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Properties

Label 888.2.bd.a.803.119
Level $888$
Weight $2$
Character 888.803
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 803.119
Character \(\chi\) \(=\) 888.803
Dual form 888.2.bd.a.491.119

$q$-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+(1.11917 - 0.864557i) q^{2} +(1.07287 + 1.35976i) q^{3} +(0.505081 - 1.93517i) 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} +(-0.697891 + 2.91770i) q^{9} +(-0.422173 - 3.12819i) q^{10} -4.51766i q^{11} +(3.17326 - 1.38940i) q^{12} +(0.278602 + 0.160851i) q^{13} +(-5.56324 + 0.750801i) q^{14} +(3.82573 - 0.556341i) q^{15} +(-3.48979 - 1.95484i) q^{16} +(2.17888 - 1.25797i) q^{17} +(1.74146 + 3.86876i) q^{18} +(-1.70805 + 2.95843i) q^{19} +(-3.17698 - 3.13598i) q^{20} +(-0.989409 - 6.80376i) q^{21} +(-3.90577 - 5.05602i) q^{22} +5.13641 q^{23} +(2.35019 - 4.29844i) q^{24} +(0.00904640 + 0.0156688i) q^{25} +(0.450868 - 0.0608480i) q^{26} +(-4.71611 + 2.18135i) q^{27} +(-5.57710 + 5.65002i) q^{28} -4.78789 q^{29} +(3.80065 - 3.93020i) q^{30} -3.44124i q^{31} +(-5.59573 + 0.829328i) q^{32} +(6.14293 - 4.84687i) q^{33} +(1.35094 - 3.29165i) 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} +(0.0801858 + 0.551404i) q^{39} +(-6.26682 - 0.763012i) q^{40} +(4.67813 + 2.70092i) q^{41} +(-6.98956 - 6.75916i) q^{42} +0.569160 q^{43} +(-8.74245 - 2.28178i) q^{44} +(4.86101 + 4.60519i) q^{45} +(5.74851 - 4.44072i) q^{46} +2.89366 q^{47} +(-1.08599 - 6.84256i) q^{48} +(4.37834 + 7.58351i) q^{49} +(0.0236711 + 0.00971494i) q^{50} +(4.04820 + 1.61310i) q^{51} +(0.451991 - 0.457900i) q^{52} +(7.24197 + 12.5435i) q^{53} +(-3.39223 + 6.51865i) q^{54} +(-8.73256 - 5.04175i) q^{55} +(-1.35696 + 11.1451i) q^{56} +(-5.85528 + 0.851481i) q^{57} +(-5.35846 + 4.13940i) q^{58} +(9.25399 - 5.34279i) q^{59} +(0.855685 - 7.68444i) q^{60} +(6.81036 + 3.93196i) q^{61} +(-2.97515 - 3.85134i) 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} +(5.51071 + 6.98428i) q^{69} +(-4.75734 + 11.5916i) q^{70} +(-7.17106 + 12.4206i) q^{71} +(8.36630 - 1.41598i) q^{72} -0.955104 q^{73} +(3.81171 - 7.71174i) q^{74} +(-0.0116002 + 0.0291116i) q^{75} +(4.86238 + 4.79963i) q^{76} +(-8.96635 + 15.5302i) q^{77} +(0.566462 + 0.547790i) q^{78} +(1.79343 + 1.03544i) q^{79} +(-7.67330 + 4.56409i) q^{80} +(-8.02590 - 4.07247i) q^{81} +(7.57072 - 1.02173i) q^{82} +(-5.63211 + 3.25170i) q^{83} +(-13.6662 - 1.52177i) q^{84} -5.61564i q^{85} +(0.636987 - 0.492072i) q^{86} +(-5.13679 - 6.51038i) q^{87} +(-11.7570 + 5.00465i) q^{88} +(-8.36289 + 4.82832i) q^{89} +(9.42174 + 0.951365i) q^{90} +(-0.638493 - 1.10590i) q^{91} +(2.59430 - 9.93984i) q^{92} +(4.67926 - 3.69202i) q^{93} +(3.23850 - 2.50174i) q^{94} +(3.81241 + 6.60328i) q^{95} +(-7.13119 - 6.71909i) q^{96} +1.67808 q^{97} +(11.4565 + 4.70191i) q^{98} +(13.1811 + 3.15283i) 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.11917 − 0.864557i 0.791372 − 0.611334i
\(3\) 1.07287 + 1.35976i 0.619423 + 0.785057i
\(4\) 0.505081 − 1.93517i 0.252540 − 0.967586i
\(5\) 1.11601 − 1.93298i 0.499095 − 0.864457i −0.500905 − 0.865502i \(-0.666999\pi\)
