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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 77.6
Character \(\chi\) \(=\) 80.77
Dual form 80.11.t.a.53.6

$q$-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+(-31.7358 - 4.10388i) q^{2} +106.445 q^{3} +(990.316 + 260.479i) q^{4} +(-2840.60 + 1302.54i) q^{5} +(-3378.13 - 436.839i) q^{6} +(-14456.7 + 14456.7i) q^{7} +(-30359.5 - 12330.7i) q^{8} -47718.4 q^{9} +(95494.1 - 29679.7i) q^{10} +(-177248. + 177248. i) q^{11} +(105415. + 27726.9i) q^{12} -625109. q^{13} +(518123. - 399466. i) q^{14} +(-302369. + 138650. i) q^{15} +(912877. + 515914. i) q^{16} +(-1.73389e6 - 1.73389e6i) q^{17} +(1.51438e6 + 195830. i) q^{18} +(2.29557e6 - 2.29557e6i) q^{19} +(-3.15238e6 + 550011. i) q^{20} +(-1.53885e6 + 1.53885e6i) q^{21} +(6.35252e6 - 4.89771e6i) q^{22} +(4.37370e6 + 4.37370e6i) q^{23} +(-3.23163e6 - 1.31254e6i) q^{24} +(6.37239e6 - 7.40000e6i) q^{25} +(1.98383e7 + 2.56537e6i) q^{26} -1.13649e7 q^{27} +(-1.80824e7 + 1.05510e7i) q^{28} +(-2.34787e7 + 2.34787e7i) q^{29} +(1.01649e7 - 3.15927e6i) q^{30} +2.62528e6 q^{31} +(-2.68536e7 - 2.01193e7i) q^{32} +(-1.88673e7 + 1.88673e7i) q^{33} +(4.79106e7 + 6.21419e7i) q^{34} +(2.22353e7 - 5.98962e7i) q^{35} +(-4.72563e7 - 1.24297e7i) q^{36} -1.50725e7 q^{37} +(-8.22723e7 + 6.34308e7i) q^{38} -6.65400e7 q^{39} +(1.02300e8 - 4.51803e6i) q^{40} +5.85671e7i q^{41} +(5.51519e7 - 4.25213e7i) q^{42} +1.56498e8i q^{43} +(-2.21702e8 + 1.29362e8i) q^{44} +(1.35549e8 - 6.21552e7i) q^{45} +(-1.20853e8 - 1.56752e8i) q^{46} +(2.61879e8 + 2.61879e8i) q^{47} +(9.71716e7 + 5.49167e7i) q^{48} -1.35518e8i q^{49} +(-2.32601e8 + 2.08693e8i) q^{50} +(-1.84565e8 - 1.84565e8i) q^{51} +(-6.19055e8 - 1.62828e8i) q^{52} -2.80824e8i q^{53} +(3.60674e8 + 4.66402e7i) q^{54} +(2.72618e8 - 7.34365e8i) q^{55} +(6.17159e8 - 2.60637e8i) q^{56} +(2.44353e8 - 2.44353e8i) q^{57} +(8.41466e8 - 6.48759e8i) q^{58} +(4.87044e8 + 4.87044e8i) q^{59} +(-3.35556e8 + 5.85462e7i) q^{60} +(-1.22483e8 - 1.22483e8i) q^{61} +(-8.33154e7 - 1.07739e7i) q^{62} +(6.89850e8 - 6.89850e8i) q^{63} +(7.69652e8 + 7.48704e8i) q^{64} +(1.77568e9 - 8.14231e8i) q^{65} +(6.76197e8 - 5.21339e8i) q^{66} +4.19845e8i q^{67} +(-1.26546e9 - 2.16874e9i) q^{68} +(4.65560e8 + 4.65560e8i) q^{69} +(-9.51459e8 + 1.80960e9i) q^{70} +8.68452e8i q^{71} +(1.44870e9 + 5.88399e8i) q^{72} +(-1.91387e9 - 1.91387e9i) q^{73} +(4.78336e8 + 6.18556e7i) q^{74} +(6.78312e8 - 7.87697e8i) q^{75} +(2.87129e9 - 