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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 43.3
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.3

$q$-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+(-1.40508 + 0.160452i) q^{2} -1.00000i q^{3} +(1.94851 - 0.450896i) q^{4} +(2.38279 - 2.38279i) q^{5} +(0.160452 + 1.40508i) q^{6} -3.14379i q^{7} +(-2.66547 + 0.946188i) q^{8} -1.00000 q^{9} +(-2.96569 + 3.73034i) q^{10} -1.73701i q^{11} +(-0.450896 - 1.94851i) q^{12} +(2.24578 - 2.24578i) q^{13} +(0.504427 + 4.41728i) q^{14} +(-2.38279 - 2.38279i) q^{15} +(3.59339 - 1.75715i) q^{16} +(-0.467334 - 0.467334i) q^{17} +(1.40508 - 0.160452i) q^{18} +(-2.42109 - 2.42109i) q^{19} +(3.56850 - 5.71728i) q^{20} -3.14379 q^{21} +(0.278707 + 2.44065i) q^{22} +(-2.91739 + 2.91739i) q^{23} +(0.946188 + 2.66547i) q^{24} -6.35539i q^{25} +(-2.79516 + 3.51584i) q^{26} +1.00000i q^{27} +(-1.41752 - 6.12571i) q^{28} +(2.16206 + 2.16206i) q^{29} +(3.73034 + 2.96569i) q^{30} +(0.222977 + 0.222977i) q^{31} +(-4.76706 + 3.04551i) q^{32} -1.73701 q^{33} +(0.731627 + 0.581658i) q^{34} +(-7.49099 - 7.49099i) q^{35} +(-1.94851 + 0.450896i) q^{36} +(5.93691 + 1.32403i) q^{37} +(3.79029 + 3.01336i) q^{38} +(-2.24578 - 2.24578i) q^{39} +(-4.09669 + 8.60583i) q^{40} +0.621619i q^{41} +(4.41728 - 0.504427i) q^{42} +(6.22159 + 6.22159i) q^{43} +(-0.783212 - 3.38459i) q^{44} +(-2.38279 + 2.38279i) q^{45} +(3.63108 - 4.56728i) q^{46} +12.9722i q^{47} +(-1.75715 - 3.59339i) q^{48} -2.88341 q^{49} +(1.01973 + 8.92984i) q^{50} +(-0.467334 + 0.467334i) q^{51} +(3.36331 - 5.38853i) q^{52} +13.8875i q^{53} +(-0.160452 - 1.40508i) q^{54} +(-4.13894 - 4.13894i) q^{55} +(2.97461 + 8.37967i) q^{56} +(-2.42109 + 2.42109i) q^{57} +(-3.38477 - 2.69096i) q^{58} +(-10.2164 - 10.2164i) q^{59} +(-5.71728 - 3.56850i) q^{60} +(0.980438 + 0.980438i) q^{61} +(-0.349078 - 0.277524i) q^{62} +3.14379i q^{63} +(6.20946 - 5.04407i) q^{64} -10.7024i q^{65} +(2.44065 - 0.278707i) q^{66} +3.02006i q^{67} +(-1.12132 - 0.699886i) q^{68} +(2.91739 + 2.91739i) q^{69} +(11.7274 + 9.32351i) q^{70} -13.7628i q^{71} +(2.66547 - 0.946188i) q^{72} -4.11600i q^{73} +(-8.55429 - 0.907788i) q^{74} -6.35539 q^{75} +(-5.80917 - 3.62585i) q^{76} -5.46080 q^{77} +(3.51584 + 2.79516i) q^{78} +(7.45631 - 7.45631i) q^{79} +(4.37537 - 12.7492i) q^{80} +1.00000 q^{81} +(-0.0997400 - 0.873426i) q^{82} -3.67031 q^{83} +(-6.12571 + 1.41752i) q^{84} -2.22712 q^{85} +(-9.74011 - 7.74358i) q^{86} +(2.16206 - 2.16206i) q^{87} +(1.64354 + 4.62996i) q^{88} +(2.57508 - 2.57508i) q^{89} +(2.96569 - 3.73034i) q^{90} +(-7.06025 - 7.06025i) q^{91} +(-4.36913 + 7.00002i) q^{92} +(0.222977 - 0.222977i) q^{93} +(-2.08142 - 18.2271i) q^{94} -11.5379 q^{95} +(3.04551 + 4.76706i) q^{96} +(-10.4856 - 10.4856i) q^{97} +(4.05142 - 0.462648i) q^{98} +1.73701i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40508 + 0.160452i −0.993543 + 0.113457i
\(3\) − 1.00000i − 0.577350i
\(4\) 1.94851 − 0.450896i 0.974255 − 0.225448i
\(5\) 2.38279 − 2.38279i 1.06562 − 1.06562i 0.0679263 − 0.997690i \(-0.478362\pi\)
