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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,2,Mod(1,882)] 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("882.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,0,1,-3,0,0,-1,0,3,3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.04280545828\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 882.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{8} +3.00000 q^{10} +3.00000 q^{11} +2.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} +2.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} +6.00000 q^{23} +4.00000 q^{25} -2.00000 q^{26} -9.00000 q^{29} -7.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -10.0000 q^{37} -2.00000 q^{38} +3.00000 q^{40} -4.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} -12.0000 q^{47} -4.00000 q^{50} +2.00000 q^{52} +3.00000 q^{53} -9.00000 q^{55} +9.00000 q^{58} +3.00000 q^{59} -4.00000 q^{61} +7.00000 q^{62} +1.00000 q^{64} -6.00000 q^{65} +2.00000 q^{67} -6.00000 q^{68} +2.00000 q^{73} +10.0000 q^{74} +2.00000 q^{76} +5.00000 q^{79} -3.00000 q^{80} -9.00000 q^{83} +18.0000 q^{85} +4.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} +6.00000 q^{92} +12.0000 q^{94} -6.00000 q^{95} -13.0000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) 0 0
\(58\) 9.00000 1.18176
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 7.00000 0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 − 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 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 882.2.a.a.1.1 1
3.2 odd 2 882.2.a.j.1.1 1
4.3 odd 2 7056.2.a.e.1.1 1
7.2 even 3 126.2.g.d.109.1 yes 2
7.3 odd 6 882.2.g.g.667.1 2
7.4 even 3 126.2.g.d.37.1 yes 2
7.5 odd 6 882.2.g.g.361.1 2
7.6 odd 2 882.2.a.e.1.1 1
12.11 even 2 7056.2.a.by.1.1 1
21.2 odd 6 126.2.g.a.109.1 yes 2
21.5 even 6 882.2.g.e.361.1 2
21.11 odd 6 126.2.g.a.37.1 ✓ 2
21.17 even 6 882.2.g.e.667.1 2
21.20 even 2 882.2.a.h.1.1 1
28.11 odd 6 1008.2.s.o.289.1 2
28.23 odd 6 1008.2.s.o.865.1 2
28.27 even 2 7056.2.a.bx.1.1 1
63.2 odd 6 1134.2.h.f.109.1 2
63.4 even 3 1134.2.h.j.541.1 2
63.11 odd 6 1134.2.e.k.919.1 2
63.16 even 3 1134.2.h.j.109.1 2
63.23 odd 6 1134.2.e.k.865.1 2
63.25 even 3 1134.2.e.g.919.1 2
63.32 odd 6 1134.2.h.f.541.1 2
63.58 even 3 1134.2.e.g.865.1 2
84.11 even 6 1008.2.s.b.289.1 2
84.23 even 6 1008.2.s.b.865.1 2
84.83 odd 2 7056.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.g.a.37.1 ✓ 2 21.11 odd 6
126.2.g.a.109.1 yes 2 21.2 odd 6
126.2.g.d.37.1 yes 2 7.4 even 3
126.2.g.d.109.1 yes 2 7.2 even 3
882.2.a.a.1.1 1 1.1 even 1 trivial
882.2.a.e.1.1 1 7.6 odd 2
882.2.a.h.1.1 1 21.20 even 2
882.2.a.j.1.1 1 3.2 odd 2
882.2.g.e.361.1 2 21.5 even 6
882.2.g.e.667.1 2 21.17 even 6
882.2.g.g.361.1 2 7.5 odd 6
882.2.g.g.667.1 2 7.3 odd 6
1008.2.s.b.289.1 2 84.11 even 6
1008.2.s.b.865.1 2 84.23 even 6
1008.2.s.o.289.1 2 28.11 odd 6
1008.2.s.o.865.1 2 28.23 odd 6
1134.2.e.g.865.1 2 63.58 even 3
1134.2.e.g.919.1 2 63.25 even 3
1134.2.e.k.865.1 2 63.23 odd 6
1134.2.e.k.919.1 2 63.11 odd 6
1134.2.h.f.109.1 2 63.2 odd 6
1134.2.h.f.541.1 2 63.32 odd 6
1134.2.h.j.109.1 2 63.16 even 3
1134.2.h.j.541.1 2 63.4 even 3
7056.2.a.e.1.1 1 4.3 odd 2
7056.2.a.h.1.1 1 84.83 odd 2
7056.2.a.bx.1.1 1 28.27 even 2
7056.2.a.by.1.1 1 12.11 even 2