LMFDB - Embedded newform 1520.2.g.a.1519.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1519,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1520.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.g (of order $2$, degree $1$, 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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 5 $ x^2 - x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1519.2
Root			$0.500000 - 2.17945i$ of defining polynomial
Character	$\chi$	$=$	1520.1519
Dual form			1520.2.g.a.1519.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 2.17945i) q^{5} -3.00000 q^{7} +3.00000 q^{9} +O(q^{10})$	$q+(-0.500000 + 2.17945i) q^{5} -3.00000 q^{7} +3.00000 q^{9} +4.35890i q^{11} -4.35890i q^{17} +4.35890i q^{19} -4.00000 q^{23} +(-4.50000 - 2.17945i) q^{25} +(1.50000 - 6.53835i) q^{35} +1.00000 q^{43} +(-1.50000 + 6.53835i) q^{45} -13.0000 q^{47} +2.00000 q^{49} +(-9.50000 - 2.17945i) q^{55} -15.0000 q^{61} -9.00000 q^{63} +13.0767i q^{73} -13.0767i q^{77} +9.00000 q^{81} -16.0000 q^{83} +(9.50000 + 2.17945i) q^{85} +(-9.50000 - 2.17945i) q^{95} +13.0767i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 6 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - q^5 - 6 * q^7 + 6 * q^9	$ 2 q - q^{5} - 6 q^{7} + 6 q^{9} - 8 q^{23} - 9 q^{25} + 3 q^{35} + 2 q^{43} - 3 q^{45} - 26 q^{47} + 4 q^{49} - 19 q^{55} - 30 q^{61} - 18 q^{63} + 18 q^{81} - 32 q^{83} + 19 q^{85} - 19 q^{95}+O(q^{100}) $ 2 * q - q^5 - 6 * q^7 + 6 * q^9 - 8 * q^23 - 9 * q^25 + 3 * q^35 + 2 * q^43 - 3 * q^45 - 26 * q^47 + 4 * q^49 - 19 * q^55 - 30 * q^61 - 18 * q^63 + 18 * q^81 - 32 * q^83 + 19 * q^85 - 19 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$		0			0
$5$	−0.500000	+	2.17945i	−0.223607	+	0.974679i
$5$	−0.500000	+	2.17945i	−0.223607	+	0.974679i
$6$		0			0
$7$	−3.00000			−1.13389			−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000			−1.13389			−0.566947	+	0.823754i	$0.691875\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$			4.35890i			1.31426i	0.753778	+	0.657129i	$0.228229\pi$
$11$			4.35890i			1.31426i	−0.753778	+	0.657129i	$0.771771\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.35890i		−	1.05719i	−0.848875	−	0.528594i	$-0.822719\pi$
$17$		−	4.35890i		−	1.05719i	0.848875	−	0.528594i	$-0.177281\pi$
$18$		0			0
$19$			4.35890i			1.00000i
$19$			4.35890i			1.00000i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−4.50000	−	2.17945i	−0.900000	−	0.435890i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.50000	−	6.53835i	0.253546	−	1.10518i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$		0			0
$45$	−1.50000	+	6.53835i	−0.223607	+	0.974679i
$46$		0			0
$47$	−13.0000			−1.89624			−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−13.0000			−1.89624			−0.948122	+	0.317905i	$0.897021\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−9.50000	−	2.17945i	−1.28098	−	0.293877i
$56$		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
$61$	−15.0000			−1.92055			−0.960277	−	0.279050i	$-0.909981\pi$
$61$	−15.0000			−1.92055			−0.960277	+	0.279050i	$0.909981\pi$
$62$		0			0
$63$	−9.00000			−1.13389
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			13.0767i			1.53051i	0.643726	+	0.765256i	$0.277388\pi$
$73$			13.0767i			1.53051i	−0.643726	+	0.765256i	$0.722612\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	13.0767i		−	1.49023i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	9.50000	+	2.17945i	1.03042	+	0.236394i
$86$		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			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.50000	−	2.17945i	−0.974679	−	0.223607i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$			13.0767i			1.31426i
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		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
$115$	2.00000	−	8.71780i	0.186501	−	0.812939i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			13.0767i			1.19874i
$120$		0			0
$121$	−8.00000			−0.727273
