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Properties

Label 888.2.j.b.371.3
Level $888$
Weight $2$
Character 888.371
Analytic conductor $7.091$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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(371,888)] 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("888.371"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 371.3
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 888.371
Dual form 888.2.j.b.371.4

$q$-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+(1.36603 - 0.366025i) q^{2} -1.73205i q^{3} +(1.73205 - 1.00000i) q^{4} -0.732051 q^{5} +(-0.633975 - 2.36603i) q^{6} -2.00000i q^{7} +(2.00000 - 2.00000i) q^{8} -3.00000 q^{9} +(-1.00000 + 0.267949i) q^{10} +1.46410i q^{11} +(-1.73205 - 3.00000i) q^{12} -2.00000i q^{13} +(-0.732051 - 2.73205i) q^{14} +1.26795i q^{15} +(2.00000 - 3.46410i) q^{16} -2.73205i q^{17} +(-4.09808 + 1.09808i) q^{18} -6.19615 q^{19} +(-1.26795 + 0.732051i) q^{20} -3.46410 q^{21} +(0.535898 + 2.00000i) q^{22} +6.00000 q^{23} +(-3.46410 - 3.46410i) q^{24} -4.46410 q^{25} +(-0.732051 - 2.73205i) q^{26} +5.19615i q^{27} +(-2.00000 - 3.46410i) q^{28} +10.1962 q^{29} +(0.464102 + 1.73205i) q^{30} +4.19615i q^{31} +(1.46410 - 5.46410i) q^{32} +2.53590 q^{33} +(-1.00000 - 3.73205i) q^{34} +1.46410i q^{35} +(-5.19615 + 3.00000i) q^{36} +1.00000i q^{37} +(-8.46410 + 2.26795i) q^{38} -3.46410 q^{39} +(-1.46410 + 1.46410i) q^{40} -9.46410i q^{41} +(-4.73205 + 1.26795i) q^{42} -3.26795 q^{43} +(1.46410 + 2.53590i) q^{44} +2.19615 q^{45} +(8.19615 - 2.19615i) q^{46} -3.46410 q^{47} +(-6.00000 - 3.46410i) q^{48} +3.00000 q^{49} +(-6.09808 + 1.63397i) q^{50} -4.73205 q^{51} +(-2.00000 - 3.46410i) q^{52} +2.00000 q^{53} +(1.90192 + 7.09808i) q^{54} -1.07180i q^{55} +(-4.00000 - 4.00000i) q^{56} +10.7321i q^{57} +(13.9282 - 3.73205i) q^{58} +3.46410i q^{59} +(1.26795 + 2.19615i) q^{60} +8.53590i q^{61} +(1.53590 + 5.73205i) q^{62} +6.00000i q^{63} -8.00000i q^{64} +1.46410i q^{65} +(3.46410 - 0.928203i) q^{66} -4.00000 q^{67} +(-2.73205 - 4.73205i) q^{68} -10.3923i q^{69} +(0.535898 + 2.00000i) q^{70} +15.4641 q^{71} +(-6.00000 + 6.00000i) q^{72} +4.92820 q^{73} +(0.366025 + 1.36603i) q^{74} +7.73205i q^{75} +(-10.7321 + 6.19615i) q^{76} +2.92820 q^{77} +(-4.73205 + 1.26795i) q^{78} -11.1244i q^{79} +(-1.46410 + 2.53590i) q^{80} +9.00000 q^{81} +(-3.46410 - 12.9282i) q^{82} +14.3923i q^{83} +(-6.00000 + 3.46410i) q^{84} +2.00000i q^{85} +(-4.46410 + 1.19615i) q^{86} -17.6603i q^{87} +(2.92820 + 2.92820i) q^{88} -9.66025i q^{89} +(3.00000 - 0.803848i) q^{90} -4.00000 q^{91} +(10.3923 - 6.00000i) q^{92} +7.26795 q^{93} +(-4.73205 + 1.26795i) q^{94} +4.53590 q^{95} +(-9.46410 - 2.53590i) q^{96} +15.8564 q^{97} +(4.09808 - 1.09808i) q^{98} -4.39230i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{5} - 6 q^{6} + 8 q^{8} - 12 q^{9} - 4 q^{10} + 4 q^{14} + 8 q^{16} - 6 q^{18} - 4 q^{19} - 12 q^{20} + 16 q^{22} + 24 q^{23} - 4 q^{25} + 4 q^{26} - 8 q^{28} + 20 q^{29} - 12 q^{30}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36603 − 0.366025i 0.965926 − 0.258819i
