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Properties

Label 888.2.bd.a.491.20
Level $888$
Weight $2$
Character 888.491
Analytic conductor $7.091$
Analytic rank $0$
Dimension $296$
Inner twists $8$

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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(491,888)] 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("888.491"); 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([3, 3, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bd (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: \(296\)
Relative dimension: \(148\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 491.20
Character \(\chi\) \(=\) 888.491
Dual form 888.2.bd.a.803.20

$q$-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+(-1.30831 - 0.536950i) q^{2} +(-1.71402 + 0.249255i) q^{3} +(1.42337 + 1.40500i) q^{4} +(1.11601 + 1.93298i) q^{5} +(2.37632 + 0.594242i) q^{6} +(3.43766 - 1.98473i) q^{7} +(-1.10780 - 2.60246i) q^{8} +(2.87574 - 0.854457i) q^{9} +(-0.422173 - 3.12819i) q^{10} -4.51766i q^{11} +(-2.78989 - 2.05342i) q^{12} +(-0.278602 + 0.160851i) q^{13} +(-5.56324 + 0.750801i) q^{14} +(-2.39467 - 3.03501i) q^{15} +(0.0519552 + 3.99966i) q^{16} +(-2.17888 - 1.25797i) q^{17} +(-4.22118 - 0.426235i) q^{18} +(-1.70805 - 2.95843i) q^{19} +(-1.12735 + 4.31934i) q^{20} +(-5.39752 + 4.25873i) q^{21} +(-2.42576 + 5.91051i) q^{22} +5.13641 q^{23} +(2.54746 + 4.18455i) q^{24} +(0.00904640 - 0.0156688i) q^{25} +(0.450868 - 0.0608480i) q^{26} +(-4.71611 + 2.18135i) q^{27} +(7.68161 + 2.00490i) q^{28} -4.78789 q^{29} +(1.50333 + 5.25656i) q^{30} -3.44124i q^{31} +(2.07965 - 5.26071i) q^{32} +(1.12605 + 7.74336i) q^{33} +(2.17518 + 2.81577i) q^{34} +(7.67292 + 4.42996i) q^{35} +(5.29375 + 2.82421i) q^{36} +(-5.46659 - 2.66765i) q^{37} +(0.646136 + 4.78770i) q^{38} +(0.437437 - 0.345145i) q^{39} +(3.79420 - 5.04572i) q^{40} +(-4.67813 + 2.70092i) q^{41} +(9.34838 - 2.67356i) q^{42} +0.569160 q^{43} +(6.34730 - 6.43029i) q^{44} +(4.86101 + 4.60519i) q^{45} +(-6.72003 - 2.75800i) q^{46} +2.89366 q^{47} +(-1.08599 - 6.84256i) q^{48} +(4.37834 - 7.58351i) q^{49} +(-0.0202489 + 0.0156423i) q^{50} +(4.04820 + 1.61310i) q^{51} +(-0.622549 - 0.162485i) q^{52} +(7.24197 - 12.5435i) q^{53} +(7.34143 - 0.321572i) q^{54} +(8.73256 - 5.04175i) q^{55} +(-8.97342 - 6.74768i) q^{56} +(3.66505 + 4.64508i) q^{57} +(6.26406 + 2.57086i) q^{58} +(-9.25399 - 5.34279i) q^{59} +(0.855685 - 7.68444i) q^{60} +(-6.81036 + 3.93196i) q^{61} +(-1.84778 + 4.50223i) q^{62} +(8.18996 - 8.64492i) q^{63} +(-5.54557 + 5.76599i) q^{64} +(-0.621845 - 0.359022i) q^{65} +(2.68458 - 10.7354i) q^{66} +(7.76424 + 13.4481i) q^{67} +(-1.33389 - 4.85188i) q^{68} +(-8.80392 + 1.28027i) q^{69} +(-7.65992 - 9.91576i) q^{70} +(-7.17106 - 12.4206i) q^{71} +(-5.40943 - 6.53744i) q^{72} -0.955104 q^{73} +(5.71962 + 6.42541i) q^{74} +(-0.0116002 + 0.0291116i) q^{75} +(1.72541 - 6.61076i) q^{76} +(-8.96635 - 15.5302i) q^{77} +(-0.757631 + 0.216676i) q^{78} +(-1.79343 + 1.03544i) q^{79} +(-7.67330 + 4.56409i) q^{80} +(7.53981 - 4.91440i) q^{81} +(7.57072 - 1.02173i) q^{82} +(5.63211 + 3.25170i) q^{83} +(-13.6662 - 1.52177i) q^{84} -5.61564i q^{85} +(-0.744640 - 0.305611i) q^{86} +(8.20655 - 1.19340i) q^{87} +(-11.7570 + 5.00465i) q^{88} +(8.36289 + 4.82832i) q^{89} +(-3.88697 - 8.63515i) q^{90} +(-0.638493 + 1.10590i) q^{91} +(7.31100 + 7.21665i) q^{92} +(0.857747 + 5.89837i) q^{93} +(-3.78582 - 1.55375i) q^{94} +(3.81241 - 6.60328i) q^{95} +(-2.25330 + 9.53534i) q^{96} +1.67808 q^{97} +(-9.80022 + 7.57066i) q^{98} +(-3.86014 - 12.9916i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 296 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9} + 10 q^{12} + 6 q^{16} - 2 q^{18} - 4 q^{19} - 6 q^{22} + 16 q^{24} - 136 q^{25} - 8 q^{27} - 2 q^{28} - 4 q^{30} + 4 q^{33} - 18 q^{34} + 20 q^{36} + 6 q^{40}+ \cdots - 16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.30831 − 0.536950i −0.925117 − 0.379681i
\(3\) −1.71402 + 0.249255i −0.989591 + 0.143907i
\(4\) 1.42337 + 1.40500i 0.711684 + 0.702500i
\(5\) 1.11601 + 1.93298i 0.499095 + 0.864457i 0.999999 − 0.00104522i \(-0.000332704\pi\)
−0.500905 + 0.865502i \(0.666999\pi\)
\(6\) 2.37632 + 0.594242i 0.970127 + 0.242598i
\(7\) 3.43766 − 1.98473i 1.29931 − 0.750159i 0.319028 − 0.947745i \(-0.396643\pi\)
0.980286 + 0.197586i \(0.0633102\pi\)
\(8\) −1.10780 − 2.60246i −0.391665 − 0.920108i
\(9\) 2.87574 − 0.854457i 0.958581 − 0.284819i
\(10\) −0.422173 − 3.12819i −0.133503 − 0.989221i
\(11\) − 4.51766i − 1.36212i −0.732225 − 0.681062i \(-0.761518\pi\)
0.732225 − 0.681062i \(-0.238482\pi\)
\(12\) −2.78989 − 2.05342i −0.805371 − 0.592771i
\(13\) −0.278602 + 0.160851i −0.0772703 + 0.0446120i −0.538137 − 0.842857i \(-0.680872\pi\)
0.460867 + 0.887469i \(0.347538\pi\)
\(14\) −5.56324 + 0.750801i −1.48684 + 0.200660i
\(15\) −2.39467 − 3.03501i −0.618301 − 0.783636i
\(16\) 0.0519552 + 3.99966i 0.0129888 + 0.999916i
\(17\) −2.17888 − 1.25797i −0.528455 − 0.305104i 0.211932 − 0.977284i \(-0.432024\pi\)
−0.740387 + 0.672181i \(0.765358\pi\)
\(18\) −4.22118 − 0.426235i −0.994941 − 0.100465i
\(19\) −1.70805 − 2.95843i −0.391854 − 0.678712i 0.600840 − 0.799369i \(-0.294833\pi\)
−0.992694 + 0.120658i \(0.961500\pi\)
\(20\) −1.12735 + 4.31934i −0.252083 + 0.965834i
\(21\) −5.39752 + 4.25873i −1.17784 + 0.929332i
\(22\) −2.42576 + 5.91051i −0.517173 + 1.26013i
\(23\) 5.13641 1.07102 0.535508 − 0.844530i \(-0.320120\pi\)
0.535508 + 0.844530i \(0.320120\pi\)
