LMFDB - Embedded newform 342.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,4,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, 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("342.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	342.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:	$20.1786532220$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +7.00000 q^{5} -15.0000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} +7.00000 q^{5} -15.0000 q^{7} +8.00000 q^{8} +14.0000 q^{10} +49.0000 q^{11} +14.0000 q^{13} -30.0000 q^{14} +16.0000 q^{16} +33.0000 q^{17} -19.0000 q^{19} +28.0000 q^{20} +98.0000 q^{22} +148.000 q^{23} -76.0000 q^{25} +28.0000 q^{26} -60.0000 q^{28} +278.000 q^{29} +94.0000 q^{31} +32.0000 q^{32} +66.0000 q^{34} -105.000 q^{35} +160.000 q^{37} -38.0000 q^{38} +56.0000 q^{40} -400.000 q^{41} +73.0000 q^{43} +196.000 q^{44} +296.000 q^{46} -173.000 q^{47} -118.000 q^{49} -152.000 q^{50} +56.0000 q^{52} -170.000 q^{53} +343.000 q^{55} -120.000 q^{56} +556.000 q^{58} +12.0000 q^{59} +419.000 q^{61} +188.000 q^{62} +64.0000 q^{64} +98.0000 q^{65} +444.000 q^{67} +132.000 q^{68} -210.000 q^{70} +952.000 q^{71} -27.0000 q^{73} +320.000 q^{74} -76.0000 q^{76} -735.000 q^{77} -556.000 q^{79} +112.000 q^{80} -800.000 q^{82} +276.000 q^{83} +231.000 q^{85} +146.000 q^{86} +392.000 q^{88} -1386.00 q^{89} -210.000 q^{91} +592.000 q^{92} -346.000 q^{94} -133.000 q^{95} +130.000 q^{97} -236.000 q^{98} +O(q^{100})\)

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
$3$	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
$7$	−15.0000	−0.809924	−0.404962	−	0.914334i	$-0.632715\pi$
$7$	−15.0000	−0.809924	−0.404962	+	0.914334i	$0.632715\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	14.0000	0.442719
$11$	49.0000	1.34310	0.671548	−	0.740961i	$-0.265630\pi$
$11$	49.0000	1.34310	0.671548	+	0.740961i	$0.265630\pi$
$12$	0	0
$13$	14.0000	0.298685	0.149342	−	0.988786i	$-0.452284\pi$
$13$	14.0000	0.298685	0.149342	+	0.988786i	$0.452284\pi$
$14$	−30.0000	−0.572703
$15$	0	0
$16$	16.0000	0.250000
$17$	33.0000	0.470804	0.235402	−	0.971898i	$-0.424359\pi$
$17$	33.0000	0.470804	0.235402	+	0.971898i	$0.424359\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	28.0000	0.313050
$21$	0	0
$22$	98.0000	0.949712
$23$	148.000	1.34174	0.670872	−	0.741573i	$-0.265920\pi$
$23$	148.000	1.34174	0.670872	+	0.741573i	$0.265920\pi$
$24$	0	0
$25$	−76.0000	−0.608000
$26$	28.0000	0.211202
$27$	0	0
$28$	−60.0000	−0.404962
$29$	278.000	1.78011	0.890057	−	0.455849i	$-0.150664\pi$
$29$	278.000	1.78011	0.890057	+	0.455849i	$0.150664\pi$
$30$	0	0
$31$	94.0000	0.544610	0.272305	−	0.962211i	$-0.412214\pi$
$31$	94.0000	0.544610	0.272305	+	0.962211i	$0.412214\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	66.0000	0.332909
$35$	−105.000	−0.507093
$36$	0	0
$37$	160.000	0.710915	0.355457	−	0.934693i	$-0.384325\pi$
$37$	160.000	0.710915	0.355457	+	0.934693i	$0.384325\pi$
$38$	−38.0000	−0.162221
$39$	0	0
$40$	56.0000	0.221359
$41$	−400.000	−1.52365	−0.761823	−	0.647785i	$-0.775696\pi$
$41$	−400.000	−1.52365	−0.761823	+	0.647785i	$0.775696\pi$
$42$	0	0
$43$	73.0000	0.258893	0.129446	−	0.991586i	$-0.458680\pi$
$43$	73.0000	0.258893	0.129446	+	0.991586i	$0.458680\pi$
$44$	196.000	0.671548
$45$	0	0
$46$	296.000	0.948757
$47$	−173.000	−0.536907	−0.268454	−	0.963293i	$-0.586513\pi$
$47$	−173.000	−0.536907	−0.268454	+	0.963293i	$0.586513\pi$
$48$	0	0
$49$	−118.000	−0.344023
$50$	−152.000	−0.429921
$51$	0	0
$52$	56.0000	0.149342
$53$	−170.000	−0.440590	−0.220295	−	0.975433i	$-0.570702\pi$
$53$	−170.000	−0.440590	−0.220295	+	0.975433i	$0.570702\pi$
$54$	0	0
$55$	343.000	0.840911
$56$	−120.000	−0.286351
$57$	0	0
$58$	556.000	1.25873
$59$	12.0000	0.0264791	0.0132396	−	0.999912i	$-0.495786\pi$
$59$	12.0000	0.0264791	0.0132396	+	0.999912i	$0.495786\pi$
$60$	0	0
$61$	419.000	0.879466	0.439733	−	0.898128i	$-0.355073\pi$
$61$	419.000	0.879466	0.439733	+	0.898128i	$0.355073\pi$
$62$	188.000	0.385097
$63$	0	0
$64$	64.0000	0.125000
$65$	98.0000	0.187006
$66$	0	0
$67$	444.000	0.809600	0.404800	−	0.914405i	$-0.367341\pi$
$67$	444.000	0.809600	0.404800	+	0.914405i	$0.367341\pi$
$68$	132.000	0.235402
$69$	0	0
$70$	−210.000	−0.358569
$71$	952.000	1.59129	0.795645	−	0.605763i	$-0.207132\pi$
$71$	952.000	1.59129	0.795645	+	0.605763i	$0.207132\pi$
$72$	0	0
$73$	−27.0000	−0.0432892	−0.0216446	−	0.999766i	$-0.506890\pi$
$73$	−27.0000	−0.0432892	−0.0216446	+	0.999766i	$0.506890\pi$
$74$	320.000	0.502692
$75$	0	0
$76$	−76.0000	−0.114708
$77$	−735.000	−1.08781
$78$	0	0
$79$	−556.000	−0.791834	−0.395917	−	0.918286i	$-0.629573\pi$
