LMFDB - Embedded newform 8001.2.a.v.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:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79341$ 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)

$f(q)$	$=$	$q-2.79341 q^{2} +5.80316 q^{4} +1.63704 q^{5} +1.00000 q^{7} -10.6238 q^{8} +O(q^{10})$	\(q-2.79341 q^{2} +5.80316 q^{4} +1.63704 q^{5} +1.00000 q^{7} -10.6238 q^{8} -4.57293 q^{10} +6.47808 q^{11} +4.75000 q^{13} -2.79341 q^{14} +18.0703 q^{16} -0.645517 q^{17} +5.60124 q^{19} +9.50000 q^{20} -18.0960 q^{22} +0.0762406 q^{23} -2.32010 q^{25} -13.2687 q^{26} +5.80316 q^{28} -1.88240 q^{29} +2.27176 q^{31} -29.2303 q^{32} +1.80319 q^{34} +1.63704 q^{35} -5.95489 q^{37} -15.6466 q^{38} -17.3916 q^{40} +8.30140 q^{41} +0.409994 q^{43} +37.5934 q^{44} -0.212971 q^{46} -5.35216 q^{47} +1.00000 q^{49} +6.48100 q^{50} +27.5650 q^{52} +10.0642 q^{53} +10.6049 q^{55} -10.6238 q^{56} +5.25833 q^{58} +8.26943 q^{59} -4.31290 q^{61} -6.34596 q^{62} +45.5117 q^{64} +7.77594 q^{65} +7.25212 q^{67} -3.74604 q^{68} -4.57293 q^{70} -3.65756 q^{71} +11.6201 q^{73} +16.6345 q^{74} +32.5049 q^{76} +6.47808 q^{77} -12.8009 q^{79} +29.5818 q^{80} -23.1892 q^{82} -10.3550 q^{83} -1.05674 q^{85} -1.14528 q^{86} -68.8218 q^{88} +4.61148 q^{89} +4.75000 q^{91} +0.442436 q^{92} +14.9508 q^{94} +9.16946 q^{95} -8.08150 q^{97} -2.79341 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * 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 $

$2$	−2.79341	−1.97524	−0.987621	−	0.156861i	$-0.949863\pi$
$2$	−2.79341	−1.97524	−0.987621	+	0.156861i	$0.949863\pi$
$3$	0	0
$3$	0	0
$4$	5.80316	2.90158
$5$	1.63704	0.732106	0.366053	−	0.930594i	$-0.380709\pi$
$5$	1.63704	0.732106	0.366053	+	0.930594i	$0.380709\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−10.6238	−3.75608
$9$	0	0
$10$	−4.57293	−1.44609
$11$	6.47808	1.95322	0.976608	−	0.215027i	$-0.0689840\pi$
$11$	6.47808	1.95322	0.976608	+	0.215027i	$0.0689840\pi$
$12$	0	0
$13$	4.75000	1.31741	0.658707	−	0.752400i	$-0.271104\pi$
$13$	4.75000	1.31741	0.658707	+	0.752400i	$0.271104\pi$
$14$	−2.79341	−0.746571
$15$	0	0
$16$	18.0703	4.51758
$17$	−0.645517	−0.156561	−0.0782804	−	0.996931i	$-0.524943\pi$
$17$	−0.645517	−0.156561	−0.0782804	+	0.996931i	$0.524943\pi$
$18$	0	0
$19$	5.60124	1.28501	0.642507	−	0.766280i	$-0.277895\pi$
$19$	5.60124	1.28501	0.642507	+	0.766280i	$0.277895\pi$
$20$	9.50000	2.12426
$21$	0	0
$22$	−18.0960	−3.85807
$23$	0.0762406	0.0158973	0.00794863	−	0.999968i	$-0.497470\pi$
$23$	0.0762406	0.0158973	0.00794863	+	0.999968i	$0.497470\pi$
$24$	0	0
$25$	−2.32010	−0.464020
$26$	−13.2687	−2.60221
$27$	0	0
$28$	5.80316	1.09669
$29$	−1.88240	−0.349553	−0.174777	−	0.984608i	$-0.555920\pi$
$29$	−1.88240	−0.349553	−0.174777	+	0.984608i	$0.555920\pi$
$30$	0	0
$31$	2.27176	0.408020	0.204010	−	0.978969i	$-0.434602\pi$
$31$	2.27176	0.408020	0.204010	+	0.978969i	$0.434602\pi$
$32$	−29.2303	−5.16724
$33$	0	0
$34$	1.80319	0.309245
$35$	1.63704	0.276710
$36$	0	0
$37$	−5.95489	−0.978978	−0.489489	−	0.872010i	$-0.662817\pi$
$37$	−5.95489	−0.978978	−0.489489	+	0.872010i	$0.662817\pi$
$38$	−15.6466	−2.53821
$39$	0	0
$40$	−17.3916	−2.74985
$41$	8.30140	1.29646	0.648230	−	0.761445i	$-0.275510\pi$
$41$	8.30140	1.29646	0.648230	+	0.761445i	$0.275510\pi$
$42$	0	0
$43$	0.409994	0.0625234	0.0312617	−	0.999511i	$-0.490047\pi$
$43$	0.409994	0.0625234	0.0312617	+	0.999511i	$0.490047\pi$
$44$	37.5934	5.66741
$45$	0	0
$46$	−0.212971	−0.0314009
$47$	−5.35216	−0.780693	−0.390346	−	0.920668i	$-0.627645\pi$
$47$	−5.35216	−0.780693	−0.390346	+	0.920668i	$0.627645\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	6.48100	0.916552
$51$	0	0
$52$	27.5650	3.82258
$53$	10.0642	1.38243	0.691214	−	0.722650i	$-0.257076\pi$
$53$	10.0642	1.38243	0.691214	+	0.722650i	$0.257076\pi$
$54$	0	0
$55$	10.6049	1.42996
$56$	−10.6238	−1.41966
$57$	0	0
$58$	5.25833	0.690452
$59$	8.26943	1.07659	0.538294	−	0.842757i	$-0.319069\pi$
$59$	8.26943	1.07659	0.538294	+	0.842757i	$0.319069\pi$
$60$	0	0
$61$	−4.31290	−0.552210	−0.276105	−	0.961128i	$-0.589044\pi$
$61$	−4.31290	−0.552210	−0.276105	+	0.961128i	$0.589044\pi$
$62$	−6.34596	−0.805938
$63$	0	0
$64$	45.5117	5.68896
$65$	7.77594	0.964487
$66$	0	0
$67$	7.25212	0.885988	0.442994	−	0.896525i	$-0.353916\pi$
$67$	7.25212	0.885988	0.442994	+	0.896525i	$0.353916\pi$
$68$	−3.74604	−0.454273
$69$	0	0
$70$	−4.57293	−0.546569
$71$	−3.65756	−0.434072	−0.217036	−	0.976164i	$-0.569639\pi$
$71$	−3.65756	−0.434072	−0.217036	+	0.976164i	$0.569639\pi$
$72$	0	0
$73$	11.6201	1.36003	0.680014	−	0.733199i	$-0.261974\pi$
$73$	11.6201	1.36003	0.680014	+	0.733199i	$0.261974\pi$
$74$	16.6345	1.93372
$75$	0	0
$76$	32.5049	3.72857
$77$	6.47808	0.738246
$78$	0	0
$79$	−12.8009	−1.44021	−0.720105	−	0.693865i	$-0.755907\pi$
