LMFDB - Embedded newform 2304.2.k.g.1729.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(577,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2304.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.k (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:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			1729.3
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2304.1729
Dual form			2304.2.k.g.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.378937i q^{7} +O(q^{10})$	$q+0.378937i q^{7} +(2.46410 - 2.46410i) q^{13} +(2.44949 - 2.44949i) q^{19} -5.00000i q^{25} -10.1769 q^{31} +(-1.53590 - 1.53590i) q^{37} +(7.34847 + 7.34847i) q^{43} +6.85641 q^{49} +(10.4641 - 10.4641i) q^{61} +(11.3137 - 11.3137i) q^{67} -13.8564i q^{73} -9.41902 q^{79} +(0.933740 + 0.933740i) q^{91} +13.8564 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{13} - 40 q^{37} - 56 q^{49} + 56 q^{61}+O(q^{100}) $ 8 * q - 8 * q^13 - 40 * q^37 - 56 * q^49 + 56 * q^61

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$5$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$6$		0			0
$7$			0.378937i			0.143225i	0.997433	+	0.0716124i	$0.0228145\pi$
$7$			0.378937i			0.143225i	−0.997433	+	0.0716124i	$0.977186\pi$
$8$		0			0
$9$		0			0
$10$		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
$13$	2.46410	−	2.46410i	0.683419	−	0.683419i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	2.46410	−	2.46410i	0.683419	−	0.683419i	0.960769	+	0.277350i	$0.0894562\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	2.44949	−	2.44949i	0.561951	−	0.561951i	−0.367910	−	0.929861i	$-0.619927\pi$
$19$	2.44949	−	2.44949i	0.561951	−	0.561951i	0.929861	+	0.367910i	$0.119927\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$		−	5.00000i		−	1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$	−10.1769			−1.82782			−0.913912	−	0.405912i	$-0.866954\pi$
$31$	−10.1769			−1.82782			−0.913912	+	0.405912i	$0.866954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.53590	−	1.53590i	−0.252500	−	0.252500i	0.569495	−	0.821995i	$-0.307139\pi$
$37$	−1.53590	−	1.53590i	−0.252500	−	0.252500i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	7.34847	+	7.34847i	1.12063	+	1.12063i	0.991647	+	0.128984i	$0.0411717\pi$
$43$	7.34847	+	7.34847i	1.12063	+	1.12063i	0.128984	+	0.991647i	$0.458828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	6.85641			0.979487
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		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
$61$	10.4641	−	10.4641i	1.33979	−	1.33979i	0.443533	−	0.896258i	$-0.353725\pi$
$61$	10.4641	−	10.4641i	1.33979	−	1.33979i	0.896258	−	0.443533i	$-0.146275\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	11.3137	−	11.3137i	1.38219	−	1.38219i	0.541468	−	0.840721i	$-0.317869\pi$
$67$	11.3137	−	11.3137i	1.38219	−	1.38219i	0.840721	−	0.541468i	$-0.182131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	13.8564i		−	1.62177i	−0.585206	−	0.810885i	$-0.698986\pi$
$73$		−	13.8564i		−	1.62177i	0.585206	−	0.810885i	$-0.301014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−9.41902			−1.05972			−0.529861	−	0.848084i	$-0.677756\pi$
$79$	−9.41902			−1.05972			−0.529861	+	0.848084i	$0.677756\pi$
$80$		0			0
$81$		0			0
$82$		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
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	0.933740	+	0.933740i	0.0978826	+	0.0978826i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.8564			1.40690			0.703452	−	0.710742i	$-0.251641\pi$
$97$	13.8564			1.40690			0.703452	+	0.710742i	$0.251641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$101$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$102$		0			0
$103$			11.6926i			1.15211i	0.817411	+	0.576055i	$0.195409\pi$
$103$			11.6926i			1.15211i	−0.817411	+	0.576055i	$0.804591\pi$
$104$		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
