LMFDB - Embedded newform 512.2.e.a.385.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [512,2,Mod(129,512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(512, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("512.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 512 = 2^{9} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	512.e (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.08834058349$
Analytic rank:	$1$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			385.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	512.385
Dual form			512.2.e.a.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 2.00000i) q^{3} +(-1.00000 - 1.00000i) q^{5} +4.00000i q^{7} -5.00000i q^{9} +O(q^{10})$	\(q+(-2.00000 + 2.00000i) q^{3} +(-1.00000 - 1.00000i) q^{5} +4.00000i q^{7} -5.00000i q^{9} +(-2.00000 - 2.00000i) q^{11} +(-3.00000 + 3.00000i) q^{13} +4.00000 q^{15} +(2.00000 - 2.00000i) q^{19} +(-8.00000 - 8.00000i) q^{21} -4.00000i q^{23} -3.00000i q^{25} +(4.00000 + 4.00000i) q^{27} +(3.00000 - 3.00000i) q^{29} -8.00000 q^{31} +8.00000 q^{33} +(4.00000 - 4.00000i) q^{35} +(1.00000 + 1.00000i) q^{37} -12.0000i q^{39} -8.00000i q^{41} +(2.00000 + 2.00000i) q^{43} +(-5.00000 + 5.00000i) q^{45} -8.00000 q^{47} -9.00000 q^{49} +(1.00000 + 1.00000i) q^{53} +4.00000i q^{55} +8.00000i q^{57} +(-6.00000 - 6.00000i) q^{59} +(-3.00000 + 3.00000i) q^{61} +20.0000 q^{63} +6.00000 q^{65} +(2.00000 - 2.00000i) q^{67} +(8.00000 + 8.00000i) q^{69} +12.0000i q^{71} +2.00000i q^{73} +(6.00000 + 6.00000i) q^{75} +(8.00000 - 8.00000i) q^{77} -1.00000 q^{81} +(-10.0000 + 10.0000i) q^{83} +12.0000i q^{87} +14.0000i q^{89} +(-12.0000 - 12.0000i) q^{91} +(16.0000 - 16.0000i) q^{93} -4.00000 q^{95} -16.0000 q^{97} +(-10.0000 + 10.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 2 q^{5}+O(q^{10}) $ 2 * q - 4 * q^3 - 2 * q^5	$ 2 q - 4 q^{3} - 2 q^{5} - 4 q^{11} - 6 q^{13} + 8 q^{15} + 4 q^{19} - 16 q^{21} + 8 q^{27} + 6 q^{29} - 16 q^{31} + 16 q^{33} + 8 q^{35} + 2 q^{37} + 4 q^{43} - 10 q^{45} - 16 q^{47} - 18 q^{49} + 2 q^{53} - 12 q^{59} - 6 q^{61} + 40 q^{63} + 12 q^{65} + 4 q^{67} + 16 q^{69} + 12 q^{75} + 16 q^{77} - 2 q^{81} - 20 q^{83} - 24 q^{91} + 32 q^{93} - 8 q^{95} - 32 q^{97} - 20 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 - 2 * q^5 - 4 * q^11 - 6 * q^13 + 8 * q^15 + 4 * q^19 - 16 * q^21 + 8 * q^27 + 6 * q^29 - 16 * q^31 + 16 * q^33 + 8 * q^35 + 2 * q^37 + 4 * q^43 - 10 * q^45 - 16 * q^47 - 18 * q^49 + 2 * q^53 - 12 * q^59 - 6 * q^61 + 40 * q^63 + 12 * q^65 + 4 * q^67 + 16 * q^69 + 12 * q^75 + 16 * q^77 - 2 * q^81 - 20 * q^83 - 24 * q^91 + 32 * q^93 - 8 * q^95 - 32 * q^97 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/512\mathbb{Z}\right)^\times$.

