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Properties

Label 80.11.i.a
Level $80$
Weight $11$
Character orbit 80.i
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(13,80)] 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("80.13"); 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, 3, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.i (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.

\(\operatorname{Tr}(f)(q) = \) \( 236 q - 2 q^{2} + 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} - 4487724 q^{9} + 2046 q^{10} - 4 q^{11} + 738232 q^{12} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} + 5924662 q^{18} - 5107040 q^{19} + 2913160 q^{20}+ \cdots - 28071956444 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −31.9615 − 1.56973i − 214.607i 1019.07 + 100.341i 2623.04 + 1698.61i −336.874 + 6859.15i −21161.1 − 21161.1i −32413.5 − 4806.72i 12992.9 −81170.0 − 58407.5i
13.2 −31.9536 − 1.72183i 388.128i 1018.07 + 110.037i 3121.22 + 153.749i 668.289 − 12402.1i 1274.43 + 1274.43i −32341.6 − 5269.03i −91594.5 −99469.5 − 10287.0i
13.3 −31.9449 + 1.87773i 324.949i 1016.95 − 119.968i −2059.26 − 2350.54i −610.167 − 10380.5i 17094.6 + 17094.6i −32261.0 + 5741.91i −46542.9 70196.6 + 71221.0i
13.4 −31.8807 + 2.76092i − 368.635i 1008.75 − 176.040i −2496.78 − 1879.29i 1017.77 + 11752.3i 12663.0 + 12663.0i −31673.7 + 8397.37i −76842.7 84787.5 + 53019.7i
13.5 −31.8587 − 3.00404i − 26.1884i 1005.95 + 191.409i 629.600 + 3060.92i −78.6710 + 834.328i 8782.74 + 8782.74i −31473.3 − 9119.97i 58363.2 −10863.1 − 99408.2i
13.6 −31.2876 + 6.71444i 412.535i 933.833 − 420.158i −1073.57 + 2934.80i −2769.94 − 12907.2i −8110.53 − 8110.53i −26396.3 + 19415.9i −111136. 13884.1 − 99031.5i
13.7 −31.2550 − 6.86473i − 228.353i 929.751 + 429.115i 1310.86 − 2836.77i −1567.58 + 7137.17i −3078.17 − 3078.17i −26113.6 − 19794.5i 6903.90 −60444.6 + 79664.6i
13.8 −31.2442 − 6.91353i 290.364i 928.406 + 432.016i 2533.30 − 1829.76i 2007.44 − 9072.21i −16042.8 − 16042.8i −26020.6 − 19916.6i −25262.4 −91801.1 + 39655.5i
13.9 −31.1873 + 7.16609i 113.551i 921.294 − 446.982i −3033.64 − 750.089i −813.717 − 3541.35i −9621.52 − 9621.52i −25529.6 + 20542.2i 46155.2 99986.3 + 1653.87i
13.10 −31.1219 + 7.44510i 73.3622i 913.141 − 463.411i 3123.99 − 79.4807i −546.189 − 2283.17i 19105.3 + 19105.3i −24968.5 + 21220.6i 53667.0 −96632.6 + 25732.0i
13.11 −31.0905 + 7.57519i − 417.831i 909.233 − 471.032i −2306.53 + 2108.44i 3165.15 + 12990.6i −15187.1 − 15187.1i −24700.3 + 21532.2i −115534. 55739.2 − 83024.9i
13.12 −31.0538 + 7.72412i − 201.522i 904.676 − 479.726i 1379.87 − 2803.85i 1556.58 + 6258.01i 4671.61 + 4671.61i −24388.2 + 21885.1i 18438.0 −21192.8 + 97728.5i
13.13 −30.7961 − 8.69488i − 71.4666i 872.798 + 535.537i −2977.19 − 949.731i −621.394 + 2200.89i −14532.1 − 14532.1i −22222.3 − 24081.3i 53941.5 83427.9 + 55134.3i
13.14 −30.7560 − 8.83561i − 445.379i 867.864 + 543.496i 2471.93 + 1911.85i −3935.20 + 13698.1i 15417.6 + 15417.6i −21889.9 − 24383.9i −139314. −59134.5 − 80641.9i
13.15 −30.3669 − 10.0922i 245.343i 820.295 + 612.938i −2236.79 + 2182.29i 2476.05 − 7450.30i 5366.20 + 5366.20i −18723.9 − 26891.6i −1144.21 89948.5 − 43695.1i
13.16 −29.2822 + 12.9055i − 135.733i 690.895 − 755.804i −2255.45 + 2163.00i 1751.71 + 3974.57i 7878.52 + 7878.52i −10476.9 + 31048.0i 40625.4 38129.9 − 92445.2i
13.17 −29.0236 + 13.4770i 161.597i 660.740 − 782.303i 462.093 − 3090.65i −2177.85 − 4690.14i −13285.8 − 13285.8i −8633.96 + 31610.1i 32935.3 28241.1 + 95929.4i
13.18 −28.7151 − 14.1224i 163.501i 625.113 + 811.054i −226.398 − 3116.79i 2309.03 − 4694.93i 9037.18 + 9037.18i −6496.12 − 32117.6i 32316.6 −37515.6 + 92696.2i
13.19 −28.4618 − 14.6261i − 233.425i 596.154 + 832.572i −2188.11 + 2231.09i −3414.10 + 6643.70i 4255.80 + 4255.80i −4790.33 − 32416.0i 4561.92 94910.0 − 31497.4i
13.20 −27.8828 + 15.7020i 72.8954i 530.896 − 875.629i 1447.28 + 2769.66i −1144.60 − 2032.52i −11191.3 − 11191.3i −1053.74 + 32751.1i 53735.3 −83843.3 − 54500.5i
See next 80 embeddings (of 236 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 13.118
Significant digits:
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Inner twists

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

Hecke kernels

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