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Properties

Label 16.22.e.a.5.3
Level $16$
Weight $22$
Character 16.5
Analytic conductor $44.716$
Analytic rank $0$
Dimension $82$
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] = [16,22,Mod(5,16)] 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("16.5"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(16, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 16.e (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: \(44.7163750859\)
Analytic rank: \(0\)
Dimension: \(82\)
Relative dimension: \(41\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 5.3
Character \(\chi\) \(=\) 16.5
Dual form 16.22.e.a.13.3

$q$-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+(-1400.35 - 369.019i) q^{2} +(109714. - 109714. i) q^{3} +(1.82480e6 + 1.03351e6i) q^{4} +(-1.97304e7 - 1.97304e7i) q^{5} +(-1.94125e8 + 1.13152e8i) q^{6} -1.02336e9i q^{7} +(-2.17397e9 - 2.12066e9i) q^{8} -1.36141e10i q^{9} +(2.03485e10 + 3.49103e10i) q^{10} +(-7.32170e10 - 7.32170e10i) q^{11} +(3.13598e11 - 8.68160e10i) q^{12} +(-4.28208e11 + 4.28208e11i) q^{13} +(-3.77640e11 + 1.43306e12i) q^{14} -4.32941e12 q^{15} +(2.26176e12 + 3.77190e12i) q^{16} +6.62050e12 q^{17} +(-5.02387e12 + 1.90645e13i) q^{18} +(-1.48987e13 + 1.48987e13i) q^{19} +(-1.56125e13 - 5.63956e13i) q^{20} +(-1.12277e14 - 1.12277e14i) q^{21} +(7.55109e13 + 1.29548e14i) q^{22} -2.73784e14i q^{23} +(-4.71183e14 + 5.84907e12i) q^{24} +3.01738e14i q^{25} +(7.57658e14 - 4.41624e14i) q^{26} +(-3.46012e14 - 3.46012e14i) q^{27} +(1.05766e15 - 1.86743e15i) q^{28} +(1.70828e15 - 1.70828e15i) q^{29} +(6.06268e15 + 1.59763e15i) q^{30} +1.42001e15 q^{31} +(-1.77534e15 - 6.11661e15i) q^{32} -1.60659e16 q^{33} +(-9.27100e15 - 2.44309e15i) q^{34} +(-2.01913e16 + 2.01913e16i) q^{35} +(1.40703e16 - 2.48431e16i) q^{36} +(-9.05710e15 - 9.05710e15i) q^{37} +(2.63613e16 - 1.53655e16i) q^{38} +9.39612e16i q^{39} +(1.05186e15 + 8.47348e16i) q^{40} +5.53453e16i q^{41} +(1.15795e17 + 1.98660e17i) q^{42} +(-8.48059e16 - 8.48059e16i) q^{43} +(-5.79360e16 - 2.09277e17i) q^{44} +(-2.68611e17 + 2.68611e17i) q^{45} +(-1.01032e17 + 3.83393e17i) q^{46} +5.38091e17 q^{47} +(6.61979e17 + 1.65685e17i) q^{48} -4.88724e17 q^{49} +(1.11347e17 - 4.22538e17i) q^{50} +(7.26363e17 - 7.26363e17i) q^{51} +(-1.22395e18 + 3.38837e17i) q^{52} +(-6.11834e17 - 6.11834e17i) q^{53} +(3.56853e17 + 6.12223e17i) q^{54} +2.88920e18i q^{55} +(-2.17021e18 + 2.22476e18i) q^{56} +3.26920e18i q^{57} +(-3.02257e18 + 1.76180e18i) q^{58} +(5.01134e18 + 5.01134e18i) q^{59} +(-7.90031e18 - 4.47449e18i) q^{60} +(7.57981e18 - 7.57981e18i) q^{61} +(-1.98850e18 - 5.24009e17i) q^{62} -1.39322e19 q^{63} +(2.28955e17 + 9.22053e18i) q^{64} +1.68974e19 