LMFDB - Embedded newform 378.4.a.h.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)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -9.00000 q^{5} +7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} -9.00000 q^{5} +7.00000 q^{7} +8.00000 q^{8} -18.0000 q^{10} -45.0000 q^{11} -16.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -66.0000 q^{17} +11.0000 q^{19} -36.0000 q^{20} -90.0000 q^{22} -27.0000 q^{23} -44.0000 q^{25} -32.0000 q^{26} +28.0000 q^{28} +12.0000 q^{29} -169.000 q^{31} +32.0000 q^{32} -132.000 q^{34} -63.0000 q^{35} +209.000 q^{37} +22.0000 q^{38} -72.0000 q^{40} -291.000 q^{41} -394.000 q^{43} -180.000 q^{44} -54.0000 q^{46} -174.000 q^{47} +49.0000 q^{49} -88.0000 q^{50} -64.0000 q^{52} -228.000 q^{53} +405.000 q^{55} +56.0000 q^{56} +24.0000 q^{58} -474.000 q^{59} -232.000 q^{61} -338.000 q^{62} +64.0000 q^{64} +144.000 q^{65} +992.000 q^{67} -264.000 q^{68} -126.000 q^{70} +153.000 q^{71} +686.000 q^{73} +418.000 q^{74} +44.0000 q^{76} -315.000 q^{77} +1046.00 q^{79} -144.000 q^{80} -582.000 q^{82} -708.000 q^{83} +594.000 q^{85} -788.000 q^{86} -360.000 q^{88} -195.000 q^{89} -112.000 q^{91} -108.000 q^{92} -348.000 q^{94} -99.0000 q^{95} -88.0000 q^{97} +98.0000 q^{98} +O(q^{100})\)

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
$5$	−9.00000	−0.804984	−0.402492	−	0.915423i	$-0.631856\pi$
$5$	−9.00000	−0.804984	−0.402492	+	0.915423i	$0.631856\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	−18.0000	−0.569210
$11$	−45.0000	−1.23346	−0.616728	−	0.787177i	$-0.711542\pi$
$11$	−45.0000	−1.23346	−0.616728	+	0.787177i	$0.711542\pi$
$12$	0	0
$13$	−16.0000	−0.341354	−0.170677	−	0.985327i	$-0.554595\pi$
$13$	−16.0000	−0.341354	−0.170677	+	0.985327i	$0.554595\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
$17$	−66.0000	−0.941609	−0.470804	+	0.882238i	$0.656036\pi$
$18$	0	0
$19$	11.0000	0.132820	0.0664098	−	0.997792i	$-0.478846\pi$
$19$	11.0000	0.132820	0.0664098	+	0.997792i	$0.478846\pi$
$20$	−36.0000	−0.402492
$21$	0	0
$22$	−90.0000	−0.872185
$23$	−27.0000	−0.244778	−0.122389	−	0.992482i	$-0.539056\pi$
$23$	−27.0000	−0.244778	−0.122389	+	0.992482i	$0.539056\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	−32.0000	−0.241374
$27$	0	0
$28$	28.0000	0.188982
$29$	12.0000	0.0768395	0.0384197	−	0.999262i	$-0.487768\pi$
$29$	12.0000	0.0768395	0.0384197	+	0.999262i	$0.487768\pi$
$30$	0	0
$31$	−169.000	−0.979139	−0.489569	−	0.871964i	$-0.662846\pi$
$31$	−169.000	−0.979139	−0.489569	+	0.871964i	$0.662846\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−132.000	−0.665818
$35$	−63.0000	−0.304256
$36$	0	0
$37$	209.000	0.928632	0.464316	−	0.885670i	$-0.346300\pi$
$37$	209.000	0.928632	0.464316	+	0.885670i	$0.346300\pi$
$38$	22.0000	0.0939177
$39$	0	0
$40$	−72.0000	−0.284605
$41$	−291.000	−1.10845	−0.554226	−	0.832366i	$-0.686986\pi$
$41$	−291.000	−1.10845	−0.554226	+	0.832366i	$0.686986\pi$
$42$	0	0
$43$	−394.000	−1.39731	−0.698656	−	0.715458i	$-0.746218\pi$
$43$	−394.000	−1.39731	−0.698656	+	0.715458i	$0.746218\pi$
$44$	−180.000	−0.616728
$45$	0	0
$46$	−54.0000	−0.173084
$47$	−174.000	−0.540011	−0.270005	−	0.962859i	$-0.587025\pi$
$47$	−174.000	−0.540011	−0.270005	+	0.962859i	$0.587025\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−88.0000	−0.248902
$51$	0	0
$52$	−64.0000	−0.170677
$53$	−228.000	−0.590910	−0.295455	−	0.955357i	$-0.595471\pi$
$53$	−228.000	−0.590910	−0.295455	+	0.955357i	$0.595471\pi$
$54$	0	0
$55$	405.000	0.992913
$56$	56.0000	0.133631
$57$	0	0
$58$	24.0000	0.0543337
$59$	−474.000	−1.04592	−0.522962	−	0.852356i	$-0.675173\pi$
$59$	−474.000	−1.04592	−0.522962	+	0.852356i	$0.675173\pi$
$60$	0	0
$61$	−232.000	−0.486960	−0.243480	−	0.969906i	$-0.578289\pi$
$61$	−232.000	−0.486960	−0.243480	+	0.969906i	$0.578289\pi$
$62$	−338.000	−0.692356
$63$	0	0
$64$	64.0000	0.125000
$65$	144.000	0.274785
$66$	0	0
$67$	992.000	1.80884	0.904419	−	0.426646i	$-0.140305\pi$
$67$	992.000	1.80884	0.904419	+	0.426646i	$0.140305\pi$
$68$	−264.000	−0.470804
$69$	0	0
$70$	−126.000	−0.215141
$71$	153.000	0.255743	0.127872	−	0.991791i	$-0.459185\pi$
$71$	153.000	0.255743	0.127872	+	0.991791i	$0.459185\pi$
$72$	0	0
$73$	686.000	1.09987	0.549933	−	0.835209i	$-0.314653\pi$
$73$	686.000	1.09987	0.549933	+	0.835209i	$0.314653\pi$
$74$	418.000	0.656642
$75$	0	0
$76$	44.0000	0.0664098
$77$	−315.000	−0.466202
$78$	0	0
$79$	1046.00	1.48967	0.744837	−	0.667247i	$-0.232527\pi$
$79$	1046.00	1.48967	0.744837	+	0.667247i	$0.232527\pi$
$80$	−144.000	−0.201246
$81$	0	0
$82$	−582.000	−0.783794
$83$	−708.000	−0.936302	−0.468151	−	0.883648i	$-0.655080\pi$
