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Properties

Label 888.2.o.a.517.9
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 517.9
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.10

$q$-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+(-1.34547 - 0.435559i) q^{2} +1.00000i q^{3} +(1.62058 + 1.17206i) q^{4} +0.598100 q^{5} +(0.435559 - 1.34547i) q^{6} -4.50845 q^{7} +(-1.66994 - 2.28283i) q^{8} -1.00000 q^{9} +(-0.804725 - 0.260508i) q^{10} +1.30412i q^{11} +(-1.17206 + 1.62058i) q^{12} +3.44826 q^{13} +(6.06599 + 1.96370i) q^{14} +0.598100i q^{15} +(1.25254 + 3.79883i) q^{16} -4.39922i q^{17} +(1.34547 + 0.435559i) q^{18} -3.50187 q^{19} +(0.969267 + 0.701010i) q^{20} -4.50845i q^{21} +(0.568023 - 1.75466i) q^{22} -3.10879i q^{23} +(2.28283 - 1.66994i) q^{24} -4.64228 q^{25} +(-4.63953 - 1.50192i) q^{26} -1.00000i q^{27} +(-7.30630 - 5.28419i) q^{28} +5.02900 q^{29} +(0.260508 - 0.804725i) q^{30} -5.05333i q^{31} +(-0.0306414 - 5.65677i) q^{32} -1.30412 q^{33} +(-1.91612 + 5.91901i) q^{34} -2.69651 q^{35} +(-1.62058 - 1.17206i) q^{36} +(3.40580 - 5.03990i) q^{37} +(4.71166 + 1.52527i) q^{38} +3.44826i q^{39} +(-0.998788 - 1.36536i) q^{40} +1.71921 q^{41} +(-1.96370 + 6.06599i) q^{42} -9.32710 q^{43} +(-1.52851 + 2.11343i) q^{44} -0.598100 q^{45} +(-1.35406 + 4.18278i) q^{46} +10.9753 q^{47} +(-3.79883 + 1.25254i) q^{48} +13.3262 q^{49} +(6.24604 + 2.02198i) q^{50} +4.39922 q^{51} +(5.58817 + 4.04158i) q^{52} -9.44849i q^{53} +(-0.435559 + 1.34547i) q^{54} +0.779996i q^{55} +(7.52883 + 10.2920i) q^{56} -3.50187i q^{57} +(-6.76637 - 2.19042i) q^{58} +3.55467 q^{59} +(-0.701010 + 0.969267i) q^{60} -7.87202 q^{61} +(-2.20102 + 6.79911i) q^{62} +4.50845 q^{63} +(-2.42263 + 7.62436i) q^{64} +2.06240 q^{65} +(1.75466 + 0.568023i) q^{66} +7.87891i q^{67} +(5.15615 - 7.12927i) q^{68} +3.10879 q^{69} +(3.62807 + 1.17449i) q^{70} +1.68222 q^{71} +(1.66994 + 2.28283i) q^{72} -3.11875 q^{73} +(-6.77757 + 5.29760i) q^{74} -4.64228i q^{75} +(-5.67505 - 4.10441i) q^{76} -5.87958i q^{77} +(1.50192 - 4.63953i) q^{78} -16.6650i q^{79} +(0.749145 + 2.27208i) q^{80} +1.00000 q^{81} +(-2.31315 - 0.748817i) q^{82} -16.7465i q^{83} +(5.28419 - 7.30630i) q^{84} -2.63117i q^{85} +(12.5493 + 4.06250i) q^{86} +5.02900i q^{87} +(2.97709 - 2.17780i) q^{88} -5.64431i q^{89} +(0.804725 + 0.260508i) q^{90} -15.5463 q^{91} +(3.64370 - 5.03804i) q^{92} +5.05333 q^{93} +(-14.7670 - 4.78040i) q^{94} -2.09447 q^{95} +(5.65677 - 0.0306414i) q^{96} +7.07085i q^{97} +(-17.9299 - 5.80432i) q^{98} -1.30412i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34547 − 0.435559i −0.951391 − 0.307986i
\(3\) 1.00000i 0.577350i
\(4\) 1.62058 + 1.17206i 0.810289 + 0.586031i
\(5\) 0.598100 0.267478 0.133739 − 0.991017i \(-0.457302\pi\)
0.133739 + 0.991017i \(0.457302\pi\)
\(6\) 0.435559 − 1.34547i 0.177816 − 0.549286i
