LMFDB - Embedded newform 378.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, 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("378.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.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:	$22.3027219822$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	378.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -7.00000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} -7.00000 q^{5} -7.00000 q^{7} +8.00000 q^{8} -14.0000 q^{10} -17.0000 q^{11} +12.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -38.0000 q^{17} -43.0000 q^{19} -28.0000 q^{20} -34.0000 q^{22} -131.000 q^{23} -76.0000 q^{25} +24.0000 q^{26} -28.0000 q^{28} -160.000 q^{29} +45.0000 q^{31} +32.0000 q^{32} -76.0000 q^{34} +49.0000 q^{35} -331.000 q^{37} -86.0000 q^{38} -56.0000 q^{40} +111.000 q^{41} +230.000 q^{43} -68.0000 q^{44} -262.000 q^{46} -282.000 q^{47} +49.0000 q^{49} -152.000 q^{50} +48.0000 q^{52} -396.000 q^{53} +119.000 q^{55} -56.0000 q^{56} -320.000 q^{58} -214.000 q^{59} +768.000 q^{61} +90.0000 q^{62} +64.0000 q^{64} -84.0000 q^{65} +388.000 q^{67} -152.000 q^{68} +98.0000 q^{70} -551.000 q^{71} +274.000 q^{73} -662.000 q^{74} -172.000 q^{76} +119.000 q^{77} +390.000 q^{79} -112.000 q^{80} +222.000 q^{82} -440.000 q^{83} +266.000 q^{85} +460.000 q^{86} -136.000 q^{88} -105.000 q^{89} -84.0000 q^{91} -524.000 q^{92} -564.000 q^{94} +301.000 q^{95} +304.000 q^{97} +98.0000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−7.00000	−0.626099	−0.313050	−	0.949737i	$-0.601351\pi$
$5$	−7.00000	−0.626099	−0.313050	+	0.949737i	$0.601351\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	−14.0000	−0.442719
$11$	−17.0000	−0.465972	−0.232986	−	0.972480i	$-0.574850\pi$
$11$	−17.0000	−0.465972	−0.232986	+	0.972480i	$0.574850\pi$
$12$	0	0
$13$	12.0000	0.256015	0.128008	−	0.991773i	$-0.459142\pi$
$13$	12.0000	0.256015	0.128008	+	0.991773i	$0.459142\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−38.0000	−0.542138	−0.271069	−	0.962560i	$-0.587377\pi$
$17$	−38.0000	−0.542138	−0.271069	+	0.962560i	$0.587377\pi$
$18$	0	0
$19$	−43.0000	−0.519204	−0.259602	−	0.965716i	$-0.583591\pi$
$19$	−43.0000	−0.519204	−0.259602	+	0.965716i	$0.583591\pi$
$20$	−28.0000	−0.313050
$21$	0	0
$22$	−34.0000	−0.329492
$23$	−131.000	−1.18763	−0.593813	−	0.804603i	$-0.702378\pi$
$23$	−131.000	−1.18763	−0.593813	+	0.804603i	$0.702378\pi$
$24$	0	0
$25$	−76.0000	−0.608000
$26$	24.0000	0.181030
$27$	0	0
$28$	−28.0000	−0.188982
$29$	−160.000	−1.02453	−0.512263	−	0.858829i	$-0.671193\pi$
$29$	−160.000	−1.02453	−0.512263	+	0.858829i	$0.671193\pi$
$30$	0	0
$31$	45.0000	0.260717	0.130359	−	0.991467i	$-0.458387\pi$
$31$	45.0000	0.260717	0.130359	+	0.991467i	$0.458387\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−76.0000	−0.383350
$35$	49.0000	0.236643
$36$	0	0
$37$	−331.000	−1.47070	−0.735352	−	0.677685i	$-0.762983\pi$
$37$	−331.000	−1.47070	−0.735352	+	0.677685i	$0.762983\pi$
$38$	−86.0000	−0.367133
$39$	0	0
$40$	−56.0000	−0.221359
$41$	111.000	0.422812	0.211406	−	0.977398i	$-0.432196\pi$
$41$	111.000	0.422812	0.211406	+	0.977398i	$0.432196\pi$
$42$	0	0
$43$	230.000	0.815690	0.407845	−	0.913051i	$-0.366280\pi$
$43$	230.000	0.815690	0.407845	+	0.913051i	$0.366280\pi$
$44$	−68.0000	−0.232986
$45$	0	0
$46$	−262.000	−0.839778
$47$	−282.000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−282.000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−152.000	−0.429921
$51$	0	0
$52$	48.0000	0.128008
$53$	−396.000	−1.02632	−0.513158	−	0.858294i	$-0.671525\pi$
$53$	−396.000	−1.02632	−0.513158	+	0.858294i	$0.671525\pi$
$54$	0	0
$55$	119.000	0.291745
$56$	−56.0000	−0.133631
$57$	0	0
$58$	−320.000	−0.724449
$59$	−214.000	−0.472211	−0.236105	−	0.971727i	$-0.575871\pi$
$59$	−214.000	−0.472211	−0.236105	+	0.971727i	$0.575871\pi$
$60$	0	0
$61$	768.000	1.61201	0.806003	−	0.591912i	$-0.201627\pi$
$61$	768.000	1.61201	0.806003	+	0.591912i	$0.201627\pi$
$62$	90.0000	0.184355
$63$	0	0
$64$	64.0000	0.125000
$65$	−84.0000	−0.160291
$66$	0	0
$67$	388.000	0.707489	0.353744	−	0.935342i	$-0.384908\pi$
$67$	388.000	0.707489	0.353744	+	0.935342i	$0.384908\pi$
$68$	−152.000	−0.271069
$69$	0	0
$70$	98.0000	0.167332
$71$	−551.000	−0.921009	−0.460505	−	0.887657i	$-0.652332\pi$
$71$	−551.000	−0.921009	−0.460505	+	0.887657i	$0.652332\pi$
$72$	0	0
$73$	274.000	0.439305	0.219653	−	0.975578i	$-0.429508\pi$
$73$	274.000	0.439305	0.219653	+	0.975578i	$0.429508\pi$
$74$	−662.000	−1.03995
$75$	0	0
$76$	−172.000	−0.259602
$77$	119.000	0.176121
$78$	0	0
$79$	390.000	0.555423	0.277712	−	0.960665i	$-0.410424\pi$
