LMFDB - Embedded newform 378.4.a.c.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:	$0$
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)

$f(q)$

$=$

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

\(q-2.00000 q^{2} +4.00000 q^{4} +7.00000 q^{5} -7.00000 q^{7} -8.00000 q^{8} -14.0000 q^{10} -28.0000 q^{11} +30.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} +47.0000 q^{17} +164.000 q^{19} +28.0000 q^{20} +56.0000 q^{22} -94.0000 q^{23} -76.0000 q^{25} -60.0000 q^{26} -28.0000 q^{28} -200.000 q^{29} +162.000 q^{31} -32.0000 q^{32} -94.0000 q^{34} -49.0000 q^{35} +137.000 q^{37} -328.000 q^{38} -56.0000 q^{40} +141.000 q^{41} +293.000 q^{43} -112.000 q^{44} +188.000 q^{46} +471.000 q^{47} +49.0000 q^{49} +152.000 q^{50} +120.000 q^{52} -306.000 q^{53} -196.000 q^{55} +56.0000 q^{56} +400.000 q^{58} +331.000 q^{59} -204.000 q^{61} -324.000 q^{62} +64.0000 q^{64} +210.000 q^{65} +928.000 q^{67} +188.000 q^{68} +98.0000 q^{70} +740.000 q^{71} +706.000 q^{73} -274.000 q^{74} +656.000 q^{76} +196.000 q^{77} -195.000 q^{79} +112.000 q^{80} -282.000 q^{82} +485.000 q^{83} +329.000 q^{85} -586.000 q^{86} +224.000 q^{88} +114.000 q^{89} -210.000 q^{91} -376.000 q^{92} -942.000 q^{94} +1148.00 q^{95} -344.000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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	$-0.398649\pi$
$5$	7.00000	0.626099	0.313050	+	0.949737i	$0.398649\pi$
$6$	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
$11$	−28.0000	−0.767483	−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−28.0000	−0.767483	−0.383742	+	0.923440i	$0.625365\pi$
$12$	0	0
$13$	30.0000	0.640039	0.320019	−	0.947411i	$-0.396311\pi$
$13$	30.0000	0.640039	0.320019	+	0.947411i	$0.396311\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	47.0000	0.670540	0.335270	−	0.942122i	$-0.391173\pi$
$17$	47.0000	0.670540	0.335270	+	0.942122i	$0.391173\pi$
$18$	0	0
$19$	164.000	1.98022	0.990110	−	0.140293i	$-0.0448045\pi$
$19$	164.000	1.98022	0.990110	+	0.140293i	$0.0448045\pi$
$20$	28.0000	0.313050
$21$	0	0
$22$	56.0000	0.542693
$23$	−94.0000	−0.852189	−0.426095	−	0.904679i	$-0.640111\pi$
$23$	−94.0000	−0.852189	−0.426095	+	0.904679i	$0.640111\pi$
$24$	0	0
$25$	−76.0000	−0.608000
$26$	−60.0000	−0.452576
$27$	0	0
$28$	−28.0000	−0.188982
$29$	−200.000	−1.28066	−0.640329	−	0.768101i	$-0.721202\pi$
$29$	−200.000	−1.28066	−0.640329	+	0.768101i	$0.721202\pi$
$30$	0	0
$31$	162.000	0.938583	0.469291	−	0.883043i	$-0.344509\pi$
$31$	162.000	0.938583	0.469291	+	0.883043i	$0.344509\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−94.0000	−0.474143
$35$	−49.0000	−0.236643
$36$	0	0
$37$	137.000	0.608721	0.304360	−	0.952557i	$-0.401557\pi$
$37$	137.000	0.608721	0.304360	+	0.952557i	$0.401557\pi$
$38$	−328.000	−1.40023
$39$	0	0
$40$	−56.0000	−0.221359
$41$	141.000	0.537085	0.268543	−	0.963268i	$-0.413458\pi$
$41$	141.000	0.537085	0.268543	+	0.963268i	$0.413458\pi$
$42$	0	0
$43$	293.000	1.03912	0.519559	−	0.854435i	$-0.326096\pi$
$43$	293.000	1.03912	0.519559	+	0.854435i	$0.326096\pi$
$44$	−112.000	−0.383742
$45$	0	0
$46$	188.000	0.602589
$47$	471.000	1.46175	0.730877	−	0.682510i	$-0.239111\pi$
$47$	471.000	1.46175	0.730877	+	0.682510i	$0.239111\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	152.000	0.429921
$51$	0	0
$52$	120.000	0.320019
$53$	−306.000	−0.793063	−0.396531	−	0.918021i	$-0.629786\pi$
$53$	−306.000	−0.793063	−0.396531	+	0.918021i	$0.629786\pi$
$54$	0	0
$55$	−196.000	−0.480521
$56$	56.0000	0.133631
$57$	0	0
$58$	400.000	0.905562
$59$	331.000	0.730382	0.365191	−	0.930933i	$-0.381004\pi$
$59$	331.000	0.730382	0.365191	+	0.930933i	$0.381004\pi$
$60$	0	0
$61$	−204.000	−0.428189	−0.214094	−	0.976813i	$-0.568680\pi$
$61$	−204.000	−0.428189	−0.214094	+	0.976813i	$0.568680\pi$
$62$	−324.000	−0.663678
$63$	0	0
$64$	64.0000	0.125000
$65$	210.000	0.400728
$66$	0	0
$67$	928.000	1.69214	0.846069	−	0.533073i	$-0.178963\pi$
$67$	928.000	1.69214	0.846069	+	0.533073i	$0.178963\pi$
$68$	188.000	0.335270
$69$	0	0
$70$	98.0000	0.167332
$71$	740.000	1.23693	0.618464	−	0.785813i	$-0.287755\pi$
$71$	740.000	1.23693	0.618464	+	0.785813i	$0.287755\pi$
$72$	0	0
$73$	706.000	1.13193	0.565966	−	0.824429i	$-0.308503\pi$
$73$	706.000	1.13193	0.565966	+	0.824429i	$0.308503\pi$
$74$	−274.000	−0.430430
$75$	0	0
$76$	656.000	0.990110
$77$	196.000	0.290081
$78$	0	0
$79$	−195.000	−0.277712	−0.138856	−	0.990313i	$-0.544342\pi$
$79$	−195.000	−0.277712	−0.138856	+	0.990313i	$0.544342\pi$
$80$	112.000	0.156525
$81$	0	0
$82$	−282.000	−0.379777
$83$	485.000	0.641394	0.320697	−	0.947182i	$-0.396083\pi$