$n$	$5$	$511$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	−2.00000	+	2.00000i	−1.15470	+	1.15470i	−0.169102	+	0.985599i	$0.554087\pi$
$3$	−2.00000	+	2.00000i	−1.15470	+	1.15470i	−0.985599	+	0.169102i	$0.945913\pi$
$4$		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
$9$		−	5.00000i		−	1.66667i
$10$		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	−0.987797	−	0.155747i	$-0.950222\pi$
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	0.155747	+	0.987797i	$0.450222\pi$
$14$		0			0
$15$	4.00000			1.03280
$16$		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
$21$	−8.00000	−	8.00000i	−1.74574	−	1.74574i
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$		−	3.00000i		−	0.600000i
$26$		0			0
$27$	4.00000	+	4.00000i	0.769800	+	0.769800i
$28$		0			0
$29$	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
$33$	8.00000			1.39262
$34$		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
$39$		−	12.0000i		−	1.92154i
$40$		0			0
$41$		−	8.00000i		−	1.24939i	−0.780869	−	0.624695i	$-0.785223\pi$
$41$		−	8.00000i		−	1.24939i	0.780869	−	0.624695i	$-0.214777\pi$
$42$		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
$45$	−5.00000	+	5.00000i	−0.745356	+	0.745356i
$46$		0			0
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		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
$57$			8.00000i			1.05963i
$58$		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	−0.872581	−	0.488470i	$-0.837555\pi$
$61$	−3.00000	+	3.00000i	−0.384111	+	0.384111i	0.488470	+	0.872581i	$0.337555\pi$
$62$		0			0
$63$	20.0000			2.51976
$64$		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
$69$	8.00000	+	8.00000i	0.963087	+	0.963087i
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		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
$75$	6.00000	+	6.00000i	0.692820	+	0.692820i
$76$		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
$81$	−1.00000			−0.111111
$82$		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
$87$			12.0000i			1.28654i
$88$		0			0
$89$			14.0000i			1.48400i	0.670402	+	0.741999i	$0.266122\pi$
$89$			14.0000i			1.48400i	−0.670402	+	0.741999i	$0.733878\pi$
$90$		0			0
$91$	−12.0000	−	12.0000i	−1.25794	−	1.25794i
$92$		0			0
$93$	16.0000	−	16.0000i	1.65912	−	1.65912i
$94$		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
$99$	−10.0000	+	10.0000i	−1.00504	+	1.00504i
$100$		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
$105$			16.0000i			1.56144i
$106$		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	$0.592774\pi$
$109$	−13.0000	+	13.0000i	−1.24517	+	1.24517i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$	−4.00000			−0.379663
$112$		0			0
$113$	2.00000			0.188144			0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000			0.188144			0.0940721	+	0.995565i	$0.470012\pi$
$114$		0			0
$115$	−4.00000	+	4.00000i	−0.373002	+	0.373002i
$116$		0			0
$117$	15.0000	+	15.0000i	1.38675	+	1.38675i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	3.00000i		−	0.272727i
$122$		0			0
$123$	16.0000	+	16.0000i	1.44267	+	1.44267i
$124$		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
$129$	−8.00000			−0.704361
$130$		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
$135$		−	8.00000i		−	0.688530i
$136$		0			0
$137$			8.00000i			0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$			8.00000i			0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		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
$141$	16.0000	−	16.0000i	1.34744	−	1.34744i
$142$		0			0
$143$	12.0000			1.00349
$144$		0			0
$145$	−6.00000			−0.498273
$146$		0			0
$147$	18.0000	−	18.0000i	1.48461	−	1.48461i
