Properties

Label 473.2.w.a
Level $473$
Weight $2$
Character orbit 473.w
Analytic conductor $3.777$
Analytic rank $0$
Dimension $504$
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(76,473)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(473, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 31]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("473.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 473 = 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 473.w (of order \(42\), degree \(12\), 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: \(504\)
Relative dimension: \(42\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 504 q - 22 q^{3} - 104 q^{4} - 22 q^{5} - 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 504 q - 22 q^{3} - 104 q^{4} - 22 q^{5} - 76 q^{9} - 24 q^{11} - 44 q^{12} + 4 q^{14} + 18 q^{15} - 108 q^{16} + 30 q^{20} - 35 q^{22} - 30 q^{23} - 68 q^{25} + 4 q^{26} - 28 q^{27} - 34 q^{31} + 77 q^{33} - 28 q^{34} + 166 q^{36} + 36 q^{37} + 62 q^{38} - 50 q^{44} - 84 q^{45} - 48 q^{47} + 160 q^{48} - 186 q^{49} + 26 q^{53} - 30 q^{55} - 14 q^{56} - 54 q^{58} - 54 q^{59} + 2 q^{60} - 36 q^{64} - 83 q^{66} - 94 q^{67} - 158 q^{69} + 14 q^{70} - 80 q^{71} + 98 q^{75} - 99 q^{77} + 144 q^{78} - 18 q^{80} - 244 q^{81} - 112 q^{82} + 184 q^{86} + 49 q^{88} + 118 q^{89} + 100 q^{91} - 64 q^{92} - 210 q^{93} + 92 q^{97} - 111 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −0.613285 2.68698i 0.369676 0.398416i −5.04180 + 2.42800i 0.144296 + 0.957343i −1.29725 0.748969i 0.842602 + 1.45943i 6.17928 + 7.74857i 0.202115 + 2.69704i 2.48387 0.974845i
76.2 −0.589874 2.58441i 1.89592 2.04332i −4.52927 + 2.18118i −0.412899 2.73941i −6.39911 3.69453i 0.296478 + 0.513515i 5.00319 + 6.27380i −0.356436 4.75631i −6.83619 + 2.68301i
76.3 −0.562314 2.46366i −2.22656 + 2.39966i −3.95148 + 1.90293i −0.542591 3.59986i 7.16395 + 4.13611i −1.18655 2.05516i 3.75900 + 4.71364i −0.576610 7.69432i −8.56371 + 3.36101i
76.4 −0.559146 2.44978i −0.669393 + 0.721434i −3.88683 + 1.87180i 0.165193 + 1.09599i 2.14164 + 1.23648i −2.34418 4.06024i 3.62542 + 4.54614i 0.151810 + 2.02577i 2.59256 1.01750i
76.5 −0.546308 2.39353i −1.62751 + 1.75404i −3.62860 + 1.74744i 0.449946 + 2.98520i 5.08747 + 2.93725i 0.763291 + 1.32206i 3.10346 + 3.89161i −0.203673 2.71783i 6.89936 2.70780i
76.6 −0.496652 2.17598i −0.725880 + 0.782313i −2.68627 + 1.29364i −0.428723 2.84439i 2.06280 + 1.19096i 2.36941 + 4.10394i 1.36589 + 1.71277i 0.139079 + 1.85588i −5.97640 + 2.34556i
76.7 −0.493610 2.16265i 0.376719 0.406007i −2.63145 + 1.26724i −0.351823 2.33419i −1.06400 0.614301i −0.524871 0.909103i 1.27337 + 1.59675i 0.201266 + 2.68571i −4.87437 + 1.91305i
76.8 −0.484408 2.12233i 2.17672 2.34595i −2.46769 + 1.18838i 0.431771 + 2.86461i −6.03330 3.48333i −1.18160 2.04659i 1.00294 + 1.25765i −0.541166 7.22136i 5.87050 2.30400i
