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 1.1
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.73205 q^{5} -2.73205 q^{6} -2.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.73205 q^{10} +4.73205 q^{11} -2.73205 q^{12} -3.46410 q^{13} -2.00000 q^{14} -4.73205 q^{15} +1.00000 q^{16} -7.73205 q^{17} +4.46410 q^{18} +1.26795 q^{19} +1.73205 q^{20} +5.46410 q^{21} +4.73205 q^{22} -4.73205 q^{23} -2.73205 q^{24} -2.00000 q^{25} -3.46410 q^{26} -4.00000 q^{27} -2.00000 q^{28} +8.66025 q^{29} -4.73205 q^{30} -1.26795 q^{31} +1.00000 q^{32} -12.9282 q^{33} -7.73205 q^{34} -3.46410 q^{35} +4.46410 q^{36} +1.26795 q^{38} +9.46410 q^{39} +1.73205 q^{40} -9.92820 q^{41} +5.46410 q^{42} +0.928203 q^{43} +4.73205 q^{44} +7.73205 q^{45} -4.73205 q^{46} +4.73205 q^{47} -2.73205 q^{48} -3.00000 q^{49} -2.00000 q^{50} +21.1244 q^{51} -3.46410 q^{52} +2.53590 q^{53} -4.00000 q^{54} +8.19615 q^{55} -2.00000 q^{56} -3.46410 q^{57} +8.66025 q^{58} -2.53590 q^{59} -4.73205 q^{60} -1.73205 q^{61} -1.26795 q^{62} -8.92820 q^{63} +1.00000 q^{64} -6.00000 q^{65} -12.9282 q^{66} +10.1962 q^{67} -7.73205 q^{68} +12.9282 q^{69} -3.46410 q^{70} +3.46410 q^{71} +4.46410 q^{72} -4.00000 q^{73} +5.46410 q^{75} +1.26795 q^{76} -9.46410 q^{77} +9.46410 q^{78} -13.2679 q^{79} +1.73205 q^{80} -2.46410 q^{81} -9.92820 q^{82} -5.66025 q^{83} +5.46410 q^{84} -13.3923 q^{85} +0.928203 q^{86} -23.6603 q^{87} +4.73205 q^{88} -6.80385 q^{89} +7.73205 q^{90} +6.92820 q^{91} -4.73205 q^{92} +3.46410 q^{93} +4.73205 q^{94} +2.19615 q^{95} -2.73205 q^{96} -7.73205 q^{97} -3.00000 q^{98} +21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9} + 6 q^{11} - 2 q^{12} - 4 q^{14} - 6 q^{15} + 2 q^{16} - 12 q^{17} + 2 q^{18} + 6 q^{19} + 4 q^{21} + 6 q^{22} - 6 q^{23} - 2 q^{24}+ \cdots + 18 q^{99}+O(q^{100}) \) 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
\(3\) −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
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.46410 1.48803
\(10\) 1.73205 0.547723
\(11\) 4.73205 1.42677 0.713384 − 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) −2.73205 −0.788675
\(13\) −3.46410 −0.960769 −0.480384 − 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) −2.00000 −0.534522
\(15\) −4.73205 −1.22181
\(16\) 1.00000 0.250000
\(17\) −7.73205 −1.87530 −0.937649 − 0.347584i \(-0.887002\pi\)
−0.937649 + 0.347584i \(0.887002\pi\)
\(18\) 4.46410 1.05220
\(19\) 1.26795 0.290887 0.145444 − 0.989367i \(-0.453539\pi\)
0.145444 + 0.989367i \(0.453539\pi\)
\(20\) 1.73205 0.387298
\(21\) 5.46410 1.19236
\(22\) 4.73205 1.00888
\(23\) −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
\(26\) −3.46410 −0.679366
\(27\) −4.00000 −0.769800
\(28\) −2.00000 −0.377964
\(29\) 8.66025 1.60817 0.804084 − 0.594515i \(-0.202656\pi\)
0.804084 + 0.594515i \(0.202656\pi\)
\(30\) −4.73205 −0.863950
\(31\) −1.26795 −0.227730 −0.113865 − 0.993496i \(-0.536323\pi\)
−0.113865 + 0.993496i \(0.536323\pi\)
\(32\) 1.00000 0.176777
\(33\) −12.9282 −2.25051
\(34\) −7.73205 −1.32604
\(35\) −3.46410 −0.585540
