Properties

Label 429.2.e.a
Level $429$
Weight $2$
Character orbit 429.e
Analytic conductor $3.426$
Analytic rank $0$
Dimension $8$
CM discriminant -39
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: 8.0.151613669376.21
Defining polynomial: \(x^{8} + 5 x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} -\beta_{2} q^{3} + ( -2 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{6} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{8} + 3 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} -\beta_{2} q^{3} + ( -2 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{6} + ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{8} + 3 q^{9} + ( 1 - \beta_{2} + 2 \beta_{5} - 2 \beta_{7} ) q^{10} + ( -\beta_{1} + \beta_{3} + \beta_{4} ) q^{11} + ( 3 + 2 \beta_{2} ) q^{12} + \beta_{5} q^{13} + ( -\beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{15} + ( -1 + 2 \beta_{2} ) q^{16} + 3 \beta_{3} q^{18} + ( -3 \beta_{1} - 2 \beta_{4} + 3 \beta_{6} ) q^{20} + ( -4 - 2 \beta_{2} + \beta_{7} ) q^{22} + 3 \beta_{3} q^{24} + ( 5 + 4 \beta_{2} ) q^{25} + ( -\beta_{1} - 3 \beta_{4} + \beta_{6} ) q^{26} -3 \beta_{2} q^{27} + ( -1 + \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{30} -\beta_{3} q^{32} + ( -2 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{33} + ( -6 - 3 \beta_{2} ) q^{36} + ( -1 + \beta_{2} - \beta_{5} + 2 \beta_{7} ) q^{39} + ( -1 + \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{40} + ( -\beta_{1} - 4 \beta_{3} - \beta_{6} ) q^{41} + ( 2 - 2 \beta_{2} + 2 \beta_{5} - 4 \beta_{7} ) q^{43} + ( 2 \beta_{1} - 4 \beta_{3} + \beta_{6} ) q^{44} + ( 3 \beta_{1} - 3 \beta_{6} ) q^{45} + ( -3 \beta_{1} + 2 \beta_{4} + 3 \beta_{6} ) q^{47} + ( -6 + \beta_{2} ) q^{48} -7 q^{49} + ( -4 \beta_{1} + 9 \beta_{3} - 4 \beta_{6} ) q^{50} + ( -1 + \beta_{2} - 3 \beta_{5} + 2 \beta_{7} ) q^{52} + ( 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{6} ) q^{54} + ( -3 + 3 \beta_{5} - 2 \beta_{7} ) q^{55} + ( 3 \beta_{1} + 2 \beta_{4} - 3 \beta_{6} ) q^{59} + ( 5 \beta_{1} + 4 \beta_{4} - 5 \beta_{6} ) q^{60} + 2 \beta_{5} q^{61} + ( 2 + 5 \beta_{2} ) q^{64} + ( -3 \beta_{1} + 4 \beta_{3} - 3 \beta_{6} ) q^{65} + ( 7 + 4 \beta_{2} + \beta_{5} + \beta_{7} ) q^{66} + ( \beta_{1} - 6 \beta_{4} - \beta_{6} ) q^{71} + ( 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{6} ) q^{72} + ( -12 - 5 \beta_{2} ) q^{75} + ( 4 \beta_{1} - \beta_{4} - 4 \beta_{6} ) q^{78} -4 \beta_{5} q^{79} + ( \beta_{1} + 4 \beta_{4} - \beta_{6} ) q^{80} + 9 q^{81} + ( 17 + \beta_{2} ) q^{82} + ( -\beta_{1} - 6 \beta_{3} - \beta_{6} ) q^{83} + ( -8 \beta_{1} + 2 \beta_{4} + 8 \beta_{6} ) q^{86} + ( 7 + 4 \beta_{2} + \beta_{5} + \beta_{7} ) q^{88} + ( 3 \beta_{1} + 4 \beta_{4} - 3 \beta_{6} ) q^{89} + ( 3 - 3 \beta_{2} + 6 \beta_{5} - 6 \beta_{7} ) q^{90} + ( -3 + 3 \beta_{2} - 4 \beta_{5} + 6 \beta_{7} ) q^{94} + ( -\beta_{1} + \beta_{3} - \beta_{6} ) q^{96} -7 \beta_{3} q^{98} + ( -3 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 16q^{4} + 24q^{9} + O(q^{10}) \) \( 8q - 16q^{4} + 24q^{9} + 24q^{12} - 8q^{16} - 28q^{22} + 40q^{25} - 48q^{36} - 48q^{48} - 56q^{49} - 32q^{55} + 16q^{64} + 60q^{66} - 96q^{75} + 72q^{81} + 136q^{82} + 60q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5 x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 3 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + \nu^{2} \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{3} + 8 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 5 \nu^{3} + 8 \nu \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} + 9 \nu^{2} \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{3} \)\()/8\)
