Newspace parameters
Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 165.m (of order \(5\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.31753163335\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
Coefficient field: | 8.0.13140625.1 |
Defining polynomial: |
\( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \)
|
\(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \)
|
\(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \)
|
\(\nu^{4}\) | \(=\) |
\( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \)
|
\(\nu^{5}\) | \(=\) |
\( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \)
|
\(\nu^{6}\) | \(=\) |
\( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \)
|
\(\nu^{7}\) | \(=\) |
\( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(56\) | \(67\) |
\(\chi(n)\) | \(\beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 |
|
0.359123 | + | 1.10527i | −0.809017 | − | 0.587785i | 0.525387 | − | 0.381716i | −0.309017 | + | 0.951057i | 0.359123 | − | 1.10527i | 3.46813 | − | 2.51974i | 2.49097 | + | 1.80980i | 0.309017 | + | 0.951057i | −1.16215 | ||||||||||||||||||||||||||
16.2 | 0.758911 | + | 2.33569i | −0.809017 | − | 0.587785i | −3.26145 | + | 2.36959i | −0.309017 | + | 0.951057i | 0.758911 | − | 2.33569i | −2.65911 | + | 1.93196i | −4.03606 | − | 2.93237i | 0.309017 | + | 0.951057i | −2.45589 | |||||||||||||||||||||||||||
31.1 | 0.359123 | − | 1.10527i | −0.809017 | + | 0.587785i | 0.525387 | + | 0.381716i | −0.309017 | − | 0.951057i | 0.359123 | + | 1.10527i | 3.46813 | + | 2.51974i | 2.49097 | − | 1.80980i | 0.309017 | − | 0.951057i | −1.16215 | |||||||||||||||||||||||||||
31.2 | 0.758911 | − | 2.33569i | −0.809017 | + | 0.587785i | −3.26145 | − | 2.36959i | −0.309017 | − | 0.951057i | 0.758911 | + | 2.33569i | −2.65911 | − | 1.93196i | −4.03606 | + | 2.93237i | 0.309017 | − | 0.951057i | −2.45589 | |||||||||||||||||||||||||||
91.1 | −2.24278 | + | 1.62947i | 0.309017 | + | 0.951057i | 1.75683 | − | 5.40697i | 0.809017 | + | 0.587785i | −2.24278 | − | 1.62947i | −0.703814 | + | 2.16612i | 3.15700 | + | 9.71623i | −0.809017 | + | 0.587785i | −2.77222 | |||||||||||||||||||||||||||
91.2 | 1.12474 | − | 0.817172i | 0.309017 | + | 0.951057i | −0.0207616 | + | 0.0638975i | 0.809017 | + | 0.587785i | 1.12474 | + | 0.817172i | 0.394797 | − | 1.21506i | 0.888090 | + | 2.73326i | −0.809017 | + | 0.587785i | 1.39026 | |||||||||||||||||||||||||||
136.1 | −2.24278 | − | 1.62947i | 0.309017 | − | 0.951057i | 1.75683 | + | 5.40697i | 0.809017 | − | 0.587785i | −2.24278 | + | 1.62947i | −0.703814 | − | 2.16612i | 3.15700 | − | 9.71623i | −0.809017 | − | 0.587785i | −2.77222 | |||||||||||||||||||||||||||
136.2 | 1.12474 | + | 0.817172i | 0.309017 | − | 0.951057i | −0.0207616 | − | 0.0638975i | 0.809017 | − | 0.587785i | 1.12474 | − | 0.817172i | 0.394797 | + | 1.21506i | 0.888090 | − | 2.73326i | −0.809017 | − | 0.587785i | 1.39026 | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 165.2.m.a | ✓ | 8 |
