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Properties

Label 429.2.bj.b
Level $429$
Weight $2$
Character orbit 429.bj
Analytic conductor $3.426$
Analytic rank $0$
Dimension $112$
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(73,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 14, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bj (of order \(20\), degree \(8\), 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: \(112\)
Relative dimension: \(14\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 28 q^{3} + 6 q^{5} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 28 q^{3} + 6 q^{5} - 28 q^{9} - 14 q^{11} + 10 q^{13} + 12 q^{14} + 4 q^{15} - 48 q^{16} - 2 q^{20} + 32 q^{22} + 30 q^{24} - 46 q^{26} + 28 q^{27} - 20 q^{29} - 24 q^{31} - 16 q^{33} - 16 q^{34} - 20 q^{35} + 12 q^{37} + 10 q^{39} - 40 q^{40} - 10 q^{41} + 28 q^{42} + 24 q^{44} - 4 q^{45} - 20 q^{46} - 62 q^{47} - 92 q^{48} - 90 q^{50} + 110 q^{52} + 68 q^{53} + 32 q^{55} - 30 q^{57} - 56 q^{58} + 16 q^{59} + 2 q^{60} + 20 q^{61} + 8 q^{66} + 12 q^{67} + 60 q^{68} - 196 q^{70} - 52 q^{71} + 30 q^{72} - 10 q^{73} - 120 q^{74} - 84 q^{78} + 40 q^{79} - 56 q^{80} - 28 q^{81} + 110 q^{83} - 30 q^{84} - 40 q^{85} + 18 q^{86} - 96 q^{89} - 44 q^{91} + 80 q^{92} + 24 q^{93} - 20 q^{94} + 100 q^{96} + 18 q^{97} + 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −2.50252 + 0.396361i −0.309017 + 0.951057i 4.20341 − 1.36577i −1.89481 − 0.300108i 0.396361 − 2.50252i 0.290317 − 0.569780i −5.46267 + 2.78337i −0.809017 − 0.587785i 4.86076
73.2 −2.34881 + 0.372015i −0.309017 + 0.951057i 3.47641 − 1.12955i 3.36131 + 0.532379i 0.372015 − 2.34881i −2.10675 + 4.13473i −3.50742 + 1.78712i −0.809017 − 0.587785i −8.09314
73.3 −1.79125 + 0.283706i −0.309017 + 0.951057i 1.22598 − 0.398345i −3.18553 − 0.504538i 0.283706 − 1.79125i 0.465361 − 0.913322i 1.14880 − 0.585343i −0.809017 − 0.587785i 5.84923
73.4 −1.73374 + 0.274598i −0.309017 + 0.951057i 1.02835 − 0.334131i 2.33112 + 0.369213i 0.274598 − 1.73374i 1.54772 − 3.03757i 1.43692 − 0.732150i −0.809017 − 0.587785i −4.14295
73.5 −1.43832 + 0.227807i −0.309017 + 0.951057i 0.114751 − 0.0372850i −0.828408 − 0.131207i 0.227807 − 1.43832i −1.40334 + 2.75420i 2.43850 − 1.24248i −0.809017 − 0.587785i 1.22140
73.6 −0.778309 + 0.123272i −0.309017 + 0.951057i −1.31154 + 0.426146i 3.15337 + 0.499445i 0.123272 − 0.778309i −0.271920 + 0.533674i 2.37250 − 1.20885i −0.809017 − 0.587785i −2.51586
73.7 0.143806 − 0.0227766i −0.309017 + 0.951057i −1.88195 + 0.611483i −1.10538 − 0.175075i −0.0227766 + 0.143806i −0.0900281 + 0.176690i −0.516165 + 0.262999i −0.809017 − 0.587785i −0.162947
73.8 0.227816 − 0.0360826i −0.309017 + 0.951057i −1.85151 + 0.601594i −1.68185 − 0.266378i −0.0360826 + 0.227816i 2.29608 − 4.50631i −0.811131 + 0.413292i −0.809017 − 0.587785i −0.392764
