LMFDB - Embedded newform 1445.2.d.e.866.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1445,2,Mod(866,1445)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1445, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1445.866");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1445 = 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1445.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$11.5383830921$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			866.4
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1445.866
Dual form			1445.2.d.e.866.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205 q^{2} +2.73205i q^{3} +1.00000 q^{4} +1.00000i q^{5} +4.73205i q^{6} +2.73205i q^{7} -1.73205 q^{8} -4.46410 q^{9} +1.73205i q^{10} -4.73205i q^{11} +2.73205i q^{12} -4.00000 q^{13} +4.73205i q^{14} -2.73205 q^{15} -5.00000 q^{16} -7.73205 q^{18} +1.46410 q^{19} +1.00000i q^{20} -7.46410 q^{21} -8.19615i q^{22} +8.19615i q^{23} -4.73205i q^{24} -1.00000 q^{25} -6.92820 q^{26} -4.00000i q^{27} +2.73205i q^{28} -3.46410i q^{29} -4.73205 q^{30} +3.26795i q^{31} -5.19615 q^{32} +12.9282 q^{33} -2.73205 q^{35} -4.46410 q^{36} -0.535898i q^{37} +2.53590 q^{38} -10.9282i q^{39} -1.73205i q^{40} +3.46410i q^{41} -12.9282 q^{42} +0.535898 q^{43} -4.73205i q^{44} -4.46410i q^{45} +14.1962i q^{46} +12.9282 q^{47} -13.6603i q^{48} -0.464102 q^{49} -1.73205 q^{50} -4.00000 q^{52} -6.00000 q^{53} -6.92820i q^{54} +4.73205 q^{55} -4.73205i q^{56} +4.00000i q^{57} -6.00000i q^{58} -2.53590 q^{59} -2.73205 q^{60} +4.92820i q^{61} +5.66025i q^{62} -12.1962i q^{63} +1.00000 q^{64} -4.00000i q^{65} +22.3923 q^{66} -10.0000 q^{67} -22.3923 q^{69} -4.73205 q^{70} +11.6603i q^{71} +7.73205 q^{72} +6.39230i q^{73} -0.928203i q^{74} -2.73205i q^{75} +1.46410 q^{76} +12.9282 q^{77} -18.9282i q^{78} -14.5885i q^{79} -5.00000i q^{80} -2.46410 q^{81} +6.00000i q^{82} -8.53590 q^{83} -7.46410 q^{84} +0.928203 q^{86} +9.46410 q^{87} +8.19615i q^{88} +4.39230 q^{89} -7.73205i q^{90} -10.9282i q^{91} +8.19615i q^{92} -8.92820 q^{93} +22.3923 q^{94} +1.46410i q^{95} -14.1962i q^{96} -4.92820i q^{97} -0.803848 q^{98} +21.1244i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 4 q^{9} - 16 q^{13} - 4 q^{15} - 20 q^{16} - 24 q^{18} - 8 q^{19} - 16 q^{21} - 4 q^{25} - 12 q^{30} + 24 q^{33} - 4 q^{35} - 4 q^{36} + 24 q^{38} - 24 q^{42} + 16 q^{43} + 24 q^{47} + 12 q^{49}+ \cdots - 24 q^{98}+O(q^{100}) $ 4 * q + 4 * q^4 - 4 * q^9 - 16 * q^13 - 4 * q^15 - 20 * q^16 - 24 * q^18 - 8 * q^19 - 16 * q^21 - 4 * q^25 - 12 * q^30 + 24 * q^33 - 4 * q^35 - 4 * q^36 + 24 * q^38 - 24 * q^42 + 16 * q^43 + 24 * q^47 + 12 * q^49 - 16 * q^52 - 24 * q^53 + 12 * q^55 - 24 * q^59 - 4 * q^60 + 4 * q^64 + 48 * q^66 - 40 * q^67 - 48 * q^69 - 12 * q^70 + 24 * q^72 - 8 * q^76 + 24 * q^77 + 4 * q^81 - 48 * q^83 - 16 * q^84 - 24 * q^86 + 24 * q^87 - 24 * q^89 - 8 * q^93 + 48 * q^94 - 24 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1445\mathbb{Z}\right)^\times$.

$n$	$581$	$1157$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	2.73205i	1.57735i	0.614810	+	0.788675i	$0.289233\pi$
$3$	2.73205i	1.57735i	−0.614810	+	0.788675i	$0.710767\pi$
$4$	1.00000	0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	4.73205i	1.93185i
$7$	2.73205i	1.03262i	0.856402	+	0.516309i	$0.172694\pi$
$7$	2.73205i	1.03262i	−0.856402	+	0.516309i	$0.827306\pi$
$8$	−1.73205	−0.612372
$9$	−4.46410	−1.48803
$10$	1.73205i	0.547723i
$11$	− 4.73205i	− 1.42677i	−0.700774	−	0.713384i	$-0.747162\pi$
$11$	− 4.73205i	− 1.42677i	0.700774	−	0.713384i	$-0.252838\pi$
$12$	2.73205i	0.788675i
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	4.73205i	1.26469i
$15$	−2.73205	−0.705412
$16$	−5.00000	−1.25000
$17$	0	0
$17$	0	0
$18$	−7.73205	−1.82246
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	1.00000i	0.223607i
$21$	−7.46410	−1.62880
$22$	− 8.19615i	− 1.74743i
$23$	8.19615i	1.70902i	0.519438	+	0.854508i	$0.326141\pi$
$23$	8.19615i	1.70902i	−0.519438	+	0.854508i	$0.673859\pi$
$24$	− 4.73205i	− 0.965926i
$25$	−1.00000	−0.200000
$26$	−6.92820	−1.35873
$27$	− 4.00000i	− 0.769800i
$28$	2.73205i	0.516309i
$29$	− 3.46410i	− 0.643268i	−0.946864	−	0.321634i	$-0.895768\pi$
$29$	− 3.46410i	− 0.643268i	0.946864	−	0.321634i	$-0.104232\pi$
$30$	−4.73205	−0.863950
$31$	3.26795i	0.586941i	0.955968	+	0.293471i	$0.0948102\pi$
$31$	3.26795i	0.586941i	−0.955968	+	0.293471i	$0.905190\pi$
$32$	−5.19615	−0.918559
$33$	12.9282	2.25051
$34$	0	0
$35$	−2.73205	−0.461801
$36$	−4.46410	−0.744017
$37$	− 0.535898i	− 0.0881012i	−0.999029	−	0.0440506i	$-0.985974\pi$
$37$	− 0.535898i	− 0.0881012i	0.999029	−	0.0440506i	$-0.0140263\pi$
$38$	2.53590	0.411377
$39$	− 10.9282i	− 1.74991i
$40$	− 1.73205i	− 0.273861i
$41$	3.46410i	0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$	3.46410i	0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$	−12.9282	−1.99487
$43$	0.535898	0.0817237	0.0408619	−	0.999165i	$-0.486990\pi$
$43$	0.535898	0.0817237	0.0408619	+	0.999165i	$0.486990\pi$
$44$	− 4.73205i	− 0.713384i
$45$	− 4.46410i	− 0.665469i
$46$	14.1962i	2.09311i
$47$	12.9282	1.88577	0.942886	−	0.333115i	$-0.108100\pi$
$47$	12.9282	1.88577	0.942886	+	0.333115i	$0.108100\pi$
$48$	− 13.6603i	− 1.97169i
$49$	−0.464102	−0.0663002
$50$	−1.73205	−0.244949
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	− 6.92820i	− 0.942809i
$55$	4.73205	0.638070
$56$	− 4.73205i	− 0.632347i
$57$	4.00000i	0.529813i
$58$	− 6.00000i	− 0.787839i
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	−2.73205	−0.352706
$61$	4.92820i	0.630992i	0.948927	+	0.315496i	$0.102171\pi$
$61$	4.92820i	0.630992i	−0.948927	+	0.315496i	$0.897829\pi$
$62$	5.66025i	0.718853i
$63$	− 12.1962i	− 1.53657i
$64$	1.00000	0.125000
