LMFDB - Embedded newform 2450.4.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
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	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +8.00000 q^{3} +4.00000 q^{4} +16.0000 q^{6} +8.00000 q^{8} +37.0000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +8.00000 q^{3} +4.00000 q^{4} +16.0000 q^{6} +8.00000 q^{8} +37.0000 q^{9} -28.0000 q^{11} +32.0000 q^{12} +18.0000 q^{13} +16.0000 q^{16} +74.0000 q^{17} +74.0000 q^{18} -80.0000 q^{19} -56.0000 q^{22} +112.000 q^{23} +64.0000 q^{24} +36.0000 q^{26} +80.0000 q^{27} +190.000 q^{29} -72.0000 q^{31} +32.0000 q^{32} -224.000 q^{33} +148.000 q^{34} +148.000 q^{36} +346.000 q^{37} -160.000 q^{38} +144.000 q^{39} -162.000 q^{41} +412.000 q^{43} -112.000 q^{44} +224.000 q^{46} +24.0000 q^{47} +128.000 q^{48} +592.000 q^{51} +72.0000 q^{52} -318.000 q^{53} +160.000 q^{54} -640.000 q^{57} +380.000 q^{58} +200.000 q^{59} +198.000 q^{61} -144.000 q^{62} +64.0000 q^{64} -448.000 q^{66} +716.000 q^{67} +296.000 q^{68} +896.000 q^{69} +392.000 q^{71} +296.000 q^{72} +538.000 q^{73} +692.000 q^{74} -320.000 q^{76} +288.000 q^{78} +240.000 q^{79} -359.000 q^{81} -324.000 q^{82} -1072.00 q^{83} +824.000 q^{86} +1520.00 q^{87} -224.000 q^{88} -810.000 q^{89} +448.000 q^{92} -576.000 q^{93} +48.0000 q^{94} +256.000 q^{96} +1354.00 q^{97} -1036.00 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	16.0000	1.08866
$7$	0	0
$7$	0	0
$8$	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	$-0.438499\pi$
$13$	18.0000	0.384023	0.192012	+	0.981393i	$0.438499\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	74.0000	1.05574	0.527872	−	0.849324i	$-0.322990\pi$
$17$	74.0000	1.05574	0.527872	+	0.849324i	$0.322990\pi$
$18$	74.0000	0.968998
$19$	−80.0000	−0.965961	−0.482980	−	0.875631i	$-0.660446\pi$
$19$	−80.0000	−0.965961	−0.482980	+	0.875631i	$0.660446\pi$
$20$	0	0
$21$	0	0
$22$	−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
$28$	0	0
$29$	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	$-0.566882\pi$
$31$	−72.0000	−0.417148	−0.208574	+	0.978007i	$0.566882\pi$
$32$	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	$-0.599840\pi$
$41$	−162.000	−0.617077	−0.308538	+	0.951212i	$0.599840\pi$
$42$	0	0
$43$	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	$-0.488143\pi$
$47$	24.0000	0.0744843	0.0372421	+	0.999306i	$0.488143\pi$
$48$	128.000	0.384900
$49$	0	0
$50$	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
$56$	0	0
$57$	−640.000	−1.48719
$58$	380.000	0.860284
$59$	200.000	0.441318	0.220659	−	0.975351i	$-0.429179\pi$
$59$	200.000	0.441318	0.220659	+	0.975351i	$0.429179\pi$
$60$	0	0
$61$	198.000	0.415595	0.207798	−	0.978172i	$-0.433370\pi$
$61$	198.000	0.415595	0.207798	+	0.978172i	$0.433370\pi$
$62$	−144.000	−0.294968
$63$	0	0
$64$	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	$-0.358059\pi$
$73$	538.000	0.862577	0.431289	+	0.902214i	$0.358059\pi$
$74$	692.000	1.08707
$75$	0	0
$76$	−320.000	−0.482980
$77$	0	0
$78$	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	$-0.750781\pi$
$83$	−1072.00	−1.41768	−0.708839	+	0.705370i	$0.750781\pi$
$84$	0	0
$85$	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	$-0.660220\pi$
$89$	−810.000	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$90$	0	0
$91$	0	0
