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Properties

Label 1368.2.p.a
Level $1368$
Weight $2$
Character orbit 1368.p
Analytic conductor $10.924$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1368,2,Mod(341,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1368.341");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.p (of order \(2\), degree \(1\), 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: \(10.9235349965\)
Analytic rank: \(0\)
Dimension: \(80\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{4} + 8 q^{16} + 80 q^{25} + 48 q^{49} - 16 q^{58} + 8 q^{64} + 32 q^{73} + 72 q^{76} - 16 q^{82}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1 −1.41138 − 0.0894099i 0 1.98401 + 0.252383i −1.64947 0 −2.97899 −2.77764 − 0.533600i 0 2.32804 + 0.147479i
341.2 −1.41138 − 0.0894099i 0 1.98401 + 0.252383i 1.64947 0 −2.97899 −2.77764 − 0.533600i 0 −2.32804 − 0.147479i
341.3 −1.41138 + 0.0894099i 0 1.98401 − 0.252383i −1.64947 0 −2.97899 −2.77764 + 0.533600i 0 2.32804 − 0.147479i
341.4 −1.41138 + 0.0894099i 0 1.98401 − 0.252383i 1.64947 0 −2.97899 −2.77764 + 0.533600i 0 −2.32804 + 0.147479i
341.5 −1.37444 − 0.333032i 0 1.77818 + 0.915465i −3.95094 0 1.26339 −2.13913 − 1.85044i 0 5.43034 + 1.31579i
341.6 −1.37444 − 0.333032i 0 1.77818 + 0.915465i 3.95094 0 1.26339 −2.13913 − 1.85044i 0 −5.43034 − 1.31579i
341.7 −1.37444 + 0.333032i 0 1.77818 − 0.915465i −3.95094 0 1.26339 −2.13913 + 1.85044i 0 5.43034 − 1.31579i
341.8 −1.37444 + 0.333032i 0 1.77818 − 0.915465i 3.95094 0 1.26339 −2.13913 + 1.85044i 0 −5.43034 + 1.31579i
341.9 −1.35713 − 0.397726i 0 1.68363 + 1.07953i −0.516799 0 3.97945 −1.85555 − 2.13470i 0 0.701366 + 0.205544i
341.10 −1.35713 − 0.397726i 0 1.68363 + 1.07953i 0.516799 0 3.97945 −1.85555 − 2.13470i 0 −0.701366 − 0.205544i
341.11 −1.35713 + 0.397726i 0 1.68363 − 1.07953i −0.516799 0 3.97945 −1.85555 + 2.13470i 0 0.701366 − 0.205544i
341.12 −1.35713 + 0.397726i 0 1.68363 − 1.07953i 0.516799 0 3.97945 −1.85555 + 2.13470i 0 −0.701366 + 0.205544i
341.13 −1.21162 − 0.729373i 0 0.936031 + 1.76744i −3.69887 0 −1.26582 0.155012 − 2.82418i 0 4.48161 + 2.69785i
341.14 −1.21162 − 0.729373i 0 0.936031 + 1.76744i 3.69887 0 −1.26582 0.155012 − 2.82418i 0 −4.48161 − 2.69785i
341.15 −1.21162 + 0.729373i 0 0.936031 − 1.76744i −3.69887 0 −1.26582 0.155012 + 2.82418i 0 4.48161 − 2.69785i
341.16 −1.21162 + 0.729373i 0 0.936031 − 1.76744i 3.69887 0 −1.26582 0.155012 + 2.82418i 0 −4.48161 + 2.69785i
341.17 −1.12729 − 0.853938i 0 0.541581 + 1.92528i −0.186730 0 −1.42880 1.03355 − 2.63283i 0 0.210500 + 0.159456i
341.18 −1.12729 − 0.853938i 0 0.541581 + 1.92528i 0.186730 0 −1.42880 1.03355 − 2.63283i 0 −0.210500 − 0.159456i
341.19 −1.12729 + 0.853938i 0 0.541581 − 1.92528i −0.186730 0 −1.42880 1.03355 + 2.63283i 0 0.210500 − 0.159456i
341.20 −1.12729 + 0.853938i 0 0.541581 − 1.92528i 0.186730 0 −1.42880 1.03355 + 2.63283i 0 −0.210500 + 0.159456i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
19.b odd 2 1 inner
24.h odd 2 1 inner
57.d even 2 1 inner
152.g odd 2 1 inner
456.p even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.p.a ✓ 80
3.b odd 2 1 inner 1368.2.p.a ✓ 80
4.b odd 2 1 5472.2.p.a 80
8.b even 2 1 inner 1368.2.p.a ✓ 80
8.d odd 2 1 5472.2.p.a 80
12.b even 2 1 5472.2.p.a 80
19.b odd 2 1 inner 1368.2.p.a ✓ 80
24.f even 2 1 5472.2.p.a 80
24.h odd 2 1 inner 1368.2.p.a ✓ 80
57.d even 2 1 inner 1368.2.p.a ✓ 80
76.d even 2 1 5472.2.p.a 80
152.b even 2 1 5472.2.p.a 80
152.g odd 2 1 inner 1368.2.p.a ✓ 80
228.b odd 2 1 5472.2.p.a 80
456.l odd 2 1 5472.2.p.a 80
456.p even 2 1 inner 1368.2.p.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.p.a ✓ 80 1.a even 1 1 trivial
1368.2.p.a ✓ 80 3.b odd 2 1 inner
1368.2.p.a ✓ 80 8.b even 2 1 inner
1368.2.p.a ✓ 80 19.b odd 2 1 inner
1368.2.p.a ✓ 80 24.h odd 2 1 inner
1368.2.p.a ✓ 80 57.d even 2 1 inner
1368.2.p.a ✓ 80 152.g odd 2 1 inner
1368.2.p.a ✓ 80 456.p even 2 1 inner
5472.2.p.a 80 4.b odd 2 1
5472.2.p.a 80 8.d odd 2 1
5472.2.p.a 80 12.b even 2 1
5472.2.p.a 80 24.f even 2 1
5472.2.p.a 80 76.d even 2 1
5472.2.p.a 80 152.b even 2 1
5472.2.p.a 80 228.b odd 2 1
5472.2.p.a 80 456.l odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1368, [\chi])\).