Properties

Label 1183.2.e.a
Level $1183$
Weight $2$
Character orbit 1183.e
Analytic conductor $9.446$
Analytic rank $0$
Dimension $2$
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
170.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 1.50000 + 2.59808i 0.500000 + 0.866025i −1.50000 + 2.59808i −3.00000 0.500000 2.59808i −3.00000 −3.00000 + 5.19615i −1.50000 2.59808i
508.1 −0.500000 0.866025i 1.50000 2.59808i 0.500000 0.866025i −1.50000 2.59808i −3.00000 0.500000 + 2.59808i −3.00000 −3.00000 5.19615i −1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.a 2
7.c even 3 1 inner 1183.2.e.a 2
7.c even 3 1 8281.2.a.i 1
7.d odd 6 1 8281.2.a.j 1
13.b even 2 1 1183.2.e.c 2
13.c even 3 1 91.2.g.a 2
13.c even 3 1 91.2.h.a yes 2
39.i odd 6 1 819.2.n.c 2
39.i odd 6 1 819.2.s.a 2
91.g even 3 1 91.2.h.a yes 2
91.g even 3 1 637.2.f.b 2
91.h even 3 1 91.2.g.a 2
91.h even 3 1 637.2.f.b 2
91.m odd 6 1 637.2.f.a 2
91.m odd 6 1 637.2.h.a 2
91.n odd 6 1 637.2.g.a 2
91.n odd 6 1 637.2.h.a 2
91.r even 6 1 1183.2.e.c 2
91.r even 6 1 8281.2.a.c 1
91.s odd 6 1 8281.2.a.g 1
91.v odd 6 1 637.2.f.a 2
91.v odd 6 1 637.2.g.a 2
273.s odd 6 1 819.2.n.c 2
273.bm odd 6 1 819.2.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 13.c even 3 1
91.2.g.a 2 91.h even 3 1
91.2.h.a yes 2 13.c even 3 1
91.2.h.a yes 2 91.g even 3 1
637.2.f.a 2 91.m odd 6 1
637.2.f.a 2 91.v odd 6 1
637.2.f.b 2 91.g even 3 1
637.2.f.b 2 91.h even 3 1
637.2.g.a 2 91.n odd 6 1
637.2.g.a 2 91.v odd 6 1
637.2.h.a 2 91.m odd 6 1
637.2.h.a 2 91.n odd 6 1
819.2.n.c 2 39.i odd 6 1
819.2.n.c 2 273.s odd 6 1
819.2.s.a 2 39.i odd 6 1
819.2.s.a 2 273.bm odd 6 1
1183.2.e.a 2 1.a even 1 1 trivial
1183.2.e.a 2 7.c even 3 1 inner
1183.2.e.c 2 13.b even 2 1
1183.2.e.c 2 91.r even 6 1
8281.2.a.c 1 91.r even 6 1
8281.2.a.g 1 91.s odd 6 1
8281.2.a.i 1 7.c even 3 1
8281.2.a.j 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \)
\( T_{3}^{2} - 3 T_{3} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 9 - 3 T + T^{2} \)
$5$ \( 9 + 3 T + T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -7 + T )^{2} \)
$31$ \( 9 + 3 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( ( -3 + T )^{2} \)
$43$ \( ( 7 + T )^{2} \)
$47$ \( 1 + T + T^{2} \)
$53$ \( 9 + 3 T + T^{2} \)
$59$ \( 16 - 4 T + T^{2} \)
$61$ \( 169 - 13 T + T^{2} \)
$67$ \( 9 - 3 T + T^{2} \)
$71$ \( ( -13 + T )^{2} \)
$73$ \( 169 - 13 T + T^{2} \)
$79$ \( 9 - 3 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( 5 + T )^{2} \)
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