LMFDB - Embedded newform 6422.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +6.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} +1.00000 q^{21} +6.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -5.00000 q^{27} +1.00000 q^{28} +9.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} +1.00000 q^{42} +8.00000 q^{43} +6.00000 q^{44} +3.00000 q^{46} +1.00000 q^{48} -6.00000 q^{49} -5.00000 q^{50} +3.00000 q^{51} -3.00000 q^{53} -5.00000 q^{54} +1.00000 q^{56} -1.00000 q^{57} +9.00000 q^{58} -9.00000 q^{59} -10.0000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} -5.00000 q^{67} +3.00000 q^{68} +3.00000 q^{69} +6.00000 q^{71} -2.00000 q^{72} +7.00000 q^{73} -2.00000 q^{74} -5.00000 q^{75} -1.00000 q^{76} +6.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +1.00000 q^{84} +8.00000 q^{86} +9.00000 q^{87} +6.00000 q^{88} +12.0000 q^{89} +3.00000 q^{92} +4.00000 q^{93} +1.00000 q^{96} +10.0000 q^{97} -6.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−2.00000	−0.471405
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	6.00000	1.27920
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	1.00000	0.188982
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	6.00000	1.04447
$34$	3.00000	0.514496
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−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
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	1.00000	0.154303
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	3.00000	0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	−5.00000	−0.707107
$51$	3.00000	0.420084
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	1.00000	0.133631
$57$	−1.00000	−0.132453
$58$	9.00000	1.18176
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	0.508001
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	3.00000	0.363803
$69$	3.00000	0.361158
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−2.00000	−0.235702
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	−2.00000	−0.232495
$75$	−5.00000	−0.577350
$76$	−1.00000	−0.114708
$77$	6.00000	0.683763
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	8.00000	0.862662
$87$	9.00000	0.964901
$88$	6.00000	0.639602
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	0	0
$92$	3.00000	0.312772
$93$	4.00000	0.414781
$94$	0	0
$95$	0	0
$96$	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	$0.330506\pi$
$98$	−6.00000	−0.606092
$99$	−12.0000	−1.20605
$100$	−5.00000	−0.500000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	3.00000	0.297044
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	−3.00000	−0.291386
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	−5.00000	−0.481125
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	1.00000	0.0944911
$113$	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
$115$	0	0
$116$	9.00000	0.835629
$117$	0	0
$118$	−9.00000	−0.828517
$119$	3.00000	0.275010
$120$	0	0
$121$	25.0000	2.27273
$122$	−10.0000	−0.905357
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	−2.00000	−0.178174
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	6.00000	0.522233
$133$	−1.00000	−0.0867110
$134$	−5.00000	−0.431934
$135$	0	0
$136$	3.00000	0.257248
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	3.00000	0.255377
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	6.00000	0.503509
$143$	0	0
$144$	−2.00000	−0.166667
$145$	0	0
$146$	7.00000	0.579324
$147$	−6.00000	−0.494872
$148$	−2.00000	−0.164399
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−5.00000	−0.408248
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	−1.00000	−0.0811107
