Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2166,2,Mod(1,2166)] 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("2166.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2166, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,-3,3,-6,-3,3] 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: yes
Analytic conductor: \(17.2955970778\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.34730 q^{5} -1.00000 q^{6} +3.57398 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.34730 q^{10} +2.71688 q^{11} -1.00000 q^{12} +5.41147 q^{13} +3.57398 q^{14} +2.34730 q^{15} +1.00000 q^{16} -3.87939 q^{17} +1.00000 q^{18} -2.34730 q^{20} -3.57398 q^{21} +2.71688 q^{22} -8.23442 q^{23} -1.00000 q^{24} +0.509800 q^{25} +5.41147 q^{26} -1.00000 q^{27} +3.57398 q^{28} +3.53209 q^{29} +2.34730 q^{30} +6.53209 q^{31} +1.00000 q^{32} -2.71688 q^{33} -3.87939 q^{34} -8.38919 q^{35} +1.00000 q^{36} +0.389185 q^{37} -5.41147 q^{39} -2.34730 q^{40} -1.94356 q^{41} -3.57398 q^{42} +5.02229 q^{43} +2.71688 q^{44} -2.34730 q^{45} -8.23442 q^{46} +2.98545 q^{47} -1.00000 q^{48} +5.77332 q^{49} +0.509800 q^{50} +3.87939 q^{51} +5.41147 q^{52} +8.30541 q^{53} -1.00000 q^{54} -6.37733 q^{55} +3.57398 q^{56} +3.53209 q^{58} +2.73143 q^{59} +2.34730 q^{60} +6.29086 q^{61} +6.53209 q^{62} +3.57398 q^{63} +1.00000 q^{64} -12.7023 q^{65} -2.71688 q^{66} -14.9368 q^{67} -3.87939 q^{68} +8.23442 q^{69} -8.38919 q^{70} +9.02229 q^{71} +1.00000 q^{72} +10.2909 q^{73} +0.389185 q^{74} -0.509800 q^{75} +9.71007 q^{77} -5.41147 q^{78} +13.0077 q^{79} -2.34730 q^{80} +1.00000 q^{81} -1.94356 q^{82} -8.17024 q^{83} -3.57398 q^{84} +9.10607 q^{85} +5.02229 q^{86} -3.53209 q^{87} +2.71688 q^{88} +11.7246 q^{89} -2.34730 q^{90} +19.3405 q^{91} -8.23442 q^{92} -6.53209 q^{93} +2.98545 q^{94} -1.00000 q^{96} -8.60401 q^{97} +5.77332 q^{98} +2.71688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 6 q^{10} - 3 q^{12} + 6 q^{13} + 3 q^{14} + 6 q^{15} + 3 q^{16} - 6 q^{17} + 3 q^{18} - 6 q^{20} - 3 q^{21} + 6 q^{23}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.34730 −1.04974 −0.524871 − 0.851182i \(-0.675887\pi\)
−0.524871 + 0.851182i \(0.675887\pi\)
\(6\) −1.00000 −0.408248
\(7\) 3.57398 1.35084 0.675418 − 0.737435i \(-0.263963\pi\)
0.675418 + 0.737435i \(0.263963\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.34730 −0.742280
\(11\) 2.71688 0.819171 0.409585 − 0.912272i \(-0.365673\pi\)
0.409585 + 0.912272i \(0.365673\pi\)
\(12\) −1.00000 −0.288675
\(13\) 5.41147 1.50087 0.750436 − 0.660943i \(-0.229843\pi\)
0.750436 + 0.660943i \(0.229843\pi\)
\(14\) 3.57398 0.955186
\(15\) 2.34730 0.606069
\(16\) 1.00000 0.250000
\(17\) −3.87939 −0.940889 −0.470445 − 0.882430i \(-0.655906\pi\)
−0.470445 + 0.882430i \(0.655906\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0
\(20\) −2.34730 −0.524871
\(21\) −3.57398 −0.779906
\(22\) 2.71688 0.579241
\(23\) −8.23442 −1.71700 −0.858498 − 0.512817i \(-0.828602\pi\)
−0.858498 + 0.512817i \(0.828602\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0.509800 0.101960
\(26\) 5.41147 1.06128
\(27\) −1.00000 −0.192450
\(28\) 3.57398 0.675418
\(29\) 3.53209 0.655892 0.327946 − 0.944696i \(-0.393644\pi\)
0.327946 + 0.944696i \(0.393644\pi\)
\(30\) 2.34730 0.428556
\(31\) 6.53209 1.17320 0.586599 − 0.809878i \(-0.300467\pi\)
0.586599 + 0.809878i \(0.300467\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.71688 −0.472948
