Newspace parameters
Level: | \( N \) | \(=\) | \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2640.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(21.0805061336\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | 6.0.350464.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 1320) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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 \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) :
\(\beta_{1}\) | \(=\) | \( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \) |
\(\beta_{2}\) | \(=\) | \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) |
\(\beta_{3}\) | \(=\) | \( ( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23 \) |
\(\beta_{4}\) | \(=\) | \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) |
\(\beta_{5}\) | \(=\) | \( ( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23 \) |
\(\nu\) | \(=\) | \( ( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \) |
\(\nu^{4}\) | \(=\) | \( 2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7 \) |
\(\nu^{5}\) | \(=\) | \( -9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times\).
\(n\) | \(661\) | \(881\) | \(991\) | \(1057\) | \(1201\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
529.1 |
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0 | − | 1.00000i | 0 | −2.17009 | + | 0.539189i | 0 | 1.70928i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||
529.2 | 0 | − | 1.00000i | 0 | −0.311108 | − | 2.21432i | 0 | − | 2.90321i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||
529.3 | 0 | − | 1.00000i | 0 | 1.48119 | + | 1.67513i | 0 | − | 0.806063i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||
529.4 | 0 | 1.00000i | 0 | −2.17009 | − | 0.539189i | 0 | − | 1.70928i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
529.5 | 0 | 1.00000i | 0 | −0.311108 | + | 2.21432i | 0 | 2.90321i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
529.6 | 0 | 1.00000i | 0 | 1.48119 | − | 1.67513i | 0 | 0.806063i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2640.2.d.g | 6 | |
4.b | odd | 2 | 1 | 1320.2.d.a | ✓ | 6 | |
5.b | even | 2 | 1 | inner | 2640.2.d.g | 6 | |
12.b | even | 2 | 1 | 3960.2.d.e | 6 | ||
20.d | odd | 2 | 1 | 1320.2.d.a | ✓ | 6 | |
20.e | even | 4 | 1 | 6600.2.a.bp | 3 | ||
20.e | even | 4 | 1 | 6600.2.a.bt | 3 | ||
60.h | even | 2 | 1 | 3960.2.d.e | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1320.2.d.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
1320.2.d.a | ✓ | 6 | 20.d | odd | 2 | 1 | |
2640.2.d.g | 6 | 1.a | even | 1 | 1 | trivial | |
2640.2.d.g | 6 | 5.b | even | 2 | 1 | inner | |
3960.2.d.e | 6 | 12.b | even | 2 | 1 | ||
3960.2.d.e | 6 | 60.h | even | 2 | 1 | ||
6600.2.a.bp | 3 | 20.e | even | 4 | 1 | ||
6600.2.a.bt | 3 | 20.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} + 12T_{7}^{4} + 32T_{7}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(2640, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( (T^{2} + 1)^{3} \)
$5$
\( T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125 \)
$7$
\( T^{6} + 12 T^{4} + 32 T^{2} + 16 \)
$11$
\( (T - 1)^{6} \)
$13$
\( T^{6} + 24 T^{4} + 176 T^{2} + \cdots + 400 \)
$17$
\( T^{6} + 68 T^{4} + 816 T^{2} + \cdots + 2704 \)
$19$
\( (T^{3} + 2 T^{2} - 20 T - 8)^{2} \)
$23$
\( T^{6} \)
$29$
\( (T^{3} - 12 T^{2} + 32 T - 16)^{2} \)
$31$
\( (T^{3} - 4 T^{2} - 8 T + 16)^{2} \)
$37$
\( T^{6} + 80 T^{4} + 768 T^{2} + \cdots + 1024 \)
$41$
\( (T^{3} - 4 T^{2} - 48 T - 80)^{2} \)
$43$
\( T^{6} + 12 T^{4} + 32 T^{2} + 16 \)
$47$
\( T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024 \)
$53$
\( T^{6} + 80 T^{4} + 640 T^{2} + \cdots + 256 \)
$59$
\( (T^{3} + 12 T^{2} + 32 T + 16)^{2} \)
$61$
\( (T + 2)^{6} \)
$67$
\( T^{6} + 192 T^{4} + 8192 T^{2} + \cdots + 65536 \)
$71$
\( (T^{3} - 24 T^{2} + 176 T - 400)^{2} \)
$73$
\( T^{6} + 104 T^{4} + 304 T^{2} + \cdots + 16 \)
$79$
\( (T^{3} + 6 T^{2} - 100 T + 200)^{2} \)
$83$
\( T^{6} + 328 T^{4} + 23216 T^{2} + \cdots + 300304 \)
$89$
\( (T^{3} - 18 T^{2} - 12 T + 488)^{2} \)
$97$
\( T^{6} + 240 T^{4} + 8960 T^{2} + \cdots + 4096 \)
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