LMFDB - Embedded newform 8034.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} -4.00000 q^{20} -1.00000 q^{21} +3.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -4.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +3.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -12.0000 q^{37} +8.00000 q^{38} -1.00000 q^{39} +4.00000 q^{40} +4.00000 q^{41} +1.00000 q^{42} -3.00000 q^{44} -4.00000 q^{45} +6.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -11.0000 q^{50} +3.00000 q^{51} +1.00000 q^{52} -9.00000 q^{53} +1.00000 q^{54} +12.0000 q^{55} -1.00000 q^{56} +8.00000 q^{57} -2.00000 q^{59} +4.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -3.00000 q^{66} -3.00000 q^{67} -3.00000 q^{68} +6.00000 q^{69} +4.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} -1.00000 q^{73} +12.0000 q^{74} -11.0000 q^{75} -8.00000 q^{76} -3.00000 q^{77} +1.00000 q^{78} -6.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} -6.00000 q^{83} -1.00000 q^{84} +12.0000 q^{85} +3.00000 q^{88} -10.0000 q^{89} +4.00000 q^{90} +1.00000 q^{91} -6.00000 q^{92} -4.00000 q^{93} +4.00000 q^{94} +32.0000 q^{95} +1.00000 q^{96} -8.00000 q^{97} +6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	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
$9$	1.00000	0.333333
$10$	4.00000	1.26491
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−1.00000	−0.235702
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−4.00000	−0.894427
$21$	−1.00000	−0.218218
$22$	3.00000	0.639602
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	11.0000	2.20000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−4.00000	−0.730297
$31$	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
$33$	3.00000	0.522233
$34$	3.00000	0.514496
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	−12.0000	−1.97279	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−12.0000	−1.97279	−0.986394	+	0.164399i	$0.947432\pi$
$38$	8.00000	1.29777
$39$	−1.00000	−0.160128
$40$	4.00000	0.632456
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	1.00000	0.154303
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−3.00000	−0.452267
$45$	−4.00000	−0.596285
$46$	6.00000	0.884652
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	−11.0000	−1.55563
$51$	3.00000	0.420084
$52$	1.00000	0.138675
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	1.00000	0.136083
$55$	12.0000	1.61808
$56$	−1.00000	−0.133631
$57$	8.00000	1.05963
$58$	0	0
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	4.00000	0.516398
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−3.00000	−0.369274
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−3.00000	−0.363803
$69$	6.00000	0.722315
$70$	4.00000	0.478091
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.00000	−0.117851
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	12.0000	1.39497
$75$	−11.0000	−1.27017
$76$	−8.00000	−0.917663
$77$	−3.00000	−0.341882
$78$	1.00000	0.113228
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−1.00000	−0.109109
$85$	12.0000	1.30158
$86$	0	0
$87$	0	0
$88$	3.00000	0.319801
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	4.00000	0.421637
$91$	1.00000	0.104828
$92$	−6.00000	−0.625543
$93$	−4.00000	−0.414781
$94$	4.00000	0.412568
$95$	32.0000	3.28313
$96$	1.00000	0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	6.00000	0.606092
$99$	−3.00000	−0.301511
$100$	11.0000	1.10000
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	−3.00000	−0.297044
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−1.00000	−0.0980581
$105$	4.00000	0.390360
$106$	9.00000	0.874157
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	−1.00000	−0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−12.0000	−1.14416
$111$	12.0000	1.13899
$112$	1.00000	0.0944911
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	−8.00000	−0.749269
$115$	24.0000	2.23801
$116$	0	0
$117$	1.00000	0.0924500
$118$	2.00000	0.184115
