Newspace parameters
Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 136.j (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.70573159530\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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
\(\beta_{1}\) | \(=\) | \( 4\zeta_{8} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{8}^{2} \) |
\(\beta_{3}\) | \(=\) | \( 4\zeta_{8}^{3} \) |
\(\zeta_{8}\) | \(=\) | \( ( \beta_1 ) / 4 \) |
\(\zeta_{8}^{2}\) | \(=\) | \( \beta_{2} \) |
\(\zeta_{8}^{3}\) | \(=\) | \( ( \beta_{3} ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).
\(n\) | \(69\) | \(103\) | \(105\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
115.1 |
|
2.00000i | −1.82843 | − | 1.82843i | −4.00000 | 0 | 3.65685 | − | 3.65685i | 0 | − | 8.00000i | − | 2.31371i | 0 | ||||||||||||||||||||||||
115.2 | 2.00000i | 3.82843 | + | 3.82843i | −4.00000 | 0 | −7.65685 | + | 7.65685i | 0 | − | 8.00000i | 20.3137i | 0 | ||||||||||||||||||||||||||
123.1 | − | 2.00000i | −1.82843 | + | 1.82843i | −4.00000 | 0 | 3.65685 | + | 3.65685i | 0 | 8.00000i | 2.31371i | 0 | ||||||||||||||||||||||||||
123.2 | − | 2.00000i | 3.82843 | − | 3.82843i | −4.00000 | 0 | −7.65685 | − | 7.65685i | 0 | 8.00000i | − | 20.3137i | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
17.c | even | 4 | 1 | inner |
136.j | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 136.3.j.a | ✓ | 4 |
4.b | odd | 2 | 1 | 544.3.n.a | 4 | ||
8.b | even | 2 | 1 | 544.3.n.a | 4 | ||
8.d | odd | 2 | 1 | CM | 136.3.j.a | ✓ | 4 |
17.c | even | 4 | 1 | inner | 136.3.j.a | ✓ | 4 |
68.f | odd | 4 | 1 | 544.3.n.a | 4 | ||
136.i | even | 4 | 1 | 544.3.n.a | 4 | ||
136.j | odd | 4 | 1 | inner | 136.3.j.a | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
136.3.j.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
136.3.j.a | ✓ | 4 | 8.d | odd | 2 | 1 | CM |
136.3.j.a | ✓ | 4 | 17.c | even | 4 | 1 | inner |
136.3.j.a | ✓ | 4 | 136.j | odd | 4 | 1 | inner |
544.3.n.a | 4 | 4.b | odd | 2 | 1 | ||
544.3.n.a | 4 | 8.b | even | 2 | 1 | ||
544.3.n.a | 4 | 68.f | odd | 4 | 1 | ||
544.3.n.a | 4 | 136.i | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4T_{3}^{3} + 8T_{3}^{2} + 56T_{3} + 196 \)
acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 4)^{2} \)
$3$
\( T^{4} - 4 T^{3} + 8 T^{2} + 56 T + 196 \)
$5$
\( T^{4} \)
$7$
\( T^{4} \)
$11$
\( T^{4} - 28 T^{3} + 392 T^{2} + \cdots + 2116 \)
$13$
\( T^{4} \)
$17$
\( T^{4} - 574 T^{2} + 83521 \)
$19$
\( (T^{2} + 1156)^{2} \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( T^{4} - 92 T^{3} + 4232 T^{2} + \cdots + 1552516 \)
$43$
\( (T^{2} + 7200)^{2} \)
$47$
\( T^{4} \)
$53$
\( T^{4} \)
$59$
\( (T^{2} + 7200)^{2} \)
$61$
\( T^{4} \)
$67$
\( (T^{2} - 14112)^{2} \)
$71$
\( T^{4} \)
$73$
\( T^{4} - 284 T^{3} + \cdots + 90364036 \)
$79$
\( T^{4} \)
$83$
\( (T^{2} + 2592)^{2} \)
$89$
\( (T^{2} - 10368)^{2} \)
$97$
\( T^{4} + 188 T^{3} + \cdots + 99640324 \)
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