LMFDB - Embedded newform 342.4.a.a.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:	$1$
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)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +11.0000 q^{5} -15.0000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} +11.0000 q^{5} -15.0000 q^{7} -8.00000 q^{8} -22.0000 q^{10} +29.0000 q^{11} -82.0000 q^{13} +30.0000 q^{14} +16.0000 q^{16} -27.0000 q^{17} -19.0000 q^{19} +44.0000 q^{20} -58.0000 q^{22} -100.000 q^{23} -4.00000 q^{25} +164.000 q^{26} -60.0000 q^{28} +118.000 q^{29} +70.0000 q^{31} -32.0000 q^{32} +54.0000 q^{34} -165.000 q^{35} +232.000 q^{37} +38.0000 q^{38} -88.0000 q^{40} -8.00000 q^{41} -287.000 q^{43} +116.000 q^{44} +200.000 q^{46} -385.000 q^{47} -118.000 q^{49} +8.00000 q^{50} -328.000 q^{52} -538.000 q^{53} +319.000 q^{55} +120.000 q^{56} -236.000 q^{58} +300.000 q^{59} -901.000 q^{61} -140.000 q^{62} +64.0000 q^{64} -902.000 q^{65} +132.000 q^{67} -108.000 q^{68} +330.000 q^{70} -472.000 q^{71} -1131.00 q^{73} -464.000 q^{74} -76.0000 q^{76} -435.000 q^{77} -52.0000 q^{79} +176.000 q^{80} +16.0000 q^{82} -276.000 q^{83} -297.000 q^{85} +574.000 q^{86} -232.000 q^{88} +1302.00 q^{89} +1230.00 q^{91} -400.000 q^{92} +770.000 q^{94} -209.000 q^{95} -1310.00 q^{97} +236.000 q^{98} +O(q^{100})\)

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
$5$	11.0000	0.983870	0.491935	−	0.870632i	$-0.336290\pi$
$5$	11.0000	0.983870	0.491935	+	0.870632i	$0.336290\pi$
$6$	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
$10$	−22.0000	−0.695701
$11$	29.0000	0.794894	0.397447	−	0.917625i	$-0.369896\pi$
$11$	29.0000	0.794894	0.397447	+	0.917625i	$0.369896\pi$
$12$	0	0
$13$	−82.0000	−1.74944	−0.874720	−	0.484629i	$-0.838954\pi$
$13$	−82.0000	−1.74944	−0.874720	+	0.484629i	$0.838954\pi$
$14$	30.0000	0.572703
$15$	0	0
$16$	16.0000	0.250000
$17$	−27.0000	−0.385204	−0.192602	−	0.981277i	$-0.561693\pi$
$17$	−27.0000	−0.385204	−0.192602	+	0.981277i	$0.561693\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	44.0000	0.491935
$21$	0	0
$22$	−58.0000	−0.562075
$23$	−100.000	−0.906584	−0.453292	−	0.891362i	$-0.649751\pi$
$23$	−100.000	−0.906584	−0.453292	+	0.891362i	$0.649751\pi$
$24$	0	0
$25$	−4.00000	−0.0320000
$26$	164.000	1.23704
$27$	0	0
$28$	−60.0000	−0.404962
$29$	118.000	0.755588	0.377794	−	0.925890i	$-0.376683\pi$
$29$	118.000	0.755588	0.377794	+	0.925890i	$0.376683\pi$
$30$	0	0
$31$	70.0000	0.405560	0.202780	−	0.979224i	$-0.435002\pi$
$31$	70.0000	0.405560	0.202780	+	0.979224i	$0.435002\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	54.0000	0.272380
$35$	−165.000	−0.796860
$36$	0	0
$37$	232.000	1.03083	0.515413	−	0.856942i	$-0.327639\pi$
$37$	232.000	1.03083	0.515413	+	0.856942i	$0.327639\pi$
$38$	38.0000	0.162221
$39$	0	0
$40$	−88.0000	−0.347851
$41$	−8.00000	−0.0304729	−0.0152365	−	0.999884i	$-0.504850\pi$
$41$	−8.00000	−0.0304729	−0.0152365	+	0.999884i	$0.504850\pi$
$42$	0	0
$43$	−287.000	−1.01784	−0.508920	−	0.860814i	$-0.669955\pi$
$43$	−287.000	−1.01784	−0.508920	+	0.860814i	$0.669955\pi$
$44$	116.000	0.397447
$45$	0	0
$46$	200.000	0.641052
$47$	−385.000	−1.19485	−0.597426	−	0.801924i	$-0.703810\pi$
$47$	−385.000	−1.19485	−0.597426	+	0.801924i	$0.703810\pi$
$48$	0	0
$49$	−118.000	−0.344023
$50$	8.00000	0.0226274
$51$	0	0
$52$	−328.000	−0.874720
$53$	−538.000	−1.39434	−0.697170	−	0.716906i	$-0.745558\pi$
$53$	−538.000	−1.39434	−0.697170	+	0.716906i	$0.745558\pi$
$54$	0	0
$55$	319.000	0.782072
$56$	120.000	0.286351
$57$	0	0
$58$	−236.000	−0.534281
$59$	300.000	0.661978	0.330989	−	0.943635i	$-0.392618\pi$
$59$	300.000	0.661978	0.330989	+	0.943635i	$0.392618\pi$
$60$	0	0
$61$	−901.000	−1.89117	−0.945584	−	0.325379i	$-0.894508\pi$
$61$	−901.000	−1.89117	−0.945584	+	0.325379i	$0.894508\pi$
$62$	−140.000	−0.286774
$63$	0	0
$64$	64.0000	0.125000
$65$	−902.000	−1.72122
$66$	0	0
$67$	132.000	0.240692	0.120346	−	0.992732i	$-0.461600\pi$
$67$	132.000	0.240692	0.120346	+	0.992732i	$0.461600\pi$
$68$	−108.000	−0.192602
$69$	0	0
$70$	330.000	0.563465
$71$	−472.000	−0.788959	−0.394480	−	0.918905i	$-0.629075\pi$
$71$	−472.000	−0.788959	−0.394480	+	0.918905i	$0.629075\pi$
$72$	0	0
$73$	−1131.00	−1.81334	−0.906668	−	0.421845i	$-0.861383\pi$
$73$	−1131.00	−1.81334	−0.906668	+	0.421845i	$0.861383\pi$
$74$	−464.000	−0.728904
$75$	0	0
$76$	−76.0000	−0.114708
$77$	−435.000	−0.643803
$78$	0	0
$79$	−52.0000	−0.0740564	−0.0370282	−	0.999314i	$-0.511789\pi$
$79$	−52.0000	−0.0740564	−0.0370282	+	0.999314i	$0.511789\pi$
$80$	176.000	0.245967
$81$	0	0
$82$	16.0000	0.0215476
$83$	−276.000	−0.364999	−0.182500	−	0.983206i	$-0.558419\pi$
