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(361,882)] 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("882.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,-1,-2,0,0,-2,0,2,-4,0,12] 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: no
Analytic conductor: \(7.04280545828\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 667.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 882.667
Dual form 882.2.g.h.361.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} -1.00000 q^{8} +(1.00000 + 1.73205i) q^{10} +(-2.00000 - 3.46410i) q^{11} +6.00000 q^{13} +(-0.500000 + 0.866025i) q^{16} +(1.00000 + 1.73205i) q^{17} +(2.00000 - 3.46410i) q^{19} +2.00000 q^{20} -4.00000 q^{22} +(4.00000 - 6.92820i) q^{23} +(0.500000 + 0.866025i) q^{25} +(3.00000 - 5.19615i) q^{26} +2.00000 q^{29} +(0.500000 + 0.866025i) q^{32} +2.00000 q^{34} +(5.00000 - 8.66025i) q^{37} +(-2.00000 - 3.46410i) q^{38} +(1.00000 - 1.73205i) q^{40} +6.00000 q^{41} -4.00000 q^{43} +(-2.00000 + 3.46410i) q^{44} +(-4.00000 - 6.92820i) q^{46} +1.00000 q^{50} +(-3.00000 - 5.19615i) q^{52} +(3.00000 + 5.19615i) q^{53} +8.00000 q^{55} +(1.00000 - 1.73205i) q^{58} +(2.00000 + 3.46410i) q^{59} +(-3.00000 + 5.19615i) q^{61} +1.00000 q^{64} +(-6.00000 + 10.3923i) q^{65} +(-2.00000 - 3.46410i) q^{67} +(1.00000 - 1.73205i) q^{68} -8.00000 q^{71} +(-5.00000 - 8.66025i) q^{73} +(-5.00000 - 8.66025i) q^{74} -4.00000 q^{76} +(-1.00000 - 1.73205i) q^{80} +(3.00000 - 5.19615i) q^{82} +4.00000 q^{83} -4.00000 q^{85} +(-2.00000 + 3.46410i) q^{86} +(2.00000 + 3.46410i) q^{88} +(-3.00000 + 5.19615i) q^{89} -8.00000 q^{92} +(4.00000 + 6.92820i) q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 12 q^{13} - q^{16} + 2 q^{17} + 4 q^{19} + 4 q^{20} - 8 q^{22} + 8 q^{23} + q^{25} + 6 q^{26} + 4 q^{29} + q^{32} + 4 q^{34} + 10 q^{37}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 0.866025i 0.353553 − 0.612372i
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −1.00000 + 1.73205i −0.447214 + 0.774597i −0.998203 − 0.0599153i \(-0.980917\pi\)
0.550990 + 0.834512i \(0.314250\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 + 1.73205i 0.316228 + 0.547723i
\(11\) −2.00000 − 3.46410i −0.603023 − 1.04447i −0.992361 − 0.123371i \(-0.960630\pi\)
0.389338 − 0.921095i \(-0.372704\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 1.00000 + 1.73205i 0.242536 + 0.420084i 0.961436 − 0.275029i \(-0.0886875\pi\)
−0.718900 + 0.695113i \(0.755354\pi\)
\(18\) 0 0
\(19\) 2.00000 − 3.46410i 0.458831 − 0.794719i −0.540068 − 0.841621i \(-0.681602\pi\)
0.998899 + 0.0469020i \(0.0149348\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 4.00000 − 6.92820i 0.834058 − 1.44463i −0.0607377 − 0.998154i \(-0.519345\pi\)
0.894795 − 0.446476i \(-0.147321\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 3.00000 − 5.19615i 0.588348 − 1.01905i
\(27\) 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 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 − 8.66025i 0.821995 − 1.42374i −0.0821995 − 0.996616i \(-0.526194\pi\)
0.904194 − 0.427121i \(-0.140472\pi\)
\(38\) −2.00000 − 3.46410i −0.324443 − 0.561951i
\(39\) 0 0
\(40\) 1.00000 − 1.73205i 0.158114 − 0.273861i
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −2.00000 + 3.46410i −0.301511 + 0.522233i
\(45\) 0 0
\(46\) −4.00000 − 6.92820i −0.589768 − 1.02151i
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −3.00000 − 5.19615i −0.416025 − 0.720577i
\(53\) 3.00000 + 5.19615i 0.412082 + 0.713746i 0.995117 − 0.0987002i \(-0.0314685\pi\)
−0.583036 + 0.812447i \(0.698135\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 1.00000 − 1.73205i 0.131306 − 0.227429i
\(59\) 2.00000 + 3.46410i 0.260378 + 0.450988i 0.966342 − 0.257260i \(-0.0828195\pi\)
−0.705965 + 0.708247i \(0.749486\pi\)
\(60\) 0 0
\(61\) −3.00000 + 5.19615i −0.384111 + 0.665299i −0.991645 − 0.128994i \(-0.958825\pi\)
