LMFDB - Embedded newform 2166.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.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:	$127.798137072$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -19.0000 q^{5} +6.00000 q^{6} +9.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -19.0000 q^{5} +6.00000 q^{6} +9.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -38.0000 q^{10} -13.0000 q^{11} +12.0000 q^{12} -38.0000 q^{13} +18.0000 q^{14} -57.0000 q^{15} +16.0000 q^{16} +99.0000 q^{17} +18.0000 q^{18} -76.0000 q^{20} +27.0000 q^{21} -26.0000 q^{22} +68.0000 q^{23} +24.0000 q^{24} +236.000 q^{25} -76.0000 q^{26} +27.0000 q^{27} +36.0000 q^{28} -130.000 q^{29} -114.000 q^{30} -262.000 q^{31} +32.0000 q^{32} -39.0000 q^{33} +198.000 q^{34} -171.000 q^{35} +36.0000 q^{36} +296.000 q^{37} -114.000 q^{39} -152.000 q^{40} +8.00000 q^{41} +54.0000 q^{42} +73.0000 q^{43} -52.0000 q^{44} -171.000 q^{45} +136.000 q^{46} -271.000 q^{47} +48.0000 q^{48} -262.000 q^{49} +472.000 q^{50} +297.000 q^{51} -152.000 q^{52} +502.000 q^{53} +54.0000 q^{54} +247.000 q^{55} +72.0000 q^{56} -260.000 q^{58} -540.000 q^{59} -228.000 q^{60} +587.000 q^{61} -524.000 q^{62} +81.0000 q^{63} +64.0000 q^{64} +722.000 q^{65} -78.0000 q^{66} -684.000 q^{67} +396.000 q^{68} +204.000 q^{69} -342.000 q^{70} -992.000 q^{71} +72.0000 q^{72} -507.000 q^{73} +592.000 q^{74} +708.000 q^{75} -117.000 q^{77} -228.000 q^{78} -980.000 q^{79} -304.000 q^{80} +81.0000 q^{81} +16.0000 q^{82} -492.000 q^{83} +108.000 q^{84} -1881.00 q^{85} +146.000 q^{86} -390.000 q^{87} -104.000 q^{88} -810.000 q^{89} -342.000 q^{90} -342.000 q^{91} +272.000 q^{92} -786.000 q^{93} -542.000 q^{94} +96.0000 q^{96} +1046.00 q^{97} -524.000 q^{98} -117.000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−19.0000	−1.69941	−0.849706	−	0.527257i	$-0.823220\pi$
$5$	−19.0000	−1.69941	−0.849706	+	0.527257i	$0.823220\pi$
$6$	6.00000	0.408248
$7$	9.00000	0.485954	0.242977	−	0.970032i	$-0.421876\pi$
$7$	9.00000	0.485954	0.242977	+	0.970032i	$0.421876\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−38.0000	−1.20167
$11$	−13.0000	−0.356332	−0.178166	−	0.984000i	$-0.557016\pi$
$11$	−13.0000	−0.356332	−0.178166	+	0.984000i	$0.557016\pi$
$12$	12.0000	0.288675
$13$	−38.0000	−0.810716	−0.405358	−	0.914158i	$-0.632853\pi$
$13$	−38.0000	−0.810716	−0.405358	+	0.914158i	$0.632853\pi$
$14$	18.0000	0.343622
$15$	−57.0000	−0.981156
$16$	16.0000	0.250000
$17$	99.0000	1.41241	0.706207	−	0.708006i	$-0.250405\pi$
$17$	99.0000	1.41241	0.706207	+	0.708006i	$0.250405\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	−76.0000	−0.849706
$21$	27.0000	0.280566
$22$	−26.0000	−0.251964
$23$	68.0000	0.616477	0.308239	−	0.951309i	$-0.400260\pi$
$23$	68.0000	0.616477	0.308239	+	0.951309i	$0.400260\pi$
$24$	24.0000	0.204124
$25$	236.000	1.88800
$26$	−76.0000	−0.573263
$27$	27.0000	0.192450
$28$	36.0000	0.242977
$29$	−130.000	−0.832427	−0.416214	−	0.909267i	$-0.636643\pi$
$29$	−130.000	−0.832427	−0.416214	+	0.909267i	$0.636643\pi$
$30$	−114.000	−0.693782
$31$	−262.000	−1.51795	−0.758977	−	0.651117i	$-0.774301\pi$
$31$	−262.000	−1.51795	−0.758977	+	0.651117i	$0.774301\pi$
$32$	32.0000	0.176777
$33$	−39.0000	−0.205728
$34$	198.000	0.998727
$35$	−171.000	−0.825836
$36$	36.0000	0.166667
$37$	296.000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	296.000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	−114.000	−0.468067
$40$	−152.000	−0.600833
$41$	8.00000	0.0304729	0.0152365	−	0.999884i	$-0.495150\pi$
$41$	8.00000	0.0304729	0.0152365	+	0.999884i	$0.495150\pi$
$42$	54.0000	0.198390
$43$	73.0000	0.258893	0.129446	−	0.991586i	$-0.458680\pi$
$43$	73.0000	0.258893	0.129446	+	0.991586i	$0.458680\pi$
$44$	−52.0000	−0.178166
$45$	−171.000	−0.566471
$46$	136.000	0.435915
$47$	−271.000	−0.841051	−0.420526	−	0.907281i	$-0.638154\pi$
$47$	−271.000	−0.841051	−0.420526	+	0.907281i	$0.638154\pi$
$48$	48.0000	0.144338
$49$	−262.000	−0.763848
$50$	472.000	1.33502
$51$	297.000	0.815457
$52$	−152.000	−0.405358
$53$	502.000	1.30104	0.650519	−	0.759490i	$-0.274551\pi$
$53$	502.000	1.30104	0.650519	+	0.759490i	$0.274551\pi$
$54$	54.0000	0.136083
$55$	247.000	0.605554
$56$	72.0000	0.171811
$57$	0	0
$58$	−260.000	−0.588615
$59$	−540.000	−1.19156	−0.595780	−	0.803148i	$-0.703157\pi$
$59$	−540.000	−1.19156	−0.595780	+	0.803148i	$0.703157\pi$
$60$	−228.000	−0.490578
$61$	587.000	1.23209	0.616046	−	0.787710i	$-0.288733\pi$
$61$	587.000	1.23209	0.616046	+	0.787710i	$0.288733\pi$
$62$	−524.000	−1.07336
$63$	81.0000	0.161985
$64$	64.0000	0.125000
$65$	722.000	1.37774
$66$	−78.0000	−0.145472
$67$	−684.000	−1.24722	−0.623611	−	0.781735i	$-0.714335\pi$
$67$	−684.000	−1.24722	−0.623611	+	0.781735i	$0.714335\pi$
$68$	396.000	0.706207
$69$	204.000	0.355923
$70$	−342.000	−0.583955
$71$	−992.000	−1.65815	−0.829076	−	0.559136i	$-0.811133\pi$
$71$	−992.000	−1.65815	−0.829076	+	0.559136i	$0.811133\pi$
$72$	72.0000	0.117851
$73$	−507.000	−0.812875	−0.406437	−	0.913679i	$-0.633229\pi$
$73$	−507.000	−0.812875	−0.406437	+	0.913679i	$0.633229\pi$
$74$	592.000	0.929981
$75$	708.000	1.09004
$76$	0	0
$77$	−117.000	−0.173161
$78$	−228.000	−0.330973
