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Properties

Label 475.2.p.h
Level $475$
Weight $2$
Character orbit 475.p
Analytic conductor $3.793$
Analytic rank $0$
Dimension $24$
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(107,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.p (of order \(12\), 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{3} - 20 q^{6} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{3} - 20 q^{6} + 12 q^{7} + 16 q^{11} + 12 q^{13} - 4 q^{16} - 16 q^{17} + 12 q^{21} + 24 q^{22} + 32 q^{26} - 4 q^{28} + 36 q^{32} + 36 q^{33} - 52 q^{36} - 38 q^{38} + 18 q^{42} - 2 q^{43} - 2 q^{47} - 96 q^{48} + 6 q^{52} - 36 q^{53} - 48 q^{57} + 56 q^{58} + 12 q^{61} - 22 q^{62} - 30 q^{63} - 40 q^{66} - 48 q^{67} + 116 q^{68} - 24 q^{71} - 96 q^{72} + 28 q^{73} + 8 q^{76} + 12 q^{77} - 84 q^{78} + 24 q^{81} - 20 q^{82} + 24 q^{83} - 12 q^{86} + 100 q^{87} - 108 q^{91} + 4 q^{92} - 12 q^{93} - 48 q^{96} + 6 q^{97} + 78 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.94595 − 0.521416i −0.106656 + 0.398044i 1.78279 + 1.02930i 0 0.415093 − 0.718962i 3.08761 + 3.08761i −0.0834685 − 0.0834685i 2.45101 + 1.41509i 0
107.2 −1.79221 − 0.480221i 0.593155 − 2.21369i 1.24936 + 0.721316i 0 −2.12612 + 3.68255i 0.320821 + 0.320821i 0.731261 + 0.731261i −1.95049 − 1.12612i 0
107.3 −0.791474 − 0.212075i −0.173661 + 0.648112i −1.15060 − 0.664297i 0 0.274896 − 0.476135i −0.921839 − 0.921839i 1.92858 + 1.92858i 2.20819 + 1.27490i 0
107.4 0.349035 + 0.0935236i −0.309771 + 1.15608i −1.61897 − 0.934714i 0 −0.216242 + 0.374542i 0.933157 + 0.933157i −0.988682 − 0.988682i 1.35751 + 0.783758i 0
107.5 1.69754 + 0.454853i 0.117117 − 0.437086i 0.942686 + 0.544260i 0 0.397620 − 0.688698i 0.671894 + 0.671894i −1.13268 − 1.13268i 2.42075 + 1.39762i 0
107.6 2.48306 + 0.665335i −0.754159 + 2.81456i 3.99088 + 2.30414i 0 −3.74525 + 6.48696i −1.09165 − 1.09165i 4.74114 + 4.74114i −4.75491 − 2.74525i 0
293.1 −1.94595 + 0.521416i −0.106656 − 0.398044i 1.78279 − 1.02930i 0 0.415093 + 0.718962i 3.08761 − 3.08761i −0.0834685 + 0.0834685i 2.45101 − 1.41509i 0
293.2 −1.79221 + 0.480221i 0.593155 + 2.21369i 1.24936 − 0.721316i 0 −2.12612 − 3.68255i 0.320821 − 0.320821i 0.731261 − 0.731261i −1.95049 + 1.12612i 0
293.3 −0.791474 + 0.212075i −0.173661 − 0.648112i −1.15060 + 0.664297i 0 0.274896 + 0.476135i −0.921839 + 0.921839i 1.92858 − 1.92858i 2.20819 − 1.27490i 0
293.4 0.349035 − 0.0935236i −0.309771 − 1.15608i −1.61897 + 0.934714i 0 −0.216242 − 0.374542i 0.933157 − 0.933157i −0.988682 + 0.988682i 1.35751 − 0.783758i 0
