Properties

Label 740.4.i.b
Level 740740
Weight 44
Character orbit 740.i
Analytic conductor 43.66143.661
Analytic rank 00
Dimension 3838
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [740,4,Mod(121,740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("740.121"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 740=22537 740 = 2^{2} \cdot 5 \cdot 37
Weight: k k == 4 4
Character orbit: [χ][\chi] == 740.i (of order 33, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [38,0,-6,0,95] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 43.661413404243.6614134042
Analytic rank: 00
Dimension: 3838
Relative dimension: 1919 over Q(ζ3)\Q(\zeta_{3})
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

qq-expansion

The algebraic qq-expansion of this newform has not been computed, but we have computed the trace expansion.

Tr(f)(q)=\operatorname{Tr}(f)(q) = 38q6q3+95q532q7187q92q11+93q13+30q15+38q1791q19230q21306q23475q25+552q27+156q29144q31484q33+160q35++721q99+O(q100) 38 q - 6 q^{3} + 95 q^{5} - 32 q^{7} - 187 q^{9} - 2 q^{11} + 93 q^{13} + 30 q^{15} + 38 q^{17} - 91 q^{19} - 230 q^{21} - 306 q^{23} - 475 q^{25} + 552 q^{27} + 156 q^{29} - 144 q^{31} - 484 q^{33} + 160 q^{35}+ \cdots + 721 q^{99}+O(q^{100}) Copy content Toggle raw display

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\iota_m(a_n) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
121.1 0 −4.92424 8.52903i 0 2.50000 + 4.33013i 0 −5.38226 9.32234i 0 −34.9962 + 60.6152i 0
121.2 0 −4.83172 8.36878i 0 2.50000 + 4.33013i 0 −1.66213 2.87889i 0 −33.1910 + 57.4886i 0
121.3 0 −3.84492 6.65960i 0 2.50000 + 4.33013i 0 −3.54678 6.14320i 0 −16.0668 + 27.8286i 0
121.4 0 −3.36414 5.82686i 0 2.50000 + 4.33013i 0 11.5801 + 20.0572i 0 −9.13484 + 15.8220i 0
121.5 0 −2.76970 4.79727i 0 2.50000 + 4.33013i 0 −14.1228 24.4613i 0 −1.84251 + 3.19132i 0
121.6 0 −2.52314 4.37020i 0 2.50000 + 4.33013i 0 −9.26690 16.0507i 0 0.767570 1.32947i 0
121.7 0 −1.62936 2.82214i 0 2.50000 + 4.33013i 0 −15.0273 26.0281i 0 8.19034 14.1861i 0
121.8 0 −1.50219 2.60187i 0 2.50000 + 4.33013i 0 4.84611 + 8.39370i 0 8.98684 15.5657i 0
121.9 0 −1.46436 2.53634i 0 2.50000 + 4.33013i 0 14.2772 + 24.7289i 0 9.21131 15.9545i 0
121.10 0 0.111910 + 0.193833i 0 2.50000 + 4.33013i 0 5.38871 + 9.33352i 0 13.4750 23.3393i 0
121.11 0 0.411849 + 0.713344i 0 2.50000 + 4.33013i 0 7.51095 + 13.0093i 0 13.1608 22.7951i 0
121.12 0 0.816236 + 1.41376i 0 2.50000 + 4.33013i 0 −6.46590 11.1993i 0 12.1675 21.0748i 0
121.13 0 1.52693 + 2.64471i 0 2.50000 + 4.33013i 0 −2.60491 4.51183i 0 8.83700 15.3061i 0
121.14 0 2.56266 + 4.43866i 0 2.50000 + 4.33013i 0 −9.93494 17.2078i 0 0.365526 0.633109i 0
121.15 0 2.83527 + 4.91084i 0 2.50000 + 4.33013i 0 −6.77514 11.7349i 0 −2.57754 + 4.46443i 0
121.16 0 3.13113 + 5.42328i 0 2.50000 + 4.33013i 0 13.8884 + 24.0555i 0 −6.10799 + 10.5794i 0
121.17 0 3.44241 + 5.96243i 0 2.50000 + 4.33013i 0 13.3083 + 23.0507i 0 −10.2003 + 17.6675i 0
121.18 0 4.24903 + 7.35954i 0 2.50000 + 4.33013i 0 −14.8184 25.6662i 0 −22.6086 + 39.1592i 0
121.19 0 4.76634 + 8.25554i 0 2.50000 + 4.33013i 0 2.80766 + 4.86301i 0 −31.9359 + 55.3147i 0
581.1 0 −4.92424 + 8.52903i 0 2.50000 4.33013i 0 −5.38226 + 9.32234i 0 −34.9962 60.6152i 0
See all 38 embeddings
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.19
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.4.i.b 38
37.c even 3 1 inner 740.4.i.b 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.4.i.b 38 1.a even 1 1 trivial
740.4.i.b 38 37.c even 3 1 inner