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Properties

Label 888.2.bd.a.491.8
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 491.8
Character \(\chi\) \(=\) 888.491
Dual form 888.2.bd.a.803.8

$q$-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+(-1.39656 + 0.222785i) q^{2} +(1.70171 + 0.322773i) q^{3} +(1.90073 - 0.622263i) q^{4} +(-1.38974 - 2.40710i) q^{5} +(-2.44844 - 0.0716546i) q^{6} +(-4.20477 + 2.42763i) q^{7} +(-2.51585 + 1.29248i) q^{8} +(2.79164 + 1.09853i) q^{9} +(2.47712 + 3.05204i) q^{10} +0.115286i q^{11} +(3.43535 - 0.445406i) q^{12} +(1.35586 - 0.782806i) q^{13} +(5.33136 - 4.32707i) q^{14} +(-1.58799 - 4.54476i) q^{15} +(3.22558 - 2.36551i) q^{16} +(-1.06173 - 0.612992i) q^{17} +(-4.14341 - 0.912225i) q^{18} +(3.57483 + 6.19178i) q^{19} +(-4.13938 - 3.71048i) q^{20} +(-7.93887 + 2.77393i) q^{21} +(-0.0256841 - 0.161004i) q^{22} +6.63386 q^{23} +(-4.69842 + 1.38738i) q^{24} +(-1.36277 + 2.36038i) q^{25} +(-1.71914 + 1.39530i) q^{26} +(4.39598 + 2.77045i) q^{27} +(-6.48153 + 7.23074i) q^{28} +9.40115 q^{29} +(3.23022 + 5.99323i) q^{30} +6.34675i q^{31} +(-3.97770 + 4.02218i) q^{32} +(-0.0372113 + 0.196184i) q^{33} +(1.61934 + 0.619540i) q^{34} +(11.6871 + 6.74755i) q^{35} +(5.98973 + 0.350884i) q^{36} +(5.96963 - 1.16768i) q^{37} +(-6.37188 - 7.85075i) q^{38} +(2.55995 - 0.894474i) q^{39} +(6.60751 + 4.25970i) q^{40} +(-0.120665 + 0.0696662i) q^{41} +(10.4691 - 5.64261i) q^{42} +2.51566 q^{43} +(0.0717385 + 0.219129i) q^{44} +(-1.23537 - 8.24643i) q^{45} +(-9.26455 + 1.47792i) q^{46} -6.36372 q^{47} +(6.25252 - 2.98429i) q^{48} +(8.28674 - 14.3531i) q^{49} +(1.37732 - 3.60000i) q^{50} +(-1.60891 - 1.38583i) q^{51} +(2.09002 - 2.33161i) q^{52} +(-1.20636 + 2.08947i) q^{53} +(-6.75644 - 2.88972i) q^{54} +(0.277507 - 0.160218i) q^{55} +(7.44092 - 11.5421i) q^{56} +(4.08478 + 11.6905i) q^{57} +(-13.1292 + 2.09443i) q^{58} +(1.46726 + 0.847126i) q^{59} +(-5.84639 - 7.65024i) q^{60} +(-6.81602 + 3.93523i) q^{61} +(-1.41396 - 8.86359i) q^{62} +(-14.4050 + 2.15798i) q^{63} +(4.65900 - 6.50337i) q^{64} +(-3.76859 - 2.17580i) q^{65} +(0.00826081 - 0.282272i) q^{66} +(-2.97660 - 5.15562i) q^{67} +(-2.39952 - 0.504458i) q^{68} +(11.2889 + 2.14123i) q^{69} +(-17.8249 - 6.81962i) q^{70} +(0.0851746 + 0.147527i) q^{71} +(-8.44316 + 0.844392i) q^{72} +12.9415 q^{73} +(-8.07678 + 2.96068i) q^{74} +(-3.08090 + 3.57682i) q^{75} +(10.6477 + 9.54445i) q^{76} +(-0.279872 - 0.484753i) q^{77} +(-3.37583 + 1.81950i) q^{78} +(5.07991 - 2.93289i) q^{79} +(-10.1768 - 