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Properties

Label 80.11.i.a.13.18
Level $80$
Weight $11$
Character 80.13
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: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.18
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.18

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-28.7151 - 14.1224i) q^{2} +163.501i q^{3} +(625.113 + 811.054i) q^{4} +(-226.398 - 3116.79i) q^{5} +(2309.03 - 4694.93i) q^{6} +(9037.18 + 9037.18i) q^{7} +(-6496.12 - 32117.6i) q^{8} +32316.6 q^{9} +(-37515.6 + 92696.2i) q^{10} +(-128414. + 128414. i) q^{11} +(-132608. + 102206. i) q^{12} +385731. i q^{13} +(-131876. - 387131. i) q^{14} +(509596. - 37016.2i) q^{15} +(-267043. + 1.01400e6i) q^{16} +(1.06906e6 - 1.06906e6i) q^{17} +(-927974. - 456389. i) q^{18} +(945739. - 945739. i) q^{19} +(2.38636e6 - 2.13197e6i) q^{20} +(-1.47758e6 + 1.47758e6i) q^{21} +(5.50095e6 - 1.87390e6i) q^{22} +(-392909. + 392909. i) q^{23} +(5.25125e6 - 1.06212e6i) q^{24} +(-9.66311e6 + 1.41127e6i) q^{25} +(5.44746e6 - 1.10763e7i) q^{26} +1.49383e7i q^{27} +(-1.68038e6 + 1.29789e7i) q^{28} +(2.00171e7 - 2.00171e7i) q^{29} +(-1.51559e7 - 6.13382e6i) q^{30} -1.53924e7 q^{31} +(2.19883e7 - 2.53459e7i) q^{32} +(-2.09958e7 - 2.09958e7i) q^{33} +(-4.57957e7 + 1.56004e7i) q^{34} +(2.61210e7 - 3.02130e7i) q^{35} +(2.02015e7 + 2.62105e7i) q^{36} -8.75551e7i q^{37} +(-4.05131e7 + 1.38008e7i) q^{38} -6.30672e7 q^{39} +(-9.86331e7 + 2.75184e7i) q^{40} +1.47737e8i q^{41} +(6.32961e7 - 2.15619e7i) q^{42} -1.24965e8 q^{43} +(-1.84424e8 - 2.38775e7i) q^{44} +(-7.31641e6 - 1.00724e8i) q^{45} +(1.68312e7 - 5.73358e6i) q^{46} +(-2.13816e8 + 2.13816e8i) q^{47} +(-1.65790e8 - 4.36616e7i) q^{48} -1.19134e8i q^{49} +(2.97408e8 + 9.59420e7i) q^{50} +(1.74791e8 + 1.74791e8i) q^{51} +(-3.12849e8 + 2.41126e8i) q^{52} +7.46145e8 q^{53} +(2.10966e8 - 4.28955e8i) q^{54} +(4.29313e8 + 3.71167e8i) q^{55} +(2.31546e8 - 3.48960e8i) q^{56} +(1.54629e8 + 1.54629e8i) q^{57} +(-8.57483e8 + 2.92103e8i) q^{58} +(7.06065e7 + 7.06065e7i) q^{59} +(3.48578e8 + 3.90171e8i) q^{60} +(9.37334e8 + 9.37334e8i) q^{61} +(4.41996e8 + 2.17379e8i) q^{62} +(2.92051e8 + 2.92051e8i) q^{63} +(-9.89343e8 + 4.17280e8i) q^{64} +(1.20224e9 - 8.73287e7i) q^{65} +(3.06384e8 + 8.99408e8i) q^{66} -3.00127e8 q^{67} +(1.53534e9 + 1.98782e8i) q^{68} +(-6.42408e7 - 6.42408e7i) q^{69} +(-1.17675e9 + 4.98677e8i) q^{70} +1.89991e9i q^{71} +(-2.09932e8 - 1.03793e9i) q^{72} +(-6.95678e8 + 6.95678e8i) q^{73} +(-1.23649e9 + 2.51415e9i) q^{74} +(-2.30743e8 - 1.57992e9i) q^{75} +(1.35824e9 + 