LMFDB - Embedded newform 4608.2.k.f.3457.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4608,2,Mod(1153,4608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4608, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4608.1153"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.k (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,0,0,-4,0,6,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$36.7950652514$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3457.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4608.3457
Dual form			4608.2.k.f.1153.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.00000i) q^{5} +4.00000i q^{7} +(-2.00000 - 2.00000i) q^{11} +(3.00000 - 3.00000i) q^{13} +(-2.00000 + 2.00000i) q^{19} +4.00000i q^{23} -3.00000i q^{25} +(3.00000 - 3.00000i) q^{29} -8.00000 q^{31} +(4.00000 - 4.00000i) q^{35} +(-1.00000 - 1.00000i) q^{37} +8.00000i q^{41} +(-2.00000 - 2.00000i) q^{43} +8.00000 q^{47} -9.00000 q^{49} +(1.00000 + 1.00000i) q^{53} +4.00000i q^{55} +(-6.00000 - 6.00000i) q^{59} +(3.00000 - 3.00000i) q^{61} -6.00000 q^{65} +(-2.00000 + 2.00000i) q^{67} -12.0000i q^{71} +2.00000i q^{73} +(8.00000 - 8.00000i) q^{77} +(-10.0000 + 10.0000i) q^{83} -14.0000i q^{89} +(12.0000 + 12.0000i) q^{91} +4.00000 q^{95} -16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 4 q^{11} + 6 q^{13} - 4 q^{19} + 6 q^{29} - 16 q^{31} + 8 q^{35} - 2 q^{37} - 4 q^{43} + 16 q^{47} - 18 q^{49} + 2 q^{53} - 12 q^{59} + 6 q^{61} - 12 q^{65} - 4 q^{67} + 16 q^{77} - 20 q^{83}+ \cdots - 32 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^11 + 6 * q^13 - 4 * q^19 + 6 * q^29 - 16 * q^31 + 8 * q^35 - 2 * q^37 - 4 * q^43 + 16 * q^47 - 18 * q^49 + 2 * q^53 - 12 * q^59 + 6 * q^61 - 12 * q^65 - 4 * q^67 + 16 * q^77 - 20 * q^83 + 24 * q^91 + 8 * q^95 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000	−	1.00000i	−0.447214	−	0.447214i	0.447214	−	0.894427i	$-0.352416\pi$
$5$	−1.00000	−	1.00000i	−0.447214	−	0.447214i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000	−	2.00000i	−0.603023	−	0.603023i	0.338091	−	0.941113i	$-0.390219\pi$
$11$	−2.00000	−	2.00000i	−0.603023	−	0.603023i	−0.941113	+	0.338091i	$0.890219\pi$
$12$		0			0
$13$	3.00000	−	3.00000i	0.832050	−	0.832050i	−0.155747	−	0.987797i	$-0.549778\pi$
$13$	3.00000	−	3.00000i	0.832050	−	0.832050i	0.987797	+	0.155747i	$0.0497784\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−2.00000	+	2.00000i	−0.458831	+	0.458831i	−0.898272	−	0.439440i	$-0.855177\pi$
$19$	−2.00000	+	2.00000i	−0.458831	+	0.458831i	0.439440	+	0.898272i	$0.355177\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$		−	3.00000i		−	0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.00000	−	3.00000i	0.557086	−	0.557086i	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	3.00000	−	3.00000i	0.557086	−	0.557086i	0.928477	+	0.371391i	$0.121119\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.00000	−	4.00000i	0.676123	−	0.676123i
$36$		0			0
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	0.620113	−	0.784512i	$-0.287087\pi$
$37$	−1.00000	−	1.00000i	−0.164399	−	0.164399i	−0.784512	+	0.620113i	$0.787087\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			8.00000i			1.24939i	0.780869	+	0.624695i	$0.214777\pi$
$41$			8.00000i			1.24939i	−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$	−2.00000	−	2.00000i	−0.304997	−	0.304997i	0.537968	−	0.842965i	$-0.319192\pi$
$43$	−2.00000	−	2.00000i	−0.304997	−	0.304997i	−0.842965	+	0.537968i	$0.819192\pi$
$44$		0			0
$45$		0			0
$46$		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
$49$	−9.00000			−1.28571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.00000	+	1.00000i	0.137361	+	0.137361i	0.772444	−	0.635083i	$-0.219034\pi$
$53$	1.00000	+	1.00000i	0.137361	+	0.137361i	−0.635083	+	0.772444i	$0.719034\pi$
$54$		0			0
$55$			4.00000i			0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	−	6.00000i	−0.781133	−	0.781133i	0.198889	−	0.980022i	$-0.436267\pi$
$59$	−6.00000	−	6.00000i	−0.781133	−	0.781133i	−0.980022	+	0.198889i	$0.936267\pi$
$60$		0			0
$61$	3.00000	−	3.00000i	0.384111	−	0.384111i	−0.488470	−	0.872581i	$-0.662445\pi$
$61$	3.00000	−	3.00000i	0.384111	−	0.384111i	0.872581	+	0.488470i	$0.162445\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000			−0.744208
$66$		0			0
$67$	−2.00000	+	2.00000i	−0.244339	+	0.244339i	−0.818642	−	0.574304i	$-0.805273\pi$
$67$	−2.00000	+	2.00000i	−0.244339	+	0.244339i	0.574304	+	0.818642i	$0.305273\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$		0			0
$73$			2.00000i			0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$			2.00000i			0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.00000	−	8.00000i	0.911685	−	0.911685i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.0000	+	10.0000i	−1.09764	+	1.09764i	−0.102957	+	0.994686i	$0.532830\pi$
$83$	−10.0000	+	10.0000i	−1.09764	+	1.09764i	−0.994686	+	0.102957i	$0.967170\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	14.0000i		−	1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$		−	14.0000i		−	1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$		0			0
