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Properties

Label 348.2.b.d
Level $348$
Weight $2$
Character orbit 348.b
Analytic conductor $2.779$
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] = [348,2,Mod(347,348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(348, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("348.347");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 348.b (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: \(2.77879399034\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{4} - 4 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{4} - 4 q^{6} + 8 q^{9} + 24 q^{13} - 8 q^{16} + 8 q^{22} + 8 q^{24} - 8 q^{25} - 40 q^{28} + 40 q^{30} - 16 q^{33} - 96 q^{34} - 28 q^{36} + 44 q^{42} + 56 q^{45} + 32 q^{49} + 16 q^{52} - 36 q^{54} + 8 q^{57} - 32 q^{58} + 16 q^{64} - 4 q^{78} - 144 q^{81} + 24 q^{82} - 40 q^{88} - 112 q^{93} + 40 q^{94} + 12 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
347.1 −1.37820 − 0.317122i −1.04908 − 1.37820i 1.79887 + 0.874114i − 3.30813i 1.00878 + 2.23212i − 2.59774i −2.20200 − 1.77516i −0.798868 + 2.89168i −1.04908 + 4.55926i
347.2 −1.37820 − 0.317122i 1.04908 − 1.37820i 1.79887 + 0.874114i 3.30813i −1.88290 + 1.56675i − 2.59774i −2.20200 − 1.77516i −0.798868 − 2.89168i 1.04908 − 4.55926i
347.3 −1.37820 + 0.317122i −1.04908 + 1.37820i 1.79887 − 0.874114i 3.30813i 1.00878 − 2.23212i 2.59774i −2.20200 + 1.77516i −0.798868 − 2.89168i −1.04908 − 4.55926i
347.4 −1.37820 + 0.317122i 1.04908 + 1.37820i 1.79887 − 0.874114i − 3.30813i −1.88290 − 1.56675i 2.59774i −2.20200 + 1.77516i −0.798868 + 2.89168i 1.04908 + 4.55926i
347.5 −1.25694 − 0.648161i −1.19168 + 1.25694i 1.15978 + 1.62939i − 1.83856i 2.31257 − 0.807485i 1.31955i −0.401655 − 2.79976i −0.159775 − 2.99574i −1.19168 + 2.31096i
347.6 −1.25694 − 0.648161i 1.19168 + 1.25694i 1.15978 + 1.62939i 1.83856i −0.683175 − 2.35229i 1.31955i −0.401655 − 2.79976i −0.159775 + 2.99574i 1.19168 − 2.31096i
347.7 −1.25694 + 0.648161i −1.19168 − 1.25694i 1.15978 − 1.62939i 1.83856i 2.31257 + 0.807485i − 1.31955i −0.401655 + 2.79976i −0.159775 + 2.99574i −1.19168 − 2.31096i
347.8 −1.25694 + 0.648161i 1.19168 − 1.25694i 1.15978 − 1.62939i − 1.83856i −0.683175 + 2.35229i − 1.31955i −0.401655 + 2.79976i −0.159775 − 2.99574i 1.19168 + 2.31096i
347.9 −0.721581 − 1.21627i −1.57459 − 0.721581i −0.958643 + 1.75528i − 1.29460i 0.258551 + 2.43581i 2.91729i 2.82664 − 0.100603i 1.95864 + 2.27238i −1.57459 + 0.934157i
347.10 −0.721581 − 1.21627i 1.57459 − 0.721581i −0.958643 + 1.75528i 1.29460i −2.01383 − 1.39445i 2.91729i 2.82664 − 0.100603i 1.95864 − 2.27238i 1.57459 − 0.934157i
347.11 −0.721581 + 1.21627i −1.57459 + 0.721581i −0.958643 − 1.75528i 1.29460i 0.258551 − 2.43581i − 2.91729i 2.82664 + 0.100603i 1.95864 − 2.27238i −1.57459 − 0.934157i
347.12 −0.721581 + 1.21627i 1.57459 + 0.721581i −0.958643 − 1.75528i − 1.29460i −2.01383 + 1.39445i − 2.91729i 2.82664 + 0.100603i 1.95864 + 2.27238i 1.57459 + 0.934157i
347.13 0.721581 − 1.21627i −1.57459 − 0.721581i −0.958643 − 1.75528i − 1.29460i −2.01383 + 1.39445i − 2.91729i −2.82664 − 0.100603i 1.95864 + 2.27238i −1.57459 − 0.934157i
347.14 0.721581 − 1.21627i 1.57459 − 0.721581i −0.958643 − 1.75528i 1.29460i 0.258551 − 2.43581i − 2.91729i −2.82664 − 0.100603i 1.95864 − 2.27238i 1.57459 + 0.934157i
347.15 0.721581 + 1.21627i −1.57459 + 0.721581i −0.958643 + 1.75528i 1.29460i −2.01383 − 1.39445i 2.91729i −2.82664 + 0.100603i 1.95864 − 2.27238i −1.57459 + 0.934157i
347.16 0.721581 + 1.21627i 1.57459 + 0.721581i −0.958643 + 1.75528i − 1.29460i 0.258551 + 2.43581i 2.91729i −2.82664 + 0.100603i 1.95864 + 2.27238i 1.57459 − 0.934157i
347.17 1.25694 − 0.648161i −1.19168 + 1.25694i 1.15978 − 1.62939i − 1.83856i −0.683175 + 2.35229i − 1.31955i 0.401655 − 2.79976i −0.159775 − 2.99574i −1.19168 − 2.31096i
347.18 1.25694 − 0.648161i 1.19168 + 1.25694i 1.15978 − 1.62939i 1.83856i 2.31257 + 0.807485i − 1.31955i 0.401655 − 2.79976i −0.159775 + 2.99574i 1.19168 + 2.31096i
347.19 1.25694 + 0.648161i −1.19168 − 1.25694i 1.15978 + 1.62939i 1.83856i −0.683175 − 2.35229i 1.31955i 0.401655 + 2.79976i −0.159775 + 2.99574i −1.19168 + 2.31096i
347.20 1.25694 + 0.648161i 1.19168 − 1.25694i 1.15978 + 1.62939i − 1.83856i 2.31257 − 0.807485i 1.31955i 0.401655 + 2.79976i −0.159775 − 2.99574i 1.19168 − 2.31096i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 347.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
29.b even 2 1 inner
87.d odd 2 1 inner
116.d odd 2 1 inner
348.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.2.b.d ✓ 24
3.b odd 2 1 inner 348.2.b.d ✓ 24
4.b odd 2 1 inner 348.2.b.d ✓ 24
12.b even 2 1 inner 348.2.b.d ✓ 24
29.b even 2 1 inner 348.2.b.d ✓ 24
87.d odd 2 1 inner 348.2.b.d ✓ 24
116.d odd 2 1 inner 348.2.b.d ✓ 24
348.b even 2 1 inner 348.2.b.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.2.b.d ✓ 24 1.a even 1 1 trivial
348.2.b.d ✓ 24 3.b odd 2 1 inner
348.2.b.d ✓ 24 4.b odd 2 1 inner
348.2.b.d ✓ 24 12.b even 2 1 inner
348.2.b.d ✓ 24 29.b even 2 1 inner
348.2.b.d ✓ 24 87.d odd 2 1 inner
348.2.b.d ✓ 24 116.d odd 2 1 inner
348.2.b.d ✓ 24 348.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 16T_{5}^{4} + 61T_{5}^{2} + 62 \) acting on \(S_{2}^{\mathrm{new}}(348, [\chi])\). Copy content Toggle raw display