LMFDB - Embedded newform 390.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,4,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("390.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	390.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:	$23.0107449022$
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	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} +24.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} +24.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} +12.0000 q^{12} +13.0000 q^{13} -48.0000 q^{14} -15.0000 q^{15} +16.0000 q^{16} +50.0000 q^{17} -18.0000 q^{18} +28.0000 q^{19} -20.0000 q^{20} +72.0000 q^{21} -208.000 q^{23} -24.0000 q^{24} +25.0000 q^{25} -26.0000 q^{26} +27.0000 q^{27} +96.0000 q^{28} +190.000 q^{29} +30.0000 q^{30} +248.000 q^{31} -32.0000 q^{32} -100.000 q^{34} -120.000 q^{35} +36.0000 q^{36} -186.000 q^{37} -56.0000 q^{38} +39.0000 q^{39} +40.0000 q^{40} -194.000 q^{41} -144.000 q^{42} +348.000 q^{43} -45.0000 q^{45} +416.000 q^{46} +260.000 q^{47} +48.0000 q^{48} +233.000 q^{49} -50.0000 q^{50} +150.000 q^{51} +52.0000 q^{52} +462.000 q^{53} -54.0000 q^{54} -192.000 q^{56} +84.0000 q^{57} -380.000 q^{58} -520.000 q^{59} -60.0000 q^{60} -506.000 q^{61} -496.000 q^{62} +216.000 q^{63} +64.0000 q^{64} -65.0000 q^{65} +772.000 q^{67} +200.000 q^{68} -624.000 q^{69} +240.000 q^{70} +780.000 q^{71} -72.0000 q^{72} -62.0000 q^{73} +372.000 q^{74} +75.0000 q^{75} +112.000 q^{76} -78.0000 q^{78} +736.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} +388.000 q^{82} +1464.00 q^{83} +288.000 q^{84} -250.000 q^{85} -696.000 q^{86} +570.000 q^{87} +406.000 q^{89} +90.0000 q^{90} +312.000 q^{91} -832.000 q^{92} +744.000 q^{93} -520.000 q^{94} -140.000 q^{95} -96.0000 q^{96} +922.000 q^{97} -466.000 q^{98} +O(q^{100})\)

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−6.00000	−0.408248
$7$	24.0000	1.29588	0.647939	−	0.761692i	$-0.275631\pi$
$7$	24.0000	1.29588	0.647939	+	0.761692i	$0.275631\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	10.0000	0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	12.0000	0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−48.0000	−0.916324
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	50.0000	0.713340	0.356670	−	0.934230i	$-0.383912\pi$
$17$	50.0000	0.713340	0.356670	+	0.934230i	$0.383912\pi$
$18$	−18.0000	−0.235702
$19$	28.0000	0.338086	0.169043	−	0.985609i	$-0.445932\pi$
$19$	28.0000	0.338086	0.169043	+	0.985609i	$0.445932\pi$
$20$	−20.0000	−0.223607
$21$	72.0000	0.748176
$22$	0	0
$23$	−208.000	−1.88570	−0.942848	−	0.333224i	$-0.891864\pi$
$23$	−208.000	−1.88570	−0.942848	+	0.333224i	$0.891864\pi$
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	−26.0000	−0.196116
$27$	27.0000	0.192450
$28$	96.0000	0.647939
$29$	190.000	1.21662	0.608312	−	0.793698i	$-0.291847\pi$
$29$	190.000	1.21662	0.608312	+	0.793698i	$0.291847\pi$
$30$	30.0000	0.182574
$31$	248.000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	248.000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−100.000	−0.504408
$35$	−120.000	−0.579534
$36$	36.0000	0.166667
$37$	−186.000	−0.826438	−0.413219	−	0.910632i	$-0.635596\pi$
$37$	−186.000	−0.826438	−0.413219	+	0.910632i	$0.635596\pi$
$38$	−56.0000	−0.239063
$39$	39.0000	0.160128
$40$	40.0000	0.158114
$41$	−194.000	−0.738969	−0.369484	−	0.929237i	$-0.620466\pi$
$41$	−194.000	−0.738969	−0.369484	+	0.929237i	$0.620466\pi$
$42$	−144.000	−0.529040
$43$	348.000	1.23417	0.617087	−	0.786895i	$-0.288313\pi$
$43$	348.000	1.23417	0.617087	+	0.786895i	$0.288313\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	416.000	1.33339
$47$	260.000	0.806913	0.403456	−	0.914999i	$-0.367809\pi$
$47$	260.000	0.806913	0.403456	+	0.914999i	$0.367809\pi$
$48$	48.0000	0.144338
$49$	233.000	0.679300
$50$	−50.0000	−0.141421
$51$	150.000	0.411847
$52$	52.0000	0.138675
$53$	462.000	1.19737	0.598685	−	0.800985i	$-0.295690\pi$
$53$	462.000	1.19737	0.598685	+	0.800985i	$0.295690\pi$
$54$	−54.0000	−0.136083
$55$	0	0
$56$	−192.000	−0.458162
$57$	84.0000	0.195194
$58$	−380.000	−0.860284
$59$	−520.000	−1.14743	−0.573714	−	0.819056i	$-0.694498\pi$
$59$	−520.000	−1.14743	−0.573714	+	0.819056i	$0.694498\pi$
$60$	−60.0000	−0.129099
$61$	−506.000	−1.06208	−0.531038	−	0.847348i	$-0.678198\pi$
$61$	−506.000	−1.06208	−0.531038	+	0.847348i	$0.678198\pi$
$62$	−496.000	−1.01600
$63$	216.000	0.431959
$64$	64.0000	0.125000
$65$	−65.0000	−0.124035
$66$	0	0
$67$	772.000	1.40768	0.703842	−	0.710357i	$-0.251466\pi$
$67$	772.000	1.40768	0.703842	+	0.710357i	$0.251466\pi$
$68$	200.000	0.356670
$69$	−624.000	−1.08871
$70$	240.000	0.409793
$71$	780.000	1.30379	0.651894	−	0.758310i	$-0.273975\pi$
$71$	780.000	1.30379	0.651894	+	0.758310i	$0.273975\pi$
$72$	−72.0000	−0.117851
$73$	−62.0000	−0.0994048	−0.0497024	−	0.998764i	$-0.515827\pi$
$73$	−62.0000	−0.0994048	−0.0497024	+	0.998764i	$0.515827\pi$
$74$	372.000	0.584380
$75$	75.0000	0.115470
$76$	112.000	0.169043
$77$	0	0
$78$	−78.0000	−0.113228
$79$	736.000	1.04818	0.524092	−	0.851662i	$-0.324405\pi$
