LMFDB - Embedded newform 512.2.e.c.385.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [512,2,Mod(129,512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("512.129"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [2,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.00000 - 3.00000i) q^{5} +3.00000i q^{9} +(-1.00000 + 1.00000i) q^{13} -8.00000 q^{17} +13.0000i q^{25} +(-7.00000 + 7.00000i) q^{29} +(-5.00000 - 5.00000i) q^{37} -8.00000i q^{41} +(9.00000 - 9.00000i) q^{45} +7.00000 q^{49} +(-5.00000 - 5.00000i) q^{53} +(-1.00000 + 1.00000i) q^{61} +6.00000 q^{65} -6.00000i q^{73} -9.00000 q^{81} +(24.0000 + 24.0000i) q^{85} -10.0000i q^{89} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} - 2 q^{13} - 16 q^{17} - 14 q^{29} - 10 q^{37} + 18 q^{45} + 14 q^{49} - 10 q^{53} - 2 q^{61} + 12 q^{65} - 18 q^{81} + 48 q^{85} + 16 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 - 2 * q^13 - 16 * q^17 - 14 * q^29 - 10 * q^37 + 18 * q^45 + 14 * q^49 - 10 * q^53 - 2 * q^61 + 12 * q^65 - 18 * q^81 + 48 * q^85 + 16 * q^97

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
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		0			0
$5$	−3.00000	−	3.00000i	−1.34164	−	1.34164i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−3.00000	−	3.00000i	−1.34164	−	1.34164i	−0.447214	−	0.894427i	$-0.647584\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$	−1.00000	+	1.00000i	−0.277350	+	0.277350i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−1.00000	+	1.00000i	−0.277350	+	0.277350i	0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−8.00000			−1.94029			−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−8.00000			−1.94029			−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$			13.0000i			2.60000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.00000	+	7.00000i	−1.29987	+	1.29987i	−0.371391	+	0.928477i	$0.621119\pi$
$29$	−7.00000	+	7.00000i	−1.29987	+	1.29987i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		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
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	9.00000	−	9.00000i	1.34164	−	1.34164i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.00000	−	5.00000i	−0.686803	−	0.686803i	0.274721	−	0.961524i	$-0.411414\pi$
$53$	−5.00000	−	5.00000i	−0.686803	−	0.686803i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	−1.00000	+	1.00000i	−0.128037	+	0.128037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−1.00000	+	1.00000i	−0.128037	+	0.128037i	0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000			0.744208
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	24.0000	+	24.0000i	2.60317	+	2.60317i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	10.0000i		−	1.06000i	−0.847998	−	0.529999i	$-0.822192\pi$
$89$		−	10.0000i		−	1.06000i	0.847998	−	0.529999i	$-0.177808\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	8.00000			0.812277			0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000			0.812277			0.406138	+	0.913812i	$0.366875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.0000	+	11.0000i	1.09454	+	1.09454i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$	11.0000	+	11.0000i	1.09454	+	1.09454i	0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$	−7.00000	+	7.00000i	−0.670478	+	0.670478i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−7.00000	+	7.00000i	−0.670478	+	0.670478i	0.287348	+	0.957826i	$0.407226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.00000	−	3.00000i	−0.277350	−	0.277350i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	11.0000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000	−	24.0000i	2.14663	−	2.14663i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$		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
$139$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$139$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	42.0000			3.48791
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	−	3.00000i	−0.245770	−	0.245770i	0.573462	−	0.819232i	$-0.305600\pi$
$149$	−3.00000	−	3.00000i	−0.245770	−	0.245770i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		−	24.0000i		−	1.94029i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−17.0000	+	17.0000i	−1.35675	+	1.35675i	−0.478852	+	0.877896i	$0.658947\pi$
$157$	−17.0000	+	17.0000i	−1.35675	+	1.35675i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$			11.0000i			0.846154i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	15.0000	−	15.0000i	1.14043	−	1.14043i	0.152057	−	0.988372i	$-0.451410\pi$
