Newspace parameters
Level: | \( N \) | \(=\) | \( 270 = 2 \cdot 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 270.j (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.35696713773\) |
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: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
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
\(\beta_{1}\) | \(=\) | \( \zeta_{24}^{2} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{24}^{4} \) |
\(\beta_{3}\) | \(=\) | \( \zeta_{24}^{6} \) |
\(\beta_{4}\) | \(=\) | \( \zeta_{24}^{7} + \zeta_{24} \) |
\(\beta_{5}\) | \(=\) | \( -\zeta_{24}^{7} + \zeta_{24} \) |
\(\beta_{6}\) | \(=\) | \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) |
\(\beta_{7}\) | \(=\) | \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) |
\(\zeta_{24}\) | \(=\) | \( ( \beta_{5} + \beta_{4} ) / 2 \) |
\(\zeta_{24}^{2}\) | \(=\) | \( \beta_1 \) |
\(\zeta_{24}^{3}\) | \(=\) | \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) |
\(\zeta_{24}^{4}\) | \(=\) | \( \beta_{2} \) |
\(\zeta_{24}^{5}\) | \(=\) | \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) |
\(\zeta_{24}^{6}\) | \(=\) | \( \beta_{3} \) |
\(\zeta_{24}^{7}\) | \(=\) | \( ( -\beta_{5} + \beta_{4} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(217\) |
\(\chi(n)\) | \(\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
89.1 |
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−0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | −3.38865 | − | 3.67656i | 0 | 1.81954 | + | 1.05051i | 2.82843 | 0 | 6.89898 | − | 1.55051i | ||||||||||||||||||||||||||||||||
89.2 | −0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | 4.97443 | − | 0.504984i | 0 | −10.3048 | − | 5.94949i | 2.82843 | 0 | −2.89898 | + | 6.44949i | |||||||||||||||||||||||||||||||||
89.3 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | 1.48967 | + | 4.77293i | 0 | −1.81954 | − | 1.05051i | −2.82843 | 0 | 6.89898 | + | 1.55051i | |||||||||||||||||||||||||||||||||
89.4 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | 2.92455 | − | 4.05549i | 0 | 10.3048 | + | 5.94949i | −2.82843 | 0 | −2.89898 | − | 6.44949i | |||||||||||||||||||||||||||||||||
179.1 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | −3.38865 | + | 3.67656i | 0 | 1.81954 | − | 1.05051i | 2.82843 | 0 | 6.89898 | + | 1.55051i | |||||||||||||||||||||||||||||||||
179.2 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | 4.97443 | + | 0.504984i | 0 | −10.3048 | + | 5.94949i | 2.82843 | 0 | −2.89898 | − | 6.44949i | |||||||||||||||||||||||||||||||||
179.3 | 0.707107 | + | 1.22474i | 0 | −1.00000 | + | 1.73205i | 1.48967 | − | 4.77293i | 0 | −1.81954 | + | 1.05051i | −2.82843 | 0 | 6.89898 | − | 1.55051i | |||||||||||||||||||||||||||||||||
179.4 | 0.707107 | + | 1.22474i | 0 | −1.00000 | + | 1.73205i | 2.92455 | + | 4.05549i | 0 | 10.3048 | − | 5.94949i | −2.82843 | 0 | −2.89898 | + | 6.44949i | |||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
45.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 270.3.j.a | 8 | |
3.b | odd | 2 | 1 | 90.3.j.a | ✓ | 8 | |
5.b | even | 2 | 1 | inner | 270.3.j.a | 8 | |
