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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(443,888)] 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("888.443"); 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([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,-4,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 443.83
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.82

$q$-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+(1.27104 + 0.620045i) q^{2} +(1.50401 - 0.859035i) q^{3} +(1.23109 + 1.57620i) q^{4} -2.29587i q^{5} +(2.44430 - 0.159312i) q^{6} -1.33692i q^{7} +(0.587448 + 2.76675i) q^{8} +(1.52412 - 2.58400i) q^{9} +(1.42354 - 2.91814i) q^{10} -2.88736i q^{11} +(3.20559 + 1.31309i) q^{12} -1.66413 q^{13} +(0.828949 - 1.69928i) q^{14} +(-1.97223 - 3.45302i) q^{15} +(-0.968839 + 3.88090i) q^{16} -0.260096 q^{17} +(3.53941 - 2.33935i) q^{18} +3.39819i q^{19} +(3.61875 - 2.82642i) q^{20} +(-1.14846 - 2.01074i) q^{21} +(1.79029 - 3.66995i) q^{22} +0.960147i q^{23} +(3.26026 + 3.65659i) q^{24} -0.271003 q^{25} +(-2.11518 - 1.03184i) q^{26} +(0.0725511 - 5.19565i) q^{27} +(2.10725 - 1.64586i) q^{28} -0.429289i q^{29} +(-0.365759 - 5.61179i) q^{30} +2.48383 q^{31} +(-3.63776 + 4.33205i) q^{32} +(-2.48034 - 4.34263i) q^{33} +(-0.330593 - 0.161271i) q^{34} -3.06938 q^{35} +(5.94924 - 0.778813i) q^{36} +(-5.73507 + 2.02705i) q^{37} +(-2.10703 + 4.31924i) q^{38} +(-2.50288 + 1.42955i) q^{39} +(6.35209 - 1.34870i) q^{40} +5.80252i q^{41} +(-0.212987 - 3.26783i) q^{42} +9.91054i q^{43} +(4.55107 - 3.55460i) q^{44} +(-5.93252 - 3.49917i) q^{45} +(-0.595334 + 1.22039i) q^{46} +5.41989 q^{47} +(1.87668 + 6.66919i) q^{48} +5.21265 q^{49} +(-0.344455 - 0.168034i) q^{50} +(-0.391188 + 0.223432i) q^{51} +(-2.04870 - 2.62301i) q^{52} +7.55415 q^{53} +(3.31375 - 6.55889i) q^{54} -6.62900 q^{55} +(3.69892 - 0.785369i) q^{56} +(2.91916 + 5.11093i) q^{57} +(0.266179 - 0.545644i) q^{58} -4.57413 q^{59} +(3.01467 - 7.35961i) q^{60} +1.49574 q^{61} +(3.15705 + 1.54009i) q^{62} +(-3.45460 - 2.03762i) q^{63} +(-7.30981 + 3.25064i) q^{64} +3.82063i q^{65} +(-0.459991 - 7.05759i) q^{66} -8.82790 q^{67} +(-0.320202 - 0.409965i) q^{68} +(0.824799 + 1.44407i) q^{69} +(-3.90131 - 1.90316i) q^{70} +7.07539 q^{71} +(8.04462 + 2.69889i) q^{72} -5.34818 q^{73} +(-8.54637 - 0.979541i) q^{74} +(-0.407592 + 0.232801i) q^{75} +(-5.35624 + 4.18347i) q^{76} -3.86016 q^{77} +(-4.06765 + 