76.9 −0.412201 1.80597i 0.743758 0.801580i −1.28968 + 0.621075i 0.351427 + 2.33156i −1.75421 1.01279i 1.62205 + 2.80947i −0.656673 0.823441i 0.134835 + 1.79925i 4.06587 1.59574i
76.10 −0.354968 1.55522i −1.13957 + 1.22817i −0.490756 + 0.236336i 0.00101781 + 0.00675270i 2.31457 + 1.33632i 0.0793577 + 0.137452i −1.44744 1.81503i 0.0144220 + 0.192448i 0.0101406 0.00397990i
76.11 −0.338969 1.48512i 1.60762 1.73261i −0.288748 + 0.139054i −0.280400 1.86033i −3.11806 1.80022i −1.28427 2.22442i −1.59515 2.00026i −0.193280 2.57914i −2.66777 + 1.04702i
76.12 −0.326253 1.42941i −1.61194 + 1.73726i −0.134823 + 0.0649274i 0.428619 + 2.84370i 3.00915 + 1.73733i −0.179222 0.310421i −1.69148 2.12106i −0.195526 2.60911i 3.92496 1.54043i
76.13 −0.322397 1.41251i −1.08303 + 1.16722i −0.0893135 + 0.0430111i −0.325539 2.15981i 1.99788 + 1.15348i −1.17225 2.03039i −1.71712 2.15320i 0.0347228 + 0.463344i −2.94580 + 1.15614i
76.14 −0.298715 1.30876i 1.76185 1.89882i 0.178324 0.0858764i −0.152231 1.00999i −3.01139 1.73863i 1.59204 + 2.75750i −1.83962 2.30681i −0.277223 3.69929i −1.27635 + 0.500931i
76.15 −0.231231 1.01309i −0.0484510 + 0.0522177i 0.829057 0.399253i 0.598452 + 3.97047i 0.0641046 + 0.0370108i −0.0616950 0.106859i −1.89197 2.37246i 0.223811 + 2.98655i 3.88406 1.52438i
76.16 −0.205438 0.900084i 0.515876 0.555982i 1.03399 0.497944i −0.139820 0.927642i −0.606412 0.350112i −1.50664 2.60957i −1.81186 2.27201i 0.181202 + 2.41798i −0.806232 + 0.316423i
76.17 −0.173875 0.761795i −1.91643 + 2.06542i 1.25184 0.602854i −0.341326 2.26455i 1.90664 + 1.10080i 1.23866 + 2.14543i −1.65129 2.07065i −0.369065 4.92483i −1.66577 + 0.653768i
76.18 −0.131533 0.576284i 0.714374 0.769912i 1.48714 0.716167i −0.616223 4.08837i −0.537652 0.310413i 1.52888 + 2.64810i −1.34542 1.68710i 0.141756 + 1.89160i −2.27501 + 0.892876i
76.19 −0.0350325 0.153487i 1.67297 1.80304i 1.77961 0.857013i 0.369634 + 2.45236i −0.335351 0.193615i 1.06688 + 1.84789i −0.390202 0.489298i −0.227910 3.04125i 0.363457 0.142647i
76.20 −0.0221475 0.0970344i −1.96696 + 2.11988i 1.79301 0.863469i 0.245703 + 1.63013i 0.249264 + 0.143913i 2.53990 + 4.39924i −0.247608 0.310491i −0.400763 5.34781i 0.152737 0.0599449i
See next 80 embeddings (of 504 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
43.h odd 42 1 inner
473.w even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 473.2.w.a 504
11.b odd 2 1 inner 473.2.w.a 504
43.h odd 42 1 inner 473.2.w.a 504
473.w even 42 1 inner 473.2.w.a 504
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
473.2.w.a 504 1.a even 1 1 trivial
473.2.w.a 504 11.b odd 2 1 inner
473.2.w.a 504 43.h odd 42 1 inner
473.2.w.a 504 473.w even 42 1 inner

Hecke kernels

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