Newspace parameters
Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 117.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(6.90322347067\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | 4.4.1520092.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} - 40x^{2} - 9x + 81 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{3} \) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \( x^{4} - x^{3} - 40x^{2} - 9x + 81 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} - \nu^{2} - 31\nu - 9 ) / 9 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + \nu^{2} + 49\nu + 9 ) / 9 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{3} + 10\nu^{2} + 22\nu - 171 ) / 9 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} + \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( 2\beta_{3} + \beta_{2} + 3\beta _1 + 40 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} + 16\beta_{2} + 26\beta _1 + 29 \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
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−5.48928 | 0 | 22.1322 | 14.0859 | 0 | 24.2643 | −77.5754 | 0 | −77.3217 | ||||||||||||||||||||||||||||||
1.2 | −3.85588 | 0 | 6.86783 | −19.5342 | 0 | −6.26434 | 4.36551 | 0 | 75.3217 | |||||||||||||||||||||||||||||||
1.3 | 3.85588 | 0 | 6.86783 | 19.5342 | 0 | −6.26434 | −4.36551 | 0 | 75.3217 | |||||||||||||||||||||||||||||||
1.4 | 5.48928 | 0 | 22.1322 | −14.0859 | 0 | 24.2643 | 77.5754 | 0 | −77.3217 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(13\) | \(1\) |
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 | 117.4.a.g | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 117.4.a.g | ✓ | 4 |
4.b | odd | 2 | 1 | 1872.4.a.bo | 4 | ||
12.b | even | 2 | 1 | 1872.4.a.bo | 4 | ||
13.b | even | 2 | 1 | 1521.4.a.ba | 4 | ||
39.d | odd | 2 | 1 | 1521.4.a.ba | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
117.4.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
117.4.a.g | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
1521.4.a.ba | 4 | 13.b | even | 2 | 1 | ||
1521.4.a.ba | 4 | 39.d | odd | 2 | 1 | ||
1872.4.a.bo | 4 | 4.b | odd | 2 | 1 | ||
1872.4.a.bo | 4 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 45T_{2}^{2} + 448 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(117))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 45T^{2} + 448 \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 580 T^{2} + 75712 \)
$7$
\( (T^{2} - 18 T - 152)^{2} \)
$11$
\( T^{4} - 752T^{2} + 7168 \)
$13$
\( (T + 13)^{4} \)
$17$
\( T^{4} - 2880 T^{2} + \cdots + 1835008 \)
$19$
\( (T^{2} - 42 T - 10976)^{2} \)
$23$
\( T^{4} - 48656 T^{2} + \cdots + 295386112 \)
$29$
\( T^{4} - 75200 T^{2} + \cdots + 71680000 \)
$31$
\( (T^{2} + 302 T + 22568)^{2} \)
$37$
\( (T^{2} - 92 T - 57532)^{2} \)
$41$
\( T^{4} - 47844 T^{2} + \cdots + 569017792 \)
$43$
\( (T^{2} - 440 T + 44672)^{2} \)
$47$
\( T^{4} - 144640 T^{2} + \cdots + 4778706688 \)
$53$
\( T^{4} - 157136 T^{2} + \cdots + 3798925312 \)
$59$
\( T^{4} - 222960 T^{2} + \cdots + 375897088 \)
$61$
\( (T^{2} - 328 T - 85876)^{2} \)
$67$
\( (T^{2} - 1526 T + 514832)^{2} \)
$71$
\( T^{4} - 931328 T^{2} + \cdots + 31867295488 \)
$73$
\( (T^{2} + 156 T - 1071308)^{2} \)
$79$
\( (T^{2} + 360 T - 698288)^{2} \)
$83$
\( T^{4} - 14640 T^{2} + \cdots + 39251968 \)
$89$
\( T^{4} - 1906852 T^{2} + \cdots + 626946109888 \)
$97$
\( (T^{2} + 172 T - 622636)^{2} \)
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