LMFDB - Embedded newform 336.4.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(1,336)] 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("336.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	336.1

$q$ -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} +14.0000 q^{5} +7.00000 q^{7} +9.00000 q^{9} -4.00000 q^{11} +54.0000 q^{13} -42.0000 q^{15} -14.0000 q^{17} -92.0000 q^{19} -21.0000 q^{21} +152.000 q^{23} +71.0000 q^{25} -27.0000 q^{27} -106.000 q^{29} +144.000 q^{31} +12.0000 q^{33} +98.0000 q^{35} +158.000 q^{37} -162.000 q^{39} -390.000 q^{41} +508.000 q^{43} +126.000 q^{45} +528.000 q^{47} +49.0000 q^{49} +42.0000 q^{51} +606.000 q^{53} -56.0000 q^{55} +276.000 q^{57} +364.000 q^{59} +678.000 q^{61} +63.0000 q^{63} +756.000 q^{65} -844.000 q^{67} -456.000 q^{69} +8.00000 q^{71} -422.000 q^{73} -213.000 q^{75} -28.0000 q^{77} -384.000 q^{79} +81.0000 q^{81} +548.000 q^{83} -196.000 q^{85} +318.000 q^{87} +1194.00 q^{89} +378.000 q^{91} -432.000 q^{93} -1288.00 q^{95} -1502.00 q^{97} -36.0000 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
$5$	14.0000	1.25220	0.626099	−	0.779744i	$-0.284651\pi$
$5$	14.0000	1.25220	0.626099	+	0.779744i	$0.284651\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−4.00000	−0.109640	−0.0548202	−	0.998496i	$-0.517459\pi$
$11$	−4.00000	−0.109640	−0.0548202	+	0.998496i	$0.517459\pi$
$12$	0	0
$13$	54.0000	1.15207	0.576035	−	0.817425i	$-0.304599\pi$
$13$	54.0000	1.15207	0.576035	+	0.817425i	$0.304599\pi$
$14$	0	0
$15$	−42.0000	−0.722957
$16$	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
$19$	−92.0000	−1.11086	−0.555428	−	0.831565i	$-0.687445\pi$
$19$	−92.0000	−1.11086	−0.555428	+	0.831565i	$0.687445\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	152.000	1.37801	0.689004	−	0.724757i	$-0.258048\pi$
$23$	152.000	1.37801	0.689004	+	0.724757i	$0.258048\pi$
$24$	0	0
$25$	71.0000	0.568000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−106.000	−0.678748	−0.339374	−	0.940651i	$-0.610215\pi$
$29$	−106.000	−0.678748	−0.339374	+	0.940651i	$0.610215\pi$
$30$	0	0
$31$	144.000	0.834296	0.417148	−	0.908839i	$-0.363030\pi$
$31$	144.000	0.834296	0.417148	+	0.908839i	$0.363030\pi$
$32$	0	0
$33$	12.0000	0.0633010
$34$	0	0
$35$	98.0000	0.473286
$36$	0	0
$37$	158.000	0.702028	0.351014	−	0.936370i	$-0.385837\pi$
$37$	158.000	0.702028	0.351014	+	0.936370i	$0.385837\pi$
$38$	0	0
$39$	−162.000	−0.665148
$40$	0	0
$41$	−390.000	−1.48556	−0.742778	−	0.669538i	$-0.766492\pi$
$41$	−390.000	−1.48556	−0.742778	+	0.669538i	$0.766492\pi$
$42$	0	0
$43$	508.000	1.80161	0.900806	−	0.434223i	$-0.142977\pi$
$43$	508.000	1.80161	0.900806	+	0.434223i	$0.142977\pi$
$44$	0	0
$45$	126.000	0.417399
$46$	0	0
$47$	528.000	1.63865	0.819327	−	0.573327i	$-0.194347\pi$
$47$	528.000	1.63865	0.819327	+	0.573327i	$0.194347\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	42.0000	0.115317
$52$	0	0
$53$	606.000	1.57058	0.785288	−	0.619131i	$-0.212515\pi$
$53$	606.000	1.57058	0.785288	+	0.619131i	$0.212515\pi$
$54$	0	0
$55$	−56.0000	−0.137292
$56$	0	0
$57$	276.000	0.641353
$58$	0	0
$59$	364.000	0.803199	0.401600	−	0.915815i	$-0.368454\pi$
$59$	364.000	0.803199	0.401600	+	0.915815i	$0.368454\pi$
$60$	0	0
$61$	678.000	1.42310	0.711549	−	0.702636i	$-0.247994\pi$
$61$	678.000	1.42310	0.711549	+	0.702636i	$0.247994\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	756.000	1.44262
$66$	0	0
$67$	−844.000	−1.53897	−0.769485	−	0.638665i	$-0.779487\pi$
$67$	−844.000	−1.53897	−0.769485	+	0.638665i	$0.779487\pi$
$68$	0	0
$69$	−456.000	−0.795593
$70$	0	0
$71$	8.00000	0.0133722	0.00668609	−	0.999978i	$-0.497872\pi$
$71$	8.00000	0.0133722	0.00668609	+	0.999978i	$0.497872\pi$
$72$	0	0
$73$	−422.000	−0.676594	−0.338297	−	0.941039i	$-0.609851\pi$
$73$	−422.000	−0.676594	−0.338297	+	0.941039i	$0.609851\pi$
$74$	0	0
$75$	−213.000	−0.327935
$76$	0	0
$77$	−28.0000	−0.0414402
$78$	0	0
$79$	−384.000	−0.546878	−0.273439	−	0.961889i	$-0.588161\pi$
$79$	−384.000	−0.546878	−0.273439	+	0.961889i	$0.588161\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	548.000	0.724709	0.362354	−	0.932040i	$-0.381973\pi$
$83$	548.000	0.724709	0.362354	+	0.932040i	$0.381973\pi$
$84$	0	0
$85$	−196.000	−0.250108
$86$	0	0
$87$	318.000	0.391876
$88$	0	0
$89$	1194.00	1.42206	0.711032	−	0.703159i	$-0.248228\pi$
$89$	1194.00	1.42206	0.711032	+	0.703159i	$0.248228\pi$
$90$	0	0
$91$	378.000	0.435441
$92$	0	0
$93$	−432.000	−0.481681
$94$	0	0
$95$	−1288.00	−1.39101
