Properties

Label 1183.1.t.a
Level $1183$
Weight $1$
Character orbit 1183.t
Analytic conductor $0.590$
Analytic rank $0$
Dimension $12$
Projective image $D_{7}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.590393909945\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
Defining polynomial: \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} - \beta_{6} ) q^{2} + ( -1 + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{4} + \beta_{10} q^{7} + ( \beta_{2} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{8} -\beta_{7} q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} - \beta_{6} ) q^{2} + ( -1 + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{4} + \beta_{10} q^{7} + ( \beta_{2} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{8} -\beta_{7} q^{9} + \beta_{2} q^{11} -\beta_{3} q^{14} + ( -1 - \beta_{4} + \beta_{7} - \beta_{9} ) q^{16} + ( -\beta_{2} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{18} + ( -1 + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{22} + \beta_{4} q^{23} - q^{25} + ( -\beta_{2} + \beta_{6} ) q^{28} + ( 1 + \beta_{4} - \beta_{7} + \beta_{9} ) q^{29} + \beta_{10} q^{32} + ( 1 - \beta_{7} + \beta_{9} ) q^{36} + \beta_{1} q^{37} + ( 1 - \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{43} + ( \beta_{2} - 2 \beta_{6} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{44} + ( -\beta_{1} + \beta_{8} + \beta_{11} ) q^{46} + ( 1 - \beta_{7} ) q^{49} + ( -\beta_{1} - \beta_{2} + \beta_{6} ) q^{50} -\beta_{5} q^{53} + ( 1 - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{56} + ( -2 \beta_{10} - \beta_{11} ) q^{58} -\beta_{6} q^{63} -\beta_{1} q^{67} + ( \beta_{1} - \beta_{8} ) q^{71} + ( -\beta_{10} - \beta_{11} ) q^{72} + ( 1 - \beta_{3} + \beta_{4} + \beta_{9} ) q^{74} + ( 1 - \beta_{3} - \beta_{5} ) q^{77} + ( -1 + \beta_{3} + \beta_{5} ) q^{79} + ( -1 + \beta_{7} ) q^{81} + ( -\beta_{2} + 2 \beta_{6} - 2 \beta_{10} - \beta_{11} ) q^{86} + ( -2 - \beta_{4} + 2 \beta_{7} - \beta_{9} ) q^{88} - q^{92} + ( \beta_{1} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{98} + ( -\beta_{2} - \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{4} - 6q^{9} + O(q^{10}) \) \( 12q + 4q^{4} - 6q^{9} - 4q^{14} - 2q^{16} + 4q^{22} - 2q^{23} - 12q^{25} + 2q^{29} + 4q^{36} - 2q^{43} + 6q^{49} - 4q^{53} - 4q^{56} + 4q^{74} + 4q^{77} - 4q^{79} - 6q^{81} - 8q^{88} - 12q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -25 \nu^{11} + 95 \nu^{9} - 361 \nu^{7} + 155 \nu^{5} - 30 \nu^{3} - 1563 \nu \)\()/559\)
\(\beta_{3}\)\(=\)\((\)\( 25 \nu^{10} - 95 \nu^{8} + 361 \nu^{6} - 155 \nu^{4} + 30 \nu^{2} + 1004 \)\()/559\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{10} - 20 \nu^{8} + 76 \nu^{6} - 139 \nu^{4} + 124 \nu^{2} - 24 \)\()/43\)
\(\beta_{5}\)\(=\)\((\)\( 45 \nu^{10} - 171 \nu^{8} + 538 \nu^{6} - 279 \nu^{4} + 54 \nu^{2} + 242 \)\()/559\)
