Newspace parameters
Level: | \( N \) | \(=\) | \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 540.n (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.31192170915\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 2x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 180) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \( x^{4} + 2x^{2} + 4 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} ) / 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 2\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/540\mathbb{Z}\right)^\times\).
\(n\) | \(217\) | \(271\) | \(461\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
179.1 |
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−0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | −2.20711 | + | 0.358719i | 0 | 1.50000 | + | 2.59808i | 2.82843 | 0 | 2.00000 | + | 2.44949i | ||||||||||||||||||||
179.2 | 0.707107 | + | 1.22474i | 0 | −1.00000 | + | 1.73205i | −0.792893 | − | 2.09077i | 0 | 1.50000 | + | 2.59808i | −2.82843 | 0 | 2.00000 | − | 2.44949i | |||||||||||||||||||||
359.1 | −0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | −2.20711 | − | 0.358719i | 0 | 1.50000 | − | 2.59808i | 2.82843 | 0 | 2.00000 | − | 2.44949i | |||||||||||||||||||||
359.2 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | −0.792893 | + | 2.09077i | 0 | 1.50000 | − | 2.59808i | −2.82843 | 0 | 2.00000 | + | 2.44949i | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
20.d | odd | 2 | 1 | inner |
180.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 540.2.n.b | 4 | |
3.b | odd | 2 | 1 | 180.2.n.a | ✓ | 4 | |
4.b | odd | 2 | 1 | 540.2.n.a | 4 | ||
5.b | even | 2 | 1 | 540.2.n.a | 4 | ||
9.c | even | 3 | 1 | 180.2.n.a | ✓ | 4 | |
9.d | odd | 6 | 1 | inner | 540.2.n.b | 4 | |
12.b | even | 2 | 1 | 180.2.n.b | yes | 4 | |
15.d | odd | 2 | 1 | 180.2.n.b | yes | 4 | |
15.e | even | 4 | 2 | 900.2.r.b | 8 | ||
20.d | odd | 2 | 1 | inner | 540.2.n.b | 4 | |
36.f | odd | 6 | 1 | 180.2.n.b | yes | 4 | |
36.h | even | 6 | 1 | 540.2.n.a | 4 | ||
45.h | odd | 6 | 1 | 540.2.n.a | 4 | ||
45.j | even | 6 | 1 | 180.2.n.b | yes | 4 | |
45.k | odd | 12 | 2 | 900.2.r.b | 8 | ||
60.h | even | 2 | 1 | 180.2.n.a | ✓ | 4 | |
60.l | odd | 4 | 2 | 900.2.r.b | 8 | ||
180.n | even | 6 | 1 | inner | 540.2.n.b | 4 | |
180.p | odd | 6 | 1 | 180.2.n.a | ✓ | 4 | |
180.x | even | 12 | 2 | 900.2.r.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
180.2.n.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
180.2.n.a | ✓ | 4 | 9.c | even | 3 | 1 | |
180.2.n.a | ✓ | 4 | 60.h | even | 2 | 1 | |
180.2.n.a | ✓ | 4 | 180.p | odd | 6 | 1 | |
180.2.n.b | yes | 4 | 12.b | even | 2 | 1 | |
180.2.n.b | yes | 4 | 15.d | odd | 2 | 1 | |
180.2.n.b | yes | 4 | 36.f | odd | 6 | 1 | |
180.2.n.b | yes | 4 | 45.j | even | 6 | 1 | |
540.2.n.a | 4 | 4.b | odd | 2 | 1 | ||
540.2.n.a | 4 | 5.b | even | 2 | 1 | ||
540.2.n.a | 4 | 36.h | even | 6 | 1 | ||
540.2.n.a | 4 | 45.h | odd | 6 | 1 | ||
540.2.n.b | 4 | 1.a | even | 1 | 1 | trivial | |
540.2.n.b | 4 | 9.d | odd | 6 | 1 | inner | |
540.2.n.b | 4 | 20.d | odd | 2 | 1 | inner | |
540.2.n.b | 4 | 180.n | even | 6 | 1 | inner | |
900.2.r.b | 8 | 15.e | even | 4 | 2 | ||
900.2.r.b | 8 | 45.k | odd | 12 | 2 | ||
900.2.r.b | 8 | 60.l | odd | 4 | 2 | ||
900.2.r.b | 8 | 180.x | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 3T_{7} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(540, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 2T^{2} + 4 \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 6 T^{3} + 17 T^{2} + 30 T + 25 \)
$7$
\( (T^{2} - 3 T + 9)^{2} \)
$11$
\( T^{4} + 18T^{2} + 324 \)
$13$
\( T^{4} - 6T^{2} + 36 \)
$17$
\( (T^{2} - 2)^{2} \)
$19$
\( (T^{2} + 54)^{2} \)
$23$
\( (T^{2} + 9 T + 27)^{2} \)
$29$
\( (T^{2} + 3 T + 3)^{2} \)
$31$
\( T^{4} - 54T^{2} + 2916 \)
$37$
\( (T^{2} + 6)^{2} \)
$41$
\( (T^{2} - 9 T + 27)^{2} \)
$43$
\( (T^{2} + 6 T + 36)^{2} \)
$47$
\( (T^{2} - 9 T + 27)^{2} \)
$53$
\( (T^{2} - 32)^{2} \)
$59$
\( T^{4} \)
$61$
\( (T^{2} + 7 T + 49)^{2} \)
$67$
\( (T^{2} - 3 T + 9)^{2} \)
$71$
\( (T^{2} - 18)^{2} \)
$73$
\( (T^{2} + 24)^{2} \)
$79$
\( T^{4} \)
$83$
\( (T^{2} + 9 T + 27)^{2} \)
$89$
\( (T^{2} + 75)^{2} \)
$97$
\( T^{4} - 216 T^{2} + 46656 \)
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