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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 77.58
Character \(\chi\) \(=\) 80.77
Dual form 80.11.t.a.53.58

$q$-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+(-1.87773 - 31.9449i) q^{2} +324.949 q^{3} +(-1016.95 + 119.968i) q^{4} +(2350.54 + 2059.26i) q^{5} +(-610.167 - 10380.5i) q^{6} +(-17094.6 + 17094.6i) q^{7} +(5741.91 + 32261.0i) q^{8} +46542.9 q^{9} +(61369.2 - 78954.5i) q^{10} +(98495.7 - 98495.7i) q^{11} +(-330456. + 38983.4i) q^{12} -449379. q^{13} +(578185. + 513986. i) q^{14} +(763807. + 669156. i) q^{15} +(1.01979e6 - 244002. i) q^{16} +(-1.31211e6 - 1.31211e6i) q^{17} +(-87395.0 - 1.48681e6i) q^{18} +(-449741. + 449741. i) q^{19} +(-2.63743e6 - 1.81218e6i) q^{20} +(-5.55488e6 + 5.55488e6i) q^{21} +(-3.33138e6 - 2.96148e6i) q^{22} +(1.73793e6 + 1.73793e6i) q^{23} +(1.86583e6 + 1.04832e7i) q^{24} +(1.28448e6 + 9.68078e6i) q^{25} +(843813. + 1.43554e7i) q^{26} -4.06385e6 q^{27} +(1.53335e7 - 1.94352e7i) q^{28} +(-3.49471e6 + 3.49471e6i) q^{29} +(1.99419e7 - 2.56562e7i) q^{30} -3.57803e7 q^{31} +(-9.70950e6 - 3.21189e7i) q^{32} +(3.20061e7 - 3.20061e7i) q^{33} +(-3.94515e7 + 4.43791e7i) q^{34} +(-7.53840e7 + 4.97929e6i) q^{35} +(-4.73317e7 + 5.58364e6i) q^{36} -1.03431e8 q^{37} +(1.52114e7 + 1.35224e7i) q^{38} -1.46025e8 q^{39} +(-5.29374e7 + 8.76550e7i) q^{40} -1.29745e7i q^{41} +(1.87881e8 + 1.67019e8i) q^{42} -1.49758e8i q^{43} +(-8.83487e7 + 1.11981e8i) q^{44} +(1.09401e8 + 9.58441e7i) q^{45} +(5.22545e7 - 5.87812e7i) q^{46} +(-1.77265e8 - 1.77265e8i) q^{47} +(3.31380e8 - 7.92882e7i) q^{48} -3.01977e8i q^{49} +(3.06839e8 - 5.92105e7i) q^{50} +(-4.26370e8 - 4.26370e8i) q^{51} +(4.56995e8 - 5.39110e7i) q^{52} +5.89702e8i q^{53} +(7.63082e6 + 1.29819e8i) q^{54} +(4.34347e8 - 2.86896e7i) q^{55} +(-6.49646e8 - 4.53334e8i) q^{56} +(-1.46143e8 + 1.46143e8i) q^{57} +(1.18200e8 + 1.05076e8i) q^{58} +(8.92910e8 + 8.92910e8i) q^{59} +(-8.57029e8 - 5.88865e8i) q^{60} +(-1.27465e8 - 1.27465e8i) q^{61} +(6.71858e7 + 1.14300e9i) q^{62} +(-7.95633e8 + 7.95633e8i) q^{63} +(-1.00780e9 + 3.70479e8i) q^{64} +(-1.05629e9 - 9.25391e8i) q^{65} +(-1.08253e9 - 9.62331e8i) q^{66} -5.64061e8i q^{67} +(1.49176e9 + 1.17694e9i) q^{68} +(5.64738e8 + 5.64738e8i) q^{69} +(3.00614e8 + 2.39878e9i) q^{70} +2.21686e9i q^{71} +(2.67245e8 + 1.50152e9i) q^{72} +(-1.37196e9 - 1.37196e9i) q^{73} +(1.94216e8 + 3.30409e9i) q^{74} +(4.17391e8 + 3.14576e9i) q^{75} +(4.03409e8 - 