1.75852e8i) q^{76} -2.32101e9 q^{77} +(1.81098e9 + 8.90663e8i) q^{78} +3.07865e9i q^{79} +(3.22089e9 + 6.02748e8i) q^{80} -5.34161e8 q^{81} +(2.08641e9 - 4.24229e9i) q^{82} +5.16023e9i q^{83} +(-2.12206e9 - 2.74744e8i) q^{84} +(-3.57405e9 - 3.08999e9i) q^{85} +(3.58839e9 + 1.76482e9i) q^{86} +(3.27281e9 + 3.27281e9i) q^{87} +(4.95856e9 + 3.29017e9i) q^{88} +7.76406e9 q^{89} +(-1.21238e9 + 2.99562e9i) q^{90} +(-3.48592e9 + 3.48592e9i) q^{91} +(-5.64283e8 - 7.30579e7i) q^{92} -2.51667e9i q^{93} +(9.15935e9 - 3.12014e9i) q^{94} +(-3.16178e9 - 2.73356e9i) q^{95} +(4.14406e9 + 3.59510e9i) q^{96} +(-5.17776e9 + 5.17776e9i) q^{97} +(-1.68246e9 + 3.42094e9i) q^{98} +(-4.14991e9 + 4.14991e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 236 q - 2 q^{2} + 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} - 4487724 q^{9} + 2046 q^{10} - 4 q^{11} + 738232 q^{12} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} + 5924662 q^{18} - 5107040 q^{19} + 2913160 q^{20}+ \cdots - 28071956444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −28.7151 − 14.1224i −0.897347 − 0.441326i
\(3\) 163.501i 0.672842i 0.941712 + 0.336421i \(0.109216\pi\)
−0.941712 + 0.336421i \(0.890784\pi\)
\(4\) 625.113 + 811.054i 0.610462 + 0.792045i
\(5\) −226.398 − 3116.79i −0.0724473 − 0.997372i
\(6\) 2309.03 − 4694.93i 0.296943 − 0.603772i
\(7\) 9037.18 + 9037.18i 0.537703 + 0.537703i 0.922854 − 0.385150i \(-0.125850\pi\)
−0.385150 + 0.922854i \(0.625850\pi\)
\(8\) −6496.12 − 32117.6i −0.198246 − 0.980152i
\(9\) 32316.6 0.547284
\(10\) −37515.6 + 92696.2i −0.375156 + 0.926962i
\(11\) −128414. + 128414.i −0.797351 + 0.797351i −0.982677 − 0.185326i \(-0.940666\pi\)
0.185326 + 0.982677i \(0.440666\pi\)
\(12\) −132608. + 102206.i −0.532921 + 0.410744i
\(13\) 385731.i 1.03889i 0.854505 + 0.519443i \(0.173860\pi\)
−0.854505 + 0.519443i \(0.826140\pi\)
\(14\) −131876. − 387131.i −0.245204 − 0.719809i
\(15\) 509596. − 37016.2i 0.671074 − 0.0487456i
\(16\) −267043. + 1.01400e6i −0.254672 + 0.967028i
\(17\) 1.06906e6 − 1.06906e6i 0.752932 − 0.752932i −0.222093 − 0.975025i \(-0.571289\pi\)
0.975025 + 0.222093i \(0.0712889\pi\)
\(18\) −927974. − 456389.i −0.491104 − 0.241531i
\(19\) 945739. − 945739.i 0.381947 − 0.381947i −0.489856 − 0.871803i \(-0.662951\pi\)
0.871803 + 0.489856i \(0.162951\pi\)
\(20\) 2.38636e6 − 2.13197e6i 0.745738 − 0.666240i
\(21\) −1.47758e6 + 1.47758e6i −0.361789 + 0.361789i
\(22\) 5.50095e6 − 1.87390e6i 1.06739 − 0.363609i
\(23\) −392909. + 392909.i −0.0610453 + 0.0610453i −0.736970 − 0.675925i \(-0.763744\pi\)
0.675925 + 0.736970i \(0.263744\pi\)