$83$	485.000	0.641394	0.320697	+	0.947182i	$0.396083\pi$
$84$	0	0
$85$	329.000	0.419824
$86$	−586.000	−0.734768
$87$	0	0
$88$	224.000	0.271346
$89$	114.000	0.135775	0.0678875	−	0.997693i	$-0.478374\pi$
$89$	114.000	0.135775	0.0678875	+	0.997693i	$0.478374\pi$
$90$	0	0
$91$	−210.000	−0.241912
$92$	−376.000	−0.426095
$93$	0	0
$94$	−942.000	−1.03362
$95$	1148.00	1.23981
$96$	0	0
$97$	−344.000	−0.360082	−0.180041	−	0.983659i	$-0.557623\pi$
$97$	−344.000	−0.360082	−0.180041	+	0.983659i	$0.557623\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	−304.000	−0.304000
$101$	898.000	0.884696	0.442348	−	0.896843i	$-0.354146\pi$
$101$	898.000	0.884696	0.442348	+	0.896843i	$0.354146\pi$
$102$	0	0
$103$	1202.00	1.14987	0.574935	−	0.818199i	$-0.305027\pi$
$103$	1202.00	1.14987	0.574935	+	0.818199i	$0.305027\pi$
$104$	−240.000	−0.226288
$105$	0	0
$106$	612.000	0.560780
$107$	−1950.00	−1.76181	−0.880905	−	0.473294i	$-0.843065\pi$
$107$	−1950.00	−1.76181	−0.880905	+	0.473294i	$0.843065\pi$
$108$	0	0
$109$	475.000	0.417401	0.208701	−	0.977980i	$-0.433077\pi$
$109$	475.000	0.417401	0.208701	+	0.977980i	$0.433077\pi$
$110$	392.000	0.339779
$111$	0	0
$112$	−112.000	−0.0944911
$113$	814.000	0.677652	0.338826	−	0.940849i	$-0.389970\pi$
$113$	814.000	0.677652	0.338826	+	0.940849i	$0.389970\pi$
$114$	0	0
$115$	−658.000	−0.533555
$116$	−800.000	−0.640329
$117$	0	0
$118$	−662.000	−0.516458
$119$	−329.000	−0.253440
$120$	0	0
$121$	−547.000	−0.410969
$122$	408.000	0.302775
$123$	0	0
$124$	648.000	0.469291
$125$	−1407.00	−1.00677
$126$	0	0
$127$	−981.000	−0.685431	−0.342715	−	0.939439i	$-0.611347\pi$
$127$	−981.000	−0.685431	−0.342715	+	0.939439i	$0.611347\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−420.000	−0.283357
$131$	2236.00	1.49130	0.745650	−	0.666338i	$-0.232139\pi$
$131$	2236.00	1.49130	0.745650	+	0.666338i	$0.232139\pi$
$132$	0	0
$133$	−1148.00	−0.748453
$134$	−1856.00	−1.19652
$135$	0	0
$136$	−376.000	−0.237072
$137$	1846.00	1.15120	0.575600	−	0.817731i	$-0.304769\pi$
$137$	1846.00	1.15120	0.575600	+	0.817731i	$0.304769\pi$
$138$	0	0
$139$	−1904.00	−1.16184	−0.580918	−	0.813962i	$-0.697306\pi$
$139$	−1904.00	−1.16184	−0.580918	+	0.813962i	$0.697306\pi$
$140$	−196.000	−0.118322
$141$	0	0
$142$	−1480.00	−0.874640
$143$	−840.000	−0.491219
$144$	0	0
$145$	−1400.00	−0.801818
$146$	−1412.00	−0.800397
$147$	0	0
$148$	548.000	0.304360
$149$	−228.000	−0.125359	−0.0626795	−	0.998034i	$-0.519965\pi$
$149$	−228.000	−0.125359	−0.0626795	+	0.998034i	$0.519965\pi$
$150$	0	0
$151$	79.0000	0.0425757	0.0212878	−	0.999773i	$-0.493223\pi$
$151$	79.0000	0.0425757	0.0212878	+	0.999773i	$0.493223\pi$
$152$	−1312.00	−0.700114
$153$	0	0
$154$	−392.000	−0.205119
$155$	1134.00	0.587646
$156$	0	0
$157$	−1156.00	−0.587636	−0.293818	−	0.955861i	$-0.594926\pi$
$157$	−1156.00	−0.587636	−0.293818	+	0.955861i	$0.594926\pi$
$158$	390.000	0.196372
$159$	0	0
$160$	−224.000	−0.110680
$161$	658.000	0.322097
$162$	0	0
$163$	1331.00	0.639583	0.319791	−	0.947488i	$-0.396387\pi$
$163$	1331.00	0.639583	0.319791	+	0.947488i	$0.396387\pi$
$164$	564.000	0.268543
$165$	0	0
$166$	−970.000	−0.453534
$167$	1111.00	0.514801	0.257400	−	0.966305i	$-0.417134\pi$
$167$	1111.00	0.514801	0.257400	+	0.966305i	$0.417134\pi$
$168$	0	0
$169$	−1297.00	−0.590350
$170$	−658.000	−0.296861
$171$	0	0
$172$	1172.00	0.519559
$173$	−1658.00	−0.728644	−0.364322	−	0.931273i	$-0.618699\pi$
$173$	−1658.00	−0.728644	−0.364322	+	0.931273i	$0.618699\pi$
$174$	0	0
$175$	532.000	0.229802
$176$	−448.000	−0.191871
$177$	0	0
$178$	−228.000	−0.0960074
$179$	−1290.00	−0.538654	−0.269327	−	0.963049i	$-0.586801\pi$
$179$	−1290.00	−0.538654	−0.269327	+	0.963049i	$0.586801\pi$
$180$	0	0
$181$	−252.000	−0.103486	−0.0517431	−	0.998660i	$-0.516478\pi$
$181$	−252.000	−0.103486	−0.0517431	+	0.998660i	$0.516478\pi$
$182$	420.000	0.171058
$183$	0	0
$184$	752.000	0.301294
$185$	959.000	0.381119
$186$	0	0
$187$	−1316.00	−0.514628
$188$	1884.00	0.730877
$189$	0	0
$190$	−2296.00	−0.876681
$191$	−1410.00	−0.534157	−0.267079	−	0.963675i	$-0.586058\pi$
$191$	−1410.00	−0.534157	−0.267079	+	0.963675i	$0.586058\pi$
$192$	0	0
$193$	335.000	0.124942	0.0624711	−	0.998047i	$-0.480102\pi$
$193$	335.000	0.124942	0.0624711	+	0.998047i	$0.480102\pi$
$194$	688.000	0.254616
$195$	0	0
$196$	196.000	0.0714286
$197$	−4960.00	−1.79383	−0.896917	−	0.442199i	$-0.854199\pi$
$197$	−4960.00	−1.79383	−0.896917	+	0.442199i	$0.854199\pi$
$198$	0	0
$199$	488.000	0.173836	0.0869181	−	0.996215i	$-0.472298\pi$
$199$	488.000	0.173836	0.0869181	+	0.996215i	$0.472298\pi$
$200$	608.000	0.214960
$201$	0	0
