LMFDB - Embedded newform 8001.2.a.u.1.15

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$2.40246$ 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)

$f(q)$	$=$	$q+2.40246 q^{2} +3.77179 q^{4} +0.666645 q^{5} +1.00000 q^{7} +4.25666 q^{8} +O(q^{10})$	\(q+2.40246 q^{2} +3.77179 q^{4} +0.666645 q^{5} +1.00000 q^{7} +4.25666 q^{8} +1.60159 q^{10} +5.05203 q^{11} +5.76961 q^{13} +2.40246 q^{14} +2.68284 q^{16} +0.396381 q^{17} +3.56828 q^{19} +2.51445 q^{20} +12.1373 q^{22} +4.24325 q^{23} -4.55558 q^{25} +13.8612 q^{26} +3.77179 q^{28} -0.519685 q^{29} +3.84273 q^{31} -2.06791 q^{32} +0.952288 q^{34} +0.666645 q^{35} -7.00515 q^{37} +8.57264 q^{38} +2.83768 q^{40} -3.06775 q^{41} -4.79677 q^{43} +19.0552 q^{44} +10.1942 q^{46} -11.2665 q^{47} +1.00000 q^{49} -10.9446 q^{50} +21.7618 q^{52} +1.29111 q^{53} +3.36791 q^{55} +4.25666 q^{56} -1.24852 q^{58} +1.96260 q^{59} -9.45403 q^{61} +9.23199 q^{62} -10.3337 q^{64} +3.84628 q^{65} -8.88556 q^{67} +1.49507 q^{68} +1.60159 q^{70} +8.51845 q^{71} -6.50942 q^{73} -16.8296 q^{74} +13.4588 q^{76} +5.05203 q^{77} +14.8928 q^{79} +1.78850 q^{80} -7.37013 q^{82} -13.6579 q^{83} +0.264246 q^{85} -11.5240 q^{86} +21.5047 q^{88} +12.6023 q^{89} +5.76961 q^{91} +16.0047 q^{92} -27.0673 q^{94} +2.37878 q^{95} +9.21904 q^{97} +2.40246 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * 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 $

$2$	2.40246	1.69879	0.849396	−	0.527755i	$-0.176966\pi$
$2$	2.40246	1.69879	0.849396	+	0.527755i	$0.176966\pi$
$3$	0	0
$3$	0	0
$4$	3.77179	1.88590
$5$	0.666645	0.298133	0.149066	−	0.988827i	$-0.452373\pi$
$5$	0.666645	0.298133	0.149066	+	0.988827i	$0.452373\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	4.25666	1.50496
$9$	0	0
$10$	1.60159	0.506466
$11$	5.05203	1.52324	0.761622	−	0.648022i	$-0.224403\pi$
$11$	5.05203	1.52324	0.761622	+	0.648022i	$0.224403\pi$
$12$	0	0
$13$	5.76961	1.60020	0.800101	−	0.599866i	$-0.204779\pi$
$13$	5.76961	1.60020	0.800101	+	0.599866i	$0.204779\pi$
$14$	2.40246	0.642083
$15$	0	0
$16$	2.68284	0.670710
$17$	0.396381	0.0961366	0.0480683	−	0.998844i	$-0.484693\pi$
$17$	0.396381	0.0961366	0.0480683	+	0.998844i	$0.484693\pi$
$18$	0	0
$19$	3.56828	0.818620	0.409310	−	0.912395i	$-0.365769\pi$
$19$	3.56828	0.818620	0.409310	+	0.912395i	$0.365769\pi$
$20$	2.51445	0.562248
$21$	0	0
$22$	12.1373	2.58768
$23$	4.24325	0.884780	0.442390	−	0.896823i	$-0.354131\pi$
$23$	4.24325	0.884780	0.442390	+	0.896823i	$0.354131\pi$
$24$	0	0
$25$	−4.55558	−0.911117
$26$	13.8612	2.71841
$27$	0	0
$28$	3.77179	0.712802
$29$	−0.519685	−0.0965030	−0.0482515	−	0.998835i	$-0.515365\pi$
$29$	−0.519685	−0.0965030	−0.0482515	+	0.998835i	$0.515365\pi$
$30$	0	0
$31$	3.84273	0.690175	0.345087	−	0.938571i	$-0.387849\pi$
$31$	3.84273	0.690175	0.345087	+	0.938571i	$0.387849\pi$
$32$	−2.06791	−0.365558
$33$	0	0
$34$	0.952288	0.163316
$35$	0.666645	0.112684
$36$	0	0
$37$	−7.00515	−1.15164	−0.575820	−	0.817577i	$-0.695317\pi$
$37$	−7.00515	−1.15164	−0.575820	+	0.817577i	$0.695317\pi$
$38$	8.57264	1.39067
$39$	0	0
$40$	2.83768	0.448677
$41$	−3.06775	−0.479102	−0.239551	−	0.970884i	$-0.577000\pi$
$41$	−3.06775	−0.479102	−0.239551	+	0.970884i	$0.577000\pi$
$42$	0	0
$43$	−4.79677	−0.731501	−0.365750	−	0.930713i	$-0.619188\pi$
$43$	−4.79677	−0.731501	−0.365750	+	0.930713i	$0.619188\pi$
$44$	19.0552	2.87268
$45$	0	0
$46$	10.1942	1.50306
$47$	−11.2665	−1.64339	−0.821696	−	0.569926i	$-0.806972\pi$
$47$	−11.2665	−1.64339	−0.821696	+	0.569926i	$0.806972\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.9446	−1.54780
$51$	0	0
$52$	21.7618	3.01781
$53$	1.29111	0.177348	0.0886740	−	0.996061i	$-0.471737\pi$
$53$	1.29111	0.177348	0.0886740	+	0.996061i	$0.471737\pi$
$54$	0	0
$55$	3.36791	0.454129
$56$	4.25666	0.568820
$57$	0	0
$58$	−1.24852	−0.163939
$59$	1.96260	0.255509	0.127755	−	0.991806i	$-0.459223\pi$
$59$	1.96260	0.255509	0.127755	+	0.991806i	$0.459223\pi$
$60$	0	0
$61$	−9.45403	−1.21046	−0.605232	−	0.796049i	$-0.706920\pi$
$61$	−9.45403	−1.21046	−0.605232	+	0.796049i	$0.706920\pi$
$62$	9.23199	1.17246
$63$	0	0
$64$	−10.3337	−1.29172
$65$	3.84628	0.477073
$66$	0	0
$67$	−8.88556	−1.08554	−0.542772	−	0.839880i	$-0.682625\pi$
$67$	−8.88556	−1.08554	−0.542772	+	0.839880i	$0.682625\pi$
$68$	1.49507	0.181304
$69$	0	0
$70$	1.60159	0.191426
$71$	8.51845	1.01095	0.505477	−	0.862840i	$-0.331317\pi$
$71$	8.51845	1.01095	0.505477	+	0.862840i	$0.331317\pi$
$72$	0	0
$73$	−6.50942	−0.761870	−0.380935	−	0.924602i	$-0.624398\pi$
$73$	−6.50942	−0.761870	−0.380935	+	0.924602i	$0.624398\pi$
$74$	−16.8296	−1.95640
$75$	0	0
$76$	13.4588	1.54383
$77$	5.05203	0.575732
$78$	0	0
$79$	14.8928	1.67557	0.837783	−	0.546004i	$-0.183852\pi$
$79$	14.8928	1.67557	0.837783	+	0.546004i	$0.183852\pi$
