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Properties

Label 16.22.e.a.13.2
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 13.2
Character \(\chi\) \(=\) 16.13
Dual form 16.22.e.a.5.2

$q$-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+(-1428.70 - 236.579i) q^{2} +(-28799.9 - 28799.9i) q^{3} +(1.98521e6 + 676002. i) q^{4} +(2.59531e7 - 2.59531e7i) q^{5} +(3.43329e7 + 4.79599e7i) q^{6} -2.01795e8i q^{7} +(-2.67634e9 - 1.43546e9i) q^{8} -8.80148e9i q^{9} +(-4.32191e10 + 3.09392e10i) q^{10} +(-4.46963e10 + 4.46963e10i) q^{11} +(-3.77051e10 - 7.66427e10i) q^{12} +(-5.15894e11 - 5.15894e11i) q^{13} +(-4.77407e10 + 2.88305e11i) q^{14} -1.49489e12 q^{15} +(3.48409e12 + 2.68401e12i) q^{16} +1.15984e13 q^{17} +(-2.08225e12 + 1.25747e13i) q^{18} +(-2.59560e13 - 2.59560e13i) q^{19} +(6.90666e13 - 3.39780e13i) q^{20} +(-5.81169e12 + 5.81169e12i) q^{21} +(7.44317e13 - 5.32833e13i) q^{22} -1.15953e14i q^{23} +(3.57372e13 + 1.18420e14i) q^{24} -8.70285e14i q^{25} +(6.15007e14 + 8.59107e14i) q^{26} +(-5.54739e14 + 5.54739e14i) q^{27} +(1.36414e14 - 4.00607e14i) q^{28} +(1.46551e15 + 1.46551e15i) q^{29} +(2.13575e15 + 3.53661e14i) q^{30} +5.11550e15 q^{31} +(-4.34273e15 - 4.65891e15i) q^{32} +2.57450e15 q^{33} +(-1.65706e16 - 2.74395e15i) q^{34} +(-5.23721e15 - 5.23721e15i) q^{35} +(5.94982e15 - 1.74728e16i) q^{36} +(1.81827e16 - 1.81827e16i) q^{37} +(3.09426e16 + 4.32239e16i) q^{38} +2.97154e16i q^{39} +(-1.06714e17 + 3.22046e16i) q^{40} -1.73804e16i q^{41} +(9.67808e15 - 6.92823e15i) q^{42} +(-1.70734e17 + 1.70734e17i) q^{43} +(-1.18946e17 + 5.85168e16i) q^{44} +(-2.28425e17 - 2.28425e17i) q^{45} +(-2.74321e16 + 1.65662e17i) q^{46} -5.78278e17 q^{47} +(-2.30421e16 - 1.77641e17i) q^{48} +5.17824e17 q^{49} +(-2.05892e17 + 1.24338e18i) q^{50} +(-3.34033e17 - 3.34033e17i) q^{51} +(-6.75413e17 - 1.37290e18i) q^{52} +(-6.56047e16 + 6.56047e16i) q^{53} +(9.23795e17 - 6.61315e17i) q^{54} +2.32001e18i q^{55} +(-2.89670e17 + 5.40074e17i) q^{56} +1.49506e18i q^{57} +(-1.74706e18 - 2.44048e18i) q^{58} +(4.63515e17 - 4.63515e17i) q^{59} +(-2.96768e18 - 1.01055e18i) q^{60} +(-1.42884e18 - 1.42884e18i) q^{61} +(-7.30851e18 - 1.21022e18i) q^{62} -1.77610e18 q^{63} +(5.10226e18 + 7.68359e18i) q^{64} -2.67780e19 q^{65} +(-3.67818e18 - 6.09073e17i) q^{66} +(8.18075e18 + 8.18075e18i) q^{67} +(2.30253e19 + 7.84055e18i) q^{68} +(-3.33944e18 + 3.33944e18i) q^{69} +(6.24338e18 + 8.72141e18i) q^{70} -4.65145e19i q^{71} +(-1.26342e19 + 2.35558e19i) q^{72} +1.37530e19i q^{73} +(-3.02792e19 + 2.16759e19i) q^{74} +(-2.50641e19 + 