LMFDB - Embedded newform 512.2.e.e.129.1

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:	$0$
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:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			129.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	512.129
Dual form			512.2.e.e.385.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{5} -3.00000i q^{9} +O(q^{10})$	$q+(1.00000 - 1.00000i) q^{5} -3.00000i q^{9} +(-5.00000 - 5.00000i) q^{13} +8.00000 q^{17} +3.00000i q^{25} +(-3.00000 - 3.00000i) q^{29} +(7.00000 - 7.00000i) q^{37} -8.00000i q^{41} +(-3.00000 - 3.00000i) q^{45} +7.00000 q^{49} +(-9.00000 + 9.00000i) q^{53} +(11.0000 + 11.0000i) q^{61} -10.0000 q^{65} +6.00000i q^{73} -9.00000 q^{81} +(8.00000 - 8.00000i) q^{85} +10.0000i q^{89} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} - 10 q^{13} + 16 q^{17} - 6 q^{29} + 14 q^{37} - 6 q^{45} + 14 q^{49} - 18 q^{53} + 22 q^{61} - 20 q^{65} - 18 q^{81} + 16 q^{85} - 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 10 * q^13 + 16 * q^17 - 6 * q^29 + 14 * q^37 - 6 * q^45 + 14 * q^49 - 18 * q^53 + 22 * q^61 - 20 * q^65 - 18 * q^81 + 16 * q^85 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$5$	$511$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

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	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$	1.00000	−	1.00000i	0.447214	−	0.447214i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	1.00000	−	1.00000i	0.447214	−	0.447214i	0.894427	+	0.447214i	$0.147584\pi$
$6$		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	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$		0			0
$13$	−5.00000	−	5.00000i	−1.38675	−	1.38675i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−5.00000	−	5.00000i	−1.38675	−	1.38675i	−0.554700	−	0.832050i	$-0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	8.00000			1.94029			0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000			1.94029			0.970143	+	0.242536i	$0.0779791\pi$
$18$		0			0
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$			3.00000i			0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000	−	3.00000i	−0.557086	−	0.557086i	0.371391	−	0.928477i	$-0.378881\pi$
$29$	−3.00000	−	3.00000i	−0.557086	−	0.557086i	−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
$37$	7.00000	−	7.00000i	1.15079	−	1.15079i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	7.00000	−	7.00000i	1.15079	−	1.15079i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		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	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	−3.00000	−	3.00000i	−0.447214	−	0.447214i
$46$		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
$53$	−9.00000	+	9.00000i	−1.23625	+	1.23625i	−0.274721	+	0.961524i	$0.588586\pi$
$53$	−9.00000	+	9.00000i	−1.23625	+	1.23625i	−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	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	11.0000	+	11.0000i	1.40841	+	1.40841i	0.768221	+	0.640184i	$0.221142\pi$
$61$	11.0000	+	11.0000i	1.40841	+	1.40841i	0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.0000			−1.24035
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		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	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		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	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	8.00000	−	8.00000i	0.867722	−	0.867722i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			10.0000i			1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$			10.0000i			1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$		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	$-0.633125\pi$
$97$	−8.00000			−0.812277			−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.00000	+	9.00000i	−0.895533	+	0.895533i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	−9.00000	+	9.00000i	−0.895533	+	0.895533i	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	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	13.0000	+	13.0000i	1.24517	+	1.24517i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	13.0000	+	13.0000i	1.24517	+	1.24517i	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
$117$	−15.0000	+	15.0000i	−1.38675	+	1.38675i
$118$		0			0
$119$		0			0
$120$		0			0
$121$			11.0000i			1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.00000	+	8.00000i	0.715542	+	0.715542i
$126$		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	$-0.250000\pi$
