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 = [3,-3,0,3,-6,0,3] 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: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) 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 \(-0.347296\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.347296 q^{3} +1.00000 q^{4} -0.120615 q^{5} -0.347296 q^{6} +1.34730 q^{7} -1.00000 q^{8} -2.87939 q^{9} +0.120615 q^{10} +4.53209 q^{11} +0.347296 q^{12} -0.773318 q^{13} -1.34730 q^{14} -0.0418891 q^{15} +1.00000 q^{16} -3.00000 q^{17} +2.87939 q^{18} -2.04189 q^{19} -0.120615 q^{20} +0.467911 q^{21} -4.53209 q^{22} +0.0564370 q^{23} -0.347296 q^{24} -4.98545 q^{25} +0.773318 q^{26} -2.04189 q^{27} +1.34730 q^{28} -5.78106 q^{29} +0.0418891 q^{30} -3.34730 q^{31} -1.00000 q^{32} +1.57398 q^{33} +3.00000 q^{34} -0.162504 q^{35} -2.87939 q^{36} +2.04189 q^{38} -0.268571 q^{39} +0.120615 q^{40} -7.14796 q^{41} -0.467911 q^{42} +9.31315 q^{43} +4.53209 q^{44} +0.347296 q^{45} -0.0564370 q^{46} +8.51754 q^{47} +0.347296 q^{48} -5.18479 q^{49} +4.98545 q^{50} -1.04189 q^{51} -0.773318 q^{52} +2.78106 q^{53} +2.04189 q^{54} -0.546637 q^{55} -1.34730 q^{56} -0.709141 q^{57} +5.78106 q^{58} -12.9932 q^{59} -0.0418891 q^{60} -11.5175 q^{61} +3.34730 q^{62} -3.87939 q^{63} +1.00000 q^{64} +0.0932736 q^{65} -1.57398 q^{66} +5.12061 q^{67} -3.00000 q^{68} +0.0196004 q^{69} +0.162504 q^{70} +13.4338 q^{71} +2.87939 q^{72} -8.71688 q^{73} -1.73143 q^{75} -2.04189 q^{76} +6.10607 q^{77} +0.268571 q^{78} +4.14796 q^{79} -0.120615 q^{80} +7.92902 q^{81} +7.14796 q^{82} -5.92127 q^{83} +0.467911 q^{84} +0.361844 q^{85} -9.31315 q^{86} -2.00774 q^{87} -4.53209 q^{88} +7.61856 q^{89} -0.347296 q^{90} -1.04189 q^{91} +0.0564370 q^{92} -1.16250 q^{93} -8.51754 q^{94} +0.246282 q^{95} -0.347296 q^{96} -10.2412 q^{97} +5.18479 q^{98} -13.0496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{9} + 6 q^{10} + 9 q^{11} - 9 q^{13} - 3 q^{14} + 3 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{19} - 6 q^{20} + 6 q^{21} - 9 q^{22} + 15 q^{23}+ \cdots - 12 q^{99}+O(q^{100}) \) 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
\(3\) 0.347296 0.200512 0.100256 − 0.994962i \(-0.468034\pi\)
0.100256 + 0.994962i \(0.468034\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.120615 −0.0539406 −0.0269703 − 0.999636i \(-0.508586\pi\)
−0.0269703 + 0.999636i \(0.508586\pi\)
\(6\) −0.347296 −0.141783
\(7\) 1.34730 0.509230 0.254615 − 0.967042i \(-0.418051\pi\)
0.254615 + 0.967042i \(0.418051\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.87939 −0.959795
\(10\) 0.120615 0.0381417
\(11\) 4.53209 1.36648 0.683238 − 0.730196i \(-0.260571\pi\)
0.683238 + 0.730196i \(0.260571\pi\)
\(12\) 0.347296 0.100256
\(13\) −0.773318 −0.214480 −0.107240 − 0.994233i \(-0.534201\pi\)
−0.107240 + 0.994233i \(0.534201\pi\)
\(14\) −1.34730 −0.360080
\(15\) −0.0418891 −0.0108157
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 2.87939 0.678678
\(19\) −2.04189 −0.468441 −0.234221 − 0.972183i \(-0.575254\pi\)
−0.234221 + 0.972183i \(0.575254\pi\)
\(20\) −0.120615 −0.0269703
\(21\) 0.467911 0.102107
\(22\) −4.53209 −0.966245
\(23\) 0.0564370 0.0117679 0.00588396 − 0.999983i \(-0.498127\pi\)
0.00588396 + 0.999983i \(0.498127\pi\)
\(24\) −0.347296 −0.0708916
\(25\) −4.98545 −0.997090
\(26\) 0.773318 0.151660
\(27\) −2.04189 −0.392962
\(28\) 1.34730 0.254615
\(29\) −5.78106 −1.07352 −0.536758 − 0.843736i \(-0.680351\pi\)
−0.536758 + 0.843736i \(0.680351\pi\)