\(7\) −4.50845 −1.70404 −0.852018 − 0.523513i \(-0.824621\pi\)
−0.852018 + 0.523513i \(0.824621\pi\)
\(8\) −1.66994 − 2.28283i −0.590411 − 0.807102i
\(9\) −1.00000 −0.333333
\(10\) −0.804725 − 0.260508i −0.254476 − 0.0823797i
\(11\) 1.30412i 0.393208i 0.980483 + 0.196604i \(0.0629914\pi\)
−0.980483 + 0.196604i \(0.937009\pi\)
\(12\) −1.17206 + 1.62058i −0.338345 + 0.467820i
\(13\) 3.44826 0.956375 0.478188 − 0.878258i \(-0.341294\pi\)
0.478188 + 0.878258i \(0.341294\pi\)
\(14\) 6.06599 + 1.96370i 1.62120 + 0.524820i
\(15\) 0.598100i 0.154429i
\(16\) 1.25254 + 3.79883i 0.313135 + 0.949709i
\(17\) − 4.39922i − 1.06697i −0.845811 − 0.533483i \(-0.820883\pi\)
0.845811 − 0.533483i \(-0.179117\pi\)
\(18\) 1.34547 + 0.435559i 0.317130 + 0.102662i
\(19\) −3.50187 −0.803384 −0.401692 − 0.915775i \(-0.631578\pi\)
−0.401692 + 0.915775i \(0.631578\pi\)
\(20\) 0.969267 + 0.701010i 0.216735 + 0.156751i
\(21\) − 4.50845i − 0.983825i
\(22\) 0.568023 − 1.75466i 0.121103 − 0.374095i
\(23\) − 3.10879i − 0.648228i −0.946018 − 0.324114i \(-0.894934\pi\)
0.946018 − 0.324114i \(-0.105066\pi\)
\(24\) 2.28283 − 1.66994i 0.465981 − 0.340874i
\(25\) −4.64228 −0.928455
\(26\) −4.63953 − 1.50192i −0.909887 − 0.294551i
\(27\) − 1.00000i − 0.192450i
\(28\) −7.30630 − 5.28419i −1.38076 − 0.998618i
\(29\) 5.02900 0.933862 0.466931 − 0.884294i \(-0.345360\pi\)
0.466931 + 0.884294i \(0.345360\pi\)
\(30\) 0.260508 − 0.804725i 0.0475620 − 0.146922i
\(31\) − 5.05333i − 0.907606i −0.891102 − 0.453803i \(-0.850067\pi\)
0.891102 − 0.453803i \(-0.149933\pi\)
\(32\) −0.0306414 − 5.65677i −0.00541668 − 0.999985i
\(33\) −1.30412 −0.227019
\(34\) −1.91612 + 5.91901i −0.328611 + 1.01510i
\(35\) −2.69651 −0.455793
\(36\) −1.62058 − 1.17206i −0.270096 − 0.195344i
\(37\) 3.40580 − 5.03990i 0.559910 − 0.828554i
\(38\) 4.71166 + 1.52527i 0.764332 + 0.247431i
\(39\) 3.44826i 0.552164i
\(40\) −0.998788 − 1.36536i −0.157922 − 0.215882i
\(41\) 1.71921 0.268496 0.134248 − 0.990948i \(-0.457138\pi\)
0.134248 + 0.990948i \(0.457138\pi\)
\(42\) −1.96370 + 6.06599i −0.303005 + 0.936002i
\(43\) −9.32710 −1.42237 −0.711185 − 0.703005i \(-0.751841\pi\)
−0.711185 + 0.703005i \(0.751841\pi\)
\(44\) −1.52851 + 2.11343i −0.230432 + 0.318612i
\(45\) −0.598100 −0.0891595
\(46\) −1.35406 + 4.18278i −0.199645 + 0.616718i
\(47\) 10.9753 1.60092 0.800458 − 0.599388i \(-0.204589\pi\)
0.800458 + 0.599388i \(0.204589\pi\)
\(48\) −3.79883 + 1.25254i −0.548314 + 0.180789i
\(49\) 13.3262 1.90374
\(50\) 6.24604 + 2.02198i 0.883324 + 0.285952i
\(51\) 4.39922 0.616014
\(52\) 5.58817 + 4.04158i 0.774940 + 0.560466i
\(53\) − 9.44849i − 1.29785i −0.760853 − 0.648925i \(-0.775219\pi\)
0.760853 − 0.648925i \(-0.224781\pi\)
\(54\) −0.435559 + 1.34547i −0.0592720 + 0.183095i
\(55\) 0.779996i 0.105175i
\(56\) 7.52883 + 10.2920i 1.00608 + 1.37533i
\(57\) − 3.50187i − 0.463834i
\(58\) −6.76637 − 2.19042i −0.888468 − 0.287617i