$109$	11.3923	−	11.3923i	1.09118	−	1.09118i	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	11.3923	−	11.3923i	1.09118	−	1.09118i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	11.0000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−21.4906			−1.90698			−0.953491	−	0.301420i	$-0.902539\pi$
$127$	−21.4906			−1.90698			−0.953491	+	0.301420i	$0.902539\pi$
$128$		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
$133$	0.928203	+	0.928203i	0.0804854	+	0.0804854i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	11.3137	+	11.3137i	0.959616	+	0.959616i	0.999216	−	0.0395994i	$-0.0126082\pi$
$139$	11.3137	+	11.3137i	0.959616	+	0.959616i	−0.0395994	+	0.999216i	$0.512608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$149$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$150$		0			0
$151$		−	19.9749i		−	1.62553i	−0.582591	−	0.812765i	$-0.697961\pi$
$151$		−	19.9749i		−	1.62553i	0.582591	−	0.812765i	$-0.302039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.39230	−	3.39230i	0.270735	−	0.270735i	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	3.39230	−	3.39230i	0.270735	−	0.270735i	0.829396	+	0.558661i	$0.188685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	17.1464	−	17.1464i	1.34301	−	1.34301i	0.449966	−	0.893045i	$-0.351436\pi$
$163$	17.1464	−	17.1464i	1.34301	−	1.34301i	0.893045	−	0.449966i	$-0.148564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$			0.856406i			0.0658774i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$	1.89469			0.143225
$176$		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
$181$	−9.53590	−	9.53590i	−0.708798	−	0.708798i	0.257485	−	0.966282i	$-0.417106\pi$
$181$	−9.53590	−	9.53590i	−0.708798	−	0.708798i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		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
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$			22.2485i			1.57715i	0.614937	+	0.788576i	$0.289182\pi$
$199$			22.2485i			1.57715i	−0.614937	+	0.788576i	$0.710818\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−11.3137	+	11.3137i	−0.778868	+	0.778868i	−0.979638	−	0.200770i	$-0.935655\pi$
$211$	−11.3137	+	11.3137i	−0.778868	+	0.778868i	0.200770	+	0.979638i	$0.435655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	3.85641i		−	0.261790i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−12.4505			−0.833749			−0.416874	−	0.908964i	$-0.636874\pi$
$223$	−12.4505			−0.833749			−0.416874	+	0.908964i	$0.636874\pi$
$224$		0			0
$225$		0			0
$226$		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
$229$	0.607695	+	0.607695i	0.0401576	+	0.0401576i	0.726900	−	0.686743i	$-0.240960\pi$
$229$	0.607695	+	0.607695i	0.0401576	+	0.0401576i	−0.686743	+	0.726900i	$0.740960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		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
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	12.0716i		−	0.768096i
$248$		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
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	0.582009	−	0.582009i	0.0361643	−	0.0361643i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$269$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$	32.0464			1.94668			0.973341	−	0.229362i	$-0.0736639\pi$
$271$	32.0464			1.94668			0.973341	+	0.229362i	$0.0736639\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	23.3923	+	23.3923i	1.40551	+	1.40551i	0.781094	+	0.624413i	$0.214662\pi$
$277$	23.3923	+	23.3923i	1.40551	+	1.40551i	0.624413	+	0.781094i	$0.285338\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−22.6274	−	22.6274i	−1.34506	−	1.34506i	−0.890941	−	0.454120i	$-0.849954\pi$
$283$	−22.6274	−	22.6274i	−1.34506	−	1.34506i	−0.454120	−	0.890941i	$-0.650046\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−2.78461	+	2.78461i	−0.160502	+	0.160502i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−11.3137	+	11.3137i	−0.645707	+	0.645707i	−0.951953	−	0.306245i	$-0.900927\pi$
