LMFDB - Embedded newform 8001.2.a.r.1.3

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 $ x^16 - 5x^15 - 13x^14 + 98x^13 + 9x^12 - 712x^11 + 565x^10 + 2282x^9 - 3082x^8 - 2747x^7 + 5821x^6 - 158x^5 - 3341x^4 + 1002x^3 + 416x^2 - 148*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.41798$ 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)

$f(q)$	$=$	$q-2.41798 q^{2} +3.84663 q^{4} -2.86533 q^{5} -1.00000 q^{7} -4.46511 q^{8} +O(q^{10})$	\(q-2.41798 q^{2} +3.84663 q^{4} -2.86533 q^{5} -1.00000 q^{7} -4.46511 q^{8} +6.92831 q^{10} +5.94265 q^{11} +5.81109 q^{13} +2.41798 q^{14} +3.10329 q^{16} +2.46675 q^{17} +4.96080 q^{19} -11.0219 q^{20} -14.3692 q^{22} -5.91940 q^{23} +3.21012 q^{25} -14.0511 q^{26} -3.84663 q^{28} -9.12150 q^{29} +1.52132 q^{31} +1.42653 q^{32} -5.96456 q^{34} +2.86533 q^{35} +10.9350 q^{37} -11.9951 q^{38} +12.7940 q^{40} -7.45599 q^{41} -6.37361 q^{43} +22.8592 q^{44} +14.3130 q^{46} +3.77187 q^{47} +1.00000 q^{49} -7.76200 q^{50} +22.3531 q^{52} -13.0229 q^{53} -17.0277 q^{55} +4.46511 q^{56} +22.0556 q^{58} +12.1352 q^{59} -13.1708 q^{61} -3.67852 q^{62} -9.65590 q^{64} -16.6507 q^{65} -2.34916 q^{67} +9.48868 q^{68} -6.92831 q^{70} -10.9342 q^{71} -9.61640 q^{73} -26.4406 q^{74} +19.0824 q^{76} -5.94265 q^{77} -12.9373 q^{79} -8.89194 q^{80} +18.0284 q^{82} +2.71284 q^{83} -7.06806 q^{85} +15.4113 q^{86} -26.5346 q^{88} +1.78782 q^{89} -5.81109 q^{91} -22.7697 q^{92} -9.12031 q^{94} -14.2143 q^{95} -5.89963 q^{97} -2.41798 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

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 $

$2$	−2.41798	−1.70977	−0.854885	−	0.518817i	$-0.826372\pi$
$2$	−2.41798	−1.70977	−0.854885	+	0.518817i	$0.826372\pi$
$3$	0	0
$3$	0	0
$4$	3.84663	1.92331
$5$	−2.86533	−1.28141	−0.640707	−	0.767785i	$-0.721359\pi$
$5$	−2.86533	−1.28141	−0.640707	+	0.767785i	$0.721359\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−4.46511	−1.57865
$9$	0	0
$10$	6.92831	2.19092
$11$	5.94265	1.79178	0.895889	−	0.444278i	$-0.146540\pi$
$11$	5.94265	1.79178	0.895889	+	0.444278i	$0.146540\pi$
$12$	0	0
$13$	5.81109	1.61171	0.805854	−	0.592115i	$-0.201707\pi$
$13$	5.81109	1.61171	0.805854	+	0.592115i	$0.201707\pi$
$14$	2.41798	0.646232
$15$	0	0
$16$	3.10329	0.775822
$17$	2.46675	0.598276	0.299138	−	0.954210i	$-0.403301\pi$
$17$	2.46675	0.598276	0.299138	+	0.954210i	$0.403301\pi$
$18$	0	0
$19$	4.96080	1.13809	0.569043	−	0.822308i	$-0.307314\pi$
$19$	4.96080	1.13809	0.569043	+	0.822308i	$0.307314\pi$
$20$	−11.0219	−2.46456
$21$	0	0
$22$	−14.3692	−3.06353
$23$	−5.91940	−1.23428	−0.617140	−	0.786853i	$-0.711709\pi$
$23$	−5.91940	−1.23428	−0.617140	+	0.786853i	$0.711709\pi$
$24$	0	0
$25$	3.21012	0.642024
$26$	−14.0511	−2.75565
$27$	0	0
$28$	−3.84663	−0.726944
$29$	−9.12150	−1.69382	−0.846910	−	0.531737i	$-0.821540\pi$
$29$	−9.12150	−1.69382	−0.846910	+	0.531737i	$0.821540\pi$
$30$	0	0
$31$	1.52132	0.273237	0.136619	−	0.990624i	$-0.456377\pi$
$31$	1.52132	0.273237	0.136619	+	0.990624i	$0.456377\pi$
$32$	1.42653	0.252177
$33$	0	0
$34$	−5.96456	−1.02291
$35$	2.86533	0.484329
$36$	0	0
$37$	10.9350	1.79770	0.898850	−	0.438256i	$-0.144404\pi$
$37$	10.9350	1.79770	0.898850	+	0.438256i	$0.144404\pi$
$38$	−11.9951	−1.94587
$39$	0	0
$40$	12.7940	2.02291
$41$	−7.45599	−1.16443	−0.582215	−	0.813035i	$-0.697814\pi$
$41$	−7.45599	−1.16443	−0.582215	+	0.813035i	$0.697814\pi$
$42$	0	0
$43$	−6.37361	−0.971967	−0.485983	−	0.873968i	$-0.661538\pi$
$43$	−6.37361	−0.971967	−0.485983	+	0.873968i	$0.661538\pi$
$44$	22.8592	3.44615
$45$	0	0
$46$	14.3130	2.11034
$47$	3.77187	0.550184	0.275092	−	0.961418i	$-0.411292\pi$
$47$	3.77187	0.550184	0.275092	+	0.961418i	$0.411292\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−7.76200	−1.09771
$51$	0	0
$52$	22.3531	3.09982
$53$	−13.0229	−1.78883	−0.894414	−	0.447240i	$-0.852407\pi$
$53$	−13.0229	−1.78883	−0.894414	+	0.447240i	$0.852407\pi$
$54$	0	0
$55$	−17.0277	−2.29601
$56$	4.46511	0.596675
$57$	0	0
$58$	22.0556	2.89604
$59$	12.1352	1.57986	0.789931	−	0.613195i	$-0.210116\pi$
$59$	12.1352	1.57986	0.789931	+	0.613195i	$0.210116\pi$
$60$	0	0
$61$	−13.1708	−1.68635	−0.843175	−	0.537639i	$-0.819316\pi$
$61$	−13.1708	−1.68635	−0.843175	+	0.537639i	$0.819316\pi$
$62$	−3.67852	−0.467173
$63$	0	0
$64$	−9.65590	−1.20699
$65$	−16.6507	−2.06527
$66$	0	0
$67$	−2.34916	−0.286995	−0.143498	−	0.989651i	$-0.545835\pi$
$67$	−2.34916	−0.286995	−0.143498	+	0.989651i	$0.545835\pi$
$68$	9.48868	1.15067
$69$	0	0
$70$	−6.92831	−0.828092
$71$	−10.9342	−1.29765	−0.648826	−	0.760937i	$-0.724740\pi$
$71$	−10.9342	−1.29765	−0.648826	+	0.760937i	$0.724740\pi$
$72$	0	0
$73$	−9.61640	−1.12551	−0.562757	−	0.826622i	$-0.690259\pi$
$73$	−9.61640	−1.12551	−0.562757	+	0.826622i	$0.690259\pi$
$74$	−26.4406	−3.07366
