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Properties

Label 888.2.o.a.517.13
Level $888$
Weight $2$
Character 888.517
Analytic conductor $7.091$
Analytic rank $0$
Dimension $76$
Inner twists $4$

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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 517.13
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.14

$q$-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+(-1.21738 - 0.719718i) q^{2} -1.00000i q^{3} +(0.964012 + 1.75234i) q^{4} -1.66630 q^{5} +(-0.719718 + 1.21738i) q^{6} +4.35248 q^{7} +(0.0876211 - 2.82707i) q^{8} -1.00000 q^{9} +(2.02852 + 1.19927i) q^{10} +1.28232i q^{11} +(1.75234 - 0.964012i) q^{12} -0.633848 q^{13} +(-5.29861 - 3.13256i) q^{14} +1.66630i q^{15} +(-2.14136 + 3.37855i) q^{16} +1.28166i q^{17} +(1.21738 + 0.719718i) q^{18} +7.67261 q^{19} +(-1.60634 - 2.91992i) q^{20} -4.35248i q^{21} +(0.922912 - 1.56107i) q^{22} +5.73709i q^{23} +(-2.82707 - 0.0876211i) q^{24} -2.22343 q^{25} +(0.771631 + 0.456192i) q^{26} +1.00000i q^{27} +(4.19585 + 7.62701i) q^{28} -2.02837 q^{29} +(1.19927 - 2.02852i) q^{30} -0.848932i q^{31} +(5.03844 - 2.57179i) q^{32} +1.28232 q^{33} +(0.922433 - 1.56026i) q^{34} -7.25256 q^{35} +(-0.964012 - 1.75234i) q^{36} +(5.01707 + 3.43934i) q^{37} +(-9.34046 - 5.52212i) q^{38} +0.633848i q^{39} +(-0.146003 + 4.71076i) q^{40} +10.7325 q^{41} +(-3.13256 + 5.29861i) q^{42} -3.66597 q^{43} +(-2.24706 + 1.23618i) q^{44} +1.66630 q^{45} +(4.12909 - 6.98420i) q^{46} +2.09458 q^{47} +(3.37855 + 2.14136i) q^{48} +11.9441 q^{49} +(2.70675 + 1.60024i) q^{50} +1.28166 q^{51} +(-0.611037 - 1.11071i) q^{52} -1.97749i q^{53} +(0.719718 - 1.21738i) q^{54} -2.13674i q^{55} +(0.381369 - 12.3048i) q^{56} -7.67261i q^{57} +(2.46929 + 1.45986i) q^{58} +9.18203 q^{59} +(-2.91992 + 1.60634i) q^{60} +3.70220 q^{61} +(-0.610991 + 1.03347i) q^{62} -4.35248 q^{63} +(-7.98465 - 0.495422i) q^{64} +1.05618 q^{65} +(-1.56107 - 0.922912i) q^{66} -9.87136i q^{67} +(-2.24590 + 1.23553i) q^{68} +5.73709 q^{69} +(8.82910 + 5.21980i) q^{70} +2.59837 q^{71} +(-0.0876211 + 2.82707i) q^{72} -6.53444 q^{73} +(-3.63231 - 7.79784i) q^{74} +2.22343i q^{75} +(7.39650 + 13.4450i) q^{76} +5.58130i q^{77} +(0.456192 - 0.771631i) q^{78} +0.443289i q^{79} +(3.56816 - 5.62969i) q^{80} +1.00000 q^{81} +(-13.0655 - 7.72437i) q^{82} -11.4371i q^{83} +(7.62701 - 4.19585i) q^{84} -2.13563i q^{85} +(4.46287 + 2.63846i) q^{86} +2.02837i q^{87} +(3.62522 + 0.112359i) q^{88} -8.99952i q^{89} +(-2.02852 - 1.19927i) q^{90} -2.75881 q^{91} +(-10.0533 + 5.53063i) q^{92} -0.848932 q^{93} +(-2.54990 - 1.50751i) q^{94} -12.7849 q^{95} +(-2.57179 - 5.03844i) q^{96} -15.2466i q^{97} +(-14.5405 - 8.59639i) q^{98} -1.28232i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21738 − 0.719718i −0.860815 − 0.508917i
\(3\) − 1.00000i − 0.577350i
\(4\) 0.964012 + 1.75234i 0.482006 + 0.876168i
\(5\) −1.66630 −0.745194 −0.372597 − 0.927993i \(-0.621533\pi\)
