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Properties

Label 80.11.i.a.13.20
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 13.20
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.20

$q$-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+(-27.8828 + 15.7020i) q^{2} +72.8954i q^{3} +(530.896 - 875.629i) q^{4} +(1447.28 + 2769.66i) q^{5} +(-1144.60 - 2032.52i) q^{6} +(-11191.3 - 11191.3i) q^{7} +(-1053.74 + 32751.1i) q^{8} +53735.3 q^{9} +(-83843.3 - 54500.5i) q^{10} +(2660.76 - 2660.76i) q^{11} +(63829.3 + 38699.9i) q^{12} +14372.9i q^{13} +(487772. + 136319. i) q^{14} +(-201895. + 105500. i) q^{15} +(-484875. - 929735. i) q^{16} +(-1.01075e6 + 1.01075e6i) q^{17} +(-1.49829e6 + 843750. i) q^{18} +(-1.29170e6 + 1.29170e6i) q^{19} +(3.19355e6 + 203118. i) q^{20} +(815797. - 815797. i) q^{21} +(-32410.1 + 115968. i) q^{22} +(7.66600e6 - 7.66600e6i) q^{23} +(-2.38740e6 - 76813.0i) q^{24} +(-5.57637e6 + 8.01695e6i) q^{25} +(-225683. - 400757. i) q^{26} +8.22146e6i q^{27} +(-1.57409e7 + 3.85802e6i) q^{28} +(1.58713e7 - 1.58713e7i) q^{29} +(3.97283e6 - 6.11179e6i) q^{30} +4.23301e7 q^{31} +(2.81183e7 + 1.83101e7i) q^{32} +(193957. + 193957. i) q^{33} +(1.23118e7 - 4.40535e7i) q^{34} +(1.47991e7 - 4.71932e7i) q^{35} +(2.85278e7 - 4.70521e7i) q^{36} +9.12151e7i q^{37} +(1.57340e7 - 5.62986e7i) q^{38} -1.04772e6 q^{39} +(-9.22342e7 + 4.44815e7i) q^{40} +8.24054e7i q^{41} +(-9.93705e6 + 3.55563e7i) q^{42} +2.17317e8 q^{43} +(-917250. - 3.74242e6i) q^{44} +(7.77701e7 + 1.48828e8i) q^{45} +(-9.33778e7 + 3.34120e8i) q^{46} +(-6.68783e7 + 6.68783e7i) q^{47} +(6.77734e7 - 3.53452e7i) q^{48} -3.19830e7i q^{49} +(2.96027e7 - 3.11095e8i) q^{50} +(-7.36794e7 - 7.36794e7i) q^{51} +(1.25853e7 + 7.63053e6i) q^{52} -7.62941e8 q^{53} +(-1.29093e8 - 2.29237e8i) q^{54} +(1.12203e7 + 3.51852e6i) q^{55} +(3.78321e8 - 3.54735e8i) q^{56} +(-9.41593e7 - 9.41593e7i) q^{57} +(-1.93325e8 + 6.91747e8i) q^{58} +(-7.63423e8 - 7.63423e8i) q^{59} +(-1.48064e7 + 2.32795e8i) q^{60} +(3.35200e8 + 3.35200e8i) q^{61} +(-1.18028e9 + 6.64666e8i) q^{62} +(-6.01370e8 - 6.01370e8i) q^{63} +(-1.07152e9 - 6.90224e7i) q^{64} +(-3.98081e7 + 2.08017e7i) q^{65} +(-8.45357e6 - 2.36255e6i) q^{66} -1.46916e8 q^{67} +(3.48440e8 + 1.42165e9i) q^{68} +(5.58816e8 + 5.58816e8i) q^{69} +(3.28386e8 + 1.54825e9i) q^{70} +2.46193e9i q^{71} +(-5.66231e7 + 1.75989e9i) q^{72} +(-2.38177e9 + 2.38177e9i) q^{73} +(-1.43226e9 - 2.54333e9i) q^{74} +(-5.84399e8 - 4.06492e8i) q^{75} +(4.45293e8 + 