$65$	− 4.00000i	− 0.496139i
$66$	22.3923	2.75630
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	−22.3923	−2.69572
$70$	−4.73205	−0.565588
$71$	11.6603i	1.38382i	0.721985	+	0.691909i	$0.243230\pi$
$71$	11.6603i	1.38382i	−0.721985	+	0.691909i	$0.756770\pi$
$72$	7.73205	0.911231
$73$	6.39230i	0.748163i	0.927396	+	0.374081i	$0.122042\pi$
$73$	6.39230i	0.748163i	−0.927396	+	0.374081i	$0.877958\pi$
$74$	− 0.928203i	− 0.107901i
$75$	− 2.73205i	− 0.315470i
$76$	1.46410	0.167944
$77$	12.9282	1.47331
$78$	− 18.9282i	− 2.14320i
$79$	− 14.5885i	− 1.64133i	−0.571410	−	0.820665i	$-0.693603\pi$
$79$	− 14.5885i	− 1.64133i	0.571410	−	0.820665i	$-0.306397\pi$
$80$	− 5.00000i	− 0.559017i
$81$	−2.46410	−0.273789
$82$	6.00000i	0.662589i
$83$	−8.53590	−0.936937	−0.468468	−	0.883480i	$-0.655194\pi$
$83$	−8.53590	−0.936937	−0.468468	+	0.883480i	$0.655194\pi$
$84$	−7.46410	−0.814400
$85$	0	0
$86$	0.928203	0.100091
$87$	9.46410	1.01466
$88$	8.19615i	0.873713i
$89$	4.39230	0.465583	0.232792	−	0.972527i	$-0.425214\pi$
$89$	4.39230	0.465583	0.232792	+	0.972527i	$0.425214\pi$
$90$	− 7.73205i	− 0.815030i
$91$	− 10.9282i	− 1.14559i
$92$	8.19615i	0.854508i
$93$	−8.92820	−0.925812
$94$	22.3923	2.30959
$95$	1.46410i	0.150214i
$96$	− 14.1962i	− 1.44889i
$97$	− 4.92820i	− 0.500383i	−0.968196	−	0.250192i	$-0.919506\pi$
$97$	− 4.92820i	− 0.500383i	0.968196	−	0.250192i	$-0.0804936\pi$
$98$	−0.803848	−0.0812009
$99$	21.1244i	2.12308i
$100$	−1.00000	−0.100000
$101$	−9.46410	−0.941713	−0.470857	−	0.882210i	$-0.656055\pi$
$101$	−9.46410	−0.941713	−0.470857	+	0.882210i	$0.656055\pi$
$102$	0	0
$103$	8.92820	0.879722	0.439861	−	0.898066i	$-0.355028\pi$
$103$	8.92820	0.879722	0.439861	+	0.898066i	$0.355028\pi$
$104$	6.92820	0.679366
$105$	− 7.46410i	− 0.728422i
$106$	−10.3923	−1.00939
$107$	17.6603i	1.70728i	0.520862	+	0.853641i	$0.325610\pi$
$107$	17.6603i	1.70728i	−0.520862	+	0.853641i	$0.674390\pi$
$108$	− 4.00000i	− 0.384900i
$109$	10.0000i	0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$	10.0000i	0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	8.19615	0.781472
$111$	1.46410	0.138966
$112$	− 13.6603i	− 1.29077i
$113$	17.3205i	1.62938i	0.579899	+	0.814688i	$0.303092\pi$
$113$	17.3205i	1.62938i	−0.579899	+	0.814688i	$0.696908\pi$
$114$	6.92820i	0.648886i
$115$	−8.19615	−0.764295
$116$	− 3.46410i	− 0.321634i
$117$	17.8564	1.65083
$118$	−4.39230	−0.404344
$119$	0	0
$120$	4.73205	0.431975
$121$	−11.3923	−1.03566
$122$	8.53590i	0.772804i
$123$	−9.46410	−0.853349
$124$	3.26795i	0.293471i
$125$	− 1.00000i	− 0.0894427i
$126$	− 21.1244i	− 1.88191i
$127$	14.3923	1.27711	0.638555	−	0.769576i	$-0.279532\pi$
$127$	14.3923	1.27711	0.638555	+	0.769576i	$0.279532\pi$
$128$	12.1244	1.07165
$129$	1.46410i	0.128907i
$130$	− 6.92820i	− 0.607644i
$131$	− 2.19615i	− 0.191879i	−0.995387	−	0.0959394i	$-0.969415\pi$
$131$	− 2.19615i	− 0.191879i	0.995387	−	0.0959394i	$-0.0305855\pi$
$132$	12.9282	1.12526
$133$	4.00000i	0.346844i
$134$	−17.3205	−1.49626
$135$	4.00000	0.344265
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	−38.7846	−3.30157
$139$	− 3.66025i	− 0.310459i	−0.987878	−	0.155229i	$-0.950388\pi$
$139$	− 3.66025i	− 0.310459i	0.987878	−	0.155229i	$-0.0496117\pi$
$140$	−2.73205	−0.230900
$141$	35.3205i	2.97452i
$142$	20.1962i	1.69482i
$143$	18.9282i	1.58286i
$144$	22.3205	1.86004
$145$	3.46410	0.287678
$146$	11.0718i	0.916308i
$147$	− 1.26795i	− 0.104579i
$148$	− 0.535898i	− 0.0440506i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	− 4.73205i	− 0.386370i
$151$	1.46410	0.119147	0.0595734	−	0.998224i	$-0.481026\pi$
$151$	1.46410	0.119147	0.0595734	+	0.998224i	$0.481026\pi$
$152$	−2.53590	−0.205689
$153$	0	0
$154$	22.3923	1.80442
$155$	−3.26795	−0.262488
$156$	− 10.9282i	− 0.874957i
$157$	8.92820	0.712548	0.356274	−	0.934381i	$-0.384047\pi$
$157$	8.92820	0.712548	0.356274	+	0.934381i	$0.384047\pi$
$158$	− 25.2679i	− 2.01021i
$159$	− 16.3923i	− 1.29999i
$160$	− 5.19615i	− 0.410792i
$161$	−22.3923	−1.76476
$162$	−4.26795	−0.335322
$163$	0.196152i	0.0153638i	0.999970	+	0.00768192i	$0.00244526\pi$
$163$	0.196152i	0.0153638i	−0.999970	+	0.00768192i	$0.997555\pi$
$164$	3.46410i	0.270501i
$165$	12.9282i	1.00646i
$166$	−14.7846	−1.14751
$167$	12.5885i	0.974124i	0.873367	+	0.487062i	$0.161931\pi$
$167$	12.5885i	0.974124i	−0.873367	+	0.487062i	$0.838069\pi$
$168$	12.9282	0.997433
$169$	3.00000	0.230769
$170$	0	0
$171$	−6.53590	−0.499813
$172$	0.535898	0.0408619
$173$	3.46410i	0.263371i	0.991292	+	0.131685i	$0.0420389\pi$
$173$	3.46410i	0.263371i	−0.991292	+	0.131685i	$0.957961\pi$
$174$	16.3923	1.24270
$175$	− 2.73205i	− 0.206524i
$176$	23.6603i	1.78346i
$177$	− 6.92820i	− 0.520756i
$178$	7.60770	0.570221
$179$	11.3205	0.846135	0.423067	−	0.906098i	$-0.360953\pi$
$179$	11.3205	0.846135	0.423067	+	0.906098i	$0.360953\pi$
$180$	− 4.46410i	− 0.332734i
$181$	2.39230i	0.177819i	0.996040	+	0.0889093i	$0.0283381\pi$
$181$	2.39230i	0.177819i	−0.996040	+	0.0889093i	$0.971662\pi$
$182$	− 18.9282i	− 1.40305i
$183$	−13.4641	−0.995295
$184$	− 14.1962i	− 1.04655i
$185$	0.535898	0.0394000
$186$	−15.4641	−1.13388
$187$	0	0
$188$	12.9282	0.942886
$189$	10.9282	0.794910
$190$	2.53590i	0.183973i
$191$	1.85641	0.134325	0.0671624	−	0.997742i	$-0.478605\pi$
$191$	1.85641	0.134325	0.0671624	+	0.997742i	$0.478605\pi$
$192$	2.73205i	0.197169i
$193$	− 16.5359i	− 1.19028i	−0.803622	−	0.595140i	$-0.797097\pi$
$193$	− 16.5359i	− 1.19028i	0.803622	−	0.595140i	$-0.202903\pi$
$194$	− 8.53590i	− 0.612842i
$195$	10.9282	0.782585
$196$	−0.464102	−0.0331501
$197$	− 17.3205i	− 1.23404i	−0.786949	−	0.617018i	$-0.788341\pi$
$197$	− 17.3205i	− 1.23404i	0.786949	−	0.617018i	$-0.211659\pi$
$198$	36.5885i	2.60023i
$199$	10.1962i	0.722786i	0.932414	+	0.361393i	$0.117699\pi$