\(59\) 3.55467 0.462779 0.231389 − 0.972861i \(-0.425673\pi\)
0.231389 + 0.972861i \(0.425673\pi\)
\(60\) −0.701010 + 0.969267i −0.0905000 + 0.125132i
\(61\) −7.87202 −1.00791 −0.503954 − 0.863730i \(-0.668122\pi\)
−0.503954 + 0.863730i \(0.668122\pi\)
\(62\) −2.20102 + 6.79911i −0.279530 + 0.863488i
\(63\) 4.50845 0.568012
\(64\) −2.42263 + 7.62436i −0.302829 + 0.953045i
\(65\) 2.06240 0.255810
\(66\) 1.75466 + 0.568023i 0.215984 + 0.0699188i
\(67\) 7.87891i 0.962562i 0.876566 + 0.481281i \(0.159828\pi\)
−0.876566 + 0.481281i \(0.840172\pi\)
\(68\) 5.15615 − 7.12927i 0.625276 − 0.864551i
\(69\) 3.10879 0.374254
\(70\) 3.62807 + 1.17449i 0.433637 + 0.140378i
\(71\) 1.68222 0.199643 0.0998213 − 0.995005i \(-0.468173\pi\)
0.0998213 + 0.995005i \(0.468173\pi\)
\(72\) 1.66994 + 2.28283i 0.196804 + 0.269034i
\(73\) −3.11875 −0.365023 −0.182511 − 0.983204i \(-0.558423\pi\)
−0.182511 + 0.983204i \(0.558423\pi\)
\(74\) −6.77757 + 5.29760i −0.787876 + 0.615834i
\(75\) − 4.64228i − 0.536044i
\(76\) −5.67505 − 4.10441i −0.650973 − 0.470808i
\(77\) − 5.87958i − 0.670041i
\(78\) 1.50192 − 4.63953i 0.170059 − 0.525323i
\(79\) − 16.6650i − 1.87496i −0.348038 − 0.937480i \(-0.613152\pi\)
0.348038 − 0.937480i \(-0.386848\pi\)
\(80\) 0.749145 + 2.27208i 0.0837569 + 0.254026i
\(81\) 1.00000 0.111111
\(82\) −2.31315 − 0.748817i −0.255444 − 0.0826930i
\(83\) − 16.7465i − 1.83816i −0.394067 − 0.919082i \(-0.628932\pi\)
0.394067 − 0.919082i \(-0.371068\pi\)
\(84\) 5.28419 − 7.30630i 0.576552 − 0.797182i
\(85\) − 2.63117i − 0.285390i
\(86\) 12.5493 + 4.06250i 1.35323 + 0.438071i
\(87\) 5.02900i 0.539165i
\(88\) 2.97709 − 2.17780i 0.317359 − 0.232155i
\(89\) − 5.64431i − 0.598296i −0.954207 − 0.299148i \(-0.903298\pi\)
0.954207 − 0.299148i \(-0.0967024\pi\)
\(90\) 0.804725 + 0.260508i 0.0848255 + 0.0274599i
\(91\) −15.5463 −1.62970
\(92\) 3.64370 − 5.03804i 0.379882 − 0.525252i
\(93\) 5.05333 0.524006
\(94\) −14.7670 − 4.78040i −1.52310 − 0.493061i
\(95\) −2.09447 −0.214888
\(96\) 5.65677 − 0.0306414i 0.577342 − 0.00312732i
\(97\) 7.07085i 0.717936i 0.933350 + 0.358968i \(0.116871\pi\)
−0.933350 + 0.358968i \(0.883129\pi\)
\(98\) −17.9299 − 5.80432i −1.81120 − 0.586325i
\(99\) − 1.30412i − 0.131069i
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.517.9 ✓ 76
4.3 odd 2 3552.2.o.a.2737.57 76
8.3 odd 2 3552.2.o.a.2737.16 76
8.5 even 2 inner 888.2.o.a.517.67 yes 76
37.36 even 2 inner 888.2.o.a.517.68 yes 76
148.147 odd 2 3552.2.o.a.2737.15 76
296.147 odd 2 3552.2.o.a.2737.58 76
296.221 even 2 inner 888.2.o.a.517.10 yes 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.9 ✓ 76 1.1 even 1 trivial
888.2.o.a.517.10 yes 76 296.221 even 2 inner
888.2.o.a.517.67 yes 76 8.5 even 2 inner
888.2.o.a.517.68 yes 76 37.36 even 2 inner
3552.2.o.a.2737.15 76 148.147 odd 2
3552.2.o.a.2737.16 76 8.3 odd 2
3552.2.o.a.2737.57 76 4.3 odd 2
3552.2.o.a.2737.58 76 296.147 odd 2