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Properties

Label 888.2.r.e.43.16
Level $888$
Weight $2$
Character 888.43
Analytic conductor $7.091$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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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 43.16
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.16

$q$-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+(-0.160452 + 1.40508i) q^{2} -1.00000i q^{3} +(-1.94851 - 0.450896i) q^{4} +(-2.38279 + 2.38279i) q^{5} +(1.40508 + 0.160452i) q^{6} +3.14379i q^{7} +(0.946188 - 2.66547i) 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} +(-4.41728 - 0.504427i) q^{14} +(2.38279 + 2.38279i) q^{15} +(3.59339 + 1.75715i) q^{16} +(-0.467334 - 0.467334i) q^{17} +(0.160452 - 1.40508i) q^{18} +(-2.42109 - 2.42109i) q^{19} +(5.71728 - 3.56850i) q^{20} +3.14379 q^{21} +(2.44065 + 0.278707i) q^{22} +(2.91739 - 2.91739i) q^{23} +(-2.66547 - 0.946188i) 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} +(-3.04551 + 4.76706i) 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} +(-0.504427 + 4.41728i) 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} +(8.92984 + 1.01973i) q^{50} +(-0.467334 + 0.467334i) q^{51} +(5.38853 - 3.36331i) q^{52} -13.8875i q^{53} +(-1.40508 - 0.160452i) q^{54} +(4.13894 + 4.13894i) q^{55} +(8.37967 + 2.97461i) q^{56} +(-2.42109 + 2.42109i) q^{57} +(3.38477 - 2.69096i) q^{58} +(-10.2164 - 10.2164i) q^{59} +(-3.56850 - 5.71728i) 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} +(0.278707 - 2.44065i) q^{66} +3.02006i q^{67} +(0.699886 + 1.12132i) q^{68} +(-2.91739 - 2.91739i) q^{69} +(11.7274 - 9.32351i) q^{70} +13.7628i q^{71} +(-0.946188 + 2.66547i) q^{72} -4.11600i q^{73} +(2.81296 - 8.12941i) q^{74} -6.35539 q^{75} +(3.62585 + 5.80917i) q^{76} +5.46080 q^{77} +(-3.51584 + 2.79516i) q^{78} +(-7.45631 + 7.45631i) q^{79} +(-12.7492 + 4.37537i) q^{80} +1.00000 q^{81} +(-0.873426 - 0.0997400i) 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} +(-4.62996 - 1.64354i) q^{88} +(2.57508 - 2.57508i) q^{89} +(2.96569 + 3.73034i) q^{90} +(-7.06025 - 7.06025i) q^{91} +(-7.00002 + 4.36913i) q^{92} +(-0.222977 + 0.222977i) q^{93} +(18.2271 + 2.08142i) q^{94} +11.5379 q^{95} +(4.76706 + 3.04551i) q^{96} +(-10.4856 - 10.4856i) q^{97} +(0.462648 - 4.05142i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.160452 + 1.40508i −0.113457 + 0.993543i
\(3\) − 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 \(0.521638\pi\)
−0.997690 + 0.0679263i \(0.978362\pi\)
\(6\) 1.40508 + 0.160452i 0.573622 + 0.0655042i
\(7\) 3.14379i 1.18824i 0.804376 + 0.594120i \(0.202500\pi\)
−0.804376 + 0.594120i \(0.797500\pi\)
\(8\) 0.946188 − 2.66547i 0.334528 − 0.942386i
\(9\) −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 −0.946263 − 0.323397i \(-0.895175\pi\)
0.323397 + 0.946263i \(0.395175\pi\)
\(14\) −4.41728 − 0.504427i −1.18057 − 0.134814i
\(15\) 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 \(0.775539\pi\)
\(18\) 0.160452 − 1.40508i 0.0378189 − 0.331181i
\(19\) −2.42109 − 2.42109i −0.555435 − 0.555435i 0.372569 − 0.928004i \(-0.378477\pi\)
−0.928004 + 0.372569i \(0.878477\pi\)
\(20\) 5.71728 − 3.56850i 1.27842 − 0.797941i
\(21\) 3.14379 0.686031
\(22\) 2.44065 + 0.278707i 0.520347 + 0.0594205i
\(23\) 2.91739 − 2.91739i 0.608319 − 0.608319i −0.334188 − 0.942507i \(-0.608462\pi\)