$131$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$132$		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	$-0.750000\pi$
$139$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−6.00000			−0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	17.0000	−	17.0000i	1.39269	−	1.39269i	0.573462	−	0.819232i	$-0.305600\pi$
$149$	17.0000	−	17.0000i	1.39269	−	1.39269i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		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
$157$	−5.00000	−	5.00000i	−0.399043	−	0.399043i	0.478852	−	0.877896i	$-0.341053\pi$
$157$	−5.00000	−	5.00000i	−0.399043	−	0.399043i	−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	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$			37.0000i			2.84615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.0000	+	11.0000i	0.836315	+	0.836315i	0.988372	−	0.152057i	$-0.0485898\pi$
$173$	11.0000	+	11.0000i	0.836315	+	0.836315i	−0.152057	+	0.988372i	$0.548590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$179$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$	1.00000	−	1.00000i	0.0743294	−	0.0743294i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	1.00000	−	1.00000i	0.0743294	−	0.0743294i	0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	14.0000i		−	1.02930i
$186$		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	$-0.168091\pi$
$193$	24.0000			1.72756			0.863779	+	0.503871i	$0.168091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.0712470	+	0.997459i	$0.522698\pi$
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		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
$205$	−8.00000	−	8.00000i	−0.558744	−	0.558744i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$211$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$212$		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
$221$	−40.0000	−	40.0000i	−2.69069	−	2.69069i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	9.00000			0.600000
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$	17.0000	−	17.0000i	1.12339	−	1.12339i	0.132164	−	0.991228i	$-0.457808\pi$
$229$	17.0000	−	17.0000i	1.12339	−	1.12339i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	26.0000i		−	1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$		−	26.0000i		−	1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$		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	$-0.417048\pi$
$241$	8.00000			0.515325			0.257663	+	0.966235i	$0.417048\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.00000	−	7.00000i	0.447214	−	0.447214i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$251$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$252$		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
$261$	−9.00000	+	9.00000i	−0.557086	+	0.557086i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$			18.0000i			1.10573i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.00000	−	3.00000i	−0.182913	−	0.182913i	0.609711	−	0.792624i	$-0.291286\pi$
$269$	−3.00000	−	3.00000i	−0.182913	−	0.182913i	−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
$277$	23.0000	−	23.0000i	1.38194	−	1.38194i	0.540758	−	0.841178i	$-0.318138\pi$
$277$	23.0000	−	23.0000i	1.38194	−	1.38194i	0.841178	−	0.540758i	$-0.181862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	10.0000i		−	0.596550i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$		−	10.0000i		−	0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		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
$293$	−15.0000	+	15.0000i	−0.876309	+	0.876309i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	−15.0000	+	15.0000i	−0.876309	+	0.876309i	0.116841	+	0.993151i	$0.462723\pi$
$294$		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
$305$	22.0000			1.25972
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		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
$317$	−3.00000	−	3.00000i	−0.168497	−	0.168497i	0.617822	−	0.786318i	$-0.288015\pi$
$317$	−3.00000	−	3.00000i	−0.168497	−	0.168497i	−0.786318	+	0.617822i	$0.788015\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	15.0000	−	15.0000i	0.832050	−	0.832050i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$331$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$332$		0			0
$333$	−21.0000	−	21.0000i	−1.15079	−	1.15079i
$334$		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	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$	13.0000	+	13.0000i	0.695874	+	0.695874i	0.963518	−	0.267644i	$-0.0862451\pi$
$349$	13.0000	+	13.0000i	0.695874	+	0.695874i	−0.267644	+	0.963518i	$0.586245\pi$
$350$		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
