Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7203,2,Mod(1,7203)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7203, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7203.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 7203 = 3 \cdot 7^{4} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 7203.a (trivial) |
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: | yes |
Analytic conductor: | \(57.5162445759\) |
Analytic rank: | \(1\) |
Dimension: | \(24\) |
Twist minimal: | no (minimal twist has level 147) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −2.76937 | −1.00000 | 5.66944 | 0.158853 | 2.76937 | 0 | −10.1620 | 1.00000 | −0.439924 | ||||||||||||||||||
1.2 | −2.59513 | −1.00000 | 4.73469 | −3.53982 | 2.59513 | 0 | −7.09686 | 1.00000 | 9.18628 | ||||||||||||||||||
1.3 | −2.35187 | −1.00000 | 3.53127 | 0.557059 | 2.35187 | 0 | −3.60134 | 1.00000 | −1.31013 | ||||||||||||||||||
1.4 | −2.23534 | −1.00000 | 2.99674 | −1.56164 | 2.23534 | 0 | −2.22804 | 1.00000 | 3.49078 | ||||||||||||||||||
1.5 | −2.16475 | −1.00000 | 2.68616 | 2.35627 | 2.16475 | 0 | −1.48537 | 1.00000 | −5.10075 | ||||||||||||||||||
1.6 | −1.95099 | −1.00000 | 1.80636 | −3.02702 | 1.95099 | 0 | 0.377796 | 1.00000 | 5.90568 | ||||||||||||||||||
1.7 | −1.73436 | −1.00000 | 1.00799 | 2.45117 | 1.73436 | 0 | 1.72050 | 1.00000 | −4.25120 | ||||||||||||||||||
1.8 | −1.30793 | −1.00000 | −0.289308 | 1.06184 | 1.30793 | 0 | 2.99426 | 1.00000 | −1.38882 | ||||||||||||||||||
1.9 | −0.871253 | −1.00000 | −1.24092 | 4.24654 | 0.871253 | 0 | 2.82366 | 1.00000 | −3.69981 | ||||||||||||||||||
1.10 | −0.839793 | −1.00000 | −1.29475 | −1.55513 | 0.839793 | 0 | 2.76691 | 1.00000 | 1.30599 | ||||||||||||||||||
1.11 | −0.777469 | −1.00000 | −1.39554 | −1.98233 | 0.777469 | 0 | 2.63993 | 1.00000 | 1.54120 | ||||||||||||||||||
1.12 | −0.635256 | −1.00000 | −1.59645 | −2.16973 | 0.635256 | 0 | 2.28467 | 1.00000 | 1.37833 | ||||||||||||||||||
1.13 | −0.411506 | −1.00000 | −1.83066 | 1.75068 | 0.411506 | 0 | 1.57634 | 1.00000 | −0.720416 | ||||||||||||||||||
1.14 | 0.0536083 | −1.00000 | −1.99713 | −2.17062 | −0.0536083 | 0 | −0.214279 | 1.00000 | −0.116363 | ||||||||||||||||||
1.15 | 0.544156 | −1.00000 | −1.70389 | 1.49980 | −0.544156 | 0 | −2.01550 | 1.00000 | 0.816125 | ||||||||||||||||||
1.16 | 0.671730 | −1.00000 | −1.54878 | −0.375440 | −0.671730 | 0 | −2.38382 | 1.00000 | −0.252195 | ||||||||||||||||||
1.17 | 0.804104 | −1.00000 | −1.35342 | −1.35338 | −0.804104 | 0 | −2.69650 | 1.00000 | −1.08826 | ||||||||||||||||||
1.18 | 1.06836 | −1.00000 | −0.858601 | 2.14514 | −1.06836 | 0 | −3.05402 | 1.00000 | 2.29179 | ||||||||||||||||||
1.19 | 1.33627 | −1.00000 | −0.214387 | −3.76441 | −1.33627 | 0 | −2.95902 | 1.00000 | −5.03026 | ||||||||||||||||||
1.20 | 1.76319 | −1.00000 | 1.10884 | 3.70870 | −1.76319 | 0 | −1.57128 | 1.00000 | 6.53915 | ||||||||||||||||||
See all 24 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(7\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 7203.2.a.i | 24 | |
7.b | odd | 2 | 1 | 7203.2.a.k | 24 | ||
49.g | even | 21 | 2 | 147.2.m.a | ✓ | 48 | |
147.n | odd | 42 | 2 | 441.2.bb.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
147.2.m.a | ✓ | 48 | 49.g | even | 21 | 2 | |
441.2.bb.c | 48 | 147.n | odd | 42 | 2 | ||
7203.2.a.i | 24 | 1.a | even | 1 | 1 | trivial | |
7203.2.a.k | 24 | 7.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7203))\):
\( T_{2}^{24} + 6 T_{2}^{23} - 15 T_{2}^{22} - 148 T_{2}^{21} + 12 T_{2}^{20} + 1536 T_{2}^{19} + 1162 T_{2}^{18} + \cdots - 41 \) |
\( T_{5}^{24} - 60 T_{5}^{22} - 8 T_{5}^{21} + 1506 T_{5}^{20} + 344 T_{5}^{19} - 20845 T_{5}^{18} + \cdots + 33013 \) |