LMFDB - Embedded newform 8001.2.a.o.1.12

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 $ x^13 - 4x^12 - 10x^11 + 53x^10 + 19x^9 - 242x^8 + 61x^7 + 467x^6 - 211x^5 - 372x^4 + 146x^3 + 116x^2 - 12x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
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			1.12
Root			$-1.82297$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82297 q^{2} +1.32323 q^{4} -0.630622 q^{5} +1.00000 q^{7} -1.23374 q^{8} +O(q^{10})$	\(q+1.82297 q^{2} +1.32323 q^{4} -0.630622 q^{5} +1.00000 q^{7} -1.23374 q^{8} -1.14961 q^{10} +2.60824 q^{11} -4.34638 q^{13} +1.82297 q^{14} -4.89553 q^{16} +1.67481 q^{17} +7.62672 q^{19} -0.834455 q^{20} +4.75474 q^{22} -4.50538 q^{23} -4.60232 q^{25} -7.92333 q^{26} +1.32323 q^{28} +1.00235 q^{29} -6.18227 q^{31} -6.45692 q^{32} +3.05313 q^{34} -0.630622 q^{35} -0.667280 q^{37} +13.9033 q^{38} +0.778025 q^{40} -8.44630 q^{41} +0.767479 q^{43} +3.45128 q^{44} -8.21318 q^{46} -1.27605 q^{47} +1.00000 q^{49} -8.38989 q^{50} -5.75124 q^{52} -9.23546 q^{53} -1.64481 q^{55} -1.23374 q^{56} +1.82725 q^{58} -3.99665 q^{59} -6.45858 q^{61} -11.2701 q^{62} -1.97973 q^{64} +2.74093 q^{65} +5.31265 q^{67} +2.21615 q^{68} -1.14961 q^{70} +9.52764 q^{71} +4.29370 q^{73} -1.21643 q^{74} +10.0919 q^{76} +2.60824 q^{77} -2.53600 q^{79} +3.08723 q^{80} -15.3974 q^{82} +4.96788 q^{83} -1.05617 q^{85} +1.39909 q^{86} -3.21789 q^{88} +0.366460 q^{89} -4.34638 q^{91} -5.96163 q^{92} -2.32620 q^{94} -4.80958 q^{95} -10.1853 q^{97} +1.82297 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * 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 $

$2$	1.82297	1.28904	0.644518	−	0.764589i	$-0.277058\pi$
$2$	1.82297	1.28904	0.644518	+	0.764589i	$0.277058\pi$
$3$	0	0
$3$	0	0
$4$	1.32323	0.661613
$5$	−0.630622	−0.282023	−0.141011	−	0.990008i	$-0.545035\pi$
$5$	−0.630622	−0.282023	−0.141011	+	0.990008i	$0.545035\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.23374	−0.436193
$9$	0	0
$10$	−1.14961	−0.363537
$11$	2.60824	0.786413	0.393206	−	0.919450i	$-0.371366\pi$
$11$	2.60824	0.786413	0.393206	+	0.919450i	$0.371366\pi$
$12$	0	0
$13$	−4.34638	−1.20547	−0.602735	−	0.797942i	$-0.705922\pi$
$13$	−4.34638	−1.20547	−0.602735	+	0.797942i	$0.705922\pi$
$14$	1.82297	0.487210
$15$	0	0
$16$	−4.89553	−1.22388
$17$	1.67481	0.406201	0.203100	−	0.979158i	$-0.434898\pi$
$17$	1.67481	0.406201	0.203100	+	0.979158i	$0.434898\pi$
$18$	0	0
$19$	7.62672	1.74969	0.874845	−	0.484403i	$-0.160963\pi$
$19$	7.62672	1.74969	0.874845	+	0.484403i	$0.160963\pi$
$20$	−0.834455	−0.186590
$21$	0	0
$22$	4.75474	1.01371
$23$	−4.50538	−0.939437	−0.469718	−	0.882816i	$-0.655645\pi$
$23$	−4.50538	−0.939437	−0.469718	+	0.882816i	$0.655645\pi$
$24$	0	0
$25$	−4.60232	−0.920463
$26$	−7.92333	−1.55389
$27$	0	0
$28$	1.32323	0.250066
$29$	1.00235	0.186131	0.0930655	−	0.995660i	$-0.470333\pi$
$29$	1.00235	0.186131	0.0930655	+	0.995660i	$0.470333\pi$
$30$	0	0
$31$	−6.18227	−1.11037	−0.555184	−	0.831727i	$-0.687352\pi$
$31$	−6.18227	−1.11037	−0.555184	+	0.831727i	$0.687352\pi$
$32$	−6.45692	−1.14143
$33$	0	0
$34$	3.05313	0.523607
$35$	−0.630622	−0.106595
$36$	0	0
$37$	−0.667280	−0.109700	−0.0548501	−	0.998495i	$-0.517468\pi$
$37$	−0.667280	−0.109700	−0.0548501	+	0.998495i	$0.517468\pi$
$38$	13.9033	2.25541
$39$	0	0
$40$	0.778025	0.123017
$41$	−8.44630	−1.31909	−0.659545	−	0.751665i	$-0.729251\pi$
$41$	−8.44630	−1.31909	−0.659545	+	0.751665i	$0.729251\pi$
$42$	0	0
$43$	0.767479	0.117039	0.0585197	−	0.998286i	$-0.481362\pi$
$43$	0.767479	0.117039	0.0585197	+	0.998286i	$0.481362\pi$
$44$	3.45128	0.520301
$45$	0	0
$46$	−8.21318	−1.21097
$47$	−1.27605	−0.186131	−0.0930656	−	0.995660i	$-0.529667\pi$
$47$	−1.27605	−0.186131	−0.0930656	+	0.995660i	$0.529667\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.38989	−1.18651
$51$	0	0
$52$	−5.75124	−0.797554
$53$	−9.23546	−1.26859	−0.634294	−	0.773092i	$-0.718709\pi$
$53$	−9.23546	−1.26859	−0.634294	+	0.773092i	$0.718709\pi$
$54$	0	0
$55$	−1.64481	−0.221786
$56$	−1.23374	−0.164866
$57$	0	0
$58$	1.82725	0.239929
$59$	−3.99665	−0.520320	−0.260160	−	0.965566i	$-0.583775\pi$
$59$	−3.99665	−0.520320	−0.260160	+	0.965566i	$0.583775\pi$
$60$	0	0
$61$	−6.45858	−0.826937	−0.413468	−	0.910518i	$-0.635683\pi$
$61$	−6.45858	−0.826937	−0.413468	+	0.910518i	$0.635683\pi$
$62$	−11.2701	−1.43130
$63$	0	0
$64$	−1.97973	−0.247467
$65$	2.74093	0.339970
$66$	0	0
$67$	5.31265	0.649043	0.324522	−	0.945878i	$-0.394797\pi$
$67$	5.31265	0.649043	0.324522	+	0.945878i	$0.394797\pi$
$68$	2.21615	0.268748
$69$	0	0
$70$	−1.14961	−0.137404
$71$	9.52764	1.13072	0.565362	−	0.824843i	$-0.308737\pi$
$71$	9.52764	1.13072	0.565362	+	0.824843i	$0.308737\pi$
$72$	0	0
$73$	4.29370	0.502539	0.251270	−	0.967917i	$-0.419152\pi$
$73$	4.29370	0.502539	0.251270	+	0.967917i	$0.419152\pi$
