Properties

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

Related objects

Downloads

Learn more about

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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 2 q^{5} + 3 \zeta_{6} q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 2 q^{5} + 3 \zeta_{6} q^{7} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} + 2 q^{12} + ( -4 + 3 \zeta_{6} ) q^{13} + ( 2 - 2 \zeta_{6} ) q^{15} + ( -4 + 4 \zeta_{6} ) q^{16} -2 \zeta_{6} q^{17} -4 \zeta_{6} q^{19} + 4 \zeta_{6} q^{20} + 3 q^{21} + ( 4 - 4 \zeta_{6} ) q^{23} - q^{25} - q^{27} + ( -6 + 6 \zeta_{6} ) q^{28} + 5 q^{31} + \zeta_{6} q^{33} + 6 \zeta_{6} q^{35} + ( 2 - 2 \zeta_{6} ) q^{36} + ( 6 - 6 \zeta_{6} ) q^{37} + ( -1 + 4 \zeta_{6} ) q^{39} + ( 8 - 8 \zeta_{6} ) q^{41} -5 \zeta_{6} q^{43} -2 q^{44} -2 \zeta_{6} q^{45} + 4 \zeta_{6} q^{48} + ( -2 + 2 \zeta_{6} ) q^{49} -2 q^{51} + ( -6 - 2 \zeta_{6} ) q^{52} + 14 q^{53} + ( -2 + 2 \zeta_{6} ) q^{55} -4 q^{57} + 4 q^{60} + 7 \zeta_{6} q^{61} + ( 3 - 3 \zeta_{6} ) q^{63} -8 q^{64} + ( -8 + 6 \zeta_{6} ) q^{65} + ( -3 + 3 \zeta_{6} ) q^{67} + ( 4 - 4 \zeta_{6} ) q^{68} -4 \zeta_{6} q^{69} -10 \zeta_{6} q^{71} -9 q^{73} + ( -1 + \zeta_{6} ) q^{75} + ( 8 - 8 \zeta_{6} ) q^{76} -3 q^{77} -11 q^{79} + ( -8 + 8 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -6 q^{83} + 6 \zeta_{6} q^{84} -4 \zeta_{6} q^{85} + ( 8 - 8 \zeta_{6} ) q^{89} + ( -9 - 3 \zeta_{6} ) q^{91} + 8 q^{92} + ( 5 - 5 \zeta_{6} ) q^{93} -8 \zeta_{6} q^{95} + 7 \zeta_{6} q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{4} + 4q^{5} + 3q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{4} + 4q^{5} + 3q^{7} - q^{9} - q^{11} + 4q^{12} - 5q^{13} + 2q^{15} - 4q^{16} - 2q^{17} - 4q^{19} + 4q^{20} + 6q^{21} + 4q^{23} - 2q^{25} - 2q^{27} - 6q^{28} + 10q^{31} + q^{33} + 6q^{35} + 2q^{36} + 6q^{37} + 2q^{39} + 8q^{41} - 5q^{43} - 4q^{44} - 2q^{45} + 4q^{48} - 2q^{49} - 4q^{51} - 14q^{52} + 28q^{53} - 2q^{55} - 8q^{57} + 8q^{60} + 7q^{61} + 3q^{63} - 16q^{64} - 10q^{65} - 3q^{67} + 4q^{68} - 4q^{69} - 10q^{71} - 18q^{73} - q^{75} + 8q^{76} - 6q^{77} - 22q^{79} - 8q^{80} - q^{81} - 12q^{83} + 6q^{84} - 4q^{85} + 8q^{89} - 21q^{91} + 16q^{92} + 5q^{93} - 8q^{95} + 7q^{97} + 2q^{99} + O(q^{100}) \)

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 1.50000 + 2.59808i 0 −0.500000 0.866025i 0
133.1 0 0.500000 + 0.866025i 1.00000 1.73205i 2.00000 0 1.50000 2.59808i 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.b 2
13.c even 3 1 inner 429.2.i.b 2
13.c even 3 1 5577.2.a.d 1
13.e even 6 1 5577.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.i.b 2 1.a even 1 1 trivial
429.2.i.b 2 13.c even 3 1 inner
5577.2.a.c 1 13.e even 6 1
5577.2.a.d 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 \(S_{2}^{\mathrm{new}}(429, [\chi])\):

\( T_{2} \)
\( T_{7}^{2} - 3 T_{7} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( 9 - 3 T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( 13 + 5 T + T^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 16 + 4 T + T^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 36 - 6 T + T^{2} \)
$41$ \( 64 - 8 T + T^{2} \)
$43$ \( 25 + 5 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -14 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 49 - 7 T + T^{2} \)
$67$ \( 9 + 3 T + T^{2} \)
$71$ \( 100 + 10 T + T^{2} \)
$73$ \( ( 9 + T )^{2} \)
$79$ \( ( 11 + T )^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( 64 - 8 T + T^{2} \)
$97$ \( 49 - 7 T + T^{2} \)
show more
show less