2.50641e19i) q^{75} +(-3.39818e19 - 6.90744e19i) q^{76} +(9.01950e18 + 9.01950e18i) q^{77} +(7.03005e18 - 4.24543e19i) q^{78} +1.96531e19 q^{79} +(1.60081e20 - 2.07644e19i) q^{80} -6.01138e19 q^{81} +(-4.11185e18 + 2.48314e19i) q^{82} +(1.47960e20 + 1.47960e20i) q^{83} +(-1.54662e19 + 7.60872e18i) q^{84} +(3.01014e20 - 3.01014e20i) q^{85} +(2.84319e20 - 2.03535e20i) q^{86} -8.44130e19i q^{87} +(1.83782e20 - 5.54627e19i) q^{88} +1.13078e20i q^{89} +(2.72311e20 + 3.80392e20i) q^{90} +(-1.04105e20 + 1.04105e20i) q^{91} +(7.83845e19 - 2.30191e20i) q^{92} +(-1.47326e20 - 1.47326e20i) q^{93} +(8.26186e20 + 1.36809e20i) q^{94} -1.34727e21 q^{95} +(-9.10599e18 + 2.59247e20i) q^{96} -8.64580e20 q^{97} +(-7.39816e20 - 1.22507e20i) q^{98} +(3.93393e20 + 3.93393e20i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1428.70 − 236.579i −0.986566 − 0.163366i
\(3\) −28799.9 − 28799.9i −0.281590 − 0.281590i 0.552153 − 0.833743i \(-0.313807\pi\)
−0.833743 + 0.552153i \(0.813807\pi\)
\(4\) 1.98521e6 + 676002.i 0.946623 + 0.322343i
\(5\) 2.59531e7 − 2.59531e7i 1.18851 − 1.18851i 0.211033 − 0.977479i \(-0.432317\pi\)
0.977479 − 0.211033i \(-0.0676827\pi\)
\(6\) 3.43329e7 + 4.79599e7i 0.231805 + 0.323810i
\(7\) − 2.01795e8i − 0.270011i −0.990845 − 0.135006i \(-0.956895\pi\)
0.990845 − 0.135006i \(-0.0431053\pi\)
\(8\) −2.67634e9 − 1.43546e9i −0.881246 − 0.472659i
\(9\) − 8.80148e9i − 0.841414i
\(10\) −4.32191e10 + 3.09392e10i −1.36671 + 0.978382i
\(11\) −4.46963e10 + 4.46963e10i −0.519575 + 0.519575i −0.917443 − 0.397868i \(-0.869750\pi\)
0.397868 + 0.917443i \(0.369750\pi\)
\(12\) −3.77051e10 − 7.66427e10i −0.175791 − 0.357329i
\(13\) −5.15894e11 − 5.15894e11i −1.03790 − 1.03790i −0.999253 − 0.0386458i \(-0.987696\pi\)
−0.0386458 − 0.999253i \(-0.512304\pi\)
\(14\) −4.77407e10 + 2.88305e11i −0.0441107 + 0.266384i
\(15\) −1.49489e12 −0.669347
\(16\) 3.48409e12 + 2.68401e12i 0.792190 + 0.610274i
\(17\) 1.15984e13 1.39536 0.697678 − 0.716412i \(-0.254217\pi\)
0.697678 + 0.716412i \(0.254217\pi\)
\(18\) −2.08225e12 + 1.25747e13i −0.137459 + 0.830110i
\(19\) −2.59560e13 − 2.59560e13i −0.971235 − 0.971235i 0.0283626 − 0.999598i \(-0.490971\pi\)
−0.999598 + 0.0283626i \(0.990971\pi\)
\(20\) 6.90666e13 − 3.39780e13i 1.50818 − 0.741964i
\(21\) −5.81169e12 + 5.81169e12i −0.0760326 + 0.0760326i
\(22\) 7.44317e13 − 5.32833e13i 0.597475 − 0.427713i
\(23\) − 1.15953e14i − 0.583633i −0.956474 − 0.291817i \(-0.905740\pi\)
0.956474 − 0.291817i \(-0.0942597\pi\)
\(24\) 3.57372e13 + 1.18420e14i 0.115054 + 0.381246i
\(25\) − 8.70285e14i − 1.82512i
