Properties

Label 429.2.a.c
Level $429$
Weight $2$
Character orbit 429.a
Self dual yes
Analytic conductor $3.426$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
−2.41421 1.00000 3.82843 −3.41421 −2.41421 0.828427 −4.41421 1.00000 8.24264
1.2 0.414214 1.00000 −1.82843 −0.585786 0.414214 −4.82843 −1.58579 1.00000 −0.242641
\(n\): 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.c 2
3.b odd 2 1 1287.2.a.g 2
4.b odd 2 1 6864.2.a.bc 2
11.b odd 2 1 4719.2.a.o 2
13.b even 2 1 5577.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.c 2 1.a even 1 1 trivial
1287.2.a.g 2 3.b odd 2 1
4719.2.a.o 2 11.b odd 2 1
5577.2.a.i 2 13.b even 2 1
6864.2.a.bc 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(429))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 2 + 4 T + T^{2} \)
$7$ \( -4 + 4 T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 14 + 8 T + T^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( -4 - 4 T + T^{2} \)
$29$ \( -18 + T^{2} \)
$31$ \( -18 + T^{2} \)
$37$ \( -28 - 4 T + T^{2} \)
$41$ \( ( 12 + T )^{2} \)
$43$ \( -46 + 4 T + T^{2} \)
$47$ \( -72 + T^{2} \)
$53$ \( -124 - 4 T + T^{2} \)
$59$ \( 56 + 16 T + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( -34 - 8 T + T^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( -56 - 8 T + T^{2} \)
$79$ \( -46 + 4 T + T^{2} \)
$83$ \( -128 + T^{2} \)
$89$ \( 194 + 28 T + T^{2} \)
$97$ \( ( -10 + T )^{2} \)
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