LMFDB - Embedded newform 6042.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$2$
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	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -1.00000 q^{20} +3.00000 q^{21} +4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} -7.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +4.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} -8.00000 q^{37} -1.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -3.00000 q^{42} -1.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} -1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} -1.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +3.00000 q^{56} -1.00000 q^{57} +1.00000 q^{58} -9.00000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +7.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -5.00000 q^{67} -4.00000 q^{68} -1.00000 q^{69} -3.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} -8.00000 q^{73} +8.00000 q^{74} +4.00000 q^{75} +1.00000 q^{76} +12.0000 q^{77} -2.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +12.0000 q^{83} +3.00000 q^{84} +4.00000 q^{85} +1.00000 q^{86} +1.00000 q^{87} +4.00000 q^{88} +5.00000 q^{89} +1.00000 q^{90} +6.00000 q^{91} +1.00000 q^{92} +7.00000 q^{93} -1.00000 q^{95} +1.00000 q^{96} +14.0000 q^{97} -2.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	3.00000	0.801784
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.00000	−0.223607
$21$	3.00000	0.654654
$22$	4.00000	0.852803
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	−1.00000	−0.182574
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	4.00000	0.685994
$35$	3.00000	0.507093
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−1.00000	−0.162221
$39$	2.00000	0.320256
$40$	1.00000	0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−3.00000	−0.462910
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−4.00000	−0.603023
$45$	−1.00000	−0.149071
$46$	−1.00000	−0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	4.00000	0.565685
$51$	4.00000	0.560112
$52$	−2.00000	−0.277350
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	1.00000	0.136083
$55$	4.00000	0.539360
$56$	3.00000	0.400892
$57$	−1.00000	−0.132453
$58$	1.00000	0.131306
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	1.00000	0.129099
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	7.00000	0.889001
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−4.00000	−0.492366
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	−4.00000	−0.485071
$69$	−1.00000	−0.120386
$70$	−3.00000	−0.358569
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.00000	−0.117851
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	8.00000	0.929981
$75$	4.00000	0.461880
$76$	1.00000	0.114708
$77$	12.0000	1.36753
$78$	−2.00000	−0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	3.00000	0.327327
$85$	4.00000	0.433861
$86$	1.00000	0.107833
$87$	1.00000	0.107211
$88$	4.00000	0.426401
$89$	5.00000	0.529999	0.264999	−	0.964249i	$-0.414628\pi$
$89$	5.00000	0.529999	0.264999	+	0.964249i	$0.414628\pi$
$90$	1.00000	0.105409
$91$	6.00000	0.628971
$92$	1.00000	0.104257
$93$	7.00000	0.725866
$94$	0	0
$95$	−1.00000	−0.102598
$96$	1.00000	0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−2.00000	−0.202031
$99$	−4.00000	−0.402015
$100$	−4.00000	−0.400000
$101$	−17.0000	−1.69156	−0.845782	−	0.533529i	$-0.820865\pi$
$101$	−17.0000	−1.69156	−0.845782	+	0.533529i	$0.820865\pi$
$102$	−4.00000	−0.396059
$103$	−3.00000	−0.295599	−0.147799	−	0.989017i	$-0.547219\pi$
$103$	−3.00000	−0.295599	−0.147799	+	0.989017i	$0.547219\pi$
$104$	2.00000	0.196116
$105$	−3.00000	−0.292770
$106$	1.00000	0.0971286
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	−1.00000	−0.0962250
$109$	−17.0000	−1.62830	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000	−1.62830	−0.814152	+	0.580651i	$0.802798\pi$
$110$	−4.00000	−0.381385
$111$	8.00000	0.759326
$112$	−3.00000	−0.283473
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	1.00000	0.0936586
