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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 43.15
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.15

$q$-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+(-0.398861 - 1.35680i) q^{2} -1.00000i q^{3} +(-1.68182 + 1.08235i) q^{4} +(2.91128 - 2.91128i) q^{5} +(-1.35680 + 0.398861i) q^{6} +4.26853i q^{7} +(2.13935 + 1.85019i) q^{8} -1.00000 q^{9} +(-5.11123 - 2.78884i) q^{10} +5.15463i q^{11} +(1.08235 + 1.68182i) q^{12} +(3.77619 - 3.77619i) q^{13} +(5.79155 - 1.70255i) q^{14} +(-2.91128 - 2.91128i) q^{15} +(1.65704 - 3.64064i) q^{16} +(2.91999 + 2.91999i) q^{17} +(0.398861 + 1.35680i) q^{18} +(3.65305 + 3.65305i) q^{19} +(-1.74523 + 8.04729i) q^{20} +4.26853 q^{21} +(6.99380 - 2.05598i) q^{22} +(0.968756 - 0.968756i) q^{23} +(1.85019 - 2.13935i) q^{24} -11.9512i q^{25} +(-6.62972 - 3.61737i) q^{26} +1.00000i q^{27} +(-4.62004 - 7.17890i) q^{28} +(2.50099 + 2.50099i) q^{29} +(-2.78884 + 5.11123i) q^{30} +(-4.49711 - 4.49711i) q^{31} +(-5.60055 - 0.796160i) q^{32} +5.15463 q^{33} +(2.79718 - 5.12652i) q^{34} +(12.4269 + 12.4269i) q^{35} +(1.68182 - 1.08235i) q^{36} +(1.89336 + 5.78059i) q^{37} +(3.49941 - 6.41353i) q^{38} +(-3.77619 - 3.77619i) q^{39} +(11.6147 - 0.841823i) q^{40} +7.36696i q^{41} +(-1.70255 - 5.79155i) q^{42} +(-2.05603 - 2.05603i) q^{43} +(-5.57911 - 8.66915i) q^{44} +(-2.91128 + 2.91128i) q^{45} +(-1.70081 - 0.928010i) q^{46} +3.22232i q^{47} +(-3.64064 - 1.65704i) q^{48} -11.2203 q^{49} +(-16.2153 + 4.76685i) q^{50} +(2.91999 - 2.91999i) q^{51} +(-2.26371 + 10.4380i) q^{52} -1.90911i q^{53} +(1.35680 - 0.398861i) q^{54} +(15.0066 + 15.0066i) q^{55} +(-7.89758 + 9.13186i) q^{56} +(3.65305 - 3.65305i) q^{57} +(2.39580 - 4.39089i) q^{58} +(-7.61165 - 7.61165i) q^{59} +(8.04729 + 1.74523i) q^{60} +(-3.90308 - 3.90308i) q^{61} +(-4.30796 + 7.89541i) q^{62} -4.26853i q^{63} +(1.15361 + 7.91639i) q^{64} -21.9872i q^{65} +(-2.05598 - 6.99380i) q^{66} -10.0449i q^{67} +(-8.07136 - 1.75045i) q^{68} +(-0.968756 - 0.968756i) q^{69} +(11.9042 - 21.8174i) q^{70} -4.15921i q^{71} +(-2.13935 - 1.85019i) q^{72} -3.97603i q^{73} +(7.08792 - 4.87457i) q^{74} -11.9512 q^{75} +(-10.0977 - 2.18990i) q^{76} -22.0027 q^{77} +(-3.61737 + 6.62972i) q^{78} +(-2.25738 + 2.25738i) q^{79} +(-5.77483 - 15.4230i) q^{80} +1.00000 q^{81} +(9.99550 - 2.93839i) q^{82} +3.32135 q^{83} +(-7.17890 + 4.62004i) q^{84} +17.0019 q^{85} +(-1.96955 + 3.60969i) q^{86} +(2.50099 - 2.50099i) q^{87} +(-9.53703 + 11.0275i) q^{88} +(0.179257 - 0.179257i) q^{89} +(5.11123 + 2.78884i) q^{90} +(16.1188 + 16.1188i) q^{91} +(-0.580739 + 2.67781i) q^{92} +(-4.49711 + 4.49711i) q^{93} +(4.37205 - 1.28526i) q^{94} +21.2702 q^{95} +(-0.796160 + 5.60055i) q^{96} +(2.60820 + 2.60820i) q^{97} +(4.47535 + 15.2238i) q^{98} -5.15463i 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.398861 − 1.35680i −0.282037 − 0.959403i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.68182 + 1.08235i −0.840910 + 0.541175i
\(5\) 2.91128 − 2.91128i 1.30197 − 1.30197i 0.374901 − 0.927065i \(-0.377677\pi\)
