Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [625,2,Mod(124,625)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(625, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("625.124");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 625 = 5^{4} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 625.e (of order \(10\), degree \(4\), not 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: | \(4.99065012633\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
124.1 | −2.35325 | − | 0.764617i | 1.24419 | − | 1.71249i | 3.33511 | + | 2.42310i | 0 | −4.23730 | + | 3.07858i | 0.973070i | −3.08683 | − | 4.24866i | −0.457541 | − | 1.40817i | 0 | ||||||
124.2 | −2.21225 | − | 0.718805i | −1.35327 | + | 1.86261i | 2.75935 | + | 2.00479i | 0 | 4.33262 | − | 3.14783i | − | 3.59425i | −1.92884 | − | 2.65482i | −0.710939 | − | 2.18805i | 0 | |||||
124.3 | −1.60102 | − | 0.520202i | −0.417528 | + | 0.574677i | 0.674615 | + | 0.490137i | 0 | 0.967418 | − | 0.702870i | 4.59110i | 1.15387 | + | 1.58816i | 0.771126 | + | 2.37328i | 0 | ||||||
124.4 | −0.476469 | − | 0.154814i | 1.81817 | − | 2.50250i | −1.41498 | − | 1.02804i | 0 | −1.25372 | + | 0.910884i | − | 0.0237879i | 1.10399 | + | 1.51951i | −2.02970 | − | 6.24676i | 0 | |||||
124.5 | 0.476469 | + | 0.154814i | −1.81817 | + | 2.50250i | −1.41498 | − | 1.02804i | 0 | −1.25372 | + | 0.910884i | 0.0237879i | −1.10399 | − | 1.51951i | −2.02970 | − | 6.24676i | 0 | ||||||
124.6 | 1.60102 | + | 0.520202i | 0.417528 | − | 0.574677i | 0.674615 | + | 0.490137i | 0 | 0.967418 | − | 0.702870i | − | 4.59110i | −1.15387 | − | 1.58816i | 0.771126 | + | 2.37328i | 0 | |||||
124.7 | 2.21225 | + | 0.718805i | 1.35327 | − | 1.86261i | 2.75935 | + | 2.00479i | 0 | 4.33262 | − | 3.14783i | 3.59425i | 1.92884 | + | 2.65482i | −0.710939 | − | 2.18805i | 0 | ||||||
124.8 | 2.35325 | + | 0.764617i | −1.24419 | + | 1.71249i | 3.33511 | + | 2.42310i | 0 | −4.23730 | + | 3.07858i | − | 0.973070i | 3.08683 | + | 4.24866i | −0.457541 | − | 1.40817i | 0 | |||||
249.1 | −1.56645 | + | 2.15604i | −0.721930 | + | 0.234569i | −1.57669 | − | 4.85257i | 0 | 0.625130 | − | 1.92395i | 2.04213i | 7.86300 | + | 2.55484i | −1.96089 | + | 1.42467i | 0 | ||||||
249.2 | −1.18361 | + | 1.62909i | 2.87758 | − | 0.934982i | −0.634989 | − | 1.95429i | 0 | −1.88274 | + | 5.79449i | 0.369971i | 0.105077 | + | 0.0341417i | 4.97921 | − | 3.61761i | 0 | ||||||
249.3 | −0.621509 | + | 0.855434i | 0.653760 | − | 0.212419i | 0.272540 | + | 0.838792i | 0 | −0.224607 | + | 0.691269i | − | 1.01199i | −2.89816 | − | 0.941671i | −2.04477 | + | 1.48561i | 0 | |||||
249.4 | −0.192059 | + | 0.264347i | −1.63142 | + | 0.530081i | 0.585041 | + | 1.80057i | 0 | 0.173205 | − | 0.533069i | 3.42409i | −1.20986 | − | 0.393106i | −0.0465016 | + | 0.0337854i | 0 | ||||||
249.5 | 0.192059 | − | 0.264347i | 1.63142 | − | 0.530081i | 0.585041 | + | 1.80057i | 0 | 0.173205 | − | 0.533069i | − | 3.42409i | 1.20986 | + | 0.393106i | −0.0465016 | + | 0.0337854i | 0 | |||||
249.6 | 0.621509 | − | 0.855434i | −0.653760 | + | 0.212419i | 0.272540 | + | 0.838792i | 0 | −0.224607 | + | 0.691269i | 1.01199i | 2.89816 | + | 0.941671i | −2.04477 | + | 1.48561i | 0 | ||||||