$80$	1.78850	0.199961
$81$	0	0
$82$	−7.37013	−0.813894
$83$	−13.6579	−1.49915	−0.749575	−	0.661919i	$-0.769742\pi$
$83$	−13.6579	−1.49915	−0.749575	+	0.661919i	$0.769742\pi$
$84$	0	0
$85$	0.264246	0.0286615
$86$	−11.5240	−1.24267
$87$	0	0
$88$	21.5047	2.29241
$89$	12.6023	1.33585	0.667923	−	0.744231i	$-0.267184\pi$
$89$	12.6023	1.33585	0.667923	+	0.744231i	$0.267184\pi$
$90$	0	0
$91$	5.76961	0.604819
$92$	16.0047	1.66860
$93$	0	0
$94$	−27.0673	−2.79178
$95$	2.37878	0.244058
$96$	0	0
$97$	9.21904	0.936052	0.468026	−	0.883715i	$-0.344965\pi$
$97$	9.21904	0.936052	0.468026	+	0.883715i	$0.344965\pi$
$98$	2.40246	0.242685
$99$	0	0
$100$	−17.1827	−1.71827
$101$	−5.22301	−0.519709	−0.259855	−	0.965648i	$-0.583675\pi$
$101$	−5.22301	−0.519709	−0.259855	+	0.965648i	$0.583675\pi$
$102$	0	0
$103$	−1.04722	−0.103185	−0.0515926	−	0.998668i	$-0.516430\pi$
$103$	−1.04722	−0.103185	−0.0515926	+	0.998668i	$0.516430\pi$
$104$	24.5592	2.40823
$105$	0	0
$106$	3.10184	0.301277
$107$	−15.4399	−1.49263	−0.746317	−	0.665590i	$-0.768180\pi$
$107$	−15.4399	−1.49263	−0.746317	+	0.665590i	$0.768180\pi$
$108$	0	0
$109$	8.22589	0.787897	0.393949	−	0.919132i	$-0.371109\pi$
$109$	8.22589	0.787897	0.393949	+	0.919132i	$0.371109\pi$
$110$	8.09126	0.771471
$111$	0	0
$112$	2.68284	0.253505
$113$	2.61402	0.245906	0.122953	−	0.992412i	$-0.460764\pi$
$113$	2.61402	0.245906	0.122953	+	0.992412i	$0.460764\pi$
$114$	0	0
$115$	2.82875	0.263782
$116$	−1.96014	−0.181995
$117$	0	0
$118$	4.71506	0.434057
$119$	0.396381	0.0363362
$120$	0	0
$121$	14.5230	1.32027
$122$	−22.7129	−2.05633
$123$	0	0
$124$	14.4940	1.30160
$125$	−6.37019	−0.569767
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−20.6905	−1.82880
$129$	0	0
$130$	9.24052	0.810447
$131$	−14.8530	−1.29771	−0.648855	−	0.760912i	$-0.724752\pi$
$131$	−14.8530	−1.29771	−0.648855	+	0.760912i	$0.724752\pi$
$132$	0	0
$133$	3.56828	0.309409
$134$	−21.3472	−1.84411
$135$	0	0
$136$	1.68726	0.144681
$137$	20.4969	1.75117	0.875583	−	0.483068i	$-0.160478\pi$
$137$	20.4969	1.75117	0.875583	+	0.483068i	$0.160478\pi$
$138$	0	0
$139$	−3.53740	−0.300039	−0.150019	−	0.988683i	$-0.547934\pi$
$139$	−3.53740	−0.300039	−0.150019	+	0.988683i	$0.547934\pi$
$140$	2.51445	0.212510
$141$	0	0
$142$	20.4652	1.71740
$143$	29.1482	2.43750
$144$	0	0
$145$	−0.346445	−0.0287707
$146$	−15.6386	−1.29426
$147$	0	0
$148$	−26.4220	−2.17187
$149$	19.9965	1.63817	0.819087	−	0.573669i	$-0.194480\pi$
$149$	19.9965	1.63817	0.819087	+	0.573669i	$0.194480\pi$
$150$	0	0
$151$	18.2975	1.48903	0.744516	−	0.667605i	$-0.232680\pi$
$151$	18.2975	1.48903	0.744516	+	0.667605i	$0.232680\pi$
$152$	15.1889	1.23199
$153$	0	0
$154$	12.1373	0.978049
$155$	2.56174	0.205764
$156$	0	0
$157$	−23.8970	−1.90718	−0.953592	−	0.301100i	$-0.902646\pi$
$157$	−23.8970	−1.90718	−0.953592	+	0.301100i	$0.902646\pi$
$158$	35.7792	2.84644
$159$	0	0
$160$	−1.37856	−0.108985
$161$	4.24325	0.334415
$162$	0	0
$163$	−14.7857	−1.15810	−0.579051	−	0.815291i	$-0.696577\pi$
$163$	−14.7857	−1.15810	−0.579051	+	0.815291i	$0.696577\pi$
$164$	−11.5709	−0.903536
$165$	0	0
$166$	−32.8125	−2.54675
$167$	−21.2454	−1.64402	−0.822010	−	0.569474i	$-0.807147\pi$
$167$	−21.2454	−1.64402	−0.822010	+	0.569474i	$0.807147\pi$
$168$	0	0
$169$	20.2884	1.56064
$170$	0.634839	0.0486899
$171$	0	0
$172$	−18.0924	−1.37954
$173$	8.03384	0.610802	0.305401	−	0.952224i	$-0.401210\pi$
$173$	8.03384	0.610802	0.305401	+	0.952224i	$0.401210\pi$
$174$	0	0
$175$	−4.55558	−0.344370
$176$	13.5538	1.02165
$177$	0	0
$178$	30.2766	2.26932
$179$	18.5927	1.38968	0.694840	−	0.719164i	$-0.255475\pi$
$179$	18.5927	1.38968	0.694840	+	0.719164i	$0.255475\pi$
$180$	0	0
$181$	10.8782	0.808569	0.404284	−	0.914633i	$-0.367521\pi$
$181$	10.8782	0.808569	0.404284	+	0.914633i	$0.367521\pi$
$182$	13.8612	1.02746
$183$	0	0
$184$	18.0621	1.33155
$185$	−4.66995	−0.343342
$186$	0	0
$187$	2.00253	0.146439
$188$	−42.4950	−3.09927
$189$	0	0
$190$	5.71491	0.414603
$191$	−19.0877	−1.38114	−0.690570	−	0.723266i	$-0.742640\pi$
$191$	−19.0877	−1.38114	−0.690570	+	0.723266i	$0.742640\pi$
$192$	0	0
$193$	−16.5306	−1.18990	−0.594951	−	0.803762i	$-0.702828\pi$
$193$	−16.5306	−1.18990	−0.594951	+	0.803762i	$0.702828\pi$
$194$	22.1483	1.59016
$195$	0	0
$196$	3.77179	0.269414
$197$	26.3357	1.87634	0.938169	−	0.346178i	$-0.112521\pi$
$197$	26.3357	1.87634	0.938169	+	0.346178i	$0.112521\pi$
$198$	0	0
$199$	−0.679810	−0.0481905	−0.0240952	−	0.999710i	$-0.507670\pi$
$199$	−0.679810	−0.0481905	−0.0240952	+	0.999710i	$0.507670\pi$
$200$	−19.3916	−1.37119
$201$	0	0
$202$	−12.5481	−0.882878
$203$	−0.519685	−0.0364747
$204$	0	0
$205$	−2.04510	−0.142836
$206$	−2.51589	−0.175290
$207$	0	0
$208$	15.4789	1.07327
$209$	18.0271	1.24696
$210$	0	0
$211$	17.0443	1.17338	0.586688	−	0.809813i	$-0.300432\pi$