\(30\) 0.0418891 0.00764786
\(31\) −3.34730 −0.601192 −0.300596 − 0.953752i \(-0.597186\pi\)
−0.300596 + 0.953752i \(0.597186\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.57398 0.273994
\(34\) 3.00000 0.514496
\(35\) −0.162504 −0.0274682
\(36\) −2.87939 −0.479898
\(37\) 0 0
\(38\) 2.04189 0.331238
\(39\) −0.268571 −0.0430057
\(40\) 0.120615 0.0190709
\(41\) −7.14796 −1.11632 −0.558162 − 0.829732i \(-0.688493\pi\)
−0.558162 + 0.829732i \(0.688493\pi\)
\(42\) −0.467911 −0.0722003
\(43\) 9.31315 1.42024 0.710121 − 0.704080i \(-0.248640\pi\)
0.710121 + 0.704080i \(0.248640\pi\)
\(44\) 4.53209 0.683238
\(45\) 0.347296 0.0517719
\(46\) −0.0564370 −0.00832118
\(47\) 8.51754 1.24241 0.621206 − 0.783648i \(-0.286643\pi\)
0.621206 + 0.783648i \(0.286643\pi\)
\(48\) 0.347296 0.0501279
\(49\) −5.18479 −0.740685
\(50\) 4.98545 0.705049
\(51\) −1.04189 −0.145894
\(52\) −0.773318 −0.107240
\(53\) 2.78106 0.382008 0.191004 − 0.981589i \(-0.438826\pi\)
0.191004 + 0.981589i \(0.438826\pi\)
\(54\) 2.04189 0.277866
\(55\) −0.546637 −0.0737085
\(56\) −1.34730 −0.180040
\(57\) −0.709141 −0.0939280
\(58\) 5.78106 0.759090
\(59\) −12.9932 −1.69157 −0.845785 − 0.533524i \(-0.820867\pi\)
−0.845785 + 0.533524i \(0.820867\pi\)
\(60\) −0.0418891 −0.00540786
\(61\) −11.5175 −1.47467 −0.737335 − 0.675527i \(-0.763916\pi\)
−0.737335 + 0.675527i \(0.763916\pi\)
\(62\) 3.34730 0.425107
\(63\) −3.87939 −0.488757
\(64\) 1.00000 0.125000
\(65\) 0.0932736 0.0115692
\(66\) −1.57398 −0.193743
\(67\) 5.12061 0.625583 0.312791 − 0.949822i \(-0.398736\pi\)
0.312791 + 0.949822i \(0.398736\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0.0196004 0.00235961
\(70\) 0.162504 0.0194229
\(71\) 13.4338 1.59429 0.797147 − 0.603785i \(-0.206341\pi\)
0.797147 + 0.603785i \(0.206341\pi\)
\(72\) 2.87939 0.339339
\(73\) −8.71688 −1.02023 −0.510117 − 0.860105i \(-0.670398\pi\)
−0.510117 + 0.860105i \(0.670398\pi\)
\(74\) 0 0
\(75\) −1.73143 −0.199928
\(76\) −2.04189 −0.234221
\(77\) 6.10607 0.695851
\(78\) 0.268571 0.0304096
\(79\) 4.14796 0.466682 0.233341 − 0.972395i \(-0.425034\pi\)
0.233341 + 0.972395i \(0.425034\pi\)
\(80\) −0.120615 −0.0134851
\(81\) 7.92902 0.881002
\(82\) 7.14796 0.789360
\(83\) −5.92127 −0.649944 −0.324972 − 0.945724i \(-0.605355\pi\)
−0.324972 + 0.945724i \(0.605355\pi\)
\(84\) 0.467911 0.0510533
\(85\) 0.361844 0.0392475
\(86\) −9.31315 −1.00426
\(87\) −2.00774 −0.215252
\(88\) −4.53209 −0.483122
\(89\) 7.61856 0.807565 0.403783 − 0.914855i \(-0.367695\pi\)
0.403783 + 0.914855i \(0.367695\pi\)
\(90\) −0.347296 −0.0366083
\(91\) −1.04189 −0.109220
\(92\) 0.0564370 0.00588396
\(93\) −1.16250 −0.120546
\(94\) −8.51754 −0.878517
\(95\) 0.246282 0.0252680
\(96\) −0.347296 −0.0354458
\(97\) −10.2412 −1.03984 −0.519920 − 0.854215i \(-0.674038\pi\)
−0.519920 + 0.854215i \(0.674038\pi\)
\(98\) 5.18479 0.523743
\(99\) −13.0496 −1.31154
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.m.1.2 3
37.9 even 9 74.2.f.a.7.1 ✓ 6
37.33 even 9 74.2.f.a.53.1 yes 6
37.36 even 2 2738.2.a.p.1.2 3
111.83 odd 18 666.2.x.c.451.1 6
111.107 odd 18 666.2.x.c.127.1 6
148.83 odd 18 592.2.bc.b.81.1 6
148.107 odd 18 592.2.bc.b.497.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.a.7.1 ✓ 6 37.9 even 9
74.2.f.a.53.1 yes 6 37.33 even 9
592.2.bc.b.81.1 6 148.83 odd 18
592.2.bc.b.497.1 6 148.107 odd 18
666.2.x.c.127.1 6 111.107 odd 18
666.2.x.c.451.1 6 111.83 odd 18
2738.2.a.m.1.2 3 1.1 even 1 trivial
2738.2.a.p.1.2 3 37.36 even 2