$79$	−556.000	−0.791834	−0.395917	+	0.918286i	$0.629573\pi$
$80$	112.000	0.156525
$81$	0	0
$82$	−800.000	−1.07738
$83$	276.000	0.364999	0.182500	−	0.983206i	$-0.441581\pi$
$83$	276.000	0.364999	0.182500	+	0.983206i	$0.441581\pi$
$84$	0	0
$85$	231.000	0.294770
$86$	146.000	0.183065
$87$	0	0
$88$	392.000	0.474856
$89$	−1386.00	−1.65074	−0.825369	−	0.564593i	$-0.809033\pi$
$89$	−1386.00	−1.65074	−0.825369	+	0.564593i	$0.809033\pi$
$90$	0	0
$91$	−210.000	−0.241912
$92$	592.000	0.670872
$93$	0	0
$94$	−346.000	−0.379651
$95$	−133.000	−0.143637
$96$	0	0
$97$	130.000	0.136077	0.0680387	−	0.997683i	$-0.478326\pi$
$97$	130.000	0.136077	0.0680387	+	0.997683i	$0.478326\pi$
$98$	−236.000	−0.243261
$99$	0	0
$100$	−304.000	−0.304000
$101$	238.000	0.234474	0.117237	−	0.993104i	$-0.462596\pi$
$101$	238.000	0.234474	0.117237	+	0.993104i	$0.462596\pi$
$102$	0	0
$103$	−1374.00	−1.31441	−0.657205	−	0.753712i	$-0.728261\pi$
$103$	−1374.00	−1.31441	−0.657205	+	0.753712i	$0.728261\pi$
$104$	112.000	0.105601
$105$	0	0
$106$	−340.000	−0.311545
$107$	−218.000	−0.196961	−0.0984806	−	0.995139i	$-0.531398\pi$
$107$	−218.000	−0.196961	−0.0984806	+	0.995139i	$0.531398\pi$
$108$	0	0
$109$	−2184.00	−1.91917	−0.959584	−	0.281423i	$-0.909194\pi$
$109$	−2184.00	−1.91917	−0.959584	+	0.281423i	$0.909194\pi$
$110$	686.000	0.594614
$111$	0	0
$112$	−240.000	−0.202481
$113$	−1334.00	−1.11055	−0.555275	−	0.831667i	$-0.687387\pi$
$113$	−1334.00	−1.11055	−0.555275	+	0.831667i	$0.687387\pi$
$114$	0	0
$115$	1036.00	0.840065
$116$	1112.00	0.890057
$117$	0	0
$118$	24.0000	0.0187236
$119$	−495.000	−0.381316
$120$	0	0
$121$	1070.00	0.803907
$122$	838.000	0.621877
$123$	0	0
$124$	376.000	0.272305
$125$	−1407.00	−1.00677
$126$	0	0
$127$	−666.000	−0.465338	−0.232669	−	0.972556i	$-0.574746\pi$
$127$	−666.000	−0.465338	−0.232669	+	0.972556i	$0.574746\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	196.000	0.132233
$131$	303.000	0.202086	0.101043	−	0.994882i	$-0.467782\pi$
$131$	303.000	0.202086	0.101043	+	0.994882i	$0.467782\pi$
$132$	0	0
$133$	285.000	0.185809
$134$	888.000	0.572474
$135$	0	0
$136$	264.000	0.166455
$137$	−583.000	−0.363570	−0.181785	−	0.983338i	$-0.558187\pi$
$137$	−583.000	−0.363570	−0.181785	+	0.983338i	$0.558187\pi$
$138$	0	0
$139$	−1467.00	−0.895175	−0.447587	−	0.894240i	$-0.647717\pi$
$139$	−1467.00	−0.895175	−0.447587	+	0.894240i	$0.647717\pi$
$140$	−420.000	−0.253546
$141$	0	0
$142$	1904.00	1.12521
$143$	686.000	0.401162
$144$	0	0
$145$	1946.00	1.11453
$146$	−54.0000	−0.0306101
$147$	0	0
$148$	640.000	0.355457
$149$	−351.000	−0.192987	−0.0964934	−	0.995334i	$-0.530763\pi$
$149$	−351.000	−0.192987	−0.0964934	+	0.995334i	$0.530763\pi$
$150$	0	0
$151$	−3100.00	−1.67069	−0.835346	−	0.549725i	$-0.814733\pi$
$151$	−3100.00	−1.67069	−0.835346	+	0.549725i	$0.814733\pi$
$152$	−152.000	−0.0811107
$153$	0	0
$154$	−1470.00	−0.769195
$155$	658.000	0.340980
$156$	0	0
$157$	−2474.00	−1.25762	−0.628811	−	0.777558i	$-0.716458\pi$
$157$	−2474.00	−1.25762	−0.628811	+	0.777558i	$0.716458\pi$
$158$	−1112.00	−0.559911
$159$	0	0
$160$	224.000	0.110680
$161$	−2220.00	−1.08671
$162$	0	0
$163$	2360.00	1.13405	0.567023	−	0.823702i	$-0.308095\pi$
$163$	2360.00	1.13405	0.567023	+	0.823702i	$0.308095\pi$
$164$	−1600.00	−0.761823
$165$	0	0
$166$	552.000	0.258093
$167$	1110.00	0.514338	0.257169	−	0.966366i	$-0.417210\pi$
$167$	1110.00	0.514338	0.257169	+	0.966366i	$0.417210\pi$
$168$	0	0
$169$	−2001.00	−0.910787
$170$	462.000	0.208434
$171$	0	0
$172$	292.000	0.129446
$173$	258.000	0.113384	0.0566918	−	0.998392i	$-0.481945\pi$
$173$	258.000	0.113384	0.0566918	+	0.998392i	$0.481945\pi$
$174$	0	0
$175$	1140.00	0.492434
$176$	784.000	0.335774
$177$	0	0
$178$	−2772.00	−1.16725
$179$	3762.00	1.57087	0.785433	−	0.618946i	$-0.212440\pi$
$179$	3762.00	1.57087	0.785433	+	0.618946i	$0.212440\pi$
$180$	0	0
$181$	706.000	0.289926	0.144963	−	0.989437i	$-0.453694\pi$
$181$	706.000	0.289926	0.144963	+	0.989437i	$0.453694\pi$
$182$	−420.000	−0.171058
$183$	0	0
$184$	1184.00	0.474378
$185$	1120.00	0.445103
$186$	0	0
$187$	1617.00	0.632336
$188$	−692.000	−0.268454
$189$	0	0
$190$	−266.000	−0.101567
$191$	−2659.00	−1.00732	−0.503661	−	0.863901i	$-0.668014\pi$
$191$	−2659.00	−1.00732	−0.503661	+	0.863901i	$0.668014\pi$
$192$	0	0
$193$	3648.00	1.36056	0.680282	−	0.732951i	$-0.261857\pi$
$193$	3648.00	1.36056	0.680282	+	0.732951i	$0.261857\pi$
$194$	260.000	0.0962212
$195$	0	0
$196$	−472.000	−0.172012
$197$	−494.000	−0.178660	−0.0893301	−	0.996002i	$-0.528473\pi$
$197$	−494.000	−0.178660	−0.0893301	+	0.996002i	$0.528473\pi$
$198$	0	0
$199$	−3679.00	−1.31054	−0.655270	−	0.755395i	$-0.727445\pi$