$173$	15.0000	−	15.0000i	1.14043	−	1.14043i	0.988372	−	0.152057i	$-0.0485898\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$179$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$180$		0			0
$181$	−19.0000	−	19.0000i	−1.41226	−	1.41226i	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−19.0000	−	19.0000i	−1.41226	−	1.41226i	−0.668965	−	0.743294i	$-0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			30.0000i			2.20564i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−24.0000			−1.72756			−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000			−1.72756			−0.863779	+	0.503871i	$0.831909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.0000	+	13.0000i	0.926212	+	0.926212i	0.997459	−	0.0712470i	$-0.0226979\pi$
$197$	13.0000	+	13.0000i	0.926212	+	0.926212i	−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−24.0000	+	24.0000i	−1.67623	+	1.67623i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$211$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.00000	−	8.00000i	0.538138	−	0.538138i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−39.0000			−2.60000
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$	13.0000	+	13.0000i	0.859064	+	0.859064i	0.991228	−	0.132164i	$-0.0421925\pi$
$229$	13.0000	+	13.0000i	0.859064	+	0.859064i	−0.132164	+	0.991228i	$0.542192\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			26.0000i			1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$			26.0000i			1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−8.00000			−0.515325			−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000			−0.515325			−0.257663	+	0.966235i	$0.582952\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−21.0000	−	21.0000i	−1.34164	−	1.34164i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.00000			0.124757			0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000			0.124757			0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−21.0000	−	21.0000i	−1.29987	−	1.29987i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$			30.0000i			1.84289i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−23.0000	+	23.0000i	−1.40233	+	1.40233i	−0.609711	+	0.792624i	$0.708714\pi$
$269$	−23.0000	+	23.0000i	−1.40233	+	1.40233i	−0.792624	+	0.609711i	$0.791286\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.00000	−	5.00000i	−0.300421	−	0.300421i	0.540758	−	0.841178i	$-0.318138\pi$
$277$	−5.00000	−	5.00000i	−0.300421	−	0.300421i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	47.0000			2.76471
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−19.0000	−	19.0000i	−1.10999	−	1.10999i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	−19.0000	−	19.0000i	−1.10999	−	1.10999i	−0.116841	−	0.993151i	$-0.537277\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.00000			0.343559
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		−	24.0000i		−	1.35656i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$		−	24.0000i		−	1.35656i	0.734803	−	0.678280i	$-0.237274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	25.0000	−	25.0000i	1.40414	−	1.40414i	0.617822	−	0.786318i	$-0.288015\pi$
$317$	25.0000	−	25.0000i	1.40414	−	1.40414i	0.786318	−	0.617822i	$-0.211985\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−13.0000	−	13.0000i	−0.721110	−	0.721110i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$		0			0
$333$	15.0000	−	15.0000i	0.821995	−	0.821995i
$334$		0			0
$335$		0			0
$336$		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
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$	−23.0000	+	23.0000i	−1.23116	+	1.23116i	−0.267644	+	0.963518i	$0.586245\pi$
$349$	−23.0000	+	23.0000i	−1.23116	+	1.23116i	−0.963518	+	0.267644i	$0.913755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−34.0000			−1.80964			−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000			−1.80964			−0.904819	+	0.425797i	$0.859994\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$			19.0000i			1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−18.0000	+	18.0000i	−0.942163	+	0.942163i
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	24.0000			1.24939
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.0000	+	11.0000i	0.569558	+	0.569558i	0.932005	−	0.362446i	$-0.118058\pi$
$373$	11.0000	+	11.0000i	0.569558	+	0.569558i	−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	14.0000i		−	0.721037i
$378$		0			0
$379$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$379$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$380$		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
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	27.0000	+	27.0000i	1.36895	+	1.36895i	0.861934	+	0.507020i	$0.169253\pi$