$75$	0	0
$76$	19.0824	2.18890
$77$	−5.94265	−0.677228
$78$	0	0
$79$	−12.9373	−1.45556	−0.727778	−	0.685813i	$-0.759447\pi$
$79$	−12.9373	−1.45556	−0.727778	+	0.685813i	$0.759447\pi$
$80$	−8.89194	−0.994149
$81$	0	0
$82$	18.0284	1.99091
$83$	2.71284	0.297772	0.148886	−	0.988854i	$-0.452431\pi$
$83$	2.71284	0.297772	0.148886	+	0.988854i	$0.452431\pi$
$84$	0	0
$85$	−7.06806	−0.766639
$86$	15.4113	1.66184
$87$	0	0
$88$	−26.5346	−2.82860
$89$	1.78782	0.189509	0.0947543	−	0.995501i	$-0.469793\pi$
$89$	1.78782	0.189509	0.0947543	+	0.995501i	$0.469793\pi$
$90$	0	0
$91$	−5.81109	−0.609168
$92$	−22.7697	−2.37391
$93$	0	0
$94$	−9.12031	−0.940688
$95$	−14.2143	−1.45836
$96$	0	0
$97$	−5.89963	−0.599017	−0.299508	−	0.954094i	$-0.596823\pi$
$97$	−5.89963	−0.599017	−0.299508	+	0.954094i	$0.596823\pi$
$98$	−2.41798	−0.244253
$99$	0	0
$100$	12.3481	1.23481
$101$	0.292411	0.0290960	0.0145480	−	0.999894i	$-0.495369\pi$
$101$	0.292411	0.0290960	0.0145480	+	0.999894i	$0.495369\pi$
$102$	0	0
$103$	−0.384246	−0.0378608	−0.0189304	−	0.999821i	$-0.506026\pi$
$103$	−0.384246	−0.0378608	−0.0189304	+	0.999821i	$0.506026\pi$
$104$	−25.9472	−2.54433
$105$	0	0
$106$	31.4890	3.05848
$107$	6.48261	0.626697	0.313349	−	0.949638i	$-0.398549\pi$
$107$	6.48261	0.626697	0.313349	+	0.949638i	$0.398549\pi$
$108$	0	0
$109$	−7.32894	−0.701985	−0.350993	−	0.936378i	$-0.614156\pi$
$109$	−7.32894	−0.701985	−0.350993	+	0.936378i	$0.614156\pi$
$110$	41.1726	3.92565
$111$	0	0
$112$	−3.10329	−0.293233
$113$	8.75155	0.823277	0.411638	−	0.911347i	$-0.364957\pi$
$113$	8.75155	0.823277	0.411638	+	0.911347i	$0.364957\pi$
$114$	0	0
$115$	16.9610	1.58162
$116$	−35.0870	−3.25775
$117$	0	0
$118$	−29.3426	−2.70120
$119$	−2.46675	−0.226127
$120$	0	0
$121$	24.3151	2.21047
$122$	31.8468	2.88327
$123$	0	0
$124$	5.85195	0.525521
$125$	5.12860	0.458716
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	20.4947	1.81149
$129$	0	0
$130$	40.2611	3.53113
$131$	−6.98949	−0.610675	−0.305337	−	0.952244i	$-0.598769\pi$
$131$	−6.98949	−0.610675	−0.305337	+	0.952244i	$0.598769\pi$
$132$	0	0
$133$	−4.96080	−0.430156
$134$	5.68022	0.490696
$135$	0	0
$136$	−11.0143	−0.944470
$137$	13.3983	1.14470	0.572349	−	0.820010i	$-0.306032\pi$
$137$	13.3983	1.14470	0.572349	+	0.820010i	$0.306032\pi$
$138$	0	0
$139$	−4.56630	−0.387308	−0.193654	−	0.981070i	$-0.562034\pi$
$139$	−4.56630	−0.387308	−0.193654	+	0.981070i	$0.562034\pi$
$140$	11.0219	0.931517
$141$	0	0
$142$	26.4387	2.21869
$143$	34.5333	2.88782
$144$	0	0
$145$	26.1361	2.17049
$146$	23.2523	1.92437
$147$	0	0
$148$	42.0628	3.45754
$149$	4.52922	0.371048	0.185524	−	0.982640i	$-0.440602\pi$
$149$	4.52922	0.371048	0.185524	+	0.982640i	$0.440602\pi$
$150$	0	0
$151$	8.08303	0.657788	0.328894	−	0.944367i	$-0.393324\pi$
$151$	8.08303	0.657788	0.328894	+	0.944367i	$0.393324\pi$
$152$	−22.1505	−1.79664
$153$	0	0
$154$	14.3692	1.15790
$155$	−4.35908	−0.350130
$156$	0	0
$157$	−22.0319	−1.75834	−0.879168	−	0.476512i	$-0.841901\pi$
$157$	−22.0319	−1.75834	−0.879168	+	0.476512i	$0.841901\pi$
$158$	31.2821	2.48867
$159$	0	0
$160$	−4.08748	−0.323144
$161$	5.91940	0.466514
$162$	0	0
$163$	8.72198	0.683158	0.341579	−	0.939853i	$-0.389038\pi$
$163$	8.72198	0.683158	0.341579	+	0.939853i	$0.389038\pi$
$164$	−28.6804	−2.23957
$165$	0	0
$166$	−6.55958	−0.509122
$167$	−2.92573	−0.226400	−0.113200	−	0.993572i	$-0.536110\pi$
$167$	−2.92573	−0.226400	−0.113200	+	0.993572i	$0.536110\pi$
$168$	0	0
$169$	20.7688	1.59760
$170$	17.0904	1.31078
$171$	0	0
$172$	−24.5169	−1.86940
$173$	−18.6048	−1.41450	−0.707248	−	0.706966i	$-0.750064\pi$
$173$	−18.6048	−1.41450	−0.707248	+	0.706966i	$0.750064\pi$
$174$	0	0
$175$	−3.21012	−0.242662
$176$	18.4418	1.39010
$177$	0	0
$178$	−4.32292	−0.324016
$179$	−4.73174	−0.353667	−0.176833	−	0.984241i	$-0.556585\pi$
$179$	−4.73174	−0.353667	−0.176833	+	0.984241i	$0.556585\pi$
$180$	0	0
$181$	−10.1816	−0.756790	−0.378395	−	0.925644i	$-0.623524\pi$
$181$	−10.1816	−0.756790	−0.378395	+	0.925644i	$0.623524\pi$
$182$	14.0511	1.04154
$183$	0	0
$184$	26.4308	1.94850
$185$	−31.3324	−2.30360
$186$	0	0
$187$	14.6591	1.07198
$188$	14.5090	1.05818
$189$	0	0
$190$	34.3700	2.49346
$191$	−19.9897	−1.44641	−0.723203	−	0.690635i	$-0.757331\pi$
$191$	−19.9897	−1.44641	−0.723203	+	0.690635i	$0.757331\pi$
$192$	0	0
$193$	−17.4744	−1.25783	−0.628916	−	0.777473i	$-0.716501\pi$
$193$	−17.4744	−1.25783	−0.628916	+	0.777473i	$0.716501\pi$
$194$	14.2652	1.02418
$195$	0	0
$196$	3.84663	0.274759
$197$	0.0269163	0.00191771	0.000958855	−	1.00000i	$-0.499695\pi$
$197$	0.0269163	0.00191771	0.000958855		1.00000i	$0.499695\pi$
$198$	0	0
$199$	−11.1642	−0.791411	−0.395706	−	0.918377i	$-0.629500\pi$
$199$	−11.1642	−0.791411	−0.395706	+	0.918377i	$0.629500\pi$
$200$	−14.3335	−1.01353
$201$	0	0
