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Properties

Label 888.2.bh.a.565.8
Level $888$
Weight $2$
Character 888.565
Analytic conductor $7.091$
Analytic rank $0$
Dimension $152$
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(565,888)] 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("888.565"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bh (of order \(6\), degree \(2\), 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: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(76\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 565.8
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.8

$q$-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+(-1.38541 + 0.283957i) q^{2} +(0.866025 - 0.500000i) q^{3} +(1.83874 - 0.786796i) q^{4} +(0.598658 - 0.345635i) q^{5} +(-1.05782 + 0.938621i) q^{6} +(-0.383557 - 0.664340i) q^{7} +(-2.32399 + 1.61216i) q^{8} +(0.500000 - 0.866025i) q^{9} +(-0.731242 + 0.648841i) q^{10} +5.71646i q^{11} +(1.19899 - 1.60075i) q^{12} +(-2.71404 + 1.56695i) q^{13} +(0.720029 + 0.811471i) q^{14} +(0.345635 - 0.598658i) q^{15} +(2.76190 - 2.89342i) q^{16} +(-2.58209 + 4.47231i) q^{17} +(-0.446792 + 1.34178i) q^{18} +(-4.80622 + 2.77487i) q^{19} +(0.828829 - 1.10655i) q^{20} +(-0.664340 - 0.383557i) q^{21} +(-1.62323 - 7.91965i) q^{22} +5.25463 q^{23} +(-1.20656 + 2.55817i) q^{24} +(-2.26107 + 3.91629i) q^{25} +(3.31511 - 2.94154i) q^{26} -1.00000i q^{27} +(-1.22796 - 0.919765i) q^{28} +8.54780i q^{29} +(-0.308854 + 0.927533i) q^{30} -0.573969 q^{31} +(-3.00477 + 4.79285i) q^{32} +(2.85823 + 4.95060i) q^{33} +(2.30731 - 6.92919i) q^{34} +(-0.459239 - 0.265141i) q^{35} +(0.237983 - 1.98579i) q^{36} +(5.92731 - 1.36640i) q^{37} +(5.87066 - 5.20911i) q^{38} +(-1.56695 + 2.71404i) q^{39} +(-0.834056 + 1.76839i) q^{40} +(-2.85874 - 4.95148i) q^{41} +(1.02930 + 0.342740i) q^{42} -6.36994i q^{43} +(4.49769 + 10.5111i) q^{44} -0.691270i q^{45} +(-7.27983 + 1.49209i) q^{46} +7.32681 q^{47} +(0.945168 - 3.88673i) q^{48} +(3.20577 - 5.55255i) q^{49} +(2.02046 - 6.06773i) q^{50} +5.16417i q^{51} +(-3.75753 + 5.01660i) q^{52} +(10.0541 + 5.80472i) q^{53} +(0.283957 + 1.38541i) q^{54} +(1.97581 + 3.42220i) q^{55} +(1.96241 + 0.925566i) q^{56} +(-2.77487 + 4.80622i) q^{57} +(-2.42721 - 11.8422i) q^{58} +(-9.48918 - 5.47858i) q^{59} +(0.164510 - 1.37272i) q^{60} +(-3.75546 + 2.16822i) q^{61} +(0.795185 - 0.162983i) q^{62} -0.767114 q^{63} +(2.80188 - 7.49330i) q^{64} +(-1.08319 + 1.87613i) q^{65} +(-5.36559 - 6.04701i) q^{66} +(-7.04564 + 4.06780i) q^{67} +(-1.22898 + 10.2550i) q^{68} +(4.55064 - 