Newspace parameters
Level: | \( N \) | \(=\) | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1008.m (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(27.4660106475\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} - x^{2} - 2x + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | no (minimal twist has level 336) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} - x^{3} - x^{2} - 2x + 4 \) :
\(\beta_{1}\) | \(=\) | \( -\nu^{3} + \nu^{2} + 3\nu + 1 \) |
\(\beta_{2}\) | \(=\) | \( \nu^{3} + \nu^{2} - \nu - 3 \) |
\(\beta_{3}\) | \(=\) | \( 2\nu^{3} - 5 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{3} + 3\beta_{2} + \beta _1 + 3 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{3} + 5 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(577\) | \(757\) | \(785\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
127.1 |
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0 | 0 | 0 | −5.58258 | 0 | − | 2.64575i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
127.2 | 0 | 0 | 0 | −5.58258 | 0 | 2.64575i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
127.3 | 0 | 0 | 0 | 3.58258 | 0 | − | 2.64575i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
127.4 | 0 | 0 | 0 | 3.58258 | 0 | 2.64575i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1008.3.m.b | 4 | |
3.b | odd | 2 | 1 | 336.3.m.c | ✓ | 4 | |
4.b | odd | 2 | 1 | inner | 1008.3.m.b | 4 | |
12.b | even | 2 | 1 | 336.3.m.c | ✓ | 4 | |
21.c | even | 2 | 1 | 2352.3.m.f | 4 | ||
24.f | even | 2 | 1 | 1344.3.m.a | 4 | ||
24.h | odd | 2 | 1 | 1344.3.m.a | 4 | ||
84.h | odd | 2 | 1 | 2352.3.m.f | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
336.3.m.c | ✓ | 4 | 3.b | odd | 2 | 1 | |
336.3.m.c | ✓ | 4 | 12.b | even | 2 | 1 | |
1008.3.m.b | 4 | 1.a | even | 1 | 1 | trivial | |
1008.3.m.b | 4 | 4.b | odd | 2 | 1 | inner | |
1344.3.m.a | 4 | 24.f | even | 2 | 1 | ||
1344.3.m.a | 4 | 24.h | odd | 2 | 1 | ||
2352.3.m.f | 4 | 21.c | even | 2 | 1 | ||
2352.3.m.f | 4 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 2T_{5} - 20 \)
acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( (T^{2} + 2 T - 20)^{2} \)
$7$
\( (T^{2} + 7)^{2} \)
$11$
\( T^{4} + 20T^{2} + 16 \)
$13$
\( (T^{2} - 16 T - 20)^{2} \)
$17$
\( (T^{2} - 26 T - 20)^{2} \)
$19$
\( T^{4} + 320T^{2} + 4096 \)
$23$
\( T^{4} + 836 T^{2} + 71824 \)
$29$
\( (T - 2)^{4} \)
$31$
\( T^{4} + 2400 T^{2} + 665856 \)
$37$
\( (T^{2} - 40 T - 356)^{2} \)
$41$
\( (T^{2} + 34 T - 3260)^{2} \)
$43$
\( T^{4} + 5120 T^{2} + \cdots + 1048576 \)
$47$
\( T^{4} + 1680 T^{2} + 112896 \)
$53$
\( (T^{2} + 8 T - 2084)^{2} \)
$59$
\( T^{4} + 8720 T^{2} + \cdots + 5837056 \)
$61$
\( (T^{2} + 132 T + 1332)^{2} \)
$67$
\( T^{4} + 5712 T^{2} + \cdots + 2822400 \)
$71$
\( T^{4} + 7460 T^{2} + \cdots + 13512976 \)
$73$
\( (T^{2} + 136 T + 3868)^{2} \)
$79$
\( T^{4} + 13200 T^{2} + \cdots + 37161216 \)
$83$
\( T^{4} + 14432 T^{2} + \cdots + 14137600 \)
$89$
\( (T^{2} - 86 T - 4220)^{2} \)
$97$
\( (T^{2} - 160 T + 5644)^{2} \)
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