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Properties

Label 97.6.h.a
Level $97$
Weight $6$
Character orbit 97.h
Analytic conductor $15.557$
Analytic rank $0$
Dimension $328$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [97,6,Mod(8,97)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(97, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([1])) 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("97.8"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 97 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 97.h (of order \(16\), degree \(8\), 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: \(15.5572305219\)
Analytic rank: \(0\)
Dimension: \(328\)
Relative dimension: \(41\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 328 q - 8 q^{2} - 96 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{7} - 8 q^{8} + 256 q^{9} - 1992 q^{10} - 8 q^{11} + 4472 q^{12} - 8 q^{13} - 5504 q^{14} + 7640 q^{15} + 2880 q^{17} - 14104 q^{18} + 13984 q^{19} + 11504 q^{20}+ \cdots + 390672 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −10.3746 − 4.29730i 7.95190 + 19.1976i 66.5381 + 66.5381i 44.7629 + 8.90389i − 233.339i −154.075 + 30.6475i −266.858 − 644.252i −133.487 + 133.487i −426.134 − 284.734i
8.2 −9.61987 − 3.98468i −10.1332 − 24.4637i 54.0368 + 54.0368i −13.8191 − 2.74879i 275.715i 241.835 − 48.1040i −176.998 − 427.310i −323.965 + 323.965i 121.985 + 81.5078i
8.3 −9.56034 − 3.96002i −5.31633 − 12.8347i 53.0910 + 53.0910i −37.9759 − 7.55388i 143.757i −181.921 + 36.1864i −170.606 − 411.878i 35.3595 − 35.3595i 333.149 + 222.603i
8.4 −9.33488 − 3.86663i 0.193073 + 0.466119i 49.5617 + 49.5617i 75.7581 + 15.0692i − 5.09771i 191.198 − 38.0316i −147.283 − 355.573i 171.647 − 171.647i −648.926 − 433.598i
8.5 −8.98497 − 3.72170i 3.74591 + 9.04342i 44.2512 + 44.2512i −100.004 − 19.8919i − 95.1960i 89.3228 − 17.7674i −113.812 − 274.767i 104.075 − 104.075i 824.497 + 550.912i
8.6 −7.86145 − 3.25632i 4.70878 + 11.3680i 28.5714 + 28.5714i 13.6132 + 2.70783i − 104.702i 37.0219 − 7.36412i −27.3726 − 66.0833i 64.7683 − 64.7683i −98.2018 − 65.6163i
8.7 −7.73908 − 3.20563i −6.81101 − 16.4432i 26.9899 + 26.9899i 82.5625 + 16.4227i 149.089i −93.3807 + 18.5746i −19.7772 − 47.7464i −52.1631 + 52.1631i −586.313 − 391.762i
8.8 −7.30587 − 3.02619i 10.8205 + 26.1229i 21.5905 + 21.5905i −32.3274 − 6.43032i − 223.595i 18.1818 − 3.61659i 4.43776 + 10.7137i −393.496 + 393.496i 216.720 + 144.808i
8.9 −7.16917 − 2.96957i −1.65377 − 3.99255i 19.9513 + 19.9513i −0.609800 − 0.121297i 33.5343i −24.6779 + 4.90875i 11.2386 + 27.1324i 158.621 − 158.621i 4.01157 + 2.68044i
8.10 −5.84992 − 2.42311i −8.52128 − 20.5722i 5.72262 + 5.72262i −63.8050 − 12.6916i 140.994i −3.31217 + 0.658832i 57.9294 + 139.854i −178.776 + 178.776i 342.501 + 228.852i
8.11 −5.66080 − 2.34478i 7.20400 + 17.3920i 3.91928 + 3.91928i 37.0679 + 7.37327i − 115.344i −151.078 + 30.0512i 62.0366 + 149.770i −78.7570 + 78.7570i −192.545 − 128.655i
8.12 −4.42586 − 1.83325i −3.76603 − 9.09200i −6.39996 − 6.39996i −8.96145 − 1.78254i 47.1440i 129.522 − 25.7635i 75.2567 + 181.686i 103.346 − 103.346i 36.3943 + 24.3179i
8.13 −4.38334 − 1.81564i 8.11822 + 19.5991i −6.71026 − 6.71026i 96.2593 + 19.1472i − 100.649i 186.642 − 37.1253i 75.3305 + 181.864i −146.393 + 146.393i −387.173 − 258.701i
8.14 −3.88645 − 1.60982i 2.98824 + 7.21424i −10.1145 − 10.1145i −78.1286 − 15.5408i − 32.8483i −241.760 + 48.0891i 74.5411 + 179.958i 128.711 − 128.711i 278.625 + 186.171i
8.15 −3.63002 − 1.50360i −10.5360 − 25.4361i −11.7112 − 11.7112i 61.6904 + 12.2710i 108.175i 65.0408 − 12.9374i 73.0182 + 176.281i −364.159 + 364.159i −205.486 − 137.302i
8.16 −3.57640 − 1.48139i 0.214320 + 0.517414i −12.0313 − 12.0313i −45.4532 − 9.04121i − 2.16797i 213.862 − 42.5397i 72.6102 + 175.297i 171.605 − 171.605i 149.165 + 99.6690i
8.17 −2.69225 − 1.11517i −0.00886495 − 0.0214019i −16.6228 − 16.6228i 72.8619 + 14.4931i 0.0675052i −105.323 + 20.9501i 61.9009 + 149.442i 171.827 − 171.827i −180.000 − 120.272i
8.18 −1.58184 − 0.655221i 9.22393 + 22.2685i −20.5545 − 20.5545i −72.2782 − 14.3770i − 41.2690i 77.7079 − 15.4571i 40.0133 + 96.6007i −238.980 + 238.980i 104.913 + 70.1004i
8.19 −1.04174 − 0.431504i −10.0554 − 24.2760i −21.7284 − 21.7284i −8.06889 − 1.60500i 29.6283i −208.217 + 41.4169i 27.0676 + 65.3470i −316.384 + 316.384i 7.71315 + 5.15376i
8.20 −1.00943 − 0.418119i −2.14107 − 5.16901i −21.7833 − 21.7833i 36.7237 + 7.30479i 6.11297i −35.3812 + 7.03776i 26.2605 + 63.3984i 149.693 − 149.693i −34.0157 − 22.7285i
See next 80 embeddings (of 328 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.41
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
97.h even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 97.6.h.a ✓ 328
97.h even 16 1 inner 97.6.h.a ✓ 328
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
97.6.h.a ✓ 328 1.a even 1 1 trivial
97.6.h.a ✓ 328 97.h even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(97, [\chi])\).