Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(43,888)] 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("888.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,2,0,0,0,-2,0,2,-72,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.23
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.23

$q$-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+(0.615974 + 1.27302i) q^{2} -1.00000i q^{3} +(-1.24115 + 1.56829i) q^{4} +(0.116713 - 0.116713i) q^{5} +(1.27302 - 0.615974i) q^{6} -4.08372i q^{7} +(-2.76098 - 0.613982i) q^{8} -1.00000 q^{9} +(0.220470 + 0.0766855i) q^{10} -0.192323i q^{11} +(1.56829 + 1.24115i) q^{12} +(-3.21807 + 3.21807i) q^{13} +(5.19865 - 2.51546i) q^{14} +(-0.116713 - 0.116713i) q^{15} +(-0.919083 - 3.89298i) q^{16} +(-2.81029 - 2.81029i) q^{17} +(-0.615974 - 1.27302i) q^{18} +(-1.79870 - 1.79870i) q^{19} +(0.0381815 + 0.327898i) q^{20} -4.08372 q^{21} +(0.244830 - 0.118466i) q^{22} +(2.66342 - 2.66342i) q^{23} +(-0.613982 + 2.76098i) q^{24} +4.97276i q^{25} +(-6.07891 - 2.11442i) q^{26} +1.00000i q^{27} +(6.40447 + 5.06852i) q^{28} +(-7.17148 - 7.17148i) q^{29} +(0.0766855 - 0.220470i) q^{30} +(-6.96071 - 6.96071i) q^{31} +(4.38970 - 3.56798i) q^{32} -0.192323 q^{33} +(1.84648 - 5.30861i) q^{34} +(-0.476623 - 0.476623i) q^{35} +(1.24115 - 1.56829i) q^{36} +(2.55443 - 5.52041i) q^{37} +(1.18183 - 3.39773i) q^{38} +(3.21807 + 3.21807i) q^{39} +(-0.393902 + 0.250583i) q^{40} -2.35739i q^{41} +(-2.51546 - 5.19865i) q^{42} +(6.40651 + 6.40651i) q^{43} +(0.301618 + 0.238702i) q^{44} +(-0.116713 + 0.116713i) q^{45} +(5.03117 + 1.74998i) q^{46} +2.36878i q^{47} +(-3.89298 + 0.919083i) q^{48} -9.67676 q^{49} +(-6.33041 + 3.06309i) q^{50} +(-2.81029 + 2.81029i) q^{51} +(-1.05276 - 9.04099i) q^{52} -1.87566i q^{53} +(-1.27302 + 0.615974i) q^{54} +(-0.0224465 - 0.0224465i) q^{55} +(-2.50733 + 11.2751i) q^{56} +(-1.79870 + 1.79870i) q^{57} +(4.71198 - 13.5469i) q^{58} +(4.29355 + 4.29355i) q^{59} +(0.327898 - 0.0381815i) q^{60} +(7.77530 + 7.77530i) q^{61} +(4.57349 - 13.1487i) q^{62} +4.08372i q^{63} +(7.24605 + 3.39039i) q^{64} +0.751180i q^{65} +(-0.118466 - 0.244830i) q^{66} +3.98715i q^{67} +(7.89535 - 0.919359i) q^{68} +(-2.66342 - 2.66342i) q^{69} +(0.313162 - 0.900337i) q^{70} -12.3057i q^{71} +(2.76098 + 0.613982i) q^{72} -1.08281i q^{73} +(8.60104 - 0.148590i) q^{74} +4.97276 q^{75} +(5.05335 - 0.588428i) q^{76} -0.785391 q^{77} +(-2.11442 + 6.07891i) q^{78} +(-1.29501 + 1.29501i) q^{79} +(-0.561630 - 0.347092i) q^{80} +1.00000 q^{81} +(3.00100 - 1.45209i) q^{82} +14.5702 q^{83} +(5.06852 - 6.40447i) q^{84} -0.655993 q^{85} +(-4.20936 + 12.1019i) q^{86} +(-7.17148 + 7.17148i) q^{87} +(-0.118083 + 0.530999i) q^{88} +(5.63591 - 5.63591i) q^{89} +(-0.220470 - 0.0766855i) q^{90} +(13.1417 + 13.1417i) q^{91} +(0.871311 + 7.48272i) q^{92} +(-6.96071 + 6.96071i) q^{93} +(-3.01550 + 1.45911i) q^{94} -0.419863 q^{95} +(-3.56798 - 4.38970i) q^{96} +(-4.76026 - 4.76026i) q^{97} +(-5.96064 - 12.3187i) q^{98} +0.192323i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 2 q^{2} - 2 q^{6} + 2 q^{8} - 72 q^{9} + 8 q^{10} - 8 q^{12} + 20 q^{14} + 12 q^{16} + 12 q^{17} - 2 q^{18} + 20 q^{19} - 22 q^{20} - 16 q^{22} + 2 q^{24} - 32 q^{26} - 8 q^{32} - 16 q^{33} + 8 q^{34}+ \cdots - 110 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.615974 + 1.27302i 0.435559 + 0.900160i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.24115 + 1.56829i −0.620576 + 0.784146i