q^{65} +(2.24979e19 + 5.92863e18i) q^{66} +(-1.45969e19 + 1.45969e19i) q^{67} +(1.20811e19 + 6.84235e18i) q^{68} +(-3.00381e19 - 3.00381e19i) q^{69} +(3.57259e19 - 2.08239e19i) q^{70} +3.61689e19i q^{71} +(-2.88709e19 + 2.95967e19i) q^{72} -1.74764e19i q^{73} +(9.34086e18 + 1.60253e19i) q^{74} +(3.31049e19 + 3.31049e19i) q^{75} +(-4.25851e19 + 1.17892e19i) q^{76} +(-7.49276e19 + 7.49276e19i) q^{77} +(3.46735e19 - 1.31578e20i) q^{78} +9.41739e18 q^{79} +(2.97958e19 - 1.19046e20i) q^{80} +6.64834e19 q^{81} +(2.04235e19 - 7.75027e19i) q^{82} +(-1.22582e20 + 1.22582e20i) q^{83} +(-8.88442e19 - 3.20924e20i) q^{84} +(-1.30625e20 - 1.30625e20i) q^{85} +(8.74629e19 + 1.50053e20i) q^{86} -3.74845e20i q^{87} +(3.90333e18 + 3.14441e20i) q^{88} -4.34193e20i q^{89} +(4.75272e20 - 2.77027e20i) q^{90} +(4.38212e20 + 4.38212e20i) q^{91} +(2.82959e20 - 4.99602e20i) q^{92} +(1.55795e20 - 1.55795e20i) q^{93} +(-7.53515e20 - 1.98566e20i) q^{94} +5.87913e20 q^{95} +(-8.65861e20 - 4.76299e20i) q^{96} -4.29145e20 q^{97} +(6.84385e20 + 1.80349e20i) q^{98} +(-9.96785e20 + 9.96785e20i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 82 q - 2 q^{2} - 2 q^{3} + 1604328 q^{4} - 2 q^{5} - 142760912 q^{6} - 5437750796 q^{8} - 101897120756 q^{10} + 67333320738 q^{11} + 366511438084 q^{12} - 2 q^{13} + 3790342566044 q^{14} + 4613203124996 q^{15}+ \cdots + 27\!\cdots\!38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1400.35 − 369.019i −0.966988 − 0.254820i
\(3\) 109714. − 109714.i 1.07273 − 1.07273i 0.0755903 − 0.997139i \(-0.475916\pi\)
0.997139 − 0.0755903i \(-0.0240841\pi\)
\(4\) 1.82480e6 + 1.03351e6i 0.870133 + 0.492816i
\(5\) −1.97304e7 − 1.97304e7i −0.903546 − 0.903546i 0.0921952 − 0.995741i \(-0.470612\pi\)
−0.995741 + 0.0921952i \(0.970612\pi\)
\(6\) −1.94125e8 + 1.13152e8i −1.31067 + 0.763964i
\(7\) − 1.02336e9i − 1.36930i −0.728870 − 0.684652i \(-0.759954\pi\)
0.728870 − 0.684652i \(-0.240046\pi\)
\(8\) −2.17397e9 − 2.12066e9i −0.715829 − 0.698275i
\(9\) − 1.36141e10i − 1.30150i
\(10\) 2.03485e10 + 3.49103e10i 0.643477 + 1.10396i
\(11\) −7.32170e10 − 7.32170e10i −0.851116 − 0.851116i 0.139154 − 0.990271i \(-0.455562\pi\)
−0.990271 + 0.139154i \(0.955562\pi\)
\(12\) 3.13598e11 − 8.68160e10i 1.46208 − 0.404759i
\(13\) −4.28208e11 + 4.28208e11i −0.861489 + 0.861489i −0.991511 − 0.130022i \(-0.958495\pi\)
0.130022 + 0.991511i \(0.458495\pi\)
\(14\) −3.77640e11 + 1.43306e12i −0.348926 + 1.32410i
\(15\) −4.32941e12 −1.93852
\(16\) 2.26176e12 + 3.77190e12i 0.514264 + 0.857632i
\(17\) 6.62050e12 0.796484 0.398242 − 0.917280i \(-0.369620\pi\)
0.398242 + 0.917280i \(0.369620\pi\)
\(18\) −5.02387e12 + 1.90645e13i −0.331648 + 1.25853i
\(19\) −1.48987e13 + 1.48987e13i −0.557488 + 0.557488i −0.928591 − 0.371104i \(-0.878979\pi\)
