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: \(1\)
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 1.1
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.73205 q^{5} -2.73205 q^{6} +4.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.73205 q^{10} -1.26795 q^{11} -2.73205 q^{12} -6.00000 q^{13} +4.00000 q^{14} -4.73205 q^{15} +1.00000 q^{16} -1.73205 q^{17} +4.46410 q^{18} -4.73205 q^{19} +1.73205 q^{20} -10.9282 q^{21} -1.26795 q^{22} -4.73205 q^{23} -2.73205 q^{24} -2.00000 q^{25} -6.00000 q^{26} -4.00000 q^{27} +4.00000 q^{28} -7.73205 q^{29} -4.73205 q^{30} -4.73205 q^{31} +1.00000 q^{32} +3.46410 q^{33} -1.73205 q^{34} +6.92820 q^{35} +4.46410 q^{36} -4.73205 q^{38} +16.3923 q^{39} +1.73205 q^{40} +6.46410 q^{41} -10.9282 q^{42} -2.53590 q^{43} -1.26795 q^{44} +7.73205 q^{45} -4.73205 q^{46} -5.66025 q^{47} -2.73205 q^{48} +9.00000 q^{49} -2.00000 q^{50} +4.73205 q^{51} -6.00000 q^{52} -9.46410 q^{53} -4.00000 q^{54} -2.19615 q^{55} +4.00000 q^{56} +12.9282 q^{57} -7.73205 q^{58} +9.46410 q^{59} -4.73205 q^{60} +2.66025 q^{61} -4.73205 q^{62} +17.8564 q^{63} +1.00000 q^{64} -10.3923 q^{65} +3.46410 q^{66} +4.19615 q^{67} -1.73205 q^{68} +12.9282 q^{69} +6.92820 q^{70} +9.46410 q^{71} +4.46410 q^{72} -8.39230 q^{73} +5.46410 q^{75} -4.73205 q^{76} -5.07180 q^{77} +16.3923 q^{78} +2.19615 q^{79} +1.73205 q^{80} -2.46410 q^{81} +6.46410 q^{82} -5.66025 q^{83} -10.9282 q^{84} -3.00000 q^{85} -2.53590 q^{86} +21.1244 q^{87} -1.26795 q^{88} -5.19615 q^{89} +7.73205 q^{90} -24.0000 q^{91} -4.73205 q^{92} +12.9282 q^{93} -5.66025 q^{94} -8.19615 q^{95} -2.73205 q^{96} +5.19615 q^{97} +9.00000 q^{98} -5.66025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 8 q^{7} + 2 q^{8} + 2 q^{9} - 6 q^{11} - 2 q^{12} - 12 q^{13} + 8 q^{14} - 6 q^{15} + 2 q^{16} + 2 q^{18} - 6 q^{19} - 8 q^{21} - 6 q^{22} - 6 q^{23} - 2 q^{24}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) −2.73205 −1.57735 −0.788675 − 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.73205 0.774597 0.387298 − 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) −2.73205 −1.11536
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.46410 1.48803
\(10\) 1.73205 0.547723
\(11\) −1.26795 −0.382301 −0.191151 − 0.981561i \(-0.561222\pi\)
−0.191151 + 0.981561i \(0.561222\pi\)
\(12\) −2.73205 −0.788675
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 4.00000 1.06904
\(15\) −4.73205 −1.22181
\(16\) 1.00000 0.250000
\(17\) −1.73205 −0.420084 −0.210042 − 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 4.46410 1.05220
\(19\) −4.73205 −1.08561 −0.542803 − 0.839860i \(-0.682637\pi\)
−0.542803 + 0.839860i \(0.682637\pi\)
\(20\) 1.73205 0.387298
\(21\) −10.9282 −2.38473
\(22\) −1.26795 −0.270328
\(23\) −4.73205 −0.986701 −0.493350 − 0.869831i \(-0.664228\pi\)
−0.493350 + 0.869831i \(0.664228\pi\)
\(24\) −2.73205 −0.557678
\(25\) −2.00000 −0.400000
\(26\) −6.00000 −1.17670
\(27\) −4.00000 −0.769800
\(28\) 4.00000 0.755929
\(29\) −7.73205 −1.43581 −0.717903 − 0.696143i \(-0.754898\pi\)
−0.717903 + 0.696143i \(0.754898\pi\)
\(30\) −4.73205 −0.863950
\(31\) −4.73205 −0.849901 −0.424951 − 0.905216i \(-0.639709\pi\)
−0.424951 + 0.905216i \(0.639709\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.46410 0.603023
