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Properties

Label 374.2.o.b
Level $374$
Weight $2$
Character orbit 374.o
Analytic conductor $2.986$
Analytic rank $0$
Dimension $80$
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] = [374,2,Mod(47,374)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(374, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([16, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("374.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 374 = 2 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 374.o (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: \(2.98640503560\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{3} + 20 q^{4} - 4 q^{5} - 6 q^{6} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{3} + 20 q^{4} - 4 q^{5} - 6 q^{6} - 2 q^{7} - 4 q^{10} - 12 q^{11} - 4 q^{12} + 32 q^{13} + 2 q^{14} - 20 q^{16} + 6 q^{17} - 16 q^{18} + 4 q^{20} - 16 q^{21} + 2 q^{22} - 12 q^{23} + 6 q^{24} + 16 q^{27} + 2 q^{28} - 2 q^{29} + 16 q^{30} + 12 q^{31} - 44 q^{33} - 16 q^{34} - 24 q^{37} - 24 q^{38} - 36 q^{39} - 6 q^{40} + 18 q^{41} - 28 q^{44} + 20 q^{45} - 8 q^{46} + 24 q^{47} + 4 q^{48} - 24 q^{50} + 4 q^{51} + 28 q^{52} - 16 q^{54} + 8 q^{56} + 54 q^{57} - 2 q^{58} - 32 q^{61} - 8 q^{62} + 46 q^{63} + 20 q^{64} - 16 q^{65} + 168 q^{67} - 6 q^{68} - 20 q^{69} + 20 q^{71} + 36 q^{72} + 4 q^{73} - 24 q^{74} - 64 q^{75} + 64 q^{78} + 18 q^{79} + 6 q^{80} - 64 q^{81} + 32 q^{82} - 4 q^{84} - 52 q^{85} - 64 q^{86} - 12 q^{88} - 48 q^{89} - 20 q^{90} + 46 q^{91} + 2 q^{92} - 114 q^{95} + 4 q^{96} - 24 q^{97} - 120 q^{98} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0.587785 + 0.809017i −1.51328 + 2.96998i −0.309017 + 0.951057i −0.206362 − 1.30292i −3.29225 + 0.521440i −1.58535 − 3.11141i −0.951057 + 0.309017i −4.76740 − 6.56176i 0.932784 − 0.932784i
47.2 0.587785 + 0.809017i −1.35776 + 2.66474i −0.309017 + 0.951057i 0.239245 + 1.51053i −2.95389 + 0.467851i 2.14191 + 4.20374i −0.951057 + 0.309017i −3.49401 − 4.80909i −1.08142 + 1.08142i
47.3 0.587785 + 0.809017i −0.677007 + 1.32870i −0.309017 + 0.951057i 0.577482 + 3.64608i −1.47288 + 0.233281i −1.80550 − 3.54349i −0.951057 + 0.309017i 0.456246 + 0.627969i −2.61030 + 2.61030i
47.4 0.587785 + 0.809017i −0.640225 + 1.25651i −0.309017 + 0.951057i −0.351700 − 2.22055i −1.39285 + 0.220606i 1.42359 + 2.79396i −0.951057 + 0.309017i 0.594422 + 0.818152i 1.58974 − 1.58974i
47.5 0.587785 + 0.809017i −0.462887 + 0.908466i −0.309017 + 0.951057i −0.607475 − 3.83544i −1.00704 + 0.159500i −1.34284 − 2.63548i −0.951057 + 0.309017i 1.15231 + 1.58602i 2.74587 − 2.74587i
47.6 0.587785 + 0.809017i −0.0443789 + 0.0870984i −0.309017 + 0.951057i 0.324133 + 2.04649i −0.0965494 + 0.0152919i 0.239671 + 0.470381i −0.951057 + 0.309017i 1.75774 + 2.41932i −1.46513 + 1.46513i
47.7 0.587785 + 0.809017i 0.270278 − 0.530450i −0.309017 + 0.951057i −0.0901303 − 0.569060i 0.588008 − 0.0931313i 0.406505 + 0.797811i −0.951057 + 0.309017i 1.55503 + 2.14031i 0.407402 − 0.407402i
