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: | 8.8.5444000000.1 |
Defining polynomial: |
\( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 75) |
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
\( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
\(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
\(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1 \)
|
\(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - \nu^{3} + 7\nu^{2} - 3\nu - 1 \)
|
\(\beta_{6}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 3\nu^{4} + 11\nu^{3} + 3\nu^{2} - 9\nu - 2 \)
|
\(\beta_{7}\) | \(=\) |
\( \nu^{7} - 4\nu^{6} - \nu^{5} + 16\nu^{4} - 5\nu^{3} - 16\nu^{2} + 5\nu + 2 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
\(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 7 \)
|
\(\nu^{5}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + 7\beta_{3} + 9\beta_{2} + 21\beta _1 + 9 \)
|
\(\nu^{6}\) | \(=\) |
\( \beta_{6} + 3\beta_{5} + 12\beta_{4} + 16\beta_{3} + 28\beta_{2} + 46\beta _1 + 33 \)
|
\(\nu^{7}\) | \(=\) |
\( \beta_{7} + 4\beta_{6} + 13\beta_{5} + 35\beta_{4} + 44\beta_{3} + 62\beta_{2} + 124\beta _1 + 64 \)
|
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 |
|
−1.53767 | −1.00000 | 0.364440 | 0 | 1.53767 | 1.68601 | 2.51496 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
1.2 | −1.35083 | −1.00000 | −0.175259 | 0 | 1.35083 | 1.59580 | 2.93840 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1.3 | −0.536547 | −1.00000 | −1.71212 | 0 | 0.536547 | −2.57318 | 1.99173 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1.4 | −0.0898194 | −1.00000 | −1.99193 | 0 | 0.0898194 | 4.36070 | 0.358553 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1.5 | 1.08982 | −1.00000 | −0.812294 | 0 | −1.08982 | −3.08724 | −3.06489 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1.6 | 1.53655 | −1.00000 | 0.360976 | 0 | −1.53655 | 1.49550 | −2.51844 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1.7 | 2.35083 | −1.00000 | 3.52640 | 0 | −2.35083 | 3.48189 | 3.58831 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1.8 | 2.53767 | −1.00000 | 4.43979 | 0 | −2.53767 | 1.04054 | 6.19138 | 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.p | 8 | |
3.b | odd | 2 | 1 | 5625.2.a.t | 8 | ||
5.b | even | 2 | 1 | 1875.2.a.m | 8 | ||
5.c | odd | 4 | 2 | 1875.2.b.h | 16 | ||
15.d | odd | 2 | 1 | 5625.2.a.bd | 8 | ||
25.d | even | 5 | 2 | 375.2.g.d | 16 | ||
25.e | even | 10 | 2 | 375.2.g.e | 16 | ||
25.f | odd | 20 | 2 | 75.2.i.a | ✓ | 16 | |
25.f | odd | 20 | 2 | 375.2.i.c | 16 | ||
75.l | even | 20 | 2 | 225.2.m.b | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.2.i.a | ✓ | 16 | 25.f | odd | 20 | 2 | |
225.2.m.b | 16 | 75.l | even | 20 | 2 | ||
375.2.g.d | 16 | 25.d | even | 5 | 2 | ||
375.2.g.e | 16 | 25.e | even | 10 | 2 | ||
375.2.i.c | 16 | 25.f | odd | 20 | 2 | ||
1875.2.a.m | 8 | 5.b | even | 2 | 1 | ||
1875.2.a.p | 8 | 1.a | even | 1 | 1 | trivial | |
1875.2.b.h | 16 | 5.c | odd | 4 | 2 | ||
5625.2.a.t | 8 | 3.b | odd | 2 | 1 | ||
5625.2.a.bd | 8 | 15.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 4T_{2}^{7} - 2T_{2}^{6} + 20T_{2}^{5} - 4T_{2}^{4} - 30T_{2}^{3} + 7T_{2}^{2} + 12T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 4 T^{7} - 2 T^{6} + 20 T^{5} + \cdots + 1 \)
$3$
\( (T + 1)^{8} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 8 T^{7} + 4 T^{6} + 108 T^{5} + \cdots + 505 \)
$11$
\( T^{8} - 2 T^{7} - 43 T^{6} + \cdots + 5281 \)
$13$
\( T^{8} - 16 T^{7} + 73 T^{6} + \cdots + 281 \)
$17$
\( T^{8} - 16 T^{7} + 42 T^{6} + \cdots - 7339 \)
$19$
\( T^{8} + 14 T^{7} + 11 T^{6} + \cdots + 2525 \)
$23$
\( T^{8} - 14 T^{7} + 11 T^{6} + \cdots - 2095 \)
$29$
\( T^{8} - 2 T^{7} - 106 T^{6} + \cdots - 395 \)
$31$
\( T^{8} + 22 T^{7} + 169 T^{6} + \cdots + 125 \)
$37$
\( T^{8} - 28 T^{7} + 204 T^{6} + \cdots + 93025 \)
$41$
\( T^{8} - 8 T^{7} - 56 T^{6} + \cdots + 4705 \)
$43$
\( T^{8} - 20 T^{7} + 6 T^{6} + \cdots + 22961 \)
$47$
\( T^{8} - 10 T^{7} - 186 T^{6} + \cdots - 6057019 \)
$53$
\( T^{8} - 44 T^{7} + 746 T^{6} + \cdots - 200995 \)
$59$
\( T^{8} - 14 T^{7} - 29 T^{6} + \cdots - 3595 \)
$61$
\( T^{8} + 20 T^{7} - 136 T^{6} + \cdots + 16604261 \)
$67$
\( T^{8} - 16 T^{7} - 138 T^{6} + \cdots - 3739 \)
$71$
\( T^{8} - 16 T^{7} - 78 T^{6} + \cdots - 159779 \)
$73$
\( T^{8} - 24 T^{7} - 34 T^{6} + \cdots - 870295 \)
$79$
\( T^{8} + 30 T^{7} + 145 T^{6} + \cdots - 1984975 \)
$83$
\( T^{8} - 12 T^{7} - 188 T^{6} + \cdots + 48541 \)
$89$
\( T^{8} - 16 T^{7} - 39 T^{6} + 1454 T^{5} + \cdots + 5 \)
$97$
\( T^{8} - 16 T^{7} - 108 T^{6} + \cdots - 14719 \)
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