LMFDB - Embedded newform 1216.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$71.7463225670$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +8.00000 q^{5} +17.0000 q^{7} -26.0000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +8.00000 q^{5} +17.0000 q^{7} -26.0000 q^{9} -70.0000 q^{11} +61.0000 q^{13} -8.00000 q^{15} +83.0000 q^{17} +19.0000 q^{19} -17.0000 q^{21} -115.000 q^{23} -61.0000 q^{25} +53.0000 q^{27} -279.000 q^{29} +72.0000 q^{31} +70.0000 q^{33} +136.000 q^{35} +34.0000 q^{37} -61.0000 q^{39} +108.000 q^{41} -192.000 q^{43} -208.000 q^{45} +392.000 q^{47} -54.0000 q^{49} -83.0000 q^{51} -131.000 q^{53} -560.000 q^{55} -19.0000 q^{57} -609.000 q^{59} -338.000 q^{61} -442.000 q^{63} +488.000 q^{65} -461.000 q^{67} +115.000 q^{69} -750.000 q^{71} +1177.00 q^{73} +61.0000 q^{75} -1190.00 q^{77} +22.0000 q^{79} +649.000 q^{81} -810.000 q^{83} +664.000 q^{85} +279.000 q^{87} -476.000 q^{89} +1037.00 q^{91} -72.0000 q^{93} +152.000 q^{95} +1426.00 q^{97} +1820.00 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.192450	−0.0962250	−	0.995360i	$-0.530677\pi$
$3$	−1.00000	−0.192450	−0.0962250	+	0.995360i	$0.530677\pi$
$4$	0	0
$5$	8.00000	0.715542	0.357771	−	0.933809i	$-0.383537\pi$
$5$	8.00000	0.715542	0.357771	+	0.933809i	$0.383537\pi$
$6$	0	0
$7$	17.0000	0.917914	0.458957	−	0.888459i	$-0.348223\pi$
$7$	17.0000	0.917914	0.458957	+	0.888459i	$0.348223\pi$
$8$	0	0
$9$	−26.0000	−0.962963
$10$	0	0
$11$	−70.0000	−1.91871	−0.959354	−	0.282204i	$-0.908934\pi$
$11$	−70.0000	−1.91871	−0.959354	+	0.282204i	$0.908934\pi$
$12$	0	0
$13$	61.0000	1.30141	0.650706	−	0.759330i	$-0.274473\pi$
$13$	61.0000	1.30141	0.650706	+	0.759330i	$0.274473\pi$
$14$	0	0
$15$	−8.00000	−0.137706
$16$	0	0
$17$	83.0000	1.18414	0.592072	−	0.805885i	$-0.298310\pi$
$17$	83.0000	1.18414	0.592072	+	0.805885i	$0.298310\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−17.0000	−0.176653
$22$	0	0
$23$	−115.000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−115.000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	−61.0000	−0.488000
$26$	0	0
$27$	53.0000	0.377772
$28$	0	0
$29$	−279.000	−1.78652	−0.893259	−	0.449543i	$-0.851587\pi$
$29$	−279.000	−1.78652	−0.893259	+	0.449543i	$0.851587\pi$
$30$	0	0
$31$	72.0000	0.417148	0.208574	−	0.978007i	$-0.433118\pi$
$31$	72.0000	0.417148	0.208574	+	0.978007i	$0.433118\pi$
$32$	0	0
$33$	70.0000	0.369256
$34$	0	0
$35$	136.000	0.656806
$36$	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
$39$	−61.0000	−0.250457
$40$	0	0
$41$	108.000	0.411385	0.205692	−	0.978617i	$-0.434055\pi$
$41$	108.000	0.411385	0.205692	+	0.978617i	$0.434055\pi$
$42$	0	0
$43$	−192.000	−0.680924	−0.340462	−	0.940258i	$-0.610583\pi$
$43$	−192.000	−0.680924	−0.340462	+	0.940258i	$0.610583\pi$
$44$	0	0
$45$	−208.000	−0.689040
$46$	0	0
$47$	392.000	1.21658	0.608288	−	0.793716i	$-0.291857\pi$
$47$	392.000	1.21658	0.608288	+	0.793716i	$0.291857\pi$
$48$	0	0
$49$	−54.0000	−0.157434
$50$	0	0
$51$	−83.0000	−0.227889
$52$	0	0
$53$	−131.000	−0.339514	−0.169757	−	0.985486i	$-0.554298\pi$
$53$	−131.000	−0.339514	−0.169757	+	0.985486i	$0.554298\pi$
$54$	0	0
$55$	−560.000	−1.37292
$56$	0	0
$57$	−19.0000	−0.0441511
$58$	0	0
$59$	−609.000	−1.34381	−0.671907	−	0.740635i	$-0.734525\pi$
$59$	−609.000	−1.34381	−0.671907	+	0.740635i	$0.734525\pi$
$60$	0	0
$61$	−338.000	−0.709450	−0.354725	−	0.934971i	$-0.615426\pi$
$61$	−338.000	−0.709450	−0.354725	+	0.934971i	$0.615426\pi$
$62$	0	0
$63$	−442.000	−0.883917
$64$	0	0
$65$	488.000	0.931215
$66$	0	0
$67$	−461.000	−0.840599	−0.420299	−	0.907386i	$-0.638075\pi$
$67$	−461.000	−0.840599	−0.420299	+	0.907386i	$0.638075\pi$
$68$	0	0
$69$	115.000	0.200643
$70$	0	0
$71$	−750.000	−1.25364	−0.626821	−	0.779163i	$-0.715644\pi$
$71$	−750.000	−1.25364	−0.626821	+	0.779163i	$0.715644\pi$
$72$	0	0
$73$	1177.00	1.88709	0.943544	−	0.331247i	$-0.107469\pi$
$73$	1177.00	1.88709	0.943544	+	0.331247i	$0.107469\pi$
$74$	0	0
$75$	61.0000	0.0939156
$76$	0	0
$77$	−1190.00	−1.76121
$78$	0	0
$79$	22.0000	0.0313316	0.0156658	−	0.999877i	$-0.495013\pi$
$79$	22.0000	0.0313316	0.0156658	+	0.999877i	$0.495013\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	0	0
$83$	−810.000	−1.07119	−0.535597	−	0.844474i	$-0.679913\pi$
$83$	−810.000	−1.07119	−0.535597	+	0.844474i	$0.679913\pi$