1.81681e9i) q^{76} -5.95549e7 q^{77} +(2.92133e7 - 1.64513e7i) q^{78} +5.30087e9i q^{79} +(1.87330e9 - 2.68853e9i) q^{80} +2.57371e9 q^{81} +(-1.29393e9 - 2.29769e9i) q^{82} +2.42231e9i q^{83} +(-2.81232e8 - 1.14744e9i) q^{84} +(-4.26229e9 - 1.33660e9i) q^{85} +(-6.05940e9 + 3.41231e9i) q^{86} +(1.15695e9 + 1.15695e9i) q^{87} +(8.43389e7 + 8.99464e7i) q^{88} -5.73832e8 q^{89} +(-4.50534e9 - 2.92860e9i) q^{90} +(1.60852e8 - 1.60852e8i) q^{91} +(-2.64272e9 - 1.07824e10i) q^{92} +3.08567e9i q^{93} +(8.14630e8 - 2.91487e9i) q^{94} +(-5.44704e9 - 1.70812e9i) q^{95} +(-1.33472e9 + 2.04970e9i) q^{96} +(-7.24215e9 + 7.24215e9i) q^{97} +(5.02197e8 + 8.91775e8i) q^{98} +(1.42977e8 - 1.42977e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −27.8828 + 15.7020i −0.871336 + 0.490687i
\(3\) 72.8954i 0.299981i 0.988687 + 0.149991i \(0.0479243\pi\)
−0.988687 + 0.149991i \(0.952076\pi\)
\(4\) 530.896 − 875.629i 0.518453 − 0.855106i
\(5\) 1447.28 + 2769.66i 0.463130 + 0.886290i
\(6\) −1144.60 − 2032.52i −0.147197 − 0.261384i
\(7\) −11191.3 − 11191.3i −0.665874 − 0.665874i 0.290884 − 0.956758i \(-0.406050\pi\)
−0.956758 + 0.290884i \(0.906050\pi\)
\(8\) −1053.74 + 32751.1i −0.0321577 + 0.999483i
\(9\) 53735.3 0.910011
\(10\) −83843.3 − 54500.5i −0.838433 − 0.545005i
\(11\) 2660.76 − 2660.76i 0.0165212 − 0.0165212i −0.698798 − 0.715319i \(-0.746281\pi\)
0.715319 + 0.698798i \(0.246281\pi\)
\(12\) 63829.3 + 38699.9i 0.256516 + 0.155526i
\(13\) 14372.9i 0.0387105i 0.999813 + 0.0193552i \(0.00616135\pi\)
−0.999813 + 0.0193552i \(0.993839\pi\)
\(14\) 487772. + 136319.i 0.906935 + 0.253464i
\(15\) −201895. + 105500.i −0.265870 + 0.138930i
\(16\) −484875. − 929735.i −0.462413 − 0.886665i
\(17\) −1.01075e6 + 1.01075e6i −0.711871 + 0.711871i −0.966926 − 0.255056i \(-0.917906\pi\)
0.255056 + 0.966926i \(0.417906\pi\)
\(18\) −1.49829e6 + 843750.i −0.792926 + 0.446530i
\(19\) −1.29170e6 + 1.29170e6i −0.521669 + 0.521669i −0.918075 − 0.396406i \(-0.870257\pi\)
0.396406 + 0.918075i \(0.370257\pi\)
\(20\) 3.19355e6 + 203118.i 0.997983 + 0.0634743i
\(21\) 815797. − 815797.i 0.199750 − 0.199750i
\(22\) −32410.1 + 115968.i −0.00628879 + 0.0225023i
\(23\) 7.66600e6 − 7.66600e6i 1.19105 − 1.19105i 0.214275 − 0.976773i \(-0.431261\pi\)
0.976773 − 0.214275i \(-0.0687390\pi\)
\(24\) −2.38740e6 − 76813.0i −0.299826 − 0.00964669i
\(25\) −5.57637e6 + 8.01695e6i −0.571021 + 0.820936i
\(26\) −225683. − 400757.i −0.0189947 − 0.0337298i
\(27\) 8.22146e6i 0.572967i
