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Properties

Label 888.2.bd.a.803.136
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 803.136
Character \(\chi\) \(=\) 888.803
Dual form 888.2.bd.a.491.136

$q$-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+(1.34708 - 0.430552i) q^{2} +(-1.71798 - 0.220366i) q^{3} +(1.62925 - 1.15998i) q^{4} +(1.48905 - 2.57912i) q^{5} +(-2.40913 + 0.442828i) q^{6} +(-1.43848 - 0.830505i) q^{7} +(1.69530 - 2.26406i) q^{8} +(2.90288 + 0.757165i) q^{9} +(0.895430 - 4.11539i) q^{10} -4.72935i q^{11} +(-3.05463 + 1.63378i) q^{12} +(3.15343 + 1.82064i) q^{13} +(-2.29532 - 0.499418i) q^{14} +(-3.12651 + 4.10272i) q^{15} +(1.30891 - 3.77978i) q^{16} +(-5.81067 + 3.35479i) q^{17} +(4.23641 - 0.229878i) q^{18} +(-2.15641 + 3.73502i) q^{19} +(-0.565675 - 5.92929i) q^{20} +(2.28825 + 1.74378i) q^{21} +(-2.03623 - 6.37082i) q^{22} -1.66390 q^{23} +(-3.41140 + 3.51601i) q^{24} +(-1.93456 - 3.35076i) q^{25} +(5.03181 + 1.09482i) q^{26} +(-4.82022 - 1.94049i) q^{27} +(-3.30701 + 0.315500i) q^{28} +9.50892 q^{29} +(-2.44522 + 6.87282i) q^{30} -8.60573i q^{31} +(0.135810 - 5.65522i) q^{32} +(-1.04219 + 8.12491i) q^{33} +(-6.38303 + 7.02097i) q^{34} +(-4.28394 + 2.47333i) q^{35} +(5.60781 - 2.13366i) q^{36} +(-5.81839 + 1.77378i) q^{37} +(-1.29674 + 5.95981i) q^{38} +(-5.01631 - 3.82271i) q^{39} +(-3.31488 - 7.74368i) q^{40} +(-2.81760 - 1.62674i) q^{41} +(3.83325 + 1.36380i) q^{42} +1.70602 q^{43} +(-5.48594 - 7.70529i) q^{44} +(6.27536 - 6.35940i) q^{45} +(-2.24140 + 0.716394i) q^{46} +5.83269 q^{47} +(-3.08161 + 6.20514i) q^{48} +(-2.12052 - 3.67285i) q^{49} +(-4.04869 - 3.68081i) q^{50} +(10.7219 - 4.48298i) q^{51} +(7.24962 - 0.691639i) q^{52} +(1.04996 + 1.81858i) q^{53} +(-7.32870 - 0.538634i) q^{54} +(-12.1976 - 7.04226i) q^{55} +(-4.31896 + 1.84884i) q^{56} +(4.52773 - 5.94147i) q^{57} +(12.8093 - 4.09409i) q^{58} +(-2.56985 + 1.48370i) q^{59} +(-0.334796 + 10.3110i) q^{60} +(4.57313 + 2.64030i) q^{61} +(-3.70522 - 11.5926i) q^{62} +(-3.54689 - 3.50002i) q^{63} +(-2.25192 - 7.67651i) q^{64} +(9.39126 - 5.42205i) q^{65} +(2.09429 + 11.3936i) q^{66} +(-2.41074 + 4.17552i) q^{67} +(-5.57555 + 12.2060i) q^{68} +(2.85853 + 0.366666i) q^{69} +(-4.70591 + 5.17624i) q^{70} +(2.75475 - 4.77137i) q^{71} +(6.63551 - 5.28866i) q^{72} +0.868664 q^{73} +(-7.07414 + 4.89455i) q^{74} +(2.58514 + 6.18284i) q^{75} +(0.819197 + 8.58666i) q^{76} +(-3.92775 + 6.80307i) q^{77} +(-8.40326 - 2.98972i) q^{78} +(6.32966 + 3.65443i) q^{79} +(-7.79947 - 