Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1002,2,Mod(1001,1002)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1002, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1002.1001");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1002 = 2 \cdot 3 \cdot 167 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1002.d (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: | \(8.00101028253\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
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.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1001.1 | − | 1.00000i | −1.63328 | + | 0.576543i | −1.00000 | 3.86291 | 0.576543 | + | 1.63328i | 2.63317 | 1.00000i | 2.33520 | − | 1.88331i | − | 3.86291i | ||||||||||
1001.2 | 1.00000i | −1.63328 | − | 0.576543i | −1.00000 | 3.86291 | 0.576543 | − | 1.63328i | 2.63317 | − | 1.00000i | 2.33520 | + | 1.88331i | 3.86291i | |||||||||||
1001.3 | − | 1.00000i | −1.44295 | − | 0.958070i | −1.00000 | −3.09196 | −0.958070 | + | 1.44295i | −4.02418 | 1.00000i | 1.16420 | + | 2.76489i | 3.09196i | |||||||||||
1001.4 | 1.00000i | −1.44295 | + | 0.958070i | −1.00000 | −3.09196 | −0.958070 | − | 1.44295i | −4.02418 | − | 1.00000i | 1.16420 | − | 2.76489i | − | 3.09196i | ||||||||||
1001.5 | − | 1.00000i | 0.387921 | + | 1.68805i | −1.00000 | 2.66678 | 1.68805 | − | 0.387921i | 2.13474 | 1.00000i | −2.69904 | + | 1.30966i | − | 2.66678i | ||||||||||
1001.6 | 1.00000i | 0.387921 | − | 1.68805i | −1.00000 | 2.66678 | 1.68805 | + | 0.387921i | 2.13474 | − | 1.00000i | −2.69904 | − | 1.30966i | 2.66678i | |||||||||||
1001.7 | − | 1.00000i | 1.16629 | − | 1.28053i | −1.00000 | 1.79742 | −1.28053 | − | 1.16629i | 2.83390 | 1.00000i | −0.279525 | − | 2.98695i | − | 1.79742i | ||||||||||
1001.8 | 1.00000i | 1.16629 | + | 1.28053i | −1.00000 | 1.79742 | −1.28053 | + | 1.16629i | 2.83390 | − | 1.00000i | −0.279525 | + | 2.98695i | 1.79742i | |||||||||||
1001.9 | − | 1.00000i | 1.70354 | + | 0.312980i | −1.00000 | 3.03132 | 0.312980 | − | 1.70354i | −1.54520 | 1.00000i | 2.80409 | + | 1.06635i | − | 3.03132i | ||||||||||
1001.10 | 1.00000i | 1.70354 | − | 0.312980i | −1.00000 | 3.03132 | 0.312980 | + | 1.70354i | −1.54520 | − | 1.00000i | 2.80409 | − | 1.06635i | 3.03132i | |||||||||||
1001.11 | − | 1.00000i | −1.63328 | + | 0.576543i | −1.00000 | −3.86291 | 0.576543 | + | 1.63328i | 2.63317 | 1.00000i | 2.33520 | − | 1.88331i | 3.86291i | |||||||||||
1001.12 | 1.00000i | −1.63328 | − | 0.576543i | −1.00000 | −3.86291 | 0.576543 | − | 1.63328i | 2.63317 | − | 1.00000i | 2.33520 | + | 1.88331i | − | 3.86291i | ||||||||||
1001.13 | − | 1.00000i | 0.387921 | + | 1.68805i | −1.00000 | −2.66678 | 1.68805 | − | 0.387921i | 2.13474 | 1.00000i | −2.69904 | + | 1.30966i | 2.66678i | |||||||||||
1001.14 | 1.00000i | 0.387921 | − | 1.68805i | −1.00000 | −2.66678 | 1.68805 | + | 0.387921i | 2.13474 | − | 1.00000i | −2.69904 | − | 1.30966i | − | 2.66678i | ||||||||||
1001.15 | − | 1.00000i | −1.32193 | − | 1.11916i | −1.00000 | 0.490913 | −1.11916 | + | 1.32193i | 2.44549 | 1.00000i | 0.494978 | + | 2.95888i | − | 0.490913i | ||||||||||
1001.16 | 1.00000i | −1.32193 | + | 1.11916i | −1.00000 | 0.490913 | −1.11916 | − | 1.32193i | 2.44549 | − | 1.00000i | 0.494978 | − | 2.95888i | 0.490913i | |||||||||||
1001.17 | − | 1.00000i | −0.183033 | + | 1.72235i | −1.00000 | 3.52388 | 1.72235 | + | 0.183033i | −5.07645 | 1.00000i | −2.93300 | − | 0.630495i | − | 3.52388i | ||||||||||
1001.18 | 1.00000i | −0.183033 | − | 1.72235i | −1.00000 | 3.52388 | 1.72235 | − | 0.183033i | −5.07645 | − | 1.00000i | −2.93300 | + | 0.630495i | 3.52388i | |||||||||||
1001.19 | − | 1.00000i | 1.67234 | − | 0.450849i | −1.00000 | 2.07212 | −0.450849 | − | 1.67234i | −2.63137 | 1.00000i | 2.59347 | − | 1.50795i | − | 2.07212i | ||||||||||
1001.20 | 1.00000i | 1.67234 | + | 0.450849i | −1.00000 | 2.07212 | −0.450849 | + | 1.67234i | −2.63137 | − | 1.00000i | 2.59347 | + | 1.50795i | 2.07212i | |||||||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
167.b | odd | 2 | 1 | inner |
501.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1002.2.d.a | ✓ | 56 |
3.b | odd | 2 | 1 | inner | 1002.2.d.a | ✓ | 56 |
167.b | odd | 2 | 1 | inner | 1002.2.d.a | ✓ | 56 |
501.c | even | 2 | 1 | inner | 1002.2.d.a | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1002.2.d.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
1002.2.d.a | ✓ | 56 | 3.b | odd | 2 | 1 | inner |
1002.2.d.a | ✓ | 56 | 167.b | odd | 2 | 1 | inner |
1002.2.d.a | ✓ | 56 | 501.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1002, [\chi])\).