$84$	0	0
$85$	664.000	0.847305
$86$	0	0
$87$	279.000	0.343815
$88$	0	0
$89$	−476.000	−0.566920	−0.283460	−	0.958984i	$-0.591482\pi$
$89$	−476.000	−0.566920	−0.283460	+	0.958984i	$0.591482\pi$
$90$	0	0
$91$	1037.00	1.19458
$92$	0	0
$93$	−72.0000	−0.0802801
$94$	0	0
$95$	152.000	0.164157
$96$	0	0
$97$	1426.00	1.49266	0.746332	−	0.665574i	$-0.231813\pi$
$97$	1426.00	1.49266	0.746332	+	0.665574i	$0.231813\pi$
$98$	0	0
$99$	1820.00	1.84765
$100$	0	0
$101$	1230.00	1.21178	0.605889	−	0.795549i	$-0.292818\pi$
$101$	1230.00	1.21178	0.605889	+	0.795549i	$0.292818\pi$
$102$	0	0
$103$	−310.000	−0.296555	−0.148278	−	0.988946i	$-0.547373\pi$
$103$	−310.000	−0.296555	−0.148278	+	0.988946i	$0.547373\pi$
$104$	0	0
$105$	−136.000	−0.126402
$106$	0	0
$107$	−1791.00	−1.61815	−0.809077	−	0.587702i	$-0.800033\pi$
$107$	−1791.00	−1.61815	−0.809077	+	0.587702i	$0.800033\pi$
$108$	0	0
$109$	779.000	0.684538	0.342269	−	0.939602i	$-0.388805\pi$
$109$	779.000	0.684538	0.342269	+	0.939602i	$0.388805\pi$
$110$	0	0
$111$	−34.0000	−0.0290733
$112$	0	0
$113$	−1430.00	−1.19047	−0.595235	−	0.803552i	$-0.702941\pi$
$113$	−1430.00	−1.19047	−0.595235	+	0.803552i	$0.702941\pi$
$114$	0	0
$115$	−920.000	−0.746004
$116$	0	0
$117$	−1586.00	−1.25321
$118$	0	0
$119$	1411.00	1.08694
$120$	0	0
$121$	3569.00	2.68144
$122$	0	0
$123$	−108.000	−0.0791710
$124$	0	0
$125$	−1488.00	−1.06473
$126$	0	0
$127$	−466.000	−0.325597	−0.162798	−	0.986659i	$-0.552052\pi$
$127$	−466.000	−0.325597	−0.162798	+	0.986659i	$0.552052\pi$
$128$	0	0
$129$	192.000	0.131044
$130$	0	0
$131$	2240.00	1.49397	0.746984	−	0.664842i	$-0.231501\pi$
$131$	2240.00	1.49397	0.746984	+	0.664842i	$0.231501\pi$
$132$	0	0
$133$	323.000	0.210584
$134$	0	0
$135$	424.000	0.270312
$136$	0	0
$137$	−1145.00	−0.714043	−0.357022	−	0.934096i	$-0.616208\pi$
$137$	−1145.00	−0.714043	−0.357022	+	0.934096i	$0.616208\pi$
$138$	0	0
$139$	−2336.00	−1.42545	−0.712723	−	0.701446i	$-0.752538\pi$
$139$	−2336.00	−1.42545	−0.712723	+	0.701446i	$0.752538\pi$
$140$	0	0
$141$	−392.000	−0.234130
$142$	0	0
$143$	−4270.00	−2.49703
$144$	0	0
$145$	−2232.00	−1.27833
$146$	0	0
$147$	54.0000	0.0302983
$148$	0	0
$149$	−3516.00	−1.93317	−0.966584	−	0.256351i	$-0.917480\pi$
$149$	−3516.00	−1.93317	−0.966584	+	0.256351i	$0.917480\pi$
$150$	0	0
$151$	390.000	0.210184	0.105092	−	0.994463i	$-0.466486\pi$
$151$	390.000	0.210184	0.105092	+	0.994463i	$0.466486\pi$
$152$	0	0
$153$	−2158.00	−1.14029
$154$	0	0
$155$	576.000	0.298487
$156$	0	0
$157$	−2306.00	−1.17222	−0.586111	−	0.810231i	$-0.699342\pi$
$157$	−2306.00	−1.17222	−0.586111	+	0.810231i	$0.699342\pi$
$158$	0	0
$159$	131.000	0.0653395
$160$	0	0
$161$	−1955.00	−0.956991
$162$	0	0
$163$	−1272.00	−0.611231	−0.305616	−	0.952155i	$-0.598862\pi$
$163$	−1272.00	−0.611231	−0.305616	+	0.952155i	$0.598862\pi$
$164$	0	0
$165$	560.000	0.264218
$166$	0	0
$167$	−1768.00	−0.819233	−0.409617	−	0.912258i	$-0.634338\pi$
$167$	−1768.00	−0.819233	−0.409617	+	0.912258i	$0.634338\pi$
$168$	0	0
$169$	1524.00	0.693673
$170$	0	0
$171$	−494.000	−0.220919
$172$	0	0
$173$	3726.00	1.63747	0.818736	−	0.574171i	$-0.194675\pi$
$173$	3726.00	1.63747	0.818736	+	0.574171i	$0.194675\pi$
$174$	0	0
$175$	−1037.00	−0.447942
$176$	0	0
$177$	609.000	0.258617
$178$	0	0
$179$	−1048.00	−0.437604	−0.218802	−	0.975769i	$-0.570215\pi$
$179$	−1048.00	−0.437604	−0.218802	+	0.975769i	$0.570215\pi$
$180$	0	0
$181$	−2678.00	−1.09975	−0.549873	−	0.835248i	$-0.685324\pi$
$181$	−2678.00	−1.09975	−0.549873	+	0.835248i	$0.685324\pi$
$182$	0	0
$183$	338.000	0.136534
$184$	0	0
$185$	272.000	0.108096
$186$	0	0
$187$	−5810.00	−2.27203
$188$	0	0
$189$	901.000	0.346762
$190$	0	0
$191$	−1443.00	−0.546659	−0.273329	−	0.961921i	$-0.588125\pi$
$191$	−1443.00	−0.546659	−0.273329	+	0.961921i	$0.588125\pi$
$192$	0	0
$193$	890.000	0.331936	0.165968	−	0.986131i	$-0.446925\pi$
$193$	890.000	0.331936	0.165968	+	0.986131i	$0.446925\pi$
$194$	0	0
$195$	−488.000	−0.179212
$196$	0	0
$197$	300.000	0.108498	0.0542490	−	0.998527i	$-0.482724\pi$
$197$	300.000	0.108498	0.0542490	+	0.998527i	$0.482724\pi$
$198$	0	0
$199$	−3339.00	−1.18942	−0.594712	−	0.803939i	$-0.702734\pi$
$199$	−3339.00	−1.18942	−0.594712	+	0.803939i	$0.702734\pi$
$200$	0	0
$201$	461.000	0.161773
$202$	0	0
$203$	−4743.00	−1.63987
$204$	0	0
$205$	864.000	0.294363
$206$	0	0
