Properties

Label 3552.1.i.d.1553.1
Level $3552$
Weight $1$
Character 3552.1553
Self dual yes
Analytic conductor $1.773$
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] = [3552,1,Mod(1553,3552)] 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("3552.1553"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3552, 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: \( N \) \(=\) \( 3552 = 2^{5} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3552.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,1,0,0,0,1,0,1,0,-1,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.77267892487\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.888.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.25233408.1
Stark unit: Root of $x^{6} - 3088158x^{5} + 7269055683x^{4} - 9522187897916x^{3} + 7269055683x^{2} - 3088158x + 1$

Embedding invariants

Embedding label 1553.1
Character \(\chi\) \(=\) 3552.1553

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{17} -1.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{33} -1.00000 q^{37} +1.00000 q^{39} +2.00000 q^{43} -1.00000 q^{51} +1.00000 q^{53} -1.00000 q^{57} -2.00000 q^{61} +1.00000 q^{63} +1.00000 q^{69} -1.00000 q^{73} +1.00000 q^{75} -1.00000 q^{77} +1.00000 q^{81} -1.00000 q^{83} -1.00000 q^{89} +1.00000 q^{91} -1.00000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3552\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(2369\) \(3073\) \(3109\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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