LMFDB - Embedded newform 2112.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,4,Mod(1,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2112.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2112.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:	$124.612033932$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +14.0000 q^{5} -32.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +14.0000 q^{5} -32.0000 q^{7} +9.00000 q^{9} +11.0000 q^{11} +38.0000 q^{13} -42.0000 q^{15} -2.00000 q^{17} -72.0000 q^{19} +96.0000 q^{21} +68.0000 q^{23} +71.0000 q^{25} -27.0000 q^{27} +54.0000 q^{29} -152.000 q^{31} -33.0000 q^{33} -448.000 q^{35} -174.000 q^{37} -114.000 q^{39} +94.0000 q^{41} +528.000 q^{43} +126.000 q^{45} -340.000 q^{47} +681.000 q^{49} +6.00000 q^{51} +438.000 q^{53} +154.000 q^{55} +216.000 q^{57} -20.0000 q^{59} -570.000 q^{61} -288.000 q^{63} +532.000 q^{65} +460.000 q^{67} -204.000 q^{69} -1092.00 q^{71} +562.000 q^{73} -213.000 q^{75} -352.000 q^{77} -16.0000 q^{79} +81.0000 q^{81} -372.000 q^{83} -28.0000 q^{85} -162.000 q^{87} -966.000 q^{89} -1216.00 q^{91} +456.000 q^{93} -1008.00 q^{95} -526.000 q^{97} +99.0000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	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
$7$	−32.0000	−1.72784	−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000	−1.72784	−0.863919	+	0.503631i	$0.831997\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	0	0
$15$	−42.0000	−0.722957
$16$	0	0
$17$	−2.00000	−0.0285336	−0.0142668	−	0.999898i	$-0.504541\pi$
$17$	−2.00000	−0.0285336	−0.0142668	+	0.999898i	$0.504541\pi$
$18$	0	0
$19$	−72.0000	−0.869365	−0.434682	−	0.900584i	$-0.643139\pi$
$19$	−72.0000	−0.869365	−0.434682	+	0.900584i	$0.643139\pi$
$20$	0	0
$21$	96.0000	0.997567
$22$	0	0
$23$	68.0000	0.616477	0.308239	−	0.951309i	$-0.400260\pi$
$23$	68.0000	0.616477	0.308239	+	0.951309i	$0.400260\pi$
$24$	0	0
$25$	71.0000	0.568000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	54.0000	0.345778	0.172889	−	0.984941i	$-0.444690\pi$
$29$	54.0000	0.345778	0.172889	+	0.984941i	$0.444690\pi$
$30$	0	0
$31$	−152.000	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
$31$	−152.000	−0.880645	−0.440323	+	0.897840i	$0.645136\pi$
$32$	0	0
$33$	−33.0000	−0.174078
$34$	0	0
$35$	−448.000	−2.16359
$36$	0	0
$37$	−174.000	−0.773120	−0.386560	−	0.922264i	$-0.626337\pi$
$37$	−174.000	−0.773120	−0.386560	+	0.922264i	$0.626337\pi$
$38$	0	0
$39$	−114.000	−0.468067
$40$	0	0
$41$	94.0000	0.358057	0.179028	−	0.983844i	$-0.442705\pi$
$41$	94.0000	0.358057	0.179028	+	0.983844i	$0.442705\pi$
$42$	0	0
$43$	528.000	1.87254	0.936270	−	0.351280i	$-0.114254\pi$
$43$	528.000	1.87254	0.936270	+	0.351280i	$0.114254\pi$
$44$	0	0
$45$	126.000	0.417399
$46$	0	0
$47$	−340.000	−1.05519	−0.527597	−	0.849495i	$-0.676907\pi$
$47$	−340.000	−1.05519	−0.527597	+	0.849495i	$0.676907\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	6.00000	0.0164739
$52$	0	0
$53$	438.000	1.13517	0.567584	−	0.823315i	$-0.307878\pi$
$53$	438.000	1.13517	0.567584	+	0.823315i	$0.307878\pi$
$54$	0	0
$55$	154.000	0.377552
$56$	0	0
$57$	216.000	0.501928
$58$	0	0
$59$	−20.0000	−0.0441318	−0.0220659	−	0.999757i	$-0.507024\pi$
$59$	−20.0000	−0.0441318	−0.0220659	+	0.999757i	$0.507024\pi$
$60$	0	0
$61$	−570.000	−1.19641	−0.598205	−	0.801343i	$-0.704119\pi$
$61$	−570.000	−1.19641	−0.598205	+	0.801343i	$0.704119\pi$
$62$	0	0
$63$	−288.000	−0.575946
$64$	0	0
$65$	532.000	1.01518
$66$	0	0
$67$	460.000	0.838775	0.419388	−	0.907807i	$-0.362245\pi$
$67$	460.000	0.838775	0.419388	+	0.907807i	$0.362245\pi$
$68$	0	0
$69$	−204.000	−0.355923
$70$	0	0
$71$	−1092.00	−1.82530	−0.912652	−	0.408738i	$-0.865969\pi$
$71$	−1092.00	−1.82530	−0.912652	+	0.408738i	$0.865969\pi$
$72$	0	0
$73$	562.000	0.901057	0.450528	−	0.892762i	$-0.351236\pi$
$73$	562.000	0.901057	0.450528	+	0.892762i	$0.351236\pi$
$74$	0	0
$75$	−213.000	−0.327935
$76$	0	0
$77$	−352.000	−0.520963
$78$	0	0
$79$	−16.0000	−0.0227866	−0.0113933	−	0.999935i	$-0.503627\pi$
$79$	−16.0000	−0.0227866	−0.0113933	+	0.999935i	$0.503627\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−372.000	−0.491955	−0.245978	−	0.969275i	$-0.579109\pi$
$83$	−372.000	−0.491955	−0.245978	+	0.969275i	$0.579109\pi$
$84$	0	0
$85$	−28.0000	−0.0357297
$86$	0	0
$87$	−162.000	−0.199635
$88$	0	0
$89$	−966.000	−1.15051	−0.575257	−	0.817973i	$-0.695098\pi$