$79$	−12.8009	−1.44021	−0.720105	+	0.693865i	$0.755907\pi$
$80$	29.5818	3.30735
$81$	0	0
$82$	−23.1892	−2.56082
$83$	−10.3550	−1.13660	−0.568302	−	0.822820i	$-0.692400\pi$
$83$	−10.3550	−1.13660	−0.568302	+	0.822820i	$0.692400\pi$
$84$	0	0
$85$	−1.05674	−0.114619
$86$	−1.14528	−0.123499
$87$	0	0
$88$	−68.8218	−7.33643
$89$	4.61148	0.488816	0.244408	−	0.969672i	$-0.421406\pi$
$89$	4.61148	0.488816	0.244408	+	0.969672i	$0.421406\pi$
$90$	0	0
$91$	4.75000	0.497936
$92$	0.442436	0.0461272
$93$	0	0
$94$	14.9508	1.54206
$95$	9.16946	0.940766
$96$	0	0
$97$	−8.08150	−0.820552	−0.410276	−	0.911961i	$-0.634568\pi$
$97$	−8.08150	−0.820552	−0.410276	+	0.911961i	$0.634568\pi$
$98$	−2.79341	−0.282177
$99$	0	0
$100$	−13.4639	−1.34639
$101$	4.65231	0.462922	0.231461	−	0.972844i	$-0.425649\pi$
$101$	4.65231	0.462922	0.231461	+	0.972844i	$0.425649\pi$
$102$	0	0
$103$	−5.57229	−0.549054	−0.274527	−	0.961579i	$-0.588521\pi$
$103$	−5.57229	−0.549054	−0.274527	+	0.961579i	$0.588521\pi$
$104$	−50.4630	−4.94831
$105$	0	0
$106$	−28.1136	−2.73063
$107$	14.8598	1.43655	0.718277	−	0.695757i	$-0.244931\pi$
$107$	14.8598	1.43655	0.718277	+	0.695757i	$0.244931\pi$
$108$	0	0
$109$	−12.5572	−1.20276	−0.601382	−	0.798961i	$-0.705383\pi$
$109$	−12.5572	−1.20276	−0.601382	+	0.798961i	$0.705383\pi$
$110$	−29.6238	−2.82452
$111$	0	0
$112$	18.0703	1.70749
$113$	−1.79519	−0.168877	−0.0844385	−	0.996429i	$-0.526910\pi$
$113$	−1.79519	−0.168877	−0.0844385	+	0.996429i	$0.526910\pi$
$114$	0	0
$115$	0.124809	0.0116385
$116$	−10.9239	−1.01426
$117$	0	0
$118$	−23.0999	−2.12652
$119$	−0.645517	−0.0591744
$120$	0	0
$121$	30.9656	2.81505
$122$	12.0477	1.09075
$123$	0	0
$124$	13.1834	1.18390
$125$	−11.9833	−1.07182
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−68.6723	−6.06983
$129$	0	0
$130$	−21.7214	−1.90509
$131$	−0.419711	−0.0366703	−0.0183352	−	0.999832i	$-0.505837\pi$
$131$	−0.419711	−0.0366703	−0.0183352	+	0.999832i	$0.505837\pi$
$132$	0	0
$133$	5.60124	0.485689
$134$	−20.2582	−1.75004
$135$	0	0
$136$	6.85783	0.588054
$137$	17.1519	1.46539	0.732694	−	0.680559i	$-0.238263\pi$
$137$	17.1519	1.46539	0.732694	+	0.680559i	$0.238263\pi$
$138$	0	0
$139$	1.89367	0.160619	0.0803094	−	0.996770i	$-0.474409\pi$
$139$	1.89367	0.160619	0.0803094	+	0.996770i	$0.474409\pi$
$140$	9.50000	0.802897
$141$	0	0
$142$	10.2171	0.857398
$143$	30.7709	2.57319
$144$	0	0
$145$	−3.08157	−0.255910
$146$	−32.4597	−2.68638
$147$	0	0
$148$	−34.5572	−2.84058
$149$	14.1732	1.16111	0.580556	−	0.814220i	$-0.302835\pi$
$149$	14.1732	1.16111	0.580556	+	0.814220i	$0.302835\pi$
$150$	0	0
$151$	−6.99988	−0.569642	−0.284821	−	0.958581i	$-0.591934\pi$
$151$	−6.99988	−0.569642	−0.284821	+	0.958581i	$0.591934\pi$
$152$	−59.5064	−4.82661
$153$	0	0
$154$	−18.0960	−1.45821
$155$	3.71896	0.298714
$156$	0	0
$157$	6.59843	0.526612	0.263306	−	0.964712i	$-0.415187\pi$
$157$	6.59843	0.526612	0.263306	+	0.964712i	$0.415187\pi$
$158$	35.7581	2.84476
$159$	0	0
$160$	−47.8512	−3.78297
$161$	0.0762406	0.00600860
$162$	0	0
$163$	−19.4556	−1.52388	−0.761940	−	0.647647i	$-0.775753\pi$
$163$	−19.4556	−1.52388	−0.761940	+	0.647647i	$0.775753\pi$
$164$	48.1743	3.76178
$165$	0	0
$166$	28.9257	2.24507
$167$	22.4412	1.73655	0.868277	−	0.496079i	$-0.165227\pi$
$167$	22.4412	1.73655	0.868277	+	0.496079i	$0.165227\pi$
$168$	0	0
$169$	9.56252	0.735579
$170$	2.95190	0.226400
$171$	0	0
$172$	2.37926	0.181417
$173$	−6.17908	−0.469786	−0.234893	−	0.972021i	$-0.575474\pi$
$173$	−6.17908	−0.469786	−0.234893	+	0.972021i	$0.575474\pi$
$174$	0	0
$175$	−2.32010	−0.175383
$176$	117.061	8.82381
$177$	0	0
$178$	−12.8818	−0.965530
$179$	−10.6465	−0.795759	−0.397880	−	0.917438i	$-0.630254\pi$
$179$	−10.6465	−0.795759	−0.397880	+	0.917438i	$0.630254\pi$
$180$	0	0
$181$	22.0777	1.64102	0.820510	−	0.571632i	$-0.193690\pi$
$181$	22.0777	1.64102	0.820510	+	0.571632i	$0.193690\pi$
$182$	−13.2687	−0.983543
$183$	0	0
$184$	−0.809964	−0.0597113
$185$	−9.74839	−0.716716
$186$	0	0
$187$	−4.18171	−0.305797
$188$	−31.0594	−2.26524
$189$	0	0
$190$	−25.6141	−1.85824
$191$	25.1450	1.81943	0.909715	−	0.415233i	$-0.136300\pi$
$191$	25.1450	1.81943	0.909715	+	0.415233i	$0.136300\pi$
$192$	0	0
$193$	−18.5403	−1.33456	−0.667279	−	0.744808i	$-0.732541\pi$
$193$	−18.5403	−1.33456	−0.667279	+	0.744808i	$0.732541\pi$
$194$	22.5750	1.62079
$195$	0	0
$196$	5.80316	0.414511
$197$	−20.8774	−1.48745	−0.743726	−	0.668485i	$-0.766943\pi$
$197$	−20.8774	−1.48745	−0.743726	+	0.668485i	$0.766943\pi$
$198$	0	0
$199$	15.1555	1.07435	0.537173	−	0.843472i	$-0.319492\pi$
$199$	15.1555	1.07435	0.537173	+	0.843472i	$0.319492\pi$
$200$	24.6483	1.74290
$201$	0	0
$202$	−12.9958	−0.914383
$203$	−1.88240	−0.132119
$204$	0	0
$205$	13.5897	0.949147
$206$	15.5657	1.08451
$207$	0	0
$208$	85.8341	5.95152
