Newspace parameters
| Level: | \( N \) | \(=\) | \( 1185 = 3 \cdot 5 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1185.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.46227263952\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.51104492.1 |
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| Defining polynomial: |
\( x^{6} - x^{5} - 7x^{4} + 5x^{3} + 11x^{2} - 6x - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{6} - x^{5} - 7x^{4} + 5x^{3} + 11x^{2} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 3 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 6\nu^{3} + 4\nu^{2} + 7\nu - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 13 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 7\beta_{3} + 8\beta_{2} + 18\beta _1 + 10 \)
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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 |
|
−2.45464 | −1.00000 | 4.02527 | −1.00000 | 2.45464 | 2.67184 | −4.97131 | 1.00000 | 2.45464 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.49884 | −1.00000 | 0.246524 | −1.00000 | 1.49884 | 2.45405 | 2.62818 | 1.00000 | 1.49884 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.674868 | −1.00000 | −1.54455 | −1.00000 | 0.674868 | −2.17877 | 2.39210 | 1.00000 | 0.674868 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.135475 | −1.00000 | −1.98165 | −1.00000 | −0.135475 | 2.87873 | −0.539414 | 1.00000 | −0.135475 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.46866 | −1.00000 | 0.156975 | −1.00000 | −1.46866 | −1.71195 | −2.70679 | 1.00000 | −1.46866 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.02421 | −1.00000 | 2.09743 | −1.00000 | −2.02421 | 4.88611 | 0.197224 | 1.00000 | −2.02421 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(79\) | \( -1 \) |
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 | 1185.2.a.j | ✓ | 6 |
| 3.b | odd | 2 | 1 | 3555.2.a.s | 6 | ||
| 5.b | even | 2 | 1 | 5925.2.a.s | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1185.2.a.j | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 3555.2.a.s | 6 | 3.b | odd | 2 | 1 | ||
| 5925.2.a.s | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1185))\):
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\( T_{2}^{6} + T_{2}^{5} - 7T_{2}^{4} - 5T_{2}^{3} + 11T_{2}^{2} + 6T_{2} - 1 \)
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\( T_{7}^{6} - 9T_{7}^{5} + 14T_{7}^{4} + 64T_{7}^{3} - 161T_{7}^{2} - 100T_{7} + 344 \)
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\( T_{11}^{6} + 8T_{11}^{5} - 5T_{11}^{4} - 134T_{11}^{3} - 167T_{11}^{2} + 170T_{11} + 46 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 7 T^{4} + \cdots - 1 \)
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| $3$ |
\( (T + 1)^{6} \)
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| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} - 9 T^{5} + \cdots + 344 \)
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| $11$ |
\( T^{6} + 8 T^{5} + \cdots + 46 \)
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| $13$ |
\( T^{6} - 8 T^{5} + \cdots + 668 \)
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| $17$ |
\( T^{6} - 2 T^{5} + \cdots - 24 \)
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| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 32 \)
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| $23$ |
\( T^{6} - 10 T^{5} + \cdots - 324 \)
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| $29$ |
\( T^{6} + 5 T^{5} + \cdots - 3438 \)
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| $31$ |
\( T^{6} - 12 T^{5} + \cdots + 8 \)
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| $37$ |
\( T^{6} - 10 T^{5} + \cdots - 6246 \)
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| $41$ |
\( T^{6} + 3 T^{5} + \cdots + 414 \)
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| $43$ |
\( T^{6} - 23 T^{5} + \cdots + 1152 \)
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| $47$ |
\( T^{6} - 9 T^{5} + \cdots - 388 \)
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| $53$ |
\( T^{6} + T^{5} + \cdots + 3096 \)
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| $59$ |
\( T^{6} - 3 T^{5} + \cdots - 14624 \)
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| $61$ |
\( T^{6} - 14 T^{5} + \cdots + 49572 \)
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| $67$ |
\( T^{6} - 17 T^{5} + \cdots - 7398 \)
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| $71$ |
\( T^{6} - 16 T^{5} + \cdots + 2270376 \)
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| $73$ |
\( T^{6} - 32 T^{5} + \cdots - 50868 \)
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| $79$ |
\( (T - 1)^{6} \)
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| $83$ |
\( T^{6} - 11 T^{5} + \cdots + 302884 \)
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| $89$ |
\( T^{6} + 4 T^{5} + \cdots - 90084 \)
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| $97$ |
\( T^{6} - 19 T^{5} + \cdots - 4276 \)
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