LMFDB - Embedded newform 350.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,4,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, 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("350.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$350 = 2 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	350.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:	$20.6506685020$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	350.1

$q$ -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} -8.00000 q^{3} +4.00000 q^{4} -16.0000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +37.0000 q^{9} -28.0000 q^{11} -32.0000 q^{12} -18.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -74.0000 q^{17} +74.0000 q^{18} +80.0000 q^{19} -56.0000 q^{21} -56.0000 q^{22} +112.000 q^{23} -64.0000 q^{24} -36.0000 q^{26} -80.0000 q^{27} +28.0000 q^{28} +190.000 q^{29} +72.0000 q^{31} +32.0000 q^{32} +224.000 q^{33} -148.000 q^{34} +148.000 q^{36} +346.000 q^{37} +160.000 q^{38} +144.000 q^{39} +162.000 q^{41} -112.000 q^{42} +412.000 q^{43} -112.000 q^{44} +224.000 q^{46} -24.0000 q^{47} -128.000 q^{48} +49.0000 q^{49} +592.000 q^{51} -72.0000 q^{52} -318.000 q^{53} -160.000 q^{54} +56.0000 q^{56} -640.000 q^{57} +380.000 q^{58} -200.000 q^{59} -198.000 q^{61} +144.000 q^{62} +259.000 q^{63} +64.0000 q^{64} +448.000 q^{66} +716.000 q^{67} -296.000 q^{68} -896.000 q^{69} +392.000 q^{71} +296.000 q^{72} -538.000 q^{73} +692.000 q^{74} +320.000 q^{76} -196.000 q^{77} +288.000 q^{78} +240.000 q^{79} -359.000 q^{81} +324.000 q^{82} +1072.00 q^{83} -224.000 q^{84} +824.000 q^{86} -1520.00 q^{87} -224.000 q^{88} +810.000 q^{89} -126.000 q^{91} +448.000 q^{92} -576.000 q^{93} -48.0000 q^{94} -256.000 q^{96} -1354.00 q^{97} +98.0000 q^{98} -1036.00 q^{99} +O(q^{100})

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−8.00000	−1.53960	−0.769800	−	0.638285i	$-0.779644\pi$
$3$	−8.00000	−1.53960	−0.769800	+	0.638285i	$0.779644\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−16.0000	−1.08866
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	37.0000	1.37037
$10$	0	0
$11$	−28.0000	−0.767483	−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−28.0000	−0.767483	−0.383742	+	0.923440i	$0.625365\pi$
$12$	−32.0000	−0.769800
$13$	−18.0000	−0.384023	−0.192012	−	0.981393i	$-0.561501\pi$
$13$	−18.0000	−0.384023	−0.192012	+	0.981393i	$0.561501\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−74.0000	−1.05574	−0.527872	−	0.849324i	$-0.677010\pi$
$17$	−74.0000	−1.05574	−0.527872	+	0.849324i	$0.677010\pi$
$18$	74.0000	0.968998
$19$	80.0000	0.965961	0.482980	−	0.875631i	$-0.339554\pi$
$19$	80.0000	0.965961	0.482980	+	0.875631i	$0.339554\pi$
$20$	0	0
$21$	−56.0000	−0.581914
$22$	−56.0000	−0.542693
$23$	112.000	1.01537	0.507687	−	0.861541i	$-0.330501\pi$
$23$	112.000	1.01537	0.507687	+	0.861541i	$0.330501\pi$
$24$	−64.0000	−0.544331
$25$	0	0
$26$	−36.0000	−0.271545
$27$	−80.0000	−0.570222
$28$	28.0000	0.188982
$29$	190.000	1.21662	0.608312	−	0.793698i	$-0.291847\pi$
$29$	190.000	1.21662	0.608312	+	0.793698i	$0.291847\pi$
$30$	0	0
$31$	72.0000	0.417148	0.208574	−	0.978007i	$-0.433118\pi$
$31$	72.0000	0.417148	0.208574	+	0.978007i	$0.433118\pi$
$32$	32.0000	0.176777
$33$	224.000	1.18162
$34$	−148.000	−0.746523
$35$	0	0
$36$	148.000	0.685185
$37$	346.000	1.53735	0.768676	−	0.639638i	$-0.220916\pi$
$37$	346.000	1.53735	0.768676	+	0.639638i	$0.220916\pi$
$38$	160.000	0.683038
$39$	144.000	0.591242
$40$	0	0
$41$	162.000	0.617077	0.308538	−	0.951212i	$-0.400160\pi$
$41$	162.000	0.617077	0.308538	+	0.951212i	$0.400160\pi$
$42$	−112.000	−0.411476
$43$	412.000	1.46115	0.730575	−	0.682833i	$-0.239252\pi$
$43$	412.000	1.46115	0.730575	+	0.682833i	$0.239252\pi$
$44$	−112.000	−0.383742
$45$	0	0
$46$	224.000	0.717978
$47$	−24.0000	−0.0744843	−0.0372421	−	0.999306i	$-0.511857\pi$
$47$	−24.0000	−0.0744843	−0.0372421	+	0.999306i	$0.511857\pi$
$48$	−128.000	−0.384900
$49$	49.0000	0.142857
$50$	0	0
$51$	592.000	1.62542
$52$	−72.0000	−0.192012
$53$	−318.000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−318.000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−160.000	−0.403208
$55$	0	0
$56$	56.0000	0.133631
$57$	−640.000	−1.48719
$58$	380.000	0.860284
$59$	−200.000	−0.441318	−0.220659	−	0.975351i	$-0.570821\pi$
$59$	−200.000	−0.441318	−0.220659	+	0.975351i	$0.570821\pi$
$60$	0	0
$61$	−198.000	−0.415595	−0.207798	−	0.978172i	$-0.566630\pi$
$61$	−198.000	−0.415595	−0.207798	+	0.978172i	$0.566630\pi$
$62$	144.000	0.294968
$63$	259.000	0.517951
$64$	64.0000	0.125000
$65$	0	0
$66$	448.000	0.835530
$67$	716.000	1.30557	0.652786	−	0.757542i	$-0.273600\pi$
$67$	716.000	1.30557	0.652786	+	0.757542i	$0.273600\pi$
$68$	−296.000	−0.527872
$69$	−896.000	−1.56327
$70$	0	0
$71$	392.000	0.655237	0.327619	−	0.944810i	$-0.393754\pi$
$71$	392.000	0.655237	0.327619	+	0.944810i	$0.393754\pi$
$72$	296.000	0.484499
$73$	−538.000	−0.862577	−0.431289	−	0.902214i	$-0.641941\pi$
$73$	−538.000	−0.862577	−0.431289	+	0.902214i	$0.641941\pi$
$74$	692.000	1.08707
$75$	0	0
$76$	320.000	0.482980