$209$	36.2853	2.50991
$210$	0	0
$211$	−2.41396	−0.166184	−0.0830919	−	0.996542i	$-0.526479\pi$
$211$	−2.41396	−0.166184	−0.0830919	+	0.996542i	$0.526479\pi$
$212$	58.4043	4.01123
$213$	0	0
$214$	−41.5097	−2.83754
$215$	0.671176	0.0457738
$216$	0	0
$217$	2.27176	0.154217
$218$	35.0775	2.37575
$219$	0	0
$220$	61.5418	4.14915
$221$	−3.06621	−0.206255
$222$	0	0
$223$	−27.1479	−1.81796	−0.908979	−	0.416842i	$-0.863137\pi$
$223$	−27.1479	−1.81796	−0.908979	+	0.416842i	$0.863137\pi$
$224$	−29.2303	−1.95303
$225$	0	0
$226$	5.01470	0.333573
$227$	−20.6978	−1.37376	−0.686880	−	0.726771i	$-0.741020\pi$
$227$	−20.6978	−1.37376	−0.686880	+	0.726771i	$0.741020\pi$
$228$	0	0
$229$	9.38781	0.620364	0.310182	−	0.950677i	$-0.399610\pi$
$229$	9.38781	0.620364	0.310182	+	0.950677i	$0.399610\pi$
$230$	−0.348643	−0.0229888
$231$	0	0
$232$	19.9982	1.31295
$233$	−10.8180	−0.708713	−0.354357	−	0.935110i	$-0.615300\pi$
$233$	−10.8180	−0.708713	−0.354357	+	0.935110i	$0.615300\pi$
$234$	0	0
$235$	−8.76170	−0.571550
$236$	47.9888	3.12381
$237$	0	0
$238$	1.80319	0.116884
$239$	−18.6532	−1.20657	−0.603286	−	0.797525i	$-0.706142\pi$
$239$	−18.6532	−1.20657	−0.603286	+	0.797525i	$0.706142\pi$
$240$	0	0
$241$	5.88263	0.378934	0.189467	−	0.981887i	$-0.439324\pi$
$241$	5.88263	0.378934	0.189467	+	0.981887i	$0.439324\pi$
$242$	−86.4997	−5.56041
$243$	0	0
$244$	−25.0284	−1.60228
$245$	1.63704	0.104587
$246$	0	0
$247$	26.6059	1.69289
$248$	−24.1347	−1.53255
$249$	0	0
$250$	33.4743	2.11710
$251$	−17.2347	−1.08784	−0.543921	−	0.839136i	$-0.683061\pi$
$251$	−17.2347	−1.08784	−0.543921	+	0.839136i	$0.683061\pi$
$252$	0	0
$253$	0.493893	0.0310508
$254$	−2.79341	−0.175274
$255$	0	0
$256$	100.807	6.30043
$257$	−2.80843	−0.175185	−0.0875925	−	0.996156i	$-0.527917\pi$
$257$	−2.80843	−0.175185	−0.0875925	+	0.996156i	$0.527917\pi$
$258$	0	0
$259$	−5.95489	−0.370019
$260$	45.1250	2.79854
$261$	0	0
$262$	1.17243	0.0724327
$263$	−11.5545	−0.712478	−0.356239	−	0.934395i	$-0.615941\pi$
$263$	−11.5545	−0.712478	−0.356239	+	0.934395i	$0.615941\pi$
$264$	0	0
$265$	16.4755	1.01208
$266$	−15.6466	−0.959354
$267$	0	0
$268$	42.0852	2.57076
$269$	−2.77245	−0.169039	−0.0845196	−	0.996422i	$-0.526936\pi$
$269$	−2.77245	−0.169039	−0.0845196	+	0.996422i	$0.526936\pi$
$270$	0	0
$271$	31.7527	1.92884	0.964421	−	0.264372i	$-0.0851645\pi$
$271$	31.7527	1.92884	0.964421	+	0.264372i	$0.0851645\pi$
$272$	−11.6647	−0.707276
$273$	0	0
$274$	−47.9124	−2.89449
$275$	−15.0298	−0.906332
$276$	0	0
$277$	11.0867	0.666135	0.333068	−	0.942903i	$-0.391916\pi$
$277$	11.0867	0.666135	0.333068	+	0.942903i	$0.391916\pi$
$278$	−5.28980	−0.317261
$279$	0	0
$280$	−17.3916	−1.03935
$281$	−30.9985	−1.84921	−0.924607	−	0.380922i	$-0.875607\pi$
$281$	−30.9985	−1.84921	−0.924607	+	0.380922i	$0.875607\pi$
$282$	0	0
$283$	−6.86487	−0.408074	−0.204037	−	0.978963i	$-0.565406\pi$
$283$	−6.86487	−0.408074	−0.204037	+	0.978963i	$0.565406\pi$
$284$	−21.2254	−1.25950
$285$	0	0
$286$	−85.9559	−5.08268
$287$	8.30140	0.490016
$288$	0	0
$289$	−16.5833	−0.975489
$290$	8.60809	0.505484
$291$	0	0
$292$	67.4332	3.94623
$293$	23.5589	1.37632	0.688162	−	0.725557i	$-0.258418\pi$
$293$	23.5589	1.37632	0.688162	+	0.725557i	$0.258418\pi$
$294$	0	0
$295$	13.5374	0.788177
$296$	63.2635	3.67712
$297$	0	0
$298$	−39.5916	−2.29348
$299$	0.362143	0.0209433
$300$	0	0
$301$	0.409994	0.0236316
$302$	19.5536	1.12518
$303$	0	0
$304$	101.216	5.80515
$305$	−7.06038	−0.404276
$306$	0	0
$307$	14.6890	0.838345	0.419172	−	0.907907i	$-0.362320\pi$
$307$	14.6890	0.838345	0.419172	+	0.907907i	$0.362320\pi$
$308$	37.5934	2.14208
$309$	0	0
$310$	−10.3886	−0.590032
$311$	29.6569	1.68169	0.840845	−	0.541275i	$-0.182058\pi$
$311$	29.6569	1.68169	0.840845	+	0.541275i	$0.182058\pi$
$312$	0	0
$313$	−9.14081	−0.516669	−0.258334	−	0.966056i	$-0.583174\pi$
$313$	−9.14081	−0.516669	−0.258334	+	0.966056i	$0.583174\pi$
$314$	−18.4321	−1.04019
$315$	0	0
$316$	−74.2855	−4.17888
$317$	−13.1590	−0.739085	−0.369542	−	0.929214i	$-0.620486\pi$
$317$	−13.1590	−0.739085	−0.369542	+	0.929214i	$0.620486\pi$
$318$	0	0
$319$	−12.1944	−0.682753
$320$	74.5044	4.16492
$321$	0	0
$322$	−0.212971	−0.0118684
$323$	−3.61570	−0.201183
$324$	0	0
$325$	−11.0205	−0.611307
$326$	54.3475	3.01003
$327$	0	0
$328$	−88.1923	−4.86961
$329$	−5.35216	−0.295074
$330$	0	0
$331$	−24.0050	−1.31944	−0.659718	−	0.751514i	$-0.729324\pi$
$331$	−24.0050	−1.31944	−0.659718	+	0.751514i	$0.729324\pi$
$332$	−60.0915	−3.29795
$333$	0	0
$334$	−62.6876	−3.43012
$335$	11.8720	0.648637
$336$	0	0
$337$	20.3680	1.10952	0.554758	−	0.832012i	$-0.312811\pi$
$337$	20.3680	1.10952	0.554758	+	0.832012i	$0.312811\pi$
$338$	−26.7121	−1.45295
$339$	0	0
$340$	−6.13241	−0.332577
$341$	14.7166	0.796951
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.35569	−0.234843
