Properties

Label 840.2.br.b
Level $840$
Weight $2$
Character orbit 840.br
Analytic conductor $6.707$
Analytic rank $0$
Dimension $72$
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] = [840,2,Mod(43,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.br (of order \(4\), degree \(2\), 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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 4 q^{6} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 4 q^{6} + 12 q^{8} - 8 q^{10} + 8 q^{12} + 4 q^{14} + 4 q^{16} - 8 q^{17} + 20 q^{20} + 28 q^{22} - 24 q^{24} + 8 q^{25} - 8 q^{28} - 8 q^{30} - 20 q^{32} - 40 q^{34} - 4 q^{36} + 16 q^{37} - 68 q^{38} + 12 q^{40} + 32 q^{43} + 40 q^{44} - 24 q^{46} + 16 q^{48} - 32 q^{50} + 16 q^{51} - 24 q^{52} + 24 q^{56} + 32 q^{58} - 52 q^{62} - 48 q^{64} + 8 q^{65} - 92 q^{68} + 8 q^{70} + 12 q^{72} + 40 q^{73} - 120 q^{74} - 24 q^{76} - 24 q^{78} - 80 q^{80} - 72 q^{81} + 8 q^{82} - 80 q^{83} + 4 q^{84} + 80 q^{85} - 8 q^{86} + 100 q^{88} - 16 q^{90} - 32 q^{92} + 56 q^{94} + 44 q^{96} + 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.40344 0.174241i 0.707107 + 0.707107i 1.93928 + 0.489074i 2.17406 0.522921i −0.869174 1.11559i −0.707107 0.707107i −2.63644 1.02429i 1.00000i −3.14228 + 0.355075i
43.2 −1.36927 + 0.353682i 0.707107 + 0.707107i 1.74982 0.968575i −1.72404 + 1.42397i −1.21831 0.718131i −0.707107 0.707107i −2.05341 + 1.94512i 1.00000i 1.85705 2.55957i
43.3 −1.33579 0.464407i −0.707107 0.707107i 1.56865 + 1.24070i −0.849377 + 2.06847i 0.616159 + 1.27293i 0.707107 + 0.707107i −1.51920 2.38580i 1.00000i 2.09520 2.36858i
43.4 −1.33450 + 0.468098i −0.707107 0.707107i 1.56177 1.24935i −0.305123 2.21515i 1.27463 + 0.612637i 0.707107 + 0.707107i −1.49936 + 2.39832i 1.00000i 1.44410 + 2.81329i
43.5 −1.31614 0.517476i −0.707107 0.707107i 1.46444 + 1.36214i 2.02721 0.943618i 0.564739 + 1.29656i 0.707107 + 0.707107i −1.22253 2.55057i 1.00000i −3.15639 + 0.192899i
43.6 −1.28725 + 0.585654i 0.707107 + 0.707107i 1.31402 1.50777i 2.10210 0.762345i −1.32434 0.496102i −0.707107 0.707107i −0.808438 + 2.71043i 1.00000i −2.25946 + 2.21243i
43.7 −1.25075 0.660017i 0.707107 + 0.707107i 1.12875 + 1.65103i −2.20216 0.387931i −0.417712 1.35112i −0.707107 0.707107i −0.322079 2.81003i 1.00000i 2.49831 + 1.93867i
43.8 −1.09266 0.897824i −0.707107 0.707107i 0.387823 + 1.96204i −1.14835 1.91867i 0.137771 + 1.40749i 0.707107 + 0.707107i 1.33781 2.49204i 1.00000i −0.467877 + 3.12747i
43.9 −0.970905 + 1.02827i 0.707107 + 0.707107i −0.114686 1.99671i 1.48489 + 1.67186i −1.41363 + 0.0405644i −0.707107 0.707107i 2.16451 + 1.82069i 1.00000i −3.16081 0.0963509i
43.10 −0.811248 + 1.15839i −0.707107 0.707107i −0.683752 1.87949i 2.19439 + 0.429702i 1.39275 0.245469i 0.707107 + 0.707107i 2.73188 + 0.732680i 1.00000i −2.27796 + 2.19337i
43.11 −0.809217 1.15981i −0.707107 0.707107i −0.690335 + 1.87708i 0.566888 + 2.16302i −0.247909 + 1.39232i 0.707107 + 0.707107i 2.73570 0.718307i 1.00000i 2.04996 2.40783i
43.12 −0.805258 + 1.16257i 0.707107 + 0.707107i −0.703118 1.87233i −0.499225 2.17963i −1.39146 + 0.252654i −0.707107 0.707107i 2.74290 + 0.690290i 1.00000i 2.93596 + 1.17478i
43.13 −0.795989 1.16893i 0.707107 + 0.707107i −0.732804 + 1.86091i 0.238905 2.22327i 0.263711 1.38941i −0.707107 0.707107i 2.75858 0.624668i 1.00000i −2.78902 + 1.49043i
43.14 −0.483261 1.32908i −0.707107 0.707107i −1.53292 + 1.28459i 1.17005 1.90551i −0.598086 + 1.28152i 0.707107 + 0.707107i 2.44812 + 1.41658i 1.00000i −3.09802 0.634239i
43.15 −0.396870 1.35738i 0.707107 + 0.707107i −1.68499 + 1.07741i 1.09785 + 1.94800i 0.679186 1.24045i −0.707107 0.707107i 2.13118 + 1.85958i 1.00000i 2.20849 2.26331i
43.16 −0.169596 + 1.40401i −0.707107 0.707107i −1.94247 0.476227i −0.100716 + 2.23380i 1.11271 0.872861i 0.707107 + 0.707107i 0.998062 2.64648i 1.00000i −3.11919 0.520249i
43.17 −0.157247 1.40544i −0.707107 0.707107i −1.95055 + 0.442004i −2.09362 + 0.785338i −0.882609 + 1.10499i 0.707107 + 0.707107i 0.927929 + 2.67188i 1.00000i 1.43296 + 2.81897i
43.18 −0.139532 + 1.40731i 0.707107 + 0.707107i −1.96106 0.392730i 1.98382 + 1.03173i −1.09378 + 0.896457i −0.707107 0.707107i 0.826326 2.70503i 1.00000i −1.72878 + 2.64789i
43.19 −0.109244 + 1.40999i −0.707107 0.707107i −1.97613 0.308065i 0.663503 2.13536i 1.07426 0.919765i 0.707107 + 0.707107i 0.650247 2.75267i 1.00000i 2.93835 + 1.16881i
43.20 0.0752007 1.41221i 0.707107 + 0.707107i −1.98869 0.212399i −1.90881 + 1.16466i 1.05176 0.945410i −0.707107 0.707107i −0.449503 + 2.79248i 1.00000i 1.50121 + 2.78323i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.br.b yes 72
5.c odd 4 1 840.2.br.a 72
8.d odd 2 1 840.2.br.a 72
40.k even 4 1 inner 840.2.br.b yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.br.a 72 5.c odd 4 1
840.2.br.a 72 8.d odd 2 1
840.2.br.b yes 72 1.a even 1 1 trivial
840.2.br.b yes 72 40.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{72} + 224 T_{13}^{69} + 6984 T_{13}^{68} + 7424 T_{13}^{67} + 25088 T_{13}^{66} + 1302784 T_{13}^{65} + 21317904 T_{13}^{64} + 43817984 T_{13}^{63} + 144166912 T_{13}^{62} + 3156340224 T_{13}^{61} + \cdots + 18\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display