LMFDB - Embedded newform 1248.2.bc.l.1087.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1248,2,Mod(31,1248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 0, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1248.31"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.bc (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,4,0,-4,0,-2,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1087.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1248.1087
Dual form			1248.2.bc.l.31.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{3} +(2.00000 + 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{7} -1.00000 q^{9} +(-3.00000 - 3.00000i) q^{11} +(-3.00000 + 2.00000i) q^{13} +(2.00000 - 2.00000i) q^{15} -2.00000i q^{17} +(-2.00000 + 2.00000i) q^{21} -6.00000 q^{23} +3.00000i q^{25} +1.00000i q^{27} -6.00000 q^{29} +(-3.00000 + 3.00000i) q^{33} -8.00000i q^{35} +(3.00000 - 3.00000i) q^{37} +(2.00000 + 3.00000i) q^{39} +(-8.00000 - 8.00000i) q^{41} -4.00000 q^{43} +(-2.00000 - 2.00000i) q^{45} +(5.00000 + 5.00000i) q^{47} +1.00000i q^{49} -2.00000 q^{51} +6.00000 q^{53} -12.0000i q^{55} +(1.00000 + 1.00000i) q^{59} +8.00000 q^{61} +(2.00000 + 2.00000i) q^{63} +(-10.0000 - 2.00000i) q^{65} +(2.00000 - 2.00000i) q^{67} +6.00000i q^{69} +(-5.00000 + 5.00000i) q^{71} +(-3.00000 + 3.00000i) q^{73} +3.00000 q^{75} +12.0000i q^{77} -12.0000i q^{79} +1.00000 q^{81} +(5.00000 - 5.00000i) q^{83} +(4.00000 - 4.00000i) q^{85} +6.00000i q^{87} +(-10.0000 + 10.0000i) q^{89} +(10.0000 + 2.00000i) q^{91} +(-11.0000 - 11.0000i) q^{97} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 4 q^{7} - 2 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - 4 q^{21} - 12 q^{23} - 12 q^{29} - 6 q^{33} + 6 q^{37} + 4 q^{39} - 16 q^{41} - 8 q^{43} - 4 q^{45} + 10 q^{47} - 4 q^{51} + 12 q^{53}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q + 4 * q^5 - 4 * q^7 - 2 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - 4 * q^21 - 12 * q^23 - 12 * q^29 - 6 * q^33 + 6 * q^37 + 4 * q^39 - 16 * q^41 - 8 * q^43 - 4 * q^45 + 10 * q^47 - 4 * q^51 + 12 * q^53 + 2 * q^59 + 16 * q^61 + 4 * q^63 - 20 * q^65 + 4 * q^67 - 10 * q^71 - 6 * q^73 + 6 * q^75 + 2 * q^81 + 10 * q^83 + 8 * q^85 - 20 * q^89 + 20 * q^91 - 22 * q^97 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	2.00000	+	2.00000i	0.894427	+	0.894427i	0.994936	−	0.100509i	$-0.0320471\pi$
$5$	2.00000	+	2.00000i	0.894427	+	0.894427i	−0.100509	+	0.994936i	$0.532047\pi$
$6$		0			0
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	0.219650	−	0.975579i	$-0.429509\pi$
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	−0.975579	+	0.219650i	$0.929509\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.00000	−	3.00000i	−0.904534	−	0.904534i	0.0912903	−	0.995824i	$-0.470901\pi$
$11$	−3.00000	−	3.00000i	−0.904534	−	0.904534i	−0.995824	+	0.0912903i	$0.970901\pi$
$12$		0			0
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$14$		0			0
$15$	2.00000	−	2.00000i	0.516398	−	0.516398i
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$	−2.00000	+	2.00000i	−0.436436	+	0.436436i
$22$		0			0
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$		0			0
$25$			3.00000i			0.600000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$		0			0
$33$	−3.00000	+	3.00000i	−0.522233	+	0.522233i
$34$		0			0
$35$		−	8.00000i		−	1.35225i
$36$		0			0
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	−0.416115	−	0.909312i	$-0.636609\pi$
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	0.909312	+	0.416115i	$0.136609\pi$
$38$		0			0
$39$	2.00000	+	3.00000i	0.320256	+	0.480384i
$40$		0			0
$41$	−8.00000	−	8.00000i	−1.24939	−	1.24939i	−0.955990	−	0.293400i	$-0.905213\pi$
$41$	−8.00000	−	8.00000i	−1.24939	−	1.24939i	−0.293400	−	0.955990i	$-0.594787\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$	−2.00000	−	2.00000i	−0.298142	−	0.298142i
$46$		0			0
$47$	5.00000	+	5.00000i	0.729325	+	0.729325i	0.970485	−	0.241160i	$-0.0775280\pi$
$47$	5.00000	+	5.00000i	0.729325	+	0.729325i	−0.241160	+	0.970485i	$0.577528\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		−	12.0000i		−	1.61808i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.00000	+	1.00000i	0.130189	+	0.130189i	0.769199	−	0.639010i	$-0.220656\pi$
$59$	1.00000	+	1.00000i	0.130189	+	0.130189i	−0.639010	+	0.769199i	$0.720656\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$	2.00000	+	2.00000i	0.251976	+	0.251976i
$64$		0			0
$65$	−10.0000	−	2.00000i	−1.24035	−	0.248069i
$66$		0			0
$67$	2.00000	−	2.00000i	0.244339	−	0.244339i	−0.574304	−	0.818642i	$-0.694727\pi$
$67$	2.00000	−	2.00000i	0.244339	−	0.244339i	0.818642	+	0.574304i	$0.194727\pi$
$68$		0			0
$69$			6.00000i			0.722315i
$70$		0			0
$71$	−5.00000	+	5.00000i	−0.593391	+	0.593391i	−0.938546	−	0.345155i	$-0.887826\pi$
$71$	−5.00000	+	5.00000i	−0.593391	+	0.593391i	0.345155	+	0.938546i	$0.387826\pi$
$72$		0			0
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	−0.860527	−	0.509404i	$-0.829866\pi$
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	0.509404	+	0.860527i	$0.329866\pi$
$74$		0			0
$75$	3.00000			0.346410
$76$		0			0
$77$			12.0000i			1.36753i
$78$		0			0
$79$		−	12.0000i		−	1.35011i	−0.737769	−	0.675053i	$-0.764121\pi$
$79$		−	12.0000i		−	1.35011i	0.737769	−	0.675053i	$-0.235879\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	5.00000	−	5.00000i	0.548821	−	0.548821i	−0.377279	−	0.926100i	$-0.623140\pi$
$83$	5.00000	−	5.00000i	0.548821	−	0.548821i	0.926100	+	0.377279i	$0.123140\pi$
$84$		0			0
$85$	4.00000	−	4.00000i	0.433861	−	0.433861i
$86$		0			0
$87$			6.00000i			0.643268i
$88$		0			0