$83$	−276.000	−0.364999	−0.182500	+	0.983206i	$0.558419\pi$
$84$	0	0
$85$	−297.000	−0.378990
$86$	574.000	0.719721
$87$	0	0
$88$	−232.000	−0.281037
$89$	1302.00	1.55069	0.775347	−	0.631536i	$-0.217575\pi$
$89$	1302.00	1.55069	0.775347	+	0.631536i	$0.217575\pi$
$90$	0	0
$91$	1230.00	1.41691
$92$	−400.000	−0.453292
$93$	0	0
$94$	770.000	0.844888
$95$	−209.000	−0.225715
$96$	0	0
$97$	−1310.00	−1.37124	−0.685620	−	0.727959i	$-0.740469\pi$
$97$	−1310.00	−1.37124	−0.685620	+	0.727959i	$0.740469\pi$
$98$	236.000	0.243261
$99$	0	0
$100$	−16.0000	−0.0160000
$101$	638.000	0.628548	0.314274	−	0.949332i	$-0.398239\pi$
$101$	638.000	0.628548	0.314274	+	0.949332i	$0.398239\pi$
$102$	0	0
$103$	786.000	0.751911	0.375956	−	0.926638i	$-0.377314\pi$
$103$	786.000	0.751911	0.375956	+	0.926638i	$0.377314\pi$
$104$	656.000	0.618520
$105$	0	0
$106$	1076.00	0.985947
$107$	1310.00	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$107$	1310.00	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$108$	0	0
$109$	−1296.00	−1.13885	−0.569423	−	0.822044i	$-0.692833\pi$
$109$	−1296.00	−1.13885	−0.569423	+	0.822044i	$0.692833\pi$
$110$	−638.000	−0.553008
$111$	0	0
$112$	−240.000	−0.202481
$113$	1130.00	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	1130.00	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	−1100.00	−0.891961
$116$	472.000	0.377794
$117$	0	0
$118$	−600.000	−0.468089
$119$	405.000	0.311986
$120$	0	0
$121$	−490.000	−0.368144
$122$	1802.00	1.33726
$123$	0	0
$124$	280.000	0.202780
$125$	−1419.00	−1.01535
$126$	0	0
$127$	750.000	0.524029	0.262015	−	0.965064i	$-0.415613\pi$
$127$	750.000	0.524029	0.262015	+	0.965064i	$0.415613\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	1804.00	1.21709
$131$	1275.00	0.850361	0.425180	−	0.905109i	$-0.360211\pi$
$131$	1275.00	0.850361	0.425180	+	0.905109i	$0.360211\pi$
$132$	0	0
$133$	285.000	0.185809
$134$	−264.000	−0.170195
$135$	0	0
$136$	216.000	0.136190
$137$	−2123.00	−1.32394	−0.661971	−	0.749529i	$-0.730280\pi$
$137$	−2123.00	−1.32394	−0.661971	+	0.749529i	$0.730280\pi$
$138$	0	0
$139$	2277.00	1.38944	0.694722	−	0.719279i	$-0.255528\pi$
$139$	2277.00	1.38944	0.694722	+	0.719279i	$0.255528\pi$
$140$	−660.000	−0.398430
$141$	0	0
$142$	944.000	0.557878
$143$	−2378.00	−1.39062
$144$	0	0
$145$	1298.00	0.743400
$146$	2262.00	1.28222
$147$	0	0
$148$	928.000	0.515413
$149$	−2235.00	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−2235.00	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−2764.00	−1.48961	−0.744805	−	0.667282i	$-0.767458\pi$
$151$	−2764.00	−1.48961	−0.744805	+	0.667282i	$0.767458\pi$
$152$	152.000	0.0811107
$153$	0	0
$154$	870.000	0.455238
$155$	770.000	0.399019
$156$	0	0
$157$	1702.00	0.865187	0.432594	−	0.901589i	$-0.357599\pi$
$157$	1702.00	0.865187	0.432594	+	0.901589i	$0.357599\pi$
$158$	104.000	0.0523658
$159$	0	0
$160$	−352.000	−0.173925
$161$	1500.00	0.734264
$162$	0	0
$163$	2840.00	1.36470	0.682350	−	0.731026i	$-0.260958\pi$
$163$	2840.00	1.36470	0.682350	+	0.731026i	$0.260958\pi$
$164$	−32.0000	−0.0152365
$165$	0	0
$166$	552.000	0.258093
$167$	−1194.00	−0.553260	−0.276630	−	0.960976i	$-0.589218\pi$
$167$	−1194.00	−0.553260	−0.276630	+	0.960976i	$0.589218\pi$
$168$	0	0
$169$	4527.00	2.06054
$170$	594.000	0.267987
$171$	0	0
$172$	−1148.00	−0.508920
$173$	3786.00	1.66384	0.831920	−	0.554896i	$-0.187242\pi$
$173$	3786.00	1.66384	0.831920	+	0.554896i	$0.187242\pi$
$174$	0	0
$175$	60.0000	0.0259176
$176$	464.000	0.198723
$177$	0	0
$178$	−2604.00	−1.09651
$179$	−2094.00	−0.874374	−0.437187	−	0.899371i	$-0.644025\pi$
$179$	−2094.00	−0.874374	−0.437187	+	0.899371i	$0.644025\pi$
$180$	0	0
$181$	−350.000	−0.143731	−0.0718655	−	0.997414i	$-0.522895\pi$
$181$	−350.000	−0.143731	−0.0718655	+	0.997414i	$0.522895\pi$
$182$	−2460.00	−1.00191
$183$	0	0
$184$	800.000	0.320526
$185$	2552.00	1.01420
$186$	0	0
$187$	−783.000	−0.306196
$188$	−1540.00	−0.597426
$189$	0	0
$190$	418.000	0.159605
$191$	1489.00	0.564085	0.282043	−	0.959402i	$-0.408988\pi$
$191$	1489.00	0.564085	0.282043	+	0.959402i	$0.408988\pi$
$192$	0	0
$193$	600.000	0.223777	0.111888	−	0.993721i	$-0.464310\pi$
$193$	600.000	0.223777	0.111888	+	0.993721i	$0.464310\pi$
$194$	2620.00	0.969614
$195$	0	0
$196$	−472.000	−0.172012
$197$	290.000	0.104881	0.0524407	−	0.998624i	$-0.483300\pi$
$197$	290.000	0.104881	0.0524407	+	0.998624i	$0.483300\pi$
$198$	0	0
$199$	4169.00	1.48509	0.742544	−	0.669797i	$-0.233619\pi$
$199$	4169.00	1.48509	0.742544	+	0.669797i	$0.233619\pi$
$200$	32.0000	0.0113137
$201$	0	0
$202$	−1276.00	−0.444451
$203$	−1770.00	−0.611969
