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Properties

Label 273.2.bt.b
Level $273$
Weight $2$
Character orbit 273.bt
Analytic conductor $2.180$
Analytic rank $0$
Dimension $40$
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] = [273,2,Mod(136,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bt (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: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{7} + 20 q^{9} - 8 q^{11} + 24 q^{12} - 18 q^{14} - 64 q^{16} + 8 q^{17} - 14 q^{19} - 14 q^{20} - 8 q^{21} + 4 q^{22} + 18 q^{24} - 24 q^{25} - 10 q^{26} - 2 q^{28} + 8 q^{29} - 8 q^{31} + 10 q^{32} - 8 q^{33} + 24 q^{34} - 22 q^{35} + 12 q^{37} + 8 q^{38} - 24 q^{39} - 30 q^{40} + 2 q^{41} + 6 q^{42} - 66 q^{43} + 28 q^{44} + 40 q^{46} + 10 q^{47} - 24 q^{48} + 38 q^{49} - 20 q^{50} + 40 q^{52} - 8 q^{53} + 42 q^{55} + 20 q^{56} - 14 q^{57} - 48 q^{58} - 26 q^{59} + 14 q^{60} - 12 q^{61} - 24 q^{62} - 4 q^{63} - 44 q^{65} - 18 q^{66} + 46 q^{67} - 4 q^{69} + 32 q^{70} - 6 q^{71} - 18 q^{72} + 10 q^{73} + 40 q^{74} + 48 q^{75} + 64 q^{76} - 24 q^{77} + 8 q^{78} + 34 q^{80} - 20 q^{81} + 24 q^{82} + 12 q^{83} + 20 q^{84} + 2 q^{85} + 12 q^{86} - 84 q^{88} - 16 q^{89} + 26 q^{91} + 236 q^{92} + 22 q^{93} + 30 q^{94} + 26 q^{96} + 62 q^{97} - 14 q^{98} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −1.75053 + 1.75053i −0.866025 + 0.500000i − 4.12868i −0.286571 + 1.06950i 0.640737 − 2.39126i 1.02891 − 2.43749i 3.72630 + 3.72630i 0.500000 − 0.866025i −1.37053 − 2.37383i
136.2 −1.59737 + 1.59737i −0.866025 + 0.500000i − 3.10321i 0.609466 − 2.27456i 0.584679 − 2.18205i 1.40839 + 2.23974i 1.76223 + 1.76223i 0.500000 − 0.866025i 2.65977 + 4.60686i
136.3 −1.03264 + 1.03264i −0.866025 + 0.500000i − 0.132693i 0.456824 − 1.70489i 0.377973 − 1.41061i −2.58707 + 0.554121i −1.92826 − 1.92826i 0.500000 − 0.866025i 1.28880 + 2.23227i
136.4 −0.783932 + 0.783932i −0.866025 + 0.500000i 0.770902i −0.994915 + 3.71307i 0.286939 − 1.07087i 0.0389254 + 2.64546i −2.17220 − 2.17220i 0.500000 − 0.866025i −2.13085 − 3.69074i
136.5 −0.326747 + 0.326747i −0.866025 + 0.500000i 1.78647i −0.562238 + 2.09830i 0.119598 − 0.446344i 2.25993 − 1.37576i −1.23722 − 1.23722i 0.500000 − 0.866025i −0.501904 − 0.869322i
136.6 −0.184060 + 0.184060i −0.866025 + 0.500000i 1.93224i 0.900201 − 3.35960i 0.0673706 − 0.251431i 0.939961 − 2.47315i −0.723769 − 0.723769i 0.500000 − 0.866025i 0.452676 + 0.784058i
136.7 1.07562 − 1.07562i −0.866025 + 0.500000i − 0.313900i −0.962166 + 3.59085i −0.393703 + 1.46932i −2.21486 − 1.44721i 1.81360 + 1.81360i 0.500000 − 0.866025i 2.82746 + 4.89730i
136.8 1.09526 − 1.09526i −0.866025 + 0.500000i − 0.399185i 0.128983 − 0.481371i −0.400893 + 1.49615i 2.51963 + 0.807124i 1.75331 + 1.75331i 0.500000 − 0.866025i −0.385956 − 0.668496i
136.9 1.57115 − 1.57115i −0.866025 + 0.500000i − 2.93701i 0.922649 − 3.44337i −0.575080 + 2.14623i −2.64488 + 0.0678683i −1.47218 − 1.47218i 0.500000 − 0.866025i −3.96043 − 6.85967i