\(34\) −1.73205 −0.297044
\(35\) 6.92820 1.17108
\(36\) 4.46410 0.744017
\(37\) 0 0
\(38\) −4.73205 −0.767640
\(39\) 16.3923 2.62487
\(40\) 1.73205 0.273861
\(41\) 6.46410 1.00952 0.504762 − 0.863259i \(-0.331580\pi\)
0.504762 + 0.863259i \(0.331580\pi\)
\(42\) −10.9282 −1.68626
\(43\) −2.53590 −0.386721 −0.193360 − 0.981128i \(-0.561939\pi\)
−0.193360 + 0.981128i \(0.561939\pi\)
\(44\) −1.26795 −0.191151
\(45\) 7.73205 1.15263
\(46\) −4.73205 −0.697703
\(47\) −5.66025 −0.825633 −0.412816 − 0.910814i \(-0.635455\pi\)
−0.412816 + 0.910814i \(0.635455\pi\)
\(48\) −2.73205 −0.394338
\(49\) 9.00000 1.28571
\(50\) −2.00000 −0.282843
\(51\) 4.73205 0.662620
\(52\) −6.00000 −0.832050
\(53\) −9.46410 −1.29999 −0.649997 − 0.759937i \(-0.725230\pi\)
−0.649997 + 0.759937i \(0.725230\pi\)
\(54\) −4.00000 −0.544331
\(55\) −2.19615 −0.296129
\(56\) 4.00000 0.534522
\(57\) 12.9282 1.71238
\(58\) −7.73205 −1.01527
\(59\) 9.46410 1.23212 0.616061 − 0.787699i \(-0.288728\pi\)
0.616061 + 0.787699i \(0.288728\pi\)
\(60\) −4.73205 −0.610905
\(61\) 2.66025 0.340611 0.170305 − 0.985391i \(-0.445525\pi\)
0.170305 + 0.985391i \(0.445525\pi\)
\(62\) −4.73205 −0.600971
\(63\) 17.8564 2.24970
\(64\) 1.00000 0.125000
\(65\) −10.3923 −1.28901
\(66\) 3.46410 0.426401
\(67\) 4.19615 0.512642 0.256321 − 0.966592i \(-0.417490\pi\)
0.256321 + 0.966592i \(0.417490\pi\)
\(68\) −1.73205 −0.210042
\(69\) 12.9282 1.55637
\(70\) 6.92820 0.828079
\(71\) 9.46410 1.12318 0.561591 − 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) 4.46410 0.526099
\(73\) −8.39230 −0.982245 −0.491122 − 0.871091i \(-0.663413\pi\)
−0.491122 + 0.871091i \(0.663413\pi\)
\(74\) 0 0
\(75\) 5.46410 0.630940
\(76\) −4.73205 −0.542803
\(77\) −5.07180 −0.577985
\(78\) 16.3923 1.85606
\(79\) 2.19615 0.247086 0.123543 − 0.992339i \(-0.460574\pi\)
0.123543 + 0.992339i \(0.460574\pi\)
\(80\) 1.73205 0.193649
\(81\) −2.46410 −0.273789
\(82\) 6.46410 0.713841
\(83\) −5.66025 −0.621294 −0.310647 − 0.950525i \(-0.600546\pi\)
−0.310647 + 0.950525i \(0.600546\pi\)
\(84\) −10.9282 −1.19236
\(85\) −3.00000 −0.325396
\(86\) −2.53590 −0.273453
\(87\) 21.1244 2.26477
\(88\) −1.26795 −0.135164
\(89\) −5.19615 −0.550791 −0.275396 − 0.961331i \(-0.588809\pi\)
−0.275396 + 0.961331i \(0.588809\pi\)
\(90\) 7.73205 0.815030
\(91\) −24.0000 −2.51588
\(92\) −4.73205 −0.493350
\(93\) 12.9282 1.34059
\(94\) −5.66025 −0.583811
\(95\) −8.19615 −0.840907
\(96\) −2.73205 −0.278839
\(97\) 5.19615 0.527589 0.263795 − 0.964579i \(-0.415026\pi\)
0.263795 + 0.964579i \(0.415026\pi\)
\(98\) 9.00000 0.909137
\(99\) −5.66025 −0.568877
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.j.1.1 2
37.23 odd 12 74.2.e.a.11.1 ✓ 4
37.29 odd 12 74.2.e.a.27.1 yes 4
37.36 even 2 2738.2.a.f.1.1 2
111.23 even 12 666.2.s.e.307.2 4
111.29 even 12 666.2.s.e.397.2 4
148.23 even 12 592.2.w.d.529.2 4
148.103 even 12 592.2.w.d.545.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.a.11.1 ✓ 4 37.23 odd 12
74.2.e.a.27.1 yes 4 37.29 odd 12
592.2.w.d.529.2 4 148.23 even 12
592.2.w.d.545.2 4 148.103 even 12
666.2.s.e.307.2 4 111.23 even 12
666.2.s.e.397.2 4 111.29 even 12
2738.2.a.f.1.1 2 37.36 even 2
2738.2.a.j.1.1 2 1.1 even 1 trivial