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Properties

Label 980.1.j.a.587.3
Level $980$
Weight $1$
Character 980.587
Analytic conductor $0.489$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -4
Inner twists $8$

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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] = [980,1,Mod(587,980)] 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("980.587"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 980.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.489083712380\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.823543000000.2

Embedding invariants

Embedding label 587.3
Root \(-0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 980.587
Dual form 980.1.j.a.783.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.707107 + 0.707107i) q^{2} +1.00000i q^{4} +(-0.923880 - 0.382683i) q^{5} +(-0.707107 + 0.707107i) q^{8} +1.00000i q^{9} +(-0.382683 - 0.923880i) q^{10} +(1.30656 + 1.30656i) q^{13} -1.00000 q^{16} +(-0.541196 + 0.541196i) q^{17} +(-0.707107 + 0.707107i) q^{18} +(0.382683 - 0.923880i) q^{20} +(0.707107 + 0.707107i) q^{25} +1.84776i q^{26} -1.41421i q^{29} +(-0.707107 - 0.707107i) q^{32} -0.765367 q^{34} -1.00000 q^{36} +(0.923880 - 0.382683i) q^{40} -0.765367i q^{41} +(0.382683 - 0.923880i) q^{45} +1.00000i q^{50} +(-1.30656 + 1.30656i) q^{52} +(1.00000 - 1.00000i) q^{53} +(1.00000 - 1.00000i) q^{58} -1.84776i q^{61} -1.00000i q^{64} +(-0.707107 - 1.70711i) q^{65} +(-0.541196 - 0.541196i) q^{68} +(-0.707107 - 0.707107i) q^{72} +(0.541196 + 0.541196i) q^{73} +(0.923880 + 0.382683i) q^{80} -1.00000 q^{81} +(0.541196 - 0.541196i) q^{82} +(0.707107 - 0.292893i) q^{85} +1.84776 q^{89} +(0.923880 - 0.382683i) q^{90} +(-1.30656 + 1.30656i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{16} - 8 q^{36} + 8 q^{53} + 8 q^{58} - 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(3\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 1.00000i 1.00000i
\(5\) −0.923880 − 0.382683i −0.923880 − 0.382683i
\(6\) 0 0
\(7\) 0 0
\(8\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(9\) 1.00000i 1.00000i
\(10\) −0.382683 − 0.923880i −0.382683 − 0.923880i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 − 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(18\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0.382683 − 0.923880i 0.382683 − 0.923880i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(26\) 1.84776i 1.84776i
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.41421i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(33\) 0 0
\(34\) −0.765367 −0.765367
\(35\) 0 0
\(36\) −1.00000 −1.00000
\(37\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.923880 − 0.382683i 0.923880 − 0.382683i
\(41\) − 0.765367i − 0.765367i −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 − 0.382683i \(-0.125000\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 0.382683 − 0.923880i 0.382683 − 0.923880i
\(46\) 0 0
\(47\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000i 1.00000i
\(51\) 0 0
\(52\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(53\) 1.00000 − 1.00000i 1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.00000 − 1.00000i 1.00000 − 1.00000i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 1.84776i − 1.84776i −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 − 0.923880i \(-0.375000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) − 1.00000i − 1.00000i
\(65\) −0.707107 − 1.70711i −0.707107 − 1.70711i
\(66\) 0 0
\(67\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) −0.541196 − 0.541196i −0.541196 − 0.541196i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(73\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 − 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(81\) −1.00000 −1.00000
\(82\) 0.541196 − 0.541196i 0.541196 − 0.541196i
\(83\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0.707107 − 0.292893i 0.707107 − 0.292893i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.84776 1.84776 0.923880 − 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(90\) 0.923880 − 0.382683i 0.923880 − 0.382683i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(98\) 0 0
\(99\) 0 0
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 980.1.j.a.587.3 ✓ 8
4.3 odd 2 CM 980.1.j.a.587.3 ✓ 8
5.3 odd 4 inner 980.1.j.a.783.4 yes 8
7.2 even 3 980.1.y.a.227.4 16
7.3 odd 6 980.1.y.a.607.1 16
7.4 even 3 980.1.y.a.607.2 16
7.5 odd 6 980.1.y.a.227.3 16
7.6 odd 2 inner 980.1.j.a.587.4 yes 8
20.3 even 4 inner 980.1.j.a.783.4 yes 8
28.3 even 6 980.1.y.a.607.1 16
28.11 odd 6 980.1.y.a.607.2 16
28.19 even 6 980.1.y.a.227.3 16
28.23 odd 6 980.1.y.a.227.4 16
28.27 even 2 inner 980.1.j.a.587.4 yes 8
35.3 even 12 980.1.y.a.803.4 16
35.13 even 4 inner 980.1.j.a.783.3 yes 8
35.18 odd 12 980.1.y.a.803.3 16
35.23 odd 12 980.1.y.a.423.1 16
35.33 even 12 980.1.y.a.423.2 16
140.3 odd 12 980.1.y.a.803.4 16
140.23 even 12 980.1.y.a.423.1 16
140.83 odd 4 inner 980.1.j.a.783.3 yes 8
140.103 odd 12 980.1.y.a.423.2 16
140.123 even 12 980.1.y.a.803.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.1.j.a.587.3 ✓ 8 1.1 even 1 trivial
980.1.j.a.587.3 ✓ 8 4.3 odd 2 CM
980.1.j.a.587.4 yes 8 7.6 odd 2 inner
980.1.j.a.587.4 yes 8 28.27 even 2 inner
980.1.j.a.783.3 yes 8 35.13 even 4 inner
980.1.j.a.783.3 yes 8 140.83 odd 4 inner
980.1.j.a.783.4 yes 8 5.3 odd 4 inner
980.1.j.a.783.4 yes 8 20.3 even 4 inner
980.1.y.a.227.3 16 7.5 odd 6
980.1.y.a.227.3 16 28.19 even 6
980.1.y.a.227.4 16 7.2 even 3
980.1.y.a.227.4 16 28.23 odd 6
980.1.y.a.423.1 16 35.23 odd 12
980.1.y.a.423.1 16 140.23 even 12
980.1.y.a.423.2 16 35.33 even 12
980.1.y.a.423.2 16 140.103 odd 12
980.1.y.a.607.1 16 7.3 odd 6
980.1.y.a.607.1 16 28.3 even 6
980.1.y.a.607.2 16 7.4 even 3
980.1.y.a.607.2 16 28.11 odd 6
980.1.y.a.803.3 16 35.18 odd 12
980.1.y.a.803.3 16 140.123 even 12
980.1.y.a.803.4 16 35.3 even 12
980.1.y.a.803.4 16 140.3 odd 12