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 1.4
Root \(-1.27006\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.27006 q^{3} +1.00000 q^{4} -1.53209 q^{5} -1.27006 q^{6} +1.94585 q^{7} -1.00000 q^{8} -1.38694 q^{9} +1.53209 q^{10} +0.441088 q^{11} +1.27006 q^{12} +2.11688 q^{13} -1.94585 q^{14} -1.94585 q^{15} +1.00000 q^{16} +3.82524 q^{17} +1.38694 q^{18} -5.94585 q^{19} -1.53209 q^{20} +2.47135 q^{21} -0.441088 q^{22} -5.71115 q^{23} -1.27006 q^{24} -2.65270 q^{25} -2.11688 q^{26} -5.57169 q^{27} +1.94585 q^{28} -2.73423 q^{29} +1.94585 q^{30} -9.87516 q^{31} -1.00000 q^{32} +0.560210 q^{33} -3.82524 q^{34} -2.98122 q^{35} -1.38694 q^{36} +5.94585 q^{38} +2.68857 q^{39} +1.53209 q^{40} +10.4012 q^{41} -2.47135 q^{42} -3.84681 q^{43} +0.441088 q^{44} +2.12491 q^{45} +5.71115 q^{46} +7.68632 q^{47} +1.27006 q^{48} -3.21367 q^{49} +2.65270 q^{50} +4.85829 q^{51} +2.11688 q^{52} +1.72994 q^{53} +5.57169 q^{54} -0.675787 q^{55} -1.94585 q^{56} -7.55161 q^{57} +2.73423 q^{58} -2.05415 q^{59} -1.94585 q^{60} +10.9355 q^{61} +9.87516 q^{62} -2.69877 q^{63} +1.00000 q^{64} -3.24324 q^{65} -0.560210 q^{66} -1.67354 q^{67} +3.82524 q^{68} -7.25353 q^{69} +2.98122 q^{70} -10.0577 q^{71} +1.38694 q^{72} -9.88570 q^{73} -3.36910 q^{75} -5.94585 q^{76} +0.858292 q^{77} -2.68857 q^{78} -13.3443 q^{79} -1.53209 q^{80} -2.91559 q^{81} -10.4012 q^{82} +3.55891 q^{83} +2.47135 q^{84} -5.86060 q^{85} +3.84681 q^{86} -3.47265 q^{87} -0.441088 q^{88} +11.8701 q^{89} -2.12491 q^{90} +4.11912 q^{91} -5.71115 q^{92} -12.5421 q^{93} -7.68632 q^{94} +9.10957 q^{95} -1.27006 q^{96} +0.593497 q^{97} +3.21367 q^{98} -0.611763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} - 3 q^{7} - 6 q^{8} + 12 q^{9} - 3 q^{11} + 3 q^{14} + 3 q^{15} + 6 q^{16} - 3 q^{17} - 12 q^{18} - 21 q^{19} + 6 q^{21} + 3 q^{22} - 21 q^{23} - 18 q^{25} - 3 q^{27} - 3 q^{28}+ \cdots - 36 q^{99}+O(q^{100}) \) 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
\(3\) 1.27006 0.733272 0.366636 − 0.930365i \(-0.380510\pi\)
0.366636 + 0.930365i \(0.380510\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.53209 −0.685171 −0.342585 − 0.939487i \(-0.611303\pi\)
−0.342585 + 0.939487i \(0.611303\pi\)
\(6\) −1.27006 −0.518501
\(7\) 1.94585 0.735462 0.367731 − 0.929932i \(-0.380135\pi\)
0.367731 + 0.929932i \(0.380135\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.38694 −0.462313
\(10\) 1.53209 0.484489
\(11\) 0.441088 0.132993 0.0664966 − 0.997787i \(-0.478818\pi\)
0.0664966 + 0.997787i \(0.478818\pi\)
\(12\) 1.27006 0.366636
\(13\) 2.11688 0.587116 0.293558 − 0.955941i \(-0.405161\pi\)
0.293558 + 0.955941i \(0.405161\pi\)
\(14\) −1.94585 −0.520050
\(15\) −1.94585 −0.502416
\(16\) 1.00000 0.250000