0.265116i) q^{78} +12.0679 q^{79} +(8.91002 + 2.22432i) q^{80} +(-4.35412 - 7.87665i) q^{81} +(-3.59782 + 7.37524i) q^{82} +5.04882i q^{83} +(1.75549 - 4.28561i) q^{84} +0.597146i q^{85} +(-6.14498 + 12.5967i) q^{86} +(-0.368774 - 0.645657i) q^{87} +(7.98861 - 1.69617i) q^{88} -15.7372 q^{89} +(-5.37083 - 8.12602i) q^{90} +2.22481i q^{91} +(-1.51339 + 1.18203i) q^{92} +(3.73572 - 2.13370i) q^{93} +(6.88891 + 3.36058i) q^{94} +7.80179 q^{95} +(-1.74986 + 9.64043i) q^{96} +4.94050i q^{97} +(6.62549 + 3.23208i) q^{98} +(-7.46094 - 4.40068i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 4 q^{3} + 24 q^{4} - 4 q^{9} - 20 q^{10} - 18 q^{12} - 56 q^{16} + 72 q^{25} - 4 q^{27} - 64 q^{28} - 16 q^{33} - 8 q^{34} + 42 q^{36} + 44 q^{40} - 68 q^{46} - 94 q^{48} - 104 q^{49} - 28 q^{58}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.27104 + 0.620045i 0.898762 + 0.438438i
\(3\) 1.50401 − 0.859035i 0.868343 − 0.495964i
\(4\) 1.23109 + 1.57620i 0.615545 + 0.788102i
\(5\) − 2.29587i − 1.02674i −0.858167 − 0.513371i \(-0.828396\pi\)
0.858167 − 0.513371i \(-0.171604\pi\)
\(6\) 2.44430 − 0.159312i 0.997883 − 0.0650388i
\(7\) − 1.33692i − 0.505307i −0.967557 − 0.252654i \(-0.918697\pi\)
0.967557 − 0.252654i \(-0.0813033\pi\)
\(8\) 0.587448 + 2.76675i 0.207694 + 0.978194i
\(9\) 1.52412 − 2.58400i 0.508040 − 0.861334i
\(10\) 1.42354 − 2.91814i 0.450163 − 0.922797i
\(11\) − 2.88736i − 0.870572i −0.900292 − 0.435286i \(-0.856647\pi\)
0.900292 − 0.435286i \(-0.143353\pi\)
\(12\) 3.20559 + 1.31309i 0.925374 + 0.379055i
\(13\) −1.66413 −0.461548 −0.230774 − 0.973007i \(-0.574126\pi\)
−0.230774 + 0.973007i \(0.574126\pi\)
\(14\) 0.828949 − 1.69928i 0.221546 − 0.454151i
\(15\) −1.97223 − 3.45302i −0.509227 − 0.891565i
\(16\) −0.968839 + 3.88090i −0.242210 + 0.970224i
\(17\) −0.260096 −0.0630826 −0.0315413 − 0.999502i \(-0.510042\pi\)
−0.0315413 + 0.999502i \(0.510042\pi\)
\(18\) 3.53941 − 2.33935i 0.834248 − 0.551390i
\(19\) 3.39819i 0.779598i 0.920900 + 0.389799i \(0.127456\pi\)
−0.920900 + 0.389799i \(0.872544\pi\)
\(20\) 3.61875 − 2.82642i 0.809178 − 0.632006i
\(21\) −1.14846 − 2.01074i −0.250614 − 0.438780i
\(22\) 1.79029 − 3.66995i 0.381692 − 0.782437i
\(23\) 0.960147i 0.200204i 0.994977 + 0.100102i \(0.0319170\pi\)
−0.994977 + 0.100102i \(0.968083\pi\)
\(24\) 3.26026 + 3.65659i 0.665499 + 0.746399i
\(25\) −0.271003 −0.0542005
\(26\) −2.11518 − 1.03184i −0.414821 − 0.202360i
\(27\) 0.0725511 − 5.19565i 0.0139625 − 0.999903i