$91$	12.0000	+	12.0000i	1.25794	+	1.25794i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000			0.410391
$96$		0			0
$97$	−16.0000			−1.62455			−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000			−1.62455			−0.812277	+	0.583272i	$0.801772\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.00000	+	1.00000i	0.0995037	+	0.0995037i	0.755106	−	0.655602i	$-0.227585\pi$
$101$	1.00000	+	1.00000i	0.0995037	+	0.0995037i	−0.655602	+	0.755106i	$0.727585\pi$
$102$		0			0
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.00000	−	2.00000i	−0.193347	−	0.193347i	0.603793	−	0.797141i	$-0.293655\pi$
$107$	−2.00000	−	2.00000i	−0.193347	−	0.193347i	−0.797141	+	0.603793i	$0.793655\pi$
$108$		0			0
$109$	13.0000	−	13.0000i	1.24517	−	1.24517i	0.287348	−	0.957826i	$-0.407226\pi$
$109$	13.0000	−	13.0000i	1.24517	−	1.24517i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$		0			0
$112$		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
$115$	4.00000	−	4.00000i	0.373002	−	0.373002i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	3.00000i		−	0.272727i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.00000	+	8.00000i	−0.715542	+	0.715542i
$126$		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
$129$		0			0
$130$		0			0
$131$	10.0000	−	10.0000i	0.873704	−	0.873704i	−0.119170	−	0.992874i	$-0.538023\pi$
$131$	10.0000	−	10.0000i	0.873704	−	0.873704i	0.992874	+	0.119170i	$0.0380233\pi$
$132$		0			0
$133$	−8.00000	−	8.00000i	−0.693688	−	0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	8.00000i		−	0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$		−	8.00000i		−	0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$		0			0
$139$	6.00000	+	6.00000i	0.508913	+	0.508913i	0.914193	−	0.405279i	$-0.132826\pi$
$139$	6.00000	+	6.00000i	0.508913	+	0.508913i	−0.405279	+	0.914193i	$0.632826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.0000			−1.00349
$144$		0			0
$145$	−6.00000			−0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.00000	−	1.00000i	−0.0819232	−	0.0819232i	0.664958	−	0.746881i	$-0.268450\pi$
$149$	−1.00000	−	1.00000i	−0.0819232	−	0.0819232i	−0.746881	+	0.664958i	$0.768450\pi$
$150$		0			0
$151$		−	20.0000i		−	1.62758i	−0.581161	−	0.813788i	$-0.697401\pi$
$151$		−	20.0000i		−	1.62758i	0.581161	−	0.813788i	$-0.302599\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.00000	+	8.00000i	0.642575	+	0.642575i
$156$		0			0
$157$	3.00000	−	3.00000i	0.239426	−	0.239426i	−0.577186	−	0.816612i	$-0.695849\pi$
$157$	3.00000	−	3.00000i	0.239426	−	0.239426i	0.816612	+	0.577186i	$0.195849\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16.0000			−1.26098
$162$		0			0
$163$	−6.00000	+	6.00000i	−0.469956	+	0.469956i	−0.901900	−	0.431944i	$-0.857828\pi$
$163$	−6.00000	+	6.00000i	−0.469956	+	0.469956i	0.431944	+	0.901900i	$0.357828\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.00000i		−	0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$		−	4.00000i		−	0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	13.0000	−	13.0000i	0.988372	−	0.988372i	−0.0115615	−	0.999933i	$-0.503680\pi$
$173$	13.0000	−	13.0000i	0.988372	−	0.988372i	0.999933	+	0.0115615i	$0.00368021\pi$
$174$		0			0
$175$	12.0000			0.907115
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.0000	−	14.0000i	1.04641	−	1.04641i	0.0475398	−	0.998869i	$-0.484862\pi$
$179$	14.0000	−	14.0000i	1.04641	−	1.04641i	0.998869	−	0.0475398i	$-0.0151381\pi$
$180$		0			0
$181$	−15.0000	−	15.0000i	−1.11494	−	1.11494i	−0.992472	−	0.122469i	$-0.960919\pi$
$181$	−15.0000	−	15.0000i	−1.11494	−	1.11494i	−0.122469	−	0.992472i	$-0.539081\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			2.00000i			0.147043i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$		0			0
$193$	−16.0000			−1.15171			−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000			−1.15171			−0.575853	+	0.817554i	$0.695330\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−17.0000	−	17.0000i	−1.21120	−	1.21120i	−0.970632	−	0.240567i	$-0.922666\pi$
$197$	−17.0000	−	17.0000i	−1.21120	−	1.21120i	−0.240567	−	0.970632i	$-0.577334\pi$
$198$		0			0
$199$			4.00000i			0.283552i	0.989899	+	0.141776i	$0.0452813\pi$
$199$			4.00000i			0.283552i	−0.989899	+	0.141776i	$0.954719\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.0000	+	12.0000i	0.842235	+	0.842235i
$204$		0			0
$205$	8.00000	−	8.00000i	0.558744	−	0.558744i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.00000			0.553372
$210$		0			0
$211$	10.0000	−	10.0000i	0.688428	−	0.688428i	−0.273456	−	0.961884i	$-0.588167\pi$
$211$	10.0000	−	10.0000i	0.688428	−	0.688428i	0.961884	+	0.273456i	$0.0881668\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4.00000i			0.272798i
$216$		0			0
$217$		−	32.0000i		−	2.17230i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	24.0000			1.60716			0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000			1.60716			0.803579	+	0.595198i	$0.202926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.0000	+	18.0000i	−1.19470	+	1.19470i	−0.218970	+	0.975731i	$0.570270\pi$
