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Properties

Label 266.2.u.d
Level $266$
Weight $2$
Character orbit 266.u
Analytic conductor $2.124$
Analytic rank $0$
Dimension $24$
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] = [266,2,Mod(43,266)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(266, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("266.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 266 = 2 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 266.u (of order \(9\), degree \(6\), 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: \(2.12402069377\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{7} + 12 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{7} + 12 q^{8} + 6 q^{9} + 9 q^{10} - 3 q^{11} + 3 q^{12} + 6 q^{13} - 24 q^{15} - 15 q^{17} - 42 q^{18} + 30 q^{19} + 6 q^{20} - 9 q^{22} - 24 q^{23} - 6 q^{25} + 6 q^{26} + 21 q^{27} + 27 q^{29} + 6 q^{30} + 3 q^{31} + 15 q^{33} - 30 q^{34} + 9 q^{35} + 6 q^{36} - 54 q^{37} + 3 q^{38} + 54 q^{39} + 9 q^{40} - 15 q^{41} + 6 q^{43} + 9 q^{44} - 30 q^{45} + 12 q^{46} + 3 q^{47} - 12 q^{49} + 33 q^{50} - 39 q^{51} - 3 q^{52} - 33 q^{53} - 18 q^{54} - 15 q^{55} - 24 q^{56} + 87 q^{57} + 12 q^{58} + 66 q^{59} + 3 q^{60} - 72 q^{61} - 39 q^{62} - 12 q^{63} - 12 q^{64} - 9 q^{65} + 30 q^{66} - 30 q^{67} - 3 q^{68} + 18 q^{69} - 9 q^{70} + 45 q^{71} - 6 q^{72} - 27 q^{73} + 6 q^{74} - 66 q^{75} + 9 q^{76} + 6 q^{77} - 18 q^{78} + 12 q^{79} + 24 q^{81} + 15 q^{82} - 9 q^{83} + 3 q^{84} + 78 q^{85} + 30 q^{86} + 45 q^{87} + 3 q^{88} - 48 q^{89} - 3 q^{90} - 3 q^{91} - 6 q^{92} - 3 q^{93} - 42 q^{94} - 3 q^{95} + 6 q^{96} - 21 q^{97} - 150 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 0.939693 − 0.342020i −0.557269 + 3.16043i 0.766044 − 0.642788i 2.10162 + 1.76347i 0.557269 + 3.16043i −0.500000 − 0.866025i 0.500000 − 0.866025i −6.85870 − 2.49636i 2.57802 + 0.938324i
43.2 0.939693 − 0.342020i −0.0993355 + 0.563360i 0.766044 − 0.642788i 1.70023 + 1.42667i 0.0993355 + 0.563360i −0.500000 − 0.866025i 0.500000 − 0.866025i 2.51157 + 0.914137i 2.08565 + 0.759113i
43.3 0.939693 − 0.342020i 0.00175394 − 0.00994711i 0.766044 − 0.642788i −2.36676 − 1.98595i −0.00175394 − 0.00994711i −0.500000 − 0.866025i 0.500000 − 0.866025i 2.81898 + 1.02603i −2.90326 − 1.05670i
43.4 0.939693 − 0.342020i 0.481203 − 2.72904i 0.766044 − 0.642788i 1.03668 + 0.869879i −0.481203 − 2.72904i −0.500000 − 0.866025i 0.500000 − 0.866025i −4.39700 − 1.60038i 1.27168 + 0.462853i
85.1 −0.173648 − 0.984808i −2.24978 + 1.88779i −0.939693 + 0.342020i 0.157758 + 0.0574193i 2.24978 + 1.88779i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.976815 − 5.53979i 0.0291525 − 0.165332i
85.2 −0.173648 − 0.984808i −0.587692 + 0.493132i −0.939693 + 0.342020i −2.10570 − 0.766411i 0.587692 + 0.493132i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.418742 + 2.37480i −0.389117 + 2.20679i
85.3 −0.173648 − 0.984808i 0.0872325 − 0.0731967i −0.939693 + 0.342020i 3.51739 + 1.28022i −0.0872325 − 0.0731967i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.518693 + 2.94165i 0.649987 − 3.68626i
