Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,8,Mod(1,304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(304, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("304.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.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: | \(94.9650477472\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{5} - 2x^{4} - 5014x^{3} + 113222x^{2} - 625803x + 567036 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 76) |
| 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
\( x^{5} - 2x^{4} - 5014x^{3} + 113222x^{2} - 625803x + 567036 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 302\nu^{4} - 759\nu^{3} - 1458172\nu^{2} + 37319919\nu - 275123626 ) / 3056594 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -440\nu^{4} + 11227\nu^{3} + 2418003\nu^{2} - 101285025\nu + 456154575 ) / 1528297 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3831\nu^{4} + 24810\nu^{3} + 19701959\nu^{2} - 484183104\nu + 2008817956 ) / 3056594 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15023\nu^{4} + 78637\nu^{3} - 74510443\nu^{2} + 1156525641\nu - 2210497044 ) / 3056594 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 7\beta_{3} - 11\beta_{2} + 7\beta _1 + 36 ) / 84 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -13\beta_{4} - 35\beta_{3} + 101\beta_{2} + 497\beta _1 + 28190 ) / 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2299\beta_{4} + 12845\beta_{3} - 18625\beta_{2} - 5691\beta _1 - 1732420 ) / 28 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -40238\beta_{4} - 148512\beta_{3} + 345409\beta_{2} + 1195040\beta _1 + 72973885 ) / 7 \)
|
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 | −73.9019 | 0 | −358.964 | 0 | 109.800 | 0 | 3274.48 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −25.4201 | 0 | −3.35619 | 0 | 173.262 | 0 | −1540.82 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −8.09048 | 0 | 276.339 | 0 | 720.800 | 0 | −2121.54 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 36.5155 | 0 | −266.507 | 0 | −1627.87 | 0 | −853.617 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 84.8970 | 0 | 72.4872 | 0 | 210.006 | 0 | 5020.49 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 304.8.a.g | 5 | |
| 4.b | odd | 2 | 1 | 76.8.a.a | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.8.a.a | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 304.8.a.g | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 14T_{3}^{4} - 7259T_{3}^{3} + 22536T_{3}^{2} + 6469524T_{3} + 47116944 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 14 T^{4} + \cdots + 47116944 \)
|
| $5$ |
\( T^{5} + \cdots + 6431464000 \)
|
| $7$ |
\( T^{5} + \cdots + 4687831896336 \)
|
| $11$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $13$ |
\( T^{5} + \cdots - 53\!\cdots\!12 \)
|
| $17$ |
\( T^{5} + \cdots - 74\!\cdots\!26 \)
|
| $19$ |
\( (T - 6859)^{5} \)
|
| $23$ |
\( T^{5} + \cdots - 17\!\cdots\!64 \)
|
| $29$ |
\( T^{5} + \cdots - 16\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 46\!\cdots\!76 \)
|
| $37$ |
\( T^{5} + \cdots - 15\!\cdots\!80 \)
|
| $41$ |
\( T^{5} + \cdots + 13\!\cdots\!60 \)
|
| $43$ |
\( T^{5} + \cdots - 14\!\cdots\!28 \)
|
| $47$ |
\( T^{5} + \cdots - 43\!\cdots\!48 \)
|
| $53$ |
\( T^{5} + \cdots - 10\!\cdots\!52 \)
|
| $59$ |
\( T^{5} + \cdots - 51\!\cdots\!76 \)
|
| $61$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 25\!\cdots\!16 \)
|
| $71$ |
\( T^{5} + \cdots - 83\!\cdots\!08 \)
|
| $73$ |
\( T^{5} + \cdots - 35\!\cdots\!38 \)
|
| $79$ |
\( T^{5} + \cdots - 11\!\cdots\!08 \)
|
| $83$ |
\( T^{5} + \cdots + 15\!\cdots\!36 \)
|
| $89$ |
\( T^{5} + \cdots - 65\!\cdots\!68 \)
|
| $97$ |
\( T^{5} + \cdots + 75\!\cdots\!84 \)
|
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