Newspace parameters
Level: | \( N \) | \(=\) | \( 1083 = 3 \cdot 19^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1083.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.64779853890\) |
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: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | no (minimal twist has level 57) |
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
\(\beta_{1}\) | \(=\) | \( \zeta_{24}^{6} \) |
\(\beta_{2}\) | \(=\) | \( 2\zeta_{24}^{4} - 1 \) |
\(\beta_{3}\) | \(=\) | \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) |
\(\beta_{4}\) | \(=\) | \( -\zeta_{24}^{7} + \zeta_{24}^{3} - \zeta_{24} \) |
\(\beta_{5}\) | \(=\) | \( \zeta_{24}^{7} - \zeta_{24}^{5} \) |
\(\beta_{6}\) | \(=\) | \( -\zeta_{24}^{7} + \zeta_{24}^{3} + \zeta_{24} \) |
\(\beta_{7}\) | \(=\) | \( \zeta_{24}^{7} + \zeta_{24}^{5} \) |
\(\zeta_{24}\) | \(=\) | \( ( \beta_{6} - \beta_{4} ) / 2 \) |
\(\zeta_{24}^{2}\) | \(=\) | \( ( \beta_{3} + \beta_1 ) / 2 \) |
\(\zeta_{24}^{3}\) | \(=\) | \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 2 \) |
\(\zeta_{24}^{4}\) | \(=\) | \( ( \beta_{2} + 1 ) / 2 \) |
\(\zeta_{24}^{5}\) | \(=\) | \( ( \beta_{7} - \beta_{5} ) / 2 \) |
\(\zeta_{24}^{6}\) | \(=\) | \( \beta_1 \) |
\(\zeta_{24}^{7}\) | \(=\) | \( ( \beta_{7} + \beta_{5} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1083\mathbb{Z}\right)^\times\).
\(n\) | \(362\) | \(724\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1082.1 |
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−1.93185 | 1.41421 | − | 1.00000i | 1.73205 | 1.41421i | −2.73205 | + | 1.93185i | −3.73205 | 0.517638 | 1.00000 | − | 2.82843i | − | 2.73205i | |||||||||||||||||||||||||||||||||||
1082.2 | −1.93185 | 1.41421 | + | 1.00000i | 1.73205 | − | 1.41421i | −2.73205 | − | 1.93185i | −3.73205 | 0.517638 | 1.00000 | + | 2.82843i | 2.73205i | ||||||||||||||||||||||||||||||||||||
1082.3 | −0.517638 | −1.41421 | − | 1.00000i | −1.73205 | − | 1.41421i | 0.732051 | + | 0.517638i | −0.267949 | 1.93185 | 1.00000 | + | 2.82843i | 0.732051i | ||||||||||||||||||||||||||||||||||||
1082.4 | −0.517638 | −1.41421 | + | 1.00000i | −1.73205 | 1.41421i | 0.732051 | − | 0.517638i | −0.267949 | 1.93185 | 1.00000 | − | 2.82843i | − | 0.732051i | ||||||||||||||||||||||||||||||||||||
1082.5 | 0.517638 | 1.41421 | − | 1.00000i | −1.73205 | 1.41421i | 0.732051 | − | 0.517638i | −0.267949 | −1.93185 | 1.00000 | − | 2.82843i | 0.732051i | |||||||||||||||||||||||||||||||||||||
1082.6 | 0.517638 | 1.41421 | + | 1.00000i | −1.73205 | − | 1.41421i | 0.732051 | + | 0.517638i | −0.267949 | −1.93185 | 1.00000 | + | 2.82843i | − | 0.732051i | |||||||||||||||||||||||||||||||||||
