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Properties

Label 80.11.t.a
Level $80$
Weight $11$
Character orbit 80.t
Analytic conductor $50.829$
Analytic rank $0$
Dimension $236$
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] = [80,11,Mod(53,80)] 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("80.53"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.t (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: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(118\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 236 q - 2 q^{2} - 4 q^{3} - 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} + 4487724 q^{9} - 2050 q^{10} - 4 q^{11} + 502036 q^{12} - 4 q^{13} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} - 5928762 q^{18} + 5107040 q^{19}+ \cdots + 28071956444 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −31.9903 − 0.785980i 381.913 1022.76 + 50.2875i −2861.33 + 1256.34i −12217.5 − 300.176i 20197.7 + 20197.7i −32679.1 − 2412.59i 86808.8 92522.5 − 37941.9i
53.2 −31.9342 − 2.05133i −192.036 1015.58 + 131.015i 3119.02 − 193.255i 6132.51 + 393.928i 18555.5 + 18555.5i −32163.1 − 6267.14i −22171.2 −99999.7 − 226.682i
53.3 −31.8729 + 2.84943i 394.599 1007.76 − 181.639i 778.709 + 3026.42i −12577.0 + 1124.38i −4379.52 − 4379.52i −31602.7 + 8660.92i 96659.1 −33443.3 − 94242.0i
53.4 −31.8649 + 2.93778i −74.0023 1006.74 − 187.224i 0.479946 − 3125.00i 2358.07 − 217.402i −2919.78 − 2919.78i −31529.6 + 8923.43i −53572.7 9165.26 + 99579.1i
53.5 −31.8295 − 3.29876i −398.362 1002.24 + 209.996i −3088.51 + 476.134i 12679.7 + 1314.10i 13876.0 + 13876.0i −31208.0 − 9990.20i 99643.6 99876.6 − 4966.86i
53.6 −31.7358 + 4.10388i 106.445 990.316 − 260.479i −2840.60 − 1302.54i −3378.13 + 436.839i −14456.7 − 14456.7i −30359.5 + 12330.7i −47718.4 95494.1 + 29679.7i
53.7 −31.5860 − 5.13086i −69.2669 971.348 + 324.127i −1327.58 + 2828.99i 2187.86 + 355.399i −13184.8 − 13184.8i −29017.9 − 15221.7i −54251.1 56448.0 − 82544.7i
53.8 −31.4961 + 5.65641i −369.764 960.010 − 356.310i 2213.06 + 2206.36i 11646.1 − 2091.54i −8095.14 − 8095.14i −28221.1 + 16652.6i 77676.5 −82182.7 − 56973.6i
53.9 −31.4526 + 5.89374i 259.387 954.528 − 370.746i 2667.14 − 1628.49i −8158.40 + 1528.76i 820.915 + 820.915i −27837.3 + 17286.7i 8232.78 −74290.5 + 66939.7i
53.10 −30.7333 − 8.91418i 137.257 865.075 + 547.925i 1320.72 + 2832.19i −4218.36 − 1223.53i 13740.1 + 13740.1i −21702.3 − 24551.0i −40209.5 −15343.4 − 98815.9i
53.11 −30.7099 + 8.99461i −435.529 862.194 − 552.447i −425.653 − 3095.88i 13375.0 − 3917.41i −13283.1 − 13283.1i −21508.8 + 24720.7i 130637. 40918.0 + 91245.4i
53.12 −30.4979 − 9.68897i 476.454 836.248 + 590.987i −1108.33 − 2921.85i −14530.9 − 4616.35i −17569.8 − 17569.8i −19777.8 − 26126.3i 167960. 5492.17 + 99849.1i
53.13 −30.1366 − 10.7603i 107.688 792.430 + 648.560i −2853.42 − 1274.22i −3245.35 − 1158.76i 6451.25 + 6451.25i −16902.4 − 28072.2i −47452.3 72281.2 + 69104.5i
53.14 −30.1354 + 10.7636i −201.343 792.289 − 648.732i −2152.30 + 2265.67i 6067.56 − 2167.18i 3246.23 + 3246.23i −16893.3 + 28077.7i −18510.0 40473.6 − 91443.4i
53.15 −29.3849 − 12.6700i 223.074 702.944 + 744.611i 2920.49 + 1111.93i −6554.99 − 2826.33i −14947.8 − 14947.8i −11221.7 − 30786.6i −9287.21 −71730.0 − 69676.4i
53.16 −29.2793 − 12.9122i −227.008 690.550 + 756.119i 2875.48 − 1223.61i 6646.63 + 2931.18i −11637.9 − 11637.9i −10455.6 − 31055.1i −7516.24 −99991.5 − 1302.28i
53.17 −28.7505 − 14.0502i −309.068 629.181 + 807.903i −698.169 − 3046.01i 8885.85 + 4342.47i 2496.00 + 2496.00i −6738.05 − 32067.7i 36473.8 −22724.5 + 97383.8i
53.18 −28.4691 + 14.6120i 288.899 596.976 − 831.983i −3046.03 + 698.103i −8224.68 + 4221.40i −8516.05 − 8516.05i −4838.38 + 32408.8i 24413.5 76516.8 − 64383.0i
53.19 −28.0073 + 15.4787i 58.4164 544.820 − 867.033i 1195.18 + 2887.42i −1636.09 + 904.210i 11070.5 + 11070.5i −1838.42 + 32716.4i −55636.5 −78167.2 − 62368.9i
53.20 −27.7151 − 15.9960i 275.712 512.253 + 886.664i 673.877 − 3051.48i −7641.37 − 4410.29i 14140.7 + 14140.7i −14.0368 − 32768.0i 16967.8 −67488.1 + 73792.6i
See next 80 embeddings (of 236 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 53.118
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.t odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.11.t.a yes 236
5.c odd 4 1 80.11.i.a ✓ 236
16.e even 4 1 80.11.i.a ✓ 236
80.t odd 4 1 inner 80.11.t.a yes 236
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.11.i.a ✓ 236 5.c odd 4 1
80.11.i.a ✓ 236 16.e even 4 1
80.11.t.a yes 236 1.a even 1 1 trivial
80.11.t.a yes 236 80.t odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(80, [\chi])\).