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,-1,0,0,0,2,0,0,0,0,-8] 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 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 882.361
Dual form 882.2.g.c.667.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{8} -4.00000 q^{13} +(-0.500000 - 0.866025i) q^{16} +(3.00000 - 5.19615i) q^{17} +(-1.00000 - 1.73205i) q^{19} +(2.50000 - 4.33013i) q^{25} +(2.00000 + 3.46410i) q^{26} +6.00000 q^{29} +(2.00000 - 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} -6.00000 q^{34} +(-1.00000 - 1.73205i) q^{37} +(-1.00000 + 1.73205i) q^{38} -6.00000 q^{41} +8.00000 q^{43} +(-6.00000 - 10.3923i) q^{47} -5.00000 q^{50} +(2.00000 - 3.46410i) q^{52} +(3.00000 - 5.19615i) q^{53} +(-3.00000 - 5.19615i) q^{58} +(-3.00000 + 5.19615i) q^{59} +(-4.00000 - 6.92820i) q^{61} -4.00000 q^{62} +1.00000 q^{64} +(2.00000 - 3.46410i) q^{67} +(3.00000 + 5.19615i) q^{68} +(-1.00000 + 1.73205i) q^{73} +(-1.00000 + 1.73205i) q^{74} +2.00000 q^{76} +(-4.00000 - 6.92820i) q^{79} +(3.00000 + 5.19615i) q^{82} +6.00000 q^{83} +(-4.00000 - 6.92820i) q^{86} +(-3.00000 - 5.19615i) q^{89} +(-6.00000 + 10.3923i) q^{94} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 2 q^{8} - 8 q^{13} - q^{16} + 6 q^{17} - 2 q^{19} + 5 q^{25} + 4 q^{26} + 12 q^{29} + 4 q^{31} - q^{32} - 12 q^{34} - 2 q^{37} - 2 q^{38} - 12 q^{41} + 16 q^{43} - 12 q^{47} - 10 q^{50}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 3.00000 − 5.19615i 0.727607 − 1.26025i −0.230285 − 0.973123i \(-0.573966\pi\)
0.957892 − 0.287129i \(-0.0927008\pi\)
\(18\) 0 0
\(19\) −1.00000 − 1.73205i −0.229416 − 0.397360i 0.728219 − 0.685344i \(-0.240348\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 2.50000 − 4.33013i 0.500000 − 0.866025i
\(26\) 2.00000 + 3.46410i 0.392232 + 0.679366i
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 2.00000 − 3.46410i 0.359211 − 0.622171i −0.628619 − 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 − 1.73205i −0.164399 − 0.284747i 0.772043 − 0.635571i \(-0.219235\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(38\) −1.00000 + 1.73205i −0.162221 + 0.280976i
\(39\) 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
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 − 10.3923i −0.875190 − 1.51587i −0.856560 − 0.516047i \(-0.827403\pi\)
−0.0186297 − 0.999826i \(-0.505930\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) 2.00000 − 3.46410i 0.277350 − 0.480384i
\(53\) 3.00000 − 5.19615i 0.412082 − 0.713746i −0.583036 − 0.812447i \(-0.698135\pi\)
0.995117 + 0.0987002i \(0.0314685\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −3.00000 − 5.19615i −0.393919 − 0.682288i
\(59\) −3.00000 + 5.19615i −0.390567 + 0.676481i −0.992524 − 0.122047i \(-0.961054\pi\)
0.601958 + 0.798528i \(0.294388\pi\)
\(60\) 0 0
\(61\) −4.00000 − 6.92820i −0.512148 − 0.887066i −0.999901 − 0.0140840i \(-0.995517\pi\)
0.487753 − 0.872982i \(-0.337817\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 3.00000 + 5.19615i 0.363803 + 0.630126i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.00000 + 1.73205i −0.117041 + 0.202721i −0.918594 − 0.395203i \(-0.870674\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) −1.00000 + 1.73205i −0.116248 + 0.201347i
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 − 6.92820i −0.450035 − 0.779484i 0.548352 − 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.00000 + 5.19615i 0.331295 + 0.573819i
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 − 6.92820i −0.431331 − 0.747087i
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 − 5.19615i −0.317999 − 0.550791i 0.662071 − 0.749441i \(-0.269678\pi\)