$96$	0	0
$97$	−1502.00	−1.57222	−0.786108	−	0.618089i	$-0.787907\pi$
$97$	−1502.00	−1.57222	−0.786108	+	0.618089i	$0.787907\pi$
$98$	0	0
$99$	−36.0000	−0.0365468
$100$	0	0
$101$	398.000	0.392104	0.196052	−	0.980594i	$-0.437188\pi$
$101$	398.000	0.392104	0.196052	+	0.980594i	$0.437188\pi$
$102$	0	0
$103$	−1160.00	−1.10969	−0.554846	−	0.831953i	$-0.687223\pi$
$103$	−1160.00	−1.10969	−0.554846	+	0.831953i	$0.687223\pi$
$104$	0	0
$105$	−294.000	−0.273252
$106$	0	0
$107$	−324.000	−0.292731	−0.146366	−	0.989231i	$-0.546758\pi$
$107$	−324.000	−0.292731	−0.146366	+	0.989231i	$0.546758\pi$
$108$	0	0
$109$	−938.000	−0.824258	−0.412129	−	0.911126i	$-0.635215\pi$
$109$	−938.000	−0.824258	−0.412129	+	0.911126i	$0.635215\pi$
$110$	0	0
$111$	−474.000	−0.405316
$112$	0	0
$113$	−622.000	−0.517813	−0.258906	−	0.965902i	$-0.583362\pi$
$113$	−622.000	−0.517813	−0.258906	+	0.965902i	$0.583362\pi$
$114$	0	0
$115$	2128.00	1.72554
$116$	0	0
$117$	486.000	0.384023
$118$	0	0
$119$	−98.0000	−0.0754928
$120$	0	0
$121$	−1315.00	−0.987979
$122$	0	0
$123$	1170.00	0.857686
$124$	0	0
$125$	−756.000	−0.540950
$126$	0	0
$127$	−1200.00	−0.838447	−0.419224	−	0.907883i	$-0.637698\pi$
$127$	−1200.00	−0.838447	−0.419224	+	0.907883i	$0.637698\pi$
$128$	0	0
$129$	−1524.00	−1.04016
$130$	0	0
$131$	1396.00	0.931062	0.465531	−	0.885032i	$-0.345863\pi$
$131$	1396.00	0.931062	0.465531	+	0.885032i	$0.345863\pi$
$132$	0	0
$133$	−644.000	−0.419864
$134$	0	0
$135$	−378.000	−0.240986
$136$	0	0
$137$	2810.00	1.75237	0.876184	−	0.481976i	$-0.160081\pi$
$137$	2810.00	1.75237	0.876184	+	0.481976i	$0.160081\pi$
$138$	0	0
$139$	−4.00000	−0.00244083	−0.00122042	−	0.999999i	$-0.500388\pi$
$139$	−4.00000	−0.00244083	−0.00122042	+	0.999999i	$0.500388\pi$
$140$	0	0
$141$	−1584.00	−0.946077
$142$	0	0
$143$	−216.000	−0.126313
$144$	0	0
$145$	−1484.00	−0.849928
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	1374.00	0.755453	0.377726	−	0.925917i	$-0.376706\pi$
$149$	1374.00	0.755453	0.377726	+	0.925917i	$0.376706\pi$
$150$	0	0
$151$	−2104.00	−1.13391	−0.566957	−	0.823747i	$-0.691880\pi$
$151$	−2104.00	−1.13391	−0.566957	+	0.823747i	$0.691880\pi$
$152$	0	0
$153$	−126.000	−0.0665784
$154$	0	0
$155$	2016.00	1.04470
$156$	0	0
$157$	3206.00	1.62972	0.814862	−	0.579655i	$-0.196813\pi$
$157$	3206.00	1.62972	0.814862	+	0.579655i	$0.196813\pi$
$158$	0	0
$159$	−1818.00	−0.906772
$160$	0	0
$161$	1064.00	0.520838
$162$	0	0
$163$	−332.000	−0.159535	−0.0797676	−	0.996813i	$-0.525418\pi$
$163$	−332.000	−0.159535	−0.0797676	+	0.996813i	$0.525418\pi$
$164$	0	0
$165$	168.000	0.0792653
$166$	0	0
$167$	−1496.00	−0.693197	−0.346599	−	0.938014i	$-0.612663\pi$
$167$	−1496.00	−0.693197	−0.346599	+	0.938014i	$0.612663\pi$
$168$	0	0
$169$	719.000	0.327264
$170$	0	0
$171$	−828.000	−0.370285
$172$	0	0
$173$	−3322.00	−1.45992	−0.729962	−	0.683487i	$-0.760462\pi$
$173$	−3322.00	−1.45992	−0.729962	+	0.683487i	$0.760462\pi$
$174$	0	0
$175$	497.000	0.214684
$176$	0	0
$177$	−1092.00	−0.463727
$178$	0	0
$179$	900.000	0.375805	0.187903	−	0.982188i	$-0.439831\pi$
$179$	900.000	0.375805	0.187903	+	0.982188i	$0.439831\pi$
$180$	0	0
$181$	1902.00	0.781075	0.390537	−	0.920587i	$-0.372289\pi$
$181$	1902.00	0.781075	0.390537	+	0.920587i	$0.372289\pi$
$182$	0	0
$183$	−2034.00	−0.821626
$184$	0	0
$185$	2212.00	0.879078
$186$	0	0
$187$	56.0000	0.0218991
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−4128.00	−1.56383	−0.781915	−	0.623385i	$-0.785757\pi$
$191$	−4128.00	−1.56383	−0.781915	+	0.623385i	$0.785757\pi$
$192$	0	0
$193$	−1342.00	−0.500514	−0.250257	−	0.968179i	$-0.580515\pi$
$193$	−1342.00	−0.500514	−0.250257	+	0.968179i	$0.580515\pi$
$194$	0	0
$195$	−2268.00	−0.832897
$196$	0	0
$197$	−3506.00	−1.26798	−0.633990	−	0.773341i	$-0.718584\pi$
$197$	−3506.00	−1.26798	−0.633990	+	0.773341i	$0.718584\pi$
$198$	0	0
$199$	−680.000	−0.242231	−0.121115	−	0.992638i	$-0.538647\pi$
$199$	−680.000	−0.242231	−0.121115	+	0.992638i	$0.538647\pi$
$200$	0	0
$201$	2532.00	0.888525
$202$	0	0
$203$	−742.000	−0.256543
$204$	0	0
$205$	−5460.00	−1.86021
$206$	0	0
$207$	1368.00	0.459336
$208$	0	0
$209$	368.000	0.121795
$210$	0	0
$211$	−5372.00	−1.75272	−0.876360	−	0.481657i	$-0.840035\pi$
$211$	−5372.00	−1.75272	−0.876360	+	0.481657i	$0.840035\pi$
$212$	0	0
$213$	−24.0000	−0.00772044
$214$	0	0
$215$	7112.00	2.25597
$216$	0	0
$217$	1008.00	0.315334
$218$	0	0
$219$	1266.00	0.390632
$220$	0	0
$221$	−756.000	−0.230109
$222$	0	0
