LMFDB - Embedded newform 80.6.n.a.63.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,6,Mod(47,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("80.47");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	80.n (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:	$12.8307055850$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			63.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	80.63
Dual form			80.6.n.a.47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-38.0000 - 41.0000i) q^{5} +243.000i q^{9} +O(q^{10})$	$q+(-38.0000 - 41.0000i) q^{5} +243.000i q^{9} +(475.000 + 475.000i) q^{13} +(-1525.00 + 1525.00i) q^{17} +(-237.000 + 3116.00i) q^{25} +8564.00i q^{29} +(-475.000 + 475.000i) q^{37} -4952.00 q^{41} +(9963.00 - 9234.00i) q^{45} -16807.0i q^{49} +(-16475.0 - 16475.0i) q^{53} +54948.0 q^{61} +(1425.00 - 37525.0i) q^{65} +(-54475.0 - 54475.0i) q^{73} -59049.0 q^{81} +(120475. + 4575.00i) q^{85} +140464. i q^{89} +(-126475. + 126475. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 76 q^{5}+O(q^{10}) $ 2 * q - 76 * q^5	$ 2 q - 76 q^{5} + 950 q^{13} - 3050 q^{17} - 474 q^{25} - 950 q^{37} - 9904 q^{41} + 19926 q^{45} - 32950 q^{53} + 109896 q^{61} + 2850 q^{65} - 108950 q^{73} - 118098 q^{81} + 240950 q^{85} - 252950 q^{97}+O(q^{100}) $ 2 * q - 76 * q^5 + 950 * q^13 - 3050 * q^17 - 474 * q^25 - 950 * q^37 - 9904 * q^41 + 19926 * q^45 - 32950 * q^53 + 109896 * q^61 + 2850 * q^65 - 108950 * q^73 - 118098 * q^81 + 240950 * q^85 - 252950 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

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
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		0			0
$5$	−38.0000	−	41.0000i	−0.679765	−	0.733430i
$5$	−38.0000	−	41.0000i	−0.679765	−	0.733430i
$6$		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
$9$			243.000i			1.00000i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	475.000	+	475.000i	0.779534	+	0.779534i	0.979752	−	0.200217i	$-0.0641648\pi$
$13$	475.000	+	475.000i	0.779534	+	0.779534i	−0.200217	+	0.979752i	$0.564165\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1525.00	+	1525.00i	−1.27982	+	1.27982i	−0.339046	+	0.940770i	$0.610104\pi$
$17$	−1525.00	+	1525.00i	−1.27982	+	1.27982i	−0.940770	+	0.339046i	$0.889896\pi$
$18$		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			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	−237.000	+	3116.00i	−0.0758400	+	0.997120i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8564.00i			1.89096i	0.325684	+	0.945479i	$0.394405\pi$
$29$			8564.00i			1.89096i	−0.325684	+	0.945479i	$0.605595\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−475.000	+	475.000i	−0.0570413	+	0.0570413i	−0.735052	−	0.678011i	$-0.762842\pi$
$37$	−475.000	+	475.000i	−0.0570413	+	0.0570413i	0.678011	+	0.735052i	$0.262842\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4952.00			−0.460067			−0.230033	−	0.973183i	$-0.573884\pi$
$41$	−4952.00			−0.460067			−0.230033	+	0.973183i	$0.573884\pi$
$42$		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
$45$	9963.00	−	9234.00i	0.733430	−	0.679765i
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		−	16807.0i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−16475.0	−	16475.0i	−0.805630	−	0.805630i	0.178339	−	0.983969i	$-0.442928\pi$
$53$	−16475.0	−	16475.0i	−0.805630	−	0.805630i	−0.983969	+	0.178339i	$0.942928\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	54948.0			1.89072			0.945360	−	0.326028i	$-0.105710\pi$
$61$	54948.0			1.89072			0.945360	+	0.326028i	$0.105710\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1425.00	−	37525.0i	0.0418342	−	1.10163i
$66$		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
$73$	−54475.0	−	54475.0i	−1.19644	−	1.19644i	−0.975226	−	0.221212i	$-0.928999\pi$
