Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(137.416565508\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 481686x^{2} + 26523040x + 36023696000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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
\( x^{4} - x^{3} - 481686x^{2} + 26523040x + 36023696000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 81\nu - 240864 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 189\nu^{2} - 316646\nu - 25909216 ) / 74 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 81\beta _1 + 240864 \)
|
| \(\nu^{3}\) | \(=\) |
\( 74\beta_{3} - 189\beta_{2} + 331955\beta _1 - 19614080 \)
|
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 |
|
−654.567 | 6561.00 | 297386. | 0 | −4.29461e6 | −359681. | −1.08863e8 | 4.30467e7 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −257.574 | 6561.00 | −64727.8 | 0 | −1.68994e6 | −8.43659e6 | 5.04329e7 | 4.30467e7 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 367.218 | 6561.00 | 3777.06 | 0 | 2.40932e6 | 1.91248e7 | −4.67450e7 | 4.30467e7 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 577.922 | 6561.00 | 202922. | 0 | 3.79175e6 | −2.79117e7 | 4.15238e7 | 4.30467e7 | 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 | 75.18.a.f | 4 | |
| 5.b | even | 2 | 1 | 15.18.a.d | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 75.18.b.f | 8 | ||
| 15.d | odd | 2 | 1 | 45.18.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.18.a.d | ✓ | 4 | 5.b | even | 2 | 1 | |
| 45.18.a.f | 4 | 15.d | odd | 2 | 1 | ||
| 75.18.a.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.18.b.f | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 33T_{2}^{3} - 481278T_{2}^{2} + 34227776T_{2} + 35780688384 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 35780688384 \)
|
| $3$ |
\( (T - 6561)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots - 16\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 10\!\cdots\!84 \)
|
| $13$ |
\( T^{4} + \cdots + 13\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots + 81\!\cdots\!36 \)
|
| $19$ |
\( T^{4} + \cdots - 60\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 41\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 79\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 42\!\cdots\!36 \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots - 45\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots - 16\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 43\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots - 10\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots - 72\!\cdots\!64 \)
|
| $73$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 54\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 14\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 25\!\cdots\!04 \)
|
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