$211$	17.0443	1.17338	0.586688	+	0.809813i	$0.300432\pi$
$212$	4.86981	0.334460
$213$	0	0
$214$	−37.0938	−2.53568
$215$	−3.19775	−0.218084
$216$	0	0
$217$	3.84273	0.260862
$218$	19.7623	1.33847
$219$	0	0
$220$	12.7031	0.856441
$221$	2.28696	0.153838
$222$	0	0
$223$	−20.7478	−1.38938	−0.694689	−	0.719311i	$-0.744458\pi$
$223$	−20.7478	−1.38938	−0.694689	+	0.719311i	$0.744458\pi$
$224$	−2.06791	−0.138168
$225$	0	0
$226$	6.28006	0.417744
$227$	10.9843	0.729051	0.364525	−	0.931193i	$-0.381231\pi$
$227$	10.9843	0.729051	0.364525	+	0.931193i	$0.381231\pi$
$228$	0	0
$229$	−3.69557	−0.244210	−0.122105	−	0.992517i	$-0.538964\pi$
$229$	−3.69557	−0.244210	−0.122105	+	0.992517i	$0.538964\pi$
$230$	6.79594	0.448111
$231$	0	0
$232$	−2.21212	−0.145233
$233$	27.0298	1.77078	0.885389	−	0.464851i	$-0.153892\pi$
$233$	27.0298	1.77078	0.885389	+	0.464851i	$0.153892\pi$
$234$	0	0
$235$	−7.51078	−0.489949
$236$	7.40253	0.481864
$237$	0	0
$238$	0.952288	0.0617277
$239$	−2.63505	−0.170447	−0.0852235	−	0.996362i	$-0.527160\pi$
$239$	−2.63505	−0.170447	−0.0852235	+	0.996362i	$0.527160\pi$
$240$	0	0
$241$	16.8404	1.08478	0.542392	−	0.840125i	$-0.317519\pi$
$241$	16.8404	1.08478	0.542392	+	0.840125i	$0.317519\pi$
$242$	34.8908	2.24287
$243$	0	0
$244$	−35.6587	−2.28281
$245$	0.666645	0.0425904
$246$	0	0
$247$	20.5876	1.30996
$248$	16.3572	1.03868
$249$	0	0
$250$	−15.3041	−0.967916
$251$	22.7185	1.43398	0.716990	−	0.697083i	$-0.245519\pi$
$251$	22.7185	1.43398	0.716990	+	0.697083i	$0.245519\pi$
$252$	0	0
$253$	21.4370	1.34774
$254$	2.40246	0.150743
$255$	0	0
$256$	−29.0406	−1.81504
$257$	16.8856	1.05330	0.526648	−	0.850084i	$-0.323449\pi$
$257$	16.8856	1.05330	0.526648	+	0.850084i	$0.323449\pi$
$258$	0	0
$259$	−7.00515	−0.435279
$260$	14.5074	0.899710
$261$	0	0
$262$	−35.6836	−2.20454
$263$	14.4874	0.893334	0.446667	−	0.894700i	$-0.352611\pi$
$263$	14.4874	0.893334	0.446667	+	0.894700i	$0.352611\pi$
$264$	0	0
$265$	0.860714	0.0528733
$266$	8.57264	0.525622
$267$	0	0
$268$	−33.5145	−2.04722
$269$	−19.0894	−1.16390	−0.581952	−	0.813223i	$-0.697711\pi$
$269$	−19.0894	−1.16390	−0.581952	+	0.813223i	$0.697711\pi$
$270$	0	0
$271$	−12.5847	−0.764467	−0.382234	−	0.924066i	$-0.624845\pi$
$271$	−12.5847	−0.764467	−0.382234	+	0.924066i	$0.624845\pi$
$272$	1.06343	0.0644798
$273$	0	0
$274$	49.2428	2.97487
$275$	−23.0149	−1.38785
$276$	0	0
$277$	−23.5452	−1.41470	−0.707348	−	0.706866i	$-0.750108\pi$
$277$	−23.5452	−1.41470	−0.707348	+	0.706866i	$0.750108\pi$
$278$	−8.49845	−0.509703
$279$	0	0
$280$	2.83768	0.169584
$281$	17.3606	1.03565	0.517824	−	0.855487i	$-0.326742\pi$
$281$	17.3606	1.03565	0.517824	+	0.855487i	$0.326742\pi$
$282$	0	0
$283$	−28.0114	−1.66511	−0.832554	−	0.553945i	$-0.813122\pi$
$283$	−28.0114	−1.66511	−0.832554	+	0.553945i	$0.813122\pi$
$284$	32.1298	1.90655
$285$	0	0
$286$	70.0273	4.14080
$287$	−3.06775	−0.181083
$288$	0	0
$289$	−16.8429	−0.990758
$290$	−0.832320	−0.0488755
$291$	0	0
$292$	−24.5522	−1.43681
$293$	−22.0696	−1.28932	−0.644661	−	0.764469i	$-0.723001\pi$
$293$	−22.0696	−1.28932	−0.644661	+	0.764469i	$0.723001\pi$
$294$	0	0
$295$	1.30836	0.0761756
$296$	−29.8185	−1.73317
$297$	0	0
$298$	48.0406	2.78292
$299$	24.4819	1.41583
$300$	0	0
$301$	−4.79677	−0.276481
$302$	43.9590	2.52956
$303$	0	0
$304$	9.57313	0.549057
$305$	−6.30249	−0.360879
$306$	0	0
$307$	−17.8522	−1.01888	−0.509439	−	0.860507i	$-0.670147\pi$
$307$	−17.8522	−1.01888	−0.509439	+	0.860507i	$0.670147\pi$
$308$	19.0552	1.08577
$309$	0	0
$310$	6.15447	0.349550
$311$	15.7490	0.893045	0.446522	−	0.894772i	$-0.352662\pi$
$311$	15.7490	0.893045	0.446522	+	0.894772i	$0.352662\pi$
$312$	0	0
$313$	−15.8596	−0.896439	−0.448219	−	0.893924i	$-0.647942\pi$
$313$	−15.8596	−0.896439	−0.448219	+	0.893924i	$0.647942\pi$
$314$	−57.4114	−3.23991
$315$	0	0
$316$	56.1724	3.15994
$317$	−28.1773	−1.58259	−0.791296	−	0.611433i	$-0.790593\pi$
$317$	−28.1773	−1.58259	−0.791296	+	0.611433i	$0.790593\pi$
$318$	0	0
$319$	−2.62546	−0.146998
$320$	−6.88894	−0.385103
$321$	0	0
$322$	10.1942	0.568102
$323$	1.41440	0.0786993
$324$	0	0
$325$	−26.2839	−1.45797
$326$	−35.5219	−1.96738
$327$	0	0
$328$	−13.0583	−0.721026
$329$	−11.2665	−0.621144
$330$	0	0
$331$	−27.8507	−1.53081	−0.765406	−	0.643547i	$-0.777462\pi$
$331$	−27.8507	−1.53081	−0.765406	+	0.643547i	$0.777462\pi$
$332$	−51.5148	−2.82724
$333$	0	0
$334$	−51.0412	−2.79285
$335$	−5.92352	−0.323636
$336$	0	0
$337$	−7.64516	−0.416458	−0.208229	−	0.978080i	$-0.566770\pi$
$337$	−7.64516	−0.416458	−0.208229	+	0.978080i	$0.566770\pi$
$338$	48.7419	2.65121
$339$	0	0
$340$	0.996680	0.0540526
$341$	19.4136	1.05130
$342$	0	0
$343$	1.00000	0.0539949
$344$	−20.4182	−1.10088
$345$	0	0
$346$	19.3010	1.03763
$347$	−22.1004	−1.18641	−0.593205	−	0.805052i	$-0.702137\pi$
