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 475.5
Character \(\chi\) \(=\) 888.475
Dual form 888.2.r.e.43.5

$q$-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+(-1.27302 + 0.615974i) q^{2} +1.00000i q^{3} +(1.24115 - 1.56829i) q^{4} +(-0.116713 - 0.116713i) q^{5} +(-0.615974 - 1.27302i) q^{6} -4.08372i q^{7} +(-0.613982 + 2.76098i) 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} +(2.51546 + 5.19865i) q^{14} +(0.116713 - 0.116713i) q^{15} +(-0.919083 - 3.89298i) q^{16} +(-2.81029 + 2.81029i) q^{17} +(1.27302 - 0.615974i) q^{18} +(-1.79870 + 1.79870i) q^{19} +(-0.327898 + 0.0381815i) q^{20} +4.08372 q^{21} +(-0.118466 - 0.244830i) q^{22} +(-2.66342 - 2.66342i) q^{23} +(-2.76098 - 0.613982i) 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} +(3.56798 + 4.38970i) 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} +(-5.19865 + 2.51546i) 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} +(3.06309 + 6.33041i) q^{50} +(-2.81029 - 2.81029i) q^{51} +(9.04099 - 1.05276i) q^{52} -1.87566i q^{53} +(0.615974 + 1.27302i) q^{54} +(0.0224465 - 0.0224465i) q^{55} +(11.2751 + 2.50733i) q^{56} +(-1.79870 - 1.79870i) q^{57} +(-4.71198 + 13.5469i) q^{58} +(4.29355 - 4.29355i) q^{59} +(-0.0381815 - 0.327898i) 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} +(0.244830 - 0.118466i) q^{66} -3.98715i q^{67} +(0.919359 + 7.89535i) q^{68} +(2.66342 - 2.66342i) q^{69} +(0.313162 - 0.900337i) q^{70} -12.3057i q^{71} +(0.613982 - 2.76098i) q^{72} +1.08281i q^{73} +(6.65226 + 5.45412i) q^{74} +4.97276 q^{75} +(0.588428 + 5.05335i) q^{76} +0.785391 q^{77} +(2.11442 - 6.07891i) q^{78} +(1.29501 + 1.29501i) q^{79} +(-0.347092 + 0.561630i) q^{80} +1.00000 q^{81} +(-1.45209 - 3.00100i) 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} +(-0.530999 - 0.118083i) q^{88} +(5.63591 + 5.63591i) q^{89} +(-0.220470 - 0.0766855i) q^{90} +(13.1417 - 13.1417i) q^{91} +(-7.48272 + 0.871311i) q^{92} +(6.96071 + 6.96071i) q^{93} +(-1.45911 - 3.01550i) q^{94} +0.419863 q^{95} +(-4.38970 + 3.56798i) q^{96} +(-4.76026 + 4.76026i) q^{97} +(12.3187 - 5.96064i) 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.27302 + 0.615974i −0.900160 + 0.435559i
\(3\) 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 \(-0.261751\pi\)
−0.732723 + 0.680527i \(0.761751\pi\)
\(6\) −0.615974 − 1.27302i −0.251470 − 0.519708i
\(7\) − 4.08372i − 1.54350i −0.635926 − 0.771750i \(-0.719381\pi\)
0.635926 − 0.771750i \(-0.280619\pi\)
\(8\) −0.613982 + 2.76098i −0.217075 + 0.976155i
\(9\) −1.00000 −0.333333
\(10\) 0.220470 + 0.0766855i 0.0697186 + 0.0242501i
\(11\) 0.192323i 0.0579874i 0.999580 + 0.0289937i \(0.00923028\pi\)
−0.999580 + 0.0289937i \(0.990770\pi\)
\(12\) 1.56829 + 1.24115i 0.452727 + 0.358290i
\(13\) 3.21807 + 3.21807i 0.892532 + 0.892532i 0.994761 − 0.102229i \(-0.0325974\pi\)
−0.102229 + 0.994761i \(0.532597\pi\)
\(14\) 2.51546 + 5.19865i 0.672286 + 1.38940i
\(15\) 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 −0.960359 − 0.278765i \(-0.910075\pi\)
0.278765 + 0.960359i \(0.410075\pi\)
\(18\) 1.27302 − 0.615974i 0.300053 − 0.145186i
\(19\) −1.79870 + 1.79870i −0.412651 + 0.412651i −0.882661 − 0.470010i \(-0.844250\pi\)
0.470010 + 0.882661i \(0.344250\pi\)
\(20\) −0.327898 + 0.0381815i −0.0733203 + 0.00853764i
\(21\) 4.08372 0.891141
\(22\) −0.118466 − 0.244830i −0.0252570 − 0.0521980i
\(23\) −2.66342 − 2.66342i −0.555360 − 0.555360i 0.372623 − 0.927983i \(-0.378459\pi\)
−0.927983 + 0.372623i \(0.878459\pi\)