$207$	2990.00	1.00396
$208$	0	0
$209$	−1330.00	−0.440182
$210$	0	0
$211$	−2149.00	−0.701153	−0.350576	−	0.936534i	$-0.614014\pi$
$211$	−2149.00	−0.701153	−0.350576	+	0.936534i	$0.614014\pi$
$212$	0	0
$213$	750.000	0.241264
$214$	0	0
$215$	−1536.00	−0.487229
$216$	0	0
$217$	1224.00	0.382906
$218$	0	0
$219$	−1177.00	−0.363170
$220$	0	0
$221$	5063.00	1.54106
$222$	0	0
$223$	−3834.00	−1.15132	−0.575658	−	0.817690i	$-0.695254\pi$
$223$	−3834.00	−1.15132	−0.575658	+	0.817690i	$0.695254\pi$
$224$	0	0
$225$	1586.00	0.469926
$226$	0	0
$227$	−329.000	−0.0961960	−0.0480980	−	0.998843i	$-0.515316\pi$
$227$	−329.000	−0.0961960	−0.0480980	+	0.998843i	$0.515316\pi$
$228$	0	0
$229$	−430.000	−0.124084	−0.0620419	−	0.998074i	$-0.519761\pi$
$229$	−430.000	−0.124084	−0.0620419	+	0.998074i	$0.519761\pi$
$230$	0	0
$231$	1190.00	0.338945
$232$	0	0
$233$	−438.000	−0.123152	−0.0615758	−	0.998102i	$-0.519613\pi$
$233$	−438.000	−0.123152	−0.0615758	+	0.998102i	$0.519613\pi$
$234$	0	0
$235$	3136.00	0.870511
$236$	0	0
$237$	−22.0000	−0.00602976
$238$	0	0
$239$	−2099.00	−0.568088	−0.284044	−	0.958811i	$-0.591676\pi$
$239$	−2099.00	−0.568088	−0.284044	+	0.958811i	$0.591676\pi$
$240$	0	0
$241$	−2996.00	−0.800786	−0.400393	−	0.916344i	$-0.631126\pi$
$241$	−2996.00	−0.800786	−0.400393	+	0.916344i	$0.631126\pi$
$242$	0	0
$243$	−2080.00	−0.549103
$244$	0	0
$245$	−432.000	−0.112651
$246$	0	0
$247$	1159.00	0.298564
$248$	0	0
$249$	810.000	0.206151
$250$	0	0
$251$	5518.00	1.38762	0.693811	−	0.720157i	$-0.255930\pi$
$251$	5518.00	1.38762	0.693811	+	0.720157i	$0.255930\pi$
$252$	0	0
$253$	8050.00	2.00039
$254$	0	0
$255$	−664.000	−0.163064
$256$	0	0
$257$	−4068.00	−0.987373	−0.493687	−	0.869640i	$-0.664351\pi$
$257$	−4068.00	−0.987373	−0.493687	+	0.869640i	$0.664351\pi$
$258$	0	0
$259$	578.000	0.138669
$260$	0	0
$261$	7254.00	1.72035
$262$	0	0
$263$	−4992.00	−1.17042	−0.585209	−	0.810883i	$-0.698988\pi$
$263$	−4992.00	−1.17042	−0.585209	+	0.810883i	$0.698988\pi$
$264$	0	0
$265$	−1048.00	−0.242936
$266$	0	0
$267$	476.000	0.109104
$268$	0	0
$269$	1970.00	0.446517	0.223258	−	0.974759i	$-0.428331\pi$
$269$	1970.00	0.446517	0.223258	+	0.974759i	$0.428331\pi$
$270$	0	0
$271$	1861.00	0.417150	0.208575	−	0.978006i	$-0.433117\pi$
$271$	1861.00	0.417150	0.208575	+	0.978006i	$0.433117\pi$
$272$	0	0
$273$	−1037.00	−0.229898
$274$	0	0
$275$	4270.00	0.936330
$276$	0	0
$277$	1268.00	0.275042	0.137521	−	0.990499i	$-0.456086\pi$
$277$	1268.00	0.275042	0.137521	+	0.990499i	$0.456086\pi$
$278$	0	0
$279$	−1872.00	−0.401698
$280$	0	0
$281$	−8912.00	−1.89198	−0.945988	−	0.324201i	$-0.894905\pi$
$281$	−8912.00	−1.89198	−0.945988	+	0.324201i	$0.894905\pi$
$282$	0	0
$283$	3302.00	0.693581	0.346791	−	0.937943i	$-0.387271\pi$
$283$	3302.00	0.693581	0.346791	+	0.937943i	$0.387271\pi$
$284$	0	0
$285$	−152.000	−0.0315919
$286$	0	0
$287$	1836.00	0.377616
$288$	0	0
$289$	1976.00	0.402198
$290$	0	0
$291$	−1426.00	−0.287263
$292$	0	0
$293$	5435.00	1.08367	0.541836	−	0.840484i	$-0.317729\pi$
$293$	5435.00	1.08367	0.541836	+	0.840484i	$0.317729\pi$
$294$	0	0
$295$	−4872.00	−0.961555
$296$	0	0
$297$	−3710.00	−0.724835
$298$	0	0
$299$	−7015.00	−1.35682
$300$	0	0
$301$	−3264.00	−0.625029
$302$	0	0
$303$	−1230.00	−0.233207
$304$	0	0
$305$	−2704.00	−0.507641
$306$	0	0
$307$	1740.00	0.323476	0.161738	−	0.986834i	$-0.448290\pi$
$307$	1740.00	0.323476	0.161738	+	0.986834i	$0.448290\pi$
$308$	0	0
$309$	310.000	0.0570721
$310$	0	0
$311$	2837.00	0.517272	0.258636	−	0.965975i	$-0.416727\pi$
$311$	2837.00	0.517272	0.258636	+	0.965975i	$0.416727\pi$
$312$	0	0
$313$	−1579.00	−0.285145	−0.142572	−	0.989784i	$-0.545537\pi$
$313$	−1579.00	−0.285145	−0.142572	+	0.989784i	$0.545537\pi$
$314$	0	0
$315$	−3536.00	−0.632479
$316$	0	0
$317$	−10065.0	−1.78330	−0.891651	−	0.452723i	$-0.850452\pi$
$317$	−10065.0	−1.78330	−0.891651	+	0.452723i	$0.850452\pi$
$318$	0	0
$319$	19530.0	3.42781
$320$	0	0
$321$	1791.00	0.311414
$322$	0	0
$323$	1577.00	0.271661
$324$	0	0
$325$	−3721.00	−0.635089
$326$	0	0
$327$	−779.000	−0.131739
$328$	0	0
$329$	6664.00	1.11671
$330$	0	0
$331$	2953.00	0.490367	0.245184	−	0.969477i	$-0.421152\pi$
$331$	2953.00	0.490367	0.245184	+	0.969477i	$0.421152\pi$
$332$	0	0
$333$	−884.000	−0.145474
$334$	0	0
$335$	−3688.00	−0.601483
$336$	0	0