$148$		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	−0.816612	−	0.577186i	$-0.804151\pi$
$157$	−3.00000	+	3.00000i	−0.239426	+	0.239426i	0.577186	+	0.816612i	$0.304151\pi$
$158$		0			0
$159$	−4.00000			−0.317221
$160$		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
$165$	−8.00000	−	8.00000i	−0.622799	−	0.622799i
$166$		0			0
$167$			4.00000i			0.309529i	0.987951	+	0.154765i	$0.0494619\pi$
$167$			4.00000i			0.309529i	−0.987951	+	0.154765i	$0.950538\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$	−10.0000	−	10.0000i	−0.764719	−	0.764719i
$172$		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
$177$	24.0000			1.80395
$178$		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	$0.0390812\pi$
$181$	15.0000	+	15.0000i	1.11494	+	1.11494i	0.122469	+	0.992472i	$0.460919\pi$
$182$		0			0
$183$		−	12.0000i		−	0.887066i
$184$		0			0
$185$		−	2.00000i		−	0.147043i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−16.0000	+	16.0000i	−1.16383	+	1.16383i
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		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
$195$	−12.0000	+	12.0000i	−0.859338	+	0.859338i
$196$		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
$201$			8.00000i			0.564276i
$202$		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
$207$	−20.0000			−1.39010
$208$		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
$213$	−24.0000	−	24.0000i	−1.64445	−	1.64445i
$214$		0			0
$215$		−	4.00000i		−	0.272798i
$216$		0			0
$217$		−	32.0000i		−	2.17230i
$218$		0			0
$219$	−4.00000	−	4.00000i	−0.270295	−	0.270295i
$220$		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
$225$	−15.0000			−1.00000
$226$		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	0.673293	−	0.739375i	$-0.264879\pi$
$229$	−1.00000	−	1.00000i	−0.0660819	−	0.0660819i	−0.739375	+	0.673293i	$0.764879\pi$
$230$		0			0
$231$			32.0000i			2.10545i
$232$		0			0
$233$			2.00000i			0.131024i	0.997852	+	0.0655122i	$0.0208681\pi$
$233$			2.00000i			0.131024i	−0.997852	+	0.0655122i	$0.979132\pi$
$234$		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	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	−10.0000	+	10.0000i	−0.641500	+	0.641500i
$244$		0			0
$245$	9.00000	+	9.00000i	0.574989	+	0.574989i
$246$		0			0
$247$			12.0000i			0.763542i
$248$		0			0
$249$		−	40.0000i		−	2.53490i
$250$		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	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$	−4.00000	+	4.00000i	−0.248548	+	0.248548i
$260$		0			0
$261$	−15.0000	−	15.0000i	−0.928477	−	0.928477i
$262$		0			0
$263$		−	20.0000i		−	1.23325i	−0.787256	−	0.616626i	$-0.788499\pi$
$263$		−	20.0000i		−	1.23325i	0.787256	−	0.616626i	$-0.211501\pi$
$264$		0			0
$265$		−	2.00000i		−	0.122859i
$266$		0			0
$267$	−28.0000	−	28.0000i	−1.71357	−	1.71357i
$268$		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
$273$	48.0000			2.90509
$274$		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
$279$			40.0000i			2.39474i
$280$		0			0
$281$			18.0000i			1.07379i	0.843649	+	0.536895i	$0.180403\pi$
$281$			18.0000i			1.07379i	−0.843649	+	0.536895i	$0.819597\pi$
$282$		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
$285$	8.00000	−	8.00000i	0.473879	−	0.473879i
$286$		0			0
$287$	32.0000			1.88890
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	32.0000	−	32.0000i	1.87587	−	1.87587i
$292$		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
$297$		−	16.0000i		−	0.928414i
$298$		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
$303$	−4.00000			−0.229794
$304$		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
$309$	8.00000	+	8.00000i	0.455104	+	0.455104i
$310$		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
