Newspace parameters
| Level: | \( N \) | \(=\) | \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6525.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(52.1023873189\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.337383424.1 |
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| Defining polynomial: |
\( x^{6} - 13x^{4} + 41x^{2} - 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 145) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 13x^{4} + 41x^{2} - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 8\nu^{2} + 5 ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 12\nu^{3} + 35\nu ) / 2 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 14\nu^{3} - 47\nu ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 6\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} + 8\beta_{2} + 27 \)
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| \(\nu^{5}\) | \(=\) |
\( 12\beta_{5} + 14\beta_{4} + 37\beta_1 \)
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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 |
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−2.77035 | 0 | 5.67486 | 0 | 0 | 1.86960 | −10.1807 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.30229 | 0 | 3.30056 | 0 | 0 | −3.91261 | −2.99427 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.156785 | 0 | −1.97542 | 0 | 0 | −1.09364 | 0.623285 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.156785 | 0 | −1.97542 | 0 | 0 | 1.09364 | −0.623285 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.30229 | 0 | 3.30056 | 0 | 0 | 3.91261 | 2.99427 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.77035 | 0 | 5.67486 | 0 | 0 | −1.86960 | 10.1807 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(29\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6525.2.a.bt | 6 | |
| 3.b | odd | 2 | 1 | 725.2.a.l | 6 | ||
| 5.b | even | 2 | 1 | inner | 6525.2.a.bt | 6 | |
| 5.c | odd | 4 | 2 | 1305.2.c.h | 6 | ||
| 15.d | odd | 2 | 1 | 725.2.a.l | 6 | ||
| 15.e | even | 4 | 2 | 145.2.b.c | ✓ | 6 | |
| 60.l | odd | 4 | 2 | 2320.2.d.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 145.2.b.c | ✓ | 6 | 15.e | even | 4 | 2 | |
| 725.2.a.l | 6 | 3.b | odd | 2 | 1 | ||
| 725.2.a.l | 6 | 15.d | odd | 2 | 1 | ||
| 1305.2.c.h | 6 | 5.c | odd | 4 | 2 | ||
| 2320.2.d.g | 6 | 60.l | odd | 4 | 2 | ||
| 6525.2.a.bt | 6 | 1.a | even | 1 | 1 | trivial | |
| 6525.2.a.bt | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6525))\):
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\( T_{2}^{6} - 13T_{2}^{4} + 41T_{2}^{2} - 1 \)
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\( T_{7}^{6} - 20T_{7}^{4} + 76T_{7}^{2} - 64 \)
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\( T_{11}^{3} - 5T_{11}^{2} - 6T_{11} + 38 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 13 T^{4} + \cdots - 1 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} - 20 T^{4} + \cdots - 64 \)
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| $11$ |
\( (T^{3} - 5 T^{2} - 6 T + 38)^{2} \)
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| $13$ |
\( T^{6} - 59 T^{4} + \cdots - 5776 \)
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| $17$ |
\( T^{6} - 48 T^{4} + \cdots - 784 \)
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| $19$ |
\( (T^{3} - 8 T^{2} + 6 T + 8)^{2} \)
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| $23$ |
\( T^{6} - 56 T^{4} + \cdots - 16 \)
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| $29$ |
\( (T - 1)^{6} \)
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| $31$ |
\( (T^{3} - 11 T^{2} + \cdots + 22)^{2} \)
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| $37$ |
\( T^{6} - 20 T^{4} + \cdots - 64 \)
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| $41$ |
\( T^{6} \)
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| $43$ |
\( T^{6} - 127 T^{4} + \cdots - 196 \)
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| $47$ |
\( T^{6} - 63 T^{4} + \cdots - 8836 \)
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| $53$ |
\( T^{6} - 187 T^{4} + \cdots - 5776 \)
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| $59$ |
\( (T + 10)^{6} \)
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| $61$ |
\( (T^{3} - 6 T^{2} - 64 T - 64)^{2} \)
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| $67$ |
\( T^{6} - 140 T^{4} + \cdots - 92416 \)
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| $71$ |
\( (T + 2)^{6} \)
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| $73$ |
\( T^{6} - 188 T^{4} + \cdots - 3136 \)
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| $79$ |
\( (T^{3} - T^{2} - 54 T - 98)^{2} \)
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| $83$ |
\( T^{6} - 48 T^{4} + \cdots - 784 \)
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| $89$ |
\( (T^{3} - 16 T^{2} + \cdots + 304)^{2} \)
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| $97$ |
\( T^{6} - 476 T^{4} + \cdots - 7744 \)
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