LMFDB - Embedded newform 2475.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$0$
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	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-5.00000 q^{2} +17.0000 q^{4} +32.0000 q^{7} -45.0000 q^{8} +O(q^{10})$

$q-5.00000 q^{2} +17.0000 q^{4} +32.0000 q^{7} -45.0000 q^{8} +11.0000 q^{11} +38.0000 q^{13} -160.000 q^{14} +89.0000 q^{16} -2.00000 q^{17} +72.0000 q^{19} -55.0000 q^{22} +68.0000 q^{23} -190.000 q^{26} +544.000 q^{28} +54.0000 q^{29} -152.000 q^{31} -85.0000 q^{32} +10.0000 q^{34} -174.000 q^{37} -360.000 q^{38} -94.0000 q^{41} +528.000 q^{43} +187.000 q^{44} -340.000 q^{46} -340.000 q^{47} +681.000 q^{49} +646.000 q^{52} -438.000 q^{53} -1440.00 q^{56} -270.000 q^{58} -20.0000 q^{59} +570.000 q^{61} +760.000 q^{62} -287.000 q^{64} +460.000 q^{67} -34.0000 q^{68} +1092.00 q^{71} -562.000 q^{73} +870.000 q^{74} +1224.00 q^{76} +352.000 q^{77} -16.0000 q^{79} +470.000 q^{82} +372.000 q^{83} -2640.00 q^{86} -495.000 q^{88} +966.000 q^{89} +1216.00 q^{91} +1156.00 q^{92} +1700.00 q^{94} +526.000 q^{97} -3405.00 q^{98} +O(q^{100})$

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 $

$2$	−5.00000	−1.76777	−0.883883	−	0.467707i	$-0.845080\pi$
$2$	−5.00000	−1.76777	−0.883883	+	0.467707i	$0.845080\pi$
$3$	0	0
$3$	0	0
$4$	17.0000	2.12500
$5$	0	0
$5$	0	0
$6$	0	0
$7$	32.0000	1.72784	0.863919	−	0.503631i	$-0.168003\pi$
$7$	32.0000	1.72784	0.863919	+	0.503631i	$0.168003\pi$
$8$	−45.0000	−1.98874
$9$	0	0
$10$	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	$0.367147\pi$
$14$	−160.000	−3.05441
$15$	0	0
$16$	89.0000	1.39062
$17$	−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	$-0.356861\pi$
$19$	72.0000	0.869365	0.434682	+	0.900584i	$0.356861\pi$
$20$	0	0
$21$	0	0
$22$	−55.0000	−0.533002
$23$	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
$25$	0	0
$26$	−190.000	−1.43316
$27$	0	0
$28$	544.000	3.67165
$29$	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	$0.645136\pi$
$32$	−85.0000	−0.469563
$33$	0	0
$34$	10.0000	0.0504408
$35$	0	0
$36$	0	0
$37$	−174.000	−0.773120	−0.386560	−	0.922264i	$-0.626337\pi$
$37$	−174.000	−0.773120	−0.386560	+	0.922264i	$0.626337\pi$
$38$	−360.000	−1.53683
$39$	0	0
$40$	0	0
$41$	−94.0000	−0.358057	−0.179028	−	0.983844i	$-0.557295\pi$
$41$	−94.0000	−0.358057	−0.179028	+	0.983844i	$0.557295\pi$
$42$	0	0
$43$	528.000	1.87254	0.936270	−	0.351280i	$-0.114254\pi$
$43$	528.000	1.87254	0.936270	+	0.351280i	$0.114254\pi$
$44$	187.000	0.640712
$45$	0	0
$46$	−340.000	−1.08979
$47$	−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
$51$	0	0
$52$	646.000	1.72277
$53$	−438.000	−1.13517	−0.567584	−	0.823315i	$-0.692122\pi$
$53$	−438.000	−1.13517	−0.567584	+	0.823315i	$0.692122\pi$
$54$	0	0
$55$	0	0
$56$	−1440.00	−3.43622
$57$	0	0
$58$	−270.000	−0.611254
$59$	−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	$-0.295881\pi$
$61$	570.000	1.19641	0.598205	+	0.801343i	$0.295881\pi$
$62$	760.000	1.55678
$63$	0	0
$64$	−287.000	−0.560547
$65$	0	0
$66$	0	0
$67$	460.000	0.838775	0.419388	−	0.907807i	$-0.362245\pi$
$67$	460.000	0.838775	0.419388	+	0.907807i	$0.362245\pi$
$68$	−34.0000	−0.0606339
$69$	0	0
$70$	0	0
$71$	1092.00	1.82530	0.912652	−	0.408738i	$-0.134031\pi$
$71$	1092.00	1.82530	0.912652	+	0.408738i	$0.134031\pi$
$72$	0	0
$73$	−562.000	−0.901057	−0.450528	−	0.892762i	$-0.648764\pi$
$73$	−562.000	−0.901057	−0.450528	+	0.892762i	$0.648764\pi$
$74$	870.000	1.36670
$75$	0	0
$76$	1224.00	1.84740
$77$	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
$81$	0	0
$82$	470.000	0.632961
$83$	372.000	0.491955	0.245978	−	0.969275i	$-0.420891\pi$
$83$	372.000	0.491955	0.245978	+	0.969275i	$0.420891\pi$
$84$	0	0
$85$	0	0
$86$	−2640.00	−3.31022
$87$	0	0