$77$	−196.000	−0.290081
$78$	288.000	0.418072
$79$	240.000	0.341799	0.170899	−	0.985288i	$-0.445333\pi$
$79$	240.000	0.341799	0.170899	+	0.985288i	$0.445333\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	324.000	0.436339
$83$	1072.00	1.41768	0.708839	−	0.705370i	$-0.249219\pi$
$83$	1072.00	1.41768	0.708839	+	0.705370i	$0.249219\pi$
$84$	−224.000	−0.290957
$85$	0	0
$86$	824.000	1.03319
$87$	−1520.00	−1.87312
$88$	−224.000	−0.271346
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	0	0
$91$	−126.000	−0.145147
$92$	448.000	0.507687
$93$	−576.000	−0.642241
$94$	−48.0000	−0.0526683
$95$	0	0
$96$	−256.000	−0.272166
$97$	−1354.00	−1.41730	−0.708649	−	0.705561i	$-0.750695\pi$
$97$	−1354.00	−1.41730	−0.708649	+	0.705561i	$0.750695\pi$
$98$	98.0000	0.101015
$99$	−1036.00	−1.05174
$100$	0	0
$101$	−1358.00	−1.33788	−0.668941	−	0.743316i	$-0.733252\pi$
$101$	−1358.00	−1.33788	−0.668941	+	0.743316i	$0.733252\pi$
$102$	1184.00	1.14935
$103$	832.000	0.795916	0.397958	−	0.917404i	$-0.369719\pi$
$103$	832.000	0.795916	0.397958	+	0.917404i	$0.369719\pi$
$104$	−144.000	−0.135773
$105$	0	0
$106$	−636.000	−0.582772
$107$	−444.000	−0.401150	−0.200575	−	0.979678i	$-0.564281\pi$
$107$	−444.000	−0.401150	−0.200575	+	0.979678i	$0.564281\pi$
$108$	−320.000	−0.285111
$109$	1870.00	1.64324	0.821622	−	0.570033i	$-0.193070\pi$
$109$	1870.00	1.64324	0.821622	+	0.570033i	$0.193070\pi$
$110$	0	0
$111$	−2768.00	−2.36691
$112$	112.000	0.0944911
$113$	−1378.00	−1.14718	−0.573590	−	0.819143i	$-0.694450\pi$
$113$	−1378.00	−1.14718	−0.573590	+	0.819143i	$0.694450\pi$
$114$	−1280.00	−1.05161
$115$	0	0
$116$	760.000	0.608312
$117$	−666.000	−0.526254
$118$	−400.000	−0.312059
$119$	−518.000	−0.399033
$120$	0	0
$121$	−547.000	−0.410969
$122$	−396.000	−0.293870
$123$	−1296.00	−0.950052
$124$	288.000	0.208574
$125$	0	0
$126$	518.000	0.366247
$127$	−1944.00	−1.35828	−0.679142	−	0.734007i	$-0.737648\pi$
$127$	−1944.00	−1.35828	−0.679142	+	0.734007i	$0.737648\pi$
$128$	128.000	0.0883883
$129$	−3296.00	−2.24959
$130$	0	0
$131$	−848.000	−0.565573	−0.282787	−	0.959183i	$-0.591259\pi$
$131$	−848.000	−0.565573	−0.282787	+	0.959183i	$0.591259\pi$
$132$	896.000	0.590809
$133$	560.000	0.365099
$134$	1432.00	0.923179
$135$	0	0
$136$	−592.000	−0.373262
$137$	2966.00	1.84965	0.924827	−	0.380389i	$-0.124210\pi$
$137$	2966.00	1.84965	0.924827	+	0.380389i	$0.124210\pi$
$138$	−1792.00	−1.10540
$139$	2800.00	1.70858	0.854291	−	0.519795i	$-0.173992\pi$
$139$	2800.00	1.70858	0.854291	+	0.519795i	$0.173992\pi$
$140$	0	0
$141$	192.000	0.114676
$142$	784.000	0.463323
$143$	504.000	0.294731
$144$	592.000	0.342593
$145$	0	0
$146$	−1076.00	−0.609934
$147$	−392.000	−0.219943
$148$	1384.00	0.768676
$149$	510.000	0.280408	0.140204	−	0.990123i	$-0.455224\pi$
$149$	510.000	0.280408	0.140204	+	0.990123i	$0.455224\pi$
$150$	0	0
$151$	592.000	0.319048	0.159524	−	0.987194i	$-0.449004\pi$
$151$	592.000	0.319048	0.159524	+	0.987194i	$0.449004\pi$
$152$	640.000	0.341519
$153$	−2738.00	−1.44676
$154$	−392.000	−0.205119
$155$	0	0
$156$	576.000	0.295621
$157$	2686.00	1.36539	0.682695	−	0.730704i	$-0.260808\pi$
$157$	2686.00	1.36539	0.682695	+	0.730704i	$0.260808\pi$
$158$	480.000	0.241688
$159$	2544.00	1.26888
$160$	0	0
$161$	784.000	0.383776
$162$	−718.000	−0.348219
$163$	1012.00	0.486294	0.243147	−	0.969989i	$-0.421820\pi$
$163$	1012.00	0.486294	0.243147	+	0.969989i	$0.421820\pi$
$164$	648.000	0.308538
$165$	0	0
$166$	2144.00	1.00245
$167$	−544.000	−0.252072	−0.126036	−	0.992026i	$-0.540225\pi$
$167$	−544.000	−0.252072	−0.126036	+	0.992026i	$0.540225\pi$
$168$	−448.000	−0.205738
$169$	−1873.00	−0.852526
$170$	0	0
$171$	2960.00	1.32372
$172$	1648.00	0.730575
$173$	−1858.00	−0.816538	−0.408269	−	0.912862i	$-0.633868\pi$
$173$	−1858.00	−0.816538	−0.408269	+	0.912862i	$0.633868\pi$
$174$	−3040.00	−1.32449
$175$	0	0
$176$	−448.000	−0.191871
$177$	1600.00	0.679454
$178$	1620.00	0.682158
$179$	−300.000	−0.125268	−0.0626342	−	0.998037i	$-0.519950\pi$
$179$	−300.000	−0.125268	−0.0626342	+	0.998037i	$0.519950\pi$
$180$	0	0
$181$	−2358.00	−0.968336	−0.484168	−	0.874975i	$-0.660878\pi$
$181$	−2358.00	−0.968336	−0.484168	+	0.874975i	$0.660878\pi$
$182$	−252.000	−0.102635
$183$	1584.00	0.639851
$184$	896.000	0.358989
$185$	0	0
$186$	−1152.00	−0.454133
$187$	2072.00	0.810265
$188$	−96.0000	−0.0372421
$189$	−560.000	−0.215524
$190$	0	0
$191$	1392.00	0.527338	0.263669	−	0.964613i	$-0.415067\pi$
$191$	1392.00	0.527338	0.263669	+	0.964613i	$0.415067\pi$
$192$	−512.000	−0.192450
$193$	−1778.00	−0.663126	−0.331563	−	0.943433i	$-0.607576\pi$
$193$	−1778.00	−0.663126	−0.331563	+	0.943433i	$0.607576\pi$
$194$	−2708.00	−1.00218
$195$	0	0
$196$	196.000	0.0714286
$197$	−1214.00	−0.439055	−0.219528	−	0.975606i	$-0.570452\pi$
$197$	−1214.00	−0.439055	−0.219528	+	0.975606i	$0.570452\pi$
$198$	−2072.00	−0.743690
$199$	1040.00	0.370471	0.185235	−	0.982694i	$-0.440695\pi$
$199$	1040.00	0.370471	0.185235	+	0.982694i	$0.440695\pi$
$200$	0	0
$201$	−5728.00	−2.01006
$202$	−2716.00	−0.946025
$203$	1330.00	0.459841
$204$	2368.00	0.812712
$205$	0	0
