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: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 643x - 1542 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 643x - 1542 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4\nu - 429 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4\beta _1 + 429 \)
|
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 | −27.0606 | 16.0000 | 0 | 108.242 | −49.0000 | −64.0000 | 489.275 | 0 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | −5.42018 | 16.0000 | 0 | 21.6807 | −49.0000 | −64.0000 | −213.622 | 0 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 23.4808 | 16.0000 | 0 | −93.9231 | −49.0000 | −64.0000 | 308.346 | 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.u | ✓ | 3 |
| 5.b | even | 2 | 1 | 350.6.a.x | yes | 3 | |
| 5.c | odd | 4 | 2 | 350.6.c.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.6.a.u | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 350.6.a.x | yes | 3 | 5.b | even | 2 | 1 | |
| 350.6.c.n | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(350))\):
|
\( T_{3}^{3} + 9T_{3}^{2} - 616T_{3} - 3444 \)
|
|
\( T_{13}^{3} + 842T_{13}^{2} + 105560T_{13} - 30165296 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( T^{3} + 9 T^{2} + \cdots - 3444 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( (T + 49)^{3} \)
|
| $11$ |
\( T^{3} + 107 T^{2} + \cdots - 21103983 \)
|
| $13$ |
\( T^{3} + 842 T^{2} + \cdots - 30165296 \)
|
| $17$ |
\( T^{3} + 251 T^{2} + \cdots + 362765088 \)
|
| $19$ |
\( T^{3} - 441 T^{2} + \cdots - 289884980 \)
|
| $23$ |
\( T^{3} + \cdots - 13661587800 \)
|
| $29$ |
\( T^{3} + \cdots + 32353852950 \)
|
| $31$ |
\( T^{3} + \cdots - 9395311072 \)
|
| $37$ |
\( T^{3} + \cdots + 199886955928 \)
|
| $41$ |
\( T^{3} + \cdots - 1366682605092 \)
|
| $43$ |
\( T^{3} + \cdots - 948878648976 \)
|
| $47$ |
\( T^{3} + \cdots - 144057481848 \)
|
| $53$ |
\( T^{3} + \cdots - 2577509373456 \)
|
| $59$ |
\( T^{3} + \cdots - 533329584480 \)
|
| $61$ |
\( T^{3} + \cdots - 28835515765000 \)
|
| $67$ |
\( T^{3} + \cdots - 3526453467747 \)
|
| $71$ |
\( T^{3} + \cdots - 128336306971662 \)
|
| $73$ |
\( T^{3} + \cdots - 8390323795700 \)
|
| $79$ |
\( T^{3} + \cdots - 10\!\cdots\!50 \)
|
| $83$ |
\( T^{3} + \cdots + 567810730049976 \)
|
| $89$ |
\( T^{3} + \cdots + 65722788999000 \)
|
| $97$ |
\( T^{3} + \cdots + 364862905632400 \)
|
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