$223$	1072.00	0.321912	0.160956	−	0.986962i	$-0.448542\pi$
$223$	1072.00	0.321912	0.160956	+	0.986962i	$0.448542\pi$
$224$	0	0
$225$	639.000	0.189333
$226$	0	0
$227$	2868.00	0.838572	0.419286	−	0.907854i	$-0.362280\pi$
$227$	2868.00	0.838572	0.419286	+	0.907854i	$0.362280\pi$
$228$	0	0
$229$	4798.00	1.38454	0.692272	−	0.721636i	$-0.256610\pi$
$229$	4798.00	1.38454	0.692272	+	0.721636i	$0.256610\pi$
$230$	0	0
$231$	84.0000	0.0239255
$232$	0	0
$233$	−5126.00	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−5126.00	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	7392.00	2.05192
$236$	0	0
$237$	1152.00	0.315740
$238$	0	0
$239$	528.000	0.142902	0.0714508	−	0.997444i	$-0.477237\pi$
$239$	528.000	0.142902	0.0714508	+	0.997444i	$0.477237\pi$
$240$	0	0
$241$	−814.000	−0.217570	−0.108785	−	0.994065i	$-0.534696\pi$
$241$	−814.000	−0.217570	−0.108785	+	0.994065i	$0.534696\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	686.000	0.178885
$246$	0	0
$247$	−4968.00	−1.27978
$248$	0	0
$249$	−1644.00	−0.418411
$250$	0	0
$251$	1932.00	0.485844	0.242922	−	0.970046i	$-0.421894\pi$
$251$	1932.00	0.485844	0.242922	+	0.970046i	$0.421894\pi$
$252$	0	0
$253$	−608.000	−0.151086
$254$	0	0
$255$	588.000	0.144400
$256$	0	0
$257$	−3294.00	−0.799510	−0.399755	−	0.916622i	$-0.630905\pi$
$257$	−3294.00	−0.799510	−0.399755	+	0.916622i	$0.630905\pi$
$258$	0	0
$259$	1106.00	0.265342
$260$	0	0
$261$	−954.000	−0.226249
$262$	0	0
$263$	7080.00	1.65997	0.829984	−	0.557787i	$-0.188350\pi$
$263$	7080.00	1.65997	0.829984	+	0.557787i	$0.188350\pi$
$264$	0	0
$265$	8484.00	1.96667
$266$	0	0
$267$	−3582.00	−0.821029
$268$	0	0
$269$	7814.00	1.77111	0.885554	−	0.464537i	$-0.153779\pi$
$269$	7814.00	1.77111	0.885554	+	0.464537i	$0.153779\pi$
$270$	0	0
$271$	−3168.00	−0.710119	−0.355060	−	0.934844i	$-0.615539\pi$
$271$	−3168.00	−0.710119	−0.355060	+	0.934844i	$0.615539\pi$
$272$	0	0
$273$	−1134.00	−0.251402
$274$	0	0
$275$	−284.000	−0.0622758
$276$	0	0
$277$	−7858.00	−1.70448	−0.852241	−	0.523150i	$-0.824757\pi$
$277$	−7858.00	−1.70448	−0.852241	+	0.523150i	$0.824757\pi$
$278$	0	0
$279$	1296.00	0.278099
$280$	0	0
$281$	6730.00	1.42875	0.714374	−	0.699764i	$-0.246712\pi$
$281$	6730.00	1.42875	0.714374	+	0.699764i	$0.246712\pi$
$282$	0	0
$283$	3020.00	0.634348	0.317174	−	0.948367i	$-0.397266\pi$
$283$	3020.00	0.634348	0.317174	+	0.948367i	$0.397266\pi$
$284$	0	0
$285$	3864.00	0.803100
$286$	0	0
$287$	−2730.00	−0.561487
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	4506.00	0.907720
$292$	0	0
$293$	−6834.00	−1.36262	−0.681308	−	0.731997i	$-0.738589\pi$
$293$	−6834.00	−1.36262	−0.681308	+	0.731997i	$0.738589\pi$
$294$	0	0
$295$	5096.00	1.00576
$296$	0	0
$297$	108.000	0.0211003
$298$	0	0
$299$	8208.00	1.58756
$300$	0	0
$301$	3556.00	0.680945
$302$	0	0
$303$	−1194.00	−0.226381
$304$	0	0
$305$	9492.00	1.78200
$306$	0	0
$307$	−2332.00	−0.433532	−0.216766	−	0.976224i	$-0.569551\pi$
$307$	−2332.00	−0.433532	−0.216766	+	0.976224i	$0.569551\pi$
$308$	0	0
$309$	3480.00	0.640681
$310$	0	0
$311$	−8840.00	−1.61180	−0.805901	−	0.592050i	$-0.798319\pi$
$311$	−8840.00	−1.61180	−0.805901	+	0.592050i	$0.798319\pi$
$312$	0	0
$313$	−1046.00	−0.188893	−0.0944464	−	0.995530i	$-0.530108\pi$
$313$	−1046.00	−0.188893	−0.0944464	+	0.995530i	$0.530108\pi$
$314$	0	0
$315$	882.000	0.157762
$316$	0	0
$317$	7542.00	1.33628	0.668140	−	0.744035i	$-0.267091\pi$
$317$	7542.00	1.33628	0.668140	+	0.744035i	$0.267091\pi$
$318$	0	0
$319$	424.000	0.0744183
$320$	0	0
$321$	972.000	0.169009
$322$	0	0
$323$	1288.00	0.221877
$324$	0	0
$325$	3834.00	0.654376
$326$	0	0
$327$	2814.00	0.475885
$328$	0	0
$329$	3696.00	0.619353
$330$	0	0
$331$	−2756.00	−0.457654	−0.228827	−	0.973467i	$-0.573489\pi$
$331$	−2756.00	−0.457654	−0.228827	+	0.973467i	$0.573489\pi$
$332$	0	0
$333$	1422.00	0.234009
$334$	0	0
$335$	−11816.0	−1.92710
$336$	0	0
$337$	3954.00	0.639134	0.319567	−	0.947564i	$-0.396463\pi$
$337$	3954.00	0.639134	0.319567	+	0.947564i	$0.396463\pi$
$338$	0	0
$339$	1866.00	0.298959
$340$	0	0
$341$	−576.000	−0.0914726
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−6384.00	−0.996241
$346$	0	0
$347$	−6900.00	−1.06747	−0.533734	−	0.845652i	$-0.679212\pi$
$347$	−6900.00	−1.06747	−0.533734	+	0.845652i	$0.679212\pi$
$348$	0	0
$349$	−2426.00	−0.372094	−0.186047	−	0.982541i	$-0.559568\pi$
$349$	−2426.00	−0.372094	−0.186047	+	0.982541i	$0.559568\pi$
$350$	0	0
$351$	−1458.00	−0.221716
$352$	0	0
$353$	−1470.00	−0.221644	−0.110822	−	0.993840i	$-0.535348\pi$