$337$	−988.000	−0.159703	−0.0798513	−	0.996807i	$-0.525445\pi$
$337$	−988.000	−0.159703	−0.0798513	+	0.996807i	$0.525445\pi$
$338$	0	0
$339$	1430.00	0.229106
$340$	0	0
$341$	−5040.00	−0.800385
$342$	0	0
$343$	−6749.00	−1.06242
$344$	0	0
$345$	920.000	0.143569
$346$	0	0
$347$	−4458.00	−0.689677	−0.344839	−	0.938662i	$-0.612066\pi$
$347$	−4458.00	−0.689677	−0.344839	+	0.938662i	$0.612066\pi$
$348$	0	0
$349$	3522.00	0.540196	0.270098	−	0.962833i	$-0.412944\pi$
$349$	3522.00	0.540196	0.270098	+	0.962833i	$0.412944\pi$
$350$	0	0
$351$	3233.00	0.491638
$352$	0	0
$353$	8809.00	1.32820	0.664102	−	0.747642i	$-0.268814\pi$
$353$	8809.00	1.32820	0.664102	+	0.747642i	$0.268814\pi$
$354$	0	0
$355$	−6000.00	−0.897034
$356$	0	0
$357$	−1411.00	−0.209182
$358$	0	0
$359$	12611.0	1.85399	0.926996	−	0.375071i	$-0.122382\pi$
$359$	12611.0	1.85399	0.926996	+	0.375071i	$0.122382\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−3569.00	−0.516044
$364$	0	0
$365$	9416.00	1.35029
$366$	0	0
$367$	−9412.00	−1.33870	−0.669349	−	0.742948i	$-0.733427\pi$
$367$	−9412.00	−1.33870	−0.669349	+	0.742948i	$0.733427\pi$
$368$	0	0
$369$	−2808.00	−0.396148
$370$	0	0
$371$	−2227.00	−0.311644
$372$	0	0
$373$	−6553.00	−0.909655	−0.454828	−	0.890579i	$-0.650299\pi$
$373$	−6553.00	−0.909655	−0.454828	+	0.890579i	$0.650299\pi$
$374$	0	0
$375$	1488.00	0.204907
$376$	0	0
$377$	−17019.0	−2.32499
$378$	0	0
$379$	14111.0	1.91249	0.956245	−	0.292568i	$-0.0945099\pi$
$379$	14111.0	1.91249	0.956245	+	0.292568i	$0.0945099\pi$
$380$	0	0
$381$	466.000	0.0626612
$382$	0	0
$383$	9902.00	1.32107	0.660533	−	0.750797i	$-0.270330\pi$
$383$	9902.00	1.32107	0.660533	+	0.750797i	$0.270330\pi$
$384$	0	0
$385$	−9520.00	−1.26022
$386$	0	0
$387$	4992.00	0.655704
$388$	0	0
$389$	−258.000	−0.0336276	−0.0168138	−	0.999859i	$-0.505352\pi$
$389$	−258.000	−0.0336276	−0.0168138	+	0.999859i	$0.505352\pi$
$390$	0	0
$391$	−9545.00	−1.23456
$392$	0	0
$393$	−2240.00	−0.287514
$394$	0	0
$395$	176.000	0.0224190
$396$	0	0
$397$	−1224.00	−0.154738	−0.0773688	−	0.997003i	$-0.524652\pi$
$397$	−1224.00	−0.154738	−0.0773688	+	0.997003i	$0.524652\pi$
$398$	0	0
$399$	−323.000	−0.0405269
$400$	0	0
$401$	12104.0	1.50734	0.753672	−	0.657251i	$-0.228281\pi$
$401$	12104.0	1.50734	0.753672	+	0.657251i	$0.228281\pi$
$402$	0	0
$403$	4392.00	0.542881
$404$	0	0
$405$	5192.00	0.637019
$406$	0	0
$407$	−2380.00	−0.289858
$408$	0	0
$409$	6076.00	0.734569	0.367285	−	0.930109i	$-0.380287\pi$
$409$	6076.00	0.734569	0.367285	+	0.930109i	$0.380287\pi$
$410$	0	0
$411$	1145.00	0.137418
$412$	0	0
$413$	−10353.0	−1.23351
$414$	0	0
$415$	−6480.00	−0.766484
$416$	0	0
$417$	2336.00	0.274327
$418$	0	0
$419$	8148.00	0.950014	0.475007	−	0.879982i	$-0.342446\pi$
$419$	8148.00	0.950014	0.475007	+	0.879982i	$0.342446\pi$
$420$	0	0
$421$	13849.0	1.60323	0.801614	−	0.597842i	$-0.203975\pi$
$421$	13849.0	1.60323	0.801614	+	0.597842i	$0.203975\pi$
$422$	0	0
$423$	−10192.0	−1.17152
$424$	0	0
$425$	−5063.00	−0.577863
$426$	0	0
$427$	−5746.00	−0.651214
$428$	0	0
$429$	4270.00	0.480554
$430$	0	0
$431$	9322.00	1.04182	0.520911	−	0.853611i	$-0.325592\pi$
$431$	9322.00	1.04182	0.520911	+	0.853611i	$0.325592\pi$
$432$	0	0
$433$	7090.00	0.786891	0.393445	−	0.919348i	$-0.371283\pi$
$433$	7090.00	0.786891	0.393445	+	0.919348i	$0.371283\pi$
$434$	0	0
$435$	2232.00	0.246014
$436$	0	0
$437$	−2185.00	−0.239182
$438$	0	0
$439$	−9740.00	−1.05892	−0.529459	−	0.848336i	$-0.677605\pi$
$439$	−9740.00	−1.05892	−0.529459	+	0.848336i	$0.677605\pi$
$440$	0	0
$441$	1404.00	0.151603
$442$	0	0
$443$	−4658.00	−0.499567	−0.249784	−	0.968302i	$-0.580359\pi$
$443$	−4658.00	−0.499567	−0.249784	+	0.968302i	$0.580359\pi$
$444$	0	0
$445$	−3808.00	−0.405655
$446$	0	0
$447$	3516.00	0.372038
$448$	0	0
$449$	14314.0	1.50450	0.752249	−	0.658879i	$-0.228969\pi$
$449$	14314.0	1.50450	0.752249	+	0.658879i	$0.228969\pi$
$450$	0	0
$451$	−7560.00	−0.789327
$452$	0	0
$453$	−390.000	−0.0404499
$454$	0	0
$455$	8296.00	0.854775
$456$	0	0
$457$	3641.00	0.372689	0.186344	−	0.982484i	$-0.440336\pi$
$457$	3641.00	0.372689	0.186344	+	0.982484i	$0.440336\pi$
$458$	0	0
$459$	4399.00	0.447337
$460$	0	0
$461$	11540.0	1.16588	0.582941	−	0.812515i	$-0.301902\pi$
$461$	11540.0	1.16588	0.582941	+	0.812515i	$0.301902\pi$
$462$	0	0
$463$	13732.0	1.37836	0.689179	−	0.724591i	$-0.257971\pi$
