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Properties

Label 16.22.e.a.13.3
Level $16$
Weight $22$
Character 16.13
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 13.3
Character \(\chi\) \(=\) 16.13
Dual form 16.22.e.a.5.3

$q$-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 \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1400.35 + 369.019i −0.966988 + 0.254820i
\(3\) 109714. + 109714.i 1.07273 + 1.07273i 0.997139 + 0.0755903i \(0.0240841\pi\)
0.0755903 + 0.997139i \(0.475916\pi\)
\(4\) 1.82480e6 − 1.03351e6i 0.870133 − 0.492816i
\(5\) −1.97304e7 + 1.97304e7i −0.903546 + 0.903546i −0.995741 − 0.0921952i \(-0.970612\pi\)
0.0921952 + 0.995741i \(0.470612\pi\)
\(6\) −1.94125e8 − 1.13152e8i −1.31067 − 0.763964i
\(7\) 1.02336e9i 1.36930i 0.728870 + 0.684652i \(0.240046\pi\)
−0.728870 + 0.684652i \(0.759954\pi\)
\(8\) −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 −0.990271 − 0.139154i \(-0.955562\pi\)
0.139154 + 0.990271i \(0.455562\pi\)
\(12\) 3.13598e11 + 8.68160e10i 1.46208 + 0.404759i
\(13\) −4.28208e11 − 4.28208e11i −0.861489 − 0.861489i 0.130022 − 0.991511i \(-0.458495\pi\)
−0.991511 + 0.130022i \(0.958495\pi\)
\(14\) −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 0.371104 − 0.928591i \(-0.378979\pi\)
−0.928591 + 0.371104i \(0.878979\pi\)
\(20\) −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 \(0.241962\pi\)
−0.724736 + 0.689027i \(0.758038\pi\)
\(24\) −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 0.975226 − 0.221212i \(-0.0710014\pi\)
−0.221212 + 0.975226i \(0.571001\pi\)
\(30\) 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 −0.844774 − 0.535124i \(-0.820265\pi\)
0.535124 + 0.844774i \(0.320265\pi\)
\(38\) 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 \(-0.895654\pi\)
0.946749 − 0.321973i \(-0.104346\pi\)
\(42\) 1.15795e17 − 1.98660e17i 1.04610 − 1.79471i
\(43\) −8.48059e16 + 8.48059e16i −0.598422 + 0.598422i −0.939892 − 0.341471i \(-0.889075\pi\)
0.341471 + 0.939892i \(0.389075\pi\)
\(44\) −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 −0.905307 − 0.424758i \(-0.860359\pi\)
0.424758 + 0.905307i \(0.360359\pi\)
\(54\) 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 0.333827 − 0.942634i \(-0.391660\pi\)
0.942634 − 0.333827i \(-0.108340\pi\)
\(60\) −7.90031e18 + 4.47449e18i −1.68677 + 0.955334i
\(61\) 7.57981e18 + 7.57981e18i 1.36049 + 1.36049i 0.873292 + 0.487198i \(0.161981\pi\)
0.487198 + 0.873292i \(0.338019\pi\)
\(62\) −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 0.0214631 − 0.999770i \(-0.493168\pi\)
−0.999770 + 0.0214631i \(0.993168\pi\)
\(68\) 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 \(-0.770846\pi\)
0.751867 − 0.659315i \(-0.229154\pi\)
\(72\) −2.88709e19 − 2.95967e19i −0.908803 − 0.931649i
\(73\) 1.74764e19i 0.475950i 0.971271 + 0.237975i \(0.0764836\pi\)
−0.971271 + 0.237975i \(0.923516\pi\)
\(74\) 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 0.124981 − 0.992159i \(-0.460113\pi\)
−0.992159 + 0.124981i \(0.960113\pi\)
\(84\) −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 \(0.264230\pi\)
−0.674799 + 0.738002i \(0.735770\pi\)
\(90\) 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.13.3 yes 82
4.3 odd 2 64.22.e.a.17.6 82
16.5 even 4 inner 16.22.e.a.5.3 ✓ 82
16.11 odd 4 64.22.e.a.49.6 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.22.e.a.5.3 ✓ 82 16.5 even 4 inner
16.22.e.a.13.3 yes 82 1.1 even 1 trivial
64.22.e.a.17.6 82 4.3 odd 2
64.22.e.a.49.6 82 16.11 odd 4