LMFDB - Embedded newform 6042.2.a.c.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{20} +4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} -10.0000 q^{41} -4.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} -4.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -1.00000 q^{53} +1.00000 q^{54} -8.00000 q^{55} -1.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} -2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -4.00000 q^{69} -1.00000 q^{72} +10.0000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} -2.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} +14.0000 q^{89} -2.00000 q^{90} +4.00000 q^{92} -8.00000 q^{93} +2.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−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	$0.589456\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.00000	0.447214
$21$	0	0
$22$	4.00000	0.852803
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−1.00000	−0.162221
$39$	2.00000	0.320256
$40$	−2.00000	−0.316228
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	2.00000	0.298142
$46$	−4.00000	−0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	−2.00000	−0.277350
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	1.00000	0.136083
$55$	−8.00000	−1.07872
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−2.00000	−0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	−2.00000	−0.258199
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−4.00000	−0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	2.00000	0.242536
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	1.00000	0.114708
$77$	0	0
$78$	−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	$0.648589\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	4.00000	0.431331
$87$	−2.00000	−0.214423
$88$	4.00000	0.426401
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	4.00000	0.417029
$93$	−8.00000	−0.829561
$94$	0	0
$95$	2.00000	0.205196
$96$	1.00000	0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	7.00000	0.707107
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	2.00000	0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	1.00000	0.0971286
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	8.00000	0.762770
$111$	2.00000	0.189832
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	1.00000	0.0936586
$115$	8.00000	0.746004
$116$	2.00000	0.185695
$117$	−2.00000	−0.184900
$118$	−12.0000	−1.10469
$119$	0	0
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	10.0000	0.901670
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	4.00000	0.352180
$130$	4.00000	0.350823
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−4.00000	−0.345547
$135$	−2.00000	−0.172133
$136$	−2.00000	−0.171499
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	4.00000	0.340503
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−10.0000	−0.827606
$147$	7.00000	0.577350
$148$	−2.00000	−0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−1.00000	−0.0811107
$153$	2.00000	0.161690
$154$	0	0
$155$	16.0000	1.28515
$156$	2.00000	0.160128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	8.00000	0.636446
$159$	1.00000	0.0793052
$160$	−2.00000	−0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−10.0000	−0.780869
$165$	8.00000	0.622799
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	1.00000	0.0764719
$172$	−4.00000	−0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	−12.0000	−0.901975
$178$	−14.0000	−1.04934
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	2.00000	0.149071
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	−4.00000	−0.294884
$185$	−4.00000	−0.294086
$186$	8.00000	0.586588
$187$	−8.00000	−0.585018
$188$	0	0
$189$	0	0
$190$	−2.00000	−0.145095
$191$	−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	$0.430717\pi$
$194$	−2.00000	−0.143592
$195$	4.00000	0.286446
$196$	−7.00000	−0.500000
$197$	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
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	1.00000	0.0707107
$201$	−4.00000	−0.282138
$202$	−10.0000	−0.703598
$203$	0	0
$204$	−2.00000	−0.140028
$205$	−20.0000	−1.39686
$206$	0	0
$207$	4.00000	0.278019
$208$	−2.00000	−0.138675
$209$	−4.00000	−0.276686
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−1.00000	−0.0686803
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−10.0000	−0.675737
$220$	−8.00000	−0.539360
$221$	−4.00000	−0.269069
$222$	−2.00000	−0.134231