$353$	−1470.00	−0.221644	−0.110822	+	0.993840i	$0.535348\pi$
$354$	0	0
$355$	112.000	0.0167446
$356$	0	0
$357$	294.000	0.0435858
$358$	0	0
$359$	−6872.00	−1.01028	−0.505140	−	0.863038i	$-0.668559\pi$
$359$	−6872.00	−1.01028	−0.505140	+	0.863038i	$0.668559\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	3945.00	0.570410
$364$	0	0
$365$	−5908.00	−0.847230
$366$	0	0
$367$	7072.00	1.00587	0.502937	−	0.864323i	$-0.332253\pi$
$367$	7072.00	1.00587	0.502937	+	0.864323i	$0.332253\pi$
$368$	0	0
$369$	−3510.00	−0.495185
$370$	0	0
$371$	4242.00	0.593622
$372$	0	0
$373$	−818.000	−0.113551	−0.0567754	−	0.998387i	$-0.518082\pi$
$373$	−818.000	−0.113551	−0.0567754	+	0.998387i	$0.518082\pi$
$374$	0	0
$375$	2268.00	0.312317
$376$	0	0
$377$	−5724.00	−0.781966
$378$	0	0
$379$	5132.00	0.695549	0.347775	−	0.937578i	$-0.386937\pi$
$379$	5132.00	0.695549	0.347775	+	0.937578i	$0.386937\pi$
$380$	0	0
$381$	3600.00	0.484078
$382$	0	0
$383$	8576.00	1.14416	0.572080	−	0.820198i	$-0.306137\pi$
$383$	8576.00	1.14416	0.572080	+	0.820198i	$0.306137\pi$
$384$	0	0
$385$	−392.000	−0.0518914
$386$	0	0
$387$	4572.00	0.600537
$388$	0	0
$389$	−3730.00	−0.486166	−0.243083	−	0.970006i	$-0.578159\pi$
$389$	−3730.00	−0.486166	−0.243083	+	0.970006i	$0.578159\pi$
$390$	0	0
$391$	−2128.00	−0.275237
$392$	0	0
$393$	−4188.00	−0.537549
$394$	0	0
$395$	−5376.00	−0.684800
$396$	0	0
$397$	6678.00	0.844230	0.422115	−	0.906542i	$-0.361288\pi$
$397$	6678.00	0.844230	0.422115	+	0.906542i	$0.361288\pi$
$398$	0	0
$399$	1932.00	0.242408
$400$	0	0
$401$	−3054.00	−0.380323	−0.190161	−	0.981753i	$-0.560901\pi$
$401$	−3054.00	−0.380323	−0.190161	+	0.981753i	$0.560901\pi$
$402$	0	0
$403$	7776.00	0.961167
$404$	0	0
$405$	1134.00	0.139133
$406$	0	0
$407$	−632.000	−0.0769707
$408$	0	0
$409$	266.000	0.0321586	0.0160793	−	0.999871i	$-0.494882\pi$
$409$	266.000	0.0321586	0.0160793	+	0.999871i	$0.494882\pi$
$410$	0	0
$411$	−8430.00	−1.01173
$412$	0	0
$413$	2548.00	0.303581
$414$	0	0
$415$	7672.00	0.907479
$416$	0	0
$417$	12.0000	0.00140921
$418$	0	0
$419$	−8844.00	−1.03116	−0.515582	−	0.856840i	$-0.672424\pi$
$419$	−8844.00	−1.03116	−0.515582	+	0.856840i	$0.672424\pi$
$420$	0	0
$421$	−4482.00	−0.518858	−0.259429	−	0.965762i	$-0.583534\pi$
$421$	−4482.00	−0.518858	−0.259429	+	0.965762i	$0.583534\pi$
$422$	0	0
$423$	4752.00	0.546218
$424$	0	0
$425$	−994.000	−0.113450
$426$	0	0
$427$	4746.00	0.537881
$428$	0	0
$429$	648.000	0.0729271
$430$	0	0
$431$	−9936.00	−1.11044	−0.555221	−	0.831703i	$-0.687366\pi$
$431$	−9936.00	−1.11044	−0.555221	+	0.831703i	$0.687366\pi$
$432$	0	0
$433$	−11758.0	−1.30497	−0.652487	−	0.757800i	$-0.726274\pi$
$433$	−11758.0	−1.30497	−0.652487	+	0.757800i	$0.726274\pi$
$434$	0	0
$435$	4452.00	0.490706
$436$	0	0
$437$	−13984.0	−1.53077
$438$	0	0
$439$	4104.00	0.446180	0.223090	−	0.974798i	$-0.428386\pi$
$439$	4104.00	0.446180	0.223090	+	0.974798i	$0.428386\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−9748.00	−1.04547	−0.522733	−	0.852496i	$-0.675088\pi$
$443$	−9748.00	−1.04547	−0.522733	+	0.852496i	$0.675088\pi$
$444$	0	0
$445$	16716.0	1.78071
$446$	0	0
$447$	−4122.00	−0.436161
$448$	0	0
$449$	−478.000	−0.0502410	−0.0251205	−	0.999684i	$-0.507997\pi$
$449$	−478.000	−0.0502410	−0.0251205	+	0.999684i	$0.507997\pi$
$450$	0	0
$451$	1560.00	0.162877
$452$	0	0
$453$	6312.00	0.654666
$454$	0	0
$455$	5292.00	0.545259
$456$	0	0
$457$	−11174.0	−1.14376	−0.571879	−	0.820338i	$-0.693785\pi$
$457$	−11174.0	−1.14376	−0.571879	+	0.820338i	$0.693785\pi$
$458$	0	0
$459$	378.000	0.0384391
$460$	0	0
$461$	−11674.0	−1.17942	−0.589710	−	0.807615i	$-0.700758\pi$
$461$	−11674.0	−1.17942	−0.589710	+	0.807615i	$0.700758\pi$
$462$	0	0
$463$	−10528.0	−1.05676	−0.528378	−	0.849009i	$-0.677199\pi$
$463$	−10528.0	−1.05676	−0.528378	+	0.849009i	$0.677199\pi$
$464$	0	0
$465$	−6048.00	−0.603160
$466$	0	0
$467$	−16604.0	−1.64527	−0.822635	−	0.568569i	$-0.807497\pi$
$467$	−16604.0	−1.64527	−0.822635	+	0.568569i	$0.807497\pi$
$468$	0	0
$469$	−5908.00	−0.581676
$470$	0	0
$471$	−9618.00	−0.940922
$472$	0	0
$473$	−2032.00	−0.197530
$474$	0	0
$475$	−6532.00	−0.630966
$476$	0	0
$477$	5454.00	0.523525
$478$	0	0
$479$	8576.00	0.818053	0.409027	−	0.912522i	$-0.365868\pi$
$479$	8576.00	0.818053	0.409027	+	0.912522i	$0.365868\pi$
$480$	0	0
$481$	8532.00	0.808785
$482$	0	0
$483$	−3192.00	−0.300706
$484$	0	0
$485$	−21028.0	−1.96873
$486$	0	0
$487$	−9704.00	−0.902937	−0.451468	−	0.892287i	$-0.649100\pi$