47.8 0.587785 + 0.809017i 0.780930 − 1.53266i −0.309017 + 0.951057i 0.360830 + 2.27819i 1.69897 − 0.269090i 1.06914 + 2.09830i −0.951057 + 0.309017i 0.0241558 + 0.0332476i −1.63101 + 1.63101i
47.9 0.587785 + 0.809017i 1.18370 − 2.32313i −0.309017 + 0.951057i −0.605034 − 3.82003i 2.57521 − 0.407874i 0.331809 + 0.651212i −0.951057 + 0.309017i −2.23246 − 3.07271i 2.73484 − 2.73484i
47.10 0.587785 + 0.809017i 1.42179 − 2.79041i −0.309017 + 0.951057i 0.222282 + 1.40343i 3.09320 − 0.489914i −1.52098 − 2.98509i −0.951057 + 0.309017i −4.00158 − 5.50770i −1.00475 + 1.00475i
81.1 −0.587785 + 0.809017i −2.79041 + 1.42179i −0.309017 − 0.951057i 1.40343 + 0.222282i 0.489914 − 3.09320i 2.98509 + 1.52098i 0.951057 + 0.309017i 4.00158 − 5.50770i −1.00475 + 1.00475i
81.2 −0.587785 + 0.809017i −2.32313 + 1.18370i −0.309017 − 0.951057i −3.82003 − 0.605034i 0.407874 − 2.57521i −0.651212 − 0.331809i 0.951057 + 0.309017i 2.23246 − 3.07271i 2.73484 − 2.73484i
81.3 −0.587785 + 0.809017i −1.53266 + 0.780930i −0.309017 − 0.951057i 2.27819 + 0.360830i 0.269090 − 1.69897i −2.09830 − 1.06914i 0.951057 + 0.309017i −0.0241558 + 0.0332476i −1.63101 + 1.63101i
81.4 −0.587785 + 0.809017i −0.530450 + 0.270278i −0.309017 − 0.951057i −0.569060 − 0.0901303i 0.0931313 − 0.588008i −0.797811 − 0.406505i 0.951057 + 0.309017i −1.55503 + 2.14031i 0.407402 − 0.407402i
81.5 −0.587785 + 0.809017i 0.0870984 − 0.0443789i −0.309017 − 0.951057i 2.04649 + 0.324133i −0.0152919 + 0.0965494i −0.470381 − 0.239671i 0.951057 + 0.309017i −1.75774 + 2.41932i −1.46513 + 1.46513i
81.6 −0.587785 + 0.809017i 0.908466 − 0.462887i −0.309017 − 0.951057i −3.83544 − 0.607475i −0.159500 + 1.00704i 2.63548 + 1.34284i 0.951057 + 0.309017i −1.15231 + 1.58602i 2.74587 − 2.74587i
81.7 −0.587785 + 0.809017i 1.25651 − 0.640225i −0.309017 − 0.951057i −2.22055 − 0.351700i −0.220606 + 1.39285i −2.79396 − 1.42359i 0.951057 + 0.309017i −0.594422 + 0.818152i 1.58974 − 1.58974i
81.8 −0.587785 + 0.809017i 1.32870 − 0.677007i −0.309017 − 0.951057i 3.64608 + 0.577482i −0.233281 + 1.47288i 3.54349 + 1.80550i 0.951057 + 0.309017i −0.456246 + 0.627969i −2.61030 + 2.61030i
81.9 −0.587785 + 0.809017i 2.66474 − 1.35776i −0.309017 − 0.951057i 1.51053 + 0.239245i −0.467851 + 2.95389i −4.20374 − 2.14191i 0.951057 + 0.309017i 3.49401 − 4.80909i −1.08142 + 1.08142i
81.10 −0.587785 + 0.809017i 2.96998 − 1.51328i −0.309017 − 0.951057i −1.30292 − 0.206362i −0.521440 + 3.29225i 3.11141 + 1.58535i 0.951057 + 0.309017i 4.76740 − 6.56176i 0.932784 − 0.932784i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
17.c even 4 1 inner
187.p even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 374.2.o.b ✓ 80
11.c even 5 1 inner 374.2.o.b ✓ 80
17.c even 4 1 inner 374.2.o.b ✓ 80
187.p even 20 1 inner 374.2.o.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
374.2.o.b ✓ 80 1.a even 1 1 trivial
374.2.o.b ✓ 80 11.c even 5 1 inner
374.2.o.b ✓ 80 17.c even 4 1 inner
374.2.o.b ✓ 80 187.p even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} - 4 T_{3}^{79} + 8 T_{3}^{78} - 28 T_{3}^{77} - 30 T_{3}^{76} + 336 T_{3}^{75} + \cdots + 492884401 \) acting on \(S_{2}^{\mathrm{new}}(374, [\chi])\). Copy content Toggle raw display