Newspace parameters
| Level: | \( N \) | \(=\) | \( 2890 = 2 \cdot 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2890.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(23.0767661842\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.37902897.1 |
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| Defining polynomial: |
\( x^{6} - 15x^{4} - x^{3} + 60x^{2} - 64 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 15x^{4} - x^{3} + 60x^{2} - 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 15\nu^{3} + \nu^{2} - 44\nu ) / 16 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 11\nu^{2} - \nu + 20 ) / 4 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{5} - 59\nu^{3} - 5\nu^{2} + 124\nu - 16 ) / 16 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{5} - 4\nu^{4} + 59\nu^{3} + 65\nu^{2} - 120\nu - 144 ) / 16 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} + 6\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( 11\beta_{5} + 11\beta_{4} + 15\beta_{3} + \beta _1 + 35 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 16\beta_{4} + \beta_{3} + 59\beta_{2} + 46\beta _1 + 20 \)
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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 | −3.14945 | 1.00000 | 1.00000 | −3.14945 | −0.922767 | 1.00000 | 6.91903 | 1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.83388 | 1.00000 | 1.00000 | −1.83388 | −0.649084 | 1.00000 | 0.363102 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.37564 | 1.00000 | 1.00000 | −1.37564 | −4.78712 | 1.00000 | −1.10761 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.27006 | 1.00000 | 1.00000 | 1.27006 | 3.49675 | 1.00000 | −1.38694 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.18117 | 1.00000 | 1.00000 | 2.18117 | 3.36596 | 1.00000 | 1.75751 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.90773 | 1.00000 | 1.00000 | 2.90773 | −0.503742 | 1.00000 | 5.45490 | 1.00000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 2890.2.a.bh | yes | 6 |
| 17.b | even | 2 | 1 | 2890.2.a.bg | ✓ | 6 | |
| 17.c | even | 4 | 2 | 2890.2.b.q | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2890.2.a.bg | ✓ | 6 | 17.b | even | 2 | 1 | |
| 2890.2.a.bh | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 2890.2.b.q | 12 | 17.c | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2890))\):
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\( T_{3}^{6} - 15T_{3}^{4} + T_{3}^{3} + 60T_{3}^{2} - 64 \)
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\( T_{7}^{6} - 24T_{7}^{4} + 10T_{7}^{3} + 87T_{7}^{2} + 72T_{7} + 17 \)
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\( T_{13}^{6} - 12T_{13}^{5} + 27T_{13}^{4} + 121T_{13}^{3} - 486T_{13}^{2} + 504T_{13} - 136 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
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| $3$ |
\( T^{6} - 15 T^{4} + \cdots - 64 \)
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| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( T^{6} - 24 T^{4} + \cdots + 17 \)
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| $11$ |
\( T^{6} - 6 T^{5} + \cdots - 296 \)
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| $13$ |
\( T^{6} - 12 T^{5} + \cdots - 136 \)
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| $17$ |
\( T^{6} \)
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| $19$ |
\( T^{6} + 9 T^{5} + \cdots + 584 \)
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| $23$ |
\( T^{6} + 6 T^{5} + \cdots - 703 \)
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| $29$ |
\( T^{6} + 9 T^{5} + \cdots + 1864 \)
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| $31$ |
\( T^{6} - 18 T^{5} + \cdots + 88768 \)
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| $37$ |
\( T^{6} - 3 T^{5} + \cdots + 136 \)
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| $41$ |
\( T^{6} - 9 T^{5} + \cdots - 20197 \)
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| $43$ |
\( T^{6} + 9 T^{5} + \cdots - 2312 \)
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| $47$ |
\( T^{6} + 9 T^{5} + \cdots - 4913 \)
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| $53$ |
\( T^{6} - 24 T^{5} + \cdots + 2368 \)
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| $59$ |
\( T^{6} - 24 T^{5} + \cdots + 1589464 \)
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| $61$ |
\( T^{6} + 3 T^{5} + \cdots - 32104 \)
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| $67$ |
\( T^{6} - 18 T^{5} + \cdots - 223624 \)
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| $71$ |
\( T^{6} + 6 T^{5} + \cdots - 42048 \)
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| $73$ |
\( T^{6} + 18 T^{5} + \cdots + 40256 \)
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| $79$ |
\( (T^{3} - 6 T^{2} + \cdots + 456)^{2} \)
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| $83$ |
\( T^{6} - 18 T^{5} + \cdots + 80648 \)
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| $89$ |
\( T^{6} - 24 T^{5} + \cdots - 242783 \)
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| $97$ |
\( T^{6} + 18 T^{5} + \cdots - 9792 \)
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