Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(209.608008215\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 157936x - 9799664 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{4}\cdot 5 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 157936x - 9799664 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( 18\nu^{2} - 1698\nu - 1894672 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 283\beta _1 + 1895238 ) / 18 \)
|
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 |
|
−2787.27 | 59049.0 | 5.67172e6 | 0 | −1.64586e8 | 1.37938e9 | −9.96329e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||
| 1.2 | 150.269 | 59049.0 | −2.07457e6 | 0 | 8.87323e6 | −6.41000e7 | −6.26880e8 | 3.48678e9 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 1935.00 | 59049.0 | 1.64708e6 | 0 | 1.14260e8 | 7.57139e8 | −8.70895e8 | 3.48678e9 | 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 | 75.22.a.f | 3 | |
| 5.b | even | 2 | 1 | 15.22.a.c | ✓ | 3 | |
| 5.c | odd | 4 | 2 | 75.22.b.f | 6 | ||
| 15.d | odd | 2 | 1 | 45.22.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.22.a.c | ✓ | 3 | 5.b | even | 2 | 1 | |
| 45.22.a.c | 3 | 15.d | odd | 2 | 1 | ||
| 75.22.a.f | 3 | 1.a | even | 1 | 1 | trivial | |
| 75.22.b.f | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 702T_{2}^{2} - 5521440T_{2} + 810456064 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} + 702 T^{2} + \cdots + 810456064 \)
$3$
\( (T - 59049)^{3} \)
$5$
\( T^{3} \)
$7$
\( T^{3} - 2072418204 T^{2} + \cdots + 66\!\cdots\!20 \)
$11$
\( T^{3} + 8467600440 T^{2} + \cdots + 28\!\cdots\!16 \)
$13$
\( T^{3} - 469447548570 T^{2} + \cdots + 86\!\cdots\!92 \)
$17$
\( T^{3} - 6865096035486 T^{2} + \cdots + 18\!\cdots\!72 \)
$19$
\( T^{3} - 33215593555044 T^{2} + \cdots - 27\!\cdots\!00 \)
$23$
\( T^{3} + 48760125154728 T^{2} + \cdots - 19\!\cdots\!20 \)
$29$
\( T^{3} + \cdots + 10\!\cdots\!00 \)
$31$
\( T^{3} + \cdots - 64\!\cdots\!00 \)
$37$
\( T^{3} + \cdots - 91\!\cdots\!60 \)
$41$
\( T^{3} + \cdots - 42\!\cdots\!00 \)
$43$
\( T^{3} + \cdots + 79\!\cdots\!76 \)
$47$
\( T^{3} + \cdots - 26\!\cdots\!32 \)
$53$
\( T^{3} + \cdots + 18\!\cdots\!40 \)
$59$
\( T^{3} + \cdots - 13\!\cdots\!00 \)
$61$
\( T^{3} + \cdots - 20\!\cdots\!12 \)
$67$
\( T^{3} + \cdots + 97\!\cdots\!12 \)
$71$
\( T^{3} + \cdots + 11\!\cdots\!32 \)
$73$
\( T^{3} + \cdots + 11\!\cdots\!80 \)
$79$
\( T^{3} + \cdots - 20\!\cdots\!00 \)
$83$
\( T^{3} + \cdots + 14\!\cdots\!88 \)
$89$
\( T^{3} + \cdots + 71\!\cdots\!00 \)
$97$
\( T^{3} + \cdots + 62\!\cdots\!12 \)
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