$206$	1664.00	0.562798
$207$	4144.00	1.39144
$208$	−288.000	−0.0960058
$209$	−2240.00	−0.741359
$210$	0	0
$211$	−3868.00	−1.26201	−0.631005	−	0.775779i	$-0.717357\pi$
$211$	−3868.00	−1.26201	−0.631005	+	0.775779i	$0.717357\pi$
$212$	−1272.00	−0.412082
$213$	−3136.00	−1.00880
$214$	−888.000	−0.283656
$215$	0	0
$216$	−640.000	−0.201604
$217$	504.000	0.157667
$218$	3740.00	1.16195
$219$	4304.00	1.32802
$220$	0	0
$221$	1332.00	0.405430
$222$	−5536.00	−1.67366
$223$	−3968.00	−1.19156	−0.595778	−	0.803149i	$-0.703156\pi$
$223$	−3968.00	−1.19156	−0.595778	+	0.803149i	$0.703156\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−2756.00	−0.811179
$227$	3936.00	1.15084	0.575422	−	0.817857i	$-0.304838\pi$
$227$	3936.00	1.15084	0.575422	+	0.817857i	$0.304838\pi$
$228$	−2560.00	−0.743597
$229$	4810.00	1.38801	0.694004	−	0.719971i	$-0.255845\pi$
$229$	4810.00	1.38801	0.694004	+	0.719971i	$0.255845\pi$
$230$	0	0
$231$	1568.00	0.446610
$232$	1520.00	0.430142
$233$	2182.00	0.613509	0.306754	−	0.951789i	$-0.400757\pi$
$233$	2182.00	0.613509	0.306754	+	0.951789i	$0.400757\pi$
$234$	−1332.00	−0.372118
$235$	0	0
$236$	−800.000	−0.220659
$237$	−1920.00	−0.526234
$238$	−1036.00	−0.282159
$239$	−3000.00	−0.811941	−0.405970	−	0.913886i	$-0.633066\pi$
$239$	−3000.00	−0.811941	−0.405970	+	0.913886i	$0.633066\pi$
$240$	0	0
$241$	2042.00	0.545796	0.272898	−	0.962043i	$-0.412018\pi$
$241$	2042.00	0.545796	0.272898	+	0.962043i	$0.412018\pi$
$242$	−1094.00	−0.290599
$243$	5032.00	1.32841
$244$	−792.000	−0.207798
$245$	0	0
$246$	−2592.00	−0.671788
$247$	−1440.00	−0.370951
$248$	576.000	0.147484
$249$	−8576.00	−2.18266
$250$	0	0
$251$	−528.000	−0.132777	−0.0663886	−	0.997794i	$-0.521148\pi$
$251$	−528.000	−0.132777	−0.0663886	+	0.997794i	$0.521148\pi$
$252$	1036.00	0.258976
$253$	−3136.00	−0.779283
$254$	−3888.00	−0.960452
$255$	0	0
$256$	256.000	0.0625000
$257$	−5634.00	−1.36747	−0.683734	−	0.729731i	$-0.739645\pi$
$257$	−5634.00	−1.36747	−0.683734	+	0.729731i	$0.739645\pi$
$258$	−6592.00	−1.59070
$259$	2422.00	0.581065
$260$	0	0
$261$	7030.00	1.66723
$262$	−1696.00	−0.399921
$263$	−168.000	−0.0393891	−0.0196945	−	0.999806i	$-0.506269\pi$
$263$	−168.000	−0.0393891	−0.0196945	+	0.999806i	$0.506269\pi$
$264$	1792.00	0.417765
$265$	0	0
$266$	1120.00	0.258164
$267$	−6480.00	−1.48528
$268$	2864.00	0.652786
$269$	−1310.00	−0.296922	−0.148461	−	0.988918i	$-0.547432\pi$
$269$	−1310.00	−0.296922	−0.148461	+	0.988918i	$0.547432\pi$
$270$	0	0
$271$	−2208.00	−0.494932	−0.247466	−	0.968897i	$-0.579598\pi$
$271$	−2208.00	−0.494932	−0.247466	+	0.968897i	$0.579598\pi$
$272$	−1184.00	−0.263936
$273$	1008.00	0.223469
$274$	5932.00	1.30790
$275$	0	0
$276$	−3584.00	−0.781636
$277$	−5294.00	−1.14832	−0.574162	−	0.818742i	$-0.694672\pi$
$277$	−5294.00	−1.14832	−0.574162	+	0.818742i	$0.694672\pi$
$278$	5600.00	1.20815
$279$	2664.00	0.571647
$280$	0	0
$281$	3242.00	0.688262	0.344131	−	0.938922i	$-0.388174\pi$
$281$	3242.00	0.688262	0.344131	+	0.938922i	$0.388174\pi$
$282$	384.000	0.0810882
$283$	1592.00	0.334398	0.167199	−	0.985923i	$-0.446528\pi$
$283$	1592.00	0.334398	0.167199	+	0.985923i	$0.446528\pi$
$284$	1568.00	0.327619
$285$	0	0
$286$	1008.00	0.208407
$287$	1134.00	0.233233
$288$	1184.00	0.242250
$289$	563.000	0.114594
$290$	0	0
$291$	10832.0	2.18207
$292$	−2152.00	−0.431289
$293$	5022.00	1.00133	0.500663	−	0.865642i	$-0.333090\pi$
$293$	5022.00	1.00133	0.500663	+	0.865642i	$0.333090\pi$
$294$	−784.000	−0.155523
$295$	0	0
$296$	2768.00	0.543536
$297$	2240.00	0.437636
$298$	1020.00	0.198279
$299$	−2016.00	−0.389927
$300$	0	0
$301$	2884.00	0.552262
$302$	1184.00	0.225601
$303$	10864.0	2.05980
$304$	1280.00	0.241490
$305$	0	0
$306$	−5476.00	−1.02301
$307$	9536.00	1.77280	0.886398	−	0.462924i	$-0.153200\pi$
$307$	9536.00	1.77280	0.886398	+	0.462924i	$0.153200\pi$
$308$	−784.000	−0.145041
$309$	−6656.00	−1.22539
$310$	0	0
$311$	−968.000	−0.176496	−0.0882480	−	0.996099i	$-0.528127\pi$
$311$	−968.000	−0.176496	−0.0882480	+	0.996099i	$0.528127\pi$
$312$	1152.00	0.209036
$313$	−3058.00	−0.552231	−0.276116	−	0.961124i	$-0.589047\pi$
$313$	−3058.00	−0.552231	−0.276116	+	0.961124i	$0.589047\pi$
$314$	5372.00	0.965476
$315$	0	0
$316$	960.000	0.170899
$317$	4986.00	0.883412	0.441706	−	0.897160i	$-0.354373\pi$
$317$	4986.00	0.883412	0.441706	+	0.897160i	$0.354373\pi$
$318$	5088.00	0.897235
$319$	−5320.00	−0.933739
$320$	0	0
$321$	3552.00	0.617612
$322$	1568.00	0.271370
$323$	−5920.00	−1.01981
$324$	−1436.00	−0.246228
$325$	0	0
$326$	2024.00	0.343862
$327$	−14960.0	−2.52994
$328$	1296.00	0.218170
$329$	−168.000	−0.0281524
$330$	0	0
$331$	8612.00	1.43009	0.715043	−	0.699081i	$-0.246407\pi$
$331$	8612.00	1.43009	0.715043	+	0.699081i	$0.246407\pi$
$332$	4288.00	0.708839
$333$	12802.0	2.10674
$334$	−1088.00	−0.178242
$335$	0	0
$336$	−896.000	−0.145479
$337$	10206.0	1.64972	0.824861	−	0.565336i	$-0.191253\pi$
$337$	10206.0	1.64972	0.824861	+	0.565336i	$0.191253\pi$
$338$	−3746.00	−0.602827
$339$	11024.0	1.76620
$340$	0	0
$341$	−2016.00	−0.320154
$342$	5920.00	0.936014
$343$	343.000	0.0539949
$344$	3296.00	0.516594