$74$	−1.21643	−0.141407
$75$	0	0
$76$	10.0919	1.15762
$77$	2.60824	0.297236
$78$	0	0
$79$	−2.53600	−0.285322	−0.142661	−	0.989772i	$-0.545566\pi$
$79$	−2.53600	−0.285322	−0.142661	+	0.989772i	$0.545566\pi$
$80$	3.08723	0.345163
$81$	0	0
$82$	−15.3974	−1.70035
$83$	4.96788	0.545296	0.272648	−	0.962114i	$-0.412101\pi$
$83$	4.96788	0.545296	0.272648	+	0.962114i	$0.412101\pi$
$84$	0	0
$85$	−1.05617	−0.114558
$86$	1.39909	0.150868
$87$	0	0
$88$	−3.21789	−0.343028
$89$	0.366460	0.0388447	0.0194223	−	0.999811i	$-0.493817\pi$
$89$	0.366460	0.0388447	0.0194223	+	0.999811i	$0.493817\pi$
$90$	0	0
$91$	−4.34638	−0.455625
$92$	−5.96163	−0.621543
$93$	0	0
$94$	−2.32620	−0.239930
$95$	−4.80958	−0.493452
$96$	0	0
$97$	−10.1853	−1.03416	−0.517081	−	0.855936i	$-0.672982\pi$
$97$	−10.1853	−1.03416	−0.517081	+	0.855936i	$0.672982\pi$
$98$	1.82297	0.184148
$99$	0	0
$100$	−6.08990	−0.608990
$101$	5.28628	0.526005	0.263002	−	0.964795i	$-0.415287\pi$
$101$	5.28628	0.526005	0.263002	+	0.964795i	$0.415287\pi$
$102$	0	0
$103$	7.16287	0.705779	0.352890	−	0.935665i	$-0.385199\pi$
$103$	7.16287	0.705779	0.352890	+	0.935665i	$0.385199\pi$
$104$	5.36231	0.525818
$105$	0	0
$106$	−16.8360	−1.63526
$107$	6.68923	0.646672	0.323336	−	0.946284i	$-0.395196\pi$
$107$	6.68923	0.646672	0.323336	+	0.946284i	$0.395196\pi$
$108$	0	0
$109$	−3.17404	−0.304018	−0.152009	−	0.988379i	$-0.548574\pi$
$109$	−3.17404	−0.304018	−0.152009	+	0.988379i	$0.548574\pi$
$110$	−2.99845	−0.285891
$111$	0	0
$112$	−4.89553	−0.462584
$113$	−2.64523	−0.248843	−0.124421	−	0.992229i	$-0.539707\pi$
$113$	−2.64523	−0.248843	−0.124421	+	0.992229i	$0.539707\pi$
$114$	0	0
$115$	2.84119	0.264943
$116$	1.32633	0.123147
$117$	0	0
$118$	−7.28578	−0.670710
$119$	1.67481	0.153529
$120$	0	0
$121$	−4.19710	−0.381555
$122$	−11.7738	−1.06595
$123$	0	0
$124$	−8.18053	−0.734634
$125$	6.05543	0.541615
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	9.30485	0.822440
$129$	0	0
$130$	4.99663	0.438233
$131$	−20.6980	−1.80840	−0.904198	−	0.427114i	$-0.859530\pi$
$131$	−20.6980	−1.80840	−0.904198	+	0.427114i	$0.859530\pi$
$132$	0	0
$133$	7.62672	0.661321
$134$	9.68481	0.836640
$135$	0	0
$136$	−2.06628	−0.177182
$137$	−13.2703	−1.13376	−0.566878	−	0.823802i	$-0.691849\pi$
$137$	−13.2703	−1.13376	−0.566878	+	0.823802i	$0.691849\pi$
$138$	0	0
$139$	−11.5563	−0.980193	−0.490096	−	0.871668i	$-0.663038\pi$
$139$	−11.5563	−0.980193	−0.490096	+	0.871668i	$0.663038\pi$
$140$	−0.834455	−0.0705243
$141$	0	0
$142$	17.3686	1.45754
$143$	−11.3364	−0.947997
$144$	0	0
$145$	−0.632102	−0.0524932
$146$	7.82729	0.647791
$147$	0	0
$148$	−0.882962	−0.0725790
$149$	−19.7584	−1.61867	−0.809334	−	0.587349i	$-0.800172\pi$
$149$	−19.7584	−1.61867	−0.809334	+	0.587349i	$0.800172\pi$
$150$	0	0
$151$	6.99658	0.569373	0.284687	−	0.958621i	$-0.408110\pi$
$151$	6.99658	0.569373	0.284687	+	0.958621i	$0.408110\pi$
$152$	−9.40940	−0.763203
$153$	0	0
$154$	4.75474	0.383148
$155$	3.89868	0.313149
$156$	0	0
$157$	8.51770	0.679787	0.339893	−	0.940464i	$-0.389609\pi$
$157$	8.51770	0.679787	0.339893	+	0.940464i	$0.389609\pi$
$158$	−4.62305	−0.367790
$159$	0	0
$160$	4.07188	0.321910
$161$	−4.50538	−0.355074
$162$	0	0
$163$	−8.81012	−0.690062	−0.345031	−	0.938591i	$-0.612132\pi$
$163$	−8.81012	−0.690062	−0.345031	+	0.938591i	$0.612132\pi$
$164$	−11.1764	−0.872727
$165$	0	0
$166$	9.05631	0.702906
$167$	−9.41023	−0.728186	−0.364093	−	0.931363i	$-0.618621\pi$
$167$	−9.41023	−0.728186	−0.364093	+	0.931363i	$0.618621\pi$
$168$	0	0
$169$	5.89104	0.453157
$170$	−1.92537	−0.147669
$171$	0	0
$172$	1.01555	0.0774348
$173$	−25.9259	−1.97111	−0.985554	−	0.169360i	$-0.945830\pi$
$173$	−25.9259	−1.97111	−0.985554	+	0.169360i	$0.945830\pi$
$174$	0	0
$175$	−4.60232	−0.347902
$176$	−12.7687	−0.962476
$177$	0	0
$178$	0.668046	0.0500721
$179$	−20.8930	−1.56162	−0.780809	−	0.624770i	$-0.785193\pi$
$179$	−20.8930	−1.56162	−0.780809	+	0.624770i	$0.785193\pi$
$180$	0	0
$181$	−17.6229	−1.30990	−0.654950	−	0.755672i	$-0.727310\pi$
$181$	−17.6229	−1.30990	−0.654950	+	0.755672i	$0.727310\pi$
$182$	−7.92333	−0.587316
$183$	0	0
$184$	5.55847	0.409776
$185$	0.420802	0.0309380
$186$	0	0
$187$	4.36830	0.319442
$188$	−1.68850	−0.123147
$189$	0	0
$190$	−8.76773	−0.636078
$191$	−4.36797	−0.316055	−0.158028	−	0.987435i	$-0.550514\pi$
$191$	−4.36797	−0.316055	−0.158028	+	0.987435i	$0.550514\pi$
$192$	0	0
$193$	−8.07471	−0.581230	−0.290615	−	0.956840i	$-0.593860\pi$
$193$	−8.07471	−0.581230	−0.290615	+	0.956840i	$0.593860\pi$
$194$	−18.5676	−1.33307
$195$	0	0
$196$	1.32323	0.0945161
$197$	−19.0457	−1.35695	−0.678474	−	0.734625i	$-0.737358\pi$
$197$	−19.0457	−1.35695	−0.678474	+	0.734625i	$0.737358\pi$
$198$	0	0
$199$	24.0658	1.70598	0.852990	−	0.521927i	$-0.174787\pi$
$199$	24.0658	1.70598	0.852990	+	0.521927i	$0.174787\pi$
