Newspace parameters
Level: | \( N \) | \(=\) | \( 24 = 2^{3} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 24.f (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.41604584014\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{8} \) |
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 \( x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 3 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} - 10\nu^{5} + 120\nu^{3} - 384\nu ) / 256 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{6} - 4\nu^{5} + 10\nu^{4} + 8\nu^{3} - 56\nu^{2} - 160\nu + 384 ) / 128 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{6} + 4\nu^{5} + 10\nu^{4} - 8\nu^{3} - 56\nu^{2} + 160\nu + 256 ) / 128 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{7} + 2\nu^{6} + 2\nu^{5} + 44\nu^{4} + 24\nu^{3} - 16\nu^{2} - 192\nu + 2048 ) / 256 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{7} - 2\nu^{6} + 2\nu^{5} - 44\nu^{4} + 24\nu^{3} + 16\nu^{2} - 192\nu - 2048 ) / 256 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 3 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta _1 + 1 \) |
\(\nu^{4}\) | \(=\) | \( -2\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{2} - 36 \) |
\(\nu^{5}\) | \(=\) | \( 2\beta_{7} + 2\beta_{6} + 18\beta_{5} - 18\beta_{4} + 4\beta_{3} - 36\beta _1 + 18 \) |
\(\nu^{6}\) | \(=\) | \( -20\beta_{7} + 20\beta_{6} - 44\beta_{5} - 44\beta_{4} - 36\beta_{2} - 208 \) |
\(\nu^{7}\) | \(=\) | \( -100\beta_{7} - 100\beta_{6} + 60\beta_{5} - 60\beta_{4} + 56\beta_{3} - 216\beta _1 + 60 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).
\(n\) | \(7\) | \(13\) | \(17\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 |
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−2.58576 | − | 1.14624i | 1.37228 | + | 5.01167i | 5.37228 | + | 5.92778i | 12.2683 | 2.19618 | − | 14.5319i | 14.0624i | −7.09677 | − | 21.4857i | −23.2337 | + | 13.7548i | −31.7228 | − | 14.0624i | ||||||||||||||||||||||||||||
11.2 | −2.58576 | + | 1.14624i | 1.37228 | − | 5.01167i | 5.37228 | − | 5.92778i | 12.2683 | 2.19618 | + | 14.5319i | − | 14.0624i | −7.09677 | + | 21.4857i | −23.2337 | − | 13.7548i | −31.7228 | + | 14.0624i | ||||||||||||||||||||||||||||
11.3 | −1.95291 | − | 2.04601i | −4.37228 | − | 2.80770i | −0.372281 | + | 7.99133i | −13.1715 | 2.79411 | + | 14.4289i | − | 26.9490i | 17.0773 | − | 14.8447i | 11.2337 | + | 24.5521i | 25.7228 | + | 26.9490i | ||||||||||||||||||||||||||||
11.4 | −1.95291 | + | 2.04601i | −4.37228 | + | 2.80770i | −0.372281 | − | 7.99133i | −13.1715 | 2.79411 | − | 14.4289i | 26.9490i | 17.0773 | + | 14.8447i | 11.2337 | − | 24.5521i | 25.7228 | − | 26.9490i | |||||||||||||||||||||||||||||
11.5 | 1.95291 | − | 2.04601i | −4.37228 | − | 2.80770i | −0.372281 | − | 7.99133i | 13.1715 | −14.2832 | + | 3.46254i | 26.9490i | −17.0773 | − | 14.8447i | 11.2337 | + | 24.5521i | 25.7228 | − | 26.9490i | |||||||||||||||||||||||||||||
11.6 | 1.95291 | + | 2.04601i | −4.37228 | + | 2.80770i | −0.372281 | + | 7.99133i | 13.1715 | −14.2832 | − | 3.46254i | − | 26.9490i | −17.0773 | + | 14.8447i | 11.2337 | − | 24.5521i | 25.7228 | + | 26.9490i | ||||||||||||||||||||||||||||