$73$	−54475.0	−	54475.0i	−1.19644	−	1.19644i	−0.221212	−	0.975226i	$-0.571001\pi$
$74$		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
$81$	−59049.0			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	120475.	+	4575.00i	1.80863	+	0.0686821i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			140464.i			1.87971i	0.341579	+	0.939853i	$0.389038\pi$
$89$			140464.i			1.87971i	−0.341579	+	0.939853i	$0.610962\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−126475.	+	126475.i	−1.36482	+	1.36482i	−0.497162	+	0.867657i	$0.665625\pi$
$97$	−126475.	+	126475.i	−1.36482	+	1.36482i	−0.867657	+	0.497162i	$0.834375\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	98002.0			0.955942			0.477971	−	0.878376i	$-0.341372\pi$
$101$	98002.0			0.955942			0.477971	+	0.878376i	$0.341372\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		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
$109$			246486.i			1.98713i	0.113269	+	0.993564i	$0.463868\pi$
$109$			246486.i			1.98713i	−0.113269	+	0.993564i	$0.536132\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−181425.	−	181425.i	−1.33660	−	1.33660i	−0.899332	−	0.437267i	$-0.855947\pi$
$113$	−181425.	−	181425.i	−1.33660	−	1.33660i	−0.437267	−	0.899332i	$-0.644053\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−115425.	+	115425.i	−0.779534	+	0.779534i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	161051.			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	136762.	−	108691.i	0.782871	−	0.622184i
$126$		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
$135$		0			0
$136$		0			0
$137$	−178575.	+	178575.i	−0.812867	+	0.812867i	−0.985063	−	0.172196i	$-0.944914\pi$
$137$	−178575.	+	178575.i	−0.812867	+	0.812867i	0.172196	+	0.985063i	$0.444914\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	351124.	−	325432.i	1.38689	−	1.28541i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	47614.0i		−	0.175699i	−0.996134	−	0.0878494i	$-0.972001\pi$
$149$		−	47614.0i		−	0.175699i	0.996134	−	0.0878494i	$-0.0279994\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−370575.	−	370575.i	−1.27982	−	1.27982i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	432475.	−	432475.i	1.40027	−	1.40027i	0.601086	−	0.799185i	$-0.294735\pi$
$157$	432475.	−	432475.i	1.40027	−	1.40027i	0.799185	−	0.601086i	$-0.205265\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			79957.0i			0.215347i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−556575.	−	556575.i	−1.41387	−	1.41387i	−0.722596	−	0.691271i	$-0.757051\pi$
$173$	−556575.	−	556575.i	−1.41387	−	1.41387i	−0.691271	−	0.722596i	$-0.742949\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	439902.			0.998067			0.499033	−	0.866583i	$-0.333689\pi$
$181$	439902.			0.998067			0.499033	+	0.866583i	$0.333689\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	37525.0	+	1425.00i	0.0806105	+	0.00306116i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	702475.	+	702475.i	1.35749	+	1.35749i	0.876997	+	0.480496i	$0.159543\pi$
$193$	702475.	+	702475.i	1.35749	+	1.35749i	0.480496	+	0.876997i	$0.340457\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−700575.	+	700575.i	−1.28614	+	1.28614i	−0.349031	+	0.937111i	$0.613489\pi$
$197$	−700575.	+	700575.i	−1.28614	+	1.28614i	−0.937111	+	0.349031i	$0.886511\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	188176.	+	203032.i	0.312737	+	0.337427i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.44875e6			−1.99532
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$	−757188.	−	57591.0i	−0.997120	−	0.0758400i
$226$		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
$229$			976564.i			1.23059i	0.788298	+	0.615293i	$0.210962\pi$
$229$			976564.i			1.23059i	−0.788298	+	0.615293i	$0.789038\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.07848e6	+	1.07848e6i	1.30143	+	1.30143i	0.927429	+	0.373999i	$0.122014\pi$
