Properties

Label 473.2.v.a
Level $473$
Weight $2$
Character orbit 473.v
Analytic conductor $3.777$
Analytic rank $0$
Dimension $1008$
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(4,473)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(473, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([14, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("473.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 473 = 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 473.v (of order \(35\), degree \(24\), 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: \(1008\)
Relative dimension: \(42\) over \(\Q(\zeta_{35})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1008 q - 21 q^{2} - 19 q^{3} + 21 q^{4} - 19 q^{5} - 38 q^{6} - 6 q^{7} - 63 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1008 q - 21 q^{2} - 19 q^{3} + 21 q^{4} - 19 q^{5} - 38 q^{6} - 6 q^{7} - 63 q^{8} + 11 q^{9} - 64 q^{10} - 24 q^{11} - 60 q^{12} + 8 q^{13} + 3 q^{14} - 7 q^{15} + 49 q^{16} - 23 q^{17} + 37 q^{18} - 76 q^{19} - 15 q^{20} - 10 q^{21} - 29 q^{22} - 106 q^{23} - 20 q^{24} + 27 q^{25} - 21 q^{26} - 13 q^{27} + 23 q^{28} - 43 q^{29} + 97 q^{30} - 25 q^{31} + 12 q^{32} - 66 q^{33} + 30 q^{34} + 85 q^{35} - 212 q^{36} - 92 q^{37} - 149 q^{38} - 79 q^{39} + 87 q^{40} - 45 q^{41} + 100 q^{42} + 26 q^{43} - 340 q^{44} + 18 q^{45} + 105 q^{46} - 27 q^{47} - 99 q^{48} - 194 q^{49} + 108 q^{50} + 53 q^{51} + 25 q^{52} - 79 q^{53} - 50 q^{54} + 44 q^{55} + 66 q^{56} - 57 q^{57} - 7 q^{58} + 71 q^{59} - 21 q^{60} - 50 q^{61} + 45 q^{62} + 161 q^{63} - 255 q^{64} + 28 q^{65} + 122 q^{66} - 14 q^{67} - 338 q^{68} - 83 q^{69} - 92 q^{70} + 23 q^{71} - 55 q^{72} - 29 q^{73} + 105 q^{74} - 54 q^{75} + 20 q^{76} + 4 q^{77} + 98 q^{78} + 96 q^{79} - 36 q^{80} - 156 q^{81} + 137 q^{82} - 4 q^{83} + 16 q^{84} - 236 q^{85} - 133 q^{86} + 20 q^{87} + 269 q^{88} + 50 q^{89} + 143 q^{90} + 23 q^{91} + 164 q^{92} + 66 q^{93} - 273 q^{94} - 85 q^{95} + 291 q^{96} - 44 q^{97} + 10 q^{98} - 129 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.64958 + 0.731237i −2.23456 + 0.838643i 4.76865 2.84914i 1.60552 0.291358i 5.30738 3.85604i 1.23939 + 3.81445i −6.75256 + 7.06263i 2.03070 1.77417i −4.04088 + 1.94599i
4.2 −2.60508 + 0.718957i 2.83141 1.06265i 4.55265 2.72008i 0.299230 0.0543022i −6.61205 + 4.80394i −0.899463 2.76826i −6.16924 + 6.45252i 4.62844 4.04375i −0.740476 + 0.356595i
4.3 −2.50139 + 0.690340i −0.708636 + 0.265955i 4.06349 2.42782i −3.57232 + 0.648280i 1.58898 1.15446i −0.217631 0.669799i −4.90186 + 5.12695i −1.82778 + 1.59688i 8.48822 4.08771i
4.4 −2.35605 + 0.650227i 0.242385 0.0909685i 3.41126 2.03813i 0.0975855 0.0177092i −0.511919 + 0.371931i −0.573559 1.76523i −3.33375 + 3.48683i −2.20874 + 1.92972i −0.218401 + 0.105176i
4.5 −2.29877 + 0.634421i 0.881746 0.330925i 3.16497 1.89098i 3.39980 0.616973i −1.81699 + 1.32012i −0.233542 0.718767i −2.77989 + 2.90754i −1.59125 + 1.39023i −7.42395 + 3.57518i
4.6 −2.20423 + 0.608330i −2.68549 + 1.00788i 2.77168 1.65600i 1.15096 0.208869i 5.30633 3.85527i −1.45526 4.47882i −1.94162 + 2.03078i 3.93682 3.43950i −2.40993 + 1.16056i
