↑

Properties

Label 888.2.bd.a.491.113
Level $888$
Weight $2$
Character 888.491
Analytic conductor $7.091$
Analytic rank $0$
Dimension $296$
Inner twists $8$

Related objects

Downloads

Learn more

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 491.113
Character \(\chi\) \(=\) 888.491
Dual form 888.2.bd.a.803.113

$q$-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+(1.04641 + 0.951329i) q^{2} +(-1.71798 + 0.220366i) q^{3} +(0.189944 + 1.99096i) q^{4} +(-1.48905 - 2.57912i) q^{5} +(-2.00735 - 1.40377i) 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} +(-0.765059 - 3.37856i) q^{12} +(-3.15343 + 1.82064i) q^{13} +(2.29532 + 0.499418i) q^{14} +(3.12651 + 4.10272i) q^{15} +(-3.92784 + 0.756344i) q^{16} +(-5.81067 - 3.35479i) q^{17} +(3.75791 + 1.96929i) q^{18} +(-2.15641 - 3.73502i) q^{19} +(4.85208 - 3.45454i) q^{20} +(-2.28825 + 1.74378i) q^{21} +(-4.49917 + 4.94884i) q^{22} +1.66390 q^{23} +(2.41356 - 4.26318i) q^{24} +(-1.93456 + 3.35076i) q^{25} +(-5.03181 - 1.09482i) q^{26} +(-4.82022 + 1.94049i) q^{27} +(1.92673 + 2.70620i) q^{28} -9.50892 q^{29} +(-0.631436 + 7.26747i) q^{30} -8.60573i q^{31} +(-4.82966 - 2.94523i) q^{32} +(-1.04219 - 8.12491i) q^{33} +(-2.88883 - 9.03835i) q^{34} +(-4.28394 - 2.47333i) q^{35} +(2.05887 + 5.63569i) q^{36} +(5.81839 + 1.77378i) q^{37} +(1.29674 - 5.95981i) q^{38} +(5.01631 - 3.82271i) q^{39} +(8.36366 + 1.00107i) q^{40} +(-2.81760 + 1.62674i) q^{41} +(-4.05336 - 0.352177i) q^{42} +1.70602 q^{43} +(-9.41595 + 0.898314i) q^{44} +(-6.27536 - 6.35940i) q^{45} +(1.74112 + 1.58291i) q^{46} -5.83269 q^{47} +(6.58126 - 2.16494i) q^{48} +(-2.12052 + 3.67285i) q^{49} +(-5.21202 + 1.66586i) q^{50} +(10.7219 + 4.48298i) q^{51} +(-4.22379 - 5.93254i) q^{52} +(-1.04996 + 1.81858i) q^{53} +(-6.88996 - 2.55507i) q^{54} +(12.1976 - 7.04226i) q^{55} +(-0.558337 + 4.66475i) q^{56} +(4.52773 + 5.94147i) q^{57} +(-9.95022 - 9.04611i) q^{58} +(-2.56985 - 1.48370i) q^{59} +(-7.57449 + 7.00404i) q^{60} +(-4.57313 + 2.64030i) q^{61} +(8.18688 - 9.00511i) q^{62} +(3.54689 - 3.50002i) q^{63} +(-2.25192 - 7.67651i) q^{64} +(9.39126 + 5.42205i) q^{65} +(6.63891 - 9.49344i) q^{66} +(-2.41074 - 4.17552i) q^{67} +(5.57555 - 12.2060i) q^{68} +(-2.85853 + 0.366666i) q^{69} +(-2.12980 - 6.66356i) q^{70} +(-2.75475 - 4.77137i) q^{71} +(-3.20698 + 7.85591i) q^{72} +0.868664 q^{73} +(4.40097 + 7.39131i) q^{74} +(2.58514 - 6.18284i) q^{75} +(7.02667 - 5.00278i) q^{76} +(3.92775 + 6.80307i) q^{77} +(8.88578 + 0.772044i) 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} +(-3.90643 - 4.22460i) q^{84} +19.9819i q^{85} +(1.78520 + 1.62299i) q^{86} +(16.3361 - 2.09544i) q^{87} +(-10.7075 - 8.01767i) q^{88} +(-4.69970 - 2.71337i) q^{89} +(-0.516708 - 12.6245i) q^{90} +(-3.02409 + 5.23789i) q^{91} +(0.316048 + 3.31275i) q^{92} +(1.89641 + 14.7844i) q^{93} +(-6.10338 - 5.54881i) q^{94} +(-6.42203 + 11.1233i) q^{95} +(8.94627 + 3.99554i) q^{96} +8.56691 q^{97} +(-5.71303 + 1.82599i) 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.04641 + 0.951329i 0.739923 + 0.672692i
\(3\) −1.71798 + 0.220366i −0.991873 + 0.127228i
