Newspace parameters
Level: | \( N \) | \(=\) | \( 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 61.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.487087452330\) |
Analytic rank: | \(0\) |
Dimension: | \(3\) |
Coefficient field: | 3.3.148.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{3} - x^{2} - 3x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
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 a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - \nu - 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + \beta _1 + 2 \) |
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 |
|
−1.48119 | 0.806063 | 0.193937 | 3.15633 | −1.19394 | 0.675131 | 2.67513 | −2.35026 | −4.67513 | |||||||||||||||||||||||||||
1.2 | 0.311108 | 2.90321 | −1.90321 | −2.52543 | 0.903212 | −3.21432 | −1.21432 | 5.42864 | −0.785680 | ||||||||||||||||||||||||||||
1.3 | 2.17009 | −1.70928 | 2.70928 | −1.63090 | −3.70928 | −0.460811 | 1.53919 | −0.0783777 | −3.53919 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(61\) | \(-1\) |
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 | 61.2.a.b | ✓ | 3 |
3.b | odd | 2 | 1 | 549.2.a.g | 3 | ||
4.b | odd | 2 | 1 | 976.2.a.f | 3 | ||
5.b | even | 2 | 1 | 1525.2.a.d | 3 | ||
5.c | odd | 4 | 2 | 1525.2.b.b | 6 | ||
7.b | odd | 2 | 1 | 2989.2.a.i | 3 | ||
8.b | even | 2 | 1 | 3904.2.a.r | 3 | ||
8.d | odd | 2 | 1 | 3904.2.a.w | 3 | ||
11.b | odd | 2 | 1 | 7381.2.a.f | 3 | ||
12.b | even | 2 | 1 | 8784.2.a.bn | 3 | ||
61.b | even | 2 | 1 | 3721.2.a.c | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
61.2.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
549.2.a.g | 3 | 3.b | odd | 2 | 1 | ||
976.2.a.f | 3 | 4.b | odd | 2 | 1 | ||
1525.2.a.d | 3 | 5.b | even | 2 | 1 | ||
1525.2.b.b | 6 | 5.c | odd | 4 | 2 | ||
2989.2.a.i | 3 | 7.b | odd | 2 | 1 | ||
3721.2.a.c | 3 | 61.b | even | 2 | 1 | ||
3904.2.a.r | 3 | 8.b | even | 2 | 1 | ||
3904.2.a.w | 3 | 8.d | odd | 2 | 1 | ||
7381.2.a.f | 3 | 11.b | odd | 2 | 1 | ||
8784.2.a.bn | 3 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(61))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - T^{2} - 3T + 1 \)
$3$
\( T^{3} - 2 T^{2} - 4 T + 4 \)
$5$
\( T^{3} + T^{2} - 9T - 13 \)
$7$
\( T^{3} + 3T^{2} - T - 1 \)
$11$
\( T^{3} - 13 T^{2} + 53 T - 67 \)
$13$
\( T^{3} + 9 T^{2} + 11 T - 37 \)
$17$
\( T^{3} + 2 T^{2} - 8 T + 4 \)
$19$
\( T^{3} - 48T - 20 \)
$23$
\( T^{3} - 5 T^{2} + 5 T + 1 \)
$29$
\( T^{3} - 4 T^{2} - 4 T + 20 \)
$31$
\( T^{3} + 2 T^{2} - 76 T + 116 \)
$37$
\( T^{3} + 6 T^{2} - 36 T - 108 \)
$41$
\( T^{3} - 3 T^{2} - 61 T + 191 \)
$43$
\( T^{3} + 14 T^{2} + 56 T + 68 \)
$47$
\( T^{3} + 4 T^{2} - 88 T + 16 \)
$53$
\( T^{3} + 2 T^{2} - 12 T - 8 \)
$59$
\( T^{3} - 29 T^{2} + 231 T - 325 \)
$61$
\( (T - 1)^{3} \)
$67$
\( T^{3} - 9 T^{2} - 85 T + 559 \)
$71$
\( T^{3} - 14 T^{2} - 12 T + 92 \)
$73$
\( T^{3} + T^{2} - 45 T - 25 \)
$79$
\( T^{3} - 13 T^{2} - 51 T + 625 \)
$83$
\( T^{3} + 8 T^{2} - 64 T - 256 \)
$89$
\( T^{3} + 4 T^{2} - 56 T + 80 \)
$97$
\( T^{3} - 10 T^{2} - 116 T + 1096 \)
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