$313$			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
$315$	−20.0000	−	20.0000i	−1.12687	−	1.12687i
$316$		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
$321$	8.00000			0.446516
$322$		0			0
$323$		0			0
$324$		0			0
$325$	9.00000	+	9.00000i	0.499230	+	0.499230i
$326$		0			0
$327$		−	52.0000i		−	2.87561i
$328$		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
$333$	5.00000	−	5.00000i	0.273998	−	0.273998i
$334$		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
$339$	−4.00000	+	4.00000i	−0.217250	+	0.217250i
$340$		0			0
$341$	16.0000	+	16.0000i	0.866449	+	0.866449i
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$		−	16.0000i		−	0.861411i
$346$		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	−0.963518	−	0.267644i	$-0.913755\pi$
$349$	−13.0000	+	13.0000i	−0.695874	+	0.695874i	0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$	−24.0000			−1.28103
$352$		0			0
$353$	14.0000			0.745145			0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000			0.745145			0.372572	+	0.928003i	$0.378476\pi$
$354$		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	$-0.823025\pi$
$359$		−	20.0000i		−	1.05556i	0.849381	−	0.527780i	$-0.176975\pi$
$360$		0			0
$361$			11.0000i			0.578947i
$362$		0			0
$363$	6.00000	+	6.00000i	0.314918	+	0.314918i
$364$		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
$369$	−40.0000			−2.08232
$370$		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
$375$		−	32.0000i		−	1.65247i
$376$		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
$381$	16.0000	−	16.0000i	0.819705	−	0.819705i
$382$		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
$387$	10.0000	−	10.0000i	0.508329	−	0.508329i
$388$		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
$393$			40.0000i			2.01773i
$394$		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
$399$	−32.0000			−1.60200
$400$		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
$405$	1.00000	+	1.00000i	0.0496904	+	0.0496904i
$406$		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
$411$	−16.0000	−	16.0000i	−0.789222	−	0.789222i
$412$		0			0
$413$	24.0000	−	24.0000i	1.18096	−	1.18096i
$414$		0			0
$415$	20.0000			0.981761
$416$		0			0
$417$	24.0000			1.17529
$418$		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	0.970829	−	0.239774i	$-0.0770732\pi$
$421$	15.0000	+	15.0000i	0.731055	+	0.731055i	−0.239774	+	0.970829i	$0.577073\pi$
$422$		0			0
$423$			40.0000i			1.94487i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−12.0000	−	12.0000i	−0.580721	−	0.580721i
$428$		0			0
$429$	−24.0000	+	24.0000i	−1.15873	+	1.15873i
$430$		0			0
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		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
$435$	12.0000	−	12.0000i	0.575356	−	0.575356i
$436$		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
$441$			45.0000i			2.14286i
$442$		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
$447$	4.00000			0.189194
$448$		0			0
$449$	−16.0000			−0.755087			−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000			−0.755087			−0.377543	+	0.925992i	$0.623231\pi$
$450$		0			0
$451$	−16.0000	+	16.0000i	−0.753411	+	0.753411i
$452$		0			0
$453$	40.0000	+	40.0000i	1.87936	+	1.87936i
$454$		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
$465$	−32.0000			−1.48396
$466$		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
$471$		−	12.0000i		−	0.552931i
$472$		0			0
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$	−6.00000	−	6.00000i	−0.275299	−	0.275299i
$476$		0			0
$477$	5.00000	−	5.00000i	0.228934	−	0.228934i
$478$		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