$79$	736.000	1.04818	0.524092	+	0.851662i	$0.324405\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	388.000	0.522530
$83$	1464.00	1.93608	0.968041	−	0.250790i	$-0.0806905\pi$
$83$	1464.00	1.93608	0.968041	+	0.250790i	$0.0806905\pi$
$84$	288.000	0.374088
$85$	−250.000	−0.319015
$86$	−696.000	−0.872693
$87$	570.000	0.702419
$88$	0	0
$89$	406.000	0.483550	0.241775	−	0.970332i	$-0.422270\pi$
$89$	406.000	0.483550	0.241775	+	0.970332i	$0.422270\pi$
$90$	90.0000	0.105409
$91$	312.000	0.359412
$92$	−832.000	−0.942848
$93$	744.000	0.829561
$94$	−520.000	−0.570573
$95$	−140.000	−0.151197
$96$	−96.0000	−0.102062
$97$	922.000	0.965102	0.482551	−	0.875868i	$-0.339710\pi$
$97$	922.000	0.965102	0.482551	+	0.875868i	$0.339710\pi$
$98$	−466.000	−0.480338
$99$	0	0
$100$	100.000	0.100000
$101$	−522.000	−0.514267	−0.257133	−	0.966376i	$-0.582778\pi$
$101$	−522.000	−0.514267	−0.257133	+	0.966376i	$0.582778\pi$
$102$	−300.000	−0.291220
$103$	992.000	0.948977	0.474489	−	0.880262i	$-0.342633\pi$
$103$	992.000	0.948977	0.474489	+	0.880262i	$0.342633\pi$
$104$	−104.000	−0.0980581
$105$	−360.000	−0.334594
$106$	−924.000	−0.846668
$107$	−1284.00	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−1284.00	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	108.000	0.0962250
$109$	2134.00	1.87523	0.937615	−	0.347675i	$-0.113029\pi$
$109$	2134.00	1.87523	0.937615	+	0.347675i	$0.113029\pi$
$110$	0	0
$111$	−558.000	−0.477144
$112$	384.000	0.323970
$113$	826.000	0.687642	0.343821	−	0.939035i	$-0.388279\pi$
$113$	826.000	0.687642	0.343821	+	0.939035i	$0.388279\pi$
$114$	−168.000	−0.138023
$115$	1040.00	0.843309
$116$	760.000	0.608312
$117$	117.000	0.0924500
$118$	1040.00	0.811354
$119$	1200.00	0.924402
$120$	120.000	0.0912871
$121$	−1331.00	−1.00000
$122$	1012.00	0.751001
$123$	−582.000	−0.426644
$124$	992.000	0.718421
$125$	−125.000	−0.0894427
$126$	−432.000	−0.305441
$127$	−2192.00	−1.53156	−0.765782	−	0.643101i	$-0.777648\pi$
$127$	−2192.00	−1.53156	−0.765782	+	0.643101i	$0.777648\pi$
$128$	−128.000	−0.0883883
$129$	1044.00	0.712551
$130$	130.000	0.0877058
$131$	−2444.00	−1.63002	−0.815012	−	0.579443i	$-0.803270\pi$
$131$	−2444.00	−1.63002	−0.815012	+	0.579443i	$0.803270\pi$
$132$	0	0
$133$	672.000	0.438119
$134$	−1544.00	−0.995383
$135$	−135.000	−0.0860663
$136$	−400.000	−0.252204
$137$	−738.000	−0.460231	−0.230115	−	0.973163i	$-0.573910\pi$
$137$	−738.000	−0.460231	−0.230115	+	0.973163i	$0.573910\pi$
$138$	1248.00	0.769832
$139$	172.000	0.104956	0.0524779	−	0.998622i	$-0.483288\pi$
$139$	172.000	0.104956	0.0524779	+	0.998622i	$0.483288\pi$
$140$	−480.000	−0.289767
$141$	780.000	0.465871
$142$	−1560.00	−0.921918
$143$	0	0
$144$	144.000	0.0833333
$145$	−950.000	−0.544091
$146$	124.000	0.0702898
$147$	699.000	0.392194
$148$	−744.000	−0.413219
$149$	−3078.00	−1.69235	−0.846173	−	0.532908i	$-0.821099\pi$
$149$	−3078.00	−1.69235	−0.846173	+	0.532908i	$0.821099\pi$
$150$	−150.000	−0.0816497
$151$	−2432.00	−1.31068	−0.655342	−	0.755332i	$-0.727476\pi$
$151$	−2432.00	−1.31068	−0.655342	+	0.755332i	$0.727476\pi$
$152$	−224.000	−0.119532
$153$	450.000	0.237780
$154$	0	0
$155$	−1240.00	−0.642575
$156$	156.000	0.0800641
$157$	54.0000	0.0274501	0.0137251	−	0.999906i	$-0.495631\pi$
$157$	54.0000	0.0274501	0.0137251	+	0.999906i	$0.495631\pi$
$158$	−1472.00	−0.741177
$159$	1386.00	0.691302
$160$	160.000	0.0790569
$161$	−4992.00	−2.44363
$162$	−162.000	−0.0785674
$163$	1532.00	0.736169	0.368084	−	0.929792i	$-0.380014\pi$
$163$	1532.00	0.736169	0.368084	+	0.929792i	$0.380014\pi$
$164$	−776.000	−0.369484
$165$	0	0
$166$	−2928.00	−1.36902
$167$	364.000	0.168666	0.0843328	−	0.996438i	$-0.473124\pi$
$167$	364.000	0.168666	0.0843328	+	0.996438i	$0.473124\pi$
$168$	−576.000	−0.264520
$169$	169.000	0.0769231
$170$	500.000	0.225578
$171$	252.000	0.112695
$172$	1392.00	0.617087
$173$	−786.000	−0.345425	−0.172712	−	0.984972i	$-0.555253\pi$
$173$	−786.000	−0.345425	−0.172712	+	0.984972i	$0.555253\pi$
$174$	−1140.00	−0.496685
$175$	600.000	0.259176
$176$	0	0
$177$	−1560.00	−0.662468
$178$	−812.000	−0.341921
$179$	−3764.00	−1.57170	−0.785851	−	0.618416i	$-0.787775\pi$
$179$	−3764.00	−1.57170	−0.785851	+	0.618416i	$0.787775\pi$
$180$	−180.000	−0.0745356
$181$	142.000	0.0583137	0.0291568	−	0.999575i	$-0.490718\pi$
$181$	142.000	0.0583137	0.0291568	+	0.999575i	$0.490718\pi$
$182$	−624.000	−0.254143
$183$	−1518.00	−0.613190
$184$	1664.00	0.666694
$185$	930.000	0.369594
$186$	−1488.00	−0.586588
$187$	0	0
$188$	1040.00	0.403456
$189$	648.000	0.249392
$190$	280.000	0.106912
$191$	−2400.00	−0.909204	−0.454602	−	0.890695i	$-0.650218\pi$
$191$	−2400.00	−0.909204	−0.454602	+	0.890695i	$0.650218\pi$
$192$	192.000	0.0721688
$193$	−3726.00	−1.38965	−0.694827	−	0.719177i	$-0.744519\pi$
$193$	−3726.00	−1.38965	−0.694827	+	0.719177i	$0.744519\pi$
$194$	−1844.00	−0.682430
$195$	−195.000	−0.0716115
$196$	932.000	0.339650
$197$	2146.00	0.776123	0.388061	−	0.921634i	$-0.373145\pi$
$197$	2146.00	0.776123	0.388061	+	0.921634i	$0.373145\pi$
$198$	0	0
$199$	−1944.00	−0.692495	−0.346248	−	0.938143i	$-0.612544\pi$
$199$	−1944.00	−0.692495	−0.346248	+	0.938143i	$0.612544\pi$
$200$	−200.000	−0.0707107
