Properties

Label 1083.2.d.b
Level $1083$
Weight $2$
Character orbit 1083.d
Analytic conductor $8.648$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{6} + \beta_{5} - \beta_1) q^{3} + \beta_{3} q^{4} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{3} - 1) q^{6} + ( - \beta_{3} - 2) q^{7} - \beta_{5} q^{8} + ( - 2 \beta_{7} - 2 \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{6} + \beta_{5} - \beta_1) q^{3} + \beta_{3} q^{4} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{3} - 1) q^{6} + ( - \beta_{3} - 2) q^{7} - \beta_{5} q^{8} + ( - 2 \beta_{7} - 2 \beta_{4} + 1) q^{9} + ( - \beta_{2} - \beta_1) q^{10} + (\beta_{7} + 3 \beta_{4}) q^{11} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{12} + (\beta_{2} + 2 \beta_1) q^{13} + (4 \beta_{6} + \beta_{5}) q^{14} + (\beta_{6} + \beta_{5} + 2 \beta_1) q^{15} + ( - 2 \beta_{3} - 1) q^{16} + ( - 2 \beta_{7} + 2 \beta_{4}) q^{17} + ( - \beta_{6} + 2 \beta_{2} + 2 \beta_1) q^{18} + (\beta_{7} - \beta_{4}) q^{20} + ( - 3 \beta_{6} - \beta_{5} + \beta_{2} + 2 \beta_1) q^{21} + ( - \beta_{2} + \beta_1) q^{22} + (\beta_{7} + 3 \beta_{4}) q^{23} + (\beta_{4} + \beta_{3} - 1) q^{24} + 3 q^{25} + ( - 4 \beta_{7} - \beta_{4}) q^{26} + ( - \beta_{6} - \beta_{5} - 5 \beta_1) q^{27} + ( - 2 \beta_{3} - 3) q^{28} + 4 \beta_{6} q^{29} + ( - 2 \beta_{7} - \beta_{3} - 1) q^{30} + (2 \beta_{2} + \beta_1) q^{31} + (5 \beta_{6} + 4 \beta_{5}) q^{32} + (\beta_{6} + 3 \beta_{5} - 2 \beta_{2} + 4 \beta_1) q^{33} + (2 \beta_{2} + 6 \beta_1) q^{34} + ( - 3 \beta_{7} - \beta_{4}) q^{35} + ( - 2 \beta_{7} + 2 \beta_{4} + \beta_{3}) q^{36} + ( - \beta_{2} + 6 \beta_1) q^{37} + (3 \beta_{7} + \beta_{4} + \beta_{3} + 2) q^{39} + (\beta_{2} - \beta_1) q^{40} + (4 \beta_{6} + 4 \beta_{5}) q^{41} + ( - 4 \beta_{7} - \beta_{4} + 3 \beta_{3} + 5) q^{42} + ( - \beta_{3} + 4) q^{43} + ( - \beta_{7} - 5 \beta_{4}) q^{44} + (\beta_{7} + \beta_{4} + 4) q^{45} + ( - \beta_{2} + \beta_1) q^{46} + (6 \beta_{7} + 2 \beta_{4}) q^{47} + ( - 3 \beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_1) q^{48} + 4 \beta_{3} q^{49} - 3 \beta_{6} q^{50} + ( - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{2}) q^{51} + (2 \beta_{2} + 3 \beta_1) q^{52} + (5 \beta_{6} + 7 \beta_{5}) q^{53} + (5 \beta_{7} + \beta_{3} + 1) q^{54} + (2 \beta_{3} - 4) q^{55} - \beta_{6} q^{56} + ( - 4 \beta_{3} - 8) q^{58} + (3 \beta_{6} - 5 \beta_{5}) q^{59} + (\beta_{6} - \beta_{5} + 2 \beta_{2}) q^{60} + (2 \beta_{3} - 7) q^{61} + ( - 5 \beta_{7} - 2 \beta_{4}) q^{62} + (6 \beta_{7} + 2 \beta_{4} - \beta_{3} - 2) q^{63} + ( - \beta_{3} - 4) q^{64} + ( - 3 \beta_{6} - \beta_{5}) q^{65} + (2 \beta_{4} - \beta_{3} + 1) q^{66} + \beta_1 q^{67} + ( - 6 \beta_{7} - 6 \beta_{4}) q^{68} + (\beta_{6} + 3 \beta_{5} - 2 \beta_{2} + 4 \beta_1) q^{69} + (3 \beta_{2} + 5 \beta_1) q^{70} + (4 \beta_{6} + 8 \beta_{5}) q^{71} + ( - \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{72} - 3 q^{73} + ( - 4 \beta_{7} + \beta_{4}) q^{74} + (3 \beta_{6} + 3 \beta_{5} - 3 \beta_1) q^{75} + ( - \beta_{7} - \beta_{4}) q^{77} + ( - 4 \beta_{6} - \beta_{5} - 3 \beta_{2} - 5 \beta_1) q^{78} + ( - 2 \beta_{2} + 7 \beta_1) q^{79} + ( - 3 \beta_{7} + \beta_{4}) q^{80} + ( - 4 \beta_{7} - 4 \beta_{4} - 7) q^{81} + ( - 4 \beta_{3} - 4) q^{82} + 4 \beta_{7} q^{83} + ( - 5 \beta_{6} - \beta_{5} + 2 \beta_{2} + 3 \beta_1) q^{84} + 4 \beta_{3} q^{85} + ( - 2 \beta_{6} + \beta_{5}) q^{86} + ( - 4 \beta_{7} + 4 \beta_{3} + 4) q^{87} + (3 \beta_{2} - 5 \beta_1) q^{88} + (3 \beta_{6} - 3 \beta_{5}) q^{89} + ( - 4 \beta_{6} - \beta_{2} - \beta_1) q^{90} + ( - 4 \beta_{2} - 7 \beta_1) q^{91} + ( - \beta_{7} - 5 \beta_{4}) q^{92} + (3 \beta_{7} - \beta_{4} + 2 \beta_{3} + 1) q^{93} + ( - 6 \beta_{2} - 10 \beta_1) q^{94} + ( - 5 \beta_{7} - 4 \beta_{4} + \beta_{3} + 9) q^{96} + ( - 2 \beta_{2} - 4 \beta_1) q^{97} + ( - 8 \beta_{6} - 4 \beta_{5}) q^{98} + (\beta_{7} + 3 \beta_{4} - 4 \beta_{3} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{6} - 16 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{6} - 16 q^{7} + 8 q^{9} - 8 q^{16} - 8 q^{24} + 24 q^{25} - 24 q^{28} - 8 q^{30} + 16 q^{39} + 40 q^{42} + 32 q^{43} + 32 q^{45} + 8 q^{54} - 32 q^{55} - 64 q^{58} - 56 q^{61} - 16 q^{63} - 32 q^{64} + 8 q^{66} - 24 q^{73} - 56 q^{81} - 32 q^{82} + 32 q^{87} + 8 q^{93} + 72 q^{96} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24}^{5} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{6} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 2 \) Copy content Toggle raw display

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
0.965926 + 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
0.258819 0.965926i
−0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 0.258819i
−0.965926 + 0.258819i
−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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1082.8
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(1083, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 14 T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 64 T^{2} + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 26 T^{2} + 121)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 78 T^{2} + 1089)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T + 13)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 112 T^{2} + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 156 T^{2} + 4356)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 196 T^{2} + 8836)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 14 T + 37)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 192 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$73$ \( (T + 3)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 122 T^{2} + 1369)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 64 T^{2} + 256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 56 T^{2} + 16)^{2} \) Copy content Toggle raw display
show more
show less