0.999999 + 0.00104522i \(0.000332704\pi\)
\(6\) 2.37632 + 0.594242i 0.970127 + 0.242598i
\(7\) −3.43766 − 1.98473i −1.29931 − 0.750159i −0.319028 − 0.947745i \(-0.603357\pi\)
−0.980286 + 0.197586i \(0.936690\pi\)
\(8\) −1.10780 − 2.60246i −0.391665 − 0.920108i
\(9\) −0.697891 + 2.91770i −0.232630 + 0.972565i
\(10\) −0.422173 − 3.12819i −0.133503 − 0.989221i
\(11\) − 4.51766i − 1.36212i −0.732225 − 0.681062i \(-0.761518\pi\)
0.732225 − 0.681062i \(-0.238482\pi\)
\(12\) 3.17326 − 1.38940i 0.916040 − 0.401087i
\(13\) 0.278602 + 0.160851i 0.0772703 + 0.0446120i 0.538137 − 0.842857i \(-0.319128\pi\)
−0.460867 + 0.887469i \(0.652462\pi\)
\(14\) −5.56324 + 0.750801i −1.48684 + 0.200660i
\(15\) 3.82573 − 0.556341i 0.987799 − 0.143647i
\(16\) −3.48979 − 1.95484i −0.872447 − 0.488709i
\(17\) 2.17888 − 1.25797i 0.528455 − 0.305104i −0.211932 − 0.977284i \(-0.567976\pi\)
0.740387 + 0.672181i \(0.234642\pi\)
\(18\) 1.74146 + 3.86876i 0.410465 + 0.911876i
\(19\) −1.70805 + 2.95843i −0.391854 + 0.678712i −0.992694 − 0.120658i \(-0.961500\pi\)
0.600840 + 0.799369i \(0.294833\pi\)
\(20\) −3.17698 − 3.13598i −0.710395 − 0.701227i
\(21\) −0.989409 − 6.80376i −0.215907 − 1.48470i
\(22\) −3.90577 − 5.05602i −0.832714 − 1.07795i
\(23\) 5.13641 1.07102 0.535508 − 0.844530i \(-0.320120\pi\)
0.535508 + 0.844530i \(0.320120\pi\)
\(24\) 2.35019 − 4.29844i 0.479731 − 0.877416i
\(25\) 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
\(28\) −5.57710 + 5.65002i −1.05397 + 1.06775i
\(29\) −4.78789 −0.889088 −0.444544 − 0.895757i \(-0.646634\pi\)
−0.444544 + 0.895757i \(0.646634\pi\)
\(30\) 3.80065 − 3.93020i 0.693901 − 0.717554i
\(31\) − 3.44124i − 0.618066i −0.951051 − 0.309033i \(-0.899995\pi\)
0.951051 − 0.309033i \(-0.100005\pi\)
\(32\) −5.59573 + 0.829328i −0.989195 + 0.146606i
\(33\) 6.14293 − 4.84687i 1.06935 − 0.843731i
\(34\) 1.35094 − 3.29165i 0.231684 − 0.564513i
\(35\) −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
\(39\) 0.0801858 + 0.551404i 0.0128400 + 0.0882954i
\(40\) −6.26682 − 0.763012i −0.990872 − 0.120643i
\(41\) 4.67813 + 2.70092i 0.730602 + 0.421813i 0.818642 − 0.574304i \(-0.194727\pi\)
−0.0880407 + 0.996117i \(0.528061\pi\)
\(42\) −6.98956 − 6.75916i −1.07851 − 1.04296i
\(43\) 0.569160 0.0867961 0.0433981 − 0.999058i \(-0.486182\pi\)
0.0433981 + 0.999058i \(0.486182\pi\)
\(44\) −8.74245 − 2.28178i −1.31797 − 0.343991i
\(45\) 4.86101 + 4.60519i 0.724636 + 0.686501i
\(46\) 5.74851 − 4.44072i 0.847572 − 0.654749i
\(47\) 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
\(50\) 0.0236711 + 0.00971494i 0.00334759 + 0.00137390i
\(51\) 4.04820 + 1.61310i 0.566861 + 0.225879i
\(52\) 0.451991 − 0.457900i 0.0626799 − 0.0634994i
\(53\) 7.24197 + 12.5435i 0.994761 + 1.72298i 0.585916 + 0.810372i \(0.300735\pi\)
0.408844 + 0.912604i \(0.365932\pi\)
\(54\) −3.39223 + 6.51865i −0.461623 + 0.887076i
\(55\) −8.73256 − 5.04175i −1.17750 − 0.679829i
\(56\) −1.35696 + 11.1451i −0.181331 + 1.48932i
\(57\) −5.85528 + 0.851481i −0.775551 + 0.112781i
\(58\) −5.35846 + 4.13940i −0.703600 + 0.543530i
\(59\) 9.25399 − 5.34279i 1.20477 − 0.695572i 0.243155 − 0.969987i \(-0.421818\pi\)
