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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(43,888)] 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("888.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,2,0,0,0,-2,0,2,-72,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 475.1
Character \(\chi\) \(=\) 888.475
Dual form 888.2.r.e.43.1

$q$-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+(-1.41405 - 0.0214268i) q^{2} +1.00000i q^{3} +(1.99908 + 0.0605971i) q^{4} +(-2.93098 - 2.93098i) q^{5} +(0.0214268 - 1.41405i) q^{6} -1.80046i q^{7} +(-2.82551 - 0.128521i) q^{8} -1.00000 q^{9} +(4.08175 + 4.20735i) q^{10} -3.07012i q^{11} +(-0.0605971 + 1.99908i) q^{12} +(-1.11924 - 1.11924i) q^{13} +(-0.0385781 + 2.54595i) q^{14} +(2.93098 - 2.93098i) q^{15} +(3.99266 + 0.242277i) q^{16} +(-0.975958 + 0.975958i) q^{17} +(1.41405 + 0.0214268i) q^{18} +(-1.75407 + 1.75407i) q^{19} +(-5.68165 - 6.03687i) q^{20} +1.80046 q^{21} +(-0.0657827 + 4.34130i) q^{22} +(0.317295 + 0.317295i) q^{23} +(0.128521 - 2.82551i) q^{24} +12.1813i q^{25} +(1.55868 + 1.60664i) q^{26} -1.00000i q^{27} +(0.109103 - 3.59927i) q^{28} +(-2.29575 + 2.29575i) q^{29} +(-4.20735 + 4.08175i) q^{30} +(-0.417203 + 0.417203i) q^{31} +(-5.64063 - 0.428142i) q^{32} +3.07012 q^{33} +(1.40097 - 1.35914i) q^{34} +(-5.27711 + 5.27711i) q^{35} +(-1.99908 - 0.0605971i) q^{36} +(6.08276 - 0.00636497i) q^{37} +(2.51793 - 2.44276i) q^{38} +(1.11924 - 1.11924i) q^{39} +(7.90480 + 8.65819i) q^{40} +11.8381i q^{41} +(-2.54595 - 0.0385781i) q^{42} +(-6.02011 + 6.02011i) q^{43} +(0.186040 - 6.13741i) q^{44} +(2.93098 + 2.93098i) q^{45} +(-0.441873 - 0.455471i) q^{46} -1.10399i q^{47} +(-0.242277 + 3.99266i) q^{48} +3.75834 q^{49} +(0.261005 - 17.2249i) q^{50} +(-0.975958 - 0.975958i) q^{51} +(-2.16963 - 2.30527i) q^{52} -12.1038i q^{53} +(-0.0214268 + 1.41405i) q^{54} +(-8.99844 + 8.99844i) q^{55} +(-0.231398 + 5.08722i) q^{56} +(-1.75407 - 1.75407i) q^{57} +(3.29550 - 3.19712i) q^{58} +(0.476225 - 0.476225i) q^{59} +(6.03687 - 5.68165i) q^{60} +(-8.77355 + 8.77355i) q^{61} +(0.598886 - 0.581008i) q^{62} +1.80046i q^{63} +(7.96696 + 0.726275i) q^{64} +6.56093i q^{65} +(-4.34130 - 0.0657827i) q^{66} -8.14981i q^{67} +(-2.01016 + 1.89188i) q^{68} +(-0.317295 + 0.317295i) q^{69} +(7.57518 - 7.34904i) q^{70} +14.0160i q^{71} +(2.82551 + 0.128521i) q^{72} -13.1483i q^{73} +(-8.60147 - 0.121333i) q^{74} -12.1813 q^{75} +(-3.61282 + 3.40024i) q^{76} -5.52763 q^{77} +(-1.60664 + 1.55868i) q^{78} +(-9.44869 - 9.44869i) q^{79} +(-10.9923 - 12.4125i) q^{80} +1.00000 q^{81} +(0.253653 - 16.7397i) q^{82} -3.88867 q^{83} +(3.59927 + 0.109103i) q^{84} +5.72102 q^{85} +(8.64174 - 8.38376i) q^{86} +(-2.29575 - 2.29575i) q^{87} +(-0.394575 + 8.67463i) q^{88} +(1.10648 + 1.10648i) q^{89} +(-4.08175 - 4.20735i) q^{90} +(-2.01515 + 2.01515i) q^{91} +(0.615072 + 0.653527i) q^{92} +(-0.417203 - 0.417203i) q^{93} +(-0.0236549 + 1.56110i) q^{94} +10.2823 q^{95} +(0.428142 - 5.64063i) q^{96} +(-4.99241 + 4.99241i) q^{97} +(-5.31448 - 0.0805290i) q^{98} +3.07012i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 2 q^{2} - 2 q^{6} + 2 q^{8} - 72 q^{9} + 8 q^{10} - 8 q^{12} + 20 q^{14} + 12 q^{16} + 12 q^{17} - 2 q^{18} + 20 q^{19} - 22 q^{20} - 16 q^{22} + 2 q^{24} - 32 q^{26} - 8 q^{32} - 16 q^{33} + 8 q^{34}+ \cdots - 110 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41405 − 0.0214268i −0.999885 − 0.0151510i
\(3\) 1.00000i 0.577350i
\(4\) 1.99908 + 0.0605971i 0.999541 + 0.0302985i
\(5\) −2.93098 − 2.93098i −1.31077 − 1.31077i −0.920846 − 0.389927i \(-0.872500\pi\)
−0.389927 − 0.920846i \(-0.627500\pi\)
\(6\) 0.0214268 − 1.41405i 0.00874744 − 0.577284i
\(7\) − 1.80046i − 0.680511i −0.940333 − 0.340255i \(-0.889486\pi\)
0.940333 − 0.340255i \(-0.110514\pi\)
\(8\) −2.82551 − 0.128521i −0.998967 − 0.0454391i
\(9\) −1.00000 −0.333333
\(10\) 4.08175 + 4.20735i 1.29076 + 1.33048i
\(11\) − 3.07012i − 0.925675i −0.886443 − 0.462837i \(-0.846831\pi\)
0.886443 − 0.462837i \(-0.153169\pi\)
\(12\) −0.0605971 + 1.99908i −0.0174929 + 0.577085i
\(13\) −1.11924 − 1.11924i −0.310421 − 0.310421i 0.534652 − 0.845073i \(-0.320443\pi\)
−0.845073 + 0.534652i \(0.820443\pi\)
\(14\) −0.0385781 + 2.54595i −0.0103104 + 0.680433i
\(15\) 2.93098 − 2.93098i 0.756775 − 0.756775i
\(16\) 3.99266 + 0.242277i 0.998164 + 0.0605693i
\(17\) −0.975958 + 0.975958i −0.236705 + 0.236705i −0.815484 − 0.578779i \(-0.803529\pi\)
0.578779 + 0.815484i \(0.303529\pi\)
\(18\) 1.41405 + 0.0214268i 0.333295 + 0.00505034i
\(19\) −1.75407 + 1.75407i −0.402411 + 0.402411i −0.879082 − 0.476671i \(-0.841843\pi\)
0.476671 + 0.879082i \(0.341843\pi\)
\(20\) −5.68165 − 6.03687i −1.27046 − 1.34989i
\(21\) 1.80046 0.392893
\(22\) −0.0657827 + 4.34130i −0.0140249 + 0.925569i
\(23\) 0.317295 + 0.317295i 0.0661607 + 0.0661607i 0.739413 − 0.673252i \(-0.235103\pi\)
−0.673252 + 0.739413i \(0.735103\pi\)