$115$	−1.00000	−0.0932505
$116$	−1.00000	−0.0928477
$117$	−2.00000	−0.184900
$118$	9.00000	0.828517
$119$	12.0000	1.10004
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	14.0000	1.26750
$123$	−2.00000	−0.180334
$124$	−7.00000	−0.628619
$125$	9.00000	0.804984
$126$	3.00000	0.267261
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	−1.00000	−0.0883883
$129$	1.00000	0.0880451
$130$	−2.00000	−0.175412
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	4.00000	0.348155
$133$	−3.00000	−0.260133
$134$	5.00000	0.431934
$135$	1.00000	0.0860663
$136$	4.00000	0.342997
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	1.00000	0.0851257
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	6.00000	0.503509
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	1.00000	0.0830455
$146$	8.00000	0.662085
$147$	−2.00000	−0.164957
$148$	−8.00000	−0.657596
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	−4.00000	−0.326599
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	−1.00000	−0.0811107
$153$	−4.00000	−0.323381
$154$	−12.0000	−0.966988
$155$	7.00000	0.562254
$156$	2.00000	0.160128
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	8.00000	0.636446
$159$	1.00000	0.0793052
$160$	1.00000	0.0790569
$161$	−3.00000	−0.236433
$162$	−1.00000	−0.0785674
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	2.00000	0.156174
$165$	−4.00000	−0.311400
$166$	−12.0000	−0.931381
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	−3.00000	−0.231455
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	1.00000	0.0764719
$172$	−1.00000	−0.0762493
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	−1.00000	−0.0758098
$175$	12.0000	0.907115
$176$	−4.00000	−0.301511
$177$	9.00000	0.676481
$178$	−5.00000	−0.374766
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−1.00000	−0.0745356
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	−6.00000	−0.444750
$183$	14.0000	1.03491
$184$	−1.00000	−0.0737210
$185$	8.00000	0.588172
$186$	−7.00000	−0.513265
$187$	16.0000	1.17004
$188$	0	0
$189$	3.00000	0.218218
$190$	1.00000	0.0725476
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	−1.00000	−0.0721688
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−14.0000	−1.00514
$195$	−2.00000	−0.143223
$196$	2.00000	0.142857
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	4.00000	0.284268
$199$	−9.00000	−0.637993	−0.318997	−	0.947756i	$-0.603346\pi$
$199$	−9.00000	−0.637993	−0.318997	+	0.947756i	$0.603346\pi$
$200$	4.00000	0.282843
$201$	5.00000	0.352673
$202$	17.0000	1.19612
$203$	3.00000	0.210559
$204$	4.00000	0.280056
$205$	−2.00000	−0.139686
$206$	3.00000	0.209020
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	−4.00000	−0.276686
$210$	3.00000	0.207020
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−1.00000	−0.0686803
$213$	6.00000	0.411113
$214$	−9.00000	−0.615227
$215$	1.00000	0.0681994
$216$	1.00000	0.0680414
$217$	21.0000	1.42557
$218$	17.0000	1.15139
$219$	8.00000	0.540590
$220$	4.00000	0.269680
$221$	8.00000	0.538138
$222$	−8.00000	−0.536925
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	3.00000	0.200446
$225$	−4.00000	−0.266667
$226$	9.00000	0.598671
$227$	13.0000	0.862840	0.431420	−	0.902151i	$-0.358013\pi$
$227$	13.0000	0.862840	0.431420	+	0.902151i	$0.358013\pi$
$228$	−1.00000	−0.0662266
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	1.00000	0.0659380
$231$	−12.0000	−0.789542
$232$	1.00000	0.0656532
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−9.00000	−0.585850
$237$	8.00000	0.519656
$238$	−12.0000	−0.777844
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	1.00000	0.0645497
$241$	−24.0000	−1.54598	−0.772988	−	0.634421i	$-0.781239\pi$
$241$	−24.0000	−1.54598	−0.772988	+	0.634421i	$0.781239\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	−2.00000	−0.127775
$246$	2.00000	0.127515
$247$	−2.00000	−0.127257
$248$	7.00000	0.444500
$249$	−12.0000	−0.760469
$250$	−9.00000	−0.569210