\(24\) 2.54746 + 4.18455i 0.519999 + 0.854167i
\(25\) 0.00904640 − 0.0156688i 0.00180928 − 0.00313376i
\(26\) 0.450868 − 0.0608480i 0.0884225 − 0.0119333i
\(27\) −4.71611 + 2.18135i −0.907616 + 0.419801i
\(28\) 7.68161 + 2.00490i 1.45169 + 0.378891i
\(29\) −4.78789 −0.889088 −0.444544 − 0.895757i \(-0.646634\pi\)
−0.444544 + 0.895757i \(0.646634\pi\)
\(30\) 1.50333 + 5.25656i 0.274469 + 0.959712i
\(31\) − 3.44124i − 0.618066i −0.951051 − 0.309033i \(-0.899995\pi\)
0.951051 − 0.309033i \(-0.100005\pi\)
\(32\) 2.07965 − 5.26071i 0.367633 − 0.929971i
\(33\) 1.12605 + 7.74336i 0.196020 + 1.34795i
\(34\) 2.17518 + 2.81577i 0.373041 + 0.482901i
\(35\) 7.67292 + 4.42996i 1.29696 + 0.748801i
\(36\) 5.29375 + 2.82421i 0.882292 + 0.470702i
\(37\) −5.46659 − 2.66765i −0.898702 − 0.438559i
\(38\) 0.646136 + 4.78770i 0.104817 + 0.776668i
\(39\) 0.437437 − 0.345145i 0.0700460 − 0.0552674i
\(40\) 3.79420 − 5.04572i 0.599916 − 0.797799i
\(41\) −4.67813 + 2.70092i −0.730602 + 0.421813i −0.818642 − 0.574304i \(-0.805273\pi\)
0.0880407 + 0.996117i \(0.471939\pi\)
\(42\) 9.34838 − 2.67356i 1.44249 − 0.412539i
\(43\) 0.569160 0.0867961 0.0433981 − 0.999058i \(-0.486182\pi\)
0.0433981 + 0.999058i \(0.486182\pi\)
\(44\) 6.34730 − 6.43029i 0.956892 − 0.969403i
\(45\) 4.86101 + 4.60519i 0.724636 + 0.686501i
\(46\) −6.72003 − 2.75800i −0.990815 − 0.406644i
\(47\) 2.89366 0.422084 0.211042 − 0.977477i \(-0.432314\pi\)
0.211042 + 0.977477i \(0.432314\pi\)
\(48\) −1.08599 − 6.84256i −0.156749 − 0.987639i
\(49\) 4.37834 − 7.58351i 0.625478 − 1.08336i
\(50\) −0.0202489 + 0.0156423i −0.00286363 + 0.00221215i
\(51\) 4.04820 + 1.61310i 0.566861 + 0.225879i
\(52\) −0.622549 − 0.162485i −0.0863320 − 0.0225327i
\(53\) 7.24197 − 12.5435i 0.994761 − 1.72298i 0.408844 − 0.912604i \(-0.365932\pi\)
0.585916 − 0.810372i \(-0.300735\pi\)
\(54\) 7.34143 − 0.321572i 0.999042 − 0.0437604i
\(55\) 8.73256 − 5.04175i 1.17750 − 0.679829i
\(56\) −8.97342 − 6.74768i −1.19912 − 0.901697i
\(57\) 3.66505 + 4.64508i 0.485447 + 0.615256i
\(58\) 6.26406 + 2.57086i 0.822511 + 0.337570i
\(59\) −9.25399 − 5.34279i −1.20477 − 0.695572i −0.243155 − 0.969987i \(-0.578182\pi\)
−0.961611 + 0.274415i \(0.911516\pi\)
\(60\) 0.855685 − 7.68444i 0.110469 − 0.992058i
\(61\) −6.81036 + 3.93196i −0.871977 + 0.503436i −0.868005 − 0.496556i \(-0.834598\pi\)
−0.00397203 + 0.999992i \(0.501264\pi\)
\(62\) −1.84778 + 4.50223i −0.234668 + 0.571783i
\(63\) 8.18996 − 8.64492i 1.03184 − 1.08916i
\(64\) −5.54557 + 5.76599i −0.693196 + 0.720749i
\(65\) −0.621845 − 0.359022i −0.0771304 − 0.0445312i
\(66\) 2.68458 − 10.7354i 0.330449 − 1.32143i
\(67\) 7.76424 + 13.4481i 0.948553 + 1.64294i 0.748477 + 0.663161i \(0.230786\pi\)
0.200076 + 0.979780i \(0.435881\pi\)
