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Properties

Label 980.2.e.d.589.3
Level $980$
Weight $2$
Character 980.589
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
CM discriminant -35
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] = [980,2,Mod(589,980)] 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("980.589"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{-21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 13x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 589.3
Root \(1.17325i\) of defining polynomial
Character \(\chi\) \(=\) 980.589
Dual form 980.2.e.d.589.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.17325i q^{3} +2.23607i q^{5} +1.62348 q^{9} +6.62348 q^{11} +5.64539i q^{13} -2.62348 q^{15} -7.99190i q^{17} -5.00000 q^{25} +5.42451i q^{27} +0.623475 q^{29} +7.77102i q^{33} -6.62348 q^{39} +3.63020i q^{45} +12.4640i q^{47} +9.37652 q^{51} +14.8105i q^{55} -12.6235 q^{65} -12.0000 q^{71} -13.4164i q^{73} -5.86627i q^{75} -15.8704 q^{79} -1.49390 q^{81} +8.94427i q^{83} +17.8704 q^{85} +0.731495i q^{87} -12.6849i q^{97} +10.7530 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{9} + 6 q^{11} + 10 q^{15} - 20 q^{25} - 18 q^{29} - 6 q^{39} + 58 q^{51} - 30 q^{65} - 48 q^{71} - 2 q^{79} + 76 q^{81} + 10 q^{85} + 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.17325i 0.677378i 0.940898 + 0.338689i \(0.109984\pi\)
−0.940898 + 0.338689i \(0.890016\pi\)
\(4\) 0 0
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.62348 0.541158
\(10\) 0 0
\(11\) 6.62348 1.99705 0.998526 − 0.0542666i \(-0.0172821\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) 0 0
\(13\) 5.64539i 1.56575i 0.622179 + 0.782875i \(0.286247\pi\)
−0.622179 + 0.782875i \(0.713753\pi\)
\(14\) 0 0
\(15\) −2.62348 −0.677378
\(16\) 0 0
\(17\) − 7.99190i − 1.93832i −0.246433 − 0.969160i \(-0.579258\pi\)
0.246433 − 0.969160i \(-0.420742\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 5.42451i 1.04395i
\(28\) 0 0
\(29\) 0.623475 0.115776 0.0578882 − 0.998323i \(-0.481563\pi\)
0.0578882 + 0.998323i \(0.481563\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 7.77102i 1.35276i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −6.62348 −1.06060
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 3.63020i 0.541158i
\(46\) 0 0
\(47\) 12.4640i 1.81807i 0.416724 + 0.909033i \(0.363178\pi\)
−0.416724 + 0.909033i \(0.636822\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 9.37652 1.31298
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 14.8105i 1.99705i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.6235 −1.56575
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) − 13.4164i − 1.57027i −0.619324 − 0.785136i \(-0.712593\pi\)
0.619324 − 0.785136i \(-0.287407\pi\)
\(74\) 0 0
\(75\) − 5.86627i − 0.677378i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −15.8704 −1.78556 −0.892781 − 0.450490i \(-0.851249\pi\)
−0.892781 + 0.450490i \(0.851249\pi\)
\(80\) 0 0
\(81\) −1.49390 −0.165989
\(82\) 0 0
\(83\) 8.94427i 0.981761i 0.871227 + 0.490881i \(0.163325\pi\)
−0.871227 + 0.490881i \(0.836675\pi\)
\(84\) 0 0
\(85\) 17.8704 1.93832
\(86\) 0 0
\(87\) 0.731495i 0.0784245i
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 12.6849i − 1.28796i −0.765043 − 0.643979i \(-0.777282\pi\)
0.765043 − 0.643979i \(-0.222718\pi\)
\(98\) 0 0
\(99\) 10.7530 1.08072
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 980.2.e.d.589.3 yes 4
5.2 odd 4 4900.2.a.bj.1.3 4
5.3 odd 4 4900.2.a.bj.1.2 4
5.4 even 2 inner 980.2.e.d.589.2 ✓ 4
7.2 even 3 980.2.q.i.949.2 8
7.3 odd 6 980.2.q.i.569.2 8
7.4 even 3 980.2.q.i.569.3 8
7.5 odd 6 980.2.q.i.949.3 8
7.6 odd 2 inner 980.2.e.d.589.2 ✓ 4
35.4 even 6 980.2.q.i.569.2 8
35.9 even 6 980.2.q.i.949.3 8
35.13 even 4 4900.2.a.bj.1.3 4
35.19 odd 6 980.2.q.i.949.2 8
35.24 odd 6 980.2.q.i.569.3 8
35.27 even 4 4900.2.a.bj.1.2 4
35.34 odd 2 CM 980.2.e.d.589.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.2.e.d.589.2 ✓ 4 5.4 even 2 inner
980.2.e.d.589.2 ✓ 4 7.6 odd 2 inner
980.2.e.d.589.3 yes 4 1.1 even 1 trivial
980.2.e.d.589.3 yes 4 35.34 odd 2 CM
980.2.q.i.569.2 8 7.3 odd 6
980.2.q.i.569.2 8 35.4 even 6
980.2.q.i.569.3 8 7.4 even 3
980.2.q.i.569.3 8 35.24 odd 6
980.2.q.i.949.2 8 7.2 even 3
980.2.q.i.949.2 8 35.19 odd 6
980.2.q.i.949.3 8 7.5 odd 6
980.2.q.i.949.3 8 35.9 even 6
4900.2.a.bj.1.2 4 5.3 odd 4
4900.2.a.bj.1.2 4 35.27 even 4
4900.2.a.bj.1.3 4 5.2 odd 4
4900.2.a.bj.1.3 4 35.13 even 4