−0.372597 + 0.927993i \(0.621533\pi\)
\(6\) −0.719718 + 1.21738i −0.293824 + 0.496992i
\(7\) 4.35248 1.64508 0.822542 − 0.568704i \(-0.192555\pi\)
0.822542 + 0.568704i \(0.192555\pi\)
\(8\) 0.0876211 − 2.82707i 0.0309787 − 0.999520i
\(9\) −1.00000 −0.333333
\(10\) 2.02852 + 1.19927i 0.641474 + 0.379242i
\(11\) 1.28232i 0.386635i 0.981136 + 0.193318i \(0.0619248\pi\)
−0.981136 + 0.193318i \(0.938075\pi\)
\(12\) 1.75234 − 0.964012i 0.505856 − 0.278286i
\(13\) −0.633848 −0.175798 −0.0878989 − 0.996129i \(-0.528015\pi\)
−0.0878989 + 0.996129i \(0.528015\pi\)
\(14\) −5.29861 − 3.13256i −1.41611 − 0.837212i
\(15\) 1.66630i 0.430238i
\(16\) −2.14136 + 3.37855i −0.535340 + 0.844637i
\(17\) 1.28166i 0.310848i 0.987848 + 0.155424i \(0.0496743\pi\)
−0.987848 + 0.155424i \(0.950326\pi\)
\(18\) 1.21738 + 0.719718i 0.286938 + 0.169639i
\(19\) 7.67261 1.76022 0.880109 − 0.474771i \(-0.157469\pi\)
0.880109 + 0.474771i \(0.157469\pi\)
\(20\) −1.60634 − 2.91992i −0.359188 − 0.652915i
\(21\) − 4.35248i − 0.949790i
\(22\) 0.922912 − 1.56107i 0.196766 − 0.332822i
\(23\) 5.73709i 1.19627i 0.801397 + 0.598133i \(0.204091\pi\)
−0.801397 + 0.598133i \(0.795909\pi\)
\(24\) −2.82707 − 0.0876211i −0.577073 − 0.0178856i
\(25\) −2.22343 −0.444686
\(26\) 0.771631 + 0.456192i 0.151329 + 0.0894665i
\(27\) 1.00000i 0.192450i
\(28\) 4.19585 + 7.62701i 0.792941 + 1.44137i
\(29\) −2.02837 −0.376659 −0.188330 − 0.982106i \(-0.560307\pi\)
−0.188330 + 0.982106i \(0.560307\pi\)
\(30\) 1.19927 − 2.02852i 0.218956 − 0.370355i
\(31\) − 0.848932i − 0.152473i −0.997090 − 0.0762363i \(-0.975710\pi\)
0.997090 − 0.0762363i \(-0.0242903\pi\)
\(32\) 5.03844 − 2.57179i 0.890679 − 0.454632i
\(33\) 1.28232 0.223224
\(34\) 0.922433 − 1.56026i 0.158196 − 0.267583i
\(35\) −7.25256 −1.22591
\(36\) −0.964012 − 1.75234i −0.160669 − 0.292056i
\(37\) 5.01707 + 3.43934i 0.824801 + 0.565423i
\(38\) −9.34046 − 5.52212i −1.51522 − 0.895806i
\(39\) 0.633848i 0.101497i
\(40\) −0.146003 + 4.71076i −0.0230852 + 0.744836i
\(41\) 10.7325 1.67613 0.838067 − 0.545567i \(-0.183686\pi\)
0.838067 + 0.545567i \(0.183686\pi\)
\(42\) −3.13256 + 5.29861i −0.483365 + 0.817594i
\(43\) −3.66597 −0.559055 −0.279528 − 0.960138i \(-0.590178\pi\)
−0.279528 + 0.960138i \(0.590178\pi\)
\(44\) −2.24706 + 1.23618i −0.338758 + 0.186361i
\(45\) 1.66630 0.248398
\(46\) 4.12909 − 6.98420i 0.608801 − 1.02976i
\(47\) 2.09458 0.305526 0.152763 − 0.988263i \(-0.451183\pi\)
0.152763 + 0.988263i \(0.451183\pi\)
\(48\) 3.37855 + 2.14136i 0.487651 + 0.309079i
\(49\) 11.9441 1.70630
\(50\) 2.70675 + 1.60024i 0.382793 + 0.226309i
\(51\) 1.28166 0.179468
\(52\) −0.611037 − 1.11071i −0.0847356 − 0.154028i
\(53\) − 1.97749i − 0.271629i −0.990734 − 0.135814i \(-0.956635\pi\)
0.990734 − 0.135814i \(-0.0433651\pi\)
\(54\) 0.719718 − 1.21738i 0.0979412 − 0.165664i
\(55\) − 2.13674i − 0.288118i
\(56\) 0.381369 − 12.3048i 0.0509626 − 1.64429i
\(57\) − 7.67261i − 1.01626i
