Newspace parameters
Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 136.n (of order \(8\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.08596546749\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
Defining polynomial: |
\( x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 |
|
0 | −1.70711 | − | 0.707107i | 0 | 0.292893 | − | 0.707107i | 0 | −1.70711 | − | 4.12132i | 0 | 0.292893 | + | 0.292893i | 0 | ||||||||||||||||||||||
25.1 | 0 | −0.292893 | + | 0.707107i | 0 | 1.70711 | + | 0.707107i | 0 | −0.292893 | + | 0.121320i | 0 | 1.70711 | + | 1.70711i | 0 | |||||||||||||||||||||||
49.1 | 0 | −0.292893 | − | 0.707107i | 0 | 1.70711 | − | 0.707107i | 0 | −0.292893 | − | 0.121320i | 0 | 1.70711 | − | 1.70711i | 0 | |||||||||||||||||||||||
121.1 | 0 | −1.70711 | + | 0.707107i | 0 | 0.292893 | + | 0.707107i | 0 | −1.70711 | + | 4.12132i | 0 | 0.292893 | − | 0.292893i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.d | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 136.2.n.a | ✓ | 4 |
3.b | odd | 2 | 1 | 1224.2.bq.a | 4 | ||
4.b | odd | 2 | 1 | 272.2.v.e | 4 | ||
17.d | even | 8 | 1 | inner | 136.2.n.a | ✓ | 4 |
17.e | odd | 16 | 2 | 2312.2.a.s | 4 | ||
17.e | odd | 16 | 2 | 2312.2.b.j | 4 | ||
51.g | odd | 8 | 1 | 1224.2.bq.a | 4 | ||
68.g | odd | 8 | 1 | 272.2.v.e | 4 | ||
68.i | even | 16 | 2 | 4624.2.a.bm | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
136.2.n.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
136.2.n.a | ✓ | 4 | 17.d | even | 8 | 1 | inner |
272.2.v.e | 4 | 4.b | odd | 2 | 1 | ||
272.2.v.e | 4 | 68.g | odd | 8 | 1 | ||
1224.2.bq.a | 4 | 3.b | odd | 2 | 1 | ||
1224.2.bq.a | 4 | 51.g | odd | 8 | 1 | ||
2312.2.a.s | 4 | 17.e | odd | 16 | 2 | ||
2312.2.b.j | 4 | 17.e | odd | 16 | 2 | ||
4624.2.a.bm | 4 | 68.i | even | 16 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 4T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \)
$5$
\( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \)
$7$
\( T^{4} + 4 T^{3} + 22 T^{2} + 12 T + 2 \)
$11$
\( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \)
$13$
\( T^{4} \)
$17$
\( (T^{2} - 2 T + 17)^{2} \)
$19$
\( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 196 \)
$23$
\( T^{4} - 4 T^{3} + 22 T^{2} + \cdots + 1058 \)
$29$
\( T^{4} + 12 T^{3} + 54 T^{2} + \cdots + 162 \)
$31$
\( T^{4} - 12 T^{3} + 54 T^{2} + \cdots + 162 \)
$37$
\( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \)
$41$
\( T^{4} + 12 T^{3} + 134 T^{2} + \cdots + 578 \)
$43$
\( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 324 \)
$47$
\( T^{4} + 216T^{2} + 8464 \)
$53$
\( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 1156 \)
$59$
\( T^{4} - 28 T^{3} + 392 T^{2} + \cdots + 8836 \)
$61$
\( T^{4} + 28 T^{3} + 438 T^{2} + \cdots + 15842 \)
$67$
\( (T^{2} - 8 T - 16)^{2} \)
$71$
\( T^{4} - 12 T^{3} + 134 T^{2} + \cdots + 578 \)
$73$
\( T^{4} - 4 T^{3} + 102 T^{2} + \cdots + 12482 \)
$79$
\( T^{4} - 36 T^{3} + 662 T^{2} + \cdots + 37538 \)
$83$
\( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 324 \)
$89$
\( T^{4} + 96T^{2} + 256 \)
$97$
\( T^{4} + 28 T^{3} + 214 T^{2} + \cdots + 1058 \)
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