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Properties

Label 64.10.i.a
Level $64$
Weight $10$
Character orbit 64.i
Analytic conductor $32.962$
Analytic rank $0$
Dimension $568$
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] = [64,10,Mod(5,64)] 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("64.5"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(64, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 64.i (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: \(32.9622935145\)
Analytic rank: \(0\)
Dimension: \(568\)
Relative dimension: \(71\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 568 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9} - 8 q^{10} - 8 q^{11} - 8 q^{12} - 8 q^{13} - 8 q^{14} - 8 q^{15} - 8 q^{16} - 8 q^{17} - 8 q^{18} - 8 q^{19} - 8 q^{20}+ \cdots + 157456 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −22.6274 + 0.00136331i −61.0037 − 40.7614i 512.000 − 0.0616962i 897.532 + 178.530i 1380.41 + 922.241i 4072.12 − 9830.97i −11585.2 + 2.09404i −5472.40 − 13211.5i −20309.1 − 4038.45i
5.2 −22.6240 − 0.390747i 0.460275 + 0.307546i 511.695 + 17.6806i 2016.41 + 401.089i −10.2931 − 7.13779i −2239.91 + 5407.61i −11569.7 − 599.949i −7532.24 − 18184.4i −45462.6 − 9862.16i
5.3 −22.5838 − 1.40431i −173.185 − 115.719i 508.056 + 63.4294i −1620.21 − 322.279i 3748.67 + 2856.57i −3384.90 + 8171.86i −11384.8 − 2145.95i 9069.94 + 21896.8i 36137.8 + 9553.57i
5.4 −22.3444 + 3.56788i 150.893 + 100.823i 486.540 − 159.444i −1900.87 − 378.107i −3731.33 − 1714.46i −4470.29 + 10792.2i −10302.6 + 5298.59i 5070.92 + 12242.3i 43822.8 + 1666.47i
5.5 −22.2108 − 4.32229i 158.701 + 106.040i 474.636 + 192.003i 198.982 + 39.5800i −3066.53 − 3041.19i 2226.24 − 5374.62i −9712.13 − 6316.04i 6409.00 + 15472.7i −4248.47 − 1739.16i
5.6 −21.8044 − 6.04721i −166.361 − 111.159i 438.862 + 263.711i 856.044 + 170.278i 2955.21 + 3429.78i −2280.04 + 5504.50i −7974.41 − 8403.96i 7787.42 + 18800.5i −17635.8 − 8889.48i
5.7 −21.6788 + 6.48313i −139.451 − 93.1781i 427.938 − 281.093i −1449.09 − 288.241i 3627.21 + 1115.91i 1958.23 − 4727.58i −7454.81 + 8868.12i 3232.03 + 7802.81i 33283.1 − 3145.91i
5.8 −21.6323 + 6.63662i 48.0959 + 32.1367i 423.911 − 287.130i −496.541 − 98.7681i −1253.70 − 375.995i −856.360 + 2067.43i −7264.58 + 9024.61i −6251.91 − 15093.4i 11396.8 − 1158.77i
5.9 −21.5257 − 6.97468i 26.8261 + 17.9246i 414.708 + 300.269i −1973.83 − 392.619i −452.431 − 572.943i 1748.32 − 4220.82i −6832.57 − 9355.95i −7134.01 − 17223.0i 39749.6 + 22218.2i
5.10 −21.1342 + 8.08360i 185.389 + 123.873i 381.311 − 341.681i 284.180 + 56.5268i −4919.39 − 1119.35i 2456.03 − 5929.39i −5296.70 + 10303.5i 11492.2 + 27744.6i −6462.86 + 1102.54i
5.11 −20.4591 − 9.66566i 174.556 + 116.634i 325.150 + 395.502i 991.245 + 197.171i −2443.90 − 4073.43i −1912.09 + 4616.18i −2829.49 − 11234.4i 9333.73 + 22533.6i −18374.2 − 13615.0i
5.12 −19.5573 + 11.3804i −208.374 − 139.231i 252.975 − 445.138i 2165.43 + 430.730i 5659.72 + 351.610i 895.911 − 2162.92i 118.330 + 11584.6i 16502.0 + 39839.4i −47251.8 + 16219.4i
5.13 −19.1446 − 12.0616i −203.663 − 136.083i 221.034 + 461.831i −101.206 − 20.1311i 2257.67 + 5061.77i 3486.79 − 8417.86i 1338.83 − 11507.6i 15427.6 + 37245.5i 1694.74 + 1606.11i
5.14 −18.5512 + 12.9558i 98.4214 + 65.7631i 176.295 − 480.691i 2139.40 + 425.554i −2677.85 + 55.1424i 688.510 − 1662.21i 2957.25 + 11201.4i −2170.36 − 5239.72i −45201.9 + 19823.1i
5.15 −18.3262 + 13.2722i −96.8286 − 64.6988i 159.700 − 486.457i 193.294 + 38.4485i 2633.19 − 99.4401i −1929.06 + 4657.16i 3529.63 + 11034.5i −2342.51 − 5655.33i −4052.64 + 1860.81i
5.16 −17.9892 − 13.7255i −12.8383 − 8.57826i 135.223 + 493.820i −684.820 − 136.219i 113.210 + 330.528i −1821.12 + 4396.58i 4345.35 − 10739.4i −7441.12 − 17964.5i 10449.7 + 11849.9i
5.17 −17.1459 − 14.7654i −84.6262 − 56.5454i 75.9632 + 506.333i 1594.75 + 317.216i 616.073 + 2219.07i 947.665 − 2287.87i 6173.78 − 9803.17i −3568.15 − 8614.27i −22659.7 − 28986.2i
5.18 −16.2141 + 15.7830i 50.2823 + 33.5976i 13.7934 − 511.814i −1948.31 − 387.543i −1345.55 + 248.852i 1690.99 − 4082.41i 7854.32 + 8516.30i −6132.84 − 14806.0i 37706.7 − 24466.6i
5.19 −14.2080 − 17.6106i 210.883 + 140.907i −108.264 + 500.423i −2350.51 − 467.546i −514.767 − 5715.77i −369.039 + 890.939i 10351.0 − 5203.42i 17084.2 + 41245.0i 25162.4 + 48036.8i
5.20 −14.0980 − 17.6988i 113.501 + 75.8391i −114.494 + 499.034i 2470.96 + 491.505i −257.877 − 3078.01i 1570.43 − 3791.36i 10446.4 − 5008.96i −401.396 − 969.056i −26136.5 − 50662.3i
See next 80 embeddings (of 568 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.71
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.10.i.a ✓ 568
64.i even 16 1 inner 64.10.i.a ✓ 568
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.10.i.a ✓ 568 1.a even 1 1 trivial
64.10.i.a ✓ 568 64.i even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(64, [\chi])\).