1.67539e9i) q^{76} -5.12486e9i q^{77} +(2.11170e9 + 2.73072e8i) q^{78} +1.25405e9i q^{79} +(-3.26512e9 - 2.76445e8i) q^{80} +1.60798e9 q^{81} +(2.40352e8 - 1.85867e9i) q^{82} +4.16831e8 q^{83} +(-1.92479e9 + 1.12311e9i) q^{84} +(7.18375e9 + 2.66682e9i) q^{85} +(6.42249e8 - 4.96658e9i) q^{86} +(-2.49920e9 + 2.49920e9i) q^{87} +(7.56676e9 - 3.19558e9i) q^{88} -4.48253e9 q^{89} +(-4.55682e9 + 1.41627e9i) q^{90} +(9.03702e9 - 9.03702e9i) q^{91} +(3.19209e9 + 5.47060e9i) q^{92} +2.79450e8 q^{93} +(-7.23621e9 - 9.38564e9i) q^{94} +(-3.53071e9 + 9.51086e9i) q^{95} +(-2.85844e9 - 2.14160e9i) q^{96} +(-3.04240e9 - 3.04240e9i) q^{97} +(-5.56148e8 + 4.30075e9i) q^{98} +(8.45800e9 - 8.45800e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.7358 − 4.10388i −0.991742 − 0.128246i
\(3\) 106.445 0.438047 0.219024 − 0.975720i \(-0.429713\pi\)
0.219024 + 0.975720i \(0.429713\pi\)
\(4\) 990.316 + 260.479i 0.967106 + 0.254374i
\(5\) −2840.60 + 1302.54i −0.908992 + 0.416814i
\(6\) −3378.13 − 436.839i −0.434430 − 0.0561779i
\(7\) −14456.7 + 14456.7i −0.860160 + 0.860160i −0.991356 − 0.131196i \(-0.958118\pi\)
0.131196 + 0.991356i \(0.458118\pi\)
\(8\) −30359.5 − 12330.7i −0.926497 − 0.376302i
\(9\) −47718.4 −0.808115
\(10\) 95494.1 − 29679.7i 0.954941 − 0.296797i
\(11\) −177248. + 177248.i −1.10057 + 1.10057i −0.106232 + 0.994341i \(0.533879\pi\)
−0.994341 + 0.106232i \(0.966121\pi\)
\(12\) 105415. + 27726.9i 0.423638 + 0.111428i
\(13\) −625109. −1.68360 −0.841800 − 0.539790i \(-0.818504\pi\)
−0.841800 + 0.539790i \(0.818504\pi\)
\(14\) 518123. − 399466.i 0.963369 − 0.742745i
\(15\) −302369. + 138650.i −0.398181 + 0.182584i
\(16\) 912877. + 515914.i 0.870587 + 0.492014i
\(17\) −1.73389e6 − 1.73389e6i −1.22117 − 1.22117i −0.967216 − 0.253955i \(-0.918268\pi\)
−0.253955 − 0.967216i \(-0.581732\pi\)
\(18\) 1.51438e6 + 195830.i 0.801442 + 0.103638i
\(19\) 2.29557e6 − 2.29557e6i 0.927090 − 0.927090i −0.0704268 − 0.997517i \(-0.522436\pi\)
0.997517 + 0.0704268i \(0.0224361\pi\)
\(20\) −3.15238e6 + 550011.i −0.985118 + 0.171878i
\(21\) −1.53885e6 + 1.53885e6i −0.376791 + 0.376791i
\(22\) 6.35252e6 − 4.89771e6i 1.23263 − 0.950341i
\(23\) 4.37370e6 + 4.37370e6i 0.679531 + 0.679531i 0.959894 − 0.280363i \(-0.0904548\pi\)
−0.280363 + 0.959894i \(0.590455\pi\)
\(24\) −3.23163e6 − 1.31254e6i −0.405849 − 0.164838i
\(25\) 6.37239e6 − 7.40000e6i 0.652533 − 0.757760i
\(26\) 1.98383e7 + 2.56537e6i 1.66970 + 0.215915i
\(27\) −1.13649e7 −0.792040
