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Properties

Label 229.2.e.b
Level $229$
Weight $2$
Character orbit 229.e
Analytic conductor $1.829$
Analytic rank $0$
Dimension $36$
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] = [229,2,Mod(95,229)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.95");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 229.e (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: \(1.82857420629\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{3} - 50 q^{4} + 3 q^{5} + 6 q^{6} + 3 q^{7} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{3} - 50 q^{4} + 3 q^{5} + 6 q^{6} + 3 q^{7} - 28 q^{9} + 18 q^{10} - 16 q^{11} - 3 q^{12} - 8 q^{14} - 10 q^{15} + 74 q^{16} - 8 q^{17} + 30 q^{18} + q^{19} + q^{20} + 27 q^{23} - 3 q^{25} - 64 q^{26} + 4 q^{27} - 45 q^{28} - 6 q^{29} - 6 q^{31} + 5 q^{33} - 9 q^{35} + 55 q^{36} - 18 q^{37} + 45 q^{38} + 15 q^{39} + 42 q^{40} - 18 q^{41} - 14 q^{42} - 48 q^{43} - 20 q^{44} - 26 q^{45} - 2 q^{46} + 18 q^{47} + 28 q^{48} + 9 q^{49} - 42 q^{50} - 19 q^{51} - 2 q^{53} - 15 q^{55} - 11 q^{56} + 90 q^{57} - 11 q^{58} + 15 q^{59} + 92 q^{60} + 6 q^{61} + 23 q^{62} - 12 q^{63} - 102 q^{64} + 39 q^{65} + 6 q^{66} - 33 q^{67} - 32 q^{68} - 84 q^{69} + 8 q^{70} + 36 q^{71} - 144 q^{72} - 45 q^{73} + 78 q^{74} - 17 q^{75} + 10 q^{76} + 45 q^{77} + 86 q^{78} - 27 q^{79} - 45 q^{80} - 30 q^{81} + 2 q^{82} - 11 q^{83} + 63 q^{85} + 72 q^{87} + 69 q^{89} - 120 q^{90} + 33 q^{91} - 66 q^{92} + 11 q^{94} + 56 q^{95} + 69 q^{96} + 5 q^{97} - 135 q^{98} - 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 − 2.70610i 1.62230 + 2.80990i −5.32298 −1.44139 + 2.49655i 7.60387 − 4.39010i 1.19188 + 0.688132i 8.99231i −3.76369 + 6.51890i 6.75593 + 3.90054i
95.2 − 2.70372i −1.13760 − 1.97038i −5.31012 1.47932 − 2.56226i −5.32737 + 3.07576i 0.443882 + 0.256275i 8.94965i −1.08827 + 1.88494i −6.92763 − 3.99967i
95.3 − 2.24099i 0.139223 + 0.241142i −3.02203 −0.0732704 + 0.126908i 0.540397 − 0.311998i −3.24693 − 1.87462i 2.29037i 1.46123 − 2.53093i 0.284400 + 0.164198i
95.4 − 2.06495i 0.397015 + 0.687651i −2.26401 0.648681 − 1.12355i 1.41996 − 0.819816i 2.99437 + 1.72880i 0.545162i 1.18476 − 2.05206i −2.32007 − 1.33949i
95.5 − 1.94789i −1.56673 − 2.71366i −1.79426 −1.86007 + 3.22174i −5.28591 + 3.05182i 0.849699 + 0.490574i − 0.400749i −3.40931 + 5.90509i 6.27558 + 3.62321i
95.6 − 1.37034i 1.36191 + 2.35890i 0.122179 1.05030 − 1.81917i 3.23249 − 1.86628i −1.97513 − 1.14034i − 2.90810i −2.20961 + 3.82716i −2.49288 − 1.43926i
95.7 − 1.14961i −1.07282 − 1.85818i 0.678394 0.952784 − 1.65027i −2.13619 + 1.23333i 1.34597 + 0.777094i − 3.07911i −0.801896 + 1.38892i −1.89717 − 1.09533i
