Newspace parameters
Level: | \( N \) | \(=\) | \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1512.j (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(12.0733807856\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.56070144.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{6} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) :
\(\beta_{1}\) | \(=\) | \( ( -5\nu^{7} - \nu^{6} - 25\nu^{5} - 46\nu^{4} - 5\nu^{3} - 132\nu^{2} + 28\nu - 92 ) / 37 \) |
\(\beta_{2}\) | \(=\) | \( ( -6\nu^{7} + 21\nu^{6} - 67\nu^{5} + 115\nu^{4} - 117\nu^{3} + 71\nu^{2} + 115\nu - 66 ) / 37 \) |
\(\beta_{3}\) | \(=\) | \( ( -6\nu^{7} + 21\nu^{6} - 67\nu^{5} + 115\nu^{4} - 117\nu^{3} + 71\nu^{2} + 41\nu - 29 ) / 37 \) |
\(\beta_{4}\) | \(=\) | \( -\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 23\nu^{2} + 15\nu - 4 \) |
\(\beta_{5}\) | \(=\) | \( ( -5\nu^{7} + 36\nu^{6} - 136\nu^{5} + 361\nu^{4} - 634\nu^{3} + 793\nu^{2} - 601\nu + 241 ) / 37 \) |
\(\beta_{6}\) | \(=\) | \( ( -42\nu^{7} + 147\nu^{6} - 543\nu^{5} + 953\nu^{4} - 1485\nu^{3} + 1163\nu^{2} - 749\nu + 56 ) / 37 \) |
\(\beta_{7}\) | \(=\) | \( ( -42\nu^{7} + 147\nu^{6} - 543\nu^{5} + 1027\nu^{4} - 1633\nu^{3} + 1681\nu^{2} - 1193\nu + 500 ) / 37 \) |
\(\nu\) | \(=\) | \( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 4 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -\beta_{7} - \beta_{6} + 6\beta_{5} + 3\beta_{4} + 14\beta_{3} - 5\beta_{2} - 10 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( -2\beta_{6} - \beta_{5} - 4\beta_{4} + 15\beta_{3} - 6\beta_{2} + 7\beta _1 + 12 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 9\beta_{7} - \beta_{6} - 40\beta_{5} - 25\beta_{4} - 42\beta_{3} + 11\beta_{2} + 10\beta _1 + 52 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( 5\beta_{7} + 12\beta_{6} - 21\beta_{5} + 7\beta_{4} - 101\beta_{3} + 32\beta_{2} - 39\beta _1 - 40 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( -23\beta_{7} + 19\beta_{6} + 84\beta_{5} + 70\beta_{4} + \beta_{3} + 3\beta_{2} - 70\beta _1 - 153 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).
\(n\) | \(757\) | \(785\) | \(1081\) | \(1135\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
323.1 |
|
−0.366025 | − | 1.36603i | 0 | −1.73205 | + | 1.00000i | −2.38587 | 0 | − | 1.00000i | 2.00000 | + | 2.00000i | 0 | 0.873288 | + | 3.25916i | |||||||||||||||||||||||||||||||||
323.2 | −0.366025 | − | 1.36603i | 0 | −1.73205 | + | 1.00000i | 4.38587 | 0 | − | 1.00000i | 2.00000 | + | 2.00000i | 0 | −1.60534 | − | 5.99121i | ||||||||||||||||||||||||||||||||||
323.3 | −0.366025 | + | 1.36603i | 0 | −1.73205 | − | 1.00000i | −2.38587 | 0 | 1.00000i | 2.00000 | − | 2.00000i | 0 | 0.873288 | − | 3.25916i | |||||||||||||||||||||||||||||||||||
323.4 | −0.366025 | + | 1.36603i | 0 | −1.73205 | − | 1.00000i | 4.38587 | 0 | 1.00000i | 2.00000 | − | 2.00000i | 0 | −1.60534 | + | 5.99121i | |||||||||||||||||||||||||||||||||||
