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Properties

Label 888.2.bd.a.803.69
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 803.69
Character \(\chi\) \(=\) 888.803
Dual form 888.2.bd.a.491.69

$q$-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+(-0.190711 + 1.40130i) q^{2} +(-0.0101877 + 1.73202i) q^{3} +(-1.92726 - 0.534486i) q^{4} +(1.50279 - 2.60290i) q^{5} +(-2.42513 - 0.344592i) q^{6} +(1.55855 + 0.899828i) q^{7} +(1.11652 - 2.59873i) q^{8} +(-2.99979 - 0.0352908i) q^{9} +(3.36084 + 2.60225i) q^{10} -3.07058i q^{11} +(0.945376 - 3.33261i) q^{12} +(-1.79669 - 1.03732i) q^{13} +(-1.55816 + 2.01238i) q^{14} +(4.49297 + 2.62938i) q^{15} +(3.42865 + 2.06019i) q^{16} +(-0.939594 + 0.542475i) q^{17} +(0.621548 - 4.19687i) q^{18} +(2.16099 - 3.74294i) q^{19} +(-4.28748 + 4.21325i) q^{20} +(-1.57440 + 2.69027i) q^{21} +(4.30280 + 0.585596i) q^{22} +8.01801 q^{23} +(4.48967 + 1.96032i) q^{24} +(-2.01674 - 3.49310i) q^{25} +(1.79625 - 2.31987i) q^{26} +(0.0916855 - 5.19534i) q^{27} +(-2.52278 - 2.56722i) q^{28} +8.21860 q^{29} +(-4.54140 + 5.79453i) q^{30} -4.14463i q^{31} +(-3.54081 + 4.41165i) q^{32} +(5.31832 + 0.0312823i) q^{33} +(-0.580976 - 1.42010i) q^{34} +(4.68433 - 2.70450i) q^{35} +(5.76251 + 1.67136i) q^{36} +(-1.00247 + 5.99959i) q^{37} +(4.83284 + 3.74200i) q^{38} +(1.81497 - 3.10135i) q^{39} +(-5.08634 - 6.81154i) q^{40} +(3.98862 + 2.30283i) q^{41} +(-3.46961 - 2.71926i) q^{42} +3.12583 q^{43} +(-1.64119 + 5.91781i) q^{44} +(-4.59991 + 7.75514i) q^{45} +(-1.52913 + 11.2356i) q^{46} -8.20124 q^{47} +(-3.60322 + 5.91750i) q^{48} +(-1.88062 - 3.25733i) q^{49} +(5.27948 - 2.15988i) q^{50} +(-0.930005 - 1.63292i) q^{51} +(2.90826 + 2.95950i) q^{52} +(3.67734 + 6.36933i) q^{53} +(7.26273 + 1.11929i) q^{54} +(-7.99244 - 4.61444i) q^{55} +(4.07856 - 3.04556i) q^{56} +(6.46083 + 3.78101i) q^{57} +(-1.56738 + 11.5167i) q^{58} +(11.3861 - 6.57374i) q^{59} +(-7.25376 - 7.46892i) q^{60} +(-6.39037 - 3.68948i) q^{61} +(5.80786 + 0.790429i) q^{62} +(-4.64356 - 2.75430i) q^{63} +(-5.50675 - 5.80308i) q^{64} +(-5.40010 + 3.11775i) q^{65} +(-1.05810 + 7.44657i) q^{66} +(3.38827 - 5.86866i) q^{67} +(2.10078 - 0.543289i) q^{68} +(-0.0816854 + 13.8874i) q^{69} +(2.89645 + 7.07991i) q^{70} +(-2.63046 + 4.55609i) q^{71} +(-3.44105 + 7.75623i) q^{72} -10.0820 q^{73} +(-8.21601 - 2.54895i) q^{74} +(6.07066 - 3.45745i) q^{75} +(-6.16533 + 6.05859i) q^{76} +(2.76300 - 4.78565i) q^{77} +(3.99977 + 3.13477i) q^{78} +(2.81621 + 1.62594i) q^{79} +(10.5150 - 5.82842i) q^{80} +(8.99751 + 0.211730i) q^{81} +(-3.98762 + 5.15006i) q^{82} +(-11.1671 + 6.44730i) q^{83} +(4.47218 - 4.34335i) q^{84} +3.26090i q^{85} +(-0.596132 + 4.38021i) q^{86} +(-0.0837290 + 14.2348i) q^{87} +(-7.97961 - 3.42838i) q^{88} +(10.4253 - 6.01902i) q^{89} +(-9.98998 - 7.92483i) q^{90} +(-1.86682 - 3.23343i) q^{91} +(-15.4528 - 4.28552i) q^{92} +(7.17859 + 0.0422245i) q^{93} +(1.56407 - 11.4924i) q^{94} +(-6.49501 - 11.2497i) q^{95} +(-7.60499 - 6.17771i) q^{96} -16.3711 q^{97} +(4.92314 - 2.01409i) q^{98} +(-0.108363 + 9.21112i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.190711 + 1.40130i −0.134853 + 0.990866i
\(3\) −0.0101877 + 1.73202i −0.00588190 + 0.999983i
\(4\) −1.92726 − 0.534486i −0.963629 − 0.267243i
\(5\) 1.50279 − 2.60290i 0.672067 − 1.16405i −0.305250 − 0.952272i \(-0.598740\pi\)
0.977317 − 0.211782i \(-0.0679267\pi\)
\(6\) −2.42513 − 0.344592i −0.990055 − 0.140679i
\(7\) 1.55855 + 0.899828i 0.589075 + 0.340103i 0.764732 − 0.644349i \(-0.222871\pi\)
−0.175656 + 0.984452i \(0.556205\pi\)
\(8\) 1.11652 − 2.59873i 0.394751 − 0.918788i
\(9\) −2.99979 − 0.0352908i −0.999931 − 0.0117636i
\(10\) 3.36084 + 2.60225i 1.06279 + 0.822905i
\(11\) − 3.07058i − 0.925816i −0.886406 − 0.462908i \(-0.846806\pi\)
0.886406 − 0.462908i \(-0.153194\pi\)
\(12\) 0.945376 − 3.33261i 0.272907 − 0.962041i
\(13\) −1.79669 − 1.03732i −0.498314 − 0.287701i 0.229703 − 0.973261i \(-0.426224\pi\)
−0.728017 + 0.685559i \(0.759558\pi\)
\(14\) −1.55816 + 2.01238i −0.416435 + 0.537831i
\(15\) 4.49297 + 2.62938i 1.16008 + 0.678902i
\(16\) 3.42865 + 2.06019i 0.857162 + 0.515047i
\(17\) −0.939594 + 0.542475i −0.227885 + 0.131569i −0.609596 − 0.792712i \(-0.708668\pi\)
0.381711 + 0.924282i \(0.375335\pi\)
\(18\) 0.621548 − 4.19687i 0.146500 − 0.989211i
\(19\) 2.16099 − 3.74294i 0.495764 − 0.858689i −0.504224 − 0.863573i \(-0.668221\pi\)
0.999988 + 0.00488395i \(0.00155462\pi\)
\(20\) −4.28748 + 4.21325i −0.958709 + 0.942111i
\(21\) −1.57440 + 2.69027i −0.343562 + 0.587065i
\(22\) 4.30280 + 0.585596i 0.917359 + 0.124849i
\(23\) 8.01801 1.67187 0.835935 − 0.548828i \(-0.184926\pi\)
0.835935 + 0.548828i \(0.184926\pi\)
\(24\) 4.48967 + 1.96032i 0.916450 + 0.400148i
\(25\) −2.01674 − 3.49310i −0.403348 − 0.698619i
\(26\) 1.79625 − 2.31987i 0.352273 − 0.454964i
\(27\) 0.0916855 − 5.19534i 0.0176449 − 0.999844i
\(28\) −2.52278 − 2.56722i −0.476760 − 0.485159i
\(29\) 8.21860 1.52616 0.763078 − 0.646306i \(-0.223687\pi\)
0.763078 + 0.646306i \(0.223687\pi\)
\(30\) −4.54140 + 5.79453i −0.829142 + 1.05793i