5.11318e8i) q^{76} +3.36749e9i q^{77} +(2.74196e8 + 4.66476e9i) q^{78} +4.96474e9i q^{79} +(2.89953e9 + 1.52648e9i) q^{80} -4.06885e9 q^{81} +(-4.14469e8 + 2.43626e7i) q^{82} -1.19817e9 q^{83} +(4.98262e9 - 6.31544e9i) q^{84} +(-3.82190e8 - 5.78617e9i) q^{85} +(-4.78400e9 + 2.81205e8i) q^{86} +(-1.13560e9 + 1.13560e9i) q^{87} +(3.74312e9 + 2.61202e9i) q^{88} +5.45732e9 q^{89} +(2.85630e9 - 3.67477e9i) q^{90} +(7.68197e9 - 7.68197e9i) q^{91} +(-1.97588e9 - 1.55889e9i) q^{92} -1.16268e10 q^{93} +(-5.32984e9 + 5.99555e9i) q^{94} +(-1.98327e9 + 1.31000e8i) q^{95} +(-3.15509e9 - 1.04370e10i) q^{96} +(4.09877e9 + 4.09877e9i) q^{97} +(-9.64662e9 + 5.67032e8i) q^{98} +(4.58427e9 - 4.58427e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.87773 − 31.9449i −0.0586791 − 0.998277i
\(3\) 324.949 1.33724 0.668619 − 0.743605i \(-0.266886\pi\)
0.668619 + 0.743605i \(0.266886\pi\)
\(4\) −1016.95 + 119.968i −0.993114 + 0.117156i
\(5\) 2350.54 + 2059.26i 0.752174 + 0.658965i
\(6\) −610.167 − 10380.5i −0.0784680 − 1.33493i
\(7\) −17094.6 + 17094.6i −1.01711 + 1.01711i −0.0172626 + 0.999851i \(0.505495\pi\)
−0.999851 + 0.0172626i \(0.994505\pi\)
\(8\) 5741.91 + 32261.0i 0.175229 + 0.984528i
\(9\) 46542.9 0.788208
\(10\) 61369.2 − 78954.5i 0.613692 − 0.789545i
\(11\) 98495.7 − 98495.7i 0.611581 − 0.611581i −0.331777 − 0.943358i \(-0.607648\pi\)
0.943358 + 0.331777i \(0.107648\pi\)
\(12\) −330456. + 38983.4i −1.32803 + 0.156666i
\(13\) −449379. −1.21031 −0.605154 − 0.796108i \(-0.706889\pi\)
−0.605154 + 0.796108i \(0.706889\pi\)
\(14\) 578185. + 513986.i 1.07504 + 0.955678i
\(15\) 763807. + 669156.i 1.00584 + 0.881193i
\(16\) 1.01979e6 − 244002.i 0.972549 − 0.232698i
\(17\) −1.31211e6 − 1.31211e6i −0.924118 − 0.924118i 0.0731997 − 0.997317i \(-0.476679\pi\)
−0.997317 + 0.0731997i \(0.976679\pi\)
\(18\) −87395.0 − 1.48681e6i −0.0462513 − 0.786850i
\(19\) −449741. + 449741.i −0.181633 + 0.181633i −0.792067 − 0.610434i \(-0.790995\pi\)
0.610434 + 0.792067i \(0.290995\pi\)
\(20\) −2.63743e6 − 1.81218e6i −0.824196 − 0.566305i
\(21\) −5.55488e6 + 5.55488e6i −1.36012 + 1.36012i
\(22\) −3.33138e6 − 2.96148e6i −0.646414 − 0.574640i
\(23\) 1.73793e6 + 1.73793e6i 0.270018 + 0.270018i 0.829107 − 0.559090i \(-0.188849\pi\)
−0.559090 + 0.829107i \(0.688849\pi\)
\(24\) 1.86583e6 + 1.04832e7i 0.234323 + 1.31655i
\(25\) 1.28448e6 + 9.68078e6i 0.131531 + 0.991312i
\(26\) 843813. + 1.43554e7i 0.0710198 + 1.20822i
\(27\) −4.06385e6 −0.283217
