LMFDB - Embedded newform 8001.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +1.41421 q^{5} +1.00000 q^{7} +2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +1.41421 q^{5} +1.00000 q^{7} +2.82843 q^{8} -2.00000 q^{10} -2.82843 q^{11} -2.00000 q^{13} -1.41421 q^{14} -4.00000 q^{16} +5.82843 q^{17} -8.24264 q^{19} +4.00000 q^{22} -2.82843 q^{23} -3.00000 q^{25} +2.82843 q^{26} +1.58579 q^{29} +4.00000 q^{31} -8.24264 q^{34} +1.41421 q^{35} +11.4853 q^{37} +11.6569 q^{38} +4.00000 q^{40} +8.65685 q^{41} -10.0000 q^{43} +4.00000 q^{46} +3.17157 q^{47} +1.00000 q^{49} +4.24264 q^{50} +7.58579 q^{53} -4.00000 q^{55} +2.82843 q^{56} -2.24264 q^{58} +7.07107 q^{59} -3.75736 q^{61} -5.65685 q^{62} +8.00000 q^{64} -2.82843 q^{65} +4.24264 q^{67} -2.00000 q^{70} -7.41421 q^{71} -10.7279 q^{73} -16.2426 q^{74} -2.82843 q^{77} -11.0000 q^{79} -5.65685 q^{80} -12.2426 q^{82} -11.6569 q^{83} +8.24264 q^{85} +14.1421 q^{86} -8.00000 q^{88} -9.17157 q^{89} -2.00000 q^{91} -4.48528 q^{94} -11.6569 q^{95} +11.2426 q^{97} -1.41421 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} - 4 q^{10} - 4 q^{13} - 8 q^{16} + 6 q^{17} - 8 q^{19} + 8 q^{22} - 6 q^{25} + 6 q^{29} + 8 q^{31} - 8 q^{34} + 6 q^{37} + 12 q^{38} + 8 q^{40} + 6 q^{41} - 20 q^{43} + 8 q^{46} + 12 q^{47} + 2 q^{49} + 18 q^{53} - 8 q^{55} + 4 q^{58} - 16 q^{61} + 16 q^{64} - 4 q^{70} - 12 q^{71} + 4 q^{73} - 24 q^{74} - 22 q^{79} - 16 q^{82} - 12 q^{83} + 8 q^{85} - 16 q^{88} - 24 q^{89} - 4 q^{91} + 8 q^{94} - 12 q^{95} + 14 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 4 * q^10 - 4 * q^13 - 8 * q^16 + 6 * q^17 - 8 * q^19 + 8 * q^22 - 6 * q^25 + 6 * q^29 + 8 * q^31 - 8 * q^34 + 6 * q^37 + 12 * q^38 + 8 * q^40 + 6 * q^41 - 20 * q^43 + 8 * q^46 + 12 * q^47 + 2 * q^49 + 18 * q^53 - 8 * q^55 + 4 * q^58 - 16 * q^61 + 16 * q^64 - 4 * q^70 - 12 * q^71 + 4 * q^73 - 24 * q^74 - 22 * q^79 - 16 * q^82 - 12 * q^83 + 8 * q^85 - 16 * q^88 - 24 * q^89 - 4 * q^91 + 8 * q^94 - 12 * q^95 + 14 * q^97

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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.82843	1.00000
$9$	0	0
$10$	−2.00000	−0.632456
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.41421	−0.377964
$15$	0	0
$16$	−4.00000	−1.00000
$17$	5.82843	1.41360	0.706801	−	0.707413i	$-0.250138\pi$
$17$	5.82843	1.41360	0.706801	+	0.707413i	$0.250138\pi$
$18$	0	0
$19$	−8.24264	−1.89099	−0.945496	−	0.325634i	$-0.894422\pi$
$19$	−8.24264	−1.89099	−0.945496	+	0.325634i	$0.894422\pi$
$20$	0	0
$21$	0	0
$22$	4.00000	0.852803
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	2.82843	0.554700
$27$	0	0
$28$	0	0
$29$	1.58579	0.294473	0.147237	−	0.989101i	$-0.452962\pi$
$29$	1.58579	0.294473	0.147237	+	0.989101i	$0.452962\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	0	0
$34$	−8.24264	−1.41360
$35$	1.41421	0.239046
$36$	0	0
$37$	11.4853	1.88817	0.944084	−	0.329704i	$-0.106949\pi$
$37$	11.4853	1.88817	0.944084	+	0.329704i	$0.106949\pi$
$38$	11.6569	1.89099
$39$	0	0
$40$	4.00000	0.632456
$41$	8.65685	1.35197	0.675987	−	0.736914i	$-0.263718\pi$
$41$	8.65685	1.35197	0.675987	+	0.736914i	$0.263718\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	3.17157	0.462621	0.231311	−	0.972880i	$-0.425699\pi$
$47$	3.17157	0.462621	0.231311	+	0.972880i	$0.425699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.24264	0.600000
$51$	0	0
$52$	0	0
$53$	7.58579	1.04199	0.520994	−	0.853560i	$-0.325561\pi$
$53$	7.58579	1.04199	0.520994	+	0.853560i	$0.325561\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	2.82843	0.377964
$57$	0	0
$58$	−2.24264	−0.294473
$59$	7.07107	0.920575	0.460287	−	0.887770i	$-0.347746\pi$
$59$	7.07107	0.920575	0.460287	+	0.887770i	$0.347746\pi$
$60$	0	0
$61$	−3.75736	−0.481081	−0.240540	−	0.970639i	$-0.577325\pi$
$61$	−3.75736	−0.481081	−0.240540	+	0.970639i	$0.577325\pi$
$62$	−5.65685	−0.718421
$63$	0	0
$64$	8.00000	1.00000
$65$	−2.82843	−0.350823
$66$	0	0
$67$	4.24264	0.518321	0.259161	−	0.965834i	$-0.416554\pi$
$67$	4.24264	0.518321	0.259161	+	0.965834i	$0.416554\pi$
$68$	0	0
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−7.41421	−0.879905	−0.439953	−	0.898021i	$-0.645005\pi$
$71$	−7.41421	−0.879905	−0.439953	+	0.898021i	$0.645005\pi$
$72$	0	0
$73$	−10.7279	−1.25561	−0.627804	−	0.778371i	$-0.716046\pi$
$73$	−10.7279	−1.25561	−0.627804	+	0.778371i	$0.716046\pi$
$74$	−16.2426	−1.88817
$75$	0	0
$76$	0	0
$77$	−2.82843	−0.322329
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	−5.65685	−0.632456
$81$	0	0
$82$	−12.2426	−1.35197
$83$	−11.6569	−1.27951	−0.639753	−	0.768581i	$-0.720963\pi$
