Properties

Label 1183.2.a.h
Level $1183$
Weight $2$
Character orbit 1183.a
Self dual yes
Analytic conductor $9.446$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

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

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
−1.48119
0.311108
2.17009
−2.48119 1.67513 4.15633 0.675131 −4.15633 1.00000 −5.35026 −0.193937 −1.67513
1.2 −0.688892 −2.21432 −1.52543 −3.21432 1.52543 1.00000 2.42864 1.90321 2.21432
1.3 1.17009 0.539189 −0.630898 −0.460811 0.630898 1.00000 −3.07838 −2.70928 −0.539189
\(n\): 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.h 3
7.b odd 2 1 8281.2.a.be 3
13.b even 2 1 1183.2.a.j 3
13.d odd 4 2 91.2.c.a 6
39.f even 4 2 819.2.c.b 6
52.f even 4 2 1456.2.k.c 6
91.b odd 2 1 8281.2.a.bi 3
91.i even 4 2 637.2.c.d 6
91.z odd 12 4 637.2.r.e 12
91.bb even 12 4 637.2.r.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.c.a 6 13.d odd 4 2
637.2.c.d 6 91.i even 4 2
637.2.r.d 12 91.bb even 12 4
637.2.r.e 12 91.z odd 12 4
819.2.c.b 6 39.f even 4 2
1183.2.a.h 3 1.a even 1 1 trivial
1183.2.a.j 3 13.b even 2 1
1456.2.k.c 6 52.f even 4 2
8281.2.a.be 3 7.b odd 2 1
8281.2.a.bi 3 91.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\):

\( T_{2}^{3} + 2 T_{2}^{2} - 2 T_{2} - 2 \)
\( T_{11}^{3} + 8 T_{11}^{2} + 18 T_{11} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 - 2 T + 2 T^{2} + T^{3} \)
$3$ \( 2 - 4 T + T^{3} \)
$5$ \( -1 - T + 3 T^{2} + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( 10 + 18 T + 8 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 34 - 8 T - 4 T^{2} + T^{3} \)
$19$ \( -193 - 59 T + T^{2} + T^{3} \)
$23$ \( -79 - 25 T + 3 T^{2} + T^{3} \)
$29$ \( 5 - 21 T + 7 T^{2} + T^{3} \)
$31$ \( -65 + 19 T + 11 T^{2} + T^{3} \)
$37$ \( -54 + 18 T + 12 T^{2} + T^{3} \)
$41$ \( -40 - 52 T + 2 T^{2} + T^{3} \)
$43$ \( 17 + 35 T - 13 T^{2} + T^{3} \)
$47$ \( 137 + 105 T + 19 T^{2} + T^{3} \)
$53$ \( 13 - 9 T - T^{2} + T^{3} \)
$59$ \( 52 - 32 T + 2 T^{2} + T^{3} \)
$61$ \( 152 + 28 T - 14 T^{2} + T^{3} \)
$67$ \( 16 + 32 T - 12 T^{2} + T^{3} \)
$71$ \( -890 - 54 T + 14 T^{2} + T^{3} \)
$73$ \( -31 - 91 T - 9 T^{2} + T^{3} \)
$79$ \( 185 - 37 T - 13 T^{2} + T^{3} \)
$83$ \( -163 - 113 T + T^{2} + T^{3} \)
$89$ \( -227 - 55 T + 3 T^{2} + T^{3} \)
$97$ \( -2183 - 175 T + 15 T^{2} + T^{3} \)
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