$83$	−708.000	−0.936302	−0.468151	+	0.883648i	$0.655080\pi$
$84$	0	0
$85$	594.000	0.757981
$86$	−788.000	−0.988049
$87$	0	0
$88$	−360.000	−0.436092
$89$	−195.000	−0.232247	−0.116123	−	0.993235i	$-0.537047\pi$
$89$	−195.000	−0.232247	−0.116123	+	0.993235i	$0.537047\pi$
$90$	0	0
$91$	−112.000	−0.129020
$92$	−108.000	−0.122389
$93$	0	0
$94$	−348.000	−0.381845
$95$	−99.0000	−0.106918
$96$	0	0
$97$	−88.0000	−0.0921139	−0.0460569	−	0.998939i	$-0.514666\pi$
$97$	−88.0000	−0.0921139	−0.0460569	+	0.998939i	$0.514666\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	−176.000	−0.176000
$101$	1206.00	1.18813	0.594067	−	0.804416i	$-0.297521\pi$
$101$	1206.00	1.18813	0.594067	+	0.804416i	$0.297521\pi$
$102$	0	0
$103$	−1177.00	−1.12595	−0.562977	−	0.826473i	$-0.690344\pi$
$103$	−1177.00	−1.12595	−0.562977	+	0.826473i	$0.690344\pi$
$104$	−128.000	−0.120687
$105$	0	0
$106$	−456.000	−0.417836
$107$	156.000	0.140945	0.0704724	−	0.997514i	$-0.477549\pi$
$107$	156.000	0.140945	0.0704724	+	0.997514i	$0.477549\pi$
$108$	0	0
$109$	11.0000	0.00966614	0.00483307	−	0.999988i	$-0.498462\pi$
$109$	11.0000	0.00966614	0.00483307	+	0.999988i	$0.498462\pi$
$110$	810.000	0.702095
$111$	0	0
$112$	112.000	0.0944911
$113$	−1038.00	−0.864131	−0.432066	−	0.901842i	$-0.642215\pi$
$113$	−1038.00	−0.864131	−0.432066	+	0.901842i	$0.642215\pi$
$114$	0	0
$115$	243.000	0.197042
$116$	48.0000	0.0384197
$117$	0	0
$118$	−948.000	−0.739580
$119$	−462.000	−0.355895
$120$	0	0
$121$	694.000	0.521412
$122$	−464.000	−0.344333
$123$	0	0
$124$	−676.000	−0.489569
$125$	1521.00	1.08834
$126$	0	0
$127$	2216.00	1.54833	0.774166	−	0.632982i	$-0.218169\pi$
$127$	2216.00	1.54833	0.774166	+	0.632982i	$0.218169\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	288.000	0.194302
$131$	−918.000	−0.612260	−0.306130	−	0.951990i	$-0.599034\pi$
$131$	−918.000	−0.612260	−0.306130	+	0.951990i	$0.599034\pi$
$132$	0	0
$133$	77.0000	0.0502011
$134$	1984.00	1.27904
$135$	0	0
$136$	−528.000	−0.332909
$137$	2448.00	1.52662	0.763309	−	0.646033i	$-0.223573\pi$
$137$	2448.00	1.52662	0.763309	+	0.646033i	$0.223573\pi$
$138$	0	0
$139$	1244.00	0.759099	0.379549	−	0.925172i	$-0.376079\pi$
$139$	1244.00	0.759099	0.379549	+	0.925172i	$0.376079\pi$
$140$	−252.000	−0.152128
$141$	0	0
$142$	306.000	0.180838
$143$	720.000	0.421045
$144$	0	0
$145$	−108.000	−0.0618546
$146$	1372.00	0.777723
$147$	0	0
$148$	836.000	0.464316
$149$	954.000	0.524528	0.262264	−	0.964996i	$-0.415531\pi$
$149$	954.000	0.524528	0.262264	+	0.964996i	$0.415531\pi$
$150$	0	0
$151$	−1942.00	−1.04661	−0.523304	−	0.852146i	$-0.675301\pi$
$151$	−1942.00	−1.04661	−0.523304	+	0.852146i	$0.675301\pi$
$152$	88.0000	0.0469588
$153$	0	0
$154$	−630.000	−0.329655
$155$	1521.00	0.788191
$156$	0	0
$157$	−2032.00	−1.03294	−0.516469	−	0.856306i	$-0.672754\pi$
$157$	−2032.00	−1.03294	−0.516469	+	0.856306i	$0.672754\pi$
$158$	2092.00	1.05336
$159$	0	0
$160$	−288.000	−0.142302
$161$	−189.000	−0.0925173
$162$	0	0
$163$	2252.00	1.08215	0.541074	−	0.840975i	$-0.318018\pi$
$163$	2252.00	1.08215	0.541074	+	0.840975i	$0.318018\pi$
$164$	−1164.00	−0.554226
$165$	0	0
$166$	−1416.00	−0.662066
$167$	−4170.00	−1.93224	−0.966121	−	0.258091i	$-0.916907\pi$
$167$	−4170.00	−1.93224	−0.966121	+	0.258091i	$0.916907\pi$
$168$	0	0
$169$	−1941.00	−0.883477
$170$	1188.00	0.535973
$171$	0	0
$172$	−1576.00	−0.698656
$173$	4353.00	1.91302	0.956510	−	0.291700i	$-0.0942207\pi$
$173$	4353.00	1.91302	0.956510	+	0.291700i	$0.0942207\pi$
$174$	0	0
$175$	−308.000	−0.133043
$176$	−720.000	−0.308364
$177$	0	0
$178$	−390.000	−0.164223
$179$	3984.00	1.66357	0.831783	−	0.555102i	$-0.187321\pi$
$179$	3984.00	1.66357	0.831783	+	0.555102i	$0.187321\pi$
$180$	0	0
$181$	650.000	0.266929	0.133464	−	0.991054i	$-0.457390\pi$
$181$	650.000	0.266929	0.133464	+	0.991054i	$0.457390\pi$
$182$	−224.000	−0.0912307
$183$	0	0
$184$	−216.000	−0.0865420
$185$	−1881.00	−0.747534
$186$	0	0
$187$	2970.00	1.16143
$188$	−696.000	−0.270005
$189$	0	0
$190$	−198.000	−0.0756023
$191$	2967.00	1.12400	0.562002	−	0.827136i	$-0.310031\pi$
$191$	2967.00	1.12400	0.562002	+	0.827136i	$0.310031\pi$
$192$	0	0
$193$	−2842.00	−1.05996	−0.529978	−	0.848011i	$-0.677800\pi$
$193$	−2842.00	−1.05996	−0.529978	+	0.848011i	$0.677800\pi$
$194$	−176.000	−0.0651343
$195$	0	0
$196$	196.000	0.0714286
$197$	−2286.00	−0.826755	−0.413378	−	0.910560i	$-0.635651\pi$
$197$	−2286.00	−0.826755	−0.413378	+	0.910560i	$0.635651\pi$
$198$	0	0
$199$	−1645.00	−0.585985	−0.292992	−	0.956115i	$-0.594651\pi$
$199$	−1645.00	−0.585985	−0.292992	+	0.956115i	$0.594651\pi$
$200$	−352.000	−0.124451
$201$	0	0
