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Properties

Label 16.11.f.a
Level $16$
Weight $11$
Character orbit 16.f
Analytic conductor $10.166$
Analytic rank $0$
Dimension $38$
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] = [16,11,Mod(3,16)] 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("16.3"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(16, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 16.f (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: \(10.1657160428\)
Analytic rank: \(0\)
Dimension: \(38\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8} - 147364 q^{10} - 45906 q^{11} - 10988 q^{12} - 2 q^{13} + 757836 q^{14} - 469032 q^{16} - 4 q^{17} + 2066926 q^{18}+ \cdots - 25616384098 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −31.9996 + 0.164143i −221.985 + 221.985i 1023.95 − 10.5050i 2579.78 − 2579.78i 7066.97 − 7139.85i 4017.27 −32764.1 + 504.230i − 39505.3i −82128.4 + 82975.3i
3.2 −30.1099 − 10.8349i 127.462 − 127.462i 789.210 + 652.475i −1086.76 + 1086.76i −5218.91 + 2456.83i −8881.75 −16693.5 − 28197.0i 26555.9i 44497.0 − 20947.2i
3.3 −29.1906 + 13.1115i 32.7265 − 32.7265i 680.178 − 765.463i −1645.29 + 1645.29i −526.211 + 1384.40i 16644.7 −9818.42 + 31262.4i 56907.0i 26454.7 − 69599.0i
3.4 −25.9865 + 18.6735i 280.546 − 280.546i 326.598 − 970.520i 3102.01 − 3102.01i −2051.63 + 12529.2i −27408.1 9635.89 + 31319.2i − 98363.4i −22685.0 + 138536.i
3.5 −19.3334 + 25.4994i −222.022 + 222.022i −276.439 − 985.981i −2711.76 + 2711.76i −1368.99 − 9953.89i −24891.4 30486.4 + 12013.3i − 39538.9i −16720.7 − 121576.i
3.6 −19.3314 − 25.5009i −196.215 + 196.215i −276.596 + 985.936i −1311.50 + 1311.50i 8796.79 + 1210.56i −1120.35 30489.3 − 12006.1i − 17952.0i 58797.6 + 8091.39i
3.7 −16.8116 − 27.2281i 183.793 − 183.793i −458.741 + 915.496i 3496.01 − 3496.01i −8094.18 − 1914.49i 26135.6 32639.4 − 2900.27i − 8510.59i −153963. − 36416.3i
3.8 −9.61264 + 30.5221i −107.144 + 107.144i −839.194 − 586.796i 3434.17 − 3434.17i −2240.31 − 4300.18i 8771.09 25977.1 − 19973.3i 36089.4i 71806.6 + 137829.i
3.9 −6.25659 + 31.3824i 196.669 − 196.669i −945.710 − 392.694i −1812.92 + 1812.92i 4941.46 + 7402.41i 22339.6 18240.6 − 27221.7i − 18308.2i −45551.0 − 68236.4i
3.10 −1.21444 − 31.9769i 287.146 − 287.146i −1021.05 + 77.6681i −3974.49 + 3974.49i −9530.78 − 8833.33i −11755.9 3723.59 + 32555.7i − 105857.i 131919. + 122265.i
3.11 2.46002 − 31.9053i −46.0373 + 46.0373i −1011.90 − 156.975i 1933.37 − 1933.37i 1355.58 + 1582.09i −25927.9 −7497.63 + 31898.7i 54810.1i −56928.5 − 66440.7i
3.12 12.2776 − 29.5510i −126.065 + 126.065i −722.523 − 725.628i −1655.31 + 1655.31i 2177.58 + 5273.11i 29862.3 −30313.9 + 12442.3i 27264.3i 28593.0 + 69239.3i
3.13 12.5336 + 29.4433i 44.0645 − 44.0645i −709.815 + 738.064i −126.270 + 126.270i 1849.69 + 745.115i −19594.2 −30627.6 − 11648.7i 55165.6i −5300.42 − 2135.18i
3.14 17.5699 + 26.7451i −286.288 + 286.288i −406.599 + 939.816i −1338.28 + 1338.28i −12686.8 − 2626.75i 19657.4 −32279.3 + 5637.94i − 104873.i −59305.9 − 12279.0i
3.15 26.0928 − 18.5248i 164.886 − 164.886i 337.664 − 966.726i 897.573 − 897.573i 1247.85 − 7356.80i 594.817 −9097.83 − 31479.7i 4674.41i 6792.79 − 40047.5i
3.16 27.4041 + 16.5231i 305.982 − 305.982i 477.973 + 905.603i 155.264 − 155.264i 13440.9 − 3329.39i 8523.80 −1864.95 + 32714.9i − 128200.i 6820.32 − 1689.43i
3.17 28.0563 − 15.3897i −317.338 + 317.338i 550.315 − 863.556i 1564.48 − 1564.48i −4019.61 + 13787.1i −15496.4 2149.97 − 32697.4i − 142358.i 19816.8 − 67970.6i
3.18 30.4523 + 9.83157i −61.4496 + 61.4496i 830.681 + 598.787i 2595.70 − 2595.70i −2475.42 + 1267.13i 6393.57 19409.1 + 26401.3i 51496.9i 104565. − 53525.2i
3.19 32.0000 + 0.0155982i −39.7294 + 39.7294i 1024.00 + 0.998283i −4096.79 + 4096.79i −1271.96 + 1270.72i −7866.10 32768.0 + 47.9176i 55892.1i −131161. + 131033.i
11.1 −31.9996 − 0.164143i −221.985 − 221.985i 1023.95 + 10.5050i 2579.78 + 2579.78i 7066.97 + 7139.85i 4017.27 −32764.1 − 504.230i 39505.3i −82128.4 − 82975.3i
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.19
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.11.f.a ✓ 38
4.b odd 2 1 64.11.f.a 38
8.b even 2 1 128.11.f.b 38
8.d odd 2 1 128.11.f.a 38
16.e even 4 1 64.11.f.a 38
16.e even 4 1 128.11.f.a 38
16.f odd 4 1 inner 16.11.f.a ✓ 38
16.f odd 4 1 128.11.f.b 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.11.f.a ✓ 38 1.a even 1 1 trivial
16.11.f.a ✓ 38 16.f odd 4 1 inner
64.11.f.a 38 4.b odd 2 1
64.11.f.a 38 16.e even 4 1
128.11.f.a 38 8.d odd 2 1
128.11.f.a 38 16.e even 4 1
128.11.f.b 38 8.b even 2 1
128.11.f.b 38 16.f odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(16, [\chi])\).