$153$	−6.00000	−0.485071
$154$	6.00000	0.483494
$155$	0	0
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−10.0000	−0.795557
$159$	−3.00000	−0.237915
$160$	0	0
$161$	3.00000	0.236433
$162$	1.00000	0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	6.00000	0.465690
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	0	0
$171$	2.00000	0.152944
$172$	8.00000	0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	9.00000	0.682288
$175$	−5.00000	−0.377964
$176$	6.00000	0.452267
$177$	−9.00000	−0.676481
$178$	12.0000	0.899438
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	3.00000	0.221163
$185$	0	0
$186$	4.00000	0.293294
$187$	18.0000	1.31629
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	1.00000	0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−12.0000	−0.852803
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	−5.00000	−0.353553
$201$	−5.00000	−0.352673
$202$	18.0000	1.26648
$203$	9.00000	0.631676
$204$	3.00000	0.210042
$205$	0	0
$206$	14.0000	0.975426
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−3.00000	−0.206041
$213$	6.00000	0.411113
$214$	−9.00000	−0.615227
$215$	0	0
$216$	−5.00000	−0.340207
$217$	4.00000	0.271538
$218$	−11.0000	−0.745014
$219$	7.00000	0.473016
$220$	0	0
$221$	0	0
$222$	−2.00000	−0.134231
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	1.00000	0.0668153
$225$	10.0000	0.666667
$226$	6.00000	0.399114
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	−1.00000	−0.0662266
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	9.00000	0.590879
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	−9.00000	−0.585850
$237$	−10.0000	−0.649570
$238$	3.00000	0.194461
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	25.0000	1.60706
$243$	16.0000	1.02640
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	0	0
$248$	4.00000	0.254000
$249$	6.00000	0.380235
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	−2.00000	−0.125988
$253$	18.0000	1.13165
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	8.00000	0.498058
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−18.0000	−1.11417
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	−1.00000	−0.0613139
$267$	12.0000	0.734388
$268$	−5.00000	−0.305424
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	9.00000	0.543710
$275$	−30.0000	−1.80907
$276$	3.00000	0.180579
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−4.00000	−0.239904
$279$	−8.00000	−0.478947
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−2.00000	−0.117851
$289$	−8.00000	−0.470588
$290$	0	0
$291$	10.0000	0.586210
$292$	7.00000	0.409644
$293$	21.0000	1.22683	0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000	1.22683	0.613417	+	0.789760i	$0.289795\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	−2.00000	−0.116248
$297$	−30.0000	−1.74078
$298$	0	0
$299$	0	0
$300$	−5.00000	−0.288675
$301$	8.00000	0.461112
$302$	10.0000	0.575435
$303$	18.0000	1.03407
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−6.00000	−0.342997
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	6.00000	0.341882
$309$	14.0000	0.796432
$310$	0	0
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	−10.0000	−0.562544
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	−3.00000	−0.168232
$319$	54.0000	3.02342
$320$	0	0
$321$	−9.00000	−0.502331
$322$	3.00000	0.167183
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	−11.0000	−0.608301
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	6.00000	0.329293
$333$	4.00000	0.219199
$334$	−12.0000	−0.656611
$335$	0	0
$336$	1.00000	0.0545545