$79$	−980.000	−1.39568	−0.697839	−	0.716254i	$-0.745855\pi$
$79$	−980.000	−1.39568	−0.697839	+	0.716254i	$0.745855\pi$
$80$	−304.000	−0.424853
$81$	81.0000	0.111111
$82$	16.0000	0.0215476
$83$	−492.000	−0.650651	−0.325325	−	0.945602i	$-0.605474\pi$
$83$	−492.000	−0.650651	−0.325325	+	0.945602i	$0.605474\pi$
$84$	108.000	0.140283
$85$	−1881.00	−2.40027
$86$	146.000	0.183065
$87$	−390.000	−0.480602
$88$	−104.000	−0.125982
$89$	−810.000	−0.964717	−0.482359	−	0.875974i	$-0.660220\pi$
$89$	−810.000	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$90$	−342.000	−0.400555
$91$	−342.000	−0.393971
$92$	272.000	0.308239
$93$	−786.000	−0.876391
$94$	−542.000	−0.594713
$95$	0	0
$96$	96.0000	0.102062
$97$	1046.00	1.09490	0.547450	−	0.836839i	$-0.315599\pi$
$97$	1046.00	1.09490	0.547450	+	0.836839i	$0.315599\pi$
$98$	−524.000	−0.540122
$99$	−117.000	−0.118777
$100$	944.000	0.944000
$101$	1682.00	1.65708	0.828541	−	0.559929i	$-0.189171\pi$
$101$	1682.00	1.65708	0.828541	+	0.559929i	$0.189171\pi$
$102$	594.000	0.576615
$103$	−978.000	−0.935584	−0.467792	−	0.883838i	$-0.654950\pi$
$103$	−978.000	−0.935584	−0.467792	+	0.883838i	$0.654950\pi$
$104$	−304.000	−0.286631
$105$	−513.000	−0.476797
$106$	1004.00	0.919973
$107$	−674.000	−0.608954	−0.304477	−	0.952520i	$-0.598482\pi$
$107$	−674.000	−0.608954	−0.304477	+	0.952520i	$0.598482\pi$
$108$	108.000	0.0962250
$109$	360.000	0.316346	0.158173	−	0.987411i	$-0.449440\pi$
$109$	360.000	0.316346	0.158173	+	0.987411i	$0.449440\pi$
$110$	494.000	0.428191
$111$	888.000	0.759326
$112$	144.000	0.121489
$113$	−1838.00	−1.53013	−0.765064	−	0.643954i	$-0.777293\pi$
$113$	−1838.00	−1.53013	−0.765064	+	0.643954i	$0.777293\pi$
$114$	0	0
$115$	−1292.00	−1.04765
$116$	−520.000	−0.416214
$117$	−342.000	−0.270239
$118$	−1080.00	−0.842560
$119$	891.000	0.686368
$120$	−456.000	−0.346891
$121$	−1162.00	−0.873028
$122$	1174.00	0.871221
$123$	24.0000	0.0175936
$124$	−1048.00	−0.758977
$125$	−2109.00	−1.50908
$126$	162.000	0.114541
$127$	1386.00	0.968406	0.484203	−	0.874956i	$-0.339110\pi$
$127$	1386.00	0.968406	0.484203	+	0.874956i	$0.339110\pi$
$128$	128.000	0.0883883
$129$	219.000	0.149472
$130$	1444.00	0.974209
$131$	−1083.00	−0.722306	−0.361153	−	0.932506i	$-0.617617\pi$
$131$	−1083.00	−0.722306	−0.361153	+	0.932506i	$0.617617\pi$
$132$	−156.000	−0.102864
$133$	0	0
$134$	−1368.00	−0.881919
$135$	−513.000	−0.327052
$136$	792.000	0.499364
$137$	499.000	0.311186	0.155593	−	0.987821i	$-0.450271\pi$
$137$	499.000	0.311186	0.155593	+	0.987821i	$0.450271\pi$
$138$	408.000	0.251676
$139$	−195.000	−0.118991	−0.0594953	−	0.998229i	$-0.518949\pi$
$139$	−195.000	−0.118991	−0.0594953	+	0.998229i	$0.518949\pi$
$140$	−684.000	−0.412918
$141$	−813.000	−0.485581
$142$	−1984.00	−1.17249
$143$	494.000	0.288884
$144$	144.000	0.0833333
$145$	2470.00	1.41464
$146$	−1014.00	−0.574789
$147$	−786.000	−0.441008
$148$	1184.00	0.657596
$149$	−1485.00	−0.816483	−0.408241	−	0.912874i	$-0.633858\pi$
$149$	−1485.00	−0.816483	−0.408241	+	0.912874i	$0.633858\pi$
$150$	1416.00	0.770773
$151$	−332.000	−0.178926	−0.0894628	−	0.995990i	$-0.528515\pi$
$151$	−332.000	−0.178926	−0.0894628	+	0.995990i	$0.528515\pi$
$152$	0	0
$153$	891.000	0.470804
$154$	−234.000	−0.122443
$155$	4978.00	2.57963
$156$	−456.000	−0.234033
$157$	3814.00	1.93879	0.969396	−	0.245502i	$-0.0789529\pi$
$157$	3814.00	1.93879	0.969396	+	0.245502i	$0.0789529\pi$
$158$	−1960.00	−0.986894
$159$	1506.00	0.751155
$160$	−608.000	−0.300416
$161$	612.000	0.299580
$162$	162.000	0.0785674
$163$	−3352.00	−1.61073	−0.805365	−	0.592780i	$-0.798031\pi$
$163$	−3352.00	−1.61073	−0.805365	+	0.592780i	$0.798031\pi$
$164$	32.0000	0.0152365
$165$	741.000	0.349617
$166$	−984.000	−0.460080
$167$	−3714.00	−1.72095	−0.860473	−	0.509496i	$-0.829832\pi$
$167$	−3714.00	−1.72095	−0.860473	+	0.509496i	$0.829832\pi$
$168$	216.000	0.0991950
$169$	−753.000	−0.342740
$170$	−3762.00	−1.69725
$171$	0	0
$172$	292.000	0.129446
$173$	1242.00	0.545824	0.272912	−	0.962039i	$-0.412013\pi$
$173$	1242.00	0.545824	0.272912	+	0.962039i	$0.412013\pi$
$174$	−780.000	−0.339837
$175$	2124.00	0.917482
$176$	−208.000	−0.0890829
$177$	−1620.00	−0.687947
$178$	−1620.00	−0.682158
$179$	−2310.00	−0.964567	−0.482284	−	0.876015i	$-0.660192\pi$
$179$	−2310.00	−0.964567	−0.482284	+	0.876015i	$0.660192\pi$
$180$	−684.000	−0.283235
$181$	−82.0000	−0.0336741	−0.0168370	−	0.999858i	$-0.505360\pi$
$181$	−82.0000	−0.0336741	−0.0168370	+	0.999858i	$0.505360\pi$
$182$	−684.000	−0.278579
$183$	1761.00	0.711349
$184$	544.000	0.217958
$185$	−5624.00	−2.23505
$186$	−1572.00	−0.619702
$187$	−1287.00	−0.503287
$188$	−1084.00	−0.420526
$189$	243.000	0.0935220
$190$	0	0
$191$	−1313.00	−0.497410	−0.248705	−	0.968579i	$-0.580005\pi$
$191$	−1313.00	−0.497410	−0.248705	+	0.968579i	$0.580005\pi$
$192$	192.000	0.0721688
$193$	2352.00	0.877206	0.438603	−	0.898681i	$-0.355474\pi$
$193$	2352.00	0.877206	0.438603	+	0.898681i	$0.355474\pi$
$194$	2092.00	0.774211
$195$	2166.00	0.795438
$196$	−1048.00	−0.381924
$197$	−3826.00	−1.38371	−0.691856	−	0.722036i	$-0.743207\pi$
$197$	−3826.00	−1.38371	−0.691856	+	0.722036i	$0.743207\pi$
$198$	−234.000	−0.0839882
$199$	2225.00	0.792593	0.396297	−	0.918123i	$-0.370295\pi$
