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Properties

Label 232.2.m.d
Level $232$
Weight $2$
Character orbit 232.m
Analytic conductor $1.853$
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] = [232,2,Mod(25,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 232.m (of order \(7\), degree \(6\), 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: \(1.85252932689\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{3} - 8 q^{5} + 5 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{3} - 8 q^{5} + 5 q^{7} - 3 q^{9} - 6 q^{11} + q^{13} - q^{15} - 16 q^{17} - 10 q^{19} - 5 q^{21} + 11 q^{23} + 10 q^{25} - 7 q^{27} - 2 q^{29} + 12 q^{31} + 13 q^{33} - 8 q^{35} + q^{37} + 34 q^{39} - 22 q^{41} + 3 q^{43} + 60 q^{45} + 9 q^{47} - 67 q^{49} - q^{51} + 19 q^{53} - 88 q^{55} - 2 q^{57} + 114 q^{59} - 11 q^{61} - 108 q^{63} + 8 q^{65} - 25 q^{67} - 84 q^{69} - 21 q^{71} + 30 q^{73} - 26 q^{75} - 22 q^{77} + 48 q^{79} + 16 q^{81} - 37 q^{83} + 8 q^{85} + 11 q^{87} - 5 q^{89} - 11 q^{91} - 18 q^{93} - 21 q^{95} + 35 q^{97} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −2.49260 + 1.20038i 0 −0.0857933 + 0.375885i 0 −1.11973 + 0.539232i 0 2.90171 − 3.63863i 0
25.2 0 0.276527 − 0.133169i 0 −0.771954 + 3.38215i 0 0.675091 − 0.325107i 0 −1.81174 + 2.27184i 0
25.3 0 0.579046 − 0.278854i 0 0.446082 − 1.95441i 0 2.74207 − 1.32051i 0 −1.61293 + 2.02256i 0
25.4 0 2.93897 − 1.41533i 0 −0.0198029 + 0.0867622i 0 −2.19840 + 1.05869i 0 4.76390 − 5.97374i 0
49.1 0 −1.75696 − 2.20315i 0 2.89986 − 1.39650i 0 1.31138 + 1.64442i 0 −1.09943 + 4.81692i 0
49.2 0 −0.909971 − 1.14107i 0 −3.43877 + 1.65602i 0 2.68724 + 3.36970i 0 0.193575 − 0.848106i 0
49.3 0 −0.484939 − 0.608094i 0 −0.283259 + 0.136410i 0 −2.96109 − 3.71309i 0 0.532950 − 2.33501i 0
49.4 0 1.40489 + 1.76167i 0 0.901590 − 0.434183i 0 0.585961 + 0.734772i 0 −0.462221 + 2.02512i 0
65.1 0 −2.49260 − 1.20038i 0 −0.0857933 − 0.375885i 0 −1.11973 − 0.539232i 0 2.90171 + 3.63863i 0
65.2 0 0.276527 + 0.133169i 0 −0.771954 − 3.38215i 0 0.675091 + 0.325107i 0 −1.81174 − 2.27184i 0
65.3 0 0.579046 + 0.278854i 0 0.446082 + 1.95441i 0 2.74207 + 1.32051i 0 −1.61293 − 2.02256i 0
65.4 0 2.93897 + 1.41533i 0 −0.0198029 − 0.0867622i 0 −2.19840 − 1.05869i 0 4.76390 + 5.97374i 0
81.1 0 −0.614208 + 2.69102i 0 −2.26076 + 2.83491i 0 0.498020 − 2.18197i 0 −4.16144 − 2.00404i 0
81.2 0 −0.224238 + 0.982450i 0 1.50248 − 1.88405i 0 0.340528 − 1.49195i 0 1.78798 + 0.861046i 0
81.3 0 0.137440 − 0.602165i 0 −1.33002 + 1.66779i 0 −1.09597 + 4.80177i 0 2.35919 + 1.13613i 0
81.4 0 0.646047 − 2.83052i 0 −1.55965 + 1.95574i 0 1.03490 − 4.53421i 0 −4.89155 − 2.35565i 0
161.1 0 −1.75696 + 2.20315i 0 2.89986 + 1.39650i 0 1.31138 − 1.64442i 0 −1.09943 − 4.81692i 0
161.2 0 −0.909971 + 1.14107i 0 −3.43877 − 1.65602i 0 2.68724 − 3.36970i 0 0.193575 + 0.848106i 0
161.3 0 −0.484939 + 0.608094i 0 −0.283259 − 0.136410i 0 −2.96109 + 3.71309i 0 0.532950 + 2.33501i 0
161.4 0 1.40489 − 1.76167i 0 0.901590 + 0.434183i 0 0.585961 − 0.734772i 0 −0.462221 − 2.02512i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.2.m.d ✓ 24
4.b odd 2 1 464.2.u.i 24
29.d even 7 1 inner 232.2.m.d ✓ 24
29.d even 7 1 6728.2.a.z 12
29.e even 14 1 6728.2.a.bb 12
116.j odd 14 1 464.2.u.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.2.m.d ✓ 24 1.a even 1 1 trivial
232.2.m.d ✓ 24 29.d even 7 1 inner
464.2.u.i 24 4.b odd 2 1
464.2.u.i 24 116.j odd 14 1
6728.2.a.z 12 29.d even 7 1
6728.2.a.bb 12 29.e even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + T_{3}^{23} + 8 T_{3}^{22} - 2 T_{3}^{21} + 43 T_{3}^{20} - 3 T_{3}^{19} + 688 T_{3}^{18} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(232, [\chi])\). Copy content Toggle raw display