LMFDB - Embedded newform 6042.2.a.j.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)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} -4.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{20} +4.00000 q^{21} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} -4.00000 q^{28} -2.00000 q^{29} -2.00000 q^{30} +1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} -8.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} +1.00000 q^{38} +2.00000 q^{40} +2.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} +2.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +1.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} -4.00000 q^{56} -1.00000 q^{57} -2.00000 q^{58} -10.0000 q^{59} -2.00000 q^{60} -4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -4.00000 q^{69} -8.00000 q^{70} +1.00000 q^{72} +6.00000 q^{73} +8.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} -16.0000 q^{77} +8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +2.00000 q^{83} +4.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} +2.00000 q^{87} +4.00000 q^{88} +12.0000 q^{89} +2.00000 q^{90} +4.00000 q^{92} +2.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−4.00000	−1.06904
$15$	−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
$21$	4.00000	0.872872
$22$	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
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−4.00000	−0.755929
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−2.00000	−0.365148
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	2.00000	0.342997
$35$	−8.00000	−1.35225
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	2.00000	0.316228
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	4.00000	0.617213
$43$	−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
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	−1.00000	−0.136083
$55$	8.00000	1.07872
$56$	−4.00000	−0.534522
$57$	−1.00000	−0.132453
$58$	−2.00000	−0.262613
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	−2.00000	−0.258199
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.00000	0.242536
$69$	−4.00000	−0.481543
$70$	−8.00000	−0.956183
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	8.00000	0.929981
$75$	1.00000	0.115470
$76$	1.00000	0.114708
$77$	−16.0000	−1.82337
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	4.00000	0.436436
$85$	4.00000	0.433861
$86$	−4.00000	−0.431331
$87$	2.00000	0.214423
$88$	4.00000	0.426401
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	9.00000	0.909137
$99$	4.00000	0.402015
$100$	−1.00000	−0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−2.00000	−0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	1.00000	0.0971286
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	−1.00000	−0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	8.00000	0.762770
$111$	−8.00000	−0.759326
$112$	−4.00000	−0.377964
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−1.00000	−0.0936586
$115$	8.00000	0.746004
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−10.0000	−0.920575
$119$	−8.00000	−0.733359
$120$	−2.00000	−0.182574
$121$	5.00000	0.454545
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−12.0000	−1.07331
$126$	−4.00000	−0.356348
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	−4.00000	−0.348155
$133$	−4.00000	−0.346844
$134$	−4.00000	−0.345547
$135$	−2.00000	−0.172133
$136$	2.00000	0.171499
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−4.00000	−0.340503
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−8.00000	−0.676123
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	6.00000	0.496564
$147$	−9.00000	−0.742307
$148$	8.00000	0.657596
$149$	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
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	1.00000	0.0811107
$153$	2.00000	0.161690
$154$	−16.0000	−1.28932
$155$	0	0
$156$	0	0
$157$	24.0000	1.91541	0.957704	−	0.287754i	$-0.0929087\pi$
$157$	24.0000	1.91541	0.957704	+	0.287754i	$0.0929087\pi$
$158$	8.00000	0.636446
$159$	−1.00000	−0.0793052
$160$	2.00000	0.158114
$161$	−16.0000	−1.26098
$162$	1.00000	0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	2.00000	0.156174
$165$	−8.00000	−0.622799
