LMFDB - Embedded newform 8001.2.a.z.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:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.69601 q^{2} +5.26849 q^{4} +3.35934 q^{5} -1.00000 q^{7} -8.81189 q^{8} +O(q^{10})$	\(q-2.69601 q^{2} +5.26849 q^{4} +3.35934 q^{5} -1.00000 q^{7} -8.81189 q^{8} -9.05682 q^{10} -2.89389 q^{11} -2.10825 q^{13} +2.69601 q^{14} +13.2200 q^{16} +6.29317 q^{17} -0.167183 q^{19} +17.6986 q^{20} +7.80197 q^{22} +4.03441 q^{23} +6.28516 q^{25} +5.68387 q^{26} -5.26849 q^{28} -0.392019 q^{29} -7.54513 q^{31} -18.0175 q^{32} -16.9665 q^{34} -3.35934 q^{35} -2.42977 q^{37} +0.450728 q^{38} -29.6021 q^{40} +0.160937 q^{41} -6.15714 q^{43} -15.2464 q^{44} -10.8768 q^{46} -8.81645 q^{47} +1.00000 q^{49} -16.9449 q^{50} -11.1073 q^{52} +2.87949 q^{53} -9.72156 q^{55} +8.81189 q^{56} +1.05689 q^{58} +3.45891 q^{59} -12.0314 q^{61} +20.3418 q^{62} +22.1354 q^{64} -7.08233 q^{65} +9.36114 q^{67} +33.1555 q^{68} +9.05682 q^{70} +12.0277 q^{71} -11.2562 q^{73} +6.55069 q^{74} -0.880801 q^{76} +2.89389 q^{77} +3.36380 q^{79} +44.4104 q^{80} -0.433889 q^{82} -17.3020 q^{83} +21.1409 q^{85} +16.5997 q^{86} +25.5006 q^{88} -14.6779 q^{89} +2.10825 q^{91} +21.2552 q^{92} +23.7693 q^{94} -0.561624 q^{95} -1.23267 q^{97} -2.69601 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * 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 $

$2$	−2.69601	−1.90637	−0.953185	−	0.302389i	$-0.902216\pi$
$2$	−2.69601	−1.90637	−0.953185	+	0.302389i	$0.902216\pi$
$3$	0	0
$3$	0	0
$4$	5.26849	2.63424
$5$	3.35934	1.50234	0.751171	−	0.660107i	$-0.229489\pi$
$5$	3.35934	1.50234	0.751171	+	0.660107i	$0.229489\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−8.81189	−3.11547
$9$	0	0
$10$	−9.05682	−2.86402
$11$	−2.89389	−0.872541	−0.436271	−	0.899816i	$-0.643701\pi$
$11$	−2.89389	−0.872541	−0.436271	+	0.899816i	$0.643701\pi$
$12$	0	0
$13$	−2.10825	−0.584723	−0.292362	−	0.956308i	$-0.594441\pi$
$13$	−2.10825	−0.584723	−0.292362	+	0.956308i	$0.594441\pi$
$14$	2.69601	0.720540
$15$	0	0
$16$	13.2200	3.30500
$17$	6.29317	1.52632	0.763160	−	0.646210i	$-0.223647\pi$
$17$	6.29317	1.52632	0.763160	+	0.646210i	$0.223647\pi$
$18$	0	0
$19$	−0.167183	−0.0383544	−0.0191772	−	0.999816i	$-0.506105\pi$
$19$	−0.167183	−0.0383544	−0.0191772	+	0.999816i	$0.506105\pi$
$20$	17.6986	3.95754
$21$	0	0
$22$	7.80197	1.66339
$23$	4.03441	0.841233	0.420616	−	0.907239i	$-0.361814\pi$
$23$	4.03441	0.841233	0.420616	+	0.907239i	$0.361814\pi$
$24$	0	0
$25$	6.28516	1.25703
$26$	5.68387	1.11470
$27$	0	0
$28$	−5.26849	−0.995651
$29$	−0.392019	−0.0727961	−0.0363981	−	0.999337i	$-0.511588\pi$
$29$	−0.392019	−0.0727961	−0.0363981	+	0.999337i	$0.511588\pi$
$30$	0	0
$31$	−7.54513	−1.35514	−0.677572	−	0.735456i	$-0.736968\pi$
$31$	−7.54513	−1.35514	−0.677572	+	0.735456i	$0.736968\pi$
$32$	−18.0175	−3.18507
$33$	0	0
$34$	−16.9665	−2.90973
$35$	−3.35934	−0.567832
$36$	0	0
$37$	−2.42977	−0.399452	−0.199726	−	0.979852i	$-0.564005\pi$
$37$	−2.42977	−0.399452	−0.199726	+	0.979852i	$0.564005\pi$
$38$	0.450728	0.0731177
$39$	0	0
$40$	−29.6021	−4.68051
$41$	0.160937	0.0251342	0.0125671	−	0.999921i	$-0.496000\pi$
$41$	0.160937	0.0251342	0.0125671	+	0.999921i	$0.496000\pi$
$42$	0	0
$43$	−6.15714	−0.938955	−0.469478	−	0.882944i	$-0.655558\pi$
$43$	−6.15714	−0.938955	−0.469478	+	0.882944i	$0.655558\pi$
$44$	−15.2464	−2.29849
$45$	0	0
$46$	−10.8768	−1.60370
$47$	−8.81645	−1.28601	−0.643006	−	0.765861i	$-0.722313\pi$
$47$	−8.81645	−1.28601	−0.643006	+	0.765861i	$0.722313\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−16.9449	−2.39637
$51$	0	0
$52$	−11.1073	−1.54030
$53$	2.87949	0.395528	0.197764	−	0.980250i	$-0.436632\pi$
$53$	2.87949	0.395528	0.197764	+	0.980250i	$0.436632\pi$
$54$	0	0
$55$	−9.72156	−1.31086
$56$	8.81189	1.17754
$57$	0	0
$58$	1.05689	0.138776
$59$	3.45891	0.450312	0.225156	−	0.974323i	$-0.427711\pi$
$59$	3.45891	0.450312	0.225156	+	0.974323i	$0.427711\pi$
$60$	0	0
$61$	−12.0314	−1.54046	−0.770232	−	0.637763i	$-0.779860\pi$
$61$	−12.0314	−1.54046	−0.770232	+	0.637763i	$0.779860\pi$
$62$	20.3418	2.58341
$63$	0	0
$64$	22.1354	2.76693
$65$	−7.08233	−0.878455
$66$	0	0
$67$	9.36114	1.14364	0.571822	−	0.820377i	$-0.306237\pi$
$67$	9.36114	1.14364	0.571822	+	0.820377i	$0.306237\pi$
$68$	33.1555	4.02070
$69$	0	0
$70$	9.05682	1.08250
$71$	12.0277	1.42742	0.713710	−	0.700441i	$-0.247013\pi$
$71$	12.0277	1.42742	0.713710	+	0.700441i	$0.247013\pi$
$72$	0	0
$73$	−11.2562	−1.31743	−0.658717	−	0.752390i	$-0.728901\pi$
$73$	−11.2562	−1.31743	−0.658717	+	0.752390i	$0.728901\pi$
$74$	6.55069	0.761503
$75$	0	0
$76$	−0.880801	−0.101035
$77$	2.89389	0.329790
$78$	0	0
$79$	3.36380	0.378457	0.189228	−	0.981933i	$-0.439401\pi$
$79$	3.36380	0.378457	0.189228	+	0.981933i	$0.439401\pi$
