Newspace parameters
Level: | \( N \) | \(=\) | \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4100.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(32.7386648287\) |
Analytic rank: | \(1\) |
Dimension: | \(4\) |
Coefficient field: | 4.4.13068.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} - 6x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 820) |
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 \( x^{4} - x^{3} - 6x^{2} - x + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{2} - 2\nu - 3 \) |
\(\beta_{2}\) | \(=\) | \( \nu^{3} - \nu^{2} - 5\nu - 1 \) |
\(\beta_{3}\) | \(=\) | \( -\nu^{3} + 2\nu^{2} + 5\nu - 2 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + \beta_{2} + 3 \) |
\(\nu^{3}\) | \(=\) | \( ( 7\beta_{3} + 9\beta_{2} - 5\beta _1 + 8 ) / 2 \) |
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 |
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0 | −2.71519 | 0 | 0 | 0 | 0.176865 | 0 | 4.37228 | 0 | ||||||||||||||||||||||||||||||
1.2 | 0 | −1.27582 | 0 | 0 | 0 | −1.60298 | 0 | −1.37228 | 0 | |||||||||||||||||||||||||||||||
1.3 | 0 | 1.27582 | 0 | 0 | 0 | 3.97526 | 0 | −1.37228 | 0 | |||||||||||||||||||||||||||||||
1.4 | 0 | 2.71519 | 0 | 0 | 0 | −3.54915 | 0 | 4.37228 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(5\) | \(1\) |
\(41\) | \(-1\) |
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 | 4100.2.a.d | 4 | |
5.b | even | 2 | 1 | 820.2.a.d | ✓ | 4 | |
5.c | odd | 4 | 2 | 4100.2.d.e | 8 | ||
15.d | odd | 2 | 1 | 7380.2.a.t | 4 | ||
20.d | odd | 2 | 1 | 3280.2.a.be | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
820.2.a.d | ✓ | 4 | 5.b | even | 2 | 1 | |
3280.2.a.be | 4 | 20.d | odd | 2 | 1 | ||
4100.2.a.d | 4 | 1.a | even | 1 | 1 | trivial | |
4100.2.d.e | 8 | 5.c | odd | 4 | 2 | ||
7380.2.a.t | 4 | 15.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 9T_{3}^{2} + 12 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4100))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - 9T^{2} + 12 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + T^{3} - 15 T^{2} - 20 T + 4 \)
$11$
\( T^{4} - 9T^{2} + 12 \)
$13$
\( T^{4} + 7 T^{3} + 3 T^{2} - 20 T - 8 \)
$17$
\( T^{4} + 8 T^{3} + 15 T^{2} - 4 T - 8 \)
$19$
\( T^{4} + 3 T^{3} - 45 T^{2} - 108 T + 108 \)
$23$
\( (T^{2} + 3 T - 6)^{2} \)
$29$
\( T^{4} - 7 T^{3} - 9 T^{2} + 128 T - 164 \)
$31$
\( T^{4} - 3 T^{3} - 57 T^{2} - 72 T + 48 \)
$37$
\( T^{4} + 9 T^{3} + 9 T^{2} - 96 T - 204 \)
$41$
\( (T - 1)^{4} \)
$43$
\( T^{4} + 5 T^{3} - 45 T^{2} + 8 T + 124 \)
$47$
\( T^{4} + 3 T^{3} - 87 T^{2} + 324 T - 348 \)
$53$
\( T^{4} + 26 T^{3} + 228 T^{2} + \cdots + 928 \)
$59$
\( T^{4} + T^{3} - 27 T^{2} + 40 T + 16 \)
$61$
\( T^{4} + 8 T^{3} - 45 T^{2} - 244 T - 68 \)
$67$
\( T^{4} + T^{3} - 123 T^{2} - 404 T - 332 \)
$71$
\( T^{4} - 2 T^{3} - 75 T^{2} - 56 T + 388 \)
$73$
\( T^{4} + 9 T^{3} - 279 T^{2} + \cdots + 20964 \)
$79$
\( T^{4} - T^{3} - 285 T^{2} + \cdots + 16204 \)
$83$
\( T^{4} + 23 T^{3} + 111 T^{2} + \cdots - 164 \)
$89$
\( T^{4} - 10 T^{3} - 171 T^{2} + \cdots - 4652 \)
$97$
\( T^{4} + 6 T^{3} - 228 T^{2} + \cdots + 3936 \)
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