Newspace parameters
| Level: | \( N \) | \(=\) | \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7350.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(58.6900454856\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2700.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 15x - 20 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 210) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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
\( x^{3} - 15x - 20 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 10 \)
|
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −1.00000 | 1.00000 | 0 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | 0 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | 0 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 7350.2.a.dp | 3 | |
| 5.b | even | 2 | 1 | 7350.2.a.do | 3 | ||
| 5.c | odd | 4 | 2 | 1470.2.g.h | 6 | ||
| 7.b | odd | 2 | 1 | 7350.2.a.dq | 3 | ||
| 7.d | odd | 6 | 2 | 1050.2.i.u | 6 | ||
| 35.c | odd | 2 | 1 | 7350.2.a.dn | 3 | ||
| 35.f | even | 4 | 2 | 1470.2.g.i | 6 | ||
| 35.i | odd | 6 | 2 | 1050.2.i.v | 6 | ||
| 35.k | even | 12 | 4 | 210.2.n.b | ✓ | 12 | |
| 35.l | odd | 12 | 4 | 1470.2.n.j | 12 | ||
| 105.w | odd | 12 | 4 | 630.2.u.f | 12 | ||
| 140.x | odd | 12 | 4 | 1680.2.di.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.2.n.b | ✓ | 12 | 35.k | even | 12 | 4 | |
| 630.2.u.f | 12 | 105.w | odd | 12 | 4 | ||
| 1050.2.i.u | 6 | 7.d | odd | 6 | 2 | ||
| 1050.2.i.v | 6 | 35.i | odd | 6 | 2 | ||
| 1470.2.g.h | 6 | 5.c | odd | 4 | 2 | ||
| 1470.2.g.i | 6 | 35.f | even | 4 | 2 | ||
| 1470.2.n.j | 12 | 35.l | odd | 12 | 4 | ||
| 1680.2.di.c | 12 | 140.x | odd | 12 | 4 | ||
| 7350.2.a.dn | 3 | 35.c | odd | 2 | 1 | ||
| 7350.2.a.do | 3 | 5.b | even | 2 | 1 | ||
| 7350.2.a.dp | 3 | 1.a | even | 1 | 1 | trivial | |
| 7350.2.a.dq | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):
|
\( T_{11}^{3} - 3T_{11}^{2} - 27T_{11} + 49 \)
|
|
\( T_{13}^{3} + 3T_{13}^{2} - 12T_{13} - 24 \)
|
|
\( T_{17}^{3} + 6T_{17}^{2} - 48T_{17} - 272 \)
|
|
\( T_{19}^{3} + 3T_{19}^{2} - 12T_{19} - 24 \)
|
|
\( T_{23}^{3} - 9T_{23}^{2} + 12T_{23} + 8 \)
|
|
\( T_{31}^{3} - 15T_{31} + 10 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{3} \)
|
| $3$ |
\( (T + 1)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} - 3 T^{2} + \cdots + 49 \)
|
| $13$ |
\( T^{3} + 3 T^{2} + \cdots - 24 \)
|
| $17$ |
\( T^{3} + 6 T^{2} + \cdots - 272 \)
|
| $19$ |
\( T^{3} + 3 T^{2} + \cdots - 24 \)
|
| $23$ |
\( T^{3} - 9 T^{2} + \cdots + 8 \)
|
| $29$ |
\( T^{3} - 12 T^{2} + \cdots - 24 \)
|
| $31$ |
\( T^{3} - 15T + 10 \)
|
| $37$ |
\( T^{3} - 9 T^{2} + \cdots + 8 \)
|
| $41$ |
\( T^{3} - 9 T^{2} + \cdots + 128 \)
|
| $43$ |
\( (T + 2)^{3} \)
|
| $47$ |
\( T^{3} - 3 T^{2} + \cdots + 404 \)
|
| $53$ |
\( T^{3} + 9 T^{2} + \cdots - 813 \)
|
| $59$ |
\( T^{3} - 12 T^{2} + \cdots + 436 \)
|
| $61$ |
\( T^{3} + 6 T^{2} + \cdots - 712 \)
|
| $67$ |
\( T^{3} + 6 T^{2} + \cdots - 392 \)
|
| $71$ |
\( (T - 6)^{3} \)
|
| $73$ |
\( (T + 4)^{3} \)
|
| $79$ |
\( T^{3} - 24 T^{2} + \cdots - 402 \)
|
| $83$ |
\( T^{3} + 12 T^{2} + \cdots - 86 \)
|
| $89$ |
\( T^{3} - 6 T^{2} + \cdots + 392 \)
|
| $97$ |
\( T^{3} - 24 T^{2} + \cdots - 112 \)
|
| show more | |
| show less |