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Properties

Label 429.2.s.a
Level $429$
Weight $2$
Character orbit 429.s
Analytic conductor $3.426$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [429,2,Mod(166,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.166");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.s (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{3} + 14 q^{4} + 6 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{3} + 14 q^{4} + 6 q^{7} - 12 q^{9} + 28 q^{12} - 4 q^{13} + 20 q^{14} - 6 q^{15} - 14 q^{16} + 10 q^{17} - 18 q^{20} + 2 q^{22} - 14 q^{23} + 4 q^{25} - 34 q^{26} - 24 q^{27} - 30 q^{28} + 4 q^{29} + 30 q^{32} + 6 q^{35} + 14 q^{36} + 12 q^{38} - 2 q^{39} + 20 q^{40} - 30 q^{41} + 10 q^{42} - 4 q^{43} - 6 q^{45} - 24 q^{46} + 14 q^{48} + 18 q^{49} - 84 q^{50} + 20 q^{51} + 40 q^{52} - 56 q^{53} - 4 q^{55} + 26 q^{56} + 48 q^{58} + 60 q^{59} - 2 q^{61} + 18 q^{62} - 6 q^{63} - 48 q^{64} - 10 q^{65} + 4 q^{66} - 42 q^{67} - 18 q^{68} + 14 q^{69} + 6 q^{71} + 2 q^{75} - 48 q^{76} + 24 q^{77} - 26 q^{78} - 20 q^{79} + 30 q^{80} - 12 q^{81} - 10 q^{82} - 30 q^{84} + 6 q^{85} - 4 q^{87} - 12 q^{88} + 12 q^{89} + 18 q^{91} + 8 q^{92} + 12 q^{93} - 22 q^{94} + 4 q^{95} + 6 q^{97} - 114 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
166.1 −2.37717 + 1.37246i 0.500000 + 0.866025i 2.76728 − 4.79308i 0.261284i −2.37717 − 1.37246i −3.66637 − 2.11678i 9.70209i −0.500000 + 0.866025i −0.358601 − 0.621115i
166.2 −1.91936 + 1.10815i 0.500000 + 0.866025i 1.45597 − 2.52182i − 0.692237i −1.91936 − 1.10815i 2.81071 + 1.62277i 2.02113i −0.500000 + 0.866025i 0.767100 + 1.32866i
166.3 −1.70879 + 0.986569i 0.500000 + 0.866025i 0.946639 − 1.63963i − 1.03130i −1.70879 − 0.986569i −0.142489 − 0.0822660i − 0.210579i −0.500000 + 0.866025i 1.01745 + 1.76227i
166.4 −1.00815 + 0.582055i 0.500000 + 0.866025i −0.322425 + 0.558456i − 1.30774i −1.00815 − 0.582055i −2.07195 − 1.19624i − 3.07889i −0.500000 + 0.866025i 0.761176 + 1.31840i
166.5 −0.761768 + 0.439807i 0.500000 + 0.866025i −0.613140 + 1.06199i 3.40975i −0.761768 − 0.439807i 4.33112 + 2.50057i − 2.83788i −0.500000 + 0.866025i −1.49963 − 2.59744i
166.6 −0.529161 + 0.305511i 0.500000 + 0.866025i −0.813326 + 1.40872i 2.30641i −0.529161 − 0.305511i −2.90273 − 1.67589i − 2.21596i −0.500000 + 0.866025i −0.704633 − 1.22046i
166.7 0.219134 − 0.126517i 0.500000 + 0.866025i −0.967987 + 1.67660i 0.770885i 0.219134 + 0.126517i 2.38762 + 1.37849i 0.995935i −0.500000 + 0.866025i 0.0975301 + 0.168927i
166.8 0.884735 − 0.510802i 0.500000 + 0.866025i −0.478163 + 0.828202i 0.892170i 0.884735 + 0.510802i −0.417948 − 0.241303i 3.02019i −0.500000 + 0.866025i 0.455722 + 0.789334i
