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Properties

Label 975.2.bt.k
Level $975$
Weight $2$
Character orbit 975.bt
Analytic conductor $7.785$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(68,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bt (of order \(12\), degree \(4\), minimal)

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: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 28 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 28 q^{6} + 40 q^{16} + 48 q^{31} + 76 q^{36} - 120 q^{46} - 32 q^{51} - 88 q^{61} - 8 q^{66} - 40 q^{76} + 32 q^{81} + 184 q^{91} + 88 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −0.646049 + 2.41109i −0.720199 + 1.57522i −3.66392 − 2.11536i 0 −3.33271 − 2.75413i 0.0914428 + 0.341269i 3.93732 − 3.93732i −1.96263 − 2.26894i 0
68.2 −0.646049 + 2.41109i 0.163899 − 1.72428i −3.66392 − 2.11536i 0 4.05150 + 1.50914i −0.0914428 − 0.341269i 3.93732 − 3.93732i −2.94627 − 0.565216i 0
68.3 −0.547991 + 2.04513i 1.38356 + 1.04200i −2.15022 − 1.24143i 0 −2.88921 + 2.25855i −1.05458 − 3.93575i 0.722908 − 0.722908i 0.828456 + 2.88334i 0
68.4 −0.547991 + 2.04513i 1.71920 − 0.210624i −2.15022 − 1.24143i 0 −0.511352 + 3.63140i 1.05458 + 3.93575i 0.722908 − 0.722908i 2.91128 − 0.724207i 0
68.5 −0.138514 + 0.516942i 1.26613 + 1.18191i 1.48401 + 0.856792i 0 −0.786355 + 0.490807i 0.648233 + 2.41924i −1.40532 + 1.40532i 0.206193 + 2.99291i 0
68.6 −0.138514 + 0.516942i 1.68746 − 0.390494i 1.48401 + 0.856792i 0 −0.0318742 + 0.926407i −0.648233 − 2.41924i −1.40532 + 1.40532i 2.69503 − 1.31788i 0
68.7 0.138514 − 0.516942i −1.68746 + 0.390494i 1.48401 + 0.856792i 0 −0.0318742 + 0.926407i 0.648233 + 2.41924i 1.40532 − 1.40532i 2.69503 − 1.31788i 0
68.8 0.138514 − 0.516942i −1.26613 − 1.18191i 1.48401 + 0.856792i 0 −0.786355 + 0.490807i −0.648233 − 2.41924i 1.40532 − 1.40532i 0.206193 + 2.99291i 0
68.9 0.547991 − 2.04513i −1.71920 + 0.210624i −2.15022 − 1.24143i 0 −0.511352 + 3.63140i −1.05458 − 3.93575i −0.722908 + 0.722908i 2.91128 − 0.724207i 0
68.10 0.547991 − 2.04513i −1.38356 − 1.04200i −2.15022 − 1.24143i 0 −2.88921 + 2.25855i 1.05458 + 3.93575i −0.722908 + 0.722908i 0.828456 + 2.88334i 0
68.11 0.646049 − 2.41109i −0.163899 + 1.72428i −3.66392 − 2.11536i 0 4.05150 + 1.50914i 0.0914428 + 0.341269i −3.93732 + 3.93732i −2.94627 − 0.565216i 0
68.12 0.646049 − 2.41109i 0.720199 − 1.57522i −3.66392 − 2.11536i 0 −3.33271 − 2.75413i −0.0914428 − 0.341269i −3.93732 + 3.93732i −1.96263 − 2.26894i 0
107.1 −2.41109 − 0.646049i −1.72428 − 0.163899i 3.66392 + 2.11536i 0 4.05150 + 1.50914i 0.341269 − 0.0914428i −3.93732 − 3.93732i 2.94627 + 0.565216i 0
