	−	0.922763i	$-0.625926\pi$
$227$	−2636.00	−0.770738	−0.385369	+	0.922763i	$0.625926\pi$
$228$	0	0
$229$	−4830.00	−1.39378	−0.696889	−	0.717179i	$-0.745433\pi$
$229$	−4830.00	−1.39378	−0.696889	+	0.717179i	$0.745433\pi$
$230$	0	0
$231$	−3744.00	−1.06639
$232$	0	0
$233$	2682.00	0.754093	0.377046	−	0.926194i	$-0.376940\pi$
$233$	2682.00	0.754093	0.377046	+	0.926194i	$0.376940\pi$
$234$	0	0
$235$	−1280.00	−0.355311
$236$	0	0
$237$	720.000	0.197338
$238$	0	0
$239$	2320.00	0.627901	0.313950	−	0.949439i	$-0.398347\pi$
$239$	2320.00	0.627901	0.313950	+	0.949439i	$0.398347\pi$
$240$	0	0
$241$	2002.00	0.535104	0.267552	−	0.963543i	$-0.413785\pi$
$241$	2002.00	0.535104	0.267552	+	0.963543i	$0.413785\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−1165.00	−0.303792
$246$	0	0
$247$	−440.000	−0.113346
$248$	0	0
$249$	3636.00	0.925390
$250$	0	0
$251$	−132.000	−0.0331943	−0.0165971	−	0.999862i	$-0.505283\pi$
$251$	−132.000	−0.0331943	−0.0165971	+	0.999862i	$0.505283\pi$
$252$	0	0
$253$	8736.00	2.17086
$254$	0	0
$255$	−210.000	−0.0515714
$256$	0	0
$257$	−7614.00	−1.84805	−0.924024	−	0.382335i	$-0.875120\pi$
$257$	−7614.00	−1.84805	−0.924024	+	0.382335i	$0.875120\pi$
$258$	0	0
$259$	−816.000	−0.195767
$260$	0	0
$261$	−2070.00	−0.490919
$262$	0	0
$263$	−4888.00	−1.14603	−0.573017	−	0.819543i	$-0.694227\pi$
$263$	−4888.00	−1.14603	−0.573017	+	0.819543i	$0.694227\pi$
$264$	0	0
$265$	−1690.00	−0.391758
$266$	0	0
$267$	−990.000	−0.226918
$268$	0	0
$269$	−1270.00	−0.287856	−0.143928	−	0.989588i	$-0.545973\pi$
$269$	−1270.00	−0.287856	−0.143928	+	0.989588i	$0.545973\pi$
$270$	0	0
$271$	1072.00	0.240293	0.120146	−	0.992756i	$-0.461664\pi$
$271$	1072.00	0.240293	0.120146	+	0.992756i	$0.461664\pi$
$272$	0	0
$273$	−1584.00	−0.351165
$274$	0	0
$275$	−1300.00	−0.285065
$276$	0	0
$277$	5394.00	1.17001	0.585007	−	0.811028i	$-0.301092\pi$
$277$	5394.00	1.17001	0.585007	+	0.811028i	$0.301092\pi$
$278$	0	0
$279$	−2592.00	−0.556197
$280$	0	0
$281$	2442.00	0.518425	0.259213	−	0.965820i	$-0.416537\pi$
$281$	2442.00	0.518425	0.259213	+	0.965820i	$0.416537\pi$
$282$	0	0
$283$	−2772.00	−0.582255	−0.291128	−	0.956684i	$-0.594030\pi$
$283$	−2772.00	−0.582255	−0.291128	+	0.956684i	$0.594030\pi$
$284$	0	0
$285$	300.000	0.0623525
$286$	0	0
$287$	−2928.00	−0.602210
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	−2598.00	−0.523359
$292$	0	0
$293$	−4542.00	−0.905619	−0.452810	−	0.891607i	$-0.649578\pi$
$293$	−4542.00	−0.905619	−0.452810	+	0.891607i	$0.649578\pi$
$294$	0	0
$295$	500.000	0.0986818
$296$	0	0
$297$	1404.00	0.274304
$298$	0	0
$299$	3696.00	0.714867
$300$	0	0
$301$	−4512.00	−0.864011
$302$	0	0
$303$	−3654.00	−0.692795
$304$	0	0
$305$	3710.00	0.696505
$306$	0	0
$307$	−5116.00	−0.951093	−0.475546	−	0.879691i	$-0.657750\pi$
$307$	−5116.00	−0.951093	−0.475546	+	0.879691i	$0.657750\pi$
$308$	0	0
$309$	264.000	0.0486034
$310$	0	0
$311$	−2808.00	−0.511984	−0.255992	−	0.966679i	$-0.582402\pi$
$311$	−2808.00	−0.511984	−0.255992	+	0.966679i	$0.582402\pi$
$312$	0	0
$313$	−7318.00	−1.32153	−0.660763	−	0.750594i	$-0.729767\pi$
$313$	−7318.00	−1.32153	−0.660763	+	0.750594i	$0.729767\pi$
$314$	0	0
$315$	1080.00	0.193178
$316$	0	0
$317$	−2246.00	−0.397943	−0.198971	−	0.980005i	$-0.563760\pi$
$317$	−2246.00	−0.397943	−0.198971	+	0.980005i	$0.563760\pi$
