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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [185,2,Mod(21,185)] 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("185.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.w (of order \(18\), degree \(6\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 176.6
Character \(\chi\) \(=\) 185.176
Dual form 185.2.w.a.41.6

$q$-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+(0.105693 - 0.0186364i) q^{2} +(0.492384 - 2.79245i) q^{3} +(-1.86856 + 0.680101i) q^{4} +(0.642788 - 0.766044i) q^{5} -0.304317i q^{6} +(-3.01176 - 2.52717i) q^{7} +(-0.370707 + 0.214028i) q^{8} +(-4.73626 - 1.72386i) q^{9} +(0.0536615 - 0.0929444i) q^{10} +(2.81064 + 4.86818i) q^{11} +(0.979097 + 5.55274i) q^{12} +(-0.765019 - 2.10187i) q^{13} +(-0.365418 - 0.210974i) q^{14} +(-1.82264 - 2.17214i) q^{15} +(3.01134 - 2.52681i) q^{16} +(2.08588 - 5.73090i) q^{17} +(-0.532714 - 0.0939318i) q^{18} +(3.23982 + 0.571267i) q^{19} +(-0.680101 + 1.86856i) q^{20} +(-8.53993 + 7.16586i) q^{21} +(0.387789 + 0.462149i) q^{22} +(-1.91949 - 1.10822i) q^{23} +(0.415132 + 1.14057i) q^{24} +(-0.173648 - 0.984808i) q^{25} +(-0.120028 - 0.207895i) q^{26} +(-2.89255 + 5.01005i) q^{27} +(7.34639 + 2.67387i) q^{28} +(1.38356 - 0.798797i) q^{29} +(-0.233121 - 0.195611i) q^{30} -2.45822i q^{31} +(0.821483 - 0.979005i) q^{32} +(14.9781 - 5.45157i) q^{33} +(0.113658 - 0.644587i) q^{34} +(-3.87185 + 0.682711i) q^{35} +10.0224 q^{36} +(5.63641 + 2.28710i) q^{37} +0.353071 q^{38} +(-6.24606 + 1.10135i) q^{39} +(-0.0743311 + 0.421553i) q^{40} +(8.92828 - 3.24963i) q^{41} +(-0.769061 + 0.916531i) q^{42} +12.5847i q^{43} +(-8.56271 - 7.18497i) q^{44} +(-4.36496 + 2.52011i) q^{45} +(-0.223528 - 0.0813577i) q^{46} +(-3.68051 + 6.37482i) q^{47} +(-5.57326 - 9.65318i) q^{48} +(1.46859 + 8.32880i) q^{49} +(-0.0367066 - 0.100851i) q^{50} +(-14.9762 - 8.64652i) q^{51} +(2.85897 + 3.40719i) q^{52} +(-2.06899 + 1.73609i) q^{53} +(-0.212352 + 0.583431i) q^{54} +(5.53589 + 0.976126i) q^{55} +(1.65737 + 0.292238i) q^{56} +(3.19047 - 8.76575i) q^{57} +(0.131345 - 0.110211i) q^{58} +(-0.823360 - 0.981242i) q^{59} +(4.88299 + 2.81920i) q^{60} +(-0.313577 - 0.861547i) q^{61} +(-0.0458124 - 0.259815i) q^{62} +(9.90800 + 17.1612i) q^{63} +(-3.86244 + 6.68995i) q^{64} +(-2.10187 - 0.765019i) q^{65} +(1.48147 - 0.855328i) q^{66} +(-2.06513 - 1.73285i) q^{67} +12.1272i q^{68} +(-4.03976 + 4.81440i) q^{69} +(-0.396502 + 0.144315i) q^{70} +(1.38565 - 7.85840i) q^{71} +(2.12472 - 0.374645i) q^{72} -5.34759 q^{73} +(0.638350 + 0.136687i) q^{74} -2.83553 q^{75} +(-6.44232 + 1.13595i) q^{76} +(3.83771 - 21.7648i) q^{77} +(-0.639636 + 0.232809i) q^{78} +(-5.87032 + 6.99598i) q^{79} -3.93102i q^{80} +(0.982962 + 0.824803i) q^{81} +(0.883091 - 0.509853i) q^{82} +(2.11925 + 0.771346i) q^{83} +(11.0839 - 19.1979i) q^{84} +(-3.04935 - 5.28163i) q^{85} +(0.234535 + 1.33011i) q^{86} +(-1.54936 - 4.25683i) q^{87} +(-2.08385 - 1.20311i) q^{88} +(-0.660110 - 0.786689i) q^{89} +(-0.414378 + 0.347704i) q^{90} +(-3.00773 + 8.26367i) q^{91} +(4.34037 + 0.765325i) q^{92} +(-6.86445 - 1.21039i) q^{93} +(-0.270198 + 0.742363i) q^{94} +(2.52013 - 2.11464i) q^{95} +(-2.32934 - 2.77600i) q^{96} +(8.14173 + 4.70063i) q^{97} +(0.310438 + 0.852923i) q^{98} +(-4.91989 - 27.9021i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 6 q^{2} - 6 q^{4} - 6 q^{7} - 18 q^{8} + 6 q^{10} + 30 q^{12} - 30 q^{13} - 18 q^{14} - 18 q^{16} - 30 q^{18} - 18 q^{19} + 30 q^{22} - 30 q^{24} - 24 q^{27} + 30 q^{28} + 18 q^{29} + 24 q^{30} - 36 q^{32}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.105693 − 0.0186364i 0.0747359 − 0.0131780i −0.136155 − 0.990688i \(-0.543475\pi\)
0.210891 + 0.977510i \(0.432363\pi\)
\(3\) 0.492384 − 2.79245i 0.284278 − 1.61222i −0.423574 − 0.905861i \(-0.639225\pi\)
0.707852 − 0.706360i \(-0.249664\pi\)
\(4\) −1.86856 + 0.680101i −0.934281 + 0.340050i
\(5\) 0.642788 − 0.766044i 0.287463 − 0.342585i
\(6\) − 0.304317i − 0.124237i
\(7\) −3.01176 − 2.52717i −1.13834 − 0.955180i −0.138956 − 0.990299i \(-0.544375\pi\)
−0.999383 + 0.0351189i \(0.988819\pi\)
\(8\) −0.370707 + 0.214028i −0.131065 + 0.0756703i
\(9\) −4.73626 − 1.72386i −1.57875 − 0.574619i
\(10\) 0.0536615 − 0.0929444i 0.0169693 − 0.0293916i
\(11\) 2.81064 + 4.86818i 0.847441 + 1.46781i 0.883485 + 0.468460i \(0.155191\pi\)
−0.0360439 + 0.999350i \(0.511476\pi\)
\(12\) 0.979097 + 5.55274i 0.282641 + 1.60294i
\(13\) −0.765019 − 2.10187i −0.212178 − 0.582954i 0.787255 − 0.616628i \(-0.211502\pi\)
−0.999433 + 0.0336733i \(0.989279\pi\)
\(14\) −0.365418 − 0.210974i −0.0976621 − 0.0563852i
\(15\) −1.82264 − 2.17214i −0.470604 − 0.560844i
\(16\) 3.01134 − 2.52681i 0.752835 − 0.631703i
\(17\) 2.08588 − 5.73090i 0.505900 − 1.38995i −0.379532 − 0.925179i \(-0.623915\pi\)
0.885432 − 0.464769i \(-0.153863\pi\)
\(18\) −0.532714 − 0.0939318i −0.125562 − 0.0221399i
\(19\) 3.23982 + 0.571267i 0.743265 + 0.131058i 0.532442 − 0.846467i \(-0.321274\pi\)
0.210823 + 0.977524i \(0.432386\pi\)
\(20\) −0.680101 + 1.86856i −0.152075 + 0.417823i
\(21\) −8.53993 + 7.16586i −1.86357 + 1.56372i
\(22\) 0.387789 + 0.462149i 0.0826770 + 0.0985306i
\(23\) −1.91949 − 1.10822i −0.400240 − 0.231079i 0.286347 − 0.958126i \(-0.407559\pi\)
−0.686588 + 0.727047i \(0.740892\pi\)
\(24\) 0.415132 + 1.14057i 0.0847384 + 0.232817i
\(25\) −0.173648 − 0.984808i −0.0347296 − 0.196962i