0.997690 − 0.0679263i \(-0.0216383\pi\)
\(6\) 0.160452 + 1.40508i 0.0655042 + 0.573622i
\(7\) − 3.14379i − 1.18824i −0.804376 − 0.594120i \(-0.797500\pi\)
0.804376 − 0.594120i \(-0.202500\pi\)
\(8\) −2.66547 + 0.946188i −0.942386 + 0.334528i
\(9\) −1.00000 −0.333333
\(10\) −2.96569 + 3.73034i −0.937835 + 1.17964i
\(11\) − 1.73701i − 0.523729i −0.965105 − 0.261865i \(-0.915663\pi\)
0.965105 − 0.261865i \(-0.0843374\pi\)
\(12\) −0.450896 − 1.94851i −0.130162 − 0.562487i
\(13\) 2.24578 − 2.24578i 0.622867 − 0.622867i −0.323397 − 0.946263i \(-0.604825\pi\)
0.946263 + 0.323397i \(0.104825\pi\)
\(14\) 0.504427 + 4.41728i 0.134814 + 1.18057i
\(15\) −2.38279 − 2.38279i −0.615234 − 0.615234i
\(16\) 3.59339 − 1.75715i 0.898346 − 0.439288i
\(17\) −0.467334 − 0.467334i −0.113345 − 0.113345i 0.648160 − 0.761505i \(-0.275539\pi\)
−0.761505 + 0.648160i \(0.775539\pi\)
\(18\) 1.40508 − 0.160452i 0.331181 − 0.0378189i
\(19\) −2.42109 − 2.42109i −0.555435 − 0.555435i 0.372569 − 0.928004i \(-0.378477\pi\)
−0.928004 + 0.372569i \(0.878477\pi\)
\(20\) 3.56850 − 5.71728i 0.797941 − 1.27842i
\(21\) −3.14379 −0.686031
\(22\) 0.278707 + 2.44065i 0.0594205 + 0.520347i
\(23\) −2.91739 + 2.91739i −0.608319 + 0.608319i −0.942507 − 0.334188i \(-0.891538\pi\)
0.334188 + 0.942507i \(0.391538\pi\)
\(24\) 0.946188 + 2.66547i 0.193140 + 0.544087i
\(25\) − 6.35539i − 1.27108i
\(26\) −2.79516 + 3.51584i −0.548177 + 0.689513i
\(27\) 1.00000i 0.192450i
\(28\) −1.41752 − 6.12571i −0.267886 − 1.15765i
\(29\) 2.16206 + 2.16206i 0.401484 + 0.401484i 0.878756 − 0.477272i \(-0.158374\pi\)
−0.477272 + 0.878756i \(0.658374\pi\)
\(30\) 3.73034 + 2.96569i 0.681064 + 0.541459i
\(31\) 0.222977 + 0.222977i 0.0400479 + 0.0400479i 0.726847 − 0.686799i \(-0.240985\pi\)
−0.686799 + 0.726847i \(0.740985\pi\)
\(32\) −4.76706 + 3.04551i −0.842706 + 0.538374i
\(33\) −1.73701 −0.302375
\(34\) 0.731627 + 0.581658i 0.125473 + 0.0997535i
\(35\) −7.49099 − 7.49099i −1.26621 − 1.26621i
\(36\) −1.94851 + 0.450896i −0.324752 + 0.0751493i
\(37\) 5.93691 + 1.32403i 0.976022 + 0.217670i
\(38\) 3.79029 + 3.01336i 0.614866 + 0.488831i
\(39\) −2.24578 − 2.24578i −0.359612 − 0.359612i
\(40\) −4.09669 + 8.60583i −0.647744 + 1.36070i
\(41\) 0.621619i 0.0970806i 0.998821 + 0.0485403i \(0.0154569\pi\)
−0.998821 + 0.0485403i \(0.984543\pi\)
\(42\) 4.41728 − 0.504427i 0.681601 − 0.0778347i
\(43\) 6.22159 + 6.22159i 0.948784 + 0.948784i 0.998751 − 0.0499672i \(-0.0159117\pi\)
−0.0499672 + 0.998751i \(0.515912\pi\)
\(44\) −0.783212 − 3.38459i −0.118074 − 0.510246i
\(45\) −2.38279 + 2.38279i −0.355206 + 0.355206i
\(46\) 3.63108 − 4.56728i 0.535373 − 0.673409i
\(47\) 12.9722i 1.89220i 0.323880 + 0.946098i \(0.395013\pi\)
−0.323880 + 0.946098i \(0.604987\pi\)
\(48\) −1.75715 − 3.59339i −0.253623 − 0.518661i
\(49\) −2.88341 −0.411915
\(50\) 1.01973 + 8.92984i 0.144212 + 1.26287i
\(51\) −0.467334 + 0.467334i −0.0654398 + 0.0654398i
\(52\) 3.36331 − 5.38853i 0.466407 − 0.747255i
\(53\) 13.8875i 1.90759i 0.300457 + 0.953795i \(0.402861\pi\)
−0.300457 + 0.953795i \(0.597139\pi\)
