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Properties

Label 473.2.k.b
Level $473$
Weight $2$
Character orbit 473.k
Analytic conductor $3.777$
Analytic rank $0$
Dimension $160$
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] = [473,2,Mod(85,473)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(473, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("473.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 473 = 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 473.k (of order \(10\), 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: \(3.77692401561\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 46 q^{4} - 20 q^{6} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 46 q^{4} - 20 q^{6} + 50 q^{9} - 8 q^{11} - 30 q^{13} - 12 q^{14} - 10 q^{15} - 2 q^{16} - 40 q^{23} + 40 q^{24} + 54 q^{25} + 10 q^{31} - 100 q^{35} - 6 q^{36} - 70 q^{38} - 40 q^{40} - 10 q^{41} - 26 q^{44} + 18 q^{47} - 88 q^{49} - 130 q^{52} - 56 q^{53} + 132 q^{56} + 20 q^{57} + 4 q^{58} + 38 q^{59} + 66 q^{60} + 30 q^{64} + 80 q^{66} - 48 q^{67} - 160 q^{68} + 100 q^{74} - 64 q^{78} - 70 q^{79} + 40 q^{81} - 20 q^{83} + 220 q^{84} + 16 q^{86} + 260 q^{90} + 18 q^{92} + 20 q^{95} + 70 q^{96} - 72 q^{97} - 152 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1 −2.13681 + 1.55248i 1.60079 − 0.520127i 1.53771 − 4.73260i 0.573396 − 0.789212i −2.61309 + 3.59660i −1.37049 + 4.21795i 2.42909 + 7.47598i −0.135064 + 0.0981298i 2.57658i
85.2 −2.12162 + 1.54145i −0.840197 + 0.272996i 1.50718 − 4.63863i −0.962135 + 1.32427i 1.36177 − 1.87431i 1.08441 − 3.33748i 2.33176 + 7.17643i −1.79565 + 1.30461i − 4.29267i
85.3 −2.09162 + 1.51965i 1.98079 − 0.643597i 1.44751 − 4.45496i 2.17039 − 2.98728i −3.16501 + 4.35627i 1.08092 − 3.32671i 2.14450 + 6.60010i 1.08225 − 0.786300i 9.54650i
85.4 −1.99938 + 1.45263i 1.53267 − 0.497995i 1.26933 − 3.90661i −2.36098 + 3.24961i −2.34098 + 3.22209i −0.120027 + 0.369405i 1.60960 + 4.95384i −0.325970 + 0.236831i − 9.92684i
85.5 −1.95113 + 1.41758i −3.01835 + 0.980720i 1.17934 − 3.62965i 1.51059 − 2.07915i 4.49894 − 6.19225i 0.957733 − 2.94760i 1.35373 + 4.16634i 5.72155 − 4.15695i 6.19808i
85.6 −1.89979 + 1.38028i −1.10873 + 0.360248i 1.08599 − 3.34234i 0.367347 − 0.505610i 1.60911 − 2.21475i −0.481036 + 1.48048i 1.09889 + 3.38204i −1.32755 + 0.964520i 1.46759i
85.7 −1.63866 + 1.19055i 3.20606 − 1.04171i 0.649744 − 1.99971i −0.377187 + 0.519153i −4.01342 + 5.52400i 0.119612 − 0.368127i 0.0642280 + 0.197674i 6.76663 − 4.91625i − 1.29977i
85.8 −1.51401 + 1.09999i 0.477062 − 0.155007i 0.464204 − 1.42867i 1.79083 − 2.46486i −0.551769 + 0.759445i 0.295616 − 0.909812i −0.287880 − 0.886005i −2.22349 + 1.61546i 5.70171i
85.9 −1.42913 + 1.03833i −1.35899 + 0.441564i 0.346266 − 1.06570i −1.62774 + 2.24039i 1.48370 − 2.04213i −0.605728 + 1.86424i −0.480079 − 1.47753i −0.775163 + 0.563189i − 4.89194i