\(58\) 2.46929 + 1.45986i 0.324234 + 0.191689i
\(59\) 9.18203 1.19540 0.597699 − 0.801721i \(-0.296082\pi\)
0.597699 + 0.801721i \(0.296082\pi\)
\(60\) −2.91992 + 1.60634i −0.376961 + 0.207377i
\(61\) 3.70220 0.474018 0.237009 − 0.971507i \(-0.423833\pi\)
0.237009 + 0.971507i \(0.423833\pi\)
\(62\) −0.610991 + 1.03347i −0.0775960 + 0.131251i
\(63\) −4.35248 −0.548361
\(64\) −7.98465 − 0.495422i −0.998081 − 0.0619277i
\(65\) 1.05618 0.131003
\(66\) −1.56107 − 0.922912i −0.192155 − 0.113603i
\(67\) − 9.87136i − 1.20598i −0.797749 − 0.602989i \(-0.793976\pi\)
0.797749 − 0.602989i \(-0.206024\pi\)
\(68\) −2.24590 + 1.23553i −0.272355 + 0.149831i
\(69\) 5.73709 0.690665
\(70\) 8.82910 + 5.21980i 1.05528 + 0.623885i
\(71\) 2.59837 0.308370 0.154185 − 0.988042i \(-0.450725\pi\)
0.154185 + 0.988042i \(0.450725\pi\)
\(72\) −0.0876211 + 2.82707i −0.0103262 + 0.333173i
\(73\) −6.53444 −0.764798 −0.382399 − 0.923997i \(-0.624902\pi\)
−0.382399 + 0.923997i \(0.624902\pi\)
\(74\) −3.63231 − 7.79784i −0.422247 − 0.906481i
\(75\) 2.22343i 0.256740i
\(76\) 7.39650 + 13.4450i 0.848436 + 1.54225i
\(77\) 5.58130i 0.636048i
\(78\) 0.456192 − 0.771631i 0.0516535 − 0.0873701i
\(79\) 0.443289i 0.0498739i 0.999689 + 0.0249369i \(0.00793850\pi\)
−0.999689 + 0.0249369i \(0.992061\pi\)
\(80\) 3.56816 − 5.62969i 0.398932 − 0.629418i
\(81\) 1.00000 0.111111
\(82\) −13.0655 − 7.72437i −1.44284 − 0.853014i
\(83\) − 11.4371i − 1.25538i −0.778461 − 0.627692i \(-0.784000\pi\)
0.778461 − 0.627692i \(-0.216000\pi\)
\(84\) 7.62701 − 4.19585i 0.832175 − 0.457805i
\(85\) − 2.13563i − 0.231642i
\(86\) 4.46287 + 2.63846i 0.481243 + 0.284513i
\(87\) 2.02837i 0.217464i
\(88\) 3.62522 + 0.112359i 0.386450 + 0.0119775i
\(89\) − 8.99952i − 0.953948i −0.878918 − 0.476974i \(-0.841734\pi\)
0.878918 − 0.476974i \(-0.158266\pi\)
\(90\) −2.02852 − 1.19927i −0.213825 − 0.126414i
\(91\) −2.75881 −0.289202
\(92\) −10.0533 + 5.53063i −1.04813 + 0.576608i
\(93\) −0.848932 −0.0880301
\(94\) −2.54990 − 1.50751i −0.263002 − 0.155488i
\(95\) −12.7849 −1.31170
\(96\) −2.57179 − 5.03844i −0.262482 − 0.514234i
\(97\) − 15.2466i − 1.54806i −0.633148 − 0.774030i \(-0.718238\pi\)
0.633148 − 0.774030i \(-0.281762\pi\)
\(98\) −14.5405 − 8.59639i −1.46881 − 0.868367i
\(99\) − 1.28232i − 0.128878i
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.517.13 ✓ 76
4.3 odd 2 3552.2.o.a.2737.68 76
8.3 odd 2 3552.2.o.a.2737.41 76
8.5 even 2 inner 888.2.o.a.517.63 yes 76
37.36 even 2 inner 888.2.o.a.517.64 yes 76
148.147 odd 2 3552.2.o.a.2737.42 76
296.147 odd 2 3552.2.o.a.2737.67 76
296.221 even 2 inner 888.2.o.a.517.14 yes 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.13 ✓ 76 1.1 even 1 trivial
888.2.o.a.517.14 yes 76 296.221 even 2 inner
888.2.o.a.517.63 yes 76 8.5 even 2 inner
888.2.o.a.517.64 yes 76 37.36 even 2 inner
3552.2.o.a.2737.41 76 8.3 odd 2
3552.2.o.a.2737.42 76 148.147 odd 2
3552.2.o.a.2737.67 76 296.147 odd 2
3552.2.o.a.2737.68 76 4.3 odd 2