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Properties

Label 980.2.q.i.569.1
Level $980$
Weight $2$
Character 980.569
Analytic conductor $7.825$
Analytic rank $0$
Dimension $8$
CM discriminant -35
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] = [980,2,Mod(569,980)] 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("980.569"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.31116960000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 569.1
Root \(-0.306808 - 1.70466i\) of defining polynomial
Character \(\chi\) \(=\) 980.569
Dual form 980.2.q.i.949.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.95256 + 1.70466i) q^{3} +(-1.93649 - 1.11803i) q^{5} +(4.31174 - 7.46815i) q^{9} +(1.81174 + 3.13802i) q^{11} +1.06281i q^{13} +7.62348 q^{15} +(4.98469 - 2.87791i) q^{17} +(2.50000 + 4.33013i) q^{25} +19.1722i q^{27} -9.62348 q^{29} +(-10.6985 - 6.17680i) q^{33} +(-1.81174 - 3.13802i) q^{39} +(-16.6993 + 9.64134i) q^{45} +(1.11171 + 0.641847i) q^{47} +(-9.81174 + 16.9944i) q^{51} -8.10234i q^{55} +(1.18826 - 2.05813i) q^{65} -12.0000 q^{71} +(-11.6190 + 6.70820i) q^{73} +(-14.7628 - 8.52330i) q^{75} +(-7.43521 + 12.8782i) q^{79} +(-19.7470 - 34.2027i) q^{81} +8.94427i q^{83} -12.8704 q^{85} +(28.4139 - 16.4048i) q^{87} +19.3931i q^{97} +31.2470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{9} - 6 q^{11} + 20 q^{15} + 20 q^{25} - 36 q^{29} + 6 q^{39} - 58 q^{51} + 30 q^{65} - 96 q^{71} + 2 q^{79} - 76 q^{81} + 20 q^{85} + 168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 0
\(3\) −2.95256 + 1.70466i −1.70466 + 0.984186i −0.763763 + 0.645497i \(0.776650\pi\)
−0.940898 + 0.338689i \(0.890016\pi\)
\(4\) 0 0
\(5\) −1.93649 − 1.11803i −0.866025 − 0.500000i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 4.31174 − 7.46815i 1.43725 − 2.48938i
\(10\) 0 0
\(11\) 1.81174 + 3.13802i 0.546259 + 0.946149i 0.998526 + 0.0542666i \(0.0172821\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 1.06281i 0.294772i 0.989079 + 0.147386i \(0.0470859\pi\)
−0.989079 + 0.147386i \(0.952914\pi\)
\(14\) 0 0
\(15\) 7.62348 1.96837
\(16\) 0 0
\(17\) 4.98469 − 2.87791i 1.20897 − 0.697997i 0.246433 − 0.969160i \(-0.420742\pi\)
0.962533 + 0.271163i \(0.0874083\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 19.1722i 3.68970i
\(28\) 0 0
\(29\) −9.62348 −1.78703 −0.893517 − 0.449029i \(-0.851770\pi\)
−0.893517 + 0.449029i \(0.851770\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) −10.6985 − 6.17680i −1.86237 − 1.07524i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) −1.81174 − 3.13802i −0.290110 − 0.502486i
\(40\) 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
\(45\) −16.6993 + 9.64134i −2.48938 + 1.43725i
\(46\) 0 0
\(47\) 1.11171 + 0.641847i 0.162160 + 0.0936230i 0.578884 − 0.815410i \(-0.303489\pi\)
−0.416724 + 0.909033i \(0.636822\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −9.81174 + 16.9944i −1.37392 + 2.37970i
\(52\) 0 0
\(53\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) − 8.10234i − 1.09252i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.18826 − 2.05813i 0.147386 − 0.255280i
\(66\) 0 0
\(67\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 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
\(73\) −11.6190 + 6.70820i −1.35990 + 0.785136i −0.989609 − 0.143782i \(-0.954074\pi\)
−0.370286 + 0.928918i \(0.620740\pi\)
\(74\) 0 0
\(75\) −14.7628 − 8.52330i −1.70466 − 0.984186i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −7.43521 + 12.8782i −0.836527 + 1.44891i 0.0562544 + 0.998416i \(0.482084\pi\)
−0.892781 + 0.450490i \(0.851249\pi\)
\(80\) 0 0
\(81\) −19.7470 − 34.2027i −2.19411 − 3.80030i
\(82\) 0 0
\(83\) 8.94427i 0.981761i 0.871227 + 0.490881i \(0.163325\pi\)
−0.871227 + 0.490881i \(0.836675\pi\)
\(84\) 0 0
\(85\) −12.8704 −1.39599
\(86\) 0 0
\(87\) 28.4139 − 16.4048i 3.04629 − 1.75878i
\(88\) 0 0
\(89\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 19.3931i 1.96907i 0.175180 + 0.984536i \(0.443949\pi\)
−0.175180 + 0.984536i \(0.556051\pi\)
\(98\) 0 0
\(99\) 31.2470 3.14044
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.q.i.569.1 8
5.4 even 2 inner 980.2.q.i.569.4 8
7.2 even 3 980.2.e.d.589.1 ✓ 4
7.3 odd 6 inner 980.2.q.i.949.1 8
7.4 even 3 inner 980.2.q.i.949.4 8
7.5 odd 6 980.2.e.d.589.4 yes 4
7.6 odd 2 inner 980.2.q.i.569.4 8
35.2 odd 12 4900.2.a.bj.1.1 4
35.4 even 6 inner 980.2.q.i.949.1 8
35.9 even 6 980.2.e.d.589.4 yes 4
35.12 even 12 4900.2.a.bj.1.4 4
35.19 odd 6 980.2.e.d.589.1 ✓ 4
35.23 odd 12 4900.2.a.bj.1.4 4
35.24 odd 6 inner 980.2.q.i.949.4 8
35.33 even 12 4900.2.a.bj.1.1 4
35.34 odd 2 CM 980.2.q.i.569.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.2.e.d.589.1 ✓ 4 7.2 even 3
980.2.e.d.589.1 ✓ 4 35.19 odd 6
980.2.e.d.589.4 yes 4 7.5 odd 6
980.2.e.d.589.4 yes 4 35.9 even 6
980.2.q.i.569.1 8 1.1 even 1 trivial
980.2.q.i.569.1 8 35.34 odd 2 CM
980.2.q.i.569.4 8 5.4 even 2 inner
980.2.q.i.569.4 8 7.6 odd 2 inner
980.2.q.i.949.1 8 7.3 odd 6 inner
980.2.q.i.949.1 8 35.4 even 6 inner
980.2.q.i.949.4 8 7.4 even 3 inner
980.2.q.i.949.4 8 35.24 odd 6 inner
4900.2.a.bj.1.1 4 35.2 odd 12
4900.2.a.bj.1.1 4 35.33 even 12
4900.2.a.bj.1.4 4 35.12 even 12
4900.2.a.bj.1.4 4 35.23 odd 12