4.47685i) q^{80} +(6.58646 + 6.13340i) q^{81} +(0.152995 - 0.124175i) q^{82} +(-6.09435 - 3.51857i) q^{83} +(-13.3636 + 10.2126i) q^{84} +3.40761i q^{85} +(-3.51326 + 0.560451i) q^{86} +(15.9980 + 3.03443i) q^{87} +(-0.149005 - 0.290043i) q^{88} +(-6.96872 - 4.02340i) q^{89} +(3.56245 + 11.2414i) q^{90} +(-3.80072 + 6.58304i) q^{91} +(12.6092 - 4.12800i) q^{92} +(-2.04856 + 10.8003i) q^{93} +(8.88729 - 1.41774i) q^{94} +(9.93618 - 17.2100i) q^{95} +(-8.06714 + 5.56069i) q^{96} -6.49834 q^{97} +(-8.37525 + 21.8910i) q^{98} +(-0.126646 + 0.321838i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39656 + 0.222785i −0.987514 + 0.157533i
\(3\) 1.70171 + 0.322773i 0.982483 + 0.186353i
\(4\) 1.90073 − 0.622263i 0.950367 − 0.311131i
\(5\) −1.38974 − 2.40710i −0.621512 − 1.07649i −0.989204 − 0.146542i \(-0.953186\pi\)
0.367693 − 0.929947i \(-0.380148\pi\)
\(6\) −2.44844 − 0.0716546i −0.999572 − 0.0292529i
\(7\) −4.20477 + 2.42763i −1.58925 + 0.917556i −0.595824 + 0.803115i \(0.703175\pi\)
−0.993430 + 0.114442i \(0.963492\pi\)
\(8\) −2.51585 + 1.29248i −0.889487 + 0.456960i
\(9\) 2.79164 + 1.09853i 0.930545 + 0.366177i
\(10\) 2.47712 + 3.05204i 0.783333 + 0.965140i
\(11\) 0.115286i 0.0347602i 0.999849 + 0.0173801i \(0.00553253\pi\)
−0.999849 + 0.0173801i \(0.994467\pi\)
\(12\) 3.43535 − 0.445406i 0.991699 − 0.128578i
\(13\) 1.35586 − 0.782806i 0.376048 − 0.217111i −0.300050 − 0.953924i \(-0.597003\pi\)
0.676097 + 0.736812i \(0.263670\pi\)
\(14\) 5.33136 − 4.32707i 1.42487 − 1.15646i
\(15\) −1.58799 − 4.54476i −0.410017 − 1.17345i
\(16\) 3.22558 − 2.36551i 0.806395 − 0.591378i
\(17\) −1.06173 − 0.612992i −0.257508 − 0.148673i 0.365689 − 0.930737i \(-0.380833\pi\)
−0.623197 + 0.782065i \(0.714167\pi\)
\(18\) −4.14341 − 0.912225i −0.976611 − 0.215014i
\(19\) 3.57483 + 6.19178i 0.820122 + 1.42049i 0.905591 + 0.424152i \(0.139428\pi\)
−0.0854693 + 0.996341i \(0.527239\pi\)
\(20\) −4.13938 − 3.71048i −0.925594 − 0.829688i
\(21\) −7.93887 + 2.77393i −1.73240 + 0.605321i
\(22\) −0.0256841 − 0.161004i −0.00547586 − 0.0343262i
\(23\) 6.63386 1.38326 0.691628 − 0.722254i \(-0.256894\pi\)
0.691628 + 0.722254i \(0.256894\pi\)
\(24\) −4.69842 + 1.38738i −0.959062 + 0.283197i
\(25\) −1.36277 + 2.36038i −0.272553 + 0.472076i
\(26\) −1.71914 + 1.39530i −0.337150 + 0.273640i
\(27\) 4.39598 + 2.77045i 0.846007 + 0.533172i
\(28\) −6.48153 + 7.23074i −1.22489 + 1.36648i
\(29\) 9.40115 1.74575 0.872875 − 0.487944i \(-0.162253\pi\)
0.872875 + 0.487944i \(0.162253\pi\)
\(30\) 3.23022 + 5.99323i 0.589755 + 1.09421i