2.62731i) q^{69} +(0.711524 + 0.236926i) q^{70} +(-4.97391 - 8.61507i) q^{71} +(0.234176 + 2.81872i) q^{72} +10.3784 q^{73} +(-7.82376 + 3.57613i) q^{74} +4.52215i q^{75} +(-6.65412 + 8.88378i) q^{76} +(3.79767 - 2.19259i) q^{77} +(1.40020 - 4.20501i) q^{78} +(-0.558627 - 0.967570i) q^{79} +(0.653366 - 2.68678i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(5.36654 + 6.04808i) q^{82} +(6.68104 + 3.85730i) q^{83} +(-1.52333 - 0.182560i) q^{84} +3.56984i q^{85} +(1.80879 + 8.82499i) q^{86} +(4.27390 + 7.40261i) q^{87} +(-9.21585 - 13.2850i) q^{88} +(2.31328 - 4.00672i) q^{89} +(0.196291 + 0.957695i) q^{90} +(2.08197 + 1.20203i) q^{91} +(9.66187 - 4.13432i) q^{92} +(-0.497072 + 0.286985i) q^{93} +(-10.1506 + 2.08050i) q^{94} +(-1.91819 + 3.32240i) q^{95} +(-0.205782 + 5.65311i) q^{96} +13.5569 q^{97} +(-2.86462 + 8.60288i) q^{98} +(4.95060 + 2.85823i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q - 2 q^{2} + 2 q^{4} + 8 q^{7} - 8 q^{8} + 76 q^{9} - 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 6 q^{22} - 16 q^{23} + 80 q^{25} + 6 q^{28} + 8 q^{30} + 8 q^{32} + 2 q^{34} + 4 q^{36} + 76 q^{38}+ \cdots + 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38541 + 0.283957i −0.979635 + 0.200788i
\(3\) 0.866025 − 0.500000i 0.500000 − 0.288675i
\(4\) 1.83874 − 0.786796i 0.919368 − 0.393398i
\(5\) 0.598658 − 0.345635i 0.267728 − 0.154573i −0.360127 − 0.932903i \(-0.617267\pi\)
0.627855 + 0.778331i \(0.283933\pi\)
\(6\) −1.05782 + 0.938621i −0.431855 + 0.383190i
\(7\) −0.383557 − 0.664340i −0.144971 − 0.251097i 0.784391 − 0.620266i \(-0.212976\pi\)
−0.929362 + 0.369169i \(0.879642\pi\)
\(8\) −2.32399 + 1.61216i −0.821655 + 0.569985i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) −0.731242 + 0.648841i −0.231239 + 0.205181i
\(11\) 5.71646i 1.72358i 0.507268 + 0.861789i \(0.330656\pi\)
−0.507268 + 0.861789i \(0.669344\pi\)
\(12\) 1.19899 − 1.60075i 0.346120 − 0.462098i
\(13\) −2.71404 + 1.56695i −0.752738 + 0.434594i −0.826682 − 0.562669i \(-0.809775\pi\)
0.0739441 + 0.997262i \(0.476441\pi\)
\(14\) 0.720029 + 0.811471i 0.192436 + 0.216875i
\(15\) 0.345635 − 0.598658i 0.0892426 − 0.154573i
\(16\) 2.76190 − 2.89342i 0.690476 − 0.723355i
\(17\) −2.58209 + 4.47231i −0.626248 + 1.08469i 0.362050 + 0.932159i \(0.382077\pi\)
−0.988298 + 0.152535i \(0.951256\pi\)
\(18\) −0.446792 + 1.34178i −0.105310 + 0.316261i
\(19\) −4.80622 + 2.77487i −1.10262 + 0.636600i −0.936909 − 0.349574i \(-0.886326\pi\)
−0.165714 + 0.986174i \(0.552993\pi\)
\(20\) 0.828829 − 1.10655i 0.185332 − 0.247433i
\(21\) −0.664340 − 0.383557i −0.144971 − 0.0836990i
