Newspace parameters
Level: | \( N \) | \(=\) | \( 111 = 3 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 111.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.886339462436\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | 4.4.6224.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{4} - 6x^{2} - 2x + 5 \) |
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 \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} - 6x^{2} - 2x + 5 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - \nu - 3 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} - \nu^{2} - 4\nu + 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + \beta _1 + 3 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) |
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 |
|
−2.44579 | 1.00000 | 3.98187 | −0.670719 | −2.44579 | 0.269264 | −4.84724 | 1.00000 | 1.64044 | ||||||||||||||||||||||||||||||
1.2 | −0.796815 | 1.00000 | −1.36509 | 0.845635 | −0.796815 | 4.63253 | 2.68135 | 1.00000 | −0.673815 | |||||||||||||||||||||||||||||||
1.3 | 1.37033 | 1.00000 | −0.122207 | 1.78220 | 1.37033 | −4.06064 | −2.90812 | 1.00000 | 2.44220 | |||||||||||||||||||||||||||||||
1.4 | 1.87228 | 1.00000 | 1.50542 | −3.95712 | 1.87228 | 3.15885 | −0.925994 | 1.00000 | −7.40882 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(37\) | \(1\) |
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 | 111.2.a.b | ✓ | 4 |
3.b | odd | 2 | 1 | 333.2.a.g | 4 | ||
4.b | odd | 2 | 1 | 1776.2.a.u | 4 | ||
5.b | even | 2 | 1 | 2775.2.a.x | 4 | ||
7.b | odd | 2 | 1 | 5439.2.a.u | 4 | ||
8.b | even | 2 | 1 | 7104.2.a.cc | 4 | ||
8.d | odd | 2 | 1 | 7104.2.a.cf | 4 | ||
12.b | even | 2 | 1 | 5328.2.a.bs | 4 | ||
15.d | odd | 2 | 1 | 8325.2.a.bv | 4 | ||
37.b | even | 2 | 1 | 4107.2.a.i | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
111.2.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
333.2.a.g | 4 | 3.b | odd | 2 | 1 | ||
1776.2.a.u | 4 | 4.b | odd | 2 | 1 | ||
2775.2.a.x | 4 | 5.b | even | 2 | 1 | ||
4107.2.a.i | 4 | 37.b | even | 2 | 1 | ||
5328.2.a.bs | 4 | 12.b | even | 2 | 1 | ||
5439.2.a.u | 4 | 7.b | odd | 2 | 1 | ||
7104.2.a.cc | 4 | 8.b | even | 2 | 1 | ||
7104.2.a.cf | 4 | 8.d | odd | 2 | 1 | ||
8325.2.a.bv | 4 | 15.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 6T_{2}^{2} + 2T_{2} + 5 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(111))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 6 T^{2} + 2 T + 5 \)
$3$
\( (T - 1)^{4} \)
$5$
\( T^{4} + 2 T^{3} - 8 T^{2} + 4 \)
$7$
\( T^{4} - 4 T^{3} - 16 T^{2} + 64 T - 16 \)
$11$
\( T^{4} - 32 T^{2} - 32 T + 64 \)
$13$
\( T^{4} - 4 T^{3} - 32 T^{2} + 144 T - 80 \)
$17$
\( T^{4} + 2 T^{3} - 24 T^{2} - 72 T - 28 \)
$19$
\( T^{4} - 8 T^{3} - 8 T^{2} + 144 T - 224 \)
$23$
\( T^{4} + 10 T^{3} - 32 T^{2} + \cdots + 652 \)
$29$
\( T^{4} + 2 T^{3} - 56 T^{2} - 40 T + 724 \)
$31$
\( T^{4} - 4 T^{3} - 16 T^{2} + 16 T + 32 \)
$37$
\( (T + 1)^{4} \)
$41$
\( T^{4} - 12 T^{3} + 304 T - 400 \)
$43$
\( T^{4} - 4 T^{3} - 128 T^{2} + \cdots + 3424 \)
$47$
\( T^{4} + 12 T^{3} + 16 T^{2} + \cdots - 128 \)
$53$
\( T^{4} - 8 T^{3} - 56 T^{2} + 320 T + 464 \)
$59$
\( T^{4} + 10 T^{3} - 176 T^{2} + \cdots - 7156 \)
$61$
\( T^{4} + 8 T^{3} - 72 T^{2} - 480 T + 656 \)
$67$
\( T^{4} + 4 T^{3} - 16 T^{2} - 64 T - 16 \)
$71$
\( T^{4} + 12 T^{3} - 48 T^{2} + \cdots + 1664 \)
$73$
\( T^{4} - 12 T^{3} - 8 T^{2} + 176 T - 32 \)
$79$
\( T^{4} + 8 T^{3} - 56 T^{2} + \cdots - 1504 \)
$83$
\( T^{4} + 20 T^{3} + 112 T^{2} + \cdots + 64 \)
$89$
\( T^{4} - 26 T^{3} + 128 T^{2} + \cdots - 5452 \)
$97$
\( T^{4} + 4 T^{3} - 272 T^{2} + \cdots + 17008 \)
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