9.00413i) q^{80} +(7.85340 + 4.39592i) q^{81} +(-4.49592 - 0.978226i) q^{82} +(7.38843 - 4.26571i) q^{83} +(5.75088 + 0.186729i) q^{84} +19.9819i q^{85} +(2.29815 - 0.734532i) q^{86} +(-16.3361 - 2.09544i) q^{87} +(-10.7075 - 8.01767i) q^{88} +(-4.69970 + 2.71337i) q^{89} +(5.71536 - 11.2685i) q^{90} +(-3.02409 - 5.23789i) q^{91} +(-2.71090 + 1.93008i) q^{92} +(-1.89641 + 14.7844i) q^{93} +(7.85710 - 2.51128i) q^{94} +(6.42203 + 11.1233i) q^{95} +(-1.47954 + 9.68561i) q^{96} +8.56691 q^{97} +(-4.43787 - 4.03463i) q^{98} +(3.58090 - 13.7287i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.34708 − 0.430552i 0.952529 − 0.304446i
\(3\) −1.71798 − 0.220366i −0.991873 − 0.127228i
\(4\) 1.62925 − 1.15998i 0.814625 − 0.579988i
\(5\) 1.48905 − 2.57912i 0.665925 − 1.15342i −0.313108 − 0.949717i \(-0.601370\pi\)
0.979034 − 0.203699i \(-0.0652963\pi\)
\(6\) −2.40913 + 0.442828i −0.983523 + 0.180784i
\(7\) −1.43848 − 0.830505i −0.543693 − 0.313901i 0.202881 − 0.979203i \(-0.434970\pi\)
−0.746574 + 0.665302i \(0.768303\pi\)
\(8\) 1.69530 − 2.26406i 0.599379 − 0.800466i
\(9\) 2.90288 + 0.757165i 0.967626 + 0.252388i
\(10\) 0.895430 − 4.11539i 0.283160 − 1.30140i
\(11\) − 4.72935i − 1.42595i −0.701188 − 0.712977i \(-0.747347\pi\)
0.701188 − 0.712977i \(-0.252653\pi\)
\(12\) −3.05463 + 1.63378i −0.881796 + 0.471632i
\(13\) 3.15343 + 1.82064i 0.874605 + 0.504953i 0.868876 − 0.495030i \(-0.164843\pi\)
0.00572914 + 0.999984i \(0.498176\pi\)
\(14\) −2.29532 − 0.499418i −0.613450 − 0.133475i
\(15\) −3.12651 + 4.10272i −0.807261 + 1.05932i
\(16\) 1.30891 − 3.77978i 0.327227 − 0.944946i
\(17\) −5.81067 + 3.35479i −1.40930 + 0.813657i −0.995320 − 0.0966312i \(-0.969193\pi\)
−0.413975 + 0.910288i \(0.635860\pi\)
\(18\) 4.23641 − 0.229878i 0.998531 − 0.0541828i
\(19\) −2.15641 + 3.73502i −0.494715 + 0.856871i −0.999981 − 0.00609196i \(-0.998061\pi\)
0.505267 + 0.862963i \(0.331394\pi\)
\(20\) −0.565675 − 5.92929i −0.126489 − 1.32583i
\(21\) 2.28825 + 1.74378i 0.499338 + 0.380524i
\(22\) −2.03623 − 6.37082i −0.434126 − 1.35826i
\(23\) −1.66390 −0.346946 −0.173473 − 0.984839i \(-0.555499\pi\)
−0.173473 + 0.984839i \(0.555499\pi\)
\(24\) −3.41140 + 3.51601i −0.696350 + 0.717703i
\(25\) −1.93456 − 3.35076i −0.386913 − 0.670152i
\(26\) 5.03181 + 1.09482i 0.986818 + 0.214713i
\(27\) −4.82022 − 1.94049i −0.927652 − 0.373447i
\(28\) −3.30701 + 0.315500i −0.624965 + 0.0596239i
\(29\) 9.50892 1.76576 0.882881 − 0.469597i \(-0.155601\pi\)
0.882881 + 0.469597i \(0.155601\pi\)
\(30\) −2.44522 + 6.87282i −0.446434 + 1.25480i
