Newspace parameters
Level: | \( N \) | \(=\) | \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 2016.e (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(54.9320212950\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{14} \) |
Twist minimal: | no (minimal twist has level 504) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) | \( 2\zeta_{24}^{6} \) |
\(\beta_{2}\) | \(=\) | \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) |
\(\beta_{3}\) | \(=\) | \( -4\zeta_{24}^{6} + 8\zeta_{24}^{2} \) |
\(\beta_{4}\) | \(=\) | \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) |
\(\beta_{5}\) | \(=\) | \( -4\zeta_{24}^{7} - \zeta_{24}^{6} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) |
\(\beta_{6}\) | \(=\) | \( 8\zeta_{24}^{4} - 4 \) |
\(\beta_{7}\) | \(=\) | \( 8\zeta_{24}^{7} + 4\zeta_{24}^{5} - 4\zeta_{24}^{3} + 4\zeta_{24} \) |
\(\zeta_{24}\) | \(=\) | \( ( \beta_{7} + 2\beta_{5} + 4\beta_{4} + 4\beta_{2} + \beta_1 ) / 16 \) |
\(\zeta_{24}^{2}\) | \(=\) | \( ( \beta_{3} + 2\beta_1 ) / 8 \) |
\(\zeta_{24}^{3}\) | \(=\) | \( ( \beta_{4} - \beta_{2} ) / 2 \) |
\(\zeta_{24}^{4}\) | \(=\) | \( ( \beta_{6} + 4 ) / 8 \) |
\(\zeta_{24}^{5}\) | \(=\) | \( ( \beta_{7} + 2\beta_{5} - 4\beta_{4} - 4\beta_{2} + \beta_1 ) / 16 \) |
\(\zeta_{24}^{6}\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\zeta_{24}^{7}\) | \(=\) | \( ( \beta_{7} - 2\beta_{5} + 4\beta_{4} - 4\beta_{2} - \beta_1 ) / 16 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(577\) | \(1765\) | \(1793\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1007.1 |
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0 | 0 | 0 | − | 6.92820i | 0 | −4.89898 | − | 5.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
1007.2 | 0 | 0 | 0 | − | 6.92820i | 0 | −4.89898 | + | 5.00000i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
1007.3 | 0 | 0 | 0 | − | 6.92820i | 0 | 4.89898 | − | 5.00000i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
1007.4 | 0 | 0 | 0 | − | 6.92820i | 0 | 4.89898 | + | 5.00000i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
1007.5 | 0 | 0 | 0 | 6.92820i | 0 | −4.89898 | − | 5.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
1007.6 | 0 | 0 | 0 | 6.92820i | 0 | −4.89898 | + | 5.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
1007.7 | 0 | 0 | 0 | 6.92820i | 0 | 4.89898 | − | 5.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
1007.8 | 0 | 0 | 0 | 6.92820i | 0 | 4.89898 | + | 5.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
56.e | even | 2 | 1 | inner |
168.e | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2016.3.e.a | 8 | |
3.b | odd | 2 | 1 | inner | 2016.3.e.a | 8 | |
4.b | odd | 2 | 1 | 504.3.e.a | ✓ | 8 | |
7.b | odd | 2 | 1 | inner | 2016.3.e.a | 8 | |
8.b | even | 2 | 1 | 504.3.e.a | ✓ | 8 | |
8.d | odd | 2 | 1 | inner | 2016.3.e.a | 8 | |
12.b | even | 2 | 1 | 504.3.e.a | ✓ | 8 | |
21.c | even | 2 | 1 | inner | 2016.3.e.a | 8 | |
24.f | even | 2 | 1 | inner | 2016.3.e.a | 8 | |
24.h | odd | 2 | 1 | 504.3.e.a | ✓ | 8 | |
28.d | even | 2 | 1 | 504.3.e.a | ✓ | 8 | |
56.e | even | 2 | 1 | inner | 2016.3.e.a | 8 | |
56.h | odd | 2 | 1 | 504.3.e.a | ✓ | 8 | |
84.h | odd | 2 | 1 | 504.3.e.a | ✓ | 8 | |
168.e | odd | 2 | 1 | inner | 2016.3.e.a | 8 | |
168.i | even | 2 | 1 | 504.3.e.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.3.e.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
504.3.e.a | ✓ | 8 | 8.b | even | 2 | 1 | |
504.3.e.a | ✓ | 8 | 12.b | even | 2 | 1 | |
504.3.e.a | ✓ | 8 | 24.h | odd | 2 | 1 | |
504.3.e.a | ✓ | 8 | 28.d | even | 2 | 1 | |
504.3.e.a | ✓ | 8 | 56.h | odd | 2 | 1 | |
504.3.e.a | ✓ | 8 | 84.h | odd | 2 | 1 | |
504.3.e.a | ✓ | 8 | 168.i | even | 2 | 1 | |
2016.3.e.a | 8 | 1.a | even | 1 | 1 | trivial | |
2016.3.e.a | 8 | 3.b | odd | 2 | 1 | inner | |
2016.3.e.a | 8 | 7.b | odd | 2 | 1 | inner | |
2016.3.e.a | 8 | 8.d | odd | 2 | 1 | inner | |
2016.3.e.a | 8 | 21.c | even | 2 | 1 | inner | |
2016.3.e.a | 8 | 24.f | even | 2 | 1 | inner | |
2016.3.e.a | 8 | 56.e | even | 2 | 1 | inner | |
2016.3.e.a | 8 | 168.e | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 48 \)
acting on \(S_{3}^{\mathrm{new}}(2016, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( (T^{2} + 48)^{4} \)
$7$
\( (T^{4} + 2 T^{2} + 2401)^{2} \)
$11$
\( (T^{2} + 98)^{4} \)
$13$
\( (T^{2} - 384)^{4} \)
$17$
\( (T^{2} - 48)^{4} \)
$19$
\( (T^{2} + 96)^{4} \)
$23$
\( (T^{2} - 1682)^{4} \)
$29$
\( (T^{2} - 338)^{4} \)
$31$
\( (T^{2} - 1536)^{4} \)
$37$
\( (T^{2} + 676)^{4} \)
$41$
\( (T^{2} - 3888)^{4} \)
$43$
\( (T + 48)^{8} \)
$47$
\( (T^{2} + 768)^{4} \)
$53$
\( (T^{2} - 2738)^{4} \)
$59$
\( (T^{2} - 4800)^{4} \)
$61$
\( (T^{2} - 864)^{4} \)
$67$
\( (T - 14)^{8} \)
$71$
\( (T^{2} - 1682)^{4} \)
$73$
\( (T^{2} + 864)^{4} \)
$79$
\( (T^{2} + 2304)^{4} \)
$83$
\( (T^{2} - 12288)^{4} \)
$89$
\( (T^{2} - 21168)^{4} \)
$97$
\( (T^{2} + 27744)^{4} \)
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