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: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.13366265625.1 |
| Defining polynomial: |
\( x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 10\nu^{4} + 41\nu^{3} - 20\nu^{2} - 40\nu + 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 10\nu^{5} - 34\nu^{4} - 21\nu^{3} + 94\nu^{2} + 8\nu - 56 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 10\nu^{5} + 30\nu^{4} + 21\nu^{3} - 62\nu^{2} - 12\nu + 20 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 10\nu^{5} - 30\nu^{4} - 21\nu^{3} + 66\nu^{2} + 12\nu - 32 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} + 3\nu^{6} + 36\nu^{5} - 30\nu^{4} - 115\nu^{3} + 60\nu^{2} + 88\nu - 24 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 30\nu^{4} - 52\nu^{3} + 61\nu^{2} + 30\nu - 28 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + 3\beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{5} + 7\beta_{4} - 2\beta_{3} - \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{7} + 10\beta_{6} - \beta_{4} + 24\beta_{2} + 22\beta _1 - 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( -2\beta_{7} + 59\beta_{5} + 46\beta_{4} - 20\beta_{3} - 2\beta_{2} - 14\beta _1 + 90 \)
|
| \(\nu^{7}\) | \(=\) |
\( -26\beta_{7} + 79\beta_{6} - \beta_{5} - 16\beta_{4} + 171\beta_{2} + 136\beta _1 - 44 \)
|
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 |
|
−2.69767 | 1.00000 | 5.27745 | 0 | −2.69767 | −3.56649 | −8.84149 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.52260 | 1.00000 | 0.318310 | 0 | −1.52260 | −0.990985 | 2.56054 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.895394 | 1.00000 | −1.19827 | 0 | −0.895394 | 5.08992 | 2.86371 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.770071 | 1.00000 | −1.40699 | 0 | −0.770071 | 3.98808 | 2.62363 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.741379 | 1.00000 | −1.45036 | 0 | 0.741379 | 1.03586 | −2.55802 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.31354 | 1.00000 | −0.274605 | 0 | 1.31354 | 4.19091 | −2.98779 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.23365 | 1.00000 | 2.98921 | 0 | 2.23365 | −1.03143 | 2.20956 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.59716 | 1.00000 | 4.74525 | 0 | 2.59716 | 3.28414 | 7.12986 | 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.o | yes | 8 |
| 3.b | odd | 2 | 1 | 5625.2.a.u | 8 | ||
| 5.b | even | 2 | 1 | 1875.2.a.n | ✓ | 8 | |
| 5.c | odd | 4 | 2 | 1875.2.b.g | 16 | ||
| 15.d | odd | 2 | 1 | 5625.2.a.bc | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1875.2.a.n | ✓ | 8 | 5.b | even | 2 | 1 | |
| 1875.2.a.o | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 1875.2.b.g | 16 | 5.c | odd | 4 | 2 | ||
| 5625.2.a.u | 8 | 3.b | odd | 2 | 1 | ||
| 5625.2.a.bc | 8 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - T_{2}^{7} - 12T_{2}^{6} + 10T_{2}^{5} + 41T_{2}^{4} - 20T_{2}^{3} - 48T_{2}^{2} + 8T_{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - T^{7} - 12 T^{6} + 10 T^{5} + \cdots + 16 \)
$3$
\( (T - 1)^{8} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 12 T^{7} + 29 T^{6} + \cdots - 1055 \)
$11$
\( T^{8} - 12 T^{7} + 2 T^{6} + 455 T^{5} + \cdots + 16 \)
$13$
\( T^{8} - 14 T^{7} + 43 T^{6} + \cdots + 8731 \)
$17$
\( T^{8} + T^{7} - 88 T^{6} - 122 T^{5} + \cdots + 8656 \)
$19$
\( T^{8} - 16 T^{7} + 51 T^{6} + \cdots - 14975 \)
$23$
\( T^{8} + 4 T^{7} - 104 T^{6} + \cdots + 28720 \)
$29$
\( T^{8} - 2 T^{7} - 126 T^{6} + \cdots + 57520 \)
$31$
\( T^{8} - 13 T^{7} - 96 T^{6} + \cdots + 801025 \)
$37$
\( T^{8} + 8 T^{7} - 31 T^{6} - 313 T^{5} + \cdots + 25 \)
$41$
\( T^{8} + 12 T^{7} - 6 T^{6} + \cdots - 48080 \)
$43$
\( T^{8} - 20 T^{7} + 6 T^{6} + \cdots - 74369 \)
$47$
\( T^{8} + 15 T^{7} - 71 T^{6} + \cdots - 255824 \)
$53$
\( T^{8} + 4 T^{7} - 159 T^{6} + \cdots + 28720 \)
$59$
\( T^{8} - 14 T^{7} - 164 T^{6} + \cdots + 11920 \)
$61$
\( T^{8} - 10 T^{7} - 186 T^{6} + \cdots - 1093919 \)
$67$
\( T^{8} - 19 T^{7} - 183 T^{6} + \cdots + 20204221 \)
$71$
\( T^{8} - 21 T^{7} - 18 T^{6} + \cdots + 67696 \)
$73$
\( T^{8} + 19 T^{7} + 21 T^{6} + \cdots - 379655 \)
$79$
\( T^{8} - 10 T^{7} - 320 T^{6} + \cdots + 6951025 \)
$83$
\( T^{8} + 27 T^{7} + 132 T^{6} + \cdots - 113744 \)
$89$
\( T^{8} + 9 T^{7} - 294 T^{6} + \cdots - 12105680 \)
$97$
\( T^{8} - 24 T^{7} + 87 T^{6} + \cdots - 401939 \)
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