$345$	0	0
$346$	17.2607	0.927942
$347$	−4.92725	−0.264509	−0.132254	−	0.991216i	$-0.542222\pi$
$347$	−4.92725	−0.264509	−0.132254	+	0.991216i	$0.542222\pi$
$348$	0	0
$349$	−0.0921158	−0.00493084	−0.00246542	−	0.999997i	$-0.500785\pi$
$349$	−0.0921158	−0.00493084	−0.00246542	+	0.999997i	$0.500785\pi$
$350$	6.48100	0.346424
$351$	0	0
$352$	−189.356	−10.0927
$353$	−8.47294	−0.450969	−0.225484	−	0.974247i	$-0.572396\pi$
$353$	−8.47294	−0.450969	−0.225484	+	0.974247i	$0.572396\pi$
$354$	0	0
$355$	−5.98757	−0.317787
$356$	26.7612	1.41834
$357$	0	0
$358$	29.7402	1.57182
$359$	18.1497	0.957906	0.478953	−	0.877840i	$-0.341016\pi$
$359$	18.1497	0.957906	0.478953	+	0.877840i	$0.341016\pi$
$360$	0	0
$361$	12.3739	0.651259
$362$	−61.6720	−3.24141
$363$	0	0
$364$	27.5650	1.44480
$365$	19.0225	0.995686
$366$	0	0
$367$	−19.3632	−1.01075	−0.505376	−	0.862899i	$-0.668646\pi$
$367$	−19.3632	−1.01075	−0.505376	+	0.862899i	$0.668646\pi$
$368$	1.37769	0.0718172
$369$	0	0
$370$	27.2313	1.41569
$371$	10.0642	0.522509
$372$	0	0
$373$	7.37194	0.381705	0.190852	−	0.981619i	$-0.438875\pi$
$373$	7.37194	0.381705	0.190852	+	0.981619i	$0.438875\pi$
$374$	11.6812	0.604023
$375$	0	0
$376$	56.8602	2.93234
$377$	−8.94141	−0.460506
$378$	0	0
$379$	0.00612186	0.000314459 0	0.000157229	−	1.00000i	$-0.499950\pi$
$379$	0.00612186	0.000314459 0	0.000157229		1.00000i	$0.499950\pi$
$380$	53.2118	2.72971
$381$	0	0
$382$	−70.2404	−3.59381
$383$	−36.2288	−1.85121	−0.925603	−	0.378495i	$-0.876442\pi$
$383$	−36.2288	−1.85121	−0.925603	+	0.378495i	$0.876442\pi$
$384$	0	0
$385$	10.6049	0.540475
$386$	51.7907	2.63608
$387$	0	0
$388$	−46.8982	−2.38090
$389$	17.0620	0.865080	0.432540	−	0.901615i	$-0.357617\pi$
$389$	17.0620	0.865080	0.432540	+	0.901615i	$0.357617\pi$
$390$	0	0
$391$	−0.0492146	−0.00248889
$392$	−10.6238	−0.536583
$393$	0	0
$394$	58.3192	2.93808
$395$	−20.9555	−1.05439
$396$	0	0
$397$	28.3501	1.42285	0.711425	−	0.702762i	$-0.248050\pi$
$397$	28.3501	1.42285	0.711425	+	0.702762i	$0.248050\pi$
$398$	−42.3356	−2.12209
$399$	0	0
$400$	−41.9250	−2.09625
$401$	−32.4139	−1.61867	−0.809336	−	0.587346i	$-0.800173\pi$
$401$	−32.4139	−1.61867	−0.809336	+	0.587346i	$0.800173\pi$
$402$	0	0
$403$	10.7909	0.537531
$404$	26.9981	1.34321
$405$	0	0
$406$	5.25833	0.260966
$407$	−38.5763	−1.91215
$408$	0	0
$409$	−34.2842	−1.69524	−0.847621	−	0.530602i	$-0.821966\pi$
$409$	−34.2842	−1.69524	−0.847621	+	0.530602i	$0.821966\pi$
$410$	−37.9617	−1.87479
$411$	0	0
$412$	−32.3369	−1.59312
$413$	8.26943	0.406912
$414$	0	0
$415$	−16.9515	−0.832116
$416$	−138.844	−6.80739
$417$	0	0
$418$	−101.360	−4.95768
$419$	−19.1663	−0.936333	−0.468167	−	0.883640i	$-0.655085\pi$
$419$	−19.1663	−0.936333	−0.468167	+	0.883640i	$0.655085\pi$
$420$	0	0
$421$	16.2901	0.793933	0.396966	−	0.917833i	$-0.370063\pi$
$421$	16.2901	0.793933	0.396966	+	0.917833i	$0.370063\pi$
$422$	6.74318	0.328253
$423$	0	0
$424$	−106.920	−5.19251
$425$	1.49766	0.0726474
$426$	0	0
$427$	−4.31290	−0.208716
$428$	86.2340	4.16828
$429$	0	0
$430$	−1.87487	−0.0904143
$431$	11.5589	0.556770	0.278385	−	0.960469i	$-0.410201\pi$
$431$	11.5589	0.556770	0.278385	+	0.960469i	$0.410201\pi$
$432$	0	0
$433$	−32.8944	−1.58080	−0.790401	−	0.612590i	$-0.790128\pi$
$433$	−32.8944	−1.58080	−0.790401	+	0.612590i	$0.790128\pi$
$434$	−6.34596	−0.304616
$435$	0	0
$436$	−72.8716	−3.48992
$437$	0.427042	0.0204282
$438$	0	0
$439$	−34.9713	−1.66909	−0.834545	−	0.550939i	$-0.814270\pi$
$439$	−34.9713	−1.66909	−0.834545	+	0.550939i	$0.814270\pi$
$440$	−112.664	−5.37105
$441$	0	0
$442$	8.56518	0.407404
$443$	1.22246	0.0580808	0.0290404	−	0.999578i	$-0.490755\pi$
$443$	1.22246	0.0580808	0.0290404	+	0.999578i	$0.490755\pi$
$444$	0	0
$445$	7.54918	0.357865
$446$	75.8353	3.59091
$447$	0	0
$448$	45.5117	2.15022
$449$	−32.8824	−1.55182	−0.775908	−	0.630846i	$-0.782708\pi$
$449$	−32.8824	−1.55182	−0.775908	+	0.630846i	$0.782708\pi$
$450$	0	0
$451$	53.7771	2.53227
$452$	−10.4178	−0.490010
$453$	0	0
$454$	57.8175	2.71351
$455$	7.77594	0.364542
$456$	0	0
$457$	38.0040	1.77775	0.888876	−	0.458148i	$-0.151487\pi$
$457$	38.0040	1.77775	0.888876	+	0.458148i	$0.151487\pi$
$458$	−26.2240	−1.22537
$459$	0	0
$460$	0.724285	0.0337700
$461$	13.1796	0.613837	0.306919	−	0.951736i	$-0.400702\pi$
$461$	13.1796	0.613837	0.306919	+	0.951736i	$0.400702\pi$
$462$	0	0
$463$	2.44809	0.113772	0.0568862	−	0.998381i	$-0.481883\pi$
$463$	2.44809	0.113772	0.0568862	+	0.998381i	$0.481883\pi$
$464$	−34.0156	−1.57914
$465$	0	0
$466$	30.2192	1.39988
$467$	36.8115	1.70343	0.851716	−	0.524003i	$-0.175562\pi$
$467$	36.8115	1.70343	0.851716	+	0.524003i	$0.175562\pi$
$468$	0	0
$469$	7.25212	0.334872
$470$	24.4750	1.12895
$471$	0	0
$472$	−87.8527	−4.04375
$473$	2.65597	0.122122
$474$	0	0
$475$	−12.9955	−0.596272
