Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2738,2,Mod(1,2738)] 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("2738.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2738, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-1,2,-1,1,-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: \(21.8630400734\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
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 \(\chi\) \(=\) 2738.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} +2.85410 q^{5} -0.618034 q^{6} +1.23607 q^{7} -1.00000 q^{8} -2.61803 q^{9} -2.85410 q^{10} -3.61803 q^{11} +0.618034 q^{12} -3.85410 q^{13} -1.23607 q^{14} +1.76393 q^{15} +1.00000 q^{16} -4.47214 q^{17} +2.61803 q^{18} +4.47214 q^{19} +2.85410 q^{20} +0.763932 q^{21} +3.61803 q^{22} +3.85410 q^{23} -0.618034 q^{24} +3.14590 q^{25} +3.85410 q^{26} -3.47214 q^{27} +1.23607 q^{28} -6.32624 q^{29} -1.76393 q^{30} -9.61803 q^{31} -1.00000 q^{32} -2.23607 q^{33} +4.47214 q^{34} +3.52786 q^{35} -2.61803 q^{36} -4.47214 q^{38} -2.38197 q^{39} -2.85410 q^{40} +7.38197 q^{41} -0.763932 q^{42} +0.763932 q^{43} -3.61803 q^{44} -7.47214 q^{45} -3.85410 q^{46} +3.23607 q^{47} +0.618034 q^{48} -5.47214 q^{49} -3.14590 q^{50} -2.76393 q^{51} -3.85410 q^{52} -8.47214 q^{53} +3.47214 q^{54} -10.3262 q^{55} -1.23607 q^{56} +2.76393 q^{57} +6.32624 q^{58} +9.23607 q^{59} +1.76393 q^{60} -8.38197 q^{61} +9.61803 q^{62} -3.23607 q^{63} +1.00000 q^{64} -11.0000 q^{65} +2.23607 q^{66} -10.0902 q^{67} -4.47214 q^{68} +2.38197 q^{69} -3.52786 q^{70} -14.9443 q^{71} +2.61803 q^{72} -4.09017 q^{73} +1.94427 q^{75} +4.47214 q^{76} -4.47214 q^{77} +2.38197 q^{78} -11.5623 q^{79} +2.85410 q^{80} +5.70820 q^{81} -7.38197 q^{82} -5.52786 q^{83} +0.763932 q^{84} -12.7639 q^{85} -0.763932 q^{86} -3.90983 q^{87} +3.61803 q^{88} +10.4721 q^{89} +7.47214 q^{90} -4.76393 q^{91} +3.85410 q^{92} -5.94427 q^{93} -3.23607 q^{94} +12.7639 q^{95} -0.618034 q^{96} -8.47214 q^{97} +5.47214 q^{98} +9.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9} + q^{10} - 5 q^{11} - q^{12} - q^{13} + 2 q^{14} + 8 q^{15} + 2 q^{16} + 3 q^{18} - q^{20} + 6 q^{21} + 5 q^{22} + q^{23}+ \cdots + 10 q^{99}+O(q^{100}) \) 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
\(3\) 0.618034 0.356822 0.178411 − 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.85410 1.27639 0.638197 − 0.769873i \(-0.279681\pi\)
0.638197 + 0.769873i \(0.279681\pi\)
\(6\) −0.618034 −0.252311
\(7\) 1.23607 0.467190 0.233595 − 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.61803 −0.872678
\(10\) −2.85410 −0.902546
\(11\) −3.61803 −1.09088 −0.545439 − 0.838150i \(-0.683637\pi\)
−0.545439 + 0.838150i \(0.683637\pi\)
\(12\) 0.618034 0.178411
\(13\) −3.85410 −1.06894 −0.534468 − 0.845189i \(-0.679488\pi\)
−0.534468 + 0.845189i \(0.679488\pi\)
\(14\) −1.23607 −0.330353
\(15\) 1.76393 0.455445
\(16\) 1.00000 0.250000
\(17\) −4.47214 −1.08465 −0.542326 − 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 2.61803 0.617077
\(19\) 4.47214 1.02598 0.512989 − 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 2.85410 0.638197
\(21\) 0.763932 0.166704
\(22\) 3.61803 0.771367
\(23\) 3.85410 0.803636 0.401818 − 0.915720i \(-0.368378\pi\)
0.401818 + 0.915720i \(0.368378\pi\)
\(24\) −0.618034 −0.126156
\(25\) 3.14590 0.629180
\(26\) 3.85410 0.755852
\(27\) −3.47214 −0.668213
\(28\) 1.23607 0.233595
\(29\) −6.32624 −1.17475 −0.587376 − 0.809314i \(-0.699839\pi\)
−0.587376 + 0.809314i \(0.699839\pi\)
\(30\) −1.76393 −0.322048
\(31\) −9.61803 −1.72745 −0.863725 − 0.503964i \(-0.831875\pi\)
