Newspace parameters
| Level: | \( N \) | \(=\) | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1690.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.4947179416\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.20439713.1 |
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|
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 15x^{4} + 22x^{3} + 70x^{2} - 56x - 91 \)
|
| 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
\( x^{6} - 2x^{5} - 15x^{4} + 22x^{3} + 70x^{2} - 56x - 91 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 6\nu + 1 ) / 4 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 5 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 10\nu^{2} + 5\nu + 20 ) / 4 \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 11\nu^{3} + 6\nu^{2} + 26\nu - 5 ) / 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 7\beta _1 + 4 \)
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| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 11\beta_{3} + 4\beta_{2} + 12\beta _1 + 34 \)
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| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} + 4\beta_{4} + 16\beta_{3} + 48\beta_{2} + 57\beta _1 + 53 \)
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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 | −3.23543 | 1.00000 | −1.00000 | 3.23543 | −4.13802 | −1.00000 | 7.46801 | 1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.79223 | 1.00000 | −1.00000 | 2.79223 | 4.83875 | −1.00000 | 4.79655 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.77061 | 1.00000 | −1.00000 | 1.77061 | −3.83691 | −1.00000 | 0.135043 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.988450 | 1.00000 | −1.00000 | −0.988450 | 3.47315 | −1.00000 | −2.02297 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.23727 | 1.00000 | −1.00000 | −2.23727 | −1.43294 | −1.00000 | 2.00539 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.57254 | 1.00000 | −1.00000 | −2.57254 | −1.90403 | −1.00000 | 3.61798 | 1.00000 | |||||||||||||||||||||||||||||||||||||
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 | 1690.2.a.v | ✓ | 6 |
| 5.b | even | 2 | 1 | 8450.2.a.cq | 6 | ||
| 13.b | even | 2 | 1 | 1690.2.a.w | yes | 6 | |
| 13.c | even | 3 | 2 | 1690.2.e.v | 12 | ||
| 13.d | odd | 4 | 2 | 1690.2.d.l | 12 | ||
| 13.e | even | 6 | 2 | 1690.2.e.u | 12 | ||
| 13.f | odd | 12 | 4 | 1690.2.l.n | 24 | ||
| 65.d | even | 2 | 1 | 8450.2.a.cp | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1690.2.a.v | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1690.2.a.w | yes | 6 | 13.b | even | 2 | 1 | |
| 1690.2.d.l | 12 | 13.d | odd | 4 | 2 | ||
| 1690.2.e.u | 12 | 13.e | even | 6 | 2 | ||
| 1690.2.e.v | 12 | 13.c | even | 3 | 2 | ||
| 1690.2.l.n | 24 | 13.f | odd | 12 | 4 | ||
| 8450.2.a.cp | 6 | 65.d | even | 2 | 1 | ||
| 8450.2.a.cq | 6 | 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1690))\):
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\( T_{3}^{6} + 2T_{3}^{5} - 15T_{3}^{4} - 22T_{3}^{3} + 70T_{3}^{2} + 56T_{3} - 91 \)
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\( T_{7}^{6} + 3T_{7}^{5} - 32T_{7}^{4} - 111T_{7}^{3} + 182T_{7}^{2} + 896T_{7} + 728 \)
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\( T_{11}^{6} + 15T_{11}^{5} + 54T_{11}^{4} - 183T_{11}^{3} - 1540T_{11}^{2} - 2772T_{11} - 728 \)
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\( T_{31}^{6} - 104T_{31}^{4} + 2000T_{31}^{2} + 896T_{31} - 64 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
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| $3$ |
\( T^{6} + 2 T^{5} + \cdots - 91 \)
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| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} + 3 T^{5} + \cdots + 728 \)
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| $11$ |
\( T^{6} + 15 T^{5} + \cdots - 728 \)
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| $13$ |
\( T^{6} \)
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| $17$ |
\( T^{6} + 3 T^{5} + \cdots - 2008 \)
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| $19$ |
\( T^{6} + T^{5} + \cdots - 3016 \)
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| $23$ |
\( T^{6} + 3 T^{5} + \cdots + 1352 \)
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| $29$ |
\( T^{6} - 7 T^{5} + \cdots + 7384 \)
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| $31$ |
\( T^{6} - 104 T^{4} + \cdots - 64 \)
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| $37$ |
\( T^{6} + 6 T^{5} + \cdots + 14848 \)
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| $41$ |
\( T^{6} + 2 T^{5} + \cdots - 91 \)
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| $43$ |
\( T^{6} + 22 T^{5} + \cdots - 14903 \)
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| $47$ |
\( T^{6} + 7 T^{5} + \cdots + 36392 \)
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| $53$ |
\( T^{6} + 16 T^{5} + \cdots - 832 \)
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| $59$ |
\( T^{6} + 15 T^{5} + \cdots + 18824 \)
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| $61$ |
\( T^{6} - 33 T^{5} + \cdots + 6728 \)
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| $67$ |
\( T^{6} - 8 T^{5} + \cdots - 11411 \)
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| $71$ |
\( (T^{3} + 20 T^{2} + 68 T + 8)^{2} \)
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| $73$ |
\( T^{6} + 21 T^{5} + \cdots - 8 \)
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| $79$ |
\( T^{6} - 20 T^{5} + \cdots - 77888 \)
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| $83$ |
\( T^{6} + 22 T^{5} + \cdots - 186641 \)
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| $89$ |
\( T^{6} + 20 T^{5} + \cdots + 3571 \)
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| $97$ |
\( T^{6} - 7 T^{5} + \cdots + 312424 \)
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