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,-3,3,0,3,-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: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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 \(-1.53209\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.18479 q^{5} +1.00000 q^{6} +0.532089 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.18479 q^{10} +1.87939 q^{11} -1.00000 q^{12} -3.87939 q^{13} -0.532089 q^{14} -1.18479 q^{15} +1.00000 q^{16} +1.16250 q^{17} -1.00000 q^{18} +1.18479 q^{20} -0.532089 q^{21} -1.87939 q^{22} -6.70233 q^{23} +1.00000 q^{24} -3.59627 q^{25} +3.87939 q^{26} -1.00000 q^{27} +0.532089 q^{28} -4.02229 q^{29} +1.18479 q^{30} -1.95811 q^{31} -1.00000 q^{32} -1.87939 q^{33} -1.16250 q^{34} +0.630415 q^{35} +1.00000 q^{36} +6.88713 q^{37} +3.87939 q^{39} -1.18479 q^{40} -8.98545 q^{41} +0.532089 q^{42} +2.42602 q^{43} +1.87939 q^{44} +1.18479 q^{45} +6.70233 q^{46} +2.04189 q^{47} -1.00000 q^{48} -6.71688 q^{49} +3.59627 q^{50} -1.16250 q^{51} -3.87939 q^{52} +12.9709 q^{53} +1.00000 q^{54} +2.22668 q^{55} -0.532089 q^{56} +4.02229 q^{58} -2.68004 q^{59} -1.18479 q^{60} -11.3473 q^{61} +1.95811 q^{62} +0.532089 q^{63} +1.00000 q^{64} -4.59627 q^{65} +1.87939 q^{66} -11.1702 q^{67} +1.16250 q^{68} +6.70233 q^{69} -0.630415 q^{70} +6.07192 q^{71} -1.00000 q^{72} -0.327696 q^{73} -6.88713 q^{74} +3.59627 q^{75} +1.00000 q^{77} -3.87939 q^{78} -16.1334 q^{79} +1.18479 q^{80} +1.00000 q^{81} +8.98545 q^{82} +11.5740 q^{83} -0.532089 q^{84} +1.37733 q^{85} -2.42602 q^{86} +4.02229 q^{87} -1.87939 q^{88} -3.55943 q^{89} -1.18479 q^{90} -2.06418 q^{91} -6.70233 q^{92} +1.95811 q^{93} -2.04189 q^{94} +1.00000 q^{96} +5.87939 q^{97} +6.71688 q^{98} +1.87939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{12} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} - 3 q^{18} + 3 q^{21} + 6 q^{23} + 3 q^{24} + 3 q^{25} + 6 q^{26} - 3 q^{27}+ \cdots + 12 q^{98}+O(q^{100}) \) 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
\(5\) 1.18479 0.529855 0.264928 − 0.964268i \(-0.414652\pi\)
0.264928 + 0.964268i \(0.414652\pi\)
\(6\) 1.00000 0.408248
\(7\) 0.532089 0.201111 0.100555 − 0.994931i \(-0.467938\pi\)
0.100555 + 0.994931i \(0.467938\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.18479 −0.374664
\(11\) 1.87939 0.566656 0.283328 − 0.959023i \(-0.408561\pi\)
0.283328 + 0.959023i \(0.408561\pi\)
\(12\) −1.00000 −0.288675
\(13\) −3.87939 −1.07595 −0.537974 − 0.842961i \(-0.680810\pi\)
−0.537974 + 0.842961i \(0.680810\pi\)
\(14\) −0.532089 −0.142207
\(15\) −1.18479 −0.305912
\(16\) 1.00000 0.250000
\(17\) 1.16250 0.281949 0.140974 − 0.990013i \(-0.454977\pi\)
0.140974 + 0.990013i \(0.454977\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0
\(20\) 1.18479 0.264928
\(21\) −0.532089 −0.116111
\(22\) −1.87939 −0.400686
\(23\) −6.70233 −1.39753 −0.698767 − 0.715350i \(-0.746267\pi\)
−0.698767 + 0.715350i \(0.746267\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.59627 −0.719253
\(26\) 3.87939 0.760810
\(27\) −1.00000 −0.192450
\(28\) 0.532089 0.100555
\(29\) −4.02229 −0.746920 −0.373460 − 0.927646i \(-0.621829\pi\)
−0.373460 + 0.927646i \(0.621829\pi\)
\(30\) 1.18479 0.216313
\(31\) −1.95811 −0.351687 −0.175844 − 0.984418i \(-0.556265\pi\)
−0.175844 + 0.984418i \(0.556265\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.87939 −0.327159
\(34\) −1.16250 −0.199368