$202$	−1796.00	−0.625575
$203$	1400.00	0.484043
$204$	0	0
$205$	987.000	0.336269
$206$	−2404.00	−0.813081
$207$	0	0
$208$	480.000	0.160010
$209$	−4592.00	−1.51979
$210$	0	0
$211$	−6124.00	−1.99807	−0.999037	−	0.0438793i	$-0.986028\pi$
$211$	−6124.00	−1.99807	−0.999037	+	0.0438793i	$0.986028\pi$
$212$	−1224.00	−0.396531
$213$	0	0
$214$	3900.00	1.24579
$215$	2051.00	0.650591
$216$	0	0
$217$	−1134.00	−0.354751
$218$	−950.000	−0.295147
$219$	0	0
$220$	−784.000	−0.240260
$221$	1410.00	0.429171
$222$	0	0
$223$	−5902.00	−1.77232	−0.886160	−	0.463380i	$-0.846636\pi$
$223$	−5902.00	−1.77232	−0.886160	+	0.463380i	$0.846636\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−1628.00	−0.479172
$227$	5832.00	1.70521	0.852607	−	0.522553i	$-0.175020\pi$
$227$	5832.00	1.70521	0.852607	+	0.522553i	$0.175020\pi$
$228$	0	0
$229$	3472.00	1.00190	0.500952	−	0.865475i	$-0.332983\pi$
$229$	3472.00	1.00190	0.500952	+	0.865475i	$0.332983\pi$
$230$	1316.00	0.377280
$231$	0	0
$232$	1600.00	0.452781
$233$	−1642.00	−0.461678	−0.230839	−	0.972992i	$-0.574147\pi$
$233$	−1642.00	−0.461678	−0.230839	+	0.972992i	$0.574147\pi$
$234$	0	0
$235$	3297.00	0.915202
$236$	1324.00	0.365191
$237$	0	0
$238$	658.000	0.179209
$239$	3054.00	0.826556	0.413278	−	0.910605i	$-0.364384\pi$
$239$	3054.00	0.826556	0.413278	+	0.910605i	$0.364384\pi$
$240$	0	0
$241$	−2914.00	−0.778868	−0.389434	−	0.921054i	$-0.627329\pi$
$241$	−2914.00	−0.778868	−0.389434	+	0.921054i	$0.627329\pi$
$242$	1094.00	0.290599
$243$	0	0
$244$	−816.000	−0.214094
$245$	343.000	0.0894427
$246$	0	0
$247$	4920.00	1.26742
$248$	−1296.00	−0.331839
$249$	0	0
$250$	2814.00	0.711892
$251$	−4347.00	−1.09315	−0.546574	−	0.837411i	$-0.684068\pi$
$251$	−4347.00	−1.09315	−0.546574	+	0.837411i	$0.684068\pi$
$252$	0	0
$253$	2632.00	0.654041
$254$	1962.00	0.484673
$255$	0	0
$256$	256.000	0.0625000
$257$	−2526.00	−0.613103	−0.306552	−	0.951854i	$-0.599175\pi$
$257$	−2526.00	−0.613103	−0.306552	+	0.951854i	$0.599175\pi$
$258$	0	0
$259$	−959.000	−0.230075
$260$	840.000	0.200364
$261$	0	0
$262$	−4472.00	−1.05451
$263$	2106.00	0.493770	0.246885	−	0.969045i	$-0.420593\pi$
$263$	2106.00	0.493770	0.246885	+	0.969045i	$0.420593\pi$
$264$	0	0
$265$	−2142.00	−0.496536
$266$	2296.00	0.529236
$267$	0	0
$268$	3712.00	0.846069
$269$	5035.00	1.14122	0.570612	−	0.821220i	$-0.306706\pi$
$269$	5035.00	1.14122	0.570612	+	0.821220i	$0.306706\pi$
$270$	0	0
$271$	5250.00	1.17681	0.588404	−	0.808567i	$-0.299757\pi$
$271$	5250.00	1.17681	0.588404	+	0.808567i	$0.299757\pi$
$272$	752.000	0.167635
$273$	0	0
$274$	−3692.00	−0.814021
$275$	2128.00	0.466630
$276$	0	0
$277$	5681.00	1.23227	0.616134	−	0.787641i	$-0.288698\pi$
$277$	5681.00	1.23227	0.616134	+	0.787641i	$0.288698\pi$
$278$	3808.00	0.821542
$279$	0	0
$280$	392.000	0.0836660
$281$	−4570.00	−0.970190	−0.485095	−	0.874462i	$-0.661215\pi$
$281$	−4570.00	−0.970190	−0.485095	+	0.874462i	$0.661215\pi$
$282$	0	0
$283$	−7628.00	−1.60225	−0.801126	−	0.598495i	$-0.795766\pi$
$283$	−7628.00	−1.60225	−0.801126	+	0.598495i	$0.795766\pi$
$284$	2960.00	0.618464
$285$	0	0
$286$	1680.00	0.347344
$287$	−987.000	−0.202999
$288$	0	0
$289$	−2704.00	−0.550377
$290$	2800.00	0.566971
$291$	0	0
$292$	2824.00	0.565966
$293$	393.000	0.0783594	0.0391797	−	0.999232i	$-0.487526\pi$
$293$	393.000	0.0783594	0.0391797	+	0.999232i	$0.487526\pi$
$294$	0	0
$295$	2317.00	0.457291
$296$	−1096.00	−0.215215
$297$	0	0
$298$	456.000	0.0886422
$299$	−2820.00	−0.545434
$300$	0	0
$301$	−2051.00	−0.392750
$302$	−158.000	−0.0301056
$303$	0	0
$304$	2624.00	0.495055
$305$	−1428.00	−0.268089
$306$	0	0
$307$	9070.00	1.68616	0.843082	−	0.537785i	$-0.180739\pi$
$307$	9070.00	1.68616	0.843082	+	0.537785i	$0.180739\pi$
$308$	784.000	0.145041
$309$	0	0
$310$	−2268.00	−0.415528
$311$	−3677.00	−0.670429	−0.335215	−	0.942142i	$-0.608809\pi$
$311$	−3677.00	−0.670429	−0.335215	+	0.942142i	$0.608809\pi$
$312$	0	0
$313$	8980.00	1.62166	0.810830	−	0.585282i	$-0.199016\pi$
$313$	8980.00	1.62166	0.810830	+	0.585282i	$0.199016\pi$
$314$	2312.00	0.415521
$315$	0	0
$316$	−780.000	−0.138856
$317$	−4806.00	−0.851520	−0.425760	−	0.904836i	$-0.639993\pi$
$317$	−4806.00	−0.851520	−0.425760	+	0.904836i	$0.639993\pi$
$318$	0	0
$319$	5600.00	0.982883
$320$	448.000	0.0782624
$321$	0	0
$322$	−1316.00	−0.227757
$323$	7708.00	1.32782
$324$	0	0
$325$	−2280.00	−0.389144
$326$	−2662.00	−0.452253
$327$	0	0
$328$	−1128.00	−0.189888
$329$	−3297.00	−0.552491
$330$	0	0
$331$	305.000	0.0506475	0.0253237	−	0.999679i	$-0.491938\pi$
$331$	305.000	0.0506475	0.0253237	+	0.999679i	$0.491938\pi$
$332$	1940.00	0.320697
$333$	0	0