$202$	−0.707043	−0.0497474
$203$	9.12150	0.640204
$204$	0	0
$205$	21.3639	1.49212
$206$	0.929098	0.0647333
$207$	0	0
$208$	18.0335	1.25040
$209$	29.4803	2.03920
$210$	0	0
$211$	7.93667	0.546383	0.273191	−	0.961960i	$-0.411921\pi$
$211$	7.93667	0.546383	0.273191	+	0.961960i	$0.411921\pi$
$212$	−50.0941	−3.44048
$213$	0	0
$214$	−15.6748	−1.07151
$215$	18.2625	1.24549
$216$	0	0
$217$	−1.52132	−0.103274
$218$	17.7212	1.20023
$219$	0	0
$220$	−65.4991	−4.41595
$221$	14.3345	0.964245
$222$	0	0
$223$	16.2035	1.08507	0.542534	−	0.840034i	$-0.317465\pi$
$223$	16.2035	1.08507	0.542534	+	0.840034i	$0.317465\pi$
$224$	−1.42653	−0.0953141
$225$	0	0
$226$	−21.1611	−1.40761
$227$	−1.13795	−0.0755283	−0.0377641	−	0.999287i	$-0.512024\pi$
$227$	−1.13795	−0.0755283	−0.0377641	+	0.999287i	$0.512024\pi$
$228$	0	0
$229$	−16.1025	−1.06409	−0.532043	−	0.846717i	$-0.678576\pi$
$229$	−16.1025	−1.06409	−0.532043	+	0.846717i	$0.678576\pi$
$230$	−41.0114	−2.70421
$231$	0	0
$232$	40.7285	2.67396
$233$	11.7016	0.766599	0.383300	−	0.923624i	$-0.374788\pi$
$233$	11.7016	0.766599	0.383300	+	0.923624i	$0.374788\pi$
$234$	0	0
$235$	−10.8077	−0.705014
$236$	46.6794	3.03857
$237$	0	0
$238$	5.96456	0.386625
$239$	−21.5712	−1.39532	−0.697661	−	0.716428i	$-0.745776\pi$
$239$	−21.5712	−1.39532	−0.697661	+	0.716428i	$0.745776\pi$
$240$	0	0
$241$	1.24666	0.0803044	0.0401522	−	0.999194i	$-0.487216\pi$
$241$	1.24666	0.0803044	0.0401522	+	0.999194i	$0.487216\pi$
$242$	−58.7935	−3.77939
$243$	0	0
$244$	−50.6632	−3.24338
$245$	−2.86533	−0.183059
$246$	0	0
$247$	28.8277	1.83426
$248$	−6.79286	−0.431347
$249$	0	0
$250$	−12.4009	−0.784299
$251$	17.4247	1.09983	0.549917	−	0.835219i	$-0.314659\pi$
$251$	17.4247	1.09983	0.549917	+	0.835219i	$0.314659\pi$
$252$	0	0
$253$	−35.1769	−2.21156
$254$	−2.41798	−0.151718
$255$	0	0
$256$	−30.2440	−1.89025
$257$	−7.74905	−0.483372	−0.241686	−	0.970354i	$-0.577700\pi$
$257$	−7.74905	−0.483372	−0.241686	+	0.970354i	$0.577700\pi$
$258$	0	0
$259$	−10.9350	−0.679467
$260$	−64.0490	−3.97215
$261$	0	0
$262$	16.9004	1.04411
$263$	23.7131	1.46221	0.731107	−	0.682262i	$-0.239004\pi$
$263$	23.7131	1.46221	0.731107	+	0.682262i	$0.239004\pi$
$264$	0	0
$265$	37.3148	2.29223
$266$	11.9951	0.735468
$267$	0	0
$268$	−9.03634	−0.551982
$269$	3.63039	0.221349	0.110674	−	0.993857i	$-0.464699\pi$
$269$	3.63039	0.221349	0.110674	+	0.993857i	$0.464699\pi$
$270$	0	0
$271$	13.2306	0.803699	0.401850	−	0.915706i	$-0.368367\pi$
$271$	13.2306	0.803699	0.401850	+	0.915706i	$0.368367\pi$
$272$	7.65504	0.464155
$273$	0	0
$274$	−32.3969	−1.95717
$275$	19.0766	1.15036
$276$	0	0
$277$	5.09376	0.306055	0.153027	−	0.988222i	$-0.451098\pi$
$277$	5.09376	0.306055	0.153027	+	0.988222i	$0.451098\pi$
$278$	11.0412	0.662208
$279$	0	0
$280$	−12.7940	−0.764588
$281$	−9.47263	−0.565090	−0.282545	−	0.959254i	$-0.591179\pi$
$281$	−9.47263	−0.565090	−0.282545	+	0.959254i	$0.591179\pi$
$282$	0	0
$283$	29.4196	1.74881	0.874406	−	0.485195i	$-0.161252\pi$
$283$	29.4196	1.74881	0.874406	+	0.485195i	$0.161252\pi$
$284$	−42.0598	−2.49579
$285$	0	0
$286$	−83.5009	−4.93751
$287$	7.45599	0.440113
$288$	0	0
$289$	−10.9151	−0.642066
$290$	−63.1966	−3.71103
$291$	0	0
$292$	−36.9907	−2.16472
$293$	12.0828	0.705885	0.352943	−	0.935645i	$-0.385181\pi$
$293$	12.0828	0.705885	0.352943	+	0.935645i	$0.385181\pi$
$294$	0	0
$295$	−34.7712	−2.02446
$296$	−48.8259	−2.83795
$297$	0	0
$298$	−10.9516	−0.634406
$299$	−34.3982	−1.98930
$300$	0	0
$301$	6.37361	0.367369
$302$	−19.5446	−1.12467
$303$	0	0
$304$	15.3948	0.882952
$305$	37.7387	2.16091
$306$	0	0
$307$	−14.4483	−0.824607	−0.412304	−	0.911046i	$-0.635276\pi$
$307$	−14.4483	−0.824607	−0.412304	+	0.911046i	$0.635276\pi$
$308$	−22.8592	−1.30252
$309$	0	0
$310$	10.5402	0.598642
$311$	8.00701	0.454036	0.227018	−	0.973891i	$-0.427102\pi$
$311$	8.00701	0.454036	0.227018	+	0.973891i	$0.427102\pi$
$312$	0	0
$313$	15.2196	0.860263	0.430131	−	0.902766i	$-0.358467\pi$
$313$	15.2196	0.860263	0.430131	+	0.902766i	$0.358467\pi$
$314$	53.2726	3.00635
$315$	0	0
$316$	−49.7648	−2.79949
$317$	19.1583	1.07604	0.538019	−	0.842933i	$-0.319173\pi$
$317$	19.1583	1.07604	0.538019	+	0.842933i	$0.319173\pi$
$318$	0	0
$319$	−54.2059	−3.03495
$320$	27.6673	1.54665
$321$	0	0
$322$	−14.3130	−0.797632
$323$	12.2371	0.680889
$324$	0	0
$325$	18.6543	1.03475
$326$	−21.0896	−1.16804
$327$	0	0
$328$	33.2918	1.83823
$329$	−3.77187	−0.207950
$330$	0	0
$331$	−34.7472	−1.90988	−0.954940	−	0.296799i	$-0.904081\pi$
$331$	−34.7472	−1.90988	−0.954940	+	0.296799i	$0.904081\pi$
$332$	10.4353	0.572710
$333$	0	0
$334$	7.07437	0.387092
$335$	6.73112	0.367760
$336$	0	0
$337$	−15.1407	−0.824765	−0.412382	−	0.911011i	$-0.635303\pi$
$337$	−15.1407	−0.824765	−0.412382	+	0.911011i	$0.635303\pi$
$338$	−50.2186	−2.73153
$339$	0	0