$166$	2.00000	0.155230
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	4.00000	0.308607
$169$	−13.0000	−1.00000
$170$	4.00000	0.306786
$171$	1.00000	0.0764719
$172$	−4.00000	−0.304997
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	2.00000	0.151620
$175$	4.00000	0.302372
$176$	4.00000	0.301511
$177$	10.0000	0.751646
$178$	12.0000	0.899438
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	2.00000	0.149071
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	0	0
$184$	4.00000	0.294884
$185$	16.0000	1.17634
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	4.00000	0.290957
$190$	2.00000	0.145095
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	4.00000	0.284268
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−1.00000	−0.0707107
$201$	4.00000	0.282138
$202$	6.00000	0.422159
$203$	8.00000	0.561490
$204$	−2.00000	−0.140028
$205$	4.00000	0.279372
$206$	8.00000	0.557386
$207$	4.00000	0.278019
$208$	0	0
$209$	4.00000	0.276686
$210$	8.00000	0.552052
$211$	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
$213$	0	0
$214$	−14.0000	−0.957020
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	2.00000	0.135457
$219$	−6.00000	−0.405442
$220$	8.00000	0.539360
$221$	0	0
$222$	−8.00000	−0.536925
$223$	18.0000	1.20537	0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000	1.20537	0.602685	+	0.797980i	$0.294098\pi$
$224$	−4.00000	−0.267261
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	−1.00000	−0.0662266
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	8.00000	0.527504
$231$	16.0000	1.05272
$232$	−2.00000	−0.131306
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	0	0
$236$	−10.0000	−0.650945
$237$	−8.00000	−0.519656
$238$	−8.00000	−0.518563
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	−2.00000	−0.129099
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	18.0000	1.14998
$246$	−2.00000	−0.127515
$247$	0	0
$248$	0	0
$249$	−2.00000	−0.126745
$250$	−12.0000	−0.758947
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−4.00000	−0.251976
$253$	16.0000	1.00591
$254$	8.00000	0.501965
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	4.00000	0.249029
$259$	−32.0000	−1.98838
$260$	0	0
$261$	−2.00000	−0.123797
$262$	−8.00000	−0.494242
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	−4.00000	−0.246183
$265$	2.00000	0.122859
$266$	−4.00000	−0.245256
$267$	−12.0000	−0.734388
$268$	−4.00000	−0.244339
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−2.00000	−0.121716
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	8.00000	0.483298
$275$	−4.00000	−0.241209
$276$	−4.00000	−0.240772
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	−8.00000	−0.478091
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	−8.00000	−0.472225
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	−10.0000	−0.586210
$292$	6.00000	0.351123
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	−9.00000	−0.524891
$295$	−20.0000	−1.16445
$296$	8.00000	0.464991
$297$	−4.00000	−0.232104
$298$	6.00000	0.347571
$299$	0	0
$300$	1.00000	0.0577350
$301$	16.0000	0.922225
$302$	4.00000	0.230174
$303$	−6.00000	−0.344691
$304$	1.00000	0.0573539
$305$	0	0
$306$	2.00000	0.114332
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	−16.0000	−0.911685
$309$	−8.00000	−0.455104
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	24.0000	1.35440
$315$	−8.00000	−0.450749
$316$	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
$319$	−8.00000	−0.447914
$320$	2.00000	0.111803
$321$	14.0000	0.781404
$322$	−16.0000	−0.891645
$323$	2.00000	0.111283
$324$	1.00000	0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	−2.00000	−0.110600
$328$	2.00000	0.110432
$329$	0	0
$330$	−8.00000	−0.440386
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	2.00000	0.109764
$333$	8.00000	0.438397
$334$	8.00000	0.437741
$335$	−8.00000	−0.437087
$336$	4.00000	0.218218
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−13.0000	−0.707107
$339$	0	0
$340$	4.00000	0.216930
$341$	0	0
$342$	1.00000	0.0540738
$343$	−8.00000	−0.431959
$344$	−4.00000	−0.215666
$345$	−8.00000	−0.430706
$346$	18.0000	0.967686
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	2.00000	0.107211