\(31\) 6.34675i 1.13991i 0.821676 + 0.569955i \(0.193039\pi\)
−0.821676 + 0.569955i \(0.806961\pi\)
\(32\) −3.97770 + 4.02218i −0.703164 + 0.711027i
\(33\) −0.0372113 + 0.196184i −0.00647766 + 0.0341513i
\(34\) 1.61934 + 0.619540i 0.277714 + 0.106250i
\(35\) 11.6871 + 6.74755i 1.97548 + 1.14054i
\(36\) 5.98973 + 0.350884i 0.998289 + 0.0584807i
\(37\) 5.96963 − 1.16768i 0.981402 − 0.191966i
\(38\) −6.37188 − 7.85075i −1.03366 − 1.27356i
\(39\) 2.55995 − 0.894474i 0.409920 − 0.143231i
\(40\) 6.60751 + 4.25970i 1.04474 + 0.673517i
\(41\) −0.120665 + 0.0696662i −0.0188448 + 0.0108800i −0.509393 − 0.860534i \(-0.670130\pi\)
0.490548 + 0.871414i \(0.336797\pi\)
\(42\) 10.4691 − 5.64261i 1.61542 − 0.870674i
\(43\) 2.51566 0.383634 0.191817 − 0.981431i \(-0.438562\pi\)
0.191817 + 0.981431i \(0.438562\pi\)
\(44\) 0.0717385 + 0.219129i 0.0108150 + 0.0330349i
\(45\) −1.23537 − 8.24643i −0.184159 − 1.22931i
\(46\) −9.26455 + 1.47792i −1.36598 + 0.217908i
\(47\) −6.36372 −0.928244 −0.464122 − 0.885771i \(-0.653630\pi\)
−0.464122 + 0.885771i \(0.653630\pi\)
\(48\) 6.25252 − 2.98429i 0.902474 − 0.430745i
\(49\) 8.28674 − 14.3531i 1.18382 − 2.05044i
\(50\) 1.37732 − 3.60000i 0.194783 − 0.509117i
\(51\) −1.60891 − 1.38583i −0.225292 − 0.194056i
\(52\) 2.09002 − 2.33161i 0.289833 − 0.323336i
\(53\) −1.20636 + 2.08947i −0.165706 + 0.287011i −0.936906 − 0.349582i \(-0.886323\pi\)
0.771200 + 0.636593i \(0.219657\pi\)
\(54\) −6.75644 − 2.88972i −0.919435 − 0.393241i
\(55\) 0.277507 − 0.160218i 0.0374190 − 0.0216039i
\(56\) 7.44092 − 11.5421i 0.994334 − 1.54238i
\(57\) 4.08478 + 11.6905i 0.541043 + 1.54844i
\(58\) −13.1292 + 2.09443i −1.72395 + 0.275013i
\(59\) 1.46726 + 0.847126i 0.191022 + 0.110286i 0.592461 − 0.805599i \(-0.298157\pi\)
−0.401439 + 0.915886i \(0.631490\pi\)
\(60\) −5.84639 − 7.65024i −0.754765 − 0.987642i
\(61\) −6.81602 + 3.93523i −0.872703 + 0.503855i −0.868245 − 0.496135i \(-0.834752\pi\)
−0.00445716 + 0.999990i \(0.501419\pi\)
\(62\) −1.41396 − 8.86359i −0.179573 − 1.12568i
\(63\) −14.4050 + 2.15798i −1.81486 + 0.271879i
\(64\) 4.65900 − 6.50337i 0.582374 − 0.812921i
\(65\) −3.76859 − 2.17580i −0.467436 − 0.269874i
\(66\) 0.00826081 − 0.282272i 0.00101684 − 0.0347453i
\(67\) −2.97660 − 5.15562i −0.363649 − 0.629859i 0.624909 − 0.780697i \(-0.285136\pi\)
−0.988558 + 0.150839i \(0.951803\pi\)
\(68\) −2.39952 − 0.504458i −0.290984 − 0.0611745i
\(69\) 11.2889 + 2.14123i 1.35902 + 0.257774i
\(70\) −17.8249 − 6.81962i −2.13049 − 0.815100i
\(71\) 0.0851746 + 0.147527i 0.0101084 + 0.0175082i 0.871035 − 0.491220i \(-0.163449\pi\)