\(31\) − 4.14463i − 0.744398i −0.928153 − 0.372199i \(-0.878604\pi\)
0.928153 − 0.372199i \(-0.121396\pi\)
\(32\) −3.54081 + 4.41165i −0.625933 + 0.779877i
\(33\) 5.31832 + 0.0312823i 0.925800 + 0.00544556i
\(34\) −0.580976 − 1.42010i −0.0996366 − 0.243546i
\(35\) 4.68433 − 2.70450i 0.791796 − 0.457144i
\(36\) 5.76251 + 1.67136i 0.960419 + 0.278560i
\(37\) −1.00247 + 5.99959i −0.164805 + 0.986326i
\(38\) 4.83284 + 3.74200i 0.783990 + 0.607033i
\(39\) 1.81497 − 3.10135i 0.290627 − 0.496613i
\(40\) −5.08634 − 6.81154i −0.804220 − 1.07700i
\(41\) 3.98862 + 2.30283i 0.622917 + 0.359642i 0.778004 − 0.628259i \(-0.216232\pi\)
−0.155087 + 0.987901i \(0.549566\pi\)
\(42\) −3.46961 − 2.71926i −0.535372 − 0.419591i
\(43\) 3.12583 0.476685 0.238342 − 0.971181i \(-0.423396\pi\)
0.238342 + 0.971181i \(0.423396\pi\)
\(44\) −1.64119 + 5.91781i −0.247418 + 0.892143i
\(45\) −4.59991 + 7.75514i −0.685714 + 1.15607i
\(46\) −1.52913 + 11.2356i −0.225457 + 1.65660i
\(47\) −8.20124 −1.19627 −0.598137 − 0.801394i \(-0.704092\pi\)
−0.598137 + 0.801394i \(0.704092\pi\)
\(48\) −3.60322 + 5.91750i −0.520079 + 0.854118i
\(49\) −1.88062 − 3.25733i −0.268660 − 0.465333i
\(50\) 5.27948 − 2.15988i 0.746631 − 0.305453i
\(51\) −0.930005 − 1.63292i −0.130227 − 0.228655i
\(52\) 2.90826 + 2.95950i 0.403303 + 0.410408i
\(53\) 3.67734 + 6.36933i 0.505121 + 0.874895i 0.999982 + 0.00592342i \(0.00188549\pi\)
−0.494861 + 0.868972i \(0.664781\pi\)
\(54\) 7.26273 + 1.11929i 0.988332 + 0.152316i
\(55\) −7.99244 − 4.61444i −1.07770 − 0.622210i
\(56\) 4.07856 − 3.04556i 0.545020 − 0.406980i
\(57\) 6.46083 + 3.78101i 0.855758 + 0.500807i
\(58\) −1.56738 + 11.5167i −0.205807 + 1.51222i
\(59\) 11.3861 − 6.57374i 1.48234 − 0.855829i 0.482540 − 0.875874i \(-0.339714\pi\)
0.999799 + 0.0200454i \(0.00638109\pi\)
\(60\) −7.25376 − 7.46892i −0.936456 − 0.964234i
\(61\) −6.39037 − 3.68948i −0.818203 − 0.472390i 0.0315934 − 0.999501i \(-0.489942\pi\)
−0.849796 + 0.527111i \(0.823275\pi\)
\(62\) 5.80786 + 0.790429i 0.737599 + 0.100385i
\(63\) −4.64356 − 2.75430i −0.585034 − 0.347009i
\(64\) −5.50675 − 5.80308i −0.688344 − 0.725385i
\(65\) −5.40010 + 3.11775i −0.669800 + 0.386709i
\(66\) −1.05810 + 7.44657i −0.130243 + 0.916609i
\(67\) 3.38827 − 5.86866i 0.413943 − 0.716971i −0.581374 − 0.813637i \(-0.697485\pi\)
0.995317 + 0.0966661i \(0.0308179\pi\)
\(68\) 2.10078 − 0.543289i 0.254758 − 0.0658834i
\(69\) −0.0816854 + 13.8874i −0.00983377 + 1.67184i
\(70\) 2.89645 + 7.07991i 0.346192 + 0.846211i
\(71\) −2.63046 + 4.55609i −0.312178 + 0.540709i −0.978834 − 0.204657i \(-0.934392\pi\)
