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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [185,2,Mod(82,185)] 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("185.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.p (of order \(12\), degree \(4\), 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: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 103.6
Character \(\chi\) \(=\) 185.103
Dual form 185.2.p.a.97.6

$q$-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+(-0.567972 - 0.983757i) q^{2} +(0.273625 + 1.02118i) q^{3} +(0.354815 - 0.614557i) q^{4} +(-1.79329 - 1.33571i) q^{5} +(0.849185 - 0.849185i) q^{6} +(-0.422839 - 1.57806i) q^{7} -3.07799 q^{8} +(1.63013 - 0.941157i) q^{9} +(-0.295475 + 2.52281i) q^{10} -1.92257i q^{11} +(0.724662 + 0.194173i) q^{12} +(0.763528 - 1.32247i) q^{13} +(-1.31226 + 1.31226i) q^{14} +(0.873315 - 2.19676i) q^{15} +(1.03858 + 1.79888i) q^{16} +(1.14334 - 0.660110i) q^{17} +(-1.85174 - 1.06910i) q^{18} +(2.17790 - 0.583566i) q^{19} +(-1.45715 + 0.628149i) q^{20} +(1.49579 - 0.863592i) q^{21} +(-1.89134 + 1.09196i) q^{22} -4.57925 q^{23} +(-0.842216 - 3.14319i) q^{24} +(1.43176 + 4.79062i) q^{25} -1.73465 q^{26} +(3.64981 + 3.64981i) q^{27} +(-1.11984 - 0.300059i) q^{28} +(0.241089 - 0.241089i) q^{29} +(-2.65710 + 0.388569i) q^{30} +(-1.00617 - 1.00617i) q^{31} +(-1.89822 + 3.28781i) q^{32} +(1.96329 - 0.526063i) q^{33} +(-1.29878 - 0.749849i) q^{34} +(-1.34955 + 3.39470i) q^{35} -1.33575i q^{36} +(1.71804 + 5.83510i) q^{37} +(-1.81107 - 1.81107i) q^{38} +(1.55940 + 0.417841i) q^{39} +(5.51972 + 4.11130i) q^{40} +(6.76304 + 3.90464i) q^{41} +(-1.69913 - 0.980993i) q^{42} +5.46568 q^{43} +(-1.18153 - 0.682155i) q^{44} +(-4.18041 - 0.489616i) q^{45} +(2.60089 + 4.50487i) q^{46} +(1.79140 - 1.79140i) q^{47} +(-1.55280 + 1.55280i) q^{48} +(3.75071 - 2.16547i) q^{49} +(3.89961 - 4.12945i) q^{50} +(0.986942 + 0.986942i) q^{51} +(-0.541822 - 0.938463i) q^{52} +(-0.870548 + 3.24893i) q^{53} +(1.51754 - 5.66352i) q^{54} +(-2.56799 + 3.44772i) q^{55} +(1.30149 + 4.85724i) q^{56} +(1.19186 + 2.06436i) q^{57} +(-0.374105 - 0.100241i) q^{58} +(-1.40987 + 5.26170i) q^{59} +(-1.04017 - 1.31614i) q^{60} +(10.4583 - 2.80230i) q^{61} +(-0.418352 + 1.56131i) q^{62} +(-2.17448 - 2.17448i) q^{63} +8.46687 q^{64} +(-3.13566 + 1.35172i) q^{65} +(-1.63261 - 1.63261i) q^{66} +(-9.45477 + 2.53340i) q^{67} -0.936868i q^{68} +(-1.25300 - 4.67625i) q^{69} +(4.10607 - 0.600463i) q^{70} +(1.65295 - 2.86299i) q^{71} +(-5.01753 + 2.89687i) q^{72} +(1.78634 - 1.78634i) q^{73} +(4.76452 - 5.00430i) q^{74} +(-4.50034 + 2.77293i) q^{75} +(0.414116 - 1.54550i) q^{76} +(-3.03392 + 0.812936i) q^{77} +(-0.474644 - 1.77140i) q^{78} +(-12.3408 + 3.30670i) q^{79} +(0.540301 - 4.61315i) q^{80} +(0.0950223 - 0.164583i) q^{81} -8.87092i q^{82} +(2.29888 - 8.57952i) q^{83} -1.22566i q^{84} +(-2.93206 - 0.343408i) q^{85} +(-3.10436 - 5.37690i) q^{86} +(0.312164 + 0.180228i) q^{87} +5.91764i q^{88} +(-16.4223 - 4.40035i) q^{89} +(1.89269 + 4.39059i) q^{90} +(-2.40978 - 0.645698i) q^{91} +(-1.62479 + 2.81421i) q^{92} +(0.752174 - 1.30280i) q^{93} +(-2.77976 - 0.744835i) q^{94} +(-4.68507 - 1.86254i) q^{95} +(-3.87685 - 1.03880i) q^{96} -15.4196i q^{97} +(-4.26060 - 2.45986i) q^{98} +(-1.80944 - 3.13404i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 68 q - 4 q^{2} - 8 q^{3} - 30 q^{4} - 10 q^{5} - 8 q^{6} - 2 q^{7} + 12 q^{8} - 6 q^{10} + 14 q^{12} - 6 q^{13} - 8 q^{15} - 26 q^{16} + 12 q^{17} + 18 q^{18} + 4 q^{19} + 48 q^{20} - 12 q^{21} + 6 q^{22}+ \cdots + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(76\) \(112\)
\(\chi(n)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.567972 − 0.983757i −0.401617 − 0.695621i 0.592304 − 0.805714i \(-0.298218\pi\)
−0.993921 + 0.110093i \(0.964885\pi\)
\(3\) 0.273625 + 1.02118i 0.157978 + 0.589581i 0.998832 + 0.0483191i \(0.0153864\pi\)
−0.840854 + 0.541262i \(0.817947\pi\)
\(4\) 0.354815 − 0.614557i 0.177407 − 0.307279i
\(5\) −1.79329 − 1.33571i −0.801983 − 0.597347i
\(6\) 0.849185 − 0.849185i 0.346678 − 0.346678i
\(7\) −0.422839 − 1.57806i −0.159818 − 0.596449i −0.998644 − 0.0520501i \(-0.983424\pi\)
0.838826 − 0.544399i \(-0.183242\pi\)
\(8\) −3.07799 −1.08823
\(9\) 1.63013 − 0.941157i 0.543377 − 0.313719i
\(10\) −0.295475 + 2.52281i −0.0934375 + 0.797781i
\(11\) − 1.92257i − 0.579676i −0.957076 − 0.289838i \(-0.906399\pi\)
0.957076 − 0.289838i \(-0.0936014\pi\)
\(12\) 0.724662 + 0.194173i 0.209192 + 0.0560528i
\(13\) 0.763528 − 1.32247i 0.211764 − 0.366787i −0.740502 − 0.672054i \(-0.765412\pi\)
0.952267 + 0.305267i \(0.0987457\pi\)
\(14\) −1.31226 + 1.31226i −0.350717 + 0.350717i
\(15\) 0.873315 − 2.19676i 0.225489 − 0.567201i
\(16\) 1.03858 + 1.79888i 0.259646 + 0.449720i
\(17\) 1.14334 − 0.660110i 0.277302 − 0.160100i −0.354899 − 0.934904i \(-0.615485\pi\)
0.632201 + 0.774804i \(0.282152\pi\)
\(18\) −1.85174 − 1.06910i −0.436459 − 0.251990i
\(19\) 2.17790 − 0.583566i 0.499644 − 0.133879i −0.000190628 − 1.00000i \(-0.500061\pi\)
0.499835 + 0.866121i \(0.333394\pi\)
\(20\) −1.45715 + 0.628149i −0.325830 + 0.140458i
\(21\) 1.49579 − 0.863592i 0.326407 − 0.188451i
\(22\) −1.89134 + 1.09196i −0.403235 + 0.232808i
\(23\) −4.57925 −0.954839 −0.477420 − 0.878675i \(-0.658428\pi\)
−0.477420 + 0.878675i \(0.658428\pi\)