$119$	−3.00000	−0.275010
$120$	−4.00000	−0.365148
$121$	−2.00000	−0.181818
$122$	−6.00000	−0.543214
$123$	−4.00000	−0.360668
$124$	4.00000	0.359211
$125$	−24.0000	−2.14663
$126$	−1.00000	−0.0890871
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	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	$0.555909\pi$
$132$	3.00000	0.261116
$133$	−8.00000	−0.693688
$134$	3.00000	0.259161
$135$	4.00000	0.344265
$136$	3.00000	0.257248
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−6.00000	−0.510754
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	−4.00000	−0.338062
$141$	4.00000	0.336861
$142$	6.00000	0.503509
$143$	−3.00000	−0.250873
$144$	1.00000	0.0833333
$145$	0	0
$146$	1.00000	0.0827606
$147$	6.00000	0.494872
$148$	−12.0000	−0.986394
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	11.0000	0.898146
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	8.00000	0.648886
$153$	−3.00000	−0.242536
$154$	3.00000	0.241747
$155$	−16.0000	−1.28515
$156$	−1.00000	−0.0800641
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	6.00000	0.477334
$159$	9.00000	0.713746
$160$	4.00000	0.316228
$161$	−6.00000	−0.472866
$162$	−1.00000	−0.0785674
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	4.00000	0.312348
$165$	−12.0000	−0.934199
$166$	6.00000	0.465690
$167$	5.00000	0.386912	0.193456	−	0.981109i	$-0.438030\pi$
$167$	5.00000	0.386912	0.193456	+	0.981109i	$0.438030\pi$
$168$	1.00000	0.0771517
$169$	1.00000	0.0769231
$170$	−12.0000	−0.920358
$171$	−8.00000	−0.611775
$172$	0	0
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	0	0
$175$	11.0000	0.831522
$176$	−3.00000	−0.226134
$177$	2.00000	0.150329
$178$	10.0000	0.749532
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	−4.00000	−0.298142
$181$	−17.0000	−1.26360	−0.631800	−	0.775131i	$-0.717684\pi$
$181$	−17.0000	−1.26360	−0.631800	+	0.775131i	$0.717684\pi$
$182$	−1.00000	−0.0741249
$183$	−6.00000	−0.443533
$184$	6.00000	0.442326
$185$	48.0000	3.52903
$186$	4.00000	0.293294
$187$	9.00000	0.658145
$188$	−4.00000	−0.291730
$189$	−1.00000	−0.0727393
$190$	−32.0000	−2.32152
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−1.00000	−0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	8.00000	0.574367
$195$	4.00000	0.286446
$196$	−6.00000	−0.428571
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	3.00000	0.213201
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	−11.0000	−0.777817
$201$	3.00000	0.211604
$202$	5.00000	0.351799
$203$	0	0
$204$	3.00000	0.210042
$205$	−16.0000	−1.11749
$206$	1.00000	0.0696733
$207$	−6.00000	−0.417029
$208$	1.00000	0.0693375
$209$	24.0000	1.66011
$210$	−4.00000	−0.276026
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−9.00000	−0.618123
$213$	6.00000	0.411113
$214$	−9.00000	−0.615227
$215$	0	0
$216$	1.00000	0.0680414
$217$	4.00000	0.271538
$218$	2.00000	0.135457
$219$	1.00000	0.0675737
$220$	12.0000	0.809040
$221$	−3.00000	−0.201802
$222$	−12.0000	−0.805387
$223$	−13.0000	−0.870544	−0.435272	−	0.900299i	$-0.643348\pi$
$223$	−13.0000	−0.870544	−0.435272	+	0.900299i	$0.643348\pi$
$224$	−1.00000	−0.0668153
$225$	11.0000	0.733333
$226$	16.0000	1.06430
$227$	−7.00000	−0.464606	−0.232303	−	0.972643i	$-0.574626\pi$
$227$	−7.00000	−0.464606	−0.232303	+	0.972643i	$0.574626\pi$
$228$	8.00000	0.529813
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−24.0000	−1.58251
$231$	3.00000	0.197386
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−1.00000	−0.0653720
$235$	16.0000	1.04372
$236$	−2.00000	−0.130189
$237$	6.00000	0.389742
$238$	3.00000	0.194461
$239$	−21.0000	−1.35838	−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000	−1.35838	−0.679189	+	0.733964i	$0.737668\pi$
$240$	4.00000	0.258199
$241$	−9.00000	−0.579741	−0.289870	−	0.957066i	$-0.593612\pi$
$241$	−9.00000	−0.579741	−0.289870	+	0.957066i	$0.593612\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	24.0000	1.53330
$246$	4.00000	0.255031
$247$	−8.00000	−0.509028