$365$	6.00000	+	6.00000i	0.314054	+	0.314054i
$366$		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
$373$	−25.0000	+	25.0000i	−1.29445	+	1.29445i	−0.362446	+	0.932005i	$0.618058\pi$
$373$	−25.0000	+	25.0000i	−1.29445	+	1.29445i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			30.0000i			1.54508i
$378$		0			0
$379$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$379$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$380$		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
$389$	7.00000	−	7.00000i	0.354914	−	0.354914i	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	7.00000	−	7.00000i	0.354914	−	0.354914i	0.861934	+	0.507020i	$0.169253\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	13.0000	+	13.0000i	0.652451	+	0.652451i	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	13.0000	+	13.0000i	0.652451	+	0.652451i	−0.301131	+	0.953583i	$0.597364\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	40.0000			1.99750			0.998752	−	0.0499376i	$-0.0159023\pi$
$401$	40.0000			1.99750			0.998752	+	0.0499376i	$0.0159023\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−9.00000	+	9.00000i	−0.447214	+	0.447214i
$406$		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	$-0.250000\pi$
$419$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$	1.00000	−	1.00000i	0.0487370	−	0.0487370i	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	1.00000	−	1.00000i	0.0487370	−	0.0487370i	0.731055	+	0.682318i	$0.239028\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			24.0000i			1.16417i
$426$		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	$-0.695653\pi$
$433$	−24.0000			−1.15337			−0.576683	+	0.816968i	$0.695653\pi$
$434$		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	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	10.0000	+	10.0000i	0.474045	+	0.474045i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−40.0000			−1.88772			−0.943858	−	0.330350i	$-0.892833\pi$
$449$	−40.0000			−1.88772			−0.943858	+	0.330350i	$0.892833\pi$
$450$		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
$461$	29.0000	+	29.0000i	1.35066	+	1.35066i	0.884918	+	0.465746i	$0.154214\pi$
$461$	29.0000	+	29.0000i	1.35066	+	1.35066i	0.465746	+	0.884918i	$0.345786\pi$
$462$		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	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		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
$477$	27.0000	+	27.0000i	1.23625	+	1.23625i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−70.0000			−3.19173
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.00000	+	8.00000i	−0.363261	+	0.363261i
$486$		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	$-0.750000\pi$
$491$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$492$		0			0
$493$	−24.0000	−	24.0000i	−1.08091	−	1.08091i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$			18.0000i			0.800989i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	27.0000	+	27.0000i	1.19675	+	1.19675i	0.975133	+	0.221621i	$0.0711348\pi$
$509$	27.0000	+	27.0000i	1.19675	+	1.19675i	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	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		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
$533$	−40.0000	+	40.0000i	−1.73259	+	1.73259i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	11.0000	+	11.0000i	0.472927	+	0.472927i	0.902861	−	0.429934i	$-0.141463\pi$
$541$	11.0000	+	11.0000i	0.472927	+	0.472927i	−0.429934	+	0.902861i	$0.641463\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	26.0000			1.11372
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$	33.0000	−	33.0000i	1.40841	−	1.40841i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.00000	−	5.00000i	−0.211857	−	0.211857i	0.593199	−	0.805056i	$-0.297865\pi$
$557$	−5.00000	−	5.00000i	−0.211857	−	0.211857i	−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	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	−14.0000	+	14.0000i	−0.588984	+	0.588984i
$566$		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	$-0.750000\pi$
$571$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$572$		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
$585$			30.0000i			1.24035i
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		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	$0.0653794\pi$
$601$			10.0000i			0.407909i	−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.0000	+	11.0000i	0.447214	+	0.447214i
$606$		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