$199$	10.1962i	0.722786i	−0.932414	+	0.361393i	$0.882301\pi$
$200$	1.73205	0.122474
$201$	− 27.3205i	− 1.92704i
$202$	−16.3923	−1.15336
$203$	9.46410	0.664250
$204$	0	0
$205$	−3.46410	−0.241943
$206$	15.4641	1.07744
$207$	− 36.5885i	− 2.54307i
$208$	20.0000	1.38675
$209$	− 6.92820i	− 0.479234i
$210$	− 12.9282i	− 0.892131i
$211$	− 10.1962i	− 0.701932i	−0.936388	−	0.350966i	$-0.885853\pi$
$211$	− 10.1962i	− 0.701932i	0.936388	−	0.350966i	$-0.114147\pi$
$212$	−6.00000	−0.412082
$213$	−31.8564	−2.18277
$214$	30.5885i	2.09098i
$215$	0.535898i	0.0365480i
$216$	6.92820i	0.471405i
$217$	−8.92820	−0.606086
$218$	17.3205i	1.17309i
$219$	−17.4641	−1.18011
$220$	4.73205	0.319035
$221$	0	0
$222$	2.53590	0.170198
$223$	26.3923	1.76736	0.883680	−	0.468092i	$-0.155058\pi$
$223$	26.3923	1.76736	0.883680	+	0.468092i	$0.155058\pi$
$224$	− 14.1962i	− 0.948520i
$225$	4.46410	0.297607
$226$	30.0000i	1.99557i
$227$	− 22.7321i	− 1.50878i	−0.656427	−	0.754390i	$-0.727933\pi$
$227$	− 22.7321i	− 1.50878i	0.656427	−	0.754390i	$-0.272067\pi$
$228$	4.00000i	0.264906i
$229$	8.39230	0.554579	0.277290	−	0.960786i	$-0.410564\pi$
$229$	8.39230	0.554579	0.277290	+	0.960786i	$0.410564\pi$
$230$	−14.1962	−0.936067
$231$	35.3205i	2.32392i
$232$	6.00000i	0.393919i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	30.9282	2.02184
$235$	12.9282i	0.843343i
$236$	−2.53590	−0.165073
$237$	39.8564	2.58895
$238$	0	0
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	13.6603	0.881766
$241$	− 5.60770i	− 0.361223i	−0.983554	−	0.180612i	$-0.942192\pi$
$241$	− 5.60770i	− 0.361223i	0.983554	−	0.180612i	$-0.0578077\pi$
$242$	−19.7321	−1.26842
$243$	− 18.7321i	− 1.20166i
$244$	4.92820i	0.315496i
$245$	− 0.464102i	− 0.0296504i
$246$	−16.3923	−1.04514
$247$	−5.85641	−0.372634
$248$	− 5.66025i	− 0.359426i
$249$	− 23.3205i	− 1.47788i
$250$	− 1.73205i	− 0.109545i
$251$	6.92820	0.437304	0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820	0.437304	0.218652	+	0.975803i	$0.429834\pi$
$252$	− 12.1962i	− 0.768285i
$253$	38.7846	2.43837
$254$	24.9282	1.56413
$255$	0	0
$256$	19.0000	1.18750
$257$	6.92820	0.432169	0.216085	−	0.976375i	$-0.430671\pi$
$257$	6.92820	0.432169	0.216085	+	0.976375i	$0.430671\pi$
$258$	2.53590i	0.157878i
$259$	1.46410	0.0909748
$260$	− 4.00000i	− 0.248069i
$261$	15.4641i	0.957204i
$262$	− 3.80385i	− 0.235002i
$263$	1.60770	0.0991347	0.0495674	−	0.998771i	$-0.484216\pi$
$263$	1.60770	0.0991347	0.0495674	+	0.998771i	$0.484216\pi$
$264$	−22.3923	−1.37815
$265$	− 6.00000i	− 0.368577i
$266$	6.92820i	0.424795i
$267$	12.0000i	0.734388i
$268$	−10.0000	−0.610847
$269$	0.928203i	0.0565935i	0.999600	+	0.0282968i	$0.00900835\pi$
$269$	0.928203i	0.0565935i	−0.999600	+	0.0282968i	$0.990992\pi$
$270$	6.92820	0.421637
$271$	2.92820	0.177876	0.0889378	−	0.996037i	$-0.471653\pi$
$271$	2.92820	0.177876	0.0889378	+	0.996037i	$0.471653\pi$
$272$	0	0
$273$	29.8564	1.80699
$274$	0	0
$275$	4.73205i	0.285353i
$276$	−22.3923	−1.34786
$277$	20.9282i	1.25745i	0.777626	+	0.628727i	$0.216424\pi$
$277$	20.9282i	1.25745i	−0.777626	+	0.628727i	$0.783576\pi$
$278$	− 6.33975i	− 0.380233i
$279$	− 14.5885i	− 0.873388i
$280$	4.73205	0.282794
$281$	12.9282	0.771232	0.385616	−	0.922659i	$-0.373989\pi$
$281$	12.9282	0.771232	0.385616	+	0.922659i	$0.373989\pi$
$282$	61.1769i	3.64303i
$283$	5.26795i	0.313147i	0.987666	+	0.156574i	$0.0500448\pi$
$283$	5.26795i	0.313147i	−0.987666	+	0.156574i	$0.949955\pi$
$284$	11.6603i	0.691909i
$285$	−4.00000	−0.236940
$286$	32.7846i	1.93859i
$287$	−9.46410	−0.558648
$288$	23.1962	1.36685
$289$	0	0
$290$	6.00000	0.352332
$291$	13.4641	0.789280
$292$	6.39230i	0.374081i
$293$	−0.928203	−0.0542262	−0.0271131	−	0.999632i	$-0.508631\pi$
$293$	−0.928203	−0.0542262	−0.0271131	+	0.999632i	$0.508631\pi$
$294$	− 2.19615i	− 0.128082i
$295$	− 2.53590i	− 0.147646i
$296$	0.928203i	0.0539507i
$297$	−18.9282	−1.09833
$298$	−10.3923	−0.602010
$299$	− 32.7846i	− 1.89598i
$300$	− 2.73205i	− 0.157735i
$301$	1.46410i	0.0843894i
$302$	2.53590	0.145925
$303$	− 25.8564i	− 1.48541i
$304$	−7.32051	−0.419860
$305$	−4.92820	−0.282188
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	12.9282	0.736653
$309$	24.3923i	1.38763i
$310$	−5.66025	−0.321481
$311$	− 16.0526i	− 0.910257i	−0.890426	−	0.455129i	$-0.849593\pi$
$311$	− 16.0526i	− 0.910257i	0.890426	−	0.455129i	$-0.150407\pi$
$312$	18.9282i	1.07160i
$313$	26.3923i	1.49178i	0.666068	+	0.745891i	$0.267976\pi$
$313$	26.3923i	1.49178i	−0.666068	+	0.745891i	$0.732024\pi$
$314$	15.4641	0.872690
$315$	12.1962	0.687175
$316$	− 14.5885i	− 0.820665i
$317$	24.9282i	1.40011i	0.714090	+	0.700054i	$0.246841\pi$
$317$	24.9282i	1.40011i	−0.714090	+	0.700054i	$0.753159\pi$
$318$	− 28.3923i	− 1.59216i
$319$	−16.3923	−0.917793
$320$	1.00000i	0.0559017i
$321$	−48.2487	−2.69298
$322$	−38.7846	−2.16138
$323$	0	0
$324$	−2.46410	−0.136895
$325$	4.00000	0.221880
$326$	0.339746i	0.0188168i
$327$	−27.3205	−1.51083
$328$	− 6.00000i	− 0.331295i
$329$	35.3205i	1.94728i
$330$	22.3923i	1.23266i
$331$	6.53590	0.359245	0.179623	−	0.983736i	$-0.442512\pi$
$331$	6.53590	0.359245	0.179623	+	0.983736i	$0.442512\pi$
$332$	−8.53590	−0.468468
$333$	2.39230i	0.131097i
$334$	21.8038i	1.19305i
$335$	− 10.0000i	− 0.546358i
$336$	37.3205	2.03600
$337$	− 6.78461i	− 0.369581i	−0.982778	−	0.184791i	$-0.940839\pi$
$337$	− 6.78461i	− 0.369581i	0.982778	−	0.184791i	$-0.0591607\pi$
$338$	5.19615	0.282633
$339$	−47.3205	−2.57010
$340$	0	0
$341$	15.4641	0.837428
$342$	−11.3205	−0.612143
$343$	17.8564i	0.964155i
$344$	−0.928203	−0.0500454
$345$	− 22.3923i	− 1.20556i
$346$	6.00000i	0.322562i
$347$	− 3.80385i	− 0.204201i	−0.994774	−	0.102101i	$-0.967444\pi$
$347$	− 3.80385i	− 0.204201i	0.994774	−	0.102101i	$-0.0325564\pi$