$80$	44.4104	4.96524
$81$	0	0
$82$	−0.433889	−0.0479150
$83$	−17.3020	−1.89914	−0.949569	−	0.313559i	$-0.898479\pi$
$83$	−17.3020	−1.89914	−0.949569	+	0.313559i	$0.898479\pi$
$84$	0	0
$85$	21.1409	2.29305
$86$	16.5997	1.79000
$87$	0	0
$88$	25.5006	2.71838
$89$	−14.6779	−1.55585	−0.777927	−	0.628355i	$-0.783729\pi$
$89$	−14.6779	−1.55585	−0.777927	+	0.628355i	$0.783729\pi$
$90$	0	0
$91$	2.10825	0.221005
$92$	21.2552	2.21601
$93$	0	0
$94$	23.7693	2.45161
$95$	−0.561624	−0.0576215
$96$	0	0
$97$	−1.23267	−0.125159	−0.0625793	−	0.998040i	$-0.519933\pi$
$97$	−1.23267	−0.125159	−0.0625793	+	0.998040i	$0.519933\pi$
$98$	−2.69601	−0.272338
$99$	0	0
$100$	33.1133	3.31133
$101$	12.2730	1.22121	0.610603	−	0.791937i	$-0.290927\pi$
$101$	12.2730	1.22121	0.610603	+	0.791937i	$0.290927\pi$
$102$	0	0
$103$	−3.50168	−0.345030	−0.172515	−	0.985007i	$-0.555189\pi$
$103$	−3.50168	−0.345030	−0.172515	+	0.985007i	$0.555189\pi$
$104$	18.5777	1.82169
$105$	0	0
$106$	−7.76314	−0.754023
$107$	11.7913	1.13991	0.569953	−	0.821677i	$-0.306961\pi$
$107$	11.7913	1.13991	0.569953	+	0.821677i	$0.306961\pi$
$108$	0	0
$109$	−8.25140	−0.790341	−0.395170	−	0.918608i	$-0.629314\pi$
$109$	−8.25140	−0.790341	−0.395170	+	0.918608i	$0.629314\pi$
$110$	26.2095	2.49897
$111$	0	0
$112$	−13.2200	−1.24917
$113$	13.2805	1.24932	0.624662	−	0.780895i	$-0.285237\pi$
$113$	13.2805	1.24932	0.624662	+	0.780895i	$0.285237\pi$
$114$	0	0
$115$	13.5530	1.26382
$116$	−2.06535	−0.191763
$117$	0	0
$118$	−9.32528	−0.858462
$119$	−6.29317	−0.576894
$120$	0	0
$121$	−2.62539	−0.238672
$122$	32.4368	2.93669
$123$	0	0
$124$	−39.7514	−3.56978
$125$	4.31730	0.386151
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−23.6424	−2.08971
$129$	0	0
$130$	19.0940	1.67466
$131$	−19.3595	−1.69145	−0.845726	−	0.533618i	$-0.820832\pi$
$131$	−19.3595	−1.69145	−0.845726	+	0.533618i	$0.820832\pi$
$132$	0	0
$133$	0.167183	0.0144966
$134$	−25.2378	−2.18021
$135$	0	0
$136$	−55.4547	−4.75520
$137$	−15.6936	−1.34079	−0.670397	−	0.742003i	$-0.733876\pi$
$137$	−15.6936	−1.34079	−0.670397	+	0.742003i	$0.733876\pi$
$138$	0	0
$139$	−7.82773	−0.663939	−0.331970	−	0.943290i	$-0.607713\pi$
$139$	−7.82773	−0.663939	−0.331970	+	0.943290i	$0.607713\pi$
$140$	−17.6986	−1.49581
$141$	0	0
$142$	−32.4267	−2.72119
$143$	6.10105	0.510195
$144$	0	0
$145$	−1.31693	−0.109365
$146$	30.3468	2.51152
$147$	0	0
$148$	−12.8012	−1.05225
$149$	15.7545	1.29066	0.645328	−	0.763905i	$-0.276721\pi$
$149$	15.7545	1.29066	0.645328	+	0.763905i	$0.276721\pi$
$150$	0	0
$151$	−1.99541	−0.162384	−0.0811922	−	0.996698i	$-0.525873\pi$
$151$	−1.99541	−0.162384	−0.0811922	+	0.996698i	$0.525873\pi$
$152$	1.47320	0.119492
$153$	0	0
$154$	−7.80197	−0.628701
$155$	−25.3466	−2.03589
$156$	0	0
$157$	4.40954	0.351919	0.175960	−	0.984397i	$-0.443697\pi$
$157$	4.40954	0.351919	0.175960	+	0.984397i	$0.443697\pi$
$158$	−9.06884	−0.721478
$159$	0	0
$160$	−60.5269	−4.78507
$161$	−4.03441	−0.317956
$162$	0	0
$163$	−9.81692	−0.768921	−0.384460	−	0.923142i	$-0.625612\pi$
$163$	−9.81692	−0.768921	−0.384460	+	0.923142i	$0.625612\pi$
$164$	0.847896	0.0662095
$165$	0	0
$166$	46.6463	3.62046
$167$	15.3402	1.18706	0.593531	−	0.804811i	$-0.297734\pi$
$167$	15.3402	1.18706	0.593531	+	0.804811i	$0.297734\pi$
$168$	0	0
$169$	−8.55528	−0.658099
$170$	−56.9962	−4.37141
$171$	0	0
$172$	−32.4388	−2.47344
$173$	−12.7598	−0.970110	−0.485055	−	0.874484i	$-0.661200\pi$
$173$	−12.7598	−0.970110	−0.485055	+	0.874484i	$0.661200\pi$
$174$	0	0
$175$	−6.28516	−0.475114
$176$	−38.2572	−2.88375
$177$	0	0
$178$	39.5718	2.96603
$179$	−1.20349	−0.0899531	−0.0449766	−	0.998988i	$-0.514321\pi$
$179$	−1.20349	−0.0899531	−0.0449766	+	0.998988i	$0.514321\pi$
$180$	0	0
$181$	−8.27628	−0.615171	−0.307585	−	0.951520i	$-0.599521\pi$
$181$	−8.27628	−0.615171	−0.307585	+	0.951520i	$0.599521\pi$
$182$	−5.68387	−0.421316
$183$	0	0
$184$	−35.5508	−2.62084
$185$	−8.16243	−0.600113
$186$	0	0
$187$	−18.2118	−1.33178
$188$	−46.4494	−3.38767
$189$	0	0
$190$	1.51415	0.109848
$191$	16.1586	1.16920	0.584598	−	0.811323i	$-0.301252\pi$
$191$	16.1586	1.16920	0.584598	+	0.811323i	$0.301252\pi$
$192$	0	0
$193$	10.3243	0.743159	0.371579	−	0.928401i	$-0.378816\pi$
$193$	10.3243	0.743159	0.371579	+	0.928401i	$0.378816\pi$
$194$	3.32329	0.238598
$195$	0	0
$196$	5.26849	0.376321
$197$	19.7255	1.40539	0.702693	−	0.711493i	$-0.251981\pi$
$197$	19.7255	1.40539	0.702693	+	0.711493i	$0.251981\pi$
$198$	0	0
$199$	1.25357	0.0888634	0.0444317	−	0.999012i	$-0.485852\pi$
$199$	1.25357	0.0888634	0.0444317	+	0.999012i	$0.485852\pi$
$200$	−55.3841	−3.91625
$201$	0	0
$202$	−33.0881	−2.32807
$203$	0.392019	0.0275144
$204$	0	0
$205$	0.540643	0.0377601
$206$	9.44056	0.657755
$207$	0	0
$208$	−27.8710	−1.93251
$209$	0.483809	0.0334658
$210$	0	0
