Newspace parameters
| Level: | \( N \) | \(=\) | \( 1519 = 7^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1519.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(12.1292760670\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 8x^{5} + 17x^{3} - 3x^{2} - 9x + 3 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 217) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 8x^{5} + 17x^{3} - 3x^{2} - 9x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} - \nu^{2} + 6\nu \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 7\nu^{4} - \nu^{3} + 11\nu^{2} + \nu - 3 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 7\nu^{4} - \nu^{3} + 12\nu^{2} + \nu - 5 \)
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| \(\beta_{6}\) | \(=\) |
\( \nu^{6} + \nu^{5} - 8\nu^{4} - 7\nu^{3} + 16\nu^{2} + 8\nu - 8 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{2} + 4\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + 6\beta_{5} - 5\beta_{4} + \beta_{3} + \beta _1 + 7 \)
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| \(\nu^{5}\) | \(=\) |
\( 7\beta_{5} - 7\beta_{4} + \beta_{3} + 6\beta_{2} + 18\beta _1 + 2 \)
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| \(\nu^{6}\) | \(=\) |
\( -7\beta_{6} + 32\beta_{5} - 24\beta_{4} + 7\beta_{3} + \beta_{2} + 10\beta _1 + 30 \)
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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 |
|
−2.75716 | 1.78161 | 5.60193 | 2.11666 | −4.91219 | 0 | −9.93111 | 0.174145 | −5.83598 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.95889 | −2.30691 | 1.83726 | 0.193728 | 4.51899 | 0 | 0.318791 | 2.32183 | −0.379493 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.20176 | −0.383711 | −0.555767 | 1.82438 | 0.461129 | 0 | 3.07142 | −2.85277 | −2.19247 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.668509 | −1.31306 | −1.55310 | −3.52580 | 0.877790 | 0 | 2.37528 | −1.27588 | 2.35703 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.250454 | 2.04580 | −1.93727 | −0.922133 | 0.512379 | 0 | −0.986105 | 1.18530 | −0.230952 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.52143 | −0.757989 | 0.314764 | 1.61060 | −1.15323 | 0 | −2.56398 | −2.42545 | 2.45042 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.81444 | 0.934251 | 1.29218 | −2.29744 | 1.69514 | 0 | −1.28430 | −2.12718 | −4.16856 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(31\) | \( -1 \) |
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 | 1519.2.a.f | 7 | |
| 7.b | odd | 2 | 1 | 1519.2.a.g | 7 | ||
| 7.d | odd | 6 | 2 | 217.2.f.a | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 217.2.f.a | ✓ | 14 | 7.d | odd | 6 | 2 | |
| 1519.2.a.f | 7 | 1.a | even | 1 | 1 | trivial | |
| 1519.2.a.g | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1519))\):
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\( T_{2}^{7} + 3T_{2}^{6} - 5T_{2}^{5} - 17T_{2}^{4} + 4T_{2}^{3} + 24T_{2}^{2} + 6T_{2} - 3 \)
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\( T_{3}^{7} - 8T_{3}^{5} + 17T_{3}^{3} + 3T_{3}^{2} - 9T_{3} - 3 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 3 T^{6} + \cdots - 3 \)
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| $3$ |
\( T^{7} - 8 T^{5} + \cdots - 3 \)
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| $5$ |
\( T^{7} + T^{6} - 14 T^{5} + \cdots + 9 \)
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| $7$ |
\( T^{7} \)
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| $11$ |
\( T^{7} + 13 T^{6} + \cdots - 1 \)
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| $13$ |
\( T^{7} - 4 T^{6} + \cdots - 203 \)
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| $17$ |
\( T^{7} + 2 T^{6} + \cdots + 479 \)
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| $19$ |
\( T^{7} + 2 T^{6} + \cdots - 183 \)
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| $23$ |
\( T^{7} + 10 T^{6} + \cdots + 2007 \)
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| $29$ |
\( T^{7} + 22 T^{6} + \cdots + 97 \)
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| $31$ |
\( (T - 1)^{7} \)
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| $37$ |
\( T^{7} + 12 T^{6} + \cdots - 353 \)
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| $41$ |
\( T^{7} - 4 T^{6} + \cdots + 10451 \)
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| $43$ |
\( T^{7} + 3 T^{6} + \cdots - 17979 \)
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| $47$ |
\( T^{7} + 10 T^{6} + \cdots + 21211 \)
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| $53$ |
\( T^{7} + 9 T^{6} + \cdots - 2692637 \)
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| $59$ |
\( T^{7} + 11 T^{6} + \cdots + 195657 \)
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| $61$ |
\( T^{7} + 13 T^{6} + \cdots + 194127 \)
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| $67$ |
\( T^{7} - 19 T^{6} + \cdots - 173567 \)
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| $71$ |
\( T^{7} + 20 T^{6} + \cdots + 5267219 \)
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| $73$ |
\( T^{7} - 3 T^{6} + \cdots - 3982183 \)
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| $79$ |
\( T^{7} + 4 T^{6} + \cdots + 1006447 \)
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| $83$ |
\( T^{7} - 20 T^{6} + \cdots - 231721 \)
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| $89$ |
\( T^{7} - 3 T^{6} + \cdots + 11331993 \)
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| $97$ |
\( T^{7} - 3 T^{6} + \cdots + 22421 \)
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