Properties

Label 983.1.b.c
Level $983$
Weight $1$
Character orbit 983.b
Self dual yes
Analytic conductor $0.491$
Analytic rank $0$
Dimension $9$
Projective image $D_{27}$
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: \(9\)
Coefficient field: \(\Q(\zeta_{54})^+\)
Defining polynomial: \( x^{9} - 9x^{7} + 27x^{5} - 30x^{3} + 9x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{27}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{27} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{7} - \beta_{2}) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{8} - \beta_{6} + \beta_{3} + \beta_1) q^{6} - \beta_{7} q^{7} + ( - \beta_{3} - \beta_1) q^{8} + ( - \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{7} - \beta_{2}) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{8} - \beta_{6} + \beta_{3} + \beta_1) q^{6} - \beta_{7} q^{7} + ( - \beta_{3} - \beta_1) q^{8} + ( - \beta_{5} + 1) q^{9} + (\beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{12} + (\beta_{8} + \beta_{6}) q^{14} + (\beta_{4} + \beta_{2} + 1) q^{16} + (\beta_{6} + \beta_{4} - \beta_1) q^{18} + \beta_{8} q^{19} + (\beta_{4} - 1) q^{21} + (\beta_{5} - \beta_{4}) q^{23} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{24} + q^{25} + (\beta_{7} + \beta_{6} - \beta_{2}) q^{27} + ( - \beta_{7} - \beta_{5} - 1) q^{28} - \beta_{3} q^{31} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{32} + ( - \beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{36} + \beta_{6} q^{37} + ( - \beta_{7} - 1) q^{38} + ( - \beta_{6} + \beta_{3}) q^{41} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{42} + ( - \beta_{8} + \beta_1) q^{43} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3}) q^{46} + \beta_{4} q^{47} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{48} + (\beta_{5} - \beta_{4} + 1) q^{49} - \beta_1 q^{50} + ( - \beta_{8} + \beta_1) q^{53} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{54} + (\beta_{6} + \beta_{4} + \beta_1) q^{56} + (\beta_{8} - \beta_{3}) q^{57} - q^{59} + (\beta_{4} + \beta_{2}) q^{62} + ( - \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2}) q^{63} + (\beta_{6} + \beta_{4} + \beta_{2} + 1) q^{64} + ( - \beta_{6} + \beta_{3}) q^{67} + ( - \beta_{3} + \beta_{2}) q^{69} + (\beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{72} + ( - \beta_{7} - \beta_{5}) q^{74} + (\beta_{7} - \beta_{2}) q^{75} + (\beta_{6} + \beta_1) q^{76} - q^{79} + ( - \beta_{8} - \beta_{5} + \beta_1 + 1) q^{81} + (\beta_{7} + \beta_{5} - \beta_{4} - \beta_{2}) q^{82} + (\beta_{6} + \beta_{4} - 1) q^{84} + (\beta_{7} - \beta_{2} - 1) q^{86} + \beta_{2} q^{89} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{92} + (\beta_{8} + \beta_{5} - \beta_{4}) q^{93} + ( - \beta_{5} - \beta_{3}) q^{94} + (\beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{96} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{4} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{4} + 9 q^{9} - 9 q^{12} + 9 q^{16} - 9 q^{21} + 9 q^{25} - 9 q^{28} + 9 q^{36} - 9 q^{38} - 9 q^{48} + 9 q^{49} - 9 q^{59} + 9 q^{64} - 9 q^{79} + 9 q^{81} - 9 q^{84} - 9 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{54} + \zeta_{54}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 5\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70 \) 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
1.98648
1.67098
1.37248
0.573606
0.116290
−0.792160
−1.19432
−1.78727
−1.94609
−1.98648 −0.573606 2.94609 0 1.13946 −1.37248 −3.86586 −0.670976 0
982.2 −1.67098 −1.98648 1.79216 0 3.31935 1.19432 −1.32368 2.94609 0
982.3 −1.37248 1.78727 0.883710 0 −2.45299 −1.67098 0.159606 2.19432 0
982.4 −0.573606 −0.116290 −0.670976 0 0.0667045 1.78727 0.958482 −0.986477 0
982.5 −0.116290 1.19432 −0.986477 0 −0.138887 0.792160 0.231007 0.426394 0
982.6 0.792160 1.94609 −0.372483 0 1.54161 −0.573606 −1.08723 2.78727 0
982.7 1.19432 −1.37248 0.426394 0 −1.63918 1.94609 −0.685068 0.883710 0
982.8 1.78727 0.792160 2.19432 0 1.41580 −1.98648 2.13456 −0.372483 0
982.9 1.94609 −1.67098 2.78727 0 −3.25187 −0.116290 3.47818 1.79216 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 982.9
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.c 9
983.b odd 2 1 CM 983.1.b.c 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
983.1.b.c 9 1.a even 1 1 trivial
983.1.b.c 9 983.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 9T_{2}^{7} + 27T_{2}^{5} - 30T_{2}^{3} + 9T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(983, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 9 T^{7} + 27 T^{5} - 30 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{9} - 9 T^{7} + 27 T^{5} - 30 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 9 T^{7} + 27 T^{5} - 30 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{9} \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} \) Copy content Toggle raw display
$19$ \( T^{9} - 9 T^{7} + 27 T^{5} - 30 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{9} - 9 T^{7} + 27 T^{5} - 30 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{9} \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T + 1)^{3} \) Copy content Toggle raw display
$37$ \( (T^{3} - 3 T + 1)^{3} \) Copy content Toggle raw display
$41$ \( (T^{3} - 3 T + 1)^{3} \) Copy content Toggle raw display
$43$ \( T^{9} - 9 T^{7} + 27 T^{5} - 30 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{9} - 9 T^{7} + 27 T^{5} - 30 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{9} - 9 T^{7} + 27 T^{5} - 30 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( (T + 1)^{9} \) Copy content Toggle raw display
$61$ \( T^{9} \) Copy content Toggle raw display
$67$ \( (T^{3} - 3 T + 1)^{3} \) Copy content Toggle raw display
$71$ \( T^{9} \) Copy content Toggle raw display
$73$ \( T^{9} \) Copy content Toggle raw display
$79$ \( (T + 1)^{9} \) Copy content Toggle raw display
$83$ \( T^{9} \) Copy content Toggle raw display
$89$ \( T^{9} - 9 T^{7} + 27 T^{5} - 30 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{9} \) Copy content Toggle raw display
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