$233$	1.07848e6	+	1.07848e6i	1.30143	+	1.30143i	0.373999	+	0.927429i	$0.377986\pi$
$234$		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
$241$	1.73905e6			1.92872			0.964360	−	0.264595i	$-0.0852384\pi$
$241$	1.73905e6			1.92872			0.964360	+	0.264595i	$0.0852384\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−689087.	+	638666.i	−0.733430	+	0.679765i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	682575.	−	682575.i	0.644640	−	0.644640i	−0.307052	−	0.951693i	$-0.599343\pi$
$257$	682575.	−	682575.i	0.644640	−	0.644640i	0.951693	+	0.307052i	$0.0993426\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2.08105e6			−1.89096
$262$		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
$265$	−49425.0	+	1.30152e6i	−0.0432347	+	1.13851i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	2.35141e6i		−	1.98129i	−0.136458	−	0.990646i	$-0.543572\pi$
$269$		−	2.35141e6i		−	1.98129i	0.136458	−	0.990646i	$-0.456428\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.58448e6	−	1.58448e6i	1.24075	−	1.24075i	0.281067	−	0.959688i	$-0.409312\pi$
$277$	1.58448e6	−	1.58448e6i	1.24075	−	1.24075i	0.959688	−	0.281067i	$-0.0906882\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−149152.			−0.112684			−0.0563421	−	0.998412i	$-0.517944\pi$
$281$	−149152.			−0.112684			−0.0563421	+	0.998412i	$0.517944\pi$
$282$		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
$289$		−	3.23139e6i		−	2.27586i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	412575.	+	412575.i	0.280759	+	0.280759i	0.833412	−	0.552653i	$-0.186384\pi$
$293$	412575.	+	412575.i	0.280759	+	0.280759i	−0.552653	+	0.833412i	$0.686384\pi$
$294$		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
$305$	−2.08802e6	−	2.25287e6i	−1.28524	−	1.38671i
$306$		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
$313$	2.40238e6	+	2.40238e6i	1.38605	+	1.38605i	0.833433	+	0.552620i	$0.186372\pi$
$313$	2.40238e6	+	2.40238e6i	1.38605	+	1.38605i	0.552620	+	0.833433i	$0.313628\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.09238e6	−	2.09238e6i	1.16948	−	1.16948i	0.187144	−	0.982333i	$-0.440077\pi$
$317$	2.09238e6	−	2.09238e6i	1.16948	−	1.16948i	0.982333	−	0.187144i	$-0.0599230\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.59268e6	+	1.36752e6i	−0.836409	+	0.718169i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−115425.	−	115425.i	−0.0570413	−	0.0570413i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−602375.	+	602375.i	−0.288930	+	0.288930i	−0.836657	−	0.547727i	$-0.815493\pi$
$337$	−602375.	+	602375.i	−0.288930	+	0.288930i	0.547727	+	0.836657i	$0.315493\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	975636.i		−	0.428770i	−0.976749	−	0.214385i	$-0.931225\pi$
$349$		−	975636.i		−	0.428770i	0.976749	−	0.214385i	$-0.0687747\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	517625.	+	517625.i	0.221095	+	0.221095i	0.808959	−	0.587865i	$-0.200031\pi$
$353$	517625.	+	517625.i	0.221095	+	0.221095i	−0.587865	+	0.808959i	$0.700031\pi$
$354$		0			0
$355$		0			0
$356$		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
$361$	−2.47610e6			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−163425.	+	4.30352e6i	−0.0642076	+	1.69080i
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$		−	1.20334e6i		−	0.460067i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.82762e6	+	1.82762e6i	0.680166	+	0.680166i	0.960038	−	0.279871i	$-0.0902918\pi$
$373$	1.82762e6	+	1.82762e6i	0.680166	+	0.680166i	−0.279871	+	0.960038i	$0.590292\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.06790e6	+	4.06790e6i	−1.47407	+	1.47407i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			5.28629e6i			1.77124i	0.464414	+	0.885618i	$0.346265\pi$
