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 1.3
Root \(1.87939\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.120615 q^{5} -1.00000 q^{6} -4.29086 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.120615 q^{10} +2.57398 q^{11} -1.00000 q^{12} +0.815207 q^{13} -4.29086 q^{14} +0.120615 q^{15} +1.00000 q^{16} -0.467911 q^{17} +1.00000 q^{18} -0.120615 q^{20} +4.29086 q^{21} +2.57398 q^{22} +5.55438 q^{23} -1.00000 q^{24} -4.98545 q^{25} +0.815207 q^{26} -1.00000 q^{27} -4.29086 q^{28} +2.34730 q^{29} +0.120615 q^{30} +5.34730 q^{31} +1.00000 q^{32} -2.57398 q^{33} -0.467911 q^{34} +0.517541 q^{35} +1.00000 q^{36} -8.51754 q^{37} -0.815207 q^{39} -0.120615 q^{40} +3.83750 q^{41} +4.29086 q^{42} +9.33275 q^{43} +2.57398 q^{44} -0.120615 q^{45} +5.55438 q^{46} -9.47565 q^{47} -1.00000 q^{48} +11.4115 q^{49} -4.98545 q^{50} +0.467911 q^{51} +0.815207 q^{52} +12.7588 q^{53} -1.00000 q^{54} -0.310460 q^{55} -4.29086 q^{56} +2.34730 q^{58} +15.0496 q^{59} +0.120615 q^{60} -1.71688 q^{61} +5.34730 q^{62} -4.29086 q^{63} +1.00000 q^{64} -0.0983261 q^{65} -2.57398 q^{66} +11.4561 q^{67} -0.467911 q^{68} -5.55438 q^{69} +0.517541 q^{70} +13.3327 q^{71} +1.00000 q^{72} +2.28312 q^{73} -8.51754 q^{74} +4.98545 q^{75} -11.0446 q^{77} -0.815207 q^{78} +4.85710 q^{79} -0.120615 q^{80} +1.00000 q^{81} +3.83750 q^{82} +3.24897 q^{83} +4.29086 q^{84} +0.0564370 q^{85} +9.33275 q^{86} -2.34730 q^{87} +2.57398 q^{88} +3.43107 q^{89} -0.120615 q^{90} -3.49794 q^{91} +5.55438 q^{92} -5.34730 q^{93} -9.47565 q^{94} -1.00000 q^{96} +3.10101 q^{97} +11.4115 q^{98} +2.57398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 6 q^{10} - 3 q^{12} + 6 q^{13} + 3 q^{14} + 6 q^{15} + 3 q^{16} - 6 q^{17} + 3 q^{18} - 6 q^{20} - 3 q^{21} + 6 q^{23}+ \cdots + 24 q^{98}+O(q^{100}) \) 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
\(5\) −0.120615 −0.0539406 −0.0269703 − 0.999636i \(-0.508586\pi\)
−0.0269703 + 0.999636i \(0.508586\pi\)
\(6\) −1.00000 −0.408248
\(7\) −4.29086 −1.62179 −0.810896 − 0.585190i \(-0.801020\pi\)
−0.810896 + 0.585190i \(0.801020\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.120615 −0.0381417
\(11\) 2.57398 0.776084 0.388042 − 0.921642i \(-0.373152\pi\)
0.388042 + 0.921642i \(0.373152\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0.815207 0.226098 0.113049 − 0.993589i \(-0.463938\pi\)
0.113049 + 0.993589i \(0.463938\pi\)
\(14\) −4.29086 −1.14678
\(15\) 0.120615 0.0311426
\(16\) 1.00000 0.250000
\(17\) −0.467911 −0.113485 −0.0567426 − 0.998389i \(-0.518071\pi\)
−0.0567426 + 0.998389i \(0.518071\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0
\(20\) −0.120615 −0.0269703
\(21\) 4.29086 0.936342
\(22\) 2.57398 0.548774
\(23\) 5.55438 1.15817 0.579084 − 0.815268i \(-0.303410\pi\)
0.579084 + 0.815268i \(0.303410\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.98545 −0.997090
\(26\) 0.815207 0.159875
\(27\) −1.00000 −0.192450
\(28\) −4.29086 −0.810896
\(29\) 2.34730 0.435882 0.217941 − 0.975962i \(-0.430066\pi\)
0.217941 + 0.975962i \(0.430066\pi\)
\(30\) 0.120615 0.0220211
\(31\) 5.34730 0.960403 0.480201 − 0.877158i \(-0.340564\pi\)
0.480201 + 0.877158i \(0.340564\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.57398 −0.448072
\(34\) −0.467911 −0.0802461
\(35\) 0.517541 0.0874804
\(36\) 1.00000 0.166667