$199$	−3679.00	−1.31054	−0.655270	+	0.755395i	$0.727445\pi$
$200$	−608.000	−0.214960
$201$	0	0
$202$	476.000	0.165798
$203$	−4170.00	−1.44176
$204$	0	0
$205$	−2800.00	−0.953954
$206$	−2748.00	−0.929428
$207$	0	0
$208$	224.000	0.0746712
$209$	−931.000	−0.308127
$210$	0	0
$211$	792.000	0.258405	0.129203	−	0.991618i	$-0.458758\pi$
$211$	792.000	0.258405	0.129203	+	0.991618i	$0.458758\pi$
$212$	−680.000	−0.220295
$213$	0	0
$214$	−436.000	−0.139273
$215$	511.000	0.162093
$216$	0	0
$217$	−1410.00	−0.441092
$218$	−4368.00	−1.35706
$219$	0	0
$220$	1372.00	0.420456
$221$	462.000	0.140622
$222$	0	0
$223$	−4636.00	−1.39215	−0.696075	−	0.717969i	$-0.745072\pi$
$223$	−4636.00	−1.39215	−0.696075	+	0.717969i	$0.745072\pi$
$224$	−480.000	−0.143176
$225$	0	0
$226$	−2668.00	−0.785278
$227$	6446.00	1.88474	0.942370	−	0.334572i	$-0.108592\pi$
$227$	6446.00	1.88474	0.942370	+	0.334572i	$0.108592\pi$
$228$	0	0
$229$	5765.00	1.66359	0.831795	−	0.555084i	$-0.187314\pi$
$229$	5765.00	1.66359	0.831795	+	0.555084i	$0.187314\pi$
$230$	2072.00	0.594016
$231$	0	0
$232$	2224.00	0.629365
$233$	5847.00	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$233$	5847.00	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$234$	0	0
$235$	−1211.00	−0.336157
$236$	48.0000	0.0132396
$237$	0	0
$238$	−990.000	−0.269631
$239$	−2823.00	−0.764036	−0.382018	−	0.924155i	$-0.624771\pi$
$239$	−2823.00	−0.764036	−0.382018	+	0.924155i	$0.624771\pi$
$240$	0	0
$241$	−6140.00	−1.64113	−0.820565	−	0.571554i	$-0.806341\pi$
$241$	−6140.00	−1.64113	−0.820565	+	0.571554i	$0.806341\pi$
$242$	2140.00	0.568448
$243$	0	0
$244$	1676.00	0.439733
$245$	−826.000	−0.215393
$246$	0	0
$247$	−266.000	−0.0685230
$248$	752.000	0.192549
$249$	0	0
$250$	−2814.00	−0.711892
$251$	3103.00	0.780317	0.390159	−	0.920748i	$-0.372420\pi$
$251$	3103.00	0.780317	0.390159	+	0.920748i	$0.372420\pi$
$252$	0	0
$253$	7252.00	1.80209
$254$	−1332.00	−0.329044
$255$	0	0
$256$	256.000	0.0625000
$257$	−2336.00	−0.566987	−0.283494	−	0.958974i	$-0.591493\pi$
$257$	−2336.00	−0.566987	−0.283494	+	0.958974i	$0.591493\pi$
$258$	0	0
$259$	−2400.00	−0.575787
$260$	392.000	0.0935031
$261$	0	0
$262$	606.000	0.142896
$263$	2739.00	0.642182	0.321091	−	0.947048i	$-0.395950\pi$
$263$	2739.00	0.642182	0.321091	+	0.947048i	$0.395950\pi$
$264$	0	0
$265$	−1190.00	−0.275853
$266$	570.000	0.131387
$267$	0	0
$268$	1776.00	0.404800
$269$	6486.00	1.47011	0.735053	−	0.678010i	$-0.237157\pi$
$269$	6486.00	1.47011	0.735053	+	0.678010i	$0.237157\pi$
$270$	0	0
$271$	−308.000	−0.0690394	−0.0345197	−	0.999404i	$-0.510990\pi$
$271$	−308.000	−0.0690394	−0.0345197	+	0.999404i	$0.510990\pi$
$272$	528.000	0.117701
$273$	0	0
$274$	−1166.00	−0.257083
$275$	−3724.00	−0.816602
$276$	0	0
$277$	2977.00	0.645742	0.322871	−	0.946443i	$-0.395352\pi$
$277$	2977.00	0.645742	0.322871	+	0.946443i	$0.395352\pi$
$278$	−2934.00	−0.632984
$279$	0	0
$280$	−840.000	−0.179284
$281$	−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
$283$	6429.00	1.35040	0.675202	−	0.737633i	$-0.264056\pi$
$283$	6429.00	1.35040	0.675202	+	0.737633i	$0.264056\pi$
$284$	3808.00	0.795645
$285$	0	0
$286$	1372.00	0.283665
$287$	6000.00	1.23404
$288$	0	0
$289$	−3824.00	−0.778343
$290$	3892.00	0.788090
$291$	0	0
$292$	−108.000	−0.0216446
$293$	−5724.00	−1.14130	−0.570648	−	0.821195i	$-0.693308\pi$
$293$	−5724.00	−1.14130	−0.570648	+	0.821195i	$0.693308\pi$
$294$	0	0
$295$	84.0000	0.0165785
$296$	1280.00	0.251346
$297$	0	0
$298$	−702.000	−0.136462
$299$	2072.00	0.400759
$300$	0	0
$301$	−1095.00	−0.209684
$302$	−6200.00	−1.18136
$303$	0	0
$304$	−304.000	−0.0573539
$305$	2933.00	0.550633
$306$	0	0
$307$	8304.00	1.54376	0.771880	−	0.635768i	$-0.219317\pi$
$307$	8304.00	1.54376	0.771880	+	0.635768i	$0.219317\pi$
$308$	−2940.00	−0.543903
$309$	0	0
$310$	1316.00	0.241109
$311$	791.000	0.144223	0.0721117	−	0.997397i	$-0.477026\pi$
$311$	791.000	0.144223	0.0721117	+	0.997397i	$0.477026\pi$
$312$	0	0
$313$	10166.0	1.83583	0.917917	−	0.396772i	$-0.129869\pi$
$313$	10166.0	1.83583	0.917917	+	0.396772i	$0.129869\pi$
$314$	−4948.00	−0.889273
$315$	0	0
$316$	−2224.00	−0.395917
$317$	−6408.00	−1.13536	−0.567680	−	0.823249i	$-0.692159\pi$
$317$	−6408.00	−1.13536	−0.567680	+	0.823249i	$0.692159\pi$
$318$	0	0
$319$	13622.0	2.39086
$320$	448.000	0.0782624
$321$	0	0
$322$	−4440.00	−0.768421
$323$	−627.000	−0.108010
$324$	0	0
$325$	−1064.00	−0.181600
$326$	4720.00	0.801891
$327$	0	0
$328$	−3200.00	−0.538690
$329$	2595.00	0.434854
$330$	0	0
$331$	2576.00	0.427764	0.213882	−	0.976860i	$-0.431389\pi$
$331$	2576.00	0.427764	0.213882	+	0.976860i	$0.431389\pi$
$332$	1104.00	0.182500
$333$	0	0
$334$	2220.00	0.363692
