Newspace parameters
Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 72.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.574922894553\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 2\nu + 3 ) / 2 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( -2\beta_{3} + 2\beta _1 + 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(55\) | \(65\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 |
|
0 | −1.18614 | + | 1.26217i | 0 | 1.68614 | + | 2.92048i | 0 | 0.686141 | − | 1.18843i | 0 | −0.186141 | − | 2.99422i | 0 | ||||||||||||||||||||||
25.2 | 0 | 1.68614 | − | 0.396143i | 0 | −1.18614 | − | 2.05446i | 0 | −2.18614 | + | 3.78651i | 0 | 2.68614 | − | 1.33591i | 0 | |||||||||||||||||||||||
49.1 | 0 | −1.18614 | − | 1.26217i | 0 | 1.68614 | − | 2.92048i | 0 | 0.686141 | + | 1.18843i | 0 | −0.186141 | + | 2.99422i | 0 | |||||||||||||||||||||||
49.2 | 0 | 1.68614 | + | 0.396143i | 0 | −1.18614 | + | 2.05446i | 0 | −2.18614 | − | 3.78651i | 0 | 2.68614 | + | 1.33591i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 72.2.i.b | ✓ | 4 |
3.b | odd | 2 | 1 | 216.2.i.b | 4 | ||
4.b | odd | 2 | 1 | 144.2.i.d | 4 | ||
8.b | even | 2 | 1 | 576.2.i.j | 4 | ||
8.d | odd | 2 | 1 | 576.2.i.l | 4 | ||
9.c | even | 3 | 1 | inner | 72.2.i.b | ✓ | 4 |
9.c | even | 3 | 1 | 648.2.a.f | 2 | ||
9.d | odd | 6 | 1 | 216.2.i.b | 4 | ||
9.d | odd | 6 | 1 | 648.2.a.g | 2 | ||
12.b | even | 2 | 1 | 432.2.i.d | 4 | ||
24.f | even | 2 | 1 | 1728.2.i.j | 4 | ||
24.h | odd | 2 | 1 | 1728.2.i.i | 4 | ||
36.f | odd | 6 | 1 | 144.2.i.d | 4 | ||
36.f | odd | 6 | 1 | 1296.2.a.n | 2 | ||
36.h | even | 6 | 1 | 432.2.i.d | 4 | ||
36.h | even | 6 | 1 | 1296.2.a.p | 2 | ||
72.j | odd | 6 | 1 | 1728.2.i.i | 4 | ||
72.j | odd | 6 | 1 | 5184.2.a.bp | 2 | ||
72.l | even | 6 | 1 | 1728.2.i.j | 4 | ||
72.l | even | 6 | 1 | 5184.2.a.bo | 2 | ||
72.n | even | 6 | 1 | 576.2.i.j | 4 | ||
72.n | even | 6 | 1 | 5184.2.a.bt | 2 | ||
72.p | odd | 6 | 1 | 576.2.i.l | 4 | ||
72.p | odd | 6 | 1 | 5184.2.a.bs | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
72.2.i.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
72.2.i.b | ✓ | 4 | 9.c | even | 3 | 1 | inner |
144.2.i.d | 4 | 4.b | odd | 2 | 1 | ||
144.2.i.d | 4 | 36.f | odd | 6 | 1 | ||
216.2.i.b | 4 | 3.b | odd | 2 | 1 | ||
216.2.i.b | 4 | 9.d | odd | 6 | 1 | ||
432.2.i.d | 4 | 12.b | even | 2 | 1 | ||
432.2.i.d | 4 | 36.h | even | 6 | 1 | ||
576.2.i.j | 4 | 8.b | even | 2 | 1 | ||
576.2.i.j | 4 | 72.n | even | 6 | 1 | ||
576.2.i.l | 4 | 8.d | odd | 2 | 1 | ||
576.2.i.l | 4 | 72.p | odd | 6 | 1 | ||
648.2.a.f | 2 | 9.c | even | 3 | 1 | ||
648.2.a.g | 2 | 9.d | odd | 6 | 1 | ||
1296.2.a.n | 2 | 36.f | odd | 6 | 1 | ||
1296.2.a.p | 2 | 36.h | even | 6 | 1 | ||
1728.2.i.i | 4 | 24.h | odd | 2 | 1 | ||
1728.2.i.i | 4 | 72.j | odd | 6 | 1 | ||
1728.2.i.j | 4 | 24.f | even | 2 | 1 | ||
1728.2.i.j | 4 | 72.l | even | 6 | 1 | ||
5184.2.a.bo | 2 | 72.l | even | 6 | 1 | ||
5184.2.a.bp | 2 | 72.j | odd | 6 | 1 | ||
5184.2.a.bs | 2 | 72.p | odd | 6 | 1 | ||
5184.2.a.bt | 2 | 72.n | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - T_{5}^{3} + 9T_{5}^{2} + 8T_{5} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - T^{3} - 2 T^{2} - 3 T + 9 \)
$5$
\( T^{4} - T^{3} + 9 T^{2} + 8 T + 64 \)
$7$
\( T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36 \)
$11$
\( (T^{2} + T + 1)^{2} \)
$13$
\( T^{4} + 5 T^{3} + 27 T^{2} - 10 T + 4 \)
$17$
\( (T^{2} + 5 T - 2)^{2} \)
$19$
\( (T^{2} - 7 T + 4)^{2} \)
$23$
\( T^{4} + 5 T^{3} + 27 T^{2} - 10 T + 4 \)
$29$
\( T^{4} - 3 T^{3} + 15 T^{2} + 18 T + 36 \)
$31$
\( T^{4} + 7 T^{3} + 45 T^{2} + 28 T + 16 \)
$37$
\( (T^{2} - 6 T - 24)^{2} \)
$41$
\( T^{4} - 12 T^{3} + 141 T^{2} - 36 T + 9 \)
$43$
\( T^{4} + 8 T^{3} + 81 T^{2} - 136 T + 289 \)
$47$
\( T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36 \)
$53$
\( (T^{2} + 10 T - 8)^{2} \)
$59$
\( (T^{2} + 7 T + 49)^{2} \)
$61$
\( T^{4} - T^{3} + 9 T^{2} + 8 T + 64 \)
$67$
\( T^{4} + 4 T^{3} + 45 T^{2} - 116 T + 841 \)
$71$
\( (T + 4)^{4} \)
$73$
\( (T^{2} + 7 T - 62)^{2} \)
$79$
\( T^{4} - 7 T^{3} + 45 T^{2} - 28 T + 16 \)
$83$
\( T^{4} + 25 T^{3} + 477 T^{2} + \cdots + 21904 \)
$89$
\( (T - 6)^{4} \)
$97$
\( T^{4} - 8 T^{3} + 81 T^{2} + 136 T + 289 \)
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