$201$	2316.00	0.812727
$202$	1044.00	0.363642
$203$	4560.00	1.57660
$204$	600.000	0.205924
$205$	970.000	0.330477
$206$	−1984.00	−0.671028
$207$	−1872.00	−0.628565
$208$	208.000	0.0693375
$209$	0	0
$210$	720.000	0.236594
$211$	−1604.00	−0.523336	−0.261668	−	0.965158i	$-0.584273\pi$
$211$	−1604.00	−0.523336	−0.261668	+	0.965158i	$0.584273\pi$
$212$	1848.00	0.598685
$213$	2340.00	0.752743
$214$	2568.00	0.820303
$215$	−1740.00	−0.551940
$216$	−216.000	−0.0680414
$217$	5952.00	1.86197
$218$	−4268.00	−1.32599
$219$	−186.000	−0.0573914
$220$	0	0
$221$	650.000	0.197845
$222$	1116.00	0.337392
$223$	40.0000	0.0120117	0.00600583	−	0.999982i	$-0.498088\pi$
$223$	40.0000	0.0120117	0.00600583	+	0.999982i	$0.498088\pi$
$224$	−768.000	−0.229081
$225$	225.000	0.0666667
$226$	−1652.00	−0.486236
$227$	1904.00	0.556709	0.278354	−	0.960478i	$-0.410211\pi$
$227$	1904.00	0.556709	0.278354	+	0.960478i	$0.410211\pi$
$228$	336.000	0.0975971
$229$	1454.00	0.419577	0.209788	−	0.977747i	$-0.432723\pi$
$229$	1454.00	0.419577	0.209788	+	0.977747i	$0.432723\pi$
$230$	−2080.00	−0.596309
$231$	0	0
$232$	−1520.00	−0.430142
$233$	1890.00	0.531408	0.265704	−	0.964055i	$-0.414396\pi$
$233$	1890.00	0.531408	0.265704	+	0.964055i	$0.414396\pi$
$234$	−234.000	−0.0653720
$235$	−1300.00	−0.360862
$236$	−2080.00	−0.573714
$237$	2208.00	0.605169
$238$	−2400.00	−0.653651
$239$	1940.00	0.525055	0.262528	−	0.964924i	$-0.415444\pi$
$239$	1940.00	0.525055	0.262528	+	0.964924i	$0.415444\pi$
$240$	−240.000	−0.0645497
$241$	−3574.00	−0.955276	−0.477638	−	0.878557i	$-0.658507\pi$
$241$	−3574.00	−0.955276	−0.477638	+	0.878557i	$0.658507\pi$
$242$	2662.00	0.707107
$243$	243.000	0.0641500
$244$	−2024.00	−0.531038
$245$	−1165.00	−0.303792
$246$	1164.00	0.301683
$247$	364.000	0.0937683
$248$	−1984.00	−0.508001
$249$	4392.00	1.11780
$250$	250.000	0.0632456
$251$	4220.00	1.06121	0.530606	−	0.847619i	$-0.321965\pi$
$251$	4220.00	1.06121	0.530606	+	0.847619i	$0.321965\pi$
$252$	864.000	0.215980
$253$	0	0
$254$	4384.00	1.08298
$255$	−750.000	−0.184184
$256$	256.000	0.0625000
$257$	−934.000	−0.226698	−0.113349	−	0.993555i	$-0.536158\pi$
$257$	−934.000	−0.226698	−0.113349	+	0.993555i	$0.536158\pi$
$258$	−2088.00	−0.503850
$259$	−4464.00	−1.07096
$260$	−260.000	−0.0620174
$261$	1710.00	0.405542
$262$	4888.00	1.15260
$263$	−5200.00	−1.21919	−0.609593	−	0.792715i	$-0.708667\pi$
$263$	−5200.00	−1.21919	−0.609593	+	0.792715i	$0.708667\pi$
$264$	0	0
$265$	−2310.00	−0.535480
$266$	−1344.00	−0.309797
$267$	1218.00	0.279177
$268$	3088.00	0.703842
$269$	4158.00	0.942445	0.471223	−	0.882014i	$-0.343813\pi$
$269$	4158.00	0.942445	0.471223	+	0.882014i	$0.343813\pi$
$270$	270.000	0.0608581
$271$	−3712.00	−0.832059	−0.416029	−	0.909351i	$-0.636579\pi$
$271$	−3712.00	−0.832059	−0.416029	+	0.909351i	$0.636579\pi$
$272$	800.000	0.178335
$273$	936.000	0.207507
$274$	1476.00	0.325432
$275$	0	0
$276$	−2496.00	−0.544353
$277$	−4178.00	−0.906252	−0.453126	−	0.891447i	$-0.649691\pi$
$277$	−4178.00	−0.906252	−0.453126	+	0.891447i	$0.649691\pi$
$278$	−344.000	−0.0742149
$279$	2232.00	0.478947
$280$	960.000	0.204896
$281$	5630.00	1.19522	0.597611	−	0.801786i	$-0.296117\pi$
$281$	5630.00	1.19522	0.597611	+	0.801786i	$0.296117\pi$
$282$	−1560.00	−0.329421
$283$	916.000	0.192405	0.0962024	−	0.995362i	$-0.469330\pi$
$283$	916.000	0.192405	0.0962024	+	0.995362i	$0.469330\pi$
$284$	3120.00	0.651894
$285$	−420.000	−0.0872935
$286$	0	0
$287$	−4656.00	−0.957613
$288$	−288.000	−0.0589256
$289$	−2413.00	−0.491146
$290$	1900.00	0.384730
$291$	2766.00	0.557202
$292$	−248.000	−0.0497024
$293$	−198.000	−0.0394788	−0.0197394	−	0.999805i	$-0.506284\pi$
$293$	−198.000	−0.0394788	−0.0197394	+	0.999805i	$0.506284\pi$
$294$	−1398.00	−0.277323
$295$	2600.00	0.513145
$296$	1488.00	0.292190
$297$	0	0
$298$	6156.00	1.19667
$299$	−2704.00	−0.522998
$300$	300.000	0.0577350
$301$	8352.00	1.59934
$302$	4864.00	0.926794
$303$	−1566.00	−0.296912
$304$	448.000	0.0845216
$305$	2530.00	0.474975
$306$	−900.000	−0.168136
$307$	7492.00	1.39280	0.696402	−	0.717652i	$-0.254783\pi$
$307$	7492.00	1.39280	0.696402	+	0.717652i	$0.254783\pi$
$308$	0	0
$309$	2976.00	0.547892
$310$	2480.00	0.454369
$311$	−8736.00	−1.59284	−0.796420	−	0.604744i	$-0.793275\pi$
$311$	−8736.00	−1.59284	−0.796420	+	0.604744i	$0.793275\pi$
$312$	−312.000	−0.0566139
$313$	−7590.00	−1.37065	−0.685323	−	0.728239i	$-0.740339\pi$
$313$	−7590.00	−1.37065	−0.685323	+	0.728239i	$0.740339\pi$
$314$	−108.000	−0.0194102
$315$	−1080.00	−0.193178
$316$	2944.00	0.524092
$317$	−3446.00	−0.610557	−0.305279	−	0.952263i	$-0.598750\pi$
$317$	−3446.00	−0.610557	−0.305279	+	0.952263i	$0.598750\pi$
$318$	−2772.00	−0.488824
$319$	0	0
$320$	−320.000	−0.0559017
$321$	−3852.00	−0.669775
$322$	9984.00	1.72791
$323$	1400.00	0.241171
$324$	324.000	0.0555556
$325$	325.000	0.0554700
$326$	−3064.00	−0.520550
$327$	6402.00	1.08266
$328$	1552.00	0.261265
$329$	6240.00	1.04566
$330$	0	0
$331$	−11348.0	−1.88442	−0.942209	−	0.335025i	$-0.891255\pi$
$331$	−11348.0	−1.88442	−0.942209	+	0.335025i	$0.891255\pi$
$332$	5856.00	0.968041
$333$	−1674.00	−0.275479
$334$	−728.000	−0.119265