$200$	5.67807	0.401500
$201$	0	0
$202$	9.63674	0.678039
$203$	1.00235	0.0703509
$204$	0	0
$205$	5.32642	0.372014
$206$	13.0577	0.909774
$207$	0	0
$208$	21.2778	1.47535
$209$	19.8923	1.37598
$210$	0	0
$211$	−11.4713	−0.789720	−0.394860	−	0.918741i	$-0.629207\pi$
$211$	−11.4713	−0.789720	−0.394860	+	0.918741i	$0.629207\pi$
$212$	−12.2206	−0.839314
$213$	0	0
$214$	12.1943	0.833583
$215$	−0.483989	−0.0330078
$216$	0	0
$217$	−6.18227	−0.419680
$218$	−5.78618	−0.391889
$219$	0	0
$220$	−2.17646	−0.146737
$221$	−7.27936	−0.489663
$222$	0	0
$223$	29.3326	1.96426	0.982129	−	0.188211i	$-0.0602687\pi$
$223$	29.3326	1.96426	0.982129	+	0.188211i	$0.0602687\pi$
$224$	−6.45692	−0.431421
$225$	0	0
$226$	−4.82219	−0.320767
$227$	16.9327	1.12386	0.561930	−	0.827185i	$-0.310059\pi$
$227$	16.9327	1.12386	0.561930	+	0.827185i	$0.310059\pi$
$228$	0	0
$229$	29.4027	1.94298	0.971492	−	0.237070i	$-0.0761872\pi$
$229$	29.4027	1.94298	0.971492	+	0.237070i	$0.0761872\pi$
$230$	5.17942	0.341521
$231$	0	0
$232$	−1.23664	−0.0811891
$233$	−3.31695	−0.217301	−0.108650	−	0.994080i	$-0.534653\pi$
$233$	−3.31695	−0.217301	−0.108650	+	0.994080i	$0.534653\pi$
$234$	0	0
$235$	0.804706	0.0524932
$236$	−5.28847	−0.344250
$237$	0	0
$238$	3.05313	0.197905
$239$	1.52032	0.0983415	0.0491707	−	0.998790i	$-0.484342\pi$
$239$	1.52032	0.0983415	0.0491707	+	0.998790i	$0.484342\pi$
$240$	0	0
$241$	−9.95634	−0.641344	−0.320672	−	0.947190i	$-0.603909\pi$
$241$	−9.95634	−0.641344	−0.320672	+	0.947190i	$0.603909\pi$
$242$	−7.65120	−0.491837
$243$	0	0
$244$	−8.54616	−0.547112
$245$	−0.630622	−0.0402890
$246$	0	0
$247$	−33.1486	−2.10920
$248$	7.62732	0.484335
$249$	0	0
$250$	11.0389	0.698160
$251$	9.73220	0.614291	0.307146	−	0.951663i	$-0.400626\pi$
$251$	9.73220	0.614291	0.307146	+	0.951663i	$0.400626\pi$
$252$	0	0
$253$	−11.7511	−0.738785
$254$	−1.82297	−0.114383
$255$	0	0
$256$	20.9219	1.30762
$257$	−27.0819	−1.68932	−0.844661	−	0.535302i	$-0.820198\pi$
$257$	−27.0819	−1.68932	−0.844661	+	0.535302i	$0.820198\pi$
$258$	0	0
$259$	−0.667280	−0.0414628
$260$	3.62686	0.224928
$261$	0	0
$262$	−37.7319	−2.33109
$263$	−4.38851	−0.270607	−0.135304	−	0.990804i	$-0.543201\pi$
$263$	−4.38851	−0.270607	−0.135304	+	0.990804i	$0.543201\pi$
$264$	0	0
$265$	5.82409	0.357771
$266$	13.9033	0.852466
$267$	0	0
$268$	7.02983	0.429415
$269$	−15.3450	−0.935604	−0.467802	−	0.883833i	$-0.654954\pi$
$269$	−15.3450	−0.935604	−0.467802	+	0.883833i	$0.654954\pi$
$270$	0	0
$271$	−4.68264	−0.284450	−0.142225	−	0.989834i	$-0.545426\pi$
$271$	−4.68264	−0.284450	−0.142225	+	0.989834i	$0.545426\pi$
$272$	−8.19907	−0.497142
$273$	0	0
$274$	−24.1913	−1.46145
$275$	−12.0039	−0.723864
$276$	0	0
$277$	−16.7745	−1.00788	−0.503942	−	0.863738i	$-0.668117\pi$
$277$	−16.7745	−1.00788	−0.503942	+	0.863738i	$0.668117\pi$
$278$	−21.0668	−1.26350
$279$	0	0
$280$	0.778025	0.0464959
$281$	−25.1133	−1.49813	−0.749066	−	0.662495i	$-0.769498\pi$
$281$	−25.1133	−1.49813	−0.749066	+	0.662495i	$0.769498\pi$
$282$	0	0
$283$	14.7143	0.874674	0.437337	−	0.899298i	$-0.355922\pi$
$283$	14.7143	0.874674	0.437337	+	0.899298i	$0.355922\pi$
$284$	12.6072	0.748101
$285$	0	0
$286$	−20.6659	−1.22200
$287$	−8.44630	−0.498569
$288$	0	0
$289$	−14.1950	−0.835001
$290$	−1.15230	−0.0676656
$291$	0	0
$292$	5.68153	0.332486
$293$	−0.0711543	−0.00415688	−0.00207844	−	0.999998i	$-0.500662\pi$
$293$	−0.0711543	−0.00415688	−0.00207844	+	0.999998i	$0.500662\pi$
$294$	0	0
$295$	2.52038	0.146742
$296$	0.823251	0.0478505
$297$	0	0
$298$	−36.0189	−2.08652
$299$	19.5821	1.13246
$300$	0	0
$301$	0.767479	0.0442367
$302$	12.7546	0.733942
$303$	0	0
$304$	−37.3368	−2.14141
$305$	4.07293	0.233215
$306$	0	0
$307$	13.3153	0.759942	0.379971	−	0.924998i	$-0.375934\pi$
$307$	13.3153	0.759942	0.379971	+	0.924998i	$0.375934\pi$
$308$	3.45128	0.196655
$309$	0	0
$310$	7.10718	0.403661
$311$	1.35589	0.0768855	0.0384427	−	0.999261i	$-0.487760\pi$
$311$	1.35589	0.0768855	0.0384427	+	0.999261i	$0.487760\pi$
$312$	0	0
$313$	14.3146	0.809108	0.404554	−	0.914514i	$-0.367427\pi$
$313$	14.3146	0.809108	0.404554	+	0.914514i	$0.367427\pi$
$314$	15.5275	0.876269
$315$	0	0
$316$	−3.35569	−0.188773
$317$	−2.01097	−0.112948	−0.0564738	−	0.998404i	$-0.517986\pi$
$317$	−2.01097	−0.112948	−0.0564738	+	0.998404i	$0.517986\pi$
$318$	0	0
$319$	2.61435	0.146376
$320$	1.24846	0.0697912
$321$	0	0
$322$	−8.21318	−0.457703
$323$	12.7733	0.710725
$324$	0	0
$325$	20.0034	1.10959
$326$	−16.0606	−0.889514
$327$	0	0
$328$	10.4205	0.575378
$329$	−1.27605	−0.0703510
$330$	0	0
$331$	−1.88305	−0.103502	−0.0517508	−	0.998660i	$-0.516480\pi$
$331$	−1.88305	−0.103502	−0.0517508	+	0.998660i	$0.516480\pi$
$332$	6.57363	0.360775
$333$	0	0
$334$	−17.1546	−0.938657
$335$	−3.35028	−0.183045
$336$	0	0
$337$	1.13790	0.0619855	0.0309928	−	0.999520i	$-0.490133\pi$
