LMFDB - Embedded newform 3024.2.r.f.2017.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1009,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

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

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

chi := DirichletCharacter("3024.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.r (of order $3$, degree $2$, 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:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2017.1
Root			$-1.18614 - 1.26217i$ of defining polynomial
Character	$\chi$	$=$	3024.2017
Dual form			3024.2.r.f.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.686141 - 1.18843i) q^{5} +(0.500000 - 0.866025i) q^{7} +O(q^{10})$	$q+(-0.686141 - 1.18843i) q^{5} +(0.500000 - 0.866025i) q^{7} +(-2.18614 + 3.78651i) q^{11} +(-1.00000 - 1.73205i) q^{13} +4.37228 q^{17} -5.00000 q^{19} +(3.68614 + 6.38458i) q^{23} +(1.55842 - 2.69927i) q^{25} +(1.37228 - 2.37686i) q^{29} +(1.00000 + 1.73205i) q^{31} -1.37228 q^{35} +2.00000 q^{37} +(5.18614 + 8.98266i) q^{41} +(4.55842 - 7.89542i) q^{43} +(-0.500000 - 0.866025i) q^{49} -2.74456 q^{53} +6.00000 q^{55} +(-3.55842 - 6.16337i) q^{59} +(7.05842 - 12.2255i) q^{61} +(-1.37228 + 2.37686i) q^{65} +(7.55842 + 13.0916i) q^{67} +10.1168 q^{71} -5.11684 q^{73} +(2.18614 + 3.78651i) q^{77} +(6.05842 - 10.4935i) q^{79} +(2.74456 - 4.75372i) q^{83} +(-3.00000 - 5.19615i) q^{85} -3.25544 q^{89} -2.00000 q^{91} +(3.43070 + 5.94215i) q^{95} +(-4.55842 + 7.89542i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q + 3 * q^5 + 2 * q^7	$ 4 q + 3 q^{5} + 2 q^{7} - 3 q^{11} - 4 q^{13} + 6 q^{17} - 20 q^{19} + 9 q^{23} - 11 q^{25} - 6 q^{29} + 4 q^{31} + 6 q^{35} + 8 q^{37} + 15 q^{41} + q^{43} - 2 q^{49} + 12 q^{53} + 24 q^{55} + 3 q^{59} + 11 q^{61} + 6 q^{65} + 13 q^{67} + 6 q^{71} + 14 q^{73} + 3 q^{77} + 7 q^{79} - 12 q^{83} - 12 q^{85} - 36 q^{89} - 8 q^{91} - 15 q^{95} - q^{97}+O(q^{100}) $ 4 * q + 3 * q^5 + 2 * q^7 - 3 * q^11 - 4 * q^13 + 6 * q^17 - 20 * q^19 + 9 * q^23 - 11 * q^25 - 6 * q^29 + 4 * q^31 + 6 * q^35 + 8 * q^37 + 15 * q^41 + q^43 - 2 * q^49 + 12 * q^53 + 24 * q^55 + 3 * q^59 + 11 * q^61 + 6 * q^65 + 13 * q^67 + 6 * q^71 + 14 * q^73 + 3 * q^77 + 7 * q^79 - 12 * q^83 - 12 * q^85 - 36 * q^89 - 8 * q^91 - 15 * q^95 - q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.686141	−	1.18843i	−0.306851	−	0.531482i	0.670820	−	0.741620i	$-0.265942\pi$
$5$	−0.686141	−	1.18843i	−0.306851	−	0.531482i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.18614	+	3.78651i	−0.659146	+	1.14167i	0.321691	+	0.946845i	$0.395749\pi$
$11$	−2.18614	+	3.78651i	−0.659146	+	1.14167i	−0.980837	+	0.194830i	$0.937584\pi$
$12$		0			0
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.37228			1.06043			0.530217	−	0.847862i	$-0.322110\pi$
$17$	4.37228			1.06043			0.530217	+	0.847862i	$0.322110\pi$
$18$		0			0
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.68614	+	6.38458i	0.768613	+	1.33128i	0.938315	+	0.345782i	$0.112386\pi$
$23$	3.68614	+	6.38458i	0.768613	+	1.33128i	−0.169701	+	0.985496i	$0.554280\pi$
$24$		0			0
$25$	1.55842	−	2.69927i	0.311684	−	0.539853i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.37228	−	2.37686i	0.254826	−	0.441372i	−0.710022	−	0.704179i	$-0.751315\pi$
$29$	1.37228	−	2.37686i	0.254826	−	0.441372i	0.964848	+	0.262807i	$0.0846484\pi$
$30$		0			0
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	0.941745	−	0.336327i	$-0.109185\pi$
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	−0.762140	+	0.647412i	$0.775851\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.37228			−0.231958
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.18614	+	8.98266i	0.809939	+	1.40286i	0.912906	+	0.408171i	$0.133833\pi$
$41$	5.18614	+	8.98266i	0.809939	+	1.40286i	−0.102966	+	0.994685i	$0.532833\pi$
$42$		0			0
$43$	4.55842	−	7.89542i	0.695153	−	1.20404i	−0.274976	−	0.961451i	$-0.588670\pi$
$43$	4.55842	−	7.89542i	0.695153	−	1.20404i	0.970129	−	0.242589i	$-0.0779967\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.0714286	−	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.74456			−0.376995			−0.188497	−	0.982074i	$-0.560362\pi$
$53$	−2.74456			−0.376995			−0.188497	+	0.982074i	$0.560362\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.55842	−	6.16337i	−0.463267	−	0.802402i	0.535854	−	0.844310i	$-0.319990\pi$
$59$	−3.55842	−	6.16337i	−0.463267	−	0.802402i	−0.999121	+	0.0419083i	$0.986656\pi$
$60$		0			0
$61$	7.05842	−	12.2255i	0.903738	−	1.56532i	0.0811364	−	0.996703i	$-0.474145\pi$
$61$	7.05842	−	12.2255i	0.903738	−	1.56532i	0.822602	−	0.568618i	$-0.192522\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.37228	+	2.37686i	−0.170211	+	0.294813i
$66$		0			0
$67$	7.55842	+	13.0916i	0.923408	+	1.59939i	0.794101	+	0.607785i	$0.207942\pi$
$67$	7.55842	+	13.0916i	0.923408	+	1.59939i	0.129307	+	0.991605i	$0.458725\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.1168			1.20065			0.600324	−	0.799757i	$-0.295038\pi$
$71$	10.1168			1.20065			0.600324	+	0.799757i	$0.295038\pi$
$72$		0			0
$73$	−5.11684			−0.598881			−0.299441	−	0.954115i	$-0.596800\pi$
$73$	−5.11684			−0.598881			−0.299441	+	0.954115i	$0.596800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.18614	+	3.78651i	0.249134	+	0.431512i
$78$		0			0
$79$	6.05842	−	10.4935i	0.681626	−	1.18061i	−0.292859	−	0.956156i	$-0.594607\pi$
$79$	6.05842	−	10.4935i	0.681626	−	1.18061i	0.974485	−	0.224455i	$-0.0720601\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.74456	−	4.75372i	0.301255	−	0.521789i	−0.675166	−	0.737666i	$-0.735928\pi$
$83$	2.74456	−	4.75372i	0.301255	−	0.521789i	0.976420	+	0.215877i	$0.0692612\pi$
$84$		0			0
