Newspace parameters
Level: | \( N \) | \(=\) | \( 1248 = 2^{5} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1248.ca (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.96533017226\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.12960000.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | no (minimal twist has level 312) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{7} + 13\nu ) / 8 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{7} + 29\nu ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{6} + 9 ) / 4 \) |
\(\beta_{4}\) | \(=\) | \( ( 3\nu^{6} - 8\nu^{4} + 24\nu^{2} - 9 ) / 8 \) |
\(\beta_{5}\) | \(=\) | \( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 \) |
\(\beta_{6}\) | \(=\) | \( ( -7\nu^{6} + 24\nu^{4} - 56\nu^{2} + 21 ) / 8 \) |
\(\beta_{7}\) | \(=\) | \( ( 3\nu^{7} - 8\nu^{5} + 22\nu^{3} - \nu ) / 2 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} + 3\beta_{4} - \beta_{3} + 3 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{7} + 2\beta_{5} \) |
\(\nu^{4}\) | \(=\) | \( ( 3\beta_{6} + 7\beta_{4} ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 5\beta_{7} + 11\beta_{5} - 5\beta_{2} + 11\beta_1 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( 4\beta_{3} - 9 \) |
\(\nu^{7}\) | \(=\) | \( ( -13\beta_{2} + 29\beta_1 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times\).
\(n\) | \(703\) | \(769\) | \(833\) | \(1093\) |
\(\chi(n)\) | \(1\) | \(1 + \beta_{4}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
0 | −0.866025 | + | 0.500000i | 0 | −2.23607 | 0 | 0.436492 | + | 0.252009i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||||||||
49.2 | 0 | −0.866025 | + | 0.500000i | 0 | 2.23607 | 0 | −3.43649 | − | 1.98406i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
49.3 | 0 | 0.866025 | − | 0.500000i | 0 | −2.23607 | 0 | −3.43649 | − | 1.98406i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
49.4 | 0 | 0.866025 | − | 0.500000i | 0 | 2.23607 | 0 | 0.436492 | + | 0.252009i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
433.1 | 0 | −0.866025 | − | 0.500000i | 0 | −2.23607 | 0 | 0.436492 | − | 0.252009i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
433.2 | 0 | −0.866025 | − | 0.500000i | 0 | 2.23607 | 0 | −3.43649 | + | 1.98406i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
433.3 | 0 | 0.866025 | + | 0.500000i | 0 | −2.23607 | 0 | −3.43649 | + | 1.98406i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
433.4 | 0 | 0.866025 | + | 0.500000i | 0 | 2.23607 | 0 | 0.436492 | − | 0.252009i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
13.e | even | 6 | 1 | inner |
104.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1248.2.ca.a | 8 | |
4.b | odd | 2 | 1 | 312.2.bk.a | ✓ | 8 | |
8.b | even | 2 | 1 | inner | 1248.2.ca.a | 8 | |
8.d | odd | 2 | 1 | 312.2.bk.a | ✓ | 8 | |
12.b | even | 2 | 1 | 936.2.dg.c | 8 | ||
13.e | even | 6 | 1 | inner | 1248.2.ca.a | 8 | |
24.f | even | 2 | 1 | 936.2.dg.c | 8 | ||
52.i | odd | 6 | 1 | 312.2.bk.a | ✓ | 8 | |
104.p | odd | 6 | 1 | 312.2.bk.a | ✓ | 8 | |
104.s | even | 6 | 1 | inner | 1248.2.ca.a | 8 | |
156.r | even | 6 | 1 | 936.2.dg.c | 8 | ||
312.ba | even | 6 | 1 | 936.2.dg.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
312.2.bk.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
312.2.bk.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
312.2.bk.a | ✓ | 8 | 52.i | odd | 6 | 1 | |
312.2.bk.a | ✓ | 8 | 104.p | odd | 6 | 1 | |
936.2.dg.c | 8 | 12.b | even | 2 | 1 | ||
936.2.dg.c | 8 | 24.f | even | 2 | 1 | ||
936.2.dg.c | 8 | 156.r | even | 6 | 1 | ||
936.2.dg.c | 8 | 312.ba | even | 6 | 1 | ||
1248.2.ca.a | 8 | 1.a | even | 1 | 1 | trivial | |
1248.2.ca.a | 8 | 8.b | even | 2 | 1 | inner | |
1248.2.ca.a | 8 | 13.e | even | 6 | 1 | inner | |
1248.2.ca.a | 8 | 104.s | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 5 \)
acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} - T^{2} + 1)^{2} \)
$5$
\( (T^{2} - 5)^{4} \)
$7$
\( (T^{4} + 6 T^{3} + 10 T^{2} - 12 T + 4)^{2} \)
$11$
\( T^{8} + 16 T^{6} + 252 T^{4} + \cdots + 16 \)
$13$
\( (T^{4} - T^{2} + 169)^{2} \)
$17$
\( (T^{4} + 4 T^{3} + 27 T^{2} - 44 T + 121)^{2} \)
$19$
\( T^{8} + 16 T^{6} + 252 T^{4} + \cdots + 16 \)
$23$
\( (T^{4} - 10 T^{3} + 90 T^{2} - 100 T + 100)^{2} \)
$29$
\( T^{8} - 102 T^{6} + 9963 T^{4} + \cdots + 194481 \)
$31$
\( (T^{2} + 20)^{4} \)
$37$
\( T^{8} + 166 T^{6} + \cdots + 35153041 \)
$41$
\( (T^{4} - 12 T^{3} + 55 T^{2} - 84 T + 49)^{2} \)
$43$
\( T^{8} - 80 T^{6} + 6300 T^{4} + \cdots + 10000 \)
$47$
\( (T^{4} + 16 T^{2} + 4)^{2} \)
$53$
\( (T^{4} + 62 T^{2} + 1)^{2} \)
$59$
\( T^{8} + 136 T^{6} + 17712 T^{4} + \cdots + 614656 \)
$61$
\( (T^{4} - T^{2} + 1)^{2} \)
$67$
\( T^{8} + 240 T^{6} + 56700 T^{4} + \cdots + 810000 \)
$71$
\( (T^{4} + 6 T^{3} - 30 T^{2} - 252 T + 1764)^{2} \)
$73$
\( (T^{4} + 190 T^{2} + 3025)^{2} \)
$79$
\( (T + 14)^{8} \)
$83$
\( (T^{4} - 64 T^{2} + 484)^{2} \)
$89$
\( (T^{4} - 24 T^{3} + 160 T^{2} + 768 T + 1024)^{2} \)
$97$
\( (T^{2} + 30 T + 300)^{4} \)
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