\(54\) −0.160452 − 1.40508i −0.0218347 − 0.191207i
\(55\) −4.13894 − 4.13894i −0.558094 − 0.558094i
\(56\) 2.97461 + 8.37967i 0.397500 + 1.11978i
\(57\) −2.42109 + 2.42109i −0.320681 + 0.320681i
\(58\) −3.38477 − 2.69096i −0.444442 − 0.353340i
\(59\) −10.2164 − 10.2164i −1.33006 − 1.33006i −0.905309 − 0.424754i \(-0.860361\pi\)
−0.424754 − 0.905309i \(-0.639639\pi\)
\(60\) −5.71728 − 3.56850i −0.738098 − 0.460692i
\(61\) 0.980438 + 0.980438i 0.125532 + 0.125532i 0.767082 − 0.641549i \(-0.221708\pi\)
−0.641549 + 0.767082i \(0.721708\pi\)
\(62\) −0.349078 − 0.277524i −0.0443330 − 0.0352456i
\(63\) 3.14379i 0.396080i
\(64\) 6.20946 − 5.04407i 0.776182 − 0.630509i
\(65\) − 10.7024i − 1.32747i
\(66\) 2.44065 − 0.278707i 0.300423 − 0.0343064i
\(67\) 3.02006i 0.368959i 0.982836 + 0.184479i \(0.0590599\pi\)
−0.982836 + 0.184479i \(0.940940\pi\)
\(68\) −1.12132 − 0.699886i −0.135980 − 0.0848736i
\(69\) 2.91739 + 2.91739i 0.351213 + 0.351213i
\(70\) 11.7274 + 9.32351i 1.40169 + 1.11437i
\(71\) − 13.7628i − 1.63334i −0.577102 − 0.816672i \(-0.695817\pi\)
0.577102 − 0.816672i \(-0.304183\pi\)
\(72\) 2.66547 − 0.946188i 0.314129 − 0.111509i
\(73\) − 4.11600i − 0.481742i −0.970557 − 0.240871i \(-0.922567\pi\)
0.970557 − 0.240871i \(-0.0774330\pi\)
\(74\) −8.55429 − 0.907788i −0.994416 − 0.105528i
\(75\) −6.35539 −0.733857
\(76\) −5.80917 − 3.62585i −0.666357 − 0.415914i
\(77\) −5.46080 −0.622316
\(78\) 3.51584 + 2.79516i 0.398091 + 0.316490i
\(79\) 7.45631 − 7.45631i 0.838900 − 0.838900i −0.149814 − 0.988714i \(-0.547867\pi\)
0.988714 + 0.149814i \(0.0478675\pi\)
\(80\) 4.37537 − 12.7492i 0.489181 − 1.42541i
\(81\) 1.00000 0.111111
\(82\) −0.0997400 − 0.873426i −0.0110144 − 0.0964538i
\(83\) −3.67031 −0.402869 −0.201435 − 0.979502i \(-0.564560\pi\)
−0.201435 + 0.979502i \(0.564560\pi\)
\(84\) −6.12571 + 1.41752i −0.668369 + 0.154664i
\(85\) −2.22712 −0.241565
\(86\) −9.74011 − 7.74358i −1.05030 − 0.835012i
\(87\) 2.16206 − 2.16206i 0.231797 − 0.231797i
\(88\) 1.64354 + 4.62996i 0.175202 + 0.493555i
\(89\) 2.57508 − 2.57508i 0.272958 − 0.272958i −0.557332 − 0.830290i \(-0.688175\pi\)
0.830290 + 0.557332i \(0.188175\pi\)
\(90\) 2.96569 − 3.73034i 0.312612 − 0.393212i
\(91\) −7.06025 − 7.06025i −0.740116 − 0.740116i
\(92\) −4.36913 + 7.00002i −0.455514 + 0.729802i
\(93\) 0.222977 − 0.222977i 0.0231216 − 0.0231216i
\(94\) −2.08142 − 18.2271i −0.214682 − 1.87998i
\(95\) −11.5379 −1.18376
\(96\) 3.04551 + 4.76706i 0.310831 + 0.486536i
\(97\) −10.4856 − 10.4856i −1.06465 − 1.06465i −0.997760 − 0.0668891i \(-0.978693\pi\)
−0.0668891 − 0.997760i \(-0.521307\pi\)
\(98\) 4.05142 − 0.462648i 0.409256 − 0.0467345i
\(99\) 1.73701i 0.174576i
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.43.3 ✓ 72
8.3 odd 2 inner 888.2.r.e.43.16 yes 72
37.31 odd 4 inner 888.2.r.e.475.16 yes 72
296.179 even 4 inner 888.2.r.e.475.3 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.3 ✓ 72 1.1 even 1 trivial
888.2.r.e.43.16 yes 72 8.3 odd 2 inner
888.2.r.e.475.3 yes 72 296.179 even 4 inner
888.2.r.e.475.16 yes 72 37.31 odd 4 inner