$89$	−966.000	−1.15051	−0.575257	+	0.817973i	$0.695098\pi$
$90$	0	0
$91$	−1216.00	−1.40079
$92$	0	0
$93$	456.000	0.508441
$94$	0	0
$95$	−1008.00	−1.08862
$96$	0	0
$97$	−526.000	−0.550590	−0.275295	−	0.961360i	$-0.588775\pi$
$97$	−526.000	−0.550590	−0.275295	+	0.961360i	$0.588775\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	0	0
$101$	−50.0000	−0.0492593	−0.0246296	−	0.999697i	$-0.507841\pi$
$101$	−50.0000	−0.0492593	−0.0246296	+	0.999697i	$0.507841\pi$
$102$	0	0
$103$	944.000	0.903059	0.451530	−	0.892256i	$-0.350879\pi$
$103$	944.000	0.903059	0.451530	+	0.892256i	$0.350879\pi$
$104$	0	0
$105$	1344.00	1.24915
$106$	0	0
$107$	−468.000	−0.422834	−0.211417	−	0.977396i	$-0.567808\pi$
$107$	−468.000	−0.422834	−0.211417	+	0.977396i	$0.567808\pi$
$108$	0	0
$109$	−154.000	−0.135326	−0.0676630	−	0.997708i	$-0.521554\pi$
$109$	−154.000	−0.135326	−0.0676630	+	0.997708i	$0.521554\pi$
$110$	0	0
$111$	522.000	0.446361
$112$	0	0
$113$	−54.0000	−0.0449548	−0.0224774	−	0.999747i	$-0.507155\pi$
$113$	−54.0000	−0.0449548	−0.0224774	+	0.999747i	$0.507155\pi$
$114$	0	0
$115$	952.000	0.771952
$116$	0	0
$117$	342.000	0.270239
$118$	0	0
$119$	64.0000	0.0493014
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−282.000	−0.206724
$124$	0	0
$125$	−756.000	−0.540950
$126$	0	0
$127$	−2224.00	−1.55392	−0.776961	−	0.629549i	$-0.783240\pi$
$127$	−2224.00	−1.55392	−0.776961	+	0.629549i	$0.783240\pi$
$128$	0	0
$129$	−1584.00	−1.08111
$130$	0	0
$131$	2772.00	1.84878	0.924392	−	0.381443i	$-0.124573\pi$
$131$	2772.00	1.84878	0.924392	+	0.381443i	$0.124573\pi$
$132$	0	0
$133$	2304.00	1.50212
$134$	0	0
$135$	−378.000	−0.240986
$136$	0	0
$137$	1130.00	0.704689	0.352345	−	0.935870i	$-0.385385\pi$
$137$	1130.00	0.704689	0.352345	+	0.935870i	$0.385385\pi$
$138$	0	0
$139$	1616.00	0.986096	0.493048	−	0.870002i	$-0.335883\pi$
$139$	1616.00	0.986096	0.493048	+	0.870002i	$0.335883\pi$
$140$	0	0
$141$	1020.00	0.609216
$142$	0	0
$143$	418.000	0.244440
$144$	0	0
$145$	756.000	0.432982
$146$	0	0
$147$	−2043.00	−1.14628
$148$	0	0
$149$	−2066.00	−1.13593	−0.567964	−	0.823053i	$-0.692269\pi$
$149$	−2066.00	−1.13593	−0.567964	+	0.823053i	$0.692269\pi$
$150$	0	0
$151$	248.000	0.133655	0.0668277	−	0.997765i	$-0.478712\pi$
$151$	248.000	0.133655	0.0668277	+	0.997765i	$0.478712\pi$
$152$	0	0
$153$	−18.0000	−0.00951120
$154$	0	0
$155$	−2128.00	−1.10274
$156$	0	0
$157$	−2366.00	−1.20272	−0.601361	−	0.798977i	$-0.705375\pi$
$157$	−2366.00	−1.20272	−0.601361	+	0.798977i	$0.705375\pi$
$158$	0	0
$159$	−1314.00	−0.655390
$160$	0	0
$161$	−2176.00	−1.06517
$162$	0	0
$163$	284.000	0.136470	0.0682350	−	0.997669i	$-0.478263\pi$
$163$	284.000	0.136470	0.0682350	+	0.997669i	$0.478263\pi$
$164$	0	0
$165$	−462.000	−0.217980
$166$	0	0
$167$	600.000	0.278020	0.139010	−	0.990291i	$-0.455608\pi$
$167$	600.000	0.278020	0.139010	+	0.990291i	$0.455608\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	0	0
$171$	−648.000	−0.289788
$172$	0	0
$173$	−138.000	−0.0606471	−0.0303235	−	0.999540i	$-0.509654\pi$
$173$	−138.000	−0.0606471	−0.0303235	+	0.999540i	$0.509654\pi$
$174$	0	0
$175$	−2272.00	−0.981412
$176$	0	0
$177$	60.0000	0.0254795
$178$	0	0
$179$	−3972.00	−1.65855	−0.829277	−	0.558838i	$-0.811248\pi$
$179$	−3972.00	−1.65855	−0.829277	+	0.558838i	$0.811248\pi$
$180$	0	0
$181$	−2230.00	−0.915771	−0.457886	−	0.889011i	$-0.651393\pi$
$181$	−2230.00	−0.915771	−0.457886	+	0.889011i	$0.651393\pi$
$182$	0	0
$183$	1710.00	0.690748
$184$	0	0
$185$	−2436.00	−0.968099
$186$	0	0
$187$	−22.0000	−0.00860320
$188$	0	0
$189$	864.000	0.332522
$190$	0	0
$191$	−772.000	−0.292461	−0.146230	−	0.989251i	$-0.546714\pi$
$191$	−772.000	−0.292461	−0.146230	+	0.989251i	$0.546714\pi$
$192$	0	0
$193$	394.000	0.146947	0.0734734	−	0.997297i	$-0.476592\pi$
$193$	394.000	0.146947	0.0734734	+	0.997297i	$0.476592\pi$
$194$	0	0
$195$	−1596.00	−0.586112
$196$	0	0
$197$	−3058.00	−1.10596	−0.552978	−	0.833196i	$-0.686509\pi$
$197$	−3058.00	−1.10596	−0.552978	+	0.833196i	$0.686509\pi$
$198$	0	0
$199$	2664.00	0.948975	0.474487	−	0.880262i	$-0.342633\pi$
$199$	2664.00	0.948975	0.474487	+	0.880262i	$0.342633\pi$
$200$	0	0
$201$	−1380.00	−0.484267
$202$	0	0
$203$	−1728.00	−0.597447
$204$	0	0
$205$	1316.00	0.448358
$206$	0	0
$207$	612.000	0.205492
$208$	0	0
$209$	−792.000	−0.262123
$210$	0	0
$211$	6000.00	1.95762	0.978808	−	0.204779i	$-0.0656477\pi$
