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,-2,2,0,2,8] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) 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 \(1.73205\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.73205 q^{5} -0.732051 q^{6} +4.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} -1.73205 q^{10} -4.73205 q^{11} +0.732051 q^{12} +6.00000 q^{13} -4.00000 q^{14} +1.26795 q^{15} +1.00000 q^{16} -1.73205 q^{17} +2.46410 q^{18} +1.26795 q^{19} +1.73205 q^{20} +2.92820 q^{21} +4.73205 q^{22} +1.26795 q^{23} -0.732051 q^{24} -2.00000 q^{25} -6.00000 q^{26} -4.00000 q^{27} +4.00000 q^{28} +4.26795 q^{29} -1.26795 q^{30} +1.26795 q^{31} -1.00000 q^{32} -3.46410 q^{33} +1.73205 q^{34} +6.92820 q^{35} -2.46410 q^{36} -1.26795 q^{38} +4.39230 q^{39} -1.73205 q^{40} -0.464102 q^{41} -2.92820 q^{42} +9.46410 q^{43} -4.73205 q^{44} -4.26795 q^{45} -1.26795 q^{46} +11.6603 q^{47} +0.732051 q^{48} +9.00000 q^{49} +2.00000 q^{50} -1.26795 q^{51} +6.00000 q^{52} -2.53590 q^{53} +4.00000 q^{54} -8.19615 q^{55} -4.00000 q^{56} +0.928203 q^{57} -4.26795 q^{58} -2.53590 q^{59} +1.26795 q^{60} +14.6603 q^{61} -1.26795 q^{62} -9.85641 q^{63} +1.00000 q^{64} +10.3923 q^{65} +3.46410 q^{66} -6.19615 q^{67} -1.73205 q^{68} +0.928203 q^{69} -6.92820 q^{70} +2.53590 q^{71} +2.46410 q^{72} +12.3923 q^{73} -1.46410 q^{75} +1.26795 q^{76} -18.9282 q^{77} -4.39230 q^{78} +8.19615 q^{79} +1.73205 q^{80} +4.46410 q^{81} +0.464102 q^{82} +11.6603 q^{83} +2.92820 q^{84} -3.00000 q^{85} -9.46410 q^{86} +3.12436 q^{87} +4.73205 q^{88} -5.19615 q^{89} +4.26795 q^{90} +24.0000 q^{91} +1.26795 q^{92} +0.928203 q^{93} -11.6603 q^{94} +2.19615 q^{95} -0.732051 q^{96} +5.19615 q^{97} -9.00000 q^{98} +11.6603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 8 q^{7} - 2 q^{8} + 2 q^{9} - 6 q^{11} - 2 q^{12} + 12 q^{13} - 8 q^{14} + 6 q^{15} + 2 q^{16} - 2 q^{18} + 6 q^{19} - 8 q^{21} + 6 q^{22} + 6 q^{23} + 2 q^{24}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) 0.732051 0.422650 0.211325 − 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.73205 0.774597 0.387298 − 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) −0.732051 −0.298858
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.46410 −0.821367
\(10\) −1.73205 −0.547723
\(11\) −4.73205 −1.42677 −0.713384 − 0.700774i \(-0.752838\pi\)
−0.713384 + 0.700774i \(0.752838\pi\)
\(12\) 0.732051 0.211325
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 1.26795 0.327383
\(16\) 1.00000 0.250000
\(17\) −1.73205 −0.420084 −0.210042 − 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 2.46410 0.580794
\(19\) 1.26795 0.290887 0.145444 − 0.989367i \(-0.453539\pi\)
0.145444 + 0.989367i \(0.453539\pi\)
\(20\) 1.73205 0.387298
\(21\) 2.92820 0.638986
\(22\) 4.73205 1.00888
\(23\) 1.26795 0.264386 0.132193 − 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(24\) −0.732051 −0.149429
\(25\) −2.00000 −0.400000
\(26\) −6.00000 −1.17670
\(27\) −4.00000 −0.769800
\(28\) 4.00000 0.755929
\(29\) 4.26795 0.792538 0.396269 − 0.918134i \(-0.370305\pi\)
0.396269 + 0.918134i \(0.370305\pi\)
\(30\) −1.26795 −0.231495
\(31\) 1.26795 0.227730 0.113865 − 0.993496i \(-0.463677\pi\)
