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Properties

Label 429.2.bb.a
Level $429$
Weight $2$
Character orbit 429.bb
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(25,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bb (of order \(10\), 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 \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 14 q^{3} + 8 q^{4} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 14 q^{3} + 8 q^{4} - 14 q^{9} + 40 q^{10} - 52 q^{12} + 5 q^{13} - 6 q^{14} + 16 q^{16} - 14 q^{17} - 36 q^{22} + 28 q^{23} + 36 q^{25} + 6 q^{26} - 14 q^{27} + 34 q^{29} - 10 q^{30} + 18 q^{35} + 8 q^{36} - 18 q^{38} + 5 q^{39} + 40 q^{40} - 6 q^{42} + 48 q^{43} - 14 q^{48} - 14 q^{49} - 4 q^{51} + 11 q^{52} + 10 q^{53} + 8 q^{55} - 68 q^{56} + 10 q^{61} - 132 q^{62} - 52 q^{64} - 42 q^{65} + 14 q^{66} + 50 q^{68} - 2 q^{69} + 10 q^{74} + 6 q^{75} - 88 q^{77} + 26 q^{78} + 48 q^{79} - 14 q^{81} - 30 q^{82} - 36 q^{87} - 10 q^{88} - 10 q^{90} - 11 q^{91} - 126 q^{92} - 34 q^{94} + 36 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −1.45613 − 2.00419i 0.309017 + 0.951057i −1.27843 + 3.93462i −2.20631 + 3.03673i 1.45613 − 2.00419i −2.63274 − 0.855429i 5.03515 − 1.63602i −0.809017 + 0.587785i 9.29887
25.2 −1.38149 − 1.90146i 0.309017 + 0.951057i −1.08900 + 3.35161i 0.382487 − 0.526448i 1.38149 − 1.90146i 0.0445990 + 0.0144911i 3.40681 − 1.10694i −0.809017 + 0.587785i −1.52942
25.3 −1.22454 − 1.68544i 0.309017 + 0.951057i −0.723159 + 2.22565i 1.88056 − 2.58836i 1.22454 − 1.68544i 3.04968 + 0.990901i 0.674037 − 0.219008i −0.809017 + 0.587785i −6.66533
25.4 −0.897576 − 1.23541i 0.309017 + 0.951057i −0.102555 + 0.315631i 0.900087 − 1.23886i 0.897576 − 1.23541i −2.72966 − 0.886920i −2.42263 + 0.787161i −0.809017 + 0.587785i −2.33840
25.5 −0.817916 − 1.12577i 0.309017 + 0.951057i 0.0196739 − 0.0605500i −1.89008 + 2.60147i 0.817916 − 1.12577i 2.89956 + 0.942124i −2.73109 + 0.887385i −0.809017 + 0.587785i 4.47458
25.6 −0.665766 − 0.916348i 0.309017 + 0.951057i 0.221584 − 0.681966i −1.07016 + 1.47295i 0.665766 − 0.916348i −1.70616 − 0.554365i −2.92691 + 0.951009i −0.809017 + 0.587785i 2.06221
25.7 −0.150436 − 0.207057i 0.309017 + 0.951057i 0.597792 − 1.83982i 0.694723 − 0.956204i 0.150436 − 0.207057i −2.82596 − 0.918210i −0.957697 + 0.311175i −0.809017 + 0.587785i −0.302500
25.8 0.150436 + 0.207057i 0.309017 + 0.951057i 0.597792 − 1.83982i −0.694723 + 0.956204i −0.150436 + 0.207057i 2.82596 + 0.918210i 0.957697 − 0.311175i −0.809017 + 0.587785i −0.302500
25.9 0.665766 + 0.916348i 0.309017 + 0.951057i 0.221584 − 0.681966i 1.07016 − 1.47295i −0.665766 + 0.916348i 1.70616 + 0.554365i 2.92691 − 0.951009i −0.809017 + 0.587785i 2.06221
