Newspace parameters
Level: | \( N \) | \(=\) | \( 66 = 2 \cdot 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 66.h (of order \(10\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.527012653340\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
Coefficient field: | 8.0.185640625.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: |
\( x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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
\( x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + \nu^{5} + 4\nu^{4} + 16\nu^{3} + 51\nu^{2} - 54\nu - 27 ) / 216 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - \nu^{5} - 4\nu^{4} - 16\nu^{3} + 21\nu^{2} - 18\nu + 27 ) / 72 \)
|
\(\beta_{4}\) | \(=\) |
\( ( -11\nu^{7} + 6\nu^{6} + 7\nu^{5} + 16\nu^{4} + 28\nu^{3} - 69\nu^{2} - 216\nu + 351 ) / 216 \)
|
\(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} + \nu^{5} + \nu^{4} + 4\nu^{3} + 3\nu^{2} + 9\nu - 81 ) / 27 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 19\nu^{7} - 30\nu^{6} - 35\nu^{5} - 8\nu^{4} + 76\nu^{3} + 165\nu^{2} + 360\nu - 1107 ) / 216 \)
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\(\beta_{7}\) | \(=\) |
\( ( 3\nu^{7} - 2\nu^{6} - 3\nu^{5} - 8\nu^{4} + 4\nu^{3} + 13\nu^{2} + 60\nu - 99 ) / 24 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + \beta_1 \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{7} + 3\beta_{4} - \beta_{3} + 3\beta_{2} + \beta_1 \)
|
\(\nu^{4}\) | \(=\) |
\( -4\beta_{7} + 2\beta_{6} - \beta_{5} - 6\beta_{4} - 4\beta_{3} + 2 \)
|
\(\nu^{5}\) | \(=\) |
\( 2\beta_{7} - 6\beta_{6} + 6\beta_{5} + 12\beta_{2} + 6\beta _1 - 3 \)
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\(\nu^{6}\) | \(=\) |
\( -2\beta_{6} - 8\beta_{5} - 6\beta_{4} - 8\beta_{3} + 12\beta_{2} + \beta _1 - 20 \)
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\(\nu^{7}\) | \(=\) |
\( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 24\beta_{4} - 19\beta_{3} + 3\beta_{2} - 19\beta _1 + 22 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).
\(n\) | \(13\) | \(23\) |
\(\chi(n)\) | \(1 - \beta_{2} - \beta_{4} + \beta_{6}\) | \(-1\) |
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 |
|
−0.309017 | − | 0.951057i | 0.809017 | − | 1.53150i | −0.809017 | + | 0.587785i | −0.897526 | − | 0.291624i | −1.70654 | − | 0.296161i | 0.897526 | + | 1.23534i | 0.809017 | + | 0.587785i | −1.69098 | − | 2.47802i | 0.943715i | ||||||||||||||||||||||||||
17.2 | −0.309017 | − | 0.951057i | 0.809017 | + | 1.53150i | −0.809017 | + | 0.587785i | 2.01556 | + | 0.654895i | 1.20654 | − | 1.24268i | −2.01556 | − | 2.77418i | 0.809017 | + | 0.587785i | −1.69098 | + | 2.47802i | − | 2.11929i | ||||||||||||||||||||||||||