$318$	0	0
$319$	11960.0	2.09916
$320$	0	0
$321$	108.000	0.0187787
$322$	0	0
$323$	−280.000	−0.0482341
$324$	0	0
$325$	−550.000	−0.0938723
$326$	0	0
$327$	−2910.00	−0.492120
$328$	0	0
$329$	−6144.00	−1.02957
$330$	0	0
$331$	−1332.00	−0.221188	−0.110594	−	0.993866i	$-0.535275\pi$
$331$	−1332.00	−0.221188	−0.110594	+	0.993866i	$0.535275\pi$
$332$	0	0
$333$	306.000	0.0503564
$334$	0	0
$335$	−420.000	−0.0684987
$336$	0	0
$337$	−11534.0	−1.86438	−0.932191	−	0.361966i	$-0.882106\pi$
$337$	−11534.0	−1.86438	−0.932191	+	0.361966i	$0.882106\pi$
$338$	0	0
$339$	−3126.00	−0.500829
$340$	0	0
$341$	14976.0	2.37829
$342$	0	0
$343$	2640.00	0.415588
$344$	0	0
$345$	−2520.00	−0.393253
$346$	0	0
$347$	−11956.0	−1.84966	−0.924830	−	0.380382i	$-0.875793\pi$
$347$	−11956.0	−1.84966	−0.924830	+	0.380382i	$0.875793\pi$
$348$	0	0
$349$	−4870.00	−0.746949	−0.373474	−	0.927640i	$-0.621834\pi$
$349$	−4870.00	−0.746949	−0.373474	+	0.927640i	$0.621834\pi$
$350$	0	0
$351$	594.000	0.0903287
$352$	0	0
$353$	10722.0	1.61664	0.808321	−	0.588742i	$-0.200377\pi$
$353$	10722.0	1.61664	0.808321	+	0.588742i	$0.200377\pi$
$354$	0	0
$355$	1640.00	0.245189
$356$	0	0
$357$	−1008.00	−0.149437
$358$	0	0
$359$	120.000	0.0176417	0.00882083	−	0.999961i	$-0.497192\pi$
$359$	120.000	0.0176417	0.00882083	+	0.999961i	$0.497192\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	−4119.00	−0.595569
$364$	0	0
$365$	190.000	0.0272467
$366$	0	0
$367$	3936.00	0.559830	0.279915	−	0.960025i	$-0.409694\pi$
$367$	3936.00	0.559830	0.279915	+	0.960025i	$0.409694\pi$
$368$	0	0
$369$	1098.00	0.154904
$370$	0	0
$371$	−8112.00	−1.13519
$372$	0	0
$373$	−3022.00	−0.419499	−0.209750	−	0.977755i	$-0.567265\pi$
$373$	−3022.00	−0.419499	−0.209750	+	0.977755i	$0.567265\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	5060.00	0.691255
$378$	0	0
$379$	13340.0	1.80799	0.903997	−	0.427539i	$-0.140619\pi$
$379$	13340.0	1.80799	0.903997	+	0.427539i	$0.140619\pi$
$380$	0	0
$381$	−5808.00	−0.780979
$382$	0	0
$383$	−1008.00	−0.134481	−0.0672407	−	0.997737i	$-0.521420\pi$
$383$	−1008.00	−0.134481	−0.0672407	+	0.997737i	$0.521420\pi$
$384$	0	0
$385$	−6240.00	−0.826026
$386$	0	0
$387$	1692.00	0.222246
$388$	0	0
$389$	−9630.00	−1.25517	−0.627584	−	0.778549i	$-0.715956\pi$
$389$	−9630.00	−1.25517	−0.627584	+	0.778549i	$0.715956\pi$
$390$	0	0
$391$	2352.00	0.304209
$392$	0	0
$393$	2196.00	0.281867
$394$	0	0
$395$	1200.00	0.152857
$396$	0	0
$397$	−7126.00	−0.900866	−0.450433	−	0.892810i	$-0.648730\pi$
$397$	−7126.00	−0.900866	−0.450433	+	0.892810i	$0.648730\pi$
$398$	0	0
$399$	1440.00	0.180677
$400$	0	0
$401$	−8718.00	−1.08568	−0.542838	−	0.839837i	$-0.682650\pi$
$401$	−8718.00	−1.08568	−0.542838	+	0.839837i	$0.682650\pi$
$402$	0	0
$403$	6336.00	0.783173
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−1768.00	−0.215323
$408$	0	0
$409$	−10870.0	−1.31415	−0.657074	−	0.753826i	$-0.728206\pi$
$409$	−10870.0	−1.31415	−0.657074	+	0.753826i	$0.728206\pi$
$410$	0	0
$411$	6642.00	0.797143
$412$	0	0
$413$	2400.00	0.285947
$414$	0	0
$415$	6060.00	0.716804
$416$	0	0
$417$	60.0000	0.00704607
$418$	0	0
$419$	9700.00	1.13097	0.565484	−	0.824759i	$-0.308689\pi$
$419$	9700.00	1.13097	0.565484	+	0.824759i	$0.308689\pi$
$420$	0	0
$421$	−862.000	−0.0997893	−0.0498947	−	0.998754i	$-0.515889\pi$