$334$	−2222.00	−0.364019
$335$	6496.00	1.05945
$336$	0	0
$337$	351.000	0.0567365	0.0283682	−	0.999598i	$-0.490969\pi$
$337$	351.000	0.0567365	0.0283682	+	0.999598i	$0.490969\pi$
$338$	2594.00	0.417441
$339$	0	0
$340$	1316.00	0.209912
$341$	−4536.00	−0.720347
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−2344.00	−0.367384
$345$	0	0
$346$	3316.00	0.515229
$347$	666.000	0.103034	0.0515169	−	0.998672i	$-0.483594\pi$
$347$	666.000	0.103034	0.0515169	+	0.998672i	$0.483594\pi$
$348$	0	0
$349$	−5396.00	−0.827625	−0.413813	−	0.910362i	$-0.635803\pi$
$349$	−5396.00	−0.827625	−0.413813	+	0.910362i	$0.635803\pi$
$350$	−1064.00	−0.162495
$351$	0	0
$352$	896.000	0.135673
$353$	−1893.00	−0.285423	−0.142711	−	0.989764i	$-0.545582\pi$
$353$	−1893.00	−0.285423	−0.142711	+	0.989764i	$0.545582\pi$
$354$	0	0
$355$	5180.00	0.774439
$356$	456.000	0.0678875
$357$	0	0
$358$	2580.00	0.380886
$359$	−7778.00	−1.14347	−0.571737	−	0.820437i	$-0.693730\pi$
$359$	−7778.00	−1.14347	−0.571737	+	0.820437i	$0.693730\pi$
$360$	0	0
$361$	20037.0	2.92127
$362$	504.000	0.0731758
$363$	0	0
$364$	−840.000	−0.120956
$365$	4942.00	0.708702
$366$	0	0
$367$	734.000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	734.000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	−1504.00	−0.213047
$369$	0	0
$370$	−1918.00	−0.269492
$371$	2142.00	0.299750
$372$	0	0
$373$	11935.0	1.65676	0.828379	−	0.560168i	$-0.189263\pi$
$373$	11935.0	1.65676	0.828379	+	0.560168i	$0.189263\pi$
$374$	2632.00	0.363897
$375$	0	0
$376$	−3768.00	−0.516808
$377$	−6000.00	−0.819670
$378$	0	0
$379$	−12581.0	−1.70513	−0.852563	−	0.522625i	$-0.824953\pi$
$379$	−12581.0	−1.70513	−0.852563	+	0.522625i	$0.824953\pi$
$380$	4592.00	0.619907
$381$	0	0
$382$	2820.00	0.377706
$383$	−7693.00	−1.02635	−0.513177	−	0.858283i	$-0.671532\pi$
$383$	−7693.00	−1.02635	−0.513177	+	0.858283i	$0.671532\pi$
$384$	0	0
$385$	1372.00	0.181620
$386$	−670.000	−0.0883474
$387$	0	0
$388$	−1376.00	−0.180041
$389$	−7352.00	−0.958255	−0.479128	−	0.877745i	$-0.659047\pi$
$389$	−7352.00	−0.958255	−0.479128	+	0.877745i	$0.659047\pi$
$390$	0	0
$391$	−4418.00	−0.571427
$392$	−392.000	−0.0505076
$393$	0	0
$394$	9920.00	1.26843
$395$	−1365.00	−0.173875
$396$	0	0
$397$	2418.00	0.305682	0.152841	−	0.988251i	$-0.451158\pi$
$397$	2418.00	0.305682	0.152841	+	0.988251i	$0.451158\pi$
$398$	−976.000	−0.122921
$399$	0	0
$400$	−1216.00	−0.152000
$401$	−14244.0	−1.77384	−0.886922	−	0.461919i	$-0.847161\pi$
$401$	−14244.0	−1.77384	−0.886922	+	0.461919i	$0.847161\pi$
$402$	0	0
$403$	4860.00	0.600729
$404$	3592.00	0.442348
$405$	0	0
$406$	−2800.00	−0.342270
$407$	−3836.00	−0.467183
$408$	0	0
$409$	−784.000	−0.0947831	−0.0473916	−	0.998876i	$-0.515091\pi$
$409$	−784.000	−0.0947831	−0.0473916	+	0.998876i	$0.515091\pi$
$410$	−1974.00	−0.237778
$411$	0	0
$412$	4808.00	0.574935
$413$	−2317.00	−0.276058
$414$	0	0
$415$	3395.00	0.401576
$416$	−960.000	−0.113144
$417$	0	0
$418$	9184.00	1.07465
$419$	−8331.00	−0.971351	−0.485675	−	0.874139i	$-0.661426\pi$
$419$	−8331.00	−0.971351	−0.485675	+	0.874139i	$0.661426\pi$
$420$	0	0
$421$	−14514.0	−1.68021	−0.840106	−	0.542423i	$-0.817507\pi$
$421$	−14514.0	−1.68021	−0.840106	+	0.542423i	$0.817507\pi$
$422$	12248.0	1.41285
$423$	0	0
$424$	2448.00	0.280390
$425$	−3572.00	−0.407688
$426$	0	0
$427$	1428.00	0.161840
$428$	−7800.00	−0.880905
$429$	0	0
$430$	−4102.00	−0.460037
$431$	10176.0	1.13726	0.568632	−	0.822592i	$-0.307473\pi$
$431$	10176.0	1.13726	0.568632	+	0.822592i	$0.307473\pi$
$432$	0	0
$433$	−7390.00	−0.820186	−0.410093	−	0.912044i	$-0.634504\pi$
$433$	−7390.00	−0.820186	−0.410093	+	0.912044i	$0.634504\pi$
$434$	2268.00	0.250847
$435$	0	0
$436$	1900.00	0.208701
$437$	−15416.0	−1.68752
$438$	0	0
$439$	−11844.0	−1.28766	−0.643831	−	0.765168i	$-0.722656\pi$
$439$	−11844.0	−1.28766	−0.643831	+	0.765168i	$0.722656\pi$
$440$	1568.00	0.169890
$441$	0	0
$442$	−2820.00	−0.303470
$443$	16364.0	1.75503	0.877514	−	0.479552i	$-0.159201\pi$
$443$	16364.0	1.75503	0.877514	+	0.479552i	$0.159201\pi$
$444$	0	0
$445$	798.000	0.0850086
$446$	11804.0	1.25322
$447$	0	0
$448$	−448.000	−0.0472456
$449$	14602.0	1.53477	0.767384	−	0.641188i	$-0.221558\pi$
$449$	14602.0	1.53477	0.767384	+	0.641188i	$0.221558\pi$
$450$	0	0
$451$	−3948.00	−0.412204
$452$	3256.00	0.338826
$453$	0	0
$454$	−11664.0	−1.20577
$455$	−1470.00	−0.151461
$456$	0	0
$457$	−11606.0	−1.18798	−0.593989	−	0.804473i	$-0.702448\pi$
$457$	−11606.0	−1.18798	−0.593989	+	0.804473i	$0.702448\pi$
$458$	−6944.00	−0.708454
$459$	0	0
$460$	−2632.00	−0.266777
$461$	−8423.00	−0.850972	−0.425486	−	0.904965i	$-0.639897\pi$