5.c | odd | 4 | 1 | 1350.3.i.a | 4 | ||
5.c | odd | 4 | 1 | 1350.3.i.c | 4 | ||
9.c | even | 3 | 1 | 90.3.j.a | ✓ | 8 | |
9.c | even | 3 | 1 | 810.3.b.a | 8 | ||
9.d | odd | 6 | 1 | inner | 270.3.j.a | 8 | |
9.d | odd | 6 | 1 | 810.3.b.a | 8 | ||
15.d | odd | 2 | 1 | 90.3.j.a | ✓ | 8 | |
15.e | even | 4 | 1 | 450.3.i.a | 4 | ||
15.e | even | 4 | 1 | 450.3.i.c | 4 | ||
45.h | odd | 6 | 1 | inner | 270.3.j.a | 8 | |
45.h | odd | 6 | 1 | 810.3.b.a | 8 | ||
45.j | even | 6 | 1 | 90.3.j.a | ✓ | 8 | |
45.j | even | 6 | 1 | 810.3.b.a | 8 | ||
45.k | odd | 12 | 1 | 450.3.i.a | 4 | ||
45.k | odd | 12 | 1 | 450.3.i.c | 4 | ||
45.l | even | 12 | 1 | 1350.3.i.a | 4 | ||
45.l | even | 12 | 1 | 1350.3.i.c | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
90.3.j.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
90.3.j.a | ✓ | 8 | 9.c | even | 3 | 1 | |
90.3.j.a | ✓ | 8 | 15.d | odd | 2 | 1 | |
90.3.j.a | ✓ | 8 | 45.j | even | 6 | 1 | |
270.3.j.a | 8 | 1.a | even | 1 | 1 | trivial | |
270.3.j.a | 8 | 5.b | even | 2 | 1 | inner | |
270.3.j.a | 8 | 9.d | odd | 6 | 1 | inner | |
270.3.j.a | 8 | 45.h | odd | 6 | 1 | inner | |
450.3.i.a | 4 | 15.e | even | 4 | 1 | ||
450.3.i.a | 4 | 45.k | odd | 12 | 1 | ||
450.3.i.c | 4 | 15.e | even | 4 | 1 | ||
450.3.i.c | 4 | 45.k | odd | 12 | 1 | ||
810.3.b.a | 8 | 9.c | even | 3 | 1 | ||
810.3.b.a | 8 | 9.d | odd | 6 | 1 | ||
810.3.b.a | 8 | 45.h | odd | 6 | 1 | ||
810.3.b.a | 8 | 45.j | even | 6 | 1 | ||
1350.3.i.a | 4 | 5.c | odd | 4 | 1 | ||
1350.3.i.a | 4 | 45.l | even | 12 | 1 | ||
1350.3.i.c | 4 | 5.c | odd | 4 | 1 | ||
1350.3.i.c | 4 | 45.l | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 146T_{7}^{6} + 20691T_{7}^{4} - 91250T_{7}^{2} + 390625 \)
acting on \(S_{3}^{\mathrm{new}}(270, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 2 T^{2} + 4)^{2} \)
$3$
\( T^{8} \)
$5$
\( T^{8} - 12 T^{7} + 78 T^{6} + \cdots + 390625 \)
$7$
\( T^{8} - 146 T^{6} + 20691 T^{4} + \cdots + 390625 \)
$11$
\( (T^{4} - 12 T^{3} - 38 T^{2} + 1032 T + 7396)^{2} \)
$13$
\( T^{8} - 660 T^{6} + \cdots + 4430766096 \)
$17$
\( (T^{4} - 1180 T^{2} + 320356)^{2} \)
$19$
\( (T^{2} + 36 T + 318)^{4} \)
$23$
\( T^{8} + 310 T^{6} + \cdots + 373301041 \)
$29$
\( (T^{4} - 18 T^{3} + 63 T^{2} + 810 T + 2025)^{2} \)
$31$
\( (T^{4} + 44 T^{3} + 2466 T^{2} + \cdots + 280900)^{2} \)
$37$
\( (T^{4} + 1460 T^{2} + 386884)^{2} \)
$41$
\( (T^{4} + 18 T^{3} + 103 T^{2} - 90 T + 25)^{2} \)
$43$
\( T^{8} - 3104 T^{6} + \cdots + 5337948160000 \)
$47$
\( T^{8} + 5446 T^{6} + \cdots + 29120366676241 \)
$53$
\( (T^{4} - 2832 T^{2} + 14400)^{2} \)
$59$
\( (T^{4} + 96 T^{3} - 1568 T^{2} + \cdots + 21529600)^{2} \)
$61$
\( (T^{4} + 10 T^{3} + 99 T^{2} + 10 T + 1)^{2} \)
$67$
\( T^{8} - 4898 T^{6} + \cdots + 30549950894401 \)
$71$
\( (T^{4} + 21084 T^{2} + 92968164)^{2} \)
$73$
\( (T^{4} + 560 T^{2} + 53824)^{2} \)
$79$
\( (T^{4} + 144 T^{3} + 20952 T^{2} + \cdots + 46656)^{2} \)
$83$
\( T^{8} + 550 T^{6} + \cdots + 1982119441 \)
$89$
\( (T^{4} + 28774 T^{2} + \cdots + 125731369)^{2} \)
$97$
\( T^{8} - 7712 T^{6} + \cdots + 87219461226496 \)
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