73.9 1.10738 − 0.175392i −0.309017 + 0.951057i −0.706578 + 0.229581i 2.85202 + 0.451715i −0.175392 + 1.10738i −1.47375 + 2.89241i −2.74016 + 1.39618i −0.809017 − 0.587785i 3.23751
73.10 1.19422 − 0.189145i −0.309017 + 0.951057i −0.511738 + 0.166274i −0.797331 − 0.126285i −0.189145 + 1.19422i −1.27410 + 2.50056i −2.73431 + 1.39320i −0.809017 − 0.587785i −0.976071
73.11 1.48140 − 0.234631i −0.309017 + 0.951057i 0.237388 − 0.0771321i 3.64808 + 0.577798i −0.234631 + 1.48140i 1.93361 − 3.79492i −2.33921 + 1.19189i −0.809017 − 0.587785i 5.53984
73.12 1.66516 − 0.263735i −0.309017 + 0.951057i 0.801076 − 0.260285i −3.99771 − 0.633176i −0.263735 + 1.66516i −0.308156 + 0.604790i −1.73905 + 0.886089i −0.809017 − 0.587785i −6.82381
73.13 2.30315 − 0.364783i −0.309017 + 0.951057i 3.26933 − 1.06227i 0.468964 + 0.0742767i −0.364783 + 2.30315i −1.08156 + 2.12269i 2.98686 − 1.52188i −0.809017 − 0.587785i 1.10719
73.14 2.47003 − 0.391214i −0.309017 + 0.951057i 4.04586 − 1.31458i −0.0637683 − 0.0100999i −0.391214 + 2.47003i 1.47652 − 2.89783i 5.02262 − 2.55915i −0.809017 − 0.587785i −0.161460
112.1 −1.24336 − 2.44023i 0.809017 − 0.587785i −3.23319 + 4.45011i −1.70076 + 3.33794i −2.44023 − 1.24336i −0.416032 − 2.62673i 9.46926 + 1.49978i 0.309017 − 0.951057i 10.2600
112.2 −1.24128 − 2.43615i 0.809017 − 0.587785i −3.21847 + 4.42985i 1.65945 − 3.25685i −2.43615 − 1.24128i 0.168914 + 1.06648i 9.38582 + 1.48657i 0.309017 − 0.951057i −9.99400
112.3 −0.943260 − 1.85125i 0.809017 − 0.587785i −1.36183 + 1.87439i −0.530412 + 1.04099i −1.85125 − 0.943260i 0.785270 + 4.95800i 0.650274 + 0.102993i 0.309017 − 0.951057i 2.42745
112.4 −0.608810 − 1.19486i 0.809017 − 0.587785i 0.118537 − 0.163152i 0.900222 − 1.76679i −1.19486 − 0.608810i −0.329220 − 2.07862i −2.91613 − 0.461869i 0.309017 − 0.951057i −2.65912
112.5 −0.558611 − 1.09633i 0.809017 − 0.587785i 0.285666 − 0.393186i −1.49041 + 2.92509i −1.09633 − 0.558611i −0.495831 − 3.13056i −3.02123 − 0.478516i 0.309017 − 0.951057i 4.03944
112.6 −0.390599 − 0.766593i 0.809017 − 0.587785i 0.740473 − 1.01917i −0.202131 + 0.396704i −0.766593 − 0.390599i 0.246699 + 1.55759i −2.77007 − 0.438736i 0.309017 − 0.951057i 0.383063
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
13.d odd 4 1 inner
143.s even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bj.b ✓ 112
11.d odd 10 1 inner 429.2.bj.b ✓ 112
13.d odd 4 1 inner 429.2.bj.b ✓ 112
143.s even 20 1 inner 429.2.bj.b ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bj.b ✓ 112 1.a even 1 1 trivial
429.2.bj.b ✓ 112 11.d odd 10 1 inner
429.2.bj.b ✓ 112 13.d odd 4 1 inner
429.2.bj.b ✓ 112 143.s even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} - 74 T_{2}^{108} - 200 T_{2}^{105} + 5879 T_{2}^{104} - 3280 T_{2}^{103} + \cdots + 495504774241 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display