Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(109.334758919\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 2473x^{2} - 31160x + 389808 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 5^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 2473x^{2} - 31160x + 389808 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 165\nu^{2} + 613\nu - 180252 ) / 2520 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11\nu^{3} - 240\nu^{2} - 20918\nu + 24417 ) / 315 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{3} - 255\nu^{2} - 6991\nu + 140544 ) / 180 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 10\beta _1 + 12 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} - \beta_{2} + 206\beta _1 + 12470 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1049\beta_{3} - 554\beta_{2} + 16250\beta _1 + 386223 ) / 15 \)
|
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 |
|
8.00000 | −61.8960 | 64.0000 | 0 | −495.168 | −343.000 | 512.000 | 1644.11 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −8.12245 | 64.0000 | 0 | −64.9796 | −343.000 | 512.000 | −2121.03 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 36.1742 | 64.0000 | 0 | 289.394 | −343.000 | 512.000 | −878.424 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 75.8442 | 64.0000 | 0 | 606.753 | −343.000 | 512.000 | 3565.34 | 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 | 350.8.a.y | yes | 4 |
| 5.b | even | 2 | 1 | 350.8.a.v | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 350.8.c.o | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.8.a.v | ✓ | 4 | 5.b | even | 2 | 1 | |
| 350.8.a.y | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 350.8.c.o | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 42T_{3}^{3} - 4597T_{3}^{2} + 135786T_{3} + 1379340 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(350))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{4} \)
|
| $3$ |
\( T^{4} - 42 T^{3} + \cdots + 1379340 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T + 343)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 395322363347625 \)
|
| $13$ |
\( T^{4} + \cdots - 705898274880784 \)
|
| $17$ |
\( T^{4} + \cdots - 49\!\cdots\!32 \)
|
| $19$ |
\( T^{4} + \cdots + 26\!\cdots\!80 \)
|
| $23$ |
\( T^{4} + \cdots + 57\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 79\!\cdots\!96 \)
|
| $31$ |
\( T^{4} + \cdots + 81\!\cdots\!88 \)
|
| $37$ |
\( T^{4} + \cdots - 36\!\cdots\!80 \)
|
| $41$ |
\( T^{4} + \cdots + 22\!\cdots\!52 \)
|
| $43$ |
\( T^{4} + \cdots + 69\!\cdots\!52 \)
|
| $47$ |
\( T^{4} + \cdots - 31\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots - 16\!\cdots\!68 \)
|
| $59$ |
\( T^{4} + \cdots + 28\!\cdots\!28 \)
|
| $61$ |
\( T^{4} + \cdots - 44\!\cdots\!28 \)
|
| $67$ |
\( T^{4} + \cdots - 14\!\cdots\!43 \)
|
| $71$ |
\( T^{4} + \cdots - 65\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 10\!\cdots\!96 \)
|
| $79$ |
\( T^{4} + \cdots - 30\!\cdots\!76 \)
|
| $83$ |
\( T^{4} + \cdots - 63\!\cdots\!92 \)
|
| $89$ |
\( T^{4} + \cdots - 76\!\cdots\!20 \)
|
| $97$ |
\( T^{4} + \cdots - 75\!\cdots\!00 \)
|
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