$85$	−3.00000	−	5.19615i	−0.325396	−	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.25544			−0.345076			−0.172538	−	0.985003i	$-0.555197\pi$
$89$	−3.25544			−0.345076			−0.172538	+	0.985003i	$0.555197\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.43070	+	5.94215i	0.351983	+	0.609652i
$96$		0			0
$97$	−4.55842	+	7.89542i	−0.462838	+	0.801658i	−0.999101	−	0.0423924i	$-0.986502\pi$
$97$	−4.55842	+	7.89542i	−0.462838	+	0.801658i	0.536263	+	0.844051i	$0.319835\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.68614	−	6.38458i	0.366785	−	0.635290i	−0.622276	−	0.782798i	$-0.713792\pi$
$101$	3.68614	−	6.38458i	0.366785	−	0.635290i	0.989061	+	0.147508i	$0.0471252\pi$
$102$		0			0
$103$	−5.00000	−	8.66025i	−0.492665	−	0.853320i	0.507300	−	0.861770i	$-0.330644\pi$
$103$	−5.00000	−	8.66025i	−0.492665	−	0.853320i	−0.999964	+	0.00844953i	$0.997310\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.62772			−0.157358			−0.0786788	−	0.996900i	$-0.525070\pi$
$107$	−1.62772			−0.157358			−0.0786788	+	0.996900i	$0.525070\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.686141	+	1.18843i	0.0645467	+	0.111798i	0.896493	−	0.443058i	$-0.146107\pi$
$113$	0.686141	+	1.18843i	0.0645467	+	0.111798i	−0.831946	+	0.554856i	$0.812773\pi$
$114$		0			0
$115$	5.05842	−	8.76144i	0.471700	−	0.817009i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.18614	−	3.78651i	0.200403	−	0.347108i
$120$		0			0
$121$	−4.05842	−	7.02939i	−0.368947	−	0.639036i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.1386			−0.996266
$126$		0			0
$127$	14.1168			1.25267			0.626334	−	0.779555i	$-0.284555\pi$
$127$	14.1168			1.25267			0.626334	+	0.779555i	$0.284555\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.68614	+	6.38458i	0.322060	+	0.557824i	0.980913	−	0.194448i	$-0.0622915\pi$
$131$	3.68614	+	6.38458i	0.322060	+	0.557824i	−0.658853	+	0.752271i	$0.728958\pi$
$132$		0			0
$133$	−2.50000	+	4.33013i	−0.216777	+	0.375470i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.18614	−	14.1788i	0.699389	−	1.21138i	−0.269289	−	0.963059i	$-0.586789\pi$
$137$	8.18614	−	14.1788i	0.699389	−	1.21138i	0.968678	−	0.248318i	$-0.0798779\pi$
$138$		0			0
$139$	−10.6168	−	18.3889i	−0.900509	−	1.55973i	−0.826835	−	0.562445i	$-0.809861\pi$
$139$	−10.6168	−	18.3889i	−0.900509	−	1.55973i	−0.0736742	−	0.997282i	$-0.523472\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.74456			0.731257
$144$		0			0
$145$	−3.76631			−0.312775
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.37228	+	12.7692i	0.603961	+	1.04609i	0.992215	+	0.124538i	$0.0397450\pi$
$149$	7.37228	+	12.7692i	0.603961	+	1.04609i	−0.388254	+	0.921552i	$0.626922\pi$
$150$		0			0
$151$	−4.05842	+	7.02939i	−0.330270	+	0.572044i	−0.982565	−	0.185921i	$-0.940473\pi$
$151$	−4.05842	+	7.02939i	−0.330270	+	0.572044i	0.652295	+	0.757965i	$0.273806\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.37228	−	2.37686i	0.110224	−	0.190914i
$156$		0			0
$157$	4.05842	+	7.02939i	0.323897	+	0.561007i	0.981289	−	0.192543i	$-0.0616734\pi$
$157$	4.05842	+	7.02939i	0.323897	+	0.561007i	−0.657391	+	0.753549i	$0.728340\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.37228			0.581017
$162$		0			0
$163$	−16.2337			−1.27152			−0.635760	−	0.771887i	$-0.719313\pi$
$163$	−16.2337			−1.27152			−0.635760	+	0.771887i	$0.719313\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.74456	−	15.1460i	−0.676675	−	1.17203i	−0.975976	−	0.217876i	$-0.930087\pi$
$167$	−8.74456	−	15.1460i	−0.676675	−	1.17203i	0.299302	−	0.954158i	$-0.403246\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	−0.957241	−	0.289292i	$-0.906580\pi$
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	0.729155	+	0.684349i	$0.239913\pi$
$174$		0			0
$175$	−1.55842	−	2.69927i	−0.117806	−	0.204045i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.7446			1.10206			0.551030	−	0.834485i	$-0.314235\pi$
$179$	14.7446			1.10206			0.551030	+	0.834485i	$0.314235\pi$
$180$		0			0
$181$	18.1168			1.34661			0.673307	−	0.739363i	$-0.264873\pi$
$181$	18.1168			1.34661			0.673307	+	0.739363i	$0.264873\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.37228	−	2.37686i	−0.100892	−	0.174750i
$186$		0			0
$187$	−9.55842	+	16.5557i	−0.698981	+	1.21067i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.941578	−	1.63086i	0.0681302	−	0.118005i	−0.829948	−	0.557841i	$-0.811630\pi$
$191$	0.941578	−	1.63086i	0.0681302	−	0.118005i	0.898078	+	0.439836i	$0.144963\pi$
$192$		0			0
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	0.964059	−	0.265689i	$-0.0855996\pi$
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	−0.712123	+	0.702055i	$0.752266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.37228	−	2.37686i	−0.0963153	−	0.166823i
$204$		0			0
$205$	7.11684	−	12.3267i	0.497062	−	0.860937i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.9307	−	18.9325i	0.756093	−	1.30959i
$210$		0			0
$211$	−8.00000	−	13.8564i	−0.550743	−	0.953914i	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−8.00000	−	13.8564i	−0.550743	−	0.953914i	0.447478	−	0.894295i	$-0.352322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.5109			−0.853235
$216$		0			0
$217$	2.00000			0.135769
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.37228	−	7.57301i	−0.294111	−	0.509416i
$222$		0			0
$223$	−2.00000	+	3.46410i	−0.133930	+	0.231973i	−0.925188	−	0.379509i	$-0.876093\pi$
$223$	−2.00000	+	3.46410i	−0.133930	+	0.231973i	0.791258	+	0.611482i	$0.209426\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.8723	−	20.5634i	0.787991	−	1.36484i	−0.139205	−	0.990264i	$-0.544455\pi$
$227$	11.8723	−	20.5634i	0.787991	−	1.36484i	0.927196	−	0.374577i	$-0.122212\pi$
$228$		0			0
$229$	10.0584	+	17.4217i	0.664679	+	1.15126i	0.979372	+	0.202065i	$0.0647651\pi$