323.5 | 1.36603 | − | 0.366025i | 0 | 1.73205 | − | 1.00000i | −1.12976 | 0 | 1.00000i | 2.00000 | − | 2.00000i | 0 | −1.54329 | + | 0.413523i | |||||||||||||||||||||||||||||||||||
323.6 | 1.36603 | − | 0.366025i | 0 | 1.73205 | − | 1.00000i | 3.12976 | 0 | 1.00000i | 2.00000 | − | 2.00000i | 0 | 4.27534 | − | 1.14557i | |||||||||||||||||||||||||||||||||||
323.7 | 1.36603 | + | 0.366025i | 0 | 1.73205 | + | 1.00000i | −1.12976 | 0 | − | 1.00000i | 2.00000 | + | 2.00000i | 0 | −1.54329 | − | 0.413523i | ||||||||||||||||||||||||||||||||||
323.8 | 1.36603 | + | 0.366025i | 0 | 1.73205 | + | 1.00000i | 3.12976 | 0 | − | 1.00000i | 2.00000 | + | 2.00000i | 0 | 4.27534 | + | 1.14557i | ||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1512.2.j.b | yes | 8 |
3.b | odd | 2 | 1 | 1512.2.j.a | ✓ | 8 | |
4.b | odd | 2 | 1 | 6048.2.j.b | 8 | ||
8.b | even | 2 | 1 | 6048.2.j.a | 8 | ||
8.d | odd | 2 | 1 | 1512.2.j.a | ✓ | 8 | |
12.b | even | 2 | 1 | 6048.2.j.a | 8 | ||
24.f | even | 2 | 1 | inner | 1512.2.j.b | yes | 8 |
24.h | odd | 2 | 1 | 6048.2.j.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1512.2.j.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
1512.2.j.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
1512.2.j.b | yes | 8 | 1.a | even | 1 | 1 | trivial |
1512.2.j.b | yes | 8 | 24.f | even | 2 | 1 | inner |
6048.2.j.a | 8 | 8.b | even | 2 | 1 | ||
6048.2.j.a | 8 | 12.b | even | 2 | 1 | ||
6048.2.j.b | 8 | 4.b | odd | 2 | 1 | ||
6048.2.j.b | 8 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 4T_{5}^{3} - 10T_{5}^{2} + 28T_{5} + 37 \)
acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \)
$3$
\( T^{8} \)
$5$
\( (T^{4} - 4 T^{3} - 10 T^{2} + 28 T + 37)^{2} \)
$7$
\( (T^{2} + 1)^{4} \)
$11$
\( T^{8} + 44 T^{6} + 510 T^{4} + \cdots + 169 \)
$13$
\( T^{8} + 96 T^{6} + 2984 T^{4} + \cdots + 1936 \)
$17$
\( (T^{2} + 4)^{4} \)
$19$
\( (T^{4} - 8 T^{3} - 46 T^{2} + 344 T + 193)^{2} \)
$23$
\( (T^{4} + 8 T^{3} - 46 T^{2} - 464 T - 827)^{2} \)
$29$
\( (T^{2} - 6 T + 6)^{4} \)
$31$
\( T^{8} + 92 T^{6} + 2310 T^{4} + \cdots + 9409 \)
$37$
\( T^{8} + 132 T^{6} + 5126 T^{4} + \cdots + 146689 \)
$41$
\( T^{8} + 212 T^{6} + 16446 T^{4} + \cdots + 6651241 \)
$43$
\( (T^{4} - 4 T^{3} - 88 T^{2} + 472 T - 524)^{2} \)
$47$
\( (T^{4} - 4 T^{3} - 112 T^{2} + 520 T - 572)^{2} \)
$53$
\( (T^{4} - 88 T^{2} + 192 T + 208)^{2} \)
$59$
\( T^{8} + 240 T^{6} + 16328 T^{4} + \cdots + 913936 \)
$61$
\( (T^{4} + 152 T^{2} + 2704)^{2} \)
$67$
\( (T^{4} - 4 T^{3} - 40 T^{2} + 184 T - 44)^{2} \)
$71$
\( (T^{4} + 16 T^{3} + 26 T^{2} - 88 T - 131)^{2} \)
$73$
\( (T^{4} - 4 T^{3} - 136 T^{2} - 584 T - 716)^{2} \)
$79$
\( T^{8} + 384 T^{6} + \cdots + 18215824 \)
$83$
\( T^{8} + 432 T^{6} + 56840 T^{4} + \cdots + 6697744 \)
$89$
\( T^{8} + 420 T^{6} + 50382 T^{4} + \cdots + 9865881 \)
$97$
\( (T^{4} + 16 T^{3} - 88 T^{2} - 448 T + 1936)^{2} \)
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