Newspace parameters
Level: | \( N \) | \(=\) | \( 270 = 2 \cdot 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 270.i (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.15596085457\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3^{2} \) |
Twist minimal: | no (minimal twist has level 90) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) | \( \zeta_{24}^{2} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{24}^{4} \) |
\(\beta_{3}\) | \(=\) | \( \zeta_{24}^{5} + \zeta_{24} \) |
\(\beta_{4}\) | \(=\) | \( \zeta_{24}^{6} \) |
\(\beta_{5}\) | \(=\) | \( \zeta_{24}^{7} + \zeta_{24}^{3} \) |
\(\beta_{6}\) | \(=\) | \( -\zeta_{24}^{5} + 2\zeta_{24} \) |
\(\beta_{7}\) | \(=\) | \( -\zeta_{24}^{7} + 2\zeta_{24}^{3} \) |
\(\zeta_{24}\) | \(=\) | \( ( \beta_{6} + \beta_{3} ) / 3 \) |
\(\zeta_{24}^{2}\) | \(=\) | \( \beta_1 \) |
\(\zeta_{24}^{3}\) | \(=\) | \( ( \beta_{7} + \beta_{5} ) / 3 \) |
\(\zeta_{24}^{4}\) | \(=\) | \( \beta_{2} \) |
\(\zeta_{24}^{5}\) | \(=\) | \( ( -\beta_{6} + 2\beta_{3} ) / 3 \) |
\(\zeta_{24}^{6}\) | \(=\) | \( \beta_{4} \) |
\(\zeta_{24}^{7}\) | \(=\) | \( ( -\beta_{7} + 2\beta_{5} ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(217\) |
\(\chi(n)\) | \(-1 + \beta_{2}\) | \(-1\) |
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 |
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−0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.917738 | + | 2.03906i | 0 | 0.389270 | + | 0.224745i | − | 1.00000i | 0 | 0.224745 | − | 2.22474i | |||||||||||||||||||||||||||||||
19.2 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.81431 | − | 1.30701i | 0 | −3.85337 | − | 2.22474i | − | 1.00000i | 0 | −2.22474 | + | 0.224745i | ||||||||||||||||||||||||||||||||
19.3 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | −2.03906 | + | 0.917738i | 0 | 3.85337 | + | 2.22474i | 1.00000i | 0 | −2.22474 | − | 0.224745i | |||||||||||||||||||||||||||||||||
19.4 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.30701 | + | 1.81431i | 0 | −0.389270 | − | 0.224745i | 1.00000i | 0 | 0.224745 | + | 2.22474i | |||||||||||||||||||||||||||||||||
199.1 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.917738 | − | 2.03906i | 0 | 0.389270 | − | 0.224745i | 1.00000i | 0 | 0.224745 | + | 2.22474i | |||||||||||||||||||||||||||||||||
199.2 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.81431 | + | 1.30701i | 0 | −3.85337 | + | 2.22474i | 1.00000i | 0 | −2.22474 | − | 0.224745i | |||||||||||||||||||||||||||||||||
199.3 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | −2.03906 | − | 0.917738i | 0 | 3.85337 | − | 2.22474i | − | 1.00000i | 0 | −2.22474 | + | 0.224745i | ||||||||||||||||||||||||||||||||
199.4 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.30701 | − | 1.81431i | 0 | −0.389270 | + | 0.224745i | − | 1.00000i | 0 | 0.224745 | − | 2.22474i | ||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
45.j | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 270.2.i.b | 8 | |
3.b | odd | 2 | 1 | 90.2.i.b | ✓ | 8 | |
4.b | odd | 2 | 1 | 2160.2.by.d | 8 | ||
5.b | even | 2 | 1 | inner | 270.2.i.b | 8 | |
5.c | odd | 4 | 1 | 1350.2.e.j | 4 | ||
5.c | odd | 4 | 1 | 1350.2.e.m | 4 | ||
9.c | even | 3 | 1 | inner | 270.2.i.b | 8 | |
9.c | even | 3 | 1 | 810.2.c.e | 4 | ||
9.d | odd | 6 | 1 | 90.2.i.b | ✓ | 8 | |
9.d | odd | 6 | 1 | 810.2.c.f | 4 | ||
12.b | even | 2 | 1 | 720.2.by.c | 8 | ||
15.d | odd | 2 | 1 | 90.2.i.b | ✓ | 8 | |
15.e | even | 4 | 1 | 450.2.e.k | 4 | ||