$476$	−3.74604	−0.171699
$477$	0	0
$478$	52.1060	2.38327
$479$	−16.1423	−0.737560	−0.368780	−	0.929517i	$-0.620224\pi$
$479$	−16.1423	−0.737560	−0.368780	+	0.929517i	$0.620224\pi$
$480$	0	0
$481$	−28.2857	−1.28972
$482$	−16.4326	−0.748486
$483$	0	0
$484$	179.698	8.16810
$485$	−13.2297	−0.600731
$486$	0	0
$487$	−13.0028	−0.589214	−0.294607	−	0.955618i	$-0.595189\pi$
$487$	−13.0028	−0.589214	−0.294607	+	0.955618i	$0.595189\pi$
$488$	45.8193	2.07414
$489$	0	0
$490$	−4.57293	−0.206584
$491$	−24.8426	−1.12113	−0.560565	−	0.828111i	$-0.689416\pi$
$491$	−24.8426	−1.12113	−0.560565	+	0.828111i	$0.689416\pi$
$492$	0	0
$493$	1.21512	0.0547263
$494$	−74.3213	−3.34387
$495$	0	0
$496$	41.0514	1.84326
$497$	−3.65756	−0.164064
$498$	0	0
$499$	−5.74183	−0.257040	−0.128520	−	0.991707i	$-0.541023\pi$
$499$	−5.74183	−0.257040	−0.128520	+	0.991707i	$0.541023\pi$
$500$	−69.5410	−3.10997
$501$	0	0
$502$	48.1436	2.14875
$503$	−7.36156	−0.328236	−0.164118	−	0.986441i	$-0.552478\pi$
$503$	−7.36156	−0.328236	−0.164118	+	0.986441i	$0.552478\pi$
$504$	0	0
$505$	7.61602	0.338908
$506$	−1.37965	−0.0613328
$507$	0	0
$508$	5.80316	0.257474
$509$	25.0910	1.11214	0.556070	−	0.831136i	$-0.312309\pi$
$509$	25.0910	1.11214	0.556070	+	0.831136i	$0.312309\pi$
$510$	0	0
$511$	11.6201	0.514042
$512$	−144.250	−6.37503
$513$	0	0
$514$	7.84511	0.346033
$515$	−9.12206	−0.401966
$516$	0	0
$517$	−34.6718	−1.52486
$518$	16.6345	0.730876
$519$	0	0
$520$	−82.6100	−3.62269
$521$	−9.28849	−0.406936	−0.203468	−	0.979082i	$-0.565221\pi$
$521$	−9.28849	−0.406936	−0.203468	+	0.979082i	$0.565221\pi$
$522$	0	0
$523$	2.49360	0.109038	0.0545188	−	0.998513i	$-0.482638\pi$
$523$	2.49360	0.109038	0.0545188	+	0.998513i	$0.482638\pi$
$524$	−2.43565	−0.106402
$525$	0	0
$526$	32.2764	1.40732
$527$	−1.46646	−0.0638799
$528$	0	0
$529$	−22.9942	−0.999747
$530$	−46.0230	−1.99911
$531$	0	0
$532$	32.5049	1.40927
$533$	39.4316	1.70797
$534$	0	0
$535$	24.3261	1.05171
$536$	−77.0450	−3.32784
$537$	0	0
$538$	7.74459	0.333893
$539$	6.47808	0.279031
$540$	0	0
$541$	21.2860	0.915156	0.457578	−	0.889170i	$-0.348717\pi$
$541$	21.2860	0.915156	0.457578	+	0.889170i	$0.348717\pi$
$542$	−88.6985	−3.80993
$543$	0	0
$544$	18.8687	0.808987
$545$	−20.5567	−0.880552
$546$	0	0
$547$	−6.21251	−0.265628	−0.132814	−	0.991141i	$-0.542401\pi$
$547$	−6.21251	−0.265628	−0.132814	+	0.991141i	$0.542401\pi$
$548$	99.5353	4.25194
$549$	0	0
$550$	41.9845	1.79022
$551$	−10.5438	−0.449181
$552$	0	0
$553$	−12.8009	−0.544348
$554$	−30.9697	−1.31578
$555$	0	0
$556$	10.9893	0.466048
$557$	−3.39391	−0.143804	−0.0719022	−	0.997412i	$-0.522907\pi$
$557$	−3.39391	−0.143804	−0.0719022	+	0.997412i	$0.522907\pi$
$558$	0	0
$559$	1.94747	0.0823692
$560$	29.5818	1.25006
$561$	0	0
$562$	86.5916	3.65265
$563$	26.3575	1.11084	0.555418	−	0.831571i	$-0.312558\pi$
$563$	26.3575	1.11084	0.555418	+	0.831571i	$0.312558\pi$
$564$	0	0
$565$	−2.93879	−0.123636
$566$	19.1764	0.806046
$567$	0	0
$568$	38.8571	1.63041
$569$	−36.8918	−1.54658	−0.773292	−	0.634050i	$-0.781392\pi$
$569$	−36.8918	−1.54658	−0.773292	+	0.634050i	$0.781392\pi$
$570$	0	0
$571$	5.78023	0.241895	0.120948	−	0.992659i	$-0.461407\pi$
$571$	5.78023	0.241895	0.120948	+	0.992659i	$0.461407\pi$
$572$	178.569	7.46632
$573$	0	0
$574$	−23.1892	−0.967900
$575$	−0.176886	−0.00737665
$576$	0	0
$577$	−23.9133	−0.995524	−0.497762	−	0.867314i	$-0.665845\pi$
$577$	−23.9133	−0.995524	−0.497762	+	0.867314i	$0.665845\pi$
$578$	46.3240	1.92683
$579$	0	0
$580$	−17.8828	−0.742544
$581$	−10.3550	−0.429596
$582$	0	0
$583$	65.1969	2.70018
$584$	−123.449	−5.10837
$585$	0	0
$586$	−65.8096	−2.71857
$587$	6.21255	0.256419	0.128210	−	0.991747i	$-0.459077\pi$
$587$	6.21255	0.256419	0.128210	+	0.991747i	$0.459077\pi$
$588$	0	0
$589$	12.7247	0.524311
$590$	−37.8155	−1.55684
$591$	0	0
$592$	−107.607	−4.42261
$593$	−3.72057	−0.152785	−0.0763927	−	0.997078i	$-0.524340\pi$
$593$	−3.72057	−0.152785	−0.0763927	+	0.997078i	$0.524340\pi$
$594$	0	0
$595$	−1.05674	−0.0433220
$596$	82.2492	3.36906
$597$	0	0
$598$	−1.01161	−0.0413680
$599$	−2.35801	−0.0963457	−0.0481729	−	0.998839i	$-0.515340\pi$
$599$	−2.35801	−0.0963457	−0.0481729	+	0.998839i	$0.515340\pi$
$600$	0	0
$601$	9.65719	0.393925	0.196963	−	0.980411i	$-0.436892\pi$
$601$	9.65719	0.393925	0.196963	+	0.980411i	$0.436892\pi$
$602$	−1.14528	−0.0466782
$603$	0	0
$604$	−40.6214	−1.65286
$605$	50.6919	2.06092
$606$	0	0
$607$	−17.1890	−0.697681	−0.348841	−	0.937182i	$-0.613425\pi$
$607$	−17.1890	−0.697681	−0.348841	+	0.937182i	$0.613425\pi$
$608$	−163.726	−6.63997
$609$	0	0
$610$	19.7226	0.798543
$611$	−25.4228	−1.02850
$612$	0	0
$613$	5.98400	0.241691	0.120846	−	0.992671i	$-0.461439\pi$
$613$	5.98400	0.241691	0.120846	+	0.992671i	$0.461439\pi$
$614$	−41.0324	−1.65593
$615$	0	0