\(22\) −1.62323 − 7.91965i −0.346074 − 1.68848i
\(23\) 5.25463 1.09567 0.547833 − 0.836588i \(-0.315453\pi\)
0.547833 + 0.836588i \(0.315453\pi\)
\(24\) −1.20656 + 2.55817i −0.246287 + 0.522184i
\(25\) −2.26107 + 3.91629i −0.452215 + 0.783259i
\(26\) 3.31511 − 2.94154i 0.650147 − 0.576884i
\(27\) − 1.00000i − 0.192450i
\(28\) −1.22796 − 0.919765i −0.232063 − 0.173819i
\(29\) 8.54780i 1.58729i 0.608383 + 0.793644i \(0.291818\pi\)
−0.608383 + 0.793644i \(0.708182\pi\)
\(30\) −0.308854 + 0.927533i −0.0563888 + 0.169344i
\(31\) −0.573969 −0.103088 −0.0515440 − 0.998671i \(-0.516414\pi\)
−0.0515440 + 0.998671i \(0.516414\pi\)
\(32\) −3.00477 + 4.79285i −0.531173 + 0.847263i
\(33\) 2.85823 + 4.95060i 0.497554 + 0.861789i
\(34\) 2.30731 − 6.92919i 0.395701 − 1.18835i
\(35\) −0.459239 − 0.265141i −0.0776255 − 0.0448171i
\(36\) 0.237983 − 1.98579i 0.0396638 − 0.330965i
\(37\) 5.92731 − 1.36640i 0.974443 − 0.224635i
\(38\) 5.87066 − 5.20911i 0.952346 − 0.845029i
\(39\) −1.56695 + 2.71404i −0.250913 + 0.434594i
\(40\) −0.834056 + 1.76839i −0.131876 + 0.279606i
\(41\) −2.85874 − 4.95148i −0.446460 − 0.773291i 0.551693 − 0.834047i \(-0.313982\pi\)
−0.998153 + 0.0607563i \(0.980649\pi\)
\(42\) 1.02930 + 0.342740i 0.158824 + 0.0528860i
\(43\) − 6.36994i − 0.971406i −0.874124 − 0.485703i \(-0.838564\pi\)
0.874124 − 0.485703i \(-0.161436\pi\)
\(44\) 4.49769 + 10.5111i 0.678052 + 1.58460i
\(45\) − 0.691270i − 0.103048i
\(46\) −7.27983 + 1.49209i −1.07335 + 0.219997i
\(47\) 7.32681 1.06872 0.534362 − 0.845256i \(-0.320552\pi\)
0.534362 + 0.845256i \(0.320552\pi\)
\(48\) 0.945168 − 3.88673i 0.136423 − 0.561001i
\(49\) 3.20577 − 5.55255i 0.457967 − 0.793222i
\(50\) 2.02046 − 6.06773i 0.285736 − 0.858107i
\(51\) 5.16417i 0.723129i
\(52\) −3.75753 + 5.01660i −0.521075 + 0.695677i
\(53\) 10.0541 + 5.80472i 1.38103 + 0.797339i 0.992282 − 0.124004i \(-0.0395735\pi\)
0.388750 + 0.921343i \(0.372907\pi\)
\(54\) 0.283957 + 1.38541i 0.0386417 + 0.188531i
\(55\) 1.97581 + 3.42220i 0.266418 + 0.461450i
\(56\) 1.96241 + 0.925566i 0.262237 + 0.123684i
\(57\) −2.77487 + 4.80622i −0.367541 + 0.636600i
\(58\) −2.42721 − 11.8422i −0.318708 − 1.55496i
\(59\) −9.48918 − 5.47858i −1.23539 − 0.713251i −0.267238 − 0.963630i \(-0.586111\pi\)
−0.968148 + 0.250380i \(0.919444\pi\)
\(60\) 0.164510 − 1.37272i 0.0212382 − 0.177217i
\(61\) −3.75546 + 2.16822i −0.480837 + 0.277612i −0.720765 − 0.693179i \(-0.756210\pi\)
0.239928 + 0.970791i \(0.422876\pi\)
\(62\) 0.795185 − 0.162983i 0.100989 − 0.0206988i
\(63\) −0.767114 −0.0966473
\(64\) 2.80188 − 7.49330i 0.350235 − 0.936662i