$335$	3108.00	0.506890
$336$	0	0
$337$	−7922.00	−1.28053	−0.640265	−	0.768154i	$-0.721176\pi$
$337$	−7922.00	−1.28053	−0.640265	+	0.768154i	$0.721176\pi$
$338$	−4002.00	−0.644024
$339$	0	0
$340$	924.000	0.147385
$341$	4606.00	0.731463
$342$	0	0
$343$	6915.00	1.08856
$344$	584.000	0.0915325
$345$	0	0
$346$	516.000	0.0801744
$347$	−2305.00	−0.356596	−0.178298	−	0.983977i	$-0.557059\pi$
$347$	−2305.00	−0.356596	−0.178298	+	0.983977i	$0.557059\pi$
$348$	0	0
$349$	−4619.00	−0.708451	−0.354226	−	0.935160i	$-0.615255\pi$
$349$	−4619.00	−0.708451	−0.354226	+	0.935160i	$0.615255\pi$
$350$	2280.00	0.348203
$351$	0	0
$352$	1568.00	0.237428
$353$	−12366.0	−1.86452	−0.932260	−	0.361788i	$-0.882166\pi$
$353$	−12366.0	−1.86452	−0.932260	+	0.361788i	$0.882166\pi$
$354$	0	0
$355$	6664.00	0.996305
$356$	−5544.00	−0.825369
$357$	0	0
$358$	7524.00	1.11077
$359$	12417.0	1.82547	0.912736	−	0.408551i	$-0.133966\pi$
$359$	12417.0	1.82547	0.912736	+	0.408551i	$0.133966\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	1412.00	0.205008
$363$	0	0
$364$	−840.000	−0.120956
$365$	−189.000	−0.0271033
$366$	0	0
$367$	5776.00	0.821539	0.410769	−	0.911739i	$-0.365260\pi$
$367$	5776.00	0.821539	0.410769	+	0.911739i	$0.365260\pi$
$368$	2368.00	0.335436
$369$	0	0
$370$	2240.00	0.314735
$371$	2550.00	0.356845
$372$	0	0
$373$	3392.00	0.470861	0.235430	−	0.971891i	$-0.424350\pi$
$373$	3392.00	0.470861	0.235430	+	0.971891i	$0.424350\pi$
$374$	3234.00	0.447129
$375$	0	0
$376$	−1384.00	−0.189825
$377$	3892.00	0.531693
$378$	0	0
$379$	5766.00	0.781476	0.390738	−	0.920502i	$-0.372220\pi$
$379$	5766.00	0.781476	0.390738	+	0.920502i	$0.372220\pi$
$380$	−532.000	−0.0718185
$381$	0	0
$382$	−5318.00	−0.712284
$383$	−8482.00	−1.13162	−0.565809	−	0.824536i	$-0.691436\pi$
$383$	−8482.00	−1.13162	−0.565809	+	0.824536i	$0.691436\pi$
$384$	0	0
$385$	−5145.00	−0.681074
$386$	7296.00	0.962064
$387$	0	0
$388$	520.000	0.0680387
$389$	1983.00	0.258463	0.129231	−	0.991614i	$-0.458749\pi$
$389$	1983.00	0.258463	0.129231	+	0.991614i	$0.458749\pi$
$390$	0	0
$391$	4884.00	0.631699
$392$	−944.000	−0.121631
$393$	0	0
$394$	−988.000	−0.126332
$395$	−3892.00	−0.495767
$396$	0	0
$397$	−4555.00	−0.575841	−0.287921	−	0.957654i	$-0.592964\pi$
$397$	−4555.00	−0.575841	−0.287921	+	0.957654i	$0.592964\pi$
$398$	−7358.00	−0.926691
$399$	0	0
$400$	−1216.00	−0.152000
$401$	11624.0	1.44757	0.723784	−	0.690026i	$-0.242401\pi$
$401$	11624.0	1.44757	0.723784	+	0.690026i	$0.242401\pi$
$402$	0	0
$403$	1316.00	0.162667
$404$	952.000	0.117237
$405$	0	0
$406$	−8340.00	−1.01948
$407$	7840.00	0.954826
$408$	0	0
$409$	−12446.0	−1.50468	−0.752341	−	0.658774i	$-0.771075\pi$
$409$	−12446.0	−1.50468	−0.752341	+	0.658774i	$0.771075\pi$
$410$	−5600.00	−0.674547
$411$	0	0
$412$	−5496.00	−0.657205
$413$	−180.000	−0.0214461
$414$	0	0
$415$	1932.00	0.228526
$416$	448.000	0.0528005
$417$	0	0
$418$	−1862.00	−0.217879
$419$	468.000	0.0545663	0.0272832	−	0.999628i	$-0.491314\pi$
$419$	468.000	0.0545663	0.0272832	+	0.999628i	$0.491314\pi$
$420$	0	0
$421$	−7894.00	−0.913848	−0.456924	−	0.889506i	$-0.651049\pi$
$421$	−7894.00	−0.913848	−0.456924	+	0.889506i	$0.651049\pi$
$422$	1584.00	0.182720
$423$	0	0
$424$	−1360.00	−0.155772
$425$	−2508.00	−0.286249
$426$	0	0
$427$	−6285.00	−0.712301
$428$	−872.000	−0.0984806
$429$	0	0
$430$	1022.00	0.114617
$431$	9234.00	1.03199	0.515993	−	0.856593i	$-0.327423\pi$
$431$	9234.00	1.03199	0.515993	+	0.856593i	$0.327423\pi$
$432$	0	0
$433$	9842.00	1.09232	0.546162	−	0.837680i	$-0.316088\pi$
$433$	9842.00	1.09232	0.546162	+	0.837680i	$0.316088\pi$
$434$	−2820.00	−0.311899
$435$	0	0
$436$	−8736.00	−0.959584
$437$	−2812.00	−0.307817
$438$	0	0
$439$	−10966.0	−1.19221	−0.596103	−	0.802908i	$-0.703285\pi$
$439$	−10966.0	−1.19221	−0.596103	+	0.802908i	$0.703285\pi$
$440$	2744.00	0.297307
$441$	0	0
$442$	924.000	0.0994348
$443$	8795.00	0.943257	0.471629	−	0.881797i	$-0.343666\pi$
$443$	8795.00	0.943257	0.471629	+	0.881797i	$0.343666\pi$
$444$	0	0
$445$	−9702.00	−1.03353
$446$	−9272.00	−0.984399
$447$	0	0
$448$	−960.000	−0.101240
$449$	−2476.00	−0.260244	−0.130122	−	0.991498i	$-0.541537\pi$
$449$	−2476.00	−0.260244	−0.130122	+	0.991498i	$0.541537\pi$
$450$	0	0
$451$	−19600.0	−2.04640
$452$	−5336.00	−0.555275
$453$	0	0
$454$	12892.0	1.33271
$455$	−1470.00	−0.151461
$456$	0	0
$457$	−13837.0	−1.41634	−0.708170	−	0.706042i	$-0.750479\pi$
$457$	−13837.0	−1.41634	−0.708170	+	0.706042i	$0.750479\pi$
$458$	11530.0	1.17634
$459$	0	0
$460$	4144.00	0.420033
$461$	−407.000	−0.0411190	−0.0205595	−	0.999789i	$-0.506545\pi$
$461$	−407.000	−0.0411190	−0.0205595	+	0.999789i	$0.506545\pi$
$462$	0	0