$616$	−68.8218	−2.77291
$617$	6.59582	0.265538	0.132769	−	0.991147i	$-0.457613\pi$
$617$	6.59582	0.265538	0.132769	+	0.991147i	$0.457613\pi$
$618$	0	0
$619$	16.8404	0.676873	0.338437	−	0.940989i	$-0.390102\pi$
$619$	16.8404	0.676873	0.338437	+	0.940989i	$0.390102\pi$
$620$	21.5817	0.866742
$621$	0	0
$622$	−82.8441	−3.32175
$623$	4.61148	0.184755
$624$	0	0
$625$	−8.01662	−0.320665
$626$	25.5341	1.02055
$627$	0	0
$628$	38.2917	1.52801
$629$	3.84398	0.153269
$630$	0	0
$631$	1.38194	0.0550144	0.0275072	−	0.999622i	$-0.491243\pi$
$631$	1.38194	0.0550144	0.0275072	+	0.999622i	$0.491243\pi$
$632$	135.994	5.40954
$633$	0	0
$634$	36.7586	1.45987
$635$	1.63704	0.0649639
$636$	0	0
$637$	4.75000	0.188202
$638$	34.0639	1.34860
$639$	0	0
$640$	−112.419	−4.44376
$641$	−23.3835	−0.923593	−0.461796	−	0.886986i	$-0.652795\pi$
$641$	−23.3835	−0.923593	−0.461796	+	0.886986i	$0.652795\pi$
$642$	0	0
$643$	−5.52497	−0.217884	−0.108942	−	0.994048i	$-0.534746\pi$
$643$	−5.52497	−0.217884	−0.108942	+	0.994048i	$0.534746\pi$
$644$	0.442436	0.0174344
$645$	0	0
$646$	10.1001	0.397384
$647$	−32.3975	−1.27368	−0.636840	−	0.770996i	$-0.719759\pi$
$647$	−32.3975	−1.27368	−0.636840	+	0.770996i	$0.719759\pi$
$648$	0	0
$649$	53.5701	2.10281
$650$	30.7848	1.20748
$651$	0	0
$652$	−112.904	−4.42166
$653$	−32.9254	−1.28847	−0.644235	−	0.764828i	$-0.722824\pi$
$653$	−32.9254	−1.28847	−0.644235	+	0.764828i	$0.722824\pi$
$654$	0	0
$655$	−0.687083	−0.0268466
$656$	150.009	5.85687
$657$	0	0
$658$	14.9508	0.582843
$659$	14.4822	0.564145	0.282072	−	0.959393i	$-0.408978\pi$
$659$	14.4822	0.564145	0.282072	+	0.959393i	$0.408978\pi$
$660$	0	0
$661$	1.90129	0.0739516	0.0369758	−	0.999316i	$-0.488228\pi$
$661$	1.90129	0.0739516	0.0369758	+	0.999316i	$0.488228\pi$
$662$	67.0559	2.60620
$663$	0	0
$664$	110.009	4.26918
$665$	9.16946	0.355576
$666$	0	0
$667$	−0.143515	−0.00555694
$668$	130.230	5.03875
$669$	0	0
$670$	−33.1634	−1.28122
$671$	−27.9393	−1.07858
$672$	0	0
$673$	−16.2876	−0.627841	−0.313920	−	0.949449i	$-0.601643\pi$
$673$	−16.2876	−0.627841	−0.313920	+	0.949449i	$0.601643\pi$
$674$	−56.8963	−2.19156
$675$	0	0
$676$	55.4928	2.13434
$677$	−21.3059	−0.818850	−0.409425	−	0.912344i	$-0.634271\pi$
$677$	−21.3059	−0.818850	−0.409425	+	0.912344i	$0.634271\pi$
$678$	0	0
$679$	−8.08150	−0.310140
$680$	11.2265	0.430518
$681$	0	0
$682$	−41.1097	−1.57417
$683$	−26.2565	−1.00468	−0.502338	−	0.864671i	$-0.667527\pi$
$683$	−26.2565	−1.00468	−0.502338	+	0.864671i	$0.667527\pi$
$684$	0	0
$685$	28.0784	1.07282
$686$	−2.79341	−0.106653
$687$	0	0
$688$	7.40872	0.282455
$689$	47.8051	1.82123
$690$	0	0
$691$	−9.62773	−0.366256	−0.183128	−	0.983089i	$-0.558622\pi$
$691$	−9.62773	−0.366256	−0.183128	+	0.983089i	$0.558622\pi$
$692$	−35.8582	−1.36312
$693$	0	0
$694$	13.7638	0.522468
$695$	3.10001	0.117590
$696$	0	0
$697$	−5.35869	−0.202975
$698$	0.257317	0.00973961
$699$	0	0
$700$	−13.4639	−0.508888
$701$	18.5426	0.700344	0.350172	−	0.936685i	$-0.386123\pi$
$701$	18.5426	0.700344	0.350172	+	0.936685i	$0.386123\pi$
$702$	0	0
$703$	−33.3548	−1.25800
$704$	294.829	11.1118
$705$	0	0
$706$	23.6684	0.890772
$707$	4.65231	0.174968
$708$	0	0
$709$	10.4568	0.392712	0.196356	−	0.980533i	$-0.437089\pi$
$709$	10.4568	0.392712	0.196356	+	0.980533i	$0.437089\pi$
$710$	16.7258	0.627706
$711$	0	0
$712$	−48.9914	−1.83603
$713$	0.173200	0.00648640
$714$	0	0
$715$	50.3732	1.88385
$716$	−61.7835	−2.30896
$717$	0	0
$718$	−50.6997	−1.89210
$719$	−32.2373	−1.20225	−0.601125	−	0.799155i	$-0.705280\pi$
$719$	−32.2373	−1.20225	−0.601125	+	0.799155i	$0.705280\pi$
$720$	0	0
$721$	−5.57229	−0.207523
$722$	−34.5655	−1.28639
$723$	0	0
$724$	128.120	4.76155
$725$	4.36736	0.162200
$726$	0	0
$727$	6.69881	0.248445	0.124223	−	0.992254i	$-0.460356\pi$
$727$	6.69881	0.248445	0.124223	+	0.992254i	$0.460356\pi$
$728$	−50.4630	−1.87028
$729$	0	0
$730$	−53.1378	−1.96672
$731$	−0.264658	−0.00978872
$732$	0	0
$733$	28.6921	1.05977	0.529883	−	0.848070i	$-0.322236\pi$
$733$	28.6921	1.05977	0.529883	+	0.848070i	$0.322236\pi$
$734$	54.0895	1.99648
$735$	0	0
$736$	−2.22854	−0.0821449
$737$	46.9799	1.73053
$738$	0	0
$739$	16.9397	0.623138	0.311569	−	0.950224i	$-0.399145\pi$
$739$	16.9397	0.623138	0.311569	+	0.950224i	$0.399145\pi$
$740$	−56.5714	−2.07961
$741$	0	0
$742$	−28.1136	−1.03208
$743$	7.74347	0.284081	0.142040	−	0.989861i	$-0.454634\pi$
$743$	7.74347	0.284081	0.142040	+	0.989861i	$0.454634\pi$
$744$	0	0
$745$	23.2021	0.850058
$746$	−20.5929	−0.753959
$747$	0	0
$748$	−24.2671	−0.887294
$749$	14.8598	0.542967
$750$	0	0
$751$	25.6251	0.935073	0.467537	−	0.883974i	$-0.345142\pi$
$751$	25.6251	0.935073	0.467537	+	0.883974i	$0.345142\pi$
$752$	−96.7153	−3.52684
$753$	0	0
$754$	24.9771	0.909611
$755$	−11.4591	−0.417039
$756$	0	0
