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Properties

Label 1368.2.e.f
Level $1368$
Weight $2$
Character orbit 1368.e
Analytic conductor $10.924$
Analytic rank $0$
Dimension $24$
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(379,1368)]
 
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([1, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1368.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.e (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: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{4} - 12 q^{16} + 24 q^{19} - 40 q^{25} - 40 q^{28} - 72 q^{49} - 104 q^{58} + 20 q^{64} + 80 q^{73} - 12 q^{76} + 56 q^{82}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1 −1.39298 − 0.244153i 0 1.88078 + 0.680200i 3.10261i 0 − 4.34495i −2.45381 − 1.40670i 0 0.757512 − 4.32187i
379.2 −1.39298 − 0.244153i 0 1.88078 + 0.680200i − 3.10261i 0 − 4.34495i −2.45381 − 1.40670i 0 −0.757512 + 4.32187i
379.3 −1.39298 + 0.244153i 0 1.88078 − 0.680200i − 3.10261i 0 4.34495i −2.45381 + 1.40670i 0 0.757512 + 4.32187i
379.4 −1.39298 + 0.244153i 0 1.88078 − 0.680200i 3.10261i 0 4.34495i −2.45381 + 1.40670i 0 −0.757512 − 4.32187i
379.5 −1.26128 − 0.639662i 0 1.18166 + 1.61359i − 1.96435i 0 1.25539i −0.458259 − 2.79106i 0 −1.25652 + 2.47760i
379.6 −1.26128 − 0.639662i 0 1.18166 + 1.61359i 1.96435i 0 1.25539i −0.458259 − 2.79106i 0 1.25652 − 2.47760i
379.7 −1.26128 + 0.639662i 0 1.18166 − 1.61359i 1.96435i 0 − 1.25539i −0.458259 + 2.79106i 0 −1.25652 − 2.47760i
379.8 −1.26128 + 0.639662i 0 1.18166 − 1.61359i − 1.96435i 0 − 1.25539i −0.458259 + 2.79106i 0 1.25652 + 2.47760i
379.9 −0.847808 − 1.13191i 0 −0.562443 + 1.91929i − 2.55248i 0 3.08957i 2.64930 − 0.990551i 0 −2.88918 + 2.16401i
379.10 −0.847808 − 1.13191i 0 −0.562443 + 1.91929i 2.55248i 0 3.08957i 2.64930 − 0.990551i 0 2.88918 − 2.16401i
379.11 −0.847808 + 1.13191i 0 −0.562443 − 1.91929i 2.55248i 0 − 3.08957i 2.64930 + 0.990551i 0 −2.88918 − 2.16401i
379.12 −0.847808 + 1.13191i 0 −0.562443 − 1.91929i − 2.55248i 0 − 3.08957i 2.64930 + 0.990551i 0 2.88918 + 2.16401i
379.13 0.847808 − 1.13191i 0 −0.562443 − 1.91929i 2.55248i 0 − 3.08957i −2.64930 − 0.990551i 0 2.88918 + 2.16401i
379.14 0.847808 − 1.13191i 0 −0.562443 − 1.91929i − 2.55248i 0 − 3.08957i −2.64930 − 0.990551i 0 −2.88918 − 2.16401i
379.15 0.847808 + 1.13191i 0 −0.562443 + 1.91929i − 2.55248i 0 3.08957i −2.64930 + 0.990551i 0 2.88918 − 2.16401i
379.16 0.847808 + 1.13191i 0 −0.562443 + 1.91929i 2.55248i 0 3.08957i −2.64930 + 0.990551i 0 −2.88918 + 2.16401i
379.17 1.26128 − 0.639662i 0 1.18166 − 1.61359i − 1.96435i 0 − 1.25539i 0.458259 − 2.79106i 0 −1.25652 − 2.47760i
379.18 1.26128 − 0.639662i 0 1.18166 − 1.61359i 1.96435i 0 − 1.25539i 0.458259 − 2.79106i 0 1.25652 + 2.47760i
379.19 1.26128 + 0.639662i 0 1.18166 + 1.61359i 1.96435i 0 1.25539i 0.458259 + 2.79106i 0 −1.25652 + 2.47760i
379.20 1.26128 + 0.639662i 0 1.18166 + 1.61359i − 1.96435i 0 1.25539i 0.458259 + 2.79106i 0 1.25652 − 2.47760i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.24
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1368, [\chi])\):

\( T_{5}^{6} + 20T_{5}^{4} + 125T_{5}^{2} + 242 \) Copy content Toggle raw display
\( T_{7}^{6} + 30T_{7}^{4} + 225T_{7}^{2} + 284 \) Copy content Toggle raw display