$340$	−27.1882	−1.47449
$341$	9.04068	0.489580
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	28.4589	1.53440
$345$	0	0
$346$	44.9860	2.41846
$347$	−9.52937	−0.511564	−0.255782	−	0.966735i	$-0.582333\pi$
$347$	−9.52937	−0.511564	−0.255782	+	0.966735i	$0.582333\pi$
$348$	0	0
$349$	−17.8724	−0.956689	−0.478345	−	0.878172i	$-0.658763\pi$
$349$	−17.8724	−0.956689	−0.478345	+	0.878172i	$0.658763\pi$
$350$	7.76200	0.414897
$351$	0	0
$352$	8.47738	0.451846
$353$	−29.9672	−1.59499	−0.797495	−	0.603325i	$-0.793842\pi$
$353$	−29.9672	−1.59499	−0.797495	+	0.603325i	$0.793842\pi$
$354$	0	0
$355$	31.3301	1.66283
$356$	6.87708	0.364485
$357$	0	0
$358$	11.4412	0.604689
$359$	−16.0299	−0.846027	−0.423013	−	0.906123i	$-0.639028\pi$
$359$	−16.0299	−0.846027	−0.423013	+	0.906123i	$0.639028\pi$
$360$	0	0
$361$	5.60956	0.295240
$362$	24.6188	1.29394
$363$	0	0
$364$	−22.3531	−1.17162
$365$	27.5542	1.44225
$366$	0	0
$367$	−30.8788	−1.61186	−0.805929	−	0.592012i	$-0.798334\pi$
$367$	−30.8788	−1.61186	−0.805929	+	0.592012i	$0.798334\pi$
$368$	−18.3696	−0.957581
$369$	0	0
$370$	75.7610	3.93863
$371$	13.0229	0.676113
$372$	0	0
$373$	5.78052	0.299304	0.149652	−	0.988739i	$-0.452185\pi$
$373$	5.78052	0.299304	0.149652	+	0.988739i	$0.452185\pi$
$374$	−35.4453	−1.83283
$375$	0	0
$376$	−16.8418	−0.868550
$377$	−53.0059	−2.72994
$378$	0	0
$379$	13.7475	0.706159	0.353080	−	0.935593i	$-0.385134\pi$
$379$	13.7475	0.706159	0.353080	+	0.935593i	$0.385134\pi$
$380$	−54.6773	−2.80488
$381$	0	0
$382$	48.3348	2.47302
$383$	−0.702358	−0.0358888	−0.0179444	−	0.999839i	$-0.505712\pi$
$383$	−0.702358	−0.0358888	−0.0179444	+	0.999839i	$0.505712\pi$
$384$	0	0
$385$	17.0277	0.867810
$386$	42.2527	2.15060
$387$	0	0
$388$	−22.6937	−1.15210
$389$	−34.6654	−1.75761	−0.878803	−	0.477184i	$-0.841658\pi$
$389$	−34.6654	−1.75761	−0.878803	+	0.477184i	$0.841658\pi$
$390$	0	0
$391$	−14.6017	−0.738440
$392$	−4.46511	−0.225522
$393$	0	0
$394$	−0.0650832	−0.00327884
$395$	37.0695	1.86517
$396$	0	0
$397$	−16.7668	−0.841501	−0.420751	−	0.907176i	$-0.638233\pi$
$397$	−16.7668	−0.841501	−0.420751	+	0.907176i	$0.638233\pi$
$398$	26.9949	1.35313
$399$	0	0
$400$	9.96192	0.498096
$401$	4.08639	0.204065	0.102032	−	0.994781i	$-0.467465\pi$
$401$	4.08639	0.204065	0.102032	+	0.994781i	$0.467465\pi$
$402$	0	0
$403$	8.84053	0.440378
$404$	1.12480	0.0559607
$405$	0	0
$406$	−22.0556	−1.09460
$407$	64.9829	3.22108
$408$	0	0
$409$	28.8726	1.42766	0.713830	−	0.700319i	$-0.246959\pi$
$409$	28.8726	1.42766	0.713830	+	0.700319i	$0.246959\pi$
$410$	−51.6575	−2.55118
$411$	0	0
$412$	−1.47805	−0.0728183
$413$	−12.1352	−0.597132
$414$	0	0
$415$	−7.77317	−0.381570
$416$	8.28970	0.406436
$417$	0	0
$418$	−71.2829	−3.48656
$419$	12.8305	0.626811	0.313406	−	0.949619i	$-0.398530\pi$
$419$	12.8305	0.626811	0.313406	+	0.949619i	$0.398530\pi$
$420$	0	0
$421$	−4.40335	−0.214606	−0.107303	−	0.994226i	$-0.534222\pi$
$421$	−4.40335	−0.214606	−0.107303	+	0.994226i	$0.534222\pi$
$422$	−19.1907	−0.934189
$423$	0	0
$424$	58.1485	2.82394
$425$	7.91857	0.384107
$426$	0	0
$427$	13.1708	0.637380
$428$	24.9362	1.20534
$429$	0	0
$430$	−44.1584	−2.12951
$431$	−12.8114	−0.617103	−0.308552	−	0.951208i	$-0.599844\pi$
$431$	−12.8114	−0.617103	−0.308552	+	0.951208i	$0.599844\pi$
$432$	0	0
$433$	−10.6612	−0.512344	−0.256172	−	0.966631i	$-0.582461\pi$
$433$	−10.6612	−0.512344	−0.256172	+	0.966631i	$0.582461\pi$
$434$	3.67852	0.176575
$435$	0	0
$436$	−28.1917	−1.35014
$437$	−29.3650	−1.40472
$438$	0	0
$439$	7.15197	0.341345	0.170672	−	0.985328i	$-0.445406\pi$
$439$	7.15197	0.341345	0.170672	+	0.985328i	$0.445406\pi$
$440$	76.0304	3.62461
$441$	0	0
$442$	−34.6606	−1.64864
$443$	−39.4432	−1.87400	−0.937000	−	0.349329i	$-0.886410\pi$
$443$	−39.4432	−1.87400	−0.937000	+	0.349329i	$0.886410\pi$
$444$	0	0
$445$	−5.12270	−0.242839
$446$	−39.1798	−1.85522
$447$	0	0
$448$	9.65590	0.456198
$449$	21.2990	1.00516	0.502582	−	0.864530i	$-0.332384\pi$
$449$	21.2990	1.00516	0.502582	+	0.864530i	$0.332384\pi$
$450$	0	0
$451$	−44.3084	−2.08640
$452$	33.6640	1.58342
$453$	0	0
$454$	2.75154	0.129136
$455$	16.6507	0.780597
$456$	0	0
$457$	28.1983	1.31906	0.659530	−	0.751678i	$-0.270755\pi$
$457$	28.1983	1.31906	0.659530	+	0.751678i	$0.270755\pi$
$458$	38.9356	1.81934
$459$	0	0
$460$	65.2428	3.04196
$461$	−33.2624	−1.54919	−0.774593	−	0.632461i	$-0.782045\pi$
$461$	−33.2624	−1.54919	−0.774593	+	0.632461i	$0.782045\pi$
$462$	0	0
$463$	24.3142	1.12998	0.564988	−	0.825099i	$-0.308881\pi$
$463$	24.3142	1.12998	0.564988	+	0.825099i	$0.308881\pi$
$464$	−28.3066	−1.31410
$465$	0	0
$466$	−28.2943	−1.31071
$467$	4.18602	0.193706	0.0968529	−	0.995299i	$-0.469122\pi$
$467$	4.18602	0.193706	0.0968529	+	0.995299i	$0.469122\pi$
$468$	0	0
$469$	2.34916	0.108474
$470$	26.1327	1.20541
$471$	0	0
