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 = [6,-6,0,6,0,0,-3] 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: \(6\)
Coefficient field: 6.6.37902897.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 15x^{4} - x^{3} + 60x^{2} - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(3.14945\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.14945 q^{3} +1.00000 q^{4} -1.53209 q^{5} +3.14945 q^{6} -4.82524 q^{7} -1.00000 q^{8} +6.91903 q^{9} +1.53209 q^{10} -1.09379 q^{11} -3.14945 q^{12} -1.76958 q^{13} +4.82524 q^{14} +4.82524 q^{15} +1.00000 q^{16} -2.94585 q^{17} -6.91903 q^{18} +0.825235 q^{19} -1.53209 q^{20} +15.1968 q^{21} +1.09379 q^{22} +0.243241 q^{23} +3.14945 q^{24} -2.65270 q^{25} +1.76958 q^{26} -12.3428 q^{27} -4.82524 q^{28} +5.57173 q^{29} -4.82524 q^{30} +5.73495 q^{31} -1.00000 q^{32} +3.44484 q^{33} +2.94585 q^{34} +7.39269 q^{35} +6.91903 q^{36} -0.825235 q^{38} +5.57320 q^{39} +1.53209 q^{40} +2.91194 q^{41} -15.1968 q^{42} -4.37987 q^{43} -1.09379 q^{44} -10.6006 q^{45} -0.243241 q^{46} +2.26498 q^{47} -3.14945 q^{48} +16.2829 q^{49} +2.65270 q^{50} +9.27780 q^{51} -1.76958 q^{52} +6.14945 q^{53} +12.3428 q^{54} +1.67579 q^{55} +4.82524 q^{56} -2.59904 q^{57} -5.57173 q^{58} -8.82524 q^{59} +4.82524 q^{60} +11.7522 q^{61} -5.73495 q^{62} -33.3859 q^{63} +1.00000 q^{64} +2.71115 q^{65} -3.44484 q^{66} +8.98400 q^{67} -2.94585 q^{68} -0.766075 q^{69} -7.39269 q^{70} -1.21864 q^{71} -6.91903 q^{72} +8.79418 q^{73} +8.35455 q^{75} +0.825235 q^{76} +5.27780 q^{77} -5.57320 q^{78} -2.15367 q^{79} -1.53209 q^{80} +18.1159 q^{81} -2.91194 q^{82} +5.09379 q^{83} +15.1968 q^{84} +4.51330 q^{85} +4.37987 q^{86} -17.5479 q^{87} +1.09379 q^{88} -8.97619 q^{89} +10.6006 q^{90} +8.53863 q^{91} +0.243241 q^{92} -18.0619 q^{93} -2.26498 q^{94} -1.26433 q^{95} +3.14945 q^{96} -4.35907 q^{97} -16.2829 q^{98} -7.56798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} - 3 q^{7} - 6 q^{8} + 12 q^{9} - 3 q^{11} + 3 q^{14} + 3 q^{15} + 6 q^{16} - 3 q^{17} - 12 q^{18} - 21 q^{19} + 6 q^{21} + 3 q^{22} - 21 q^{23} - 18 q^{25} - 3 q^{27} - 3 q^{28}+ \cdots - 36 q^{99}+O(q^{100}) \) 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
\(3\) −3.14945 −1.81834 −0.909168 − 0.416431i \(-0.863281\pi\)
−0.909168 + 0.416431i \(0.863281\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.53209 −0.685171 −0.342585 − 0.939487i \(-0.611303\pi\)
−0.342585 + 0.939487i \(0.611303\pi\)
\(6\) 3.14945 1.28576
\(7\) −4.82524 −1.82377 −0.911884 − 0.410448i \(-0.865372\pi\)
−0.911884 + 0.410448i \(0.865372\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.91903 2.30634
\(10\) 1.53209 0.484489
\(11\) −1.09379 −0.329791 −0.164895 − 0.986311i \(-0.552729\pi\)
−0.164895 + 0.986311i \(0.552729\pi\)
\(12\) −3.14945 −0.909168
\(13\) −1.76958 −0.490793 −0.245396 − 0.969423i \(-0.578918\pi\)
−0.245396 + 0.969423i \(0.578918\pi\)
\(14\) 4.82524 1.28960
\(15\) 4.82524 1.24587
\(16\) 1.00000 0.250000
\(17\) −2.94585 −0.714474 −0.357237 − 0.934014i \(-0.616281\pi\)
−0.357237 + 0.934014i \(0.616281\pi\)
\(18\) −6.91903 −1.63083
\(19\) 0.825235 0.189322 0.0946610 − 0.995510i \(-0.469823\pi\)
0.0946610 + 0.995510i \(0.469823\pi\)
\(20\) −1.53209 −0.342585
\(21\) 15.1968 3.31622
\(22\) 1.09379 0.233197
\(23\) 0.243241 0.0507192 0.0253596 − 0.999678i \(-0.491927\pi\)
0.0253596 + 0.999678i \(0.491927\pi\)
\(24\) 3.14945 0.642879
\(25\) −2.65270 −0.530541
\(26\) 1.76958 0.347043
\(27\) −12.3428 −2.37537
\(28\) −4.82524 −0.911884
