Newspace parameters
Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 810.h (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(22.0709014132\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.3317760000.8 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 30) |
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 in terms of a root \(\nu\) of \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{6} + 14\nu^{4} - 7\nu^{2} - 36 ) / 63 \) |
\(\beta_{2}\) | \(=\) | \( ( 4\nu^{6} + 7\nu^{4} + 28\nu^{2} + 144 ) / 63 \) |
\(\beta_{3}\) | \(=\) | \( ( 4\nu^{7} + 7\nu^{5} - 35\nu^{3} + 81\nu ) / 189 \) |
\(\beta_{4}\) | \(=\) | \( ( -5\nu^{7} + 7\nu^{5} - 35\nu^{3} - 180\nu ) / 189 \) |
\(\beta_{5}\) | \(=\) | \( ( -8\nu^{6} - 14\nu^{4} + 7\nu^{2} - 162 ) / 63 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{7} + 13\nu ) / 21 \) |
\(\beta_{7}\) | \(=\) | \( ( 19\nu^{7} + 49\nu^{5} + 133\nu^{3} + 684\nu ) / 189 \) |
\(\nu\) | \(=\) | \( ( \beta_{6} - \beta_{4} + \beta_{3} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{5} + 2\beta_{2} - 2 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{7} - \beta_{6} - 7\beta_{3} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{2} + 4\beta_1 \) |
\(\nu^{5}\) | \(=\) | \( ( 5\beta_{7} + 19\beta_{4} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( -7\beta_{5} - 7\beta _1 - 22 \) |
\(\nu^{7}\) | \(=\) | \( ( -29\beta_{6} - 13\beta_{4} + 13\beta_{3} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).
\(n\) | \(487\) | \(731\) |
\(\chi(n)\) | \(1\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
431.1 |
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−1.22474 | − | 0.707107i | 0 | 1.00000 | + | 1.73205i | −1.93649 | + | 1.11803i | 0 | −5.74342 | + | 9.94789i | − | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||
431.2 | −1.22474 | − | 0.707107i | 0 | 1.00000 | + | 1.73205i | 1.93649 | − | 1.11803i | 0 | 3.74342 | − | 6.48379i | − | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||
431.3 | 1.22474 | + | 0.707107i | 0 | 1.00000 | + | 1.73205i | −1.93649 | + | 1.11803i | 0 | 3.74342 | − | 6.48379i | 2.82843i | 0 | −3.16228 | |||||||||||||||||||||||||||||||||||
431.4 | 1.22474 | + | 0.707107i | 0 | 1.00000 | + | 1.73205i | 1.93649 | − | 1.11803i | 0 | −5.74342 | + | 9.94789i | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||||
701.1 | −1.22474 | + | 0.707107i | 0 | 1.00000 | − | 1.73205i | −1.93649 | − | 1.11803i | 0 | −5.74342 | − | 9.94789i | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||||
701.2 | −1.22474 | + | 0.707107i | 0 | 1.00000 | − | 1.73205i | 1.93649 | + | 1.11803i | 0 | 3.74342 | + | 6.48379i | 2.82843i | 0 | −3.16228 | |||||||||||||||||||||||||||||||||||
701.3 | 1.22474 | − | 0.707107i | 0 | 1.00000 | − | 1.73205i | −1.93649 | − | 1.11803i | 0 | 3.74342 | + | 6.48379i | − | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||
701.4 | 1.22474 | − | 0.707107i | 0 | 1.00000 | − | 1.73205i | 1.93649 | + | 1.11803i | 0 | −5.74342 | − | 9.94789i | − | 2.82843i | 0 | 3.16228 | ||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