\(28\) −1.80824e7 + 1.05510e7i −1.05067 + 0.613063i
\(29\) −2.34787e7 + 2.34787e7i −1.14468 + 1.14468i −0.157094 + 0.987584i \(0.550213\pi\)
−0.987584 + 0.157094i \(0.949787\pi\)
\(30\) 1.01649e7 − 3.15927e6i 0.418309 − 0.130011i
\(31\) 2.62528e6 0.0916997 0.0458498 − 0.998948i \(-0.485400\pi\)
0.0458498 + 0.998948i \(0.485400\pi\)
\(32\) −2.68536e7 − 2.01193e7i −0.800299 − 0.599601i
\(33\) −1.88673e7 + 1.88673e7i −0.482103 + 0.482103i
\(34\) 4.79106e7 + 6.21419e7i 1.05448 + 1.36770i
\(35\) 2.22353e7 − 5.98962e7i 0.423352 − 1.14040i
\(36\) −4.72563e7 − 1.24297e7i −0.781532 − 0.205564i
\(37\) −1.50725e7 −0.217358 −0.108679 − 0.994077i \(-0.534662\pi\)
−0.108679 + 0.994077i \(0.534662\pi\)
\(38\) −8.22723e7 + 6.34308e7i −1.03833 + 0.800539i
\(39\) −6.65400e7 −0.737496
\(40\) 1.02300e8 − 4.51803e6i 0.999026 − 0.0441214i
\(41\) 5.85671e7i 0.505516i 0.967530 + 0.252758i \(0.0813376\pi\)
−0.967530 + 0.252758i \(0.918662\pi\)
\(42\) 5.51519e7 − 4.25213e7i 0.422001 − 0.325357i
\(43\) 1.56498e8i 1.06455i 0.846571 + 0.532276i \(0.178663\pi\)
−0.846571 + 0.532276i \(0.821337\pi\)
\(44\) −2.21702e8 + 1.29362e8i −1.34433 + 0.784413i
\(45\) 1.35549e8 − 6.21552e7i 0.734570 − 0.336833i
\(46\) −1.20853e8 − 1.56752e8i −0.586773 − 0.761067i
\(47\) 2.61879e8 + 2.61879e8i 1.14186 + 1.14186i 0.988111 + 0.153745i \(0.0491335\pi\)
0.153745 + 0.988111i \(0.450867\pi\)
\(48\) 9.71716e7 + 5.49167e7i 0.381358 + 0.215525i
\(49\) − 1.35518e8i − 0.479750i
\(50\) −2.32601e8 + 2.08693e8i −0.744324 + 0.667818i
\(51\) −1.84565e8 − 1.84565e8i −0.534931 − 0.534931i
\(52\) −6.19055e8 − 1.62828e8i −1.62822 − 0.428265i
\(53\) − 2.80824e8i − 0.671513i −0.941949 − 0.335757i \(-0.891008\pi\)
0.941949 − 0.335757i \(-0.108992\pi\)
\(54\) 3.60674e8 + 4.66402e7i 0.785499 + 0.101576i
\(55\) 2.72618e8 − 7.34365e8i 0.541678 − 1.45915i
\(56\) 6.17159e8 − 2.60637e8i 1.12062 − 0.473256i
\(57\) 2.44353e8 − 2.44353e8i 0.406109 − 0.406109i
\(58\) 8.41466e8 − 6.48759e8i 1.28203 − 0.988425i
\(59\) 4.87044e8 + 4.87044e8i 0.681253 + 0.681253i 0.960282 − 0.279030i \(-0.0900128\pi\)
−0.279030 + 0.960282i \(0.590013\pi\)
\(60\) −3.35556e8 + 5.85462e7i −0.431528 + 0.0752909i
\(61\) −1.22483e8 − 1.22483e8i −0.145019 − 0.145019i 0.630870 − 0.775889i \(-0.282698\pi\)
−0.775889 + 0.630870i \(0.782698\pi\)
\(62\) −8.33154e7 − 1.07739e7i −0.0909425 − 0.0117601i
\(63\) 6.89850e8 − 6.89850e8i 0.695108 − 0.695108i
\(64\) 7.69652e8 + 7.48704e8i 0.716794 + 0.697285i
\(65\) 1.77568e9 − 8.14231e8i 1.53038 − 0.701747i
