LMFDB - Embedded newform 8041.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$1$
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	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} +2.00000 q^{5} +4.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} +2.00000 q^{5} +4.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} -4.00000 q^{10} +1.00000 q^{11} -4.00000 q^{12} -2.00000 q^{13} -6.00000 q^{14} -4.00000 q^{15} -4.00000 q^{16} +1.00000 q^{17} -2.00000 q^{18} -8.00000 q^{19} +4.00000 q^{20} -6.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} -1.00000 q^{25} +4.00000 q^{26} +4.00000 q^{27} +6.00000 q^{28} -3.00000 q^{29} +8.00000 q^{30} +8.00000 q^{31} +8.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} +6.00000 q^{35} +2.00000 q^{36} +6.00000 q^{37} +16.0000 q^{38} +4.00000 q^{39} +11.0000 q^{41} +12.0000 q^{42} -1.00000 q^{43} +2.00000 q^{44} +2.00000 q^{45} +8.00000 q^{46} -5.00000 q^{47} +8.00000 q^{48} +2.00000 q^{49} +2.00000 q^{50} -2.00000 q^{51} -4.00000 q^{52} +1.00000 q^{53} -8.00000 q^{54} +2.00000 q^{55} +16.0000 q^{57} +6.00000 q^{58} -3.00000 q^{59} -8.00000 q^{60} +2.00000 q^{61} -16.0000 q^{62} +3.00000 q^{63} -8.00000 q^{64} -4.00000 q^{65} +4.00000 q^{66} -9.00000 q^{67} +2.00000 q^{68} +8.00000 q^{69} -12.0000 q^{70} +2.00000 q^{71} -7.00000 q^{73} -12.0000 q^{74} +2.00000 q^{75} -16.0000 q^{76} +3.00000 q^{77} -8.00000 q^{78} +4.00000 q^{79} -8.00000 q^{80} -11.0000 q^{81} -22.0000 q^{82} -12.0000 q^{84} +2.00000 q^{85} +2.00000 q^{86} +6.00000 q^{87} +11.0000 q^{89} -4.00000 q^{90} -6.00000 q^{91} -8.00000 q^{92} -16.0000 q^{93} +10.0000 q^{94} -16.0000 q^{95} -16.0000 q^{96} -8.00000 q^{97} -4.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	2.00000	1.00000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	4.00000	1.63299
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−4.00000	−1.15470
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−6.00000	−1.60357
$15$	−4.00000	−1.03280
$16$	−4.00000	−1.00000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.00000	−0.471405
$19$	−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
$21$	−6.00000	−1.30931
$22$	−2.00000	−0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	4.00000	0.784465
$27$	4.00000	0.769800
$28$	6.00000	1.13389
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	8.00000	1.46059
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	8.00000	1.41421
$33$	−2.00000	−0.348155
$34$	−2.00000	−0.342997
$35$	6.00000	1.01419
$36$	2.00000	0.333333
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	16.0000	2.59554
$39$	4.00000	0.640513
$40$	0	0
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	12.0000	1.85164
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	2.00000	0.301511
$45$	2.00000	0.298142
$46$	8.00000	1.17954
$47$	−5.00000	−0.729325	−0.364662	−	0.931140i	$-0.618816\pi$
$47$	−5.00000	−0.729325	−0.364662	+	0.931140i	$0.618816\pi$
$48$	8.00000	1.15470
$49$	2.00000	0.285714
$50$	2.00000	0.282843
$51$	−2.00000	−0.280056
$52$	−4.00000	−0.554700
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	−8.00000	−1.08866
$55$	2.00000	0.269680
$56$	0	0
$57$	16.0000	2.11925
$58$	6.00000	0.787839
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	−8.00000	−1.03280
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−16.0000	−2.03200
$63$	3.00000	0.377964
$64$	−8.00000	−1.00000
$65$	−4.00000	−0.496139
$66$	4.00000	0.492366
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	2.00000	0.242536
$69$	8.00000	0.963087
$70$	−12.0000	−1.43427
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−12.0000	−1.39497
$75$	2.00000	0.230940
$76$	−16.0000	−1.83533
$77$	3.00000	0.341882
$78$	−8.00000	−0.905822
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−8.00000	−0.894427
$81$	−11.0000	−1.22222
$82$	−22.0000	−2.42949
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−12.0000	−1.30931