\(37\) −8.51754 −1.40028 −0.700138 − 0.714008i \(-0.746878\pi\)
−0.700138 + 0.714008i \(0.746878\pi\)
\(38\) 0 0
\(39\) −0.815207 −0.130538
\(40\) −0.120615 −0.0190709
\(41\) 3.83750 0.599316 0.299658 − 0.954047i \(-0.403127\pi\)
0.299658 + 0.954047i \(0.403127\pi\)
\(42\) 4.29086 0.662094
\(43\) 9.33275 1.42323 0.711615 − 0.702569i \(-0.247964\pi\)
0.711615 + 0.702569i \(0.247964\pi\)
\(44\) 2.57398 0.388042
\(45\) −0.120615 −0.0179802
\(46\) 5.55438 0.818948
\(47\) −9.47565 −1.38217 −0.691083 − 0.722775i \(-0.742866\pi\)
−0.691083 + 0.722775i \(0.742866\pi\)
\(48\) −1.00000 −0.144338
\(49\) 11.4115 1.63021
\(50\) −4.98545 −0.705049
\(51\) 0.467911 0.0655207
\(52\) 0.815207 0.113049
\(53\) 12.7588 1.75255 0.876276 − 0.481810i \(-0.160020\pi\)
0.876276 + 0.481810i \(0.160020\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.310460 −0.0418624
\(56\) −4.29086 −0.573390
\(57\) 0 0
\(58\) 2.34730 0.308215
\(59\) 15.0496 1.95929 0.979647 − 0.200726i \(-0.0643300\pi\)
0.979647 + 0.200726i \(0.0643300\pi\)
\(60\) 0.120615 0.0155713
\(61\) −1.71688 −0.219824 −0.109912 − 0.993941i \(-0.535057\pi\)
−0.109912 + 0.993941i \(0.535057\pi\)
\(62\) 5.34730 0.679107
\(63\) −4.29086 −0.540597
\(64\) 1.00000 0.125000
\(65\) −0.0983261 −0.0121958
\(66\) −2.57398 −0.316835
\(67\) 11.4561 1.39958 0.699790 − 0.714349i \(-0.253277\pi\)
0.699790 + 0.714349i \(0.253277\pi\)
\(68\) −0.467911 −0.0567426
\(69\) −5.55438 −0.668668
\(70\) 0.517541 0.0618580
\(71\) 13.3327 1.58231 0.791153 − 0.611618i \(-0.209481\pi\)
0.791153 + 0.611618i \(0.209481\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.28312 0.267219 0.133609 − 0.991034i \(-0.457343\pi\)
0.133609 + 0.991034i \(0.457343\pi\)
\(74\) −8.51754 −0.990144
\(75\) 4.98545 0.575670
\(76\) 0 0
\(77\) −11.0446 −1.25865
\(78\) −0.815207 −0.0923041
\(79\) 4.85710 0.546466 0.273233 − 0.961948i \(-0.411907\pi\)
0.273233 + 0.961948i \(0.411907\pi\)
\(80\) −0.120615 −0.0134851
\(81\) 1.00000 0.111111
\(82\) 3.83750 0.423781
\(83\) 3.24897 0.356621 0.178310 − 0.983974i \(-0.442937\pi\)
0.178310 + 0.983974i \(0.442937\pi\)
\(84\) 4.29086 0.468171
\(85\) 0.0564370 0.00612145
\(86\) 9.33275 1.00638
\(87\) −2.34730 −0.251657
\(88\) 2.57398 0.274387
\(89\) 3.43107 0.363693 0.181847 − 0.983327i \(-0.441793\pi\)
0.181847 + 0.983327i \(0.441793\pi\)
\(90\) −0.120615 −0.0127139
\(91\) −3.49794 −0.366684
\(92\) 5.55438 0.579084
\(93\) −5.34730 −0.554489
\(94\) −9.47565 −0.977339
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 3.10101 0.314860 0.157430 − 0.987530i \(-0.449679\pi\)
0.157430 + 0.987530i \(0.449679\pi\)
\(98\) 11.4115 1.15273
\(99\) 2.57398 0.258695
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.1.3 3
3.2 odd 2 6498.2.a.bp.1.1 3
19.9 even 9 114.2.i.c.43.1 ✓ 6
19.17 even 9 114.2.i.c.61.1 yes 6
19.18 odd 2 2166.2.a.p.1.3 3
57.17 odd 18 342.2.u.b.289.1 6
57.47 odd 18 342.2.u.b.271.1 6
57.56 even 2 6498.2.a.bu.1.1 3
76.47 odd 18 912.2.bo.d.385.1 6
76.55 odd 18 912.2.bo.d.289.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.c.43.1 ✓ 6 19.9 even 9
114.2.i.c.61.1 yes 6 19.17 even 9
342.2.u.b.271.1 6 57.47 odd 18
342.2.u.b.289.1 6 57.17 odd 18
912.2.bo.d.289.1 6 76.55 odd 18
912.2.bo.d.385.1 6 76.47 odd 18
2166.2.a.p.1.3 3 19.18 odd 2
2166.2.a.r.1.3 3 1.1 even 1 trivial
6498.2.a.bp.1.1 3 3.2 odd 2
6498.2.a.bu.1.1 3 57.56 even 2