Properties

Label 637.2.bc.b
Level $637$
Weight $2$
Character orbit 637.bc
Analytic conductor $5.086$
Analytic rank $0$
Dimension $32$
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] = [637,2,Mod(31,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([2, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bc (of order \(12\), degree \(4\), not 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 91)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + 12 q^{3} + 6 q^{5} - 16 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + 12 q^{3} + 6 q^{5} - 16 q^{8} + 8 q^{9} - 10 q^{11} - 44 q^{15} + 12 q^{16} - 4 q^{18} - 12 q^{19} - 8 q^{22} + 12 q^{24} - 24 q^{26} + 16 q^{29} - 24 q^{31} + 4 q^{32} - 48 q^{33} - 8 q^{37} - 6 q^{39} + 132 q^{40} - 42 q^{44} + 24 q^{45} + 12 q^{46} - 30 q^{47} + 88 q^{50} - 36 q^{52} - 12 q^{53} - 78 q^{54} + 40 q^{57} + 26 q^{58} + 54 q^{59} + 16 q^{60} + 48 q^{61} - 8 q^{65} - 12 q^{66} + 16 q^{67} + 48 q^{68} - 36 q^{71} + 22 q^{72} - 66 q^{73} + 12 q^{74} - 176 q^{78} - 32 q^{79} - 138 q^{80} + 16 q^{81} - 84 q^{85} + 42 q^{86} + 24 q^{87} + 60 q^{89} + 48 q^{92} + 6 q^{93} + 72 q^{94} + 42 q^{96} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −2.60347 0.697597i 0.657528 + 0.379624i 4.55935 + 2.63234i −0.654162 + 2.44137i −1.44703 1.44703i 0 −6.22207 6.22207i −1.21177 2.09885i 3.40618 5.89968i
31.2 −1.84208 0.493585i −2.29307 1.32391i 1.41759 + 0.818448i 0.885596 3.30509i 3.57057 + 3.57057i 0 0.489646 + 0.489646i 2.00545 + 3.47355i −3.26268 + 5.65113i
31.3 −1.58070 0.423548i 1.71590 + 0.990678i 0.587174 + 0.339005i 0.742954 2.77274i −2.29273 2.29273i 0 1.52975 + 1.52975i 0.462886 + 0.801742i −2.34878 + 4.06820i
31.4 −0.746505 0.200025i 0.421869 + 0.243566i −1.21479 0.701360i −0.472319 + 1.76272i −0.266208 0.266208i 0 1.85952 + 1.85952i −1.38135 2.39257i 0.705177 1.22140i
31.5 0.694471 + 0.186083i −1.44134 0.832160i −1.28439 0.741542i 0.501414 1.87130i −0.846120 0.846120i 0 −1.77076 1.77076i −0.115021 0.199221i 0.696434 1.20626i
31.6 0.788958 + 0.211401i 2.60482 + 1.50389i −1.15429 0.666428i −0.814012 + 3.03793i 1.73717 + 1.73717i 0 −1.92491 1.92491i 3.02338 + 5.23665i −1.28444 + 2.22472i
31.7 1.53492 + 0.411280i −0.436133 0.251802i 0.454770 + 0.262561i 0.0206096 0.0769162i −0.565867 0.565867i 0 −1.65723 1.65723i −1.37319 2.37844i 0.0632682 0.109584i
31.8 2.38839 + 0.639966i 1.77043 + 1.02216i 3.56278 + 2.05697i 0.423894 1.58199i 3.57432 + 3.57432i 0 3.69606 + 3.69606i 0.589612 + 1.02124i 2.02484 3.50713i
411.1 −2.60347 + 0.697597i 0.657528 0.379624i 4.55935 2.63234i −0.654162 2.44137i −1.44703 + 1.44703i 0 −6.22207 + 6.22207i −1.21177 + 2.09885i 3.40618 + 5.89968i
411.2 −1.84208 + 0.493585i −2.29307 + 1.32391i 1.41759 0.818448i 0.885596 + 3.30509i 3.57057 3.57057i 0 0.489646 0.489646i 2.00545 3.47355i −3.26268 5.65113i
411.3 −1.58070 + 0.423548i 1.71590 0.990678i 0.587174 0.339005i 0.742954 + 2.77274i −2.29273 + 2.29273i 0 1.52975 1.52975i 0.462886 0.801742i −2.34878 4.06820i
