Newspace parameters
| Level: | \( N \) | \(=\) | \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4004.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(31.9721009693\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.463341.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 5\nu + 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 6\nu + 5 \)
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| \(\beta_{4}\) | \(=\) |
\( -2\nu^{4} + 5\nu^{3} + 10\nu^{2} - 17\nu - 6 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 2\beta_{2} + 7\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} - 5\beta_{3} + 10\beta_{2} + 14\beta _1 + 17 \)
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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 |
|
0 | −2.09892 | 0 | −2.18167 | 0 | −1.00000 | 0 | 1.40545 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.538379 | 0 | 3.82180 | 0 | −1.00000 | 0 | −2.71015 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.23175 | 0 | −0.600512 | 0 | −1.00000 | 0 | −1.48280 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.46689 | 0 | 1.17328 | 0 | −1.00000 | 0 | −0.848245 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.93866 | 0 | −2.21290 | 0 | −1.00000 | 0 | 5.63574 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(1\) |
| \(13\) | \(-1\) |
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 | 4004.2.a.f | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4004.2.a.f | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 3T_{3}^{4} - 4T_{3}^{3} + 14T_{3}^{2} - 3T_{3} - 6 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} - 3 T^{4} + \cdots - 6 \)
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| $5$ |
\( T^{5} - 13 T^{3} + \cdots + 13 \)
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| $7$ |
\( (T + 1)^{5} \)
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| $11$ |
\( (T + 1)^{5} \)
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| $13$ |
\( (T - 1)^{5} \)
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| $17$ |
\( T^{5} + 9 T^{4} + \cdots + 134 \)
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| $19$ |
\( T^{5} + 4 T^{4} + \cdots + 99 \)
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| $23$ |
\( T^{5} + 4 T^{4} + \cdots + 587 \)
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| $29$ |
\( T^{5} + 8 T^{4} + \cdots + 811 \)
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| $31$ |
\( T^{5} - 7 T^{4} + \cdots + 1873 \)
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| $37$ |
\( T^{5} + 10 T^{4} + \cdots + 202 \)
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| $41$ |
\( T^{5} + 18 T^{4} + \cdots + 142 \)
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| $43$ |
\( T^{5} + 22 T^{4} + \cdots - 9143 \)
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| $47$ |
\( T^{5} - 12 T^{4} + \cdots + 1331 \)
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| $53$ |
\( T^{5} - 7 T^{4} + \cdots + 507 \)
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| $59$ |
\( T^{5} + T^{4} + \cdots + 432 \)
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| $61$ |
\( T^{5} + 26 T^{4} + \cdots - 18838 \)
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| $67$ |
\( T^{5} + 8 T^{4} + \cdots - 4854 \)
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| $71$ |
\( T^{5} - 18 T^{4} + \cdots - 3488 \)
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| $73$ |
\( T^{5} + 25 T^{4} + \cdots - 1151 \)
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| $79$ |
\( T^{5} + 6 T^{4} + \cdots + 12507 \)
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| $83$ |
\( T^{5} - 11 T^{4} + \cdots + 3897 \)
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| $89$ |
\( T^{5} + 14 T^{4} + \cdots + 101751 \)
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| $97$ |
\( T^{5} + 33 T^{4} + \cdots + 41 \)
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