$487$	−9704.00	−0.902937	−0.451468	+	0.892287i	$0.649100\pi$
$488$	0	0
$489$	996.000	0.0921077
$490$	0	0
$491$	4092.00	0.376109	0.188054	−	0.982159i	$-0.439782\pi$
$491$	4092.00	0.376109	0.188054	+	0.982159i	$0.439782\pi$
$492$	0	0
$493$	1484.00	0.135570
$494$	0	0
$495$	−504.000	−0.0457639
$496$	0	0
$497$	56.0000	0.00505421
$498$	0	0
$499$	−17884.0	−1.60440	−0.802202	−	0.597052i	$-0.796338\pi$
$499$	−17884.0	−1.60440	−0.802202	+	0.597052i	$0.796338\pi$
$500$	0	0
$501$	4488.00	0.400218
$502$	0	0
$503$	7704.00	0.682911	0.341456	−	0.939898i	$-0.389080\pi$
$503$	7704.00	0.682911	0.341456	+	0.939898i	$0.389080\pi$
$504$	0	0
$505$	5572.00	0.490992
$506$	0	0
$507$	−2157.00	−0.188946
$508$	0	0
$509$	14358.0	1.25031	0.625154	−	0.780501i	$-0.285036\pi$
$509$	14358.0	1.25031	0.625154	+	0.780501i	$0.285036\pi$
$510$	0	0
$511$	−2954.00	−0.255729
$512$	0	0
$513$	2484.00	0.213784
$514$	0	0
$515$	−16240.0	−1.38955
$516$	0	0
$517$	−2112.00	−0.179663
$518$	0	0
$519$	9966.00	0.842888
$520$	0	0
$521$	5082.00	0.427344	0.213672	−	0.976905i	$-0.431458\pi$
$521$	5082.00	0.427344	0.213672	+	0.976905i	$0.431458\pi$
$522$	0	0
$523$	1756.00	0.146816	0.0734078	−	0.997302i	$-0.476613\pi$
$523$	1756.00	0.146816	0.0734078	+	0.997302i	$0.476613\pi$
$524$	0	0
$525$	−1491.00	−0.123948
$526$	0	0
$527$	−2016.00	−0.166638
$528$	0	0
$529$	10937.0	0.898907
$530$	0	0
$531$	3276.00	0.267733
$532$	0	0
$533$	−21060.0	−1.71146
$534$	0	0
$535$	−4536.00	−0.366558
$536$	0	0
$537$	−2700.00	−0.216971
$538$	0	0
$539$	−196.000	−0.0156629
$540$	0	0
$541$	16230.0	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	16230.0	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	−5706.00	−0.450954
$544$	0	0
$545$	−13132.0	−1.03213
$546$	0	0
$547$	−17676.0	−1.38167	−0.690833	−	0.723014i	$-0.742756\pi$
$547$	−17676.0	−1.38167	−0.690833	+	0.723014i	$0.742756\pi$
$548$	0	0
$549$	6102.00	0.474366
$550$	0	0
$551$	9752.00	0.753991
$552$	0	0
$553$	−2688.00	−0.206701
$554$	0	0
$555$	−6636.00	−0.507536
$556$	0	0
$557$	−12250.0	−0.931866	−0.465933	−	0.884820i	$-0.654281\pi$
$557$	−12250.0	−0.931866	−0.465933	+	0.884820i	$0.654281\pi$
$558$	0	0
$559$	27432.0	2.07558
$560$	0	0
$561$	−168.000	−0.0126434
$562$	0	0
$563$	10052.0	0.752471	0.376236	−	0.926524i	$-0.377218\pi$
$563$	10052.0	0.752471	0.376236	+	0.926524i	$0.377218\pi$
$564$	0	0
$565$	−8708.00	−0.648404
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	25674.0	1.89158	0.945791	−	0.324776i	$-0.105289\pi$
$569$	25674.0	1.89158	0.945791	+	0.324776i	$0.105289\pi$
$570$	0	0
$571$	−3732.00	−0.273519	−0.136759	−	0.990604i	$-0.543669\pi$
$571$	−3732.00	−0.273519	−0.136759	+	0.990604i	$0.543669\pi$
$572$	0	0
$573$	12384.0	0.902878
$574$	0	0
$575$	10792.0	0.782709
$576$	0	0
$577$	−1214.00	−0.0875901	−0.0437950	−	0.999041i	$-0.513945\pi$
$577$	−1214.00	−0.0875901	−0.0437950	+	0.999041i	$0.513945\pi$
$578$	0	0
$579$	4026.00	0.288972
$580$	0	0
$581$	3836.00	0.273914
$582$	0	0
$583$	−2424.00	−0.172199
$584$	0	0
$585$	6804.00	0.480873
$586$	0	0
$587$	−7108.00	−0.499793	−0.249897	−	0.968273i	$-0.580397\pi$
$587$	−7108.00	−0.499793	−0.249897	+	0.968273i	$0.580397\pi$
$588$	0	0
$589$	−13248.0	−0.926782
$590$	0	0
$591$	10518.0	0.732069
$592$	0	0
$593$	6162.00	0.426717	0.213358	−	0.976974i	$-0.431560\pi$
$593$	6162.00	0.426717	0.213358	+	0.976974i	$0.431560\pi$
$594$	0	0
$595$	−1372.00	−0.0945320
$596$	0	0
$597$	2040.00	0.139852
$598$	0	0
$599$	−2472.00	−0.168620	−0.0843098	−	0.996440i	$-0.526869\pi$
$599$	−2472.00	−0.168620	−0.0843098	+	0.996440i	$0.526869\pi$
$600$	0	0
$601$	−13750.0	−0.933235	−0.466617	−	0.884459i	$-0.654528\pi$
$601$	−13750.0	−0.933235	−0.466617	+	0.884459i	$0.654528\pi$
$602$	0	0
$603$	−7596.00	−0.512990
$604$	0	0
$605$	−18410.0	−1.23715
$606$	0	0
$607$	11376.0	0.760688	0.380344	−	0.924845i	$-0.375806\pi$
$607$	11376.0	0.760688	0.380344	+	0.924845i	$0.375806\pi$
$608$	0	0
$609$	2226.00	0.148115
$610$	0	0
$611$	28512.0	1.88784
$612$	0	0
$613$	20382.0	1.34294	0.671469	−	0.741032i	$-0.265664\pi$
$613$	20382.0	1.34294	0.671469	+	0.741032i	$0.265664\pi$
$614$	0	0
$615$	16380.0	1.07399
$616$	0	0
$617$	21178.0	1.38184	0.690919	−	0.722932i	$-0.257206\pi$
$617$	21178.0	1.38184	0.690919	+	0.722932i	$0.257206\pi$
$618$	0	0
$619$	4700.00	0.305184	0.152592	−	0.988289i	$-0.451238\pi$
$619$	4700.00	0.305184	0.152592	+	0.988289i	$0.451238\pi$
$620$	0	0
$621$	−4104.00	−0.265198
$622$	0	0
$623$	8358.00	0.537490
$624$	0	0
$625$	−19459.0	−1.24538
$626$	0	0