\(5\) 0.116713 − 0.116713i 0.0521956 − 0.0521956i −0.680527 − 0.732723i \(-0.738249\pi\)
0.732723 + 0.680527i \(0.238249\pi\)
\(6\) 1.27302 − 0.615974i 0.519708 − 0.251470i
\(7\) − 4.08372i − 1.54350i −0.635926 − 0.771750i \(-0.719381\pi\)
0.635926 − 0.771750i \(-0.280619\pi\)
\(8\) −2.76098 − 0.613982i −0.976155 − 0.217075i
\(9\) −1.00000 −0.333333
\(10\) 0.220470 + 0.0766855i 0.0697186 + 0.0242501i
\(11\) − 0.192323i − 0.0579874i −0.999580 − 0.0289937i \(-0.990770\pi\)
0.999580 − 0.0289937i \(-0.00923028\pi\)
\(12\) 1.56829 + 1.24115i 0.452727 + 0.358290i
\(13\) −3.21807 + 3.21807i −0.892532 + 0.892532i −0.994761 − 0.102229i \(-0.967403\pi\)
0.102229 + 0.994761i \(0.467403\pi\)
\(14\) 5.19865 − 2.51546i 1.38940 − 0.672286i
\(15\) −0.116713 − 0.116713i −0.0301351 − 0.0301351i
\(16\) −0.919083 − 3.89298i −0.229771 − 0.973245i
\(17\) −2.81029 − 2.81029i −0.681595 − 0.681595i 0.278765 − 0.960359i \(-0.410075\pi\)
−0.960359 + 0.278765i \(0.910075\pi\)
\(18\) −0.615974 − 1.27302i −0.145186 − 0.300053i
\(19\) −1.79870 − 1.79870i −0.412651 − 0.412651i 0.470010 − 0.882661i \(-0.344250\pi\)
−0.882661 + 0.470010i \(0.844250\pi\)
\(20\) 0.0381815 + 0.327898i 0.00853764 + 0.0733203i
\(21\) −4.08372 −0.891141
\(22\) 0.244830 − 0.118466i 0.0521980 − 0.0252570i
\(23\) 2.66342 − 2.66342i 0.555360 − 0.555360i −0.372623 − 0.927983i \(-0.621541\pi\)
0.927983 + 0.372623i \(0.121541\pi\)
\(24\) −0.613982 + 2.76098i −0.125329 + 0.563583i
\(25\) 4.97276i 0.994551i
\(26\) −6.07891 − 2.11442i −1.19217 − 0.414671i
\(27\) 1.00000i 0.192450i
\(28\) 6.40447 + 5.06852i 1.21033 + 0.957860i
\(29\) −7.17148 − 7.17148i −1.33171 − 1.33171i −0.903840 − 0.427872i \(-0.859264\pi\)
−0.427872 − 0.903840i \(-0.640736\pi\)
\(30\) 0.0766855 − 0.220470i 0.0140008 − 0.0402521i
\(31\) −6.96071 − 6.96071i −1.25018 − 1.25018i −0.955639 − 0.294541i \(-0.904833\pi\)
−0.294541 − 0.955639i \(-0.595167\pi\)
\(32\) 4.38970 − 3.56798i 0.775997 − 0.630736i
\(33\) −0.192323 −0.0334791
\(34\) 1.84648 − 5.30861i 0.316669 − 0.910419i
\(35\) −0.476623 − 0.476623i −0.0805639 − 0.0805639i
\(36\) 1.24115 − 1.56829i 0.206859 − 0.261382i
\(37\) 2.55443 − 5.52041i 0.419946 − 0.907549i
\(38\) 1.18183 − 3.39773i 0.191718 − 0.551185i
\(39\) 3.21807 + 3.21807i 0.515304 + 0.515304i
\(40\) −0.393902 + 0.250583i −0.0622813 + 0.0396206i
\(41\) − 2.35739i − 0.368162i −0.982911 − 0.184081i \(-0.941069\pi\)
0.982911 − 0.184081i \(-0.0589309\pi\)
\(42\) −2.51546 − 5.19865i −0.388145 − 0.802169i
\(43\) 6.40651 + 6.40651i 0.976984 + 0.976984i 0.999741 − 0.0227571i \(-0.00724442\pi\)
−0.0227571 + 0.999741i \(0.507244\pi\)
\(44\) 0.301618 + 0.238702i 0.0454706 + 0.0359856i
\(45\) −0.116713 + 0.116713i −0.0173985 + 0.0173985i
\(46\) 5.03117 + 1.74998i 0.741806 + 0.258021i
\(47\) 2.36878i 0.345522i 0.984964 + 0.172761i \(0.0552689\pi\)
−0.984964 + 0.172761i \(0.944731\pi\)
\(48\) −3.89298 + 0.919083i −0.561903 + 0.132658i
\(49\) −9.67676 −1.38239
\(50\) −6.33041 + 3.06309i −0.895255 + 0.433186i
\(51\) −2.81029 + 2.81029i −0.393519 + 0.393519i
