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Properties

Label 639.2.j.c
Level $639$
Weight $2$
Character orbit 639.j
Analytic conductor $5.102$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [639,2,Mod(37,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.j (of order \(7\), degree \(6\), 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: \(5.10244068916\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{7})\)
Twist minimal: no (minimal twist has level 71)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{2} - 7 q^{4} + 10 q^{5} - 3 q^{7} - 17 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{2} - 7 q^{4} + 10 q^{5} - 3 q^{7} - 17 q^{8} - 22 q^{10} - 7 q^{11} + q^{13} + 2 q^{14} - 15 q^{16} + 24 q^{17} + q^{19} - 8 q^{20} + 19 q^{22} - 21 q^{23} + 12 q^{25} + 32 q^{26} + 13 q^{28} + 11 q^{29} - 30 q^{31} + 5 q^{32} + 15 q^{34} + 44 q^{35} + 37 q^{37} - 24 q^{38} + 10 q^{40} - 16 q^{41} - 43 q^{43} - 33 q^{44} - 54 q^{46} + 16 q^{47} - 8 q^{49} - 20 q^{50} - 53 q^{52} - 65 q^{53} + 13 q^{55} + 21 q^{56} - 12 q^{58} - 30 q^{59} - 18 q^{61} - 8 q^{62} + 17 q^{64} - 14 q^{65} + 29 q^{67} + 13 q^{68} + 64 q^{70} + 5 q^{71} - 61 q^{73} + 35 q^{74} - 58 q^{76} + 82 q^{77} + 55 q^{79} + 22 q^{80} + 18 q^{82} - 45 q^{83} - 22 q^{85} + 89 q^{86} + 112 q^{88} + 8 q^{89} + 35 q^{91} - 17 q^{92} + q^{94} - 50 q^{95} - 26 q^{97} - 7 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.44855 + 1.81642i 0 −0.756051 − 3.31248i 3.98144 0 2.24758 + 2.81838i 2.92560 + 1.40890i 0 −5.76729 + 7.23196i
37.2 −0.599543 + 0.751803i 0 0.239286 + 1.04838i 0.464535 0 −0.437866 − 0.549067i −2.66437 − 1.28309i 0 −0.278509 + 0.349239i
37.3 −0.191306 + 0.239890i 0 0.424093 + 1.85807i 0.883903 0 −2.40928 − 3.02114i −1.07975 − 0.519982i 0 −0.169096 + 0.212040i
37.4 1.18346 − 1.48402i 0 −0.356677 − 1.56270i 0.529055 0 0.212498 + 0.266464i 0.679118 + 0.327046i 0 0.626117 − 0.785126i
37.5 1.55593 − 1.95108i 0 −0.940734 − 4.12162i −2.61195 0 −1.63739 − 2.05323i −5.00855 − 2.41199i 0 −4.06401 + 5.09611i
91.1 −0.471544 − 2.06597i 0 −2.24394 + 1.08063i −0.618957 0 0.245991 − 1.07776i 0.648183 + 0.812795i 0 0.291866 + 1.27875i
91.2 −0.0919395 − 0.402813i 0 1.64813 − 0.793699i 2.73724 0 0.0252603 − 0.110672i −0.986458 − 1.23698i 0 −0.251661 − 1.10260i
91.3 0.0812940 + 0.356172i 0 1.68169 − 0.809858i −3.28338 0 −0.0815359 + 0.357232i 0.880722 + 1.10439i 0 −0.266919 − 1.16945i
91.4 0.469495 + 2.05699i 0 −2.20886 + 1.06373i −0.419938 0 −0.760462 + 3.33180i −0.594137 − 0.745024i 0 −0.197159 − 0.863809i
91.5 0.512695 + 2.24626i 0 −2.98090 + 1.43552i 3.13999 0 0.916758 − 4.01658i −1.87978 − 2.35717i 0 1.60985 + 7.05323i
172.1 −2.35521 − 1.13421i 0 3.01359 + 3.77892i 1.56041 0 −2.07157 + 0.997613i −1.64817 − 7.22110i 0 −3.67509 − 1.76983i
172.2 −0.196395 − 0.0945787i 0 −1.21735 − 1.52651i −3.36610 0 −0.907124 + 0.436848i 0.191717 + 0.839967i 0 0.661083 + 0.318361i
172.3 0.154599 + 0.0744510i 0 −1.22862 − 1.54064i −0.566787 0 −1.38741 + 0.668143i −0.151607 − 0.664234i 0 −0.0876248 − 0.0421979i
172.4 0.774825 + 0.373136i 0 −0.785856 − 0.985433i 3.62688 0 2.52089 − 1.21399i −0.623933 − 2.73363i 0 2.81020 + 1.35332i
172.5 2.12218 + 1.02199i 0 2.21220 + 2.77401i −1.05635 0 2.02367 − 0.974546i 0.811409 + 3.55501i 0 −2.24176 − 1.07957i
190.1 −1.44855 − 1.81642i 0 −0.756051 + 3.31248i 3.98144 0 2.24758 − 2.81838i 2.92560 − 1.40890i 0 −5.76729 − 7.23196i
190.2 −0.599543 − 0.751803i 0 0.239286 − 1.04838i 0.464535 0 −0.437866 + 0.549067i −2.66437 + 1.28309i 0 −0.278509 − 0.349239i
190.3 −0.191306 − 0.239890i 0 0.424093 − 1.85807i 0.883903 0 −2.40928 + 3.02114i −1.07975 + 0.519982i 0 −0.169096 − 0.212040i
190.4 1.18346 + 1.48402i 0 −0.356677 + 1.56270i 0.529055 0 0.212498 − 0.266464i 0.679118 − 0.327046i 0 0.626117 + 0.785126i
190.5 1.55593 + 1.95108i 0 −0.940734 + 4.12162i −2.61195 0 −1.63739 + 2.05323i −5.00855 + 2.41199i 0 −4.06401 − 5.09611i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.j.c 30
3.b odd 2 1 71.2.d.a ✓ 30
71.d even 7 1 inner 639.2.j.c 30
213.k odd 14 1 71.2.d.a ✓ 30
213.k odd 14 1 5041.2.a.l 15
213.l even 14 1 5041.2.a.m 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.2.d.a ✓ 30 3.b odd 2 1
71.2.d.a ✓ 30 213.k odd 14 1
639.2.j.c 30 1.a even 1 1 trivial
639.2.j.c 30 71.d even 7 1 inner
5041.2.a.l 15 213.k odd 14 1
5041.2.a.m 15 213.l even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - 3 T_{2}^{29} + 13 T_{2}^{28} - 19 T_{2}^{27} + 69 T_{2}^{26} - 109 T_{2}^{25} + 511 T_{2}^{24} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(639, [\chi])\). Copy content Toggle raw display