Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(1,666)] 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("666.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,-1,0,-2] 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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 666.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.85410 q^{5} +1.23607 q^{7} -1.00000 q^{8} -2.85410 q^{10} +3.61803 q^{11} +3.85410 q^{13} -1.23607 q^{14} +1.00000 q^{16} -4.47214 q^{17} -4.47214 q^{19} +2.85410 q^{20} -3.61803 q^{22} +3.85410 q^{23} +3.14590 q^{25} -3.85410 q^{26} +1.23607 q^{28} -6.32624 q^{29} +9.61803 q^{31} -1.00000 q^{32} +4.47214 q^{34} +3.52786 q^{35} -1.00000 q^{37} +4.47214 q^{38} -2.85410 q^{40} -7.38197 q^{41} -0.763932 q^{43} +3.61803 q^{44} -3.85410 q^{46} -3.23607 q^{47} -5.47214 q^{49} -3.14590 q^{50} +3.85410 q^{52} +8.47214 q^{53} +10.3262 q^{55} -1.23607 q^{56} +6.32624 q^{58} +9.23607 q^{59} +8.38197 q^{61} -9.61803 q^{62} +1.00000 q^{64} +11.0000 q^{65} -10.0902 q^{67} -4.47214 q^{68} -3.52786 q^{70} +14.9443 q^{71} -4.09017 q^{73} +1.00000 q^{74} -4.47214 q^{76} +4.47214 q^{77} +11.5623 q^{79} +2.85410 q^{80} +7.38197 q^{82} +5.52786 q^{83} -12.7639 q^{85} +0.763932 q^{86} -3.61803 q^{88} +10.4721 q^{89} +4.76393 q^{91} +3.85410 q^{92} +3.23607 q^{94} -12.7639 q^{95} +8.47214 q^{97} +5.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + 5 q^{11} + q^{13} + 2 q^{14} + 2 q^{16} - q^{20} - 5 q^{22} + q^{23} + 13 q^{25} - q^{26} - 2 q^{28} + 3 q^{29} + 17 q^{31} - 2 q^{32}+ \cdots + 2 q^{98}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.85410 1.27639 0.638197 − 0.769873i \(-0.279681\pi\)
0.638197 + 0.769873i \(0.279681\pi\)
\(6\) 0 0
\(7\) 1.23607 0.467190 0.233595 − 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.85410 −0.902546
\(11\) 3.61803 1.09088 0.545439 − 0.838150i \(-0.316363\pi\)
0.545439 + 0.838150i \(0.316363\pi\)
\(12\) 0 0
\(13\) 3.85410 1.06894 0.534468 − 0.845189i \(-0.320512\pi\)
0.534468 + 0.845189i \(0.320512\pi\)
\(14\) −1.23607 −0.330353
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.47214 −1.08465 −0.542326 − 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) −4.47214 −1.02598 −0.512989 − 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 2.85410 0.638197
\(21\) 0 0
\(22\) −3.61803 −0.771367
\(23\) 3.85410 0.803636 0.401818 − 0.915720i \(-0.368378\pi\)
0.401818 + 0.915720i \(0.368378\pi\)
\(24\) 0 0
\(25\) 3.14590 0.629180
\(26\) −3.85410 −0.755852
\(27\) 0 0
\(28\) 1.23607 0.233595
\(29\) −6.32624 −1.17475 −0.587376 − 0.809314i \(-0.699839\pi\)
−0.587376 + 0.809314i \(0.699839\pi\)
\(30\) 0 0
\(31\) 9.61803 1.72745 0.863725 − 0.503964i \(-0.168125\pi\)
0.863725 + 0.503964i \(0.168125\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.47214 0.766965
\(35\) 3.52786 0.596318
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 4.47214 0.725476
\(39\) 0 0
\(40\) −2.85410 −0.451273