$757$	15.7826	0.573629	0.286814	−	0.957986i	$-0.407404\pi$
$757$	15.7826	0.573629	0.286814	+	0.957986i	$0.407404\pi$
$758$	−0.0171009	−0.000621132 0
$759$	0	0
$760$	−97.4144	−3.53359
$761$	−27.0152	−0.979301	−0.489650	−	0.871919i	$-0.662876\pi$
$761$	−27.0152	−0.979301	−0.489650	+	0.871919i	$0.662876\pi$
$762$	0	0
$763$	−12.5572	−0.454602
$764$	145.921	5.27922
$765$	0	0
$766$	101.202	3.65658
$767$	39.2798	1.41831
$768$	0	0
$769$	−12.5918	−0.454071	−0.227036	−	0.973886i	$-0.572903\pi$
$769$	−12.5918	−0.454071	−0.227036	+	0.973886i	$0.572903\pi$
$770$	−29.6238	−1.06757
$771$	0	0
$772$	−107.592	−3.87233
$773$	−25.5403	−0.918621	−0.459311	−	0.888276i	$-0.651904\pi$
$773$	−25.5403	−0.918621	−0.459311	+	0.888276i	$0.651904\pi$
$774$	0	0
$775$	−5.27071	−0.189330
$776$	85.8562	3.08206
$777$	0	0
$778$	−47.6613	−1.70874
$779$	46.4981	1.66597
$780$	0	0
$781$	−23.6940	−0.847837
$782$	0.137477	0.00491615
$783$	0	0
$784$	18.0703	0.645369
$785$	10.8019	0.385536
$786$	0	0
$787$	20.8753	0.744123	0.372061	−	0.928208i	$-0.378651\pi$
$787$	20.8753	0.744123	0.372061	+	0.928208i	$0.378651\pi$
$788$	−121.155	−4.31596
$789$	0	0
$790$	58.5375	2.08267
$791$	−1.79519	−0.0638295
$792$	0	0
$793$	−20.4863	−0.727489
$794$	−79.1935	−2.81047
$795$	0	0
$796$	87.9498	3.11730
$797$	37.7310	1.33650	0.668251	−	0.743936i	$-0.267043\pi$
$797$	37.7310	1.33650	0.668251	+	0.743936i	$0.267043\pi$
$798$	0	0
$799$	3.45491	0.122226
$800$	67.8173	2.39770
$801$	0	0
$802$	90.5454	3.19727
$803$	75.2759	2.65643
$804$	0	0
$805$	0.124809	0.00439893
$806$	−30.1433	−1.06175
$807$	0	0
$808$	−49.4252	−1.73877
$809$	40.8855	1.43746	0.718728	−	0.695291i	$-0.244724\pi$
$809$	40.8855	1.43746	0.718728	+	0.695291i	$0.244724\pi$
$810$	0	0
$811$	21.0579	0.739442	0.369721	−	0.929143i	$-0.379453\pi$
$811$	21.0579	0.739442	0.369721	+	0.929143i	$0.379453\pi$
$812$	−10.9239	−0.383353
$813$	0	0
$814$	107.759	3.77697
$815$	−31.8496	−1.11564
$816$	0	0
$817$	2.29647	0.0803434
$818$	95.7698	3.34851
$819$	0	0
$820$	78.8633	2.75402
$821$	−11.9078	−0.415584	−0.207792	−	0.978173i	$-0.566628\pi$
$821$	−11.9078	−0.415584	−0.207792	+	0.978173i	$0.566628\pi$
$822$	0	0
$823$	23.9178	0.833721	0.416860	−	0.908971i	$-0.363130\pi$
$823$	23.9178	0.833721	0.416860	+	0.908971i	$0.363130\pi$
$824$	59.1988	2.06229
$825$	0	0
$826$	−23.0999	−0.803750
$827$	20.4240	0.710212	0.355106	−	0.934826i	$-0.384445\pi$
$827$	20.4240	0.710212	0.355106	+	0.934826i	$0.384445\pi$
$828$	0	0
$829$	−19.7342	−0.685396	−0.342698	−	0.939446i	$-0.611341\pi$
$829$	−19.7342	−0.685396	−0.342698	+	0.939446i	$0.611341\pi$
$830$	47.3525	1.64363
$831$	0	0
$832$	216.181	7.49471
$833$	−0.645517	−0.0223658
$834$	0	0
$835$	36.7372	1.27134
$836$	210.569	7.28270
$837$	0	0
$838$	53.5393	1.84948
$839$	0.908480	0.0313642	0.0156821	−	0.999877i	$-0.495008\pi$
$839$	0.908480	0.0313642	0.0156821	+	0.999877i	$0.495008\pi$
$840$	0	0
$841$	−25.4566	−0.877813
$842$	−45.5051	−1.56821
$843$	0	0
$844$	−14.0086	−0.482195
$845$	15.6542	0.538522
$846$	0	0
$847$	30.9656	1.06399
$848$	181.864	6.24523
$849$	0	0
$850$	−4.18359	−0.143496
$851$	−0.454004	−0.0155631
$852$	0	0
$853$	9.77125	0.334561	0.167281	−	0.985909i	$-0.446501\pi$
$853$	9.77125	0.334561	0.167281	+	0.985909i	$0.446501\pi$
$854$	12.0477	0.412264
$855$	0	0
$856$	−157.868	−5.39581
$857$	54.9123	1.87577	0.937884	−	0.346950i	$-0.112783\pi$
$857$	54.9123	1.87577	0.937884	+	0.346950i	$0.112783\pi$
$858$	0	0
$859$	33.3227	1.13696	0.568478	−	0.822698i	$-0.307532\pi$
$859$	33.3227	1.13696	0.568478	+	0.822698i	$0.307532\pi$
$860$	3.89494	0.132816
$861$	0	0
$862$	−32.2887	−1.09976
$863$	−13.4052	−0.456320	−0.228160	−	0.973624i	$-0.573271\pi$
$863$	−13.4052	−0.456320	−0.228160	+	0.973624i	$0.573271\pi$
$864$	0	0
$865$	−10.1154	−0.343934
$866$	91.8875	3.12247
$867$	0	0
$868$	13.1834	0.447473
$869$	−82.9251	−2.81304
$870$	0	0
$871$	34.4476	1.16721
$872$	133.405	4.51768
$873$	0	0
$874$	−1.19290	−0.0403506
$875$	−11.9833	−0.405109
$876$	0	0
$877$	24.6955	0.833909	0.416954	−	0.908927i	$-0.363097\pi$
$877$	24.6955	0.833909	0.416954	+	0.908927i	$0.363097\pi$
$878$	97.6894	3.29686
$879$	0	0
$880$	191.634	6.45997
$881$	−43.6844	−1.47177	−0.735883	−	0.677109i	$-0.763233\pi$
$881$	−43.6844	−1.47177	−0.735883	+	0.677109i	$0.763233\pi$
$882$	0	0
$883$	−22.9644	−0.772814	−0.386407	−	0.922328i	$-0.626284\pi$
$883$	−22.9644	−0.772814	−0.386407	+	0.922328i	$0.626284\pi$
$884$	−17.7937	−0.598466
$885$	0	0
$886$	−3.41483	−0.114724
$887$	35.8678	1.20432	0.602162	−	0.798374i	$-0.294306\pi$
$887$	35.8678	1.20432	0.602162	+	0.798374i	$0.294306\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−21.0880	−0.706870
$891$	0	0
$892$	−157.544	−5.27495
$893$	−29.9788	−1.00320
$894$	0	0
$895$	−17.4288	−0.582581
$896$	−68.6723	−2.29418
$897$	0	0
$898$	91.8542	3.06521
$899$	−4.27636	−0.142625