$461$	−8423.00	−0.850972	−0.425486	+	0.904965i	$0.639897\pi$
$462$	0	0
$463$	9043.00	0.907697	0.453849	−	0.891079i	$-0.350051\pi$
$463$	9043.00	0.907697	0.453849	+	0.891079i	$0.350051\pi$
$464$	−3200.00	−0.320164
$465$	0	0
$466$	3284.00	0.326456
$467$	18892.0	1.87199	0.935993	−	0.352019i	$-0.114505\pi$
$467$	18892.0	1.87199	0.935993	+	0.352019i	$0.114505\pi$
$468$	0	0
$469$	−6496.00	−0.639568
$470$	−6594.00	−0.647146
$471$	0	0
$472$	−2648.00	−0.258229
$473$	−8204.00	−0.797506
$474$	0	0
$475$	−12464.0	−1.20397
$476$	−1316.00	−0.126720
$477$	0	0
$478$	−6108.00	−0.584463
$479$	9209.00	0.878434	0.439217	−	0.898381i	$-0.355256\pi$
$479$	9209.00	0.878434	0.439217	+	0.898381i	$0.355256\pi$
$480$	0	0
$481$	4110.00	0.389605
$482$	5828.00	0.550743
$483$	0	0
$484$	−2188.00	−0.205485
$485$	−2408.00	−0.225447
$486$	0	0
$487$	13256.0	1.23344	0.616721	−	0.787181i	$-0.288460\pi$
$487$	13256.0	1.23344	0.616721	+	0.787181i	$0.288460\pi$
$488$	1632.00	0.151388
$489$	0	0
$490$	−686.000	−0.0632456
$491$	4146.00	0.381072	0.190536	−	0.981680i	$-0.438977\pi$
$491$	4146.00	0.381072	0.190536	+	0.981680i	$0.438977\pi$
$492$	0	0
$493$	−9400.00	−0.858732
$494$	−9840.00	−0.896199
$495$	0	0
$496$	2592.00	0.234646
$497$	−5180.00	−0.467515
$498$	0	0
$499$	−2885.00	−0.258818	−0.129409	−	0.991591i	$-0.541308\pi$
$499$	−2885.00	−0.258818	−0.129409	+	0.991591i	$0.541308\pi$
$500$	−5628.00	−0.503384
$501$	0	0
$502$	8694.00	0.772973
$503$	20595.0	1.82562	0.912809	−	0.408387i	$-0.133909\pi$
$503$	20595.0	1.82562	0.912809	+	0.408387i	$0.133909\pi$
$504$	0	0
$505$	6286.00	0.553908
$506$	−5264.00	−0.462477
$507$	0	0
$508$	−3924.00	−0.342715
$509$	−1401.00	−0.122000	−0.0610002	−	0.998138i	$-0.519429\pi$
$509$	−1401.00	−0.122000	−0.0610002	+	0.998138i	$0.519429\pi$
$510$	0	0
$511$	−4942.00	−0.427830
$512$	−512.000	−0.0441942
$513$	0	0
$514$	5052.00	0.433530
$515$	8414.00	0.719932
$516$	0	0
$517$	−13188.0	−1.12187
$518$	1918.00	0.162687
$519$	0	0
$520$	−1680.00	−0.141679
$521$	−3453.00	−0.290362	−0.145181	−	0.989405i	$-0.546376\pi$
$521$	−3453.00	−0.290362	−0.145181	+	0.989405i	$0.546376\pi$
$522$	0	0
$523$	5798.00	0.484759	0.242379	−	0.970182i	$-0.422072\pi$
$523$	5798.00	0.484759	0.242379	+	0.970182i	$0.422072\pi$
$524$	8944.00	0.745650
$525$	0	0
$526$	−4212.00	−0.349148
$527$	7614.00	0.629357
$528$	0	0
$529$	−3331.00	−0.273773
$530$	4284.00	0.351104
$531$	0	0
$532$	−4592.00	−0.374226
$533$	4230.00	0.343755
$534$	0	0
$535$	−13650.0	−1.10307
$536$	−7424.00	−0.598261
$537$	0	0
$538$	−10070.0	−0.806968
$539$	−1372.00	−0.109640
$540$	0	0
$541$	−18855.0	−1.49841	−0.749205	−	0.662338i	$-0.769564\pi$
$541$	−18855.0	−1.49841	−0.749205	+	0.662338i	$0.769564\pi$
$542$	−10500.0	−0.832128
$543$	0	0
$544$	−1504.00	−0.118536
$545$	3325.00	0.261335
$546$	0	0
$547$	−2259.00	−0.176577	−0.0882887	−	0.996095i	$-0.528140\pi$
$547$	−2259.00	−0.176577	−0.0882887	+	0.996095i	$0.528140\pi$
$548$	7384.00	0.575600
$549$	0	0
$550$	−4256.00	−0.329957
$551$	−32800.0	−2.53598
$552$	0	0
$553$	1365.00	0.104965
$554$	−11362.0	−0.871345
$555$	0	0
$556$	−7616.00	−0.580918
$557$	−11564.0	−0.879681	−0.439841	−	0.898076i	$-0.644965\pi$
$557$	−11564.0	−0.879681	−0.439841	+	0.898076i	$0.644965\pi$
$558$	0	0
$559$	8790.00	0.665076
$560$	−784.000	−0.0591608
$561$	0	0
$562$	9140.00	0.686028
$563$	18692.0	1.39924	0.699622	−	0.714514i	$-0.253352\pi$
$563$	18692.0	1.39924	0.699622	+	0.714514i	$0.253352\pi$
$564$	0	0
$565$	5698.00	0.424277
$566$	15256.0	1.13296
$567$	0	0
$568$	−5920.00	−0.437320
$569$	21342.0	1.57241	0.786207	−	0.617964i	$-0.212042\pi$
$569$	21342.0	1.57241	0.786207	+	0.617964i	$0.212042\pi$
$570$	0	0
$571$	14181.0	1.03933	0.519664	−	0.854371i	$-0.326057\pi$
$571$	14181.0	1.03933	0.519664	+	0.854371i	$0.326057\pi$
$572$	−3360.00	−0.245610
$573$	0	0
$574$	1974.00	0.143542
$575$	7144.00	0.518131
$576$	0	0
$577$	−7424.00	−0.535642	−0.267821	−	0.963469i	$-0.586304\pi$
$577$	−7424.00	−0.535642	−0.267821	+	0.963469i	$0.586304\pi$
$578$	5408.00	0.389175
$579$	0	0
$580$	−5600.00	−0.400909
$581$	−3395.00	−0.242424
$582$	0	0
$583$	8568.00	0.608663
$584$	−5648.00	−0.400198
$585$	0	0
$586$	−786.000	−0.0554085
$587$	−15556.0	−1.09381	−0.546904	−	0.837196i	$-0.684194\pi$
$587$	−15556.0	−1.09381	−0.546904	+	0.837196i	$0.684194\pi$
$588$	0	0
$589$	26568.0	1.85860
$590$	−4634.00	−0.323354
$591$	0	0
$592$	2192.00	0.152180
$593$	24003.0	1.66220	0.831100	−	0.556122i	$-0.187711\pi$
$593$	24003.0	1.66220	0.831100	+	0.556122i	$0.187711\pi$
$594$	0	0
$595$	−2303.00	−0.158679
$596$	−912.000	−0.0626795
$597$	0	0
$598$	5640.00	0.385680
