LMFDB - Embedded newform 8001.2.a.j.1.2

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			1.2
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)

$f(q)$	$=$	$q+1.41421 q^{2} -1.41421 q^{5} +1.00000 q^{7} -2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} -1.41421 q^{5} +1.00000 q^{7} -2.82843 q^{8} -2.00000 q^{10} +2.82843 q^{11} -2.00000 q^{13} +1.41421 q^{14} -4.00000 q^{16} +0.171573 q^{17} +0.242641 q^{19} +4.00000 q^{22} +2.82843 q^{23} -3.00000 q^{25} -2.82843 q^{26} +4.41421 q^{29} +4.00000 q^{31} +0.242641 q^{34} -1.41421 q^{35} -5.48528 q^{37} +0.343146 q^{38} +4.00000 q^{40} -2.65685 q^{41} -10.0000 q^{43} +4.00000 q^{46} +8.82843 q^{47} +1.00000 q^{49} -4.24264 q^{50} +10.4142 q^{53} -4.00000 q^{55} -2.82843 q^{56} +6.24264 q^{58} -7.07107 q^{59} -12.2426 q^{61} +5.65685 q^{62} +8.00000 q^{64} +2.82843 q^{65} -4.24264 q^{67} -2.00000 q^{70} -4.58579 q^{71} +14.7279 q^{73} -7.75736 q^{74} +2.82843 q^{77} -11.0000 q^{79} +5.65685 q^{80} -3.75736 q^{82} -0.343146 q^{83} -0.242641 q^{85} -14.1421 q^{86} -8.00000 q^{88} -14.8284 q^{89} -2.00000 q^{91} +12.4853 q^{94} -0.343146 q^{95} +2.75736 q^{97} +1.41421 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	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	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	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
$17$	0.171573	0.0416125	0.0208063	−	0.999784i	$-0.493377\pi$
$17$	0.171573	0.0416125	0.0208063	+	0.999784i	$0.493377\pi$
$18$	0	0
$19$	0.242641	0.0556656	0.0278328	−	0.999613i	$-0.491139\pi$
$19$	0.242641	0.0556656	0.0278328	+	0.999613i	$0.491139\pi$
$20$	0	0
$21$	0	0
$22$	4.00000	0.852803
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	−2.82843	−0.554700
$27$	0	0
$28$	0	0
$29$	4.41421	0.819699	0.409849	−	0.912153i	$-0.365581\pi$
$29$	4.41421	0.819699	0.409849	+	0.912153i	$0.365581\pi$
$30$	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
$34$	0.242641	0.0416125
$35$	−1.41421	−0.239046
$36$	0	0
$37$	−5.48528	−0.901775	−0.450887	−	0.892581i	$-0.648892\pi$
$37$	−5.48528	−0.901775	−0.450887	+	0.892581i	$0.648892\pi$
$38$	0.343146	0.0556656
$39$	0	0
$40$	4.00000	0.632456
$41$	−2.65685	−0.414931	−0.207465	−	0.978242i	$-0.566521\pi$
$41$	−2.65685	−0.414931	−0.207465	+	0.978242i	$0.566521\pi$
$42$	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
$47$	8.82843	1.28776	0.643879	−	0.765127i	$-0.277324\pi$
$47$	8.82843	1.28776	0.643879	+	0.765127i	$0.277324\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.24264	−0.600000
$51$	0	0
$52$	0	0
$53$	10.4142	1.43050	0.715251	−	0.698868i	$-0.246312\pi$
$53$	10.4142	1.43050	0.715251	+	0.698868i	$0.246312\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	−2.82843	−0.377964
$57$	0	0
$58$	6.24264	0.819699
$59$	−7.07107	−0.920575	−0.460287	−	0.887770i	$-0.652254\pi$
$59$	−7.07107	−0.920575	−0.460287	+	0.887770i	$0.652254\pi$
$60$	0	0
$61$	−12.2426	−1.56751	−0.783755	−	0.621070i	$-0.786698\pi$
$61$	−12.2426	−1.56751	−0.783755	+	0.621070i	$0.786698\pi$
$62$	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	$-0.583446\pi$
$67$	−4.24264	−0.518321	−0.259161	+	0.965834i	$0.583446\pi$
$68$	0	0
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−4.58579	−0.544233	−0.272116	−	0.962264i	$-0.587724\pi$
$71$	−4.58579	−0.544233	−0.272116	+	0.962264i	$0.587724\pi$
$72$	0	0
$73$	14.7279	1.72377	0.861886	−	0.507101i	$-0.169283\pi$
$73$	14.7279	1.72377	0.861886	+	0.507101i	$0.169283\pi$
$74$	−7.75736	−0.901775
$75$	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
$82$	−3.75736	−0.414931
$83$	−0.343146	−0.0376651	−0.0188326	−	0.999823i	$-0.505995\pi$
$83$	−0.343146	−0.0376651	−0.0188326	+	0.999823i	$0.505995\pi$
$84$	0	0
$85$	−0.242641	−0.0263181
$86$	−14.1421	−1.52499
$87$	0	0
$88$	−8.00000	−0.852803
$89$	−14.8284	−1.57181	−0.785905	−	0.618347i	$-0.787803\pi$
$89$	−14.8284	−1.57181	−0.785905	+	0.618347i	$0.787803\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	12.4853	1.28776
$95$	−0.343146	−0.0352060
$96$	0	0
$97$	2.75736	0.279967	0.139984	−	0.990154i	$-0.455295\pi$
$97$	2.75736	0.279967	0.139984	+	0.990154i	$0.455295\pi$
$98$	1.41421	0.142857
$99$	0	0
$100$	0	0
$101$	−2.82843	−0.281439	−0.140720	−	0.990050i	$-0.544942\pi$
$101$	−2.82843	−0.281439	−0.140720	+	0.990050i	$0.544942\pi$
$102$	0	0