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.00000	−	8.71780i	0.626099	−	0.779744i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	21.7945i		−	1.90419i	−0.305796	−	0.952097i	$-0.598923\pi$
$131$		−	21.7945i		−	1.90419i	0.305796	−	0.952097i	$-0.401077\pi$
$132$		0			0
$133$		−	13.0767i		−	1.13389i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.35890i			0.372406i	0.982511	+	0.186203i	$0.0596182\pi$
$137$			4.35890i			0.372406i	−0.982511	+	0.186203i	$0.940382\pi$
$138$		0			0
$139$			21.7945i			1.84858i	0.381685	+	0.924292i	$0.375344\pi$
$139$			21.7945i			1.84858i	−0.381685	+	0.924292i	$0.624656\pi$
$140$		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
$149$	−11.0000			−0.901155			−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000			−0.901155			−0.450578	+	0.892737i	$0.648782\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		−	13.0767i		−	1.05719i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			17.4356i			1.39151i	0.718278	+	0.695756i	$0.244931\pi$
$157$			17.4356i			1.39151i	−0.718278	+	0.695756i	$0.755069\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.0000			0.945732
$162$		0			0
$163$	24.0000			1.87983			0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000			1.87983			0.939913	+	0.341415i	$0.110906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$			13.0767i			1.00000i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	13.5000	+	6.53835i	1.02050	+	0.494253i
$176$		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
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	19.0000			1.38942
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	21.7945i		−	1.57699i	−0.615038	−	0.788497i	$-0.710860\pi$
$191$		−	21.7945i		−	1.57699i	0.615038	−	0.788497i	$-0.289140\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			17.4356i			1.24223i	0.783718	+	0.621117i	$0.213321\pi$
$197$			17.4356i			1.24223i	−0.783718	+	0.621117i	$0.786679\pi$
$198$		0			0
$199$			13.0767i			0.926982i	0.886102	+	0.463491i	$0.153403\pi$
$199$			13.0767i			0.926982i	−0.886102	+	0.463491i	$0.846597\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−12.0000			−0.834058
$208$		0			0
$209$	−19.0000			−1.31426
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.500000	+	2.17945i	−0.0340997	+	0.148637i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−13.5000	−	6.53835i	−0.900000	−	0.435890i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	21.0000			1.38772			0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000			1.38772			0.693860	+	0.720110i	$0.255909\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			30.5123i			1.99893i	0.0327561	+	0.999463i	$0.489572\pi$
$233$			30.5123i			1.99893i	−0.0327561	+	0.999463i	$0.510428\pi$
$234$		0			0
$235$	6.50000	−	28.3328i	0.424013	−	1.84823i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			30.5123i			1.97368i	0.161712	+	0.986838i	$0.448299\pi$
$239$			30.5123i			1.97368i	−0.161712	+	0.986838i	$0.551701\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.00000	+	4.35890i	−0.0638877	+	0.278480i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	21.7945i		−	1.37566i	−0.725874	−	0.687828i	$-0.758564\pi$
$251$		−	21.7945i		−	1.37566i	0.725874	−	0.687828i	$-0.241436\pi$
$252$		0			0
$253$		−	17.4356i		−	1.09617i
$254$		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			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.0000			−0.678289			−0.339145	−	0.940734i	$-0.610138\pi$
$263$	−11.0000			−0.678289			−0.339145	+	0.940734i	$0.610138\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		−	26.1534i		−	1.58871i	−0.607457	−	0.794353i	$-0.707810\pi$
$271$		−	26.1534i		−	1.58871i	0.607457	−	0.794353i	$-0.292190\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.50000	−	19.6150i	0.572872	−	1.18283i
$276$		0			0
$277$		−	4.35890i		−	0.261901i	−0.991389	−	0.130950i	$-0.958197\pi$
$277$		−	4.35890i		−	0.261901i	0.991389	−	0.130950i	$-0.0418029\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	31.0000			1.84276			0.921379	−	0.388664i	$-0.127063\pi$
$283$	31.0000			1.84276			0.921379	+	0.388664i	$0.127063\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.00000			−0.117647
