LMFDB - Embedded newform 448.2.m.b.113.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,2,Mod(113,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.113"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.m (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$3.57729801055$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			113.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	448.113
Dual form			448.2.m.b.337.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 2.00000i) q^{3} +(-2.00000 - 2.00000i) q^{5} -1.00000i q^{7} -5.00000i q^{9} +(3.00000 + 3.00000i) q^{11} -8.00000 q^{15} -6.00000 q^{17} +(4.00000 - 4.00000i) q^{19} +(-2.00000 - 2.00000i) q^{21} -2.00000i q^{23} +3.00000i q^{25} +(-4.00000 - 4.00000i) q^{27} +(-1.00000 + 1.00000i) q^{29} -4.00000 q^{31} +12.0000 q^{33} +(-2.00000 + 2.00000i) q^{35} +(3.00000 + 3.00000i) q^{37} +2.00000i q^{41} +(5.00000 + 5.00000i) q^{43} +(-10.0000 + 10.0000i) q^{45} +8.00000 q^{47} -1.00000 q^{49} +(-12.0000 + 12.0000i) q^{51} +(7.00000 + 7.00000i) q^{53} -12.0000i q^{55} -16.0000i q^{57} +(-2.00000 - 2.00000i) q^{59} +(6.00000 - 6.00000i) q^{61} -5.00000 q^{63} +(5.00000 - 5.00000i) q^{67} +(-4.00000 - 4.00000i) q^{69} +8.00000i q^{71} -10.0000i q^{73} +(6.00000 + 6.00000i) q^{75} +(3.00000 - 3.00000i) q^{77} +14.0000 q^{79} -1.00000 q^{81} +(12.0000 + 12.0000i) q^{85} +4.00000i q^{87} +6.00000i q^{89} +(-8.00000 + 8.00000i) q^{93} -16.0000 q^{95} -6.00000 q^{97} +(15.0000 - 15.0000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 4 q^{5} + 6 q^{11} - 16 q^{15} - 12 q^{17} + 8 q^{19} - 4 q^{21} - 8 q^{27} - 2 q^{29} - 8 q^{31} + 24 q^{33} - 4 q^{35} + 6 q^{37} + 10 q^{43} - 20 q^{45} + 16 q^{47} - 2 q^{49} - 24 q^{51}+ \cdots + 30 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 - 4 * q^5 + 6 * q^11 - 16 * q^15 - 12 * q^17 + 8 * q^19 - 4 * q^21 - 8 * q^27 - 2 * q^29 - 8 * q^31 + 24 * q^33 - 4 * q^35 + 6 * q^37 + 10 * q^43 - 20 * q^45 + 16 * q^47 - 2 * q^49 - 24 * q^51 + 14 * q^53 - 4 * q^59 + 12 * q^61 - 10 * q^63 + 10 * q^67 - 8 * q^69 + 12 * q^75 + 6 * q^77 + 28 * q^79 - 2 * q^81 + 24 * q^85 - 16 * q^93 - 32 * q^95 - 12 * q^97 + 30 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	2.00000	−	2.00000i	1.15470	−	1.15470i	0.169102	−	0.985599i	$-0.445913\pi$
$3$	2.00000	−	2.00000i	1.15470	−	1.15470i	0.985599	−	0.169102i	$-0.0540867\pi$
$4$		0			0
$5$	−2.00000	−	2.00000i	−0.894427	−	0.894427i	0.100509	−	0.994936i	$-0.467953\pi$
$5$	−2.00000	−	2.00000i	−0.894427	−	0.894427i	−0.994936	+	0.100509i	$0.967953\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$		−	5.00000i		−	1.66667i
$10$		0			0
$11$	3.00000	+	3.00000i	0.904534	+	0.904534i	0.995824	−	0.0912903i	$-0.0290991\pi$
$11$	3.00000	+	3.00000i	0.904534	+	0.904534i	−0.0912903	+	0.995824i	$0.529099\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$	−8.00000			−2.06559
$16$		0			0
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$	4.00000	−	4.00000i	0.917663	−	0.917663i	−0.0791961	−	0.996859i	$-0.525235\pi$
$19$	4.00000	−	4.00000i	0.917663	−	0.917663i	0.996859	+	0.0791961i	$0.0252353\pi$
$20$		0			0
$21$	−2.00000	−	2.00000i	−0.436436	−	0.436436i
$22$		0			0
$23$		−	2.00000i		−	0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$		−	2.00000i		−	0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$		0			0
$25$			3.00000i			0.600000i
$26$		0			0
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		0			0
$29$	−1.00000	+	1.00000i	−0.185695	+	0.185695i	−0.793832	−	0.608137i	$-0.791917\pi$
$29$	−1.00000	+	1.00000i	−0.185695	+	0.185695i	0.608137	+	0.793832i	$0.291917\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$	12.0000			2.08893
$34$		0			0
$35$	−2.00000	+	2.00000i	−0.338062	+	0.338062i
$36$		0			0
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	0.909312	−	0.416115i	$-0.136609\pi$
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	−0.416115	+	0.909312i	$0.636609\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.00000i			0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$			2.00000i			0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$		0			0
$43$	5.00000	+	5.00000i	0.762493	+	0.762493i	0.976772	−	0.214280i	$-0.0687403\pi$
$43$	5.00000	+	5.00000i	0.762493	+	0.762493i	−0.214280	+	0.976772i	$0.568740\pi$
$44$		0			0
$45$	−10.0000	+	10.0000i	−1.49071	+	1.49071i
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−12.0000	+	12.0000i	−1.68034	+	1.68034i
$52$		0			0
$53$	7.00000	+	7.00000i	0.961524	+	0.961524i	0.999287	−	0.0377628i	$-0.0120231\pi$
$53$	7.00000	+	7.00000i	0.961524	+	0.961524i	−0.0377628	+	0.999287i	$0.512023\pi$
$54$		0			0
$55$		−	12.0000i		−	1.61808i
$56$		0			0
$57$		−	16.0000i		−	2.11925i
$58$		0			0
$59$	−2.00000	−	2.00000i	−0.260378	−	0.260378i	0.564830	−	0.825208i	$-0.308942\pi$
$59$	−2.00000	−	2.00000i	−0.260378	−	0.260378i	−0.825208	+	0.564830i	$0.808942\pi$
$60$		0			0
$61$	6.00000	−	6.00000i	0.768221	−	0.768221i	−0.209572	−	0.977793i	$-0.567207\pi$
$61$	6.00000	−	6.00000i	0.768221	−	0.768221i	0.977793	+	0.209572i	$0.0672070\pi$
$62$		0			0
$63$	−5.00000			−0.629941
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.00000	−	5.00000i	0.610847	−	0.610847i	−0.332320	−	0.943167i	$-0.607831\pi$
$67$	5.00000	−	5.00000i	0.610847	−	0.610847i	0.943167	+	0.332320i	$0.107831\pi$
$68$		0			0
$69$	−4.00000	−	4.00000i	−0.481543	−	0.481543i
$70$		0			0
$71$			8.00000i			0.949425i	0.880141	+	0.474713i	$0.157448\pi$
$71$			8.00000i			0.949425i	−0.880141	+	0.474713i	$0.842552\pi$
$72$		0			0
$73$		−	10.0000i		−	1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$		−	10.0000i		−	1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$		0			0
