Properties

Label 1183.1.d.b
Level $1183$
Weight $1$
Character orbit 1183.d
Self dual yes
Analytic conductor $0.590$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -7
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1183.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.590393909945\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.1655595487.1

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} - q^{7} + ( 1 + \beta_{2} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} - q^{7} + ( 1 + \beta_{2} ) q^{8} + q^{9} -\beta_{2} q^{11} -\beta_{1} q^{14} + \beta_{1} q^{16} + \beta_{1} q^{18} + ( -1 - \beta_{2} ) q^{22} + ( -1 + \beta_{1} - \beta_{2} ) q^{23} + q^{25} + ( -1 - \beta_{2} ) q^{28} -\beta_{1} q^{29} + q^{32} + ( 1 + \beta_{2} ) q^{36} + ( 1 - \beta_{1} + \beta_{2} ) q^{37} -\beta_{1} q^{43} + ( -1 - \beta_{1} ) q^{44} + ( 1 - \beta_{1} ) q^{46} + q^{49} + \beta_{1} q^{50} + ( -1 + \beta_{1} - \beta_{2} ) q^{53} + ( -1 - \beta_{2} ) q^{56} + ( -2 - \beta_{2} ) q^{58} - q^{63} + ( 1 - \beta_{1} + \beta_{2} ) q^{67} + ( 1 - \beta_{1} + \beta_{2} ) q^{71} + ( 1 + \beta_{2} ) q^{72} + ( -1 + \beta_{1} ) q^{74} + \beta_{2} q^{77} + \beta_{2} q^{79} + q^{81} + ( -2 - \beta_{2} ) q^{86} + ( -1 - \beta_{1} ) q^{88} - q^{92} + \beta_{1} q^{98} -\beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 2q^{4} - 3q^{7} + 2q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} + 2q^{4} - 3q^{7} + 2q^{8} + 3q^{9} + q^{11} - q^{14} + q^{16} + q^{18} - 2q^{22} - q^{23} + 3q^{25} - 2q^{28} - q^{29} + 3q^{32} + 2q^{36} + q^{37} - q^{43} - 4q^{44} + 2q^{46} + 3q^{49} + q^{50} - q^{53} - 2q^{56} - 5q^{58} - 3q^{63} + q^{67} + q^{71} + 2q^{72} - 2q^{74} - q^{77} - q^{79} + 3q^{81} - 5q^{86} - 4q^{88} - 3q^{92} + q^{98} + q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
846.1
−1.24698
0.445042
1.80194
−1.24698 0 0.554958 0 0 −1.00000 0.554958 1.00000 0
846.2 0.445042 0 −0.801938 0 0 −1.00000 −0.801938 1.00000 0
846.3 1.80194 0 2.24698 0 0 −1.00000 2.24698 1.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.1.d.b yes 3
7.b odd 2 1 CM 1183.1.d.b yes 3
13.b even 2 1 1183.1.d.a 3
13.c even 3 2 1183.1.n.a 6
13.d odd 4 2 1183.1.b.a 6
13.e even 6 2 1183.1.n.b 6
13.f odd 12 4 1183.1.t.a 12
91.b odd 2 1 1183.1.d.a 3
91.i even 4 2 1183.1.b.a 6
91.n odd 6 2 1183.1.n.a 6
91.t odd 6 2 1183.1.n.b 6
91.bc even 12 4 1183.1.t.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.1.b.a 6 13.d odd 4 2
1183.1.b.a 6 91.i even 4 2
1183.1.d.a 3 13.b even 2 1
1183.1.d.a 3 91.b odd 2 1
1183.1.d.b yes 3 1.a even 1 1 trivial
1183.1.d.b yes 3 7.b odd 2 1 CM
1183.1.n.a 6 13.c even 3 2
1183.1.n.a 6 91.n odd 6 2
1183.1.n.b 6 13.e even 6 2
1183.1.n.b 6 91.t odd 6 2
1183.1.t.a 12 13.f odd 12 4
1183.1.t.a 12 91.bc even 12 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 2 T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1183, [\chi])\).

Hecke characteristic polynomials

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