$79$	390.000	0.555423	0.277712	+	0.960665i	$0.410424\pi$
$80$	−112.000	−0.156525
$81$	0	0
$82$	222.000	0.298973
$83$	−440.000	−0.581883	−0.290941	−	0.956741i	$-0.593968\pi$
$83$	−440.000	−0.581883	−0.290941	+	0.956741i	$0.593968\pi$
$84$	0	0
$85$	266.000	0.339432
$86$	460.000	0.576780
$87$	0	0
$88$	−136.000	−0.164746
$89$	−105.000	−0.125056	−0.0625280	−	0.998043i	$-0.519916\pi$
$89$	−105.000	−0.125056	−0.0625280	+	0.998043i	$0.519916\pi$
$90$	0	0
$91$	−84.0000	−0.0967648
$92$	−524.000	−0.593813
$93$	0	0
$94$	−564.000	−0.618853
$95$	301.000	0.325073
$96$	0	0
$97$	304.000	0.318212	0.159106	−	0.987262i	$-0.449139\pi$
$97$	304.000	0.318212	0.159106	+	0.987262i	$0.449139\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	−304.000	−0.304000
$101$	−286.000	−0.281763	−0.140882	−	0.990026i	$-0.544994\pi$
$101$	−286.000	−0.281763	−0.140882	+	0.990026i	$0.544994\pi$
$102$	0	0
$103$	725.000	0.693557	0.346779	−	0.937947i	$-0.387276\pi$
$103$	725.000	0.693557	0.346779	+	0.937947i	$0.387276\pi$
$104$	96.0000	0.0905151
$105$	0	0
$106$	−792.000	−0.725715
$107$	−876.000	−0.791459	−0.395730	−	0.918367i	$-0.629508\pi$
$107$	−876.000	−0.791459	−0.395730	+	0.918367i	$0.629508\pi$
$108$	0	0
$109$	−281.000	−0.246926	−0.123463	−	0.992349i	$-0.539400\pi$
$109$	−281.000	−0.246926	−0.123463	+	0.992349i	$0.539400\pi$
$110$	238.000	0.206295
$111$	0	0
$112$	−112.000	−0.0944911
$113$	1094.00	0.910751	0.455376	−	0.890299i	$-0.349505\pi$
$113$	1094.00	0.910751	0.455376	+	0.890299i	$0.349505\pi$
$114$	0	0
$115$	917.000	0.743571
$116$	−640.000	−0.512263
$117$	0	0
$118$	−428.000	−0.333903
$119$	266.000	0.204909
$120$	0	0
$121$	−1042.00	−0.782870
$122$	1536.00	1.13986
$123$	0	0
$124$	180.000	0.130359
$125$	1407.00	1.00677
$126$	0	0
$127$	540.000	0.377301	0.188651	−	0.982044i	$-0.439589\pi$
$127$	540.000	0.377301	0.188651	+	0.982044i	$0.439589\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−168.000	−0.113343
$131$	1886.00	1.25787	0.628934	−	0.777459i	$-0.283492\pi$
$131$	1886.00	1.25787	0.628934	+	0.777459i	$0.283492\pi$
$132$	0	0
$133$	301.000	0.196241
$134$	776.000	0.500270
$135$	0	0
$136$	−304.000	−0.191675
$137$	−1684.00	−1.05017	−0.525087	−	0.851049i	$-0.675967\pi$
$137$	−1684.00	−1.05017	−0.525087	+	0.851049i	$0.675967\pi$
$138$	0	0
$139$	364.000	0.222116	0.111058	−	0.993814i	$-0.464576\pi$
$139$	364.000	0.222116	0.111058	+	0.993814i	$0.464576\pi$
$140$	196.000	0.118322
$141$	0	0
$142$	−1102.00	−0.651252
$143$	−204.000	−0.119296
$144$	0	0
$145$	1120.00	0.641455
$146$	548.000	0.310636
$147$	0	0
$148$	−1324.00	−0.735352
$149$	2334.00	1.28328	0.641640	−	0.767006i	$-0.278254\pi$
$149$	2334.00	1.28328	0.641640	+	0.767006i	$0.278254\pi$
$150$	0	0
$151$	1510.00	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	1510.00	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	−344.000	−0.183566
$153$	0	0
$154$	238.000	0.124536
$155$	−315.000	−0.163235
$156$	0	0
$157$	1508.00	0.766570	0.383285	−	0.923630i	$-0.374793\pi$
$157$	1508.00	0.766570	0.383285	+	0.923630i	$0.374793\pi$
$158$	780.000	0.392743
$159$	0	0
$160$	−224.000	−0.110680
$161$	917.000	0.448880
$162$	0	0
$163$	−2512.00	−1.20709	−0.603543	−	0.797330i	$-0.706245\pi$
$163$	−2512.00	−1.20709	−0.603543	+	0.797330i	$0.706245\pi$
$164$	444.000	0.211406
$165$	0	0
$166$	−880.000	−0.411453
$167$	3686.00	1.70797	0.853986	−	0.520296i	$-0.174179\pi$
$167$	3686.00	1.70797	0.853986	+	0.520296i	$0.174179\pi$
$168$	0	0
$169$	−2053.00	−0.934456
$170$	532.000	0.240015
$171$	0	0
$172$	920.000	0.407845
$173$	−1897.00	−0.833678	−0.416839	−	0.908980i	$-0.636862\pi$
$173$	−1897.00	−0.833678	−0.416839	+	0.908980i	$0.636862\pi$
$174$	0	0
$175$	532.000	0.229802
$176$	−272.000	−0.116493
$177$	0	0
$178$	−210.000	−0.0884279
$179$	−1176.00	−0.491052	−0.245526	−	0.969390i	$-0.578961\pi$
$179$	−1176.00	−0.491052	−0.245526	+	0.969390i	$0.578961\pi$
$180$	0	0
$181$	−4518.00	−1.85536	−0.927680	−	0.373375i	$-0.878200\pi$
$181$	−4518.00	−1.85536	−0.927680	+	0.373375i	$0.878200\pi$
$182$	−168.000	−0.0684230
$183$	0	0
$184$	−1048.00	−0.419889
$185$	2317.00	0.920807
$186$	0	0
$187$	646.000	0.252621
$188$	−1128.00	−0.437595
$189$	0	0
$190$	602.000	0.229861
$191$	−1785.00	−0.676220	−0.338110	−	0.941107i	$-0.609788\pi$
$191$	−1785.00	−0.676220	−0.338110	+	0.941107i	$0.609788\pi$
$192$	0	0
$193$	2558.00	0.954036	0.477018	−	0.878894i	$-0.341718\pi$
$193$	2558.00	0.954036	0.477018	+	0.878894i	$0.341718\pi$
$194$	608.000	0.225010
$195$	0	0
$196$	196.000	0.0714286
$197$	4366.00	1.57901	0.789504	−	0.613745i	$-0.210338\pi$
$197$	4366.00	1.57901	0.789504	+	0.613745i	$0.210338\pi$
$198$	0	0
$199$	1577.00	0.561762	0.280881	−	0.959743i	$-0.409373\pi$