$83$	−11.6569	−1.27951	−0.639753	+	0.768581i	$0.720963\pi$
$84$	0	0
$85$	8.24264	0.894040
$86$	14.1421	1.52499
$87$	0	0
$88$	−8.00000	−0.852803
$89$	−9.17157	−0.972185	−0.486092	−	0.873907i	$-0.661578\pi$
$89$	−9.17157	−0.972185	−0.486092	+	0.873907i	$0.661578\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	−4.48528	−0.462621
$95$	−11.6569	−1.19597
$96$	0	0
$97$	11.2426	1.14152	0.570759	−	0.821118i	$-0.306649\pi$
$97$	11.2426	1.14152	0.570759	+	0.821118i	$0.306649\pi$
$98$	−1.41421	−0.142857
$99$	0	0
$100$	0	0
$101$	2.82843	0.281439	0.140720	−	0.990050i	$-0.455058\pi$
$101$	2.82843	0.281439	0.140720	+	0.990050i	$0.455058\pi$
$102$	0	0
$103$	−16.4853	−1.62434	−0.812172	−	0.583419i	$-0.801715\pi$
$103$	−16.4853	−1.62434	−0.812172	+	0.583419i	$0.801715\pi$
$104$	−5.65685	−0.554700
$105$	0	0
$106$	−10.7279	−1.04199
$107$	1.41421	0.136717	0.0683586	−	0.997661i	$-0.478224\pi$
$107$	1.41421	0.136717	0.0683586	+	0.997661i	$0.478224\pi$
$108$	0	0
$109$	0.242641	0.0232408	0.0116204	−	0.999932i	$-0.496301\pi$
$109$	0.242641	0.0232408	0.0116204	+	0.999932i	$0.496301\pi$
$110$	5.65685	0.539360
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−0.343146	−0.0322804	−0.0161402	−	0.999870i	$-0.505138\pi$
$113$	−0.343146	−0.0322804	−0.0161402	+	0.999870i	$0.505138\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	−10.0000	−0.920575
$119$	5.82843	0.534291
$120$	0	0
$121$	−3.00000	−0.272727
$122$	5.31371	0.481081
$123$	0	0
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−11.3137	−1.00000
$129$	0	0
$130$	4.00000	0.350823
$131$	20.6569	1.80480	0.902399	−	0.430902i	$-0.141804\pi$
$131$	20.6569	1.80480	0.902399	+	0.430902i	$0.141804\pi$
$132$	0	0
$133$	−8.24264	−0.714728
$134$	−6.00000	−0.518321
$135$	0	0
$136$	16.4853	1.41360
$137$	5.31371	0.453981	0.226990	−	0.973897i	$-0.427111\pi$
$137$	5.31371	0.453981	0.226990	+	0.973897i	$0.427111\pi$
$138$	0	0
$139$	−3.24264	−0.275037	−0.137519	−	0.990499i	$-0.543913\pi$
$139$	−3.24264	−0.275037	−0.137519	+	0.990499i	$0.543913\pi$
$140$	0	0
$141$	0	0
$142$	10.4853	0.879905
$143$	5.65685	0.473050
$144$	0	0
$145$	2.24264	0.186241
$146$	15.1716	1.25561
$147$	0	0
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−4.24264	−0.345261	−0.172631	−	0.984987i	$-0.555227\pi$
$151$	−4.24264	−0.345261	−0.172631	+	0.984987i	$0.555227\pi$
$152$	−23.3137	−1.89099
$153$	0	0
$154$	4.00000	0.322329
$155$	5.65685	0.454369
$156$	0	0
$157$	6.72792	0.536947	0.268473	−	0.963287i	$-0.413481\pi$
$157$	6.72792	0.536947	0.268473	+	0.963287i	$0.413481\pi$
$158$	15.5563	1.23760
$159$	0	0
$160$	0	0
$161$	−2.82843	−0.222911
$162$	0	0
$163$	11.4853	0.899597	0.449798	−	0.893130i	$-0.351496\pi$
$163$	11.4853	0.899597	0.449798	+	0.893130i	$0.351496\pi$
$164$	0	0
$165$	0	0
$166$	16.4853	1.27951
$167$	−19.4142	−1.50232	−0.751158	−	0.660122i	$-0.770505\pi$
$167$	−19.4142	−1.50232	−0.751158	+	0.660122i	$0.770505\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−11.6569	−0.894040
$171$	0	0
$172$	0	0
$173$	−13.0711	−0.993775	−0.496887	−	0.867815i	$-0.665524\pi$
$173$	−13.0711	−0.993775	−0.496887	+	0.867815i	$0.665524\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	11.3137	0.852803
$177$	0	0
$178$	12.9706	0.972185
$179$	−10.9706	−0.819978	−0.409989	−	0.912090i	$-0.634468\pi$
$179$	−10.9706	−0.819978	−0.409989	+	0.912090i	$0.634468\pi$
$180$	0	0
$181$	−13.2426	−0.984318	−0.492159	−	0.870505i	$-0.663792\pi$
$181$	−13.2426	−0.984318	−0.492159	+	0.870505i	$0.663792\pi$
$182$	2.82843	0.209657
$183$	0	0
$184$	−8.00000	−0.589768
$185$	16.2426	1.19418
$186$	0	0
$187$	−16.4853	−1.20552
$188$	0	0
$189$	0	0
$190$	16.4853	1.19597
$191$	21.5563	1.55976	0.779881	−	0.625927i	$-0.215279\pi$
$191$	21.5563	1.55976	0.779881	+	0.625927i	$0.215279\pi$
$192$	0	0
$193$	19.2132	1.38300	0.691498	−	0.722378i	$-0.256951\pi$
$193$	19.2132	1.38300	0.691498	+	0.722378i	$0.256951\pi$
$194$	−15.8995	−1.14152
$195$	0	0
$196$	0	0
$197$	3.89949	0.277828	0.138914	−	0.990304i	$-0.455639\pi$
$197$	3.89949	0.277828	0.138914	+	0.990304i	$0.455639\pi$
$198$	0	0
$199$	0.485281	0.0344007	0.0172003	−	0.999852i	$-0.494525\pi$
$199$	0.485281	0.0344007	0.0172003	+	0.999852i	$0.494525\pi$
$200$	−8.48528	−0.600000
$201$	0	0
$202$	−4.00000	−0.281439
$203$	1.58579	0.111300
$204$	0	0
$205$	12.2426	0.855063
$206$	23.3137	1.62434
$207$	0	0
$208$	8.00000	0.554700
$209$	23.3137	1.61264
$210$	0	0
$211$	−18.4853	−1.27258	−0.636290	−	0.771450i	$-0.719532\pi$
$211$	−18.4853	−1.27258	−0.636290	+	0.771450i	$0.719532\pi$
$212$	0	0
$213$	0	0
$214$	−2.00000	−0.136717
$215$	−14.1421	−0.964486
$216$	0	0
$217$	4.00000	0.271538
$218$	−0.343146	−0.0232408
$219$	0	0
$220$	0	0