$599$	−8052.00	−0.549242	−0.274621	−	0.961553i	$-0.588552\pi$
$599$	−8052.00	−0.549242	−0.274621	+	0.961553i	$0.588552\pi$
$600$	0	0
$601$	−4138.00	−0.280853	−0.140426	−	0.990091i	$-0.544847\pi$
$601$	−4138.00	−0.280853	−0.140426	+	0.990091i	$0.544847\pi$
$602$	4102.00	0.277716
$603$	0	0
$604$	316.000	0.0212878
$605$	−3829.00	−0.257307
$606$	0	0
$607$	−10482.0	−0.700908	−0.350454	−	0.936580i	$-0.613973\pi$
$607$	−10482.0	−0.700908	−0.350454	+	0.936580i	$0.613973\pi$
$608$	−5248.00	−0.350057
$609$	0	0
$610$	2856.00	0.189567
$611$	14130.0	0.935579
$612$	0	0
$613$	−18762.0	−1.23620	−0.618100	−	0.786100i	$-0.712097\pi$
$613$	−18762.0	−1.23620	−0.618100	+	0.786100i	$0.712097\pi$
$614$	−18140.0	−1.19230
$615$	0	0
$616$	−1568.00	−0.102559
$617$	6734.00	0.439385	0.219693	−	0.975569i	$-0.429495\pi$
$617$	6734.00	0.439385	0.219693	+	0.975569i	$0.429495\pi$
$618$	0	0
$619$	−4832.00	−0.313755	−0.156878	−	0.987618i	$-0.550143\pi$
$619$	−4832.00	−0.313755	−0.156878	+	0.987618i	$0.550143\pi$
$620$	4536.00	0.293823
$621$	0	0
$622$	7354.00	0.474065
$623$	−798.000	−0.0513181
$624$	0	0
$625$	−349.000	−0.0223360
$626$	−17960.0	−1.14669
$627$	0	0
$628$	−4624.00	−0.293818
$629$	6439.00	0.408171
$630$	0	0
$631$	−18265.0	−1.15233	−0.576163	−	0.817335i	$-0.695451\pi$
$631$	−18265.0	−1.15233	−0.576163	+	0.817335i	$0.695451\pi$
$632$	1560.00	0.0981859
$633$	0	0
$634$	9612.00	0.602116
$635$	−6867.00	−0.429147
$636$	0	0
$637$	1470.00	0.0914341
$638$	−11200.0	−0.695004
$639$	0	0
$640$	−896.000	−0.0553399
$641$	19316.0	1.19023	0.595114	−	0.803641i	$-0.297107\pi$
$641$	19316.0	1.19023	0.595114	+	0.803641i	$0.297107\pi$
$642$	0	0
$643$	−27610.0	−1.69336	−0.846681	−	0.532100i	$-0.821403\pi$
$643$	−27610.0	−1.69336	−0.846681	+	0.532100i	$0.821403\pi$
$644$	2632.00	0.161049
$645$	0	0
$646$	−15416.0	−0.938908
$647$	11544.0	0.701455	0.350728	−	0.936478i	$-0.385934\pi$
$647$	11544.0	0.701455	0.350728	+	0.936478i	$0.385934\pi$
$648$	0	0
$649$	−9268.00	−0.560556
$650$	4560.00	0.275166
$651$	0	0
$652$	5324.00	0.319791
$653$	4890.00	0.293048	0.146524	−	0.989207i	$-0.453191\pi$
$653$	4890.00	0.293048	0.146524	+	0.989207i	$0.453191\pi$
$654$	0	0
$655$	15652.0	0.933701
$656$	2256.00	0.134271
$657$	0	0
$658$	6594.00	0.390670
$659$	31232.0	1.84617	0.923085	−	0.384596i	$-0.125659\pi$
$659$	31232.0	1.84617	0.923085	+	0.384596i	$0.125659\pi$
$660$	0	0
$661$	−9002.00	−0.529708	−0.264854	−	0.964288i	$-0.585324\pi$
$661$	−9002.00	−0.529708	−0.264854	+	0.964288i	$0.585324\pi$
$662$	−610.000	−0.0358132
$663$	0	0
$664$	−3880.00	−0.226767
$665$	−8036.00	−0.468606
$666$	0	0
$667$	18800.0	1.09136
$668$	4444.00	0.257400
$669$	0	0
$670$	−12992.0	−0.749141
$671$	5712.00	0.328628
$672$	0	0
$673$	−4978.00	−0.285123	−0.142562	−	0.989786i	$-0.545534\pi$
$673$	−4978.00	−0.285123	−0.142562	+	0.989786i	$0.545534\pi$
$674$	−702.000	−0.0401187
$675$	0	0
$676$	−5188.00	−0.295175
$677$	−27738.0	−1.57468	−0.787340	−	0.616519i	$-0.788542\pi$
$677$	−27738.0	−1.57468	−0.787340	+	0.616519i	$0.788542\pi$
$678$	0	0
$679$	2408.00	0.136098
$680$	−2632.00	−0.148430
$681$	0	0
$682$	9072.00	0.509362
$683$	−8490.00	−0.475638	−0.237819	−	0.971309i	$-0.576433\pi$
$683$	−8490.00	−0.475638	−0.237819	+	0.971309i	$0.576433\pi$
$684$	0	0
$685$	12922.0	0.720765
$686$	686.000	0.0381802
$687$	0	0
$688$	4688.00	0.259780
$689$	−9180.00	−0.507591
$690$	0	0
$691$	−910.000	−0.0500985	−0.0250492	−	0.999686i	$-0.507974\pi$
$691$	−910.000	−0.0500985	−0.0250492	+	0.999686i	$0.507974\pi$
$692$	−6632.00	−0.364322
$693$	0	0
$694$	−1332.00	−0.0728559
$695$	−13328.0	−0.727424
$696$	0	0
$697$	6627.00	0.360137
$698$	10792.0	0.585220
$699$	0	0
$700$	2128.00	0.114901
$701$	14830.0	0.799032	0.399516	−	0.916726i	$-0.369178\pi$
$701$	14830.0	0.799032	0.399516	+	0.916726i	$0.369178\pi$
$702$	0	0
$703$	22468.0	1.20540
$704$	−1792.00	−0.0959354
$705$	0	0
$706$	3786.00	0.201824
$707$	−6286.00	−0.334384
$708$	0	0
$709$	36323.0	1.92403	0.962016	−	0.272993i	$-0.0880134\pi$
$709$	36323.0	1.92403	0.962016	+	0.272993i	$0.0880134\pi$
$710$	−10360.0	−0.547611
$711$	0	0
$712$	−912.000	−0.0480037
$713$	−15228.0	−0.799850
$714$	0	0
$715$	−5880.00	−0.307552
$716$	−5160.00	−0.269327
$717$	0	0
$718$	15556.0	0.808558
$719$	−15009.0	−0.778500	−0.389250	−	0.921132i	$-0.627266\pi$
$719$	−15009.0	−0.778500	−0.389250	+	0.921132i	$0.627266\pi$
$720$	0	0
$721$	−8414.00	−0.434610
$722$	−40074.0	−2.06565
$723$	0	0
$724$	−1008.00	−0.0517431
$725$	15200.0	0.778640
$726$	0	0
$727$	13864.0	0.707273	0.353636	−	0.935383i	$-0.384945\pi$
$727$	13864.0	0.707273	0.353636	+	0.935383i	$0.384945\pi$
$728$	1680.00	0.0855288
$729$	0	0
$730$	−9884.00	−0.501128
