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Properties

Label 348.6.h.a
Level $348$
Weight $6$
Character orbit 348.h
Analytic conductor $55.814$
Analytic rank $0$
Dimension $26$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [348,6,Mod(289,348)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.289");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 348.h (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: \(55.8135692949\)
Analytic rank: \(0\)
Dimension: \(26\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 120 q^{5} - 76 q^{7} - 2106 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 120 q^{5} - 76 q^{7} - 2106 q^{9} - 680 q^{13} - 8124 q^{23} + 24358 q^{25} - 766 q^{29} - 9432 q^{33} - 6268 q^{35} - 9720 q^{45} + 97038 q^{49} + 15408 q^{51} + 11872 q^{53} - 27324 q^{57} - 135864 q^{59} + 6156 q^{63} - 7156 q^{65} + 22228 q^{67} - 105892 q^{71} + 170586 q^{81} + 177896 q^{83} - 23796 q^{87} - 62324 q^{91} - 7380 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0 − 9.00000i 0 −83.1556 0 −53.2701 0 −81.0000 0
289.2 0 9.00000i 0 −83.1556 0 −53.2701 0 −81.0000 0
289.3 0 − 9.00000i 0 60.6339 0 240.855 0 −81.0000 0
289.4 0 9.00000i 0 60.6339 0 240.855 0 −81.0000 0
289.5 0 − 9.00000i 0 −27.8512 0 −221.784 0 −81.0000 0
289.6 0 9.00000i 0 −27.8512 0 −221.784 0 −81.0000 0
289.7 0 − 9.00000i 0 67.4769 0 −45.8398 0 −81.0000 0
289.8 0 9.00000i 0 67.4769 0 −45.8398 0 −81.0000 0
289.9 0 − 9.00000i 0 −67.0270 0 −116.843 0 −81.0000 0
289.10 0 9.00000i 0 −67.0270 0 −116.843 0 −81.0000 0
289.11 0 − 9.00000i 0 87.7704 0 97.4446 0 −81.0000 0
289.12 0 9.00000i 0 87.7704 0 97.4446 0 −81.0000 0
289.13 0 − 9.00000i 0 −27.4108 0 148.772 0 −81.0000 0
289.14 0 9.00000i 0 −27.4108 0 148.772 0 −81.0000 0
289.15 0 − 9.00000i 0 1.81526 0 75.9981 0 −81.0000 0
289.16 0 9.00000i 0 1.81526 0 75.9981 0 −81.0000 0
289.17 0 − 9.00000i 0 24.2797 0 −218.867 0 −81.0000 0
289.18 0 9.00000i 0 24.2797 0 −218.867 0 −81.0000 0
289.19 0 − 9.00000i 0 −98.1842 0 170.728 0 −81.0000 0
289.20 0 9.00000i 0 −98.1842 0 170.728 0 −81.0000 0
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.6.h.a ✓ 26
3.b odd 2 1 1044.6.h.c 26
29.b even 2 1 inner 348.6.h.a ✓ 26
87.d odd 2 1 1044.6.h.c 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.6.h.a ✓ 26 1.a even 1 1 trivial
348.6.h.a ✓ 26 29.b even 2 1 inner
1044.6.h.c 26 3.b odd 2 1
1044.6.h.c 26 87.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(348, [\chi])\).