$221$	−11.6569	−0.784125
$222$	0	0
$223$	−11.2426	−0.752863	−0.376431	−	0.926444i	$-0.622849\pi$
$223$	−11.2426	−0.752863	−0.376431	+	0.926444i	$0.622849\pi$
$224$	0	0
$225$	0	0
$226$	0.485281	0.0322804
$227$	−20.1421	−1.33688	−0.668440	−	0.743766i	$-0.733038\pi$
$227$	−20.1421	−1.33688	−0.668440	+	0.743766i	$0.733038\pi$
$228$	0	0
$229$	26.2132	1.73222	0.866109	−	0.499856i	$-0.166614\pi$
$229$	26.2132	1.73222	0.866109	+	0.499856i	$0.166614\pi$
$230$	5.65685	0.373002
$231$	0	0
$232$	4.48528	0.294473
$233$	−15.7279	−1.03037	−0.515185	−	0.857079i	$-0.672277\pi$
$233$	−15.7279	−1.03037	−0.515185	+	0.857079i	$0.672277\pi$
$234$	0	0
$235$	4.48528	0.292587
$236$	0	0
$237$	0	0
$238$	−8.24264	−0.534291
$239$	−28.0711	−1.81577	−0.907883	−	0.419223i	$-0.862302\pi$
$239$	−28.0711	−1.81577	−0.907883	+	0.419223i	$0.862302\pi$
$240$	0	0
$241$	25.2426	1.62602	0.813011	−	0.582249i	$-0.197827\pi$
$241$	25.2426	1.62602	0.813011	+	0.582249i	$0.197827\pi$
$242$	4.24264	0.272727
$243$	0	0
$244$	0	0
$245$	1.41421	0.0903508
$246$	0	0
$247$	16.4853	1.04893
$248$	11.3137	0.718421
$249$	0	0
$250$	16.0000	1.01193
$251$	−6.17157	−0.389546	−0.194773	−	0.980848i	$-0.562397\pi$
$251$	−6.17157	−0.389546	−0.194773	+	0.980848i	$0.562397\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	1.41421	0.0887357
$255$	0	0
$256$	0	0
$257$	−4.58579	−0.286053	−0.143027	−	0.989719i	$-0.545683\pi$
$257$	−4.58579	−0.286053	−0.143027	+	0.989719i	$0.545683\pi$
$258$	0	0
$259$	11.4853	0.713661
$260$	0	0
$261$	0	0
$262$	−29.2132	−1.80480
$263$	−7.41421	−0.457180	−0.228590	−	0.973523i	$-0.573412\pi$
$263$	−7.41421	−0.457180	−0.228590	+	0.973523i	$0.573412\pi$
$264$	0	0
$265$	10.7279	0.659011
$266$	11.6569	0.714728
$267$	0	0
$268$	0	0
$269$	−13.9706	−0.851800	−0.425900	−	0.904770i	$-0.640043\pi$
$269$	−13.9706	−0.851800	−0.425900	+	0.904770i	$0.640043\pi$
$270$	0	0
$271$	1.75736	0.106752	0.0533760	−	0.998574i	$-0.483002\pi$
$271$	1.75736	0.106752	0.0533760	+	0.998574i	$0.483002\pi$
$272$	−23.3137	−1.41360
$273$	0	0
$274$	−7.51472	−0.453981
$275$	8.48528	0.511682
$276$	0	0
$277$	−3.51472	−0.211179	−0.105589	−	0.994410i	$-0.533673\pi$
$277$	−3.51472	−0.211179	−0.105589	+	0.994410i	$0.533673\pi$
$278$	4.58579	0.275037
$279$	0	0
$280$	4.00000	0.239046
$281$	−6.55635	−0.391119	−0.195560	−	0.980692i	$-0.562652\pi$
$281$	−6.55635	−0.391119	−0.195560	+	0.980692i	$0.562652\pi$
$282$	0	0
$283$	−28.2132	−1.67710	−0.838550	−	0.544824i	$-0.816596\pi$
$283$	−28.2132	−1.67710	−0.838550	+	0.544824i	$0.816596\pi$
$284$	0	0
$285$	0	0
$286$	−8.00000	−0.473050
$287$	8.65685	0.510998
$288$	0	0
$289$	16.9706	0.998268
$290$	−3.17157	−0.186241
$291$	0	0
$292$	0	0
$293$	15.5563	0.908812	0.454406	−	0.890795i	$-0.349852\pi$
$293$	15.5563	0.908812	0.454406	+	0.890795i	$0.349852\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	32.4853	1.88817
$297$	0	0
$298$	−8.48528	−0.491539
$299$	5.65685	0.327144
$300$	0	0
$301$	−10.0000	−0.576390
$302$	6.00000	0.345261
$303$	0	0
$304$	32.9706	1.89099
$305$	−5.31371	−0.304262
$306$	0	0
$307$	−15.2426	−0.869943	−0.434972	−	0.900444i	$-0.643242\pi$
$307$	−15.2426	−0.869943	−0.434972	+	0.900444i	$0.643242\pi$
$308$	0	0
$309$	0	0
$310$	−8.00000	−0.454369
$311$	4.24264	0.240578	0.120289	−	0.992739i	$-0.461618\pi$
$311$	4.24264	0.240578	0.120289	+	0.992739i	$0.461618\pi$
$312$	0	0
$313$	−9.72792	−0.549855	−0.274927	−	0.961465i	$-0.588654\pi$
$313$	−9.72792	−0.549855	−0.274927	+	0.961465i	$0.588654\pi$
$314$	−9.51472	−0.536947
$315$	0	0
$316$	0	0
$317$	8.14214	0.457308	0.228654	−	0.973508i	$-0.426568\pi$
$317$	8.14214	0.457308	0.228654	+	0.973508i	$0.426568\pi$
$318$	0	0
$319$	−4.48528	−0.251128
$320$	11.3137	0.632456
$321$	0	0
$322$	4.00000	0.222911
$323$	−48.0416	−2.67311
$324$	0	0
$325$	6.00000	0.332820
$326$	−16.2426	−0.899597
$327$	0	0
$328$	24.4853	1.35197
$329$	3.17157	0.174854
$330$	0	0
$331$	14.7279	0.809520	0.404760	−	0.914423i	$-0.367355\pi$
$331$	14.7279	0.809520	0.404760	+	0.914423i	$0.367355\pi$
$332$	0	0
$333$	0	0
$334$	27.4558	1.50232
$335$	6.00000	0.327815
$336$	0	0
$337$	−17.7574	−0.967305	−0.483652	−	0.875260i	$-0.660690\pi$
$337$	−17.7574	−0.967305	−0.483652	+	0.875260i	$0.660690\pi$
$338$	12.7279	0.692308
$339$	0	0
$340$	0	0
$341$	−11.3137	−0.612672
$342$	0	0
$343$	1.00000	0.0539949
$344$	−28.2843	−1.52499
$345$	0	0
$346$	18.4853	0.993775
$347$	7.92893	0.425647	0.212824	−	0.977091i	$-0.431734\pi$
$347$	7.92893	0.425647	0.212824	+	0.977091i	$0.431734\pi$
$348$	0	0
$349$	−12.2132	−0.653758	−0.326879	−	0.945066i	$-0.605997\pi$
$349$	−12.2132	−0.653758	−0.326879	+	0.945066i	$0.605997\pi$
$350$	4.24264	0.226779