$345$	0	0
$346$	−3716.00	−0.577380
$347$	−2004.00	−0.310030	−0.155015	−	0.987912i	$-0.549543\pi$
$347$	−2004.00	−0.310030	−0.155015	+	0.987912i	$0.549543\pi$
$348$	−6080.00	−0.936558
$349$	1330.00	0.203992	0.101996	−	0.994785i	$-0.467477\pi$
$349$	1330.00	0.203992	0.101996	+	0.994785i	$0.467477\pi$
$350$	0	0
$351$	1440.00	0.218979
$352$	−896.000	−0.135673
$353$	−978.000	−0.147461	−0.0737304	−	0.997278i	$-0.523490\pi$
$353$	−978.000	−0.147461	−0.0737304	+	0.997278i	$0.523490\pi$
$354$	3200.00	0.480447
$355$	0	0
$356$	3240.00	0.482359
$357$	4144.00	0.614352
$358$	−600.000	−0.0885782
$359$	−9680.00	−1.42309	−0.711547	−	0.702638i	$-0.752005\pi$
$359$	−9680.00	−1.42309	−0.711547	+	0.702638i	$0.752005\pi$
$360$	0	0
$361$	−459.000	−0.0669194
$362$	−4716.00	−0.684717
$363$	4376.00	0.632728
$364$	−504.000	−0.0725736
$365$	0	0
$366$	3168.00	0.452443
$367$	8656.00	1.23117	0.615585	−	0.788070i	$-0.288920\pi$
$367$	8656.00	1.23117	0.615585	+	0.788070i	$0.288920\pi$
$368$	1792.00	0.253844
$369$	5994.00	0.845624
$370$	0	0
$371$	−2226.00	−0.311504
$372$	−2304.00	−0.321121
$373$	−5278.00	−0.732666	−0.366333	−	0.930484i	$-0.619387\pi$
$373$	−5278.00	−0.732666	−0.366333	+	0.930484i	$0.619387\pi$
$374$	4144.00	0.572944
$375$	0	0
$376$	−192.000	−0.0263342
$377$	−3420.00	−0.467212
$378$	−1120.00	−0.152398
$379$	6340.00	0.859272	0.429636	−	0.903002i	$-0.358642\pi$
$379$	6340.00	0.859272	0.429636	+	0.903002i	$0.358642\pi$
$380$	0	0
$381$	15552.0	2.09122
$382$	2784.00	0.372884
$383$	6232.00	0.831437	0.415718	−	0.909493i	$-0.363530\pi$
$383$	6232.00	0.831437	0.415718	+	0.909493i	$0.363530\pi$
$384$	−1024.00	−0.136083
$385$	0	0
$386$	−3556.00	−0.468901
$387$	15244.0	2.00232
$388$	−5416.00	−0.708649
$389$	−14810.0	−1.93033	−0.965163	−	0.261649i	$-0.915734\pi$
$389$	−14810.0	−1.93033	−0.965163	+	0.261649i	$0.915734\pi$
$390$	0	0
$391$	−8288.00	−1.07197
$392$	392.000	0.0505076
$393$	6784.00	0.870757
$394$	−2428.00	−0.310459
$395$	0	0
$396$	−4144.00	−0.525868
$397$	−5154.00	−0.651566	−0.325783	−	0.945445i	$-0.605628\pi$
$397$	−5154.00	−0.651566	−0.325783	+	0.945445i	$0.605628\pi$
$398$	2080.00	0.261962
$399$	−4480.00	−0.562107
$400$	0	0
$401$	3282.00	0.408716	0.204358	−	0.978896i	$-0.434489\pi$
$401$	3282.00	0.408716	0.204358	+	0.978896i	$0.434489\pi$
$402$	−11456.0	−1.42133
$403$	−1296.00	−0.160194
$404$	−5432.00	−0.668941
$405$	0	0
$406$	2660.00	0.325157
$407$	−9688.00	−1.17989
$408$	4736.00	0.574674
$409$	5810.00	0.702411	0.351205	−	0.936298i	$-0.385772\pi$
$409$	5810.00	0.702411	0.351205	+	0.936298i	$0.385772\pi$
$410$	0	0
$411$	−23728.0	−2.84773
$412$	3328.00	0.397958
$413$	−1400.00	−0.166803
$414$	8288.00	0.983896
$415$	0	0
$416$	−576.000	−0.0678864
$417$	−22400.0	−2.63053
$418$	−4480.00	−0.524220
$419$	13560.0	1.58102	0.790512	−	0.612446i	$-0.209814\pi$
$419$	13560.0	1.58102	0.790512	+	0.612446i	$0.209814\pi$
$420$	0	0
$421$	−738.000	−0.0854345	−0.0427172	−	0.999087i	$-0.513601\pi$
$421$	−738.000	−0.0854345	−0.0427172	+	0.999087i	$0.513601\pi$
$422$	−7736.00	−0.892376
$423$	−888.000	−0.102071
$424$	−2544.00	−0.291386
$425$	0	0
$426$	−6272.00	−0.713332
$427$	−1386.00	−0.157080
$428$	−1776.00	−0.200575
$429$	−4032.00	−0.453769
$430$	0	0
$431$	1272.00	0.142158	0.0710790	−	0.997471i	$-0.477356\pi$
$431$	1272.00	0.142158	0.0710790	+	0.997471i	$0.477356\pi$
$432$	−1280.00	−0.142556
$433$	5062.00	0.561811	0.280906	−	0.959735i	$-0.409365\pi$
$433$	5062.00	0.561811	0.280906	+	0.959735i	$0.409365\pi$
$434$	1008.00	0.111487
$435$	0	0
$436$	7480.00	0.821622
$437$	8960.00	0.980812
$438$	8608.00	0.939055
$439$	5640.00	0.613172	0.306586	−	0.951843i	$-0.400813\pi$
$439$	5640.00	0.613172	0.306586	+	0.951843i	$0.400813\pi$
$440$	0	0
$441$	1813.00	0.195767
$442$	2664.00	0.286682
$443$	−13388.0	−1.43585	−0.717927	−	0.696119i	$-0.754909\pi$
$443$	−13388.0	−1.43585	−0.717927	+	0.696119i	$0.754909\pi$
$444$	−11072.0	−1.18345
$445$	0	0
$446$	−7936.00	−0.842557
$447$	−4080.00	−0.431717
$448$	448.000	0.0472456
$449$	−3230.00	−0.339495	−0.169747	−	0.985488i	$-0.554295\pi$
$449$	−3230.00	−0.339495	−0.169747	+	0.985488i	$0.554295\pi$
$450$	0	0
$451$	−4536.00	−0.473596
$452$	−5512.00	−0.573590
$453$	−4736.00	−0.491207
$454$	7872.00	0.813769
$455$	0	0
$456$	−5120.00	−0.525803
$457$	10646.0	1.08971	0.544857	−	0.838529i	$-0.316584\pi$
$457$	10646.0	1.08971	0.544857	+	0.838529i	$0.316584\pi$
$458$	9620.00	0.981470
$459$	5920.00	0.602009
$460$	0	0
$461$	7282.00	0.735698	0.367849	−	0.929886i	$-0.380094\pi$
$461$	7282.00	0.735698	0.367849	+	0.929886i	$0.380094\pi$
$462$	3136.00	0.315801
$463$	−12688.0	−1.27357	−0.636783	−	0.771043i	$-0.719735\pi$
$463$	−12688.0	−1.27357	−0.636783	+	0.771043i	$0.719735\pi$
$464$	3040.00	0.304156
$465$	0	0
$466$	4364.00	0.433816
$467$	2816.00	0.279034	0.139517	−	0.990220i	$-0.455445\pi$
$467$	2816.00	0.279034	0.139517	+	0.990220i	$0.455445\pi$
$468$	−2664.00	−0.263127
$469$	5012.00	0.493460
$470$	0	0
$471$	−21488.0	−2.10215
$472$	−1600.00	−0.156030
$473$	−11536.0	−1.12141
$474$	−3840.00	−0.372103
$475$	0	0
$476$	−2072.00	−0.199517
$477$	−11766.0	−1.12941
$478$	−6000.00	−0.574129
