Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,10,Mod(1,350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.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: | \(180.262542657\) |
| 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: |
\( x^{5} - x^{4} - 65463x^{3} + 820957x^{2} + 682043930x + 11121327000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2}\cdot 5^{3} \) |
| 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
\( x^{5} - x^{4} - 65463x^{3} + 820957x^{2} + 682043930x + 11121327000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 18\nu - 26189 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 554\nu^{3} + 89581\nu^{2} - 25490630\nu - 921576150 ) / 20655 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{4} + 79\nu^{3} - 222919\nu^{2} + 1971665\nu + 1200945825 ) / 11475 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 18\beta _1 + 26189 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{4} + 36\beta_{3} - 59\beta_{2} + 44631\beta _1 - 462206 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2770\beta_{4} - 711\beta_{3} + 56895\beta_{2} - 2377514\beta _1 + 1168398535 \)
|
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 |
|
16.0000 | −221.017 | 256.000 | 0 | −3536.28 | 2401.00 | 4096.00 | 29165.7 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 16.0000 | −73.4713 | 256.000 | 0 | −1175.54 | 2401.00 | 4096.00 | −14285.0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 16.0000 | 1.85943 | 256.000 | 0 | 29.7509 | 2401.00 | 4096.00 | −19679.5 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 16.0000 | 155.563 | 256.000 | 0 | 2489.01 | 2401.00 | 4096.00 | 4516.84 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 16.0000 | 233.066 | 256.000 | 0 | 3729.06 | 2401.00 | 4096.00 | 34636.9 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 350.10.a.t | yes | 5 |
| 5.b | even | 2 | 1 | 350.10.a.q | ✓ | 5 | |
| 5.c | odd | 4 | 2 | 350.10.c.p | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.10.a.q | ✓ | 5 | 5.b | even | 2 | 1 | |
| 350.10.a.t | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 350.10.c.p | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 96T_{3}^{4} - 61777T_{3}^{3} + 4481592T_{3}^{2} + 580630176T_{3} - 1094737896 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(350))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{5} \)
|
| $3$ |
\( T^{5} + \cdots - 1094737896 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( (T - 2401)^{5} \)
|
| $11$ |
\( T^{5} + \cdots - 20\!\cdots\!43 \)
|
| $13$ |
\( T^{5} + \cdots + 10\!\cdots\!12 \)
|
| $17$ |
\( T^{5} + \cdots + 15\!\cdots\!72 \)
|
| $19$ |
\( T^{5} + \cdots + 11\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 13\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots - 85\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 49\!\cdots\!96 \)
|
| $37$ |
\( T^{5} + \cdots - 55\!\cdots\!48 \)
|
| $41$ |
\( T^{5} + \cdots + 37\!\cdots\!16 \)
|
| $43$ |
\( T^{5} + \cdots - 21\!\cdots\!32 \)
|
| $47$ |
\( T^{5} + \cdots - 46\!\cdots\!36 \)
|
| $53$ |
\( T^{5} + \cdots - 50\!\cdots\!72 \)
|
| $59$ |
\( T^{5} + \cdots - 12\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots + 20\!\cdots\!67 \)
|
| $71$ |
\( T^{5} + \cdots + 24\!\cdots\!16 \)
|
| $73$ |
\( T^{5} + \cdots + 41\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots - 71\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 41\!\cdots\!76 \)
|
| $89$ |
\( T^{5} + \cdots + 18\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 61\!\cdots\!00 \)
|
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