Properties

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

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(1\)

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
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i 0.500000 + 0.866025i −0.500000 0.866025i 0.866025 1.50000i −1.73205 −2.59808 0.500000i −1.73205 1.00000 1.73205i 1.50000 + 2.59808i
170.2 0.866025 1.50000i 0.500000 + 0.866025i −0.500000 0.866025i −0.866025 + 1.50000i 1.73205 2.59808 + 0.500000i 1.73205 1.00000 1.73205i 1.50000 + 2.59808i
508.1 −0.866025 1.50000i 0.500000 0.866025i −0.500000 + 0.866025i 0.866025 + 1.50000i −1.73205 −2.59808 + 0.500000i −1.73205 1.00000 + 1.73205i 1.50000 2.59808i
508.2 0.866025 + 1.50000i 0.500000 0.866025i −0.500000 + 0.866025i −0.866025 1.50000i 1.73205 2.59808 0.500000i 1.73205 1.00000 + 1.73205i 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
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.e 4
7.c even 3 1 inner 1183.2.e.e 4
7.c even 3 1 8281.2.a.s 2
7.d odd 6 1 8281.2.a.w 2
13.b even 2 1 inner 1183.2.e.e 4
13.f odd 12 1 91.2.k.a 2
13.f odd 12 1 91.2.u.a yes 2
39.k even 12 1 819.2.bm.a 2
39.k even 12 1 819.2.do.c 2
91.r even 6 1 inner 1183.2.e.e 4
91.r even 6 1 8281.2.a.s 2
91.s odd 6 1 8281.2.a.w 2
91.w even 12 1 637.2.k.b 2
91.w even 12 1 637.2.q.b 2
91.x odd 12 1 91.2.u.a yes 2
91.x odd 12 1 637.2.q.c 2
91.ba even 12 1 637.2.q.b 2
91.ba even 12 1 637.2.u.a 2
91.bc even 12 1 637.2.k.b 2
91.bc even 12 1 637.2.u.a 2
91.bd odd 12 1 91.2.k.a 2
91.bd odd 12 1 637.2.q.c 2
273.bv even 12 1 819.2.do.c 2
273.bw even 12 1 819.2.bm.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 13.f odd 12 1
91.2.k.a 2 91.bd odd 12 1
91.2.u.a yes 2 13.f odd 12 1
91.2.u.a yes 2 91.x odd 12 1
637.2.k.b 2 91.w even 12 1
637.2.k.b 2 91.bc even 12 1
637.2.q.b 2 91.w even 12 1
637.2.q.b 2 91.ba even 12 1
637.2.q.c 2 91.x odd 12 1
637.2.q.c 2 91.bd odd 12 1
637.2.u.a 2 91.ba even 12 1
637.2.u.a 2 91.bc even 12 1
819.2.bm.a 2 39.k even 12 1
819.2.bm.a 2 273.bw even 12 1
819.2.do.c 2 39.k even 12 1
819.2.do.c 2 273.bv even 12 1
1183.2.e.e 4 1.a even 1 1 trivial
1183.2.e.e 4 7.c even 3 1 inner
1183.2.e.e 4 13.b even 2 1 inner
1183.2.e.e 4 91.r even 6 1 inner
8281.2.a.s 2 7.c even 3 1
8281.2.a.s 2 91.r even 6 1
8281.2.a.w 2 7.d odd 6 1
8281.2.a.w 2 91.s odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 3 T_{2}^{2} + 9 \)
\( T_{3}^{2} - T_{3} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 3 T^{2} + T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( 9 + 3 T^{2} + T^{4} \)
$7$ \( 49 - 13 T^{2} + T^{4} \)
$11$ \( 729 + 27 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 36 - 6 T + T^{2} )^{2} \)
$19$ \( 9 + 3 T^{2} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( -3 + T )^{4} \)
$31$ \( 9 + 3 T^{2} + T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( -27 + T^{2} )^{2} \)
$43$ \( ( 11 + T )^{4} \)
$47$ \( 5625 + 75 T^{2} + T^{4} \)
$53$ \( ( 81 - 9 T + T^{2} )^{2} \)
$59$ \( 144 + 12 T^{2} + T^{4} \)
$61$ \( ( 49 + 7 T + T^{2} )^{2} \)
$67$ \( 5625 + 75 T^{2} + T^{4} \)
$71$ \( ( -3 + T^{2} )^{2} \)
$73$ \( 5625 + 75 T^{2} + T^{4} \)
$79$ \( ( 25 - 5 T + T^{2} )^{2} \)
$83$ \( ( -12 + T^{2} )^{2} \)
$89$ \( 2304 + 48 T^{2} + T^{4} \)
$97$ \( ( -27 + T^{2} )^{2} \)
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