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Properties

Label 80.11.i.a.13.3
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 13.3
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.3

$q$-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+(-31.9449 + 1.87773i) q^{2} +324.949i q^{3} +(1016.95 - 119.968i) q^{4} +(-2059.26 - 2350.54i) q^{5} +(-610.167 - 10380.5i) q^{6} +(17094.6 + 17094.6i) q^{7} +(-32261.0 + 5741.91i) q^{8} -46542.9 q^{9} +(70196.6 + 71221.0i) q^{10} +(98495.7 - 98495.7i) q^{11} +(38983.4 + 330456. i) q^{12} -449379. i q^{13} +(-578185. - 513986. i) q^{14} +(763807. - 669156. i) q^{15} +(1.01979e6 - 244002. i) q^{16} +(-1.31211e6 + 1.31211e6i) q^{17} +(1.48681e6 - 87395.0i) q^{18} +(449741. - 449741. i) q^{19} +(-2.37616e6 - 2.14334e6i) q^{20} +(-5.55488e6 + 5.55488e6i) q^{21} +(-2.96148e6 + 3.33138e6i) 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} +4.06385e6i q^{27} +(1.94352e7 + 1.53335e7i) q^{28} +(3.49471e6 - 3.49471e6i) q^{29} +(-2.31432e7 + 2.28103e7i) q^{30} -3.57803e7 q^{31} +(-3.21189e7 + 9.70950e6i) q^{32} +(3.20061e7 + 3.20061e7i) q^{33} +(3.94515e7 - 4.43791e7i) q^{34} +(4.97929e6 - 7.53840e7i) q^{35} +(-4.73317e7 + 5.58364e6i) q^{36} +1.03431e8i q^{37} +(-1.35224e7 + 1.52114e7i) q^{38} +1.46025e8 q^{39} +(7.99306e7 + 6.40068e7i) q^{40} -1.29745e7i q^{41} +(1.67019e8 - 1.87881e8i) q^{42} +1.49758e8 q^{43} +(8.83487e7 - 1.11981e8i) q^{44} +(9.58441e7 + 1.09401e8i) q^{45} +(5.22545e7 - 5.87812e7i) q^{46} +(-1.77265e8 + 1.77265e8i) q^{47} +(7.92882e7 + 3.31380e8i) q^{48} +3.01977e8i q^{49} +(2.28546e7 - 3.11663e8i) q^{50} +(-4.26370e8 - 4.26370e8i) q^{51} +(-5.39110e7 - 4.56995e8i) q^{52} -5.89702e8 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} +(-1.05076e8 + 1.18200e8i) q^{58} +(-8.92910e8 - 8.92910e8i) q^{59} +(6.96475e8 - 7.72129e8i) q^{60} +(-1.27465e8 - 1.27465e8i) q^{61} +(1.14300e9 - 6.71858e7i) 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} -5.64061e8 q^{67} +(-1.17694e9 + 1.49176e9i) q^{68} +(-5.64738e8 - 5.64738e8i) q^{69} +(-1.75118e7 + 2.41748e9i) q^{70} +2.21686e9i q^{71} +(1.50152e9 - 2.67245e8i) q^{72} +(1.37196e9 - 1.37196e9i) q^{73} +(-1.94216e8 - 3.30409e9i) q^{74} +(-3.14576e9 - 4.17391e8i) q^{75} +(4.03409e8 - 5.11318e8i) q^{76} +3.36749e9 q^{77} +(-4.66476e9 + 2.74196e8i) q^{78} -4.96474e9i q^{79} +(-2.67356e9 - 1.89460e9i) q^{80} -4.06885e9 q^{81} +(2.43626e7 + 4.14469e8i) q^{82} -1.19817e9i q^{83} +(-4.98262e9 + 6.31544e9i) q^{84} +(5.78617e9 + 3.82190e8i) q^{85} +(-4.78400e9 + 2.81205e8i) q^{86} +(1.13560e9 + 1.13560e9i) q^{87} +(-2.61202e9 + 3.74312e9i) q^{88} -5.45732e9 q^{89} +(-3.26715e9 - 3.31483e9i) q^{90} +(7.68197e9 - 7.68197e9i) q^{91} +(-1.55889e9 + 1.97588e9i) q^{92} -1.16268e10i 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} +(-5.67032e8 - 9.64662e9i) q^{98} +(-4.58427e9 + 4.58427e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.9449 + 1.87773i −0.998277 + 0.0586791i