$463$	−17741.0	−1.78076	−0.890382	−	0.455213i	$-0.849563\pi$
$463$	−17741.0	−1.78076	−0.890382	+	0.455213i	$0.849563\pi$
$464$	4448.00	0.445028
$465$	0	0
$466$	11694.0	1.16248
$467$	−16765.0	−1.66122	−0.830612	−	0.556851i	$-0.812009\pi$
$467$	−16765.0	−1.66122	−0.830612	+	0.556851i	$0.812009\pi$
$468$	0	0
$469$	−6660.00	−0.655715
$470$	−2422.00	−0.237699
$471$	0	0
$472$	96.0000	0.00936178
$473$	3577.00	0.347718
$474$	0	0
$475$	1444.00	0.139485
$476$	−1980.00	−0.190658
$477$	0	0
$478$	−5646.00	−0.540255
$479$	824.000	0.0786003	0.0393001	−	0.999227i	$-0.487487\pi$
$479$	824.000	0.0786003	0.0393001	+	0.999227i	$0.487487\pi$
$480$	0	0
$481$	2240.00	0.212339
$482$	−12280.0	−1.16045
$483$	0	0
$484$	4280.00	0.401953
$485$	910.000	0.0851979
$486$	0	0
$487$	−668.000	−0.0621560	−0.0310780	−	0.999517i	$-0.509894\pi$
$487$	−668.000	−0.0621560	−0.0310780	+	0.999517i	$0.509894\pi$
$488$	3352.00	0.310938
$489$	0	0
$490$	−1652.00	−0.152306
$491$	−15080.0	−1.38605	−0.693025	−	0.720913i	$-0.743723\pi$
$491$	−15080.0	−1.38605	−0.693025	+	0.720913i	$0.743723\pi$
$492$	0	0
$493$	9174.00	0.838086
$494$	−532.000	−0.0484531
$495$	0	0
$496$	1504.00	0.136152
$497$	−14280.0	−1.28882
$498$	0	0
$499$	10915.0	0.979203	0.489602	−	0.871946i	$-0.337142\pi$
$499$	10915.0	0.979203	0.489602	+	0.871946i	$0.337142\pi$
$500$	−5628.00	−0.503384
$501$	0	0
$502$	6206.00	0.551768
$503$	16728.0	1.48283	0.741416	−	0.671046i	$-0.234154\pi$
$503$	16728.0	1.48283	0.741416	+	0.671046i	$0.234154\pi$
$504$	0	0
$505$	1666.00	0.146804
$506$	14504.0	1.27427
$507$	0	0
$508$	−2664.00	−0.232669
$509$	−17754.0	−1.54604	−0.773018	−	0.634384i	$-0.781254\pi$
$509$	−17754.0	−1.54604	−0.773018	+	0.634384i	$0.781254\pi$
$510$	0	0
$511$	405.000	0.0350609
$512$	512.000	0.0441942
$513$	0	0
$514$	−4672.00	−0.400920
$515$	−9618.00	−0.822951
$516$	0	0
$517$	−8477.00	−0.721118
$518$	−4800.00	−0.407143
$519$	0	0
$520$	784.000	0.0661167
$521$	2584.00	0.217288	0.108644	−	0.994081i	$-0.465349\pi$
$521$	2584.00	0.217288	0.108644	+	0.994081i	$0.465349\pi$
$522$	0	0
$523$	−2158.00	−0.180426	−0.0902130	−	0.995922i	$-0.528755\pi$
$523$	−2158.00	−0.180426	−0.0902130	+	0.995922i	$0.528755\pi$
$524$	1212.00	0.101043
$525$	0	0
$526$	5478.00	0.454092
$527$	3102.00	0.256405
$528$	0	0
$529$	9737.00	0.800279
$530$	−2380.00	−0.195058
$531$	0	0
$532$	1140.00	0.0929046
$533$	−5600.00	−0.455090
$534$	0	0
$535$	−1526.00	−0.123317
$536$	3552.00	0.286237
$537$	0	0
$538$	12972.0	1.03952
$539$	−5782.00	−0.462056
$540$	0	0
$541$	−14137.0	−1.12347	−0.561735	−	0.827317i	$-0.689866\pi$
$541$	−14137.0	−1.12347	−0.561735	+	0.827317i	$0.689866\pi$
$542$	−616.000	−0.0488182
$543$	0	0
$544$	1056.00	0.0832273
$545$	−15288.0	−1.20159
$546$	0	0
$547$	10222.0	0.799015	0.399507	−	0.916730i	$-0.369181\pi$
$547$	10222.0	0.799015	0.399507	+	0.916730i	$0.369181\pi$
$548$	−2332.00	−0.181785
$549$	0	0
$550$	−7448.00	−0.577425
$551$	−5282.00	−0.408386
$552$	0	0
$553$	8340.00	0.641325
$554$	5954.00	0.456609
$555$	0	0
$556$	−5868.00	−0.447587
$557$	−10387.0	−0.790146	−0.395073	−	0.918650i	$-0.629281\pi$
$557$	−10387.0	−0.790146	−0.395073	+	0.918650i	$0.629281\pi$
$558$	0	0
$559$	1022.00	0.0773274
$560$	−1680.00	−0.126773
$561$	0	0
$562$	−9140.00	−0.686028
$563$	10404.0	0.778821	0.389411	−	0.921064i	$-0.372679\pi$
$563$	10404.0	0.778821	0.389411	+	0.921064i	$0.372679\pi$
$564$	0	0
$565$	−9338.00	−0.695314
$566$	12858.0	0.954880
$567$	0	0
$568$	7616.00	0.562606
$569$	−4258.00	−0.313716	−0.156858	−	0.987621i	$-0.550137\pi$
$569$	−4258.00	−0.313716	−0.156858	+	0.987621i	$0.550137\pi$
$570$	0	0
$571$	6440.00	0.471989	0.235994	−	0.971754i	$-0.424165\pi$
$571$	6440.00	0.471989	0.235994	+	0.971754i	$0.424165\pi$
$572$	2744.00	0.200581
$573$	0	0
$574$	12000.0	0.872596
$575$	−11248.0	−0.815781
$576$	0	0
$577$	−14869.0	−1.07280	−0.536399	−	0.843964i	$-0.680216\pi$
$577$	−14869.0	−1.07280	−0.536399	+	0.843964i	$0.680216\pi$
$578$	−7648.00	−0.550372
$579$	0	0
$580$	7784.00	0.557264
$581$	−4140.00	−0.295622
$582$	0	0
$583$	−8330.00	−0.591755
$584$	−216.000	−0.0153050
$585$	0	0
$586$	−11448.0	−0.807018
$587$	−1041.00	−0.0731970	−0.0365985	−	0.999330i	$-0.511652\pi$
$587$	−1041.00	−0.0731970	−0.0365985	+	0.999330i	$0.511652\pi$
$588$	0	0
$589$	−1786.00	−0.124942
$590$	168.000	0.0117228
$591$	0	0
$592$	2560.00	0.177729
$593$	15662.0	1.08459	0.542294	−	0.840188i	$-0.317556\pi$
$593$	15662.0	1.08459	0.542294	+	0.840188i	$0.317556\pi$
$594$	0	0
$595$	−3465.00	−0.238741
$596$	−1404.00	−0.0964934
$597$	0	0
$598$	4144.00	0.283379
$599$	−18900.0	−1.28920	−0.644602	−	0.764518i	$-0.722977\pi$
$599$	−18900.0	−1.28920	−0.644602	+	0.764518i	$0.722977\pi$
$600$	0	0
