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Properties

Label 429.2.bd.a
Level $429$
Weight $2$
Character orbit 429.bd
Analytic conductor $3.426$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [429,2,Mod(76,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bd (of order \(12\), degree \(4\), 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: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 28 q^{3} + 4 q^{5} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 28 q^{3} + 4 q^{5} - 28 q^{9} - 12 q^{11} + 24 q^{14} - 8 q^{15} + 68 q^{16} - 8 q^{20} - 10 q^{22} - 12 q^{23} - 48 q^{26} + 56 q^{27} + 4 q^{31} + 12 q^{33} + 32 q^{34} + 36 q^{37} - 12 q^{42} + 24 q^{44} + 4 q^{45} - 8 q^{47} + 68 q^{48} - 24 q^{49} - 56 q^{53} - 2 q^{55} + 72 q^{56} - 88 q^{58} + 56 q^{59} - 44 q^{60} + 20 q^{66} + 92 q^{67} + 12 q^{69} - 216 q^{70} + 72 q^{71} - 12 q^{75} - 12 q^{78} + 44 q^{80} - 28 q^{81} - 156 q^{82} - 120 q^{86} - 6 q^{88} - 56 q^{89} + 92 q^{91} + 32 q^{92} - 20 q^{93} - 44 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −2.55489 − 0.684581i −0.500000 − 0.866025i 4.32676 + 2.49806i −2.81573 − 2.81573i 0.684581 + 2.55489i −2.71459 + 0.727372i −5.60366 − 5.60366i −0.500000 + 0.866025i 5.26630 + 9.12149i
76.2 −2.52014 − 0.675270i −0.500000 − 0.866025i 4.16307 + 2.40355i 0.961072 + 0.961072i 0.675270 + 2.52014i 1.60148 − 0.429116i −5.17874 − 5.17874i −0.500000 + 0.866025i −1.77305 − 3.07102i
76.3 −2.00245 − 0.536555i −0.500000 − 0.866025i 1.98987 + 1.14885i 1.79777 + 1.79777i 0.536555 + 2.00245i −3.46337 + 0.928006i −0.436407 − 0.436407i −0.500000 + 0.866025i −2.63535 − 4.56456i
76.4 −1.43745 − 0.385163i −0.500000 − 0.866025i 0.185859 + 0.107306i −1.55279 − 1.55279i 0.385163 + 1.43745i −2.05994 + 0.551959i 1.87874 + 1.87874i −0.500000 + 0.866025i 1.63398 + 2.83014i
76.5 −1.42470 − 0.381747i −0.500000 − 0.866025i 0.151984 + 0.0877479i −0.971227 − 0.971227i 0.381747 + 1.42470i 4.17142 − 1.11773i 1.90287 + 1.90287i −0.500000 + 0.866025i 1.01294 + 1.75447i
76.6 −0.567668 − 0.152106i −0.500000 − 0.866025i −1.43294 − 0.827308i 0.319197 + 0.319197i 0.152106 + 0.567668i 2.56095 − 0.686204i 1.51872 + 1.51872i −0.500000 + 0.866025i −0.132646 − 0.229750i
76.7 −0.390674 − 0.104681i −0.500000 − 0.866025i −1.59038 − 0.918208i 1.89569 + 1.89569i 0.104681 + 0.390674i −1.61789 + 0.433512i 1.09719 + 1.09719i −0.500000 + 0.866025i −0.542154 − 0.939038i
76.8 0.390674 + 0.104681i −0.500000 − 0.866025i −1.59038 − 0.918208i 1.89569 + 1.89569i −0.104681 − 0.390674i 1.61789 − 0.433512i −1.09719 − 1.09719i −0.500000 + 0.866025i 0.542154 + 0.939038i
76.9 0.567668 + 0.152106i −0.500000 − 0.866025i −1.43294 − 0.827308i 0.319197 + 0.319197i −0.152106 − 0.567668i −2.56095 + 0.686204i −1.51872 − 1.51872i −0.500000 + 0.866025i 0.132646 + 0.229750i