\(3\) 324.949i 1.33724i 0.743605 + 0.668619i \(0.233114\pi\)
−0.743605 + 0.668619i \(0.766886\pi\)
\(4\) 1016.95 − 119.968i 0.993114 − 0.117156i
\(5\) −2059.26 − 2350.54i −0.658965 − 0.752174i
\(6\) −610.167 − 10380.5i −0.0784680 − 1.33493i
\(7\) 17094.6 + 17094.6i 1.01711 + 1.01711i 0.999851 + 0.0172626i \(0.00549512\pi\)
0.0172626 + 0.999851i \(0.494505\pi\)
\(8\) −32261.0 + 5741.91i −0.984528 + 0.175229i
\(9\) −46542.9 −0.788208
\(10\) 70196.6 + 71221.0i 0.701966 + 0.712210i
\(11\) 98495.7 − 98495.7i 0.611581 − 0.611581i −0.331777 − 0.943358i \(-0.607648\pi\)
0.943358 + 0.331777i \(0.107648\pi\)
\(12\) 38983.4 + 330456.i 0.156666 + 1.32803i
\(13\) − 449379.i − 1.21031i −0.796108 − 0.605154i \(-0.793111\pi\)
0.796108 − 0.605154i \(-0.206889\pi\)
\(14\) −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 −0.997317 − 0.0731997i \(-0.976679\pi\)
0.0731997 + 0.997317i \(0.476679\pi\)
\(18\) 1.48681e6 − 87395.0i 0.786850 − 0.0462513i
\(19\) 449741. − 449741.i 0.181633 − 0.181633i −0.610434 − 0.792067i \(-0.709005\pi\)
0.792067 + 0.610434i \(0.209005\pi\)
\(20\) −2.37616e6 − 2.14334e6i −0.742549 − 0.669792i
\(21\) −5.55488e6 + 5.55488e6i −1.36012 + 1.36012i
\(22\) −2.96148e6 + 3.33138e6i −0.574640 + 0.646414i
\(23\) −1.73793e6 + 1.73793e6i −0.270018 + 0.270018i −0.829107 − 0.559090i \(-0.811151\pi\)
0.559090 + 0.829107i \(0.311151\pi\)
\(24\) −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
\(27\) 4.06385e6i 0.283217i
\(28\) 1.94352e7 + 1.53335e7i 1.12927 + 0.890948i
\(29\) 3.49471e6 − 3.49471e6i 0.170381 − 0.170381i −0.616766 − 0.787147i \(-0.711557\pi\)
0.787147 + 0.616766i \(0.211557\pi\)
\(30\) −2.31432e7 + 2.28103e7i −0.952395 + 0.938697i
\(31\) −3.57803e7 −1.24979 −0.624893 − 0.780710i \(-0.714858\pi\)
−0.624893 + 0.780710i \(0.714858\pi\)
\(32\) −3.21189e7 + 9.70950e6i −0.957219 + 0.289366i
\(33\) 3.20061e7 + 3.20061e7i 0.817829 + 0.817829i
\(34\) 3.94515e7 − 4.43791e7i 0.868299 − 0.976752i
\(35\) 4.97929e6 − 7.53840e7i 0.0948041 − 1.43529i
\(36\) −4.73317e7 + 5.58364e6i −0.782780 + 0.0923433i
\(37\) 1.03431e8i 1.49157i 0.666189 + 0.745783i \(0.267924\pi\)
−0.666189 + 0.745783i \(0.732076\pi\)
\(38\) −1.35224e7 + 1.52114e7i −0.170662 + 0.191978i
\(39\) 1.46025e8 1.61847
\(40\) 7.99306e7 + 6.40068e7i 0.780572 + 0.625066i
\(41\) − 1.29745e7i − 0.111988i −0.998431 − 0.0559940i \(-0.982167\pi\)
0.998431 − 0.0559940i \(-0.0178328\pi\)
\(42\) 1.67019e8 − 1.87881e8i 1.27797 − 1.43759i
\(43\) 1.49758e8 1.01870 0.509352 − 0.860558i \(-0.329885\pi\)
0.509352 + 0.860558i \(0.329885\pi\)
\(44\) 8.83487e7 − 1.11981e8i 0.535719 − 0.679019i
\(45\) 9.58441e7 + 1.09401e8i 0.519401 + 0.592869i
\(46\) 5.22545e7 − 5.87812e7i 0.253708 − 0.285397i
\(47\) −1.77265e8 + 1.77265e8i −0.772917 + 0.772917i −0.978615 − 0.205698i \(-0.934053\pi\)
0.205698 + 0.978615i \(0.434053\pi\)
\(48\) 7.92882e7 + 3.31380e8i 0.311173 + 1.30053i
\(49\) 3.01977e8i 1.06904i
\(50\) 2.28546e7 − 3.11663e8i 0.0731349 − 0.997322i
\(51\) −4.26370e8 − 4.26370e8i −1.23577 − 1.23577i