$472$	−54.1848	−2.49406
$473$	−37.8762	−1.74155
$474$	0	0
$475$	15.9248	0.730678
$476$	−9.48868	−0.434913
$477$	0	0
$478$	52.1586	2.38568
$479$	24.6994	1.12854	0.564271	−	0.825589i	$-0.309157\pi$
$479$	24.6994	1.12854	0.564271	+	0.825589i	$0.309157\pi$
$480$	0	0
$481$	63.5442	2.89737
$482$	−3.01440	−0.137302
$483$	0	0
$484$	93.5313	4.25142
$485$	16.9044	0.767589
$486$	0	0
$487$	−3.19849	−0.144937	−0.0724686	−	0.997371i	$-0.523088\pi$
$487$	−3.19849	−0.144937	−0.0724686	+	0.997371i	$0.523088\pi$
$488$	58.8091	2.66216
$489$	0	0
$490$	6.92831	0.312989
$491$	−14.4712	−0.653075	−0.326538	−	0.945184i	$-0.605882\pi$
$491$	−14.4712	−0.653075	−0.326538	+	0.945184i	$0.605882\pi$
$492$	0	0
$493$	−22.5005	−1.01337
$494$	−69.7048	−3.13617
$495$	0	0
$496$	4.72109	0.211983
$497$	10.9342	0.490466
$498$	0	0
$499$	9.59004	0.429309	0.214655	−	0.976690i	$-0.431137\pi$
$499$	9.59004	0.429309	0.214655	+	0.976690i	$0.431137\pi$
$500$	19.7278	0.882255
$501$	0	0
$502$	−42.1325	−1.88046
$503$	3.29941	0.147114	0.0735568	−	0.997291i	$-0.476565\pi$
$503$	3.29941	0.147114	0.0735568	+	0.997291i	$0.476565\pi$
$504$	0	0
$505$	−0.837854	−0.0372840
$506$	85.0571	3.78125
$507$	0	0
$508$	3.84663	0.170666
$509$	25.4914	1.12988	0.564942	−	0.825131i	$-0.308898\pi$
$509$	25.4914	1.12988	0.564942	+	0.825131i	$0.308898\pi$
$510$	0	0
$511$	9.61640	0.425404
$512$	32.1400	1.42040
$513$	0	0
$514$	18.7370	0.826455
$515$	1.10099	0.0485154
$516$	0	0
$517$	22.4149	0.985807
$518$	26.4406	1.16173
$519$	0	0
$520$	74.3472	3.26034
$521$	−14.4182	−0.631672	−0.315836	−	0.948814i	$-0.602285\pi$
$521$	−14.4182	−0.631672	−0.315836	+	0.948814i	$0.602285\pi$
$522$	0	0
$523$	−15.8407	−0.692667	−0.346333	−	0.938112i	$-0.612573\pi$
$523$	−15.8407	−0.692667	−0.346333	+	0.938112i	$0.612573\pi$
$524$	−26.8860	−1.17452
$525$	0	0
$526$	−57.3379	−2.50005
$527$	3.75272	0.163471
$528$	0	0
$529$	12.0393	0.523447
$530$	−90.2264	−3.91919
$531$	0	0
$532$	−19.0824	−0.827325
$533$	−43.3275	−1.87672
$534$	0	0
$535$	−18.5748	−0.803059
$536$	10.4892	0.453066
$537$	0	0
$538$	−8.77820	−0.378455
$539$	5.94265	0.255968
$540$	0	0
$541$	−29.1139	−1.25170	−0.625851	−	0.779943i	$-0.715248\pi$
$541$	−29.1139	−1.25170	−0.625851	+	0.779943i	$0.715248\pi$
$542$	−31.9912	−1.37414
$543$	0	0
$544$	3.51890	0.150872
$545$	20.9998	0.899535
$546$	0	0
$547$	−33.0139	−1.41157	−0.705786	−	0.708425i	$-0.749406\pi$
$547$	−33.0139	−1.41157	−0.705786	+	0.708425i	$0.749406\pi$
$548$	51.5384	2.20161
$549$	0	0
$550$	−46.1269	−1.96686
$551$	−45.2500	−1.92771
$552$	0	0
$553$	12.9373	0.550149
$554$	−12.3166	−0.523283
$555$	0	0
$556$	−17.5649	−0.744916
$557$	21.0505	0.891940	0.445970	−	0.895048i	$-0.352859\pi$
$557$	21.0505	0.891940	0.445970	+	0.895048i	$0.352859\pi$
$558$	0	0
$559$	−37.0376	−1.56653
$560$	8.89194	0.375753
$561$	0	0
$562$	22.9046	0.966174
$563$	18.6899	0.787686	0.393843	−	0.919178i	$-0.371145\pi$
$563$	18.6899	0.787686	0.393843	+	0.919178i	$0.371145\pi$
$564$	0	0
$565$	−25.0761	−1.05496
$566$	−71.1359	−2.99007
$567$	0	0
$568$	48.8224	2.04854
$569$	15.5497	0.651877	0.325938	−	0.945391i	$-0.394320\pi$
$569$	15.5497	0.651877	0.325938	+	0.945391i	$0.394320\pi$
$570$	0	0
$571$	10.8384	0.453571	0.226786	−	0.973945i	$-0.427178\pi$
$571$	10.8384	0.453571	0.226786	+	0.973945i	$0.427178\pi$
$572$	132.837	5.55419
$573$	0	0
$574$	−18.0284	−0.752493
$575$	−19.0020	−0.792437
$576$	0	0
$577$	26.5432	1.10501	0.552505	−	0.833510i	$-0.313672\pi$
$577$	26.5432	1.10501	0.552505	+	0.833510i	$0.313672\pi$
$578$	26.3926	1.09779
$579$	0	0
$580$	100.536	4.17452
$581$	−2.71284	−0.112547
$582$	0	0
$583$	−77.3904	−3.20518
$584$	42.9383	1.77680
$585$	0	0
$586$	−29.2160	−1.20690
$587$	−22.6806	−0.936128	−0.468064	−	0.883695i	$-0.655048\pi$
$587$	−22.6806	−0.936128	−0.468064	+	0.883695i	$0.655048\pi$
$588$	0	0
$589$	7.54697	0.310967
$590$	84.0762	3.46136
$591$	0	0
$592$	33.9344	1.39470
$593$	31.6754	1.30075	0.650376	−	0.759612i	$-0.274611\pi$
$593$	31.6754	1.30075	0.650376	+	0.759612i	$0.274611\pi$
$594$	0	0
$595$	7.06806	0.289762
$596$	17.4222	0.713641
$597$	0	0
$598$	83.1741	3.40124
$599$	13.2647	0.541980	0.270990	−	0.962582i	$-0.412649\pi$
$599$	13.2647	0.541980	0.270990	+	0.962582i	$0.412649\pi$
$600$	0	0
$601$	−28.7624	−1.17324	−0.586622	−	0.809861i	$-0.699542\pi$
$601$	−28.7624	−1.17324	−0.586622	+	0.809861i	$0.699542\pi$
$602$	−15.4113	−0.628116
$603$	0	0
$604$	31.0924	1.26513
$605$	−69.6709	−2.83253
$606$	0	0
$607$	13.7938	0.559873	0.279937	−	0.960018i	$-0.409687\pi$
$607$	13.7938	0.559873	0.279937	+	0.960018i	$0.409687\pi$
$608$	7.07673	0.286999
$609$	0	0
$610$	−91.2515	−3.69467
$611$	21.9187	0.886735
$612$	0	0
$613$	−2.67053	−0.107862	−0.0539308	−	0.998545i	$-0.517175\pi$
$613$	−2.67053	−0.107862	−0.0539308	+	0.998545i	$0.517175\pi$
$614$	34.9357	1.40989
$615$	0	0
