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: \(0\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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 \(-0.319190\pi\)
0.537974 + 0.842961i \(0.319190\pi\)
\(14\) 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 \(-0.378171\pi\)
0.373460 + 0.927646i \(0.378171\pi\)
\(30\) 1.18479 0.216313
\(31\) 1.95811 0.351687 0.175844 − 0.984418i \(-0.443735\pi\)
0.175844 + 0.984418i \(0.443735\pi\)
\(32\) 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 \(-0.691555\pi\)
−0.566118 + 0.824324i \(0.691555\pi\)
\(38\) 0 0
\(39\) 3.87939 0.621199
\(40\) 1.18479 0.187332
\(41\) 8.98545 1.40329 0.701646 − 0.712526i \(-0.252449\pi\)
0.701646 + 0.712526i \(0.252449\pi\)
\(42\) 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 \(-0.849887\pi\)
−0.890845 + 0.454307i \(0.849887\pi\)
\(54\) 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 \(-0.444183\pi\)
0.174456 + 0.984665i \(0.444183\pi\)
\(60\) 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 \(-0.260966\pi\)
0.682331 + 0.731043i \(0.260966\pi\)
\(68\) 1.16250 0.140974
\(69\) −6.70233 −0.806866
\(70\) 0.630415 0.0753490
\(71\) −6.07192 −0.720604 −0.360302 − 0.932836i \(-0.617326\pi\)
−0.360302 + 0.932836i \(0.617326\pi\)
\(72\) 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 \(-0.137931\pi\)
0.907575 + 0.419890i \(0.137931\pi\)
\(80\) 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 \(-0.439589\pi\)
0.188649 + 0.982044i \(0.439589\pi\)
\(90\) 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 \(-0.596480\pi\)
−0.298481 + 0.954416i \(0.596480\pi\)
\(98\) −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.u.1.2 3
3.2 odd 2 6498.2.a.bn.1.2 3
19.2 odd 18 114.2.i.d.61.1 yes 6
19.10 odd 18 114.2.i.d.43.1 ✓ 6
19.18 odd 2 2166.2.a.o.1.2 3
57.2 even 18 342.2.u.a.289.1 6
57.29 even 18 342.2.u.a.271.1 6
57.56 even 2 6498.2.a.bs.1.2 3
76.59 even 18 912.2.bo.f.289.1 6
76.67 even 18 912.2.bo.f.385.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.43.1 ✓ 6 19.10 odd 18
114.2.i.d.61.1 yes 6 19.2 odd 18
342.2.u.a.271.1 6 57.29 even 18
342.2.u.a.289.1 6 57.2 even 18
912.2.bo.f.289.1 6 76.59 even 18
912.2.bo.f.385.1 6 76.67 even 18
2166.2.a.o.1.2 3 19.18 odd 2
2166.2.a.u.1.2 3 1.1 even 1 trivial
6498.2.a.bn.1.2 3 3.2 odd 2
6498.2.a.bs.1.2 3 57.56 even 2