Newspace parameters
Level: | \( N \) | \(=\) | \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3744.m (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(29.8959905168\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{13})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 2x^{3} - x^{2} + 2x + 27 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{41}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | no (minimal twist has level 936) |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \( x^{4} - 2x^{3} - x^{2} + 2x + 27 \) :
\(\beta_{1}\) | \(=\) | \( ( 4\nu^{3} - 6\nu^{2} + 14\nu - 6 ) / 21 \) |
\(\beta_{2}\) | \(=\) | \( ( -4\nu^{3} + 6\nu^{2} + 28\nu - 15 ) / 21 \) |
\(\beta_{3}\) | \(=\) | \( 2\nu^{2} - 2\nu - 2 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + \beta_{2} + \beta _1 + 3 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 3\beta_{3} - 4\beta_{2} + 17\beta _1 + 8 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).
\(n\) | \(703\) | \(2017\) | \(2081\) | \(2341\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1585.1 |
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0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
1585.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
1585.3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
1585.4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
312.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-78}) \) |
3.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
24.h | odd | 2 | 1 | inner |
39.d | odd | 2 | 1 | inner |
104.e | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3744.2.m.e | 4 | |
3.b | odd | 2 | 1 | inner | 3744.2.m.e | 4 | |
4.b | odd | 2 | 1 | 936.2.m.e | ✓ | 4 | |
8.b | even | 2 | 1 | inner | 3744.2.m.e | 4 | |
8.d | odd | 2 | 1 | 936.2.m.e | ✓ | 4 | |
12.b | even | 2 | 1 | 936.2.m.e | ✓ | 4 | |
13.b | even | 2 | 1 | inner | 3744.2.m.e | 4 | |
24.f | even | 2 | 1 | 936.2.m.e | ✓ | 4 | |
24.h | odd | 2 | 1 | inner | 3744.2.m.e | 4 | |
39.d | odd | 2 | 1 | inner | 3744.2.m.e | 4 | |
52.b | odd | 2 | 1 | 936.2.m.e | ✓ | 4 | |
104.e | even | 2 | 1 | inner | 3744.2.m.e | 4 | |
104.h | odd | 2 | 1 | 936.2.m.e | ✓ | 4 | |
156.h | even | 2 | 1 | 936.2.m.e | ✓ | 4 | |
312.b | odd | 2 | 1 | CM | 3744.2.m.e | 4 | |
312.h | even | 2 | 1 | 936.2.m.e | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
936.2.m.e | ✓ | 4 | 4.b | odd | 2 | 1 | |
936.2.m.e | ✓ | 4 | 8.d | odd | 2 | 1 | |
936.2.m.e | ✓ | 4 | 12.b | even | 2 | 1 | |
936.2.m.e | ✓ | 4 | 24.f | even | 2 | 1 | |
936.2.m.e | ✓ | 4 | 52.b | odd | 2 | 1 | |
936.2.m.e | ✓ | 4 | 104.h | odd | 2 | 1 | |
936.2.m.e | ✓ | 4 | 156.h | even | 2 | 1 | |
936.2.m.e | ✓ | 4 | 312.h | even | 2 | 1 | |
3744.2.m.e | 4 | 1.a | even | 1 | 1 | trivial | |
3744.2.m.e | 4 | 3.b | odd | 2 | 1 | inner | |
3744.2.m.e | 4 | 8.b | even | 2 | 1 | inner | |
3744.2.m.e | 4 | 13.b | even | 2 | 1 | inner | |
3744.2.m.e | 4 | 24.h | odd | 2 | 1 | inner | |
3744.2.m.e | 4 | 39.d | odd | 2 | 1 | inner | |
3744.2.m.e | 4 | 104.e | even | 2 | 1 | inner | |
3744.2.m.e | 4 | 312.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{2}^{\mathrm{new}}(3744, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( T^{4} \)
$7$
\( T^{4} \)
$11$
\( T^{4} \)
$13$
\( (T^{2} - 13)^{2} \)
$17$
\( T^{4} \)
$19$
\( (T^{2} - 52)^{2} \)
$23$
\( T^{4} \)
$29$
\( (T^{2} + 104)^{2} \)
$31$
\( T^{4} \)
$37$
\( (T^{2} - 52)^{2} \)
$41$
\( (T^{2} + 8)^{2} \)
$43$
\( T^{4} \)
$47$
\( (T^{2} + 32)^{2} \)
$53$
\( (T^{2} + 104)^{2} \)
$59$
\( T^{4} \)
$61$
\( T^{4} \)
$67$
\( (T^{2} - 52)^{2} \)
$71$
\( (T^{2} + 128)^{2} \)
$73$
\( T^{4} \)
$79$
\( (T + 2)^{4} \)
$83$
\( T^{4} \)
$89$
\( (T^{2} + 200)^{2} \)
$97$
\( T^{4} \)
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