Newspace parameters
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 48 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(195.870727717\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 2 x^{5} + \cdots - 16\!\cdots\!84 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{36}\cdot 3^{15}\cdot 5^{6}\cdot 7^{6} \) |
| 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
\( x^{6} - 2 x^{5} + \cdots - 16\!\cdots\!84 \)
:
| \(\beta_{1}\) | \(=\) |
\( 18\nu - 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\!\cdots\!57 \nu^{5} + \cdots + 10\!\cdots\!48 ) / 58\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 14\!\cdots\!41 \nu^{5} + \cdots + 77\!\cdots\!24 ) / 14\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\!\cdots\!01 \nu^{5} + \cdots - 43\!\cdots\!36 ) / 14\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\!\cdots\!31 \nu^{5} + \cdots + 72\!\cdots\!84 ) / 58\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 6 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - 44372\beta_{2} + 47374132105\beta _1 + 30292256447970046868844 ) / 324 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2424649383 \beta_{5} + 9339567283 \beta_{4} + 20557041921 \beta_{3} + \cdots + 15\!\cdots\!16 ) / 648 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 22\!\cdots\!05 \beta_{5} + \cdots + 21\!\cdots\!56 ) / 1296 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 76\!\cdots\!51 \beta_{5} + \cdots + 66\!\cdots\!16 ) / 648 \)
|
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 |
|
8.38861e6 | −2.71254e11 | 7.03687e13 | −7.17251e15 | −2.27544e18 | −2.73687e19 | 5.90296e20 | 4.69899e22 | −6.01674e22 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 8.38861e6 | −1.15220e11 | 7.03687e13 | −3.66016e15 | −9.66537e17 | −2.73687e19 | 5.90296e20 | −1.33131e22 | −3.07037e22 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 8.38861e6 | 2.85805e9 | 7.03687e13 | −4.96190e16 | 2.39750e16 | −2.73687e19 | 5.90296e20 | −2.65806e22 | −4.16234e23 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 8.38861e6 | 4.57489e10 | 7.03687e13 | 4.17176e16 | 3.83770e17 | −2.73687e19 | 5.90296e20 | −2.44958e22 | 3.49953e23 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 8.38861e6 | 1.82392e11 | 7.03687e13 | −1.56844e16 | 1.53001e18 | −2.73687e19 | 5.90296e20 | 6.67788e21 | −1.31570e23 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 8.38861e6 | 2.46824e11 | 7.03687e13 | 3.56349e15 | 2.07051e18 | −2.73687e19 | 5.90296e20 | 3.43332e22 | 2.98927e22 | |||||||||||||||||||||||||||||||||||||
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 | 14.48.a.c | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.48.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 91348200216 T_{3}^{5} + \cdots + 18\!\cdots\!00 \)
acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(14))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8388608)^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
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| $5$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
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| $7$ |
\( (T + 27\!\cdots\!43)^{6} \)
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| $11$ |
\( T^{6} + \cdots + 79\!\cdots\!00 \)
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| $13$ |
\( T^{6} + \cdots + 80\!\cdots\!64 \)
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| $17$ |
\( T^{6} + \cdots - 22\!\cdots\!44 \)
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| $19$ |
\( T^{6} + \cdots - 25\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots - 62\!\cdots\!00 \)
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| $29$ |
\( T^{6} + \cdots + 18\!\cdots\!96 \)
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| $31$ |
\( T^{6} + \cdots - 11\!\cdots\!76 \)
|
| $37$ |
\( T^{6} + \cdots + 59\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots + 15\!\cdots\!04 \)
|
| $43$ |
\( T^{6} + \cdots + 16\!\cdots\!64 \)
|
| $47$ |
\( T^{6} + \cdots + 17\!\cdots\!96 \)
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| $53$ |
\( T^{6} + \cdots - 33\!\cdots\!84 \)
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| $59$ |
\( T^{6} + \cdots + 14\!\cdots\!04 \)
|
| $61$ |
\( T^{6} + \cdots - 85\!\cdots\!84 \)
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| $67$ |
\( T^{6} + \cdots + 12\!\cdots\!76 \)
|
| $71$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 10\!\cdots\!44 \)
|
| $79$ |
\( T^{6} + \cdots + 14\!\cdots\!44 \)
|
| $83$ |
\( T^{6} + \cdots + 57\!\cdots\!16 \)
|
| $89$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
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| $97$ |
\( T^{6} + \cdots + 79\!\cdots\!00 \)
|
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