$85$	2.00000	0.216930
$86$	2.00000	0.215666
$87$	6.00000	0.643268
$88$	0	0
$89$	11.0000	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$90$	−4.00000	−0.421637
$91$	−6.00000	−0.628971
$92$	−8.00000	−0.834058
$93$	−16.0000	−1.65912
$94$	10.0000	1.03142
$95$	−16.0000	−1.64157
$96$	−16.0000	−1.63299
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−4.00000	−0.404061
$99$	1.00000	0.100504
$100$	−2.00000	−0.200000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	4.00000	0.396059
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	−12.0000	−1.17108
$106$	−2.00000	−0.194257
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	8.00000	0.769800
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	−4.00000	−0.381385
$111$	−12.0000	−1.13899
$112$	−12.0000	−1.13389
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	−32.0000	−2.99707
$115$	−8.00000	−0.746004
$116$	−6.00000	−0.557086
$117$	−2.00000	−0.184900
$118$	6.00000	0.552345
$119$	3.00000	0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.00000	−0.362143
$123$	−22.0000	−1.98367
$124$	16.0000	1.43684
$125$	−12.0000	−1.07331
$126$	−6.00000	−0.534522
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	8.00000	0.701646
$131$	5.00000	0.436852	0.218426	−	0.975854i	$-0.429908\pi$
$131$	5.00000	0.436852	0.218426	+	0.975854i	$0.429908\pi$
$132$	−4.00000	−0.348155
$133$	−24.0000	−2.08106
$134$	18.0000	1.55496
$135$	8.00000	0.688530
$136$	0	0
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	−16.0000	−1.36201
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	12.0000	1.01419
$141$	10.0000	0.842152
$142$	−4.00000	−0.335673
$143$	−2.00000	−0.167248
$144$	−4.00000	−0.333333
$145$	−6.00000	−0.498273
$146$	14.0000	1.15865
$147$	−4.00000	−0.329914
$148$	12.0000	0.986394
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−4.00000	−0.326599
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	−6.00000	−0.483494
$155$	16.0000	1.28515
$156$	8.00000	0.640513
$157$	23.0000	1.83560	0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000	1.83560	0.917800	+	0.397043i	$0.129964\pi$
$158$	−8.00000	−0.636446
$159$	−2.00000	−0.158610
$160$	16.0000	1.26491
$161$	−12.0000	−0.945732
$162$	22.0000	1.72848
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	22.0000	1.71791
$165$	−4.00000	−0.311400
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	−8.00000	−0.611775
$172$	−2.00000	−0.152499
$173$	−3.00000	−0.228086	−0.114043	−	0.993476i	$-0.536380\pi$
$173$	−3.00000	−0.228086	−0.114043	+	0.993476i	$0.536380\pi$
$174$	−12.0000	−0.909718
$175$	−3.00000	−0.226779
$176$	−4.00000	−0.301511
$177$	6.00000	0.450988
$178$	−22.0000	−1.64897
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	4.00000	0.298142
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	12.0000	0.889499
$183$	−4.00000	−0.295689
$184$	0	0
$185$	12.0000	0.882258
$186$	32.0000	2.34635
$187$	1.00000	0.0731272
$188$	−10.0000	−0.729325
$189$	12.0000	0.872872
$190$	32.0000	2.32152
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	16.0000	1.15470
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	16.0000	1.14873
$195$	8.00000	0.572892
$196$	4.00000	0.285714
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	−2.00000	−0.142134
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	18.0000	1.26962
$202$	0	0
$203$	−9.00000	−0.631676
$204$	−4.00000	−0.280056
$205$	22.0000	1.53655
$206$	−2.00000	−0.139347
$207$	−4.00000	−0.278019
$208$	8.00000	0.554700
$209$	−8.00000	−0.553372
$210$	24.0000	1.65616
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	2.00000	0.137361
$213$	−4.00000	−0.274075
$214$	6.00000	0.410152
$215$	−2.00000	−0.136399
$216$	0	0
$217$	24.0000	1.62923
$218$	22.0000	1.49003
$219$	14.0000	0.946032
$220$	4.00000	0.269680
$221$	−2.00000	−0.134535
$222$	24.0000	1.61077
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	24.0000	1.60357
$225$	−1.00000	−0.0666667