−0.860927 + 0.508729i \(0.830116\pi\)
\(72\) −8.44316 + 0.844392i −0.995036 + 0.0995126i
\(73\) 12.9415 1.51469 0.757344 − 0.653017i \(-0.226497\pi\)
0.757344 + 0.653017i \(0.226497\pi\)
\(74\) −8.07678 + 2.96068i −0.938907 + 0.344172i
\(75\) −3.08090 + 3.57682i −0.355751 + 0.413015i
\(76\) 10.6477 + 9.54445i 1.22138 + 1.09482i
\(77\) −0.279872 − 0.484753i −0.0318944 − 0.0552428i
\(78\) −3.37583 + 1.81950i −0.382238 + 0.206018i
\(79\) 5.07991 − 2.93289i 0.571534 − 0.329976i −0.186228 − 0.982507i \(-0.559626\pi\)
0.757762 + 0.652531i \(0.226293\pi\)
\(80\) −10.1768 − 4.47685i −1.13780 − 0.500527i
\(81\) 6.58646 + 6.13340i 0.731829 + 0.681489i
\(82\) 0.152995 − 0.124175i 0.0168955 − 0.0137129i
\(83\) −6.09435 − 3.51857i −0.668941 − 0.386213i 0.126734 − 0.991937i \(-0.459551\pi\)
−0.795675 + 0.605723i \(0.792884\pi\)
\(84\) −13.3636 + 10.2126i −1.45809 + 1.11428i
\(85\) 3.40761i 0.369607i
\(86\) −3.51326 + 0.560451i −0.378844 + 0.0604349i
\(87\) 15.9980 + 3.03443i 1.71517 + 0.325325i
\(88\) −0.149005 − 0.290043i −0.0158840 − 0.0309187i
\(89\) −6.96872 − 4.02340i −0.738683 − 0.426479i 0.0829071 − 0.996557i \(-0.473580\pi\)
−0.821590 + 0.570078i \(0.806913\pi\)
\(90\) 3.56245 + 11.2414i 0.375515 + 1.18494i
\(91\) −3.80072 + 6.58304i −0.398424 + 0.690090i
\(92\) 12.6092 − 4.12800i 1.31460 − 0.430374i
\(93\) −2.04856 + 10.8003i −0.212426 + 1.11994i
\(94\) 8.88729 − 1.41774i 0.916654 − 0.146229i
\(95\) 9.93618 − 17.2100i 1.01943 − 1.76570i
\(96\) −8.06714 + 5.56069i −0.823349 + 0.567536i
\(97\) −6.49834 −0.659806 −0.329903 − 0.944015i \(-0.607016\pi\)
−0.329903 + 0.944015i \(0.607016\pi\)
\(98\) −8.37525 + 21.8910i −0.846028 + 2.21132i
\(99\) −0.126646 + 0.321838i −0.0127284 + 0.0323459i
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.491.8 ✓ 296
3.2 odd 2 inner 888.2.bd.a.491.141 yes 296
8.3 odd 2 inner 888.2.bd.a.491.57 yes 296
24.11 even 2 inner 888.2.bd.a.491.92 yes 296
37.26 even 3 inner 888.2.bd.a.803.92 yes 296
111.26 odd 6 inner 888.2.bd.a.803.57 yes 296
296.211 odd 6 inner 888.2.bd.a.803.141 yes 296
888.803 even 6 inner 888.2.bd.a.803.8 yes 296
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bd.a.491.8 ✓ 296 1.1 even 1 trivial
888.2.bd.a.491.57 yes 296 8.3 odd 2 inner
888.2.bd.a.491.92 yes 296 24.11 even 2 inner
888.2.bd.a.491.141 yes 296 3.2 odd 2 inner
888.2.bd.a.803.8 yes 296 888.803 even 6 inner
888.2.bd.a.803.57 yes 296 111.26 odd 6 inner
888.2.bd.a.803.92 yes 296 37.26 even 3 inner
888.2.bd.a.803.141 yes 296 296.211 odd 6 inner