0.927065 − 0.374901i \(-0.122323\pi\)
\(6\) −1.35680 + 0.398861i −0.553912 + 0.162834i
\(7\) 4.26853i 1.61335i 0.590994 + 0.806676i \(0.298736\pi\)
−0.590994 + 0.806676i \(0.701264\pi\)
\(8\) 2.13935 + 1.85019i 0.756373 + 0.654140i
\(9\) −1.00000 −0.333333
\(10\) −5.11123 − 2.78884i −1.61631 − 0.881908i
\(11\) 5.15463i 1.55418i 0.629390 + 0.777089i \(0.283305\pi\)
−0.629390 + 0.777089i \(0.716695\pi\)
\(12\) 1.08235 + 1.68182i 0.312448 + 0.485500i
\(13\) 3.77619 − 3.77619i 1.04733 − 1.04733i 0.0485049 − 0.998823i \(-0.484554\pi\)
0.998823 − 0.0485049i \(-0.0154456\pi\)
\(14\) 5.79155 − 1.70255i 1.54786 − 0.455026i
\(15\) −2.91128 − 2.91128i −0.751690 − 0.751690i
\(16\) 1.65704 − 3.64064i 0.414259 − 0.910159i
\(17\) 2.91999 + 2.91999i 0.708202 + 0.708202i 0.966157 − 0.257955i \(-0.0830485\pi\)
−0.257955 + 0.966157i \(0.583049\pi\)
\(18\) 0.398861 + 1.35680i 0.0940124 + 0.319801i
\(19\) 3.65305 + 3.65305i 0.838068 + 0.838068i 0.988604 − 0.150536i \(-0.0481000\pi\)
−0.150536 + 0.988604i \(0.548100\pi\)
\(20\) −1.74523 + 8.04729i −0.390244 + 1.79943i
\(21\) 4.26853 0.931469
\(22\) 6.99380 − 2.05598i 1.49108 − 0.438336i
\(23\) 0.968756 − 0.968756i 0.202000 − 0.202000i −0.598857 − 0.800856i \(-0.704378\pi\)
0.800856 + 0.598857i \(0.204378\pi\)
\(24\) 1.85019 − 2.13935i 0.377668 − 0.436692i
\(25\) − 11.9512i − 2.39023i
\(26\) −6.62972 − 3.61737i −1.30020 − 0.709424i
\(27\) 1.00000i 0.192450i
\(28\) −4.62004 − 7.17890i −0.873106 − 1.35668i
\(29\) 2.50099 + 2.50099i 0.464422 + 0.464422i 0.900102 − 0.435680i \(-0.143492\pi\)
−0.435680 + 0.900102i \(0.643492\pi\)
\(30\) −2.78884 + 5.11123i −0.509170 + 0.933179i
\(31\) −4.49711 − 4.49711i −0.807705 − 0.807705i 0.176581 − 0.984286i \(-0.443496\pi\)
−0.984286 + 0.176581i \(0.943496\pi\)
\(32\) −5.60055 − 0.796160i −0.990046 − 0.140743i
\(33\) 5.15463 0.897305
\(34\) 2.79718 − 5.12652i 0.479712 − 0.879191i
\(35\) 12.4269 + 12.4269i 2.10053 + 2.10053i
\(36\) 1.68182 − 1.08235i 0.280303 − 0.180392i
\(37\) 1.89336 + 5.78059i 0.311267 + 0.950322i
\(38\) 3.49941 − 6.41353i 0.567679 − 1.04041i
\(39\) −3.77619 − 3.77619i −0.604675 − 0.604675i
\(40\) 11.6147 − 0.841823i 1.83644 − 0.133104i
\(41\) 7.36696i 1.15053i 0.817969 + 0.575263i \(0.195100\pi\)
−0.817969 + 0.575263i \(0.804900\pi\)
\(42\) −1.70255 − 5.79155i −0.262709 − 0.893655i
\(43\) −2.05603 − 2.05603i −0.313542 − 0.313542i 0.532738 − 0.846280i \(-0.321163\pi\)
−0.846280 + 0.532738i \(0.821163\pi\)
\(44\) −5.57911 − 8.66915i −0.841083 − 1.30692i
\(45\) −2.91128 + 2.91128i −0.433989 + 0.433989i
\(46\) −1.70081 − 0.928010i −0.250770 − 0.136828i
\(47\) 3.22232i 0.470024i 0.971992 + 0.235012i \(0.0755130\pi\)
−0.971992 + 0.235012i \(0.924487\pi\)
\(48\) −3.64064 − 1.65704i −0.525481 − 0.239172i
\(49\) −11.2203 −1.60291
\(50\) −16.2153 + 4.76685i −2.29320 + 0.674134i
\(51\) 2.91999 − 2.91999i 0.408881 − 0.408881i
\(52\) −2.26371 + 10.4380i −0.313921 + 1.44750i
\(53\) − 1.90911i − 0.262237i −0.991367 − 0.131118i \(-0.958143\pi\)
0.991367 − 0.131118i \(-0.0418568\pi\)