\(29\) 5.57173 1.03464 0.517322 − 0.855791i \(-0.326929\pi\)
0.517322 + 0.855791i \(0.326929\pi\)
\(30\) −4.82524 −0.880963
\(31\) 5.73495 1.03003 0.515013 − 0.857182i \(-0.327787\pi\)
0.515013 + 0.857182i \(0.327787\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.44484 0.599670
\(34\) 2.94585 0.505209
\(35\) 7.39269 1.24959
\(36\) 6.91903 1.15317
\(37\) 0 0
\(38\) −0.825235 −0.133871
\(39\) 5.57320 0.892426
\(40\) 1.53209 0.242245
\(41\) 2.91194 0.454768 0.227384 − 0.973805i \(-0.426983\pi\)
0.227384 + 0.973805i \(0.426983\pi\)
\(42\) −15.1968 −2.34492
\(43\) −4.37987 −0.667924 −0.333962 − 0.942587i \(-0.608386\pi\)
−0.333962 + 0.942587i \(0.608386\pi\)
\(44\) −1.09379 −0.164895
\(45\) −10.6006 −1.58024
\(46\) −0.243241 −0.0358639
\(47\) 2.26498 0.330382 0.165191 − 0.986262i \(-0.447176\pi\)
0.165191 + 0.986262i \(0.447176\pi\)
\(48\) −3.14945 −0.454584
\(49\) 16.2829 2.32613
\(50\) 2.65270 0.375149
\(51\) 9.27780 1.29915
\(52\) −1.76958 −0.245396
\(53\) 6.14945 0.844692 0.422346 − 0.906435i \(-0.361207\pi\)
0.422346 + 0.906435i \(0.361207\pi\)
\(54\) 12.3428 1.67964
\(55\) 1.67579 0.225963
\(56\) 4.82524 0.644799
\(57\) −2.59904 −0.344251
\(58\) −5.57173 −0.731604
\(59\) −8.82524 −1.14895 −0.574474 − 0.818523i \(-0.694793\pi\)
−0.574474 + 0.818523i \(0.694793\pi\)
\(60\) 4.82524 0.622935
\(61\) 11.7522 1.50472 0.752360 − 0.658752i \(-0.228915\pi\)
0.752360 + 0.658752i \(0.228915\pi\)
\(62\) −5.73495 −0.728339
\(63\) −33.3859 −4.20623
\(64\) 1.00000 0.125000
\(65\) 2.71115 0.336277
\(66\) −3.44484 −0.424031
\(67\) 8.98400 1.09757 0.548785 − 0.835963i \(-0.315091\pi\)
0.548785 + 0.835963i \(0.315091\pi\)
\(68\) −2.94585 −0.357237
\(69\) −0.766075 −0.0922245
\(70\) −7.39269 −0.883595
\(71\) −1.21864 −0.144626 −0.0723132 − 0.997382i \(-0.523038\pi\)
−0.0723132 + 0.997382i \(0.523038\pi\)
\(72\) −6.91903 −0.815415
\(73\) 8.79418 1.02928 0.514640 − 0.857406i \(-0.327925\pi\)
0.514640 + 0.857406i \(0.327925\pi\)
\(74\) 0 0
\(75\) 8.35455 0.964701
\(76\) 0.825235 0.0946610
\(77\) 5.27780 0.601462
\(78\) −5.57320 −0.631040
\(79\) −2.15367 −0.242307 −0.121153 − 0.992634i \(-0.538659\pi\)
−0.121153 + 0.992634i \(0.538659\pi\)
\(80\) −1.53209 −0.171293
\(81\) 18.1159 2.01287
\(82\) −2.91194 −0.321570
\(83\) 5.09379 0.559116 0.279558 − 0.960129i \(-0.409812\pi\)
0.279558 + 0.960129i \(0.409812\pi\)
\(84\) 15.1968 1.65811
\(85\) 4.51330 0.489537
\(86\) 4.37987 0.472294
\(87\) −17.5479 −1.88133
\(88\) 1.09379 0.116599
\(89\) −8.97619 −0.951474 −0.475737 − 0.879587i \(-0.657819\pi\)
−0.475737 + 0.879587i \(0.657819\pi\)
\(90\) 10.6006 1.11740
\(91\) 8.53863 0.895092
\(92\) 0.243241 0.0253596
\(93\) −18.0619 −1.87293
\(94\) −2.26498 −0.233615
\(95\) −1.26433 −0.129718
\(96\) 3.14945 0.321439
\(97\) −4.35907 −0.442597 −0.221298 − 0.975206i \(-0.571029\pi\)
−0.221298 + 0.975206i \(0.571029\pi\)
\(98\) −16.2829 −1.64482
\(99\) −7.56798 −0.760610
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.q.1.1 6
37.21 even 18 74.2.f.b.71.1 yes 12
37.30 even 18 74.2.f.b.49.1 ✓ 12
37.36 even 2 2738.2.a.t.1.1 6
111.95 odd 18 666.2.x.g.145.2 12
111.104 odd 18 666.2.x.g.271.2 12
148.67 odd 18 592.2.bc.d.49.2 12
148.95 odd 18 592.2.bc.d.145.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.49.1 ✓ 12 37.30 even 18
74.2.f.b.71.1 yes 12 37.21 even 18
592.2.bc.d.49.2 12 148.67 odd 18
592.2.bc.d.145.2 12 148.95 odd 18
666.2.x.g.145.2 12 111.95 odd 18
666.2.x.g.271.2 12 111.104 odd 18
2738.2.a.q.1.1 6 1.1 even 1 trivial
2738.2.a.t.1.1 6 37.36 even 2