$211$	6000.00	1.95762	0.978808	+	0.204779i	$0.0656477\pi$
$212$	0	0
$213$	3276.00	1.05384
$214$	0	0
$215$	7392.00	2.34479
$216$	0	0
$217$	4864.00	1.52161
$218$	0	0
$219$	−1686.00	−0.520225
$220$	0	0
$221$	−76.0000	−0.0231326
$222$	0	0
$223$	−560.000	−0.168163	−0.0840816	−	0.996459i	$-0.526796\pi$
$223$	−560.000	−0.168163	−0.0840816	+	0.996459i	$0.526796\pi$
$224$	0	0
$225$	639.000	0.189333
$226$	0	0
$227$	−5292.00	−1.54732	−0.773662	−	0.633599i	$-0.781577\pi$
$227$	−5292.00	−1.54732	−0.773662	+	0.633599i	$0.781577\pi$
$228$	0	0
$229$	5322.00	1.53575	0.767877	−	0.640597i	$-0.221313\pi$
$229$	5322.00	1.53575	0.767877	+	0.640597i	$0.221313\pi$
$230$	0	0
$231$	1056.00	0.300778
$232$	0	0
$233$	−3954.00	−1.11174	−0.555869	−	0.831270i	$-0.687615\pi$
$233$	−3954.00	−1.11174	−0.555869	+	0.831270i	$0.687615\pi$
$234$	0	0
$235$	−4760.00	−1.32131
$236$	0	0
$237$	48.0000	0.0131558
$238$	0	0
$239$	−3360.00	−0.909374	−0.454687	−	0.890651i	$-0.650249\pi$
$239$	−3360.00	−0.909374	−0.454687	+	0.890651i	$0.650249\pi$
$240$	0	0
$241$	−3278.00	−0.876160	−0.438080	−	0.898936i	$-0.644341\pi$
$241$	−3278.00	−0.876160	−0.438080	+	0.898936i	$0.644341\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	9534.00	2.48614
$246$	0	0
$247$	−2736.00	−0.704808
$248$	0	0
$249$	1116.00	0.284031
$250$	0	0
$251$	−2092.00	−0.526079	−0.263040	−	0.964785i	$-0.584725\pi$
$251$	−2092.00	−0.526079	−0.263040	+	0.964785i	$0.584725\pi$
$252$	0	0
$253$	748.000	0.185875
$254$	0	0
$255$	84.0000	0.0206286
$256$	0	0
$257$	658.000	0.159708	0.0798539	−	0.996807i	$-0.474555\pi$
$257$	658.000	0.159708	0.0798539	+	0.996807i	$0.474555\pi$
$258$	0	0
$259$	5568.00	1.33583
$260$	0	0
$261$	486.000	0.115259
$262$	0	0
$263$	−5104.00	−1.19668	−0.598339	−	0.801243i	$-0.704172\pi$
$263$	−5104.00	−1.19668	−0.598339	+	0.801243i	$0.704172\pi$
$264$	0	0
$265$	6132.00	1.42146
$266$	0	0
$267$	2898.00	0.664250
$268$	0	0
$269$	4238.00	0.960578	0.480289	−	0.877110i	$-0.340532\pi$
$269$	4238.00	0.960578	0.480289	+	0.877110i	$0.340532\pi$
$270$	0	0
$271$	−3376.00	−0.756743	−0.378372	−	0.925654i	$-0.623516\pi$
$271$	−3376.00	−0.756743	−0.378372	+	0.925654i	$0.623516\pi$
$272$	0	0
$273$	3648.00	0.808744
$274$	0	0
$275$	781.000	0.171258
$276$	0	0
$277$	−2074.00	−0.449872	−0.224936	−	0.974374i	$-0.572217\pi$
$277$	−2074.00	−0.449872	−0.224936	+	0.974374i	$0.572217\pi$
$278$	0	0
$279$	−1368.00	−0.293548
$280$	0	0
$281$	702.000	0.149031	0.0745157	−	0.997220i	$-0.476259\pi$
$281$	702.000	0.149031	0.0745157	+	0.997220i	$0.476259\pi$
$282$	0	0
$283$	−4912.00	−1.03176	−0.515880	−	0.856661i	$-0.672535\pi$
$283$	−4912.00	−1.03176	−0.515880	+	0.856661i	$0.672535\pi$
$284$	0	0
$285$	3024.00	0.628513
$286$	0	0
$287$	−3008.00	−0.618664
$288$	0	0
$289$	−4909.00	−0.999186
$290$	0	0
$291$	1578.00	0.317883
$292$	0	0
$293$	3486.00	0.695066	0.347533	−	0.937668i	$-0.387019\pi$
$293$	3486.00	0.695066	0.347533	+	0.937668i	$0.387019\pi$
$294$	0	0
$295$	−280.000	−0.0552618
$296$	0	0
$297$	−297.000	−0.0580259
$298$	0	0
$299$	2584.00	0.499788
$300$	0	0
$301$	−16896.0	−3.23545
$302$	0	0
$303$	150.000	0.0284399
$304$	0	0
$305$	−7980.00	−1.49814
$306$	0	0
$307$	−8360.00	−1.55417	−0.777085	−	0.629395i	$-0.783303\pi$
$307$	−8360.00	−1.55417	−0.777085	+	0.629395i	$0.783303\pi$
$308$	0	0
$309$	−2832.00	−0.521381
$310$	0	0
$311$	−5532.00	−1.00865	−0.504326	−	0.863513i	$-0.668259\pi$
$311$	−5532.00	−1.00865	−0.504326	+	0.863513i	$0.668259\pi$
$312$	0	0
$313$	4826.00	0.871507	0.435753	−	0.900066i	$-0.356482\pi$
$313$	4826.00	0.871507	0.435753	+	0.900066i	$0.356482\pi$
$314$	0	0
$315$	−4032.00	−0.721198
$316$	0	0
$317$	−7570.00	−1.34124	−0.670621	−	0.741800i	$-0.733972\pi$
$317$	−7570.00	−1.34124	−0.670621	+	0.741800i	$0.733972\pi$
$318$	0	0
$319$	594.000	0.104256
$320$	0	0
$321$	1404.00	0.244123
$322$	0	0
$323$	144.000	0.0248061
$324$	0	0
$325$	2698.00	0.460487
$326$	0	0
$327$	462.000	0.0781305
$328$	0	0
$329$	10880.0	1.82320
$330$	0	0
$331$	−3676.00	−0.610427	−0.305213	−	0.952284i	$-0.598728\pi$
$331$	−3676.00	−0.610427	−0.305213	+	0.952284i	$0.598728\pi$
$332$	0	0
$333$	−1566.00	−0.257707
$334$	0	0
$335$	6440.00	1.05031
$336$	0	0
$337$	−5686.00	−0.919098	−0.459549	−	0.888152i	$-0.651989\pi$
$337$	−5686.00	−0.919098	−0.459549	+	0.888152i	$0.651989\pi$
$338$	0	0
$339$	162.000	0.0259547
$340$	0	0
$341$	−1672.00	−0.265525
$342$	0	0
$343$	−10816.0	−1.70265
$344$	0	0
