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Properties

Label 16.22.e.a.13.39
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 13.39
Character \(\chi\) \(=\) 16.13
Dual form 16.22.e.a.5.39

$q$-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+(1411.03 + 325.802i) q^{2} +(113962. + 113962. i) q^{3} +(1.88486e6 + 919432. i) q^{4} +(-1.51881e7 + 1.51881e7i) q^{5} +(1.23674e8 + 1.97932e8i) q^{6} +8.49060e8i q^{7} +(2.36004e9 + 1.91144e9i) q^{8} +1.55141e10i q^{9} +(-2.63791e10 + 1.64825e10i) q^{10} +(7.62819e10 - 7.62819e10i) q^{11} +(1.10022e11 + 3.19581e11i) q^{12} +(-1.54464e11 - 1.54464e11i) q^{13} +(-2.76625e11 + 1.19805e12i) q^{14} -3.46171e12 q^{15} +(2.70734e12 + 3.46600e12i) q^{16} -7.35734e12 q^{17} +(-5.05453e12 + 2.18909e13i) q^{18} +(2.38833e13 + 2.38833e13i) q^{19} +(-4.25917e13 + 1.46630e13i) q^{20} +(-9.67603e13 + 9.67603e13i) q^{21} +(1.32489e14 - 8.27832e13i) q^{22} -3.02861e14i q^{23} +(5.11235e13 + 4.86784e14i) q^{24} +1.54828e13i q^{25} +(-1.67629e14 - 2.68278e14i) q^{26} +(-5.75937e14 + 5.75937e14i) q^{27} +(-7.80653e14 + 1.60036e15i) q^{28} +(-2.24375e15 - 2.24375e15i) q^{29} +(-4.88458e15 - 1.12783e15i) q^{30} +1.96706e15 q^{31} +(2.69090e15 + 5.77268e15i) q^{32} +1.73864e16 q^{33} +(-1.03814e16 - 2.39703e15i) q^{34} +(-1.28956e16 - 1.28956e16i) q^{35} +(-1.42642e16 + 2.92420e16i) q^{36} +(1.01576e16 - 1.01576e16i) q^{37} +(2.59188e16 + 4.14812e16i) q^{38} -3.52059e16i q^{39} +(-6.48754e16 + 6.81341e15i) q^{40} +1.04510e17i q^{41} +(-1.68056e17 + 1.05007e17i) q^{42} +(1.15526e17 - 1.15526e17i) q^{43} +(2.13916e17 - 7.36445e16i) q^{44} +(-2.35630e17 - 2.35630e17i) q^{45} +(9.86726e16 - 4.27346e17i) q^{46} +2.44346e17 q^{47} +(-8.64583e16 + 7.03523e17i) q^{48} -1.62358e17 q^{49} +(-5.04432e15 + 2.18467e16i) q^{50} +(-8.38454e17 - 8.38454e17i) q^{51} +(-1.49124e17 - 4.33162e17i) q^{52} +(-1.54978e18 + 1.54978e18i) q^{53} +(-1.00031e18 + 6.25023e17i) q^{54} +2.31715e18i q^{55} +(-1.62292e18 + 2.00382e18i) q^{56} +5.44355e18i q^{57} +(-2.43498e18 - 3.89701e18i) q^{58} +(-3.95171e18 + 3.95171e18i) q^{59} +(-6.52484e18 - 3.18281e18i) q^{60} +(2.70500e18 + 2.70500e18i) q^{61} +(2.77559e18 + 6.40873e17i) q^{62} -1.31724e19 q^{63} +(1.91620e18 + 9.02213e18i) q^{64} +4.69201e18 q^{65} +(2.45327e19 + 5.66452e18i) q^{66} +(-1.17691e19 - 1.17691e19i) q^{67} +(-1.38675e19 - 6.76457e18i) q^{68} +(3.45145e19 - 3.45145e19i) q^{69} +(-1.39946e19 - 2.23975e19i) q^{70} -4.58056e19i q^{71} +(-2.96543e19 + 3.66140e19i) q^{72} +2.24666e19i q^{73} +(1.76421e19 - 1.10233e19i) q^{74} +(-1.76444e18 + 1.76444e18i) q^{75} +(2.30575e19 + 6.69756e19i) q^{76} +(6.47679e19 + 6.47679e19i) q^{77} +(1.14701e19 - 4.96766e19i) q^{78} +6.54965e19 q^{79} +(-9.37610e19 - 1.15226e19i) q^{80} +3.10139e19 q^{81} +(-3.40494e19 + 1.47466e20i) q^{82} +(5.01083e19 + 5.01083e19i) q^{83} +(-2.71344e20 + 9.34149e19i) q^{84} +(1.11744e20 - 1.11744e20i) q^{85} +(2.00650e20 - 1.25372e20i) q^{86} -5.11402e20i q^{87} +(3.25836e20 - 3.42203e19i) q^{88} +2.41666e20i q^{89} +(-2.55712e20 - 4.09249e20i) q^{90} +(1.31149e20 - 1.31149e20i) q^{91} +(2.78460e20 - 5.70850e20i) q^{92} +(2.24170e20 + 2.24170e20i) q^{93} +(3.44780e20 + 7.96084e19i) q^{94} -7.25481e20 q^{95} +(-3.51204e20 + 9.64524e20i) q^{96} +8.33355e20 q^{97} +(-2.29091e20 - 5.28964e19i) q^{98} +(1.18345e21 + 1.18345e21i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1411.03 + 325.802i 0.974364 + 0.224977i
\(3\) 113962. + 113962.i 1.11426 + 1.11426i 0.992568 + 0.121689i \(0.0388309\pi\)
0.121689 + 0.992568i \(0.461169\pi\)
\(4\) 1.88486e6 + 919432.i 0.898771 + 0.438419i
\(5\) −1.51881e7 + 1.51881e7i −0.695532 + 0.695532i −0.963444 − 0.267911i \(-0.913667\pi\)
0.267911 + 0.963444i \(0.413667\pi\)
\(6\) 1.23674e8 + 1.97932e8i 0.835010 + 1.33637i
\(7\) 8.49060e8i 1.13608i 0.823001 + 0.568040i \(0.192298\pi\)
−0.823001 + 0.568040i \(0.807702\pi\)
\(8\) 2.36004e9 + 1.91144e9i 0.777095 + 0.629383i
\(9\) 1.55141e10i 1.48314i
\(10\) −2.63791e10 + 1.64825e10i −0.834180 + 0.521223i
\(11\) 7.62819e10 − 7.62819e10i 0.886744 − 0.886744i −0.107465 − 0.994209i \(-0.534273\pi\)
0.994209 + 0.107465i \(0.0342734\pi\)
\(12\) 1.10022e11 + 3.19581e11i 0.512950 + 1.48997i
\(13\) −1.54464e11 − 1.54464e11i −0.310758 − 0.310758i 0.534445 − 0.845203i \(-0.320521\pi\)
−0.845203 + 0.534445i \(0.820521\pi\)
\(14\) −2.76625e11 + 1.19805e12i −0.255592 + 1.10696i
\(15\) −3.46171e12 −1.55000
\(16\) 2.70734e12 + 3.46600e12i 0.615577 + 0.788077i
\(17\) −7.35734e12 −0.885130 −0.442565 − 0.896736i \(-0.645931\pi\)
−0.442565 + 0.896736i \(0.645931\pi\)
\(18\) −5.05453e12 + 2.18909e13i −0.333672 + 1.44512i
\(19\) 2.38833e13 + 2.38833e13i 0.893678 + 0.893678i 0.994867 − 0.101189i \(-0.0322647\pi\)
−0.101189 + 0.994867i \(0.532265\pi\)
\(20\) −4.25917e13 + 1.46630e13i −0.930059 + 0.320189i
\(21\) −9.67603e13 + 9.67603e13i −1.26589 + 1.26589i
\(22\) 1.32489e14 − 8.27832e13i 1.06351 − 0.664514i
\(23\) − 3.02861e14i − 1.52441i −0.647338 − 0.762203i \(-0.724118\pi\)
0.647338 − 0.762203i \(-0.275882\pi\)
\(24\) 5.11235e13 + 4.86784e14i 0.164590 + 1.56718i
\(25\) 1.54828e13i 0.0324698i
