Newspace parameters
Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 273.bl (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.17991597518\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} - x^{2} - 2x + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \( x^{4} - x^{3} - x^{2} - 2x + 4 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + \beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{3} + \beta_{2} + \beta _1 + 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).
\(n\) | \(92\) | \(106\) | \(157\) |
\(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
88.1 |
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−0.395644 | − | 0.228425i | 1.00000 | −0.895644 | − | 1.55130i | −1.50000 | + | 0.866025i | −0.395644 | − | 0.228425i | − | 2.64575i | 1.73205i | 1.00000 | 0.791288 | |||||||||||||||||||||
88.2 | 1.89564 | + | 1.09445i | 1.00000 | 1.39564 | + | 2.41733i | −1.50000 | + | 0.866025i | 1.89564 | + | 1.09445i | 2.64575i | 1.73205i | 1.00000 | −3.79129 | |||||||||||||||||||||||
121.1 | −0.395644 | + | 0.228425i | 1.00000 | −0.895644 | + | 1.55130i | −1.50000 | − | 0.866025i | −0.395644 | + | 0.228425i | 2.64575i | − | 1.73205i | 1.00000 | 0.791288 | ||||||||||||||||||||||
121.2 | 1.89564 | − | 1.09445i | 1.00000 | 1.39564 | − | 2.41733i | −1.50000 | − | 0.866025i | 1.89564 | − | 1.09445i | − | 2.64575i | − | 1.73205i | 1.00000 | −3.79129 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.u | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 273.2.bl.b | yes | 4 |
3.b | odd | 2 | 1 | 819.2.do.d | 4 | ||
7.c | even | 3 | 1 | 273.2.t.b | ✓ | 4 | |
13.e | even | 6 | 1 | 273.2.t.b | ✓ | 4 | |
21.h | odd | 6 | 1 | 819.2.bm.d | 4 | ||
39.h | odd | 6 | 1 | 819.2.bm.d | 4 | ||
91.u | even | 6 | 1 | inner | 273.2.bl.b | yes | 4 |
273.x | odd | 6 | 1 | 819.2.do.d | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.t.b | ✓ | 4 | 7.c | even | 3 | 1 | |
273.2.t.b | ✓ | 4 | 13.e | even | 6 | 1 | |
273.2.bl.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
273.2.bl.b | yes | 4 | 91.u | even | 6 | 1 | inner |
819.2.bm.d | 4 | 21.h | odd | 6 | 1 | ||
819.2.bm.d | 4 | 39.h | odd | 6 | 1 | ||
819.2.do.d | 4 | 3.b | odd | 2 | 1 | ||
819.2.do.d | 4 | 273.x | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 3T_{2}^{3} + 2T_{2}^{2} + 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 3 T^{3} + 2 T^{2} + 3 T + 1 \)
$3$
\( (T - 1)^{4} \)
$5$
\( (T^{2} + 3 T + 3)^{2} \)
$7$
\( (T^{2} + 7)^{2} \)
$11$
\( (T^{2} + 12)^{2} \)
$13$
\( (T^{2} + 2 T + 13)^{2} \)
$17$
\( (T^{2} + T + 1)^{2} \)
$19$
\( (T^{2} + 28)^{2} \)
$23$
\( T^{4} - 8 T^{3} + 69 T^{2} + 40 T + 25 \)
$29$
\( (T^{2} - 7 T + 49)^{2} \)
$31$
\( T^{4} + 12 T^{3} + 53 T^{2} + 60 T + 25 \)
$37$
\( T^{4} + 6 T^{3} - 13 T^{2} - 150 T + 625 \)
$41$
\( T^{4} - 6 T^{3} - 13 T^{2} + 150 T + 625 \)
$43$
\( T^{4} + 21T^{2} + 441 \)
$47$
\( T^{4} - 12 T^{3} + 53 T^{2} - 60 T + 25 \)
$53$
\( T^{4} + 6 T^{3} + 111 T^{2} + \cdots + 5625 \)
$59$
\( T^{4} + 24 T^{3} + 233 T^{2} + \cdots + 1681 \)
$61$
\( (T^{2} - 8 T - 68)^{2} \)
$67$
\( T^{4} + 248 T^{2} + 10000 \)
$71$
\( T^{4} - 12 T^{3} - 3 T^{2} + \cdots + 2601 \)
$73$
\( (T^{2} - 15 T + 75)^{2} \)
$79$
\( T^{4} - 12 T^{3} + 129 T^{2} + \cdots + 225 \)
$83$
\( (T^{2} + 12)^{2} \)
$89$
\( (T^{2} + 27 T + 243)^{2} \)
$97$
\( T^{4} - 18 T^{3} + 107 T^{2} + 18 T + 1 \)
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