$481$	−6.00000			−0.273576
$482$		0			0
$483$	−32.0000	+	32.0000i	−1.45605	+	1.45605i
$484$		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
$489$			24.0000i			1.08532i
$490$		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
$495$	20.0000			0.898933
$496$		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
$501$	−8.00000	−	8.00000i	−0.357414	−	0.357414i
$502$		0			0
$503$			28.0000i			1.24846i	0.781241	+	0.624229i	$0.214587\pi$
$503$			28.0000i			1.24846i	−0.781241	+	0.624229i	$0.785413\pi$
$504$		0			0
$505$		−	2.00000i		−	0.0889988i
$506$		0			0
$507$	10.0000	+	10.0000i	0.444116	+	0.444116i
$508$		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
$513$	16.0000			0.706417
$514$		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
$519$			52.0000i			2.28255i
$520$		0			0
$521$		−	24.0000i		−	1.05146i	−0.850652	−	0.525730i	$-0.823792\pi$
$521$		−	24.0000i		−	1.05146i	0.850652	−	0.525730i	$-0.176208\pi$
$522$		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
$525$	−24.0000	+	24.0000i	−1.04745	+	1.04745i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	−30.0000	+	30.0000i	−1.30189	+	1.30189i
$532$		0			0
$533$	24.0000	+	24.0000i	1.03956	+	1.03956i
$534$		0			0
$535$			4.00000i			0.172935i
$536$		0			0
$537$			56.0000i			2.41658i
$538$		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
$543$	−60.0000			−2.57485
$544$		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
$549$	15.0000	+	15.0000i	0.640184	+	0.640184i
$550$		0			0
$551$		−	12.0000i		−	0.511217i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	4.00000	+	4.00000i	0.169791	+	0.169791i
$556$		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
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$			24.0000i			1.00613i	0.864248	+	0.503066i	$0.167795\pi$
$569$			24.0000i			1.00613i	−0.864248	+	0.503066i	$0.832205\pi$
$570$		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
$573$	48.0000	−	48.0000i	2.00523	−	2.00523i
$574$		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
$579$	32.0000	−	32.0000i	1.32987	−	1.32987i
$580$		0			0
$581$	−40.0000	−	40.0000i	−1.65948	−	1.65948i
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$		0			0
$585$		−	30.0000i		−	1.24035i
$586$		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
$591$	68.0000			2.79715
$592$		0			0
$593$	−14.0000			−0.574911			−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000			−0.574911			−0.287456	+	0.957794i	$0.592809\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−8.00000	−	8.00000i	−0.327418	−	0.327418i
$598$		0			0
$599$		−	28.0000i		−	1.14405i	−0.820237	−	0.572024i	$-0.806158\pi$
$599$		−	28.0000i		−	1.14405i	0.820237	−	0.572024i	$-0.193842\pi$
$600$		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
$603$	−10.0000	−	10.0000i	−0.407231	−	0.407231i
$604$		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
$609$	−48.0000			−1.94506
$610$		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
$615$		−	32.0000i		−	1.29036i
$616$		0			0
$617$		−	2.00000i		−	0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$		−	2.00000i		−	0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$		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
$621$	16.0000	−	16.0000i	0.642058	−	0.642058i
$622$		0			0
$623$	−56.0000			−2.24359
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	16.0000	−	16.0000i	0.638978	−	0.638978i
$628$		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
$633$		−	40.0000i		−	1.58986i
$634$		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
$639$	60.0000			2.37356
$640$		0			0
$641$	16.0000			0.631962			0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000			0.631962			0.315981	+	0.948766i	$0.397666\pi$