$92$	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	$-0.249305\pi$
$97$	1354.00	1.41730	0.708649	+	0.705561i	$0.249305\pi$
$98$	0	0
$99$	−1036.00	−1.05174
$100$	0	0
$101$	1358.00	1.33788	0.668941	−	0.743316i	$-0.266748\pi$
$101$	1358.00	1.33788	0.668941	+	0.743316i	$0.266748\pi$
$102$	1184.00	1.14935
$103$	−832.000	−0.795916	−0.397958	−	0.917404i	$-0.630281\pi$
$103$	−832.000	−0.795916	−0.397958	+	0.917404i	$0.630281\pi$
$104$	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
$112$	0	0
$113$	−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
$119$	0	0
$120$	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
$126$	0	0
$127$	−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	$-0.408741\pi$
$131$	848.000	0.565573	0.282787	+	0.959183i	$0.408741\pi$
$132$	−896.000	−0.590809
$133$	0	0
$134$	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	$-0.826008\pi$
$139$	−2800.00	−1.70858	−0.854291	+	0.519795i	$0.826008\pi$
$140$	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
$147$	0	0
$148$	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
$154$	0	0
$155$	0	0
$156$	576.000	0.295621
$157$	−2686.00	−1.36539	−0.682695	−	0.730704i	$-0.739192\pi$
$157$	−2686.00	−1.36539	−0.682695	+	0.730704i	$0.739192\pi$
$158$	480.000	0.241688
$159$	−2544.00	−1.26888
$160$	0	0
$161$	0	0
$162$	−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	$-0.459775\pi$
$167$	544.000	0.252072	0.126036	+	0.992026i	$0.459775\pi$
$168$	0	0
$169$	−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	$-0.366132\pi$
$173$	1858.00	0.816538	0.408269	+	0.912862i	$0.366132\pi$
$174$	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	$-0.339122\pi$
$181$	2358.00	0.968336	0.484168	+	0.874975i	$0.339122\pi$
$182$	0	0
$183$	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
$189$	0	0
$190$	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
$196$	0	0
$197$	−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	$-0.559305\pi$
$199$	−1040.00	−0.370471	−0.185235	+	0.982694i	$0.559305\pi$
$200$	0	0
$201$	5728.00	2.01006
$202$	2716.00	0.946025
$203$	0	0
$204$	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
$217$	0	0
$218$	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	$-0.296844\pi$
$223$	3968.00	1.19156	0.595778	+	0.803149i	$0.296844\pi$
$224$	0	0
$225$	0	0
$226$	−2756.00	−0.811179
$227$	−3936.00	−1.15084	−0.575422	−	0.817857i	$-0.695162\pi$
$227$	−3936.00	−1.15084	−0.575422	+	0.817857i	$0.695162\pi$
$228$	−2560.00	−0.743597
$229$	−4810.00	−1.38801	−0.694004	−	0.719971i	$-0.744155\pi$
$229$	−4810.00	−1.38801	−0.694004	+	0.719971i	$0.744155\pi$
$230$	0	0
$231$	0	0
$232$	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
$238$	0	0
$239$	−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	$-0.587982\pi$
$241$	−2042.00	−0.545796	−0.272898	+	0.962043i	$0.587982\pi$
$242$	−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	$-0.478852\pi$
$251$	528.000	0.132777	0.0663886	+	0.997794i	$0.478852\pi$
$252$	0	0
$253$	−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	$-0.260355\pi$
$257$	5634.00	1.36747	0.683734	+	0.729731i	$0.260355\pi$
$258$	6592.00	1.59070
$259$	0	0
$260$	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
$266$	0	0
$267$	−6480.00	−1.48528
$268$	2864.00	0.652786
$269$	1310.00	0.296922	0.148461	−	0.988918i	$-0.452568\pi$
