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Properties

Label 471.2.e.b
Level $471$
Weight $2$
Character orbit 471.e
Analytic conductor $3.761$
Analytic rank $0$
Dimension $22$
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] = [471,2,Mod(169,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.e (of order \(3\), 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: \(3.76095393520\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{2} + 11 q^{3} + 30 q^{4} - 4 q^{5} + q^{6} + 4 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{2} + 11 q^{3} + 30 q^{4} - 4 q^{5} + q^{6} + 4 q^{7} - 11 q^{9} - 5 q^{10} + 15 q^{12} + 3 q^{13} - 14 q^{14} + 4 q^{15} + 54 q^{16} - q^{17} - q^{18} - 22 q^{19} - 7 q^{20} + 2 q^{21} - 22 q^{22} - 10 q^{23} - 15 q^{25} - 10 q^{26} - 22 q^{27} - 38 q^{28} + 22 q^{29} + 5 q^{30} - 6 q^{31} + 32 q^{32} + 17 q^{34} - 11 q^{35} - 15 q^{36} + 8 q^{37} + 14 q^{38} + 6 q^{39} + 32 q^{40} - 7 q^{42} + q^{43} - 12 q^{44} + 8 q^{45} + 24 q^{46} + 7 q^{47} + 27 q^{48} + 22 q^{49} + 13 q^{50} + q^{51} + 17 q^{52} + 30 q^{53} - 2 q^{54} + 31 q^{55} - 82 q^{56} + 22 q^{57} - 90 q^{58} - 16 q^{59} + 7 q^{60} + 8 q^{61} - 28 q^{62} - 2 q^{63} - 32 q^{64} - 68 q^{65} + 22 q^{66} - 38 q^{67} - 8 q^{68} - 5 q^{69} + 43 q^{70} + 45 q^{71} - 4 q^{73} + 3 q^{74} - 30 q^{75} - 33 q^{76} + 21 q^{77} - 20 q^{78} + 26 q^{79} - 12 q^{80} - 11 q^{81} + 16 q^{82} + 8 q^{83} - 19 q^{84} - 28 q^{85} - 16 q^{86} + 11 q^{87} - 65 q^{88} + 15 q^{89} + 10 q^{90} - 3 q^{91} - 18 q^{92} - 12 q^{93} - 28 q^{94} - 5 q^{95} + 16 q^{96} - 35 q^{97} + 90 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 −2.48879 0.500000 + 0.866025i 4.19406 −1.91720 − 3.32070i −1.24439 − 2.15535i 3.28748 −5.46054 −0.500000 + 0.866025i 4.77151 + 8.26450i
169.2 −2.44281 0.500000 + 0.866025i 3.96732 0.189821 + 0.328779i −1.22140 − 2.11553i −0.758173 −4.80578 −0.500000 + 0.866025i −0.463696 − 0.803145i
169.3 −2.18058 0.500000 + 0.866025i 2.75495 1.44529 + 2.50331i −1.09029 − 1.88844i −2.39860 −1.64623 −0.500000 + 0.866025i −3.15157 − 5.45869i
169.4 −0.685532 0.500000 + 0.866025i −1.53005 −0.295622 − 0.512033i −0.342766 − 0.593688i −1.98864 2.41996 −0.500000 + 0.866025i 0.202658 + 0.351015i
169.5 −0.479742 0.500000 + 0.866025i −1.76985 1.73849 + 3.01115i −0.239871 − 0.415469i 2.28663 1.80855 −0.500000 + 0.866025i −0.834027 − 1.44458i
169.6 0.267371 0.500000 + 0.866025i −1.92851 −0.990152 − 1.71499i 0.133686 + 0.231550i 4.13452 −1.05037 −0.500000 + 0.866025i −0.264738 − 0.458540i
169.7 0.537815 0.500000 + 0.866025i −1.71075 −0.249194 − 0.431617i 0.268908 + 0.465762i 1.57818 −1.99570 −0.500000 + 0.866025i −0.134021 − 0.232130i
169.8 1.32447 0.500000 + 0.866025i −0.245772 −2.05322 − 3.55628i 0.662236 + 1.14703i −3.69381 −2.97446 −0.500000 + 0.866025i −2.71944 − 4.71020i
169.9 2.06264 0.500000 + 0.866025i 2.25447 1.20043 + 2.07920i 1.03132 + 1.78630i 1.31301 0.524880 −0.500000 + 0.866025i 2.47604 + 4.28863i
169.10 2.33676 0.500000 + 0.866025i 3.46043 −1.34656 − 2.33230i 1.16838 + 2.02369i 2.66781 3.41268 −0.500000 + 0.866025i −3.14657 − 5.45003i
169.11 2.74840 0.500000 + 0.866025i 5.55371 0.277926 + 0.481381i 1.37420 + 2.38018i −4.42840 9.76701 −0.500000 + 0.866025i 0.763851 + 1.32303i
301.1 −2.48879 0.500000 − 0.866025i 4.19406 −1.91720 + 3.32070i −1.24439 + 2.15535i 3.28748 −5.46054 −0.500000 − 0.866025i 4.77151 − 8.26450i
301.2 −2.44281 0.500000 − 0.866025i 3.96732 0.189821 − 0.328779i −1.22140 + 2.11553i −0.758173 −4.80578 −0.500000 − 0.866025i −0.463696 + 0.803145i
301.3 −2.18058 0.500000 − 0.866025i 2.75495 1.44529 − 2.50331i −1.09029 + 1.88844i −2.39860 −1.64623 −0.500000 − 0.866025i −3.15157 + 5.45869i
301.4 −0.685532 0.500000 − 0.866025i −1.53005 −0.295622 + 0.512033i −0.342766 + 0.593688i −1.98864 2.41996 −0.500000 − 0.866025i 0.202658 − 0.351015i
301.5 −0.479742 0.500000 − 0.866025i −1.76985 1.73849 − 3.01115i −0.239871 + 0.415469i 2.28663 1.80855 −0.500000 − 0.866025i −0.834027 + 1.44458i
301.6 0.267371 0.500000 − 0.866025i −1.92851 −0.990152 + 1.71499i 0.133686 − 0.231550i 4.13452 −1.05037 −0.500000 − 0.866025i −0.264738 + 0.458540i
301.7 0.537815 0.500000 − 0.866025i −1.71075 −0.249194 + 0.431617i 0.268908 − 0.465762i 1.57818 −1.99570 −0.500000 − 0.866025i −0.134021 + 0.232130i
301.8 1.32447 0.500000 − 0.866025i −0.245772 −2.05322 + 3.55628i 0.662236 − 1.14703i −3.69381 −2.97446 −0.500000 − 0.866025i −2.71944 + 4.71020i
301.9 2.06264 0.500000 − 0.866025i 2.25447 1.20043 − 2.07920i 1.03132 − 1.78630i 1.31301 0.524880 −0.500000 − 0.866025i 2.47604 − 4.28863i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.e.b ✓ 22
157.c even 3 1 inner 471.2.e.b ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.e.b ✓ 22 1.a even 1 1 trivial
471.2.e.b ✓ 22 157.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} - T_{2}^{10} - 18 T_{2}^{9} + 17 T_{2}^{8} + 112 T_{2}^{7} - 100 T_{2}^{6} - 270 T_{2}^{5} + \cdots + 11 \) acting on \(S_{2}^{\mathrm{new}}(471, [\chi])\). Copy content Toggle raw display