\(54\) 1.35680 − 0.398861i 0.184637 − 0.0542781i
\(55\) 15.0066 + 15.0066i 2.02349 + 2.02349i
\(56\) −7.89758 + 9.13186i −1.05536 + 1.22030i
\(57\) 3.65305 − 3.65305i 0.483859 − 0.483859i
\(58\) 2.39580 − 4.39089i 0.314584 − 0.576552i
\(59\) −7.61165 − 7.61165i −0.990953 − 0.990953i 0.00900646 − 0.999959i \(-0.497133\pi\)
−0.999959 + 0.00900646i \(0.997133\pi\)
\(60\) 8.04729 + 1.74523i 1.03890 + 0.225308i
\(61\) −3.90308 − 3.90308i −0.499738 − 0.499738i 0.411618 − 0.911356i \(-0.364964\pi\)
−0.911356 + 0.411618i \(0.864964\pi\)
\(62\) −4.30796 + 7.89541i −0.547112 + 1.00272i
\(63\) − 4.26853i − 0.537784i
\(64\) 1.15361 + 7.91639i 0.144201 + 0.989548i
\(65\) − 21.9872i − 2.72717i
\(66\) −2.05598 − 6.99380i −0.253074 − 0.860878i
\(67\) − 10.0449i − 1.22718i −0.789626 − 0.613588i \(-0.789726\pi\)
0.789626 − 0.613588i \(-0.210274\pi\)
\(68\) −8.07136 − 1.75045i −0.978796 − 0.212273i
\(69\) −0.968756 − 0.968756i −0.116624 − 0.116624i
\(70\) 11.9042 − 21.8174i 1.42283 − 2.60768i
\(71\) − 4.15921i − 0.493608i −0.969065 − 0.246804i \(-0.920620\pi\)
0.969065 − 0.246804i \(-0.0793803\pi\)
\(72\) −2.13935 − 1.85019i −0.252124 − 0.218047i
\(73\) − 3.97603i − 0.465359i −0.972553 − 0.232680i \(-0.925251\pi\)
0.972553 − 0.232680i \(-0.0747493\pi\)
\(74\) 7.08792 − 4.87457i 0.823954 − 0.566657i
\(75\) −11.9512 −1.38000
\(76\) −10.0977 − 2.18990i −1.15828 − 0.251198i
\(77\) −22.0027 −2.50744
\(78\) −3.61737 + 6.62972i −0.409586 + 0.750668i
\(79\) −2.25738 + 2.25738i −0.253976 + 0.253976i −0.822598 − 0.568623i \(-0.807476\pi\)
0.568623 + 0.822598i \(0.307476\pi\)
\(80\) −5.77483 − 15.4230i −0.645645 − 1.72435i
\(81\) 1.00000 0.111111
\(82\) 9.99550 − 2.93839i 1.10382 − 0.324491i
\(83\) 3.32135 0.364566 0.182283 − 0.983246i \(-0.441651\pi\)
0.182283 + 0.983246i \(0.441651\pi\)
\(84\) −7.17890 + 4.62004i −0.783282 + 0.504088i
\(85\) 17.0019 1.84411
\(86\) −1.96955 + 3.60969i −0.212382 + 0.389243i
\(87\) 2.50099 − 2.50099i 0.268134 − 0.268134i
\(88\) −9.53703 + 11.0275i −1.01665 + 1.17554i
\(89\) 0.179257 − 0.179257i 0.0190012 − 0.0190012i −0.697542 − 0.716544i \(-0.745723\pi\)
0.716544 + 0.697542i \(0.245723\pi\)
\(90\) 5.11123 + 2.78884i 0.538771 + 0.293969i
\(91\) 16.1188 + 16.1188i 1.68971 + 1.68971i
\(92\) −0.580739 + 2.67781i −0.0605463 + 0.279181i
\(93\) −4.49711 + 4.49711i −0.466329 + 0.466329i
\(94\) 4.37205 − 1.28526i 0.450943 − 0.132564i
\(95\) 21.2702 2.18227
\(96\) −0.796160 + 5.60055i −0.0812578 + 0.571603i
\(97\) 2.60820 + 2.60820i 0.264823 + 0.264823i 0.827010 − 0.562187i \(-0.190040\pi\)
−0.562187 + 0.827010i \(0.690040\pi\)
\(98\) 4.47535 + 15.2238i 0.452079 + 1.53783i
\(99\) − 5.15463i − 0.518060i
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.43.15 ✓ 72
8.3 odd 2 inner 888.2.r.e.43.33 yes 72
37.31 odd 4 inner 888.2.r.e.475.33 yes 72
296.179 even 4 inner 888.2.r.e.475.15 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.15 ✓ 72 1.1 even 1 trivial
888.2.r.e.43.33 yes 72 8.3 odd 2 inner
888.2.r.e.475.15 yes 72 296.179 even 4 inner
888.2.r.e.475.33 yes 72 37.31 odd 4 inner