$204$	0	0
$205$	−88.0000	−0.0299814
$206$	−1572.00	−0.531682
$207$	0	0
$208$	−1312.00	−0.437360
$209$	−551.000	−0.182361
$210$	0	0
$211$	−1368.00	−0.446337	−0.223168	−	0.974780i	$-0.571640\pi$
$211$	−1368.00	−0.446337	−0.223168	+	0.974780i	$0.571640\pi$
$212$	−2152.00	−0.697170
$213$	0	0
$214$	−2620.00	−0.836914
$215$	−3157.00	−1.00142
$216$	0	0
$217$	−1050.00	−0.328473
$218$	2592.00	0.805286
$219$	0	0
$220$	1276.00	0.391036
$221$	2214.00	0.673890
$222$	0	0
$223$	2540.00	0.762740	0.381370	−	0.924423i	$-0.375452\pi$
$223$	2540.00	0.762740	0.381370	+	0.924423i	$0.375452\pi$
$224$	480.000	0.143176
$225$	0	0
$226$	−2260.00	−0.665190
$227$	5974.00	1.74673	0.873366	−	0.487064i	$-0.161932\pi$
$227$	5974.00	1.74673	0.873366	+	0.487064i	$0.161932\pi$
$228$	0	0
$229$	−355.000	−0.102441	−0.0512207	−	0.998687i	$-0.516311\pi$
$229$	−355.000	−0.102441	−0.0512207	+	0.998687i	$0.516311\pi$
$230$	2200.00	0.630712
$231$	0	0
$232$	−944.000	−0.267141
$233$	−237.000	−0.0666369	−0.0333184	−	0.999445i	$-0.510608\pi$
$233$	−237.000	−0.0666369	−0.0333184	+	0.999445i	$0.510608\pi$
$234$	0	0
$235$	−4235.00	−1.17558
$236$	1200.00	0.330989
$237$	0	0
$238$	−810.000	−0.220607
$239$	−1635.00	−0.442508	−0.221254	−	0.975216i	$-0.571015\pi$
$239$	−1635.00	−0.442508	−0.221254	+	0.975216i	$0.571015\pi$
$240$	0	0
$241$	−164.000	−0.0438347	−0.0219174	−	0.999760i	$-0.506977\pi$
$241$	−164.000	−0.0438347	−0.0219174	+	0.999760i	$0.506977\pi$
$242$	980.000	0.260317
$243$	0	0
$244$	−3604.00	−0.945584
$245$	−1298.00	−0.338474
$246$	0	0
$247$	1558.00	0.401349
$248$	−560.000	−0.143387
$249$	0	0
$250$	2838.00	0.717964
$251$	2099.00	0.527839	0.263920	−	0.964545i	$-0.414985\pi$
$251$	2099.00	0.527839	0.263920	+	0.964545i	$0.414985\pi$
$252$	0	0
$253$	−2900.00	−0.720638
$254$	−1500.00	−0.370545
$255$	0	0
$256$	256.000	0.0625000
$257$	−5536.00	−1.34368	−0.671841	−	0.740696i	$-0.734496\pi$
$257$	−5536.00	−1.34368	−0.671841	+	0.740696i	$0.734496\pi$
$258$	0	0
$259$	−3480.00	−0.834891
$260$	−3608.00	−0.860610
$261$	0	0
$262$	−2550.00	−0.601296
$263$	−2073.00	−0.486033	−0.243016	−	0.970022i	$-0.578137\pi$
$263$	−2073.00	−0.486033	−0.243016	+	0.970022i	$0.578137\pi$
$264$	0	0
$265$	−5918.00	−1.37185
$266$	−570.000	−0.131387
$267$	0	0
$268$	528.000	0.120346
$269$	−1482.00	−0.335908	−0.167954	−	0.985795i	$-0.553716\pi$
$269$	−1482.00	−0.335908	−0.167954	+	0.985795i	$0.553716\pi$
$270$	0	0
$271$	−7268.00	−1.62915	−0.814575	−	0.580058i	$-0.803030\pi$
$271$	−7268.00	−1.62915	−0.814575	+	0.580058i	$0.803030\pi$
$272$	−432.000	−0.0963009
$273$	0	0
$274$	4246.00	0.936169
$275$	−116.000	−0.0254366
$276$	0	0
$277$	−4583.00	−0.994100	−0.497050	−	0.867722i	$-0.665584\pi$
$277$	−4583.00	−0.994100	−0.497050	+	0.867722i	$0.665584\pi$
$278$	−4554.00	−0.982485
$279$	0	0
$280$	1320.00	0.281732
$281$	−5762.00	−1.22325	−0.611623	−	0.791149i	$-0.709483\pi$
$281$	−5762.00	−1.22325	−0.611623	+	0.791149i	$0.709483\pi$
$282$	0	0
$283$	3909.00	0.821081	0.410541	−	0.911842i	$-0.365340\pi$
$283$	3909.00	0.821081	0.410541	+	0.911842i	$0.365340\pi$
$284$	−1888.00	−0.394480
$285$	0	0
$286$	4756.00	0.983315
$287$	120.000	0.0246808
$288$	0	0
$289$	−4184.00	−0.851618
$290$	−2596.00	−0.525663
$291$	0	0
$292$	−4524.00	−0.906668
$293$	8148.00	1.62461	0.812306	−	0.583232i	$-0.198212\pi$
$293$	8148.00	1.62461	0.812306	+	0.583232i	$0.198212\pi$
$294$	0	0
$295$	3300.00	0.651300
$296$	−1856.00	−0.364452
$297$	0	0
$298$	4470.00	0.868927
$299$	8200.00	1.58601
$300$	0	0
$301$	4305.00	0.824372
$302$	5528.00	1.05331
$303$	0	0
$304$	−304.000	−0.0573539
$305$	−9911.00	−1.86066
$306$	0	0
$307$	600.000	0.111543	0.0557717	−	0.998444i	$-0.482238\pi$
$307$	600.000	0.111543	0.0557717	+	0.998444i	$0.482238\pi$
$308$	−1740.00	−0.321902
$309$	0	0
$310$	−1540.00	−0.282149
$311$	4963.00	0.904906	0.452453	−	0.891788i	$-0.350549\pi$
$311$	4963.00	0.904906	0.452453	+	0.891788i	$0.350549\pi$
$312$	0	0
$313$	5462.00	0.986359	0.493180	−	0.869927i	$-0.335835\pi$
$313$	5462.00	0.986359	0.493180	+	0.869927i	$0.335835\pi$
$314$	−3404.00	−0.611780
$315$	0	0
$316$	−208.000	−0.0370282
$317$	984.000	0.174344	0.0871718	−	0.996193i	$-0.472217\pi$
$317$	984.000	0.174344	0.0871718	+	0.996193i	$0.472217\pi$
$318$	0	0
$319$	3422.00	0.600612
$320$	704.000	0.122984
$321$	0	0
$322$	−3000.00	−0.519203
$323$	513.000	0.0883718
$324$	0	0
$325$	328.000	0.0559821
$326$	−5680.00	−0.964988
$327$	0	0
$328$	64.0000	0.0107738
$329$	5775.00	0.967739
$330$	0	0
$331$	−2632.00	−0.437063	−0.218531	−	0.975830i	$-0.570127\pi$
$331$	−2632.00	−0.437063	−0.218531	+	0.975830i	$0.570127\pi$
$332$	−1104.00	−0.182500
$333$	0	0
