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Properties

Label 888.2.bd.a.491.19
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 491.19
Character \(\chi\) \(=\) 888.491
Dual form 888.2.bd.a.803.19

$q$-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+(-1.30891 + 0.535487i) q^{2} +(-0.0101877 - 1.73202i) q^{3} +(1.42651 - 1.40181i) q^{4} +(-1.50279 - 2.60290i) q^{5} +(0.940809 + 2.26161i) 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} +(-2.44250 - 2.45646i) q^{12} +(1.79669 - 1.03732i) q^{13} +(1.55816 - 2.01238i) q^{14} +(-4.49297 + 2.62938i) q^{15} +(0.0698494 - 3.99939i) q^{16} +(-0.939594 - 0.542475i) q^{17} +(3.90757 - 1.65254i) q^{18} +(2.16099 + 3.74294i) q^{19} +(-5.79252 - 1.60644i) q^{20} +(1.57440 + 2.69027i) q^{21} +(-1.64426 - 4.01913i) q^{22} -8.01801 q^{23} +(4.51242 + 1.90737i) q^{24} +(-2.01674 + 3.49310i) q^{25} +(-1.79625 + 2.31987i) q^{26} +(0.0916855 + 5.19534i) q^{27} +(-0.961891 + 3.46840i) q^{28} -8.21860 q^{29} +(4.47292 - 5.84755i) q^{30} -4.14463i q^{31} +(2.05019 + 5.27226i) q^{32} +(5.31832 - 0.0312823i) q^{33} +(1.52033 + 0.206912i) q^{34} +(4.68433 + 2.70450i) q^{35} +(-4.22976 + 4.25549i) q^{36} +(1.00247 + 5.99959i) q^{37} +(-4.83284 - 3.74200i) q^{38} +(-1.81497 - 3.10135i) q^{39} +(8.44213 - 0.999128i) q^{40} +(3.98862 - 2.30283i) q^{41} +(-3.50135 - 2.67826i) q^{42} +3.12583 q^{43} +(4.30438 + 4.38021i) q^{44} +(4.59991 + 7.75514i) q^{45} +(10.4949 - 4.29354i) q^{46} +8.20124 q^{47} +(-6.92774 - 0.0802359i) q^{48} +(-1.88062 + 3.25733i) q^{49} +(0.769231 - 5.65210i) q^{50} +(-0.930005 + 1.63292i) q^{51} +(1.10887 - 3.99838i) q^{52} +(-3.67734 + 6.36933i) q^{53} +(-2.90205 - 6.75116i) q^{54} +(7.99244 - 4.61444i) q^{55} +(-0.598250 - 5.05492i) q^{56} +(6.46083 - 3.78101i) q^{57} +(10.7574 - 4.40095i) q^{58} +(11.3861 + 6.57374i) q^{59} +(-2.72337 + 10.0491i) q^{60} +(6.39037 - 3.68948i) q^{61} +(2.21940 + 5.42497i) q^{62} +(4.64356 - 2.75430i) q^{63} +(-5.50675 - 5.80308i) q^{64} +(-5.40010 - 3.11775i) q^{65} +(-6.94446 + 2.88883i) q^{66} +(3.38827 + 5.86866i) q^{67} +(-2.10078 + 0.543289i) q^{68} +(0.0816854 + 13.8874i) q^{69} +(-7.57961 - 1.03156i) q^{70} +(2.63046 + 4.55609i) q^{71} +(3.25763 - 7.83504i) q^{72} -10.0820 q^{73} +(-4.52485 - 7.31613i) q^{74} +(6.07066 + 3.45745i) q^{75} +(8.32956 + 2.31004i) q^{76} +(-2.76300 - 4.78565i) q^{77} +(4.03637 + 3.08750i) 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} +(6.01714 + 1.63068i) q^{84} +3.26090i q^{85} +(-4.09144 + 1.67384i) q^{86} +(0.0837290 + 14.2348i) q^{87} +(-7.97961 - 3.42838i) q^{88} +(10.4253 + 6.01902i) q^{89} +(-10.1737 - 7.68761i) q^{90} +(-1.86682 + 3.23343i) q^{91} +(-11.4378 + 11.2397i) q^{92} +(-7.17859 + 0.0422245i) q^{93} +(-10.7347 + 4.39166i) q^{94} +(6.49501 - 11.2497i) q^{95} +(9.11077 - 3.60469i) q^{96} -16.3711 q^{97} +(0.717312 - 5.27061i) 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.30891 + 0.535487i −0.925541 + 0.378646i
\(3\) −0.0101877 − 1.73202i −0.00588190 − 0.999983i
\(4\) 1.42651 − 1.40181i 0.713254 − 0.700906i
\(5\) −1.50279 − 2.60290i −0.672067 − 1.16405i −0.977317 − 0.211782i \(-0.932073\pi\)