$389$	27.0000	+	27.0000i	1.36895	+	1.36895i	0.507020	+	0.861934i	$0.330747\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	25.0000	−	25.0000i	1.25471	−	1.25471i	0.301131	−	0.953583i	$-0.402636\pi$
$397$	25.0000	−	25.0000i	1.25471	−	1.25471i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−40.0000			−1.99750			−0.998752	−	0.0499376i	$-0.984098\pi$
$401$	−40.0000			−1.99750			−0.998752	+	0.0499376i	$0.984098\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	27.0000	+	27.0000i	1.34164	+	1.34164i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			40.0000i			1.97787i	0.148340	+	0.988936i	$0.452607\pi$
$409$			40.0000i			1.97787i	−0.148340	+	0.988936i	$0.547393\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	29.0000	+	29.0000i	1.41337	+	1.41337i	0.731055	+	0.682318i	$0.239028\pi$
$421$	29.0000	+	29.0000i	1.41337	+	1.41337i	0.682318	+	0.731055i	$0.260972\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	104.000i		−	5.04474i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	24.0000			1.15337			0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000			1.15337			0.576683	+	0.816968i	$0.304347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$			21.0000i			1.00000i
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	−30.0000	+	30.0000i	−1.42214	+	1.42214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	40.0000			1.88772			0.943858	−	0.330350i	$-0.107167\pi$
$449$	40.0000			1.88772			0.943858	+	0.330350i	$0.107167\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		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
$461$	9.00000	−	9.00000i	0.419172	−	0.419172i	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	9.00000	−	9.00000i	0.419172	−	0.419172i	0.884918	+	0.465746i	$0.154214\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	15.0000	−	15.0000i	0.686803	−	0.686803i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	10.0000			0.455961
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−24.0000	−	24.0000i	−1.08978	−	1.08978i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$491$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$492$		0			0
$493$	56.0000	−	56.0000i	2.52211	−	2.52211i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		−	66.0000i		−	2.93696i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.0000	+	17.0000i	−0.753512	+	0.753512i	−0.975133	−	0.221621i	$-0.928865\pi$
$509$	−17.0000	+	17.0000i	−0.753512	+	0.753512i	0.221621	+	0.975133i	$0.428865\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	40.0000i		−	1.75243i	−0.481919	−	0.876216i	$-0.660060\pi$
$521$		−	40.0000i		−	1.75243i	0.481919	−	0.876216i	$-0.339940\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	8.00000	+	8.00000i	0.346518	+	0.346518i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	31.0000	−	31.0000i	1.33279	−	1.33279i	0.429934	−	0.902861i	$-0.358537\pi$
$541$	31.0000	−	31.0000i	1.33279	−	1.33279i	0.902861	−	0.429934i	$-0.141463\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	42.0000			1.79908
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$	−3.00000	−	3.00000i	−0.128037	−	0.128037i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−33.0000	+	33.0000i	−1.39825	+	1.39825i	−0.593199	+	0.805056i	$0.702135\pi$
$557$	−33.0000	+	33.0000i	−1.39825	+	1.39825i	−0.805056	+	0.593199i	$0.797865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	42.0000	+	42.0000i	1.76695	+	1.76695i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			40.0000i			1.67689i	0.544988	+	0.838444i	$0.316534\pi$
$569$			40.0000i			1.67689i	−0.544988	+	0.838444i	$0.683466\pi$
$570$		0			0
$571$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$571$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$			18.0000i			0.744208i
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−46.0000			−1.88899			−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000			−1.88899			−0.944497	+	0.328521i	$0.893450\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		−	10.0000i		−	0.407909i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$		−	10.0000i		−	0.407909i	0.978980	−	0.203954i	$-0.0653794\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−33.0000	+	33.0000i	−1.34164	+	1.34164i
$606$		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
$611$		0			0
$612$		0			0
