Newspace parameters
| Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
| Weight: | \( k \) | \(=\) | \( 30 \) |
| Character orbit: | \([\chi]\) | \(=\) | 32.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(170.489735626\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - 2 x^{5} - 13837273804 x^{4} + 292564574486272 x^{3} + \cdots + 15\!\cdots\!73 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{74}\cdot 3^{10}\cdot 5^{3} \) |
| 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
\( x^{6} - 2 x^{5} - 13837273804 x^{4} + 292564574486272 x^{3} + \cdots + 15\!\cdots\!73 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 23\!\cdots\!08 \nu^{5} + \cdots - 17\!\cdots\!12 ) / 49\!\cdots\!43 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 90\!\cdots\!60 \nu^{5} + \cdots - 68\!\cdots\!40 ) / 14\!\cdots\!29 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 51\!\cdots\!20 \nu^{5} + \cdots + 36\!\cdots\!28 ) / 16\!\cdots\!81 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 38\!\cdots\!88 \nu^{5} + \cdots - 27\!\cdots\!24 ) / 36\!\cdots\!67 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 22\!\cdots\!08 \nu^{5} + \cdots + 16\!\cdots\!56 ) / 36\!\cdots\!67 \)
|
| \(\nu\) | \(=\) |
\( ( 5\beta_{3} - 261\beta_{2} + 368640\beta _1 + 23592960 ) / 70778880 \)
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| \(\nu^{2}\) | \(=\) |
\( ( - 1521 \beta_{5} + 20673 \beta_{4} + 20210560 \beta_{3} + 59923584 \beta_{2} + \cdots + 34\!\cdots\!40 ) / 754974720 \)
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| \(\nu^{3}\) | \(=\) |
\( ( - 19978569 \beta_{5} - 60655666023 \beta_{4} - 1844291021440 \beta_{3} - 36611021559168 \beta_{2} + \cdots - 16\!\cdots\!20 ) / 1132462080 \)
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| \(\nu^{4}\) | \(=\) |
\( ( - 4646158470087 \beta_{5} + \cdots + 92\!\cdots\!00 ) / 251658240 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 12\!\cdots\!50 \beta_{5} + \cdots - 14\!\cdots\!40 ) / 70778880 \)
|
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 |
|
0 | −1.56376e7 | 0 | −5.35826e9 | 0 | 3.47372e12 | 0 | 1.75904e14 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −8.95536e6 | 0 | −1.80943e9 | 0 | −2.14818e12 | 0 | 1.15682e13 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −6.04355e6 | 0 | 1.73275e10 | 0 | 6.92451e11 | 0 | −3.21058e13 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 6.04355e6 | 0 | 1.73275e10 | 0 | −6.92451e11 | 0 | −3.21058e13 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 8.95536e6 | 0 | −1.80943e9 | 0 | 2.14818e12 | 0 | 1.15682e13 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.56376e7 | 0 | −5.35826e9 | 0 | −3.47372e12 | 0 | 1.75904e14 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 32.30.a.b | ✓ | 6 |
| 4.b | odd | 2 | 1 | inner | 32.30.a.b | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.30.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 32.30.a.b | ✓ | 6 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 361257614972928 T_{3}^{4} + \cdots - 71\!\cdots\!00 \)
acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(32))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} + \cdots - 71\!\cdots\!00 \)
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| $5$ |
\( (T^{3} + \cdots - 16\!\cdots\!00)^{2} \)
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| $7$ |
\( T^{6} + \cdots - 26\!\cdots\!00 \)
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| $11$ |
\( T^{6} + \cdots - 61\!\cdots\!00 \)
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| $13$ |
\( (T^{3} + \cdots - 63\!\cdots\!52)^{2} \)
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| $17$ |
\( (T^{3} + \cdots + 33\!\cdots\!88)^{2} \)
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| $19$ |
\( T^{6} + \cdots - 59\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots - 82\!\cdots\!00 \)
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| $29$ |
\( (T^{3} + \cdots + 53\!\cdots\!00)^{2} \)
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| $31$ |
\( T^{6} + \cdots - 27\!\cdots\!00 \)
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| $37$ |
\( (T^{3} + \cdots + 49\!\cdots\!80)^{2} \)
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| $41$ |
\( (T^{3} + \cdots + 65\!\cdots\!00)^{2} \)
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| $43$ |
\( T^{6} + \cdots - 17\!\cdots\!00 \)
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| $47$ |
\( T^{6} + \cdots - 56\!\cdots\!00 \)
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| $53$ |
\( (T^{3} + \cdots - 26\!\cdots\!44)^{2} \)
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| $59$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
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| $61$ |
\( (T^{3} + \cdots - 23\!\cdots\!00)^{2} \)
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| $67$ |
\( T^{6} + \cdots - 44\!\cdots\!00 \)
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| $71$ |
\( T^{6} + \cdots - 50\!\cdots\!00 \)
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| $73$ |
\( (T^{3} + \cdots - 18\!\cdots\!76)^{2} \)
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| $79$ |
\( T^{6} + \cdots - 32\!\cdots\!00 \)
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| $83$ |
\( T^{6} + \cdots - 26\!\cdots\!00 \)
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| $89$ |
\( (T^{3} + \cdots - 55\!\cdots\!00)^{2} \)
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| $97$ |
\( (T^{3} + \cdots - 43\!\cdots\!12)^{2} \)
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