LMFDB - Embedded newform 880.2.bd.h.593.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,2,Mod(417,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("880.417");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.bd (of order $4$, degree $2$, not 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:	$7.02683537787$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			593.2
Root			$-0.396143 + 1.68614i$ of defining polynomial
Character	$\chi$	$=$	880.593
Dual form			880.2.bd.h.417.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.29000 - 1.29000i) q^{3} +(2.12819 + 0.686141i) q^{5} +0.328185i q^{9} +O(q^{10})$	$q+(-1.29000 - 1.29000i) q^{3} +(2.12819 + 0.686141i) q^{5} +0.328185i q^{9} +3.31662 q^{11} +(-1.86025 - 3.63048i) q^{15} +(0.618183 + 0.618183i) q^{23} +(4.05842 + 2.92048i) q^{25} +(-3.44663 + 3.44663i) q^{27} +9.30506 q^{31} +(-4.27844 - 4.27844i) q^{33} +(5.51480 - 5.51480i) q^{37} +(-0.225181 + 0.698442i) q^{45} +(2.68338 - 2.68338i) q^{47} -7.00000i q^{49} +(-9.63325 - 9.63325i) q^{53} +(7.05842 + 2.27567i) q^{55} -14.6487i q^{59} +(-9.17506 + 9.17506i) q^{67} -1.59491i q^{69} +12.8614 q^{71} +(-1.46794 - 9.00277i) q^{75} +9.87685 q^{81} +18.8614i q^{89} +(-12.0035 - 12.0035i) q^{93} +(9.79026 - 9.79026i) q^{97} +1.08847i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3}+O(q^{10}) $ 8 * q - 2 * q^3	$ 8 q - 2 q^{3} + 16 q^{15} + 18 q^{23} - 2 q^{25} - 26 q^{27} + 22 q^{33} + 14 q^{37} + 18 q^{45} + 48 q^{47} - 24 q^{53} + 22 q^{55} + 26 q^{67} - 12 q^{71} - 64 q^{75} - 100 q^{81} + 18 q^{93} + 34 q^{97}+O(q^{100}) $ 8 * q - 2 * q^3 + 16 * q^15 + 18 * q^23 - 2 * q^25 - 26 * q^27 + 22 * q^33 + 14 * q^37 + 18 * q^45 + 48 * q^47 - 24 * q^53 + 22 * q^55 + 26 * q^67 - 12 * q^71 - 64 * q^75 - 100 * q^81 + 18 * q^93 + 34 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.29000	−	1.29000i	−0.744780	−	0.744780i	0.228714	−	0.973494i	$-0.426548\pi$
$3$	−1.29000	−	1.29000i	−0.744780	−	0.744780i	−0.973494	+	0.228714i	$0.926548\pi$
$4$		0			0
$5$	2.12819	+	0.686141i	0.951757	+	0.306851i
$5$	2.12819	+	0.686141i	0.951757	+	0.306851i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$			0.328185i			0.109395i
$10$		0			0
$11$	3.31662			1.00000
$11$	3.31662			1.00000
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$	−1.86025	−	3.63048i	−0.480313	−	0.937387i
$16$		0			0
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.618183	+	0.618183i	0.128900	+	0.128900i	0.768613	−	0.639713i	$-0.220947\pi$
$23$	0.618183	+	0.618183i	0.128900	+	0.128900i	−0.639713	+	0.768613i	$0.720947\pi$
$24$		0			0
$25$	4.05842	+	2.92048i	0.811684	+	0.584096i
$26$		0			0
$27$	−3.44663	+	3.44663i	−0.663305	+	0.663305i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	9.30506			1.67124			0.835619	−	0.549309i	$-0.185109\pi$
$31$	9.30506			1.67124			0.835619	+	0.549309i	$0.185109\pi$
$32$		0			0
$33$	−4.27844	−	4.27844i	−0.744780	−	0.744780i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.51480	−	5.51480i	0.906628	−	0.906628i	−0.0893706	−	0.995998i	$-0.528486\pi$
$37$	5.51480	−	5.51480i	0.906628	−	0.906628i	0.995998	+	0.0893706i	$0.0284856\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	−0.225181	+	0.698442i	−0.0335681	+	0.104118i
$46$		0			0
$47$	2.68338	−	2.68338i	0.391411	−	0.391411i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	2.68338	−	2.68338i	0.391411	−	0.391411i	0.875190	+	0.483779i	$0.160736\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.63325	−	9.63325i	−1.32323	−	1.32323i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	−9.63325	−	9.63325i	−1.32323	−	1.32323i	−0.412082	−	0.911147i	$-0.635198\pi$
$54$		0			0
