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 1.1
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.53209 q^{5} -1.00000 q^{6} +3.71688 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.53209 q^{10} -5.29086 q^{11} -1.00000 q^{12} -0.226682 q^{13} +3.71688 q^{14} +3.53209 q^{15} +1.00000 q^{16} -1.65270 q^{17} +1.00000 q^{18} -3.53209 q^{20} -3.71688 q^{21} -5.29086 q^{22} +8.68004 q^{23} -1.00000 q^{24} +7.47565 q^{25} -0.226682 q^{26} -1.00000 q^{27} +3.71688 q^{28} +0.120615 q^{29} +3.53209 q^{30} +3.12061 q^{31} +1.00000 q^{32} +5.29086 q^{33} -1.65270 q^{34} -13.1284 q^{35} +1.00000 q^{36} +5.12836 q^{37} +0.226682 q^{39} -3.53209 q^{40} +7.10607 q^{41} -3.71688 q^{42} -5.35504 q^{43} -5.29086 q^{44} -3.53209 q^{45} +8.68004 q^{46} -2.50980 q^{47} -1.00000 q^{48} +6.81521 q^{49} +7.47565 q^{50} +1.65270 q^{51} -0.226682 q^{52} +5.93582 q^{53} -1.00000 q^{54} +18.6878 q^{55} +3.71688 q^{56} +0.120615 q^{58} +0.218941 q^{59} +3.53209 q^{60} -1.57398 q^{61} +3.12061 q^{62} +3.71688 q^{63} +1.00000 q^{64} +0.800660 q^{65} +5.29086 q^{66} +15.4807 q^{67} -1.65270 q^{68} -8.68004 q^{69} -13.1284 q^{70} -1.35504 q^{71} +1.00000 q^{72} +2.42602 q^{73} +5.12836 q^{74} -7.47565 q^{75} -19.6655 q^{77} +0.226682 q^{78} -2.86484 q^{79} -3.53209 q^{80} +1.00000 q^{81} +7.10607 q^{82} +1.92127 q^{83} -3.71688 q^{84} +5.83750 q^{85} -5.35504 q^{86} -0.120615 q^{87} -5.29086 q^{88} -12.1557 q^{89} -3.53209 q^{90} -0.842549 q^{91} +8.68004 q^{92} -3.12061 q^{93} -2.50980 q^{94} -1.00000 q^{96} +17.5030 q^{97} +6.81521 q^{98} -5.29086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 6 q^{10} - 3 q^{12} + 6 q^{13} + 3 q^{14} + 6 q^{15} + 3 q^{16} - 6 q^{17} + 3 q^{18} - 6 q^{20} - 3 q^{21} + 6 q^{23}+ \cdots + 24 q^{98}+O(q^{100}) \) 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
\(5\) −3.53209 −1.57960 −0.789799 − 0.613366i \(-0.789815\pi\)
−0.789799 + 0.613366i \(0.789815\pi\)
\(6\) −1.00000 −0.408248
\(7\) 3.71688 1.40485 0.702425 − 0.711758i \(-0.252101\pi\)
0.702425 + 0.711758i \(0.252101\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.53209 −1.11694
\(11\) −5.29086 −1.59525 −0.797627 − 0.603151i \(-0.793912\pi\)
−0.797627 + 0.603151i \(0.793912\pi\)
\(12\) −1.00000 −0.288675
\(13\) −0.226682 −0.0628702 −0.0314351 − 0.999506i \(-0.510008\pi\)
−0.0314351 + 0.999506i \(0.510008\pi\)
\(14\) 3.71688 0.993378
\(15\) 3.53209 0.911981
\(16\) 1.00000 0.250000
\(17\) −1.65270 −0.400840 −0.200420 − 0.979710i \(-0.564231\pi\)
−0.200420 + 0.979710i \(0.564231\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0
\(20\) −3.53209 −0.789799
\(21\) −3.71688 −0.811090
\(22\) −5.29086 −1.12802
\(23\) 8.68004 1.80991 0.904957 − 0.425503i \(-0.139903\pi\)
0.904957 + 0.425503i \(0.139903\pi\)
\(24\) −1.00000 −0.204124
\(25\) 7.47565 1.49513
\(26\) −0.226682 −0.0444559
\(27\) −1.00000 −0.192450
\(28\) 3.71688 0.702425
\(29\) 0.120615 0.0223976 0.0111988 − 0.999937i \(-0.496435\pi\)
0.0111988 + 0.999937i \(0.496435\pi\)
\(30\) 3.53209 0.644868
\(31\) 3.12061 0.560479 0.280239 − 0.959930i \(-0.409586\pi\)
0.280239 + 0.959930i \(0.409586\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.29086 0.921020
\(34\) −1.65270 −0.283436
\(35\) −13.1284 −2.21910
\(36\) 1.00000 0.166667
\(37\) 5.12836 0.843096 0.421548 − 0.906806i \(-0.361487\pi\)