$226$	4.00000	0.266076
$227$	−13.0000	−0.862840	−0.431420	−	0.902151i	$-0.641987\pi$
$227$	−13.0000	−0.862840	−0.431420	+	0.902151i	$0.641987\pi$
$228$	32.0000	2.11925
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	16.0000	1.05501
$231$	−6.00000	−0.394771
$232$	0	0
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	4.00000	0.261488
$235$	−10.0000	−0.652328
$236$	−6.00000	−0.390567
$237$	−8.00000	−0.519656
$238$	−6.00000	−0.388922
$239$	−14.0000	−0.905585	−0.452792	−	0.891616i	$-0.649572\pi$
$239$	−14.0000	−0.905585	−0.452792	+	0.891616i	$0.649572\pi$
$240$	16.0000	1.03280
$241$	−27.0000	−1.73922	−0.869611	−	0.493737i	$-0.835631\pi$
$241$	−27.0000	−1.73922	−0.869611	+	0.493737i	$0.835631\pi$
$242$	−2.00000	−0.128565
$243$	10.0000	0.641500
$244$	4.00000	0.256074
$245$	4.00000	0.255551
$246$	44.0000	2.80534
$247$	16.0000	1.01806
$248$	0	0
$249$	0	0
$250$	24.0000	1.51789
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	6.00000	0.377964
$253$	−4.00000	−0.251478
$254$	−32.0000	−2.00786
$255$	−4.00000	−0.250490
$256$	16.0000	1.00000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	−4.00000	−0.249029
$259$	18.0000	1.11847
$260$	−8.00000	−0.496139
$261$	−3.00000	−0.185695
$262$	−10.0000	−0.617802
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	48.0000	2.94307
$267$	−22.0000	−1.34638
$268$	−18.0000	−1.09952
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	−16.0000	−0.973729
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	−4.00000	−0.242536
$273$	12.0000	0.726273
$274$	30.0000	1.81237
$275$	−1.00000	−0.0603023
$276$	16.0000	0.963087
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	−14.0000	−0.839664
$279$	8.00000	0.478947
$280$	0	0
$281$	−32.0000	−1.90896	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000	−1.90896	−0.954480	+	0.298275i	$0.903589\pi$
$282$	−20.0000	−1.19098
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	4.00000	0.237356
$285$	32.0000	1.89552
$286$	4.00000	0.236525
$287$	33.0000	1.94793
$288$	8.00000	0.471405
$289$	1.00000	0.0588235
$290$	12.0000	0.704664
$291$	16.0000	0.937937
$292$	−14.0000	−0.819288
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	8.00000	0.466569
$295$	−6.00000	−0.349334
$296$	0	0
$297$	4.00000	0.232104
$298$	−12.0000	−0.695141
$299$	8.00000	0.462652
$300$	4.00000	0.230940
$301$	−3.00000	−0.172917
$302$	4.00000	0.230174
$303$	0	0
$304$	32.0000	1.83533
$305$	4.00000	0.229039
$306$	−2.00000	−0.114332
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	6.00000	0.341882
$309$	−2.00000	−0.113776
$310$	−32.0000	−1.81748
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	−46.0000	−2.59593
$315$	6.00000	0.338062
$316$	8.00000	0.450035
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	4.00000	0.224309
$319$	−3.00000	−0.167968
$320$	−16.0000	−0.894427
$321$	6.00000	0.334887
$322$	24.0000	1.33747
$323$	−8.00000	−0.445132
$324$	−22.0000	−1.22222
$325$	2.00000	0.110940
$326$	−4.00000	−0.221540
$327$	22.0000	1.21660
$328$	0	0
$329$	−15.0000	−0.826977
$330$	8.00000	0.440386
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	24.0000	1.31322
$335$	−18.0000	−0.983445
$336$	24.0000	1.30931
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	18.0000	0.979071
$339$	4.00000	0.217250
$340$	4.00000	0.216930
$341$	8.00000	0.433224
$342$	16.0000	0.865181
$343$	−15.0000	−0.809924
$344$	0	0
$345$	16.0000	0.861411
$346$	6.00000	0.322562
$347$	7.00000	0.375780	0.187890	−	0.982190i	$-0.439835\pi$
$347$	7.00000	0.375780	0.187890	+	0.982190i	$0.439835\pi$
$348$	12.0000	0.643268
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	6.00000	0.320713
$351$	−8.00000	−0.427008
$352$	8.00000	0.426401
$353$	−25.0000	−1.33062	−0.665308	−	0.746569i	$-0.731700\pi$
$353$	−25.0000	−1.33062	−0.665308	+	0.746569i	$0.731700\pi$
$354$	−12.0000	−0.637793
$355$	4.00000	0.212298
$356$	22.0000	1.16600
$357$	−6.00000	−0.317554
$358$	−10.0000	−0.528516