\(28\) −1.57409e7 + 3.85802e6i −0.914617 + 0.224168i
\(29\) 1.58713e7 − 1.58713e7i 0.773790 − 0.773790i −0.204977 − 0.978767i \(-0.565712\pi\)
0.978767 + 0.204977i \(0.0657120\pi\)
\(30\) 3.97283e6 − 6.11179e6i 0.163491 − 0.251514i
\(31\) 4.23301e7 1.47857 0.739283 − 0.673395i \(-0.235165\pi\)
0.739283 + 0.673395i \(0.235165\pi\)
\(32\) 2.81183e7 + 1.83101e7i 0.837992 + 0.545683i
\(33\) 193957. + 193957.i 0.00495605 + 0.00495605i
\(34\) 1.23118e7 − 4.40535e7i 0.270973 − 0.969584i
\(35\) 1.47991e7 − 4.71932e7i 0.281771 − 0.898544i
\(36\) 2.85278e7 − 4.70521e7i 0.471798 − 0.778156i
\(37\) 9.12151e7i 1.31540i 0.753280 + 0.657700i \(0.228471\pi\)
−0.753280 + 0.657700i \(0.771529\pi\)
\(38\) 1.57340e7 − 5.62986e7i 0.198573 − 0.710525i
\(39\) −1.04772e6 −0.0116124
\(40\) −9.22342e7 + 4.44815e7i −0.900725 + 0.434390i
\(41\) 8.24054e7i 0.711273i 0.934624 + 0.355636i \(0.115736\pi\)
−0.934624 + 0.355636i \(0.884264\pi\)
\(42\) −9.93705e6 + 3.55563e7i −0.0760345 + 0.272063i
\(43\) 2.17317e8 1.47826 0.739132 − 0.673561i \(-0.235236\pi\)
0.739132 + 0.673561i \(0.235236\pi\)
\(44\) −917250. − 3.74242e6i −0.00556192 − 0.0226929i
\(45\) 7.77701e7 + 1.48828e8i 0.421454 + 0.806534i
\(46\) −9.33778e7 + 3.34120e8i −0.453372 + 1.62224i
\(47\) −6.68783e7 + 6.68783e7i −0.291606 + 0.291606i −0.837714 − 0.546109i \(-0.816109\pi\)
0.546109 + 0.837714i \(0.316109\pi\)
\(48\) 6.77734e7 − 3.53452e7i 0.265983 − 0.138715i
\(49\) − 3.19830e7i − 0.113224i
\(50\) 2.96027e7 − 3.11095e8i 0.0947285 − 0.995503i
\(51\) −7.36794e7 − 7.36794e7i −0.213548 − 0.213548i
\(52\) 1.25853e7 + 7.63053e6i 0.0331016 + 0.0200696i
\(53\) −7.62941e8 −1.82436 −0.912182 − 0.409785i \(-0.865604\pi\)
−0.912182 + 0.409785i \(0.865604\pi\)
\(54\) −1.29093e8 − 2.29237e8i −0.281147 − 0.499247i
\(55\) 1.12203e7 + 3.51852e6i 0.0222941 + 0.00699111i
\(56\) 3.78321e8 − 3.54735e8i 0.686942 − 0.644116i
\(57\) −9.41593e7 − 9.41593e7i −0.156491 − 0.156491i
\(58\) −1.93325e8 + 6.91747e8i −0.294543 + 1.05392i
\(59\) −7.63423e8 − 7.63423e8i −1.06784 − 1.06784i −0.997525 − 0.0703132i \(-0.977600\pi\)
−0.0703132 − 0.997525i \(-0.522400\pi\)
\(60\) −1.48064e7 + 2.32795e8i −0.0190411 + 0.299376i
\(61\) 3.35200e8 + 3.35200e8i 0.396875 + 0.396875i 0.877129 − 0.480254i \(-0.159456\pi\)
−0.480254 + 0.877129i \(0.659456\pi\)
\(62\) −1.18028e9 + 6.64666e8i −1.28833 + 0.725512i
\(63\) −6.01370e8 − 6.01370e8i −0.605953 − 0.605953i
\(64\) −1.07152e9 − 6.90224e7i −0.997932 − 0.0642821i
\(65\) −3.98081e7 + 2.08017e7i −0.0343087 + 0.0179280i
