Newspace parameters
Level: | \( N \) | \(=\) | \( 55 = 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 55.g (of order \(5\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.439177211117\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
Coefficient field: | 8.0.13140625.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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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 \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).
\(n\) | \(12\) | \(46\) |
\(\chi(n)\) | \(1\) | \(\beta_{7}\) |
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 |
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−0.647481 | − | 1.99274i | −1.54765 | − | 1.12443i | −1.93376 | + | 1.40496i | −0.309017 | + | 0.951057i | −1.23863 | + | 3.81211i | 2.48141 | − | 1.80285i | 0.661536 | + | 0.480634i | 0.203814 | + | 0.627276i | 2.09529 | ||||||||||||||||||||||||||
16.2 | 0.147481 | + | 0.453901i | −0.261370 | − | 0.189896i | 1.43376 | − | 1.04169i | −0.309017 | + | 0.951057i | 0.0476470 | − | 0.146642i | −2.17239 | + | 1.57833i | 1.45650 | + | 1.05821i | −0.894797 | − | 2.75390i | −0.477260 | |||||||||||||||||||||||||||
26.1 | −1.09676 | − | 0.796845i | 0.177837 | − | 0.547326i | −0.0501062 | − | 0.154211i | 0.809017 | − | 0.587785i | −0.631180 | + | 0.458579i | −1.12773 | − | 3.47080i | −0.905781 | + | 2.78771i | 2.15911 | + | 1.56869i | −1.35567 | |||||||||||||||||||||||||||
26.2 | 0.596764 | + | 0.433574i | −0.868820 | + | 2.67395i | −0.449894 | − | 1.38463i | 0.809017 | − | 0.587785i | −1.67784 | + | 1.21902i | 0.318714 | + | 0.980901i | 0.787747 | − | 2.42443i | −3.96813 | − | 2.88301i | 0.737640 | |||||||||||||||||||||||||||
31.1 | −0.647481 | + | 1.99274i | −1.54765 | + | 1.12443i | −1.93376 | − | 1.40496i | −0.309017 | − | 0.951057i | −1.23863 | − | 3.81211i | 2.48141 | + | 1.80285i | 0.661536 | − | 0.480634i | 0.203814 | − | 0.627276i | 2.09529 | |||||||||||||||||||||||||||
31.2 | 0.147481 | − | 0.453901i | −0.261370 | + | 0.189896i | 1.43376 | + | 1.04169i | −0.309017 | − | 0.951057i | 0.0476470 | + | 0.146642i | −2.17239 | − | 1.57833i | 1.45650 | − | 1.05821i | −0.894797 | + | 2.75390i | −0.477260 | |||||||||||||||||||||||||||
36.1 | −1.09676 | + | 0.796845i | 0.177837 | + | 0.547326i | −0.0501062 | + | 0.154211i | 0.809017 | + | 0.587785i | −0.631180 | − | 0.458579i | −1.12773 | + | 3.47080i | −0.905781 | − | 2.78771i | 2.15911 | − | 1.56869i | −1.35567 | |||||||||||||||||||||||||||
36.2 | 0.596764 | − | 0.433574i | −0.868820 | − | 2.67395i | −0.449894 | + | 1.38463i | 0.809017 | + | 0.587785i | −1.67784 | − | 1.21902i | 0.318714 | − | 0.980901i | 0.787747 | + | 2.42443i | −3.96813 | + | 2.88301i | 0.737640 | |||||||||||||||||||||||||||
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 | 55.2.g.b | ✓ | 8 |
3.b | odd | 2 | 1 | 495.2.n.e | 8 | ||
4.b | odd | 2 | 1 | 880.2.bo.h | 8 | ||
5.b | even | 2 | 1 | 275.2.h.a | 8 | ||
5.c | odd | 4 | 2 | 275.2.z.a | 16 | ||
11.b | odd | 2 | 1 | 605.2.g.k | 8 | ||
11.c | even | 5 | 1 | inner | 55.2.g.b | ✓ | 8 |
11.c | even | 5 | 1 | 605.2.a.j | 4 | ||
11.c | even | 5 | 2 | 605.2.g.m | 8 | ||
