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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 443.16
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.13

$q$-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} +(-6.03265 + 6.13246i) 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 \(0.0813033\pi\)
−0.967557 + 0.252654i \(0.918697\pi\)
\(8\) −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 \(0.143353\pi\)
−0.900292 + 0.435286i \(0.856647\pi\)
\(12\) 3.20559 − 1.31309i 0.925374 − 0.379055i
\(13\) 1.66413 0.461548 0.230774 − 0.973007i \(-0.425874\pi\)
0.230774 + 0.973007i \(0.425874\pi\)
\(14\) −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 \(-0.489958\pi\)
0.0315413 + 0.999502i \(0.489958\pi\)
\(18\) −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 \(-0.571603\pi\)
−0.223055 + 0.974806i \(0.571603\pi\)
\(32\) 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 \(-0.850318\pi\)
0.891459 − 0.453101i \(-0.149682\pi\)
\(42\) 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 \(-0.403764\pi\)
0.297750 + 0.954644i \(0.403764\pi\)
\(60\) −3.01467 − 7.35961i −0.389192 − 0.950121i
\(61\) −1.49574 −0.191510 −0.0957550 − 0.995405i \(-0.530527\pi\)
−0.0957550 + 0.995405i \(0.530527\pi\)
\(62\) 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 \(0.601327\pi\)
\(74\) −6.03265 + 6.13246i −0.701282 + 0.712884i
\(75\) −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 \(-0.737532\pi\)
−0.678873 + 0.734255i \(0.737532\pi\)
\(80\) −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 \(-0.910630\pi\)
0.960844 − 0.277090i \(-0.0893700\pi\)
\(84\) 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 \(-0.186005\pi\)
0.834070 + 0.551658i \(0.186005\pi\)
\(90\) −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.443.16 yes 96
3.2 odd 2 inner 888.2.c.d.443.81 yes 96
8.3 odd 2 inner 888.2.c.d.443.14 yes 96
24.11 even 2 inner 888.2.c.d.443.83 yes 96
37.36 even 2 inner 888.2.c.d.443.82 yes 96
111.110 odd 2 inner 888.2.c.d.443.15 yes 96
296.147 odd 2 inner 888.2.c.d.443.84 yes 96
888.443 even 2 inner 888.2.c.d.443.13 ✓ 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.13 ✓ 96 888.443 even 2 inner
888.2.c.d.443.14 yes 96 8.3 odd 2 inner
888.2.c.d.443.15 yes 96 111.110 odd 2 inner
888.2.c.d.443.16 yes 96 1.1 even 1 trivial
888.2.c.d.443.81 yes 96 3.2 odd 2 inner
888.2.c.d.443.82 yes 96 37.36 even 2 inner
888.2.c.d.443.83 yes 96 24.11 even 2 inner
888.2.c.d.443.84 yes 96 296.147 odd 2 inner