$75$	6.00000	+	6.00000i	0.692820	+	0.692820i
$76$		0			0
$77$	3.00000	−	3.00000i	0.341882	−	0.341882i
$78$		0			0
$79$	14.0000			1.57512			0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000			1.57512			0.787562	+	0.616236i	$0.211343\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	12.0000	+	12.0000i	1.30158	+	1.30158i
$86$		0			0
$87$			4.00000i			0.428845i
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−8.00000	+	8.00000i	−0.829561	+	0.829561i
$94$		0			0
$95$	−16.0000			−1.64157
$96$		0			0
$97$	−6.00000			−0.609208			−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000			−0.609208			−0.304604	+	0.952479i	$0.598524\pi$
$98$		0			0
$99$	15.0000	−	15.0000i	1.50756	−	1.50756i
$100$		0			0
$101$	2.00000	+	2.00000i	0.199007	+	0.199007i	0.799574	−	0.600567i	$-0.205058\pi$
$101$	2.00000	+	2.00000i	0.199007	+	0.199007i	−0.600567	+	0.799574i	$0.705058\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$			8.00000i			0.780720i
$106$		0			0
$107$	−5.00000	−	5.00000i	−0.483368	−	0.483368i	0.422837	−	0.906206i	$-0.361034\pi$
$107$	−5.00000	−	5.00000i	−0.483368	−	0.483368i	−0.906206	+	0.422837i	$0.861034\pi$
$108$		0			0
$109$	−3.00000	+	3.00000i	−0.287348	+	0.287348i	−0.836031	−	0.548683i	$-0.815129\pi$
$109$	−3.00000	+	3.00000i	−0.287348	+	0.287348i	0.548683	+	0.836031i	$0.315129\pi$
$110$		0			0
$111$	12.0000			1.13899
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$	−4.00000	+	4.00000i	−0.373002	+	0.373002i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			6.00000i			0.550019i
$120$		0			0
$121$			7.00000i			0.636364i
$122$		0			0
$123$	4.00000	+	4.00000i	0.360668	+	0.360668i
$124$		0			0
$125$	−4.00000	+	4.00000i	−0.357771	+	0.357771i
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$	20.0000			1.76090
$130$		0			0
$131$	−12.0000	+	12.0000i	−1.04844	+	1.04844i	−0.0496797	+	0.998765i	$0.515820\pi$
$131$	−12.0000	+	12.0000i	−1.04844	+	1.04844i	−0.998765	+	0.0496797i	$0.984180\pi$
$132$		0			0
$133$	−4.00000	−	4.00000i	−0.346844	−	0.346844i
$134$		0			0
$135$			16.0000i			1.37706i
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	12.0000	+	12.0000i	1.01783	+	1.01783i	0.999838	+	0.0179885i	$0.00572623\pi$
$139$	12.0000	+	12.0000i	1.01783	+	1.01783i	0.0179885	+	0.999838i	$0.494274\pi$
$140$		0			0
$141$	16.0000	−	16.0000i	1.34744	−	1.34744i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	4.00000			0.332182
$146$		0			0
$147$	−2.00000	+	2.00000i	−0.164957	+	0.164957i
$148$		0			0
$149$	−13.0000	−	13.0000i	−1.06500	−	1.06500i	−0.997735	−	0.0672664i	$-0.978572\pi$
$149$	−13.0000	−	13.0000i	−1.06500	−	1.06500i	−0.0672664	−	0.997735i	$-0.521428\pi$
$150$		0			0
$151$			6.00000i			0.488273i	0.969741	+	0.244137i	$0.0785045\pi$
$151$			6.00000i			0.488273i	−0.969741	+	0.244137i	$0.921495\pi$
$152$		0			0
$153$			30.0000i			2.42536i
$154$		0			0
$155$	8.00000	+	8.00000i	0.642575	+	0.642575i
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$	28.0000			2.22054
$160$		0			0
$161$	−2.00000			−0.157622
$162$		0			0
$163$	5.00000	−	5.00000i	0.391630	−	0.391630i	−0.483638	−	0.875268i	$-0.660685\pi$
$163$	5.00000	−	5.00000i	0.391630	−	0.391630i	0.875268	+	0.483638i	$0.160685\pi$
$164$		0			0
$165$	−24.0000	−	24.0000i	−1.86840	−	1.86840i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$	−20.0000	−	20.0000i	−1.52944	−	1.52944i
$172$		0			0
$173$	16.0000	−	16.0000i	1.21646	−	1.21646i	0.247593	−	0.968864i	$-0.420360\pi$
$173$	16.0000	−	16.0000i	1.21646	−	1.21646i	0.968864	−	0.247593i	$-0.0796397\pi$
$174$		0			0
$175$	3.00000			0.226779
$176$		0			0
$177$	−8.00000			−0.601317
$178$		0			0
$179$	−5.00000	+	5.00000i	−0.373718	+	0.373718i	−0.868829	−	0.495112i	$-0.835127\pi$
$179$	−5.00000	+	5.00000i	−0.373718	+	0.373718i	0.495112	+	0.868829i	$0.335127\pi$
$180$		0			0
$181$	4.00000	+	4.00000i	0.297318	+	0.297318i	0.839962	−	0.542645i	$-0.182577\pi$
$181$	4.00000	+	4.00000i	0.297318	+	0.297318i	−0.542645	+	0.839962i	$0.682577\pi$
$182$		0			0
$183$		−	24.0000i		−	1.77413i
$184$		0			0
$185$		−	12.0000i		−	0.882258i
$186$		0			0
$187$	−18.0000	−	18.0000i	−1.31629	−	1.31629i
$188$		0			0
$189$	−4.00000	+	4.00000i	−0.290957	+	0.290957i
$190$		0			0
$191$	−22.0000			−1.59186			−0.795932	−	0.605386i	$-0.793019\pi$
$191$	−22.0000			−1.59186			−0.795932	+	0.605386i	$0.793019\pi$
$192$		0			0
$193$	−24.0000			−1.72756			−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000			−1.72756			−0.863779	+	0.503871i	$0.831909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.0000	−	11.0000i	−0.783718	−	0.783718i	0.196738	−	0.980456i	$-0.436965\pi$
$197$	−11.0000	−	11.0000i	−0.783718	−	0.783718i	−0.980456	+	0.196738i	$0.936965\pi$
$198$		0			0
$199$			20.0000i			1.41776i	0.705328	+	0.708881i	$0.250800\pi$
$199$			20.0000i			1.41776i	−0.705328	+	0.708881i	$0.749200\pi$
$200$		0			0
$201$		−	20.0000i		−	1.41069i
$202$		0			0
$203$	1.00000	+	1.00000i	0.0701862	+	0.0701862i
$204$		0			0
$205$	4.00000	−	4.00000i	0.279372	−	0.279372i
$206$		0			0
$207$	−10.0000			−0.695048
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	9.00000	−	9.00000i	0.619586	−	0.619586i	−0.325840	−	0.945425i	$-0.605647\pi$
$211$	9.00000	−	9.00000i	0.619586	−	0.619586i	0.945425	+	0.325840i	$0.105647\pi$
$212$		0			0
$213$	16.0000	+	16.0000i	1.09630	+	1.09630i
$214$		0			0
$215$		−	20.0000i		−	1.36399i
$216$		0			0
$217$			4.00000i			0.271538i
$218$		0			0
$219$	−20.0000	−	20.0000i	−1.35147	−	1.35147i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.00000			−0.267860			−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000			−0.267860			−0.133930	+	0.990991i	$0.542760\pi$
