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: | \( 2^{8} \) |
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 \( x^{4} - x^{3} - x^{2} - 2x + 4 \) :
\(\beta_{1}\) | \(=\) | \( 2\nu^{3} - 5 \) |
\(\beta_{2}\) | \(=\) | \( -4\nu^{3} + 4\nu^{2} + 12\nu + 4 \) |
\(\beta_{3}\) | \(=\) | \( -4\nu^{3} - 4\nu^{2} + 4\nu + 12 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} + 4\beta _1 + 4 ) / 16 \) |
\(\nu^{2}\) | \(=\) | \( ( -3\beta_{3} + \beta_{2} - 4\beta _1 + 12 ) / 16 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta _1 + 5 ) / 2 \) |
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 | 2.00000 | 0 | − | 2.64575i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
127.2 | 0 | 0 | 0 | 2.00000 | 0 | − | 2.64575i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
127.3 | 0 | 0 | 0 | 2.00000 | 0 | 2.64575i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
127.4 | 0 | 0 | 0 | 2.00000 | 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.e | 4 | |
3.b | odd | 2 | 1 | 336.3.m.b | ✓ | 4 | |
4.b | odd | 2 | 1 | inner | 1008.3.m.e | 4 | |
12.b | even | 2 | 1 | 336.3.m.b | ✓ | 4 | |
21.c | even | 2 | 1 | 2352.3.m.i | 4 | ||
24.f | even | 2 | 1 | 1344.3.m.b | 4 | ||
24.h | odd | 2 | 1 | 1344.3.m.b | 4 | ||
84.h | odd | 2 | 1 | 2352.3.m.i | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
336.3.m.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
336.3.m.b | ✓ | 4 | 12.b | even | 2 | 1 | |
1008.3.m.e | 4 | 1.a | even | 1 | 1 | trivial | |
1008.3.m.e | 4 | 4.b | odd | 2 | 1 | inner | |
1344.3.m.b | 4 | 24.f | even | 2 | 1 | ||
1344.3.m.b | 4 | 24.h | odd | 2 | 1 | ||
2352.3.m.i | 4 | 21.c | even | 2 | 1 | ||
2352.3.m.i | 4 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 2 \)
acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( (T - 2)^{4} \)
$7$
\( (T^{2} + 7)^{2} \)
$11$
\( T^{4} + 320T^{2} + 4096 \)
$13$
\( (T^{2} - 4 T - 332)^{2} \)
$17$
\( (T^{2} + 28 T - 140)^{2} \)
$19$
\( T^{4} + 992 T^{2} + 160000 \)
$23$
\( T^{4} + 1088 T^{2} + 102400 \)
$29$
\( (T^{2} - 28 T - 1148)^{2} \)
$31$
\( (T^{2} + 768)^{2} \)
$37$
\( (T^{2} - 52 T - 668)^{2} \)
$41$
\( (T^{2} - 20 T - 236)^{2} \)
$43$
\( T^{4} + 2432 T^{2} + 102400 \)
$47$
\( T^{4} + 9600 T^{2} + \cdots + 10653696 \)
$53$
\( (T^{2} + 20 T - 1244)^{2} \)
$59$
\( T^{4} + 1760T^{2} + 256 \)
$61$
\( (T^{2} + 12 T - 300)^{2} \)
$67$
\( (T^{2} + 3072)^{2} \)
$71$
\( T^{4} + 11840 T^{2} + \cdots + 25563136 \)
$73$
\( (T^{2} - 116 T + 2020)^{2} \)
$79$
\( (T^{2} + 9408)^{2} \)
$83$
\( T^{4} + 15200 T^{2} + \cdots + 45373696 \)
$89$
\( (T^{2} + 76 T - 1580)^{2} \)
$97$
\( (T^{2} + 44 T - 4892)^{2} \)
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