Properties

Label 1183.2.c.c
Level $1183$
Weight $2$
Character orbit 1183.c
Analytic conductor $9.446$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{3} ) q^{2} + ( -1 + \beta_{2} ) q^{3} + 3 \beta_{2} q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{6} + \beta_{3} q^{7} + ( -4 \beta_{1} + \beta_{3} ) q^{8} + ( -1 - 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{3} ) q^{2} + ( -1 + \beta_{2} ) q^{3} + 3 \beta_{2} q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{6} + \beta_{3} q^{7} + ( -4 \beta_{1} + \beta_{3} ) q^{8} + ( -1 - 3 \beta_{2} ) q^{9} + ( -2 + 3 \beta_{2} ) q^{10} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{11} + ( 3 - 6 \beta_{2} ) q^{12} + ( 1 - \beta_{2} ) q^{14} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{15} + ( 5 - 3 \beta_{2} ) q^{16} + ( -5 - 4 \beta_{2} ) q^{17} + ( 5 \beta_{1} - 2 \beta_{3} ) q^{18} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{19} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{20} + ( \beta_{1} - \beta_{3} ) q^{21} -3 \beta_{2} q^{22} + ( -2 - 4 \beta_{2} ) q^{23} + ( 9 \beta_{1} - 5 \beta_{3} ) q^{24} + ( 3 + 3 \beta_{2} ) q^{25} + ( 1 + 2 \beta_{2} ) q^{27} + 3 \beta_{1} q^{28} + ( -1 - 5 \beta_{2} ) q^{29} + ( 5 - 8 \beta_{2} ) q^{30} + ( -6 \beta_{1} - 5 \beta_{3} ) q^{31} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{32} + 3 \beta_{1} q^{33} + ( 3 \beta_{1} + \beta_{3} ) q^{34} + ( 1 - \beta_{2} ) q^{35} + ( -9 + 6 \beta_{2} ) q^{36} + 4 \beta_{3} q^{37} -3 \beta_{2} q^{38} + ( 5 - 9 \beta_{2} ) q^{40} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{41} + ( -2 + 3 \beta_{2} ) q^{42} + ( 2 + 9 \beta_{2} ) q^{43} -9 \beta_{3} q^{44} + ( 5 \beta_{1} - 2 \beta_{3} ) q^{45} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{46} + ( 2 \beta_{1} + \beta_{3} ) q^{47} + ( -8 + 11 \beta_{2} ) q^{48} - q^{49} -3 \beta_{1} q^{50} + ( 1 + 3 \beta_{2} ) q^{51} + ( 7 + 2 \beta_{2} ) q^{53} + ( -3 \beta_{1} + \beta_{3} ) q^{54} -3 \beta_{2} q^{55} + ( -1 + 4 \beta_{2} ) q^{56} + 3 \beta_{1} q^{57} + ( 9 \beta_{1} - 4 \beta_{3} ) q^{58} + ( 2 \beta_{1} + \beta_{3} ) q^{59} + ( 15 \beta_{1} - 9 \beta_{3} ) q^{60} -6 q^{61} + ( 1 - 7 \beta_{2} ) q^{62} + ( -3 \beta_{1} - \beta_{3} ) q^{63} + ( 1 + 6 \beta_{2} ) q^{64} + ( -3 + 6 \beta_{2} ) q^{66} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{67} + ( -12 - 3 \beta_{2} ) q^{68} + ( -2 + 6 \beta_{2} ) q^{69} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{70} + ( -10 \beta_{1} - 2 \beta_{3} ) q^{71} + ( -11 \beta_{1} + 11 \beta_{3} ) q^{72} -2 \beta_{3} q^{73} + ( 4 - 4 \beta_{2} ) q^{74} -3 \beta_{2} q^{75} -9 \beta_{3} q^{76} + ( 3 + 3 \beta_{2} ) q^{77} + 4 q^{79} + ( 11 \beta_{1} - 8 \beta_{3} ) q^{80} + ( 4 + 6 \beta_{2} ) q^{81} -2 q^{82} + ( 6 \beta_{1} + 3 \beta_{3} ) q^{83} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{84} + ( 3 \beta_{1} + \beta_{3} ) q^{85} + ( -16 \beta_{1} + 7 \beta_{3} ) q^{86} + ( -4 + 9 \beta_{2} ) q^{87} + ( -9 + 3 \beta_{2} ) q^{88} + ( 5 \beta_{1} + 13 \beta_{3} ) q^{89} + ( -7 + 12 \beta_{2} ) q^{90} + ( -12 + 6 \beta_{2} ) q^{92} + ( 7 \beta_{1} - \beta_{3} ) q^{93} + ( -1 + 3 \beta_{2} ) q^{94} -3 \beta_{2} q^{95} + ( -12 \beta_{1} + 9 \beta_{3} ) q^{96} + ( 3 \beta_{1} - 14 \beta_{3} ) q^{97} + ( -\beta_{1} + \beta_{3} ) q^{98} + ( 3 \beta_{1} + 12 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{3} - 6q^{4} + 2q^{9} + O(q^{10}) \) \( 4q - 6q^{3} - 6q^{4} + 2q^{9} - 14q^{10} + 24q^{12} + 6q^{14} + 26q^{16} - 12q^{17} + 6q^{22} + 6q^{25} + 6q^{29} + 36q^{30} + 6q^{35} - 48q^{36} + 6q^{38} + 38q^{40} - 14q^{42} - 10q^{43} - 54q^{48} - 4q^{49} - 2q^{51} + 24q^{53} + 6q^{55} - 12q^{56} - 24q^{61} + 18q^{62} - 8q^{64} - 24q^{66} - 42q^{68} - 20q^{69} + 24q^{74} + 6q^{75} + 6q^{77} + 16q^{79} + 4q^{81} - 8q^{82} - 34q^{87} - 42q^{88} - 52q^{90} - 60q^{92} - 10q^{94} + 6q^{95} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.61803i
0.618034i
0.618034i
1.61803i
2.61803i −2.61803 −4.85410 2.61803i 6.85410i 1.00000i 7.47214i 3.85410 −6.85410
337.2 0.381966i −0.381966 1.85410 0.381966i 0.145898i 1.00000i 1.47214i −2.85410 −0.145898
337.3 0.381966i −0.381966 1.85410 0.381966i 0.145898i 1.00000i 1.47214i −2.85410 −0.145898
337.4 2.61803i −2.61803 −4.85410 2.61803i 6.85410i 1.00000i 7.47214i 3.85410 −6.85410
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.c.c 4
13.b even 2 1 inner 1183.2.c.c 4
13.d odd 4 1 1183.2.a.c 2
13.d odd 4 1 1183.2.a.g 2
13.f odd 12 2 91.2.f.a 4
39.k even 12 2 819.2.o.c 4
52.l even 12 2 1456.2.s.h 4
91.i even 4 1 8281.2.a.n 2
91.i even 4 1 8281.2.a.bb 2
91.w even 12 2 637.2.g.c 4
91.x odd 12 2 637.2.h.g 4
91.ba even 12 2 637.2.h.f 4
91.bc even 12 2 637.2.f.c 4
91.bd odd 12 2 637.2.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 13.f odd 12 2
637.2.f.c 4 91.bc even 12 2
637.2.g.b 4 91.bd odd 12 2
637.2.g.c 4 91.w even 12 2
637.2.h.f 4 91.ba even 12 2
637.2.h.g 4 91.x odd 12 2
819.2.o.c 4 39.k even 12 2
1183.2.a.c 2 13.d odd 4 1
1183.2.a.g 2 13.d odd 4 1
1183.2.c.c 4 1.a even 1 1 trivial
1183.2.c.c 4 13.b even 2 1 inner
1456.2.s.h 4 52.l even 12 2
8281.2.a.n 2 91.i even 4 1
8281.2.a.bb 2 91.i even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 7 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 7 T^{2} + T^{4} \)
$3$ \( ( 1 + 3 T + T^{2} )^{2} \)
$5$ \( 1 + 7 T^{2} + T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( 81 + 27 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -11 + 6 T + T^{2} )^{2} \)
$19$ \( 81 + 27 T^{2} + T^{4} \)
$23$ \( ( -20 + T^{2} )^{2} \)
$29$ \( ( -29 - 3 T + T^{2} )^{2} \)
$31$ \( 1681 + 98 T^{2} + T^{4} \)
$37$ \( ( 16 + T^{2} )^{2} \)
$41$ \( 16 + 28 T^{2} + T^{4} \)
$43$ \( ( -95 + 5 T + T^{2} )^{2} \)
$47$ \( ( 5 + T^{2} )^{2} \)
$53$ \( ( 31 - 12 T + T^{2} )^{2} \)
$59$ \( ( 5 + T^{2} )^{2} \)
$61$ \( ( 6 + T )^{4} \)
$67$ \( 81 + 162 T^{2} + T^{4} \)
$71$ \( 13456 + 268 T^{2} + T^{4} \)
$73$ \( ( 4 + T^{2} )^{2} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( ( 45 + T^{2} )^{2} \)
$89$ \( 6241 + 283 T^{2} + T^{4} \)
$97$ \( 52441 + 503 T^{2} + T^{4} \)
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