\(24\) −0.842216 − 3.14319i −0.171917 − 0.641601i
\(25\) 1.43176 + 4.79062i 0.286352 + 0.958124i
\(26\) −1.73465 −0.340193
\(27\) 3.64981 + 3.64981i 0.702407 + 0.702407i
\(28\) −1.11984 − 0.300059i −0.211629 − 0.0567058i
\(29\) 0.241089 − 0.241089i 0.0447691 − 0.0447691i −0.684368 − 0.729137i \(-0.739922\pi\)
0.729137 + 0.684368i \(0.239922\pi\)
\(30\) −2.65710 + 0.388569i −0.485117 + 0.0709426i
\(31\) −1.00617 − 1.00617i −0.180714 − 0.180714i 0.610953 − 0.791667i \(-0.290787\pi\)
−0.791667 + 0.610953i \(0.790787\pi\)
\(32\) −1.89822 + 3.28781i −0.335560 + 0.581208i
\(33\) 1.96329 − 0.526063i 0.341766 − 0.0915758i
\(34\) −1.29878 − 0.749849i −0.222738 − 0.128598i
\(35\) −1.34955 + 3.39470i −0.228116 + 0.573809i
\(36\) − 1.33575i − 0.222624i
\(37\) 1.71804 + 5.83510i 0.282443 + 0.959284i
\(38\) −1.81107 − 1.81107i −0.293795 − 0.293795i
\(39\) 1.55940 + 0.417841i 0.249704 + 0.0669081i
\(40\) 5.51972 + 4.11130i 0.872744 + 0.650053i
\(41\) 6.76304 + 3.90464i 1.05621 + 0.609803i 0.924381 − 0.381470i \(-0.124582\pi\)
0.131828 + 0.991273i \(0.457915\pi\)
\(42\) −1.69913 − 0.980993i −0.262181 − 0.151370i
\(43\) 5.46568 0.833509 0.416754 − 0.909019i \(-0.363168\pi\)
0.416754 + 0.909019i \(0.363168\pi\)
\(44\) −1.18153 − 0.682155i −0.178122 − 0.102839i
\(45\) −4.18041 − 0.489616i −0.623178 − 0.0729877i
\(46\) 2.60089 + 4.50487i 0.383480 + 0.664207i
\(47\) 1.79140 − 1.79140i 0.261302 − 0.261302i −0.564281 − 0.825583i \(-0.690846\pi\)
0.825583 + 0.564281i \(0.190846\pi\)
\(48\) −1.55280 + 1.55280i −0.224128 + 0.224128i
\(49\) 3.75071 − 2.16547i 0.535816 − 0.309353i
\(50\) 3.89961 − 4.12945i 0.551488 − 0.583992i
\(51\) 0.986942 + 0.986942i 0.138200 + 0.138200i
\(52\) −0.541822 − 0.938463i −0.0751372 − 0.130141i
\(53\) −0.870548 + 3.24893i −0.119579 + 0.446275i −0.999589 − 0.0286809i \(-0.990869\pi\)
0.880010 + 0.474956i \(0.157536\pi\)
\(54\) 1.51754 − 5.66352i 0.206511 − 0.770708i
\(55\) −2.56799 + 3.44772i −0.346268 + 0.464890i
\(56\) 1.30149 + 4.85724i 0.173919 + 0.649076i
\(57\) 1.19186 + 2.06436i 0.157865 + 0.273431i
\(58\) −0.374105 − 0.100241i −0.0491223 − 0.0131623i
\(59\) −1.40987 + 5.26170i −0.183549 + 0.685016i 0.811387 + 0.584509i \(0.198713\pi\)
−0.994936 + 0.100506i \(0.967954\pi\)
\(60\) −1.04017 − 1.31614i −0.134285 − 0.169914i
\(61\) 10.4583 − 2.80230i 1.33905 − 0.358797i 0.482968 − 0.875638i \(-0.339559\pi\)
0.856081 + 0.516841i \(0.172892\pi\)
\(62\) −0.418352 + 1.56131i −0.0531307 + 0.198287i
\(63\) −2.17448 − 2.17448i −0.273959 − 0.273959i
\(64\) 8.46687 1.05836
\(65\) −3.13566 + 1.35172i −0.388931 + 0.167660i
\(66\) −1.63261 − 1.63261i −0.200961 − 0.200961i
\(67\) −9.45477 + 2.53340i −1.15508 + 0.309504i −0.785001 − 0.619495i \(-0.787338\pi\)