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	−3.00000	−0.188982
$253$	−4.00000	−0.251478
$254$	−7.00000	−0.439219
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−1.00000	−0.0622573
$259$	24.0000	1.49129
$260$	2.00000	0.124035
$261$	−1.00000	−0.0618984
$262$	10.0000	0.617802
$263$	13.0000	0.801614	0.400807	−	0.916162i	$-0.368730\pi$
$263$	13.0000	0.801614	0.400807	+	0.916162i	$0.368730\pi$
$264$	−4.00000	−0.246183
$265$	1.00000	0.0614295
$266$	3.00000	0.183942
$267$	−5.00000	−0.305995
$268$	−5.00000	−0.305424
$269$	−19.0000	−1.15845	−0.579225	−	0.815168i	$-0.696645\pi$
$269$	−19.0000	−1.15845	−0.579225	+	0.815168i	$0.696645\pi$
$270$	−1.00000	−0.0608581
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	−4.00000	−0.242536
$273$	−6.00000	−0.363137
$274$	13.0000	0.785359
$275$	16.0000	0.964836
$276$	−1.00000	−0.0601929
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−6.00000	−0.359856
$279$	−7.00000	−0.419079
$280$	−3.00000	−0.179284
$281$	−13.0000	−0.775515	−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000	−0.775515	−0.387757	+	0.921761i	$0.626750\pi$
$282$	0	0
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	−6.00000	−0.356034
$285$	1.00000	0.0592349
$286$	−8.00000	−0.473050
$287$	−6.00000	−0.354169
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	−1.00000	−0.0587220
$291$	−14.0000	−0.820695
$292$	−8.00000	−0.468165
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	2.00000	0.116642
$295$	9.00000	0.524000
$296$	8.00000	0.464991
$297$	4.00000	0.232104
$298$	−12.0000	−0.695141
$299$	−2.00000	−0.115663
$300$	4.00000	0.230940
$301$	3.00000	0.172917
$302$	17.0000	0.978240
$303$	17.0000	0.976624
$304$	1.00000	0.0573539
$305$	14.0000	0.801638
$306$	4.00000	0.228665
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	12.0000	0.683763
$309$	3.00000	0.170664
$310$	−7.00000	−0.397573
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	−2.00000	−0.113228
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−2.00000	−0.112867
$315$	3.00000	0.169031
$316$	−8.00000	−0.450035
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−1.00000	−0.0560772
$319$	4.00000	0.223957
$320$	−1.00000	−0.0559017
$321$	−9.00000	−0.502331
$322$	3.00000	0.167183
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	8.00000	0.443760
$326$	−17.0000	−0.941543
$327$	17.0000	0.940102
$328$	−2.00000	−0.110432
$329$	0	0
$330$	4.00000	0.220193
$331$	30.0000	1.64895	0.824475	−	0.565899i	$-0.191471\pi$
$331$	30.0000	1.64895	0.824475	+	0.565899i	$0.191471\pi$
$332$	12.0000	0.658586
$333$	−8.00000	−0.438397
$334$	2.00000	0.109435
$335$	5.00000	0.273179
$336$	3.00000	0.163663
$337$	11.0000	0.599208	0.299604	−	0.954064i	$-0.403145\pi$
$337$	11.0000	0.599208	0.299604	+	0.954064i	$0.403145\pi$
$338$	9.00000	0.489535
$339$	9.00000	0.488813
$340$	4.00000	0.216930
$341$	28.0000	1.51629
$342$	−1.00000	−0.0540738
$343$	15.0000	0.809924
$344$	1.00000	0.0539164
$345$	1.00000	0.0538382
$346$	−8.00000	−0.430083
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	1.00000	0.0536056
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−12.0000	−0.641427
$351$	2.00000	0.106752
$352$	4.00000	0.213201
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−9.00000	−0.478345
$355$	6.00000	0.318447
$356$	5.00000	0.264999
$357$	−12.0000	−0.635107
$358$	0	0
$359$	7.00000	0.369446	0.184723	−	0.982791i	$-0.440861\pi$
$359$	7.00000	0.369446	0.184723	+	0.982791i	$0.440861\pi$
$360$	1.00000	0.0527046
$361$	1.00000	0.0526316
$362$	−7.00000	−0.367912
$363$	−5.00000	−0.262432
$364$	6.00000	0.314485
$365$	8.00000	0.418739
$366$	−14.0000	−0.731792
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	1.00000	0.0521286
$369$	2.00000	0.104116
$370$	−8.00000	−0.415900
$371$	3.00000	0.155752
$372$	7.00000	0.362933
$373$	5.00000	0.258890	0.129445	−	0.991587i	$-0.458680\pi$
$373$	5.00000	0.258890	0.129445	+	0.991587i	$0.458680\pi$
$374$	−16.0000	−0.827340
$375$	−9.00000	−0.464758
$376$	0	0
$377$	2.00000	0.103005
$378$	−3.00000	−0.154303