$345$	−2856.00	−0.445687
$346$	0	0
$347$	1652.00	0.255574	0.127787	−	0.991802i	$-0.459213\pi$
$347$	1652.00	0.255574	0.127787	+	0.991802i	$0.459213\pi$
$348$	0	0
$349$	6990.00	1.07211	0.536055	−	0.844183i	$-0.319914\pi$
$349$	6990.00	1.07211	0.536055	+	0.844183i	$0.319914\pi$
$350$	0	0
$351$	−1026.00	−0.156022
$352$	0	0
$353$	−8094.00	−1.22040	−0.610199	−	0.792249i	$-0.708910\pi$
$353$	−8094.00	−1.22040	−0.610199	+	0.792249i	$0.708910\pi$
$354$	0	0
$355$	−15288.0	−2.28564
$356$	0	0
$357$	−192.000	−0.0284642
$358$	0	0
$359$	1024.00	0.150542	0.0752711	−	0.997163i	$-0.476018\pi$
$359$	1024.00	0.150542	0.0752711	+	0.997163i	$0.476018\pi$
$360$	0	0
$361$	−1675.00	−0.244205
$362$	0	0
$363$	−363.000	−0.0524864
$364$	0	0
$365$	7868.00	1.12830
$366$	0	0
$367$	−13664.0	−1.94347	−0.971737	−	0.236066i	$-0.924142\pi$
$367$	−13664.0	−1.94347	−0.971737	+	0.236066i	$0.924142\pi$
$368$	0	0
$369$	846.000	0.119352
$370$	0	0
$371$	−14016.0	−1.96139
$372$	0	0
$373$	1958.00	0.271800	0.135900	−	0.990723i	$-0.456607\pi$
$373$	1958.00	0.271800	0.135900	+	0.990723i	$0.456607\pi$
$374$	0	0
$375$	2268.00	0.312317
$376$	0	0
$377$	2052.00	0.280327
$378$	0	0
$379$	−6124.00	−0.829997	−0.414998	−	0.909822i	$-0.636218\pi$
$379$	−6124.00	−0.829997	−0.414998	+	0.909822i	$0.636218\pi$
$380$	0	0
$381$	6672.00	0.897157
$382$	0	0
$383$	5612.00	0.748720	0.374360	−	0.927283i	$-0.377862\pi$
$383$	5612.00	0.748720	0.374360	+	0.927283i	$0.377862\pi$
$384$	0	0
$385$	−4928.00	−0.652348
$386$	0	0
$387$	4752.00	0.624180
$388$	0	0
$389$	−12450.0	−1.62273	−0.811363	−	0.584543i	$-0.801274\pi$
$389$	−12450.0	−1.62273	−0.811363	+	0.584543i	$0.801274\pi$
$390$	0	0
$391$	−136.000	−0.0175903
$392$	0	0
$393$	−8316.00	−1.06740
$394$	0	0
$395$	−224.000	−0.0285333
$396$	0	0
$397$	−14830.0	−1.87480	−0.937401	−	0.348252i	$-0.886775\pi$
$397$	−14830.0	−1.87480	−0.937401	+	0.348252i	$0.886775\pi$
$398$	0	0
$399$	−6912.00	−0.867250
$400$	0	0
$401$	−3358.00	−0.418181	−0.209090	−	0.977896i	$-0.567050\pi$
$401$	−3358.00	−0.418181	−0.209090	+	0.977896i	$0.567050\pi$
$402$	0	0
$403$	−5776.00	−0.713953
$404$	0	0
$405$	1134.00	0.139133
$406$	0	0
$407$	−1914.00	−0.233104
$408$	0	0
$409$	10698.0	1.29335	0.646677	−	0.762764i	$-0.276158\pi$
$409$	10698.0	1.29335	0.646677	+	0.762764i	$0.276158\pi$
$410$	0	0
$411$	−3390.00	−0.406852
$412$	0	0
$413$	640.000	0.0762526
$414$	0	0
$415$	−5208.00	−0.616026
$416$	0	0
$417$	−4848.00	−0.569323
$418$	0	0
$419$	2044.00	0.238320	0.119160	−	0.992875i	$-0.461980\pi$
$419$	2044.00	0.238320	0.119160	+	0.992875i	$0.461980\pi$
$420$	0	0
$421$	−3070.00	−0.355398	−0.177699	−	0.984085i	$-0.556865\pi$
$421$	−3070.00	−0.355398	−0.177699	+	0.984085i	$0.556865\pi$
$422$	0	0
$423$	−3060.00	−0.351731
$424$	0	0
$425$	−142.000	−0.0162071
$426$	0	0
$427$	18240.0	2.06720
$428$	0	0
$429$	−1254.00	−0.141127
$430$	0	0
$431$	−12600.0	−1.40817	−0.704084	−	0.710116i	$-0.748642\pi$
$431$	−12600.0	−1.40817	−0.704084	+	0.710116i	$0.748642\pi$
$432$	0	0
$433$	−9902.00	−1.09898	−0.549492	−	0.835499i	$-0.685179\pi$
$433$	−9902.00	−1.09898	−0.549492	+	0.835499i	$0.685179\pi$
$434$	0	0
$435$	−2268.00	−0.249982
$436$	0	0
$437$	−4896.00	−0.535944
$438$	0	0
$439$	11440.0	1.24374	0.621869	−	0.783121i	$-0.286373\pi$
$439$	11440.0	1.24374	0.621869	+	0.783121i	$0.286373\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	0	0
$443$	5180.00	0.555551	0.277776	−	0.960646i	$-0.410403\pi$
$443$	5180.00	0.555551	0.277776	+	0.960646i	$0.410403\pi$
$444$	0	0
$445$	−13524.0	−1.44067
$446$	0	0
$447$	6198.00	0.655829
$448$	0	0
$449$	10826.0	1.13789	0.568943	−	0.822377i	$-0.307353\pi$
$449$	10826.0	1.13789	0.568943	+	0.822377i	$0.307353\pi$
$450$	0	0
$451$	1034.00	0.107958
$452$	0	0
$453$	−744.000	−0.0771659
$454$	0	0
$455$	−17024.0	−1.75406
$456$	0	0
$457$	−15798.0	−1.61707	−0.808533	−	0.588451i	$-0.799738\pi$
$457$	−15798.0	−1.61707	−0.808533	+	0.588451i	$0.799738\pi$
$458$	0	0
$459$	54.0000	0.00549129
$460$	0	0
$461$	3894.00	0.393409	0.196705	−	0.980463i	$-0.436976\pi$
$461$	3894.00	0.393409	0.196705	+	0.980463i	$0.436976\pi$
$462$	0	0
$463$	−15992.0	−1.60521	−0.802604	−	0.596512i	$-0.796553\pi$
$463$	−15992.0	−1.60521	−0.802604	+	0.596512i	$0.796553\pi$
$464$	0	0
$465$	6384.00	0.636669
$466$	0	0
$467$	−11844.0	−1.17361	−0.586804	−	0.809729i	$-0.699614\pi$
$467$	−11844.0	−1.17361	−0.586804	+	0.809729i	$0.699614\pi$
$468$	0	0
$469$	−14720.0	−1.44927
