Newspace parameters
| Level: | \( N \) | \(=\) | \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8450.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(67.4735897080\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.148.1 |
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|
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| Defining polynomial: |
\( x^{3} - x^{2} - 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 130) |
| 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
\( x^{3} - x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
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 | −2.21432 | 1.00000 | 0 | −2.21432 | −3.90321 | 1.00000 | 1.90321 | 0 | |||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0.539189 | 1.00000 | 0 | 0.539189 | 0.709275 | 1.00000 | −2.70928 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.67513 | 1.00000 | 0 | 1.67513 | −1.80606 | 1.00000 | −0.193937 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 8450.2.a.ca | 3 | |
| 5.b | even | 2 | 1 | 8450.2.a.bt | 3 | ||
| 5.c | odd | 4 | 2 | 1690.2.b.b | 6 | ||
| 13.b | even | 2 | 1 | 8450.2.a.bu | 3 | ||
| 13.e | even | 6 | 2 | 650.2.e.k | 6 | ||
| 65.d | even | 2 | 1 | 8450.2.a.cb | 3 | ||
| 65.f | even | 4 | 2 | 1690.2.c.b | 6 | ||
| 65.h | odd | 4 | 2 | 1690.2.b.c | 6 | ||
| 65.k | even | 4 | 2 | 1690.2.c.c | 6 | ||
| 65.l | even | 6 | 2 | 650.2.e.j | 6 | ||
| 65.r | odd | 12 | 4 | 130.2.n.a | ✓ | 12 | |
| 195.bf | even | 12 | 4 | 1170.2.bp.h | 12 | ||
| 260.bg | even | 12 | 4 | 1040.2.dh.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.2.n.a | ✓ | 12 | 65.r | odd | 12 | 4 | |
| 650.2.e.j | 6 | 65.l | even | 6 | 2 | ||
| 650.2.e.k | 6 | 13.e | even | 6 | 2 | ||
| 1040.2.dh.b | 12 | 260.bg | even | 12 | 4 | ||
| 1170.2.bp.h | 12 | 195.bf | even | 12 | 4 | ||
| 1690.2.b.b | 6 | 5.c | odd | 4 | 2 | ||
| 1690.2.b.c | 6 | 65.h | odd | 4 | 2 | ||
| 1690.2.c.b | 6 | 65.f | even | 4 | 2 | ||
| 1690.2.c.c | 6 | 65.k | even | 4 | 2 | ||
| 8450.2.a.bt | 3 | 5.b | even | 2 | 1 | ||
| 8450.2.a.bu | 3 | 13.b | even | 2 | 1 | ||
| 8450.2.a.ca | 3 | 1.a | even | 1 | 1 | trivial | |
| 8450.2.a.cb | 3 | 65.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):
|
\( T_{3}^{3} - 4T_{3} + 2 \)
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\( T_{7}^{3} + 5T_{7}^{2} + 3T_{7} - 5 \)
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\( T_{11}^{3} + 3T_{11}^{2} - 27T_{11} - 31 \)
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\( T_{17}^{3} + 4T_{17}^{2} + 2T_{17} - 2 \)
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\( T_{31}^{3} + 6T_{31}^{2} - 58T_{31} - 218 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{3} \)
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| $3$ |
\( T^{3} - 4T + 2 \)
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| $5$ |
\( T^{3} \)
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| $7$ |
\( T^{3} + 5 T^{2} + \cdots - 5 \)
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| $11$ |
\( T^{3} + 3 T^{2} + \cdots - 31 \)
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| $13$ |
\( T^{3} \)
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| $17$ |
\( T^{3} + 4 T^{2} + \cdots - 2 \)
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| $19$ |
\( T^{3} - 13 T^{2} + \cdots - 5 \)
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| $23$ |
\( T^{3} - 40T - 76 \)
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| $29$ |
\( T^{3} + 14 T^{2} + \cdots - 152 \)
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| $31$ |
\( T^{3} + 6 T^{2} + \cdots - 218 \)
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| $37$ |
\( T^{3} + 5 T^{2} + \cdots - 107 \)
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| $41$ |
\( T^{3} + 2 T^{2} + \cdots - 20 \)
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| $43$ |
\( (T + 2)^{3} \)
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| $47$ |
\( T^{3} + 7 T^{2} + \cdots + 1 \)
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| $53$ |
\( T^{3} + 9 T^{2} + \cdots - 13 \)
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| $59$ |
\( T^{3} + 8 T^{2} + \cdots - 272 \)
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| $61$ |
\( T^{3} - 4 T^{2} + \cdots + 610 \)
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| $67$ |
\( T^{3} + 12 T^{2} + \cdots + 16 \)
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| $71$ |
\( T^{3} - 20 T^{2} + \cdots + 464 \)
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| $73$ |
\( T^{3} + 6 T^{2} + \cdots + 270 \)
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| $79$ |
\( T^{3} + 14 T^{2} + \cdots - 158 \)
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| $83$ |
\( T^{3} + 4 T^{2} + \cdots - 46 \)
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| $89$ |
\( T^{3} + 7 T^{2} + \cdots - 19 \)
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| $97$ |
\( T^{3} + 26 T^{2} + \cdots + 466 \)
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