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Properties

Label 473.2.f.b
Level $473$
Weight $2$
Character orbit 473.f
Analytic conductor $3.777$
Analytic rank $0$
Dimension $92$
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] = [473,2,Mod(130,473)]
 
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([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("473.130");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 473 = 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 473.f (of order \(5\), 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: \(92\)
Relative dimension: \(23\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 7 q^{3} - 26 q^{4} - 15 q^{5} - 2 q^{6} - 4 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q - 7 q^{3} - 26 q^{4} - 15 q^{5} - 2 q^{6} - 4 q^{7} - 26 q^{9} + 4 q^{10} + 74 q^{12} + 7 q^{13} - 10 q^{14} - q^{15} - 32 q^{16} - 6 q^{17} - 8 q^{18} - 10 q^{19} - 39 q^{20} - 2 q^{21} + 95 q^{22} + 60 q^{23} + 80 q^{24} - 38 q^{25} - 11 q^{26} - 28 q^{27} - 38 q^{28} + 11 q^{29} - 60 q^{30} - 25 q^{31} - 20 q^{32} + 28 q^{33} - 38 q^{34} + 6 q^{35} - 52 q^{36} - 39 q^{37} - 31 q^{38} - q^{39} + 2 q^{40} + q^{41} - 91 q^{42} + 92 q^{43} + 70 q^{44} + 182 q^{45} + 14 q^{46} - 16 q^{47} - 73 q^{48} - 47 q^{49} - 92 q^{50} + 26 q^{51} - 71 q^{52} - 82 q^{53} + 84 q^{54} + 21 q^{55} + 102 q^{56} + 60 q^{57} + 42 q^{58} - 63 q^{59} - 87 q^{60} - 12 q^{61} + 9 q^{62} - 79 q^{63} - 128 q^{64} + 32 q^{65} + 116 q^{66} + 56 q^{67} + 18 q^{68} - 43 q^{69} - 88 q^{70} - 35 q^{71} - 39 q^{72} + 39 q^{73} - 48 q^{74} - 114 q^{75} + 114 q^{76} + 65 q^{77} + 32 q^{78} - 43 q^{79} + 7 q^{80} - 97 q^{81} - 40 q^{82} - 8 q^{83} - 36 q^{84} - 44 q^{85} + 76 q^{87} + 30 q^{88} + 160 q^{89} + 90 q^{90} + 141 q^{91} - 95 q^{92} - 46 q^{93} - 110 q^{94} - 10 q^{95} - 199 q^{96} - 92 q^{97} + 80 q^{98} + 62 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
130.1 −0.797864 + 2.45557i 1.99989 − 1.45301i −3.77522 − 2.74286i 1.27781 + 3.93271i 1.97232 + 6.07018i 2.16372 + 1.57204i 5.56974 − 4.04665i 0.961286 − 2.95854i −10.6766
130.2 −0.782035 + 2.40686i −2.28837 + 1.66260i −3.56334 − 2.58892i 0.834718 + 2.56900i −2.21205 − 6.80800i −0.327125 − 0.237671i 4.92303 − 3.57679i 1.54536 − 4.75614i −6.83598
130.3 −0.767517 + 2.36218i −1.93312 + 1.40449i −3.37276 − 2.45045i −1.08211 − 3.33038i −1.83395 − 5.64433i −0.0807336 − 0.0586564i 4.35828 − 3.16647i 0.837290 − 2.57691i 8.69748
130.4 −0.719695 + 2.21499i −0.956899 + 0.695228i −2.77021 − 2.01267i 0.316857 + 0.975187i −0.851250 − 2.61988i 2.82493 + 2.05243i 2.68339 − 1.94960i −0.494737 + 1.52264i −2.38807
130.5 −0.672304 + 2.06914i 1.59553 − 1.15922i −2.21131 − 1.60661i −0.853653 − 2.62727i 1.32591 + 4.08073i 2.77418 + 2.01556i 1.29074 − 0.937776i 0.274880 − 0.845995i 6.01011
130.6 −0.463120 + 1.42534i −0.895034 + 0.650281i −0.199068 − 0.144631i −0.583724 − 1.79652i −0.512360 − 1.57688i −3.04110 − 2.20949i −2.12659 + 1.54505i −0.548829 + 1.68912i 2.83097