$248$	−4.00000	−0.254000
$249$	6.00000	0.380235
$250$	24.0000	1.51789
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	1.00000	0.0629941
$253$	18.0000	1.13165
$254$	−13.0000	−0.815693
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	−4.00000	−0.248069
$261$	0	0
$262$	4.00000	0.247121
$263$	−11.0000	−0.678289	−0.339145	−	0.940734i	$-0.610138\pi$
$263$	−11.0000	−0.678289	−0.339145	+	0.940734i	$0.610138\pi$
$264$	−3.00000	−0.184637
$265$	36.0000	2.21146
$266$	8.00000	0.490511
$267$	10.0000	0.611990
$268$	−3.00000	−0.183254
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−4.00000	−0.243432
$271$	−26.0000	−1.57939	−0.789694	−	0.613501i	$-0.789761\pi$
$271$	−26.0000	−1.57939	−0.789694	+	0.613501i	$0.789761\pi$
$272$	−3.00000	−0.181902
$273$	−1.00000	−0.0605228
$274$	−2.00000	−0.120824
$275$	−33.0000	−1.98997
$276$	6.00000	0.361158
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−5.00000	−0.299880
$279$	4.00000	0.239474
$280$	4.00000	0.239046
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	−4.00000	−0.238197
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	−6.00000	−0.356034
$285$	−32.0000	−1.89552
$286$	3.00000	0.177394
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	8.00000	0.468968
$292$	−1.00000	−0.0585206
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	−6.00000	−0.349927
$295$	8.00000	0.465778
$296$	12.0000	0.697486
$297$	3.00000	0.174078
$298$	3.00000	0.173785
$299$	−6.00000	−0.346989
$300$	−11.0000	−0.635085
$301$	0	0
$302$	4.00000	0.230174
$303$	5.00000	0.287242
$304$	−8.00000	−0.458831
$305$	−24.0000	−1.37424
$306$	3.00000	0.171499
$307$	9.00000	0.513657	0.256829	−	0.966457i	$-0.417322\pi$
$307$	9.00000	0.513657	0.256829	+	0.966457i	$0.417322\pi$
$308$	−3.00000	−0.170941
$309$	1.00000	0.0568880
$310$	16.0000	0.908739
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	1.00000	0.0566139
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	5.00000	0.282166
$315$	−4.00000	−0.225374
$316$	−6.00000	−0.337526
$317$	15.0000	0.842484	0.421242	−	0.906948i	$-0.361594\pi$
$317$	15.0000	0.842484	0.421242	+	0.906948i	$0.361594\pi$
$318$	−9.00000	−0.504695
$319$	0	0
$320$	−4.00000	−0.223607
$321$	−9.00000	−0.502331
$322$	6.00000	0.334367
$323$	24.0000	1.33540
$324$	1.00000	0.0555556
$325$	11.0000	0.610170
$326$	10.0000	0.553849
$327$	2.00000	0.110600
$328$	−4.00000	−0.220863
$329$	−4.00000	−0.220527
$330$	12.0000	0.660578
$331$	5.00000	0.274825	0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000	0.274825	0.137412	+	0.990514i	$0.456121\pi$
$332$	−6.00000	−0.329293
$333$	−12.0000	−0.657596
$334$	−5.00000	−0.273588
$335$	12.0000	0.655630
$336$	−1.00000	−0.0545545
$337$	15.0000	0.817102	0.408551	−	0.912735i	$-0.366034\pi$
$337$	15.0000	0.817102	0.408551	+	0.912735i	$0.366034\pi$
$338$	−1.00000	−0.0543928
$339$	16.0000	0.869001
$340$	12.0000	0.650791
$341$	−12.0000	−0.649836
$342$	8.00000	0.432590
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−24.0000	−1.29212
$346$	13.0000	0.698884
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	−11.0000	−0.587975
$351$	−1.00000	−0.0533761
$352$	3.00000	0.159901
$353$	37.0000	1.96931	0.984656	−	0.174509i	$-0.0558337\pi$
$353$	37.0000	1.96931	0.984656	+	0.174509i	$0.0558337\pi$
$354$	−2.00000	−0.106299
$355$	24.0000	1.27379
$356$	−10.0000	−0.529999
$357$	3.00000	0.158777
$358$	5.00000	0.264258
$359$	17.0000	0.897226	0.448613	−	0.893726i	$-0.351918\pi$
$359$	17.0000	0.897226	0.448613	+	0.893726i	$0.351918\pi$
$360$	4.00000	0.210819
$361$	45.0000	2.36842
$362$	17.0000	0.893500
$363$	2.00000	0.104973
$364$	1.00000	0.0524142
$365$	4.00000	0.209370
$366$	6.00000	0.313625
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	−6.00000	−0.312772
$369$	4.00000	0.208232
$370$	−48.0000	−2.49540
$371$	−9.00000	−0.467257
$372$	−4.00000	−0.207390
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−9.00000	−0.465379
$375$	24.0000	1.23935
$376$	4.00000	0.206284
$377$	0	0