$307$	−11.3137	+	11.3137i	−0.645707	+	0.645707i	0.306245	+	0.951953i	$0.400927\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			22.0000i			1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$			22.0000i			1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−12.3205	−	12.3205i	−0.683419	−	0.683419i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	22.6274	+	22.6274i	1.24372	+	1.24372i	0.958448	+	0.285267i	$0.0920824\pi$
$331$	22.6274	+	22.6274i	1.24372	+	1.24372i	0.285267	+	0.958448i	$0.407918\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−13.8564			−0.754807			−0.377403	−	0.926049i	$-0.623183\pi$
$337$	−13.8564			−0.754807			−0.377403	+	0.926049i	$0.623183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			5.25071i			0.283512i
$344$		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
$349$	−24.3205	+	24.3205i	−1.30185	+	1.30185i	−0.374701	+	0.927146i	$0.622255\pi$
$349$	−24.3205	+	24.3205i	−1.30185	+	1.30185i	−0.927146	+	0.374701i	$0.877745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		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
$361$			7.00000i			0.368421i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	29.7728			1.55413			0.777064	−	0.629421i	$-0.216708\pi$
$367$	29.7728			1.55413			0.777064	+	0.629421i	$0.216708\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−22.4641	−	22.4641i	−1.16315	−	1.16315i	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−22.4641	−	22.4641i	−1.16315	−	1.16315i	−0.179364	−	0.983783i	$-0.557404\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−26.9444	−	26.9444i	−1.38404	−	1.38404i	−0.837307	−	0.546734i	$-0.815871\pi$
$379$	−26.9444	−	26.9444i	−1.38404	−	1.38404i	−0.546734	−	0.837307i	$-0.684129\pi$
$380$		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
$389$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$389$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−27.3923	+	27.3923i	−1.37478	+	1.37478i	−0.521575	+	0.853206i	$0.674655\pi$
$397$	−27.3923	+	27.3923i	−1.37478	+	1.37478i	−0.853206	+	0.521575i	$0.825345\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−25.0769	+	25.0769i	−1.24917	+	1.24917i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			13.8564i			0.685155i	0.939490	+	0.342578i	$0.111300\pi$
$409$			13.8564i			0.685155i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		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
$421$	28.3205	+	28.3205i	1.38026	+	1.38026i	0.844150	+	0.536107i	$0.180106\pi$
$421$	28.3205	+	28.3205i	1.38026	+	1.38026i	0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3.96524	+	3.96524i	0.191891	+	0.191891i
$428$		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
$433$	−41.5692			−1.99769			−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	−41.5692			−1.99769			−0.998845	+	0.0480569i	$0.984697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	41.8444i		−	1.99712i	−0.0536078	−	0.998562i	$-0.517072\pi$
$439$		−	41.8444i		−	1.99712i	0.0536078	−	0.998562i	$-0.482928\pi$
$440$		0			0
$441$		0			0
$442$		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
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			41.5692i			1.94453i	0.233890	+	0.972263i	$0.424854\pi$
$457$			41.5692i			1.94453i	−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$	41.0865			1.90945			0.954726	−	0.297486i	$-0.0961480\pi$
$463$	41.0865			1.90945			0.954726	+	0.297486i	$0.0961480\pi$
$464$		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
$469$	4.28719	+	4.28719i	0.197964	+	0.197964i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−12.2474	−	12.2474i	−0.561951	−	0.561951i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−7.56922			−0.345127
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			33.5622i			1.52085i	0.649427	+	0.760424i	$0.275009\pi$
$487$			33.5622i			1.52085i	−0.649427	+	0.760424i	$0.724991\pi$
$488$		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
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−22.6274	+	22.6274i	−1.01294	+	1.01294i	−0.0130272	+	0.999915i	$0.504147\pi$
