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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: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 77.51
Character \(\chi\) \(=\) 80.77
Dual form 80.11.t.a.53.51

$q$-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+(-7.44510 - 31.1219i) q^{2} +73.3622 q^{3} +(-913.141 + 463.411i) q^{4} +(79.4807 - 3123.99i) q^{5} +(-546.189 - 2283.17i) q^{6} +(-19105.3 + 19105.3i) q^{7} +(21220.6 + 24968.5i) q^{8} -53667.0 q^{9} +(-97816.1 + 20784.8i) q^{10} +(-101838. + 101838. i) q^{11} +(-66990.0 + 33996.8i) q^{12} +459634. q^{13} +(736832. + 452351. i) q^{14} +(5830.88 - 229183. i) q^{15} +(619077. - 846319. i) q^{16} +(-1.57208e6 - 1.57208e6i) q^{17} +(399556. + 1.67022e6i) q^{18} +(-583978. + 583978. i) q^{19} +(1.37511e6 + 2.88947e6i) q^{20} +(-1.40160e6 + 1.40160e6i) q^{21} +(3.92757e6 + 2.41119e6i) 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} -8.26909e6 q^{27} +(8.59221e6 - 2.62994e7i) q^{28} +(4.75144e6 - 4.75144e6i) q^{29} +(-7.17600e6 + 1.52482e6i) q^{30} +3.46404e7 q^{31} +(-3.09481e7 - 1.29659e7i) q^{32} +(-7.47103e6 + 7.47103e6i) q^{33} +(-3.72217e7 + 6.06303e7i) q^{34} +(5.81661e7 + 6.12031e7i) q^{35} +(4.90055e7 - 2.48699e7i) q^{36} +1.05736e8 q^{37} +(2.25223e7 + 1.38267e7i) q^{38} +3.37198e7 q^{39} +(7.96880e7 - 6.43085e7i) q^{40} -1.27310e8i q^{41} +(5.40556e7 + 3.31854e7i) q^{42} +1.79676e8i q^{43} +(4.57995e7 - 1.40185e8i) q^{44} +(-4.26549e6 + 1.67655e8i) q^{45} +(1.08996e8 - 1.77543e8i) q^{46} +(2.82333e7 + 2.82333e7i) q^{47} +(4.54168e7 - 6.20878e7i) q^{48} -4.47547e8i q^{49} +(5.71571e7 + 3.07228e8i) q^{50} +(-1.15331e8 - 1.15331e8i) q^{51} +(-4.19711e8 + 2.12999e8i) q^{52} -4.31755e7i 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} +(-1.83249e8 - 1.12499e8i) q^{58} +(-2.15274e8 - 2.15274e8i) q^{59} +(1.00881e8 + 2.11978e8i) q^{60} +(-8.91616e8 - 8.91616e8i) q^{61} +(-2.57901e8 - 1.07807e9i) 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} -5.32104e8i q^{67} +(2.16405e9 + 7.07011e8i) q^{68} +(3.37723e8 + 3.37723e8i) q^{69} +(1.47170e9 - 2.26590e9i) q^{70} -4.99197e8i q^{71} +(-1.13885e9 - 1.33998e9i) q^{72} +(8.77316e8 + 8.77316e8i) q^{73} +(-7.87212e8 - 3.29069e9i) q^{74} +(-7.15501e8 - 3.64312e7i) q^{75} +(2.62633e8 - 8.03876e8i) q^{76} -3.89127e9i q^{77} +(-2.51047e8 - 1.04942e9i) q^{78} -1.04248e9i q^{79} +(-2.59469e9 - 2.00126e9i) q^{80} +2.56234e9 q^{81} +(-3.96212e9 + 9.47834e8i) q^{82} +1.28366e9 q^{83} +(6.30344e8 - 1.92938e9i) q^{84} +(-5.03611e9 + 4.78621e9i) q^{85} +(5.59184e9 - 1.33770e9i) q^{86} +(3.48576e8 - 3.48576e8i) q^{87} +(-4.70379e9 - 3.81675e8i) q^{88} +1.06839e10 q^{89} +(5.24950e9 - 1.11546e9i) q^{90} +(-8.78143e9 + 8.78143e9i) q^{91} +(-6.33695e9 - 2.07033e9i) q^{92} +2.54129e9 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} +(-1.39285e10 + 3.33203e9i) q^{98} +(5.46532e9 - 5.46532e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 236 q - 2 q^{2} - 4 q^{3} - 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} + 4487724 q^{9} - 2050 q^{10} - 4 q^{11} + 502036 q^{12} - 4 q^{13} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} - 5928762 q^{18} + 5107040 q^{19}+ \cdots + 28071956444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.44510 − 31.1219i −0.232659 − 0.972558i
