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Properties

Label 888.2.o.a.517.20
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 517.20
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.19

$q$-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+(-1.02048 + 0.979091i) q^{2} -1.00000i q^{3} +(0.0827621 - 1.99829i) q^{4} -3.42335 q^{5} +(0.979091 + 1.02048i) q^{6} +1.31476 q^{7} +(1.87205 + 2.12024i) q^{8} -1.00000 q^{9} +(3.49346 - 3.35177i) q^{10} -2.36153i q^{11} +(-1.99829 - 0.0827621i) q^{12} +4.49633 q^{13} +(-1.34169 + 1.28727i) q^{14} +3.42335i q^{15} +(-3.98630 - 0.330765i) q^{16} -3.85196i q^{17} +(1.02048 - 0.979091i) q^{18} -3.36545 q^{19} +(-0.283323 + 6.84083i) q^{20} -1.31476i q^{21} +(2.31215 + 2.40989i) q^{22} +4.50535i q^{23} +(2.12024 - 1.87205i) q^{24} +6.71931 q^{25} +(-4.58842 + 4.40231i) q^{26} +1.00000i q^{27} +(0.108812 - 2.62727i) q^{28} -7.97545 q^{29} +(-3.35177 - 3.49346i) q^{30} -3.81255i q^{31} +(4.39179 - 3.56541i) q^{32} -2.36153 q^{33} +(3.77142 + 3.93085i) q^{34} -4.50088 q^{35} +(-0.0827621 + 1.99829i) q^{36} +(5.68460 + 2.16457i) q^{37} +(3.43438 - 3.29508i) q^{38} -4.49633i q^{39} +(-6.40867 - 7.25834i) q^{40} -2.54834 q^{41} +(1.28727 + 1.34169i) q^{42} -11.0092 q^{43} +(-4.71900 - 0.195445i) q^{44} +3.42335 q^{45} +(-4.41115 - 4.59763i) q^{46} -6.41848 q^{47} +(-0.330765 + 3.98630i) q^{48} -5.27141 q^{49} +(-6.85693 + 6.57882i) q^{50} -3.85196 q^{51} +(0.372125 - 8.98495i) q^{52} +4.59896i q^{53} +(-0.979091 - 1.02048i) q^{54} +8.08432i q^{55} +(2.46129 + 2.78761i) q^{56} +3.36545i q^{57} +(8.13880 - 7.80869i) q^{58} -6.65249 q^{59} +(6.84083 + 0.283323i) q^{60} -1.28295 q^{61} +(3.73283 + 3.89063i) q^{62} -1.31476 q^{63} +(-0.990877 + 7.93840i) q^{64} -15.3925 q^{65} +(2.40989 - 2.31215i) q^{66} +0.397943i q^{67} +(-7.69732 - 0.318796i) q^{68} +4.50535 q^{69} +(4.59306 - 4.40677i) q^{70} -0.221498 q^{71} +(-1.87205 - 2.12024i) q^{72} -13.7728 q^{73} +(-7.92033 + 3.35683i) q^{74} -6.71931i q^{75} +(-0.278532 + 6.72514i) q^{76} -3.10484i q^{77} +(4.40231 + 4.58842i) q^{78} -0.202285i q^{79} +(13.6465 + 1.13232i) q^{80} +1.00000 q^{81} +(2.60053 - 2.49506i) q^{82} +1.43044i q^{83} +(-2.62727 - 0.108812i) q^{84} +13.1866i q^{85} +(11.2347 - 10.7790i) q^{86} +7.97545i q^{87} +(5.00701 - 4.42089i) q^{88} -12.6722i q^{89} +(-3.49346 + 3.35177i) q^{90} +5.91159 q^{91} +(9.00299 + 0.372872i) q^{92} -3.81255 q^{93} +(6.54993 - 6.28427i) q^{94} +11.5211 q^{95} +(-3.56541 - 4.39179i) q^{96} +16.3634i q^{97} +(5.37937 - 5.16119i) q^{98} +2.36153i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.02048 + 0.979091i −0.721589 + 0.692322i
\(3\) − 1.00000i − 0.577350i
\(4\) 0.0827621 − 1.99829i 0.0413810 − 0.999143i
\(5\) −3.42335 −1.53097 −0.765484 − 0.643455i \(-0.777500\pi\)
−0.765484 + 0.643455i \(0.777500\pi\)
\(6\) 0.979091 + 1.02048i 0.399712 + 0.416610i