0.305250 − 0.952272i \(-0.401260\pi\)
\(6\) 0.940809 + 2.26161i 0.384084 + 0.923298i
\(7\) −1.55855 + 0.899828i −0.589075 + 0.340103i −0.764732 − 0.644349i \(-0.777129\pi\)
0.175656 + 0.984452i \(0.443795\pi\)
\(8\) −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 \(0.153194\pi\)
−0.886406 + 0.462908i \(0.846806\pi\)
\(12\) −2.44250 − 2.45646i −0.705089 − 0.709119i
\(13\) 1.79669 − 1.03732i 0.498314 − 0.287701i −0.229703 − 0.973261i \(-0.573776\pi\)
0.728017 + 0.685559i \(0.240442\pi\)
\(14\) 1.55816 − 2.01238i 0.416435 − 0.537831i
\(15\) −4.49297 + 2.62938i −1.16008 + 0.678902i
\(16\) 0.0698494 − 3.99939i 0.0174624 − 0.999848i
\(17\) −0.939594 − 0.542475i −0.227885 − 0.131569i 0.381711 − 0.924282i \(-0.375335\pi\)
−0.609596 + 0.792712i \(0.708668\pi\)
\(18\) 3.90757 − 1.65254i 0.921023 − 0.389508i
\(19\) 2.16099 + 3.74294i 0.495764 + 0.858689i 0.999988 − 0.00488395i \(-0.00155462\pi\)
−0.504224 + 0.863573i \(0.668221\pi\)
\(20\) −5.79252 − 1.60644i −1.29525 − 0.359211i
\(21\) 1.57440 + 2.69027i 0.343562 + 0.587065i
\(22\) −1.64426 − 4.01913i −0.350557 − 0.856881i
\(23\) −8.01801 −1.67187 −0.835935 − 0.548828i \(-0.815074\pi\)
−0.835935 + 0.548828i \(0.815074\pi\)
\(24\) 4.51242 + 1.90737i 0.921094 + 0.389340i
\(25\) −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
\(28\) −0.961891 + 3.46840i −0.181780 + 0.655466i
\(29\) −8.21860 −1.52616 −0.763078 − 0.646306i \(-0.776313\pi\)
−0.763078 + 0.646306i \(0.776313\pi\)
\(30\) 4.47292 − 5.84755i 0.816639 − 1.06761i
\(31\) − 4.14463i − 0.744398i −0.928153 − 0.372199i \(-0.878604\pi\)
0.928153 − 0.372199i \(-0.121396\pi\)
\(32\) 2.05019 + 5.27226i 0.362426 + 0.932012i
\(33\) 5.31832 − 0.0312823i 0.925800 − 0.00544556i
\(34\) 1.52033 + 0.206912i 0.260735 + 0.0354852i
\(35\) 4.68433 + 2.70450i 0.791796 + 0.457144i
\(36\) −4.22976 + 4.25549i −0.704959 + 0.709248i
\(37\) 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
\(40\) 8.44213 − 0.999128i 1.33482 − 0.157976i
\(41\) 3.98862 − 2.30283i 0.622917 − 0.359642i −0.155087 − 0.987901i \(-0.549566\pi\)
0.778004 + 0.628259i \(0.216232\pi\)
\(42\) −3.50135 − 2.67826i −0.540271 − 0.413264i
\(43\) 3.12583 0.476685 0.238342 − 0.971181i \(-0.423396\pi\)
0.238342 + 0.971181i \(0.423396\pi\)
\(44\) 4.30438 + 4.38021i 0.648910 + 0.660342i
\(45\) 4.59991 + 7.75514i 0.685714 + 1.15607i
\(46\) 10.4949 − 4.29354i 1.54739 − 0.633047i
\(47\) 8.20124 1.19627 0.598137 − 0.801394i \(-0.295908\pi\)
0.598137 + 0.801394i \(0.295908\pi\)
\(48\) −6.92774 − 0.0802359i −0.999933 − 0.0115810i
\(49\) −1.88062 + 3.25733i −0.268660 + 0.465333i
\(50\) 0.769231 − 5.65210i 0.108786 − 0.799327i
\(51\) −0.930005 + 1.63292i −0.130227 + 0.228655i
\(52\) 1.10887 − 3.99838i 0.153772 − 0.554475i
\(53\) −3.67734 + 6.36933i −0.505121 + 0.874895i 0.494861 + 0.868972i \(0.335219\pi\)
−0.999982 + 0.00592342i \(0.998115\pi\)
\(54\) −2.90205 − 6.75116i −0.394918 − 0.918716i
\(55\) 7.99244 − 4.61444i 1.07770 − 0.622210i
\(56\) −0.598250 − 5.05492i −0.0799446 − 0.675491i
\(57\) 6.46083 − 3.78101i 0.855758 − 0.500807i