$463$	13732.0	1.37836	0.689179	+	0.724591i	$0.257971\pi$
$464$	0	0
$465$	−576.000	−0.0574438
$466$	0	0
$467$	15978.0	1.58324	0.791621	−	0.611013i	$-0.209238\pi$
$467$	15978.0	1.58324	0.791621	+	0.611013i	$0.209238\pi$
$468$	0	0
$469$	−7837.00	−0.771597
$470$	0	0
$471$	2306.00	0.225594
$472$	0	0
$473$	13440.0	1.30649
$474$	0	0
$475$	−1159.00	−0.111955
$476$	0	0
$477$	3406.00	0.326939
$478$	0	0
$479$	4300.00	0.410171	0.205086	−	0.978744i	$-0.434253\pi$
$479$	4300.00	0.410171	0.205086	+	0.978744i	$0.434253\pi$
$480$	0	0
$481$	2074.00	0.196603
$482$	0	0
$483$	1955.00	0.184173
$484$	0	0
$485$	11408.0	1.06806
$486$	0	0
$487$	12326.0	1.14691	0.573454	−	0.819238i	$-0.305603\pi$
$487$	12326.0	1.14691	0.573454	+	0.819238i	$0.305603\pi$
$488$	0	0
$489$	1272.00	0.117632
$490$	0	0
$491$	−7236.00	−0.665084	−0.332542	−	0.943088i	$-0.607906\pi$
$491$	−7236.00	−0.665084	−0.332542	+	0.943088i	$0.607906\pi$
$492$	0	0
$493$	−23157.0	−2.11549
$494$	0	0
$495$	14560.0	1.32207
$496$	0	0
$497$	−12750.0	−1.15074
$498$	0	0
$499$	−1148.00	−0.102989	−0.0514945	−	0.998673i	$-0.516398\pi$
$499$	−1148.00	−0.102989	−0.0514945	+	0.998673i	$0.516398\pi$
$500$	0	0
$501$	1768.00	0.157662
$502$	0	0
$503$	−5739.00	−0.508726	−0.254363	−	0.967109i	$-0.581866\pi$
$503$	−5739.00	−0.508726	−0.254363	+	0.967109i	$0.581866\pi$
$504$	0	0
$505$	9840.00	0.867078
$506$	0	0
$507$	−1524.00	−0.133497
$508$	0	0
$509$	−8442.00	−0.735138	−0.367569	−	0.929996i	$-0.619810\pi$
$509$	−8442.00	−0.735138	−0.367569	+	0.929996i	$0.619810\pi$
$510$	0	0
$511$	20009.0	1.73218
$512$	0	0
$513$	1007.00	0.0866669
$514$	0	0
$515$	−2480.00	−0.212198
$516$	0	0
$517$	−27440.0	−2.33425
$518$	0	0
$519$	−3726.00	−0.315131
$520$	0	0
$521$	744.000	0.0625628	0.0312814	−	0.999511i	$-0.490041\pi$
$521$	744.000	0.0625628	0.0312814	+	0.999511i	$0.490041\pi$
$522$	0	0
$523$	10013.0	0.837166	0.418583	−	0.908179i	$-0.362527\pi$
$523$	10013.0	0.837166	0.418583	+	0.908179i	$0.362527\pi$
$524$	0	0
$525$	1037.00	0.0862065
$526$	0	0
$527$	5976.00	0.493963
$528$	0	0
$529$	1058.00	0.0869565
$530$	0	0
$531$	15834.0	1.29404
$532$	0	0
$533$	6588.00	0.535381
$534$	0	0
$535$	−14328.0	−1.15786
$536$	0	0
$537$	1048.00	0.0842170
$538$	0	0
$539$	3780.00	0.302071
$540$	0	0
$541$	−14582.0	−1.15883	−0.579417	−	0.815031i	$-0.696720\pi$
$541$	−14582.0	−1.15883	−0.579417	+	0.815031i	$0.696720\pi$
$542$	0	0
$543$	2678.00	0.211646
$544$	0	0
$545$	6232.00	0.489816
$546$	0	0
$547$	17924.0	1.40105	0.700526	−	0.713627i	$-0.252949\pi$
$547$	17924.0	1.40105	0.700526	+	0.713627i	$0.252949\pi$
$548$	0	0
$549$	8788.00	0.683174
$550$	0	0
$551$	−5301.00	−0.409855
$552$	0	0
$553$	374.000	0.0287597
$554$	0	0
$555$	−272.000	−0.0208032
$556$	0	0
$557$	−11960.0	−0.909805	−0.454903	−	0.890541i	$-0.650326\pi$
$557$	−11960.0	−0.909805	−0.454903	+	0.890541i	$0.650326\pi$
$558$	0	0
$559$	−11712.0	−0.886162
$560$	0	0
$561$	5810.00	0.437252
$562$	0	0
$563$	1348.00	0.100908	0.0504542	−	0.998726i	$-0.483933\pi$
$563$	1348.00	0.100908	0.0504542	+	0.998726i	$0.483933\pi$
$564$	0	0
$565$	−11440.0	−0.851831
$566$	0	0
$567$	11033.0	0.817182
$568$	0	0
$569$	−9820.00	−0.723508	−0.361754	−	0.932274i	$-0.617822\pi$
$569$	−9820.00	−0.723508	−0.361754	+	0.932274i	$0.617822\pi$
$570$	0	0
$571$	−20904.0	−1.53206	−0.766029	−	0.642806i	$-0.777770\pi$
$571$	−20904.0	−1.53206	−0.766029	+	0.642806i	$0.777770\pi$
$572$	0	0
$573$	1443.00	0.105205
$574$	0	0
$575$	7015.00	0.508775
$576$	0	0
$577$	−15093.0	−1.08896	−0.544480	−	0.838774i	$-0.683273\pi$
$577$	−15093.0	−1.08896	−0.544480	+	0.838774i	$0.683273\pi$
$578$	0	0
$579$	−890.000	−0.0638811
$580$	0	0
$581$	−13770.0	−0.983263
$582$	0	0
$583$	9170.00	0.651428
$584$	0	0
$585$	−12688.0	−0.896725
$586$	0	0
$587$	−16740.0	−1.17706	−0.588530	−	0.808476i	$-0.700293\pi$
$587$	−16740.0	−1.17706	−0.588530	+	0.808476i	$0.700293\pi$
$588$	0	0
$589$	1368.00	0.0957003
$590$	0	0
$591$	−300.000	−0.0208805
$592$	0	0
$593$	3042.00	0.210658	0.105329	−	0.994437i	$-0.466410\pi$
$593$	3042.00	0.210658	0.105329	+	0.994437i	$0.466410\pi$
$594$	0	0
$595$	11288.0	0.777753
$596$	0	0
$597$	3339.00	0.228905
$598$	0	0
$599$	−27816.0	−1.89738	−0.948690	−	0.316207i	$-0.897591\pi$
$599$	−27816.0	−1.89738	−0.948690	+	0.316207i	$0.897591\pi$
$600$	0	0
$601$	19320.0	1.31128	0.655640	−	0.755074i	$-0.272399\pi$