\(26\) 6.15007e14 + 8.59107e14i 0.854398 + 1.19351i
\(27\) −5.54739e14 + 5.54739e14i −0.518524 + 0.518524i
\(28\) 1.36414e14 − 4.00607e14i 0.0870362 − 0.255599i
\(29\) 1.46551e15 + 1.46551e15i 0.646859 + 0.646859i 0.952232 − 0.305374i \(-0.0987815\pi\)
−0.305374 + 0.952232i \(0.598781\pi\)
\(30\) 2.13575e15 + 3.53661e14i 0.660355 + 0.109349i
\(31\) 5.11550e15 1.12096 0.560480 − 0.828168i \(-0.310617\pi\)
0.560480 + 0.828168i \(0.310617\pi\)
\(32\) −4.34273e15 − 4.65891e15i −0.681849 − 0.731493i
\(33\) 2.57450e15 0.292614
\(34\) −1.65706e16 − 2.74395e15i −1.37661 − 0.227954i
\(35\) −5.23721e15 − 5.23721e15i −0.320912 − 0.320912i
\(36\) 5.94982e15 − 1.74728e16i 0.271224 − 0.796502i
\(37\) 1.81827e16 − 1.81827e16i 0.621640 − 0.621640i −0.324310 − 0.945951i \(-0.605132\pi\)
0.945951 + 0.324310i \(0.105132\pi\)
\(38\) 3.09426e16 + 4.32239e16i 0.799520 + 1.11685i
\(39\) 2.97154e16i 0.584525i
\(40\) −1.06714e17 + 3.22046e16i −1.60913 + 0.485611i
\(41\) − 1.73804e16i − 0.202223i −0.994875 − 0.101111i \(-0.967760\pi\)
0.994875 − 0.101111i \(-0.0322398\pi\)
\(42\) 9.67808e15 − 6.92823e15i 0.0874323 − 0.0625900i
\(43\) −1.70734e17 + 1.70734e17i −1.20476 + 1.20476i −0.232057 + 0.972702i \(0.574546\pi\)
−0.972702 + 0.232057i \(0.925454\pi\)
\(44\) −1.18946e17 + 5.85168e16i −0.659322 + 0.324360i
\(45\) −2.28425e17 − 2.28425e17i −1.00003 − 1.00003i
\(46\) −2.74321e16 + 1.65662e17i −0.0953459 + 0.575792i
\(47\) −5.78278e17 −1.60365 −0.801825 − 0.597559i \(-0.796137\pi\)
−0.801825 + 0.597559i \(0.796137\pi\)
\(48\) −2.30421e16 − 1.77641e17i −0.0512257 − 0.394921i
\(49\) 5.17824e17 0.927094
\(50\) −2.05892e17 + 1.24338e18i −0.298163 + 1.80060i
\(51\) −3.34033e17 − 3.34033e17i −0.392919 − 0.392919i
\(52\) −6.75413e17 − 1.37290e18i −0.647940 − 1.31706i
\(53\) −6.56047e16 + 6.56047e16i −0.0515274 + 0.0515274i −0.732401 − 0.680874i \(-0.761600\pi\)
0.680874 + 0.732401i \(0.261600\pi\)
\(54\) 9.23795e17 − 6.61315e17i 0.596268 − 0.426849i
\(55\) 2.32001e18i 1.23504i
\(56\) −2.89670e17 + 5.40074e17i −0.127623 + 0.237946i
\(57\) 1.49506e18i 0.546981i
\(58\) −1.74706e18 − 2.44048e18i −0.532493 − 0.743843i
\(59\) 4.63515e17 − 4.63515e17i 0.118064 − 0.118064i −0.645606 − 0.763670i \(-0.723395\pi\)
0.763670 + 0.645606i \(0.223395\pi\)
\(60\) −2.96768e18 − 1.01055e18i −0.633619 − 0.215759i
\(61\) −1.42884e18 − 1.42884e18i −0.256460 − 0.256460i 0.567153 − 0.823613i \(-0.308045\pi\)
−0.823613 + 0.567153i \(0.808045\pi\)
\(62\) −7.30851e18 − 1.21022e18i −1.10590 − 0.183127i
\(63\) −1.77610e18 −0.227191
\(64\) 5.10226e18 + 7.68359e18i 0.553188 + 0.833057i