$199$	2225.00	0.792593	0.396297	+	0.918123i	$0.370295\pi$
$200$	1888.00	0.667509
$201$	−2052.00	−0.720084
$202$	3364.00	1.17173
$203$	−1170.00	−0.404522
$204$	1188.00	0.407729
$205$	−152.000	−0.0517861
$206$	−1956.00	−0.661558
$207$	612.000	0.205492
$208$	−608.000	−0.202679
$209$	0	0
$210$	−1026.00	−0.337146
$211$	−1632.00	−0.532472	−0.266236	−	0.963908i	$-0.585780\pi$
$211$	−1632.00	−0.532472	−0.266236	+	0.963908i	$0.585780\pi$
$212$	2008.00	0.650519
$213$	−2976.00	−0.957334
$214$	−1348.00	−0.430595
$215$	−1387.00	−0.439966
$216$	216.000	0.0680414
$217$	−2358.00	−0.737657
$218$	720.000	0.223691
$219$	−1521.00	−0.469313
$220$	988.000	0.302777
$221$	−3762.00	−1.14507
$222$	1776.00	0.536925
$223$	−1148.00	−0.344734	−0.172367	−	0.985033i	$-0.555142\pi$
$223$	−1148.00	−0.344734	−0.172367	+	0.985033i	$0.555142\pi$
$224$	288.000	0.0859054
$225$	2124.00	0.629333
$226$	−3676.00	−1.08196
$227$	−5794.00	−1.69410	−0.847051	−	0.531511i	$-0.821624\pi$
$227$	−5794.00	−1.69410	−0.847051	+	0.531511i	$0.821624\pi$
$228$	0	0
$229$	2525.00	0.728632	0.364316	−	0.931275i	$-0.381303\pi$
$229$	2525.00	0.728632	0.364316	+	0.931275i	$0.381303\pi$
$230$	−2584.00	−0.740800
$231$	−351.000	−0.0999745
$232$	−1040.00	−0.294308
$233$	−1227.00	−0.344993	−0.172497	−	0.985010i	$-0.555183\pi$
$233$	−1227.00	−0.344993	−0.172497	+	0.985010i	$0.555183\pi$
$234$	−684.000	−0.191088
$235$	5149.00	1.42929
$236$	−2160.00	−0.595780
$237$	−2940.00	−0.805795
$238$	1782.00	0.485336
$239$	675.000	0.182687	0.0913433	−	0.995819i	$-0.470884\pi$
$239$	675.000	0.182687	0.0913433	+	0.995819i	$0.470884\pi$
$240$	−912.000	−0.245289
$241$	6068.00	1.62188	0.810942	−	0.585126i	$-0.198955\pi$
$241$	6068.00	1.62188	0.810942	+	0.585126i	$0.198955\pi$
$242$	−2324.00	−0.617324
$243$	243.000	0.0641500
$244$	2348.00	0.616046
$245$	4978.00	1.29809
$246$	48.0000	0.0124405
$247$	0	0
$248$	−2096.00	−0.536678
$249$	−1476.00	−0.375653
$250$	−4218.00	−1.06708
$251$	−3763.00	−0.946289	−0.473144	−	0.880985i	$-0.656881\pi$
$251$	−3763.00	−0.946289	−0.473144	+	0.880985i	$0.656881\pi$
$252$	324.000	0.0809924
$253$	−884.000	−0.219670
$254$	2772.00	0.684767
$255$	−5643.00	−1.38580
$256$	256.000	0.0625000
$257$	−4304.00	−1.04465	−0.522327	−	0.852745i	$-0.674936\pi$
$257$	−4304.00	−1.04465	−0.522327	+	0.852745i	$0.674936\pi$
$258$	438.000	0.105693
$259$	2664.00	0.639123
$260$	2888.00	0.688870
$261$	−1170.00	−0.277476
$262$	−2166.00	−0.510748
$263$	−8247.00	−1.93358	−0.966791	−	0.255570i	$-0.917737\pi$
$263$	−8247.00	−1.93358	−0.966791	+	0.255570i	$0.917737\pi$
$264$	−312.000	−0.0727359
$265$	−9538.00	−2.21100
$266$	0	0
$267$	−2430.00	−0.556980
$268$	−2736.00	−0.623611
$269$	5430.00	1.23075	0.615377	−	0.788233i	$-0.289004\pi$
$269$	5430.00	1.23075	0.615377	+	0.788233i	$0.289004\pi$
$270$	−1026.00	−0.231261
$271$	5452.00	1.22209	0.611043	−	0.791597i	$-0.290750\pi$
$271$	5452.00	1.22209	0.611043	+	0.791597i	$0.290750\pi$
$272$	1584.00	0.353103
$273$	−1026.00	−0.227459
$274$	998.000	0.220042
$275$	−3068.00	−0.672754
$276$	816.000	0.177962
$277$	−6311.00	−1.36892	−0.684461	−	0.729050i	$-0.739962\pi$
$277$	−6311.00	−1.36892	−0.684461	+	0.729050i	$0.739962\pi$
$278$	−390.000	−0.0841390
$279$	−2358.00	−0.505985
$280$	−1368.00	−0.291977
$281$	6158.00	1.30731	0.653657	−	0.756791i	$-0.273234\pi$
$281$	6158.00	1.30731	0.653657	+	0.756791i	$0.273234\pi$
$282$	−1626.00	−0.343358
$283$	5853.00	1.22942	0.614708	−	0.788755i	$-0.289274\pi$
$283$	5853.00	1.22942	0.614708	+	0.788755i	$0.289274\pi$
$284$	−3968.00	−0.829076
$285$	0	0
$286$	988.000	0.204272
$287$	72.0000	0.0148085
$288$	288.000	0.0589256
$289$	4888.00	0.994911
$290$	4940.00	1.00030
$291$	3138.00	0.632140
$292$	−2028.00	−0.406437
$293$	5892.00	1.17479	0.587396	−	0.809299i	$-0.300153\pi$
$293$	5892.00	1.17479	0.587396	+	0.809299i	$0.300153\pi$
$294$	−1572.00	−0.311840
$295$	10260.0	2.02495
$296$	2368.00	0.464991
$297$	−351.000	−0.0685760
$298$	−2970.00	−0.577341
$299$	−2584.00	−0.499788
$300$	2832.00	0.545019
$301$	657.000	0.125810
$302$	−664.000	−0.126520
$303$	5046.00	0.956717
$304$	0	0
$305$	−11153.0	−2.09383
$306$	1782.00	0.332909
$307$	−1824.00	−0.339092	−0.169546	−	0.985522i	$-0.554230\pi$
$307$	−1824.00	−0.339092	−0.169546	+	0.985522i	$0.554230\pi$
$308$	−468.000	−0.0865804
$309$	−2934.00	−0.540160
$310$	9956.00	1.82407
$311$	9757.00	1.77900	0.889500	−	0.456936i	$-0.151053\pi$
$311$	9757.00	1.77900	0.889500	+	0.456936i	$0.151053\pi$
$312$	−912.000	−0.165487
$313$	−5722.00	−1.03331	−0.516656	−	0.856193i	$-0.672823\pi$
$313$	−5722.00	−1.03331	−0.516656	+	0.856193i	$0.672823\pi$
$314$	7628.00	1.37093
$315$	−1539.00	−0.275279
$316$	−3920.00	−0.697839
$317$	−10224.0	−1.81147	−0.905737	−	0.423841i	$-0.860682\pi$
$317$	−10224.0	−1.81147	−0.905737	+	0.423841i	$0.860682\pi$
$318$	3012.00	0.531146
$319$	1690.00	0.296620
$320$	−1216.00	−0.212426
$321$	−2022.00	−0.351580
$322$	1224.00	0.211835
$323$	0	0
$324$	324.000	0.0555556
$325$	−8968.00	−1.53063
$326$	−6704.00	−1.13896
$327$	1080.00	0.182643
$328$	64.0000	0.0107738
$329$	−2439.00	−0.408713
$330$	1482.00	0.247216
$331$	−1712.00	−0.284290	−0.142145	−	0.989846i	$-0.545400\pi$