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−6.00000	−0.399114
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	−1.00000	−0.0662266
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	12.0000	0.781133
$237$	8.00000	0.519656
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−2.00000	−0.129099
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	−14.0000	−0.894427
$246$	−10.0000	−0.637577
$247$	−2.00000	−0.127257
$248$	−8.00000	−0.508001
$249$	0	0
$250$	12.0000	0.758947
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	−16.0000	−1.00393
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	−4.00000	−0.248069
$261$	2.00000	0.123797
$262$	4.00000	0.247121
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	−4.00000	−0.246183
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−14.0000	−0.856786
$268$	4.00000	0.244339
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	2.00000	0.121716
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	10.0000	0.604122
$275$	4.00000	0.241209
$276$	−4.00000	−0.240772
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	12.0000	0.719712
$279$	8.00000	0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	−8.00000	−0.473050
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	−2.00000	−0.117242
$292$	10.0000	0.585206
$293$	34.0000	1.98630	0.993151	−	0.116841i	$-0.0372769\pi$
$293$	34.0000	1.98630	0.993151	+	0.116841i	$0.0372769\pi$
$294$	−7.00000	−0.408248
$295$	24.0000	1.39733
$296$	2.00000	0.116248
$297$	4.00000	0.232104
$298$	−6.00000	−0.347571
$299$	−8.00000	−0.462652
$300$	1.00000	0.0577350
$301$	0	0
$302$	−16.0000	−0.920697
$303$	−10.0000	−0.574485
$304$	1.00000	0.0573539
$305$	−4.00000	−0.229039
$306$	−2.00000	−0.114332
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	0	0
$309$	0	0
$310$	−16.0000	−0.908739
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	−2.00000	−0.113228
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	−1.00000	−0.0560772
$319$	−8.00000	−0.447914
$320$	2.00000	0.111803
$321$	−12.0000	−0.669775
$322$	0	0
$323$	2.00000	0.111283
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	4.00000	0.221540
$327$	−10.0000	−0.553001
$328$	10.0000	0.552158
$329$	0	0
$330$	−8.00000	−0.440386
$331$	−24.0000	−1.31916	−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000	−1.31916	−0.659580	+	0.751635i	$0.729266\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	8.00000	0.437741
$335$	8.00000	0.437087
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	9.00000	0.489535
$339$	−6.00000	−0.325875
$340$	4.00000	0.216930
$341$	−32.0000	−1.73290
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	4.00000	0.215666
$345$	−8.00000	−0.430706
$346$	−2.00000	−0.107521
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−2.00000	−0.107211
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	4.00000	0.213201
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	14.0000	0.741999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	28.0000	1.47778	0.738892	−	0.673824i	$-0.235349\pi$
$359$	28.0000	1.47778	0.738892	+	0.673824i	$0.235349\pi$
$360$	−2.00000	−0.105409
$361$	1.00000	0.0526316
$362$	14.0000	0.735824
$363$	−5.00000	−0.262432
$364$	0	0
$365$	20.0000	1.04685
$366$	−2.00000	−0.104542
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	4.00000	0.208514
$369$	−10.0000	−0.520579
$370$	4.00000	0.207950
$371$	0	0
$372$	−8.00000	−0.414781
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	8.00000	0.413670
$375$	12.0000	0.619677
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	2.00000	0.102598
$381$	−16.0000	−0.819705
$382$	20.0000	1.02329
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	−4.00000	−0.202548
$391$	8.00000	0.404577
$392$	7.00000	0.353553
$393$	4.00000	0.201773
$394$	−22.0000	−1.10834
$395$	−16.0000	−0.805047
$396$	−4.00000	−0.201008
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	4.00000	0.199502
$403$	−16.0000	−0.797017
$404$	10.0000	0.497519
$405$	2.00000	0.0993808
$406$	0	0
$407$	8.00000	0.396545
$408$	2.00000	0.0990148
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	20.0000	0.987730
$411$	10.0000	0.493264
$412$	0	0
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	2.00000	0.0980581
$417$	12.0000	0.587643
$418$	4.00000	0.195646
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	−16.0000	−0.778868
$423$	0	0
$424$	1.00000	0.0485643
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	−8.00000	−0.386244
$430$	8.00000	0.385794