$269$	1310.00	0.296922	0.148461	+	0.988918i	$0.452568\pi$
$270$	0	0
$271$	2208.00	0.494932	0.247466	−	0.968897i	$-0.420402\pi$
$271$	2208.00	0.494932	0.247466	+	0.968897i	$0.420402\pi$
$272$	1184.00	0.263936
$273$	0	0
$274$	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	$-0.553472\pi$
$283$	−1592.00	−0.334398	−0.167199	+	0.985923i	$0.553472\pi$
$284$	1568.00	0.327619
$285$	0	0
$286$	−1008.00	−0.208407
$287$	0	0
$288$	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	$-0.666910\pi$
$293$	−5022.00	−1.00133	−0.500663	+	0.865642i	$0.666910\pi$
$294$	0	0
$295$	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
$301$	0	0
$302$	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	$-0.846800\pi$
$307$	−9536.00	−1.77280	−0.886398	+	0.462924i	$0.846800\pi$
$308$	0	0
$309$	−6656.00	−1.22539
$310$	0	0
$311$	968.000	0.176496	0.0882480	−	0.996099i	$-0.471873\pi$
$311$	968.000	0.176496	0.0882480	+	0.996099i	$0.471873\pi$
$312$	1152.00	0.209036
$313$	3058.00	0.552231	0.276116	−	0.961124i	$-0.410953\pi$
$313$	3058.00	0.552231	0.276116	+	0.961124i	$0.410953\pi$
$314$	−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
$322$	0	0
$323$	−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
$329$	0	0
$330$	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
$336$	0	0
$337$	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
$343$	0	0
$344$	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	$-0.532523\pi$
$349$	−1330.00	−0.203992	−0.101996	+	0.994785i	$0.532523\pi$
$350$	0	0
$351$	1440.00	0.218979
$352$	−896.000	−0.135673
$353$	978.000	0.147461	0.0737304	−	0.997278i	$-0.476510\pi$
$353$	978.000	0.147461	0.0737304	+	0.997278i	$0.476510\pi$
$354$	3200.00	0.480447
$355$	0	0
$356$	−3240.00	−0.482359
$357$	0	0
$358$	−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
$364$	0	0
$365$	0	0
$366$	3168.00	0.452443
$367$	−8656.00	−1.23117	−0.615585	−	0.788070i	$-0.711080\pi$
$367$	−8656.00	−1.23117	−0.615585	+	0.788070i	$0.711080\pi$
$368$	1792.00	0.253844
$369$	−5994.00	−0.845624
$370$	0	0
$371$	0	0
$372$	−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
$378$	0	0
$379$	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	$-0.636470\pi$
$383$	−6232.00	−0.831437	−0.415718	+	0.909493i	$0.636470\pi$
$384$	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
$392$	0	0
$393$	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	$-0.394372\pi$
$397$	5154.00	0.651566	0.325783	+	0.945445i	$0.394372\pi$
$398$	−2080.00	−0.261962
$399$	0	0
$400$	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
$406$	0	0
$407$	−9688.00	−1.17989
$408$	4736.00	0.574674
$409$	−5810.00	−0.702411	−0.351205	−	0.936298i	$-0.614228\pi$
$409$	−5810.00	−0.702411	−0.351205	+	0.936298i	$0.614228\pi$
$410$	0	0
$411$	23728.0	2.84773
$412$	−3328.00	−0.397958
$413$	0	0
$414$	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	$-0.790186\pi$
$419$	−13560.0	−1.58102	−0.790512	+	0.612446i	$0.790186\pi$
$420$	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
$427$	0	0
$428$	−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	$-0.590635\pi$
$433$	−5062.00	−0.561811	−0.280906	+	0.959735i	$0.590635\pi$
$434$	0	0
$435$	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	$-0.599187\pi$
$439$	−5640.00	−0.613172	−0.306586	+	0.951843i	$0.599187\pi$
$440$	0	0
$441$	0	0
$442$	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