0.421548 + 0.906806i \(0.361487\pi\)
\(38\) 0 0
\(39\) 0.226682 0.0362981
\(40\) −3.53209 −0.558472
\(41\) 7.10607 1.10978 0.554891 − 0.831923i \(-0.312760\pi\)
0.554891 + 0.831923i \(0.312760\pi\)
\(42\) −3.71688 −0.573527
\(43\) −5.35504 −0.816636 −0.408318 − 0.912840i \(-0.633884\pi\)
−0.408318 + 0.912840i \(0.633884\pi\)
\(44\) −5.29086 −0.797627
\(45\) −3.53209 −0.526533
\(46\) 8.68004 1.27980
\(47\) −2.50980 −0.366092 −0.183046 − 0.983104i \(-0.558596\pi\)
−0.183046 + 0.983104i \(0.558596\pi\)
\(48\) −1.00000 −0.144338
\(49\) 6.81521 0.973601
\(50\) 7.47565 1.05722
\(51\) 1.65270 0.231425
\(52\) −0.226682 −0.0314351
\(53\) 5.93582 0.815348 0.407674 − 0.913128i \(-0.366340\pi\)
0.407674 + 0.913128i \(0.366340\pi\)
\(54\) −1.00000 −0.136083
\(55\) 18.6878 2.51986
\(56\) 3.71688 0.496689
\(57\) 0 0
\(58\) 0.120615 0.0158375
\(59\) 0.218941 0.0285037 0.0142518 − 0.999898i \(-0.495463\pi\)
0.0142518 + 0.999898i \(0.495463\pi\)
\(60\) 3.53209 0.455991
\(61\) −1.57398 −0.201527 −0.100764 − 0.994910i \(-0.532129\pi\)
−0.100764 + 0.994910i \(0.532129\pi\)
\(62\) 3.12061 0.396318
\(63\) 3.71688 0.468283
\(64\) 1.00000 0.125000
\(65\) 0.800660 0.0993096
\(66\) 5.29086 0.651260
\(67\) 15.4807 1.89127 0.945635 − 0.325231i \(-0.105442\pi\)
0.945635 + 0.325231i \(0.105442\pi\)
\(68\) −1.65270 −0.200420
\(69\) −8.68004 −1.04495
\(70\) −13.1284 −1.56914
\(71\) −1.35504 −0.160813 −0.0804067 − 0.996762i \(-0.525622\pi\)
−0.0804067 + 0.996762i \(0.525622\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.42602 0.283944 0.141972 − 0.989871i \(-0.454656\pi\)
0.141972 + 0.989871i \(0.454656\pi\)
\(74\) 5.12836 0.596159
\(75\) −7.47565 −0.863214
\(76\) 0 0
\(77\) −19.6655 −2.24109
\(78\) 0.226682 0.0256666
\(79\) −2.86484 −0.322319 −0.161160 − 0.986928i \(-0.551523\pi\)
−0.161160 + 0.986928i \(0.551523\pi\)
\(80\) −3.53209 −0.394900
\(81\) 1.00000 0.111111
\(82\) 7.10607 0.784734
\(83\) 1.92127 0.210887 0.105444 − 0.994425i \(-0.466374\pi\)
0.105444 + 0.994425i \(0.466374\pi\)
\(84\) −3.71688 −0.405545
\(85\) 5.83750 0.633165
\(86\) −5.35504 −0.577449
\(87\) −0.120615 −0.0129313
\(88\) −5.29086 −0.564008
\(89\) −12.1557 −1.28850 −0.644251 − 0.764814i \(-0.722831\pi\)
−0.644251 + 0.764814i \(0.722831\pi\)
\(90\) −3.53209 −0.372315
\(91\) −0.842549 −0.0883231
\(92\) 8.68004 0.904957
\(93\) −3.12061 −0.323593
\(94\) −2.50980 −0.258866
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 17.5030 1.77716 0.888580 − 0.458722i \(-0.151693\pi\)
0.888580 + 0.458722i \(0.151693\pi\)
\(98\) 6.81521 0.688440
\(99\) −5.29086 −0.531751
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.1.1 3
3.2 odd 2 6498.2.a.bp.1.3 3
19.4 even 9 114.2.i.c.73.1 yes 6
19.5 even 9 114.2.i.c.25.1 ✓ 6
19.18 odd 2 2166.2.a.p.1.1 3
57.5 odd 18 342.2.u.b.253.1 6
57.23 odd 18 342.2.u.b.73.1 6
57.56 even 2 6498.2.a.bu.1.3 3
76.23 odd 18 912.2.bo.d.529.1 6
76.43 odd 18 912.2.bo.d.481.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.c.25.1 ✓ 6 19.5 even 9
114.2.i.c.73.1 yes 6 19.4 even 9
342.2.u.b.73.1 6 57.23 odd 18
342.2.u.b.253.1 6 57.5 odd 18
912.2.bo.d.481.1 6 76.43 odd 18
912.2.bo.d.529.1 6 76.23 odd 18
2166.2.a.p.1.1 3 19.18 odd 2
2166.2.a.r.1.1 3 1.1 even 1 trivial
6498.2.a.bp.1.3 3 3.2 odd 2
6498.2.a.bu.1.3 3 57.56 even 2