$199$	1577.00	0.561762	0.280881	+	0.959743i	$0.409373\pi$
$200$	−608.000	−0.214960
$201$	0	0
$202$	−572.000	−0.199237
$203$	1120.00	0.387234
$204$	0	0
$205$	−777.000	−0.264722
$206$	1450.00	0.490419
$207$	0	0
$208$	192.000	0.0640039
$209$	731.000	0.241935
$210$	0	0
$211$	1814.00	0.591853	0.295926	−	0.955211i	$-0.404372\pi$
$211$	1814.00	0.591853	0.295926	+	0.955211i	$0.404372\pi$
$212$	−1584.00	−0.513158
$213$	0	0
$214$	−1752.00	−0.559646
$215$	−1610.00	−0.510703
$216$	0	0
$217$	−315.000	−0.0985419
$218$	−562.000	−0.174603
$219$	0	0
$220$	476.000	0.145872
$221$	−456.000	−0.138796
$222$	0	0
$223$	−2617.00	−0.785862	−0.392931	−	0.919568i	$-0.628539\pi$
$223$	−2617.00	−0.785862	−0.392931	+	0.919568i	$0.628539\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	2188.00	0.643998
$227$	3762.00	1.09997	0.549984	−	0.835175i	$-0.314634\pi$
$227$	3762.00	1.09997	0.549984	+	0.835175i	$0.314634\pi$
$228$	0	0
$229$	−3476.00	−1.00306	−0.501530	−	0.865141i	$-0.667229\pi$
$229$	−3476.00	−1.00306	−0.501530	+	0.865141i	$0.667229\pi$
$230$	1834.00	0.525784
$231$	0	0
$232$	−1280.00	−0.362225
$233$	−2498.00	−0.702358	−0.351179	−	0.936308i	$-0.614219\pi$
$233$	−2498.00	−0.702358	−0.351179	+	0.936308i	$0.614219\pi$
$234$	0	0
$235$	1974.00	0.547956
$236$	−856.000	−0.236105
$237$	0	0
$238$	532.000	0.144893
$239$	3840.00	1.03928	0.519642	−	0.854384i	$-0.326065\pi$
$239$	3840.00	1.03928	0.519642	+	0.854384i	$0.326065\pi$
$240$	0	0
$241$	−1654.00	−0.442089	−0.221045	−	0.975264i	$-0.570947\pi$
$241$	−1654.00	−0.442089	−0.221045	+	0.975264i	$0.570947\pi$
$242$	−2084.00	−0.553573
$243$	0	0
$244$	3072.00	0.806003
$245$	−343.000	−0.0894427
$246$	0	0
$247$	−516.000	−0.132924
$248$	360.000	0.0921775
$249$	0	0
$250$	2814.00	0.711892
$251$	5400.00	1.35795	0.678974	−	0.734162i	$-0.262425\pi$
$251$	5400.00	1.35795	0.678974	+	0.734162i	$0.262425\pi$
$252$	0	0
$253$	2227.00	0.553400
$254$	1080.00	0.266792
$255$	0	0
$256$	256.000	0.0625000
$257$	−7653.00	−1.85751	−0.928757	−	0.370689i	$-0.879121\pi$
$257$	−7653.00	−1.85751	−0.928757	+	0.370689i	$0.879121\pi$
$258$	0	0
$259$	2317.00	0.555874
$260$	−336.000	−0.0801455
$261$	0	0
$262$	3772.00	0.889446
$263$	−6975.00	−1.63535	−0.817675	−	0.575680i	$-0.804737\pi$
$263$	−6975.00	−1.63535	−0.817675	+	0.575680i	$0.804737\pi$
$264$	0	0
$265$	2772.00	0.642576
$266$	602.000	0.138763
$267$	0	0
$268$	1552.00	0.353744
$269$	−1435.00	−0.325255	−0.162627	−	0.986688i	$-0.551997\pi$
$269$	−1435.00	−0.325255	−0.162627	+	0.986688i	$0.551997\pi$
$270$	0	0
$271$	−3696.00	−0.828472	−0.414236	−	0.910169i	$-0.635951\pi$
$271$	−3696.00	−0.828472	−0.414236	+	0.910169i	$0.635951\pi$
$272$	−608.000	−0.135535
$273$	0	0
$274$	−3368.00	−0.742585
$275$	1292.00	0.283311
$276$	0	0
$277$	−3859.00	−0.837057	−0.418529	−	0.908204i	$-0.637454\pi$
$277$	−3859.00	−0.837057	−0.418529	+	0.908204i	$0.637454\pi$
$278$	728.000	0.157059
$279$	0	0
$280$	392.000	0.0836660
$281$	−1622.00	−0.344343	−0.172172	−	0.985067i	$-0.555078\pi$
$281$	−1622.00	−0.344343	−0.172172	+	0.985067i	$0.555078\pi$
$282$	0	0
$283$	8572.00	1.80054	0.900269	−	0.435333i	$-0.143369\pi$
$283$	8572.00	1.80054	0.900269	+	0.435333i	$0.143369\pi$
$284$	−2204.00	−0.460505
$285$	0	0
$286$	−408.000	−0.0843551
$287$	−777.000	−0.159808
$288$	0	0
$289$	−3469.00	−0.706086
$290$	2240.00	0.453577
$291$	0	0
$292$	1096.00	0.219653
$293$	−2454.00	−0.489298	−0.244649	−	0.969612i	$-0.578673\pi$
$293$	−2454.00	−0.489298	−0.244649	+	0.969612i	$0.578673\pi$
$294$	0	0
$295$	1498.00	0.295651
$296$	−2648.00	−0.519973
$297$	0	0
$298$	4668.00	0.907416
$299$	−1572.00	−0.304051
$300$	0	0
$301$	−1610.00	−0.308302
$302$	3020.00	0.575435
$303$	0	0
$304$	−688.000	−0.129801
$305$	−5376.00	−1.00927
$306$	0	0
$307$	−263.000	−0.0488932	−0.0244466	−	0.999701i	$-0.507782\pi$
$307$	−263.000	−0.0488932	−0.0244466	+	0.999701i	$0.507782\pi$
$308$	476.000	0.0880604
$309$	0	0
$310$	−630.000	−0.115425
$311$	2372.00	0.432488	0.216244	−	0.976339i	$-0.430619\pi$
$311$	2372.00	0.432488	0.216244	+	0.976339i	$0.430619\pi$
$312$	0	0
$313$	124.000	0.0223926	0.0111963	−	0.999937i	$-0.496436\pi$
$313$	124.000	0.0223926	0.0111963	+	0.999937i	$0.496436\pi$
$314$	3016.00	0.542047
$315$	0	0
$316$	1560.00	0.277712
$317$	−4950.00	−0.877034	−0.438517	−	0.898723i	$-0.644496\pi$
$317$	−4950.00	−0.877034	−0.438517	+	0.898723i	$0.644496\pi$
$318$	0	0
$319$	2720.00	0.477401
$320$	−448.000	−0.0782624
$321$	0	0
$322$	1834.00	0.317406
$323$	1634.00	0.281480
$324$	0	0
$325$	−912.000	−0.155657
$326$	−5024.00	−0.853539
$327$	0	0
$328$	888.000	0.149487
$329$	1974.00	0.330791
$330$	0	0
$331$	−1108.00	−0.183992	−0.0919958	−	0.995759i	$-0.529325\pi$