$55$	7.05842	+	2.27567i	0.951757	+	0.306851i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	14.6487i		−	1.90710i	−0.301239	−	0.953549i	$-0.597400\pi$
$59$		−	14.6487i		−	1.90710i	0.301239	−	0.953549i	$-0.402600\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−9.17506	+	9.17506i	−1.12091	+	1.12091i	−0.129307	+	0.991605i	$0.541275\pi$
$67$	−9.17506	+	9.17506i	−1.12091	+	1.12091i	−0.991605	+	0.129307i	$0.958725\pi$
$68$		0			0
$69$		−	1.59491i		−	0.192004i
$70$		0			0
$71$	12.8614			1.52637			0.763184	−	0.646181i	$-0.223635\pi$
$71$	12.8614			1.52637			0.763184	+	0.646181i	$0.223635\pi$
$72$		0			0
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$	−1.46794	−	9.00277i	−0.169503	−	1.03955i
$76$		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
$81$	9.87685			1.09743
$82$		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
$89$			18.8614i			1.99931i	0.0263586	+	0.999653i	$0.491609\pi$
$89$			18.8614i			1.99931i	−0.0263586	+	0.999653i	$0.508391\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−12.0035	−	12.0035i	−1.24471	−	1.24471i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9.79026	−	9.79026i	0.994051	−	0.994051i	−0.00593185	−	0.999982i	$-0.501888\pi$
$97$	9.79026	−	9.79026i	0.994051	−	0.994051i	0.999982	+	0.00593185i	$0.00188818\pi$
$98$		0			0
$99$			1.08847i			0.109395i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−11.9499	−	11.9499i	−1.17746	−	1.17746i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	−11.9499	−	11.9499i	−1.17746	−	1.17746i	−0.197066	−	0.980390i	$-0.563141\pi$
$104$		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
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−14.2282			−1.35048
$112$		0			0
$113$	14.9503	+	14.9503i	1.40640	+	1.40640i	0.777422	+	0.628979i	$0.216527\pi$
$113$	14.9503	+	14.9503i	1.40640	+	1.40640i	0.628979	+	0.777422i	$0.283473\pi$
$114$		0			0
$115$	0.891452	+	1.73977i	0.0831284	+	0.162235i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.63325	+	9.00000i	0.593296	+	0.804984i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−9.69998	+	4.97023i	−0.834841	+	0.427769i
$136$		0			0
$137$	−15.3080	+	15.3080i	−1.30785	+	1.30785i	−0.384893	+	0.922961i	$0.625762\pi$
$137$	−15.3080	+	15.3080i	−1.30785	+	1.30785i	−0.922961	+	0.384893i	$0.874238\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−6.92309			−0.583030
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−9.02998	+	9.02998i	−0.744780	+	0.744780i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	19.8030	+	6.38458i	1.59061	+	0.512822i
$156$		0			0
$157$	−17.5302	+	17.5302i	−1.39907	+	1.39907i	−0.596316	+	0.802749i	$0.703370\pi$
$157$	−17.5302	+	17.5302i	−1.39907	+	1.39907i	−0.802749	+	0.596316i	$0.796630\pi$
$158$		0			0
$159$			24.8537i			1.97103i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.94987	−	1.94987i	−0.152726	−	0.152726i	0.626608	−	0.779334i	$-0.284443\pi$
$163$	−1.94987	−	1.94987i	−0.152726	−	0.152726i	−0.779334	+	0.626608i	$0.784443\pi$
$164$		0			0
$165$	−6.16974	−	12.0410i	−0.480313	−	0.937387i
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		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
$175$		0			0
$176$		0			0
$177$	−18.8968	+	18.8968i	−1.42037	+	1.42037i
$178$		0			0
$179$			24.8614i			1.85823i	0.369792	+	0.929114i	$0.379429\pi$
$179$			24.8614i			1.85823i	−0.369792	+	0.929114i	$0.620571\pi$
$180$		0			0
$181$	−26.6256			−1.97906			−0.989532	−	0.144316i	$-0.953902\pi$
$181$	−26.6256			−1.97906			−0.989532	+	0.144316i	$0.953902\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.5205	−	7.95264i	1.14109	−	0.584690i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.38219			0.100012			0.0500060	−	0.998749i	$-0.484076\pi$
$191$	1.38219			0.100012			0.0500060	+	0.998749i	$0.484076\pi$