$334$	2388.00	0.391214
$335$	1452.00	0.236810
$336$	0	0
$337$	−11234.0	−1.81589	−0.907945	−	0.419089i	$-0.862349\pi$
$337$	−11234.0	−1.81589	−0.907945	+	0.419089i	$0.862349\pi$
$338$	−9054.00	−1.45702
$339$	0	0
$340$	−1188.00	−0.189495
$341$	2030.00	0.322377
$342$	0	0
$343$	6915.00	1.08856
$344$	2296.00	0.359861
$345$	0	0
$346$	−7572.00	−1.17651
$347$	3211.00	0.496759	0.248380	−	0.968663i	$-0.420102\pi$
$347$	3211.00	0.496759	0.248380	+	0.968663i	$0.420102\pi$
$348$	0	0
$349$	2341.00	0.359057	0.179528	−	0.983753i	$-0.442543\pi$
$349$	2341.00	0.359057	0.179528	+	0.983753i	$0.442543\pi$
$350$	−120.000	−0.0183265
$351$	0	0
$352$	−928.000	−0.140519
$353$	−6366.00	−0.959853	−0.479926	−	0.877309i	$-0.659337\pi$
$353$	−6366.00	−0.959853	−0.479926	+	0.877309i	$0.659337\pi$
$354$	0	0
$355$	−5192.00	−0.776233
$356$	5208.00	0.775347
$357$	0	0
$358$	4188.00	0.618276
$359$	7989.00	1.17449	0.587247	−	0.809408i	$-0.300212\pi$
$359$	7989.00	1.17449	0.587247	+	0.809408i	$0.300212\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	700.000	0.101633
$363$	0	0
$364$	4920.00	0.708456
$365$	−12441.0	−1.78409
$366$	0	0
$367$	−10544.0	−1.49971	−0.749853	−	0.661604i	$-0.769876\pi$
$367$	−10544.0	−1.49971	−0.749853	+	0.661604i	$0.769876\pi$
$368$	−1600.00	−0.226646
$369$	0	0
$370$	−5104.00	−0.717147
$371$	8070.00	1.12931
$372$	0	0
$373$	7616.00	1.05722	0.528608	−	0.848866i	$-0.322714\pi$
$373$	7616.00	1.05722	0.528608	+	0.848866i	$0.322714\pi$
$374$	1566.00	0.216513
$375$	0	0
$376$	3080.00	0.422444
$377$	−9676.00	−1.32186
$378$	0	0
$379$	5958.00	0.807498	0.403749	−	0.914870i	$-0.367707\pi$
$379$	5958.00	0.807498	0.403749	+	0.914870i	$0.367707\pi$
$380$	−836.000	−0.112858
$381$	0	0
$382$	−2978.00	−0.398868
$383$	12382.0	1.65193	0.825967	−	0.563719i	$-0.190630\pi$
$383$	12382.0	1.65193	0.825967	+	0.563719i	$0.190630\pi$
$384$	0	0
$385$	−4785.00	−0.633419
$386$	−1200.00	−0.158234
$387$	0	0
$388$	−5240.00	−0.685620
$389$	−7989.00	−1.04128	−0.520641	−	0.853776i	$-0.674307\pi$
$389$	−7989.00	−1.04128	−0.520641	+	0.853776i	$0.674307\pi$
$390$	0	0
$391$	2700.00	0.349220
$392$	944.000	0.121631
$393$	0	0
$394$	−580.000	−0.0741624
$395$	−572.000	−0.0728619
$396$	0	0
$397$	7061.00	0.892648	0.446324	−	0.894871i	$-0.352733\pi$
$397$	7061.00	0.892648	0.446324	+	0.894871i	$0.352733\pi$
$398$	−8338.00	−1.05012
$399$	0	0
$400$	−64.0000	−0.00800000
$401$	14512.0	1.80722	0.903609	−	0.428357i	$-0.140908\pi$
$401$	14512.0	1.80722	0.903609	+	0.428357i	$0.140908\pi$
$402$	0	0
$403$	−5740.00	−0.709503
$404$	2552.00	0.314274
$405$	0	0
$406$	3540.00	0.432727
$407$	6728.00	0.819397
$408$	0	0
$409$	12634.0	1.52741	0.763705	−	0.645565i	$-0.223378\pi$
$409$	12634.0	1.52741	0.763705	+	0.645565i	$0.223378\pi$
$410$	176.000	0.0212000
$411$	0	0
$412$	3144.00	0.375956
$413$	−4500.00	−0.536151
$414$	0	0
$415$	−3036.00	−0.359112
$416$	2624.00	0.309260
$417$	0	0
$418$	1102.00	0.128949
$419$	8268.00	0.964005	0.482003	−	0.876170i	$-0.339910\pi$
$419$	8268.00	0.964005	0.482003	+	0.876170i	$0.339910\pi$
$420$	0	0
$421$	−1534.00	−0.177583	−0.0887917	−	0.996050i	$-0.528301\pi$
$421$	−1534.00	−0.177583	−0.0887917	+	0.996050i	$0.528301\pi$
$422$	2736.00	0.315608
$423$	0	0
$424$	4304.00	0.492973
$425$	108.000	0.0123265
$426$	0	0
$427$	13515.0	1.53170
$428$	5240.00	0.591787
$429$	0	0
$430$	6314.00	0.708112
$431$	−14358.0	−1.60464	−0.802321	−	0.596893i	$-0.796402\pi$
$431$	−14358.0	−1.60464	−0.802321	+	0.596893i	$0.796402\pi$
$432$	0	0
$433$	−4534.00	−0.503210	−0.251605	−	0.967830i	$-0.580958\pi$
$433$	−4534.00	−0.503210	−0.251605	+	0.967830i	$0.580958\pi$
$434$	2100.00	0.232265
$435$	0	0
$436$	−5184.00	−0.569423
$437$	1900.00	0.207985
$438$	0	0
$439$	−12766.0	−1.38790	−0.693950	−	0.720023i	$-0.744131\pi$
$439$	−12766.0	−1.38790	−0.693950	+	0.720023i	$0.744131\pi$
$440$	−2552.00	−0.276504
$441$	0	0
$442$	−4428.00	−0.476512
$443$	16711.0	1.79224	0.896121	−	0.443809i	$-0.146373\pi$
$443$	16711.0	1.79224	0.896121	+	0.443809i	$0.146373\pi$
$444$	0	0
$445$	14322.0	1.52568
$446$	−5080.00	−0.539339
$447$	0	0
$448$	−960.000	−0.101240
$449$	−3932.00	−0.413280	−0.206640	−	0.978417i	$-0.566253\pi$
$449$	−3932.00	−0.413280	−0.206640	+	0.978417i	$0.566253\pi$
$450$	0	0
$451$	−232.000	−0.0242227
$452$	4520.00	0.470360
$453$	0	0
$454$	−11948.0	−1.23513
$455$	13530.0	1.39406
$456$	0	0
$457$	4451.00	0.455600	0.227800	−	0.973708i	$-0.426847\pi$
$457$	4451.00	0.455600	0.227800	+	0.973708i	$0.426847\pi$
$458$	710.000	0.0724369
$459$	0	0
$460$	−4400.00	−0.445981
$461$	−9619.00	−0.971804	−0.485902	−	0.874013i	$-0.661509\pi$
$461$	−9619.00	−0.971804	−0.485902	+	0.874013i	$0.661509\pi$