$103$	0.485281	0.0478162	0.0239081	−	0.999714i	$-0.492389\pi$
$103$	0.485281	0.0478162	0.0239081	+	0.999714i	$0.492389\pi$
$104$	5.65685	0.554700
$105$	0	0
$106$	14.7279	1.43050
$107$	−1.41421	−0.136717	−0.0683586	−	0.997661i	$-0.521776\pi$
$107$	−1.41421	−0.136717	−0.0683586	+	0.997661i	$0.521776\pi$
$108$	0	0
$109$	−8.24264	−0.789502	−0.394751	−	0.918788i	$-0.629169\pi$
$109$	−8.24264	−0.789502	−0.394751	+	0.918788i	$0.629169\pi$
$110$	−5.65685	−0.539360
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−11.6569	−1.09658	−0.548292	−	0.836287i	$-0.684722\pi$
$113$	−11.6569	−1.09658	−0.548292	+	0.836287i	$0.684722\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	−10.0000	−0.920575
$119$	0.171573	0.0157281
$120$	0	0
$121$	−3.00000	−0.272727
$122$	−17.3137	−1.56751
$123$	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
$131$	9.34315	0.816314	0.408157	−	0.912912i	$-0.366172\pi$
$131$	9.34315	0.816314	0.408157	+	0.912912i	$0.366172\pi$
$132$	0	0
$133$	0.242641	0.0210396
$134$	−6.00000	−0.518321
$135$	0	0
$136$	−0.485281	−0.0416125
$137$	−17.3137	−1.47921	−0.739605	−	0.673041i	$-0.764988\pi$
$137$	−17.3137	−1.47921	−0.739605	+	0.673041i	$0.764988\pi$
$138$	0	0
$139$	5.24264	0.444675	0.222337	−	0.974970i	$-0.428631\pi$
$139$	5.24264	0.444675	0.222337	+	0.974970i	$0.428631\pi$
$140$	0	0
$141$	0	0
$142$	−6.48528	−0.544233
$143$	−5.65685	−0.473050
$144$	0	0
$145$	−6.24264	−0.518423
$146$	20.8284	1.72377
$147$	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	$-0.444773\pi$
$151$	4.24264	0.345261	0.172631	+	0.984987i	$0.444773\pi$
$152$	−0.686292	−0.0556656
$153$	0	0
$154$	4.00000	0.322329
$155$	−5.65685	−0.454369
$156$	0	0
$157$	−18.7279	−1.49465	−0.747325	−	0.664458i	$-0.768662\pi$
$157$	−18.7279	−1.49465	−0.747325	+	0.664458i	$0.768662\pi$
$158$	−15.5563	−1.23760
$159$	0	0
$160$	0	0
$161$	2.82843	0.222911
$162$	0	0
$163$	−5.48528	−0.429640	−0.214820	−	0.976654i	$-0.568917\pi$
$163$	−5.48528	−0.429640	−0.214820	+	0.976654i	$0.568917\pi$
$164$	0	0
$165$	0	0
$166$	−0.485281	−0.0376651
$167$	−16.5858	−1.28345	−0.641723	−	0.766936i	$-0.721780\pi$
$167$	−16.5858	−1.28345	−0.641723	+	0.766936i	$0.721780\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−0.343146	−0.0263181
$171$	0	0
$172$	0	0
$173$	1.07107	0.0814318	0.0407159	−	0.999171i	$-0.487036\pi$
$173$	1.07107	0.0814318	0.0407159	+	0.999171i	$0.487036\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	−11.3137	−0.852803
$177$	0	0
$178$	−20.9706	−1.57181
$179$	22.9706	1.71690	0.858450	−	0.512897i	$-0.171428\pi$
$179$	22.9706	1.71690	0.858450	+	0.512897i	$0.171428\pi$
$180$	0	0
$181$	−4.75736	−0.353612	−0.176806	−	0.984246i	$-0.556576\pi$
$181$	−4.75736	−0.353612	−0.176806	+	0.984246i	$0.556576\pi$
$182$	−2.82843	−0.209657
$183$	0	0
$184$	−8.00000	−0.589768
$185$	7.75736	0.570332
$186$	0	0
$187$	0.485281	0.0354873
$188$	0	0
$189$	0	0
$190$	−0.485281	−0.0352060
$191$	−9.55635	−0.691473	−0.345737	−	0.938332i	$-0.612371\pi$
$191$	−9.55635	−0.691473	−0.345737	+	0.938332i	$0.612371\pi$
$192$	0	0
$193$	−23.2132	−1.67092	−0.835461	−	0.549549i	$-0.814800\pi$
$193$	−23.2132	−1.67092	−0.835461	+	0.549549i	$0.814800\pi$
$194$	3.89949	0.279967
$195$	0	0
$196$	0	0
$197$	−15.8995	−1.13279	−0.566396	−	0.824133i	$-0.691663\pi$
$197$	−15.8995	−1.13279	−0.566396	+	0.824133i	$0.691663\pi$
$198$	0	0
$199$	−16.4853	−1.16861	−0.584305	−	0.811534i	$-0.698633\pi$
$199$	−16.4853	−1.16861	−0.584305	+	0.811534i	$0.698633\pi$
$200$	8.48528	0.600000
$201$	0	0
$202$	−4.00000	−0.281439
$203$	4.41421	0.309817
$204$	0	0
$205$	3.75736	0.262425
$206$	0.686292	0.0478162
$207$	0	0
$208$	8.00000	0.554700
$209$	0.686292	0.0474718
$210$	0	0
$211$	−1.51472	−0.104278	−0.0521388	−	0.998640i	$-0.516604\pi$
$211$	−1.51472	−0.104278	−0.0521388	+	0.998640i	$0.516604\pi$
$212$	0	0
$213$	0	0
$214$	−2.00000	−0.136717
$215$	14.1421	0.964486
$216$	0	0
$217$	4.00000	0.271538
$218$	−11.6569	−0.789502
$219$	0	0
$220$	0	0
$221$	−0.343146	−0.0230825
$222$	0	0
$223$	−2.75736	−0.184646	−0.0923232	−	0.995729i	$-0.529429\pi$
$223$	−2.75736	−0.184646	−0.0923232	+	0.995729i	$0.529429\pi$
$224$	0	0
$225$	0	0
$226$	−16.4853	−1.09658
$227$	8.14214	0.540413	0.270206	−	0.962802i	$-0.412908\pi$
$227$	8.14214	0.540413	0.270206	+	0.962802i	$0.412908\pi$
$228$	0	0
$229$	−16.2132	−1.07140	−0.535699	−	0.844409i	$-0.679952\pi$
