Properties

Label 2475.2.a.g
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
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] = [2475,2,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, 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("2475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{4} - 4 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} - 4 q^{7} - 3 q^{8} - q^{11} + 2 q^{13} - 4 q^{14} - q^{16} - 2 q^{17} - q^{22} + 8 q^{23} + 2 q^{26} + 4 q^{28} + 6 q^{29} - 8 q^{31} + 5 q^{32} - 2 q^{34} - 6 q^{37} + 2 q^{41} + q^{44} + 8 q^{46} + 8 q^{47} + 9 q^{49} - 2 q^{52} + 6 q^{53} + 12 q^{56} + 6 q^{58} + 4 q^{59} + 6 q^{61} - 8 q^{62} + 7 q^{64} + 4 q^{67} + 2 q^{68} + 14 q^{73} - 6 q^{74} + 4 q^{77} - 4 q^{79} + 2 q^{82} + 12 q^{83} + 3 q^{88} + 6 q^{89} - 8 q^{91} - 8 q^{92} + 8 q^{94} - 2 q^{97} + 9 q^{98}+O(q^{100}) \) 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
1.00000 0 −1.00000 0 0 −4.00000 −3.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 2475.2.a.g 1
3.b odd 2 1 825.2.a.a 1
5.b even 2 1 99.2.a.b 1
5.c odd 4 2 2475.2.c.d 2
15.d odd 2 1 33.2.a.a 1
15.e even 4 2 825.2.c.a 2
20.d odd 2 1 1584.2.a.o 1
33.d even 2 1 9075.2.a.q 1
35.c odd 2 1 4851.2.a.b 1
40.e odd 2 1 6336.2.a.n 1
40.f even 2 1 6336.2.a.x 1
45.h odd 6 2 891.2.e.e 2
45.j even 6 2 891.2.e.g 2
55.d odd 2 1 1089.2.a.j 1
60.h even 2 1 528.2.a.g 1
105.g even 2 1 1617.2.a.j 1
120.i odd 2 1 2112.2.a.bb 1
120.m even 2 1 2112.2.a.j 1
165.d even 2 1 363.2.a.b 1
165.o odd 10 4 363.2.e.e 4
165.r even 10 4 363.2.e.g 4
195.e odd 2 1 5577.2.a.a 1
255.h odd 2 1 9537.2.a.m 1
660.g odd 2 1 5808.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 15.d odd 2 1
99.2.a.b 1 5.b even 2 1
363.2.a.b 1 165.d even 2 1
363.2.e.e 4 165.o odd 10 4
363.2.e.g 4 165.r even 10 4
528.2.a.g 1 60.h even 2 1
825.2.a.a 1 3.b odd 2 1
825.2.c.a 2 15.e even 4 2
891.2.e.e 2 45.h odd 6 2
891.2.e.g 2 45.j even 6 2
1089.2.a.j 1 55.d odd 2 1
1584.2.a.o 1 20.d odd 2 1
1617.2.a.j 1 105.g even 2 1
2112.2.a.j 1 120.m even 2 1
2112.2.a.bb 1 120.i odd 2 1
2475.2.a.g 1 1.a even 1 1 trivial
2475.2.c.d 2 5.c odd 4 2
4851.2.a.b 1 35.c odd 2 1
5577.2.a.a 1 195.e odd 2 1
5808.2.a.t 1 660.g odd 2 1
6336.2.a.n 1 40.e odd 2 1
6336.2.a.x 1 40.f even 2 1
9075.2.a.q 1 33.d even 2 1
9537.2.a.m 1 255.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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