$462$	0	0
$463$	16603.0	1.66654	0.833269	−	0.552868i	$-0.186467\pi$
$463$	16603.0	1.66654	0.833269	+	0.552868i	$0.186467\pi$
$464$	1888.00	0.188897
$465$	0	0
$466$	474.000	0.0471194
$467$	−12857.0	−1.27399	−0.636993	−	0.770870i	$-0.719822\pi$
$467$	−12857.0	−1.27399	−0.636993	+	0.770870i	$0.719822\pi$
$468$	0	0
$469$	−1980.00	−0.194942
$470$	8470.00	0.831260
$471$	0	0
$472$	−2400.00	−0.234044
$473$	−8323.00	−0.809074
$474$	0	0
$475$	76.0000	0.00734130
$476$	1620.00	0.155993
$477$	0	0
$478$	3270.00	0.312900
$479$	−11072.0	−1.05614	−0.528072	−	0.849200i	$-0.677085\pi$
$479$	−11072.0	−1.05614	−0.528072	+	0.849200i	$0.677085\pi$
$480$	0	0
$481$	−19024.0	−1.80337
$482$	328.000	0.0309958
$483$	0	0
$484$	−1960.00	−0.184072
$485$	−14410.0	−1.34912
$486$	0	0
$487$	11284.0	1.04995	0.524976	−	0.851117i	$-0.324074\pi$
$487$	11284.0	1.04995	0.524976	+	0.851117i	$0.324074\pi$
$488$	7208.00	0.668629
$489$	0	0
$490$	2596.00	0.239337
$491$	11984.0	1.10149	0.550744	−	0.834674i	$-0.314344\pi$
$491$	11984.0	1.10149	0.550744	+	0.834674i	$0.314344\pi$
$492$	0	0
$493$	−3186.00	−0.291055
$494$	−3116.00	−0.283796
$495$	0	0
$496$	1120.00	0.101390
$497$	7080.00	0.638997
$498$	0	0
$499$	−18701.0	−1.67770	−0.838849	−	0.544364i	$-0.816771\pi$
$499$	−18701.0	−1.67770	−0.838849	+	0.544364i	$0.816771\pi$
$500$	−5676.00	−0.507677
$501$	0	0
$502$	−4198.00	−0.373239
$503$	−21888.0	−1.94023	−0.970117	−	0.242638i	$-0.921987\pi$
$503$	−21888.0	−1.94023	−0.970117	+	0.242638i	$0.921987\pi$
$504$	0	0
$505$	7018.00	0.618410
$506$	5800.00	0.509568
$507$	0	0
$508$	3000.00	0.262015
$509$	14238.0	1.23986	0.619930	−	0.784657i	$-0.287161\pi$
$509$	14238.0	1.23986	0.619930	+	0.784657i	$0.287161\pi$
$510$	0	0
$511$	16965.0	1.46866
$512$	−512.000	−0.0441942
$513$	0	0
$514$	11072.0	0.950126
$515$	8646.00	0.739783
$516$	0	0
$517$	−11165.0	−0.949780
$518$	6960.00	0.590357
$519$	0	0
$520$	7216.00	0.608543
$521$	12992.0	1.09249	0.546247	−	0.837624i	$-0.316056\pi$
$521$	12992.0	1.09249	0.546247	+	0.837624i	$0.316056\pi$
$522$	0	0
$523$	−9070.00	−0.758324	−0.379162	−	0.925330i	$-0.623788\pi$
$523$	−9070.00	−0.758324	−0.379162	+	0.925330i	$0.623788\pi$
$524$	5100.00	0.425180
$525$	0	0
$526$	4146.00	0.343677
$527$	−1890.00	−0.156223
$528$	0	0
$529$	−2167.00	−0.178105
$530$	11836.0	0.970043
$531$	0	0
$532$	1140.00	0.0929046
$533$	656.000	0.0533105
$534$	0	0
$535$	14410.0	1.16448
$536$	−1056.00	−0.0850975
$537$	0	0
$538$	2964.00	0.237523
$539$	−3422.00	−0.273462
$540$	0	0
$541$	−2473.00	−0.196530	−0.0982649	−	0.995160i	$-0.531329\pi$
$541$	−2473.00	−0.196530	−0.0982649	+	0.995160i	$0.531329\pi$
$542$	14536.0	1.15198
$543$	0	0
$544$	864.000	0.0680950
$545$	−14256.0	−1.12048
$546$	0	0
$547$	−11282.0	−0.881871	−0.440936	−	0.897539i	$-0.645353\pi$
$547$	−11282.0	−0.881871	−0.440936	+	0.897539i	$0.645353\pi$
$548$	−8492.00	−0.661971
$549$	0	0
$550$	232.000	0.0179864
$551$	−2242.00	−0.173344
$552$	0	0
$553$	780.000	0.0599801
$554$	9166.00	0.702935
$555$	0	0
$556$	9108.00	0.694722
$557$	−23975.0	−1.82379	−0.911897	−	0.410419i	$-0.865382\pi$
$557$	−23975.0	−1.82379	−0.911897	+	0.410419i	$0.865382\pi$
$558$	0	0
$559$	23534.0	1.78065
$560$	−2640.00	−0.199215
$561$	0	0
$562$	11524.0	0.864965
$563$	−17892.0	−1.33936	−0.669678	−	0.742651i	$-0.733568\pi$
$563$	−17892.0	−1.33936	−0.669678	+	0.742651i	$0.733568\pi$
$564$	0	0
$565$	12430.0	0.925547
$566$	−7818.00	−0.580592
$567$	0	0
$568$	3776.00	0.278939
$569$	−10778.0	−0.794090	−0.397045	−	0.917799i	$-0.629964\pi$
$569$	−10778.0	−0.794090	−0.397045	+	0.917799i	$0.629964\pi$
$570$	0	0
$571$	8984.00	0.658439	0.329220	−	0.944253i	$-0.393214\pi$
$571$	8984.00	0.658439	0.329220	+	0.944253i	$0.393214\pi$
$572$	−9512.00	−0.695309
$573$	0	0
$574$	−240.000	−0.0174519
$575$	400.000	0.0290107
$576$	0	0
$577$	9539.00	0.688239	0.344119	−	0.938926i	$-0.388178\pi$
$577$	9539.00	0.688239	0.344119	+	0.938926i	$0.388178\pi$
$578$	8368.00	0.602185
$579$	0	0
$580$	5192.00	0.371700
$581$	4140.00	0.295622
$582$	0	0
$583$	−15602.0	−1.10835
$584$	9048.00	0.641111
$585$	0	0
$586$	−16296.0	−1.14877
$587$	−21789.0	−1.53208	−0.766038	−	0.642796i	$-0.777774\pi$
$587$	−21789.0	−1.53208	−0.766038	+	0.642796i	$0.777774\pi$
$588$	0	0
$589$	−1330.00	−0.0930419
$590$	−6600.00	−0.460538
$591$	0	0
$592$	3712.00	0.257707
$593$	−5474.00	−0.379073	−0.189536	−	0.981874i	$-0.560699\pi$
$593$	−5474.00	−0.379073	−0.189536	+	0.981874i	$0.560699\pi$
$594$	0	0
$595$	4455.00	0.306953
$596$	−8940.00	−0.614424
$597$	0	0
$598$	−16400.0	−1.12148
$599$	17748.0	1.21062	0.605312	−	0.795988i	$-0.293048\pi$