136.10 1.93326 − 1.93326i −0.866025 + 0.500000i − 5.47495i −0.212233 + 0.792066i −0.707621 + 2.64088i 2.21517 − 1.44673i −6.71797 − 6.71797i 0.500000 − 0.866025i 1.12096 + 1.94157i
145.1 −1.96855 + 1.96855i 0.866025 + 0.500000i − 5.75037i −1.75415 + 0.470023i −2.68909 + 0.720539i 2.27503 + 1.35065i 7.38279 + 7.38279i 0.500000 + 0.866025i 2.52787 − 4.37840i
145.2 −1.65256 + 1.65256i 0.866025 + 0.500000i − 3.46192i 2.69306 − 0.721604i −2.25744 + 0.604879i −2.54381 − 0.727358i 2.41591 + 2.41591i 0.500000 + 0.866025i −3.25795 + 5.64294i
145.3 −1.22934 + 1.22934i 0.866025 + 0.500000i − 1.02253i −1.95691 + 0.524353i −1.67930 + 0.449968i −1.82011 − 1.92021i −1.20163 − 1.20163i 0.500000 + 0.866025i 1.76110 − 3.05031i
145.4 −0.507981 + 0.507981i 0.866025 + 0.500000i 1.48391i 1.10096 − 0.295002i −0.693915 + 0.185934i 0.718657 + 2.54628i −1.76976 − 1.76976i 0.500000 + 0.866025i −0.409414 + 0.709125i
145.5 −0.240784 + 0.240784i 0.866025 + 0.500000i 1.88405i 2.06336 − 0.552877i −0.328916 + 0.0881329i 1.35855 − 2.27032i −0.935214 − 0.935214i 0.500000 + 0.866025i −0.363700 + 0.629948i
145.6 0.465913 − 0.465913i 0.866025 + 0.500000i 1.56585i −3.81958 + 1.02345i 0.636449 − 0.170536i −2.61148 + 0.424440i 1.66138 + 1.66138i 0.500000 + 0.866025i −1.30275 + 2.25643i
145.7 0.837153 − 0.837153i 0.866025 + 0.500000i 0.598351i −1.58368 + 0.424345i 1.14357 − 0.306419i 2.46703 + 0.955901i 2.17522 + 2.17522i 0.500000 + 0.866025i −0.970539 + 1.68102i
145.8 0.884731 − 0.884731i 0.866025 + 0.500000i 0.434503i 3.68041 − 0.986163i 1.20856 − 0.323834i −2.53212 − 0.767050i 2.15388 + 2.15388i 0.500000 + 0.866025i 2.38368 − 4.12866i
145.9 1.55234 − 1.55234i 0.866025 + 0.500000i − 2.81950i −0.926472 + 0.248247i 2.12053 − 0.568195i 0.619609 − 2.57218i −1.27214 − 1.27214i 0.500000 + 0.866025i −1.05283 + 1.82356i
145.10 1.85908 − 1.85908i 0.866025 + 0.500000i − 4.91234i 0.502992 − 0.134776i 2.53955 − 0.680470i −1.89546 + 1.84587i −5.41427 − 5.41427i 0.500000 + 0.866025i 0.684542 − 1.18566i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.ba even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bt.b ✓ 40
3.b odd 2 1 819.2.et.d 40
7.d odd 6 1 273.2.cg.b yes 40
13.f odd 12 1 273.2.cg.b yes 40
21.g even 6 1 819.2.gh.d 40
39.k even 12 1 819.2.gh.d 40
91.ba even 12 1 inner 273.2.bt.b ✓ 40
273.bs odd 12 1 819.2.et.d 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bt.b ✓ 40 1.a even 1 1 trivial
273.2.bt.b ✓ 40 91.ba even 12 1 inner
273.2.cg.b yes 40 7.d odd 6 1
273.2.cg.b yes 40 13.f odd 12 1
819.2.et.d 40 3.b odd 2 1
819.2.et.d 40 273.bs odd 12 1
819.2.gh.d 40 21.g even 6 1
819.2.gh.d 40 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 168 T_{2}^{36} - 2 T_{2}^{35} + 8 T_{2}^{33} + 10786 T_{2}^{32} - 308 T_{2}^{31} + \cdots + 59049 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display