$335$	−3860.00	−0.629535
$336$	1152.00	0.187044
$337$	−702.000	−0.113473	−0.0567365	−	0.998389i	$-0.518069\pi$
$337$	−702.000	−0.113473	−0.0567365	+	0.998389i	$0.518069\pi$
$338$	−338.000	−0.0543928
$339$	2478.00	0.397010
$340$	−1000.00	−0.159508
$341$	0	0
$342$	−504.000	−0.0796877
$343$	−2640.00	−0.415588
$344$	−2784.00	−0.436347
$345$	3120.00	0.486885
$346$	1572.00	0.244252
$347$	2708.00	0.418943	0.209471	−	0.977815i	$-0.432826\pi$
$347$	2708.00	0.418943	0.209471	+	0.977815i	$0.432826\pi$
$348$	2280.00	0.351209
$349$	8990.00	1.37886	0.689432	−	0.724350i	$-0.257860\pi$
$349$	8990.00	1.37886	0.689432	+	0.724350i	$0.257860\pi$
$350$	−1200.00	−0.183265
$351$	351.000	0.0533761
$352$	0	0
$353$	8358.00	1.26020	0.630101	−	0.776513i	$-0.283013\pi$
$353$	8358.00	1.26020	0.630101	+	0.776513i	$0.283013\pi$
$354$	3120.00	0.468435
$355$	−3900.00	−0.583072
$356$	1624.00	0.241775
$357$	3600.00	0.533704
$358$	7528.00	1.11136
$359$	−4196.00	−0.616870	−0.308435	−	0.951245i	$-0.599805\pi$
$359$	−4196.00	−0.616870	−0.308435	+	0.951245i	$0.599805\pi$
$360$	360.000	0.0527046
$361$	−6075.00	−0.885698
$362$	−284.000	−0.0412340
$363$	−3993.00	−0.577350
$364$	1248.00	0.179706
$365$	310.000	0.0444552
$366$	3036.00	0.433591
$367$	696.000	0.0989943	0.0494971	−	0.998774i	$-0.484238\pi$
$367$	696.000	0.0989943	0.0494971	+	0.998774i	$0.484238\pi$
$368$	−3328.00	−0.471424
$369$	−1746.00	−0.246323
$370$	−1860.00	−0.261343
$371$	11088.0	1.55164
$372$	2976.00	0.414781
$373$	9438.00	1.31014	0.655069	−	0.755569i	$-0.272640\pi$
$373$	9438.00	1.31014	0.655069	+	0.755569i	$0.272640\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	−2080.00	−0.285287
$377$	2470.00	0.337431
$378$	−1296.00	−0.176347
$379$	−5676.00	−0.769278	−0.384639	−	0.923067i	$-0.625674\pi$
$379$	−5676.00	−0.769278	−0.384639	+	0.923067i	$0.625674\pi$
$380$	−560.000	−0.0755984
$381$	−6576.00	−0.884249
$382$	4800.00	0.642904
$383$	10556.0	1.40832	0.704160	−	0.710042i	$-0.251324\pi$
$383$	10556.0	1.40832	0.704160	+	0.710042i	$0.251324\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	7452.00	0.982634
$387$	3132.00	0.411391
$388$	3688.00	0.482551
$389$	2814.00	0.366775	0.183387	−	0.983041i	$-0.441294\pi$
$389$	2814.00	0.366775	0.183387	+	0.983041i	$0.441294\pi$
$390$	390.000	0.0506370
$391$	−10400.0	−1.34514
$392$	−1864.00	−0.240169
$393$	−7332.00	−0.941095
$394$	−4292.00	−0.548802
$395$	−3680.00	−0.468762
$396$	0	0
$397$	−9618.00	−1.21590	−0.607952	−	0.793974i	$-0.708009\pi$
$397$	−9618.00	−1.21590	−0.607952	+	0.793974i	$0.708009\pi$
$398$	3888.00	0.489668
$399$	2016.00	0.252948
$400$	400.000	0.0500000
$401$	1182.00	0.147198	0.0735988	−	0.997288i	$-0.476552\pi$
$401$	1182.00	0.147198	0.0735988	+	0.997288i	$0.476552\pi$
$402$	−4632.00	−0.574684
$403$	3224.00	0.398508
$404$	−2088.00	−0.257133
$405$	−405.000	−0.0496904
$406$	−9120.00	−1.11482
$407$	0	0
$408$	−1200.00	−0.145610
$409$	−7774.00	−0.939852	−0.469926	−	0.882706i	$-0.655719\pi$
$409$	−7774.00	−0.939852	−0.469926	+	0.882706i	$0.655719\pi$
$410$	−1940.00	−0.233682
$411$	−2214.00	−0.265714
$412$	3968.00	0.474489
$413$	−12480.0	−1.48693
$414$	3744.00	0.444463
$415$	−7320.00	−0.865843
$416$	−416.000	−0.0490290
$417$	516.000	0.0605962
$418$	0	0
$419$	−156.000	−0.0181888	−0.00909439	−	0.999959i	$-0.502895\pi$
$419$	−156.000	−0.0181888	−0.00909439	+	0.999959i	$0.502895\pi$
$420$	−1440.00	−0.167297
$421$	−1298.00	−0.150263	−0.0751314	−	0.997174i	$-0.523938\pi$
$421$	−1298.00	−0.150263	−0.0751314	+	0.997174i	$0.523938\pi$
$422$	3208.00	0.370054
$423$	2340.00	0.268971
$424$	−3696.00	−0.423334
$425$	1250.00	0.142668
$426$	−4680.00	−0.532269
$427$	−12144.0	−1.37632
$428$	−5136.00	−0.580042
$429$	0	0
$430$	3480.00	0.390280
$431$	−1716.00	−0.191779	−0.0958896	−	0.995392i	$-0.530570\pi$
$431$	−1716.00	−0.191779	−0.0958896	+	0.995392i	$0.530570\pi$
$432$	432.000	0.0481125
$433$	5122.00	0.568470	0.284235	−	0.958755i	$-0.408260\pi$
$433$	5122.00	0.568470	0.284235	+	0.958755i	$0.408260\pi$
$434$	−11904.0	−1.31661
$435$	−2850.00	−0.314131
$436$	8536.00	0.937615
$437$	−5824.00	−0.637528
$438$	372.000	0.0405818
$439$	288.000	0.0313109	0.0156555	−	0.999877i	$-0.495017\pi$
$439$	288.000	0.0313109	0.0156555	+	0.999877i	$0.495017\pi$
$440$	0	0
$441$	2097.00	0.226433
$442$	−1300.00	−0.139897
$443$	10620.0	1.13899	0.569494	−	0.821996i	$-0.307139\pi$
$443$	10620.0	1.13899	0.569494	+	0.821996i	$0.307139\pi$
$444$	−2232.00	−0.238572
$445$	−2030.00	−0.216250
$446$	−80.0000	−0.00849352
$447$	−9234.00	−0.977077
$448$	1536.00	0.161985
$449$	−2850.00	−0.299554	−0.149777	−	0.988720i	$-0.547856\pi$
$449$	−2850.00	−0.299554	−0.149777	+	0.988720i	$0.547856\pi$
$450$	−450.000	−0.0471405
$451$	0	0
$452$	3304.00	0.343821
$453$	−7296.00	−0.756724
$454$	−3808.00	−0.393653
$455$	−1560.00	−0.160734
$456$	−672.000	−0.0690116
$457$	754.000	0.0771786	0.0385893	−	0.999255i	$-0.487714\pi$
$457$	754.000	0.0771786	0.0385893	+	0.999255i	$0.487714\pi$
$458$	−2908.00	−0.296685
$459$	1350.00	0.137282
$460$	4160.00	0.421654
$461$	16538.0	1.67083	0.835414	−	0.549622i	$-0.185228\pi$
$461$	16538.0	1.67083	0.835414	+	0.549622i	$0.185228\pi$
