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Properties

Label 63.6.s.a
Level $63$
Weight $6$
Character orbit 63.s
Analytic conductor $10.104$
Analytic rank $0$
Dimension $76$
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(47,63)] 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("63.47"); 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([1, 5])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 3 q^{2} - 3 q^{3} + 577 q^{4} - 6 q^{5} - 96 q^{6} - 30 q^{7} - 81 q^{9} - 6 q^{10} - 3 q^{12} - 543 q^{13} - 123 q^{14} + 234 q^{15} - 8223 q^{16} + 801 q^{17} + 5721 q^{18} - 6 q^{19} - 96 q^{20}+ \cdots - 144540 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −9.58962 + 5.53657i −0.877361 + 15.5637i 45.3072 − 78.4743i 21.5696 −77.7562 − 154.108i 61.1717 + 114.302i 649.045i −241.460 − 27.3101i −206.844 + 119.422i
47.2 −9.06532 + 5.23386i 7.40328 − 13.7183i 38.7866 − 67.1804i 85.8214 4.68654 + 163.108i 39.9810 − 123.323i 477.048i −133.383 − 203.121i −777.998 + 449.178i
47.3 −8.69257 + 5.01866i −15.4559 − 2.02881i 34.3739 − 59.5373i −49.6832 144.533 − 59.9322i 56.4374 − 116.713i 368.849i 234.768 + 62.7140i 431.875 − 249.343i
47.4 −8.37859 + 4.83738i 9.11067 − 12.6489i 30.8005 − 53.3480i −104.481 −15.1468 + 150.052i −115.003 + 59.8436i 286.382i −76.9914 − 230.481i 875.403 − 505.414i
47.5 −7.50494 + 4.33298i 14.2230 + 6.38014i 21.5494 − 37.3246i 15.7794 −134.388 + 13.7454i −117.294 − 55.2192i 96.1815i 161.588 + 181.489i −118.424 + 68.3718i
47.6 −7.40857 + 4.27734i −9.98088 − 11.9742i 20.5913 − 35.6651i 38.9005 125.162 + 46.0202i −32.6607 + 125.460i 78.5539i −43.7639 + 239.027i −288.197 + 166.391i
47.7 −7.18813 + 4.15007i 15.2650 + 3.15899i 18.4461 − 31.9497i −13.4718 −122.837 + 40.6437i 121.599 + 44.9527i 40.6067i 223.042 + 96.4440i 96.8369 − 55.9088i
47.8 −6.85527 + 3.95789i −11.6841 + 10.3190i 15.3298 − 26.5520i −25.8967 39.2562 − 116.984i −120.641 + 47.4627i − 10.6099i 30.0365 − 241.136i 177.529 − 102.497i
47.9 −5.95180 + 3.43627i −2.44206 + 15.3960i 7.61597 − 13.1912i 82.4113 −38.3702 − 100.025i −27.2714 − 126.741i − 115.239i −231.073 − 75.1959i −490.496 + 283.188i
47.10 −5.22444 + 3.01633i 1.37473 + 15.5277i 2.19649 − 3.80443i −106.554 −54.0189 − 76.9769i 99.6671 − 82.9064i − 166.544i −239.220 + 42.6928i 556.683 − 321.401i
47.11 −4.52842 + 2.61449i −15.1040 + 3.85599i −2.32894 + 4.03383i 57.3397 58.3159 − 56.9508i 128.059 + 20.1980i − 191.683i 213.263 − 116.482i −259.658 + 149.914i
47.12 −4.44864 + 2.56843i 6.98225 − 13.9373i −2.80638 + 4.86080i −22.3974 4.73533 + 79.9354i 125.358 + 33.0507i − 193.211i −145.496 − 194.627i 99.6378 − 57.5259i
47.13 −4.43807 + 2.56232i −5.73757 − 14.4941i −2.86900 + 4.96925i −12.0147 62.6024 + 49.6246i −44.5801 − 121.736i − 193.394i −177.161 + 166.322i 53.3223 − 30.7856i
47.14 −3.42420 + 1.97696i 13.4136 − 7.94196i −8.18323 + 14.1738i 96.1527 −30.2300 + 53.7131i −67.3980 + 110.745i − 191.237i 116.850 − 213.061i −329.246 + 190.090i
47.15 −2.69688 + 1.55704i 8.07896 + 13.3316i −11.1512 + 19.3145i −14.7675 −42.5458 − 23.3743i −76.3428 + 104.780i − 169.103i −112.461 + 215.410i 39.8260 − 22.9936i
47.16 −1.69409 + 0.978081i −13.0902 − 8.46437i −14.0867 + 24.3989i −99.5152 30.4548 + 1.53606i 54.4830 + 117.638i − 117.709i 99.7087 + 221.601i 168.587 − 97.3339i
47.17 −1.45110 + 0.837793i 14.6627 − 5.29211i −14.5962 + 25.2814i −41.1571 −16.8433 + 19.9637i −56.9736 − 116.452i − 102.533i 186.987 − 155.193i 59.7231 − 34.4811i
47.18 −1.22697 + 0.708392i −14.9068 + 4.55938i −14.9964 + 25.9745i −39.2064 15.0604 − 16.1541i −128.843 − 14.3719i − 87.8303i 201.424 − 135.931i 48.1052 − 27.7735i
47.19 0.0316775 − 0.0182890i 12.3725 + 9.48272i −15.9993 + 27.7117i 62.5776 0.565358 + 0.0741087i 87.2452 − 95.8920i 2.34094i 63.1559 + 234.649i 1.98230 − 1.14448i
47.20 0.0675663 − 0.0390094i −14.2360 − 6.35113i −15.9970 + 27.7075i 92.3091 −1.20963 + 0.126215i −119.454 − 50.3751i 4.99273i 162.326 + 180.829i 6.23698 − 3.60092i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.38
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.s.a yes 76
3.b odd 2 1 189.6.s.a 76
7.d odd 6 1 63.6.i.a ✓ 76
9.c even 3 1 189.6.i.a 76
9.d odd 6 1 63.6.i.a ✓ 76
21.g even 6 1 189.6.i.a 76
63.k odd 6 1 189.6.s.a 76
63.s even 6 1 inner 63.6.s.a yes 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.i.a ✓ 76 7.d odd 6 1
63.6.i.a ✓ 76 9.d odd 6 1
63.6.s.a yes 76 1.a even 1 1 trivial
63.6.s.a yes 76 63.s even 6 1 inner
189.6.i.a 76 9.c even 3 1
189.6.i.a 76 21.g even 6 1
189.6.s.a 76 3.b odd 2 1
189.6.s.a 76 63.k odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(63, [\chi])\).