LMFDB - Embedded newform 504.6.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,6,Mod(1,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("504.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	504.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-48,0,-98] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$80.8334451857$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.54138$ of defining polynomial
Character	$\chi$	$=$	504.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+36.8276 q^{5} -49.0000 q^{7} +266.124 q^{11} -1051.24 q^{13} +1254.54 q^{17} +1284.05 q^{19} -982.814 q^{23} -1768.73 q^{25} -3199.78 q^{29} -1617.35 q^{31} -1804.55 q^{35} +2135.46 q^{37} -8931.56 q^{41} -14702.7 q^{43} +6417.60 q^{47} +2401.00 q^{49} +36936.3 q^{53} +9800.73 q^{55} -29523.6 q^{59} +31991.3 q^{61} -38714.7 q^{65} -29023.3 q^{67} -49999.1 q^{71} -12542.0 q^{73} -13040.1 q^{77} +28218.2 q^{79} -110571. q^{83} +46201.6 q^{85} +56586.5 q^{89} +51510.9 q^{91} +47288.7 q^{95} -108767. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 48 q^{5} - 98 q^{7} - 368 q^{11} - 156 q^{13} + 3312 q^{17} + 3736 q^{19} - 1552 q^{23} + 2302 q^{25} - 1728 q^{29} - 3624 q^{31} + 2352 q^{35} + 6996 q^{37} - 26160 q^{41} - 30184 q^{43} + 11424 q^{47}+ \cdots + 66652 q^{97}+O(q^{100}) $ 2 * q - 48 * q^5 - 98 * q^7 - 368 * q^11 - 156 * q^13 + 3312 * q^17 + 3736 * q^19 - 1552 * q^23 + 2302 * q^25 - 1728 * q^29 - 3624 * q^31 + 2352 * q^35 + 6996 * q^37 - 26160 * q^41 - 30184 * q^43 + 11424 * q^47 + 4802 * q^49 + 17376 * q^53 + 63592 * q^55 - 15008 * q^59 + 35564 * q^61 - 114656 * q^65 - 70504 * q^67 - 40752 * q^71 - 53892 * q^73 + 18032 * q^77 - 9744 * q^79 - 31360 * q^83 - 128328 * q^85 - 35952 * q^89 + 7644 * q^91 - 160704 * q^95 + 66652 * q^97