$347$	−22.1004	−1.18641	−0.593205	+	0.805052i	$0.702137\pi$
$348$	0	0
$349$	7.12427	0.381353	0.190677	−	0.981653i	$-0.438932\pi$
$349$	7.12427	0.381353	0.190677	+	0.981653i	$0.438932\pi$
$350$	−10.9446	−0.585013
$351$	0	0
$352$	−10.4471	−0.556833
$353$	14.0285	0.746662	0.373331	−	0.927698i	$-0.378216\pi$
$353$	14.0285	0.746662	0.373331	+	0.927698i	$0.378216\pi$
$354$	0	0
$355$	5.67878	0.301399
$356$	47.5334	2.51927
$357$	0	0
$358$	44.6680	2.36078
$359$	−6.03937	−0.318746	−0.159373	−	0.987218i	$-0.550947\pi$
$359$	−6.03937	−0.318746	−0.159373	+	0.987218i	$0.550947\pi$
$360$	0	0
$361$	−6.26736	−0.329861
$362$	26.1343	1.37359
$363$	0	0
$364$	21.7618	1.14063
$365$	−4.33948	−0.227139
$366$	0	0
$367$	−12.6586	−0.660775	−0.330387	−	0.943845i	$-0.607179\pi$
$367$	−12.6586	−0.660775	−0.330387	+	0.943845i	$0.607179\pi$
$368$	11.3840	0.593431
$369$	0	0
$370$	−11.2194	−0.583267
$371$	1.29111	0.0670312
$372$	0	0
$373$	16.3249	0.845269	0.422635	−	0.906300i	$-0.361105\pi$
$373$	16.3249	0.845269	0.422635	+	0.906300i	$0.361105\pi$
$374$	4.81099	0.248770
$375$	0	0
$376$	−47.9577	−2.47323
$377$	−2.99838	−0.154424
$378$	0	0
$379$	16.1793	0.831074	0.415537	−	0.909576i	$-0.363594\pi$
$379$	16.1793	0.831074	0.415537	+	0.909576i	$0.363594\pi$
$380$	8.97226	0.460267
$381$	0	0
$382$	−45.8574	−2.34627
$383$	14.1963	0.725399	0.362699	−	0.931906i	$-0.381855\pi$
$383$	14.1963	0.725399	0.362699	+	0.931906i	$0.381855\pi$
$384$	0	0
$385$	3.36791	0.171645
$386$	−39.7141	−2.02140
$387$	0	0
$388$	34.7723	1.76530
$389$	−1.32405	−0.0671318	−0.0335659	−	0.999437i	$-0.510686\pi$
$389$	−1.32405	−0.0671318	−0.0335659	+	0.999437i	$0.510686\pi$
$390$	0	0
$391$	1.68195	0.0850597
$392$	4.25666	0.214994
$393$	0	0
$394$	63.2702	3.18751
$395$	9.92819	0.499541
$396$	0	0
$397$	−33.9588	−1.70434	−0.852172	−	0.523262i	$-0.824715\pi$
$397$	−33.9588	−1.70434	−0.852172	+	0.523262i	$0.824715\pi$
$398$	−1.63321	−0.0818656
$399$	0	0
$400$	−12.2219	−0.611095
$401$	3.27288	0.163440	0.0817198	−	0.996655i	$-0.473959\pi$
$401$	3.27288	0.163440	0.0817198	+	0.996655i	$0.473959\pi$
$402$	0	0
$403$	22.1710	1.10442
$404$	−19.7001	−0.980118
$405$	0	0
$406$	−1.24852	−0.0619630
$407$	−35.3902	−1.75423
$408$	0	0
$409$	7.93648	0.392434	0.196217	−	0.980561i	$-0.437134\pi$
$409$	7.93648	0.392434	0.196217	+	0.980561i	$0.437134\pi$
$410$	−4.91326	−0.242649
$411$	0	0
$412$	−3.94988	−0.194597
$413$	1.96260	0.0965733
$414$	0	0
$415$	−9.10498	−0.446946
$416$	−11.9310	−0.584966
$417$	0	0
$418$	43.3092	2.11832
$419$	21.1144	1.03151	0.515754	−	0.856737i	$-0.327512\pi$
$419$	21.1144	1.03151	0.515754	+	0.856737i	$0.327512\pi$
$420$	0	0
$421$	−30.0789	−1.46596	−0.732978	−	0.680252i	$-0.761870\pi$
$421$	−30.0789	−1.46596	−0.732978	+	0.680252i	$0.761870\pi$
$422$	40.9481	1.99332
$423$	0	0
$424$	5.49582	0.266901
$425$	−1.80575	−0.0875916
$426$	0	0
$427$	−9.45403	−0.457513
$428$	−58.2362	−2.81495
$429$	0	0
$430$	−7.68244	−0.370480
$431$	16.0258	0.771938	0.385969	−	0.922512i	$-0.373867\pi$
$431$	16.0258	0.771938	0.385969	+	0.922512i	$0.373867\pi$
$432$	0	0
$433$	7.23386	0.347637	0.173819	−	0.984778i	$-0.444389\pi$
$433$	7.23386	0.347637	0.173819	+	0.984778i	$0.444389\pi$
$434$	9.23199	0.443150
$435$	0	0
$436$	31.0264	1.48589
$437$	15.1411	0.724298
$438$	0	0
$439$	−10.4196	−0.497300	−0.248650	−	0.968593i	$-0.579987\pi$
$439$	−10.4196	−0.497300	−0.248650	+	0.968593i	$0.579987\pi$
$440$	14.3360	0.683444
$441$	0	0
$442$	5.49433	0.261339
$443$	9.43052	0.448057	0.224029	−	0.974583i	$-0.428079\pi$
$443$	9.43052	0.448057	0.224029	+	0.974583i	$0.428079\pi$
$444$	0	0
$445$	8.40129	0.398259
$446$	−49.8457	−2.36026
$447$	0	0
$448$	−10.3337	−0.488223
$449$	6.74268	0.318207	0.159103	−	0.987262i	$-0.449140\pi$
$449$	6.74268	0.318207	0.159103	+	0.987262i	$0.449140\pi$
$450$	0	0
$451$	−15.4983	−0.729788
$452$	9.85954	0.463754
$453$	0	0
$454$	26.3892	1.23851
$455$	3.84628	0.180316
$456$	0	0
$457$	5.04899	0.236182	0.118091	−	0.993003i	$-0.462323\pi$
$457$	5.04899	0.236182	0.118091	+	0.993003i	$0.462323\pi$
$458$	−8.87844	−0.414862
$459$	0	0
$460$	10.6694	0.497466
$461$	12.8408	0.598056	0.299028	−	0.954244i	$-0.403338\pi$
$461$	12.8408	0.598056	0.299028	+	0.954244i	$0.403338\pi$
$462$	0	0
$463$	−7.31274	−0.339852	−0.169926	−	0.985457i	$-0.554353\pi$
$463$	−7.31274	−0.339852	−0.169926	+	0.985457i	$0.554353\pi$
$464$	−1.39423	−0.0647256
$465$	0	0
$466$	64.9378	3.00818
$467$	20.4933	0.948317	0.474159	−	0.880439i	$-0.342752\pi$
$467$	20.4933	0.948317	0.474159	+	0.880439i	$0.342752\pi$
$468$	0	0
$469$	−8.88556	−0.410297
$470$	−18.0443	−0.832322
$471$	0	0
$472$	8.35412	0.384530
$473$	−24.2334	−1.11425
$474$	0	0
$475$	−16.2556	−0.745858
$476$	1.49507	0.0685263
$477$	0	0
$478$	−6.33058	−0.289554
$479$	−42.2742	−1.93156	−0.965778	−	0.259371i	$-0.916485\pi$