$88$	−495.000	−0.599627
$89$	966.000	1.15051	0.575257	−	0.817973i	$-0.304902\pi$
$89$	966.000	1.15051	0.575257	+	0.817973i	$0.304902\pi$
$90$	0	0
$91$	1216.00	1.40079
$92$	1156.00	1.31001
$93$	0	0
$94$	1700.00	1.86534
$95$	0	0
$96$	0	0
$97$	526.000	0.550590	0.275295	−	0.961360i	$-0.411225\pi$
$97$	526.000	0.550590	0.275295	+	0.961360i	$0.411225\pi$
$98$	−3405.00	−3.50976
$99$	0	0
$100$	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	$-0.649121\pi$
$103$	−944.000	−0.903059	−0.451530	+	0.892256i	$0.649121\pi$
$104$	−1710.00	−1.61230
$105$	0	0
$106$	2190.00	2.00671
$107$	468.000	0.422834	0.211417	−	0.977396i	$-0.432192\pi$
$107$	468.000	0.422834	0.211417	+	0.977396i	$0.432192\pi$
$108$	0	0
$109$	154.000	0.135326	0.0676630	−	0.997708i	$-0.478446\pi$
$109$	154.000	0.135326	0.0676630	+	0.997708i	$0.478446\pi$
$110$	0	0
$111$	0	0
$112$	2848.00	2.40277
$113$	−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
$115$	0	0
$116$	918.000	0.734777
$117$	0	0
$118$	100.000	0.0780148
$119$	−64.0000	−0.0493014
$120$	0	0
$121$	121.000	0.0909091
$122$	−2850.00	−2.11497
$123$	0	0
$124$	−2584.00	−1.87137
$125$	0	0
$126$	0	0
$127$	2224.00	1.55392	0.776961	−	0.629549i	$-0.216760\pi$
$127$	2224.00	1.55392	0.776961	+	0.629549i	$0.216760\pi$
$128$	2115.00	1.46048
$129$	0	0
$130$	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
$134$	−2300.00	−1.48276
$135$	0	0
$136$	90.0000	0.0567459
$137$	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	$-0.664117\pi$
$139$	−1616.00	−0.986096	−0.493048	+	0.870002i	$0.664117\pi$
$140$	0	0
$141$	0	0
$142$	−5460.00	−3.22671
$143$	418.000	0.244440
$144$	0	0
$145$	0	0
$146$	2810.00	1.59286
$147$	0	0
$148$	−2958.00	−1.64288
$149$	−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	$0.478712\pi$
$152$	−3240.00	−1.72894
$153$	0	0
$154$	−1760.00	−0.920941
$155$	0	0
$156$	0	0
$157$	−2366.00	−1.20272	−0.601361	−	0.798977i	$-0.705375\pi$
$157$	−2366.00	−1.20272	−0.601361	+	0.798977i	$0.705375\pi$
$158$	80.0000	0.0402814
$159$	0	0
$160$	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	$0.478263\pi$
$164$	−1598.00	−0.760871
$165$	0	0
$166$	−1860.00	−0.869663
$167$	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
$171$	0	0
$172$	8976.00	3.97915
$173$	138.000	0.0606471	0.0303235	−	0.999540i	$-0.490346\pi$
$173$	138.000	0.0606471	0.0303235	+	0.999540i	$0.490346\pi$
$174$	0	0
$175$	0	0
$176$	979.000	0.419289
$177$	0	0
$178$	−4830.00	−2.03384
$179$	−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	$-0.348607\pi$
$181$	2230.00	0.915771	0.457886	+	0.889011i	$0.348607\pi$
$182$	−6080.00	−2.47626
$183$	0	0
$184$	−3060.00	−1.22601
$185$	0	0
$186$	0	0
$187$	−22.0000	−0.00860320
$188$	−5780.00	−2.24229
$189$	0	0
$190$	0	0
$191$	772.000	0.292461	0.146230	−	0.989251i	$-0.453286\pi$
$191$	772.000	0.292461	0.146230	+	0.989251i	$0.453286\pi$
$192$	0	0
$193$	−394.000	−0.146947	−0.0734734	−	0.997297i	$-0.523408\pi$
$193$	−394.000	−0.146947	−0.0734734	+	0.997297i	$0.523408\pi$
$194$	−2630.00	−0.973314
$195$	0	0
$196$	11577.0	4.21902
$197$	3058.00	1.10596	0.552978	−	0.833196i	$-0.313491\pi$
$197$	3058.00	1.10596	0.552978	+	0.833196i	$0.313491\pi$
$198$	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
$201$	0	0
$202$	250.000	0.0870789
$203$	1728.00	0.597447
$204$	0	0
$205$	0	0
$206$	4720.00	1.59640
$207$	0	0
$208$	3382.00	1.12740
$209$	792.000	0.262123
$210$	0	0
$211$	−6000.00	−1.95762	−0.978808	−	0.204779i	$-0.934352\pi$
$211$	−6000.00	−1.95762	−0.978808	+	0.204779i	$0.934352\pi$
$212$	−7446.00	−2.41223
$213$	0	0
$214$	−2340.00	−0.747472
$215$	0	0
$216$	0	0
$217$	−4864.00	−1.52161
$218$	−770.000	−0.239225
$219$	0	0
$220$	0	0
$221$	−76.0000	−0.0231326
$222$	0	0
$223$	560.000	0.168163	0.0840816	−	0.996459i	$-0.473204\pi$
