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Properties

Label 880.1.y.d.527.2
Level $880$
Weight $1$
Character 880.527
Analytic conductor $0.439$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -11
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] = [880,1,Mod(527,880)] 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("880.527"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 880.y (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2] 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: no
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

Embedding invariants

Embedding label 527.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 880.527
Dual form 880.1.y.d.703.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.36603 + 1.36603i) q^{3} +(-0.866025 + 0.500000i) q^{5} +2.73205i q^{9} -1.00000i q^{11} +(-1.86603 - 0.500000i) q^{15} +(-0.366025 - 0.366025i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-2.36603 + 2.36603i) q^{27} -1.00000i q^{31} +(1.36603 - 1.36603i) q^{33} +(1.36603 + 1.36603i) q^{37} +(-1.36603 - 2.36603i) q^{45} +(1.00000 - 1.00000i) q^{47} -1.00000i q^{49} +(-1.00000 + 1.00000i) q^{53} +(0.500000 + 0.866025i) q^{55} -1.00000 q^{59} +(0.366025 - 0.366025i) q^{67} -1.00000i q^{69} -1.73205i q^{71} +(1.86603 - 0.500000i) q^{75} -3.73205 q^{81} +1.00000i q^{89} +(1.36603 - 1.36603i) q^{93} +(0.366025 + 0.366025i) q^{97} +2.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{15} + 2 q^{23} + 2 q^{25} - 6 q^{27} + 2 q^{33} + 2 q^{37} - 2 q^{45} + 4 q^{47} - 4 q^{53} + 2 q^{55} - 4 q^{59} - 2 q^{67} + 4 q^{75} - 8 q^{81} + 2 q^{93} - 2 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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
\(3\) 1.36603 + 1.36603i 1.36603 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(6\) 0 0
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 2.73205i 2.73205i
\(10\) 0 0
\(11\) − 1.00000i − 1.00000i
\(12\) 0 0
\(13\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) −1.86603 − 0.500000i −1.86603 − 0.500000i
\(16\) 0 0
\(17\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.366025 − 0.366025i −0.366025 − 0.366025i 0.500000 − 0.866025i \(-0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(26\) 0 0
\(27\) −2.36603 + 2.36603i −2.36603 + 2.36603i
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(32\) 0 0
\(33\) 1.36603 − 1.36603i 1.36603 − 1.36603i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.36603 + 1.36603i 1.36603 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) −1.36603 − 2.36603i −1.36603 − 2.36603i
\(46\) 0 0
\(47\) 1.00000 − 1.00000i 1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(48\) 0 0
\(49\) − 1.00000i − 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.366025 − 0.366025i 0.366025 − 0.366025i −0.500000 − 0.866025i \(-0.666667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 0 0
\(69\) − 1.00000i − 1.00000i
\(70\) 0 0
\(71\) − 1.73205i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(72\) 0 0
\(73\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) 1.86603 − 0.500000i 1.86603 − 0.500000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −3.73205 −3.73205
\(82\) 0 0
\(83\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.36603 − 1.36603i 1.36603 − 1.36603i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.366025 + 0.366025i 0.366025 + 0.366025i 0.866025 − 0.500000i \(-0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 0 0
\(99\) 2.73205 2.73205
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 880.1.y.d.527.2 yes 4
4.3 odd 2 880.1.y.c.527.1 ✓ 4
5.3 odd 4 880.1.y.c.703.1 yes 4
8.3 odd 2 3520.1.y.d.1407.2 4
8.5 even 2 3520.1.y.c.1407.1 4
11.10 odd 2 CM 880.1.y.d.527.2 yes 4
20.3 even 4 inner 880.1.y.d.703.2 yes 4
40.3 even 4 3520.1.y.c.703.1 4
40.13 odd 4 3520.1.y.d.703.2 4
44.43 even 2 880.1.y.c.527.1 ✓ 4
55.43 even 4 880.1.y.c.703.1 yes 4
88.21 odd 2 3520.1.y.c.1407.1 4
88.43 even 2 3520.1.y.d.1407.2 4
220.43 odd 4 inner 880.1.y.d.703.2 yes 4
440.43 odd 4 3520.1.y.c.703.1 4
440.373 even 4 3520.1.y.d.703.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
880.1.y.c.527.1 ✓ 4 4.3 odd 2
880.1.y.c.527.1 ✓ 4 44.43 even 2
880.1.y.c.703.1 yes 4 5.3 odd 4
880.1.y.c.703.1 yes 4 55.43 even 4
880.1.y.d.527.2 yes 4 1.1 even 1 trivial
880.1.y.d.527.2 yes 4 11.10 odd 2 CM
880.1.y.d.703.2 yes 4 20.3 even 4 inner
880.1.y.d.703.2 yes 4 220.43 odd 4 inner
3520.1.y.c.703.1 4 40.3 even 4
3520.1.y.c.703.1 4 440.43 odd 4
3520.1.y.c.1407.1 4 8.5 even 2
3520.1.y.c.1407.1 4 88.21 odd 2
3520.1.y.d.703.2 4 40.13 odd 4
3520.1.y.d.703.2 4 440.373 even 4
3520.1.y.d.1407.2 4 8.3 odd 2
3520.1.y.d.1407.2 4 88.43 even 2