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−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	$0.677293\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	10.0000	0.478913
$437$	4.00000	0.191346
$438$	10.0000	0.477818
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	8.00000	0.381385
$441$	−7.00000	−0.333333
$442$	4.00000	0.190261
$443$	32.0000	1.52037	0.760183	−	0.649709i	$-0.225109\pi$
$443$	32.0000	1.52037	0.760183	+	0.649709i	$0.225109\pi$
$444$	2.00000	0.0949158
$445$	28.0000	1.32733
$446$	−4.00000	−0.189405
$447$	−6.00000	−0.283790
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	1.00000	0.0471405
$451$	40.0000	1.88353
$452$	6.00000	0.282216
$453$	−16.0000	−0.751746
$454$	−28.0000	−1.31411
$455$	0	0
$456$	1.00000	0.0468293
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	−6.00000	−0.280362
$459$	−2.00000	−0.0933520
$460$	8.00000	0.373002
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	2.00000	0.0928477
$465$	−16.0000	−0.741982
$466$	10.0000	0.463241
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	0	0
$471$	10.0000	0.460776
$472$	−12.0000	−0.552345
$473$	16.0000	0.735681
$474$	−8.00000	−0.367452
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−1.00000	−0.0457869
$478$	−12.0000	−0.548867
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	2.00000	0.0912871
$481$	4.00000	0.182384
$482$	30.0000	1.36646
$483$	0	0
$484$	5.00000	0.227273
$485$	4.00000	0.181631
$486$	1.00000	0.0453609
$487$	−36.0000	−1.63132	−0.815658	−	0.578535i	$-0.803625\pi$
$487$	−36.0000	−1.63132	−0.815658	+	0.578535i	$0.803625\pi$
$488$	2.00000	0.0905357
$489$	4.00000	0.180886
$490$	14.0000	0.632456
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	10.0000	0.450835
$493$	4.00000	0.180151
$494$	2.00000	0.0899843
$495$	−8.00000	−0.359573
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−12.0000	−0.536656
$501$	8.00000	0.357414
$502$	−8.00000	−0.357057
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	16.0000	0.711287
$507$	9.00000	0.399704
$508$	16.0000	0.709885
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	18.0000	0.793946
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	4.00000	0.175412
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−2.00000	−0.0875376
$523$	40.0000	1.74908	0.874539	−	0.484955i	$-0.161164\pi$
$523$	40.0000	1.74908	0.874539	+	0.484955i	$0.161164\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−4.00000	−0.174408
$527$	16.0000	0.696971
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	2.00000	0.0868744
$531$	12.0000	0.520756
$532$	0	0
$533$	20.0000	0.866296
$534$	14.0000	0.605839
$535$	24.0000	1.03761
$536$	−4.00000	−0.172774
$537$	−12.0000	−0.517838
$538$	−26.0000	−1.12094
$539$	28.0000	1.20605
$540$	−2.00000	−0.0860663
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−24.0000	−1.03089
$543$	14.0000	0.600798
$544$	−2.00000	−0.0857493
$545$	20.0000	0.856706
$546$	0	0
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	−10.0000	−0.427179
$549$	−2.00000	−0.0853579
$550$	−4.00000	−0.170561
$551$	2.00000	0.0852029
$552$	4.00000	0.170251
$553$	0	0
$554$	10.0000	0.424859
$555$	4.00000	0.169791
$556$	−12.0000	−0.508913
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	−8.00000	−0.338667
$559$	8.00000	0.338364
$560$	0	0
$561$	8.00000	0.337760
$562$	10.0000	0.421825
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	−4.00000	−0.168133
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	2.00000	0.0837708
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	8.00000	0.334497
$573$	20.0000	0.835512
$574$	0	0
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	13.0000	0.540729
$579$	−6.00000	−0.249351
$580$	4.00000	0.166091
$581$	0	0
$582$	2.00000	0.0829027
$583$	4.00000	0.165663
$584$	−10.0000	−0.413803
$585$	−4.00000	−0.165380
$586$	−34.0000	−1.40453
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	7.00000	0.288675
$589$	8.00000	0.329634
$590$	−24.0000	−0.988064
$591$	−22.0000	−0.904959
$592$	−2.00000	−0.0821995
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	8.00000	0.327144
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	−1.00000	−0.0408248
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	16.0000	0.651031
$605$	10.0000	0.406558
$606$	10.0000	0.406222
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	4.00000	0.161955
$611$	0	0
$612$	2.00000	0.0808452
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−32.0000	−1.29141