$421$	−862.000	−0.0997893	−0.0498947	+	0.998754i	$0.515889\pi$
$422$	0	0
$423$	2304.00	0.264833
$424$	0	0
$425$	−350.000	−0.0399470
$426$	0	0
$427$	17808.0	2.01824
$428$	0	0
$429$	−3432.00	−0.386244
$430$	0	0
$431$	15792.0	1.76490	0.882452	−	0.470402i	$-0.155891\pi$
$431$	15792.0	1.76490	0.882452	+	0.470402i	$0.155891\pi$
$432$	0	0
$433$	11602.0	1.28766	0.643830	−	0.765169i	$-0.277345\pi$
$433$	11602.0	1.28766	0.643830	+	0.765169i	$0.277345\pi$
$434$	0	0
$435$	−3450.00	−0.380264
$436$	0	0
$437$	−3360.00	−0.367805
$438$	0	0
$439$	−440.000	−0.0478361	−0.0239181	−	0.999714i	$-0.507614\pi$
$439$	−440.000	−0.0478361	−0.0239181	+	0.999714i	$0.507614\pi$
$440$	0	0
$441$	2097.00	0.226433
$442$	0	0
$443$	10188.0	1.09266	0.546328	−	0.837571i	$-0.316025\pi$
$443$	10188.0	1.09266	0.546328	+	0.837571i	$0.316025\pi$
$444$	0	0
$445$	−1650.00	−0.175770
$446$	0	0
$447$	−3990.00	−0.422194
$448$	0	0
$449$	−13310.0	−1.39897	−0.699485	−	0.714647i	$-0.746587\pi$
$449$	−13310.0	−1.39897	−0.699485	+	0.714647i	$0.746587\pi$
$450$	0	0
$451$	−6344.00	−0.662367
$452$	0	0
$453$	3624.00	0.375873
$454$	0	0
$455$	−2640.00	−0.272011
$456$	0	0
$457$	3226.00	0.330210	0.165105	−	0.986276i	$-0.447204\pi$
$457$	3226.00	0.330210	0.165105	+	0.986276i	$0.447204\pi$
$458$	0	0
$459$	378.000	0.0384391
$460$	0	0
$461$	−6582.00	−0.664977	−0.332488	−	0.943107i	$-0.607888\pi$
$461$	−6582.00	−0.664977	−0.332488	+	0.943107i	$0.607888\pi$
$462$	0	0
$463$	15072.0	1.51286	0.756431	−	0.654073i	$-0.226941\pi$
$463$	15072.0	1.51286	0.756431	+	0.654073i	$0.226941\pi$
$464$	0	0
$465$	−4320.00	−0.430828
$466$	0	0
$467$	−476.000	−0.0471663	−0.0235831	−	0.999722i	$-0.507507\pi$
$467$	−476.000	−0.0471663	−0.0235831	+	0.999722i	$0.507507\pi$
$468$	0	0
$469$	−2016.00	−0.198487
$470$	0	0
$471$	−10542.0	−1.03132
$472$	0	0
$473$	−9776.00	−0.950319
$474$	0	0
$475$	500.000	0.0482980
$476$	0	0
$477$	3042.00	0.291999
$478$	0	0
$479$	−19680.0	−1.87725	−0.938624	−	0.344941i	$-0.887899\pi$
$479$	−19680.0	−1.87725	−0.938624	+	0.344941i	$0.887899\pi$
$480$	0	0
$481$	−748.000	−0.0709062
$482$	0	0
$483$	−12096.0	−1.13952
$484$	0	0
$485$	−4330.00	−0.405392
$486$	0	0
$487$	−5944.00	−0.553077	−0.276538	−	0.961003i	$-0.589187\pi$
$487$	−5944.00	−0.553077	−0.276538	+	0.961003i	$0.589187\pi$
$488$	0	0
$489$	−6204.00	−0.573731
$490$	0	0
$491$	−10772.0	−0.990089	−0.495044	−	0.868868i	$-0.664848\pi$
$491$	−10772.0	−0.990089	−0.495044	+	0.868868i	$0.664848\pi$
$492$	0	0
$493$	3220.00	0.294161
$494$	0	0
$495$	2340.00	0.212475
$496$	0	0
$497$	7872.00	0.710478
$498$	0	0
$499$	−8140.00	−0.730253	−0.365127	−	0.930958i	$-0.618974\pi$
$499$	−8140.00	−0.730253	−0.365127	+	0.930958i	$0.618974\pi$
$500$	0	0
$501$	72.0000	0.00642060
$502$	0	0
$503$	−13768.0	−1.22045	−0.610223	−	0.792229i	$-0.708920\pi$
$503$	−13768.0	−1.22045	−0.610223	+	0.792229i	$0.708920\pi$
$504$	0	0
$505$	−6090.00	−0.536637
$506$	0	0
$507$	5139.00	0.450160
$508$	0	0
$509$	−22150.0	−1.92884	−0.964422	−	0.264368i	$-0.914837\pi$
$509$	−22150.0	−1.92884	−0.964422	+	0.264368i	$0.914837\pi$
$510$	0	0
$511$	912.000	0.0789521
$512$	0	0
$513$	−540.000	−0.0464748
$514$	0	0
$515$	440.000	0.0376480
$516$	0	0
$517$	−13312.0	−1.13242
$518$	0	0
$519$	−1854.00	−0.156805
$520$	0	0
$521$	1562.00	0.131348	0.0656741	−	0.997841i	$-0.479080\pi$