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	36.8276	0.658793	0.329396	−	0.944192i	$-0.393155\pi$
$5$	36.8276	0.658793	0.329396	+	0.944192i	$0.393155\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	266.124	0.663137	0.331568	−	0.943431i	$-0.392422\pi$
$11$	266.124	0.663137	0.331568	+	0.943431i	$0.392422\pi$
$12$	0	0
$13$	−1051.24	−1.72522	−0.862610	−	0.505870i	$-0.831172\pi$
$13$	−1051.24	−1.72522	−0.862610	+	0.505870i	$0.831172\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1254.54	1.05284	0.526419	−	0.850225i	$-0.323534\pi$
$17$	1254.54	1.05284	0.526419	+	0.850225i	$0.323534\pi$
$18$	0	0
$19$	1284.05	0.816018	0.408009	−	0.912978i	$-0.366223\pi$
$19$	1284.05	0.816018	0.408009	+	0.912978i	$0.366223\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−982.814	−0.387393	−0.193696	−	0.981062i	$-0.562048\pi$
$23$	−982.814	−0.387393	−0.193696	+	0.981062i	$0.562048\pi$
$24$	0	0
$25$	−1768.73	−0.565992
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3199.78	−0.706521	−0.353261	−	0.935525i	$-0.614927\pi$
$29$	−3199.78	−0.706521	−0.353261	+	0.935525i	$0.614927\pi$
$30$	0	0
$31$	−1617.35	−0.302274	−0.151137	−	0.988513i	$-0.548293\pi$
$31$	−1617.35	−0.302274	−0.151137	+	0.988513i	$0.548293\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1804.55	−0.249000
$36$	0	0
$37$	2135.46	0.256441	0.128220	−	0.991746i	$-0.459073\pi$
$37$	2135.46	0.256441	0.128220	+	0.991746i	$0.459073\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8931.56	−0.829789	−0.414894	−	0.909870i	$-0.636181\pi$
$41$	−8931.56	−0.829789	−0.414894	+	0.909870i	$0.636181\pi$
$42$	0	0
$43$	−14702.7	−1.21262	−0.606312	−	0.795227i	$-0.707352\pi$
$43$	−14702.7	−1.21262	−0.606312	+	0.795227i	$0.707352\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6417.60	0.423768	0.211884	−	0.977295i	$-0.432040\pi$
$47$	6417.60	0.423768	0.211884	+	0.977295i	$0.432040\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	36936.3	1.80619	0.903097	−	0.429437i	$-0.141288\pi$
$53$	36936.3	1.80619	0.903097	+	0.429437i	$0.141288\pi$
$54$	0	0
$55$	9800.73	0.436870
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−29523.6	−1.10418	−0.552089	−	0.833785i	$-0.686169\pi$
$59$	−29523.6	−1.10418	−0.552089	+	0.833785i	$0.686169\pi$
$60$	0	0
$61$	31991.3	1.10080	0.550399	−	0.834902i	$-0.314476\pi$
$61$	31991.3	1.10080	0.550399	+	0.834902i	$0.314476\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−38714.7	−1.13656
$66$	0	0
$67$	−29023.3	−0.789876	−0.394938	−	0.918708i	$-0.629234\pi$
$67$	−29023.3	−0.789876	−0.394938	+	0.918708i	$0.629234\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−49999.1	−1.17711	−0.588553	−	0.808458i	$-0.700302\pi$
$71$	−49999.1	−1.17711	−0.588553	+	0.808458i	$0.700302\pi$
$72$	0	0
$73$	−12542.0	−0.275461	−0.137731	−	0.990470i	$-0.543981\pi$
$73$	−12542.0	−0.275461	−0.137731	+	0.990470i	$0.543981\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13040.1	−0.250642
$78$	0	0
$79$	28218.2	0.508700	0.254350	−	0.967112i	$-0.418138\pi$
$79$	28218.2	0.508700	0.254350	+	0.967112i	$0.418138\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−110571.	−1.76176	−0.880879	−	0.473341i	$-0.843048\pi$
$83$	−110571.	−1.76176	−0.880879	+	0.473341i	$0.843048\pi$
$84$	0	0
$85$	46201.6	0.693602
$86$	0	0
$87$	0	0
$88$	0	0
$89$	56586.5	0.757247	0.378624	−	0.925551i	$-0.376397\pi$
$89$	56586.5	0.757247	0.378624	+	0.925551i	$0.376397\pi$
$90$	0	0
$91$	51510.9	0.652072
$92$	0	0
$93$	0	0
$94$	0	0
$95$	47288.7	0.537586
$96$	0	0
$97$	−108767.	−1.17373	−0.586866	−	0.809684i	$-0.699639\pi$
$97$	−108767.	−1.17373	−0.586866	+	0.809684i	$0.699639\pi$
$98$	0	0
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.6.a.l.1.2	✓	2
3.2	odd	2		504.6.a.r.1.1	yes	2
4.3	odd	2		1008.6.a.bh.1.2		2
12.11	even	2		1008.6.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.6.a.l.1.2	✓	2	1.1	even	1	trivial
504.6.a.r.1.1	yes	2	3.2	odd	2
1008.6.a.bh.1.2		2	4.3	odd	2
1008.6.a.bs.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	504.a (trivial)

\(f(q)\)	\(=\)	\(q+36.8276 q^{5} -49.0000 q^{7} +266.124 q^{11} -1051.24 q^{13} +1254.54 q^{17} +1284.05 q^{19} -982.814 q^{23} -1768.73 q^{25} -3199.78 q^{29} -1617.35 q^{31} -1804.55 q^{35} +2135.46 q^{37} -8931.56 q^{41} -14702.7 q^{43} +6417.60 q^{47} +2401.00 q^{49} +36936.3 q^{53} +9800.73 q^{55} -29523.6 q^{59} +31991.3 q^{61} -38714.7 q^{65} -29023.3 q^{67} -49999.1 q^{71} -12542.0 q^{73} -13040.1 q^{77} +28218.2 q^{79} -110571. q^{83} +46201.6 q^{85} +56586.5 q^{89} +51510.9 q^{91} +47288.7 q^{95} -108767. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 48 q^{5} - 98 q^{7} - 368 q^{11} - 156 q^{13} + 3312 q^{17} + 3736 q^{19} - 1552 q^{23} + 2302 q^{25} - 1728 q^{29} - 3624 q^{31} + 2352 q^{35} + 6996 q^{37} - 26160 q^{41} - 30184 q^{43} + 11424 q^{47}+ \cdots + 66652 q^{97}+O(q^{100}) \) 2 * q - 48 * q^5 - 98 * q^7 - 368 * q^11 - 156 * q^13 + 3312 * q^17 + 3736 * q^19 - 1552 * q^23 + 2302 * q^25 - 1728 * q^29 - 3624 * q^31 + 2352 * q^35 + 6996 * q^37 - 26160 * q^41 - 30184 * q^43 + 11424 * q^47 + 4802 * q^49 + 17376 * q^53 + 63592 * q^55 - 15008 * q^59 + 35564 * q^61 - 114656 * q^65 - 70504 * q^67 - 40752 * q^71 - 53892 * q^73 + 18032 * q^77 - 9744 * q^79 - 31360 * q^83 - 128328 * q^85 - 35952 * q^89 + 7644 * q^91 - 160704 * q^95 + 66652 * q^97

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