$499$	−22.6274	+	22.6274i	−1.01294	+	1.01294i	−0.999915	+	0.0130272i	$0.995853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$509$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$	5.25071			0.232278
$512$		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
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−31.8434	−	31.8434i	−1.39241	−	1.39241i	−0.819885	−	0.572528i	$-0.805963\pi$
$523$	−31.8434	−	31.8434i	−1.39241	−	1.39241i	−0.572528	−	0.819885i	$-0.694037\pi$
$524$		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
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−26.4641	+	26.4641i	−1.13778	+	1.13778i	−0.148933	+	0.988847i	$0.547584\pi$
$541$	−26.4641	+	26.4641i	−1.13778	+	1.13778i	−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−17.1464	+	17.1464i	−0.733128	+	0.733128i	−0.971238	−	0.238110i	$-0.923472\pi$
$547$	−17.1464	+	17.1464i	−0.733128	+	0.733128i	0.238110	+	0.971238i	$0.423472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	3.56922i		−	0.151779i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$	36.2147			1.53172
$560$		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
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−11.3137	−	11.3137i	−0.473464	−	0.473464i	0.429570	−	0.903034i	$-0.358665\pi$
$571$	−11.3137	−	11.3137i	−0.473464	−	0.473464i	−0.903034	+	0.429570i	$0.858665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−13.8564			−0.576850			−0.288425	−	0.957503i	$-0.593132\pi$
$577$	−13.8564			−0.576850			−0.288425	+	0.957503i	$0.593132\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		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
$589$	−24.9282	+	24.9282i	−1.02715	+	1.02715i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		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
$601$			41.5692i			1.69564i	0.530281	+	0.847822i	$0.322086\pi$
$601$			41.5692i			1.69564i	−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	17.7012			0.718471			0.359235	−	0.933247i	$-0.383038\pi$
$607$	17.7012			0.718471			0.359235	+	0.933247i	$0.383038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−29.2487	−	29.2487i	−1.18134	−	1.18134i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$	−29.2487	−	29.2487i	−1.18134	−	1.18134i	−0.201948	−	0.979396i	$-0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	22.6274	+	22.6274i	0.909473	+	0.909473i	0.996230	−	0.0867568i	$-0.0276503\pi$
$619$	22.6274	+	22.6274i	0.909473	+	0.909473i	−0.0867568	+	0.996230i	$0.527650\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	13.9663i		−	0.555988i	−0.960583	−	0.277994i	$-0.910330\pi$
$631$		−	13.9663i		−	0.555988i	0.960583	−	0.277994i	$-0.0896696\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	16.8949	−	16.8949i	0.669400	−	0.669400i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	22.0454	−	22.0454i	0.869386	−	0.869386i	−0.123018	−	0.992404i	$-0.539257\pi$
$643$	22.0454	−	22.0454i	0.869386	−	0.869386i	0.992404	+	0.123018i	$0.0392574\pi$
$644$		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
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		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
$661$	36.3205	+	36.3205i	1.41270	+	1.41270i	0.739014	+	0.673690i	$0.235292\pi$
$661$	36.3205	+	36.3205i	1.41270	+	1.41270i	0.673690	+	0.739014i	$0.264708\pi$
$662$		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
$673$	−13.8564			−0.534125			−0.267063	−	0.963679i	$-0.586053\pi$
$673$	−13.8564			−0.534125			−0.267063	+	0.963679i	$0.586053\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$			5.25071i			0.201504i
$680$		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
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−36.7423	+	36.7423i	−1.39774	+	1.39774i	−0.591274	+	0.806471i	$0.701375\pi$
$691$	−36.7423	+	36.7423i	−1.39774	+	1.39774i	−0.806471	+	0.591274i	$0.798625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$701$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$702$		0			0
$703$	−7.52433			−0.283786
