Properties

Label 429.2.p.a
Level $429$
Weight $2$
Character orbit 429.p
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.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.151613669376.7
Defining polynomial: \(x^{8} + 12 x^{6} + 95 x^{4} + 588 x^{2} + 2401\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} + \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} -2 \beta_{6} q^{5} -\beta_{4} q^{7} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} + \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} -2 \beta_{6} q^{5} -\beta_{4} q^{7} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{9} + ( \beta_{3} - \beta_{7} ) q^{11} + ( -2 + 2 \beta_{6} ) q^{12} -\beta_{4} q^{13} + ( 4 - 4 \beta_{2} + 2 \beta_{3} ) q^{15} + ( -4 + 4 \beta_{2} ) q^{16} + ( 2 \beta_{1} + \beta_{3} - \beta_{6} ) q^{17} + 2 \beta_{4} q^{19} + ( 4 \beta_{3} - 4 \beta_{6} ) q^{20} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{21} + \beta_{3} q^{23} -3 q^{25} + ( -5 - \beta_{6} ) q^{27} + ( -2 \beta_{4} + 2 \beta_{5} ) q^{28} + ( \beta_{3} + 2 \beta_{7} ) q^{29} + q^{31} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{33} + ( 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} ) q^{35} + ( -2 + 2 \beta_{2} - 4 \beta_{3} ) q^{36} + ( -8 + 8 \beta_{2} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{39} + ( 2 \beta_{3} + 4 \beta_{7} ) q^{41} + \beta_{4} q^{43} + ( -2 \beta_{1} - 2 \beta_{3} + 4 \beta_{6} - 2 \beta_{7} ) q^{44} + ( 8 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{45} -7 \beta_{6} q^{47} + ( -4 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{48} + ( 6 - 6 \beta_{2} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{51} + ( -2 \beta_{4} + 2 \beta_{5} ) q^{52} + 3 \beta_{6} q^{53} + ( 6 - 6 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{55} + ( 4 \beta_{1} + 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{57} + ( 3 \beta_{3} - 3 \beta_{6} ) q^{59} + ( 8 + 4 \beta_{6} ) q^{60} -\beta_{4} q^{61} + ( 2 \beta_{3} - \beta_{4} + \beta_{5} + 4 \beta_{7} ) q^{63} -8 q^{64} + ( 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} ) q^{65} + ( 13 - 13 \beta_{2} ) q^{67} + ( -2 \beta_{3} - 4 \beta_{7} ) q^{68} + ( 2 \beta_{2} - \beta_{3} + \beta_{6} ) q^{69} + ( -2 \beta_{3} + 2 \beta_{6} ) q^{71} -3 \beta_{5} q^{73} + ( 3 - 3 \beta_{2} - 3 \beta_{3} ) q^{75} + ( 4 \beta_{4} - 4 \beta_{5} ) q^{76} + ( -3 \beta_{1} - 3 \beta_{3} - 5 \beta_{6} - 3 \beta_{7} ) q^{77} + \beta_{5} q^{79} + 8 \beta_{3} q^{80} + ( 7 - 7 \beta_{2} - 4 \beta_{3} ) q^{81} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{83} + ( -4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{84} + 4 \beta_{4} q^{85} + ( 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{87} -2 \beta_{3} q^{89} + ( 13 - 13 \beta_{2} ) q^{91} + 2 \beta_{6} q^{92} + ( -1 + \beta_{2} + \beta_{3} ) q^{93} + ( -8 \beta_{1} - 4 \beta_{3} + 4 \beta_{6} ) q^{95} + 11 \beta_{2} q^{97} + ( -6 - \beta_{1} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 8q^{4} + 4q^{9} + O(q^{10}) \) \( 8q - 4q^{3} + 8q^{4} + 4q^{9} - 16q^{12} + 16q^{15} - 16q^{16} - 24q^{25} - 40q^{27} + 8q^{31} + 12q^{33} - 8q^{36} - 32q^{37} + 32q^{45} - 16q^{48} + 24q^{49} + 24q^{55} + 64q^{60} - 64q^{64} + 52q^{67} + 8q^{69} + 12q^{75} + 28q^{81} + 52q^{91} - 4q^{93} + 44q^{97} - 48q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 12 x^{6} + 95 x^{4} + 588 x^{2} + 2401\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 12 \nu^{6} + 95 \nu^{4} + 1140 \nu^{2} + 7056 \)\()/4655\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{7} + 475 \nu^{5} + 1045 \nu^{3} + 6468 \nu \)\()/32585\)
\(\beta_{4}\)\(=\)\((\)\( 23 \nu^{6} + 570 \nu^{4} + 2185 \nu^{2} + 13524 \)\()/4655\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} - 18 \)\()/95\)
\(\beta_{6}\)\(=\)\((\)\( 12 \nu^{7} + 95 \nu^{5} + 209 \nu^{3} + 2401 \nu \)\()/6517\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - 12 \nu^{5} - 95 \nu^{3} - 588 \nu \)\()/343\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + 6 \beta_{2} - 6\)
\(\nu^{3}\)\(=\)\(-5 \beta_{7} - 7 \beta_{6} - 5 \beta_{3} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(12 \beta_{4} - 23 \beta_{2}\)
\(\nu^{5}\)\(=\)\(11 \beta_{7} + 95 \beta_{3}\)
\(\nu^{6}\)\(=\)\(-95 \beta_{5} - 18\)
\(\nu^{7}\)\(=\)\(665 \beta_{6} - 665 \beta_{3} - 113 \beta_{1}\)