$331$	−1108.00	−0.183992	−0.0919958	+	0.995759i	$0.529325\pi$
$332$	−1760.00	−0.290941
$333$	0	0
$334$	7372.00	1.20772
$335$	−2716.00	−0.442958
$336$	0	0
$337$	−6669.00	−1.07799	−0.538996	−	0.842308i	$-0.681196\pi$
$337$	−6669.00	−1.07799	−0.538996	+	0.842308i	$0.681196\pi$
$338$	−4106.00	−0.660760
$339$	0	0
$340$	1064.00	0.169716
$341$	−765.000	−0.121487
$342$	0	0
$343$	−343.000	−0.0539949
$344$	1840.00	0.288390
$345$	0	0
$346$	−3794.00	−0.589499
$347$	−11601.0	−1.79474	−0.897369	−	0.441280i	$-0.854524\pi$
$347$	−11601.0	−1.79474	−0.897369	+	0.441280i	$0.854524\pi$
$348$	0	0
$349$	−7610.00	−1.16720	−0.583602	−	0.812040i	$-0.698357\pi$
$349$	−7610.00	−1.16720	−0.583602	+	0.812040i	$0.698357\pi$
$350$	1064.00	0.162495
$351$	0	0
$352$	−544.000	−0.0823730
$353$	9237.00	1.39274	0.696368	−	0.717685i	$-0.254798\pi$
$353$	9237.00	1.39274	0.696368	+	0.717685i	$0.254798\pi$
$354$	0	0
$355$	3857.00	0.576643
$356$	−420.000	−0.0625280
$357$	0	0
$358$	−2352.00	−0.347226
$359$	−8188.00	−1.20375	−0.601875	−	0.798590i	$-0.705579\pi$
$359$	−8188.00	−1.20375	−0.601875	+	0.798590i	$0.705579\pi$
$360$	0	0
$361$	−5010.00	−0.730427
$362$	−9036.00	−1.31194
$363$	0	0
$364$	−336.000	−0.0483824
$365$	−1918.00	−0.275049
$366$	0	0
$367$	−5719.00	−0.813432	−0.406716	−	0.913555i	$-0.633326\pi$
$367$	−5719.00	−0.813432	−0.406716	+	0.913555i	$0.633326\pi$
$368$	−2096.00	−0.296906
$369$	0	0
$370$	4634.00	0.651109
$371$	2772.00	0.387911
$372$	0	0
$373$	−9881.00	−1.37163	−0.685816	−	0.727775i	$-0.740555\pi$
$373$	−9881.00	−1.37163	−0.685816	+	0.727775i	$0.740555\pi$
$374$	1292.00	0.178630
$375$	0	0
$376$	−2256.00	−0.309426
$377$	−1920.00	−0.262295
$378$	0	0
$379$	−1142.00	−0.154777	−0.0773887	−	0.997001i	$-0.524658\pi$
$379$	−1142.00	−0.154777	−0.0773887	+	0.997001i	$0.524658\pi$
$380$	1204.00	0.162537
$381$	0	0
$382$	−3570.00	−0.478160
$383$	−12710.0	−1.69569	−0.847847	−	0.530241i	$-0.822101\pi$
$383$	−12710.0	−1.69569	−0.847847	+	0.530241i	$0.822101\pi$
$384$	0	0
$385$	−833.000	−0.110269
$386$	5116.00	0.674605
$387$	0	0
$388$	1216.00	0.159106
$389$	−4852.00	−0.632407	−0.316203	−	0.948691i	$-0.602408\pi$
$389$	−4852.00	−0.632407	−0.316203	+	0.948691i	$0.602408\pi$
$390$	0	0
$391$	4978.00	0.643857
$392$	392.000	0.0505076
$393$	0	0
$394$	8732.00	1.11653
$395$	−2730.00	−0.347750
$396$	0	0
$397$	−13494.0	−1.70591	−0.852953	−	0.521988i	$-0.825190\pi$
$397$	−13494.0	−1.70591	−0.852953	+	0.521988i	$0.825190\pi$
$398$	3154.00	0.397225
$399$	0	0
$400$	−1216.00	−0.152000
$401$	−1416.00	−0.176338	−0.0881692	−	0.996106i	$-0.528102\pi$
$401$	−1416.00	−0.176338	−0.0881692	+	0.996106i	$0.528102\pi$
$402$	0	0
$403$	540.000	0.0667477
$404$	−1144.00	−0.140882
$405$	0	0
$406$	2240.00	0.273816
$407$	5627.00	0.685307
$408$	0	0
$409$	−4798.00	−0.580063	−0.290032	−	0.957017i	$-0.593666\pi$
$409$	−4798.00	−0.580063	−0.290032	+	0.957017i	$0.593666\pi$
$410$	−1554.00	−0.187187
$411$	0	0
$412$	2900.00	0.346779
$413$	1498.00	0.178479
$414$	0	0
$415$	3080.00	0.364316
$416$	384.000	0.0452576
$417$	0	0
$418$	1462.00	0.171074
$419$	7962.00	0.928327	0.464164	−	0.885749i	$-0.346355\pi$
$419$	7962.00	0.928327	0.464164	+	0.885749i	$0.346355\pi$
$420$	0	0
$421$	9597.00	1.11100	0.555498	−	0.831518i	$-0.312528\pi$
$421$	9597.00	1.11100	0.555498	+	0.831518i	$0.312528\pi$
$422$	3628.00	0.418503
$423$	0	0
$424$	−3168.00	−0.362858
$425$	2888.00	0.329620
$426$	0	0
$427$	−5376.00	−0.609281
$428$	−3504.00	−0.395730
$429$	0	0
$430$	−3220.00	−0.361121
$431$	−11877.0	−1.32737	−0.663683	−	0.748014i	$-0.731008\pi$
$431$	−11877.0	−1.32737	−0.663683	+	0.748014i	$0.731008\pi$
$432$	0	0
$433$	−14032.0	−1.55736	−0.778678	−	0.627424i	$-0.784109\pi$
$433$	−14032.0	−1.55736	−0.778678	+	0.627424i	$0.784109\pi$
$434$	−630.000	−0.0696796
$435$	0	0
$436$	−1124.00	−0.123463
$437$	5633.00	0.616620
$438$	0	0
$439$	5112.00	0.555769	0.277884	−	0.960615i	$-0.410367\pi$
$439$	5112.00	0.555769	0.277884	+	0.960615i	$0.410367\pi$
$440$	952.000	0.103147
$441$	0	0
$442$	−912.000	−0.0981435
$443$	9169.00	0.983369	0.491684	−	0.870774i	$-0.336381\pi$
$443$	9169.00	0.983369	0.491684	+	0.870774i	$0.336381\pi$
$444$	0	0
$445$	735.000	0.0782974
$446$	−5234.00	−0.555689
$447$	0	0
$448$	−448.000	−0.0472456
$449$	7682.00	0.807430	0.403715	−	0.914885i	$-0.367719\pi$
$449$	7682.00	0.807430	0.403715	+	0.914885i	$0.367719\pi$
$450$	0	0
$451$	−1887.00	−0.197019
$452$	4376.00	0.455376
$453$	0	0
$454$	7524.00	0.777795
$455$	588.000	0.0605843
$456$	0	0
$457$	3973.00	0.406672	0.203336	−	0.979109i	$-0.434822\pi$
$457$	3973.00	0.406672	0.203336	+	0.979109i	$0.434822\pi$
$458$	−6952.00	−0.709270
$459$	0	0
$460$	3668.00	0.371786
$461$	15767.0	1.59293	0.796467	−	0.604682i	$-0.206700\pi$