$389$			5.28629e6i			1.77124i	−0.464414	+	0.885618i	$0.653735\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	3.26762e6	−	3.26762e6i	1.04053	−	1.04053i	0.0413901	−	0.999143i	$-0.486821\pi$
$397$	3.26762e6	−	3.26762e6i	1.04053	−	1.04053i	0.999143	−	0.0413901i	$-0.0131787\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.59200e6			0.494405			0.247202	−	0.968964i	$-0.420489\pi$
$401$	1.59200e6			0.494405			0.247202	+	0.968964i	$0.420489\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	2.24386e6	+	2.42101e6i	0.679765	+	0.733430i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			4.58449e6i			1.35513i	0.735461	+	0.677567i	$0.236966\pi$
$409$			4.58449e6i			1.35513i	−0.735461	+	0.677567i	$0.763034\pi$
$410$		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			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−4.18485e6			−1.15073			−0.575367	−	0.817895i	$-0.695141\pi$
$421$	−4.18485e6			−1.15073			−0.575367	+	0.817895i	$0.695141\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.39048e6	−	5.11332e6i	−1.17907	−	1.37319i
$426$		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
$433$	−4.15847e6	−	4.15847e6i	−1.06589	−	1.06589i	−0.997670	−	0.0682249i	$-0.978266\pi$
$433$	−4.15847e6	−	4.15847e6i	−1.06589	−	1.06589i	−0.0682249	−	0.997670i	$-0.521734\pi$
$434$		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
$441$	4.08410e6			1.00000
$442$		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
$445$	5.75902e6	−	5.33763e6i	1.37863	−	1.27776i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	8.48961e6i		−	1.98734i	−0.112340	−	0.993670i	$-0.535835\pi$
$449$		−	8.48961e6i		−	1.98734i	0.112340	−	0.993670i	$-0.464165\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6.23762e6	+	6.23762e6i	−1.39710	+	1.39710i	−0.588893	+	0.808211i	$0.700436\pi$
$457$	−6.23762e6	+	6.23762e6i	−1.39710	+	1.39710i	−0.808211	+	0.588893i	$0.799564\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.86580e6			1.50466			0.752331	−	0.658785i	$-0.228929\pi$
$461$	6.86580e6			1.50466			0.752331	+	0.658785i	$0.228929\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		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
$477$	4.00342e6	−	4.00342e6i	0.805630	−	0.805630i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−451250.			−0.0889313
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.99152e6	+	379425.i	1.92876	+	0.0732439i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−1.30601e7	−	1.30601e7i	−2.42008	−	2.42008i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		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
$505$	−3.72408e6	−	4.01808e6i	−0.649816	−	0.701117i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			5.12076e6i			0.876073i	0.898957	+	0.438037i	$0.144326\pi$
$509$			5.12076e6i			0.876073i	−0.898957	+	0.438037i	$0.855674\pi$
$510$		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
$521$	7.27410e6			1.17405			0.587023	−	0.809570i	$-0.300300\pi$
$521$	7.27410e6			1.17405			0.587023	+	0.809570i	$0.300300\pi$
$522$		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
$529$			6.43634e6i			1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.35220e6	−	2.35220e6i	−0.358638	−	0.358638i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.25380e6			−1.21244			−0.606221	−	0.795297i	$-0.707315\pi$
$541$	−8.25380e6			−1.21244			−0.606221	+	0.795297i	$0.707315\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.01059e7	−	9.36647e6i	1.45742	−	1.35078i
$546$		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
$549$			1.33524e7i			1.89072i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.07248e6	+	7.07248e6i	−0.965903	+	0.965903i	−0.999438	−	0.0335347i	$-0.989324\pi$
$557$	−7.07248e6	+	7.07248e6i	−0.965903	+	0.965903i	0.0335347	+	0.999438i	$0.489324\pi$
$558$		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