$521$	1562.00	0.131348	0.0656741	+	0.997841i	$0.479080\pi$
$522$	0	0
$523$	668.000	0.0558501	0.0279250	−	0.999610i	$-0.491110\pi$
$523$	668.000	0.0558501	0.0279250	+	0.999610i	$0.491110\pi$
$524$	0	0
$525$	1800.00	0.149635
$526$	0	0
$527$	4032.00	0.333276
$528$	0	0
$529$	16057.0	1.31972
$530$	0	0
$531$	−900.000	−0.0735531
$532$	0	0
$533$	−2684.00	−0.218118
$534$	0	0
$535$	180.000	0.0145459
$536$	0	0
$537$	10020.0	0.805205
$538$	0	0
$539$	−12116.0	−0.968225
$540$	0	0
$541$	6138.00	0.487788	0.243894	−	0.969802i	$-0.421575\pi$
$541$	6138.00	0.487788	0.243894	+	0.969802i	$0.421575\pi$
$542$	0	0
$543$	−534.000	−0.0422028
$544$	0	0
$545$	−4850.00	−0.381195
$546$	0	0
$547$	10484.0	0.819494	0.409747	−	0.912199i	$-0.365617\pi$
$547$	10484.0	0.819494	0.409747	+	0.912199i	$0.365617\pi$
$548$	0	0
$549$	−6678.00	−0.519144
$550$	0	0
$551$	−4600.00	−0.355656
$552$	0	0
$553$	5760.00	0.442930
$554$	0	0
$555$	510.000	0.0390059
$556$	0	0
$557$	−3606.00	−0.274311	−0.137155	−	0.990550i	$-0.543796\pi$
$557$	−3606.00	−0.274311	−0.137155	+	0.990550i	$0.543796\pi$
$558$	0	0
$559$	−4136.00	−0.312941
$560$	0	0
$561$	−2184.00	−0.164365
$562$	0	0
$563$	−12252.0	−0.917159	−0.458579	−	0.888654i	$-0.651641\pi$
$563$	−12252.0	−0.917159	−0.458579	+	0.888654i	$0.651641\pi$
$564$	0	0
$565$	−5210.00	−0.387940
$566$	0	0
$567$	−1944.00	−0.143986
$568$	0	0
$569$	−14550.0	−1.07200	−0.536000	−	0.844218i	$-0.680065\pi$
$569$	−14550.0	−1.07200	−0.536000	+	0.844218i	$0.680065\pi$
$570$	0	0
$571$	25468.0	1.86655	0.933277	−	0.359157i	$-0.116936\pi$
$571$	25468.0	1.86655	0.933277	+	0.359157i	$0.116936\pi$
$572$	0	0
$573$	5664.00	0.412944
$574$	0	0
$575$	−4200.00	−0.304612
$576$	0	0
$577$	12866.0	0.928282	0.464141	−	0.885761i	$-0.346363\pi$
$577$	12866.0	0.928282	0.464141	+	0.885761i	$0.346363\pi$
$578$	0	0
$579$	−5766.00	−0.413863
$580$	0	0
$581$	29088.0	2.07706
$582$	0	0
$583$	−17576.0	−1.24858
$584$	0	0
$585$	990.000	0.0699683
$586$	0	0
$587$	14844.0	1.04374	0.521872	−	0.853024i	$-0.325234\pi$
$587$	14844.0	1.04374	0.521872	+	0.853024i	$0.325234\pi$
$588$	0	0
$589$	−5760.00	−0.402948
$590$	0	0
$591$	7578.00	0.527440
$592$	0	0
$593$	20402.0	1.41283	0.706416	−	0.707797i	$-0.250311\pi$
$593$	20402.0	1.41283	0.706416	+	0.707797i	$0.250311\pi$
$594$	0	0
$595$	−1680.00	−0.115753
$596$	0	0
$597$	3480.00	0.238571
$598$	0	0
$599$	10760.0	0.733959	0.366980	−	0.930229i	$-0.380392\pi$
$599$	10760.0	0.733959	0.366980	+	0.930229i	$0.380392\pi$
$600$	0	0
$601$	14282.0	0.969343	0.484671	−	0.874696i	$-0.338939\pi$
$601$	14282.0	0.969343	0.484671	+	0.874696i	$0.338939\pi$
$602$	0	0
$603$	756.000	0.0510559
$604$	0	0
$605$	−6865.00	−0.461326
$606$	0	0
$607$	11056.0	0.739290	0.369645	−	0.929173i	$-0.379479\pi$
$607$	11056.0	0.739290	0.369645	+	0.929173i	$0.379479\pi$
$608$	0	0
$609$	−16560.0	−1.10188
$610$	0	0
$611$	−5632.00	−0.372907
$612$	0	0
$613$	16418.0	1.08176	0.540878	−	0.841101i	$-0.318092\pi$
$613$	16418.0	1.08176	0.540878	+	0.841101i	$0.318092\pi$
$614$	0	0
$615$	1830.00	0.119988
$616$	0	0
$617$	−10374.0	−0.676891	−0.338445	−	0.940986i	$-0.609901\pi$
$617$	−10374.0	−0.676891	−0.338445	+	0.940986i	$0.609901\pi$
$618$	0	0
$619$	5260.00	0.341546	0.170773	−	0.985310i	$-0.445373\pi$
$619$	5260.00	0.341546	0.170773	+	0.985310i	$0.445373\pi$
$620$	0	0
$621$	4536.00	0.293113
$622$	0	0