$211$	1.81620	0.125032	0.0625162	−	0.998044i	$-0.480088\pi$
$211$	1.81620	0.125032	0.0625162	+	0.998044i	$0.480088\pi$
$212$	15.1706	1.04192
$213$	0	0
$214$	−31.7894	−2.17308
$215$	−20.6839	−1.41063
$216$	0	0
$217$	7.54513	0.512197
$218$	22.2459	1.50668
$219$	0	0
$220$	−51.2179	−3.45311
$221$	−13.2676	−0.892474
$222$	0	0
$223$	0.622163	0.0416631	0.0208316	−	0.999783i	$-0.493369\pi$
$223$	0.622163	0.0416631	0.0208316	+	0.999783i	$0.493369\pi$
$224$	18.0175	1.20384
$225$	0	0
$226$	−35.8044	−2.38167
$227$	−25.3816	−1.68464	−0.842319	−	0.538980i	$-0.818810\pi$
$227$	−25.3816	−1.68464	−0.842319	+	0.538980i	$0.818810\pi$
$228$	0	0
$229$	−0.702452	−0.0464193	−0.0232097	−	0.999731i	$-0.507389\pi$
$229$	−0.702452	−0.0464193	−0.0232097	+	0.999731i	$0.507389\pi$
$230$	−36.5389	−2.40931
$231$	0	0
$232$	3.45443	0.226794
$233$	−16.1341	−1.05698	−0.528490	−	0.848939i	$-0.677242\pi$
$233$	−16.1341	−1.05698	−0.528490	+	0.848939i	$0.677242\pi$
$234$	0	0
$235$	−29.6175	−1.93203
$236$	18.2232	1.18623
$237$	0	0
$238$	16.9665	1.09977
$239$	−8.34664	−0.539899	−0.269949	−	0.962874i	$-0.587007\pi$
$239$	−8.34664	−0.539899	−0.269949	+	0.962874i	$0.587007\pi$
$240$	0	0
$241$	11.7059	0.754043	0.377021	−	0.926205i	$-0.376948\pi$
$241$	11.7059	0.754043	0.377021	+	0.926205i	$0.376948\pi$
$242$	7.07809	0.454997
$243$	0	0
$244$	−63.3873	−4.05796
$245$	3.35934	0.214620
$246$	0	0
$247$	0.352463	0.0224267
$248$	66.4868	4.22192
$249$	0	0
$250$	−11.6395	−0.736146
$251$	−17.8668	−1.12774	−0.563871	−	0.825863i	$-0.690689\pi$
$251$	−17.8668	−1.12774	−0.563871	+	0.825863i	$0.690689\pi$
$252$	0	0
$253$	−11.6751	−0.734010
$254$	2.69601	0.169163
$255$	0	0
$256$	19.4694	1.21683
$257$	9.30841	0.580643	0.290321	−	0.956929i	$-0.406238\pi$
$257$	9.30841	0.580643	0.290321	+	0.956929i	$0.406238\pi$
$258$	0	0
$259$	2.42977	0.150979
$260$	−37.3131	−2.31406
$261$	0	0
$262$	52.1936	3.22453
$263$	−12.0727	−0.744435	−0.372217	−	0.928146i	$-0.621402\pi$
$263$	−12.0727	−0.744435	−0.372217	+	0.928146i	$0.621402\pi$
$264$	0	0
$265$	9.67318	0.594219
$266$	−0.450728	−0.0276359
$267$	0	0
$268$	49.3190	3.01264
$269$	27.9575	1.70460	0.852299	−	0.523056i	$-0.175208\pi$
$269$	27.9575	1.70460	0.852299	+	0.523056i	$0.175208\pi$
$270$	0	0
$271$	−11.6402	−0.707089	−0.353545	−	0.935418i	$-0.615024\pi$
$271$	−11.6402	−0.707089	−0.353545	+	0.935418i	$0.615024\pi$
$272$	83.1957	5.04448
$273$	0	0
$274$	42.3101	2.55605
$275$	−18.1886	−1.09681
$276$	0	0
$277$	26.0313	1.56407	0.782036	−	0.623234i	$-0.214181\pi$
$277$	26.0313	1.56407	0.782036	+	0.623234i	$0.214181\pi$
$278$	21.1037	1.26571
$279$	0	0
$280$	29.6021	1.76906
$281$	−17.6693	−1.05406	−0.527031	−	0.849846i	$-0.676695\pi$
$281$	−17.6693	−1.05406	−0.527031	+	0.849846i	$0.676695\pi$
$282$	0	0
$283$	−9.81022	−0.583157	−0.291579	−	0.956547i	$-0.594180\pi$
$283$	−9.81022	−0.583157	−0.291579	+	0.956547i	$0.594180\pi$
$284$	63.3676	3.76017
$285$	0	0
$286$	−16.4485	−0.972620
$287$	−0.160937	−0.00949982
$288$	0	0
$289$	22.6041	1.32965
$290$	3.55045	0.208490
$291$	0	0
$292$	−59.3030	−3.47044
$293$	−18.0827	−1.05641	−0.528203	−	0.849118i	$-0.677134\pi$
$293$	−18.0827	−1.05641	−0.528203	+	0.849118i	$0.677134\pi$
$294$	0	0
$295$	11.6197	0.676523
$296$	21.4109	1.24448
$297$	0	0
$298$	−42.4743	−2.46047
$299$	−8.50554	−0.491888
$300$	0	0
$301$	6.15714	0.354892
$302$	5.37966	0.309565
$303$	0	0
$304$	−2.21016	−0.126761
$305$	−40.4176	−2.31431
$306$	0	0
$307$	4.05704	0.231547	0.115774	−	0.993276i	$-0.463065\pi$
$307$	4.05704	0.231547	0.115774	+	0.993276i	$0.463065\pi$
$308$	15.2464	0.868746
$309$	0	0
$310$	68.3349	3.88116
$311$	−13.5710	−0.769543	−0.384772	−	0.923012i	$-0.625720\pi$
$311$	−13.5710	−0.769543	−0.384772	+	0.923012i	$0.625720\pi$
$312$	0	0
$313$	23.3786	1.32144	0.660719	−	0.750633i	$-0.270251\pi$
$313$	23.3786	1.32144	0.660719	+	0.750633i	$0.270251\pi$
$314$	−11.8882	−0.670888
$315$	0	0
$316$	17.7221	0.996947
$317$	31.8794	1.79053	0.895263	−	0.445537i	$-0.146987\pi$
$317$	31.8794	1.79053	0.895263	+	0.445537i	$0.146987\pi$
$318$	0	0
$319$	1.13446	0.0635176
$320$	74.3603	4.15687
$321$	0	0
$322$	10.8768	0.606142
$323$	−1.05211	−0.0585411
$324$	0	0
$325$	−13.2507	−0.735016
$326$	26.4666	1.46585
$327$	0	0
$328$	−1.41816	−0.0783048
$329$	8.81645	0.486067
$330$	0	0
$331$	−32.8969	−1.80818	−0.904090	−	0.427343i	$-0.859450\pi$
$331$	−32.8969	−1.80818	−0.904090	+	0.427343i	$0.859450\pi$
$332$	−91.1552	−5.00279
$333$	0	0
$334$	−41.3574	−2.26298
$335$	31.4472	1.71815
$336$	0	0
$337$	12.2361	0.666544	0.333272	−	0.942831i	$-0.391847\pi$
$337$	12.2361	0.666544	0.333272	+	0.942831i	$0.391847\pi$
$338$	23.0652	1.25458
$339$	0	0
$340$	111.381	6.04046
$341$	21.8348	1.18242
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	54.2560	2.92529
$345$	0	0
$346$	34.4006	1.84939