$348$	9.46410	0.507329
$349$	−10.7846	−0.577287	−0.288643	−	0.957437i	$-0.593204\pi$
$349$	−10.7846	−0.577287	−0.288643	+	0.957437i	$0.593204\pi$
$350$	− 4.73205i	− 0.252939i
$351$	16.0000i	0.854017i
$352$	24.5885i	1.31057i
$353$	26.7846	1.42560	0.712800	−	0.701367i	$-0.247427\pi$
$353$	26.7846	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$354$	− 12.0000i	− 0.637793i
$355$	−11.6603	−0.618862
$356$	4.39230	0.232792
$357$	0	0
$358$	19.6077	1.03630
$359$	21.4641	1.13283	0.566416	−	0.824119i	$-0.308330\pi$
$359$	21.4641	1.13283	0.566416	+	0.824119i	$0.308330\pi$
$360$	7.73205i	0.407515i
$361$	−16.8564	−0.887179
$362$	4.14359i	0.217782i
$363$	− 31.1244i	− 1.63361i
$364$	− 10.9282i	− 0.572793i
$365$	−6.39230	−0.334589
$366$	−23.3205	−1.21898
$367$	7.80385i	0.407358i	0.979038	+	0.203679i	$0.0652898\pi$
$367$	7.80385i	0.407358i	−0.979038	+	0.203679i	$0.934710\pi$
$368$	− 40.9808i	− 2.13627i
$369$	− 15.4641i	− 0.805029i
$370$	0.928203	0.0482550
$371$	− 16.3923i	− 0.851046i
$372$	−8.92820	−0.462906
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	0	0
$375$	2.73205	0.141082
$376$	−22.3923	−1.15479
$377$	13.8564i	0.713641i
$378$	18.9282	0.973562
$379$	17.8038i	0.914522i	0.889332	+	0.457261i	$0.151170\pi$
$379$	17.8038i	0.914522i	−0.889332	+	0.457261i	$0.848830\pi$
$380$	1.46410i	0.0751068i
$381$	39.3205i	2.01445i
$382$	3.21539	0.164514
$383$	−15.4641	−0.790179	−0.395089	−	0.918643i	$-0.629286\pi$
$383$	−15.4641	−0.790179	−0.395089	+	0.918643i	$0.629286\pi$
$384$	33.1244i	1.69037i
$385$	12.9282i	0.658882i
$386$	− 28.6410i	− 1.45779i
$387$	−2.39230	−0.121608
$388$	− 4.92820i	− 0.250192i
$389$	4.39230	0.222699	0.111349	−	0.993781i	$-0.464483\pi$
$389$	4.39230	0.222699	0.111349	+	0.993781i	$0.464483\pi$
$390$	18.9282	0.958467
$391$	0	0
$392$	0.803848	0.0406004
$393$	6.00000	0.302660
$394$	− 30.0000i	− 1.51138i
$395$	14.5885	0.734025
$396$	21.1244i	1.06154i
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	17.6603i	0.885229i
$399$	−10.9282	−0.547094
$400$	5.00000	0.250000
$401$	− 12.9282i	− 0.645604i	−0.946467	−	0.322802i	$-0.895375\pi$
$401$	− 12.9282i	− 0.645604i	0.946467	−	0.322802i	$-0.104625\pi$
$402$	− 47.3205i	− 2.36013i
$403$	− 13.0718i	− 0.651153i
$404$	−9.46410	−0.470857
$405$	− 2.46410i	− 0.122442i
$406$	16.3923	0.813536
$407$	−2.53590	−0.125700
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	8.92820	0.439861
$413$	− 6.92820i	− 0.340915i
$414$	− 63.3731i	− 3.11462i
$415$	− 8.53590i	− 0.419011i
$416$	20.7846	1.01905
$417$	10.0000	0.489702
$418$	− 12.0000i	− 0.586939i
$419$	− 38.1962i	− 1.86600i	−0.359871	−	0.933002i	$-0.617179\pi$
$419$	− 38.1962i	− 1.86600i	0.359871	−	0.933002i	$-0.382821\pi$
$420$	− 7.46410i	− 0.364211i
$421$	5.46410	0.266304	0.133152	−	0.991096i	$-0.457490\pi$
$421$	5.46410	0.266304	0.133152	+	0.991096i	$0.457490\pi$
$422$	− 17.6603i	− 0.859688i
$423$	−57.7128	−2.80609
$424$	10.3923	0.504695
$425$	0	0
$426$	−55.1769	−2.67333
$427$	−13.4641	−0.651574
$428$	17.6603i	0.853641i
$429$	−51.7128	−2.49672
$430$	0.928203i	0.0447619i
$431$	− 9.80385i	− 0.472235i	−0.971725	−	0.236117i	$-0.924125\pi$
$431$	− 9.80385i	− 0.472235i	0.971725	−	0.236117i	$-0.0758750\pi$
$432$	20.0000i	0.962250i
$433$	17.8564	0.858124	0.429062	−	0.903275i	$-0.358844\pi$
$433$	17.8564	0.858124	0.429062	+	0.903275i	$0.358844\pi$
$434$	−15.4641	−0.742301
$435$	9.46410i	0.453769i
$436$	10.0000i	0.478913i
$437$	12.0000i	0.574038i
$438$	−30.2487	−1.44534
$439$	− 20.0526i	− 0.957056i	−0.878072	−	0.478528i	$-0.841170\pi$
$439$	− 20.0526i	− 0.957056i	0.878072	−	0.478528i	$-0.158830\pi$
$440$	−8.19615	−0.390736
$441$	2.07180	0.0986570
$442$	0	0
$443$	−12.9282	−0.614237	−0.307119	−	0.951671i	$-0.599365\pi$
$443$	−12.9282	−0.614237	−0.307119	+	0.951671i	$0.599365\pi$
$444$	1.46410	0.0694832
$445$	4.39230i	0.208215i
$446$	45.7128	2.16456
$447$	− 16.3923i	− 0.775329i
$448$	2.73205i	0.129077i
$449$	34.3923i	1.62307i	0.584302	+	0.811537i	$0.301369\pi$
$449$	34.3923i	1.62307i	−0.584302	+	0.811537i	$0.698631\pi$
$450$	7.73205	0.364492
$451$	16.3923	0.771883
$452$	17.3205i	0.814688i
$453$	4.00000i	0.187936i
$454$	− 39.3731i	− 1.84787i
$455$	10.9282	0.512322
$456$	− 6.92820i	− 0.324443i
$457$	36.7846	1.72071	0.860356	−	0.509694i	$-0.170241\pi$
$457$	36.7846	1.72071	0.860356	+	0.509694i	$0.170241\pi$
$458$	14.5359	0.679218
$459$	0	0
$460$	−8.19615	−0.382148
$461$	24.9282	1.16102	0.580511	−	0.814252i	$-0.302853\pi$
$461$	24.9282	1.16102	0.580511	+	0.814252i	$0.302853\pi$
$462$	61.1769i	2.84621i
$463$	−23.8564	−1.10870	−0.554351	−	0.832283i	$-0.687033\pi$
$463$	−23.8564	−1.10870	−0.554351	+	0.832283i	$0.687033\pi$
$464$	17.3205i	0.804084i
$465$	− 8.92820i	− 0.414036i
$466$	10.3923i	0.481414i
$467$	−1.60770	−0.0743953	−0.0371976	−	0.999308i	$-0.511843\pi$
$467$	−1.60770	−0.0743953	−0.0371976	+	0.999308i	$0.511843\pi$
$468$	17.8564	0.825413
$469$	− 27.3205i	− 1.26154i
$470$	22.3923i	1.03288i
$471$	24.3923i	1.12394i
$472$	4.39230	0.202172
$473$	− 2.53590i	− 0.116601i
$474$	69.0333	3.17081
$475$	−1.46410	−0.0671776
$476$	0	0
$477$	26.7846	1.22638
$478$	−36.0000	−1.64660
$479$	− 11.6603i	− 0.532771i	−0.963867	−	0.266385i	$-0.914171\pi$
$479$	− 11.6603i	− 0.532771i	0.963867	−	0.266385i	$-0.0858294\pi$
$480$	14.1962	0.647963
$481$	2.14359i	0.0977395i
$482$	− 9.71281i	− 0.442407i
$483$	− 61.1769i	− 2.78365i
$484$	−11.3923	−0.517832
$485$	4.92820	0.223778
$486$	− 32.4449i	− 1.47173i
$487$	− 24.9808i	− 1.13199i	−0.824410	−	0.565993i	$-0.808493\pi$
$487$	− 24.9808i	− 1.13199i	0.824410	−	0.565993i	$-0.191507\pi$
$488$	− 8.53590i	− 0.386402i
$489$	−0.535898	−0.0242342
$490$	− 0.803848i	− 0.0363141i
$491$	19.6077	0.884883	0.442441	−	0.896797i	$-0.354112\pi$
$491$	19.6077	0.884883	0.442441	+	0.896797i	$0.354112\pi$