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−10.4853	−0.556501
$356$	0	0
$357$	0	0
$358$	15.5147	0.819978
$359$	25.2426	1.33226	0.666128	−	0.745838i	$-0.267951\pi$
$359$	25.2426	1.33226	0.666128	+	0.745838i	$0.267951\pi$
$360$	0	0
$361$	48.9411	2.57585
$362$	18.7279	0.984318
$363$	0	0
$364$	0	0
$365$	−15.1716	−0.794116
$366$	0	0
$367$	2.48528	0.129731	0.0648653	−	0.997894i	$-0.479338\pi$
$367$	2.48528	0.129731	0.0648653	+	0.997894i	$0.479338\pi$
$368$	11.3137	0.589768
$369$	0	0
$370$	−22.9706	−1.19418
$371$	7.58579	0.393834
$372$	0	0
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	23.3137	1.20552
$375$	0	0
$376$	8.97056	0.462621
$377$	−3.17157	−0.163344
$378$	0	0
$379$	−4.97056	−0.255321	−0.127660	−	0.991818i	$-0.540747\pi$
$379$	−4.97056	−0.255321	−0.127660	+	0.991818i	$0.540747\pi$
$380$	0	0
$381$	0	0
$382$	−30.4853	−1.55976
$383$	−8.65685	−0.442345	−0.221172	−	0.975235i	$-0.570988\pi$
$383$	−8.65685	−0.442345	−0.221172	+	0.975235i	$0.570988\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	−27.1716	−1.38300
$387$	0	0
$388$	0	0
$389$	−27.5563	−1.39716	−0.698581	−	0.715531i	$-0.746185\pi$
$389$	−27.5563	−1.39716	−0.698581	+	0.715531i	$0.746185\pi$
$390$	0	0
$391$	−16.4853	−0.833697
$392$	2.82843	0.142857
$393$	0	0
$394$	−5.51472	−0.277828
$395$	−15.5563	−0.782725
$396$	0	0
$397$	8.97056	0.450220	0.225110	−	0.974333i	$-0.427726\pi$
$397$	8.97056	0.450220	0.225110	+	0.974333i	$0.427726\pi$
$398$	−0.686292	−0.0344007
$399$	0	0
$400$	12.0000	0.600000
$401$	−10.4142	−0.520061	−0.260031	−	0.965600i	$-0.583733\pi$
$401$	−10.4142	−0.520061	−0.260031	+	0.965600i	$0.583733\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	−2.24264	−0.111300
$407$	−32.4853	−1.61024
$408$	0	0
$409$	−9.51472	−0.470473	−0.235236	−	0.971938i	$-0.575586\pi$
$409$	−9.51472	−0.470473	−0.235236	+	0.971938i	$0.575586\pi$
$410$	−17.3137	−0.855063
$411$	0	0
$412$	0	0
$413$	7.07107	0.347945
$414$	0	0
$415$	−16.4853	−0.809231
$416$	0	0
$417$	0	0
$418$	−32.9706	−1.61264
$419$	31.4558	1.53672	0.768359	−	0.640019i	$-0.221073\pi$
$419$	31.4558	1.53672	0.768359	+	0.640019i	$0.221073\pi$
$420$	0	0
$421$	22.2426	1.08404	0.542020	−	0.840366i	$-0.317660\pi$
$421$	22.2426	1.08404	0.542020	+	0.840366i	$0.317660\pi$
$422$	26.1421	1.27258
$423$	0	0
$424$	21.4558	1.04199
$425$	−17.4853	−0.848161
$426$	0	0
$427$	−3.75736	−0.181831
$428$	0	0
$429$	0	0
$430$	20.0000	0.964486
$431$	−30.0416	−1.44705	−0.723527	−	0.690296i	$-0.757480\pi$
$431$	−30.0416	−1.44705	−0.723527	+	0.690296i	$0.757480\pi$
$432$	0	0
$433$	−18.2426	−0.876685	−0.438343	−	0.898808i	$-0.644434\pi$
$433$	−18.2426	−0.876685	−0.438343	+	0.898808i	$0.644434\pi$
$434$	−5.65685	−0.271538
$435$	0	0
$436$	0	0
$437$	23.3137	1.11525
$438$	0	0
$439$	8.27208	0.394805	0.197402	−	0.980323i	$-0.436749\pi$
$439$	8.27208	0.394805	0.197402	+	0.980323i	$0.436749\pi$
$440$	−11.3137	−0.539360
$441$	0	0
$442$	16.4853	0.784125
$443$	−7.07107	−0.335957	−0.167978	−	0.985791i	$-0.553724\pi$
$443$	−7.07107	−0.335957	−0.167978	+	0.985791i	$0.553724\pi$
$444$	0	0
$445$	−12.9706	−0.614864
$446$	15.8995	0.752863
$447$	0	0
$448$	8.00000	0.377964
$449$	−30.3431	−1.43198	−0.715991	−	0.698110i	$-0.754025\pi$
$449$	−30.3431	−1.43198	−0.715991	+	0.698110i	$0.754025\pi$
$450$	0	0
$451$	−24.4853	−1.15297
$452$	0	0
$453$	0	0
$454$	28.4853	1.33688
$455$	−2.82843	−0.132599
$456$	0	0
$457$	−25.0000	−1.16945	−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000	−1.16945	−0.584725	+	0.811231i	$0.698798\pi$
$458$	−37.0711	−1.73222
$459$	0	0
$460$	0	0
$461$	−15.5563	−0.724531	−0.362266	−	0.932075i	$-0.617997\pi$
$461$	−15.5563	−0.724531	−0.362266	+	0.932075i	$0.617997\pi$
$462$	0	0
$463$	−6.97056	−0.323950	−0.161975	−	0.986795i	$-0.551786\pi$
$463$	−6.97056	−0.323950	−0.161975	+	0.986795i	$0.551786\pi$
$464$	−6.34315	−0.294473
$465$	0	0
$466$	22.2426	1.03037
$467$	−13.4558	−0.622662	−0.311331	−	0.950302i	$-0.600775\pi$
$467$	−13.4558	−0.622662	−0.311331	+	0.950302i	$0.600775\pi$
$468$	0	0
$469$	4.24264	0.195907
$470$	−6.34315	−0.292587
$471$	0	0
$472$	20.0000	0.920575
$473$	28.2843	1.30051
$474$	0	0
$475$	24.7279	1.13459
$476$	0	0
$477$	0	0
$478$	39.6985	1.81577
$479$	11.1421	0.509097	0.254549	−	0.967060i	$-0.418073\pi$
$479$	11.1421	0.509097	0.254549	+	0.967060i	$0.418073\pi$
$480$	0	0
$481$	−22.9706	−1.04737
$482$	−35.6985	−1.62602
$483$	0	0
$484$	0	0
$485$	15.8995	0.721959
$486$	0	0
$487$	38.7279	1.75493	0.877465	−	0.479641i	$-0.159233\pi$
$487$	38.7279	1.75493	0.877465	+	0.479641i	$0.159233\pi$
$488$	−10.6274	−0.481081
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	−5.65685	−0.255290	−0.127645	−	0.991820i	$-0.540742\pi$