\(24\) 0.128521 − 2.82551i 0.0262343 − 0.576754i
\(25\) 12.1813i 2.43625i
\(26\) 1.55868 + 1.60664i 0.305682 + 0.315089i
\(27\) − 1.00000i − 0.192450i
\(28\) 0.109103 − 3.59927i 0.0206185 − 0.680198i
\(29\) −2.29575 + 2.29575i −0.426310 + 0.426310i −0.887369 − 0.461059i \(-0.847469\pi\)
0.461059 + 0.887369i \(0.347469\pi\)
\(30\) −4.20735 + 4.08175i −0.768154 + 0.745222i
\(31\) −0.417203 + 0.417203i −0.0749319 + 0.0749319i −0.743579 − 0.668648i \(-0.766874\pi\)
0.668648 + 0.743579i \(0.266874\pi\)
\(32\) −5.64063 − 0.428142i −0.997132 − 0.0756855i
\(33\) 3.07012 0.534439
\(34\) 1.40097 − 1.35914i 0.240264 − 0.233091i
\(35\) −5.27711 + 5.27711i −0.891995 + 0.891995i
\(36\) −1.99908 − 0.0605971i −0.333180 − 0.0100995i
\(37\) 6.08276 − 0.00636497i 0.999999 − 0.00104640i
\(38\) 2.51793 − 2.44276i 0.408462 − 0.396268i
\(39\) 1.11924 − 1.11924i 0.179222 − 0.179222i
\(40\) 7.90480 + 8.65819i 1.24986 + 1.36898i
\(41\) 11.8381i 1.84881i 0.381415 + 0.924404i \(0.375437\pi\)
−0.381415 + 0.924404i \(0.624563\pi\)
\(42\) −2.54595 − 0.0385781i −0.392848 − 0.00595272i
\(43\) −6.02011 + 6.02011i −0.918059 + 0.918059i −0.996888 − 0.0788294i \(-0.974882\pi\)
0.0788294 + 0.996888i \(0.474882\pi\)
\(44\) 0.186040 − 6.13741i 0.0280466 − 0.925250i
\(45\) 2.93098 + 2.93098i 0.436924 + 0.436924i
\(46\) −0.441873 − 0.455471i −0.0651507 − 0.0671555i
\(47\) − 1.10399i − 0.161033i −0.996753 − 0.0805166i \(-0.974343\pi\)
0.996753 − 0.0805166i \(-0.0256570\pi\)
\(48\) −0.242277 + 3.99266i −0.0349697 + 0.576290i
\(49\) 3.75834 0.536905
\(50\) 0.261005 − 17.2249i 0.0369117 − 2.43597i
\(51\) −0.975958 − 0.975958i −0.136662 − 0.136662i
\(52\) −2.16963 − 2.30527i −0.300873 − 0.319684i
\(53\) − 12.1038i − 1.66258i −0.555840 − 0.831289i \(-0.687603\pi\)
0.555840 − 0.831289i \(-0.312397\pi\)
\(54\) −0.0214268 + 1.41405i −0.00291581 + 0.192428i
\(55\) −8.99844 + 8.99844i −1.21335 + 1.21335i
\(56\) −0.231398 + 5.08722i −0.0309218 + 0.679808i
\(57\) −1.75407 − 1.75407i −0.232332 − 0.232332i
\(58\) 3.29550 − 3.19712i 0.432720 − 0.419802i
\(59\) 0.476225 − 0.476225i 0.0619992 − 0.0619992i −0.675427 − 0.737427i \(-0.736041\pi\)
0.737427 + 0.675427i \(0.236041\pi\)
\(60\) 6.03687 − 5.68165i 0.779357 − 0.733498i
\(61\) −8.77355 + 8.77355i −1.12334 + 1.12334i −0.132102 + 0.991236i \(0.542173\pi\)
−0.991236 + 0.132102i \(0.957827\pi\)
\(62\) 0.598886 − 0.581008i 0.0760586 − 0.0737880i
\(63\) 1.80046i 0.226837i
