Newspace parameters
Level: | \( N \) | \(=\) | \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1872.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(14.9479952584\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.3317760000.8 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
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 \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{6} + 14\nu^{4} + 56\nu^{2} + 27 ) / 63 \) |
\(\beta_{2}\) | \(=\) | \( ( -4\nu^{7} - 7\nu^{5} + 35\nu^{3} - 81\nu ) / 189 \) |
\(\beta_{3}\) | \(=\) | \( ( 10\nu^{7} + 49\nu^{5} + 133\nu^{3} + 801\nu ) / 189 \) |
\(\beta_{4}\) | \(=\) | \( ( -2\nu^{7} + \nu^{5} - 5\nu^{3} - 63\nu ) / 27 \) |
\(\beta_{5}\) | \(=\) | \( ( -4\nu^{7} - 7\nu^{5} - 19\nu^{3} - 81\nu ) / 27 \) |
\(\beta_{6}\) | \(=\) | \( ( -2\nu^{6} - 8\nu^{4} + 4\nu^{2} - 36 ) / 9 \) |
\(\beta_{7}\) | \(=\) | \( ( 17\nu^{6} + 14\nu^{4} + 56\nu^{2} + 423 ) / 63 \) |
\(\nu\) | \(=\) | \( ( \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{7} + \beta_{6} + 3\beta _1 - 4 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( -\beta_{5} + 7\beta_{2} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( -3\beta_{7} - 4\beta_{6} + 5\beta _1 + 2 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( -5\beta_{5} + 19\beta_{4} + 5\beta_{3} - 19\beta_{2} ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( 7\beta_{7} - 7\beta _1 - 44 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( -29\beta_{5} - 13\beta_{4} - 29\beta_{3} - 13\beta_{2} ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).
\(n\) | \(145\) | \(209\) | \(469\) | \(703\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
287.1 |
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0 | 0 | 0 | − | 1.41421i | 0 | − | 3.96812i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
287.2 | 0 | 0 | 0 | − | 1.41421i | 0 | − | 0.504017i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
287.3 | 0 | 0 | 0 | − | 1.41421i | 0 | 0.504017i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
287.4 | 0 | 0 | 0 | − | 1.41421i | 0 | 3.96812i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
287.5 | 0 | 0 | 0 | 1.41421i | 0 | − | 3.96812i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
287.6 | 0 | 0 | 0 | 1.41421i | 0 | − | 0.504017i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
287.7 | 0 | 0 | 0 | 1.41421i | 0 | 0.504017i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
287.8 | 0 | 0 | 0 | 1.41421i | 0 | 3.96812i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1872.2.d.d | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 1872.2.d.d | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 1872.2.d.d | ✓ | 8 |
8.b | even | 2 | 1 | 7488.2.d.g | 8 | ||
8.d | odd | 2 | 1 | 7488.2.d.g | 8 | ||
12.b | even | 2 | 1 | inner | 1872.2.d.d | ✓ | 8 |
24.f | even | 2 | 1 | 7488.2.d.g | 8 | ||
24.h | odd | 2 | 1 | 7488.2.d.g | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1872.2.d.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
1872.2.d.d | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
1872.2.d.d | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
1872.2.d.d | ✓ | 8 | 12.b | even | 2 | 1 | inner |
7488.2.d.g | 8 | 8.b | even | 2 | 1 | ||
7488.2.d.g | 8 | 8.d | odd | 2 | 1 | ||
7488.2.d.g | 8 | 24.f | even | 2 | 1 | ||
7488.2.d.g | 8 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( (T^{2} + 2)^{4} \)
$7$
\( (T^{4} + 16 T^{2} + 4)^{2} \)
$11$
\( (T^{2} - 10)^{4} \)
$13$
\( (T - 1)^{8} \)
$17$
\( (T^{2} + 30)^{4} \)
$19$
\( (T^{4} + 96 T^{2} + 1764)^{2} \)
$23$
\( (T^{4} - 32 T^{2} + 16)^{2} \)
$29$
\( (T^{2} + 18)^{4} \)
$31$
\( (T^{4} + 64 T^{2} + 484)^{2} \)
$37$
\( (T^{2} - 4 T - 56)^{4} \)
$41$
\( (T^{4} + 244 T^{2} + 13924)^{2} \)
$43$
\( (T^{2} + 12)^{4} \)
$47$
\( (T^{4} - 212 T^{2} + 7396)^{2} \)
$53$
\( (T^{4} + 76 T^{2} + 484)^{2} \)
$59$
\( (T^{2} - 54)^{4} \)
$61$
\( (T^{2} - 60)^{4} \)
$67$
\( (T^{4} + 96 T^{2} + 1764)^{2} \)
$71$
\( (T^{4} - 188 T^{2} + 196)^{2} \)
$73$
\( (T^{2} + 4 T - 56)^{4} \)
$79$
\( (T^{4} + 64 T^{2} + 64)^{2} \)
$83$
\( (T^{2} - 54)^{4} \)
$89$
\( (T^{4} + 244 T^{2} + 13924)^{2} \)
$97$
\( (T^{2} - 12 T - 24)^{4} \)
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