$223$	560.000	0.168163	0.0840816	+	0.996459i	$0.473204\pi$
$224$	−2720.00	−0.811329
$225$	0	0
$226$	270.000	0.0794696
$227$	5292.00	1.54732	0.773662	−	0.633599i	$-0.218423\pi$
$227$	5292.00	1.54732	0.773662	+	0.633599i	$0.218423\pi$
$228$	0	0
$229$	−5322.00	−1.53575	−0.767877	−	0.640597i	$-0.778687\pi$
$229$	−5322.00	−1.53575	−0.767877	+	0.640597i	$0.778687\pi$
$230$	0	0
$231$	0	0
$232$	−2430.00	−0.687661
$233$	−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
$235$	0	0
$236$	−340.000	−0.0937801
$237$	0	0
$238$	320.000	0.0871534
$239$	3360.00	0.909374	0.454687	−	0.890651i	$-0.349751\pi$
$239$	3360.00	0.909374	0.454687	+	0.890651i	$0.349751\pi$
$240$	0	0
$241$	−3278.00	−0.876160	−0.438080	−	0.898936i	$-0.644341\pi$
$241$	−3278.00	−0.876160	−0.438080	+	0.898936i	$0.644341\pi$
$242$	−605.000	−0.160706
$243$	0	0
$244$	9690.00	2.54237
$245$	0	0
$246$	0	0
$247$	2736.00	0.704808
$248$	6840.00	1.75137
$249$	0	0
$250$	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
$254$	−11120.0	−2.74697
$255$	0	0
$256$	−8279.00	−2.02124
$257$	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
$261$	0	0
$262$	−13860.0	−3.26822
$263$	−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
$265$	0	0
$266$	−11520.0	−2.65540
$267$	0	0
$268$	7820.00	1.78240
$269$	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	$0.623516\pi$
$272$	−178.000	−0.0396795
$273$	0	0
$274$	−5650.00	−1.24573
$275$	0	0
$276$	0	0
$277$	−2074.00	−0.449872	−0.224936	−	0.974374i	$-0.572217\pi$
$277$	−2074.00	−0.449872	−0.224936	+	0.974374i	$0.572217\pi$
$278$	8080.00	1.74319
$279$	0	0
$280$	0	0
$281$	−702.000	−0.149031	−0.0745157	−	0.997220i	$-0.523741\pi$
$281$	−702.000	−0.149031	−0.0745157	+	0.997220i	$0.523741\pi$
$282$	0	0
$283$	−4912.00	−1.03176	−0.515880	−	0.856661i	$-0.672535\pi$
$283$	−4912.00	−1.03176	−0.515880	+	0.856661i	$0.672535\pi$
$284$	18564.0	3.87877
$285$	0	0
$286$	−2090.00	−0.432113
$287$	−3008.00	−0.618664
$288$	0	0
$289$	−4909.00	−0.999186
$290$	0	0
$291$	0	0
$292$	−9554.00	−1.91475
$293$	−3486.00	−0.695066	−0.347533	−	0.937668i	$-0.612981\pi$
$293$	−3486.00	−0.695066	−0.347533	+	0.937668i	$0.612981\pi$
$294$	0	0
$295$	0	0
$296$	7830.00	1.53753
$297$	0	0
$298$	10330.0	2.00806
$299$	2584.00	0.499788
$300$	0	0
$301$	16896.0	3.23545
$302$	−1240.00	−0.236271
$303$	0	0
$304$	6408.00	1.20896
$305$	0	0
$306$	0	0
$307$	−8360.00	−1.55417	−0.777085	−	0.629395i	$-0.783303\pi$
$307$	−8360.00	−1.55417	−0.777085	+	0.629395i	$0.783303\pi$
$308$	5984.00	1.10705
$309$	0	0
$310$	0	0
$311$	5532.00	1.00865	0.504326	−	0.863513i	$-0.331741\pi$
$311$	5532.00	1.00865	0.504326	+	0.863513i	$0.331741\pi$
$312$	0	0
$313$	−4826.00	−0.871507	−0.435753	−	0.900066i	$-0.643518\pi$
$313$	−4826.00	−0.871507	−0.435753	+	0.900066i	$0.643518\pi$
$314$	11830.0	2.12613
$315$	0	0
$316$	−272.000	−0.0484215
$317$	7570.00	1.34124	0.670621	−	0.741800i	$-0.266028\pi$
$317$	7570.00	1.34124	0.670621	+	0.741800i	$0.266028\pi$
$318$	0	0
$319$	594.000	0.104256
$320$	0	0
$321$	0	0
$322$	−10880.0	−1.88298
$323$	−144.000	−0.0248061
$324$	0	0
$325$	0	0
$326$	−1420.00	−0.241247
$327$	0	0
$328$	4230.00	0.712081
$329$	−10880.0	−1.82320
$330$	0	0
$331$	3676.00	0.610427	0.305213	−	0.952284i	$-0.401272\pi$
$331$	3676.00	0.610427	0.305213	+	0.952284i	$0.401272\pi$
$332$	6324.00	1.04541
$333$	0	0
$334$	−3000.00	−0.491475
$335$	0	0
$336$	0	0
$337$	5686.00	0.919098	0.459549	−	0.888152i	$-0.348011\pi$
$337$	5686.00	0.919098	0.459549	+	0.888152i	$0.348011\pi$
$338$	3765.00	0.605885
$339$	0	0
$340$	0	0
$341$	−1672.00	−0.265525
$342$	0	0
$343$	10816.0	1.70265
$344$	−23760.0	−3.72399
$345$	0	0
$346$	−690.000	−0.107210
$347$	−1652.00	−0.255574	−0.127787	−	0.991802i	$-0.540787\pi$
$347$	−1652.00	−0.255574	−0.127787	+	0.991802i	$0.540787\pi$