$229$	10.0584	+	17.4217i	0.664679	+	1.15126i	−0.314693	+	0.949194i	$0.601902\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.7446			0.769412			0.384706	−	0.923039i	$-0.374303\pi$
$233$	11.7446			0.769412			0.384706	+	0.923039i	$0.374303\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.43070	+	16.3345i	0.610021	+	1.05659i	0.991236	+	0.132102i	$0.0421725\pi$
$239$	9.43070	+	16.3345i	0.610021	+	1.05659i	−0.381215	+	0.924487i	$0.624494\pi$
$240$		0			0
$241$	−0.441578	+	0.764836i	−0.0284445	+	0.0492674i	−0.879897	−	0.475164i	$-0.842389\pi$
$241$	−0.441578	+	0.764836i	−0.0284445	+	0.0492674i	0.851453	+	0.524431i	$0.175722\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.686141	+	1.18843i	−0.0438359	+	0.0759260i
$246$		0			0
$247$	5.00000	+	8.66025i	0.318142	+	0.551039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.00000			−0.568075			−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000			−0.568075			−0.284037	+	0.958813i	$0.591674\pi$
$252$		0			0
$253$	−32.2337			−2.02651
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.9307	+	18.9325i	0.681839	+	1.18098i	0.974419	+	0.224738i	$0.0721527\pi$
$257$	10.9307	+	18.9325i	0.681839	+	1.18098i	−0.292581	+	0.956241i	$0.594514\pi$
$258$		0			0
$259$	1.00000	−	1.73205i	0.0621370	−	0.107624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.68614	+	11.5807i	−0.412285	+	0.714099i	−0.995139	−	0.0984781i	$-0.968603\pi$
$263$	−6.68614	+	11.5807i	−0.412285	+	0.714099i	0.582854	+	0.812577i	$0.301936\pi$
$264$		0			0
$265$	1.88316	+	3.26172i	0.115681	+	0.200366i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.37228			0.449496			0.224748	−	0.974417i	$-0.427844\pi$
$269$	7.37228			0.449496			0.224748	+	0.974417i	$0.427844\pi$
$270$		0			0
$271$	18.2337			1.10762			0.553809	−	0.832644i	$-0.313174\pi$
$271$	18.2337			1.10762			0.553809	+	0.832644i	$0.313174\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.81386	+	11.8020i	0.410891	+	0.711684i
$276$		0			0
$277$	−11.1168	+	19.2549i	−0.667946	+	1.15692i	0.310531	+	0.950563i	$0.399493\pi$
$277$	−11.1168	+	19.2549i	−0.667946	+	1.15692i	−0.978477	+	0.206354i	$0.933840\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.31386	−	9.20387i	0.316998	−	0.549057i	−0.662862	−	0.748742i	$-0.730658\pi$
$281$	5.31386	−	9.20387i	0.316998	−	0.549057i	0.979860	+	0.199685i	$0.0639917\pi$
$282$		0			0
$283$	4.94158	+	8.55906i	0.293746	+	0.508784i	0.974692	−	0.223550i	$-0.0717646\pi$
$283$	4.94158	+	8.55906i	0.293746	+	0.508784i	−0.680946	+	0.732333i	$0.738431\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.3723			0.612256
$288$		0			0
$289$	2.11684			0.124520
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.31386	−	4.00772i	−0.135177	−	0.234134i	0.790488	−	0.612478i	$-0.209827\pi$
$293$	−2.31386	−	4.00772i	−0.135177	−	0.234134i	−0.925665	+	0.378344i	$0.876494\pi$
$294$		0			0
$295$	−4.88316	+	8.45787i	−0.284308	+	0.492436i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.37228	−	12.7692i	0.426350	−	0.738460i
$300$		0			0
$301$	−4.55842	−	7.89542i	−0.262743	−	0.455084i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−19.3723			−1.10925
$306$		0			0
$307$	13.0000			0.741949			0.370975	−	0.928643i	$-0.379024\pi$
$307$	13.0000			0.741949			0.370975	+	0.928643i	$0.379024\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.1168	+	22.7190i	0.743788	+	1.28828i	0.950759	+	0.309931i	$0.100306\pi$
$311$	13.1168	+	22.7190i	0.743788	+	1.28828i	−0.206971	+	0.978347i	$0.566361\pi$
$312$		0			0
$313$	1.44158	−	2.49689i	0.0814828	−	0.141132i	−0.822404	−	0.568904i	$-0.807368\pi$
$313$	1.44158	−	2.49689i	0.0814828	−	0.141132i	0.903887	+	0.427771i	$0.140701\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.00000	+	5.19615i	−0.168497	+	0.291845i	−0.937892	−	0.346929i	$-0.887225\pi$
$317$	−3.00000	+	5.19615i	−0.168497	+	0.291845i	0.769395	+	0.638774i	$0.220558\pi$
$318$		0			0
$319$	6.00000	+	10.3923i	0.335936	+	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−21.8614			−1.21640
$324$		0			0
$325$	−6.23369			−0.345783
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−6.11684	+	10.5947i	−0.336212	+	0.582337i	−0.983717	−	0.179725i	$-0.942479\pi$
$331$	−6.11684	+	10.5947i	−0.336212	+	0.582337i	0.647505	+	0.762061i	$0.275813\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10.3723	−	17.9653i	0.566698	−	0.981550i
$336$		0			0
$337$	−4.55842	−	7.89542i	−0.248313	−	0.430091i	0.714745	−	0.699385i	$-0.246543\pi$
$337$	−4.55842	−	7.89542i	−0.248313	−	0.430091i	−0.963058	+	0.269294i	$0.913210\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.74456			−0.473545
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.55842	−	6.16337i	−0.191026	−	0.330867i	0.754564	−	0.656226i	$-0.227848\pi$
$347$	−3.55842	−	6.16337i	−0.191026	−	0.330867i	−0.945591	+	0.325359i	$0.894515\pi$
$348$		0			0
$349$	11.0000	−	19.0526i	0.588817	−	1.01986i	−0.405571	−	0.914063i	$-0.632927\pi$
$349$	11.0000	−	19.0526i	0.588817	−	1.01986i	0.994388	−	0.105797i	$-0.0337393\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.81386	+	6.60580i	−0.202991	+	0.351591i	−0.949491	−	0.313795i	$-0.898400\pi$
$353$	−3.81386	+	6.60580i	−0.202991	+	0.351591i	0.746500	+	0.665386i	$0.231733\pi$
$354$		0			0
$355$	−6.94158	−	12.0232i	−0.368421	−	0.638123i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.86141			−0.362131			−0.181066	−	0.983471i	$-0.557955\pi$
$359$	−6.86141			−0.362131			−0.181066	+	0.983471i	$0.557955\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.51087	+	6.08101i	0.183768	+	0.318295i
$366$		0			0
$367$	11.1168	−	19.2549i	0.580295	−	1.00510i	−0.415150	−	0.909753i	$-0.636271\pi$
$367$	11.1168	−	19.2549i	0.580295	−	1.00510i	0.995444	−	0.0953465i	$-0.0303959\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.37228	+	2.37686i	−0.0712453	+	0.123400i