$347$	4.98147	0.267419	0.133710	−	0.991021i	$-0.457311\pi$
$347$	4.98147	0.267419	0.133710	+	0.991021i	$0.457311\pi$
$348$	0	0
$349$	−7.29876	−0.390693	−0.195347	−	0.980734i	$-0.562583\pi$
$349$	−7.29876	−0.390693	−0.195347	+	0.980734i	$0.562583\pi$
$350$	16.9449	0.905742
$351$	0	0
$352$	52.1406	2.77911
$353$	12.2191	0.650356	0.325178	−	0.945653i	$-0.394576\pi$
$353$	12.2191	0.650356	0.325178	+	0.945653i	$0.394576\pi$
$354$	0	0
$355$	40.4050	2.14447
$356$	−77.3303	−4.09850
$357$	0	0
$358$	3.24463	0.171484
$359$	−36.0746	−1.90395	−0.951973	−	0.306181i	$-0.900949\pi$
$359$	−36.0746	−1.90395	−0.951973	+	0.306181i	$0.900949\pi$
$360$	0	0
$361$	−18.9720	−0.998529
$362$	22.3130	1.17274
$363$	0	0
$364$	11.1073	0.582180
$365$	−37.8133	−1.97924
$366$	0	0
$367$	−25.0223	−1.30616	−0.653078	−	0.757291i	$-0.726523\pi$
$367$	−25.0223	−1.30616	−0.653078	+	0.757291i	$0.726523\pi$
$368$	53.3348	2.78027
$369$	0	0
$370$	22.0060	1.14404
$371$	−2.87949	−0.149496
$372$	0	0
$373$	−12.9187	−0.668905	−0.334452	−	0.942413i	$-0.608551\pi$
$373$	−12.9187	−0.668905	−0.334452	+	0.942413i	$0.608551\pi$
$374$	49.0992	2.53886
$375$	0	0
$376$	77.6896	4.00653
$377$	0.826474	0.0425656
$378$	0	0
$379$	1.07672	0.0553076	0.0276538	−	0.999618i	$-0.491196\pi$
$379$	1.07672	0.0553076	0.0276538	+	0.999618i	$0.491196\pi$
$380$	−2.95891	−0.151789
$381$	0	0
$382$	−43.5638	−2.22892
$383$	−14.0723	−0.719060	−0.359530	−	0.933133i	$-0.617063\pi$
$383$	−14.0723	−0.719060	−0.359530	+	0.933133i	$0.617063\pi$
$384$	0	0
$385$	9.72156	0.495457
$386$	−27.8344	−1.41674
$387$	0	0
$388$	−6.49430	−0.329698
$389$	−0.896424	−0.0454505	−0.0227253	−	0.999742i	$-0.507234\pi$
$389$	−0.896424	−0.0454505	−0.0227253	+	0.999742i	$0.507234\pi$
$390$	0	0
$391$	25.3892	1.28399
$392$	−8.81189	−0.445067
$393$	0	0
$394$	−53.1803	−2.67918
$395$	11.3001	0.568571
$396$	0	0
$397$	15.5645	0.781158	0.390579	−	0.920569i	$-0.372275\pi$
$397$	15.5645	0.781158	0.390579	+	0.920569i	$0.372275\pi$
$398$	−3.37965	−0.169406
$399$	0	0
$400$	83.0898	4.15449
$401$	21.5171	1.07451	0.537257	−	0.843419i	$-0.319461\pi$
$401$	21.5171	1.07451	0.537257	+	0.843419i	$0.319461\pi$
$402$	0	0
$403$	15.9070	0.792385
$404$	64.6600	3.21696
$405$	0	0
$406$	−1.05689	−0.0524525
$407$	7.03149	0.348538
$408$	0	0
$409$	−0.413177	−0.0204303	−0.0102151	−	0.999948i	$-0.503252\pi$
$409$	−0.413177	−0.0204303	−0.0102151	+	0.999948i	$0.503252\pi$
$410$	−1.45758	−0.0719848
$411$	0	0
$412$	−18.4485	−0.908894
$413$	−3.45891	−0.170202
$414$	0	0
$415$	−58.1232	−2.85316
$416$	37.9854	1.86239
$417$	0	0
$418$	−1.30436	−0.0637982
$419$	−36.4195	−1.77921	−0.889605	−	0.456730i	$-0.849020\pi$
$419$	−36.4195	−1.77921	−0.889605	+	0.456730i	$0.849020\pi$
$420$	0	0
$421$	−13.2124	−0.643934	−0.321967	−	0.946751i	$-0.604344\pi$
$421$	−13.2124	−0.643934	−0.321967	+	0.946751i	$0.604344\pi$
$422$	−4.89650	−0.238358
$423$	0	0
$424$	−25.3737	−1.23226
$425$	39.5536	1.91863
$426$	0	0
$427$	12.0314	0.582241
$428$	62.1222	3.00279
$429$	0	0
$430$	55.7642	2.68919
$431$	−33.8851	−1.63219	−0.816095	−	0.577918i	$-0.803865\pi$
$431$	−33.8851	−1.63219	−0.816095	+	0.577918i	$0.803865\pi$
$432$	0	0
$433$	−4.57030	−0.219635	−0.109817	−	0.993952i	$-0.535027\pi$
$433$	−4.57030	−0.219635	−0.109817	+	0.993952i	$0.535027\pi$
$434$	−20.3418	−0.976436
$435$	0	0
$436$	−43.4724	−2.08195
$437$	−0.674485	−0.0322650
$438$	0	0
$439$	−15.1264	−0.721945	−0.360973	−	0.932576i	$-0.617555\pi$
$439$	−15.1264	−0.721945	−0.360973	+	0.932576i	$0.617555\pi$
$440$	85.6653	4.08393
$441$	0	0
$442$	35.7696	1.70139
$443$	21.4687	1.02001	0.510005	−	0.860171i	$-0.329643\pi$
$443$	21.4687	1.02001	0.510005	+	0.860171i	$0.329643\pi$
$444$	0	0
$445$	−49.3080	−2.33743
$446$	−1.67736	−0.0794253
$447$	0	0
$448$	−22.1354	−1.04580
$449$	−27.3231	−1.28945	−0.644727	−	0.764413i	$-0.723029\pi$
$449$	−27.3231	−1.28945	−0.644727	+	0.764413i	$0.723029\pi$
$450$	0	0
$451$	−0.465735	−0.0219306
$452$	69.9681	3.29102
$453$	0	0
$454$	68.4292	3.21154
$455$	7.08233	0.332025
$456$	0	0
$457$	17.8679	0.835826	0.417913	−	0.908487i	$-0.362762\pi$
$457$	17.8679	0.835826	0.417913	+	0.908487i	$0.362762\pi$
$458$	1.89382	0.0884924
$459$	0	0
$460$	71.4036	3.32921
$461$	5.37461	0.250320	0.125160	−	0.992137i	$-0.460056\pi$
$461$	5.37461	0.250320	0.125160	+	0.992137i	$0.460056\pi$
$462$	0	0
$463$	21.8586	1.01586	0.507928	−	0.861399i	$-0.330411\pi$
$463$	21.8586	1.01586	0.507928	+	0.861399i	$0.330411\pi$
$464$	−5.18249	−0.240591
$465$	0	0
$466$	43.4978	2.01500
$467$	32.8761	1.52133	0.760663	−	0.649147i	$-0.224874\pi$
$467$	32.8761	1.52133	0.760663	+	0.649147i	$0.224874\pi$
$468$	0	0
$469$	−9.36114	−0.432257
$470$	79.8491	3.68316
$471$	0	0
$472$	−30.4796	−1.40294
$473$	17.8181	0.819277
$474$	0	0
$475$	−1.05077	−0.0482127
$476$	−33.1555	−1.51968
$477$	0	0
$478$	22.5026	1.02925