$601$	6100.00	0.414017	0.207008	−	0.978339i	$-0.433627\pi$
$601$	6100.00	0.414017	0.207008	+	0.978339i	$0.433627\pi$
$602$	−2190.00	−0.148269
$603$	0	0
$604$	−12400.0	−0.835346
$605$	7490.00	0.503325
$606$	0	0
$607$	−5902.00	−0.394654	−0.197327	−	0.980338i	$-0.563226\pi$
$607$	−5902.00	−0.394654	−0.197327	+	0.980338i	$0.563226\pi$
$608$	−608.000	−0.0405554
$609$	0	0
$610$	5866.00	0.389356
$611$	−2422.00	−0.160366
$612$	0	0
$613$	15901.0	1.04769	0.523846	−	0.851813i	$-0.324497\pi$
$613$	15901.0	1.04769	0.523846	+	0.851813i	$0.324497\pi$
$614$	16608.0	1.09160
$615$	0	0
$616$	−5880.00	−0.384597
$617$	30429.0	1.98545	0.992727	−	0.120385i	$-0.0384130\pi$
$617$	30429.0	1.98545	0.992727	+	0.120385i	$0.0384130\pi$
$618$	0	0
$619$	−22484.0	−1.45995	−0.729974	−	0.683475i	$-0.760468\pi$
$619$	−22484.0	−1.45995	−0.729974	+	0.683475i	$0.760468\pi$
$620$	2632.00	0.170490
$621$	0	0
$622$	1582.00	0.101981
$623$	20790.0	1.33697
$624$	0	0
$625$	−349.000	−0.0223360
$626$	20332.0	1.29813
$627$	0	0
$628$	−9896.00	−0.628811
$629$	5280.00	0.334702
$630$	0	0
$631$	29885.0	1.88542	0.942712	−	0.333607i	$-0.108266\pi$
$631$	29885.0	1.88542	0.942712	+	0.333607i	$0.108266\pi$
$632$	−4448.00	−0.279956
$633$	0	0
$634$	−12816.0	−0.802821
$635$	−4662.00	−0.291348
$636$	0	0
$637$	−1652.00	−0.102755
$638$	27244.0	1.69060
$639$	0	0
$640$	896.000	0.0553399
$641$	−4038.00	−0.248817	−0.124408	−	0.992231i	$-0.539703\pi$
$641$	−4038.00	−0.248817	−0.124408	+	0.992231i	$0.539703\pi$
$642$	0	0
$643$	19993.0	1.22620	0.613100	−	0.790005i	$-0.289922\pi$
$643$	19993.0	1.22620	0.613100	+	0.790005i	$0.289922\pi$
$644$	−8880.00	−0.543356
$645$	0	0
$646$	−1254.00	−0.0763746
$647$	17077.0	1.03766	0.518830	−	0.854877i	$-0.326368\pi$
$647$	17077.0	1.03766	0.518830	+	0.854877i	$0.326368\pi$
$648$	0	0
$649$	588.000	0.0355640
$650$	−2128.00	−0.128411
$651$	0	0
$652$	9440.00	0.567023
$653$	−17631.0	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−17631.0	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	0	0
$655$	2121.00	0.126526
$656$	−6400.00	−0.380912
$657$	0	0
$658$	5190.00	0.307488
$659$	−12014.0	−0.710165	−0.355083	−	0.934835i	$-0.615547\pi$
$659$	−12014.0	−0.710165	−0.355083	+	0.934835i	$0.615547\pi$
$660$	0	0
$661$	10852.0	0.638569	0.319284	−	0.947659i	$-0.396558\pi$
$661$	10852.0	0.638569	0.319284	+	0.947659i	$0.396558\pi$
$662$	5152.00	0.302475
$663$	0	0
$664$	2208.00	0.129047
$665$	1995.00	0.116335
$666$	0	0
$667$	41144.0	2.38846
$668$	4440.00	0.257169
$669$	0	0
$670$	6216.00	0.358425
$671$	20531.0	1.18121
$672$	0	0
$673$	−1708.00	−0.0978285	−0.0489142	−	0.998803i	$-0.515576\pi$
$673$	−1708.00	−0.0978285	−0.0489142	+	0.998803i	$0.515576\pi$
$674$	−15844.0	−0.905472
$675$	0	0
$676$	−8004.00	−0.455394
$677$	−17902.0	−1.01629	−0.508146	−	0.861271i	$-0.669669\pi$
$677$	−17902.0	−1.01629	−0.508146	+	0.861271i	$0.669669\pi$
$678$	0	0
$679$	−1950.00	−0.110212
$680$	1848.00	0.104217
$681$	0	0
$682$	9212.00	0.517222
$683$	−2938.00	−0.164597	−0.0822983	−	0.996608i	$-0.526226\pi$
$683$	−2938.00	−0.164597	−0.0822983	+	0.996608i	$0.526226\pi$
$684$	0	0
$685$	−4081.00	−0.227631
$686$	13830.0	0.769726
$687$	0	0
$688$	1168.00	0.0647232
$689$	−2380.00	−0.131598
$690$	0	0
$691$	519.000	0.0285726	0.0142863	−	0.999898i	$-0.495452\pi$
$691$	519.000	0.0285726	0.0142863	+	0.999898i	$0.495452\pi$
$692$	1032.00	0.0566918
$693$	0	0
$694$	−4610.00	−0.252152
$695$	−10269.0	−0.560468
$696$	0	0
$697$	−13200.0	−0.717340
$698$	−9238.00	−0.500951
$699$	0	0
$700$	4560.00	0.246217
$701$	4942.00	0.266272	0.133136	−	0.991098i	$-0.457495\pi$
$701$	4942.00	0.266272	0.133136	+	0.991098i	$0.457495\pi$
$702$	0	0
$703$	−3040.00	−0.163095
$704$	3136.00	0.167887
$705$	0	0
$706$	−24732.0	−1.31842
$707$	−3570.00	−0.189906
$708$	0	0
$709$	19302.0	1.02243	0.511214	−	0.859453i	$-0.329196\pi$
$709$	19302.0	1.02243	0.511214	+	0.859453i	$0.329196\pi$
$710$	13328.0	0.704494
$711$	0	0
$712$	−11088.0	−0.583624
$713$	13912.0	0.730727
$714$	0	0
$715$	4802.00	0.251167
$716$	15048.0	0.785433
$717$	0	0
$718$	24834.0	1.29080
$719$	22973.0	1.19158	0.595792	−	0.803139i	$-0.296838\pi$
$719$	22973.0	1.19158	0.595792	+	0.803139i	$0.296838\pi$
$720$	0	0
$721$	20610.0	1.06457
$722$	722.000	0.0372161
$723$	0	0
$724$	2824.00	0.144963
$725$	−21128.0	−1.08231
$726$	0	0
$727$	−32429.0	−1.65437	−0.827184	−	0.561932i	$-0.810058\pi$
$727$	−32429.0	−1.65437	−0.827184	+	0.561932i	$0.810058\pi$
$728$	−1680.00	−0.0855288
$729$	0	0
$730$	−378.000	−0.0191649
$731$	2409.00	0.121888
$732$	0	0
$733$	2682.00	0.135146	0.0675729	−	0.997714i	$-0.478474\pi$
$733$	2682.00	0.135146	0.0675729	+	0.997714i	$0.478474\pi$
$734$	11552.0	0.580916
$735$	0	0