Character values

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

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
230.1
−0.662382 2.56149i
1.88713 + 1.85439i
0.662382 + 2.56149i
−1.88713 1.85439i
−0.662382 + 2.56149i
1.88713 1.85439i
0.662382 2.56149i
−1.88713 + 1.85439i
0 −1.72474 + 0.158919i 1.00000 + 1.73205i 2.82843i 0 −3.12250 + 1.80278i 0 2.94949 0.548188i 0
230.2 0 −1.72474 + 0.158919i 1.00000 + 1.73205i 2.82843i 0 3.12250 1.80278i 0 2.94949 0.548188i 0
230.3 0 0.724745 + 1.57313i 1.00000 + 1.73205i 2.82843i 0 −3.12250 + 1.80278i 0 −1.94949 + 2.28024i 0
230.4 0 0.724745 + 1.57313i 1.00000 + 1.73205i 2.82843i 0 3.12250 1.80278i 0 −1.94949 + 2.28024i 0
263.1 0 −1.72474 0.158919i 1.00000 1.73205i 2.82843i 0 −3.12250 1.80278i 0 2.94949 + 0.548188i 0
263.2 0 −1.72474 0.158919i 1.00000 1.73205i 2.82843i 0 3.12250 + 1.80278i 0 2.94949 + 0.548188i 0
263.3 0 0.724745 1.57313i 1.00000 1.73205i 2.82843i 0 −3.12250 1.80278i 0 −1.94949 2.28024i 0
263.4 0 0.724745 1.57313i 1.00000 1.73205i 2.82843i 0 3.12250 + 1.80278i 0 −1.94949 2.28024i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.4
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.c even 3 1 inner
33.d even 2 1 inner
39.i odd 6 1 inner
143.k odd 6 1 inner
429.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.p.a 8
3.b odd 2 1 inner 429.2.p.a 8
11.b odd 2 1 inner 429.2.p.a 8
13.c even 3 1 inner 429.2.p.a 8
33.d even 2 1 inner 429.2.p.a 8
39.i odd 6 1 inner 429.2.p.a 8
143.k odd 6 1 inner 429.2.p.a 8
429.p even 6 1 inner 429.2.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.p.a 8 1.a even 1 1 trivial
429.2.p.a 8 3.b odd 2 1 inner
429.2.p.a 8 11.b odd 2 1 inner
429.2.p.a 8 13.c even 3 1 inner
429.2.p.a 8 33.d even 2 1 inner
429.2.p.a 8 39.i odd 6 1 inner
429.2.p.a 8 143.k odd 6 1 inner
429.2.p.a 8 429.p even 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 9 + 6 T + T^{2} + 2 T^{3} + T^{4} )^{2} \)
$5$ \( ( 8 + T^{2} )^{4} \)
$7$ \( ( 169 - 13 T^{2} + T^{4} )^{2} \)
$11$ \( 14641 + 484 T^{2} - 105 T^{4} + 4 T^{6} + T^{8} \)
$13$ \( ( 169 - 13 T^{2} + T^{4} )^{2} \)
$17$ \( ( 676 + 26 T^{2} + T^{4} )^{2} \)
$19$ \( ( 2704 - 52 T^{2} + T^{4} )^{2} \)
$23$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$29$ \( ( 676 + 26 T^{2} + T^{4} )^{2} \)
$31$ \( ( -1 + T )^{8} \)
$37$ \( ( 64 + 8 T + T^{2} )^{4} \)
$41$ \( ( 10816 + 104 T^{2} + T^{4} )^{2} \)
$43$ \( ( 169 - 13 T^{2} + T^{4} )^{2} \)
$47$ \( ( 98 + T^{2} )^{4} \)
$53$ \( ( 18 + T^{2} )^{4} \)
$59$ \( ( 324 - 18 T^{2} + T^{4} )^{2} \)
$61$ \( ( 169 - 13 T^{2} + T^{4} )^{2} \)
$67$ \( ( 169 - 13 T + T^{2} )^{4} \)
$71$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$73$ \( ( 117 + T^{2} )^{4} \)
$79$ \( ( 13 + T^{2} )^{4} \)
$83$ \( ( -26 + T^{2} )^{4} \)
$89$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$97$ \( ( 121 - 11 T + T^{2} )^{4} \)
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