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Properties

Label 975.2.k.e
Level $975$
Weight $2$
Character orbit 975.k
Analytic conductor $7.785$
Analytic rank $0$
Dimension $40$
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(307,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.k (of order \(4\), degree \(2\), 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: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 48 q^{4} + 24 q^{11} + 64 q^{16} + 20 q^{19} + 12 q^{21} + 24 q^{24} - 56 q^{26} + 4 q^{31} - 80 q^{34} - 16 q^{39} - 40 q^{41} - 152 q^{44} - 24 q^{46} + 184 q^{49} - 32 q^{59} - 64 q^{61} - 48 q^{64} - 32 q^{66} - 16 q^{69} - 88 q^{71} + 16 q^{76} - 40 q^{81} - 24 q^{84} - 48 q^{86} + 80 q^{89} + 68 q^{91} - 8 q^{96} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 − 2.72289i 0.707107 − 0.707107i −5.41413 0 −1.92537 − 1.92537i 5.14187 9.29631i − 1.00000i 0
307.2 − 2.52343i −0.707107 + 0.707107i −4.36771 0 1.78434 + 1.78434i 1.25275 5.97474i − 1.00000i 0
307.3 − 2.32208i 0.707107 − 0.707107i −3.39207 0 −1.64196 − 1.64196i −3.79321 3.23251i − 1.00000i 0
307.4 − 2.26576i −0.707107 + 0.707107i −3.13367 0 1.60214 + 1.60214i −0.785318 2.56864i − 1.00000i 0
307.5 − 1.82478i 0.707107 − 0.707107i −1.32981 0 −1.29031 − 1.29031i 3.97622 − 1.22295i − 1.00000i 0
307.6 − 1.37366i −0.707107 + 0.707107i 0.113053 0 0.971326 + 0.971326i 3.35143 − 2.90262i − 1.00000i 0
307.7 − 1.06014i 0.707107 − 0.707107i 0.876099 0 −0.749634 − 0.749634i −0.213817 − 3.04907i − 1.00000i 0
307.8 − 1.01772i −0.707107 + 0.707107i 0.964247 0 0.719636 + 0.719636i 4.35318 − 3.01677i − 1.00000i 0
307.9 − 0.507438i −0.707107 + 0.707107i 1.74251 0 0.358813 + 0.358813i −3.27968 − 1.89909i − 1.00000i 0
307.10 − 0.241882i −0.707107 + 0.707107i 1.94149 0 0.171036 + 0.171036i −4.02393 − 0.953375i − 1.00000i 0
307.11 0.241882i 0.707107 − 0.707107i 1.94149 0 0.171036 + 0.171036i 4.02393 0.953375i − 1.00000i 0
307.12 0.507438i 0.707107 − 0.707107i 1.74251 0 0.358813 + 0.358813i 3.27968 1.89909i − 1.00000i 0
307.13 1.01772i 0.707107 − 0.707107i 0.964247 0 0.719636 + 0.719636i −4.35318 3.01677i − 1.00000i 0
307.14 1.06014i −0.707107 + 0.707107i 0.876099 0 −0.749634 − 0.749634i 0.213817 3.04907i − 1.00000i 0
307.15 1.37366i 0.707107 − 0.707107i 0.113053 0 0.971326 + 0.971326i −3.35143 2.90262i − 1.00000i 0
307.16 1.82478i −0.707107 + 0.707107i −1.32981 0 −1.29031 − 1.29031i −3.97622 1.22295i − 1.00000i 0
307.17 2.26576i 0.707107 − 0.707107i −3.13367 0 1.60214 + 1.60214i 0.785318 − 2.56864i − 1.00000i 0
307.18 2.32208i −0.707107 + 0.707107i −3.39207 0 −1.64196 − 1.64196i 3.79321 − 3.23251i − 1.00000i 0
307.19 2.52343i 0.707107 − 0.707107i −4.36771 0 1.78434 + 1.78434i −1.25275 − 5.97474i − 1.00000i 0
307.20 2.72289i −0.707107 + 0.707107i −5.41413 0 −1.92537 − 1.92537i −5.14187 − 9.29631i − 1.00000i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
65.f even 4 1 inner
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.k.e ✓ 40
5.b even 2 1 inner 975.2.k.e ✓ 40
5.c odd 4 2 975.2.t.e yes 40
13.d odd 4 1 975.2.t.e yes 40
65.f even 4 1 inner 975.2.k.e ✓ 40
65.g odd 4 1 975.2.t.e yes 40
65.k even 4 1 inner 975.2.k.e ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.k.e ✓ 40 1.a even 1 1 trivial
975.2.k.e ✓ 40 5.b even 2 1 inner
975.2.k.e ✓ 40 65.f even 4 1 inner
975.2.k.e ✓ 40 65.k even 4 1 inner
975.2.t.e yes 40 5.c odd 4 2
975.2.t.e yes 40 13.d odd 4 1
975.2.t.e yes 40 65.g odd 4 1

Hecke kernels

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

\( T_{2}^{20} + 32 T_{2}^{18} + 428 T_{2}^{16} + 3108 T_{2}^{14} + 13334 T_{2}^{12} + 34536 T_{2}^{10} + \cdots + 144 \) Copy content Toggle raw display
\( T_{7}^{20} - 116 T_{7}^{18} + 5717 T_{7}^{16} - 155916 T_{7}^{14} + 2561090 T_{7}^{12} - 25737852 T_{7}^{10} + \cdots + 9865881 \) Copy content Toggle raw display