$491$	−5.65685	−0.255290	−0.127645	+	0.991820i	$0.540742\pi$
$492$	0	0
$493$	9.24264	0.416268
$494$	−23.3137	−1.04893
$495$	0	0
$496$	−16.0000	−0.718421
$497$	−7.41421	−0.332573
$498$	0	0
$499$	−1.27208	−0.0569460	−0.0284730	−	0.999595i	$-0.509064\pi$
$499$	−1.27208	−0.0569460	−0.0284730	+	0.999595i	$0.509064\pi$
$500$	0	0
$501$	0	0
$502$	8.72792	0.389546
$503$	32.6569	1.45610	0.728049	−	0.685526i	$-0.240428\pi$
$503$	32.6569	1.45610	0.728049	+	0.685526i	$0.240428\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	−11.3137	−0.502956
$507$	0	0
$508$	0	0
$509$	34.7990	1.54244	0.771219	−	0.636570i	$-0.219647\pi$
$509$	34.7990	1.54244	0.771219	+	0.636570i	$0.219647\pi$
$510$	0	0
$511$	−10.7279	−0.474575
$512$	22.6274	1.00000
$513$	0	0
$514$	6.48528	0.286053
$515$	−23.3137	−1.02732
$516$	0	0
$517$	−8.97056	−0.394525
$518$	−16.2426	−0.713661
$519$	0	0
$520$	−8.00000	−0.350823
$521$	4.97056	0.217764	0.108882	−	0.994055i	$-0.465273\pi$
$521$	4.97056	0.217764	0.108882	+	0.994055i	$0.465273\pi$
$522$	0	0
$523$	−39.6985	−1.73589	−0.867947	−	0.496657i	$-0.834561\pi$
$523$	−39.6985	−1.73589	−0.867947	+	0.496657i	$0.834561\pi$
$524$	0	0
$525$	0	0
$526$	10.4853	0.457180
$527$	23.3137	1.01556
$528$	0	0
$529$	−15.0000	−0.652174
$530$	−15.1716	−0.659011
$531$	0	0
$532$	0	0
$533$	−17.3137	−0.749940
$534$	0	0
$535$	2.00000	0.0864675
$536$	12.0000	0.518321
$537$	0	0
$538$	19.7574	0.851800
$539$	−2.82843	−0.121829
$540$	0	0
$541$	−16.4853	−0.708758	−0.354379	−	0.935102i	$-0.615308\pi$
$541$	−16.4853	−0.708758	−0.354379	+	0.935102i	$0.615308\pi$
$542$	−2.48528	−0.106752
$543$	0	0
$544$	0	0
$545$	0.343146	0.0146987
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	0	0
$550$	−12.0000	−0.511682
$551$	−13.0711	−0.556846
$552$	0	0
$553$	−11.0000	−0.467768
$554$	4.97056	0.211179
$555$	0	0
$556$	0	0
$557$	0.686292	0.0290791	0.0145396	−	0.999894i	$-0.495372\pi$
$557$	0.686292	0.0290791	0.0145396	+	0.999894i	$0.495372\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	−5.65685	−0.239046
$561$	0	0
$562$	9.27208	0.391119
$563$	−21.1716	−0.892275	−0.446138	−	0.894964i	$-0.647201\pi$
$563$	−21.1716	−0.892275	−0.446138	+	0.894964i	$0.647201\pi$
$564$	0	0
$565$	−0.485281	−0.0204159
$566$	39.8995	1.67710
$567$	0	0
$568$	−20.9706	−0.879905
$569$	8.14214	0.341336	0.170668	−	0.985329i	$-0.445407\pi$
$569$	8.14214	0.341336	0.170668	+	0.985329i	$0.445407\pi$
$570$	0	0
$571$	30.2426	1.26562	0.632808	−	0.774309i	$-0.281902\pi$
$571$	30.2426	1.26562	0.632808	+	0.774309i	$0.281902\pi$
$572$	0	0
$573$	0	0
$574$	−12.2426	−0.510998
$575$	8.48528	0.353861
$576$	0	0
$577$	−30.2426	−1.25902	−0.629509	−	0.776993i	$-0.716744\pi$
$577$	−30.2426	−1.25902	−0.629509	+	0.776993i	$0.716744\pi$
$578$	−24.0000	−0.998268
$579$	0	0
$580$	0	0
$581$	−11.6569	−0.483608
$582$	0	0
$583$	−21.4558	−0.888610
$584$	−30.3431	−1.25561
$585$	0	0
$586$	−22.0000	−0.908812
$587$	19.6274	0.810110	0.405055	−	0.914292i	$-0.367252\pi$
$587$	19.6274	0.810110	0.405055	+	0.914292i	$0.367252\pi$
$588$	0	0
$589$	−32.9706	−1.35853
$590$	−14.1421	−0.582223
$591$	0	0
$592$	−45.9411	−1.88817
$593$	−20.4853	−0.841230	−0.420615	−	0.907239i	$-0.638186\pi$
$593$	−20.4853	−0.841230	−0.420615	+	0.907239i	$0.638186\pi$
$594$	0	0
$595$	8.24264	0.337915
$596$	0	0
$597$	0	0
$598$	−8.00000	−0.327144
$599$	−5.44365	−0.222422	−0.111211	−	0.993797i	$-0.535473\pi$
$599$	−5.44365	−0.222422	−0.111211	+	0.993797i	$0.535473\pi$
$600$	0	0
$601$	−28.2132	−1.15084	−0.575420	−	0.817858i	$-0.695162\pi$
$601$	−28.2132	−1.15084	−0.575420	+	0.817858i	$0.695162\pi$
$602$	14.1421	0.576390
$603$	0	0
$604$	0	0
$605$	−4.24264	−0.172488
$606$	0	0
$607$	−25.6985	−1.04307	−0.521535	−	0.853230i	$-0.674640\pi$
$607$	−25.6985	−1.04307	−0.521535	+	0.853230i	$0.674640\pi$
$608$	0	0
$609$	0	0
$610$	7.51472	0.304262
$611$	−6.34315	−0.256616
$612$	0	0
$613$	−42.9706	−1.73556	−0.867782	−	0.496944i	$-0.834455\pi$
$613$	−42.9706	−1.73556	−0.867782	+	0.496944i	$0.834455\pi$
$614$	21.5563	0.869943
$615$	0	0
$616$	−8.00000	−0.322329
$617$	−15.0416	−0.605553	−0.302777	−	0.953062i	$-0.597914\pi$
$617$	−15.0416	−0.605553	−0.302777	+	0.953062i	$0.597914\pi$
$618$	0	0
$619$	2.75736	0.110828	0.0554138	−	0.998463i	$-0.482352\pi$
$619$	2.75736	0.110828	0.0554138	+	0.998463i	$0.482352\pi$
$620$	0	0
$621$	0	0
$622$	−6.00000	−0.240578
$623$	−9.17157	−0.367451
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	13.7574	0.549855
$627$	0	0
$628$	0	0
$629$	66.9411	2.66912
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	−31.1127	−1.23760
$633$	0	0
$634$	−11.5147	−0.457308
$635$	−1.41421	−0.0561214
