Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(56.1343369345\) |
| 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: |
\( x^{5} - x^{4} - 915x^{3} - 2649x^{2} + 122688x - 432576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 5^{3} \) |
| Twist minimal: | no (minimal twist has level 70) |
| 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
\( x^{5} - x^{4} - 915x^{3} - 2649x^{2} + 122688x - 432576 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 7\nu^{3} + 693\nu^{2} + 1161\nu - 29394 ) / 30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 23\nu^{3} - 1083\nu^{2} - 18561\nu + 111504 ) / 120 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 23\nu^{3} - 1203\nu^{2} - 17841\nu + 155304 ) / 120 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 6\beta _1 + 365 \)
|
| \(\nu^{3}\) | \(=\) |
\( -13\beta_{4} + 17\beta_{3} + \beta_{2} + 658\beta _1 + 2008 \)
|
| \(\nu^{4}\) | \(=\) |
\( -784\beta_{4} + 812\beta_{3} - 23\beta_{2} + 9925\beta _1 + 237607 \)
|
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 |
|
4.00000 | −27.2375 | 16.0000 | 0 | −108.950 | 49.0000 | 64.0000 | 498.879 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −3.91835 | 16.0000 | 0 | −15.6734 | 49.0000 | 64.0000 | −227.647 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −1.91234 | 16.0000 | 0 | −7.64937 | 49.0000 | 64.0000 | −239.343 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 22.6958 | 16.0000 | 0 | 90.7832 | 49.0000 | 64.0000 | 272.099 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | 24.3723 | 16.0000 | 0 | 97.4894 | 49.0000 | 64.0000 | 351.011 | 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 | 350.6.a.z | 5 | |
| 5.b | even | 2 | 1 | 350.6.a.y | 5 | ||
| 5.c | odd | 4 | 2 | 70.6.c.d | ✓ | 10 | |
| 15.e | even | 4 | 2 | 630.6.g.h | 10 | ||
| 20.e | even | 4 | 2 | 560.6.g.d | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.6.c.d | ✓ | 10 | 5.c | odd | 4 | 2 | |
| 350.6.a.y | 5 | 5.b | even | 2 | 1 | ||
| 350.6.a.z | 5 | 1.a | even | 1 | 1 | trivial | |
| 560.6.g.d | 10 | 20.e | even | 4 | 2 | ||
| 630.6.g.h | 10 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(350))\):
|
\( T_{3}^{5} - 14T_{3}^{4} - 837T_{3}^{3} + 10668T_{3}^{2} + 82386T_{3} + 112896 \)
|
|
\( T_{13}^{5} - 456T_{13}^{4} - 1454059T_{13}^{3} + 376896800T_{13}^{2} + 514353680586T_{13} - 12714955073672 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{5} \)
|
| $3$ |
\( T^{5} - 14 T^{4} + \cdots + 112896 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( (T - 49)^{5} \)
|
| $11$ |
\( T^{5} + \cdots + 4772453962752 \)
|
| $13$ |
\( T^{5} + \cdots - 12714955073672 \)
|
| $17$ |
\( T^{5} + \cdots - 849591109344752 \)
|
| $19$ |
\( T^{5} + \cdots + 11\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 23088345840000 \)
|
| $29$ |
\( T^{5} + \cdots + 94\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 76\!\cdots\!16 \)
|
| $37$ |
\( T^{5} + \cdots - 22\!\cdots\!72 \)
|
| $41$ |
\( T^{5} + \cdots - 26\!\cdots\!84 \)
|
| $43$ |
\( T^{5} + \cdots + 58\!\cdots\!52 \)
|
| $47$ |
\( T^{5} + \cdots + 18\!\cdots\!56 \)
|
| $53$ |
\( T^{5} + \cdots + 11\!\cdots\!32 \)
|
| $59$ |
\( T^{5} + \cdots + 32\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 38\!\cdots\!92 \)
|
| $71$ |
\( T^{5} + \cdots - 26\!\cdots\!24 \)
|
| $73$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots + 12\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 42\!\cdots\!76 \)
|
| $89$ |
\( T^{5} + \cdots + 24\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
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