Properties

Label 1183.1.n.b
Level $1183$
Weight $1$
Character orbit 1183.n
Analytic conductor $0.590$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1183.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.590393909945\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.64827.1
Defining polynomial: \(x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{2} + ( -\beta_{3} - \beta_{4} - \beta_{5} ) q^{4} -\beta_{5} q^{7} + ( -1 - \beta_{3} ) q^{8} -\beta_{5} q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{2} + ( -\beta_{3} - \beta_{4} - \beta_{5} ) q^{4} -\beta_{5} q^{7} + ( -1 - \beta_{3} ) q^{8} -\beta_{5} q^{9} + \beta_{4} q^{11} + ( -1 + \beta_{2} - \beta_{3} ) q^{14} + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{16} + ( -1 + \beta_{2} - \beta_{3} ) q^{18} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{22} + \beta_{1} q^{23} + q^{25} + ( -1 + \beta_{4} + \beta_{5} ) q^{28} + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{29} + \beta_{5} q^{32} + ( -1 + \beta_{4} + \beta_{5} ) q^{36} + \beta_{1} q^{37} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{43} + ( 2 - \beta_{2} + \beta_{3} ) q^{44} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{46} + ( -1 + \beta_{5} ) q^{49} + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{50} -\beta_{2} q^{53} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{56} + ( -\beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{58} + ( -1 + \beta_{5} ) q^{63} + \beta_{1} q^{67} + ( -\beta_{1} + \beta_{2} ) q^{71} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{72} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{74} + \beta_{3} q^{77} + \beta_{3} q^{79} + ( -1 + \beta_{5} ) q^{81} + ( 2 + \beta_{3} ) q^{86} + ( 2 - \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{88} - q^{92} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{98} + \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + q^{2} - 2q^{4} - 3q^{7} - 4q^{8} - 3q^{9} + O(q^{10}) \) \( 6q + q^{2} - 2q^{4} - 3q^{7} - 4q^{8} - 3q^{9} + q^{11} - 2q^{14} - q^{16} - 2q^{18} + 2q^{22} + q^{23} + 6q^{25} - 2q^{28} + q^{29} + 3q^{32} - 2q^{36} + q^{37} + q^{43} + 8q^{44} + 2q^{46} - 3q^{49} + q^{50} - 2q^{53} + 2q^{56} - 5q^{58} - 3q^{63} + q^{67} + q^{71} + 2q^{72} + 2q^{74} - 2q^{77} - 2q^{79} - 3q^{81} + 10q^{86} + 4q^{88} - 6q^{92} + q^{98} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6 \)\()/13\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18 \)\()/13\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2 \)\()/13\)
\(\beta_{5}\)\(=\)\((\)\( -6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10 \)\()/13\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{2}\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}\)

Character values

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

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
0.900969 + 1.56052i
−0.623490 1.07992i
0.222521 + 0.385418i
0.900969 1.56052i
−0.623490 + 1.07992i
0.222521 0.385418i
−0.623490 1.07992i 0 −0.277479 + 0.480608i 0 0 −0.500000 + 0.866025i −0.554958 −0.500000 + 0.866025i 0
146.2 0.222521 + 0.385418i 0 0.400969 0.694498i 0 0 −0.500000 + 0.866025i 0.801938 −0.500000 + 0.866025i 0
146.3 0.900969 + 1.56052i 0 −1.12349 + 1.94594i 0 0 −0.500000 + 0.866025i −2.24698 −0.500000 + 0.866025i 0
867.1 −0.623490 + 1.07992i 0 −0.277479 0.480608i 0 0 −0.500000 0.866025i −0.554958 −0.500000 0.866025i 0
867.2 0.222521 0.385418i 0 0.400969 + 0.694498i 0 0 −0.500000 0.866025i 0.801938 −0.500000 0.866025i 0
867.3 0.900969 1.56052i 0 −1.12349 1.94594i 0 0 −0.500000 0.866025i −2.24698 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 867.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
13.c even 3 1 inner
91.n odd 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 3 T_{2}^{4} + 5 T_{2}^{2} - 2 T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1183, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( T^{6} \)
$7$ \( ( 1 + T + T^{2} )^{3} \)
$11$ \( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( T^{6} \)
$19$ \( T^{6} \)
$23$ \( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} \)
$29$ \( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} \)
$31$ \( T^{6} \)
$37$ \( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} \)
$41$ \( T^{6} \)
$43$ \( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( ( -1 - 2 T + T^{2} + T^{3} )^{2} \)
$59$ \( T^{6} \)
$61$ \( T^{6} \)
$67$ \( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} \)
$71$ \( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} \)
$73$ \( T^{6} \)
$79$ \( ( -1 - 2 T + T^{2} + T^{3} )^{2} \)
$83$ \( T^{6} \)
$89$ \( T^{6} \)
$97$ \( T^{6} \)
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