\(26\) −0.120028 − 0.207895i −0.0235395 − 0.0407716i
\(27\) −2.89255 + 5.01005i −0.556672 + 0.964184i
\(28\) 7.34639 + 2.67387i 1.38834 + 0.505313i
\(29\) 1.38356 − 0.798797i 0.256920 − 0.148333i −0.366009 − 0.930611i \(-0.619276\pi\)
0.622929 + 0.782279i \(0.285943\pi\)
\(30\) −0.233121 − 0.195611i −0.0425618 − 0.0357136i
\(31\) − 2.45822i − 0.441509i −0.975329 − 0.220755i \(-0.929148\pi\)
0.975329 − 0.220755i \(-0.0708520\pi\)
\(32\) 0.821483 − 0.979005i 0.145219 − 0.173065i
\(33\) 14.9781 − 5.45157i 2.60735 − 0.948996i
\(34\) 0.113658 − 0.644587i 0.0194922 − 0.110546i
\(35\) −3.87185 + 0.682711i −0.654461 + 0.115399i
\(36\) 10.0224 1.67040
\(37\) 5.63641 + 2.28710i 0.926621 + 0.375997i
\(38\) 0.353071 0.0572757
\(39\) −6.24606 + 1.10135i −1.00017 + 0.176357i
\(40\) −0.0743311 + 0.421553i −0.0117528 + 0.0666533i
\(41\) 8.92828 − 3.24963i 1.39436 − 0.507507i 0.467863 − 0.883801i \(-0.345024\pi\)
0.926500 + 0.376294i \(0.122802\pi\)
\(42\) −0.769061 + 0.916531i −0.118669 + 0.141424i
\(43\) 12.5847i 1.91916i 0.281444 + 0.959578i \(0.409187\pi\)
−0.281444 + 0.959578i \(0.590813\pi\)
\(44\) −8.56271 − 7.18497i −1.29088 − 1.08317i
\(45\) −4.36496 + 2.52011i −0.650690 + 0.375676i
\(46\) −0.223528 − 0.0813577i −0.0329575 − 0.0119955i
\(47\) −3.68051 + 6.37482i −0.536857 + 0.929864i 0.462214 + 0.886768i \(0.347055\pi\)
−0.999071 + 0.0430951i \(0.986278\pi\)
\(48\) −5.57326 − 9.65318i −0.804431 − 1.39332i
\(49\) 1.46859 + 8.32880i 0.209799 + 1.18983i
\(50\) −0.0367066 − 0.100851i −0.00519110 − 0.0142624i
\(51\) −14.9762 − 8.64652i −2.09709 − 1.21075i
\(52\) 2.85897 + 3.40719i 0.396468 + 0.472492i
\(53\) −2.06899 + 1.73609i −0.284197 + 0.238470i −0.773731 − 0.633515i \(-0.781612\pi\)
0.489533 + 0.871985i \(0.337167\pi\)
\(54\) −0.212352 + 0.583431i −0.0288974 + 0.0793950i
\(55\) 5.53589 + 0.976126i 0.746459 + 0.131621i
\(56\) 1.65737 + 0.292238i 0.221475 + 0.0390520i
\(57\) 3.19047 − 8.76575i 0.422588 − 1.16105i
\(58\) 0.131345 − 0.110211i 0.0172464 − 0.0144715i
\(59\) −0.823360 − 0.981242i −0.107192 − 0.127747i 0.709781 − 0.704423i \(-0.248794\pi\)
−0.816973 + 0.576676i \(0.804350\pi\)
\(60\) 4.88299 + 2.81920i 0.630392 + 0.363957i
\(61\) −0.313577 − 0.861547i −0.0401495 − 0.110310i 0.917998 − 0.396585i \(-0.129805\pi\)
−0.958147 + 0.286276i \(0.907583\pi\)
\(62\) −0.0458124 − 0.259815i −0.00581819 − 0.0329966i
\(63\) 9.90800 + 17.1612i 1.24829 + 2.16210i
\(64\) −3.86244 + 6.68995i −0.482805 + 0.836244i
\(65\) −2.10187 − 0.765019i −0.260705 − 0.0948889i
\(66\) 1.48147 − 0.855328i 0.182356 − 0.105284i
\(67\) −2.06513 − 1.73285i −0.252296 − 0.211701i 0.507864 − 0.861437i \(-0.330435\pi\)
−0.760160 + 0.649736i \(0.774880\pi\)
\(68\) 12.1272i 1.47063i