$224$		0			0
$225$	15.0000			1.00000
$226$		0			0
$227$	−4.00000	+	4.00000i	−0.265489	+	0.265489i	−0.827280	−	0.561790i	$-0.810113\pi$
$227$	−4.00000	+	4.00000i	−0.265489	+	0.265489i	0.561790	+	0.827280i	$0.310113\pi$
$228$		0			0
$229$	12.0000	+	12.0000i	0.792982	+	0.792982i	0.981978	−	0.188996i	$-0.0605232\pi$
$229$	12.0000	+	12.0000i	0.792982	+	0.792982i	−0.188996	+	0.981978i	$0.560523\pi$
$230$		0			0
$231$		−	12.0000i		−	0.789542i
$232$		0			0
$233$		−	8.00000i		−	0.524097i	−0.965055	−	0.262049i	$-0.915602\pi$
$233$		−	8.00000i		−	0.524097i	0.965055	−	0.262049i	$-0.0843981\pi$
$234$		0			0
$235$	−16.0000	−	16.0000i	−1.04372	−	1.04372i
$236$		0			0
$237$	28.0000	−	28.0000i	1.81880	−	1.81880i
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	−22.0000			−1.41714			−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000			−1.41714			−0.708572	+	0.705638i	$0.750660\pi$
$242$		0			0
$243$	10.0000	−	10.0000i	0.641500	−	0.641500i
$244$		0			0
$245$	2.00000	+	2.00000i	0.127775	+	0.127775i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.00000	−	4.00000i	−0.252478	−	0.252478i	0.569508	−	0.821986i	$-0.307134\pi$
$251$	−4.00000	−	4.00000i	−0.252478	−	0.252478i	−0.821986	+	0.569508i	$0.807134\pi$
$252$		0			0
$253$	6.00000	−	6.00000i	0.377217	−	0.377217i
$254$		0			0
$255$	48.0000			3.00588
$256$		0			0
$257$	−10.0000			−0.623783			−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000			−0.623783			−0.311891	+	0.950118i	$0.600963\pi$
$258$		0			0
$259$	3.00000	−	3.00000i	0.186411	−	0.186411i
$260$		0			0
$261$	5.00000	+	5.00000i	0.309492	+	0.309492i
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$		−	28.0000i		−	1.72003i
$266$		0			0
$267$	12.0000	+	12.0000i	0.734388	+	0.734388i
$268$		0			0
$269$	−16.0000	+	16.0000i	−0.975537	+	0.975537i	−0.999708	−	0.0241706i	$-0.992306\pi$
$269$	−16.0000	+	16.0000i	−0.975537	+	0.975537i	0.0241706	+	0.999708i	$0.492306\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.00000	+	9.00000i	−0.542720	+	0.542720i
$276$		0			0
$277$	1.00000	+	1.00000i	0.0600842	+	0.0600842i	0.736510	−	0.676426i	$-0.236472\pi$
$277$	1.00000	+	1.00000i	0.0600842	+	0.0600842i	−0.676426	+	0.736510i	$0.736472\pi$
$278$		0			0
$279$			20.0000i			1.19737i
$280$		0			0
$281$			24.0000i			1.43172i	0.698244	+	0.715860i	$0.253965\pi$
$281$			24.0000i			1.43172i	−0.698244	+	0.715860i	$0.746035\pi$
$282$		0			0
$283$	6.00000	+	6.00000i	0.356663	+	0.356663i	0.862581	−	0.505918i	$-0.168846\pi$
$283$	6.00000	+	6.00000i	0.356663	+	0.356663i	−0.505918	+	0.862581i	$0.668846\pi$
$284$		0			0
$285$	−32.0000	+	32.0000i	−1.89552	+	1.89552i
$286$		0			0
$287$	2.00000			0.118056
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$	−12.0000	+	12.0000i	−0.703452	+	0.703452i
$292$		0			0
$293$	−10.0000	−	10.0000i	−0.584206	−	0.584206i	0.351850	−	0.936056i	$-0.385553\pi$
$293$	−10.0000	−	10.0000i	−0.584206	−	0.584206i	−0.936056	+	0.351850i	$0.885553\pi$
$294$		0			0
$295$			8.00000i			0.465778i
$296$		0			0
$297$		−	24.0000i		−	1.39262i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	5.00000	−	5.00000i	0.288195	−	0.288195i
$302$		0			0
$303$	8.00000			0.459588
$304$		0			0
$305$	−24.0000			−1.37424
$306$		0			0
$307$	8.00000	−	8.00000i	0.456584	−	0.456584i	−0.440948	−	0.897532i	$-0.645358\pi$
$307$	8.00000	−	8.00000i	0.456584	−	0.456584i	0.897532	+	0.440948i	$0.145358\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			4.00000i			0.226819i	0.993548	+	0.113410i	$0.0361772\pi$
$311$			4.00000i			0.226819i	−0.993548	+	0.113410i	$0.963823\pi$
$312$		0			0
$313$		−	2.00000i		−	0.113047i	−0.998401	−	0.0565233i	$-0.981998\pi$
$313$		−	2.00000i		−	0.113047i	0.998401	−	0.0565233i	$-0.0180015\pi$
$314$		0			0
$315$	10.0000	+	10.0000i	0.563436	+	0.563436i
$316$		0			0
$317$	−15.0000	+	15.0000i	−0.842484	+	0.842484i	−0.989181	−	0.146697i	$-0.953136\pi$
$317$	−15.0000	+	15.0000i	−0.842484	+	0.842484i	0.146697	+	0.989181i	$0.453136\pi$
$318$		0			0
$319$	−6.00000			−0.335936
$320$		0			0
$321$	−20.0000			−1.11629
$322$		0			0
$323$	−24.0000	+	24.0000i	−1.33540	+	1.33540i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			12.0000i			0.663602i
$328$		0			0
$329$		−	8.00000i		−	0.441054i
$330$		0			0
$331$	15.0000	+	15.0000i	0.824475	+	0.824475i	0.986746	−	0.162272i	$-0.0518821\pi$
$331$	15.0000	+	15.0000i	0.824475	+	0.824475i	−0.162272	+	0.986746i	$0.551882\pi$
$332$		0			0
$333$	15.0000	−	15.0000i	0.821995	−	0.821995i
$334$		0			0
$335$	−20.0000			−1.09272
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$		0			0
$339$	−24.0000	+	24.0000i	−1.30350	+	1.30350i
$340$		0			0
$341$	−12.0000	−	12.0000i	−0.649836	−	0.649836i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			16.0000i			0.861411i
$346$		0			0
$347$	13.0000	+	13.0000i	0.697877	+	0.697877i	0.963952	−	0.266076i	$-0.0857271\pi$
$347$	13.0000	+	13.0000i	0.697877	+	0.697877i	−0.266076	+	0.963952i	$0.585727\pi$
$348$		0			0
$349$	16.0000	−	16.0000i	0.856460	−	0.856460i	−0.134459	−	0.990919i	$-0.542930\pi$
$349$	16.0000	−	16.0000i	0.856460	−	0.856460i	0.990919	+	0.134459i	$0.0429296\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$		0			0
$355$	16.0000	−	16.0000i	0.849192	−	0.849192i
$356$		0			0
$357$	12.0000	+	12.0000i	0.635107	+	0.635107i
$358$		0			0
$359$		−	22.0000i		−	1.16112i	−0.814219	−	0.580558i	$-0.802835\pi$
$359$		−	22.0000i		−	1.16112i	0.814219	−	0.580558i	$-0.197165\pi$
$360$		0			0
$361$		−	13.0000i		−	0.684211i
$362$		0			0
$363$	14.0000	+	14.0000i	0.734809	+	0.734809i
$364$		0			0
