Newspace parameters
Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 136.p (of order \(8\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.70573159530\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
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: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{8}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).
\(n\) | \(69\) | \(103\) | \(105\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(\zeta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
|
1.41421 | + | 1.41421i | 4.12132 | − | 1.70711i | 4.00000i | 0 | 8.24264 | + | 3.41421i | 0 | −5.65685 | + | 5.65685i | 7.70711 | − | 7.70711i | 0 | ||||||||||||||||||||
43.1 | 1.41421 | − | 1.41421i | 4.12132 | + | 1.70711i | − | 4.00000i | 0 | 8.24264 | − | 3.41421i | 0 | −5.65685 | − | 5.65685i | 7.70711 | + | 7.70711i | 0 | ||||||||||||||||||||
59.1 | −1.41421 | + | 1.41421i | −0.121320 | + | 0.292893i | − | 4.00000i | 0 | −0.242641 | − | 0.585786i | 0 | 5.65685 | + | 5.65685i | 6.29289 | + | 6.29289i | 0 | ||||||||||||||||||||
83.1 | −1.41421 | − | 1.41421i | −0.121320 | − | 0.292893i | 4.00000i | 0 | −0.242641 | + | 0.585786i | 0 | 5.65685 | − | 5.65685i | 6.29289 | − | 6.29289i | 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.d | even | 8 | 1 | inner |
136.p | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 136.3.p.b | ✓ | 4 |
8.d | odd | 2 | 1 | CM | 136.3.p.b | ✓ | 4 |
17.d | even | 8 | 1 | inner | 136.3.p.b | ✓ | 4 |
136.p | odd | 8 | 1 | inner | 136.3.p.b | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
136.3.p.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
136.3.p.b | ✓ | 4 | 8.d | odd | 2 | 1 | CM |
136.3.p.b | ✓ | 4 | 17.d | even | 8 | 1 | inner |
136.3.p.b | ✓ | 4 | 136.p | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 8T_{3}^{3} + 18T_{3}^{2} + 4T_{3} + 2 \)
acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 16 \)
$3$
\( T^{4} - 8 T^{3} + 18 T^{2} + 4 T + 2 \)
$5$
\( T^{4} \)
$7$
\( T^{4} \)
$11$
\( T^{4} + 4 T^{3} + 54 T^{2} + \cdots + 4418 \)
$13$
\( T^{4} \)
$17$
\( T^{4} - 48 T^{3} + 1152 T^{2} + \cdots + 83521 \)
$19$
\( (T^{2} + 24 T + 288)^{2} \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( T^{4} - 96 T^{3} + 3362 T^{2} + \cdots + 30248642 \)
$43$
\( T^{4} - 28 T^{3} + 392 T^{2} + \cdots + 12264004 \)
$47$
\( T^{4} \)
$53$
\( T^{4} \)
$59$
\( T^{4} - 164 T^{3} + 13448 T^{2} + \cdots + 56644 \)
$61$
\( T^{4} \)
$67$
\( (T^{2} + 168 T + 5134)^{2} \)
$71$
\( T^{4} \)
$73$
\( T^{4} + 48 T^{3} + \cdots + 152670338 \)
$79$
\( T^{4} \)
$83$
\( T^{4} - 316 T^{3} + \cdots + 125126596 \)
$89$
\( T^{4} + 31684 T^{2} + \cdots + 29964676 \)
$97$
\( T^{4} + 52 T^{3} + 11334 T^{2} + \cdots + 7001282 \)
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