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	4.00000	0.210235
$363$	−2.00000	−0.104973
$364$	−12.0000	−0.628971
$365$	−14.0000	−0.732793
$366$	8.00000	0.418167
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	16.0000	0.834058
$369$	11.0000	0.572637
$370$	−24.0000	−1.24770
$371$	3.00000	0.155752
$372$	−32.0000	−1.65912
$373$	38.0000	1.96757	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000	1.96757	0.983783	+	0.179364i	$0.0574041\pi$
$374$	−2.00000	−0.103418
$375$	24.0000	1.23935
$376$	0	0
$377$	6.00000	0.309016
$378$	−24.0000	−1.23443
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	−32.0000	−1.64157
$381$	−32.0000	−1.63941
$382$	−16.0000	−0.818631
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	12.0000	0.610784
$387$	−1.00000	−0.0508329
$388$	−16.0000	−0.812277
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	−16.0000	−0.810191
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−10.0000	−0.504433
$394$	−52.0000	−2.61972
$395$	8.00000	0.402524
$396$	2.00000	0.100504
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	32.0000	1.60402
$399$	48.0000	2.40301
$400$	4.00000	0.200000
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	−36.0000	−1.79552
$403$	−16.0000	−0.797017
$404$	0	0
$405$	−22.0000	−1.09319
$406$	18.0000	0.893325
$407$	6.00000	0.297409
$408$	0	0
$409$	−12.0000	−0.593362	−0.296681	−	0.954977i	$-0.595880\pi$
$409$	−12.0000	−0.593362	−0.296681	+	0.954977i	$0.595880\pi$
$410$	−44.0000	−2.17301
$411$	30.0000	1.47979
$412$	2.00000	0.0985329
$413$	−9.00000	−0.442861
$414$	8.00000	0.393179
$415$	0	0
$416$	−16.0000	−0.784465
$417$	−14.0000	−0.685583
$418$	16.0000	0.782586
$419$	−40.0000	−1.95413	−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000	−1.95413	−0.977064	+	0.212946i	$0.931694\pi$
$420$	−24.0000	−1.17108
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	2.00000	0.0973585
$423$	−5.00000	−0.243108
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	8.00000	0.387601
$427$	6.00000	0.290360
$428$	−6.00000	−0.290021
$429$	4.00000	0.193122
$430$	4.00000	0.192897
$431$	21.0000	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$431$	21.0000	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$432$	−16.0000	−0.769800
$433$	15.0000	0.720854	0.360427	−	0.932787i	$-0.382631\pi$
$433$	15.0000	0.720854	0.360427	+	0.932787i	$0.382631\pi$
$434$	−48.0000	−2.30407
$435$	12.0000	0.575356
$436$	−22.0000	−1.05361
$437$	32.0000	1.53077
$438$	−28.0000	−1.33789
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	4.00000	0.190261
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−24.0000	−1.13899
$445$	22.0000	1.04290
$446$	32.0000	1.51524
$447$	−12.0000	−0.567581
$448$	−24.0000	−1.13389
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	2.00000	0.0942809
$451$	11.0000	0.517970
$452$	−4.00000	−0.188144
$453$	4.00000	0.187936
$454$	26.0000	1.22024
$455$	−12.0000	−0.562569
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−30.0000	−1.40181
$459$	4.00000	0.186704
$460$	−16.0000	−0.746004
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	12.0000	0.558291
$463$	31.0000	1.44069	0.720346	−	0.693615i	$-0.243983\pi$
$463$	31.0000	1.44069	0.720346	+	0.693615i	$0.243983\pi$
$464$	12.0000	0.557086
$465$	−32.0000	−1.48396
$466$	6.00000	0.277945
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	−4.00000	−0.184900
$469$	−27.0000	−1.24674
$470$	20.0000	0.922531
$471$	−46.0000	−2.11957
$472$	0	0
$473$	−1.00000	−0.0459800
$474$	16.0000	0.734904
$475$	8.00000	0.367065
$476$	6.00000	0.275010
$477$	1.00000	0.0457869
$478$	28.0000	1.28069
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	−32.0000	−1.46059
$481$	−12.0000	−0.547153
$482$	54.0000	2.45963
$483$	24.0000	1.09204
$484$	2.00000	0.0909091
$485$	−16.0000	−0.726523
$486$	−20.0000	−0.907218
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	−8.00000	−0.361403