29.1 | 0.809017 | + | 0.587785i | −0.309017 | − | 1.70426i | 0.309017 | + | 0.951057i | 0.442723 | + | 0.609356i | 0.751740 | − | 1.56041i | −0.442723 | + | 0.143849i | −0.309017 | + | 0.951057i | −2.80902 | + | 1.05329i | 0.753205i | |||||||||||||||||||||||||||
29.2 | 0.809017 | + | 0.587785i | −0.309017 | + | 1.70426i | 0.309017 | + | 0.951057i | −1.56076 | − | 2.14820i | −1.25174 | + | 1.19714i | 1.56076 | − | 0.507121i | −0.309017 | + | 0.951057i | −2.80902 | − | 1.05329i | − | 2.65532i | ||||||||||||||||||||||||||
35.1 | −0.309017 | + | 0.951057i | 0.809017 | − | 1.53150i | −0.809017 | − | 0.587785i | 2.01556 | − | 0.654895i | 1.20654 | + | 1.24268i | −2.01556 | + | 2.77418i | 0.809017 | − | 0.587785i | −1.69098 | − | 2.47802i | 2.11929i | |||||||||||||||||||||||||||
35.2 | −0.309017 | + | 0.951057i | 0.809017 | + | 1.53150i | −0.809017 | − | 0.587785i | −0.897526 | + | 0.291624i | −1.70654 | + | 0.296161i | 0.897526 | − | 1.23534i | 0.809017 | − | 0.587785i | −1.69098 | + | 2.47802i | − | 0.943715i | ||||||||||||||||||||||||||
41.1 | 0.809017 | − | 0.587785i | −0.309017 | − | 1.70426i | 0.309017 | − | 0.951057i | −1.56076 | + | 2.14820i | −1.25174 | − | 1.19714i | 1.56076 | + | 0.507121i | −0.309017 | − | 0.951057i | −2.80902 | + | 1.05329i | 2.65532i | |||||||||||||||||||||||||||
41.2 | 0.809017 | − | 0.587785i | −0.309017 | + | 1.70426i | 0.309017 | − | 0.951057i | 0.442723 | − | 0.609356i | 0.751740 | + | 1.56041i | −0.442723 | − | 0.143849i | −0.309017 | − | 0.951057i | −2.80902 | − | 1.05329i | − | 0.753205i | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
33.f | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 66.2.h.b | yes | 8 |
3.b | odd | 2 | 1 | 66.2.h.a | ✓ | 8 | |
4.b | odd | 2 | 1 | 528.2.bn.a | 8 | ||
11.b | odd | 2 | 1 | 726.2.h.d | 8 | ||
11.c | even | 5 | 1 | 726.2.b.c | 8 | ||
11.c | even | 5 | 1 | 726.2.h.f | 8 | ||
11.c | even | 5 | 1 | 726.2.h.h | 8 | ||
11.c | even | 5 | 1 | 726.2.h.j | 8 | ||
11.d | odd | 10 | 1 | 66.2.h.a | ✓ | 8 | |
11.d | odd | 10 | 1 | 726.2.b.e | 8 | ||
11.d | odd | 10 | 1 | 726.2.h.a | 8 | ||
11.d | odd | 10 | 1 | 726.2.h.c | 8 | ||
12.b | even | 2 | 1 | 528.2.bn.b | 8 | ||
33.d | even | 2 | 1 | 726.2.h.j | 8 | ||
33.f | even | 10 | 1 | inner | 66.2.h.b | yes | 8 |
33.f | even | 10 | 1 | 726.2.b.c | 8 | ||
33.f | even | 10 | 1 | 726.2.h.f | 8 | ||
33.f | even | 10 | 1 | 726.2.h.h | 8 | ||
33.h | odd | 10 | 1 | 726.2.b.e | 8 | ||
33.h | odd | 10 | 1 | 726.2.h.a | 8 | ||
33.h | odd | 10 | 1 | 726.2.h.c | 8 | ||
33.h | odd | 10 | 1 | 726.2.h.d | 8 | ||
44.g | even | 10 | 1 | 528.2.bn.b | 8 | ||
132.n | odd | 10 | 1 | 528.2.bn.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