$378$	1.00000	0.0514344
$379$	−23.0000	−1.18143	−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000	−1.18143	−0.590715	+	0.806880i	$0.701154\pi$
$380$	32.0000	1.64157
$381$	−13.0000	−0.666010
$382$	3.00000	0.153493
$383$	22.0000	1.12415	0.562074	−	0.827087i	$-0.310004\pi$
$383$	22.0000	1.12415	0.562074	+	0.827087i	$0.310004\pi$
$384$	1.00000	0.0510310
$385$	12.0000	0.611577
$386$	18.0000	0.916176
$387$	0	0
$388$	−8.00000	−0.406138
$389$	39.0000	1.97738	0.988689	−	0.149979i	$-0.0479205\pi$
$389$	39.0000	1.97738	0.988689	+	0.149979i	$0.0479205\pi$
$390$	−4.00000	−0.202548
$391$	18.0000	0.910299
$392$	6.00000	0.303046
$393$	4.00000	0.201773
$394$	18.0000	0.906827
$395$	24.0000	1.20757
$396$	−3.00000	−0.150756
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	−17.0000	−0.852133
$399$	8.00000	0.400501
$400$	11.0000	0.550000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	−3.00000	−0.149626
$403$	4.00000	0.199254
$404$	−5.00000	−0.248759
$405$	−4.00000	−0.198762
$406$	0	0
$407$	36.0000	1.78445
$408$	−3.00000	−0.148522
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	16.0000	0.790184
$411$	−2.00000	−0.0986527
$412$	−1.00000	−0.0492665
$413$	−2.00000	−0.0984136
$414$	6.00000	0.294884
$415$	24.0000	1.17811
$416$	−1.00000	−0.0490290
$417$	−5.00000	−0.244851
$418$	−24.0000	−1.17388
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	4.00000	0.195180
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	4.00000	0.194717
$423$	−4.00000	−0.194487
$424$	9.00000	0.437079
$425$	−33.0000	−1.60074
$426$	−6.00000	−0.290701
$427$	6.00000	0.290360
$428$	9.00000	0.435031
$429$	3.00000	0.144841
$430$	0	0
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	−1.00000	−0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	48.0000	2.29615
$438$	−1.00000	−0.0477818
$439$	3.00000	0.143182	0.0715911	−	0.997434i	$-0.477192\pi$
$439$	3.00000	0.143182	0.0715911	+	0.997434i	$0.477192\pi$
$440$	−12.0000	−0.572078
$441$	−6.00000	−0.285714
$442$	3.00000	0.142695
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	12.0000	0.569495
$445$	40.0000	1.89618
$446$	13.0000	0.615568
$447$	3.00000	0.141895
$448$	1.00000	0.0472456
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	−11.0000	−0.518545
$451$	−12.0000	−0.565058
$452$	−16.0000	−0.752577
$453$	4.00000	0.187936
$454$	7.00000	0.328526
$455$	−4.00000	−0.187523
$456$	−8.00000	−0.374634
$457$	−30.0000	−1.40334	−0.701670	−	0.712502i	$-0.747562\pi$
$457$	−30.0000	−1.40334	−0.701670	+	0.712502i	$0.747562\pi$
$458$	−14.0000	−0.654177
$459$	3.00000	0.140028
$460$	24.0000	1.11901
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	−3.00000	−0.139573
$463$	30.0000	1.39422	0.697109	−	0.716965i	$-0.254469\pi$
$463$	30.0000	1.39422	0.697109	+	0.716965i	$0.254469\pi$
$464$	0	0
$465$	16.0000	0.741982
$466$	6.00000	0.277945
$467$	5.00000	0.231372	0.115686	−	0.993286i	$-0.463093\pi$
$467$	5.00000	0.231372	0.115686	+	0.993286i	$0.463093\pi$
$468$	1.00000	0.0462250
$469$	−3.00000	−0.138527
$470$	−16.0000	−0.738025
$471$	5.00000	0.230388
$472$	2.00000	0.0920575
$473$	0	0
$474$	−6.00000	−0.275589
$475$	−88.0000	−4.03772
$476$	−3.00000	−0.137505
$477$	−9.00000	−0.412082
$478$	21.0000	0.960518
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	−4.00000	−0.182574
$481$	−12.0000	−0.547153
$482$	9.00000	0.409939
$483$	6.00000	0.273009
$484$	−2.00000	−0.0909091
$485$	32.0000	1.45305
$486$	1.00000	0.0453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−6.00000	−0.271607
$489$	10.0000	0.452216
$490$	−24.0000	−1.08421
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	−4.00000	−0.180334
$493$	0	0
$494$	8.00000	0.359937
$495$	12.0000	0.539360
$496$	4.00000	0.179605
$497$	−6.00000	−0.269137
$498$	−6.00000	−0.268866
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−24.0000	−1.07331
$501$	−5.00000	−0.223384
$502$	24.0000	1.07117
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	−1.00000	−0.0445435
$505$	20.0000	0.889988
