Properties

Label 1143.2.a.g
Level $1143$
Weight $2$
Character orbit 1143.a
Self dual yes
Analytic conductor $9.127$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
Defining polynomial: \(x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 7 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 381)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{4} -\beta_{3} q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( -2 \beta_{1} - \beta_{2} ) q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{4} -\beta_{3} q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( -2 \beta_{1} - \beta_{2} ) q^{8} + ( -\beta_{3} - \beta_{4} ) q^{10} + ( -3 + \beta_{1} - \beta_{4} ) q^{11} + ( -1 - \beta_{1} + \beta_{2} ) q^{13} + ( -3 + \beta_{1} - \beta_{4} ) q^{14} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{16} + ( -2 + 2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( 2 - 2 \beta_{2} - 2 \beta_{4} ) q^{19} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{20} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{22} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{23} + ( -2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{25} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{26} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{28} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{29} + ( -1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{31} + ( -4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{32} + ( 2 - 4 \beta_{1} + 2 \beta_{4} ) q^{34} + ( -2 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{35} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{37} + ( 2 + 2 \beta_{2} + 2 \beta_{4} ) q^{38} + ( -4 + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{40} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{41} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{43} + ( -3 + \beta_{1} - 2 \beta_{3} - 3 \beta_{4} ) q^{44} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{46} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{47} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{49} + ( 3 + \beta_{1} + 3 \beta_{2} + \beta_{4} ) q^{50} + ( -2 \beta_{1} - 3 \beta_{3} - \beta_{4} ) q^{52} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{53} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{55} + ( 3 - \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{56} + ( -2 + 5 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{58} + ( -7 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{59} + ( -5 + 6 \beta_{1} - \beta_{3} - \beta_{4} ) q^{61} + ( 6 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{62} + ( -2 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{64} + ( -2 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{65} + ( 2 \beta_{1} + 4 \beta_{2} + \beta_{3} + 5 \beta_{4} ) q^{67} + ( 2 + 6 \beta_{1} + 2 \beta_{3} ) q^{68} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{70} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{71} + ( -7 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{73} + ( 3 + 5 \beta_{1} + \beta_{2} + 6 \beta_{3} + 3 \beta_{4} ) q^{74} + ( -6 - 2 \beta_{2} + 2 \beta_{4} ) q^{76} + ( -1 + 5 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{77} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{80} + ( -4 - 8 \beta_{1} - 6 \beta_{3} - 2 \beta_{4} ) q^{82} + ( -1 - 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{83} + ( -6 + 2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 3 + 7 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{86} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{88} + ( -2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{89} + ( 6 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{91} + ( 2 - 6 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{92} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{94} + ( -4 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{95} + ( -6 + 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} ) q^{97} + ( -1 + 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 2q^{2} + 6q^{4} - q^{5} - 6q^{8} + O(q^{10}) \) \( 5q - 2q^{2} + 6q^{4} - q^{5} - 6q^{8} - 2q^{10} - 14q^{11} - 5q^{13} - 14q^{14} + 8q^{16} - 4q^{17} + 4q^{19} - 6q^{20} - 2q^{22} - 15q^{23} - 6q^{25} - 12q^{26} - 2q^{28} - 9q^{29} + 3q^{31} - 14q^{32} + 4q^{34} - 4q^{35} - 5q^{37} + 16q^{38} - 14q^{40} - 4q^{41} + 10q^{43} - 18q^{44} - 4q^{46} + 4q^{47} - 9q^{49} + 24q^{50} - 8q^{52} - 3q^{53} + 4q^{55} + 10q^{56} + 6q^{58} - 23q^{59} - 15q^{61} + 24q^{62} - 3q^{65} + 18q^{67} + 24q^{68} + 4q^{70} - 12q^{71} - 43q^{73} + 36q^{74} - 32q^{76} - 4q^{77} + 16q^{79} - 4q^{80} - 44q^{82} - 11q^{83} - 24q^{85} + 28q^{86} - 14q^{88} - 9q^{89} + 26q^{91} - 14q^{92} - 14q^{94} - 16q^{95} - 20q^{97} + 10q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 7 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 2 \nu^{2} + 7 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 6\)