$599$	17748.0	1.21062	0.605312	+	0.795988i	$0.293048\pi$
$600$	0	0
$601$	−8972.00	−0.608944	−0.304472	−	0.952521i	$-0.598480\pi$
$601$	−8972.00	−0.608944	−0.304472	+	0.952521i	$0.598480\pi$
$602$	−8610.00	−0.582919
$603$	0	0
$604$	−11056.0	−0.744805
$605$	−5390.00	−0.362206
$606$	0	0
$607$	−17710.0	−1.18423	−0.592114	−	0.805854i	$-0.701707\pi$
$607$	−17710.0	−1.18423	−0.592114	+	0.805854i	$0.701707\pi$
$608$	608.000	0.0405554
$609$	0	0
$610$	19822.0	1.31569
$611$	31570.0	2.09032
$612$	0	0
$613$	11557.0	0.761473	0.380736	−	0.924684i	$-0.375671\pi$
$613$	11557.0	0.761473	0.380736	+	0.924684i	$0.375671\pi$
$614$	−1200.00	−0.0788731
$615$	0	0
$616$	3480.00	0.227619
$617$	4473.00	0.291858	0.145929	−	0.989295i	$-0.453383\pi$
$617$	4473.00	0.291858	0.145929	+	0.989295i	$0.453383\pi$
$618$	0	0
$619$	1948.00	0.126489	0.0632445	−	0.997998i	$-0.479855\pi$
$619$	1948.00	0.126489	0.0632445	+	0.997998i	$0.479855\pi$
$620$	3080.00	0.199509
$621$	0	0
$622$	−9926.00	−0.639865
$623$	−19530.0	−1.25594
$624$	0	0
$625$	−15109.0	−0.966976
$626$	−10924.0	−0.697461
$627$	0	0
$628$	6808.00	0.432594
$629$	−6264.00	−0.397078
$630$	0	0
$631$	12653.0	0.798269	0.399135	−	0.916892i	$-0.369311\pi$
$631$	12653.0	0.798269	0.399135	+	0.916892i	$0.369311\pi$
$632$	416.000	0.0261829
$633$	0	0
$634$	−1968.00	−0.123280
$635$	8250.00	0.515577
$636$	0	0
$637$	9676.00	0.601848
$638$	−6844.00	−0.424697
$639$	0	0
$640$	−1408.00	−0.0869626
$641$	11874.0	0.731661	0.365831	−	0.930681i	$-0.380785\pi$
$641$	11874.0	0.731661	0.365831	+	0.930681i	$0.380785\pi$
$642$	0	0
$643$	−26783.0	−1.64264	−0.821321	−	0.570467i	$-0.806762\pi$
$643$	−26783.0	−1.64264	−0.821321	+	0.570467i	$0.806762\pi$
$644$	6000.00	0.367132
$645$	0	0
$646$	−1026.00	−0.0624883
$647$	24593.0	1.49436	0.747180	−	0.664622i	$-0.231407\pi$
$647$	24593.0	1.49436	0.747180	+	0.664622i	$0.231407\pi$
$648$	0	0
$649$	8700.00	0.526202
$650$	−656.000	−0.0395853
$651$	0	0
$652$	11360.0	0.682350
$653$	3261.00	0.195425	0.0977127	−	0.995215i	$-0.468847\pi$
$653$	3261.00	0.195425	0.0977127	+	0.995215i	$0.468847\pi$
$654$	0	0
$655$	14025.0	0.836644
$656$	−128.000	−0.00761823
$657$	0	0
$658$	−11550.0	−0.684295
$659$	20714.0	1.22444	0.612218	−	0.790689i	$-0.290278\pi$
$659$	20714.0	1.22444	0.612218	+	0.790689i	$0.290278\pi$
$660$	0	0
$661$	−30572.0	−1.79896	−0.899480	−	0.436961i	$-0.856055\pi$
$661$	−30572.0	−1.79896	−0.899480	+	0.436961i	$0.856055\pi$
$662$	5264.00	0.309050
$663$	0	0
$664$	2208.00	0.129047
$665$	3135.00	0.182812
$666$	0	0
$667$	−11800.0	−0.685004
$668$	−4776.00	−0.276630
$669$	0	0
$670$	−2904.00	−0.167450
$671$	−26129.0	−1.50328
$672$	0	0
$673$	−9772.00	−0.559707	−0.279854	−	0.960043i	$-0.590286\pi$
$673$	−9772.00	−0.559707	−0.279854	+	0.960043i	$0.590286\pi$
$674$	22468.0	1.28403
$675$	0	0
$676$	18108.0	1.03027
$677$	−12350.0	−0.701106	−0.350553	−	0.936543i	$-0.614006\pi$
$677$	−12350.0	−0.701106	−0.350553	+	0.936543i	$0.614006\pi$
$678$	0	0
$679$	19650.0	1.11060
$680$	2376.00	0.133993
$681$	0	0
$682$	−4060.00	−0.227955
$683$	8686.00	0.486619	0.243309	−	0.969949i	$-0.421767\pi$
$683$	8686.00	0.486619	0.243309	+	0.969949i	$0.421767\pi$
$684$	0	0
$685$	−23353.0	−1.30259
$686$	−13830.0	−0.769726
$687$	0	0
$688$	−4592.00	−0.254460
$689$	44116.0	2.43931
$690$	0	0
$691$	−22449.0	−1.23589	−0.617945	−	0.786221i	$-0.712035\pi$
$691$	−22449.0	−1.23589	−0.617945	+	0.786221i	$0.712035\pi$
$692$	15144.0	0.831920
$693$	0	0
$694$	−6422.00	−0.351262
$695$	25047.0	1.36703
$696$	0	0
$697$	216.000	0.0117383
$698$	−4682.00	−0.253892
$699$	0	0
$700$	240.000	0.0129588
$701$	16430.0	0.885239	0.442619	−	0.896710i	$-0.354049\pi$
$701$	16430.0	0.885239	0.442619	+	0.896710i	$0.354049\pi$
$702$	0	0
$703$	−4408.00	−0.236488
$704$	1856.00	0.0993617
$705$	0	0
$706$	12732.0	0.678718
$707$	−9570.00	−0.509076
$708$	0	0
$709$	−15882.0	−0.841271	−0.420635	−	0.907230i	$-0.638193\pi$
$709$	−15882.0	−0.841271	−0.420635	+	0.907230i	$0.638193\pi$
$710$	10384.0	0.548880
$711$	0	0
$712$	−10416.0	−0.548253
$713$	−7000.00	−0.367675
$714$	0	0
$715$	−26158.0	−1.36819
$716$	−8376.00	−0.437187
$717$	0	0
$718$	−15978.0	−0.830493
$719$	−19079.0	−0.989606	−0.494803	−	0.869005i	$-0.664760\pi$
$719$	−19079.0	−0.989606	−0.494803	+	0.869005i	$0.664760\pi$
$720$	0	0
$721$	−11790.0	−0.608991
$722$	−722.000	−0.0372161
$723$	0	0
$724$	−1400.00	−0.0718655
$725$	−472.000	−0.0241788
$726$	0	0
$727$	17275.0	0.881285	0.440643	−	0.897683i	$-0.354751\pi$
$727$	17275.0	0.881285	0.440643	+	0.897683i	$0.354751\pi$
$728$	−9840.00	−0.500954
$729$	0	0
$730$	24882.0	1.26154
$731$	7749.00	0.392075
$732$	0	0
$733$	−26358.0	−1.32818	−0.664089	−	0.747653i	$-0.731181\pi$