$89$	−10.0000	+	10.0000i	−1.06000	+	1.06000i	−0.0619166	+	0.998081i	$0.519721\pi$
$89$	−10.0000	+	10.0000i	−1.06000	+	1.06000i	−0.998081	+	0.0619166i	$0.980279\pi$
$90$		0			0
$91$	10.0000	+	2.00000i	1.04828	+	0.209657i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−11.0000	−	11.0000i	−1.11688	−	1.11688i	−0.992196	−	0.124684i	$-0.960208\pi$
$97$	−11.0000	−	11.0000i	−1.11688	−	1.11688i	−0.124684	−	0.992196i	$-0.539792\pi$
$98$		0			0
$99$	3.00000	+	3.00000i	0.301511	+	0.301511i
$100$		0			0
$101$			2.00000i			0.199007i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$			2.00000i			0.199007i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$		0			0
$103$	−20.0000			−1.97066			−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−20.0000			−1.97066			−0.985329	+	0.170664i	$0.945409\pi$
$104$		0			0
$105$	−8.00000			−0.780720
$106$		0			0
$107$		−	14.0000i		−	1.35343i	−0.736245	−	0.676716i	$-0.763403\pi$
$107$		−	14.0000i		−	1.35343i	0.736245	−	0.676716i	$-0.236597\pi$
$108$		0			0
$109$	3.00000	+	3.00000i	0.287348	+	0.287348i	0.836031	−	0.548683i	$-0.184871\pi$
$109$	3.00000	+	3.00000i	0.287348	+	0.287348i	−0.548683	+	0.836031i	$0.684871\pi$
$110$		0			0
$111$	−3.00000	−	3.00000i	−0.284747	−	0.284747i
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$	−12.0000	−	12.0000i	−1.11901	−	1.11901i
$116$		0			0
$117$	3.00000	−	2.00000i	0.277350	−	0.184900i
$118$		0			0
$119$	−4.00000	+	4.00000i	−0.366679	+	0.366679i
$120$		0			0
$121$			7.00000i			0.636364i
$122$		0			0
$123$	−8.00000	+	8.00000i	−0.721336	+	0.721336i
$124$		0			0
$125$	4.00000	−	4.00000i	0.357771	−	0.357771i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$			4.00000i			0.352180i
$130$		0			0
$131$			22.0000i			1.92215i	0.276289	+	0.961074i	$0.410895\pi$
$131$			22.0000i			1.92215i	−0.276289	+	0.961074i	$0.589105\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−2.00000	+	2.00000i	−0.172133	+	0.172133i
$136$		0			0
$137$	−12.0000	+	12.0000i	−1.02523	+	1.02523i	−0.0255558	+	0.999673i	$0.508136\pi$
$137$	−12.0000	+	12.0000i	−1.02523	+	1.02523i	−0.999673	+	0.0255558i	$0.991864\pi$
$138$		0			0
$139$			8.00000i			0.678551i	0.940687	+	0.339276i	$0.110182\pi$
$139$			8.00000i			0.678551i	−0.940687	+	0.339276i	$0.889818\pi$
$140$		0			0
$141$	5.00000	−	5.00000i	0.421076	−	0.421076i
$142$		0			0
$143$	15.0000	+	3.00000i	1.25436	+	0.250873i
$144$		0			0
$145$	−12.0000	−	12.0000i	−0.996546	−	0.996546i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$	8.00000	+	8.00000i	0.655386	+	0.655386i	0.954285	−	0.298899i	$-0.0966194\pi$
$149$	8.00000	+	8.00000i	0.655386	+	0.655386i	−0.298899	+	0.954285i	$0.596619\pi$
$150$		0			0
$151$	−12.0000	−	12.0000i	−0.976546	−	0.976546i	0.0231850	−	0.999731i	$-0.492619\pi$
$151$	−12.0000	−	12.0000i	−0.976546	−	0.976546i	−0.999731	+	0.0231850i	$0.992619\pi$
$152$		0			0
$153$			2.00000i			0.161690i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	20.0000			1.59617			0.798087	−	0.602542i	$-0.205846\pi$
$157$	20.0000			1.59617			0.798087	+	0.602542i	$0.205846\pi$
$158$		0			0
$159$		−	6.00000i		−	0.475831i
$160$		0			0
$161$	12.0000	+	12.0000i	0.945732	+	0.945732i
$162$		0			0
$163$	−12.0000	−	12.0000i	−0.939913	−	0.939913i	0.0583818	−	0.998294i	$-0.481406\pi$
$163$	−12.0000	−	12.0000i	−0.939913	−	0.939913i	−0.998294	+	0.0583818i	$0.981406\pi$
$164$		0			0
$165$	−12.0000			−0.934199
$166$		0			0
$167$	13.0000	+	13.0000i	1.00597	+	1.00597i	0.999982	+	0.00598813i	$0.00190609\pi$
$167$	13.0000	+	13.0000i	1.00597	+	1.00597i	0.00598813	+	0.999982i	$0.498094\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$	6.00000	−	6.00000i	0.453557	−	0.453557i
$176$		0			0
$177$	1.00000	−	1.00000i	0.0751646	−	0.0751646i
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$			26.0000i			1.93256i	0.257485	+	0.966282i	$0.417106\pi$
$181$			26.0000i			1.93256i	−0.257485	+	0.966282i	$0.582894\pi$
$182$		0			0
$183$		−	8.00000i		−	0.591377i
$184$		0			0
$185$	12.0000			0.882258
$186$		0			0
$187$	−6.00000	+	6.00000i	−0.438763	+	0.438763i
$188$		0			0
$189$	2.00000	−	2.00000i	0.145479	−	0.145479i
$190$		0			0
$191$			16.0000i			1.15772i	0.815427	+	0.578860i	$0.196502\pi$
$191$			16.0000i			1.15772i	−0.815427	+	0.578860i	$0.803498\pi$
$192$		0			0
$193$	17.0000	−	17.0000i	1.22369	−	1.22369i	0.257375	−	0.966312i	$-0.417142\pi$
$193$	17.0000	−	17.0000i	1.22369	−	1.22369i	0.966312	−	0.257375i	$-0.0828576\pi$
$194$		0			0
$195$	−2.00000	+	10.0000i	−0.143223	+	0.716115i
$196$		0			0
$197$	−14.0000	−	14.0000i	−0.997459	−	0.997459i	0.00253808	−	0.999997i	$-0.499192\pi$
$197$	−14.0000	−	14.0000i	−0.997459	−	0.997459i	−0.999997	+	0.00253808i	$0.999192\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$		0			0
$201$	−2.00000	−	2.00000i	−0.141069	−	0.141069i
$202$		0			0
$203$	12.0000	+	12.0000i	0.842235	+	0.842235i
$204$		0			0
$205$		−	32.0000i		−	2.23498i
$206$		0			0
$207$	6.00000			0.417029
$208$		0			0
$209$		0			0
$210$		0			0
$211$			4.00000i			0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$			4.00000i			0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$		0			0
$213$	5.00000	+	5.00000i	0.342594	+	0.342594i
$214$		0			0
$215$	−8.00000	−	8.00000i	−0.545595	−	0.545595i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	3.00000	+	3.00000i	0.202721	+	0.202721i
$220$		0			0
$221$	4.00000	+	6.00000i	0.269069	+	0.403604i
$222$		0			0
$223$	4.00000	−	4.00000i	0.267860	−	0.267860i	−0.560378	−	0.828237i	$-0.689344\pi$
$223$	4.00000	−	4.00000i	0.267860	−	0.267860i	0.828237	+	0.560378i	$0.189344\pi$
$224$		0			0
$225$		−	3.00000i		−	0.200000i
$226$		0			0
$227$	7.00000	−	7.00000i	0.464606	−	0.464606i	−0.435556	−	0.900162i	$-0.643448\pi$
