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Properties

Label 475.2.bb.b
Level $475$
Weight $2$
Character orbit 475.bb
Analytic conductor $3.793$
Analytic rank $0$
Dimension $96$
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] = [475,2,Mod(32,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.bb (of order \(36\), 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(8\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 12 q^{2} + 12 q^{3} - 12 q^{6} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 12 q^{2} + 12 q^{3} - 12 q^{6} + 18 q^{8} - 12 q^{11} + 18 q^{12} + 12 q^{13} + 12 q^{16} + 30 q^{17} + 24 q^{21} + 24 q^{22} - 48 q^{26} + 18 q^{27} - 36 q^{31} - 18 q^{32} - 90 q^{33} + 24 q^{36} - 54 q^{38} + 12 q^{41} - 24 q^{42} - 48 q^{43} - 36 q^{46} + 24 q^{47} - 60 q^{48} - 96 q^{51} + 30 q^{53} + 66 q^{57} - 120 q^{58} - 48 q^{61} - 60 q^{62} + 126 q^{63} + 72 q^{66} - 108 q^{67} - 18 q^{68} - 24 q^{71} - 48 q^{72} - 6 q^{73} + 60 q^{76} - 168 q^{77} + 138 q^{78} - 120 q^{81} - 60 q^{82} - 180 q^{86} + 6 q^{87} + 198 q^{88} - 24 q^{91} + 72 q^{92} + 90 q^{93} - 192 q^{96} + 72 q^{97} + 90 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −0.215520 + 2.46340i −0.236608 − 0.507407i −4.05227 − 0.714525i 0 1.30094 − 0.473503i 2.99058 − 0.801323i 1.35349 − 5.05128i 1.72688 − 2.05802i 0
32.2 −0.184204 + 2.10547i 1.05623 + 2.26510i −2.42944 − 0.428375i 0 −4.96365 + 1.80662i −2.23442 + 0.598711i 0.255410 − 0.953204i −2.08669 + 2.48682i 0
32.3 −0.148765 + 1.70039i −1.25242 − 2.68582i −0.899578 − 0.158620i 0 4.75325 − 1.73004i −2.71511 + 0.727512i −0.480007 + 1.79141i −3.71671 + 4.42940i 0
32.4 −0.0741107 + 0.847089i −0.344068 − 0.737855i 1.25755 + 0.221740i 0 0.650528 − 0.236773i 3.83889 − 1.02863i −0.721192 + 2.69152i 1.50231 − 1.79039i 0
32.5 −0.0260387 + 0.297624i 0.818952 + 1.75625i 1.88171 + 0.331797i 0 −0.544026 + 0.198009i 0.323332 − 0.0866366i −0.302398 + 1.12856i −0.485360 + 0.578430i 0
32.6 0.119667 − 1.36780i 0.753423 + 1.61572i 0.113072 + 0.0199376i 0 2.30013 − 0.837180i 0.254507 − 0.0681949i 0.751529 − 2.80474i −0.114542 + 0.136506i 0
32.7 0.140400 − 1.60478i −0.303112 − 0.650026i −0.585979 − 0.103324i 0 −1.08570 + 0.395164i −1.05988 + 0.283994i 0.585783 − 2.18617i 1.59771 − 1.90407i 0
32.8 0.230116 − 2.63024i −0.901235 − 1.93270i −4.89558 − 0.863222i 0 −5.29086 + 1.92571i −1.39789 + 0.374564i −2.03032 + 7.57725i −0.994759 + 1.18551i 0
143.1 −1.31842 + 1.88289i −0.0291608 + 0.333309i −1.12303 − 3.08549i 0 −0.589140 − 0.494347i −0.282738 − 1.05519i 2.84973 + 0.763583i 2.84418 + 0.501505i 0
143.2 −0.583359 + 0.833123i 0.155202 − 1.77397i 0.330254 + 0.907365i 0 1.38740 + 1.16417i −0.267922 − 0.999898i −2.91340 − 0.780644i −0.168464 − 0.0297048i 0