$331$	−1712.00	−0.284290	−0.142145	+	0.989846i	$0.545400\pi$
$332$	−1968.00	−0.325325
$333$	2664.00	0.438397
$334$	−7428.00	−1.21689
$335$	12996.0	2.11954
$336$	432.000	0.0701415
$337$	−934.000	−0.150974	−0.0754870	−	0.997147i	$-0.524051\pi$
$337$	−934.000	−0.150974	−0.0754870	+	0.997147i	$0.524051\pi$
$338$	−1506.00	−0.242354
$339$	−5514.00	−0.883420
$340$	−7524.00	−1.20014
$341$	3406.00	0.540895
$342$	0	0
$343$	−5445.00	−0.857150
$344$	584.000	0.0915325
$345$	−3876.00	−0.604860
$346$	2484.00	0.385956
$347$	−6491.00	−1.00419	−0.502097	−	0.864811i	$-0.667438\pi$
$347$	−6491.00	−1.00419	−0.502097	+	0.864811i	$0.667438\pi$
$348$	−1560.00	−0.240301
$349$	3205.00	0.491575	0.245788	−	0.969324i	$-0.420953\pi$
$349$	3205.00	0.491575	0.245788	+	0.969324i	$0.420953\pi$
$350$	4248.00	0.648758
$351$	−1026.00	−0.156022
$352$	−416.000	−0.0629911
$353$	−3522.00	−0.531040	−0.265520	−	0.964105i	$-0.585544\pi$
$353$	−3522.00	−0.531040	−0.265520	+	0.964105i	$0.585544\pi$
$354$	−3240.00	−0.486452
$355$	18848.0	2.81788
$356$	−3240.00	−0.482359
$357$	2673.00	0.396275
$358$	−4620.00	−0.682052
$359$	−7365.00	−1.08276	−0.541379	−	0.840779i	$-0.682097\pi$
$359$	−7365.00	−1.08276	−0.541379	+	0.840779i	$0.682097\pi$
$360$	−1368.00	−0.200278
$361$	0	0
$362$	−164.000	−0.0238112
$363$	−3486.00	−0.504043
$364$	−1368.00	−0.196985
$365$	9633.00	1.38141
$366$	3522.00	0.503000
$367$	10624.0	1.51109	0.755543	−	0.655099i	$-0.227373\pi$
$367$	10624.0	1.51109	0.755543	+	0.655099i	$0.227373\pi$
$368$	1088.00	0.154119
$369$	72.0000	0.0101576
$370$	−11248.0	−1.58042
$371$	4518.00	0.632245
$372$	−3144.00	−0.438196
$373$	−2528.00	−0.350925	−0.175462	−	0.984486i	$-0.556142\pi$
$373$	−2528.00	−0.350925	−0.175462	+	0.984486i	$0.556142\pi$
$374$	−2574.00	−0.355878
$375$	−6327.00	−0.871266
$376$	−2168.00	−0.297357
$377$	4940.00	0.674862
$378$	486.000	0.0661300
$379$	6810.00	0.922972	0.461486	−	0.887148i	$-0.347317\pi$
$379$	6810.00	0.922972	0.461486	+	0.887148i	$0.347317\pi$
$380$	0	0
$381$	4158.00	0.559110
$382$	−2626.00	−0.351722
$383$	8222.00	1.09693	0.548466	−	0.836173i	$-0.315212\pi$
$383$	8222.00	1.09693	0.548466	+	0.836173i	$0.315212\pi$
$384$	384.000	0.0510310
$385$	2223.00	0.294272
$386$	4704.00	0.620278
$387$	657.000	0.0862976
$388$	4184.00	0.547450
$389$	1725.00	0.224835	0.112418	−	0.993661i	$-0.464141\pi$
$389$	1725.00	0.224835	0.112418	+	0.993661i	$0.464141\pi$
$390$	4332.00	0.562460
$391$	6732.00	0.870721
$392$	−2096.00	−0.270061
$393$	−3249.00	−0.417024
$394$	−7652.00	−0.978432
$395$	18620.0	2.37183
$396$	−468.000	−0.0593886
$397$	11669.0	1.47519	0.737595	−	0.675244i	$-0.235961\pi$
$397$	11669.0	1.47519	0.737595	+	0.675244i	$0.235961\pi$
$398$	4450.00	0.560448
$399$	0	0
$400$	3776.00	0.472000
$401$	−9592.00	−1.19452	−0.597259	−	0.802049i	$-0.703744\pi$
$401$	−9592.00	−1.19452	−0.597259	+	0.802049i	$0.703744\pi$
$402$	−4104.00	−0.509176
$403$	9956.00	1.23063
$404$	6728.00	0.828541
$405$	−1539.00	−0.188824
$406$	−2340.00	−0.286040
$407$	−3848.00	−0.468644
$408$	2376.00	0.288308
$409$	−9130.00	−1.10379	−0.551894	−	0.833914i	$-0.686095\pi$
$409$	−9130.00	−1.10379	−0.551894	+	0.833914i	$0.686095\pi$
$410$	−304.000	−0.0366183
$411$	1497.00	0.179663
$412$	−3912.00	−0.467792
$413$	−4860.00	−0.579044
$414$	1224.00	0.145305
$415$	9348.00	1.10572
$416$	−1216.00	−0.143316
$417$	−585.000	−0.0686992
$418$	0	0
$419$	−3660.00	−0.426737	−0.213368	−	0.976972i	$-0.568443\pi$
$419$	−3660.00	−0.426737	−0.213368	+	0.976972i	$0.568443\pi$
$420$	−2052.00	−0.238398
$421$	13438.0	1.55565	0.777824	−	0.628482i	$-0.216323\pi$
$421$	13438.0	1.55565	0.777824	+	0.628482i	$0.216323\pi$
$422$	−3264.00	−0.376514
$423$	−2439.00	−0.280350
$424$	4016.00	0.459986
$425$	23364.0	2.66664
$426$	−5952.00	−0.676937
$427$	5283.00	0.598741
$428$	−2696.00	−0.304477
$429$	1482.00	0.166787
$430$	−2774.00	−0.311103
$431$	1458.00	0.162945	0.0814726	−	0.996676i	$-0.474038\pi$
$431$	1458.00	0.162945	0.0814726	+	0.996676i	$0.474038\pi$
$432$	432.000	0.0481125
$433$	13342.0	1.48078	0.740388	−	0.672180i	$-0.234642\pi$
$433$	13342.0	1.48078	0.740388	+	0.672180i	$0.234642\pi$
$434$	−4716.00	−0.521602
$435$	7410.00	0.816741
$436$	1440.00	0.158173
$437$	0	0
$438$	−3042.00	−0.331855
$439$	−4490.00	−0.488146	−0.244073	−	0.969757i	$-0.578484\pi$
$439$	−4490.00	−0.488146	−0.244073	+	0.969757i	$0.578484\pi$
$440$	1976.00	0.214096
$441$	−2358.00	−0.254616
$442$	−7524.00	−0.809684
$443$	2713.00	0.290967	0.145484	−	0.989361i	$-0.453526\pi$
$443$	2713.00	0.290967	0.145484	+	0.989361i	$0.453526\pi$
$444$	3552.00	0.379663
$445$	15390.0	1.63945
$446$	−2296.00	−0.243764
$447$	−4455.00	−0.471397
$448$	576.000	0.0607443
$449$	−2140.00	−0.224928	−0.112464	−	0.993656i	$-0.535874\pi$
$449$	−2140.00	−0.224928	−0.112464	+	0.993656i	$0.535874\pi$
$450$	4248.00	0.445006
$451$	−104.000	−0.0108585
$452$	−7352.00	−0.765064
$453$	−996.000	−0.103303
$454$	−11588.0	−1.19791
$455$	6498.00	0.669519
$456$	0	0
$457$	−2221.00	−0.227339	−0.113670	−	0.993519i	$-0.536261\pi$
$457$	−2221.00	−0.227339	−0.113670	+	0.993519i	$0.536261\pi$
$458$	5050.00	0.515221
$459$	2673.00	0.271819
$460$	−5168.00	−0.523824
$461$	16907.0	1.70811	0.854054	−	0.520185i	$-0.174137\pi$
