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(517,888)] 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("888.517"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.o (of order \(2\), degree \(1\), 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: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 517.71
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.72

$q$-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+(1.39216 - 0.248794i) q^{2} +1.00000i q^{3} +(1.87620 - 0.692722i) q^{4} -1.62102 q^{5} +(0.248794 + 1.39216i) q^{6} -0.682716 q^{7} +(2.43962 - 1.43117i) q^{8} -1.00000 q^{9} +(-2.25671 + 0.403300i) q^{10} +1.65068i q^{11} +(0.692722 + 1.87620i) q^{12} +6.88190 q^{13} +(-0.950447 + 0.169856i) q^{14} -1.62102i q^{15} +(3.04027 - 2.59937i) q^{16} +5.88377i q^{17} +(-1.39216 + 0.248794i) q^{18} +5.78573 q^{19} +(-3.04135 + 1.12291i) q^{20} -0.682716i q^{21} +(0.410680 + 2.29801i) q^{22} +4.15036i q^{23} +(1.43117 + 2.43962i) q^{24} -2.37231 q^{25} +(9.58069 - 1.71218i) q^{26} -1.00000i q^{27} +(-1.28091 + 0.472932i) q^{28} +5.64652 q^{29} +(-0.403300 - 2.25671i) q^{30} -9.48802i q^{31} +(3.58583 - 4.37514i) q^{32} -1.65068 q^{33} +(1.46385 + 8.19113i) q^{34} +1.10669 q^{35} +(-1.87620 + 0.692722i) q^{36} +(-0.243550 + 6.07788i) q^{37} +(8.05464 - 1.43946i) q^{38} +6.88190i q^{39} +(-3.95467 + 2.31994i) q^{40} +2.20152 q^{41} +(-0.169856 - 0.950447i) q^{42} -5.62302 q^{43} +(1.14346 + 3.09701i) q^{44} +1.62102 q^{45} +(1.03259 + 5.77796i) q^{46} -2.88131 q^{47} +(2.59937 + 3.04027i) q^{48} -6.53390 q^{49} +(-3.30262 + 0.590217i) q^{50} -5.88377 q^{51} +(12.9118 - 4.76724i) q^{52} -4.56700i q^{53} +(-0.248794 - 1.39216i) q^{54} -2.67578i q^{55} +(-1.66557 + 0.977080i) q^{56} +5.78573i q^{57} +(7.86084 - 1.40482i) q^{58} -6.60560 q^{59} +(-1.12291 - 3.04135i) q^{60} +0.499819 q^{61} +(-2.36057 - 13.2088i) q^{62} +0.682716 q^{63} +(3.90353 - 6.98301i) q^{64} -11.1557 q^{65} +(-2.29801 + 0.410680i) q^{66} -10.9456i q^{67} +(4.07581 + 11.0391i) q^{68} -4.15036 q^{69} +(1.54069 - 0.275339i) q^{70} -1.34402 q^{71} +(-2.43962 + 1.43117i) q^{72} -9.65894 q^{73} +(1.17308 + 8.52196i) q^{74} -2.37231i q^{75} +(10.8552 - 4.00790i) q^{76} -1.12695i q^{77} +(1.71218 + 9.58069i) q^{78} -3.25589i q^{79} +(-4.92833 + 4.21363i) q^{80} +1.00000 q^{81} +(3.06486 - 0.547726i) q^{82} +12.4949i q^{83} +(-0.472932 - 1.28091i) q^{84} -9.53768i q^{85} +(-7.82812 + 1.39897i) q^{86} +5.64652i q^{87} +(2.36240 + 4.02704i) q^{88} -10.7654i q^{89} +(2.25671 - 0.403300i) q^{90} -4.69838 q^{91} +(2.87505 + 7.78693i) q^{92} +9.48802 q^{93} +(-4.01124 + 0.716854i) q^{94} -9.37875 q^{95} +(4.37514 + 3.58583i) q^{96} +11.0332i q^{97} +(-9.09621 + 1.62560i) q^{98} -1.65068i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q + 4 q^{4} + 8 q^{7} - 76 q^{9} + 4 q^{16} + 84 q^{25} + 12 q^{28} + 8 q^{30} - 12 q^{34} - 4 q^{36} + 8 q^{38} - 56 q^{40} - 8 q^{41} - 24 q^{44} - 44 q^{46} - 8 q^{48} + 60 q^{49} - 40 q^{58} + 32 q^{62}+ \cdots - 56 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.39216 − 0.248794i 0.984404 − 0.175924i
