Properties

Label 1183.2.e.d
Level $1183$
Weight $2$
Character orbit 1183.e
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.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: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} - \beta_{3} ) q^{2} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{3} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( -1 + 3 \beta_{2} ) q^{6} + ( 1 - 2 \beta_{3} ) q^{7} + ( 1 - 4 \beta_{2} ) q^{8} + ( -2 - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} - \beta_{3} ) q^{2} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{3} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( -1 + 3 \beta_{2} ) q^{6} + ( 1 - 2 \beta_{3} ) q^{7} + ( 1 - 4 \beta_{2} ) q^{8} + ( -2 - 2 \beta_{3} ) q^{9} + ( -3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{10} + 3 \beta_{3} q^{11} + ( 6 + 3 \beta_{1} + 6 \beta_{3} ) q^{12} + ( -3 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{14} + 5 q^{15} + ( -5 - 3 \beta_{1} - 5 \beta_{3} ) q^{16} + ( 4 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{17} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{18} + ( 3 + 3 \beta_{3} ) q^{19} + ( -6 + 3 \beta_{2} ) q^{20} + ( 2 - 6 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{21} + ( 3 - 3 \beta_{2} ) q^{22} + ( 5 + 2 \beta_{1} + 5 \beta_{3} ) q^{23} + ( -6 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} ) q^{24} + ( -1 - 2 \beta_{2} ) q^{27} + ( 9 \beta_{1} + 3 \beta_{2} ) q^{28} + ( -2 - 4 \beta_{2} ) q^{29} + ( -5 - 5 \beta_{1} - 5 \beta_{3} ) q^{30} -5 \beta_{3} q^{31} + ( 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{32} + ( -3 + 6 \beta_{1} - 3 \beta_{3} ) q^{33} + ( -1 - 3 \beta_{2} ) q^{34} + ( -3 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{35} -6 \beta_{2} q^{36} + ( -5 + 6 \beta_{1} - 5 \beta_{3} ) q^{37} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{38} + ( 7 + 6 \beta_{1} + 7 \beta_{3} ) q^{40} + ( -2 - 4 \beta_{2} ) q^{41} + ( -1 + 6 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} ) q^{42} -8 q^{43} -9 \beta_{1} q^{44} + ( -4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{45} + ( -9 \beta_{1} - 9 \beta_{2} - 7 \beta_{3} ) q^{46} + ( -1 - 4 \beta_{1} - \beta_{3} ) q^{47} + ( -1 + 13 \beta_{2} ) q^{48} + ( -3 - 8 \beta_{3} ) q^{49} + ( 13 - 6 \beta_{1} + 13 \beta_{3} ) q^{51} + ( -4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{53} + ( -1 - 3 \beta_{1} - \beta_{3} ) q^{54} + ( 3 + 6 \beta_{2} ) q^{55} + ( 1 - 8 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{56} + ( -3 - 6 \beta_{2} ) q^{57} + ( -2 - 6 \beta_{1} - 2 \beta_{3} ) q^{58} + ( 4 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{59} + ( 15 \beta_{1} + 15 \beta_{2} ) q^{60} + ( -3 - 3 \beta_{3} ) q^{61} + ( -5 + 5 \beta_{2} ) q^{62} + ( -6 - 2 \beta_{3} ) q^{63} + ( -1 - 6 \beta_{2} ) q^{64} + ( -9 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{66} + 3 \beta_{3} q^{67} + ( -12 + 3 \beta_{1} - 12 \beta_{3} ) q^{68} + ( -1 - 12 \beta_{2} ) q^{69} + ( -2 - 9 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{70} + ( -4 - 8 \beta_{2} ) q^{71} + ( -2 - 8 \beta_{1} - 2 \beta_{3} ) q^{72} + ( 6 \beta_{1} + 6 \beta_{2} - 7 \beta_{3} ) q^{73} + ( -7 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{74} + 9 \beta_{2} q^{76} + ( 6 + 9 \beta_{3} ) q^{77} + ( -7 + 6 \beta_{1} - 7 \beta_{3} ) q^{79} + ( -13 \beta_{1} - 13 \beta_{2} - \beta_{3} ) q^{80} -11 \beta_{3} q^{81} + ( -2 - 6 \beta_{1} - 2 \beta_{3} ) q^{82} + ( 18 + 3 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{84} + ( -13 - 6 \beta_{2} ) q^{85} + ( 8 + 8 \beta_{1} + 8 \beta_{3} ) q^{86} -10 \beta_{3} q^{87} + ( 12 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} ) q^{88} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{89} + ( -2 + 6 \beta_{2} ) q^{90} + ( -6 + 21 \beta_{2} ) q^{92} + ( 5 - 10 \beta_{1} + 5 \beta_{3} ) q^{93} + ( 9 \beta_{1} + 9 \beta_{2} + 5 \beta_{3} ) q^{94} + ( 6 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{95} + 15 \beta_{1} q^{96} + ( 10 + 12 \beta_{2} ) q^{97} + ( -5 + 3 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} ) q^{98} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{2} - 3q^{4} - 10q^{6} + 8q^{7} + 12q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 3q^{2} - 3q^{4} - 10q^{6} + 8q^{7} + 12q^{8} - 4q^{9} + 5q^{10} - 6q^{11} + 15q^{12} - 15q^{14} + 20q^{15} - 13q^{16} + 6q^{17} - 6q^{18} + 6q^{19} - 30q^{20} + 18q^{22} + 12q^{23} + 20q^{24} + 3q^{28} - 15q^{30} + 10q^{31} - 15q^{32} + 2q^{34} + 12q^{36} - 4q^{37} + 9q^{38} + 20q^{40} - 20q^{42} - 32q^{43} - 9q^{44} + 23q^{46} - 6q^{47} - 30q^{48} + 4q^{49} + 20q^{51} + 6q^{53} - 5q^{54} + 24q^{56} - 10q^{58} + 6q^{59} - 15q^{60} - 6q^{61} - 30q^{62} - 20q^{63} + 8q^{64} + 15q^{66} - 6q^{67} - 21q^{68} + 20q^{69} - 5q^{70} - 12q^{72} + 8q^{73} + 9q^{74} - 18q^{76} + 6q^{77} - 8q^{79} + 15q^{80} + 22q^{81} - 10q^{82} + 75q^{84} - 40q^{85} + 24q^{86} + 20q^{87} - 18q^{88} - 20q^{90} - 66q^{92} - 19q^{94} + 15q^{96} + 16q^{97} - 39q^{98} + 24q^{99} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(1\)