$627$	−1104.00	−0.0703182
$628$	0	0
$629$	−2212.00	−0.140220
$630$	0	0
$631$	21736.0	1.37131	0.685655	−	0.727927i	$-0.259516\pi$
$631$	21736.0	1.37131	0.685655	+	0.727927i	$0.259516\pi$
$632$	0	0
$633$	16116.0	1.01193
$634$	0	0
$635$	−16800.0	−1.04990
$636$	0	0
$637$	2646.00	0.164581
$638$	0	0
$639$	72.0000	0.00445740
$640$	0	0
$641$	−13022.0	−0.802399	−0.401200	−	0.915991i	$-0.631407\pi$
$641$	−13022.0	−0.802399	−0.401200	+	0.915991i	$0.631407\pi$
$642$	0	0
$643$	−3308.00	−0.202885	−0.101442	−	0.994841i	$-0.532346\pi$
$643$	−3308.00	−0.202885	−0.101442	+	0.994841i	$0.532346\pi$
$644$	0	0
$645$	−21336.0	−1.30249
$646$	0	0
$647$	13800.0	0.838538	0.419269	−	0.907862i	$-0.362286\pi$
$647$	13800.0	0.838538	0.419269	+	0.907862i	$0.362286\pi$
$648$	0	0
$649$	−1456.00	−0.0880632
$650$	0	0
$651$	−3024.00	−0.182058
$652$	0	0
$653$	−2682.00	−0.160727	−0.0803635	−	0.996766i	$-0.525608\pi$
$653$	−2682.00	−0.160727	−0.0803635	+	0.996766i	$0.525608\pi$
$654$	0	0
$655$	19544.0	1.16587
$656$	0	0
$657$	−3798.00	−0.225531
$658$	0	0
$659$	−23836.0	−1.40898	−0.704491	−	0.709713i	$-0.748825\pi$
$659$	−23836.0	−1.40898	−0.704491	+	0.709713i	$0.748825\pi$
$660$	0	0
$661$	−11282.0	−0.663871	−0.331936	−	0.943302i	$-0.607702\pi$
$661$	−11282.0	−0.663871	−0.331936	+	0.943302i	$0.607702\pi$
$662$	0	0
$663$	2268.00	0.132853
$664$	0	0
$665$	−9016.00	−0.525753
$666$	0	0
$667$	−16112.0	−0.935321
$668$	0	0
$669$	−3216.00	−0.185856
$670$	0	0
$671$	−2712.00	−0.156029
$672$	0	0
$673$	−13726.0	−0.786179	−0.393089	−	0.919500i	$-0.628594\pi$
$673$	−13726.0	−0.786179	−0.393089	+	0.919500i	$0.628594\pi$
$674$	0	0
$675$	−1917.00	−0.109312
$676$	0	0
$677$	4974.00	0.282373	0.141186	−	0.989983i	$-0.454908\pi$
$677$	4974.00	0.282373	0.141186	+	0.989983i	$0.454908\pi$
$678$	0	0
$679$	−10514.0	−0.594242
$680$	0	0
$681$	−8604.00	−0.484150
$682$	0	0
$683$	8988.00	0.503538	0.251769	−	0.967787i	$-0.418988\pi$
$683$	8988.00	0.503538	0.251769	+	0.967787i	$0.418988\pi$
$684$	0	0
$685$	39340.0	2.19431
$686$	0	0
$687$	−14394.0	−0.799367
$688$	0	0
$689$	32724.0	1.80941
$690$	0	0
$691$	−10172.0	−0.560002	−0.280001	−	0.960000i	$-0.590335\pi$
$691$	−10172.0	−0.560002	−0.280001	+	0.960000i	$0.590335\pi$
$692$	0	0
$693$	−252.000	−0.0138134
$694$	0	0
$695$	−56.0000	−0.00305640
$696$	0	0
$697$	5460.00	0.296718
$698$	0	0
$699$	15378.0	0.832116
$700$	0	0
$701$	27446.0	1.47877	0.739387	−	0.673280i	$-0.235115\pi$
$701$	27446.0	1.47877	0.739387	+	0.673280i	$0.235115\pi$
$702$	0	0
$703$	−14536.0	−0.779852
$704$	0	0
$705$	−22176.0	−1.18468
$706$	0	0
$707$	2786.00	0.148201
$708$	0	0
$709$	3998.00	0.211774	0.105887	−	0.994378i	$-0.466232\pi$
$709$	3998.00	0.211774	0.105887	+	0.994378i	$0.466232\pi$
$710$	0	0
$711$	−3456.00	−0.182293
$712$	0	0
$713$	21888.0	1.14967
$714$	0	0
$715$	−3024.00	−0.158169
$716$	0	0
$717$	−1584.00	−0.0825043
$718$	0	0
$719$	−25872.0	−1.34195	−0.670976	−	0.741480i	$-0.734124\pi$
$719$	−25872.0	−1.34195	−0.670976	+	0.741480i	$0.734124\pi$
$720$	0	0
$721$	−8120.00	−0.419424
$722$	0	0
$723$	2442.00	0.125614
$724$	0	0
$725$	−7526.00	−0.385529
$726$	0	0
$727$	−12088.0	−0.616670	−0.308335	−	0.951278i	$-0.599772\pi$
$727$	−12088.0	−0.616670	−0.308335	+	0.951278i	$0.599772\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−7112.00	−0.359845
$732$	0	0
$733$	7974.00	0.401810	0.200905	−	0.979611i	$-0.435612\pi$
$733$	7974.00	0.401810	0.200905	+	0.979611i	$0.435612\pi$
$734$	0	0
$735$	−2058.00	−0.103280
$736$	0	0
$737$	3376.00	0.168733
$738$	0	0
$739$	31764.0	1.58113	0.790567	−	0.612376i	$-0.209786\pi$
$739$	31764.0	1.58113	0.790567	+	0.612376i	$0.209786\pi$
$740$	0	0
$741$	14904.0	0.738883
$742$	0	0
$743$	−888.000	−0.0438460	−0.0219230	−	0.999760i	$-0.506979\pi$
$743$	−888.000	−0.0438460	−0.0219230	+	0.999760i	$0.506979\pi$
$744$	0	0
$745$	19236.0	0.945977
$746$	0	0
$747$	4932.00	0.241570
$748$	0	0
$749$	−2268.00	−0.110642
$750$	0	0
$751$	34656.0	1.68391	0.841954	−	0.539549i	$-0.181405\pi$
$751$	34656.0	1.68391	0.841954	+	0.539549i	$0.181405\pi$
$752$	0	0
$753$	−5796.00	−0.280502
$754$	0	0
$755$	−29456.0	−1.41989
$756$	0	0
$757$	−22866.0	−1.09786	−0.548929	−	0.835869i	$-0.684964\pi$
$757$	−22866.0	−1.09786	−0.548929	+	0.835869i	$0.684964\pi$
$758$	0	0
$759$	1824.00	0.0872293
$760$	0	0
$761$	22570.0	1.07511	0.537557	−	0.843227i	$-0.319347\pi$
$761$	22570.0	1.07511	0.537557	+	0.843227i	$0.319347\pi$
$762$	0	0
$763$	−6566.00	−0.311540
$764$	0	0
$765$	−1764.00	−0.0833694
$766$	0	0
$767$	19656.0	0.925342
