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: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.32716729.1 |
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| Defining polynomial: |
\( x^{6} - x^{5} - 7x^{4} + 3x^{3} + 8x^{2} - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{6} - x^{5} - 7x^{4} + 3x^{3} + 8x^{2} - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 3\nu^{2} + 7\nu \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 3 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 9\nu^{2} + 6\nu - 3 \)
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| \(\beta_{5}\) | \(=\) |
\( 2\nu^{5} - 3\nu^{4} - 12\nu^{3} + 11\nu^{2} + 8\nu - 2 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta_{2} + 2\beta _1 + 4 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 3\beta_{3} - 4\beta_{2} + \beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{5} + 7\beta_{4} + 11\beta_{3} - 11\beta_{2} + 14\beta _1 + 20 ) / 2 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 16\beta_{5} + 25\beta_{4} + 43\beta_{3} - 55\beta_{2} + 22\beta _1 + 22 ) / 2 \)
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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 |
|
−2.50742 | −1.00000 | 4.28717 | −1.00000 | 2.50742 | −4.28717 | −5.73491 | 1.00000 | 2.50742 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.66801 | −1.00000 | 0.782242 | −1.00000 | 1.66801 | −0.782242 | 2.03123 | 1.00000 | 1.66801 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.278809 | −1.00000 | −1.92227 | −1.00000 | −0.278809 | 1.92227 | −1.09356 | 1.00000 | −0.278809 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.551334 | −1.00000 | −1.69603 | −1.00000 | −0.551334 | 1.69603 | −2.03775 | 1.00000 | −0.551334 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.94013 | −1.00000 | 1.76410 | −1.00000 | −1.94013 | −1.76410 | −0.457680 | 1.00000 | −1.94013 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.40516 | −1.00000 | 3.78478 | −1.00000 | −2.40516 | −3.78478 | 4.29268 | 1.00000 | −2.40516 | |||||||||||||||||||||||||||||||||||||
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.l | ✓ | 6 |
| 3.b | odd | 2 | 1 | 3555.2.a.r | 6 | ||
| 5.b | even | 2 | 1 | 5925.2.a.q | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1185.2.a.l | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 3555.2.a.r | 6 | 3.b | odd | 2 | 1 | ||
| 5925.2.a.q | 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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\( T_{2}^{6} - T_{2}^{5} - 9T_{2}^{4} + 9T_{2}^{3} + 17T_{2}^{2} - 16T_{2} + 3 \)
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\( T_{7}^{6} + 7T_{7}^{5} + 3T_{7}^{4} - 51T_{7}^{3} - 43T_{7}^{2} + 90T_{7} + 73 \)
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\( T_{11}^{6} - 8T_{11}^{5} - 14T_{11}^{4} + 306T_{11}^{3} - 1048T_{11}^{2} + 1366T_{11} - 599 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} - 9 T^{4} + \cdots + 3 \)
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| $3$ |
\( (T + 1)^{6} \)
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| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} + 7 T^{5} + \cdots + 73 \)
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| $11$ |
\( T^{6} - 8 T^{5} + \cdots - 599 \)
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| $13$ |
\( T^{6} + 16 T^{5} + \cdots - 823 \)
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| $17$ |
\( T^{6} + 2 T^{5} + \cdots + 1933 \)
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| $19$ |
\( T^{6} + 2 T^{5} + \cdots - 16 \)
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| $23$ |
\( T^{6} + 4 T^{5} + \cdots - 1153 \)
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| $29$ |
\( T^{6} - 11 T^{5} + \cdots - 27625 \)
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| $31$ |
\( T^{6} + 12 T^{5} + \cdots + 4828 \)
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| $37$ |
\( T^{6} + 16 T^{5} + \cdots - 68 \)
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| $41$ |
\( T^{6} + 11 T^{5} + \cdots - 108 \)
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| $43$ |
\( T^{6} + 3 T^{5} + \cdots - 657 \)
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| $47$ |
\( T^{6} + 5 T^{5} + \cdots - 156 \)
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| $53$ |
\( T^{6} - T^{5} + \cdots - 140292 \)
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| $59$ |
\( T^{6} + 15 T^{5} + \cdots - 884 \)
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| $61$ |
\( T^{6} + 4 T^{5} + \cdots + 10704 \)
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| $67$ |
\( T^{6} + 13 T^{5} + \cdots - 1060 \)
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| $71$ |
\( T^{6} - 212 T^{4} + \cdots - 63120 \)
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| $73$ |
\( T^{6} + 28 T^{5} + \cdots - 71135 \)
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| $79$ |
\( (T + 1)^{6} \)
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| $83$ |
\( T^{6} + 21 T^{5} + \cdots - 15075 \)
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| $89$ |
\( T^{6} + 20 T^{5} + \cdots + 56796 \)
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| $97$ |
\( T^{6} + 31 T^{5} + \cdots + 1152905 \)
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