\(58\) 10.7574 − 4.40095i 1.41252 − 0.577873i
\(59\) 11.3861 + 6.57374i 1.48234 + 0.855829i 0.999799 − 0.0200454i \(-0.00638109\pi\)
0.482540 + 0.875874i \(0.339714\pi\)
\(60\) −2.72337 + 10.0491i −0.351586 + 1.29734i
\(61\) 6.39037 − 3.68948i 0.818203 − 0.472390i −0.0315934 − 0.999501i \(-0.510058\pi\)
0.849796 + 0.527111i \(0.176725\pi\)
\(62\) 2.21940 + 5.42497i 0.281864 + 0.688971i
\(63\) 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
\(66\) −6.94446 + 2.88883i −0.854804 + 0.355591i
\(67\) 3.38827 + 5.86866i 0.413943 + 0.716971i 0.995317 − 0.0966661i \(-0.0308179\pi\)
−0.581374 + 0.813637i \(0.697485\pi\)
\(68\) −2.10078 + 0.543289i −0.254758 + 0.0658834i
\(69\) 0.0816854 + 13.8874i 0.00983377 + 1.67184i
\(70\) −7.57961 − 1.03156i −0.905936 − 0.123295i
\(71\) 2.63046 + 4.55609i 0.312178 + 0.540709i 0.978834 − 0.204657i \(-0.0656080\pi\)
−0.666655 + 0.745366i \(0.732275\pi\)
\(72\) 3.25763 − 7.83504i 0.383915 − 0.923368i
\(73\) −10.0820 −1.18001 −0.590004 − 0.807400i \(-0.700874\pi\)
−0.590004 + 0.807400i \(0.700874\pi\)
\(74\) −4.52485 − 7.31613i −0.526003 − 0.850483i
\(75\) 6.07066 + 3.45745i 0.700980 + 0.399232i
\(76\) 8.32956 + 2.31004i 0.955466 + 0.264979i
\(77\) −2.76300 − 4.78565i −0.314873 − 0.545376i
\(78\) 4.03637 + 3.08750i 0.457028 + 0.349591i
\(79\) −2.81621 + 1.62594i −0.316848 + 0.182932i −0.649987 − 0.759946i \(-0.725226\pi\)
0.333139 + 0.942878i \(0.391892\pi\)
\(80\) −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 −0.259607 − 0.965714i \(-0.583593\pi\)
−0.966137 + 0.258031i \(0.916926\pi\)
\(84\) 6.01714 + 1.63068i 0.656524 + 0.177922i
\(85\) 3.26090i 0.353694i
\(86\) −4.09144 + 1.67384i −0.441191 + 0.180495i
\(87\) 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 0.937549 − 0.347853i \(-0.113089\pi\)
0.167525 + 0.985868i \(0.446422\pi\)
\(90\) −10.1737 − 7.68761i −1.07240 − 0.810346i
\(91\) −1.86682 + 3.23343i −0.195696 + 0.338956i
\(92\) −11.4378 + 11.2397i −1.19247 + 1.17182i
\(93\) −7.17859 + 0.0422245i −0.744385 + 0.00437847i
\(94\) −10.7347 + 4.39166i −1.10720 + 0.452965i
\(95\) 6.49501 − 11.2497i 0.666374 − 1.15419i
\(96\) 9.11077 − 3.60469i 0.929864 − 0.367902i
\(97\) −16.3711 −1.66223 −0.831117 − 0.556097i \(-0.812298\pi\)
−0.831117 + 0.556097i \(0.812298\pi\)
\(98\) 0.717312 − 5.27061i 0.0724595 − 0.532412i
\(99\) −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.491.19 ✓ 296
3.2 odd 2 inner 888.2.bd.a.491.130 yes 296
8.3 odd 2 inner 888.2.bd.a.491.69 yes 296
24.11 even 2 inner 888.2.bd.a.491.80 yes 296
37.26 even 3 inner 888.2.bd.a.803.80 yes 296
111.26 odd 6 inner 888.2.bd.a.803.69 yes 296
296.211 odd 6 inner 888.2.bd.a.803.130 yes 296
888.803 even 6 inner 888.2.bd.a.803.19 yes 296
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bd.a.491.19 ✓ 296 1.1 even 1 trivial
888.2.bd.a.491.69 yes 296 8.3 odd 2 inner
888.2.bd.a.491.80 yes 296 24.11 even 2 inner
888.2.bd.a.491.130 yes 296 3.2 odd 2 inner
888.2.bd.a.803.19 yes 296 888.803 even 6 inner
888.2.bd.a.803.69 yes 296 111.26 odd 6 inner
888.2.bd.a.803.80 yes 296 37.26 even 3 inner
888.2.bd.a.803.130 yes 296 296.211 odd 6 inner