$616$	26.5346	1.06911
$617$	−3.24977	−0.130831	−0.0654155	−	0.997858i	$-0.520837\pi$
$617$	−3.24977	−0.130831	−0.0654155	+	0.997858i	$0.520837\pi$
$618$	0	0
$619$	−39.3418	−1.58128	−0.790640	−	0.612282i	$-0.790252\pi$
$619$	−39.3418	−1.58128	−0.790640	+	0.612282i	$0.790252\pi$
$620$	−16.7678	−0.673410
$621$	0	0
$622$	−19.3608	−0.776297
$623$	−1.78782	−0.0716275
$624$	0	0
$625$	−30.7457	−1.22983
$626$	−36.8007	−1.47085
$627$	0	0
$628$	−84.7484	−3.38183
$629$	26.9739	1.07552
$630$	0	0
$631$	11.7580	0.468077	0.234039	−	0.972227i	$-0.424806\pi$
$631$	11.7580	0.468077	0.234039	+	0.972227i	$0.424806\pi$
$632$	57.7663	2.29782
$633$	0	0
$634$	−46.3244	−1.83978
$635$	−2.86533	−0.113707
$636$	0	0
$637$	5.81109	0.230244
$638$	131.069	5.18906
$639$	0	0
$640$	−58.7241	−2.32127
$641$	26.1532	1.03299	0.516494	−	0.856291i	$-0.327237\pi$
$641$	26.1532	1.03299	0.516494	+	0.856291i	$0.327237\pi$
$642$	0	0
$643$	−42.2728	−1.66708	−0.833539	−	0.552461i	$-0.813689\pi$
$643$	−42.2728	−1.66708	−0.833539	+	0.552461i	$0.813689\pi$
$644$	22.7697	0.897253
$645$	0	0
$646$	−29.5890	−1.16416
$647$	18.8312	0.740331	0.370165	−	0.928966i	$-0.379301\pi$
$647$	18.8312	0.740331	0.370165	+	0.928966i	$0.379301\pi$
$648$	0	0
$649$	72.1150	2.83076
$650$	−45.1057	−1.76919
$651$	0	0
$652$	33.5502	1.31393
$653$	42.9175	1.67949	0.839746	−	0.542979i	$-0.182704\pi$
$653$	42.9175	1.67949	0.839746	+	0.542979i	$0.182704\pi$
$654$	0	0
$655$	20.0272	0.782527
$656$	−23.1381	−0.903391
$657$	0	0
$658$	9.12031	0.355547
$659$	−36.1838	−1.40952	−0.704760	−	0.709446i	$-0.748945\pi$
$659$	−36.1838	−1.40952	−0.704760	+	0.709446i	$0.748945\pi$
$660$	0	0
$661$	−10.4284	−0.405620	−0.202810	−	0.979218i	$-0.565007\pi$
$661$	−10.4284	−0.405620	−0.202810	+	0.979218i	$0.565007\pi$
$662$	84.0181	3.26546
$663$	0	0
$664$	−12.1131	−0.470080
$665$	14.2143	0.551208
$666$	0	0
$667$	53.9938	2.09065
$668$	−11.2542	−0.435439
$669$	0	0
$670$	−16.2757	−0.628785
$671$	−78.2696	−3.02156
$672$	0	0
$673$	1.54766	0.0596579	0.0298290	−	0.999555i	$-0.490504\pi$
$673$	1.54766	0.0596579	0.0298290	+	0.999555i	$0.490504\pi$
$674$	36.6098	1.41016
$675$	0	0
$676$	79.8899	3.07269
$677$	13.1548	0.505580	0.252790	−	0.967521i	$-0.418652\pi$
$677$	13.1548	0.505580	0.252790	+	0.967521i	$0.418652\pi$
$678$	0	0
$679$	5.89963	0.226407
$680$	31.5597	1.21026
$681$	0	0
$682$	−21.8602	−0.837070
$683$	−6.32001	−0.241828	−0.120914	−	0.992663i	$-0.538583\pi$
$683$	−6.32001	−0.241828	−0.120914	+	0.992663i	$0.538583\pi$
$684$	0	0
$685$	−38.3907	−1.46683
$686$	2.41798	0.0923189
$687$	0	0
$688$	−19.7791	−0.754073
$689$	−75.6771	−2.88307
$690$	0	0
$691$	−12.5225	−0.476379	−0.238190	−	0.971219i	$-0.576554\pi$
$691$	−12.5225	−0.476379	−0.238190	+	0.971219i	$0.576554\pi$
$692$	−71.5657	−2.72052
$693$	0	0
$694$	23.0418	0.874656
$695$	13.0840	0.496303
$696$	0	0
$697$	−18.3921	−0.696650
$698$	43.2152	1.63572
$699$	0	0
$700$	−12.3481	−0.466715
$701$	−32.2849	−1.21939	−0.609693	−	0.792638i	$-0.708707\pi$
$701$	−32.2849	−1.21939	−0.609693	+	0.792638i	$0.708707\pi$
$702$	0	0
$703$	54.2463	2.04594
$704$	−57.3816	−2.16265
$705$	0	0
$706$	72.4600	2.72707
$707$	−0.292411	−0.0109972
$708$	0	0
$709$	24.5194	0.920846	0.460423	−	0.887700i	$-0.347698\pi$
$709$	24.5194	0.920846	0.460423	+	0.887700i	$0.347698\pi$
$710$	−75.7556	−2.84306
$711$	0	0
$712$	−7.98281	−0.299169
$713$	−9.00530	−0.337251
$714$	0	0
$715$	−98.9494	−3.70050
$716$	−18.2012	−0.680212
$717$	0	0
$718$	38.7600	1.44651
$719$	4.97860	0.185670	0.0928352	−	0.995681i	$-0.470407\pi$
$719$	4.97860	0.185670	0.0928352	+	0.995681i	$0.470407\pi$
$720$	0	0
$721$	0.384246	0.0143101
$722$	−13.5638	−0.504793
$723$	0	0
$724$	−39.1647	−1.45554
$725$	−29.2811	−1.08747
$726$	0	0
$727$	−5.39878	−0.200230	−0.100115	−	0.994976i	$-0.531921\pi$
$727$	−5.39878	−0.200230	−0.100115	+	0.994976i	$0.531921\pi$
$728$	25.9472	0.961666
$729$	0	0
$730$	−66.6254	−2.46592
$731$	−15.7221	−0.581504
$732$	0	0
$733$	46.1590	1.70492	0.852460	−	0.522792i	$-0.175110\pi$
$733$	46.1590	1.70492	0.852460	+	0.522792i	$0.175110\pi$
$734$	74.6642	2.75591
$735$	0	0
$736$	−8.44420	−0.311257
$737$	−13.9602	−0.514232
$738$	0	0
$739$	−7.62697	−0.280563	−0.140281	−	0.990112i	$-0.544801\pi$
$739$	−7.62697	−0.280563	−0.140281	+	0.990112i	$0.544801\pi$
$740$	−120.524	−4.43055
$741$	0	0
$742$	−31.4890	−1.15600
$743$	−29.1648	−1.06995	−0.534977	−	0.844867i	$-0.679680\pi$
$743$	−29.1648	−1.06995	−0.534977	+	0.844867i	$0.679680\pi$
$744$	0	0
$745$	−12.9777	−0.475466
$746$	−13.9772	−0.511741
$747$	0	0
$748$	56.3880	2.06175
$749$	−6.48261	−0.236869
$750$	0	0
$751$	−16.1853	−0.590609	−0.295304	−	0.955403i	$-0.595421\pi$
$751$	−16.1853	−0.590609	−0.295304	+	0.955403i	$0.595421\pi$
$752$	11.7052	0.426844
$753$	0	0
$754$	128.167	4.66757
$755$	−23.1606	−0.842899
$756$	0	0
