Newspace parameters
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(159.816725712\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - 2 x^{5} - 18150291 x^{4} + 1485708004 x^{3} + 82508901381652 x^{2} + \cdots - 13\!\cdots\!48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{31}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 56) |
| 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
\( x^{6} - 2 x^{5} - 18150291 x^{4} + 1485708004 x^{3} + 82508901381652 x^{2} + \cdots - 13\!\cdots\!48 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1641724279 \nu^{5} + 4451440996944 \nu^{4} + \cdots + 62\!\cdots\!98 ) / 95\!\cdots\!66 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1641724279 \nu^{5} + 4451440996944 \nu^{4} + \cdots - 76\!\cdots\!80 ) / 31\!\cdots\!22 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 4749130364 \nu^{5} - 19953760999536 \nu^{4} + \cdots - 52\!\cdots\!49 ) / 67\!\cdots\!19 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 216401459815 \nu^{5} - 46633036605840 \nu^{4} + \cdots + 91\!\cdots\!54 ) / 95\!\cdots\!66 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} - 239\beta _1 + 24200310 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 321\beta_{5} + 1750\beta_{4} - 363\beta_{3} + 114274\beta_{2} + 36191197\beta _1 - 5785528544 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 826377 \beta_{5} - 10838047 \beta_{4} + 39126143 \beta_{3} - 447378412 \beta_{2} + \cdots + 876843997280852 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1433729403 \beta_{5} + 28617407946 \beta_{4} - 8422424597 \beta_{3} + 1836188281482 \beta_{2} + \cdots - 11\!\cdots\!96 ) / 8 \)
|
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 | −6174.26 | 0 | −150875. | 0 | −823543. | 0 | 2.37726e7 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −5990.17 | 0 | 337573. | 0 | −823543. | 0 | 2.15333e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1119.43 | 0 | 165118. | 0 | −823543. | 0 | −1.30958e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 570.890 | 0 | −37697.4 | 0 | −823543. | 0 | −1.40230e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 5328.00 | 0 | −204876. | 0 | −823543. | 0 | 1.40387e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 6432.97 | 0 | 147546. | 0 | −823543. | 0 | 2.70342e7 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 112.16.a.l | 6 | |
| 4.b | odd | 2 | 1 | 56.16.a.c | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.16.a.c | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 112.16.a.l | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 952 T_{3}^{5} - 72223544 T_{3}^{4} - 57689714880 T_{3}^{3} + \cdots - 81\!\cdots\!28 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(112))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} + \cdots - 81\!\cdots\!28 \)
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| $5$ |
\( T^{6} + \cdots - 95\!\cdots\!00 \)
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| $7$ |
\( (T + 823543)^{6} \)
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| $11$ |
\( T^{6} + \cdots + 81\!\cdots\!80 \)
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| $13$ |
\( T^{6} + \cdots - 47\!\cdots\!84 \)
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| $17$ |
\( T^{6} + \cdots - 14\!\cdots\!20 \)
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| $19$ |
\( T^{6} + \cdots - 69\!\cdots\!24 \)
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| $23$ |
\( T^{6} + \cdots + 54\!\cdots\!12 \)
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| $29$ |
\( T^{6} + \cdots - 88\!\cdots\!88 \)
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| $31$ |
\( T^{6} + \cdots - 58\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots - 15\!\cdots\!04 \)
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| $41$ |
\( T^{6} + \cdots - 47\!\cdots\!64 \)
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| $43$ |
\( T^{6} + \cdots - 31\!\cdots\!44 \)
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| $47$ |
\( T^{6} + \cdots + 15\!\cdots\!92 \)
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| $53$ |
\( T^{6} + \cdots - 12\!\cdots\!52 \)
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| $59$ |
\( T^{6} + \cdots + 28\!\cdots\!36 \)
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| $61$ |
\( T^{6} + \cdots + 40\!\cdots\!80 \)
|
| $67$ |
\( T^{6} + \cdots - 54\!\cdots\!36 \)
|
| $71$ |
\( T^{6} + \cdots - 35\!\cdots\!64 \)
|
| $73$ |
\( T^{6} + \cdots + 32\!\cdots\!80 \)
|
| $79$ |
\( T^{6} + \cdots + 60\!\cdots\!28 \)
|
| $83$ |
\( T^{6} + \cdots - 35\!\cdots\!24 \)
|
| $89$ |
\( T^{6} + \cdots + 19\!\cdots\!84 \)
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| $97$ |
\( T^{6} + \cdots - 90\!\cdots\!32 \)
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