$613$	1.00000	−	1.00000i	0.0403896	−	0.0403896i	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	1.00000	−	1.00000i	0.0403896	−	0.0403896i	0.727013	+	0.686624i	$0.240908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	38.0000i		−	1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$		−	38.0000i		−	1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	56.0000	−	56.0000i	2.23287	−	2.23287i
$630$		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
$637$	−35.0000	−	35.0000i	−1.38675	−	1.38675i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.00000			−0.315981			−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000			−0.315981			−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		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
$653$	−35.0000	−	35.0000i	−1.36966	−	1.36966i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	−35.0000	−	35.0000i	−1.36966	−	1.36966i	−0.508729	−	0.860927i	$-0.669885\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	18.0000			0.702247
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	−31.0000	+	31.0000i	−1.20576	+	1.20576i	−0.233373	+	0.972387i	$0.574976\pi$
$661$	−31.0000	+	31.0000i	−1.20576	+	1.20576i	−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	$-0.346929\pi$
$673$	24.0000			0.925132			0.462566	+	0.886585i	$0.346929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−25.0000	+	25.0000i	−0.960828	+	0.960828i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	−25.0000	+	25.0000i	−0.960828	+	0.960828i	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	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	−8.00000	−	8.00000i	−0.305664	−	0.305664i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	90.0000			3.42873
$690$		0			0
$691$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$691$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$692$		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
$701$	−21.0000	−	21.0000i	−0.793159	−	0.793159i	0.188847	−	0.982006i	$-0.439525\pi$
$701$	−21.0000	−	21.0000i	−0.793159	−	0.793159i	−0.982006	+	0.188847i	$0.939525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	7.00000	−	7.00000i	0.262891	−	0.262891i	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	7.00000	−	7.00000i	0.262891	−	0.262891i	0.826227	+	0.563337i	$0.190483\pi$
$710$		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
$725$	9.00000	−	9.00000i	0.334252	−	0.334252i
$726$		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
$733$	29.0000	+	29.0000i	1.07114	+	1.07114i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	29.0000	+	29.0000i	1.07114	+	1.07114i	0.0738717	+	0.997268i	$0.476464\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$739$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		−	34.0000i		−	1.24566i
$746$		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
$757$	17.0000	−	17.0000i	0.617876	−	0.617876i	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	17.0000	−	17.0000i	0.617876	−	0.617876i	0.944986	+	0.327111i	$0.106075\pi$
$758$		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
$765$	−24.0000	−	24.0000i	−0.867722	−	0.867722i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	24.0000			0.865462			0.432731	−	0.901523i	$-0.357550\pi$
$769$	24.0000			0.865462			0.432731	+	0.901523i	$0.357550\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	39.0000	−	39.0000i	1.40273	−	1.40273i	0.611448	−	0.791285i	$-0.290588\pi$
$773$	39.0000	−	39.0000i	1.40273	−	1.40273i	0.791285	−	0.611448i	$-0.209412\pi$
$774$		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
$785$	−10.0000			−0.356915
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	110.000i		−	3.90621i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−37.0000	−	37.0000i	−1.31061	−	1.31061i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−37.0000	−	37.0000i	−1.31061	−	1.31061i	−0.389640	−	0.920967i	$-0.627401\pi$
$798$		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	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		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
$821$	39.0000	−	39.0000i	1.36111	−	1.36111i	0.488603	−	0.872506i	$-0.337507\pi$
$821$	39.0000	−	39.0000i	1.36111	−	1.36111i	0.872506	−	0.488603i	$-0.162493\pi$
$822$		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	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	−37.0000	−	37.0000i	−1.28506	−	1.28506i	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−37.0000	−	37.0000i	−1.28506	−	1.28506i	−0.347314	−	0.937749i	$-0.612906\pi$
$830$		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