$601$	19320.0	1.31128	0.655640	+	0.755074i	$0.272399\pi$
$602$	0	0
$603$	11986.0	0.809465
$604$	0	0
$605$	28552.0	1.91868
$606$	0	0
$607$	−17926.0	−1.19867	−0.599336	−	0.800498i	$-0.704569\pi$
$607$	−17926.0	−1.19867	−0.599336	+	0.800498i	$0.704569\pi$
$608$	0	0
$609$	4743.00	0.315593
$610$	0	0
$611$	23912.0	1.58327
$612$	0	0
$613$	−18294.0	−1.20536	−0.602682	−	0.797982i	$-0.705901\pi$
$613$	−18294.0	−1.20536	−0.602682	+	0.797982i	$0.705901\pi$
$614$	0	0
$615$	−864.000	−0.0566502
$616$	0	0
$617$	22538.0	1.47058	0.735288	−	0.677755i	$-0.237047\pi$
$617$	22538.0	1.47058	0.735288	+	0.677755i	$0.237047\pi$
$618$	0	0
$619$	−17558.0	−1.14009	−0.570045	−	0.821614i	$-0.693074\pi$
$619$	−17558.0	−1.14009	−0.570045	+	0.821614i	$0.693074\pi$
$620$	0	0
$621$	−6095.00	−0.393855
$622$	0	0
$623$	−8092.00	−0.520384
$624$	0	0
$625$	−4279.00	−0.273856
$626$	0	0
$627$	1330.00	0.0847131
$628$	0	0
$629$	2822.00	0.178888
$630$	0	0
$631$	12440.0	0.784831	0.392416	−	0.919788i	$-0.371639\pi$
$631$	12440.0	0.784831	0.392416	+	0.919788i	$0.371639\pi$
$632$	0	0
$633$	2149.00	0.134937
$634$	0	0
$635$	−3728.00	−0.232978
$636$	0	0
$637$	−3294.00	−0.204887
$638$	0	0
$639$	19500.0	1.20721
$640$	0	0
$641$	−10842.0	−0.668071	−0.334035	−	0.942561i	$-0.608410\pi$
$641$	−10842.0	−0.668071	−0.334035	+	0.942561i	$0.608410\pi$
$642$	0	0
$643$	322.000	0.0197487	0.00987437	−	0.999951i	$-0.496857\pi$
$643$	322.000	0.0197487	0.00987437	+	0.999951i	$0.496857\pi$
$644$	0	0
$645$	1536.00	0.0937674
$646$	0	0
$647$	−2995.00	−0.181987	−0.0909935	−	0.995851i	$-0.529004\pi$
$647$	−2995.00	−0.181987	−0.0909935	+	0.995851i	$0.529004\pi$
$648$	0	0
$649$	42630.0	2.57839
$650$	0	0
$651$	−1224.00	−0.0736902
$652$	0	0
$653$	−14652.0	−0.878066	−0.439033	−	0.898471i	$-0.644679\pi$
$653$	−14652.0	−0.878066	−0.439033	+	0.898471i	$0.644679\pi$
$654$	0	0
$655$	17920.0	1.06900
$656$	0	0
$657$	−30602.0	−1.81720
$658$	0	0
$659$	20613.0	1.21847	0.609233	−	0.792992i	$-0.291478\pi$
$659$	20613.0	1.21847	0.609233	+	0.792992i	$0.291478\pi$
$660$	0	0
$661$	7875.00	0.463392	0.231696	−	0.972788i	$-0.425573\pi$
$661$	7875.00	0.463392	0.231696	+	0.972788i	$0.425573\pi$
$662$	0	0
$663$	−5063.00	−0.296577
$664$	0	0
$665$	2584.00	0.150682
$666$	0	0
$667$	32085.0	1.86257
$668$	0	0
$669$	3834.00	0.221571
$670$	0	0
$671$	23660.0	1.36123
$672$	0	0
$673$	6216.00	0.356031	0.178016	−	0.984028i	$-0.443032\pi$
$673$	6216.00	0.356031	0.178016	+	0.984028i	$0.443032\pi$
$674$	0	0
$675$	−3233.00	−0.184353
$676$	0	0
$677$	−13961.0	−0.792563	−0.396281	−	0.918129i	$-0.629699\pi$
$677$	−13961.0	−0.792563	−0.396281	+	0.918129i	$0.629699\pi$
$678$	0	0
$679$	24242.0	1.37014
$680$	0	0
$681$	329.000	0.0185129
$682$	0	0
$683$	−7788.00	−0.436310	−0.218155	−	0.975914i	$-0.570004\pi$
$683$	−7788.00	−0.436310	−0.218155	+	0.975914i	$0.570004\pi$
$684$	0	0
$685$	−9160.00	−0.510928
$686$	0	0
$687$	430.000	0.0238799
$688$	0	0
$689$	−7991.00	−0.441847
$690$	0	0
$691$	−11782.0	−0.648637	−0.324319	−	0.945948i	$-0.605135\pi$
$691$	−11782.0	−0.648637	−0.324319	+	0.945948i	$0.605135\pi$
$692$	0	0
$693$	30940.0	1.69598
$694$	0	0
$695$	−18688.0	−1.01997
$696$	0	0
$697$	8964.00	0.487139
$698$	0	0
$699$	438.000	0.0237005
$700$	0	0
$701$	22700.0	1.22306	0.611532	−	0.791220i	$-0.290554\pi$
$701$	22700.0	1.22306	0.611532	+	0.791220i	$0.290554\pi$
$702$	0	0
$703$	646.000	0.0346577
$704$	0	0
$705$	−3136.00	−0.167530
$706$	0	0
$707$	20910.0	1.11231
$708$	0	0
$709$	−12162.0	−0.644222	−0.322111	−	0.946702i	$-0.604392\pi$
$709$	−12162.0	−0.644222	−0.322111	+	0.946702i	$0.604392\pi$
$710$	0	0
$711$	−572.000	−0.0301711
$712$	0	0
$713$	−8280.00	−0.434907
$714$	0	0
$715$	−34160.0	−1.78673
$716$	0	0
$717$	2099.00	0.109329
$718$	0	0
$719$	−2199.00	−0.114060	−0.0570298	−	0.998372i	$-0.518163\pi$
$719$	−2199.00	−0.114060	−0.0570298	+	0.998372i	$0.518163\pi$
$720$	0	0
$721$	−5270.00	−0.272212
$722$	0	0
$723$	2996.00	0.154111
$724$	0	0
$725$	17019.0	0.871820
$726$	0	0
$727$	18965.0	0.967501	0.483750	−	0.875206i	$-0.339274\pi$
$727$	18965.0	0.967501	0.483750	+	0.875206i	$0.339274\pi$
$728$	0	0
$729$	−15443.0	−0.784586
$730$	0	0
$731$	−15936.0	−0.806312
$732$	0	0
$733$	5552.00	0.279765	0.139883	−	0.990168i	$-0.455328\pi$
$733$	5552.00	0.279765	0.139883	+	0.990168i	$0.455328\pi$
$734$	0	0
$735$	432.000	0.0216797
$736$	0	0
$737$	32270.0	1.61286
