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Properties

Label 980.2.q.e.949.1
Level $980$
Weight $2$
Character 980.949
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
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(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 = [4,0,0,0,2] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 949.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 980.949
Dual form 980.2.q.e.569.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.23205 + 1.86603i) q^{5} +(-1.50000 - 2.59808i) q^{9} +4.00000i q^{13} +(3.46410 + 2.00000i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(-6.92820 + 4.00000i) q^{23} +(-1.96410 - 4.59808i) q^{25} -2.00000 q^{29} +(-4.00000 + 6.92820i) q^{31} +(-6.92820 + 4.00000i) q^{37} -6.00000 q^{41} -8.00000i q^{43} +(6.69615 + 0.401924i) q^{45} +(-6.92820 + 4.00000i) q^{47} +(2.00000 - 3.46410i) q^{59} +(-3.00000 - 5.19615i) q^{61} +(-7.46410 - 4.92820i) q^{65} +(-6.92820 - 4.00000i) q^{67} +12.0000 q^{71} +(3.46410 + 2.00000i) q^{73} +(-2.00000 - 3.46410i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-8.00000 + 4.00000i) q^{85} +(5.00000 + 8.66025i) q^{89} +(8.92820 + 0.535898i) q^{95} +12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 6 q^{9} - 8 q^{19} + 6 q^{25} - 8 q^{29} - 16 q^{31} - 24 q^{41} + 6 q^{45} + 8 q^{59} - 12 q^{61} - 16 q^{65} + 48 q^{71} - 8 q^{79} - 18 q^{81} - 32 q^{85} + 20 q^{89} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)

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
\(3\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) −1.23205 + 1.86603i −0.550990 + 0.834512i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.50000 − 2.59808i −0.500000 − 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410 + 2.00000i 0.840168 + 0.485071i 0.857321 − 0.514782i \(-0.172127\pi\)
−0.0171533 + 0.999853i \(0.505460\pi\)
\(18\) 0 0
\(19\) −2.00000 − 3.46410i −0.458831 − 0.794719i 0.540068 − 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.92820 + 4.00000i −1.44463 + 0.834058i −0.998154 − 0.0607377i \(-0.980655\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(24\) 0 0
\(25\) −1.96410 − 4.59808i −0.392820 − 0.919615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.00000 + 6.92820i −0.718421 + 1.24434i 0.243204 + 0.969975i \(0.421802\pi\)
−0.961625 + 0.274367i \(0.911532\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.92820 + 4.00000i −1.13899 + 0.657596i −0.946180 − 0.323640i \(-0.895093\pi\)
−0.192809 + 0.981236i \(0.561760\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 − 0.609994i \(-0.791172\pi\)
0.792406 − 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 6.69615 + 0.401924i 0.998203 + 0.0599153i
\(46\) 0 0
\(47\) −6.92820 + 4.00000i −1.01058 + 0.583460i −0.911362 − 0.411606i \(-0.864968\pi\)
−0.0992202 + 0.995066i \(0.531635\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000 − 3.46410i 0.260378 − 0.450988i −0.705965 − 0.708247i \(-0.749486\pi\)
0.966342 + 0.257260i \(0.0828195\pi\)
\(60\) 0 0