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.48913			−0.282704
$378$		0			0
$379$	−9.11684			−0.468301			−0.234150	−	0.972200i	$-0.575231\pi$
$379$	−9.11684			−0.468301			−0.234150	+	0.972200i	$0.575231\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.6277	+	18.4077i	0.543051	+	0.940592i	0.998727	+	0.0504462i	$0.0160643\pi$
$383$	10.6277	+	18.4077i	0.543051	+	0.940592i	−0.455676	+	0.890146i	$0.650602\pi$
$384$		0			0
$385$	3.00000	−	5.19615i	0.152894	−	0.264820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.4891	+	30.2921i	−0.886734	+	1.53587i	−0.0430204	+	0.999074i	$0.513698\pi$
$389$	−17.4891	+	30.2921i	−0.886734	+	1.53587i	−0.843713	+	0.536794i	$0.819635\pi$
$390$		0			0
$391$	16.1168	+	27.9152i	0.815064	+	1.41173i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.6277			−0.836631
$396$		0			0
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.127719	−	0.221215i	−0.00637797	−	0.0110470i	0.862819	−	0.505513i	$-0.168697\pi$
$401$	−0.127719	−	0.221215i	−0.00637797	−	0.0110470i	−0.869197	+	0.494466i	$0.835364\pi$
$402$		0			0
$403$	2.00000	−	3.46410i	0.0996271	−	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.37228	+	7.57301i	−0.216726	+	0.375380i
$408$		0			0
$409$	−14.6753	−	25.4183i	−0.725645	−	1.25685i	−0.958708	−	0.284393i	$-0.908208\pi$
$409$	−14.6753	−	25.4183i	−0.725645	−	1.25685i	0.233063	−	0.972462i	$-0.425125\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.11684			−0.350197
$414$		0			0
$415$	−7.53262			−0.369762
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.8030	−	23.9075i	−0.674320	−	1.16796i	−0.976667	−	0.214759i	$-0.931104\pi$
$419$	−13.8030	−	23.9075i	−0.674320	−	1.16796i	0.302347	−	0.953198i	$-0.402230\pi$
$420$		0			0
$421$	0.116844	−	0.202380i	0.00569463	−	0.00986338i	−0.863164	−	0.504924i	$-0.831521\pi$
$421$	0.116844	−	0.202380i	0.00569463	−	0.00986338i	0.868859	+	0.495060i	$0.164854\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.81386	−	11.8020i	0.330521	−	0.572479i
$426$		0			0
$427$	−7.05842	−	12.2255i	−0.341581	−	0.591636i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−29.4891			−1.42044			−0.710221	−	0.703979i	$-0.751405\pi$
$431$	−29.4891			−1.42044			−0.710221	+	0.703979i	$0.751405\pi$
$432$		0			0
$433$	−2.88316			−0.138556			−0.0692778	−	0.997597i	$-0.522069\pi$
$433$	−2.88316			−0.138556			−0.0692778	+	0.997597i	$0.522069\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−18.4307	−	31.9229i	−0.881660	−	1.52708i
$438$		0			0
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	−0.754642	−	0.656136i	$-0.772190\pi$
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	0.945552	+	0.325471i	$0.105523\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.4416	−	19.8174i	0.543606	−	0.941553i	−0.455087	−	0.890447i	$-0.650392\pi$
$443$	11.4416	−	19.8174i	0.543606	−	0.941553i	0.998693	−	0.0511061i	$-0.0162747\pi$
$444$		0			0
$445$	2.23369	+	3.86886i	0.105887	+	0.183402i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.0000			−1.55737			−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000			−1.55737			−0.778683	+	0.627417i	$0.784112\pi$
$450$		0			0
$451$	−45.3505			−2.13547
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.37228	+	2.37686i	0.0643335	+	0.111429i
$456$		0			0
$457$	−16.7337	+	28.9836i	−0.782769	+	1.35580i	0.147554	+	0.989054i	$0.452860\pi$
$457$	−16.7337	+	28.9836i	−0.782769	+	1.35580i	−0.930323	+	0.366742i	$0.880473\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.4307	−	26.7268i	0.718680	−	1.24479i	−0.242844	−	0.970065i	$-0.578080\pi$
$461$	15.4307	−	26.7268i	0.718680	−	1.24479i	0.961523	−	0.274724i	$-0.0885865\pi$
$462$		0			0
$463$	−2.94158	−	5.09496i	−0.136707	−	0.236783i	0.789541	−	0.613697i	$-0.210318\pi$
$463$	−2.94158	−	5.09496i	−0.136707	−	0.236783i	−0.926248	+	0.376914i	$0.876985\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−30.0951			−1.39263			−0.696317	−	0.717734i	$-0.745179\pi$
$467$	−30.0951			−1.39263			−0.696317	+	0.717734i	$0.745179\pi$
$468$		0			0
$469$	15.1168			0.698031
$470$		0			0
$471$		0			0
$472$		0			0
$473$	19.9307	+	34.5210i	0.916415	+	1.58728i
$474$		0			0
$475$	−7.79211	+	13.4963i	−0.357527	+	0.619254i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.6277	+	18.4077i	−0.485593	+	0.841072i	−0.999863	−	0.0165568i	$-0.994730\pi$
$479$	−10.6277	+	18.4077i	−0.485593	+	0.841072i	0.514270	+	0.857628i	$0.328063\pi$
$480$		0			0
$481$	−2.00000	−	3.46410i	−0.0911922	−	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.5109			0.568090
$486$		0			0
$487$	16.3505			0.740913			0.370457	−	0.928850i	$-0.379201\pi$
$487$	16.3505			0.740913			0.370457	+	0.928850i	$0.379201\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.81386	+	16.9981i	0.442893	+	0.767114i	0.997903	−	0.0647303i	$-0.0206187\pi$
$491$	9.81386	+	16.9981i	0.442893	+	0.767114i	−0.555010	+	0.831844i	$0.687285\pi$
$492$		0			0
$493$	6.00000	−	10.3923i	0.270226	−	0.468046i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.05842	−	8.76144i	0.226901	−	0.393004i
$498$		0			0
$499$	0.441578	+	0.764836i	0.0197677	+	0.0342387i	0.875740	−	0.482783i	$-0.160374\pi$
$499$	0.441578	+	0.764836i	0.0197677	+	0.0342387i	−0.855972	+	0.517022i	$0.827041\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.23369			0.0995952			0.0497976	−	0.998759i	$-0.484142\pi$
$503$	2.23369			0.0995952			0.0497976	+	0.998759i	$0.484142\pi$
$504$		0			0
$505$	−10.1168			−0.450194
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.48913	+	14.7036i	0.376274	+	0.651725i	0.990517	−	0.137392i	$-0.0438718\pi$
$509$	8.48913	+	14.7036i	0.376274	+	0.651725i	−0.614243	+	0.789117i	$0.710539\pi$
$510$		0			0