$227$	−18.0000	+	18.0000i	−1.19470	+	1.19470i	−0.975731	+	0.218970i	$0.929730\pi$
$228$		0			0
$229$	1.00000	+	1.00000i	0.0660819	+	0.0660819i	0.739375	−	0.673293i	$-0.235121\pi$
$229$	1.00000	+	1.00000i	0.0660819	+	0.0660819i	−0.673293	+	0.739375i	$0.735121\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.00000i		−	0.131024i	−0.997852	−	0.0655122i	$-0.979132\pi$
$233$		−	2.00000i		−	0.131024i	0.997852	−	0.0655122i	$-0.0208681\pi$
$234$		0			0
$235$	−8.00000	−	8.00000i	−0.521862	−	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.00000	+	9.00000i	0.574989	+	0.574989i
$246$		0			0
$247$			12.0000i			0.763542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.0000	−	14.0000i	−0.883672	−	0.883672i	0.110234	−	0.993906i	$-0.464840\pi$
$251$	−14.0000	−	14.0000i	−0.883672	−	0.883672i	−0.993906	+	0.110234i	$0.964840\pi$
$252$		0			0
$253$	8.00000	−	8.00000i	0.502956	−	0.502956i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−0.561405	+	0.827541i	$0.689739\pi$
$258$		0			0
$259$	4.00000	−	4.00000i	0.248548	−	0.248548i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			20.0000i			1.23325i	0.787256	+	0.616626i	$0.211501\pi$
$263$			20.0000i			1.23325i	−0.787256	+	0.616626i	$0.788499\pi$
$264$		0			0
$265$		−	2.00000i		−	0.122859i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.00000	−	3.00000i	0.182913	−	0.182913i	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	3.00000	−	3.00000i	0.182913	−	0.182913i	0.792624	+	0.609711i	$0.208714\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.00000	+	6.00000i	−0.361814	+	0.361814i
$276$		0			0
$277$	15.0000	+	15.0000i	0.901263	+	0.901263i	0.995545	−	0.0942828i	$-0.0300558\pi$
$277$	15.0000	+	15.0000i	0.901263	+	0.901263i	−0.0942828	+	0.995545i	$0.530056\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$	−14.0000	−	14.0000i	−0.832214	−	0.832214i	0.155606	−	0.987819i	$-0.450267\pi$
$283$	−14.0000	−	14.0000i	−0.832214	−	0.832214i	−0.987819	+	0.155606i	$0.950267\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−32.0000			−1.88890
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−17.0000	−	17.0000i	−0.993151	−	0.993151i	0.00682610	−	0.999977i	$-0.497827\pi$
$293$	−17.0000	−	17.0000i	−0.993151	−	0.993151i	−0.999977	+	0.00682610i	$0.997827\pi$
$294$		0			0
$295$			12.0000i			0.698667i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	+	12.0000i	0.693978	+	0.693978i
$300$		0			0
$301$	8.00000	−	8.00000i	0.461112	−	0.461112i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.00000			−0.343559
$306$		0			0
$307$	6.00000	−	6.00000i	0.342438	−	0.342438i	−0.514845	−	0.857283i	$-0.672151\pi$
$307$	6.00000	−	6.00000i	0.342438	−	0.342438i	0.857283	+	0.514845i	$0.172151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	4.00000i		−	0.226819i	−0.993548	−	0.113410i	$-0.963823\pi$
$311$		−	4.00000i		−	0.226819i	0.993548	−	0.113410i	$-0.0361772\pi$
$312$		0			0
$313$			8.00000i			0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$			8.00000i			0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.0000	+	13.0000i	−0.730153	+	0.730153i	−0.970650	−	0.240497i	$-0.922690\pi$
$317$	−13.0000	+	13.0000i	−0.730153	+	0.730153i	0.240497	+	0.970650i	$0.422690\pi$
$318$		0			0
$319$	−12.0000			−0.671871
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−9.00000	−	9.00000i	−0.499230	−	0.499230i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			32.0000i			1.76422i
$330$		0			0
$331$	−22.0000	−	22.0000i	−1.20923	−	1.20923i	−0.971277	−	0.237953i	$-0.923524\pi$
$331$	−22.0000	−	22.0000i	−1.20923	−	1.20923i	−0.237953	−	0.971277i	$-0.576476\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.00000			0.218543
$336$		0			0
$337$	−18.0000			−0.980522			−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000			−0.980522			−0.490261	+	0.871576i	$0.663099\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.0000	+	16.0000i	0.866449	+	0.866449i
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.00000	+	2.00000i	0.107366	+	0.107366i	0.758749	−	0.651383i	$-0.225811\pi$
$347$	2.00000	+	2.00000i	0.107366	+	0.107366i	−0.651383	+	0.758749i	$0.725811\pi$
$348$		0			0
$349$	13.0000	−	13.0000i	0.695874	−	0.695874i	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	13.0000	−	13.0000i	0.695874	−	0.695874i	0.963518	+	0.267644i	$0.0862451\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.0000			−0.745145			−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000			−0.745145			−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$	−12.0000	+	12.0000i	−0.636894	+	0.636894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			20.0000i			1.05556i	0.849381	+	0.527780i	$0.176975\pi$
$359$			20.0000i			1.05556i	−0.849381	+	0.527780i	$0.823025\pi$
$360$		0			0
$361$			11.0000i			0.578947i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.00000	−	2.00000i	0.104685	−	0.104685i
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.00000	+	4.00000i	−0.207670	+	0.207670i