$192$		0			0
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		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
$199$			19.8997i			1.41066i	0.708881	+	0.705328i	$0.249200\pi$
$199$			19.8997i			1.41066i	−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	23.6716			1.66967
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.202878	+	0.202878i	−0.0141010	+	0.0141010i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	−16.5912	−	16.5912i	−1.13681	−	1.13681i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−20.2047	−	20.2047i	−1.35300	−	1.35300i	−0.882281	−	0.470723i	$-0.843993\pi$
$223$	−20.2047	−	20.2047i	−1.35300	−	1.35300i	−0.470723	−	0.882281i	$-0.656007\pi$
$224$		0			0
$225$	−0.958459	+	1.33191i	−0.0638973	+	0.0887943i
$226$		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
$229$		−	10.5947i		−	0.700116i	−0.936728	−	0.350058i	$-0.886162\pi$
$229$		−	10.5947i		−	0.700116i	0.936728	−	0.350058i	$-0.113838\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$	7.55192	−	3.86957i	0.492633	−	0.252423i
$236$		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
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−2.40121	−	2.40121i	−0.154038	−	0.154038i
$244$		0			0
$245$	4.80298	−	14.8974i	0.306851	−	0.951757i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.861407			−0.0543715			−0.0271858	−	0.999630i	$-0.508655\pi$
$251$	−0.861407			−0.0543715			−0.0271858	+	0.999630i	$0.508655\pi$
$252$		0			0
$253$	2.05028	+	2.05028i	0.128900	+	0.128900i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.26650	+	4.26650i	−0.266137	+	0.266137i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	−4.26650	+	4.26650i	−0.266137	+	0.266137i	0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$	−13.8917	−	27.1112i	−0.853358	−	1.66543i
$266$		0			0
$267$	24.3312	−	24.3312i	1.48904	−	1.48904i
$268$		0			0
$269$			13.2665i			0.808873i	0.914566	+	0.404436i	$0.132532\pi$
$269$			13.2665i			0.808873i	−0.914566	+	0.404436i	$0.867468\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	13.4603	+	9.68614i	0.811684	+	0.584096i
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$			3.05379i			0.182825i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		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
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$	−25.2588			−1.48070
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$	10.0511	−	31.1753i	0.585196	−	1.81509i
$296$		0			0
$297$	−11.4312	+	11.4312i	−0.663305	+	0.663305i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		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
$309$			30.8306i			1.75389i
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	23.8649	+	23.8649i	1.34892	+	1.34892i	0.886831	+	0.462093i	$0.152902\pi$
$313$	23.8649	+	23.8649i	1.34892	+	1.34892i	0.462093	+	0.886831i	$0.347098\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	22.6902	−	22.6902i	1.27441	−	1.27441i	0.330661	−	0.943750i	$-0.392728\pi$
$317$	22.6902	−	22.6902i	1.27441	−	1.27441i	0.943750	−	0.330661i	$-0.107272\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−25.3360			−1.39259			−0.696295	−	0.717756i	$-0.745169\pi$
$331$	−25.3360			−1.39259			−0.696295	+	0.717756i	$0.745169\pi$
$332$		0			0
$333$	1.80988	+	1.80988i	0.0991807	+	0.0991807i
$334$		0			0
$335$	−25.8217	+	13.2309i	−1.41079	+	0.722883i
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$		−	38.5716i		−	2.09492i
$340$		0			0
$341$	30.8614			1.67124
$342$		0			0
$343$		0			0
$344$		0			0
$345$	1.09433	−	3.39427i	0.0589168	−	0.182742i
$346$		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