$615$	20.0000	0.806478
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	16.0000	0.642575
$621$	−4.00000	−0.160514
$622$	−24.0000	−0.962312
$623$	0	0
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	−10.0000	−0.399680
$627$	4.00000	0.159745
$628$	−10.0000	−0.399043
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	8.00000	0.318223
$633$	−16.0000	−0.635943
$634$	−10.0000	−0.397151
$635$	32.0000	1.26988
$636$	1.00000	0.0396526
$637$	14.0000	0.554700
$638$	8.00000	0.316723
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	12.0000	0.473602
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	−2.00000	−0.0786889
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−1.00000	−0.0392837
$649$	−48.0000	−1.88416
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	10.0000	0.391031
$655$	−8.00000	−0.312586
$656$	−10.0000	−0.390434
$657$	10.0000	0.390137
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	8.00000	0.311400
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	24.0000	0.932786
$663$	4.00000	0.155347
$664$	0	0
$665$	0	0
$666$	2.00000	0.0774984
$667$	8.00000	0.309761
$668$	−8.00000	−0.309529
$669$	−4.00000	−0.154649
$670$	−8.00000	−0.309067
$671$	8.00000	0.308837
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	34.0000	1.30963
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	−4.00000	−0.153393
$681$	−28.0000	−1.07296
$682$	32.0000	1.22534
$683$	−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
$685$	−20.0000	−0.764161
$686$	0	0
$687$	−6.00000	−0.228914
$688$	−4.00000	−0.152499
$689$	2.00000	0.0761939
$690$	8.00000	0.304555
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−24.0000	−0.910372
$696$	2.00000	0.0758098
$697$	−20.0000	−0.757554
$698$	10.0000	0.378506
$699$	10.0000	0.378235
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	−2.00000	−0.0754851
$703$	−2.00000	−0.0754314
$704$	−4.00000	−0.150756
$705$	0	0
$706$	2.00000	0.0752710
$707$	0	0
$708$	−12.0000	−0.450988
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	−14.0000	−0.524672
$713$	32.0000	1.19841
$714$	0	0
$715$	16.0000	0.598366
$716$	12.0000	0.448461
$717$	−12.0000	−0.448148
$718$	−28.0000	−1.04495
$719$	44.0000	1.64092	0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000	1.64092	0.820462	+	0.571702i	$0.193717\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	30.0000	1.11571
$724$	−14.0000	−0.520306
$725$	−2.00000	−0.0742781
$726$	5.00000	0.185567
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−20.0000	−0.740233
$731$	−8.00000	−0.295891
$732$	2.00000	0.0739221
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−24.0000	−0.885856
$735$	14.0000	0.516398
$736$	−4.00000	−0.147442
$737$	−16.0000	−0.589368
$738$	10.0000	0.368105
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	−4.00000	−0.147043
$741$	2.00000	0.0734718
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	8.00000	0.293294
$745$	12.0000	0.439646
$746$	−2.00000	−0.0732252
$747$	0	0
$748$	−8.00000	−0.292509
$749$	0	0
$750$	−12.0000	−0.438178
$751$	−44.0000	−1.60558	−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000	−1.60558	−0.802791	+	0.596260i	$0.796653\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	4.00000	0.145671
$755$	32.0000	1.16460
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	−20.0000	−0.726433
$759$	16.0000	0.580763
$760$	−2.00000	−0.0725476
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	−20.0000	−0.723575
$765$	4.00000	0.144620
$766$	8.00000	0.289052
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	6.00000	0.215945
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	4.00000	0.143777
$775$	−8.00000	−0.287368
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−26.0000	−0.932145
$779$	−10.0000	−0.358287
$780$	4.00000	0.143223
$781$	0	0
$782$	−8.00000	−0.286079
$783$	−2.00000	−0.0714742
$784$	−7.00000	−0.250000
$785$	−20.0000	−0.713831
$786$	−4.00000	−0.142675
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	22.0000	0.783718
$789$	−4.00000	−0.142404
$790$	16.0000	0.569254
$791$	0	0
$792$	4.00000	0.142134
$793$	4.00000	0.142044
$794$	−6.00000	−0.212932
$795$	2.00000	0.0709327
$796$	0	0
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	14.0000	0.494666
$802$	−30.0000	−1.05934
$803$	−40.0000	−1.41157
$804$	−4.00000	−0.141069
$805$	0	0
$806$	16.0000	0.563576
$807$	−26.0000	−0.915243
$808$	−10.0000	−0.351799
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	−2.00000	−0.0702728