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	24.0000	1.29967
$342$	2.00000	0.108148
$343$	−13.0000	−0.701934
$344$	8.00000	0.431331
$345$	0	0
$346$	6.00000	0.322562
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	9.00000	0.482451
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	−5.00000	−0.267261
$351$	0	0
$352$	6.00000	0.319801
$353$	15.0000	0.798369	0.399185	−	0.916871i	$-0.369293\pi$
$353$	15.0000	0.798369	0.399185	+	0.916871i	$0.369293\pi$
$354$	−9.00000	−0.478345
$355$	0	0
$356$	12.0000	0.635999
$357$	3.00000	0.158777
$358$	0	0
$359$	−21.0000	−1.10834	−0.554169	−	0.832404i	$-0.686964\pi$
$359$	−21.0000	−1.10834	−0.554169	+	0.832404i	$0.686964\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.00000	0.105118
$363$	25.0000	1.31216
$364$	0	0
$365$	0	0
$366$	−10.0000	−0.522708
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	0	0
$371$	−3.00000	−0.155752
$372$	4.00000	0.207390
$373$	23.0000	1.19089	0.595447	−	0.803394i	$-0.296975\pi$
$373$	23.0000	1.19089	0.595447	+	0.803394i	$0.296975\pi$
$374$	18.0000	0.930758
$375$	0	0
$376$	0	0
$377$	0	0
$378$	−5.00000	−0.257172
$379$	7.00000	0.359566	0.179783	−	0.983706i	$-0.442460\pi$
$379$	7.00000	0.359566	0.179783	+	0.983706i	$0.442460\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	3.00000	0.153493
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−16.0000	−0.813326
$388$	10.0000	0.507673
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	−6.00000	−0.303046
$393$	0	0
$394$	0	0
$395$	0	0
$396$	−12.0000	−0.603023
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	11.0000	0.551380
$399$	−1.00000	−0.0500626
$400$	−5.00000	−0.250000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	−5.00000	−0.249377
$403$	0	0
$404$	18.0000	0.895533
$405$	0	0
$406$	9.00000	0.446663
$407$	−12.0000	−0.594818
$408$	3.00000	0.148522
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	14.0000	0.689730
$413$	−9.00000	−0.442861
$414$	−6.00000	−0.294884
$415$	0	0
$416$	0	0
$417$	−4.00000	−0.195881
$418$	−6.00000	−0.293470
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−17.0000	−0.828529	−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000	−0.828529	−0.414265	+	0.910156i	$0.635961\pi$
$422$	5.00000	0.243396
$423$	0	0
$424$	−3.00000	−0.145693
$425$	−15.0000	−0.727607
$426$	6.00000	0.290701
$427$	−10.0000	−0.483934
$428$	−9.00000	−0.435031
$429$	0	0
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	−5.00000	−0.240563
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−11.0000	−0.526804
$437$	−3.00000	−0.143509
$438$	7.00000	0.334473
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−26.0000	−1.23114
$447$	0	0
$448$	1.00000	0.0472456
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	10.0000	0.471405
$451$	0	0
$452$	6.00000	0.282216
$453$	10.0000	0.469841
$454$	15.0000	0.703985
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	22.0000	1.02799
$459$	−15.0000	−0.700140
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	6.00000	0.279145
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	−6.00000	−0.277945
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	−5.00000	−0.230879
$470$	0	0
$471$	−22.0000	−1.01371
$472$	−9.00000	−0.414259
$473$	48.0000	2.20704
$474$	−10.0000	−0.459315
$475$	5.00000	0.229416
$476$	3.00000	0.137505
$477$	6.00000	0.274721
$478$	21.0000	0.960518
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	0	0
$482$	−8.00000	−0.364390
$483$	3.00000	0.136505
$484$	25.0000	1.13636
$485$	0	0
$486$	16.0000	0.725775
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−10.0000	−0.452679
$489$	−20.0000	−0.904431