0.961611 + 0.274415i \(0.0884842\pi\)
\(60\) 0.855685 − 7.68444i 0.110469 − 0.992058i
\(61\) 6.81036 + 3.93196i 0.871977 + 0.503436i 0.868005 − 0.496556i \(-0.165402\pi\)
0.00397203 + 0.999992i \(0.498736\pi\)
\(62\) −2.97515 − 3.85134i −0.377845 − 0.489120i
\(63\) 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 0.200076 − 0.979780i \(-0.435881\pi\)
0.748477 − 0.663161i \(-0.230786\pi\)
\(68\) −1.33389 − 4.85188i −0.161758 − 0.588377i
\(69\) 5.51071 + 6.98428i 0.663411 + 0.840808i
\(70\) −4.75734 + 11.5916i −0.568611 + 1.38546i
\(71\) −7.17106 + 12.4206i −0.851048 + 1.47406i 0.0292159 + 0.999573i \(0.490699\pi\)
−0.880264 + 0.474485i \(0.842634\pi\)
\(72\) 8.36630 − 1.41598i 0.985978 − 0.166875i
\(73\) −0.955104 −0.111786 −0.0558932 − 0.998437i \(-0.517801\pi\)
−0.0558932 + 0.998437i \(0.517801\pi\)
\(74\) 3.81171 − 7.71174i 0.443102 − 0.896471i
\(75\) −0.0116002 + 0.0291116i −0.00133948 + 0.00336151i
\(76\) 4.86238 + 4.79963i 0.557753 + 0.550555i
\(77\) −8.96635 + 15.5302i −1.02181 + 1.76983i
\(78\) 0.566462 + 0.547790i 0.0641392 + 0.0620250i
\(79\) 1.79343 + 1.03544i 0.201777 + 0.116496i 0.597484 − 0.801881i \(-0.296167\pi\)
−0.395707 + 0.918377i \(0.629501\pi\)
\(80\) −7.67330 + 4.56409i −0.857902 + 0.510281i
\(81\) −8.02590 − 4.07247i −0.891766 − 0.452496i
\(82\) 7.57072 − 1.02173i 0.836047 − 0.112831i
\(83\) −5.63211 + 3.25170i −0.618204 + 0.356920i −0.776169 − 0.630524i \(-0.782840\pi\)
0.157965 + 0.987445i \(0.449507\pi\)
\(84\) −13.6662 − 1.52177i −1.49110 − 0.166039i
\(85\) − 5.61564i − 0.609102i
\(86\) 0.636987 − 0.492072i 0.0686881 − 0.0530615i
\(87\) −5.13679 − 6.51038i −0.550722 − 0.697985i
\(88\) −11.7570 + 5.00465i −1.25330 + 0.533497i
\(89\) −8.36289 + 4.82832i −0.886465 + 0.511801i −0.872784 − 0.488106i \(-0.837688\pi\)
−0.0136802 + 0.999906i \(0.504355\pi\)
\(90\) 9.42174 + 0.951365i 0.993139 + 0.100283i
\(91\) −0.638493 − 1.10590i −0.0669323 − 0.115930i
\(92\) 2.59430 − 9.93984i 0.270475 − 1.03630i
\(93\) 4.67926 − 3.69202i 0.485217 − 0.382844i
\(94\) 3.23850 − 2.50174i 0.334026 − 0.258035i
\(95\) 3.81241 + 6.60328i 0.391145 + 0.677482i
\(96\) −7.13119 − 6.71909i −0.727824 − 0.685764i
\(97\) 1.67808 0.170384 0.0851918 − 0.996365i \(-0.472850\pi\)
0.0851918 + 0.996365i \(0.472850\pi\)
\(98\) 11.4565 + 4.70191i 1.15728 + 0.474964i
\(99\) 13.1811 + 3.15283i 1.32476 + 0.316872i
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.803.119 yes 296
3.2 odd 2 inner 888.2.bd.a.803.30 yes 296
8.3 odd 2 inner 888.2.bd.a.803.129 yes 296
24.11 even 2 inner 888.2.bd.a.803.20 yes 296
37.10 even 3 inner 888.2.bd.a.491.20 ✓ 296
111.47 odd 6 inner 888.2.bd.a.491.129 yes 296
296.195 odd 6 inner 888.2.bd.a.491.30 yes 296
888.491 even 6 inner 888.2.bd.a.491.119 yes 296
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bd.a.491.20 ✓ 296 37.10 even 3 inner
888.2.bd.a.491.30 yes 296 296.195 odd 6 inner
888.2.bd.a.491.119 yes 296 888.491 even 6 inner
888.2.bd.a.491.129 yes 296 111.47 odd 6 inner
888.2.bd.a.803.20 yes 296 24.11 even 2 inner
888.2.bd.a.803.30 yes 296 3.2 odd 2 inner
888.2.bd.a.803.119 yes 296 1.1 even 1 trivial
888.2.bd.a.803.129 yes 296 8.3 odd 2 inner