\(28\) 1.53335e7 − 1.94352e7i 0.890948 − 1.12927i
\(29\) −3.49471e6 + 3.49471e6i −0.170381 + 0.170381i −0.787147 − 0.616766i \(-0.788443\pi\)
0.616766 + 0.787147i \(0.288443\pi\)
\(30\) 1.99419e7 − 2.56562e7i 0.820653 − 1.05581i
\(31\) −3.57803e7 −1.24979 −0.624893 − 0.780710i \(-0.714858\pi\)
−0.624893 + 0.780710i \(0.714858\pi\)
\(32\) −9.70950e6 − 3.21189e7i −0.289366 − 0.957219i
\(33\) 3.20061e7 − 3.20061e7i 0.817829 − 0.817829i
\(34\) −3.94515e7 + 4.43791e7i −0.868299 + 0.976752i
\(35\) −7.53840e7 + 4.97929e6i −1.43529 + 0.0948041i
\(36\) −4.73317e7 + 5.58364e6i −0.782780 + 0.0923433i
\(37\) −1.03431e8 −1.49157 −0.745783 − 0.666189i \(-0.767924\pi\)
−0.745783 + 0.666189i \(0.767924\pi\)
\(38\) 1.52114e7 + 1.35224e7i 0.191978 + 0.170662i
\(39\) −1.46025e8 −1.61847
\(40\) −5.29374e7 + 8.76550e7i −0.516966 + 0.856006i
\(41\) − 1.29745e7i − 0.111988i −0.998431 − 0.0559940i \(-0.982167\pi\)
0.998431 − 0.0559940i \(-0.0178328\pi\)
\(42\) 1.87881e8 + 1.67019e8i 1.43759 + 1.27797i
\(43\) − 1.49758e8i − 1.01870i −0.860558 − 0.509352i \(-0.829885\pi\)
0.860558 − 0.509352i \(-0.170115\pi\)
\(44\) −8.83487e7 + 1.11981e8i −0.535719 + 0.679019i
\(45\) 1.09401e8 + 9.58441e7i 0.592869 + 0.519401i
\(46\) 5.22545e7 − 5.87812e7i 0.253708 − 0.285397i
\(47\) −1.77265e8 − 1.77265e8i −0.772917 − 0.772917i 0.205698 − 0.978615i \(-0.434053\pi\)
−0.978615 + 0.205698i \(0.934053\pi\)
\(48\) 3.31380e8 − 7.92882e7i 1.30053 − 0.311173i
\(49\) − 3.01977e8i − 1.06904i
\(50\) 3.06839e8 − 5.92105e7i 0.981886 − 0.189473i
\(51\) −4.26370e8 − 4.26370e8i −1.23577 − 1.23577i
\(52\) 4.56995e8 − 5.39110e7i 1.20197 − 0.141795i
\(53\) 5.89702e8i 1.41011i 0.709152 + 0.705055i \(0.249078\pi\)
−0.709152 + 0.705055i \(0.750922\pi\)
\(54\) 7.63082e6 + 1.29819e8i 0.0166189 + 0.282729i
\(55\) 4.34347e8 − 2.86896e7i 0.863025 − 0.0570048i
\(56\) −6.49646e8 − 4.53334e8i −1.17960 − 0.823149i
\(57\) −1.46143e8 + 1.46143e8i −0.242887 + 0.242887i
\(58\) 1.18200e8 + 1.05076e8i 0.180085 + 0.160090i
\(59\) 8.92910e8 + 8.92910e8i 1.24896 + 1.24896i 0.956181 + 0.292777i \(0.0945795\pi\)
0.292777 + 0.956181i \(0.405420\pi\)
\(60\) −8.57029e8 − 5.88865e8i −1.10215 − 0.757285i
\(61\) −1.27465e8 − 1.27465e8i −0.150918 − 0.150918i 0.627610 − 0.778528i \(-0.284033\pi\)
−0.778528 + 0.627610i \(0.784033\pi\)
\(62\) 6.71858e7 + 1.14300e9i 0.0733363 + 1.24763i
\(63\) −7.95633e8 + 7.95633e8i −0.801697 + 0.801697i
\(64\) −1.00780e9 + 3.70479e8i −0.938590 + 0.345036i
\(65\) −1.05629e9 − 9.25391e8i −0.910362 − 0.797551i
