Properties

Label 232.2.k.a
Level $232$
Weight $2$
Character orbit 232.k
Analytic conductor $1.853$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [232,2,Mod(75,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.75");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 232.k (of order \(4\), degree \(2\), 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: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{3} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{3} - 6 q^{8} - 6 q^{10} + 4 q^{11} + 10 q^{12} - 24 q^{16} + 6 q^{18} - 4 q^{19} - 20 q^{20} - 16 q^{24} + 24 q^{25} - 22 q^{26} + 32 q^{27} - 20 q^{30} - 10 q^{32} + 24 q^{36} - 14 q^{40} - 8 q^{41} - 12 q^{43} - 42 q^{44} - 28 q^{46} + 66 q^{48} - 40 q^{49} - 34 q^{50} + 24 q^{52} - 16 q^{54} + 48 q^{56} - 24 q^{58} - 8 q^{59} - 46 q^{60} - 48 q^{65} + 10 q^{66} + 24 q^{68} + 56 q^{70} + 8 q^{72} - 24 q^{73} - 12 q^{74} - 92 q^{75} - 4 q^{76} - 64 q^{78} - 24 q^{82} - 8 q^{83} + 64 q^{84} + 52 q^{88} - 28 q^{90} + 92 q^{94} - 24 q^{97} + 44 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1 −1.40475 0.163373i 1.93275 1.93275i 1.94662 + 0.458994i −1.59089 −3.03077 + 2.39926i 2.39513i −2.65952 0.962795i 4.47101i 2.23479 + 0.259908i
75.2 −1.35234 0.413719i 0.209418 0.209418i 1.65767 + 1.11898i 3.96424 −0.369846 + 0.196565i 0.571667i −1.77880 2.19906i 2.91229i −5.36101 1.64008i
75.3 −1.33549 0.465261i −1.34923 + 1.34923i 1.56706 + 1.24270i −3.43757 2.42962 1.17413i 2.52803i −1.51462 2.38871i 0.640818i 4.59083 + 1.59937i
75.4 −1.31449 + 0.521645i 1.04162 1.04162i 1.45577 1.37139i 0.904075 −0.825843 + 1.91255i 4.02517i −1.19822 + 2.56208i 0.830062i −1.18840 + 0.471606i
75.5 −1.28732 + 0.585494i −0.370846 + 0.370846i 1.31439 1.50744i −1.70947 0.260270 0.694526i 1.28453i −0.809448 + 2.71013i 2.72495i 2.20063 1.00088i
75.6 −1.21406 + 0.725302i −2.04577 + 2.04577i 0.947874 1.76112i 3.27541 0.999882 3.96748i 4.14816i 0.126568 + 2.82559i 5.37035i −3.97654 + 2.37566i
75.7 −1.02641 0.972871i 0.672064 0.672064i 0.107046 + 1.99713i −2.40299 −1.34365 + 0.0359837i 5.06036i 1.83308 2.15402i 2.09666i 2.46645 + 2.33779i
75.8 −0.927188 1.06786i −2.16838 + 2.16838i −0.280644 + 1.98021i 1.14784 4.32601 + 0.305026i 1.02191i 2.37480 1.53634i 6.40371i −1.06427 1.22573i
75.9 −0.725302 + 1.21406i −2.04577 + 2.04577i −0.947874 1.76112i −3.27541 −0.999882 3.96748i 4.14816i 2.82559 + 0.126568i 5.37035i 2.37566 3.97654i
75.10 −0.655900 1.25291i 2.08616 2.08616i −1.13959 + 1.64357i 1.86530 −3.98209 1.24547i 0.180541i 2.80671 + 0.349791i 5.70411i −1.22345 2.33706i
75.11 −0.585494 + 1.28732i −0.370846 + 0.370846i −1.31439 1.50744i 1.70947 −0.260270 0.694526i 1.28453i 2.71013 0.809448i 2.72495i −1.00088 + 2.20063i
75.12 −0.525953 1.31277i −0.551041 + 0.551041i −1.44675 + 1.38091i 1.29489 1.01321 + 0.433570i 2.20860i 2.57375 + 1.17295i 2.39271i −0.681053 1.69990i
75.13 −0.521645 + 1.31449i 1.04162 1.04162i −1.45577 1.37139i −0.904075 0.825843 + 1.91255i 4.02517i 2.56208 1.19822i 0.830062i 0.471606 1.18840i
75.14 0.116162 1.40943i −0.888085 + 0.888085i −1.97301 0.327446i −1.40028 1.14854 + 1.35486i 2.21207i −0.690704 + 2.74280i 1.42261i −0.162659 + 1.97360i
75.15 0.163373 + 1.40475i 1.93275 1.93275i −1.94662 + 0.458994i 1.59089 3.03077 + 2.39926i 2.39513i −0.962795 2.65952i 4.47101i 0.259908 + 2.23479i
75.16 0.413719 + 1.35234i 0.209418 0.209418i −1.65767 + 1.11898i −3.96424 0.369846 + 0.196565i 0.571667i −2.19906 1.77880i 2.91229i −1.64008 5.36101i
75.17 0.465261 + 1.33549i −1.34923 + 1.34923i −1.56706 + 1.24270i 3.43757 −2.42962 1.17413i 2.52803i −2.38871 1.51462i 0.640818i 1.59937 + 4.59083i
75.18 0.479876 1.33031i 0.970955 0.970955i −1.53944 1.27677i 3.05723 −0.825731 1.75761i 0.737668i −2.43723 + 1.43524i 1.11449i 1.46709 4.06706i
75.19 0.626398 1.26792i 1.19755 1.19755i −1.21525 1.58845i −0.582650 −0.768258 2.26855i 2.78101i −2.77526 + 0.545842i 0.131741i −0.364971 + 0.738755i
75.20 0.948490 1.04898i −1.73717 + 1.73717i −0.200733 1.98990i −2.75213 0.174574 + 3.46995i 3.96898i −2.27777 1.67684i 3.03551i −2.61037 + 2.88694i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
29.c odd 4 1 inner
232.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.2.k.a 56
4.b odd 2 1 928.2.q.a 56
8.b even 2 1 928.2.q.a 56
8.d odd 2 1 inner 232.2.k.a 56
29.c odd 4 1 inner 232.2.k.a 56
116.e even 4 1 928.2.q.a 56
232.k even 4 1 inner 232.2.k.a 56
232.l odd 4 1 928.2.q.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.2.k.a 56 1.a even 1 1 trivial
232.2.k.a 56 8.d odd 2 1 inner
232.2.k.a 56 29.c odd 4 1 inner
232.2.k.a 56 232.k even 4 1 inner
928.2.q.a 56 4.b odd 2 1
928.2.q.a 56 8.b even 2 1
928.2.q.a 56 116.e even 4 1
928.2.q.a 56 232.l odd 4 1

Hecke kernels

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