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−3.00000			−0.172917
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.50000	−	32.6917i	0.429449	−	1.87192i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			4.35890i			0.247170i	0.992334	+	0.123585i	$0.0394392\pi$
$311$			4.35890i			0.247170i	−0.992334	+	0.123585i	$0.960561\pi$
$312$		0			0
$313$		−	34.8712i		−	1.97104i	−0.169570	−	0.985518i	$-0.554238\pi$
$313$		−	34.8712i		−	1.97104i	0.169570	−	0.985518i	$-0.445762\pi$
$314$		0			0
$315$	4.50000	−	19.6150i	0.253546	−	1.10518i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	19.0000			1.05719
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	39.0000			2.15014
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	37.0000			1.98626			0.993132	−	0.116999i	$-0.0373274\pi$
$347$	37.0000			1.98626			0.993132	+	0.116999i	$0.0373274\pi$
$348$		0			0
$349$	35.0000			1.87351			0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000			1.87351			0.936754	+	0.349990i	$0.113815\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			34.8712i			1.85601i	0.372572	+	0.928003i	$0.378476\pi$
$353$			34.8712i			1.85601i	−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			21.7945i			1.15027i	0.818059	+	0.575135i	$0.195050\pi$
$359$			21.7945i			1.15027i	−0.818059	+	0.575135i	$0.804950\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−28.5000	−	6.53835i	−1.49176	−	0.342233i
$366$		0			0
$367$	28.0000			1.46159			0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000			1.46159			0.730794	+	0.682598i	$0.239150\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		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		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	28.5000	+	6.53835i	1.45249	+	0.333225i
$386$		0			0
$387$	3.00000			0.152499
$388$		0			0
$389$	−25.0000			−1.26755			−0.633775	−	0.773517i	$-0.718496\pi$
$389$	−25.0000			−1.26755			−0.633775	+	0.773517i	$0.718496\pi$
$390$		0			0
$391$			17.4356i			0.881756i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			39.2301i			1.96890i	0.175660	+	0.984451i	$0.443794\pi$
$397$			39.2301i			1.96890i	−0.175660	+	0.984451i	$0.556206\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−4.50000	+	19.6150i	−0.223607	+	0.974679i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	8.00000	−	34.8712i	0.392705	−	1.71176i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			8.71780i			0.425892i	0.977064	+	0.212946i	$0.0683059\pi$
$419$			8.71780i			0.425892i	−0.977064	+	0.212946i	$0.931694\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	−39.0000			−1.89624
$424$		0			0
$425$	−9.50000	+	19.6150i	−0.460818	+	0.951469i
$426$		0			0
$427$	45.0000			2.17770
$428$		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
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	17.4356i		−	0.834058i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	6.00000			0.285714
$442$		0			0
$443$	29.0000			1.37783			0.688916	−	0.724841i	$-0.258087\pi$
$443$	29.0000			1.37783			0.688916	+	0.724841i	$0.258087\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	39.2301i		−	1.83511i	−0.397613	−	0.917553i	$-0.630161\pi$
$457$		−	39.2301i		−	1.83511i	0.397613	−	0.917553i	$-0.369839\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−37.0000			−1.72326			−0.861631	−	0.507535i	$-0.830557\pi$
$461$	−37.0000			−1.72326			−0.861631	+	0.507535i	$0.830557\pi$
$462$		0			0
$463$	41.0000			1.90543			0.952716	−	0.303863i	$-0.0982765\pi$
$463$	41.0000			1.90543			0.952716	+	0.303863i	$0.0982765\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−43.0000			−1.98980			−0.994901	−	0.100853i	$-0.967843\pi$
$467$	−43.0000			−1.98980			−0.994901	+	0.100853i	$0.967843\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			4.35890i			0.200422i
$474$		0			0
$475$	9.50000	−	19.6150i	0.435890	−	0.900000i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			43.5890i			1.99163i	0.0913823	+	0.995816i	$0.470871\pi$
$479$			43.5890i			1.99163i	−0.0913823	+	0.995816i	$0.529129\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			43.5890i			1.96714i	0.180517	+	0.983572i	$0.442223\pi$
