LMFDB - Embedded newform 4608.2.k.s.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.s.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	0.894427	−	0.447214i	$-0.147584\pi$
$5$	1.00000	+	1.00000i	0.447214	+	0.447214i	−0.447214	+	0.894427i	$0.647584\pi$
$6$		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	0.941113	−	0.338091i	$-0.109781\pi$
$11$	2.00000	+	2.00000i	0.603023	+	0.603023i	−0.338091	+	0.941113i	$0.609781\pi$
$12$		0			0
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	−0.987797	−	0.155747i	$-0.950222\pi$
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	0.155747	+	0.987797i	$0.450222\pi$
$14$		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	−0.439440	−	0.898272i	$-0.644823\pi$
$19$	2.00000	−	2.00000i	0.458831	−	0.458831i	0.898272	+	0.439440i	$0.144823\pi$
$20$		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	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−3.00000	+	3.00000i	−0.557086	+	0.557086i	0.371391	+	0.928477i	$0.378881\pi$
$30$		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	0.784512	−	0.620113i	$-0.212913\pi$
$37$	1.00000	+	1.00000i	0.164399	+	0.164399i	−0.620113	+	0.784512i	$0.712913\pi$
$38$		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	0.842965	−	0.537968i	$-0.180808\pi$
$43$	2.00000	+	2.00000i	0.304997	+	0.304997i	−0.537968	+	0.842965i	$0.680808\pi$
$44$		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	0.635083	−	0.772444i	$-0.280966\pi$
$53$	−1.00000	−	1.00000i	−0.137361	−	0.137361i	−0.772444	+	0.635083i	$0.780966\pi$
$54$		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	0.980022	−	0.198889i	$-0.0637332\pi$
$59$	6.00000	+	6.00000i	0.781133	+	0.781133i	−0.198889	+	0.980022i	$0.563733\pi$
$60$		0			0
$61$	−3.00000	+	3.00000i	−0.384111	+	0.384111i	−0.872581	−	0.488470i	$-0.837555\pi$
$61$	−3.00000	+	3.00000i	−0.384111	+	0.384111i	0.488470	+	0.872581i	$0.337555\pi$
$62$		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	−0.574304	−	0.818642i	$-0.694727\pi$
$67$	2.00000	−	2.00000i	0.244339	−	0.244339i	0.818642	+	0.574304i	$0.194727\pi$
$68$		0			0
$69$		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	$-0.467170\pi$
$83$	10.0000	−	10.0000i	1.09764	−	1.09764i	0.994686	−	0.102957i	$-0.0328303\pi$
$84$		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	0.655602	−	0.755106i	$-0.272415\pi$
$101$	−1.00000	−	1.00000i	−0.0995037	−	0.0995037i	−0.755106	+	0.655602i	$0.772415\pi$
$102$		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	0.797141	−	0.603793i	$-0.206345\pi$
$107$	2.00000	+	2.00000i	0.193347	+	0.193347i	−0.603793	+	0.797141i	$0.706345\pi$
$108$		0			0
$109$	−13.0000	+	13.0000i	−1.24517	+	1.24517i	−0.287348	+	0.957826i	$0.592774\pi$
$109$	−13.0000	+	13.0000i	−1.24517	+	1.24517i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		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	−0.992874	−	0.119170i	$-0.961977\pi$
$131$	−10.0000	+	10.0000i	−0.873704	+	0.873704i	0.119170	+	0.992874i	$0.461977\pi$
$132$		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	0.405279	−	0.914193i	$-0.367174\pi$
$139$	−6.00000	−	6.00000i	−0.508913	−	0.508913i	−0.914193	+	0.405279i	$0.867174\pi$
$140$		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	0.746881	−	0.664958i	$-0.231550\pi$
$149$	1.00000	+	1.00000i	0.0819232	+	0.0819232i	−0.664958	+	0.746881i	$0.731550\pi$
$150$		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	−0.816612	−	0.577186i	$-0.804151\pi$
$157$	−3.00000	+	3.00000i	−0.239426	+	0.239426i	0.577186	+	0.816612i	$0.304151\pi$
$158$		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	−0.431944	−	0.901900i	$-0.642172\pi$
$163$	6.00000	−	6.00000i	0.469956	−	0.469956i	0.901900	+	0.431944i	$0.142172\pi$
$164$		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	−0.999933	−	0.0115615i	$-0.996320\pi$
$173$	−13.0000	+	13.0000i	−0.988372	+	0.988372i	0.0115615	+	0.999933i	$0.496320\pi$
$174$		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	$0.515138\pi$