\(65\) −1.08319 + 1.87613i −0.134353 + 0.232706i
\(66\) −5.36559 − 6.04701i −0.660458 − 0.744335i
\(67\) −7.04564 + 4.06780i −0.860762 + 0.496961i −0.864267 − 0.503033i \(-0.832217\pi\)
0.00350570 + 0.999994i \(0.498884\pi\)
\(68\) −1.22898 + 10.2550i −0.149036 + 1.24360i
\(69\) 4.55064 − 2.62731i 0.547833 − 0.316291i
\(70\) 0.711524 + 0.236926i 0.0850433 + 0.0283181i
\(71\) −4.97391 − 8.61507i −0.590295 − 1.02242i −0.994192 − 0.107617i \(-0.965678\pi\)
0.403898 − 0.914804i \(-0.367655\pi\)
\(72\) 0.234176 + 2.81872i 0.0275979 + 0.332189i
\(73\) 10.3784 1.21470 0.607350 − 0.794434i \(-0.292233\pi\)
0.607350 + 0.794434i \(0.292233\pi\)
\(74\) −7.82376 + 3.57613i −0.909494 + 0.415717i
\(75\) 4.52215i 0.522172i
\(76\) −6.65412 + 8.88378i −0.763280 + 1.01904i
\(77\) 3.79767 − 2.19259i 0.432785 − 0.249868i
\(78\) 1.40020 − 4.20501i 0.158542 − 0.476123i
\(79\) −0.558627 − 0.967570i −0.0628504 − 0.108860i 0.832888 − 0.553442i \(-0.186686\pi\)
−0.895738 + 0.444581i \(0.853352\pi\)
\(80\) 0.653366 − 2.68678i 0.0730486 − 0.300391i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 5.36654 + 6.04808i 0.592635 + 0.667899i
\(83\) 6.68104 + 3.85730i 0.733339 + 0.423393i 0.819642 − 0.572875i \(-0.194172\pi\)
−0.0863034 + 0.996269i \(0.527505\pi\)
\(84\) −1.52333 − 0.182560i −0.166209 − 0.0199189i
\(85\) 3.56984i 0.387204i
\(86\) 1.80879 + 8.82499i 0.195047 + 0.951623i
\(87\) 4.27390 + 7.40261i 0.458210 + 0.793644i
\(88\) −9.21585 − 13.2850i −0.982413 − 1.41619i
\(89\) 2.31328 − 4.00672i 0.245207 − 0.424711i −0.716983 − 0.697091i \(-0.754477\pi\)
0.962190 + 0.272380i \(0.0878107\pi\)
\(90\) 0.196291 + 0.957695i 0.0206909 + 0.100950i
\(91\) 2.08197 + 1.20203i 0.218250 + 0.126007i
\(92\) 9.66187 − 4.13432i 1.00732 − 0.431033i
\(93\) −0.497072 + 0.286985i −0.0515440 + 0.0297589i
\(94\) −10.1506 + 2.08050i −1.04696 + 0.214587i
\(95\) −1.91819 + 3.32240i −0.196802 + 0.340871i
\(96\) −0.205782 + 5.65311i −0.0210025 + 0.576968i
\(97\) 13.5569 1.37650 0.688250 − 0.725474i \(-0.258379\pi\)
0.688250 + 0.725474i \(0.258379\pi\)
\(98\) −2.86462 + 8.60288i −0.289371 + 0.869022i
\(99\) 4.95060 + 2.85823i 0.497554 + 0.287263i
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.bh.a.565.8 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.46 yes 152
37.26 even 3 inner 888.2.bh.a.877.46 yes 152
296.285 even 6 inner 888.2.bh.a.877.8 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.8 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.46 yes 152 8.5 even 2 inner
888.2.bh.a.877.8 yes 152 296.285 even 6 inner
888.2.bh.a.877.46 yes 152 37.26 even 3 inner