$479$	−42.2742	−1.93156	−0.965778	+	0.259371i	$0.916485\pi$
$480$	0	0
$481$	−40.4170	−1.84286
$482$	40.4583	1.84282
$483$	0	0
$484$	54.7777	2.48989
$485$	6.14583	0.279068
$486$	0	0
$487$	11.3695	0.515202	0.257601	−	0.966251i	$-0.417068\pi$
$487$	11.3695	0.515202	0.257601	+	0.966251i	$0.417068\pi$
$488$	−40.2426	−1.82170
$489$	0	0
$490$	1.60159	0.0723523
$491$	8.86723	0.400172	0.200086	−	0.979778i	$-0.435878\pi$
$491$	8.86723	0.400172	0.200086	+	0.979778i	$0.435878\pi$
$492$	0	0
$493$	−0.205993	−0.00927747
$494$	49.4608	2.22534
$495$	0	0
$496$	10.3094	0.462907
$497$	8.51845	0.382105
$498$	0	0
$499$	4.93677	0.221000	0.110500	−	0.993876i	$-0.464755\pi$
$499$	4.93677	0.221000	0.110500	+	0.993876i	$0.464755\pi$
$500$	−24.0270	−1.07452
$501$	0	0
$502$	54.5802	2.43604
$503$	−23.9458	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$503$	−23.9458	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$504$	0	0
$505$	−3.48190	−0.154942
$506$	51.5015	2.28952
$507$	0	0
$508$	3.77179	0.167346
$509$	21.1558	0.937716	0.468858	−	0.883273i	$-0.344666\pi$
$509$	21.1558	0.937716	0.468858	+	0.883273i	$0.344666\pi$
$510$	0	0
$511$	−6.50942	−0.287960
$512$	−28.3877	−1.25457
$513$	0	0
$514$	40.5669	1.78933
$515$	−0.698122	−0.0307629
$516$	0	0
$517$	−56.9188	−2.50329
$518$	−16.8296	−0.739449
$519$	0	0
$520$	16.3723	0.717973
$521$	−28.8671	−1.26469	−0.632346	−	0.774686i	$-0.717908\pi$
$521$	−28.8671	−1.26469	−0.632346	+	0.774686i	$0.717908\pi$
$522$	0	0
$523$	13.2821	0.580785	0.290393	−	0.956908i	$-0.406214\pi$
$523$	13.2821	0.580785	0.290393	+	0.956908i	$0.406214\pi$
$524$	−56.0223	−2.44735
$525$	0	0
$526$	34.8054	1.51759
$527$	1.52319	0.0663510
$528$	0	0
$529$	−4.99479	−0.217165
$530$	2.06783	0.0898207
$531$	0	0
$532$	13.4588	0.583514
$533$	−17.6997	−0.766659
$534$	0	0
$535$	−10.2930	−0.445004
$536$	−37.8228	−1.63370
$537$	0	0
$538$	−45.8616	−1.97723
$539$	5.05203	0.217606
$540$	0	0
$541$	35.5192	1.52709	0.763545	−	0.645754i	$-0.223457\pi$
$541$	35.5192	1.52709	0.763545	+	0.645754i	$0.223457\pi$
$542$	−30.2342	−1.29867
$543$	0	0
$544$	−0.819679	−0.0351435
$545$	5.48375	0.234898
$546$	0	0
$547$	−1.19586	−0.0511314	−0.0255657	−	0.999673i	$-0.508139\pi$
$547$	−1.19586	−0.0511314	−0.0255657	+	0.999673i	$0.508139\pi$
$548$	77.3100	3.30252
$549$	0	0
$550$	−55.2924	−2.35767
$551$	−1.85438	−0.0789993
$552$	0	0
$553$	14.8928	0.633304
$554$	−56.5664	−2.40327
$555$	0	0
$556$	−13.3424	−0.565842
$557$	−0.640333	−0.0271318	−0.0135659	−	0.999908i	$-0.504318\pi$
$557$	−0.640333	−0.0271318	−0.0135659	+	0.999908i	$0.504318\pi$
$558$	0	0
$559$	−27.6755	−1.17055
$560$	1.78850	0.0755781
$561$	0	0
$562$	41.7081	1.75935
$563$	−0.399140	−0.0168217	−0.00841086	−	0.999965i	$-0.502677\pi$
$563$	−0.399140	−0.0168217	−0.00841086	+	0.999965i	$0.502677\pi$
$564$	0	0
$565$	1.74262	0.0733127
$566$	−67.2962	−2.82867
$567$	0	0
$568$	36.2601	1.52144
$569$	10.0883	0.422924	0.211462	−	0.977386i	$-0.432177\pi$
$569$	10.0883	0.422924	0.211462	+	0.977386i	$0.432177\pi$
$570$	0	0
$571$	5.05274	0.211451	0.105725	−	0.994395i	$-0.466284\pi$
$571$	5.05274	0.211451	0.105725	+	0.994395i	$0.466284\pi$
$572$	109.941	4.59687
$573$	0	0
$574$	−7.37013	−0.307623
$575$	−19.3305	−0.806138
$576$	0	0
$577$	−16.5648	−0.689601	−0.344800	−	0.938676i	$-0.612053\pi$
$577$	−16.5648	−0.689601	−0.344800	+	0.938676i	$0.612053\pi$
$578$	−40.4643	−1.68309
$579$	0	0
$580$	−1.30672	−0.0542586
$581$	−13.6579	−0.566626
$582$	0	0
$583$	6.52273	0.270144
$584$	−27.7084	−1.14658
$585$	0	0
$586$	−53.0213	−2.19029
$587$	15.6730	0.646896	0.323448	−	0.946246i	$-0.395158\pi$
$587$	15.6730	0.646896	0.323448	+	0.946246i	$0.395158\pi$
$588$	0	0
$589$	13.7119	0.564991
$590$	3.14328	0.129407
$591$	0	0
$592$	−18.7937	−0.772417
$593$	−2.34319	−0.0962235	−0.0481117	−	0.998842i	$-0.515320\pi$
$593$	−2.34319	−0.0962235	−0.0481117	+	0.998842i	$0.515320\pi$
$594$	0	0
$595$	0.264246	0.0108330
$596$	75.4225	3.08943
$597$	0	0
$598$	58.8167	2.40519
$599$	19.0363	0.777802	0.388901	−	0.921280i	$-0.372855\pi$
$599$	19.0363	0.777802	0.388901	+	0.921280i	$0.372855\pi$
$600$	0	0
$601$	23.0581	0.940560	0.470280	−	0.882517i	$-0.344153\pi$
$601$	23.0581	0.940560	0.470280	+	0.882517i	$0.344153\pi$
$602$	−11.5240	−0.469684
$603$	0	0
$604$	69.0145	2.80816
$605$	9.68168	0.393616
$606$	0	0
$607$	16.5040	0.669878	0.334939	−	0.942240i	$-0.391284\pi$
$607$	16.5040	0.669878	0.334939	+	0.942240i	$0.391284\pi$
$608$	−7.37887	−0.299253
$609$	0	0
$610$	−15.1414	−0.613059
$611$	−65.0034	−2.62976
$612$	0	0
$613$	−13.9856	−0.564872	−0.282436	−	0.959286i	$-0.591142\pi$
$613$	−13.9856	−0.564872	−0.282436	+	0.959286i	$0.591142\pi$
$614$	−42.8891	−1.73086
$615$	0	0
$616$	21.5047	0.866451
$617$	−43.4723	−1.75013	−0.875064	−	0.484007i	$-0.839181\pi$