$506$	−18.0000	−0.800198
$507$	−1.00000	−0.0444116
$508$	13.0000	0.576782
$509$	−17.0000	−0.753512	−0.376756	−	0.926313i	$-0.622960\pi$
$509$	−17.0000	−0.753512	−0.376756	+	0.926313i	$0.622960\pi$
$510$	12.0000	0.531369
$511$	−1.00000	−0.0442374
$512$	−1.00000	−0.0441942
$513$	8.00000	0.353209
$514$	−20.0000	−0.882162
$515$	4.00000	0.176261
$516$	0	0
$517$	12.0000	0.527759
$518$	12.0000	0.527250
$519$	13.0000	0.570637
$520$	4.00000	0.175412
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	0	0
$523$	−39.0000	−1.70535	−0.852675	−	0.522441i	$-0.825022\pi$
$523$	−39.0000	−1.70535	−0.852675	+	0.522441i	$0.825022\pi$
$524$	−4.00000	−0.174741
$525$	−11.0000	−0.480079
$526$	11.0000	0.479623
$527$	−12.0000	−0.522728
$528$	3.00000	0.130558
$529$	13.0000	0.565217
$530$	−36.0000	−1.56374
$531$	−2.00000	−0.0867926
$532$	−8.00000	−0.346844
$533$	4.00000	0.173259
$534$	−10.0000	−0.432742
$535$	−36.0000	−1.55642
$536$	3.00000	0.129580
$537$	5.00000	0.215766
$538$	0	0
$539$	18.0000	0.775315
$540$	4.00000	0.172133
$541$	31.0000	1.33279	0.666397	−	0.745597i	$-0.267836\pi$
$541$	31.0000	1.33279	0.666397	+	0.745597i	$0.267836\pi$
$542$	26.0000	1.11680
$543$	17.0000	0.729540
$544$	3.00000	0.128624
$545$	8.00000	0.342682
$546$	1.00000	0.0427960
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	2.00000	0.0854358
$549$	6.00000	0.256074
$550$	33.0000	1.40712
$551$	0	0
$552$	−6.00000	−0.255377
$553$	−6.00000	−0.255146
$554$	−18.0000	−0.764747
$555$	−48.0000	−2.03749
$556$	5.00000	0.212047
$557$	−40.0000	−1.69485	−0.847427	−	0.530912i	$-0.821850\pi$
$557$	−40.0000	−1.69485	−0.847427	+	0.530912i	$0.821850\pi$
$558$	−4.00000	−0.169334
$559$	0	0
$560$	−4.00000	−0.169031
$561$	−9.00000	−0.379980
$562$	−26.0000	−1.09674
$563$	−2.00000	−0.0842900	−0.0421450	−	0.999112i	$-0.513419\pi$
$563$	−2.00000	−0.0842900	−0.0421450	+	0.999112i	$0.513419\pi$
$564$	4.00000	0.168430
$565$	64.0000	2.69250
$566$	−8.00000	−0.336265
$567$	1.00000	0.0419961
$568$	6.00000	0.251754
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	32.0000	1.34033
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	−3.00000	−0.125436
$573$	3.00000	0.125327
$574$	−4.00000	−0.166957
$575$	−66.0000	−2.75239
$576$	1.00000	0.0416667
$577$	−33.0000	−1.37381	−0.686904	−	0.726748i	$-0.741031\pi$
$577$	−33.0000	−1.37381	−0.686904	+	0.726748i	$0.741031\pi$
$578$	8.00000	0.332756
$579$	18.0000	0.748054
$580$	0	0
$581$	−6.00000	−0.248922
$582$	−8.00000	−0.331611
$583$	27.0000	1.11823
$584$	1.00000	0.0413803
$585$	−4.00000	−0.165380
$586$	16.0000	0.660954
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	6.00000	0.247436
$589$	−32.0000	−1.31854
$590$	−8.00000	−0.329355
$591$	18.0000	0.740421
$592$	−12.0000	−0.493197
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	−3.00000	−0.123091
$595$	12.0000	0.491952
$596$	−3.00000	−0.122885
$597$	−17.0000	−0.695764
$598$	6.00000	0.245358
$599$	39.0000	1.59350	0.796748	−	0.604311i	$-0.206552\pi$
$599$	39.0000	1.59350	0.796748	+	0.604311i	$0.206552\pi$
$600$	11.0000	0.449073
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	0	0
$603$	−3.00000	−0.122169
$604$	−4.00000	−0.162758
$605$	8.00000	0.325246
$606$	−5.00000	−0.203111
$607$	46.0000	1.86708	0.933541	−	0.358470i	$-0.116702\pi$
$607$	46.0000	1.86708	0.933541	+	0.358470i	$0.116702\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	24.0000	0.971732
$611$	−4.00000	−0.161823
$612$	−3.00000	−0.121268
$613$	−25.0000	−1.00974	−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000	−1.00974	−0.504870	+	0.863195i	$0.668460\pi$
$614$	−9.00000	−0.363210
$615$	16.0000	0.645182
$616$	3.00000	0.120873
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−1.00000	−0.0402259
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−16.0000	−0.642575
$621$	6.00000	0.240772
$622$	0	0
$623$	−10.0000	−0.400642
$624$	−1.00000	−0.0400320
$625$	41.0000	1.64000
$626$	6.00000	0.239808
$627$	−24.0000	−0.958468
$628$	−5.00000	−0.199522