85.4 −0.173648 − 0.984808i 1.98419 − 1.66494i −0.939693 + 0.342020i −3.62248 − 1.31848i −1.98419 − 1.66494i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.644068 − 3.65269i −0.669408 + 3.79640i
99.1 0.939693 + 0.342020i −0.557269 − 3.16043i 0.766044 + 0.642788i 2.10162 − 1.76347i 0.557269 − 3.16043i −0.500000 + 0.866025i 0.500000 + 0.866025i −6.85870 + 2.49636i 2.57802 − 0.938324i
99.2 0.939693 + 0.342020i −0.0993355 − 0.563360i 0.766044 + 0.642788i 1.70023 − 1.42667i 0.0993355 − 0.563360i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.51157 − 0.914137i 2.08565 − 0.759113i
99.3 0.939693 + 0.342020i 0.00175394 + 0.00994711i 0.766044 + 0.642788i −2.36676 + 1.98595i −0.00175394 + 0.00994711i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.81898 − 1.02603i −2.90326 + 1.05670i
99.4 0.939693 + 0.342020i 0.481203 + 2.72904i 0.766044 + 0.642788i 1.03668 − 0.869879i −0.481203 + 2.72904i −0.500000 + 0.866025i 0.500000 + 0.866025i −4.39700 + 1.60038i 1.27168 − 0.462853i
169.1 −0.173648 + 0.984808i −2.24978 − 1.88779i −0.939693 − 0.342020i 0.157758 − 0.0574193i 2.24978 − 1.88779i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.976815 + 5.53979i 0.0291525 + 0.165332i
169.2 −0.173648 + 0.984808i −0.587692 − 0.493132i −0.939693 − 0.342020i −2.10570 + 0.766411i 0.587692 − 0.493132i −0.500000 − 0.866025i 0.500000 − 0.866025i −0.418742 − 2.37480i −0.389117 − 2.20679i
169.3 −0.173648 + 0.984808i 0.0872325 + 0.0731967i −0.939693 − 0.342020i 3.51739 − 1.28022i −0.0872325 + 0.0731967i −0.500000 − 0.866025i 0.500000 − 0.866025i −0.518693 − 2.94165i 0.649987 + 3.68626i
169.4 −0.173648 + 0.984808i 1.98419 + 1.66494i −0.939693 − 0.342020i −3.62248 + 1.31848i −1.98419 + 1.66494i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.644068 + 3.65269i −0.669408 − 3.79640i
225.1 −0.766044 − 0.642788i −2.11749 + 0.770704i 0.173648 + 0.984808i 0.208129 − 1.18036i 2.11749 + 0.770704i −0.500000 − 0.866025i 0.500000 − 0.866025i 1.59165 − 1.33555i −0.918156 + 0.770424i
225.2 −0.766044 − 0.642788i −1.83827 + 0.669077i 0.173648 + 0.984808i −0.413853 + 2.34708i 1.83827 + 0.669077i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.633454 − 0.531531i 1.82570 − 1.53195i
225.3 −0.766044 − 0.642788i 2.09366 − 0.762029i 0.173648 + 0.984808i 0.535985 − 3.03972i −2.09366 − 0.762029i −0.500000 − 0.866025i 0.500000 − 0.866025i 1.50458 − 1.26249i −2.36448 + 1.98404i
225.4 −0.766044 − 0.642788i 2.80180 − 1.01977i 0.173648 + 0.984808i −0.749009 + 4.24784i −2.80180 − 1.01977i −0.500000 − 0.866025i 0.500000 − 0.866025i 4.51201 − 3.78603i 3.30423 − 2.77258i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 266.2.u.d ✓ 24
19.e even 9 1 inner 266.2.u.d ✓ 24
19.e even 9 1 5054.2.a.bl 12
19.f odd 18 1 5054.2.a.bm 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.u.d ✓ 24 1.a even 1 1 trivial
266.2.u.d ✓ 24 19.e even 9 1 inner
5054.2.a.bl 12 19.e even 9 1
5054.2.a.bm 12 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 3 T_{3}^{22} - 13 T_{3}^{21} - 15 T_{3}^{20} + 69 T_{3}^{19} + 722 T_{3}^{18} - 15 T_{3}^{17} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(266, [\chi])\). Copy content Toggle raw display