\(4\) 0.189944 + 1.99096i 0.0949722 + 0.995480i
\(5\) −1.48905 − 2.57912i −0.665925 − 1.15342i −0.979034 − 0.203699i \(-0.934704\pi\)
0.313108 − 0.949717i \(-0.398630\pi\)
\(6\) −2.00735 − 1.40377i −0.819495 − 0.573086i
\(7\) 1.43848 − 0.830505i 0.543693 − 0.313901i −0.202881 − 0.979203i \(-0.565030\pi\)
0.746574 + 0.665302i \(0.231697\pi\)
\(8\) −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 \(0.252653\pi\)
−0.701188 + 0.712977i \(0.747347\pi\)
\(12\) −0.765059 − 3.37856i −0.220854 − 0.975307i
\(13\) −3.15343 + 1.82064i −0.874605 + 0.504953i −0.868876 − 0.495030i \(-0.835157\pi\)
−0.00572914 + 0.999984i \(0.501824\pi\)
\(14\) 2.29532 + 0.499418i 0.613450 + 0.133475i
\(15\) 3.12651 + 4.10272i 0.807261 + 1.05932i
\(16\) −3.92784 + 0.756344i −0.981961 + 0.189086i
\(17\) −5.81067 − 3.35479i −1.40930 − 0.813657i −0.413975 − 0.910288i \(-0.635860\pi\)
−0.995320 + 0.0966312i \(0.969193\pi\)
\(18\) 3.75791 + 1.96929i 0.885748 + 0.464166i
\(19\) −2.15641 − 3.73502i −0.494715 − 0.856871i 0.505267 − 0.862963i \(-0.331394\pi\)
−0.999981 + 0.00609196i \(0.998061\pi\)
\(20\) 4.85208 − 3.45454i 1.08496 − 0.772458i
\(21\) −2.28825 + 1.74378i −0.499338 + 0.380524i
\(22\) −4.49917 + 4.94884i −0.959227 + 1.05510i
\(23\) 1.66390 0.346946 0.173473 − 0.984839i \(-0.444501\pi\)
0.173473 + 0.984839i \(0.444501\pi\)
\(24\) 2.41356 − 4.26318i 0.492666 − 0.870218i
\(25\) −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
\(28\) 1.92673 + 2.70620i 0.364118 + 0.511424i
\(29\) −9.50892 −1.76576 −0.882881 − 0.469597i \(-0.844399\pi\)
−0.882881 + 0.469597i \(0.844399\pi\)
\(30\) −0.631436 + 7.26747i −0.115284 + 1.32685i
\(31\) − 8.60573i − 1.54563i −0.634629 − 0.772817i \(-0.718847\pi\)
0.634629 − 0.772817i \(-0.281153\pi\)
\(32\) −4.82966 − 2.94523i −0.853772 − 0.520648i
\(33\) −1.04219 − 8.12491i −0.181421 − 1.41437i
\(34\) −2.88883 − 9.03835i −0.495430 − 1.55006i
\(35\) −4.28394 − 2.47333i −0.724118 − 0.418070i
\(36\) 2.05887 + 5.63569i 0.343145 + 0.939282i
\(37\) 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
\(40\) 8.36366 + 1.00107i 1.32241 + 0.158283i
\(41\) −2.81760 + 1.62674i −0.440034 + 0.254054i −0.703612 − 0.710584i \(-0.748431\pi\)
0.263578 + 0.964638i \(0.415097\pi\)
\(42\) −4.05336 − 0.352177i −0.625447 − 0.0543421i
\(43\) 1.70602 0.260166 0.130083 − 0.991503i \(-0.458476\pi\)
0.130083 + 0.991503i \(0.458476\pi\)
\(44\) −9.41595 + 0.898314i −1.41951 + 0.135426i
\(45\) −6.27536 − 6.35940i −0.935475 − 0.948004i
\(46\) 1.74112 + 1.58291i 0.256714 + 0.233388i
\(47\) −5.83269 −0.850785 −0.425392 − 0.905009i \(-0.639864\pi\)
−0.425392 + 0.905009i \(0.639864\pi\)
\(48\) 6.58126 − 2.16494i 0.949924 − 0.312482i
\(49\) −2.12052 + 3.67285i −0.302932 + 0.524693i
\(50\) −5.21202 + 1.66586i −0.737092 + 0.235588i
\(51\) 10.7219 + 4.48298i 1.50136 + 0.627743i
\(52\) −4.22379 − 5.93254i −0.585734 − 0.822695i
\(53\) −1.04996 + 1.81858i −0.144223 + 0.249801i −0.929083 − 0.369872i \(-0.879402\pi\)
0.784860 + 0.619673i \(0.212735\pi\)
\(54\) −6.88996 − 2.55507i −0.937605 − 0.347702i
\(55\) 12.1976 − 7.04226i 1.64472 − 0.949578i
\(56\) −0.558337 + 4.66475i −0.0746109 + 0.623354i