−0.980071 + 0.198650i \(0.936344\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −6.00000 + 10.3923i −0.618853 + 1.07188i
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 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.c.361.1 2
3.2 odd 2 98.2.c.b.67.1 2
7.2 even 3 inner 882.2.g.c.667.1 2
7.3 odd 6 882.2.a.i.1.1 1
7.4 even 3 126.2.a.b.1.1 1
7.5 odd 6 882.2.g.d.667.1 2
7.6 odd 2 882.2.g.d.361.1 2
12.11 even 2 784.2.i.c.753.1 2
21.2 odd 6 98.2.c.b.79.1 2
21.5 even 6 98.2.c.a.79.1 2
21.11 odd 6 14.2.a.a.1.1 ✓ 1
21.17 even 6 98.2.a.a.1.1 1
21.20 even 2 98.2.c.a.67.1 2
28.3 even 6 7056.2.a.bd.1.1 1
28.11 odd 6 1008.2.a.h.1.1 1
35.4 even 6 3150.2.a.i.1.1 1
35.18 odd 12 3150.2.g.j.2899.1 2
35.32 odd 12 3150.2.g.j.2899.2 2
56.11 odd 6 4032.2.a.r.1.1 1
56.53 even 6 4032.2.a.w.1.1 1
63.4 even 3 1134.2.f.f.379.1 2
63.11 odd 6 1134.2.f.l.757.1 2
63.25 even 3 1134.2.f.f.757.1 2
63.32 odd 6 1134.2.f.l.379.1 2
84.11 even 6 112.2.a.c.1.1 1
84.23 even 6 784.2.i.c.177.1 2
84.47 odd 6 784.2.i.i.177.1 2
84.59 odd 6 784.2.a.b.1.1 1
84.83 odd 2 784.2.i.i.753.1 2
105.17 odd 12 2450.2.c.c.99.1 2
105.32 even 12 350.2.c.d.99.1 2
105.38 odd 12 2450.2.c.c.99.2 2
105.53 even 12 350.2.c.d.99.2 2
105.59 even 6 2450.2.a.t.1.1 1
105.74 odd 6 350.2.a.f.1.1 1
168.11 even 6 448.2.a.a.1.1 1
168.53 odd 6 448.2.a.g.1.1 1
168.59 odd 6 3136.2.a.z.1.1 1
168.101 even 6 3136.2.a.e.1.1 1
231.32 even 6 1694.2.a.e.1.1 1
273.116 odd 6 2366.2.a.j.1.1 1
273.200 even 12 2366.2.d.b.337.2 2
273.242 even 12 2366.2.d.b.337.1 2
336.11 even 12 1792.2.b.g.897.1 2
336.53 odd 12 1792.2.b.c.897.2 2
336.179 even 12 1792.2.b.g.897.2 2
336.221 odd 12 1792.2.b.c.897.1 2
357.305 odd 6 4046.2.a.f.1.1 1
399.284 even 6 5054.2.a.c.1.1 1
420.179 even 6 2800.2.a.g.1.1 1
420.263 odd 12 2800.2.g.h.449.2 2
420.347 odd 12 2800.2.g.h.449.1 2
483.137 even 6 7406.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 ✓ 1 21.11 odd 6
98.2.a.a.1.1 1 21.17 even 6
98.2.c.a.67.1 2 21.20 even 2
98.2.c.a.79.1 2 21.5 even 6
98.2.c.b.67.1 2 3.2 odd 2
98.2.c.b.79.1 2 21.2 odd 6
112.2.a.c.1.1 1 84.11 even 6
126.2.a.b.1.1 1 7.4 even 3
350.2.a.f.1.1 1 105.74 odd 6
350.2.c.d.99.1 2 105.32 even 12
350.2.c.d.99.2 2 105.53 even 12
448.2.a.a.1.1 1 168.11 even 6
448.2.a.g.1.1 1 168.53 odd 6
784.2.a.b.1.1 1 84.59 odd 6
784.2.i.c.177.1 2 84.23 even 6
784.2.i.c.753.1 2 12.11 even 2
784.2.i.i.177.1 2 84.47 odd 6
784.2.i.i.753.1 2 84.83 odd 2
882.2.a.i.1.1 1 7.3 odd 6
882.2.g.c.361.1 2 1.1 even 1 trivial
882.2.g.c.667.1 2 7.2 even 3 inner
882.2.g.d.361.1 2 7.6 odd 2
882.2.g.d.667.1 2 7.5 odd 6
1008.2.a.h.1.1 1 28.11 odd 6
1134.2.f.f.379.1 2 63.4 even 3
1134.2.f.f.757.1 2 63.25 even 3
1134.2.f.l.379.1 2 63.32 odd 6
1134.2.f.l.757.1 2 63.11 odd 6
1694.2.a.e.1.1 1 231.32 even 6
1792.2.b.c.897.1 2 336.221 odd 12
1792.2.b.c.897.2 2 336.53 odd 12
1792.2.b.g.897.1 2 336.11 even 12
1792.2.b.g.897.2 2 336.179 even 12
2366.2.a.j.1.1 1 273.116 odd 6
2366.2.d.b.337.1 2 273.242 even 12
2366.2.d.b.337.2 2 273.200 even 12
2450.2.a.t.1.1 1 105.59 even 6
2450.2.c.c.99.1 2 105.17 odd 12
2450.2.c.c.99.2 2 105.38 odd 12
2800.2.a.g.1.1 1 420.179 even 6
2800.2.g.h.449.1 2 420.347 odd 12
2800.2.g.h.449.2 2 420.263 odd 12
3136.2.a.e.1.1 1 168.101 even 6
3136.2.a.z.1.1 1 168.59 odd 6
3150.2.a.i.1.1 1 35.4 even 6
3150.2.g.j.2899.1 2 35.18 odd 12
3150.2.g.j.2899.2 2 35.32 odd 12
4032.2.a.r.1.1 1 56.11 odd 6
4032.2.a.w.1.1 1 56.53 even 6
4046.2.a.f.1.1 1 357.305 odd 6
5054.2.a.c.1.1 1 399.284 even 6
7056.2.a.bd.1.1 1 28.3 even 6
7406.2.a.a.1.1 1 483.137 even 6