$479$	−15.8690	−0.725073	−0.362536	−	0.931970i	$-0.618089\pi$
$479$	−15.8690	−0.725073	−0.362536	+	0.931970i	$0.618089\pi$
$480$	0	0
$481$	5.12256	0.233569
$482$	−31.5592	−1.43748
$483$	0	0
$484$	−13.8318	−0.628720
$485$	−4.14095	−0.188031
$486$	0	0
$487$	8.98925	0.407342	0.203671	−	0.979039i	$-0.434713\pi$
$487$	8.98925	0.407342	0.203671	+	0.979039i	$0.434713\pi$
$488$	106.019	4.79927
$489$	0	0
$490$	−9.05682	−0.409146
$491$	−1.39367	−0.0628953	−0.0314476	−	0.999505i	$-0.510012\pi$
$491$	−1.39367	−0.0628953	−0.0314476	+	0.999505i	$0.510012\pi$
$492$	0	0
$493$	−2.46705	−0.111110
$494$	−0.950246	−0.0427536
$495$	0	0
$496$	−99.7465	−4.47875
$497$	−12.0277	−0.539514
$498$	0	0
$499$	−5.96722	−0.267129	−0.133565	−	0.991040i	$-0.542642\pi$
$499$	−5.96722	−0.267129	−0.133565	+	0.991040i	$0.542642\pi$
$500$	22.7456	1.01722
$501$	0	0
$502$	48.1691	2.14989
$503$	−28.1974	−1.25726	−0.628630	−	0.777704i	$-0.716384\pi$
$503$	−28.1974	−1.25726	−0.628630	+	0.777704i	$0.716384\pi$
$504$	0	0
$505$	41.2291	1.83467
$506$	31.4763	1.39929
$507$	0	0
$508$	−5.26849	−0.233751
$509$	−19.4653	−0.862786	−0.431393	−	0.902164i	$-0.641978\pi$
$509$	−19.4653	−0.862786	−0.431393	+	0.902164i	$0.641978\pi$
$510$	0	0
$511$	11.2562	0.497944
$512$	−5.20488	−0.230026
$513$	0	0
$514$	−25.0956	−1.10692
$515$	−11.7633	−0.518354
$516$	0	0
$517$	25.5139	1.12210
$518$	−6.55069	−0.287821
$519$	0	0
$520$	62.4087	2.73680
$521$	18.0600	0.791221	0.395611	−	0.918418i	$-0.370533\pi$
$521$	18.0600	0.791221	0.395611	+	0.918418i	$0.370533\pi$
$522$	0	0
$523$	−0.634346	−0.0277380	−0.0138690	−	0.999904i	$-0.504415\pi$
$523$	−0.634346	−0.0277380	−0.0138690	+	0.999904i	$0.504415\pi$
$524$	−101.996	−4.45570
$525$	0	0
$526$	32.5482	1.41917
$527$	−47.4828	−2.06838
$528$	0	0
$529$	−6.72353	−0.292328
$530$	−26.0790	−1.13280
$531$	0	0
$532$	0.880801	0.0381876
$533$	−0.339296	−0.0146965
$534$	0	0
$535$	39.6109	1.71253
$536$	−82.4893	−3.56299
$537$	0	0
$538$	−75.3737	−3.24959
$539$	−2.89389	−0.124649
$540$	0	0
$541$	12.8668	0.553186	0.276593	−	0.960987i	$-0.410795\pi$
$541$	12.8668	0.553186	0.276593	+	0.960987i	$0.410795\pi$
$542$	31.3820	1.34797
$543$	0	0
$544$	−113.387	−4.86144
$545$	−27.7193	−1.18736
$546$	0	0
$547$	25.5679	1.09320	0.546602	−	0.837393i	$-0.315921\pi$
$547$	25.5679	1.09320	0.546602	+	0.837393i	$0.315921\pi$
$548$	−82.6814	−3.53198
$549$	0	0
$550$	49.0367	2.09093
$551$	0.0655389	0.00279205
$552$	0	0
$553$	−3.36380	−0.143043
$554$	−70.1808	−2.98170
$555$	0	0
$556$	−41.2403	−1.74898
$557$	−40.7860	−1.72816	−0.864079	−	0.503357i	$-0.832098\pi$
$557$	−40.7860	−1.72816	−0.864079	+	0.503357i	$0.832098\pi$
$558$	0	0
$559$	12.9808	0.549029
$560$	−44.4104	−1.87668
$561$	0	0
$562$	47.6367	2.00943
$563$	−10.0286	−0.422657	−0.211329	−	0.977415i	$-0.567779\pi$
$563$	−10.0286	−0.422657	−0.211329	+	0.977415i	$0.567779\pi$
$564$	0	0
$565$	44.6137	1.87691
$566$	26.4485	1.11171
$567$	0	0
$568$	−105.986	−4.44709
$569$	29.2372	1.22569	0.612844	−	0.790204i	$-0.290025\pi$
$569$	29.2372	1.22569	0.612844	+	0.790204i	$0.290025\pi$
$570$	0	0
$571$	10.8647	0.454672	0.227336	−	0.973816i	$-0.426998\pi$
$571$	10.8647	0.454672	0.227336	+	0.973816i	$0.426998\pi$
$572$	32.1433	1.34398
$573$	0	0
$574$	0.433889	0.0181102
$575$	25.3569	1.05746
$576$	0	0
$577$	−3.53035	−0.146971	−0.0734853	−	0.997296i	$-0.523412\pi$
$577$	−3.53035	−0.146971	−0.0734853	+	0.997296i	$0.523412\pi$
$578$	−60.9408	−2.53480
$579$	0	0
$580$	−6.93821	−0.288093
$581$	17.3020	0.717807
$582$	0	0
$583$	−8.33293	−0.345115
$584$	99.1881	4.10443
$585$	0	0
$586$	48.7513	2.01390
$587$	−30.2595	−1.24894	−0.624472	−	0.781048i	$-0.714686\pi$
$587$	−30.2595	−1.24894	−0.624472	+	0.781048i	$0.714686\pi$
$588$	0	0
$589$	1.26142	0.0519758
$590$	−31.3268	−1.28970
$591$	0	0
$592$	−32.1215	−1.32019
$593$	31.0070	1.27330	0.636652	−	0.771151i	$-0.280319\pi$
$593$	31.0070	1.27330	0.636652	+	0.771151i	$0.280319\pi$
$594$	0	0
$595$	−21.1409	−0.866693
$596$	83.0022	3.39990
$597$	0	0
$598$	22.9311	0.937721
$599$	−35.9127	−1.46735	−0.733676	−	0.679499i	$-0.762197\pi$
$599$	−35.9127	−1.46735	−0.733676	+	0.679499i	$0.762197\pi$
$600$	0	0
$601$	−46.4045	−1.89288	−0.946440	−	0.322879i	$-0.895349\pi$
$601$	−46.4045	−1.89288	−0.946440	+	0.322879i	$0.895349\pi$
$602$	−16.5997	−0.676555
$603$	0	0
$604$	−10.5128	−0.427760
$605$	−8.81958	−0.358567
$606$	0	0
$607$	−13.7737	−0.559058	−0.279529	−	0.960137i	$-0.590178\pi$
$607$	−13.7737	−0.559058	−0.279529	+	0.960137i	$0.590178\pi$
$608$	3.01222	0.122162
$609$	0	0
$610$	108.966	4.41192
$611$	18.5873	0.751961
$612$	0	0
$613$	−33.4089	−1.34937	−0.674686	−	0.738105i	$-0.735721\pi$
$613$	−33.4089	−1.34937	−0.674686	+	0.738105i	$0.735721\pi$
$614$	−10.9378	−0.441415
$615$	0	0
$616$	−25.5006	−1.02745
