Newspace parameters
| Level: | \( N \) | \(=\) | \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1638.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(96.6451285894\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{5} - x^{4} - 417x^{3} + 287x^{2} + 37482x + 69612 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 417x^{3} + 287x^{2} + 37482x + 69612 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 198\nu^{3} - 571\nu^{2} + 48160\nu + 61926 ) / 1958 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -19\nu^{4} - 154\nu^{3} + 4975\nu^{2} + 28716\nu - 70324 ) / 1958 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 9\nu^{4} + 176\nu^{3} - 1223\nu^{2} - 38438\nu - 159294 ) / 1958 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( 2\beta_{4} + \beta_{3} + \beta_{2} + 167 \)
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| \(\nu^{3}\) | \(=\) |
\( -3\beta_{4} - 2\beta_{3} - 11\beta_{2} + 241\beta _1 + 32 \)
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| \(\nu^{4}\) | \(=\) |
\( 548\beta_{4} + 175\beta_{3} + 351\beta_{2} - 442\beta _1 + 39767 \)
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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 |
|
−2.00000 | 0 | 4.00000 | −21.5099 | 0 | −7.00000 | −8.00000 | 0 | 43.0197 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −13.1795 | 0 | −7.00000 | −8.00000 | 0 | 26.3590 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | −5.97094 | 0 | −7.00000 | −8.00000 | 0 | 11.9419 | ||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0 | 4.00000 | 10.3893 | 0 | −7.00000 | −8.00000 | 0 | −20.7786 | ||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 0 | 4.00000 | 11.2710 | 0 | −7.00000 | −8.00000 | 0 | −22.5420 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 1638.4.a.bh | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1638.4.a.bk | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1638.4.a.bh | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1638.4.a.bk | yes | 5 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1638))\):
|
\( T_{5}^{5} + 19T_{5}^{4} - 273T_{5}^{3} - 4173T_{5}^{2} + 20786T_{5} + 198212 \)
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\( T_{11}^{5} + 53T_{11}^{4} - 2026T_{11}^{3} - 111952T_{11}^{2} + 1043848T_{11} + 58414384 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{5} \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} + 19 T^{4} + \cdots + 198212 \)
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| $7$ |
\( (T + 7)^{5} \)
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| $11$ |
\( T^{5} + 53 T^{4} + \cdots + 58414384 \)
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| $13$ |
\( (T - 13)^{5} \)
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| $17$ |
\( T^{5} + 5 T^{4} + \cdots + 70916864 \)
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| $19$ |
\( T^{5} + \cdots + 1384749936 \)
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| $23$ |
\( T^{5} + \cdots + 24732659076 \)
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| $29$ |
\( T^{5} + \cdots + 4726387344 \)
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| $31$ |
\( T^{5} + \cdots + 3615488736 \)
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| $37$ |
\( T^{5} - 391 T^{4} + \cdots + 116283328 \)
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| $41$ |
\( T^{5} + \cdots + 213178464128 \)
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| $43$ |
\( T^{5} + \cdots - 3506167579664 \)
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| $47$ |
\( T^{5} + \cdots + 41762111328 \)
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| $53$ |
\( T^{5} + \cdots + 1500672800928 \)
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| $59$ |
\( T^{5} + \cdots - 1344396681312 \)
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| $61$ |
\( T^{5} + \cdots + 10873148452208 \)
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| $67$ |
\( T^{5} + \cdots - 117948857122304 \)
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| $71$ |
\( T^{5} + \cdots - 71435986501376 \)
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| $73$ |
\( T^{5} + \cdots + 45654633816276 \)
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| $79$ |
\( T^{5} + \cdots + 218715318368 \)
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| $83$ |
\( T^{5} + \cdots + 92976296071656 \)
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| $89$ |
\( T^{5} + \cdots - 7174022685336 \)
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| $97$ |
\( T^{5} + \cdots + 12\!\cdots\!08 \)
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