$629$	36.0000	1.43541
$630$	4.00000	0.159364
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	6.00000	0.238667
$633$	4.00000	0.158986
$634$	−15.0000	−0.595726
$635$	−52.0000	−2.06356
$636$	9.00000	0.356873
$637$	−6.00000	−0.237729
$638$	0	0
$639$	−6.00000	−0.237356
$640$	4.00000	0.158114
$641$	41.0000	1.61940	0.809701	−	0.586842i	$-0.199629\pi$
$641$	41.0000	1.61940	0.809701	+	0.586842i	$0.199629\pi$
$642$	9.00000	0.355202
$643$	−24.0000	−0.946468	−0.473234	−	0.880937i	$-0.656913\pi$
$643$	−24.0000	−0.946468	−0.473234	+	0.880937i	$0.656913\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−20.0000	−0.786281	−0.393141	−	0.919478i	$-0.628611\pi$
$647$	−20.0000	−0.786281	−0.393141	+	0.919478i	$0.628611\pi$
$648$	−1.00000	−0.0392837
$649$	6.00000	0.235521
$650$	−11.0000	−0.431455
$651$	−4.00000	−0.156772
$652$	−10.0000	−0.391630
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−2.00000	−0.0782062
$655$	16.0000	0.625172
$656$	4.00000	0.156174
$657$	−1.00000	−0.0390137
$658$	4.00000	0.155936
$659$	−23.0000	−0.895953	−0.447976	−	0.894045i	$-0.647855\pi$
$659$	−23.0000	−0.895953	−0.447976	+	0.894045i	$0.647855\pi$
$660$	−12.0000	−0.467099
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−5.00000	−0.194331
$663$	3.00000	0.116510
$664$	6.00000	0.232845
$665$	32.0000	1.24091
$666$	12.0000	0.464991
$667$	0	0
$668$	5.00000	0.193456
$669$	13.0000	0.502609
$670$	−12.0000	−0.463600
$671$	−18.0000	−0.694882
$672$	1.00000	0.0385758
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	−15.0000	−0.577778
$675$	−11.0000	−0.423390
$676$	1.00000	0.0384615
$677$	−32.0000	−1.22986	−0.614930	−	0.788582i	$-0.710816\pi$
$677$	−32.0000	−1.22986	−0.614930	+	0.788582i	$0.710816\pi$
$678$	−16.0000	−0.614476
$679$	−8.00000	−0.307012
$680$	−12.0000	−0.460179
$681$	7.00000	0.268241
$682$	12.0000	0.459504
$683$	49.0000	1.87493	0.937466	−	0.348076i	$-0.113165\pi$
$683$	49.0000	1.87493	0.937466	+	0.348076i	$0.113165\pi$
$684$	−8.00000	−0.305888
$685$	−8.00000	−0.305664
$686$	13.0000	0.496342
$687$	−14.0000	−0.534133
$688$	0	0
$689$	−9.00000	−0.342873
$690$	24.0000	0.913664
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	−13.0000	−0.494186
$693$	−3.00000	−0.113961
$694$	−4.00000	−0.151838
$695$	−20.0000	−0.758643
$696$	0	0
$697$	−12.0000	−0.454532
$698$	18.0000	0.681310
$699$	6.00000	0.226941
$700$	11.0000	0.415761
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	1.00000	0.0377426
$703$	96.0000	3.62071
$704$	−3.00000	−0.113067
$705$	−16.0000	−0.602595
$706$	−37.0000	−1.39251
$707$	−5.00000	−0.188044
$708$	2.00000	0.0751646
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−24.0000	−0.900704
$711$	−6.00000	−0.225018
$712$	10.0000	0.374766
$713$	−24.0000	−0.898807
$714$	−3.00000	−0.112272
$715$	12.0000	0.448775
$716$	−5.00000	−0.186859
$717$	21.0000	0.784259
$718$	−17.0000	−0.634434
$719$	35.0000	1.30528	0.652640	−	0.757668i	$-0.273661\pi$
$719$	35.0000	1.30528	0.652640	+	0.757668i	$0.273661\pi$
$720$	−4.00000	−0.149071
$721$	−1.00000	−0.0372419
$722$	−45.0000	−1.67473
$723$	9.00000	0.334714
$724$	−17.0000	−0.631800
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	1.00000	0.0370879	0.0185440	−	0.999828i	$-0.494097\pi$
$727$	1.00000	0.0370879	0.0185440	+	0.999828i	$0.494097\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	0	0
$732$	−6.00000	−0.221766
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	−2.00000	−0.0738213
$735$	−24.0000	−0.885253
$736$	6.00000	0.221163
$737$	9.00000	0.331519
$738$	−4.00000	−0.147242
$739$	−22.0000	−0.809283	−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000	−0.809283	−0.404642	+	0.914475i	$0.632604\pi$
$740$	48.0000	1.76452
$741$	8.00000	0.293887
$742$	9.00000	0.330400
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	4.00000	0.146647
$745$	12.0000	0.439646
$746$	−10.0000	−0.366126
$747$	−6.00000	−0.219529
$748$	9.00000	0.329073
$749$	9.00000	0.328853
$750$	−24.0000	−0.876356