$900$	0	0
$901$	−6.49663	−0.216434
$902$	−150.222	−5.00184
$903$	0	0
$904$	19.0717	0.634315
$905$	36.1420	1.20140
$906$	0	0
$907$	46.3210	1.53806	0.769032	−	0.639210i	$-0.220739\pi$
$907$	46.3210	1.53806	0.769032	+	0.639210i	$0.220739\pi$
$908$	−120.113	−3.98607
$909$	0	0
$910$	−21.7214	−0.720058
$911$	12.2609	0.406221	0.203111	−	0.979156i	$-0.434895\pi$
$911$	12.2609	0.406221	0.203111	+	0.979156i	$0.434895\pi$
$912$	0	0
$913$	−67.0803	−2.22004
$914$	−106.161	−3.51149
$915$	0	0
$916$	54.4789	1.80004
$917$	−0.419711	−0.0138601
$918$	0	0
$919$	16.5157	0.544803	0.272402	−	0.962184i	$-0.412182\pi$
$919$	16.5157	0.544803	0.272402	+	0.962184i	$0.412182\pi$
$920$	−1.32594	−0.0437151
$921$	0	0
$922$	−36.8162	−1.21248
$923$	−17.3734	−0.571853
$924$	0	0
$925$	13.8159	0.454265
$926$	−6.83852	−0.224728
$927$	0	0
$928$	55.0232	1.80622
$929$	−45.0217	−1.47712	−0.738558	−	0.674190i	$-0.764493\pi$
$929$	−45.0217	−1.47712	−0.738558	+	0.674190i	$0.764493\pi$
$930$	0	0
$931$	5.60124	0.183573
$932$	−62.7788	−2.05639
$933$	0	0
$934$	−102.830	−3.36469
$935$	−6.84563	−0.223876
$936$	0	0
$937$	41.6320	1.36006	0.680029	−	0.733185i	$-0.261967\pi$
$937$	41.6320	1.36006	0.680029	+	0.733185i	$0.261967\pi$
$938$	−20.2582	−0.661453
$939$	0	0
$940$	−50.8455	−1.65840
$941$	−14.6293	−0.476902	−0.238451	−	0.971155i	$-0.576640\pi$
$941$	−14.6293	−0.476902	−0.238451	+	0.971155i	$0.576640\pi$
$942$	0	0
$943$	0.632903	0.0206102
$944$	149.431	4.86358
$945$	0	0
$946$	−7.41923	−0.241220
$947$	48.5519	1.57772	0.788862	−	0.614570i	$-0.210671\pi$
$947$	48.5519	1.57772	0.788862	+	0.614570i	$0.210671\pi$
$948$	0	0
$949$	55.1955	1.79172
$950$	36.3017	1.17778
$951$	0	0
$952$	6.85783	0.222264
$953$	−25.4440	−0.824211	−0.412106	−	0.911136i	$-0.635207\pi$
$953$	−25.4440	−0.824211	−0.412106	+	0.911136i	$0.635207\pi$
$954$	0	0
$955$	41.1634	1.33202
$956$	−108.247	−3.50096
$957$	0	0
$958$	45.0921	1.45686
$959$	17.1519	0.553864
$960$	0	0
$961$	−25.8391	−0.833520
$962$	79.0137	2.54751
$963$	0	0
$964$	34.1379	1.09951
$965$	−30.3512	−0.977039
$966$	0	0
$967$	−55.0393	−1.76995	−0.884973	−	0.465643i	$-0.845823\pi$
$967$	−55.0393	−1.76995	−0.884973	+	0.465643i	$0.845823\pi$
$968$	−328.972	−10.5736
$969$	0	0
$970$	36.9561	1.18659
$971$	−30.6876	−0.984813	−0.492406	−	0.870365i	$-0.663883\pi$
$971$	−30.6876	−0.984813	−0.492406	+	0.870365i	$0.663883\pi$
$972$	0	0
$973$	1.89367	0.0607082
$974$	36.3223	1.16384
$975$	0	0
$976$	−77.9354	−2.49465
$977$	8.85827	0.283401	0.141701	−	0.989910i	$-0.454743\pi$
$977$	8.85827	0.283401	0.141701	+	0.989910i	$0.454743\pi$
$978$	0	0
$979$	29.8736	0.954763
$980$	9.50000	0.303466
$981$	0	0
$982$	69.3956	2.21450
$983$	31.5119	1.00507	0.502537	−	0.864556i	$-0.332400\pi$
$983$	31.5119	1.00507	0.502537	+	0.864556i	$0.332400\pi$
$984$	0	0
$985$	−34.1771	−1.08897
$986$	−3.39434	−0.108098
$987$	0	0
$988$	154.398	4.91207
$989$	0.0312581	0.000993951 0
$990$	0	0
$991$	−9.28778	−0.295036	−0.147518	−	0.989059i	$-0.547128\pi$
$991$	−9.28778	−0.295036	−0.147518	+	0.989059i	$0.547128\pi$
$992$	−66.4042	−2.10834
$993$	0	0
$994$	10.2171	0.324066
$995$	24.8102	0.786535
$996$	0	0
$997$	−13.4491	−0.425937	−0.212969	−	0.977059i	$-0.568313\pi$
$997$	−13.4491	−0.425937	−0.212969	+	0.977059i	$0.568313\pi$
$998$	16.0393	0.507715
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.1		19
3.2	odd	2		2667.2.a.q.1.19	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.19	✓	19	3.2	odd	2
8001.2.a.v.1.1		19	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)

\(f(q)\)	\(=\)	\(q-2.79341 q^{2} +5.80316 q^{4} +1.63704 q^{5} +1.00000 q^{7} -10.6238 q^{8} +O(q^{10})\)	\(q-2.79341 q^{2} +5.80316 q^{4} +1.63704 q^{5} +1.00000 q^{7} -10.6238 q^{8} -4.57293 q^{10} +6.47808 q^{11} +4.75000 q^{13} -2.79341 q^{14} +18.0703 q^{16} -0.645517 q^{17} +5.60124 q^{19} +9.50000 q^{20} -18.0960 q^{22} +0.0762406 q^{23} -2.32010 q^{25} -13.2687 q^{26} +5.80316 q^{28} -1.88240 q^{29} +2.27176 q^{31} -29.2303 q^{32} +1.80319 q^{34} +1.63704 q^{35} -5.95489 q^{37} -15.6466 q^{38} -17.3916 q^{40} +8.30140 q^{41} +0.409994 q^{43} +37.5934 q^{44} -0.212971 q^{46} -5.35216 q^{47} +1.00000 q^{49} +6.48100 q^{50} +27.5650 q^{52} +10.0642 q^{53} +10.6049 q^{55} -10.6238 q^{56} +5.25833 q^{58} +8.26943 q^{59} -4.31290 q^{61} -6.34596 q^{62} +45.5117 q^{64} +7.77594 q^{65} +7.25212 q^{67} -3.74604 q^{68} -4.57293 q^{70} -3.65756 q^{71} +11.6201 q^{73} +16.6345 q^{74} +32.5049 q^{76} +6.47808 q^{77} -12.8009 q^{79} +29.5818 q^{80} -23.1892 q^{82} -10.3550 q^{83} -1.05674 q^{85} -1.14528 q^{86} -68.8218 q^{88} +4.61148 q^{89} +4.75000 q^{91} +0.442436 q^{92} +14.9508 q^{94} +9.16946 q^{95} -8.08150 q^{97} -2.79341 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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