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Properties

Label 63.6.f.a
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,-12] 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 12 q^{2} - 240 q^{4} - 129 q^{5} + 144 q^{6} - 735 q^{7} + 1578 q^{8} + 666 q^{9} - 834 q^{11} - 2412 q^{12} - 165 q^{13} - 588 q^{14} + 3987 q^{15} - 3840 q^{16} + 5196 q^{17} + 2871 q^{18} - 894 q^{19}+ \cdots + 416466 q^{99}+O(q^{100}) \) 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 −5.49210 − 9.51260i 4.70599 + 14.8611i −44.3264 + 76.7756i 40.1193 − 69.4887i 115.522 − 126.385i −24.5000 − 42.4352i 622.286 −198.707 + 139.873i −881.358
22.2 −5.09242 − 8.82033i 15.1000 − 3.87171i −35.8655 + 62.1208i −39.1601 + 67.8273i −111.045 − 113.471i −24.5000 − 42.4352i 404.653 213.020 − 116.925i 797.679
22.3 −4.79080 − 8.29790i −15.5621 − 0.906717i −29.9034 + 51.7943i −31.9444 + 55.3294i 67.0308 + 133.476i −24.5000 − 42.4352i 266.434 241.356 + 28.2208i 612.157
22.4 −3.39411 − 5.87878i 0.457406 − 15.5817i −7.04003 + 12.1937i 2.98739 − 5.17432i −93.1541 + 50.1972i −24.5000 − 42.4352i −121.645 −242.582 − 14.2544i −40.5582
22.5 −3.01642 − 5.22459i −6.20790 + 14.2990i −2.19756 + 3.80629i 9.43880 − 16.3485i 93.4321 − 10.6981i −24.5000 − 42.4352i −166.536 −165.924 − 177.534i −113.885
22.6 −2.08756 − 3.61576i 15.1236 + 3.77846i 7.28421 − 12.6166i 32.0946 − 55.5895i −17.9094 − 62.5710i −24.5000 − 42.4352i −194.428 214.446 + 114.288i −267.997
22.7 −1.77063 − 3.06682i 7.83869 + 13.4742i 9.72974 − 16.8524i −45.3198 + 78.4962i 27.4436 − 47.8977i −24.5000 − 42.4352i −182.231 −120.110 + 211.241i 320.978
22.8 0.0345737 + 0.0598835i −11.5599 − 10.4580i 15.9976 − 27.7087i 24.0990 − 41.7407i 0.226590 − 1.05382i −24.5000 − 42.4352i 4.42511 24.2624 + 241.786i 3.33277
22.9 0.117370 + 0.203290i −14.3090 + 6.18483i 15.9724 − 27.6651i −32.0449 + 55.5034i −2.93676 − 2.18297i −24.5000 − 42.4352i 15.0104 166.496 − 176.998i −15.0444
22.10 1.13248 + 1.96150i 14.8712 − 4.67422i 13.4350 − 23.2701i 4.60366 − 7.97377i 26.0097 + 23.8764i −24.5000 − 42.4352i 133.338 199.303 − 139.022i 20.8541
22.11 2.37672 + 4.11660i −3.67112 − 15.1500i 4.70238 − 8.14475i −51.8406 + 89.7906i 53.6414 − 51.1199i −24.5000 − 42.4352i 196.815 −216.046 + 111.235i −492.843
22.12 3.05887 + 5.29812i −11.9804 + 9.97350i −2.71339 + 4.69973i 45.5993 − 78.9802i −89.4872 − 32.9658i −24.5000 − 42.4352i 162.568 44.0585 − 238.972i 557.929
22.13 3.57508 + 6.19221i 10.6843 − 11.3510i −9.56235 + 16.5625i 13.8812 − 24.0429i 108.485 + 25.5786i −24.5000 − 42.4352i 92.0604 −14.6919 − 242.555i 198.505
22.14 4.50870 + 7.80929i 9.94289 + 12.0058i −24.6567 + 42.7066i −12.7186 + 22.0293i −48.9272 + 131.777i −24.5000 − 42.4352i −156.122 −45.2778 + 238.744i −229.378
22.15 4.84025 + 8.38356i −15.4337 − 2.19130i −30.8561 + 53.4443i −24.2948 + 42.0798i −56.3320 − 139.996i −24.5000 − 42.4352i −287.628 233.396 + 67.6396i −470.371
43.1 −5.49210 + 9.51260i 4.70599 − 14.8611i −44.3264 − 76.7756i 40.1193 + 69.4887i 115.522 + 126.385i −24.5000 + 42.4352i 622.286 −198.707 − 139.873i −881.358
43.2 −5.09242 + 8.82033i 15.1000 + 3.87171i −35.8655 − 62.1208i −39.1601 − 67.8273i −111.045 + 113.471i −24.5000 + 42.4352i 404.653 213.020 + 116.925i 797.679
43.3 −4.79080 + 8.29790i −15.5621 + 0.906717i −29.9034 − 51.7943i −31.9444 − 55.3294i 67.0308 − 133.476i −24.5000 + 42.4352i 266.434 241.356 − 28.2208i 612.157
43.4 −3.39411 + 5.87878i 0.457406 + 15.5817i −7.04003 − 12.1937i 2.98739 + 5.17432i −93.1541 − 50.1972i −24.5000 + 42.4352i −121.645 −242.582 + 14.2544i −40.5582
43.5 −3.01642 + 5.22459i −6.20790 − 14.2990i −2.19756 − 3.80629i 9.43880 + 16.3485i 93.4321 + 10.6981i −24.5000 + 42.4352i −166.536 −165.924 + 177.534i −113.885
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.a ✓ 30
3.b odd 2 1 189.6.f.b 30
9.c even 3 1 inner 63.6.f.a ✓ 30
9.c even 3 1 567.6.a.i 15
9.d odd 6 1 189.6.f.b 30
9.d odd 6 1 567.6.a.f 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.f.a ✓ 30 1.a even 1 1 trivial
63.6.f.a ✓ 30 9.c even 3 1 inner
189.6.f.b 30 3.b odd 2 1
189.6.f.b 30 9.d odd 6 1
567.6.a.f 15 9.d odd 6 1
567.6.a.i 15 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 12 T_{2}^{29} + 432 T_{2}^{28} + 3570 T_{2}^{27} + 93528 T_{2}^{26} + 647757 T_{2}^{25} + \cdots + 18\!\cdots\!16 \) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display