$337$	1.13790	0.0619855	0.0309928	+	0.999520i	$0.490133\pi$
$338$	10.7392	0.584136
$339$	0	0
$340$	−1.39755	−0.0757930
$341$	−16.1248	−0.873208
$342$	0	0
$343$	1.00000	0.0539949
$344$	−0.946870	−0.0510518
$345$	0	0
$346$	−47.2621	−2.54083
$347$	−10.1578	−0.545299	−0.272649	−	0.962113i	$-0.587900\pi$
$347$	−10.1578	−0.545299	−0.272649	+	0.962113i	$0.587900\pi$
$348$	0	0
$349$	1.60321	0.0858179	0.0429089	−	0.999079i	$-0.486337\pi$
$349$	1.60321	0.0858179	0.0429089	+	0.999079i	$0.486337\pi$
$350$	−8.38989	−0.448458
$351$	0	0
$352$	−16.8412	−0.897638
$353$	2.04982	0.109101	0.0545505	−	0.998511i	$-0.482627\pi$
$353$	2.04982	0.109101	0.0545505	+	0.998511i	$0.482627\pi$
$354$	0	0
$355$	−6.00835	−0.318890
$356$	0.484909	0.0257001
$357$	0	0
$358$	−38.0874	−2.01298
$359$	7.82477	0.412976	0.206488	−	0.978449i	$-0.433797\pi$
$359$	7.82477	0.412976	0.206488	+	0.978449i	$0.433797\pi$
$360$	0	0
$361$	39.1669	2.06141
$362$	−32.1261	−1.68851
$363$	0	0
$364$	−5.75124	−0.301447
$365$	−2.70770	−0.141728
$366$	0	0
$367$	19.6337	1.02487	0.512435	−	0.858726i	$-0.328744\pi$
$367$	19.6337	1.02487	0.512435	+	0.858726i	$0.328744\pi$
$368$	22.0562	1.14976
$369$	0	0
$370$	0.767110	0.0398801
$371$	−9.23546	−0.479481
$372$	0	0
$373$	12.5065	0.647561	0.323780	−	0.946132i	$-0.395046\pi$
$373$	12.5065	0.647561	0.323780	+	0.946132i	$0.395046\pi$
$374$	7.96328	0.411772
$375$	0	0
$376$	1.57432	0.0811892
$377$	−4.35658	−0.224375
$378$	0	0
$379$	34.4032	1.76717	0.883587	−	0.468266i	$-0.155121\pi$
$379$	34.4032	1.76717	0.883587	+	0.468266i	$0.155121\pi$
$380$	−6.36416	−0.326474
$381$	0	0
$382$	−7.96269	−0.407407
$383$	11.1574	0.570115	0.285057	−	0.958510i	$-0.407987\pi$
$383$	11.1574	0.570115	0.285057	+	0.958510i	$0.407987\pi$
$384$	0	0
$385$	−1.64481	−0.0838274
$386$	−14.7200	−0.749226
$387$	0	0
$388$	−13.4775	−0.684215
$389$	−5.94990	−0.301672	−0.150836	−	0.988559i	$-0.548197\pi$
$389$	−5.94990	−0.301672	−0.150836	+	0.988559i	$0.548197\pi$
$390$	0	0
$391$	−7.54565	−0.381600
$392$	−1.23374	−0.0623133
$393$	0	0
$394$	−34.7197	−1.74915
$395$	1.59926	0.0804673
$396$	0	0
$397$	6.47617	0.325029	0.162515	−	0.986706i	$-0.448039\pi$
$397$	6.47617	0.325029	0.162515	+	0.986706i	$0.448039\pi$
$398$	43.8713	2.19907
$399$	0	0
$400$	22.5308	1.12654
$401$	3.27212	0.163402	0.0817010	−	0.996657i	$-0.473965\pi$
$401$	3.27212	0.163402	0.0817010	+	0.996657i	$0.473965\pi$
$402$	0	0
$403$	26.8705	1.33852
$404$	6.99494	0.348011
$405$	0	0
$406$	1.82725	0.0906848
$407$	−1.74042	−0.0862696
$408$	0	0
$409$	12.9834	0.641990	0.320995	−	0.947081i	$-0.395983\pi$
$409$	12.9834	0.641990	0.320995	+	0.947081i	$0.395983\pi$
$410$	9.70992	0.479539
$411$	0	0
$412$	9.47810	0.466952
$413$	−3.99665	−0.196662
$414$	0	0
$415$	−3.13286	−0.153786
$416$	28.0642	1.37596
$417$	0	0
$418$	36.2631	1.77369
$419$	−0.228730	−0.0111742	−0.00558710	−	0.999984i	$-0.501778\pi$
$419$	−0.228730	−0.0111742	−0.00558710	+	0.999984i	$0.501778\pi$
$420$	0	0
$421$	13.6768	0.666566	0.333283	−	0.942827i	$-0.391843\pi$
$421$	13.6768	0.666566	0.333283	+	0.942827i	$0.391843\pi$
$422$	−20.9119	−1.01798
$423$	0	0
$424$	11.3942	0.553350
$425$	−7.70800	−0.373893
$426$	0	0
$427$	−6.45858	−0.312553
$428$	8.85135	0.427846
$429$	0	0
$430$	−0.882299	−0.0425482
$431$	16.7015	0.804482	0.402241	−	0.915534i	$-0.368231\pi$
$431$	16.7015	0.804482	0.402241	+	0.915534i	$0.368231\pi$
$432$	0	0
$433$	31.0013	1.48983	0.744913	−	0.667161i	$-0.232491\pi$
$433$	31.0013	1.48983	0.744913	+	0.667161i	$0.232491\pi$
$434$	−11.2701	−0.540982
$435$	0	0
$436$	−4.19996	−0.201142
$437$	−34.3613	−1.64372
$438$	0	0
$439$	−31.0227	−1.48063	−0.740317	−	0.672258i	$-0.765325\pi$
$439$	−31.0227	−1.48063	−0.740317	+	0.672258i	$0.765325\pi$
$440$	2.02927	0.0967418
$441$	0	0
$442$	−13.2701	−0.631193
$443$	−17.4735	−0.830191	−0.415096	−	0.909778i	$-0.636252\pi$
$443$	−17.4735	−0.830191	−0.415096	+	0.909778i	$0.636252\pi$
$444$	0	0
$445$	−0.231098	−0.0109551
$446$	53.4725	2.53200
$447$	0	0
$448$	−1.97973	−0.0935336
$449$	37.0459	1.74831	0.874153	−	0.485651i	$-0.161417\pi$
$449$	37.0459	1.74831	0.874153	+	0.485651i	$0.161417\pi$
$450$	0	0
$451$	−22.0299	−1.03735
$452$	−3.50024	−0.164637
$453$	0	0
$454$	30.8677	1.44870
$455$	2.74093	0.128497
$456$	0	0
$457$	29.9331	1.40021	0.700106	−	0.714039i	$-0.253136\pi$
$457$	29.9331	1.40021	0.700106	+	0.714039i	$0.253136\pi$
$458$	53.6003	2.50458
$459$	0	0
$460$	3.75954	0.175289
$461$	−0.774046	−0.0360509	−0.0180255	−	0.999838i	$-0.505738\pi$
$461$	−0.774046	−0.0360509	−0.0180255	+	0.999838i	$0.505738\pi$
$462$	0	0
$463$	−18.6451	−0.866509	−0.433255	−	0.901272i	$-0.642635\pi$
$463$	−18.6451	−0.866509	−0.433255	+	0.901272i	$0.642635\pi$
$464$	−4.90701	−0.227802
$465$	0	0
$466$	−6.04670	−0.280108
$467$	24.7279	1.14427	0.572136	−	0.820158i	$-0.306115\pi$
$467$	24.7279	1.14427	0.572136	+	0.820158i	$0.306115\pi$
