↑

Properties

Label 888.2.bd.a.491.141
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 491.141
Character \(\chi\) \(=\) 888.491
Dual form 888.2.bd.a.803.141

$q$-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} +(-0.571326 + 1.63511i) q^{3} +(1.90073 - 0.622263i) q^{4} +(1.38974 + 2.40710i) q^{5} +(-0.433610 + 2.41081i) q^{6} +(-4.20477 + 2.42763i) q^{7} +(2.51585 - 1.29248i) q^{8} +(-2.34717 - 1.86836i) q^{9} +(2.47712 + 3.05204i) q^{10} -0.115286i q^{11} +(-0.0684698 + 3.46342i) q^{12} +(1.35586 - 0.782806i) q^{13} +(-5.33136 + 4.32707i) q^{14} +(-4.72988 + 0.897141i) q^{15} +(3.22558 - 2.36551i) q^{16} +(1.06173 + 0.612992i) q^{17} +(-3.69420 - 2.08636i) q^{18} +(3.57483 + 6.19178i) q^{19} +(4.13938 + 3.71048i) q^{20} +(-1.56714 - 8.26223i) q^{21} +(-0.0256841 - 0.161004i) q^{22} -6.63386 q^{23} +(0.675977 + 4.85212i) 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} +(-6.40566 + 2.30665i) q^{30} +6.34675i q^{31} +(3.97770 - 4.02218i) q^{32} +(0.188506 + 0.0658661i) q^{33} +(1.61934 + 0.619540i) q^{34} +(-11.6871 - 6.74755i) q^{35} +(-5.62396 - 2.09070i) q^{36} +(5.96963 - 1.16768i) q^{37} +(6.37188 + 7.85075i) q^{38} +(0.505337 + 2.66422i) q^{39} +(6.60751 + 4.25970i) q^{40} +(0.120665 - 0.0696662i) q^{41} +(-4.02930 - 11.1895i) 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} +(2.02502 + 6.62565i) 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} +(5.52202 - 4.84844i) q^{54} +(0.277507 - 0.160218i) q^{55} +(-7.44092 + 11.5421i) q^{56} +(-12.1666 + 2.30771i) q^{57} +(-13.1292 + 2.09443i) q^{58} +(-1.46726 - 0.847126i) q^{59} +(-8.43198 + 4.64845i) 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} +(0.277933 + 0.0499894i) q^{66} +(-2.97660 - 5.15562i) q^{67} +(2.39952 + 0.504458i) q^{68} +(3.79010 - 10.8471i) q^{69} +(-17.8249 - 6.81962i) q^{70} +(-0.0851746 - 0.147527i) q^{71} +(-8.31995 - 1.66684i) 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} +(1.29928 + 3.60815i) q^{78} +(5.07991 - 2.93289i) q^{79} +(10.1768 + 4.47685i) q^{80} +(2.01845 + 8.77074i) q^{81} +(0.152995 - 0.124175i) q^{82} +(6.09435 + 3.51857i) q^{83} +(-8.12000 - 14.7291i) q^{84} +3.40761i q^{85} +(3.51326 - 0.560451i) q^{86} +(5.37112 - 15.3719i) q^{87} +(-0.149005 - 0.290043i) q^{88} +(6.96872 + 4.02340i) q^{89} +(-0.111911 - 11.7918i) q^{90} +(-3.80072 + 6.58304i) q^{91} +(-12.6092 + 4.12800i) q^{92} +(-10.3776 - 3.62606i) q^{93} +(8.88729 - 1.41774i) q^{94} +(-9.93618 + 17.2100i) q^{95} +(4.30414 + 8.80195i) q^{96} -6.49834 q^{97} +(8.37525 - 21.8910i) q^{98} +(-0.215397 + 0.270597i) 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
\(3\) −0.571326 + 1.63511i −0.329855 + 0.944032i
\(4\) 1.90073 − 0.622263i 0.950367 − 0.311131i
\(5\) 1.38974 + 2.40710i 0.621512 + 1.07649i 0.989204 + 0.146542i \(0.0468144\pi\)
−0.367693 + 0.929947i \(0.619852\pi\)
\(6\) −0.433610 + 2.41081i −0.177021 + 0.984207i
\(7\) −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
\(9\) −2.34717 − 1.86836i −0.782391 − 0.622787i
\(10\) 2.47712 + 3.05204i 0.783333 + 0.965140i
\(11\) − 0.115286i − 0.0347602i −0.999849 − 0.0173801i \(-0.994467\pi\)
0.999849 − 0.0173801i \(-0.00553253\pi\)
\(12\) −0.0684698 + 3.46342i −0.0197655 + 0.999805i
\(13\) 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
\(15\) −4.72988 + 0.897141i −1.22125 + 0.231641i
\(16\) 3.22558 − 2.36551i 0.806395 − 0.591378i
\(17\) 1.06173 + 0.612992i 0.257508 + 0.148673i 0.623197 − 0.782065i \(-0.285833\pi\)
−0.365689 + 0.930737i \(0.619167\pi\)
\(18\) −3.69420 − 2.08636i −0.870731 − 0.491759i
\(19\) 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
\(21\) −1.56714 − 8.26223i −0.341979 − 1.80297i
\(22\) −0.0256841 − 0.161004i −0.00547586 − 0.0343262i
\(23\) −6.63386 −1.38326 −0.691628 − 0.722254i \(-0.743106\pi\)
−0.691628 + 0.722254i \(0.743106\pi\)
\(24\) 0.675977 + 4.85212i 0.137983 + 0.990435i
\(25\) −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 \(-0.837747\pi\)
