Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [176,6,Mod(1,176)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(176, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("176.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 176 = 2^{4} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 176.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: | \(28.2275522871\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 158x^{2} - 78x + 2316 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 88) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 2x^{3} - 158x^{2} - 78x + 2316 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 52\nu^{2} + 222\nu - 3918 ) / 222 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 10\nu^{3} - 76\nu^{2} - 888\nu + 3549 ) / 111 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -17\nu^{3} - 4\nu^{2} + 2886\nu + 3990 ) / 222 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 3\beta _1 + 3 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{3} - \beta_{2} + 31\beta _1 + 633 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 33\beta_{3} + 85\beta_{2} + 251\beta _1 + 1119 ) / 4 \)
|
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 | −13.8972 | 0 | −4.29272 | 0 | 204.438 | 0 | −49.8668 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −8.08220 | 0 | −88.6814 | 0 | −238.927 | 0 | −177.678 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 5.39087 | 0 | 32.7380 | 0 | 53.5703 | 0 | −213.939 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 29.5886 | 0 | 41.2361 | 0 | −77.0813 | 0 | 632.483 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 176.6.a.l | 4 | |
| 4.b | odd | 2 | 1 | 88.6.a.c | ✓ | 4 | |
| 8.b | even | 2 | 1 | 704.6.a.u | 4 | ||
| 8.d | odd | 2 | 1 | 704.6.a.x | 4 | ||
| 12.b | even | 2 | 1 | 792.6.a.l | 4 | ||
| 44.c | even | 2 | 1 | 968.6.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.6.a.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 176.6.a.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 704.6.a.u | 4 | 8.b | even | 2 | 1 | ||
| 704.6.a.x | 4 | 8.d | odd | 2 | 1 | ||
| 792.6.a.l | 4 | 12.b | even | 2 | 1 | ||
| 968.6.a.d | 4 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 13T_{3}^{3} - 497T_{3}^{2} - 423T_{3} + 17916 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(176))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 13 T^{3} + \cdots + 17916 \)
|
| $5$ |
\( T^{4} + 19 T^{3} + \cdots + 513918 \)
|
| $7$ |
\( T^{4} + 58 T^{3} + \cdots + 201696448 \)
|
| $11$ |
\( (T - 121)^{4} \)
|
| $13$ |
\( T^{4} + \cdots - 522545202432 \)
|
| $17$ |
\( T^{4} + \cdots + 1669425881616 \)
|
| $19$ |
\( T^{4} + \cdots - 12485832626432 \)
|
| $23$ |
\( T^{4} + \cdots - 2129166090528 \)
|
| $29$ |
\( T^{4} + \cdots + 254878076716032 \)
|
| $31$ |
\( T^{4} + \cdots + 586842385422744 \)
|
| $37$ |
\( T^{4} + \cdots - 38775318298614 \)
|
| $41$ |
\( T^{4} + \cdots + 40933180410496 \)
|
| $43$ |
\( T^{4} + \cdots + 17\!\cdots\!08 \)
|
| $47$ |
\( T^{4} + \cdots - 36\!\cdots\!08 \)
|
| $53$ |
\( T^{4} + \cdots + 37\!\cdots\!12 \)
|
| $59$ |
\( T^{4} + \cdots + 49\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 22\!\cdots\!52 \)
|
| $67$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots - 17\!\cdots\!56 \)
|
| $73$ |
\( T^{4} + \cdots - 40\!\cdots\!12 \)
|
| $79$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
|
| $83$ |
\( T^{4} + \cdots - 15\!\cdots\!72 \)
|
| $89$ |
\( T^{4} + \cdots - 46\!\cdots\!54 \)
|
| $97$ |
\( T^{4} + \cdots - 54\!\cdots\!98 \)
|
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