0.371104 + 0.928591i \(0.378979\pi\)
\(20\) −1.56125e13 − 5.63956e13i −0.340923 − 1.23149i
\(21\) −1.12277e14 − 1.12277e14i −1.46889 − 1.46889i
\(22\) 7.55109e13 + 1.29548e14i 0.606138 + 1.03990i
\(23\) − 2.73784e14i − 1.37805i −0.724736 − 0.689027i \(-0.758038\pi\)
0.724736 − 0.689027i \(-0.241962\pi\)
\(24\) −4.71183e14 + 5.84907e12i −1.51695 + 0.0188308i
\(25\) 3.01738e14i 0.632790i
\(26\) 7.57658e14 − 4.41624e14i 1.05257 − 0.613525i
\(27\) −3.46012e14 − 3.46012e14i −0.323424 − 0.323424i
\(28\) 1.05766e15 − 1.86743e15i 0.674816 − 1.19148i
\(29\) 1.70828e15 − 1.70828e15i 0.754013 − 0.754013i −0.221212 − 0.975226i \(-0.571001\pi\)
0.975226 + 0.221212i \(0.0710014\pi\)
\(30\) 6.06268e15 + 1.59763e15i 1.87453 + 0.493974i
\(31\) 1.42001e15 0.311166 0.155583 − 0.987823i \(-0.450274\pi\)
0.155583 + 0.987823i \(0.450274\pi\)
\(32\) −1.77534e15 − 6.11661e15i −0.278746 − 0.960365i
\(33\) −1.60659e16 −1.82604
\(34\) −9.27100e15 − 2.44309e15i −0.770191 − 0.202960i
\(35\) −2.01913e16 + 2.01913e16i −1.23723 + 1.23723i
\(36\) 1.40703e16 − 2.48431e16i 0.641399 − 1.13248i
\(37\) −9.05710e15 − 9.05710e15i −0.309650 − 0.309650i 0.535124 − 0.844774i \(-0.320265\pi\)
−0.844774 + 0.535124i \(0.820265\pi\)
\(38\) 2.63613e16 − 1.53655e16i 0.681143 − 0.397025i
\(39\) 9.39612e16i 1.84829i
\(40\) 1.05186e15 + 8.47348e16i 0.0158609 + 1.27771i
\(41\) 5.53453e16i 0.643947i 0.946749 + 0.321973i \(0.104346\pi\)
−0.946749 + 0.321973i \(0.895654\pi\)
\(42\) 1.15795e17 + 1.98660e17i 1.04610 + 1.79471i
\(43\) −8.48059e16 − 8.48059e16i −0.598422 − 0.598422i 0.341471 − 0.939892i \(-0.389075\pi\)
−0.939892 + 0.341471i \(0.889075\pi\)
\(44\) −5.79360e16 − 2.09277e17i −0.321141 − 1.16003i
\(45\) −2.68611e17 + 2.68611e17i −1.17596 + 1.17596i
\(46\) −1.01032e17 + 3.83393e17i −0.351156 + 1.33256i
\(47\) 5.38091e17 1.49220 0.746102 − 0.665832i \(-0.231923\pi\)
0.746102 + 0.665832i \(0.231923\pi\)
\(48\) 6.61979e17 + 1.65685e17i 1.47167 + 0.368341i
\(49\) −4.88724e17 −0.874994
\(50\) 1.11347e17 − 4.22538e17i 0.161248 − 0.611900i
\(51\) 7.26363e17 − 7.26363e17i 0.854411 − 0.854411i
\(52\) −1.22395e18 + 3.38837e17i −1.17417 + 0.325055i
\(53\) −6.11834e17 − 6.11834e17i −0.480548 − 0.480548i 0.424758 − 0.905307i \(-0.360359\pi\)
−0.905307 + 0.424758i \(0.860359\pi\)
\(54\) 3.56853e17 + 6.12223e17i 0.230332 + 0.395162i
\(55\) 2.88920e18i 1.53805i
\(56\) −2.17021e18 + 2.22476e18i −0.956151 + 0.980188i
\(57\) 3.26920e18i 1.19607i
\(58\) −3.02257e18 + 1.76180e18i −0.921260 + 0.536984i
\(59\) 5.01134e18 + 5.01134e18i 1.27646 + 1.27646i 0.942634 + 0.333827i \(0.108340\pi\)
0.333827 + 0.942634i \(0.391660\pi\)
\(60\) −7.90031e18 − 4.47449e18i −1.68677 − 0.955334i
\(61\) 7.57981e18 − 7.57981e18i 1.36049 − 1.36049i 0.487198 − 0.873292i \(-0.338019\pi\)
