Newspace parameters
Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 810.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(22.0709014132\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.3317760000.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{7} + 14\nu^{5} - 25\nu^{3} - 78\nu ) / 72 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} - 14\nu^{5} + 61\nu^{3} - 30\nu ) / 36 \) |
\(\beta_{4}\) | \(=\) | \( ( 5\nu^{7} - 22\nu^{5} - 96\nu^{4} + 5\nu^{3} + 384\nu^{2} + 30\nu + 144 ) / 144 \) |
\(\beta_{5}\) | \(=\) | \( ( 7\nu^{7} + 6\nu^{6} - 50\nu^{5} - 36\nu^{4} + 55\nu^{3} - 42\nu^{2} - 102\nu + 180 ) / 144 \) |
\(\beta_{6}\) | \(=\) | \( ( 7\nu^{7} - 12\nu^{6} - 50\nu^{5} + 72\nu^{4} + 55\nu^{3} + 84\nu^{2} - 102\nu - 360 ) / 144 \) |
\(\beta_{7}\) | \(=\) | \( ( 17\nu^{7} - 94\nu^{5} + 65\nu^{3} + 246\nu ) / 144 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} - \beta_{6} - 2\beta_{5} - 2\beta_1 ) / 6 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} - \beta_{5} + \beta_{2} + 4 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{7} - \beta_{6} - 2\beta_{5} + 2\beta_{3} + 2\beta_1 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( \beta_{7} + 4\beta_{6} - 4\beta_{5} - 3\beta_{4} + 4\beta_{2} + \beta _1 + 19 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 7\beta_{7} - 5\beta_{6} - 10\beta_{5} + 5\beta_{3} + 17\beta_1 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( 6\beta_{7} + 15\beta_{6} - 15\beta_{5} - 18\beta_{4} + 31\beta_{2} + 6\beta _1 + 82 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( 47\beta_{7} - 19\beta_{6} - 38\beta_{5} + 20\beta_{3} + 96\beta_1 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).
\(n\) | \(487\) | \(731\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
161.1 |
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− | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | −6.46891 | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||
161.2 | − | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | −5.01793 | 2.82843i | 0 | −3.16228 | |||||||||||||||||||||||||||||||||||||||||
161.3 | − | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | −2.45930 | 2.82843i | 0 | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||
161.4 | − | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | 9.94613 | 2.82843i | 0 | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||
161.5 | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | −2.45930 | − | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||||||||||
161.6 | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | 9.94613 | − | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||||||||||
161.7 | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | −6.46891 | − | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||
161.8 | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | −5.01793 | − | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 810.3.d.b | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 810.3.d.b | ✓ | 8 |
9.c | even | 3 | 2 | 810.3.h.c | 16 | ||
9.d | odd | 6 | 2 | 810.3.h.c | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
810.3.d.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
810.3.d.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
810.3.h.c | 16 | 9.c | even | 3 | 2 | ||
810.3.h.c | 16 | 9.d | odd | 6 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 4T_{7}^{3} - 78T_{7}^{2} - 524T_{7} - 794 \)
acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2)^{4} \)
$3$
\( T^{8} \)
$5$
\( (T^{2} + 5)^{4} \)
$7$
\( (T^{4} + 4 T^{3} - 78 T^{2} - 524 T - 794)^{2} \)
$11$
\( T^{8} + 552 T^{6} + 46098 T^{4} + \cdots + 6561 \)
$13$
\( (T^{4} - 44 T^{3} + 516 T^{2} - 704 T - 8744)^{2} \)
$17$
\( T^{8} + 2964 T^{6} + \cdots + 282400839396 \)
$19$
\( (T^{4} + 40 T^{3} - 186 T^{2} + \cdots + 52369)^{2} \)
$23$
\( T^{8} + 2280 T^{6} + \cdots + 2361960000 \)
$29$
\( T^{8} + 4728 T^{6} + \cdots + 1227661784001 \)
$31$
\( (T^{4} + 76 T^{3} + 1722 T^{2} + \cdots - 3719)^{2} \)
$37$
\( (T^{4} - 68 T^{3} - 2262 T^{2} + \cdots - 88874)^{2} \)
$41$
\( T^{8} + 3384 T^{6} + \cdots + 50305555521 \)
$43$
\( (T^{4} + 16 T^{3} - 3138 T^{2} + \cdots + 868726)^{2} \)
$47$
\( T^{8} + 11316 T^{6} + \cdots + 7727654977956 \)
$53$
\( T^{8} + 15276 T^{6} + \cdots + 1421755755876 \)
$59$
\( T^{8} + 10056 T^{6} + \cdots + 2097749586321 \)
$61$
\( (T^{4} - 116 T^{3} + 1680 T^{2} + \cdots - 4500956)^{2} \)
$67$
\( (T^{4} - 92 T^{3} - 7860 T^{2} + \cdots + 7322584)^{2} \)
$71$
\( T^{8} + 16056 T^{6} + \cdots + 57836161890081 \)
$73$
\( (T^{4} - 140 T^{3} - 174 T^{2} + \cdots + 1972054)^{2} \)
$79$
\( (T^{4} - 92 T^{3} - 6672 T^{2} + \cdots + 12736516)^{2} \)
$83$
\( T^{8} + \cdots + 184192832358756 \)
$89$
\( T^{8} + 36792 T^{6} + \cdots + 16\!\cdots\!61 \)
$97$
\( (T^{4} + 256 T^{3} + 18462 T^{2} + \cdots + 3143446)^{2} \)
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