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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 37.10
Character \(\chi\) \(=\) 80.37
Dual form 80.11.i.a.13.10

$q$-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+(-31.1219 - 7.44510i) q^{2} -73.3622i q^{3} +(913.141 + 463.411i) q^{4} +(3123.99 + 79.4807i) q^{5} +(-546.189 + 2283.17i) q^{6} +(19105.3 - 19105.3i) q^{7} +(-24968.5 - 21220.6i) q^{8} +53667.0 q^{9} +(-96632.6 - 25732.0i) q^{10} +(-101838. - 101838. i) q^{11} +(33996.8 - 66990.0i) q^{12} -459634. i q^{13} +(-736832. + 452351. i) q^{14} +(5830.88 - 229183. i) q^{15} +(619077. + 846319. i) q^{16} +(-1.57208e6 - 1.57208e6i) q^{17} +(-1.67022e6 - 399556. i) q^{18} +(583978. + 583978. i) q^{19} +(2.81581e6 + 1.52027e6i) q^{20} +(-1.40160e6 - 1.40160e6i) q^{21} +(2.41119e6 + 3.92757e6i) q^{22} +(-4.60350e6 - 4.60350e6i) q^{23} +(-1.55679e6 + 1.83174e6i) q^{24} +(9.75299e6 + 496594. i) q^{25} +(-3.42202e6 + 1.43047e7i) q^{26} -8.26909e6i q^{27} +(2.62994e7 - 8.59221e6i) q^{28} +(-4.75144e6 - 4.75144e6i) q^{29} +(-1.88776e6 + 7.08918e6i) q^{30} +3.46404e7 q^{31} +(-1.29659e7 - 3.09481e7i) q^{32} +(-7.47103e6 + 7.47103e6i) q^{33} +(3.72217e7 + 6.06303e7i) q^{34} +(6.12031e7 - 5.81661e7i) q^{35} +(4.90055e7 + 2.48699e7i) q^{36} +1.05736e8i q^{37} +(-1.38267e7 - 2.25223e7i) q^{38} -3.37198e7 q^{39} +(-7.63147e7 - 6.82775e7i) q^{40} +1.27310e8i q^{41} +(3.31854e7 + 5.40556e7i) q^{42} -1.79676e8 q^{43} +(-4.57995e7 - 1.40185e8i) q^{44} +(1.67655e8 + 4.26549e6i) q^{45} +(1.08996e8 + 1.77543e8i) q^{46} +(2.82333e7 + 2.82333e7i) q^{47} +(6.20878e7 - 4.54168e7i) q^{48} -4.47547e8i q^{49} +(-2.99834e8 - 8.80669e7i) q^{50} +(-1.15331e8 + 1.15331e8i) q^{51} +(2.12999e8 - 4.19711e8i) q^{52} +4.31755e7 q^{53} +(-6.15642e7 + 2.57350e8i) q^{54} +(-3.10046e8 - 3.26234e8i) q^{55} +(-8.82456e8 + 7.16041e7i) q^{56} +(4.28419e7 - 4.28419e7i) q^{57} +(1.12499e8 + 1.83249e8i) q^{58} +(2.15274e8 - 2.15274e8i) q^{59} +(1.11530e8 - 2.06574e8i) q^{60} +(-8.91616e8 + 8.91616e8i) q^{61} +(-1.07807e9 - 2.57901e8i) q^{62} +(1.02532e9 - 1.02532e9i) q^{63} +(1.73111e8 + 1.05970e9i) q^{64} +(3.65320e7 - 1.43589e9i) q^{65} +(2.88135e8 - 1.76890e8i) q^{66} -5.32104e8 q^{67} +(-7.07011e8 - 2.16405e9i) q^{68} +(-3.37723e8 + 3.37723e8i) q^{69} +(-2.33781e9 + 1.35457e9i) q^{70} +4.99197e8i q^{71} +(-1.33998e9 - 1.13885e9i) q^{72} +(-8.77316e8 - 8.77316e8i) q^{73} +(7.87212e8 - 3.29069e9i) q^{74} +(3.64312e7 - 7.15501e8i) q^{75} +(2.62633e8 + 8.03876e8i) q^{76} -3.89127e9 