Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [154,6,Mod(1,154)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(154, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("154.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 154 = 2 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 154.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: | \(24.6991082512\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{337}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 84 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{337})\). 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 |
|
4.00000 | −5.67878 | 16.0000 | 14.3939 | −22.7151 | −49.0000 | 64.0000 | −210.751 | 57.5756 | ||||||||||||||||||||||||
| 1.2 | 4.00000 | 12.6788 | 16.0000 | −77.3939 | 50.7151 | −49.0000 | 64.0000 | −82.2485 | −309.576 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 154.6.a.e | ✓ | 2 |
| 7.b | odd | 2 | 1 | 1078.6.a.i | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.6.a.e | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 1078.6.a.i | 2 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 7T_{3} - 72 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(154))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{2} \)
|
| $3$ |
\( T^{2} - 7T - 72 \)
|
| $5$ |
\( T^{2} + 63T - 1114 \)
|
| $7$ |
\( (T + 49)^{2} \)
|
| $11$ |
\( (T - 121)^{2} \)
|
| $13$ |
\( T^{2} + 546T - 47128 \)
|
| $17$ |
\( T^{2} + 1540 T - 1784972 \)
|
| $19$ |
\( T^{2} + 2562 T + 546048 \)
|
| $23$ |
\( T^{2} + 2359 T - 792624 \)
|
| $29$ |
\( T^{2} + 2256 T - 1964164 \)
|
| $31$ |
\( T^{2} + 6601 T - 578264 \)
|
| $37$ |
\( T^{2} + 4555 T - 8225678 \)
|
| $41$ |
\( T^{2} - 8344 T - 22011284 \)
|
| $43$ |
\( T^{2} + 4036 T - 210530624 \)
|
| $47$ |
\( T^{2} + 28000 T + 191147200 \)
|
| $53$ |
\( T^{2} - 28384 T + 190250076 \)
|
| $59$ |
\( T^{2} - 20265 T - 144553032 \)
|
| $61$ |
\( T^{2} + 57736 T + 272559724 \)
|
| $67$ |
\( T^{2} + 11073 T + 17240148 \)
|
| $71$ |
\( T^{2} - 53467 T + 593965864 \)
|
| $73$ |
\( T^{2} + \cdots - 2644873548 \)
|
| $79$ |
\( T^{2} + \cdots - 2700270048 \)
|
| $83$ |
\( T^{2} + \cdots - 7071885312 \)
|
| $89$ |
\( T^{2} - 2191 T - 40912158 \)
|
| $97$ |
\( T^{2} + \cdots - 22740312746 \)
|
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