Properties

Label 2475.2.a.bh
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + (\beta_{3} - \beta_1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + (\beta_{3} - \beta_1) q^{7} + \beta_{3} q^{8} + q^{11} + ( - 3 \beta_{3} - 3 \beta_1) q^{13} + ( - \beta_{2} - 3) q^{14} + ( - 2 \beta_{2} - 1) q^{16} + (\beta_{3} + 3 \beta_1) q^{17} - 2 q^{19} + \beta_1 q^{22} - 4 \beta_{3} q^{23} + ( - 3 \beta_{2} - 3) q^{26} + ( - 3 \beta_{3} - 3 \beta_1) q^{28} + ( - 2 \beta_{2} - 6) q^{29} + ( - 2 \beta_{2} - 4) q^{31} + ( - 4 \beta_{3} - 5 \beta_1) q^{32} + (3 \beta_{2} + 5) q^{34} + (2 \beta_{3} - 2 \beta_1) q^{37} - 2 \beta_1 q^{38} + ( - 2 \beta_{2} - 6) q^{41} + (7 \beta_{3} + 5 \beta_1) q^{43} + \beta_{2} q^{44} + 4 q^{46} + (2 \beta_{3} + 2 \beta_1) q^{47} - q^{49} + (3 \beta_{3} - 3 \beta_1) q^{52} + (6 \beta_{3} + 2 \beta_1) q^{53} + ( - \beta_{2} + 3) q^{56} + ( - 2 \beta_{3} - 10 \beta_1) q^{58} + (4 \beta_{2} - 6) q^{59} + (4 \beta_{2} + 2) q^{61} + ( - 2 \beta_{3} - 8 \beta_1) q^{62} + ( - \beta_{2} - 4) q^{64} + ( - 4 \beta_{3} - 8 \beta_1) q^{67} + (\beta_{3} + 5 \beta_1) q^{68} - 6 q^{71} + (\beta_{3} + 5 \beta_1) q^{73} + ( - 2 \beta_{2} - 6) q^{74} - 2 \beta_{2} q^{76} + (\beta_{3} - \beta_1) q^{77} + ( - 8 \beta_{2} - 2) q^{79} + ( - 2 \beta_{3} - 10 \beta_1) q^{82} + ( - 5 \beta_{3} - 5 \beta_1) q^{83} + (5 \beta_{2} + 3) q^{86} + \beta_{3} q^{88} + (2 \beta_{2} - 12) q^{89} + 6 \beta_{2} q^{91} + (8 \beta_{3} + 4 \beta_1) q^{92} + (2 \beta_{2} + 2) q^{94} + ( - 8 \beta_{3} - 4 \beta_1) q^{97} - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{11} - 12 q^{14} - 4 q^{16} - 8 q^{19} - 12 q^{26} - 24 q^{29} - 16 q^{31} + 20 q^{34} - 24 q^{41} + 16 q^{46} - 4 q^{49} + 12 q^{56} - 24 q^{59} + 8 q^{61} - 16 q^{64} - 24 q^{71} - 24 q^{74} - 8 q^{79} + 12 q^{86} - 48 q^{89} + 8 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) 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
−1.93185
−0.517638
0.517638
1.93185
−1.93185 0 1.73205 0 0 2.44949 0.517638 0 0
1.2 −0.517638 0 −1.73205 0 0 2.44949 1.93185 0 0
1.3 0.517638 0 −1.73205 0 0 −2.44949 −1.93185 0 0
1.4 1.93185 0 1.73205 0 0 −2.44949 −0.517638 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.bh 4
3.b odd 2 1 2475.2.a.bg 4
5.b even 2 1 inner 2475.2.a.bh 4
5.c odd 4 2 495.2.c.c yes 4
15.d odd 2 1 2475.2.a.bg 4
15.e even 4 2 495.2.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.c.b 4 15.e even 4 2
495.2.c.c yes 4 5.c odd 4 2
2475.2.a.bg 4 3.b odd 2 1
2475.2.a.bg 4 15.d odd 2 1
2475.2.a.bh 4 1.a even 1 1 trivial
2475.2.a.bh 4 5.b even 2 1 inner

Hecke kernels

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

\( T_{2}^{4} - 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 6 \) Copy content Toggle raw display
\( T_{29}^{2} + 12T_{29} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 28T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T + 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 64T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 156T^{2} + 4356 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 112T^{2} + 64 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 192T^{2} + 2304 \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 84T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 188)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 24 T + 132)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 192T^{2} + 2304 \) Copy content Toggle raw display
show more
show less