4.7 −2.15618 + 0.595068i 2.20094 0.826027i 2.57811 1.54035i −2.29238 + 0.416005i −4.25409 + 3.09078i 0.928872 + 2.85877i −1.55075 + 1.62195i 1.90261 1.66226i 4.69523 2.26110i
4.8 −1.92141 + 0.530276i 0.216379 0.0812083i 1.69374 1.01196i 1.47051 0.266858i −0.372690 + 0.270775i 0.937210 + 2.88444i 0.0371585 0.0388648i −2.21899 + 1.93867i −2.68395 + 1.29252i
4.9 −1.66037 + 0.458232i −1.88126 + 0.706049i 0.829940 0.495866i 3.17696 0.576534i 2.80005 2.03435i −0.186938 0.575337i 1.22984 1.28631i 0.781425 0.682711i −5.01074 + 2.41304i
4.10 −1.55494 + 0.429136i −2.73531 + 1.02658i 0.516785 0.308764i −2.57482 + 0.467261i 3.81270 2.77009i 0.285364 + 0.878262i 1.55840 1.62996i 4.16883 3.64219i 3.80317 1.83151i
4.11 −1.53787 + 0.424427i 0.517766 0.194321i 0.468024 0.279631i −3.90061 + 0.707856i −0.713785 + 0.518595i −0.197839 0.608886i 1.60392 1.67757i −2.02889 + 1.77259i 5.69821 2.74411i
4.12 −1.39350 + 0.384582i 3.03112 1.13760i 0.0770500 0.0460352i 2.34716 0.425946i −3.78637 + 2.75096i 1.13017 + 3.47830i 1.90833 1.99596i 5.63434 4.92257i −3.10696 + 1.49623i
4.13 −1.30207 + 0.359348i −1.43507 + 0.538592i −0.150647 + 0.0900074i −1.18882 + 0.215738i 1.67502 1.21697i 1.09100 + 3.35776i 2.03071 2.12395i −0.489863 + 0.427980i 1.47039 0.708105i
4.14 −1.26189 + 0.348258i 1.92898 0.723959i −0.245827 + 0.146875i 2.19533 0.398394i −2.18203 + 1.58534i −0.592754 1.82431i 2.06834 2.16331i 0.937643 0.819194i −2.63152 + 1.26727i
4.15 −1.22164 + 0.337151i −0.665373 + 0.249719i −0.338163 + 0.202043i −0.914590 + 0.165974i 0.728654 0.529398i −1.37743 4.23930i 2.09658 2.19285i −1.87885 + 1.64150i 1.06134 0.511115i
4.16 −0.739140 + 0.203990i 1.12386 0.421792i −1.21218 + 0.724245i −2.05879 + 0.373616i −0.744649 + 0.541019i 1.02720 + 3.16140i 1.80801 1.89103i −1.17406 + 1.02575i 1.44552 0.696126i
4.17 −0.669119 + 0.184665i −1.55099 + 0.582096i −1.30328 + 0.778672i −0.0591621 + 0.0107363i 0.930304 0.675905i −0.562464 1.73109i 1.68763 1.76513i −0.192484 + 0.168168i 0.0376039 0.0181091i
4.18 −0.410819 + 0.113379i −3.17237 + 1.19061i −1.56098 + 0.932642i 2.69162 0.488457i 1.16828 0.848805i 0.728507 + 2.24211i 1.12457 1.17621i 6.38717 5.58030i −1.05039 + 0.505840i
4.19 −0.261439 + 0.0721526i 1.82397 0.684548i −1.65375 + 0.988072i 1.99651 0.362314i −0.427466 + 0.310572i −0.337623 1.03910i 0.735914 0.769706i 0.599052 0.523376i −0.495825 + 0.238777i
4.20 −0.179531 + 0.0495474i 2.12349 0.796961i −1.68712 + 1.00801i −1.95877 + 0.355463i −0.341746 + 0.248293i 0.490319 + 1.50905i 0.510357 0.533792i 1.61487 1.41087i 0.334047 0.160869i
See next 80 embeddings (of 1008 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
43.e even 7 1 inner
473.v even 35 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 473.2.v.a 1008
11.c even 5 1 inner 473.2.v.a 1008
43.e even 7 1 inner 473.2.v.a 1008
473.v even 35 1 inner 473.2.v.a 1008
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
473.2.v.a 1008 1.a even 1 1 trivial
473.2.v.a 1008 11.c even 5 1 inner
473.2.v.a 1008 43.e even 7 1 inner
473.2.v.a 1008 473.v even 35 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(473, [\chi])\).