$227$	7.00000	−	7.00000i	0.464606	−	0.464606i	0.900162	+	0.435556i	$0.143448\pi$
$228$		0			0
$229$	15.0000	−	15.0000i	0.991228	−	0.991228i	−0.00873396	−	0.999962i	$-0.502780\pi$
$229$	15.0000	−	15.0000i	0.991228	−	0.991228i	0.999962	+	0.00873396i	$0.00278014\pi$
$230$		0			0
$231$	12.0000			0.789542
$232$		0			0
$233$		−	22.0000i		−	1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$		−	22.0000i		−	1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$		0			0
$235$			20.0000i			1.30466i
$236$		0			0
$237$	−12.0000			−0.779484
$238$		0			0
$239$	1.00000	−	1.00000i	0.0646846	−	0.0646846i	−0.674024	−	0.738709i	$-0.735436\pi$
$239$	1.00000	−	1.00000i	0.0646846	−	0.0646846i	0.738709	+	0.674024i	$0.235436\pi$
$240$		0			0
$241$	15.0000	−	15.0000i	0.966235	−	0.966235i	−0.0332133	−	0.999448i	$-0.510574\pi$
$241$	15.0000	−	15.0000i	0.966235	−	0.966235i	0.999448	+	0.0332133i	$0.0105741\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−2.00000	+	2.00000i	−0.127775	+	0.127775i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−5.00000	−	5.00000i	−0.316862	−	0.316862i
$250$		0			0
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$		0			0
$253$	18.0000	+	18.0000i	1.13165	+	1.13165i
$254$		0			0
$255$	−4.00000	−	4.00000i	−0.250490	−	0.250490i
$256$		0			0
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$	−12.0000			−0.745644
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$			6.00000i			0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$			6.00000i			0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$		0			0
$265$	12.0000	+	12.0000i	0.737154	+	0.737154i
$266$		0			0
$267$	10.0000	+	10.0000i	0.611990	+	0.611990i
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$	−10.0000	−	10.0000i	−0.607457	−	0.607457i	0.334824	−	0.942281i	$-0.391323\pi$
$271$	−10.0000	−	10.0000i	−0.607457	−	0.607457i	−0.942281	+	0.334824i	$0.891323\pi$
$272$		0			0
$273$	2.00000	−	10.0000i	0.121046	−	0.605228i
$274$		0			0
$275$	9.00000	−	9.00000i	0.542720	−	0.542720i
$276$		0			0
$277$		−	8.00000i		−	0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$		−	8.00000i		−	0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.00000	−	4.00000i	0.238620	−	0.238620i	−0.577659	−	0.816279i	$-0.696033\pi$
$281$	4.00000	−	4.00000i	0.238620	−	0.238620i	0.816279	+	0.577659i	$0.196033\pi$
$282$		0			0
$283$	16.0000			0.951101			0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000			0.951101			0.475551	+	0.879688i	$0.342249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			32.0000i			1.88890i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	−11.0000	+	11.0000i	−0.644831	+	0.644831i
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$			4.00000i			0.232889i
$296$		0			0
$297$	3.00000	−	3.00000i	0.174078	−	0.174078i
$298$		0			0
$299$	18.0000	−	12.0000i	1.04097	−	0.693978i
$300$		0			0
$301$	8.00000	+	8.00000i	0.461112	+	0.461112i
$302$		0			0
$303$	2.00000			0.114897
$304$		0			0
$305$	16.0000	+	16.0000i	0.916157	+	0.916157i
$306$		0			0
$307$	−2.00000	−	2.00000i	−0.114146	−	0.114146i	0.647727	−	0.761873i	$-0.275720\pi$
$307$	−2.00000	−	2.00000i	−0.114146	−	0.114146i	−0.761873	+	0.647727i	$0.775720\pi$
$308$		0			0
$309$			20.0000i			1.13776i
$310$		0			0
$311$	−18.0000			−1.02069			−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000			−1.02069			−0.510343	+	0.859971i	$0.670482\pi$
$312$		0			0
$313$	26.0000			1.46961			0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000			1.46961			0.734803	+	0.678280i	$0.237274\pi$
$314$		0			0
$315$			8.00000i			0.450749i
$316$		0			0
$317$	−18.0000	−	18.0000i	−1.01098	−	1.01098i	−0.999939	−	0.0110417i	$-0.996485\pi$
$317$	−18.0000	−	18.0000i	−1.01098	−	1.01098i	−0.0110417	−	0.999939i	$-0.503515\pi$
$318$		0			0
$319$	18.0000	+	18.0000i	1.00781	+	1.00781i
$320$		0			0
$321$	−14.0000			−0.781404
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−6.00000	−	9.00000i	−0.332820	−	0.499230i
$326$		0			0
$327$	3.00000	−	3.00000i	0.165900	−	0.165900i
$328$		0			0
$329$		−	20.0000i		−	1.10264i
$330$		0			0
$331$	14.0000	−	14.0000i	0.769510	−	0.769510i	−0.208511	−	0.978020i	$-0.566862\pi$
$331$	14.0000	−	14.0000i	0.769510	−	0.769510i	0.978020	+	0.208511i	$0.0668616\pi$
$332$		0			0
$333$	−3.00000	+	3.00000i	−0.164399	+	0.164399i
$334$		0			0
$335$	8.00000			0.437087
$336$		0			0
$337$			16.0000i			0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$			16.0000i			0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$		0			0
$339$			2.00000i			0.108625i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$		0			0
$345$	−12.0000	+	12.0000i	−0.646058	+	0.646058i
$346$		0			0
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	−3.00000	+	3.00000i	−0.160586	+	0.160586i	−0.782826	−	0.622240i	$-0.786223\pi$
$349$	−3.00000	+	3.00000i	−0.160586	+	0.160586i	0.622240	+	0.782826i	$0.286223\pi$
$350$		0			0
$351$	−2.00000	−	3.00000i	−0.106752	−	0.160128i
$352$		0			0
$353$	−16.0000	−	16.0000i	−0.851594	−	0.851594i	0.138735	−	0.990329i	$-0.455696\pi$
$353$	−16.0000	−	16.0000i	−0.851594	−	0.851594i	−0.990329	+	0.138735i	$0.955696\pi$
$354$		0			0
$355$	−20.0000			−1.06149
$356$		0			0
$357$	4.00000	+	4.00000i	0.211702	+	0.211702i
$358$		0			0
$359$	−17.0000	−	17.0000i	−0.897226	−	0.897226i	0.0979643	−	0.995190i	$-0.468767\pi$
$359$	−17.0000	−	17.0000i	−0.897226	−	0.897226i	−0.995190	+	0.0979643i	$0.968767\pi$
$360$		0			0
$361$			19.0000i			1.00000i
$362$		0			0
$363$	7.00000			0.367405
$364$		0			0
$365$	−12.0000			−0.628109
$366$		0			0
$367$			28.0000i			1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$			28.0000i			1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$		0			0