\(3\) 1.00000i 0.577350i
\(4\) 1.87620 − 0.692722i 0.938101 − 0.346361i
\(5\) −1.62102 −0.724940 −0.362470 − 0.931995i \(-0.618067\pi\)
−0.362470 + 0.931995i \(0.618067\pi\)
\(6\) 0.248794 + 1.39216i 0.101570 + 0.568346i
\(7\) −0.682716 −0.258042 −0.129021 − 0.991642i \(-0.541184\pi\)
−0.129021 + 0.991642i \(0.541184\pi\)
\(8\) 2.43962 − 1.43117i 0.862537 − 0.505994i
\(9\) −1.00000 −0.333333
\(10\) −2.25671 + 0.403300i −0.713634 + 0.127535i
\(11\) 1.65068i 0.497699i 0.968542 + 0.248850i \(0.0800525\pi\)
−0.968542 + 0.248850i \(0.919948\pi\)
\(12\) 0.692722 + 1.87620i 0.199972 + 0.541613i
\(13\) 6.88190 1.90870 0.954348 − 0.298698i \(-0.0965522\pi\)
0.954348 + 0.298698i \(0.0965522\pi\)
\(14\) −0.950447 + 0.169856i −0.254018 + 0.0453959i
\(15\) − 1.62102i − 0.418545i
\(16\) 3.04027 − 2.59937i 0.760068 − 0.649843i
\(17\) 5.88377i 1.42702i 0.700644 + 0.713511i \(0.252896\pi\)
−0.700644 + 0.713511i \(0.747104\pi\)
\(18\) −1.39216 + 0.248794i −0.328135 + 0.0586414i
\(19\) 5.78573 1.32734 0.663668 − 0.748027i \(-0.268999\pi\)
0.663668 + 0.748027i \(0.268999\pi\)
\(20\) −3.04135 + 1.12291i −0.680068 + 0.251091i
\(21\) − 0.682716i − 0.148981i
\(22\) 0.410680 + 2.29801i 0.0875573 + 0.489937i
\(23\) 4.15036i 0.865411i 0.901535 + 0.432705i \(0.142441\pi\)
−0.901535 + 0.432705i \(0.857559\pi\)
\(24\) 1.43117 + 2.43962i 0.292136 + 0.497986i
\(25\) −2.37231 −0.474461
\(26\) 9.58069 − 1.71218i 1.87893 − 0.335786i
\(27\) − 1.00000i − 0.192450i
\(28\) −1.28091 + 0.472932i −0.242070 + 0.0893758i
\(29\) 5.64652 1.04853 0.524266 − 0.851554i \(-0.324340\pi\)
0.524266 + 0.851554i \(0.324340\pi\)
\(30\) −0.403300 − 2.25671i −0.0736321 − 0.412017i
\(31\) − 9.48802i − 1.70410i −0.523461 − 0.852050i \(-0.675359\pi\)
0.523461 − 0.852050i \(-0.324641\pi\)
\(32\) 3.58583 − 4.37514i 0.633891 − 0.773423i
\(33\) −1.65068 −0.287347
\(34\) 1.46385 + 8.19113i 0.251048 + 1.40477i
\(35\) 1.10669 0.187065
\(36\) −1.87620 + 0.692722i −0.312700 + 0.115454i
\(37\) −0.243550 + 6.07788i −0.0400393 + 0.999198i
\(38\) 8.05464 − 1.43946i 1.30664 − 0.233511i
\(39\) 6.88190i 1.10199i
\(40\) −3.95467 + 2.31994i −0.625288 + 0.366815i
\(41\) 2.20152 0.343820 0.171910 − 0.985113i \(-0.445006\pi\)
0.171910 + 0.985113i \(0.445006\pi\)
\(42\) −0.169856 − 0.950447i −0.0262093 − 0.146657i
\(43\) −5.62302 −0.857502 −0.428751 − 0.903423i \(-0.641046\pi\)
−0.428751 + 0.903423i \(0.641046\pi\)
\(44\) 1.14346 + 3.09701i 0.172383 + 0.466892i
\(45\) 1.62102 0.241647
\(46\) 1.03259 + 5.77796i 0.152247 + 0.851914i
\(47\) −2.88131 −0.420282 −0.210141 − 0.977671i \(-0.567392\pi\)
−0.210141 + 0.977671i \(0.567392\pi\)
\(48\) 2.59937 + 3.04027i 0.375187 + 0.438826i
\(49\) −6.53390 −0.933414
\(50\) −3.30262 + 0.590217i −0.467062 + 0.0834693i
\(51\) −5.88377 −0.823892
\(52\) 12.9118 − 4.76724i 1.79055 − 0.661098i
\(53\) − 4.56700i − 0.627326i −0.949534 − 0.313663i \(-0.898444\pi\)
0.949534 − 0.313663i \(-0.101556\pi\)
\(54\) −0.248794 − 1.39216i −0.0338566 − 0.189449i
\(55\) − 2.67578i − 0.360802i