\(\beta_{7}\)\(=\)\( \nu^{4} + \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{6} - \beta_{4} + \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{7} - \beta_{5} + \beta_{2} - 6\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-3 \beta_{4} - 3 \beta_{3} + 4 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-\beta_{5} + 9 \beta_{2}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-10 \beta_{6} - 3 \beta_{4} + 3 \beta_{3}\)\()/2\)

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
−0.752986 1.19709i
0.752986 1.19709i
1.19709 0.752986i
−1.19709 0.752986i
1.19709 + 0.752986i
−1.19709 + 0.752986i
−0.752986 + 1.19709i
0.752986 + 1.19709i
2.39417i −1.73205 −3.73205 −4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
428.2 2.39417i −1.73205 −3.73205 4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
428.3 1.50597i 1.73205 −0.267949 −1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.4 1.50597i 1.73205 −0.267949 1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.5 1.50597i 1.73205 −0.267949 −1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.6 1.50597i 1.73205 −0.267949 1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.7 2.39417i −1.73205 −3.73205 −4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
428.8 2.39417i −1.73205 −3.73205 4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
\(n\): 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
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
33.d even 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.a 8
3.b odd 2 1 inner 429.2.e.a 8
11.b odd 2 1 inner 429.2.e.a 8
13.b even 2 1 inner 429.2.e.a 8
33.d even 2 1 inner 429.2.e.a 8
39.d odd 2 1 CM 429.2.e.a 8
143.d odd 2 1 inner 429.2.e.a 8
429.e even 2 1 inner 429.2.e.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.e.a 8 1.a even 1 1 trivial
429.2.e.a 8 3.b odd 2 1 inner
429.2.e.a 8 11.b odd 2 1 inner
429.2.e.a 8 13.b even 2 1 inner
429.2.e.a 8 33.d even 2 1 inner
429.2.e.a 8 39.d odd 2 1 CM
429.2.e.a 8 143.d odd 2 1 inner
429.2.e.a 8 429.e even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 8 T_{2}^{2} + 13 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 13 + 8 T^{2} + T^{4} )^{2} \)
$3$ \( ( -3 + T^{2} )^{4} \)
$5$ \( ( 52 - 20 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( 14641 - 190 T^{4} + T^{8} \)
$13$ \( ( 13 + T^{2} )^{4} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( ( 6292 + 164 T^{2} + T^{4} )^{2} \)
$43$ \( ( 156 + T^{2} )^{4} \)
$47$ \( ( 8788 - 188 T^{2} + T^{4} )^{2} \)
$53$ \( T^{8} \)
$59$ \( ( 52 - 236 T^{2} + T^{4} )^{2} \)
$61$ \( ( 52 + T^{2} )^{4} \)
$67$ \( T^{8} \)
$71$ \( ( 6292 - 284 T^{2} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( ( 208 + T^{2} )^{4} \)
$83$ \( ( 27508 + 332 T^{2} + T^{4} )^{2} \)
$89$ \( ( 6292 - 356 T^{2} + T^{4} )^{2} \)
$97$ \( T^{8} \)
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