$369$	8.00000	+	8.00000i	0.416463	+	0.416463i
$370$		0			0
$371$	−12.0000	−	12.0000i	−0.623009	−	0.623009i
$372$		0			0
$373$	−38.0000			−1.96757			−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−38.0000			−1.96757			−0.983783	+	0.179364i	$0.942596\pi$
$374$		0			0
$375$	−4.00000	−	4.00000i	−0.206559	−	0.206559i
$376$		0			0
$377$	18.0000	−	12.0000i	0.927047	−	0.618031i
$378$		0			0
$379$	4.00000	−	4.00000i	0.205466	−	0.205466i	−0.596871	−	0.802337i	$-0.703590\pi$
$379$	4.00000	−	4.00000i	0.205466	−	0.205466i	0.802337	+	0.596871i	$0.203590\pi$
$380$		0			0
$381$		−	8.00000i		−	0.409852i
$382$		0			0
$383$	−21.0000	+	21.0000i	−1.07305	+	1.07305i	−0.0759373	+	0.997113i	$0.524195\pi$
$383$	−21.0000	+	21.0000i	−1.07305	+	1.07305i	−0.997113	+	0.0759373i	$0.975805\pi$
$384$		0			0
$385$	−24.0000	+	24.0000i	−1.22315	+	1.22315i
$386$		0			0
$387$	4.00000			0.203331
$388$		0			0
$389$			10.0000i			0.507020i	0.967333	+	0.253510i	$0.0815851\pi$
$389$			10.0000i			0.507020i	−0.967333	+	0.253510i	$0.918415\pi$
$390$		0			0
$391$			12.0000i			0.606866i
$392$		0			0
$393$	22.0000			1.10975
$394$		0			0
$395$	24.0000	−	24.0000i	1.20757	−	1.20757i
$396$		0			0
$397$	13.0000	−	13.0000i	0.652451	−	0.652451i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	13.0000	−	13.0000i	0.652451	−	0.652451i	0.953583	+	0.301131i	$0.0973643\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−22.0000	+	22.0000i	−1.09863	+	1.09863i	−0.104056	+	0.994571i	$0.533182\pi$
$401$	−22.0000	+	22.0000i	−1.09863	+	1.09863i	−0.994571	+	0.104056i	$0.966818\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	2.00000	+	2.00000i	0.0993808	+	0.0993808i
$406$		0			0
$407$	−18.0000			−0.892227
$408$		0			0
$409$	−3.00000	−	3.00000i	−0.148340	−	0.148340i	0.629036	−	0.777376i	$-0.283450\pi$
$409$	−3.00000	−	3.00000i	−0.148340	−	0.148340i	−0.777376	+	0.629036i	$0.783450\pi$
$410$		0			0
$411$	12.0000	+	12.0000i	0.591916	+	0.591916i
$412$		0			0
$413$		−	4.00000i		−	0.196827i
$414$		0			0
$415$	20.0000			0.981761
$416$		0			0
$417$	8.00000			0.391762
$418$		0			0
$419$			26.0000i			1.27018i	0.772437	+	0.635092i	$0.219038\pi$
$419$			26.0000i			1.27018i	−0.772437	+	0.635092i	$0.780962\pi$
$420$		0			0
$421$	−3.00000	−	3.00000i	−0.146211	−	0.146211i	0.630212	−	0.776423i	$-0.282968\pi$
$421$	−3.00000	−	3.00000i	−0.146211	−	0.146211i	−0.776423	+	0.630212i	$0.782968\pi$
$422$		0			0
$423$	−5.00000	−	5.00000i	−0.243108	−	0.243108i
$424$		0			0
$425$	6.00000			0.291043
$426$		0			0
$427$	−16.0000	−	16.0000i	−0.774294	−	0.774294i
$428$		0			0
$429$	3.00000	−	15.0000i	0.144841	−	0.724207i
$430$		0			0
$431$	−1.00000	+	1.00000i	−0.0481683	+	0.0481683i	−0.730781	−	0.682612i	$-0.760844\pi$
$431$	−1.00000	+	1.00000i	−0.0481683	+	0.0481683i	0.682612	+	0.730781i	$0.260844\pi$
$432$		0			0
$433$		−	18.0000i		−	0.865025i	−0.901628	−	0.432512i	$-0.857627\pi$
$433$		−	18.0000i		−	0.865025i	0.901628	−	0.432512i	$-0.142373\pi$
$434$		0			0
$435$	−12.0000	+	12.0000i	−0.575356	+	0.575356i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$		−	1.00000i		−	0.0476190i
$442$		0			0
$443$			18.0000i			0.855206i	0.903967	+	0.427603i	$0.140642\pi$
$443$			18.0000i			0.855206i	−0.903967	+	0.427603i	$0.859358\pi$
$444$		0			0
$445$	−40.0000			−1.89618
$446$		0			0
$447$	8.00000	−	8.00000i	0.378387	−	0.378387i
$448$		0			0
$449$	−2.00000	+	2.00000i	−0.0943858	+	0.0943858i	−0.752723	−	0.658337i	$-0.771260\pi$
$449$	−2.00000	+	2.00000i	−0.0943858	+	0.0943858i	0.658337	+	0.752723i	$0.271260\pi$
$450$		0			0
$451$			48.0000i			2.26023i
$452$		0			0
$453$	−12.0000	+	12.0000i	−0.563809	+	0.563809i
$454$		0			0
$455$	16.0000	+	24.0000i	0.750092	+	1.12514i
$456$		0			0
$457$	−5.00000	−	5.00000i	−0.233890	−	0.233890i	0.580424	−	0.814314i	$-0.302887\pi$
$457$	−5.00000	−	5.00000i	−0.233890	−	0.233890i	−0.814314	+	0.580424i	$0.802887\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$	14.0000	+	14.0000i	0.652045	+	0.652045i	0.953485	−	0.301440i	$-0.0974673\pi$
$461$	14.0000	+	14.0000i	0.652045	+	0.652045i	−0.301440	+	0.953485i	$0.597467\pi$
$462$		0			0
$463$	−22.0000	−	22.0000i	−1.02243	−	1.02243i	−0.999743	−	0.0226840i	$-0.992779\pi$
$463$	−22.0000	−	22.0000i	−1.02243	−	1.02243i	−0.0226840	−	0.999743i	$-0.507221\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−38.0000			−1.75843			−0.879215	−	0.476425i	$-0.841932\pi$
$467$	−38.0000			−1.75843			−0.879215	+	0.476425i	$0.841932\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$		−	20.0000i		−	0.921551i
$472$		0			0
$473$	12.0000	+	12.0000i	0.551761	+	0.551761i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	13.0000	+	13.0000i	0.593985	+	0.593985i	0.938705	−	0.344720i	$-0.112026\pi$
$479$	13.0000	+	13.0000i	0.593985	+	0.593985i	−0.344720	+	0.938705i	$0.612026\pi$
$480$		0			0
$481$	−3.00000	+	15.0000i	−0.136788	+	0.683941i
$482$		0			0
$483$	12.0000	−	12.0000i	0.546019	−	0.546019i
$484$		0			0
$485$		−	44.0000i		−	1.99794i
$486$		0			0
$487$	8.00000	−	8.00000i	0.362515	−	0.362515i	−0.502223	−	0.864738i	$-0.667484\pi$
$487$	8.00000	−	8.00000i	0.362515	−	0.362515i	0.864738	+	0.502223i	$0.167484\pi$
$488$		0			0
$489$	−12.0000	+	12.0000i	−0.542659	+	0.542659i
$490$		0			0
$491$	−2.00000			−0.0902587			−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000			−0.0902587			−0.0451294	+	0.998981i	$0.514370\pi$
$492$		0			0
$493$			12.0000i			0.540453i
$494$		0			0
$495$			12.0000i			0.539360i
$496$		0			0
$497$	20.0000			0.897123
$498$		0			0
$499$	30.0000	−	30.0000i	1.34298	−	1.34298i	0.449911	−	0.893073i	$-0.351456\pi$
$499$	30.0000	−	30.0000i	1.34298	−	1.34298i	0.893073	−	0.449911i	$-0.148544\pi$