$365$	−20.0000	+	20.0000i	−1.04685	+	1.04685i
$366$		0			0
$367$	16.0000			0.835193			0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000			0.835193			0.417597	+	0.908633i	$0.362873\pi$
$368$		0			0
$369$	10.0000			0.520579
$370$		0			0
$371$	7.00000	−	7.00000i	0.363422	−	0.363422i
$372$		0			0
$373$	5.00000	+	5.00000i	0.258890	+	0.258890i	0.824603	−	0.565712i	$-0.191399\pi$
$373$	5.00000	+	5.00000i	0.258890	+	0.258890i	−0.565712	+	0.824603i	$0.691399\pi$
$374$		0			0
$375$			16.0000i			0.826236i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−17.0000	−	17.0000i	−0.873231	−	0.873231i	0.119592	−	0.992823i	$-0.461841\pi$
$379$	−17.0000	−	17.0000i	−0.873231	−	0.873231i	−0.992823	+	0.119592i	$0.961841\pi$
$380$		0			0
$381$	−16.0000	+	16.0000i	−0.819705	+	0.819705i
$382$		0			0
$383$	16.0000			0.817562			0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000			0.817562			0.408781	+	0.912633i	$0.365954\pi$
$384$		0			0
$385$	−12.0000			−0.611577
$386$		0			0
$387$	25.0000	−	25.0000i	1.27082	−	1.27082i
$388$		0			0
$389$	19.0000	+	19.0000i	0.963338	+	0.963338i	0.999351	−	0.0360131i	$-0.0114658\pi$
$389$	19.0000	+	19.0000i	0.963338	+	0.963338i	−0.0360131	+	0.999351i	$0.511466\pi$
$390$		0			0
$391$			12.0000i			0.606866i
$392$		0			0
$393$			48.0000i			2.42128i
$394$		0			0
$395$	−28.0000	−	28.0000i	−1.40883	−	1.40883i
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$	−16.0000			−0.801002
$400$		0			0
$401$	−4.00000			−0.199750			−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000			−0.199750			−0.0998752	+	0.995000i	$0.531844\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	2.00000	+	2.00000i	0.0993808	+	0.0993808i
$406$		0			0
$407$			18.0000i			0.892227i
$408$		0			0
$409$			10.0000i			0.494468i	0.968956	+	0.247234i	$0.0795217\pi$
$409$			10.0000i			0.494468i	−0.968956	+	0.247234i	$0.920478\pi$
$410$		0			0
$411$	24.0000	+	24.0000i	1.18383	+	1.18383i
$412$		0			0
$413$	−2.00000	+	2.00000i	−0.0984136	+	0.0984136i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	48.0000			2.35057
$418$		0			0
$419$	10.0000	−	10.0000i	0.488532	−	0.488532i	−0.419311	−	0.907843i	$-0.637728\pi$
$419$	10.0000	−	10.0000i	0.488532	−	0.488532i	0.907843	+	0.419311i	$0.137728\pi$
$420$		0			0
$421$	−25.0000	−	25.0000i	−1.21843	−	1.21843i	−0.968183	−	0.250242i	$-0.919490\pi$
$421$	−25.0000	−	25.0000i	−1.21843	−	1.21843i	−0.250242	−	0.968183i	$-0.580510\pi$
$422$		0			0
$423$		−	40.0000i		−	1.94487i
$424$		0			0
$425$		−	18.0000i		−	0.873128i
$426$		0			0
$427$	−6.00000	−	6.00000i	−0.290360	−	0.290360i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$	8.00000	−	8.00000i	0.383571	−	0.383571i
$436$		0			0
$437$	−8.00000	−	8.00000i	−0.382692	−	0.382692i
$438$		0			0
$439$			24.0000i			1.14546i	0.819745	+	0.572729i	$0.194115\pi$
$439$			24.0000i			1.14546i	−0.819745	+	0.572729i	$0.805885\pi$
$440$		0			0
$441$			5.00000i			0.238095i
$442$		0			0
$443$	17.0000	+	17.0000i	0.807694	+	0.807694i	0.984284	−	0.176590i	$-0.0565067\pi$
$443$	17.0000	+	17.0000i	0.807694	+	0.807694i	−0.176590	+	0.984284i	$0.556507\pi$
$444$		0			0
$445$	12.0000	−	12.0000i	0.568855	−	0.568855i
$446$		0			0
$447$	−52.0000			−2.45952
$448$		0			0
$449$	34.0000			1.60456			0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000			1.60456			0.802280	+	0.596948i	$0.203620\pi$
$450$		0			0
$451$	−6.00000	+	6.00000i	−0.282529	+	0.282529i
$452$		0			0
$453$	12.0000	+	12.0000i	0.563809	+	0.563809i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	26.0000i		−	1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$		−	26.0000i		−	1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$		0			0
$459$	24.0000	+	24.0000i	1.12022	+	1.12022i
$460$		0			0
$461$	−28.0000	+	28.0000i	−1.30409	+	1.30409i	−0.378481	+	0.925609i	$0.623553\pi$
$461$	−28.0000	+	28.0000i	−1.30409	+	1.30409i	−0.925609	+	0.378481i	$0.876447\pi$
$462$		0			0
$463$	−14.0000			−0.650635			−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000			−0.650635			−0.325318	+	0.945605i	$0.605471\pi$
$464$		0			0
$465$	32.0000			1.48396
$466$		0			0
$467$	16.0000	−	16.0000i	0.740392	−	0.740392i	−0.232262	−	0.972653i	$-0.574613\pi$
$467$	16.0000	−	16.0000i	0.740392	−	0.740392i	0.972653	+	0.232262i	$0.0746126\pi$
$468$		0			0
$469$	−5.00000	−	5.00000i	−0.230879	−	0.230879i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			30.0000i			1.37940i
$474$		0			0
$475$	12.0000	+	12.0000i	0.550598	+	0.550598i
$476$		0			0
$477$	35.0000	−	35.0000i	1.60254	−	1.60254i
$478$		0			0
$479$	8.00000			0.365529			0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000			0.365529			0.182765	+	0.983157i	$0.441495\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−4.00000	+	4.00000i	−0.182006	+	0.182006i
$484$		0			0
$485$	12.0000	+	12.0000i	0.544892	+	0.544892i
$486$		0			0
$487$			2.00000i			0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$			2.00000i			0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$		0			0
$489$		−	20.0000i		−	0.904431i
$490$		0			0
$491$	11.0000	+	11.0000i	0.496423	+	0.496423i	0.910323	−	0.413900i	$-0.135834\pi$
$491$	11.0000	+	11.0000i	0.496423	+	0.496423i	−0.413900	+	0.910323i	$0.635834\pi$
$492$		0			0
$493$	6.00000	−	6.00000i	0.270226	−	0.270226i
$494$		0			0
$495$	−60.0000			−2.69680
$496$		0			0
$497$	8.00000			0.358849
$498$		0			0
$499$	15.0000	−	15.0000i	0.671492	−	0.671492i	−0.286568	−	0.958060i	$-0.592514\pi$
$499$	15.0000	−	15.0000i	0.671492	−	0.671492i	0.958060	+	0.286568i	$0.0925144\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8.00000i		−	0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$		−	8.00000i		−	0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$		0			0
