Newspace parameters
Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 72.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.96185790339\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-6}, \sqrt{10})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 2x^{2} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{2} \) |
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 \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} - 2x^{2} + 16 \) :
\(\beta_{1}\) | \(=\) | \( 2\nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 1 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} - 2\nu ) / 4 \) |
\(\nu\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 1 \) |
\(\nu^{3}\) | \(=\) | \( 4\beta_{3} + \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(55\) | \(65\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
|
−1.58114 | − | 1.22474i | 0 | 1.00000 | + | 3.87298i | − | 4.89898i | 0 | − | 7.74597i | 3.16228 | − | 7.34847i | 0 | −6.00000 | + | 7.74597i | ||||||||||||||||||||
19.2 | −1.58114 | + | 1.22474i | 0 | 1.00000 | − | 3.87298i | 4.89898i | 0 | 7.74597i | 3.16228 | + | 7.34847i | 0 | −6.00000 | − | 7.74597i | |||||||||||||||||||||||
19.3 | 1.58114 | − | 1.22474i | 0 | 1.00000 | − | 3.87298i | − | 4.89898i | 0 | 7.74597i | −3.16228 | − | 7.34847i | 0 | −6.00000 | − | 7.74597i | ||||||||||||||||||||||
19.4 | 1.58114 | + | 1.22474i | 0 | 1.00000 | + | 3.87298i | 4.89898i | 0 | − | 7.74597i | −3.16228 | + | 7.34847i | 0 | −6.00000 | + | 7.74597i | ||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 72.3.b.c | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 72.3.b.c | ✓ | 4 |
4.b | odd | 2 | 1 | 288.3.b.c | 4 | ||
8.b | even | 2 | 1 | 288.3.b.c | 4 | ||
8.d | odd | 2 | 1 | inner | 72.3.b.c | ✓ | 4 |
12.b | even | 2 | 1 | 288.3.b.c | 4 | ||
16.e | even | 4 | 2 | 2304.3.g.y | 8 | ||
16.f | odd | 4 | 2 | 2304.3.g.y | 8 | ||
24.f | even | 2 | 1 | inner | 72.3.b.c | ✓ | 4 |
24.h | odd | 2 | 1 | 288.3.b.c | 4 | ||
48.i | odd | 4 | 2 | 2304.3.g.y | 8 | ||
48.k | even | 4 | 2 | 2304.3.g.y | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
72.3.b.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
72.3.b.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
72.3.b.c | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
72.3.b.c | ✓ | 4 | 24.f | even | 2 | 1 | inner |
288.3.b.c | 4 | 4.b | odd | 2 | 1 | ||
288.3.b.c | 4 | 8.b | even | 2 | 1 | ||
288.3.b.c | 4 | 12.b | even | 2 | 1 | ||
288.3.b.c | 4 | 24.h | odd | 2 | 1 | ||
2304.3.g.y | 8 | 16.e | even | 4 | 2 | ||
2304.3.g.y | 8 | 16.f | odd | 4 | 2 | ||
2304.3.g.y | 8 | 48.i | odd | 4 | 2 | ||
2304.3.g.y | 8 | 48.k | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 24 \)
acting on \(S_{3}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2T^{2} + 16 \)
$3$
\( T^{4} \)
$5$
\( (T^{2} + 24)^{2} \)
$7$
\( (T^{2} + 60)^{2} \)
$11$
\( (T^{2} - 160)^{2} \)
$13$
\( (T^{2} + 240)^{2} \)
$17$
\( (T^{2} - 640)^{2} \)
$19$
\( (T - 8)^{4} \)
$23$
\( (T^{2} + 1536)^{2} \)
$29$
\( (T^{2} + 600)^{2} \)
$31$
\( (T^{2} + 60)^{2} \)
$37$
\( (T^{2} + 2160)^{2} \)
$41$
\( (T^{2} - 640)^{2} \)
$43$
\( (T + 40)^{4} \)
$47$
\( (T^{2} + 1536)^{2} \)
$53$
\( (T^{2} + 216)^{2} \)
$59$
\( (T^{2} - 640)^{2} \)
$61$
\( (T^{2} + 240)^{2} \)
$67$
\( (T - 80)^{4} \)
$71$
\( T^{4} \)
$73$
\( (T + 10)^{4} \)
$79$
\( (T^{2} + 2940)^{2} \)
$83$
\( (T^{2} - 19360)^{2} \)
$89$
\( (T^{2} - 2560)^{2} \)
$97$
\( (T - 50)^{4} \)
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