411.4 −0.746505 + 0.200025i 0.421869 0.243566i −1.21479 + 0.701360i −0.472319 1.76272i −0.266208 + 0.266208i 0 1.85952 1.85952i −1.38135 + 2.39257i 0.705177 + 1.22140i
411.5 0.694471 0.186083i −1.44134 + 0.832160i −1.28439 + 0.741542i 0.501414 + 1.87130i −0.846120 + 0.846120i 0 −1.77076 + 1.77076i −0.115021 + 0.199221i 0.696434 + 1.20626i
411.6 0.788958 0.211401i 2.60482 1.50389i −1.15429 + 0.666428i −0.814012 3.03793i 1.73717 1.73717i 0 −1.92491 + 1.92491i 3.02338 5.23665i −1.28444 2.22472i
411.7 1.53492 0.411280i −0.436133 + 0.251802i 0.454770 0.262561i 0.0206096 + 0.0769162i −0.565867 + 0.565867i 0 −1.65723 + 1.65723i −1.37319 + 2.37844i 0.0632682 + 0.109584i
411.8 2.38839 0.639966i 1.77043 1.02216i 3.56278 2.05697i 0.423894 + 1.58199i 3.57432 3.57432i 0 3.69606 3.69606i 0.589612 1.02124i 2.02484 + 3.50713i
460.1 −0.639966 2.38839i 1.77043 1.02216i −3.56278 + 2.05697i 1.58199 0.423894i −3.57432 3.57432i 0 3.69606 + 3.69606i 0.589612 1.02124i −2.02484 3.50713i
460.2 −0.411280 1.53492i −0.436133 + 0.251802i −0.454770 + 0.262561i 0.0769162 0.0206096i 0.565867 + 0.565867i 0 −1.65723 1.65723i −1.37319 + 2.37844i −0.0632682 0.109584i
460.3 −0.211401 0.788958i 2.60482 1.50389i 1.15429 0.666428i −3.03793 + 0.814012i −1.73717 1.73717i 0 −1.92491 1.92491i 3.02338 5.23665i 1.28444 + 2.22472i
460.4 −0.186083 0.694471i −1.44134 + 0.832160i 1.28439 0.741542i 1.87130 0.501414i 0.846120 + 0.846120i 0 −1.77076 1.77076i −0.115021 + 0.199221i −0.696434 1.20626i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
13.d odd 4 1 inner
91.bb even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.bc.b 32
7.b odd 2 1 91.2.bb.a 32
7.c even 3 1 91.2.bb.a 32
7.c even 3 1 637.2.i.a 32
7.d odd 6 1 637.2.i.a 32
7.d odd 6 1 inner 637.2.bc.b 32
13.d odd 4 1 inner 637.2.bc.b 32
21.c even 2 1 819.2.fn.e 32
21.h odd 6 1 819.2.fn.e 32
91.i even 4 1 91.2.bb.a 32
91.z odd 12 1 91.2.bb.a 32
91.z odd 12 1 637.2.i.a 32
91.bb even 12 1 637.2.i.a 32
91.bb even 12 1 inner 637.2.bc.b 32
273.o odd 4 1 819.2.fn.e 32
273.cd even 12 1 819.2.fn.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.bb.a 32 7.b odd 2 1
91.2.bb.a 32 7.c even 3 1
91.2.bb.a 32 91.i even 4 1
91.2.bb.a 32 91.z odd 12 1
637.2.i.a 32 7.c even 3 1
637.2.i.a 32 7.d odd 6 1
637.2.i.a 32 91.z odd 12 1
637.2.i.a 32 91.bb even 12 1
637.2.bc.b 32 1.a even 1 1 trivial
637.2.bc.b 32 7.d odd 6 1 inner
637.2.bc.b 32 13.d odd 4 1 inner
637.2.bc.b 32 91.bb even 12 1 inner
819.2.fn.e 32 21.c even 2 1
819.2.fn.e 32 21.h odd 6 1
819.2.fn.e 32 273.o odd 4 1
819.2.fn.e 32 273.cd even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 2 T_{2}^{31} + 2 T_{2}^{30} + 12 T_{2}^{29} - 37 T_{2}^{28} - 118 T_{2}^{27} - 90 T_{2}^{26} - 516 T_{2}^{25} + 1641 T_{2}^{24} + 4354 T_{2}^{23} + 2866 T_{2}^{22} + 17542 T_{2}^{21} - 1650 T_{2}^{20} - 49422 T_{2}^{19} + \cdots + 50625 \) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\). Copy content Toggle raw display