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: | \(\Q(\zeta_{14})^+\) |
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| Defining polynomial: |
\( x^{3} - x^{2} - 2x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 338) |
| 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 \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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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 | −2.69202 | 1.00000 | 0 | 2.69202 | −1.60388 | −1.00000 | 4.24698 | 0 | |||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.35690 | 1.00000 | 0 | 2.35690 | 4.49396 | −1.00000 | 2.55496 | 0 | ||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 2.04892 | 1.00000 | 0 | −2.04892 | 1.10992 | −1.00000 | 1.19806 | 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.bn | 3 | |
| 5.b | even | 2 | 1 | 338.2.a.h | yes | 3 | |
| 13.b | even | 2 | 1 | 8450.2.a.bx | 3 | ||
| 15.d | odd | 2 | 1 | 3042.2.a.z | 3 | ||
| 20.d | odd | 2 | 1 | 2704.2.a.w | 3 | ||
| 65.d | even | 2 | 1 | 338.2.a.g | ✓ | 3 | |
| 65.g | odd | 4 | 2 | 338.2.b.d | 6 | ||
| 65.l | even | 6 | 2 | 338.2.c.i | 6 | ||
| 65.n | even | 6 | 2 | 338.2.c.h | 6 | ||
| 65.s | odd | 12 | 4 | 338.2.e.e | 12 | ||
| 195.e | odd | 2 | 1 | 3042.2.a.bi | 3 | ||
| 195.n | even | 4 | 2 | 3042.2.b.n | 6 | ||
| 260.g | odd | 2 | 1 | 2704.2.a.v | 3 | ||
| 260.u | even | 4 | 2 | 2704.2.f.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 338.2.a.g | ✓ | 3 | 65.d | even | 2 | 1 | |
| 338.2.a.h | yes | 3 | 5.b | even | 2 | 1 | |
| 338.2.b.d | 6 | 65.g | odd | 4 | 2 | ||
| 338.2.c.h | 6 | 65.n | even | 6 | 2 | ||
| 338.2.c.i | 6 | 65.l | even | 6 | 2 | ||
| 338.2.e.e | 12 | 65.s | odd | 12 | 4 | ||
| 2704.2.a.v | 3 | 260.g | odd | 2 | 1 | ||
| 2704.2.a.w | 3 | 20.d | odd | 2 | 1 | ||
| 2704.2.f.m | 6 | 260.u | even | 4 | 2 | ||
| 3042.2.a.z | 3 | 15.d | odd | 2 | 1 | ||
| 3042.2.a.bi | 3 | 195.e | odd | 2 | 1 | ||
| 3042.2.b.n | 6 | 195.n | even | 4 | 2 | ||
| 8450.2.a.bn | 3 | 1.a | even | 1 | 1 | trivial | |
| 8450.2.a.bx | 3 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):
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\( T_{3}^{3} + 3T_{3}^{2} - 4T_{3} - 13 \)
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\( T_{7}^{3} - 4T_{7}^{2} - 4T_{7} + 8 \)
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\( T_{11}^{3} - 3T_{11}^{2} - 4T_{11} + 13 \)
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\( T_{17}^{3} + 5T_{17}^{2} - 22T_{17} - 97 \)
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\( T_{31}^{3} + 16T_{31}^{2} + 76T_{31} + 104 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{3} \)
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| $3$ |
\( T^{3} + 3 T^{2} + \cdots - 13 \)
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| $5$ |
\( T^{3} \)
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| $7$ |
\( T^{3} - 4 T^{2} + \cdots + 8 \)
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| $11$ |
\( T^{3} - 3 T^{2} + \cdots + 13 \)
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| $13$ |
\( T^{3} \)
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| $17$ |
\( T^{3} + 5 T^{2} + \cdots - 97 \)
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| $19$ |
\( T^{3} - T^{2} + \cdots - 13 \)
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| $23$ |
\( T^{3} - 28T - 56 \)
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| $29$ |
\( T^{3} + 10 T^{2} + \cdots - 104 \)
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| $31$ |
\( T^{3} + 16 T^{2} + \cdots + 104 \)
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| $37$ |
\( T^{3} - 14 T^{2} + \cdots - 56 \)
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| $41$ |
\( T^{3} - 7 T^{2} + \cdots + 91 \)
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| $43$ |
\( T^{3} + 11 T^{2} + \cdots - 1 \)
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| $47$ |
\( T^{3} - 2 T^{2} + \cdots + 8 \)
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| $53$ |
\( T^{3} - 28T + 56 \)
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| $59$ |
\( T^{3} + 3 T^{2} + \cdots + 127 \)
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| $61$ |
\( T^{3} + 4 T^{2} + \cdots - 64 \)
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| $67$ |
\( T^{3} + 21 T^{2} + \cdots - 287 \)
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| $71$ |
\( T^{3} + 12 T^{2} + \cdots + 8 \)
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| $73$ |
\( T^{3} + T^{2} + \cdots + 251 \)
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| $79$ |
\( T^{3} + 18 T^{2} + \cdots - 232 \)
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| $83$ |
\( T^{3} - T^{2} + \cdots - 13 \)
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| $89$ |
\( T^{3} - 25 T^{2} + \cdots + 757 \)
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| $97$ |
\( T^{3} + 23 T^{2} + \cdots - 883 \)
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