\(\beta_{6}\)\(=\)\((\)\( 70 \nu^{11} - 266 \nu^{9} + 899 \nu^{7} - 434 \nu^{5} + 84 \nu^{3} + 1246 \nu \)\()/559\)
\(\beta_{7}\)\(=\)\((\)\( 114 \nu^{10} - 545 \nu^{8} + 2071 \nu^{6} - 2831 \nu^{4} + 3379 \nu^{2} - 95 \)\()/559\)
\(\beta_{8}\)\(=\)\((\)\( 114 \nu^{11} - 545 \nu^{9} + 2071 \nu^{7} - 2831 \nu^{5} + 3379 \nu^{3} - 95 \nu \)\()/559\)
\(\beta_{9}\)\(=\)\((\)\( -128 \nu^{10} + 710 \nu^{8} - 2698 \nu^{6} + 4483 \nu^{4} - 4402 \nu^{2} + 852 \)\()/559\)
\(\beta_{10}\)\(=\)\((\)\( -242 \nu^{11} + 1255 \nu^{9} - 4769 \nu^{7} + 7314 \nu^{5} - 7781 \nu^{3} + 1506 \nu \)\()/559\)
\(\beta_{11}\)\(=\)\((\)\( -317 \nu^{11} + 1540 \nu^{9} - 5852 \nu^{7} + 8338 \nu^{5} - 9548 \nu^{3} + 1848 \nu \)\()/559\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{11} + 3 \beta_{8} + \beta_{2}\)
\(\nu^{4}\)\(=\)\(3 \beta_{9} + 2 \beta_{7} + 4 \beta_{4} - 2\)
\(\nu^{5}\)\(=\)\(4 \beta_{11} - \beta_{10} + 9 \beta_{8} - 9 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-5 \beta_{5} + 9 \beta_{3} - 14\)
\(\nu^{7}\)\(=\)\(-5 \beta_{6} - 14 \beta_{2} - 28 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-28 \beta_{9} - 14 \beta_{7} - 19 \beta_{5} - 47 \beta_{4} + 28 \beta_{3} - 28\)
\(\nu^{9}\)\(=\)\(-47 \beta_{11} + 19 \beta_{10} - 89 \beta_{8} - 19 \beta_{6} - 47 \beta_{2}\)
\(\nu^{10}\)\(=\)\(-89 \beta_{9} - 42 \beta_{7} - 155 \beta_{4} + 42\)
\(\nu^{11}\)\(=\)\(-155 \beta_{11} + 66 \beta_{10} - 286 \beta_{8} + 286 \beta_{1}\)

Character values

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

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-1\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
699.1
0.385418 0.222521i
−1.56052 + 0.900969i
−1.07992 + 0.623490i
1.07992 0.623490i
1.56052 0.900969i
−0.385418 + 0.222521i
0.385418 + 0.222521i
−1.56052 0.900969i
−1.07992 0.623490i
1.07992 + 0.623490i
1.56052 + 0.900969i
−0.385418 0.222521i
−1.56052 + 0.900969i 0 1.12349 1.94594i 0 0 0.866025 + 0.500000i 2.24698i −0.500000 + 0.866025i 0
699.2 −1.07992 + 0.623490i 0 0.277479 0.480608i 0 0 −0.866025 0.500000i 0.554958i −0.500000 + 0.866025i 0
699.3 −0.385418 + 0.222521i 0 −0.400969 + 0.694498i 0 0 0.866025 + 0.500000i 0.801938i −0.500000 + 0.866025i 0
699.4 0.385418 0.222521i 0 −0.400969 + 0.694498i 0 0 −0.866025 0.500000i 0.801938i −0.500000 + 0.866025i 0
699.5 1.07992 0.623490i 0 0.277479 0.480608i 0 0 0.866025 + 0.500000i 0.554958i −0.500000 + 0.866025i 0
699.6 1.56052 0.900969i 0 1.12349 1.94594i 0 0 −0.866025 0.500000i 2.24698i −0.500000 + 0.866025i 0
1161.1 −1.56052 0.900969i 0 1.12349 + 1.94594i 0 0 0.866025 0.500000i 2.24698i −0.500000 0.866025i 0
1161.2 −1.07992 0.623490i 0 0.277479 + 0.480608i 0 0 −0.866025 + 0.500000i 0.554958i −0.500000 0.866025i 0
1161.3 −0.385418 0.222521i 0 −0.400969 0.694498i 0 0 0.866025 0.500000i 0.801938i −0.500000 0.866025i 0
1161.4 0.385418 + 0.222521i 0 −0.400969 0.694498i 0 0 −0.866025 + 0.500000i 0.801938i −0.500000 0.866025i 0