\(3\) − 1.73205i − 1.00000i
\(4\) 1.73205 − 1.00000i 0.866025 − 0.500000i
\(5\) −0.732051 −0.327383 −0.163692 − 0.986512i \(-0.552340\pi\)
−0.163692 + 0.986512i \(0.552340\pi\)
\(6\) −0.633975 − 2.36603i −0.258819 − 0.965926i
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) 2.00000 − 2.00000i 0.707107 − 0.707107i
\(9\) −3.00000 −1.00000
\(10\) −1.00000 + 0.267949i −0.316228 + 0.0847330i
\(11\) 1.46410i 0.441443i 0.975337 + 0.220722i \(0.0708412\pi\)
−0.975337 + 0.220722i \(0.929159\pi\)
\(12\) −1.73205 − 3.00000i −0.500000 − 0.866025i
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) −0.732051 − 2.73205i −0.195649 − 0.730171i
\(15\) 1.26795i 0.327383i
\(16\) 2.00000 − 3.46410i 0.500000 − 0.866025i
\(17\) − 2.73205i − 0.662620i −0.943522 − 0.331310i \(-0.892509\pi\)
0.943522 − 0.331310i \(-0.107491\pi\)
\(18\) −4.09808 + 1.09808i −0.965926 + 0.258819i
\(19\) −6.19615 −1.42149 −0.710747 − 0.703447i \(-0.751643\pi\)
−0.710747 + 0.703447i \(0.751643\pi\)
\(20\) −1.26795 + 0.732051i −0.283522 + 0.163692i
\(21\) −3.46410 −0.755929
\(22\) 0.535898 + 2.00000i 0.114254 + 0.426401i
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −3.46410 − 3.46410i −0.707107 − 0.707107i
\(25\) −4.46410 −0.892820
\(26\) −0.732051 − 2.73205i −0.143567 − 0.535799i
\(27\) 5.19615i 1.00000i
\(28\) −2.00000 − 3.46410i −0.377964 − 0.654654i
\(29\) 10.1962 1.89338 0.946689 − 0.322149i \(-0.104405\pi\)
0.946689 + 0.322149i \(0.104405\pi\)
\(30\) 0.464102 + 1.73205i 0.0847330 + 0.316228i
\(31\) 4.19615i 0.753651i 0.926284 + 0.376826i \(0.122984\pi\)
−0.926284 + 0.376826i \(0.877016\pi\)
\(32\) 1.46410 − 5.46410i 0.258819 − 0.965926i
\(33\) 2.53590 0.441443
\(34\) −1.00000 − 3.73205i −0.171499 − 0.640041i
\(35\) 1.46410i 0.247478i
\(36\) −5.19615 + 3.00000i −0.866025 + 0.500000i
\(37\) 1.00000i 0.164399i
\(38\) −8.46410 + 2.26795i −1.37306 + 0.367910i
\(39\) −3.46410 −0.554700
\(40\) −1.46410 + 1.46410i −0.231495 + 0.231495i
\(41\) − 9.46410i − 1.47804i −0.673681 − 0.739022i \(-0.735288\pi\)
0.673681 − 0.739022i \(-0.264712\pi\)
\(42\) −4.73205 + 1.26795i −0.730171 + 0.195649i
\(43\) −3.26795 −0.498358 −0.249179 − 0.968458i \(-0.580161\pi\)
−0.249179 + 0.968458i \(0.580161\pi\)
\(44\) 1.46410 + 2.53590i 0.220722 + 0.382301i
\(45\) 2.19615 0.327383
\(46\) 8.19615 − 2.19615i 1.20846 − 0.323805i
\(47\) −3.46410 −0.505291 −0.252646 − 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) −6.00000 − 3.46410i −0.866025 − 0.500000i
\(49\) 3.00000 0.428571
\(50\) −6.09808 + 1.63397i −0.862398 + 0.231079i
