Properties

Label 1143.2.a.e
Level $1143$
Weight $2$
Character orbit 1143.a
Self dual yes
Analytic conductor $9.127$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1143.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.12690095103\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 127)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( 2 + \beta_{1} - \beta_{2} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} ) q^{7} + ( 2 - 2 \beta_{1} + 3 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( 2 + \beta_{1} - \beta_{2} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} ) q^{7} + ( 2 - 2 \beta_{1} + 3 \beta_{2} ) q^{8} + ( 1 - 2 \beta_{2} ) q^{10} + ( -2 \beta_{1} - \beta_{2} ) q^{11} + ( -1 + 2 \beta_{1} - 3 \beta_{2} ) q^{13} + ( -\beta_{1} + 2 \beta_{2} ) q^{14} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{16} + ( 6 + \beta_{1} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} ) q^{19} + ( -1 - \beta_{1} ) q^{20} + ( 5 - \beta_{1} + \beta_{2} ) q^{22} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{23} + ( 1 + 3 \beta_{1} - 4 \beta_{2} ) q^{25} + ( -2 + 6 \beta_{1} - 5 \beta_{2} ) q^{26} + ( 2 - \beta_{1} + \beta_{2} ) q^{28} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{29} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{31} -3 \beta_{1} q^{32} + ( 4 - 5 \beta_{1} - \beta_{2} ) q^{34} + ( -4 - 2 \beta_{1} + 3 \beta_{2} ) q^{35} + ( 6 \beta_{1} - 4 \beta_{2} ) q^{37} + ( 2 - 3 \beta_{1} + 2 \beta_{2} ) q^{38} + ( -1 + 5 \beta_{2} ) q^{40} + ( 4 + 4 \beta_{2} ) q^{41} + ( -3 - 6 \beta_{1} + 6 \beta_{2} ) q^{43} + ( 6 - 3 \beta_{1} + 4 \beta_{2} ) q^{44} + ( 3 - 6 \beta_{1} + 3 \beta_{2} ) q^{46} + ( 1 - 4 \beta_{1} - 2 \beta_{2} ) q^{47} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{49} + ( -1 + 6 \beta_{1} - 7 \beta_{2} ) q^{50} + ( -7 + 9 \beta_{1} - 5 \beta_{2} ) q^{52} + ( -1 + 6 \beta_{1} - 7 \beta_{2} ) q^{53} + ( -1 - 2 \beta_{1} - 5 \beta_{2} ) q^{55} + ( 3 - 2 \beta_{1} - 2 \beta_{2} ) q^{56} + ( -6 + 5 \beta_{1} - 4 \beta_{2} ) q^{58} + ( 2 \beta_{1} + \beta_{2} ) q^{59} + ( -1 + 2 \beta_{1} + 6 \beta_{2} ) q^{61} + ( 5 - 4 \beta_{2} ) q^{62} + ( 4 + 3 \beta_{1} - 3 \beta_{2} ) q^{64} + ( 3 + \beta_{1} - 6 \beta_{2} ) q^{65} + ( -1 + \beta_{1} - \beta_{2} ) q^{67} + ( 3 - 10 \beta_{1} + 4 \beta_{2} ) q^{68} + ( -3 - \beta_{1} + 5 \beta_{2} ) q^{70} + ( -1 + 6 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 1 - 5 \beta_{1} + 7 \beta_{2} ) q^{73} + ( -8 + 10 \beta_{1} - 10 \beta_{2} ) q^{74} + ( 4 - 5 \beta_{1} + 3 \beta_{2} ) q^{76} + ( 1 + 4 \beta_{2} ) q^{77} + ( 3 + 7 \beta_{1} ) q^{79} + ( -4 - 2 \beta_{1} + 5 \beta_{2} ) q^{80} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{82} + ( -4 + 6 \beta_{1} + 5 \beta_{2} ) q^{83} + ( 13 + 7 \beta_{1} - 5 \beta_{2} ) q^{85} + ( 3 - 9 \beta_{1} + 12 \beta_{2} ) q^{86} + ( -2 - 11 \beta_{1} + 5 \beta_{2} ) q^{88} + ( 11 + 5 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -4 + \beta_{1} + 3 \beta_{2} ) q^{91} + ( 6 - 10 \beta_{1} + 5 \beta_{2} ) q^{92} + ( 11 - 3 \beta_{1} + 2 \beta_{2} ) q^{94} + \beta_{2} q^{95} + ( -5 - 3 \beta_{1} + 6 \beta_{2} ) q^{97} + ( -4 + 7 \beta_{1} - 3 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + 6q^{5} - 3q^{7} + 6q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + 6q^{5} - 3q^{7} + 6q^{8} + 3q^{10} - 3q^{13} + 3q^{16} + 18q^{17} + 3q^{19} - 3q^{20} + 15q^{22} + 9q^{23} + 3q^{25} - 6q^{26} + 6q^{28} - 3q^{29} + 12q^{31} + 12q^{34} - 12q^{35} + 6q^{38} - 3q^{40} + 12q^{41} - 9q^{43} + 18q^{44} + 9q^{46} + 3q^{47} - 12q^{49} - 3q^{50} - 21q^{52} - 3q^{53} - 3q^{55} + 9q^{56} - 18q^{58} - 3q^{61} + 15q^{62} + 12q^{64} + 9q^{65} - 3q^{67} + 9q^{68} - 9q^{70} - 3q^{71} + 3q^{73} - 24q^{74} + 12q^{76} + 3q^{77} + 9q^{79} - 12q^{80} - 12q^{83} + 39q^{85} + 9q^{86} - 6q^{88} + 33q^{89} - 12q^{91} + 18q^{92} + 33q^{94} - 15q^{97} - 12q^{98} + O(q^{100}) \)