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	6.34315	0.251128
$639$	0	0
$640$	−16.0000	−0.632456
$641$	19.2426	0.760039	0.380019	−	0.924979i	$-0.375917\pi$
$641$	19.2426	0.760039	0.380019	+	0.924979i	$0.375917\pi$
$642$	0	0
$643$	−17.6985	−0.697960	−0.348980	−	0.937130i	$-0.613472\pi$
$643$	−17.6985	−0.697960	−0.348980	+	0.937130i	$0.613472\pi$
$644$	0	0
$645$	0	0
$646$	67.9411	2.67311
$647$	−37.7990	−1.48603	−0.743016	−	0.669274i	$-0.766605\pi$
$647$	−37.7990	−1.48603	−0.743016	+	0.669274i	$0.766605\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	−8.48528	−0.332820
$651$	0	0
$652$	0	0
$653$	−7.75736	−0.303569	−0.151784	−	0.988414i	$-0.548502\pi$
$653$	−7.75736	−0.303569	−0.151784	+	0.988414i	$0.548502\pi$
$654$	0	0
$655$	29.2132	1.14145
$656$	−34.6274	−1.35197
$657$	0	0
$658$	−4.48528	−0.174854
$659$	−11.3137	−0.440720	−0.220360	−	0.975419i	$-0.570723\pi$
$659$	−11.3137	−0.440720	−0.220360	+	0.975419i	$0.570723\pi$
$660$	0	0
$661$	17.2721	0.671806	0.335903	−	0.941897i	$-0.390959\pi$
$661$	17.2721	0.671806	0.335903	+	0.941897i	$0.390959\pi$
$662$	−20.8284	−0.809520
$663$	0	0
$664$	−32.9706	−1.27951
$665$	−11.6569	−0.452033
$666$	0	0
$667$	−4.48528	−0.173671
$668$	0	0
$669$	0	0
$670$	−8.48528	−0.327815
$671$	10.6274	0.410267
$672$	0	0
$673$	−8.45584	−0.325949	−0.162974	−	0.986630i	$-0.552109\pi$
$673$	−8.45584	−0.325949	−0.162974	+	0.986630i	$0.552109\pi$
$674$	25.1127	0.967305
$675$	0	0
$676$	0	0
$677$	−19.2843	−0.741155	−0.370577	−	0.928802i	$-0.620840\pi$
$677$	−19.2843	−0.741155	−0.370577	+	0.928802i	$0.620840\pi$
$678$	0	0
$679$	11.2426	0.431453
$680$	23.3137	0.894040
$681$	0	0
$682$	16.0000	0.612672
$683$	25.4558	0.974041	0.487020	−	0.873391i	$-0.338084\pi$
$683$	25.4558	0.974041	0.487020	+	0.873391i	$0.338084\pi$
$684$	0	0
$685$	7.51472	0.287123
$686$	−1.41421	−0.0539949
$687$	0	0
$688$	40.0000	1.52499
$689$	−15.1716	−0.577991
$690$	0	0
$691$	−38.2132	−1.45370	−0.726849	−	0.686797i	$-0.759016\pi$
$691$	−38.2132	−1.45370	−0.726849	+	0.686797i	$0.759016\pi$
$692$	0	0
$693$	0	0
$694$	−11.2132	−0.425647
$695$	−4.58579	−0.173949
$696$	0	0
$697$	50.4558	1.91115
$698$	17.2721	0.653758
$699$	0	0
$700$	0	0
$701$	1.79899	0.0679469	0.0339735	−	0.999423i	$-0.489184\pi$
$701$	1.79899	0.0679469	0.0339735	+	0.999423i	$0.489184\pi$
$702$	0	0
$703$	−94.6690	−3.57051
$704$	−22.6274	−0.852803
$705$	0	0
$706$	8.48528	0.319348
$707$	2.82843	0.106374
$708$	0	0
$709$	−9.00000	−0.338002	−0.169001	−	0.985616i	$-0.554054\pi$
$709$	−9.00000	−0.338002	−0.169001	+	0.985616i	$0.554054\pi$
$710$	14.8284	0.556501
$711$	0	0
$712$	−25.9411	−0.972185
$713$	−11.3137	−0.423702
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	0	0
$718$	−35.6985	−1.33226
$719$	43.9706	1.63983	0.819913	−	0.572489i	$-0.194022\pi$
$719$	43.9706	1.63983	0.819913	+	0.572489i	$0.194022\pi$
$720$	0	0
$721$	−16.4853	−0.613944
$722$	−69.2132	−2.57585
$723$	0	0
$724$	0	0
$725$	−4.75736	−0.176684
$726$	0	0
$727$	−23.7279	−0.880020	−0.440010	−	0.897993i	$-0.645025\pi$
$727$	−23.7279	−0.880020	−0.440010	+	0.897993i	$0.645025\pi$
$728$	−5.65685	−0.209657
$729$	0	0
$730$	21.4558	0.794116
$731$	−58.2843	−2.15572
$732$	0	0
$733$	−34.1838	−1.26261	−0.631303	−	0.775536i	$-0.717480\pi$
$733$	−34.1838	−1.26261	−0.631303	+	0.775536i	$0.717480\pi$
$734$	−3.51472	−0.129731
$735$	0	0
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	24.5147	0.901789	0.450894	−	0.892577i	$-0.351105\pi$
$739$	24.5147	0.901789	0.450894	+	0.892577i	$0.351105\pi$
$740$	0	0
$741$	0	0
$742$	−10.7279	−0.393834
$743$	51.0416	1.87254	0.936268	−	0.351287i	$-0.114256\pi$
$743$	51.0416	1.87254	0.936268	+	0.351287i	$0.114256\pi$
$744$	0	0
$745$	8.48528	0.310877
$746$	11.3137	0.414224
$747$	0	0
$748$	0	0
$749$	1.41421	0.0516742
$750$	0	0
$751$	8.72792	0.318486	0.159243	−	0.987239i	$-0.449095\pi$
$751$	8.72792	0.318486	0.159243	+	0.987239i	$0.449095\pi$
$752$	−12.6863	−0.462621
$753$	0	0
$754$	4.48528	0.163344
$755$	−6.00000	−0.218362
$756$	0	0
$757$	11.4853	0.417440	0.208720	−	0.977975i	$-0.433070\pi$
$757$	11.4853	0.417440	0.208720	+	0.977975i	$0.433070\pi$
$758$	7.02944	0.255321
$759$	0	0
$760$	−32.9706	−1.19597
$761$	−28.9289	−1.04867	−0.524336	−	0.851511i	$-0.675687\pi$
$761$	−28.9289	−1.04867	−0.524336	+	0.851511i	$0.675687\pi$
$762$	0	0
$763$	0.242641	0.00878418
$764$	0	0
$765$	0	0
$766$	12.2426	0.442345
$767$	−14.1421	−0.510643
$768$	0	0
$769$	−14.2132	−0.512541	−0.256271	−	0.966605i	$-0.582494\pi$
$769$	−14.2132	−0.512541	−0.256271	+	0.966605i	$0.582494\pi$
$770$	5.65685	0.203859
$771$	0	0
$772$	0	0
$773$	43.6274	1.56917	0.784585	−	0.620022i	$-0.212876\pi$