\(51\) −4.73205 −0.662620
\(52\) −2.00000 − 3.46410i −0.277350 − 0.480384i
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.90192 + 7.09808i 0.258819 + 0.965926i
\(55\) − 1.07180i − 0.144521i
\(56\) −4.00000 − 4.00000i −0.534522 − 0.534522i
\(57\) 10.7321i 1.42149i
\(58\) 13.9282 − 3.73205i 1.82886 − 0.490042i
\(59\) 3.46410i 0.450988i 0.974245 + 0.225494i \(0.0723995\pi\)
−0.974245 + 0.225494i \(0.927600\pi\)
\(60\) 1.26795 + 2.19615i 0.163692 + 0.283522i
\(61\) 8.53590i 1.09291i 0.837489 + 0.546455i \(0.184023\pi\)
−0.837489 + 0.546455i \(0.815977\pi\)
\(62\) 1.53590 + 5.73205i 0.195059 + 0.727971i
\(63\) 6.00000i 0.755929i
\(64\) − 8.00000i − 1.00000i
\(65\) 1.46410i 0.181599i
\(66\) 3.46410 − 0.928203i 0.426401 − 0.114254i
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.73205 − 4.73205i −0.331310 − 0.573845i
\(69\) − 10.3923i − 1.25109i
\(70\) 0.535898 + 2.00000i 0.0640521 + 0.239046i
\(71\) 15.4641 1.83525 0.917626 − 0.397446i \(-0.130103\pi\)
0.917626 + 0.397446i \(0.130103\pi\)
\(72\) −6.00000 + 6.00000i −0.707107 + 0.707107i
\(73\) 4.92820 0.576803 0.288401 − 0.957510i \(-0.406876\pi\)
0.288401 + 0.957510i \(0.406876\pi\)
\(74\) 0.366025 + 1.36603i 0.0425496 + 0.158797i
\(75\) 7.73205i 0.892820i
\(76\) −10.7321 + 6.19615i −1.23105 + 0.710747i
\(77\) 2.92820 0.333700
\(78\) −4.73205 + 1.26795i −0.535799 + 0.143567i
\(79\) − 11.1244i − 1.25159i −0.779988 − 0.625794i \(-0.784775\pi\)
0.779988 − 0.625794i \(-0.215225\pi\)
\(80\) −1.46410 + 2.53590i −0.163692 + 0.283522i
\(81\) 9.00000 1.00000
\(82\) −3.46410 − 12.9282i −0.382546 − 1.42768i
\(83\) 14.3923i 1.57976i 0.613261 + 0.789880i \(0.289857\pi\)
−0.613261 + 0.789880i \(0.710143\pi\)
\(84\) −6.00000 + 3.46410i −0.654654 + 0.377964i
\(85\) 2.00000i 0.216930i
\(86\) −4.46410 + 1.19615i −0.481376 + 0.128984i
\(87\) − 17.6603i − 1.89338i
\(88\) 2.92820 + 2.92820i 0.312148 + 0.312148i
\(89\) − 9.66025i − 1.02398i −0.858990 − 0.511992i \(-0.828908\pi\)
0.858990 − 0.511992i \(-0.171092\pi\)
\(90\) 3.00000 − 0.803848i 0.316228 − 0.0847330i
\(91\) −4.00000 −0.419314
\(92\) 10.3923 − 6.00000i 1.08347 − 0.625543i
\(93\) 7.26795 0.753651
\(94\) −4.73205 + 1.26795i −0.488074 + 0.130779i
\(95\) 4.53590 0.465373
\(96\) −9.46410 − 2.53590i −0.965926 − 0.258819i
\(97\) 15.8564 1.60997 0.804987 − 0.593292i \(-0.202172\pi\)
0.804987 + 0.593292i \(0.202172\pi\)
\(98\) 4.09808 − 1.09808i 0.413968 − 0.110922i
\(99\) − 4.39230i − 0.441443i
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.j.b.371.3 yes 4
3.2 odd 2 888.2.j.a.371.2 yes 4
8.3 odd 2 888.2.j.a.371.1 ✓ 4
24.11 even 2 inner 888.2.j.b.371.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.j.a.371.1 ✓ 4 8.3 odd 2
888.2.j.a.371.2 yes 4 3.2 odd 2
888.2.j.b.371.3 yes 4 1.1 even 1 trivial
888.2.j.b.371.4 yes 4 24.11 even 2 inner