$470$	0	0
$471$	7098.00	0.694392
$472$	0	0
$473$	5808.00	0.564592
$474$	0	0
$475$	−5112.00	−0.493799
$476$	0	0
$477$	3942.00	0.378389
$478$	0	0
$479$	14936.0	1.42472	0.712362	−	0.701812i	$-0.247625\pi$
$479$	14936.0	1.42472	0.712362	+	0.701812i	$0.247625\pi$
$480$	0	0
$481$	−6612.00	−0.626780
$482$	0	0
$483$	6528.00	0.614978
$484$	0	0
$485$	−7364.00	−0.689447
$486$	0	0
$487$	−2056.00	−0.191306	−0.0956532	−	0.995415i	$-0.530494\pi$
$487$	−2056.00	−0.191306	−0.0956532	+	0.995415i	$0.530494\pi$
$488$	0	0
$489$	−852.000	−0.0787909
$490$	0	0
$491$	17852.0	1.64083	0.820417	−	0.571766i	$-0.193741\pi$
$491$	17852.0	1.64083	0.820417	+	0.571766i	$0.193741\pi$
$492$	0	0
$493$	−108.000	−0.00986628
$494$	0	0
$495$	1386.00	0.125851
$496$	0	0
$497$	34944.0	3.15383
$498$	0	0
$499$	−4508.00	−0.404420	−0.202210	−	0.979342i	$-0.564812\pi$
$499$	−4508.00	−0.404420	−0.202210	+	0.979342i	$0.564812\pi$
$500$	0	0
$501$	−1800.00	−0.160515
$502$	0	0
$503$	−5912.00	−0.524062	−0.262031	−	0.965059i	$-0.584392\pi$
$503$	−5912.00	−0.524062	−0.262031	+	0.965059i	$0.584392\pi$
$504$	0	0
$505$	−700.000	−0.0616824
$506$	0	0
$507$	2259.00	0.197881
$508$	0	0
$509$	11406.0	0.993246	0.496623	−	0.867966i	$-0.334573\pi$
$509$	11406.0	0.993246	0.496623	+	0.867966i	$0.334573\pi$
$510$	0	0
$511$	−17984.0	−1.55688
$512$	0	0
$513$	1944.00	0.167309
$514$	0	0
$515$	13216.0	1.13081
$516$	0	0
$517$	−3740.00	−0.318153
$518$	0	0
$519$	414.000	0.0350146
$520$	0	0
$521$	−1542.00	−0.129667	−0.0648333	−	0.997896i	$-0.520652\pi$
$521$	−1542.00	−0.129667	−0.0648333	+	0.997896i	$0.520652\pi$
$522$	0	0
$523$	7504.00	0.627394	0.313697	−	0.949523i	$-0.398432\pi$
$523$	7504.00	0.627394	0.313697	+	0.949523i	$0.398432\pi$
$524$	0	0
$525$	6816.00	0.566618
$526$	0	0
$527$	304.000	0.0251280
$528$	0	0
$529$	−7543.00	−0.619956
$530$	0	0
$531$	−180.000	−0.0147106
$532$	0	0
$533$	3572.00	0.290282
$534$	0	0
$535$	−6552.00	−0.529472
$536$	0	0
$537$	11916.0	0.957567
$538$	0	0
$539$	7491.00	0.598627
$540$	0	0
$541$	−1018.00	−0.0809006	−0.0404503	−	0.999182i	$-0.512879\pi$
$541$	−1018.00	−0.0809006	−0.0404503	+	0.999182i	$0.512879\pi$
$542$	0	0
$543$	6690.00	0.528721
$544$	0	0
$545$	−2156.00	−0.169455
$546$	0	0
$547$	−7904.00	−0.617826	−0.308913	−	0.951090i	$-0.599965\pi$
$547$	−7904.00	−0.617826	−0.308913	+	0.951090i	$0.599965\pi$
$548$	0	0
$549$	−5130.00	−0.398803
$550$	0	0
$551$	−3888.00	−0.300607
$552$	0	0
$553$	512.000	0.0393715
$554$	0	0
$555$	7308.00	0.558932
$556$	0	0
$557$	22934.0	1.74460	0.872302	−	0.488967i	$-0.162626\pi$
$557$	22934.0	1.74460	0.872302	+	0.488967i	$0.162626\pi$
$558$	0	0
$559$	20064.0	1.51810
$560$	0	0
$561$	66.0000	0.00496706
$562$	0	0
$563$	−14020.0	−1.04951	−0.524754	−	0.851254i	$-0.675843\pi$
$563$	−14020.0	−1.04951	−0.524754	+	0.851254i	$0.675843\pi$
$564$	0	0
$565$	−756.000	−0.0562923
$566$	0	0
$567$	−2592.00	−0.191982
$568$	0	0
$569$	4230.00	0.311653	0.155827	−	0.987784i	$-0.450196\pi$
$569$	4230.00	0.311653	0.155827	+	0.987784i	$0.450196\pi$
$570$	0	0
$571$	8536.00	0.625605	0.312803	−	0.949818i	$-0.398732\pi$
$571$	8536.00	0.625605	0.312803	+	0.949818i	$0.398732\pi$
$572$	0	0
$573$	2316.00	0.168852
$574$	0	0
$575$	4828.00	0.350159
$576$	0	0
$577$	−11982.0	−0.864501	−0.432251	−	0.901754i	$-0.642280\pi$
$577$	−11982.0	−0.864501	−0.432251	+	0.901754i	$0.642280\pi$
$578$	0	0
$579$	−1182.00	−0.0848398
$580$	0	0
$581$	11904.0	0.850019
$582$	0	0
$583$	4818.00	0.342266
$584$	0	0
$585$	4788.00	0.338392
$586$	0	0
$587$	20396.0	1.43413	0.717064	−	0.697007i	$-0.245486\pi$
$587$	20396.0	1.43413	0.717064	+	0.697007i	$0.245486\pi$
$588$	0	0
$589$	10944.0	0.765602
$590$	0	0
$591$	9174.00	0.638524
$592$	0	0
$593$	12518.0	0.866868	0.433434	−	0.901185i	$-0.357302\pi$
$593$	12518.0	0.866868	0.433434	+	0.901185i	$0.357302\pi$
$594$	0	0
$595$	896.000	0.0617352
$596$	0	0
$597$	−7992.00	−0.547891
$598$	0	0
$599$	−25292.0	−1.72521	−0.862607	−	0.505875i	$-0.831170\pi$
$599$	−25292.0	−1.72521	−0.862607	+	0.505875i	$0.831170\pi$
$600$	0	0
$601$	15962.0	1.08337	0.541683	−	0.840583i	$-0.317787\pi$
$601$	15962.0	1.08337	0.541683	+	0.840583i	$0.317787\pi$
$602$	0	0
$603$	4140.00	0.279592
$604$	0	0
$605$	1694.00	0.113836
$606$	0	0
$607$	−1600.00	−0.106988	−0.0534942	−	0.998568i	$-0.517036\pi$
$607$	−1600.00	−0.106988	−0.0534942	+	0.998568i	$0.517036\pi$
$608$	0	0
$609$	5184.00	0.344936
$610$	0	0
$611$	−12920.0	−0.855462