$642$		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
$645$	8.00000	+	8.00000i	0.315000	+	0.315000i
$646$		0			0
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$		0			0
$649$			24.0000i			0.942082i
$650$		0			0
$651$	64.0000	+	64.0000i	2.50836	+	2.50836i
$652$		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
$657$	10.0000			0.390137
$658$		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	0.294445	−	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
$669$	−48.0000	+	48.0000i	−1.85579	+	1.85579i
$670$		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
$675$	12.0000	−	12.0000i	0.461880	−	0.461880i
$676$		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
$681$		−	72.0000i		−	2.75905i
$682$		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
$687$	4.00000			0.152610
$688$		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
$693$	−40.0000	−	40.0000i	−1.51947	−	1.51947i
$694$		0			0
$695$			12.0000i			0.455186i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−4.00000	−	4.00000i	−0.151294	−	0.151294i
$700$		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
$705$	−32.0000			−1.20519
$706$		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
$717$	−32.0000	+	32.0000i	−1.19506	+	1.19506i
$718$		0			0
$719$	−48.0000			−1.79010			−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000			−1.79010			−0.895049	+	0.445968i	$0.852860\pi$
$720$		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
$729$		−	43.0000i		−	1.59259i
$730$		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
$735$	−36.0000			−1.32788
$736$		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
$741$	−24.0000	−	24.0000i	−0.881662	−	0.881662i
$742$		0			0
$743$		−	12.0000i		−	0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$		−	12.0000i		−	0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$		0			0
$745$			2.00000i			0.0732743i
$746$		0			0
$747$	50.0000	+	50.0000i	1.82940	+	1.82940i
$748$		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
$753$	56.0000			2.04075
$754$		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
$759$		−	32.0000i		−	1.16153i
$760$		0			0
$761$		−	24.0000i		−	0.869999i	−0.900431	−	0.435000i	$-0.856748\pi$
$761$		−	24.0000i		−	0.869999i	0.900431	−	0.435000i	$-0.143252\pi$
$762$		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
$771$	−36.0000	+	36.0000i	−1.29651	+	1.29651i
$772$		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
$777$		−	16.0000i		−	0.573997i
$778$		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
$783$	24.0000			0.857690
$784$		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
$789$	40.0000	+	40.0000i	1.42404	+	1.42404i
$790$		0			0
$791$			8.00000i			0.284447i
$792$		0			0
$793$		−	18.0000i		−	0.639199i
$794$		0			0
$795$	4.00000	+	4.00000i	0.141865	+	0.141865i
$796$		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
$801$	70.0000			2.47333
$802$		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
$807$			12.0000i			0.422420i
$808$		0			0
$809$		−	40.0000i		−	1.40633i	−0.711029	−	0.703163i	$-0.751771\pi$
$809$		−	40.0000i		−	1.40633i	0.711029	−	0.703163i	$-0.248229\pi$
$810$		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
$813$	16.0000	−	16.0000i	0.561144	−	0.561144i
$814$		0			0
$815$	−12.0000			−0.420342
$816$		0			0
$817$	8.00000			0.279885
$818$		0			0
$819$	−60.0000	+	60.0000i	−2.09657	+	2.09657i
$820$		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
$825$		−	24.0000i		−	0.835573i
$826$		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	−0.444346	−	0.895855i	$-0.646564\pi$
$829$	13.0000	−	13.0000i	0.451509	−	0.451509i	0.895855	+	0.444346i	$0.146564\pi$