\(31\) − 8.60573i − 1.54563i −0.634629 − 0.772817i \(-0.718847\pi\)
0.634629 − 0.772817i \(-0.281153\pi\)
\(32\) 0.135810 − 5.65522i 0.0240081 − 0.999712i
\(33\) −1.04219 + 8.12491i −0.181421 + 1.41437i
\(34\) −6.38303 + 7.02097i −1.09468 + 1.20409i
\(35\) −4.28394 + 2.47333i −0.724118 + 0.418070i
\(36\) 5.60781 − 2.13366i 0.934634 − 0.355610i
\(37\) −5.81839 + 1.77378i −0.956538 + 0.291608i
\(38\) −1.29674 + 5.95981i −0.210359 + 0.966809i
\(39\) −5.01631 − 3.82271i −0.803253 − 0.612124i
\(40\) −3.31488 − 7.74368i −0.524129 − 1.22438i
\(41\) −2.81760 − 1.62674i −0.440034 − 0.254054i 0.263578 − 0.964638i \(-0.415097\pi\)
−0.703612 + 0.710584i \(0.748431\pi\)
\(42\) 3.83325 + 1.36380i 0.591483 + 0.210438i
\(43\) 1.70602 0.260166 0.130083 − 0.991503i \(-0.458476\pi\)
0.130083 + 0.991503i \(0.458476\pi\)
\(44\) −5.48594 − 7.70529i −0.827036 − 1.16162i
\(45\) 6.27536 − 6.35940i 0.935475 − 0.948004i
\(46\) −2.24140 + 0.716394i −0.330477 + 0.105627i
\(47\) 5.83269 0.850785 0.425392 − 0.905009i \(-0.360136\pi\)
0.425392 + 0.905009i \(0.360136\pi\)
\(48\) −3.08161 + 6.20514i −0.444792 + 0.895634i
\(49\) −2.12052 − 3.67285i −0.302932 − 0.524693i
\(50\) −4.04869 − 3.68081i −0.572571 − 0.520546i
\(51\) 10.7219 − 4.48298i 1.50136 − 0.627743i
\(52\) 7.24962 − 0.691639i 1.00534 − 0.0959131i
\(53\) 1.04996 + 1.81858i 0.144223 + 0.249801i 0.929083 − 0.369872i \(-0.120598\pi\)
−0.784860 + 0.619673i \(0.787265\pi\)
\(54\) −7.32870 − 0.538634i −0.997310 − 0.0732987i
\(55\) −12.1976 − 7.04226i −1.64472 − 0.949578i
\(56\) −4.31896 + 1.84884i −0.577146 + 0.247062i
\(57\) 4.52773 − 5.94147i 0.599713 − 0.786966i
\(58\) 12.8093 − 4.09409i 1.68194 − 0.537580i
\(59\) −2.56985 + 1.48370i −0.334566 + 0.193162i −0.657866 − 0.753135i \(-0.728541\pi\)
0.323300 + 0.946296i \(0.395208\pi\)
\(60\) −0.334796 + 10.3110i −0.0432220 + 1.33115i
\(61\) 4.57313 + 2.64030i 0.585529 + 0.338056i 0.763328 − 0.646011i \(-0.223564\pi\)
−0.177798 + 0.984067i \(0.556897\pi\)
\(62\) −3.70522 − 11.5926i −0.470563 − 1.47226i
\(63\) −3.54689 − 3.50002i −0.446867 − 0.440961i
\(64\) −2.25192 − 7.67651i −0.281490 − 0.959564i
\(65\) 9.39126 − 5.42205i 1.16484 − 0.672522i
\(66\) 2.09429 + 11.3936i 0.257789 + 1.40246i
\(67\) −2.41074 + 4.17552i −0.294518 + 0.510121i −0.974873 − 0.222763i \(-0.928493\pi\)
0.680354 + 0.732883i \(0.261826\pi\)
\(68\) −5.57555 + 12.2060i −0.676135 + 1.48020i
\(69\) 2.85853 + 0.366666i 0.344127 + 0.0441413i
\(70\) −4.70591 + 5.17624i −0.562464 + 0.618679i
\(71\) 2.75475 − 4.77137i 0.326929 − 0.566257i −0.654972 − 0.755653i \(-0.727320\pi\)