$372$		0			0
$373$	15.0000	+	15.0000i	0.776671	+	0.776671i	0.979263	−	0.202593i	$-0.0649367\pi$
$373$	15.0000	+	15.0000i	0.776671	+	0.776671i	−0.202593	+	0.979263i	$0.564937\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	18.0000i		−	0.927047i
$378$		0			0
$379$	−18.0000	−	18.0000i	−0.924598	−	0.924598i	0.0727522	−	0.997350i	$-0.476822\pi$
$379$	−18.0000	−	18.0000i	−0.924598	−	0.924598i	−0.997350	+	0.0727522i	$0.976822\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	−16.0000			−0.815436
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.00000	+	1.00000i	0.0507020	+	0.0507020i	0.732003	−	0.681301i	$-0.238586\pi$
$389$	1.00000	+	1.00000i	0.0507020	+	0.0507020i	−0.681301	+	0.732003i	$0.738586\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.00000	+	3.00000i	−0.150566	+	0.150566i	−0.778371	−	0.627805i	$-0.783954\pi$
$397$	−3.00000	+	3.00000i	−0.150566	+	0.150566i	0.627805	+	0.778371i	$0.283954\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−24.0000	+	24.0000i	−1.19553	+	1.19553i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			4.00000i			0.198273i
$408$		0			0
$409$			8.00000i			0.395575i	0.980245	+	0.197787i	$0.0633755\pi$
$409$			8.00000i			0.395575i	−0.980245	+	0.197787i	$0.936624\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	24.0000	−	24.0000i	1.18096	−	1.18096i
$414$		0			0
$415$	20.0000			0.981761
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.00000	+	6.00000i	−0.293119	+	0.293119i	−0.838311	−	0.545192i	$-0.816457\pi$
$419$	−6.00000	+	6.00000i	−0.293119	+	0.293119i	0.545192	+	0.838311i	$0.316457\pi$
$420$		0			0
$421$	−15.0000	−	15.0000i	−0.731055	−	0.731055i	0.239774	−	0.970829i	$-0.422927\pi$
$421$	−15.0000	−	15.0000i	−0.731055	−	0.731055i	−0.970829	+	0.239774i	$0.922927\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	12.0000	+	12.0000i	0.580721	+	0.580721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.0000			−0.770693			−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$		0			0
$433$	32.0000			1.53782			0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000			1.53782			0.768911	+	0.639356i	$0.220799\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.00000	−	8.00000i	−0.382692	−	0.382692i
$438$		0			0
$439$			4.00000i			0.190910i	0.995434	+	0.0954548i	$0.0304305\pi$
$439$			4.00000i			0.190910i	−0.995434	+	0.0954548i	$0.969569\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.00000	+	6.00000i	0.285069	+	0.285069i	0.835127	−	0.550058i	$-0.185394\pi$
$443$	6.00000	+	6.00000i	0.285069	+	0.285069i	−0.550058	+	0.835127i	$0.685394\pi$
$444$		0			0
$445$	−14.0000	+	14.0000i	−0.663664	+	0.663664i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	16.0000			0.755087			0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000			0.755087			0.377543	+	0.925992i	$0.376769\pi$
$450$		0			0
$451$	16.0000	−	16.0000i	0.753411	−	0.753411i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	24.0000i		−	1.12514i
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	19.0000	−	19.0000i	0.884918	−	0.884918i	−0.109111	−	0.994030i	$-0.534800\pi$
$461$	19.0000	−	19.0000i	0.884918	−	0.884918i	0.994030	+	0.109111i	$0.0348005\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.00000	−	2.00000i	0.0925490	−	0.0925490i	−0.659317	−	0.751865i	$-0.729154\pi$
$467$	2.00000	−	2.00000i	0.0925490	−	0.0925490i	0.751865	+	0.659317i	$0.229154\pi$
$468$		0			0
$469$	−8.00000	−	8.00000i	−0.369406	−	0.369406i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			8.00000i			0.367840i
$474$		0			0
$475$	6.00000	+	6.00000i	0.275299	+	0.275299i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000			−0.365529			−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000			−0.365529			−0.182765	+	0.983157i	$0.558505\pi$
$480$		0			0
$481$	−6.00000			−0.273576
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.0000	+	16.0000i	0.726523	+	0.726523i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−14.0000	−	14.0000i	−0.631811	−	0.631811i	0.316711	−	0.948522i	$-0.397421\pi$
$491$	−14.0000	−	14.0000i	−0.631811	−	0.631811i	−0.948522	+	0.316711i	$0.897421\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	48.0000			2.15309
$498$		0			0
$499$	2.00000	−	2.00000i	0.0895323	−	0.0895323i	−0.660922	−	0.750454i	$-0.729835\pi$
$499$	2.00000	−	2.00000i	0.0895323	−	0.0895323i	0.750454	+	0.660922i	$0.229835\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	28.0000i		−	1.24846i	−0.781241	−	0.624229i	$-0.785413\pi$
$503$		−	28.0000i		−	1.24846i	0.781241	−	0.624229i	$-0.214587\pi$
$504$		0			0
$505$		−	2.00000i		−	0.0889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−19.0000	+	19.0000i	−0.842160	+	0.842160i	−0.989140	−	0.146979i	$-0.953045\pi$
$509$	−19.0000	+	19.0000i	−0.842160	+	0.842160i	0.146979	+	0.989140i	$0.453045\pi$
$510$		0			0
$511$	−8.00000			−0.353899
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.00000	+	4.00000i	−0.176261	+	0.176261i
$516$		0			0