$348$	0	0
$349$	−6990.00	−1.07211	−0.536055	−	0.844183i	$-0.680086\pi$
$349$	−6990.00	−1.07211	−0.536055	+	0.844183i	$0.680086\pi$
$350$	0	0
$351$	0	0
$352$	−935.000	−0.141579
$353$	−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
$355$	0	0
$356$	16422.0	2.44484
$357$	0	0
$358$	19860.0	2.93194
$359$	−1024.00	−0.150542	−0.0752711	−	0.997163i	$-0.523982\pi$
$359$	−1024.00	−0.150542	−0.0752711	+	0.997163i	$0.523982\pi$
$360$	0	0
$361$	−1675.00	−0.244205
$362$	−11150.0	−1.61887
$363$	0	0
$364$	20672.0	2.97667
$365$	0	0
$366$	0	0
$367$	13664.0	1.94347	0.971737	−	0.236066i	$-0.0758581\pi$
$367$	13664.0	1.94347	0.971737	+	0.236066i	$0.0758581\pi$
$368$	6052.00	0.857289
$369$	0	0
$370$	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	$0.456607\pi$
$374$	110.000	0.0152085
$375$	0	0
$376$	15300.0	2.09850
$377$	2052.00	0.280327
$378$	0	0
$379$	6124.00	0.829997	0.414998	−	0.909822i	$-0.363782\pi$
$379$	6124.00	0.829997	0.414998	+	0.909822i	$0.363782\pi$
$380$	0	0
$381$	0	0
$382$	−3860.00	−0.517002
$383$	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
$385$	0	0
$386$	1970.00	0.259768
$387$	0	0
$388$	8942.00	1.17000
$389$	−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
$392$	−30645.0	−3.94849
$393$	0	0
$394$	−15290.0	−1.95507
$395$	0	0
$396$	0	0
$397$	−14830.0	−1.87480	−0.937401	−	0.348252i	$-0.886775\pi$
$397$	−14830.0	−1.87480	−0.937401	+	0.348252i	$0.886775\pi$
$398$	−13320.0	−1.67757
$399$	0	0
$400$	0	0
$401$	3358.00	0.418181	0.209090	−	0.977896i	$-0.432950\pi$
$401$	3358.00	0.418181	0.209090	+	0.977896i	$0.432950\pi$
$402$	0	0
$403$	−5776.00	−0.713953
$404$	−850.000	−0.104676
$405$	0	0
$406$	−8640.00	−1.05615
$407$	−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
$411$	0	0
$412$	−16048.0	−1.91900
$413$	−640.000	−0.0762526
$414$	0	0
$415$	0	0
$416$	−3230.00	−0.380682
$417$	0	0
$418$	−3960.00	−0.463373
$419$	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	$-0.443135\pi$
$421$	3070.00	0.355398	0.177699	+	0.984085i	$0.443135\pi$
$422$	30000.0	3.46061
$423$	0	0
$424$	19710.0	2.25755
$425$	0	0
$426$	0	0
$427$	18240.0	2.06720
$428$	7956.00	0.898523
$429$	0	0
$430$	0	0
$431$	12600.0	1.40817	0.704084	−	0.710116i	$-0.251358\pi$
$431$	12600.0	1.40817	0.704084	+	0.710116i	$0.251358\pi$
$432$	0	0
$433$	9902.00	1.09898	0.549492	−	0.835499i	$-0.314821\pi$
$433$	9902.00	1.09898	0.549492	+	0.835499i	$0.314821\pi$
$434$	24320.0	2.68986
$435$	0	0
$436$	2618.00	0.287568
$437$	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
$441$	0	0
$442$	380.000	0.0408931
$443$	−5180.00	−0.555551	−0.277776	−	0.960646i	$-0.589597\pi$
$443$	−5180.00	−0.555551	−0.277776	+	0.960646i	$0.589597\pi$
$444$	0	0
$445$	0	0
$446$	−2800.00	−0.297273
$447$	0	0
$448$	−9184.00	−0.968534
$449$	−10826.0	−1.13789	−0.568943	−	0.822377i	$-0.692647\pi$
$449$	−10826.0	−1.13789	−0.568943	+	0.822377i	$0.692647\pi$
$450$	0	0
$451$	−1034.00	−0.107958
$452$	−918.000	−0.0955290
$453$	0	0
$454$	−26460.0	−2.73531
$455$	0	0
$456$	0	0
$457$	15798.0	1.61707	0.808533	−	0.588451i	$-0.200262\pi$
$457$	15798.0	1.61707	0.808533	+	0.588451i	$0.200262\pi$
$458$	26610.0	2.71486
$459$	0	0
$460$	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	$-0.203447\pi$
$463$	15992.0	1.60521	0.802604	+	0.596512i	$0.203447\pi$
$464$	4806.00	0.480847
$465$	0	0
$466$	19770.0	1.96530
$467$	11844.0	1.17361	0.586804	−	0.809729i	$-0.300386\pi$
$467$	11844.0	1.17361	0.586804	+	0.809729i	$0.300386\pi$
$468$	0	0
$469$	14720.0	1.44927
$470$	0	0
$471$	0	0
$472$	900.000	0.0877666
$473$	5808.00	0.564592
$474$	0	0
$475$	0	0
$476$	−1088.00	−0.104766
$477$	0	0
$478$	−16800.0	−1.60756
$479$	−14936.0	−1.42472	−0.712362	−	0.701812i	$-0.752375\pi$
$479$	−14936.0	−1.42472	−0.712362	+	0.701812i	$0.752375\pi$
