Newspace parameters
Level: | \( N \) | \(=\) | \( 26 = 2 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 26.c (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.53404966015\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{217})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} + 55x^{2} + 54x + 2916 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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 \( x^{4} - x^{3} + 55x^{2} + 54x + 2916 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + 55\nu^{2} - 55\nu + 2916 ) / 2970 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} + 109 ) / 55 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + 54\beta_{2} + \beta _1 - 55 \) |
\(\nu^{3}\) | \(=\) | \( 55\beta_{3} - 109 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).
\(n\) | \(15\) |
\(\chi(n)\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 |
|
−1.00000 | − | 1.73205i | −2.93273 | − | 5.07964i | −2.00000 | + | 3.46410i | −10.8655 | −5.86546 | + | 10.1593i | 14.9327 | − | 25.8642i | 8.00000 | −3.70181 | + | 6.41172i | 10.8655 | + | 18.8195i | ||||||||||||||||
3.2 | −1.00000 | − | 1.73205i | 4.43273 | + | 7.67771i | −2.00000 | + | 3.46410i | 3.86546 | 8.86546 | − | 15.3554i | 7.56727 | − | 13.1069i | 8.00000 | −25.7982 | + | 44.6838i | −3.86546 | − | 6.69517i | |||||||||||||||||
9.1 | −1.00000 | + | 1.73205i | −2.93273 | + | 5.07964i | −2.00000 | − | 3.46410i | −10.8655 | −5.86546 | − | 10.1593i | 14.9327 | + | 25.8642i | 8.00000 | −3.70181 | − | 6.41172i | 10.8655 | − | 18.8195i | |||||||||||||||||
9.2 | −1.00000 | + | 1.73205i | 4.43273 | − | 7.67771i | −2.00000 | − | 3.46410i | 3.86546 | 8.86546 | + | 15.3554i | 7.56727 | + | 13.1069i | 8.00000 | −25.7982 | − | 44.6838i | −3.86546 | + | 6.69517i | |||||||||||||||||
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 | 26.4.c.b | ✓ | 4 |
3.b | odd | 2 | 1 | 234.4.h.h | 4 | ||
4.b | odd | 2 | 1 | 208.4.i.d | 4 | ||
13.b | even | 2 | 1 | 338.4.c.j | 4 | ||
13.c | even | 3 | 1 | inner | 26.4.c.b | ✓ | 4 |
13.c | even | 3 | 1 | 338.4.a.h | 2 | ||
13.d | odd | 4 | 2 | 338.4.e.f | 8 | ||
13.e | even | 6 | 1 | 338.4.a.g | 2 | ||
13.e | even | 6 | 1 | 338.4.c.j | 4 | ||
13.f | odd | 12 | 2 | 338.4.b.e | 4 | ||
13.f | odd | 12 | 2 | 338.4.e.f | 8 | ||
39.i | odd | 6 | 1 | 234.4.h.h | 4 | ||
52.j | odd | 6 | 1 | 208.4.i.d | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
26.4.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
26.4.c.b | ✓ | 4 | 13.c | even | 3 | 1 | inner |
208.4.i.d | 4 | 4.b | odd | 2 | 1 | ||
208.4.i.d | 4 | 52.j | odd | 6 | 1 | ||
234.4.h.h | 4 | 3.b | odd | 2 | 1 | ||
234.4.h.h | 4 | 39.i | odd | 6 | 1 | ||
338.4.a.g | 2 | 13.e | even | 6 | 1 | ||
338.4.a.h | 2 | 13.c | even | 3 | 1 | ||
338.4.b.e | 4 | 13.f | odd | 12 | 2 | ||
338.4.c.j | 4 | 13.b | even | 2 | 1 | ||
338.4.c.j | 4 | 13.e | even | 6 | 1 | ||
338.4.e.f | 8 | 13.d | odd | 4 | 2 | ||
338.4.e.f | 8 | 13.f | odd | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 3T_{3}^{3} + 61T_{3}^{2} + 156T_{3} + 2704 \)
acting on \(S_{4}^{\mathrm{new}}(26, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2 T + 4)^{2} \)
$3$
\( T^{4} - 3 T^{3} + 61 T^{2} + \cdots + 2704 \)
$5$
\( (T^{2} + 7 T - 42)^{2} \)
$7$
\( T^{4} - 45 T^{3} + 1573 T^{2} + \cdots + 204304 \)
$11$
\( T^{4} + 5 T^{3} + 2677 T^{2} + \cdots + 7033104 \)
$13$
\( T^{4} + 35 T^{3} + 4212 T^{2} + \cdots + 4826809 \)
$17$
\( T^{4} - 130 T^{3} + 16147 T^{2} + \cdots + 567009 \)
$19$
\( T^{4} + 3 T^{3} + 61 T^{2} + \cdots + 2704 \)
$23$
\( T^{4} + 33 T^{3} + 1305 T^{2} + \cdots + 46656 \)
$29$
\( T^{4} - 198 T^{3} + 37215 T^{2} + \cdots + 3956121 \)
$31$
\( (T^{2} - 280 T - 11648)^{2} \)
$37$
\( T^{4} - 702 T^{3} + \cdots + 14965362889 \)
$41$
\( T^{4} + 242 T^{3} + \cdots + 49829481 \)
$43$
\( T^{4} - 93 T^{3} + \cdots + 1007681536 \)
$47$
\( (T^{2} - 224 T - 157584)^{2} \)
$53$
\( (T^{2} + 535 T + 71502)^{2} \)
$59$
\( T^{4} - 389 T^{3} + \cdots + 1237069584 \)
$61$
\( T^{4} + 654 T^{3} + \cdots + 2639596129 \)
$67$
\( T^{4} - 107 T^{3} + \cdots + 45137551936 \)
$71$
\( T^{4} - 569 T^{3} + \cdots + 3764558736 \)
$73$
\( (T^{2} + 623 T + 84826)^{2} \)
$79$
\( (T^{2} - 760 T + 19408)^{2} \)
$83$
\( (T^{2} + 1764 T + 707616)^{2} \)
$89$
\( T^{4} - 871 T^{3} + \cdots + 18913400676 \)
$97$
\( T^{4} - 879 T^{3} + \cdots + 10397065156 \)
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