\(26\) −1.67629e14 − 2.68278e14i −0.232878 − 0.372704i
\(27\) −5.75937e14 + 5.75937e14i −0.538339 + 0.538339i
\(28\) −7.80653e14 + 1.60036e15i −0.498080 + 1.02108i
\(29\) −2.24375e15 − 2.24375e15i −0.990365 − 0.990365i 0.00958900 − 0.999954i \(-0.496948\pi\)
−0.999954 + 0.00958900i \(0.996948\pi\)
\(30\) −4.88458e15 − 1.12783e15i −1.51027 − 0.348715i
\(31\) 1.96706e15 0.431043 0.215522 − 0.976499i \(-0.430855\pi\)
0.215522 + 0.976499i \(0.430855\pi\)
\(32\) 2.69090e15 + 5.77268e15i 0.422497 + 0.906364i
\(33\) 1.73864e16 1.97612
\(34\) −1.03814e16 − 2.39703e15i −0.862439 − 0.199134i
\(35\) −1.28956e16 − 1.28956e16i −0.790181 − 0.790181i
\(36\) −1.42642e16 + 2.92420e16i −0.650236 + 1.33300i
\(37\) 1.01576e16 − 1.01576e16i 0.347275 − 0.347275i −0.511819 − 0.859094i \(-0.671028\pi\)
0.859094 + 0.511819i \(0.171028\pi\)
\(38\) 2.59188e16 + 4.14812e16i 0.669711 + 1.07182i
\(39\) − 3.52059e16i − 0.692528i
\(40\) −6.48754e16 + 6.81341e15i −0.978251 + 0.102739i
\(41\) 1.04510e17i 1.21598i 0.793945 + 0.607989i \(0.208024\pi\)
−0.793945 + 0.607989i \(0.791976\pi\)
\(42\) −1.68056e17 + 1.05007e17i −1.51823 + 0.948638i
\(43\) 1.15526e17 − 1.15526e17i 0.815195 − 0.815195i −0.170212 − 0.985407i \(-0.554445\pi\)
0.985407 + 0.170212i \(0.0544453\pi\)
\(44\) 2.13916e17 − 7.36445e16i 1.18574 − 0.408214i
\(45\) −2.35630e17 − 2.35630e17i −1.03157 − 1.03157i
\(46\) 9.86726e16 − 4.27346e17i 0.342957 − 1.48533i
\(47\) 2.44346e17 0.677607 0.338803 − 0.940857i \(-0.389978\pi\)
0.338803 + 0.940857i \(0.389978\pi\)
\(48\) −8.64583e16 + 7.03523e17i −0.192209 + 1.56403i
\(49\) −1.62358e17 −0.290679
\(50\) −5.04432e15 + 2.18467e16i −0.00730496 + 0.0316374i
\(51\) −8.38454e17 − 8.38454e17i −0.986262 − 0.986262i
\(52\) −1.49124e17 − 4.33162e17i −0.143058 − 0.415542i
\(53\) −1.54978e18 + 1.54978e18i −1.21723 + 1.21723i −0.248636 + 0.968597i \(0.579982\pi\)
−0.968597 + 0.248636i \(0.920018\pi\)
\(54\) −1.00031e18 + 6.25023e17i −0.645652 + 0.403424i
\(55\) 2.31715e18i 1.23352i
\(56\) −1.62292e18 + 2.00382e18i −0.715030 + 0.882843i
\(57\) 5.44355e18i 1.99157i
\(58\) −2.43498e18 − 3.89701e18i −0.742167 − 1.18779i
\(59\) −3.95171e18 + 3.95171e18i −1.00656 + 1.00656i −0.00657896 + 0.999978i \(0.502094\pi\)
−0.999978 + 0.00657896i \(0.997906\pi\)
\(60\) −6.52484e18 − 3.18281e18i −1.39310 − 0.679551i
\(61\) 2.70500e18 + 2.70500e18i 0.485517 + 0.485517i 0.906888 − 0.421372i \(-0.138451\pi\)
−0.421372 + 0.906888i \(0.638451\pi\)
\(62\) 2.77559e18 + 6.40873e17i 0.419993 + 0.0969749i
\(63\) −1.31724e19 −1.68496
\(64\) 1.91620e18 + 9.02213e18i 0.207754 + 0.978181i
