Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(80.4231967139\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 1651x^{2} + 4960x + 346125 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3\cdot 5^{2} \) |
| 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
\( x^{4} - x^{3} - 1651x^{2} + 4960x + 346125 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 16\nu^{2} + 52\nu - 13225 \)
|
| \(\beta_{3}\) | \(=\) |
\( 8\nu^{3} + 48\nu^{2} - 9668\nu - 17367 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 13\beta _1 + 13212 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} + 2456\beta _1 - 19852 ) / 8 \)
|
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 |
|
−143.467 | 729.000 | 12390.7 | 0 | −104587. | −115746. | −602380. | 531441. | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −43.4185 | 729.000 | −6306.83 | 0 | −31652.1 | 200629. | 629518. | 531441. | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 83.9533 | 729.000 | −1143.84 | 0 | 61202.0 | 179519. | −783775. | 531441. | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 154.932 | 729.000 | 15811.9 | 0 | 112945. | −563397. | 1.18057e6 | 531441. | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
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 | 75.14.a.h | yes | 4 |
| 5.b | even | 2 | 1 | 75.14.a.g | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 75.14.b.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.14.a.g | ✓ | 4 | 5.b | even | 2 | 1 | |
| 75.14.a.h | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 75.14.b.g | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 52T_{2}^{3} - 25408T_{2}^{2} + 942784T_{2} + 81022464 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 52 T^{3} + \cdots + 81022464 \)
|
| $3$ |
\( (T - 729)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 23\!\cdots\!25 \)
|
| $11$ |
\( T^{4} + \cdots - 89\!\cdots\!36 \)
|
| $13$ |
\( T^{4} + \cdots + 13\!\cdots\!09 \)
|
| $17$ |
\( T^{4} + \cdots + 36\!\cdots\!96 \)
|
| $19$ |
\( T^{4} + \cdots + 89\!\cdots\!25 \)
|
| $23$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 83\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 35\!\cdots\!75 \)
|
| $37$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 22\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 60\!\cdots\!61 \)
|
| $47$ |
\( T^{4} + \cdots + 44\!\cdots\!36 \)
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| $53$ |
\( T^{4} + \cdots - 50\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 16\!\cdots\!99 \)
|
| $67$ |
\( T^{4} + \cdots + 36\!\cdots\!21 \)
|
| $71$ |
\( T^{4} + \cdots + 38\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 27\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 23\!\cdots\!36 \)
|
| $89$ |
\( T^{4} + \cdots + 77\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 69\!\cdots\!39 \)
|
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