\(3\) 73.3622 0.301902 0.150951 − 0.988541i \(-0.451766\pi\)
0.150951 + 0.988541i \(0.451766\pi\)
\(4\) −913.141 + 463.411i −0.891739 + 0.452550i
\(5\) 79.4807 − 3123.99i 0.0254338 − 0.999677i
\(6\) −546.189 − 2283.17i −0.0702403 − 0.293617i
\(7\) −19105.3 + 19105.3i −1.13674 + 1.13674i −0.147714 + 0.989030i \(0.547192\pi\)
−0.989030 + 0.147714i \(0.952808\pi\)
\(8\) 21220.6 + 24968.5i 0.647602 + 0.761978i
\(9\) −53667.0 −0.908855
\(10\) −97816.1 + 20784.8i −0.978161 + 0.207848i
\(11\) −101838. + 101838.i −0.632332 + 0.632332i −0.948652 − 0.316321i \(-0.897552\pi\)
0.316321 + 0.948652i \(0.397552\pi\)
\(12\) −66990.0 + 33996.8i −0.269218 + 0.136626i
\(13\) 459634. 1.23793 0.618964 − 0.785419i \(-0.287553\pi\)
0.618964 + 0.785419i \(0.287553\pi\)
\(14\) 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 \(-0.536268\pi\)
\(18\) 399556. + 1.67022e6i 0.211454 + 0.883915i
\(19\) −583978. + 583978.i −0.235846 + 0.235846i −0.815128 − 0.579282i \(-0.803333\pi\)
0.579282 + 0.815128i \(0.303333\pi\)
\(20\) 1.37511e6 + 2.88947e6i 0.429723 + 0.902961i
\(21\) −1.40160e6 + 1.40160e6i −0.343185 + 0.343185i
\(22\) 3.92757e6 + 2.41119e6i 0.762097 + 0.467862i
\(23\) 4.60350e6 + 4.60350e6i 0.715235 + 0.715235i 0.967625 − 0.252391i \(-0.0812168\pi\)
−0.252391 + 0.967625i \(0.581217\pi\)
\(24\) 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
\(27\) −8.26909e6 −0.576287
\(28\) 8.59221e6 − 2.62994e7i 0.499246 − 1.52811i
\(29\) 4.75144e6 − 4.75144e6i 0.231652 − 0.231652i −0.581730 − 0.813382i \(-0.697624\pi\)
0.813382 + 0.581730i \(0.197624\pi\)
\(30\) −7.17600e6 + 1.52482e6i −0.295309 + 0.0627498i
\(31\) 3.46404e7 1.20997 0.604984 − 0.796237i \(-0.293179\pi\)
0.604984 + 0.796237i \(0.293179\pi\)
\(32\) −3.09481e7 − 1.29659e7i −0.922325 − 0.386414i
\(33\) −7.47103e6 + 7.47103e6i −0.190902 + 0.190902i
\(34\) −3.72217e7 + 6.06303e7i −0.819223 + 1.33443i
\(35\) 5.81661e7 + 6.12031e7i 1.10746 + 1.16529i
\(36\) 4.90055e7 − 2.48699e7i 0.810462 − 0.411302i
\(37\) 1.05736e8 1.52480 0.762399 − 0.647107i \(-0.224021\pi\)
0.762399 + 0.647107i \(0.224021\pi\)
\(38\) 2.25223e7 + 1.38267e7i 0.284246 + 0.174502i
\(39\) 3.37198e7 0.373733
\(40\) 7.96880e7 − 6.43085e7i 0.778203 − 0.628013i
\(41\) − 1.27310e8i − 1.09886i −0.835540 − 0.549430i \(-0.814845\pi\)
0.835540 − 0.549430i \(-0.185155\pi\)
\(42\) 5.40556e7 + 3.31854e7i 0.413613 + 0.253922i
\(43\) 1.79676e8i 1.22221i 0.791549 + 0.611106i \(0.209275\pi\)
−0.791549 + 0.611106i \(0.790725\pi\)
\(44\) 4.57995e7 − 1.40185e8i 0.277714 − 0.850037i
\(45\) −4.26549e6 + 1.67655e8i −0.0231157 + 0.908561i
\(46\) 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 \(0.722257\pi\)
\(48\) 4.54168e7 − 6.20878e7i 0.178242 − 0.243669i
\(49\) − 4.47547e8i − 1.58437i
