Newspace parameters
Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 261.g (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.08409549276\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.40960000.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 7x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 3^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{5} + 2\nu ) / 3 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{6} + 8\nu^{2} ) / 3 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} + 5\nu^{3} ) / 3 \) |
\(\beta_{4}\) | \(=\) | \( ( 2\nu^{5} + 13\nu ) / 3 \) |
\(\beta_{5}\) | \(=\) | \( \nu^{4} + 4 \) |
\(\beta_{6}\) | \(=\) | \( ( -5\nu^{6} - 31\nu^{2} ) / 3 \) |
\(\beta_{7}\) | \(=\) | \( ( 5\nu^{7} + 34\nu^{3} ) / 3 \) |
\(\nu\) | \(=\) | \( ( \beta_{4} - 2\beta_1 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} + 5\beta_{2} ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{7} - 5\beta_{3} ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{5} - 4 \) |
\(\nu^{5}\) | \(=\) | \( ( -2\beta_{4} + 13\beta_1 ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( -8\beta_{6} - 31\beta_{2} ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( -5\beta_{7} + 34\beta_{3} ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).
\(n\) | \(118\) | \(146\) |
\(\chi(n)\) | \(-\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 |
|
−1.85123 | − | 1.85123i | 0 | 4.85410i | 0.540182 | 0 | −1.00000 | 5.28360 | − | 5.28360i | 0 | −1.00000 | − | 1.00000i | ||||||||||||||||||||||||||||||||||||
17.2 | −0.270091 | − | 0.270091i | 0 | − | 1.85410i | 3.70246 | 0 | −1.00000 | −1.04096 | + | 1.04096i | 0 | −1.00000 | − | 1.00000i | ||||||||||||||||||||||||||||||||||||
17.3 | 0.270091 | + | 0.270091i | 0 | − | 1.85410i | −3.70246 | 0 | −1.00000 | 1.04096 | − | 1.04096i | 0 | −1.00000 | − | 1.00000i | ||||||||||||||||||||||||||||||||||||
17.4 | 1.85123 | + | 1.85123i | 0 | 4.85410i | −0.540182 | 0 | −1.00000 | −5.28360 | + | 5.28360i | 0 | −1.00000 | − | 1.00000i | |||||||||||||||||||||||||||||||||||||
215.1 | −1.85123 | + | 1.85123i | 0 | − | 4.85410i | 0.540182 | 0 | −1.00000 | 5.28360 | + | 5.28360i | 0 | −1.00000 | + | 1.00000i | ||||||||||||||||||||||||||||||||||||
215.2 | −0.270091 | + | 0.270091i | 0 | 1.85410i | 3.70246 | 0 | −1.00000 | −1.04096 | − | 1.04096i | 0 | −1.00000 | + | 1.00000i | |||||||||||||||||||||||||||||||||||||
215.3 | 0.270091 | − | 0.270091i | 0 | 1.85410i | −3.70246 | 0 | −1.00000 | 1.04096 | + | 1.04096i | 0 | −1.00000 | + | 1.00000i | |||||||||||||||||||||||||||||||||||||
215.4 | 1.85123 | − | 1.85123i | 0 | − | 4.85410i | −0.540182 | 0 | −1.00000 | −5.28360 | − | 5.28360i | 0 | −1.00000 | + | 1.00000i | ||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
29.c | odd | 4 | 1 | inner |
87.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 261.2.g.b | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 261.2.g.b | ✓ | 8 |
29.c | odd | 4 | 1 | inner | 261.2.g.b | ✓ | 8 |
87.f | even | 4 | 1 | inner | 261.2.g.b | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
261.2.g.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
261.2.g.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
261.2.g.b | ✓ | 8 | 29.c | odd | 4 | 1 | inner |
261.2.g.b | ✓ | 8 | 87.f | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 47T_{2}^{4} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(261, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 47T^{4} + 1 \)
$3$
\( T^{8} \)
$5$
\( (T^{4} - 14 T^{2} + 4)^{2} \)
$7$
\( (T + 1)^{8} \)
$11$
\( (T^{4} + 25)^{2} \)
$13$
\( (T^{2} + 25)^{4} \)
$17$
\( T^{8} + 3122 T^{4} + 14641 \)
$19$
\( (T^{4} - 6 T^{3} + 18 T^{2} + 108 T + 324)^{2} \)
$23$
\( (T^{4} + 14 T^{2} + 4)^{2} \)
$29$
\( T^{8} - 1198 T^{4} + 707281 \)
$31$
\( (T^{4} - 6 T^{3} + 18 T^{2} + 108 T + 324)^{2} \)
$37$
\( (T^{4} - 6 T^{3} + 18 T^{2} + 108 T + 324)^{2} \)
$41$
\( (T^{4} + 1296)^{2} \)
$43$
\( (T^{4} - 4 T^{3} + 8 T^{2} + 352 T + 7744)^{2} \)
$47$
\( T^{8} + 21602 T^{4} + \cdots + 25411681 \)
$53$
\( (T^{4} + 54 T^{2} + 324)^{2} \)
$59$
\( (T^{4} + 116 T^{2} + 484)^{2} \)
$61$
\( (T^{4} + 2 T^{3} + 2 T^{2} - 44 T + 484)^{2} \)
$67$
\( (T^{4} + 218 T^{2} + 361)^{2} \)
$71$
\( (T^{4} - 126 T^{2} + 324)^{2} \)
$73$
\( (T^{4} + 22 T^{3} + 242 T^{2} + 836 T + 1444)^{2} \)
$79$
\( (T^{4} - 26 T^{3} + 338 T^{2} - 1612 T + 3844)^{2} \)
$83$
\( (T^{2} + 90)^{4} \)
$89$
\( T^{8} + 103682T^{4} + 1 \)
$97$
\( (T^{4} - 30 T^{3} + 450 T^{2} - 2700 T + 8100)^{2} \)
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