$461$	15767.0	1.59293	0.796467	+	0.604682i	$0.206700\pi$
$462$	0	0
$463$	15478.0	1.55361	0.776807	−	0.629738i	$-0.216838\pi$
$463$	15478.0	1.55361	0.776807	+	0.629738i	$0.216838\pi$
$464$	−2560.00	−0.256132
$465$	0	0
$466$	−4996.00	−0.496642
$467$	13598.0	1.34741	0.673705	−	0.739000i	$-0.264702\pi$
$467$	13598.0	1.34741	0.673705	+	0.739000i	$0.264702\pi$
$468$	0	0
$469$	−2716.00	−0.267406
$470$	3948.00	0.387463
$471$	0	0
$472$	−1712.00	−0.166952
$473$	−3910.00	−0.380089
$474$	0	0
$475$	3268.00	0.315676
$476$	1064.00	0.102455
$477$	0	0
$478$	7680.00	0.734885
$479$	11824.0	1.12788	0.563938	−	0.825817i	$-0.309286\pi$
$479$	11824.0	1.12788	0.563938	+	0.825817i	$0.309286\pi$
$480$	0	0
$481$	−3972.00	−0.376523
$482$	−3308.00	−0.312604
$483$	0	0
$484$	−4168.00	−0.391435
$485$	−2128.00	−0.199232
$486$	0	0
$487$	−9694.00	−0.902006	−0.451003	−	0.892522i	$-0.648934\pi$
$487$	−9694.00	−0.902006	−0.451003	+	0.892522i	$0.648934\pi$
$488$	6144.00	0.569930
$489$	0	0
$490$	−686.000	−0.0632456
$491$	−3309.00	−0.304141	−0.152070	−	0.988370i	$-0.548594\pi$
$491$	−3309.00	−0.304141	−0.152070	+	0.988370i	$0.548594\pi$
$492$	0	0
$493$	6080.00	0.555435
$494$	−1032.00	−0.0939917
$495$	0	0
$496$	720.000	0.0651793
$497$	3857.00	0.348109
$498$	0	0
$499$	7762.00	0.696342	0.348171	−	0.937431i	$-0.386803\pi$
$499$	7762.00	0.696342	0.348171	+	0.937431i	$0.386803\pi$
$500$	5628.00	0.503384
$501$	0	0
$502$	10800.0	0.960214
$503$	11040.0	0.978627	0.489313	−	0.872108i	$-0.337247\pi$
$503$	11040.0	0.978627	0.489313	+	0.872108i	$0.337247\pi$
$504$	0	0
$505$	2002.00	0.176412
$506$	4454.00	0.391313
$507$	0	0
$508$	2160.00	0.188651
$509$	−16230.0	−1.41332	−0.706662	−	0.707551i	$-0.749800\pi$
$509$	−16230.0	−1.41332	−0.706662	+	0.707551i	$0.749800\pi$
$510$	0	0
$511$	−1918.00	−0.166042
$512$	512.000	0.0441942
$513$	0	0
$514$	−15306.0	−1.31346
$515$	−5075.00	−0.434235
$516$	0	0
$517$	4794.00	0.407814
$518$	4634.00	0.393062
$519$	0	0
$520$	−672.000	−0.0566714
$521$	20265.0	1.70408	0.852040	−	0.523477i	$-0.175365\pi$
$521$	20265.0	1.70408	0.852040	+	0.523477i	$0.175365\pi$
$522$	0	0
$523$	4673.00	0.390700	0.195350	−	0.980734i	$-0.437416\pi$
$523$	4673.00	0.390700	0.195350	+	0.980734i	$0.437416\pi$
$524$	7544.00	0.628934
$525$	0	0
$526$	−13950.0	−1.15637
$527$	−1710.00	−0.141345
$528$	0	0
$529$	4994.00	0.410455
$530$	5544.00	0.454370
$531$	0	0
$532$	1204.00	0.0981203
$533$	1332.00	0.108246
$534$	0	0
$535$	6132.00	0.495532
$536$	3104.00	0.250135
$537$	0	0
$538$	−2870.00	−0.229990
$539$	−833.000	−0.0665674
$540$	0	0
$541$	−2547.00	−0.202411	−0.101205	−	0.994866i	$-0.532270\pi$
$541$	−2547.00	−0.202411	−0.101205	+	0.994866i	$0.532270\pi$
$542$	−7392.00	−0.585818
$543$	0	0
$544$	−1216.00	−0.0958374
$545$	1967.00	0.154600
$546$	0	0
$547$	24876.0	1.94446	0.972231	−	0.234022i	$-0.0751889\pi$
$547$	24876.0	1.94446	0.972231	+	0.234022i	$0.0751889\pi$
$548$	−6736.00	−0.525087
$549$	0	0
$550$	2584.00	0.200331
$551$	6880.00	0.531938
$552$	0	0
$553$	−2730.00	−0.209930
$554$	−7718.00	−0.591889
$555$	0	0
$556$	1456.00	0.111058
$557$	−14266.0	−1.08522	−0.542612	−	0.839983i	$-0.682565\pi$
$557$	−14266.0	−1.08522	−0.542612	+	0.839983i	$0.682565\pi$
$558$	0	0
$559$	2760.00	0.208829
$560$	784.000	0.0591608
$561$	0	0
$562$	−3244.00	−0.243487
$563$	−5768.00	−0.431780	−0.215890	−	0.976418i	$-0.569265\pi$
$563$	−5768.00	−0.431780	−0.215890	+	0.976418i	$0.569265\pi$
$564$	0	0
$565$	−7658.00	−0.570220
$566$	17144.0	1.27317
$567$	0	0
$568$	−4408.00	−0.325626
$569$	1374.00	0.101232	0.0506161	−	0.998718i	$-0.483882\pi$
$569$	1374.00	0.101232	0.0506161	+	0.998718i	$0.483882\pi$
$570$	0	0
$571$	13614.0	0.997773	0.498886	−	0.866667i	$-0.333743\pi$
$571$	13614.0	0.997773	0.498886	+	0.866667i	$0.333743\pi$
$572$	−816.000	−0.0596480
$573$	0	0
$574$	−1554.00	−0.113001
$575$	9956.00	0.722076
$576$	0	0
$577$	−1574.00	−0.113564	−0.0567820	−	0.998387i	$-0.518084\pi$
$577$	−1574.00	−0.113564	−0.0567820	+	0.998387i	$0.518084\pi$
$578$	−6938.00	−0.499278
$579$	0	0
$580$	4480.00	0.320727
$581$	3080.00	0.219931
$582$	0	0
$583$	6732.00	0.478235
$584$	2192.00	0.155318
$585$	0	0
$586$	−4908.00	−0.345986
$587$	6034.00	0.424276	0.212138	−	0.977240i	$-0.431957\pi$
$587$	6034.00	0.424276	0.212138	+	0.977240i	$0.431957\pi$
$588$	0	0
$589$	−1935.00	−0.135366
$590$	2996.00	0.209057
$591$	0	0
$592$	−5296.00	−0.367676
$593$	9441.00	0.653787	0.326893	−	0.945061i	$-0.393998\pi$
$593$	9441.00	0.653787	0.326893	+	0.945061i	$0.393998\pi$
$594$	0	0
$595$	−1862.00	−0.128293
$596$	9336.00	0.641640
$597$	0	0
$598$	−3144.00	−0.214996
$599$	−5457.00	−0.372232	−0.186116	−	0.982528i	$-0.559590\pi$
