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: \(0\)
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 1.5
Root \(-2.18117\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.18117 q^{3} +1.00000 q^{4} -1.87939 q^{5} +2.18117 q^{6} -4.09926 q^{7} +1.00000 q^{8} +1.75751 q^{9} -1.87939 q^{10} +3.34175 q^{11} +2.18117 q^{12} +1.93869 q^{13} -4.09926 q^{14} -4.09926 q^{15} +1.00000 q^{16} +4.44656 q^{17} +1.75751 q^{18} -0.0992633 q^{19} -1.87939 q^{20} -8.94120 q^{21} +3.34175 q^{22} +9.52292 q^{23} +2.18117 q^{24} -1.46791 q^{25} +1.93869 q^{26} -2.71008 q^{27} -4.09926 q^{28} +0.774576 q^{29} -4.09926 q^{30} +10.3914 q^{31} +1.00000 q^{32} +7.28893 q^{33} +4.44656 q^{34} +7.70409 q^{35} +1.75751 q^{36} -0.0992633 q^{38} +4.22861 q^{39} -1.87939 q^{40} +5.69443 q^{41} -8.94120 q^{42} -1.11986 q^{43} +3.34175 q^{44} -3.30304 q^{45} +9.52292 q^{46} -7.11448 q^{47} +2.18117 q^{48} +9.80396 q^{49} -1.46791 q^{50} +9.69871 q^{51} +1.93869 q^{52} +0.818828 q^{53} -2.71008 q^{54} -6.28044 q^{55} -4.09926 q^{56} -0.216510 q^{57} +0.774576 q^{58} +8.09926 q^{59} -4.09926 q^{60} -10.7009 q^{61} +10.3914 q^{62} -7.20451 q^{63} +1.00000 q^{64} -3.64354 q^{65} +7.28893 q^{66} +13.1663 q^{67} +4.44656 q^{68} +20.7711 q^{69} +7.70409 q^{70} -2.97316 q^{71} +1.75751 q^{72} -2.55740 q^{73} -3.20177 q^{75} -0.0992633 q^{76} -13.6987 q^{77} +4.22861 q^{78} -5.43563 q^{79} -1.87939 q^{80} -11.1837 q^{81} +5.69443 q^{82} +0.658250 q^{83} -8.94120 q^{84} -8.35680 q^{85} -1.11986 q^{86} +1.68948 q^{87} +3.34175 q^{88} +3.52523 q^{89} -3.30304 q^{90} -7.94718 q^{91} +9.52292 q^{92} +22.6655 q^{93} -7.11448 q^{94} +0.186554 q^{95} +2.18117 q^{96} +3.62024 q^{97} +9.80396 q^{98} +5.87317 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} - 3 q^{7} + 6 q^{8} + 12 q^{9} - 3 q^{11} - 3 q^{14} - 3 q^{15} + 6 q^{16} + 3 q^{17} + 12 q^{18} + 21 q^{19} + 6 q^{21} - 3 q^{22} + 21 q^{23} - 18 q^{25} - 3 q^{27} - 3 q^{28}+ \cdots - 36 q^{99}+O(q^{100}) \) 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
\(3\) 2.18117 1.25930 0.629650 − 0.776879i \(-0.283198\pi\)
0.629650 + 0.776879i \(0.283198\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.87939 −0.840487 −0.420243 − 0.907411i \(-0.638055\pi\)
−0.420243 + 0.907411i \(0.638055\pi\)
\(6\) 2.18117 0.890460
\(7\) −4.09926 −1.54938 −0.774688 − 0.632344i \(-0.782093\pi\)
−0.774688 + 0.632344i \(0.782093\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.75751 0.585838
\(10\) −1.87939 −0.594314
\(11\) 3.34175 1.00758 0.503788 − 0.863827i \(-0.331939\pi\)
0.503788 + 0.863827i \(0.331939\pi\)
\(12\) 2.18117 0.629650
\(13\) 1.93869 0.537695 0.268847 − 0.963183i \(-0.413357\pi\)
0.268847 + 0.963183i \(0.413357\pi\)
\(14\) −4.09926 −1.09557
\(15\) −4.09926 −1.05843
\(16\) 1.00000 0.250000
\(17\) 4.44656 1.07845 0.539225 − 0.842162i \(-0.318717\pi\)
0.539225 + 0.842162i \(0.318717\pi\)
\(18\) 1.75751 0.414250
\(19\) −0.0992633 −0.0227726 −0.0113863 − 0.999935i \(-0.503624\pi\)
−0.0113863 + 0.999935i \(0.503624\pi\)
\(20\) −1.87939 −0.420243
\(21\) −8.94120 −1.95113
\(22\) 3.34175 0.712464
\(23\) 9.52292 1.98567 0.992833 − 0.119507i \(-0.0381315\pi\)
0.992833 + 0.119507i \(0.0381315\pi\)
\(24\) 2.18117 0.445230
\(25\) −1.46791 −0.293582
\(26\) 1.93869 0.380208
\(27\) −2.71008 −0.521555
\(28\) −4.09926 −0.774688
\(29\) 0.774576 0.143835 0.0719175 − 0.997411i \(-0.477088\pi\)
0.0719175 + 0.997411i \(0.477088\pi\)