0.607535 + 0.794293i \(0.292159\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 + 10.3923i −0.744208 + 1.28901i
\(66\) 0 0
\(67\) −2.00000 − 3.46410i −0.244339 − 0.423207i 0.717607 − 0.696449i \(-0.245238\pi\)
−0.961946 + 0.273241i \(0.911904\pi\)
\(68\) 1.00000 − 1.73205i 0.121268 − 0.210042i
\(69\) 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
\(73\) −5.00000 − 8.66025i −0.585206 − 1.01361i −0.994850 − 0.101361i \(-0.967680\pi\)
0.409644 − 0.912245i \(-0.365653\pi\)
\(74\) −5.00000 − 8.66025i −0.581238 − 1.00673i
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) −1.00000 − 1.73205i −0.111803 − 0.193649i
\(81\) 0 0
\(82\) 3.00000 − 5.19615i 0.331295 − 0.573819i
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −2.00000 + 3.46410i −0.215666 + 0.373544i
\(87\) 0 0
\(88\) 2.00000 + 3.46410i 0.213201 + 0.369274i
\(89\) −3.00000 + 5.19615i −0.317999 + 0.550791i −0.980071 − 0.198650i \(-0.936344\pi\)
0.662071 + 0.749441i \(0.269678\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 + 6.92820i 0.410391 + 0.710819i
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 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.g.h.667.1 2
3.2 odd 2 294.2.e.c.79.1 2
7.2 even 3 126.2.a.a.1.1 1
7.3 odd 6 882.2.g.j.361.1 2
7.4 even 3 inner 882.2.g.h.361.1 2
7.5 odd 6 882.2.a.b.1.1 1
7.6 odd 2 882.2.g.j.667.1 2
12.11 even 2 2352.2.q.i.961.1 2
21.2 odd 6 42.2.a.a.1.1 ✓ 1
21.5 even 6 294.2.a.g.1.1 1
21.11 odd 6 294.2.e.c.67.1 2
21.17 even 6 294.2.e.a.67.1 2
21.20 even 2 294.2.e.a.79.1 2
28.19 even 6 7056.2.a.k.1.1 1
28.23 odd 6 1008.2.a.j.1.1 1
35.2 odd 12 3150.2.g.r.2899.1 2
35.9 even 6 3150.2.a.bo.1.1 1
35.23 odd 12 3150.2.g.r.2899.2 2
56.37 even 6 4032.2.a.e.1.1 1
56.51 odd 6 4032.2.a.m.1.1 1
63.2 odd 6 1134.2.f.g.757.1 2
63.16 even 3 1134.2.f.j.757.1 2
63.23 odd 6 1134.2.f.g.379.1 2
63.58 even 3 1134.2.f.j.379.1 2
84.11 even 6 2352.2.q.i.1537.1 2
84.23 even 6 336.2.a.d.1.1 1
84.47 odd 6 2352.2.a.l.1.1 1
84.59 odd 6 2352.2.q.n.1537.1 2
84.83 odd 2 2352.2.q.n.961.1 2
105.2 even 12 1050.2.g.a.799.2 2
105.23 even 12 1050.2.g.a.799.1 2
105.44 odd 6 1050.2.a.i.1.1 1
105.89 even 6 7350.2.a.f.1.1 1
168.5 even 6 9408.2.a.n.1.1 1
168.107 even 6 1344.2.a.i.1.1 1
168.131 odd 6 9408.2.a.bw.1.1 1
168.149 odd 6 1344.2.a.q.1.1 1
231.65 even 6 5082.2.a.d.1.1 1
273.233 odd 6 7098.2.a.f.1.1 1
336.107 even 12 5376.2.c.e.2689.1 2
336.149 odd 12 5376.2.c.bc.2689.2 2
336.275 even 12 5376.2.c.e.2689.2 2
336.317 odd 12 5376.2.c.bc.2689.1 2
420.359 even 6 8400.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 ✓ 1 21.2 odd 6
126.2.a.a.1.1 1 7.2 even 3
294.2.a.g.1.1 1 21.5 even 6
294.2.e.a.67.1 2 21.17 even 6
294.2.e.a.79.1 2 21.20 even 2
294.2.e.c.67.1 2 21.11 odd 6
294.2.e.c.79.1 2 3.2 odd 2
336.2.a.d.1.1 1 84.23 even 6
882.2.a.b.1.1 1 7.5 odd 6
882.2.g.h.361.1 2 7.4 even 3 inner
882.2.g.h.667.1 2 1.1 even 1 trivial
882.2.g.j.361.1 2 7.3 odd 6
882.2.g.j.667.1 2 7.6 odd 2
1008.2.a.j.1.1 1 28.23 odd 6
1050.2.a.i.1.1 1 105.44 odd 6
1050.2.g.a.799.1 2 105.23 even 12
1050.2.g.a.799.2 2 105.2 even 12
1134.2.f.g.379.1 2 63.23 odd 6
1134.2.f.g.757.1 2 63.2 odd 6
1134.2.f.j.379.1 2 63.58 even 3
1134.2.f.j.757.1 2 63.16 even 3
1344.2.a.i.1.1 1 168.107 even 6
1344.2.a.q.1.1 1 168.149 odd 6
2352.2.a.l.1.1 1 84.47 odd 6
2352.2.q.i.961.1 2 12.11 even 2
2352.2.q.i.1537.1 2 84.11 even 6
2352.2.q.n.961.1 2 84.83 odd 2
2352.2.q.n.1537.1 2 84.59 odd 6
3150.2.a.bo.1.1 1 35.9 even 6
3150.2.g.r.2899.1 2 35.2 odd 12
3150.2.g.r.2899.2 2 35.23 odd 12
4032.2.a.e.1.1 1 56.37 even 6
4032.2.a.m.1.1 1 56.51 odd 6
5082.2.a.d.1.1 1 231.65 even 6
5376.2.c.e.2689.1 2 336.107 even 12
5376.2.c.e.2689.2 2 336.275 even 12
5376.2.c.bc.2689.1 2 336.317 odd 12
5376.2.c.bc.2689.2 2 336.149 odd 12
7056.2.a.k.1.1 1 28.19 even 6
7098.2.a.f.1.1 1 273.233 odd 6
7350.2.a.f.1.1 1 105.89 even 6
8400.2.a.k.1.1 1 420.359 even 6
9408.2.a.n.1.1 1 168.5 even 6
9408.2.a.bw.1.1 1 168.131 odd 6