Properties

Label 1183.2.c.f
Level $1183$
Weight $2$
Character orbit 1183.c
Analytic conductor $9.446$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} - \nu^{2} + 2 \nu - 2 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} - 3 \nu^{3} + 6 \nu^{2} - 2 \nu + 4 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} + 3 \nu^{3} - 6 \nu^{2} + 10 \nu - 8 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{5} - 2 \nu^{4} + 5 \nu^{3} - 6 \nu^{2} + 2 \nu - 12 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{5} + 2 \beta_{3} - 5 \beta_{2} - \beta_{1} + 4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(4 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} + 5\)\()/2\)

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
−0.671462 1.24464i
1.40680 + 0.144584i
0.264658 + 1.38923i
0.264658 1.38923i
1.40680 0.144584i
−0.671462 + 1.24464i
2.34292i −1.14637 −3.48929 1.34292i 2.68585i 1.00000i 3.48929i −1.68585 3.14637
337.2 1.81361i −3.10278 −1.28917 2.81361i 5.62721i 1.00000i 1.28917i 6.62721 5.10278
337.3 0.470683i 2.24914 1.77846 0.529317i 1.05863i 1.00000i 1.77846i 2.05863 −0.249141
337.4 0.470683i 2.24914 1.77846 0.529317i 1.05863i 1.00000i 1.77846i 2.05863 −0.249141
337.5 1.81361i −3.10278 −1.28917 2.81361i 5.62721i 1.00000i 1.28917i 6.62721 5.10278
337.6 2.34292i −1.14637 −3.48929 1.34292i 2.68585i 1.00000i 3.48929i −1.68585 3.14637
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
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.f 6
13.b even 2 1 inner 1183.2.c.f 6
13.d odd 4 1 91.2.a.d 3
13.d odd 4 1 1183.2.a.i 3
39.f even 4 1 819.2.a.i 3
52.f even 4 1 1456.2.a.t 3
65.g odd 4 1 2275.2.a.m 3
91.i even 4 1 637.2.a.j 3
91.i even 4 1 8281.2.a.bg 3
91.z odd 12 2 637.2.e.j 6
91.bb even 12 2 637.2.e.i 6
104.j odd 4 1 5824.2.a.by 3
104.m even 4 1 5824.2.a.bs 3
273.o odd 4 1 5733.2.a.x 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 13.d odd 4 1
637.2.a.j 3 91.i even 4 1
637.2.e.i 6 91.bb even 12 2
637.2.e.j 6 91.z odd 12 2
819.2.a.i 3 39.f even 4 1
1183.2.a.i 3 13.d odd 4 1
1183.2.c.f 6 1.a even 1 1 trivial
1183.2.c.f 6 13.b even 2 1 inner
1456.2.a.t 3 52.f even 4 1
2275.2.a.m 3 65.g odd 4 1
5733.2.a.x 3 273.o odd 4 1
5824.2.a.bs 3 104.m even 4 1
5824.2.a.by 3 104.j odd 4 1
8281.2.a.bg 3 91.i even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 9 T_{2}^{4} + 20 T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 20 T^{2} + 9 T^{4} + T^{6} \)
$3$ \( ( -8 - 6 T + 2 T^{2} + T^{3} )^{2} \)
$5$ \( 4 + 17 T^{2} + 10 T^{4} + T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( 64 + 68 T^{2} + 16 T^{4} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( ( 4 - 10 T + 4 T^{2} + T^{3} )^{2} \)
$19$ \( 16 + 33 T^{2} + 14 T^{4} + T^{6} \)
$23$ \( ( -136 + T + 10 T^{2} + T^{3} )^{2} \)
$29$ \( ( -454 + 185 T - 24 T^{2} + T^{3} )^{2} \)
$31$ \( 256 + 233 T^{2} + 54 T^{4} + T^{6} \)
$37$ \( 15376 + 3364 T^{2} + 116 T^{4} + T^{6} \)
$41$ \( 64 + 752 T^{2} + 60 T^{4} + T^{6} \)
$43$ \( ( -628 - 71 T + 10 T^{2} + T^{3} )^{2} \)
$47$ \( 295936 + 14945 T^{2} + 222 T^{4} + T^{6} \)
$53$ \( ( -22 - 35 T - 8 T^{2} + T^{3} )^{2} \)
$59$ \( 473344 + 29840 T^{2} + 328 T^{4} + T^{6} \)
$61$ \( ( 2 + T )^{6} \)
$67$ \( 952576 + 38800 T^{2} + 392 T^{4} + T^{6} \)
$71$ \( 256 + 292 T^{2} + 80 T^{4} + T^{6} \)
$73$ \( 75076 + 15281 T^{2} + 298 T^{4} + T^{6} \)
$79$ \( ( -16 + 5 T + 14 T^{2} + T^{3} )^{2} \)
$83$ \( 10679824 + 151873 T^{2} + 686 T^{4} + T^{6} \)
$89$ \( 178084 + 10713 T^{2} + 194 T^{4} + T^{6} \)
$97$ \( 484 + 401 T^{2} + 42 T^{4} + T^{6} \)
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