$612$	0	0
$613$	−2162.00	−0.142451	−0.0712254	−	0.997460i	$-0.522691\pi$
$613$	−2162.00	−0.142451	−0.0712254	+	0.997460i	$0.522691\pi$
$614$	0	0
$615$	−3948.00	−0.258860
$616$	0	0
$617$	−18126.0	−1.18270	−0.591350	−	0.806415i	$-0.701405\pi$
$617$	−18126.0	−1.18270	−0.591350	+	0.806415i	$0.701405\pi$
$618$	0	0
$619$	−17348.0	−1.12645	−0.563227	−	0.826302i	$-0.690440\pi$
$619$	−17348.0	−1.12645	−0.563227	+	0.826302i	$0.690440\pi$
$620$	0	0
$621$	−1836.00	−0.118641
$622$	0	0
$623$	30912.0	1.98790
$624$	0	0
$625$	−19459.0	−1.24538
$626$	0	0
$627$	2376.00	0.151337
$628$	0	0
$629$	348.000	0.0220599
$630$	0	0
$631$	10096.0	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	10096.0	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−18000.0	−1.13023
$634$	0	0
$635$	−31136.0	−1.94582
$636$	0	0
$637$	25878.0	1.60961
$638$	0	0
$639$	−9828.00	−0.608435
$640$	0	0
$641$	8922.00	0.549763	0.274881	−	0.961478i	$-0.411361\pi$
$641$	8922.00	0.549763	0.274881	+	0.961478i	$0.411361\pi$
$642$	0	0
$643$	14644.0	0.898138	0.449069	−	0.893497i	$-0.351756\pi$
$643$	14644.0	0.898138	0.449069	+	0.893497i	$0.351756\pi$
$644$	0	0
$645$	−22176.0	−1.35377
$646$	0	0
$647$	6932.00	0.421213	0.210607	−	0.977571i	$-0.432456\pi$
$647$	6932.00	0.421213	0.210607	+	0.977571i	$0.432456\pi$
$648$	0	0
$649$	−220.000	−0.0133062
$650$	0	0
$651$	−14592.0	−0.878503
$652$	0	0
$653$	5942.00	0.356093	0.178046	−	0.984022i	$-0.443022\pi$
$653$	5942.00	0.356093	0.178046	+	0.984022i	$0.443022\pi$
$654$	0	0
$655$	38808.0	2.31504
$656$	0	0
$657$	5058.00	0.300352
$658$	0	0
$659$	−484.000	−0.0286100	−0.0143050	−	0.999898i	$-0.504554\pi$
$659$	−484.000	−0.0286100	−0.0143050	+	0.999898i	$0.504554\pi$
$660$	0	0
$661$	17114.0	1.00705	0.503523	−	0.863982i	$-0.332037\pi$
$661$	17114.0	1.00705	0.503523	+	0.863982i	$0.332037\pi$
$662$	0	0
$663$	228.000	0.0133556
$664$	0	0
$665$	32256.0	1.88095
$666$	0	0
$667$	3672.00	0.213164
$668$	0	0
$669$	1680.00	0.0970890
$670$	0	0
$671$	−6270.00	−0.360731
$672$	0	0
$673$	16154.0	0.925247	0.462623	−	0.886555i	$-0.346908\pi$
$673$	16154.0	0.925247	0.462623	+	0.886555i	$0.346908\pi$
$674$	0	0
$675$	−1917.00	−0.109312
$676$	0	0
$677$	3390.00	0.192449	0.0962247	−	0.995360i	$-0.469323\pi$
$677$	3390.00	0.192449	0.0962247	+	0.995360i	$0.469323\pi$
$678$	0	0
$679$	16832.0	0.951330
$680$	0	0
$681$	15876.0	0.893347
$682$	0	0
$683$	25540.0	1.43084	0.715418	−	0.698697i	$-0.246236\pi$
$683$	25540.0	1.43084	0.715418	+	0.698697i	$0.246236\pi$
$684$	0	0
$685$	15820.0	0.882410
$686$	0	0
$687$	−15966.0	−0.886668
$688$	0	0
$689$	16644.0	0.920299
$690$	0	0
$691$	−12476.0	−0.686844	−0.343422	−	0.939181i	$-0.611586\pi$
$691$	−12476.0	−0.686844	−0.343422	+	0.939181i	$0.611586\pi$
$692$	0	0
$693$	−3168.00	−0.173654
$694$	0	0
$695$	22624.0	1.23479
$696$	0	0
$697$	−188.000	−0.0102167
$698$	0	0
$699$	11862.0	0.641863
$700$	0	0
$701$	20806.0	1.12102	0.560508	−	0.828149i	$-0.310606\pi$
$701$	20806.0	1.12102	0.560508	+	0.828149i	$0.310606\pi$
$702$	0	0
$703$	12528.0	0.672123
$704$	0	0
$705$	14280.0	0.762859
$706$	0	0
$707$	1600.00	0.0851120
$708$	0	0
$709$	−14198.0	−0.752069	−0.376035	−	0.926606i	$-0.622713\pi$
$709$	−14198.0	−0.752069	−0.376035	+	0.926606i	$0.622713\pi$
$710$	0	0
$711$	−144.000	−0.00759553
$712$	0	0
$713$	−10336.0	−0.542898
$714$	0	0
$715$	5852.00	0.306087
$716$	0	0
$717$	10080.0	0.525027
$718$	0	0
$719$	4596.00	0.238389	0.119195	−	0.992871i	$-0.461969\pi$
$719$	4596.00	0.238389	0.119195	+	0.992871i	$0.461969\pi$
$720$	0	0
$721$	−30208.0	−1.56034
$722$	0	0
$723$	9834.00	0.505851
$724$	0	0
$725$	3834.00	0.196402
$726$	0	0
$727$	19560.0	0.997855	0.498927	−	0.866644i	$-0.333727\pi$
$727$	19560.0	0.997855	0.498927	+	0.866644i	$0.333727\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1056.00	−0.0534303
$732$	0	0
$733$	1638.00	0.0825388	0.0412694	−	0.999148i	$-0.486860\pi$
$733$	1638.00	0.0825388	0.0412694	+	0.999148i	$0.486860\pi$
$734$	0	0
$735$	−28602.0	−1.43538
$736$	0	0
$737$	5060.00	0.252900
$738$	0	0
$739$	15592.0	0.776131	0.388066	−	0.921632i	$-0.373143\pi$
$739$	15592.0	0.776131	0.388066	+	0.921632i	$0.373143\pi$
$740$	0	0
$741$	8208.00	0.406921
$742$	0	0
$743$	592.000	0.0292307	0.0146153	−	0.999893i	$-0.495348\pi$
$743$	592.000	0.0292307	0.0146153	+	0.999893i	$0.495348\pi$
$744$	0	0
$745$	−28924.0	−1.42241
$746$	0	0
$747$	−3348.00	−0.163985
$748$	0	0
$749$	14976.0	0.730589