$736$	4736.00	0.237189
$737$	21756.0	1.08737
$738$	0	0
$739$	−15835.0	−0.788227	−0.394114	−	0.919062i	$-0.628948\pi$
$739$	−15835.0	−0.788227	−0.394114	+	0.919062i	$0.628948\pi$
$740$	4480.00	0.222551
$741$	0	0
$742$	5100.00	0.252327
$743$	16876.0	0.833271	0.416636	−	0.909074i	$-0.363209\pi$
$743$	16876.0	0.833271	0.416636	+	0.909074i	$0.363209\pi$
$744$	0	0
$745$	−2457.00	−0.120829
$746$	6784.00	0.332949
$747$	0	0
$748$	6468.00	0.316168
$749$	3270.00	0.159524
$750$	0	0
$751$	−35296.0	−1.71501	−0.857503	−	0.514479i	$-0.827985\pi$
$751$	−35296.0	−1.71501	−0.857503	+	0.514479i	$0.827985\pi$
$752$	−2768.00	−0.134227
$753$	0	0
$754$	7784.00	0.375964
$755$	−21700.0	−1.04602
$756$	0	0
$757$	18259.0	0.876664	0.438332	−	0.898813i	$-0.355569\pi$
$757$	18259.0	0.876664	0.438332	+	0.898813i	$0.355569\pi$
$758$	11532.0	0.552587
$759$	0	0
$760$	−1064.00	−0.0507833
$761$	−13455.0	−0.640924	−0.320462	−	0.947261i	$-0.603838\pi$
$761$	−13455.0	−0.640924	−0.320462	+	0.947261i	$0.603838\pi$
$762$	0	0
$763$	32760.0	1.55438
$764$	−10636.0	−0.503661
$765$	0	0
$766$	−16964.0	−0.800175
$767$	168.000	0.00790890
$768$	0	0
$769$	−16061.0	−0.753153	−0.376576	−	0.926386i	$-0.622899\pi$
$769$	−16061.0	−0.753153	−0.376576	+	0.926386i	$0.622899\pi$
$770$	−10290.0	−0.481592
$771$	0	0
$772$	14592.0	0.680282
$773$	−4680.00	−0.217759	−0.108880	−	0.994055i	$-0.534726\pi$
$773$	−4680.00	−0.217759	−0.108880	+	0.994055i	$0.534726\pi$
$774$	0	0
$775$	−7144.00	−0.331123
$776$	1040.00	0.0481106
$777$	0	0
$778$	3966.00	0.182761
$779$	7600.00	0.349548
$780$	0	0
$781$	46648.0	2.13726
$782$	9768.00	0.446679
$783$	0	0
$784$	−1888.00	−0.0860058
$785$	−17318.0	−0.787396
$786$	0	0
$787$	−37760.0	−1.71029	−0.855145	−	0.518388i	$-0.826532\pi$
$787$	−37760.0	−1.71029	−0.855145	+	0.518388i	$0.826532\pi$
$788$	−1976.00	−0.0893301
$789$	0	0
$790$	−7784.00	−0.350560
$791$	20010.0	0.899461
$792$	0	0
$793$	5866.00	0.262683
$794$	−9110.00	−0.407181
$795$	0	0
$796$	−14716.0	−0.655270
$797$	−22008.0	−0.978122	−0.489061	−	0.872250i	$-0.662660\pi$
$797$	−22008.0	−0.978122	−0.489061	+	0.872250i	$0.662660\pi$
$798$	0	0
$799$	−5709.00	−0.252778
$800$	−2432.00	−0.107480
$801$	0	0
$802$	23248.0	1.02359
$803$	−1323.00	−0.0581415
$804$	0	0
$805$	−15540.0	−0.680389
$806$	2632.00	0.115023
$807$	0	0
$808$	1904.00	0.0828991
$809$	12615.0	0.548232	0.274116	−	0.961697i	$-0.411615\pi$
$809$	12615.0	0.548232	0.274116	+	0.961697i	$0.411615\pi$
$810$	0	0
$811$	45402.0	1.96582	0.982910	−	0.184087i	$-0.0589329\pi$
$811$	45402.0	1.96582	0.982910	+	0.184087i	$0.0589329\pi$
$812$	−16680.0	−0.720878
$813$	0	0
$814$	15680.0	0.675164
$815$	16520.0	0.710025
$816$	0	0
$817$	−1387.00	−0.0593941
$818$	−24892.0	−1.06397
$819$	0	0
$820$	−11200.0	−0.476977
$821$	−1335.00	−0.0567501	−0.0283750	−	0.999597i	$-0.509033\pi$
$821$	−1335.00	−0.0567501	−0.0283750	+	0.999597i	$0.509033\pi$
$822$	0	0
$823$	−559.000	−0.0236762	−0.0118381	−	0.999930i	$-0.503768\pi$
$823$	−559.000	−0.0236762	−0.0118381	+	0.999930i	$0.503768\pi$
$824$	−10992.0	−0.464714
$825$	0	0
$826$	−360.000	−0.0151647
$827$	−13856.0	−0.582612	−0.291306	−	0.956630i	$-0.594090\pi$
$827$	−13856.0	−0.582612	−0.291306	+	0.956630i	$0.594090\pi$
$828$	0	0
$829$	18228.0	0.763673	0.381836	−	0.924230i	$-0.375292\pi$
$829$	18228.0	0.763673	0.381836	+	0.924230i	$0.375292\pi$
$830$	3864.00	0.161592
$831$	0	0
$832$	896.000	0.0373356
$833$	−3894.00	−0.161968
$834$	0	0
$835$	7770.00	0.322026
$836$	−3724.00	−0.154064
$837$	0	0
$838$	936.000	0.0385842
$839$	13414.0	0.551970	0.275985	−	0.961162i	$-0.410996\pi$
$839$	13414.0	0.551970	0.275985	+	0.961162i	$0.410996\pi$
$840$	0	0
$841$	52895.0	2.16881
$842$	−15788.0	−0.646188
$843$	0	0
$844$	3168.00	0.129203
$845$	−14007.0	−0.570243
$846$	0	0
$847$	−16050.0	−0.651103
$848$	−2720.00	−0.110148
$849$	0	0
$850$	−5016.00	−0.202409
$851$	23680.0	0.953866
$852$	0	0
$853$	−44718.0	−1.79498	−0.897488	−	0.441038i	$-0.854610\pi$
$853$	−44718.0	−1.79498	−0.897488	+	0.441038i	$0.854610\pi$
$854$	−12570.0	−0.503673
$855$	0	0
$856$	−1744.00	−0.0696363
$857$	33924.0	1.35218	0.676092	−	0.736817i	$-0.263672\pi$
$857$	33924.0	1.35218	0.676092	+	0.736817i	$0.263672\pi$
$858$	0	0
$859$	−16427.0	−0.652482	−0.326241	−	0.945287i	$-0.605782\pi$
$859$	−16427.0	−0.652482	−0.326241	+	0.945287i	$0.605782\pi$
$860$	2044.00	0.0810463
$861$	0	0
$862$	18468.0	0.729725
$863$	−23292.0	−0.918736	−0.459368	−	0.888246i	$-0.651924\pi$
$863$	−23292.0	−0.918736	−0.459368	+	0.888246i	$0.651924\pi$
$864$	0	0
$865$	1806.00	0.0709894
$866$	19684.0	0.772390
$867$	0	0
$868$	−5640.00	−0.220546
$869$	−27244.0	−1.06351
$870$	0	0
$871$	6216.00	0.241815
$872$	−17472.0	−0.678528