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.529715	+	0.529715i	0.0281939	+	0.0281939i	0.721063	−	0.692869i	$-0.243654\pi$
$353$	0.529715	+	0.529715i	0.0281939	+	0.0281939i	−0.692869	+	0.721063i	$0.743654\pi$
$354$		0			0
$355$	27.3716	+	8.82473i	1.45273	+	0.468368i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	−14.1900	−	14.1900i	−0.744780	−	0.744780i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−23.4505	+	23.4505i	−1.22411	+	1.22411i	−0.257948	+	0.966159i	$0.583046\pi$
$367$	−23.4505	+	23.4505i	−1.22411	+	1.22411i	−0.966159	+	0.257948i	$0.916954\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$	3.05310	−	20.1668i	0.157661	−	1.04141i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	6.72582i		−	0.345482i	−0.984967	−	0.172741i	$-0.944738\pi$
$379$		−	6.72582i		−	0.345482i	0.984967	−	0.172741i	$-0.0552624\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	26.0305	+	26.0305i	1.33010	+	1.33010i	0.905279	+	0.424818i	$0.139662\pi$
$383$	26.0305	+	26.0305i	1.33010	+	1.33010i	0.424818	+	0.905279i	$0.360338\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			5.25106i			0.266239i	0.991100	+	0.133120i	$0.0424994\pi$
$389$			5.25106i			0.266239i	−0.991100	+	0.133120i	$0.957501\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	18.8997	−	18.8997i	0.948551	−	0.948551i	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	18.8997	−	18.8997i	0.948551	−	0.948551i	0.998740	+	0.0501886i	$0.0159822\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	26.5330			1.32499			0.662497	−	0.749064i	$-0.269497\pi$
$401$	26.5330			1.32499			0.662497	+	0.749064i	$0.269497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	21.0199	+	6.77691i	1.04449	+	0.336747i
$406$		0			0
$407$	18.2905	−	18.2905i	0.906628	−	0.906628i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	39.4947			1.94813
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			24.0000i			1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$			24.0000i			1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$		0			0
$421$	−39.7995			−1.93971			−0.969854	−	0.243685i	$-0.921644\pi$
$421$	−39.7995			−1.93971			−0.969854	+	0.243685i	$0.921644\pi$
$422$		0			0
$423$	0.880645	+	0.880645i	0.0428184	+	0.0428184i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−25.2702	−	25.2702i	−1.21441	−	1.21441i	−0.969561	−	0.244848i	$-0.921262\pi$
$433$	−25.2702	−	25.2702i	−1.21441	−	1.21441i	−0.244848	−	0.969561i	$-0.578738\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	2.29730			0.109395
$442$		0			0
$443$	28.7615	+	28.7615i	1.36650	+	1.36650i	0.865373	+	0.501129i	$0.167082\pi$
$443$	28.7615	+	28.7615i	1.36650	+	1.36650i	0.501129	+	0.865373i	$0.332918\pi$
$444$		0			0
$445$	−12.9416	+	40.1407i	−0.613490	+	1.90285i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			5.13859i			0.242505i	0.992622	+	0.121253i	$0.0386911\pi$
$449$			5.13859i			0.242505i	−0.992622	+	0.121253i	$0.961309\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−15.7105	−	15.7105i	−0.730130	−	0.730130i	0.240515	−	0.970645i	$-0.422684\pi$
$463$	−15.7105	−	15.7105i	−0.730130	−	0.730130i	−0.970645	+	0.240515i	$0.922684\pi$
$464$		0			0
$465$	−17.3097	−	33.7819i	−0.802718	−	1.56660i
$466$		0			0
$467$	−29.9979	+	29.9979i	−1.38814	+	1.38814i	−0.558906	+	0.829231i	$0.688779\pi$
$467$	−29.9979	+	29.9979i	−1.38814	+	1.38814i	−0.829231	+	0.558906i	$0.811221\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	45.2279			2.08399
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	3.16149	−	3.16149i	0.144755	−	0.144755i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	27.5531	−	14.1181i	1.25112	−	0.641069i
$486$		0			0
$487$	31.1905	−	31.1905i	1.41338	−	1.41338i	0.682362	−	0.731014i	$-0.260953\pi$