143.3 −0.259809 + 0.371046i −0.264820 + 3.02691i 0.613866 + 1.68658i 0 −1.05432 − 0.884679i −0.508516 − 1.89781i −1.66034 − 0.444888i −6.13763 − 1.08223i 0
143.4 −0.0545146 + 0.0778549i −0.111718 + 1.27695i 0.680951 + 1.87090i 0 −0.0933263 − 0.0783101i 0.883627 + 3.29774i −0.366390 − 0.0981738i 1.33631 + 0.235628i 0
143.5 0.425033 − 0.607009i 0.144020 − 1.64615i 0.496232 + 1.36339i 0 −0.938018 − 0.787091i 0.706367 + 2.63620i 2.47005 + 0.661847i 0.265340 + 0.0467866i 0
143.6 0.983828 − 1.40505i 0.215709 − 2.46557i −0.322212 − 0.885271i 0 −3.25203 − 2.72877i −0.0743036 − 0.277305i 1.75276 + 0.469650i −3.07806 − 0.542745i 0
143.7 1.06046 − 1.51450i −0.101890 + 1.16461i −0.485079 − 1.33274i 0 1.65575 + 1.38934i −1.17582 − 4.38823i 1.03888 + 0.278366i 1.60848 + 0.283619i 0
143.8 1.34445 − 1.92007i −0.165797 + 1.89507i −1.19510 − 3.28350i 0 3.41577 + 2.86617i 0.719307 + 2.68449i −3.38309 − 0.906496i −0.609382 − 0.107451i 0
193.1 −0.215520 − 2.46340i −0.236608 + 0.507407i −4.05227 + 0.714525i 0 1.30094 + 0.473503i 2.99058 + 0.801323i 1.35349 + 5.05128i 1.72688 + 2.05802i 0
193.2 −0.184204 − 2.10547i 1.05623 − 2.26510i −2.42944 + 0.428375i 0 −4.96365 − 1.80662i −2.23442 − 0.598711i 0.255410 + 0.953204i −2.08669 − 2.48682i 0
193.3 −0.148765 − 1.70039i −1.25242 + 2.68582i −0.899578 + 0.158620i 0 4.75325 + 1.73004i −2.71511 − 0.727512i −0.480007 − 1.79141i −3.71671 − 4.42940i 0
193.4 −0.0741107 − 0.847089i −0.344068 + 0.737855i 1.25755 − 0.221740i 0 0.650528 + 0.236773i 3.83889 + 1.02863i −0.721192 − 2.69152i 1.50231 + 1.79039i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.f odd 18 1 inner
95.r even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.bb.b 96
5.b even 2 1 95.2.r.a ✓ 96
5.c odd 4 1 95.2.r.a ✓ 96
5.c odd 4 1 inner 475.2.bb.b 96
15.d odd 2 1 855.2.dl.a 96
15.e even 4 1 855.2.dl.a 96
19.f odd 18 1 inner 475.2.bb.b 96
95.o odd 18 1 95.2.r.a ✓ 96
95.r even 36 1 95.2.r.a ✓ 96
95.r even 36 1 inner 475.2.bb.b 96
285.bf even 18 1 855.2.dl.a 96
285.bj odd 36 1 855.2.dl.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.r.a ✓ 96 5.b even 2 1
95.2.r.a ✓ 96 5.c odd 4 1
95.2.r.a ✓ 96 95.o odd 18 1
95.2.r.a ✓ 96 95.r even 36 1
475.2.bb.b 96 1.a even 1 1 trivial
475.2.bb.b 96 5.c odd 4 1 inner
475.2.bb.b 96 19.f odd 18 1 inner
475.2.bb.b 96 95.r even 36 1 inner
855.2.dl.a 96 15.d odd 2 1
855.2.dl.a 96 15.e even 4 1
855.2.dl.a 96 285.bf even 18 1
855.2.dl.a 96 285.bj odd 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 12 T_{2}^{95} + 72 T_{2}^{94} - 294 T_{2}^{93} + 897 T_{2}^{92} - 1962 T_{2}^{91} + \cdots + 531441 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display