$517$	−16.0000	−	16.0000i	−0.703679	−	0.703679i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			24.0000i			1.05146i	0.850652	+	0.525730i	$0.176208\pi$
$521$			24.0000i			1.05146i	−0.850652	+	0.525730i	$0.823792\pi$
$522$		0			0
$523$	18.0000	+	18.0000i	0.787085	+	0.787085i	0.981015	−	0.193930i	$-0.0621236\pi$
$523$	18.0000	+	18.0000i	0.787085	+	0.787085i	−0.193930	+	0.981015i	$0.562124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.0000	+	24.0000i	1.03956	+	1.03956i
$534$		0			0
$535$			4.00000i			0.172935i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.0000	+	18.0000i	0.775315	+	0.775315i
$540$		0			0
$541$	3.00000	−	3.00000i	0.128980	−	0.128980i	−0.639670	−	0.768650i	$-0.720929\pi$
$541$	3.00000	−	3.00000i	0.128980	−	0.128980i	0.768650	+	0.639670i	$0.220929\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−26.0000			−1.11372
$546$		0			0
$547$	30.0000	−	30.0000i	1.28271	−	1.28271i	0.343586	−	0.939121i	$-0.388358\pi$
$547$	30.0000	−	30.0000i	1.28271	−	1.28271i	0.939121	−	0.343586i	$-0.111642\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			12.0000i			0.511217i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	13.0000	−	13.0000i	0.550828	−	0.550828i	−0.375852	−	0.926680i	$-0.622650\pi$
$557$	13.0000	−	13.0000i	0.550828	−	0.550828i	0.926680	+	0.375852i	$0.122650\pi$
$558$		0			0
$559$	−12.0000			−0.507546
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.00000	−	2.00000i	0.0842900	−	0.0842900i	−0.663705	−	0.747995i	$-0.731017\pi$
$563$	2.00000	−	2.00000i	0.0842900	−	0.0842900i	0.747995	+	0.663705i	$0.231017\pi$
$564$		0			0
$565$	2.00000	+	2.00000i	0.0841406	+	0.0841406i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	24.0000i		−	1.00613i	−0.864248	−	0.503066i	$-0.832205\pi$
$569$		−	24.0000i		−	1.00613i	0.864248	−	0.503066i	$-0.167795\pi$
$570$		0			0
$571$	−6.00000	−	6.00000i	−0.251092	−	0.251092i	0.570326	−	0.821418i	$-0.306817\pi$
$571$	−6.00000	−	6.00000i	−0.251092	−	0.251092i	−0.821418	+	0.570326i	$0.806817\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	12.0000			0.500435
$576$		0			0
$577$	30.0000			1.24892			0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000			1.24892			0.624458	+	0.781058i	$0.285320\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−40.0000	−	40.0000i	−1.65948	−	1.65948i
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.0000	−	10.0000i	−0.412744	−	0.412744i	0.469949	−	0.882693i	$-0.344272\pi$
$587$	−10.0000	−	10.0000i	−0.412744	−	0.412744i	−0.882693	+	0.469949i	$0.844272\pi$
$588$		0			0
$589$	16.0000	−	16.0000i	0.659269	−	0.659269i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.0000			0.574911			0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000			0.574911			0.287456	+	0.957794i	$0.407191\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			28.0000i			1.14405i	0.820237	+	0.572024i	$0.193842\pi$
$599$			28.0000i			1.14405i	−0.820237	+	0.572024i	$0.806158\pi$
$600$		0			0
$601$			14.0000i			0.571072i	0.958368	+	0.285536i	$0.0921716\pi$
$601$			14.0000i			0.571072i	−0.958368	+	0.285536i	$0.907828\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.00000	+	3.00000i	−0.121967	+	0.121967i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.0000	−	24.0000i	0.970936	−	0.970936i
$612$		0			0
$613$	17.0000	+	17.0000i	0.686624	+	0.686624i	0.961484	−	0.274861i	$-0.0886317\pi$
$613$	17.0000	+	17.0000i	0.686624	+	0.686624i	−0.274861	+	0.961484i	$0.588632\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2.00000i			0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$			2.00000i			0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$		0			0
$619$	18.0000	+	18.0000i	0.723481	+	0.723481i	0.969313	−	0.245831i	$-0.0790610\pi$
$619$	18.0000	+	18.0000i	0.723481	+	0.723481i	−0.245831	+	0.969313i	$0.579061\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	56.0000			2.24359
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	4.00000i		−	0.159237i	−0.996825	−	0.0796187i	$-0.974630\pi$
$631$		−	4.00000i		−	0.159237i	0.996825	−	0.0796187i	$-0.0253703\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.00000	+	8.00000i	0.317470	+	0.317470i
$636$		0			0
$637$	−27.0000	+	27.0000i	−1.06978	+	1.06978i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−16.0000			−0.631962			−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000			−0.631962			−0.315981	+	0.948766i	$0.602334\pi$
$642$		0			0
$643$	−18.0000	+	18.0000i	−0.709851	+	0.709851i	−0.966504	−	0.256653i	$-0.917380\pi$
$643$	−18.0000	+	18.0000i	−0.709851	+	0.709851i	0.256653	+	0.966504i	$0.417380\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	12.0000i		−	0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$		−	12.0000i		−	0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$		0			0
$649$			24.0000i			0.942082i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−29.0000	+	29.0000i	−1.13486	+	1.13486i	−0.145499	+	0.989358i	$0.546479\pi$
$653$	−29.0000	+	29.0000i	−1.13486	+	1.13486i	−0.989358	+	0.145499i	$0.953521\pi$
$654$		0			0
$655$	−20.0000			−0.781465
$656$		0			0
$657$		0			0
$658$		0			0