1082.7 | 1.93185 | −1.41421 | − | 1.00000i | 1.73205 | − | 1.41421i | −2.73205 | − | 1.93185i | −3.73205 | −0.517638 | 1.00000 | + | 2.82843i | − | 2.73205i | |||||||||||||||||||||||||||||||||||
1082.8 | 1.93185 | −1.41421 | + | 1.00000i | 1.73205 | 1.41421i | −2.73205 | + | 1.93185i | −3.73205 | −0.517638 | 1.00000 | − | 2.82843i | 2.73205i | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
57.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1083.2.d.b | 8 | |
3.b | odd | 2 | 1 | inner | 1083.2.d.b | 8 | |
19.b | odd | 2 | 1 | inner | 1083.2.d.b | 8 | |
19.c | even | 3 | 1 | 57.2.f.a | ✓ | 8 | |
19.d | odd | 6 | 1 | 57.2.f.a | ✓ | 8 | |
57.d | even | 2 | 1 | inner | 1083.2.d.b | 8 | |
57.f | even | 6 | 1 | 57.2.f.a | ✓ | 8 | |
57.h | odd | 6 | 1 | 57.2.f.a | ✓ | 8 | |
76.f | even | 6 | 1 | 912.2.bn.m | 8 | ||
76.g | odd | 6 | 1 | 912.2.bn.m | 8 | ||
228.m | even | 6 | 1 | 912.2.bn.m | 8 | ||
228.n | odd | 6 | 1 | 912.2.bn.m | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
57.2.f.a | ✓ | 8 | 19.c | even | 3 | 1 | |
57.2.f.a | ✓ | 8 | 19.d | odd | 6 | 1 | |
57.2.f.a | ✓ | 8 | 57.f | even | 6 | 1 | |
57.2.f.a | ✓ | 8 | 57.h | odd | 6 | 1 | |
912.2.bn.m | 8 | 76.f | even | 6 | 1 | ||
912.2.bn.m | 8 | 76.g | odd | 6 | 1 | ||
912.2.bn.m | 8 | 228.m | even | 6 | 1 | ||
912.2.bn.m | 8 | 228.n | odd | 6 | 1 | ||
1083.2.d.b | 8 | 1.a | even | 1 | 1 | trivial | |
1083.2.d.b | 8 | 3.b | odd | 2 | 1 | inner | |
1083.2.d.b | 8 | 19.b | odd | 2 | 1 | inner | |
1083.2.d.b | 8 | 57.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 4T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1083, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 4 T^{2} + 1)^{2} \)
$3$
\( (T^{4} - 2 T^{2} + 9)^{2} \)
$5$
\( (T^{2} + 2)^{4} \)
$7$
\( (T^{2} + 4 T + 1)^{4} \)
$11$
\( (T^{4} + 28 T^{2} + 4)^{2} \)
$13$
\( (T^{4} + 14 T^{2} + 1)^{2} \)
$17$
\( (T^{2} + 24)^{4} \)
$19$
\( T^{8} \)
$23$
\( (T^{4} + 28 T^{2} + 4)^{2} \)
$29$
\( (T^{4} - 64 T^{2} + 256)^{2} \)
$31$
\( (T^{4} + 26 T^{2} + 121)^{2} \)
$37$
\( (T^{4} + 78 T^{2} + 1089)^{2} \)
$41$
\( (T^{2} - 32)^{4} \)
$43$
\( (T^{2} - 8 T + 13)^{4} \)
$47$
\( (T^{4} + 112 T^{2} + 64)^{2} \)
$53$
\( (T^{4} - 156 T^{2} + 4356)^{2} \)
$59$
\( (T^{4} - 196 T^{2} + 8836)^{2} \)
$61$
\( (T^{2} + 14 T + 37)^{4} \)
$67$
\( (T^{2} + 1)^{4} \)
$71$
\( (T^{4} - 192 T^{2} + 2304)^{2} \)
$73$
\( (T + 3)^{8} \)
$79$
\( (T^{4} + 122 T^{2} + 1369)^{2} \)
$83$
\( (T^{4} + 64 T^{2} + 256)^{2} \)
$89$
\( (T^{2} - 54)^{4} \)
$97$
\( (T^{4} + 56 T^{2} + 16)^{2} \)
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