$599$	−5457.00	−0.372232	−0.186116	+	0.982528i	$0.559590\pi$
$600$	0	0
$601$	−10348.0	−0.702336	−0.351168	−	0.936313i	$-0.614215\pi$
$601$	−10348.0	−0.702336	−0.351168	+	0.936313i	$0.614215\pi$
$602$	−3220.00	−0.218002
$603$	0	0
$604$	6040.00	0.406894
$605$	7294.00	0.490154
$606$	0	0
$607$	−16692.0	−1.11616	−0.558079	−	0.829788i	$-0.688461\pi$
$607$	−16692.0	−1.11616	−0.558079	+	0.829788i	$0.688461\pi$
$608$	−1376.00	−0.0917832
$609$	0	0
$610$	−10752.0	−0.713665
$611$	−3384.00	−0.224062
$612$	0	0
$613$	6393.00	0.421225	0.210612	−	0.977570i	$-0.432454\pi$
$613$	6393.00	0.421225	0.210612	+	0.977570i	$0.432454\pi$
$614$	−526.000	−0.0345727
$615$	0	0
$616$	952.000	0.0622681
$617$	−19766.0	−1.28971	−0.644853	−	0.764306i	$-0.723082\pi$
$617$	−19766.0	−1.28971	−0.644853	+	0.764306i	$0.723082\pi$
$618$	0	0
$619$	3295.00	0.213954	0.106977	−	0.994262i	$-0.465883\pi$
$619$	3295.00	0.213954	0.106977	+	0.994262i	$0.465883\pi$
$620$	−1260.00	−0.0816174
$621$	0	0
$622$	4744.00	0.305815
$623$	735.000	0.0472667
$624$	0	0
$625$	−349.000	−0.0223360
$626$	248.000	0.0158340
$627$	0	0
$628$	6032.00	0.383285
$629$	12578.0	0.797325
$630$	0	0
$631$	10598.0	0.668621	0.334310	−	0.942463i	$-0.391497\pi$
$631$	10598.0	0.668621	0.334310	+	0.942463i	$0.391497\pi$
$632$	3120.00	0.196372
$633$	0	0
$634$	−9900.00	−0.620157
$635$	−3780.00	−0.236228
$636$	0	0
$637$	588.000	0.0365736
$638$	5440.00	0.337573
$639$	0	0
$640$	−896.000	−0.0553399
$641$	18394.0	1.13342	0.566708	−	0.823919i	$-0.308217\pi$
$641$	18394.0	1.13342	0.566708	+	0.823919i	$0.308217\pi$
$642$	0	0
$643$	29585.0	1.81449	0.907246	−	0.420600i	$-0.138180\pi$
$643$	29585.0	1.81449	0.907246	+	0.420600i	$0.138180\pi$
$644$	3668.00	0.224440
$645$	0	0
$646$	3268.00	0.199037
$647$	−5118.00	−0.310988	−0.155494	−	0.987837i	$-0.549697\pi$
$647$	−5118.00	−0.310988	−0.155494	+	0.987837i	$0.549697\pi$
$648$	0	0
$649$	3638.00	0.220037
$650$	−1824.00	−0.110066
$651$	0	0
$652$	−10048.0	−0.603543
$653$	23262.0	1.39405	0.697023	−	0.717048i	$-0.254507\pi$
$653$	23262.0	1.39405	0.697023	+	0.717048i	$0.254507\pi$
$654$	0	0
$655$	−13202.0	−0.787549
$656$	1776.00	0.105703
$657$	0	0
$658$	3948.00	0.233904
$659$	−7301.00	−0.431573	−0.215786	−	0.976441i	$-0.569232\pi$
$659$	−7301.00	−0.431573	−0.215786	+	0.976441i	$0.569232\pi$
$660$	0	0
$661$	−13520.0	−0.795563	−0.397781	−	0.917480i	$-0.630220\pi$
$661$	−13520.0	−0.795563	−0.397781	+	0.917480i	$0.630220\pi$
$662$	−2216.00	−0.130102
$663$	0	0
$664$	−3520.00	−0.205727
$665$	−2107.00	−0.122866
$666$	0	0
$667$	20960.0	1.21675
$668$	14744.0	0.853986
$669$	0	0
$670$	−5432.00	−0.313219
$671$	−13056.0	−0.751149
$672$	0	0
$673$	24578.0	1.40774	0.703872	−	0.710326i	$-0.251453\pi$
$673$	24578.0	1.40774	0.703872	+	0.710326i	$0.251453\pi$
$674$	−13338.0	−0.762256
$675$	0	0
$676$	−8212.00	−0.467228
$677$	−25245.0	−1.43315	−0.716576	−	0.697509i	$-0.754292\pi$
$677$	−25245.0	−1.43315	−0.716576	+	0.697509i	$0.754292\pi$
$678$	0	0
$679$	−2128.00	−0.120273
$680$	2128.00	0.120007
$681$	0	0
$682$	−1530.00	−0.0859043
$683$	−7755.00	−0.434461	−0.217230	−	0.976120i	$-0.569702\pi$
$683$	−7755.00	−0.434461	−0.217230	+	0.976120i	$0.569702\pi$
$684$	0	0
$685$	11788.0	0.657513
$686$	−686.000	−0.0381802
$687$	0	0
$688$	3680.00	0.203923
$689$	−4752.00	−0.262753
$690$	0	0
$691$	−24508.0	−1.34924	−0.674622	−	0.738163i	$-0.735694\pi$
$691$	−24508.0	−1.34924	−0.674622	+	0.738163i	$0.735694\pi$
$692$	−7588.00	−0.416839
$693$	0	0
$694$	−23202.0	−1.26907
$695$	−2548.00	−0.139066
$696$	0	0
$697$	−4218.00	−0.229223
$698$	−15220.0	−0.825337
$699$	0	0
$700$	2128.00	0.114901
$701$	7400.00	0.398708	0.199354	−	0.979928i	$-0.436116\pi$
$701$	7400.00	0.398708	0.199354	+	0.979928i	$0.436116\pi$
$702$	0	0
$703$	14233.0	0.763596
$704$	−1088.00	−0.0582465
$705$	0	0
$706$	18474.0	0.984813
$707$	2002.00	0.106496
$708$	0	0
$709$	24695.0	1.30810	0.654048	−	0.756453i	$-0.273069\pi$
$709$	24695.0	1.30810	0.654048	+	0.756453i	$0.273069\pi$
$710$	7714.00	0.407748
$711$	0	0
$712$	−840.000	−0.0442139
$713$	−5895.00	−0.309635
$714$	0	0
$715$	1428.00	0.0746911
$716$	−4704.00	−0.245526
$717$	0	0
$718$	−16376.0	−0.851180
$719$	28410.0	1.47359	0.736797	−	0.676114i	$-0.236337\pi$
$719$	28410.0	1.47359	0.736797	+	0.676114i	$0.236337\pi$
$720$	0	0
$721$	−5075.00	−0.262140
$722$	−10020.0	−0.516490
$723$	0	0
$724$	−18072.0	−0.927680
$725$	12160.0	0.622912
$726$	0	0
$727$	−37256.0	−1.90062	−0.950308	−	0.311310i	$-0.899232\pi$
$727$	−37256.0	−1.90062	−0.950308	+	0.311310i	$0.899232\pi$
$728$	−672.000	−0.0342115
$729$	0	0
$730$	−3836.00	−0.194489
$731$	−8740.00	−0.442217
$732$	0	0
$733$	−14694.0	−0.740430	−0.370215	−	0.928946i	$-0.620716\pi$