$448$	0	0
$449$	−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	$-0.619906\pi$
$461$	−7282.00	−0.735698	−0.367849	+	0.929886i	$0.619906\pi$
$462$	0	0
$463$	−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	$-0.544555\pi$
$467$	−2816.00	−0.279034	−0.139517	+	0.990220i	$0.544555\pi$
$468$	2664.00	0.263127
$469$	0	0
$470$	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
$476$	0	0
$477$	−11766.0	−1.12941
$478$	−6000.00	−0.574129
$479$	3160.00	0.301428	0.150714	−	0.988577i	$-0.451843\pi$
$479$	3160.00	0.301428	0.150714	+	0.988577i	$0.451843\pi$
$480$	0	0
$481$	6228.00	0.590379
$482$	−4084.00	−0.385936
$483$	0	0
$484$	−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
$497$	0	0
$498$	−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	$-0.521347\pi$
$503$	−1512.00	−0.134029	−0.0670147	+	0.997752i	$0.521347\pi$
$504$	0	0
$505$	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	$-0.328407\pi$
$509$	11790.0	1.02668	0.513342	+	0.858184i	$0.328407\pi$
$510$	0	0
$511$	0	0
$512$	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
$518$	0	0
$519$	14864.0	1.25714
$520$	0	0
$521$	−1362.00	−0.114530	−0.0572652	−	0.998359i	$-0.518238\pi$
$521$	−1362.00	−0.114530	−0.0572652	+	0.998359i	$0.518238\pi$
$522$	14060.0	1.17891
$523$	6968.00	0.582580	0.291290	−	0.956635i	$-0.405916\pi$
$523$	6968.00	0.582580	0.291290	+	0.956635i	$0.405916\pi$
$524$	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
$532$	0	0
$533$	−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
$539$	0	0
$540$	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
$546$	0	0
$547$	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
$553$	0	0
$554$	−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	$-0.115824\pi$
$563$	24968.0	1.86905	0.934526	+	0.355896i	$0.115824\pi$
$564$	768.000	0.0573380
$565$	0	0
$566$	−3184.00	−0.236455
$567$	0	0
$568$	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
$574$	0	0
$575$	0	0
$576$	2368.00	0.171296
$577$	−8366.00	−0.603607	−0.301803	−	0.953370i	$-0.597589\pi$
$577$	−8366.00	−0.603607	−0.301803	+	0.953370i	$0.597589\pi$
$578$	1126.00	0.0810301
$579$	−14224.0	−1.02095
$580$	0	0
$581$	0	0
$582$	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	$-0.245678\pi$
$587$	20384.0	1.43328	0.716642	+	0.697441i	$0.245678\pi$
$588$	0	0
$589$	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	$-0.394733\pi$
$593$	9378.00	0.649424	0.324712	+	0.945813i	$0.394733\pi$
$594$	−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	$-0.582610\pi$
$601$	−7562.00	−0.513245	−0.256623	+	0.966512i	$0.582610\pi$
$602$	0	0
$603$	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	$-0.531724\pi$
$607$	−2976.00	−0.198999	−0.0994993	+	0.995038i	$0.531724\pi$
$608$	−2560.00	−0.170759
$609$	0	0
$610$	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
$616$	0	0
$617$	−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	$-0.699178\pi$
$619$	−18040.0	−1.17139	−0.585694	+	0.810532i	$0.699178\pi$
$620$	0	0
$621$	8960.00	0.578989
$622$	1936.00	0.124801
$623$	0	0
$624$	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
$637$	0	0
$638$	−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	$-0.796215\pi$
$643$	−26152.0	−1.60394	−0.801971	+	0.597363i	$0.796215\pi$
$644$	0	0
$645$	0	0
$646$	−11840.0	−0.721112