$659$	22.0000	−	22.0000i	0.856998	−	0.856998i	−0.133985	−	0.990983i	$-0.542777\pi$
$659$	22.0000	−	22.0000i	0.856998	−	0.856998i	0.990983	+	0.133985i	$0.0427774\pi$
$660$		0			0
$661$	17.0000	+	17.0000i	0.661223	+	0.661223i	0.955668	−	0.294445i	$-0.0951348\pi$
$661$	17.0000	+	17.0000i	0.661223	+	0.661223i	−0.294445	+	0.955668i	$0.595135\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			16.0000i			0.620453i
$666$		0			0
$667$	12.0000	+	12.0000i	0.464642	+	0.464642i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.0000			−0.463255
$672$		0			0
$673$	16.0000			0.616755			0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000			0.616755			0.308377	+	0.951264i	$0.400214\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	33.0000	+	33.0000i	1.26829	+	1.26829i	0.946969	+	0.321324i	$0.104128\pi$
$677$	33.0000	+	33.0000i	1.26829	+	1.26829i	0.321324	+	0.946969i	$0.395872\pi$
$678$		0			0
$679$		−	64.0000i		−	2.45609i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.00000	−	6.00000i	−0.229584	−	0.229584i	0.582935	−	0.812519i	$-0.301904\pi$
$683$	−6.00000	−	6.00000i	−0.229584	−	0.229584i	−0.812519	+	0.582935i	$0.801904\pi$
$684$		0			0
$685$	−8.00000	+	8.00000i	−0.305664	+	0.305664i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000			0.228582
$690$		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
$693$		0			0
$694$		0			0
$695$		−	12.0000i		−	0.455186i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−19.0000	+	19.0000i	−0.717620	+	0.717620i	−0.968117	−	0.250497i	$-0.919406\pi$
$701$	−19.0000	+	19.0000i	−0.717620	+	0.717620i	0.250497	+	0.968117i	$0.419406\pi$
$702$		0			0
$703$	4.00000			0.150863
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.00000	+	4.00000i	−0.150435	+	0.150435i
$708$		0			0
$709$	15.0000	+	15.0000i	0.563337	+	0.563337i	0.930254	−	0.366917i	$-0.119587\pi$
$709$	15.0000	+	15.0000i	0.563337	+	0.563337i	−0.366917	+	0.930254i	$0.619587\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	32.0000i		−	1.19841i
$714$		0			0
$715$	12.0000	+	12.0000i	0.448775	+	0.448775i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.00000	−	9.00000i	−0.334252	−	0.334252i
$726$		0			0
$727$			20.0000i			0.741759i	0.928681	+	0.370879i	$0.120944\pi$
$727$			20.0000i			0.741759i	−0.928681	+	0.370879i	$0.879056\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−19.0000	+	19.0000i	−0.701781	+	0.701781i	−0.964793	−	0.263012i	$-0.915284\pi$
$733$	−19.0000	+	19.0000i	−0.701781	+	0.701781i	0.263012	+	0.964793i	$0.415284\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.00000			0.294684
$738$		0			0
$739$	−6.00000	+	6.00000i	−0.220714	+	0.220714i	−0.808799	−	0.588085i	$-0.799882\pi$
$739$	−6.00000	+	6.00000i	−0.220714	+	0.220714i	0.588085	+	0.808799i	$0.299882\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			12.0000i			0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$			12.0000i			0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$		0			0
$745$			2.00000i			0.0732743i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.00000	−	8.00000i	0.292314	−	0.292314i
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−20.0000	+	20.0000i	−0.727875	+	0.727875i
$756$		0			0
$757$	1.00000	+	1.00000i	0.0363456	+	0.0363456i	0.725046	−	0.688700i	$-0.241818\pi$
$757$	1.00000	+	1.00000i	0.0363456	+	0.0363456i	−0.688700	+	0.725046i	$0.741818\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			24.0000i			0.869999i	0.900431	+	0.435000i	$0.143252\pi$
$761$			24.0000i			0.869999i	−0.900431	+	0.435000i	$0.856748\pi$
$762$		0			0
$763$	52.0000	+	52.0000i	1.88253	+	1.88253i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−36.0000			−1.29988
$768$		0			0
$769$	16.0000			0.576975			0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000			0.576975			0.288487	+	0.957484i	$0.406848\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15.0000	−	15.0000i	−0.539513	−	0.539513i	0.383873	−	0.923386i	$-0.374590\pi$
$773$	−15.0000	−	15.0000i	−0.539513	−	0.539513i	−0.923386	+	0.383873i	$0.874590\pi$
$774$		0			0
$775$			24.0000i			0.862105i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16.0000	−	16.0000i	−0.573259	−	0.573259i
$780$		0			0
$781$	−24.0000	+	24.0000i	−0.858788	+	0.858788i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.00000			−0.214149
$786$		0			0
$787$	18.0000	−	18.0000i	0.641631	−	0.641631i	−0.309326	−	0.950956i	$-0.600103\pi$
$787$	18.0000	−	18.0000i	0.641631	−	0.641631i	0.950956	+	0.309326i	$0.100103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	8.00000i		−	0.284447i
$792$		0			0
$793$		−	18.0000i		−	0.639199i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.0000	+	19.0000i	−0.673015	+	0.673015i	−0.958410	−	0.285395i	$-0.907875\pi$
$797$	−19.0000	+	19.0000i	−0.673015	+	0.673015i	0.285395	+	0.958410i	$0.407875\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.00000	−	4.00000i	0.141157	−	0.141157i
$804$		0			0
$805$	16.0000	+	16.0000i	0.563926	+	0.563926i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			40.0000i			1.40633i	0.711029	+	0.703163i	$0.248229\pi$
