Properties

Label 315.2.a.b
Level $315$
Weight $2$
Character orbit 315.a
Self dual yes
Analytic conductor $2.515$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

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: yes
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{4} + q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} + q^{5} + q^{7} + 3 q^{11} + 5 q^{13} + 4 q^{16} - 3 q^{17} + 2 q^{19} - 2 q^{20} + 6 q^{23} + q^{25} - 2 q^{28} - 3 q^{29} - 4 q^{31} + q^{35} + 2 q^{37} + 12 q^{41} - 10 q^{43} - 6 q^{44} - 9 q^{47} + q^{49} - 10 q^{52} - 12 q^{53} + 3 q^{55} + 8 q^{61} - 8 q^{64} + 5 q^{65} - 4 q^{67} + 6 q^{68} + 2 q^{73} - 4 q^{76} + 3 q^{77} - q^{79} + 4 q^{80} - 12 q^{83} - 3 q^{85} + 12 q^{89} + 5 q^{91} - 12 q^{92} + 2 q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
0
0 0 −2.00000 1.00000 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 315.2.a.b 1
3.b odd 2 1 35.2.a.a 1
4.b odd 2 1 5040.2.a.v 1
5.b even 2 1 1575.2.a.f 1
5.c odd 4 2 1575.2.d.c 2
7.b odd 2 1 2205.2.a.e 1
12.b even 2 1 560.2.a.b 1
15.d odd 2 1 175.2.a.b 1
15.e even 4 2 175.2.b.a 2
21.c even 2 1 245.2.a.c 1
21.g even 6 2 245.2.e.b 2
21.h odd 6 2 245.2.e.a 2
24.f even 2 1 2240.2.a.u 1
24.h odd 2 1 2240.2.a.k 1
33.d even 2 1 4235.2.a.c 1
39.d odd 2 1 5915.2.a.f 1
60.h even 2 1 2800.2.a.z 1
60.l odd 4 2 2800.2.g.l 2
84.h odd 2 1 3920.2.a.ba 1
105.g even 2 1 1225.2.a.e 1
105.k odd 4 2 1225.2.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 3.b odd 2 1
175.2.a.b 1 15.d odd 2 1
175.2.b.a 2 15.e even 4 2
245.2.a.c 1 21.c even 2 1
245.2.e.a 2 21.h odd 6 2
245.2.e.b 2 21.g even 6 2
315.2.a.b 1 1.a even 1 1 trivial
560.2.a.b 1 12.b even 2 1
1225.2.a.e 1 105.g even 2 1
1225.2.b.d 2 105.k odd 4 2
1575.2.a.f 1 5.b even 2 1
1575.2.d.c 2 5.c odd 4 2
2205.2.a.e 1 7.b odd 2 1
2240.2.a.k 1 24.h odd 2 1
2240.2.a.u 1 24.f even 2 1
2800.2.a.z 1 60.h even 2 1
2800.2.g.l 2 60.l odd 4 2
3920.2.a.ba 1 84.h odd 2 1
4235.2.a.c 1 33.d even 2 1
5040.2.a.v 1 4.b odd 2 1
5915.2.a.f 1 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T - 5 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 3 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 12 \) Copy content Toggle raw display
$43$ \( T + 10 \) Copy content Toggle raw display
$47$ \( T + 9 \) Copy content Toggle raw display
$53$ \( T + 12 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 1 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 12 \) Copy content Toggle raw display
$97$ \( T + 1 \) Copy content Toggle raw display
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