\(7\) 1.31476 0.496932 0.248466 − 0.968641i \(-0.420074\pi\)
0.248466 + 0.968641i \(0.420074\pi\)
\(8\) 1.87205 + 2.12024i 0.661869 + 0.749620i
\(9\) −1.00000 −0.333333
\(10\) 3.49346 − 3.35177i 1.10473 − 1.05992i
\(11\) − 2.36153i − 0.712027i −0.934481 − 0.356013i \(-0.884136\pi\)
0.934481 − 0.356013i \(-0.115864\pi\)
\(12\) −1.99829 − 0.0827621i −0.576856 − 0.0238914i
\(13\) 4.49633 1.24706 0.623528 − 0.781801i \(-0.285699\pi\)
0.623528 + 0.781801i \(0.285699\pi\)
\(14\) −1.34169 + 1.28727i −0.358581 + 0.344037i
\(15\) 3.42335i 0.883905i
\(16\) −3.98630 − 0.330765i −0.996575 − 0.0826912i
\(17\) − 3.85196i − 0.934238i −0.884195 − 0.467119i \(-0.845292\pi\)
0.884195 − 0.467119i \(-0.154708\pi\)
\(18\) 1.02048 − 0.979091i 0.240530 − 0.230774i
\(19\) −3.36545 −0.772088 −0.386044 − 0.922480i \(-0.626159\pi\)
−0.386044 + 0.922480i \(0.626159\pi\)
\(20\) −0.283323 + 6.84083i −0.0633530 + 1.52966i
\(21\) − 1.31476i − 0.286904i
\(22\) 2.31215 + 2.40989i 0.492952 + 0.513790i
\(23\) 4.50535i 0.939431i 0.882818 + 0.469716i \(0.155644\pi\)
−0.882818 + 0.469716i \(0.844356\pi\)
\(24\) 2.12024 − 1.87205i 0.432793 − 0.382130i
\(25\) 6.71931 1.34386
\(26\) −4.58842 + 4.40231i −0.899862 + 0.863365i
\(27\) 1.00000i 0.192450i
\(28\) 0.108812 − 2.62727i 0.0205636 − 0.496507i
\(29\) −7.97545 −1.48100 −0.740502 − 0.672054i \(-0.765412\pi\)
−0.740502 + 0.672054i \(0.765412\pi\)
\(30\) −3.35177 − 3.49346i −0.611946 − 0.637816i
\(31\) − 3.81255i − 0.684753i −0.939563 − 0.342377i \(-0.888768\pi\)
0.939563 − 0.342377i \(-0.111232\pi\)
\(32\) 4.39179 − 3.56541i 0.776367 − 0.630282i
\(33\) −2.36153 −0.411089
\(34\) 3.77142 + 3.93085i 0.646793 + 0.674136i
\(35\) −4.50088 −0.760787
\(36\) −0.0827621 + 1.99829i −0.0137937 + 0.333048i
\(37\) 5.68460 + 2.16457i 0.934542 + 0.355854i
\(38\) 3.43438 − 3.29508i 0.557130 − 0.534533i
\(39\) − 4.49633i − 0.719989i
\(40\) −6.40867 − 7.25834i −1.01330 − 1.14764i
\(41\) −2.54834 −0.397984 −0.198992 − 0.980001i \(-0.563767\pi\)
−0.198992 + 0.980001i \(0.563767\pi\)
\(42\) 1.28727 + 1.34169i 0.198630 + 0.207027i
\(43\) −11.0092 −1.67888 −0.839442 − 0.543449i \(-0.817118\pi\)
−0.839442 + 0.543449i \(0.817118\pi\)
\(44\) −4.71900 − 0.195445i −0.711417 − 0.0294644i
\(45\) 3.42335 0.510323
\(46\) −4.41115 − 4.59763i −0.650389 − 0.677883i
\(47\) −6.41848 −0.936231 −0.468115 − 0.883667i \(-0.655067\pi\)
−0.468115 + 0.883667i \(0.655067\pi\)
\(48\) −0.330765 + 3.98630i −0.0477418 + 0.575373i
\(49\) −5.27141 −0.753058
\(50\) −6.85693 + 6.57882i −0.969716 + 0.930385i
\(51\) −3.85196 −0.539382
\(52\) 0.372125 − 8.98495i 0.0516045 − 1.24599i
\(53\) 4.59896i 0.631715i 0.948807 + 0.315858i \(0.102292\pi\)
−0.948807 + 0.315858i \(0.897708\pi\)
\(54\) −0.979091 − 1.02048i −0.133237 − 0.138870i
\(55\) 8.08432i 1.09009i
\(56\) 2.46129 + 2.78761i 0.328904 + 0.372510i
\(57\) 3.36545i 0.445765i
\(58\) 8.13880 − 7.80869i 1.06868 − 1.02533i