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	4.00000	0.213201
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	10.0000	0.531494
$355$	0	0
$356$	12.0000	0.635999
$357$	8.00000	0.423405
$358$	16.0000	0.845626
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	2.00000	0.105409
$361$	1.00000	0.0526316
$362$	−22.0000	−1.15629
$363$	−5.00000	−0.262432
$364$	0	0
$365$	12.0000	0.628109
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	4.00000	0.208514
$369$	2.00000	0.104116
$370$	16.0000	0.831800
$371$	−4.00000	−0.207670
$372$	0	0
$373$	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
$377$	0	0
$378$	4.00000	0.205738
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	2.00000	0.102598
$381$	−8.00000	−0.409852
$382$	−12.0000	−0.613973
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−1.00000	−0.0510310
$385$	−32.0000	−1.63087
$386$	26.0000	1.32337
$387$	−4.00000	−0.203331
$388$	10.0000	0.507673
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	9.00000	0.454569
$393$	8.00000	0.403547
$394$	18.0000	0.906827
$395$	16.0000	0.805047
$396$	4.00000	0.201008
$397$	−32.0000	−1.60603	−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000	−1.60603	−0.803017	+	0.595956i	$0.796773\pi$
$398$	−16.0000	−0.802008
$399$	4.00000	0.200250
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	6.00000	0.298511
$405$	2.00000	0.0993808
$406$	8.00000	0.397033
$407$	32.0000	1.58618
$408$	−2.00000	−0.0990148
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	4.00000	0.197546
$411$	−8.00000	−0.394611
$412$	8.00000	0.394132
$413$	40.0000	1.96827
$414$	4.00000	0.196589
$415$	4.00000	0.196352
$416$	0	0
$417$	−8.00000	−0.391762
$418$	4.00000	0.195646
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	8.00000	0.390360
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	1.00000	0.0485643
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	−14.0000	−0.676716
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−1.00000	−0.0481125
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	2.00000	0.0957826
$437$	4.00000	0.191346
$438$	−6.00000	−0.286691
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	8.00000	0.381385
$441$	9.00000	0.428571
$442$	0	0
$443$	2.00000	0.0950229	0.0475114	−	0.998871i	$-0.484871\pi$
$443$	2.00000	0.0950229	0.0475114	+	0.998871i	$0.484871\pi$
$444$	−8.00000	−0.379663
$445$	24.0000	1.13771
$446$	18.0000	0.852325
$447$	−6.00000	−0.283790
$448$	−4.00000	−0.188982
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	−1.00000	−0.0471405
$451$	8.00000	0.376705
$452$	0	0
$453$	−4.00000	−0.187936
$454$	14.0000	0.657053
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−14.0000	−0.654177
$459$	−2.00000	−0.0933520
$460$	8.00000	0.373002
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	16.0000	0.744387
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−16.0000	−0.741186
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	−24.0000	−1.10586
$472$	−10.0000	−0.460287
$473$	−16.0000	−0.735681
$474$	−8.00000	−0.367452
$475$	−1.00000	−0.0458831
$476$	−8.00000	−0.366679
$477$	1.00000	0.0457869
$478$	−20.0000	−0.914779
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	−2.00000	−0.0912871
$481$	0	0
$482$	10.0000	0.455488
$483$	16.0000	0.728025
$484$	5.00000	0.227273
$485$	20.0000	0.908153
$486$	−1.00000	−0.0453609
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	18.0000	0.813157
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	−2.00000	−0.0901670
$493$	−4.00000	−0.180151
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	0	0
$498$	−2.00000	−0.0896221
$499$	−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
$502$	−2.00000	−0.0892644
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	−4.00000	−0.178174
$505$	12.0000	0.533993
$506$	16.0000	0.711287
$507$	13.0000	0.577350
$508$	8.00000	0.354943
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	−4.00000	−0.177123
$511$	−24.0000	−1.06170
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	14.0000	0.617514
$515$	16.0000	0.705044
$516$	4.00000	0.176090
$517$	0	0
$518$	−32.0000	−1.40600
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	−2.00000	−0.0875376