15.e | even | 4 | 1 | 450.2.e.n | 4 | ||
20.d | odd | 2 | 1 | 2160.2.by.d | 8 | ||
36.f | odd | 6 | 1 | 2160.2.by.d | 8 | ||
36.h | even | 6 | 1 | 720.2.by.c | 8 | ||
45.h | odd | 6 | 1 | 90.2.i.b | ✓ | 8 | |
45.h | odd | 6 | 1 | 810.2.c.f | 4 | ||
45.j | even | 6 | 1 | inner | 270.2.i.b | 8 | |
45.j | even | 6 | 1 | 810.2.c.e | 4 | ||
45.k | odd | 12 | 1 | 1350.2.e.j | 4 | ||
45.k | odd | 12 | 1 | 1350.2.e.m | 4 | ||
45.k | odd | 12 | 1 | 4050.2.a.bm | 2 | ||
45.k | odd | 12 | 1 | 4050.2.a.bz | 2 | ||
45.l | even | 12 | 1 | 450.2.e.k | 4 | ||
45.l | even | 12 | 1 | 450.2.e.n | 4 | ||
45.l | even | 12 | 1 | 4050.2.a.bq | 2 | ||
45.l | even | 12 | 1 | 4050.2.a.bs | 2 | ||
60.h | even | 2 | 1 | 720.2.by.c | 8 | ||
180.n | even | 6 | 1 | 720.2.by.c | 8 | ||
180.p | odd | 6 | 1 | 2160.2.by.d | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
90.2.i.b | ✓ | 8 | 3.b | odd | 2 | 1 | |
90.2.i.b | ✓ | 8 | 9.d | odd | 6 | 1 | |
90.2.i.b | ✓ | 8 | 15.d | odd | 2 | 1 | |
90.2.i.b | ✓ | 8 | 45.h | odd | 6 | 1 | |
270.2.i.b | 8 | 1.a | even | 1 | 1 | trivial | |
270.2.i.b | 8 | 5.b | even | 2 | 1 | inner | |
270.2.i.b | 8 | 9.c | even | 3 | 1 | inner | |
270.2.i.b | 8 | 45.j | even | 6 | 1 | inner | |
450.2.e.k | 4 | 15.e | even | 4 | 1 | ||
450.2.e.k | 4 | 45.l | even | 12 | 1 | ||
450.2.e.n | 4 | 15.e | even | 4 | 1 | ||
450.2.e.n | 4 | 45.l | even | 12 | 1 | ||
720.2.by.c | 8 | 12.b | even | 2 | 1 | ||
720.2.by.c | 8 | 36.h | even | 6 | 1 | ||
720.2.by.c | 8 | 60.h | even | 2 | 1 | ||
720.2.by.c | 8 | 180.n | even | 6 | 1 | ||
810.2.c.e | 4 | 9.c | even | 3 | 1 | ||
810.2.c.e | 4 | 45.j | even | 6 | 1 | ||
810.2.c.f | 4 | 9.d | odd | 6 | 1 | ||
810.2.c.f | 4 | 45.h | odd | 6 | 1 | ||
1350.2.e.j | 4 | 5.c | odd | 4 | 1 | ||
1350.2.e.j | 4 | 45.k | odd | 12 | 1 | ||
1350.2.e.m | 4 | 5.c | odd | 4 | 1 | ||
1350.2.e.m | 4 | 45.k | odd | 12 | 1 | ||
2160.2.by.d | 8 | 4.b | odd | 2 | 1 | ||
2160.2.by.d | 8 | 20.d | odd | 2 | 1 | ||
2160.2.by.d | 8 | 36.f | odd | 6 | 1 | ||
2160.2.by.d | 8 | 180.p | odd | 6 | 1 | ||
4050.2.a.bm | 2 | 45.k | odd | 12 | 1 | ||
4050.2.a.bq | 2 | 45.l | even | 12 | 1 | ||
4050.2.a.bs | 2 | 45.l | even | 12 | 1 | ||
4050.2.a.bz | 2 | 45.k | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 20T_{7}^{6} + 396T_{7}^{4} - 80T_{7}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(270, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{2} \)
$3$
\( T^{8} \)
$5$
\( T^{8} - 4 T^{7} + 8 T^{6} + 8 T^{5} + \cdots + 625 \)
$7$
\( T^{8} - 20 T^{6} + 396 T^{4} + \cdots + 16 \)
$11$
\( (T^{4} + 2 T^{3} + 9 T^{2} - 10 T + 25)^{2} \)
$13$
\( (T^{4} - 6 T^{2} + 36)^{2} \)
$17$
\( (T^{4} + 50 T^{2} + 529)^{2} \)
$19$
\( (T^{2} - 6 T + 3)^{4} \)
$23$
\( T^{8} - 56 T^{6} + 2736 T^{4} + \cdots + 160000 \)
$29$
\( (T^{2} - 6 T + 36)^{4} \)
$31$
\( (T^{4} + 8 T^{3} + 54 T^{2} + 80 T + 100)^{2} \)
$37$
\( (T^{2} + 64)^{4} \)
$41$
\( (T^{2} + T + 1)^{4} \)
$43$
\( T^{8} - 62 T^{6} + 3483 T^{4} + \cdots + 130321 \)
$47$
\( T^{8} - 20 T^{6} + 396 T^{4} + \cdots + 16 \)
$53$
\( (T^{4} + 84 T^{2} + 900)^{2} \)
$59$
\( (T^{4} + 2 T^{3} + 153 T^{2} - 298 T + 22201)^{2} \)
$61$
\( (T^{4} + 4 T^{3} + 18 T^{2} - 8 T + 4)^{2} \)
$67$
\( T^{8} - 110 T^{6} + 10251 T^{4} + \cdots + 3418801 \)
$71$
\( (T^{2} - 6)^{4} \)
$73$
\( (T^{4} + 242 T^{2} + 5041)^{2} \)
$79$
\( (T^{4} + 54 T^{2} + 2916)^{2} \)
$83$
\( (T^{4} - 16 T^{2} + 256)^{2} \)
$89$
\( (T^{2} - 16 T + 40)^{4} \)
$97$
\( (T^{4} - 169 T^{2} + 28561)^{2} \)
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