107.2 −2.41109 − 0.646049i 1.57522 + 0.720199i 3.66392 + 2.11536i 0 −3.33271 − 2.75413i −0.341269 + 0.0914428i −3.93732 − 3.93732i 1.96263 + 2.26894i 0
107.3 −2.04513 − 0.547991i −0.210624 − 1.71920i 2.15022 + 1.24143i 0 −0.511352 + 3.63140i −3.93575 + 1.05458i −0.722908 − 0.722908i −2.91128 + 0.724207i 0
107.4 −2.04513 − 0.547991i 1.04200 − 1.38356i 2.15022 + 1.24143i 0 −2.88921 + 2.25855i 3.93575 − 1.05458i −0.722908 − 0.722908i −0.828456 − 2.88334i 0
107.5 −0.516942 − 0.138514i −0.390494 − 1.68746i −1.48401 − 0.856792i 0 −0.0318742 + 0.926407i 2.41924 − 0.648233i 1.40532 + 1.40532i −2.69503 + 1.31788i 0
107.6 −0.516942 − 0.138514i 1.18191 − 1.26613i −1.48401 − 0.856792i 0 −0.786355 + 0.490807i −2.41924 + 0.648233i 1.40532 + 1.40532i −0.206193 − 2.99291i 0
107.7 0.516942 + 0.138514i −1.18191 + 1.26613i −1.48401 − 0.856792i 0 −0.786355 + 0.490807i 2.41924 − 0.648233i −1.40532 − 1.40532i −0.206193 − 2.99291i 0
107.8 0.516942 + 0.138514i 0.390494 + 1.68746i −1.48401 − 0.856792i 0 −0.0318742 + 0.926407i −2.41924 + 0.648233i −1.40532 − 1.40532i −2.69503 + 1.31788i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
13.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
39.i odd 6 1 inner
65.n even 6 1 inner
65.q odd 12 2 inner
195.x odd 6 1 inner
195.bl even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bt.k ✓ 48
3.b odd 2 1 inner 975.2.bt.k ✓ 48
5.b even 2 1 inner 975.2.bt.k ✓ 48
5.c odd 4 2 inner 975.2.bt.k ✓ 48
13.c even 3 1 inner 975.2.bt.k ✓ 48
15.d odd 2 1 inner 975.2.bt.k ✓ 48
15.e even 4 2 inner 975.2.bt.k ✓ 48
39.i odd 6 1 inner 975.2.bt.k ✓ 48
65.n even 6 1 inner 975.2.bt.k ✓ 48
65.q odd 12 2 inner 975.2.bt.k ✓ 48
195.x odd 6 1 inner 975.2.bt.k ✓ 48
195.bl even 12 2 inner 975.2.bt.k ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bt.k ✓ 48 1.a even 1 1 trivial
975.2.bt.k ✓ 48 3.b odd 2 1 inner
975.2.bt.k ✓ 48 5.b even 2 1 inner
975.2.bt.k ✓ 48 5.c odd 4 2 inner
975.2.bt.k ✓ 48 13.c even 3 1 inner
975.2.bt.k ✓ 48 15.d odd 2 1 inner
975.2.bt.k ✓ 48 15.e even 4 2 inner
975.2.bt.k ✓ 48 39.i odd 6 1 inner
975.2.bt.k ✓ 48 65.n even 6 1 inner
975.2.bt.k ✓ 48 65.q odd 12 2 inner
975.2.bt.k ✓ 48 195.x odd 6 1 inner
975.2.bt.k ✓ 48 195.bl even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{24} - 59T_{2}^{20} + 2696T_{2}^{16} - 46187T_{2}^{12} + 612449T_{2}^{8} - 50240T_{2}^{4} + 4096 \) Copy content Toggle raw display
\( T_{7}^{24} - 315T_{7}^{20} + 88374T_{7}^{16} - 3417727T_{7}^{12} + 117690966T_{7}^{8} - 1833819T_{7}^{4} + 28561 \) Copy content Toggle raw display
\( T_{59}^{12} + 110T_{59}^{10} + 8923T_{59}^{8} + 332622T_{59}^{6} + 9166689T_{59}^{4} + 26763048T_{59}^{2} + 70963776 \) Copy content Toggle raw display