0.981901 + 0.189396i \(0.0606529\pi\)
\(72\) 6.63551 − 5.28866i 0.782003 − 0.623275i
\(73\) 0.868664 0.101669 0.0508347 − 0.998707i \(-0.483812\pi\)
0.0508347 + 0.998707i \(0.483812\pi\)
\(74\) −7.07414 + 4.89455i −0.822352 + 0.568979i
\(75\) 2.58514 + 6.18284i 0.298506 + 0.713933i
\(76\) 0.819197 + 8.58666i 0.0939684 + 0.984958i
\(77\) −3.92775 + 6.80307i −0.447609 + 0.775281i
\(78\) −8.40326 − 2.98972i −0.951481 − 0.338519i
\(79\) 6.32966 + 3.65443i 0.712142 + 0.411155i 0.811854 − 0.583861i \(-0.198459\pi\)
−0.0997117 + 0.995016i \(0.531792\pi\)
\(80\) −7.79947 − 9.00413i −0.872007 − 1.00669i
\(81\) 7.85340 + 4.39592i 0.872600 + 0.488435i
\(82\) −4.49592 − 0.978226i −0.496492 − 0.108027i
\(83\) 7.38843 − 4.26571i 0.810985 − 0.468223i −0.0363127 − 0.999340i \(-0.511561\pi\)
0.847298 + 0.531118i \(0.178228\pi\)
\(84\) 5.75088 + 0.186729i 0.627472 + 0.0203738i
\(85\) 19.9819i 2.16734i
\(86\) 2.29815 − 0.734532i 0.247816 − 0.0792066i
\(87\) −16.3361 − 2.09544i −1.75141 − 0.224655i
\(88\) −10.7075 − 8.01767i −1.14143 − 0.854686i
\(89\) −4.69970 + 2.71337i −0.498167 + 0.287617i −0.727956 − 0.685624i \(-0.759530\pi\)
0.229789 + 0.973240i \(0.426196\pi\)
\(90\) 5.71536 − 11.2685i 0.602452 − 1.18780i
\(91\) −3.02409 − 5.23789i −0.317011 − 0.549080i
\(92\) −2.71090 + 1.93008i −0.282631 + 0.201225i
\(93\) −1.89641 + 14.7844i −0.196648 + 1.53307i
\(94\) 7.85710 − 2.51128i 0.810397 − 0.259018i
\(95\) 6.42203 + 11.1233i 0.658886 + 1.14122i
\(96\) −1.47954 + 9.68561i −0.151004 + 0.988533i
\(97\) 8.56691 0.869838 0.434919 − 0.900470i \(-0.356777\pi\)
0.434919 + 0.900470i \(0.356777\pi\)
\(98\) −4.43787 − 4.03463i −0.448292 − 0.407559i
\(99\) 3.58090 − 13.7287i 0.359894 − 1.37979i
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.803.136 yes 296
3.2 odd 2 inner 888.2.bd.a.803.13 yes 296
8.3 odd 2 inner 888.2.bd.a.803.113 yes 296
24.11 even 2 inner 888.2.bd.a.803.36 yes 296
37.10 even 3 inner 888.2.bd.a.491.36 yes 296
111.47 odd 6 inner 888.2.bd.a.491.113 yes 296
296.195 odd 6 inner 888.2.bd.a.491.13 ✓ 296
888.491 even 6 inner 888.2.bd.a.491.136 yes 296
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bd.a.491.13 ✓ 296 296.195 odd 6 inner
888.2.bd.a.491.36 yes 296 37.10 even 3 inner
888.2.bd.a.491.113 yes 296 111.47 odd 6 inner
888.2.bd.a.491.136 yes 296 888.491 even 6 inner
888.2.bd.a.803.13 yes 296 3.2 odd 2 inner
888.2.bd.a.803.36 yes 296 24.11 even 2 inner
888.2.bd.a.803.113 yes 296 8.3 odd 2 inner
888.2.bd.a.803.136 yes 296 1.1 even 1 trivial