$773$	43.6274	1.56917	0.784585	+	0.620022i	$0.212876\pi$
$774$	0	0
$775$	−12.0000	−0.431053
$776$	31.7990	1.14152
$777$	0	0
$778$	38.9706	1.39716
$779$	−71.3553	−2.55657
$780$	0	0
$781$	20.9706	0.750386
$782$	23.3137	0.833697
$783$	0	0
$784$	−4.00000	−0.142857
$785$	9.51472	0.339595
$786$	0	0
$787$	20.7279	0.738871	0.369435	−	0.929256i	$-0.379551\pi$
$787$	20.7279	0.738871	0.369435	+	0.929256i	$0.379551\pi$
$788$	0	0
$789$	0	0
$790$	22.0000	0.782725
$791$	−0.343146	−0.0122009
$792$	0	0
$793$	7.51472	0.266855
$794$	−12.6863	−0.450220
$795$	0	0
$796$	0	0
$797$	47.3137	1.67594	0.837969	−	0.545718i	$-0.183743\pi$
$797$	47.3137	1.67594	0.837969	+	0.545718i	$0.183743\pi$
$798$	0	0
$799$	18.4853	0.653962
$800$	0	0
$801$	0	0
$802$	14.7279	0.520061
$803$	30.3431	1.07079
$804$	0	0
$805$	−4.00000	−0.140981
$806$	11.3137	0.398508
$807$	0	0
$808$	8.00000	0.281439
$809$	3.89949	0.137099	0.0685495	−	0.997648i	$-0.478163\pi$
$809$	3.89949	0.137099	0.0685495	+	0.997648i	$0.478163\pi$
$810$	0	0
$811$	−3.02944	−0.106378	−0.0531890	−	0.998584i	$-0.516939\pi$
$811$	−3.02944	−0.106378	−0.0531890	+	0.998584i	$0.516939\pi$
$812$	0	0
$813$	0	0
$814$	45.9411	1.61024
$815$	16.2426	0.568955
$816$	0	0
$817$	82.4264	2.88373
$818$	13.4558	0.470473
$819$	0	0
$820$	0	0
$821$	28.7574	1.00364	0.501819	−	0.864972i	$-0.332664\pi$
$821$	28.7574	1.00364	0.501819	+	0.864972i	$0.332664\pi$
$822$	0	0
$823$	21.9706	0.765846	0.382923	−	0.923780i	$-0.374918\pi$
$823$	21.9706	0.765846	0.382923	+	0.923780i	$0.374918\pi$
$824$	−46.6274	−1.62434
$825$	0	0
$826$	−10.0000	−0.347945
$827$	9.38478	0.326341	0.163170	−	0.986598i	$-0.447828\pi$
$827$	9.38478	0.326341	0.163170	+	0.986598i	$0.447828\pi$
$828$	0	0
$829$	−5.45584	−0.189489	−0.0947446	−	0.995502i	$-0.530203\pi$
$829$	−5.45584	−0.189489	−0.0947446	+	0.995502i	$0.530203\pi$
$830$	23.3137	0.809231
$831$	0	0
$832$	−16.0000	−0.554700
$833$	5.82843	0.201943
$834$	0	0
$835$	−27.4558	−0.950149
$836$	0	0
$837$	0	0
$838$	−44.4853	−1.53672
$839$	−51.2132	−1.76808	−0.884038	−	0.467415i	$-0.845185\pi$
$839$	−51.2132	−1.76808	−0.884038	+	0.467415i	$0.845185\pi$
$840$	0	0
$841$	−26.4853	−0.913286
$842$	−31.4558	−1.08404
$843$	0	0
$844$	0	0
$845$	−12.7279	−0.437854
$846$	0	0
$847$	−3.00000	−0.103081
$848$	−30.3431	−1.04199
$849$	0	0
$850$	24.7279	0.848161
$851$	−32.4853	−1.11358
$852$	0	0
$853$	1.72792	0.0591629	0.0295815	−	0.999562i	$-0.490583\pi$
$853$	1.72792	0.0591629	0.0295815	+	0.999562i	$0.490583\pi$
$854$	5.31371	0.181831
$855$	0	0
$856$	4.00000	0.136717
$857$	−34.2843	−1.17113	−0.585564	−	0.810626i	$-0.699127\pi$
$857$	−34.2843	−1.17113	−0.585564	+	0.810626i	$0.699127\pi$
$858$	0	0
$859$	21.1838	0.722781	0.361390	−	0.932415i	$-0.382302\pi$
$859$	21.1838	0.722781	0.361390	+	0.932415i	$0.382302\pi$
$860$	0	0
$861$	0	0
$862$	42.4853	1.44705
$863$	−25.5858	−0.870950	−0.435475	−	0.900201i	$-0.643420\pi$
$863$	−25.5858	−0.870950	−0.435475	+	0.900201i	$0.643420\pi$
$864$	0	0
$865$	−18.4853	−0.628518
$866$	25.7990	0.876685
$867$	0	0
$868$	0	0
$869$	31.1127	1.05543
$870$	0	0
$871$	−8.48528	−0.287513
$872$	0.686292	0.0232408
$873$	0	0
$874$	−32.9706	−1.11525
$875$	−11.3137	−0.382473
$876$	0	0
$877$	−39.4853	−1.33332	−0.666662	−	0.745361i	$-0.732277\pi$
$877$	−39.4853	−1.33332	−0.666662	+	0.745361i	$0.732277\pi$
$878$	−11.6985	−0.394805
$879$	0	0
$880$	16.0000	0.539360
$881$	−47.6985	−1.60700	−0.803501	−	0.595303i	$-0.797032\pi$
$881$	−47.6985	−1.60700	−0.803501	+	0.595303i	$0.797032\pi$
$882$	0	0
$883$	−35.9411	−1.20952	−0.604758	−	0.796410i	$-0.706730\pi$
$883$	−35.9411	−1.20952	−0.604758	+	0.796410i	$0.706730\pi$
$884$	0	0
$885$	0	0
$886$	10.0000	0.335957
$887$	−18.3431	−0.615903	−0.307951	−	0.951402i	$-0.599643\pi$
$887$	−18.3431	−0.615903	−0.307951	+	0.951402i	$0.599643\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	18.3431	0.614864
$891$	0	0
$892$	0	0
$893$	−26.1421	−0.874813
$894$	0	0
$895$	−15.5147	−0.518600
$896$	−11.3137	−0.377964
$897$	0	0
$898$	42.9117	1.43198
$899$	6.34315	0.211556
$900$	0	0
$901$	44.2132	1.47296
$902$	34.6274	1.15297
$903$	0	0
$904$	−0.970563	−0.0322804
$905$	−18.7279	−0.622537
$906$	0	0
$907$	−51.9117	−1.72370	−0.861850	−	0.507164i	$-0.830694\pi$
$907$	−51.9117	−1.72370	−0.861850	+	0.507164i	$0.830694\pi$
$908$	0	0
$909$	0	0
$910$	4.00000	0.132599
$911$	29.3137	0.971206	0.485603	−	0.874179i	$-0.338600\pi$
$911$	29.3137	0.971206	0.485603	+	0.874179i	$0.338600\pi$
$912$	0	0
$913$	32.9706	1.09117
$914$	35.3553	1.16945
$915$	0	0
$916$	0	0
$917$	20.6569	0.682149
$918$	0	0
$919$	−35.4264	−1.16861	−0.584305	−	0.811534i	$-0.698633\pi$
$919$	−35.4264	−1.16861	−0.584305	+	0.811534i	$0.698633\pi$