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	−1.00000	−0.0512989
$381$	−7.00000	−0.358621
$382$	20.0000	1.02329
$383$	10.0000	0.510976	0.255488	−	0.966812i	$-0.417764\pi$
$383$	10.0000	0.510976	0.255488	+	0.966812i	$0.417764\pi$
$384$	1.00000	0.0510310
$385$	−12.0000	−0.611577
$386$	−6.00000	−0.305392
$387$	−1.00000	−0.0508329
$388$	14.0000	0.710742
$389$	−19.0000	−0.963338	−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000	−0.963338	−0.481669	+	0.876353i	$0.659969\pi$
$390$	2.00000	0.101274
$391$	−4.00000	−0.202289
$392$	−2.00000	−0.101015
$393$	10.0000	0.504433
$394$	−22.0000	−1.10834
$395$	8.00000	0.402524
$396$	−4.00000	−0.201008
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	9.00000	0.451129
$399$	3.00000	0.150188
$400$	−4.00000	−0.200000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−5.00000	−0.249377
$403$	14.0000	0.697390
$404$	−17.0000	−0.845782
$405$	−1.00000	−0.0496904
$406$	−3.00000	−0.148888
$407$	32.0000	1.58618
$408$	−4.00000	−0.198030
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	2.00000	0.0987730
$411$	13.0000	0.641243
$412$	−3.00000	−0.147799
$413$	27.0000	1.32858
$414$	−1.00000	−0.0491473
$415$	−12.0000	−0.589057
$416$	2.00000	0.0980581
$417$	−6.00000	−0.293821
$418$	4.00000	0.195646
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	−3.00000	−0.146385
$421$	−39.0000	−1.90074	−0.950372	−	0.311116i	$-0.899297\pi$
$421$	−39.0000	−1.90074	−0.950372	+	0.311116i	$0.899297\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	1.00000	0.0485643
$425$	16.0000	0.776114
$426$	−6.00000	−0.290701
$427$	42.0000	2.03252
$428$	9.00000	0.435031
$429$	−8.00000	−0.386244
$430$	−1.00000	−0.0482243
$431$	3.00000	0.144505	0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000	0.144505	0.0722525	+	0.997386i	$0.476981\pi$
$432$	−1.00000	−0.0481125
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	−21.0000	−1.00803
$435$	−1.00000	−0.0479463
$436$	−17.0000	−0.814152
$437$	1.00000	0.0478365
$438$	−8.00000	−0.382255
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	−4.00000	−0.190693
$441$	2.00000	0.0952381
$442$	−8.00000	−0.380521
$443$	−37.0000	−1.75792	−0.878962	−	0.476893i	$-0.841763\pi$
$443$	−37.0000	−1.75792	−0.878962	+	0.476893i	$0.841763\pi$
$444$	8.00000	0.379663
$445$	−5.00000	−0.237023
$446$	2.00000	0.0947027
$447$	−12.0000	−0.567581
$448$	−3.00000	−0.141737
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	4.00000	0.188562
$451$	−8.00000	−0.376705
$452$	−9.00000	−0.423324
$453$	17.0000	0.798730
$454$	−13.0000	−0.610120
$455$	−6.00000	−0.281284
$456$	1.00000	0.0468293
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	15.0000	0.700904
$459$	4.00000	0.186704
$460$	−1.00000	−0.0466252
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	12.0000	0.558291
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	−1.00000	−0.0464238
$465$	−7.00000	−0.324617
$466$	−2.00000	−0.0926482
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	−2.00000	−0.0924500
$469$	15.0000	0.692636
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	9.00000	0.414259
$473$	4.00000	0.183920
$474$	−8.00000	−0.367452
$475$	−4.00000	−0.183533
$476$	12.0000	0.550019
$477$	−1.00000	−0.0457869
$478$	−12.0000	−0.548867
$479$	39.0000	1.78196	0.890978	−	0.454047i	$-0.150020\pi$
$479$	39.0000	1.78196	0.890978	+	0.454047i	$0.150020\pi$
$480$	−1.00000	−0.0456435
$481$	16.0000	0.729537
$482$	24.0000	1.09317
$483$	3.00000	0.136505
$484$	5.00000	0.227273
$485$	−14.0000	−0.635707
$486$	1.00000	0.0453609
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	14.0000	0.633750
$489$	−17.0000	−0.768767
$490$	2.00000	0.0903508
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	−2.00000	−0.0901670
$493$	4.00000	0.180151
$494$	2.00000	0.0899843
$495$	4.00000	0.179787
$496$	−7.00000	−0.314309
$497$	18.0000	0.807410
$498$	12.0000	0.537733
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	9.00000	0.402492
$501$	2.00000	0.0893534
$502$	−20.0000	−0.892644
