Newspace parameters
| Level: | \( N \) | \(=\) | \( 5720 = 2^{3} \cdot 5 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5720.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(45.6744299562\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 2x^{6} - 13x^{5} + 15x^{4} + 50x^{3} - 11x^{2} - 28x + 10 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{7} - 2x^{6} - 13x^{5} + 15x^{4} + 50x^{3} - 11x^{2} - 28x + 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( -4\nu^{6} + 6\nu^{5} + 55\nu^{4} - 33\nu^{3} - 216\nu^{2} - 60\nu + 82 \)
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| \(\beta_{4}\) | \(=\) |
\( -7\nu^{6} + 11\nu^{5} + 95\nu^{4} - 62\nu^{3} - 370\nu^{2} - 98\nu + 137 \)
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| \(\beta_{5}\) | \(=\) |
\( -8\nu^{6} + 12\nu^{5} + 110\nu^{4} - 65\nu^{3} - 433\nu^{2} - 128\nu + 164 \)
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| \(\beta_{6}\) | \(=\) |
\( 10\nu^{6} - 15\nu^{5} - 138\nu^{4} + 82\nu^{3} + 545\nu^{2} + 157\nu - 207 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{3} + \beta_{2} + 9\beta _1 + 4 \)
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| \(\nu^{4}\) | \(=\) |
\( -2\beta_{6} - \beta_{5} - 3\beta_{3} + 9\beta_{2} + 15\beta _1 + 32 \)
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| \(\nu^{5}\) | \(=\) |
\( -5\beta_{6} + 6\beta_{5} + 2\beta_{4} - 28\beta_{3} + 15\beta_{2} + 84\beta _1 + 63 \)
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| \(\nu^{6}\) | \(=\) |
\( -35\beta_{6} - 13\beta_{5} + 3\beta_{4} - 67\beta_{3} + 84\beta_{2} + 189\beta _1 + 306 \)
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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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0 | −3.46011 | 0 | 1.00000 | 0 | 1.85879 | 0 | 8.97235 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.68685 | 0 | 1.00000 | 0 | −3.36386 | 0 | 4.21918 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.05350 | 0 | 1.00000 | 0 | −4.65418 | 0 | 1.21686 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.518227 | 0 | 1.00000 | 0 | 0.155278 | 0 | −2.73144 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.506310 | 0 | 1.00000 | 0 | 3.43169 | 0 | −2.74365 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.84661 | 0 | 1.00000 | 0 | −0.347722 | 0 | 0.409962 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.37839 | 0 | 1.00000 | 0 | −4.08000 | 0 | 2.65674 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 5720.2.a.u | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5720.2.a.u | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5720))\):
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\( T_{3}^{7} + 5T_{3}^{6} - 4T_{3}^{5} - 45T_{3}^{4} - 25T_{3}^{3} + 90T_{3}^{2} + 90T_{3} + 22 \)
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\( T_{7}^{7} + 7T_{7}^{6} - 8T_{7}^{5} - 117T_{7}^{4} - 51T_{7}^{3} + 408T_{7}^{2} + 80T_{7} - 22 \)
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\( T_{17}^{7} - 5T_{17}^{6} - 64T_{17}^{5} + 280T_{17}^{4} + 987T_{17}^{3} - 3975T_{17}^{2} + 164T_{17} + 268 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
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| $3$ |
\( T^{7} + 5 T^{6} + \cdots + 22 \)
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| $5$ |
\( (T - 1)^{7} \)
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| $7$ |
\( T^{7} + 7 T^{6} + \cdots - 22 \)
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| $11$ |
\( (T - 1)^{7} \)
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| $13$ |
\( (T + 1)^{7} \)
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| $17$ |
\( T^{7} - 5 T^{6} + \cdots + 268 \)
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| $19$ |
\( T^{7} + 12 T^{6} + \cdots - 16847 \)
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| $23$ |
\( T^{7} + 17 T^{6} + \cdots + 80 \)
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| $29$ |
\( T^{7} + 8 T^{6} + \cdots - 384568 \)
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| $31$ |
\( T^{7} + 12 T^{6} + \cdots - 75328 \)
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| $37$ |
\( T^{7} + 11 T^{6} + \cdots + 157348 \)
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| $41$ |
\( T^{7} - 14 T^{6} + \cdots - 7666 \)
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| $43$ |
\( T^{7} + 5 T^{6} + \cdots + 2888 \)
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| $47$ |
\( T^{7} + 9 T^{6} + \cdots - 117688 \)
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| $53$ |
\( T^{7} + 13 T^{6} + \cdots + 608 \)
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| $59$ |
\( T^{7} + 40 T^{6} + \cdots + 1698400 \)
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| $61$ |
\( T^{7} + 6 T^{6} + \cdots + 759160 \)
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| $67$ |
\( T^{7} + 23 T^{6} + \cdots - 505552 \)
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| $71$ |
\( T^{7} + 8 T^{6} + \cdots - 2200 \)
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| $73$ |
\( T^{7} - 15 T^{6} + \cdots + 323152 \)
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| $79$ |
\( T^{7} + 6 T^{6} + \cdots - 32896 \)
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| $83$ |
\( T^{7} + 21 T^{6} + \cdots + 19072 \)
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| $89$ |
\( T^{7} - 16 T^{6} + \cdots - 4120192 \)
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| $97$ |
\( T^{7} - 7 T^{6} + \cdots + 16672 \)
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