$479$	−3160.00	−0.301428	−0.150714	−	0.988577i	$-0.548157\pi$
$479$	−3160.00	−0.301428	−0.150714	+	0.988577i	$0.548157\pi$
$480$	0	0
$481$	−6228.00	−0.590379
$482$	4084.00	0.385936
$483$	−6272.00	−0.590861
$484$	−2188.00	−0.205485
$485$	0	0
$486$	10064.0	0.939326
$487$	14176.0	1.31905	0.659523	−	0.751684i	$-0.270758\pi$
$487$	14176.0	1.31905	0.659523	+	0.751684i	$0.270758\pi$
$488$	−1584.00	−0.146935
$489$	−8096.00	−0.748699
$490$	0	0
$491$	−11268.0	−1.03568	−0.517839	−	0.855478i	$-0.673263\pi$
$491$	−11268.0	−1.03568	−0.517839	+	0.855478i	$0.673263\pi$
$492$	−5184.00	−0.475026
$493$	−14060.0	−1.28444
$494$	−2880.00	−0.262302
$495$	0	0
$496$	1152.00	0.104287
$497$	2744.00	0.247656
$498$	−17152.0	−1.54337
$499$	−4460.00	−0.400114	−0.200057	−	0.979784i	$-0.564113\pi$
$499$	−4460.00	−0.400114	−0.200057	+	0.979784i	$0.564113\pi$
$500$	0	0
$501$	4352.00	0.388090
$502$	−1056.00	−0.0938876
$503$	1512.00	0.134029	0.0670147	−	0.997752i	$-0.478653\pi$
$503$	1512.00	0.134029	0.0670147	+	0.997752i	$0.478653\pi$
$504$	2072.00	0.183123
$505$	0	0
$506$	−6272.00	−0.551036
$507$	14984.0	1.31255
$508$	−7776.00	−0.679142
$509$	−11790.0	−1.02668	−0.513342	−	0.858184i	$-0.671593\pi$
$509$	−11790.0	−1.02668	−0.513342	+	0.858184i	$0.671593\pi$
$510$	0	0
$511$	−3766.00	−0.326024
$512$	512.000	0.0441942
$513$	−6400.00	−0.550813
$514$	−11268.0	−0.966946
$515$	0	0
$516$	−13184.0	−1.12479
$517$	672.000	0.0571654
$518$	4844.00	0.410875
$519$	14864.0	1.25714
$520$	0	0
$521$	1362.00	0.114530	0.0572652	−	0.998359i	$-0.481762\pi$
$521$	1362.00	0.114530	0.0572652	+	0.998359i	$0.481762\pi$
$522$	14060.0	1.17891
$523$	−6968.00	−0.582580	−0.291290	−	0.956635i	$-0.594084\pi$
$523$	−6968.00	−0.582580	−0.291290	+	0.956635i	$0.594084\pi$
$524$	−3392.00	−0.282787
$525$	0	0
$526$	−336.000	−0.0278523
$527$	−5328.00	−0.440401
$528$	3584.00	0.295405
$529$	377.000	0.0309855
$530$	0	0
$531$	−7400.00	−0.604770
$532$	2240.00	0.182549
$533$	−2916.00	−0.236972
$534$	−12960.0	−1.05025
$535$	0	0
$536$	5728.00	0.461589
$537$	2400.00	0.192863
$538$	−2620.00	−0.209956
$539$	−1372.00	−0.109640
$540$	0	0
$541$	7062.00	0.561218	0.280609	−	0.959822i	$-0.409464\pi$
$541$	7062.00	0.561218	0.280609	+	0.959822i	$0.409464\pi$
$542$	−4416.00	−0.349969
$543$	18864.0	1.49085
$544$	−2368.00	−0.186631
$545$	0	0
$546$	2016.00	0.158016
$547$	8196.00	0.640650	0.320325	−	0.947308i	$-0.396208\pi$
$547$	8196.00	0.640650	0.320325	+	0.947308i	$0.396208\pi$
$548$	11864.0	0.924827
$549$	−7326.00	−0.569519
$550$	0	0
$551$	15200.0	1.17521
$552$	−7168.00	−0.552700
$553$	1680.00	0.129188
$554$	−10588.0	−0.811987
$555$	0	0
$556$	11200.0	0.854291
$557$	7466.00	0.567944	0.283972	−	0.958833i	$-0.408348\pi$
$557$	7466.00	0.567944	0.283972	+	0.958833i	$0.408348\pi$
$558$	5328.00	0.404215
$559$	−7416.00	−0.561115
$560$	0	0
$561$	−16576.0	−1.24749
$562$	6484.00	0.486674
$563$	−24968.0	−1.86905	−0.934526	−	0.355896i	$-0.884176\pi$
$563$	−24968.0	−1.86905	−0.934526	+	0.355896i	$0.884176\pi$
$564$	768.000	0.0573380
$565$	0	0
$566$	3184.00	0.236455
$567$	−2513.00	−0.186131
$568$	3136.00	0.231661
$569$	14250.0	1.04990	0.524948	−	0.851134i	$-0.324085\pi$
$569$	14250.0	1.04990	0.524948	+	0.851134i	$0.324085\pi$
$570$	0	0
$571$	6372.00	0.467005	0.233503	−	0.972356i	$-0.424981\pi$
$571$	6372.00	0.467005	0.233503	+	0.972356i	$0.424981\pi$
$572$	2016.00	0.147366
$573$	−11136.0	−0.811890
$574$	2268.00	0.164921
$575$	0	0
$576$	2368.00	0.171296
$577$	8366.00	0.603607	0.301803	−	0.953370i	$-0.402411\pi$
$577$	8366.00	0.603607	0.301803	+	0.953370i	$0.402411\pi$
$578$	1126.00	0.0810301
$579$	14224.0	1.02095
$580$	0	0
$581$	7504.00	0.535832
$582$	21664.0	1.54296
$583$	8904.00	0.632532
$584$	−4304.00	−0.304967
$585$	0	0
$586$	10044.0	0.708044
$587$	−20384.0	−1.43328	−0.716642	−	0.697441i	$-0.754322\pi$
$587$	−20384.0	−1.43328	−0.716642	+	0.697441i	$0.754322\pi$
$588$	−1568.00	−0.109971
$589$	5760.00	0.402948
$590$	0	0
$591$	9712.00	0.675970
$592$	5536.00	0.384338
$593$	−9378.00	−0.649424	−0.324712	−	0.945813i	$-0.605267\pi$
$593$	−9378.00	−0.649424	−0.324712	+	0.945813i	$0.605267\pi$
$594$	4480.00	0.309456
$595$	0	0
$596$	2040.00	0.140204
$597$	−8320.00	−0.570377
$598$	−4032.00	−0.275720
$599$	−9000.00	−0.613907	−0.306953	−	0.951725i	$-0.599310\pi$
$599$	−9000.00	−0.613907	−0.306953	+	0.951725i	$0.599310\pi$
$600$	0	0
$601$	7562.00	0.513245	0.256623	−	0.966512i	$-0.417390\pi$
$601$	7562.00	0.513245	0.256623	+	0.966512i	$0.417390\pi$
$602$	5768.00	0.390509
$603$	26492.0	1.78912
$604$	2368.00	0.159524
$605$	0	0
$606$	21728.0	1.45650
$607$	2976.00	0.198999	0.0994993	−	0.995038i	$-0.468276\pi$
$607$	2976.00	0.198999	0.0994993	+	0.995038i	$0.468276\pi$
$608$	2560.00	0.170759
$609$	−10640.0	−0.707971
$610$	0	0
$611$	432.000	0.0286037
$612$	−10952.0	−0.723380
$613$	−4278.00	−0.281871	−0.140935	−	0.990019i	$-0.545011\pi$
$613$	−4278.00	−0.281871	−0.140935	+	0.990019i	$0.545011\pi$
$614$	19072.0	1.25356
$615$	0	0
$616$	−1568.00	−0.102559
$617$	−18794.0	−1.22629	−0.613143	−	0.789972i	$-0.710095\pi$
$617$	−18794.0	−1.22629	−0.613143	+	0.789972i	$0.710095\pi$
