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.
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 |
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 \(S_{2}^{\mathrm{new}}(232, [\chi])\).