$492$	−9.46410	−0.426675
$493$	0	0
$494$	−10.1436	−0.456382
$495$	−21.1244	−0.949469
$496$	− 16.3397i	− 0.733676i
$497$	−31.8564	−1.42896
$498$	− 40.3923i	− 1.81002i
$499$	15.6603i	0.701049i	0.936554	+	0.350525i	$0.113997\pi$
$499$	15.6603i	0.701049i	−0.936554	+	0.350525i	$0.886003\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	−34.3923	−1.53653
$502$	12.0000	0.535586
$503$	− 15.1244i	− 0.674362i	−0.941440	−	0.337181i	$-0.890527\pi$
$503$	− 15.1244i	− 0.674362i	0.941440	−	0.337181i	$-0.109473\pi$
$504$	21.1244i	0.940954i
$505$	− 9.46410i	− 0.421147i
$506$	67.1769	2.98638
$507$	8.19615i	0.364004i
$508$	14.3923	0.638555
$509$	19.8564	0.880120	0.440060	−	0.897968i	$-0.354957\pi$
$509$	19.8564	0.880120	0.440060	+	0.897968i	$0.354957\pi$
$510$	0	0
$511$	−17.4641	−0.772566
$512$	8.66025	0.382733
$513$	− 5.85641i	− 0.258567i
$514$	12.0000	0.529297
$515$	8.92820i	0.393424i
$516$	1.46410i	0.0644535i
$517$	− 61.1769i	− 2.69056i
$518$	2.53590	0.111421
$519$	−9.46410	−0.415428
$520$	6.92820i	0.303822i
$521$	− 4.14359i	− 0.181534i	−0.995872	−	0.0907671i	$-0.971068\pi$
$521$	− 4.14359i	− 0.181534i	0.995872	−	0.0907671i	$-0.0289319\pi$
$522$	26.7846i	1.17233i
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	− 2.19615i	− 0.0959394i
$525$	7.46410	0.325760
$526$	2.78461	0.121415
$527$	0	0
$528$	−64.6410	−2.81314
$529$	−44.1769	−1.92074
$530$	− 10.3923i	− 0.451413i
$531$	11.3205	0.491268
$532$	4.00000i	0.173422i
$533$	− 13.8564i	− 0.600188i
$534$	20.7846i	0.899438i
$535$	−17.6603	−0.763519
$536$	17.3205	0.748132
$537$	30.9282i	1.33465i
$538$	1.60770i	0.0693127i
$539$	2.19615i	0.0945950i
$540$	4.00000	0.172133
$541$	39.1769i	1.68435i	0.539207	+	0.842174i	$0.318724\pi$
$541$	39.1769i	1.68435i	−0.539207	+	0.842174i	$0.681276\pi$
$542$	5.07180	0.217852
$543$	−6.53590	−0.280482
$544$	0	0
$545$	−10.0000	−0.428353
$546$	51.7128	2.21310
$547$	− 39.9090i	− 1.70638i	−0.521597	−	0.853192i	$-0.674663\pi$
$547$	− 39.9090i	− 1.70638i	0.521597	−	0.853192i	$-0.325337\pi$
$548$	0	0
$549$	− 22.0000i	− 0.938937i
$550$	8.19615i	0.349485i
$551$	− 5.07180i	− 0.216066i
$552$	38.7846	1.65078
$553$	39.8564	1.69487
$554$	36.2487i	1.54006i
$555$	1.46410i	0.0621477i
$556$	− 3.66025i	− 0.155229i
$557$	−6.92820	−0.293557	−0.146779	−	0.989169i	$-0.546891\pi$
$557$	−6.92820	−0.293557	−0.146779	+	0.989169i	$0.546891\pi$
$558$	− 25.2679i	− 1.06968i
$559$	−2.14359	−0.0906643
$560$	13.6603	0.577251
$561$	0	0
$562$	22.3923	0.944562
$563$	−27.4641	−1.15747	−0.578737	−	0.815514i	$-0.696454\pi$
$563$	−27.4641	−1.15747	−0.578737	+	0.815514i	$0.696454\pi$
$564$	35.3205i	1.48726i
$565$	−17.3205	−0.728679
$566$	9.12436i	0.383525i
$567$	− 6.73205i	− 0.282720i
$568$	− 20.1962i	− 0.847412i
$569$	−40.6410	−1.70376	−0.851880	−	0.523737i	$-0.824537\pi$
$569$	−40.6410	−1.70376	−0.851880	+	0.523737i	$0.824537\pi$
$570$	−6.92820	−0.290191
$571$	36.4449i	1.52517i	0.646888	+	0.762585i	$0.276070\pi$
$571$	36.4449i	1.52517i	−0.646888	+	0.762585i	$0.723930\pi$
$572$	18.9282i	0.791428i
$573$	5.07180i	0.211877i
$574$	−16.3923	−0.684202
$575$	− 8.19615i	− 0.341803i
$576$	−4.46410	−0.186004
$577$	−38.6410	−1.60865	−0.804323	−	0.594192i	$-0.797472\pi$
$577$	−38.6410	−1.60865	−0.804323	+	0.594192i	$0.797472\pi$
$578$	0	0
$579$	45.1769	1.87749
$580$	3.46410	0.143839
$581$	− 23.3205i	− 0.967498i
$582$	23.3205	0.966666
$583$	28.3923i	1.17589i
$584$	− 11.0718i	− 0.458154i
$585$	17.8564i	0.738272i
$586$	−1.60770	−0.0664133
$587$	−46.3923	−1.91482	−0.957408	−	0.288740i	$-0.906764\pi$
$587$	−46.3923	−1.91482	−0.957408	+	0.288740i	$0.906764\pi$
$588$	− 1.26795i	− 0.0522893i
$589$	4.78461i	0.197146i
$590$	− 4.39230i	− 0.180828i
$591$	47.3205	1.94651
$592$	2.67949i	0.110126i
$593$	−19.8564	−0.815405	−0.407702	−	0.913115i	$-0.633670\pi$
$593$	−19.8564	−0.815405	−0.407702	+	0.913115i	$0.633670\pi$
$594$	−32.7846	−1.34517
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−27.8564	−1.14009
$598$	− 56.7846i	− 2.32210i
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	4.73205i	0.193185i
$601$	− 8.24871i	− 0.336472i	−0.985747	−	0.168236i	$-0.946193\pi$
$601$	− 8.24871i	− 0.336472i	0.985747	−	0.168236i	$-0.0538071\pi$
$602$	2.53590i	0.103356i
$603$	44.6410	1.81792
$604$	1.46410	0.0595734
$605$	− 11.3923i	− 0.463163i
$606$	− 44.7846i	− 1.81925i
$607$	− 21.6603i	− 0.879163i	−0.898203	−	0.439581i	$-0.855127\pi$
$607$	− 21.6603i	− 0.879163i	0.898203	−	0.439581i	$-0.144873\pi$
$608$	−7.60770	−0.308533
$609$	25.8564i	1.04775i
$610$	−8.53590	−0.345608
$611$	−51.7128	−2.09208
$612$	0	0
$613$	15.8564	0.640434	0.320217	−	0.947344i	$-0.396244\pi$
$613$	15.8564	0.640434	0.320217	+	0.947344i	$0.396244\pi$
$614$	−17.3205	−0.698999
$615$	− 9.46410i	− 0.381629i
$616$	−22.3923	−0.902212
$617$	− 27.4641i	− 1.10566i	−0.833293	−	0.552832i	$-0.813547\pi$
$617$	− 27.4641i	− 1.10566i	0.833293	−	0.552832i	$-0.186453\pi$
$618$	42.2487i	1.69949i
$619$	− 38.5885i	− 1.55100i	−0.631347	−	0.775501i	$-0.717498\pi$
$619$	− 38.5885i	− 1.55100i	0.631347	−	0.775501i	$-0.282502\pi$
$620$	−3.26795	−0.131244
$621$	32.7846	1.31560
$622$	− 27.8038i	− 1.11483i
$623$	12.0000i	0.480770i
$624$	54.6410i	2.18739i
$625$	1.00000	0.0400000
$626$	45.7128i	1.82705i
$627$	18.9282	0.755920
$628$	8.92820	0.356274
$629$	0	0
$630$	21.1244	0.841614
$631$	32.3923	1.28952	0.644759	−	0.764386i	$-0.276958\pi$
$631$	32.3923	1.28952	0.644759	+	0.764386i	$0.276958\pi$
$632$	25.2679i	1.00511i
$633$	27.8564	1.10719
$634$	43.1769i	1.71477i
$635$	14.3923i	0.571141i
$636$	− 16.3923i	− 0.649997i
$637$	1.85641	0.0735535
$638$	−28.3923	−1.12406
$639$	− 52.0526i	− 2.05917i
$640$	12.1244i	0.479257i
$641$	31.1769i	1.23141i	0.787975	+	0.615707i	$0.211130\pi$
$641$	31.1769i	1.23141i	−0.787975	+	0.615707i	$0.788870\pi$