$768$	0	0
$769$	−1790.00	−0.0839389	−0.0419695	−	0.999119i	$-0.513363\pi$
$769$	−1790.00	−0.0839389	−0.0419695	+	0.999119i	$0.513363\pi$
$770$	0	0
$771$	9882.00	0.461597
$772$	0	0
$773$	2990.00	0.139124	0.0695620	−	0.997578i	$-0.477840\pi$
$773$	2990.00	0.139124	0.0695620	+	0.997578i	$0.477840\pi$
$774$	0	0
$775$	10224.0	0.473880
$776$	0	0
$777$	−3318.00	−0.153195
$778$	0	0
$779$	35880.0	1.65024
$780$	0	0
$781$	−32.0000	−0.00146613
$782$	0	0
$783$	2862.00	0.130625
$784$	0	0
$785$	44884.0	2.04074
$786$	0	0
$787$	30756.0	1.39305	0.696527	−	0.717531i	$-0.254728\pi$
$787$	30756.0	1.39305	0.696527	+	0.717531i	$0.254728\pi$
$788$	0	0
$789$	−21240.0	−0.958383
$790$	0	0
$791$	−4354.00	−0.195715
$792$	0	0
$793$	36612.0	1.63951
$794$	0	0
$795$	−25452.0	−1.13546
$796$	0	0
$797$	15126.0	0.672259	0.336129	−	0.941816i	$-0.390882\pi$
$797$	15126.0	0.672259	0.336129	+	0.941816i	$0.390882\pi$
$798$	0	0
$799$	−7392.00	−0.327297
$800$	0	0
$801$	10746.0	0.474022
$802$	0	0
$803$	1688.00	0.0741821
$804$	0	0
$805$	14896.0	0.652193
$806$	0	0
$807$	−23442.0	−1.02255
$808$	0	0
$809$	−6502.00	−0.282569	−0.141284	−	0.989969i	$-0.545123\pi$
$809$	−6502.00	−0.282569	−0.141284	+	0.989969i	$0.545123\pi$
$810$	0	0
$811$	8252.00	0.357296	0.178648	−	0.983913i	$-0.442828\pi$
$811$	8252.00	0.357296	0.178648	+	0.983913i	$0.442828\pi$
$812$	0	0
$813$	9504.00	0.409987
$814$	0	0
$815$	−4648.00	−0.199770
$816$	0	0
$817$	−46736.0	−2.00133
$818$	0	0
$819$	3402.00	0.145147
$820$	0	0
$821$	−21986.0	−0.934612	−0.467306	−	0.884096i	$-0.654775\pi$
$821$	−21986.0	−0.934612	−0.467306	+	0.884096i	$0.654775\pi$
$822$	0	0
$823$	−3736.00	−0.158237	−0.0791183	−	0.996865i	$-0.525210\pi$
$823$	−3736.00	−0.158237	−0.0791183	+	0.996865i	$0.525210\pi$
$824$	0	0
$825$	852.000	0.0359549
$826$	0	0
$827$	23820.0	1.00158	0.500788	−	0.865570i	$-0.333044\pi$
$827$	23820.0	1.00158	0.500788	+	0.865570i	$0.333044\pi$
$828$	0	0
$829$	7942.00	0.332735	0.166367	−	0.986064i	$-0.446796\pi$
$829$	7942.00	0.332735	0.166367	+	0.986064i	$0.446796\pi$
$830$	0	0
$831$	23574.0	0.984083
$832$	0	0
$833$	−686.000	−0.0285336
$834$	0	0
$835$	−20944.0	−0.868020
$836$	0	0
$837$	−3888.00	−0.160560
$838$	0	0
$839$	−21016.0	−0.864783	−0.432391	−	0.901686i	$-0.642330\pi$
$839$	−21016.0	−0.864783	−0.432391	+	0.901686i	$0.642330\pi$
$840$	0	0
$841$	−13153.0	−0.539301
$842$	0	0
$843$	−20190.0	−0.824888
$844$	0	0
$845$	10066.0	0.409800
$846$	0	0
$847$	−9205.00	−0.373421
$848$	0	0
$849$	−9060.00	−0.366241
$850$	0	0
$851$	24016.0	0.967401
$852$	0	0
$853$	24878.0	0.998601	0.499300	−	0.866429i	$-0.333590\pi$
$853$	24878.0	0.998601	0.499300	+	0.866429i	$0.333590\pi$
$854$	0	0
$855$	−11592.0	−0.463670
$856$	0	0
$857$	−6390.00	−0.254700	−0.127350	−	0.991858i	$-0.540647\pi$
$857$	−6390.00	−0.254700	−0.127350	+	0.991858i	$0.540647\pi$
$858$	0	0
$859$	46444.0	1.84476	0.922380	−	0.386284i	$-0.126241\pi$
$859$	46444.0	1.84476	0.922380	+	0.386284i	$0.126241\pi$
$860$	0	0
$861$	8190.00	0.324175
$862$	0	0
$863$	25408.0	1.00220	0.501100	−	0.865389i	$-0.332929\pi$
$863$	25408.0	1.00220	0.501100	+	0.865389i	$0.332929\pi$
$864$	0	0
$865$	−46508.0	−1.82811
$866$	0	0
$867$	14151.0	0.554317
$868$	0	0
$869$	1536.00	0.0599600
$870$	0	0
$871$	−45576.0	−1.77300
$872$	0	0
$873$	−13518.0	−0.524072
$874$	0	0
$875$	−5292.00	−0.204460
$876$	0	0
$877$	1078.00	0.0415068	0.0207534	−	0.999785i	$-0.493394\pi$
$877$	1078.00	0.0415068	0.0207534	+	0.999785i	$0.493394\pi$
$878$	0	0
$879$	20502.0	0.786707
$880$	0	0
$881$	−45006.0	−1.72110	−0.860551	−	0.509364i	$-0.829881\pi$
$881$	−45006.0	−1.72110	−0.860551	+	0.509364i	$0.829881\pi$
$882$	0	0
$883$	−4028.00	−0.153514	−0.0767571	−	0.997050i	$-0.524457\pi$
$883$	−4028.00	−0.153514	−0.0767571	+	0.997050i	$0.524457\pi$
$884$	0	0
$885$	−15288.0	−0.580679
$886$	0	0
$887$	29304.0	1.10928	0.554640	−	0.832090i	$-0.312856\pi$
$887$	29304.0	1.10928	0.554640	+	0.832090i	$0.312856\pi$
$888$	0	0
$889$	−8400.00	−0.316903
$890$	0	0
$891$	−324.000	−0.0121823
$892$	0	0
$893$	−48576.0	−1.82031
$894$	0	0
$895$	12600.0	0.470583
$896$	0	0
$897$	−24624.0	−0.916579
$898$	0	0
$899$	−15264.0	−0.566277
$900$	0	0
$901$	−8484.00	−0.313699
$902$	0	0
$903$	−10668.0	−0.393144
$904$	0	0
$905$	26628.0	0.978060
$906$	0	0
$907$	−50916.0	−1.86399	−0.931995	−	0.362472i	$-0.881933\pi$
$907$	−50916.0	−1.86399	−0.931995	+	0.362472i	$0.881933\pi$
$908$	0	0
$909$	3582.00	0.130701
$910$	0	0
$911$	24432.0	0.888549	0.444275	−	0.895891i	$-0.353461\pi$