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	27.0000	1.21602
$494$	0	0
$495$	0	0
$496$	4.00000	0.179605
$497$	6.00000	0.269137
$498$	6.00000	0.268866
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	6.00000	0.267793
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	18.0000	0.800198
$507$	0	0
$508$	2.00000	0.0887357
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	1.00000	0.0441942
$513$	5.00000	0.220755
$514$	12.0000	0.529297
$515$	0	0
$516$	8.00000	0.352180
$517$	0	0
$518$	−2.00000	−0.0878750
$519$	6.00000	0.263371
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	−18.0000	−0.787839
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	0	0
$525$	−5.00000	−0.218218
$526$	24.0000	1.04645
$527$	12.0000	0.522728
$528$	6.00000	0.261116
$529$	−14.0000	−0.608696
$530$	0	0
$531$	18.0000	0.781133
$532$	−1.00000	−0.0433555
$533$	0	0
$534$	12.0000	0.519291
$535$	0	0
$536$	−5.00000	−0.215967
$537$	0	0
$538$	−6.00000	−0.258678
$539$	−36.0000	−1.55063
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−11.0000	−0.472490
$543$	2.00000	0.0858282
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	9.00000	0.384461
$549$	20.0000	0.853579
$550$	−30.0000	−1.27920
$551$	−9.00000	−0.383413
$552$	3.00000	0.127688
$553$	−10.0000	−0.425243
$554$	8.00000	0.339887
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	−8.00000	−0.338667
$559$	0	0
$560$	0	0
$561$	18.0000	0.759961
$562$	0	0
$563$	−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
$565$	0	0
$566$	−22.0000	−0.924729
$567$	1.00000	0.0419961
$568$	6.00000	0.251754
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	3.00000	0.125327
$574$	0	0
$575$	−15.0000	−0.625543
$576$	−2.00000	−0.0833333
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	−8.00000	−0.332756
$579$	−14.0000	−0.581820
$580$	0	0
$581$	6.00000	0.248922
$582$	10.0000	0.414513
$583$	−18.0000	−0.745484
$584$	7.00000	0.289662
$585$	0	0
$586$	21.0000	0.867502
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−6.00000	−0.247436
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	−30.0000	−1.23091
$595$	0	0
$596$	0	0
$597$	11.0000	0.450200
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	−5.00000	−0.204124
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	8.00000	0.326056
$603$	10.0000	0.407231
$604$	10.0000	0.406894
$605$	0	0
$606$	18.0000	0.731200
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−1.00000	−0.0405554
$609$	9.00000	0.364698
$610$	0	0
$611$	0	0
$612$	−6.00000	−0.242536
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	6.00000	0.241747
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	14.0000	0.563163
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	−21.0000	−0.842023
$623$	12.0000	0.480770
$624$	0	0
$625$	25.0000	1.00000
$626$	−19.0000	−0.759393
$627$	−6.00000	−0.239617
$628$	−22.0000	−0.877896
$629$	−6.00000	−0.239236
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−10.0000	−0.397779
$633$	5.00000	0.198732
$634$	9.00000	0.357436
$635$	0	0
$636$	−3.00000	−0.118958
$637$	0	0
$638$	54.0000	2.13788
$639$	−12.0000	−0.474713
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	−9.00000	−0.355202
$643$	22.0000	0.867595	0.433798	−	0.901010i	$-0.357173\pi$
$643$	22.0000	0.867595	0.433798	+	0.901010i	$0.357173\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	−3.00000	−0.118033
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	1.00000	0.0392837
$649$	−54.0000	−2.11969
$650$	0	0
$651$	4.00000	0.156772
$652$	−20.0000	−0.783260
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	−11.0000	−0.430134
$655$	0	0