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	−4.00000	−0.145865
$753$	24.0000	0.874609
$754$	0	0
$755$	16.0000	0.582300
$756$	−1.00000	−0.0363696
$757$	28.0000	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$758$	23.0000	0.835398
$759$	−18.0000	−0.653359
$760$	−32.0000	−1.16076
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	13.0000	0.470940
$763$	−2.00000	−0.0724049
$764$	−3.00000	−0.108536
$765$	12.0000	0.433861
$766$	−22.0000	−0.794892
$767$	−2.00000	−0.0722158
$768$	−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	$0.528735\pi$
$770$	−12.0000	−0.432450
$771$	−20.0000	−0.720282
$772$	−18.0000	−0.647834
$773$	7.00000	0.251773	0.125886	−	0.992045i	$-0.459823\pi$
$773$	7.00000	0.251773	0.125886	+	0.992045i	$0.459823\pi$
$774$	0	0
$775$	44.0000	1.58053
$776$	8.00000	0.287183
$777$	12.0000	0.430498
$778$	−39.0000	−1.39822
$779$	−32.0000	−1.14652
$780$	4.00000	0.143223
$781$	18.0000	0.644091
$782$	−18.0000	−0.643679
$783$	0	0
$784$	−6.00000	−0.214286
$785$	20.0000	0.713831
$786$	−4.00000	−0.142675
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−18.0000	−0.641223
$789$	11.0000	0.391610
$790$	−24.0000	−0.853882
$791$	−16.0000	−0.568895
$792$	3.00000	0.106600
$793$	6.00000	0.213066
$794$	26.0000	0.922705
$795$	−36.0000	−1.27679
$796$	17.0000	0.602549
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	−8.00000	−0.283197
$799$	12.0000	0.424529
$800$	−11.0000	−0.388909
$801$	−10.0000	−0.353333
$802$	10.0000	0.353112
$803$	3.00000	0.105868
$804$	3.00000	0.105802
$805$	24.0000	0.845889
$806$	−4.00000	−0.140894
$807$	0	0
$808$	5.00000	0.175899
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	4.00000	0.140546
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	26.0000	0.911860
$814$	−36.0000	−1.26180
$815$	40.0000	1.40114
$816$	3.00000	0.105021
$817$	0	0
$818$	−12.0000	−0.419570
$819$	1.00000	0.0349428
$820$	−16.0000	−0.558744
$821$	51.0000	1.77991	0.889956	−	0.456046i	$-0.150735\pi$
$821$	51.0000	1.77991	0.889956	+	0.456046i	$0.150735\pi$
$822$	2.00000	0.0697580
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	1.00000	0.0348367
$825$	33.0000	1.14891
$826$	2.00000	0.0695889
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	−6.00000	−0.208514
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−24.0000	−0.833052
$831$	−18.0000	−0.624413
$832$	1.00000	0.0346688
$833$	18.0000	0.623663
$834$	5.00000	0.173136
$835$	−20.0000	−0.692129
$836$	24.0000	0.830057
$837$	−4.00000	−0.138260
$838$	−21.0000	−0.725433
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	−4.00000	−0.138013
$841$	−29.0000	−1.00000
$842$	−13.0000	−0.448010
$843$	−26.0000	−0.895488
$844$	−4.00000	−0.137686
$845$	−4.00000	−0.137604
$846$	4.00000	0.137523
$847$	−2.00000	−0.0687208
$848$	−9.00000	−0.309061
$849$	−8.00000	−0.274559
$850$	33.0000	1.13189
$851$	72.0000	2.46813
$852$	6.00000	0.205557
$853$	−47.0000	−1.60925	−0.804625	−	0.593784i	$-0.797633\pi$
$853$	−47.0000	−1.60925	−0.804625	+	0.593784i	$0.797633\pi$
$854$	−6.00000	−0.205316
$855$	32.0000	1.09438
$856$	−9.00000	−0.307614
$857$	45.0000	1.53717	0.768585	−	0.639747i	$-0.220961\pi$
$857$	45.0000	1.53717	0.768585	+	0.639747i	$0.220961\pi$
$858$	−3.00000	−0.102418
$859$	−18.0000	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$859$	−18.0000	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	−40.0000	−1.36241
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	1.00000	0.0340207
$865$	52.0000	1.76805
$866$	−16.0000	−0.543702
$867$	8.00000	0.271694
$868$	4.00000	0.135769
$869$	18.0000	0.610608
$870$	0	0
$871$	−3.00000	−0.101651
$872$	2.00000	0.0677285
$873$	−8.00000	−0.270759
$874$	−48.0000	−1.62362
$875$	−24.0000	−0.811348
$876$	1.00000	0.0337869
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	−3.00000	−0.101245
$879$	16.0000	0.539667
$880$	12.0000	0.404520
$881$	−4.00000	−0.134763	−0.0673817	−	0.997727i	$-0.521465\pi$
$881$	−4.00000	−0.134763	−0.0673817	+	0.997727i	$0.521465\pi$