$617$	−43.4723	−1.75013	−0.875064	+	0.484007i	$0.839181\pi$
$618$	0	0
$619$	40.2578	1.61810	0.809049	−	0.587741i	$-0.199983\pi$
$619$	40.2578	1.61810	0.809049	+	0.587741i	$0.199983\pi$
$620$	9.66235	0.388049
$621$	0	0
$622$	37.8363	1.51710
$623$	12.6023	0.504902
$624$	0	0
$625$	18.5313	0.741251
$626$	−38.1020	−1.52286
$627$	0	0
$628$	−90.1344	−3.59675
$629$	−2.77671	−0.110715
$630$	0	0
$631$	0.191328	0.00761664	0.00380832	−	0.999993i	$-0.498788\pi$
$631$	0.191328	0.00761664	0.00380832	+	0.999993i	$0.498788\pi$
$632$	63.3933	2.52165
$633$	0	0
$634$	−67.6946	−2.68850
$635$	0.666645	0.0264550
$636$	0	0
$637$	5.76961	0.228600
$638$	−6.30756	−0.249719
$639$	0	0
$640$	−13.7932	−0.545226
$641$	−16.7190	−0.660361	−0.330181	−	0.943918i	$-0.607110\pi$
$641$	−16.7190	−0.660361	−0.330181	+	0.943918i	$0.607110\pi$
$642$	0	0
$643$	−27.9484	−1.10218	−0.551088	−	0.834447i	$-0.685787\pi$
$643$	−27.9484	−1.10218	−0.551088	+	0.834447i	$0.685787\pi$
$644$	16.0047	0.630673
$645$	0	0
$646$	3.39803	0.133694
$647$	20.8215	0.818578	0.409289	−	0.912405i	$-0.365777\pi$
$647$	20.8215	0.818578	0.409289	+	0.912405i	$0.365777\pi$
$648$	0	0
$649$	9.91512	0.389202
$650$	−63.1460	−2.47679
$651$	0	0
$652$	−55.7685	−2.18406
$653$	−4.58740	−0.179519	−0.0897594	−	0.995963i	$-0.528610\pi$
$653$	−4.58740	−0.179519	−0.0897594	+	0.995963i	$0.528610\pi$
$654$	0	0
$655$	−9.90167	−0.386890
$656$	−8.23028	−0.321338
$657$	0	0
$658$	−27.0673	−1.05519
$659$	12.5704	0.489674	0.244837	−	0.969564i	$-0.421266\pi$
$659$	12.5704	0.489674	0.244837	+	0.969564i	$0.421266\pi$
$660$	0	0
$661$	25.8123	1.00398	0.501991	−	0.864873i	$-0.332601\pi$
$661$	25.8123	1.00398	0.501991	+	0.864873i	$0.332601\pi$
$662$	−66.9101	−2.60053
$663$	0	0
$664$	−58.1370	−2.25615
$665$	2.37878	0.0922451
$666$	0	0
$667$	−2.20515	−0.0853839
$668$	−80.1333	−3.10045
$669$	0	0
$670$	−14.2310	−0.549791
$671$	−47.7620	−1.84383
$672$	0	0
$673$	−46.6132	−1.79681	−0.898403	−	0.439172i	$-0.855272\pi$
$673$	−46.6132	−1.79681	−0.898403	+	0.439172i	$0.855272\pi$
$674$	−18.3672	−0.707476
$675$	0	0
$676$	76.5235	2.94321
$677$	15.3276	0.589087	0.294544	−	0.955638i	$-0.404832\pi$
$677$	15.3276	0.589087	0.294544	+	0.955638i	$0.404832\pi$
$678$	0	0
$679$	9.21904	0.353794
$680$	1.12480	0.0431342
$681$	0	0
$682$	46.6403	1.78595
$683$	−1.55964	−0.0596778	−0.0298389	−	0.999555i	$-0.509499\pi$
$683$	−1.55964	−0.0596778	−0.0298389	+	0.999555i	$0.509499\pi$
$684$	0	0
$685$	13.6641	0.522080
$686$	2.40246	0.0917262
$687$	0	0
$688$	−12.8690	−0.490625
$689$	7.44921	0.283792
$690$	0	0
$691$	−13.6905	−0.520810	−0.260405	−	0.965499i	$-0.583856\pi$
$691$	−13.6905	−0.520810	−0.260405	+	0.965499i	$0.583856\pi$
$692$	30.3020	1.15191
$693$	0	0
$694$	−53.0951	−2.01546
$695$	−2.35819	−0.0894514
$696$	0	0
$697$	−1.21600	−0.0460592
$698$	17.1157	0.647840
$699$	0	0
$700$	−17.1827	−0.649446
$701$	−35.3818	−1.33635	−0.668176	−	0.744003i	$-0.732925\pi$
$701$	−35.3818	−1.33635	−0.668176	+	0.744003i	$0.732925\pi$
$702$	0	0
$703$	−24.9964	−0.942756
$704$	−52.2063	−1.96760
$705$	0	0
$706$	33.7028	1.26842
$707$	−5.22301	−0.196432
$708$	0	0
$709$	2.44584	0.0918554	0.0459277	−	0.998945i	$-0.485376\pi$
$709$	2.44584	0.0918554	0.0459277	+	0.998945i	$0.485376\pi$
$710$	13.6430	0.512014
$711$	0	0
$712$	53.6438	2.01039
$713$	16.3057	0.610653
$714$	0	0
$715$	19.4315	0.726698
$716$	70.1277	2.62079
$717$	0	0
$718$	−14.5093	−0.541483
$719$	14.0093	0.522459	0.261229	−	0.965277i	$-0.415872\pi$
$719$	14.0093	0.522459	0.261229	+	0.965277i	$0.415872\pi$
$720$	0	0
$721$	−1.04722	−0.0390004
$722$	−15.0571	−0.560366
$723$	0	0
$724$	41.0303	1.52488
$725$	2.36747	0.0879255
$726$	0	0
$727$	5.51126	0.204401	0.102201	−	0.994764i	$-0.467412\pi$
$727$	5.51126	0.204401	0.102201	+	0.994764i	$0.467412\pi$
$728$	24.5592	0.910226
$729$	0	0
$730$	−10.4254	−0.385861
$731$	−1.90135	−0.0703240
$732$	0	0
$733$	−7.42286	−0.274170	−0.137085	−	0.990559i	$-0.543773\pi$
$733$	−7.42286	−0.274170	−0.137085	+	0.990559i	$0.543773\pi$
$734$	−30.4118	−1.12252
$735$	0	0
$736$	−8.77465	−0.323438
$737$	−44.8901	−1.65355
$738$	0	0
$739$	31.5583	1.16089	0.580445	−	0.814299i	$-0.302879\pi$
$739$	31.5583	1.16089	0.580445	+	0.814299i	$0.302879\pi$
$740$	−17.6141	−0.647507
$741$	0	0
$742$	3.10184	0.113872
$743$	34.9462	1.28205	0.641027	−	0.767519i	$-0.278509\pi$
$743$	34.9462	1.28205	0.641027	+	0.767519i	$0.278509\pi$
$744$	0	0
$745$	13.3306	0.488394
$746$	39.2198	1.43594
$747$	0	0
$748$	7.55313	0.276170
$749$	−15.4399	−0.564163
$750$	0	0
$751$	47.6101	1.73732	0.868659	−	0.495410i	$-0.164982\pi$
$751$	47.6101	1.73732	0.868659	+	0.495410i	$0.164982\pi$
$752$	−30.2263	−1.10224
$753$	0	0
$754$	−7.20347	−0.262335
$755$	12.1980	0.443929
$756$	0	0
$757$	−7.55653	−0.274647	−0.137323	−	0.990526i	$-0.543850\pi$
