Newspace parameters
Level: | \( N \) | \(=\) | \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3724.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(29.7362897127\) |
Analytic rank: | \(0\) |
Dimension: | \(3\) |
Coefficient field: | 3.3.733.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{3} - x^{2} - 7x + 8 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 532) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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 \( x^{3} - x^{2} - 7x + 8 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 5 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 5 \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
0 | −2.69639 | 0 | −1.42586 | 0 | 0 | 0 | 4.27053 | 0 | |||||||||||||||||||||||||||
1.2 | 0 | 1.17819 | 0 | −3.43366 | 0 | 0 | 0 | −1.61186 | 0 | ||||||||||||||||||||||||||||
1.3 | 0 | 2.51820 | 0 | 2.85952 | 0 | 0 | 0 | 3.34132 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(7\) | \(-1\) |
\(19\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3724.2.a.i | 3 | |
7.b | odd | 2 | 1 | 532.2.a.e | ✓ | 3 | |
21.c | even | 2 | 1 | 4788.2.a.o | 3 | ||
28.d | even | 2 | 1 | 2128.2.a.r | 3 | ||
56.e | even | 2 | 1 | 8512.2.a.bl | 3 | ||
56.h | odd | 2 | 1 | 8512.2.a.bn | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
532.2.a.e | ✓ | 3 | 7.b | odd | 2 | 1 | |
2128.2.a.r | 3 | 28.d | even | 2 | 1 | ||
3724.2.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
4788.2.a.o | 3 | 21.c | even | 2 | 1 | ||
8512.2.a.bl | 3 | 56.e | even | 2 | 1 | ||
8512.2.a.bn | 3 | 56.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - T_{3}^{2} - 7T_{3} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3724))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} \)
$3$
\( T^{3} - T^{2} - 7T + 8 \)
$5$
\( T^{3} + 2 T^{2} - 9 T - 14 \)
$7$
\( T^{3} \)
$11$
\( T^{3} + 3 T^{2} - 7 T - 20 \)
$13$
\( T^{3} + 8 T^{2} + 11 T - 16 \)
$17$
\( T^{3} + 9 T^{2} + 17 T - 14 \)
$19$
\( (T - 1)^{3} \)
$23$
\( T^{3} - 4 T^{2} - 5 T + 4 \)
$29$
\( T^{3} - 11 T^{2} + 15 T + 2 \)
$31$
\( T^{3} - 5 T^{2} - 17 T - 4 \)
$37$
\( T^{3} - 4 T^{2} - 19 T + 50 \)
$41$
\( T^{3} + 11 T^{2} - 13 T - 280 \)
$43$
\( T^{3} + 4 T^{2} - 92 T + 32 \)
$47$
\( T^{3} - 2 T^{2} - 75 T - 184 \)
$53$
\( T^{3} - 9 T^{2} - 39 T + 322 \)
$59$
\( T^{3} - 12 T^{2} - 45 T + 108 \)
$61$
\( T^{3} - 2 T^{2} - 157 T + 202 \)
$67$
\( T^{3} + 9 T^{2} + 17 T - 14 \)
$71$
\( T^{3} - 55T - 58 \)
$73$
\( T^{3} - 21 T^{2} + 83 T + 302 \)
$79$
\( T^{3} - 6 T^{2} - 148 T + 1016 \)
$83$
\( T^{3} - 7 T^{2} - 45 T + 100 \)
$89$
\( T^{3} - 18 T^{2} + 68 T - 64 \)
$97$
\( T^{3} + 28 T^{2} + 39 T - 2536 \)
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