\(65\) 4.69201e18 0.432284
\(66\) 2.45327e19 + 5.66452e18i 1.92546 + 0.444582i
\(67\) −1.17691e19 − 1.17691e19i −0.788783 − 0.788783i 0.192512 − 0.981295i \(-0.438337\pi\)
−0.981295 + 0.192512i \(0.938337\pi\)
\(68\) −1.38675e19 − 6.76457e18i −0.795529 − 0.388058i
\(69\) 3.45145e19 − 3.45145e19i 1.69858 − 1.69858i
\(70\) −1.39946e19 − 2.23975e19i −0.592151 − 0.947696i
\(71\) − 4.58056e19i − 1.66996i −0.550279 − 0.834981i \(-0.685479\pi\)
0.550279 − 0.834981i \(-0.314521\pi\)
\(72\) −2.96543e19 + 3.66140e19i −0.933461 + 1.15254i
\(73\) 2.24666e19i 0.611853i 0.952055 + 0.305927i \(0.0989662\pi\)
−0.952055 + 0.305927i \(0.901034\pi\)
\(74\) 1.76421e19 − 1.10233e19i 0.416501 − 0.260243i
\(75\) −1.76444e18 + 1.76444e18i −0.0361797 + 0.0361797i
\(76\) 2.30575e19 + 6.69756e19i 0.411406 + 1.19502i
\(77\) 6.47679e19 + 6.47679e19i 1.00741 + 1.00741i
\(78\) 1.14701e19 − 4.96766e19i 0.155803 − 0.674774i
\(79\) 6.54965e19 0.778277 0.389138 − 0.921179i \(-0.372773\pi\)
0.389138 + 0.921179i \(0.372773\pi\)
\(80\) −9.37610e19 − 1.15226e19i −0.976286 − 0.119979i
\(81\) 3.10139e19 0.283442
\(82\) −3.40494e19 + 1.47466e20i −0.273567 + 1.18481i
\(83\) 5.01083e19 + 5.01083e19i 0.354478 + 0.354478i 0.861773 − 0.507295i \(-0.169354\pi\)
−0.507295 + 0.861773i \(0.669354\pi\)
\(84\) −2.71344e20 + 9.34149e19i −1.69273 + 0.582752i
\(85\) 1.11744e20 − 1.11744e20i 0.615637 − 0.615637i
\(86\) 2.00650e20 − 1.25372e20i 0.977697 − 0.610897i
\(87\) − 5.11402e20i − 2.20704i
\(88\) 3.25836e20 − 3.42203e19i 1.24719 − 0.130983i
\(89\) 2.41666e20i 0.821526i 0.911742 + 0.410763i \(0.134738\pi\)
−0.911742 + 0.410763i \(0.865262\pi\)
\(90\) −2.55712e20 − 4.09249e20i −0.773045 − 1.23720i
\(91\) 1.31149e20 − 1.31149e20i 0.353046 − 0.353046i
\(92\) 2.78460e20 − 5.70850e20i 0.668329 − 1.37009i
\(93\) 2.24170e20 + 2.24170e20i 0.480293 + 0.480293i
\(94\) 3.44780e20 + 7.96084e19i 0.660236 + 0.152446i
\(95\) −7.25481e20 −1.24316
\(96\) −3.51204e20 + 9.64524e20i −0.539153 + 1.48069i
\(97\) 8.33355e20 1.14743 0.573716 − 0.819054i \(-0.305501\pi\)
0.573716 + 0.819054i \(0.305501\pi\)
\(98\) −2.29091e20 − 5.28964e19i −0.283227 − 0.0653962i
\(99\) 1.18345e21 + 1.18345e21i 1.31516 + 1.31516i
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.13.39 yes 82
4.3 odd 2 64.22.e.a.17.5 82
16.5 even 4 inner 16.22.e.a.5.39 ✓ 82
16.11 odd 4 64.22.e.a.49.5 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.22.e.a.5.39 ✓ 82 16.5 even 4 inner
16.22.e.a.13.39 yes 82 1.1 even 1 trivial
64.22.e.a.17.5 82 4.3 odd 2
64.22.e.a.49.5 82 16.11 odd 4