11.7 | 2.58576 | − | 1.14624i | 1.37228 | + | 5.01167i | 5.37228 | − | 5.92778i | −12.2683 | 9.29295 | + | 11.3860i | − | 14.0624i | 7.09677 | − | 21.4857i | −23.2337 | + | 13.7548i | −31.7228 | + | 14.0624i | ||||||||||||||||||||||||||||
11.8 | 2.58576 | + | 1.14624i | 1.37228 | − | 5.01167i | 5.37228 | + | 5.92778i | −12.2683 | 9.29295 | − | 11.3860i | 14.0624i | 7.09677 | + | 21.4857i | −23.2337 | − | 13.7548i | −31.7228 | − | 14.0624i | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
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 | 24.4.f.b | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 24.4.f.b | ✓ | 8 |
4.b | odd | 2 | 1 | 96.4.f.b | 8 | ||
8.b | even | 2 | 1 | 96.4.f.b | 8 | ||
8.d | odd | 2 | 1 | inner | 24.4.f.b | ✓ | 8 |
12.b | even | 2 | 1 | 96.4.f.b | 8 | ||
16.e | even | 4 | 2 | 768.4.c.v | 16 | ||
16.f | odd | 4 | 2 | 768.4.c.v | 16 | ||
24.f | even | 2 | 1 | inner | 24.4.f.b | ✓ | 8 |
24.h | odd | 2 | 1 | 96.4.f.b | 8 | ||
48.i | odd | 4 | 2 | 768.4.c.v | 16 | ||
48.k | even | 4 | 2 | 768.4.c.v | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
24.4.f.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
24.4.f.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
24.4.f.b | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
24.4.f.b | ✓ | 8 | 24.f | even | 2 | 1 | inner |
96.4.f.b | 8 | 4.b | odd | 2 | 1 | ||
96.4.f.b | 8 | 8.b | even | 2 | 1 | ||
96.4.f.b | 8 | 12.b | even | 2 | 1 | ||
96.4.f.b | 8 | 24.h | odd | 2 | 1 | ||
768.4.c.v | 16 | 16.e | even | 4 | 2 | ||
768.4.c.v | 16 | 16.f | odd | 4 | 2 | ||
768.4.c.v | 16 | 48.i | odd | 4 | 2 | ||
768.4.c.v | 16 | 48.k | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 324T_{5}^{2} + 26112 \)
acting on \(S_{4}^{\mathrm{new}}(24, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 10 T^{6} + 120 T^{4} + \cdots + 4096 \)
$3$
\( (T^{4} + 6 T^{3} + 30 T^{2} + 162 T + 729)^{2} \)
$5$
\( (T^{4} - 324 T^{2} + 26112)^{2} \)
$7$
\( (T^{4} + 924 T^{2} + 143616)^{2} \)
$11$
\( (T^{4} + 484 T^{2} + 352)^{2} \)
$13$
\( (T^{4} + 5808 T^{2} + 574464)^{2} \)
$17$
\( (T^{4} + 2464 T^{2} + 90112)^{2} \)
$19$
\( (T^{2} - 46 T - 3464)^{4} \)
$23$
\( (T^{4} - 17472 T^{2} + 1671168)^{2} \)
$29$
\( (T^{4} - 43620 T^{2} + \cdots + 448108032)^{2} \)
$31$
\( (T^{4} + 20988 T^{2} + 41505024)^{2} \)
$37$
\( (T^{4} + 77616 T^{2} + \cdots + 1494180864)^{2} \)
$41$
\( (T^{4} + 193600 T^{2} + \cdots + 3151126528)^{2} \)
$43$
\( (T^{2} + 38 T - 49832)^{4} \)
$47$
\( (T^{4} - 321792 T^{2} + \cdots + 427819008)^{2} \)
$53$
\( (T^{4} - 177156 T^{2} + \cdots + 7033554432)^{2} \)
$59$
\( (T^{4} + 21604 T^{2} + 296032)^{2} \)
$61$
\( (T^{4} + 550704 T^{2} + \cdots + 18820015104)^{2} \)
$67$
\( (T^{2} + 374 T + 32296)^{4} \)
$71$
\( (T^{4} - 654912 T^{2} + \cdots + 103614087168)^{2} \)
$73$
\( (T^{2} - 268 T - 704876)^{4} \)
$79$
\( (T^{4} + 1028412 T^{2} + \cdots + 99653562624)^{2} \)
$83$
\( (T^{4} + 396484 T^{2} + \cdots + 2128431712)^{2} \)
$89$
\( (T^{4} + 2644576 T^{2} + \cdots + 147293673472)^{2} \)
$97$
\( (T^{2} + 968 T - 33044)^{4} \)
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