$229$	−16.2132	−1.07140	−0.535699	+	0.844409i	$0.679952\pi$
$230$	−5.65685	−0.373002
$231$	0	0
$232$	−12.4853	−0.819699
$233$	9.72792	0.637297	0.318649	−	0.947873i	$-0.396771\pi$
$233$	9.72792	0.637297	0.318649	+	0.947873i	$0.396771\pi$
$234$	0	0
$235$	−12.4853	−0.814450
$236$	0	0
$237$	0	0
$238$	0.242641	0.0157281
$239$	−13.9289	−0.900988	−0.450494	−	0.892780i	$-0.648752\pi$
$239$	−13.9289	−0.900988	−0.450494	+	0.892780i	$0.648752\pi$
$240$	0	0
$241$	16.7574	1.07944	0.539718	−	0.841846i	$-0.318531\pi$
$241$	16.7574	1.07944	0.539718	+	0.841846i	$0.318531\pi$
$242$	−4.24264	−0.272727
$243$	0	0
$244$	0	0
$245$	−1.41421	−0.0903508
$246$	0	0
$247$	−0.485281	−0.0308777
$248$	−11.3137	−0.718421
$249$	0	0
$250$	16.0000	1.01193
$251$	−11.8284	−0.746604	−0.373302	−	0.927710i	$-0.621774\pi$
$251$	−11.8284	−0.746604	−0.373302	+	0.927710i	$0.621774\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	−1.41421	−0.0887357
$255$	0	0
$256$	0	0
$257$	−7.41421	−0.462486	−0.231243	−	0.972896i	$-0.574279\pi$
$257$	−7.41421	−0.462486	−0.231243	+	0.972896i	$0.574279\pi$
$258$	0	0
$259$	−5.48528	−0.340839
$260$	0	0
$261$	0	0
$262$	13.2132	0.816314
$263$	−4.58579	−0.282772	−0.141386	−	0.989955i	$-0.545156\pi$
$263$	−4.58579	−0.282772	−0.141386	+	0.989955i	$0.545156\pi$
$264$	0	0
$265$	−14.7279	−0.904729
$266$	0.343146	0.0210396
$267$	0	0
$268$	0	0
$269$	19.9706	1.21763	0.608813	−	0.793313i	$-0.291646\pi$
$269$	19.9706	1.21763	0.608813	+	0.793313i	$0.291646\pi$
$270$	0	0
$271$	10.2426	0.622196	0.311098	−	0.950378i	$-0.399303\pi$
$271$	10.2426	0.622196	0.311098	+	0.950378i	$0.399303\pi$
$272$	−0.686292	−0.0416125
$273$	0	0
$274$	−24.4853	−1.47921
$275$	−8.48528	−0.511682
$276$	0	0
$277$	−20.4853	−1.23084	−0.615421	−	0.788199i	$-0.711014\pi$
$277$	−20.4853	−1.23084	−0.615421	+	0.788199i	$0.711014\pi$
$278$	7.41421	0.444675
$279$	0	0
$280$	4.00000	0.239046
$281$	24.5563	1.46491	0.732454	−	0.680816i	$-0.238375\pi$
$281$	24.5563	1.46491	0.732454	+	0.680816i	$0.238375\pi$
$282$	0	0
$283$	14.2132	0.844887	0.422444	−	0.906389i	$-0.361172\pi$
$283$	14.2132	0.844887	0.422444	+	0.906389i	$0.361172\pi$
$284$	0	0
$285$	0	0
$286$	−8.00000	−0.473050
$287$	−2.65685	−0.156829
$288$	0	0
$289$	−16.9706	−0.998268
$290$	−8.82843	−0.518423
$291$	0	0
$292$	0	0
$293$	−15.5563	−0.908812	−0.454406	−	0.890795i	$-0.650148\pi$
$293$	−15.5563	−0.908812	−0.454406	+	0.890795i	$0.650148\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	15.5147	0.901775
$297$	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
$304$	−0.970563	−0.0556656
$305$	17.3137	0.991380
$306$	0	0
$307$	−6.75736	−0.385663	−0.192831	−	0.981232i	$-0.561767\pi$
$307$	−6.75736	−0.385663	−0.192831	+	0.981232i	$0.561767\pi$
$308$	0	0
$309$	0	0
$310$	−8.00000	−0.454369
$311$	−4.24264	−0.240578	−0.120289	−	0.992739i	$-0.538382\pi$
$311$	−4.24264	−0.240578	−0.120289	+	0.992739i	$0.538382\pi$
$312$	0	0
$313$	15.7279	0.888995	0.444497	−	0.895780i	$-0.353382\pi$
$313$	15.7279	0.888995	0.444497	+	0.895780i	$0.353382\pi$
$314$	−26.4853	−1.49465
$315$	0	0
$316$	0	0
$317$	−20.1421	−1.13130	−0.565648	−	0.824647i	$-0.691374\pi$
$317$	−20.1421	−1.13130	−0.565648	+	0.824647i	$0.691374\pi$
$318$	0	0
$319$	12.4853	0.699042
$320$	−11.3137	−0.632456
$321$	0	0
$322$	4.00000	0.222911
$323$	0.0416306	0.00231639
$324$	0	0
$325$	6.00000	0.332820
$326$	−7.75736	−0.429640
$327$	0	0
$328$	7.51472	0.414931
$329$	8.82843	0.486727
$330$	0	0
$331$	−10.7279	−0.589660	−0.294830	−	0.955550i	$-0.595263\pi$
$331$	−10.7279	−0.589660	−0.294830	+	0.955550i	$0.595263\pi$
$332$	0	0
$333$	0	0
$334$	−23.4558	−1.28345
$335$	6.00000	0.327815
$336$	0	0
$337$	−26.2426	−1.42953	−0.714764	−	0.699366i	$-0.753466\pi$
$337$	−26.2426	−1.42953	−0.714764	+	0.699366i	$0.753466\pi$
$338$	−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
$346$	1.51472	0.0814318
$347$	22.0711	1.18484	0.592418	−	0.805630i	$-0.298173\pi$
$347$	22.0711	1.18484	0.592418	+	0.805630i	$0.298173\pi$
$348$	0	0
$349$	30.2132	1.61728	0.808638	−	0.588307i	$-0.200205\pi$
$349$	30.2132	1.61728	0.808638	+	0.588307i	$0.200205\pi$
$350$	−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
$355$	6.48528	0.344203
$356$	0	0
$357$	0	0
$358$	32.4853	1.71690
$359$	16.7574	0.884420	0.442210	−	0.896912i	$-0.354195\pi$
$359$	16.7574	0.884420	0.442210	+	0.896912i	$0.354195\pi$
$360$	0	0
$361$	−18.9411	−0.996901