$617$	26.6250	1.07188	0.535941	−	0.844256i	$-0.319957\pi$
$617$	26.6250	1.07188	0.535941	+	0.844256i	$0.319957\pi$
$618$	0	0
$619$	−3.87967	−0.155937	−0.0779686	−	0.996956i	$-0.524843\pi$
$619$	−3.87967	−0.155937	−0.0779686	+	0.996956i	$0.524843\pi$
$620$	−133.538	−5.36303
$621$	0	0
$622$	36.5877	1.46703
$623$	14.6779	0.588058
$624$	0	0
$625$	−16.9225	−0.676902
$626$	−63.0291	−2.51915
$627$	0	0
$628$	23.2316	0.927041
$629$	−15.2910	−0.609691
$630$	0	0
$631$	24.1518	0.961467	0.480733	−	0.876867i	$-0.340370\pi$
$631$	24.1518	0.961467	0.480733	+	0.876867i	$0.340370\pi$
$632$	−29.6414	−1.17907
$633$	0	0
$634$	−85.9473	−3.41340
$635$	−3.35934	−0.133311
$636$	0	0
$637$	−2.10825	−0.0835319
$638$	−3.05852	−0.121088
$639$	0	0
$640$	−79.4228	−3.13946
$641$	3.52432	0.139202	0.0696012	−	0.997575i	$-0.477827\pi$
$641$	3.52432	0.139202	0.0696012	+	0.997575i	$0.477827\pi$
$642$	0	0
$643$	19.9282	0.785890	0.392945	−	0.919562i	$-0.371456\pi$
$643$	19.9282	0.785890	0.392945	+	0.919562i	$0.371456\pi$
$644$	−21.2552	−0.837574
$645$	0	0
$646$	2.83651	0.111601
$647$	−38.7661	−1.52405	−0.762027	−	0.647546i	$-0.775795\pi$
$647$	−38.7661	−1.52405	−0.762027	+	0.647546i	$0.775795\pi$
$648$	0	0
$649$	−10.0097	−0.392916
$650$	35.7240	1.40121
$651$	0	0
$652$	−51.7203	−2.02552
$653$	23.5705	0.922386	0.461193	−	0.887300i	$-0.347421\pi$
$653$	23.5705	0.922386	0.461193	+	0.887300i	$0.347421\pi$
$654$	0	0
$655$	−65.0353	−2.54114
$656$	2.12759	0.0830683
$657$	0	0
$658$	−23.7693	−0.926623
$659$	−7.43377	−0.289579	−0.144789	−	0.989463i	$-0.546250\pi$
$659$	−7.43377	−0.289579	−0.144789	+	0.989463i	$0.546250\pi$
$660$	0	0
$661$	3.67368	0.142890	0.0714449	−	0.997445i	$-0.477239\pi$
$661$	3.67368	0.142890	0.0714449	+	0.997445i	$0.477239\pi$
$662$	88.6906	3.44706
$663$	0	0
$664$	152.463	5.91671
$665$	0.561624	0.0217789
$666$	0	0
$667$	−1.58157	−0.0612385
$668$	80.8197	3.12701
$669$	0	0
$670$	−84.7822	−3.27542
$671$	34.8176	1.34412
$672$	0	0
$673$	45.7468	1.76341	0.881706	−	0.471800i	$-0.156396\pi$
$673$	45.7468	1.76341	0.881706	+	0.471800i	$0.156396\pi$
$674$	−32.9888	−1.27068
$675$	0	0
$676$	−45.0734	−1.73359
$677$	49.9400	1.91935	0.959674	−	0.281114i	$-0.0907039\pi$
$677$	49.9400	1.91935	0.959674	+	0.281114i	$0.0907039\pi$
$678$	0	0
$679$	1.23267	0.0473055
$680$	−186.291	−7.14395
$681$	0	0
$682$	−58.8668	−2.25413
$683$	31.8312	1.21799	0.608993	−	0.793176i	$-0.291574\pi$
$683$	31.8312	1.21799	0.608993	+	0.793176i	$0.291574\pi$
$684$	0	0
$685$	−52.7201	−2.01433
$686$	2.69601	0.102934
$687$	0	0
$688$	−81.3973	−3.10324
$689$	−6.07068	−0.231275
$690$	0	0
$691$	15.3903	0.585475	0.292737	−	0.956193i	$-0.405434\pi$
$691$	15.3903	0.585475	0.292737	+	0.956193i	$0.405434\pi$
$692$	−67.2249	−2.55551
$693$	0	0
$694$	−13.4301	−0.509800
$695$	−26.2960	−0.997464
$696$	0	0
$697$	1.01281	0.0383628
$698$	19.6775	0.744806
$699$	0	0
$700$	−33.1133	−1.25157
$701$	−16.6857	−0.630210	−0.315105	−	0.949057i	$-0.602040\pi$
$701$	−16.6857	−0.630210	−0.315105	+	0.949057i	$0.602040\pi$
$702$	0	0
$703$	0.406216	0.0153207
$704$	−64.0575	−2.41426
$705$	0	0
$706$	−32.9428	−1.23982
$707$	−12.2730	−0.461573
$708$	0	0
$709$	13.3407	0.501022	0.250511	−	0.968114i	$-0.419401\pi$
$709$	13.3407	0.501022	0.250511	+	0.968114i	$0.419401\pi$
$710$	−108.932	−4.08816
$711$	0	0
$712$	129.340	4.84722
$713$	−30.4401	−1.13999
$714$	0	0
$715$	20.4955	0.766488
$716$	−6.34058	−0.236958
$717$	0	0
$718$	97.2577	3.62962
$719$	20.3493	0.758903	0.379451	−	0.925212i	$-0.376113\pi$
$719$	20.3493	0.758903	0.379451	+	0.925212i	$0.376113\pi$
$720$	0	0
$721$	3.50168	0.130409
$722$	51.1489	1.90356
$723$	0	0
$724$	−43.6035	−1.62051
$725$	−2.46390	−0.0915071
$726$	0	0
$727$	6.19761	0.229856	0.114928	−	0.993374i	$-0.463336\pi$
$727$	6.19761	0.229856	0.114928	+	0.993374i	$0.463336\pi$
$728$	−18.5777	−0.688534
$729$	0	0
$730$	101.945	3.77316
$731$	−38.7480	−1.43315
$732$	0	0
$733$	−20.4599	−0.755703	−0.377851	−	0.925866i	$-0.623337\pi$
$733$	−20.4599	−0.755703	−0.377851	+	0.925866i	$0.623337\pi$
$734$	67.4606	2.49001
$735$	0	0
$736$	−72.6899	−2.67939
$737$	−27.0901	−0.997877
$738$	0	0
$739$	−40.7478	−1.49893	−0.749466	−	0.662042i	$-0.769690\pi$
$739$	−40.7478	−1.49893	−0.749466	+	0.662042i	$0.769690\pi$
$740$	−43.0036	−1.58085
$741$	0	0
$742$	7.76314	0.284994
$743$	−4.75496	−0.174443	−0.0872213	−	0.996189i	$-0.527799\pi$
$743$	−4.75496	−0.174443	−0.0872213	+	0.996189i	$0.527799\pi$
$744$	0	0
$745$	52.9246	1.93901
$746$	34.8290	1.27518
$747$	0	0
$748$	−95.9485	−3.50822
$749$	−11.7913	−0.430844
$750$	0	0
$751$	30.3761	1.10844	0.554220	−	0.832370i	$-0.313017\pi$
$751$	30.3761	1.10844	0.554220	+	0.832370i	$0.313017\pi$
$752$	−116.553	−4.25026
$753$	0	0
$754$	−2.22819	−0.0811457
$755$	−6.70327	−0.243957
$756$	0	0
$757$	2.83567	0.103064	0.0515321	−	0.998671i	$-0.483590\pi$