$511$	−2.55842	+	4.43132i	−0.113178	+	0.196030i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.86141	+	11.8843i	−0.302350	+	0.523685i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.86141			−0.169171			−0.0845856	−	0.996416i	$-0.526957\pi$
$521$	−3.86141			−0.169171			−0.0845856	+	0.996416i	$0.526957\pi$
$522$		0			0
$523$	17.8832			0.781976			0.390988	−	0.920396i	$-0.372133\pi$
$523$	17.8832			0.781976			0.390988	+	0.920396i	$0.372133\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.37228	+	7.57301i	0.190460	+	0.329886i
$528$		0			0
$529$	−15.6753	+	27.1504i	−0.681533	+	1.18045i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10.3723	−	17.9653i	0.449273	−	0.778164i
$534$		0			0
$535$	1.11684	+	1.93443i	0.0482854	+	0.0836327i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.37228			0.188327
$540$		0			0
$541$	28.2337			1.21386			0.606931	−	0.794755i	$-0.292401\pi$
$541$	28.2337			1.21386			0.606931	+	0.794755i	$0.292401\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.60597	−	16.6380i	−0.411475	−	0.712695i
$546$		0			0
$547$	0.441578	−	0.764836i	0.0188805	−	0.0327020i	−0.856431	−	0.516262i	$-0.827323\pi$
$547$	0.441578	−	0.764836i	0.0188805	−	0.0327020i	0.875311	+	0.483560i	$0.160656\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.86141	+	11.8843i	−0.292306	+	0.506288i
$552$		0			0
$553$	−6.05842	−	10.4935i	−0.257630	−	0.446229i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.51087			−0.275875			−0.137937	−	0.990441i	$-0.544047\pi$
$557$	−6.51087			−0.275875			−0.137937	+	0.990441i	$0.544047\pi$
$558$		0			0
$559$	−18.2337			−0.771203
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.50000	−	2.59808i	−0.0632175	−	0.109496i	0.832684	−	0.553748i	$-0.186803\pi$
$563$	−1.50000	−	2.59808i	−0.0632175	−	0.109496i	−0.895902	+	0.444252i	$0.853470\pi$
$564$		0			0
$565$	0.941578	−	1.63086i	0.0396125	−	0.0686108i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.558422	+	0.967215i	−0.0234103	+	0.0405478i	−0.877493	−	0.479589i	$-0.840786\pi$
$569$	−0.558422	+	0.967215i	−0.0234103	+	0.0405478i	0.854083	+	0.520137i	$0.174119\pi$
$570$		0			0
$571$	14.6753	+	25.4183i	0.614141	+	1.06372i	0.990535	+	0.137263i	$0.0438306\pi$
$571$	14.6753	+	25.4183i	0.614141	+	1.06372i	−0.376394	+	0.926460i	$0.622836\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	22.9783			0.958259
$576$		0			0
$577$	27.1168			1.12889			0.564444	−	0.825471i	$-0.309090\pi$
$577$	27.1168			1.12889			0.564444	+	0.825471i	$0.309090\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.74456	−	4.75372i	−0.113864	−	0.197218i
$582$		0			0
$583$	6.00000	−	10.3923i	0.248495	−	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.24456	−	7.35180i	0.175192	−	0.303441i	−0.765036	−	0.643988i	$-0.777279\pi$
$587$	4.24456	−	7.35180i	0.175192	−	0.303441i	0.940228	+	0.340547i	$0.110612\pi$
$588$		0			0
$589$	−5.00000	−	8.66025i	−0.206021	−	0.356840i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.25544			−0.133685			−0.0668424	−	0.997764i	$-0.521292\pi$
$593$	−3.25544			−0.133685			−0.0668424	+	0.997764i	$0.521292\pi$
$594$		0			0
$595$	−6.00000			−0.245976
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	0.509631	−	0.860393i	$-0.329782\pi$
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	−0.999938	+	0.0111569i	$0.996449\pi$
$600$		0			0
$601$	−3.44158	+	5.96099i	−0.140385	+	0.243154i	−0.927642	−	0.373472i	$-0.878167\pi$
$601$	−3.44158	+	5.96099i	−0.140385	+	0.243154i	0.787257	+	0.616625i	$0.211501\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.56930	+	9.64630i	−0.226424	+	0.392178i
$606$		0			0
$607$	−6.11684	−	10.5947i	−0.248275	−	0.430025i	0.714772	−	0.699357i	$-0.246530\pi$
$607$	−6.11684	−	10.5947i	−0.248275	−	0.430025i	−0.963047	+	0.269332i	$0.913197\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.76631			−0.0713407			−0.0356703	−	0.999364i	$-0.511357\pi$
$613$	−1.76631			−0.0713407			−0.0356703	+	0.999364i	$0.511357\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.93070	−	8.54023i	−0.198503	−	0.343817i	0.749540	−	0.661959i	$-0.230275\pi$
$617$	−4.93070	−	8.54023i	−0.198503	−	0.343817i	−0.948043	+	0.318142i	$0.896941\pi$
$618$		0			0
$619$	−11.7337	+	20.3233i	−0.471617	+	0.816864i	−0.999473	−	0.0324697i	$-0.989663\pi$
$619$	−11.7337	+	20.3233i	−0.471617	+	0.816864i	0.527856	+	0.849334i	$0.322996\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.62772	+	2.81929i	−0.0652132	+	0.112953i
$624$		0			0
$625$	−0.149468	−	0.258886i	−0.00597872	−	0.0103555i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.74456			0.348669
$630$		0			0
$631$	−14.3505			−0.571286			−0.285643	−	0.958336i	$-0.592207\pi$
$631$	−14.3505			−0.571286			−0.285643	+	0.958336i	$0.592207\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.68614	−	16.7769i	−0.384383	−	0.665770i
$636$		0			0
$637$	−1.00000	+	1.73205i	−0.0396214	+	0.0686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.1060	+	40.0207i	−0.912631	+	1.58072i	−0.102298	+	0.994754i	$0.532619\pi$
$641$	−23.1060	+	40.0207i	−0.912631	+	1.58072i	−0.810333	+	0.585969i	$0.800714\pi$
$642$		0			0
$643$	−12.6753	−	21.9542i	−0.499864	−	0.865789i	0.500136	−	0.865947i	$-0.333283\pi$
$643$	−12.6753	−	21.9542i	−0.499864	−	0.865789i	−1.00000		0.000157386i	$0.999950\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17.4891			−0.687568			−0.343784	−	0.939049i	$-0.611709\pi$
$647$	−17.4891			−0.687568			−0.343784	+	0.939049i	$0.611709\pi$
$648$		0			0
$649$	31.1168			1.22144
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.62772	−	13.2116i	−0.298496	−	0.517010i	0.677296	−	0.735710i	$-0.263152\pi$
$653$	−7.62772	−	13.2116i	−0.298496	−	0.517010i	−0.975792	+	0.218701i	$0.929818\pi$
$654$		0			0
$655$	5.05842	−	8.76144i	0.197649	−	0.342338i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.62772	−	8.01544i	0.180270	−	0.312237i	−0.761702	−	0.647927i	$-0.775636\pi$