\(56\) −1.66557 + 0.977080i −0.222571 + 0.130568i
\(57\) 5.78573i 0.766338i
\(58\) 7.86084 − 1.40482i 1.03218 − 0.184462i
\(59\) −6.60560 −0.859975 −0.429988 − 0.902835i \(-0.641482\pi\)
−0.429988 + 0.902835i \(0.641482\pi\)
\(60\) −1.12291 − 3.04135i −0.144967 − 0.392637i
\(61\) 0.499819 0.0639953 0.0319976 − 0.999488i \(-0.489813\pi\)
0.0319976 + 0.999488i \(0.489813\pi\)
\(62\) −2.36057 − 13.2088i −0.299792 − 1.67752i
\(63\) 0.682716 0.0860141
\(64\) 3.90353 − 6.98301i 0.487941 − 0.872877i
\(65\) −11.1557 −1.38369
\(66\) −2.29801 + 0.410680i −0.282865 + 0.0505512i
\(67\) − 10.9456i − 1.33722i −0.743612 − 0.668611i \(-0.766889\pi\)
0.743612 − 0.668611i \(-0.233111\pi\)
\(68\) 4.07581 + 11.0391i 0.494265 + 1.33869i
\(69\) −4.15036 −0.499645
\(70\) 1.54069 − 0.275339i 0.184148 − 0.0329093i
\(71\) −1.34402 −0.159506 −0.0797531 − 0.996815i \(-0.525413\pi\)
−0.0797531 + 0.996815i \(0.525413\pi\)
\(72\) −2.43962 + 1.43117i −0.287512 + 0.168665i
\(73\) −9.65894 −1.13049 −0.565246 − 0.824922i \(-0.691219\pi\)
−0.565246 + 0.824922i \(0.691219\pi\)
\(74\) 1.17308 + 8.52196i 0.136368 + 0.990658i
\(75\) − 2.37231i − 0.273930i
\(76\) 10.8552 − 4.00790i 1.24518 − 0.459738i
\(77\) − 1.12695i − 0.128427i
\(78\) 1.71218 + 9.58069i 0.193866 + 1.08480i
\(79\) − 3.25589i − 0.366316i −0.983083 − 0.183158i \(-0.941368\pi\)
0.983083 − 0.183158i \(-0.0586321\pi\)
\(80\) −4.92833 + 4.21363i −0.551004 + 0.471098i
\(81\) 1.00000 0.111111
\(82\) 3.06486 − 0.547726i 0.338458 − 0.0604862i
\(83\) 12.4949i 1.37149i 0.727842 + 0.685745i \(0.240523\pi\)
−0.727842 + 0.685745i \(0.759477\pi\)
\(84\) −0.472932 − 1.28091i −0.0516011 − 0.139759i
\(85\) − 9.53768i − 1.03451i
\(86\) −7.82812 + 1.39897i −0.844128 + 0.150855i
\(87\) 5.64652i 0.605371i
\(88\) 2.36240 + 4.02704i 0.251833 + 0.429284i
\(89\) − 10.7654i − 1.14113i −0.821254 − 0.570563i \(-0.806725\pi\)
0.821254 − 0.570563i \(-0.193275\pi\)
\(90\) 2.25671 − 0.403300i 0.237878 − 0.0425115i
\(91\) −4.69838 −0.492524
\(92\) 2.87505 + 7.78693i 0.299745 + 0.811843i
\(93\) 9.48802 0.983862
\(94\) −4.01124 + 0.716854i −0.413728 + 0.0739379i
\(95\) −9.37875 −0.962240
\(96\) 4.37514 + 3.58583i 0.446536 + 0.365977i
\(97\) 11.0332i 1.12026i 0.828406 + 0.560128i \(0.189248\pi\)
−0.828406 + 0.560128i \(0.810752\pi\)
\(98\) −9.09621 + 1.62560i −0.918856 + 0.164210i
\(99\) − 1.65068i − 0.165900i
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.o.a.517.71 yes 76
4.3 odd 2 3552.2.o.a.2737.29 76
8.3 odd 2 3552.2.o.a.2737.28 76
8.5 even 2 inner 888.2.o.a.517.5 ✓ 76
37.36 even 2 inner 888.2.o.a.517.6 yes 76
148.147 odd 2 3552.2.o.a.2737.27 76
296.147 odd 2 3552.2.o.a.2737.30 76
296.221 even 2 inner 888.2.o.a.517.72 yes 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.5 ✓ 76 8.5 even 2 inner
888.2.o.a.517.6 yes 76 37.36 even 2 inner
888.2.o.a.517.71 yes 76 1.1 even 1 trivial
888.2.o.a.517.72 yes 76 296.221 even 2 inner
3552.2.o.a.2737.27 76 148.147 odd 2
3552.2.o.a.2737.28 76 8.3 odd 2
3552.2.o.a.2737.29 76 4.3 odd 2
3552.2.o.a.2737.30 76 296.147 odd 2