$733$	−26358.0	−1.32818	−0.664089	+	0.747653i	$0.731181\pi$
$734$	21088.0	1.06045
$735$	0	0
$736$	3200.00	0.160263
$737$	3828.00	0.191325
$738$	0	0
$739$	−1675.00	−0.0833774	−0.0416887	−	0.999131i	$-0.513274\pi$
$739$	−1675.00	−0.0833774	−0.0416887	+	0.999131i	$0.513274\pi$
$740$	10208.0	0.507099
$741$	0	0
$742$	−16140.0	−0.798542
$743$	−26428.0	−1.30491	−0.652456	−	0.757827i	$-0.726261\pi$
$743$	−26428.0	−1.30491	−0.652456	+	0.757827i	$0.726261\pi$
$744$	0	0
$745$	−24585.0	−1.20903
$746$	−15232.0	−0.747565
$747$	0	0
$748$	−3132.00	−0.153098
$749$	−19650.0	−0.958605
$750$	0	0
$751$	−21328.0	−1.03631	−0.518156	−	0.855286i	$-0.673381\pi$
$751$	−21328.0	−1.03631	−0.518156	+	0.855286i	$0.673381\pi$
$752$	−6160.00	−0.298713
$753$	0	0
$754$	19352.0	0.934693
$755$	−30404.0	−1.46558
$756$	0	0
$757$	−16229.0	−0.779198	−0.389599	−	0.920985i	$-0.627386\pi$
$757$	−16229.0	−0.779198	−0.389599	+	0.920985i	$0.627386\pi$
$758$	−11916.0	−0.570988
$759$	0	0
$760$	1672.00	0.0798024
$761$	4965.00	0.236506	0.118253	−	0.992983i	$-0.462271\pi$
$761$	4965.00	0.236506	0.118253	+	0.992983i	$0.462271\pi$
$762$	0	0
$763$	19440.0	0.922379
$764$	5956.00	0.282043
$765$	0	0
$766$	−24764.0	−1.16809
$767$	−24600.0	−1.15809
$768$	0	0
$769$	35275.0	1.65416	0.827080	−	0.562084i	$-0.190000\pi$
$769$	35275.0	1.65416	0.827080	+	0.562084i	$0.190000\pi$
$770$	9570.00	0.447895
$771$	0	0
$772$	2400.00	0.111888
$773$	−4776.00	−0.222226	−0.111113	−	0.993808i	$-0.535442\pi$
$773$	−4776.00	−0.222226	−0.111113	+	0.993808i	$0.535442\pi$
$774$	0	0
$775$	−280.000	−0.0129779
$776$	10480.0	0.484807
$777$	0	0
$778$	15978.0	0.736297
$779$	152.000	0.00699097
$780$	0	0
$781$	−13688.0	−0.627138
$782$	−5400.00	−0.246936
$783$	0	0
$784$	−1888.00	−0.0860058
$785$	18722.0	0.851232
$786$	0	0
$787$	21136.0	0.957328	0.478664	−	0.877998i	$-0.341121\pi$
$787$	21136.0	0.957328	0.478664	+	0.877998i	$0.341121\pi$
$788$	1160.00	0.0524407
$789$	0	0
$790$	1144.00	0.0515211
$791$	−16950.0	−0.761912
$792$	0	0
$793$	73882.0	3.30848
$794$	−14122.0	−0.631198
$795$	0	0
$796$	16676.0	0.742544
$797$	−38256.0	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−38256.0	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	0	0
$799$	10395.0	0.460261
$800$	128.000	0.00565685
$801$	0	0
$802$	−29024.0	−1.27790
$803$	−32799.0	−1.44141
$804$	0	0
$805$	16500.0	0.722421
$806$	11480.0	0.501694
$807$	0	0
$808$	−5104.00	−0.222225
$809$	−6189.00	−0.268966	−0.134483	−	0.990916i	$-0.542937\pi$
$809$	−6189.00	−0.268966	−0.134483	+	0.990916i	$0.542937\pi$
$810$	0	0
$811$	−3030.00	−0.131193	−0.0655966	−	0.997846i	$-0.520895\pi$
$811$	−3030.00	−0.131193	−0.0655966	+	0.997846i	$0.520895\pi$
$812$	−7080.00	−0.305984
$813$	0	0
$814$	−13456.0	−0.579401
$815$	31240.0	1.34269
$816$	0	0
$817$	5453.00	0.233508
$818$	−25268.0	−1.08004
$819$	0	0
$820$	−352.000	−0.0149907
$821$	−38571.0	−1.63963	−0.819816	−	0.572628i	$-0.805924\pi$
$821$	−38571.0	−1.63963	−0.819816	+	0.572628i	$0.805924\pi$
$822$	0	0
$823$	−14287.0	−0.605120	−0.302560	−	0.953130i	$-0.597841\pi$
$823$	−14287.0	−0.605120	−0.302560	+	0.953130i	$0.597841\pi$
$824$	−6288.00	−0.265841
$825$	0	0
$826$	9000.00	0.379116
$827$	−23464.0	−0.986606	−0.493303	−	0.869858i	$-0.664211\pi$
$827$	−23464.0	−0.986606	−0.493303	+	0.869858i	$0.664211\pi$
$828$	0	0
$829$	−38700.0	−1.62136	−0.810679	−	0.585490i	$-0.800902\pi$
$829$	−38700.0	−1.62136	−0.810679	+	0.585490i	$0.800902\pi$
$830$	6072.00	0.253930
$831$	0	0
$832$	−5248.00	−0.218680
$833$	3186.00	0.132519
$834$	0	0
$835$	−13134.0	−0.544336
$836$	−2204.00	−0.0911805
$837$	0	0
$838$	−16536.0	−0.681655
$839$	21758.0	0.895315	0.447658	−	0.894205i	$-0.352258\pi$
$839$	21758.0	0.895315	0.447658	+	0.894205i	$0.352258\pi$
$840$	0	0
$841$	−10465.0	−0.429087
$842$	3068.00	0.125570
$843$	0	0
$844$	−5472.00	−0.223168
$845$	49797.0	2.02730
$846$	0	0
$847$	7350.00	0.298169
$848$	−8608.00	−0.348585
$849$	0	0
$850$	−216.000	−0.00871616
$851$	−23200.0	−0.934531
$852$	0	0
$853$	22770.0	0.913986	0.456993	−	0.889470i	$-0.348926\pi$
$853$	22770.0	0.913986	0.456993	+	0.889470i	$0.348926\pi$
$854$	−27030.0	−1.08308
$855$	0	0
$856$	−10480.0	−0.418457
$857$	38484.0	1.53394	0.766971	−	0.641682i	$-0.221763\pi$
$857$	38484.0	1.53394	0.766971	+	0.641682i	$0.221763\pi$
$858$	0	0
$859$	23653.0	0.939499	0.469750	−	0.882800i	$-0.344344\pi$
$859$	23653.0	0.939499	0.469750	+	0.882800i	$0.344344\pi$
$860$	−12628.0	−0.500711
$861$	0	0
$862$	28716.0	1.13465
$863$	−29988.0	−1.18285	−0.591427	−	0.806358i	$-0.701435\pi$
$863$	−29988.0	−1.18285	−0.591427	+	0.806358i	$0.701435\pi$
$864$	0	0
$865$	41646.0	1.63700