$613$	−35.0000	−	35.0000i	−1.41364	−	1.41364i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	−35.0000	−	35.0000i	−1.41364	−	1.41364i	−0.686624	−	0.727013i	$-0.740908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			38.0000i			1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$			38.0000i			1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−79.0000			−3.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	40.0000	+	40.0000i	1.59490	+	1.59490i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−7.00000	+	7.00000i	−0.277350	+	0.277350i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.00000			0.315981			0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000			0.315981			0.157991	+	0.987441i	$0.449498\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	−	9.00000i	0.352197	−	0.352197i	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	9.00000	−	9.00000i	0.352197	−	0.352197i	0.860927	+	0.508729i	$0.169885\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	18.0000			0.702247
$658$		0			0
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		0			0
$661$	−19.0000	−	19.0000i	−0.739014	−	0.739014i	0.233373	−	0.972387i	$-0.425024\pi$
$661$	−19.0000	−	19.0000i	−0.739014	−	0.739014i	−0.972387	+	0.233373i	$0.925024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−24.0000			−0.925132			−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000			−0.925132			−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	27.0000	+	27.0000i	1.03769	+	1.03769i	0.999261	+	0.0384331i	$0.0122367\pi$
$677$	27.0000	+	27.0000i	1.03769	+	1.03769i	0.0384331	+	0.999261i	$0.487763\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	−24.0000	+	24.0000i	−0.916993	+	0.916993i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.0000			0.380970
$690$		0			0
$691$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$691$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$			64.0000i			2.42417i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	31.0000	−	31.0000i	1.17085	−	1.17085i	0.188847	−	0.982006i	$-0.439525\pi$
$701$	31.0000	−	31.0000i	1.17085	−	1.17085i	0.982006	−	0.188847i	$-0.0604752\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−37.0000	−	37.0000i	−1.38956	−	1.38956i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−37.0000	−	37.0000i	−1.38956	−	1.38956i	−0.563337	−	0.826227i	$-0.690483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−91.0000	−	91.0000i	−3.37966	−	3.37966i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	25.0000	−	25.0000i	0.923396	−	0.923396i	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	25.0000	−	25.0000i	0.923396	−	0.923396i	0.997268	+	0.0738717i	$0.0235355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$			18.0000i			0.659469i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−35.0000	−	35.0000i	−1.27210	−	1.27210i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−35.0000	−	35.0000i	−1.27210	−	1.27210i	−0.327111	−	0.944986i	$-0.606075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			40.0000i			1.45000i	0.688749	+	0.724999i	$0.258160\pi$
$761$			40.0000i			1.45000i	−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−72.0000	+	72.0000i	−2.60317	+	2.60317i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−24.0000			−0.865462			−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000			−0.865462			−0.432731	+	0.901523i	$0.642450\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.00000	−	5.00000i	−0.179838	−	0.179838i	0.611448	−	0.791285i	$-0.290588\pi$
$773$	−5.00000	−	5.00000i	−0.179838	−	0.179838i	−0.791285	+	0.611448i	$0.790588\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	102.000			3.64054
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	2.00000i		−	0.0710221i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.0000	−	15.0000i	0.531327	−	0.531327i	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	15.0000	−	15.0000i	0.531327	−	0.531327i	0.920967	+	0.389640i	$0.127401\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	30.0000			1.06000
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			56.0000i			1.96886i	0.175791	+	0.984428i	$0.443752\pi$
$809$			56.0000i			1.96886i	−0.175791	+	0.984428i	$0.556248\pi$
$810$		0			0
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11.0000	+	11.0000i	0.383903	+	0.383903i	0.872506	−	0.488603i	$-0.162493\pi$
$821$	11.0000	+	11.0000i	0.383903	+	0.383903i	−0.488603	+	0.872506i	$0.662493\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$	−17.0000	+	17.0000i	−0.590434	+	0.590434i	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−17.0000	+	17.0000i	−0.590434	+	0.590434i	0.347314	+	0.937749i	$0.387094\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−56.0000			−1.94029
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$		−	69.0000i		−	2.37931i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	33.0000	−	33.0000i	1.13523	−	1.13523i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−5.00000	−	5.00000i	−0.171197	−	0.171197i	0.616308	−	0.787505i	$-0.288628\pi$