3.b | odd | 2 | 1 | 495.2.n.d | 8 | ||
5.b | even | 2 | 1 | 825.2.n.k | 8 | ||
5.c | odd | 4 | 2 | 825.2.bx.h | 16 | ||
11.c | even | 5 | 1 | inner | 165.2.m.a | ✓ | 8 |
11.c | even | 5 | 1 | 1815.2.a.x | 4 | ||
11.d | odd | 10 | 1 | 1815.2.a.o | 4 | ||
33.f | even | 10 | 1 | 5445.2.a.bv | 4 | ||
33.h | odd | 10 | 1 | 495.2.n.d | 8 | ||
33.h | odd | 10 | 1 | 5445.2.a.be | 4 | ||
55.h | odd | 10 | 1 | 9075.2.a.dj | 4 | ||
55.j | even | 10 | 1 | 825.2.n.k | 8 | ||
55.j | even | 10 | 1 | 9075.2.a.cl | 4 | ||
55.k | odd | 20 | 2 | 825.2.bx.h | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
165.2.m.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
165.2.m.a | ✓ | 8 | 11.c | even | 5 | 1 | inner |
495.2.n.d | 8 | 3.b | odd | 2 | 1 | ||
495.2.n.d | 8 | 33.h | odd | 10 | 1 | ||
825.2.n.k | 8 | 5.b | even | 2 | 1 | ||
825.2.n.k | 8 | 55.j | even | 10 | 1 | ||
825.2.bx.h | 16 | 5.c | odd | 4 | 2 | ||
825.2.bx.h | 16 | 55.k | odd | 20 | 2 | ||
1815.2.a.o | 4 | 11.d | odd | 10 | 1 | ||
1815.2.a.x | 4 | 11.c | even | 5 | 1 | ||
5445.2.a.be | 4 | 33.h | odd | 10 | 1 | ||
5445.2.a.bv | 4 | 33.f | even | 10 | 1 | ||
9075.2.a.cl | 4 | 55.j | even | 10 | 1 | ||
9075.2.a.dj | 4 | 55.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 3T_{2}^{6} + 5T_{2}^{5} + 24T_{2}^{4} - 85T_{2}^{3} + 177T_{2}^{2} - 165T_{2} + 121 \)
acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 3 T^{6} + 5 T^{5} + 24 T^{4} + \cdots + 121 \)
$3$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$5$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$7$
\( T^{8} - T^{7} - 3 T^{6} + 7 T^{5} + \cdots + 1681 \)
$11$
\( T^{8} + 3 T^{7} + 8 T^{6} + \cdots + 14641 \)
$13$
\( T^{8} - 6 T^{7} + 39 T^{6} - 87 T^{5} + \cdots + 1 \)
$17$
\( (T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625)^{2} \)
$19$
\( T^{8} - 6 T^{7} + 9 T^{6} + 123 T^{5} + \cdots + 961 \)
$23$
\( (T^{4} + 5 T^{3} - T^{2} - 5 T - 1)^{2} \)
$29$
\( T^{8} + 17 T^{6} + 95 T^{5} + \cdots + 290521 \)
$31$
\( T^{8} - 3 T^{7} + 11 T^{6} + \cdots + 19321 \)
$37$
\( T^{8} + 19 T^{7} + 255 T^{6} + \cdots + 185761 \)
$41$
\( T^{8} + 25 T^{7} + 327 T^{6} + \cdots + 4289041 \)
$43$
\( (T^{4} + 2 T^{3} - 92 T^{2} - 63 T + 1861)^{2} \)
$47$
\( T^{8} - 15 T^{7} + 123 T^{6} + \cdots + 121 \)
$53$
\( T^{8} - 7 T^{7} - 14 T^{6} + \cdots + 1615441 \)
$59$
\( T^{8} - 35 T^{7} + 633 T^{6} + \cdots + 5285401 \)
$61$
\( T^{8} - 21 T^{7} + 242 T^{6} + \cdots + 3575881 \)
$67$
\( (T^{4} + 13 T^{3} - 136 T^{2} - 1768 T - 3379)^{2} \)
$71$
\( T^{8} - 25 T^{7} + 347 T^{6} + \cdots + 5527201 \)
$73$
\( T^{8} - T^{7} + 30 T^{6} + \cdots + 84621601 \)
$79$
\( T^{8} - 30 T^{7} + 487 T^{6} + \cdots + 1437601 \)
$83$
\( T^{8} - 11 T^{7} + \cdots + 149352841 \)
$89$
\( (T^{4} - 16 T^{3} + 45 T^{2} + 132 T - 271)^{2} \)
$97$
\( T^{8} - 5 T^{7} - 20 T^{6} + 225 T^{5} + \cdots + 625 \)
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