Properties

Label 888.1.i.b.221.1
Level $888$
Weight $1$
Character 888.221
Self dual yes
Analytic conductor $0.443$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -888
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] = [888,1,Mod(221,888)] 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("888.221"); S:= CuspForms(chi, 1); 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, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 888.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.443169731218\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.888.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.18925056.1
Stark unit: Root of $x^{6} - 70x^{5} - 37x^{4} - 5116x^{3} - 37x^{2} - 70x + 1$

Embedding invariants

Embedding label 221.1
Character \(\chi\) \(=\) 888.221

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -1.00000 q^{21} +1.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -1.00000 q^{38} +1.00000 q^{39} +1.00000 q^{42} -2.00000 q^{43} -1.00000 q^{44} -1.00000 q^{46} +1.00000 q^{48} -1.00000 q^{50} +1.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +1.00000 q^{57} -2.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} +1.00000 q^{68} +1.00000 q^{69} -1.00000 q^{72} -1.00000 q^{73} +1.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} +1.00000 q^{77} -1.00000 q^{78} +1.00000 q^{81} -1.00000 q^{83} -1.00000 q^{84} +2.00000 q^{86} +1.00000 q^{88} +1.00000 q^{89} -1.00000 q^{91} +1.00000 q^{92} -1.00000 q^{96} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −1.00000
\(3\) 1.00000 1.00000
\(4\) 1.00000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −1.00000 −1.00000
\(7\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) −1.00000 −1.00000
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 1.00000 1.00000
\(13\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 1.00000 1.00000
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) −1.00000 −1.00000
\(19\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) −1.00000 −1.00000
\(22\) 1.00000 1.00000
\(23\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) −1.00000 −1.00000
\(25\) 1.00000 1.00000
\(26\) −1.00000 −1.00000
\(27\) 1.00000 1.00000
\(28\) −1.00000 −1.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) −1.00000 −1.00000
\(34\) −1.00000 −1.00000
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) −1.00000 −1.00000
\(38\) −1.00000 −1.00000
\(39\) 1.00000 1.00000
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 1.00000 1.00000
\(43\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(44\) −1.00000 −1.00000
\(45\) 0 0
\(46\) −1.00000 −1.00000
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000 1.00000
\(49\) 0 0
\(50\) −1.00000 −1.00000
\(51\) 1.00000 1.00000
\(52\) 1.00000 1.00000
\(53\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) 1.00000 1.00000
\(57\) 1.00000 1.00000
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −1.00000 −1.00000
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 1.00000 1.00000
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 1.00000 1.00000
\(69\) 1.00000 1.00000
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −1.00000 −1.00000
\(73\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 1.00000 1.00000
\(75\) 1.00000 1.00000
\(76\) 1.00000 1.00000
\(77\) 1.00000 1.00000
\(78\) −1.00000 −1.00000
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) −1.00000 −1.00000
\(85\) 0 0
\(86\) 2.00000 2.00000
\(87\) 0 0
\(88\) 1.00000 1.00000
\(89\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) −1.00000 −1.00000
\(92\) 1.00000 1.00000
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −1.00000 −1.00000
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 888.1.i.b.221.1 yes 1
3.2 odd 2 888.1.i.c.221.1 yes 1
4.3 odd 2 3552.1.i.b.1553.1 1
8.3 odd 2 3552.1.i.c.1553.1 1
8.5 even 2 888.1.i.a.221.1 ✓ 1
12.11 even 2 3552.1.i.d.1553.1 1
24.5 odd 2 888.1.i.d.221.1 yes 1
24.11 even 2 3552.1.i.a.1553.1 1
37.36 even 2 888.1.i.d.221.1 yes 1
111.110 odd 2 888.1.i.a.221.1 ✓ 1
148.147 odd 2 3552.1.i.a.1553.1 1
296.147 odd 2 3552.1.i.d.1553.1 1
296.221 even 2 888.1.i.c.221.1 yes 1
444.443 even 2 3552.1.i.c.1553.1 1
888.221 odd 2 CM 888.1.i.b.221.1 yes 1
888.443 even 2 3552.1.i.b.1553.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.1.i.a.221.1 ✓ 1 8.5 even 2
888.1.i.a.221.1 ✓ 1 111.110 odd 2
888.1.i.b.221.1 yes 1 1.1 even 1 trivial
888.1.i.b.221.1 yes 1 888.221 odd 2 CM
888.1.i.c.221.1 yes 1 3.2 odd 2
888.1.i.c.221.1 yes 1 296.221 even 2
888.1.i.d.221.1 yes 1 24.5 odd 2
888.1.i.d.221.1 yes 1 37.36 even 2
3552.1.i.a.1553.1 1 24.11 even 2
3552.1.i.a.1553.1 1 148.147 odd 2
3552.1.i.b.1553.1 1 4.3 odd 2
3552.1.i.b.1553.1 1 888.443 even 2
3552.1.i.c.1553.1 1 8.3 odd 2
3552.1.i.c.1553.1 1 444.443 even 2
3552.1.i.d.1553.1 1 12.11 even 2
3552.1.i.d.1553.1 1 296.147 odd 2