$853$	−5.00000	−	5.00000i	−0.171197	−	0.171197i	−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			8.00000i			0.273275i	0.990621	+	0.136637i	$0.0436295\pi$
$857$			8.00000i			0.273275i	−0.990621	+	0.136637i	$0.956370\pi$
$858$		0			0
$859$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$859$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−90.0000			−3.06009
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$			24.0000i			0.812277i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.0000	+	23.0000i	−0.776655	+	0.776655i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−23.0000	+	23.0000i	−0.776655	+	0.776655i	0.202606	+	0.979260i	$0.435059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−50.0000			−1.68454			−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000			−1.68454			−0.842271	+	0.539054i	$0.818782\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	40.0000	+	40.0000i	1.33259	+	1.33259i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			114.000i			3.78949i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$	−33.0000	+	33.0000i	−1.09454	+	1.09454i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	65.0000	−	65.0000i	2.13719	−	2.13719i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	40.0000			1.31236			0.656179	−	0.754606i	$-0.272172\pi$
$929$	40.0000			1.31236			0.656179	+	0.754606i	$0.272172\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	38.0000i		−	1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$		−	38.0000i		−	1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−39.0000	+	39.0000i	−1.27136	+	1.27136i	−0.325991	+	0.945373i	$0.605698\pi$
$941$	−39.0000	+	39.0000i	−1.27136	+	1.27136i	−0.945373	+	0.325991i	$0.894302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$	6.00000	+	6.00000i	0.194768	+	0.194768i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	56.0000i		−	1.81402i	−0.421111	−	0.907009i	$-0.638360\pi$
$953$		−	56.0000i		−	1.81402i	0.421111	−	0.907009i	$-0.361640\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	72.0000	+	72.0000i	2.31776	+	2.31776i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.00000			−0.255943			−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000			−0.255943			−0.127971	+	0.991778i	$0.540847\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−21.0000	−	21.0000i	−0.670478	−	0.670478i
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		−	78.0000i		−	2.48529i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−37.0000	−	37.0000i	−1.17180	−	1.17180i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−37.0000	−	37.0000i	−1.17180	−	1.17180i	−0.190022	−	0.981780i	$-0.560856\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.c.129.1	✓	2	16.3	odd	4	inner
512.2.e.c.129.1	✓	2	16.13	even	4	inner
512.2.e.c.385.1	yes	2	1.1	even	1	trivial
512.2.e.c.385.1	yes	2	4.3	odd	2	CM
512.2.e.f.129.1	yes	2	16.5	even	4
512.2.e.f.129.1	yes	2	16.11	odd	4
512.2.e.f.385.1	yes	2	8.3	odd	2
512.2.e.f.385.1	yes	2	8.5	even	2
1024.2.a.d.1.1		2	32.13	even	8
1024.2.a.d.1.1		2	32.19	odd	8
1024.2.a.d.1.2		2	32.3	odd	8
1024.2.a.d.1.2		2	32.29	even	8
1024.2.b.d.513.1		2	32.5	even	8
1024.2.b.d.513.1		2	32.27	odd	8
1024.2.b.d.513.2		2	32.11	odd	8
1024.2.b.d.513.2		2	32.21	even	8
4608.2.k.b.1153.1		2	48.5	odd	4
4608.2.k.b.1153.1		2	48.11	even	4
4608.2.k.b.3457.1		2	24.5	odd	2
4608.2.k.b.3457.1		2	24.11	even	2
4608.2.k.w.1153.1		2	48.29	odd	4
4608.2.k.w.1153.1		2	48.35	even	4
4608.2.k.w.3457.1		2	3.2	odd	2
4608.2.k.w.3457.1		2	12.11	even	2
9216.2.a.g.1.1		2	96.29	odd	8
9216.2.a.g.1.1		2	96.35	even	8
9216.2.a.g.1.2		2	96.77	odd	8
9216.2.a.g.1.2		2	96.83	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:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.00000 - 3.00000i) q^{5} +3.00000i q^{9} +(-1.00000 + 1.00000i) q^{13} -8.00000 q^{17} +13.0000i q^{25} +(-7.00000 + 7.00000i) q^{29} +(-5.00000 - 5.00000i) q^{37} -8.00000i q^{41} +(9.00000 - 9.00000i) q^{45} +7.00000 q^{49} +(-5.00000 - 5.00000i) q^{53} +(-1.00000 + 1.00000i) q^{61} +6.00000 q^{65} -6.00000i q^{73} -9.00000 q^{81} +(24.0000 + 24.0000i) q^{85} -10.0000i q^{89} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} - 2 q^{13} - 16 q^{17} - 14 q^{29} - 10 q^{37} + 18 q^{45} + 14 q^{49} - 10 q^{53} - 2 q^{61} + 12 q^{65} - 18 q^{81} + 48 q^{85} + 16 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 - 2 * q^13 - 16 * q^17 - 14 * q^29 - 10 * q^37 + 18 * q^45 + 14 * q^49 - 10 * q^53 - 2 * q^61 + 12 * q^65 - 18 * q^81 + 48 * q^85 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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