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Properties

Label 429.2.bg.a
Level $429$
Weight $2$
Character orbit 429.bg
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(16,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 12, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bg (of order \(15\), 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 \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9} + 3 q^{11} - 120 q^{12} + q^{13} + 6 q^{14} - 2 q^{15} - 16 q^{16} - 20 q^{17} - 40 q^{20} + 15 q^{22} + 4 q^{23} - 18 q^{24} - 26 q^{25} + 49 q^{26} + 28 q^{27} - 6 q^{28} - 10 q^{29} - 10 q^{30} - 4 q^{31} - 34 q^{32} - 8 q^{33} + 96 q^{34} - 12 q^{35} + 20 q^{36} + 20 q^{37} - 62 q^{38} - 4 q^{39} - 20 q^{40} - 50 q^{41} + 8 q^{42} + 44 q^{43} - 12 q^{44} + 2 q^{45} + 42 q^{46} + 18 q^{47} - 34 q^{48} + 48 q^{49} - 37 q^{50} + 10 q^{51} - 19 q^{52} - 46 q^{53} + 48 q^{55} - 12 q^{56} - 40 q^{58} - 18 q^{59} - 80 q^{60} + 2 q^{61} - 49 q^{62} + 68 q^{64} + 24 q^{65} - 20 q^{66} - 52 q^{67} - 56 q^{68} + 11 q^{69} + 220 q^{70} - 54 q^{71} + 18 q^{72} - 50 q^{73} - 64 q^{74} - 28 q^{75} + 28 q^{76} + 84 q^{77} + 2 q^{78} - 68 q^{79} + 34 q^{80} + 14 q^{81} + 51 q^{82} - 72 q^{83} - 9 q^{84} + 14 q^{85} - 10 q^{86} - 60 q^{87} - 22 q^{88} + 120 q^{89} + 20 q^{90} - 43 q^{91} + 122 q^{92} - 2 q^{93} + 30 q^{94} - 16 q^{95} - 68 q^{96} + 51 q^{97} - 24 q^{98} + 14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.80037 + 1.99951i 0.104528 − 0.994522i −0.547661 − 5.21064i −0.759776 + 2.33835i 1.80037 + 1.99951i 0.0209204 + 0.199044i 7.05123 + 5.12302i −0.978148 − 0.207912i −3.30768 − 5.72907i
16.2 −1.68093 + 1.86686i 0.104528 − 0.994522i −0.450589 − 4.28707i 1.24954 − 3.84568i 1.68093 + 1.86686i 0.0277297 + 0.263830i 4.69608 + 3.41190i −0.978148 − 0.207912i 5.07897 + 8.79703i
16.3 −1.46128 + 1.62292i 0.104528 − 0.994522i −0.289460 − 2.75403i −0.299446 + 0.921600i 1.46128 + 1.62292i −0.235463 − 2.24028i 1.35900 + 0.987368i −0.978148 − 0.207912i −1.05810 − 1.83269i
16.4 −0.990763 + 1.10035i 0.104528 − 0.994522i −0.0201101 − 0.191335i −0.983732 + 3.02762i 0.990763 + 1.10035i 0.365713 + 3.47953i −2.16532 − 1.57319i −0.978148 − 0.207912i −2.35680 − 4.08210i
16.5 −0.946617 + 1.05132i 0.104528 − 0.994522i −0.000142744 − 0.00135811i 0.776847 − 2.39089i 0.946617 + 1.05132i −0.131132 − 1.24764i −2.28746 − 1.66194i −0.978148 − 0.207912i 1.77822 + 3.07997i
16.6 −0.491663 + 0.546047i 0.104528 − 0.994522i 0.152622 + 1.45210i −0.00147074 + 0.00452647i 0.491663 + 0.546047i −0.428655 − 4.07838i −2.05685 − 1.49439i −0.978148 − 0.207912i −0.00174856 − 0.00302859i
16.7 −0.0978317 + 0.108653i 0.104528 − 0.994522i 0.206822 + 1.96778i −0.621977 + 1.91425i 0.0978317 + 0.108653i 0.00250914 + 0.0238729i −0.470608 − 0.341917i −0.978148 − 0.207912i −0.147140 − 0.254854i
16.8 0.244102 − 0.271103i 0.104528 − 0.994522i 0.195146 + 1.85669i 0.211790 − 0.651823i −0.244102 − 0.271103i 0.465929 + 4.43302i 1.14126 + 0.829171i −0.978148 − 0.207912i −0.125013 − 0.216528i