$733$	−14694.0	−0.740430	−0.370215	+	0.928946i	$0.620716\pi$
$734$	−11438.0	−0.575183
$735$	0	0
$736$	−4192.00	−0.209945
$737$	−6596.00	−0.329670
$738$	0	0
$739$	4494.00	0.223700	0.111850	−	0.993725i	$-0.464322\pi$
$739$	4494.00	0.223700	0.111850	+	0.993725i	$0.464322\pi$
$740$	9268.00	0.460403
$741$	0	0
$742$	5544.00	0.274295
$743$	633.000	0.0312551	0.0156275	−	0.999878i	$-0.495025\pi$
$743$	633.000	0.0312551	0.0156275	+	0.999878i	$0.495025\pi$
$744$	0	0
$745$	−16338.0	−0.803460
$746$	−19762.0	−0.969890
$747$	0	0
$748$	2584.00	0.126311
$749$	6132.00	0.299143
$750$	0	0
$751$	−17742.0	−0.862070	−0.431035	−	0.902335i	$-0.641852\pi$
$751$	−17742.0	−0.862070	−0.431035	+	0.902335i	$0.641852\pi$
$752$	−4512.00	−0.218797
$753$	0	0
$754$	−3840.00	−0.185470
$755$	−10570.0	−0.509512
$756$	0	0
$757$	−7734.00	−0.371330	−0.185665	−	0.982613i	$-0.559444\pi$
$757$	−7734.00	−0.371330	−0.185665	+	0.982613i	$0.559444\pi$
$758$	−2284.00	−0.109444
$759$	0	0
$760$	2408.00	0.114931
$761$	−17318.0	−0.824937	−0.412468	−	0.910972i	$-0.635333\pi$
$761$	−17318.0	−0.824937	−0.412468	+	0.910972i	$0.635333\pi$
$762$	0	0
$763$	1967.00	0.0933292
$764$	−7140.00	−0.338110
$765$	0	0
$766$	−25420.0	−1.19904
$767$	−2568.00	−0.120893
$768$	0	0
$769$	−16796.0	−0.787619	−0.393810	−	0.919192i	$-0.628843\pi$
$769$	−16796.0	−0.787619	−0.393810	+	0.919192i	$0.628843\pi$
$770$	−1666.00	−0.0779720
$771$	0	0
$772$	10232.0	0.477018
$773$	−38677.0	−1.79963	−0.899816	−	0.436270i	$-0.856299\pi$
$773$	−38677.0	−1.79963	−0.899816	+	0.436270i	$0.856299\pi$
$774$	0	0
$775$	−3420.00	−0.158516
$776$	2432.00	0.112505
$777$	0	0
$778$	−9704.00	−0.447179
$779$	−4773.00	−0.219526
$780$	0	0
$781$	9367.00	0.429165
$782$	9956.00	0.455276
$783$	0	0
$784$	784.000	0.0357143
$785$	−10556.0	−0.479949
$786$	0	0
$787$	40896.0	1.85233	0.926166	−	0.377117i	$-0.123084\pi$
$787$	40896.0	1.85233	0.926166	+	0.377117i	$0.123084\pi$
$788$	17464.0	0.789504
$789$	0	0
$790$	−5460.00	−0.245896
$791$	−7658.00	−0.344232
$792$	0	0
$793$	9216.00	0.412698
$794$	−26988.0	−1.20626
$795$	0	0
$796$	6308.00	0.280881
$797$	2151.00	0.0955989	0.0477995	−	0.998857i	$-0.484779\pi$
$797$	2151.00	0.0955989	0.0477995	+	0.998857i	$0.484779\pi$
$798$	0	0
$799$	10716.0	0.474474
$800$	−2432.00	−0.107480
$801$	0	0
$802$	−2832.00	−0.124690
$803$	−4658.00	−0.204704
$804$	0	0
$805$	−6419.00	−0.281044
$806$	1080.00	0.0471977
$807$	0	0
$808$	−2288.00	−0.0996183
$809$	−41656.0	−1.81032	−0.905159	−	0.425074i	$-0.860248\pi$
$809$	−41656.0	−1.81032	−0.905159	+	0.425074i	$0.860248\pi$
$810$	0	0
$811$	−8045.00	−0.348333	−0.174167	−	0.984716i	$-0.555723\pi$
$811$	−8045.00	−0.348333	−0.174167	+	0.984716i	$0.555723\pi$
$812$	4480.00	0.193617
$813$	0	0
$814$	11254.0	0.484585
$815$	17584.0	0.755755
$816$	0	0
$817$	−9890.00	−0.423510
$818$	−9596.00	−0.410167
$819$	0	0
$820$	−3108.00	−0.132361
$821$	−9350.00	−0.397463	−0.198732	−	0.980054i	$-0.563682\pi$
$821$	−9350.00	−0.397463	−0.198732	+	0.980054i	$0.563682\pi$
$822$	0	0
$823$	15598.0	0.660647	0.330323	−	0.943868i	$-0.392842\pi$
$823$	15598.0	0.660647	0.330323	+	0.943868i	$0.392842\pi$
$824$	5800.00	0.245209
$825$	0	0
$826$	2996.00	0.126204
$827$	957.000	0.0402396	0.0201198	−	0.999798i	$-0.493595\pi$
$827$	957.000	0.0402396	0.0201198	+	0.999798i	$0.493595\pi$
$828$	0	0
$829$	19054.0	0.798278	0.399139	−	0.916890i	$-0.369309\pi$
$829$	19054.0	0.798278	0.399139	+	0.916890i	$0.369309\pi$
$830$	6160.00	0.257611
$831$	0	0
$832$	768.000	0.0320019
$833$	−1862.00	−0.0774484
$834$	0	0
$835$	−25802.0	−1.06936
$836$	2924.00	0.120967
$837$	0	0
$838$	15924.0	0.656427
$839$	−25400.0	−1.04518	−0.522590	−	0.852584i	$-0.675034\pi$
$839$	−25400.0	−1.04518	−0.522590	+	0.852584i	$0.675034\pi$
$840$	0	0
$841$	1211.00	0.0496535
$842$	19194.0	0.785593
$843$	0	0
$844$	7256.00	0.295926
$845$	14371.0	0.585062
$846$	0	0
$847$	7294.00	0.295897
$848$	−6336.00	−0.256579
$849$	0	0
$850$	5776.00	0.233077
$851$	43361.0	1.74665
$852$	0	0
$853$	416.000	0.0166982	0.00834910	−	0.999965i	$-0.497342\pi$
$853$	416.000	0.0166982	0.00834910	+	0.999965i	$0.497342\pi$
$854$	−10752.0	−0.430827
$855$	0	0
$856$	−7008.00	−0.279823
$857$	−36261.0	−1.44534	−0.722668	−	0.691196i	$-0.757084\pi$
$857$	−36261.0	−1.44534	−0.722668	+	0.691196i	$0.757084\pi$
$858$	0	0
$859$	−3979.00	−0.158046	−0.0790231	−	0.996873i	$-0.525180\pi$
$859$	−3979.00	−0.158046	−0.0790231	+	0.996873i	$0.525180\pi$
$860$	−6440.00	−0.255351
$861$	0	0
$862$	−23754.0	−0.938590
$863$	−23032.0	−0.908480	−0.454240	−	0.890879i	$-0.650089\pi$
$863$	−23032.0	−0.908480	−0.454240	+	0.890879i	$0.650089\pi$
$864$	0	0
$865$	13279.0	0.521965
$866$	−28064.0	−1.10122
$867$	0	0
