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Properties

Label 888.2.j.b.371.1
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 371.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 888.371
Dual form 888.2.j.b.371.2

$q$-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+(-0.366025 - 1.36603i) q^{2} -1.73205i q^{3} +(-1.73205 + 1.00000i) q^{4} +2.73205 q^{5} +(-2.36603 + 0.633975i) q^{6} +2.00000i q^{7} +(2.00000 + 2.00000i) q^{8} -3.00000 q^{9} +(-1.00000 - 3.73205i) q^{10} +5.46410i q^{11} +(1.73205 + 3.00000i) q^{12} +2.00000i q^{13} +(2.73205 - 0.732051i) q^{14} -4.73205i q^{15} +(2.00000 - 3.46410i) q^{16} -0.732051i q^{17} +(1.09808 + 4.09808i) q^{18} +4.19615 q^{19} +(-4.73205 + 2.73205i) q^{20} +3.46410 q^{21} +(7.46410 - 2.00000i) q^{22} +6.00000 q^{23} +(3.46410 - 3.46410i) q^{24} +2.46410 q^{25} +(2.73205 - 0.732051i) q^{26} +5.19615i q^{27} +(-2.00000 - 3.46410i) q^{28} -0.196152 q^{29} +(-6.46410 + 1.73205i) q^{30} +6.19615i q^{31} +(-5.46410 - 1.46410i) q^{32} +9.46410 q^{33} +(-1.00000 + 0.267949i) q^{34} +5.46410i q^{35} +(5.19615 - 3.00000i) q^{36} -1.00000i q^{37} +(-1.53590 - 5.73205i) q^{38} +3.46410 q^{39} +(5.46410 + 5.46410i) q^{40} +2.53590i q^{41} +(-1.26795 - 4.73205i) q^{42} -6.73205 q^{43} +(-5.46410 - 9.46410i) q^{44} -8.19615 q^{45} +(-2.19615 - 8.19615i) q^{46} +3.46410 q^{47} +(-6.00000 - 3.46410i) q^{48} +3.00000 q^{49} +(-0.901924 - 3.36603i) q^{50} -1.26795 q^{51} +(-2.00000 - 3.46410i) q^{52} +2.00000 q^{53} +(7.09808 - 1.90192i) q^{54} +14.9282i q^{55} +(-4.00000 + 4.00000i) q^{56} -7.26795i q^{57} +(0.0717968 + 0.267949i) q^{58} +3.46410i q^{59} +(4.73205 + 8.19615i) q^{60} -15.4641i q^{61} +(8.46410 - 2.26795i) q^{62} -6.00000i q^{63} +8.00000i q^{64} +5.46410i q^{65} +(-3.46410 - 12.9282i) q^{66} -4.00000 q^{67} +(0.732051 + 1.26795i) q^{68} -10.3923i q^{69} +(7.46410 - 2.00000i) q^{70} +8.53590 q^{71} +(-6.00000 - 6.00000i) q^{72} -8.92820 q^{73} +(-1.36603 + 0.366025i) q^{74} -4.26795i q^{75} +(-7.26795 + 4.19615i) q^{76} -10.9282 q^{77} +(-1.26795 - 4.73205i) q^{78} -13.1244i q^{79} +(5.46410 - 9.46410i) q^{80} +9.00000 q^{81} +(3.46410 - 0.928203i) q^{82} +6.39230i q^{83} +(-6.00000 + 3.46410i) q^{84} -2.00000i q^{85} +(2.46410 + 9.19615i) q^{86} +0.339746i q^{87} +(-10.9282 + 10.9282i) q^{88} -7.66025i q^{89} +(3.00000 + 11.1962i) q^{90} -4.00000 q^{91} +(-10.3923 + 6.00000i) q^{92} +10.7321 q^{93} +(-1.26795 - 4.73205i) q^{94} +11.4641 q^{95} +(-2.53590 + 9.46410i) q^{96} -11.8564 q^{97} +(-1.09808 - 4.09808i) q^{98} -16.3923i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.366025 − 1.36603i −0.258819 − 0.965926i