$505$		−	8.00000i		−	0.355995i
$506$		0			0
$507$	26.0000	+	26.0000i	1.15470	+	1.15470i
$508$		0			0
$509$	−12.0000	+	12.0000i	−0.531891	+	0.531891i	−0.921135	−	0.389244i	$-0.872736\pi$
$509$	−12.0000	+	12.0000i	−0.531891	+	0.531891i	0.389244	+	0.921135i	$0.372736\pi$
$510$		0			0
$511$	−10.0000			−0.442374
$512$		0			0
$513$	−32.0000			−1.41283
$514$		0			0
$515$		0			0
$516$		0			0
$517$	24.0000	+	24.0000i	1.05552	+	1.05552i
$518$		0			0
$519$		−	64.0000i		−	2.80929i
$520$		0			0
$521$			6.00000i			0.262865i	0.991325	+	0.131432i	$0.0419576\pi$
$521$			6.00000i			0.262865i	−0.991325	+	0.131432i	$0.958042\pi$
$522$		0			0
$523$	−26.0000	−	26.0000i	−1.13690	−	1.13690i	−0.989003	−	0.147898i	$-0.952749\pi$
$523$	−26.0000	−	26.0000i	−1.13690	−	1.13690i	−0.147898	−	0.989003i	$-0.547251\pi$
$524$		0			0
$525$	6.00000	−	6.00000i	0.261861	−	0.261861i
$526$		0			0
$527$	24.0000			1.04546
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$	−10.0000	+	10.0000i	−0.433963	+	0.433963i
$532$		0			0
$533$		0			0
$534$		0			0
$535$			20.0000i			0.864675i
$536$		0			0
$537$			20.0000i			0.863064i
$538$		0			0
$539$	−3.00000	−	3.00000i	−0.129219	−	0.129219i
$540$		0			0
$541$	−29.0000	+	29.0000i	−1.24681	+	1.24681i	−0.289685	+	0.957122i	$0.593551\pi$
$541$	−29.0000	+	29.0000i	−1.24681	+	1.24681i	−0.957122	+	0.289685i	$0.906449\pi$
$542$		0			0
$543$	16.0000			0.686626
$544$		0			0
$545$	12.0000			0.514024
$546$		0			0
$547$	−19.0000	+	19.0000i	−0.812381	+	0.812381i	−0.984990	−	0.172609i	$-0.944780\pi$
$547$	−19.0000	+	19.0000i	−0.812381	+	0.812381i	0.172609	+	0.984990i	$0.444780\pi$
$548$		0			0
$549$	−30.0000	−	30.0000i	−1.28037	−	1.28037i
$550$		0			0
$551$			8.00000i			0.340811i
$552$		0			0
$553$		−	14.0000i		−	0.595341i
$554$		0			0
$555$	−24.0000	−	24.0000i	−1.01874	−	1.01874i
$556$		0			0
$557$	15.0000	−	15.0000i	0.635570	−	0.635570i	−0.313889	−	0.949460i	$-0.601632\pi$
$557$	15.0000	−	15.0000i	0.635570	−	0.635570i	0.949460	+	0.313889i	$0.101632\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−72.0000			−3.03984
$562$		0			0
$563$	22.0000	−	22.0000i	0.927189	−	0.927189i	−0.0703340	−	0.997523i	$-0.522407\pi$
$563$	22.0000	−	22.0000i	0.927189	−	0.927189i	0.997523	+	0.0703340i	$0.0224065\pi$
$564$		0			0
$565$	24.0000	+	24.0000i	1.00969	+	1.00969i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$		−	46.0000i		−	1.92842i	−0.265139	−	0.964210i	$-0.585418\pi$
$569$		−	46.0000i		−	1.92842i	0.265139	−	0.964210i	$-0.414582\pi$
$570$		0			0
$571$	21.0000	+	21.0000i	0.878823	+	0.878823i	0.993413	−	0.114590i	$-0.0365555\pi$
$571$	21.0000	+	21.0000i	0.878823	+	0.878823i	−0.114590	+	0.993413i	$0.536555\pi$
$572$		0			0
$573$	−44.0000	+	44.0000i	−1.83813	+	1.83813i
$574$		0			0
$575$	6.00000			0.250217
$576$		0			0
$577$	−26.0000			−1.08239			−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000			−1.08239			−0.541197	+	0.840896i	$0.682029\pi$
$578$		0			0
$579$	−48.0000	+	48.0000i	−1.99481	+	1.99481i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			42.0000i			1.73946i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.00000	−	8.00000i	−0.330195	−	0.330195i	0.522465	−	0.852661i	$-0.325012\pi$
$587$	−8.00000	−	8.00000i	−0.330195	−	0.330195i	−0.852661	+	0.522465i	$0.825012\pi$
$588$		0			0
$589$	−16.0000	+	16.0000i	−0.659269	+	0.659269i
$590$		0			0
$591$	−44.0000			−1.80992
$592$		0			0
$593$	−22.0000			−0.903432			−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000			−0.903432			−0.451716	+	0.892162i	$0.649188\pi$
$594$		0			0
$595$	12.0000	−	12.0000i	0.491952	−	0.491952i
$596$		0			0
$597$	40.0000	+	40.0000i	1.63709	+	1.63709i
$598$		0			0
$599$			8.00000i			0.326871i	0.986554	+	0.163436i	$0.0522576\pi$
$599$			8.00000i			0.326871i	−0.986554	+	0.163436i	$0.947742\pi$
$600$		0			0
$601$		−	10.0000i		−	0.407909i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$		−	10.0000i		−	0.407909i	0.978980	−	0.203954i	$-0.0653794\pi$
$602$		0			0
$603$	−25.0000	−	25.0000i	−1.01808	−	1.01808i
$604$		0			0
$605$	14.0000	−	14.0000i	0.569181	−	0.569181i
$606$		0			0
$607$	−20.0000			−0.811775			−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000			−0.811775			−0.405887	+	0.913923i	$0.633038\pi$
$608$		0			0
$609$	4.00000			0.162088
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−15.0000	−	15.0000i	−0.605844	−	0.605844i	0.336013	−	0.941857i	$-0.390921\pi$
$613$	−15.0000	−	15.0000i	−0.605844	−	0.605844i	−0.941857	+	0.336013i	$0.890921\pi$
$614$		0			0
$615$		−	16.0000i		−	0.645182i
$616$		0			0
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$		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
$621$	−8.00000	+	8.00000i	−0.321029	+	0.321029i
$622$		0			0
$623$	6.00000			0.240385
$624$		0			0
$625$	31.0000			1.24000
$626$		0			0
$627$	48.0000	−	48.0000i	1.91694	−	1.91694i
$628$		0			0
$629$	−18.0000	−	18.0000i	−0.717707	−	0.717707i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		−	36.0000i		−	1.43087i
$634$		0			0
$635$	16.0000	+	16.0000i	0.634941	+	0.634941i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	40.0000			1.58238
$640$		0			0
$641$	44.0000			1.73790			0.868948	−	0.494904i	$-0.164797\pi$
$641$	44.0000			1.73790			0.868948	+	0.494904i	$0.164797\pi$
$642$		0			0
$643$	−26.0000	+	26.0000i	−1.02534	+	1.02534i	−0.0256694	+	0.999670i	$0.508172\pi$
$643$	−26.0000	+	26.0000i	−1.02534	+	1.02534i	−0.999670	+	0.0256694i	$0.991828\pi$
$644$		0			0
$645$	−40.0000	−	40.0000i	−1.57500	−	1.57500i
$646$		0			0
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$		0			0
