Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,12,Mod(1,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 240.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: | \(184.402363334\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1801}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 450 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | no (minimal twist has level 15) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 128\sqrt{1801}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 243.000 | 0 | −3125.00 | 0 | −79941.2 | 0 | 59049.0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 243.000 | 0 | −3125.00 | 0 | 72157.2 | 0 | 59049.0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
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 | 240.12.a.m | 2 | |
| 4.b | odd | 2 | 1 | 15.12.a.c | ✓ | 2 | |
| 12.b | even | 2 | 1 | 45.12.a.c | 2 | ||
| 20.d | odd | 2 | 1 | 75.12.a.c | 2 | ||
| 20.e | even | 4 | 2 | 75.12.b.d | 4 | ||
| 60.h | even | 2 | 1 | 225.12.a.i | 2 | ||
| 60.l | odd | 4 | 2 | 225.12.b.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.12.a.c | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 45.12.a.c | 2 | 12.b | even | 2 | 1 | ||
| 75.12.a.c | 2 | 20.d | odd | 2 | 1 | ||
| 75.12.b.d | 4 | 20.e | even | 4 | 2 | ||
| 225.12.a.i | 2 | 60.h | even | 2 | 1 | ||
| 225.12.b.i | 4 | 60.l | odd | 4 | 2 | ||
| 240.12.a.m | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 7784T_{7} - 5768338800 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(240))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 243)^{2} \)
|
| $5$ |
\( (T + 3125)^{2} \)
|
| $7$ |
\( T^{2} + \cdots - 5768338800 \)
|
| $11$ |
\( T^{2} + \cdots - 410180426688 \)
|
| $13$ |
\( T^{2} + \cdots - 2221874407708 \)
|
| $17$ |
\( T^{2} + \cdots + 15658933919076 \)
|
| $19$ |
\( T^{2} + \cdots + 69890598801680 \)
|
| $23$ |
\( T^{2} + \cdots - 889077131006400 \)
|
| $29$ |
\( T^{2} + \cdots + 21\!\cdots\!00 \)
|
| $31$ |
\( T^{2} + \cdots - 17\!\cdots\!00 \)
|
| $37$ |
\( T^{2} + \cdots + 50\!\cdots\!20 \)
|
| $41$ |
\( T^{2} + \cdots - 12\!\cdots\!80 \)
|
| $43$ |
\( T^{2} + \cdots + 25\!\cdots\!56 \)
|
| $47$ |
\( T^{2} + \cdots + 10\!\cdots\!84 \)
|
| $53$ |
\( T^{2} + \cdots - 43\!\cdots\!40 \)
|
| $59$ |
\( T^{2} + \cdots - 25\!\cdots\!20 \)
|
| $61$ |
\( T^{2} + \cdots + 10\!\cdots\!16 \)
|
| $67$ |
\( T^{2} + \cdots + 12\!\cdots\!96 \)
|
| $71$ |
\( T^{2} + \cdots - 20\!\cdots\!16 \)
|
| $73$ |
\( T^{2} + \cdots - 20\!\cdots\!60 \)
|
| $79$ |
\( T^{2} + \cdots - 60\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots + 43\!\cdots\!88 \)
|
| $89$ |
\( T^{2} + \cdots + 33\!\cdots\!20 \)
|
| $97$ |
\( T^{2} + \cdots + 34\!\cdots\!76 \)
|
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