−0.863725 + 0.503964i \(0.831875\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.23607 −0.389249
\(34\) 4.47214 0.766965
\(35\) 3.52786 0.596318
\(36\) −2.61803 −0.436339
\(37\) 0 0
\(38\) −4.47214 −0.725476
\(39\) −2.38197 −0.381420
\(40\) −2.85410 −0.451273
\(41\) 7.38197 1.15287 0.576435 − 0.817143i \(-0.304444\pi\)
0.576435 + 0.817143i \(0.304444\pi\)
\(42\) −0.763932 −0.117877
\(43\) 0.763932 0.116499 0.0582493 − 0.998302i \(-0.481448\pi\)
0.0582493 + 0.998302i \(0.481448\pi\)
\(44\) −3.61803 −0.545439
\(45\) −7.47214 −1.11388
\(46\) −3.85410 −0.568256
\(47\) 3.23607 0.472029 0.236015 − 0.971750i \(-0.424159\pi\)
0.236015 + 0.971750i \(0.424159\pi\)
\(48\) 0.618034 0.0892055
\(49\) −5.47214 −0.781734
\(50\) −3.14590 −0.444897
\(51\) −2.76393 −0.387028
\(52\) −3.85410 −0.534468
\(53\) −8.47214 −1.16374 −0.581869 − 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(54\) 3.47214 0.472498
\(55\) −10.3262 −1.39239
\(56\) −1.23607 −0.165177
\(57\) 2.76393 0.366092
\(58\) 6.32624 0.830676
\(59\) 9.23607 1.20243 0.601217 − 0.799086i \(-0.294683\pi\)
0.601217 + 0.799086i \(0.294683\pi\)
\(60\) 1.76393 0.227723
\(61\) −8.38197 −1.07320 −0.536600 − 0.843836i \(-0.680292\pi\)
−0.536600 + 0.843836i \(0.680292\pi\)
\(62\) 9.61803 1.22149
\(63\) −3.23607 −0.407706
\(64\) 1.00000 0.125000
\(65\) −11.0000 −1.36438
\(66\) 2.23607 0.275241
\(67\) −10.0902 −1.23271 −0.616355 − 0.787468i \(-0.711391\pi\)
−0.616355 + 0.787468i \(0.711391\pi\)
\(68\) −4.47214 −0.542326
\(69\) 2.38197 0.286755
\(70\) −3.52786 −0.421660
\(71\) −14.9443 −1.77356 −0.886779 − 0.462193i \(-0.847063\pi\)
−0.886779 + 0.462193i \(0.847063\pi\)
\(72\) 2.61803 0.308538
\(73\) −4.09017 −0.478718 −0.239359 − 0.970931i \(-0.576937\pi\)
−0.239359 + 0.970931i \(0.576937\pi\)
\(74\) 0 0
\(75\) 1.94427 0.224505
\(76\) 4.47214 0.512989
\(77\) −4.47214 −0.509647
\(78\) 2.38197 0.269705
\(79\) −11.5623 −1.30086 −0.650431 − 0.759566i \(-0.725411\pi\)
−0.650431 + 0.759566i \(0.725411\pi\)
\(80\) 2.85410 0.319098
\(81\) 5.70820 0.634245
\(82\) −7.38197 −0.815202
\(83\) −5.52786 −0.606762 −0.303381 − 0.952869i \(-0.598115\pi\)
−0.303381 + 0.952869i \(0.598115\pi\)
\(84\) 0.763932 0.0833518
\(85\) −12.7639 −1.38444
\(86\) −0.763932 −0.0823769
\(87\) −3.90983 −0.419178
\(88\) 3.61803 0.385684
\(89\) 10.4721 1.11004 0.555022 − 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 7.47214 0.787632
\(91\) −4.76393 −0.499396
\(92\) 3.85410 0.401818
\(93\) −5.94427 −0.616392
\(94\) −3.23607 −0.333775
\(95\) 12.7639 1.30955
\(96\) −0.618034 −0.0630778
\(97\) −8.47214 −0.860215 −0.430108 − 0.902778i \(-0.641524\pi\)
−0.430108 + 0.902778i \(0.641524\pi\)
\(98\) 5.47214 0.552769
\(99\) 9.47214 0.951985
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 2738.2.a.g.1.2 2
37.36 even 2 74.2.a.b.1.2 ✓ 2
111.110 odd 2 666.2.a.i.1.2 2
148.147 odd 2 592.2.a.g.1.1 2
185.73 odd 4 1850.2.b.j.149.2 4
185.147 odd 4 1850.2.b.j.149.3 4
185.184 even 2 1850.2.a.t.1.1 2
259.258 odd 2 3626.2.a.s.1.1 2
296.147 odd 2 2368.2.a.u.1.2 2
296.221 even 2 2368.2.a.y.1.1 2
407.406 odd 2 8954.2.a.j.1.2 2
444.443 even 2 5328.2.a.bc.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.2 ✓ 2 37.36 even 2
592.2.a.g.1.1 2 148.147 odd 2
666.2.a.i.1.2 2 111.110 odd 2
1850.2.a.t.1.1 2 185.184 even 2
1850.2.b.j.149.2 4 185.73 odd 4
1850.2.b.j.149.3 4 185.147 odd 4
2368.2.a.u.1.2 2 296.147 odd 2
2368.2.a.y.1.1 2 296.221 even 2
2738.2.a.g.1.2 2 1.1 even 1 trivial
3626.2.a.s.1.1 2 259.258 odd 2
5328.2.a.bc.1.2 2 444.443 even 2
8954.2.a.j.1.2 2 407.406 odd 2