$649$		−	12.0000i		−	0.471041i
$650$		0			0
$651$	8.00000	+	8.00000i	0.313545	+	0.313545i
$652$		0			0
$653$	25.0000	−	25.0000i	0.978326	−	0.978326i	−0.0214444	−	0.999770i	$-0.506827\pi$
$653$	25.0000	−	25.0000i	0.978326	−	0.978326i	0.999770	+	0.0214444i	$0.00682650\pi$
$654$		0			0
$655$	48.0000			1.87552
$656$		0			0
$657$	−50.0000			−1.95069
$658$		0			0
$659$	−17.0000	+	17.0000i	−0.662226	+	0.662226i	−0.955904	−	0.293678i	$-0.905121\pi$
$659$	−17.0000	+	17.0000i	−0.662226	+	0.662226i	0.293678	+	0.955904i	$0.405121\pi$
$660$		0			0
$661$	30.0000	+	30.0000i	1.16686	+	1.16686i	0.982940	+	0.183924i	$0.0588801\pi$
$661$	30.0000	+	30.0000i	1.16686	+	1.16686i	0.183924	+	0.982940i	$0.441120\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			16.0000i			0.620453i
$666$		0			0
$667$	2.00000	+	2.00000i	0.0774403	+	0.0774403i
$668$		0			0
$669$	−8.00000	+	8.00000i	−0.309298	+	0.309298i
$670$		0			0
$671$	36.0000			1.38976
$672$		0			0
$673$	26.0000			1.00223			0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000			1.00223			0.501113	+	0.865382i	$0.332924\pi$
$674$		0			0
$675$	12.0000	−	12.0000i	0.461880	−	0.461880i
$676$		0			0
$677$	8.00000	+	8.00000i	0.307465	+	0.307465i	0.843925	−	0.536460i	$-0.180239\pi$
$677$	8.00000	+	8.00000i	0.307465	+	0.307465i	−0.536460	+	0.843925i	$0.680239\pi$
$678$		0			0
$679$			6.00000i			0.230259i
$680$		0			0
$681$			16.0000i			0.613121i
$682$		0			0
$683$	−5.00000	−	5.00000i	−0.191320	−	0.191320i	0.604946	−	0.796266i	$-0.293195\pi$
$683$	−5.00000	−	5.00000i	−0.191320	−	0.191320i	−0.796266	+	0.604946i	$0.793195\pi$
$684$		0			0
$685$	24.0000	−	24.0000i	0.916993	−	0.916993i
$686$		0			0
$687$	48.0000			1.83131
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−12.0000	+	12.0000i	−0.456502	+	0.456502i	−0.897505	−	0.441004i	$-0.854623\pi$
$691$	−12.0000	+	12.0000i	−0.456502	+	0.456502i	0.441004	+	0.897505i	$0.354623\pi$
$692$		0			0
$693$	−15.0000	−	15.0000i	−0.569803	−	0.569803i
$694$		0			0
$695$		−	48.0000i		−	1.82074i
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$		0			0
$699$	−16.0000	−	16.0000i	−0.605176	−	0.605176i
$700$		0			0
$701$	13.0000	−	13.0000i	0.491003	−	0.491003i	−0.417619	−	0.908622i	$-0.637135\pi$
$701$	13.0000	−	13.0000i	0.491003	−	0.491003i	0.908622	+	0.417619i	$0.137135\pi$
$702$		0			0
$703$	24.0000			0.905177
$704$		0			0
$705$	−64.0000			−2.41038
$706$		0			0
$707$	2.00000	−	2.00000i	0.0752177	−	0.0752177i
$708$		0			0
$709$	−31.0000	−	31.0000i	−1.16423	−	1.16423i	−0.983540	−	0.180689i	$-0.942167\pi$
$709$	−31.0000	−	31.0000i	−1.16423	−	1.16423i	−0.180689	−	0.983540i	$-0.557833\pi$
$710$		0			0
$711$		−	70.0000i		−	2.62521i
$712$		0			0
$713$			8.00000i			0.299602i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	16.0000	−	16.0000i	0.597531	−	0.597531i
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−44.0000	+	44.0000i	−1.63638	+	1.63638i
$724$		0			0
$725$	−3.00000	−	3.00000i	−0.111417	−	0.111417i
$726$		0			0
$727$		−	44.0000i		−	1.63187i	−0.578144	−	0.815935i	$-0.696223\pi$
$727$		−	44.0000i		−	1.63187i	0.578144	−	0.815935i	$-0.303777\pi$
$728$		0			0
$729$		−	43.0000i		−	1.59259i
$730$		0			0
$731$	−30.0000	−	30.0000i	−1.10959	−	1.10959i
$732$		0			0
$733$	−2.00000	+	2.00000i	−0.0738717	+	0.0738717i	−0.743077	−	0.669206i	$-0.766635\pi$
$733$	−2.00000	+	2.00000i	−0.0738717	+	0.0738717i	0.669206	+	0.743077i	$0.266635\pi$
$734$		0			0
$735$	8.00000			0.295084
$736$		0			0
$737$	30.0000			1.10506
$738$		0			0
$739$	5.00000	−	5.00000i	0.183928	−	0.183928i	−0.609137	−	0.793065i	$-0.708484\pi$
$739$	5.00000	−	5.00000i	0.183928	−	0.183928i	0.793065	+	0.609137i	$0.208484\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			6.00000i			0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$			6.00000i			0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$		0			0
$745$			52.0000i			1.90513i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.00000	+	5.00000i	−0.182696	+	0.182696i
$750$		0			0
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$		0			0
$753$	−16.0000			−0.583072
$754$		0			0
$755$	12.0000	−	12.0000i	0.436725	−	0.436725i
$756$		0			0
$757$	33.0000	+	33.0000i	1.19941	+	1.19941i	0.974345	+	0.225061i	$0.0722580\pi$
$757$	33.0000	+	33.0000i	1.19941	+	1.19941i	0.225061	+	0.974345i	$0.427742\pi$
$758$		0			0
$759$		−	24.0000i		−	0.871145i
$760$		0			0
$761$			18.0000i			0.652499i	0.945284	+	0.326250i	$0.105785\pi$
$761$			18.0000i			0.652499i	−0.945284	+	0.326250i	$0.894215\pi$
$762$		0			0
$763$	3.00000	+	3.00000i	0.108607	+	0.108607i
$764$		0			0
$765$	60.0000	−	60.0000i	2.16930	−	2.16930i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$	−20.0000	+	20.0000i	−0.720282	+	0.720282i
$772$		0			0
$773$	−30.0000	−	30.0000i	−1.07903	−	1.07903i	−0.996597	−	0.0824280i	$-0.973733\pi$
$773$	−30.0000	−	30.0000i	−1.07903	−	1.07903i	−0.0824280	−	0.996597i	$-0.526267\pi$
$774$		0			0
$775$		−	12.0000i		−	0.431053i
$776$		0			0
$777$		−	12.0000i		−	0.430498i
$778$		0			0
$779$	8.00000	+	8.00000i	0.286630	+	0.286630i
$780$		0			0
$781$	−24.0000	+	24.0000i	−0.858788	+	0.858788i
$782$		0			0
$783$	8.00000			0.285897
$784$		0			0
$785$		0			0
$786$		0			0
$787$	12.0000	−	12.0000i	0.427754	−	0.427754i	−0.460109	−	0.887863i	$-0.652190\pi$
$787$	12.0000	−	12.0000i	0.427754	−	0.427754i	0.887863	+	0.460109i	$0.152190\pi$
$788$		0			0
$789$	−48.0000	−	48.0000i	−1.70885	−	1.70885i
$790$		0			0
$791$			12.0000i			0.426671i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−56.0000	−	56.0000i	−1.98612	−	1.98612i
$796$		0			0
$797$	8.00000	−	8.00000i	0.283375	−	0.283375i	−0.551079	−	0.834453i	$-0.685784\pi$