\(69\) −4.03976 + 4.81440i −0.486330 + 0.579586i
\(70\) −0.396502 + 0.144315i −0.0473910 + 0.0172489i
\(71\) 1.38565 − 7.85840i 0.164446 − 0.932620i −0.785188 − 0.619258i \(-0.787434\pi\)
0.949634 − 0.313362i \(-0.101455\pi\)
\(72\) 2.12472 − 0.374645i 0.250401 − 0.0441524i
\(73\) −5.34759 −0.625888 −0.312944 − 0.949772i \(-0.601315\pi\)
−0.312944 + 0.949772i \(0.601315\pi\)
\(74\) 0.638350 + 0.136687i 0.0742067 + 0.0158895i
\(75\) −2.83553 −0.327419
\(76\) −6.44232 + 1.13595i −0.738985 + 0.130303i
\(77\) 3.83771 − 21.7648i 0.437348 − 2.48032i
\(78\) −0.639636 + 0.232809i −0.0724246 + 0.0263604i
\(79\) −5.87032 + 6.99598i −0.660463 + 0.787109i −0.987452 − 0.157919i \(-0.949521\pi\)
0.326989 + 0.945028i \(0.393966\pi\)
\(80\) − 3.93102i − 0.439502i
\(81\) 0.982962 + 0.824803i 0.109218 + 0.0916447i
\(82\) 0.883091 − 0.509853i 0.0975211 − 0.0563038i
\(83\) 2.11925 + 0.771346i 0.232618 + 0.0846662i 0.455699 − 0.890134i \(-0.349389\pi\)
−0.223081 + 0.974800i \(0.571611\pi\)
\(84\) 11.0839 − 19.1979i 1.20935 − 2.09466i
\(85\) −3.04935 − 5.28163i −0.330748 − 0.572873i
\(86\) 0.234535 + 1.33011i 0.0252905 + 0.143430i
\(87\) −1.54936 − 4.25683i −0.166109 − 0.456380i
\(88\) −2.08385 − 1.20311i −0.222139 − 0.128252i
\(89\) −0.660110 − 0.786689i −0.0699715 − 0.0833888i 0.729923 − 0.683529i \(-0.239556\pi\)
−0.799895 + 0.600140i \(0.795111\pi\)
\(90\) −0.414378 + 0.347704i −0.0436792 + 0.0366512i
\(91\) −3.00773 + 8.26367i −0.315296 + 0.866268i
\(92\) 4.34037 + 0.765325i 0.452515 + 0.0797907i
\(93\) −6.86445 − 1.21039i −0.711811 − 0.125511i
\(94\) −0.270198 + 0.742363i −0.0278688 + 0.0765689i
\(95\) 2.52013 − 2.11464i 0.258560 − 0.216958i
\(96\) −2.32934 − 2.77600i −0.237737 − 0.283324i
\(97\) 8.14173 + 4.70063i 0.826668 + 0.477277i 0.852710 − 0.522384i \(-0.174957\pi\)
−0.0260428 + 0.999661i \(0.508291\pi\)
\(98\) 0.310438 + 0.852923i 0.0313590 + 0.0861582i
\(99\) −4.91989 − 27.9021i −0.494468 − 2.80427i
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 185.2.w.a.176.6 yes 72
5.2 odd 4 925.2.ba.b.324.6 72
5.3 odd 4 925.2.ba.c.324.7 72
5.4 even 2 925.2.bb.b.176.7 72
37.2 odd 36 6845.2.a.u.1.18 36
37.4 even 18 inner 185.2.w.a.41.6 ✓ 72
37.35 odd 36 6845.2.a.t.1.19 36
185.4 even 18 925.2.bb.b.226.7 72
185.78 odd 36 925.2.ba.b.374.6 72
185.152 odd 36 925.2.ba.c.374.7 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.6 ✓ 72 37.4 even 18 inner
185.2.w.a.176.6 yes 72 1.1 even 1 trivial
925.2.ba.b.324.6 72 5.2 odd 4
925.2.ba.b.374.6 72 185.78 odd 36
925.2.ba.c.324.7 72 5.3 odd 4
925.2.ba.c.374.7 72 185.152 odd 36
925.2.bb.b.176.7 72 5.4 even 2
925.2.bb.b.226.7 72 185.4 even 18
6845.2.a.t.1.19 36 37.35 odd 36
6845.2.a.u.1.18 36 37.2 odd 36