$659$	4.62772	−	8.01544i	0.180270	−	0.312237i	0.941973	+	0.335690i	$0.108969\pi$
$660$		0			0
$661$	−4.94158	−	8.55906i	−0.192205	−	0.332909i	0.753776	−	0.657132i	$-0.228231\pi$
$661$	−4.94158	−	8.55906i	−0.192205	−	0.332909i	−0.945981	+	0.324223i	$0.894897\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.86141			0.266074
$666$		0			0
$667$	20.2337			0.783452
$668$		0			0
$669$		0			0
$670$		0			0
$671$	30.8614	+	53.4535i	1.19139	+	2.06355i
$672$		0			0
$673$	10.0584	−	17.4217i	0.387724	−	0.671557i	−0.604419	−	0.796666i	$-0.706595\pi$
$673$	10.0584	−	17.4217i	0.387724	−	0.671557i	0.992143	+	0.125109i	$0.0399281\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−17.2337	+	29.8496i	−0.662344	+	1.14721i	0.317654	+	0.948207i	$0.397105\pi$
$677$	−17.2337	+	29.8496i	−0.662344	+	1.14721i	−0.979998	+	0.199007i	$0.936228\pi$
$678$		0			0
$679$	4.55842	+	7.89542i	0.174936	+	0.302998i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−44.8397			−1.71574			−0.857871	−	0.513865i	$-0.828213\pi$
$683$	−44.8397			−1.71574			−0.857871	+	0.513865i	$0.828213\pi$
$684$		0			0
$685$	−22.4674			−0.858434
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.74456	+	4.75372i	0.104560	+	0.181102i
$690$		0			0
$691$	−2.94158	+	5.09496i	−0.111903	+	0.193822i	−0.916537	−	0.399949i	$-0.869028\pi$
$691$	−2.94158	+	5.09496i	−0.111903	+	0.193822i	0.804635	+	0.593770i	$0.202361\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−14.5693	+	25.2348i	−0.552645	+	0.957209i
$696$		0			0
$697$	22.6753	+	39.2747i	0.858887	+	1.48764i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.76631			0.142252			0.0711258	−	0.997467i	$-0.477341\pi$
$701$	3.76631			0.142252			0.0711258	+	0.997467i	$0.477341\pi$
$702$		0			0
$703$	−10.0000			−0.377157
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.68614	−	6.38458i	−0.138632	−	0.240117i
$708$		0			0
$709$	−22.0000	+	38.1051i	−0.826227	+	1.43107i	0.0747503	+	0.997202i	$0.476184\pi$
$709$	−22.0000	+	38.1051i	−0.826227	+	1.43107i	−0.900978	+	0.433865i	$0.857149\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.37228	+	12.7692i	−0.276094	+	0.478209i
$714$		0			0
$715$	−6.00000	−	10.3923i	−0.224387	−	0.388650i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.74456			−0.326117			−0.163059	−	0.986616i	$-0.552136\pi$
$719$	−8.74456			−0.326117			−0.163059	+	0.986616i	$0.552136\pi$
$720$		0			0
$721$	−10.0000			−0.372419
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.27719	−	7.40830i	−0.158851	−	0.275138i
$726$		0			0
$727$	−0.883156	+	1.52967i	−0.0327544	+	0.0567324i	−0.881938	−	0.471366i	$-0.843761\pi$
$727$	−0.883156	+	1.52967i	−0.0327544	+	0.0567324i	0.849183	+	0.528098i	$0.177095\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	19.9307	−	34.5210i	0.737164	−	1.27680i
$732$		0			0
$733$	11.9416	+	20.6834i	0.441072	+	0.763960i	0.997769	−	0.0667560i	$-0.0212649\pi$
$733$	11.9416	+	20.6834i	0.441072	+	0.763960i	−0.556697	+	0.830716i	$0.687932\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−66.0951			−2.43464
$738$		0			0
$739$	−9.11684			−0.335369			−0.167684	−	0.985841i	$-0.553629\pi$
$739$	−9.11684			−0.335369			−0.167684	+	0.985841i	$0.553629\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21.8614	−	37.8651i	−0.802017	−	1.38913i	−0.918286	−	0.395917i	$-0.870427\pi$
$743$	−21.8614	−	37.8651i	−0.802017	−	1.38913i	0.116269	−	0.993218i	$-0.462906\pi$
$744$		0			0
$745$	10.1168	−	17.5229i	0.370652	−	0.641989i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.813859	+	1.40965i	−0.0297378	+	0.0515073i
$750$		0			0
$751$	0.0584220	+	0.101190i	0.00213185	+	0.00369247i	0.867089	−	0.498153i	$-0.165988\pi$
$751$	0.0584220	+	0.101190i	0.00213185	+	0.00369247i	−0.864958	+	0.501845i	$0.832655\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.1386			0.405375
$756$		0			0
$757$	11.7663			0.427654			0.213827	−	0.976872i	$-0.431407\pi$
$757$	11.7663			0.427654			0.213827	+	0.976872i	$0.431407\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.25544	+	10.8347i	0.226759	+	0.392759i	0.956846	−	0.290596i	$-0.0938536\pi$
$761$	6.25544	+	10.8347i	0.226759	+	0.392759i	−0.730086	+	0.683355i	$0.760520\pi$
$762$		0			0
$763$	7.00000	−	12.1244i	0.253417	−	0.438931i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.11684	+	12.3267i	−0.256974	+	0.445093i
$768$		0			0
$769$	5.00000	+	8.66025i	0.180305	+	0.312297i	0.941984	−	0.335657i	$-0.108958\pi$
$769$	5.00000	+	8.66025i	0.180305	+	0.312297i	−0.761680	+	0.647954i	$0.775625\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11.1386			0.400627			0.200314	−	0.979732i	$-0.435804\pi$
$773$	11.1386			0.400627			0.200314	+	0.979732i	$0.435804\pi$
$774$		0			0
$775$	6.23369			0.223921
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−25.9307	−	44.9133i	−0.929064	−	1.60919i
$780$		0			0
$781$	−22.1168	+	38.3075i	−0.791403	+	1.37075i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.56930	−	9.64630i	0.198777	−	0.344291i
$786$		0			0
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	0.828176	−	0.560469i	$-0.189379\pi$
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	−0.899468	+	0.436987i	$0.856046\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.37228			0.0487927
$792$		0			0
$793$	−28.2337			−1.00261
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.4307	−	31.9229i	−0.652849	−	1.13077i	−0.982428	−	0.186640i	$-0.940240\pi$
$797$	−18.4307	−	31.9229i	−0.652849	−	1.13077i	0.329579	−	0.944128i	$-0.393093\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11.1861	−	19.3750i	0.394750	−	0.683728i
$804$		0			0
$805$	−5.05842	−	8.76144i	−0.178286	−	0.308800i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−21.8614			−0.768606			−0.384303	−	0.923207i	$-0.625558\pi$
