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Properties

Label 97.6.k.a
Level $97$
Weight $6$
Character orbit 97.k
Analytic conductor $15.557$
Analytic rank $0$
Dimension $640$
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(2,97)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(97, base_ring=CyclotomicField(48)) chi = DirichletCharacter(H, H._module([17])) 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("97.2"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 97 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 97.k (of order \(48\), degree \(16\), 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: \(640\)
Relative dimension: \(40\) over \(\Q(\zeta_{48})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 16 q^{2} + 72 q^{3} - 16 q^{4} - 16 q^{5} - 24 q^{6} - 16 q^{7} - 16 q^{8} - 280 q^{9} - 3984 q^{10} - 16 q^{11} - 4496 q^{12} - 16 q^{13} - 11008 q^{14} - 7664 q^{15} - 24 q^{16} - 2904 q^{17}+ \cdots - 1255776 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −10.5278 − 1.38602i 24.6151 − 18.8878i 78.0046 + 20.9013i 2.94026 + 5.96225i −285.322 + 164.731i 122.547 − 8.03214i −478.318 − 198.126i 186.260 − 695.130i −22.6908 − 66.8449i
2.2 −10.4339 − 1.37365i 2.96604 − 2.27592i 76.0706 + 20.3831i 14.7671 + 29.9447i −34.0738 + 19.6725i −135.246 + 8.86447i −454.585 − 188.295i −59.2755 + 221.219i −112.945 − 332.726i
2.3 −10.2671 − 1.35169i −10.5830 + 8.12059i 72.6768 + 19.4737i −9.64301 − 19.5541i 119.633 − 69.0701i 177.633 − 11.6427i −413.701 − 171.360i −16.8380 + 62.8402i 72.5748 + 213.798i
2.4 −9.59368 − 1.26303i −23.7394 + 18.2159i 59.5338 + 15.9520i 31.0907 + 63.0456i 250.755 − 144.774i −89.1773 + 5.84499i −264.924 − 109.735i 168.847 − 630.147i −218.645 − 644.108i
2.5 −8.95128 − 1.17846i 9.54367 − 7.32311i 47.8271 + 12.8152i −38.1307 − 77.3215i −94.0581 + 54.3044i −75.6527 + 4.95854i −146.091 − 60.5129i −25.4394 + 94.9413i 250.199 + 737.062i
2.6 −8.56885 − 1.12811i −14.9675 + 11.4850i 41.2430 + 11.0510i −38.3472 − 77.7605i 141.211 − 81.5282i −21.2512 + 1.39287i −85.4212 − 35.3826i 29.2291 − 109.084i 240.869 + 709.578i
2.7 −7.92337 − 1.04313i 5.32344 − 4.08482i 30.7820 + 8.24802i 42.3505 + 85.8783i −46.4406 + 26.8125i 187.048 − 12.2598i 0.975199 + 0.403941i −51.2397 + 191.229i −245.976 − 724.623i
2.8 −7.53692 − 0.992255i 13.3403 − 10.2364i 24.9110 + 6.67489i −7.25114 − 14.7039i −110.702 + 63.9138i 27.4174 − 1.79703i 43.6160 + 18.0664i 10.2872 − 38.3924i 40.0613 + 118.017i
2.9 −7.49445 − 0.986663i −11.6059 + 8.90554i 24.2837 + 6.50679i 22.2947 + 45.2091i 95.7668 − 55.2910i −1.79553 + 0.117685i 47.9061 + 19.8434i −7.50421 + 28.0061i −122.480 − 360.815i
2.10 −6.12833 − 0.806809i −7.11727 + 5.46127i 5.99580 + 1.60657i 4.39771 + 8.91768i 48.0231 − 27.7262i −197.859 + 12.9684i 147.294 + 61.0112i −42.0630 + 156.981i −19.7558 − 58.1986i
2.11 −6.12651 − 0.806570i 18.8531 − 14.4665i 5.97390 + 1.60070i 26.2500 + 53.2298i −127.172 + 73.4229i −133.670 + 8.76122i 147.380 + 61.0467i 83.2676 − 310.759i −117.887 − 347.285i
2.12 −5.19156 − 0.683482i 10.2619 − 7.87425i −4.42448 − 1.18554i −24.1827 − 49.0377i −58.6573 + 33.8658i 206.724 − 13.5494i 176.968 + 73.3026i −19.5898 + 73.1102i 92.0297 + 271.111i
2.13 −5.06084 − 0.666273i −21.0211 + 16.1300i −5.74140 − 1.53840i −23.3379 − 47.3246i 117.131 − 67.6258i −58.3412 + 3.82388i 178.942 + 74.1202i 118.814 − 443.420i 86.5785 + 255.052i
2.14 −4.28506 − 0.564138i −6.71998 + 5.15642i −12.8662 − 3.44748i −11.0997 − 22.5080i 31.7044 − 18.3046i 112.717 − 7.38789i 180.965 + 74.9580i −44.3236 + 165.418i 34.8653 + 102.710i
2.15 −2.71724 − 0.357732i 15.5736 − 11.9500i −23.6542 − 6.33812i 10.9580 + 22.2205i −46.5920 + 26.8999i −98.7476 + 6.47226i 143.033 + 59.2462i 36.8398 − 137.488i −21.8264 − 64.2986i
2.16 −2.56821 − 0.338112i −21.4712 + 16.4754i −24.4282 − 6.54552i 17.9792 + 36.4583i 60.7132 − 35.0528i 206.320 − 13.5229i 137.106 + 56.7912i 126.680 − 472.776i −33.8475 − 99.7116i
2.17 −2.36332 − 0.311137i −8.69061 + 6.66854i −25.4212 − 6.81158i 45.5411 + 92.3482i 22.6135 − 13.0559i −92.9581 + 6.09280i 128.431 + 53.1980i −31.8358 + 118.813i −78.8952 − 232.418i
2.18 −1.78422 − 0.234897i 1.84682 − 1.41711i −27.7814 − 7.44399i −33.1729 − 67.2680i −3.62800 + 2.09463i −215.781 + 14.1431i 101.024 + 41.8453i −61.4905 + 229.486i 43.3867 + 127.813i
2.19 −1.36295 − 0.179435i 21.8908 − 16.7974i −29.0842 − 7.79309i −35.1195 − 71.2154i −32.8500 + 18.9659i 34.8614 − 2.28493i 78.8838 + 32.6748i 134.161 − 500.696i 35.0875 + 103.364i
2.20 −0.366237 − 0.0482160i 1.88695 − 1.44790i −30.7778 − 8.24689i 7.97879 + 16.1794i −0.760881 + 0.439295i 49.2297 − 3.22668i 21.7952 + 9.02788i −61.4289 + 229.256i −2.14202 − 6.31018i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
97.k even 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 97.6.k.a ✓ 640
97.k even 48 1 inner 97.6.k.a ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
97.6.k.a ✓ 640 1.a even 1 1 trivial
97.6.k.a ✓ 640 97.k even 48 1 inner

Hecke kernels

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