$462$	0	0
$463$	800.000	0.0803005	0.0401503	−	0.999194i	$-0.487216\pi$
$463$	800.000	0.0803005	0.0401503	+	0.999194i	$0.487216\pi$
$464$	3040.00	0.304156
$465$	−3720.00	−0.370991
$466$	−3780.00	−0.375762
$467$	6692.00	0.663102	0.331551	−	0.943437i	$-0.392428\pi$
$467$	6692.00	0.663102	0.331551	+	0.943437i	$0.392428\pi$
$468$	468.000	0.0462250
$469$	18528.0	1.82419
$470$	2600.00	0.255168
$471$	162.000	0.0158483
$472$	4160.00	0.405677
$473$	0	0
$474$	−4416.00	−0.427919
$475$	700.000	0.0676173
$476$	4800.00	0.462201
$477$	4158.00	0.399123
$478$	−3880.00	−0.371270
$479$	9980.00	0.951979	0.475989	−	0.879451i	$-0.342090\pi$
$479$	9980.00	0.951979	0.475989	+	0.879451i	$0.342090\pi$
$480$	480.000	0.0456435
$481$	−2418.00	−0.229213
$482$	7148.00	0.675482
$483$	−14976.0	−1.41083
$484$	−5324.00	−0.500000
$485$	−4610.00	−0.431607
$486$	−486.000	−0.0453609
$487$	5344.00	0.497248	0.248624	−	0.968600i	$-0.420022\pi$
$487$	5344.00	0.497248	0.248624	+	0.968600i	$0.420022\pi$
$488$	4048.00	0.375501
$489$	4596.00	0.425027
$490$	2330.00	0.214814
$491$	−11044.0	−1.01509	−0.507545	−	0.861626i	$-0.669447\pi$
$491$	−11044.0	−1.01509	−0.507545	+	0.861626i	$0.669447\pi$
$492$	−2328.00	−0.213322
$493$	9500.00	0.867867
$494$	−728.000	−0.0663042
$495$	0	0
$496$	3968.00	0.359211
$497$	18720.0	1.68955
$498$	−8784.00	−0.790403
$499$	−18844.0	−1.69053	−0.845264	−	0.534349i	$-0.820557\pi$
$499$	−18844.0	−1.69053	−0.845264	+	0.534349i	$0.820557\pi$
$500$	−500.000	−0.0447214
$501$	1092.00	0.0973792
$502$	−8440.00	−0.750390
$503$	−3424.00	−0.303516	−0.151758	−	0.988418i	$-0.548493\pi$
$503$	−3424.00	−0.303516	−0.151758	+	0.988418i	$0.548493\pi$
$504$	−1728.00	−0.152721
$505$	2610.00	0.229987
$506$	0	0
$507$	507.000	0.0444116
$508$	−8768.00	−0.765782
$509$	9114.00	0.793656	0.396828	−	0.917893i	$-0.370111\pi$
$509$	9114.00	0.793656	0.396828	+	0.917893i	$0.370111\pi$
$510$	1500.00	0.130237
$511$	−1488.00	−0.128817
$512$	−512.000	−0.0441942
$513$	756.000	0.0650647
$514$	1868.00	0.160300
$515$	−4960.00	−0.424396
$516$	4176.00	0.356275
$517$	0	0
$518$	8928.00	0.757285
$519$	−2358.00	−0.199431
$520$	520.000	0.0438529
$521$	−20782.0	−1.74755	−0.873777	−	0.486326i	$-0.838337\pi$
$521$	−20782.0	−1.74755	−0.873777	+	0.486326i	$0.838337\pi$
$522$	−3420.00	−0.286761
$523$	−13460.0	−1.12536	−0.562681	−	0.826674i	$-0.690230\pi$
$523$	−13460.0	−1.12536	−0.562681	+	0.826674i	$0.690230\pi$
$524$	−9776.00	−0.815012
$525$	1800.00	0.149635
$526$	10400.0	0.862094
$527$	12400.0	1.02496
$528$	0	0
$529$	31097.0	2.55585
$530$	4620.00	0.378641
$531$	−4680.00	−0.382476
$532$	2688.00	0.219059
$533$	−2522.00	−0.204953
$534$	−2436.00	−0.197408
$535$	6420.00	0.518805
$536$	−6176.00	−0.497691
$537$	−11292.0	−0.907422
$538$	−8316.00	−0.666409
$539$	0	0
$540$	−540.000	−0.0430331
$541$	−10922.0	−0.867973	−0.433987	−	0.900919i	$-0.642893\pi$
$541$	−10922.0	−0.867973	−0.433987	+	0.900919i	$0.642893\pi$
$542$	7424.00	0.588354
$543$	426.000	0.0336674
$544$	−1600.00	−0.126102
$545$	−10670.0	−0.838629
$546$	−1872.00	−0.146729
$547$	−14380.0	−1.12403	−0.562015	−	0.827127i	$-0.689974\pi$
$547$	−14380.0	−1.12403	−0.562015	+	0.827127i	$0.689974\pi$
$548$	−2952.00	−0.230115
$549$	−4554.00	−0.354025
$550$	0	0
$551$	5320.00	0.411324
$552$	4992.00	0.384916
$553$	17664.0	1.35832
$554$	8356.00	0.640817
$555$	2790.00	0.213385
$556$	688.000	0.0524779
$557$	13250.0	1.00794	0.503968	−	0.863722i	$-0.331873\pi$
$557$	13250.0	1.00794	0.503968	+	0.863722i	$0.331873\pi$
$558$	−4464.00	−0.338667
$559$	4524.00	0.342298
$560$	−1920.00	−0.144884
$561$	0	0
$562$	−11260.0	−0.845150
$563$	−17988.0	−1.34654	−0.673272	−	0.739395i	$-0.735112\pi$
$563$	−17988.0	−1.34654	−0.673272	+	0.739395i	$0.735112\pi$
$564$	3120.00	0.232936
$565$	−4130.00	−0.307523
$566$	−1832.00	−0.136051
$567$	1944.00	0.143986
$568$	−6240.00	−0.460959
$569$	−13886.0	−1.02308	−0.511539	−	0.859260i	$-0.670924\pi$
$569$	−13886.0	−1.02308	−0.511539	+	0.859260i	$0.670924\pi$
$570$	840.000	0.0617258
$571$	11732.0	0.859840	0.429920	−	0.902867i	$-0.358542\pi$
$571$	11732.0	0.859840	0.429920	+	0.902867i	$0.358542\pi$
$572$	0	0
$573$	−7200.00	−0.524929
$574$	9312.00	0.677135
$575$	−5200.00	−0.377139
$576$	576.000	0.0416667
$577$	−13822.0	−0.997257	−0.498629	−	0.866816i	$-0.666163\pi$
$577$	−13822.0	−0.997257	−0.498629	+	0.866816i	$0.666163\pi$
$578$	4826.00	0.347293
$579$	−11178.0	−0.802317
$580$	−3800.00	−0.272046
$581$	35136.0	2.50893
$582$	−5532.00	−0.394001
$583$	0	0
$584$	496.000	0.0351449
$585$	−585.000	−0.0413449
$586$	396.000	0.0279157
$587$	−23216.0	−1.63241	−0.816207	−	0.577760i	$-0.803927\pi$
$587$	−23216.0	−1.63241	−0.816207	+	0.577760i	$0.803927\pi$
$588$	2796.00	0.196097
$589$	6944.00	0.485777
$590$	−5200.00	−0.362848
$591$	6438.00	0.448095
$592$	−2976.00	−0.206610
$593$	−4258.00	−0.294865	−0.147433	−	0.989072i	$-0.547101\pi$
$593$	−4258.00	−0.294865	−0.147433	+	0.989072i	$0.547101\pi$
$594$	0	0
$595$	−6000.00	−0.413405
$596$	−12312.0	−0.846173
$597$	−5832.00	−0.399812
$598$	5408.00	0.369815
$599$	15056.0	1.02700	0.513499	−	0.858090i	$-0.328349\pi$
$599$	15056.0	1.02700	0.513499	+	0.858090i	$0.328349\pi$