$362$	−6.72792	−0.353612
$363$	0	0
$364$	0	0
$365$	−20.8284	−1.09021
$366$	0	0
$367$	−14.4853	−0.756126	−0.378063	−	0.925780i	$-0.623410\pi$
$367$	−14.4853	−0.756126	−0.378063	+	0.925780i	$0.623410\pi$
$368$	−11.3137	−0.589768
$369$	0	0
$370$	10.9706	0.570332
$371$	10.4142	0.540679
$372$	0	0
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	0.686292	0.0354873
$375$	0	0
$376$	−24.9706	−1.28776
$377$	−8.82843	−0.454687
$378$	0	0
$379$	28.9706	1.48812	0.744059	−	0.668114i	$-0.232898\pi$
$379$	28.9706	1.48812	0.744059	+	0.668114i	$0.232898\pi$
$380$	0	0
$381$	0	0
$382$	−13.5147	−0.691473
$383$	2.65685	0.135759	0.0678795	−	0.997694i	$-0.478377\pi$
$383$	2.65685	0.135759	0.0678795	+	0.997694i	$0.478377\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	−32.8284	−1.67092
$387$	0	0
$388$	0	0
$389$	3.55635	0.180314	0.0901570	−	0.995928i	$-0.471263\pi$
$389$	3.55635	0.180314	0.0901570	+	0.995928i	$0.471263\pi$
$390$	0	0
$391$	0.485281	0.0245417
$392$	−2.82843	−0.142857
$393$	0	0
$394$	−22.4853	−1.13279
$395$	15.5563	0.782725
$396$	0	0
$397$	−24.9706	−1.25324	−0.626618	−	0.779326i	$-0.715561\pi$
$397$	−24.9706	−1.25324	−0.626618	+	0.779326i	$0.715561\pi$
$398$	−23.3137	−1.16861
$399$	0	0
$400$	12.0000	0.600000
$401$	−7.58579	−0.378816	−0.189408	−	0.981898i	$-0.560657\pi$
$401$	−7.58579	−0.378816	−0.189408	+	0.981898i	$0.560657\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	6.24264	0.309817
$407$	−15.5147	−0.769036
$408$	0	0
$409$	−26.4853	−1.30961	−0.654806	−	0.755797i	$-0.727250\pi$
$409$	−26.4853	−1.30961	−0.654806	+	0.755797i	$0.727250\pi$
$410$	5.31371	0.262425
$411$	0	0
$412$	0	0
$413$	−7.07107	−0.347945
$414$	0	0
$415$	0.485281	0.0238215
$416$	0	0
$417$	0	0
$418$	0.970563	0.0474718
$419$	−19.4558	−0.950480	−0.475240	−	0.879856i	$-0.657639\pi$
$419$	−19.4558	−0.950480	−0.475240	+	0.879856i	$0.657639\pi$
$420$	0	0
$421$	13.7574	0.670493	0.335246	−	0.942131i	$-0.391180\pi$
$421$	13.7574	0.670493	0.335246	+	0.942131i	$0.391180\pi$
$422$	−2.14214	−0.104278
$423$	0	0
$424$	−29.4558	−1.43050
$425$	−0.514719	−0.0249675
$426$	0	0
$427$	−12.2426	−0.592463
$428$	0	0
$429$	0	0
$430$	20.0000	0.964486
$431$	18.0416	0.869035	0.434517	−	0.900663i	$-0.356919\pi$
$431$	18.0416	0.869035	0.434517	+	0.900663i	$0.356919\pi$
$432$	0	0
$433$	−9.75736	−0.468909	−0.234454	−	0.972127i	$-0.575330\pi$
$433$	−9.75736	−0.468909	−0.234454	+	0.972127i	$0.575330\pi$
$434$	5.65685	0.271538
$435$	0	0
$436$	0	0
$437$	0.686292	0.0328298
$438$	0	0
$439$	33.7279	1.60975	0.804873	−	0.593447i	$-0.202233\pi$
$439$	33.7279	1.60975	0.804873	+	0.593447i	$0.202233\pi$
$440$	11.3137	0.539360
$441$	0	0
$442$	−0.485281	−0.0230825
$443$	7.07107	0.335957	0.167978	−	0.985791i	$-0.446276\pi$
$443$	7.07107	0.335957	0.167978	+	0.985791i	$0.446276\pi$
$444$	0	0
$445$	20.9706	0.994100
$446$	−3.89949	−0.184646
$447$	0	0
$448$	8.00000	0.377964
$449$	−41.6569	−1.96591	−0.982954	−	0.183850i	$-0.941144\pi$
$449$	−41.6569	−1.96591	−0.982954	+	0.183850i	$0.941144\pi$
$450$	0	0
$451$	−7.51472	−0.353854
$452$	0	0
$453$	0	0
$454$	11.5147	0.540413
$455$	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	$0.698798\pi$
$458$	−22.9289	−1.07140
$459$	0	0
$460$	0	0
$461$	15.5563	0.724531	0.362266	−	0.932075i	$-0.382003\pi$
$461$	15.5563	0.724531	0.362266	+	0.932075i	$0.382003\pi$
$462$	0	0
$463$	26.9706	1.25343	0.626714	−	0.779249i	$-0.284399\pi$
$463$	26.9706	1.25343	0.626714	+	0.779249i	$0.284399\pi$
$464$	−17.6569	−0.819699
$465$	0	0
$466$	13.7574	0.637297
$467$	37.4558	1.73325	0.866625	−	0.498960i	$-0.166285\pi$
$467$	37.4558	1.73325	0.866625	+	0.498960i	$0.166285\pi$
$468$	0	0
$469$	−4.24264	−0.195907
$470$	−17.6569	−0.814450
$471$	0	0
$472$	20.0000	0.920575
$473$	−28.2843	−1.30051
$474$	0	0
$475$	−0.727922	−0.0333994
$476$	0	0
$477$	0	0
$478$	−19.6985	−0.900988
$479$	−17.1421	−0.783244	−0.391622	−	0.920126i	$-0.628086\pi$
$479$	−17.1421	−0.783244	−0.391622	+	0.920126i	$0.628086\pi$
$480$	0	0
$481$	10.9706	0.500215
$482$	23.6985	1.07944
$483$	0	0
$484$	0	0
$485$	−3.89949	−0.177067
$486$	0	0
$487$	13.2721	0.601415	0.300708	−	0.953716i	$-0.402777\pi$
$487$	13.2721	0.601415	0.300708	+	0.953716i	$0.402777\pi$
$488$	34.6274	1.56751
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	5.65685	0.255290	0.127645	−	0.991820i	$-0.459258\pi$
$491$	5.65685	0.255290	0.127645	+	0.991820i	$0.459258\pi$
