Newspace parameters
Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 117.g (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.90322347067\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 13) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} + 4 ) / 5 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) |
\(\nu^{3}\) | \(=\) | \( 5\beta_{3} - 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).
\(n\) | \(28\) | \(92\) |
\(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
55.1 |
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−2.28078 | − | 3.95042i | 0 | −6.40388 | + | 11.0918i | −2.80776 | 0 | −4.78078 | + | 8.28055i | 21.9309 | 0 | 6.40388 | + | 11.0918i | ||||||||||||||||||||||
55.2 | −0.219224 | − | 0.379706i | 0 | 3.90388 | − | 6.76172i | 17.8078 | 0 | −2.71922 | + | 4.70983i | −6.93087 | 0 | −3.90388 | − | 6.76172i | |||||||||||||||||||||||
100.1 | −2.28078 | + | 3.95042i | 0 | −6.40388 | − | 11.0918i | −2.80776 | 0 | −4.78078 | − | 8.28055i | 21.9309 | 0 | 6.40388 | − | 11.0918i | |||||||||||||||||||||||
100.2 | −0.219224 | + | 0.379706i | 0 | 3.90388 | + | 6.76172i | 17.8078 | 0 | −2.71922 | − | 4.70983i | −6.93087 | 0 | −3.90388 | + | 6.76172i | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 117.4.g.d | 4 | |
3.b | odd | 2 | 1 | 13.4.c.b | ✓ | 4 | |
12.b | even | 2 | 1 | 208.4.i.e | 4 | ||
13.c | even | 3 | 1 | inner | 117.4.g.d | 4 | |
13.c | even | 3 | 1 | 1521.4.a.t | 2 | ||
13.e | even | 6 | 1 | 1521.4.a.l | 2 | ||
39.d | odd | 2 | 1 | 169.4.c.f | 4 | ||
39.f | even | 4 | 2 | 169.4.e.g | 8 | ||
39.h | odd | 6 | 1 | 169.4.a.j | 2 | ||
39.h | odd | 6 | 1 | 169.4.c.f | 4 | ||
39.i | odd | 6 | 1 | 13.4.c.b | ✓ | 4 | |
39.i | odd | 6 | 1 | 169.4.a.f | 2 | ||
39.k | even | 12 | 2 | 169.4.b.e | 4 | ||
39.k | even | 12 | 2 | 169.4.e.g | 8 | ||
156.p | even | 6 | 1 | 208.4.i.e | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.4.c.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
13.4.c.b | ✓ | 4 | 39.i | odd | 6 | 1 | |
117.4.g.d | 4 | 1.a | even | 1 | 1 | trivial | |
117.4.g.d | 4 | 13.c | even | 3 | 1 | inner | |
169.4.a.f | 2 | 39.i | odd | 6 | 1 | ||
169.4.a.j | 2 | 39.h | odd | 6 | 1 | ||
169.4.b.e | 4 | 39.k | even | 12 | 2 | ||
169.4.c.f | 4 | 39.d | odd | 2 | 1 | ||
169.4.c.f | 4 | 39.h | odd | 6 | 1 | ||
169.4.e.g | 8 | 39.f | even | 4 | 2 | ||
169.4.e.g | 8 | 39.k | even | 12 | 2 | ||
208.4.i.e | 4 | 12.b | even | 2 | 1 | ||
208.4.i.e | 4 | 156.p | even | 6 | 1 | ||
1521.4.a.l | 2 | 13.e | even | 6 | 1 | ||
1521.4.a.t | 2 | 13.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 5T_{2}^{3} + 23T_{2}^{2} + 10T_{2} + 4 \)
acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \)
$3$
\( T^{4} \)
$5$
\( (T^{2} - 15 T - 50)^{2} \)
$7$
\( T^{4} + 15 T^{3} + 173 T^{2} + \cdots + 2704 \)
$11$
\( T^{4} - 17 T^{3} + 1173 T^{2} + \cdots + 781456 \)
$13$
\( T^{4} - 125 T^{3} + 7956 T^{2} + \cdots + 4826809 \)
$17$
\( T^{4} - 70 T^{3} + 4763 T^{2} + \cdots + 18769 \)
$19$
\( T^{4} - 141 T^{3} + \cdots + 23658496 \)
$23$
\( T^{4} - 145 T^{3} + 20397 T^{2} + \cdots + 394384 \)
$29$
\( T^{4} - 34 T^{3} + \cdots + 225330121 \)
$31$
\( (T^{2} + 140 T - 37600)^{2} \)
$37$
\( T^{4} - 190 T^{3} + 35303 T^{2} + \cdots + 635209 \)
$41$
\( T^{4} - 538 T^{3} + \cdots + 4992976921 \)
$43$
\( T^{4} + 455 T^{3} + \cdots + 138485824 \)
$47$
\( (T^{2} + 60 T - 82400)^{2} \)
$53$
\( (T^{2} + 545 T - 41450)^{2} \)
$59$
\( T^{4} + 809 T^{3} + \cdots + 22729783696 \)
$61$
\( T^{4} + 502 T^{3} + \cdots + 11448786001 \)
$67$
\( T^{4} - 475 T^{3} + \cdots + 449948944 \)
$71$
\( T^{4} - 127 T^{3} + \cdots + 1833894976 \)
$73$
\( (T^{2} - 585 T + 54850)^{2} \)
$79$
\( (T^{2} - 240 T + 7600)^{2} \)
$83$
\( (T^{2} + 260 T - 25600)^{2} \)
$89$
\( T^{4} - 921 T^{3} + \cdots + 21215087716 \)
$97$
\( T^{4} - 415 T^{3} + \cdots + 795268001284 \)
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