$618$	−13312.0	−0.866484
$619$	18040.0	1.17139	0.585694	−	0.810532i	$-0.300822\pi$
$619$	18040.0	1.17139	0.585694	+	0.810532i	$0.300822\pi$
$620$	0	0
$621$	−8960.00	−0.578989
$622$	−1936.00	−0.124801
$623$	5670.00	0.364629
$624$	2304.00	0.147811
$625$	0	0
$626$	−6116.00	−0.390486
$627$	17920.0	1.14140
$628$	10744.0	0.682695
$629$	−25604.0	−1.62305
$630$	0	0
$631$	−21688.0	−1.36828	−0.684141	−	0.729350i	$-0.739823\pi$
$631$	−21688.0	−1.36828	−0.684141	+	0.729350i	$0.739823\pi$
$632$	1920.00	0.120844
$633$	30944.0	1.94299
$634$	9972.00	0.624667
$635$	0	0
$636$	10176.0	0.634441
$637$	−882.000	−0.0548605
$638$	−10640.0	−0.660253
$639$	14504.0	0.897918
$640$	0	0
$641$	−10558.0	−0.650571	−0.325285	−	0.945616i	$-0.605460\pi$
$641$	−10558.0	−0.650571	−0.325285	+	0.945616i	$0.605460\pi$
$642$	7104.00	0.436717
$643$	26152.0	1.60394	0.801971	−	0.597363i	$-0.203785\pi$
$643$	26152.0	1.60394	0.801971	+	0.597363i	$0.203785\pi$
$644$	3136.00	0.191888
$645$	0	0
$646$	−11840.0	−0.721112
$647$	−25584.0	−1.55458	−0.777288	−	0.629145i	$-0.783405\pi$
$647$	−25584.0	−1.55458	−0.777288	+	0.629145i	$0.783405\pi$
$648$	−2872.00	−0.174109
$649$	5600.00	0.338705
$650$	0	0
$651$	−4032.00	−0.242744
$652$	4048.00	0.243147
$653$	−15198.0	−0.910787	−0.455393	−	0.890290i	$-0.650501\pi$
$653$	−15198.0	−0.910787	−0.455393	+	0.890290i	$0.650501\pi$
$654$	−29920.0	−1.78894
$655$	0	0
$656$	2592.00	0.154269
$657$	−19906.0	−1.18205
$658$	−336.000	−0.0199068
$659$	−6100.00	−0.360580	−0.180290	−	0.983613i	$-0.557704\pi$
$659$	−6100.00	−0.360580	−0.180290	+	0.983613i	$0.557704\pi$
$660$	0	0
$661$	−2318.00	−0.136399	−0.0681995	−	0.997672i	$-0.521725\pi$
$661$	−2318.00	−0.136399	−0.0681995	+	0.997672i	$0.521725\pi$
$662$	17224.0	1.01122
$663$	−10656.0	−0.624200
$664$	8576.00	0.501225
$665$	0	0
$666$	25604.0	1.48969
$667$	21280.0	1.23533
$668$	−2176.00	−0.126036
$669$	31744.0	1.83452
$670$	0	0
$671$	5544.00	0.318962
$672$	−1792.00	−0.102869
$673$	10222.0	0.585482	0.292741	−	0.956192i	$-0.405433\pi$
$673$	10222.0	0.585482	0.292741	+	0.956192i	$0.405433\pi$
$674$	20412.0	1.16653
$675$	0	0
$676$	−7492.00	−0.426263
$677$	−25434.0	−1.44388	−0.721941	−	0.691955i	$-0.756750\pi$
$677$	−25434.0	−1.44388	−0.721941	+	0.691955i	$0.756750\pi$
$678$	22048.0	1.24889
$679$	−9478.00	−0.535688
$680$	0	0
$681$	−31488.0	−1.77184
$682$	−4032.00	−0.226383
$683$	8532.00	0.477991	0.238996	−	0.971021i	$-0.423182\pi$
$683$	8532.00	0.477991	0.238996	+	0.971021i	$0.423182\pi$
$684$	11840.0	0.661862
$685$	0	0
$686$	686.000	0.0381802
$687$	−38480.0	−2.13698
$688$	6592.00	0.365287
$689$	5724.00	0.316498
$690$	0	0
$691$	20672.0	1.13806	0.569030	−	0.822317i	$-0.307319\pi$
$691$	20672.0	1.13806	0.569030	+	0.822317i	$0.307319\pi$
$692$	−7432.00	−0.408269
$693$	−7252.00	−0.397519
$694$	−4008.00	−0.219224
$695$	0	0
$696$	−12160.0	−0.662247
$697$	−11988.0	−0.651475
$698$	2660.00	0.144244
$699$	−17456.0	−0.944559
$700$	0	0
$701$	−21458.0	−1.15614	−0.578072	−	0.815985i	$-0.696195\pi$
$701$	−21458.0	−1.15614	−0.578072	+	0.815985i	$0.696195\pi$
$702$	2880.00	0.154841
$703$	27680.0	1.48502
$704$	−1792.00	−0.0959354
$705$	0	0
$706$	−1956.00	−0.104271
$707$	−9506.00	−0.505672
$708$	6400.00	0.339727
$709$	−9850.00	−0.521755	−0.260878	−	0.965372i	$-0.584012\pi$
$709$	−9850.00	−0.521755	−0.260878	+	0.965372i	$0.584012\pi$
$710$	0	0
$711$	8880.00	0.468391
$712$	6480.00	0.341079
$713$	8064.00	0.423561
$714$	8288.00	0.434413
$715$	0	0
$716$	−1200.00	−0.0626342
$717$	24000.0	1.25006
$718$	−19360.0	−1.00628
$719$	−18840.0	−0.977209	−0.488605	−	0.872505i	$-0.662494\pi$
$719$	−18840.0	−0.977209	−0.488605	+	0.872505i	$0.662494\pi$
$720$	0	0
$721$	5824.00	0.300828
$722$	−918.000	−0.0473191
$723$	−16336.0	−0.840308
$724$	−9432.00	−0.484168
$725$	0	0
$726$	8752.00	0.447407
$727$	−37504.0	−1.91327	−0.956634	−	0.291291i	$-0.905915\pi$
$727$	−37504.0	−1.91327	−0.956634	+	0.291291i	$0.905915\pi$
$728$	−1008.00	−0.0513173
$729$	−30563.0	−1.55276
$730$	0	0
$731$	−30488.0	−1.54260
$732$	6336.00	0.319925
$733$	−13338.0	−0.672101	−0.336051	−	0.941844i	$-0.609091\pi$
$733$	−13338.0	−0.672101	−0.336051	+	0.941844i	$0.609091\pi$
$734$	17312.0	0.870569
$735$	0	0
$736$	3584.00	0.179495
$737$	−20048.0	−1.00200
$738$	11988.0	0.597946
$739$	17100.0	0.851196	0.425598	−	0.904912i	$-0.360064\pi$
$739$	17100.0	0.851196	0.425598	+	0.904912i	$0.360064\pi$
$740$	0	0
$741$	11520.0	0.571117
$742$	−4452.00	−0.220267
$743$	19632.0	0.969352	0.484676	−	0.874694i	$-0.338938\pi$
$743$	19632.0	0.969352	0.484676	+	0.874694i	$0.338938\pi$
$744$	−4608.00	−0.227067
$745$	0	0
$746$	−10556.0	−0.518073
$747$	39664.0	1.94274
$748$	8288.00	0.405133
$749$	−3108.00	−0.151621
$750$	0	0
$751$	33912.0	1.64776	0.823879	−	0.566766i	$-0.191805\pi$
$751$	33912.0	1.64776	0.823879	+	0.566766i	$0.191805\pi$
$752$	−384.000	−0.0186211
$753$	4224.00	0.204424
$754$	−6840.00	−0.330369
$755$	0	0
$756$	−2240.00	−0.107762
$757$	31386.0	1.50693	0.753463	−	0.657490i	$-0.228382\pi$
$757$	31386.0	1.50693	0.753463	+	0.657490i	$0.228382\pi$
$758$	12680.0	0.607597
$759$	25088.0	1.19978
$760$	0	0