q^{77} +(1.04942e9 + 2.51047e8i) q^{78} -1.04248e9i q^{79} +(1.86672e9 + 2.69310e9i) q^{80} +2.56234e9 q^{81} +(9.47834e8 - 3.96212e9i) q^{82} -1.28366e9i q^{83} +(-6.30344e8 - 1.92938e9i) q^{84} +(-4.78621e9 - 5.03611e9i) q^{85} +(5.59184e9 + 1.33770e9i) q^{86} +(-3.48576e8 + 3.48576e8i) q^{87} +(3.81675e8 + 4.70379e9i) q^{88} -1.06839e10 q^{89} +(-5.18598e9 - 1.38096e9i) q^{90} +(-8.78143e9 - 8.78143e9i) q^{91} +(-2.07033e9 - 6.33695e9i) q^{92} -2.54129e9i q^{93} +(-6.68473e8 - 1.08887e9i) q^{94} +(1.77793e9 + 1.87076e9i) q^{95} +(-2.27042e9 + 9.51207e8i) q^{96} +(4.24598e9 + 4.24598e9i) q^{97} +(-3.33203e9 + 1.39285e10i) q^{98} +(-5.46532e9 - 5.46532e9i) 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 \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.1219 − 7.44510i −0.972558 − 0.232659i
\(3\) − 73.3622i − 0.301902i −0.988541 − 0.150951i \(-0.951766\pi\)
0.988541 − 0.150951i \(-0.0482336\pi\)
\(4\) 913.141 + 463.411i 0.891739 + 0.452550i
\(5\) 3123.99 + 79.4807i 0.999677 + 0.0254338i
\(6\) −546.189 + 2283.17i −0.0702403 + 0.293617i
\(7\) 19105.3 − 19105.3i 1.13674 − 1.13674i 0.147714 − 0.989030i \(-0.452808\pi\)
0.989030 − 0.147714i \(-0.0471916\pi\)
\(8\) −24968.5 − 21220.6i −0.761978 − 0.647602i
\(9\) 53667.0 0.908855
\(10\) −96632.6 − 25732.0i −0.966326 − 0.257320i
\(11\) −101838. − 101838.i −0.632332 − 0.632332i 0.316321 − 0.948652i \(-0.397552\pi\)
−0.948652 + 0.316321i \(0.897552\pi\)
\(12\) 33996.8 − 66990.0i 0.136626 − 0.269218i
\(13\) − 459634.i − 1.23793i −0.785419 − 0.618964i \(-0.787553\pi\)
0.785419 − 0.618964i \(-0.212447\pi\)
\(14\) −736832. + 452351.i −1.37002 + 0.841076i
\(15\) 5830.88 − 229183.i 0.00767852 − 0.301804i
\(16\) 619077. + 846319.i 0.590398 + 0.807112i
\(17\) −1.57208e6 − 1.57208e6i −1.10721 − 1.10721i −0.993516 − 0.113693i \(-0.963732\pi\)
−0.113693 − 0.993516i \(-0.536268\pi\)
\(18\) −1.67022e6 − 399556.i −0.883915 − 0.211454i
\(19\) 583978. + 583978.i 0.235846 + 0.235846i 0.815128 − 0.579282i \(-0.196667\pi\)
−0.579282 + 0.815128i \(0.696667\pi\)
\(20\) 2.81581e6 + 1.52027e6i 0.879941 + 0.475084i
\(21\) −1.40160e6 − 1.40160e6i −0.343185 − 0.343185i
\(22\) 2.41119e6 + 3.92757e6i 0.467862 + 0.762097i
\(23\) −4.60350e6 − 4.60350e6i −0.715235 − 0.715235i 0.252391 − 0.967625i \(-0.418783\pi\)
−0.967625 + 0.252391i \(0.918783\pi\)
\(24\) −1.55679e6 + 1.83174e6i −0.195512 + 0.230043i
\(25\) 9.75299e6 + 496594.i 0.998706 + 0.0508512i
\(26\) −3.42202e6 + 1.43047e7i −0.288016 + 1.20396i
\(27\) − 8.26909e6i − 0.576287i
