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,-4] 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: \(\Q(\zeta_{12})^+\)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} -1.73205 q^{5} +0.732051 q^{6} -2.00000 q^{7} +1.00000 q^{8} -2.46410 q^{9} -1.73205 q^{10} +1.26795 q^{11} +0.732051 q^{12} +3.46410 q^{13} -2.00000 q^{14} -1.26795 q^{15} +1.00000 q^{16} -4.26795 q^{17} -2.46410 q^{18} +4.73205 q^{19} -1.73205 q^{20} -1.46410 q^{21} +1.26795 q^{22} -1.26795 q^{23} +0.732051 q^{24} -2.00000 q^{25} +3.46410 q^{26} -4.00000 q^{27} -2.00000 q^{28} -8.66025 q^{29} -1.26795 q^{30} -4.73205 q^{31} +1.00000 q^{32} +0.928203 q^{33} -4.26795 q^{34} +3.46410 q^{35} -2.46410 q^{36} +4.73205 q^{38} +2.53590 q^{39} -1.73205 q^{40} +3.92820 q^{41} -1.46410 q^{42} -12.9282 q^{43} +1.26795 q^{44} +4.26795 q^{45} -1.26795 q^{46} +1.26795 q^{47} +0.732051 q^{48} -3.00000 q^{49} -2.00000 q^{50} -3.12436 q^{51} +3.46410 q^{52} +9.46410 q^{53} -4.00000 q^{54} -2.19615 q^{55} -2.00000 q^{56} +3.46410 q^{57} -8.66025 q^{58} -9.46410 q^{59} -1.26795 q^{60} +1.73205 q^{61} -4.73205 q^{62} +4.92820 q^{63} +1.00000 q^{64} -6.00000 q^{65} +0.928203 q^{66} -0.196152 q^{67} -4.26795 q^{68} -0.928203 q^{69} +3.46410 q^{70} -3.46410 q^{71} -2.46410 q^{72} -4.00000 q^{73} -1.46410 q^{75} +4.73205 q^{76} -2.53590 q^{77} +2.53590 q^{78} -16.7321 q^{79} -1.73205 q^{80} +4.46410 q^{81} +3.92820 q^{82} +11.6603 q^{83} -1.46410 q^{84} +7.39230 q^{85} -12.9282 q^{86} -6.33975 q^{87} +1.26795 q^{88} -17.1962 q^{89} +4.26795 q^{90} -6.92820 q^{91} -1.26795 q^{92} -3.46410 q^{93} +1.26795 q^{94} -8.19615 q^{95} +0.732051 q^{96} -4.26795 q^{97} -3.00000 q^{98} -3.12436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9} + 6 q^{11} - 2 q^{12} - 4 q^{14} - 6 q^{15} + 2 q^{16} - 12 q^{17} + 2 q^{18} + 6 q^{19} + 4 q^{21} + 6 q^{22} - 6 q^{23} - 2 q^{24}+ \cdots + 18 q^{99}+O(q^{100}) \) 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 \(-0.626592\pi\)
−0.387298 + 0.921954i \(0.626592\pi\)
\(6\) 0.732051 0.298858
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.46410 −0.821367
\(10\) −1.73205 −0.547723
\(11\) 1.26795 0.382301 0.191151 − 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0.732051 0.211325
\(13\) 3.46410 0.960769 0.480384 − 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) −2.00000 −0.534522
\(15\) −1.26795 −0.327383
\(16\) 1.00000 0.250000
\(17\) −4.26795 −1.03513 −0.517565 − 0.855644i \(-0.673161\pi\)
−0.517565 + 0.855644i \(0.673161\pi\)
\(18\) −2.46410 −0.580794
\(19\) 4.73205 1.08561 0.542803 − 0.839860i \(-0.317363\pi\)
0.542803 + 0.839860i \(0.317363\pi\)
\(20\) −1.73205 −0.387298
\(21\) −1.46410 −0.319493
\(22\) 1.26795 0.270328
\(23\) −1.26795 −0.264386 −0.132193 − 0.991224i \(-0.542202\pi\)
−0.132193 + 0.991224i \(0.542202\pi\)
\(24\) 0.732051 0.149429
\(25\) −2.00000 −0.400000
\(26\) 3.46410 0.679366
\(27\) −4.00000 −0.769800
\(28\) −2.00000 −0.377964
\(29\) −8.66025 −1.60817 −0.804084 − 0.594515i \(-0.797344\pi\)
−0.804084 + 0.594515i \(0.797344\pi\)
\(30\) −1.26795 −0.231495
\(31\) −4.73205 −0.849901 −0.424951 − 0.905216i \(-0.639709\pi\)
−0.424951 + 0.905216i \(0.639709\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.928203 0.161579
