Properties

Label 429.2.a.f
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.

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

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
−1.48119
0.311108
2.17009
−1.48119 1.00000 0.193937 −1.67513 −1.48119 −4.15633 2.67513 1.00000 2.48119
1.2 0.311108 1.00000 −1.90321 2.21432 0.311108 1.52543 −1.21432 1.00000 0.688892
1.3 2.17009 1.00000 2.70928 −0.539189 2.17009 0.630898 1.53919 1.00000 −1.17009
\(n\): 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.f 3
3.b odd 2 1 1287.2.a.i 3
4.b odd 2 1 6864.2.a.bp 3
11.b odd 2 1 4719.2.a.t 3
13.b even 2 1 5577.2.a.k 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.f 3 1.a even 1 1 trivial
1287.2.a.i 3 3.b odd 2 1
4719.2.a.t 3 11.b odd 2 1
5577.2.a.k 3 13.b even 2 1
6864.2.a.bp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(429))\):

\( T_{2}^{3} - T_{2}^{2} - 3 T_{2} + 1 \)
\( T_{5}^{3} - 4 T_{5} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T - T^{2} + T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( -2 - 4 T + T^{3} \)
$7$ \( 4 - 8 T + 2 T^{2} + T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( 26 - 2 T - 8 T^{2} + T^{3} \)
$19$ \( 100 - 16 T - 6 T^{2} + T^{3} \)
$23$ \( -4 + 12 T - 8 T^{2} + T^{3} \)
$29$ \( -10 - 14 T - 2 T^{2} + T^{3} \)
$31$ \( 2 + 8 T + 6 T^{2} + T^{3} \)
$37$ \( -16 + 32 T + 12 T^{2} + T^{3} \)
$41$ \( 100 + 20 T - 12 T^{2} + T^{3} \)
$43$ \( 74 - 74 T - 2 T^{2} + T^{3} \)
$47$ \( 104 - 36 T - 2 T^{2} + T^{3} \)
$53$ \( 296 - 148 T + 2 T^{2} + T^{3} \)
$59$ \( -80 - 64 T - 8 T^{2} + T^{3} \)
$61$ \( -32 - 16 T + 4 T^{2} + T^{3} \)
$67$ \( 74 - 36 T - 2 T^{2} + T^{3} \)
$71$ \( 184 - 52 T - 2 T^{2} + T^{3} \)
$73$ \( -412 - 84 T + 4 T^{2} + T^{3} \)
$79$ \( -10 + 42 T - 14 T^{2} + T^{3} \)
$83$ \( -16 - 8 T + 4 T^{2} + T^{3} \)
$89$ \( -1690 - 168 T + 10 T^{2} + T^{3} \)
$97$ \( -16 + 32 T + 12 T^{2} + T^{3} \)
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