\(66\) −1.08253e9 − 9.62331e8i −0.864410 − 0.768431i
\(67\) − 5.64061e8i − 0.417784i −0.977939 − 0.208892i \(-0.933014\pi\)
0.977939 − 0.208892i \(-0.0669858\pi\)
\(68\) 1.49176e9 + 1.17694e9i 1.02602 + 0.809488i
\(69\) 5.64738e8 + 5.64738e8i 0.361078 + 0.361078i
\(70\) 3.00614e8 + 2.39878e9i 0.178862 + 1.42725i
\(71\) 2.21686e9i 1.22870i 0.789032 + 0.614352i \(0.210582\pi\)
−0.789032 + 0.614352i \(0.789418\pi\)
\(72\) 2.67245e8 + 1.50152e9i 0.138117 + 0.776013i
\(73\) −1.37196e9 − 1.37196e9i −0.661800 − 0.661800i 0.294004 − 0.955804i \(-0.405012\pi\)
−0.955804 + 0.294004i \(0.905012\pi\)
\(74\) 1.94216e8 + 3.30409e9i 0.0875237 + 1.48900i
\(75\) 4.17391e8 + 3.14576e9i 0.175888 + 1.32562i
\(76\) 4.03409e8 − 5.11318e8i 0.159103 − 0.201662i
\(77\) 3.36749e9i 1.24409i
\(78\) 2.74196e8 + 4.66476e9i 0.0949705 + 1.61568i
\(79\) 4.96474e9i 1.61347i 0.590913 + 0.806735i \(0.298768\pi\)
−0.590913 + 0.806735i \(0.701232\pi\)
\(80\) 2.89953e9 + 1.52648e9i 0.884866 + 0.465846i
\(81\) −4.06885e9 −1.16694
\(82\) −4.14469e8 + 2.43626e7i −0.111795 + 0.00657136i
\(83\) −1.19817e9 −0.304178 −0.152089 − 0.988367i \(-0.548600\pi\)
−0.152089 + 0.988367i \(0.548600\pi\)
\(84\) 4.98262e9 − 6.31544e9i 1.19141 − 1.51010i
\(85\) −3.82190e8 − 5.78617e9i −0.0861361 − 1.30406i
\(86\) −4.78400e9 + 2.81205e8i −1.01695 + 0.0597766i
\(87\) −1.13560e9 + 1.13560e9i −0.227840 + 0.227840i
\(88\) 3.74312e9 + 2.61202e9i 0.709285 + 0.494951i
\(89\) 5.45732e9 0.977304 0.488652 − 0.872479i \(-0.337489\pi\)
0.488652 + 0.872479i \(0.337489\pi\)
\(90\) 2.85630e9 − 3.67477e9i 0.483717 − 0.622326i
\(91\) 7.68197e9 − 7.68197e9i 1.23102 − 1.23102i
\(92\) −1.97588e9 − 1.55889e9i −0.299792 − 0.236524i
\(93\) −1.16268e10 −1.67126
\(94\) −5.32984e9 + 5.99555e9i −0.726231 + 0.816939i
\(95\) −1.98327e9 + 1.31000e8i −0.256309 + 0.0169298i
\(96\) −3.15509e9 − 1.04370e10i −0.386951 − 1.28003i
\(97\) 4.09877e9 + 4.09877e9i 0.477304 + 0.477304i 0.904268 − 0.426964i \(-0.140417\pi\)
−0.426964 + 0.904268i \(0.640417\pi\)
\(98\) −9.64662e9 + 5.67032e8i −1.06720 + 0.0627303i
\(99\) 4.58427e9 − 4.58427e9i 0.482053 − 0.482053i
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.77.58 yes 236
5.3 odd 4 80.11.i.a.13.3 ✓ 236
16.5 even 4 80.11.i.a.37.3 yes 236
80.53 odd 4 inner 80.11.t.a.53.58 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.3 ✓ 236 5.3 odd 4
80.11.i.a.37.3 yes 236 16.5 even 4
80.11.t.a.53.58 yes 236 80.53 odd 4 inner
80.11.t.a.77.58 yes 236 1.1 even 1 trivial