$623$	−7920.00	−0.509323
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	3120.00	0.198725
$628$	0	0
$629$	−476.000	−0.0301739
$630$	0	0
$631$	21352.0	1.34708	0.673542	−	0.739149i	$-0.264772\pi$
$631$	21352.0	1.34708	0.673542	+	0.739149i	$0.264772\pi$
$632$	0	0
$633$	−13404.0	−0.841645
$634$	0	0
$635$	−9680.00	−0.604943
$636$	0	0
$637$	−5126.00	−0.318838
$638$	0	0
$639$	−2952.00	−0.182753
$640$	0	0
$641$	−29118.0	−1.79422	−0.897108	−	0.441812i	$-0.854336\pi$
$641$	−29118.0	−1.79422	−0.897108	+	0.441812i	$0.854336\pi$
$642$	0	0
$643$	−5772.00	−0.354005	−0.177003	−	0.984210i	$-0.556640\pi$
$643$	−5772.00	−0.354005	−0.177003	+	0.984210i	$0.556640\pi$
$644$	0	0
$645$	2820.00	0.172151
$646$	0	0
$647$	−14264.0	−0.866732	−0.433366	−	0.901218i	$-0.642674\pi$
$647$	−14264.0	−0.866732	−0.433366	+	0.901218i	$0.642674\pi$
$648$	0	0
$649$	5200.00	0.314511
$650$	0	0
$651$	−20736.0	−1.24840
$652$	0	0
$653$	−6902.00	−0.413623	−0.206812	−	0.978381i	$-0.566309\pi$
$653$	−6902.00	−0.413623	−0.206812	+	0.978381i	$0.566309\pi$
$654$	0	0
$655$	3660.00	0.218333
$656$	0	0
$657$	−342.000	−0.0203085
$658$	0	0
$659$	−20140.0	−1.19051	−0.595253	−	0.803539i	$-0.702948\pi$
$659$	−20140.0	−1.19051	−0.595253	+	0.803539i	$0.702948\pi$
$660$	0	0
$661$	3218.00	0.189358	0.0946790	−	0.995508i	$-0.469818\pi$
$661$	3218.00	0.189358	0.0946790	+	0.995508i	$0.469818\pi$
$662$	0	0
$663$	−924.000	−0.0541255
$664$	0	0
$665$	2400.00	0.139952
$666$	0	0
$667$	38640.0	2.24310
$668$	0	0
$669$	−18096.0	−1.04579
$670$	0	0
$671$	38584.0	2.21985
$672$	0	0
$673$	−7518.00	−0.430606	−0.215303	−	0.976547i	$-0.569074\pi$
$673$	−7518.00	−0.430606	−0.215303	+	0.976547i	$0.569074\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	18114.0	1.02833	0.514164	−	0.857692i	$-0.328102\pi$
$677$	18114.0	1.02833	0.514164	+	0.857692i	$0.328102\pi$
$678$	0	0
$679$	−20784.0	−1.17469
$680$	0	0
$681$	7908.00	0.444986
$682$	0	0
$683$	23868.0	1.33716	0.668582	−	0.743638i	$-0.266901\pi$
$683$	23868.0	1.33716	0.668582	+	0.743638i	$0.266901\pi$
$684$	0	0
$685$	11070.0	0.617464
$686$	0	0
$687$	14490.0	0.804699
$688$	0	0
$689$	−7436.00	−0.411160
$690$	0	0
$691$	−172.000	−0.00946916	−0.00473458	−	0.999989i	$-0.501507\pi$
$691$	−172.000	−0.00946916	−0.00473458	+	0.999989i	$0.501507\pi$
$692$	0	0
$693$	11232.0	0.615683
$694$	0	0
$695$	100.000	0.00545787
$696$	0	0
$697$	−1708.00	−0.0928194
$698$	0	0
$699$	−8046.00	−0.435376
$700$	0	0
$701$	22138.0	1.19278	0.596391	−	0.802694i	$-0.296601\pi$
$701$	22138.0	1.19278	0.596391	+	0.802694i	$0.296601\pi$
$702$	0	0
$703$	680.000	0.0364818
$704$	0	0
$705$	3840.00	0.205139
$706$	0	0
$707$	−29232.0	−1.55500
$708$	0	0
$709$	−3070.00	−0.162618	−0.0813091	−	0.996689i	$-0.525910\pi$
$709$	−3070.00	−0.162618	−0.0813091	+	0.996689i	$0.525910\pi$
$710$	0	0
$711$	−2160.00	−0.113933
$712$	0	0
$713$	48384.0	2.54137
$714$	0	0
$715$	−5720.00	−0.299183
$716$	0	0
$717$	−6960.00	−0.362519
$718$	0	0
$719$	15600.0	0.809154	0.404577	−	0.914504i	$-0.367419\pi$
$719$	15600.0	0.809154	0.404577	+	0.914504i	$0.367419\pi$
$720$	0	0
$721$	2112.00	0.109092
$722$	0	0
$723$	−6006.00	−0.308943
$724$	0	0
$725$	−5750.00	−0.294551
$726$	0	0
$727$	20696.0	1.05581	0.527904	−	0.849304i	$-0.322978\pi$
$727$	20696.0	1.05581	0.527904	+	0.849304i	$0.322978\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−2632.00	−0.133171