$738$	0	0
$739$	13136.0	0.653878	0.326939	−	0.945046i	$-0.393983\pi$
$739$	13136.0	0.653878	0.326939	+	0.945046i	$0.393983\pi$
$740$	0	0
$741$	−1159.00	−0.0574587
$742$	0	0
$743$	13416.0	0.662430	0.331215	−	0.943555i	$-0.392541\pi$
$743$	13416.0	0.662430	0.331215	+	0.943555i	$0.392541\pi$
$744$	0	0
$745$	−28128.0	−1.38326
$746$	0	0
$747$	21060.0	1.03152
$748$	0	0
$749$	−30447.0	−1.48533
$750$	0	0
$751$	33348.0	1.62035	0.810177	−	0.586185i	$-0.199371\pi$
$751$	33348.0	1.62035	0.810177	+	0.586185i	$0.199371\pi$
$752$	0	0
$753$	−5518.00	−0.267048
$754$	0	0
$755$	3120.00	0.150395
$756$	0	0
$757$	−10606.0	−0.509223	−0.254611	−	0.967043i	$-0.581948\pi$
$757$	−10606.0	−0.509223	−0.254611	+	0.967043i	$0.581948\pi$
$758$	0	0
$759$	−8050.00	−0.384976
$760$	0	0
$761$	−29061.0	−1.38431	−0.692155	−	0.721749i	$-0.743339\pi$
$761$	−29061.0	−1.38431	−0.692155	+	0.721749i	$0.743339\pi$
$762$	0	0
$763$	13243.0	0.628347
$764$	0	0
$765$	−17264.0	−0.815923
$766$	0	0
$767$	−37149.0	−1.74886
$768$	0	0
$769$	−15955.0	−0.748182	−0.374091	−	0.927392i	$-0.622045\pi$
$769$	−15955.0	−0.748182	−0.374091	+	0.927392i	$0.622045\pi$
$770$	0	0
$771$	4068.00	0.190020
$772$	0	0
$773$	36763.0	1.71057	0.855287	−	0.518155i	$-0.173381\pi$
$773$	36763.0	1.71057	0.855287	+	0.518155i	$0.173381\pi$
$774$	0	0
$775$	−4392.00	−0.203568
$776$	0	0
$777$	−578.000	−0.0266868
$778$	0	0
$779$	2052.00	0.0943781
$780$	0	0
$781$	52500.0	2.40537
$782$	0	0
$783$	−14787.0	−0.674897
$784$	0	0
$785$	−18448.0	−0.838774
$786$	0	0
$787$	31655.0	1.43377	0.716886	−	0.697190i	$-0.245567\pi$
$787$	31655.0	1.43377	0.716886	+	0.697190i	$0.245567\pi$
$788$	0	0
$789$	4992.00	0.225247
$790$	0	0
$791$	−24310.0	−1.09275
$792$	0	0
$793$	−20618.0	−0.923287
$794$	0	0
$795$	1048.00	0.0467531
$796$	0	0
$797$	12945.0	0.575327	0.287663	−	0.957732i	$-0.407122\pi$
$797$	12945.0	0.575327	0.287663	+	0.957732i	$0.407122\pi$
$798$	0	0
$799$	32536.0	1.44060
$800$	0	0
$801$	12376.0	0.545923
$802$	0	0
$803$	−82390.0	−3.62077
$804$	0	0
$805$	−15640.0	−0.684767
$806$	0	0
$807$	−1970.00	−0.0859322
$808$	0	0
$809$	−20263.0	−0.880605	−0.440302	−	0.897850i	$-0.645129\pi$
$809$	−20263.0	−0.880605	−0.440302	+	0.897850i	$0.645129\pi$
$810$	0	0
$811$	29477.0	1.27630	0.638149	−	0.769913i	$-0.279700\pi$
$811$	29477.0	1.27630	0.638149	+	0.769913i	$0.279700\pi$
$812$	0	0
$813$	−1861.00	−0.0802806
$814$	0	0
$815$	−10176.0	−0.437362
$816$	0	0
$817$	−3648.00	−0.156215
$818$	0	0
$819$	−26962.0	−1.15034
$820$	0	0
$821$	1756.00	0.0746466	0.0373233	−	0.999303i	$-0.488117\pi$
$821$	1756.00	0.0746466	0.0373233	+	0.999303i	$0.488117\pi$
$822$	0	0
$823$	−1721.00	−0.0728922	−0.0364461	−	0.999336i	$-0.511604\pi$
$823$	−1721.00	−0.0728922	−0.0364461	+	0.999336i	$0.511604\pi$
$824$	0	0
$825$	−4270.00	−0.180197
$826$	0	0
$827$	3063.00	0.128792	0.0643960	−	0.997924i	$-0.479488\pi$
$827$	3063.00	0.128792	0.0643960	+	0.997924i	$0.479488\pi$
$828$	0	0
$829$	25467.0	1.06695	0.533477	−	0.845814i	$-0.320885\pi$
$829$	25467.0	1.06695	0.533477	+	0.845814i	$0.320885\pi$
$830$	0	0
$831$	−1268.00	−0.0529319
$832$	0	0
$833$	−4482.00	−0.186425
$834$	0	0
$835$	−14144.0	−0.586196
$836$	0	0
$837$	3816.00	0.157587
$838$	0	0
$839$	−2976.00	−0.122459	−0.0612294	−	0.998124i	$-0.519502\pi$
$839$	−2976.00	−0.122459	−0.0612294	+	0.998124i	$0.519502\pi$
$840$	0	0
$841$	53452.0	2.19164
$842$	0	0
$843$	8912.00	0.364111
$844$	0	0
$845$	12192.0	0.496352
$846$	0	0
$847$	60673.0	2.46133
$848$	0	0
$849$	−3302.00	−0.133480
$850$	0	0
$851$	−3910.00	−0.157501
$852$	0	0
$853$	−2126.00	−0.0853375	−0.0426687	−	0.999089i	$-0.513586\pi$
$853$	−2126.00	−0.0853375	−0.0426687	+	0.999089i	$0.513586\pi$
$854$	0	0
$855$	−3952.00	−0.158077
$856$	0	0
$857$	−4684.00	−0.186701	−0.0933503	−	0.995633i	$-0.529758\pi$
$857$	−4684.00	−0.186701	−0.0933503	+	0.995633i	$0.529758\pi$
$858$	0	0
$859$	1370.00	0.0544165	0.0272083	−	0.999630i	$-0.491338\pi$
$859$	1370.00	0.0544165	0.0272083	+	0.999630i	$0.491338\pi$
$860$	0	0
$861$	−1836.00	−0.0726721
$862$	0	0
$863$	−30630.0	−1.20818	−0.604089	−	0.796917i	$-0.706463\pi$
$863$	−30630.0	−1.20818	−0.604089	+	0.796917i	$0.706463\pi$
$864$	0	0
$865$	29808.0	1.17168
$866$	0	0
$867$	−1976.00	−0.0774031
$868$	0	0
$869$	−1540.00	−0.0601161
$870$	0	0
$871$	−28121.0	−1.09397
$872$	0	0
$873$	−37076.0	−1.43738
$874$	0	0