\(59\) −6.65249 −0.866081 −0.433040 − 0.901375i \(-0.642559\pi\)
−0.433040 + 0.901375i \(0.642559\pi\)
\(60\) 6.84083 + 0.283323i 0.883148 + 0.0365769i
\(61\) −1.28295 −0.164265 −0.0821323 − 0.996621i \(-0.526173\pi\)
−0.0821323 + 0.996621i \(0.526173\pi\)
\(62\) 3.73283 + 3.89063i 0.474070 + 0.494110i
\(63\) −1.31476 −0.165644
\(64\) −0.990877 + 7.93840i −0.123860 + 0.992300i
\(65\) −15.3925 −1.90920
\(66\) 2.40989 − 2.31215i 0.296637 − 0.284606i
\(67\) 0.397943i 0.0486165i 0.999705 + 0.0243083i \(0.00773832\pi\)
−0.999705 + 0.0243083i \(0.992262\pi\)
\(68\) −7.69732 − 0.318796i −0.933438 − 0.0386597i
\(69\) 4.50535 0.542381
\(70\) 4.59306 − 4.40677i 0.548976 − 0.526710i
\(71\) −0.221498 −0.0262870 −0.0131435 − 0.999914i \(-0.504184\pi\)
−0.0131435 + 0.999914i \(0.504184\pi\)
\(72\) −1.87205 − 2.12024i −0.220623 − 0.249873i
\(73\) −13.7728 −1.61198 −0.805990 − 0.591929i \(-0.798367\pi\)
−0.805990 + 0.591929i \(0.798367\pi\)
\(74\) −7.92033 + 3.35683i −0.920720 + 0.390224i
\(75\) − 6.71931i − 0.775879i
\(76\) −0.278532 + 6.72514i −0.0319498 + 0.771427i
\(77\) − 3.10484i − 0.353829i
\(78\) 4.40231 + 4.58842i 0.498464 + 0.519536i
\(79\) − 0.202285i − 0.0227589i −0.999935 − 0.0113794i \(-0.996378\pi\)
0.999935 − 0.0113794i \(-0.00362227\pi\)
\(80\) 13.6465 + 1.13232i 1.52572 + 0.126598i
\(81\) 1.00000 0.111111
\(82\) 2.60053 − 2.49506i 0.287181 − 0.275533i
\(83\) 1.43044i 0.157011i 0.996914 + 0.0785054i \(0.0250148\pi\)
−0.996914 + 0.0785054i \(0.974985\pi\)
\(84\) −2.62727 − 0.108812i −0.286658 − 0.0118724i
\(85\) 13.1866i 1.43029i
\(86\) 11.2347 − 10.7790i 1.21146 − 1.16233i
\(87\) 7.97545i 0.855058i
\(88\) 5.00701 − 4.42089i 0.533749 − 0.471268i
\(89\) − 12.6722i − 1.34325i −0.740893 − 0.671624i \(-0.765597\pi\)
0.740893 − 0.671624i \(-0.234403\pi\)
\(90\) −3.49346 + 3.35177i −0.368243 + 0.353307i
\(91\) 5.91159 0.619703
\(92\) 9.00299 + 0.372872i 0.938626 + 0.0388746i
\(93\) −3.81255 −0.395343
\(94\) 6.54993 − 6.28427i 0.675574 − 0.648173i
\(95\) 11.5211 1.18204
\(96\) −3.56541 − 4.39179i −0.363893 − 0.448235i
\(97\) 16.3634i 1.66145i 0.556680 + 0.830727i \(0.312075\pi\)
−0.556680 + 0.830727i \(0.687925\pi\)
\(98\) 5.37937 − 5.16119i 0.543399 − 0.521359i
\(99\) 2.36153i 0.237342i
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.517.20 yes 76
4.3 odd 2 3552.2.o.a.2737.62 76
8.3 odd 2 3552.2.o.a.2737.39 76
8.5 even 2 inner 888.2.o.a.517.58 yes 76
37.36 even 2 inner 888.2.o.a.517.57 yes 76
148.147 odd 2 3552.2.o.a.2737.40 76
296.147 odd 2 3552.2.o.a.2737.61 76
296.221 even 2 inner 888.2.o.a.517.19 ✓ 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.19 ✓ 76 296.221 even 2 inner
888.2.o.a.517.20 yes 76 1.1 even 1 trivial
888.2.o.a.517.57 yes 76 37.36 even 2 inner
888.2.o.a.517.58 yes 76 8.5 even 2 inner
3552.2.o.a.2737.39 76 8.3 odd 2
3552.2.o.a.2737.40 76 148.147 odd 2
3552.2.o.a.2737.61 76 296.147 odd 2
3552.2.o.a.2737.62 76 4.3 odd 2