$179$	−14.0000	+	14.0000i	−1.04641	+	1.04641i	−0.998869	+	0.0475398i	$0.984862\pi$
$180$		0			0
$181$	15.0000	+	15.0000i	1.11494	+	1.11494i	0.992472	+	0.122469i	$0.0390812\pi$
$181$	15.0000	+	15.0000i	1.11494	+	1.11494i	0.122469	+	0.992472i	$0.460919\pi$
$182$		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	$0.0773335\pi$
$197$	17.0000	+	17.0000i	1.21120	+	1.21120i	0.240567	+	0.970632i	$0.422666\pi$
$198$		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	−0.961884	−	0.273456i	$-0.911833\pi$
$211$	−10.0000	+	10.0000i	−0.688428	+	0.688428i	0.273456	+	0.961884i	$0.411833\pi$
$212$		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	$-0.429730\pi$
$227$	18.0000	−	18.0000i	1.19470	−	1.19470i	0.975731	−	0.218970i	$-0.0702698\pi$
$228$		0			0
$229$	−1.00000	−	1.00000i	−0.0660819	−	0.0660819i	0.673293	−	0.739375i	$-0.264879\pi$
$229$	−1.00000	−	1.00000i	−0.0660819	−	0.0660819i	−0.739375	+	0.673293i	$0.764879\pi$
$230$		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	0.993906	−	0.110234i	$-0.0351599\pi$
$251$	14.0000	+	14.0000i	0.883672	+	0.883672i	−0.110234	+	0.993906i	$0.535160\pi$
$252$		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	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−3.00000	+	3.00000i	−0.182913	+	0.182913i	0.609711	+	0.792624i	$0.291286\pi$
$270$		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	0.0942828	−	0.995545i	$-0.469944\pi$
$277$	−15.0000	−	15.0000i	−0.901263	−	0.901263i	−0.995545	+	0.0942828i	$0.969944\pi$
$278$		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	0.987819	−	0.155606i	$-0.0497329\pi$
$283$	14.0000	+	14.0000i	0.832214	+	0.832214i	−0.155606	+	0.987819i	$0.549733\pi$
$284$		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	0.999977	−	0.00682610i	$-0.00217283\pi$
$293$	17.0000	+	17.0000i	0.993151	+	0.993151i	−0.00682610	+	0.999977i	$0.502173\pi$
$294$		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	−0.857283	−	0.514845i	$-0.827849\pi$
$307$	−6.00000	+	6.00000i	−0.342438	+	0.342438i	0.514845	+	0.857283i	$0.327849\pi$
$308$		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	−0.240497	−	0.970650i	$-0.577310\pi$
$317$	13.0000	−	13.0000i	0.730153	−	0.730153i	0.970650	+	0.240497i	$0.0773105\pi$
$318$		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	$0.0764763\pi$
$331$	22.0000	+	22.0000i	1.20923	+	1.20923i	0.237953	+	0.971277i	$0.423524\pi$
$332$		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	0.651383	−	0.758749i	$-0.274189\pi$
$347$	−2.00000	−	2.00000i	−0.107366	−	0.107366i	−0.758749	+	0.651383i	$0.774189\pi$
$348$		0			0
$349$	−13.0000	+	13.0000i	−0.695874	+	0.695874i	−0.963518	−	0.267644i	$-0.913755\pi$
$349$	−13.0000	+	13.0000i	−0.695874	+	0.695874i	0.267644	+	0.963518i	$0.413755\pi$
$350$		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	0.202593	−	0.979263i	$-0.435063\pi$
$373$	−15.0000	−	15.0000i	−0.776671	−	0.776671i	−0.979263	+	0.202593i	$0.935063\pi$
$374$		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	0.997350	−	0.0727522i	$-0.0231782\pi$
$379$	18.0000	+	18.0000i	0.924598	+	0.924598i	−0.0727522	+	0.997350i	$0.523178\pi$
$380$		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	0.681301	−	0.732003i	$-0.261414\pi$
$389$	−1.00000	−	1.00000i	−0.0507020	−	0.0507020i	−0.732003	+	0.681301i	$0.761414\pi$
$390$		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	−0.627805	−	0.778371i	$-0.716046\pi$
$397$	3.00000	−	3.00000i	0.150566	−	0.150566i	0.778371	+	0.627805i	$0.216046\pi$
$398$		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	−0.545192	−	0.838311i	$-0.683543\pi$
$419$	6.00000	−	6.00000i	0.293119	−	0.293119i	0.838311	+	0.545192i	$0.183543\pi$
$420$		0			0
$421$	15.0000	+	15.0000i	0.731055	+	0.731055i	0.970829	−	0.239774i	$-0.0770732\pi$
$421$	15.0000	+	15.0000i	0.731055	+	0.731055i	−0.239774	+	0.970829i	$0.577073\pi$
$422$		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	0.550058	−	0.835127i	$-0.314606\pi$