$491$			43.5890i			1.96714i	−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−28.5000	−	6.53835i	−1.28098	−	0.293877i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			21.7945i			0.975656i	0.872940	+	0.487828i	$0.162211\pi$
$499$			21.7945i			0.975656i	−0.872940	+	0.487828i	$0.837789\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−44.0000			−1.96186			−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−44.0000			−1.96186			−0.980932	+	0.194354i	$0.937739\pi$
$504$		0			0
$505$	−5.00000	+	21.7945i	−0.222497	+	0.969842i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		−	39.2301i		−	1.73544i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	56.6657i		−	2.49215i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.00000			−0.304348
$530$		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
$539$			8.71780i			0.375502i
$540$		0			0
$541$	25.0000			1.07483			0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000			1.07483			0.537417	+	0.843317i	$0.319400\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	−45.0000			−1.92055
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	4.35890i		−	0.184692i	−0.995727	−	0.0923462i	$-0.970563\pi$
$557$		−	4.35890i		−	0.184692i	0.995727	−	0.0923462i	$-0.0294367\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−27.0000			−1.13389
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$			26.1534i			1.09449i	0.836974	+	0.547243i	$0.184323\pi$
$571$			26.1534i			1.09449i	−0.836974	+	0.547243i	$0.815677\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18.0000	+	8.71780i	0.750652	+	0.363557i
$576$		0			0
$577$			47.9479i			1.99610i	0.0624458	+	0.998048i	$0.480110\pi$
$577$			47.9479i			1.99610i	−0.0624458	+	0.998048i	$0.519890\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	48.0000			1.99138
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.00000			−0.288921			−0.144460	−	0.989511i	$-0.546145\pi$
$587$	−7.00000			−0.288921			−0.144460	+	0.989511i	$0.546145\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	34.8712i		−	1.43199i	−0.698106	−	0.715994i	$-0.745974\pi$
$593$		−	34.8712i		−	1.43199i	0.698106	−	0.715994i	$-0.254026\pi$
$594$		0			0
$595$	−28.5000	−	6.53835i	−1.16839	−	0.268046i
$596$		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
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.00000	−	17.4356i	0.162623	−	0.708858i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			30.5123i			1.23238i	0.787598	+	0.616190i	$0.211325\pi$
$613$			30.5123i			1.23238i	−0.787598	+	0.616190i	$0.788675\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	47.9479i		−	1.93031i	−0.261680	−	0.965155i	$-0.584277\pi$
$617$		−	47.9479i		−	1.93031i	0.261680	−	0.965155i	$-0.415723\pi$
$618$		0			0
$619$		−	43.5890i		−	1.75199i	−0.482321	−	0.875995i	$-0.660206\pi$
$619$		−	43.5890i		−	1.75199i	0.482321	−	0.875995i	$-0.339794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15.5000	+	19.6150i	0.620000	+	0.784602i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	47.9479i		−	1.90878i	−0.298570	−	0.954388i	$-0.596510\pi$
$631$		−	47.9479i		−	1.90878i	0.298570	−	0.954388i	$-0.403490\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	−49.0000			−1.93237			−0.966186	−	0.257847i	$-0.916987\pi$
$643$	−49.0000			−1.93237			−0.966186	+	0.257847i	$0.916987\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17.0000			−0.668339			−0.334169	−	0.942513i	$-0.608456\pi$
$647$	−17.0000			−0.668339			−0.334169	+	0.942513i	$0.608456\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			30.5123i			1.19404i	0.802227	+	0.597019i	$0.203648\pi$
$653$			30.5123i			1.19404i	−0.802227	+	0.597019i	$0.796352\pi$
$654$		0			0
$655$	47.5000	+	10.8972i	1.85598	+	0.425791i
$656$		0			0
$657$			39.2301i			1.53051i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	28.5000	+	6.53835i	1.10518	+	0.253546i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	65.3835i		−	2.52410i