85.10 −1.33997 + 0.973545i −2.34454 + 0.761787i 0.229696 − 0.706930i 2.01577 − 2.77447i 2.39998 − 3.30329i −1.40653 + 4.32884i −0.643203 − 1.97957i 2.48949 − 1.80872i 5.68015i
85.11 −1.31995 + 0.959001i 1.64419 − 0.534230i 0.204555 − 0.629555i −0.724107 + 0.996648i −1.65793 + 2.28194i 0.510027 − 1.56970i −0.674612 − 2.07624i −0.00908497 + 0.00660062i − 2.00995i
85.12 −1.16317 + 0.845094i −0.212112 + 0.0689192i 0.0207520 − 0.0638681i 1.22222 − 1.68224i 0.188479 − 0.259419i 0.187229 − 0.576232i −0.858748 − 2.64296i −2.38681 + 1.73412i 2.98962i
85.13 −1.14863 + 0.834528i −2.12603 + 0.690787i 0.00487863 − 0.0150149i −1.51199 + 2.08107i 1.86553 − 2.56769i 0.860537 − 2.64846i −0.870548 − 2.67927i 1.61574 − 1.17391i − 3.65218i
85.14 −1.10124 + 0.800095i 2.57329 − 0.836113i −0.0454641 + 0.139924i 0.0699231 − 0.0962410i −2.16483 + 2.97964i −1.37461 + 4.23063i −0.903156 − 2.77963i 3.49569 − 2.53977i 0.161929i
85.15 −0.937751 + 0.681316i 0.998402 − 0.324400i −0.202848 + 0.624303i −1.00076 + 1.37742i −0.715233 + 0.984434i 1.48720 − 4.57712i −0.951505 − 2.92843i −1.53548 + 1.11559i − 1.97351i
85.16 −0.698305 + 0.507348i −0.00995135 + 0.00323339i −0.387807 + 1.19355i −1.35624 + 1.86671i 0.00530862 − 0.00730669i −1.32202 + 4.06875i −0.868193 − 2.67202i −2.42696 + 1.76329i − 1.99162i
85.17 −0.515998 + 0.374894i 2.26436 − 0.735736i −0.492326 + 1.51522i 2.49185 − 3.42974i −0.892582 + 1.22853i −0.132335 + 0.407285i −0.708197 − 2.17961i 2.15898 − 1.56859i 2.70391i
85.18 −0.472226 + 0.343093i −2.52446 + 0.820247i −0.512749 + 1.57808i 0.355184 − 0.488869i 0.910696 − 1.25347i 0.740248 − 2.27825i −0.660042 − 2.03140i 3.27304 − 2.37800i 0.352718i
85.19 −0.268424 + 0.195022i −1.71036 + 0.555728i −0.584016 + 1.79742i 1.86901 − 2.57248i 0.350722 − 0.482727i 0.135245 − 0.416242i −0.398829 − 1.22747i 0.189430 − 0.137629i 1.05501i
85.20 −0.0857211 + 0.0622800i 2.70919 − 0.880268i −0.614565 + 1.89144i 0.616736 − 0.848865i −0.177411 + 0.244186i 1.21555 − 3.74108i −0.130603 − 0.401953i 4.13777 − 3.00627i 0.111176i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
43.b odd 2 1 inner
473.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 473.2.k.b ✓ 160
11.d odd 10 1 inner 473.2.k.b ✓ 160
43.b odd 2 1 inner 473.2.k.b ✓ 160
473.k even 10 1 inner 473.2.k.b ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
473.2.k.b ✓ 160 1.a even 1 1 trivial
473.2.k.b ✓ 160 11.d odd 10 1 inner
473.2.k.b ✓ 160 43.b odd 2 1 inner
473.2.k.b ✓ 160 473.k even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} + 63 T_{2}^{158} + 2134 T_{2}^{156} + 51634 T_{2}^{154} + 1001913 T_{2}^{152} + \cdots + 97\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(473, [\chi])\). Copy content Toggle raw display