$809$			40.0000i			1.40633i	−0.711029	+	0.703163i	$0.751771\pi$
$810$		0			0
$811$	−22.0000	−	22.0000i	−0.772524	−	0.772524i	0.206023	−	0.978547i	$-0.433948\pi$
$811$	−22.0000	−	22.0000i	−0.772524	−	0.772524i	−0.978547	+	0.206023i	$0.933948\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.0000			0.420342
$816$		0			0
$817$	8.00000			0.279885
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.0000	−	15.0000i	−0.523504	−	0.523504i	0.395124	−	0.918628i	$-0.370702\pi$
$821$	−15.0000	−	15.0000i	−0.523504	−	0.523504i	−0.918628	+	0.395124i	$0.870702\pi$
$822$		0			0
$823$			52.0000i			1.81261i	0.422628	+	0.906303i	$0.361108\pi$
$823$			52.0000i			1.81261i	−0.422628	+	0.906303i	$0.638892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14.0000	+	14.0000i	0.486828	+	0.486828i	0.907304	−	0.420476i	$-0.138137\pi$
$827$	14.0000	+	14.0000i	0.486828	+	0.486828i	−0.420476	+	0.907304i	$0.638137\pi$
$828$		0			0
$829$	−13.0000	+	13.0000i	−0.451509	+	0.451509i	−0.895855	−	0.444346i	$-0.853436\pi$
$829$	−13.0000	+	13.0000i	−0.451509	+	0.451509i	0.444346	+	0.895855i	$0.353436\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−4.00000	+	4.00000i	−0.138426	+	0.138426i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	36.0000i		−	1.24286i	−0.783470	−	0.621429i	$-0.786552\pi$
$839$		−	36.0000i		−	1.24286i	0.783470	−	0.621429i	$-0.213448\pi$
$840$		0			0
$841$			11.0000i			0.379310i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.00000	+	5.00000i	−0.172005	+	0.172005i
$846$		0			0
$847$	12.0000			0.412325
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.00000	−	4.00000i	0.137118	−	0.137118i
$852$		0			0
$853$	−1.00000	−	1.00000i	−0.0342393	−	0.0342393i	0.689780	−	0.724019i	$-0.257707\pi$
$853$	−1.00000	−	1.00000i	−0.0342393	−	0.0342393i	−0.724019	+	0.689780i	$0.757707\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	24.0000i		−	0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$		−	24.0000i		−	0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$		0			0
$859$	−18.0000	−	18.0000i	−0.614152	−	0.614152i	0.329873	−	0.944025i	$-0.392994\pi$
$859$	−18.0000	−	18.0000i	−0.614152	−	0.614152i	−0.944025	+	0.329873i	$0.892994\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		0			0
$865$	−26.0000			−0.884027
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$			12.0000i			0.406604i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−32.0000	−	32.0000i	−1.08180	−	1.08180i
$876$		0			0
$877$	−19.0000	+	19.0000i	−0.641584	+	0.641584i	−0.950945	−	0.309360i	$-0.899885\pi$
$877$	−19.0000	+	19.0000i	−0.641584	+	0.641584i	0.309360	+	0.950945i	$0.399885\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	−34.0000	+	34.0000i	−1.14419	+	1.14419i	−0.156516	+	0.987675i	$0.550026\pi$
$883$	−34.0000	+	34.0000i	−1.14419	+	1.14419i	−0.987675	+	0.156516i	$0.949974\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	36.0000i		−	1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$		−	36.0000i		−	1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$		0			0
$889$		−	32.0000i		−	1.07325i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16.0000	+	16.0000i	−0.535420	+	0.535420i
$894$		0			0
$895$	−28.0000			−0.935937
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24.0000	+	24.0000i	−0.800445	+	0.800445i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			30.0000i			0.997234i
$906$		0			0
$907$	−10.0000	−	10.0000i	−0.332045	−	0.332045i	0.521318	−	0.853363i	$-0.325441\pi$
$907$	−10.0000	−	10.0000i	−0.332045	−	0.332045i	−0.853363	+	0.521318i	$0.825441\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−56.0000			−1.85536			−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000			−1.85536			−0.927681	+	0.373373i	$0.878201\pi$
$912$		0			0
$913$	40.0000			1.32381
$914$		0			0
$915$		0			0
$916$		0			0
$917$	40.0000	+	40.0000i	1.32092	+	1.32092i
$918$		0			0
$919$		−	4.00000i		−	0.131948i	−0.997821	−	0.0659739i	$-0.978985\pi$
$919$		−	4.00000i		−	0.131948i	0.997821	−	0.0659739i	$-0.0210154\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−36.0000	−	36.0000i	−1.18495	−	1.18495i
$924$		0			0
$925$	−3.00000	+	3.00000i	−0.0986394	+	0.0986394i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	48.0000			1.57483			0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000			1.57483			0.787414	+	0.616424i	$0.211419\pi$
$930$		0			0
$931$	18.0000	−	18.0000i	0.589926	−	0.589926i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			2.00000i			0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$			2.00000i			0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.0000	+	13.0000i	−0.423788	+	0.423788i	−0.886506	−	0.462718i	$-0.846874\pi$
$941$	−13.0000	+	13.0000i	−0.423788	+	0.423788i	0.462718	+	0.886506i	$0.346874\pi$
$942$		0			0
$943$	−32.0000			−1.04206
$944$		0			0
$945$		0			0
$946$		0			0
$947$	10.0000	−	10.0000i	0.324956	−	0.324956i	−0.525708	−	0.850665i	$-0.676200\pi$
$947$	10.0000	−	10.0000i	0.324956	−	0.324956i	0.850665	+	0.525708i	$0.176200\pi$
$948$		0			0