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	−9.50000	−	2.17945i	−0.362976	−	0.0832725i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			39.2301i			1.49238i	0.665731	+	0.746191i	$0.268120\pi$
$691$			39.2301i			1.49238i	−0.665731	+	0.746191i	$0.731880\pi$
$692$		0			0
$693$		−	39.2301i		−	1.49023i
$694$		0			0
$695$	−47.5000	−	10.8972i	−1.80178	−	0.413356i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	50.0000			1.88847			0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000			1.88847			0.944237	+	0.329267i	$0.106802\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−30.0000			−1.12827
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		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
$719$			21.7945i			0.812798i	0.913696	+	0.406399i	$0.133216\pi$
$719$			21.7945i			0.812798i	−0.913696	+	0.406399i	$0.866784\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−37.0000			−1.37225			−0.686127	−	0.727482i	$-0.740691\pi$
$727$	−37.0000			−1.37225			−0.686127	+	0.727482i	$0.740691\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		−	4.35890i		−	0.161220i
$732$		0			0
$733$			52.3068i			1.93200i	0.258551	+	0.965998i	$0.416755\pi$
$733$			52.3068i			1.93200i	−0.258551	+	0.965998i	$0.583245\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$			30.5123i			1.12241i	0.827676	+	0.561206i	$0.189663\pi$
$739$			30.5123i			1.12241i	−0.827676	+	0.561206i	$0.810337\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	5.50000	−	23.9739i	0.201504	−	0.878337i
$746$		0			0
$747$	−48.0000			−1.75623
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	47.9479i		−	1.74270i	−0.490666	−	0.871348i	$-0.663246\pi$
$757$		−	47.9479i		−	1.74270i	0.490666	−	0.871348i	$-0.336754\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−55.0000			−1.99375			−0.996874	−	0.0790050i	$-0.974826\pi$
$761$	−55.0000			−1.99375			−0.996874	+	0.0790050i	$0.974826\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	28.5000	+	6.53835i	1.03042	+	0.236394i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−51.0000			−1.83911			−0.919554	−	0.392965i	$-0.871449\pi$
$769$	−51.0000			−1.83911			−0.919554	+	0.392965i	$0.871449\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		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
$785$	−38.0000	−	8.71780i	−1.35628	−	0.311152i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$			56.6657i			2.00469i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−57.0000			−2.01149
$804$		0			0
$805$	−6.00000	+	26.1534i	−0.211472	+	0.921786i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.00000			−0.175791			−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000			−0.175791			−0.0878953	+	0.996130i	$0.528014\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−12.0000	+	52.3068i	−0.420342	+	1.83223i
$816$		0			0
$817$			4.35890i			0.152499i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−53.0000			−1.84971			−0.924856	−	0.380317i	$-0.875815\pi$
$821$	−53.0000			−1.84971			−0.924856	+	0.380317i	$0.875815\pi$
$822$		0			0
$823$	9.00000			0.313720			0.156860	−	0.987621i	$-0.449863\pi$
$823$	9.00000			0.313720			0.156860	+	0.987621i	$0.449863\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	8.71780i		−	0.302054i
$834$		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
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.50000	−	28.3328i	0.223607	−	0.974679i
$846$		0			0
$847$	24.0000			0.824650
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			52.3068i			1.79095i	0.445112	+	0.895475i	$0.353164\pi$
$853$			52.3068i			1.79095i	−0.445112	+	0.895475i	$0.646836\pi$
$854$		0			0
$855$	−28.5000	−	6.53835i	−0.974679	−	0.223607i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		−	56.6657i		−	1.93341i	−0.255897	−	0.966704i	$-0.582371\pi$
$859$		−	56.6657i		−	1.93341i	0.255897	−	0.966704i	$-0.417629\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−21.0000	+	26.1534i	−0.709930	+	0.884146i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−35.0000			−1.17918			−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000			−1.17918			−0.589590	+	0.807703i	$0.700711\pi$
$882$		0			0