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	−8.00000	−0.349482
$525$	−4.00000	−0.174574
$526$	−16.0000	−0.697633
$527$	0	0
$528$	−4.00000	−0.174078
$529$	−7.00000	−0.304348
$530$	2.00000	0.0868744
$531$	−10.0000	−0.433963
$532$	−4.00000	−0.173422
$533$	0	0
$534$	−12.0000	−0.519291
$535$	−28.0000	−1.21055
$536$	−4.00000	−0.172774
$537$	−16.0000	−0.690451
$538$	−18.0000	−0.776035
$539$	36.0000	1.55063
$540$	−2.00000	−0.0860663
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	12.0000	0.515444
$543$	22.0000	0.944110
$544$	2.00000	0.0857493
$545$	4.00000	0.171341
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	8.00000	0.341743
$549$	0	0
$550$	−4.00000	−0.170561
$551$	−2.00000	−0.0852029
$552$	−4.00000	−0.170251
$553$	−32.0000	−1.36078
$554$	−16.0000	−0.679775
$555$	−16.0000	−0.679162
$556$	8.00000	0.339276
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	0	0
$560$	−8.00000	−0.338062
$561$	−8.00000	−0.337760
$562$	0	0
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	0	0
$566$	24.0000	1.00880
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	−2.00000	−0.0837708
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	−8.00000	−0.333914
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	46.0000	1.91501	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000	1.91501	0.957503	+	0.288425i	$0.0931316\pi$
$578$	−13.0000	−0.540729
$579$	−26.0000	−1.08052
$580$	−4.00000	−0.166091
$581$	−8.00000	−0.331896
$582$	−10.0000	−0.414513
$583$	4.00000	0.165663
$584$	6.00000	0.248282
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	−9.00000	−0.371154
$589$	0	0
$590$	−20.0000	−0.823387
$591$	−18.0000	−0.740421
$592$	8.00000	0.328798
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	−4.00000	−0.164122
$595$	−16.0000	−0.655936
$596$	6.00000	0.245770
$597$	16.0000	0.654836
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	1.00000	0.0408248
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	16.0000	0.652111
$603$	−4.00000	−0.162893
$604$	4.00000	0.162758
$605$	10.0000	0.406558
$606$	−6.00000	−0.243733
$607$	−6.00000	−0.243532	−0.121766	−	0.992559i	$-0.538856\pi$
$607$	−6.00000	−0.243532	−0.121766	+	0.992559i	$0.538856\pi$
$608$	1.00000	0.0405554
$609$	−8.00000	−0.324176
$610$	0	0
$611$	0	0
$612$	2.00000	0.0808452
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	4.00000	0.161427
$615$	−4.00000	−0.161296
$616$	−16.0000	−0.644658
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	−8.00000	−0.321807
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	8.00000	0.320771
$623$	−48.0000	−1.92308
$624$	0	0
$625$	−19.0000	−0.760000
$626$	14.0000	0.559553
$627$	−4.00000	−0.159745
$628$	24.0000	0.957704
$629$	16.0000	0.637962
$630$	−8.00000	−0.318728
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	8.00000	0.318223
$633$	−4.00000	−0.158986
$634$	−2.00000	−0.0794301
$635$	16.0000	0.634941
$636$	−1.00000	−0.0396526
$637$	0	0
$638$	−8.00000	−0.316723
$639$	0	0
$640$	2.00000	0.0790569
$641$	50.0000	1.97488	0.987441	−	0.157991i	$-0.0505015\pi$
$641$	50.0000	1.97488	0.987441	+	0.157991i	$0.0505015\pi$
$642$	14.0000	0.552536
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−16.0000	−0.630488
$645$	8.00000	0.315000
$646$	2.00000	0.0786889
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	1.00000	0.0392837
$649$	−40.0000	−1.57014
$650$	0	0
$651$	0	0
$652$	12.0000	0.469956
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	−2.00000	−0.0782062
$655$	−16.0000	−0.625172
$656$	2.00000	0.0780869
$657$	6.00000	0.234082
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	−8.00000	−0.311400
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	2.00000	0.0776151
$665$	−8.00000	−0.310227
$666$	8.00000	0.309994
$667$	−8.00000	−0.309761
$668$	8.00000	0.309529
$669$	−18.0000	−0.695920
$670$	−8.00000	−0.309067
$671$	0	0
$672$	4.00000	0.154303
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	−18.0000	−0.693334
$675$	1.00000	0.0384900
$676$	−13.0000	−0.500000
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	0	0
$679$	−40.0000	−1.53506
$680$	4.00000	0.153393
$681$	−14.0000	−0.536481
$682$	0	0
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	1.00000	0.0382360
$685$	16.0000	0.611329