$647$	25584.0	1.55458	0.777288	−	0.629145i	$-0.216595\pi$
$647$	25584.0	1.55458	0.777288	+	0.629145i	$0.216595\pi$
$648$	−2872.00	−0.174109
$649$	−5600.00	−0.338705
$650$	0	0
$651$	0	0
$652$	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
$658$	0	0
$659$	−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	$-0.478275\pi$
$661$	2318.00	0.136399	0.0681995	+	0.997672i	$0.478275\pi$
$662$	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
$672$	0	0
$673$	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	$-0.243250\pi$
$677$	25434.0	1.44388	0.721941	+	0.691955i	$0.243250\pi$
$678$	−22048.0	−1.24889
$679$	0	0
$680$	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
$686$	0	0
$687$	−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	$-0.692681\pi$
$691$	−20672.0	−1.13806	−0.569030	+	0.822317i	$0.692681\pi$
$692$	7432.00	0.408269
$693$	0	0
$694$	−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
$707$	0	0
$708$	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
$714$	0	0
$715$	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	$-0.337506\pi$
$719$	18840.0	0.977209	0.488605	+	0.872505i	$0.337506\pi$
$720$	0	0
$721$	0	0
$722$	−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	$-0.0940849\pi$
$727$	37504.0	1.91327	0.956634	+	0.291291i	$0.0940849\pi$
$728$	0	0
$729$	−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	$-0.390909\pi$
$733$	13338.0	0.672101	0.336051	+	0.941844i	$0.390909\pi$
$734$	−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
$742$	0	0
$743$	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
$749$	0	0
$750$	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
$756$	0	0
$757$	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	$-0.192254\pi$
$761$	34558.0	1.64616	0.823079	+	0.567927i	$0.192254\pi$
$762$	−31104.0	−1.47871
$763$	0	0
$764$	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	$-0.869769\pi$
$769$	−39130.0	−1.83493	−0.917467	+	0.397812i	$0.869769\pi$
$770$	0	0
$771$	45072.0	2.10535
$772$	−7112.00	−0.331563
$773$	−25982.0	−1.20894	−0.604468	−	0.796629i	$-0.706614\pi$
$773$	−25982.0	−1.20894	−0.604468	+	0.796629i	$0.706614\pi$
$774$	30488.0	1.41585
$775$	0	0
$776$	10832.0	0.501090
$777$	0	0
$778$	−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
$784$	0	0
$785$	0	0
$786$	13568.0	0.615718
$787$	35424.0	1.60448	0.802242	−	0.596999i	$-0.203640\pi$
$787$	35424.0	1.60448	0.802242	+	0.596999i	$0.203640\pi$
$788$	−4856.00	−0.219528
$789$	−1344.00	−0.0606434
$790$	0	0
$791$	0	0
$792$	−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	$-0.738075\pi$
$797$	−30606.0	−1.36025	−0.680126	+	0.733096i	$0.738075\pi$
$798$	0	0
$799$	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	$-0.434993\pi$
$811$	9368.00	0.405616	0.202808	+	0.979218i	$0.434993\pi$
$812$	0	0
$813$	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
$819$	0	0
$820$	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
$826$	0	0
$827$	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	$-0.447013\pi$
$829$	7910.00	0.331394	0.165697	+	0.986177i	$0.447013\pi$
$830$	0	0
$831$	−42352.0	−1.76796
$832$	1152.00	0.0480029
$833$	0	0
$834$	−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	$-0.369593\pi$
$839$	19360.0	0.796641	0.398320	+	0.917246i	$0.369593\pi$
$840$	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
$847$	0	0