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	−45.0000	−1.75295	−0.876476	−	0.481446i	$-0.840112\pi$
$659$	−45.0000	−1.75295	−0.876476	+	0.481446i	$0.840112\pi$
$660$	0	0
$661$	13.0000	0.505641	0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000	0.505641	0.252821	+	0.967513i	$0.418642\pi$
$662$	1.00000	0.0388661
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	4.00000	0.154997
$667$	27.0000	1.04544
$668$	−12.0000	−0.464294
$669$	−26.0000	−1.00522
$670$	0	0
$671$	−60.0000	−2.31627
$672$	1.00000	0.0385758
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	−4.00000	−0.154074
$675$	25.0000	0.962250
$676$	0	0
$677$	−33.0000	−1.26829	−0.634147	−	0.773213i	$-0.718648\pi$
$677$	−33.0000	−1.26829	−0.634147	+	0.773213i	$0.718648\pi$
$678$	6.00000	0.230429
$679$	10.0000	0.383765
$680$	0	0
$681$	15.0000	0.574801
$682$	24.0000	0.919007
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	−13.0000	−0.496342
$687$	22.0000	0.839352
$688$	8.00000	0.304997
$689$	0	0
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	6.00000	0.228086
$693$	−12.0000	−0.455842
$694$	18.0000	0.683271
$695$	0	0
$696$	9.00000	0.341144
$697$	0	0
$698$	10.0000	0.378506
$699$	−6.00000	−0.226941
$700$	−5.00000	−0.188982
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	6.00000	0.226134
$705$	0	0
$706$	15.0000	0.564532
$707$	18.0000	0.676960
$708$	−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	$0.439871\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	12.0000	0.449719
$713$	12.0000	0.449404
$714$	3.00000	0.112272
$715$	0	0
$716$	0	0
$717$	21.0000	0.784259
$718$	−21.0000	−0.783713
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	1.00000	0.0372161
$723$	−8.00000	−0.297523
$724$	2.00000	0.0743294
$725$	−45.0000	−1.67126
$726$	25.0000	0.927837
$727$	−37.0000	−1.37225	−0.686127	−	0.727482i	$-0.740691\pi$
$727$	−37.0000	−1.37225	−0.686127	+	0.727482i	$0.740691\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	24.0000	0.887672
$732$	−10.0000	−0.369611
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	3.00000	0.110581
$737$	−30.0000	−1.10506
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	0	0
$742$	−3.00000	−0.110133
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	23.0000	0.842090
$747$	−12.0000	−0.439057
$748$	18.0000	0.658145
$749$	−9.00000	−0.328853
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	0	0
$755$	0	0
$756$	−5.00000	−0.181848
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	7.00000	0.254251
$759$	18.0000	0.653359
$760$	0	0
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	2.00000	0.0724524
$763$	−11.0000	−0.398227
$764$	3.00000	0.108536
$765$	0	0
$766$	−18.0000	−0.650366
$767$	0	0
$768$	1.00000	0.0360844
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	−14.0000	−0.503871
$773$	−51.0000	−1.83434	−0.917171	−	0.398493i	$-0.869533\pi$
$773$	−51.0000	−1.83434	−0.917171	+	0.398493i	$0.869533\pi$
$774$	−16.0000	−0.575108
$775$	−20.0000	−0.718421
$776$	10.0000	0.358979
$777$	−2.00000	−0.0717496
$778$	18.0000	0.645331
$779$	0	0
$780$	0	0
$781$	36.0000	1.28818
$782$	9.00000	0.321839
$783$	−45.0000	−1.60817
$784$	−6.00000	−0.214286
$785$	0	0
$786$	0	0
$787$	31.0000	1.10503	0.552515	−	0.833503i	$-0.313668\pi$
$787$	31.0000	1.10503	0.552515	+	0.833503i	$0.313668\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	6.00000	0.213335
$792$	−12.0000	−0.426401
$793$	0	0
$794$	−20.0000	−0.709773
$795$	0	0
$796$	11.0000	0.389885
$797$	−39.0000	−1.38145	−0.690725	−	0.723117i	$-0.742709\pi$
$797$	−39.0000	−1.38145	−0.690725	+	0.723117i	$0.742709\pi$
$798$	−1.00000	−0.0353996
