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Properties

Label 63.6.h.a
Level $63$
Weight $6$
Character orbit 63.h
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(25,63)] 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("63.25"); 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([4, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.h (of order \(3\), 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 \(\Q(\zeta_{3})\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 2 q^{2} - q^{3} + 1150 q^{4} + 101 q^{5} - 116 q^{6} + 28 q^{7} - 72 q^{8} - 271 q^{9} - 66 q^{10} + 191 q^{11} - 1550 q^{12} + 179 q^{13} + 416 q^{14} - 1828 q^{15} + 16318 q^{16} + 2043 q^{17}+ \cdots + 54410 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −11.1043 −1.53149 + 15.5130i 91.3051 0.957516 − 1.65847i 17.0061 − 172.261i 126.695 − 27.4837i −658.540 −238.309 − 47.5163i −10.6325 + 18.4161i
25.2 −10.6425 −12.5736 − 9.21442i 81.2627 31.3897 − 54.3686i 133.814 + 98.0644i −95.8859 + 87.2519i −524.278 73.1891 + 231.716i −334.065 + 578.617i
25.3 −9.83212 15.2873 − 3.04928i 64.6706 −23.4211 + 40.5666i −150.307 + 29.9809i 24.7311 + 127.261i −321.222 224.404 − 93.2306i 230.279 − 398.856i
25.4 −9.45352 3.54709 − 15.1795i 57.3690 −21.3620 + 37.0001i −33.5324 + 143.500i −64.6321 − 112.382i −239.826 −217.836 − 107.686i 201.946 − 349.781i
25.5 −8.74637 13.9336 + 6.98958i 44.4989 40.7256 − 70.5388i −121.869 − 61.1335i −112.974 − 63.5913i −109.320 145.291 + 194.780i −356.201 + 616.958i
25.6 −8.44601 −15.2683 − 3.14312i 39.3351 −31.6926 + 54.8932i 128.956 + 26.5469i 91.2699 − 92.0696i −61.9527 223.242 + 95.9802i 267.676 − 463.629i
25.7 −7.88198 9.53651 − 12.3311i 30.1257 38.5225 − 66.7230i −75.1666 + 97.1932i 129.582 + 3.92924i 14.7734 −61.1100 − 235.190i −303.634 + 525.909i
25.8 −7.81572 −10.2037 + 11.7849i 29.0855 4.49720 − 7.78938i 79.7493 − 92.1076i −112.708 − 64.0621i 22.7791 −34.7687 − 240.500i −35.1488 + 60.8796i
25.9 −7.37520 4.78138 + 14.8371i 22.3935 −44.2047 + 76.5647i −35.2636 − 109.426i −110.483 + 67.8278i 70.8497 −197.277 + 141.883i 326.018 − 564.680i
25.10 −6.52174 −13.5230 + 7.75429i 10.5330 22.2890 − 38.6057i 88.1933 − 50.5714i 54.1928 + 117.772i 140.002 122.742 − 209.722i −145.363 + 251.776i
25.11 −5.58795 −5.95585 − 14.4058i −0.774815 −19.8780 + 34.4298i 33.2810 + 80.4990i 16.5636 + 128.579i 183.144 −172.056 + 171.598i 111.077 − 192.392i
25.12 −5.43445 10.0747 + 11.8954i −2.46680 4.09061 − 7.08514i −54.7506 − 64.6447i 125.102 − 34.0067i 187.308 −39.9998 + 239.685i −22.2302 + 38.5038i
25.13 −4.09285 −12.1227 − 9.80006i −15.2486 49.4436 − 85.6388i 49.6162 + 40.1101i 36.0973 − 124.515i 193.381 50.9177 + 237.606i −202.365 + 350.507i
25.14 −3.70846 15.1614 − 3.62361i −18.2473 −34.3261 + 59.4546i −56.2256 + 13.4380i 10.4345 − 129.221i 186.340 216.739 − 109.878i 127.297 − 220.485i
25.15 −3.14164 5.14370 − 14.7154i −22.1301 14.2505 − 24.6826i −16.1597 + 46.2304i −128.689 − 15.6849i 170.057 −190.085 − 151.383i −44.7699 + 77.5437i
25.16 −1.95843 15.5855 − 0.302128i −28.1645 8.48525 − 14.6969i −30.5232 + 0.591697i −45.7039 + 121.318i 117.828 242.817 − 9.41764i −16.6178 + 28.7828i
25.17 −1.06968 −15.5828 − 0.418972i −30.8558 −22.7653 + 39.4307i 16.6686 + 0.448165i −128.436 − 17.6428i 67.2354 242.649 + 13.0575i 24.3516 − 42.1782i
25.18 −1.01524 −6.16476 + 14.3177i −30.9693 25.8821 − 44.8291i 6.25870 − 14.5358i 19.6392 − 128.146i 63.9288 −166.991 − 176.530i −26.2765 + 45.5122i
25.19 −0.964960 3.10998 + 15.2751i −31.0689 33.8862 − 58.6927i −3.00101 − 14.7398i −51.0811 + 119.154i 60.8589 −223.656 + 95.0104i −32.6989 + 56.6361i
25.20 −0.744106 −8.02344 + 13.3650i −31.4463 −48.6549 + 84.2728i 5.97029 − 9.94501i 116.701 + 56.4623i 47.2108 −114.249 − 214.467i 36.2044 − 62.7079i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.38
Significant digits:
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Inner twists

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

Hecke kernels

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