$642$	−83.5692	−3.29821
$643$	− 24.1962i	− 0.954203i	−0.878848	−	0.477102i	$-0.841687\pi$
$643$	− 24.1962i	− 0.954203i	0.878848	−	0.477102i	$-0.158313\pi$
$644$	−22.3923	−0.882380
$645$	−1.46410	−0.0576489
$646$	0	0
$647$	−38.7846	−1.52478	−0.762390	−	0.647118i	$-0.775974\pi$
$647$	−38.7846	−1.52478	−0.762390	+	0.647118i	$0.775974\pi$
$648$	4.26795	0.167661
$649$	12.0000i	0.471041i
$650$	6.92820	0.271746
$651$	− 24.3923i	− 0.956010i
$652$	0.196152i	0.00768192i
$653$	25.6077i	1.00211i	0.865416	+	0.501053i	$0.167054\pi$
$653$	25.6077i	1.00211i	−0.865416	+	0.501053i	$0.832946\pi$
$654$	−47.3205	−1.85038
$655$	2.19615	0.0858108
$656$	− 17.3205i	− 0.676252i
$657$	− 28.5359i	− 1.11329i
$658$	61.1769i	2.38492i
$659$	−32.7846	−1.27711	−0.638554	−	0.769577i	$-0.720467\pi$
$659$	−32.7846	−1.27711	−0.638554	+	0.769577i	$0.720467\pi$
$660$	12.9282i	0.503230i
$661$	8.14359	0.316749	0.158375	−	0.987379i	$-0.449375\pi$
$661$	8.14359	0.316749	0.158375	+	0.987379i	$0.449375\pi$
$662$	11.3205	0.439984
$663$	0	0
$664$	14.7846	0.573754
$665$	−4.00000	−0.155113
$666$	4.14359i	0.160561i
$667$	28.3923	1.09935
$668$	12.5885i	0.487062i
$669$	72.1051i	2.78774i
$670$	− 17.3205i	− 0.669150i
$671$	23.3205	0.900278
$672$	38.7846	1.49615
$673$	− 23.4641i	− 0.904475i	−0.891898	−	0.452237i	$-0.850626\pi$
$673$	− 23.4641i	− 0.904475i	0.891898	−	0.452237i	$-0.149374\pi$
$674$	− 11.7513i	− 0.452643i
$675$	4.00000i	0.153960i
$676$	3.00000	0.115385
$677$	2.78461i	0.107021i	0.998567	+	0.0535106i	$0.0170411\pi$
$677$	2.78461i	0.107021i	−0.998567	+	0.0535106i	$0.982959\pi$
$678$	−81.9615	−3.14771
$679$	13.4641	0.516705
$680$	0	0
$681$	62.1051	2.37987
$682$	26.7846	1.02564
$683$	− 30.8372i	− 1.17995i	−0.807421	−	0.589976i	$-0.799137\pi$
$683$	− 30.8372i	− 1.17995i	0.807421	−	0.589976i	$-0.200863\pi$
$684$	−6.53590	−0.249906
$685$	0	0
$686$	30.9282i	1.18084i
$687$	22.9282i	0.874766i
$688$	−2.67949	−0.102155
$689$	24.0000	0.914327
$690$	− 38.7846i	− 1.47650i
$691$	38.9808i	1.48290i	0.671009	+	0.741449i	$0.265861\pi$
$691$	38.9808i	1.48290i	−0.671009	+	0.741449i	$0.734139\pi$
$692$	3.46410i	0.131685i
$693$	−57.7128	−2.19233
$694$	− 6.58846i	− 0.250094i
$695$	3.66025	0.138841
$696$	−16.3923	−0.621349
$697$	0	0
$698$	−18.6795	−0.707029
$699$	−16.3923	−0.620014
$700$	− 2.73205i	− 0.103262i
$701$	11.3205	0.427570	0.213785	−	0.976881i	$-0.431421\pi$
$701$	11.3205	0.427570	0.213785	+	0.976881i	$0.431421\pi$
$702$	27.7128i	1.04595i
$703$	− 0.784610i	− 0.0295921i
$704$	− 4.73205i	− 0.178346i
$705$	−35.3205	−1.33025
$706$	46.3923	1.74600
$707$	− 25.8564i	− 0.972430i
$708$	− 6.92820i	− 0.260378i
$709$	4.53590i	0.170349i	0.996366	+	0.0851746i	$0.0271448\pi$
$709$	4.53590i	0.170349i	−0.996366	+	0.0851746i	$0.972855\pi$
$710$	−20.1962	−0.757948
$711$	65.1244i	2.44235i
$712$	−7.60770	−0.285110
$713$	−26.7846	−1.00309
$714$	0	0
$715$	−18.9282	−0.707875
$716$	11.3205	0.423067
$717$	− 56.7846i	− 2.12066i
$718$	37.1769	1.38743
$719$	− 5.41154i	− 0.201816i	−0.994896	−	0.100908i	$-0.967825\pi$
$719$	− 5.41154i	− 0.201816i	0.994896	−	0.100908i	$-0.0321749\pi$
$720$	22.3205i	0.831836i
$721$	24.3923i	0.908417i
$722$	−29.1962	−1.08657
$723$	15.3205	0.569776
$724$	2.39230i	0.0889093i
$725$	3.46410i	0.128654i
$726$	− 53.9090i	− 2.00075i
$727$	0.143594	0.00532559	0.00266279	−	0.999996i	$-0.499152\pi$
$727$	0.143594	0.00532559	0.00266279	+	0.999996i	$0.499152\pi$
$728$	18.9282i	0.701526i
$729$	43.7846	1.62165
$730$	−11.0718	−0.409786
$731$	0	0
$732$	−13.4641	−0.497648
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	13.5167i	0.498909i
$735$	1.26795	0.0467690
$736$	− 42.5885i	− 1.56983i
$737$	47.3205i	1.74307i
$738$	− 26.7846i	− 0.985955i
$739$	11.6077	0.426996	0.213498	−	0.976944i	$-0.431514\pi$
$739$	11.6077	0.426996	0.213498	+	0.976944i	$0.431514\pi$
$740$	0.535898	0.0197000
$741$	− 16.0000i	− 0.587775i
$742$	− 28.3923i	− 1.04231i
$743$	− 22.7321i	− 0.833958i	−0.908916	−	0.416979i	$-0.863089\pi$
$743$	− 22.7321i	− 0.833958i	0.908916	−	0.416979i	$-0.136911\pi$
$744$	15.4641	0.566941
$745$	− 6.00000i	− 0.219823i
$746$	34.6410	1.26830
$747$	38.1051	1.39419
$748$	0	0
$749$	−48.2487	−1.76297
$750$	4.73205	0.172790
$751$	43.6603i	1.59319i	0.604516	+	0.796593i	$0.293366\pi$
$751$	43.6603i	1.59319i	−0.604516	+	0.796593i	$0.706634\pi$
$752$	−64.6410	−2.35722
$753$	18.9282i	0.689782i
$754$	24.0000i	0.874028i
$755$	1.46410i	0.0532841i
$756$	10.9282	0.397455
$757$	−30.6410	−1.11367	−0.556833	−	0.830624i	$-0.687984\pi$
$757$	−30.6410	−1.11367	−0.556833	+	0.830624i	$0.687984\pi$
$758$	30.8372i	1.12006i
$759$	105.962i	3.84616i
$760$	− 2.53590i	− 0.0919867i
$761$	−28.3923	−1.02922	−0.514610	−	0.857424i	$-0.672063\pi$
$761$	−28.3923	−1.02922	−0.514610	+	0.857424i	$0.672063\pi$
$762$	68.1051i	2.46719i
$763$	−27.3205	−0.989069
$764$	1.85641	0.0671624
$765$	0	0
$766$	−26.7846	−0.967767
$767$	10.1436	0.366264
$768$	51.9090i	1.87310i
$769$	36.3923	1.31234	0.656170	−	0.754613i	$-0.272175\pi$
$769$	36.3923	1.31234	0.656170	+	0.754613i	$0.272175\pi$
$770$	22.3923i	0.806963i
$771$	18.9282i	0.681683i
$772$	− 16.5359i	− 0.595140i
$773$	46.6410	1.67756	0.838780	−	0.544470i	$-0.183269\pi$
$773$	46.6410	1.67756	0.838780	+	0.544470i	$0.183269\pi$
$774$	−4.14359	−0.148938
$775$	− 3.26795i	− 0.117388i
$776$	8.53590i	0.306421i
$777$	4.00000i	0.143499i
$778$	7.60770	0.272749
$779$	5.07180i	0.181716i
$780$	10.9282	0.391292
$781$	55.1769	1.97439
$782$	0	0
$783$	−13.8564	−0.495188
$784$	2.32051	0.0828753
$785$	8.92820i	0.318661i
$786$	10.3923	0.370681
$787$	43.9090i	1.56519i	0.622534	+	0.782593i	$0.286103\pi$
$787$	43.9090i	1.56519i	−0.622534	+	0.782593i	$0.713897\pi$
$788$	− 17.3205i	− 0.617018i
$789$	4.39230i	0.156370i
$790$	25.2679	0.898993