11.d | odd | 10 | 1 | 605.2.a.k | 4 | ||
11.d | odd | 10 | 2 | 605.2.g.e | 8 | ||
11.d | odd | 10 | 1 | 605.2.g.k | 8 | ||
33.f | even | 10 | 1 | 5445.2.a.bi | 4 | ||
33.h | odd | 10 | 1 | 495.2.n.e | 8 | ||
33.h | odd | 10 | 1 | 5445.2.a.bp | 4 | ||
44.g | even | 10 | 1 | 9680.2.a.cm | 4 | ||
44.h | odd | 10 | 1 | 880.2.bo.h | 8 | ||
44.h | odd | 10 | 1 | 9680.2.a.cn | 4 | ||
55.h | odd | 10 | 1 | 3025.2.a.w | 4 | ||
55.j | even | 10 | 1 | 275.2.h.a | 8 | ||
55.j | even | 10 | 1 | 3025.2.a.bd | 4 | ||
55.k | odd | 20 | 2 | 275.2.z.a | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
55.2.g.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
55.2.g.b | ✓ | 8 | 11.c | even | 5 | 1 | inner |
275.2.h.a | 8 | 5.b | even | 2 | 1 | ||
275.2.h.a | 8 | 55.j | even | 10 | 1 | ||
275.2.z.a | 16 | 5.c | odd | 4 | 2 | ||
275.2.z.a | 16 | 55.k | odd | 20 | 2 | ||
495.2.n.e | 8 | 3.b | odd | 2 | 1 | ||
495.2.n.e | 8 | 33.h | odd | 10 | 1 | ||
605.2.a.j | 4 | 11.c | even | 5 | 1 | ||
605.2.a.k | 4 | 11.d | odd | 10 | 1 | ||
605.2.g.e | 8 | 11.d | odd | 10 | 2 | ||
605.2.g.k | 8 | 11.b | odd | 2 | 1 | ||
605.2.g.k | 8 | 11.d | odd | 10 | 1 | ||
605.2.g.m | 8 | 11.c | even | 5 | 2 | ||
880.2.bo.h | 8 | 4.b | odd | 2 | 1 | ||
880.2.bo.h | 8 | 44.h | odd | 10 | 1 | ||
3025.2.a.w | 4 | 55.h | odd | 10 | 1 | ||
3025.2.a.bd | 4 | 55.j | even | 10 | 1 | ||
5445.2.a.bi | 4 | 33.f | even | 10 | 1 | ||
5445.2.a.bp | 4 | 33.h | odd | 10 | 1 | ||
9680.2.a.cm | 4 | 44.g | even | 10 | 1 | ||
9680.2.a.cn | 4 | 44.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} + 5T_{2}^{6} + 2T_{2}^{5} - T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(55, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 2 T^{7} + 5 T^{6} + 2 T^{5} + \cdots + 1 \)
$3$
\( T^{8} + 5 T^{7} + 18 T^{6} + 35 T^{5} + \cdots + 1 \)
$5$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$7$
\( T^{8} + T^{7} + 7 T^{6} - 17 T^{5} + \cdots + 961 \)
$11$
\( T^{8} - 3 T^{7} + 18 T^{6} + \cdots + 14641 \)
$13$
\( T^{8} + 2 T^{7} + 21 T^{6} + \cdots + 19321 \)
$17$
\( T^{8} + 13 T^{7} + 96 T^{6} + \cdots + 361 \)
$19$
\( T^{8} - 15 T^{7} + 135 T^{6} + \cdots + 625 \)
$23$
\( (T^{4} - 5 T^{3} + 4 T^{2} + 10 T - 11)^{2} \)
$29$
\( T^{8} + 9 T^{7} + 99 T^{6} + \cdots + 203401 \)
$31$
\( T^{8} + 10 T^{7} + 125 T^{6} + \cdots + 390625 \)
$37$
\( T^{8} - 24 T^{7} + 359 T^{6} + \cdots + 1324801 \)
$41$
\( T^{8} - 8 T^{7} + 93 T^{6} + \cdots + 101761 \)
$43$
\( (T^{4} + 19 T^{3} + 121 T^{2} + 289 T + 211)^{2} \)
$47$
\( T^{8} + 23 T^{6} + 90 T^{5} + \cdots + 28561 \)
$53$
\( T^{8} - 13 T^{7} + 85 T^{6} + \cdots + 885481 \)
$59$
\( T^{8} + 27 T^{7} + 385 T^{6} + \cdots + 687241 \)
$61$
\( T^{8} - 6 T^{7} + 10 T^{6} + \cdots + 28561 \)
$67$
\( (T^{4} + 19 T^{3} + 22 T^{2} - 1014 T - 4079)^{2} \)
$71$
\( T^{8} + 20 T^{7} + 213 T^{6} + \cdots + 17161 \)
$73$
\( T^{8} - 13 T^{7} + 56 T^{6} + \cdots + 121 \)
$79$
\( T^{8} - 37 T^{7} + 698 T^{6} + \cdots + 45954841 \)
$83$
\( T^{8} - 27 T^{7} + 493 T^{6} + \cdots + 2886601 \)
$89$
\( (T^{4} + 8 T^{3} - 102 T^{2} - 472 T + 1861)^{2} \)
$97$
\( T^{8} - 24 T^{7} + 749 T^{6} + \cdots + 9066121 \)
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