166.9 1.30176 − 0.751572i 0.500000 + 0.866025i 0.129721 − 0.224683i − 4.13566i 1.30176 + 0.751572i 3.62040 + 2.09024i 2.61631i −0.500000 + 0.866025i −3.10825 − 5.38364i
166.10 1.56256 − 0.902143i 0.500000 + 0.866025i 0.627724 − 1.08725i 3.28340i 1.56256 + 0.902143i −1.35933 − 0.784811i 1.34338i −0.500000 + 0.866025i 2.96210 + 5.13050i
166.11 2.15823 − 1.24606i 0.500000 + 0.866025i 2.10532 − 3.64652i 1.90483i 2.15823 + 1.24606i 1.37703 + 0.795028i − 5.50915i −0.500000 + 0.866025i 2.37353 + 4.11108i
166.12 2.17798 − 1.25745i 0.500000 + 0.866025i 2.16238 − 3.74536i − 2.19769i 2.17798 + 1.25745i −0.966057 − 0.557753i − 5.84658i −0.500000 + 0.866025i −2.76349 − 4.78651i
199.1 −2.37717 − 1.37246i 0.500000 − 0.866025i 2.76728 + 4.79308i − 0.261284i −2.37717 + 1.37246i −3.66637 + 2.11678i − 9.70209i −0.500000 − 0.866025i −0.358601 + 0.621115i
199.2 −1.91936 − 1.10815i 0.500000 − 0.866025i 1.45597 + 2.52182i 0.692237i −1.91936 + 1.10815i 2.81071 − 1.62277i − 2.02113i −0.500000 − 0.866025i 0.767100 − 1.32866i
199.3 −1.70879 − 0.986569i 0.500000 − 0.866025i 0.946639 + 1.63963i 1.03130i −1.70879 + 0.986569i −0.142489 + 0.0822660i 0.210579i −0.500000 − 0.866025i 1.01745 − 1.76227i
199.4 −1.00815 − 0.582055i 0.500000 − 0.866025i −0.322425 − 0.558456i 1.30774i −1.00815 + 0.582055i −2.07195 + 1.19624i 3.07889i −0.500000 − 0.866025i 0.761176 − 1.31840i
199.5 −0.761768 − 0.439807i 0.500000 − 0.866025i −0.613140 − 1.06199i − 3.40975i −0.761768 + 0.439807i 4.33112 − 2.50057i 2.83788i −0.500000 − 0.866025i −1.49963 + 2.59744i
199.6 −0.529161 − 0.305511i 0.500000 − 0.866025i −0.813326 − 1.40872i − 2.30641i −0.529161 + 0.305511i −2.90273 + 1.67589i 2.21596i −0.500000 − 0.866025i −0.704633 + 1.22046i
199.7 0.219134 + 0.126517i 0.500000 − 0.866025i −0.967987 − 1.67660i − 0.770885i 0.219134 − 0.126517i 2.38762 − 1.37849i − 0.995935i −0.500000 − 0.866025i 0.0975301 − 0.168927i
199.8 0.884735 + 0.510802i 0.500000 − 0.866025i −0.478163 − 0.828202i − 0.892170i 0.884735 − 0.510802i −0.417948 + 0.241303i − 3.02019i −0.500000 − 0.866025i 0.455722 − 0.789334i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 166.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.s.a ✓ 24
13.e even 6 1 inner 429.2.s.a ✓ 24
13.f odd 12 1 5577.2.a.z 12
13.f odd 12 1 5577.2.a.be 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.s.a ✓ 24 1.a even 1 1 trivial
429.2.s.a ✓ 24 13.e even 6 1 inner
5577.2.a.z 12 13.f odd 12 1
5577.2.a.be 12 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 19 T_{2}^{22} + 229 T_{2}^{20} - 6 T_{2}^{19} - 1682 T_{2}^{18} + 54 T_{2}^{17} + \cdots + 1089 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display