\(66\) −8.45357e6 − 2.36255e6i −0.00675026 − 0.00188652i
\(67\) −1.46916e8 −0.108816 −0.0544082 − 0.998519i \(-0.517327\pi\)
−0.0544082 + 0.998519i \(0.517327\pi\)
\(68\) 3.48440e8 + 1.42165e9i 0.239653 + 0.977797i
\(69\) 5.58816e8 + 5.58816e8i 0.357292 + 0.357292i
\(70\) 3.28386e8 + 1.54825e9i 0.195386 + 0.921195i
\(71\) 2.46193e9i 1.36453i 0.731103 + 0.682267i \(0.239006\pi\)
−0.731103 + 0.682267i \(0.760994\pi\)
\(72\) −5.66231e7 + 1.75989e9i −0.0292638 + 0.909541i
\(73\) −2.38177e9 + 2.38177e9i −1.14891 + 1.14891i −0.162140 + 0.986768i \(0.551840\pi\)
−0.986768 + 0.162140i \(0.948160\pi\)
\(74\) −1.43226e9 − 2.54333e9i −0.645449 − 1.14616i
\(75\) −5.84399e8 − 4.06492e8i −0.246265 − 0.171295i
\(76\) 4.45293e8 + 1.81681e9i 0.175621 + 0.716543i
\(77\) −5.95549e7 −0.0220021
\(78\) 2.92133e7 − 1.64513e7i 0.0101183 − 0.00569805i
\(79\) 5.30087e9i 1.72271i 0.508006 + 0.861353i \(0.330383\pi\)
−0.508006 + 0.861353i \(0.669617\pi\)
\(80\) 1.87330e9 − 2.68853e9i 0.571685 − 0.820473i
\(81\) 2.57371e9 0.738132
\(82\) −1.29393e9 − 2.29769e9i −0.349012 − 0.619758i
\(83\) 2.42231e9i 0.614948i 0.951556 + 0.307474i \(0.0994837\pi\)
−0.951556 + 0.307474i \(0.900516\pi\)
\(84\) −2.81232e8 − 1.14744e9i −0.0672463 − 0.274368i
\(85\) −4.26229e9 − 1.33660e9i −0.960613 − 0.301235i
\(86\) −6.05940e9 + 3.41231e9i −1.28806 + 0.725364i
\(87\) 1.15695e9 + 1.15695e9i 0.232122 + 0.232122i
\(88\) 8.43389e7 + 8.99464e7i 0.0159814 + 0.0170440i
\(89\) −5.73832e8 −0.102763 −0.0513813 − 0.998679i \(-0.516362\pi\)
−0.0513813 + 0.998679i \(0.516362\pi\)
\(90\) −4.50534e9 − 2.92860e9i −0.762984 − 0.495960i
\(91\) 1.60852e8 − 1.60852e8i 0.0257763 − 0.0257763i
\(92\) −2.64272e9 − 1.07824e10i −0.400970 − 1.63598i
\(93\) 3.08567e9i 0.443542i
\(94\) 8.14630e8 − 2.91487e9i 0.111000 − 0.397174i
\(95\) −5.44704e9 − 1.70812e9i −0.703951 − 0.220749i
\(96\) −1.33472e9 + 2.04970e9i −0.163695 + 0.251382i
\(97\) −7.24215e9 + 7.24215e9i −0.843351 + 0.843351i −0.989293 − 0.145942i \(-0.953379\pi\)
0.145942 + 0.989293i \(0.453379\pi\)
\(98\) 5.02197e8 + 8.91775e8i 0.0555576 + 0.0986563i
\(99\) 1.42977e8 − 1.42977e8i 0.0150345 − 0.0150345i
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.13.20 ✓ 236
5.2 odd 4 80.11.t.a.77.40 yes 236
16.5 even 4 80.11.t.a.53.40 yes 236
80.37 odd 4 inner 80.11.i.a.37.20 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.20 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.20 yes 236 80.37 odd 4 inner
80.11.t.a.53.40 yes 236 16.5 even 4
80.11.t.a.77.40 yes 236 5.2 odd 4