$848$	−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	$-0.495541\pi$
$853$	698.000	0.0280177	0.0140088	+	0.999902i	$0.495541\pi$
$854$	0	0
$855$	0	0
$856$	−3552.00	−0.141828
$857$	−23406.0	−0.932945	−0.466472	−	0.884536i	$-0.654475\pi$
$857$	−23406.0	−0.932945	−0.466472	+	0.884536i	$0.654475\pi$
$858$	−8064.00	−0.320863
$859$	−7280.00	−0.289162	−0.144581	−	0.989493i	$-0.546183\pi$
$859$	−7280.00	−0.289162	−0.144581	+	0.989493i	$0.546183\pi$
$860$	0	0
$861$	0	0
$862$	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
$868$	0	0
$869$	−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	$-0.664510\pi$
$881$	−25842.0	−0.988240	−0.494120	+	0.869394i	$0.664510\pi$
$882$	0	0
$883$	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	$-0.582470\pi$
$887$	−13536.0	−0.512395	−0.256198	+	0.966624i	$0.582470\pi$
$888$	22144.0	0.836829
$889$	0	0
$890$	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
$896$	0	0
$897$	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
$903$	0	0
$904$	−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
$917$	0	0
$918$	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
$924$	0	0
$925$	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	$-0.421470\pi$
$929$	13830.0	0.488426	0.244213	+	0.969722i	$0.421470\pi$
$930$	0	0
$931$	0	0
$932$	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	$-0.638417\pi$
$937$	−24166.0	−0.842549	−0.421275	+	0.906933i	$0.638417\pi$
$938$	0	0
$939$	24464.0	0.850216
$940$	0	0
$941$	10838.0	0.375461	0.187730	−	0.982221i	$-0.439887\pi$
$941$	10838.0	0.375461	0.187730	+	0.982221i	$0.439887\pi$
$942$	−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
$952$	0	0
$953$	−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
$959$	0	0
$960$	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
$966$	0	0
$967$	−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	$-0.624739\pi$
$971$	−23112.0	−0.763851	−0.381926	+	0.924193i	$0.624739\pi$
$972$	−20128.0	−0.664204
$973$	0	0
$974$	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	$-0.579909\pi$
$983$	−15312.0	−0.496823	−0.248411	+	0.968655i	$0.579909\pi$
$984$	−10368.0	−0.335894
$985$	0	0
$986$	28120.0	0.908239
$987$	0	0
$988$	−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
$994$	0	0
$995$	0	0
$996$	−34304.0	−1.09133
$997$	−28606.0	−0.908687	−0.454344	−	0.890827i	$-0.650126\pi$
$997$	−28606.0	−0.908687	−0.454344	+	0.890827i	$0.650126\pi$
$998$	−8920.00	−0.282924
$999$	27680.0	0.876633

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.4.a.a.1.1	✓	1	35.34	odd	2
98.4.a.a.1.1		1	5.4	even	2
98.4.c.d.67.1		2	35.19	odd	6
98.4.c.d.79.1		2	35.24	odd	6
98.4.c.f.67.1		2	35.9	even	6
98.4.c.f.79.1		2	35.4	even	6
112.4.a.a.1.1		1	140.139	even	2
126.4.a.h.1.1		1	105.104	even	2
350.4.a.l.1.1		1	7.6	odd	2
350.4.c.b.99.1		2	35.13	even	4
350.4.c.b.99.2		2	35.27	even	4
448.4.a.b.1.1		1	280.69	odd	2
448.4.a.o.1.1		1	280.139	even	2
784.4.a.s.1.1		1	20.19	odd	2
882.4.a.i.1.1		1	15.14	odd	2
882.4.g.b.361.1		2	105.89	even	6
882.4.g.b.667.1		2	105.59	even	6
882.4.g.k.361.1		2	105.44	odd	6
882.4.g.k.667.1		2	105.74	odd	6
1008.4.a.s.1.1		1	420.419	odd	2
1694.4.a.g.1.1		1	385.384	even	2
2366.4.a.h.1.1		1	455.454	odd	2
2450.4.a.bo.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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more