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	3.00000	0.133631
$505$	17.0000	0.756490
$506$	4.00000	0.177822
$507$	9.00000	0.399704
$508$	7.00000	0.310575
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	4.00000	0.177123
$511$	24.0000	1.06170
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	6.00000	0.264649
$515$	3.00000	0.132196
$516$	1.00000	0.0440225
$517$	0	0
$518$	−24.0000	−1.05450
$519$	−8.00000	−0.351161
$520$	−2.00000	−0.0877058
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	1.00000	0.0437688
$523$	−32.0000	−1.39926	−0.699631	−	0.714504i	$-0.746652\pi$
$523$	−32.0000	−1.39926	−0.699631	+	0.714504i	$0.746652\pi$
$524$	−10.0000	−0.436852
$525$	−12.0000	−0.523723
$526$	−13.0000	−0.566827
$527$	28.0000	1.21970
$528$	4.00000	0.174078
$529$	−22.0000	−0.956522
$530$	−1.00000	−0.0434372
$531$	−9.00000	−0.390567
$532$	−3.00000	−0.130066
$533$	−4.00000	−0.173259
$534$	5.00000	0.216371
$535$	−9.00000	−0.389104
$536$	5.00000	0.215967
$537$	0	0
$538$	19.0000	0.819148
$539$	−8.00000	−0.344584
$540$	1.00000	0.0430331
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	15.0000	0.644305
$543$	−7.00000	−0.300399
$544$	4.00000	0.171499
$545$	17.0000	0.728200
$546$	6.00000	0.256776
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−13.0000	−0.555332
$549$	−14.0000	−0.597505
$550$	−16.0000	−0.682242
$551$	−1.00000	−0.0426014
$552$	1.00000	0.0425628
$553$	24.0000	1.02058
$554$	−2.00000	−0.0849719
$555$	−8.00000	−0.339581
$556$	6.00000	0.254457
$557$	35.0000	1.48300	0.741499	−	0.670954i	$-0.234115\pi$
$557$	35.0000	1.48300	0.741499	+	0.670954i	$0.234115\pi$
$558$	7.00000	0.296334
$559$	2.00000	0.0845910
$560$	3.00000	0.126773
$561$	−16.0000	−0.675521
$562$	13.0000	0.548372
$563$	30.0000	1.26435	0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000	1.26435	0.632175	+	0.774826i	$0.282163\pi$
$564$	0	0
$565$	9.00000	0.378633
$566$	32.0000	1.34506
$567$	−3.00000	−0.125988
$568$	6.00000	0.251754
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	−1.00000	−0.0418854
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	8.00000	0.334497
$573$	20.0000	0.835512
$574$	6.00000	0.250435
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	1.00000	0.0415945
$579$	−6.00000	−0.249351
$580$	1.00000	0.0415227
$581$	−36.0000	−1.49353
$582$	14.0000	0.580319
$583$	4.00000	0.165663
$584$	8.00000	0.331042
$585$	2.00000	0.0826898
$586$	26.0000	1.07405
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	−2.00000	−0.0824786
$589$	−7.00000	−0.288430
$590$	−9.00000	−0.370524
$591$	−22.0000	−0.904959
$592$	−8.00000	−0.328798
$593$	−48.0000	−1.97112	−0.985562	−	0.169316i	$-0.945844\pi$
$593$	−48.0000	−1.97112	−0.985562	+	0.169316i	$0.945844\pi$
$594$	−4.00000	−0.164122
$595$	−12.0000	−0.491952
$596$	12.0000	0.491539
$597$	9.00000	0.368345
$598$	2.00000	0.0817861
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	−4.00000	−0.163299
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	−3.00000	−0.122271
$603$	−5.00000	−0.203616
$604$	−17.0000	−0.691720
$605$	−5.00000	−0.203279
$606$	−17.0000	−0.690578
$607$	26.0000	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$607$	26.0000	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$608$	−1.00000	−0.0405554
$609$	−3.00000	−0.121566
$610$	−14.0000	−0.566843
$611$	0	0
$612$	−4.00000	−0.161690
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	22.0000	0.887848
$615$	2.00000	0.0806478
$616$	−12.0000	−0.483494
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−3.00000	−0.120678
$619$	5.00000	0.200967	0.100483	−	0.994939i	$-0.467961\pi$
$619$	5.00000	0.200967	0.100483	+	0.994939i	$0.467961\pi$
$620$	7.00000	0.281127
$621$	−1.00000	−0.0401286
$622$	−6.00000	−0.240578
$623$	−15.0000	−0.600962
$624$	2.00000	0.0800641
$625$	11.0000	0.440000
$626$	−22.0000	−0.879297
$627$	4.00000	0.159745
$628$	2.00000	0.0798087
$629$	32.0000	1.27592
$630$	−3.00000	−0.119523
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	8.00000	0.318223
$633$	−4.00000	−0.158986