\(3\) − 1.73205i − 1.00000i
\(4\) −1.73205 + 1.00000i −0.866025 + 0.500000i
\(5\) 2.73205 1.22181 0.610905 − 0.791704i \(-0.290806\pi\)
0.610905 + 0.791704i \(0.290806\pi\)
\(6\) −2.36603 + 0.633975i −0.965926 + 0.258819i
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) −3.00000 −1.00000
\(10\) −1.00000 − 3.73205i −0.316228 − 1.18018i
\(11\) 5.46410i 1.64749i 0.566961 + 0.823744i \(0.308119\pi\)
−0.566961 + 0.823744i \(0.691881\pi\)
\(12\) 1.73205 + 3.00000i 0.500000 + 0.866025i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 2.73205 − 0.732051i 0.730171 − 0.195649i
\(15\) − 4.73205i − 1.22181i
\(16\) 2.00000 − 3.46410i 0.500000 − 0.866025i
\(17\) − 0.732051i − 0.177548i −0.996052 − 0.0887742i \(-0.971705\pi\)
0.996052 − 0.0887742i \(-0.0282950\pi\)
\(18\) 1.09808 + 4.09808i 0.258819 + 0.965926i
\(19\) 4.19615 0.962663 0.481332 − 0.876539i \(-0.340153\pi\)
0.481332 + 0.876539i \(0.340153\pi\)
\(20\) −4.73205 + 2.73205i −1.05812 + 0.610905i
\(21\) 3.46410 0.755929
\(22\) 7.46410 − 2.00000i 1.59135 − 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
\(25\) 2.46410 0.492820
\(26\) 2.73205 − 0.732051i 0.535799 − 0.143567i
\(27\) 5.19615i 1.00000i
\(28\) −2.00000 − 3.46410i −0.377964 − 0.654654i
\(29\) −0.196152 −0.0364246 −0.0182123 − 0.999834i \(-0.505797\pi\)
−0.0182123 + 0.999834i \(0.505797\pi\)
\(30\) −6.46410 + 1.73205i −1.18018 + 0.316228i
\(31\) 6.19615i 1.11286i 0.830894 + 0.556431i \(0.187830\pi\)
−0.830894 + 0.556431i \(0.812170\pi\)
\(32\) −5.46410 − 1.46410i −0.965926 − 0.258819i
\(33\) 9.46410 1.64749
\(34\) −1.00000 + 0.267949i −0.171499 + 0.0459529i
\(35\) 5.46410i 0.923602i
\(36\) 5.19615 − 3.00000i 0.866025 − 0.500000i
\(37\) − 1.00000i − 0.164399i
\(38\) −1.53590 − 5.73205i −0.249156 − 0.929861i
\(39\) 3.46410 0.554700
\(40\) 5.46410 + 5.46410i 0.863950 + 0.863950i
\(41\) 2.53590i 0.396041i 0.980198 + 0.198020i \(0.0634512\pi\)
−0.980198 + 0.198020i \(0.936549\pi\)
\(42\) −1.26795 − 4.73205i −0.195649 − 0.730171i
\(43\) −6.73205 −1.02663 −0.513314 − 0.858201i \(-0.671582\pi\)
−0.513314 + 0.858201i \(0.671582\pi\)
\(44\) −5.46410 − 9.46410i −0.823744 − 1.42677i
\(45\) −8.19615 −1.22181
\(46\) −2.19615 − 8.19615i −0.323805 − 1.20846i
\(47\) 3.46410 0.505291 0.252646 − 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) −6.00000 − 3.46410i −0.866025 − 0.500000i
\(49\) 3.00000 0.428571
\(50\) −0.901924 − 3.36603i −0.127551 − 0.476028i
\(51\) −1.26795 −0.177548
\(52\) −2.00000 − 3.46410i −0.277350 − 0.480384i
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 7.09808 − 1.90192i 0.965926 − 0.258819i
\(55\) 14.9282i 2.01292i
