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Properties

Label 475.2.v.b
Level $475$
Weight $2$
Character orbit 475.v
Analytic conductor $3.793$
Analytic rank $0$
Dimension $368$
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(37,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(20))
 
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("475.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.v (of order \(20\), degree \(8\), 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: \(368\)
Relative dimension: \(46\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 368 q - 20 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 368 q - 20 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9} - 12 q^{11} + 56 q^{16} + 20 q^{17} + 10 q^{19} - 52 q^{20} + 32 q^{23} - 16 q^{25} - 32 q^{26} - 40 q^{28} - 60 q^{30} - 84 q^{35} - 96 q^{36} + 16 q^{38} - 100 q^{39} - 20 q^{42} + 56 q^{43} - 160 q^{44} + 100 q^{45} - 56 q^{47} - 20 q^{54} + 20 q^{55} + 70 q^{57} - 72 q^{58} - 12 q^{61} + 72 q^{62} - 160 q^{63} + 40 q^{64} + 36 q^{66} + 176 q^{68} - 48 q^{73} - 32 q^{76} - 156 q^{77} + 288 q^{80} + 16 q^{81} - 108 q^{82} + 136 q^{83} - 88 q^{85} + 152 q^{87} - 200 q^{92} - 164 q^{93} + 46 q^{95} + 44 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −2.57245 + 0.407436i 0.914041 − 1.79391i 4.54939 − 1.47819i 2.23046 + 0.158330i −1.62042 + 4.98715i −1.55502 − 1.55502i −6.45953 + 3.29130i −0.619276 − 0.852360i −5.80225 + 0.501471i
37.2 −2.56186 + 0.405759i 1.49095 − 2.92615i 4.49639 − 1.46097i −1.87524 + 1.21798i −2.63229 + 8.10135i 2.46351 + 2.46351i −6.30417 + 3.21213i −4.57605 − 6.29840i 4.30991 − 3.88119i
37.3 −2.55470 + 0.404625i 0.406101 − 0.797018i 4.46067 − 1.44936i −0.401861 − 2.19966i −0.714974 + 2.20046i −0.946400 − 0.946400i −6.19998 + 3.15905i 1.29304 + 1.77971i 1.91667 + 5.45688i
37.4 −2.54706 + 0.403414i −0.717023 + 1.40724i 4.42264 − 1.43700i −2.22091 − 0.259925i 1.25860 − 3.87357i 0.483664 + 0.483664i −6.08954 + 3.10278i 0.297161 + 0.409007i 5.76164 − 0.233904i
37.5 −2.31915 + 0.367317i −0.747180 + 1.46642i 3.34143 − 1.08569i 1.50980 + 1.64940i 1.19418 − 3.67531i −3.47120 − 3.47120i −3.16620 + 1.61326i 0.171237 + 0.235687i −4.10730 − 3.27063i
37.6 −2.19210 + 0.347194i 0.285120 − 0.559580i 2.78263 − 0.904130i −0.769943 + 2.09933i −0.430728 + 1.32565i −0.449429 − 0.449429i −1.83084 + 0.932859i 1.53152 + 2.10796i 0.958914 − 4.86925i
37.7 −2.18623 + 0.346265i −1.27487 + 2.50207i 2.75760 − 0.895997i −0.0802985 + 2.23463i 1.92078 − 5.91156i 1.40321 + 1.40321i −1.77404 + 0.903916i −2.87172 − 3.95259i −0.598222 − 4.91321i
37.8 −2.11642 + 0.335208i −0.0397833 + 0.0780792i 2.46476 − 0.800848i 1.86279 − 1.23693i 0.0580255 − 0.178584i 2.22146 + 2.22146i −1.12951 + 0.575515i 1.75884 + 2.42084i −3.52782 + 3.24229i