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
1.87939
−0.347296
−1.53209
−0.879385 0 −1.22668 2.34730 0 −1.34730 2.83750 0 −2.06418
1.2 1.34730 0 −0.184793 3.53209 0 −2.53209 −2.94356 0 4.75877
1.3 2.53209 0 4.41147 0.120615 0 0.879385 6.10607 0 0.305407
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(127\) \(1\)

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 1143.2.a.e 3
3.b odd 2 1 127.2.a.a 3
12.b even 2 1 2032.2.a.k 3
15.d odd 2 1 3175.2.a.h 3
21.c even 2 1 6223.2.a.e 3
24.f even 2 1 8128.2.a.w 3
24.h odd 2 1 8128.2.a.bd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
127.2.a.a 3 3.b odd 2 1
1143.2.a.e 3 1.a even 1 1 trivial
2032.2.a.k 3 12.b even 2 1
3175.2.a.h 3 15.d odd 2 1
6223.2.a.e 3 21.c even 2 1
8128.2.a.w 3 24.f even 2 1
8128.2.a.bd 3 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1143))\):

\( T_{2}^{3} - 3 T_{2}^{2} + 3 \)
\( T_{5}^{3} - 6 T_{5}^{2} + 9 T_{5} - 1 \)
\( T_{7}^{3} + 3 T_{7}^{2} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 3 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( -1 + 9 T - 6 T^{2} + T^{3} \)
$7$ \( -3 + 3 T^{2} + T^{3} \)
$11$ \( 37 - 21 T + T^{3} \)
$13$ \( -37 - 18 T + 3 T^{2} + T^{3} \)
$17$ \( -199 + 105 T - 18 T^{2} + T^{3} \)
$19$ \( 1 - 3 T^{2} + T^{3} \)
$23$ \( 9 + 18 T - 9 T^{2} + T^{3} \)
$29$ \( -3 - 18 T + 3 T^{2} + T^{3} \)
$31$ \( -17 + 27 T - 12 T^{2} + T^{3} \)
$37$ \( 296 - 84 T + T^{3} \)
$41$ \( 192 - 12 T^{2} + T^{3} \)
$43$ \( -513 - 81 T + 9 T^{2} + T^{3} \)
$47$ \( 379 - 81 T - 3 T^{2} + T^{3} \)
$53$ \( -57 - 126 T + 3 T^{2} + T^{3} \)
$59$ \( -37 - 21 T + T^{3} \)
$61$ \( -307 - 153 T + 3 T^{2} + T^{3} \)
$67$ \( -1 + 3 T^{2} + T^{3} \)
$71$ \( -867 - 153 T + 3 T^{2} + T^{3} \)
$73$ \( 269 - 114 T - 3 T^{2} + T^{3} \)
$79$ \( 71 - 120 T - 9 T^{2} + T^{3} \)
$83$ \( -2649 - 225 T + 12 T^{2} + T^{3} \)
$89$ \( -597 + 306 T - 33 T^{2} + T^{3} \)
$97$ \( -37 - 6 T + 15 T^{2} + T^{3} \)
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