$757$	−7.55653	−0.274647	−0.137323	+	0.990526i	$0.543850\pi$
$758$	38.8700	1.41182
$759$	0	0
$760$	10.1256	0.367296
$761$	2.24334	0.0813209	0.0406605	−	0.999173i	$-0.487054\pi$
$761$	2.24334	0.0813209	0.0406605	+	0.999173i	$0.487054\pi$
$762$	0	0
$763$	8.22589	0.297797
$764$	−71.9950	−2.60469
$765$	0	0
$766$	34.1061	1.23230
$767$	11.3234	0.408866
$768$	0	0
$769$	−31.4607	−1.13450	−0.567250	−	0.823546i	$-0.691993\pi$
$769$	−31.4607	−1.13450	−0.567250	+	0.823546i	$0.691993\pi$
$770$	8.09126	0.291589
$771$	0	0
$772$	−62.3501	−2.24403
$773$	−7.68396	−0.276373	−0.138186	−	0.990406i	$-0.544127\pi$
$773$	−7.68396	−0.276373	−0.138186	+	0.990406i	$0.544127\pi$
$774$	0	0
$775$	−17.5059	−0.628830
$776$	39.2423	1.40872
$777$	0	0
$778$	−3.18096	−0.114043
$779$	−10.9466	−0.392202
$780$	0	0
$781$	43.0354	1.53993
$782$	4.04080	0.144499
$783$	0	0
$784$	2.68284	0.0958157
$785$	−15.9308	−0.568595
$786$	0	0
$787$	−4.63829	−0.165337	−0.0826686	−	0.996577i	$-0.526344\pi$
$787$	−4.63829	−0.165337	−0.0826686	+	0.996577i	$0.526344\pi$
$788$	99.3326	3.53858
$789$	0	0
$790$	23.8520	0.848617
$791$	2.61402	0.0929438
$792$	0	0
$793$	−54.5461	−1.93699
$794$	−81.5845	−2.89533
$795$	0	0
$796$	−2.56410	−0.0908823
$797$	−50.7223	−1.79668	−0.898338	−	0.439304i	$-0.855225\pi$
$797$	−50.7223	−1.79668	−0.898338	+	0.439304i	$0.855225\pi$
$798$	0	0
$799$	−4.46584	−0.157990
$800$	9.42052	0.333066
$801$	0	0
$802$	7.86294	0.277650
$803$	−32.8858	−1.16051
$804$	0	0
$805$	2.82875	0.0997002
$806$	53.2650	1.87618
$807$	0	0
$808$	−22.2326	−0.782139
$809$	8.25319	0.290167	0.145083	−	0.989419i	$-0.453655\pi$
$809$	8.25319	0.290167	0.145083	+	0.989419i	$0.453655\pi$
$810$	0	0
$811$	−42.5309	−1.49346	−0.746731	−	0.665127i	$-0.768378\pi$
$811$	−42.5309	−1.49346	−0.746731	+	0.665127i	$0.768378\pi$
$812$	−1.96014	−0.0687876
$813$	0	0
$814$	−85.0234	−2.98007
$815$	−9.85679	−0.345268
$816$	0	0
$817$	−17.1162	−0.598821
$818$	19.0670	0.666664
$819$	0	0
$820$	−7.71369	−0.269374
$821$	−34.7449	−1.21260	−0.606302	−	0.795235i	$-0.707348\pi$
$821$	−34.7449	−1.21260	−0.606302	+	0.795235i	$0.707348\pi$
$822$	0	0
$823$	6.45981	0.225175	0.112587	−	0.993642i	$-0.464086\pi$
$823$	6.45981	0.225175	0.112587	+	0.993642i	$0.464086\pi$
$824$	−4.45764	−0.155289
$825$	0	0
$826$	4.71506	0.164058
$827$	32.2022	1.11978	0.559889	−	0.828567i	$-0.310844\pi$
$827$	32.2022	1.11978	0.559889	+	0.828567i	$0.310844\pi$
$828$	0	0
$829$	17.8107	0.618593	0.309296	−	0.950966i	$-0.399906\pi$
$829$	17.8107	0.618593	0.309296	+	0.950966i	$0.399906\pi$
$830$	−21.8743	−0.759269
$831$	0	0
$832$	−59.6216	−2.06701
$833$	0.396381	0.0137338
$834$	0	0
$835$	−14.1632	−0.490136
$836$	67.9943	2.35163
$837$	0	0
$838$	50.7265	1.75232
$839$	−11.6047	−0.400639	−0.200319	−	0.979731i	$-0.564198\pi$
$839$	−11.6047	−0.400639	−0.200319	+	0.979731i	$0.564198\pi$
$840$	0	0
$841$	−28.7299	−0.990687
$842$	−72.2633	−2.49036
$843$	0	0
$844$	64.2875	2.21287
$845$	13.5251	0.465279
$846$	0	0
$847$	14.5230	0.499016
$848$	3.46385	0.118949
$849$	0	0
$850$	−4.33823	−0.148800
$851$	−29.7246	−1.01895
$852$	0	0
$853$	34.3397	1.17577	0.587884	−	0.808945i	$-0.299961\pi$
$853$	34.3397	1.17577	0.587884	+	0.808945i	$0.299961\pi$
$854$	−22.7129	−0.777219
$855$	0	0
$856$	−65.7225	−2.24635
$857$	−14.3054	−0.488664	−0.244332	−	0.969692i	$-0.578569\pi$
$857$	−14.3054	−0.488664	−0.244332	+	0.969692i	$0.578569\pi$
$858$	0	0
$859$	−49.4636	−1.68768	−0.843838	−	0.536597i	$-0.819709\pi$
$859$	−49.4636	−1.68768	−0.843838	+	0.536597i	$0.819709\pi$
$860$	−12.0612	−0.411285
$861$	0	0
$862$	38.5014	1.31136
$863$	39.0263	1.32847	0.664235	−	0.747524i	$-0.268757\pi$
$863$	39.0263	1.32847	0.664235	+	0.747524i	$0.268757\pi$
$864$	0	0
$865$	5.35572	0.182100
$866$	17.3790	0.590564
$867$	0	0
$868$	14.4940	0.491958
$869$	75.2386	2.55229
$870$	0	0
$871$	−51.2662	−1.73709
$872$	35.0148	1.18575
$873$	0	0
$874$	36.3759	1.23043
$875$	−6.37019	−0.215352
$876$	0	0
$877$	−32.8794	−1.11026	−0.555129	−	0.831764i	$-0.687331\pi$
$877$	−32.8794	−1.11026	−0.555129	+	0.831764i	$0.687331\pi$
$878$	−25.0326	−0.844809
$879$	0	0
$880$	9.03557	0.304589
$881$	23.5044	0.791884	0.395942	−	0.918275i	$-0.370418\pi$
$881$	23.5044	0.791884	0.395942	+	0.918275i	$0.370418\pi$
$882$	0	0
$883$	3.78292	0.127305	0.0636527	−	0.997972i	$-0.479725\pi$
$883$	3.78292	0.127305	0.0636527	+	0.997972i	$0.479725\pi$
$884$	8.62596	0.290122
$885$	0	0
$886$	22.6564	0.761156
$887$	−16.0603	−0.539253	−0.269627	−	0.962965i	$-0.586900\pi$
$887$	−16.0603	−0.539253	−0.269627	+	0.962965i	$0.586900\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	20.1837	0.676560
$891$	0	0
$892$	−78.2565	−2.62022
$893$	−40.2021	−1.34531
$894$	0	0
$895$	12.3947	0.414309
$896$	−20.6905	−0.691222
$897$	0	0
$898$	16.1990	0.540567
$899$	−1.99701	−0.0666040
$900$	0	0