293.5 1.69754 − 0.454853i 0.117117 + 0.437086i 0.942686 − 0.544260i 0 0.397620 + 0.688698i 0.671894 − 0.671894i −1.13268 + 1.13268i 2.42075 − 1.39762i 0
293.6 2.48306 − 0.665335i −0.754159 − 2.81456i 3.99088 − 2.30414i 0 −3.74525 − 6.48696i −1.09165 + 1.09165i 4.74114 − 4.74114i −4.75491 + 2.74525i 0
407.1 −0.521416 − 1.94595i −0.398044 + 0.106656i −1.78279 + 1.02930i 0 0.415093 + 0.718962i 3.08761 + 3.08761i 0.0834685 + 0.0834685i −2.45101 + 1.41509i 0
407.2 −0.480221 − 1.79221i 2.21369 − 0.593155i −1.24936 + 0.721316i 0 −2.12612 − 3.68255i 0.320821 + 0.320821i −0.731261 − 0.731261i 1.95049 − 1.12612i 0
407.3 −0.212075 − 0.791474i −0.648112 + 0.173661i 1.15060 − 0.664297i 0 0.274896 + 0.476135i −0.921839 − 0.921839i −1.92858 − 1.92858i −2.20819 + 1.27490i 0
407.4 0.0935236 + 0.349035i −1.15608 + 0.309771i 1.61897 − 0.934714i 0 −0.216242 − 0.374542i 0.933157 + 0.933157i 0.988682 + 0.988682i −1.35751 + 0.783758i 0
407.5 0.454853 + 1.69754i 0.437086 − 0.117117i −0.942686 + 0.544260i 0 0.397620 + 0.688698i 0.671894 + 0.671894i 1.13268 + 1.13268i −2.42075 + 1.39762i 0
407.6 0.665335 + 2.48306i −2.81456 + 0.754159i −3.99088 + 2.30414i 0 −3.74525 − 6.48696i −1.09165 − 1.09165i −4.74114 − 4.74114i 4.75491 − 2.74525i 0
468.1 −0.521416 + 1.94595i −0.398044 − 0.106656i −1.78279 − 1.02930i 0 0.415093 − 0.718962i 3.08761 − 3.08761i 0.0834685 − 0.0834685i −2.45101 − 1.41509i 0
468.2 −0.480221 + 1.79221i 2.21369 + 0.593155i −1.24936 − 0.721316i 0 −2.12612 + 3.68255i 0.320821 − 0.320821i −0.731261 + 0.731261i 1.95049 + 1.12612i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.p.h 24
5.b even 2 1 95.2.l.c ✓ 24
5.c odd 4 1 95.2.l.c ✓ 24
5.c odd 4 1 inner 475.2.p.h 24
15.d odd 2 1 855.2.cj.e 24
15.e even 4 1 855.2.cj.e 24
19.d odd 6 1 inner 475.2.p.h 24
95.h odd 6 1 95.2.l.c ✓ 24
95.l even 12 1 95.2.l.c ✓ 24
95.l even 12 1 inner 475.2.p.h 24
285.q even 6 1 855.2.cj.e 24
285.w odd 12 1 855.2.cj.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.l.c ✓ 24 5.b even 2 1
95.2.l.c ✓ 24 5.c odd 4 1
95.2.l.c ✓ 24 95.h odd 6 1
95.2.l.c ✓ 24 95.l even 12 1
475.2.p.h 24 1.a even 1 1 trivial
475.2.p.h 24 5.c odd 4 1 inner
475.2.p.h 24 19.d odd 6 1 inner
475.2.p.h 24 95.l even 12 1 inner
855.2.cj.e 24 15.d odd 2 1
855.2.cj.e 24 15.e even 4 1
855.2.cj.e 24 285.q even 6 1
855.2.cj.e 24 285.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\):

\( T_{2}^{24} - 41 T_{2}^{20} - 36 T_{2}^{19} + 204 T_{2}^{17} + 1443 T_{2}^{16} + 1476 T_{2}^{15} + \cdots + 625 \) Copy content Toggle raw display
\( T_{3}^{24} + 6 T_{3}^{23} + 18 T_{3}^{22} + 36 T_{3}^{21} + 3 T_{3}^{20} - 168 T_{3}^{19} - 414 T_{3}^{18} + \cdots + 1 \) Copy content Toggle raw display