$443$	−6.00000	−	6.00000i	−0.285069	−	0.285069i	−0.835127	+	0.550058i	$0.814606\pi$
$444$		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	−0.994030	−	0.109111i	$-0.965200\pi$
$461$	−19.0000	+	19.0000i	−0.884918	+	0.884918i	0.109111	+	0.994030i	$0.465200\pi$
$462$		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	−0.751865	−	0.659317i	$-0.770846\pi$
$467$	−2.00000	+	2.00000i	−0.0925490	+	0.0925490i	0.659317	+	0.751865i	$0.270846\pi$
$468$		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	0.948522	−	0.316711i	$-0.102579\pi$
$491$	14.0000	+	14.0000i	0.631811	+	0.631811i	−0.316711	+	0.948522i	$0.602579\pi$
$492$		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	−0.750454	−	0.660922i	$-0.770165\pi$
$499$	−2.00000	+	2.00000i	−0.0895323	+	0.0895323i	0.660922	+	0.750454i	$0.270165\pi$
$500$		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	−0.146979	−	0.989140i	$-0.546955\pi$
$509$	19.0000	−	19.0000i	0.842160	−	0.842160i	0.989140	+	0.146979i	$0.0469551\pi$
$510$		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	0.193930	−	0.981015i	$-0.437876\pi$
$523$	−18.0000	−	18.0000i	−0.787085	−	0.787085i	−0.981015	+	0.193930i	$0.937876\pi$
$524$		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	−0.768650	−	0.639670i	$-0.779071\pi$
$541$	−3.00000	+	3.00000i	−0.128980	+	0.128980i	0.639670	+	0.768650i	$0.279071\pi$
$542$		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	$0.611642\pi$
$547$	−30.0000	+	30.0000i	−1.28271	+	1.28271i	−0.939121	+	0.343586i	$0.888358\pi$
$548$		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	−0.926680	−	0.375852i	$-0.877350\pi$
$557$	−13.0000	+	13.0000i	−0.550828	+	0.550828i	0.375852	+	0.926680i	$0.377350\pi$
$558$		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	−0.747995	−	0.663705i	$-0.768983\pi$
$563$	−2.00000	+	2.00000i	−0.0842900	+	0.0842900i	0.663705	+	0.747995i	$0.268983\pi$
$564$		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	0.821418	−	0.570326i	$-0.193183\pi$
$571$	6.00000	+	6.00000i	0.251092	+	0.251092i	−0.570326	+	0.821418i	$0.693183\pi$
$572$		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	0.882693	−	0.469949i	$-0.155728\pi$
$587$	10.0000	+	10.0000i	0.412744	+	0.412744i	−0.469949	+	0.882693i	$0.655728\pi$
$588$		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	0.274861	−	0.961484i	$-0.411368\pi$
$613$	−17.0000	−	17.0000i	−0.686624	−	0.686624i	−0.961484	+	0.274861i	$0.911368\pi$
$614$		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	0.245831	−	0.969313i	$-0.420939\pi$
$619$	−18.0000	−	18.0000i	−0.723481	−	0.723481i	−0.969313	+	0.245831i	$0.920939\pi$
$620$		0			0
$621$		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	−0.256653	−	0.966504i	$-0.582620\pi$
$643$	18.0000	−	18.0000i	0.709851	−	0.709851i	0.966504	+	0.256653i	$0.0826197\pi$
$644$		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	$-0.453521\pi$
$653$	29.0000	−	29.0000i	1.13486	−	1.13486i	0.989358	−	0.145499i	$-0.0464789\pi$
$654$		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	−0.990983	−	0.133985i	$-0.957223\pi$
$659$	−22.0000	+	22.0000i	−0.856998	+	0.856998i	0.133985	+	0.990983i	$0.457223\pi$
$660$		0			0
$661$	−17.0000	−	17.0000i	−0.661223	−	0.661223i	0.294445	−	0.955668i	$-0.404865\pi$
$661$	−17.0000	−	17.0000i	−0.661223	−	0.661223i	−0.955668	+	0.294445i	$0.904865\pi$
$662$		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	$-0.895872\pi$
$677$	−33.0000	−	33.0000i	−1.26829	−	1.26829i	−0.321324	−	0.946969i	$-0.604128\pi$
$678$		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	0.812519	−	0.582935i	$-0.198096\pi$
$683$	6.00000	+	6.00000i	0.229584	+	0.229584i	−0.582935	+	0.812519i	$0.698096\pi$
$684$		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	−0.583711	−	0.811962i	$-0.698400\pi$
$691$	6.00000	−	6.00000i	0.228251	−	0.228251i	0.811962	+	0.583711i	$0.198400\pi$
$692$		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	−0.250497	−	0.968117i	$-0.580594\pi$