\(65\) −2.67780e19 −2.46711
\(66\) −3.67818e18 − 6.09073e17i −0.288683 − 0.0478033i
\(67\) 8.18075e18 + 8.18075e18i 0.548286 + 0.548286i 0.925945 − 0.377658i \(-0.123271\pi\)
−0.377658 + 0.925945i \(0.623271\pi\)
\(68\) 2.30253e19 + 7.84055e18i 1.32088 + 0.449783i
\(69\) −3.33944e18 + 3.33944e18i −0.164345 + 0.164345i
\(70\) 6.24338e18 + 8.72141e18i 0.264174 + 0.369026i
\(71\) − 4.65145e19i − 1.69580i −0.530154 − 0.847901i \(-0.677866\pi\)
0.530154 − 0.847901i \(-0.322134\pi\)
\(72\) −1.26342e19 + 2.35558e19i −0.397701 + 0.741492i
\(73\) 1.37530e19i 0.374547i 0.982308 + 0.187274i \(0.0599651\pi\)
−0.982308 + 0.187274i \(0.940035\pi\)
\(74\) −3.02792e19 + 2.16759e19i −0.714844 + 0.511734i
\(75\) −2.50641e19 + 2.50641e19i −0.513936 + 0.513936i
\(76\) −3.39818e19 − 6.90744e19i −0.606323 − 1.23246i
\(77\) 9.01950e18 + 9.01950e18i 0.140291 + 0.140291i
\(78\) 7.03005e18 − 4.24543e19i 0.0954916 − 0.576672i
\(79\) 1.96531e19 0.233532 0.116766 − 0.993159i \(-0.462747\pi\)
0.116766 + 0.993159i \(0.462747\pi\)
\(80\) 1.60081e20 − 2.07644e19i 1.66685 − 0.216209i
\(81\) −6.01138e19 −0.549391
\(82\) −4.11185e18 + 2.48314e19i −0.0330363 + 0.199506i
\(83\) 1.47960e20 + 1.47960e20i 1.04670 + 1.04670i 0.998855 + 0.0478502i \(0.0152370\pi\)
0.0478502 + 0.998855i \(0.484763\pi\)
\(84\) −1.54662e19 + 7.60872e18i −0.0964828 + 0.0474656i
\(85\) 3.01014e20 − 3.01014e20i 1.65840 − 1.65840i
\(86\) 2.84319e20 − 2.03535e20i 1.38539 − 0.991757i
\(87\) − 8.44130e19i − 0.364298i
\(88\) 1.83782e20 − 5.54627e19i 0.703454 − 0.212291i
\(89\) 1.13078e20i 0.384401i 0.981356 + 0.192200i \(0.0615624\pi\)
−0.981356 + 0.192200i \(0.938438\pi\)
\(90\) 2.72311e20 + 3.80392e20i 0.823224 + 1.14997i
\(91\) −1.04105e20 + 1.04105e20i −0.280244 + 0.280244i
\(92\) 7.83845e19 − 2.30191e20i 0.188130 − 0.552480i
\(93\) −1.47326e20 − 1.47326e20i −0.315651 − 0.315651i
\(94\) 8.26186e20 + 1.36809e20i 1.58211 + 0.261982i
\(95\) −1.34727e21 −2.30865
\(96\) −9.10599e18 + 2.59247e20i −0.0139791 + 0.397984i
\(97\) −8.64580e20 −1.19042 −0.595212 − 0.803569i \(-0.702932\pi\)
−0.595212 + 0.803569i \(0.702932\pi\)
\(98\) −7.39816e20 − 1.22507e20i −0.914639 − 0.151456i
\(99\) 3.93393e20 + 3.93393e20i 0.437177 + 0.437177i
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.13.2 yes 82
4.3 odd 2 64.22.e.a.17.26 82
16.5 even 4 inner 16.22.e.a.5.2 ✓ 82
16.11 odd 4 64.22.e.a.49.26 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.22.e.a.5.2 ✓ 82 16.5 even 4 inner
16.22.e.a.13.2 yes 82 1.1 even 1 trivial
64.22.e.a.17.26 82 4.3 odd 2
64.22.e.a.49.26 82 16.11 odd 4