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	−44.0000	−1.98367
$493$	−3.00000	−0.135113
$494$	−32.0000	−1.43975
$495$	2.00000	0.0898933
$496$	−32.0000	−1.43684
$497$	6.00000	0.269137
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−24.0000	−1.07331
$501$	24.0000	1.07224
$502$	−8.00000	−0.357057
$503$	33.0000	1.47140	0.735699	−	0.677309i	$-0.236854\pi$
$503$	33.0000	1.47140	0.735699	+	0.677309i	$0.236854\pi$
$504$	0	0
$505$	0	0
$506$	8.00000	0.355643
$507$	18.0000	0.799408
$508$	32.0000	1.41977
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	8.00000	0.354246
$511$	−21.0000	−0.928985
$512$	−32.0000	−1.41421
$513$	−32.0000	−1.41283
$514$	−42.0000	−1.85254
$515$	2.00000	0.0881305
$516$	4.00000	0.176090
$517$	−5.00000	−0.219900
$518$	−36.0000	−1.58175
$519$	6.00000	0.263371
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	6.00000	0.262613
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	10.0000	0.436852
$525$	6.00000	0.261861
$526$	48.0000	2.09290
$527$	8.00000	0.348485
$528$	8.00000	0.348155
$529$	−7.00000	−0.304348
$530$	−4.00000	−0.173749
$531$	−3.00000	−0.130189
$532$	−48.0000	−2.08106
$533$	−22.0000	−0.952926
$534$	44.0000	1.90407
$535$	−6.00000	−0.259403
$536$	0	0
$537$	−10.0000	−0.431532
$538$	20.0000	0.862261
$539$	2.00000	0.0861461
$540$	16.0000	0.688530
$541$	43.0000	1.84871	0.924357	−	0.381528i	$-0.124602\pi$
$541$	43.0000	1.84871	0.924357	+	0.381528i	$0.124602\pi$
$542$	44.0000	1.88996
$543$	4.00000	0.171656
$544$	8.00000	0.342997
$545$	−22.0000	−0.942376
$546$	−24.0000	−1.02711
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−30.0000	−1.28154
$549$	2.00000	0.0853579
$550$	2.00000	0.0852803
$551$	24.0000	1.02243
$552$	0	0
$553$	12.0000	0.510292
$554$	−2.00000	−0.0849719
$555$	−24.0000	−1.01874
$556$	14.0000	0.593732
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	−16.0000	−0.677334
$559$	2.00000	0.0845910
$560$	−24.0000	−1.01419
$561$	−2.00000	−0.0844401
$562$	64.0000	2.69968
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	20.0000	0.842152
$565$	−4.00000	−0.168281
$566$	24.0000	1.00880
$567$	−33.0000	−1.38587
$568$	0	0
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	−64.0000	−2.68067
$571$	31.0000	1.29731	0.648655	−	0.761083i	$-0.275332\pi$
$571$	31.0000	1.29731	0.648655	+	0.761083i	$0.275332\pi$
$572$	−4.00000	−0.167248
$573$	−16.0000	−0.668410
$574$	−66.0000	−2.75479
$575$	4.00000	0.166812
$576$	−8.00000	−0.333333
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	−2.00000	−0.0831890
$579$	12.0000	0.498703
$580$	−12.0000	−0.498273
$581$	0	0
$582$	−32.0000	−1.32644
$583$	1.00000	0.0414158
$584$	0	0
$585$	−4.00000	−0.165380
$586$	−8.00000	−0.330477
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	−8.00000	−0.329914
$589$	−64.0000	−2.63707
$590$	12.0000	0.494032
$591$	−52.0000	−2.13899
$592$	−24.0000	−0.986394
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	−8.00000	−0.328244
$595$	6.00000	0.245976
$596$	12.0000	0.491539
$597$	32.0000	1.30967
$598$	−16.0000	−0.654289
$599$	−43.0000	−1.75693	−0.878466	−	0.477805i	$-0.841433\pi$
$599$	−43.0000	−1.75693	−0.878466	+	0.477805i	$0.841433\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	6.00000	0.244542
$603$	−9.00000	−0.366508
$604$	−4.00000	−0.162758
$605$	2.00000	0.0813116
$606$	0	0
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	−64.0000	−2.59554
$609$	18.0000	0.729397
$610$	−8.00000	−0.323911
$611$	10.0000	0.404557
$612$	2.00000	0.0808452
$613$	42.0000	1.69636	0.848182	−	0.529705i	$-0.177697\pi$
$613$	42.0000	1.69636	0.848182	+	0.529705i	$0.177697\pi$
$614$	−4.00000	−0.161427
$615$	−44.0000	−1.77425
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	4.00000	0.160904
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	32.0000	1.28515
$621$	−16.0000	−0.642058
$622$	60.0000	2.40578
$623$	33.0000	1.32212
$624$	−16.0000	−0.640513
$625$	−19.0000	−0.760000
$626$	4.00000	0.159872
$627$	16.0000	0.638978