$202$	2412.00	0.840137
$203$	84.0000	0.0290426
$204$	0	0
$205$	2619.00	0.892287
$206$	−2354.00	−0.796170
$207$	0	0
$208$	−256.000	−0.0853385
$209$	−495.000	−0.163827
$210$	0	0
$211$	−2878.00	−0.939003	−0.469502	−	0.882932i	$-0.655566\pi$
$211$	−2878.00	−0.939003	−0.469502	+	0.882932i	$0.655566\pi$
$212$	−912.000	−0.295455
$213$	0	0
$214$	312.000	0.0996630
$215$	3546.00	1.12481
$216$	0	0
$217$	−1183.00	−0.370080
$218$	22.0000	0.00683499
$219$	0	0
$220$	1620.00	0.496456
$221$	1056.00	0.321422
$222$	0	0
$223$	4349.00	1.30597	0.652983	−	0.757372i	$-0.273517\pi$
$223$	4349.00	1.30597	0.652983	+	0.757372i	$0.273517\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−2076.00	−0.611033
$227$	4266.00	1.24733	0.623666	−	0.781691i	$-0.285643\pi$
$227$	4266.00	1.24733	0.623666	+	0.781691i	$0.285643\pi$
$228$	0	0
$229$	−1060.00	−0.305881	−0.152941	−	0.988235i	$-0.548874\pi$
$229$	−1060.00	−0.305881	−0.152941	+	0.988235i	$0.548874\pi$
$230$	486.000	0.139330
$231$	0	0
$232$	96.0000	0.0271668
$233$	−2814.00	−0.791207	−0.395604	−	0.918421i	$-0.629465\pi$
$233$	−2814.00	−0.791207	−0.395604	+	0.918421i	$0.629465\pi$
$234$	0	0
$235$	1566.00	0.434700
$236$	−1896.00	−0.522962
$237$	0	0
$238$	−924.000	−0.251656
$239$	−1200.00	−0.324776	−0.162388	−	0.986727i	$-0.551920\pi$
$239$	−1200.00	−0.324776	−0.162388	+	0.986727i	$0.551920\pi$
$240$	0	0
$241$	−5650.00	−1.51016	−0.755080	−	0.655633i	$-0.772402\pi$
$241$	−5650.00	−1.51016	−0.755080	+	0.655633i	$0.772402\pi$
$242$	1388.00	0.368694
$243$	0	0
$244$	−928.000	−0.243480
$245$	−441.000	−0.114998
$246$	0	0
$247$	−176.000	−0.0453385
$248$	−1352.00	−0.346178
$249$	0	0
$250$	3042.00	0.769572
$251$	−2160.00	−0.543179	−0.271590	−	0.962413i	$-0.587549\pi$
$251$	−2160.00	−0.543179	−0.271590	+	0.962413i	$0.587549\pi$
$252$	0	0
$253$	1215.00	0.301923
$254$	4432.00	1.09484
$255$	0	0
$256$	256.000	0.0625000
$257$	2457.00	0.596356	0.298178	−	0.954510i	$-0.403621\pi$
$257$	2457.00	0.596356	0.298178	+	0.954510i	$0.403621\pi$
$258$	0	0
$259$	1463.00	0.350990
$260$	576.000	0.137392
$261$	0	0
$262$	−1836.00	−0.432933
$263$	2889.00	0.677351	0.338676	−	0.940903i	$-0.390021\pi$
$263$	2889.00	0.677351	0.338676	+	0.940903i	$0.390021\pi$
$264$	0	0
$265$	2052.00	0.475673
$266$	154.000	0.0354975
$267$	0	0
$268$	3968.00	0.904419
$269$	−7221.00	−1.63670	−0.818350	−	0.574721i	$-0.805111\pi$
$269$	−7221.00	−1.63670	−0.818350	+	0.574721i	$0.805111\pi$
$270$	0	0
$271$	2504.00	0.561281	0.280641	−	0.959813i	$-0.409453\pi$
$271$	2504.00	0.561281	0.280641	+	0.959813i	$0.409453\pi$
$272$	−1056.00	−0.235402
$273$	0	0
$274$	4896.00	1.07948
$275$	1980.00	0.434176
$276$	0	0
$277$	−2455.00	−0.532515	−0.266257	−	0.963902i	$-0.585787\pi$
$277$	−2455.00	−0.532515	−0.266257	+	0.963902i	$0.585787\pi$
$278$	2488.00	0.536764
$279$	0	0
$280$	−504.000	−0.107571
$281$	−8070.00	−1.71322	−0.856612	−	0.515961i	$-0.827435\pi$
$281$	−8070.00	−1.71322	−0.856612	+	0.515961i	$0.827435\pi$
$282$	0	0
$283$	1244.00	0.261301	0.130650	−	0.991429i	$-0.458293\pi$
$283$	1244.00	0.261301	0.130650	+	0.991429i	$0.458293\pi$
$284$	612.000	0.127872
$285$	0	0
$286$	1440.00	0.297724
$287$	−2037.00	−0.418956
$288$	0	0
$289$	−557.000	−0.113373
$290$	−216.000	−0.0437378
$291$	0	0
$292$	2744.00	0.549933
$293$	−3450.00	−0.687888	−0.343944	−	0.938990i	$-0.611763\pi$
$293$	−3450.00	−0.687888	−0.343944	+	0.938990i	$0.611763\pi$
$294$	0	0
$295$	4266.00	0.841953
$296$	1672.00	0.328321
$297$	0	0
$298$	1908.00	0.370898
$299$	432.000	0.0835559
$300$	0	0
$301$	−2758.00	−0.528134
$302$	−3884.00	−0.740063
$303$	0	0
$304$	176.000	0.0332049
$305$	2088.00	0.391995
$306$	0	0
$307$	−2761.00	−0.513285	−0.256643	−	0.966506i	$-0.582616\pi$
$307$	−2761.00	−0.513285	−0.256643	+	0.966506i	$0.582616\pi$
$308$	−1260.00	−0.233101
$309$	0	0
$310$	3042.00	0.557335
$311$	−7008.00	−1.27777	−0.638886	−	0.769301i	$-0.720605\pi$
$311$	−7008.00	−1.27777	−0.638886	+	0.769301i	$0.720605\pi$
$312$	0	0
$313$	−160.000	−0.0288937	−0.0144469	−	0.999896i	$-0.504599\pi$
$313$	−160.000	−0.0288937	−0.0144469	+	0.999896i	$0.504599\pi$
$314$	−4064.00	−0.730397
$315$	0	0
$316$	4184.00	0.744837
$317$	−750.000	−0.132884	−0.0664420	−	0.997790i	$-0.521165\pi$
$317$	−750.000	−0.132884	−0.0664420	+	0.997790i	$0.521165\pi$
$318$	0	0
$319$	−540.000	−0.0947780
$320$	−576.000	−0.100623
$321$	0	0
$322$	−378.000	−0.0654196
$323$	−726.000	−0.125064
$324$	0	0
$325$	704.000	0.120157
$326$	4504.00	0.765195
$327$	0	0
$328$	−2328.00	−0.391897
$329$	−1218.00	−0.204105
$330$	0	0
$331$	3188.00	0.529391	0.264695	−	0.964332i	$-0.414729\pi$
$331$	3188.00	0.529391	0.264695	+	0.964332i	$0.414729\pi$