$500$		0			0
$501$	13.0000	−	13.0000i	0.580797	−	0.580797i
$502$		0			0
$503$		−	26.0000i		−	1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$		−	26.0000i		−	1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$		0			0
$505$	−4.00000	+	4.00000i	−0.177998	+	0.177998i
$506$		0			0
$507$	−12.0000	−	5.00000i	−0.532939	−	0.222058i
$508$		0			0
$509$	2.00000	+	2.00000i	0.0886484	+	0.0886484i	0.750040	−	0.661392i	$-0.230034\pi$
$509$	2.00000	+	2.00000i	0.0886484	+	0.0886484i	−0.661392	+	0.750040i	$0.730034\pi$
$510$		0			0
$511$	12.0000			0.530849
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−40.0000	−	40.0000i	−1.76261	−	1.76261i
$516$		0			0
$517$		−	30.0000i		−	1.31940i
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	26.0000			1.13908			0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000			1.13908			0.569540	+	0.821963i	$0.307121\pi$
$522$		0			0
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$		0			0
$525$	−6.00000	−	6.00000i	−0.261861	−	0.261861i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$	−1.00000	−	1.00000i	−0.0433963	−	0.0433963i
$532$		0			0
$533$	40.0000	+	8.00000i	1.73259	+	0.346518i
$534$		0			0
$535$	28.0000	−	28.0000i	1.21055	−	1.21055i
$536$		0			0
$537$			4.00000i			0.172613i
$538$		0			0
$539$	3.00000	−	3.00000i	0.129219	−	0.129219i
$540$		0			0
$541$	−25.0000	+	25.0000i	−1.07483	+	1.07483i	−0.0778705	+	0.996963i	$0.524812\pi$
$541$	−25.0000	+	25.0000i	−1.07483	+	1.07483i	−0.996963	+	0.0778705i	$0.975188\pi$
$542$		0			0
$543$	26.0000			1.11577
$544$		0			0
$545$			12.0000i			0.514024i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	−8.00000			−0.341432
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−24.0000	+	24.0000i	−1.02058	+	1.02058i
$554$		0			0
$555$		−	12.0000i		−	0.509372i
$556$		0			0
$557$	−12.0000	+	12.0000i	−0.508456	+	0.508456i	−0.914052	−	0.405596i	$-0.867064\pi$
$557$	−12.0000	+	12.0000i	−0.508456	+	0.508456i	0.405596	+	0.914052i	$0.367064\pi$
$558$		0			0
$559$	12.0000	−	8.00000i	0.507546	−	0.338364i
$560$		0			0
$561$	6.00000	+	6.00000i	0.253320	+	0.253320i
$562$		0			0
$563$	28.0000			1.18006			0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000			1.18006			0.590030	+	0.807382i	$0.299116\pi$
$564$		0			0
$565$	−4.00000	−	4.00000i	−0.168281	−	0.168281i
$566$		0			0
$567$	−2.00000	−	2.00000i	−0.0839921	−	0.0839921i
$568$		0			0
$569$		−	30.0000i		−	1.25767i	−0.777541	−	0.628833i	$-0.783533\pi$
$569$		−	30.0000i		−	1.25767i	0.777541	−	0.628833i	$-0.216467\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$	16.0000			0.668410
$574$		0			0
$575$		−	18.0000i		−	0.750652i
$576$		0			0
$577$	−31.0000	−	31.0000i	−1.29055	−	1.29055i	−0.934448	−	0.356098i	$-0.884107\pi$
$577$	−31.0000	−	31.0000i	−1.29055	−	1.29055i	−0.356098	−	0.934448i	$-0.615893\pi$
$578$		0			0
$579$	−17.0000	−	17.0000i	−0.706496	−	0.706496i
$580$		0			0
$581$	−20.0000			−0.829740
$582$		0			0
$583$	−18.0000	−	18.0000i	−0.745484	−	0.745484i
$584$		0			0
$585$	10.0000	+	2.00000i	0.413449	+	0.0826898i
$586$		0			0
$587$	11.0000	−	11.0000i	0.454019	−	0.454019i	−0.442667	−	0.896686i	$-0.645968\pi$
$587$	11.0000	−	11.0000i	0.454019	−	0.454019i	0.896686	+	0.442667i	$0.145968\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−14.0000	+	14.0000i	−0.575883	+	0.575883i
$592$		0			0
$593$	−6.00000	+	6.00000i	−0.246390	+	0.246390i	−0.819488	−	0.573097i	$-0.805742\pi$
$593$	−6.00000	+	6.00000i	−0.246390	+	0.246390i	0.573097	+	0.819488i	$0.305742\pi$
$594$		0			0
$595$	−16.0000			−0.655936
$596$		0			0
$597$		−	20.0000i		−	0.818546i
$598$		0			0
$599$			2.00000i			0.0817178i	0.999165	+	0.0408589i	$0.0130094\pi$
$599$			2.00000i			0.0817178i	−0.999165	+	0.0408589i	$0.986991\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$	−2.00000	+	2.00000i	−0.0814463	+	0.0814463i
$604$		0			0
$605$	−14.0000	+	14.0000i	−0.569181	+	0.569181i
$606$		0			0
$607$		−	36.0000i		−	1.46119i	−0.682808	−	0.730597i	$-0.739242\pi$
$607$		−	36.0000i		−	1.46119i	0.682808	−	0.730597i	$-0.260758\pi$
$608$		0			0
$609$	12.0000	−	12.0000i	0.486265	−	0.486265i
$610$		0			0
$611$	−25.0000	−	5.00000i	−1.01139	−	0.202278i
$612$		0			0
$613$	21.0000	+	21.0000i	0.848182	+	0.848182i	0.989906	−	0.141724i	$-0.0452646\pi$
$613$	21.0000	+	21.0000i	0.848182	+	0.848182i	−0.141724	+	0.989906i	$0.545265\pi$
$614$		0			0
$615$	−32.0000			−1.29036
$616$		0			0
$617$	−20.0000	−	20.0000i	−0.805170	−	0.805170i	0.178729	−	0.983898i	$-0.442802\pi$
$617$	−20.0000	−	20.0000i	−0.805170	−	0.805170i	−0.983898	+	0.178729i	$0.942802\pi$
$618$		0			0
$619$	−18.0000	−	18.0000i	−0.723481	−	0.723481i	0.245831	−	0.969313i	$-0.420939\pi$
$619$	−18.0000	−	18.0000i	−0.723481	−	0.723481i	−0.969313	+	0.245831i	$0.920939\pi$
$620$		0			0
$621$		−	6.00000i		−	0.240772i
$622$		0			0
$623$	40.0000			1.60257
$624$		0			0
$625$	31.0000			1.24000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.00000	−	6.00000i	−0.239236	−	0.239236i
$630$		0			0
$631$	−20.0000	−	20.0000i	−0.796187	−	0.796187i	0.186305	−	0.982492i	$-0.440349\pi$
$631$	−20.0000	−	20.0000i	−0.796187	−	0.796187i	−0.982492	+	0.186305i	$0.940349\pi$
$632$		0			0
$633$	4.00000			0.158986
$634$		0			0
$635$	16.0000	+	16.0000i	0.634941	+	0.634941i
$636$		0			0
$637$	−2.00000	−	3.00000i	−0.0792429	−	0.118864i
$638$		0			0
$639$	5.00000	−	5.00000i	0.197797	−	0.197797i
$640$		0			0
$641$			30.0000i			1.18493i	0.805597	+	0.592464i	$0.201845\pi$
$641$			30.0000i			1.18493i	−0.805597	+	0.592464i	$0.798155\pi$
$642$		0			0
$643$	−34.0000	+	34.0000i	−1.34083	+	1.34083i	−0.445594	+	0.895235i	$0.647007\pi$
