Properties

Label 983.1.b.b
Level $983$
Weight $1$
Character orbit 983.b
Self dual yes
Analytic conductor $0.491$
Analytic rank $0$
Dimension $3$
Projective image $D_{9}$
CM discriminant -983
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 983 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 983.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.490580907418\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.933714431521.1
Artin image: $D_9$
Artin field: Galois closure of 9.1.933714431521.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_1 - 1) q^{6} + \beta_{2} q^{7} + ( - \beta_1 - 1) q^{8} + ( - \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_1 - 1) q^{6} + \beta_{2} q^{7} + ( - \beta_1 - 1) q^{8} + ( - \beta_{2} + \beta_1 + 1) q^{9} + (\beta_1 + 2) q^{12} + ( - \beta_1 - 1) q^{14} + (\beta_1 + 1) q^{16} + ( - \beta_{2} - 1) q^{18} - \beta_1 q^{19} + ( - \beta_{2} + \beta_1 + 2) q^{21} + ( - \beta_{2} + \beta_1) q^{23} + ( - \beta_{2} - \beta_1 - 1) q^{24} + q^{25} + (\beta_{2} - 1) q^{27} + (\beta_1 + 2) q^{28} - q^{31} + ( - \beta_{2} - 1) q^{32} + (\beta_{2} + \beta_1) q^{36} - q^{37} + (\beta_{2} + 2) q^{38} - q^{41} + ( - \beta_{2} - \beta_1 - 1) q^{42} - \beta_1 q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{46} + ( - \beta_{2} + \beta_1) q^{47} + (\beta_{2} + \beta_1 + 1) q^{48} + ( - \beta_{2} + \beta_1 + 1) q^{49} - \beta_1 q^{50} - \beta_1 q^{53} - q^{54} + ( - \beta_{2} - \beta_1 - 1) q^{56} + ( - \beta_1 - 1) q^{57} + 2 q^{59} + \beta_1 q^{62} + (2 \beta_{2} - 1) q^{63} + \beta_1 q^{64} - q^{67} + (\beta_{2} - 1) q^{69} + ( - \beta_1 - 2) q^{72} + \beta_1 q^{74} + \beta_{2} q^{75} + ( - 2 \beta_1 - 1) q^{76} + 2 q^{79} + ( - \beta_{2} + 1) q^{81} + \beta_1 q^{82} + (2 \beta_{2} + \beta_1 + 1) q^{84} + (\beta_{2} + 2) q^{86} + \beta_{2} q^{89} + (\beta_1 - 1) q^{92} - \beta_{2} q^{93} + ( - \beta_{2} + \beta_1 - 1) q^{94} + ( - \beta_1 - 2) q^{96} + ( - \beta_{2} - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{4} - 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{4} - 3 q^{6} - 3 q^{8} + 3 q^{9} + 6 q^{12} - 3 q^{14} + 3 q^{16} - 3 q^{18} + 6 q^{21} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 6 q^{28} - 3 q^{31} - 3 q^{32} - 3 q^{37} + 6 q^{38} - 3 q^{41} - 3 q^{42} - 3 q^{46} + 3 q^{48} + 3 q^{49} - 3 q^{54} - 3 q^{56} - 3 q^{57} + 6 q^{59} - 3 q^{63} - 3 q^{67} - 3 q^{69} - 6 q^{72} - 3 q^{76} + 6 q^{79} + 3 q^{81} + 3 q^{84} + 6 q^{86} - 3 q^{92} - 3 q^{94} - 6 q^{96} - 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/983\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
982.1
1.87939
−0.347296
−1.53209
−1.87939 1.53209 2.53209 0 −2.87939 1.53209 −2.87939 1.34730 0
982.2 0.347296 −1.87939 −0.879385 0 −0.652704 −1.87939 −0.652704 2.53209 0
982.3 1.53209 0.347296 1.34730 0 0.532089 0.347296 0.532089 −0.879385 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
983.b odd 2 1 CM by \(\Q(\sqrt{-983}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 983.1.b.b 3
983.b odd 2 1 CM 983.1.b.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
983.1.b.b 3 1.a even 1 1 trivial
983.1.b.b 3 983.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(983, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$23$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$29$ \( T^{3} \) Copy content Toggle raw display
$31$ \( (T + 1)^{3} \) Copy content Toggle raw display
$37$ \( (T + 1)^{3} \) Copy content Toggle raw display
$41$ \( (T + 1)^{3} \) Copy content Toggle raw display
$43$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$47$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$53$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$59$ \( (T - 2)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} \) Copy content Toggle raw display
$67$ \( (T + 1)^{3} \) Copy content Toggle raw display
$71$ \( T^{3} \) Copy content Toggle raw display
$73$ \( T^{3} \) Copy content Toggle raw display
$79$ \( (T - 2)^{3} \) Copy content Toggle raw display
$83$ \( T^{3} \) Copy content Toggle raw display
$89$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$97$ \( T^{3} \) Copy content Toggle raw display
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