$634$	2.00000	0.0794301
$635$	−7.00000	−0.277787
$636$	1.00000	0.0396526
$637$	−4.00000	−0.158486
$638$	−4.00000	−0.158362
$639$	−6.00000	−0.237356
$640$	1.00000	0.0395285
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	9.00000	0.355202
$643$	37.0000	1.45914	0.729569	−	0.683907i	$-0.239721\pi$
$643$	37.0000	1.45914	0.729569	+	0.683907i	$0.239721\pi$
$644$	−3.00000	−0.118217
$645$	−1.00000	−0.0393750
$646$	4.00000	0.157378
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−1.00000	−0.0392837
$649$	36.0000	1.41312
$650$	−8.00000	−0.313786
$651$	−21.0000	−0.823055
$652$	17.0000	0.665771
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−17.0000	−0.664753
$655$	10.0000	0.390732
$656$	2.00000	0.0780869
$657$	−8.00000	−0.312110
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	−4.00000	−0.155700
$661$	−12.0000	−0.466746	−0.233373	−	0.972387i	$-0.574976\pi$
$661$	−12.0000	−0.466746	−0.233373	+	0.972387i	$0.574976\pi$
$662$	−30.0000	−1.16598
$663$	−8.00000	−0.310694
$664$	−12.0000	−0.465690
$665$	3.00000	0.116335
$666$	8.00000	0.309994
$667$	−1.00000	−0.0387202
$668$	−2.00000	−0.0773823
$669$	2.00000	0.0773245
$670$	−5.00000	−0.193167
$671$	56.0000	2.16186
$672$	−3.00000	−0.115728
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	−11.0000	−0.423704
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	−9.00000	−0.345643
$679$	−42.0000	−1.61181
$680$	−4.00000	−0.153393
$681$	−13.0000	−0.498161
$682$	−28.0000	−1.07218
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	1.00000	0.0382360
$685$	13.0000	0.496704
$686$	−15.0000	−0.572703
$687$	15.0000	0.572286
$688$	−1.00000	−0.0381246
$689$	2.00000	0.0761939
$690$	−1.00000	−0.0380693
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	8.00000	0.304114
$693$	12.0000	0.455842
$694$	−18.0000	−0.683271
$695$	−6.00000	−0.227593
$696$	−1.00000	−0.0379049
$697$	−8.00000	−0.303022
$698$	−2.00000	−0.0757011
$699$	−2.00000	−0.0756469
$700$	12.0000	0.453557
$701$	−27.0000	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$701$	−27.0000	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$702$	−2.00000	−0.0754851
$703$	−8.00000	−0.301726
$704$	−4.00000	−0.150756
$705$	0	0
$706$	14.0000	0.526897
$707$	51.0000	1.91805
$708$	9.00000	0.338241
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	−6.00000	−0.225176
$711$	−8.00000	−0.300023
$712$	−5.00000	−0.187383
$713$	−7.00000	−0.262152
$714$	12.0000	0.449089
$715$	−8.00000	−0.299183
$716$	0	0
$717$	−12.0000	−0.448148
$718$	−7.00000	−0.261238
$719$	41.0000	1.52904	0.764521	−	0.644599i	$-0.222976\pi$
$719$	41.0000	1.52904	0.764521	+	0.644599i	$0.222976\pi$
$720$	−1.00000	−0.0372678
$721$	9.00000	0.335178
$722$	−1.00000	−0.0372161
$723$	24.0000	0.892570
$724$	7.00000	0.260153
$725$	4.00000	0.148556
$726$	5.00000	0.185567
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	4.00000	0.147945
$732$	14.0000	0.517455
$733$	41.0000	1.51437	0.757185	−	0.653201i	$-0.226574\pi$
$733$	41.0000	1.51437	0.757185	+	0.653201i	$0.226574\pi$
$734$	−24.0000	−0.885856
$735$	2.00000	0.0737711
$736$	−1.00000	−0.0368605
$737$	20.0000	0.736709
$738$	−2.00000	−0.0736210
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	8.00000	0.294086
$741$	2.00000	0.0734718
$742$	−3.00000	−0.110133
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	−7.00000	−0.256632
$745$	−12.0000	−0.439646
$746$	−5.00000	−0.183063
$747$	12.0000	0.439057
$748$	16.0000	0.585018
$749$	−27.0000	−0.986559
$750$	9.00000	0.328634
$751$	−14.0000	−0.510867	−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000	−0.510867	−0.255434	+	0.966827i	$0.582218\pi$
$752$	0	0
$753$	−20.0000	−0.728841
$754$	−2.00000	−0.0728357
$755$	17.0000	0.618693
$756$	3.00000	0.109109
$757$	−9.00000	−0.327111	−0.163555	−	0.986534i	$-0.552296\pi$
$757$	−9.00000	−0.327111	−0.163555	+	0.986534i	$0.552296\pi$
$758$	−20.0000	−0.726433
$759$	4.00000	0.145191
$760$	1.00000	0.0362738
$761$	−50.0000	−1.81250	−0.906249	−	0.422744i	$-0.861067\pi$
