Newspace parameters
| Level: | \( N \) | \(=\) | \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2900.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(23.1566165862\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.2370465.1 |
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|
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| Defining polynomial: |
\( x^{5} - 9x^{3} - x^{2} + 9x - 3 \)
|
| 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
\( x^{5} - 9x^{3} - x^{2} + 9x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( -\nu^{4} + 9\nu^{2} + 2\nu - 7 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 9\nu^{2} - 9\nu + 7 \)
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| \(\beta_{4}\) | \(=\) |
\( -2\nu^{4} - \nu^{3} + 17\nu^{2} + 11\nu - 10 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} - \beta_{3} + \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 7\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( -9\beta_{4} - 9\beta_{3} + 8\beta_{2} + 2\beta _1 + 29 \)
|
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 | −2.89768 | 0 | 0 | 0 | 1.71851 | 0 | 5.39654 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.672645 | 0 | 0 | 0 | −3.37701 | 0 | −2.54755 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.434833 | 0 | 0 | 0 | 3.15620 | 0 | −2.81092 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.31654 | 0 | 0 | 0 | −0.257197 | 0 | −1.26672 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.68862 | 0 | 0 | 0 | 2.75949 | 0 | 4.22866 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 2900.2.a.k | yes | 5 |
| 5.b | even | 2 | 1 | 2900.2.a.i | ✓ | 5 | |
| 5.c | odd | 4 | 2 | 2900.2.c.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2900.2.a.i | ✓ | 5 | 5.b | even | 2 | 1 | |
| 2900.2.a.k | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 2900.2.c.i | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2900))\):
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\( T_{3}^{5} - 9T_{3}^{3} + T_{3}^{2} + 9T_{3} + 3 \)
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\( T_{7}^{5} - 4T_{7}^{4} - 8T_{7}^{3} + 47T_{7}^{2} - 38T_{7} - 13 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} - 9 T^{3} + \cdots + 3 \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} - 4 T^{4} + \cdots - 13 \)
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| $11$ |
\( T^{5} - 4 T^{4} + \cdots - 129 \)
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| $13$ |
\( T^{5} - 12 T^{4} + \cdots + 555 \)
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| $17$ |
\( T^{5} - 5 T^{4} + \cdots + 387 \)
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| $19$ |
\( T^{5} + 4 T^{4} + \cdots + 1 \)
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| $23$ |
\( T^{5} - 9 T^{4} + \cdots + 2421 \)
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| $29$ |
\( (T + 1)^{5} \)
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| $31$ |
\( T^{5} - 3 T^{4} + \cdots - 1665 \)
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| $37$ |
\( T^{5} - 4 T^{4} + \cdots + 659 \)
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| $41$ |
\( T^{5} - 5 T^{4} + \cdots + 2715 \)
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| $43$ |
\( T^{5} - 4 T^{4} + \cdots - 415 \)
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| $47$ |
\( T^{5} - 5 T^{4} + \cdots + 3 \)
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| $53$ |
\( T^{5} - 16 T^{4} + \cdots + 135 \)
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| $59$ |
\( T^{5} + 23 T^{4} + \cdots - 3405 \)
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| $61$ |
\( T^{5} - 5 T^{4} + \cdots - 9575 \)
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| $67$ |
\( T^{5} - 5 T^{4} + \cdots - 14479 \)
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| $71$ |
\( T^{5} - 23 T^{4} + \cdots + 48951 \)
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| $73$ |
\( T^{5} - 18 T^{4} + \cdots + 1395 \)
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| $79$ |
\( T^{5} - 4 T^{4} + \cdots - 32525 \)
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| $83$ |
\( T^{5} - 9 T^{4} + \cdots + 6561 \)
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| $89$ |
\( T^{5} + 6 T^{4} + \cdots + 30717 \)
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| $97$ |
\( T^{5} - 21 T^{4} + \cdots + 1395 \)
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