\(28\) 2.10725 − 1.64586i 0.398234 − 0.311039i
\(29\) − 0.429289i − 0.0797170i −0.999205 − 0.0398585i \(-0.987309\pi\)
0.999205 − 0.0398585i \(-0.0126907\pi\)
\(30\) −0.365759 − 5.61179i −0.0667781 − 1.02457i
\(31\) 2.48383 0.446110 0.223055 − 0.974806i \(-0.428397\pi\)
0.223055 + 0.974806i \(0.428397\pi\)
\(32\) −3.63776 + 4.33205i −0.643072 + 0.765806i
\(33\) −2.48034 − 4.34263i −0.431772 − 0.755955i
\(34\) −0.330593 − 0.161271i −0.0566962 − 0.0276578i
\(35\) −3.06938 −0.518821
\(36\) 5.94924 − 0.778813i 0.991540 − 0.129802i
\(37\) −5.73507 + 2.02705i −0.942840 + 0.333245i
\(38\) −2.10703 + 4.31924i −0.341805 + 0.700673i
\(39\) −2.50288 + 1.42955i −0.400782 + 0.228911i
\(40\) 6.35209 − 1.34870i 1.00435 − 0.213248i
\(41\) 5.80252i 0.906202i 0.891459 + 0.453101i \(0.149682\pi\)
−0.891459 + 0.453101i \(0.850318\pi\)
\(42\) −0.212987 − 3.26783i −0.0328646 − 0.504237i
\(43\) 9.91054i 1.51134i 0.654950 + 0.755672i \(0.272689\pi\)
−0.654950 + 0.755672i \(0.727311\pi\)
\(44\) 4.55107 − 3.55460i 0.686100 − 0.535876i
\(45\) −5.93252 − 3.49917i −0.884368 − 0.521626i
\(46\) −0.595334 + 1.22039i −0.0877772 + 0.179936i
\(47\) 5.41989 0.790573 0.395286 − 0.918558i \(-0.370645\pi\)
0.395286 + 0.918558i \(0.370645\pi\)
\(48\) 1.87668 + 6.66919i 0.270875 + 0.962615i
\(49\) 5.21265 0.744664
\(50\) −0.344455 − 0.168034i −0.0487133 − 0.0237636i
\(51\) −0.391188 + 0.223432i −0.0547773 + 0.0312867i
\(52\) −2.04870 − 2.62301i −0.284103 − 0.363747i
\(53\) 7.55415 1.03764 0.518821 − 0.854883i \(-0.326371\pi\)
0.518821 + 0.854883i \(0.326371\pi\)
\(54\) 3.31375 − 6.55889i 0.450944 − 0.892552i
\(55\) −6.62900 −0.893854
\(56\) 3.69892 − 0.785369i 0.494289 − 0.104949i
\(57\) 2.91916 + 5.11093i 0.386652 + 0.676959i
\(58\) 0.266179 − 0.545644i 0.0349510 − 0.0716466i
\(59\) −4.57413 −0.595501 −0.297750 − 0.954644i \(-0.596236\pi\)
−0.297750 + 0.954644i \(0.596236\pi\)
\(60\) 3.01467 − 7.35961i 0.389192 − 0.950121i
\(61\) 1.49574 0.191510 0.0957550 − 0.995405i \(-0.469473\pi\)
0.0957550 + 0.995405i \(0.469473\pi\)
\(62\) 3.15705 + 1.54009i 0.400946 + 0.195591i
\(63\) −3.45460 − 2.03762i −0.435238 − 0.256716i
\(64\) −7.30981 + 3.25064i −0.913726 + 0.406330i
\(65\) 3.82063i 0.473891i
\(66\) −0.459991 − 7.05759i −0.0566209 − 0.868729i
\(67\) −8.82790 −1.07850 −0.539250 − 0.842146i \(-0.681292\pi\)
−0.539250 + 0.842146i \(0.681292\pi\)
\(68\) −0.320202 − 0.409965i −0.0388301 − 0.0497155i
\(69\) 0.824799 + 1.44407i 0.0992942 + 0.173846i