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	13.2487	+	13.2487i	0.497566	+	0.497566i	0.910679	−	0.413114i	$-0.135559\pi$
$709$	13.2487	+	13.2487i	0.497566	+	0.497566i	−0.413114	+	0.910679i	$0.635559\pi$
$710$		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
$721$	−4.43078			−0.165011
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	53.1581i		−	1.97153i	−0.168144	−	0.985763i	$-0.553777\pi$
$727$		−	53.1581i		−	1.97153i	0.168144	−	0.985763i	$-0.446223\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−35.3923	+	35.3923i	−1.30724	+	1.30724i	−0.383849	+	0.923396i	$0.625402\pi$
$733$	−35.3923	+	35.3923i	−1.30724	+	1.30724i	−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	11.3137	−	11.3137i	0.416181	−	0.416181i	−0.467704	−	0.883885i	$-0.654919\pi$
$739$	11.3137	−	11.3137i	0.416181	−	0.416181i	0.883885	+	0.467704i	$0.154919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−24.5221			−0.894824			−0.447412	−	0.894328i	$-0.647654\pi$
$751$	−24.5221			−0.894824			−0.447412	+	0.894328i	$0.647654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−37.2487	−	37.2487i	−1.35383	−	1.35383i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$	−37.2487	−	37.2487i	−1.35383	−	1.35383i	−0.472493	−	0.881334i	$-0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	4.31697	+	4.31697i	0.156285	+	0.156285i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$			50.8845i			1.82782i
$776$		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
$785$		0			0
$786$		0			0
$787$	−2.44949	+	2.44949i	−0.0873149	+	0.0873149i	−0.749415	−	0.662100i	$-0.769665\pi$
$787$	−2.44949	+	2.44949i	−0.0873149	+	0.0873149i	0.662100	+	0.749415i	$0.269665\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	51.5692i		−	1.83128i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−7.34847	−	7.34847i	−0.258040	−	0.258040i	0.566217	−	0.824256i	$-0.308407\pi$
$811$	−7.34847	−	7.34847i	−0.258040	−	0.258040i	−0.824256	+	0.566217i	$0.808407\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	36.0000			1.25948
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$821$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$822$		0			0
$823$			53.9160i			1.87939i	0.342010	+	0.939696i	$0.388892\pi$
$823$			53.9160i			1.87939i	−0.342010	+	0.939696i	$0.611108\pi$
$824$		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
$829$	40.3205	−	40.3205i	1.40039	−	1.40039i	0.601566	−	0.798823i	$-0.294544\pi$
$829$	40.3205	−	40.3205i	1.40039	−	1.40039i	0.798823	−	0.601566i	$-0.205456\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		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
$841$			29.0000i			1.00000i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	4.16831			0.143225
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	25.5359	+	25.5359i	0.874333	+	0.874333i	0.992941	−	0.118609i	$-0.0378434\pi$
$853$	25.5359	+	25.5359i	0.874333	+	0.874333i	−0.118609	+	0.992941i	$0.537843\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−12.2474	−	12.2474i	−0.417878	−	0.417878i	0.466594	−	0.884472i	$-0.345481\pi$
$859$	−12.2474	−	12.2474i	−0.417878	−	0.417878i	−0.884472	+	0.466594i	$0.845481\pi$
$860$		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
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	55.7563i		−	1.88923i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	41.2487	−	41.2487i	1.39287	−	1.39287i	0.574049	−	0.818821i	$-0.305372\pi$
$877$	41.2487	−	41.2487i	1.39287	−	1.39287i	0.818821	−	0.574049i	$-0.194628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	41.6413	−	41.6413i	1.40134	−	1.40134i	0.605487	−	0.795855i	$-0.292978\pi$
$883$	41.6413	−	41.6413i	1.40134	−	1.40134i	0.795855	−	0.605487i	$-0.207022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		−	8.14359i		−	0.273127i
$890$		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
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	31.8434	+	31.8434i	1.05734	+	1.05734i	0.998253	+	0.0590889i	$0.0188195\pi$
$907$	31.8434	+	31.8434i	1.05734	+	1.05734i	0.0590889	+	0.998253i	$0.481180\pi$