$643$	−34.0000	+	34.0000i	−1.34083	+	1.34083i	−0.895235	+	0.445594i	$0.852993\pi$
$644$		0			0
$645$	−8.00000	+	8.00000i	−0.315000	+	0.315000i
$646$		0			0
$647$	−8.00000			−0.314512			−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000			−0.314512			−0.157256	+	0.987558i	$0.550265\pi$
$648$		0			0
$649$		−	6.00000i		−	0.235521i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−30.0000			−1.17399			−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000			−1.17399			−0.586995	+	0.809590i	$0.699689\pi$
$654$		0			0
$655$	−44.0000	+	44.0000i	−1.71922	+	1.71922i
$656$		0			0
$657$	3.00000	−	3.00000i	0.117041	−	0.117041i
$658$		0			0
$659$		−	44.0000i		−	1.71400i	−0.515319	−	0.856998i	$-0.672327\pi$
$659$		−	44.0000i		−	1.71400i	0.515319	−	0.856998i	$-0.327673\pi$
$660$		0			0
$661$	−17.0000	+	17.0000i	−0.661223	+	0.661223i	−0.955668	−	0.294445i	$-0.904865\pi$
$661$	−17.0000	+	17.0000i	−0.661223	+	0.661223i	0.294445	+	0.955668i	$0.404865\pi$
$662$		0			0
$663$	6.00000	−	4.00000i	0.233021	−	0.155347i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	36.0000			1.39393
$668$		0			0
$669$	−4.00000	−	4.00000i	−0.154649	−	0.154649i
$670$		0			0
$671$	−24.0000	−	24.0000i	−0.926510	−	0.926510i
$672$		0			0
$673$		−	14.0000i		−	0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$		−	14.0000i		−	0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$		0			0
$675$	−3.00000			−0.115470
$676$		0			0
$677$	−14.0000			−0.538064			−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000			−0.538064			−0.269032	+	0.963131i	$0.586704\pi$
$678$		0			0
$679$			44.0000i			1.68857i
$680$		0			0
$681$	−7.00000	−	7.00000i	−0.268241	−	0.268241i
$682$		0			0
$683$	7.00000	+	7.00000i	0.267848	+	0.267848i	0.828232	−	0.560385i	$-0.189347\pi$
$683$	7.00000	+	7.00000i	0.267848	+	0.267848i	−0.560385	+	0.828232i	$0.689347\pi$
$684$		0			0
$685$	−48.0000			−1.83399
$686$		0			0
$687$	−15.0000	−	15.0000i	−0.572286	−	0.572286i
$688$		0			0
$689$	−18.0000	+	12.0000i	−0.685745	+	0.457164i
$690$		0			0
$691$	−6.00000	+	6.00000i	−0.228251	+	0.228251i	−0.811962	−	0.583711i	$-0.801600\pi$
$691$	−6.00000	+	6.00000i	−0.228251	+	0.228251i	0.583711	+	0.811962i	$0.301600\pi$
$692$		0			0
$693$		−	12.0000i		−	0.455842i
$694$		0			0
$695$	−16.0000	+	16.0000i	−0.606915	+	0.606915i
$696$		0			0
$697$	−16.0000	+	16.0000i	−0.606043	+	0.606043i
$698$		0			0
$699$	−22.0000			−0.832116
$700$		0			0
$701$			50.0000i			1.88847i	0.329267	+	0.944237i	$0.393198\pi$
$701$			50.0000i			1.88847i	−0.329267	+	0.944237i	$0.606802\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	20.0000			0.753244
$706$		0			0
$707$	4.00000	−	4.00000i	0.150435	−	0.150435i
$708$		0			0
$709$	−3.00000	+	3.00000i	−0.112667	+	0.112667i	−0.761193	−	0.648526i	$-0.775386\pi$
$709$	−3.00000	+	3.00000i	−0.112667	+	0.112667i	0.648526	+	0.761193i	$0.275386\pi$
$710$		0			0
$711$			12.0000i			0.450035i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	24.0000	+	36.0000i	0.897549	+	1.34632i
$716$		0			0
$717$	−1.00000	−	1.00000i	−0.0373457	−	0.0373457i
$718$		0			0
$719$	−10.0000			−0.372937			−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000			−0.372937			−0.186469	+	0.982461i	$0.559704\pi$
$720$		0			0
$721$	40.0000	+	40.0000i	1.48968	+	1.48968i
$722$		0			0
$723$	−15.0000	−	15.0000i	−0.557856	−	0.557856i
$724$		0			0
$725$		−	18.0000i		−	0.668503i
$726$		0			0
$727$	−36.0000			−1.33517			−0.667583	−	0.744535i	$-0.732671\pi$
$727$	−36.0000			−1.33517			−0.667583	+	0.744535i	$0.732671\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			8.00000i			0.295891i
$732$		0			0
$733$	−35.0000	−	35.0000i	−1.29275	−	1.29275i	−0.933076	−	0.359678i	$-0.882887\pi$
$733$	−35.0000	−	35.0000i	−1.29275	−	1.29275i	−0.359678	−	0.933076i	$-0.617113\pi$
$734$		0			0
$735$	2.00000	+	2.00000i	0.0737711	+	0.0737711i
$736$		0			0
$737$	−12.0000			−0.442026
$738$		0			0
$739$	10.0000	+	10.0000i	0.367856	+	0.367856i	0.866695	−	0.498839i	$-0.166240\pi$
$739$	10.0000	+	10.0000i	0.367856	+	0.367856i	−0.498839	+	0.866695i	$0.666240\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.00000	−	5.00000i	0.183432	−	0.183432i	−0.609417	−	0.792850i	$-0.708597\pi$
$743$	5.00000	−	5.00000i	0.183432	−	0.183432i	0.792850	+	0.609417i	$0.208597\pi$
$744$		0			0
$745$			32.0000i			1.17239i
$746$		0			0
$747$	−5.00000	+	5.00000i	−0.182940	+	0.182940i
$748$		0			0
$749$	−28.0000	+	28.0000i	−1.02310	+	1.02310i
$750$		0			0
$751$	12.0000			0.437886			0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000			0.437886			0.218943	+	0.975738i	$0.429739\pi$
$752$		0			0
$753$		−	2.00000i		−	0.0728841i
$754$		0			0
$755$		−	48.0000i		−	1.74690i
$756$		0			0
$757$	−12.0000			−0.436147			−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000			−0.436147			−0.218074	+	0.975932i	$0.569977\pi$
$758$		0			0
$759$	18.0000	−	18.0000i	0.653359	−	0.653359i
$760$		0			0
$761$	8.00000	−	8.00000i	0.290000	−	0.290000i	−0.547080	−	0.837080i	$-0.684261\pi$
$761$	8.00000	−	8.00000i	0.290000	−	0.290000i	0.837080	+	0.547080i	$0.184261\pi$
$762$		0			0
$763$		−	12.0000i		−	0.434429i
$764$		0			0
$765$	−4.00000	+	4.00000i	−0.144620	+	0.144620i
$766$		0			0
$767$	−5.00000	−	1.00000i	−0.180540	−	0.0361079i
$768$		0			0
$769$	−31.0000	−	31.0000i	−1.11789	−	1.11789i	−0.992051	−	0.125838i	$-0.959838\pi$
$769$	−31.0000	−	31.0000i	−1.11789	−	1.11789i	−0.125838	−	0.992051i	$-0.540162\pi$
$770$		0			0
$771$	6.00000			0.216085
$772$		0			0
$773$	−8.00000	−	8.00000i	−0.287740	−	0.287740i	0.548446	−	0.836186i	$-0.315220\pi$
$773$	−8.00000	−	8.00000i	−0.287740	−	0.287740i	−0.836186	+	0.548446i	$0.815220\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			12.0000i			0.430498i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	30.0000			1.07348