$600$	−600.000	−0.0408248
$601$	−1990.00	−0.135065	−0.0675323	−	0.997717i	$-0.521513\pi$
$601$	−1990.00	−0.135065	−0.0675323	+	0.997717i	$0.521513\pi$
$602$	−16704.0	−1.13090
$603$	6948.00	0.469228
$604$	−9728.00	−0.655342
$605$	6655.00	0.447214
$606$	3132.00	0.209949
$607$	−3000.00	−0.200603	−0.100302	−	0.994957i	$-0.531981\pi$
$607$	−3000.00	−0.200603	−0.100302	+	0.994957i	$0.531981\pi$
$608$	−896.000	−0.0597658
$609$	13680.0	0.910249
$610$	−5060.00	−0.335858
$611$	3380.00	0.223797
$612$	1800.00	0.118890
$613$	−6106.00	−0.402315	−0.201157	−	0.979559i	$-0.564470\pi$
$613$	−6106.00	−0.402315	−0.201157	+	0.979559i	$0.564470\pi$
$614$	−14984.0	−0.984862
$615$	2910.00	0.190801
$616$	0	0
$617$	−7906.00	−0.515857	−0.257928	−	0.966164i	$-0.583040\pi$
$617$	−7906.00	−0.515857	−0.257928	+	0.966164i	$0.583040\pi$
$618$	−5952.00	−0.387418
$619$	−7188.00	−0.466737	−0.233368	−	0.972388i	$-0.574975\pi$
$619$	−7188.00	−0.466737	−0.233368	+	0.972388i	$0.574975\pi$
$620$	−4960.00	−0.321288
$621$	−5616.00	−0.362902
$622$	17472.0	1.12631
$623$	9744.00	0.626621
$624$	624.000	0.0400320
$625$	625.000	0.0400000
$626$	15180.0	0.969193
$627$	0	0
$628$	216.000	0.0137251
$629$	−9300.00	−0.589531
$630$	2160.00	0.136598
$631$	−23280.0	−1.46872	−0.734360	−	0.678760i	$-0.762518\pi$
$631$	−23280.0	−1.46872	−0.734360	+	0.678760i	$0.762518\pi$
$632$	−5888.00	−0.370589
$633$	−4812.00	−0.302148
$634$	6892.00	0.431729
$635$	10960.0	0.684936
$636$	5544.00	0.345651
$637$	3029.00	0.188404
$638$	0	0
$639$	7020.00	0.434596
$640$	640.000	0.0395285
$641$	28194.0	1.73728	0.868640	−	0.495444i	$-0.164995\pi$
$641$	28194.0	1.73728	0.868640	+	0.495444i	$0.164995\pi$
$642$	7704.00	0.473602
$643$	10948.0	0.671457	0.335729	−	0.941959i	$-0.391018\pi$
$643$	10948.0	0.671457	0.335729	+	0.941959i	$0.391018\pi$
$644$	−19968.0	−1.22182
$645$	−5220.00	−0.318662
$646$	−2800.00	−0.170533
$647$	−23368.0	−1.41992	−0.709962	−	0.704240i	$-0.751288\pi$
$647$	−23368.0	−1.41992	−0.709962	+	0.704240i	$0.751288\pi$
$648$	−648.000	−0.0392837
$649$	0	0
$650$	−650.000	−0.0392232
$651$	17856.0	1.07501
$652$	6128.00	0.368084
$653$	3070.00	0.183979	0.0919896	−	0.995760i	$-0.470677\pi$
$653$	3070.00	0.183979	0.0919896	+	0.995760i	$0.470677\pi$
$654$	−12804.0	−0.765560
$655$	12220.0	0.728969
$656$	−3104.00	−0.184742
$657$	−558.000	−0.0331349
$658$	−12480.0	−0.739394
$659$	19308.0	1.14132	0.570662	−	0.821185i	$-0.306687\pi$
$659$	19308.0	1.14132	0.570662	+	0.821185i	$0.306687\pi$
$660$	0	0
$661$	13414.0	0.789325	0.394663	−	0.918826i	$-0.370861\pi$
$661$	13414.0	0.789325	0.394663	+	0.918826i	$0.370861\pi$
$662$	22696.0	1.33249
$663$	1950.00	0.114226
$664$	−11712.0	−0.684509
$665$	−3360.00	−0.195933
$666$	3348.00	0.194793
$667$	−39520.0	−2.29418
$668$	1456.00	0.0843328
$669$	120.000	0.00693493
$670$	7720.00	0.445149
$671$	0	0
$672$	−2304.00	−0.132260
$673$	−3614.00	−0.206998	−0.103499	−	0.994630i	$-0.533004\pi$
$673$	−3614.00	−0.206998	−0.103499	+	0.994630i	$0.533004\pi$
$674$	1404.00	0.0802375
$675$	675.000	0.0384900
$676$	676.000	0.0384615
$677$	3166.00	0.179733	0.0898665	−	0.995954i	$-0.471356\pi$
$677$	3166.00	0.179733	0.0898665	+	0.995954i	$0.471356\pi$
$678$	−4956.00	−0.280729
$679$	22128.0	1.25065
$680$	2000.00	0.112789
$681$	5712.00	0.321416
$682$	0	0
$683$	−2528.00	−0.141627	−0.0708135	−	0.997490i	$-0.522560\pi$
$683$	−2528.00	−0.141627	−0.0708135	+	0.997490i	$0.522560\pi$
$684$	1008.00	0.0563477
$685$	3690.00	0.205821
$686$	5280.00	0.293865
$687$	4362.00	0.242243
$688$	5568.00	0.308544
$689$	6006.00	0.332091
$690$	−6240.00	−0.344279
$691$	−33244.0	−1.83019	−0.915095	−	0.403238i	$-0.867885\pi$
$691$	−33244.0	−1.83019	−0.915095	+	0.403238i	$0.867885\pi$
$692$	−3144.00	−0.172712
$693$	0	0
$694$	−5416.00	−0.296237
$695$	−860.000	−0.0469376
$696$	−4560.00	−0.248342
$697$	−9700.00	−0.527136
$698$	−17980.0	−0.975004
$699$	5670.00	0.306808
$700$	2400.00	0.129588
$701$	19766.0	1.06498	0.532490	−	0.846436i	$-0.321256\pi$
$701$	19766.0	1.06498	0.532490	+	0.846436i	$0.321256\pi$
$702$	−702.000	−0.0377426
$703$	−5208.00	−0.279407
$704$	0	0
$705$	−3900.00	−0.208344
$706$	−16716.0	−0.891098
$707$	−12528.0	−0.666427
$708$	−6240.00	−0.331234
$709$	28190.0	1.49323	0.746613	−	0.665258i	$-0.231679\pi$
$709$	28190.0	1.49323	0.746613	+	0.665258i	$0.231679\pi$
$710$	7800.00	0.412294
$711$	6624.00	0.349394
$712$	−3248.00	−0.170961
$713$	−51584.0	−2.70945
$714$	−7200.00	−0.377385
$715$	0	0
$716$	−15056.0	−0.785851
$717$	5820.00	0.303141
$718$	8392.00	0.436193
$719$	−7592.00	−0.393788	−0.196894	−	0.980425i	$-0.563086\pi$
$719$	−7592.00	−0.393788	−0.196894	+	0.980425i	$0.563086\pi$
$720$	−720.000	−0.0372678
$721$	23808.0	1.22976
$722$	12150.0	0.626283
$723$	−10722.0	−0.551529
$724$	568.000	0.0291568
$725$	4750.00	0.243325
$726$	7986.00	0.408248
$727$	−15208.0	−0.775837	−0.387919	−	0.921694i	$-0.626806\pi$
$727$	−15208.0	−0.775837	−0.387919	+	0.921694i	$0.626806\pi$
$728$	−2496.00	−0.127071
$729$	729.000	0.0370370
$730$	−620.000	−0.0314346
$731$	17400.0	0.880386
$732$	−6072.00	−0.306595
$733$	29270.0	1.47491	0.737457	−	0.675394i	$-0.236026\pi$