$757$	2.83567	0.103064	0.0515321	+	0.998671i	$0.483590\pi$
$758$	−2.90286	−0.105437
$759$	0	0
$760$	4.94897	0.179518
$761$	10.8152	0.392052	0.196026	−	0.980599i	$-0.437196\pi$
$761$	10.8152	0.392052	0.196026	+	0.980599i	$0.437196\pi$
$762$	0	0
$763$	8.25140	0.298721
$764$	85.1314	3.07994
$765$	0	0
$766$	37.9391	1.37079
$767$	−7.29226	−0.263308
$768$	0	0
$769$	−12.8275	−0.462571	−0.231286	−	0.972886i	$-0.574293\pi$
$769$	−12.8275	−0.462571	−0.231286	+	0.972886i	$0.574293\pi$
$770$	−26.2095	−0.944524
$771$	0	0
$772$	54.3934	1.95766
$773$	34.8329	1.25285	0.626427	−	0.779480i	$-0.284517\pi$
$773$	34.8329	1.25285	0.626427	+	0.779480i	$0.284517\pi$
$774$	0	0
$775$	−47.4223	−1.70346
$776$	10.8621	0.389928
$777$	0	0
$778$	2.41677	0.0866455
$779$	−0.0269060	−0.000964006 0
$780$	0	0
$781$	−34.8067	−1.24548
$782$	−68.4498	−2.44776
$783$	0	0
$784$	13.2200	0.472142
$785$	14.8131	0.528703
$786$	0	0
$787$	7.06256	0.251753	0.125876	−	0.992046i	$-0.459826\pi$
$787$	7.06256	0.251753	0.125876	+	0.992046i	$0.459826\pi$
$788$	103.924	3.70213
$789$	0	0
$790$	−30.4653	−1.08391
$791$	−13.2805	−0.472200
$792$	0	0
$793$	25.3652	0.900745
$794$	−41.9620	−1.48918
$795$	0	0
$796$	6.60443	0.234088
$797$	−4.18736	−0.148324	−0.0741619	−	0.997246i	$-0.523628\pi$
$797$	−4.18736	−0.148324	−0.0741619	+	0.997246i	$0.523628\pi$
$798$	0	0
$799$	−55.4835	−1.96286
$800$	−113.243	−4.00374
$801$	0	0
$802$	−58.0104	−2.04842
$803$	32.5741	1.14952
$804$	0	0
$805$	−13.5530	−0.477679
$806$	−42.8855	−1.51058
$807$	0	0
$808$	−108.148	−3.80463
$809$	−5.20778	−0.183096	−0.0915479	−	0.995801i	$-0.529181\pi$
$809$	−5.20778	−0.183096	−0.0915479	+	0.995801i	$0.529181\pi$
$810$	0	0
$811$	40.9326	1.43734	0.718668	−	0.695353i	$-0.244752\pi$
$811$	40.9326	1.43734	0.718668	+	0.695353i	$0.244752\pi$
$812$	2.06535	0.0724795
$813$	0	0
$814$	−18.9570	−0.664442
$815$	−32.9784	−1.15518
$816$	0	0
$817$	1.02937	0.0360131
$818$	1.11393	0.0389476
$819$	0	0
$820$	2.84837	0.0994694
$821$	−9.27823	−0.323813	−0.161906	−	0.986806i	$-0.551764\pi$
$821$	−9.27823	−0.323813	−0.161906	+	0.986806i	$0.551764\pi$
$822$	0	0
$823$	−33.4733	−1.16681	−0.583404	−	0.812182i	$-0.698279\pi$
$823$	−33.4733	−1.16681	−0.583404	+	0.812182i	$0.698279\pi$
$824$	30.8564	1.07493
$825$	0	0
$826$	9.32528	0.324468
$827$	−11.0635	−0.384716	−0.192358	−	0.981325i	$-0.561613\pi$
$827$	−11.0635	−0.384716	−0.192358	+	0.981325i	$0.561613\pi$
$828$	0	0
$829$	−18.5732	−0.645075	−0.322538	−	0.946557i	$-0.604536\pi$
$829$	−18.5732	−0.645075	−0.322538	+	0.946557i	$0.604536\pi$
$830$	156.701	5.43917
$831$	0	0
$832$	−46.6670	−1.61789
$833$	6.29317	0.218046
$834$	0	0
$835$	51.5330	1.78337
$836$	2.54894	0.0881571
$837$	0	0
$838$	98.1875	3.39183
$839$	−43.0577	−1.48652	−0.743258	−	0.669004i	$-0.766721\pi$
$839$	−43.0577	−1.48652	−0.743258	+	0.669004i	$0.766721\pi$
$840$	0	0
$841$	−28.8463	−0.994701
$842$	35.6209	1.22758
$843$	0	0
$844$	9.56862	0.329366
$845$	−28.7401	−0.988690
$846$	0	0
$847$	2.62539	0.0902095
$848$	38.0668	1.30722
$849$	0	0
$850$	−106.637	−3.65762
$851$	−9.80269	−0.336032
$852$	0	0
$853$	17.7530	0.607850	0.303925	−	0.952696i	$-0.401703\pi$
$853$	17.7530	0.607850	0.303925	+	0.952696i	$0.401703\pi$
$854$	−32.4368	−1.10997
$855$	0	0
$856$	−103.903	−3.55134
$857$	2.85537	0.0975377	0.0487688	−	0.998810i	$-0.484470\pi$
$857$	2.85537	0.0975377	0.0487688	+	0.998810i	$0.484470\pi$
$858$	0	0
$859$	19.0355	0.649485	0.324742	−	0.945803i	$-0.394722\pi$
$859$	19.0355	0.649485	0.324742	+	0.945803i	$0.394722\pi$
$860$	−108.973	−3.71595
$861$	0	0
$862$	91.3548	3.11156
$863$	24.5348	0.835173	0.417586	−	0.908637i	$-0.362876\pi$
$863$	24.5348	0.835173	0.417586	+	0.908637i	$0.362876\pi$
$864$	0	0
$865$	−42.8645	−1.45744
$866$	12.3216	0.418705
$867$	0	0
$868$	39.7514	1.34925
$869$	−9.73446	−0.330219
$870$	0	0
$871$	−19.7356	−0.668716
$872$	72.7104	2.46229
$873$	0	0
$874$	1.81842	0.0615090
$875$	−4.31730	−0.145951
$876$	0	0
$877$	0.731536	0.0247022	0.0123511	−	0.999924i	$-0.496068\pi$
$877$	0.731536	0.0247022	0.0123511	+	0.999924i	$0.496068\pi$
$878$	40.7810	1.37629
$879$	0	0
$880$	−128.519	−4.33237
$881$	24.2382	0.816605	0.408302	−	0.912847i	$-0.366121\pi$
$881$	24.2382	0.816605	0.408302	+	0.912847i	$0.366121\pi$
$882$	0	0
$883$	−24.7304	−0.832245	−0.416122	−	0.909309i	$-0.636611\pi$
$883$	−24.7304	−0.832245	−0.416122	+	0.909309i	$0.636611\pi$
$884$	−69.9001	−2.35099
$885$	0	0
$886$	−57.8800	−1.94452
$887$	2.92462	0.0981992	0.0490996	−	0.998794i	$-0.484365\pi$
$887$	2.92462	0.0981992	0.0490996	+	0.998794i	$0.484365\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	132.935	4.45600
$891$	0	0
$892$	3.27786	0.109751
$893$	1.47396	0.0493242
$894$	0	0
$895$	−4.04293	−0.135140
$896$	23.6424	0.789836
$897$	0	0
$898$	73.6633	2.45818
$899$	2.95783	0.0986493
$900$	0	0