$875$	−25296.0	−0.977327
$876$	0	0
$877$	−19617.0	−0.755324	−0.377662	−	0.925944i	$-0.623272\pi$
$877$	−19617.0	−0.755324	−0.377662	+	0.925944i	$0.623272\pi$
$878$	0	0
$879$	−5435.00	−0.208553
$880$	0	0
$881$	1614.00	0.0617220	0.0308610	−	0.999524i	$-0.490175\pi$
$881$	1614.00	0.0617220	0.0308610	+	0.999524i	$0.490175\pi$
$882$	0	0
$883$	35258.0	1.34374	0.671872	−	0.740667i	$-0.265490\pi$
$883$	35258.0	1.34374	0.671872	+	0.740667i	$0.265490\pi$
$884$	0	0
$885$	4872.00	0.185051
$886$	0	0
$887$	−2166.00	−0.0819923	−0.0409961	−	0.999159i	$-0.513053\pi$
$887$	−2166.00	−0.0819923	−0.0409961	+	0.999159i	$0.513053\pi$
$888$	0	0
$889$	−7922.00	−0.298870
$890$	0	0
$891$	−45430.0	−1.70815
$892$	0	0
$893$	7448.00	0.279102
$894$	0	0
$895$	−8384.00	−0.313124
$896$	0	0
$897$	7015.00	0.261119
$898$	0	0
$899$	−20088.0	−0.745242
$900$	0	0
$901$	−10873.0	−0.402033
$902$	0	0
$903$	3264.00	0.120287
$904$	0	0
$905$	−21424.0	−0.786915
$906$	0	0
$907$	−30419.0	−1.11361	−0.556806	−	0.830642i	$-0.687973\pi$
$907$	−30419.0	−1.11361	−0.556806	+	0.830642i	$0.687973\pi$
$908$	0	0
$909$	−31980.0	−1.16690
$910$	0	0
$911$	−32864.0	−1.19521	−0.597603	−	0.801792i	$-0.703880\pi$
$911$	−32864.0	−1.19521	−0.597603	+	0.801792i	$0.703880\pi$
$912$	0	0
$913$	56700.0	2.05531
$914$	0	0
$915$	2704.00	0.0976956
$916$	0	0
$917$	38080.0	1.37133
$918$	0	0
$919$	−20225.0	−0.725964	−0.362982	−	0.931796i	$-0.618241\pi$
$919$	−20225.0	−0.725964	−0.362982	+	0.931796i	$0.618241\pi$
$920$	0	0
$921$	−1740.00	−0.0622529
$922$	0	0
$923$	−45750.0	−1.63151
$924$	0	0
$925$	−2074.00	−0.0737218
$926$	0	0
$927$	8060.00	0.285572
$928$	0	0
$929$	−29415.0	−1.03883	−0.519416	−	0.854522i	$-0.673850\pi$
$929$	−29415.0	−1.03883	−0.519416	+	0.854522i	$0.673850\pi$
$930$	0	0
$931$	−1026.00	−0.0361179
$932$	0	0
$933$	−2837.00	−0.0995490
$934$	0	0
$935$	−46480.0	−1.62573
$936$	0	0
$937$	3697.00	0.128896	0.0644481	−	0.997921i	$-0.479471\pi$
$937$	3697.00	0.128896	0.0644481	+	0.997921i	$0.479471\pi$
$938$	0	0
$939$	1579.00	0.0548762
$940$	0	0
$941$	55245.0	1.91385	0.956926	−	0.290331i	$-0.0937653\pi$
$941$	55245.0	1.91385	0.956926	+	0.290331i	$0.0937653\pi$
$942$	0	0
$943$	−12420.0	−0.428898
$944$	0	0
$945$	7208.00	0.248123
$946$	0	0
$947$	54296.0	1.86313	0.931564	−	0.363576i	$-0.118444\pi$
$947$	54296.0	1.86313	0.931564	+	0.363576i	$0.118444\pi$
$948$	0	0
$949$	71797.0	2.45588
$950$	0	0
$951$	10065.0	0.343197
$952$	0	0
$953$	−25770.0	−0.875941	−0.437971	−	0.898989i	$-0.644303\pi$
$953$	−25770.0	−0.875941	−0.437971	+	0.898989i	$0.644303\pi$
$954$	0	0
$955$	−11544.0	−0.391157
$956$	0	0
$957$	−19530.0	−0.659682
$958$	0	0
$959$	−19465.0	−0.655430
$960$	0	0
$961$	−24607.0	−0.825988
$962$	0	0
$963$	46566.0	1.55822
$964$	0	0
$965$	7120.00	0.237514
$966$	0	0
$967$	−20296.0	−0.674949	−0.337474	−	0.941335i	$-0.609573\pi$
$967$	−20296.0	−0.674949	−0.337474	+	0.941335i	$0.609573\pi$
$968$	0	0
$969$	−1577.00	−0.0522813
$970$	0	0
$971$	34476.0	1.13943	0.569715	−	0.821842i	$-0.307053\pi$
$971$	34476.0	1.13943	0.569715	+	0.821842i	$0.307053\pi$
$972$	0	0
$973$	−39712.0	−1.30844
$974$	0	0
$975$	3721.00	0.122223
$976$	0	0
$977$	39952.0	1.30827	0.654134	−	0.756379i	$-0.273033\pi$
$977$	39952.0	1.30827	0.654134	+	0.756379i	$0.273033\pi$
$978$	0	0
$979$	33320.0	1.08775
$980$	0	0
$981$	−20254.0	−0.659185
$982$	0	0
$983$	56942.0	1.84758	0.923788	−	0.382904i	$-0.125076\pi$
$983$	56942.0	1.84758	0.923788	+	0.382904i	$0.125076\pi$
$984$	0	0
$985$	2400.00	0.0776349
$986$	0	0
$987$	−6664.00	−0.214911
$988$	0	0
$989$	22080.0	0.709912
$990$	0	0
$991$	−45772.0	−1.46720	−0.733600	−	0.679581i	$-0.762161\pi$
$991$	−45772.0	−1.46720	−0.733600	+	0.679581i	$0.762161\pi$
$992$	0	0
$993$	−2953.00	−0.0943712
$994$	0	0
$995$	−26712.0	−0.851083
$996$	0	0
$997$	24916.0	0.791472	0.395736	−	0.918364i	$-0.370490\pi$
$997$	24916.0	0.791472	0.395736	+	0.918364i	$0.370490\pi$
$998$	0	0
$999$	1802.00	0.0570698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.c.1.1		1
4.3	odd	2		1216.4.a.d.1.1		1
8.3	odd	2		608.4.a.a.1.1	✓	1
8.5	even	2		608.4.a.b.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.a.1.1	✓	1	8.3	odd	2
608.4.a.b.1.1	yes	1	8.5	even	2
1216.4.a.c.1.1		1	1.1	even	1	trivial
1216.4.a.d.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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