$830$		0			0
$831$	60.0000			2.08138
$832$		0			0
$833$		0			0
$834$		0			0
$835$	4.00000	−	4.00000i	0.138426	−	0.138426i
$836$		0			0
$837$	−32.0000	−	32.0000i	−1.10608	−	1.10608i
$838$		0			0
$839$			36.0000i			1.24286i	0.783470	+	0.621429i	$0.213448\pi$
$839$			36.0000i			1.24286i	−0.783470	+	0.621429i	$0.786552\pi$
$840$		0			0
$841$			11.0000i			0.379310i
$842$		0			0
$843$	−36.0000	−	36.0000i	−1.23991	−	1.23991i
$844$		0			0
$845$	−5.00000	+	5.00000i	−0.172005	+	0.172005i
$846$		0			0
$847$	12.0000			0.412325
$848$		0			0
$849$	−56.0000			−1.92192
$850$		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
$855$			20.0000i			0.683986i
$856$		0			0
$857$			24.0000i			0.819824i	0.912125	+	0.409912i	$0.134441\pi$
$857$			24.0000i			0.819824i	−0.912125	+	0.409912i	$0.865559\pi$
$858$		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
$861$	−64.0000	+	64.0000i	−2.18111	+	2.18111i
$862$		0			0
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$	−26.0000			−0.884027
$866$		0			0
$867$	34.0000	−	34.0000i	1.15470	−	1.15470i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			12.0000i			0.406604i
$872$		0			0
$873$			80.0000i			2.70759i
$874$		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
$879$	68.0000			2.29358
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		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
$885$	−24.0000	−	24.0000i	−0.806751	−	0.806751i
$886$		0			0
$887$			36.0000i			1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$			36.0000i			1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$		0			0
$889$		−	32.0000i		−	1.07325i
$890$		0			0
$891$	2.00000	+	2.00000i	0.0670025	+	0.0670025i
$892$		0			0
$893$	−16.0000	+	16.0000i	−0.535420	+	0.535420i
$894$		0			0
$895$	−28.0000			−0.935937
$896$		0			0
$897$	−48.0000			−1.60267
$898$		0			0
$899$	−24.0000	+	24.0000i	−0.800445	+	0.800445i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		−	32.0000i		−	1.06489i
$904$		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
$909$	5.00000	−	5.00000i	0.165840	−	0.165840i
$910$		0			0
$911$	56.0000			1.85536			0.927681	−	0.373373i	$-0.121799\pi$
$911$	56.0000			1.85536			0.927681	+	0.373373i	$0.121799\pi$
$912$		0			0
$913$	40.0000			1.32381
$914$		0			0
$915$	−12.0000	+	12.0000i	−0.396708	+	0.396708i
$916$		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
$921$		−	24.0000i		−	0.790827i
$922$		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
$927$	−20.0000			−0.656886
$928$		0			0
$929$	−48.0000			−1.57483			−0.787414	−	0.616424i	$-0.788581\pi$
$929$	−48.0000			−1.57483			−0.787414	+	0.616424i	$0.788581\pi$
$930$		0			0
$931$	−18.0000	+	18.0000i	−0.589926	+	0.589926i
$932$		0			0
$933$	−8.00000	−	8.00000i	−0.261908	−	0.261908i
$934$		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
$939$	−16.0000	−	16.0000i	−0.522140	−	0.522140i
$940$		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
$945$	32.0000			1.04096
$946$		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
$951$		−	52.0000i		−	1.68622i
$952$		0			0
$953$			24.0000i			0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$			24.0000i			0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$		0			0
$955$	24.0000	+	24.0000i	0.776622	+	0.776622i
$956$		0			0
$957$	24.0000	−	24.0000i	0.775810	−	0.775810i
$958$		0			0
$959$	−32.0000			−1.03333
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	−10.0000	+	10.0000i	−0.322245	+	0.322245i
$964$		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
$975$	−36.0000			−1.15292
$976$		0			0
$977$	32.0000			1.02377			0.511885	−	0.859054i	$-0.328947\pi$
$977$	32.0000			1.02377			0.511885	+	0.859054i	$0.328947\pi$