\(50\) 5.71571e7 + 3.07228e8i 0.182903 + 0.983131i
\(51\) −1.15331e8 − 1.15331e8i −0.334269 − 0.334269i
\(52\) −4.19711e8 + 2.12999e8i −1.10391 + 0.560224i
\(53\) − 4.31755e7i − 0.103242i −0.998667 − 0.0516211i \(-0.983561\pi\)
0.998667 − 0.0516211i \(-0.0164388\pi\)
\(54\) 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
\(58\) −1.83249e8 − 1.12499e8i −0.279191 − 0.171399i
\(59\) −2.15274e8 − 2.15274e8i −0.301115 − 0.301115i 0.540335 − 0.841450i \(-0.318297\pi\)
−0.841450 + 0.540335i \(0.818297\pi\)
\(60\) 1.00881e8 + 2.11978e8i 0.129734 + 0.272606i
\(61\) −8.91616e8 − 8.91616e8i −1.05567 − 1.05567i −0.998356 − 0.0573151i \(-0.981746\pi\)
−0.0573151 − 0.998356i \(-0.518254\pi\)
\(62\) −2.57901e8 − 1.07807e9i −0.281511 − 1.17676i
\(63\) 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
\(67\) − 5.32104e8i − 0.394115i −0.980392 − 0.197057i \(-0.936861\pi\)
0.980392 − 0.197057i \(-0.0631385\pi\)
\(68\) 2.16405e9 + 7.07011e8i 1.48841 + 0.486275i
\(69\) 3.37723e8 + 3.37723e8i 0.215931 + 0.215931i
\(70\) 1.47170e9 − 2.26590e9i 0.875649 − 1.34819i
\(71\) − 4.99197e8i − 0.276681i −0.990385 − 0.138341i \(-0.955823\pi\)
0.990385 − 0.138341i \(-0.0441769\pi\)
\(72\) −1.13885e9 − 1.33998e9i −0.588577 − 0.692528i
\(73\) 8.77316e8 + 8.77316e8i 0.423196 + 0.423196i 0.886303 − 0.463107i \(-0.153265\pi\)
−0.463107 + 0.886303i \(0.653265\pi\)
\(74\) −7.87212e8 − 3.29069e9i −0.354759 − 1.48296i
\(75\) −7.15501e8 − 3.64312e7i −0.301511 − 0.0153521i
\(76\) 2.62633e8 − 8.03876e8i 0.103581 − 0.317045i
\(77\) − 3.89127e9i − 1.43760i
\(78\) −2.51047e8 − 1.04942e9i −0.0869525 − 0.363477i
\(79\) − 1.04248e9i − 0.338791i −0.985548 − 0.169396i \(-0.945818\pi\)
0.985548 − 0.169396i \(-0.0541816\pi\)
\(80\) −2.59469e9 − 2.00126e9i −0.791835 − 0.610735i
\(81\) 2.56234e9 0.734873
\(82\) −3.96212e9 + 9.47834e8i −1.06871 + 0.255660i
\(83\) 1.28366e9 0.325882 0.162941 − 0.986636i \(-0.447902\pi\)
0.162941 + 0.986636i \(0.447902\pi\)
\(84\) 6.30344e8 − 1.92938e9i 0.150723 − 0.461340i
\(85\) −5.03611e9 + 4.78621e9i −1.13501 + 1.07869i
\(86\) 5.59184e9 − 1.33770e9i 1.18867 − 0.284359i
\(87\) 3.48576e8 − 3.48576e8i 0.0699361 − 0.0699361i
\(88\) −4.70379e9 − 3.81675e8i −0.891323 − 0.0723236i
\(89\) 1.06839e10 1.91329 0.956647 − 0.291251i \(-0.0940715\pi\)
0.956647 + 0.291251i \(0.0940715\pi\)
\(90\) 5.24950e9 − 1.11546e9i 0.889007 − 0.188904i
\(91\) −8.78143e9 + 8.78143e9i −1.40721 + 1.40721i
\(92\) −6.33695e9 − 2.07033e9i −0.961482 − 0.314124i
\(93\) 2.54129e9 0.365292
\(94\) 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 \(0.636308\pi\)
\(98\) −1.39285e10 + 3.33203e9i −1.54090 + 0.368620i
\(99\) 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.t.a.77.51 yes 236
5.3 odd 4 80.11.i.a.13.10 ✓ 236
16.5 even 4 80.11.i.a.37.10 yes 236
80.53 odd 4 inner 80.11.t.a.53.51 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.10 ✓ 236 5.3 odd 4
80.11.i.a.37.10 yes 236 16.5 even 4
80.11.t.a.53.51 yes 236 80.53 odd 4 inner
80.11.t.a.77.51 yes 236 1.1 even 1 trivial