\(70\) −3.90131 − 1.90316i −0.466296 − 0.227471i
\(71\) 7.07539 0.839694 0.419847 − 0.907595i \(-0.362084\pi\)
0.419847 + 0.907595i \(0.362084\pi\)
\(72\) 8.04462 + 2.69889i 0.948068 + 0.318067i
\(73\) −5.34818 −0.625957 −0.312978 − 0.949760i \(-0.601327\pi\)
−0.312978 + 0.949760i \(0.601327\pi\)
\(74\) −8.54637 − 0.979541i −0.993496 − 0.113869i
\(75\) −0.407592 + 0.232801i −0.0470646 + 0.0268815i
\(76\) −5.35624 + 4.18347i −0.614403 + 0.479877i
\(77\) −3.86016 −0.439907
\(78\) −4.06765 + 0.265116i −0.460570 + 0.0300185i
\(79\) 12.0679 1.35775 0.678873 − 0.734255i \(-0.262468\pi\)
0.678873 + 0.734255i \(0.262468\pi\)
\(80\) 8.91002 + 2.22432i 0.996170 + 0.248687i
\(81\) −4.35412 − 7.87665i −0.483791 − 0.875183i
\(82\) −3.59782 + 7.37524i −0.397313 + 0.814459i
\(83\) 5.04882i 0.554180i 0.960844 + 0.277090i \(0.0893700\pi\)
−0.960844 + 0.277090i \(0.910630\pi\)
\(84\) 1.75549 − 4.28561i 0.191539 − 0.467598i
\(85\) 0.597146i 0.0647696i
\(86\) −6.14498 + 12.5967i −0.662630 + 1.35834i
\(87\) −0.368774 − 0.645657i −0.0395368 − 0.0692217i
\(88\) 7.98861 − 1.69617i 0.851588 − 0.180813i
\(89\) −15.7372 −1.66814 −0.834070 − 0.551658i \(-0.813995\pi\)
−0.834070 + 0.551658i \(0.813995\pi\)
\(90\) −5.37083 − 8.12602i −0.566135 − 0.856558i
\(91\) 2.22481i 0.233223i
\(92\) −1.51339 + 1.18203i −0.157782 + 0.123235i
\(93\) 3.73572 − 2.13370i 0.387376 − 0.221254i
\(94\) 6.88891 + 3.36058i 0.710536 + 0.346617i
\(95\) 7.80179 0.800447
\(96\) −1.74986 + 9.64043i −0.178595 + 0.983923i
\(97\) 4.94050i 0.501632i 0.968035 + 0.250816i \(0.0806990\pi\)
−0.968035 + 0.250816i \(0.919301\pi\)
\(98\) 6.62549 + 3.23208i 0.669276 + 0.326489i
\(99\) −7.46094 − 4.40068i −0.749853 − 0.442285i
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.c.d.443.83 yes 96
3.2 odd 2 inner 888.2.c.d.443.14 yes 96
8.3 odd 2 inner 888.2.c.d.443.81 yes 96
24.11 even 2 inner 888.2.c.d.443.16 yes 96
37.36 even 2 inner 888.2.c.d.443.13 ✓ 96
111.110 odd 2 inner 888.2.c.d.443.84 yes 96
296.147 odd 2 inner 888.2.c.d.443.15 yes 96
888.443 even 2 inner 888.2.c.d.443.82 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.13 ✓ 96 37.36 even 2 inner
888.2.c.d.443.14 yes 96 3.2 odd 2 inner
888.2.c.d.443.15 yes 96 296.147 odd 2 inner
888.2.c.d.443.16 yes 96 24.11 even 2 inner
888.2.c.d.443.81 yes 96 8.3 odd 2 inner
888.2.c.d.443.82 yes 96 888.443 even 2 inner
888.2.c.d.443.83 yes 96 1.1 even 1 trivial
888.2.c.d.443.84 yes 96 111.110 odd 2 inner