$799$	0	0
$800$	−5.00000	−0.176777
$801$	−24.0000	−0.847998
$802$	0	0
$803$	42.0000	1.48215
$804$	−5.00000	−0.176336
$805$	0	0
$806$	0	0
$807$	−6.00000	−0.211210
$808$	18.0000	0.633238
$809$	9.00000	0.316423	0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000	0.316423	0.158212	+	0.987405i	$0.449427\pi$
$810$	0	0
$811$	−11.0000	−0.386262	−0.193131	−	0.981173i	$-0.561864\pi$
$811$	−11.0000	−0.386262	−0.193131	+	0.981173i	$0.561864\pi$
$812$	9.00000	0.315838
$813$	−11.0000	−0.385787
$814$	−12.0000	−0.420600
$815$	0	0
$816$	3.00000	0.105021
$817$	−8.00000	−0.279885
$818$	−32.0000	−1.11885
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	9.00000	0.313911
$823$	41.0000	1.42917	0.714585	−	0.699549i	$-0.246616\pi$
$823$	41.0000	1.42917	0.714585	+	0.699549i	$0.246616\pi$
$824$	14.0000	0.487713
$825$	−30.0000	−1.04447
$826$	−9.00000	−0.313150
$827$	−33.0000	−1.14752	−0.573761	−	0.819023i	$-0.694516\pi$
$827$	−33.0000	−1.14752	−0.573761	+	0.819023i	$0.694516\pi$
$828$	−6.00000	−0.208514
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	0	0
$833$	−18.0000	−0.623663
$834$	−4.00000	−0.138509
$835$	0	0
$836$	−6.00000	−0.207514
$837$	−20.0000	−0.691301
$838$	−12.0000	−0.414533
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−17.0000	−0.585859
$843$	0	0
$844$	5.00000	0.172107
$845$	0	0
$846$	0	0
$847$	25.0000	0.859010
$848$	−3.00000	−0.103020
$849$	−22.0000	−0.755038
$850$	−15.0000	−0.514496
$851$	−6.00000	−0.205677
$852$	6.00000	0.205557
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	−9.00000	−0.307614
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	0	0
$862$	−6.00000	−0.204361
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	2.00000	0.0679628
$867$	−8.00000	−0.271694
$868$	4.00000	0.135769
$869$	−60.0000	−2.03536
$870$	0	0
$871$	0	0
$872$	−11.0000	−0.372507
$873$	−20.0000	−0.676897
$874$	−3.00000	−0.101477
$875$	0	0
$876$	7.00000	0.236508
$877$	−23.0000	−0.776655	−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000	−0.776655	−0.388327	+	0.921521i	$0.626947\pi$
$878$	−28.0000	−0.944954
$879$	21.0000	0.708312
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	12.0000	0.404061
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	0	0
$885$	0	0
$886$	−18.0000	−0.604722
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	−2.00000	−0.0671156
$889$	2.00000	0.0670778
$890$	0	0
$891$	6.00000	0.201008
$892$	−26.0000	−0.870544
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	18.0000	0.600668
$899$	36.0000	1.20067
$900$	10.0000	0.333333
$901$	−9.00000	−0.299833
$902$	0	0
$903$	8.00000	0.266223
$904$	6.00000	0.199557
$905$	0	0
$906$	10.0000	0.332228
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	15.0000	0.497792
$909$	−36.0000	−1.19404
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−1.00000	−0.0331133
$913$	36.0000	1.19143
$914$	−17.0000	−0.562310
$915$	0	0
$916$	22.0000	0.726900
$917$	0	0
$918$	−15.0000	−0.495074
$919$	−7.00000	−0.230909	−0.115454	−	0.993313i	$-0.536832\pi$
$919$	−7.00000	−0.230909	−0.115454	+	0.993313i	$0.536832\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	12.0000	0.395199
$923$	0	0
$924$	6.00000	0.197386
$925$	10.0000	0.328798
$926$	4.00000	0.131448
$927$	−28.0000	−0.919641
$928$	9.00000	0.295439
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	−6.00000	−0.196537
$933$	−21.0000	−0.687509
$934$	18.0000	0.588978
$935$	0	0
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	−5.00000	−0.163256
$939$	−19.0000	−0.620042
$940$	0	0
$941$	−21.0000	−0.684580	−0.342290	−	0.939594i	$-0.611203\pi$
$941$	−21.0000	−0.684580	−0.342290	+	0.939594i	$0.611203\pi$