$732$	0	0
$733$	30778.0	1.55090	0.775451	−	0.631408i	$-0.217522\pi$
$733$	30778.0	1.55090	0.775451	+	0.631408i	$0.217522\pi$
$734$	0	0
$735$	3495.00	0.175395
$736$	0	0
$737$	−4368.00	−0.218314
$738$	0	0
$739$	−11740.0	−0.584388	−0.292194	−	0.956359i	$-0.594385\pi$
$739$	−11740.0	−0.584388	−0.292194	+	0.956359i	$0.594385\pi$
$740$	0	0
$741$	1320.00	0.0654405
$742$	0	0
$743$	2632.00	0.129958	0.0649789	−	0.997887i	$-0.479302\pi$
$743$	2632.00	0.129958	0.0649789	+	0.997887i	$0.479302\pi$
$744$	0	0
$745$	−6650.00	−0.327030
$746$	0	0
$747$	−10908.0	−0.534274
$748$	0	0
$749$	864.000	0.0421494
$750$	0	0
$751$	−20528.0	−0.997440	−0.498720	−	0.866763i	$-0.666196\pi$
$751$	−20528.0	−0.997440	−0.498720	+	0.866763i	$0.666196\pi$
$752$	0	0
$753$	396.000	0.0191647
$754$	0	0
$755$	6040.00	0.291150
$756$	0	0
$757$	−21646.0	−1.03928	−0.519642	−	0.854384i	$-0.673934\pi$
$757$	−21646.0	−1.03928	−0.519642	+	0.854384i	$0.673934\pi$
$758$	0	0
$759$	−26208.0	−1.25335
$760$	0	0
$761$	18282.0	0.870857	0.435428	−	0.900223i	$-0.356597\pi$
$761$	18282.0	0.870857	0.435428	+	0.900223i	$0.356597\pi$
$762$	0	0
$763$	−23280.0	−1.10458
$764$	0	0
$765$	630.000	0.0297748
$766$	0	0
$767$	2200.00	0.103569
$768$	0	0
$769$	−24190.0	−1.13435	−0.567174	−	0.823598i	$-0.691963\pi$
$769$	−24190.0	−1.13435	−0.567174	+	0.823598i	$0.691963\pi$
$770$	0	0
$771$	22842.0	1.06697
$772$	0	0
$773$	25698.0	1.19572	0.597861	−	0.801600i	$-0.296018\pi$
$773$	25698.0	1.19572	0.597861	+	0.801600i	$0.296018\pi$
$774$	0	0
$775$	−7200.00	−0.333718
$776$	0	0
$777$	2448.00	0.113026
$778$	0	0
$779$	2440.00	0.112223
$780$	0	0
$781$	17056.0	0.781449
$782$	0	0
$783$	6210.00	0.283432
$784$	0	0
$785$	−17570.0	−0.798854
$786$	0	0
$787$	−33436.0	−1.51444	−0.757220	−	0.653160i	$-0.773443\pi$
$787$	−33436.0	−1.51444	−0.757220	+	0.653160i	$0.773443\pi$
$788$	0	0
$789$	14664.0	0.661663
$790$	0	0
$791$	−25008.0	−1.12412
$792$	0	0
$793$	16324.0	0.730999
$794$	0	0
$795$	5070.00	0.226182
$796$	0	0
$797$	37594.0	1.67083	0.835413	−	0.549623i	$-0.185229\pi$
$797$	37594.0	1.67083	0.835413	+	0.549623i	$0.185229\pi$
$798$	0	0
$799$	−3584.00	−0.158689
$800$	0	0
$801$	2970.00	0.131011
$802$	0	0
$803$	1976.00	0.0868388
$804$	0	0
$805$	−20160.0	−0.882667
$806$	0	0
$807$	3810.00	0.166194
$808$	0	0
$809$	4730.00	0.205560	0.102780	−	0.994704i	$-0.467226\pi$
$809$	4730.00	0.205560	0.102780	+	0.994704i	$0.467226\pi$
$810$	0	0
$811$	8748.00	0.378772	0.189386	−	0.981903i	$-0.439350\pi$
$811$	8748.00	0.378772	0.189386	+	0.981903i	$0.439350\pi$
$812$	0	0
$813$	−3216.00	−0.138733
$814$	0	0
$815$	−10340.0	−0.444410
$816$	0	0
$817$	3760.00	0.161011
$818$	0	0
$819$	4752.00	0.202745
$820$	0	0
$821$	−44142.0	−1.87645	−0.938226	−	0.346024i	$-0.887532\pi$
$821$	−44142.0	−1.87645	−0.938226	+	0.346024i	$0.887532\pi$
$822$	0	0
$823$	3992.00	0.169079	0.0845397	−	0.996420i	$-0.473058\pi$
$823$	3992.00	0.169079	0.0845397	+	0.996420i	$0.473058\pi$
$824$	0	0
$825$	3900.00	0.164583
$826$	0	0
$827$	14444.0	0.607336	0.303668	−	0.952778i	$-0.401789\pi$
$827$	14444.0	0.607336	0.303668	+	0.952778i	$0.401789\pi$
$828$	0	0
$829$	−42150.0	−1.76590	−0.882949	−	0.469468i	$-0.844446\pi$
$829$	−42150.0	−1.76590	−0.882949	+	0.469468i	$0.844446\pi$
$830$	0	0
$831$	−16182.0	−0.675508
$832$	0	0