$468$	0	0
$469$	5.31265	0.245315
$470$	1.46696	0.0676657
$471$	0	0
$472$	4.93083	0.226960
$473$	2.00177	0.0920413
$474$	0	0
$475$	−35.1006	−1.61052
$476$	2.21615	0.101577
$477$	0	0
$478$	2.77150	0.126766
$479$	35.4021	1.61756	0.808782	−	0.588109i	$-0.200127\pi$
$479$	35.4021	1.61756	0.808782	+	0.588109i	$0.200127\pi$
$480$	0	0
$481$	2.90025	0.132240
$482$	−18.1501	−0.826716
$483$	0	0
$484$	−5.55371	−0.252441
$485$	6.42309	0.291658
$486$	0	0
$487$	−15.3543	−0.695768	−0.347884	−	0.937538i	$-0.613100\pi$
$487$	−15.3543	−0.695768	−0.347884	+	0.937538i	$0.613100\pi$
$488$	7.96822	0.360704
$489$	0	0
$490$	−1.14961	−0.0519339
$491$	0.585211	0.0264102	0.0132051	−	0.999913i	$-0.495797\pi$
$491$	0.585211	0.0264102	0.0132051	+	0.999913i	$0.495797\pi$
$492$	0	0
$493$	1.67874	0.0756065
$494$	−60.4290	−2.71883
$495$	0	0
$496$	30.2655	1.35896
$497$	9.52764	0.427373
$498$	0	0
$499$	−20.9905	−0.939662	−0.469831	−	0.882756i	$-0.655685\pi$
$499$	−20.9905	−0.939662	−0.469831	+	0.882756i	$0.655685\pi$
$500$	8.01270	0.358339
$501$	0	0
$502$	17.7415	0.791843
$503$	5.24106	0.233687	0.116844	−	0.993150i	$-0.462722\pi$
$503$	5.24106	0.233687	0.116844	+	0.993150i	$0.462722\pi$
$504$	0	0
$505$	−3.33365	−0.148345
$506$	−21.4219	−0.952321
$507$	0	0
$508$	−1.32323	−0.0587086
$509$	−10.4086	−0.461353	−0.230677	−	0.973030i	$-0.574094\pi$
$509$	−10.4086	−0.461353	−0.230677	+	0.973030i	$0.574094\pi$
$510$	0	0
$511$	4.29370	0.189942
$512$	19.5304	0.863130
$513$	0	0
$514$	−49.3695	−2.17760
$515$	−4.51707	−0.199046
$516$	0	0
$517$	−3.32824	−0.146376
$518$	−1.21643	−0.0534470
$519$	0	0
$520$	−3.38159	−0.148293
$521$	−6.18793	−0.271098	−0.135549	−	0.990771i	$-0.543280\pi$
$521$	−6.18793	−0.271098	−0.135549	+	0.990771i	$0.543280\pi$
$522$	0	0
$523$	40.4704	1.76965	0.884823	−	0.465927i	$-0.154279\pi$
$523$	40.4704	1.76965	0.884823	+	0.465927i	$0.154279\pi$
$524$	−27.3882	−1.19646
$525$	0	0
$526$	−8.00014	−0.348823
$527$	−10.3541	−0.451033
$528$	0	0
$529$	−2.70154	−0.117458
$530$	10.6171	0.461180
$531$	0	0
$532$	10.0919	0.437538
$533$	36.7108	1.59012
$534$	0	0
$535$	−4.21838	−0.182376
$536$	−6.55443	−0.283108
$537$	0	0
$538$	−27.9736	−1.20603
$539$	2.60824	0.112345
$540$	0	0
$541$	21.7350	0.934461	0.467231	−	0.884135i	$-0.345252\pi$
$541$	21.7350	0.934461	0.467231	+	0.884135i	$0.345252\pi$
$542$	−8.53631	−0.366666
$543$	0	0
$544$	−10.8141	−0.463651
$545$	2.00162	0.0857399
$546$	0	0
$547$	−11.4270	−0.488582	−0.244291	−	0.969702i	$-0.578555\pi$
$547$	−11.4270	−0.488582	−0.244291	+	0.969702i	$0.578555\pi$
$548$	−17.5596	−0.750107
$549$	0	0
$550$	−21.8828	−0.933087
$551$	7.64461	0.325671
$552$	0	0
$553$	−2.53600	−0.107842
$554$	−30.5795	−1.29920
$555$	0	0
$556$	−15.2916	−0.648508
$557$	−32.5753	−1.38026	−0.690129	−	0.723686i	$-0.742446\pi$
$557$	−32.5753	−1.38026	−0.690129	+	0.723686i	$0.742446\pi$
$558$	0	0
$559$	−3.33576	−0.141087
$560$	3.08723	0.130459
$561$	0	0
$562$	−45.7808	−1.93115
$563$	23.5920	0.994282	0.497141	−	0.867670i	$-0.334383\pi$
$563$	23.5920	0.994282	0.497141	+	0.867670i	$0.334383\pi$
$564$	0	0
$565$	1.66814	0.0701793
$566$	26.8237	1.12749
$567$	0	0
$568$	−11.7546	−0.493214
$569$	−23.4064	−0.981248	−0.490624	−	0.871371i	$-0.663231\pi$
$569$	−23.4064	−0.981248	−0.490624	+	0.871371i	$0.663231\pi$
$570$	0	0
$571$	−33.5499	−1.40402	−0.702010	−	0.712167i	$-0.747714\pi$
$571$	−33.5499	−1.40402	−0.702010	+	0.712167i	$0.747714\pi$
$572$	−15.0006	−0.627207
$573$	0	0
$574$	−15.3974	−0.642673
$575$	20.7352	0.864717
$576$	0	0
$577$	−20.4275	−0.850406	−0.425203	−	0.905098i	$-0.639797\pi$
$577$	−20.4275	−0.850406	−0.425203	+	0.905098i	$0.639797\pi$
$578$	−25.8771	−1.07635
$579$	0	0
$580$	−0.836413	−0.0347301
$581$	4.96788	0.206103
$582$	0	0
$583$	−24.0883	−0.997634
$584$	−5.29731	−0.219204
$585$	0	0
$586$	−0.129712	−0.00535836
$587$	22.6860	0.936350	0.468175	−	0.883636i	$-0.344912\pi$
$587$	22.6860	0.936350	0.468175	+	0.883636i	$0.344912\pi$
$588$	0	0
$589$	−47.1504	−1.94280
$590$	4.59457	0.189156
$591$	0	0
$592$	3.26669	0.134260
$593$	3.77769	0.155131	0.0775656	−	0.996987i	$-0.475285\pi$
$593$	3.77769	0.155131	0.0775656	+	0.996987i	$0.475285\pi$
$594$	0	0
$595$	−1.05617	−0.0432988
$596$	−26.1447	−1.07093
$597$	0	0
$598$	35.6976	1.45978
$599$	23.7126	0.968869	0.484435	−	0.874827i	$-0.339025\pi$
$599$	23.7126	0.968869	0.484435	+	0.874827i	$0.339025\pi$
$600$	0	0
$601$	27.5139	1.12231	0.561157	−	0.827709i	$-0.310356\pi$
$601$	27.5139	1.12231	0.561157	+	0.827709i	$0.310356\pi$
$602$	1.39909	0.0570227
$603$	0	0
$604$	9.25804	0.376704
$605$	2.64679	0.107607
$606$	0	0
$607$	48.7476	1.97860	0.989301	−	0.145887i	$-0.0466035\pi$
$607$	48.7476	1.97860	0.989301	+	0.145887i	$0.0466035\pi$
$608$	−49.2451	−1.99715
$609$	0	0
$610$	7.42483	0.300622
$611$	5.54621	0.224375
$612$	0	0
$613$	−20.2080	−0.816193	−0.408096	−	0.912939i	$-0.633807\pi$
