Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(20.6506685020\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.51960.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 70x + 82 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 5 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 70x + 82 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 47 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 47 \)
|
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 |
|
2.00000 | −6.44221 | 4.00000 | 0 | −12.8844 | −7.00000 | 8.00000 | 14.5021 | 0 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | 3.17488 | 4.00000 | 0 | 6.34975 | −7.00000 | 8.00000 | −16.9202 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 10.2673 | 4.00000 | 0 | 20.5347 | −7.00000 | 8.00000 | 78.4181 | 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 | 350.4.a.x | 3 | |
| 5.b | even | 2 | 1 | 350.4.a.w | 3 | ||
| 5.c | odd | 4 | 2 | 70.4.c.b | ✓ | 6 | |
| 7.b | odd | 2 | 1 | 2450.4.a.cg | 3 | ||
| 15.e | even | 4 | 2 | 630.4.g.j | 6 | ||
| 20.e | even | 4 | 2 | 560.4.g.e | 6 | ||
| 35.c | odd | 2 | 1 | 2450.4.a.cf | 3 | ||
| 35.f | even | 4 | 2 | 490.4.c.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.4.c.b | ✓ | 6 | 5.c | odd | 4 | 2 | |
| 350.4.a.w | 3 | 5.b | even | 2 | 1 | ||
| 350.4.a.x | 3 | 1.a | even | 1 | 1 | trivial | |
| 490.4.c.c | 6 | 35.f | even | 4 | 2 | ||
| 560.4.g.e | 6 | 20.e | even | 4 | 2 | ||
| 630.4.g.j | 6 | 15.e | even | 4 | 2 | ||
| 2450.4.a.cf | 3 | 35.c | odd | 2 | 1 | ||
| 2450.4.a.cg | 3 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(350))\):
|
\( T_{3}^{3} - 7T_{3}^{2} - 54T_{3} + 210 \)
|
|
\( T_{11}^{3} - 19T_{11}^{2} - 1560T_{11} - 600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( T^{3} - 7 T^{2} + \cdots + 210 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( (T + 7)^{3} \)
|
| $11$ |
\( T^{3} - 19 T^{2} + \cdots - 600 \)
|
| $13$ |
\( T^{3} - 41 T^{2} + \cdots + 28802 \)
|
| $17$ |
\( T^{3} - 155 T^{2} + \cdots + 8324 \)
|
| $19$ |
\( T^{3} - 156 T^{2} + \cdots + 1196840 \)
|
| $23$ |
\( T^{3} - 50 T^{2} + \cdots + 1575000 \)
|
| $29$ |
\( T^{3} - 91 T^{2} + \cdots - 204564 \)
|
| $31$ |
\( T^{3} - 170 T^{2} + \cdots + 2033232 \)
|
| $37$ |
\( T^{3} + 118 T^{2} + \cdots + 130280 \)
|
| $41$ |
\( T^{3} + 6 T^{2} + \cdots - 7243272 \)
|
| $43$ |
\( T^{3} + 216 T^{2} + \cdots + 11182208 \)
|
| $47$ |
\( T^{3} - 79 T^{2} + \cdots - 13015016 \)
|
| $53$ |
\( T^{3} + 742 T^{2} + \cdots - 42049416 \)
|
| $59$ |
\( T^{3} - 590 T^{2} + \cdots + 88782692 \)
|
| $61$ |
\( T^{3} - 352 T^{2} + \cdots - 3430824 \)
|
| $67$ |
\( T^{3} + 1258 T^{2} + \cdots + 69582336 \)
|
| $71$ |
\( T^{3} - 724 T^{2} + \cdots + 55149600 \)
|
| $73$ |
\( T^{3} + 1326 T^{2} + \cdots - 67887496 \)
|
| $79$ |
\( T^{3} - 1037 T^{2} + \cdots + 276622388 \)
|
| $83$ |
\( T^{3} + 580 T^{2} + \cdots - 91140456 \)
|
| $89$ |
\( T^{3} + 1048 T^{2} + \cdots - 210826480 \)
|
| $97$ |
\( T^{3} + 2821 T^{2} + \cdots + 696704100 \)
|
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