$761$	−34558.0	−1.64616	−0.823079	−	0.567927i	$-0.807746\pi$
$761$	−34558.0	−1.64616	−0.823079	+	0.567927i	$0.807746\pi$
$762$	31104.0	1.47871
$763$	13090.0	0.621088
$764$	5568.00	0.263669
$765$	0	0
$766$	12464.0	0.587915
$767$	3600.00	0.169476
$768$	−2048.00	−0.0962250
$769$	39130.0	1.83493	0.917467	−	0.397812i	$-0.130231\pi$
$769$	39130.0	1.83493	0.917467	+	0.397812i	$0.130231\pi$
$770$	0	0
$771$	45072.0	2.10535
$772$	−7112.00	−0.331563
$773$	25982.0	1.20894	0.604468	−	0.796629i	$-0.293386\pi$
$773$	25982.0	1.20894	0.604468	+	0.796629i	$0.293386\pi$
$774$	30488.0	1.41585
$775$	0	0
$776$	−10832.0	−0.501090
$777$	−19376.0	−0.894608
$778$	−29620.0	−1.36495
$779$	12960.0	0.596072
$780$	0	0
$781$	−10976.0	−0.502884
$782$	−16576.0	−0.758001
$783$	−15200.0	−0.693747
$784$	784.000	0.0357143
$785$	0	0
$786$	13568.0	0.615718
$787$	−35424.0	−1.60448	−0.802242	−	0.596999i	$-0.796360\pi$
$787$	−35424.0	−1.60448	−0.802242	+	0.596999i	$0.796360\pi$
$788$	−4856.00	−0.219528
$789$	1344.00	0.0606434
$790$	0	0
$791$	−9646.00	−0.433593
$792$	−8288.00	−0.371845
$793$	3564.00	0.159598
$794$	−10308.0	−0.460727
$795$	0	0
$796$	4160.00	0.185235
$797$	30606.0	1.36025	0.680126	−	0.733096i	$-0.261925\pi$
$797$	30606.0	1.36025	0.680126	+	0.733096i	$0.261925\pi$
$798$	−8960.00	−0.397469
$799$	1776.00	0.0786362
$800$	0	0
$801$	29970.0	1.32202
$802$	6564.00	0.289006
$803$	15064.0	0.662014
$804$	−22912.0	−1.00503
$805$	0	0
$806$	−2592.00	−0.113275
$807$	10480.0	0.457142
$808$	−10864.0	−0.473013
$809$	16810.0	0.730542	0.365271	−	0.930901i	$-0.380976\pi$
$809$	16810.0	0.730542	0.365271	+	0.930901i	$0.380976\pi$
$810$	0	0
$811$	−9368.00	−0.405616	−0.202808	−	0.979218i	$-0.565007\pi$
$811$	−9368.00	−0.405616	−0.202808	+	0.979218i	$0.565007\pi$
$812$	5320.00	0.229920
$813$	17664.0	0.761997
$814$	−19376.0	−0.834310
$815$	0	0
$816$	9472.00	0.406356
$817$	32960.0	1.41141
$818$	11620.0	0.496679
$819$	−4662.00	−0.198905
$820$	0	0
$821$	34382.0	1.46156	0.730780	−	0.682614i	$-0.239157\pi$
$821$	34382.0	1.46156	0.730780	+	0.682614i	$0.239157\pi$
$822$	−47456.0	−2.01365
$823$	4472.00	0.189410	0.0947048	−	0.995505i	$-0.469809\pi$
$823$	4472.00	0.189410	0.0947048	+	0.995505i	$0.469809\pi$
$824$	6656.00	0.281399
$825$	0	0
$826$	−2800.00	−0.117947
$827$	1716.00	0.0721538	0.0360769	−	0.999349i	$-0.488514\pi$
$827$	1716.00	0.0721538	0.0360769	+	0.999349i	$0.488514\pi$
$828$	16576.0	0.695720
$829$	−7910.00	−0.331394	−0.165697	−	0.986177i	$-0.552987\pi$
$829$	−7910.00	−0.331394	−0.165697	+	0.986177i	$0.552987\pi$
$830$	0	0
$831$	42352.0	1.76796
$832$	−1152.00	−0.0480029
$833$	−3626.00	−0.150820
$834$	−44800.0	−1.86007
$835$	0	0
$836$	−8960.00	−0.370680
$837$	−5760.00	−0.237867
$838$	27120.0	1.11795
$839$	−19360.0	−0.796641	−0.398320	−	0.917246i	$-0.630407\pi$
$839$	−19360.0	−0.796641	−0.398320	+	0.917246i	$0.630407\pi$
$840$	0	0
$841$	11711.0	0.480175
$842$	−1476.00	−0.0604113
$843$	−25936.0	−1.05965
$844$	−15472.0	−0.631005
$845$	0	0
$846$	−1776.00	−0.0721751
$847$	−3829.00	−0.155332
$848$	−5088.00	−0.206041
$849$	−12736.0	−0.514839
$850$	0	0
$851$	38752.0	1.56099
$852$	−12544.0	−0.504402
$853$	−698.000	−0.0280177	−0.0140088	−	0.999902i	$-0.504459\pi$
$853$	−698.000	−0.0280177	−0.0140088	+	0.999902i	$0.504459\pi$
$854$	−2772.00	−0.111072
$855$	0	0
$856$	−3552.00	−0.141828
$857$	23406.0	0.932945	0.466472	−	0.884536i	$-0.345525\pi$
$857$	23406.0	0.932945	0.466472	+	0.884536i	$0.345525\pi$
$858$	−8064.00	−0.320863
$859$	7280.00	0.289162	0.144581	−	0.989493i	$-0.453817\pi$
$859$	7280.00	0.289162	0.144581	+	0.989493i	$0.453817\pi$
$860$	0	0
$861$	−9072.00	−0.359086
$862$	2544.00	0.100521
$863$	−9808.00	−0.386869	−0.193435	−	0.981113i	$-0.561963\pi$
$863$	−9808.00	−0.386869	−0.193435	+	0.981113i	$0.561963\pi$
$864$	−2560.00	−0.100802
$865$	0	0
$866$	10124.0	0.397260
$867$	−4504.00	−0.176429
$868$	2016.00	0.0788335
$869$	−6720.00	−0.262325
$870$	0	0
$871$	−12888.0	−0.501370
$872$	14960.0	0.580974
$873$	−50098.0	−1.94222
$874$	17920.0	0.693539
$875$	0	0
$876$	17216.0	0.664012
$877$	8066.00	0.310570	0.155285	−	0.987870i	$-0.450370\pi$
$877$	8066.00	0.310570	0.155285	+	0.987870i	$0.450370\pi$
$878$	11280.0	0.433578
$879$	−40176.0	−1.54164
$880$	0	0
$881$	25842.0	0.988240	0.494120	−	0.869394i	$-0.335490\pi$
$881$	25842.0	0.988240	0.494120	+	0.869394i	$0.335490\pi$
$882$	3626.00	0.138428
$883$	5692.00	0.216932	0.108466	−	0.994100i	$-0.465406\pi$
$883$	5692.00	0.216932	0.108466	+	0.994100i	$0.465406\pi$
$884$	5328.00	0.202715
$885$	0	0
$886$	−26776.0	−1.01530
$887$	13536.0	0.512395	0.256198	−	0.966624i	$-0.417530\pi$
$887$	13536.0	0.512395	0.256198	+	0.966624i	$0.417530\pi$
$888$	−22144.0	−0.836829
$889$	−13608.0	−0.513383
$890$	0	0
$891$	10052.0	0.377951
$892$	−15872.0	−0.595778
$893$	−1920.00	−0.0719489
$894$	−8160.00	−0.305270
$895$	0	0
$896$	896.000	0.0334077
$897$	16128.0	0.600332
$898$	−6460.00	−0.240059
$899$	13680.0	0.507512
$900$	0	0
$901$	23532.0	0.870105
$902$	−9072.00	−0.334883
$903$	−23072.0	−0.850264
$904$	−11024.0	−0.405589
$905$	0	0
$906$	−9472.00	−0.347336
$907$	−17004.0	−0.622501	−0.311251	−	0.950328i	$-0.600748\pi$