$908$		0			0
$909$		0			0
$910$		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
$919$			14.7241i			0.485705i	0.970063	+	0.242852i	$0.0780831\pi$
$919$			14.7241i			0.485705i	−0.970063	+	0.242852i	$0.921917\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−7.67949	+	7.67949i	−0.252500	+	0.252500i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	16.7947	−	16.7947i	0.550424	−	0.550424i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	26.0000i		−	0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$		−	26.0000i		−	0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$		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
$949$	−34.1436	−	34.1436i	−1.10835	−	1.10835i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	72.5692			2.34094
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	27.4992i		−	0.884314i	−0.896938	−	0.442157i	$-0.854213\pi$
$967$		−	27.4992i		−	0.884314i	0.896938	−	0.442157i	$-0.145787\pi$
$968$		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
$973$	−4.28719	+	4.28719i	−0.137441	+	0.137441i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−62.9561			−1.99987			−0.999933	−	0.0116052i	$-0.996306\pi$
$991$	−62.9561			−1.99987			−0.999933	+	0.0116052i	$0.996306\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	26.1769	+	26.1769i	0.829031	+	0.829031i	0.987383	−	0.158352i	$-0.0506179\pi$
$997$	26.1769	+	26.1769i	0.829031	+	0.829031i	−0.158352	+	0.987383i	$0.550618\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.k.g.1729.3	yes	8
3.2	odd	2	CM	2304.2.k.g.1729.3	yes	8
4.3	odd	2	inner	2304.2.k.g.1729.2	yes	8
8.3	odd	2		2304.2.k.j.1729.2	yes	8
8.5	even	2		2304.2.k.j.1729.3	yes	8
12.11	even	2	inner	2304.2.k.g.1729.2	yes	8
16.3	odd	4	inner	2304.2.k.g.577.3	yes	8
16.5	even	4		2304.2.k.j.577.2	yes	8
16.11	odd	4		2304.2.k.j.577.3	yes	8
16.13	even	4	inner	2304.2.k.g.577.2	✓	8
24.5	odd	2		2304.2.k.j.1729.3	yes	8
24.11	even	2		2304.2.k.j.1729.2	yes	8
32.3	odd	8		9216.2.a.bf.1.2		4
32.13	even	8		9216.2.a.bf.1.3		4
32.19	odd	8		9216.2.a.bg.1.2		4
32.29	even	8		9216.2.a.bg.1.3		4
48.5	odd	4		2304.2.k.j.577.2	yes	8
48.11	even	4		2304.2.k.j.577.3	yes	8
48.29	odd	4	inner	2304.2.k.g.577.2	✓	8
48.35	even	4	inner	2304.2.k.g.577.3	yes	8
96.29	odd	8		9216.2.a.bg.1.3		4
96.35	even	8		9216.2.a.bf.1.2		4
96.77	odd	8		9216.2.a.bf.1.3		4
96.83	even	8		9216.2.a.bg.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2304.2.k.g.577.2	✓	8	16.13	even	4	inner
2304.2.k.g.577.2	✓	8	48.29	odd	4	inner
2304.2.k.g.577.3	yes	8	16.3	odd	4	inner
2304.2.k.g.577.3	yes	8	48.35	even	4	inner
2304.2.k.g.1729.2	yes	8	4.3	odd	2	inner
2304.2.k.g.1729.2	yes	8	12.11	even	2	inner
2304.2.k.g.1729.3	yes	8	1.1	even	1	trivial
2304.2.k.g.1729.3	yes	8	3.2	odd	2	CM
2304.2.k.j.577.2	yes	8	16.5	even	4
2304.2.k.j.577.2	yes	8	48.5	odd	4
2304.2.k.j.577.3	yes	8	16.11	odd	4
2304.2.k.j.577.3	yes	8	48.11	even	4
2304.2.k.j.1729.2	yes	8	8.3	odd	2
2304.2.k.j.1729.2	yes	8	24.11	even	2
2304.2.k.j.1729.3	yes	8	8.5	even	2
2304.2.k.j.1729.3	yes	8	24.5	odd	2
9216.2.a.bf.1.2		4	32.3	odd	8
9216.2.a.bf.1.2		4	96.35	even	8
9216.2.a.bf.1.3		4	32.13	even	8
9216.2.a.bf.1.3		4	96.77	odd	8
9216.2.a.bg.1.2		4	32.19	odd	8
9216.2.a.bg.1.2		4	96.83	even	8
9216.2.a.bg.1.3		4	32.29	even	8
9216.2.a.bg.1.3		4	96.29	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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+0.378937i q^{7} +O(q^{10})\)	\(q+0.378937i q^{7} +(2.46410 - 2.46410i) q^{13} +(2.44949 - 2.44949i) q^{19} -5.00000i q^{25} -10.1769 q^{31} +(-1.53590 - 1.53590i) q^{37} +(7.34847 + 7.34847i) q^{43} +6.85641 q^{49} +(10.4641 - 10.4641i) q^{61} +(11.3137 - 11.3137i) q^{67} -13.8564i q^{73} -9.41902 q^{79} +(0.933740 + 0.933740i) q^{91} +13.8564 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{13} - 40 q^{37} - 56 q^{49} + 56 q^{61}+O(q^{100}) \) 8 * q - 8 * q^13 - 40 * q^37 - 56 * q^49 + 56 * q^61

Introduction

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