$883$	−51.0000			−1.71629			−0.858143	−	0.513410i	$-0.828382\pi$
$883$	−51.0000			−1.71629			−0.858143	+	0.513410i	$0.828382\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			39.2301i			1.31426i
$892$		0			0
$893$		−	56.6657i		−	1.89624i
$894$		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
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	30.0000			0.995037
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		−	69.7424i		−	2.30814i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			65.3835i			2.15915i
$918$		0			0
$919$		−	8.71780i		−	0.287574i	−0.989609	−	0.143787i	$-0.954072\pi$
$919$		−	8.71780i		−	0.287574i	0.989609	−	0.143787i	$-0.0459280\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	50.0000			1.64045			0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000			1.64045			0.820223	+	0.572043i	$0.193849\pi$
$930$		0			0
$931$			8.71780i			0.285714i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9.50000	+	41.4095i	−0.310683	+	1.35424i
$936$		0			0
$937$			39.2301i			1.28159i	0.767712	+	0.640796i	$0.221395\pi$
$937$			39.2301i			1.28159i	−0.767712	+	0.640796i	$0.778605\pi$
$938$		0			0
$939$		0			0
$940$		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
$947$	8.00000			0.259965			0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000			0.259965			0.129983	+	0.991516i	$0.458508\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	47.5000	+	10.8972i	1.53706	+	0.352627i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	13.0767i		−	0.422269i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	12.0000			0.385894			0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000			0.385894			0.192947	+	0.981209i	$0.438195\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		−	65.3835i		−	2.09610i
$974$		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
$985$	−38.0000	−	8.71780i	−1.21078	−	0.277772i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.00000			−0.127193
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−28.5000	−	6.53835i	−0.903511	−	0.207280i
$996$		0			0
$997$			4.35890i			0.138048i	0.997615	+	0.0690239i	$0.0219885\pi$
$997$			4.35890i			0.138048i	−0.997615	+	0.0690239i	$0.978012\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.g.a.1519.2	yes	2
4.3	odd	2		1520.2.g.b.1519.2	yes	2
5.4	even	2		1520.2.g.b.1519.1	yes	2
19.18	odd	2	CM	1520.2.g.a.1519.2	yes	2
20.19	odd	2	inner	1520.2.g.a.1519.1	✓	2
76.75	even	2		1520.2.g.b.1519.2	yes	2
95.94	odd	2		1520.2.g.b.1519.1	yes	2
380.379	even	2	inner	1520.2.g.a.1519.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1520.2.g.a.1519.1	✓	2	20.19	odd	2	inner
1520.2.g.a.1519.1	✓	2	380.379	even	2	inner
1520.2.g.a.1519.2	yes	2	1.1	even	1	trivial
1520.2.g.a.1519.2	yes	2	19.18	odd	2	CM
1520.2.g.b.1519.1	yes	2	5.4	even	2
1520.2.g.b.1519.1	yes	2	95.94	odd	2
1520.2.g.b.1519.2	yes	2	4.3	odd	2
1520.2.g.b.1519.2	yes	2	76.75	even	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 2.17945i) q^{5} -3.00000 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+(-0.500000 + 2.17945i) q^{5} -3.00000 q^{7} +3.00000 q^{9} +4.35890i q^{11} -4.35890i q^{17} +4.35890i q^{19} -4.00000 q^{23} +(-4.50000 - 2.17945i) q^{25} +(1.50000 - 6.53835i) q^{35} +1.00000 q^{43} +(-1.50000 + 6.53835i) q^{45} -13.0000 q^{47} +2.00000 q^{49} +(-9.50000 - 2.17945i) q^{55} -15.0000 q^{61} -9.00000 q^{63} +13.0767i q^{73} -13.0767i q^{77} +9.00000 q^{81} -16.0000 q^{83} +(9.50000 + 2.17945i) q^{85} +(-9.50000 - 2.17945i) q^{95} +13.0767i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - q^5 - 6 * q^7 + 6 * q^9	\( 2 q - q^{5} - 6 q^{7} + 6 q^{9} - 8 q^{23} - 9 q^{25} + 3 q^{35} + 2 q^{43} - 3 q^{45} - 26 q^{47} + 4 q^{49} - 19 q^{55} - 30 q^{61} - 18 q^{63} + 18 q^{81} - 32 q^{83} + 19 q^{85} - 19 q^{95}+O(q^{100}) \) 2 * q - q^5 - 6 * q^7 + 6 * q^9 - 8 * q^23 - 9 * q^25 + 3 * q^35 + 2 * q^43 - 3 * q^45 - 26 * q^47 + 4 * q^49 - 19 * q^55 - 30 * q^61 - 18 * q^63 + 18 * q^81 - 32 * q^83 + 19 * q^85 - 19 * q^95

Introduction

L-functions

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