130.7 −0.429723 + 1.32255i 0.529278 − 0.384543i 0.0535547 + 0.0389098i 0.924434 + 2.84512i 0.281135 + 0.865244i −2.94300 − 2.13822i −2.32453 + 1.68887i −0.794789 + 2.44611i −4.16006
130.8 −0.372257 + 1.14569i −2.25593 + 1.63903i 0.444003 + 0.322587i 1.04802 + 3.22549i −1.03803 − 3.19474i −0.0952081 − 0.0691727i −2.48403 + 1.80476i 1.47575 − 4.54190i −4.08554
130.9 −0.240022 + 0.738712i 0.391712 − 0.284595i 1.12995 + 0.820956i 0.114874 + 0.353545i 0.116214 + 0.357671i 2.30529 + 1.67489i −2.13443 + 1.55076i −0.854607 + 2.63021i −0.288740
130.10 −0.146728 + 0.451584i 2.33128 − 1.69378i 1.43564 + 1.04305i 1.29817 + 3.99534i 0.422816 + 1.30129i −1.95504 − 1.42042i −1.44995 + 1.05345i 1.63895 − 5.04417i −1.99471
130.11 −0.0807533 + 0.248533i 1.46932 − 1.06752i 1.56279 + 1.13543i 0.196542 + 0.604894i 0.146662 + 0.451381i −0.0714702 − 0.0519261i −0.831222 + 0.603918i 0.0922433 − 0.283896i −0.166208
130.12 −0.00948041 + 0.0291777i −0.878443 + 0.638227i 1.61727 + 1.17502i −0.507421 − 1.56168i −0.0102940 − 0.0316816i −3.37356 − 2.45103i −0.0992567 + 0.0721142i −0.562721 + 1.73188i 0.0503768
130.13 0.126986 − 0.390821i 1.79159 − 1.30167i 1.48142 + 1.07631i −0.813210 − 2.50280i −0.281213 − 0.865485i −1.99388 − 1.44864i 1.27367 − 0.925375i 0.588409 − 1.81094i −1.08142
130.14 0.130473 − 0.401554i −1.81404 + 1.31797i 1.47381 + 1.07079i 0.842796 + 2.59386i 0.292555 + 0.900392i 4.11327 + 2.98847i 1.30543 − 0.948454i 0.626619 − 1.92854i 1.15154
130.15 0.143423 − 0.441409i −2.76942 + 2.01210i 1.44376 + 1.04895i −0.381524 − 1.17421i 0.490964 + 1.51103i −1.84771 − 1.34244i 1.42106 − 1.03246i 2.69410 − 8.29158i −0.573026
130.16 0.372009 − 1.14493i −0.0445493 + 0.0323670i 0.445570 + 0.323725i 0.357270 + 1.09956i 0.0204850 + 0.0630465i 0.623968 + 0.453339i 2.48426 − 1.80492i −0.926114 + 2.85029i 1.39183
130.17 0.525771 − 1.61816i 0.226435 − 0.164515i −0.723960 − 0.525988i −0.503067 − 1.54828i −0.147157 − 0.452904i 3.94684 + 2.86755i 1.52120 − 1.10522i −0.902843 + 2.77867i −2.76986
130.18 0.571005 − 1.75737i −2.31514 + 1.68205i −1.14428 − 0.831369i 1.23822 + 3.81085i 1.63403 + 5.02903i −2.91753 − 2.11971i 0.875405 − 0.636019i 1.60354 − 4.93520i 7.40412
130.19 0.579805 − 1.78446i 2.61795 − 1.90205i −1.23007 − 0.893700i 0.559416 + 1.72171i −1.87623 − 5.77444i −2.17954 − 1.58353i 0.727926 − 0.528869i 2.30882 − 7.10581i 3.39666
130.20 0.679460 − 2.09116i 1.31964 − 0.958773i −2.29326 − 1.66615i −0.761721 − 2.34434i −1.10831 − 3.41102i 0.0277951 + 0.0201943i −1.48467 + 1.07867i −0.104853 + 0.322704i −5.41995
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 130.23
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 473.2.f.b ✓ 92
11.c even 5 1 inner 473.2.f.b ✓ 92
11.c even 5 1 5203.2.a.w 46
11.d odd 10 1 5203.2.a.v 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
473.2.f.b ✓ 92 1.a even 1 1 trivial
473.2.f.b ✓ 92 11.c even 5 1 inner
5203.2.a.v 46 11.d odd 10 1
5203.2.a.w 46 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{92} + 36 T_{2}^{90} + 741 T_{2}^{88} + 4 T_{2}^{87} + 11508 T_{2}^{86} + 156 T_{2}^{85} + \cdots + 63744256 \) acting on \(S_{2}^{\mathrm{new}}(473, [\chi])\). Copy content Toggle raw display