\(52\) −1.05276 − 9.04099i −0.145992 − 1.25376i
\(53\) − 1.87566i − 0.257642i −0.991668 − 0.128821i \(-0.958881\pi\)
0.991668 − 0.128821i \(-0.0411192\pi\)
\(54\) −1.27302 + 0.615974i −0.173236 + 0.0838234i
\(55\) −0.0224465 − 0.0224465i −0.00302669 − 0.00302669i
\(56\) −2.50733 + 11.2751i −0.335056 + 1.50670i
\(57\) −1.79870 + 1.79870i −0.238244 + 0.238244i
\(58\) 4.71198 − 13.5469i 0.618714 − 1.77879i
\(59\) 4.29355 + 4.29355i 0.558973 + 0.558973i 0.929015 − 0.370042i \(-0.120657\pi\)
−0.370042 + 0.929015i \(0.620657\pi\)
\(60\) 0.327898 − 0.0381815i 0.0423315 − 0.00492921i
\(61\) 7.77530 + 7.77530i 0.995525 + 0.995525i 0.999990 − 0.00446497i \(-0.00142125\pi\)
−0.00446497 + 0.999990i \(0.501421\pi\)
\(62\) 4.57349 − 13.1487i 0.580834 − 1.66989i
\(63\) 4.08372i 0.514500i
\(64\) 7.24605 + 3.39039i 0.905756 + 0.423799i
\(65\) 0.751180i 0.0931725i
\(66\) −0.118466 − 0.244830i −0.0145821 − 0.0301365i
\(67\) 3.98715i 0.487108i 0.969887 + 0.243554i \(0.0783133\pi\)
−0.969887 + 0.243554i \(0.921687\pi\)
\(68\) 7.89535 − 0.919359i 0.957451 − 0.111489i
\(69\) −2.66342 − 2.66342i −0.320637 − 0.320637i
\(70\) 0.313162 − 0.900337i 0.0374300 − 0.107611i
\(71\) − 12.3057i − 1.46041i −0.683226 − 0.730207i \(-0.739424\pi\)
0.683226 − 0.730207i \(-0.260576\pi\)
\(72\) 2.76098 + 0.613982i 0.325385 + 0.0723585i
\(73\) − 1.08281i − 0.126733i −0.997990 − 0.0633666i \(-0.979816\pi\)
0.997990 − 0.0633666i \(-0.0201837\pi\)
\(74\) 8.60104 − 0.148590i 0.999851 − 0.0172732i
\(75\) 4.97276 0.574204
\(76\) 5.05335 − 0.588428i 0.579659 − 0.0674973i
\(77\) −0.785391 −0.0895036
\(78\) −2.11442 + 6.07891i −0.239410 + 0.688301i
\(79\) −1.29501 + 1.29501i −0.145701 + 0.145701i −0.776194 − 0.630494i \(-0.782852\pi\)
0.630494 + 0.776194i \(0.282852\pi\)
\(80\) −0.561630 − 0.347092i −0.0627921 − 0.0388061i
\(81\) 1.00000 0.111111
\(82\) 3.00100 − 1.45209i 0.331405 − 0.160356i
\(83\) 14.5702 1.59929 0.799643 − 0.600476i \(-0.205022\pi\)
0.799643 + 0.600476i \(0.205022\pi\)
\(84\) 5.06852 − 6.40447i 0.553021 − 0.698785i
\(85\) −0.655993 −0.0711525
\(86\) −4.20936 + 12.1019i −0.453907 + 1.30498i
\(87\) −7.17148 + 7.17148i −0.768864 + 0.768864i
\(88\) −0.118083 + 0.530999i −0.0125876 + 0.0566047i
\(89\) 5.63591 − 5.63591i 0.597406 − 0.597406i −0.342216 − 0.939621i \(-0.611177\pi\)
0.939621 + 0.342216i \(0.111177\pi\)
\(90\) −0.220470 − 0.0766855i −0.0232395 − 0.00808337i
\(91\) 13.1417 + 13.1417i 1.37762 + 1.37762i
\(92\) 0.871311 + 7.48272i 0.0908404 + 0.780127i
\(93\) −6.96071 + 6.96071i −0.721792 + 0.721792i
\(94\) −3.01550 + 1.45911i −0.311025 + 0.150495i
\(95\) −0.419863 −0.0430771
\(96\) −3.56798 − 4.38970i −0.364156 − 0.448022i
\(97\) −4.76026 − 4.76026i −0.483332 − 0.483332i 0.422862 − 0.906194i \(-0.361025\pi\)
−0.906194 + 0.422862i \(0.861025\pi\)
\(98\) −5.96064 − 12.3187i −0.602115 − 1.24438i
\(99\) 0.192323i 0.0193291i
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 888.2.r.e.43.23 yes 72
8.3 odd 2 inner 888.2.r.e.43.5 ✓ 72
37.31 odd 4 inner 888.2.r.e.475.5 yes 72
296.179 even 4 inner 888.2.r.e.475.23 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.5 ✓ 72 8.3 odd 2 inner
888.2.r.e.43.23 yes 72 1.1 even 1 trivial
888.2.r.e.475.5 yes 72 37.31 odd 4 inner
888.2.r.e.475.23 yes 72 296.179 even 4 inner