−0.370083 + 0.928999i \(0.620671\pi\)
\(68\) − 0.936868i − 0.113612i
\(69\) −1.25300 − 4.67625i −0.150843 − 0.562955i
\(70\) 4.10607 − 0.600463i 0.490769 − 0.0717691i
\(71\) 1.65295 − 2.86299i 0.196169 − 0.339774i −0.751114 − 0.660172i \(-0.770483\pi\)
0.947283 + 0.320398i \(0.103817\pi\)
\(72\) −5.01753 + 2.89687i −0.591321 + 0.341399i
\(73\) 1.78634 − 1.78634i 0.209075 − 0.209075i −0.594799 − 0.803874i \(-0.702768\pi\)
0.803874 + 0.594799i \(0.202768\pi\)
\(74\) 4.76452 − 5.00430i 0.553864 − 0.581739i
\(75\) −4.50034 + 2.77293i −0.519654 + 0.320190i
\(76\) 0.414116 − 1.54550i 0.0475023 − 0.177281i
\(77\) −3.03392 + 0.812936i −0.345747 + 0.0926427i
\(78\) −0.474644 − 1.77140i −0.0537429 − 0.200571i
\(79\) −12.3408 + 3.30670i −1.38844 + 0.372032i −0.874181 − 0.485600i \(-0.838601\pi\)
−0.514263 + 0.857633i \(0.671934\pi\)
\(80\) 0.540301 − 4.61315i 0.0604074 − 0.515766i
\(81\) 0.0950223 − 0.164583i 0.0105580 − 0.0182871i
\(82\) − 8.87092i − 0.979629i
\(83\) 2.29888 − 8.57952i 0.252334 − 0.941725i −0.717220 − 0.696847i \(-0.754586\pi\)
0.969554 − 0.244878i \(-0.0787478\pi\)
\(84\) − 1.22566i − 0.133731i
\(85\) −2.93206 − 0.343408i −0.318027 − 0.0372479i
\(86\) −3.10436 − 5.37690i −0.334751 − 0.579806i
\(87\) 0.312164 + 0.180228i 0.0334675 + 0.0193225i
\(88\) 5.91764i 0.630823i
\(89\) −16.4223 − 4.40035i −1.74076 − 0.466436i −0.758146 − 0.652085i \(-0.773894\pi\)
−0.982616 + 0.185650i \(0.940561\pi\)
\(90\) 1.89269 + 4.39059i 0.199507 + 0.462809i
\(91\) −2.40978 − 0.645698i −0.252613 − 0.0676876i
\(92\) −1.62479 + 2.81421i −0.169396 + 0.293402i
\(93\) 0.752174 − 1.30280i 0.0779968 − 0.135094i
\(94\) −2.77976 − 0.744835i −0.286711 − 0.0768239i
\(95\) −4.68507 − 1.86254i −0.480678 − 0.191092i
\(96\) −3.87685 − 1.03880i −0.395680 − 0.106022i
\(97\) − 15.4196i − 1.56563i −0.622256 − 0.782814i \(-0.713784\pi\)
0.622256 − 0.782814i \(-0.286216\pi\)
\(98\) −4.26060 − 2.45986i −0.430385 − 0.248483i
\(99\) −1.80944 − 3.13404i −0.181855 − 0.314983i
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.p.a.103.6 yes 68
5.2 odd 4 185.2.u.a.177.12 yes 68
5.3 odd 4 925.2.y.b.732.6 68
5.4 even 2 925.2.t.b.843.12 68
37.23 odd 12 185.2.u.a.23.12 yes 68
185.23 even 12 925.2.t.b.282.12 68
185.97 even 12 inner 185.2.p.a.97.6 ✓ 68
185.134 odd 12 925.2.y.b.393.6 68
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.p.a.97.6 ✓ 68 185.97 even 12 inner
185.2.p.a.103.6 yes 68 1.1 even 1 trivial
185.2.u.a.23.12 yes 68 37.23 odd 12
185.2.u.a.177.12 yes 68 5.2 odd 4
925.2.t.b.282.12 68 185.23 even 12
925.2.t.b.843.12 68 5.4 even 2
925.2.y.b.393.6 68 185.134 odd 12
925.2.y.b.732.6 68 5.3 odd 4