$733$	29270.0	1.47491	0.737457	+	0.675394i	$0.236026\pi$
$734$	−1392.00	−0.0699995
$735$	−3495.00	−0.175395
$736$	6656.00	0.333347
$737$	0	0
$738$	3492.00	0.174177
$739$	−8268.00	−0.411561	−0.205780	−	0.978598i	$-0.565973\pi$
$739$	−8268.00	−0.411561	−0.205780	+	0.978598i	$0.565973\pi$
$740$	3720.00	0.184797
$741$	1092.00	0.0541371
$742$	−22176.0	−1.09718
$743$	24028.0	1.18641	0.593204	−	0.805052i	$-0.297863\pi$
$743$	24028.0	1.18641	0.593204	+	0.805052i	$0.297863\pi$
$744$	−5952.00	−0.293294
$745$	15390.0	0.756840
$746$	−18876.0	−0.926407
$747$	13176.0	0.645361
$748$	0	0
$749$	−30816.0	−1.50333
$750$	750.000	0.0365148
$751$	−30040.0	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−30040.0	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	4160.00	0.201728
$753$	12660.0	0.612691
$754$	−4940.00	−0.238600
$755$	12160.0	0.586156
$756$	2592.00	0.124696
$757$	−15698.0	−0.753703	−0.376852	−	0.926274i	$-0.622993\pi$
$757$	−15698.0	−0.753703	−0.376852	+	0.926274i	$0.622993\pi$
$758$	11352.0	0.543962
$759$	0	0
$760$	1120.00	0.0534561
$761$	9486.00	0.451862	0.225931	−	0.974143i	$-0.427458\pi$
$761$	9486.00	0.451862	0.225931	+	0.974143i	$0.427458\pi$
$762$	13152.0	0.625258
$763$	51216.0	2.43007
$764$	−9600.00	−0.454602
$765$	−2250.00	−0.106338
$766$	−21112.0	−0.995832
$767$	−6760.00	−0.318239
$768$	768.000	0.0360844
$769$	−18638.0	−0.873997	−0.436998	−	0.899462i	$-0.643958\pi$
$769$	−18638.0	−0.873997	−0.436998	+	0.899462i	$0.643958\pi$
$770$	0	0
$771$	−2802.00	−0.130884
$772$	−14904.0	−0.694827
$773$	29242.0	1.36062	0.680312	−	0.732923i	$-0.261844\pi$
$773$	29242.0	1.36062	0.680312	+	0.732923i	$0.261844\pi$
$774$	−6264.00	−0.290898
$775$	6200.00	0.287368
$776$	−7376.00	−0.341215
$777$	−13392.0	−0.618321
$778$	−5628.00	−0.259349
$779$	−5432.00	−0.249835
$780$	−780.000	−0.0358057
$781$	0	0
$782$	20800.0	0.951159
$783$	5130.00	0.234140
$784$	3728.00	0.169825
$785$	−270.000	−0.0122761
$786$	14664.0	0.665455
$787$	8708.00	0.394418	0.197209	−	0.980362i	$-0.436812\pi$
$787$	8708.00	0.394418	0.197209	+	0.980362i	$0.436812\pi$
$788$	8584.00	0.388061
$789$	−15600.0	−0.703897
$790$	7360.00	0.331465
$791$	19824.0	0.891100
$792$	0	0
$793$	−6578.00	−0.294567
$794$	19236.0	0.859773
$795$	−6930.00	−0.309159
$796$	−7776.00	−0.346248
$797$	2870.00	0.127554	0.0637770	−	0.997964i	$-0.479685\pi$
$797$	2870.00	0.127554	0.0637770	+	0.997964i	$0.479685\pi$
$798$	−4032.00	−0.178861
$799$	13000.0	0.575603
$800$	−800.000	−0.0353553
$801$	3654.00	0.161183
$802$	−2364.00	−0.104084
$803$	0	0
$804$	9264.00	0.406363
$805$	24960.0	1.09283
$806$	−6448.00	−0.281788
$807$	12474.0	0.544121
$808$	4176.00	0.181821
$809$	1346.00	0.0584955	0.0292477	−	0.999572i	$-0.490689\pi$
$809$	1346.00	0.0584955	0.0292477	+	0.999572i	$0.490689\pi$
$810$	810.000	0.0351364
$811$	−41724.0	−1.80657	−0.903285	−	0.429042i	$-0.858851\pi$
$811$	−41724.0	−1.80657	−0.903285	+	0.429042i	$0.858851\pi$
$812$	18240.0	0.788299
$813$	−11136.0	−0.480389
$814$	0	0
$815$	−7660.00	−0.329225
$816$	2400.00	0.102962
$817$	9744.00	0.417258
$818$	15548.0	0.664576
$819$	2808.00	0.119804
$820$	3880.00	0.165238
$821$	11034.0	0.469049	0.234525	−	0.972110i	$-0.424647\pi$
$821$	11034.0	0.469049	0.234525	+	0.972110i	$0.424647\pi$
$822$	4428.00	0.187888
$823$	−25504.0	−1.08021	−0.540105	−	0.841597i	$-0.681616\pi$
$823$	−25504.0	−1.08021	−0.540105	+	0.841597i	$0.681616\pi$
$824$	−7936.00	−0.335514
$825$	0	0
$826$	24960.0	1.05142
$827$	−40464.0	−1.70142	−0.850708	−	0.525638i	$-0.823826\pi$
$827$	−40464.0	−1.70142	−0.850708	+	0.525638i	$0.823826\pi$
$828$	−7488.00	−0.314283
$829$	−13114.0	−0.549419	−0.274709	−	0.961527i	$-0.588582\pi$
$829$	−13114.0	−0.549419	−0.274709	+	0.961527i	$0.588582\pi$
$830$	14640.0	0.612243
$831$	−12534.0	−0.523225
$832$	832.000	0.0346688
$833$	11650.0	0.484572
$834$	−1032.00	−0.0428480
$835$	−1820.00	−0.0754296
$836$	0	0
$837$	6696.00	0.276520
$838$	312.000	0.0128614
$839$	9268.00	0.381367	0.190683	−	0.981652i	$-0.438930\pi$
$839$	9268.00	0.381367	0.190683	+	0.981652i	$0.438930\pi$
$840$	2880.00	0.118297
$841$	11711.0	0.480175
$842$	2596.00	0.106252
$843$	16890.0	0.690062
$844$	−6416.00	−0.261668
$845$	−845.000	−0.0344010
$846$	−4680.00	−0.190191
$847$	−31944.0	−1.29588
$848$	7392.00	0.299342
$849$	2748.00	0.111085
$850$	−2500.00	−0.100882
$851$	38688.0	1.55841
$852$	9360.00	0.376371
$853$	46198.0	1.85438	0.927192	−	0.374587i	$-0.122215\pi$
$853$	46198.0	1.85438	0.927192	+	0.374587i	$0.122215\pi$
$854$	24288.0	0.973206
$855$	−1260.00	−0.0503989
$856$	10272.0	0.410152
$857$	4386.00	0.174823	0.0874113	−	0.996172i	$-0.472141\pi$
$857$	4386.00	0.174823	0.0874113	+	0.996172i	$0.472141\pi$
$858$	0	0
$859$	39620.0	1.57371	0.786855	−	0.617138i	$-0.211708\pi$
$859$	39620.0	1.57371	0.786855	+	0.617138i	$0.211708\pi$
$860$	−6960.00	−0.275970
$861$	−13968.0	−0.552878
$862$	3432.00	0.135608
$863$	−7316.00	−0.288574	−0.144287	−	0.989536i	$-0.546089\pi$
$863$	−7316.00	−0.288574	−0.144287	+	0.989536i	$0.546089\pi$
$864$	−864.000	−0.0340207
$865$	3930.00	0.154479
$866$	−10244.0	−0.401969
$867$	−7239.00	−0.283563
$868$	23808.0	0.930986
$869$	0	0
$870$	5700.00	0.222124
$871$	10036.0	0.390421