$480$	0	0
$481$	−6612.00	−0.626780
$482$	16390.0	1.54885
$483$	0	0
$484$	2057.00	0.193182
$485$	0	0
$486$	0	0
$487$	2056.00	0.191306	0.0956532	−	0.995415i	$-0.469506\pi$
$487$	2056.00	0.191306	0.0956532	+	0.995415i	$0.469506\pi$
$488$	−25650.0	−2.37935
$489$	0	0
$490$	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
$494$	−13680.0	−1.24594
$495$	0	0
$496$	−13528.0	−1.22465
$497$	34944.0	3.15383
$498$	0	0
$499$	4508.00	0.404420	0.202210	−	0.979342i	$-0.435188\pi$
$499$	4508.00	0.404420	0.202210	+	0.979342i	$0.435188\pi$
$500$	0	0
$501$	0	0
$502$	10460.0	0.929985
$503$	−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
$505$	0	0
$506$	−3740.00	−0.328584
$507$	0	0
$508$	37808.0	3.30208
$509$	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
$512$	24475.0	2.11260
$513$	0	0
$514$	−3290.00	−0.282326
$515$	0	0
$516$	0	0
$517$	−3740.00	−0.318153
$518$	27840.0	2.36143
$519$	0	0
$520$	0	0
$521$	1542.00	0.129667	0.0648333	−	0.997896i	$-0.479348\pi$
$521$	1542.00	0.129667	0.0648333	+	0.997896i	$0.479348\pi$
$522$	0	0
$523$	7504.00	0.627394	0.313697	−	0.949523i	$-0.398432\pi$
$523$	7504.00	0.627394	0.313697	+	0.949523i	$0.398432\pi$
$524$	47124.0	3.92867
$525$	0	0
$526$	25520.0	2.11545
$527$	304.000	0.0251280
$528$	0	0
$529$	−7543.00	−0.619956
$530$	0	0
$531$	0	0
$532$	39168.0	3.19201
$533$	−3572.00	−0.290282
$534$	0	0
$535$	0	0
$536$	−20700.0	−1.66810
$537$	0	0
$538$	−21190.0	−1.69808
$539$	7491.00	0.598627
$540$	0	0
$541$	1018.00	0.0809006	0.0404503	−	0.999182i	$-0.487121\pi$
$541$	1018.00	0.0809006	0.0404503	+	0.999182i	$0.487121\pi$
$542$	16880.0	1.33775
$543$	0	0
$544$	170.000	0.0133983
$545$	0	0
$546$	0	0
$547$	−7904.00	−0.617826	−0.308913	−	0.951090i	$-0.599965\pi$
$547$	−7904.00	−0.617826	−0.308913	+	0.951090i	$0.599965\pi$
$548$	19210.0	1.49746
$549$	0	0
$550$	0	0
$551$	3888.00	0.300607
$552$	0	0
$553$	−512.000	−0.0393715
$554$	10370.0	0.795269
$555$	0	0
$556$	−27472.0	−2.09545
$557$	−22934.0	−1.74460	−0.872302	−	0.488967i	$-0.837374\pi$
$557$	−22934.0	−1.74460	−0.872302	+	0.488967i	$0.837374\pi$
$558$	0	0
$559$	20064.0	1.51810
$560$	0	0
$561$	0	0
$562$	3510.00	0.263453
$563$	14020.0	1.04951	0.524754	−	0.851254i	$-0.324157\pi$
$563$	14020.0	1.04951	0.524754	+	0.851254i	$0.324157\pi$
$564$	0	0
$565$	0	0
$566$	24560.0	1.82391
$567$	0	0
$568$	−49140.0	−3.63005
$569$	−4230.00	−0.311653	−0.155827	−	0.987784i	$-0.549804\pi$
$569$	−4230.00	−0.311653	−0.155827	+	0.987784i	$0.549804\pi$
$570$	0	0
$571$	−8536.00	−0.625605	−0.312803	−	0.949818i	$-0.601268\pi$
$571$	−8536.00	−0.625605	−0.312803	+	0.949818i	$0.601268\pi$
$572$	7106.00	0.519435
$573$	0	0
$574$	15040.0	1.09365
$575$	0	0
$576$	0	0
$577$	11982.0	0.864501	0.432251	−	0.901754i	$-0.357720\pi$
$577$	11982.0	0.864501	0.432251	+	0.901754i	$0.357720\pi$
$578$	24545.0	1.76633
$579$	0	0
$580$	0	0
$581$	11904.0	0.850019
$582$	0	0
$583$	−4818.00	−0.342266
$584$	25290.0	1.79197
$585$	0	0
$586$	17430.0	1.22871
$587$	−20396.0	−1.43413	−0.717064	−	0.697007i	$-0.754514\pi$
$587$	−20396.0	−1.43413	−0.717064	+	0.697007i	$0.754514\pi$
$588$	0	0
$589$	−10944.0	−0.765602
$590$	0	0
$591$	0	0
$592$	−15486.0	−1.07512
$593$	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
$595$	0	0
$596$	−35122.0	−2.41385
$597$	0	0
$598$	−12920.0	−0.883509
$599$	25292.0	1.72521	0.862607	−	0.505875i	$-0.168830\pi$
$599$	25292.0	1.72521	0.862607	+	0.505875i	$0.168830\pi$
$600$	0	0
$601$	15962.0	1.08337	0.541683	−	0.840583i	$-0.317787\pi$
$601$	15962.0	1.08337	0.541683	+	0.840583i	$0.317787\pi$
$602$	−84480.0	−5.71951
$603$	0	0
$604$	4216.00	0.284018
$605$	0	0
$606$	0	0
$607$	1600.00	0.106988	0.0534942	−	0.998568i	$-0.482964\pi$
$607$	1600.00	0.106988	0.0534942	+	0.998568i	$0.482964\pi$
$608$	−6120.00	−0.408222
$609$	0	0