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	−8.00000	−0.280400
$815$	−8.00000	−0.280228
$816$	−2.00000	−0.0700140
$817$	−4.00000	−0.139942
$818$	30.0000	1.04893
$819$	0	0
$820$	−20.0000	−0.698430
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	−10.0000	−0.348790
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	4.00000	0.139010
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	−2.00000	−0.0693375
$833$	−14.0000	−0.485071
$834$	−12.0000	−0.415526
$835$	−16.0000	−0.553703
$836$	−4.00000	−0.138343
$837$	−8.00000	−0.276520
$838$	24.0000	0.829066
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	6.00000	0.206774
$843$	10.0000	0.344418
$844$	16.0000	0.550743
$845$	−18.0000	−0.619219
$846$	0	0
$847$	0	0
$848$	−1.00000	−0.0343401
$849$	−4.00000	−0.137280
$850$	2.00000	0.0685994
$851$	−8.00000	−0.274236
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	−12.0000	−0.410152
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	8.00000	0.273115
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−24.0000	−0.817443
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	1.00000	0.0340207
$865$	4.00000	0.136004
$866$	22.0000	0.747590
$867$	13.0000	0.441503
$868$	0	0
$869$	32.0000	1.08553
$870$	4.00000	0.135613
$871$	−8.00000	−0.271070
$872$	−10.0000	−0.338643
$873$	2.00000	0.0676897
$874$	−4.00000	−0.135302
$875$	0	0
$876$	−10.0000	−0.337869
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	4.00000	0.134993
$879$	−34.0000	−1.14679
$880$	−8.00000	−0.269680
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	7.00000	0.235702
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	−4.00000	−0.134535
$885$	−24.0000	−0.806751
$886$	−32.0000	−1.07506
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	−28.0000	−0.938562
$891$	−4.00000	−0.134005
$892$	4.00000	0.133930
$893$	0	0
$894$	6.00000	0.200670
$895$	24.0000	0.802232
$896$	0	0
$897$	8.00000	0.267112
$898$	−30.0000	−1.00111
$899$	16.0000	0.533630
$900$	−1.00000	−0.0333333
$901$	−2.00000	−0.0666297
$902$	−40.0000	−1.33185
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−28.0000	−0.930751
$906$	16.0000	0.531564
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	28.0000	0.929213
$909$	10.0000	0.331679
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	−34.0000	−1.12462
$915$	4.00000	0.132236
$916$	6.00000	0.198246
$917$	0	0
$918$	2.00000	0.0660098
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	−8.00000	−0.263752
$921$	−32.0000	−1.05444
$922$	18.0000	0.592798
$923$	0	0
$924$	0	0
$925$	2.00000	0.0657596
$926$	−24.0000	−0.788689
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	16.0000	0.524661
$931$	−7.00000	−0.229416
$932$	−10.0000	−0.327561
$933$	−24.0000	−0.785725
$934$	28.0000	0.916188
$935$	−16.0000	−0.523256
$936$	2.00000	0.0653720
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	−10.0000	−0.325818
$943$	−40.0000	−1.30258
$944$	12.0000	0.390567
$945$	0	0
$946$	−16.0000	−0.520205
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	8.00000	0.259828
$949$	−20.0000	−0.649227
$950$	1.00000	0.0324443
$951$	−10.0000	−0.324272
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	1.00000	0.0323762
$955$	−40.0000	−1.29437
$956$	12.0000	0.388108
$957$	8.00000	0.258603
$958$	36.0000	1.16311
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	33.0000	1.06452
$962$	−4.00000	−0.128965
$963$	12.0000	0.386695
$964$	−30.0000	−0.966235
$965$	12.0000	0.386294
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−5.00000	−0.160706
$969$	−2.00000	−0.0642493
$970$	−4.00000	−0.128432
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	36.0000	1.15351
$975$	−2.00000	−0.0640513
$976$	−2.00000	−0.0640184
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	−4.00000	−0.127906
$979$	−56.0000	−1.78977
$980$	−14.0000	−0.447214
$981$	10.0000	0.319275
$982$	24.0000	0.765871
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	−10.0000	−0.318788
$985$	44.0000	1.40196
$986$	−4.00000	−0.127386
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	−16.0000	−0.508770
$990$	8.00000	0.254257
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	−8.00000	−0.254000
$993$	24.0000	0.761617
$994$	0	0
$995$	0	0
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	4.00000	0.126618
$999$	2.00000	0.0632772

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.c.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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