$872$	−17072.0	−0.662994
$873$	8298.00	0.321701
$874$	11648.0	0.450800
$875$	−3000.00	−0.115907
$876$	−744.000	−0.0286957
$877$	4094.00	0.157633	0.0788167	−	0.996889i	$-0.474886\pi$
$877$	4094.00	0.157633	0.0788167	+	0.996889i	$0.474886\pi$
$878$	−576.000	−0.0221402
$879$	−594.000	−0.0227931
$880$	0	0
$881$	31466.0	1.20331	0.601655	−	0.798756i	$-0.294508\pi$
$881$	31466.0	1.20331	0.601655	+	0.798756i	$0.294508\pi$
$882$	−4194.00	−0.160113
$883$	9676.00	0.368769	0.184385	−	0.982854i	$-0.440971\pi$
$883$	9676.00	0.368769	0.184385	+	0.982854i	$0.440971\pi$
$884$	2600.00	0.0989225
$885$	7800.00	0.296265
$886$	−21240.0	−0.805386
$887$	33800.0	1.27947	0.639737	−	0.768594i	$-0.279043\pi$
$887$	33800.0	1.27947	0.639737	+	0.768594i	$0.279043\pi$
$888$	4464.00	0.168696
$889$	−52608.0	−1.98472
$890$	4060.00	0.152912
$891$	0	0
$892$	160.000	0.00600583
$893$	7280.00	0.272806
$894$	18468.0	0.690897
$895$	18820.0	0.702886
$896$	−3072.00	−0.114541
$897$	−8112.00	−0.301953
$898$	5700.00	0.211817
$899$	47120.0	1.74810
$900$	900.000	0.0333333
$901$	23100.0	0.854132
$902$	0	0
$903$	25056.0	0.923379
$904$	−6608.00	−0.243118
$905$	−710.000	−0.0260787
$906$	14592.0	0.535085
$907$	26804.0	0.981270	0.490635	−	0.871365i	$-0.336765\pi$
$907$	26804.0	0.981270	0.490635	+	0.871365i	$0.336765\pi$
$908$	7616.00	0.278354
$909$	−4698.00	−0.171422
$910$	3120.00	0.113656
$911$	24288.0	0.883312	0.441656	−	0.897184i	$-0.354391\pi$
$911$	24288.0	0.883312	0.441656	+	0.897184i	$0.354391\pi$
$912$	1344.00	0.0487986
$913$	0	0
$914$	−1508.00	−0.0545735
$915$	7590.00	0.274227
$916$	5816.00	0.209788
$917$	−58656.0	−2.11231
$918$	−2700.00	−0.0970733
$919$	2456.00	0.0881567	0.0440783	−	0.999028i	$-0.485965\pi$
$919$	2456.00	0.0881567	0.0440783	+	0.999028i	$0.485965\pi$
$920$	−8320.00	−0.298155
$921$	22476.0	0.804136
$922$	−33076.0	−1.18145
$923$	10140.0	0.361606
$924$	0	0
$925$	−4650.00	−0.165288
$926$	−1600.00	−0.0567811
$927$	8928.00	0.316326
$928$	−6080.00	−0.215071
$929$	−17066.0	−0.602710	−0.301355	−	0.953512i	$-0.597439\pi$
$929$	−17066.0	−0.602710	−0.301355	+	0.953512i	$0.597439\pi$
$930$	7440.00	0.262330
$931$	6524.00	0.229662
$932$	7560.00	0.265704
$933$	−26208.0	−0.919626
$934$	−13384.0	−0.468884
$935$	0	0
$936$	−936.000	−0.0326860
$937$	−45526.0	−1.58727	−0.793634	−	0.608396i	$-0.791813\pi$
$937$	−45526.0	−1.58727	−0.793634	+	0.608396i	$0.791813\pi$
$938$	−37056.0	−1.28989
$939$	−22770.0	−0.791343
$940$	−5200.00	−0.180431
$941$	−49518.0	−1.71545	−0.857726	−	0.514107i	$-0.828123\pi$
$941$	−49518.0	−1.71545	−0.857726	+	0.514107i	$0.828123\pi$
$942$	−324.000	−0.0112065
$943$	40352.0	1.39347
$944$	−8320.00	−0.286857
$945$	−3240.00	−0.111531
$946$	0	0
$947$	−43648.0	−1.49775	−0.748875	−	0.662711i	$-0.769406\pi$
$947$	−43648.0	−1.49775	−0.748875	+	0.662711i	$0.769406\pi$
$948$	8832.00	0.302584
$949$	−806.000	−0.0275699
$950$	−1400.00	−0.0478126
$951$	−10338.0	−0.352505
$952$	−9600.00	−0.326825
$953$	−44878.0	−1.52544	−0.762718	−	0.646731i	$-0.776136\pi$
$953$	−44878.0	−1.52544	−0.762718	+	0.646731i	$0.776136\pi$
$954$	−8316.00	−0.282223
$955$	12000.0	0.406608
$956$	7760.00	0.262528
$957$	0	0
$958$	−19960.0	−0.673151
$959$	−17712.0	−0.596403
$960$	−960.000	−0.0322749
$961$	31713.0	1.06452
$962$	4836.00	0.162078
$963$	−11556.0	−0.386695
$964$	−14296.0	−0.477638
$965$	18630.0	0.621472
$966$	29952.0	0.997608
$967$	56400.0	1.87560	0.937798	−	0.347181i	$-0.112861\pi$
$967$	56400.0	1.87560	0.937798	+	0.347181i	$0.112861\pi$
$968$	10648.0	0.353553
$969$	4200.00	0.139240
$970$	9220.00	0.305192
$971$	−56620.0	−1.87129	−0.935645	−	0.352943i	$-0.885181\pi$
$971$	−56620.0	−1.87129	−0.935645	+	0.352943i	$0.885181\pi$
$972$	972.000	0.0320750
$973$	4128.00	0.136010
$974$	−10688.0	−0.351607
$975$	975.000	0.0320256
$976$	−8096.00	−0.265519
$977$	39774.0	1.30244	0.651220	−	0.758889i	$-0.274258\pi$
$977$	39774.0	1.30244	0.651220	+	0.758889i	$0.274258\pi$
$978$	−9192.00	−0.300540
$979$	0	0
$980$	−4660.00	−0.151896
$981$	19206.0	0.625077
$982$	22088.0	0.717776
$983$	−41532.0	−1.34757	−0.673787	−	0.738926i	$-0.735333\pi$
$983$	−41532.0	−1.34757	−0.673787	+	0.738926i	$0.735333\pi$
$984$	4656.00	0.150841
$985$	−10730.0	−0.347093
$986$	−19000.0	−0.613675
$987$	18720.0	0.603712
$988$	1456.00	0.0468841
$989$	−72384.0	−2.32728
$990$	0	0
$991$	29384.0	0.941891	0.470945	−	0.882162i	$-0.343913\pi$
$991$	29384.0	0.941891	0.470945	+	0.882162i	$0.343913\pi$
$992$	−7936.00	−0.254000
$993$	−34044.0	−1.08797
$994$	−37440.0	−1.19469
$995$	9720.00	0.309693
$996$	17568.0	0.558899
$997$	30238.0	0.960529	0.480264	−	0.877124i	$-0.340541\pi$
$997$	30238.0	0.960529	0.480264	+	0.877124i	$0.340541\pi$
$998$	37688.0	1.19538
$999$	−5022.00	−0.159048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.4.a.e.1.1	✓	1
3.2	odd	2		1170.4.a.q.1.1		1
5.4	even	2		1950.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.4.a.e.1.1	✓	1	1.1	even	1	trivial
1170.4.a.q.1.1		1	3.2	odd	2
1950.4.a.i.1.1		1	5.4	even	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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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