66.2.h.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
66.2.h.a | ✓ | 8 | 11.d | odd | 10 | 1 | |
66.2.h.b | yes | 8 | 1.a | even | 1 | 1 | trivial |
66.2.h.b | yes | 8 | 33.f | even | 10 | 1 | inner |
528.2.bn.a | 8 | 4.b | odd | 2 | 1 | ||
528.2.bn.a | 8 | 132.n | odd | 10 | 1 | ||
528.2.bn.b | 8 | 12.b | even | 2 | 1 | ||
528.2.bn.b | 8 | 44.g | even | 10 | 1 | ||
726.2.b.c | 8 | 11.c | even | 5 | 1 | ||
726.2.b.c | 8 | 33.f | even | 10 | 1 | ||
726.2.b.e | 8 | 11.d | odd | 10 | 1 | ||
726.2.b.e | 8 | 33.h | odd | 10 | 1 | ||
726.2.h.a | 8 | 11.d | odd | 10 | 1 | ||
726.2.h.a | 8 | 33.h | odd | 10 | 1 | ||
726.2.h.c | 8 | 11.d | odd | 10 | 1 | ||
726.2.h.c | 8 | 33.h | odd | 10 | 1 | ||
726.2.h.d | 8 | 11.b | odd | 2 | 1 | ||
726.2.h.d | 8 | 33.h | odd | 10 | 1 | ||
726.2.h.f | 8 | 11.c | even | 5 | 1 | ||
726.2.h.f | 8 | 33.f | even | 10 | 1 | ||
726.2.h.h | 8 | 11.c | even | 5 | 1 | ||
726.2.h.h | 8 | 33.f | even | 10 | 1 | ||
726.2.h.j | 8 | 11.c | even | 5 | 1 | ||
726.2.h.j | 8 | 33.d | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 2T_{5}^{6} - 15T_{5}^{5} + 19T_{5}^{4} + 30T_{5}^{3} - 8T_{5}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(66, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$3$
\( (T^{4} - T^{3} + 5 T^{2} - 3 T + 9)^{2} \)
$5$
\( T^{8} - 2 T^{6} - 15 T^{5} + 19 T^{4} + \cdots + 16 \)
$7$
\( T^{8} + 2 T^{6} - 25 T^{5} + 59 T^{4} + \cdots + 16 \)
$11$
\( T^{8} + T^{7} + 15 T^{6} - 11 T^{5} + \cdots + 14641 \)
$13$
\( T^{8} - 8 T^{6} - 120 T^{5} + \cdots + 4096 \)
$17$
\( T^{8} + 10 T^{7} + 60 T^{6} + \cdots + 144400 \)
$19$
\( T^{8} + 15 T^{7} + 100 T^{6} + \cdots + 400 \)
$23$
\( T^{8} + 88 T^{6} + 2544 T^{4} + \cdots + 30976 \)
$29$
\( T^{8} - 23 T^{7} + 328 T^{6} + \cdots + 15376 \)
$31$
\( T^{8} - 13 T^{7} + 168 T^{6} + \cdots + 1175056 \)
$37$
\( T^{8} - 6 T^{7} + 32 T^{6} + \cdots + 30976 \)
$41$
\( T^{8} - 2 T^{7} - 32 T^{6} + 91 T^{5} + \cdots + 16 \)
$43$
\( T^{8} + 113 T^{6} + 4549 T^{4} + \cdots + 430336 \)
$47$
\( T^{8} - 10 T^{7} - 8 T^{6} + \cdots + 1048576 \)
$53$
\( T^{8} - 15 T^{7} + 100 T^{6} + \cdots + 2310400 \)
$59$
\( T^{8} + 25 T^{7} + 182 T^{6} + \cdots + 844561 \)
$61$
\( T^{8} + 10 T^{7} + 112 T^{6} + \cdots + 4096 \)
$67$
\( (T^{4} + T^{3} - 149 T^{2} - 284 T + 3076)^{2} \)
$71$
\( (T^{4} - 20 T^{3} + 120 T^{2} - 120 T + 80)^{2} \)
$73$
\( T^{8} - 5 T^{7} - 50 T^{6} + \cdots + 24025 \)
$79$
\( T^{8} - 10 T^{7} - 48 T^{6} + \cdots + 55696 \)
$83$
\( T^{8} + 21 T^{7} + 252 T^{6} + \cdots + 737881 \)
$89$
\( T^{8} + 513 T^{6} + \cdots + 12702096 \)
$97$
\( T^{8} - 9 T^{7} + 282 T^{6} + \cdots + 2070721 \)
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