$610$	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	$0.522691\pi$
$614$	41800.0	2.74741
$615$	0	0
$616$	−15840.0	−1.03606
$617$	−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	$-0.309560\pi$
$619$	17348.0	1.12645	0.563227	+	0.826302i	$0.309560\pi$
$620$	0	0
$621$	0	0
$622$	−27660.0	−1.78306
$623$	30912.0	1.98790
$624$	0	0
$625$	0	0
$626$	24130.0	1.54062
$627$	0	0
$628$	−40222.0	−2.55578
$629$	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	$0.396829\pi$
$632$	720.000	0.0453166
$633$	0	0
$634$	−37850.0	−2.37100
$635$	0	0
$636$	0	0
$637$	25878.0	1.60961
$638$	−2970.00	−0.184300
$639$	0	0
$640$	0	0
$641$	−8922.00	−0.549763	−0.274881	−	0.961478i	$-0.588639\pi$
$641$	−8922.00	−0.549763	−0.274881	+	0.961478i	$0.588639\pi$
$642$	0	0
$643$	14644.0	0.898138	0.449069	−	0.893497i	$-0.351756\pi$
$643$	14644.0	0.898138	0.449069	+	0.893497i	$0.351756\pi$
$644$	36992.0	2.26349
$645$	0	0
$646$	720.000	0.0438514
$647$	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
$651$	0	0
$652$	4828.00	0.289999
$653$	−5942.00	−0.356093	−0.178046	−	0.984022i	$-0.556978\pi$
$653$	−5942.00	−0.356093	−0.178046	+	0.984022i	$0.556978\pi$
$654$	0	0
$655$	0	0
$656$	−8366.00	−0.497923
$657$	0	0
$658$	54400.0	3.22300
$659$	−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	$-0.667963\pi$
$661$	−17114.0	−1.00705	−0.503523	+	0.863982i	$0.667963\pi$
$662$	−18380.0	−1.07909
$663$	0	0
$664$	−16740.0	−0.978370
$665$	0	0
$666$	0	0
$667$	3672.00	0.213164
$668$	10200.0	0.590793
$669$	0	0
$670$	0	0
$671$	6270.00	0.360731
$672$	0	0
$673$	−16154.0	−0.925247	−0.462623	−	0.886555i	$-0.653092\pi$
$673$	−16154.0	−0.925247	−0.462623	+	0.886555i	$0.653092\pi$
$674$	−28430.0	−1.62475
$675$	0	0
$676$	−12801.0	−0.728323
$677$	−3390.00	−0.192449	−0.0962247	−	0.995360i	$-0.530677\pi$
$677$	−3390.00	−0.192449	−0.0962247	+	0.995360i	$0.530677\pi$
$678$	0	0
$679$	16832.0	0.951330
$680$	0	0
$681$	0	0
$682$	8360.00	0.469386
$683$	−25540.0	−1.43084	−0.715418	−	0.698697i	$-0.753764\pi$
$683$	−25540.0	−1.43084	−0.715418	+	0.698697i	$0.753764\pi$
$684$	0	0
$685$	0	0
$686$	−54080.0	−3.00989
$687$	0	0
$688$	46992.0	2.60400
$689$	−16644.0	−0.920299
$690$	0	0
$691$	12476.0	0.686844	0.343422	−	0.939181i	$-0.388414\pi$
$691$	12476.0	0.686844	0.343422	+	0.939181i	$0.388414\pi$
$692$	2346.00	0.128875
$693$	0	0
$694$	8260.00	0.451794
$695$	0	0
$696$	0	0
$697$	188.000	0.0102167
$698$	34950.0	1.89524
$699$	0	0
$700$	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
$704$	−3157.00	−0.169011
$705$	0	0
$706$	40470.0	2.15738
$707$	−1600.00	−0.0851120
$708$	0	0
$709$	14198.0	0.752069	0.376035	−	0.926606i	$-0.377287\pi$
$709$	14198.0	0.752069	0.376035	+	0.926606i	$0.377287\pi$
$710$	0	0
$711$	0	0
$712$	−43470.0	−2.28807
$713$	−10336.0	−0.542898
$714$	0	0
$715$	0	0
$716$	−67524.0	−3.52443
$717$	0	0
$718$	5120.00	0.266124
$719$	−4596.00	−0.238389	−0.119195	−	0.992871i	$-0.538031\pi$
$719$	−4596.00	−0.238389	−0.119195	+	0.992871i	$0.538031\pi$
$720$	0	0
$721$	−30208.0	−1.56034
$722$	8375.00	0.431697
$723$	0	0
$724$	37910.0	1.94601
$725$	0	0
$726$	0	0
$727$	−19560.0	−0.997855	−0.498927	−	0.866644i	$-0.666273\pi$
$727$	−19560.0	−0.997855	−0.498927	+	0.866644i	$0.666273\pi$
$728$	−54720.0	−2.78579
$729$	0	0
$730$	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	$0.486860\pi$
$734$	−68320.0	−3.43561
$735$	0	0
$736$	−5780.00	−0.289475
$737$	5060.00	0.252900
$738$	0	0
$739$	−15592.0	−0.776131	−0.388066	−	0.921632i	$-0.626857\pi$
$739$	−15592.0	−0.776131	−0.388066	+	0.921632i	$0.626857\pi$
$740$	0	0
$741$	0	0
$742$	70080.0	3.46727
$743$	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
$745$	0	0
$746$	−9790.00	−0.480479
$747$	0	0
$748$	−374.000	−0.0182818
$749$	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	$0.0811186\pi$
