Newspace parameters
Level: | \( N \) | \(=\) | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 156.u (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.24566627153\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{12}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).
\(n\) | \(53\) | \(79\) | \(145\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(\zeta_{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 |
|
0 | 0.866025 | + | 1.50000i | 0 | 0 | 0 | 2.23205 | − | 0.598076i | 0 | −1.50000 | + | 2.59808i | 0 | ||||||||||||||||||||||||
89.1 | 0 | −0.866025 | − | 1.50000i | 0 | 0 | 0 | −1.23205 | − | 4.59808i | 0 | −1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||
137.1 | 0 | 0.866025 | − | 1.50000i | 0 | 0 | 0 | 2.23205 | + | 0.598076i | 0 | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||
149.1 | 0 | −0.866025 | + | 1.50000i | 0 | 0 | 0 | −1.23205 | + | 4.59808i | 0 | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
13.f | odd | 12 | 1 | inner |
39.k | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 156.2.u.a | ✓ | 4 |
3.b | odd | 2 | 1 | CM | 156.2.u.a | ✓ | 4 |
4.b | odd | 2 | 1 | 624.2.cn.a | 4 | ||
12.b | even | 2 | 1 | 624.2.cn.a | 4 | ||
13.f | odd | 12 | 1 | inner | 156.2.u.a | ✓ | 4 |
39.k | even | 12 | 1 | inner | 156.2.u.a | ✓ | 4 |
52.l | even | 12 | 1 | 624.2.cn.a | 4 | ||
156.v | odd | 12 | 1 | 624.2.cn.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
156.2.u.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
156.2.u.a | ✓ | 4 | 3.b | odd | 2 | 1 | CM |
156.2.u.a | ✓ | 4 | 13.f | odd | 12 | 1 | inner |
156.2.u.a | ✓ | 4 | 39.k | even | 12 | 1 | inner |
624.2.cn.a | 4 | 4.b | odd | 2 | 1 | ||
624.2.cn.a | 4 | 12.b | even | 2 | 1 | ||
624.2.cn.a | 4 | 52.l | even | 12 | 1 | ||
624.2.cn.a | 4 | 156.v | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 3T^{2} + 9 \)
$5$
\( T^{4} \)
$7$
\( T^{4} - 2 T^{3} + 17 T^{2} - 88 T + 121 \)
$11$
\( T^{4} \)
$13$
\( T^{4} + 23T^{2} + 169 \)
$17$
\( T^{4} \)
$19$
\( T^{4} - 2 T^{3} + 50 T^{2} - 364 T + 676 \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} - 22 T^{3} + 242 T^{2} + \cdots + 3481 \)
$37$
\( T^{4} - 22 T^{3} + 122 T^{2} + \cdots + 676 \)
$41$
\( T^{4} \)
$43$
\( (T^{2} + 21 T + 147)^{2} \)
$47$
\( T^{4} \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( T^{4} + 243 T^{2} + 59049 \)
$67$
\( T^{4} - 32 T^{3} + 377 T^{2} + \cdots + 11881 \)
$71$
\( T^{4} \)
$73$
\( T^{4} - 14 T^{3} + 98 T^{2} + \cdots + 9409 \)
$79$
\( (T^{2} - 27)^{2} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} - 10 T^{3} + 221 T^{2} + \cdots + 27889 \)
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