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Properties

Label 63.6.f.b
Level $63$
Weight $6$
Character orbit 63.f
Analytic conductor $10.104$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,6,Mod(22,63)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("63.22"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 4 q^{2} + 20 q^{3} - 240 q^{4} + 71 q^{5} - 266 q^{6} + 735 q^{7} + 42 q^{8} - 442 q^{9} + 134 q^{11} - 1400 q^{12} + 165 q^{13} - 196 q^{14} + 647 q^{15} - 3840 q^{16} - 2184 q^{17} + 9235 q^{18}+ \cdots - 518014 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −5.32234 − 9.21856i −5.57188 − 14.5586i −40.6546 + 70.4158i 9.68441 − 16.7739i −104.554 + 128.851i 24.5000 + 42.4352i 524.880 −180.908 + 162.238i −206.175
22.2 −4.38794 − 7.60013i −6.48542 + 14.1753i −22.5080 + 38.9850i −26.4033 + 45.7318i 136.192 − 12.9103i 24.5000 + 42.4352i 114.226 −158.879 − 183.866i 463.424
22.3 −4.21162 − 7.29473i 13.4591 − 7.86457i −19.4754 + 33.7324i 25.8420 − 44.7596i −114.055 − 65.0583i 24.5000 + 42.4352i 58.5489 119.297 − 211.701i −435.346
22.4 −3.81386 − 6.60581i 12.4507 + 9.37976i −13.0911 + 22.6745i −6.17916 + 10.7026i 14.4756 − 118.020i 24.5000 + 42.4352i −44.3762 67.0401 + 233.569i 94.2659
22.5 −2.21406 − 3.83487i −11.6210 − 10.3900i 6.19585 − 10.7315i −30.8296 + 53.3984i −14.1145 + 67.5692i 24.5000 + 42.4352i −196.572 27.0967 + 241.485i 273.035
22.6 −0.876872 − 1.51879i −13.7798 + 7.28808i 14.4622 − 25.0492i 4.30330 − 7.45354i 23.1522 + 14.5379i 24.5000 + 42.4352i −106.846 136.768 − 200.857i −15.0938
22.7 −0.772339 − 1.33773i 2.43538 + 15.3970i 14.8070 − 25.6464i 33.3433 − 57.7523i 18.7162 − 15.1496i 24.5000 + 42.4352i −95.1738 −231.138 + 74.9954i −103.009
22.8 −0.195179 − 0.338060i 5.35520 − 14.6397i 15.9238 − 27.5808i 48.2760 − 83.6165i −5.99433 + 1.04699i 24.5000 + 42.4352i −24.9234 −185.644 − 156.797i −37.6898
22.9 1.29488 + 2.24280i 14.4289 + 5.89965i 12.6466 − 21.9045i −20.9318 + 36.2550i 5.45202 + 40.0006i 24.5000 + 42.4352i 148.376 173.388 + 170.251i −108.417
22.10 2.01180 + 3.48454i −0.357568 − 15.5844i 7.90534 − 13.6924i −7.29857 + 12.6415i 53.5849 − 32.5985i 24.5000 + 42.4352i 192.371 −242.744 + 11.1449i −58.7330
22.11 3.08078 + 5.33606i −14.8938 − 4.60155i −2.98236 + 5.16561i 9.09655 − 15.7557i −21.3303 − 93.6507i 24.5000 + 42.4352i 160.418 200.651 + 137.069i 112.098
22.12 3.31910 + 5.74885i −7.74251 + 13.5297i −6.03282 + 10.4491i −41.6346 + 72.1133i −103.478 + 0.395999i 24.5000 + 42.4352i 132.328 −123.107 − 209.508i −552.758
22.13 3.78148 + 6.54972i 14.7418 + 5.06742i −12.5992 + 21.8225i 51.7579 − 89.6473i 22.5558 + 115.717i 24.5000 + 42.4352i 51.4396 191.643 + 149.406i 782.887
22.14 4.84484 + 8.39151i 13.9648 − 6.92717i −30.9450 + 53.5982i −35.2811 + 61.1087i 125.786 + 83.6243i 24.5000 + 42.4352i −289.624 147.029 − 193.473i −683.726
22.15 5.46133 + 9.45930i −6.38387 + 14.2213i −43.6522 + 75.6079i 21.7547 − 37.6803i −169.388 + 17.2804i 24.5000 + 42.4352i −604.072 −161.492 − 181.574i 475.239
43.1 −5.32234 + 9.21856i −5.57188 + 14.5586i −40.6546 − 70.4158i 9.68441 + 16.7739i −104.554 − 128.851i 24.5000 − 42.4352i 524.880 −180.908 − 162.238i −206.175
43.2 −4.38794 + 7.60013i −6.48542 − 14.1753i −22.5080 − 38.9850i −26.4033 − 45.7318i 136.192 + 12.9103i 24.5000 − 42.4352i 114.226 −158.879 + 183.866i 463.424
43.3 −4.21162 + 7.29473i 13.4591 + 7.86457i −19.4754 − 33.7324i 25.8420 + 44.7596i −114.055 + 65.0583i 24.5000 − 42.4352i 58.5489 119.297 + 211.701i −435.346
43.4 −3.81386 + 6.60581i 12.4507 − 9.37976i −13.0911 − 22.6745i −6.17916 − 10.7026i 14.4756 + 118.020i 24.5000 − 42.4352i −44.3762 67.0401 − 233.569i 94.2659
43.5 −2.21406 + 3.83487i −11.6210 + 10.3900i 6.19585 + 10.7315i −30.8296 − 53.3984i −14.1145 − 67.5692i 24.5000 − 42.4352i −196.572 27.0967 − 241.485i 273.035
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.15
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.f.b ✓ 30
3.b odd 2 1 189.6.f.a 30
9.c even 3 1 inner 63.6.f.b ✓ 30
9.c even 3 1 567.6.a.g 15
9.d odd 6 1 189.6.f.a 30
9.d odd 6 1 567.6.a.h 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.f.b ✓ 30 1.a even 1 1 trivial
63.6.f.b ✓ 30 9.c even 3 1 inner
189.6.f.a 30 3.b odd 2 1
189.6.f.a 30 9.d odd 6 1
567.6.a.g 15 9.c even 3 1
567.6.a.h 15 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - 4 T_{2}^{29} + 368 T_{2}^{28} - 1294 T_{2}^{27} + 81464 T_{2}^{26} - 268975 T_{2}^{25} + \cdots + 91\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display