0.113865 + 0.993496i \(0.463677\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.46410 −0.603023
\(34\) 1.73205 0.297044
\(35\) 6.92820 1.17108
\(36\) −2.46410 −0.410684
\(37\) 0 0
\(38\) −1.26795 −0.205689
\(39\) 4.39230 0.703332
\(40\) −1.73205 −0.273861
\(41\) −0.464102 −0.0724805 −0.0362402 − 0.999343i \(-0.511538\pi\)
−0.0362402 + 0.999343i \(0.511538\pi\)
\(42\) −2.92820 −0.451832
\(43\) 9.46410 1.44326 0.721631 − 0.692278i \(-0.243393\pi\)
0.721631 + 0.692278i \(0.243393\pi\)
\(44\) −4.73205 −0.713384
\(45\) −4.26795 −0.636228
\(46\) −1.26795 −0.186949
\(47\) 11.6603 1.70082 0.850411 − 0.526118i \(-0.176353\pi\)
0.850411 + 0.526118i \(0.176353\pi\)
\(48\) 0.732051 0.105662
\(49\) 9.00000 1.28571
\(50\) 2.00000 0.282843
\(51\) −1.26795 −0.177548
\(52\) 6.00000 0.832050
\(53\) −2.53590 −0.348332 −0.174166 − 0.984716i \(-0.555723\pi\)
−0.174166 + 0.984716i \(0.555723\pi\)
\(54\) 4.00000 0.544331
\(55\) −8.19615 −1.10517
\(56\) −4.00000 −0.534522
\(57\) 0.928203 0.122944
\(58\) −4.26795 −0.560409
\(59\) −2.53590 −0.330146 −0.165073 − 0.986281i \(-0.552786\pi\)
−0.165073 + 0.986281i \(0.552786\pi\)
\(60\) 1.26795 0.163692
\(61\) 14.6603 1.87705 0.938527 − 0.345207i \(-0.112191\pi\)
0.938527 + 0.345207i \(0.112191\pi\)
\(62\) −1.26795 −0.161030
\(63\) −9.85641 −1.24179
\(64\) 1.00000 0.125000
\(65\) 10.3923 1.28901
\(66\) 3.46410 0.426401
\(67\) −6.19615 −0.756980 −0.378490 − 0.925605i \(-0.623557\pi\)
−0.378490 + 0.925605i \(0.623557\pi\)
\(68\) −1.73205 −0.210042
\(69\) 0.928203 0.111743
\(70\) −6.92820 −0.828079
\(71\) 2.53590 0.300956 0.150478 − 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 2.46410 0.290397
\(73\) 12.3923 1.45041 0.725205 − 0.688533i \(-0.241745\pi\)
0.725205 + 0.688533i \(0.241745\pi\)
\(74\) 0 0
\(75\) −1.46410 −0.169060
\(76\) 1.26795 0.145444
\(77\) −18.9282 −2.15707
\(78\) −4.39230 −0.497331
\(79\) 8.19615 0.922139 0.461070 − 0.887364i \(-0.347466\pi\)
0.461070 + 0.887364i \(0.347466\pi\)
\(80\) 1.73205 0.193649
\(81\) 4.46410 0.496011
\(82\) 0.464102 0.0512514
\(83\) 11.6603 1.27988 0.639940 − 0.768425i \(-0.278959\pi\)
0.639940 + 0.768425i \(0.278959\pi\)
\(84\) 2.92820 0.319493
\(85\) −3.00000 −0.325396
\(86\) −9.46410 −1.02054
\(87\) 3.12436 0.334966
\(88\) 4.73205 0.504438
\(89\) −5.19615 −0.550791 −0.275396 − 0.961331i \(-0.588809\pi\)
−0.275396 + 0.961331i \(0.588809\pi\)
\(90\) 4.26795 0.449881
\(91\) 24.0000 2.51588
\(92\) 1.26795 0.132193
\(93\) 0.928203 0.0962502
\(94\) −11.6603 −1.20266
\(95\) 2.19615 0.225320
\(96\) −0.732051 −0.0747146
\(97\) 5.19615 0.527589 0.263795 − 0.964579i \(-0.415026\pi\)
0.263795 + 0.964579i \(0.415026\pi\)
\(98\) −9.00000 −0.909137
\(99\) 11.6603 1.17190
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.f.1.2 2
37.23 odd 12 74.2.e.a.11.2 ✓ 4
37.29 odd 12 74.2.e.a.27.2 yes 4
37.36 even 2 2738.2.a.j.1.2 2
111.23 even 12 666.2.s.e.307.1 4
111.29 even 12 666.2.s.e.397.1 4
148.23 even 12 592.2.w.d.529.1 4
148.103 even 12 592.2.w.d.545.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.a.11.2 ✓ 4 37.23 odd 12
74.2.e.a.27.2 yes 4 37.29 odd 12
592.2.w.d.529.1 4 148.23 even 12
592.2.w.d.545.1 4 148.103 even 12
666.2.s.e.307.1 4 111.23 even 12
666.2.s.e.397.1 4 111.29 even 12
2738.2.a.f.1.2 2 1.1 even 1 trivial
2738.2.a.j.1.2 2 37.36 even 2