\(56\) −4.00000 + 4.00000i −0.534522 + 0.534522i
\(57\) − 7.26795i − 0.962663i
\(58\) 0.0717968 + 0.267949i 0.00942738 + 0.0351835i
\(59\) 3.46410i 0.450988i 0.974245 + 0.225494i \(0.0723995\pi\)
−0.974245 + 0.225494i \(0.927600\pi\)
\(60\) 4.73205 + 8.19615i 0.610905 + 1.05812i
\(61\) − 15.4641i − 1.97998i −0.141154 − 0.989988i \(-0.545081\pi\)
0.141154 − 0.989988i \(-0.454919\pi\)
\(62\) 8.46410 − 2.26795i 1.07494 − 0.288030i
\(63\) − 6.00000i − 0.755929i
\(64\) 8.00000i 1.00000i
\(65\) 5.46410i 0.677738i
\(66\) −3.46410 − 12.9282i −0.426401 − 1.59135i
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0.732051 + 1.26795i 0.0887742 + 0.153761i
\(69\) − 10.3923i − 1.25109i
\(70\) 7.46410 − 2.00000i 0.892131 − 0.239046i
\(71\) 8.53590 1.01302 0.506512 − 0.862233i \(-0.330934\pi\)
0.506512 + 0.862233i \(0.330934\pi\)
\(72\) −6.00000 − 6.00000i −0.707107 − 0.707107i
\(73\) −8.92820 −1.04497 −0.522484 − 0.852649i \(-0.674994\pi\)
−0.522484 + 0.852649i \(0.674994\pi\)
\(74\) −1.36603 + 0.366025i −0.158797 + 0.0425496i
\(75\) − 4.26795i − 0.492820i
\(76\) −7.26795 + 4.19615i −0.833691 + 0.481332i
\(77\) −10.9282 −1.24538
\(78\) −1.26795 − 4.73205i −0.143567 − 0.535799i
\(79\) − 13.1244i − 1.47661i −0.674470 − 0.738303i \(-0.735628\pi\)
0.674470 − 0.738303i \(-0.264372\pi\)
\(80\) 5.46410 − 9.46410i 0.610905 − 1.05812i
\(81\) 9.00000 1.00000
\(82\) 3.46410 − 0.928203i 0.382546 − 0.102503i
\(83\) 6.39230i 0.701647i 0.936442 + 0.350823i \(0.114098\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(84\) −6.00000 + 3.46410i −0.654654 + 0.377964i
\(85\) − 2.00000i − 0.216930i
\(86\) 2.46410 + 9.19615i 0.265711 + 0.991647i
\(87\) 0.339746i 0.0364246i
\(88\) −10.9282 + 10.9282i −1.16495 + 1.16495i
\(89\) − 7.66025i − 0.811985i −0.913876 − 0.405993i \(-0.866926\pi\)
0.913876 − 0.405993i \(-0.133074\pi\)
\(90\) 3.00000 + 11.1962i 0.316228 + 1.18018i
\(91\) −4.00000 −0.419314
\(92\) −10.3923 + 6.00000i −1.08347 + 0.625543i
\(93\) 10.7321 1.11286
\(94\) −1.26795 − 4.73205i −0.130779 − 0.488074i
\(95\) 11.4641 1.17619
\(96\) −2.53590 + 9.46410i −0.258819 + 0.965926i
\(97\) −11.8564 −1.20384 −0.601918 − 0.798558i \(-0.705597\pi\)
−0.601918 + 0.798558i \(0.705597\pi\)
\(98\) −1.09808 − 4.09808i −0.110922 − 0.413968i
\(99\) − 16.3923i − 1.64749i
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.371.1 yes 4
3.2 odd 2 888.2.j.a.371.4 yes 4
8.3 odd 2 888.2.j.a.371.3 ✓ 4
24.11 even 2 inner 888.2.j.b.371.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.j.a.371.3 ✓ 4 8.3 odd 2
888.2.j.a.371.4 yes 4 3.2 odd 2
888.2.j.b.371.1 yes 4 1.1 even 1 trivial
888.2.j.b.371.2 yes 4 24.11 even 2 inner