16.9 0.502391 − 0.557961i 0.104528 − 0.994522i 0.150132 + 1.42841i −0.0412359 + 0.126911i −0.502391 − 0.557961i −0.139017 − 1.32266i 2.08726 + 1.51649i −0.978148 − 0.207912i 0.0500950 + 0.0867670i
16.10 0.844998 − 0.938465i 0.104528 − 0.994522i 0.0423614 + 0.403042i 1.13050 − 3.47932i −0.844998 − 0.938465i −0.100820 − 0.959240i 2.45734 + 1.78536i −0.978148 − 0.207912i −2.30995 − 4.00095i
16.11 1.06574 − 1.18363i 0.104528 − 0.994522i −0.0561096 − 0.533847i −1.11978 + 3.44633i −1.06574 − 1.18363i 0.311034 + 2.95930i 1.88542 + 1.36983i −0.978148 − 0.207912i 2.88578 + 4.99832i
16.12 1.44079 − 1.60016i 0.104528 − 0.994522i −0.275575 − 2.62192i 0.0727615 − 0.223937i −1.44079 − 1.60016i −0.0981000 − 0.933359i −1.10854 − 0.805401i −0.978148 − 0.207912i −0.253500 − 0.439075i
16.13 1.65656 − 1.83979i 0.104528 − 0.994522i −0.431601 − 4.10641i −0.543715 + 1.67338i −1.65656 − 1.83979i −0.506656 − 4.82051i −4.26418 − 3.09811i −0.978148 − 0.207912i 2.17798 + 3.77238i
16.14 1.71487 − 1.90455i 0.104528 − 0.994522i −0.477496 − 4.54307i 1.12068 − 3.44910i −1.71487 − 1.90455i 0.446007 + 4.24348i −5.32461 − 3.86856i −0.978148 − 0.207912i −4.64718 − 8.04915i
256.1 −0.273007 − 2.59749i 0.978148 + 0.207912i −4.71612 + 1.00244i 0.822674 + 0.597708i 0.273007 − 2.59749i 3.34488 − 0.710976i 2.27719 + 7.00848i 0.913545 + 0.406737i 1.32794 − 2.30007i
256.2 −0.258546 − 2.45990i 0.978148 + 0.207912i −4.02797 + 0.856172i −0.872789 − 0.634118i 0.258546 − 2.45990i −3.11026 + 0.661107i 1.61884 + 4.98227i 0.913545 + 0.406737i −1.33421 + 2.31092i
256.3 −0.220383 − 2.09680i 0.978148 + 0.207912i −2.39171 + 0.508373i 3.27922 + 2.38249i 0.220383 − 2.09680i −0.946033 + 0.201086i 0.290016 + 0.892577i 0.913545 + 0.406737i 4.27293 − 7.40092i
256.4 −0.140589 − 1.33762i 0.978148 + 0.207912i 0.186846 − 0.0397154i −0.768940 − 0.558668i 0.140589 − 1.33762i 3.61727 − 0.768874i −0.910638 − 2.80265i 0.913545 + 0.406737i −0.639178 + 1.10709i
256.5 −0.134448 − 1.27919i 0.978148 + 0.207912i 0.338055 − 0.0718558i −1.25069 − 0.908681i 0.134448 − 1.27919i 1.33154 − 0.283029i −0.932303 − 2.86933i 0.913545 + 0.406737i −0.994219 + 1.72204i
256.6 −0.0790719 − 0.752319i 0.978148 + 0.207912i 1.39656 − 0.296849i 2.62007 + 1.90359i 0.0790719 − 0.752319i −0.683673 + 0.145319i −0.801274 − 2.46607i 0.913545 + 0.406737i 1.22494 − 2.12165i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.c even 3 1 inner
143.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bg.a ✓ 112
11.c even 5 1 inner 429.2.bg.a ✓ 112
13.c even 3 1 inner 429.2.bg.a ✓ 112
143.q even 15 1 inner 429.2.bg.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bg.a ✓ 112 1.a even 1 1 trivial
429.2.bg.a ✓ 112 11.c even 5 1 inner
429.2.bg.a ✓ 112 13.c even 3 1 inner
429.2.bg.a ✓ 112 143.q even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} - 24 T_{2}^{110} - 8 T_{2}^{109} + 249 T_{2}^{108} + 218 T_{2}^{107} - 1094 T_{2}^{106} + \cdots + 152587890625 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display