$461$	16907.0	1.70811	0.854054	+	0.520185i	$0.174137\pi$
$462$	−702.000	−0.0706926
$463$	−2237.00	−0.224540	−0.112270	−	0.993678i	$-0.535812\pi$
$463$	−2237.00	−0.224540	−0.112270	+	0.993678i	$0.535812\pi$
$464$	−2080.00	−0.208107
$465$	14934.0	1.48935
$466$	−2454.00	−0.243947
$467$	−311.000	−0.0308166	−0.0154083	−	0.999881i	$-0.504905\pi$
$467$	−311.000	−0.0308166	−0.0154083	+	0.999881i	$0.504905\pi$
$468$	−1368.00	−0.135119
$469$	−6156.00	−0.606093
$470$	10298.0	1.01066
$471$	11442.0	1.11936
$472$	−4320.00	−0.421280
$473$	−949.000	−0.0922517
$474$	−5880.00	−0.569783
$475$	0	0
$476$	3564.00	0.343184
$477$	4518.00	0.433679
$478$	1350.00	0.129179
$479$	12640.0	1.20571	0.602856	−	0.797850i	$-0.294029\pi$
$479$	12640.0	1.20571	0.602856	+	0.797850i	$0.294029\pi$
$480$	−1824.00	−0.173445
$481$	−11248.0	−1.06625
$482$	12136.0	1.14685
$483$	1836.00	0.172963
$484$	−4648.00	−0.436514
$485$	−19874.0	−1.86068
$486$	486.000	0.0453609
$487$	−13324.0	−1.23977	−0.619885	−	0.784693i	$-0.712821\pi$
$487$	−13324.0	−1.23977	−0.619885	+	0.784693i	$0.712821\pi$
$488$	4696.00	0.435611
$489$	−10056.0	−0.929955
$490$	9956.00	0.917890
$491$	5552.00	0.510302	0.255151	−	0.966901i	$-0.417875\pi$
$491$	5552.00	0.510302	0.255151	+	0.966901i	$0.417875\pi$
$492$	96.0000	0.00879678
$493$	−12870.0	−1.17573
$494$	0	0
$495$	2223.00	0.201851
$496$	−4192.00	−0.379489
$497$	−8928.00	−0.805786
$498$	−2952.00	−0.265627
$499$	−15125.0	−1.35689	−0.678445	−	0.734651i	$-0.737346\pi$
$499$	−15125.0	−1.35689	−0.678445	+	0.734651i	$0.737346\pi$
$500$	−8436.00	−0.754539
$501$	−11142.0	−0.993589
$502$	−7526.00	−0.669127
$503$	−1632.00	−0.144667	−0.0723333	−	0.997381i	$-0.523045\pi$
$503$	−1632.00	−0.144667	−0.0723333	+	0.997381i	$0.523045\pi$
$504$	648.000	0.0572703
$505$	−31958.0	−2.81606
$506$	−1768.00	−0.155330
$507$	−2259.00	−0.197881
$508$	5544.00	0.484203
$509$	−4410.00	−0.384027	−0.192014	−	0.981392i	$-0.561502\pi$
$509$	−4410.00	−0.384027	−0.192014	+	0.981392i	$0.561502\pi$
$510$	−11286.0	−0.979907
$511$	−4563.00	−0.395020
$512$	512.000	0.0441942
$513$	0	0
$514$	−8608.00	−0.738682
$515$	18582.0	1.58994
$516$	876.000	0.0747359
$517$	3523.00	0.299693
$518$	5328.00	0.451928
$519$	3726.00	0.315131
$520$	5776.00	0.487105
$521$	2488.00	0.209215	0.104608	−	0.994514i	$-0.466641\pi$
$521$	2488.00	0.209215	0.104608	+	0.994514i	$0.466641\pi$
$522$	−2340.00	−0.196205
$523$	17662.0	1.47668	0.738342	−	0.674427i	$-0.235609\pi$
$523$	17662.0	1.47668	0.738342	+	0.674427i	$0.235609\pi$
$524$	−4332.00	−0.361153
$525$	6372.00	0.529708
$526$	−16494.0	−1.36725
$527$	−25938.0	−2.14398
$528$	−624.000	−0.0514320
$529$	−7543.00	−0.619956
$530$	−19076.0	−1.56341
$531$	−4860.00	−0.397187
$532$	0	0
$533$	−304.000	−0.0247049
$534$	−4860.00	−0.393844
$535$	12806.0	1.03486
$536$	−5472.00	−0.440960
$537$	−6930.00	−0.556893
$538$	10860.0	0.870275
$539$	3406.00	0.272183
$540$	−2052.00	−0.163526
$541$	10487.0	0.833404	0.416702	−	0.909043i	$-0.363186\pi$
$541$	10487.0	0.833404	0.416702	+	0.909043i	$0.363186\pi$
$542$	10904.0	0.864146
$543$	−246.000	−0.0194418
$544$	3168.00	0.249682
$545$	−6840.00	−0.537603
$546$	−2052.00	−0.160838
$547$	−13174.0	−1.02976	−0.514881	−	0.857262i	$-0.672164\pi$
$547$	−13174.0	−1.02976	−0.514881	+	0.857262i	$0.672164\pi$
$548$	1996.00	0.155593
$549$	5283.00	0.410698
$550$	−6136.00	−0.475709
$551$	0	0
$552$	1632.00	0.125838
$553$	−8820.00	−0.678236
$554$	−12622.0	−0.967974
$555$	−16872.0	−1.29041
$556$	−780.000	−0.0594953
$557$	−14321.0	−1.08941	−0.544704	−	0.838628i	$-0.683358\pi$
$557$	−14321.0	−1.08941	−0.544704	+	0.838628i	$0.683358\pi$
$558$	−4716.00	−0.357785
$559$	−2774.00	−0.209889
$560$	−2736.00	−0.206459
$561$	−3861.00	−0.290573
$562$	12316.0	0.924411
$563$	7212.00	0.539875	0.269937	−	0.962878i	$-0.412997\pi$
$563$	7212.00	0.539875	0.269937	+	0.962878i	$0.412997\pi$
$564$	−3252.00	−0.242791
$565$	34922.0	2.60032
$566$	11706.0	0.869328
$567$	729.000	0.0539949
$568$	−7936.00	−0.586245
$569$	−3490.00	−0.257133	−0.128566	−	0.991701i	$-0.541038\pi$
$569$	−3490.00	−0.257133	−0.128566	+	0.991701i	$0.541038\pi$
$570$	0	0
$571$	6152.00	0.450881	0.225441	−	0.974257i	$-0.427618\pi$
$571$	6152.00	0.450881	0.225441	+	0.974257i	$0.427618\pi$
$572$	1976.00	0.144442
$573$	−3939.00	−0.287180
$574$	144.000	0.0104712
$575$	16048.0	1.16391
$576$	576.000	0.0416667
$577$	−20941.0	−1.51089	−0.755446	−	0.655210i	$-0.772580\pi$
$577$	−20941.0	−1.51089	−0.755446	+	0.655210i	$0.772580\pi$
$578$	9776.00	0.703509
$579$	7056.00	0.506455
$580$	9880.00	0.707318
$581$	−4428.00	−0.316187
$582$	6276.00	0.446991
$583$	−6526.00	−0.463601
$584$	−4056.00	−0.287395
$585$	6498.00	0.459247
$586$	11784.0	0.830704
$587$	−23331.0	−1.64050	−0.820250	−	0.572005i	$-0.806166\pi$
$587$	−23331.0	−1.64050	−0.820250	+	0.572005i	$0.806166\pi$
$588$	−3144.00	−0.220504
$589$	0	0
$590$	20520.0	1.43186
$591$	−11478.0	−0.798886
$592$	4736.00	0.328798
$593$	−18542.0	−1.28403	−0.642014	−	0.766693i	$-0.721901\pi$
$593$	−18542.0	−1.28403	−0.642014	+	0.766693i	$0.721901\pi$
$594$	−702.000	−0.0484906
$595$	−16929.0	−1.16642
$596$	−5940.00	−0.408241
$597$	6675.00	0.457604
$598$	−5168.00	−0.353403
$599$	2700.00	0.184172	0.0920860	−	0.995751i	$-0.470647\pi$