$628$	46.0000	1.83560
$629$	6.00000	0.239236
$630$	−12.0000	−0.478091
$631$	−27.0000	−1.07485	−0.537427	−	0.843311i	$-0.680603\pi$
$631$	−27.0000	−1.07485	−0.537427	+	0.843311i	$0.680603\pi$
$632$	0	0
$633$	2.00000	0.0794929
$634$	−24.0000	−0.953162
$635$	32.0000	1.26988
$636$	−4.00000	−0.158610
$637$	−4.00000	−0.158486
$638$	6.00000	0.237542
$639$	2.00000	0.0791188
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−12.0000	−0.473602
$643$	6.00000	0.236617	0.118308	−	0.992977i	$-0.462253\pi$
$643$	6.00000	0.236617	0.118308	+	0.992977i	$0.462253\pi$
$644$	−24.0000	−0.945732
$645$	4.00000	0.157500
$646$	16.0000	0.629512
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	−4.00000	−0.156893
$651$	−48.0000	−1.88127
$652$	4.00000	0.156652
$653$	−8.00000	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$653$	−8.00000	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$654$	−44.0000	−1.72054
$655$	10.0000	0.390732
$656$	−44.0000	−1.71791
$657$	−7.00000	−0.273096
$658$	30.0000	1.16952
$659$	−22.0000	−0.856998	−0.428499	−	0.903542i	$-0.640958\pi$
$659$	−22.0000	−0.856998	−0.428499	+	0.903542i	$0.640958\pi$
$660$	−8.00000	−0.311400
$661$	−29.0000	−1.12797	−0.563985	−	0.825785i	$-0.690732\pi$
$661$	−29.0000	−1.12797	−0.563985	+	0.825785i	$0.690732\pi$
$662$	16.0000	0.621858
$663$	4.00000	0.155347
$664$	0	0
$665$	−48.0000	−1.86136
$666$	−12.0000	−0.464991
$667$	12.0000	0.464642
$668$	−24.0000	−0.928588
$669$	32.0000	1.23719
$670$	36.0000	1.39080
$671$	2.00000	0.0772091
$672$	−48.0000	−1.85164
$673$	11.0000	0.424019	0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000	0.424019	0.212009	+	0.977268i	$0.431999\pi$
$674$	18.0000	0.693334
$675$	−4.00000	−0.153960
$676$	−18.0000	−0.692308
$677$	−23.0000	−0.883962	−0.441981	−	0.897024i	$-0.645724\pi$
$677$	−23.0000	−0.883962	−0.441981	+	0.897024i	$0.645724\pi$
$678$	−8.00000	−0.307238
$679$	−24.0000	−0.921035
$680$	0	0
$681$	26.0000	0.996322
$682$	−16.0000	−0.612672
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	−16.0000	−0.611775
$685$	−30.0000	−1.14624
$686$	30.0000	1.14541
$687$	−30.0000	−1.14457
$688$	4.00000	0.152499
$689$	−2.00000	−0.0761939
$690$	−32.0000	−1.21822
$691$	38.0000	1.44559	0.722794	−	0.691063i	$-0.242858\pi$
$691$	38.0000	1.44559	0.722794	+	0.691063i	$0.242858\pi$
$692$	−6.00000	−0.228086
$693$	3.00000	0.113961
$694$	−14.0000	−0.531433
$695$	14.0000	0.531050
$696$	0	0
$697$	11.0000	0.416655
$698$	−16.0000	−0.605609
$699$	6.00000	0.226941
$700$	−6.00000	−0.226779
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	16.0000	0.603881
$703$	−48.0000	−1.81035
$704$	−8.00000	−0.301511
$705$	20.0000	0.753244
$706$	50.0000	1.88177
$707$	0	0
$708$	12.0000	0.450988
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	−8.00000	−0.300235
$711$	4.00000	0.150012
$712$	0	0
$713$	−32.0000	−1.19841
$714$	12.0000	0.449089
$715$	−4.00000	−0.149592
$716$	10.0000	0.373718
$717$	28.0000	1.04568
$718$	4.00000	0.149279
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	−8.00000	−0.298142
$721$	3.00000	0.111726
$722$	−90.0000	−3.34945
$723$	54.0000	2.00828
$724$	−4.00000	−0.148659
$725$	3.00000	0.111417
$726$	4.00000	0.148454
$727$	13.0000	0.482143	0.241072	−	0.970507i	$-0.422501\pi$
$727$	13.0000	0.482143	0.241072	+	0.970507i	$0.422501\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	28.0000	1.03633
$731$	−1.00000	−0.0369863
$732$	−8.00000	−0.295689
$733$	44.0000	1.62518	0.812589	−	0.582838i	$-0.198058\pi$
$733$	44.0000	1.62518	0.812589	+	0.582838i	$0.198058\pi$
$734$	−56.0000	−2.06700
$735$	−8.00000	−0.295084
$736$	−32.0000	−1.17954
$737$	−9.00000	−0.331519
$738$	−22.0000	−0.809831
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	24.0000	0.882258
$741$	−32.0000	−1.17555
$742$	−6.00000	−0.220267
$743$	−33.0000	−1.21065	−0.605326	−	0.795977i	$-0.706957\pi$
$743$	−33.0000	−1.21065	−0.605326	+	0.795977i	$0.706957\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	−76.0000	−2.78256