$565$	−544275.	+	1.43326e7i	−0.0717294	+	1.88887i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			3.96659e6i			0.513613i	0.966463	+	0.256807i	$0.0826703\pi$
$569$			3.96659e6i			0.513613i	−0.966463	+	0.256807i	$0.917330\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−9.47762e6	+	9.47762e6i	−1.18511	+	1.18511i	−0.206712	+	0.978402i	$0.566276\pi$
$577$	−9.47762e6	+	9.47762e6i	−1.18511	+	1.18511i	−0.978402	+	0.206712i	$0.933724\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	9.11858e6	+	346275.i	1.10163	+	0.0418342i
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.39743e6	+	9.39743e6i	1.09742	+	1.09742i	0.994712	+	0.102706i	$0.0327502\pi$
$593$	9.39743e6	+	9.39743e6i	1.09742	+	1.09742i	0.102706	+	0.994712i	$0.467250\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	9.16325e6			1.03482			0.517408	−	0.855739i	$-0.326897\pi$
$601$	9.16325e6			1.03482			0.517408	+	0.855739i	$0.326897\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.11994e6	−	6.60309e6i	−0.679765	−	0.733430i
$606$		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
$613$	−1.30233e7	−	1.30233e7i	−1.39982	−	1.39982i	−0.800552	−	0.599263i	$-0.795460\pi$
$613$	−1.30233e7	−	1.30233e7i	−1.39982	−	1.39982i	−0.599263	−	0.800552i	$-0.704540\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.21693e7	+	1.21693e7i	−1.28693	+	1.28693i	−0.350282	+	0.936644i	$0.613914\pi$
$617$	−1.21693e7	+	1.21693e7i	−1.28693	+	1.28693i	−0.936644	+	0.350282i	$0.886086\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−9.65329e6	−	1.47698e6i	−0.988497	−	0.151243i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.44875e6i		−	0.146005i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	7.98333e6	−	7.98333e6i	0.779534	−	0.779534i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.48270e7			1.42531			0.712655	−	0.701514i	$-0.247492\pi$
$641$	1.48270e7			1.42531			0.712655	+	0.701514i	$0.247492\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.73532e6	+	4.73532e6i	0.434577	+	0.434577i	0.890182	−	0.455605i	$-0.150577\pi$
$653$	4.73532e6	+	4.73532e6i	0.434577	+	0.434577i	−0.455605	+	0.890182i	$0.650577\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.32374e7	−	1.32374e7i	1.19644	−	1.19644i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−2.07531e7			−1.84747			−0.923737	−	0.383027i	$-0.874881\pi$
$661$	−2.07531e7			−1.84747			−0.923737	+	0.383027i	$0.874881\pi$
$662$		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
$673$	801325.	+	801325.i	0.0681979	+	0.0681979i	0.740383	−	0.672185i	$-0.234644\pi$
$673$	801325.	+	801325.i	0.0681979	+	0.0681979i	−0.672185	+	0.740383i	$0.734644\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.42762e6	−	9.42762e6i	0.790552	−	0.790552i	−0.191032	−	0.981584i	$-0.561183\pi$
$677$	9.42762e6	−	9.42762e6i	0.790552	−	0.790552i	0.981584	+	0.191032i	$0.0611833\pi$
$678$		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
$685$	1.41074e7	+	535725.i	1.14874	+	0.0436230i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1.56513e7i		−	1.25603i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	7.55180e6	−	7.55180e6i	0.588801	−	0.588801i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.51373e7			−1.16346			−0.581731	−	0.813381i	$-0.697624\pi$
$701$	−1.51373e7			−1.16346			−0.581731	+	0.813381i	$0.697624\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	2.64712e7i		−	1.97769i	−0.148942	−	0.988846i	$-0.547587\pi$
$709$		−	2.64712e7i		−	1.97769i	0.148942	−	0.988846i	$-0.452413\pi$
$710$		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
$725$	−2.66854e7	−	2.02967e6i	−1.88551	−	0.143410i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	1.43489e7i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	8.30762e6	+	8.30762e6i	0.571106	+	0.571106i	0.932438	−	0.361331i	$-0.117678\pi$