$752$	−30260.0	−1.46738
$753$	0	0
$754$	−10260.0	−0.495553
$755$	0	0
$756$	0	0
$757$	−10958.0	−0.526123	−0.263062	−	0.964779i	$-0.584732\pi$
$757$	−10958.0	−0.526123	−0.263062	+	0.964779i	$0.584732\pi$
$758$	−30620.0	−1.46724
$759$	0	0
$760$	0	0
$761$	8970.00	0.427283	0.213641	−	0.976912i	$-0.431468\pi$
$761$	8970.00	0.427283	0.213641	+	0.976912i	$0.431468\pi$
$762$	0	0
$763$	4928.00	0.233821
$764$	13124.0	0.621479
$765$	0	0
$766$	−28060.0	−1.32356
$767$	−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
$771$	0	0
$772$	−6698.00	−0.312262
$773$	26346.0	1.22587	0.612936	−	0.790132i	$-0.289988\pi$
$773$	26346.0	1.22587	0.612936	+	0.790132i	$0.289988\pi$
$774$	0	0
$775$	0	0
$776$	−23670.0	−1.09498
$777$	0	0
$778$	62250.0	2.86860
$779$	−6768.00	−0.311282
$780$	0	0
$781$	12012.0	0.550350
$782$	680.000	0.0310956
$783$	0	0
$784$	60609.0	2.76098
$785$	0	0
$786$	0	0
$787$	16040.0	0.726511	0.363256	−	0.931690i	$-0.381665\pi$
$787$	16040.0	0.726511	0.363256	+	0.931690i	$0.381665\pi$
$788$	51986.0	2.35016
$789$	0	0
$790$	0	0
$791$	−1728.00	−0.0776746
$792$	0	0
$793$	21660.0	0.969948
$794$	74150.0	3.31421
$795$	0	0
$796$	45288.0	2.01657
$797$	32810.0	1.45821	0.729103	−	0.684404i	$-0.239938\pi$
$797$	32810.0	1.45821	0.729103	+	0.684404i	$0.239938\pi$
$798$	0	0
$799$	680.000	0.0301085
$800$	0	0
$801$	0	0
$802$	−16790.0	−0.739246
$803$	−6182.00	−0.271679
$804$	0	0
$805$	0	0
$806$	28880.0	1.26210
$807$	0	0
$808$	2250.00	0.0979638
$809$	−18918.0	−0.822153	−0.411076	−	0.911601i	$-0.634847\pi$
$809$	−18918.0	−0.822153	−0.411076	+	0.911601i	$0.634847\pi$
$810$	0	0
$811$	−8552.00	−0.370285	−0.185143	−	0.982712i	$-0.559275\pi$
$811$	−8552.00	−0.370285	−0.185143	+	0.982712i	$0.559275\pi$
$812$	29376.0	1.26958
$813$	0	0
$814$	9570.00	0.412074
$815$	0	0
$816$	0	0
$817$	38016.0	1.62792
$818$	−53490.0	−2.28635
$819$	0	0
$820$	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	$-0.612846\pi$
$823$	−16392.0	−0.694276	−0.347138	+	0.937814i	$0.612846\pi$
$824$	42480.0	1.79595
$825$	0	0
$826$	3200.00	0.134797
$827$	−13876.0	−0.583453	−0.291727	−	0.956502i	$-0.594230\pi$
$827$	−13876.0	−0.583453	−0.291727	+	0.956502i	$0.594230\pi$
$828$	0	0
$829$	−24554.0	−1.02870	−0.514352	−	0.857579i	$-0.671968\pi$
$829$	−24554.0	−1.02870	−0.514352	+	0.857579i	$0.671968\pi$
$830$	0	0
$831$	0	0
$832$	−10906.0	−0.454444
$833$	−1362.00	−0.0566513
$834$	0	0
$835$	0	0
$836$	13464.0	0.557012
$837$	0	0
$838$	−10220.0	−0.421294
$839$	−19900.0	−0.818861	−0.409430	−	0.912341i	$-0.634273\pi$
$839$	−19900.0	−0.818861	−0.409430	+	0.912341i	$0.634273\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	−15350.0	−0.628261
$843$	0	0
$844$	−102000.	−4.15993
$845$	0	0
$846$	0	0
$847$	3872.00	0.157076
$848$	−38982.0	−1.57859
$849$	0	0
$850$	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	$0.809185\pi$
$854$	−91200.0	−3.65433
$855$	0	0
$856$	−21060.0	−0.840907
$857$	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	$-0.175102\pi$
$859$	42924.0	1.70495	0.852473	+	0.522772i	$0.175102\pi$
$860$	0	0
$861$	0	0
$862$	−63000.0	−2.48931
$863$	−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
$865$	0	0
$866$	−49510.0	−1.94275
$867$	0	0
$868$	−82688.0	−3.23343
$869$	−176.000	−0.00687042
$870$	0	0
$871$	17480.0	0.680008
$872$	−6930.00	−0.269128
$873$	0	0
$874$	−24480.0	−0.947424
$875$	0	0
$876$	0	0
$877$	−25746.0	−0.991312	−0.495656	−	0.868519i	$-0.665072\pi$
$877$	−25746.0	−0.991312	−0.495656	+	0.868519i	$0.665072\pi$
$878$	−57200.0	−2.19864
$879$	0	0
$880$	0	0
$881$	24550.0	0.938831	0.469416	−	0.882977i	$-0.344465\pi$
$881$	24550.0	0.938831	0.469416	+	0.882977i	$0.344465\pi$
$882$	0	0
$883$	19436.0	0.740740	0.370370	−	0.928884i	$-0.379231\pi$