$761$	−50.0000	−1.81250	−0.906249	+	0.422744i	$0.861067\pi$
$762$	7.00000	0.253583
$763$	51.0000	1.84632
$764$	−20.0000	−0.723575
$765$	4.00000	0.144620
$766$	−10.0000	−0.361315
$767$	18.0000	0.649942
$768$	−1.00000	−0.0360844
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	12.0000	0.432450
$771$	6.00000	0.216085
$772$	6.00000	0.215945
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	1.00000	0.0359443
$775$	28.0000	1.00579
$776$	−14.0000	−0.502571
$777$	−24.0000	−0.860995
$778$	19.0000	0.681183
$779$	2.00000	0.0716574
$780$	−2.00000	−0.0716115
$781$	24.0000	0.858788
$782$	4.00000	0.143040
$783$	1.00000	0.0357371
$784$	2.00000	0.0714286
$785$	−2.00000	−0.0713831
$786$	−10.0000	−0.356688
$787$	−15.0000	−0.534692	−0.267346	−	0.963601i	$-0.586147\pi$
$787$	−15.0000	−0.534692	−0.267346	+	0.963601i	$0.586147\pi$
$788$	22.0000	0.783718
$789$	−13.0000	−0.462812
$790$	−8.00000	−0.284627
$791$	27.0000	0.960009
$792$	4.00000	0.142134
$793$	28.0000	0.994309
$794$	0	0
$795$	−1.00000	−0.0354663
$796$	−9.00000	−0.318997
$797$	−32.0000	−1.13350	−0.566749	−	0.823890i	$-0.691799\pi$
$797$	−32.0000	−1.13350	−0.566749	+	0.823890i	$0.691799\pi$
$798$	−3.00000	−0.106199
$799$	0	0
$800$	4.00000	0.141421
$801$	5.00000	0.176666
$802$	6.00000	0.211867
$803$	32.0000	1.12926
$804$	5.00000	0.176336
$805$	3.00000	0.105736
$806$	−14.0000	−0.493129
$807$	19.0000	0.668832
$808$	17.0000	0.598058
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	1.00000	0.0351364
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	3.00000	0.105279
$813$	15.0000	0.526073
$814$	−32.0000	−1.12160
$815$	−17.0000	−0.595484
$816$	4.00000	0.140028
$817$	−1.00000	−0.0349856
$818$	−24.0000	−0.839140
$819$	6.00000	0.209657
$820$	−2.00000	−0.0698430
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	−13.0000	−0.453427
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	3.00000	0.104510
$825$	−16.0000	−0.557048
$826$	−27.0000	−0.939450
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	1.00000	0.0347524
$829$	1.00000	0.0347314	0.0173657	−	0.999849i	$-0.494472\pi$
$829$	1.00000	0.0347314	0.0173657	+	0.999849i	$0.494472\pi$
$830$	12.0000	0.416526
$831$	−2.00000	−0.0693792
$832$	−2.00000	−0.0693375
$833$	−8.00000	−0.277184
$834$	6.00000	0.207763
$835$	2.00000	0.0692129
$836$	−4.00000	−0.138343
$837$	7.00000	0.241955
$838$	−12.0000	−0.414533
$839$	−53.0000	−1.82976	−0.914882	−	0.403722i	$-0.867716\pi$
$839$	−53.0000	−1.82976	−0.914882	+	0.403722i	$0.867716\pi$
$840$	3.00000	0.103510
$841$	−28.0000	−0.965517
$842$	39.0000	1.34403
$843$	13.0000	0.447744
$844$	4.00000	0.137686
$845$	9.00000	0.309609
$846$	0	0
$847$	−15.0000	−0.515406
$848$	−1.00000	−0.0343401
$849$	32.0000	1.09824
$850$	−16.0000	−0.548795
$851$	−8.00000	−0.274236
$852$	6.00000	0.205557
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	−42.0000	−1.43721
$855$	−1.00000	−0.0341993
$856$	−9.00000	−0.307614
$857$	25.0000	0.853984	0.426992	−	0.904255i	$-0.359573\pi$
$857$	25.0000	0.853984	0.426992	+	0.904255i	$0.359573\pi$
$858$	8.00000	0.273115
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	1.00000	0.0340997
$861$	6.00000	0.204479
$862$	−3.00000	−0.102180
$863$	−33.0000	−1.12333	−0.561667	−	0.827364i	$-0.689840\pi$
$863$	−33.0000	−1.12333	−0.561667	+	0.827364i	$0.689840\pi$
$864$	1.00000	0.0340207
$865$	−8.00000	−0.272008
$866$	22.0000	0.747590
$867$	1.00000	0.0339618
$868$	21.0000	0.712786
$869$	32.0000	1.08553
$870$	1.00000	0.0339032
$871$	10.0000	0.338837
$872$	17.0000	0.575693
$873$	14.0000	0.473828
$874$	−1.00000	−0.0338255
$875$	−27.0000	−0.912767
$876$	8.00000	0.270295
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	16.0000	0.539974
$879$	26.0000	0.876958
$880$	4.00000	0.134840
$881$	47.0000	1.58347	0.791735	−	0.610865i	$-0.209178\pi$
$881$	47.0000	1.58347	0.791735	+	0.610865i	$0.209178\pi$
$882$	−2.00000	−0.0673435
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	8.00000	0.269069
$885$	−9.00000	−0.302532
$886$	37.0000	1.24304