0.873292 − 0.487198i \(-0.161981\pi\)
\(62\) −1.98850e18 − 5.24009e17i −0.300894 − 0.0792914i
\(63\) −1.39322e19 −1.78214
\(64\) 2.28955e17 + 9.22053e18i 0.0248233 + 0.999692i
\(65\) 1.68974e19 1.55679
\(66\) 2.24979e19 + 5.92863e18i 1.76575 + 0.465311i
\(67\) −1.45969e19 + 1.45969e19i −0.978307 + 0.978307i −0.999770 − 0.0214631i \(-0.993168\pi\)
0.0214631 + 0.999770i \(0.493168\pi\)
\(68\) 1.20811e19 + 6.84235e18i 0.693047 + 0.392520i
\(69\) −3.00381e19 − 3.00381e19i −1.47828 − 1.47828i
\(70\) 3.57259e19 − 2.08239e19i 1.51166 − 0.881115i
\(71\) 3.61689e19i 1.31863i 0.751867 + 0.659315i \(0.229154\pi\)
−0.751867 + 0.659315i \(0.770846\pi\)
\(72\) −2.88709e19 + 2.95967e19i −0.908803 + 0.931649i
\(73\) − 1.74764e19i − 0.475950i −0.971271 − 0.237975i \(-0.923516\pi\)
0.971271 − 0.237975i \(-0.0764836\pi\)
\(74\) 9.34086e18 + 1.60253e19i 0.220523 + 0.378333i
\(75\) 3.31049e19 + 3.31049e19i 0.678812 + 0.678812i
\(76\) −4.25851e19 + 1.17892e19i −0.759828 + 0.210350i
\(77\) −7.49276e19 + 7.49276e19i −1.16544 + 1.16544i
\(78\) 3.46735e19 − 1.31578e20i 0.470981 − 1.78727i
\(79\) 9.41739e18 0.111904 0.0559521 − 0.998433i \(-0.482181\pi\)
0.0559521 + 0.998433i \(0.482181\pi\)
\(80\) 2.97958e19 − 1.19046e20i 0.310248 − 1.23957i
\(81\) 6.64834e19 0.607604
\(82\) 2.04235e19 − 7.75027e19i 0.164091 − 0.622689i
\(83\) −1.22582e20 + 1.22582e20i −0.867178 + 0.867178i −0.992159 − 0.124981i \(-0.960113\pi\)
0.124981 + 0.992159i \(0.460113\pi\)
\(84\) −8.88442e19 − 3.20924e20i −0.554238 − 2.00203i
\(85\) −1.30625e20 − 1.30625e20i −0.719659 − 0.719659i
\(86\) 8.74629e19 + 1.50053e20i 0.426177 + 0.731157i
\(87\) − 3.74845e20i − 1.61770i
\(88\) 3.90333e18 + 3.14441e20i 0.0149406 + 1.20357i
\(89\) − 4.34193e20i − 1.47600i −0.674799 − 0.738002i \(-0.735770\pi\)
0.674799 − 0.738002i \(-0.264230\pi\)
\(90\) 4.75272e20 − 2.77027e20i 1.43680 − 0.837482i
\(91\) 4.38212e20 + 4.38212e20i 1.17964 + 1.17964i
\(92\) 2.82959e20 − 4.99602e20i 0.679127 − 1.19909i
\(93\) 1.55795e20 − 1.55795e20i 0.333797 − 0.333797i
\(94\) −7.53515e20 − 1.98566e20i −1.44294 − 0.380244i
\(95\) 5.87913e20 1.00743
\(96\) −8.65861e20 − 4.76299e20i −1.32923 − 0.731193i
\(97\) −4.29145e20 −0.590882 −0.295441 − 0.955361i \(-0.595467\pi\)
−0.295441 + 0.955361i \(0.595467\pi\)
\(98\) 6.84385e20 + 1.80349e20i 0.846109 + 0.222966i
\(99\) −9.96785e20 + 9.96785e20i −1.10772 + 1.10772i
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 16.22.e.a.5.3 ✓ 82
4.3 odd 2 64.22.e.a.49.6 82
16.3 odd 4 64.22.e.a.17.6 82
16.13 even 4 inner 16.22.e.a.13.3 yes 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.22.e.a.5.3 ✓ 82 1.1 even 1 trivial
16.22.e.a.13.3 yes 82 16.13 even 4 inner
64.22.e.a.17.6 82 16.3 odd 4
64.22.e.a.49.6 82 4.3 odd 2