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
0.809017 1.40126i
−0.309017 + 0.535233i
0.809017 + 1.40126i
−0.309017 0.535233i
−1.30902 + 2.26728i 1.11803 + 1.93649i −2.42705 4.20378i 1.11803 1.93649i −5.85410 2.00000 + 1.73205i 7.47214 −1.00000 + 1.73205i 2.92705 + 5.06980i
170.2 −0.190983 + 0.330792i −1.11803 1.93649i 0.927051 + 1.60570i −1.11803 + 1.93649i 0.854102 2.00000 + 1.73205i −1.47214 −1.00000 + 1.73205i −0.427051 0.739674i
508.1 −1.30902 2.26728i 1.11803 1.93649i −2.42705 + 4.20378i 1.11803 + 1.93649i −5.85410 2.00000 1.73205i 7.47214 −1.00000 1.73205i 2.92705 5.06980i
508.2 −0.190983 0.330792i −1.11803 + 1.93649i 0.927051 1.60570i −1.11803 1.93649i 0.854102 2.00000 1.73205i −1.47214 −1.00000 1.73205i −0.427051 + 0.739674i
\(n\): 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.d 4
7.c even 3 1 inner 1183.2.e.d 4
7.c even 3 1 8281.2.a.z 2
7.d odd 6 1 8281.2.a.ba 2
13.b even 2 1 91.2.e.b 4
39.d odd 2 1 819.2.j.c 4
52.b odd 2 1 1456.2.r.j 4
91.b odd 2 1 637.2.e.h 4
91.r even 6 1 91.2.e.b 4
91.r even 6 1 637.2.a.f 2
91.s odd 6 1 637.2.a.e 2
91.s odd 6 1 637.2.e.h 4
273.w odd 6 1 819.2.j.c 4
273.w odd 6 1 5733.2.a.v 2
273.ba even 6 1 5733.2.a.w 2
364.bl odd 6 1 1456.2.r.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.b 4 13.b even 2 1
91.2.e.b 4 91.r even 6 1
637.2.a.e 2 91.s odd 6 1
637.2.a.f 2 91.r even 6 1
637.2.e.h 4 91.b odd 2 1
637.2.e.h 4 91.s odd 6 1
819.2.j.c 4 39.d odd 2 1
819.2.j.c 4 273.w odd 6 1
1183.2.e.d 4 1.a even 1 1 trivial
1183.2.e.d 4 7.c even 3 1 inner
1456.2.r.j 4 52.b odd 2 1
1456.2.r.j 4 364.bl odd 6 1
5733.2.a.v 2 273.w odd 6 1
5733.2.a.w 2 273.ba even 6 1
8281.2.a.z 2 7.c even 3 1
8281.2.a.ba 2 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 \(S_{2}^{\mathrm{new}}(1183, [\chi])\):

\( T_{2}^{4} + 3 T_{2}^{3} + 8 T_{2}^{2} + 3 T_{2} + 1 \)
\( T_{3}^{4} + 5 T_{3}^{2} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 8 T^{2} + 3 T^{3} + T^{4} \)
$3$ \( 25 + 5 T^{2} + T^{4} \)
$5$ \( 25 + 5 T^{2} + T^{4} \)
$7$ \( ( 7 - 4 T + T^{2} )^{2} \)
$11$ \( ( 9 + 3 T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4} \)
$19$ \( ( 9 - 3 T + T^{2} )^{2} \)
$23$ \( 961 - 372 T + 113 T^{2} - 12 T^{3} + T^{4} \)
$29$ \( ( -20 + T^{2} )^{2} \)
$31$ \( ( 25 - 5 T + T^{2} )^{2} \)
$37$ \( 1681 - 164 T + 57 T^{2} + 4 T^{3} + T^{4} \)
$41$ \( ( -20 + T^{2} )^{2} \)
$43$ \( ( 8 + T )^{4} \)
$47$ \( 121 - 66 T + 47 T^{2} + 6 T^{3} + T^{4} \)
$53$ \( 121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( 121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4} \)
$61$ \( ( 9 + 3 T + T^{2} )^{2} \)
$67$ \( ( 9 + 3 T + T^{2} )^{2} \)
$71$ \( ( -80 + T^{2} )^{2} \)
$73$ \( 841 + 232 T + 93 T^{2} - 8 T^{3} + T^{4} \)
$79$ \( 841 - 232 T + 93 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( T^{4} \)
$89$ \( 25 + 5 T^{2} + T^{4} \)
$97$ \( ( -164 - 8 T + T^{2} )^{2} \)
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