\(28\) 2.62994e7 − 8.59221e6i 1.52811 − 0.499246i
\(29\) −4.75144e6 − 4.75144e6i −0.231652 − 0.231652i 0.581730 − 0.813382i \(-0.302376\pi\)
−0.813382 + 0.581730i \(0.802376\pi\)
\(30\) −1.88776e6 + 7.08918e6i −0.0776854 + 0.291736i
\(31\) 3.46404e7 1.20997 0.604984 − 0.796237i \(-0.293179\pi\)
0.604984 + 0.796237i \(0.293179\pi\)
\(32\) −1.29659e7 − 3.09481e7i −0.386414 − 0.922325i
\(33\) −7.47103e6 + 7.47103e6i −0.190902 + 0.190902i
\(34\) 3.72217e7 + 6.06303e7i 0.819223 + 1.33443i
\(35\) 6.12031e7 − 5.81661e7i 1.16529 − 1.10746i
\(36\) 4.90055e7 + 2.48699e7i 0.810462 + 0.411302i
\(37\) 1.05736e8i 1.52480i 0.647107 + 0.762399i \(0.275979\pi\)
−0.647107 + 0.762399i \(0.724021\pi\)
\(38\) −1.38267e7 − 2.25223e7i −0.174502 − 0.284246i
\(39\) −3.37198e7 −0.373733
\(40\) −7.63147e7 − 6.82775e7i −0.745261 − 0.666773i
\(41\) 1.27310e8i 1.09886i 0.835540 + 0.549430i \(0.185155\pi\)
−0.835540 + 0.549430i \(0.814845\pi\)
\(42\) 3.31854e7 + 5.40556e7i 0.253922 + 0.413613i
\(43\) −1.79676e8 −1.22221 −0.611106 − 0.791549i \(-0.709275\pi\)
−0.611106 + 0.791549i \(0.709275\pi\)
\(44\) −4.57995e7 − 1.40185e8i −0.277714 − 0.850037i
\(45\) 1.67655e8 + 4.26549e6i 0.908561 + 0.0231157i
\(46\) 1.08996e8 + 1.77543e8i 0.529201 + 0.862013i
\(47\) 2.82333e7 + 2.82333e7i 0.123104 + 0.123104i 0.765975 − 0.642871i \(-0.222257\pi\)
−0.642871 + 0.765975i \(0.722257\pi\)
\(48\) 6.20878e7 − 4.54168e7i 0.243669 − 0.178242i
\(49\) − 4.47547e8i − 1.58437i
\(50\) −2.99834e8 − 8.80669e7i −0.959469 − 0.281814i
\(51\) −1.15331e8 + 1.15331e8i −0.334269 + 0.334269i
\(52\) 2.12999e8 − 4.19711e8i 0.560224 − 1.10391i
\(53\) 4.31755e7 0.103242 0.0516211 − 0.998667i \(-0.483561\pi\)
0.0516211 + 0.998667i \(0.483561\pi\)
\(54\) −6.15642e7 + 2.57350e8i −0.134079 + 0.560473i
\(55\) −3.10046e8 − 3.26234e8i −0.616045 − 0.648210i
\(56\) −8.82456e8 + 7.16041e7i −1.60233 + 0.130016i
\(57\) 4.28419e7 − 4.28419e7i 0.0712024 − 0.0712024i
\(58\) 1.12499e8 + 1.83249e8i 0.171399 + 0.279191i
\(59\) 2.15274e8 − 2.15274e8i 0.301115 − 0.301115i −0.540335 − 0.841450i \(-0.681703\pi\)
0.841450 + 0.540335i \(0.181703\pi\)
\(60\) 1.11530e8 − 2.06574e8i 0.143429 − 0.265656i
\(61\) −8.91616e8 + 8.91616e8i −1.05567 + 1.05567i −0.0573151 + 0.998356i \(0.518254\pi\)
−0.998356 + 0.0573151i \(0.981746\pi\)
\(62\) −1.07807e9 − 2.57901e8i −1.17676 − 0.281511i
\(63\) 1.02532e9 − 1.02532e9i 1.03314 − 1.03314i
\(64\) 1.73111e8 + 1.05970e9i 0.161222 + 0.986918i
\(65\) 3.65320e7 − 1.43589e9i 0.0314853 − 1.23753i