$911$	24432.0	0.888549	0.444275	+	0.895891i	$0.353461\pi$
$912$	0	0
$913$	−2192.00	−0.0794574
$914$	0	0
$915$	−28476.0	−1.02884
$916$	0	0
$917$	9772.00	0.351908
$918$	0	0
$919$	20360.0	0.730810	0.365405	−	0.930849i	$-0.380930\pi$
$919$	20360.0	0.730810	0.365405	+	0.930849i	$0.380930\pi$
$920$	0	0
$921$	6996.00	0.250300
$922$	0	0
$923$	432.000	0.0154057
$924$	0	0
$925$	11218.0	0.398752
$926$	0	0
$927$	−10440.0	−0.369897
$928$	0	0
$929$	23202.0	0.819411	0.409706	−	0.912218i	$-0.365631\pi$
$929$	23202.0	0.819411	0.409706	+	0.912218i	$0.365631\pi$
$930$	0	0
$931$	−4508.00	−0.158694
$932$	0	0
$933$	26520.0	0.930574
$934$	0	0
$935$	784.000	0.0274220
$936$	0	0
$937$	−1990.00	−0.0693815	−0.0346908	−	0.999398i	$-0.511045\pi$
$937$	−1990.00	−0.0693815	−0.0346908	+	0.999398i	$0.511045\pi$
$938$	0	0
$939$	3138.00	0.109057
$940$	0	0
$941$	−51130.0	−1.77130	−0.885648	−	0.464356i	$-0.846286\pi$
$941$	−51130.0	−1.77130	−0.885648	+	0.464356i	$0.846286\pi$
$942$	0	0
$943$	−59280.0	−2.04711
$944$	0	0
$945$	−2646.00	−0.0910840
$946$	0	0
$947$	47044.0	1.61428	0.807141	−	0.590359i	$-0.201014\pi$
$947$	47044.0	1.61428	0.807141	+	0.590359i	$0.201014\pi$
$948$	0	0
$949$	−22788.0	−0.779483
$950$	0	0
$951$	−22626.0	−0.771502
$952$	0	0
$953$	46858.0	1.59274	0.796369	−	0.604811i	$-0.206751\pi$
$953$	46858.0	1.59274	0.796369	+	0.604811i	$0.206751\pi$
$954$	0	0
$955$	−57792.0	−1.95823
$956$	0	0
$957$	−1272.00	−0.0429654
$958$	0	0
$959$	19670.0	0.662333
$960$	0	0
$961$	−9055.00	−0.303951
$962$	0	0
$963$	−2916.00	−0.0975771
$964$	0	0
$965$	−18788.0	−0.626743
$966$	0	0
$967$	−30632.0	−1.01867	−0.509337	−	0.860567i	$-0.670110\pi$
$967$	−30632.0	−1.01867	−0.509337	+	0.860567i	$0.670110\pi$
$968$	0	0
$969$	−3864.00	−0.128101
$970$	0	0
$971$	3804.00	0.125722	0.0628611	−	0.998022i	$-0.479978\pi$
$971$	3804.00	0.125722	0.0628611	+	0.998022i	$0.479978\pi$
$972$	0	0
$973$	−28.0000	−0.000922548 0
$974$	0	0
$975$	−11502.0	−0.377804
$976$	0	0
$977$	−49326.0	−1.61523	−0.807614	−	0.589711i	$-0.799242\pi$
$977$	−49326.0	−1.61523	−0.807614	+	0.589711i	$0.799242\pi$
$978$	0	0
$979$	−4776.00	−0.155916
$980$	0	0
$981$	−8442.00	−0.274753
$982$	0	0
$983$	−11112.0	−0.360547	−0.180274	−	0.983617i	$-0.557698\pi$
$983$	−11112.0	−0.360547	−0.180274	+	0.983617i	$0.557698\pi$
$984$	0	0
$985$	−49084.0	−1.58776
$986$	0	0
$987$	−11088.0	−0.357584
$988$	0	0
$989$	77216.0	2.48263
$990$	0	0
$991$	−13616.0	−0.436455	−0.218227	−	0.975898i	$-0.570027\pi$
$991$	−13616.0	−0.436455	−0.218227	+	0.975898i	$0.570027\pi$
$992$	0	0
$993$	8268.00	0.264227
$994$	0	0
$995$	−9520.00	−0.303321
$996$	0	0
$997$	−56674.0	−1.80028	−0.900142	−	0.435596i	$-0.856538\pi$
$997$	−56674.0	−1.80028	−0.900142	+	0.435596i	$0.856538\pi$
$998$	0	0
$999$	−4266.00	−0.135105

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.a.e.1.1		1
3.2	odd	2		1008.4.a.d.1.1		1
4.3	odd	2		84.4.a.b.1.1	✓	1
7.6	odd	2		2352.4.a.v.1.1		1
8.3	odd	2		1344.4.a.b.1.1		1
8.5	even	2		1344.4.a.p.1.1		1
12.11	even	2		252.4.a.a.1.1		1
20.3	even	4		2100.4.k.g.1849.2		2
20.7	even	4		2100.4.k.g.1849.1		2
20.19	odd	2		2100.4.a.g.1.1		1
28.3	even	6		588.4.i.h.373.1		2
28.11	odd	6		588.4.i.a.373.1		2
28.19	even	6		588.4.i.h.361.1		2
28.23	odd	6		588.4.i.a.361.1		2
28.27	even	2		588.4.a.a.1.1		1
84.11	even	6		1764.4.k.n.1549.1		2
84.23	even	6		1764.4.k.n.361.1		2
84.47	odd	6		1764.4.k.c.361.1		2
84.59	odd	6		1764.4.k.c.1549.1		2
84.83	odd	2		1764.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.a.b.1.1	✓	1	4.3	odd	2
252.4.a.a.1.1		1	12.11	even	2
336.4.a.e.1.1		1	1.1	even	1	trivial
588.4.a.a.1.1		1	28.27	even	2
588.4.i.a.361.1		2	28.23	odd	6
588.4.i.a.373.1		2	28.11	odd	6
588.4.i.h.361.1		2	28.19	even	6
588.4.i.h.373.1		2	28.3	even	6
1008.4.a.d.1.1		1	3.2	odd	2
1344.4.a.b.1.1		1	8.3	odd	2
1344.4.a.p.1.1		1	8.5	even	2
1764.4.a.l.1.1		1	84.83	odd	2
1764.4.k.c.361.1		2	84.47	odd	6
1764.4.k.c.1549.1		2	84.59	odd	6
1764.4.k.n.361.1		2	84.23	even	6
1764.4.k.n.1549.1		2	84.11	even	6
2100.4.a.g.1.1		1	20.19	odd	2
2100.4.k.g.1849.1		2	20.7	even	4
2100.4.k.g.1849.2		2	20.3	even	4
2352.4.a.v.1.1		1	7.6	odd	2

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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

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Fields

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	336.4.a.e.1.1

Level	$336$
Weight	$4$
Character	336.1
Self dual	yes
Analytic conductor	$19.825$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$