Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 2 q^{5} -\zeta_{6} q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 2 q^{5} -\zeta_{6} q^{7} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} + 2 q^{12} + ( 4 - \zeta_{6} ) q^{13} + ( 2 - 2 \zeta_{6} ) q^{15} + ( -4 + 4 \zeta_{6} ) q^{16} -2 \zeta_{6} q^{17} + 4 \zeta_{6} q^{19} + 4 \zeta_{6} q^{20} - q^{21} + ( -4 + 4 \zeta_{6} ) q^{23} - q^{25} - q^{27} + ( 2 - 2 \zeta_{6} ) q^{28} + ( 8 - 8 \zeta_{6} ) q^{29} -3 q^{31} -\zeta_{6} q^{33} -2 \zeta_{6} q^{35} + ( 2 - 2 \zeta_{6} ) q^{36} + ( -10 + 10 \zeta_{6} ) q^{37} + ( 3 - 4 \zeta_{6} ) q^{39} + ( 8 - 8 \zeta_{6} ) q^{41} + 7 \zeta_{6} q^{43} + 2 q^{44} -2 \zeta_{6} q^{45} + 8 q^{47} + 4 \zeta_{6} q^{48} + ( 6 - 6 \zeta_{6} ) q^{49} -2 q^{51} + ( 2 + 6 \zeta_{6} ) q^{52} -10 q^{53} + ( 2 - 2 \zeta_{6} ) q^{55} + 4 q^{57} -8 \zeta_{6} q^{59} + 4 q^{60} -13 \zeta_{6} q^{61} + ( -1 + \zeta_{6} ) q^{63} -8 q^{64} + ( 8 - 2 \zeta_{6} ) q^{65} + ( -11 + 11 \zeta_{6} ) q^{67} + ( 4 - 4 \zeta_{6} ) q^{68} + 4 \zeta_{6} q^{69} + 6 \zeta_{6} q^{71} -13 q^{73} + ( -1 + \zeta_{6} ) q^{75} + ( -8 + 8 \zeta_{6} ) q^{76} - q^{77} -7 q^{79} + ( -8 + 8 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -6 q^{83} -2 \zeta_{6} q^{84} -4 \zeta_{6} q^{85} -8 \zeta_{6} q^{87} + ( -8 + 8 \zeta_{6} ) q^{89} + ( -1 - 3 \zeta_{6} ) q^{91} -8 q^{92} + ( -3 + 3 \zeta_{6} ) q^{93} + 8 \zeta_{6} q^{95} + 7 \zeta_{6} q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{4} + 4q^{5} - q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{4} + 4q^{5} - q^{7} - q^{9} + q^{11} + 4q^{12} + 7q^{13} + 2q^{15} - 4q^{16} - 2q^{17} + 4q^{19} + 4q^{20} - 2q^{21} - 4q^{23} - 2q^{25} - 2q^{27} + 2q^{28} + 8q^{29} - 6q^{31} - q^{33} - 2q^{35} + 2q^{36} - 10q^{37} + 2q^{39} + 8q^{41} + 7q^{43} + 4q^{44} - 2q^{45} + 16q^{47} + 4q^{48} + 6q^{49} - 4q^{51} + 10q^{52} - 20q^{53} + 2q^{55} + 8q^{57} - 8q^{59} + 8q^{60} - 13q^{61} - q^{63} - 16q^{64} + 14q^{65} - 11q^{67} + 4q^{68} + 4q^{69} + 6q^{71} - 26q^{73} - q^{75} - 8q^{76} - 2q^{77} - 14q^{79} - 8q^{80} - q^{81} - 12q^{83} - 2q^{84} - 4q^{85} - 8q^{87} - 8q^{89} - 5q^{91} - 16q^{92} - 3q^{93} + 8q^{95} + 7q^{97} - 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 1.00000 + 1.73205i 2.00000 0 −0.500000 0.866025i 0 −0.500000 0.866025i 0
133.1 0 0.500000 + 0.866025i 1.00000 1.73205i 2.00000 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.i.a 2
13.c even 3 1 inner 429.2.i.a 2
13.c even 3 1 5577.2.a.e 1
13.e even 6 1 5577.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.i.a 2 1.a even 1 1 trivial
429.2.i.a 2 13.c even 3 1 inner
5577.2.a.b 1 13.e even 6 1
5577.2.a.e 1 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\):

\( T_{2} \)
\( T_{7}^{2} + T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( 13 - 7 T + T^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 16 - 4 T + T^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( 64 - 8 T + T^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( 100 + 10 T + T^{2} \)
$41$ \( 64 - 8 T + T^{2} \)
$43$ \( 49 - 7 T + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( ( 10 + T )^{2} \)
$59$ \( 64 + 8 T + T^{2} \)
$61$ \( 169 + 13 T + T^{2} \)
$67$ \( 121 + 11 T + T^{2} \)
$71$ \( 36 - 6 T + T^{2} \)
$73$ \( ( 13 + T )^{2} \)
$79$ \( ( 7 + T )^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( 64 + 8 T + T^{2} \)
$97$ \( 49 - 7 T + T^{2} \)
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