$757$	29.7380	1.08085	0.540424	−	0.841393i	$-0.318264\pi$
$757$	29.7380	1.08085	0.540424	+	0.841393i	$0.318264\pi$
$758$	−33.2411	−1.20737
$759$	0	0
$760$	63.4686	2.30225
$761$	−30.9215	−1.12090	−0.560451	−	0.828188i	$-0.689372\pi$
$761$	−30.9215	−1.12090	−0.560451	+	0.828188i	$0.689372\pi$
$762$	0	0
$763$	7.32894	0.265326
$764$	−76.8931	−2.78189
$765$	0	0
$766$	1.69829	0.0613617
$767$	70.5185	2.54628
$768$	0	0
$769$	8.67272	0.312746	0.156373	−	0.987698i	$-0.450020\pi$
$769$	8.67272	0.312746	0.156373	+	0.987698i	$0.450020\pi$
$770$	−41.1726	−1.48376
$771$	0	0
$772$	−67.2174	−2.41921
$773$	−46.3828	−1.66827	−0.834137	−	0.551557i	$-0.814034\pi$
$773$	−46.3828	−1.66827	−0.834137	+	0.551557i	$0.814034\pi$
$774$	0	0
$775$	4.88362	0.175425
$776$	26.3425	0.945641
$777$	0	0
$778$	83.8203	3.00510
$779$	−36.9877	−1.32522
$780$	0	0
$781$	−64.9782	−2.32510
$782$	35.3066	1.26256
$783$	0	0
$784$	3.10329	0.110832
$785$	63.1286	2.25316
$786$	0	0
$787$	3.41934	0.121886	0.0609431	−	0.998141i	$-0.480589\pi$
$787$	3.41934	0.121886	0.0609431	+	0.998141i	$0.480589\pi$
$788$	0.103537	0.00368836
$789$	0	0
$790$	−89.6334	−3.18901
$791$	−8.75155	−0.311169
$792$	0	0
$793$	−76.5368	−2.71790
$794$	40.5418	1.43877
$795$	0	0
$796$	−42.9446	−1.52213
$797$	−25.0547	−0.887482	−0.443741	−	0.896155i	$-0.646349\pi$
$797$	−25.0547	−0.887482	−0.443741	+	0.896155i	$0.646349\pi$
$798$	0	0
$799$	9.30427	0.329161
$800$	4.57933	0.161904
$801$	0	0
$802$	−9.88081	−0.348904
$803$	−57.1469	−2.01667
$804$	0	0
$805$	−16.9610	−0.597798
$806$	−21.3762	−0.752946
$807$	0	0
$808$	−1.30565	−0.0459325
$809$	−31.7389	−1.11588	−0.557941	−	0.829881i	$-0.688408\pi$
$809$	−31.7389	−1.11588	−0.557941	+	0.829881i	$0.688408\pi$
$810$	0	0
$811$	11.1326	0.390920	0.195460	−	0.980712i	$-0.437380\pi$
$811$	11.1326	0.390920	0.195460	+	0.980712i	$0.437380\pi$
$812$	35.0870	1.23131
$813$	0	0
$814$	−157.127	−5.50731
$815$	−24.9914	−0.875409
$816$	0	0
$817$	−31.6182	−1.10618
$818$	−69.8135	−2.44097
$819$	0	0
$820$	82.1789	2.86981
$821$	−30.4868	−1.06400	−0.531998	−	0.846746i	$-0.678559\pi$
$821$	−30.4868	−1.06400	−0.531998	+	0.846746i	$0.678559\pi$
$822$	0	0
$823$	51.0907	1.78091	0.890454	−	0.455073i	$-0.150387\pi$
$823$	51.0907	1.78091	0.890454	+	0.455073i	$0.150387\pi$
$824$	1.71570	0.0597692
$825$	0	0
$826$	29.3426	1.02096
$827$	33.8562	1.17730	0.588648	−	0.808389i	$-0.299660\pi$
$827$	33.8562	1.17730	0.588648	+	0.808389i	$0.299660\pi$
$828$	0	0
$829$	−31.5681	−1.09640	−0.548202	−	0.836346i	$-0.684687\pi$
$829$	−31.5681	−1.09640	−0.548202	+	0.836346i	$0.684687\pi$
$830$	18.7954	0.652397
$831$	0	0
$832$	−56.1113	−1.94531
$833$	2.46675	0.0854679
$834$	0	0
$835$	8.38320	0.290113
$836$	113.400	3.92202
$837$	0	0
$838$	−31.0239	−1.07170
$839$	38.0449	1.31346	0.656728	−	0.754128i	$-0.271940\pi$
$839$	38.0449	1.31346	0.656728	+	0.754128i	$0.271940\pi$
$840$	0	0
$841$	54.2017	1.86902
$842$	10.6472	0.366927
$843$	0	0
$844$	30.5294	1.05087
$845$	−59.5095	−2.04719
$846$	0	0
$847$	−24.3151	−0.835478
$848$	−40.4137	−1.38781
$849$	0	0
$850$	−19.1469	−0.656735
$851$	−64.7286	−2.21887
$852$	0	0
$853$	−49.6692	−1.70064	−0.850320	−	0.526266i	$-0.823592\pi$
$853$	−49.6692	−1.70064	−0.850320	+	0.526266i	$0.823592\pi$
$854$	−31.8468	−1.08977
$855$	0	0
$856$	−28.9455	−0.989338
$857$	51.1798	1.74827	0.874135	−	0.485683i	$-0.161429\pi$
$857$	51.1798	1.74827	0.874135	+	0.485683i	$0.161429\pi$
$858$	0	0
$859$	22.3730	0.763357	0.381678	−	0.924295i	$-0.375346\pi$
$859$	22.3730	0.763357	0.381678	+	0.924295i	$0.375346\pi$
$860$	70.2490	2.39547
$861$	0	0
$862$	30.9777	1.05510
$863$	36.7882	1.25229	0.626143	−	0.779708i	$-0.284633\pi$
$863$	36.7882	1.25229	0.626143	+	0.779708i	$0.284633\pi$
$864$	0	0
$865$	53.3089	1.81256
$866$	25.7786	0.875991
$867$	0	0
$868$	−5.85195	−0.198628
$869$	−76.8817	−2.60803
$870$	0	0
$871$	−13.6512	−0.462553
$872$	32.7245	1.10819
$873$	0	0
$874$	71.0039	2.40174
$875$	−5.12860	−0.173378
$876$	0	0
$877$	8.02579	0.271012	0.135506	−	0.990777i	$-0.456734\pi$
$877$	8.02579	0.271012	0.135506	+	0.990777i	$0.456734\pi$
$878$	−17.2933	−0.583621
$879$	0	0
$880$	−52.8417	−1.78129
$881$	−42.4618	−1.43058	−0.715288	−	0.698830i	$-0.753704\pi$
$881$	−42.4618	−1.43058	−0.715288	+	0.698830i	$0.753704\pi$
$882$	0	0
$883$	54.1159	1.82114	0.910572	−	0.413350i	$-0.135641\pi$
$883$	54.1159	1.82114	0.910572	+	0.413350i	$0.135641\pi$
$884$	55.1396	1.85455
$885$	0	0
$886$	95.3727	3.20411
$887$	−8.66097	−0.290807	−0.145403	−	0.989372i	$-0.546448\pi$
$887$	−8.66097	−0.290807	−0.145403	+	0.989372i	$0.546448\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	12.3866	0.415199
$891$	0	0
$892$	62.3289	2.08693
$893$	18.7115	0.626157
$894$	0	0
$895$	13.5580	0.453194
$896$	−20.4947	−0.684680
$897$	0	0
$898$	−51.5007	−1.71860
$899$	−13.8767	−0.462814
