Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 76 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(320.605553540\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{6} - 3 x^{5} + \cdots - 67\!\cdots\!50 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{58}\cdot 3^{36}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 13\cdot 19 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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
\( x^{6} - 3 x^{5} + \cdots - 67\!\cdots\!50 \)
:
| \(\beta_{1}\) | \(=\) |
\( 72\nu - 36 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 49\!\cdots\!53 \nu^{5} + \cdots + 28\!\cdots\!58 ) / 10\!\cdots\!92 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 49\!\cdots\!53 \nu^{5} + \cdots - 99\!\cdots\!10 ) / 10\!\cdots\!92 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 28\!\cdots\!31 \nu^{5} + \cdots - 25\!\cdots\!70 ) / 33\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13\!\cdots\!79 \nu^{5} + \cdots + 74\!\cdots\!30 ) / 26\!\cdots\!48 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 36 ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 28859335836\beta _1 + 66455170111740773427552 ) / 5184 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 927 \beta_{5} - 5496670 \beta_{4} + 13763504113 \beta_{3} + 13847230069 \beta_{2} + \cdots + 23\!\cdots\!88 ) / 46656 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 19416283325423 \beta_{5} + \cdots + 10\!\cdots\!72 ) / 419904 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 13\!\cdots\!96 \beta_{5} + \cdots + 53\!\cdots\!84 ) / 1889568 \)
|
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 |
|
−3.67444e11 | 0 | 9.72365e22 | −2.43619e26 | 0 | 1.21530e31 | −2.18473e34 | 0 | 8.95165e37 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.20777e11 | 0 | 1.09637e22 | 2.30711e25 | 0 | 1.96027e31 | 5.92020e33 | 0 | −5.09358e36 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −9.60898e10 | 0 | −2.85457e22 | 1.49650e26 | 0 | −3.16100e31 | 6.37312e33 | 0 | −1.43799e37 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.25000e11 | 0 | −2.21540e22 | −1.80927e26 | 0 | 2.76229e31 | −7.49160e33 | 0 | −2.26158e37 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.89652e11 | 0 | 4.61192e22 | 2.47463e26 | 0 | −6.83079e31 | 2.41577e33 | 0 | 7.16781e37 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 3.26741e11 | 0 | 6.89808e22 | 4.33451e25 | 0 | 4.24638e31 | 1.01949e34 | 0 | 1.41626e37 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
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 | 9.76.a.c | 6 | |
| 3.b | odd | 2 | 1 | 1.76.a.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.76.a.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 9.76.a.c | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 57080822040 T_{2}^{5} + \cdots - 92\!\cdots\!16 \)
acting on \(S_{76}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 92\!\cdots\!16 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
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| $7$ |
\( T^{6} + \cdots + 60\!\cdots\!44 \)
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| $11$ |
\( T^{6} + \cdots - 76\!\cdots\!36 \)
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| $13$ |
\( T^{6} + \cdots + 87\!\cdots\!16 \)
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| $17$ |
\( T^{6} + \cdots - 35\!\cdots\!36 \)
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| $19$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!96 \)
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| $29$ |
\( T^{6} + \cdots + 99\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 14\!\cdots\!36 \)
|
| $37$ |
\( T^{6} + \cdots - 18\!\cdots\!96 \)
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| $41$ |
\( T^{6} + \cdots + 16\!\cdots\!64 \)
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| $43$ |
\( T^{6} + \cdots + 52\!\cdots\!56 \)
|
| $47$ |
\( T^{6} + \cdots + 11\!\cdots\!24 \)
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| $53$ |
\( T^{6} + \cdots + 83\!\cdots\!36 \)
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| $59$ |
\( T^{6} + \cdots - 41\!\cdots\!00 \)
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| $61$ |
\( T^{6} + \cdots + 57\!\cdots\!64 \)
|
| $67$ |
\( T^{6} + \cdots + 21\!\cdots\!64 \)
|
| $71$ |
\( T^{6} + \cdots + 14\!\cdots\!64 \)
|
| $73$ |
\( T^{6} + \cdots - 66\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 13\!\cdots\!76 \)
|
| $89$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 36\!\cdots\!24 \)
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