$332$	−2832.00	−0.468151
$333$	0	0
$334$	−8340.00	−1.36630
$335$	−8928.00	−1.45609
$336$	0	0
$337$	−8917.00	−1.44136	−0.720682	−	0.693265i	$-0.756171\pi$
$337$	−8917.00	−1.44136	−0.720682	+	0.693265i	$0.756171\pi$
$338$	−3882.00	−0.624713
$339$	0	0
$340$	2376.00	0.378990
$341$	7605.00	1.20772
$342$	0	0
$343$	343.000	0.0539949
$344$	−3152.00	−0.494025
$345$	0	0
$346$	8706.00	1.35271
$347$	7491.00	1.15890	0.579449	−	0.815008i	$-0.303268\pi$
$347$	7491.00	1.15890	0.579449	+	0.815008i	$0.303268\pi$
$348$	0	0
$349$	−7450.00	−1.14266	−0.571331	−	0.820719i	$-0.693573\pi$
$349$	−7450.00	−1.14266	−0.571331	+	0.820719i	$0.693573\pi$
$350$	−616.000	−0.0940760
$351$	0	0
$352$	−1440.00	−0.218046
$353$	−9225.00	−1.39093	−0.695463	−	0.718561i	$-0.744801\pi$
$353$	−9225.00	−1.39093	−0.695463	+	0.718561i	$0.744801\pi$
$354$	0	0
$355$	−1377.00	−0.205869
$356$	−780.000	−0.116123
$357$	0	0
$358$	7968.00	1.17632
$359$	−3660.00	−0.538071	−0.269035	−	0.963130i	$-0.586705\pi$
$359$	−3660.00	−0.538071	−0.269035	+	0.963130i	$0.586705\pi$
$360$	0	0
$361$	−6738.00	−0.982359
$362$	1300.00	0.188747
$363$	0	0
$364$	−448.000	−0.0645098
$365$	−6174.00	−0.885375
$366$	0	0
$367$	−10213.0	−1.45263	−0.726314	−	0.687363i	$-0.758768\pi$
$367$	−10213.0	−1.45263	−0.726314	+	0.687363i	$0.758768\pi$
$368$	−432.000	−0.0611944
$369$	0	0
$370$	−3762.00	−0.528587
$371$	−1596.00	−0.223343
$372$	0	0
$373$	−8629.00	−1.19784	−0.598918	−	0.800811i	$-0.704402\pi$
$373$	−8629.00	−1.19784	−0.598918	+	0.800811i	$0.704402\pi$
$374$	5940.00	0.821257
$375$	0	0
$376$	−1392.00	−0.190923
$377$	−192.000	−0.0262295
$378$	0	0
$379$	1226.00	0.166162	0.0830810	−	0.996543i	$-0.473524\pi$
$379$	1226.00	0.166162	0.0830810	+	0.996543i	$0.473524\pi$
$380$	−396.000	−0.0534589
$381$	0	0
$382$	5934.00	0.794790
$383$	−10998.0	−1.46729	−0.733644	−	0.679534i	$-0.762182\pi$
$383$	−10998.0	−1.46729	−0.733644	+	0.679534i	$0.762182\pi$
$384$	0	0
$385$	2835.00	0.375286
$386$	−5684.00	−0.749503
$387$	0	0
$388$	−352.000	−0.0460569
$389$	2772.00	0.361301	0.180650	−	0.983547i	$-0.442180\pi$
$389$	2772.00	0.361301	0.180650	+	0.983547i	$0.442180\pi$
$390$	0	0
$391$	1782.00	0.230485
$392$	392.000	0.0505076
$393$	0	0
$394$	−4572.00	−0.584604
$395$	−9414.00	−1.19916
$396$	0	0
$397$	−6118.00	−0.773435	−0.386717	−	0.922198i	$-0.626391\pi$
$397$	−6118.00	−0.773435	−0.386717	+	0.922198i	$0.626391\pi$
$398$	−3290.00	−0.414354
$399$	0	0
$400$	−704.000	−0.0880000
$401$	14076.0	1.75292	0.876461	−	0.481472i	$-0.159898\pi$
$401$	14076.0	1.75292	0.876461	+	0.481472i	$0.159898\pi$
$402$	0	0
$403$	2704.00	0.334233
$404$	4824.00	0.594067
$405$	0	0
$406$	168.000	0.0205362
$407$	−9405.00	−1.14543
$408$	0	0
$409$	542.000	0.0655261	0.0327631	−	0.999463i	$-0.489569\pi$
$409$	542.000	0.0655261	0.0327631	+	0.999463i	$0.489569\pi$
$410$	5238.00	0.630942
$411$	0	0
$412$	−4708.00	−0.562977
$413$	−3318.00	−0.395322
$414$	0	0
$415$	6372.00	0.753709
$416$	−512.000	−0.0603434
$417$	0	0
$418$	−990.000	−0.115843
$419$	6090.00	0.710062	0.355031	−	0.934855i	$-0.384470\pi$
$419$	6090.00	0.710062	0.355031	+	0.934855i	$0.384470\pi$
$420$	0	0
$421$	−14335.0	−1.65949	−0.829745	−	0.558143i	$-0.811514\pi$
$421$	−14335.0	−1.65949	−0.829745	+	0.558143i	$0.811514\pi$
$422$	−5756.00	−0.663976
$423$	0	0
$424$	−1824.00	−0.208918
$425$	2904.00	0.331446
$426$	0	0
$427$	−1624.00	−0.184054
$428$	624.000	0.0704724
$429$	0	0
$430$	7092.00	0.795364
$431$	16755.0	1.87253	0.936264	−	0.351296i	$-0.114259\pi$
$431$	16755.0	1.87253	0.936264	+	0.351296i	$0.114259\pi$
$432$	0	0
$433$	7436.00	0.825292	0.412646	−	0.910892i	$-0.364605\pi$
$433$	7436.00	0.825292	0.412646	+	0.910892i	$0.364605\pi$
$434$	−2366.00	−0.261686
$435$	0	0
$436$	44.0000	0.00483307
$437$	−297.000	−0.0325113
$438$	0	0
$439$	1352.00	0.146987	0.0734937	−	0.997296i	$-0.476585\pi$
$439$	1352.00	0.146987	0.0734937	+	0.997296i	$0.476585\pi$
$440$	3240.00	0.351048
$441$	0	0
$442$	2112.00	0.227280
$443$	−147.000	−0.0157656	−0.00788282	−	0.999969i	$-0.502509\pi$
$443$	−147.000	−0.0157656	−0.00788282	+	0.999969i	$0.502509\pi$
$444$	0	0
$445$	1755.00	0.186955
$446$	8698.00	0.923458
$447$	0	0
$448$	448.000	0.0472456
$449$	10638.0	1.11813	0.559063	−	0.829125i	$-0.311161\pi$
$449$	10638.0	1.11813	0.559063	+	0.829125i	$0.311161\pi$
$450$	0	0
$451$	13095.0	1.36723
$452$	−4152.00	−0.432066
$453$	0	0
$454$	8532.00	0.881997
$455$	1008.00	0.103859
$456$	0	0
$457$	−3499.00	−0.358154	−0.179077	−	0.983835i	$-0.557311\pi$
$457$	−3499.00	−0.358154	−0.179077	+	0.983835i	$0.557311\pi$
$458$	−2120.00	−0.216291
$459$	0	0
$460$	972.000	0.0985212
$461$	−4119.00	−0.416141	−0.208070	−	0.978114i	$-0.566718\pi$