$492$	0	0
$493$	0.757359	0.0341097
$494$	−0.686292	−0.0308777
$495$	0	0
$496$	−16.0000	−0.718421
$497$	−4.58579	−0.205701
$498$	0	0
$499$	−26.7279	−1.19651	−0.598253	−	0.801307i	$-0.704138\pi$
$499$	−26.7279	−1.19651	−0.598253	+	0.801307i	$0.704138\pi$
$500$	0	0
$501$	0	0
$502$	−16.7279	−0.746604
$503$	21.3431	0.951644	0.475822	−	0.879542i	$-0.342151\pi$
$503$	21.3431	0.951644	0.475822	+	0.879542i	$0.342151\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	11.3137	0.502956
$507$	0	0
$508$	0	0
$509$	−4.79899	−0.212711	−0.106356	−	0.994328i	$-0.533918\pi$
$509$	−4.79899	−0.212711	−0.106356	+	0.994328i	$0.533918\pi$
$510$	0	0
$511$	14.7279	0.651525
$512$	−22.6274	−1.00000
$513$	0	0
$514$	−10.4853	−0.462486
$515$	−0.686292	−0.0302416
$516$	0	0
$517$	24.9706	1.09820
$518$	−7.75736	−0.340839
$519$	0	0
$520$	−8.00000	−0.350823
$521$	−28.9706	−1.26922	−0.634612	−	0.772831i	$-0.718840\pi$
$521$	−28.9706	−1.26922	−0.634612	+	0.772831i	$0.718840\pi$
$522$	0	0
$523$	19.6985	0.861355	0.430677	−	0.902506i	$-0.358275\pi$
$523$	19.6985	0.861355	0.430677	+	0.902506i	$0.358275\pi$
$524$	0	0
$525$	0	0
$526$	−6.48528	−0.282772
$527$	0.686292	0.0298953
$528$	0	0
$529$	−15.0000	−0.652174
$530$	−20.8284	−0.904729
$531$	0	0
$532$	0	0
$533$	5.31371	0.230162
$534$	0	0
$535$	2.00000	0.0864675
$536$	12.0000	0.518321
$537$	0	0
$538$	28.2426	1.21763
$539$	2.82843	0.121829
$540$	0	0
$541$	0.485281	0.0208639	0.0104319	−	0.999946i	$-0.496679\pi$
$541$	0.485281	0.0208639	0.0104319	+	0.999946i	$0.496679\pi$
$542$	14.4853	0.622196
$543$	0	0
$544$	0	0
$545$	11.6569	0.499325
$546$	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
$551$	1.07107	0.0456290
$552$	0	0
$553$	−11.0000	−0.467768
$554$	−28.9706	−1.23084
$555$	0	0
$556$	0	0
$557$	23.3137	0.987834	0.493917	−	0.869509i	$-0.335565\pi$
$557$	23.3137	0.987834	0.493917	+	0.869509i	$0.335565\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	5.65685	0.239046
$561$	0	0
$562$	34.7279	1.46491
$563$	−26.8284	−1.13068	−0.565342	−	0.824857i	$-0.691256\pi$
$563$	−26.8284	−1.13068	−0.565342	+	0.824857i	$0.691256\pi$
$564$	0	0
$565$	16.4853	0.693541
$566$	20.1005	0.844887
$567$	0	0
$568$	12.9706	0.544233
$569$	−20.1421	−0.844402	−0.422201	−	0.906502i	$-0.638742\pi$
$569$	−20.1421	−0.844402	−0.422201	+	0.906502i	$0.638742\pi$
$570$	0	0
$571$	21.7574	0.910517	0.455259	−	0.890359i	$-0.349547\pi$
$571$	21.7574	0.910517	0.455259	+	0.890359i	$0.349547\pi$
$572$	0	0
$573$	0	0
$574$	−3.75736	−0.156829
$575$	−8.48528	−0.353861
$576$	0	0
$577$	−21.7574	−0.905771	−0.452885	−	0.891569i	$-0.649605\pi$
$577$	−21.7574	−0.905771	−0.452885	+	0.891569i	$0.649605\pi$
$578$	−24.0000	−0.998268
$579$	0	0
$580$	0	0
$581$	−0.343146	−0.0142361
$582$	0	0
$583$	29.4558	1.21994
$584$	−41.6569	−1.72377
$585$	0	0
$586$	−22.0000	−0.908812
$587$	−25.6274	−1.05776	−0.528878	−	0.848698i	$-0.677387\pi$
$587$	−25.6274	−1.05776	−0.528878	+	0.848698i	$0.677387\pi$
$588$	0	0
$589$	0.970563	0.0399913
$590$	14.1421	0.582223
$591$	0	0
$592$	21.9411	0.901775
$593$	−3.51472	−0.144332	−0.0721661	−	0.997393i	$-0.522991\pi$
$593$	−3.51472	−0.144332	−0.0721661	+	0.997393i	$0.522991\pi$
$594$	0	0
$595$	−0.242641	−0.00994730
$596$	0	0
$597$	0	0
$598$	−8.00000	−0.327144
$599$	−36.5563	−1.49365	−0.746826	−	0.665020i	$-0.768423\pi$
$599$	−36.5563	−1.49365	−0.746826	+	0.665020i	$0.768423\pi$
$600$	0	0
$601$	14.2132	0.579769	0.289884	−	0.957062i	$-0.406383\pi$
$601$	14.2132	0.579769	0.289884	+	0.957062i	$0.406383\pi$
$602$	−14.1421	−0.576390
$603$	0	0
$604$	0	0
$605$	4.24264	0.172488
$606$	0	0
$607$	33.6985	1.36778	0.683890	−	0.729585i	$-0.260287\pi$
$607$	33.6985	1.36778	0.683890	+	0.729585i	$0.260287\pi$
$608$	0	0
$609$	0	0
$610$	24.4853	0.991380
$611$	−17.6569	−0.714320
$612$	0	0
$613$	−9.02944	−0.364696	−0.182348	−	0.983234i	$-0.558370\pi$
$613$	−9.02944	−0.364696	−0.182348	+	0.983234i	$0.558370\pi$
$614$	−9.55635	−0.385663
$615$	0	0
$616$	−8.00000	−0.322329
$617$	33.0416	1.33021	0.665103	−	0.746752i	$-0.268388\pi$
$617$	33.0416	1.33021	0.665103	+	0.746752i	$0.268388\pi$
$618$	0	0
$619$	11.2426	0.451880	0.225940	−	0.974141i	$-0.427455\pi$
$619$	11.2426	0.451880	0.225940	+	0.974141i	$0.427455\pi$
$620$	0	0
$621$	0	0
$622$	−6.00000	−0.240578
$623$	−14.8284	−0.594088
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	22.2426	0.888995