$613$	−20.2080	−0.816193	−0.408096	+	0.912939i	$0.633807\pi$
$614$	24.2733	0.979593
$615$	0	0
$616$	−3.21789	−0.129652
$617$	−4.52887	−0.182326	−0.0911628	−	0.995836i	$-0.529058\pi$
$617$	−4.52887	−0.182326	−0.0911628	+	0.995836i	$0.529058\pi$
$618$	0	0
$619$	−3.54068	−0.142312	−0.0711560	−	0.997465i	$-0.522669\pi$
$619$	−3.54068	−0.142312	−0.0711560	+	0.997465i	$0.522669\pi$
$620$	5.15883	0.207184
$621$	0	0
$622$	2.47175	0.0991081
$623$	0.366460	0.0146819
$624$	0	0
$625$	19.1929	0.767715
$626$	26.0951	1.04297
$627$	0	0
$628$	11.2708	0.449756
$629$	−1.11757	−0.0445603
$630$	0	0
$631$	3.26907	0.130140	0.0650698	−	0.997881i	$-0.479273\pi$
$631$	3.26907	0.130140	0.0650698	+	0.997881i	$0.479273\pi$
$632$	3.12876	0.124455
$633$	0	0
$634$	−3.66595	−0.145593
$635$	0.630622	0.0250255
$636$	0	0
$637$	−4.34638	−0.172210
$638$	4.76589	0.188684
$639$	0	0
$640$	−5.86784	−0.231947
$641$	−24.7808	−0.978781	−0.489390	−	0.872065i	$-0.662781\pi$
$641$	−24.7808	−0.978781	−0.489390	+	0.872065i	$0.662781\pi$
$642$	0	0
$643$	4.30104	0.169616	0.0848082	−	0.996397i	$-0.472972\pi$
$643$	4.30104	0.169616	0.0848082	+	0.996397i	$0.472972\pi$
$644$	−5.96163	−0.234921
$645$	0	0
$646$	23.2854	0.916150
$647$	−36.0020	−1.41538	−0.707692	−	0.706522i	$-0.750263\pi$
$647$	−36.0020	−1.41538	−0.707692	+	0.706522i	$0.750263\pi$
$648$	0	0
$649$	−10.4242	−0.409186
$650$	36.4657	1.43030
$651$	0	0
$652$	−11.6578	−0.456554
$653$	−33.7337	−1.32010	−0.660051	−	0.751221i	$-0.729465\pi$
$653$	−33.7337	−1.32010	−0.660051	+	0.751221i	$0.729465\pi$
$654$	0	0
$655$	13.0526	0.510009
$656$	41.3491	1.61441
$657$	0	0
$658$	−2.32620	−0.0906849
$659$	35.5995	1.38676	0.693380	−	0.720572i	$-0.256121\pi$
$659$	35.5995	1.38676	0.693380	+	0.720572i	$0.256121\pi$
$660$	0	0
$661$	−7.65803	−0.297863	−0.148931	−	0.988848i	$-0.547583\pi$
$661$	−7.65803	−0.297863	−0.148931	+	0.988848i	$0.547583\pi$
$662$	−3.43274	−0.133417
$663$	0	0
$664$	−6.12908	−0.237855
$665$	−4.80958	−0.186508
$666$	0	0
$667$	−4.51595	−0.174858
$668$	−12.4519	−0.481777
$669$	0	0
$670$	−6.10746	−0.235952
$671$	−16.8455	−0.650314
$672$	0	0
$673$	−5.29132	−0.203965	−0.101983	−	0.994786i	$-0.532519\pi$
$673$	−5.29132	−0.203965	−0.101983	+	0.994786i	$0.532519\pi$
$674$	2.07437	0.0799016
$675$	0	0
$676$	7.79518	0.299814
$677$	−45.8433	−1.76190	−0.880951	−	0.473208i	$-0.843096\pi$
$677$	−45.8433	−1.76190	−0.880951	+	0.473208i	$0.843096\pi$
$678$	0	0
$679$	−10.1853	−0.390877
$680$	1.30304	0.0499694
$681$	0	0
$682$	−29.3951	−1.12560
$683$	38.6903	1.48044	0.740222	−	0.672362i	$-0.234720\pi$
$683$	38.6903	1.48044	0.740222	+	0.672362i	$0.234720\pi$
$684$	0	0
$685$	8.36853	0.319745
$686$	1.82297	0.0696014
$687$	0	0
$688$	−3.75721	−0.143242
$689$	40.1409	1.52924
$690$	0	0
$691$	31.5628	1.20071	0.600353	−	0.799735i	$-0.295027\pi$
$691$	31.5628	1.20071	0.600353	+	0.799735i	$0.295027\pi$
$692$	−34.3058	−1.30411
$693$	0	0
$694$	−18.5174	−0.702910
$695$	7.28766	0.276437
$696$	0	0
$697$	−14.1459	−0.535816
$698$	2.92261	0.110622
$699$	0	0
$700$	−6.08990	−0.230177
$701$	−7.21972	−0.272685	−0.136342	−	0.990662i	$-0.543535\pi$
$701$	−7.21972	−0.272685	−0.136342	+	0.990662i	$0.543535\pi$
$702$	0	0
$703$	−5.08916	−0.191941
$704$	−5.16361	−0.194611
$705$	0	0
$706$	3.73676	0.140635
$707$	5.28628	0.198811
$708$	0	0
$709$	−20.6634	−0.776030	−0.388015	−	0.921653i	$-0.626839\pi$
$709$	−20.6634	−0.776030	−0.388015	+	0.921653i	$0.626839\pi$
$710$	−10.9530	−0.411060
$711$	0	0
$712$	−0.452116	−0.0169438
$713$	27.8535	1.04312
$714$	0	0
$715$	7.14898	0.267357
$716$	−27.6462	−1.03319
$717$	0	0
$718$	14.2643	0.532340
$719$	8.94913	0.333746	0.166873	−	0.985978i	$-0.446633\pi$
$719$	8.94913	0.333746	0.166873	+	0.985978i	$0.446633\pi$
$720$	0	0
$721$	7.16287	0.266759
$722$	71.4001	2.65724
$723$	0	0
$724$	−23.3191	−0.866647
$725$	−4.61311	−0.171327
$726$	0	0
$727$	−34.3031	−1.27223	−0.636117	−	0.771593i	$-0.719460\pi$
$727$	−34.3031	−1.27223	−0.636117	+	0.771593i	$0.719460\pi$
$728$	5.36231	0.198740
$729$	0	0
$730$	−4.93606	−0.182692
$731$	1.28538	0.0475415
$732$	0	0
$733$	23.7264	0.876354	0.438177	−	0.898889i	$-0.355624\pi$
$733$	23.7264	0.876354	0.438177	+	0.898889i	$0.355624\pi$
$734$	35.7916	1.32109
$735$	0	0
$736$	29.0909	1.07230
$737$	13.8566	0.510416
$738$	0	0
$739$	19.8370	0.729715	0.364858	−	0.931063i	$-0.381118\pi$
$739$	19.8370	0.729715	0.364858	+	0.931063i	$0.381118\pi$
$740$	0.556815	0.0204689
$741$	0	0
$742$	−16.8360	−0.618069
$743$	−16.6001	−0.608999	−0.304500	−	0.952512i	$-0.598489\pi$
$743$	−16.6001	−0.608999	−0.304500	+	0.952512i	$0.598489\pi$
$744$	0	0
$745$	12.4601	0.456501
$746$	22.7990	0.834729
$747$	0	0
$748$	5.78024	0.211347
$749$	6.68923	0.244419
$750$	0	0
$751$	−34.3894	−1.25489	−0.627444	−	0.778662i	$-0.715899\pi$
$751$	−34.3894	−1.25489	−0.627444	+	0.778662i	$0.715899\pi$