$461$	−4119.00	−0.416141	−0.208070	+	0.978114i	$0.566718\pi$
$462$	0	0
$463$	−13318.0	−1.33680	−0.668402	−	0.743801i	$-0.733021\pi$
$463$	−13318.0	−1.33680	−0.668402	+	0.743801i	$0.733021\pi$
$464$	192.000	0.0192099
$465$	0	0
$466$	−5628.00	−0.559468
$467$	3306.00	0.327588	0.163794	−	0.986495i	$-0.447627\pi$
$467$	3306.00	0.327588	0.163794	+	0.986495i	$0.447627\pi$
$468$	0	0
$469$	6944.00	0.683676
$470$	3132.00	0.307380
$471$	0	0
$472$	−3792.00	−0.369790
$473$	17730.0	1.72352
$474$	0	0
$475$	−484.000	−0.0467525
$476$	−1848.00	−0.177947
$477$	0	0
$478$	−2400.00	−0.229652
$479$	−2376.00	−0.226643	−0.113322	−	0.993558i	$-0.536149\pi$
$479$	−2376.00	−0.226643	−0.113322	+	0.993558i	$0.536149\pi$
$480$	0	0
$481$	−3344.00	−0.316992
$482$	−11300.0	−1.06784
$483$	0	0
$484$	2776.00	0.260706
$485$	792.000	0.0741502
$486$	0	0
$487$	14186.0	1.31998	0.659989	−	0.751276i	$-0.270561\pi$
$487$	14186.0	1.31998	0.659989	+	0.751276i	$0.270561\pi$
$488$	−1856.00	−0.172166
$489$	0	0
$490$	−882.000	−0.0813157
$491$	10599.0	0.974188	0.487094	−	0.873350i	$-0.338057\pi$
$491$	10599.0	0.974188	0.487094	+	0.873350i	$0.338057\pi$
$492$	0	0
$493$	−792.000	−0.0723527
$494$	−352.000	−0.0320592
$495$	0	0
$496$	−2704.00	−0.244785
$497$	1071.00	0.0966618
$498$	0	0
$499$	−3022.00	−0.271109	−0.135554	−	0.990770i	$-0.543282\pi$
$499$	−3022.00	−0.271109	−0.135554	+	0.990770i	$0.543282\pi$
$500$	6084.00	0.544170
$501$	0	0
$502$	−4320.00	−0.384086
$503$	−8244.00	−0.730779	−0.365389	−	0.930855i	$-0.619064\pi$
$503$	−8244.00	−0.730779	−0.365389	+	0.930855i	$0.619064\pi$
$504$	0	0
$505$	−10854.0	−0.956429
$506$	2430.00	0.213491
$507$	0	0
$508$	8864.00	0.774166
$509$	8406.00	0.732003	0.366001	−	0.930614i	$-0.380727\pi$
$509$	8406.00	0.732003	0.366001	+	0.930614i	$0.380727\pi$
$510$	0	0
$511$	4802.00	0.415710
$512$	512.000	0.0441942
$513$	0	0
$514$	4914.00	0.421687
$515$	10593.0	0.906375
$516$	0	0
$517$	7830.00	0.666079
$518$	2926.00	0.248187
$519$	0	0
$520$	1152.00	0.0971510
$521$	−7005.00	−0.589049	−0.294525	−	0.955644i	$-0.595161\pi$
$521$	−7005.00	−0.589049	−0.294525	+	0.955644i	$0.595161\pi$
$522$	0	0
$523$	18695.0	1.56305	0.781525	−	0.623874i	$-0.214442\pi$
$523$	18695.0	1.56305	0.781525	+	0.623874i	$0.214442\pi$
$524$	−3672.00	−0.306130
$525$	0	0
$526$	5778.00	0.478960
$527$	11154.0	0.921966
$528$	0	0
$529$	−11438.0	−0.940084
$530$	4104.00	0.336352
$531$	0	0
$532$	308.000	0.0251006
$533$	4656.00	0.378375
$534$	0	0
$535$	−1404.00	−0.113458
$536$	7936.00	0.639521
$537$	0	0
$538$	−14442.0	−1.15732
$539$	−2205.00	−0.176208
$540$	0	0
$541$	6113.00	0.485801	0.242901	−	0.970051i	$-0.421901\pi$
$541$	6113.00	0.485801	0.242901	+	0.970051i	$0.421901\pi$
$542$	5008.00	0.396886
$543$	0	0
$544$	−2112.00	−0.166455
$545$	−99.0000	−0.00778109
$546$	0	0
$547$	2036.00	0.159146	0.0795732	−	0.996829i	$-0.474644\pi$
$547$	2036.00	0.159146	0.0795732	+	0.996829i	$0.474644\pi$
$548$	9792.00	0.763309
$549$	0	0
$550$	3960.00	0.307009
$551$	132.000	0.0102058
$552$	0	0
$553$	7322.00	0.563044
$554$	−4910.00	−0.376545
$555$	0	0
$556$	4976.00	0.379549
$557$	−11262.0	−0.856708	−0.428354	−	0.903611i	$-0.640906\pi$
$557$	−11262.0	−0.856708	−0.428354	+	0.903611i	$0.640906\pi$
$558$	0	0
$559$	6304.00	0.476978
$560$	−1008.00	−0.0760639
$561$	0	0
$562$	−16140.0	−1.21143
$563$	11028.0	0.825532	0.412766	−	0.910837i	$-0.364563\pi$
$563$	11028.0	0.825532	0.412766	+	0.910837i	$0.364563\pi$
$564$	0	0
$565$	9342.00	0.695612
$566$	2488.00	0.184768
$567$	0	0
$568$	1224.00	0.0904188
$569$	−10686.0	−0.787312	−0.393656	−	0.919258i	$-0.628790\pi$
$569$	−10686.0	−0.787312	−0.393656	+	0.919258i	$0.628790\pi$
$570$	0	0
$571$	2018.00	0.147900	0.0739498	−	0.997262i	$-0.476440\pi$
$571$	2018.00	0.147900	0.0739498	+	0.997262i	$0.476440\pi$
$572$	2880.00	0.210522
$573$	0	0
$574$	−4074.00	−0.296246
$575$	1188.00	0.0861618
$576$	0	0
$577$	−10006.0	−0.721933	−0.360966	−	0.932579i	$-0.617553\pi$
$577$	−10006.0	−0.721933	−0.360966	+	0.932579i	$0.617553\pi$
$578$	−1114.00	−0.0801666
$579$	0	0
$580$	−432.000	−0.0309273
$581$	−4956.00	−0.353889
$582$	0	0
$583$	10260.0	0.728861
$584$	5488.00	0.388861
$585$	0	0
$586$	−6900.00	−0.486410
$587$	−13254.0	−0.931944	−0.465972	−	0.884799i	$-0.654295\pi$
$587$	−13254.0	−0.931944	−0.465972	+	0.884799i	$0.654295\pi$
$588$	0	0
$589$	−1859.00	−0.130049
$590$	8532.00	0.595351
$591$	0	0
$592$	3344.00	0.232158
$593$	23187.0	1.60569	0.802847	−	0.596186i	$-0.203318\pi$
$593$	23187.0	1.60569	0.802847	+	0.596186i	$0.203318\pi$
$594$	0	0
$595$	4158.00	0.286490
$596$	3816.00	0.262264
$597$	0	0
$598$	864.000	0.0590829