\(68\) −1.33389 − 4.85188i −0.161758 − 0.588377i
\(69\) −8.80392 + 1.28027i −1.05987 + 0.154127i
\(70\) −7.65992 − 9.91576i −0.915535 − 1.18516i
\(71\) −7.17106 − 12.4206i −0.851048 − 1.47406i −0.880264 − 0.474485i \(-0.842634\pi\)
0.0292159 − 0.999573i \(-0.490699\pi\)
\(72\) −5.40943 − 6.53744i −0.637507 − 0.770444i
\(73\) −0.955104 −0.111786 −0.0558932 − 0.998437i \(-0.517801\pi\)
−0.0558932 + 0.998437i \(0.517801\pi\)
\(74\) 5.71962 + 6.42541i 0.664893 + 0.746939i
\(75\) −0.0116002 + 0.0291116i −0.00133948 + 0.00336151i
\(76\) 1.72541 − 6.61076i 0.197918 − 0.758306i
\(77\) −8.96635 − 15.5302i −1.02181 − 1.76983i
\(78\) −0.757631 + 0.216676i −0.0857848 + 0.0245337i
\(79\) −1.79343 + 1.03544i −0.201777 + 0.116496i −0.597484 − 0.801881i \(-0.703833\pi\)
0.395707 + 0.918377i \(0.370499\pi\)
\(80\) −7.67330 + 4.56409i −0.857902 + 0.510281i
\(81\) 7.53981 − 4.91440i 0.837756 − 0.546044i
\(82\) 7.57072 − 1.02173i 0.836047 − 0.112831i
\(83\) 5.63211 + 3.25170i 0.618204 + 0.356920i 0.776169 − 0.630524i \(-0.217160\pi\)
−0.157965 + 0.987445i \(0.550493\pi\)
\(84\) −13.6662 − 1.52177i −1.49110 − 0.166039i
\(85\) − 5.61564i − 0.609102i
\(86\) −0.744640 − 0.305611i −0.0802966 − 0.0329549i
\(87\) 8.20655 − 1.19340i 0.879834 − 0.127946i
\(88\) −11.7570 + 5.00465i −1.25330 + 0.533497i
\(89\) 8.36289 + 4.82832i 0.886465 + 0.511801i 0.872784 − 0.488106i \(-0.162312\pi\)
0.0136802 + 0.999906i \(0.495645\pi\)
\(90\) −3.88697 − 8.63515i −0.409722 − 0.910225i
\(91\) −0.638493 + 1.10590i −0.0669323 + 0.115930i
\(92\) 7.31100 + 7.21665i 0.762225 + 0.752388i
\(93\) 0.857747 + 5.89837i 0.0889442 + 0.611632i
\(94\) −3.78582 − 1.55375i −0.390477 − 0.160257i
\(95\) 3.81241 − 6.60328i 0.391145 − 0.677482i
\(96\) −2.25330 + 9.53534i −0.229977 + 0.973196i
\(97\) 1.67808 0.170384 0.0851918 − 0.996365i \(-0.472850\pi\)
0.0851918 + 0.996365i \(0.472850\pi\)
\(98\) −9.80022 + 7.57066i −0.989971 + 0.764752i
\(99\) −3.86014 − 12.9916i −0.387959 − 1.30571i
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.bd.a.491.20 ✓ 296
3.2 odd 2 inner 888.2.bd.a.491.129 yes 296
8.3 odd 2 inner 888.2.bd.a.491.30 yes 296
24.11 even 2 inner 888.2.bd.a.491.119 yes 296
37.26 even 3 inner 888.2.bd.a.803.119 yes 296
111.26 odd 6 inner 888.2.bd.a.803.30 yes 296
296.211 odd 6 inner 888.2.bd.a.803.129 yes 296
888.803 even 6 inner 888.2.bd.a.803.20 yes 296
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bd.a.491.20 ✓ 296 1.1 even 1 trivial
888.2.bd.a.491.30 yes 296 8.3 odd 2 inner
888.2.bd.a.491.119 yes 296 24.11 even 2 inner
888.2.bd.a.491.129 yes 296 3.2 odd 2 inner
888.2.bd.a.803.20 yes 296 888.803 even 6 inner
888.2.bd.a.803.30 yes 296 111.26 odd 6 inner
888.2.bd.a.803.119 yes 296 37.26 even 3 inner
888.2.bd.a.803.129 yes 296 296.211 odd 6 inner