$701$	19.0000	−	19.0000i	0.717620	−	0.717620i	0.968117	+	0.250497i	$0.0805941\pi$
$702$		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	0.366917	−	0.930254i	$-0.380413\pi$
$709$	−15.0000	−	15.0000i	−0.563337	−	0.563337i	−0.930254	+	0.366917i	$0.880413\pi$
$710$		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	−0.263012	−	0.964793i	$-0.584716\pi$
$733$	19.0000	−	19.0000i	0.701781	−	0.701781i	0.964793	+	0.263012i	$0.0847158\pi$
$734$		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	−0.588085	−	0.808799i	$-0.700118\pi$
$739$	6.00000	−	6.00000i	0.220714	−	0.220714i	0.808799	+	0.588085i	$0.200118\pi$
$740$		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	0.688700	−	0.725046i	$-0.258182\pi$
$757$	−1.00000	−	1.00000i	−0.0363456	−	0.0363456i	−0.725046	+	0.688700i	$0.758182\pi$
$758$		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	0.923386	−	0.383873i	$-0.125410\pi$
$773$	15.0000	+	15.0000i	0.539513	+	0.539513i	−0.383873	+	0.923386i	$0.625410\pi$
$774$		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	−0.950956	−	0.309326i	$-0.899897\pi$
$787$	−18.0000	+	18.0000i	−0.641631	+	0.641631i	0.309326	+	0.950956i	$0.399897\pi$
$788$		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	−0.285395	−	0.958410i	$-0.592125\pi$
$797$	19.0000	−	19.0000i	0.673015	−	0.673015i	0.958410	+	0.285395i	$0.0921249\pi$
$798$		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	0.978547	−	0.206023i	$-0.0660521\pi$
$811$	22.0000	+	22.0000i	0.772524	+	0.772524i	−0.206023	+	0.978547i	$0.566052\pi$
$812$		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	0.918628	−	0.395124i	$-0.129298\pi$
$821$	15.0000	+	15.0000i	0.523504	+	0.523504i	−0.395124	+	0.918628i	$0.629298\pi$
$822$		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	0.420476	−	0.907304i	$-0.361863\pi$
$827$	−14.0000	−	14.0000i	−0.486828	−	0.486828i	−0.907304	+	0.420476i	$0.861863\pi$
$828$		0			0
$829$	13.0000	−	13.0000i	0.451509	−	0.451509i	−0.444346	−	0.895855i	$-0.646564\pi$
$829$	13.0000	−	13.0000i	0.451509	−	0.451509i	0.895855	+	0.444346i	$0.146564\pi$
$830$		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	0.724019	−	0.689780i	$-0.242293\pi$
$853$	1.00000	+	1.00000i	0.0342393	+	0.0342393i	−0.689780	+	0.724019i	$0.742293\pi$
$854$		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	0.944025	−	0.329873i	$-0.107006\pi$
$859$	18.0000	+	18.0000i	0.614152	+	0.614152i	−0.329873	+	0.944025i	$0.607006\pi$
$860$		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	−0.309360	−	0.950945i	$-0.600115\pi$
$877$	19.0000	−	19.0000i	0.641584	−	0.641584i	0.950945	+	0.309360i	$0.100115\pi$
$878$		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	$-0.449974\pi$
$883$	34.0000	−	34.0000i	1.14419	−	1.14419i	0.987675	−	0.156516i	$-0.0500262\pi$
$884$		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	0.853363	−	0.521318i	$-0.174559\pi$
$907$	10.0000	+	10.0000i	0.332045	+	0.332045i	−0.521318	+	0.853363i	$0.674559\pi$
$908$		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	−0.462718	−	0.886506i	$-0.653126\pi$
$941$	13.0000	−	13.0000i	0.423788	−	0.423788i	0.886506	+	0.462718i	$0.153126\pi$
$942$		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	−0.850665	−	0.525708i	$-0.823800\pi$
$947$	−10.0000	+	10.0000i	−0.324956	+	0.324956i	0.525708	+	0.850665i	$0.323800\pi$
$948$		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	0.895116	−	0.445834i	$-0.147093\pi$
$971$	14.0000	+	14.0000i	0.449281	+	0.449281i	−0.445834	+	0.895116i	$0.647093\pi$
$972$		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	0.428491	−	0.903546i	$-0.359045\pi$
$997$	−15.0000	−	15.0000i	−0.475055	−	0.475055i	−0.903546	+	0.428491i	$0.859045\pi$
$998$		0			0
$999$		0			0

Twists

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

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

Modular forms

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Database

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