$731$	13771.0	0.696770
$732$	0	0
$733$	11532.0	0.581097	0.290549	−	0.956860i	$-0.406162\pi$
$733$	11532.0	0.581097	0.290549	+	0.956860i	$0.406162\pi$
$734$	−1468.00	−0.0738213
$735$	0	0
$736$	3008.00	0.150647
$737$	−25984.0	−1.29869
$738$	0	0
$739$	28164.0	1.40193	0.700967	−	0.713194i	$-0.252752\pi$
$739$	28164.0	1.40193	0.700967	+	0.713194i	$0.252752\pi$
$740$	3836.00	0.190560
$741$	0	0
$742$	−4284.00	−0.211955
$743$	10896.0	0.538002	0.269001	−	0.963140i	$-0.413307\pi$
$743$	10896.0	0.538002	0.269001	+	0.963140i	$0.413307\pi$
$744$	0	0
$745$	−1596.00	−0.0784871
$746$	−23870.0	−1.17151
$747$	0	0
$748$	−5264.00	−0.257314
$749$	13650.0	0.665901
$750$	0	0
$751$	−33600.0	−1.63260	−0.816299	−	0.577629i	$-0.803978\pi$
$751$	−33600.0	−1.63260	−0.816299	+	0.577629i	$0.803978\pi$
$752$	7536.00	0.365438
$753$	0	0
$754$	12000.0	0.579594
$755$	553.000	0.0266566
$756$	0	0
$757$	−15213.0	−0.730417	−0.365209	−	0.930926i	$-0.619002\pi$
$757$	−15213.0	−0.730417	−0.365209	+	0.930926i	$0.619002\pi$
$758$	25162.0	1.20571
$759$	0	0
$760$	−9184.00	−0.438340
$761$	−22039.0	−1.04982	−0.524910	−	0.851158i	$-0.675901\pi$
$761$	−22039.0	−1.04982	−0.524910	+	0.851158i	$0.675901\pi$
$762$	0	0
$763$	−3325.00	−0.157763
$764$	−5640.00	−0.267079
$765$	0	0
$766$	15386.0	0.725743
$767$	9930.00	0.467473
$768$	0	0
$769$	23056.0	1.08117	0.540586	−	0.841289i	$-0.318203\pi$
$769$	23056.0	1.08117	0.540586	+	0.841289i	$0.318203\pi$
$770$	−2744.00	−0.128425
$771$	0	0
$772$	1340.00	0.0624711
$773$	7681.00	0.357395	0.178698	−	0.983904i	$-0.442812\pi$
$773$	7681.00	0.357395	0.178698	+	0.983904i	$0.442812\pi$
$774$	0	0
$775$	−12312.0	−0.570658
$776$	2752.00	0.127308
$777$	0	0
$778$	14704.0	0.677589
$779$	23124.0	1.06355
$780$	0	0
$781$	−20720.0	−0.949321
$782$	8836.00	0.404060
$783$	0	0
$784$	784.000	0.0357143
$785$	−8092.00	−0.367918
$786$	0	0
$787$	20286.0	0.918828	0.459414	−	0.888222i	$-0.348059\pi$
$787$	20286.0	0.918828	0.459414	+	0.888222i	$0.348059\pi$
$788$	−19840.0	−0.896917
$789$	0	0
$790$	2730.00	0.122948
$791$	−5698.00	−0.256128
$792$	0	0
$793$	−6120.00	−0.274057
$794$	−4836.00	−0.216150
$795$	0	0
$796$	1952.00	0.0869181
$797$	−19494.0	−0.866390	−0.433195	−	0.901300i	$-0.642614\pi$
$797$	−19494.0	−0.866390	−0.433195	+	0.901300i	$0.642614\pi$
$798$	0	0
$799$	22137.0	0.980164
$800$	2432.00	0.107480
$801$	0	0
$802$	28488.0	1.25430
$803$	−19768.0	−0.868739
$804$	0	0
$805$	4606.00	0.201665
$806$	−9720.00	−0.424780
$807$	0	0
$808$	−7184.00	−0.312787
$809$	−7898.00	−0.343237	−0.171619	−	0.985163i	$-0.554900\pi$
$809$	−7898.00	−0.343237	−0.171619	+	0.985163i	$0.554900\pi$
$810$	0	0
$811$	5626.00	0.243595	0.121798	−	0.992555i	$-0.461134\pi$
$811$	5626.00	0.243595	0.121798	+	0.992555i	$0.461134\pi$
$812$	5600.00	0.242022
$813$	0	0
$814$	7672.00	0.330348
$815$	9317.00	0.400442
$816$	0	0
$817$	48052.0	2.05768
$818$	1568.00	0.0670218
$819$	0	0
$820$	3948.00	0.168134
$821$	25136.0	1.06852	0.534258	−	0.845321i	$-0.320591\pi$
$821$	25136.0	1.06852	0.534258	+	0.845321i	$0.320591\pi$
$822$	0	0
$823$	−41129.0	−1.74200	−0.871000	−	0.491282i	$-0.836528\pi$
$823$	−41129.0	−1.74200	−0.871000	+	0.491282i	$0.836528\pi$
$824$	−9616.00	−0.406540
$825$	0	0
$826$	4634.00	0.195203
$827$	38238.0	1.60782	0.803909	−	0.594752i	$-0.202750\pi$
$827$	38238.0	1.60782	0.803909	+	0.594752i	$0.202750\pi$
$828$	0	0
$829$	21412.0	0.897068	0.448534	−	0.893766i	$-0.351946\pi$
$829$	21412.0	0.897068	0.448534	+	0.893766i	$0.351946\pi$
$830$	−6790.00	−0.283957
$831$	0	0
$832$	1920.00	0.0800048
$833$	2303.00	0.0957914
$834$	0	0
$835$	7777.00	0.322316
$836$	−18368.0	−0.759893
$837$	0	0
$838$	16662.0	0.686849
$839$	10199.0	0.419676	0.209838	−	0.977736i	$-0.432706\pi$
$839$	10199.0	0.419676	0.209838	+	0.977736i	$0.432706\pi$
$840$	0	0
$841$	15611.0	0.640084
$842$	29028.0	1.18809
$843$	0	0
$844$	−24496.0	−0.999037
$845$	−9079.00	−0.369618
$846$	0	0
$847$	3829.00	0.155332
$848$	−4896.00	−0.198266
$849$	0	0
$850$	7144.00	0.288279
$851$	−12878.0	−0.518745
$852$	0	0
$853$	29612.0	1.18862	0.594312	−	0.804235i	$-0.297425\pi$
$853$	29612.0	1.18862	0.594312	+	0.804235i	$0.297425\pi$
$854$	−2856.00	−0.114438
$855$	0	0
$856$	15600.0	0.622894
$857$	−10611.0	−0.422946	−0.211473	−	0.977384i	$-0.567826\pi$
$857$	−10611.0	−0.422946	−0.211473	+	0.977384i	$0.567826\pi$
$858$	0	0
$859$	−2170.00	−0.0861926	−0.0430963	−	0.999071i	$-0.513722\pi$
$859$	−2170.00	−0.0861926	−0.0430963	+	0.999071i	$0.513722\pi$
$860$	8204.00	0.325295
$861$	0	0
$862$	−20352.0	−0.804167
$863$	−45566.0	−1.79732	−0.898659	−	0.438649i	$-0.855457\pi$
$863$	−45566.0	−1.79732	−0.898659	+	0.438649i	$0.855457\pi$
