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Properties

Label 96.6.o.a
Level $96$
Weight $6$
Character orbit 96.o
Analytic conductor $15.397$
Analytic rank $0$
Dimension $312$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [96,6,Mod(11,96)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(96, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([4, 5, 4])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("96.11"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 96.o (of order \(8\), degree \(4\), 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: \(15.3968467020\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(78\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9} + 192 q^{10} - 4 q^{12} - 8 q^{13} - 8 q^{15} - 6696 q^{16} - 4 q^{18} - 8 q^{19} - 4 q^{21} + 5576 q^{22} - 11024 q^{24} - 8 q^{25} + 7460 q^{27}+ \cdots + 339516 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −5.65577 − 0.110924i −1.66250 − 15.4996i 31.9754 + 1.25472i −43.4566 − 18.0003i 7.68343 + 87.8463i 56.6824 − 56.6824i −180.706 − 10.6432i −237.472 + 51.5359i 243.784 + 106.626i
11.2 −5.64922 + 0.293734i 15.5875 + 0.174756i 31.8274 − 3.31874i −102.440 − 42.4320i −88.1085 + 3.59134i −65.5079 + 65.5079i −178.825 + 28.0971i 242.939 + 5.44800i 591.170 + 209.618i
11.3 −5.58086 − 0.924104i 6.74983 + 14.0513i 30.2921 + 10.3146i −19.2402 − 7.96957i −24.6850 − 84.6561i 71.2494 − 71.2494i −159.524 − 85.5573i −151.879 + 189.688i 100.012 + 62.2571i
11.4 −5.57531 + 0.957052i −15.4283 − 2.22908i 30.1681 − 10.6717i −24.8298 − 10.2848i 88.1506 − 2.33781i 36.1711 − 36.1711i −157.983 + 88.3705i 233.062 + 68.7818i 148.277 + 33.5777i
11.5 −5.55856 − 1.04998i −2.52416 + 15.3827i 29.7951 + 11.6727i 18.4880 + 7.65797i 30.1822 − 82.8555i −168.744 + 168.744i −153.362 − 96.1675i −230.257 − 77.6568i −94.7257 − 61.9791i
11.6 −5.55235 − 1.08231i 15.2176 − 3.38008i 29.6572 + 12.0187i 59.8212 + 24.7787i −88.1517 + 2.29719i 157.272 − 157.272i −151.659 − 98.8306i 220.150 − 102.873i −305.330 − 202.325i
11.7 −5.52072 + 1.23357i 8.52214 − 13.0527i 28.9566 − 13.6204i 43.6201 + 18.0681i −30.9468 + 82.5730i −129.216 + 129.216i −143.059 + 110.914i −97.7463 − 222.474i −263.103 − 45.9400i
11.8 −5.49139 − 1.35816i −13.2973 − 8.13523i 28.3108 + 14.9164i 88.6591 + 36.7238i 61.9718 + 62.7336i −66.1750 + 66.1750i −135.207 − 120.362i 110.636 + 216.353i −436.985 − 322.078i
11.9 −5.33954 + 1.86796i −10.2631 + 11.7332i 25.0215 − 19.9481i 64.8477 + 26.8608i 32.8833 − 81.8211i 53.5751 − 53.5751i −96.3409 + 153.253i −32.3363 − 240.839i −396.432 − 22.2916i
11.10 −5.20306 + 2.21994i 13.2034 + 8.28682i 22.1437 − 23.1010i 51.7868 + 21.4508i −87.0941 − 13.8062i −36.2549 + 36.2549i −63.9324 + 169.354i 105.657 + 218.828i −317.070 + 3.35378i
11.11 −4.99058 − 2.66347i −10.7907 + 11.2499i 17.8119 + 26.5845i −78.3856 − 32.4684i 83.8158 − 27.4027i 81.6131 − 81.6131i −18.0846 − 180.114i −10.1198 − 242.789i 304.712 + 370.814i
11.12 −4.89295 + 2.83885i −10.4113 + 11.6019i 15.8819 − 27.7807i −73.2819 − 30.3544i 18.0059 − 86.3237i −129.352 + 129.352i 1.15577 + 181.016i −26.2092 − 241.582i 444.736 − 59.5138i
11.13 −4.87450 − 2.87041i 13.4885 − 7.81407i 15.5214 + 27.9836i 8.96462 + 3.71327i −88.1794 − 0.627978i −89.1993 + 89.1993i 4.66538 − 180.959i 120.881 − 210.801i −33.0394 − 43.8325i
11.14 −4.75091 − 3.07064i −13.9105 + 7.03543i 13.1424 + 29.1767i 24.4184 + 10.1144i 87.6910 + 9.28947i 33.8384 − 33.8384i 27.1528 − 178.971i 144.005 − 195.733i −84.9520 − 123.033i
11.15 −4.71337 − 3.12797i −0.892396 − 15.5629i 12.4317 + 29.4865i 13.0568 + 5.40829i −44.4740 + 76.1450i 30.1232 − 30.1232i 33.6378 − 177.867i −241.407 + 27.7765i −44.6244 − 66.3324i
11.16 −4.47443 + 3.46114i 10.4026 − 11.6097i 8.04102 − 30.9732i −27.7552 − 11.4966i −6.36272 + 87.9518i 67.8781 − 67.8781i 71.2238 + 166.419i −26.5722 − 241.543i 163.980 − 44.6241i
11.17 −4.47401 + 3.46168i 1.30422 + 15.5338i 8.03359 − 30.9752i −47.4329 − 19.6473i −59.6081 − 64.9836i 130.212 − 130.212i 71.2836 + 166.393i −239.598 + 40.5191i 280.228 − 76.2948i
11.18 −4.36540 − 3.59767i −12.1487 − 9.76779i 6.11347 + 31.4106i −79.7682 − 33.0411i 17.8925 + 86.3473i −126.145 + 126.145i 86.3174 − 159.114i 52.1804 + 237.331i 229.349 + 431.217i
11.19 −4.33770 + 3.63103i −6.67206 − 14.0884i 5.63129 − 31.5006i 81.5741 + 33.7891i 80.0968 + 36.8849i 162.825 − 162.825i 89.9527 + 157.088i −153.967 + 187.998i −476.533 + 149.631i
11.20 −4.31982 + 3.65229i −9.11643 − 12.6448i 5.32161 − 31.5544i −5.13521 − 2.12707i 85.5637 + 21.3273i −137.882 + 137.882i 92.2574 + 155.745i −76.7813 + 230.551i 29.9518 − 9.56668i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.78
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.6.o.a ✓ 312
3.b odd 2 1 inner 96.6.o.a ✓ 312
32.h odd 8 1 inner 96.6.o.a ✓ 312
96.o even 8 1 inner 96.6.o.a ✓ 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.6.o.a ✓ 312 1.a even 1 1 trivial
96.6.o.a ✓ 312 3.b odd 2 1 inner
96.6.o.a ✓ 312 32.h odd 8 1 inner
96.6.o.a ✓ 312 96.o even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(96, [\chi])\).