$599$	2700.00	0.184172	0.0920860	+	0.995751i	$0.470647\pi$
$600$	5664.00	0.385386
$601$	−17452.0	−1.18450	−0.592248	−	0.805756i	$-0.701759\pi$
$601$	−17452.0	−1.18450	−0.592248	+	0.805756i	$0.701759\pi$
$602$	1314.00	0.0889612
$603$	−6156.00	−0.415741
$604$	−1328.00	−0.0894628
$605$	22078.0	1.48363
$606$	10092.0	0.676501
$607$	−5114.00	−0.341962	−0.170981	−	0.985274i	$-0.554694\pi$
$607$	−5114.00	−0.341962	−0.170981	+	0.985274i	$0.554694\pi$
$608$	0	0
$609$	−3510.00	−0.233551
$610$	−22306.0	−1.48056
$611$	10298.0	0.681853
$612$	3564.00	0.235402
$613$	19333.0	1.27382	0.636911	−	0.770938i	$-0.280212\pi$
$613$	19333.0	1.27382	0.636911	+	0.770938i	$0.280212\pi$
$614$	−3648.00	−0.239774
$615$	−456.000	−0.0298987
$616$	−936.000	−0.0612216
$617$	1599.00	0.104333	0.0521664	−	0.998638i	$-0.483387\pi$
$617$	1599.00	0.104333	0.0521664	+	0.998638i	$0.483387\pi$
$618$	−5868.00	−0.381951
$619$	3580.00	0.232459	0.116230	−	0.993222i	$-0.462919\pi$
$619$	3580.00	0.232459	0.116230	+	0.993222i	$0.462919\pi$
$620$	19912.0	1.28981
$621$	1836.00	0.118641
$622$	19514.0	1.25794
$623$	−7290.00	−0.468808
$624$	−1824.00	−0.117017
$625$	10571.0	0.676544
$626$	−11444.0	−0.730662
$627$	0	0
$628$	15256.0	0.969396
$629$	29304.0	1.85759
$630$	−3078.00	−0.194652
$631$	18917.0	1.19346	0.596730	−	0.802442i	$-0.296466\pi$
$631$	18917.0	1.19346	0.596730	+	0.802442i	$0.296466\pi$
$632$	−7840.00	−0.493447
$633$	−4896.00	−0.307423
$634$	−20448.0	−1.28091
$635$	−26334.0	−1.64572
$636$	6024.00	0.375577
$637$	9956.00	0.619264
$638$	3380.00	0.209742
$639$	−8928.00	−0.552717
$640$	−2432.00	−0.150208
$641$	8058.00	0.496524	0.248262	−	0.968693i	$-0.420141\pi$
$641$	8058.00	0.496524	0.248262	+	0.968693i	$0.420141\pi$
$642$	−4044.00	−0.248604
$643$	4153.00	0.254710	0.127355	−	0.991857i	$-0.459351\pi$
$643$	4153.00	0.254710	0.127355	+	0.991857i	$0.459351\pi$
$644$	2448.00	0.149790
$645$	−4161.00	−0.254014
$646$	0	0
$647$	17519.0	1.06452	0.532259	−	0.846582i	$-0.321343\pi$
$647$	17519.0	1.06452	0.532259	+	0.846582i	$0.321343\pi$
$648$	648.000	0.0392837
$649$	7020.00	0.424590
$650$	−17936.0	−1.08232
$651$	−7074.00	−0.425886
$652$	−13408.0	−0.805365
$653$	−3957.00	−0.237135	−0.118568	−	0.992946i	$-0.537830\pi$
$653$	−3957.00	−0.237135	−0.118568	+	0.992946i	$0.537830\pi$
$654$	2160.00	0.129148
$655$	20577.0	1.22750
$656$	128.000	0.00761823
$657$	−4563.00	−0.270958
$658$	−4878.00	−0.289003
$659$	14890.0	0.880170	0.440085	−	0.897956i	$-0.354948\pi$
$659$	14890.0	0.880170	0.440085	+	0.897956i	$0.354948\pi$
$660$	2964.00	0.174808
$661$	−11812.0	−0.695058	−0.347529	−	0.937669i	$-0.612979\pi$
$661$	−11812.0	−0.695058	−0.347529	+	0.937669i	$0.612979\pi$
$662$	−3424.00	−0.201023
$663$	−11286.0	−0.661104
$664$	−3936.00	−0.230040
$665$	0	0
$666$	5328.00	0.309994
$667$	−8840.00	−0.513173
$668$	−14856.0	−0.860473
$669$	−3444.00	−0.199032
$670$	25992.0	1.49874
$671$	−7631.00	−0.439034
$672$	864.000	0.0495975
$673$	−9308.00	−0.533131	−0.266565	−	0.963817i	$-0.585889\pi$
$673$	−9308.00	−0.533131	−0.266565	+	0.963817i	$0.585889\pi$
$674$	−1868.00	−0.106755
$675$	6372.00	0.363346
$676$	−3012.00	−0.171370
$677$	−9814.00	−0.557138	−0.278569	−	0.960416i	$-0.589860\pi$
$677$	−9814.00	−0.557138	−0.278569	+	0.960416i	$0.589860\pi$
$678$	−11028.0	−0.624672
$679$	9414.00	0.532071
$680$	−15048.0	−0.848624
$681$	−17382.0	−0.978091
$682$	6812.00	0.382471
$683$	29942.0	1.67745	0.838725	−	0.544555i	$-0.183301\pi$
$683$	29942.0	1.67745	0.838725	+	0.544555i	$0.183301\pi$
$684$	0	0
$685$	−9481.00	−0.528833
$686$	−10890.0	−0.606096
$687$	7575.00	0.420676
$688$	1168.00	0.0647232
$689$	−19076.0	−1.05477
$690$	−7752.00	−0.427701
$691$	−25353.0	−1.39576	−0.697882	−	0.716212i	$-0.745874\pi$
$691$	−25353.0	−1.39576	−0.697882	+	0.716212i	$0.745874\pi$
$692$	4968.00	0.272912
$693$	−1053.00	−0.0577203
$694$	−12982.0	−0.710072
$695$	3705.00	0.202214
$696$	−3120.00	−0.169919
$697$	792.000	0.0430404
$698$	6410.00	0.347596
$699$	−3681.00	−0.199182
$700$	8496.00	0.458741
$701$	−5038.00	−0.271445	−0.135722	−	0.990747i	$-0.543335\pi$
$701$	−5038.00	−0.271445	−0.135722	+	0.990747i	$0.543335\pi$
$702$	−2052.00	−0.110324
$703$	0	0
$704$	−832.000	−0.0445414
$705$	15447.0	0.825202
$706$	−7044.00	−0.375502
$707$	15138.0	0.805266
$708$	−6480.00	−0.343974
$709$	30150.0	1.59705	0.798524	−	0.601963i	$-0.205615\pi$
$709$	30150.0	1.59705	0.798524	+	0.601963i	$0.205615\pi$
$710$	37696.0	1.99254
$711$	−8820.00	−0.465226
$712$	−6480.00	−0.341079
$713$	−17816.0	−0.935785
$714$	5346.00	0.280209
$715$	−9386.00	−0.490932
$716$	−9240.00	−0.482284
$717$	2025.00	0.105474
$718$	−14730.0	−0.765625
$719$	−425.000	−0.0220443	−0.0110221	−	0.999939i	$-0.503509\pi$
$719$	−425.000	−0.0220443	−0.0110221	+	0.999939i	$0.503509\pi$
$720$	−2736.00	−0.141618
$721$	−8802.00	−0.454651
$722$	0	0
$723$	18204.0	0.936396
$724$	−328.000	−0.0168370
$725$	−30680.0	−1.57162
$726$	−6972.00	−0.356412
$727$	−18941.0	−0.966276	−0.483138	−	0.875544i	$-0.660503\pi$
$727$	−18941.0	−0.966276	−0.483138	+	0.875544i	$0.660503\pi$
$728$	−2736.00	−0.139290
$729$	729.000	0.0370370
$730$	19266.0	0.976804
$731$	7227.00	0.365664
$732$	7044.00	0.355674
$733$	−27702.0	−1.39590	−0.697951	−	0.716145i	$-0.745905\pi$