$901$	18.1211	0.603702
$902$	1.25563	0.0418078
$903$	0	0
$904$	−117.026	−3.89223
$905$	−27.8028	−0.924197
$906$	0	0
$907$	22.1789	0.736440	0.368220	−	0.929739i	$-0.379967\pi$
$907$	22.1789	0.736440	0.368220	+	0.929739i	$0.379967\pi$
$908$	−133.723	−4.43774
$909$	0	0
$910$	−19.0940	−0.632961
$911$	−4.10731	−0.136081	−0.0680406	−	0.997683i	$-0.521675\pi$
$911$	−4.10731	−0.136081	−0.0680406	+	0.997683i	$0.521675\pi$
$912$	0	0
$913$	50.0700	1.65708
$914$	−48.1721	−1.59339
$915$	0	0
$916$	−3.70086	−0.122280
$917$	19.3595	0.639309
$918$	0	0
$919$	18.2568	0.602238	0.301119	−	0.953587i	$-0.402640\pi$
$919$	18.2568	0.602238	0.301119	+	0.953587i	$0.402640\pi$
$920$	−119.427	−3.93739
$921$	0	0
$922$	−14.4900	−0.477203
$923$	−25.3573	−0.834646
$924$	0	0
$925$	−15.2715	−0.502124
$926$	−58.9312	−1.93660
$927$	0	0
$928$	7.06320	0.231861
$929$	−48.7368	−1.59900	−0.799501	−	0.600665i	$-0.794903\pi$
$929$	−48.7368	−1.59900	−0.799501	+	0.600665i	$0.794903\pi$
$930$	0	0
$931$	−0.167183	−0.00547920
$932$	−85.0023	−2.78434
$933$	0	0
$934$	−88.6345	−2.90021
$935$	−61.1795	−2.00078
$936$	0	0
$937$	−18.6686	−0.609875	−0.304938	−	0.952372i	$-0.598636\pi$
$937$	−18.6686	−0.609875	−0.304938	+	0.952372i	$0.598636\pi$
$938$	25.2378	0.824042
$939$	0	0
$940$	−156.039	−5.08944
$941$	−51.5302	−1.67984	−0.839918	−	0.542713i	$-0.817397\pi$
$941$	−51.5302	−1.67984	−0.839918	+	0.542713i	$0.817397\pi$
$942$	0	0
$943$	0.649287	0.0211437
$944$	45.7268	1.48828
$945$	0	0
$946$	−48.0378	−1.56184
$947$	−27.8818	−0.906036	−0.453018	−	0.891501i	$-0.649653\pi$
$947$	−27.8818	−0.906036	−0.453018	+	0.891501i	$0.649653\pi$
$948$	0	0
$949$	23.7308	0.770335
$950$	2.83290	0.0919113
$951$	0	0
$952$	55.4547	1.79730
$953$	−39.2006	−1.26983	−0.634916	−	0.772582i	$-0.718965\pi$
$953$	−39.2006	−1.26983	−0.634916	+	0.772582i	$0.718965\pi$
$954$	0	0
$955$	54.2822	1.75653
$956$	−43.9741	−1.42223
$957$	0	0
$958$	42.7830	1.38226
$959$	15.6936	0.506772
$960$	0	0
$961$	25.9289	0.836417
$962$	−13.8105	−0.445268
$963$	0	0
$964$	61.6724	1.98633
$965$	34.6828	1.11648
$966$	0	0
$967$	31.0201	0.997539	0.498770	−	0.866735i	$-0.333785\pi$
$967$	31.0201	0.997539	0.498770	+	0.866735i	$0.333785\pi$
$968$	23.1347	0.743576
$969$	0	0
$970$	11.1641	0.358456
$971$	−7.85796	−0.252174	−0.126087	−	0.992019i	$-0.540242\pi$
$971$	−7.85796	−0.252174	−0.126087	+	0.992019i	$0.540242\pi$
$972$	0	0
$973$	7.82773	0.250945
$974$	−24.2351	−0.776544
$975$	0	0
$976$	−159.055	−5.09123
$977$	−33.1002	−1.05897	−0.529484	−	0.848320i	$-0.677615\pi$
$977$	−33.1002	−1.05897	−0.529484	+	0.848320i	$0.677615\pi$
$978$	0	0
$979$	42.4762	1.35755
$980$	17.6986	0.565362
$981$	0	0
$982$	3.75734	0.119902
$983$	−5.35215	−0.170707	−0.0853535	−	0.996351i	$-0.527202\pi$
$983$	−5.35215	−0.170707	−0.0853535	+	0.996351i	$0.527202\pi$
$984$	0	0
$985$	66.2647	2.11137
$986$	6.65119	0.211817
$987$	0	0
$988$	1.85695	0.0590774
$989$	−24.8404	−0.789880
$990$	0	0
$991$	31.1999	0.991097	0.495549	−	0.868580i	$-0.334967\pi$
$991$	31.1999	0.991097	0.495549	+	0.868580i	$0.334967\pi$
$992$	135.944	4.31623
$993$	0	0
$994$	32.4267	1.02851
$995$	4.21118	0.133503
$996$	0	0
$997$	−62.0455	−1.96500	−0.982501	−	0.186257i	$-0.940364\pi$
$997$	−62.0455	−1.96500	−0.982501	+	0.186257i	$0.940364\pi$
$998$	16.0877	0.509247
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.1	✓	32
3.2	odd	2	inner	8001.2.a.z.1.32	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.1	✓	32	1.1	even	1	trivial
8001.2.a.z.1.32	yes	32	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.69601 q^{2} +5.26849 q^{4} +3.35934 q^{5} -1.00000 q^{7} -8.81189 q^{8} +O(q^{10})\)	\(q-2.69601 q^{2} +5.26849 q^{4} +3.35934 q^{5} -1.00000 q^{7} -8.81189 q^{8} -9.05682 q^{10} -2.89389 q^{11} -2.10825 q^{13} +2.69601 q^{14} +13.2200 q^{16} +6.29317 q^{17} -0.167183 q^{19} +17.6986 q^{20} +7.80197 q^{22} +4.03441 q^{23} +6.28516 q^{25} +5.68387 q^{26} -5.26849 q^{28} -0.392019 q^{29} -7.54513 q^{31} -18.0175 q^{32} -16.9665 q^{34} -3.35934 q^{35} -2.42977 q^{37} +0.450728 q^{38} -29.6021 q^{40} +0.160937 q^{41} -6.15714 q^{43} -15.2464 q^{44} -10.8768 q^{46} -8.81645 q^{47} +1.00000 q^{49} -16.9449 q^{50} -11.1073 q^{52} +2.87949 q^{53} -9.72156 q^{55} +8.81189 q^{56} +1.05689 q^{58} +3.45891 q^{59} -12.0314 q^{61} +20.3418 q^{62} +22.1354 q^{64} -7.08233 q^{65} +9.36114 q^{67} +33.1555 q^{68} +9.05682 q^{70} +12.0277 q^{71} -11.2562 q^{73} +6.55069 q^{74} -0.880801 q^{76} +2.89389 q^{77} +3.36380 q^{79} +44.4104 q^{80} -0.433889 q^{82} -17.3020 q^{83} +21.1409 q^{85} +16.5997 q^{86} +25.5006 q^{88} -14.6779 q^{89} +2.10825 q^{91} +21.2552 q^{92} +23.7693 q^{94} -0.561624 q^{95} -1.23267 q^{97} -2.69601 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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