249.7 | 1.18361 | − | 1.62909i | −2.87758 | + | 0.934982i | −0.634989 | − | 1.95429i | 0 | −1.88274 | + | 5.79449i | − | 0.369971i | −0.105077 | − | 0.0341417i | 4.97921 | − | 3.61761i | 0 | |||||
249.8 | 1.56645 | − | 2.15604i | 0.721930 | − | 0.234569i | −1.57669 | − | 4.85257i | 0 | 0.625130 | − | 1.92395i | − | 2.04213i | −7.86300 | − | 2.55484i | −1.96089 | + | 1.42467i | 0 | |||||
374.1 | −1.56645 | − | 2.15604i | −0.721930 | − | 0.234569i | −1.57669 | + | 4.85257i | 0 | 0.625130 | + | 1.92395i | − | 2.04213i | 7.86300 | − | 2.55484i | −1.96089 | − | 1.42467i | 0 | |||||
374.2 | −1.18361 | − | 1.62909i | 2.87758 | + | 0.934982i | −0.634989 | + | 1.95429i | 0 | −1.88274 | − | 5.79449i | − | 0.369971i | 0.105077 | − | 0.0341417i | 4.97921 | + | 3.61761i | 0 | |||||
374.3 | −0.621509 | − | 0.855434i | 0.653760 | + | 0.212419i | 0.272540 | − | 0.838792i | 0 | −0.224607 | − | 0.691269i | 1.01199i | −2.89816 | + | 0.941671i | −2.04477 | − | 1.48561i | 0 | ||||||
374.4 | −0.192059 | − | 0.264347i | −1.63142 | − | 0.530081i | 0.585041 | − | 1.80057i | 0 | 0.173205 | + | 0.533069i | − | 3.42409i | −1.20986 | + | 0.393106i | −0.0465016 | − | 0.0337854i | 0 | |||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
25.e | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 625.2.e.k | 32 | |
5.b | even | 2 | 1 | inner | 625.2.e.k | 32 | |
5.c | odd | 4 | 1 | 625.2.d.n | 16 | ||
5.c | odd | 4 | 1 | 625.2.d.p | 16 | ||
25.d | even | 5 | 1 | 625.2.b.d | 16 | ||
25.d | even | 5 | 2 | 625.2.e.j | 32 | ||
25.d | even | 5 | 1 | inner | 625.2.e.k | 32 | |
25.e | even | 10 | 1 | 625.2.b.d | 16 | ||
25.e | even | 10 | 2 | 625.2.e.j | 32 | ||
25.e | even | 10 | 1 | inner | 625.2.e.k | 32 | |
25.f | odd | 20 | 1 | 625.2.a.e | ✓ | 8 | |
25.f | odd | 20 | 1 | 625.2.a.g | yes | 8 | |
25.f | odd | 20 | 2 | 625.2.d.m | 16 | ||
25.f | odd | 20 | 1 | 625.2.d.n | 16 | ||
25.f | odd | 20 | 1 | 625.2.d.p | 16 | ||
25.f | odd | 20 | 2 | 625.2.d.q | 16 | ||
75.l | even | 20 | 1 | 5625.2.a.s | 8 | ||
75.l | even | 20 | 1 | 5625.2.a.be | 8 | ||
100.l | even | 20 | 1 | 10000.2.a.be | 8 | ||
100.l | even | 20 | 1 | 10000.2.a.bn | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
625.2.a.e | ✓ | 8 | 25.f | odd | 20 | 1 | |
625.2.a.g | yes | 8 | 25.f | odd | 20 | 1 | |
625.2.b.d | 16 | 25.d | even | 5 | 1 | ||
625.2.b.d | 16 | 25.e | even | 10 | 1 | ||
625.2.d.m | 16 | 25.f | odd | 20 | 2 | ||
625.2.d.n | 16 | 5.c | odd | 4 | 1 | ||
625.2.d.n | 16 | 25.f | odd | 20 | 1 | ||
625.2.d.p | 16 | 5.c | odd | 4 | 1 | ||
625.2.d.p | 16 | 25.f | odd | 20 | 1 | ||
625.2.d.q | 16 | 25.f | odd | 20 | 2 | ||
625.2.e.j | 32 | 25.d | even | 5 | 2 | ||
625.2.e.j | 32 | 25.e | even | 10 | 2 | ||
625.2.e.k | 32 | 1.a | even | 1 | 1 | trivial | |
625.2.e.k | 32 | 5.b | even | 2 | 1 | inner | |
625.2.e.k | 32 | 25.d | even | 5 | 1 | inner | |
625.2.e.k | 32 | 25.e | even | 10 | 1 | inner | |
5625.2.a.s | 8 | 75.l | even | 20 | 1 | ||
5625.2.a.be | 8 | 75.l | even | 20 | 1 | ||
10000.2.a.be | 8 | 100.l | even | 20 | 1 | ||
10000.2.a.bn | 8 | 100.l | even | 20 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):
\( T_{2}^{32} - 16 T_{2}^{30} + 160 T_{2}^{28} - 1340 T_{2}^{26} + 10170 T_{2}^{24} - 57088 T_{2}^{22} + \cdots + 6561 \) |
\( T_{3}^{32} - 9 T_{3}^{30} + 125 T_{3}^{28} - 1320 T_{3}^{26} + 12600 T_{3}^{24} - 49887 T_{3}^{22} + \cdots + 707281 \) |