$809$	−21.8614			−0.768606			−0.384303	+	0.923207i	$0.625558\pi$
$810$		0			0
$811$	−24.8832			−0.873766			−0.436883	−	0.899518i	$-0.643918\pi$
$811$	−24.8832			−0.873766			−0.436883	+	0.899518i	$0.643918\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	11.1386	+	19.2926i	0.390168	+	0.675791i
$816$		0			0
$817$	−22.7921	+	39.4771i	−0.797395	+	1.38113i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19.1168	+	33.1113i	−0.667182	+	1.15559i	0.311506	+	0.950244i	$0.399167\pi$
$821$	−19.1168	+	33.1113i	−0.667182	+	1.15559i	−0.978689	+	0.205350i	$0.934167\pi$
$822$		0			0
$823$	11.1168	+	19.2549i	0.387509	+	0.671185i	0.992114	−	0.125341i	$-0.0400023\pi$
$823$	11.1168	+	19.2549i	0.387509	+	0.671185i	−0.604605	+	0.796525i	$0.706669\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$		0			0
$829$	−48.2337			−1.67523			−0.837613	−	0.546265i	$-0.816049\pi$
$829$	−48.2337			−1.67523			−0.837613	+	0.546265i	$0.816049\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.18614	−	3.78651i	−0.0757453	−	0.131195i
$834$		0			0
$835$	−12.0000	+	20.7846i	−0.415277	+	0.719281i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8.74456	−	15.1460i	0.301896	−	0.522899i	−0.674670	−	0.738120i	$-0.735714\pi$
$839$	8.74456	−	15.1460i	0.301896	−	0.522899i	0.976565	+	0.215221i	$0.0690472\pi$
$840$		0			0
$841$	10.7337	+	18.5913i	0.370127	+	0.641079i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.3505			−0.424871
$846$		0			0
$847$	−8.11684			−0.278898
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.37228	+	12.7692i	0.252719	+	0.437721i
$852$		0			0
$853$	8.94158	−	15.4873i	0.306154	−	0.530274i	−0.671364	−	0.741128i	$-0.734291\pi$
$853$	8.94158	−	15.4873i	0.306154	−	0.530274i	0.977518	+	0.210854i	$0.0676245\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.9783	−	44.9956i	0.887400	−	1.53702i	0.0444624	−	0.999011i	$-0.485843\pi$
$857$	25.9783	−	44.9956i	0.887400	−	1.53702i	0.842938	−	0.538011i	$-0.180824\pi$
$858$		0			0
$859$	25.5584	+	44.2685i	0.872042	+	1.51042i	0.859881	+	0.510495i	$0.170538\pi$
$859$	25.5584	+	44.2685i	0.872042	+	1.51042i	0.0121615	+	0.999926i	$0.496129\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−18.8614			−0.642050			−0.321025	−	0.947071i	$-0.604027\pi$
$863$	−18.8614			−0.642050			−0.321025	+	0.947071i	$0.604027\pi$
$864$		0			0
$865$	8.23369			0.279954
$866$		0			0
$867$		0			0
$868$		0			0
$869$	26.4891	+	45.8805i	0.898582	+	1.55639i
$870$		0			0
$871$	15.1168	−	26.1831i	0.512215	−	0.887182i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.56930	+	9.64630i	−0.188277	+	0.326105i
$876$		0			0
$877$	−22.3505	−	38.7123i	−0.754724	−	1.30722i	−0.945512	−	0.325589i	$-0.894438\pi$
$877$	−22.3505	−	38.7123i	−0.754724	−	1.30722i	0.190788	−	0.981631i	$-0.438896\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.2337			−0.479545			−0.239773	−	0.970829i	$-0.577073\pi$
$881$	−14.2337			−0.479545			−0.239773	+	0.970829i	$0.577073\pi$
$882$		0			0
$883$	−11.3505			−0.381976			−0.190988	−	0.981592i	$-0.561169\pi$
$883$	−11.3505			−0.381976			−0.190988	+	0.981592i	$0.561169\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	15.8614	+	27.4728i	0.532574	+	0.922445i	0.999277	+	0.0380308i	$0.0121085\pi$
$887$	15.8614	+	27.4728i	0.532574	+	0.922445i	−0.466703	+	0.884414i	$0.654558\pi$
$888$		0			0
$889$	7.05842	−	12.2255i	0.236732	−	0.410032i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−10.1168	−	17.5229i	−0.338169	−	0.585726i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.48913			0.183073
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.4307	−	21.5306i	−0.413211	−	0.715702i
$906$		0			0
$907$	−4.44158	+	7.69304i	−0.147480	+	0.255443i	−0.930296	−	0.366811i	$-0.880450\pi$
$907$	−4.44158	+	7.69304i	−0.147480	+	0.255443i	0.782815	+	0.622254i	$0.213783\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	21.6861	−	37.5615i	0.718494	−	1.24447i	−0.243103	−	0.970001i	$-0.578165\pi$
$911$	21.6861	−	37.5615i	0.718494	−	1.24447i	0.961596	−	0.274467i	$-0.0885015\pi$
$912$		0			0
$913$	12.0000	+	20.7846i	0.397142	+	0.687870i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.37228			0.243454
$918$		0			0
$919$	29.8832			0.985754			0.492877	−	0.870099i	$-0.335945\pi$
$919$	29.8832			0.985754			0.492877	+	0.870099i	$0.335945\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.1168	−	17.5229i	−0.333000	−	0.576773i
$924$		0			0
$925$	3.11684	−	5.39853i	0.102481	−	0.177503i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4.88316	−	8.45787i	0.160211	−	0.277494i	−0.774733	−	0.632288i	$-0.782116\pi$
$929$	4.88316	−	8.45787i	0.160211	−	0.277494i	0.934944	+	0.354794i	$0.115449\pi$
$930$		0			0
$931$	2.50000	+	4.33013i	0.0819342	+	0.141914i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	26.2337			0.857933
$936$		0			0
$937$	−38.4674			−1.25667			−0.628337	−	0.777941i	$-0.716264\pi$
$937$	−38.4674			−1.25667			−0.628337	+	0.777941i	$0.716264\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.941578	+	1.63086i	0.0306946	+	0.0531645i	0.880965	−	0.473182i	$-0.156895\pi$
$941$	0.941578	+	1.63086i	0.0306946	+	0.0531645i	−0.850270	+	0.526347i	$0.823561\pi$
$942$		0			0
$943$	−38.2337	+	66.2227i	−1.24506	+	2.15651i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−8.44158	+	14.6212i	−0.274314	+	0.475127i	−0.969962	−	0.243257i	$-0.921784\pi$
$947$	−8.44158	+	14.6212i	−0.274314	+	0.475127i	0.695648	+	0.718383i	$0.255118\pi$
$948$		0			0
$949$	5.11684	+	8.86263i	0.166100	+	0.287693i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−10.8832			−0.352540			−0.176270	−	0.984342i	$-0.556403\pi$
$953$	−10.8832			−0.352540			−0.176270	+	0.984342i	$0.556403\pi$
$954$		0			0
$955$	−2.58422			−0.0836234
