Newspace parameters
Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 324.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.82836056527\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{2}\cdot 3^{4} \) |
Twist minimal: | no (minimal twist has level 36) |
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 \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) :
\(\beta_{1}\) | \(=\) | \( -\nu^{3} + \nu^{2} - \nu + 3 \) |
\(\beta_{2}\) | \(=\) | \( -\nu^{3} + \nu^{2} + 5\nu + 2 \) |
\(\beta_{3}\) | \(=\) | \( 4\nu^{3} + 5\nu^{2} - 5\nu - 21 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} - \beta _1 + 1 ) / 6 \) |
\(\nu^{2}\) | \(=\) | \( ( 2\beta_{3} + 3\beta_{2} + 5\beta _1 + 21 ) / 18 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{3} - 5\beta _1 + 36 ) / 9 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).
\(n\) | \(163\) | \(245\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
161.1 |
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0 | 0 | 0 | − | 7.57301i | 0 | 9.11684 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
161.2 | 0 | 0 | 0 | − | 2.37686i | 0 | −8.11684 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
161.3 | 0 | 0 | 0 | 2.37686i | 0 | −8.11684 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
161.4 | 0 | 0 | 0 | 7.57301i | 0 | 9.11684 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 324.3.c.b | 4 | |
3.b | odd | 2 | 1 | inner | 324.3.c.b | 4 | |
4.b | odd | 2 | 1 | 1296.3.e.e | 4 | ||
9.c | even | 3 | 1 | 36.3.g.a | ✓ | 4 | |
9.c | even | 3 | 1 | 108.3.g.a | 4 | ||
9.d | odd | 6 | 1 | 36.3.g.a | ✓ | 4 | |
9.d | odd | 6 | 1 | 108.3.g.a | 4 | ||
12.b | even | 2 | 1 | 1296.3.e.e | 4 | ||
36.f | odd | 6 | 1 | 144.3.q.b | 4 | ||
36.f | odd | 6 | 1 | 432.3.q.b | 4 | ||
36.h | even | 6 | 1 | 144.3.q.b | 4 | ||
36.h | even | 6 | 1 | 432.3.q.b | 4 | ||
45.h | odd | 6 | 1 | 900.3.p.a | 4 | ||
45.h | odd | 6 | 1 | 2700.3.p.b | 4 | ||
45.j | even | 6 | 1 | 900.3.p.a | 4 | ||
45.j | even | 6 | 1 | 2700.3.p.b | 4 | ||
45.k | odd | 12 | 2 | 900.3.u.a | 8 | ||
45.k | odd | 12 | 2 | 2700.3.u.b | 8 | ||
45.l | even | 12 | 2 | 900.3.u.a | 8 | ||
45.l | even | 12 | 2 | 2700.3.u.b | 8 | ||
72.j | odd | 6 | 1 | 576.3.q.d | 4 | ||
72.j | odd | 6 | 1 | 1728.3.q.g | 4 | ||
72.l | even | 6 | 1 | 576.3.q.g | 4 | ||
72.l | even | 6 | 1 | 1728.3.q.h | 4 | ||
72.n | even | 6 | 1 | 576.3.q.d | 4 | ||
72.n | even | 6 | 1 | 1728.3.q.g | 4 | ||
72.p | odd | 6 | 1 | 576.3.q.g | 4 | ||
72.p | odd | 6 | 1 | 1728.3.q.h | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
36.3.g.a | ✓ | 4 | 9.c | even | 3 | 1 | |
36.3.g.a | ✓ | 4 | 9.d | odd | 6 | 1 | |
108.3.g.a | 4 | 9.c | even | 3 | 1 | ||
108.3.g.a | 4 | 9.d | odd | 6 | 1 | ||
144.3.q.b | 4 | 36.f | odd | 6 | 1 | ||
144.3.q.b | 4 | 36.h | even | 6 | 1 | ||
324.3.c.b | 4 | 1.a | even | 1 | 1 | trivial | |
324.3.c.b | 4 | 3.b | odd | 2 | 1 | inner | |
432.3.q.b | 4 | 36.f | odd | 6 | 1 | ||
432.3.q.b | 4 | 36.h | even | 6 | 1 | ||
576.3.q.d | 4 | 72.j | odd | 6 | 1 | ||
576.3.q.d | 4 | 72.n | even | 6 | 1 | ||
576.3.q.g | 4 | 72.l | even | 6 | 1 | ||
576.3.q.g | 4 | 72.p | odd | 6 | 1 | ||
900.3.p.a | 4 | 45.h | odd | 6 | 1 | ||
900.3.p.a | 4 | 45.j | even | 6 | 1 | ||
900.3.u.a | 8 | 45.k | odd | 12 | 2 | ||
900.3.u.a | 8 | 45.l | even | 12 | 2 | ||
1296.3.e.e | 4 | 4.b | odd | 2 | 1 | ||
1296.3.e.e | 4 | 12.b | even | 2 | 1 | ||
1728.3.q.g | 4 | 72.j | odd | 6 | 1 | ||
1728.3.q.g | 4 | 72.n | even | 6 | 1 | ||
1728.3.q.h | 4 | 72.l | even | 6 | 1 | ||
1728.3.q.h | 4 | 72.p | odd | 6 | 1 | ||
2700.3.p.b | 4 | 45.h | odd | 6 | 1 | ||
2700.3.p.b | 4 | 45.j | even | 6 | 1 | ||
2700.3.u.b | 8 | 45.k | odd | 12 | 2 | ||
2700.3.u.b | 8 | 45.l | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 63T_{5}^{2} + 324 \)
acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 63T^{2} + 324 \)
$7$
\( (T^{2} - T - 74)^{2} \)
$11$
\( T^{4} + 414T^{2} + 81 \)
$13$
\( (T^{2} + 5 T - 68)^{2} \)
$17$
\( T^{4} + 387 T^{2} + 20736 \)
$19$
\( (T^{2} - T - 74)^{2} \)
$23$
\( T^{4} + 1683 T^{2} + 627264 \)
$29$
\( T^{4} + 3087 T^{2} + 777924 \)
$31$
\( (T^{2} - 7 T - 656)^{2} \)
$37$
\( (T^{2} + 32 T - 932)^{2} \)
$41$
\( T^{4} + 3222 T^{2} + \cdots + 2424249 \)
$43$
\( (T - 23)^{4} \)
$47$
\( T^{4} + 1539 T^{2} + 104976 \)
$53$
\( T^{4} + 4032 T^{2} + \cdots + 1327104 \)
$59$
\( T^{4} + 5814 T^{2} + 68121 \)
$61$
\( (T^{2} + 41 T - 248)^{2} \)
$67$
\( (T^{2} + 116 T + 691)^{2} \)
$71$
\( T^{4} + 1548 T^{2} + 331776 \)
$73$
\( (T^{2} - 43 T - 206)^{2} \)
$79$
\( (T^{2} + 83 T - 134)^{2} \)
$83$
\( T^{4} + 1539 T^{2} + 104976 \)
$89$
\( T^{4} + 24768 T^{2} + \cdots + 84934656 \)
$97$
\( (T^{2} - 196 T + 9307)^{2} \)
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