1161.5 1.07992 + 0.623490i 0 0.277479 + 0.480608i 0 0 0.866025 0.500000i 0.554958i −0.500000 0.866025i 0
1161.6 1.56052 + 0.900969i 0 1.12349 + 1.94594i 0 0 −0.866025 + 0.500000i 2.24698i −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1161.6
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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner
91.b odd 2 1 inner
91.n odd 6 1 inner
91.t 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.t.a 12
7.b odd 2 1 CM 1183.1.t.a 12
13.b even 2 1 inner 1183.1.t.a 12
13.c even 3 1 1183.1.b.a 6
13.c even 3 1 inner 1183.1.t.a 12
13.d odd 4 1 1183.1.n.a 6
13.d odd 4 1 1183.1.n.b 6
13.e even 6 1 1183.1.b.a 6
13.e even 6 1 inner 1183.1.t.a 12
13.f odd 12 1 1183.1.d.a 3
13.f odd 12 1 1183.1.d.b yes 3
13.f odd 12 1 1183.1.n.a 6
13.f odd 12 1 1183.1.n.b 6
91.b odd 2 1 inner 1183.1.t.a 12
91.i even 4 1 1183.1.n.a 6
91.i even 4 1 1183.1.n.b 6
91.n odd 6 1 1183.1.b.a 6
91.n odd 6 1 inner 1183.1.t.a 12
91.t odd 6 1 1183.1.b.a 6
91.t odd 6 1 inner 1183.1.t.a 12
91.bc even 12 1 1183.1.d.a 3
91.bc even 12 1 1183.1.d.b yes 3
91.bc even 12 1 1183.1.n.a 6
91.bc even 12 1 1183.1.n.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.1.b.a 6 13.c even 3 1
1183.1.b.a 6 13.e even 6 1
1183.1.b.a 6 91.n odd 6 1
1183.1.b.a 6 91.t odd 6 1
1183.1.d.a 3 13.f odd 12 1
1183.1.d.a 3 91.bc even 12 1
1183.1.d.b yes 3 13.f odd 12 1
1183.1.d.b yes 3 91.bc even 12 1
1183.1.n.a 6 13.d odd 4 1
1183.1.n.a 6 13.f odd 12 1
1183.1.n.a 6 91.i even 4 1
1183.1.n.a 6 91.bc even 12 1
1183.1.n.b 6 13.d odd 4 1
1183.1.n.b 6 13.f odd 12 1
1183.1.n.b 6 91.i even 4 1
1183.1.n.b 6 91.bc even 12 1
1183.1.t.a 12 1.a even 1 1 trivial
1183.1.t.a 12 7.b odd 2 1 CM
1183.1.t.a 12 13.b even 2 1 inner
1183.1.t.a 12 13.c even 3 1 inner
1183.1.t.a 12 13.e even 6 1 inner
1183.1.t.a 12 91.b odd 2 1 inner
1183.1.t.a 12 91.n odd 6 1 inner
1183.1.t.a 12 91.t odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1183, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( T^{12} \)
$7$ \( ( 1 - T^{2} + T^{4} )^{3} \)
$11$ \( 1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( T^{12} \)
$19$ \( T^{12} \)
$23$ \( ( 1 + 2 T + 5 T^{2} + 3 T^{4} + T^{5} + T^{6} )^{2} \)
$29$ \( ( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2} \)
$31$ \( T^{12} \)
$37$ \( 1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12} \)
$41$ \( T^{12} \)
$43$ \( ( 1 + 2 T + 5 T^{2} + 3 T^{4} + T^{5} + T^{6} )^{2} \)
$47$ \( T^{12} \)
$53$ \( ( -1 - 2 T + T^{2} + T^{3} )^{4} \)
$59$ \( T^{12} \)
$61$ \( T^{12} \)
$67$ \( 1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12} \)
$71$ \( 1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12} \)
$73$ \( T^{12} \)
$79$ \( ( -1 - 2 T + T^{2} + T^{3} )^{4} \)
$83$ \( T^{12} \)
$89$ \( T^{12} \)
$97$ \( T^{12} \)
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