$942$	−22.0000	−0.716799
$943$	0	0
$944$	−9.00000	−0.292925
$945$	0	0
$946$	48.0000	1.56061
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	−10.0000	−0.324785
$949$	0	0
$950$	5.00000	0.162221
$951$	9.00000	0.291845
$952$	3.00000	0.0972306
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	21.0000	0.679189
$957$	54.0000	1.74557
$958$	−36.0000	−1.16311
$959$	9.00000	0.290625
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	18.0000	0.580042
$964$	−8.00000	−0.257663
$965$	0	0
$966$	3.00000	0.0965234
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	25.0000	0.803530
$969$	−3.00000	−0.0963739
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	16.0000	0.513200
$973$	−4.00000	−0.128234
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	−20.0000	−0.639529
$979$	72.0000	2.30113
$980$	0	0
$981$	22.0000	0.702406
$982$	−36.0000	−1.14881
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	0	0
$985$	0	0
$986$	27.0000	0.859855
$987$	0	0
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	4.00000	0.127000
$993$	1.00000	0.0317340
$994$	6.00000	0.190308
$995$	0	0
$996$	6.00000	0.190117
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	4.00000	0.126618
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.h.1.1		1
13.12	even	2		38.2.a.a.1.1	✓	1
39.38	odd	2		342.2.a.e.1.1		1
52.51	odd	2		304.2.a.c.1.1		1
65.12	odd	4		950.2.b.b.799.1		2
65.38	odd	4		950.2.b.b.799.2		2
65.64	even	2		950.2.a.d.1.1		1
91.90	odd	2		1862.2.a.b.1.1		1
104.51	odd	2		1216.2.a.m.1.1		1
104.77	even	2		1216.2.a.e.1.1		1
143.142	odd	2		4598.2.a.p.1.1		1
156.155	even	2		2736.2.a.n.1.1		1
195.194	odd	2		8550.2.a.m.1.1		1
247.12	odd	6		722.2.c.c.429.1		2
247.25	even	18		722.2.e.f.245.1		6
247.51	odd	18		722.2.e.e.245.1		6
247.64	even	6		722.2.c.e.429.1		2
247.90	odd	18		722.2.e.e.595.1		6
247.103	odd	6		722.2.c.c.653.1		2
247.116	odd	18		722.2.e.e.99.1		6
247.129	odd	18		722.2.e.e.415.1		6
247.142	even	18		722.2.e.f.423.1		6
247.155	odd	18		722.2.e.e.389.1		6
247.168	even	18		722.2.e.f.389.1		6
247.181	odd	18		722.2.e.e.423.1		6
247.194	even	18		722.2.e.f.415.1		6
247.207	even	18		722.2.e.f.99.1		6
247.220	even	6		722.2.c.e.653.1		2
247.233	even	18		722.2.e.f.595.1		6
247.246	odd	2		722.2.a.e.1.1		1
260.259	odd	2		7600.2.a.n.1.1		1
741.740	even	2		6498.2.a.f.1.1		1
988.987	even	2		5776.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.a.a.1.1	✓	1	13.12	even	2
304.2.a.c.1.1		1	52.51	odd	2
342.2.a.e.1.1		1	39.38	odd	2
722.2.a.e.1.1		1	247.246	odd	2
722.2.c.c.429.1		2	247.12	odd	6
722.2.c.c.653.1		2	247.103	odd	6
722.2.c.e.429.1		2	247.64	even	6
722.2.c.e.653.1		2	247.220	even	6
722.2.e.e.99.1		6	247.116	odd	18
722.2.e.e.245.1		6	247.51	odd	18
722.2.e.e.389.1		6	247.155	odd	18
722.2.e.e.415.1		6	247.129	odd	18
722.2.e.e.423.1		6	247.181	odd	18
722.2.e.e.595.1		6	247.90	odd	18
722.2.e.f.99.1		6	247.207	even	18
722.2.e.f.245.1		6	247.25	even	18
722.2.e.f.389.1		6	247.168	even	18
722.2.e.f.415.1		6	247.194	even	18
722.2.e.f.423.1		6	247.142	even	18
722.2.e.f.595.1		6	247.233	even	18
950.2.a.d.1.1		1	65.64	even	2
950.2.b.b.799.1		2	65.12	odd	4
950.2.b.b.799.2		2	65.38	odd	4
1216.2.a.e.1.1		1	104.77	even	2
1216.2.a.m.1.1		1	104.51	odd	2
1862.2.a.b.1.1		1	91.90	odd	2
2736.2.a.n.1.1		1	156.155	even	2
4598.2.a.p.1.1		1	143.142	odd	2
5776.2.a.m.1.1		1	988.987	even	2
6422.2.a.h.1.1		1	1.1	even	1	trivial
6498.2.a.f.1.1		1	741.740	even	2
7600.2.a.n.1.1		1	260.259	odd	2
8550.2.a.m.1.1		1	195.194	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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