$747$	0	0
$748$	2.00000	0.0731272
$749$	−9.00000	−0.328853
$750$	−48.0000	−1.75271
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	20.0000	0.729325
$753$	−8.00000	−0.291536
$754$	−12.0000	−0.437014
$755$	−4.00000	−0.145575
$756$	24.0000	0.872872
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	20.0000	0.726433
$759$	8.00000	0.290382
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	64.0000	2.31848
$763$	−33.0000	−1.19468
$764$	16.0000	0.578860
$765$	2.00000	0.0723102
$766$	32.0000	1.15621
$767$	6.00000	0.216647
$768$	−32.0000	−1.15470
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	−12.0000	−0.432450
$771$	−42.0000	−1.51259
$772$	−12.0000	−0.431889
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	2.00000	0.0718885
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−36.0000	−1.29149
$778$	−30.0000	−1.07555
$779$	−88.0000	−3.15293
$780$	16.0000	0.572892
$781$	2.00000	0.0715656
$782$	8.00000	0.286079
$783$	−12.0000	−0.428845
$784$	−8.00000	−0.285714
$785$	46.0000	1.64181
$786$	20.0000	0.713376
$787$	−23.0000	−0.819861	−0.409931	−	0.912117i	$-0.634447\pi$
$787$	−23.0000	−0.819861	−0.409931	+	0.912117i	$0.634447\pi$
$788$	52.0000	1.85242
$789$	48.0000	1.70885
$790$	−16.0000	−0.569254
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−4.00000	−0.142044
$794$	36.0000	1.27759
$795$	−4.00000	−0.141865
$796$	−32.0000	−1.13421
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−96.0000	−3.39836
$799$	−5.00000	−0.176887
$800$	−8.00000	−0.282843
$801$	11.0000	0.388666
$802$	56.0000	1.97743
$803$	−7.00000	−0.247025
$804$	36.0000	1.26962
$805$	−24.0000	−0.845889
$806$	32.0000	1.12715
$807$	20.0000	0.704033
$808$	0	0
$809$	−19.0000	−0.668004	−0.334002	−	0.942572i	$-0.608399\pi$
$809$	−19.0000	−0.668004	−0.334002	+	0.942572i	$0.608399\pi$
$810$	44.0000	1.54600
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	−18.0000	−0.631676
$813$	44.0000	1.54315
$814$	−12.0000	−0.420600
$815$	4.00000	0.140114
$816$	8.00000	0.280056
$817$	8.00000	0.279885
$818$	24.0000	0.839140
$819$	−6.00000	−0.209657
$820$	44.0000	1.53655
$821$	−26.0000	−0.907406	−0.453703	−	0.891153i	$-0.649897\pi$
$821$	−26.0000	−0.907406	−0.453703	+	0.891153i	$0.649897\pi$
$822$	−60.0000	−2.09274
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	18.0000	0.626300
$827$	−15.0000	−0.521601	−0.260801	−	0.965393i	$-0.583986\pi$
$827$	−15.0000	−0.521601	−0.260801	+	0.965393i	$0.583986\pi$
$828$	−8.00000	−0.278019
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	16.0000	0.554700
$833$	2.00000	0.0692959
$834$	28.0000	0.969561
$835$	−24.0000	−0.830554
$836$	−16.0000	−0.553372
$837$	32.0000	1.10608
$838$	80.0000	2.76355
$839$	−34.0000	−1.17381	−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000	−1.17381	−0.586905	+	0.809656i	$0.699654\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−34.0000	−1.17172
$843$	64.0000	2.20428
$844$	−2.00000	−0.0688428
$845$	−18.0000	−0.619219
$846$	10.0000	0.343807
$847$	3.00000	0.103081
$848$	−4.00000	−0.137361
$849$	24.0000	0.823678
$850$	2.00000	0.0685994
$851$	−24.0000	−0.822709
$852$	−8.00000	−0.274075
$853$	1.00000	0.0342393	0.0171197	−	0.999853i	$-0.494550\pi$
$853$	1.00000	0.0342393	0.0171197	+	0.999853i	$0.494550\pi$
$854$	−12.0000	−0.410632
$855$	−16.0000	−0.547188
$856$	0	0
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	−8.00000	−0.273115
$859$	1.00000	0.0341196	0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000	0.0341196	0.0170598	+	0.999854i	$0.494569\pi$
$860$	−4.00000	−0.136399
$861$	−66.0000	−2.24927
$862$	−42.0000	−1.43053
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	32.0000	1.08866
$865$	−6.00000	−0.204006
$866$	−30.0000	−1.01944
$867$	−2.00000	−0.0679236
$868$	48.0000	1.62923
$869$	4.00000	0.135691
$870$	−24.0000	−0.813676
$871$	18.0000	0.609907
$872$	0	0
$873$	−8.00000	−0.270759