$868$	−1260.00	−0.0492710
$869$	−6630.00	−0.258812
$870$	0	0
$871$	4656.00	0.181128
$872$	−2248.00	−0.0873015
$873$	0	0
$874$	11266.0	0.436016
$875$	−9849.00	−0.380522
$876$	0	0
$877$	−15818.0	−0.609049	−0.304524	−	0.952505i	$-0.598498\pi$
$877$	−15818.0	−0.609049	−0.304524	+	0.952505i	$0.598498\pi$
$878$	10224.0	0.392988
$879$	0	0
$880$	1904.00	0.0729362
$881$	−8871.00	−0.339241	−0.169621	−	0.985509i	$-0.554254\pi$
$881$	−8871.00	−0.339241	−0.169621	+	0.985509i	$0.554254\pi$
$882$	0	0
$883$	21968.0	0.837239	0.418620	−	0.908162i	$-0.362514\pi$
$883$	21968.0	0.837239	0.418620	+	0.908162i	$0.362514\pi$
$884$	−1824.00	−0.0693979
$885$	0	0
$886$	18338.0	0.695347
$887$	−41208.0	−1.55990	−0.779949	−	0.625843i	$-0.784755\pi$
$887$	−41208.0	−1.55990	−0.779949	+	0.625843i	$0.784755\pi$
$888$	0	0
$889$	−3780.00	−0.142606
$890$	1470.00	0.0553646
$891$	0	0
$892$	−10468.0	−0.392931
$893$	12126.0	0.454402
$894$	0	0
$895$	8232.00	0.307447
$896$	−896.000	−0.0334077
$897$	0	0
$898$	15364.0	0.570939
$899$	−7200.00	−0.267112
$900$	0	0
$901$	15048.0	0.556406
$902$	−3774.00	−0.139313
$903$	0	0
$904$	8752.00	0.321999
$905$	31626.0	1.16164
$906$	0	0
$907$	−78.0000	−0.00285551	−0.00142775	−	0.999999i	$-0.500454\pi$
$907$	−78.0000	−0.00285551	−0.00142775	+	0.999999i	$0.500454\pi$
$908$	15048.0	0.549984
$909$	0	0
$910$	1176.00	0.0428396
$911$	44400.0	1.61475	0.807375	−	0.590038i	$-0.200887\pi$
$911$	44400.0	1.61475	0.807375	+	0.590038i	$0.200887\pi$
$912$	0	0
$913$	7480.00	0.271141
$914$	7946.00	0.287561
$915$	0	0
$916$	−13904.0	−0.501530
$917$	−13202.0	−0.475429
$918$	0	0
$919$	−5108.00	−0.183349	−0.0916743	−	0.995789i	$-0.529222\pi$
$919$	−5108.00	−0.183349	−0.0916743	+	0.995789i	$0.529222\pi$
$920$	7336.00	0.262892
$921$	0	0
$922$	31534.0	1.12637
$923$	−6612.00	−0.235793
$924$	0	0
$925$	25156.0	0.894188
$926$	30956.0	1.09857
$927$	0	0
$928$	−5120.00	−0.181112
$929$	534.000	0.0188590	0.00942948	−	0.999956i	$-0.496998\pi$
$929$	534.000	0.0188590	0.00942948	+	0.999956i	$0.496998\pi$
$930$	0	0
$931$	−2107.00	−0.0741720
$932$	−9992.00	−0.351179
$933$	0	0
$934$	27196.0	0.952763
$935$	−4522.00	−0.158166
$936$	0	0
$937$	740.000	0.0258002	0.0129001	−	0.999917i	$-0.495894\pi$
$937$	740.000	0.0258002	0.0129001	+	0.999917i	$0.495894\pi$
$938$	−5432.00	−0.189084
$939$	0	0
$940$	7896.00	0.273978
$941$	38645.0	1.33878	0.669389	−	0.742912i	$-0.266556\pi$
$941$	38645.0	1.33878	0.669389	+	0.742912i	$0.266556\pi$
$942$	0	0
$943$	−14541.0	−0.502142
$944$	−3424.00	−0.118053
$945$	0	0
$946$	−7820.00	−0.268763
$947$	−45067.0	−1.54644	−0.773221	−	0.634137i	$-0.781356\pi$
$947$	−45067.0	−1.54644	−0.773221	+	0.634137i	$0.781356\pi$
$948$	0	0
$949$	3288.00	0.112469
$950$	6536.00	0.223217
$951$	0	0
$952$	2128.00	0.0724463
$953$	11080.0	0.376617	0.188309	−	0.982110i	$-0.439699\pi$
$953$	11080.0	0.376617	0.188309	+	0.982110i	$0.439699\pi$
$954$	0	0
$955$	12495.0	0.423381
$956$	15360.0	0.519642
$957$	0	0
$958$	23648.0	0.797528
$959$	11788.0	0.396928
$960$	0	0
$961$	−27766.0	−0.932026
$962$	−7944.00	−0.266242
$963$	0	0
$964$	−6616.00	−0.221045
$965$	−17906.0	−0.597321
$966$	0	0
$967$	−14434.0	−0.480006	−0.240003	−	0.970772i	$-0.577148\pi$
$967$	−14434.0	−0.480006	−0.240003	+	0.970772i	$0.577148\pi$
$968$	−8336.00	−0.276786
$969$	0	0
$970$	−4256.00	−0.140878
$971$	59712.0	1.97348	0.986740	−	0.162308i	$-0.0518939\pi$
$971$	59712.0	1.97348	0.986740	+	0.162308i	$0.0518939\pi$
$972$	0	0
$973$	−2548.00	−0.0839518
$974$	−19388.0	−0.637815
$975$	0	0
$976$	12288.0	0.403001
$977$	49956.0	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$977$	49956.0	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$978$	0	0
$979$	1785.00	0.0582726
$980$	−1372.00	−0.0447214
$981$	0	0
$982$	−6618.00	−0.215060
$983$	−10782.0	−0.349840	−0.174920	−	0.984583i	$-0.555967\pi$
$983$	−10782.0	−0.349840	−0.174920	+	0.984583i	$0.555967\pi$
$984$	0	0
$985$	−30562.0	−0.988616
$986$	12160.0	0.392752
$987$	0	0
$988$	−2064.00	−0.0664621
$989$	−30130.0	−0.968734
$990$	0	0
$991$	−26776.0	−0.858292	−0.429146	−	0.903235i	$-0.641185\pi$
$991$	−26776.0	−0.858292	−0.429146	+	0.903235i	$0.641185\pi$
$992$	1440.00	0.0460888
$993$	0	0
$994$	7714.00	0.246150
$995$	−11039.0	−0.351718
$996$	0	0
$997$	−22210.0	−0.705514	−0.352757	−	0.935715i	$-0.614756\pi$
$997$	−22210.0	−0.705514	−0.352757	+	0.935715i	$0.614756\pi$
$998$	15524.0	0.492388
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.a.i.1.1	yes	1
3.2	odd	2		378.4.a.d.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.a.d.1.1	✓	1	3.2	odd	2
378.4.a.i.1.1	yes	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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