Properties

Label 429.2.a.g
Level $429$
Weight $2$
Character orbit 429.a
Self dual yes
Analytic conductor $3.426$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.42558224671\)
Analytic rank: \(0\)
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: yes
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{2} + q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} ) q^{5} + ( 1 + \beta_{2} ) q^{6} + ( -1 + \beta_{1} - \beta_{2} ) q^{7} + ( 4 - 3 \beta_{1} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{2} + q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} ) q^{5} + ( 1 + \beta_{2} ) q^{6} + ( -1 + \beta_{1} - \beta_{2} ) q^{7} + ( 4 - 3 \beta_{1} ) q^{8} + q^{9} + ( 2 \beta_{1} + \beta_{2} ) q^{10} - q^{11} + ( 2 - \beta_{1} + \beta_{2} ) q^{12} - q^{13} + ( -5 + 3 \beta_{1} - \beta_{2} ) q^{14} + ( 1 + \beta_{1} ) q^{15} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{16} -\beta_{2} q^{17} + ( 1 + \beta_{2} ) q^{18} + ( 1 - \beta_{1} + \beta_{2} ) q^{19} + ( -1 + \beta_{1} ) q^{20} + ( -1 + \beta_{1} - \beta_{2} ) q^{21} + ( -1 - \beta_{2} ) q^{22} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{23} + ( 4 - 3 \beta_{1} ) q^{24} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{25} + ( -1 - \beta_{2} ) q^{26} + q^{27} + ( -9 + 5 \beta_{1} - 3 \beta_{2} ) q^{28} + ( 2 - 4 \beta_{1} - 3 \beta_{2} ) q^{29} + ( 2 \beta_{1} + \beta_{2} ) q^{30} + ( 5 - \beta_{1} ) q^{31} + ( 5 - 4 \beta_{1} + 3 \beta_{2} ) q^{32} - q^{33} + ( -3 + \beta_{1} ) q^{34} + 2 q^{35} + ( 2 - \beta_{1} + \beta_{2} ) q^{36} -4 \beta_{2} q^{37} + ( 5 - 3 \beta_{1} + \beta_{2} ) q^{38} - q^{39} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{40} + ( -1 - \beta_{1} + \beta_{2} ) q^{41} + ( -5 + 3 \beta_{1} - \beta_{2} ) q^{42} + ( -4 + 6 \beta_{1} + 3 \beta_{2} ) q^{43} + ( -2 + \beta_{1} - \beta_{2} ) q^{44} + ( 1 + \beta_{1} ) q^{45} + ( -9 + 5 \beta_{1} + \beta_{2} ) q^{46} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{48} + ( 1 - 4 \beta_{1} + 2 \beta_{2} ) q^{49} + ( -2 + 5 \beta_{1} - 2 \beta_{2} ) q^{50} -\beta_{2} q^{51} + ( -2 + \beta_{1} - \beta_{2} ) q^{52} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 1 + \beta_{2} ) q^{54} + ( -1 - \beta_{1} ) q^{55} + ( -13 + 7 \beta_{1} - 7 \beta_{2} ) q^{56} + ( 1 - \beta_{1} + \beta_{2} ) q^{57} + ( -3 - 5 \beta_{1} + 2 \beta_{2} ) q^{58} + ( 10 - 2 \beta_{1} ) q^{59} + ( -1 + \beta_{1} ) q^{60} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{61} + ( 6 - 2 \beta_{1} + 5 \beta_{2} ) q^{62} + ( -1 + \beta_{1} - \beta_{2} ) q^{63} + ( 12 - 3 \beta_{1} + \beta_{2} ) q^{64} + ( -1 - \beta_{1} ) q^{65} + ( -1 - \beta_{2} ) q^{66} + ( 3 - 7 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{68} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{69} + ( 2 + 2 \beta_{2} ) q^{70} + ( 2 + 4 \beta_{1} + 8 \beta_{2} ) q^{71} + ( 4 - 3 \beta_{1} ) q^{72} + ( -7 - 3 \beta_{1} - 5 \beta_{2} ) q^{73} + ( -12 + 4 \beta_{1} ) q^{74} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{75} + ( 9 - 5 \beta_{1} + 3 \beta_{2} ) q^{76} + ( 1 - \beta_{1} + \beta_{2} ) q^{77} + ( -1 - \beta_{2} ) q^{78} + ( -6 - 4 \beta_{1} - \beta_{2} ) q^{79} + ( -7 - 3 \beta_{1} - 2 \beta_{2} ) q^{80} + q^{81} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{82} + ( 6 - 2 \beta_{1} ) q^{83} + ( -9 + 5 \beta_{1} - 3 \beta_{2} ) q^{84} + ( 1 - \beta_{1} - \beta_{2} ) q^{85} + ( -1 + 9 \beta_{1} - 4 \beta_{2} ) q^{86} + ( 2 - 4 \beta_{1} - 3 \beta_{2} ) q^{87} + ( -4 + 3 \beta_{1} ) q^{88} + ( 1 - \beta_{1} ) q^{89} + ( 2 \beta_{1} + \beta_{2} ) q^{90} + ( 1 - \beta_{1} + \beta_{2} ) q^{91} + ( -13 + 7 \beta_{1} - 3 \beta_{2} ) q^{92} + ( 5 - \beta_{1} ) q^{93} + ( 4 - 6 \beta_{1} + 6 \beta_{2} ) q^{94} -2 q^{95} + ( 5 - 4 \beta_{1} + 3 \beta_{2} ) q^{96} + ( -4 + 4 \beta_{2} ) q^{97} + ( 11 - 10 \beta_{1} + \beta_{2} ) q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 5q^{4} + 4q^{5} + 3q^{6} - 2q^{7} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 5q^{4} + 4q^{5} + 3q^{6} - 2q^{7} + 9q^{8} + 3q^{9} + 2q^{10} - 3q^{11} + 5q^{12} - 3q^{13} - 12q^{14} + 4q^{15} + 5q^{16} + 3q^{18} + 2q^{19} - 2q^{20} - 2q^{21} - 3q^{22} + 4q^{23} + 9q^{24} - 3q^{25} - 3q^{26} + 3q^{27} - 22q^{28} + 2q^{29} + 2q^{30} + 14q^{31} + 11q^{32} - 3q^{33} - 8q^{34} + 6q^{35} + 5q^{36} + 12q^{38} - 3q^{39} - 8q^{40} - 4q^{41} - 12q^{42} - 6q^{43} - 5q^{44} + 4q^{45} - 22q^{46} + 14q^{47} + 5q^{48} - q^{49} - q^{50} - 5q^{52} - 2q^{53} + 3q^{54} - 4q^{55} - 32q^{56} + 2q^{57} - 14q^{58} + 28q^{59} - 2q^{60} - 8q^{61} + 16q^{62} - 2q^{63} + 33q^{64} - 4q^{65} - 3q^{66} + 2q^{67} - 10q^{68} + 4q^{69} + 6q^{70} + 10q^{71} + 9q^{72} - 24q^{73} - 32q^{74} - 3q^{75} + 22q^{76} + 2q^{77} - 3q^{78} - 22q^{79} - 24q^{80} + 3q^{81} + 6q^{82} + 16q^{83} - 22q^{84} + 2q^{85} + 6q^{86} + 2q^{87} - 9q^{88} + 2q^{89} + 2q^{90} + 2q^{91} - 32q^{92} + 14q^{93} + 6q^{94} - 6q^{95} + 11q^{96} - 12q^{97} + 23q^{98} - 3q^{99} + O(q^{100}) \)