$627$	0	0
$628$	0	0
$629$	−0.941125	−0.0375251
$630$	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
$634$	−28.4853	−1.13130
$635$	1.41421	0.0561214
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	17.6569	0.699042
$639$	0	0
$640$	−16.0000	−0.632456
$641$	10.7574	0.424890	0.212445	−	0.977173i	$-0.431857\pi$
$641$	10.7574	0.424890	0.212445	+	0.977173i	$0.431857\pi$
$642$	0	0
$643$	41.6985	1.64443	0.822214	−	0.569179i	$-0.192739\pi$
$643$	41.6985	1.64443	0.822214	+	0.569179i	$0.192739\pi$
$644$	0	0
$645$	0	0
$646$	0.0588745	0.00231639
$647$	1.79899	0.0707256	0.0353628	−	0.999375i	$-0.488741\pi$
$647$	1.79899	0.0707256	0.0353628	+	0.999375i	$0.488741\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	8.48528	0.332820
$651$	0	0
$652$	0	0
$653$	−16.2426	−0.635624	−0.317812	−	0.948154i	$-0.602948\pi$
$653$	−16.2426	−0.635624	−0.317812	+	0.948154i	$0.602948\pi$
$654$	0	0
$655$	−13.2132	−0.516283
$656$	10.6274	0.414931
$657$	0	0
$658$	12.4853	0.486727
$659$	11.3137	0.440720	0.220360	−	0.975419i	$-0.429277\pi$
$659$	11.3137	0.440720	0.220360	+	0.975419i	$0.429277\pi$
$660$	0	0
$661$	42.7279	1.66192	0.830962	−	0.556330i	$-0.187791\pi$
$661$	42.7279	1.66192	0.830962	+	0.556330i	$0.187791\pi$
$662$	−15.1716	−0.589660
$663$	0	0
$664$	0.970563	0.0376651
$665$	−0.343146	−0.0133066
$666$	0	0
$667$	12.4853	0.483432
$668$	0	0
$669$	0	0
$670$	8.48528	0.327815
$671$	−34.6274	−1.33678
$672$	0	0
$673$	42.4558	1.63655	0.818276	−	0.574825i	$-0.194930\pi$
$673$	42.4558	1.63655	0.818276	+	0.574825i	$0.194930\pi$
$674$	−37.1127	−1.42953
$675$	0	0
$676$	0	0
$677$	37.2843	1.43295	0.716475	−	0.697612i	$-0.245754\pi$
$677$	37.2843	1.43295	0.716475	+	0.697612i	$0.245754\pi$
$678$	0	0
$679$	2.75736	0.105818
$680$	0.686292	0.0263181
$681$	0	0
$682$	16.0000	0.612672
$683$	−25.4558	−0.974041	−0.487020	−	0.873391i	$-0.661916\pi$
$683$	−25.4558	−0.974041	−0.487020	+	0.873391i	$0.661916\pi$
$684$	0	0
$685$	24.4853	0.935535
$686$	1.41421	0.0539949
$687$	0	0
$688$	40.0000	1.52499
$689$	−20.8284	−0.793500
$690$	0	0
$691$	4.21320	0.160278	0.0801389	−	0.996784i	$-0.474464\pi$
$691$	4.21320	0.160278	0.0801389	+	0.996784i	$0.474464\pi$
$692$	0	0
$693$	0	0
$694$	31.2132	1.18484
$695$	−7.41421	−0.281237
$696$	0	0
$697$	−0.455844	−0.0172663
$698$	42.7279	1.61728
$699$	0	0
$700$	0	0
$701$	−37.7990	−1.42765	−0.713824	−	0.700325i	$-0.753038\pi$
$701$	−37.7990	−1.42765	−0.713824	+	0.700325i	$0.753038\pi$
$702$	0	0
$703$	−1.33095	−0.0501978
$704$	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	$0.554054\pi$
$710$	9.17157	0.344203
$711$	0	0
$712$	41.9411	1.57181
$713$	11.3137	0.423702
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	0	0
$718$	23.6985	0.884420
$719$	10.0294	0.374035	0.187017	−	0.982357i	$-0.440118\pi$
$719$	10.0294	0.374035	0.187017	+	0.982357i	$0.440118\pi$
$720$	0	0
$721$	0.485281	0.0180728
$722$	−26.7868	−0.996901
$723$	0	0
$724$	0	0
$725$	−13.2426	−0.491819
$726$	0	0
$727$	1.72792	0.0640851	0.0320425	−	0.999487i	$-0.489799\pi$
$727$	1.72792	0.0640851	0.0320425	+	0.999487i	$0.489799\pi$
$728$	5.65685	0.209657
$729$	0	0
$730$	−29.4558	−1.09021
$731$	−1.71573	−0.0634585
$732$	0	0
$733$	42.1838	1.55809	0.779046	−	0.626966i	$-0.215704\pi$
$733$	42.1838	1.55809	0.779046	+	0.626966i	$0.215704\pi$
$734$	−20.4853	−0.756126
$735$	0	0
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	41.4853	1.52606	0.763030	−	0.646363i	$-0.223711\pi$
$739$	41.4853	1.52606	0.763030	+	0.646363i	$0.223711\pi$
$740$	0	0
$741$	0	0
$742$	14.7279	0.540679
$743$	2.95837	0.108532	0.0542660	−	0.998527i	$-0.482718\pi$
$743$	2.95837	0.108532	0.0542660	+	0.998527i	$0.482718\pi$
$744$	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
$751$	−16.7279	−0.610411	−0.305205	−	0.952287i	$-0.598725\pi$
$751$	−16.7279	−0.610411	−0.305205	+	0.952287i	$0.598725\pi$
$752$	−35.3137	−1.28776
$753$	0	0
$754$	−12.4853	−0.454687
$755$	−6.00000	−0.218362
$756$	0	0
$757$	−5.48528	−0.199366	−0.0996830	−	0.995019i	$-0.531783\pi$
$757$	−5.48528	−0.199366	−0.0996830	+	0.995019i	$0.531783\pi$
$758$	40.9706	1.48812
$759$	0	0
$760$	0.970563	0.0352060
$761$	−43.0711	−1.56132	−0.780662	−	0.624953i	$-0.785118\pi$
$761$	−43.0711	−1.56132	−0.780662	+	0.624953i	$0.785118\pi$
$762$	0	0
$763$	−8.24264	−0.298404
$764$	0	0
$765$	0	0
$766$	3.75736	0.135759
$767$	14.1421	0.510643
$768$	0	0