$599$	27255.0	1.85911	0.929557	−	0.368679i	$-0.120190\pi$
$599$	27255.0	1.85911	0.929557	+	0.368679i	$0.120190\pi$
$600$	0	0
$601$	−21976.0	−1.49155	−0.745773	−	0.666200i	$-0.767920\pi$
$601$	−21976.0	−1.49155	−0.745773	+	0.666200i	$0.767920\pi$
$602$	−5516.00	−0.373447
$603$	0	0
$604$	−7768.00	−0.523304
$605$	−6246.00	−0.419729
$606$	0	0
$607$	13052.0	0.872758	0.436379	−	0.899763i	$-0.356261\pi$
$607$	13052.0	0.872758	0.436379	+	0.899763i	$0.356261\pi$
$608$	352.000	0.0234794
$609$	0	0
$610$	4176.00	0.277182
$611$	2784.00	0.184335
$612$	0	0
$613$	−2923.00	−0.192592	−0.0962960	−	0.995353i	$-0.530700\pi$
$613$	−2923.00	−0.192592	−0.0962960	+	0.995353i	$0.530700\pi$
$614$	−5522.00	−0.362948
$615$	0	0
$616$	−2520.00	−0.164827
$617$	3714.00	0.242334	0.121167	−	0.992632i	$-0.461336\pi$
$617$	3714.00	0.242334	0.121167	+	0.992632i	$0.461336\pi$
$618$	0	0
$619$	17705.0	1.14963	0.574817	−	0.818282i	$-0.305073\pi$
$619$	17705.0	1.14963	0.574817	+	0.818282i	$0.305073\pi$
$620$	6084.00	0.394096
$621$	0	0
$622$	−14016.0	−0.903522
$623$	−1365.00	−0.0877810
$624$	0	0
$625$	−8189.00	−0.524096
$626$	−320.000	−0.0204309
$627$	0	0
$628$	−8128.00	−0.516469
$629$	−13794.0	−0.874408
$630$	0	0
$631$	−19978.0	−1.26040	−0.630199	−	0.776433i	$-0.717027\pi$
$631$	−19978.0	−1.26040	−0.630199	+	0.776433i	$0.717027\pi$
$632$	8368.00	0.526679
$633$	0	0
$634$	−1500.00	−0.0939631
$635$	−19944.0	−1.24638
$636$	0	0
$637$	−784.000	−0.0487649
$638$	−1080.00	−0.0670182
$639$	0	0
$640$	−1152.00	−0.0711512
$641$	11502.0	0.708739	0.354369	−	0.935105i	$-0.384696\pi$
$641$	11502.0	0.708739	0.354369	+	0.935105i	$0.384696\pi$
$642$	0	0
$643$	31439.0	1.92820	0.964100	−	0.265538i	$-0.0855496\pi$
$643$	31439.0	1.92820	0.964100	+	0.265538i	$0.0855496\pi$
$644$	−756.000	−0.0462587
$645$	0	0
$646$	−1452.00	−0.0884337
$647$	−10650.0	−0.647132	−0.323566	−	0.946206i	$-0.604882\pi$
$647$	−10650.0	−0.647132	−0.323566	+	0.946206i	$0.604882\pi$
$648$	0	0
$649$	21330.0	1.29010
$650$	1408.00	0.0849635
$651$	0	0
$652$	9008.00	0.541074
$653$	9774.00	0.585737	0.292868	−	0.956153i	$-0.405390\pi$
$653$	9774.00	0.585737	0.292868	+	0.956153i	$0.405390\pi$
$654$	0	0
$655$	8262.00	0.492860
$656$	−4656.00	−0.277113
$657$	0	0
$658$	−2436.00	−0.144324
$659$	−13857.0	−0.819108	−0.409554	−	0.912286i	$-0.634316\pi$
$659$	−13857.0	−0.819108	−0.409554	+	0.912286i	$0.634316\pi$
$660$	0	0
$661$	3908.00	0.229960	0.114980	−	0.993368i	$-0.463320\pi$
$661$	3908.00	0.229960	0.114980	+	0.993368i	$0.463320\pi$
$662$	6376.00	0.374336
$663$	0	0
$664$	−5664.00	−0.331033
$665$	−693.000	−0.0404111
$666$	0	0
$667$	−324.000	−0.0188086
$668$	−16680.0	−0.966121
$669$	0	0
$670$	−17856.0	−1.02961
$671$	10440.0	0.600643
$672$	0	0
$673$	26570.0	1.52184	0.760920	−	0.648846i	$-0.224748\pi$
$673$	26570.0	1.52184	0.760920	+	0.648846i	$0.224748\pi$
$674$	−17834.0	−1.01920
$675$	0	0
$676$	−7764.00	−0.441739
$677$	1725.00	0.0979278	0.0489639	−	0.998801i	$-0.484408\pi$
$677$	1725.00	0.0979278	0.0489639	+	0.998801i	$0.484408\pi$
$678$	0	0
$679$	−616.000	−0.0348158
$680$	4752.00	0.267987
$681$	0	0
$682$	15210.0	0.853990
$683$	−31071.0	−1.74070	−0.870350	−	0.492433i	$-0.836108\pi$
$683$	−31071.0	−1.74070	−0.870350	+	0.492433i	$0.836108\pi$
$684$	0	0
$685$	−22032.0	−1.22890
$686$	686.000	0.0381802
$687$	0	0
$688$	−6304.00	−0.349328
$689$	3648.00	0.201709
$690$	0	0
$691$	19964.0	1.09908	0.549541	−	0.835466i	$-0.314802\pi$
$691$	19964.0	1.09908	0.549541	+	0.835466i	$0.314802\pi$
$692$	17412.0	0.956510
$693$	0	0
$694$	14982.0	0.819465
$695$	−11196.0	−0.611063
$696$	0	0
$697$	19206.0	1.04373
$698$	−14900.0	−0.807985
$699$	0	0
$700$	−1232.00	−0.0665217
$701$	25884.0	1.39462	0.697308	−	0.716772i	$-0.254381\pi$
$701$	25884.0	1.39462	0.697308	+	0.716772i	$0.254381\pi$
$702$	0	0
$703$	2299.00	0.123341
$704$	−2880.00	−0.154182
$705$	0	0
$706$	−18450.0	−0.983534
$707$	8442.00	0.449072
$708$	0	0
$709$	−23821.0	−1.26180	−0.630900	−	0.775864i	$-0.717314\pi$
$709$	−23821.0	−1.26180	−0.630900	+	0.775864i	$0.717314\pi$
$710$	−2754.00	−0.145572
$711$	0	0
$712$	−1560.00	−0.0821116
$713$	4563.00	0.239671
$714$	0	0
$715$	−6480.00	−0.338935
$716$	15936.0	0.831783
$717$	0	0
$718$	−7320.00	−0.380474
$719$	28254.0	1.46550	0.732751	−	0.680497i	$-0.238236\pi$
$719$	28254.0	1.46550	0.732751	+	0.680497i	$0.238236\pi$
$720$	0	0
$721$	−8239.00	−0.425571
$722$	−13476.0	−0.694633
$723$	0	0
$724$	2600.00	0.133464
$725$	−528.000	−0.0270475
$726$	0	0
$727$	−17152.0	−0.875010	−0.437505	−	0.899216i	$-0.644138\pi$
$727$	−17152.0	−0.875010	−0.437505	+	0.899216i	$0.644138\pi$
$728$	−896.000	−0.0456153
$729$	0	0
$730$	−12348.0	−0.626055
$731$	26004.0	1.31572
