Newspace parameters
Level: | \( N \) | \(=\) | \( 134 = 2 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 134.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(1.06999538709\) |
Analytic rank: | \(0\) |
Dimension: | \(3\) |
Coefficient field: | \(\Q(\zeta_{18})^+\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{3} - 3x - 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 2 \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
1.00000 | −0.532089 | 1.00000 | 0.184793 | −0.532089 | 3.75877 | 1.00000 | −2.71688 | 0.184793 | |||||||||||||||||||||||||||
1.2 | 1.00000 | 0.652704 | 1.00000 | 1.22668 | 0.652704 | −3.06418 | 1.00000 | −2.57398 | 1.22668 | ||||||||||||||||||||||||||||
1.3 | 1.00000 | 2.87939 | 1.00000 | −4.41147 | 2.87939 | −0.694593 | 1.00000 | 5.29086 | −4.41147 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(67\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 134.2.a.b | ✓ | 3 |
3.b | odd | 2 | 1 | 1206.2.a.n | 3 | ||
4.b | odd | 2 | 1 | 1072.2.a.h | 3 | ||
5.b | even | 2 | 1 | 3350.2.a.i | 3 | ||
5.c | odd | 4 | 2 | 3350.2.c.l | 6 | ||
7.b | odd | 2 | 1 | 6566.2.a.bb | 3 | ||
8.b | even | 2 | 1 | 4288.2.a.s | 3 | ||
8.d | odd | 2 | 1 | 4288.2.a.x | 3 | ||
12.b | even | 2 | 1 | 9648.2.a.bp | 3 | ||
67.b | odd | 2 | 1 | 8978.2.a.g | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
134.2.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
1072.2.a.h | 3 | 4.b | odd | 2 | 1 | ||
1206.2.a.n | 3 | 3.b | odd | 2 | 1 | ||
3350.2.a.i | 3 | 5.b | even | 2 | 1 | ||
3350.2.c.l | 6 | 5.c | odd | 4 | 2 | ||
4288.2.a.s | 3 | 8.b | even | 2 | 1 | ||
4288.2.a.x | 3 | 8.d | odd | 2 | 1 | ||
6566.2.a.bb | 3 | 7.b | odd | 2 | 1 | ||
8978.2.a.g | 3 | 67.b | odd | 2 | 1 | ||
9648.2.a.bp | 3 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 3T_{3}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(134))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{3} \)
$3$
\( T^{3} - 3T^{2} + 1 \)
$5$
\( T^{3} + 3 T^{2} - 6 T + 1 \)
$7$
\( T^{3} - 12T - 8 \)
$11$
\( T^{3} + 3 T^{2} - 24 T - 53 \)
$13$
\( T^{3} + 3 T^{2} - 18 T - 3 \)
$17$
\( T^{3} + 3 T^{2} - 18 T - 3 \)
$19$
\( T^{3} - 6 T^{2} - 36 T + 152 \)
$23$
\( T^{3} + 3 T^{2} - 36 T + 51 \)
$29$
\( (T + 4)^{3} \)
$31$
\( T^{3} - 12 T^{2} + 36 T - 8 \)
$37$
\( T^{3} - 84T - 136 \)
$41$
\( T^{3} - 12T - 8 \)
$43$
\( T^{3} - 3 T^{2} - 60 T + 53 \)
$47$
\( T^{3} - 21 T^{2} + 144 T - 321 \)
$53$
\( T^{3} + 9 T^{2} + 18 T - 9 \)
$59$
\( T^{3} - 12T + 8 \)
$61$
\( T^{3} - 15 T^{2} + 66 T - 89 \)
$67$
\( (T + 1)^{3} \)
$71$
\( T^{3} - 9 T^{2} - 12 T + 179 \)
$73$
\( T^{3} - 9 T^{2} - 54 T - 27 \)
$79$
\( T^{3} + 6 T^{2} - 24 T + 8 \)
$83$
\( T^{3} - 18T^{2} + 648 \)
$89$
\( T^{3} - 3 T^{2} - 126 T - 321 \)
$97$
\( T^{3} - 18 T^{2} + 24 T + 584 \)
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