\(30\) −4.09926 −0.748420
\(31\) 10.3914 1.86636 0.933179 − 0.359413i \(-0.117023\pi\)
0.933179 + 0.359413i \(0.117023\pi\)
\(32\) 1.00000 0.176777
\(33\) 7.28893 1.26884
\(34\) 4.44656 0.762579
\(35\) 7.70409 1.30223
\(36\) 1.75751 0.292919
\(37\) 0 0
\(38\) −0.0992633 −0.0161026
\(39\) 4.22861 0.677119
\(40\) −1.87939 −0.297157
\(41\) 5.69443 0.889320 0.444660 − 0.895699i \(-0.353324\pi\)
0.444660 + 0.895699i \(0.353324\pi\)
\(42\) −8.94120 −1.37966
\(43\) −1.11986 −0.170777 −0.0853884 − 0.996348i \(-0.527213\pi\)
−0.0853884 + 0.996348i \(0.527213\pi\)
\(44\) 3.34175 0.503788
\(45\) −3.30304 −0.492389
\(46\) 9.52292 1.40408
\(47\) −7.11448 −1.03775 −0.518877 − 0.854849i \(-0.673650\pi\)
−0.518877 + 0.854849i \(0.673650\pi\)
\(48\) 2.18117 0.314825
\(49\) 9.80396 1.40057
\(50\) −1.46791 −0.207594
\(51\) 9.69871 1.35809
\(52\) 1.93869 0.268847
\(53\) 0.818828 0.112475 0.0562373 − 0.998417i \(-0.482090\pi\)
0.0562373 + 0.998417i \(0.482090\pi\)
\(54\) −2.71008 −0.368795
\(55\) −6.28044 −0.846854
\(56\) −4.09926 −0.547787
\(57\) −0.216510 −0.0286775
\(58\) 0.774576 0.101707
\(59\) 8.09926 1.05443 0.527217 − 0.849731i \(-0.323235\pi\)
0.527217 + 0.849731i \(0.323235\pi\)
\(60\) −4.09926 −0.529213
\(61\) −10.7009 −1.37011 −0.685056 − 0.728491i \(-0.740222\pi\)
−0.685056 + 0.728491i \(0.740222\pi\)
\(62\) 10.3914 1.31971
\(63\) −7.20451 −0.907683
\(64\) 1.00000 0.125000
\(65\) −3.64354 −0.451925
\(66\) 7.28893 0.897206
\(67\) 13.1663 1.60852 0.804260 − 0.594278i \(-0.202562\pi\)
0.804260 + 0.594278i \(0.202562\pi\)
\(68\) 4.44656 0.539225
\(69\) 20.7711 2.50055
\(70\) 7.70409 0.920815
\(71\) −2.97316 −0.352849 −0.176425 − 0.984314i \(-0.556453\pi\)
−0.176425 + 0.984314i \(0.556453\pi\)
\(72\) 1.75751 0.207125
\(73\) −2.55740 −0.299321 −0.149660 − 0.988737i \(-0.547818\pi\)
−0.149660 + 0.988737i \(0.547818\pi\)
\(74\) 0 0
\(75\) −3.20177 −0.369708
\(76\) −0.0992633 −0.0113863
\(77\) −13.6987 −1.56111
\(78\) 4.22861 0.478796
\(79\) −5.43563 −0.611556 −0.305778 − 0.952103i \(-0.598917\pi\)
−0.305778 + 0.952103i \(0.598917\pi\)
\(80\) −1.87939 −0.210122
\(81\) −11.1837 −1.24263
\(82\) 5.69443 0.628844
\(83\) 0.658250 0.0722523 0.0361262 − 0.999347i \(-0.488498\pi\)
0.0361262 + 0.999347i \(0.488498\pi\)
\(84\) −8.94120 −0.975565
\(85\) −8.35680 −0.906422
\(86\) −1.11986 −0.120757
\(87\) 1.68948 0.181132
\(88\) 3.34175 0.356232
\(89\) 3.52523 0.373673 0.186837 − 0.982391i \(-0.440176\pi\)
0.186837 + 0.982391i \(0.440176\pi\)
\(90\) −3.30304 −0.348171
\(91\) −7.94718 −0.833091
\(92\) 9.52292 0.992833
\(93\) 22.6655 2.35030
\(94\) −7.11448 −0.733802
\(95\) 0.186554 0.0191400
\(96\) 2.18117 0.222615
\(97\) 3.62024 0.367580 0.183790 − 0.982966i \(-0.441163\pi\)
0.183790 + 0.982966i \(0.441163\pi\)
\(98\) 9.80396 0.990349
\(99\) 5.87317 0.590276
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.t.1.5 6
37.9 even 9 74.2.f.b.7.1 ✓ 12
37.33 even 9 74.2.f.b.53.1 yes 12
37.36 even 2 2738.2.a.q.1.5 6
111.83 odd 18 666.2.x.g.451.1 12
111.107 odd 18 666.2.x.g.127.1 12
148.83 odd 18 592.2.bc.d.81.2 12
148.107 odd 18 592.2.bc.d.497.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.7.1 ✓ 12 37.9 even 9
74.2.f.b.53.1 yes 12 37.33 even 9
592.2.bc.d.81.2 12 148.83 odd 18
592.2.bc.d.497.2 12 148.107 odd 18
666.2.x.g.127.1 12 111.107 odd 18
666.2.x.g.451.1 12 111.83 odd 18
2738.2.a.q.1.5 6 37.36 even 2
2738.2.a.t.1.5 6 1.1 even 1 trivial