$949$	6.00000	+	6.00000i	0.194768	+	0.194768i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	24.0000i		−	0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$		−	24.0000i		−	0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$		0			0
$955$	−24.0000	−	24.0000i	−0.776622	−	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	32.0000			1.03333
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.0000	+	16.0000i	0.515058	+	0.515058i
$966$		0			0
$967$			20.0000i			0.643157i	0.946883	+	0.321578i	$0.104213\pi$
$967$			20.0000i			0.643157i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−14.0000	−	14.0000i	−0.449281	−	0.449281i	0.445834	−	0.895116i	$-0.352907\pi$
$971$	−14.0000	−	14.0000i	−0.449281	−	0.449281i	−0.895116	+	0.445834i	$0.852907\pi$
$972$		0			0
$973$	−24.0000	+	24.0000i	−0.769405	+	0.769405i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−32.0000			−1.02377			−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000			−1.02377			−0.511885	+	0.859054i	$0.671053\pi$
$978$		0			0
$979$	−28.0000	+	28.0000i	−0.894884	+	0.894884i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			28.0000i			0.893061i	0.894768	+	0.446531i	$0.147341\pi$
$983$			28.0000i			0.893061i	−0.894768	+	0.446531i	$0.852659\pi$
$984$		0			0
$985$			34.0000i			1.08333i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.00000	−	8.00000i	0.254385	−	0.254385i
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.00000	−	4.00000i	0.126809	−	0.126809i
$996$		0			0
$997$	15.0000	+	15.0000i	0.475055	+	0.475055i	0.903546	−	0.428491i	$-0.140955\pi$
$997$	15.0000	+	15.0000i	0.475055	+	0.475055i	−0.428491	+	0.903546i	$0.640955\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.k.f.3457.1		2
3.2	odd	2		512.2.e.h.385.1	yes	2
4.3	odd	2		4608.2.k.j.3457.1		2
8.3	odd	2		4608.2.k.o.3457.1		2
8.5	even	2		4608.2.k.s.3457.1		2
12.11	even	2		512.2.e.b.385.1	yes	2
16.3	odd	4		4608.2.k.j.1153.1		2
16.5	even	4		4608.2.k.s.1153.1		2
16.11	odd	4		4608.2.k.o.1153.1		2
16.13	even	4	inner	4608.2.k.f.1153.1		2
24.5	odd	2		512.2.e.a.385.1	yes	2
24.11	even	2		512.2.e.g.385.1	yes	2
32.3	odd	8		9216.2.a.b.1.2		2
32.13	even	8		9216.2.a.u.1.1		2
32.19	odd	8		9216.2.a.b.1.1		2
32.29	even	8		9216.2.a.u.1.2		2
48.5	odd	4		512.2.e.a.129.1	✓	2
48.11	even	4		512.2.e.g.129.1	yes	2
48.29	odd	4		512.2.e.h.129.1	yes	2
48.35	even	4		512.2.e.b.129.1	yes	2
96.5	odd	8		1024.2.b.a.513.2		2
96.11	even	8		1024.2.b.f.513.2		2
96.29	odd	8		1024.2.a.f.1.2		2
96.35	even	8		1024.2.a.a.1.1		2
96.53	odd	8		1024.2.b.a.513.1		2
96.59	even	8		1024.2.b.f.513.1		2
96.77	odd	8		1024.2.a.f.1.1		2
96.83	even	8		1024.2.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.a.129.1	✓	2	48.5	odd	4
512.2.e.a.385.1	yes	2	24.5	odd	2
512.2.e.b.129.1	yes	2	48.35	even	4
512.2.e.b.385.1	yes	2	12.11	even	2
512.2.e.g.129.1	yes	2	48.11	even	4
512.2.e.g.385.1	yes	2	24.11	even	2
512.2.e.h.129.1	yes	2	48.29	odd	4
512.2.e.h.385.1	yes	2	3.2	odd	2
1024.2.a.a.1.1		2	96.35	even	8
1024.2.a.a.1.2		2	96.83	even	8
1024.2.a.f.1.1		2	96.77	odd	8
1024.2.a.f.1.2		2	96.29	odd	8
1024.2.b.a.513.1		2	96.53	odd	8
1024.2.b.a.513.2		2	96.5	odd	8
1024.2.b.f.513.1		2	96.59	even	8
1024.2.b.f.513.2		2	96.11	even	8
4608.2.k.f.1153.1		2	16.13	even	4	inner
4608.2.k.f.3457.1		2	1.1	even	1	trivial
4608.2.k.j.1153.1		2	16.3	odd	4
4608.2.k.j.3457.1		2	4.3	odd	2
4608.2.k.o.1153.1		2	16.11	odd	4
4608.2.k.o.3457.1		2	8.3	odd	2
4608.2.k.s.1153.1		2	16.5	even	4
4608.2.k.s.3457.1		2	8.5	even	2
9216.2.a.b.1.1		2	32.19	odd	8
9216.2.a.b.1.2		2	32.3	odd	8
9216.2.a.u.1.1		2	32.13	even	8
9216.2.a.u.1.2		2	32.29	even	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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{5} +4.00000i q^{7} +(-2.00000 - 2.00000i) q^{11} +(3.00000 - 3.00000i) q^{13} +(-2.00000 + 2.00000i) q^{19} +4.00000i q^{23} -3.00000i q^{25} +(3.00000 - 3.00000i) q^{29} -8.00000 q^{31} +(4.00000 - 4.00000i) q^{35} +(-1.00000 - 1.00000i) q^{37} +8.00000i q^{41} +(-2.00000 - 2.00000i) q^{43} +8.00000 q^{47} -9.00000 q^{49} +(1.00000 + 1.00000i) q^{53} +4.00000i q^{55} +(-6.00000 - 6.00000i) q^{59} +(3.00000 - 3.00000i) q^{61} -6.00000 q^{65} +(-2.00000 + 2.00000i) q^{67} -12.0000i q^{71} +2.00000i q^{73} +(8.00000 - 8.00000i) q^{77} +(-10.0000 + 10.0000i) q^{83} -14.0000i q^{89} +(12.0000 + 12.0000i) q^{91} +4.00000 q^{95} -16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 4 q^{11} + 6 q^{13} - 4 q^{19} + 6 q^{29} - 16 q^{31} + 8 q^{35} - 2 q^{37} - 4 q^{43} + 16 q^{47} - 18 q^{49} + 2 q^{53} - 12 q^{59} + 6 q^{61} - 12 q^{65} - 4 q^{67} + 16 q^{77} - 20 q^{83}+ \cdots - 32 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^11 + 6 * q^13 - 4 * q^19 + 6 * q^29 - 16 * q^31 + 8 * q^35 - 2 * q^37 - 4 * q^43 + 16 * q^47 - 18 * q^49 + 2 * q^53 - 12 * q^59 + 6 * q^61 - 12 * q^65 - 4 * q^67 + 16 * q^77 - 20 * q^83 + 24 * q^91 + 8 * q^95 - 32 * q^97

Introduction

L-functions

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Groups

Database

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