$750$	0	0
$751$	39832.0	1.93541	0.967703	−	0.252092i	$-0.0811186\pi$
$751$	39832.0	1.93541	0.967703	+	0.252092i	$0.0811186\pi$
$752$	0	0
$753$	6276.00	0.303732
$754$	0	0
$755$	3472.00	0.167363
$756$	0	0
$757$	−10958.0	−0.526123	−0.263062	−	0.964779i	$-0.584732\pi$
$757$	−10958.0	−0.526123	−0.263062	+	0.964779i	$0.584732\pi$
$758$	0	0
$759$	−2244.00	−0.107315
$760$	0	0
$761$	−8970.00	−0.427283	−0.213641	−	0.976912i	$-0.568532\pi$
$761$	−8970.00	−0.427283	−0.213641	+	0.976912i	$0.568532\pi$
$762$	0	0
$763$	4928.00	0.233821
$764$	0	0
$765$	−252.000	−0.0119099
$766$	0	0
$767$	−760.000	−0.0357784
$768$	0	0
$769$	−10054.0	−0.471465	−0.235732	−	0.971818i	$-0.575749\pi$
$769$	−10054.0	−0.471465	−0.235732	+	0.971818i	$0.575749\pi$
$770$	0	0
$771$	−1974.00	−0.0922074
$772$	0	0
$773$	−26346.0	−1.22587	−0.612936	−	0.790132i	$-0.710012\pi$
$773$	−26346.0	−1.22587	−0.612936	+	0.790132i	$0.710012\pi$
$774$	0	0
$775$	−10792.0	−0.500207
$776$	0	0
$777$	−16704.0	−0.771239
$778$	0	0
$779$	−6768.00	−0.311282
$780$	0	0
$781$	−12012.0	−0.550350
$782$	0	0
$783$	−1458.00	−0.0665449
$784$	0	0
$785$	−33124.0	−1.50605
$786$	0	0
$787$	16040.0	0.726511	0.363256	−	0.931690i	$-0.381665\pi$
$787$	16040.0	0.726511	0.363256	+	0.931690i	$0.381665\pi$
$788$	0	0
$789$	15312.0	0.690902
$790$	0	0
$791$	1728.00	0.0776746
$792$	0	0
$793$	−21660.0	−0.969948
$794$	0	0
$795$	−18396.0	−0.820678
$796$	0	0
$797$	−32810.0	−1.45821	−0.729103	−	0.684404i	$-0.760062\pi$
$797$	−32810.0	−1.45821	−0.729103	+	0.684404i	$0.760062\pi$
$798$	0	0
$799$	680.000	0.0301085
$800$	0	0
$801$	−8694.00	−0.383505
$802$	0	0
$803$	6182.00	0.271679
$804$	0	0
$805$	−30464.0	−1.33381
$806$	0	0
$807$	−12714.0	−0.554590
$808$	0	0
$809$	18918.0	0.822153	0.411076	−	0.911601i	$-0.365153\pi$
$809$	18918.0	0.822153	0.411076	+	0.911601i	$0.365153\pi$
$810$	0	0
$811$	8552.00	0.370285	0.185143	−	0.982712i	$-0.440725\pi$
$811$	8552.00	0.370285	0.185143	+	0.982712i	$0.440725\pi$
$812$	0	0
$813$	10128.0	0.436906
$814$	0	0
$815$	3976.00	0.170887
$816$	0	0
$817$	−38016.0	−1.62792
$818$	0	0
$819$	−10944.0	−0.466928
$820$	0	0
$821$	46430.0	1.97371	0.986856	−	0.161600i	$-0.0516654\pi$
$821$	46430.0	1.97371	0.986856	+	0.161600i	$0.0516654\pi$
$822$	0	0
$823$	16392.0	0.694276	0.347138	−	0.937814i	$-0.387154\pi$
$823$	16392.0	0.694276	0.347138	+	0.937814i	$0.387154\pi$
$824$	0	0
$825$	−2343.00	−0.0988761
$826$	0	0
$827$	13876.0	0.583453	0.291727	−	0.956502i	$-0.405770\pi$
$827$	13876.0	0.583453	0.291727	+	0.956502i	$0.405770\pi$
$828$	0	0
$829$	24554.0	1.02870	0.514352	−	0.857579i	$-0.328032\pi$
$829$	24554.0	1.02870	0.514352	+	0.857579i	$0.328032\pi$
$830$	0	0
$831$	6222.00	0.259734
$832$	0	0
$833$	−1362.00	−0.0566513
$834$	0	0
$835$	8400.00	0.348137
$836$	0	0
$837$	4104.00	0.169480
$838$	0	0
$839$	19900.0	0.818861	0.409430	−	0.912341i	$-0.365727\pi$
$839$	19900.0	0.818861	0.409430	+	0.912341i	$0.365727\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	−2106.00	−0.0860433
$844$	0	0
$845$	−10542.0	−0.429178
$846$	0	0
$847$	−3872.00	−0.157076
$848$	0	0
$849$	14736.0	0.595687
$850$	0	0
$851$	−11832.0	−0.476611
$852$	0	0
$853$	−41138.0	−1.65128	−0.825638	−	0.564200i	$-0.809185\pi$
$853$	−41138.0	−1.65128	−0.825638	+	0.564200i	$0.809185\pi$
$854$	0	0
$855$	−9072.00	−0.362872
$856$	0	0
$857$	19910.0	0.793597	0.396799	−	0.917906i	$-0.370121\pi$
$857$	19910.0	0.793597	0.396799	+	0.917906i	$0.370121\pi$
$858$	0	0
$859$	−42924.0	−1.70495	−0.852473	−	0.522772i	$-0.824898\pi$
$859$	−42924.0	−1.70495	−0.852473	+	0.522772i	$0.824898\pi$
$860$	0	0
$861$	9024.00	0.357186
$862$	0	0
$863$	−46236.0	−1.82374	−0.911872	−	0.410474i	$-0.865363\pi$
$863$	−46236.0	−1.82374	−0.911872	+	0.410474i	$0.865363\pi$
$864$	0	0
$865$	−1932.00	−0.0759422
$866$	0	0
$867$	14727.0	0.576880
$868$	0	0
$869$	−176.000	−0.00687042
$870$	0	0
$871$	17480.0	0.680008
$872$	0	0
$873$	−4734.00	−0.183530
$874$	0	0
$875$	24192.0	0.934673
$876$	0	0
$877$	−25746.0	−0.991312	−0.495656	−	0.868519i	$-0.665072\pi$
$877$	−25746.0	−0.991312	−0.495656	+	0.868519i	$0.665072\pi$
$878$	0	0
$879$	−10458.0	−0.401296
$880$	0	0
$881$	−24550.0	−0.938831	−0.469416	−	0.882977i	$-0.655535\pi$
$881$	−24550.0	−0.938831	−0.469416	+	0.882977i	$0.655535\pi$
$882$	0	0
$883$	19436.0	0.740740	0.370370	−	0.928884i	$-0.379231\pi$
$883$	19436.0	0.740740	0.370370	+	0.928884i	$0.379231\pi$
$884$	0	0
$885$	840.000	0.0319054