\(66\) 6.76197e8 − 5.21339e8i 0.539950 − 0.416294i
\(67\) 4.19845e8i 0.310967i 0.987838 + 0.155484i \(0.0496936\pi\)
−0.987838 + 0.155484i \(0.950306\pi\)
\(68\) −1.26546e9 − 2.16874e9i −0.870367 − 1.49164i
\(69\) 4.65560e8 + 4.65560e8i 0.297667 + 0.297667i
\(70\) −9.51459e8 + 1.80960e9i −0.566109 + 1.07669i
\(71\) 8.68452e8i 0.481342i 0.970607 + 0.240671i \(0.0773676\pi\)
−0.970607 + 0.240671i \(0.922632\pi\)
\(72\) 1.44870e9 + 5.88399e8i 0.748716 + 0.304095i
\(73\) −1.91387e9 − 1.91387e9i −0.923206 − 0.923206i 0.0740483 − 0.997255i \(-0.476408\pi\)
−0.997255 + 0.0740483i \(0.976408\pi\)
\(74\) 4.78336e8 + 6.18556e7i 0.215563 + 0.0278754i
\(75\) 6.78312e8 − 7.87697e8i 0.285840 − 0.331935i
\(76\) 2.87129e9 − 1.67539e9i 1.13242 − 0.660766i
\(77\) − 5.12486e9i − 1.89334i
\(78\) 2.11170e9 + 2.73072e8i 0.731406 + 0.0945811i
\(79\) 1.25405e9i 0.407550i 0.979018 + 0.203775i \(0.0653211\pi\)
−0.979018 + 0.203775i \(0.934679\pi\)
\(80\) −3.26512e9 − 2.76445e8i −0.996435 − 0.0843643i
\(81\) 1.60798e9 0.461164
\(82\) 2.40352e8 − 1.85867e9i 0.0648305 − 0.501341i
\(83\) 4.16831e8 0.105820 0.0529102 − 0.998599i \(-0.483150\pi\)
0.0529102 + 0.998599i \(0.483150\pi\)
\(84\) −1.92479e9 + 1.12311e9i −0.460242 + 0.268551i
\(85\) 7.18375e9 + 2.66682e9i 1.61904 + 0.601034i
\(86\) 6.42249e8 − 4.96658e9i 0.136525 − 1.05576i
\(87\) −2.49920e9 + 2.49920e9i −0.501423 + 0.501423i
\(88\) 7.56676e9 − 3.19558e9i 1.43383 − 0.605531i
\(89\) −4.48253e9 −0.802736 −0.401368 − 0.915917i \(-0.631465\pi\)
−0.401368 + 0.915917i \(0.631465\pi\)
\(90\) −4.55682e9 + 1.41627e9i −0.771702 + 0.239846i
\(91\) 9.03702e9 − 9.03702e9i 1.44817 − 1.44817i
\(92\) 3.19209e9 + 5.47060e9i 0.484323 + 0.830034i
\(93\) 2.79450e8 0.0401688
\(94\) −7.23621e9 − 9.38564e9i −0.985988 − 1.27887i
\(95\) −3.53071e9 + 9.51086e9i −0.456294 + 1.22914i
\(96\) −2.85844e9 − 2.14160e9i −0.350569 − 0.262653i
\(97\) −3.04240e9 − 3.04240e9i −0.354289 − 0.354289i 0.507414 − 0.861703i \(-0.330602\pi\)
−0.861703 + 0.507414i \(0.830602\pi\)
\(98\) −5.56148e8 + 4.30075e9i −0.0615262 + 0.475789i
\(99\) 8.45800e9 − 8.45800e9i 0.889389 − 0.889389i
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.77.6 yes 236
5.3 odd 4 80.11.i.a.13.54 ✓ 236
16.5 even 4 80.11.i.a.37.54 yes 236
80.53 odd 4 inner 80.11.t.a.53.6 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.54 ✓ 236 5.3 odd 4
80.11.i.a.37.54 yes 236 16.5 even 4
80.11.t.a.53.6 yes 236 80.53 odd 4 inner
80.11.t.a.77.6 yes 236 1.1 even 1 trivial