$769$	28.2132	1.01739	0.508697	−	0.860946i	$-0.330127\pi$
$769$	28.2132	1.01739	0.508697	+	0.860946i	$0.330127\pi$
$770$	−5.65685	−0.203859
$771$	0	0
$772$	0	0
$773$	−1.62742	−0.0585341	−0.0292671	−	0.999572i	$-0.509317\pi$
$773$	−1.62742	−0.0585341	−0.0292671	+	0.999572i	$0.509317\pi$
$774$	0	0
$775$	−12.0000	−0.431053
$776$	−7.79899	−0.279967
$777$	0	0
$778$	5.02944	0.180314
$779$	−0.644661	−0.0230974
$780$	0	0
$781$	−12.9706	−0.464123
$782$	0.686292	0.0245417
$783$	0	0
$784$	−4.00000	−0.142857
$785$	26.4853	0.945300
$786$	0	0
$787$	−4.72792	−0.168532	−0.0842661	−	0.996443i	$-0.526855\pi$
$787$	−4.72792	−0.168532	−0.0842661	+	0.996443i	$0.526855\pi$
$788$	0	0
$789$	0	0
$790$	22.0000	0.782725
$791$	−11.6569	−0.414470
$792$	0	0
$793$	24.4853	0.869498
$794$	−35.3137	−1.25324
$795$	0	0
$796$	0	0
$797$	24.6863	0.874433	0.437217	−	0.899356i	$-0.355964\pi$
$797$	24.6863	0.874433	0.437217	+	0.899356i	$0.355964\pi$
$798$	0	0
$799$	1.51472	0.0535869
$800$	0	0
$801$	0	0
$802$	−10.7279	−0.378816
$803$	41.6569	1.47004
$804$	0	0
$805$	−4.00000	−0.140981
$806$	−11.3137	−0.398508
$807$	0	0
$808$	8.00000	0.281439
$809$	−15.8995	−0.558996	−0.279498	−	0.960146i	$-0.590168\pi$
$809$	−15.8995	−0.558996	−0.279498	+	0.960146i	$0.590168\pi$
$810$	0	0
$811$	−36.9706	−1.29821	−0.649106	−	0.760698i	$-0.724857\pi$
$811$	−36.9706	−1.29821	−0.649106	+	0.760698i	$0.724857\pi$
$812$	0	0
$813$	0	0
$814$	−21.9411	−0.769036
$815$	7.75736	0.271728
$816$	0	0
$817$	−2.42641	−0.0848892
$818$	−37.4558	−1.30961
$819$	0	0
$820$	0	0
$821$	37.2426	1.29978	0.649889	−	0.760030i	$-0.274816\pi$
$821$	37.2426	1.29978	0.649889	+	0.760030i	$0.274816\pi$
$822$	0	0
$823$	−11.9706	−0.417268	−0.208634	−	0.977994i	$-0.566902\pi$
$823$	−11.9706	−0.417268	−0.208634	+	0.977994i	$0.566902\pi$
$824$	−1.37258	−0.0478162
$825$	0	0
$826$	−10.0000	−0.347945
$827$	−27.3848	−0.952262	−0.476131	−	0.879374i	$-0.657961\pi$
$827$	−27.3848	−0.952262	−0.476131	+	0.879374i	$0.657961\pi$
$828$	0	0
$829$	45.4558	1.57875	0.789373	−	0.613913i	$-0.210406\pi$
$829$	45.4558	1.57875	0.789373	+	0.613913i	$0.210406\pi$
$830$	0.686292	0.0238215
$831$	0	0
$832$	−16.0000	−0.554700
$833$	0.171573	0.00594465
$834$	0	0
$835$	23.4558	0.811723
$836$	0	0
$837$	0	0
$838$	−27.5147	−0.950480
$839$	−8.78680	−0.303354	−0.151677	−	0.988430i	$-0.548467\pi$
$839$	−8.78680	−0.303354	−0.151677	+	0.988430i	$0.548467\pi$
$840$	0	0
$841$	−9.51472	−0.328094
$842$	19.4558	0.670493
$843$	0	0
$844$	0	0
$845$	12.7279	0.437854
$846$	0	0
$847$	−3.00000	−0.103081
$848$	−41.6569	−1.43050
$849$	0	0
$850$	−0.727922	−0.0249675
$851$	−15.5147	−0.531838
$852$	0	0
$853$	−23.7279	−0.812429	−0.406214	−	0.913778i	$-0.633151\pi$
$853$	−23.7279	−0.812429	−0.406214	+	0.913778i	$0.633151\pi$
$854$	−17.3137	−0.592463
$855$	0	0
$856$	4.00000	0.136717
$857$	22.2843	0.761216	0.380608	−	0.924736i	$-0.375715\pi$
$857$	22.2843	0.761216	0.380608	+	0.924736i	$0.375715\pi$
$858$	0	0
$859$	−55.1838	−1.88285	−0.941423	−	0.337228i	$-0.890511\pi$
$859$	−55.1838	−1.88285	−0.941423	+	0.337228i	$0.890511\pi$
$860$	0	0
$861$	0	0
$862$	25.5147	0.869035
$863$	−28.4142	−0.967231	−0.483616	−	0.875281i	$-0.660677\pi$
$863$	−28.4142	−0.967231	−0.483616	+	0.875281i	$0.660677\pi$
$864$	0	0
$865$	−1.51472	−0.0515020
$866$	−13.7990	−0.468909
$867$	0	0
$868$	0	0
$869$	−31.1127	−1.05543
$870$	0	0
$871$	8.48528	0.287513
$872$	23.3137	0.789502
$873$	0	0
$874$	0.970563	0.0328298
$875$	11.3137	0.382473
$876$	0	0
$877$	−22.5147	−0.760268	−0.380134	−	0.924931i	$-0.624122\pi$
$877$	−22.5147	−0.760268	−0.380134	+	0.924931i	$0.624122\pi$
$878$	47.6985	1.60975
$879$	0	0
$880$	16.0000	0.539360
$881$	11.6985	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$881$	11.6985	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$882$	0	0
$883$	31.9411	1.07490	0.537452	−	0.843294i	$-0.319387\pi$
$883$	31.9411	1.07490	0.537452	+	0.843294i	$0.319387\pi$
$884$	0	0
$885$	0	0
$886$	10.0000	0.335957
$887$	−29.6569	−0.995780	−0.497890	−	0.867240i	$-0.665892\pi$
$887$	−29.6569	−0.995780	−0.497890	+	0.867240i	$0.665892\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	29.6569	0.994100
$891$	0	0
$892$	0	0
$893$	2.14214	0.0716838
$894$	0	0
$895$	−32.4853	−1.08586
$896$	11.3137	0.377964
$897$	0	0
$898$	−58.9117	−1.96591
$899$	17.6569	0.588889
$900$	0	0
$901$	1.78680	0.0595268