$733$	−27702.0	−1.39590	−0.697951	+	0.716145i	$0.745905\pi$
$734$	21248.0	1.06850
$735$	14934.0	0.749454
$736$	2176.00	0.108979
$737$	8892.00	0.444425
$738$	144.000	0.00718254
$739$	4685.00	0.233208	0.116604	−	0.993179i	$-0.462799\pi$
$739$	4685.00	0.233208	0.116604	+	0.993179i	$0.462799\pi$
$740$	−22496.0	−1.11753
$741$	0	0
$742$	9036.00	0.447065
$743$	36172.0	1.78603	0.893016	−	0.450025i	$-0.148585\pi$
$743$	36172.0	1.78603	0.893016	+	0.450025i	$0.148585\pi$
$744$	−6288.00	−0.309851
$745$	28215.0	1.38754
$746$	−5056.00	−0.248141
$747$	−4428.00	−0.216884
$748$	−5148.00	−0.251644
$749$	−6066.00	−0.295924
$750$	−12654.0	−0.616078
$751$	−2912.00	−0.141492	−0.0707459	−	0.997494i	$-0.522538\pi$
$751$	−2912.00	−0.141492	−0.0707459	+	0.997494i	$0.522538\pi$
$752$	−4336.00	−0.210263
$753$	−11289.0	−0.546340
$754$	9880.00	0.477199
$755$	6308.00	0.304068
$756$	972.000	0.0467610
$757$	27259.0	1.30878	0.654389	−	0.756158i	$-0.272926\pi$
$757$	27259.0	1.30878	0.654389	+	0.756158i	$0.272926\pi$
$758$	13620.0	0.652639
$759$	−2652.00	−0.126827
$760$	0	0
$761$	147.000	0.00700229	0.00350115	−	0.999994i	$-0.498886\pi$
$761$	147.000	0.00700229	0.00350115	+	0.999994i	$0.498886\pi$
$762$	8316.00	0.395350
$763$	3240.00	0.153730
$764$	−5252.00	−0.248705
$765$	−16929.0	−0.800091
$766$	16444.0	0.775647
$767$	20520.0	0.966016
$768$	768.000	0.0360844
$769$	−14645.0	−0.686752	−0.343376	−	0.939198i	$-0.611570\pi$
$769$	−14645.0	−0.686752	−0.343376	+	0.939198i	$0.611570\pi$
$770$	4446.00	0.208081
$771$	−12912.0	−0.603131
$772$	9408.00	0.438603
$773$	1632.00	0.0759366	0.0379683	−	0.999279i	$-0.487911\pi$
$773$	1632.00	0.0759366	0.0379683	+	0.999279i	$0.487911\pi$
$774$	1314.00	0.0610216
$775$	−61832.0	−2.86590
$776$	8368.00	0.387105
$777$	7992.00	0.368998
$778$	3450.00	0.158983
$779$	0	0
$780$	8664.00	0.397719
$781$	12896.0	0.590852
$782$	13464.0	0.615693
$783$	−3510.00	−0.160201
$784$	−4192.00	−0.190962
$785$	−72466.0	−3.29481
$786$	−6498.00	−0.294880
$787$	15776.0	0.714554	0.357277	−	0.933999i	$-0.383705\pi$
$787$	15776.0	0.714554	0.357277	+	0.933999i	$0.383705\pi$
$788$	−15304.0	−0.691856
$789$	−24741.0	−1.11635
$790$	37240.0	1.67714
$791$	−16542.0	−0.743572
$792$	−936.000	−0.0419941
$793$	−22306.0	−0.998877
$794$	23338.0	1.04312
$795$	−28614.0	−1.27652
$796$	8900.00	0.396297
$797$	17256.0	0.766925	0.383462	−	0.923557i	$-0.374732\pi$
$797$	17256.0	0.766925	0.383462	+	0.923557i	$0.374732\pi$
$798$	0	0
$799$	−26829.0	−1.18791
$800$	7552.00	0.333754
$801$	−7290.00	−0.321572
$802$	−19184.0	−0.844652
$803$	6591.00	0.289653
$804$	−8208.00	−0.360042
$805$	−11628.0	−0.509110
$806$	19912.0	0.870186
$807$	16290.0	0.710576
$808$	13456.0	0.585867
$809$	25845.0	1.12319	0.561596	−	0.827412i	$-0.310188\pi$
$809$	25845.0	1.12319	0.561596	+	0.827412i	$0.310188\pi$
$810$	−3078.00	−0.133518
$811$	−10962.0	−0.474634	−0.237317	−	0.971432i	$-0.576268\pi$
$811$	−10962.0	−0.474634	−0.237317	+	0.971432i	$0.576268\pi$
$812$	−4680.00	−0.202261
$813$	16356.0	0.705572
$814$	−7696.00	−0.331382
$815$	63688.0	2.73729
$816$	4752.00	0.203864
$817$	0	0
$818$	−18260.0	−0.780496
$819$	−3078.00	−0.131324
$820$	−608.000	−0.0258930
$821$	627.000	0.0266534	0.0133267	−	0.999911i	$-0.495758\pi$
$821$	627.000	0.0266534	0.0133267	+	0.999911i	$0.495758\pi$
$822$	2994.00	0.127041
$823$	38153.0	1.61595	0.807977	−	0.589214i	$-0.200563\pi$
$823$	38153.0	1.61595	0.807977	+	0.589214i	$0.200563\pi$
$824$	−7824.00	−0.330779
$825$	−9204.00	−0.388415
$826$	−9720.00	−0.409446
$827$	−23744.0	−0.998379	−0.499190	−	0.866493i	$-0.666369\pi$
$827$	−23744.0	−0.998379	−0.499190	+	0.866493i	$0.666369\pi$
$828$	2448.00	0.102746
$829$	18060.0	0.756634	0.378317	−	0.925676i	$-0.376503\pi$
$829$	18060.0	0.756634	0.378317	+	0.925676i	$0.376503\pi$
$830$	18696.0	0.781865
$831$	−18933.0	−0.790347
$832$	−2432.00	−0.101339
$833$	−25938.0	−1.07887
$834$	−1170.00	−0.0485777
$835$	70566.0	2.92460
$836$	0	0
$837$	−7074.00	−0.292130
$838$	−7320.00	−0.301748
$839$	8590.00	0.353468	0.176734	−	0.984259i	$-0.443447\pi$
$839$	8590.00	0.353468	0.176734	+	0.984259i	$0.443447\pi$
$840$	−4104.00	−0.168573
$841$	−7489.00	−0.307065
$842$	26876.0	1.10001
$843$	18474.0	0.754779
$844$	−6528.00	−0.266236
$845$	14307.0	0.582457
$846$	−4878.00	−0.198238
$847$	−10458.0	−0.424252
$848$	8032.00	0.325259
$849$	17559.0	0.709804
$850$	46728.0	1.88560
$851$	20128.0	0.810786
$852$	−11904.0	−0.478667
$853$	−222.000	−0.00891106	−0.00445553	−	0.999990i	$-0.501418\pi$
$853$	−222.000	−0.00891106	−0.00445553	+	0.999990i	$0.501418\pi$
$854$	10566.0	0.423374
$855$	0	0
$856$	−5392.00	−0.215298
$857$	16956.0	0.675853	0.337926	−	0.941173i	$-0.390274\pi$
$857$	16956.0	0.675853	0.337926	+	0.941173i	$0.390274\pi$
$858$	2964.00	0.117936
$859$	11245.0	0.446652	0.223326	−	0.974744i	$-0.428308\pi$
$859$	11245.0	0.446652	0.223326	+	0.974744i	$0.428308\pi$
$860$	−5548.00	−0.219983
$861$	216.000	0.00854966
$862$	2916.00	0.115220
$863$	−12468.0	−0.491791	−0.245896	−	0.969296i	$-0.579082\pi$
$863$	−12468.0	−0.491791	−0.245896	+	0.969296i	$0.579082\pi$
$864$	864.000	0.0340207
$865$	−23598.0	−0.927579
$866$	26684.0	1.04707
$867$	14664.0	0.574412
$868$	−9432.00	−0.368828
$869$	12740.0	0.497324