$882$	6.00000	0.202031
$883$	19.0000	0.639401	0.319700	−	0.947519i	$-0.396418\pi$
$883$	19.0000	0.639401	0.319700	+	0.947519i	$0.396418\pi$
$884$	−3.00000	−0.100901
$885$	−8.00000	−0.268917
$886$	−4.00000	−0.134383
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	−12.0000	−0.402694
$889$	13.0000	0.436006
$890$	−40.0000	−1.34080
$891$	−3.00000	−0.100504
$892$	−13.0000	−0.435272
$893$	32.0000	1.07084
$894$	−3.00000	−0.100335
$895$	20.0000	0.668526
$896$	−1.00000	−0.0334077
$897$	6.00000	0.200334
$898$	3.00000	0.100111
$899$	0	0
$900$	11.0000	0.366667
$901$	27.0000	0.899500
$902$	12.0000	0.399556
$903$	0	0
$904$	16.0000	0.532152
$905$	68.0000	2.26040
$906$	−4.00000	−0.132891
$907$	−23.0000	−0.763702	−0.381851	−	0.924224i	$-0.624713\pi$
$907$	−23.0000	−0.763702	−0.381851	+	0.924224i	$0.624713\pi$
$908$	−7.00000	−0.232303
$909$	−5.00000	−0.165840
$910$	4.00000	0.132599
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	8.00000	0.264906
$913$	18.0000	0.595713
$914$	30.0000	0.992312
$915$	24.0000	0.793416
$916$	14.0000	0.462573
$917$	−4.00000	−0.132092
$918$	−3.00000	−0.0990148
$919$	−33.0000	−1.08857	−0.544285	−	0.838901i	$-0.683199\pi$
$919$	−33.0000	−1.08857	−0.544285	+	0.838901i	$0.683199\pi$
$920$	−24.0000	−0.791257
$921$	−9.00000	−0.296560
$922$	30.0000	0.987997
$923$	−6.00000	−0.197492
$924$	3.00000	0.0986928
$925$	−132.000	−4.34013
$926$	−30.0000	−0.985861
$927$	−1.00000	−0.0328443
$928$	0	0
$929$	−4.00000	−0.131236	−0.0656179	−	0.997845i	$-0.520902\pi$
$929$	−4.00000	−0.131236	−0.0656179	+	0.997845i	$0.520902\pi$
$930$	−16.0000	−0.524661
$931$	48.0000	1.57314
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−5.00000	−0.163605
$935$	−36.0000	−1.17733
$936$	−1.00000	−0.0326860
$937$	40.0000	1.30674	0.653372	−	0.757037i	$-0.273354\pi$
$937$	40.0000	1.30674	0.653372	+	0.757037i	$0.273354\pi$
$938$	3.00000	0.0979535
$939$	6.00000	0.195803
$940$	16.0000	0.521862
$941$	61.0000	1.98854	0.994272	−	0.106883i	$-0.0340871\pi$
$941$	61.0000	1.98854	0.994272	+	0.106883i	$0.0340871\pi$
$942$	−5.00000	−0.162909
$943$	−24.0000	−0.781548
$944$	−2.00000	−0.0650945
$945$	4.00000	0.130120
$946$	0	0
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	6.00000	0.194871
$949$	−1.00000	−0.0324614
$950$	88.0000	2.85510
$951$	−15.0000	−0.486408
$952$	3.00000	0.0972306
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	9.00000	0.291386
$955$	12.0000	0.388311
$956$	−21.0000	−0.679189
$957$	0	0
$958$	−14.0000	−0.452319
$959$	2.00000	0.0645834
$960$	4.00000	0.129099
$961$	−15.0000	−0.483871
$962$	12.0000	0.386896
$963$	9.00000	0.290021
$964$	−9.00000	−0.289870
$965$	72.0000	2.31776
$966$	−6.00000	−0.193047
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	2.00000	0.0642824
$969$	−24.0000	−0.770991
$970$	−32.0000	−1.02746
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	−1.00000	−0.0320750
$973$	5.00000	0.160293
$974$	−20.0000	−0.640841
$975$	−11.0000	−0.352282
$976$	6.00000	0.192055
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−10.0000	−0.319765
$979$	30.0000	0.958804
$980$	24.0000	0.766652
$981$	−2.00000	−0.0638551
$982$	33.0000	1.05307
$983$	−11.0000	−0.350846	−0.175423	−	0.984493i	$-0.556129\pi$
$983$	−11.0000	−0.350846	−0.175423	+	0.984493i	$0.556129\pi$
$984$	4.00000	0.127515
$985$	72.0000	2.29411
$986$	0	0
$987$	4.00000	0.127321
$988$	−8.00000	−0.254514
$989$	0	0
$990$	−12.0000	−0.381385
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	−4.00000	−0.127000
$993$	−5.00000	−0.158670
$994$	6.00000	0.190308
$995$	−68.0000	−2.15574
$996$	6.00000	0.190117
$997$	−34.0000	−1.07679	−0.538395	−	0.842692i	$-0.680969\pi$
$997$	−34.0000	−1.07679	−0.538395	+	0.842692i	$0.680969\pi$
$998$	−20.0000	−0.633089
$999$	12.0000	0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.a.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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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