$902$	−10.6274	−0.353854
$903$	0	0
$904$	32.9706	1.09658
$905$	6.72792	0.223644
$906$	0	0
$907$	49.9117	1.65729	0.828645	−	0.559774i	$-0.189112\pi$
$907$	49.9117	1.65729	0.828645	+	0.559774i	$0.189112\pi$
$908$	0	0
$909$	0	0
$910$	4.00000	0.132599
$911$	6.68629	0.221527	0.110763	−	0.993847i	$-0.464670\pi$
$911$	6.68629	0.221527	0.110763	+	0.993847i	$0.464670\pi$
$912$	0	0
$913$	−0.970563	−0.0321209
$914$	−35.3553	−1.16945
$915$	0	0
$916$	0	0
$917$	9.34315	0.308538
$918$	0	0
$919$	49.4264	1.63043	0.815213	−	0.579161i	$-0.196620\pi$
$919$	49.4264	1.63043	0.815213	+	0.579161i	$0.196620\pi$
$920$	11.3137	0.373002
$921$	0	0
$922$	22.0000	0.724531
$923$	9.17157	0.301886
$924$	0	0
$925$	16.4558	0.541065
$926$	38.1421	1.25343
$927$	0	0
$928$	0	0
$929$	13.3726	0.438740	0.219370	−	0.975642i	$-0.429600\pi$
$929$	13.3726	0.438740	0.219370	+	0.975642i	$0.429600\pi$
$930$	0	0
$931$	0.242641	0.00795223
$932$	0	0
$933$	0	0
$934$	52.9706	1.73325
$935$	−0.686292	−0.0224441
$936$	0	0
$937$	−7.51472	−0.245495	−0.122748	−	0.992438i	$-0.539171\pi$
$937$	−7.51472	−0.245495	−0.122748	+	0.992438i	$0.539171\pi$
$938$	−6.00000	−0.195907
$939$	0	0
$940$	0	0
$941$	29.4853	0.961193	0.480596	−	0.876942i	$-0.340420\pi$
$941$	29.4853	0.961193	0.480596	+	0.876942i	$0.340420\pi$
$942$	0	0
$943$	−7.51472	−0.244713
$944$	28.2843	0.920575
$945$	0	0
$946$	−40.0000	−1.30051
$947$	61.2426	1.99012	0.995059	−	0.0992833i	$-0.0316550\pi$
$947$	61.2426	1.99012	0.995059	+	0.0992833i	$0.0316550\pi$
$948$	0	0
$949$	−29.4558	−0.956177
$950$	−1.02944	−0.0333994
$951$	0	0
$952$	−0.485281	−0.0157281
$953$	61.1543	1.98098	0.990492	−	0.137574i	$-0.0439304\pi$
$953$	61.1543	1.98098	0.990492	+	0.137574i	$0.0439304\pi$
$954$	0	0
$955$	13.5147	0.437326
$956$	0	0
$957$	0	0
$958$	−24.2426	−0.783244
$959$	−17.3137	−0.559089
$960$	0	0
$961$	−15.0000	−0.483871
$962$	15.5147	0.500215
$963$	0	0
$964$	0	0
$965$	32.8284	1.05678
$966$	0	0
$967$	−40.1838	−1.29222	−0.646111	−	0.763243i	$-0.723606\pi$
$967$	−40.1838	−1.29222	−0.646111	+	0.763243i	$0.723606\pi$
$968$	8.48528	0.272727
$969$	0	0
$970$	−5.51472	−0.177067
$971$	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
$973$	5.24264	0.168071
$974$	18.7696	0.601415
$975$	0	0
$976$	48.9706	1.56751
$977$	−12.6863	−0.405870	−0.202935	−	0.979192i	$-0.565048\pi$
$977$	−12.6863	−0.405870	−0.202935	+	0.979192i	$0.565048\pi$
$978$	0	0
$979$	−41.9411	−1.34044
$980$	0	0
$981$	0	0
$982$	8.00000	0.255290
$983$	26.6569	0.850222	0.425111	−	0.905141i	$-0.360235\pi$
$983$	26.6569	0.850222	0.425111	+	0.905141i	$0.360235\pi$
$984$	0	0
$985$	22.4853	0.716441
$986$	1.07107	0.0341097
$987$	0	0
$988$	0	0
$989$	−28.2843	−0.899388
$990$	0	0
$991$	−9.75736	−0.309953	−0.154976	−	0.987918i	$-0.549530\pi$
$991$	−9.75736	−0.309953	−0.154976	+	0.987918i	$0.549530\pi$
$992$	0	0
$993$	0	0
$994$	−6.48528	−0.205701
$995$	23.3137	0.739094
$996$	0	0
$997$	2.54416	0.0805742	0.0402871	−	0.999188i	$-0.487173\pi$
$997$	2.54416	0.0805742	0.0402871	+	0.999188i	$0.487173\pi$
$998$	−37.7990	−1.19651
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.g.1.1	✓	2	3.2	odd	2
8001.2.a.j.1.2		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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.41421 q^{5} +1.00000 q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} -1.41421 q^{5} +1.00000 q^{7} -2.82843 q^{8} -2.00000 q^{10} +2.82843 q^{11} -2.00000 q^{13} +1.41421 q^{14} -4.00000 q^{16} +0.171573 q^{17} +0.242641 q^{19} +4.00000 q^{22} +2.82843 q^{23} -3.00000 q^{25} -2.82843 q^{26} +4.41421 q^{29} +4.00000 q^{31} +0.242641 q^{34} -1.41421 q^{35} -5.48528 q^{37} +0.343146 q^{38} +4.00000 q^{40} -2.65685 q^{41} -10.0000 q^{43} +4.00000 q^{46} +8.82843 q^{47} +1.00000 q^{49} -4.24264 q^{50} +10.4142 q^{53} -4.00000 q^{55} -2.82843 q^{56} +6.24264 q^{58} -7.07107 q^{59} -12.2426 q^{61} +5.65685 q^{62} +8.00000 q^{64} +2.82843 q^{65} -4.24264 q^{67} -2.00000 q^{70} -4.58579 q^{71} +14.7279 q^{73} -7.75736 q^{74} +2.82843 q^{77} -11.0000 q^{79} +5.65685 q^{80} -3.75736 q^{82} -0.343146 q^{83} -0.242641 q^{85} -14.1421 q^{86} -8.00000 q^{88} -14.8284 q^{89} -2.00000 q^{91} +12.4853 q^{94} -0.343146 q^{95} +2.75736 q^{97} +1.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 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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