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Properties

Label 882.2.a.b.1.1
Level $882$
Weight $2$
Character 882.1
Self dual yes
Analytic conductor $7.043$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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] = [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,-2,0,0,-1,0,2,4,0,-6] 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 42)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{8} +2.00000 q^{10} +4.00000 q^{11} -6.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} +4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} -8.00000 q^{23} -1.00000 q^{25} +6.00000 q^{26} +2.00000 q^{29} -1.00000 q^{32} -2.00000 q^{34} -10.0000 q^{37} -4.00000 q^{38} +2.00000 q^{40} -6.00000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +8.00000 q^{46} +1.00000 q^{50} -6.00000 q^{52} -6.00000 q^{53} -8.00000 q^{55} -2.00000 q^{58} +4.00000 q^{59} -6.00000 q^{61} +1.00000 q^{64} +12.0000 q^{65} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{71} -10.0000 q^{73} +10.0000 q^{74} +4.00000 q^{76} -2.00000 q^{80} +6.00000 q^{82} -4.00000 q^{83} -4.00000 q^{85} +4.00000 q^{86} -4.00000 q^{88} -6.00000 q^{89} -8.00000 q^{92} -8.00000 q^{95} +14.0000 q^{97} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 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 882.2.a.b.1.1 1
3.2 odd 2 294.2.a.g.1.1 1
4.3 odd 2 7056.2.a.k.1.1 1
7.2 even 3 882.2.g.j.361.1 2
7.3 odd 6 882.2.g.h.667.1 2
7.4 even 3 882.2.g.j.667.1 2
7.5 odd 6 882.2.g.h.361.1 2
7.6 odd 2 126.2.a.a.1.1 1
12.11 even 2 2352.2.a.l.1.1 1
15.14 odd 2 7350.2.a.f.1.1 1
21.2 odd 6 294.2.e.a.67.1 2
21.5 even 6 294.2.e.c.67.1 2
21.11 odd 6 294.2.e.a.79.1 2
21.17 even 6 294.2.e.c.79.1 2
21.20 even 2 42.2.a.a.1.1 ✓ 1
24.5 odd 2 9408.2.a.n.1.1 1
24.11 even 2 9408.2.a.bw.1.1 1
28.27 even 2 1008.2.a.j.1.1 1
35.13 even 4 3150.2.g.r.2899.2 2
35.27 even 4 3150.2.g.r.2899.1 2
35.34 odd 2 3150.2.a.bo.1.1 1
56.13 odd 2 4032.2.a.e.1.1 1
56.27 even 2 4032.2.a.m.1.1 1
63.13 odd 6 1134.2.f.j.379.1 2
63.20 even 6 1134.2.f.g.757.1 2
63.34 odd 6 1134.2.f.j.757.1 2
63.41 even 6 1134.2.f.g.379.1 2
84.11 even 6 2352.2.q.n.961.1 2
84.23 even 6 2352.2.q.n.1537.1 2
84.47 odd 6 2352.2.q.i.1537.1 2
84.59 odd 6 2352.2.q.i.961.1 2
84.83 odd 2 336.2.a.d.1.1 1
105.62 odd 4 1050.2.g.a.799.2 2
105.83 odd 4 1050.2.g.a.799.1 2
105.104 even 2 1050.2.a.i.1.1 1
168.83 odd 2 1344.2.a.i.1.1 1
168.125 even 2 1344.2.a.q.1.1 1
231.230 odd 2 5082.2.a.d.1.1 1
273.272 even 2 7098.2.a.f.1.1 1
336.83 odd 4 5376.2.c.e.2689.2 2
336.125 even 4 5376.2.c.bc.2689.1 2
336.251 odd 4 5376.2.c.e.2689.1 2
336.293 even 4 5376.2.c.bc.2689.2 2
420.419 odd 2 8400.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 ✓ 1 21.20 even 2
126.2.a.a.1.1 1 7.6 odd 2
294.2.a.g.1.1 1 3.2 odd 2
294.2.e.a.67.1 2 21.2 odd 6
294.2.e.a.79.1 2 21.11 odd 6
294.2.e.c.67.1 2 21.5 even 6
294.2.e.c.79.1 2 21.17 even 6
336.2.a.d.1.1 1 84.83 odd 2
882.2.a.b.1.1 1 1.1 even 1 trivial
882.2.g.h.361.1 2 7.5 odd 6
882.2.g.h.667.1 2 7.3 odd 6
882.2.g.j.361.1 2 7.2 even 3
882.2.g.j.667.1 2 7.4 even 3
1008.2.a.j.1.1 1 28.27 even 2
1050.2.a.i.1.1 1 105.104 even 2
1050.2.g.a.799.1 2 105.83 odd 4
1050.2.g.a.799.2 2 105.62 odd 4
1134.2.f.g.379.1 2 63.41 even 6
1134.2.f.g.757.1 2 63.20 even 6
1134.2.f.j.379.1 2 63.13 odd 6
1134.2.f.j.757.1 2 63.34 odd 6
1344.2.a.i.1.1 1 168.83 odd 2
1344.2.a.q.1.1 1 168.125 even 2
2352.2.a.l.1.1 1 12.11 even 2
2352.2.q.i.961.1 2 84.59 odd 6
2352.2.q.i.1537.1 2 84.47 odd 6
2352.2.q.n.961.1 2 84.11 even 6
2352.2.q.n.1537.1 2 84.23 even 6
3150.2.a.bo.1.1 1 35.34 odd 2
3150.2.g.r.2899.1 2 35.27 even 4
3150.2.g.r.2899.2 2 35.13 even 4
4032.2.a.e.1.1 1 56.13 odd 2
4032.2.a.m.1.1 1 56.27 even 2
5082.2.a.d.1.1 1 231.230 odd 2
5376.2.c.e.2689.1 2 336.251 odd 4
5376.2.c.e.2689.2 2 336.83 odd 4
5376.2.c.bc.2689.1 2 336.125 even 4
5376.2.c.bc.2689.2 2 336.293 even 4
7056.2.a.k.1.1 1 4.3 odd 2
7098.2.a.f.1.1 1 273.272 even 2
7350.2.a.f.1.1 1 15.14 odd 2
8400.2.a.k.1.1 1 420.419 odd 2
9408.2.a.n.1.1 1 24.5 odd 2
9408.2.a.bw.1.1 1 24.11 even 2