\(24\) 5.25125e6 − 1.06212e6i 0.659487 − 0.133388i
\(25\) −9.66311e6 + 1.41127e6i −0.989503 + 0.144514i
\(26\) 5.44746e6 − 1.10763e7i 0.458487 − 0.932241i
\(27\) 1.49383e7i 1.04108i
\(28\) −1.68038e6 + 1.29789e7i −0.0976379 + 0.754133i
\(29\) 2.00171e7 − 2.00171e7i 0.975913 − 0.975913i −0.0238036 − 0.999717i \(-0.507578\pi\)
0.999717 + 0.0238036i \(0.00757765\pi\)
\(30\) −1.51559e7 − 6.13382e6i −0.623698 − 0.252421i
\(31\) −1.53924e7 −0.537649 −0.268825 − 0.963189i \(-0.586635\pi\)
−0.268825 + 0.963189i \(0.586635\pi\)
\(32\) 2.19883e7 − 2.53459e7i 0.655304 − 0.755366i
\(33\) −2.09958e7 − 2.09958e7i −0.536491 − 0.536491i
\(34\) −4.57957e7 + 1.56004e7i −1.00793 + 0.343353i
\(35\) 2.61210e7 − 3.02130e7i 0.497335 − 0.575246i
\(36\) 2.02015e7 + 2.62105e7i 0.334096 + 0.433474i
\(37\) − 8.75551e7i − 1.26262i −0.775531 − 0.631310i \(-0.782518\pi\)
0.775531 − 0.631310i \(-0.217482\pi\)
\(38\) −4.05131e7 + 1.38008e7i −0.511302 + 0.174176i
\(39\) −6.30672e7 −0.699005
\(40\) −9.86331e7 + 2.75184e7i −0.963214 + 0.268734i
\(41\) 1.47737e8i 1.27518i 0.770377 + 0.637589i \(0.220068\pi\)
−0.770377 + 0.637589i \(0.779932\pi\)
\(42\) 6.32961e7 − 2.15619e7i 0.484318 − 0.164983i
\(43\) −1.24965e8 −0.850056 −0.425028 − 0.905180i \(-0.639736\pi\)
−0.425028 + 0.905180i \(0.639736\pi\)
\(44\) −1.84424e8 − 2.38775e7i −1.11829 − 0.144786i
\(45\) −7.31641e6 − 1.00724e8i −0.0396493 − 0.545846i
\(46\) 1.68312e7 − 5.73358e6i 0.0817197 − 0.0278379i
\(47\) −2.13816e8 + 2.13816e8i −0.932289 + 0.932289i −0.997849 − 0.0655594i \(-0.979117\pi\)
0.0655594 + 0.997849i \(0.479117\pi\)
\(48\) −1.65790e8 − 4.36616e7i −0.650656 − 0.171354i
\(49\) − 1.19134e8i − 0.421750i
\(50\) 2.97408e8 + 9.59420e7i 0.951705 + 0.307014i
\(51\) 1.74791e8 + 1.74791e8i 0.506604 + 0.506604i
\(52\) −3.12849e8 + 2.41126e8i −0.822844 + 0.634200i
\(53\) 7.46145e8 1.78420 0.892101 − 0.451836i \(-0.149231\pi\)
0.892101 + 0.451836i \(0.149231\pi\)
\(54\) 2.10966e8 − 4.28955e8i 0.459455 − 0.934207i
\(55\) 4.29313e8 + 3.71167e8i 0.853022 + 0.737490i
\(56\) 2.31546e8 − 3.48960e8i 0.420434 − 0.633629i
\(57\) 1.54629e8 + 1.54629e8i 0.256990 + 0.256990i
\(58\) −8.57483e8 + 2.92103e8i −1.30643 + 0.445036i
\(59\) 7.06065e7 + 7.06065e7i 0.0987608 + 0.0987608i 0.754761 − 0.656000i \(-0.227753\pi\)
−0.656000 + 0.754761i \(0.727753\pi\)
\(60\) 3.48578e8 + 3.90171e8i 0.448274 + 0.501763i
\(61\) 9.37334e8 + 9.37334e8i 1.10980 + 1.10980i 0.993176 + 0.116625i \(0.0372076\pi\)
0.116625 + 0.993176i \(0.462792\pi\)
\(62\) 4.41996e8 + 2.17379e8i 0.482458 + 0.237279i
\(63\) 2.92051e8 + 2.92051e8i 0.294277 + 0.294277i