\(34\) −3.87939 −0.665309
\(35\) −8.38919 −1.41803
\(36\) 1.00000 0.166667
\(37\) 0.389185 0.0639817 0.0319908 − 0.999488i \(-0.489815\pi\)
0.0319908 + 0.999488i \(0.489815\pi\)
\(38\) 0 0
\(39\) −5.41147 −0.866529
\(40\) −2.34730 −0.371140
\(41\) −1.94356 −0.303534 −0.151767 − 0.988416i \(-0.548496\pi\)
−0.151767 + 0.988416i \(0.548496\pi\)
\(42\) −3.57398 −0.551477
\(43\) 5.02229 0.765892 0.382946 − 0.923771i \(-0.374910\pi\)
0.382946 + 0.923771i \(0.374910\pi\)
\(44\) 2.71688 0.409585
\(45\) −2.34730 −0.349914
\(46\) −8.23442 −1.21410
\(47\) 2.98545 0.435473 0.217736 − 0.976008i \(-0.430133\pi\)
0.217736 + 0.976008i \(0.430133\pi\)
\(48\) −1.00000 −0.144338
\(49\) 5.77332 0.824760
\(50\) 0.509800 0.0720966
\(51\) 3.87939 0.543223
\(52\) 5.41147 0.750436
\(53\) 8.30541 1.14084 0.570418 − 0.821355i \(-0.306781\pi\)
0.570418 + 0.821355i \(0.306781\pi\)
\(54\) −1.00000 −0.136083
\(55\) −6.37733 −0.859918
\(56\) 3.57398 0.477593
\(57\) 0 0
\(58\) 3.53209 0.463786
\(59\) 2.73143 0.355602 0.177801 − 0.984066i \(-0.443102\pi\)
0.177801 + 0.984066i \(0.443102\pi\)
\(60\) 2.34730 0.303035
\(61\) 6.29086 0.805462 0.402731 − 0.915318i \(-0.368061\pi\)
0.402731 + 0.915318i \(0.368061\pi\)
\(62\) 6.53209 0.829576
\(63\) 3.57398 0.450279
\(64\) 1.00000 0.125000
\(65\) −12.7023 −1.57553
\(66\) −2.71688 −0.334425
\(67\) −14.9368 −1.82482 −0.912408 − 0.409283i \(-0.865779\pi\)
−0.912408 + 0.409283i \(0.865779\pi\)
\(68\) −3.87939 −0.470445
\(69\) 8.23442 0.991308
\(70\) −8.38919 −1.00270
\(71\) 9.02229 1.07075 0.535374 − 0.844615i \(-0.320171\pi\)
0.535374 + 0.844615i \(0.320171\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.2909 1.20445 0.602227 − 0.798325i \(-0.294280\pi\)
0.602227 + 0.798325i \(0.294280\pi\)
\(74\) 0.389185 0.0452419
\(75\) −0.509800 −0.0588667
\(76\) 0 0
\(77\) 9.71007 1.10657
\(78\) −5.41147 −0.612729
\(79\) 13.0077 1.46349 0.731743 − 0.681581i \(-0.238707\pi\)
0.731743 + 0.681581i \(0.238707\pi\)
\(80\) −2.34730 −0.262436
\(81\) 1.00000 0.111111
\(82\) −1.94356 −0.214631
\(83\) −8.17024 −0.896801 −0.448400 − 0.893833i \(-0.648006\pi\)
−0.448400 + 0.893833i \(0.648006\pi\)
\(84\) −3.57398 −0.389953
\(85\) 9.10607 0.987692
\(86\) 5.02229 0.541567
\(87\) −3.53209 −0.378680
\(88\) 2.71688 0.289621
\(89\) 11.7246 1.24281 0.621404 − 0.783491i \(-0.286563\pi\)
0.621404 + 0.783491i \(0.286563\pi\)
\(90\) −2.34730 −0.247427
\(91\) 19.3405 2.02743
\(92\) −8.23442 −0.858498
\(93\) −6.53209 −0.677346
\(94\) 2.98545 0.307926
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −8.60401 −0.873605 −0.436802 − 0.899558i \(-0.643889\pi\)
−0.436802 + 0.899558i \(0.643889\pi\)
\(98\) 5.77332 0.583193
\(99\) 2.71688 0.273057
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 2166.2.a.r.1.2 3
3.2 odd 2 6498.2.a.bp.1.2 3
19.6 even 9 114.2.i.c.55.1 ✓ 6
19.16 even 9 114.2.i.c.85.1 yes 6
19.18 odd 2 2166.2.a.p.1.2 3
57.35 odd 18 342.2.u.b.199.1 6
57.44 odd 18 342.2.u.b.55.1 6
57.56 even 2 6498.2.a.bu.1.2 3
76.35 odd 18 912.2.bo.d.769.1 6
76.63 odd 18 912.2.bo.d.625.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.c.55.1 ✓ 6 19.6 even 9
114.2.i.c.85.1 yes 6 19.16 even 9
342.2.u.b.55.1 6 57.44 odd 18
342.2.u.b.199.1 6 57.35 odd 18
912.2.bo.d.625.1 6 76.63 odd 18
912.2.bo.d.769.1 6 76.35 odd 18
2166.2.a.p.1.2 3 19.18 odd 2
2166.2.a.r.1.2 3 1.1 even 1 trivial
6498.2.a.bp.1.2 3 3.2 odd 2
6498.2.a.bu.1.2 3 57.56 even 2