$487$	31.1905	−	31.1905i	1.41338	−	1.41338i	0.731014	−	0.682362i	$-0.239047\pi$
$488$		0			0
$489$			5.03066i			0.227495i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.746842	+	2.31647i	−0.0335681	+	0.104118i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			19.8997i			0.890835i	0.895323	+	0.445418i	$0.146945\pi$
$499$			19.8997i			0.890835i	−0.895323	+	0.445418i	$0.853055\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	16.7700	−	16.7700i	0.744780	−	0.744780i
$508$		0			0
$509$		−	37.3128i		−	1.65386i	−0.562303	−	0.826931i	$-0.690085\pi$
$509$		−	37.3128i		−	1.65386i	0.562303	−	0.826931i	$-0.309915\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−17.2324	−	33.6309i	−0.759349	−	1.48196i
$516$		0			0
$517$	8.89975	−	8.89975i	0.391411	−	0.391411i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−34.5484			−1.51359			−0.756797	−	0.653650i	$-0.773237\pi$
$521$	−34.5484			−1.51359			−0.756797	+	0.653650i	$0.773237\pi$
$522$		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
$529$		−	22.2357i		−	0.966770i
$530$		0			0
$531$	4.80749			0.208627
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	32.0711	−	32.0711i	1.38397	−	1.38397i
$538$		0			0
$539$		−	23.2164i		−	1.00000i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	34.3469	+	34.3469i	1.47397	+	1.47397i
$544$		0			0
$545$		0			0
$546$		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
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−30.2803	−	9.76252i	−1.28533	−	0.414396i
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		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
$565$	21.5591	+	42.0750i	0.906997	+	1.77011i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−1.78303	−	1.78303i	−0.0744870	−	0.0744870i
$574$		0			0
$575$	0.703455	+	4.31424i	0.0293361	+	0.179916i
$576$		0			0
$577$	−1.80571	+	1.80571i	−0.0751725	+	0.0751725i	−0.743693	−	0.668521i	$-0.766928\pi$
$577$	−1.80571	+	1.80571i	−0.0751725	+	0.0751725i	0.668521	+	0.743693i	$0.266928\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−31.9499	−	31.9499i	−1.32323	−	1.32323i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−27.3166	+	27.3166i	−1.12748	+	1.12748i	−0.136892	+	0.990586i	$0.543711\pi$
$587$	−27.3166	+	27.3166i	−1.12748	+	1.12748i	−0.990586	+	0.136892i	$0.956289\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	25.6706	−	25.6706i	1.05063	−	1.05063i
$598$		0			0
$599$		−	36.0000i		−	1.47092i	−0.677568	−	0.735460i	$-0.736966\pi$
$599$		−	36.0000i		−	1.47092i	0.677568	−	0.735460i	$-0.263034\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−3.01112	−	3.01112i	−0.122622	−	0.122622i
$604$		0			0
$605$	23.4101	+	7.54755i	0.951757	+	0.306851i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.2665	+	34.2665i	−1.37952	+	1.37952i	−0.534089	+	0.845428i	$0.679345\pi$
$617$	−34.2665	+	34.2665i	−1.37952	+	1.37952i	−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$		−	42.5842i		−	1.71160i	−0.517303	−	0.855802i	$-0.673064\pi$
$619$		−	42.5842i		−	1.71160i	0.517303	−	0.855802i	$-0.326936\pi$
$620$		0			0
$621$	−4.26130			−0.171000
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.94158	+	23.7051i	0.317663	+	0.948204i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	46.5842			1.85449			0.927244	−	0.374457i	$-0.122171\pi$
$631$	46.5842			1.85449			0.927244	+	0.374457i	$0.122171\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$			4.22093i			0.166977i
$640$		0			0
$641$	50.5793			1.99776			0.998882	−	0.0472793i	$-0.0150551\pi$
$641$	50.5793			1.99776			0.998882	+	0.0472793i	$0.0150551\pi$
$642$		0			0
$643$	−12.8842	−	12.8842i	−0.508101	−	0.508101i	0.405842	−	0.913943i	$-0.366978\pi$