95.8 − 0.826336i −0.542050 − 0.938858i 1.31717 −0.464606 + 0.804721i −0.775812 + 0.447915i −3.68261 − 2.12615i − 2.74110i 0.912363 − 1.58026i 0.664970 + 0.383920i
95.9 − 0.617158i 0.971316 + 1.68237i 1.61912 −1.78360 + 3.08928i 1.03829 − 0.599455i −0.446000 − 0.257498i − 2.23357i −0.386909 + 0.670146i 1.90657 + 1.10076i
95.10 0.126650i 1.01172 + 1.75235i 1.98396 0.302515 − 0.523972i −0.221934 + 0.128134i 1.09468 + 0.632011i 0.504567i −0.547149 + 0.947690i 0.0663608 + 0.0383134i
95.11 0.375352i −0.722760 − 1.25186i 1.85911 −0.932320 + 1.61483i 0.469886 − 0.271289i 2.25908 + 1.30428i 1.44852i 0.455237 − 0.788493i −0.606128 − 0.349948i
95.12 0.869471i −1.52145 − 2.63523i 1.24402 1.36104 − 2.35740i 2.29125 − 1.32285i −2.92714 − 1.68999i 2.82058i −3.12961 + 5.42064i 2.04969 + 1.18339i
95.13 0.998441i 0.0318259 + 0.0551241i 1.00312 0.995972 − 1.72507i −0.0550382 + 0.0317763i −0.898894 − 0.518977i 2.99843i 1.49797 − 2.59457i 1.72238 + 0.994419i
95.14 1.83848i 1.40283 + 2.42977i −1.38001 1.27503 − 2.20842i −4.46709 + 2.57907i −1.49244 − 0.861659i 1.13983i −2.43586 + 4.21903i 4.06013 + 2.34412i
95.15 2.09051i −1.54137 − 2.66974i −2.37022 −0.771187 + 1.33573i 5.58111 − 3.22225i 2.02287 + 1.16790i − 0.773955i −3.25166 + 5.63205i −2.79236 − 1.61217i
95.16 2.24295i −0.348574 − 0.603748i −3.03082 1.75035 − 3.03169i 1.35418 − 0.781834i 4.45033 + 2.56940i − 2.31209i 1.25699 − 2.17717i 6.79993 + 3.92594i
95.17 2.63021i 1.04488 + 1.80978i −4.91801 −0.730483 + 1.26523i −4.76010 + 2.74825i 2.75891 + 1.59286i − 7.67499i −0.683535 + 1.18392i −3.32783 − 1.92133i
95.18 2.72297i −0.529652 − 0.917384i −5.41458 −0.259072 + 0.448725i 2.49801 − 1.44223i −3.24253 − 1.87207i − 9.29782i 0.938938 − 1.62629i −1.22187 − 0.705446i
135.1 − 2.72297i −0.529652 + 0.917384i −5.41458 −0.259072 − 0.448725i 2.49801 + 1.44223i −3.24253 + 1.87207i 9.29782i 0.938938 + 1.62629i −1.22187 + 0.705446i
135.2 − 2.63021i 1.04488 − 1.80978i −4.91801 −0.730483 − 1.26523i −4.76010 − 2.74825i 2.75891 − 1.59286i 7.67499i −0.683535 − 1.18392i −3.32783 + 1.92133i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
229.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 229.2.e.b ✓ 36
229.e even 6 1 inner 229.2.e.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.2.e.b ✓ 36 1.a even 1 1 trivial
229.2.e.b ✓ 36 229.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 61 T_{2}^{34} + 1695 T_{2}^{32} + 28415 T_{2}^{30} + 320850 T_{2}^{28} + 2580243 T_{2}^{26} + \cdots + 18225 \) acting on \(S_{2}^{\mathrm{new}}(229, [\chi])\). Copy content Toggle raw display