$686$	−8.00000	−0.305441
$687$	14.0000	0.534133
$688$	−4.00000	−0.152499
$689$	0	0
$690$	−8.00000	−0.304555
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	18.0000	0.684257
$693$	−16.0000	−0.607790
$694$	−24.0000	−0.911028
$695$	16.0000	0.606915
$696$	2.00000	0.0758098
$697$	4.00000	0.151511
$698$	24.0000	0.908413
$699$	16.0000	0.605176
$700$	4.00000	0.151186
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	4.00000	0.150756
$705$	0	0
$706$	4.00000	0.150542
$707$	−24.0000	−0.902613
$708$	10.0000	0.375823
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	12.0000	0.449719
$713$	0	0
$714$	8.00000	0.299392
$715$	0	0
$716$	16.0000	0.597948
$717$	20.0000	0.746914
$718$	20.0000	0.746393
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	2.00000	0.0745356
$721$	−32.0000	−1.19174
$722$	1.00000	0.0372161
$723$	−10.0000	−0.371904
$724$	−22.0000	−0.817624
$725$	2.00000	0.0742781
$726$	−5.00000	−0.185567
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	−8.00000	−0.295891
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−28.0000	−1.03350
$735$	−18.0000	−0.663940
$736$	4.00000	0.147442
$737$	−16.0000	−0.589368
$738$	2.00000	0.0736210
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	16.0000	0.588172
$741$	0	0
$742$	−4.00000	−0.146845
$743$	28.0000	1.02722	0.513610	−	0.858024i	$-0.328308\pi$
$743$	28.0000	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	2.00000	0.0732252
$747$	2.00000	0.0731762
$748$	8.00000	0.292509
$749$	56.0000	2.04620
$750$	12.0000	0.438178
$751$	30.0000	1.09472	0.547358	−	0.836899i	$-0.315634\pi$
$751$	30.0000	1.09472	0.547358	+	0.836899i	$0.315634\pi$
$752$	0	0
$753$	2.00000	0.0728841
$754$	0	0
$755$	8.00000	0.291150
$756$	4.00000	0.145479
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−12.0000	−0.435860
$759$	−16.0000	−0.580763
$760$	2.00000	0.0725476
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	−8.00000	−0.289809
$763$	−8.00000	−0.289619
$764$	−12.0000	−0.434145
$765$	4.00000	0.144620
$766$	−24.0000	−0.867155
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	−32.0000	−1.15320
$771$	−14.0000	−0.504198
$772$	26.0000	0.935760
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	10.0000	0.358979
$777$	32.0000	1.14799
$778$	10.0000	0.358517
$779$	2.00000	0.0716574
$780$	0	0
$781$	0	0
$782$	8.00000	0.286079
$783$	2.00000	0.0714742
$784$	9.00000	0.321429
$785$	48.0000	1.71319
$786$	8.00000	0.285351
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	18.0000	0.641223
$789$	16.0000	0.569615
$790$	16.0000	0.569254
$791$	0	0
$792$	4.00000	0.142134
$793$	0	0
$794$	−32.0000	−1.13564
$795$	−2.00000	−0.0709327
$796$	−16.0000	−0.567105
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	4.00000	0.141598
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	12.0000	0.423999
$802$	18.0000	0.635602
$803$	24.0000	0.846942
$804$	4.00000	0.141069
$805$	−32.0000	−1.12785
$806$	0	0
$807$	18.0000	0.633630
$808$	6.00000	0.211079
$809$	44.0000	1.54696	0.773479	−	0.633822i	$-0.218515\pi$
$809$	44.0000	1.54696	0.773479	+	0.633822i	$0.218515\pi$
$810$	2.00000	0.0702728
$811$	−24.0000	−0.842754	−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000	−0.842754	−0.421377	+	0.906886i	$0.638453\pi$
$812$	8.00000	0.280745
$813$	−12.0000	−0.420858
$814$	32.0000	1.12160
$815$	24.0000	0.840683
$816$	−2.00000	−0.0700140
$817$	−4.00000	−0.139942
$818$	−14.0000	−0.489499
$819$	0	0
$820$	4.00000	0.139686
$821$	−26.0000	−0.907406	−0.453703	−	0.891153i	$-0.649897\pi$
$821$	−26.0000	−0.907406	−0.453703	+	0.891153i	$0.649897\pi$
$822$	−8.00000	−0.279032
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	8.00000	0.278693
$825$	4.00000	0.139262
$826$	40.0000	1.39178
$827$	−24.0000	−0.834562	−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000	−0.834562	−0.417281	+	0.908778i	$0.637017\pi$
$828$	4.00000	0.139010
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	4.00000	0.138842
$831$	16.0000	0.555034
$832$	0	0
$833$	18.0000	0.623663
$834$	−8.00000	−0.277017
$835$	16.0000	0.553703
$836$	4.00000	0.138343
$837$	0	0
$838$	14.0000	0.483622
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	8.00000	0.276026
$841$	−25.0000	−0.862069