\(64\) 7.96696 + 0.726275i 0.995871 + 0.0907844i
\(65\) 6.56093i 0.813783i
\(66\) −4.34130 − 0.0657827i −0.534377 − 0.00809728i
\(67\) − 8.14981i − 0.995658i −0.867275 − 0.497829i \(-0.834131\pi\)
0.867275 − 0.497829i \(-0.165869\pi\)
\(68\) −2.01016 + 1.89188i −0.243768 + 0.229424i
\(69\) −0.317295 + 0.317295i −0.0381979 + 0.0381979i
\(70\) 7.57518 − 7.34904i 0.905407 − 0.878378i
\(71\) 14.0160i 1.66339i 0.555231 + 0.831696i \(0.312630\pi\)
−0.555231 + 0.831696i \(0.687370\pi\)
\(72\) 2.82551 + 0.128521i 0.332989 + 0.0151464i
\(73\) − 13.1483i − 1.53889i −0.638710 − 0.769447i \(-0.720532\pi\)
0.638710 − 0.769447i \(-0.279468\pi\)
\(74\) −8.60147 − 0.121333i −0.999901 − 0.0141047i
\(75\) −12.1813 −1.40657
\(76\) −3.61282 + 3.40024i −0.414419 + 0.390034i
\(77\) −5.52763 −0.629932
\(78\) −1.60664 + 1.55868i −0.181916 + 0.176486i
\(79\) −9.44869 − 9.44869i −1.06306 − 1.06306i −0.997873 − 0.0651873i \(-0.979236\pi\)
−0.0651873 − 0.997873i \(-0.520764\pi\)
\(80\) −10.9923 − 12.4125i −1.22897 − 1.38776i
\(81\) 1.00000 0.111111
\(82\) 0.253653 − 16.7397i 0.0280113 − 1.84860i
\(83\) −3.88867 −0.426837 −0.213418 − 0.976961i \(-0.568460\pi\)
−0.213418 + 0.976961i \(0.568460\pi\)
\(84\) 3.59927 + 0.109103i 0.392713 + 0.0119041i
\(85\) 5.72102 0.620532
\(86\) 8.64174 − 8.38376i 0.931863 − 0.904044i
\(87\) −2.29575 − 2.29575i −0.246130 − 0.246130i
\(88\) −0.394575 + 8.67463i −0.0420618 + 0.924719i
\(89\) 1.10648 + 1.10648i 0.117287 + 0.117287i 0.763314 − 0.646027i \(-0.223571\pi\)
−0.646027 + 0.763314i \(0.723571\pi\)
\(90\) −4.08175 − 4.20735i −0.430254 − 0.443494i
\(91\) −2.01515 + 2.01515i −0.211245 + 0.211245i
\(92\) 0.615072 + 0.653527i 0.0641257 + 0.0681349i
\(93\) −0.417203 − 0.417203i −0.0432620 − 0.0432620i
\(94\) −0.0236549 + 1.56110i −0.00243982 + 0.161015i
\(95\) 10.2823 1.05494
\(96\) 0.428142 − 5.64063i 0.0436970 − 0.575694i
\(97\) −4.99241 + 4.99241i −0.506903 + 0.506903i −0.913574 − 0.406672i \(-0.866689\pi\)
0.406672 + 0.913574i \(0.366689\pi\)
\(98\) −5.31448 − 0.0805290i −0.536844 − 0.00813466i
\(99\) 3.07012i 0.308558i
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.r.e.475.1 yes 72
8.3 odd 2 inner 888.2.r.e.475.18 yes 72
37.6 odd 4 inner 888.2.r.e.43.18 yes 72
296.43 even 4 inner 888.2.r.e.43.1 ✓ 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.1 ✓ 72 296.43 even 4 inner
888.2.r.e.43.18 yes 72 37.6 odd 4 inner
888.2.r.e.475.1 yes 72 1.1 even 1 trivial
888.2.r.e.475.18 yes 72 8.3 odd 2 inner