$907$	−17004.0	−0.622501	−0.311251	+	0.950328i	$0.600748\pi$
$908$	15744.0	0.575422
$909$	−50246.0	−1.83339
$910$	0	0
$911$	−14568.0	−0.529813	−0.264906	−	0.964274i	$-0.585341\pi$
$911$	−14568.0	−0.529813	−0.264906	+	0.964274i	$0.585341\pi$
$912$	−10240.0	−0.371799
$913$	−30016.0	−1.08804
$914$	21292.0	0.770544
$915$	0	0
$916$	19240.0	0.694004
$917$	−5936.00	−0.213767
$918$	11840.0	0.425684
$919$	−1400.00	−0.0502522	−0.0251261	−	0.999684i	$-0.507999\pi$
$919$	−1400.00	−0.0502522	−0.0251261	+	0.999684i	$0.507999\pi$
$920$	0	0
$921$	−76288.0	−2.72940
$922$	14564.0	0.520217
$923$	−7056.00	−0.251626
$924$	6272.00	0.223305
$925$	0	0
$926$	−25376.0	−0.900548
$927$	30784.0	1.09070
$928$	6080.00	0.215071
$929$	−13830.0	−0.488426	−0.244213	−	0.969722i	$-0.578530\pi$
$929$	−13830.0	−0.488426	−0.244213	+	0.969722i	$0.578530\pi$
$930$	0	0
$931$	3920.00	0.137994
$932$	8728.00	0.306754
$933$	7744.00	0.271733
$934$	5632.00	0.197307
$935$	0	0
$936$	−5328.00	−0.186059
$937$	24166.0	0.842549	0.421275	−	0.906933i	$-0.361583\pi$
$937$	24166.0	0.842549	0.421275	+	0.906933i	$0.361583\pi$
$938$	10024.0	0.348929
$939$	24464.0	0.850216
$940$	0	0
$941$	−10838.0	−0.375461	−0.187730	−	0.982221i	$-0.560113\pi$
$941$	−10838.0	−0.375461	−0.187730	+	0.982221i	$0.560113\pi$
$942$	−42976.0	−1.48645
$943$	18144.0	0.626564
$944$	−3200.00	−0.110330
$945$	0	0
$946$	−23072.0	−0.792955
$947$	40916.0	1.40400	0.702002	−	0.712175i	$-0.252290\pi$
$947$	40916.0	1.40400	0.702002	+	0.712175i	$0.252290\pi$
$948$	−7680.00	−0.263117
$949$	9684.00	0.331250
$950$	0	0
$951$	−39888.0	−1.36010
$952$	−4144.00	−0.141080
$953$	−56618.0	−1.92449	−0.962244	−	0.272189i	$-0.912253\pi$
$953$	−56618.0	−1.92449	−0.962244	+	0.272189i	$0.912253\pi$
$954$	−23532.0	−0.798613
$955$	0	0
$956$	−12000.0	−0.405970
$957$	42560.0	1.43759
$958$	−6320.00	−0.213142
$959$	20762.0	0.699103
$960$	0	0
$961$	−24607.0	−0.825988
$962$	−12456.0	−0.417461
$963$	−16428.0	−0.549725
$964$	8168.00	0.272898
$965$	0	0
$966$	−12544.0	−0.417802
$967$	−17504.0	−0.582100	−0.291050	−	0.956708i	$-0.594005\pi$
$967$	−17504.0	−0.582100	−0.291050	+	0.956708i	$0.594005\pi$
$968$	−4376.00	−0.145300
$969$	47360.0	1.57010
$970$	0	0
$971$	23112.0	0.763851	0.381926	−	0.924193i	$-0.375261\pi$
$971$	23112.0	0.763851	0.381926	+	0.924193i	$0.375261\pi$
$972$	20128.0	0.664204
$973$	19600.0	0.645783
$974$	28352.0	0.932707
$975$	0	0
$976$	−3168.00	−0.103899
$977$	−23874.0	−0.781778	−0.390889	−	0.920438i	$-0.627832\pi$
$977$	−23874.0	−0.781778	−0.390889	+	0.920438i	$0.627832\pi$
$978$	−16192.0	−0.529410
$979$	−22680.0	−0.740404
$980$	0	0
$981$	69190.0	2.25185
$982$	−22536.0	−0.732335
$983$	15312.0	0.496823	0.248411	−	0.968655i	$-0.420091\pi$
$983$	15312.0	0.496823	0.248411	+	0.968655i	$0.420091\pi$
$984$	−10368.0	−0.335894
$985$	0	0
$986$	−28120.0	−0.908239
$987$	1344.00	0.0433435
$988$	−5760.00	−0.185476
$989$	46144.0	1.48361
$990$	0	0
$991$	−16528.0	−0.529797	−0.264899	−	0.964276i	$-0.585339\pi$
$991$	−16528.0	−0.529797	−0.264899	+	0.964276i	$0.585339\pi$
$992$	2304.00	0.0737420
$993$	−68896.0	−2.20176
$994$	5488.00	0.175120
$995$	0	0
$996$	−34304.0	−1.09133
$997$	28606.0	0.908687	0.454344	−	0.890827i	$-0.349874\pi$
$997$	28606.0	0.908687	0.454344	+	0.890827i	$0.349874\pi$
$998$	−8920.00	−0.282924
$999$	−27680.0	−0.876633

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.4.a.l.1.1		1
5.2	odd	4		350.4.c.b.99.2		2
5.3	odd	4		350.4.c.b.99.1		2
5.4	even	2		14.4.a.a.1.1	✓	1
7.6	odd	2		2450.4.a.bo.1.1		1
15.14	odd	2		126.4.a.h.1.1		1
20.19	odd	2		112.4.a.a.1.1		1
35.4	even	6		98.4.c.d.79.1		2
35.9	even	6		98.4.c.d.67.1		2
35.19	odd	6		98.4.c.f.67.1		2
35.24	odd	6		98.4.c.f.79.1		2
35.34	odd	2		98.4.a.a.1.1		1
40.19	odd	2		448.4.a.o.1.1		1
40.29	even	2		448.4.a.b.1.1		1
55.54	odd	2		1694.4.a.g.1.1		1
60.59	even	2		1008.4.a.s.1.1		1
65.64	even	2		2366.4.a.h.1.1		1
105.44	odd	6		882.4.g.b.361.1		2
105.59	even	6		882.4.g.k.667.1		2
105.74	odd	6		882.4.g.b.667.1		2
105.89	even	6		882.4.g.k.361.1		2
105.104	even	2		882.4.a.i.1.1		1
140.139	even	2		784.4.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.4.a.a.1.1	✓	1	5.4	even	2
98.4.a.a.1.1		1	35.34	odd	2
98.4.c.d.67.1		2	35.9	even	6
98.4.c.d.79.1		2	35.4	even	6
98.4.c.f.67.1		2	35.19	odd	6
98.4.c.f.79.1		2	35.24	odd	6
112.4.a.a.1.1		1	20.19	odd	2
126.4.a.h.1.1		1	15.14	odd	2
350.4.a.l.1.1		1	1.1	even	1	trivial
350.4.c.b.99.1		2	5.3	odd	4
350.4.c.b.99.2		2	5.2	odd	4
448.4.a.b.1.1		1	40.29	even	2
448.4.a.o.1.1		1	40.19	odd	2
784.4.a.s.1.1		1	140.139	even	2
882.4.a.i.1.1		1	105.104	even	2
882.4.g.b.361.1		2	105.44	odd	6
882.4.g.b.667.1		2	105.74	odd	6
882.4.g.k.361.1		2	105.89	even	6
882.4.g.k.667.1		2	105.59	even	6
1008.4.a.s.1.1		1	60.59	even	2
1694.4.a.g.1.1		1	55.54	odd	2
2366.4.a.h.1.1		1	65.64	even	2
2450.4.a.bo.1.1		1	7.6	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

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

Label	350.4.a.l.1.1

Level	$350$
Weight	$4$
Character	350.1
Self dual	yes
Analytic conductor	$20.651$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$