−0.872875 + 0.487944i \(0.837747\pi\)
\(30\) −6.40566 + 2.30665i −1.16951 + 0.421135i
\(31\) 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
\(33\) 0.188506 + 0.0658661i 0.0328147 + 0.0114658i
\(34\) 1.61934 + 0.619540i 0.277714 + 0.106250i
\(35\) −11.6871 − 6.74755i −1.97548 − 1.14054i
\(36\) −5.62396 − 2.09070i −0.937327 − 0.348450i
\(37\) 5.96963 − 1.16768i 0.981402 − 0.191966i
\(38\) 6.37188 + 7.85075i 1.03366 + 1.27356i
\(39\) 0.505337 + 2.66422i 0.0809186 + 0.426616i
\(40\) 6.60751 + 4.25970i 1.04474 + 0.673517i
\(41\) 0.120665 − 0.0696662i 0.0188448 − 0.0108800i −0.490548 − 0.871414i \(-0.663203\pi\)
0.509393 + 0.860534i \(0.329870\pi\)
\(42\) −4.02930 − 11.1895i −0.621735 − 1.72658i
\(43\) 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 \(-0.346370\pi\)
0.464122 + 0.885771i \(0.346370\pi\)
\(48\) 2.02502 + 6.62565i 0.292286 + 0.956331i
\(49\) 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 −0.771200 − 0.636593i \(-0.780343\pi\)
0.936906 + 0.349582i \(0.113677\pi\)
\(54\) 5.52202 − 4.84844i 0.751451 − 0.659789i
\(55\) 0.277507 − 0.160218i 0.0374190 − 0.0216039i
\(56\) −7.44092 + 11.5421i −0.994334 + 1.54238i
\(57\) −12.1666 + 2.30771i −1.61151 + 0.305664i
\(58\) −13.1292 + 2.09443i −1.72395 + 0.275013i
\(59\) −1.46726 − 0.847126i −0.191022 − 0.110286i 0.401439 − 0.915886i \(-0.368510\pi\)
−0.592461 + 0.805599i \(0.701843\pi\)
\(60\) −8.43198 + 4.64845i −1.08856 + 0.600113i
\(61\) −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
\(66\) 0.277933 + 0.0499894i 0.0342112 + 0.00615327i
\(67\) −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
\(69\) 3.79010 − 10.8471i 0.456274 − 1.30584i
\(70\) −17.8249 − 6.81962i −2.13049 − 0.815100i
\(71\) −0.0851746 − 0.147527i −0.0101084 − 0.0175082i 0.860927 − 0.508729i \(-0.169884\pi\)
−0.871035 + 0.491220i \(0.836551\pi\)
\(72\) −8.31995 − 1.66684i −0.980516 − 0.196439i
\(73\) 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
\(78\) 1.29928 + 3.60815i 0.147114 + 0.408542i
\(79\) 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
\(81\) 2.01845 + 8.77074i 0.224272 + 0.974527i
\(82\) 0.152995 − 0.124175i 0.0168955 − 0.0137129i
\(83\) 6.09435 + 3.51857i 0.668941 + 0.386213i 0.795675 − 0.605723i \(-0.207116\pi\)
−0.126734 + 0.991937i \(0.540449\pi\)
\(84\) −8.12000 − 14.7291i −0.885965 − 1.60708i
\(85\) 3.40761i 0.369607i
\(86\) 3.51326 − 0.560451i 0.378844 − 0.0604349i
\(87\) 5.37112 − 15.3719i 0.575844 − 1.64804i
\(88\) −0.149005 − 0.290043i −0.0158840 − 0.0309187i
\(89\) 6.96872 + 4.02340i 0.738683 + 0.426479i 0.821590 − 0.570078i \(-0.193087\pi\)
−0.0829071 + 0.996557i \(0.526420\pi\)
\(90\) −0.111911 − 11.7918i −0.0117964 − 1.24297i
\(91\) −3.80072 + 6.58304i −0.398424 + 0.690090i
\(92\) −12.6092 + 4.12800i −1.31460 + 0.430374i
\(93\) −10.3776 − 3.62606i −1.07611 − 0.376005i
\(94\) 8.88729 − 1.41774i 0.916654 − 0.146229i
\(95\) −9.93618 + 17.2100i −1.01943 + 1.76570i
\(96\) 4.30414 + 8.80195i 0.439290 + 0.898345i
\(97\) −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
\(99\) −0.215397 + 0.270597i −0.0216482 + 0.0271961i
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.491.141 yes 296
3.2 odd 2 inner 888.2.bd.a.491.8 ✓ 296
8.3 odd 2 inner 888.2.bd.a.491.92 yes 296
24.11 even 2 inner 888.2.bd.a.491.57 yes 296
37.26 even 3 inner 888.2.bd.a.803.57 yes 296
111.26 odd 6 inner 888.2.bd.a.803.92 yes 296
296.211 odd 6 inner 888.2.bd.a.803.8 yes 296
888.803 even 6 inner 888.2.bd.a.803.141 yes 296
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bd.a.491.8 ✓ 296 3.2 odd 2 inner
888.2.bd.a.491.57 yes 296 24.11 even 2 inner
888.2.bd.a.491.92 yes 296 8.3 odd 2 inner
888.2.bd.a.491.141 yes 296 1.1 even 1 trivial
888.2.bd.a.803.8 yes 296 296.211 odd 6 inner
888.2.bd.a.803.57 yes 296 37.26 even 3 inner
888.2.bd.a.803.92 yes 296 111.26 odd 6 inner
888.2.bd.a.803.141 yes 296 888.803 even 6 inner