$732$	0	0
$733$	254.000	0.0127991	0.00639953	−	0.999980i	$-0.497963\pi$
$733$	254.000	0.0127991	0.00639953	+	0.999980i	$0.497963\pi$
$734$	−20426.0	−1.02716
$735$	0	0
$736$	−864.000	−0.0432710
$737$	−44640.0	−2.23112
$738$	0	0
$739$	−12382.0	−0.616345	−0.308173	−	0.951330i	$-0.599717\pi$
$739$	−12382.0	−0.616345	−0.308173	+	0.951330i	$0.599717\pi$
$740$	−7524.00	−0.373767
$741$	0	0
$742$	−3192.00	−0.157927
$743$	−34671.0	−1.71192	−0.855959	−	0.517043i	$-0.827033\pi$
$743$	−34671.0	−1.71192	−0.855959	+	0.517043i	$0.827033\pi$
$744$	0	0
$745$	−8586.00	−0.422237
$746$	−17258.0	−0.846998
$747$	0	0
$748$	11880.0	0.580716
$749$	1092.00	0.0532721
$750$	0	0
$751$	−28150.0	−1.36779	−0.683894	−	0.729582i	$-0.739715\pi$
$751$	−28150.0	−1.36779	−0.683894	+	0.729582i	$0.739715\pi$
$752$	−2784.00	−0.135003
$753$	0	0
$754$	−384.000	−0.0185470
$755$	17478.0	0.842503
$756$	0	0
$757$	−26638.0	−1.27896	−0.639481	−	0.768807i	$-0.720851\pi$
$757$	−26638.0	−1.27896	−0.639481	+	0.768807i	$0.720851\pi$
$758$	2452.00	0.117494
$759$	0	0
$760$	−792.000	−0.0378011
$761$	−26154.0	−1.24584	−0.622918	−	0.782287i	$-0.714053\pi$
$761$	−26154.0	−1.24584	−0.622918	+	0.782287i	$0.714053\pi$
$762$	0	0
$763$	77.0000	0.00365346
$764$	11868.0	0.562002
$765$	0	0
$766$	−21996.0	−1.03753
$767$	7584.00	0.357030
$768$	0	0
$769$	36416.0	1.70767	0.853833	−	0.520548i	$-0.174272\pi$
$769$	36416.0	1.70767	0.853833	+	0.520548i	$0.174272\pi$
$770$	5670.00	0.265367
$771$	0	0
$772$	−11368.0	−0.529978
$773$	16917.0	0.787144	0.393572	−	0.919294i	$-0.371239\pi$
$773$	16917.0	0.787144	0.393572	+	0.919294i	$0.371239\pi$
$774$	0	0
$775$	7436.00	0.344657
$776$	−704.000	−0.0325672
$777$	0	0
$778$	5544.00	0.255478
$779$	−3201.00	−0.147224
$780$	0	0
$781$	−6885.00	−0.315448
$782$	3564.00	0.162977
$783$	0	0
$784$	784.000	0.0357143
$785$	18288.0	0.831499
$786$	0	0
$787$	−20032.0	−0.907324	−0.453662	−	0.891174i	$-0.649883\pi$
$787$	−20032.0	−0.907324	−0.453662	+	0.891174i	$0.649883\pi$
$788$	−9144.00	−0.413378
$789$	0	0
$790$	−18828.0	−0.847937
$791$	−7266.00	−0.326611
$792$	0	0
$793$	3712.00	0.166226
$794$	−12236.0	−0.546901
$795$	0	0
$796$	−6580.00	−0.292992
$797$	−34095.0	−1.51532	−0.757658	−	0.652652i	$-0.773656\pi$
$797$	−34095.0	−1.51532	−0.757658	+	0.652652i	$0.773656\pi$
$798$	0	0
$799$	11484.0	0.508479
$800$	−1408.00	−0.0622254
$801$	0	0
$802$	28152.0	1.23950
$803$	−30870.0	−1.35664
$804$	0	0
$805$	1701.00	0.0744750
$806$	5408.00	0.236338
$807$	0	0
$808$	9648.00	0.420069
$809$	−37164.0	−1.61510	−0.807550	−	0.589798i	$-0.799207\pi$
$809$	−37164.0	−1.61510	−0.807550	+	0.589798i	$0.799207\pi$
$810$	0	0
$811$	17525.0	0.758799	0.379399	−	0.925233i	$-0.376131\pi$
$811$	17525.0	0.758799	0.379399	+	0.925233i	$0.376131\pi$
$812$	336.000	0.0145213
$813$	0	0
$814$	−18810.0	−0.809939
$815$	−20268.0	−0.871113
$816$	0	0
$817$	−4334.00	−0.185591
$818$	1084.00	0.0463340
$819$	0	0
$820$	10476.0	0.446144
$821$	2778.00	0.118091	0.0590456	−	0.998255i	$-0.481194\pi$
$821$	2778.00	0.118091	0.0590456	+	0.998255i	$0.481194\pi$
$822$	0	0
$823$	−5470.00	−0.231679	−0.115840	−	0.993268i	$-0.536956\pi$
$823$	−5470.00	−0.231679	−0.115840	+	0.993268i	$0.536956\pi$
$824$	−9416.00	−0.398085
$825$	0	0
$826$	−6636.00	−0.279535
$827$	−27711.0	−1.16518	−0.582591	−	0.812765i	$-0.697961\pi$
$827$	−27711.0	−1.16518	−0.582591	+	0.812765i	$0.697961\pi$
$828$	0	0
$829$	−6550.00	−0.274416	−0.137208	−	0.990542i	$-0.543813\pi$
$829$	−6550.00	−0.274416	−0.137208	+	0.990542i	$0.543813\pi$
$830$	12744.0	0.532953
$831$	0	0
$832$	−1024.00	−0.0426692
$833$	−3234.00	−0.134516
$834$	0	0
$835$	37530.0	1.55542
$836$	−1980.00	−0.0819136
$837$	0	0
$838$	12180.0	0.502090
$839$	7068.00	0.290840	0.145420	−	0.989370i	$-0.453547\pi$
$839$	7068.00	0.290840	0.145420	+	0.989370i	$0.453547\pi$
$840$	0	0
$841$	−24245.0	−0.994096
$842$	−28670.0	−1.17344
$843$	0	0
$844$	−11512.0	−0.469502
$845$	17469.0	0.711186
$846$	0	0
$847$	4858.00	0.197075
$848$	−3648.00	−0.147727
$849$	0	0
$850$	5808.00	0.234368
$851$	−5643.00	−0.227309
$852$	0	0
$853$	41528.0	1.66693	0.833465	−	0.552572i	$-0.186354\pi$
$853$	41528.0	1.66693	0.833465	+	0.552572i	$0.186354\pi$
$854$	−3248.00	−0.130146
$855$	0	0
$856$	1248.00	0.0498315
$857$	−32751.0	−1.30543	−0.652715	−	0.757604i	$-0.726370\pi$
$857$	−32751.0	−1.30543	−0.652715	+	0.757604i	$0.726370\pi$
$858$	0	0
$859$	15995.0	0.635323	0.317661	−	0.948204i	$-0.397102\pi$
$859$	15995.0	0.635323	0.317661	+	0.948204i	$0.397102\pi$
$860$	14184.0	0.562407
$861$	0	0
$862$	33510.0	1.32408
$863$	15528.0	0.612490	0.306245	−	0.951953i	$-0.400927\pi$
$863$	15528.0	0.612490	0.306245	+	0.951953i	$0.400927\pi$
