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: \(0\)
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 \(-0.491414\pi\)
0.0269703 + 0.999636i \(0.491414\pi\)
\(6\) 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 \(-0.465799\pi\)
0.107240 + 0.994233i \(0.465799\pi\)
\(14\) 1.34730 0.360080
\(15\) 0.0418891 0.0108157
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −2.87939 −0.678678
\(19\) 2.04189 0.468441 0.234221 − 0.972183i \(-0.424746\pi\)
0.234221 + 0.972183i \(0.424746\pi\)
\(20\) 0.120615 0.0269703
\(21\) 0.467911 0.102107
\(22\) 4.53209 0.966245
\(23\) −0.0564370 −0.0117679 −0.00588396 − 0.999983i \(-0.501873\pi\)
−0.00588396 + 0.999983i \(0.501873\pi\)
\(24\) 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 \(-0.319649\pi\)
0.536758 + 0.843736i \(0.319649\pi\)
\(30\) 0.0418891 0.00764786
\(31\) 3.34730 0.601192 0.300596 − 0.953752i \(-0.402814\pi\)
0.300596 + 0.953752i \(0.402814\pi\)
\(32\) 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 \(-0.751360\pi\)
−0.710121 + 0.704080i \(0.751360\pi\)
\(44\) 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 \(-0.179133\pi\)
0.845785 + 0.533524i \(0.179133\pi\)
\(60\) 0.0418891 0.00540786
\(61\) 11.5175 1.47467 0.737335 − 0.675527i \(-0.236084\pi\)
0.737335 + 0.675527i \(0.236084\pi\)
\(62\) 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 \(-0.574966\pi\)
−0.233341 + 0.972395i \(0.574966\pi\)
\(80\) 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 \(-0.632305\pi\)
−0.403783 + 0.914855i \(0.632305\pi\)
\(90\) −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 \(-0.325962\pi\)
0.519920 + 0.854215i \(0.325962\pi\)
\(98\) −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.p.1.2 3
37.4 even 18 74.2.f.a.53.1 yes 6
37.28 even 18 74.2.f.a.7.1 ✓ 6
37.36 even 2 2738.2.a.m.1.2 3
111.41 odd 18 666.2.x.c.127.1 6
111.65 odd 18 666.2.x.c.451.1 6
148.115 odd 18 592.2.bc.b.497.1 6
148.139 odd 18 592.2.bc.b.81.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.a.7.1 ✓ 6 37.28 even 18
74.2.f.a.53.1 yes 6 37.4 even 18
592.2.bc.b.81.1 6 148.139 odd 18
592.2.bc.b.497.1 6 148.115 odd 18
666.2.x.c.127.1 6 111.41 odd 18
666.2.x.c.451.1 6 111.65 odd 18
2738.2.a.m.1.2 3 37.36 even 2
2738.2.a.p.1.2 3 1.1 even 1 trivial