$833$	−3262.00	−0.135680
$834$	0	0
$835$	120.000	0.00497338
$836$	0	0
$837$	7776.00	0.321121
$838$	0	0
$839$	13400.0	0.551394	0.275697	−	0.961245i	$-0.411091\pi$
$839$	13400.0	0.551394	0.275697	+	0.961245i	$0.411091\pi$
$840$	0	0
$841$	28511.0	1.16901
$842$	0	0
$843$	−7326.00	−0.299313
$844$	0	0
$845$	8565.00	0.348692
$846$	0	0
$847$	−32952.0	−1.33677
$848$	0	0
$849$	8316.00	0.336165
$850$	0	0
$851$	−5712.00	−0.230088
$852$	0	0
$853$	8658.00	0.347531	0.173766	−	0.984787i	$-0.444406\pi$
$853$	8658.00	0.347531	0.173766	+	0.984787i	$0.444406\pi$
$854$	0	0
$855$	−900.000	−0.0359992
$856$	0	0
$857$	42826.0	1.70701	0.853505	−	0.521084i	$-0.174472\pi$
$857$	42826.0	1.70701	0.853505	+	0.521084i	$0.174472\pi$
$858$	0	0
$859$	35900.0	1.42595	0.712976	−	0.701189i	$-0.247347\pi$
$859$	35900.0	1.42595	0.712976	+	0.701189i	$0.247347\pi$
$860$	0	0
$861$	8784.00	0.347686
$862$	0	0
$863$	−3088.00	−0.121804	−0.0609019	−	0.998144i	$-0.519398\pi$
$863$	−3088.00	−0.121804	−0.0609019	+	0.998144i	$0.519398\pi$
$864$	0	0
$865$	−3090.00	−0.121460
$866$	0	0
$867$	14151.0	0.554317
$868$	0	0
$869$	12480.0	0.487175
$870$	0	0
$871$	−1848.00	−0.0718910
$872$	0	0
$873$	7794.00	0.302161
$874$	0	0
$875$	3000.00	0.115907
$876$	0	0
$877$	35274.0	1.35817	0.679087	−	0.734058i	$-0.262376\pi$
$877$	35274.0	1.35817	0.679087	+	0.734058i	$0.262376\pi$
$878$	0	0
$879$	13626.0	0.522860
$880$	0	0
$881$	25042.0	0.957646	0.478823	−	0.877911i	$-0.341064\pi$
$881$	25042.0	0.957646	0.478823	+	0.877911i	$0.341064\pi$
$882$	0	0
$883$	−12572.0	−0.479141	−0.239570	−	0.970879i	$-0.577007\pi$
$883$	−12572.0	−0.479141	−0.239570	+	0.970879i	$0.577007\pi$
$884$	0	0
$885$	−1500.00	−0.0569740
$886$	0	0
$887$	−21864.0	−0.827645	−0.413823	−	0.910358i	$-0.635807\pi$
$887$	−21864.0	−0.827645	−0.413823	+	0.910358i	$0.635807\pi$
$888$	0	0
$889$	−46464.0	−1.75293
$890$	0	0
$891$	−4212.00	−0.158370
$892$	0	0
$893$	5120.00	0.191864
$894$	0	0
$895$	16700.0	0.623709
$896$	0	0
$897$	−11088.0	−0.412729
$898$	0	0
$899$	66240.0	2.45743
$900$	0	0
$901$	−4732.00	−0.174968
$902$	0	0
$903$	13536.0	0.498837
$904$	0	0
$905$	−890.000	−0.0326902
$906$	0	0
$907$	−31236.0	−1.14352	−0.571761	−	0.820420i	$-0.693740\pi$
$907$	−31236.0	−1.14352	−0.571761	+	0.820420i	$0.693740\pi$
$908$	0	0
$909$	10962.0	0.399985
$910$	0	0
$911$	8272.00	0.300838	0.150419	−	0.988622i	$-0.451938\pi$
$911$	8272.00	0.300838	0.150419	+	0.988622i	$0.451938\pi$
$912$	0	0
$913$	63024.0	2.28455
$914$	0	0
$915$	−11130.0	−0.402127
$916$	0	0
$917$	17568.0	0.632657
$918$	0	0
$919$	20200.0	0.725067	0.362533	−	0.931971i	$-0.381912\pi$
$919$	20200.0	0.725067	0.362533	+	0.931971i	$0.381912\pi$
$920$	0	0
$921$	15348.0	0.549114
$922$	0	0
$923$	7216.00	0.257332
$924$	0	0
$925$	850.000	0.0302139
$926$	0	0
$927$	−792.000	−0.0280612
$928$	0	0
$929$	31010.0	1.09516	0.547581	−	0.836753i	$-0.315549\pi$
$929$	31010.0	1.09516	0.547581	+	0.836753i	$0.315549\pi$
$930$	0	0
$931$	4660.00	0.164044
$932$	0	0
$933$	8424.00	0.295594
$934$	0	0
$935$	−3640.00	−0.127316
$936$	0	0
$937$	−39174.0	−1.36580	−0.682902	−	0.730510i	$-0.739283\pi$
$937$	−39174.0	−1.36580	−0.682902	+	0.730510i	$0.739283\pi$
$938$	0	0
$939$	21954.0	0.762984
$940$	0	0
$941$	4138.00	0.143353	0.0716764	−	0.997428i	$-0.477165\pi$
$941$	4138.00	0.143353	0.0716764	+	0.997428i	$0.477165\pi$