$841$		−	11.0000i		−	0.379310i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	37.0000	+	37.0000i	1.27284	+	1.27284i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−41.0000	+	41.0000i	−1.40381	+	1.40381i	−0.616308	+	0.787505i	$0.711372\pi$
$853$	−41.0000	+	41.0000i	−1.40381	+	1.40381i	−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	$-0.750000\pi$
$859$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$860$		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
$865$	22.0000			0.748022
$866$		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
$877$	−35.0000	−	35.0000i	−1.18187	−	1.18187i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−35.0000	−	35.0000i	−1.18187	−	1.18187i	−0.202606	−	0.979260i	$-0.564941\pi$
$878$		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	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		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
$901$	−72.0000	+	72.0000i	−2.39867	+	2.39867i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2.00000i		−	0.0664822i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$	27.0000	+	27.0000i	0.895533	+	0.895533i
$910$		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
$925$	21.0000	+	21.0000i	0.690476	+	0.690476i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−40.0000			−1.31236			−0.656179	−	0.754606i	$-0.727828\pi$
$929$	−40.0000			−1.31236			−0.656179	+	0.754606i	$0.727828\pi$
$930$		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	$0.213153\pi$
$937$			38.0000i			1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−19.0000	−	19.0000i	−0.619382	−	0.619382i	0.325991	−	0.945373i	$-0.394302\pi$
$941$	−19.0000	−	19.0000i	−0.619382	−	0.619382i	−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	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$	30.0000	−	30.0000i	0.973841	−	0.973841i
$950$		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
$965$	24.0000	−	24.0000i	0.772587	−	0.772587i
$966$		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	$-0.750000\pi$
$971$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	8.00000			0.255943			0.127971	−	0.991778i	$-0.459153\pi$
$977$	8.00000			0.255943			0.127971	+	0.991778i	$0.459153\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	39.0000	−	39.0000i	1.24517	−	1.24517i
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$			30.0000i			0.955879i
$986$		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
$997$	−25.0000	+	25.0000i	−0.791758	+	0.791758i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−25.0000	+	25.0000i	−0.791758	+	0.791758i	0.190022	+	0.981780i	$0.439144\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.d.129.1	✓	2	8.3	odd	2
512.2.e.d.129.1	✓	2	8.5	even	2
512.2.e.d.385.1	yes	2	16.3	odd	4
512.2.e.d.385.1	yes	2	16.13	even	4
512.2.e.e.129.1	yes	2	1.1	even	1	trivial
512.2.e.e.129.1	yes	2	4.3	odd	2	CM
512.2.e.e.385.1	yes	2	16.5	even	4	inner
512.2.e.e.385.1	yes	2	16.11	odd	4	inner
1024.2.a.c.1.1		2	32.11	odd	8
1024.2.a.c.1.1		2	32.21	even	8
1024.2.a.c.1.2		2	32.5	even	8
1024.2.a.c.1.2		2	32.27	odd	8
1024.2.b.c.513.1		2	32.13	even	8
1024.2.b.c.513.1		2	32.19	odd	8
1024.2.b.c.513.2		2	32.3	odd	8
1024.2.b.c.513.2		2	32.29	even	8
4608.2.k.g.1153.1		2	3.2	odd	2
4608.2.k.g.1153.1		2	12.11	even	2
4608.2.k.g.3457.1		2	48.5	odd	4
4608.2.k.g.3457.1		2	48.11	even	4
4608.2.k.r.1153.1		2	24.5	odd	2
4608.2.k.r.1153.1		2	24.11	even	2
4608.2.k.r.3457.1		2	48.29	odd	4
4608.2.k.r.3457.1		2	48.35	even	4
9216.2.a.p.1.1		2	96.5	odd	8
9216.2.a.p.1.1		2	96.59	even	8
9216.2.a.p.1.2		2	96.11	even	8
9216.2.a.p.1.2		2	96.53	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)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{5} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{5} -3.00000i q^{9} +(-5.00000 - 5.00000i) q^{13} +8.00000 q^{17} +3.00000i q^{25} +(-3.00000 - 3.00000i) q^{29} +(7.00000 - 7.00000i) q^{37} -8.00000i q^{41} +(-3.00000 - 3.00000i) q^{45} +7.00000 q^{49} +(-9.00000 + 9.00000i) q^{53} +(11.0000 + 11.0000i) q^{61} -10.0000 q^{65} +6.00000i q^{73} -9.00000 q^{81} +(8.00000 - 8.00000i) q^{85} +10.0000i q^{89} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} - 10 q^{13} + 16 q^{17} - 6 q^{29} + 14 q^{37} - 6 q^{45} + 14 q^{49} - 18 q^{53} + 22 q^{61} - 20 q^{65} - 18 q^{81} + 16 q^{85} - 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 10 * q^13 + 16 * q^17 - 6 * q^29 + 14 * q^37 - 6 * q^45 + 14 * q^49 - 18 * q^53 + 22 * q^61 - 20 * q^65 - 18 * q^81 + 16 * q^85 - 16 * q^97

Introduction

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