$900$	0	0
$901$	−32.1242	−1.07021
$902$	107.137	3.56727
$903$	0	0
$904$	−39.0766	−1.29967
$905$	29.1736	0.969762
$906$	0	0
$907$	25.1110	0.833798	0.416899	−	0.908953i	$-0.363117\pi$
$907$	25.1110	0.833798	0.416899	+	0.908953i	$0.363117\pi$
$908$	−4.37726	−0.145265
$909$	0	0
$910$	−40.2611	−1.33464
$911$	−10.3906	−0.344255	−0.172128	−	0.985075i	$-0.555064\pi$
$911$	−10.3906	−0.344255	−0.172128	+	0.985075i	$0.555064\pi$
$912$	0	0
$913$	16.1214	0.533542
$914$	−68.1829	−2.25529
$915$	0	0
$916$	−61.9405	−2.04657
$917$	6.98949	0.230813
$918$	0	0
$919$	35.9077	1.18448	0.592242	−	0.805760i	$-0.298243\pi$
$919$	35.9077	1.18448	0.592242	+	0.805760i	$0.298243\pi$
$920$	−75.7328	−2.49684
$921$	0	0
$922$	80.4279	2.64875
$923$	−63.5397	−2.09143
$924$	0	0
$925$	35.1026	1.15417
$926$	−58.7912	−1.93200
$927$	0	0
$928$	−13.0121	−0.427143
$929$	−13.2349	−0.434222	−0.217111	−	0.976147i	$-0.569663\pi$
$929$	−13.2349	−0.434222	−0.217111	+	0.976147i	$0.569663\pi$
$930$	0	0
$931$	4.96080	0.162584
$932$	45.0118	1.47441
$933$	0	0
$934$	−10.1217	−0.331192
$935$	−42.0031	−1.37365
$936$	0	0
$937$	−58.2756	−1.90378	−0.951890	−	0.306441i	$-0.900862\pi$
$937$	−58.2756	−1.90378	−0.951890	+	0.306441i	$0.900862\pi$
$938$	−5.68022	−0.185466
$939$	0	0
$940$	−41.5730	−1.35596
$941$	−1.11789	−0.0364421	−0.0182210	−	0.999834i	$-0.505800\pi$
$941$	−1.11789	−0.0364421	−0.0182210	+	0.999834i	$0.505800\pi$
$942$	0	0
$943$	44.1350	1.43723
$944$	37.6589	1.22569
$945$	0	0
$946$	91.5838	2.97765
$947$	−14.8068	−0.481157	−0.240579	−	0.970630i	$-0.577337\pi$
$947$	−14.8068	−0.481157	−0.240579	+	0.970630i	$0.577337\pi$
$948$	0	0
$949$	−55.8818	−1.81400
$950$	−38.5058	−1.24929
$951$	0	0
$952$	11.0143	0.356976
$953$	−43.8606	−1.42079	−0.710393	−	0.703806i	$-0.751483\pi$
$953$	−43.8606	−1.42079	−0.710393	+	0.703806i	$0.751483\pi$
$954$	0	0
$955$	57.2772	1.85345
$956$	−82.9762	−2.68364
$957$	0	0
$958$	−59.7226	−1.92955
$959$	−13.3983	−0.432655
$960$	0	0
$961$	−28.6856	−0.925341
$962$	−153.649	−4.95383
$963$	0	0
$964$	4.79544	0.154451
$965$	50.0698	1.61181
$966$	0	0
$967$	24.8577	0.799371	0.399686	−	0.916652i	$-0.369119\pi$
$967$	24.8577	0.799371	0.399686	+	0.916652i	$0.369119\pi$
$968$	−108.570	−3.48956
$969$	0	0
$970$	−40.8745	−1.31240
$971$	10.5828	0.339617	0.169808	−	0.985477i	$-0.445685\pi$
$971$	10.5828	0.339617	0.169808	+	0.985477i	$0.445685\pi$
$972$	0	0
$973$	4.56630	0.146389
$974$	7.73388	0.247809
$975$	0	0
$976$	−40.8728	−1.30831
$977$	−27.6129	−0.883416	−0.441708	−	0.897159i	$-0.645627\pi$
$977$	−27.6129	−0.883416	−0.441708	+	0.897159i	$0.645627\pi$
$978$	0	0
$979$	10.6244	0.339557
$980$	−11.0219	−0.352080
$981$	0	0
$982$	34.9910	1.11661
$983$	3.15604	0.100662	0.0503310	−	0.998733i	$-0.483972\pi$
$983$	3.15604	0.100662	0.0503310	+	0.998733i	$0.483972\pi$
$984$	0	0
$985$	−0.0771242	−0.00245738
$986$	54.4057	1.73263
$987$	0	0
$988$	110.889	3.52786
$989$	37.7279	1.19968
$990$	0	0
$991$	−24.4911	−0.777987	−0.388993	−	0.921241i	$-0.627177\pi$
$991$	−24.4911	−0.777987	−0.388993	+	0.921241i	$0.627177\pi$
$992$	2.17021	0.0689042
$993$	0	0
$994$	−26.4387	−0.838584
$995$	31.9892	1.01413
$996$	0	0
$997$	5.17107	0.163769	0.0818847	−	0.996642i	$-0.473906\pi$
$997$	5.17107	0.163769	0.0818847	+	0.996642i	$0.473906\pi$
$998$	−23.1885	−0.734020
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.r.1.3		16
3.2	odd	2		2667.2.a.o.1.14	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.o.1.14	✓	16	3.2	odd	2
8001.2.a.r.1.3		16	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)

\(f(q)\)	\(=\)	\(q-2.41798 q^{2} +3.84663 q^{4} -2.86533 q^{5} -1.00000 q^{7} -4.46511 q^{8} +O(q^{10})\)	\(q-2.41798 q^{2} +3.84663 q^{4} -2.86533 q^{5} -1.00000 q^{7} -4.46511 q^{8} +6.92831 q^{10} +5.94265 q^{11} +5.81109 q^{13} +2.41798 q^{14} +3.10329 q^{16} +2.46675 q^{17} +4.96080 q^{19} -11.0219 q^{20} -14.3692 q^{22} -5.91940 q^{23} +3.21012 q^{25} -14.0511 q^{26} -3.84663 q^{28} -9.12150 q^{29} +1.52132 q^{31} +1.42653 q^{32} -5.96456 q^{34} +2.86533 q^{35} +10.9350 q^{37} -11.9951 q^{38} +12.7940 q^{40} -7.45599 q^{41} -6.37361 q^{43} +22.8592 q^{44} +14.3130 q^{46} +3.77187 q^{47} +1.00000 q^{49} -7.76200 q^{50} +22.3531 q^{52} -13.0229 q^{53} -17.0277 q^{55} +4.46511 q^{56} +22.0556 q^{58} +12.1352 q^{59} -13.1708 q^{61} -3.67852 q^{62} -9.65590 q^{64} -16.6507 q^{65} -2.34916 q^{67} +9.48868 q^{68} -6.92831 q^{70} -10.9342 q^{71} -9.61640 q^{73} -26.4406 q^{74} +19.0824 q^{76} -5.94265 q^{77} -12.9373 q^{79} -8.89194 q^{80} +18.0284 q^{82} +2.71284 q^{83} -7.06806 q^{85} +15.4113 q^{86} -26.5346 q^{88} +1.78782 q^{89} -5.81109 q^{91} -22.7697 q^{92} -9.12031 q^{94} -14.2143 q^{95} -5.89963 q^{97} -2.41798 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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