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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,2,Mod(289,1008)] 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("1008.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 289.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.289
Dual form 1008.2.s.j.865.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 2.59808i) q^{7} -7.00000 q^{13} +(-3.50000 + 6.06218i) q^{19} +(2.50000 + 4.33013i) q^{25} +(-3.50000 - 6.06218i) q^{31} +(0.500000 - 0.866025i) q^{37} -5.00000 q^{43} +(-6.50000 + 2.59808i) q^{49} +(-7.00000 + 12.1244i) q^{61} +(5.50000 + 9.52628i) q^{67} +(3.50000 + 6.06218i) q^{73} +(-6.50000 + 11.2583i) q^{79} +(-3.50000 - 18.1865i) q^{91} +14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{7} - 14 q^{13} - 7 q^{19} + 5 q^{25} - 7 q^{31} + q^{37} - 10 q^{43} - 13 q^{49} - 14 q^{61} + 11 q^{67} + 7 q^{73} - 13 q^{79} - 7 q^{91} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −7.00000 −1.94145 −0.970725 − 0.240192i \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −3.50000 + 6.06218i −0.802955 + 1.39076i 0.114708 + 0.993399i \(0.463407\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 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
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −3.50000 − 6.06218i −0.628619 − 1.08880i −0.987829 − 0.155543i \(-0.950287\pi\)
0.359211 − 0.933257i \(-0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 − 0.866025i 0.0821995 − 0.142374i −0.821995 − 0.569495i \(-0.807139\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 − 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(61\) −7.00000 + 12.1244i −0.896258 + 1.55236i −0.0640184 + 0.997949i \(0.520392\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.50000 + 9.52628i 0.671932 + 1.16382i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 3.50000 + 6.06218i 0.409644 + 0.709524i 0.994850 − 0.101361i \(-0.0323196\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.50000 + 11.2583i −0.731307 + 1.26666i 0.225018 + 0.974355i \(0.427756\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) −3.50000 − 18.1865i −0.366900 − 1.90647i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 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 1008.2.s.j.289.1 2
3.2 odd 2 CM 1008.2.s.j.289.1 2
4.3 odd 2 63.2.e.a.37.1 ✓ 2
7.2 even 3 7056.2.a.y.1.1 1
7.4 even 3 inner 1008.2.s.j.865.1 2
7.5 odd 6 7056.2.a.bf.1.1 1
12.11 even 2 63.2.e.a.37.1 ✓ 2
21.2 odd 6 7056.2.a.y.1.1 1
21.5 even 6 7056.2.a.bf.1.1 1
21.11 odd 6 inner 1008.2.s.j.865.1 2
28.3 even 6 441.2.e.c.361.1 2
28.11 odd 6 63.2.e.a.46.1 yes 2
28.19 even 6 441.2.a.e.1.1 1
28.23 odd 6 441.2.a.d.1.1 1
28.27 even 2 441.2.e.c.226.1 2
36.7 odd 6 567.2.h.c.352.1 2
36.11 even 6 567.2.h.c.352.1 2
36.23 even 6 567.2.g.d.541.1 2
36.31 odd 6 567.2.g.d.541.1 2
84.11 even 6 63.2.e.a.46.1 yes 2
84.23 even 6 441.2.a.d.1.1 1
84.47 odd 6 441.2.a.e.1.1 1
84.59 odd 6 441.2.e.c.361.1 2
84.83 odd 2 441.2.e.c.226.1 2
252.11 even 6 567.2.g.d.109.1 2
252.67 odd 6 567.2.h.c.298.1 2
252.95 even 6 567.2.h.c.298.1 2
252.151 odd 6 567.2.g.d.109.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.e.a.37.1 ✓ 2 4.3 odd 2
63.2.e.a.37.1 ✓ 2 12.11 even 2
63.2.e.a.46.1 yes 2 28.11 odd 6
63.2.e.a.46.1 yes 2 84.11 even 6
441.2.a.d.1.1 1 28.23 odd 6
441.2.a.d.1.1 1 84.23 even 6
441.2.a.e.1.1 1 28.19 even 6
441.2.a.e.1.1 1 84.47 odd 6
441.2.e.c.226.1 2 28.27 even 2
441.2.e.c.226.1 2 84.83 odd 2
441.2.e.c.361.1 2 28.3 even 6
441.2.e.c.361.1 2 84.59 odd 6
567.2.g.d.109.1 2 252.11 even 6
567.2.g.d.109.1 2 252.151 odd 6
567.2.g.d.541.1 2 36.23 even 6
567.2.g.d.541.1 2 36.31 odd 6
567.2.h.c.298.1 2 252.67 odd 6
567.2.h.c.298.1 2 252.95 even 6
567.2.h.c.352.1 2 36.7 odd 6
567.2.h.c.352.1 2 36.11 even 6
1008.2.s.j.289.1 2 1.1 even 1 trivial
1008.2.s.j.289.1 2 3.2 odd 2 CM
1008.2.s.j.865.1 2 7.4 even 3 inner
1008.2.s.j.865.1 2 21.11 odd 6 inner
7056.2.a.y.1.1 1 7.2 even 3
7056.2.a.y.1.1 1 21.2 odd 6
7056.2.a.bf.1.1 1 7.5 odd 6
7056.2.a.bf.1.1 1 21.5 even 6