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Properties

Label 232.2.c.a
Level $232$
Weight $2$
Character orbit 232.c
Analytic conductor $1.853$
Analytic rank $0$
Dimension $28$
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(117,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 232.c (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: \(1.85252932689\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{4} - 4 q^{6} + 6 q^{8} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{4} - 4 q^{6} + 6 q^{8} - 28 q^{9} + 2 q^{10} - 2 q^{12} - 8 q^{14} + 8 q^{15} - 12 q^{16} + 10 q^{18} - 8 q^{20} + 4 q^{22} + 24 q^{24} - 28 q^{25} + 18 q^{26} + 12 q^{30} - 4 q^{31} + 10 q^{32} + 8 q^{33} - 16 q^{34} - 36 q^{38} - 16 q^{39} + 6 q^{40} - 8 q^{41} + 24 q^{42} - 18 q^{44} - 8 q^{46} - 4 q^{47} + 46 q^{48} + 12 q^{49} + 22 q^{50} - 16 q^{52} - 8 q^{54} + 16 q^{55} - 16 q^{56} + 8 q^{57} - 2 q^{60} - 8 q^{62} - 40 q^{63} - 28 q^{64} - 26 q^{66} - 20 q^{68} - 4 q^{70} + 32 q^{71} - 48 q^{72} + 16 q^{73} + 20 q^{74} + 32 q^{76} - 4 q^{78} + 36 q^{79} + 16 q^{80} + 36 q^{81} - 44 q^{82} + 52 q^{84} + 44 q^{86} - 12 q^{87} + 40 q^{88} - 64 q^{90} - 36 q^{92} + 16 q^{95} - 80 q^{96} - 16 q^{97} - 20 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1 −1.37243 − 0.341228i 1.19718i 1.76713 + 0.936622i − 0.150647i 0.408510 − 1.64304i 1.09774 −2.10566 − 1.88844i 1.56677 −0.0514050 + 0.206753i
117.2 −1.37243 + 0.341228i − 1.19718i 1.76713 − 0.936622i 0.150647i 0.408510 + 1.64304i 1.09774 −2.10566 + 1.88844i 1.56677 −0.0514050 − 0.206753i
117.3 −1.22168 − 0.712394i − 2.26221i 0.984988 + 1.74063i − 2.04842i −1.61158 + 2.76368i −2.47025 0.0366789 − 2.82819i −2.11758 −1.45928 + 2.50251i
117.4 −1.22168 + 0.712394i 2.26221i 0.984988 − 1.74063i 2.04842i −1.61158 − 2.76368i −2.47025 0.0366789 + 2.82819i −2.11758 −1.45928 − 2.50251i
117.5 −1.08062 − 0.912283i − 2.23689i 0.335479 + 1.97166i 3.66406i −2.04067 + 2.41722i 4.20502 1.43619 − 2.43667i −2.00366 3.34266 − 3.95946i
117.6 −1.08062 + 0.912283i 2.23689i 0.335479 − 1.97166i − 3.66406i −2.04067 − 2.41722i 4.20502 1.43619 + 2.43667i −2.00366 3.34266 + 3.95946i
117.7 −0.940555 − 1.05610i 3.40990i −0.230713 + 1.98665i 2.45831i 3.60121 − 3.20719i 1.26791 2.31511 − 1.62490i −8.62739 2.59624 − 2.31218i
117.8 −0.940555 + 1.05610i − 3.40990i −0.230713 − 1.98665i − 2.45831i 3.60121 + 3.20719i 1.26791 2.31511 + 1.62490i −8.62739 2.59624 + 2.31218i
117.9 −0.822909 − 1.15014i 1.52199i −0.645640 + 1.89292i − 3.63995i 1.75050 − 1.25246i −0.333827 2.70843 − 0.815126i 0.683547 −4.18645 + 2.99535i
117.10 −0.822909 + 1.15014i − 1.52199i −0.645640 − 1.89292i 3.63995i 1.75050 + 1.25246i −0.333827 2.70843 + 0.815126i 0.683547 −4.18645 − 2.99535i
117.11 −0.501043 − 1.32248i 0.332857i −1.49791 + 1.32524i 1.62252i 0.440198 − 0.166776i −4.97022 2.50312 + 1.31696i 2.88921 2.14575 − 0.812953i
117.12 −0.501043 + 1.32248i − 0.332857i −1.49791 − 1.32524i − 1.62252i 0.440198 + 0.166776i −4.97022 2.50312 − 1.31696i 2.88921 2.14575 + 0.812953i
117.13 −0.0524431 − 1.41324i − 2.00445i −1.99450 + 0.148230i − 3.49720i −2.83277 + 0.105120i 3.01620 0.314082 + 2.81093i −1.01782 −4.94239 + 0.183404i
117.14 −0.0524431 + 1.41324i 2.00445i −1.99450 − 0.148230i 3.49720i −2.83277 − 0.105120i 3.01620 0.314082 − 2.81093i −1.01782 −4.94239 − 0.183404i
117.15 0.162431 − 1.40485i 0.965990i −1.94723 − 0.456384i 1.65510i 1.35708 + 0.156907i 3.23127 −0.957443 + 2.66145i 2.06686 2.32517 + 0.268839i
117.16 0.162431 + 1.40485i − 0.965990i −1.94723 + 0.456384i − 1.65510i 1.35708 − 0.156907i 3.23127 −0.957443 − 2.66145i 2.06686 2.32517 − 0.268839i
117.17 0.198372 − 1.40023i − 3.03841i −1.92130 − 0.555533i 2.64734i −4.25448 − 0.602735i −2.83096 −1.15901 + 2.58006i −6.23196 3.70689 + 0.525158i
117.18 0.198372 + 1.40023i 3.03841i −1.92130 + 0.555533i − 2.64734i −4.25448 + 0.602735i −2.83096 −1.15901 − 2.58006i −6.23196 3.70689 − 0.525158i
117.19 0.828221 − 1.14632i 0.718996i −0.628101 − 1.89881i − 2.64667i 0.824199 + 0.595487i −2.40239 −2.69685 − 0.852630i 2.48305 −3.03393 − 2.19203i
117.20 0.828221 + 1.14632i − 0.718996i −0.628101 + 1.89881i 2.64667i 0.824199 − 0.595487i −2.40239 −2.69685 + 0.852630i 2.48305 −3.03393 + 2.19203i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.2.c.a ✓ 28
4.b odd 2 1 928.2.c.a 28
8.b even 2 1 inner 232.2.c.a ✓ 28
8.d odd 2 1 928.2.c.a 28
16.e even 4 1 7424.2.a.v 14
16.e even 4 1 7424.2.a.z 14
16.f odd 4 1 7424.2.a.u 14
16.f odd 4 1 7424.2.a.y 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.2.c.a ✓ 28 1.a even 1 1 trivial
232.2.c.a ✓ 28 8.b even 2 1 inner
928.2.c.a 28 4.b odd 2 1
928.2.c.a 28 8.d odd 2 1
7424.2.a.u 14 16.f odd 4 1
7424.2.a.v 14 16.e even 4 1
7424.2.a.y 14 16.f odd 4 1
7424.2.a.z 14 16.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(232, [\chi])\).