Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [416,4,Mod(1,416)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 416 = 2^{5} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 416.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: | \(24.5447945624\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 66x^{3} - 139x^{2} + 283x + 676 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 66x^{3} - 139x^{2} + 283x + 676 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 24\nu^{4} - 103\nu^{3} - 1211\nu^{2} + 657\nu + 4446 ) / 163 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -28\nu^{4} + 93\nu^{3} + 1603\nu^{2} + 293\nu - 7143 ) / 163 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -50\nu^{4} + 201\nu^{3} + 2781\nu^{2} - 1165\nu - 14560 ) / 163 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -82\nu^{4} + 284\nu^{3} + 4776\nu^{2} - 574\nu - 24074 ) / 163 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} + 3\beta_{3} + \beta_{2} + 4\beta _1 + 55 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -53\beta_{4} + 81\beta_{3} + 68\beta_{2} + 67\beta _1 + 560 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -301\beta_{4} + 623\beta_{3} + 338\beta_{2} + 691\beta _1 + 7158 ) / 4 \)
|
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 | −9.48781 | 0 | −10.6440 | 0 | 7.29306 | 0 | 63.0185 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.34006 | 0 | 15.7284 | 0 | −29.0676 | 0 | 26.8765 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.323589 | 0 | 10.9246 | 0 | 6.41598 | 0 | −26.8953 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.240222 | 0 | −18.1950 | 0 | 23.9357 | 0 | −26.9423 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 5.91123 | 0 | −2.81407 | 0 | −29.5771 | 0 | 7.94266 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 416.4.a.h | ✓ | 5 |
| 4.b | odd | 2 | 1 | 416.4.a.i | yes | 5 | |
| 8.b | even | 2 | 1 | 832.4.a.bg | 5 | ||
| 8.d | odd | 2 | 1 | 832.4.a.bf | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 416.4.a.h | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 416.4.a.i | yes | 5 | 4.b | odd | 2 | 1 | |
| 832.4.a.bf | 5 | 8.d | odd | 2 | 1 | ||
| 832.4.a.bg | 5 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 11T_{3}^{4} - 29T_{3}^{3} - 415T_{3}^{2} - 32T_{3} + 32 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(416))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 11 T^{4} + \cdots + 32 \)
|
| $5$ |
\( T^{5} + 5 T^{4} + \cdots + 93644 \)
|
| $7$ |
\( T^{5} + 21 T^{4} + \cdots - 962904 \)
|
| $11$ |
\( T^{5} - 18 T^{4} + \cdots - 93856576 \)
|
| $13$ |
\( (T - 13)^{5} \)
|
| $17$ |
\( T^{5} - 123 T^{4} + \cdots - 89830068 \)
|
| $19$ |
\( T^{5} + \cdots + 4528084672 \)
|
| $23$ |
\( T^{5} + 92 T^{4} + \cdots - 44302336 \)
|
| $29$ |
\( T^{5} + \cdots + 247555437664 \)
|
| $31$ |
\( T^{5} + \cdots - 92054386688 \)
|
| $37$ |
\( T^{5} + \cdots + 4626715132 \)
|
| $41$ |
\( T^{5} + \cdots + 245417766912 \)
|
| $43$ |
\( T^{5} + \cdots + 260119797184 \)
|
| $47$ |
\( T^{5} + \cdots - 95050280376 \)
|
| $53$ |
\( T^{5} + \cdots + 8027052256256 \)
|
| $59$ |
\( T^{5} + \cdots + 5206076073024 \)
|
| $61$ |
\( T^{5} + \cdots + 6052484202496 \)
|
| $67$ |
\( T^{5} + \cdots - 2854340132288 \)
|
| $71$ |
\( T^{5} + \cdots + 5105537071224 \)
|
| $73$ |
\( T^{5} + \cdots - 4847877719264 \)
|
| $79$ |
\( T^{5} + \cdots + 1514902929408 \)
|
| $83$ |
\( T^{5} + \cdots + 208379470739968 \)
|
| $89$ |
\( T^{5} + \cdots + 9206645570784 \)
|
| $97$ |
\( T^{5} + \cdots + 839229027548768 \)
|
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