$733$	8.30762e6	+	8.30762e6i	0.571106	+	0.571106i	−0.361331	+	0.932438i	$0.617678\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		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
$745$	−1.95217e6	+	1.80933e6i	−0.128863	+	0.119434i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.41927e7	−	1.41927e7i	0.900170	−	0.900170i	−0.0952804	−	0.995450i	$-0.530375\pi$
$757$	1.41927e7	−	1.41927e7i	0.900170	−	0.900170i	0.995450	+	0.0952804i	$0.0303748\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.95198e7			1.22184			0.610919	−	0.791693i	$-0.290800\pi$
$761$	1.95198e7			1.22184			0.610919	+	0.791693i	$0.290800\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.11172e6	+	2.92754e7i	−0.0686821	+	1.80863i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			2.57727e7i			1.57161i	0.618477	+	0.785803i	$0.287750\pi$
$769$			2.57727e7i			1.57161i	−0.618477	+	0.785803i	$0.712250\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.39845e7	+	1.39845e7i	0.841778	+	0.841778i	0.989090	−	0.147312i	$-0.0470622\pi$
$773$	1.39845e7	+	1.39845e7i	0.841778	+	0.841778i	−0.147312	+	0.989090i	$0.547062\pi$
$774$		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
$785$	−3.41655e7	−	1.29742e6i	−1.97885	−	0.0751464i
$786$		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
$793$	2.61003e7	+	2.61003e7i	1.47388	+	1.47388i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.37794e7	+	2.37794e7i	−1.32604	+	1.32604i	−0.417241	+	0.908796i	$0.637003\pi$
$797$	−2.37794e7	+	2.37794e7i	−1.32604	+	1.32604i	−0.908796	+	0.417241i	$0.862997\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−3.41328e7			−1.87971
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	2.36197e7i		−	1.26883i	−0.772992	−	0.634415i	$-0.781241\pi$
$809$		−	2.36197e7i		−	1.26883i	0.772992	−	0.634415i	$-0.218759\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.14631e7			1.11131			0.555655	−	0.831413i	$-0.312467\pi$
$821$	2.14631e7			1.11131			0.555655	+	0.831413i	$0.312467\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		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
$829$			7.96819e6i			0.402692i	0.979520	+	0.201346i	$0.0645316\pi$
$829$			7.96819e6i			0.402692i	−0.979520	+	0.201346i	$0.935468\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.56307e7	+	2.56307e7i	1.27982	+	1.27982i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−5.28309e7			−2.57572
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.27824e6	−	3.03837e6i	0.157942	−	0.146386i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.71365e7	+	1.71365e7i	0.806397	+	0.806397i	0.984087	−	0.177690i	$-0.0568623\pi$
$853$	1.71365e7	+	1.71365e7i	0.806397	+	0.806397i	−0.177690	+	0.984087i	$0.556862\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.03762e6	−	3.03762e6i	0.141280	−	0.141280i	−0.632929	−	0.774210i	$-0.718148\pi$
$857$	3.03762e6	−	3.03762e6i	0.141280	−	0.141280i	0.774210	+	0.632929i	$0.218148\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	−1.66972e6	+	4.39694e7i	−0.0758760	+	1.99807i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−3.07334e7	−	3.07334e7i	−1.36482	−	1.36482i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3.13287e7	−	3.13287e7i	1.37544	−	1.37544i	0.523289	−	0.852155i	$-0.324705\pi$
$877$	3.13287e7	−	3.13287e7i	1.37544	−	1.37544i	0.852155	−	0.523289i	$-0.175295\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.33972e7			−0.581531			−0.290765	−	0.956794i	$-0.593910\pi$
$881$	−1.33972e7			−0.581531			−0.290765	+	0.956794i	$0.593910\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		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
$891$		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
$901$	5.02487e7			2.06212
