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Properties

Label 80.10.q.a
Level $80$
Weight $10$
Character orbit 80.q
Analytic conductor $41.203$
Analytic rank $0$
Dimension $212$
Inner twists $4$

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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] = [80,10,Mod(29,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.29"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 80.q (of order \(4\), 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: \(41.2028668931\)
Analytic rank: \(0\)
Dimension: \(212\)
Relative dimension: \(106\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 212 q - 4 q^{4} - 2 q^{5} - 4 q^{6} + 17812 q^{10} - 4 q^{11} + 108500 q^{14} - 4 q^{15} - 534208 q^{16} + 480884 q^{19} + 2372984 q^{20} - 78736 q^{21} - 4888112 q^{24} - 7927240 q^{26} - 4 q^{29} - 34132064 q^{30}+ \cdots + 771348172 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −22.6274 − 0.0159624i 51.5285 − 51.5285i 511.999 + 0.722376i −883.755 − 1082.64i −1166.78 + 1165.13i 1973.74 −11585.2 − 24.5182i 14372.6i 19979.8 + 24511.4i
29.2 −22.6212 + 0.528982i 109.589 − 109.589i 511.440 − 23.9325i −430.266 + 1329.66i −2421.07 + 2537.01i 12550.3 −11556.8 + 811.924i − 4336.57i 9029.77 − 30306.2i
29.3 −22.5970 + 1.17210i −90.3471 + 90.3471i 509.252 − 52.9719i −476.958 + 1313.63i 1935.68 − 2147.47i −6610.86 −11445.5 + 1793.90i 3357.81i 9238.12 − 30243.3i
29.4 −22.4389 + 2.91447i −31.0461 + 31.0461i 495.012 − 130.795i 1381.88 + 208.635i 606.158 − 787.124i 3903.52 −10726.3 + 4377.61i 17755.3i −31616.0 − 654.082i
29.5 −22.4073 + 3.14872i 15.4750 − 15.4750i 492.171 − 141.108i −1297.71 − 518.731i −298.027 + 395.480i −8769.00 −10583.9 + 4711.56i 19204.0i 30711.4 + 7537.22i
29.6 −22.3218 + 3.70652i −184.162 + 184.162i 484.523 − 165.472i 1397.54 + 4.06959i 3428.23 − 4793.43i −4698.49 −10202.1 + 5489.53i − 48148.5i −31210.6 + 5089.16i
29.7 −22.0219 − 5.19957i 112.080 − 112.080i 457.929 + 229.009i 980.598 + 995.768i −3050.99 + 1885.45i −6639.94 −8893.72 − 7424.25i − 5440.91i −16417.1 − 27027.4i
29.8 −21.9530 − 5.48332i −33.4851 + 33.4851i 451.867 + 240.750i 782.435 − 1157.98i 918.706 − 551.488i 5858.30 −8599.71 − 7762.91i 17440.5i −23526.4 + 21130.8i
29.9 −21.9219 − 5.60642i −154.741 + 154.741i 449.136 + 245.806i −1249.37 − 626.262i 4259.76 − 2524.67i 2175.39 −8467.81 − 7906.58i − 28206.7i 23877.4 + 20733.3i
29.10 −21.8788 + 5.77221i 158.152 − 158.152i 445.363 − 252.578i 807.774 − 1140.45i −2547.29 + 4373.07i 3847.35 −8286.08 + 8096.83i − 30341.3i −11090.2 + 29614.3i
29.11 −21.7913 + 6.09410i 188.670 − 188.670i 437.724 − 265.597i −1142.05 + 805.508i −2961.60 + 5261.15i −8687.40 −7920.01 + 8455.25i − 51509.9i 19978.0 − 24512.9i
29.12 −21.0565 + 8.28407i −96.5305 + 96.5305i 374.748 − 348.866i −1354.58 + 343.873i 1232.92 − 2832.26i 10222.4 −5000.84 + 10450.3i 1046.72i 25673.9 − 18462.2i
29.13 −20.8701 − 8.74303i 125.002 − 125.002i 359.119 + 364.935i 722.421 − 1196.34i −3701.70 + 1515.90i −4152.59 −4304.19 − 10756.0i − 11568.0i −25536.6 + 18651.6i
29.14 −20.4780 + 9.62562i −128.797 + 128.797i 326.695 − 394.226i 200.862 − 1383.03i 1397.76 − 3877.27i 6973.65 −2895.38 + 11217.6i − 13494.5i 9199.29 + 30255.1i
29.15 −20.3763 − 9.83893i −98.0596 + 98.0596i 318.391 + 400.963i 237.582 − 1377.20i 2962.90 − 1033.29i −10470.5 −2542.60 − 11302.8i 451.630i −18391.2 + 25724.7i
29.16 −20.2536 − 10.0892i −147.091 + 147.091i 308.414 + 408.687i 303.650 + 1364.16i 4463.15 − 1495.08i 6644.50 −2123.15 − 11389.0i − 23588.3i 7613.31 − 30692.6i
29.17 −20.0088 − 10.5663i 47.8279 − 47.8279i 288.707 + 422.838i −1077.67 + 889.809i −1462.35 + 451.618i 1025.94 −1308.86 − 11511.1i 15108.0i 30964.8 − 6417.12i
29.18 −19.7925 + 10.9662i 48.6440 − 48.6440i 271.485 − 434.097i 1255.67 − 613.526i −429.346 + 1496.23i −11079.3 −612.967 + 11569.0i 14950.5i −18124.8 + 25913.1i
29.19 −19.3748 − 11.6883i 179.082 − 179.082i 238.765 + 452.918i −1236.30 − 651.681i −5562.84 + 1376.50i 1576.93 667.839 − 11566.0i − 44457.5i 16336.0 + 27076.5i
29.20 −19.3384 + 11.7485i 96.8997 − 96.8997i 235.945 − 454.394i 804.961 + 1142.44i −735.454 + 3012.31i 1696.61 775.676 + 11559.2i 903.893i −28988.6 − 12635.8i
See next 80 embeddings (of 212 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.106
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.10.q.a ✓ 212
5.b even 2 1 inner 80.10.q.a ✓ 212
16.e even 4 1 inner 80.10.q.a ✓ 212
80.q even 4 1 inner 80.10.q.a ✓ 212
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.10.q.a ✓ 212 1.a even 1 1 trivial
80.10.q.a ✓ 212 5.b even 2 1 inner
80.10.q.a ✓ 212 16.e even 4 1 inner
80.10.q.a ✓ 212 80.q even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(80, [\chi])\).