$920$	−11.3137	−0.373002
$921$	0	0
$922$	22.0000	0.724531
$923$	14.8284	0.488084
$924$	0	0
$925$	−34.4558	−1.13290
$926$	9.85786	0.323950
$927$	0	0
$928$	0	0
$929$	58.6274	1.92350	0.961752	−	0.273923i	$-0.0883214\pi$
$929$	58.6274	1.92350	0.961752	+	0.273923i	$0.0883214\pi$
$930$	0	0
$931$	−8.24264	−0.270142
$932$	0	0
$933$	0	0
$934$	19.0294	0.622662
$935$	−23.3137	−0.762440
$936$	0	0
$937$	−24.4853	−0.799899	−0.399950	−	0.916537i	$-0.630972\pi$
$937$	−24.4853	−0.799899	−0.399950	+	0.916537i	$0.630972\pi$
$938$	−6.00000	−0.195907
$939$	0	0
$940$	0	0
$941$	12.5147	0.407968	0.203984	−	0.978974i	$-0.434611\pi$
$941$	12.5147	0.407968	0.203984	+	0.978974i	$0.434611\pi$
$942$	0	0
$943$	−24.4853	−0.797350
$944$	−28.2843	−0.920575
$945$	0	0
$946$	−40.0000	−1.30051
$947$	52.7574	1.71438	0.857192	−	0.514997i	$-0.172207\pi$
$947$	52.7574	1.71438	0.857192	+	0.514997i	$0.172207\pi$
$948$	0	0
$949$	21.4558	0.696486
$950$	−34.9706	−1.13459
$951$	0	0
$952$	16.4853	0.534291
$953$	−49.1543	−1.59226	−0.796132	−	0.605122i	$-0.793124\pi$
$953$	−49.1543	−1.59226	−0.796132	+	0.605122i	$0.793124\pi$
$954$	0	0
$955$	30.4853	0.986481
$956$	0	0
$957$	0	0
$958$	−15.7574	−0.509097
$959$	5.31371	0.171589
$960$	0	0
$961$	−15.0000	−0.483871
$962$	32.4853	1.04737
$963$	0	0
$964$	0	0
$965$	27.1716	0.874684
$966$	0	0
$967$	36.1838	1.16359	0.581796	−	0.813335i	$-0.302350\pi$
$967$	36.1838	1.16359	0.581796	+	0.813335i	$0.302350\pi$
$968$	−8.48528	−0.272727
$969$	0	0
$970$	−22.4853	−0.721959
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−3.24264	−0.103954
$974$	−54.7696	−1.75493
$975$	0	0
$976$	15.0294	0.481081
$977$	−35.3137	−1.12979	−0.564893	−	0.825164i	$-0.691082\pi$
$977$	−35.3137	−1.12979	−0.564893	+	0.825164i	$0.691082\pi$
$978$	0	0
$979$	25.9411	0.829082
$980$	0	0
$981$	0	0
$982$	8.00000	0.255290
$983$	15.3431	0.489370	0.244685	−	0.969603i	$-0.421315\pi$
$983$	15.3431	0.489370	0.244685	+	0.969603i	$0.421315\pi$
$984$	0	0
$985$	5.51472	0.175714
$986$	−13.0711	−0.416268
$987$	0	0
$988$	0	0
$989$	28.2843	0.899388
$990$	0	0
$991$	−18.2426	−0.579497	−0.289748	−	0.957103i	$-0.593572\pi$
$991$	−18.2426	−0.579497	−0.289748	+	0.957103i	$0.593572\pi$
$992$	0	0
$993$	0	0
$994$	10.4853	0.332573
$995$	0.686292	0.0217569
$996$	0	0
$997$	53.4558	1.69296	0.846482	−	0.532418i	$-0.178716\pi$
$997$	53.4558	1.69296	0.846482	+	0.532418i	$0.178716\pi$
$998$	1.79899	0.0569460
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.j.1.1		2
3.2	odd	2		2667.2.a.g.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.g.1.2	✓	2	3.2	odd	2
8001.2.a.j.1.1		2	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.41421 q^{5} +1.00000 q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +1.41421 q^{5} +1.00000 q^{7} +2.82843 q^{8} -2.00000 q^{10} -2.82843 q^{11} -2.00000 q^{13} -1.41421 q^{14} -4.00000 q^{16} +5.82843 q^{17} -8.24264 q^{19} +4.00000 q^{22} -2.82843 q^{23} -3.00000 q^{25} +2.82843 q^{26} +1.58579 q^{29} +4.00000 q^{31} -8.24264 q^{34} +1.41421 q^{35} +11.4853 q^{37} +11.6569 q^{38} +4.00000 q^{40} +8.65685 q^{41} -10.0000 q^{43} +4.00000 q^{46} +3.17157 q^{47} +1.00000 q^{49} +4.24264 q^{50} +7.58579 q^{53} -4.00000 q^{55} +2.82843 q^{56} -2.24264 q^{58} +7.07107 q^{59} -3.75736 q^{61} -5.65685 q^{62} +8.00000 q^{64} -2.82843 q^{65} +4.24264 q^{67} -2.00000 q^{70} -7.41421 q^{71} -10.7279 q^{73} -16.2426 q^{74} -2.82843 q^{77} -11.0000 q^{79} -5.65685 q^{80} -12.2426 q^{82} -11.6569 q^{83} +8.24264 q^{85} +14.1421 q^{86} -8.00000 q^{88} -9.17157 q^{89} -2.00000 q^{91} -4.48528 q^{94} -11.6569 q^{95} +11.2426 q^{97} -1.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} - 4 q^{10} - 4 q^{13} - 8 q^{16} + 6 q^{17} - 8 q^{19} + 8 q^{22} - 6 q^{25} + 6 q^{29} + 8 q^{31} - 8 q^{34} + 6 q^{37} + 12 q^{38} + 8 q^{40} + 6 q^{41} - 20 q^{43} + 8 q^{46} + 12 q^{47} + 2 q^{49} + 18 q^{53} - 8 q^{55} + 4 q^{58} - 16 q^{61} + 16 q^{64} - 4 q^{70} - 12 q^{71} + 4 q^{73} - 24 q^{74} - 22 q^{79} - 16 q^{82} - 12 q^{83} + 8 q^{85} - 16 q^{88} - 24 q^{89} - 4 q^{91} + 8 q^{94} - 12 q^{95} + 14 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 4 * q^10 - 4 * q^13 - 8 * q^16 + 6 * q^17 - 8 * q^19 + 8 * q^22 - 6 * q^25 + 6 * q^29 + 8 * q^31 - 8 * q^34 + 6 * q^37 + 12 * q^38 + 8 * q^40 + 6 * q^41 - 20 * q^43 + 8 * q^46 + 12 * q^47 + 2 * q^49 + 18 * q^53 - 8 * q^55 + 4 * q^58 - 16 * q^61 + 16 * q^64 - 4 * q^70 - 12 * q^71 + 4 * q^73 - 24 * q^74 - 22 * q^79 - 16 * q^82 - 12 * q^83 + 8 * q^85 - 16 * q^88 - 24 * q^89 - 4 * q^91 + 8 * q^94 - 12 * q^95 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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