$797$	8.00000	−	8.00000i	0.283375	−	0.283375i	0.834453	+	0.551079i	$0.185784\pi$
$798$		0			0
$799$	−48.0000			−1.69812
$800$		0			0
$801$	30.0000			1.06000
$802$		0			0
$803$	30.0000	−	30.0000i	1.05868	−	1.05868i
$804$		0			0
$805$	4.00000	+	4.00000i	0.140981	+	0.140981i
$806$		0			0
$807$			64.0000i			2.25291i
$808$		0			0
$809$			38.0000i			1.33601i	0.744157	+	0.668004i	$0.232851\pi$
$809$			38.0000i			1.33601i	−0.744157	+	0.668004i	$0.767149\pi$
$810$		0			0
$811$	−30.0000	−	30.0000i	−1.05344	−	1.05344i	−0.998489	−	0.0549536i	$-0.982499\pi$
$811$	−30.0000	−	30.0000i	−1.05344	−	1.05344i	−0.0549536	−	0.998489i	$-0.517501\pi$
$812$		0			0
$813$	−32.0000	+	32.0000i	−1.12229	+	1.12229i
$814$		0			0
$815$	−20.0000			−0.700569
$816$		0			0
$817$	40.0000			1.39942
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.00000	−	7.00000i	−0.244302	−	0.244302i	0.574325	−	0.818627i	$-0.305264\pi$
$821$	−7.00000	−	7.00000i	−0.244302	−	0.244302i	−0.818627	+	0.574325i	$0.805264\pi$
$822$		0			0
$823$			40.0000i			1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$			40.0000i			1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$		0			0
$825$			36.0000i			1.25336i
$826$		0			0
$827$	13.0000	+	13.0000i	0.452054	+	0.452054i	0.896036	−	0.443982i	$-0.146434\pi$
$827$	13.0000	+	13.0000i	0.452054	+	0.452054i	−0.443982	+	0.896036i	$0.646434\pi$
$828$		0			0
$829$	−2.00000	+	2.00000i	−0.0694629	+	0.0694629i	−0.740985	−	0.671522i	$-0.765641\pi$
$829$	−2.00000	+	2.00000i	−0.0694629	+	0.0694629i	0.671522	+	0.740985i	$0.265641\pi$
$830$		0			0
$831$	4.00000			0.138758
$832$		0			0
$833$	6.00000			0.207888
$834$		0			0
$835$		0			0
$836$		0			0
$837$	16.0000	+	16.0000i	0.553041	+	0.553041i
$838$		0			0
$839$			24.0000i			0.828572i	0.910147	+	0.414286i	$0.135969\pi$
$839$			24.0000i			0.828572i	−0.910147	+	0.414286i	$0.864031\pi$
$840$		0			0
$841$			27.0000i			0.931034i
$842$		0			0
$843$	48.0000	+	48.0000i	1.65321	+	1.65321i
$844$		0			0
$845$	26.0000	−	26.0000i	0.894427	−	0.894427i
$846$		0			0
$847$	7.00000			0.240523
$848$		0			0
$849$	24.0000			0.823678
$850$		0			0
$851$	6.00000	−	6.00000i	0.205677	−	0.205677i
$852$		0			0
$853$	−4.00000	−	4.00000i	−0.136957	−	0.136957i	0.635304	−	0.772262i	$-0.280875\pi$
$853$	−4.00000	−	4.00000i	−0.136957	−	0.136957i	−0.772262	+	0.635304i	$0.780875\pi$
$854$		0			0
$855$			80.0000i			2.73594i
$856$		0			0
$857$		−	18.0000i		−	0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$		−	18.0000i		−	0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$		0			0
$859$	4.00000	+	4.00000i	0.136478	+	0.136478i	0.772046	−	0.635567i	$-0.219234\pi$
$859$	4.00000	+	4.00000i	0.136478	+	0.136478i	−0.635567	+	0.772046i	$0.719234\pi$
$860$		0			0
$861$	4.00000	−	4.00000i	0.136320	−	0.136320i
$862$		0			0
$863$	−6.00000			−0.204242			−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000			−0.204242			−0.102121	+	0.994772i	$0.532563\pi$
$864$		0			0
$865$	−64.0000			−2.17607
$866$		0			0
$867$	38.0000	−	38.0000i	1.29055	−	1.29055i
$868$		0			0
$869$	42.0000	+	42.0000i	1.42475	+	1.42475i
$870$		0			0
$871$		0			0
$872$		0			0
$873$			30.0000i			1.01535i
$874$		0			0
$875$	4.00000	+	4.00000i	0.135225	+	0.135225i
$876$		0			0
$877$	−21.0000	+	21.0000i	−0.709120	+	0.709120i	−0.966350	−	0.257230i	$-0.917190\pi$
$877$	−21.0000	+	21.0000i	−0.709120	+	0.709120i	0.257230	+	0.966350i	$0.417190\pi$
$878$		0			0
$879$	−40.0000			−1.34917
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	17.0000	−	17.0000i	0.572096	−	0.572096i	−0.360618	−	0.932714i	$-0.617434\pi$
$883$	17.0000	−	17.0000i	0.572096	−	0.572096i	0.932714	+	0.360618i	$0.117434\pi$
$884$		0			0
$885$	16.0000	+	16.0000i	0.537834	+	0.537834i
$886$		0			0
$887$			24.0000i			0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$			24.0000i			0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$		0			0
$889$			8.00000i			0.268311i
$890$		0			0
$891$	−3.00000	−	3.00000i	−0.100504	−	0.100504i
$892$		0			0
$893$	32.0000	−	32.0000i	1.07084	−	1.07084i
$894$		0			0
$895$	20.0000			0.668526
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.00000	−	4.00000i	0.133407	−	0.133407i
$900$		0			0
$901$	−42.0000	−	42.0000i	−1.39922	−	1.39922i
$902$		0			0
$903$		−	20.0000i		−	0.665558i
$904$		0			0
$905$		−	16.0000i		−	0.531858i
$906$		0			0
$907$	9.00000	+	9.00000i	0.298840	+	0.298840i	0.840559	−	0.541719i	$-0.182226\pi$
$907$	9.00000	+	9.00000i	0.298840	+	0.298840i	−0.541719	+	0.840559i	$0.682226\pi$
$908$		0			0
$909$	10.0000	−	10.0000i	0.331679	−	0.331679i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−48.0000	+	48.0000i	−1.58683	+	1.58683i
$916$		0			0
$917$	12.0000	+	12.0000i	0.396275	+	0.396275i
$918$		0			0
$919$		−	56.0000i		−	1.84727i	−0.383274	−	0.923635i	$-0.625203\pi$
$919$		−	56.0000i		−	1.84727i	0.383274	−	0.923635i	$-0.374797\pi$
$920$		0			0
$921$		−	32.0000i		−	1.05444i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−9.00000	+	9.00000i	−0.295918	+	0.295918i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$	−4.00000	+	4.00000i	−0.131095	+	0.131095i
$932$		0			0
$933$	8.00000	+	8.00000i	0.261908	+	0.261908i
$934$		0			0
$935$			72.0000i			2.35465i
$936$		0			0
$937$			50.0000i			1.63343i	0.577042	+	0.816714i	$0.304207\pi$
$937$			50.0000i			1.63343i	−0.577042	+	0.816714i	$0.695793\pi$
$938$		0			0
$939$	−4.00000	−	4.00000i	−0.130535	−	0.130535i
$940$		0			0
$941$	26.0000	−	26.0000i	0.847576	−	0.847576i	−0.142254	−	0.989830i	$-0.545435\pi$
$941$	26.0000	−	26.0000i	0.847576	−	0.847576i	0.989830	+	0.142254i	$0.0454351\pi$
$942$		0			0