9.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 810.3.h.a | 8 | |
3.b | odd | 2 | 1 | inner | 810.3.h.a | 8 | |
9.c | even | 3 | 1 | 30.3.d.a | ✓ | 4 | |
9.c | even | 3 | 1 | inner | 810.3.h.a | 8 | |
9.d | odd | 6 | 1 | 30.3.d.a | ✓ | 4 | |
9.d | odd | 6 | 1 | inner | 810.3.h.a | 8 | |
36.f | odd | 6 | 1 | 240.3.l.c | 4 | ||
36.h | even | 6 | 1 | 240.3.l.c | 4 | ||
45.h | odd | 6 | 1 | 150.3.d.c | 4 | ||
45.j | even | 6 | 1 | 150.3.d.c | 4 | ||
45.k | odd | 12 | 2 | 150.3.b.b | 8 | ||
45.l | even | 12 | 2 | 150.3.b.b | 8 | ||
72.j | odd | 6 | 1 | 960.3.l.e | 4 | ||
72.l | even | 6 | 1 | 960.3.l.f | 4 | ||
72.n | even | 6 | 1 | 960.3.l.e | 4 | ||
72.p | odd | 6 | 1 | 960.3.l.f | 4 | ||
180.n | even | 6 | 1 | 1200.3.l.u | 4 | ||
180.p | odd | 6 | 1 | 1200.3.l.u | 4 | ||
180.v | odd | 12 | 2 | 1200.3.c.k | 8 | ||
180.x | even | 12 | 2 | 1200.3.c.k | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
30.3.d.a | ✓ | 4 | 9.c | even | 3 | 1 | |
30.3.d.a | ✓ | 4 | 9.d | odd | 6 | 1 | |
150.3.b.b | 8 | 45.k | odd | 12 | 2 | ||
150.3.b.b | 8 | 45.l | even | 12 | 2 | ||
150.3.d.c | 4 | 45.h | odd | 6 | 1 | ||
150.3.d.c | 4 | 45.j | even | 6 | 1 | ||
240.3.l.c | 4 | 36.f | odd | 6 | 1 | ||
240.3.l.c | 4 | 36.h | even | 6 | 1 | ||
810.3.h.a | 8 | 1.a | even | 1 | 1 | trivial | |
810.3.h.a | 8 | 3.b | odd | 2 | 1 | inner | |
810.3.h.a | 8 | 9.c | even | 3 | 1 | inner | |
810.3.h.a | 8 | 9.d | odd | 6 | 1 | inner | |
960.3.l.e | 4 | 72.j | odd | 6 | 1 | ||
960.3.l.e | 4 | 72.n | even | 6 | 1 | ||
960.3.l.f | 4 | 72.l | even | 6 | 1 | ||
960.3.l.f | 4 | 72.p | odd | 6 | 1 | ||
1200.3.c.k | 8 | 180.v | odd | 12 | 2 | ||
1200.3.c.k | 8 | 180.x | even | 12 | 2 | ||
1200.3.l.u | 4 | 180.n | even | 6 | 1 | ||
1200.3.l.u | 4 | 180.p | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 4T_{7}^{3} + 102T_{7}^{2} - 344T_{7} + 7396 \)
acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 2 T^{2} + 4)^{2} \)
$3$
\( T^{8} \)
$5$
\( (T^{4} - 5 T^{2} + 25)^{2} \)
$7$
\( (T^{4} + 4 T^{3} + 102 T^{2} - 344 T + 7396)^{2} \)
$11$
\( (T^{4} - 72 T^{2} + 5184)^{2} \)
$13$
\( (T^{2} - 10 T + 100)^{4} \)
$17$
\( (T^{4} + 936 T^{2} + 11664)^{2} \)
$19$
\( (T^{2} - 16 T - 296)^{4} \)
$23$
\( T^{8} - 396 T^{6} + \cdots + 688747536 \)
$29$
\( (T^{4} - 720 T^{2} + 518400)^{2} \)
$31$
\( (T^{2} + 8 T + 64)^{4} \)
$37$
\( (T^{2} + 44 T - 956)^{4} \)
$41$
\( T^{8} - 2664 T^{6} + \cdots + 892616806656 \)
$43$
\( (T^{4} + 28 T^{3} + 1398 T^{2} + \cdots + 376996)^{2} \)
$47$
\( T^{8} + \cdots + 269042006250000 \)
$53$
\( (T^{4} + 936 T^{2} + 11664)^{2} \)
$59$
\( T^{8} - 6624 T^{6} + \cdots + 12280707219456 \)
$61$
\( (T^{4} - 32 T^{3} + 2208 T^{2} + \cdots + 1401856)^{2} \)
$67$
\( (T^{4} - 164 T^{3} + 20982 T^{2} + \cdots + 34975396)^{2} \)
$71$
\( (T^{4} + 5040 T^{2} + 1166400)^{2} \)
$73$
\( (T^{2} - 100 T + 1060)^{4} \)
$79$
\( (T^{4} - 56 T^{3} + 2712 T^{2} + \cdots + 179776)^{2} \)
$83$
\( T^{8} - 684 T^{6} + 467532 T^{4} + \cdots + 104976 \)
$89$
\( (T^{4} + 3744 T^{2} + 186624)^{2} \)
$97$
\( (T^{4} + 148 T^{3} + 17868 T^{2} + \cdots + 16289296)^{2} \)
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