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Properties

Label 124.2.m.a
Level $124$
Weight $2$
Character orbit 124.m
Analytic conductor $0.990$
Analytic rank $0$
Dimension $24$
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] = [124,2,Mod(9,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 124.m (of order \(15\), degree \(8\), 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: \(0.990144985064\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{3} + q^{5} - 9 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{3} + q^{5} - 9 q^{7} + 2 q^{9} + 8 q^{11} + 2 q^{13} - q^{15} - q^{17} - q^{19} + 12 q^{21} - 21 q^{23} - 21 q^{25} - 43 q^{27} - 24 q^{29} - 29 q^{31} - 5 q^{33} + 10 q^{35} - 8 q^{37} - 17 q^{39} + 9 q^{41} - 2 q^{43} - 28 q^{45} - 38 q^{47} + 22 q^{49} + 69 q^{51} + 59 q^{53} - q^{55} - 7 q^{57} - 13 q^{59} + 36 q^{61} + 118 q^{63} + 94 q^{65} + 29 q^{67} + 52 q^{69} + 55 q^{71} + 55 q^{73} + 79 q^{75} - 38 q^{77} - 31 q^{79} - 98 q^{81} - 59 q^{83} - 60 q^{85} - 36 q^{87} - 57 q^{89} - 48 q^{91} - 124 q^{93} - 3 q^{95} - 19 q^{97} - 31 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −2.64136 − 0.561439i 0 1.90456 + 3.29879i 0 3.89674 − 1.73494i 0 3.92094 + 1.74572i 0
9.2 0 −1.51965 − 0.323012i 0 −1.97996 − 3.42939i 0 −0.778853 + 0.346768i 0 −0.535638 − 0.238481i 0
9.3 0 1.60019 + 0.340130i 0 0.744532 + 1.28957i 0 −1.14809 + 0.511161i 0 −0.295724 − 0.131665i 0
41.1 0 −0.315933 − 3.00590i 0 0.0424613 + 0.0735452i 0 −0.792927 − 0.168542i 0 −6.00118 + 1.27559i 0
41.2 0 0.0335616 + 0.319317i 0 −0.0844474 − 0.146267i 0 3.66512 + 0.779045i 0 2.83361 − 0.602301i 0
41.3 0 0.242445 + 2.30671i 0 0.955532 + 1.65503i 0 −4.11204 − 0.874041i 0 −2.32769 + 0.494766i 0
45.1 0 −0.656753 + 0.729398i 0 −1.71648 + 2.97302i 0 −0.125619 + 1.19518i 0 0.212888 + 2.02550i 0
45.2 0 0.152628 − 0.169510i 0 1.18151 − 2.04643i 0 0.200653 − 1.90909i 0 0.308147 + 2.93182i 0
45.3 0 2.25593 − 2.50547i 0 −0.443179 + 0.767609i 0 −0.381716 + 3.63179i 0 −0.874547 − 8.32076i 0
49.1 0 −2.69421 + 1.19954i 0 −0.841471 − 1.45747i 0 −3.10886 − 3.45274i 0 3.81250 − 4.23421i 0
49.2 0 1.12756 − 0.502020i 0 2.05783 + 3.56426i 0 −2.43319 − 2.70233i 0 −0.988035 + 1.09732i 0
49.3 0 1.91560 − 0.852881i 0 −1.32088 − 2.28784i 0 0.618772 + 0.687216i 0 0.934734 − 1.03813i 0
69.1 0 −2.64136 + 0.561439i 0 1.90456 − 3.29879i 0 3.89674 + 1.73494i 0 3.92094 − 1.74572i 0
69.2 0 −1.51965 + 0.323012i 0 −1.97996 + 3.42939i 0 −0.778853 − 0.346768i 0 −0.535638 + 0.238481i 0
69.3 0 1.60019 − 0.340130i 0 0.744532 − 1.28957i 0 −1.14809 − 0.511161i 0 −0.295724 + 0.131665i 0
81.1 0 −2.69421 − 1.19954i 0 −0.841471 + 1.45747i 0 −3.10886 + 3.45274i 0 3.81250 + 4.23421i 0
81.2 0 1.12756 + 0.502020i 0 2.05783 − 3.56426i 0 −2.43319 + 2.70233i 0 −0.988035 − 1.09732i 0
81.3 0 1.91560 + 0.852881i 0 −1.32088 + 2.28784i 0 0.618772 − 0.687216i 0 0.934734 + 1.03813i 0
113.1 0 −0.656753 − 0.729398i 0 −1.71648 − 2.97302i 0 −0.125619 − 1.19518i 0 0.212888 − 2.02550i 0
113.2 0 0.152628 + 0.169510i 0 1.18151 + 2.04643i 0 0.200653 + 1.90909i 0 0.308147 − 2.93182i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.2.m.a ✓ 24
4.b odd 2 1 496.2.bg.d 24
31.g even 15 1 inner 124.2.m.a ✓ 24
31.g even 15 1 3844.2.a.n 12
31.h odd 30 1 3844.2.a.m 12
124.n odd 30 1 496.2.bg.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.m.a ✓ 24 1.a even 1 1 trivial
124.2.m.a ✓ 24 31.g even 15 1 inner
496.2.bg.d 24 4.b odd 2 1
496.2.bg.d 24 124.n odd 30 1
3844.2.a.m 12 31.h odd 30 1
3844.2.a.n 12 31.g even 15 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(124, [\chi])\).