\(35\) 0.630415 0.106560
\(36\) 1.00000 0.166667
\(37\) 6.88713 1.13224 0.566118 − 0.824324i \(-0.308445\pi\)
0.566118 + 0.824324i \(0.308445\pi\)
\(38\) 0 0
\(39\) 3.87939 0.621199
\(40\) −1.18479 −0.187332
\(41\) −8.98545 −1.40329 −0.701646 − 0.712526i \(-0.747551\pi\)
−0.701646 + 0.712526i \(0.747551\pi\)
\(42\) 0.532089 0.0821031
\(43\) 2.42602 0.369965 0.184982 − 0.982742i \(-0.440777\pi\)
0.184982 + 0.982742i \(0.440777\pi\)
\(44\) 1.87939 0.283328
\(45\) 1.18479 0.176618
\(46\) 6.70233 0.988205
\(47\) 2.04189 0.297840 0.148920 − 0.988849i \(-0.452420\pi\)
0.148920 + 0.988849i \(0.452420\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.71688 −0.959554
\(50\) 3.59627 0.508589
\(51\) −1.16250 −0.162783
\(52\) −3.87939 −0.537974
\(53\) 12.9709 1.78169 0.890845 − 0.454307i \(-0.150113\pi\)
0.890845 + 0.454307i \(0.150113\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.22668 0.300246
\(56\) −0.532089 −0.0711034
\(57\) 0 0
\(58\) 4.02229 0.528152
\(59\) −2.68004 −0.348912 −0.174456 − 0.984665i \(-0.555817\pi\)
−0.174456 + 0.984665i \(0.555817\pi\)
\(60\) −1.18479 −0.152956
\(61\) −11.3473 −1.45287 −0.726436 − 0.687234i \(-0.758825\pi\)
−0.726436 + 0.687234i \(0.758825\pi\)
\(62\) 1.95811 0.248680
\(63\) 0.532089 0.0670369
\(64\) 1.00000 0.125000
\(65\) −4.59627 −0.570097
\(66\) 1.87939 0.231336
\(67\) −11.1702 −1.36466 −0.682331 − 0.731043i \(-0.739034\pi\)
−0.682331 + 0.731043i \(0.739034\pi\)
\(68\) 1.16250 0.140974
\(69\) 6.70233 0.806866
\(70\) −0.630415 −0.0753490
\(71\) 6.07192 0.720604 0.360302 − 0.932836i \(-0.382674\pi\)
0.360302 + 0.932836i \(0.382674\pi\)
\(72\) −1.00000 −0.117851
\(73\) −0.327696 −0.0383539 −0.0191770 − 0.999816i \(-0.506105\pi\)
−0.0191770 + 0.999816i \(0.506105\pi\)
\(74\) −6.88713 −0.800612
\(75\) 3.59627 0.415261
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) −3.87939 −0.439254
\(79\) −16.1334 −1.81515 −0.907575 − 0.419890i \(-0.862069\pi\)
−0.907575 + 0.419890i \(0.862069\pi\)
\(80\) 1.18479 0.132464
\(81\) 1.00000 0.111111
\(82\) 8.98545 0.992277
\(83\) 11.5740 1.27041 0.635205 − 0.772344i \(-0.280916\pi\)
0.635205 + 0.772344i \(0.280916\pi\)
\(84\) −0.532089 −0.0580557
\(85\) 1.37733 0.149392
\(86\) −2.42602 −0.261605
\(87\) 4.02229 0.431235
\(88\) −1.87939 −0.200343
\(89\) −3.55943 −0.377299 −0.188649 − 0.982044i \(-0.560411\pi\)
−0.188649 + 0.982044i \(0.560411\pi\)
\(90\) −1.18479 −0.124888
\(91\) −2.06418 −0.216385
\(92\) −6.70233 −0.698767
\(93\) 1.95811 0.203047
\(94\) −2.04189 −0.210605
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 5.87939 0.596961 0.298481 − 0.954416i \(-0.403520\pi\)
0.298481 + 0.954416i \(0.403520\pi\)
\(98\) 6.71688 0.678507
\(99\) 1.87939 0.188885
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.o.1.2 3
3.2 odd 2 6498.2.a.bs.1.2 3
19.9 even 9 114.2.i.d.43.1 ✓ 6
19.17 even 9 114.2.i.d.61.1 yes 6
19.18 odd 2 2166.2.a.u.1.2 3
57.17 odd 18 342.2.u.a.289.1 6
57.47 odd 18 342.2.u.a.271.1 6
57.56 even 2 6498.2.a.bn.1.2 3
76.47 odd 18 912.2.bo.f.385.1 6
76.55 odd 18 912.2.bo.f.289.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.43.1 ✓ 6 19.9 even 9
114.2.i.d.61.1 yes 6 19.17 even 9
342.2.u.a.271.1 6 57.47 odd 18
342.2.u.a.289.1 6 57.17 odd 18
912.2.bo.f.289.1 6 76.55 odd 18
912.2.bo.f.385.1 6 76.47 odd 18
2166.2.a.o.1.2 3 1.1 even 1 trivial
2166.2.a.u.1.2 3 19.18 odd 2
6498.2.a.bn.1.2 3 57.56 even 2
6498.2.a.bs.1.2 3 3.2 odd 2