$791$	−47.3205	−1.68252
$792$	− 36.5885i	− 1.30011i
$793$	− 19.7128i	− 0.700023i
$794$	− 24.2487i	− 0.860555i
$795$	16.3923	0.581375
$796$	10.1962i	0.361393i
$797$	24.9282	0.883002	0.441501	−	0.897261i	$-0.354446\pi$
$797$	24.9282	0.883002	0.441501	+	0.897261i	$0.354446\pi$
$798$	−18.9282	−0.670051
$799$	0	0
$800$	5.19615	0.183712
$801$	−19.6077	−0.692804
$802$	− 22.3923i	− 0.790700i
$803$	30.2487	1.06745
$804$	− 27.3205i	− 0.963520i
$805$	− 22.3923i	− 0.789225i
$806$	− 22.6410i	− 0.797496i
$807$	−2.53590	−0.0892679
$808$	16.3923	0.576679
$809$	55.8564i	1.96381i	0.189383	+	0.981903i	$0.439351\pi$
$809$	55.8564i	1.96381i	−0.189383	+	0.981903i	$0.560649\pi$
$810$	− 4.26795i	− 0.149960i
$811$	44.8372i	1.57445i	0.616668	+	0.787223i	$0.288482\pi$
$811$	44.8372i	1.57445i	−0.616668	+	0.787223i	$0.711518\pi$
$812$	9.46410	0.332125
$813$	8.00000i	0.280572i
$814$	−4.39230	−0.153950
$815$	−0.196152	−0.00687092
$816$	0	0
$817$	0.784610	0.0274500
$818$	45.0333	1.57455
$819$	48.7846i	1.70467i
$820$	−3.46410	−0.120972
$821$	− 24.9282i	− 0.870000i	−0.900430	−	0.435000i	$-0.856748\pi$
$821$	− 24.9282i	− 0.870000i	0.900430	−	0.435000i	$-0.143252\pi$
$822$	0	0
$823$	8.98076i	0.313050i	0.987674	+	0.156525i	$0.0500291\pi$
$823$	8.98076i	0.313050i	−0.987674	+	0.156525i	$0.949971\pi$
$824$	−15.4641	−0.538718
$825$	−12.9282	−0.450102
$826$	− 12.0000i	− 0.417533i
$827$	15.8038i	0.549554i	0.961508	+	0.274777i	$0.0886040\pi$
$827$	15.8038i	0.549554i	−0.961508	+	0.274777i	$0.911396\pi$
$828$	− 36.5885i	− 1.27154i
$829$	17.7128	0.615191	0.307596	−	0.951517i	$-0.400476\pi$
$829$	17.7128	0.615191	0.307596	+	0.951517i	$0.400476\pi$
$830$	− 14.7846i	− 0.513181i
$831$	−57.1769	−1.98345
$832$	−4.00000	−0.138675
$833$	0	0
$834$	17.3205	0.599760
$835$	−12.5885	−0.435642
$836$	− 6.92820i	− 0.239617i
$837$	13.0718	0.451827
$838$	− 66.1577i	− 2.28538i
$839$	− 19.2679i	− 0.665203i	−0.943067	−	0.332602i	$-0.892074\pi$
$839$	− 19.2679i	− 0.665203i	0.943067	−	0.332602i	$-0.107926\pi$
$840$	12.9282i	0.446065i
$841$	17.0000	0.586207
$842$	9.46410	0.326154
$843$	35.3205i	1.21650i
$844$	− 10.1962i	− 0.350966i
$845$	3.00000i	0.103203i
$846$	−99.9615	−3.43675
$847$	− 31.1244i	− 1.06945i
$848$	30.0000	1.03020
$849$	−14.3923	−0.493943
$850$	0	0
$851$	4.39230	0.150566
$852$	−31.8564	−1.09138
$853$	− 23.1769i	− 0.793562i	−0.917913	−	0.396781i	$-0.870127\pi$
$853$	− 23.1769i	− 0.793562i	0.917913	−	0.396781i	$-0.129873\pi$
$854$	−23.3205	−0.798011
$855$	− 6.53590i	− 0.223523i
$856$	− 30.5885i	− 1.04549i
$857$	− 31.1769i	− 1.06498i	−0.846435	−	0.532492i	$-0.821256\pi$
$857$	− 31.1769i	− 1.06498i	0.846435	−	0.532492i	$-0.178744\pi$
$858$	−89.5692	−3.05784
$859$	25.4641	0.868824	0.434412	−	0.900714i	$-0.356956\pi$
$859$	25.4641	0.868824	0.434412	+	0.900714i	$0.356956\pi$
$860$	0.535898i	0.0182740i
$861$	− 25.8564i	− 0.881184i
$862$	− 16.9808i	− 0.578367i
$863$	23.0718	0.785373	0.392687	−	0.919672i	$-0.371546\pi$
$863$	23.0718	0.785373	0.392687	+	0.919672i	$0.371546\pi$
$864$	20.7846i	0.707107i
$865$	−3.46410	−0.117783
$866$	30.9282	1.05098
$867$	0	0
$868$	−8.92820	−0.303043
$869$	−69.0333	−2.34180
$870$	16.3923i	0.555751i
$871$	40.0000	1.35535
$872$	− 17.3205i	− 0.586546i
$873$	22.0000i	0.744587i
$874$	20.7846i	0.703050i
$875$	2.73205	0.0923602
$876$	−17.4641	−0.590057
$877$	1.21539i	0.0410408i	0.999789	+	0.0205204i	$0.00653231\pi$
$877$	1.21539i	0.0410408i	−0.999789	+	0.0205204i	$0.993468\pi$
$878$	− 34.7321i	− 1.17215i
$879$	− 2.53590i	− 0.0855337i
$880$	−23.6603	−0.797587
$881$	− 41.3205i	− 1.39212i	−0.717982	−	0.696062i	$-0.754934\pi$
$881$	− 41.3205i	− 1.39212i	0.717982	−	0.696062i	$-0.245066\pi$
$882$	3.58846	0.120830
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	0	0
$885$	6.92820	0.232889
$886$	−22.3923	−0.752284
$887$	3.12436i	0.104906i	0.998623	+	0.0524528i	$0.0167039\pi$
$887$	3.12436i	0.104906i	−0.998623	+	0.0524528i	$0.983296\pi$
$888$	−2.53590	−0.0850992
$889$	39.3205i	1.31877i
$890$	7.60770i	0.255011i
$891$	11.6603i	0.390633i
$892$	26.3923	0.883680
$893$	18.9282	0.633408
$894$	− 28.3923i	− 0.949581i
$895$	11.3205i	0.378403i
$896$	33.1244i	1.10661i
$897$	89.5692	2.99063
$898$	59.5692i	1.98785i
$899$	11.3205	0.377560
$900$	4.46410	0.148803
$901$	0	0
$902$	28.3923	0.945360
$903$	−4.00000	−0.133112
$904$	− 30.0000i	− 0.997785i
$905$	−2.39230	−0.0795229
$906$	6.92820i	0.230174i
$907$	− 30.7321i	− 1.02044i	−0.860044	−	0.510220i	$-0.829564\pi$
$907$	− 30.7321i	− 1.02044i	0.860044	−	0.510220i	$-0.170436\pi$
$908$	− 22.7321i	− 0.754390i
$909$	42.2487	1.40130
$910$	18.9282	0.627464
$911$	− 36.3397i	− 1.20399i	−0.798500	−	0.601995i	$-0.794373\pi$
$911$	− 36.3397i	− 1.20399i	0.798500	−	0.601995i	$-0.205627\pi$
$912$	− 20.0000i	− 0.662266i
$913$	40.3923i	1.33679i
$914$	63.7128	2.10743
$915$	− 13.4641i	− 0.445109i
$916$	8.39230	0.277290
$917$	6.00000	0.198137
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	14.1962	0.468033
$921$	− 27.3205i	− 0.900241i
$922$	43.1769	1.42196
$923$	− 46.6410i	− 1.53521i
$924$	35.3205i	1.16196i
$925$	0.535898i	0.0176202i
$926$	−41.3205	−1.35788
$927$	−39.8564	−1.30906
$928$	18.0000i	0.590879i
$929$	3.46410i	0.113653i	0.998384	+	0.0568267i	$0.0180983\pi$
$929$	3.46410i	0.113653i	−0.998384	+	0.0568267i	$0.981902\pi$
$930$	− 15.4641i	− 0.507088i
$931$	−0.679492	−0.0222694
$932$	6.00000i	0.196537i
$933$	43.8564	1.43579
$934$	−2.78461	−0.0911152
$935$	0	0
$936$	−30.9282	−1.01092
$937$	32.6410	1.06634	0.533168	−	0.846010i	$-0.321001\pi$
$937$	32.6410	1.06634	0.533168	+	0.846010i	$0.321001\pi$
$938$	− 47.3205i	− 1.54507i
$939$	−72.1051	−2.35306
$940$	12.9282i	0.421671i
$941$	− 48.2487i	− 1.57286i	−0.617677	−	0.786432i	$-0.711926\pi$
$941$	− 48.2487i	− 1.57286i	0.617677	−	0.786432i	$-0.288074\pi$