\(36\) 4.46410 0.744017
\(37\) 0 0
\(38\) 1.26795 0.205689
\(39\) 9.46410 1.51547
\(40\) 1.73205 0.273861
\(41\) −9.92820 −1.55052 −0.775262 − 0.631639i \(-0.782382\pi\)
−0.775262 + 0.631639i \(0.782382\pi\)
\(42\) 5.46410 0.843129
\(43\) 0.928203 0.141550 0.0707748 − 0.997492i \(-0.477453\pi\)
0.0707748 + 0.997492i \(0.477453\pi\)
\(44\) 4.73205 0.713384
\(45\) 7.73205 1.15263
\(46\) −4.73205 −0.697703
\(47\) 4.73205 0.690241 0.345120 − 0.938558i \(-0.387838\pi\)
0.345120 + 0.938558i \(0.387838\pi\)
\(48\) −2.73205 −0.394338
\(49\) −3.00000 −0.428571
\(50\) −2.00000 −0.282843
\(51\) 21.1244 2.95800
\(52\) −3.46410 −0.480384
\(53\) 2.53590 0.348332 0.174166 − 0.984716i \(-0.444277\pi\)
0.174166 + 0.984716i \(0.444277\pi\)
\(54\) −4.00000 −0.544331
\(55\) 8.19615 1.10517
\(56\) −2.00000 −0.267261
\(57\) −3.46410 −0.458831
\(58\) 8.66025 1.13715
\(59\) −2.53590 −0.330146 −0.165073 − 0.986281i \(-0.552786\pi\)
−0.165073 + 0.986281i \(0.552786\pi\)
\(60\) −4.73205 −0.610905
\(61\) −1.73205 −0.221766 −0.110883 − 0.993833i \(-0.535368\pi\)
−0.110883 + 0.993833i \(0.535368\pi\)
\(62\) −1.26795 −0.161030
\(63\) −8.92820 −1.12485
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) −12.9282 −1.59135
\(67\) 10.1962 1.24566 0.622829 − 0.782358i \(-0.285983\pi\)
0.622829 + 0.782358i \(0.285983\pi\)
\(68\) −7.73205 −0.937649
\(69\) 12.9282 1.55637
\(70\) −3.46410 −0.414039
\(71\) 3.46410 0.411113 0.205557 − 0.978645i \(-0.434100\pi\)
0.205557 + 0.978645i \(0.434100\pi\)
\(72\) 4.46410 0.526099
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 5.46410 0.630940
\(76\) 1.26795 0.145444
\(77\) −9.46410 −1.07853
\(78\) 9.46410 1.07160
\(79\) −13.2679 −1.49276 −0.746380 − 0.665520i \(-0.768210\pi\)
−0.746380 + 0.665520i \(0.768210\pi\)
\(80\) 1.73205 0.193649
\(81\) −2.46410 −0.273789
\(82\) −9.92820 −1.09639
\(83\) −5.66025 −0.621294 −0.310647 − 0.950525i \(-0.600546\pi\)
−0.310647 + 0.950525i \(0.600546\pi\)
\(84\) 5.46410 0.596182
\(85\) −13.3923 −1.45260
\(86\) 0.928203 0.100091
\(87\) −23.6603 −2.53665
\(88\) 4.73205 0.504438
\(89\) −6.80385 −0.721206 −0.360603 − 0.932719i \(-0.617429\pi\)
−0.360603 + 0.932719i \(0.617429\pi\)
\(90\) 7.73205 0.815030
\(91\) 6.92820 0.726273
\(92\) −4.73205 −0.493350
\(93\) 3.46410 0.359211
\(94\) 4.73205 0.488074
\(95\) 2.19615 0.225320
\(96\) −2.73205 −0.278839
\(97\) −7.73205 −0.785071 −0.392535 − 0.919737i \(-0.628402\pi\)
−0.392535 + 0.919737i \(0.628402\pi\)
\(98\) −3.00000 −0.303046
\(99\) 21.1244 2.12308
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.1.1 2
37.8 odd 12 74.2.e.b.27.2 yes 4
37.14 odd 12 74.2.e.b.11.2 ✓ 4
37.36 even 2 2738.2.a.e.1.1 2
111.8 even 12 666.2.s.a.397.1 4
111.14 even 12 666.2.s.a.307.1 4
148.51 even 12 592.2.w.e.529.2 4
148.119 even 12 592.2.w.e.545.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.2 ✓ 4 37.14 odd 12
74.2.e.b.27.2 yes 4 37.8 odd 12
592.2.w.e.529.2 4 148.51 even 12
592.2.w.e.545.2 4 148.119 even 12
666.2.s.a.307.1 4 111.14 even 12
666.2.s.a.397.1 4 111.8 even 12
2738.2.a.e.1.1 2 37.36 even 2
2738.2.a.i.1.1 2 1.1 even 1 trivial