Properties

Label 2475.2.a.bi
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + 2 \beta_{2} q^{7} + (2 \beta_{2} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + 2 \beta_{2} q^{7} + (2 \beta_{2} + \beta_1) q^{8} + q^{11} + (2 \beta_{3} + 4) q^{14} + (\beta_{3} + 4) q^{16} + (2 \beta_{2} - 2 \beta_1) q^{17} + 4 q^{19} + \beta_1 q^{22} + (\beta_{2} - \beta_1) q^{23} + 6 \beta_1 q^{28} + ( - 2 \beta_{3} - 4) q^{29} + (\beta_{3} + 1) q^{31} + ( - 2 \beta_{2} + 3 \beta_1) q^{32} - 4 q^{34} + ( - 5 \beta_{2} + 3 \beta_1) q^{37} + 4 \beta_1 q^{38} + ( - 2 \beta_{3} - 4) q^{41} + 2 \beta_{2} q^{43} + (\beta_{3} + 2) q^{44} - 2 q^{46} + ( - 2 \beta_{2} + 4 \beta_1) q^{47} + 5 q^{49} - 4 \beta_1 q^{53} + (2 \beta_{3} + 16) q^{56} + ( - 4 \beta_{2} - 6 \beta_1) q^{58} + (\beta_{3} + 5) q^{59} + ( - 2 \beta_{3} + 4) q^{61} + (2 \beta_{2} + 2 \beta_1) q^{62} - \beta_{3} q^{64} + (\beta_{2} + 3 \beta_1) q^{67} - 4 \beta_{2} q^{68} + (3 \beta_{3} + 3) q^{71} - 4 \beta_{2} q^{73} + ( - 2 \beta_{3} + 2) q^{74} + (4 \beta_{3} + 8) q^{76} + 2 \beta_{2} q^{77} + ( - 2 \beta_{3} + 6) q^{79} + ( - 4 \beta_{2} - 6 \beta_1) q^{82} + (2 \beta_{2} - 4 \beta_1) q^{83} + (2 \beta_{3} + 4) q^{86} + (2 \beta_{2} + \beta_1) q^{88} + ( - \beta_{3} + 1) q^{89} - 2 \beta_{2} q^{92} + (2 \beta_{3} + 12) q^{94} + ( - \beta_{2} + 3 \beta_1) q^{97} + 5 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} + 4 q^{11} + 12 q^{14} + 14 q^{16} + 16 q^{19} - 12 q^{29} + 2 q^{31} - 16 q^{34} - 12 q^{41} + 6 q^{44} - 8 q^{46} + 20 q^{49} + 60 q^{56} + 18 q^{59} + 20 q^{61} + 2 q^{64} + 6 q^{71} + 12 q^{74} + 24 q^{76} + 28 q^{79} + 12 q^{86} + 6 q^{89} + 44 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 5\beta_1 \) 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
−2.52434
−0.792287
0.792287
2.52434
−2.52434 0 4.37228 0 0 −3.46410 −5.98844 0 0
1.2 −0.792287 0 −1.37228 0 0 3.46410 2.67181 0 0
1.3 0.792287 0 −1.37228 0 0 −3.46410 −2.67181 0 0
1.4 2.52434 0 4.37228 0 0 3.46410 5.98844 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.2.a.bi 4
3.b odd 2 1 275.2.a.h 4
5.b even 2 1 inner 2475.2.a.bi 4
5.c odd 4 2 495.2.c.a 4
12.b even 2 1 4400.2.a.cc 4
15.d odd 2 1 275.2.a.h 4
15.e even 4 2 55.2.b.a 4
33.d even 2 1 3025.2.a.ba 4
60.h even 2 1 4400.2.a.cc 4
60.l odd 4 2 880.2.b.h 4
165.d even 2 1 3025.2.a.ba 4
165.l odd 4 2 605.2.b.c 4
165.u odd 20 8 605.2.j.j 16
165.v even 20 8 605.2.j.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a 4 15.e even 4 2
275.2.a.h 4 3.b odd 2 1
275.2.a.h 4 15.d odd 2 1
495.2.c.a 4 5.c odd 4 2
605.2.b.c 4 165.l odd 4 2
605.2.j.i 16 165.v even 20 8
605.2.j.j 16 165.u odd 20 8
880.2.b.h 4 60.l odd 4 2
2475.2.a.bi 4 1.a even 1 1 trivial
2475.2.a.bi 4 5.b even 2 1 inner
3025.2.a.ba 4 33.d even 2 1
3025.2.a.ba 4 165.d even 2 1
4400.2.a.cc 4 12.b even 2 1
4400.2.a.cc 4 60.h even 2 1

Hecke kernels

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

\( T_{2}^{4} - 7T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display
\( T_{29}^{2} + 6T_{29} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 7T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 28T^{2} + 64 \) Copy content Toggle raw display
$19$ \( (T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 7T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 123T^{2} + 144 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 112T^{2} + 1024 \) Copy content Toggle raw display
$59$ \( (T^{2} - 9 T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 10 T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 87T^{2} + 36 \) Copy content Toggle raw display
$71$ \( (T^{2} - 3 T - 72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 51T^{2} + 576 \) Copy content Toggle raw display
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