Properties

Label 639.2.v.b
Level $639$
Weight $2$
Character orbit 639.v
Analytic conductor $5.102$
Analytic rank $0$
Dimension $144$
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(10,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([0, 34]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.v (of order \(35\), degree \(24\), 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: \(144\)
Relative dimension: \(6\) over \(\Q(\zeta_{35})\)
Twist minimal: no (minimal twist has level 213)
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 3 q^{2} - 7 q^{4} - 2 q^{5} + 4 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 3 q^{2} - 7 q^{4} - 2 q^{5} + 4 q^{7} + 8 q^{8} - 13 q^{10} - 5 q^{11} + 8 q^{13} + q^{14} - 61 q^{16} + 28 q^{17} - 2 q^{19} + 26 q^{20} - 3 q^{22} + 20 q^{23} - 30 q^{25} + 38 q^{26} + 25 q^{28} - 75 q^{29} + 3 q^{31} + 69 q^{32} - 130 q^{34} + 64 q^{35} - 24 q^{37} + q^{38} + 154 q^{40} + 9 q^{41} - 31 q^{43} + 51 q^{44} - 143 q^{46} + 27 q^{47} - 8 q^{49} + 59 q^{50} - 84 q^{52} - 22 q^{53} + 177 q^{55} - 9 q^{56} - 113 q^{58} - 112 q^{59} - 11 q^{61} - 99 q^{62} + 88 q^{64} - 17 q^{65} + 116 q^{67} - 302 q^{68} + 94 q^{70} - 31 q^{71} + 44 q^{73} + 63 q^{74} - 32 q^{76} - 17 q^{77} - 22 q^{79} - 75 q^{80} + 15 q^{82} - 102 q^{83} + 31 q^{85} + 50 q^{86} - 97 q^{88} + 14 q^{89} - 102 q^{91} + 59 q^{92} - 28 q^{94} + 154 q^{95} - 133 q^{97} - 225 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.47146 + 0.682079i 0 3.92597 2.34566i 1.04351 3.21160i 0 −0.142926 3.18251i −4.55937 + 4.76873i 0 −0.388430 + 8.64907i
10.2 −1.02477 + 0.282819i 0 −0.746726 + 0.446148i 0.474671 1.46089i 0 0.0904452 + 2.01392i 2.10836 2.20517i 0 −0.0732629 + 1.63132i
10.3 0.155567 0.0429338i 0 −1.69454 + 1.01244i −1.12939 + 3.47591i 0 −0.145691 3.24406i −0.443198 + 0.463549i 0 −0.0264622 + 0.589228i
10.4 0.667655 0.184261i 0 −1.30509 + 0.779753i 0.108964 0.335356i 0 0.0586866 + 1.30676i −1.68495 + 1.76232i 0 0.0109571 0.243980i
10.5 1.98208 0.547020i 0 1.91253 1.14268i −0.531712 + 1.63644i 0 0.152517 + 3.39604i 0.323819 0.338688i 0 −0.158731 + 3.53442i
10.6 2.14279 0.591372i 0 2.52492 1.50857i 0.276863 0.852096i 0 −0.192490 4.28613i 1.44593 1.51232i 0 0.0893526 1.98959i
19.1 −0.837190 1.95870i 0 −1.75351 + 1.83403i 0.0242415 + 0.0746077i 0 1.16020 + 1.01363i 1.07175 + 0.402234i 0 0.125840 0.109943i
19.2 −0.343458 0.803560i 0 0.854380 0.893611i 0.451067 + 1.38824i 0 −2.05235 1.79308i −2.64783 0.993749i 0 0.960614 0.839263i
19.3 −0.201911 0.472394i 0 1.19974 1.25483i −1.20057 3.69498i 0 −1.29544 1.13179i −1.79697 0.674413i 0 −1.50308 + 1.31320i
19.4 0.331164 + 0.774798i 0 0.891484 0.932419i 0.918886 + 2.82804i 0 3.26260 + 2.85045i 2.59541 + 0.974075i 0 −1.88686 + 1.64850i
19.5 0.524120 + 1.22624i 0 0.153162 0.160195i 0.181820 + 0.559585i 0 −1.62566 1.42029i 2.77375 + 1.04101i 0 −0.590890 + 0.516245i
19.6 0.899759 + 2.10509i 0 −2.23972 + 2.34256i −0.983544 3.02704i 0 3.56293 + 3.11284i −2.65983 0.998252i 0 5.48724 4.79405i
64.1 −2.47146 0.682079i 0 3.92597 + 2.34566i 1.04351 + 3.21160i 0 −0.142926 + 3.18251i −4.55937 4.76873i 0 −0.388430 8.64907i
64.2 −1.02477 0.282819i 0 −0.746726 0.446148i 0.474671 + 1.46089i 0 0.0904452 2.01392i 2.10836 + 2.20517i 0 −0.0732629 1.63132i
64.3 0.155567 + 0.0429338i 0 −1.69454 1.01244i −1.12939 3.47591i 0 −0.145691 + 3.24406i −0.443198 0.463549i 0 −0.0264622 0.589228i
64.4 0.667655 + 0.184261i 0 −1.30509 0.779753i 0.108964 + 0.335356i 0 0.0586866 1.30676i −1.68495 1.76232i 0 0.0109571 + 0.243980i
64.5 1.98208 + 0.547020i 0 1.91253 + 1.14268i −0.531712 1.63644i 0 0.152517 3.39604i 0.323819 + 0.338688i 0 −0.158731 3.53442i
64.6 2.14279 + 0.591372i 0 2.52492 + 1.50857i 0.276863 + 0.852096i 0 −0.192490 + 4.28613i 1.44593 + 1.51232i 0 0.0893526 + 1.98959i
73.1 −0.0862756 + 1.92108i 0 −1.69114 0.152205i −1.35159 4.15978i 0 −1.07120 0.295631i −0.0779628 + 0.575544i 0 8.10785 2.23763i
73.2 −0.0835467 + 1.86031i 0 −1.46183 0.131568i 0.284549 + 0.875753i 0 3.93254 + 1.08531i −0.133047 + 0.982191i 0 −1.65295 + 0.456184i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.g even 35 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.v.b 144
3.b odd 2 1 213.2.m.b 144
71.g even 35 1 inner 639.2.v.b 144
213.o odd 70 1 213.2.m.b 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.2.m.b 144 3.b odd 2 1
213.2.m.b 144 213.o odd 70 1
639.2.v.b 144 1.a even 1 1 trivial
639.2.v.b 144 71.g even 35 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} - 3 T_{2}^{143} + 2 T_{2}^{142} + 3 T_{2}^{141} - 9 T_{2}^{140} - 58 T_{2}^{139} - 86 T_{2}^{138} + 660 T_{2}^{137} - 827 T_{2}^{136} + 4695 T_{2}^{135} + 4082 T_{2}^{134} - 25220 T_{2}^{133} + 45962 T_{2}^{132} - 139132 T_{2}^{131} + \cdots + 49014001 \) acting on \(S_{2}^{\mathrm{new}}(639, [\chi])\). Copy content Toggle raw display