\(34\) −4.26795 −0.731947
\(35\) 3.46410 0.585540
\(36\) −2.46410 −0.410684
\(37\) 0 0
\(38\) 4.73205 0.767640
\(39\) 2.53590 0.406069
\(40\) −1.73205 −0.273861
\(41\) 3.92820 0.613482 0.306741 − 0.951793i \(-0.400761\pi\)
0.306741 + 0.951793i \(0.400761\pi\)
\(42\) −1.46410 −0.225916
\(43\) −12.9282 −1.97153 −0.985766 − 0.168122i \(-0.946230\pi\)
−0.985766 + 0.168122i \(0.946230\pi\)
\(44\) 1.26795 0.191151
\(45\) 4.26795 0.636228
\(46\) −1.26795 −0.186949
\(47\) 1.26795 0.184949 0.0924747 − 0.995715i \(-0.470522\pi\)
0.0924747 + 0.995715i \(0.470522\pi\)
\(48\) 0.732051 0.105662
\(49\) −3.00000 −0.428571
\(50\) −2.00000 −0.282843
\(51\) −3.12436 −0.437497
\(52\) 3.46410 0.480384
\(53\) 9.46410 1.29999 0.649997 − 0.759937i \(-0.274770\pi\)
0.649997 + 0.759937i \(0.274770\pi\)
\(54\) −4.00000 −0.544331
\(55\) −2.19615 −0.296129
\(56\) −2.00000 −0.267261
\(57\) 3.46410 0.458831
\(58\) −8.66025 −1.13715
\(59\) −9.46410 −1.23212 −0.616061 − 0.787699i \(-0.711272\pi\)
−0.616061 + 0.787699i \(0.711272\pi\)
\(60\) −1.26795 −0.163692
\(61\) 1.73205 0.221766 0.110883 − 0.993833i \(-0.464632\pi\)
0.110883 + 0.993833i \(0.464632\pi\)
\(62\) −4.73205 −0.600971
\(63\) 4.92820 0.620895
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 0.928203 0.114254
\(67\) −0.196152 −0.0239638 −0.0119819 − 0.999928i \(-0.503814\pi\)
−0.0119819 + 0.999928i \(0.503814\pi\)
\(68\) −4.26795 −0.517565
\(69\) −0.928203 −0.111743
\(70\) 3.46410 0.414039
\(71\) −3.46410 −0.411113 −0.205557 − 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) −2.46410 −0.290397
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) −1.46410 −0.169060
\(76\) 4.73205 0.542803
\(77\) −2.53590 −0.288992
\(78\) 2.53590 0.287134
\(79\) −16.7321 −1.88250 −0.941251 − 0.337707i \(-0.890349\pi\)
−0.941251 + 0.337707i \(0.890349\pi\)
\(80\) −1.73205 −0.193649
\(81\) 4.46410 0.496011
\(82\) 3.92820 0.433797
\(83\) 11.6603 1.27988 0.639940 − 0.768425i \(-0.278959\pi\)
0.639940 + 0.768425i \(0.278959\pi\)
\(84\) −1.46410 −0.159747
\(85\) 7.39230 0.801808
\(86\) −12.9282 −1.39408
\(87\) −6.33975 −0.679692
\(88\) 1.26795 0.135164
\(89\) −17.1962 −1.82279 −0.911394 − 0.411534i \(-0.864993\pi\)
−0.911394 + 0.411534i \(0.864993\pi\)
\(90\) 4.26795 0.449881
\(91\) −6.92820 −0.726273
\(92\) −1.26795 −0.132193
\(93\) −3.46410 −0.359211
\(94\) 1.26795 0.130779
\(95\) −8.19615 −0.840907
\(96\) 0.732051 0.0747146
\(97\) −4.26795 −0.433345 −0.216672 − 0.976244i \(-0.569520\pi\)
−0.216672 + 0.976244i \(0.569520\pi\)
\(98\) −3.00000 −0.303046
\(99\) −3.12436 −0.314010
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.i.1.2 2
37.23 odd 12 74.2.e.b.11.1 ✓ 4
37.29 odd 12 74.2.e.b.27.1 yes 4
37.36 even 2 2738.2.a.e.1.2 2
111.23 even 12 666.2.s.a.307.2 4
111.29 even 12 666.2.s.a.397.2 4
148.23 even 12 592.2.w.e.529.1 4
148.103 even 12 592.2.w.e.545.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.1 ✓ 4 37.23 odd 12
74.2.e.b.27.1 yes 4 37.29 odd 12
592.2.w.e.529.1 4 148.23 even 12
592.2.w.e.545.1 4 148.103 even 12
666.2.s.a.307.2 4 111.23 even 12
666.2.s.a.397.2 4 111.29 even 12
2738.2.a.e.1.2 2 37.36 even 2
2738.2.a.i.1.2 2 1.1 even 1 trivial