$782$		0			0
$783$		−	6.00000i		−	0.214423i
$784$		0			0
$785$	40.0000	+	40.0000i	1.42766	+	1.42766i
$786$		0			0
$787$	−18.0000	−	18.0000i	−0.641631	−	0.641631i	0.309326	−	0.950956i	$-0.399897\pi$
$787$	−18.0000	−	18.0000i	−0.641631	−	0.641631i	−0.950956	+	0.309326i	$0.899897\pi$
$788$		0			0
$789$	6.00000			0.213606
$790$		0			0
$791$	4.00000	+	4.00000i	0.142224	+	0.142224i
$792$		0			0
$793$	−24.0000	+	16.0000i	−0.852265	+	0.568177i
$794$		0			0
$795$	12.0000	−	12.0000i	0.425596	−	0.425596i
$796$		0			0
$797$		−	14.0000i		−	0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$		−	14.0000i		−	0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$		0			0
$799$	10.0000	−	10.0000i	0.353775	−	0.353775i
$800$		0			0
$801$	10.0000	−	10.0000i	0.353333	−	0.353333i
$802$		0			0
$803$	18.0000			0.635206
$804$		0			0
$805$			48.0000i			1.69178i
$806$		0			0
$807$		−	10.0000i		−	0.352017i
$808$		0			0
$809$	42.0000			1.47664			0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000			1.47664			0.738321	+	0.674450i	$0.235619\pi$
$810$		0			0
$811$	−12.0000	+	12.0000i	−0.421377	+	0.421377i	−0.885678	−	0.464301i	$-0.846306\pi$
$811$	−12.0000	+	12.0000i	−0.421377	+	0.421377i	0.464301	+	0.885678i	$0.346306\pi$
$812$		0			0
$813$	−10.0000	+	10.0000i	−0.350715	+	0.350715i
$814$		0			0
$815$		−	48.0000i		−	1.68137i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−10.0000	−	2.00000i	−0.349428	−	0.0698857i
$820$		0			0
$821$	−24.0000	−	24.0000i	−0.837606	−	0.837606i	0.150938	−	0.988543i	$-0.451771\pi$
$821$	−24.0000	−	24.0000i	−0.837606	−	0.837606i	−0.988543	+	0.150938i	$0.951771\pi$
$822$		0			0
$823$	8.00000			0.278862			0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000			0.278862			0.139431	+	0.990232i	$0.455473\pi$
$824$		0			0
$825$	−9.00000	−	9.00000i	−0.313340	−	0.313340i
$826$		0			0
$827$	−15.0000	−	15.0000i	−0.521601	−	0.521601i	0.396454	−	0.918055i	$-0.370241\pi$
$827$	−15.0000	−	15.0000i	−0.521601	−	0.521601i	−0.918055	+	0.396454i	$0.870241\pi$
$828$		0			0
$829$			2.00000i			0.0694629i	0.999397	+	0.0347314i	$0.0110576\pi$
$829$			2.00000i			0.0694629i	−0.999397	+	0.0347314i	$0.988942\pi$
$830$		0			0
$831$	−8.00000			−0.277517
$832$		0			0
$833$	2.00000			0.0692959
$834$		0			0
$835$			52.0000i			1.79953i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.00000	+	1.00000i	0.0345238	+	0.0345238i	0.724158	−	0.689634i	$-0.242229\pi$
$839$	1.00000	+	1.00000i	0.0345238	+	0.0345238i	−0.689634	+	0.724158i	$0.742229\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	−4.00000	−	4.00000i	−0.137767	−	0.137767i
$844$		0			0
$845$	34.0000	−	14.0000i	1.16964	−	0.481615i
$846$		0			0
$847$	14.0000	−	14.0000i	0.481046	−	0.481046i
$848$		0			0
$849$		−	16.0000i		−	0.549119i
$850$		0			0
$851$	−18.0000	+	18.0000i	−0.617032	+	0.617032i
$852$		0			0
$853$	−23.0000	+	23.0000i	−0.787505	+	0.787505i	−0.981085	−	0.193580i	$-0.937990\pi$
$853$	−23.0000	+	23.0000i	−0.787505	+	0.787505i	0.193580	+	0.981085i	$0.437990\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	30.0000i		−	1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$		−	30.0000i		−	1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$		0			0
$859$			40.0000i			1.36478i	0.730987	+	0.682391i	$0.239060\pi$
$859$			40.0000i			1.36478i	−0.730987	+	0.682391i	$0.760940\pi$
$860$		0			0
$861$	32.0000			1.09056
$862$		0			0
$863$	5.00000	−	5.00000i	0.170202	−	0.170202i	−0.616866	−	0.787068i	$-0.711598\pi$
$863$	5.00000	−	5.00000i	0.170202	−	0.170202i	0.787068	+	0.616866i	$0.211598\pi$
$864$		0			0
$865$	12.0000	−	12.0000i	0.408012	−	0.408012i
$866$		0			0
$867$		−	13.0000i		−	0.441503i
$868$		0			0
$869$	−36.0000	+	36.0000i	−1.22122	+	1.22122i
$870$		0			0
$871$	−2.00000	+	10.0000i	−0.0677674	+	0.338837i
$872$		0			0
$873$	11.0000	+	11.0000i	0.372294	+	0.372294i
$874$		0			0
$875$	−16.0000			−0.540899
$876$		0			0
$877$	15.0000	+	15.0000i	0.506514	+	0.506514i	0.913455	−	0.406941i	$-0.133404\pi$
$877$	15.0000	+	15.0000i	0.506514	+	0.506514i	−0.406941	+	0.913455i	$0.633404\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	42.0000i		−	1.41502i	−0.706705	−	0.707508i	$-0.749819\pi$
$881$		−	42.0000i		−	1.41502i	0.706705	−	0.707508i	$-0.250181\pi$
$882$		0			0
$883$	24.0000			0.807664			0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000			0.807664			0.403832	+	0.914833i	$0.367678\pi$
$884$		0			0
$885$	4.00000			0.134459
$886$		0			0
$887$		−	24.0000i		−	0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$		−	24.0000i		−	0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$		0			0
$889$	−16.0000	−	16.0000i	−0.536623	−	0.536623i
$890$		0			0
$891$	−3.00000	−	3.00000i	−0.100504	−	0.100504i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−8.00000	−	8.00000i	−0.267411	−	0.267411i
$896$		0			0
$897$	−12.0000	−	18.0000i	−0.400668	−	0.601003i
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	12.0000i		−	0.399778i
$902$		0			0
$903$	8.00000	−	8.00000i	0.266223	−	0.266223i
$904$		0			0
$905$	−52.0000	+	52.0000i	−1.72854	+	1.72854i
$906$		0			0
$907$	20.0000			0.664089			0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000			0.664089			0.332045	+	0.943264i	$0.392262\pi$
$908$		0			0
$909$		−	2.00000i		−	0.0663358i
$910$		0			0
$911$			32.0000i			1.06021i	0.847933	+	0.530104i	$0.177847\pi$
$911$			32.0000i			1.06021i	−0.847933	+	0.530104i	$0.822153\pi$
$912$		0			0
$913$	−30.0000			−0.992855
$914$		0			0
$915$	16.0000	−	16.0000i	0.528944	−	0.528944i
$916$		0			0
$917$	44.0000	−	44.0000i	1.45301	−	1.45301i
$918$		0			0