\(66\) 2.88135e8 − 1.76890e8i 0.230079 − 0.141248i
\(67\) −5.32104e8 −0.394115 −0.197057 − 0.980392i \(-0.563139\pi\)
−0.197057 + 0.980392i \(0.563139\pi\)
\(68\) −7.07011e8 − 2.16405e9i −0.486275 − 1.48841i
\(69\) −3.37723e8 + 3.37723e8i −0.215931 + 0.215931i
\(70\) −2.33781e9 + 1.35457e9i −1.39097 + 0.805959i
\(71\) 4.99197e8i 0.276681i 0.990385 + 0.138341i \(0.0441769\pi\)
−0.990385 + 0.138341i \(0.955823\pi\)
\(72\) −1.33998e9 − 1.13885e9i −0.692528 − 0.588577i
\(73\) −8.77316e8 − 8.77316e8i −0.423196 − 0.423196i 0.463107 − 0.886303i \(-0.346735\pi\)
−0.886303 + 0.463107i \(0.846735\pi\)
\(74\) 7.87212e8 − 3.29069e9i 0.354759 − 1.48296i
\(75\) 3.64312e7 − 7.15501e8i 0.0153521 − 0.301511i
\(76\) 2.62633e8 + 8.03876e8i 0.103581 + 0.317045i
\(77\) −3.89127e9 −1.43760
\(78\) 1.04942e9 + 2.51047e8i 0.363477 + 0.0869525i
\(79\) − 1.04248e9i − 0.338791i −0.985548 − 0.169396i \(-0.945818\pi\)
0.985548 − 0.169396i \(-0.0541816\pi\)
\(80\) 1.86672e9 + 2.69310e9i 0.569679 + 0.821867i
\(81\) 2.56234e9 0.734873
\(82\) 9.47834e8 − 3.96212e9i 0.255660 − 1.06871i
\(83\) − 1.28366e9i − 0.325882i −0.986636 − 0.162941i \(-0.947902\pi\)
0.986636 − 0.162941i \(-0.0520980\pi\)
\(84\) −6.30344e8 − 1.92938e9i −0.150723 − 0.461340i
\(85\) −4.78621e9 − 5.03611e9i −1.07869 − 1.13501i
\(86\) 5.59184e9 + 1.33770e9i 1.18867 + 0.284359i
\(87\) −3.48576e8 + 3.48576e8i −0.0699361 + 0.0699361i
\(88\) 3.81675e8 + 4.70379e9i 0.0723236 + 0.891323i
\(89\) −1.06839e10 −1.91329 −0.956647 − 0.291251i \(-0.905929\pi\)
−0.956647 + 0.291251i \(0.905929\pi\)
\(90\) −5.18598e9 − 1.38096e9i −0.878251 − 0.233867i
\(91\) −8.78143e9 − 8.78143e9i −1.40721 − 1.40721i
\(92\) −2.07033e9 − 6.33695e9i −0.314124 − 0.961482i
\(93\) − 2.54129e9i − 0.365292i
\(94\) −6.68473e8 − 1.08887e9i −0.0910845 − 0.148367i
\(95\) 1.77793e9 + 1.87076e9i 0.229771 + 0.241768i
\(96\) −2.27042e9 + 9.51207e8i −0.278452 + 0.116659i
\(97\) 4.24598e9 + 4.24598e9i 0.494447 + 0.494447i 0.909704 − 0.415257i \(-0.136308\pi\)
−0.415257 + 0.909704i \(0.636308\pi\)
\(98\) −3.33203e9 + 1.39285e10i −0.368620 + 1.54090i
\(99\) −5.46532e9 − 5.46532e9i −0.574698 − 0.574698i
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.37.10 yes 236
5.3 odd 4 80.11.t.a.53.51 yes 236
16.13 even 4 80.11.t.a.77.51 yes 236
80.13 odd 4 inner 80.11.i.a.13.10 ✓ 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.10 ✓ 236 80.13 odd 4 inner
80.11.i.a.37.10 yes 236 1.1 even 1 trivial
80.11.t.a.53.51 yes 236 5.3 odd 4
80.11.t.a.77.51 yes 236 16.13 even 4