$864$	0	0
$865$	−11606.0	−0.456203
$866$	14780.0	0.579959
$867$	0	0
$868$	−4536.00	−0.177375
$869$	5460.00	0.213139
$870$	0	0
$871$	27840.0	1.08303
$872$	−3800.00	−0.147574
$873$	0	0
$874$	30832.0	1.19326
$875$	9849.00	0.380522
$876$	0	0
$877$	17473.0	0.672772	0.336386	−	0.941724i	$-0.390795\pi$
$877$	17473.0	0.672772	0.336386	+	0.941724i	$0.390795\pi$
$878$	23688.0	0.910514
$879$	0	0
$880$	−3136.00	−0.120130
$881$	−5970.00	−0.228302	−0.114151	−	0.993463i	$-0.536415\pi$
$881$	−5970.00	−0.228302	−0.114151	+	0.993463i	$0.536415\pi$
$882$	0	0
$883$	21959.0	0.836896	0.418448	−	0.908241i	$-0.362574\pi$
$883$	21959.0	0.836896	0.418448	+	0.908241i	$0.362574\pi$
$884$	5640.00	0.214586
$885$	0	0
$886$	−32728.0	−1.24099
$887$	39867.0	1.50914	0.754568	−	0.656222i	$-0.227847\pi$
$887$	39867.0	1.50914	0.754568	+	0.656222i	$0.227847\pi$
$888$	0	0
$889$	6867.00	0.259068
$890$	−1596.00	−0.0601102
$891$	0	0
$892$	−23608.0	−0.886160
$893$	77244.0	2.89459
$894$	0	0
$895$	−9030.00	−0.337251
$896$	896.000	0.0334077
$897$	0	0
$898$	−29204.0	−1.08525
$899$	−32400.0	−1.20200
$900$	0	0
$901$	−14382.0	−0.531780
$902$	7896.00	0.291472
$903$	0	0
$904$	−6512.00	−0.239586
$905$	−1764.00	−0.0647926
$906$	0	0
$907$	−29841.0	−1.09245	−0.546226	−	0.837638i	$-0.683936\pi$
$907$	−29841.0	−1.09245	−0.546226	+	0.837638i	$0.683936\pi$
$908$	23328.0	0.852607
$909$	0	0
$910$	2940.00	0.107099
$911$	−33708.0	−1.22590	−0.612951	−	0.790121i	$-0.710018\pi$
$911$	−33708.0	−1.22590	−0.612951	+	0.790121i	$0.710018\pi$
$912$	0	0
$913$	−13580.0	−0.492259
$914$	23212.0	0.840027
$915$	0	0
$916$	13888.0	0.500952
$917$	−15652.0	−0.563658
$918$	0	0
$919$	−18329.0	−0.657909	−0.328954	−	0.944346i	$-0.606696\pi$
$919$	−18329.0	−0.657909	−0.328954	+	0.944346i	$0.606696\pi$
$920$	5264.00	0.188640
$921$	0	0
$922$	16846.0	0.601728
$923$	22200.0	0.791681
$924$	0	0
$925$	−10412.0	−0.370102
$926$	−18086.0	−0.641839
$927$	0	0
$928$	6400.00	0.226390
$929$	−25455.0	−0.898979	−0.449489	−	0.893286i	$-0.648394\pi$
$929$	−25455.0	−0.898979	−0.449489	+	0.893286i	$0.648394\pi$
$930$	0	0
$931$	8036.00	0.282889
$932$	−6568.00	−0.230839
$933$	0	0
$934$	−37784.0	−1.32369
$935$	−9212.00	−0.322208
$936$	0	0
$937$	13664.0	0.476396	0.238198	−	0.971217i	$-0.423443\pi$
$937$	13664.0	0.476396	0.238198	+	0.971217i	$0.423443\pi$
$938$	12992.0	0.452243
$939$	0	0
$940$	13188.0	0.457601
$941$	38863.0	1.34633	0.673166	−	0.739492i	$-0.264934\pi$
$941$	38863.0	1.34633	0.673166	+	0.739492i	$0.264934\pi$
$942$	0	0
$943$	−13254.0	−0.457698
$944$	5296.00	0.182595
$945$	0	0
$946$	16408.0	0.563922
$947$	10876.0	0.373202	0.186601	−	0.982436i	$-0.440253\pi$
$947$	10876.0	0.373202	0.186601	+	0.982436i	$0.440253\pi$
$948$	0	0
$949$	21180.0	0.724480
$950$	24928.0	0.851338
$951$	0	0
$952$	2632.00	0.0896046
$953$	−6166.00	−0.209587	−0.104793	−	0.994494i	$-0.533418\pi$
$953$	−6166.00	−0.209587	−0.104793	+	0.994494i	$0.533418\pi$
$954$	0	0
$955$	−9870.00	−0.334435
$956$	12216.0	0.413278
$957$	0	0
$958$	−18418.0	−0.621147
$959$	−12922.0	−0.435113
$960$	0	0
$961$	−3547.00	−0.119063
$962$	−8220.00	−0.275492
$963$	0	0
$964$	−11656.0	−0.389434
$965$	2345.00	0.0782261
$966$	0	0
$967$	2432.00	0.0808768	0.0404384	−	0.999182i	$-0.487125\pi$
$967$	2432.00	0.0808768	0.0404384	+	0.999182i	$0.487125\pi$
$968$	4376.00	0.145300
$969$	0	0
$970$	4816.00	0.159415
$971$	24897.0	0.822845	0.411423	−	0.911445i	$-0.365032\pi$
$971$	24897.0	0.822845	0.411423	+	0.911445i	$0.365032\pi$
$972$	0	0
$973$	13328.0	0.439133
$974$	−26512.0	−0.872176
$975$	0	0
$976$	−3264.00	−0.107047
$977$	13944.0	0.456610	0.228305	−	0.973590i	$-0.426682\pi$
$977$	13944.0	0.456610	0.228305	+	0.973590i	$0.426682\pi$
$978$	0	0
$979$	−3192.00	−0.104205
$980$	1372.00	0.0447214
$981$	0	0
$982$	−8292.00	−0.269459
$983$	24507.0	0.795170	0.397585	−	0.917565i	$-0.369848\pi$
$983$	24507.0	0.795170	0.397585	+	0.917565i	$0.369848\pi$
$984$	0	0
$985$	−34720.0	−1.12312
$986$	18800.0	0.607215
$987$	0	0
$988$	19680.0	0.633709
$989$	−27542.0	−0.885525
$990$	0	0
$991$	−43903.0	−1.40729	−0.703645	−	0.710552i	$-0.748446\pi$
$991$	−43903.0	−1.40729	−0.703645	+	0.710552i	$0.748446\pi$
$992$	−5184.00	−0.165920
$993$	0	0
$994$	10360.0	0.330583
$995$	3416.00	0.108839
$996$	0	0
$997$	53948.0	1.71369	0.856846	−	0.515573i	$-0.172421\pi$
$997$	53948.0	1.71369	0.856846	+	0.515573i	$0.172421\pi$
$998$	5770.00	0.183012
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.a.c.1.1	✓	1	1.1	even	1	trivial
378.4.a.j.1.1	yes	1	3.2	odd	2

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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