Newspace parameters
| Level: | \( N \) | \(=\) | \( 1875 = 3 \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1875.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(14.9719503790\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{15})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 4x^{2} + 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{15} + \zeta_{15}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
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 |
|
−1.95630 | −1.00000 | 1.82709 | 0 | 1.95630 | 4.57433 | 0.338261 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −0.209057 | −1.00000 | −1.95630 | 0 | 0.209057 | 0.591023 | 0.827091 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1.33826 | −1.00000 | −0.209057 | 0 | −1.33826 | 1.27977 | −2.95630 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 1.82709 | −1.00000 | 1.33826 | 0 | −1.82709 | −1.44512 | −1.20906 | 1.00000 | 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 | 1875.2.a.g | yes | 4 |
| 3.b | odd | 2 | 1 | 5625.2.a.j | 4 | ||
| 5.b | even | 2 | 1 | 1875.2.a.f | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 1875.2.b.d | 8 | ||
| 15.d | odd | 2 | 1 | 5625.2.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1875.2.a.f | ✓ | 4 | 5.b | even | 2 | 1 | |
| 1875.2.a.g | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 1875.2.b.d | 8 | 5.c | odd | 4 | 2 | ||
| 5625.2.a.j | 4 | 3.b | odd | 2 | 1 | ||
| 5625.2.a.m | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} - 4T_{2}^{2} + 4T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{3} - 4 T^{2} + 4 T + 1 \)
$3$
\( (T + 1)^{4} \)
$5$
\( T^{4} \)
$7$
\( T^{4} - 5 T^{3} + 10 T - 5 \)
$11$
\( T^{4} + 6 T^{3} + 6 T^{2} - 9 T - 9 \)
$13$
\( T^{4} - 7 T^{3} - 6 T^{2} + 92 T - 89 \)
$17$
\( T^{4} + 7 T^{3} - 16 T^{2} - 202 T - 359 \)
$19$
\( T^{4} + 9 T^{3} + 6 T^{2} - 36 T - 9 \)
$23$
\( T^{4} - 10 T^{3} + 20 T^{2} + 10 T - 5 \)
$29$
\( T^{4} + 28 T^{3} + 284 T^{2} + \cdots + 1801 \)
$31$
\( T^{4} + 10 T^{3} - 125 T - 125 \)
$37$
\( T^{4} + 10 T^{3} - 90 T^{2} + \cdots - 1475 \)
$41$
\( T^{4} - 70 T^{2} - 135 T + 145 \)
$43$
\( T^{4} - T^{3} - 79 T^{2} - 341 T - 419 \)
$47$
\( T^{4} + 23 T^{3} + 89 T^{2} + \cdots - 6089 \)
$53$
\( T^{4} - 145 T^{2} - 270 T + 2995 \)
$59$
\( T^{4} - 4 T^{3} - 154 T^{2} + \cdots + 1531 \)
$61$
\( T^{4} + 43 T^{3} + 669 T^{2} + \cdots + 10261 \)
$67$
\( T^{4} - 8 T^{3} - 61 T^{2} + 458 T + 151 \)
$71$
\( T^{4} + 27 T^{3} + 134 T^{2} + \cdots + 271 \)
$73$
\( T^{4} - 15 T^{3} - 60 T^{2} + \cdots + 2745 \)
$79$
\( T^{4} - 10 T^{3} - 105 T^{2} + \cdots - 3155 \)
$83$
\( T^{4} - 3 T^{3} - 246 T^{2} + \cdots + 13491 \)
$89$
\( T^{4} + 9 T^{3} - 4 T^{2} - 96 T + 61 \)
$97$
\( T^{4} - 13 T^{3} - 216 T^{2} + \cdots + 14701 \)
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