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	−8.00000	−0.268462
$889$	−21.0000	−0.704317
$890$	5.00000	0.167600
$891$	−4.00000	−0.134005
$892$	−2.00000	−0.0669650
$893$	0	0
$894$	12.0000	0.401340
$895$	0	0
$896$	3.00000	0.100223
$897$	2.00000	0.0667781
$898$	−6.00000	−0.200223
$899$	7.00000	0.233463
$900$	−4.00000	−0.133333
$901$	4.00000	0.133259
$902$	8.00000	0.266371
$903$	−3.00000	−0.0998337
$904$	9.00000	0.299336
$905$	−7.00000	−0.232688
$906$	−17.0000	−0.564787
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	13.0000	0.431420
$909$	−17.0000	−0.563854
$910$	6.00000	0.198898
$911$	27.0000	0.894550	0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000	0.894550	0.447275	+	0.894397i	$0.352395\pi$
$912$	−1.00000	−0.0331133
$913$	−48.0000	−1.58857
$914$	38.0000	1.25693
$915$	−14.0000	−0.462826
$916$	−15.0000	−0.495614
$917$	30.0000	0.990687
$918$	−4.00000	−0.132020
$919$	−6.00000	−0.197922	−0.0989609	−	0.995091i	$-0.531552\pi$
$919$	−6.00000	−0.197922	−0.0989609	+	0.995091i	$0.531552\pi$
$920$	1.00000	0.0329690
$921$	22.0000	0.724925
$922$	−12.0000	−0.395199
$923$	12.0000	0.394985
$924$	−12.0000	−0.394771
$925$	32.0000	1.05215
$926$	−6.00000	−0.197172
$927$	−3.00000	−0.0985329
$928$	1.00000	0.0328266
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	7.00000	0.229539
$931$	2.00000	0.0655474
$932$	2.00000	0.0655122
$933$	−6.00000	−0.196431
$934$	−8.00000	−0.261768
$935$	−16.0000	−0.523256
$936$	2.00000	0.0653720
$937$	3.00000	0.0980057	0.0490029	−	0.998799i	$-0.484396\pi$
$937$	3.00000	0.0980057	0.0490029	+	0.998799i	$0.484396\pi$
$938$	−15.0000	−0.489767
$939$	−22.0000	−0.717943
$940$	0	0
$941$	−45.0000	−1.46696	−0.733479	−	0.679712i	$-0.762105\pi$
$941$	−45.0000	−1.46696	−0.733479	+	0.679712i	$0.762105\pi$
$942$	2.00000	0.0651635
$943$	2.00000	0.0651290
$944$	−9.00000	−0.292925
$945$	−3.00000	−0.0975900
$946$	−4.00000	−0.130051
$947$	−40.0000	−1.29983	−0.649913	−	0.760009i	$-0.725195\pi$
$947$	−40.0000	−1.29983	−0.649913	+	0.760009i	$0.725195\pi$
$948$	8.00000	0.259828
$949$	16.0000	0.519382
$950$	4.00000	0.129777
$951$	2.00000	0.0648544
$952$	−12.0000	−0.388922
$953$	13.0000	0.421111	0.210556	−	0.977582i	$-0.432473\pi$
$953$	13.0000	0.421111	0.210556	+	0.977582i	$0.432473\pi$
$954$	1.00000	0.0323762
$955$	20.0000	0.647185
$956$	12.0000	0.388108
$957$	−4.00000	−0.129302
$958$	−39.0000	−1.26003
$959$	39.0000	1.25938
$960$	1.00000	0.0322749
$961$	18.0000	0.580645
$962$	−16.0000	−0.515861
$963$	9.00000	0.290021
$964$	−24.0000	−0.772988
$965$	−6.00000	−0.193147
$966$	−3.00000	−0.0965234
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−5.00000	−0.160706
$969$	4.00000	0.128499
$970$	14.0000	0.449513
$971$	51.0000	1.63667	0.818334	−	0.574743i	$-0.194898\pi$
$971$	51.0000	1.63667	0.818334	+	0.574743i	$0.194898\pi$
$972$	−1.00000	−0.0320750
$973$	−18.0000	−0.577054
$974$	0	0
$975$	−8.00000	−0.256205
$976$	−14.0000	−0.448129
$977$	−56.0000	−1.79160	−0.895799	−	0.444459i	$-0.853396\pi$
$977$	−56.0000	−1.79160	−0.895799	+	0.444459i	$0.853396\pi$
$978$	17.0000	0.543600
$979$	−20.0000	−0.639203
$980$	−2.00000	−0.0638877
$981$	−17.0000	−0.542768
$982$	−33.0000	−1.05307
$983$	−27.0000	−0.861166	−0.430583	−	0.902551i	$-0.641692\pi$
$983$	−27.0000	−0.861166	−0.430583	+	0.902551i	$0.641692\pi$
$984$	2.00000	0.0637577
$985$	−22.0000	−0.700978
$986$	−4.00000	−0.127386
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	−1.00000	−0.0317982
$990$	−4.00000	−0.127128
$991$	−18.0000	−0.571789	−0.285894	−	0.958261i	$-0.592291\pi$
$991$	−18.0000	−0.571789	−0.285894	+	0.958261i	$0.592291\pi$
$992$	7.00000	0.222250
$993$	−30.0000	−0.952021
$994$	−18.0000	−0.570925
$995$	9.00000	0.285319
$996$	−12.0000	−0.380235
$997$	−19.0000	−0.601736	−0.300868	−	0.953666i	$-0.597276\pi$
$997$	−19.0000	−0.601736	−0.300868	+	0.953666i	$0.597276\pi$
$998$	4.00000	0.126618
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.b.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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