$643$	−12.8842	−	12.8842i	−0.508101	−	0.508101i	−0.913943	+	0.405842i	$0.866978\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10.5505	+	10.5505i	−0.414785	+	0.414785i	−0.883402	−	0.468617i	$-0.844753\pi$
$647$	−10.5505	+	10.5505i	−0.414785	+	0.414785i	0.468617	+	0.883402i	$0.344753\pi$
$648$		0			0
$649$		−	48.5842i		−	1.90710i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.5989	−	11.5989i	−0.453902	−	0.453902i	0.442746	−	0.896647i	$-0.354005\pi$
$653$	−11.5989	−	11.5989i	−0.453902	−	0.453902i	−0.896647	+	0.442746i	$0.854005\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−36.5842			−1.42296			−0.711481	−	0.702706i	$-0.751975\pi$
$661$	−36.5842			−1.42296			−0.711481	+	0.702706i	$0.751975\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			52.1279i			2.01538i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$	−24.0537	+	3.92206i	−0.925828	+	0.150960i
$676$		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
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.2164	+	11.2164i	0.429183	+	0.429183i	0.888350	−	0.459167i	$-0.151852\pi$
$683$	11.2164	+	11.2164i	0.429183	+	0.429183i	−0.459167	+	0.888350i	$0.651852\pi$
$684$		0			0
$685$	−43.0820	+	22.0750i	−1.64608	+	0.843443i
$686$		0			0
$687$	−13.6671	+	13.6671i	−0.521433	+	0.521433i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−34.5842			−1.31565			−0.657823	−	0.753173i	$-0.728522\pi$
$691$	−34.5842			−1.31565			−0.657823	+	0.753173i	$0.728522\pi$
$692$		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		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−14.7337	−	4.75022i	−0.554903	−	0.178903i
$706$		0			0
$707$		0			0
$708$		0			0
$709$			52.5842i			1.97484i	0.158114	+	0.987421i	$0.449459\pi$
$709$			52.5842i			1.97484i	−0.158114	+	0.987421i	$0.550541\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.75223	+	5.75223i	0.215423	+	0.215423i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			11.1386i			0.415399i	0.978193	+	0.207700i	$0.0665977\pi$
$719$			11.1386i			0.415399i	−0.978193	+	0.207700i	$0.933402\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−16.4956	+	16.4956i	−0.611787	+	0.611787i	−0.943411	−	0.331625i	$-0.892403\pi$
$727$	−16.4956	+	16.4956i	−0.611787	+	0.611787i	0.331625	+	0.943411i	$0.392403\pi$
$728$		0			0
$729$		−	23.4354i		−	0.867979i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$	−25.4134	+	13.0217i	−0.937387	+	0.480313i
$736$		0			0
$737$	−30.4302	+	30.4302i	−1.12091	+	1.12091i
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−54.5842			−1.99181			−0.995903	−	0.0904254i	$-0.971177\pi$
$751$	−54.5842			−1.99181			−0.995903	+	0.0904254i	$0.971177\pi$
$752$		0			0
$753$	1.11121	+	1.11121i	0.0404948	+	0.0404948i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	38.8997	−	38.8997i	1.41384	−	1.41384i	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.8997	−	38.8997i	1.41384	−	1.41384i	0.723269	−	0.690567i	$-0.242639\pi$
$758$		0			0
$759$		−	5.28971i		−	0.192004i
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	11.0075			0.396427
$772$		0			0
$773$	20.3668	+	20.3668i	0.732541	+	0.732541i	0.971123	−	0.238581i	$-0.0766824\pi$
$773$	20.3668	+	20.3668i	0.732541	+	0.732541i	−0.238581	+	0.971123i	$0.576682\pi$
$774$		0			0
$775$	37.7639	+	27.1753i	1.35652	+	0.976164i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	42.6565			1.52637
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−49.3360	+	25.2796i	−1.76088	+	0.902266i
$786$		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
$793$		0			0
$794$		0			0
$795$	−17.0532	+	52.8936i	−0.604813	+	1.87594i
$796$		0			0
$797$	8.26970	−	8.26970i	0.292928	−	0.292928i	−0.545308	−	0.838236i	$-0.683587\pi$