\(57\) 4.52773 + 5.94147i 0.599713 + 0.786966i
\(58\) −9.95022 − 9.04611i −1.30653 − 1.18781i
\(59\) −2.56985 − 1.48370i −0.334566 − 0.193162i 0.323300 − 0.946296i \(-0.395208\pi\)
−0.657866 + 0.753135i \(0.728541\pi\)
\(60\) −7.57449 + 7.00404i −0.977863 + 0.904218i
\(61\) −4.57313 + 2.64030i −0.585529 + 0.338056i −0.763328 − 0.646011i \(-0.776436\pi\)
0.177798 + 0.984067i \(0.443103\pi\)
\(62\) 8.18688 − 9.00511i 1.03974 − 1.14365i
\(63\) 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
\(66\) 6.63891 − 9.49344i 0.817194 − 1.16856i
\(67\) −2.41074 − 4.17552i −0.294518 − 0.510121i 0.680354 − 0.732883i \(-0.261826\pi\)
−0.974873 + 0.222763i \(0.928493\pi\)
\(68\) 5.57555 − 12.2060i 0.676135 − 1.48020i
\(69\) −2.85853 + 0.366666i −0.344127 + 0.0441413i
\(70\) −2.12980 − 6.66356i −0.254560 − 0.796448i
\(71\) −2.75475 − 4.77137i −0.326929 − 0.566257i 0.654972 − 0.755653i \(-0.272680\pi\)
−0.981901 + 0.189396i \(0.939347\pi\)
\(72\) −3.20698 + 7.85591i −0.377946 + 0.925828i
\(73\) 0.868664 0.101669 0.0508347 − 0.998707i \(-0.483812\pi\)
0.0508347 + 0.998707i \(0.483812\pi\)
\(74\) 4.40097 + 7.39131i 0.511603 + 0.859222i
\(75\) 2.58514 − 6.18284i 0.298506 − 0.713933i
\(76\) 7.02667 − 5.00278i 0.806014 − 0.573858i
\(77\) 3.92775 + 6.80307i 0.447609 + 0.775281i
\(78\) 8.88578 + 0.772044i 1.00612 + 0.0874167i
\(79\) −6.32966 + 3.65443i −0.712142 + 0.411155i −0.811854 − 0.583861i \(-0.801541\pi\)
0.0997117 + 0.995016i \(0.468208\pi\)
\(80\) 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 0.847298 − 0.531118i \(-0.178228\pi\)
−0.0363127 + 0.999340i \(0.511561\pi\)
\(84\) −3.90643 − 4.22460i −0.426227 − 0.460942i
\(85\) 19.9819i 2.16734i
\(86\) 1.78520 + 1.62299i 0.192503 + 0.175011i
\(87\) 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 0.229789 − 0.973240i \(-0.426196\pi\)
−0.727956 + 0.685624i \(0.759530\pi\)
\(90\) −0.516708 − 12.6245i −0.0544658 − 1.33074i
\(91\) −3.02409 + 5.23789i −0.317011 + 0.549080i
\(92\) 0.316048 + 3.31275i 0.0329503 + 0.345378i
\(93\) 1.89641 + 14.7844i 0.196648 + 1.53307i
\(94\) −6.10338 − 5.54881i −0.629515 − 0.572316i
\(95\) −6.42203 + 11.1233i −0.658886 + 1.14122i
\(96\) 8.94627 + 3.99554i 0.913075 + 0.407793i
\(97\) 8.56691 0.869838 0.434919 − 0.900470i \(-0.356777\pi\)
0.434919 + 0.900470i \(0.356777\pi\)
\(98\) −5.71303 + 1.82599i −0.577103 + 0.184453i
\(99\) 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.491.113 yes 296
3.2 odd 2 inner 888.2.bd.a.491.36 yes 296
8.3 odd 2 inner 888.2.bd.a.491.136 yes 296
24.11 even 2 inner 888.2.bd.a.491.13 ✓ 296
37.26 even 3 inner 888.2.bd.a.803.13 yes 296
111.26 odd 6 inner 888.2.bd.a.803.136 yes 296
296.211 odd 6 inner 888.2.bd.a.803.36 yes 296
888.803 even 6 inner 888.2.bd.a.803.113 yes 296
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bd.a.491.13 ✓ 296 24.11 even 2 inner
888.2.bd.a.491.36 yes 296 3.2 odd 2 inner
888.2.bd.a.491.113 yes 296 1.1 even 1 trivial
888.2.bd.a.491.136 yes 296 8.3 odd 2 inner
888.2.bd.a.803.13 yes 296 37.26 even 3 inner
888.2.bd.a.803.36 yes 296 296.211 odd 6 inner
888.2.bd.a.803.113 yes 296 888.803 even 6 inner
888.2.bd.a.803.136 yes 296 111.26 odd 6 inner