Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [644,2,Mod(183,644)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(644, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("644.183");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 644 = 2^{2} \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 644.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: | \(5.14236589017\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
183.1 | −1.41198 | − | 0.0793747i | − | 1.23564i | 1.98740 | + | 0.224152i | − | 3.38329i | −0.0980783 | + | 1.74470i | 1.00000 | −2.78838 | − | 0.474248i | 1.47320 | −0.268548 | + | 4.77716i | ||||||
183.2 | −1.41198 | + | 0.0793747i | 1.23564i | 1.98740 | − | 0.224152i | 3.38329i | −0.0980783 | − | 1.74470i | 1.00000 | −2.78838 | + | 0.474248i | 1.47320 | −0.268548 | − | 4.77716i | ||||||||
183.3 | −1.38455 | − | 0.288145i | 2.18249i | 1.83394 | + | 0.797901i | 1.38768i | 0.628873 | − | 3.02176i | 1.00000 | −2.30927 | − | 1.63317i | −1.76325 | 0.399855 | − | 1.92132i | ||||||||
183.4 | −1.38455 | + | 0.288145i | − | 2.18249i | 1.83394 | − | 0.797901i | − | 1.38768i | 0.628873 | + | 3.02176i | 1.00000 | −2.30927 | + | 1.63317i | −1.76325 | 0.399855 | + | 1.92132i | ||||||
183.5 | −1.28172 | − | 0.597664i | − | 0.481494i | 1.28560 | + | 1.53207i | 0.116571i | −0.287772 | + | 0.617139i | 1.00000 | −0.732108 | − | 2.73204i | 2.76816 | 0.0696705 | − | 0.149412i | |||||||
183.6 | −1.28172 | + | 0.597664i | 0.481494i | 1.28560 | − | 1.53207i | − | 0.116571i | −0.287772 | − | 0.617139i | 1.00000 | −0.732108 | + | 2.73204i | 2.76816 | 0.0696705 | + | 0.149412i | |||||||
183.7 | −1.13221 | − | 0.847404i | 2.81043i | 0.563811 | + | 1.91888i | − | 1.87060i | 2.38157 | − | 3.18201i | 1.00000 | 0.987717 | − | 2.65036i | −4.89854 | −1.58515 | + | 2.11792i | |||||||
183.8 | −1.13221 | + | 0.847404i | − | 2.81043i | 0.563811 | − | 1.91888i | 1.87060i | 2.38157 | + | 3.18201i | 1.00000 | 0.987717 | + | 2.65036i | −4.89854 | −1.58515 | − | 2.11792i | |||||||
183.9 | −0.805628 | − | 1.16231i | − | 2.66076i | −0.701926 | + | 1.87278i | − | 2.47812i | −3.09262 | + | 2.14358i | 1.00000 | 2.74224 | − | 0.692909i | −4.07963 | −2.88034 | + | 1.99644i | ||||||
183.10 | −0.805628 | + | 1.16231i | 2.66076i | −0.701926 | − | 1.87278i | 2.47812i | −3.09262 | − | 2.14358i | 1.00000 | 2.74224 | + | 0.692909i | −4.07963 | −2.88034 | − | 1.99644i | ||||||||
183.11 | −0.766428 | − | 1.18852i | − | 1.47961i | −0.825175 | + | 1.82184i | 0.833230i | −1.75855 | + | 1.13401i | 1.00000 | 2.79773 | − | 0.415566i | 0.810767 | 0.990314 | − | 0.638611i | |||||||
183.12 | −0.766428 | + | 1.18852i | 1.47961i | −0.825175 | − | 1.82184i | − | 0.833230i | −1.75855 | − | 1.13401i | 1.00000 | 2.79773 | + | 0.415566i | 0.810767 | 0.990314 | + | 0.638611i | |||||||
183.13 | −0.413125 | − | 1.35253i | 2.02108i | −1.65866 | + | 1.11752i | 1.88306i | 2.73356 | − | 0.834958i | 1.00000 | 2.19671 | + | 1.78170i | −1.08476 | 2.54689 | − | 0.777941i | ||||||||
183.14 | −0.413125 | + | 1.35253i | − | 2.02108i | −1.65866 | − | 1.11752i | − | 1.88306i | 2.73356 | + | 0.834958i | 1.00000 | 2.19671 | − | 1.78170i | −1.08476 | 2.54689 | + | 0.777941i | ||||||
183.15 | −0.340486 | − | 1.37261i | 0.155030i | −1.76814 | + | 0.934711i | − | 1.40532i | 0.212796 | − | 0.0527855i | 1.00000 | 1.88502 | + | 2.10872i | 2.97597 | −1.92896 | + | 0.478490i | |||||||
183.16 | −0.340486 | + | 1.37261i | − | 0.155030i | −1.76814 | − | 0.934711i | 1.40532i | 0.212796 | + | 0.0527855i | 1.00000 | 1.88502 | − | 2.10872i | 2.97597 | −1.92896 | − | 0.478490i | |||||||
183.17 | −0.104849 | − | 1.41032i | − | 2.22315i | −1.97801 | + | 0.295741i | 3.61192i | −3.13536 | + | 0.233094i | 1.00000 | 0.624482 | + | 2.75863i | −1.94239 | 5.09396 | − | 0.378705i | |||||||
183.18 | −0.104849 | + | 1.41032i | 2.22315i | −1.97801 | − | 0.295741i | − | 3.61192i | −3.13536 | − | 0.233094i | 1.00000 | 0.624482 | − | 2.75863i | −1.94239 | 5.09396 | + | 0.378705i | |||||||
183.19 | 0.236103 | − | 1.39437i | 3.31913i | −1.88851 | − | 0.658429i | − | 3.46030i | 4.62808 | + | 0.783658i | 1.00000 | −1.36397 | + | 2.47782i | −8.01664 | −4.82492 | − | 0.816988i | |||||||
183.20 | 0.236103 | + | 1.39437i | − | 3.31913i | −1.88851 | + | 0.658429i | 3.46030i | 4.62808 | − | 0.783658i | 1.00000 | −1.36397 | − | 2.47782i | −8.01664 | −4.82492 | + | 0.816988i | |||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
92.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 644.2.c.b | yes | 36 |
4.b | odd | 2 | 1 | 644.2.c.a | ✓ | 36 | |
23.b | odd | 2 | 1 | 644.2.c.a | ✓ | 36 | |
92.b | even | 2 | 1 | inner | 644.2.c.b | yes | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
644.2.c.a | ✓ | 36 | 4.b | odd | 2 | 1 | |
644.2.c.a | ✓ | 36 | 23.b | odd | 2 | 1 | |
644.2.c.b | yes | 36 | 1.a | even | 1 | 1 | trivial |
644.2.c.b | yes | 36 | 92.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{18} + 4 T_{11}^{17} - 108 T_{11}^{16} - 344 T_{11}^{15} + 4870 T_{11}^{14} + 11200 T_{11}^{13} + \cdots - 61341696 \) acting on \(S_{2}^{\mathrm{new}}(644, [\chi])\).