$886$	0	0
$887$	−22912.0	−0.867316	−0.433658	−	0.901077i	$-0.642777\pi$
$887$	−22912.0	−0.867316	−0.433658	+	0.901077i	$0.642777\pi$
$888$	0	0
$889$	71168.0	2.68492
$890$	0	0
$891$	891.000	0.0335013
$892$	0	0
$893$	24480.0	0.917348
$894$	0	0
$895$	−55608.0	−2.07684
$896$	0	0
$897$	−7752.00	−0.288553
$898$	0	0
$899$	−8208.00	−0.304507
$900$	0	0
$901$	−876.000	−0.0323904
$902$	0	0
$903$	50688.0	1.86799
$904$	0	0
$905$	−31220.0	−1.14673
$906$	0	0
$907$	39900.0	1.46070	0.730352	−	0.683071i	$-0.239356\pi$
$907$	39900.0	1.46070	0.730352	+	0.683071i	$0.239356\pi$
$908$	0	0
$909$	−450.000	−0.0164198
$910$	0	0
$911$	29460.0	1.07141	0.535704	−	0.844406i	$-0.320046\pi$
$911$	29460.0	1.07141	0.535704	+	0.844406i	$0.320046\pi$
$912$	0	0
$913$	−4092.00	−0.148330
$914$	0	0
$915$	23940.0	0.864953
$916$	0	0
$917$	−88704.0	−3.19440
$918$	0	0
$919$	29368.0	1.05415	0.527073	−	0.849820i	$-0.323289\pi$
$919$	29368.0	1.05415	0.527073	+	0.849820i	$0.323289\pi$
$920$	0	0
$921$	25080.0	0.897301
$922$	0	0
$923$	−41496.0	−1.47980
$924$	0	0
$925$	−12354.0	−0.439132
$926$	0	0
$927$	8496.00	0.301020
$928$	0	0
$929$	33954.0	1.19913	0.599567	−	0.800325i	$-0.295340\pi$
$929$	33954.0	1.19913	0.599567	+	0.800325i	$0.295340\pi$
$930$	0	0
$931$	−49032.0	−1.72606
$932$	0	0
$933$	16596.0	0.582346
$934$	0	0
$935$	−308.000	−0.0107729
$936$	0	0
$937$	−2854.00	−0.0995049	−0.0497525	−	0.998762i	$-0.515843\pi$
$937$	−2854.00	−0.0995049	−0.0497525	+	0.998762i	$0.515843\pi$
$938$	0	0
$939$	−14478.0	−0.503165
$940$	0	0
$941$	6294.00	0.218043	0.109022	−	0.994039i	$-0.465228\pi$
$941$	6294.00	0.218043	0.109022	+	0.994039i	$0.465228\pi$
$942$	0	0
$943$	6392.00	0.220734
$944$	0	0
$945$	12096.0	0.416384
$946$	0	0
$947$	−2268.00	−0.0778248	−0.0389124	−	0.999243i	$-0.512389\pi$
$947$	−2268.00	−0.0778248	−0.0389124	+	0.999243i	$0.512389\pi$
$948$	0	0
$949$	21356.0	0.730501
$950$	0	0
$951$	22710.0	0.774366
$952$	0	0
$953$	26566.0	0.902998	0.451499	−	0.892272i	$-0.350889\pi$
$953$	26566.0	0.902998	0.451499	+	0.892272i	$0.350889\pi$
$954$	0	0
$955$	−10808.0	−0.366218
$956$	0	0
$957$	−1782.00	−0.0601921
$958$	0	0
$959$	−36160.0	−1.21759
$960$	0	0
$961$	−6687.00	−0.224464
$962$	0	0
$963$	−4212.00	−0.140945
$964$	0	0
$965$	5516.00	0.184007
$966$	0	0
$967$	11176.0	0.371661	0.185830	−	0.982582i	$-0.440503\pi$
$967$	11176.0	0.371661	0.185830	+	0.982582i	$0.440503\pi$
$968$	0	0
$969$	−432.000	−0.0143218
$970$	0	0
$971$	42316.0	1.39854	0.699271	−	0.714856i	$-0.253508\pi$
$971$	42316.0	1.39854	0.699271	+	0.714856i	$0.253508\pi$
$972$	0	0
$973$	−51712.0	−1.70381
$974$	0	0
$975$	−8094.00	−0.265862
$976$	0	0
$977$	−45054.0	−1.47534	−0.737669	−	0.675163i	$-0.764073\pi$
$977$	−45054.0	−1.47534	−0.737669	+	0.675163i	$0.764073\pi$
$978$	0	0
$979$	−10626.0	−0.346893
$980$	0	0
$981$	−1386.00	−0.0451086
$982$	0	0
$983$	−12300.0	−0.399094	−0.199547	−	0.979888i	$-0.563947\pi$
$983$	−12300.0	−0.399094	−0.199547	+	0.979888i	$0.563947\pi$
$984$	0	0
$985$	−42812.0	−1.38488
$986$	0	0
$987$	−32640.0	−1.05263
$988$	0	0
$989$	35904.0	1.15438
$990$	0	0
$991$	36280.0	1.16294	0.581469	−	0.813568i	$-0.302478\pi$
$991$	36280.0	1.16294	0.581469	+	0.813568i	$0.302478\pi$
$992$	0	0
$993$	11028.0	0.352430
$994$	0	0
$995$	37296.0	1.18830
$996$	0	0
$997$	−3290.00	−0.104509	−0.0522544	−	0.998634i	$-0.516641\pi$
$997$	−3290.00	−0.104509	−0.0522544	+	0.998634i	$0.516641\pi$
$998$	0	0
$999$	4698.00	0.148787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.4.a.l.1.1		1
4.3	odd	2		2112.4.a.y.1.1		1
8.3	odd	2		528.4.a.a.1.1		1
8.5	even	2		33.4.a.a.1.1	✓	1
24.5	odd	2		99.4.a.b.1.1		1
24.11	even	2		1584.4.a.t.1.1		1
40.13	odd	4		825.4.c.a.199.2		2
40.29	even	2		825.4.a.i.1.1		1
40.37	odd	4		825.4.c.a.199.1		2
56.13	odd	2		1617.4.a.a.1.1		1
88.21	odd	2		363.4.a.h.1.1		1
120.29	odd	2		2475.4.a.b.1.1		1
264.197	even	2		1089.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.a.1.1	✓	1	8.5	even	2
99.4.a.b.1.1		1	24.5	odd	2
363.4.a.h.1.1		1	88.21	odd	2
528.4.a.a.1.1		1	8.3	odd	2
825.4.a.i.1.1		1	40.29	even	2
825.4.c.a.199.1		2	40.37	odd	4
825.4.c.a.199.2		2	40.13	odd	4
1089.4.a.a.1.1		1	264.197	even	2
1584.4.a.t.1.1		1	24.11	even	2
1617.4.a.a.1.1		1	56.13	odd	2
2112.4.a.l.1.1		1	1.1	even	1	trivial
2112.4.a.y.1.1		1	4.3	odd	2
2475.4.a.b.1.1		1	120.29	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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