Properties

Label 429.2.e.b
Level $429$
Weight $2$
Character orbit 429.e
Analytic conductor $3.426$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.2
Defining polynomial: \(x^{8} - 25 x^{4} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 25 x^{4} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{6} \)\(/125\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} - 25 \)\()/25\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} - 5 \nu^{3} + 25 \nu \)\()/25\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + 50 \nu^{2} \)\()/125\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{7} + 5 \nu^{5} + 25 \nu^{3} + 125 \nu \)\()/125\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{5} + 5 \nu^{3} + 25 \nu \)\()/25\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{7} + 5 \nu^{5} - 25 \nu^{3} + 125 \nu \)\()/125\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(5 \beta_{4} + 5 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{6} - 5 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(25 \beta_{2} + 25\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(25 \beta_{7} - 25 \beta_{6} + 25 \beta_{5} - 25 \beta_{3}\)\()/4\)
\(\nu^{6}\)\(=\)\(125 \beta_{1}\)
\(\nu^{7}\)\(=\)\((\)\(125 \beta_{7} + 125 \beta_{6} - 125 \beta_{5} - 125 \beta_{3}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
428.1
−2.15988 + 0.578737i
2.15988 0.578737i
0.578737 2.15988i
−0.578737 + 2.15988i
0.578737 + 2.15988i
−0.578737 2.15988i
−2.15988 0.578737i
2.15988 + 0.578737i
1.00000i 1.73205i 1.00000 −3.16228 −1.73205 −3.46410 3.00000i −3.00000 3.16228i
428.2 1.00000i 1.73205i 1.00000 3.16228 −1.73205 −3.46410 3.00000i −3.00000 3.16228i
428.3 1.00000i 1.73205i 1.00000 −3.16228 1.73205 3.46410 3.00000i −3.00000 3.16228i
428.4 1.00000i 1.73205i 1.00000 3.16228 1.73205 3.46410 3.00000i −3.00000 3.16228i
428.5 1.00000i 1.73205i 1.00000 −3.16228 1.73205 3.46410 3.00000i −3.00000 3.16228i
428.6 1.00000i 1.73205i 1.00000 3.16228 1.73205 3.46410 3.00000i −3.00000 3.16228i
428.7 1.00000i 1.73205i 1.00000 −3.16228 −1.73205 −3.46410 3.00000i −3.00000 3.16228i
428.8 1.00000i 1.73205i 1.00000 3.16228 −1.73205 −3.46410 3.00000i −3.00000 3.16228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 428.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
33.d even 2 1 inner
39.d odd 2 1 inner
143.d odd 2 1 inner
429.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.e.b 8
3.b odd 2 1 inner 429.2.e.b 8
11.b odd 2 1 inner 429.2.e.b 8
13.b even 2 1 inner 429.2.e.b 8
33.d even 2 1 inner 429.2.e.b 8
39.d odd 2 1 inner 429.2.e.b 8
143.d odd 2 1 inner 429.2.e.b 8
429.e even 2 1 inner 429.2.e.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.e.b 8 1.a even 1 1 trivial
429.2.e.b 8 3.b odd 2 1 inner
429.2.e.b 8 11.b odd 2 1 inner
429.2.e.b 8 13.b even 2 1 inner
429.2.e.b 8 33.d even 2 1 inner
429.2.e.b 8 39.d odd 2 1 inner
429.2.e.b 8 143.d odd 2 1 inner
429.2.e.b 8 429.e even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( ( 3 + T^{2} )^{4} \)
$5$ \( ( -10 + T^{2} )^{4} \)
$7$ \( ( -12 + T^{2} )^{4} \)
$11$ \( ( 121 - 18 T^{2} + T^{4} )^{2} \)
$13$ \( ( 169 + 14 T^{2} + T^{4} )^{2} \)
$17$ \( ( -30 + T^{2} )^{4} \)
$19$ \( ( -12 + T^{2} )^{4} \)
$23$ \( ( 48 + T^{2} )^{4} \)
$29$ \( ( -30 + T^{2} )^{4} \)
$31$ \( ( 30 + T^{2} )^{4} \)
$37$ \( T^{8} \)
$41$ \( ( 4 + T^{2} )^{4} \)
$43$ \( ( 90 + T^{2} )^{4} \)
$47$ \( ( -40 + T^{2} )^{4} \)
$53$ \( ( 48 + T^{2} )^{4} \)
$59$ \( ( -40 + T^{2} )^{4} \)
$61$ \( ( 160 + T^{2} )^{4} \)
$67$ \( ( 30 + T^{2} )^{4} \)
$71$ \( ( -40 + T^{2} )^{4} \)
$73$ \( ( -48 + T^{2} )^{4} \)
$79$ \( ( 10 + T^{2} )^{4} \)
$83$ \( ( 4 + T^{2} )^{4} \)
$89$ \( ( -10 + T^{2} )^{4} \)
$97$ \( ( 120 + T^{2} )^{4} \)
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