$797$	8.26970	−	8.26970i	0.292928	−	0.292928i	0.838236	+	0.545308i	$0.183587\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−6.19004			−0.218714
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	17.1137	−	17.1137i	0.602432	−	0.602432i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.81182	−	5.48760i	−0.0984939	−	0.192222i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	38.9305	+	38.9305i	1.35703	+	1.35703i	0.877555	+	0.479477i	$0.159174\pi$
$823$	38.9305	+	38.9305i	1.35703	+	1.35703i	0.479477	+	0.877555i	$0.340826\pi$
$824$		0			0
$825$	−4.86861	−	29.8588i	−0.169503	−	1.03955i
$826$		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
$829$		−	28.5842i		−	0.992771i	−0.868102	−	0.496385i	$-0.834660\pi$
$829$		−	28.5842i		−	0.992771i	0.868102	−	0.496385i	$-0.165340\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−32.0711	+	32.0711i	−1.10854	+	1.10854i
$838$		0			0
$839$		−	57.2126i		−	1.97520i	−0.156999	−	0.987599i	$-0.550182\pi$
$839$		−	57.2126i		−	1.97520i	0.156999	−	0.987599i	$-0.449818\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.91983	+	27.6665i	−0.306851	+	0.951757i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.81831			0.233729
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$			58.5842i			1.99887i	0.0336436	+	0.999434i	$0.489289\pi$
$859$			58.5842i			1.99887i	−0.0336436	+	0.999434i	$0.510711\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	41.2164	+	41.2164i	1.40302	+	1.40302i	0.790295	+	0.612727i	$0.209928\pi$
$863$	41.2164	+	41.2164i	1.40302	+	1.40302i	0.612727	+	0.790295i	$0.290072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−21.9300	+	21.9300i	−0.744780	+	0.744780i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	3.21302	+	3.21302i	0.108744	+	0.108744i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−42.8614			−1.44404			−0.722019	−	0.691873i	$-0.756786\pi$
$881$	−42.8614			−1.44404			−0.722019	+	0.691873i	$0.756786\pi$
$882$		0			0
$883$	18.0501	+	18.0501i	0.607435	+	0.607435i	0.942275	−	0.334840i	$-0.108682\pi$
$883$	18.0501	+	18.0501i	0.607435	+	0.607435i	−0.334840	+	0.942275i	$0.608682\pi$
$884$		0			0
$885$	−53.1818	+	27.2502i	−1.78769	+	0.916004i
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	32.7578			1.09743
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−17.0584	+	52.9099i	−0.570200	+	1.76858i
$896$		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
$905$	−56.6644	−	18.2689i	−1.88359	−	0.607278i
$906$		0			0
$907$	25.8496	−	25.8496i	0.858323	−	0.858323i	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	25.8496	−	25.8496i	0.858323	−	0.858323i	0.991140	+	0.132818i	$0.0424025\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.63325			0.219769			0.109885	−	0.993944i	$-0.464952\pi$
$911$	6.63325			0.219769			0.109885	+	0.993944i	$0.464952\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	38.4873	−	6.27552i	1.26545	−	0.206338i
$926$		0			0
$927$	3.92177	−	3.92177i	0.128808	−	0.128808i
$928$		0			0
$929$		−	53.0660i		−	1.74104i	−0.492134	−	0.870519i	$-0.663783\pi$
$929$		−	53.0660i		−	1.74104i	0.492134	−	0.870519i	$-0.336217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	15.4800	+	15.4800i	0.506791	+	0.506791i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		−	61.5713i		−	2.00930i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26.2888	−	26.2888i	0.854271	−	0.854271i	−0.136385	−	0.990656i	$-0.543548\pi$
$947$	26.2888	−	26.2888i	0.854271	−	0.854271i	0.990656	+	0.136385i	$0.0435483\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−58.5407			−1.89831
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$	2.94158	+	0.948380i	0.0951872	+	0.0306888i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	55.5842			1.79304