$842$	−26.0000	−0.896019
$843$	0	0
$844$	4.00000	0.137686
$845$	−26.0000	−0.894427
$846$	0	0
$847$	−20.0000	−0.687208
$848$	1.00000	0.0343401
$849$	−24.0000	−0.823678
$850$	−2.00000	−0.0685994
$851$	32.0000	1.09695
$852$	0	0
$853$	−32.0000	−1.09566	−0.547830	−	0.836590i	$-0.684546\pi$
$853$	−32.0000	−1.09566	−0.547830	+	0.836590i	$0.684546\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	−14.0000	−0.478510
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−8.00000	−0.272798
$861$	8.00000	0.272639
$862$	−8.00000	−0.272481
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	−1.00000	−0.0340207
$865$	36.0000	1.22404
$866$	30.0000	1.01944
$867$	13.0000	0.441503
$868$	0	0
$869$	32.0000	1.08553
$870$	4.00000	0.135613
$871$	0	0
$872$	2.00000	0.0677285
$873$	10.0000	0.338449
$874$	4.00000	0.135302
$875$	48.0000	1.62270
$876$	−6.00000	−0.202721
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−22.0000	−0.742464
$879$	2.00000	0.0674583
$880$	8.00000	0.269680
$881$	−20.0000	−0.673817	−0.336909	−	0.941537i	$-0.609381\pi$
$881$	−20.0000	−0.673817	−0.336909	+	0.941537i	$0.609381\pi$
$882$	9.00000	0.303046
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	20.0000	0.672293
$886$	2.00000	0.0671913
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	−8.00000	−0.268462
$889$	−32.0000	−1.07325
$890$	24.0000	0.804482
$891$	4.00000	0.134005
$892$	18.0000	0.602685
$893$	0	0
$894$	−6.00000	−0.200670
$895$	32.0000	1.06964
$896$	−4.00000	−0.133631
$897$	0	0
$898$	−12.0000	−0.400445
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	2.00000	0.0666297
$902$	8.00000	0.266371
$903$	−16.0000	−0.532447
$904$	0	0
$905$	−44.0000	−1.46261
$906$	−4.00000	−0.132891
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	14.0000	0.464606
$909$	6.00000	0.199007
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−1.00000	−0.0331133
$913$	8.00000	0.264761
$914$	18.0000	0.595387
$915$	0	0
$916$	−14.0000	−0.462573
$917$	32.0000	1.05673
$918$	−2.00000	−0.0660098
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	8.00000	0.263752
$921$	−4.00000	−0.131804
$922$	30.0000	0.987997
$923$	0	0
$924$	16.0000	0.526361
$925$	−8.00000	−0.263038
$926$	14.0000	0.460069
$927$	8.00000	0.262754
$928$	−2.00000	−0.0656532
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	−16.0000	−0.524097
$933$	−8.00000	−0.261908
$934$	−12.0000	−0.392652
$935$	16.0000	0.523256
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	16.0000	0.522419
$939$	−14.0000	−0.456873
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−24.0000	−0.781962
$943$	8.00000	0.260516
$944$	−10.0000	−0.325472
$945$	8.00000	0.260240
$946$	−16.0000	−0.520205
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	−1.00000	−0.0324443
$951$	2.00000	0.0648544
$952$	−8.00000	−0.259281
$953$	−48.0000	−1.55487	−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000	−1.55487	−0.777436	+	0.628962i	$0.783480\pi$
$954$	1.00000	0.0323762
$955$	−24.0000	−0.776622
$956$	−20.0000	−0.646846
$957$	8.00000	0.258603
$958$	−16.0000	−0.516937
$959$	−32.0000	−1.03333
$960$	−2.00000	−0.0645497
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−14.0000	−0.451144
$964$	10.0000	0.322078
$965$	52.0000	1.67394
$966$	16.0000	0.514792
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	5.00000	0.160706
$969$	−2.00000	−0.0642493
$970$	20.0000	0.642161
$971$	−10.0000	−0.320915	−0.160458	−	0.987043i	$-0.551297\pi$
$971$	−10.0000	−0.320915	−0.160458	+	0.987043i	$0.551297\pi$
$972$	−1.00000	−0.0320750
$973$	−32.0000	−1.02587
$974$	−34.0000	−1.08943
$975$	0	0
$976$	0	0
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	−12.0000	−0.383718
$979$	48.0000	1.53409
$980$	18.0000	0.574989
$981$	2.00000	0.0638551
$982$	−18.0000	−0.574403
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	−2.00000	−0.0637577
$985$	36.0000	1.14706
$986$	−4.00000	−0.127386
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	8.00000	0.254257
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	−32.0000	−1.01447
$996$	−2.00000	−0.0633724
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	−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.j.1.1	✓	1

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