\(41\) −7.38197 −1.15287 −0.576435 − 0.817143i \(-0.695556\pi\)
−0.576435 + 0.817143i \(0.695556\pi\)
\(42\) 0 0
\(43\) −0.763932 −0.116499 −0.0582493 − 0.998302i \(-0.518552\pi\)
−0.0582493 + 0.998302i \(0.518552\pi\)
\(44\) 3.61803 0.545439
\(45\) 0 0
\(46\) −3.85410 −0.568256
\(47\) −3.23607 −0.472029 −0.236015 − 0.971750i \(-0.575841\pi\)
−0.236015 + 0.971750i \(0.575841\pi\)
\(48\) 0 0
\(49\) −5.47214 −0.781734
\(50\) −3.14590 −0.444897
\(51\) 0 0
\(52\) 3.85410 0.534468
\(53\) 8.47214 1.16374 0.581869 − 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) 10.3262 1.39239
\(56\) −1.23607 −0.165177
\(57\) 0 0
\(58\) 6.32624 0.830676
\(59\) 9.23607 1.20243 0.601217 − 0.799086i \(-0.294683\pi\)
0.601217 + 0.799086i \(0.294683\pi\)
\(60\) 0 0
\(61\) 8.38197 1.07320 0.536600 − 0.843836i \(-0.319708\pi\)
0.536600 + 0.843836i \(0.319708\pi\)
\(62\) −9.61803 −1.22149
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 11.0000 1.36438
\(66\) 0 0
\(67\) −10.0902 −1.23271 −0.616355 − 0.787468i \(-0.711391\pi\)
−0.616355 + 0.787468i \(0.711391\pi\)
\(68\) −4.47214 −0.542326
\(69\) 0 0
\(70\) −3.52786 −0.421660
\(71\) 14.9443 1.77356 0.886779 − 0.462193i \(-0.152937\pi\)
0.886779 + 0.462193i \(0.152937\pi\)
\(72\) 0 0
\(73\) −4.09017 −0.478718 −0.239359 − 0.970931i \(-0.576937\pi\)
−0.239359 + 0.970931i \(0.576937\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −4.47214 −0.512989
\(77\) 4.47214 0.509647
\(78\) 0 0
\(79\) 11.5623 1.30086 0.650431 − 0.759566i \(-0.274589\pi\)
0.650431 + 0.759566i \(0.274589\pi\)
\(80\) 2.85410 0.319098
\(81\) 0 0
\(82\) 7.38197 0.815202
\(83\) 5.52786 0.606762 0.303381 − 0.952869i \(-0.401885\pi\)
0.303381 + 0.952869i \(0.401885\pi\)
\(84\) 0 0
\(85\) −12.7639 −1.38444
\(86\) 0.763932 0.0823769
\(87\) 0 0
\(88\) −3.61803 −0.385684
\(89\) 10.4721 1.11004 0.555022 − 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) 4.76393 0.499396
\(92\) 3.85410 0.401818
\(93\) 0 0
\(94\) 3.23607 0.333775
\(95\) −12.7639 −1.30955
\(96\) 0 0
\(97\) 8.47214 0.860215 0.430108 − 0.902778i \(-0.358476\pi\)
0.430108 + 0.902778i \(0.358476\pi\)
\(98\) 5.47214 0.552769
\(99\) 0 0
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 666.2.a.i.1.2 2
3.2 odd 2 74.2.a.b.1.2 ✓ 2
4.3 odd 2 5328.2.a.bc.1.2 2
12.11 even 2 592.2.a.g.1.1 2
15.2 even 4 1850.2.b.j.149.3 4
15.8 even 4 1850.2.b.j.149.2 4
15.14 odd 2 1850.2.a.t.1.1 2
21.20 even 2 3626.2.a.s.1.1 2
24.5 odd 2 2368.2.a.y.1.1 2
24.11 even 2 2368.2.a.u.1.2 2
33.32 even 2 8954.2.a.j.1.2 2
111.110 odd 2 2738.2.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.2 ✓ 2 3.2 odd 2
592.2.a.g.1.1 2 12.11 even 2
666.2.a.i.1.2 2 1.1 even 1 trivial
1850.2.a.t.1.1 2 15.14 odd 2
1850.2.b.j.149.2 4 15.8 even 4
1850.2.b.j.149.3 4 15.2 even 4
2368.2.a.u.1.2 2 24.11 even 2
2368.2.a.y.1.1 2 24.5 odd 2
2738.2.a.g.1.2 2 111.110 odd 2
3626.2.a.s.1.1 2 21.20 even 2
5328.2.a.bc.1.2 2 4.3 odd 2
8954.2.a.j.1.2 2 33.32 even 2