$752$	6.24694	0.227802
$753$	0	0
$754$	−7.94192	−0.289228
$755$	−4.41220	−0.160576
$756$	0	0
$757$	44.4907	1.61704	0.808522	−	0.588466i	$-0.200268\pi$
$757$	44.4907	1.61704	0.808522	+	0.588466i	$0.200268\pi$
$758$	62.7161	2.27795
$759$	0	0
$760$	5.93378	0.215241
$761$	23.9855	0.869474	0.434737	−	0.900557i	$-0.356841\pi$
$761$	23.9855	0.869474	0.434737	+	0.900557i	$0.356841\pi$
$762$	0	0
$763$	−3.17404	−0.114908
$764$	−5.77981	−0.209106
$765$	0	0
$766$	20.3396	0.734898
$767$	17.3710	0.627229
$768$	0	0
$769$	−31.5927	−1.13926	−0.569632	−	0.821900i	$-0.692914\pi$
$769$	−31.5927	−1.13926	−0.569632	+	0.821900i	$0.692914\pi$
$770$	−2.99845	−0.108056
$771$	0	0
$772$	−10.6847	−0.384549
$773$	−28.0666	−1.00949	−0.504743	−	0.863270i	$-0.668413\pi$
$773$	−28.0666	−1.00949	−0.504743	+	0.863270i	$0.668413\pi$
$774$	0	0
$775$	28.4528	1.02205
$776$	12.5661	0.451095
$777$	0	0
$778$	−10.8465	−0.388866
$779$	−64.4176	−2.30800
$780$	0	0
$781$	24.8504	0.889216
$782$	−13.7555	−0.491896
$783$	0	0
$784$	−4.89553	−0.174840
$785$	−5.37145	−0.191715
$786$	0	0
$787$	7.58794	0.270481	0.135240	−	0.990813i	$-0.456819\pi$
$787$	7.58794	0.270481	0.135240	+	0.990813i	$0.456819\pi$
$788$	−25.2017	−0.897773
$789$	0	0
$790$	2.91540	0.103725
$791$	−2.64523	−0.0940537
$792$	0	0
$793$	28.0715	0.996847
$794$	11.8059	0.418975
$795$	0	0
$796$	31.8445	1.12870
$797$	−23.3421	−0.826819	−0.413409	−	0.910545i	$-0.635662\pi$
$797$	−23.3421	−0.826819	−0.413409	+	0.910545i	$0.635662\pi$
$798$	0	0
$799$	−2.13714	−0.0756066
$800$	29.7168	1.05065
$801$	0	0
$802$	5.96499	0.210631
$803$	11.1990	0.395203
$804$	0	0
$805$	2.84119	0.100139
$806$	48.9842	1.72539
$807$	0	0
$808$	−6.52190	−0.229440
$809$	6.17157	0.216981	0.108490	−	0.994098i	$-0.465398\pi$
$809$	6.17157	0.216981	0.108490	+	0.994098i	$0.465398\pi$
$810$	0	0
$811$	−3.28794	−0.115455	−0.0577276	−	0.998332i	$-0.518385\pi$
$811$	−3.28794	−0.115455	−0.0577276	+	0.998332i	$0.518385\pi$
$812$	1.32633	0.0465450
$813$	0	0
$814$	−3.17274	−0.111205
$815$	5.55586	0.194613
$816$	0	0
$817$	5.85335	0.204783
$818$	23.6685	0.827548
$819$	0	0
$820$	7.04806	0.246129
$821$	−11.6698	−0.407278	−0.203639	−	0.979046i	$-0.565277\pi$
$821$	−11.6698	−0.407278	−0.203639	+	0.979046i	$0.565277\pi$
$822$	0	0
$823$	−20.2596	−0.706206	−0.353103	−	0.935584i	$-0.614873\pi$
$823$	−20.2596	−0.706206	−0.353103	+	0.935584i	$0.614873\pi$
$824$	−8.83713	−0.307856
$825$	0	0
$826$	−7.28578	−0.253505
$827$	−18.1291	−0.630412	−0.315206	−	0.949023i	$-0.602074\pi$
$827$	−18.1291	−0.630412	−0.315206	+	0.949023i	$0.602074\pi$
$828$	0	0
$829$	28.8821	1.00312	0.501559	−	0.865123i	$-0.332760\pi$
$829$	28.8821	1.00312	0.501559	+	0.865123i	$0.332760\pi$
$830$	−5.71111	−0.198236
$831$	0	0
$832$	8.60467	0.298313
$833$	1.67481	0.0580287
$834$	0	0
$835$	5.93430	0.205365
$836$	26.3220	0.910365
$837$	0	0
$838$	−0.416968	−0.0144039
$839$	−15.1979	−0.524689	−0.262344	−	0.964974i	$-0.584496\pi$
$839$	−15.1979	−0.524689	−0.262344	+	0.964974i	$0.584496\pi$
$840$	0	0
$841$	−27.9953	−0.965355
$842$	24.9324	0.859228
$843$	0	0
$844$	−15.1792	−0.522489
$845$	−3.71502	−0.127801
$846$	0	0
$847$	−4.19710	−0.144214
$848$	45.2124	1.55260
$849$	0	0
$850$	−14.0515	−0.481961
$851$	3.00635	0.103056
$852$	0	0
$853$	−1.96381	−0.0672397	−0.0336199	−	0.999435i	$-0.510704\pi$
$853$	−1.96381	−0.0672397	−0.0336199	+	0.999435i	$0.510704\pi$
$854$	−11.7738	−0.402892
$855$	0	0
$856$	−8.25277	−0.282074
$857$	26.9219	0.919634	0.459817	−	0.888014i	$-0.347915\pi$
$857$	26.9219	0.919634	0.459817	+	0.888014i	$0.347915\pi$
$858$	0	0
$859$	19.4044	0.662070	0.331035	−	0.943618i	$-0.392602\pi$
$859$	19.4044	0.662070	0.331035	+	0.943618i	$0.392602\pi$
$860$	−0.640427	−0.0218384
$861$	0	0
$862$	30.4463	1.03701
$863$	11.8491	0.403347	0.201673	−	0.979453i	$-0.435362\pi$
$863$	11.8491	0.403347	0.201673	+	0.979453i	$0.435362\pi$
$864$	0	0
$865$	16.3494	0.555898
$866$	56.5144	1.92044
$867$	0	0
$868$	−8.18053	−0.277665
$869$	−6.61448	−0.224381
$870$	0	0
$871$	−23.0908	−0.782402
$872$	3.91594	0.132610
$873$	0	0
$874$	−62.6396	−2.11882
$875$	6.05543	0.204711
$876$	0	0
$877$	12.8788	0.434885	0.217442	−	0.976073i	$-0.430229\pi$
$877$	12.8788	0.434885	0.217442	+	0.976073i	$0.430229\pi$
$878$	−56.5536	−1.90859
$879$	0	0
$880$	8.05222	0.271440
$881$	−41.4610	−1.39686	−0.698428	−	0.715680i	$-0.746117\pi$
$881$	−41.4610	−1.39686	−0.698428	+	0.715680i	$0.746117\pi$
$882$	0	0
$883$	55.6753	1.87362	0.936812	−	0.349833i	$-0.113762\pi$
$883$	55.6753	1.87362	0.936812	+	0.349833i	$0.113762\pi$
$884$	−9.63223	−0.323967
$885$	0	0
$886$	−31.8537	−1.07015
$887$	14.9769	0.502875	0.251437	−	0.967874i	$-0.419097\pi$
$887$	14.9769	0.502875	0.251437	+	0.967874i	$0.419097\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−0.421284	−0.0141215
$891$	0	0
$892$	38.8137	1.29958
