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Properties

Label 308.2.w.a
Level $308$
Weight $2$
Character orbit 308.w
Analytic conductor $2.459$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,2,Mod(13,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.w (of order \(10\), degree \(4\), 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: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 5 q^{7} + 14 q^{9} - 4 q^{11} - 6 q^{15} + 4 q^{23} + 32 q^{25} - 20 q^{29} + 15 q^{35} - 28 q^{37} - 20 q^{39} - 15 q^{49} + 60 q^{51} - 56 q^{53} + 80 q^{57} - 80 q^{63} - 88 q^{67} + 32 q^{71}+ \cdots + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −1.89651 − 2.61033i 0 −1.37174 + 0.445705i 0 −1.29885 + 2.30499i 0 −2.28999 + 7.04787i 0
13.2 0 −1.09677 − 1.50958i 0 4.07202 − 1.32308i 0 1.58132 + 2.12119i 0 −0.148869 + 0.458171i 0
13.3 0 −0.525576 − 0.723393i 0 −2.65575 + 0.862904i 0 2.56006 − 0.667889i 0 0.679983 − 2.09277i 0
13.4 0 −0.488294 − 0.672078i 0 0.892371 − 0.289949i 0 −2.59757 − 0.502635i 0 0.713792 − 2.19683i 0
13.5 0 0.488294 + 0.672078i 0 −0.892371 + 0.289949i 0 1.28073 + 2.31511i 0 0.713792 − 2.19683i 0
13.6 0 0.525576 + 0.723393i 0 2.65575 − 0.862904i 0 −0.155903 − 2.64115i 0 0.679983 − 2.09277i 0
13.7 0 1.09677 + 1.50958i 0 −4.07202 + 1.32308i 0 −2.50602 − 0.848437i 0 −0.148869 + 0.458171i 0
13.8 0 1.89651 + 2.61033i 0 1.37174 − 0.445705i 0 −1.79081 + 1.94756i 0 −2.28999 + 7.04787i 0
41.1 0 −2.74247 + 0.891082i 0 1.27787 − 1.75883i 0 −0.297393 + 2.62898i 0 4.30005 − 3.12417i 0
41.2 0 −2.44572 + 0.794664i 0 −1.36739 + 1.88205i 0 −2.17692 − 1.50366i 0 2.92302 − 2.12370i 0
41.3 0 −1.51783 + 0.493173i 0 −0.260986 + 0.359216i 0 2.51498 − 0.821504i 0 −0.366459 + 0.266248i 0
41.4 0 −0.359392 + 0.116773i 0 1.61768 − 2.22654i 0 0.904543 − 2.48632i 0 −2.31152 + 1.67942i 0
41.5 0 0.359392 − 0.116773i 0 −1.61768 + 2.22654i 0 −0.729634 + 2.54315i 0 −2.31152 + 1.67942i 0
41.6 0 1.51783 − 0.493173i 0 0.260986 − 0.359216i 0 1.55179 + 2.14288i 0 −0.366459 + 0.266248i 0
41.7 0 2.44572 − 0.794664i 0 1.36739 − 1.88205i 0 −2.64500 − 0.0630766i 0 2.92302 − 2.12370i 0
41.8 0 2.74247 − 0.891082i 0 −1.27787 + 1.75883i 0 1.30468 − 2.30170i 0 4.30005 − 3.12417i 0
237.1 0 −1.89651 + 2.61033i 0 −1.37174 − 0.445705i 0 −1.29885 − 2.30499i 0 −2.28999 − 7.04787i 0
237.2 0 −1.09677 + 1.50958i 0 4.07202 + 1.32308i 0 1.58132 − 2.12119i 0 −0.148869 − 0.458171i 0
237.3 0 −0.525576 + 0.723393i 0 −2.65575 − 0.862904i 0 2.56006 + 0.667889i 0 0.679983 + 2.09277i 0
237.4 0 −0.488294 + 0.672078i 0 0.892371 + 0.289949i 0 −2.59757 + 0.502635i 0 0.713792 + 2.19683i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.d odd 10 1 inner
77.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.w.a ✓ 32
7.b odd 2 1 inner 308.2.w.a ✓ 32
11.c even 5 1 3388.2.c.c 32
11.d odd 10 1 inner 308.2.w.a ✓ 32
11.d odd 10 1 3388.2.c.c 32
77.j odd 10 1 3388.2.c.c 32
77.l even 10 1 inner 308.2.w.a ✓ 32
77.l even 10 1 3388.2.c.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.w.a ✓ 32 1.a even 1 1 trivial
308.2.w.a ✓ 32 7.b odd 2 1 inner
308.2.w.a ✓ 32 11.d odd 10 1 inner
308.2.w.a ✓ 32 77.l even 10 1 inner
3388.2.c.c 32 11.c even 5 1
3388.2.c.c 32 11.d odd 10 1
3388.2.c.c 32 77.j odd 10 1
3388.2.c.c 32 77.l even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(308, [\chi])\).