\(24\) −2.76098 − 0.613982i −0.563583 − 0.125329i
\(25\) − 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 0.427872 − 0.903840i \(-0.359264\pi\)
0.903840 − 0.427872i \(-0.140736\pi\)
\(30\) −0.0766855 + 0.220470i −0.0140008 + 0.0402521i
\(31\) 6.96071 − 6.96071i 1.25018 − 1.25018i 0.294541 − 0.955639i \(-0.404833\pi\)
0.955639 − 0.294541i \(-0.0951666\pi\)
\(32\) 3.56798 + 4.38970i 0.630736 + 0.775997i
\(33\) −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 \(0.0589309\pi\)
−0.982911 + 0.184081i \(0.941069\pi\)
\(42\) −5.19865 + 2.51546i −0.802169 + 0.388145i
\(43\) 6.40651 − 6.40651i 0.976984 − 0.976984i −0.0227571 − 0.999741i \(-0.507244\pi\)
0.999741 + 0.0227571i \(0.00724442\pi\)
\(44\) 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
\(50\) 3.06309 + 6.33041i 0.433186 + 0.895255i
\(51\) −2.81029 − 2.81029i −0.393519 − 0.393519i
\(52\) 9.04099 − 1.05276i 1.25376 − 0.145992i
\(53\) − 1.87566i − 0.257642i −0.991668 − 0.128821i \(-0.958881\pi\)
0.991668 − 0.128821i \(-0.0411192\pi\)
\(54\) 0.615974 + 1.27302i 0.0838234 + 0.173236i
\(55\) 0.0224465 − 0.0224465i 0.00302669 − 0.00302669i
\(56\) 11.2751 + 2.50733i 1.50670 + 0.335056i
\(57\) −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 −0.370042 − 0.929015i \(-0.620657\pi\)
0.929015 + 0.370042i \(0.120657\pi\)
\(60\) −0.0381815 − 0.327898i −0.00492921 − 0.0423315i
\(61\) −7.77530 + 7.77530i −0.995525 + 0.995525i −0.999990 − 0.00446497i \(-0.998579\pi\)
0.00446497 + 0.999990i \(0.498579\pi\)
\(62\) −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
\(66\) 0.244830 − 0.118466i 0.0301365 − 0.0145821i
\(67\) − 3.98715i − 0.487108i −0.969887 − 0.243554i \(-0.921687\pi\)
0.969887 − 0.243554i \(-0.0783133\pi\)
\(68\) 0.919359 + 7.89535i 0.111489 + 0.957451i
\(69\) 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 \(-0.260576\pi\)
\(72\) 0.613982 − 2.76098i 0.0723585 − 0.325385i
\(73\) 1.08281i 0.126733i 0.997990 + 0.0633666i \(0.0201837\pi\)
−0.997990 + 0.0633666i \(0.979816\pi\)
\(74\) 6.65226 + 5.45412i 0.773310 + 0.634028i
\(75\) 4.97276 0.574204
\(76\) 0.588428 + 5.05335i 0.0674973 + 0.579659i
\(77\) 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 \(-0.217148\pi\)
−0.630494 + 0.776194i \(0.717148\pi\)
\(80\) −0.347092 + 0.561630i −0.0388061 + 0.0627921i
\(81\) 1.00000 0.111111
\(82\) −1.45209 − 3.00100i −0.160356 − 0.331405i
\(83\) 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
\(88\) −0.530999 − 0.118083i −0.0566047 − 0.0125876i
\(89\) 5.63591 + 5.63591i 0.597406 + 0.597406i 0.939621 − 0.342216i \(-0.111177\pi\)
−0.342216 + 0.939621i \(0.611177\pi\)
\(90\) −0.220470 − 0.0766855i −0.0232395 − 0.00808337i
\(91\) 13.1417 − 13.1417i 1.37762 − 1.37762i
\(92\) −7.48272 + 0.871311i −0.780127 + 0.0908404i
\(93\) 6.96071 + 6.96071i 0.721792 + 0.721792i
\(94\) −1.45911 − 3.01550i −0.150495 − 0.311025i
\(95\) 0.419863 0.0430771
\(96\) −4.38970 + 3.56798i −0.448022 + 0.364156i
\(97\) −4.76026 + 4.76026i −0.483332 + 0.483332i −0.906194 − 0.422862i \(-0.861025\pi\)
0.422862 + 0.906194i \(0.361025\pi\)
\(98\) 12.3187 − 5.96064i 1.24438 − 0.602115i
\(99\) − 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.475.5 yes 72
8.3 odd 2 inner 888.2.r.e.475.23 yes 72
37.6 odd 4 inner 888.2.r.e.43.23 yes 72
296.43 even 4 inner 888.2.r.e.43.5 ✓ 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.5 ✓ 72 296.43 even 4 inner
888.2.r.e.43.23 yes 72 37.6 odd 4 inner
888.2.r.e.475.5 yes 72 1.1 even 1 trivial
888.2.r.e.475.23 yes 72 8.3 odd 2 inner