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Properties

Label 1216.2.m.a
Level $1216$
Weight $2$
Character orbit 1216.m
Analytic conductor $9.710$
Analytic rank $0$
Dimension $76$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(303,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.303");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.m (of order \(4\), degree \(2\), not 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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{5} + 8 q^{7} + 4 q^{11} - 8 q^{17} - 6 q^{19} + 8 q^{23} + 8 q^{39} + 4 q^{43} + 4 q^{45} + 44 q^{49} + 8 q^{55} + 28 q^{61} - 32 q^{77} - 52 q^{81} - 36 q^{83} - 56 q^{85} + 120 q^{87} - 16 q^{93}+ \cdots - 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
303.1 0 −2.33030 − 2.33030i 0 1.33252 − 1.33252i 0 −2.39899 0 7.86060i 0
303.2 0 −2.19409 − 2.19409i 0 −1.29836 + 1.29836i 0 4.71096 0 6.62806i 0
303.3 0 −2.03444 − 2.03444i 0 2.23466 − 2.23466i 0 0.725567 0 5.27790i 0
303.4 0 −1.96469 − 1.96469i 0 −0.884107 + 0.884107i 0 0.721104 0 4.72004i 0
303.5 0 −1.96187 − 1.96187i 0 −3.06749 + 3.06749i 0 −1.97168 0 4.69788i 0
303.6 0 −1.65943 − 1.65943i 0 −0.668279 + 0.668279i 0 −1.43332 0 2.50744i 0
303.7 0 −1.63122 − 1.63122i 0 0.837183 − 0.837183i 0 −1.95985 0 2.32176i 0
303.8 0 −1.42780 − 1.42780i 0 0.391081 − 0.391081i 0 3.36082 0 1.07723i 0
303.9 0 −1.34292 − 1.34292i 0 2.53450 − 2.53450i 0 2.70706 0 0.606865i 0
303.10 0 −1.08919 − 1.08919i 0 −0.770023 + 0.770023i 0 −1.47744 0 − 0.627346i 0
303.11 0 −1.07779 − 1.07779i 0 −1.37400 + 1.37400i 0 4.00321 0 − 0.676721i 0
303.12 0 −1.03801 − 1.03801i 0 2.13321 − 2.13321i 0 −3.87204 0 − 0.845075i 0
303.13 0 −1.02920 − 1.02920i 0 −2.13462 + 2.13462i 0 −4.26829 0 − 0.881504i 0
303.14 0 −0.623611 − 0.623611i 0 2.23509 − 2.23509i 0 0.609161 0 − 2.22222i 0
303.15 0 −0.599104 − 0.599104i 0 −1.69205 + 1.69205i 0 1.26000 0 − 2.28215i 0
303.16 0 −0.549728 − 0.549728i 0 1.49999 − 1.49999i 0 3.48044 0 − 2.39560i 0
303.17 0 −0.308215 − 0.308215i 0 0.114298 − 0.114298i 0 −4.05189 0 − 2.81001i 0
303.18 0 −0.105438 − 0.105438i 0 0.199404 − 0.199404i 0 −0.290770 0 − 2.97777i 0
303.19 0 −0.101486 − 0.101486i 0 −2.62302 + 2.62302i 0 2.14594 0 − 2.97940i 0
303.20 0 0.101486 + 0.101486i 0 −2.62302 + 2.62302i 0 2.14594 0 − 2.97940i 0
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 303.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner
19.b odd 2 1 inner
304.m even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.m.a 76
4.b odd 2 1 304.2.m.a ✓ 76
16.e even 4 1 304.2.m.a ✓ 76
16.f odd 4 1 inner 1216.2.m.a 76
19.b odd 2 1 inner 1216.2.m.a 76
76.d even 2 1 304.2.m.a ✓ 76
304.j odd 4 1 304.2.m.a ✓ 76
304.m even 4 1 inner 1216.2.m.a 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.m.a ✓ 76 4.b odd 2 1
304.2.m.a ✓ 76 16.e even 4 1
304.2.m.a ✓ 76 76.d even 2 1
304.2.m.a ✓ 76 304.j odd 4 1
1216.2.m.a 76 1.a even 1 1 trivial
1216.2.m.a 76 16.f odd 4 1 inner
1216.2.m.a 76 19.b odd 2 1 inner
1216.2.m.a 76 304.m even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1216, [\chi])\).