Properties

Label 1183.2.a.l
Level $1183$
Weight $2$
Character orbit 1183.a
Self dual yes
Analytic conductor $9.446$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.27004.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} + 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} + 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 6 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 7 \beta_{1} + 1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.22001
−0.231361
0.710287
2.74108
−2.22001 0.549551 2.92843 4.22001 −1.22001 1.00000 −2.06113 −2.69799 −9.36845
1.2 −0.231361 −3.32225 −1.94647 2.23136 0.768639 1.00000 0.913059 8.03736 −0.516249
1.3 0.710287 2.40788 −1.49549 1.28971 1.71029 1.00000 −2.48280 2.79790 0.916066
1.4 2.74108 1.36482 5.51353 −0.741082 3.74108 1.00000 9.63087 −1.13727 −2.03137
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.a.l 4
7.b odd 2 1 8281.2.a.bt 4
13.b even 2 1 1183.2.a.k 4
13.d odd 4 2 1183.2.c.g 8
13.e even 6 2 91.2.f.c 8
39.h odd 6 2 819.2.o.h 8
52.i odd 6 2 1456.2.s.q 8
91.b odd 2 1 8281.2.a.bp 4
91.k even 6 2 637.2.h.h 8
91.l odd 6 2 637.2.h.i 8
91.p odd 6 2 637.2.g.j 8
91.t odd 6 2 637.2.f.i 8
91.u even 6 2 637.2.g.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 13.e even 6 2
637.2.f.i 8 91.t odd 6 2
637.2.g.j 8 91.p odd 6 2
637.2.g.k 8 91.u even 6 2
637.2.h.h 8 91.k even 6 2
637.2.h.i 8 91.l odd 6 2
819.2.o.h 8 39.h odd 6 2
1183.2.a.k 4 13.b even 2 1
1183.2.a.l 4 1.a even 1 1 trivial
1183.2.c.g 8 13.d odd 4 2
1456.2.s.q 8 52.i odd 6 2
8281.2.a.bp 4 91.b odd 2 1
8281.2.a.bt 4 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\):

\( T_{2}^{4} - T_{2}^{3} - 6 T_{2}^{2} + 3 T_{2} + 1 \)
\( T_{11}^{4} - T_{11}^{3} - 9 T_{11}^{2} + 16 T_{11} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T - 6 T^{2} - T^{3} + T^{4} \)
$3$ \( -6 + 16 T - 9 T^{2} - T^{3} + T^{4} \)
$5$ \( -9 + T + 12 T^{2} - 7 T^{3} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( -6 + 16 T - 9 T^{2} - T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( -53 - 60 T - 12 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( 500 - 55 T^{2} + T^{3} + T^{4} \)
$23$ \( 72 - 56 T - 38 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( -5 - 21 T - 22 T^{2} - T^{3} + T^{4} \)
$31$ \( 54 + 26 T - 13 T^{2} - 4 T^{3} + T^{4} \)
$37$ \( -16 + 44 T + 17 T^{2} - 10 T^{3} + T^{4} \)
$41$ \( -564 - 226 T + 149 T^{2} - 22 T^{3} + T^{4} \)
$43$ \( -2 + 8 T - 3 T^{2} - 3 T^{3} + T^{4} \)
$47$ \( 100 + 12 T - 51 T^{2} + 2 T^{3} + T^{4} \)
$53$ \( -1389 + 890 T - 140 T^{2} - 2 T^{3} + T^{4} \)
$59$ \( -706 + 550 T - 77 T^{2} - 8 T^{3} + T^{4} \)
$61$ \( -100 + 176 T - 15 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 11010 + 634 T - 211 T^{2} - 6 T^{3} + T^{4} \)
$71$ \( -2032 + 712 T - 16 T^{2} - 14 T^{3} + T^{4} \)
$73$ \( 1772 + 88 T - 119 T^{2} - 8 T^{3} + T^{4} \)
$79$ \( -7680 - 1024 T + 134 T^{2} + 26 T^{3} + T^{4} \)
$83$ \( -426 + 442 T - 97 T^{2} + T^{4} \)
$89$ \( -108 + 184 T - 71 T^{2} - T^{3} + T^{4} \)
$97$ \( 15692 - 668 T - 305 T^{2} + 3 T^{3} + T^{4} \)
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