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−17.4131			−0.558812			−0.279406	−	0.960173i	$-0.590138\pi$
$971$	−17.4131			−0.558812			−0.279406	+	0.960173i	$0.590138\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−43.4514	+	43.4514i	−1.39013	+	1.39013i	−0.565134	+	0.824999i	$0.691176\pi$
$977$	−43.4514	+	43.4514i	−1.39013	+	1.39013i	−0.824999	+	0.565134i	$0.808824\pi$
$978$		0			0
$979$			62.5562i			1.99931i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−41.5105	−	41.5105i	−1.32398	−	1.32398i	−0.910525	−	0.413453i	$-0.864323\pi$
$983$	−41.5105	−	41.5105i	−1.32398	−	1.32398i	−0.413453	−	0.910525i	$-0.635677\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−59.6992			−1.89641			−0.948205	−	0.317660i	$-0.897103\pi$
$991$	−59.6992			−1.89641			−0.948205	+	0.317660i	$0.897103\pi$
$992$		0			0
$993$	32.6833	+	32.6833i	1.03717	+	1.03717i
$994$		0			0
$995$	−13.6540	+	42.3505i	−0.432862	+	1.34260i
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$			38.0150i			1.20274i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.bd.h.593.2		8
4.3	odd	2		220.2.k.b.153.3	✓	8
5.2	odd	4	inner	880.2.bd.h.417.2		8
11.10	odd	2	CM	880.2.bd.h.593.2		8
12.11	even	2		1980.2.y.b.1693.1		8
20.3	even	4		1100.2.k.b.857.2		8
20.7	even	4		220.2.k.b.197.3	yes	8
20.19	odd	2		1100.2.k.b.593.2		8
44.43	even	2		220.2.k.b.153.3	✓	8
55.32	even	4	inner	880.2.bd.h.417.2		8
60.47	odd	4		1980.2.y.b.1297.1		8
132.131	odd	2		1980.2.y.b.1693.1		8
220.43	odd	4		1100.2.k.b.857.2		8
220.87	odd	4		220.2.k.b.197.3	yes	8
220.219	even	2		1100.2.k.b.593.2		8
660.527	even	4		1980.2.y.b.1297.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.k.b.153.3	✓	8	4.3	odd	2
220.2.k.b.153.3	✓	8	44.43	even	2
220.2.k.b.197.3	yes	8	20.7	even	4
220.2.k.b.197.3	yes	8	220.87	odd	4
880.2.bd.h.417.2		8	5.2	odd	4	inner
880.2.bd.h.417.2		8	55.32	even	4	inner
880.2.bd.h.593.2		8	1.1	even	1	trivial
880.2.bd.h.593.2		8	11.10	odd	2	CM
1100.2.k.b.593.2		8	20.19	odd	2
1100.2.k.b.593.2		8	220.219	even	2
1100.2.k.b.857.2		8	20.3	even	4
1100.2.k.b.857.2		8	220.43	odd	4
1980.2.y.b.1297.1		8	60.47	odd	4
1980.2.y.b.1297.1		8	660.527	even	4
1980.2.y.b.1693.1		8	12.11	even	2
1980.2.y.b.1693.1		8	132.131	odd	2

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:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.bd (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.29000 - 1.29000i) q^{3} +(2.12819 + 0.686141i) q^{5} +0.328185i q^{9} +O(q^{10})\)	\(q+(-1.29000 - 1.29000i) q^{3} +(2.12819 + 0.686141i) q^{5} +0.328185i q^{9} +3.31662 q^{11} +(-1.86025 - 3.63048i) q^{15} +(0.618183 + 0.618183i) q^{23} +(4.05842 + 2.92048i) q^{25} +(-3.44663 + 3.44663i) q^{27} +9.30506 q^{31} +(-4.27844 - 4.27844i) q^{33} +(5.51480 - 5.51480i) q^{37} +(-0.225181 + 0.698442i) q^{45} +(2.68338 - 2.68338i) q^{47} -7.00000i q^{49} +(-9.63325 - 9.63325i) q^{53} +(7.05842 + 2.27567i) q^{55} -14.6487i q^{59} +(-9.17506 + 9.17506i) q^{67} -1.59491i q^{69} +12.8614 q^{71} +(-1.46794 - 9.00277i) q^{75} +9.87685 q^{81} +18.8614i q^{89} +(-12.0035 - 12.0035i) q^{93} +(9.79026 - 9.79026i) q^{97} +1.08847i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3}+O(q^{10}) \) 8 * q - 2 * q^3	\( 8 q - 2 q^{3} + 16 q^{15} + 18 q^{23} - 2 q^{25} - 26 q^{27} + 22 q^{33} + 14 q^{37} + 18 q^{45} + 48 q^{47} - 24 q^{53} + 22 q^{55} + 26 q^{67} - 12 q^{71} - 64 q^{75} - 100 q^{81} + 18 q^{93} + 34 q^{97}+O(q^{100}) \) 8 * q - 2 * q^3 + 16 * q^15 + 18 * q^23 - 2 * q^25 - 26 * q^27 + 22 * q^33 + 14 * q^37 + 18 * q^45 + 48 * q^47 - 24 * q^53 + 22 * q^55 + 26 * q^67 - 12 * q^71 - 64 * q^75 - 100 * q^81 + 18 * q^93 + 34 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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