$893$	−9.73208	−0.325672
$894$	0	0
$895$	13.1756	0.440412
$896$	9.30485	0.310853
$897$	0	0
$898$	67.5337	2.25363
$899$	−6.19677	−0.206674
$900$	0	0
$901$	−15.4676	−0.515302
$902$	−40.1600	−1.33718
$903$	0	0
$904$	3.26354	0.108544
$905$	11.1134	0.369422
$906$	0	0
$907$	−8.74858	−0.290492	−0.145246	−	0.989396i	$-0.546397\pi$
$907$	−8.74858	−0.290492	−0.145246	+	0.989396i	$0.546397\pi$
$908$	22.4057	0.743560
$909$	0	0
$910$	4.99663	0.165637
$911$	44.8105	1.48464	0.742319	−	0.670047i	$-0.233726\pi$
$911$	44.8105	1.48464	0.742319	+	0.670047i	$0.233726\pi$
$912$	0	0
$913$	12.9574	0.428828
$914$	54.5672	1.80492
$915$	0	0
$916$	38.9064	1.28550
$917$	−20.6980	−0.683509
$918$	0	0
$919$	−47.4135	−1.56403	−0.782013	−	0.623262i	$-0.785807\pi$
$919$	−47.4135	−1.56403	−0.782013	+	0.623262i	$0.785807\pi$
$920$	−3.50530	−0.115566
$921$	0	0
$922$	−1.41106	−0.0464709
$923$	−41.4108	−1.36305
$924$	0	0
$925$	3.07103	0.100975
$926$	−33.9894	−1.11696
$927$	0	0
$928$	−6.47207	−0.212456
$929$	46.1776	1.51504	0.757519	−	0.652814i	$-0.226412\pi$
$929$	46.1776	1.51504	0.757519	+	0.652814i	$0.226412\pi$
$930$	0	0
$931$	7.62672	0.249956
$932$	−4.38907	−0.143769
$933$	0	0
$934$	45.0783	1.47501
$935$	−2.75475	−0.0900898
$936$	0	0
$937$	27.9526	0.913170	0.456585	−	0.889680i	$-0.349072\pi$
$937$	27.9526	0.913170	0.456585	+	0.889680i	$0.349072\pi$
$938$	9.68481	0.316220
$939$	0	0
$940$	1.06481	0.0347302
$941$	−50.7404	−1.65409	−0.827045	−	0.562136i	$-0.809980\pi$
$941$	−50.7404	−1.65409	−0.827045	+	0.562136i	$0.809980\pi$
$942$	0	0
$943$	38.0538	1.23920
$944$	19.5657	0.636809
$945$	0	0
$946$	3.64916	0.118645
$947$	27.3407	0.888452	0.444226	−	0.895915i	$-0.353479\pi$
$947$	27.3407	0.888452	0.444226	+	0.895915i	$0.353479\pi$
$948$	0	0
$949$	−18.6621	−0.605796
$950$	−63.9873	−2.07602
$951$	0	0
$952$	−2.06628	−0.0669686
$953$	13.4298	0.435035	0.217517	−	0.976056i	$-0.430204\pi$
$953$	13.4298	0.435035	0.217517	+	0.976056i	$0.430204\pi$
$954$	0	0
$955$	2.75454	0.0891348
$956$	2.01173	0.0650639
$957$	0	0
$958$	64.5370	2.08510
$959$	−13.2703	−0.428520
$960$	0	0
$961$	7.22046	0.232918
$962$	5.28708	0.170462
$963$	0	0
$964$	−13.1745	−0.424322
$965$	5.09209	0.163920
$966$	0	0
$967$	28.4687	0.915493	0.457747	−	0.889083i	$-0.348657\pi$
$967$	28.4687	0.915493	0.457747	+	0.889083i	$0.348657\pi$
$968$	5.17814	0.166432
$969$	0	0
$970$	11.7091	0.375957
$971$	34.5802	1.10973	0.554866	−	0.831940i	$-0.312769\pi$
$971$	34.5802	1.10973	0.554866	+	0.831940i	$0.312769\pi$
$972$	0	0
$973$	−11.5563	−0.370478
$974$	−27.9904	−0.896870
$975$	0	0
$976$	31.6182	1.01207
$977$	27.6629	0.885013	0.442507	−	0.896765i	$-0.354089\pi$
$977$	27.6629	0.885013	0.442507	+	0.896765i	$0.354089\pi$
$978$	0	0
$979$	0.955814	0.0305479
$980$	−0.834455	−0.0266557
$981$	0	0
$982$	1.06682	0.0340437
$983$	37.8392	1.20688	0.603442	−	0.797407i	$-0.293796\pi$
$983$	37.8392	1.20688	0.603442	+	0.797407i	$0.293796\pi$
$984$	0	0
$985$	12.0106	0.382690
$986$	3.06029	0.0974595
$987$	0	0
$988$	−43.8631	−1.39547
$989$	−3.45778	−0.109951
$990$	0	0
$991$	29.1907	0.927272	0.463636	−	0.886026i	$-0.346544\pi$
$991$	29.1907	0.927272	0.463636	+	0.886026i	$0.346544\pi$
$992$	39.9184	1.26741
$993$	0	0
$994$	17.3686	0.550899
$995$	−15.1764	−0.481126
$996$	0	0
$997$	6.89333	0.218314	0.109157	−	0.994025i	$-0.465185\pi$
$997$	6.89333	0.218314	0.109157	+	0.994025i	$0.465185\pi$
$998$	−38.2650	−1.21126
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.12		13
3.2	odd	2		2667.2.a.l.1.2	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.2	✓	13	3.2	odd	2
8001.2.a.o.1.12		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.82297 q^{2} +1.32323 q^{4} -0.630622 q^{5} +1.00000 q^{7} -1.23374 q^{8} +O(q^{10})\)	\(q+1.82297 q^{2} +1.32323 q^{4} -0.630622 q^{5} +1.00000 q^{7} -1.23374 q^{8} -1.14961 q^{10} +2.60824 q^{11} -4.34638 q^{13} +1.82297 q^{14} -4.89553 q^{16} +1.67481 q^{17} +7.62672 q^{19} -0.834455 q^{20} +4.75474 q^{22} -4.50538 q^{23} -4.60232 q^{25} -7.92333 q^{26} +1.32323 q^{28} +1.00235 q^{29} -6.18227 q^{31} -6.45692 q^{32} +3.05313 q^{34} -0.630622 q^{35} -0.667280 q^{37} +13.9033 q^{38} +0.778025 q^{40} -8.44630 q^{41} +0.767479 q^{43} +3.45128 q^{44} -8.21318 q^{46} -1.27605 q^{47} +1.00000 q^{49} -8.38989 q^{50} -5.75124 q^{52} -9.23546 q^{53} -1.64481 q^{55} -1.23374 q^{56} +1.82725 q^{58} -3.99665 q^{59} -6.45858 q^{61} -11.2701 q^{62} -1.97973 q^{64} +2.74093 q^{65} +5.31265 q^{67} +2.21615 q^{68} -1.14961 q^{70} +9.52764 q^{71} +4.29370 q^{73} -1.21643 q^{74} +10.0919 q^{76} +2.60824 q^{77} -2.53600 q^{79} +3.08723 q^{80} -15.3974 q^{82} +4.96788 q^{83} -1.05617 q^{85} +1.39909 q^{86} -3.21789 q^{88} +0.366460 q^{89} -4.34638 q^{91} -5.96163 q^{92} -2.32620 q^{94} -4.80958 q^{95} -10.1853 q^{97} +1.82297 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more