76.10 1.42470 + 0.381747i −0.500000 − 0.866025i 0.151984 + 0.0877479i −0.971227 − 0.971227i −0.381747 − 1.42470i −4.17142 + 1.11773i −1.90287 − 1.90287i −0.500000 + 0.866025i −1.01294 − 1.75447i
76.11 1.43745 + 0.385163i −0.500000 − 0.866025i 0.185859 + 0.107306i −1.55279 − 1.55279i −0.385163 − 1.43745i 2.05994 − 0.551959i −1.87874 − 1.87874i −0.500000 + 0.866025i −1.63398 − 2.83014i
76.12 2.00245 + 0.536555i −0.500000 − 0.866025i 1.98987 + 1.14885i 1.79777 + 1.79777i −0.536555 − 2.00245i 3.46337 − 0.928006i 0.436407 + 0.436407i −0.500000 + 0.866025i 2.63535 + 4.56456i
76.13 2.52014 + 0.675270i −0.500000 − 0.866025i 4.16307 + 2.40355i 0.961072 + 0.961072i −0.675270 − 2.52014i −1.60148 + 0.429116i 5.17874 + 5.17874i −0.500000 + 0.866025i 1.77305 + 3.07102i
76.14 2.55489 + 0.684581i −0.500000 − 0.866025i 4.32676 + 2.49806i −2.81573 − 2.81573i −0.684581 − 2.55489i 2.71459 − 0.727372i 5.60366 + 5.60366i −0.500000 + 0.866025i −5.26630 − 9.12149i
175.1 −2.55489 + 0.684581i −0.500000 + 0.866025i 4.32676 − 2.49806i −2.81573 + 2.81573i 0.684581 − 2.55489i −2.71459 − 0.727372i −5.60366 + 5.60366i −0.500000 − 0.866025i 5.26630 − 9.12149i
175.2 −2.52014 + 0.675270i −0.500000 + 0.866025i 4.16307 − 2.40355i 0.961072 − 0.961072i 0.675270 − 2.52014i 1.60148 + 0.429116i −5.17874 + 5.17874i −0.500000 − 0.866025i −1.77305 + 3.07102i
175.3 −2.00245 + 0.536555i −0.500000 + 0.866025i 1.98987 − 1.14885i 1.79777 − 1.79777i 0.536555 − 2.00245i −3.46337 − 0.928006i −0.436407 + 0.436407i −0.500000 − 0.866025i −2.63535 + 4.56456i
175.4 −1.43745 + 0.385163i −0.500000 + 0.866025i 0.185859 − 0.107306i −1.55279 + 1.55279i 0.385163 − 1.43745i −2.05994 − 0.551959i 1.87874 − 1.87874i −0.500000 − 0.866025i 1.63398 − 2.83014i
175.5 −1.42470 + 0.381747i −0.500000 + 0.866025i 0.151984 − 0.0877479i −0.971227 + 0.971227i 0.381747 − 1.42470i 4.17142 + 1.11773i 1.90287 − 1.90287i −0.500000 − 0.866025i 1.01294 − 1.75447i
175.6 −0.567668 + 0.152106i −0.500000 + 0.866025i −1.43294 + 0.827308i 0.319197 − 0.319197i 0.152106 − 0.567668i 2.56095 + 0.686204i 1.51872 − 1.51872i −0.500000 − 0.866025i −0.132646 + 0.229750i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
13.f odd 12 1 inner
143.o even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bd.a ✓ 56
11.b odd 2 1 inner 429.2.bd.a ✓ 56
13.f odd 12 1 inner 429.2.bd.a ✓ 56
143.o even 12 1 inner 429.2.bd.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bd.a ✓ 56 1.a even 1 1 trivial
429.2.bd.a ✓ 56 11.b odd 2 1 inner
429.2.bd.a ✓ 56 13.f odd 12 1 inner
429.2.bd.a ✓ 56 143.o even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 127 T_{2}^{52} + 10741 T_{2}^{48} - 2826 T_{2}^{46} - 507170 T_{2}^{44} + 189582 T_{2}^{42} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display