25.10 0.817916 + 1.12577i 0.309017 + 0.951057i 0.0196739 − 0.0605500i 1.89008 − 2.60147i −0.817916 + 1.12577i −2.89956 − 0.942124i 2.73109 − 0.887385i −0.809017 + 0.587785i 4.47458
25.11 0.897576 + 1.23541i 0.309017 + 0.951057i −0.102555 + 0.315631i −0.900087 + 1.23886i −0.897576 + 1.23541i 2.72966 + 0.886920i 2.42263 − 0.787161i −0.809017 + 0.587785i −2.33840
25.12 1.22454 + 1.68544i 0.309017 + 0.951057i −0.723159 + 2.22565i −1.88056 + 2.58836i −1.22454 + 1.68544i −3.04968 − 0.990901i −0.674037 + 0.219008i −0.809017 + 0.587785i −6.66533
25.13 1.38149 + 1.90146i 0.309017 + 0.951057i −1.08900 + 3.35161i −0.382487 + 0.526448i −1.38149 + 1.90146i −0.0445990 − 0.0144911i −3.40681 + 1.10694i −0.809017 + 0.587785i −1.52942
25.14 1.45613 + 2.00419i 0.309017 + 0.951057i −1.27843 + 3.93462i 2.20631 − 3.03673i −1.45613 + 2.00419i 2.63274 + 0.855429i −5.03515 + 1.63602i −0.809017 + 0.587785i 9.29887
64.1 −2.53943 − 0.825112i −0.809017 + 0.587785i 4.14988 + 3.01506i −1.45528 + 0.472849i 2.53943 − 0.825112i −0.367708 + 0.506107i −4.91166 − 6.76032i 0.309017 − 0.951057i 4.08574
64.2 −2.00472 − 0.651372i −0.809017 + 0.587785i 1.97658 + 1.43607i −2.60228 + 0.845530i 2.00472 − 0.651372i 1.43209 − 1.97110i −0.549095 − 0.755765i 0.309017 − 0.951057i 5.76758
64.3 −1.81881 − 0.590967i −0.809017 + 0.587785i 1.34080 + 0.974145i 2.40393 − 0.781084i 1.81881 − 0.590967i 2.67159 − 3.67713i 0.385208 + 0.530193i 0.309017 − 0.951057i −4.83389
64.4 −1.36111 − 0.442251i −0.809017 + 0.587785i 0.0389958 + 0.0283321i −1.28374 + 0.417113i 1.36111 − 0.442251i −2.73757 + 3.76794i 1.64187 + 2.25985i 0.309017 − 0.951057i 1.93178
64.5 −1.23441 − 0.401083i −0.809017 + 0.587785i −0.255144 − 0.185373i 3.32808 − 1.08136i 1.23441 − 0.401083i −1.47987 + 2.03686i 1.76641 + 2.43126i 0.309017 − 0.951057i −4.54192
64.6 −0.605812 − 0.196840i −0.809017 + 0.587785i −1.28977 − 0.937075i −4.01620 + 1.30494i 0.605812 − 0.196840i 0.707733 − 0.974111i 1.34573 + 1.85224i 0.309017 − 0.951057i 2.68992
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.b even 2 1 inner
143.n even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bb.a ✓ 56
11.c even 5 1 inner 429.2.bb.a ✓ 56
13.b even 2 1 inner 429.2.bb.a ✓ 56
143.n even 10 1 inner 429.2.bb.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bb.a ✓ 56 1.a even 1 1 trivial
429.2.bb.a ✓ 56 11.c even 5 1 inner
429.2.bb.a ✓ 56 13.b even 2 1 inner
429.2.bb.a ✓ 56 143.n even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 18 T_{2}^{54} + 213 T_{2}^{52} - 2056 T_{2}^{50} + 17785 T_{2}^{48} - 125036 T_{2}^{46} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display