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.18614	−	14.1788i	−0.264344	−	0.457858i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.80298	−	8.31901i	0.154614	−	0.267799i
$966$		0			0
$967$	24.0584	+	41.6704i	0.773667	+	1.34003i	0.935541	+	0.353219i	$0.114913\pi$
$967$	24.0584	+	41.6704i	0.773667	+	1.34003i	−0.161874	+	0.986811i	$0.551754\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	7.37228			0.236588			0.118294	−	0.992979i	$-0.462258\pi$
$971$	7.37228			0.236588			0.118294	+	0.992979i	$0.462258\pi$
$972$		0			0
$973$	−21.2337			−0.680721
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.4416	+	19.8174i	0.366049	+	0.634015i	0.988944	−	0.148291i	$-0.0473771\pi$
$977$	11.4416	+	19.8174i	0.366049	+	0.634015i	−0.622895	+	0.782305i	$0.714044\pi$
$978$		0			0
$979$	7.11684	−	12.3267i	0.227455	−	0.393964i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	25.3723	−	43.9461i	0.809250	−	1.40166i	−0.104134	−	0.994563i	$-0.533207\pi$
$983$	25.3723	−	43.9461i	0.809250	−	1.40166i	0.913384	−	0.407099i	$-0.133460\pi$
$984$		0			0
$985$	−4.11684	−	7.13058i	−0.131174	−	0.227199i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	67.2119			2.13722
$990$		0			0
$991$	20.4674			0.650168			0.325084	−	0.945685i	$-0.394607\pi$
$991$	20.4674			0.650168			0.325084	+	0.945685i	$0.394607\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.86141	−	11.8843i	−0.217521	−	0.376758i
$996$		0			0
$997$	−6.05842	+	10.4935i	−0.191872	+	0.332332i	−0.945871	−	0.324544i	$-0.894789\pi$
$997$	−6.05842	+	10.4935i	−0.191872	+	0.332332i	0.753999	+	0.656876i	$0.228123\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.r.f.2017.1		4
3.2	odd	2		1008.2.r.f.673.1		4
4.3	odd	2		378.2.f.c.127.1		4
9.2	odd	6		9072.2.a.bm.1.1		2
9.4	even	3	inner	3024.2.r.f.1009.1		4
9.5	odd	6		1008.2.r.f.337.1		4
9.7	even	3		9072.2.a.bb.1.2		2
12.11	even	2		126.2.f.d.43.2	✓	4
28.3	even	6		2646.2.h.l.667.1		4
28.11	odd	6		2646.2.h.k.667.2		4
28.19	even	6		2646.2.e.m.2125.2		4
28.23	odd	6		2646.2.e.n.2125.1		4
28.27	even	2		2646.2.f.j.883.2		4
36.7	odd	6		1134.2.a.n.1.2		2
36.11	even	6		1134.2.a.k.1.1		2
36.23	even	6		126.2.f.d.85.2	yes	4
36.31	odd	6		378.2.f.c.253.1		4
84.11	even	6		882.2.h.m.79.1		4
84.23	even	6		882.2.e.l.655.2		4
84.47	odd	6		882.2.e.k.655.1		4
84.59	odd	6		882.2.h.n.79.2		4
84.83	odd	2		882.2.f.k.295.1		4
252.23	even	6		882.2.h.m.67.1		4
252.31	even	6		2646.2.e.m.1549.2		4
252.59	odd	6		882.2.e.k.373.2		4
252.67	odd	6		2646.2.e.n.1549.1		4
252.83	odd	6		7938.2.a.bh.1.2		2
252.95	even	6		882.2.e.l.373.1		4
252.103	even	6		2646.2.h.l.361.1		4
252.131	odd	6		882.2.h.n.67.2		4
252.139	even	6		2646.2.f.j.1765.2		4
252.167	odd	6		882.2.f.k.589.1		4
252.223	even	6		7938.2.a.bs.1.1		2
252.247	odd	6		2646.2.h.k.361.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.d.43.2	✓	4	12.11	even	2
126.2.f.d.85.2	yes	4	36.23	even	6
378.2.f.c.127.1		4	4.3	odd	2
378.2.f.c.253.1		4	36.31	odd	6
882.2.e.k.373.2		4	252.59	odd	6
882.2.e.k.655.1		4	84.47	odd	6
882.2.e.l.373.1		4	252.95	even	6
882.2.e.l.655.2		4	84.23	even	6
882.2.f.k.295.1		4	84.83	odd	2
882.2.f.k.589.1		4	252.167	odd	6
882.2.h.m.67.1		4	252.23	even	6
882.2.h.m.79.1		4	84.11	even	6
882.2.h.n.67.2		4	252.131	odd	6
882.2.h.n.79.2		4	84.59	odd	6
1008.2.r.f.337.1		4	9.5	odd	6
1008.2.r.f.673.1		4	3.2	odd	2
1134.2.a.k.1.1		2	36.11	even	6
1134.2.a.n.1.2		2	36.7	odd	6
2646.2.e.m.1549.2		4	252.31	even	6
2646.2.e.m.2125.2		4	28.19	even	6
2646.2.e.n.1549.1		4	252.67	odd	6
2646.2.e.n.2125.1		4	28.23	odd	6
2646.2.f.j.883.2		4	28.27	even	2
2646.2.f.j.1765.2		4	252.139	even	6
2646.2.h.k.361.2		4	252.247	odd	6
2646.2.h.k.667.2		4	28.11	odd	6
2646.2.h.l.361.1		4	252.103	even	6
2646.2.h.l.667.1		4	28.3	even	6
3024.2.r.f.1009.1		4	9.4	even	3	inner
3024.2.r.f.2017.1		4	1.1	even	1	trivial
7938.2.a.bh.1.2		2	252.83	odd	6
7938.2.a.bs.1.1		2	252.223	even	6
9072.2.a.bb.1.2		2	9.7	even	3
9072.2.a.bm.1.1		2	9.2	odd	6

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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.r (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.686141 - 1.18843i) q^{5} +(0.500000 - 0.866025i) q^{7} +O(q^{10})\)	\(q+(-0.686141 - 1.18843i) q^{5} +(0.500000 - 0.866025i) q^{7} +(-2.18614 + 3.78651i) q^{11} +(-1.00000 - 1.73205i) q^{13} +4.37228 q^{17} -5.00000 q^{19} +(3.68614 + 6.38458i) q^{23} +(1.55842 - 2.69927i) q^{25} +(1.37228 - 2.37686i) q^{29} +(1.00000 + 1.73205i) q^{31} -1.37228 q^{35} +2.00000 q^{37} +(5.18614 + 8.98266i) q^{41} +(4.55842 - 7.89542i) q^{43} +(-0.500000 - 0.866025i) q^{49} -2.74456 q^{53} +6.00000 q^{55} +(-3.55842 - 6.16337i) q^{59} +(7.05842 - 12.2255i) q^{61} +(-1.37228 + 2.37686i) q^{65} +(7.55842 + 13.0916i) q^{67} +10.1168 q^{71} -5.11684 q^{73} +(2.18614 + 3.78651i) q^{77} +(6.05842 - 10.4935i) q^{79} +(2.74456 - 4.75372i) q^{83} +(-3.00000 - 5.19615i) q^{85} -3.25544 q^{89} -2.00000 q^{91} +(3.43070 + 5.94215i) q^{95} +(-4.55842 + 7.89542i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q + 3 * q^5 + 2 * q^7	\( 4 q + 3 q^{5} + 2 q^{7} - 3 q^{11} - 4 q^{13} + 6 q^{17} - 20 q^{19} + 9 q^{23} - 11 q^{25} - 6 q^{29} + 4 q^{31} + 6 q^{35} + 8 q^{37} + 15 q^{41} + q^{43} - 2 q^{49} + 12 q^{53} + 24 q^{55} + 3 q^{59} + 11 q^{61} + 6 q^{65} + 13 q^{67} + 6 q^{71} + 14 q^{73} + 3 q^{77} + 7 q^{79} - 12 q^{83} - 12 q^{85} - 36 q^{89} - 8 q^{91} - 15 q^{95} - q^{97}+O(q^{100}) \) 4 * q + 3 * q^5 + 2 * q^7 - 3 * q^11 - 4 * q^13 + 6 * q^17 - 20 * q^19 + 9 * q^23 - 11 * q^25 - 6 * q^29 + 4 * q^31 + 6 * q^35 + 8 * q^37 + 15 * q^41 + q^43 - 2 * q^49 + 12 * q^53 + 24 * q^55 + 3 * q^59 + 11 * q^61 + 6 * q^65 + 13 * q^67 + 6 * q^71 + 14 * q^73 + 3 * q^77 + 7 * q^79 - 12 * q^83 - 12 * q^85 - 36 * q^89 - 8 * q^91 - 15 * q^95 - q^97

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