$943$	4.00000			0.130258
$944$		0			0
$945$	16.0000			0.520480
$946$		0			0
$947$	−13.0000	+	13.0000i	−0.422443	+	0.422443i	−0.886044	−	0.463601i	$-0.846557\pi$
$947$	−13.0000	+	13.0000i	−0.422443	+	0.422443i	0.463601	+	0.886044i	$0.346557\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$			60.0000i			1.94563i
$952$		0			0
$953$		−	16.0000i		−	0.518291i	−0.965838	−	0.259145i	$-0.916559\pi$
$953$		−	16.0000i		−	0.518291i	0.965838	−	0.259145i	$-0.0834409\pi$
$954$		0			0
$955$	44.0000	+	44.0000i	1.42381	+	1.42381i
$956$		0			0
$957$	−12.0000	+	12.0000i	−0.387905	+	0.387905i
$958$		0			0
$959$	12.0000			0.387500
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$	−25.0000	+	25.0000i	−0.805614	+	0.805614i
$964$		0			0
$965$	48.0000	+	48.0000i	1.54517	+	1.54517i
$966$		0			0
$967$			38.0000i			1.22200i	0.791632	+	0.610999i	$0.209232\pi$
$967$			38.0000i			1.22200i	−0.791632	+	0.610999i	$0.790768\pi$
$968$		0			0
$969$			96.0000i			3.08396i
$970$		0			0
$971$	−26.0000	−	26.0000i	−0.834380	−	0.834380i	0.153733	−	0.988112i	$-0.450871\pi$
$971$	−26.0000	−	26.0000i	−0.834380	−	0.834380i	−0.988112	+	0.153733i	$0.950871\pi$
$972$		0			0
$973$	12.0000	−	12.0000i	0.384702	−	0.384702i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−34.0000			−1.08776			−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000			−1.08776			−0.543878	+	0.839164i	$0.683045\pi$
$978$		0			0
$979$	−18.0000	+	18.0000i	−0.575282	+	0.575282i
$980$		0			0
$981$	15.0000	+	15.0000i	0.478913	+	0.478913i
$982$		0			0
$983$		−	48.0000i		−	1.53096i	−0.643458	−	0.765481i	$-0.722501\pi$
$983$		−	48.0000i		−	1.53096i	0.643458	−	0.765481i	$-0.277499\pi$
$984$		0			0
$985$			44.0000i			1.40196i
$986$		0			0
$987$	−16.0000	−	16.0000i	−0.509286	−	0.509286i
$988$		0			0
$989$	10.0000	−	10.0000i	0.317982	−	0.317982i
$990$		0			0
$991$	10.0000			0.317660			0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000			0.317660			0.158830	+	0.987306i	$0.449228\pi$
$992$		0			0
$993$	60.0000			1.90404
$994$		0			0
$995$	40.0000	−	40.0000i	1.26809	−	1.26809i
$996$		0			0
$997$	30.0000	+	30.0000i	0.950110	+	0.950110i	0.998813	−	0.0487037i	$-0.0155090\pi$
$997$	30.0000	+	30.0000i	0.950110	+	0.950110i	−0.0487037	+	0.998813i	$0.515509\pi$
$998$		0			0
$999$		−	24.0000i		−	0.759326i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.m.b.113.1		2
4.3	odd	2		112.2.m.a.85.1	yes	2
8.3	odd	2		896.2.m.d.225.1		2
8.5	even	2		896.2.m.a.225.1		2
16.3	odd	4		112.2.m.a.29.1	✓	2
16.5	even	4		896.2.m.a.673.1		2
16.11	odd	4		896.2.m.d.673.1		2
16.13	even	4	inner	448.2.m.b.337.1		2
28.3	even	6		784.2.x.b.373.1		4
28.11	odd	6		784.2.x.g.373.1		4
28.19	even	6		784.2.x.b.165.1		4
28.23	odd	6		784.2.x.g.165.1		4
28.27	even	2		784.2.m.c.197.1		2
32.3	odd	8		7168.2.a.q.1.1		2
32.13	even	8		7168.2.a.i.1.1		2
32.19	odd	8		7168.2.a.q.1.2		2
32.29	even	8		7168.2.a.i.1.2		2
112.3	even	12		784.2.x.b.765.1		4
112.19	even	12		784.2.x.b.557.1		4
112.51	odd	12		784.2.x.g.557.1		4
112.67	odd	12		784.2.x.g.765.1		4
112.83	even	4		784.2.m.c.589.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.a.29.1	✓	2	16.3	odd	4
112.2.m.a.85.1	yes	2	4.3	odd	2
448.2.m.b.113.1		2	1.1	even	1	trivial
448.2.m.b.337.1		2	16.13	even	4	inner
784.2.m.c.197.1		2	28.27	even	2
784.2.m.c.589.1		2	112.83	even	4
784.2.x.b.165.1		4	28.19	even	6
784.2.x.b.373.1		4	28.3	even	6
784.2.x.b.557.1		4	112.19	even	12
784.2.x.b.765.1		4	112.3	even	12
784.2.x.g.165.1		4	28.23	odd	6
784.2.x.g.373.1		4	28.11	odd	6
784.2.x.g.557.1		4	112.51	odd	12
784.2.x.g.765.1		4	112.67	odd	12
896.2.m.a.225.1		2	8.5	even	2
896.2.m.a.673.1		2	16.5	even	4
896.2.m.d.225.1		2	8.3	odd	2
896.2.m.d.673.1		2	16.11	odd	4
7168.2.a.i.1.1		2	32.13	even	8
7168.2.a.i.1.2		2	32.29	even	8
7168.2.a.q.1.1		2	32.3	odd	8
7168.2.a.q.1.2		2	32.19	odd	8

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 2.00000i) q^{3} +(-2.00000 - 2.00000i) q^{5} -1.00000i q^{7} -5.00000i q^{9} +(3.00000 + 3.00000i) q^{11} -8.00000 q^{15} -6.00000 q^{17} +(4.00000 - 4.00000i) q^{19} +(-2.00000 - 2.00000i) q^{21} -2.00000i q^{23} +3.00000i q^{25} +(-4.00000 - 4.00000i) q^{27} +(-1.00000 + 1.00000i) q^{29} -4.00000 q^{31} +12.0000 q^{33} +(-2.00000 + 2.00000i) q^{35} +(3.00000 + 3.00000i) q^{37} +2.00000i q^{41} +(5.00000 + 5.00000i) q^{43} +(-10.0000 + 10.0000i) q^{45} +8.00000 q^{47} -1.00000 q^{49} +(-12.0000 + 12.0000i) q^{51} +(7.00000 + 7.00000i) q^{53} -12.0000i q^{55} -16.0000i q^{57} +(-2.00000 - 2.00000i) q^{59} +(6.00000 - 6.00000i) q^{61} -5.00000 q^{63} +(5.00000 - 5.00000i) q^{67} +(-4.00000 - 4.00000i) q^{69} +8.00000i q^{71} -10.0000i q^{73} +(6.00000 + 6.00000i) q^{75} +(3.00000 - 3.00000i) q^{77} +14.0000 q^{79} -1.00000 q^{81} +(12.0000 + 12.0000i) q^{85} +4.00000i q^{87} +6.00000i q^{89} +(-8.00000 + 8.00000i) q^{93} -16.0000 q^{95} -6.00000 q^{97} +(15.0000 - 15.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 4 q^{5} + 6 q^{11} - 16 q^{15} - 12 q^{17} + 8 q^{19} - 4 q^{21} - 8 q^{27} - 2 q^{29} - 8 q^{31} + 24 q^{33} - 4 q^{35} + 6 q^{37} + 10 q^{43} - 20 q^{45} + 16 q^{47} - 2 q^{49} - 24 q^{51}+ \cdots + 30 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 - 4 * q^5 + 6 * q^11 - 16 * q^15 - 12 * q^17 + 8 * q^19 - 4 * q^21 - 8 * q^27 - 2 * q^29 - 8 * q^31 + 24 * q^33 - 4 * q^35 + 6 * q^37 + 10 * q^43 - 20 * q^45 + 16 * q^47 - 2 * q^49 - 24 * q^51 + 14 * q^53 - 4 * q^59 + 12 * q^61 - 10 * q^63 + 10 * q^67 - 8 * q^69 + 12 * q^75 + 6 * q^77 + 28 * q^79 - 2 * q^81 + 24 * q^85 - 16 * q^93 - 32 * q^95 - 12 * q^97 + 30 * q^99

Introduction

L-functions

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