$901$	0.511773	0.0170496
$902$	−37.2341	−1.23976
$903$	0	0
$904$	11.1270	0.370078
$905$	7.25189	0.241061
$906$	0	0
$907$	−3.13873	−0.104220	−0.0521099	−	0.998641i	$-0.516595\pi$
$907$	−3.13873	−0.104220	−0.0521099	+	0.998641i	$0.516595\pi$
$908$	41.4304	1.37491
$909$	0	0
$910$	9.24052	0.306320
$911$	−5.56286	−0.184306	−0.0921530	−	0.995745i	$-0.529375\pi$
$911$	−5.56286	−0.184306	−0.0921530	+	0.995745i	$0.529375\pi$
$912$	0	0
$913$	−69.0001	−2.28357
$914$	12.1300	0.401224
$915$	0	0
$916$	−13.9389	−0.460555
$917$	−14.8530	−0.490488
$918$	0	0
$919$	−55.9312	−1.84500	−0.922501	−	0.385995i	$-0.873858\pi$
$919$	−55.9312	−1.84500	−0.922501	+	0.385995i	$0.873858\pi$
$920$	12.0410	0.396980
$921$	0	0
$922$	30.8494	1.01597
$923$	49.1481	1.61773
$924$	0	0
$925$	31.9126	1.04928
$926$	−17.5685	−0.577338
$927$	0	0
$928$	1.07466	0.0352774
$929$	−15.0146	−0.492612	−0.246306	−	0.969192i	$-0.579217\pi$
$929$	−15.0146	−0.492612	−0.246306	+	0.969192i	$0.579217\pi$
$930$	0	0
$931$	3.56828	0.116946
$932$	101.951	3.33950
$933$	0	0
$934$	49.2343	1.61099
$935$	1.33498	0.0436584
$936$	0	0
$937$	6.36422	0.207910	0.103955	−	0.994582i	$-0.466850\pi$
$937$	6.36422	0.207910	0.103955	+	0.994582i	$0.466850\pi$
$938$	−21.3472	−0.697010
$939$	0	0
$940$	−28.3291	−0.923994
$941$	40.4525	1.31871	0.659357	−	0.751830i	$-0.270829\pi$
$941$	40.4525	1.31871	0.659357	+	0.751830i	$0.270829\pi$
$942$	0	0
$943$	−13.0172	−0.423899
$944$	5.26535	0.171372
$945$	0	0
$946$	−58.2197	−1.89289
$947$	48.6835	1.58200	0.791000	−	0.611816i	$-0.209561\pi$
$947$	48.6835	1.58200	0.791000	+	0.611816i	$0.209561\pi$
$948$	0	0
$949$	−37.5568	−1.21915
$950$	−39.0534	−1.26706
$951$	0	0
$952$	1.68726	0.0546844
$953$	−2.87131	−0.0930110	−0.0465055	−	0.998918i	$-0.514809\pi$
$953$	−2.87131	−0.0930110	−0.0465055	+	0.998918i	$0.514809\pi$
$954$	0	0
$955$	−12.7247	−0.411763
$956$	−9.93885	−0.321445
$957$	0	0
$958$	−101.562	−3.28131
$959$	20.4969	0.661878
$960$	0	0
$961$	−16.2334	−0.523659
$962$	−97.1000	−3.13063
$963$	0	0
$964$	63.5185	2.04579
$965$	−11.0201	−0.354749
$966$	0	0
$967$	45.1711	1.45261	0.726303	−	0.687375i	$-0.241237\pi$
$967$	45.1711	1.45261	0.726303	+	0.687375i	$0.241237\pi$
$968$	61.8193	1.98695
$969$	0	0
$970$	14.7651	0.474079
$971$	30.5373	0.979990	0.489995	−	0.871725i	$-0.336999\pi$
$971$	30.5373	0.979990	0.489995	+	0.871725i	$0.336999\pi$
$972$	0	0
$973$	−3.53740	−0.113404
$974$	27.3147	0.875221
$975$	0	0
$976$	−25.3637	−0.811871
$977$	51.3484	1.64278	0.821391	−	0.570365i	$-0.193198\pi$
$977$	51.3484	1.64278	0.821391	+	0.570365i	$0.193198\pi$
$978$	0	0
$979$	63.6674	2.03482
$980$	2.51445	0.0803211
$981$	0	0
$982$	21.3031	0.679810
$983$	−17.9373	−0.572111	−0.286055	−	0.958213i	$-0.592344\pi$
$983$	−17.9373	−0.572111	−0.286055	+	0.958213i	$0.592344\pi$
$984$	0	0
$985$	17.5565	0.559398
$986$	−0.494890	−0.0157605
$987$	0	0
$988$	77.6521	2.47044
$989$	−20.3539	−0.647217
$990$	0	0
$991$	1.31878	0.0418924	0.0209462	−	0.999781i	$-0.493332\pi$
$991$	1.31878	0.0418924	0.0209462	+	0.999781i	$0.493332\pi$
$992$	−7.94641	−0.252299
$993$	0	0
$994$	20.4652	0.649116
$995$	−0.453193	−0.0143672
$996$	0	0
$997$	−4.64514	−0.147113	−0.0735565	−	0.997291i	$-0.523435\pi$
$997$	−4.64514	−0.147113	−0.0735565	+	0.997291i	$0.523435\pi$
$998$	11.8604	0.375433
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.15		18
3.2	odd	2		2667.2.a.p.1.4	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.4	✓	18	3.2	odd	2
8001.2.a.u.1.15		18	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)

\(f(q)\)	\(=\)	\(q+2.40246 q^{2} +3.77179 q^{4} +0.666645 q^{5} +1.00000 q^{7} +4.25666 q^{8} +O(q^{10})\)	\(q+2.40246 q^{2} +3.77179 q^{4} +0.666645 q^{5} +1.00000 q^{7} +4.25666 q^{8} +1.60159 q^{10} +5.05203 q^{11} +5.76961 q^{13} +2.40246 q^{14} +2.68284 q^{16} +0.396381 q^{17} +3.56828 q^{19} +2.51445 q^{20} +12.1373 q^{22} +4.24325 q^{23} -4.55558 q^{25} +13.8612 q^{26} +3.77179 q^{28} -0.519685 q^{29} +3.84273 q^{31} -2.06791 q^{32} +0.952288 q^{34} +0.666645 q^{35} -7.00515 q^{37} +8.57264 q^{38} +2.83768 q^{40} -3.06775 q^{41} -4.79677 q^{43} +19.0552 q^{44} +10.1942 q^{46} -11.2665 q^{47} +1.00000 q^{49} -10.9446 q^{50} +21.7618 q^{52} +1.29111 q^{53} +3.36791 q^{55} +4.25666 q^{56} -1.24852 q^{58} +1.96260 q^{59} -9.45403 q^{61} +9.23199 q^{62} -10.3337 q^{64} +3.84628 q^{65} -8.88556 q^{67} +1.49507 q^{68} +1.60159 q^{70} +8.51845 q^{71} -6.50942 q^{73} -16.8296 q^{74} +13.4588 q^{76} +5.05203 q^{77} +14.8928 q^{79} +1.78850 q^{80} -7.37013 q^{82} -13.6579 q^{83} +0.264246 q^{85} -11.5240 q^{86} +21.5047 q^{88} +12.6023 q^{89} +5.76961 q^{91} +16.0047 q^{92} -27.0673 q^{94} +2.37878 q^{95} +9.21904 q^{97} +2.40246 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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