0.666655 + 0.745366i \(0.267725\pi\)
\(72\) −3.44105 + 7.75623i −0.405532 + 0.914081i
\(73\) −10.0820 −1.18001 −0.590004 − 0.807400i \(-0.700874\pi\)
−0.590004 + 0.807400i \(0.700874\pi\)
\(74\) −8.21601 − 2.54895i −0.955092 − 0.296309i
\(75\) 6.07066 − 3.45745i 0.700980 − 0.399232i
\(76\) −6.16533 + 6.05859i −0.707212 + 0.694968i
\(77\) 2.76300 − 4.78565i 0.314873 − 0.545376i
\(78\) 3.99977 + 3.13477i 0.452884 + 0.354943i
\(79\) 2.81621 + 1.62594i 0.316848 + 0.182932i 0.649987 − 0.759946i \(-0.274774\pi\)
−0.333139 + 0.942878i \(0.608108\pi\)
\(80\) 10.5150 − 5.82842i 1.17561 − 0.651637i
\(81\) 8.99751 + 0.211730i 0.999723 + 0.0235256i
\(82\) −3.98762 + 5.15006i −0.440359 + 0.568729i
\(83\) −11.1671 + 6.44730i −1.22574 + 0.707683i −0.966137 − 0.258031i \(-0.916926\pi\)
−0.259607 + 0.965714i \(0.583593\pi\)
\(84\) 4.47218 − 4.34335i 0.487955 − 0.473898i
\(85\) 3.26090i 0.353694i
\(86\) −0.596132 + 4.38021i −0.0642825 + 0.472330i
\(87\) −0.0837290 + 14.2348i −0.00897669 + 1.52613i
\(88\) −7.97961 − 3.42838i −0.850629 − 0.365467i
\(89\) 10.4253 − 6.01902i 1.10507 − 0.638015i 0.167525 − 0.985868i \(-0.446422\pi\)
0.937549 + 0.347853i \(0.113089\pi\)
\(90\) −9.98998 − 7.92483i −1.05304 − 0.835350i
\(91\) −1.86682 − 3.23343i −0.195696 − 0.338956i
\(92\) −15.4528 − 4.28552i −1.61106 − 0.446796i
\(93\) 7.17859 + 0.0422245i 0.744385 + 0.00437847i
\(94\) 1.56407 − 11.4924i 0.161322 − 1.18535i
\(95\) −6.49501 − 11.2497i −0.666374 − 1.15419i
\(96\) −7.60499 − 6.17771i −0.776182 − 0.630509i
\(97\) −16.3711 −1.66223 −0.831117 − 0.556097i \(-0.812298\pi\)
−0.831117 + 0.556097i \(0.812298\pi\)
\(98\) 4.92314 − 2.01409i 0.497312 − 0.203454i
\(99\) −0.108363 + 9.21112i −0.0108909 + 0.925752i
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.803.69 yes 296
3.2 odd 2 inner 888.2.bd.a.803.80 yes 296
8.3 odd 2 inner 888.2.bd.a.803.19 yes 296
24.11 even 2 inner 888.2.bd.a.803.130 yes 296
37.10 even 3 inner 888.2.bd.a.491.130 yes 296
111.47 odd 6 inner 888.2.bd.a.491.19 ✓ 296
296.195 odd 6 inner 888.2.bd.a.491.80 yes 296
888.491 even 6 inner 888.2.bd.a.491.69 yes 296
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bd.a.491.19 ✓ 296 111.47 odd 6 inner
888.2.bd.a.491.69 yes 296 888.491 even 6 inner
888.2.bd.a.491.80 yes 296 296.195 odd 6 inner
888.2.bd.a.491.130 yes 296 37.10 even 3 inner
888.2.bd.a.803.19 yes 296 8.3 odd 2 inner
888.2.bd.a.803.69 yes 296 1.1 even 1 trivial
888.2.bd.a.803.80 yes 296 3.2 odd 2 inner
888.2.bd.a.803.130 yes 296 24.11 even 2 inner