Newspace parameters
| Level: | \( N \) | \(=\) | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2450.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(144.554679514\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{29})\) |
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|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 17x^{2} + 18x + 23 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 17x^{2} + 18x + 23 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 25\nu - 13 ) / 21 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 9\nu^{2} - 23\nu - 88 ) / 21 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{3} + 6\nu^{2} + 92\nu - 47 ) / 21 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 2\beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + 19 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( 7\beta_{3} + 3\beta_{2} - 23\beta _1 + 14 \)
|
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 | −5.92921 | 4.00000 | 0 | 11.8584 | 0 | −8.00000 | 8.15549 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −1.68657 | 4.00000 | 0 | 3.37313 | 0 | −8.00000 | −24.1555 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 1.68657 | 4.00000 | 0 | −3.37313 | 0 | −8.00000 | −24.1555 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 5.92921 | 4.00000 | 0 | −11.8584 | 0 | −8.00000 | 8.15549 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2450.4.a.cl | ✓ | 4 |
| 5.b | even | 2 | 1 | 2450.4.a.cr | yes | 4 | |
| 7.b | odd | 2 | 1 | inner | 2450.4.a.cl | ✓ | 4 |
| 35.c | odd | 2 | 1 | 2450.4.a.cr | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2450.4.a.cl | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 2450.4.a.cl | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 2450.4.a.cr | yes | 4 | 5.b | even | 2 | 1 | |
| 2450.4.a.cr | yes | 4 | 35.c | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):
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\( T_{3}^{4} - 38T_{3}^{2} + 100 \)
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\( T_{11}^{2} + 26T_{11} - 1687 \)
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\( T_{19}^{4} - 9652T_{19}^{2} + 20884900 \)
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\( T_{23}^{2} - 22T_{23} - 1735 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( T^{4} - 38T^{2} + 100 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
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| $11$ |
\( (T^{2} + 26 T - 1687)^{2} \)
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| $13$ |
\( T^{4} - 342T^{2} + 8100 \)
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| $17$ |
\( T^{4} - 1240 T^{2} + 179776 \)
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| $19$ |
\( T^{4} - 9652 T^{2} + 20884900 \)
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| $23$ |
\( (T^{2} - 22 T - 1735)^{2} \)
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| $29$ |
\( (T^{2} - 92 T - 41993)^{2} \)
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| $31$ |
\( T^{4} - 54294 T^{2} + 70190884 \)
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| $37$ |
\( (T^{2} - 132 T - 545)^{2} \)
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| $41$ |
\( T^{4} - 31630 T^{2} + 248566756 \)
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| $43$ |
\( (T^{2} - 8381)^{2} \)
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| $47$ |
\( T^{4} - 3972 T^{2} + 16900 \)
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| $53$ |
\( (T^{2} - 622 T + 61196)^{2} \)
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| $59$ |
\( T^{4} - 123390 T^{2} + 188622756 \)
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| $61$ |
\( T^{4} + \cdots + 3710028100 \)
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| $67$ |
\( (T^{2} - 530 T + 70109)^{2} \)
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| $71$ |
\( (T^{2} + 704 T + 88379)^{2} \)
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| $73$ |
\( T^{4} - 2430 T^{2} + 1406596 \)
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| $79$ |
\( (T^{2} + 1326 T + 272065)^{2} \)
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| $83$ |
\( T^{4} + \cdots + 1676512219204 \)
|
| $89$ |
\( T^{4} + \cdots + 2411777940100 \)
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| $97$ |
\( T^{4} + \cdots + 423022761604 \)
|
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