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
0.311108
2.17009
−1.48119
−1.21432 1.00000 −0.525428 1.31111 −1.21432 1.52543 3.06668 1.00000 −1.59210
1.2 1.53919 1.00000 0.369102 3.17009 1.53919 0.630898 −2.51026 1.00000 4.87936
1.3 2.67513 1.00000 5.15633 −0.481194 2.67513 −4.15633 8.44358 1.00000 −1.28726
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

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 429.2.a.g 3
3.b odd 2 1 1287.2.a.h 3
4.b odd 2 1 6864.2.a.bs 3
11.b odd 2 1 4719.2.a.q 3
13.b even 2 1 5577.2.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.g 3 1.a even 1 1 trivial
1287.2.a.h 3 3.b odd 2 1
4719.2.a.q 3 11.b odd 2 1
5577.2.a.j 3 13.b even 2 1
6864.2.a.bs 3 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(429))\):

\( T_{2}^{3} - 3 T_{2}^{2} - T_{2} + 5 \)
\( T_{5}^{3} - 4 T_{5}^{2} + 2 T_{5} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 5 - T - 3 T^{2} + T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( 2 + 2 T - 4 T^{2} + T^{3} \)
$7$ \( 4 - 8 T + 2 T^{2} + T^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( -2 - 4 T + T^{3} \)
$19$ \( -4 - 8 T - 2 T^{2} + T^{3} \)
$23$ \( 68 - 40 T - 4 T^{2} + T^{3} \)
$29$ \( 178 - 64 T - 2 T^{2} + T^{3} \)
$31$ \( -86 + 62 T - 14 T^{2} + T^{3} \)
$37$ \( -128 - 64 T + T^{3} \)
$41$ \( -20 - 4 T + 4 T^{2} + T^{3} \)
$43$ \( -734 - 108 T + 6 T^{2} + T^{3} \)
$47$ \( 296 + 12 T - 14 T^{2} + T^{3} \)
$53$ \( 232 - 84 T + 2 T^{2} + T^{3} \)
$59$ \( -688 + 248 T - 28 T^{2} + T^{3} \)
$61$ \( -368 - 72 T + 8 T^{2} + T^{3} \)
$67$ \( 698 - 150 T - 2 T^{2} + T^{3} \)
$71$ \( 2056 - 212 T - 10 T^{2} + T^{3} \)
$73$ \( -556 + 92 T + 24 T^{2} + T^{3} \)
$79$ \( 134 + 112 T + 22 T^{2} + T^{3} \)
$83$ \( -80 + 72 T - 16 T^{2} + T^{3} \)
$89$ \( 2 - 2 T - 2 T^{2} + T^{3} \)
$97$ \( -64 - 16 T + 12 T^{2} + T^{3} \)
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