\(61\) −3.00000 − 5.19615i −0.384111 − 0.665299i 0.607535 − 0.794293i \(-0.292159\pi\)
−0.991645 + 0.128994i \(0.958825\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.46410 − 4.92820i −0.925808 − 0.611268i
\(66\) 0 0
\(67\) −6.92820 − 4.00000i −0.846415 − 0.488678i 0.0130248 − 0.999915i \(-0.495854\pi\)
−0.859440 + 0.511237i \(0.829187\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 3.46410 + 2.00000i 0.405442 + 0.234082i 0.688830 − 0.724923i \(-0.258125\pi\)
−0.283387 + 0.959006i \(0.591458\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.00000 − 3.46410i −0.225018 − 0.389742i 0.731307 − 0.682048i \(-0.238911\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −8.00000 + 4.00000i −0.867722 + 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.00000 + 8.66025i 0.529999 + 0.917985i 0.999388 + 0.0349934i \(0.0111410\pi\)
−0.469389 + 0.882992i \(0.655526\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.92820 + 0.535898i 0.916014 + 0.0549820i
\(96\) 0 0
\(97\) 12.0000i 1.21842i 0.793011 + 0.609208i \(0.208512\pi\)
−0.793011 + 0.609208i \(0.791488\pi\)
\(98\) 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 980.2.q.e.949.1 4
5.4 even 2 inner 980.2.q.e.949.2 4
7.2 even 3 inner 980.2.q.e.569.2 4
7.3 odd 6 140.2.e.b.29.2 yes 2
7.4 even 3 980.2.e.a.589.1 2
7.5 odd 6 980.2.q.d.569.1 4
7.6 odd 2 980.2.q.d.949.2 4
21.17 even 6 1260.2.k.b.1009.1 2
28.3 even 6 560.2.g.c.449.2 2
35.3 even 12 700.2.a.h.1.1 1
35.4 even 6 980.2.e.a.589.2 2
35.9 even 6 inner 980.2.q.e.569.1 4
35.17 even 12 700.2.a.f.1.1 1
35.18 odd 12 4900.2.a.l.1.1 1
35.19 odd 6 980.2.q.d.569.2 4
35.24 odd 6 140.2.e.b.29.1 ✓ 2
35.32 odd 12 4900.2.a.m.1.1 1
35.34 odd 2 980.2.q.d.949.1 4
56.3 even 6 2240.2.g.c.449.1 2
56.45 odd 6 2240.2.g.d.449.1 2
84.59 odd 6 5040.2.t.g.1009.1 2
105.17 odd 12 6300.2.a.g.1.1 1
105.38 odd 12 6300.2.a.y.1.1 1
105.59 even 6 1260.2.k.b.1009.2 2
140.3 odd 12 2800.2.a.o.1.1 1
140.59 even 6 560.2.g.c.449.1 2
140.87 odd 12 2800.2.a.s.1.1 1
280.59 even 6 2240.2.g.c.449.2 2
280.269 odd 6 2240.2.g.d.449.2 2
420.59 odd 6 5040.2.t.g.1009.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.e.b.29.1 ✓ 2 35.24 odd 6
140.2.e.b.29.2 yes 2 7.3 odd 6
560.2.g.c.449.1 2 140.59 even 6
560.2.g.c.449.2 2 28.3 even 6
700.2.a.f.1.1 1 35.17 even 12
700.2.a.h.1.1 1 35.3 even 12
980.2.e.a.589.1 2 7.4 even 3
980.2.e.a.589.2 2 35.4 even 6
980.2.q.d.569.1 4 7.5 odd 6
980.2.q.d.569.2 4 35.19 odd 6
980.2.q.d.949.1 4 35.34 odd 2
980.2.q.d.949.2 4 7.6 odd 2
980.2.q.e.569.1 4 35.9 even 6 inner
980.2.q.e.569.2 4 7.2 even 3 inner
980.2.q.e.949.1 4 1.1 even 1 trivial
980.2.q.e.949.2 4 5.4 even 2 inner
1260.2.k.b.1009.1 2 21.17 even 6
1260.2.k.b.1009.2 2 105.59 even 6
2240.2.g.c.449.1 2 56.3 even 6
2240.2.g.c.449.2 2 280.59 even 6
2240.2.g.d.449.1 2 56.45 odd 6
2240.2.g.d.449.2 2 280.269 odd 6
2800.2.a.o.1.1 1 140.3 odd 12
2800.2.a.s.1.1 1 140.87 odd 12
4900.2.a.l.1.1 1 35.18 odd 12
4900.2.a.m.1.1 1 35.32 odd 12
5040.2.t.g.1009.1 2 84.59 odd 6
5040.2.t.g.1009.2 2 420.59 odd 6
6300.2.a.g.1.1 1 105.17 odd 12
6300.2.a.y.1.1 1 105.38 odd 12