Properties

Label 429.2.a.h
Level $429$
Weight $2$
Character orbit 429.a
Self dual yes
Analytic conductor $3.426$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.8468.1
Defining polynomial: \(x^{4} - x^{3} - 5 x^{2} + 3 x + 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

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

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
1.31743
−0.704624
2.27841
−1.89122
−2.58181 −1.00000 4.66573 −2.71878 2.58181 −4.30059 −6.88240 1.00000 7.01937
1.2 −1.79888 −1.00000 1.23597 3.97216 1.79888 3.17328 1.37440 1.00000 −7.14544
1.3 −0.0872450 −1.00000 −1.99239 −0.477194 0.0872450 0.435561 0.348316 1.00000 0.0416328
1.4 2.46793 −1.00000 4.09069 −0.776183 −2.46793 2.69175 5.15968 1.00000 −1.91557
\(n\): 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.h 4
3.b odd 2 1 1287.2.a.m 4
4.b odd 2 1 6864.2.a.bz 4
11.b odd 2 1 4719.2.a.z 4
13.b even 2 1 5577.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.h 4 1.a even 1 1 trivial
1287.2.a.m 4 3.b odd 2 1
4719.2.a.z 4 11.b odd 2 1
5577.2.a.m 4 13.b even 2 1
6864.2.a.bz 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(429))\):

\( T_{2}^{4} + 2 T_{2}^{3} - 6 T_{2}^{2} - 12 T_{2} - 1 \)
\( T_{5}^{4} - 12 T_{5}^{2} - 14 T_{5} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 12 T - 6 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( -4 - 14 T - 12 T^{2} + T^{4} \)
$7$ \( -16 + 44 T - 16 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( 412 - 162 T - 26 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( -64 - 180 T + 104 T^{2} - 18 T^{3} + T^{4} \)
$23$ \( -128 + 148 T - 44 T^{2} + T^{4} \)
$29$ \( -116 - 382 T - 30 T^{2} + 10 T^{3} + T^{4} \)
$31$ \( -968 + 458 T - 20 T^{2} - 12 T^{3} + T^{4} \)
$37$ \( 32 + 64 T - 40 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( -88 + 140 T - 52 T^{2} - 2 T^{3} + T^{4} \)
$43$ \( -4664 + 270 T + 202 T^{2} - 28 T^{3} + T^{4} \)
$47$ \( 128 - 136 T - 84 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( 16 - 224 T - 80 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( -11456 + 2704 T - 96 T^{2} - 16 T^{3} + T^{4} \)
$61$ \( -2816 - 1088 T - 80 T^{2} + 10 T^{3} + T^{4} \)
$67$ \( 1648 + 18 T - 88 T^{2} + T^{4} \)
$71$ \( 128 + 136 T - 76 T^{2} - 10 T^{3} + T^{4} \)
$73$ \( -88 + 292 T - 108 T^{2} + 6 T^{3} + T^{4} \)
$79$ \( 4016 - 1382 T - 250 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( 3392 - 624 T - 120 T^{2} + 8 T^{3} + T^{4} \)
$89$ \( -44 + 58 T - 8 T^{2} - 6 T^{3} + T^{4} \)
$97$ \( -12832 + 3712 T - 248 T^{2} - 10 T^{3} + T^{4} \)
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