37.9 −1.80338 + 0.285628i −0.799518 + 1.56914i 1.26849 − 0.412157i 1.04833 − 1.97510i 0.993645 − 3.05813i −1.19367 − 1.19367i 1.08386 − 0.552256i −0.0596228 − 0.0820638i −1.32639 + 3.86128i
37.10 −1.75877 + 0.278561i 0.504456 − 0.990050i 1.11355 − 0.361814i −1.72301 − 1.42521i −0.611430 + 1.88179i 3.33517 + 3.33517i 1.31552 − 0.670293i 1.03763 + 1.42818i 3.42739 + 2.02664i
37.11 −1.71767 + 0.272052i 1.09467 − 2.14841i 0.974270 − 0.316560i 1.73396 + 1.41188i −1.29580 + 3.98806i −0.0569728 − 0.0569728i 1.51171 − 0.770256i −1.65399 − 2.27653i −3.36247 − 1.95341i
37.12 −1.65481 + 0.262096i −1.44781 + 2.84149i 0.767584 − 0.249403i −1.02329 − 1.98818i 1.65111 − 5.08159i 1.83846 + 1.83846i 1.78081 − 0.907369i −4.21456 − 5.80085i 2.21444 + 3.02187i
37.13 −1.57132 + 0.248873i 1.15941 − 2.27547i 0.504996 − 0.164083i −1.27443 − 1.83734i −1.25550 + 3.86404i −2.09047 − 2.09047i 2.08234 − 1.06101i −2.07019 − 2.84937i 2.45981 + 2.56988i
37.14 −1.33793 + 0.211908i 0.355500 − 0.697707i −0.156950 + 0.0509962i −1.42986 + 1.71916i −0.327785 + 1.00882i −0.837725 − 0.837725i 2.61312 − 1.33145i 1.40294 + 1.93098i 1.54875 − 2.60312i
37.15 −1.29182 + 0.204604i −0.845875 + 1.66012i −0.275185 + 0.0894130i −2.23541 − 0.0540536i 0.753049 − 2.31765i −3.34393 − 3.34393i 2.66792 − 1.35938i −0.277151 − 0.381465i 2.89881 − 0.387547i
37.16 −1.05752 + 0.167494i 0.331611 − 0.650822i −0.811826 + 0.263778i 2.01749 − 0.964220i −0.241675 + 0.743798i −1.56082 − 1.56082i 2.72234 − 1.38710i 1.44975 + 1.99541i −1.97203 + 1.35760i
37.17 −0.966688 + 0.153108i −0.726863 + 1.42655i −0.991070 + 0.322018i −0.468136 + 2.18652i 0.484233 − 1.49032i 2.36650 + 2.36650i 2.65287 − 1.35171i 0.256642 + 0.353238i 0.117768 − 2.18535i
37.18 −0.914725 + 0.144878i −0.357678 + 0.701983i −1.08638 + 0.352987i 1.97349 + 1.05134i 0.225475 − 0.693941i 1.44720 + 1.44720i 2.59297 − 1.32118i 1.39851 + 1.92488i −1.95752 − 0.675772i
37.19 −0.589567 + 0.0933783i 1.41446 − 2.77603i −1.56324 + 0.507928i −1.72206 + 1.42636i −0.574697 + 1.76873i −0.725916 − 0.725916i 1.93792 − 0.987420i −3.94228 − 5.42608i 0.882079 − 1.00174i
37.20 −0.551555 + 0.0873577i −1.48135 + 2.90732i −1.60553 + 0.521669i 2.17187 + 0.531969i 0.563071 − 1.73295i −1.59825 − 1.59825i 1.83510 − 0.935029i −4.49473 − 6.18647i −1.24438 − 0.103681i
See next 80 embeddings (of 368 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner
25.f odd 20 1 inner
475.v even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.v.b ✓ 368
19.b odd 2 1 inner 475.2.v.b ✓ 368
25.f odd 20 1 inner 475.2.v.b ✓ 368
475.v even 20 1 inner 475.2.v.b ✓ 368
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.v.b ✓ 368 1.a even 1 1 trivial
475.2.v.b ✓ 368 19.b odd 2 1 inner
475.2.v.b ✓ 368 25.f odd 20 1 inner
475.2.v.b ✓ 368 475.v even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{368} + 10 T_{2}^{366} - 298 T_{2}^{364} - 3320 T_{2}^{362} + 49843 T_{2}^{360} + \cdots + 57\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display