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
2.71457
−1.51908
−1.15351
1.71250
0.245526
−2.65432 0 5.04540 −0.121872 0 −0.0602522 −8.08346 0 0.323487
1.2 −1.82669 0 1.33679 0.563416 0 3.34577 1.21147 0 −1.02918
1.3 −0.484093 0 −1.76565 −2.26452 0 1.63760 1.82293 0 1.09624
1.4 0.779856 0 −1.39182 2.98063 0 −2.49235 −2.64513 0 2.32446
1.5 2.18524 0 2.77529 −2.15766 0 −2.43077 1.69419 0 −4.71500
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.g 5
3.b odd 2 1 381.2.a.d 5
12.b even 2 1 6096.2.a.bf 5
15.d odd 2 1 9525.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.a.d 5 3.b odd 2 1
1143.2.a.g 5 1.a even 1 1 trivial
6096.2.a.bf 5 12.b even 2 1
9525.2.a.j 5 15.d odd 2 1

Hecke kernels

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

\( T_{2}^{5} + 2 T_{2}^{4} - 6 T_{2}^{3} - 10 T_{2}^{2} + 5 T_{2} + 4 \)
\( T_{5}^{5} + T_{5}^{4} - 9 T_{5}^{3} - 11 T_{5}^{2} + 7 T_{5} + 1 \)
\( T_{7}^{5} - 13 T_{7}^{3} - 4 T_{7}^{2} + 33 T_{7} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 5 T - 10 T^{2} - 6 T^{3} + 2 T^{4} + T^{5} \)
$3$ \( T^{5} \)
$5$ \( 1 + 7 T - 11 T^{2} - 9 T^{3} + T^{4} + T^{5} \)
$7$ \( 2 + 33 T - 4 T^{2} - 13 T^{3} + T^{5} \)
$11$ \( -8 + 33 T + 96 T^{2} + 63 T^{3} + 14 T^{4} + T^{5} \)
$13$ \( 19 - 9 T - 33 T^{2} - 3 T^{3} + 5 T^{4} + T^{5} \)
$17$ \( -64 + 304 T - 64 T^{2} - 32 T^{3} + 4 T^{4} + T^{5} \)
$19$ \( -256 + 192 T + 64 T^{2} - 40 T^{3} - 4 T^{4} + T^{5} \)
$23$ \( -592 - 448 T + 4 T^{2} + 64 T^{3} + 15 T^{4} + T^{5} \)
$29$ \( 2279 + 29 T - 347 T^{2} - 31 T^{3} + 9 T^{4} + T^{5} \)
$31$ \( 1840 + 1424 T - 4 T^{2} - 100 T^{3} - 3 T^{4} + T^{5} \)
$37$ \( 907 - 669 T - 653 T^{2} - 91 T^{3} + 5 T^{4} + T^{5} \)
$41$ \( 2368 + 1424 T - 320 T^{2} - 116 T^{3} + 4 T^{4} + T^{5} \)
$43$ \( -11272 + 1661 T + 746 T^{2} - 89 T^{3} - 10 T^{4} + T^{5} \)
$47$ \( -152 + 581 T + 114 T^{2} - 49 T^{3} - 4 T^{4} + T^{5} \)
$53$ \( -211 - 475 T - 305 T^{2} - 55 T^{3} + 3 T^{4} + T^{5} \)
$59$ \( 1520 - 2496 T - 284 T^{2} + 128 T^{3} + 23 T^{4} + T^{5} \)
$61$ \( 119213 + 7395 T - 2891 T^{2} - 191 T^{3} + 15 T^{4} + T^{5} \)
$67$ \( 9836 - 18595 T + 3982 T^{2} - 137 T^{3} - 18 T^{4} + T^{5} \)
$71$ \( 106 - 135 T - 106 T^{2} + 19 T^{3} + 12 T^{4} + T^{5} \)
$73$ \( 1369 + 9735 T + 4317 T^{2} + 665 T^{3} + 43 T^{4} + T^{5} \)
$79$ \( -256 - 464 T + 64 T^{2} + 64 T^{3} - 16 T^{4} + T^{5} \)
$83$ \( -13120 - 10976 T - 2744 T^{2} - 176 T^{3} + 11 T^{4} + T^{5} \)
$89$ \( -13159 + 7699 T - 671 T^{2} - 153 T^{3} + 9 T^{4} + T^{5} \)
$97$ \( -5120 - 6512 T - 1280 T^{2} + 28 T^{3} + 20 T^{4} + T^{5} \)
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