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Properties

Label 975.2.w.k
Level $975$
Weight $2$
Character orbit 975.w
Analytic conductor $7.785$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(49,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.w (of order \(6\), degree \(2\), not 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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{4} + 12 q^{9} + 18 q^{11} - 20 q^{14} - 8 q^{16} + 18 q^{19} - 10 q^{26} + 32 q^{29} + 16 q^{36} + 6 q^{39} + 36 q^{41} - 90 q^{46} - 62 q^{49} + 50 q^{56} - 108 q^{59} - 22 q^{61} - 32 q^{64} + 60 q^{66} + 2 q^{69} - 66 q^{71} - 10 q^{74} - 54 q^{76} - 36 q^{79} - 12 q^{81} + 78 q^{84} + 96 q^{89} - 58 q^{91} + 60 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.29268 − 2.23898i 0.866025 − 0.500000i −2.34202 + 4.05650i 0 −2.23898 − 1.29268i 1.93439 − 3.35046i 6.93919 0.500000 − 0.866025i 0
49.2 −1.18025 − 2.04426i −0.866025 + 0.500000i −1.78599 + 3.09343i 0 2.04426 + 1.18025i −2.27152 + 3.93440i 3.71068 0.500000 − 0.866025i 0
49.3 −1.01982 − 1.76638i −0.866025 + 0.500000i −1.08006 + 1.87073i 0 1.76638 + 1.01982i 2.59182 − 4.48917i 0.326607 0.500000 − 0.866025i 0
49.4 −0.793724 − 1.37477i 0.866025 − 0.500000i −0.259994 + 0.450324i 0 −1.37477 − 0.793724i −0.534623 + 0.925994i −2.34944 0.500000 − 0.866025i 0
49.5 −0.417047 − 0.722346i 0.866025 − 0.500000i 0.652144 − 1.12955i 0 −0.722346 − 0.417047i −1.18426 + 2.05120i −2.75609 0.500000 − 0.866025i 0
49.6 −0.303374 − 0.525459i −0.866025 + 0.500000i 0.815929 − 1.41323i 0 0.525459 + 0.303374i −0.970816 + 1.68150i −2.20362 0.500000 − 0.866025i 0
49.7 0.303374 + 0.525459i 0.866025 − 0.500000i 0.815929 − 1.41323i 0 0.525459 + 0.303374i 0.970816 − 1.68150i 2.20362 0.500000 − 0.866025i 0
49.8 0.417047 + 0.722346i −0.866025 + 0.500000i 0.652144 − 1.12955i 0 −0.722346 − 0.417047i 1.18426 − 2.05120i 2.75609 0.500000 − 0.866025i 0
49.9 0.793724 + 1.37477i −0.866025 + 0.500000i −0.259994 + 0.450324i 0 −1.37477 − 0.793724i 0.534623 − 0.925994i 2.34944 0.500000 − 0.866025i 0
49.10 1.01982 + 1.76638i 0.866025 − 0.500000i −1.08006 + 1.87073i 0 1.76638 + 1.01982i −2.59182 + 4.48917i −0.326607 0.500000 − 0.866025i 0
49.11 1.18025 + 2.04426i 0.866025 − 0.500000i −1.78599 + 3.09343i 0 2.04426 + 1.18025i 2.27152 − 3.93440i −3.71068 0.500000 − 0.866025i 0
49.12 1.29268 + 2.23898i −0.866025 + 0.500000i −2.34202 + 4.05650i 0 −2.23898 − 1.29268i −1.93439 + 3.35046i −6.93919 0.500000 − 0.866025i 0
199.1 −1.29268 + 2.23898i 0.866025 + 0.500000i −2.34202 − 4.05650i 0 −2.23898 + 1.29268i 1.93439 + 3.35046i 6.93919 0.500000 + 0.866025i 0
199.2 −1.18025 + 2.04426i −0.866025 − 0.500000i −1.78599 − 3.09343i 0 2.04426 − 1.18025i −2.27152 − 3.93440i 3.71068 0.500000 + 0.866025i 0
199.3 −1.01982 + 1.76638i −0.866025 − 0.500000i −1.08006 − 1.87073i 0 1.76638 − 1.01982i 2.59182 + 4.48917i 0.326607 0.500000 + 0.866025i 0
199.4 −0.793724 + 1.37477i 0.866025 + 0.500000i −0.259994 − 0.450324i 0 −1.37477 + 0.793724i −0.534623 − 0.925994i −2.34944 0.500000 + 0.866025i 0
199.5 −0.417047 + 0.722346i 0.866025 + 0.500000i 0.652144 + 1.12955i 0 −0.722346 + 0.417047i −1.18426 − 2.05120i −2.75609 0.500000 + 0.866025i 0
199.6 −0.303374 + 0.525459i −0.866025 − 0.500000i 0.815929 + 1.41323i 0 0.525459 − 0.303374i −0.970816 − 1.68150i −2.20362 0.500000 + 0.866025i 0
199.7 0.303374 − 0.525459i 0.866025 + 0.500000i 0.815929 + 1.41323i 0 0.525459 − 0.303374i 0.970816 + 1.68150i 2.20362 0.500000 + 0.866025i 0
199.8 0.417047 − 0.722346i −0.866025 − 0.500000i 0.652144 + 1.12955i 0 −0.722346 + 0.417047i 1.18426 + 2.05120i 2.75609 0.500000 + 0.866025i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.e even 6 1 inner
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.w.k 24
5.b even 2 1 inner 975.2.w.k 24
5.c odd 4 1 975.2.bc.k ✓ 12
5.c odd 4 1 975.2.bc.l yes 12
13.e even 6 1 inner 975.2.w.k 24
65.l even 6 1 inner 975.2.w.k 24
65.r odd 12 1 975.2.bc.k ✓ 12
65.r odd 12 1 975.2.bc.l yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.w.k 24 1.a even 1 1 trivial
975.2.w.k 24 5.b even 2 1 inner
975.2.w.k 24 13.e even 6 1 inner
975.2.w.k 24 65.l even 6 1 inner
975.2.bc.k ✓ 12 5.c odd 4 1
975.2.bc.k ✓ 12 65.r odd 12 1
975.2.bc.l yes 12 5.c odd 4 1
975.2.bc.l yes 12 65.r odd 12 1

Hecke kernels

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

\( T_{2}^{24} + 20 T_{2}^{22} + 250 T_{2}^{20} + 1960 T_{2}^{18} + 11275 T_{2}^{16} + 45512 T_{2}^{14} + \cdots + 10000 \) Copy content Toggle raw display
\( T_{7}^{24} + 73 T_{7}^{22} + 3374 T_{7}^{20} + 95445 T_{7}^{18} + 1967470 T_{7}^{16} + \cdots + 40282095616 \) Copy content Toggle raw display