0.942507 + 0.334188i \(0.108462\pi\)
\(24\) −2.66547 − 0.946188i −0.544087 − 0.193140i
\(25\) − 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 0.477272 − 0.878756i \(-0.341626\pi\)
−0.878756 + 0.477272i \(0.841626\pi\)
\(30\) −3.73034 + 2.96569i −0.681064 + 0.541459i
\(31\) −0.222977 − 0.222977i −0.0400479 − 0.0400479i 0.686799 − 0.726847i \(-0.259015\pi\)
−0.726847 + 0.686799i \(0.759015\pi\)
\(32\) −3.04551 + 4.76706i −0.538374 + 0.842706i
\(33\) −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 \(0.984543\pi\)
\(42\) −0.504427 + 4.41728i −0.0778347 + 0.681601i
\(43\) 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 \(-0.604987\pi\)
0.323880 − 0.946098i \(-0.395013\pi\)
\(48\) 1.75715 − 3.59339i 0.253623 − 0.518661i
\(49\) −2.88341 −0.411915
\(50\) 8.92984 + 1.01973i 1.26287 + 0.144212i
\(51\) −0.467334 + 0.467334i −0.0654398 + 0.0654398i
\(52\) 5.38853 − 3.36331i 0.747255 − 0.466407i
\(53\) − 13.8875i − 1.90759i −0.300457 − 0.953795i \(-0.597139\pi\)
0.300457 − 0.953795i \(-0.402861\pi\)
\(54\) −1.40508 − 0.160452i −0.191207 − 0.0218347i
\(55\) 4.13894 + 4.13894i 0.558094 + 0.558094i
\(56\) 8.37967 + 2.97461i 1.11978 + 0.397500i
\(57\) −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 \(-0.639639\pi\)
\(60\) −3.56850 − 5.71728i −0.460692 − 0.738098i
\(61\) −0.980438 − 0.980438i −0.125532 − 0.125532i 0.641549 − 0.767082i \(-0.278292\pi\)
−0.767082 + 0.641549i \(0.778292\pi\)
\(62\) 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
\(66\) 0.278707 − 2.44065i 0.0343064 − 0.300423i
\(67\) 3.02006i 0.368959i 0.982836 + 0.184479i \(0.0590599\pi\)
−0.982836 + 0.184479i \(0.940940\pi\)
\(68\) 0.699886 + 1.12132i 0.0848736 + 0.135980i
\(69\) −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 \(0.304183\pi\)
−0.577102 + 0.816672i \(0.695817\pi\)
\(72\) −0.946188 + 2.66547i −0.111509 + 0.314129i
\(73\) − 4.11600i − 0.481742i −0.970557 − 0.240871i \(-0.922567\pi\)
0.970557 − 0.240871i \(-0.0774330\pi\)
\(74\) 2.81296 − 8.12941i 0.327000 − 0.945024i
\(75\) −6.35539 −0.733857
\(76\) 3.62585 + 5.80917i 0.415914 + 0.666357i
\(77\) 5.46080 0.622316
\(78\) −3.51584 + 2.79516i −0.398091 + 0.316490i
\(79\) −7.45631 + 7.45631i −0.838900 + 0.838900i −0.988714 − 0.149814i \(-0.952133\pi\)
0.149814 + 0.988714i \(0.452133\pi\)
\(80\) −12.7492 + 4.37537i −1.42541 + 0.489181i
\(81\) 1.00000 0.111111
\(82\) −0.873426 − 0.0997400i −0.0964538 − 0.0110144i
\(83\) −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
\(88\) −4.62996 − 1.64354i −0.493555 − 0.175202i
\(89\) 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
\(92\) −7.00002 + 4.36913i −0.729802 + 0.455514i
\(93\) −0.222977 + 0.222977i −0.0231216 + 0.0231216i
\(94\) 18.2271 + 2.08142i 1.87998 + 0.214682i
\(95\) 11.5379 1.18376
\(96\) 4.76706 + 3.04551i 0.486536 + 0.310831i
\(97\) −10.4856 − 10.4856i −1.06465 − 1.06465i −0.997760 − 0.0668891i \(-0.978693\pi\)
−0.0668891 − 0.997760i \(-0.521307\pi\)
\(98\) 0.462648 − 4.05142i 0.0467345 − 0.409256i
\(99\) 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.43.16 yes 72
8.3 odd 2 inner 888.2.r.e.43.3 ✓ 72
37.31 odd 4 inner 888.2.r.e.475.3 yes 72
296.179 even 4 inner 888.2.r.e.475.16 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.3 ✓ 72 8.3 odd 2 inner
888.2.r.e.43.16 yes 72 1.1 even 1 trivial
888.2.r.e.475.3 yes 72 37.31 odd 4 inner
888.2.r.e.475.16 yes 72 296.179 even 4 inner