$874$	−64.0000	−2.16483
$875$	−36.0000	−1.21702
$876$	28.0000	0.946032
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	2.00000	0.0674967
$879$	−8.00000	−0.269833
$880$	−8.00000	−0.269680
$881$	−44.0000	−1.48240	−0.741199	−	0.671286i	$-0.765742\pi$
$881$	−44.0000	−1.48240	−0.741199	+	0.671286i	$0.765742\pi$
$882$	−4.00000	−0.134687
$883$	−41.0000	−1.37976	−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000	−1.37976	−0.689880	+	0.723924i	$0.742337\pi$
$884$	−4.00000	−0.134535
$885$	12.0000	0.403376
$886$	8.00000	0.268765
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	48.0000	1.60987
$890$	−44.0000	−1.47488
$891$	−11.0000	−0.368514
$892$	−32.0000	−1.07144
$893$	40.0000	1.33855
$894$	24.0000	0.802680
$895$	10.0000	0.334263
$896$	0	0
$897$	−16.0000	−0.534224
$898$	−36.0000	−1.20134
$899$	−24.0000	−0.800445
$900$	−2.00000	−0.0666667
$901$	1.00000	0.0333148
$902$	−22.0000	−0.732520
$903$	6.00000	0.199667
$904$	0	0
$905$	−4.00000	−0.132964
$906$	−8.00000	−0.265782
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	−26.0000	−0.862840
$909$	0	0
$910$	24.0000	0.795592
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	−64.0000	−2.11925
$913$	0	0
$914$	−36.0000	−1.19077
$915$	−8.00000	−0.264472
$916$	30.0000	0.991228
$917$	15.0000	0.495344
$918$	−8.00000	−0.264039
$919$	26.0000	0.857661	0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000	0.857661	0.428830	+	0.903385i	$0.358926\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	−4.00000	−0.131662
$924$	−12.0000	−0.394771
$925$	−6.00000	−0.197279
$926$	−62.0000	−2.03745
$927$	1.00000	0.0328443
$928$	−24.0000	−0.787839
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	64.0000	2.09864
$931$	−16.0000	−0.524379
$932$	−6.00000	−0.196537
$933$	60.0000	1.96431
$934$	−42.0000	−1.37428
$935$	2.00000	0.0654070
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	54.0000	1.76316
$939$	4.00000	0.130535
$940$	−20.0000	−0.652328
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	92.0000	2.99752
$943$	−44.0000	−1.43284
$944$	12.0000	0.390567
$945$	24.0000	0.780720
$946$	2.00000	0.0650256
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	−16.0000	−0.519656
$949$	14.0000	0.454459
$950$	−16.0000	−0.519109
$951$	−24.0000	−0.778253
$952$	0	0
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	−2.00000	−0.0647524
$955$	16.0000	0.517748
$956$	−28.0000	−0.905585
$957$	6.00000	0.193952
$958$	42.0000	1.35696
$959$	−45.0000	−1.45313
$960$	32.0000	1.03280
$961$	33.0000	1.06452
$962$	24.0000	0.773791
$963$	−3.00000	−0.0966736
$964$	−54.0000	−1.73922
$965$	−12.0000	−0.386294
$966$	−48.0000	−1.54437
$967$	34.0000	1.09337	0.546683	−	0.837340i	$-0.315890\pi$
$967$	34.0000	1.09337	0.546683	+	0.837340i	$0.315890\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	32.0000	1.02746
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	20.0000	0.641500
$973$	21.0000	0.673229
$974$	−24.0000	−0.769010
$975$	−4.00000	−0.128103
$976$	−8.00000	−0.256074
$977$	7.00000	0.223950	0.111975	−	0.993711i	$-0.464282\pi$
$977$	7.00000	0.223950	0.111975	+	0.993711i	$0.464282\pi$
$978$	8.00000	0.255812
$979$	11.0000	0.351562
$980$	8.00000	0.255551
$981$	−11.0000	−0.351203
$982$	−12.0000	−0.382935
$983$	−60.0000	−1.91370	−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000	−1.91370	−0.956851	+	0.290578i	$0.906153\pi$
$984$	0	0
$985$	52.0000	1.65686
$986$	6.00000	0.191079
$987$	30.0000	0.954911
$988$	32.0000	1.01806
$989$	4.00000	0.127193
$990$	−4.00000	−0.127128
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	64.0000	2.03200
$993$	16.0000	0.507745
$994$	−12.0000	−0.380617
$995$	−32.0000	−1.01447
$996$	0	0
$997$	−5.00000	−0.158352	−0.0791758	−	0.996861i	$-0.525229\pi$
$997$	−5.00000	−0.158352	−0.0791758	+	0.996861i	$0.525229\pi$
$998$	40.0000	1.26618
$999$	24.0000	0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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