$942$	0	0
$943$	−20496.0	−0.707785
$944$	0	0
$945$	−3240.00	−0.111531
$946$	0	0
$947$	−23676.0	−0.812425	−0.406213	−	0.913779i	$-0.633151\pi$
$947$	−23676.0	−0.812425	−0.406213	+	0.913779i	$0.633151\pi$
$948$	0	0
$949$	836.000	0.0285961
$950$	0	0
$951$	6738.00	0.229752
$952$	0	0
$953$	18922.0	0.643173	0.321586	−	0.946880i	$-0.395784\pi$
$953$	18922.0	0.643173	0.321586	+	0.946880i	$0.395784\pi$
$954$	0	0
$955$	9440.00	0.319865
$956$	0	0
$957$	−35880.0	−1.21195
$958$	0	0
$959$	53136.0	1.78921
$960$	0	0
$961$	53153.0	1.78420
$962$	0	0
$963$	−324.000	−0.0108419
$964$	0	0
$965$	−9610.00	−0.320577
$966$	0	0
$967$	39656.0	1.31877	0.659385	−	0.751805i	$-0.270817\pi$
$967$	39656.0	1.31877	0.659385	+	0.751805i	$0.270817\pi$
$968$	0	0
$969$	840.000	0.0278480
$970$	0	0
$971$	33228.0	1.09818	0.549092	−	0.835762i	$-0.314974\pi$
$971$	33228.0	1.09818	0.549092	+	0.835762i	$0.314974\pi$
$972$	0	0
$973$	480.000	0.0158151
$974$	0	0
$975$	1650.00	0.0541972
$976$	0	0
$977$	−974.000	−0.0318946	−0.0159473	−	0.999873i	$-0.505076\pi$
$977$	−974.000	−0.0318946	−0.0159473	+	0.999873i	$0.505076\pi$
$978$	0	0
$979$	−17160.0	−0.560200
$980$	0	0
$981$	8730.00	0.284126
$982$	0	0
$983$	−13608.0	−0.441534	−0.220767	−	0.975327i	$-0.570856\pi$
$983$	−13608.0	−0.441534	−0.220767	+	0.975327i	$0.570856\pi$
$984$	0	0
$985$	12630.0	0.408554
$986$	0	0
$987$	18432.0	0.594425
$988$	0	0
$989$	−31584.0	−1.01548
$990$	0	0
$991$	13472.0	0.431839	0.215919	−	0.976411i	$-0.430725\pi$
$991$	13472.0	0.431839	0.215919	+	0.976411i	$0.430725\pi$
$992$	0	0
$993$	3996.00	0.127703
$994$	0	0
$995$	5800.00	0.184796
$996$	0	0
$997$	3234.00	0.102730	0.0513650	−	0.998680i	$-0.483643\pi$
$997$	3234.00	0.102730	0.0513650	+	0.998680i	$0.483643\pi$
$998$	0	0
$999$	−918.000	−0.0290733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.a.b.1.1		1
4.3	odd	2		960.4.a.ba.1.1		1
8.3	odd	2		240.4.a.e.1.1		1
8.5	even	2		15.4.a.a.1.1	✓	1
24.5	odd	2		45.4.a.c.1.1		1
24.11	even	2		720.4.a.n.1.1		1
40.3	even	4		1200.4.f.b.49.1		2
40.13	odd	4		75.4.b.b.49.1		2
40.19	odd	2		1200.4.a.t.1.1		1
40.27	even	4		1200.4.f.b.49.2		2
40.29	even	2		75.4.a.b.1.1		1
40.37	odd	4		75.4.b.b.49.2		2
56.13	odd	2		735.4.a.e.1.1		1
72.5	odd	6		405.4.e.i.136.1		2
72.13	even	6		405.4.e.g.136.1		2
72.29	odd	6		405.4.e.i.271.1		2
72.61	even	6		405.4.e.g.271.1		2
88.21	odd	2		1815.4.a.e.1.1		1
120.29	odd	2		225.4.a.f.1.1		1
120.53	even	4		225.4.b.e.199.2		2
120.77	even	4		225.4.b.e.199.1		2
168.125	even	2		2205.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.a.1.1	✓	1	8.5	even	2
45.4.a.c.1.1		1	24.5	odd	2
75.4.a.b.1.1		1	40.29	even	2
75.4.b.b.49.1		2	40.13	odd	4
75.4.b.b.49.2		2	40.37	odd	4
225.4.a.f.1.1		1	120.29	odd	2
225.4.b.e.199.1		2	120.77	even	4
225.4.b.e.199.2		2	120.53	even	4
240.4.a.e.1.1		1	8.3	odd	2
405.4.e.g.136.1		2	72.13	even	6
405.4.e.g.271.1		2	72.61	even	6
405.4.e.i.136.1		2	72.5	odd	6
405.4.e.i.271.1		2	72.29	odd	6
720.4.a.n.1.1		1	24.11	even	2
735.4.a.e.1.1		1	56.13	odd	2
960.4.a.b.1.1		1	1.1	even	1	trivial
960.4.a.ba.1.1		1	4.3	odd	2
1200.4.a.t.1.1		1	40.19	odd	2
1200.4.f.b.49.1		2	40.3	even	4
1200.4.f.b.49.2		2	40.27	even	4
1815.4.a.e.1.1		1	88.21	odd	2
2205.4.a.l.1.1		1	168.125	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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