$864$	0	0
$865$	−39177.0	−1.53995
$866$	14872.0	0.583569
$867$	0	0
$868$	−4732.00	−0.185040
$869$	−47070.0	−1.83745
$870$	0	0
$871$	−15872.0	−0.617454
$872$	88.0000	0.00341750
$873$	0	0
$874$	−594.000	−0.0229890
$875$	10647.0	0.411353
$876$	0	0
$877$	40214.0	1.54838	0.774191	−	0.632953i	$-0.218157\pi$
$877$	40214.0	1.54838	0.774191	+	0.632953i	$0.218157\pi$
$878$	2704.00	0.103936
$879$	0	0
$880$	6480.00	0.248228
$881$	24435.0	0.934434	0.467217	−	0.884143i	$-0.345257\pi$
$881$	24435.0	0.934434	0.467217	+	0.884143i	$0.345257\pi$
$882$	0	0
$883$	−18736.0	−0.714062	−0.357031	−	0.934093i	$-0.616211\pi$
$883$	−18736.0	−0.714062	−0.357031	+	0.934093i	$0.616211\pi$
$884$	4224.00	0.160711
$885$	0	0
$886$	−294.000	−0.0111480
$887$	−23940.0	−0.906231	−0.453115	−	0.891452i	$-0.649687\pi$
$887$	−23940.0	−0.906231	−0.453115	+	0.891452i	$0.649687\pi$
$888$	0	0
$889$	15512.0	0.585215
$890$	3510.00	0.132197
$891$	0	0
$892$	17396.0	0.652983
$893$	−1914.00	−0.0717240
$894$	0	0
$895$	−35856.0	−1.33914
$896$	896.000	0.0334077
$897$	0	0
$898$	21276.0	0.790634
$899$	−2028.00	−0.0752365
$900$	0	0
$901$	15048.0	0.556406
$902$	26190.0	0.966776
$903$	0	0
$904$	−8304.00	−0.305517
$905$	−5850.00	−0.214874
$906$	0	0
$907$	31358.0	1.14799	0.573994	−	0.818859i	$-0.305393\pi$
$907$	31358.0	1.14799	0.573994	+	0.818859i	$0.305393\pi$
$908$	17064.0	0.623666
$909$	0	0
$910$	2016.00	0.0734393
$911$	9408.00	0.342153	0.171076	−	0.985258i	$-0.445276\pi$
$911$	9408.00	0.342153	0.171076	+	0.985258i	$0.445276\pi$
$912$	0	0
$913$	31860.0	1.15489
$914$	−6998.00	−0.253253
$915$	0	0
$916$	−4240.00	−0.152941
$917$	−6426.00	−0.231412
$918$	0	0
$919$	−6748.00	−0.242215	−0.121108	−	0.992639i	$-0.538645\pi$
$919$	−6748.00	−0.242215	−0.121108	+	0.992639i	$0.538645\pi$
$920$	1944.00	0.0696650
$921$	0	0
$922$	−8238.00	−0.294256
$923$	−2448.00	−0.0872989
$924$	0	0
$925$	−9196.00	−0.326879
$926$	−26636.0	−0.945263
$927$	0	0
$928$	384.000	0.0135834
$929$	−18246.0	−0.644383	−0.322192	−	0.946675i	$-0.604420\pi$
$929$	−18246.0	−0.644383	−0.322192	+	0.946675i	$0.604420\pi$
$930$	0	0
$931$	539.000	0.0189742
$932$	−11256.0	−0.395604
$933$	0	0
$934$	6612.00	0.231639
$935$	−26730.0	−0.934935
$936$	0	0
$937$	19748.0	0.688516	0.344258	−	0.938875i	$-0.388131\pi$
$937$	19748.0	0.688516	0.344258	+	0.938875i	$0.388131\pi$
$938$	13888.0	0.483432
$939$	0	0
$940$	6264.00	0.217350
$941$	55875.0	1.93568	0.967839	−	0.251571i	$-0.0809470\pi$
$941$	55875.0	1.93568	0.967839	+	0.251571i	$0.0809470\pi$
$942$	0	0
$943$	7857.00	0.271325
$944$	−7584.00	−0.261481
$945$	0	0
$946$	35460.0	1.21871
$947$	37977.0	1.30315	0.651577	−	0.758583i	$-0.274108\pi$
$947$	37977.0	1.30315	0.651577	+	0.758583i	$0.274108\pi$
$948$	0	0
$949$	−10976.0	−0.375444
$950$	−968.000	−0.0330590
$951$	0	0
$952$	−3696.00	−0.125828
$953$	28332.0	0.963026	0.481513	−	0.876439i	$-0.340087\pi$
$953$	28332.0	0.963026	0.481513	+	0.876439i	$0.340087\pi$
$954$	0	0
$955$	−26703.0	−0.904805
$956$	−4800.00	−0.162388
$957$	0	0
$958$	−4752.00	−0.160261
$959$	17136.0	0.577008
$960$	0	0
$961$	−1230.00	−0.0412876
$962$	−6688.00	−0.224147
$963$	0	0
$964$	−22600.0	−0.755080
$965$	25578.0	0.853249
$966$	0	0
$967$	−50902.0	−1.69276	−0.846380	−	0.532580i	$-0.821222\pi$
$967$	−50902.0	−1.69276	−0.846380	+	0.532580i	$0.821222\pi$
$968$	5552.00	0.184347
$969$	0	0
$970$	1584.00	0.0524321
$971$	54444.0	1.79937	0.899686	−	0.436537i	$-0.143795\pi$
$971$	54444.0	1.79937	0.899686	+	0.436537i	$0.143795\pi$
$972$	0	0
$973$	8708.00	0.286912
$974$	28372.0	0.933365
$975$	0	0
$976$	−3712.00	−0.121740
$977$	−34836.0	−1.14074	−0.570370	−	0.821388i	$-0.693200\pi$
$977$	−34836.0	−1.14074	−0.570370	+	0.821388i	$0.693200\pi$
$978$	0	0
$979$	8775.00	0.286466
$980$	−1764.00	−0.0574989
$981$	0	0
$982$	21198.0	0.688855
$983$	34974.0	1.13479	0.567394	−	0.823446i	$-0.307952\pi$
$983$	34974.0	1.13479	0.567394	+	0.823446i	$0.307952\pi$
$984$	0	0
$985$	20574.0	0.665525
$986$	−1584.00	−0.0511611
$987$	0	0
$988$	−704.000	−0.0226693
$989$	10638.0	0.342031
$990$	0	0
$991$	−36844.0	−1.18102	−0.590509	−	0.807031i	$-0.701073\pi$
$991$	−36844.0	−1.18102	−0.590509	+	0.807031i	$0.701073\pi$
$992$	−5408.00	−0.173089
$993$	0	0
$994$	2142.00	0.0683502
$995$	14805.0	0.471709
$996$	0	0
$997$	−45970.0	−1.46027	−0.730133	−	0.683305i	$-0.760542\pi$
$997$	−45970.0	−1.46027	−0.730133	+	0.683305i	$0.760542\pi$
$998$	−6044.00	−0.191703
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.a.e.1.1	✓	1	3.2	odd	2
378.4.a.h.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

Related objects

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