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Properties

Label 980.1.p.c.79.4
Level $980$
Weight $1$
Character 980.79
Analytic conductor $0.489$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -4
Inner twists $16$

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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(79,980)] 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("980.79"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 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.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0] 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: \(0.489083712380\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of \(\Q(\sqrt[4]{700})\)

Embedding invariants

Embedding label 79.4
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 980.79
Dual form 980.1.p.c.459.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.258819 + 0.965926i) q^{5} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +(-0.258819 + 0.965926i) q^{10} -1.41421i q^{13} +(-0.500000 + 0.866025i) q^{16} +(-1.22474 + 0.707107i) q^{17} +(0.866025 - 0.500000i) q^{18} +(-0.707107 + 0.707107i) q^{20} +(-0.866025 + 0.500000i) q^{25} +(0.707107 - 1.22474i) q^{26} +(-0.866025 + 0.500000i) q^{32} -1.41421 q^{34} +1.00000 q^{36} +(-1.73205 - 1.00000i) q^{37} +(-0.965926 + 0.258819i) q^{40} +1.41421 q^{41} +(0.965926 + 0.258819i) q^{45} -1.00000 q^{50} +(1.22474 - 0.707107i) q^{52} +(0.707107 - 1.22474i) q^{61} -1.00000 q^{64} +(1.36603 - 0.366025i) q^{65} +(-1.22474 - 0.707107i) q^{68} +(0.866025 + 0.500000i) q^{72} +(1.22474 - 0.707107i) q^{73} +(-1.00000 - 1.73205i) q^{74} +(-0.965926 - 0.258819i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(1.22474 + 0.707107i) q^{82} +(-1.00000 - 1.00000i) q^{85} +(-0.707107 + 1.22474i) q^{89} +(0.707107 + 0.707107i) q^{90} +1.41421i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{9} - 4 q^{16} + 8 q^{36} - 8 q^{50} - 8 q^{64} + 4 q^{65} - 8 q^{74} - 4 q^{81} - 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(3\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(5\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000i 1.00000i
\(9\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(10\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) − 1.41421i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 − 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(18\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(26\) 0.707107 − 1.22474i 0.707107 − 1.22474i
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(33\) 0 0
\(34\) −1.41421 −1.41421
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) −1.73205 − 1.00000i −1.73205 − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 − 0.500000i \(-0.833333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(41\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.00000 −1.00000
\(51\) 0 0
\(52\) 1.22474 − 0.707107i 1.22474 − 0.707107i
\(53\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0.707107 − 1.22474i 0.707107 − 1.22474i −0.258819 − 0.965926i \(-0.583333\pi\)
0.965926 − 0.258819i \(-0.0833333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 1.36603 − 0.366025i 1.36603 − 0.366025i
\(66\) 0 0
\(67\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) −1.22474 − 0.707107i −1.22474 − 0.707107i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(73\) 1.22474 − 0.707107i 1.22474 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(74\) −1.00000 − 1.73205i −1.00000 − 1.73205i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) −0.965926 − 0.258819i −0.965926 − 0.258819i
\(81\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(82\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −1.00000 − 1.00000i −1.00000 − 1.00000i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(90\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 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.p.c.79.4 8
4.3 odd 2 CM 980.1.p.c.79.4 8
5.4 even 2 inner 980.1.p.c.79.1 8
7.2 even 3 980.1.f.e.99.2 yes 4
7.3 odd 6 inner 980.1.p.c.459.2 8
7.4 even 3 inner 980.1.p.c.459.1 8
7.5 odd 6 980.1.f.e.99.1 ✓ 4
7.6 odd 2 inner 980.1.p.c.79.3 8
20.19 odd 2 inner 980.1.p.c.79.1 8
28.3 even 6 inner 980.1.p.c.459.2 8
28.11 odd 6 inner 980.1.p.c.459.1 8
28.19 even 6 980.1.f.e.99.1 ✓ 4
28.23 odd 6 980.1.f.e.99.2 yes 4
28.27 even 2 inner 980.1.p.c.79.3 8
35.4 even 6 inner 980.1.p.c.459.4 8
35.9 even 6 980.1.f.e.99.4 yes 4
35.19 odd 6 980.1.f.e.99.3 yes 4
35.24 odd 6 inner 980.1.p.c.459.3 8
35.34 odd 2 inner 980.1.p.c.79.2 8
140.19 even 6 980.1.f.e.99.3 yes 4
140.39 odd 6 inner 980.1.p.c.459.4 8
140.59 even 6 inner 980.1.p.c.459.3 8
140.79 odd 6 980.1.f.e.99.4 yes 4
140.139 even 2 inner 980.1.p.c.79.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.1.f.e.99.1 ✓ 4 7.5 odd 6
980.1.f.e.99.1 ✓ 4 28.19 even 6
980.1.f.e.99.2 yes 4 7.2 even 3
980.1.f.e.99.2 yes 4 28.23 odd 6
980.1.f.e.99.3 yes 4 35.19 odd 6
980.1.f.e.99.3 yes 4 140.19 even 6
980.1.f.e.99.4 yes 4 35.9 even 6
980.1.f.e.99.4 yes 4 140.79 odd 6
980.1.p.c.79.1 8 5.4 even 2 inner
980.1.p.c.79.1 8 20.19 odd 2 inner
980.1.p.c.79.2 8 35.34 odd 2 inner
980.1.p.c.79.2 8 140.139 even 2 inner
980.1.p.c.79.3 8 7.6 odd 2 inner
980.1.p.c.79.3 8 28.27 even 2 inner
980.1.p.c.79.4 8 1.1 even 1 trivial
980.1.p.c.79.4 8 4.3 odd 2 CM
980.1.p.c.459.1 8 7.4 even 3 inner
980.1.p.c.459.1 8 28.11 odd 6 inner
980.1.p.c.459.2 8 7.3 odd 6 inner
980.1.p.c.459.2 8 28.3 even 6 inner
980.1.p.c.459.3 8 35.24 odd 6 inner
980.1.p.c.459.3 8 140.59 even 6 inner
980.1.p.c.459.4 8 35.4 even 6 inner
980.1.p.c.459.4 8 140.39 odd 6 inner