$942$	42.2487i	1.37654i
$943$	−28.3923	−0.924581
$944$	12.6795	0.412682
$945$	10.9282i	0.355494i
$946$	− 4.39230i	− 0.142806i
$947$	8.19615i	0.266339i	0.991093	+	0.133170i	$0.0425155\pi$
$947$	8.19615i	0.266339i	−0.991093	+	0.133170i	$0.957485\pi$
$948$	39.8564	1.29448
$949$	− 25.5692i	− 0.830012i
$950$	−2.53590	−0.0822754
$951$	−68.1051	−2.20846
$952$	0	0
$953$	−8.78461	−0.284561	−0.142281	−	0.989826i	$-0.545444\pi$
$953$	−8.78461	−0.284561	−0.142281	+	0.989826i	$0.545444\pi$
$954$	46.3923	1.50201
$955$	1.85641i	0.0600719i
$956$	−20.7846	−0.672222
$957$	− 44.7846i	− 1.44768i
$958$	− 20.1962i	− 0.652508i
$959$	0	0
$960$	−2.73205	−0.0881766
$961$	20.3205	0.655500
$962$	3.71281i	0.119706i
$963$	− 78.8372i	− 2.54049i
$964$	− 5.60770i	− 0.180612i
$965$	16.5359	0.532309
$966$	− 105.962i	− 3.40926i
$967$	−39.1769	−1.25984	−0.629922	−	0.776658i	$-0.716913\pi$
$967$	−39.1769	−1.25984	−0.629922	+	0.776658i	$0.716913\pi$
$968$	19.7321	0.634212
$969$	0	0
$970$	8.53590	0.274071
$971$	−54.2487	−1.74092	−0.870462	−	0.492236i	$-0.836180\pi$
$971$	−54.2487	−1.74092	−0.870462	+	0.492236i	$0.836180\pi$
$972$	− 18.7321i	− 0.600831i
$973$	10.0000	0.320585
$974$	− 43.2679i	− 1.38639i
$975$	10.9282i	0.349983i
$976$	− 24.6410i	− 0.788740i
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−0.928203	−0.0296807
$979$	− 20.7846i	− 0.664279i
$980$	− 0.464102i	− 0.0148252i
$981$	− 44.6410i	− 1.42528i
$982$	33.9615	1.08376
$983$	58.7321i	1.87326i	0.350318	+	0.936631i	$0.386073\pi$
$983$	58.7321i	1.87326i	−0.350318	+	0.936631i	$0.613927\pi$
$984$	16.3923	0.522568
$985$	17.3205	0.551877
$986$	0	0
$987$	−96.4974	−3.07155
$988$	−5.85641	−0.186317
$989$	4.39230i	0.139667i
$990$	−36.5885	−1.16286
$991$	− 2.98076i	− 0.0946870i	−0.998879	−	0.0473435i	$-0.984924\pi$
$991$	− 2.98076i	− 0.0946870i	0.998879	−	0.0473435i	$-0.0150755\pi$
$992$	− 16.9808i	− 0.539140i
$993$	17.8564i	0.566656i
$994$	−55.1769	−1.75011
$995$	−10.1962	−0.323240
$996$	− 23.3205i	− 0.738939i
$997$	2.39230i	0.0757651i	0.999282	+	0.0378825i	$0.0120613\pi$
$997$	2.39230i	0.0757651i	−0.999282	+	0.0378825i	$0.987939\pi$
$998$	27.1244i	0.858606i
$999$	−2.14359	−0.0678203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1445.2.d.e.866.4		4
17.4	even	4		85.2.a.c.1.1	✓	2
17.13	even	4		1445.2.a.g.1.1		2
17.16	even	2	inner	1445.2.d.e.866.3		4
51.38	odd	4		765.2.a.g.1.2		2
68.55	odd	4		1360.2.a.k.1.1		2
85.4	even	4		425.2.a.e.1.2		2
85.38	odd	4		425.2.b.d.324.4		4
85.64	even	4		7225.2.a.l.1.2		2
85.72	odd	4		425.2.b.d.324.1		4
119.55	odd	4		4165.2.a.t.1.1		2
136.21	even	4		5440.2.a.bb.1.1		2
136.123	odd	4		5440.2.a.bl.1.2		2
255.89	odd	4		3825.2.a.v.1.1		2
340.259	odd	4		6800.2.a.bg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.c.1.1	✓	2	17.4	even	4
425.2.a.e.1.2		2	85.4	even	4
425.2.b.d.324.1		4	85.72	odd	4
425.2.b.d.324.4		4	85.38	odd	4
765.2.a.g.1.2		2	51.38	odd	4
1360.2.a.k.1.1		2	68.55	odd	4
1445.2.a.g.1.1		2	17.13	even	4
1445.2.d.e.866.3		4	17.16	even	2	inner
1445.2.d.e.866.4		4	1.1	even	1	trivial
3825.2.a.v.1.1		2	255.89	odd	4
4165.2.a.t.1.1		2	119.55	odd	4
5440.2.a.bb.1.1		2	136.21	even	4
5440.2.a.bl.1.2		2	136.123	odd	4
6800.2.a.bg.1.2		2	340.259	odd	4
7225.2.a.l.1.2		2	85.64	even	4

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:	\( N \)	\(=\)	\( 1445 = 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1445.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +2.73205i q^{3} +1.00000 q^{4} +1.00000i q^{5} +4.73205i q^{6} +2.73205i q^{7} -1.73205 q^{8} -4.46410 q^{9} +1.73205i q^{10} -4.73205i q^{11} +2.73205i q^{12} -4.00000 q^{13} +4.73205i q^{14} -2.73205 q^{15} -5.00000 q^{16} -7.73205 q^{18} +1.46410 q^{19} +1.00000i q^{20} -7.46410 q^{21} -8.19615i q^{22} +8.19615i q^{23} -4.73205i q^{24} -1.00000 q^{25} -6.92820 q^{26} -4.00000i q^{27} +2.73205i q^{28} -3.46410i q^{29} -4.73205 q^{30} +3.26795i q^{31} -5.19615 q^{32} +12.9282 q^{33} -2.73205 q^{35} -4.46410 q^{36} -0.535898i q^{37} +2.53590 q^{38} -10.9282i q^{39} -1.73205i q^{40} +3.46410i q^{41} -12.9282 q^{42} +0.535898 q^{43} -4.73205i q^{44} -4.46410i q^{45} +14.1962i q^{46} +12.9282 q^{47} -13.6603i q^{48} -0.464102 q^{49} -1.73205 q^{50} -4.00000 q^{52} -6.00000 q^{53} -6.92820i q^{54} +4.73205 q^{55} -4.73205i q^{56} +4.00000i q^{57} -6.00000i q^{58} -2.53590 q^{59} -2.73205 q^{60} +4.92820i q^{61} +5.66025i q^{62} -12.1962i q^{63} +1.00000 q^{64} -4.00000i q^{65} +22.3923 q^{66} -10.0000 q^{67} -22.3923 q^{69} -4.73205 q^{70} +11.6603i q^{71} +7.73205 q^{72} +6.39230i q^{73} -0.928203i q^{74} -2.73205i q^{75} +1.46410 q^{76} +12.9282 q^{77} -18.9282i q^{78} -14.5885i q^{79} -5.00000i q^{80} -2.46410 q^{81} +6.00000i q^{82} -8.53590 q^{83} -7.46410 q^{84} +0.928203 q^{86} +9.46410 q^{87} +8.19615i q^{88} +4.39230 q^{89} -7.73205i q^{90} -10.9282i q^{91} +8.19615i q^{92} -8.92820 q^{93} +22.3923 q^{94} +1.46410i q^{95} -14.1962i q^{96} -4.92820i q^{97} -0.803848 q^{98} +21.1244i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{9} - 16 q^{13} - 4 q^{15} - 20 q^{16} - 24 q^{18} - 8 q^{19} - 16 q^{21} - 4 q^{25} - 12 q^{30} + 24 q^{33} - 4 q^{35} - 4 q^{36} + 24 q^{38} - 24 q^{42} + 16 q^{43} + 24 q^{47} + 12 q^{49}+ \cdots - 24 q^{98}+O(q^{100}) \) 4 * q + 4 * q^4 - 4 * q^9 - 16 * q^13 - 4 * q^15 - 20 * q^16 - 24 * q^18 - 8 * q^19 - 16 * q^21 - 4 * q^25 - 12 * q^30 + 24 * q^33 - 4 * q^35 - 4 * q^36 + 24 * q^38 - 24 * q^42 + 16 * q^43 + 24 * q^47 + 12 * q^49 - 16 * q^52 - 24 * q^53 + 12 * q^55 - 24 * q^59 - 4 * q^60 + 4 * q^64 + 48 * q^66 - 40 * q^67 - 48 * q^69 - 12 * q^70 + 24 * q^72 - 8 * q^76 + 24 * q^77 + 4 * q^81 - 48 * q^83 - 16 * q^84 - 24 * q^86 + 24 * q^87 - 24 * q^89 - 8 * q^93 + 48 * q^94 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more