$870$	14820.0	0.577523
$871$	25992.0	1.01114
$872$	2880.00	0.111845
$873$	9414.00	0.364966
$874$	0	0
$875$	−18981.0	−0.733343
$876$	−6084.00	−0.234657
$877$	−8194.00	−0.315498	−0.157749	−	0.987479i	$-0.550424\pi$
$877$	−8194.00	−0.315498	−0.157749	+	0.987479i	$0.550424\pi$
$878$	−8980.00	−0.345171
$879$	17676.0	0.678267
$880$	3952.00	0.151389
$881$	−28833.0	−1.10262	−0.551310	−	0.834300i	$-0.685872\pi$
$881$	−28833.0	−1.10262	−0.551310	+	0.834300i	$0.685872\pi$
$882$	−4716.00	−0.180041
$883$	29413.0	1.12098	0.560491	−	0.828161i	$-0.310613\pi$
$883$	29413.0	1.12098	0.560491	+	0.828161i	$0.310613\pi$
$884$	−15048.0	−0.572533
$885$	30780.0	1.16911
$886$	5426.00	0.205745
$887$	−24544.0	−0.929095	−0.464547	−	0.885548i	$-0.653783\pi$
$887$	−24544.0	−0.929095	−0.464547	+	0.885548i	$0.653783\pi$
$888$	7104.00	0.268462
$889$	12474.0	0.470601
$890$	30780.0	1.15927
$891$	−1053.00	−0.0395924
$892$	−4592.00	−0.172367
$893$	0	0
$894$	−8910.00	−0.333328
$895$	43890.0	1.63920
$896$	1152.00	0.0429527
$897$	−7752.00	−0.288553
$898$	−4280.00	−0.159048
$899$	34060.0	1.26359
$900$	8496.00	0.314667
$901$	49698.0	1.83760
$902$	−208.000	−0.00767810
$903$	1971.00	0.0726365
$904$	−14704.0	−0.540982
$905$	1558.00	0.0572262
$906$	−1992.00	−0.0730461
$907$	26326.0	0.963771	0.481886	−	0.876234i	$-0.339952\pi$
$907$	26326.0	0.963771	0.481886	+	0.876234i	$0.339952\pi$
$908$	−23176.0	−0.847051
$909$	15138.0	0.552361
$910$	12996.0	0.473421
$911$	42738.0	1.55431	0.777153	−	0.629311i	$-0.216663\pi$
$911$	42738.0	1.55431	0.777153	+	0.629311i	$0.216663\pi$
$912$	0	0
$913$	6396.00	0.231847
$914$	−4442.00	−0.160753
$915$	−33459.0	−1.20887
$916$	10100.0	0.364316
$917$	−9747.00	−0.351008
$918$	5346.00	0.192205
$919$	−38680.0	−1.38840	−0.694198	−	0.719784i	$-0.744241\pi$
$919$	−38680.0	−1.38840	−0.694198	+	0.719784i	$0.744241\pi$
$920$	−10336.0	−0.370400
$921$	−5472.00	−0.195775
$922$	33814.0	1.20781
$923$	37696.0	1.34429
$924$	−1404.00	−0.0499872
$925$	69856.0	2.48308
$926$	−4474.00	−0.158774
$927$	−8802.00	−0.311861
$928$	−4160.00	−0.147154
$929$	32270.0	1.13966	0.569830	−	0.821763i	$-0.307009\pi$
$929$	32270.0	1.13966	0.569830	+	0.821763i	$0.307009\pi$
$930$	29868.0	1.05313
$931$	0	0
$932$	−4908.00	−0.172497
$933$	29271.0	1.02711
$934$	−622.000	−0.0217906
$935$	24453.0	0.855293
$936$	−2736.00	−0.0955438
$937$	30989.0	1.08043	0.540217	−	0.841526i	$-0.318342\pi$
$937$	30989.0	1.08043	0.540217	+	0.841526i	$0.318342\pi$
$938$	−12312.0	−0.428573
$939$	−17166.0	−0.596583
$940$	20596.0	0.714646
$941$	8318.00	0.288161	0.144080	−	0.989566i	$-0.453978\pi$
$941$	8318.00	0.288161	0.144080	+	0.989566i	$0.453978\pi$
$942$	22884.0	0.791509
$943$	544.000	0.0187859
$944$	−8640.00	−0.297890
$945$	−4617.00	−0.158932
$946$	−1898.00	−0.0652318
$947$	−46716.0	−1.60303	−0.801513	−	0.597977i	$-0.795971\pi$
$947$	−46716.0	−1.60303	−0.801513	+	0.597977i	$0.795971\pi$
$948$	−11760.0	−0.402898
$949$	19266.0	0.659010
$950$	0	0
$951$	−30672.0	−1.04585
$952$	7128.00	0.242668
$953$	50212.0	1.70674	0.853372	−	0.521303i	$-0.174554\pi$
$953$	50212.0	1.70674	0.853372	+	0.521303i	$0.174554\pi$
$954$	9036.00	0.306658
$955$	24947.0	0.845305
$956$	2700.00	0.0913433
$957$	5070.00	0.171254
$958$	25280.0	0.852568
$959$	4491.00	0.151222
$960$	−3648.00	−0.122644
$961$	38853.0	1.30419
$962$	−22496.0	−0.753950
$963$	−6066.00	−0.202985
$964$	24272.0	0.810942
$965$	−44688.0	−1.49073
$966$	3672.00	0.122303
$967$	−12336.0	−0.410237	−0.205118	−	0.978737i	$-0.565758\pi$
$967$	−12336.0	−0.410237	−0.205118	+	0.978737i	$0.565758\pi$
$968$	−9296.00	−0.308662
$969$	0	0
$970$	−39748.0	−1.31570
$971$	−40532.0	−1.33958	−0.669791	−	0.742550i	$-0.733616\pi$
$971$	−40532.0	−1.33958	−0.669791	+	0.742550i	$0.733616\pi$
$972$	972.000	0.0320750
$973$	−1755.00	−0.0578240
$974$	−26648.0	−0.876650
$975$	−26904.0	−0.883710
$976$	9392.00	0.308023
$977$	19426.0	0.636124	0.318062	−	0.948070i	$-0.396968\pi$
$977$	19426.0	0.636124	0.318062	+	0.948070i	$0.396968\pi$
$978$	−20112.0	−0.657578
$979$	10530.0	0.343759
$980$	19912.0	0.649046
$981$	3240.00	0.105449
$982$	11104.0	0.360838
$983$	−34428.0	−1.11707	−0.558536	−	0.829480i	$-0.688637\pi$
$983$	−34428.0	−1.11707	−0.558536	+	0.829480i	$0.688637\pi$
$984$	192.000	0.00622026
$985$	72694.0	2.35150
$986$	−25740.0	−0.831368
$987$	−7317.00	−0.235970
$988$	0	0
$989$	4964.00	0.159602
$990$	4446.00	0.142730
$991$	−10072.0	−0.322853	−0.161427	−	0.986885i	$-0.551610\pi$
$991$	−10072.0	−0.322853	−0.161427	+	0.986885i	$0.551610\pi$
$992$	−8384.00	−0.268339
$993$	−5136.00	−0.164135
$994$	−17856.0	−0.569777
$995$	−42275.0	−1.34694
$996$	−5904.00	−0.187827
$997$	12589.0	0.399897	0.199949	−	0.979806i	$-0.435922\pi$
$997$	12589.0	0.399897	0.199949	+	0.979806i	$0.435922\pi$
$998$	−30250.0	−0.959466
$999$	7992.00	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.h.1.1		1
19.18	odd	2		114.4.a.a.1.1	✓	1
57.56	even	2		342.4.a.e.1.1		1
76.75	even	2		912.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.a.a.1.1	✓	1	19.18	odd	2
342.4.a.e.1.1		1	57.56	even	2
912.4.a.e.1.1		1	76.75	even	2
2166.4.a.h.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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more