$866$	9068.00	0.355824
$867$	0	0
$868$	−4200.00	−0.164237
$869$	−1508.00	−0.0588670
$870$	0	0
$871$	−10824.0	−0.421076
$872$	10368.0	0.402643
$873$	0	0
$874$	−3800.00	−0.147067
$875$	21285.0	0.822359
$876$	0	0
$877$	−29726.0	−1.14456	−0.572278	−	0.820060i	$-0.693940\pi$
$877$	−29726.0	−1.14456	−0.572278	+	0.820060i	$0.693940\pi$
$878$	25532.0	0.981393
$879$	0	0
$880$	5104.00	0.195518
$881$	28713.0	1.09803	0.549016	−	0.835812i	$-0.315003\pi$
$881$	28713.0	1.09803	0.549016	+	0.835812i	$0.315003\pi$
$882$	0	0
$883$	−2771.00	−0.105608	−0.0528038	−	0.998605i	$-0.516816\pi$
$883$	−2771.00	−0.105608	−0.0528038	+	0.998605i	$0.516816\pi$
$884$	8856.00	0.336945
$885$	0	0
$886$	−33422.0	−1.26731
$887$	−23000.0	−0.870648	−0.435324	−	0.900274i	$-0.643366\pi$
$887$	−23000.0	−0.870648	−0.435324	+	0.900274i	$0.643366\pi$
$888$	0	0
$889$	−11250.0	−0.424424
$890$	−28644.0	−1.07882
$891$	0	0
$892$	10160.0	0.381370
$893$	7315.00	0.274118
$894$	0	0
$895$	−23034.0	−0.860270
$896$	1920.00	0.0715878
$897$	0	0
$898$	7864.00	0.292233
$899$	8260.00	0.306437
$900$	0	0
$901$	14526.0	0.537105
$902$	464.000	0.0171281
$903$	0	0
$904$	−9040.00	−0.332595
$905$	−3850.00	−0.141413
$906$	0	0
$907$	42074.0	1.54029	0.770146	−	0.637868i	$-0.220183\pi$
$907$	42074.0	1.54029	0.770146	+	0.637868i	$0.220183\pi$
$908$	23896.0	0.873366
$909$	0	0
$910$	−27060.0	−0.985748
$911$	−6462.00	−0.235012	−0.117506	−	0.993072i	$-0.537490\pi$
$911$	−6462.00	−0.235012	−0.117506	+	0.993072i	$0.537490\pi$
$912$	0	0
$913$	−8004.00	−0.290136
$914$	−8902.00	−0.322158
$915$	0	0
$916$	−1420.00	−0.0512207
$917$	−19125.0	−0.688728
$918$	0	0
$919$	24776.0	0.889320	0.444660	−	0.895700i	$-0.353325\pi$
$919$	24776.0	0.889320	0.444660	+	0.895700i	$0.353325\pi$
$920$	8800.00	0.315356
$921$	0	0
$922$	19238.0	0.687169
$923$	38704.0	1.38024
$924$	0	0
$925$	−928.000	−0.0329864
$926$	−33206.0	−1.17842
$927$	0	0
$928$	−3776.00	−0.133570
$929$	−15022.0	−0.530523	−0.265261	−	0.964177i	$-0.585458\pi$
$929$	−15022.0	−0.530523	−0.265261	+	0.964177i	$0.585458\pi$
$930$	0	0
$931$	2242.00	0.0789244
$932$	−948.000	−0.0333184
$933$	0	0
$934$	25714.0	0.900843
$935$	−8613.00	−0.301257
$936$	0	0
$937$	26285.0	0.916429	0.458214	−	0.888842i	$-0.348489\pi$
$937$	26285.0	0.916429	0.458214	+	0.888842i	$0.348489\pi$
$938$	3960.00	0.137845
$939$	0	0
$940$	−16940.0	−0.587789
$941$	18622.0	0.645122	0.322561	−	0.946549i	$-0.395456\pi$
$941$	18622.0	0.645122	0.322561	+	0.946549i	$0.395456\pi$
$942$	0	0
$943$	800.000	0.0276263
$944$	4800.00	0.165494
$945$	0	0
$946$	16646.0	0.572102
$947$	3660.00	0.125590	0.0627952	−	0.998026i	$-0.479999\pi$
$947$	3660.00	0.125590	0.0627952	+	0.998026i	$0.479999\pi$
$948$	0	0
$949$	92742.0	3.17232
$950$	−152.000	−0.00519109
$951$	0	0
$952$	−3240.00	−0.110304
$953$	30044.0	1.02122	0.510609	−	0.859813i	$-0.329420\pi$
$953$	30044.0	1.02122	0.510609	+	0.859813i	$0.329420\pi$
$954$	0	0
$955$	16379.0	0.554986
$956$	−6540.00	−0.221254
$957$	0	0
$958$	22144.0	0.746806
$959$	31845.0	1.07229
$960$	0	0
$961$	−24891.0	−0.835521
$962$	38048.0	1.27517
$963$	0	0
$964$	−656.000	−0.0219174
$965$	6600.00	0.220167
$966$	0	0
$967$	−6288.00	−0.209109	−0.104555	−	0.994519i	$-0.533342\pi$
$967$	−6288.00	−0.209109	−0.104555	+	0.994519i	$0.533342\pi$
$968$	3920.00	0.130159
$969$	0	0
$970$	28820.0	0.953974
$971$	−39028.0	−1.28987	−0.644937	−	0.764236i	$-0.723117\pi$
$971$	−39028.0	−1.28987	−0.644937	+	0.764236i	$0.723117\pi$
$972$	0	0
$973$	−34155.0	−1.12534
$974$	−22568.0	−0.742429
$975$	0	0
$976$	−14416.0	−0.472792
$977$	−1678.00	−0.0549478	−0.0274739	−	0.999623i	$-0.508746\pi$
$977$	−1678.00	−0.0549478	−0.0274739	+	0.999623i	$0.508746\pi$
$978$	0	0
$979$	37758.0	1.23264
$980$	−5192.00	−0.169237
$981$	0	0
$982$	−23968.0	−0.778869
$983$	39972.0	1.29696	0.648479	−	0.761233i	$-0.275406\pi$
$983$	39972.0	1.29696	0.648479	+	0.761233i	$0.275406\pi$
$984$	0	0
$985$	3190.00	0.103190
$986$	6372.00	0.205807
$987$	0	0
$988$	6232.00	0.200674
$989$	28700.0	0.922757
$990$	0	0
$991$	−23672.0	−0.758795	−0.379398	−	0.925234i	$-0.623869\pi$
$991$	−23672.0	−0.758795	−0.379398	+	0.925234i	$0.623869\pi$
$992$	−2240.00	−0.0716936
$993$	0	0
$994$	−14160.0	−0.451839
$995$	45859.0	1.46113
$996$	0	0
$997$	−36899.0	−1.17212	−0.586060	−	0.810268i	$-0.699322\pi$
$997$	−36899.0	−1.17212	−0.586060	+	0.810268i	$0.699322\pi$
$998$	37402.0	1.18631
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.a.d.1.1	✓	1	3.2	odd	2
342.4.a.a.1.1		1	1.1	even	1	trivial
912.4.a.f.1.1		1	12.11	even	2
2166.4.a.a.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

Related objects

Downloads

Learn more