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Properties

Label 975.2.t.e
Level $975$
Weight $2$
Character orbit 975.t
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(268,975)]
 
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, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.268");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.t (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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
268.1 −2.72289 0.707107 + 0.707107i 5.41413 0 −1.92537 − 1.92537i − 5.14187i −9.29631 1.00000i 0
268.2 −2.52343 −0.707107 − 0.707107i 4.36771 0 1.78434 + 1.78434i − 1.25275i −5.97474 1.00000i 0
268.3 −2.32208 0.707107 + 0.707107i 3.39207 0 −1.64196 − 1.64196i 3.79321i −3.23251 1.00000i 0
268.4 −2.26576 −0.707107 − 0.707107i 3.13367 0 1.60214 + 1.60214i 0.785318i −2.56864 1.00000i 0
268.5 −1.82478 0.707107 + 0.707107i 1.32981 0 −1.29031 − 1.29031i − 3.97622i 1.22295 1.00000i 0
268.6 −1.37366 −0.707107 − 0.707107i −0.113053 0 0.971326 + 0.971326i − 3.35143i 2.90262 1.00000i 0
268.7 −1.06014 0.707107 + 0.707107i −0.876099 0 −0.749634 − 0.749634i 0.213817i 3.04907 1.00000i 0
268.8 −1.01772 −0.707107 − 0.707107i −0.964247 0 0.719636 + 0.719636i − 4.35318i 3.01677 1.00000i 0
268.9 −0.507438 −0.707107 − 0.707107i −1.74251 0 0.358813 + 0.358813i 3.27968i 1.89909 1.00000i 0
268.10 −0.241882 −0.707107 − 0.707107i −1.94149 0 0.171036 + 0.171036i 4.02393i 0.953375 1.00000i 0
268.11 0.241882 0.707107 + 0.707107i −1.94149 0 0.171036 + 0.171036i − 4.02393i −0.953375 1.00000i 0
268.12 0.507438 0.707107 + 0.707107i −1.74251 0 0.358813 + 0.358813i − 3.27968i −1.89909 1.00000i 0
268.13 1.01772 0.707107 + 0.707107i −0.964247 0 0.719636 + 0.719636i 4.35318i −3.01677 1.00000i 0
268.14 1.06014 −0.707107 − 0.707107i −0.876099 0 −0.749634 − 0.749634i − 0.213817i −3.04907 1.00000i 0
268.15 1.37366 0.707107 + 0.707107i −0.113053 0 0.971326 + 0.971326i 3.35143i −2.90262 1.00000i 0
268.16 1.82478 −0.707107 − 0.707107i 1.32981 0 −1.29031 − 1.29031i 3.97622i −1.22295 1.00000i 0
268.17 2.26576 0.707107 + 0.707107i 3.13367 0 1.60214 + 1.60214i − 0.785318i 2.56864 1.00000i 0
268.18 2.32208 −0.707107 − 0.707107i 3.39207 0 −1.64196 − 1.64196i − 3.79321i 3.23251 1.00000i 0
268.19 2.52343 0.707107 + 0.707107i 4.36771 0 1.78434 + 1.78434i 1.25275i 5.97474 1.00000i 0
268.20 2.72289 −0.707107 − 0.707107i 5.41413 0 −1.92537 − 1.92537i 5.14187i 9.29631 1.00000i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 268.20
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.t.e yes 40
5.b even 2 1 inner 975.2.t.e yes 40
5.c odd 4 2 975.2.k.e ✓ 40
13.d odd 4 1 975.2.k.e ✓ 40
65.f even 4 1 inner 975.2.t.e yes 40
65.g odd 4 1 975.2.k.e ✓ 40
65.k even 4 1 inner 975.2.t.e yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.k.e ✓ 40 5.c odd 4 2
975.2.k.e ✓ 40 13.d odd 4 1
975.2.k.e ✓ 40 65.g odd 4 1
975.2.t.e yes 40 1.a even 1 1 trivial
975.2.t.e yes 40 5.b even 2 1 inner
975.2.t.e yes 40 65.f even 4 1 inner
975.2.t.e yes 40 65.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 32 T_{2}^{18} + 428 T_{2}^{16} - 3108 T_{2}^{14} + 13334 T_{2}^{12} - 34536 T_{2}^{10} + \cdots + 144 \) acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\). Copy content Toggle raw display