$919$		−	20.0000i		−	0.659739i	−0.944027	−	0.329870i	$-0.892995\pi$
$919$		−	20.0000i		−	0.659739i	0.944027	−	0.329870i	$-0.107005\pi$
$920$		0			0
$921$	−2.00000	+	2.00000i	−0.0659022	+	0.0659022i
$922$		0			0
$923$	5.00000	−	25.0000i	0.164577	−	0.822885i
$924$		0			0
$925$	9.00000	+	9.00000i	0.295918	+	0.295918i
$926$		0			0
$927$	20.0000			0.656886
$928$		0			0
$929$	24.0000	+	24.0000i	0.787414	+	0.787414i	0.981070	−	0.193655i	$-0.0620343\pi$
$929$	24.0000	+	24.0000i	0.787414	+	0.787414i	−0.193655	+	0.981070i	$0.562034\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			18.0000i			0.589294i
$934$		0			0
$935$	−24.0000			−0.784884
$936$		0			0
$937$	−4.00000			−0.130674			−0.0653372	−	0.997863i	$-0.520812\pi$
$937$	−4.00000			−0.130674			−0.0653372	+	0.997863i	$0.520812\pi$
$938$		0			0
$939$		−	26.0000i		−	0.848478i
$940$		0			0
$941$	8.00000	+	8.00000i	0.260793	+	0.260793i	0.825376	−	0.564583i	$-0.190963\pi$
$941$	8.00000	+	8.00000i	0.260793	+	0.260793i	−0.564583	+	0.825376i	$0.690963\pi$
$942$		0			0
$943$	48.0000	+	48.0000i	1.56310	+	1.56310i
$944$		0			0
$945$	8.00000			0.260240
$946$		0			0
$947$	13.0000	+	13.0000i	0.422443	+	0.422443i	0.886044	−	0.463601i	$-0.153443\pi$
$947$	13.0000	+	13.0000i	0.422443	+	0.422443i	−0.463601	+	0.886044i	$0.653443\pi$
$948$		0			0
$949$	3.00000	−	15.0000i	0.0973841	−	0.486921i
$950$		0			0
$951$	−18.0000	+	18.0000i	−0.583690	+	0.583690i
$952$		0			0
$953$			6.00000i			0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$			6.00000i			0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$		0			0
$955$	−32.0000	+	32.0000i	−1.03550	+	1.03550i
$956$		0			0
$957$	18.0000	−	18.0000i	0.581857	−	0.581857i
$958$		0			0
$959$	48.0000			1.55000
$960$		0			0
$961$			31.0000i			1.00000i
$962$		0			0
$963$			14.0000i			0.451144i
$964$		0			0
$965$	68.0000			2.18900
$966$		0			0
$967$	16.0000	−	16.0000i	0.514525	−	0.514525i	−0.401384	−	0.915910i	$-0.631471\pi$
$967$	16.0000	−	16.0000i	0.514525	−	0.514525i	0.915910	+	0.401384i	$0.131471\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			12.0000i			0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$			12.0000i			0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$		0			0
$973$	16.0000	−	16.0000i	0.512936	−	0.512936i
$974$		0			0
$975$	−9.00000	+	6.00000i	−0.288231	+	0.192154i
$976$		0			0
$977$	−22.0000	−	22.0000i	−0.703842	−	0.703842i	0.261391	−	0.965233i	$-0.415819\pi$
$977$	−22.0000	−	22.0000i	−0.703842	−	0.703842i	−0.965233	+	0.261391i	$0.915819\pi$
$978$		0			0
$979$	60.0000			1.91761
$980$		0			0
$981$	−3.00000	−	3.00000i	−0.0957826	−	0.0957826i
$982$		0			0
$983$	33.0000	+	33.0000i	1.05254	+	1.05254i	0.998541	+	0.0539954i	$0.0171956\pi$
$983$	33.0000	+	33.0000i	1.05254	+	1.05254i	0.0539954	+	0.998541i	$0.482804\pi$
$984$		0			0
$985$		−	56.0000i		−	1.78431i
$986$		0			0
$987$	−20.0000			−0.636607
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$			8.00000i			0.254128i	0.991894	+	0.127064i	$0.0405554\pi$
$991$			8.00000i			0.254128i	−0.991894	+	0.127064i	$0.959445\pi$
$992$		0			0
$993$	−14.0000	−	14.0000i	−0.444277	−	0.444277i
$994$		0			0
$995$	40.0000	+	40.0000i	1.26809	+	1.26809i
$996$		0			0
$997$	14.0000			0.443384			0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000			0.443384			0.221692	+	0.975117i	$0.428842\pi$
$998$		0			0
$999$	3.00000	+	3.00000i	0.0949158	+	0.0949158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.2.bc.l.1087.1	yes	2
4.3	odd	2		1248.2.bc.m.1087.1	yes	2
13.5	odd	4		1248.2.bc.m.31.1	yes	2
52.31	even	4	inner	1248.2.bc.l.31.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.bc.l.31.1	✓	2	52.31	even	4	inner
1248.2.bc.l.1087.1	yes	2	1.1	even	1	trivial
1248.2.bc.m.31.1	yes	2	13.5	odd	4
1248.2.bc.m.1087.1	yes	2	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.bc (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(2.00000 + 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{7} -1.00000 q^{9} +(-3.00000 - 3.00000i) q^{11} +(-3.00000 + 2.00000i) q^{13} +(2.00000 - 2.00000i) q^{15} -2.00000i q^{17} +(-2.00000 + 2.00000i) q^{21} -6.00000 q^{23} +3.00000i q^{25} +1.00000i q^{27} -6.00000 q^{29} +(-3.00000 + 3.00000i) q^{33} -8.00000i q^{35} +(3.00000 - 3.00000i) q^{37} +(2.00000 + 3.00000i) q^{39} +(-8.00000 - 8.00000i) q^{41} -4.00000 q^{43} +(-2.00000 - 2.00000i) q^{45} +(5.00000 + 5.00000i) q^{47} +1.00000i q^{49} -2.00000 q^{51} +6.00000 q^{53} -12.0000i q^{55} +(1.00000 + 1.00000i) q^{59} +8.00000 q^{61} +(2.00000 + 2.00000i) q^{63} +(-10.0000 - 2.00000i) q^{65} +(2.00000 - 2.00000i) q^{67} +6.00000i q^{69} +(-5.00000 + 5.00000i) q^{71} +(-3.00000 + 3.00000i) q^{73} +3.00000 q^{75} +12.0000i q^{77} -12.0000i q^{79} +1.00000 q^{81} +(5.00000 - 5.00000i) q^{83} +(4.00000 - 4.00000i) q^{85} +6.00000i q^{87} +(-10.0000 + 10.0000i) q^{89} +(10.0000 + 2.00000i) q^{91} +(-11.0000 - 11.0000i) q^{97} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 4 q^{7} - 2 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - 4 q^{21} - 12 q^{23} - 12 q^{29} - 6 q^{33} + 6 q^{37} + 4 q^{39} - 16 q^{41} - 8 q^{43} - 4 q^{45} + 10 q^{47} - 4 q^{51} + 12 q^{53}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q + 4 * q^5 - 4 * q^7 - 2 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - 4 * q^21 - 12 * q^23 - 12 * q^29 - 6 * q^33 + 6 * q^37 + 4 * q^39 - 16 * q^41 - 8 * q^43 - 4 * q^45 + 10 * q^47 - 4 * q^51 + 12 * q^53 + 2 * q^59 + 16 * q^61 + 4 * q^63 - 20 * q^65 + 4 * q^67 - 10 * q^71 - 6 * q^73 + 6 * q^75 + 2 * q^81 + 10 * q^83 + 8 * q^85 - 20 * q^89 + 20 * q^91 - 22 * q^97 + 6 * q^99

Introduction

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