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.67163e7	−	1.80360e7i	−0.678450	−	0.732012i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$			2.38145e7i			0.955942i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		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
$925$	−1.36753e6	−	1.59268e6i	−0.0525510	−	0.0612030i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	4.76633e7i		−	1.81194i	−0.423337	−	0.905972i	$-0.639141\pi$
$929$		−	4.76633e7i		−	1.81194i	0.423337	−	0.905972i	$-0.360859\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	2.07905e7	−	2.07905e7i	0.773598	−	0.773598i	−0.205135	−	0.978734i	$-0.565763\pi$
$937$	2.07905e7	−	2.07905e7i	0.773598	−	0.773598i	0.978734	+	0.205135i	$0.0657635\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.85570e6			0.178763			0.0893816	−	0.995997i	$-0.471511\pi$
$941$	4.85570e6			0.178763			0.0893816	+	0.995997i	$0.471511\pi$
$942$		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
$949$		−	5.17512e7i		−	1.86533i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−3.89894e7	−	3.89894e7i	−1.39064	−	1.39064i	−0.823890	−	0.566749i	$-0.808201\pi$
$953$	−3.89894e7	−	3.89894e7i	−1.39064	−	1.39064i	−0.566749	−	0.823890i	$-0.691799\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.86292e7			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.10742e6	−	5.54955e7i	0.0728507	−	1.91840i
$966$		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
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.17587e7	+	4.17587e7i	−1.39962	+	1.39962i	−0.598491	+	0.801130i	$0.704233\pi$
$977$	−4.17587e7	+	4.17587e7i	−1.39962	+	1.39962i	−0.801130	+	0.598491i	$0.795767\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−5.98961e7			−1.98713
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$	5.53454e7	+	2.10172e6i	1.81757	+	0.0690216i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	7.53237e6	−	7.53237e6i	0.239990	−	0.239990i	−0.576856	−	0.816846i	$-0.695720\pi$
$997$	7.53237e6	−	7.53237e6i	0.239990	−	0.239990i	0.816846	+	0.576856i	$0.195720\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.n.a.63.1	yes	2
4.3	odd	2	CM	80.6.n.a.63.1	yes	2
5.2	odd	4	inner	80.6.n.a.47.1	✓	2
5.3	odd	4		400.6.n.a.207.1		2
5.4	even	2		400.6.n.a.143.1		2
20.3	even	4		400.6.n.a.207.1		2
20.7	even	4	inner	80.6.n.a.47.1	✓	2
20.19	odd	2		400.6.n.a.143.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.n.a.47.1	✓	2	5.2	odd	4	inner
80.6.n.a.47.1	✓	2	20.7	even	4	inner
80.6.n.a.63.1	yes	2	1.1	even	1	trivial
80.6.n.a.63.1	yes	2	4.3	odd	2	CM
400.6.n.a.143.1		2	5.4	even	2
400.6.n.a.143.1		2	20.19	odd	2
400.6.n.a.207.1		2	5.3	odd	4
400.6.n.a.207.1		2	20.3	even	4

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:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-38.0000 - 41.0000i) q^{5} +243.000i q^{9} +O(q^{10})\)	\(q+(-38.0000 - 41.0000i) q^{5} +243.000i q^{9} +(475.000 + 475.000i) q^{13} +(-1525.00 + 1525.00i) q^{17} +(-237.000 + 3116.00i) q^{25} +8564.00i q^{29} +(-475.000 + 475.000i) q^{37} -4952.00 q^{41} +(9963.00 - 9234.00i) q^{45} -16807.0i q^{49} +(-16475.0 - 16475.0i) q^{53} +54948.0 q^{61} +(1425.00 - 37525.0i) q^{65} +(-54475.0 - 54475.0i) q^{73} -59049.0 q^{81} +(120475. + 4575.00i) q^{85} +140464. i q^{89} +(-126475. + 126475. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 76 q^{5}+O(q^{10}) \) 2 * q - 76 * q^5	\( 2 q - 76 q^{5} + 950 q^{13} - 3050 q^{17} - 474 q^{25} - 950 q^{37} - 9904 q^{41} + 19926 q^{45} - 32950 q^{53} + 109896 q^{61} + 2850 q^{65} - 108950 q^{73} - 118098 q^{81} + 240950 q^{85} - 252950 q^{97}+O(q^{100}) \) 2 * q - 76 * q^5 + 950 * q^13 - 3050 * q^17 - 474 * q^25 - 950 * q^37 - 9904 * q^41 + 19926 * q^45 - 32950 * q^53 + 109896 * q^61 + 2850 * q^65 - 108950 * q^73 - 118098 * q^81 + 240950 * q^85 - 252950 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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