$883$	19436.0	0.740740	0.370370	+	0.928884i	$0.379231\pi$
$884$	−1292.00	−0.0491569
$885$	0	0
$886$	25900.0	0.982085
$887$	−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
$891$	0	0
$892$	9520.00	0.357347
$893$	−24480.0	−0.917348
$894$	0	0
$895$	0	0
$896$	67680.0	2.52347
$897$	0	0
$898$	54130.0	2.01152
$899$	−8208.00	−0.304507
$900$	0	0
$901$	876.000	0.0323904
$902$	5170.00	0.190845
$903$	0	0
$904$	2430.00	0.0894033
$905$	0	0
$906$	0	0
$907$	39900.0	1.46070	0.730352	−	0.683071i	$-0.239356\pi$
$907$	39900.0	1.46070	0.730352	+	0.683071i	$0.239356\pi$
$908$	89964.0	3.28806
$909$	0	0
$910$	0	0
$911$	−29460.0	−1.07141	−0.535704	−	0.844406i	$-0.679954\pi$
$911$	−29460.0	−1.07141	−0.535704	+	0.844406i	$0.679954\pi$
$912$	0	0
$913$	4092.00	0.148330
$914$	−78990.0	−2.85860
$915$	0	0
$916$	−90474.0	−3.26348
$917$	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
$921$	0	0
$922$	−19470.0	−0.695456
$923$	41496.0	1.47980
$924$	0	0
$925$	0	0
$926$	−79960.0	−2.83763
$927$	0	0
$928$	−4590.00	−0.162364
$929$	−33954.0	−1.19913	−0.599567	−	0.800325i	$-0.704660\pi$
$929$	−33954.0	−1.19913	−0.599567	+	0.800325i	$0.704660\pi$
$930$	0	0
$931$	49032.0	1.72606
$932$	−67218.0	−2.36245
$933$	0	0
$934$	−59220.0	−2.07467
$935$	0	0
$936$	0	0
$937$	2854.00	0.0995049	0.0497525	−	0.998762i	$-0.484157\pi$
$937$	2854.00	0.0995049	0.0497525	+	0.998762i	$0.484157\pi$
$938$	−73600.0	−2.56197
$939$	0	0
$940$	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
$944$	−1780.00	−0.0613708
$945$	0	0
$946$	−29040.0	−0.998067
$947$	2268.00	0.0778248	0.0389124	−	0.999243i	$-0.487611\pi$
$947$	2268.00	0.0778248	0.0389124	+	0.999243i	$0.487611\pi$
$948$	0	0
$949$	−21356.0	−0.730501
$950$	0	0
$951$	0	0
$952$	2880.00	0.0980476
$953$	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
$955$	0	0
$956$	57120.0	1.93242
$957$	0	0
$958$	74680.0	2.51858
$959$	36160.0	1.21759
$960$	0	0
$961$	−6687.00	−0.224464
$962$	33060.0	1.10800
$963$	0	0
$964$	−55726.0	−1.86184
$965$	0	0
$966$	0	0
$967$	−11176.0	−0.371661	−0.185830	−	0.982582i	$-0.559497\pi$
$967$	−11176.0	−0.371661	−0.185830	+	0.982582i	$0.559497\pi$
$968$	−5445.00	−0.180794
$969$	0	0
$970$	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
$974$	−10280.0	−0.338185
$975$	0	0
$976$	50730.0	1.66376
$977$	−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
$981$	0	0
$982$	−89260.0	−2.90061
$983$	−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
$985$	0	0
$986$	540.000	0.0174413
$987$	0	0
$988$	46512.0	1.49772
$989$	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	$0.302478\pi$
$992$	12920.0	0.413519
$993$	0	0
$994$	−174720.	−5.57523
$995$	0	0
$996$	0	0
$997$	−3290.00	−0.104509	−0.0522544	−	0.998634i	$-0.516641\pi$
$997$	−3290.00	−0.104509	−0.0522544	+	0.998634i	$0.516641\pi$
$998$	−22540.0	−0.714921
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.b.1.1		1
3.2	odd	2		825.4.a.i.1.1		1
5.4	even	2		99.4.a.b.1.1		1
15.2	even	4		825.4.c.a.199.2		2
15.8	even	4		825.4.c.a.199.1		2
15.14	odd	2		33.4.a.a.1.1	✓	1
20.19	odd	2		1584.4.a.t.1.1		1
55.54	odd	2		1089.4.a.a.1.1		1
60.59	even	2		528.4.a.a.1.1		1
105.104	even	2		1617.4.a.a.1.1		1
120.29	odd	2		2112.4.a.l.1.1		1
120.59	even	2		2112.4.a.y.1.1		1
165.164	even	2		363.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.a.1.1	✓	1	15.14	odd	2
99.4.a.b.1.1		1	5.4	even	2
363.4.a.h.1.1		1	165.164	even	2
528.4.a.a.1.1		1	60.59	even	2
825.4.a.i.1.1		1	3.2	odd	2
825.4.c.a.199.1		2	15.8	even	4
825.4.c.a.199.2		2	15.2	even	4
1089.4.a.a.1.1		1	55.54	odd	2
1584.4.a.t.1.1		1	20.19	odd	2
1617.4.a.a.1.1		1	105.104	even	2
2112.4.a.l.1.1		1	120.29	odd	2
2112.4.a.y.1.1		1	120.59	even	2
2475.4.a.b.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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