LMFDB - Newform orbit 8325.2.a.g

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8325,2,Mod(1,8325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8325 = 3^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,0,2,0,0,5,0,0,0,-3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.4754596827$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - 2 q^{2} + 2 q^{4} + 5 q^{7} - 3 q^{11} + 2 q^{13} - 10 q^{14} - 4 q^{16} - 4 q^{17} - 4 q^{19} + 6 q^{22} - 2 q^{23} - 4 q^{26} + 10 q^{28} - 2 q^{29} + 8 q^{32} + 8 q^{34} + q^{37} + 8 q^{38} - 7 q^{41}+ \cdots - 36 q^{98}+O(q^{100}) $

q - 2 * q^2 + 2 * q^4 + 5 * q^7 - 3 * q^11 + 2 * q^13 - 10 * q^14 - 4 * q^16 - 4 * q^17 - 4 * q^19 + 6 * q^22 - 2 * q^23 - 4 * q^26 + 10 * q^28 - 2 * q^29 + 8 * q^32 + 8 * q^34 + q^37 + 8 * q^38 - 7 * q^41 + 10 * q^43 - 6 * q^44 + 4 * q^46 + 11 * q^47 + 18 * q^49 + 4 * q^52 - 3 * q^53 + 4 * q^58 - 4 * q^61 - 8 * q^64 - 16 * q^67 - 8 * q^68 + 15 * q^71 - 11 * q^73 - 2 * q^74 - 8 * q^76 - 15 * q^77 - 12 * q^79 + 14 * q^82 - 3 * q^83 - 20 * q^86 + 4 * q^89 + 10 * q^91 - 4 * q^92 - 22 * q^94 - 8 * q^97 - 36 * q^98

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−2.00000

2.00000

5.00000

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$5$	$ +1 $
$37$	$ -1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8325.2.a.g		1
3.b	odd	2	1		925.2.a.d		1
5.b	even	2	1		1665.2.a.f		1
15.d	odd	2	1		185.2.a.a	✓	1
15.e	even	4	2		925.2.b.a		2
60.h	even	2	1		2960.2.a.e		1
105.g	even	2	1		9065.2.a.b		1
555.b	odd	2	1		6845.2.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
185.2.a.a	✓	1	15.d	odd	2	1
925.2.a.d		1	3.b	odd	2	1
925.2.b.a		2	15.e	even	4	2
1665.2.a.f		1	5.b	even	2	1
2960.2.a.e		1	60.h	even	2	1
6845.2.a.e		1	555.b	odd	2	1
8325.2.a.g		1	1.a	even	1	1	trivial
9065.2.a.b		1	105.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8325))$:

$ T_{2} + 2 $

T2 + 2

$ T_{7} - 5 $

T7 - 5

$ T_{11} + 3 $

T11 + 3

$ T_{13} - 2 $

T13 - 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 5 $ T - 5
$11$	$ T + 3 $ T + 3
$13$	$ T - 2 $ T - 2
$17$	$ T + 4 $ T + 4
$19$	$ T + 4 $ T + 4
$23$	$ T + 2 $ T + 2
$29$	$ T + 2 $ T + 2
$31$	$ T $ T
$37$	$ T - 1 $ T - 1
$41$	$ T + 7 $ T + 7
$43$	$ T - 10 $ T - 10
$47$	$ T - 11 $ T - 11
$53$	$ T + 3 $ T + 3
$59$	$ T $ T
$61$	$ T + 4 $ T + 4
$67$	$ T + 16 $ T + 16
$71$	$ T - 15 $ T - 15
$73$	$ T + 11 $ T + 11
$79$	$ T + 12 $ T + 12
$83$	$ T + 3 $ T + 3
$89$	$ T - 4 $ T - 4
$97$	$ T + 8 $ T + 8
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Level:	\( N \)	\(=\)	\( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8325.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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