$873$	0	0
$874$	−5624.00	−0.217660
$875$	21105.0	0.815405
$876$	0	0
$877$	−43598.0	−1.67868	−0.839339	−	0.543609i	$-0.817057\pi$
$877$	−43598.0	−1.67868	−0.839339	+	0.543609i	$0.817057\pi$
$878$	−21932.0	−0.843017
$879$	0	0
$880$	5488.00	0.210228
$881$	−39123.0	−1.49613	−0.748063	−	0.663627i	$-0.769016\pi$
$881$	−39123.0	−1.49613	−0.748063	+	0.663627i	$0.769016\pi$
$882$	0	0
$883$	−4115.00	−0.156830	−0.0784149	−	0.996921i	$-0.524986\pi$
$883$	−4115.00	−0.156830	−0.0784149	+	0.996921i	$0.524986\pi$
$884$	1848.00	0.0703110
$885$	0	0
$886$	17590.0	0.666984
$887$	−13384.0	−0.506641	−0.253321	−	0.967382i	$-0.581523\pi$
$887$	−13384.0	−0.506641	−0.253321	+	0.967382i	$0.581523\pi$
$888$	0	0
$889$	9990.00	0.376888
$890$	−19404.0	−0.730813
$891$	0	0
$892$	−18544.0	−0.696075
$893$	3287.00	0.123175
$894$	0	0
$895$	26334.0	0.983518
$896$	−1920.00	−0.0715878
$897$	0	0
$898$	−4952.00	−0.184020
$899$	26132.0	0.969467
$900$	0	0
$901$	−5610.00	−0.207432
$902$	−39200.0	−1.44703
$903$	0	0
$904$	−10672.0	−0.392639
$905$	4942.00	0.181522
$906$	0	0
$907$	−36718.0	−1.34421	−0.672106	−	0.740454i	$-0.734610\pi$
$907$	−36718.0	−1.34421	−0.672106	+	0.740454i	$0.734610\pi$
$908$	25784.0	0.942370
$909$	0	0
$910$	−2940.00	−0.107099
$911$	−46614.0	−1.69527	−0.847635	−	0.530580i	$-0.821974\pi$
$911$	−46614.0	−1.69527	−0.847635	+	0.530580i	$0.821974\pi$
$912$	0	0
$913$	13524.0	0.490229
$914$	−27674.0	−1.00150
$915$	0	0
$916$	23060.0	0.831795
$917$	−4545.00	−0.163674
$918$	0	0
$919$	11192.0	0.401730	0.200865	−	0.979619i	$-0.435625\pi$
$919$	11192.0	0.401730	0.200865	+	0.979619i	$0.435625\pi$
$920$	8288.00	0.297008
$921$	0	0
$922$	−814.000	−0.0290756
$923$	13328.0	0.475294
$924$	0	0
$925$	−12160.0	−0.432236
$926$	−35482.0	−1.25919
$927$	0	0
$928$	8896.00	0.314683
$929$	−542.000	−0.0191415	−0.00957074	−	0.999954i	$-0.503047\pi$
$929$	−542.000	−0.0191415	−0.00957074	+	0.999954i	$0.503047\pi$
$930$	0	0
$931$	2242.00	0.0789244
$932$	23388.0	0.821995
$933$	0	0
$934$	−33530.0	−1.17466
$935$	11319.0	0.395905
$936$	0	0
$937$	39053.0	1.36159	0.680793	−	0.732476i	$-0.261635\pi$
$937$	39053.0	1.36159	0.680793	+	0.732476i	$0.261635\pi$
$938$	−13320.0	−0.463660
$939$	0	0
$940$	−4844.00	−0.168079
$941$	33398.0	1.15701	0.578504	−	0.815680i	$-0.303637\pi$
$941$	33398.0	1.15701	0.578504	+	0.815680i	$0.303637\pi$
$942$	0	0
$943$	−59200.0	−2.04434
$944$	192.000	0.00661978
$945$	0	0
$946$	7154.00	0.245874
$947$	54084.0	1.85585	0.927927	−	0.372762i	$-0.121589\pi$
$947$	54084.0	1.85585	0.927927	+	0.372762i	$0.121589\pi$
$948$	0	0
$949$	−378.000	−0.0129298
$950$	2888.00	0.0986306
$951$	0	0
$952$	−3960.00	−0.134815
$953$	30484.0	1.03617	0.518087	−	0.855328i	$-0.326644\pi$
$953$	30484.0	1.03617	0.518087	+	0.855328i	$0.326644\pi$
$954$	0	0
$955$	−18613.0	−0.630683
$956$	−11292.0	−0.382018
$957$	0	0
$958$	1648.00	0.0555788
$959$	8745.00	0.294464
$960$	0	0
$961$	−20955.0	−0.703400
$962$	4480.00	0.150147
$963$	0	0
$964$	−24560.0	−0.820565
$965$	25536.0	0.851848
$966$	0	0
$967$	13584.0	0.451739	0.225870	−	0.974158i	$-0.427478\pi$
$967$	13584.0	0.451739	0.225870	+	0.974158i	$0.427478\pi$
$968$	8560.00	0.284224
$969$	0	0
$970$	1820.00	0.0602440
$971$	−43892.0	−1.45063	−0.725315	−	0.688417i	$-0.758306\pi$
$971$	−43892.0	−1.45063	−0.725315	+	0.688417i	$0.758306\pi$
$972$	0	0
$973$	22005.0	0.725024
$974$	−1336.00	−0.0439509
$975$	0	0
$976$	6704.00	0.219867
$977$	−30542.0	−1.00013	−0.500064	−	0.865988i	$-0.666690\pi$
$977$	−30542.0	−1.00013	−0.500064	+	0.865988i	$0.666690\pi$
$978$	0	0
$979$	−67914.0	−2.21710
$980$	−3304.00	−0.107696
$981$	0	0
$982$	−30160.0	−0.980086
$983$	2868.00	0.0930570	0.0465285	−	0.998917i	$-0.485184\pi$
$983$	2868.00	0.0930570	0.0465285	+	0.998917i	$0.485184\pi$
$984$	0	0
$985$	−3458.00	−0.111859
$986$	18348.0	0.592616
$987$	0	0
$988$	−1064.00	−0.0342615
$989$	10804.0	0.347368
$990$	0	0
$991$	−23696.0	−0.759564	−0.379782	−	0.925076i	$-0.624001\pi$
$991$	−23696.0	−0.759564	−0.379782	+	0.925076i	$0.624001\pi$
$992$	3008.00	0.0962743
$993$	0	0
$994$	−28560.0	−0.911336
$995$	−25753.0	−0.820528
$996$	0	0
$997$	−46811.0	−1.48698	−0.743490	−	0.668747i	$-0.766831\pi$
$997$	−46811.0	−1.48698	−0.743490	+	0.668747i	$0.766831\pi$
$998$	21830.0	0.692401
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.4.a.c.1.1		1
3.2	odd	2		114.4.a.b.1.1	✓	1
12.11	even	2		912.4.a.b.1.1		1
57.56	even	2		2166.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.a.b.1.1	✓	1	3.2	odd	2
342.4.a.c.1.1		1	1.1	even	1	trivial
912.4.a.b.1.1		1	12.11	even	2
2166.4.a.d.1.1		1	57.56	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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