$978$		0			0
$979$	28.0000	−	28.0000i	0.894884	−	0.894884i
$980$		0			0
$981$	65.0000	+	65.0000i	2.07529	+	2.07529i
$982$		0			0
$983$		−	28.0000i		−	0.893061i	−0.894768	−	0.446531i	$-0.852659\pi$
$983$		−	28.0000i		−	0.893061i	0.894768	−	0.446531i	$-0.147341\pi$
$984$		0			0
$985$			34.0000i			1.08333i
$986$		0			0
$987$	64.0000	+	64.0000i	2.03714	+	2.03714i
$988$		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
$993$	−88.0000			−2.79260
$994$		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
$999$			8.00000i			0.253109i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.a.129.1	✓	2	16.13	even	4	inner
512.2.e.a.385.1	yes	2	1.1	even	1	trivial
512.2.e.b.129.1	yes	2	16.11	odd	4
512.2.e.b.385.1	yes	2	8.3	odd	2
512.2.e.g.129.1	yes	2	16.3	odd	4
512.2.e.g.385.1	yes	2	4.3	odd	2
512.2.e.h.129.1	yes	2	16.5	even	4
512.2.e.h.385.1	yes	2	8.5	even	2
1024.2.a.a.1.1		2	32.19	odd	8
1024.2.a.a.1.2		2	32.3	odd	8
1024.2.a.f.1.1		2	32.29	even	8
1024.2.a.f.1.2		2	32.13	even	8
1024.2.b.a.513.1		2	32.5	even	8
1024.2.b.a.513.2		2	32.21	even	8
1024.2.b.f.513.1		2	32.11	odd	8
1024.2.b.f.513.2		2	32.27	odd	8
4608.2.k.f.1153.1		2	48.5	odd	4
4608.2.k.f.3457.1		2	24.5	odd	2
4608.2.k.j.1153.1		2	48.11	even	4
4608.2.k.j.3457.1		2	24.11	even	2
4608.2.k.o.1153.1		2	48.35	even	4
4608.2.k.o.3457.1		2	12.11	even	2
4608.2.k.s.1153.1		2	48.29	odd	4
4608.2.k.s.3457.1		2	3.2	odd	2
9216.2.a.b.1.1		2	96.35	even	8
9216.2.a.b.1.2		2	96.83	even	8
9216.2.a.u.1.1		2	96.29	odd	8
9216.2.a.u.1.2		2	96.77	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 2.00000i) q^{3} +(-1.00000 - 1.00000i) q^{5} +4.00000i q^{7} -5.00000i q^{9} +O(q^{10})\)	\(q+(-2.00000 + 2.00000i) q^{3} +(-1.00000 - 1.00000i) q^{5} +4.00000i q^{7} -5.00000i q^{9} +(-2.00000 - 2.00000i) q^{11} +(-3.00000 + 3.00000i) q^{13} +4.00000 q^{15} +(2.00000 - 2.00000i) q^{19} +(-8.00000 - 8.00000i) q^{21} -4.00000i q^{23} -3.00000i q^{25} +(4.00000 + 4.00000i) q^{27} +(3.00000 - 3.00000i) q^{29} -8.00000 q^{31} +8.00000 q^{33} +(4.00000 - 4.00000i) q^{35} +(1.00000 + 1.00000i) q^{37} -12.0000i q^{39} -8.00000i q^{41} +(2.00000 + 2.00000i) q^{43} +(-5.00000 + 5.00000i) q^{45} -8.00000 q^{47} -9.00000 q^{49} +(1.00000 + 1.00000i) q^{53} +4.00000i q^{55} +8.00000i q^{57} +(-6.00000 - 6.00000i) q^{59} +(-3.00000 + 3.00000i) q^{61} +20.0000 q^{63} +6.00000 q^{65} +(2.00000 - 2.00000i) q^{67} +(8.00000 + 8.00000i) q^{69} +12.0000i q^{71} +2.00000i q^{73} +(6.00000 + 6.00000i) q^{75} +(8.00000 - 8.00000i) q^{77} -1.00000 q^{81} +(-10.0000 + 10.0000i) q^{83} +12.0000i q^{87} +14.0000i q^{89} +(-12.0000 - 12.0000i) q^{91} +(16.0000 - 16.0000i) q^{93} -4.00000 q^{95} -16.0000 q^{97} +(-10.0000 + 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 2 q^{5}+O(q^{10}) \) 2 * q - 4 * q^3 - 2 * q^5	\( 2 q - 4 q^{3} - 2 q^{5} - 4 q^{11} - 6 q^{13} + 8 q^{15} + 4 q^{19} - 16 q^{21} + 8 q^{27} + 6 q^{29} - 16 q^{31} + 16 q^{33} + 8 q^{35} + 2 q^{37} + 4 q^{43} - 10 q^{45} - 16 q^{47} - 18 q^{49} + 2 q^{53} - 12 q^{59} - 6 q^{61} + 40 q^{63} + 12 q^{65} + 4 q^{67} + 16 q^{69} + 12 q^{75} + 16 q^{77} - 2 q^{81} - 20 q^{83} - 24 q^{91} + 32 q^{93} - 8 q^{95} - 32 q^{97} - 20 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 - 2 * q^5 - 4 * q^11 - 6 * q^13 + 8 * q^15 + 4 * q^19 - 16 * q^21 + 8 * q^27 + 6 * q^29 - 16 * q^31 + 16 * q^33 + 8 * q^35 + 2 * q^37 + 4 * q^43 - 10 * q^45 - 16 * q^47 - 18 * q^49 + 2 * q^53 - 12 * q^59 - 6 * q^61 + 40 * q^63 + 12 * q^65 + 4 * q^67 + 16 * q^69 + 12 * q^75 + 16 * q^77 - 2 * q^81 - 20 * q^83 - 24 * q^91 + 32 * q^93 - 8 * q^95 - 32 * q^97 - 20 * q^99

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