\(64\) −9.89343e8 + 4.17280e8i −0.921397 + 0.388622i
\(65\) 1.20224e9 − 8.73287e7i 1.03616 − 0.0752645i
\(66\) 3.06384e8 + 8.99408e8i 0.244651 + 0.718186i
\(67\) −3.00127e8 −0.222296 −0.111148 − 0.993804i \(-0.535453\pi\)
−0.111148 + 0.993804i \(0.535453\pi\)
\(68\) 1.53534e9 + 1.98782e8i 1.05599 + 0.136720i
\(69\) −6.42408e7 − 6.42408e7i −0.0410738 − 0.0410738i
\(70\) −1.17675e9 + 4.98677e8i −0.700153 + 0.296708i
\(71\) 1.89991e9i 1.05303i 0.850166 + 0.526515i \(0.176502\pi\)
−0.850166 + 0.526515i \(0.823498\pi\)
\(72\) −2.09932e8 − 1.03793e9i −0.108497 − 0.536422i
\(73\) −6.95678e8 + 6.95678e8i −0.335578 + 0.335578i −0.854700 − 0.519122i \(-0.826259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(74\) −1.23649e9 + 2.51415e9i −0.557227 + 1.13301i
\(75\) −2.30743e8 − 1.57992e9i −0.0972350 − 0.665779i
\(76\) 1.35824e9 + 1.75852e8i 0.535684 + 0.0693552i
\(77\) −2.32101e9 −0.857477
\(78\) 1.81098e9 + 8.90663e8i 0.627250 + 0.308489i
\(79\) 3.07865e9i 1.00052i 0.865876 + 0.500259i \(0.166762\pi\)
−0.865876 + 0.500259i \(0.833238\pi\)
\(80\) 3.22089e9 + 6.02748e8i 0.982937 + 0.183944i
\(81\) −5.34161e8 −0.153196
\(82\) 2.08641e9 − 4.24229e9i 0.562769 − 1.14428i
\(83\) 5.16023e9i 1.31002i 0.755619 + 0.655011i \(0.227336\pi\)
−0.755619 + 0.655011i \(0.772664\pi\)
\(84\) −2.12206e9 − 2.74744e8i −0.507412 − 0.0656948i
\(85\) −3.57405e9 − 3.08999e9i −0.805502 − 0.696406i
\(86\) 3.58839e9 + 1.76482e9i 0.762795 + 0.375152i
\(87\) 3.27281e9 + 3.27281e9i 0.656635 + 0.656635i
\(88\) 4.95856e9 + 3.29017e9i 0.939597 + 0.623454i
\(89\) 7.76406e9 1.39040 0.695198 − 0.718818i \(-0.255317\pi\)
0.695198 + 0.718818i \(0.255317\pi\)
\(90\) −1.21238e9 + 2.99562e9i −0.205317 + 0.507311i
\(91\) −3.48592e9 + 3.48592e9i −0.558612 + 0.558612i
\(92\) −5.64283e8 − 7.30579e7i −0.0856165 − 0.0110848i
\(93\) − 2.51667e9i − 0.361753i
\(94\) 9.15935e9 − 3.12014e9i 1.24803 − 0.425143i
\(95\) −3.16178e9 − 2.73356e9i −0.408615 − 0.353273i
\(96\) 4.14406e9 + 3.59510e9i 0.508241 + 0.440915i
\(97\) −5.17776e9 + 5.17776e9i −0.602953 + 0.602953i −0.941095 − 0.338142i \(-0.890202\pi\)
0.338142 + 0.941095i \(0.390202\pi\)
\(98\) −1.68246e9 + 3.42094e9i −0.186129 + 0.378456i
\(99\) −4.14991e9 + 4.14991e9i −0.436378 + 0.436378i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 80.11.i.a.13.18 ✓ 236
5.2 odd 4 80.11.t.a.77.76 yes 236
16.5 even 4 80.11.t.a.53.76 yes 236
80.37 odd 4 inner 80.11.i.a.37.18 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.18 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.18 yes 236 80.37 odd 4 inner
80.11.t.a.53.76 yes 236 16.5 even 4
80.11.t.a.77.76 yes 236 5.2 odd 4