\(52\) −5.39110e7 − 4.56995e8i −0.141795 − 1.20197i
\(53\) −5.89702e8 −1.41011 −0.705055 − 0.709152i \(-0.749078\pi\)
−0.705055 + 0.709152i \(0.749078\pi\)
\(54\) −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
\(58\) −1.05076e8 + 1.18200e8i −0.160090 + 0.180085i
\(59\) −8.92910e8 − 8.92910e8i −1.24896 − 1.24896i −0.956181 − 0.292777i \(-0.905420\pi\)
−0.292777 − 0.956181i \(-0.594580\pi\)
\(60\) 6.96475e8 − 7.72129e8i 0.895672 − 0.992965i
\(61\) −1.27465e8 − 1.27465e8i −0.150918 − 0.150918i 0.627610 − 0.778528i \(-0.284033\pi\)
−0.778528 + 0.627610i \(0.784033\pi\)
\(62\) 1.14300e9 − 6.71858e7i 1.24763 − 0.0733363i
\(63\) −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
\(67\) −5.64061e8 −0.417784 −0.208892 − 0.977939i \(-0.566986\pi\)
−0.208892 + 0.977939i \(0.566986\pi\)
\(68\) −1.17694e9 + 1.49176e9i −0.809488 + 1.02602i
\(69\) −5.64738e8 − 5.64738e8i −0.361078 − 0.361078i
\(70\) −1.75118e7 + 2.41748e9i −0.0104194 + 1.43838i
\(71\) 2.21686e9i 1.22870i 0.789032 + 0.614352i \(0.210582\pi\)
−0.789032 + 0.614352i \(0.789418\pi\)
\(72\) 1.50152e9 − 2.67245e8i 0.776013 − 0.138117i
\(73\) 1.37196e9 − 1.37196e9i 0.661800 − 0.661800i −0.294004 − 0.955804i \(-0.594988\pi\)
0.955804 + 0.294004i \(0.0949879\pi\)
\(74\) −1.94216e8 − 3.30409e9i −0.0875237 − 1.48900i
\(75\) −3.14576e9 − 4.17391e8i −1.32562 − 0.175888i
\(76\) 4.03409e8 − 5.11318e8i 0.159103 − 0.201662i
\(77\) 3.36749e9 1.24409
\(78\) −4.66476e9 + 2.74196e8i −1.61568 + 0.0949705i
\(79\) − 4.96474e9i − 1.61347i −0.590913 − 0.806735i \(-0.701232\pi\)
0.590913 − 0.806735i \(-0.298768\pi\)
\(80\) −2.67356e9 − 1.89460e9i −0.815905 − 0.578186i
\(81\) −4.06885e9 −1.16694
\(82\) 2.43626e7 + 4.14469e8i 0.00657136 + 0.111795i
\(83\) − 1.19817e9i − 0.304178i −0.988367 − 0.152089i \(-0.951400\pi\)
0.988367 − 0.152089i \(-0.0486000\pi\)
\(84\) −4.98262e9 + 6.31544e9i −1.19141 + 1.51010i
\(85\) 5.78617e9 + 3.82190e8i 1.30406 + 0.0861361i
\(86\) −4.78400e9 + 2.81205e8i −1.01695 + 0.0597766i
\(87\) 1.13560e9 + 1.13560e9i 0.227840 + 0.227840i
\(88\) −2.61202e9 + 3.74312e9i −0.494951 + 0.709285i
\(89\) −5.45732e9 −0.977304 −0.488652 − 0.872479i \(-0.662511\pi\)
−0.488652 + 0.872479i \(0.662511\pi\)
\(90\) −3.26715e9 − 3.31483e9i −0.553295 − 0.561370i
\(91\) 7.68197e9 − 7.68197e9i 1.23102 − 1.23102i
\(92\) −1.55889e9 + 1.97588e9i −0.236524 + 0.299792i
\(93\) − 1.16268e10i − 1.67126i
\(94\) 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 −0.426964 − 0.904268i \(-0.640417\pi\)
0.904268 + 0.426964i \(0.140417\pi\)
\(98\) −5.67032e8 − 9.64662e9i −0.0627303 − 1.06720i
\(99\) −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.i.a.13.3 ✓ 236
5.2 odd 4 80.11.t.a.77.58 yes 236
16.5 even 4 80.11.t.a.53.58 yes 236
80.37 odd 4 inner 80.11.i.a.37.3 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.3 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.3 yes 236 80.37 odd 4 inner
80.11.t.a.53.58 yes 236 16.5 even 4
80.11.t.a.77.58 yes 236 5.2 odd 4