\(17\) 3.82524 0.927756 0.463878 − 0.885899i \(-0.346458\pi\)
0.463878 + 0.885899i \(0.346458\pi\)
\(18\) 1.38694 0.326905
\(19\) −5.94585 −1.36407 −0.682036 − 0.731319i \(-0.738905\pi\)
−0.682036 + 0.731319i \(0.738905\pi\)
\(20\) −1.53209 −0.342585
\(21\) 2.47135 0.539294
\(22\) −0.441088 −0.0940404
\(23\) −5.71115 −1.19086 −0.595429 − 0.803408i \(-0.703018\pi\)
−0.595429 + 0.803408i \(0.703018\pi\)
\(24\) −1.27006 −0.259251
\(25\) −2.65270 −0.530541
\(26\) −2.11688 −0.415153
\(27\) −5.57169 −1.07227
\(28\) 1.94585 0.367731
\(29\) −2.73423 −0.507735 −0.253867 − 0.967239i \(-0.581703\pi\)
−0.253867 + 0.967239i \(0.581703\pi\)
\(30\) 1.94585 0.355262
\(31\) −9.87516 −1.77363 −0.886816 − 0.462123i \(-0.847088\pi\)
−0.886816 + 0.462123i \(0.847088\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.560210 0.0975201
\(34\) −3.82524 −0.656022
\(35\) −2.98122 −0.503917
\(36\) −1.38694 −0.231156
\(37\) 0 0
\(38\) 5.94585 0.964544
\(39\) 2.68857 0.430515
\(40\) 1.53209 0.242245
\(41\) 10.4012 1.62440 0.812198 − 0.583382i \(-0.198271\pi\)
0.812198 + 0.583382i \(0.198271\pi\)
\(42\) −2.47135 −0.381338
\(43\) −3.84681 −0.586633 −0.293317 − 0.956015i \(-0.594759\pi\)
−0.293317 + 0.956015i \(0.594759\pi\)
\(44\) 0.441088 0.0664966
\(45\) 2.12491 0.316763
\(46\) 5.71115 0.842063
\(47\) 7.68632 1.12116 0.560582 − 0.828099i \(-0.310577\pi\)
0.560582 + 0.828099i \(0.310577\pi\)
\(48\) 1.27006 0.183318
\(49\) −3.21367 −0.459095
\(50\) 2.65270 0.375149
\(51\) 4.85829 0.680297
\(52\) 2.11688 0.293558
\(53\) 1.72994 0.237625 0.118813 − 0.992917i \(-0.462091\pi\)
0.118813 + 0.992917i \(0.462091\pi\)
\(54\) 5.57169 0.758211
\(55\) −0.675787 −0.0911231
\(56\) −1.94585 −0.260025
\(57\) −7.55161 −1.00023
\(58\) 2.73423 0.359023
\(59\) −2.05415 −0.267428 −0.133714 − 0.991020i \(-0.542690\pi\)
−0.133714 + 0.991020i \(0.542690\pi\)
\(60\) −1.94585 −0.251208
\(61\) 10.9355 1.40015 0.700077 − 0.714068i \(-0.253149\pi\)
0.700077 + 0.714068i \(0.253149\pi\)
\(62\) 9.87516 1.25415
\(63\) −2.69877 −0.340014
\(64\) 1.00000 0.125000
\(65\) −3.24324 −0.402275
\(66\) −0.560210 −0.0689571
\(67\) −1.67354 −0.204455 −0.102228 − 0.994761i \(-0.532597\pi\)
−0.102228 + 0.994761i \(0.532597\pi\)
\(68\) 3.82524 0.463878
\(69\) −7.25353 −0.873222
\(70\) 2.98122 0.356323
\(71\) −10.0577 −1.19363 −0.596813 − 0.802381i \(-0.703567\pi\)
−0.596813 + 0.802381i \(0.703567\pi\)
\(72\) 1.38694 0.163452
\(73\) −9.88570 −1.15703 −0.578517 − 0.815671i \(-0.696368\pi\)
−0.578517 + 0.815671i \(0.696368\pi\)
\(74\) 0 0
\(75\) −3.36910 −0.389030
\(76\) −5.94585 −0.682036
\(77\) 0.858292 0.0978115
\(78\) −2.68857 −0.304420
\(79\) −13.3443 −1.50135 −0.750674 − 0.660673i \(-0.770271\pi\)
−0.750674 + 0.660673i \(0.770271\pi\)
\(80\) −1.53209 −0.171293
\(81\) −2.91559 −0.323954
\(82\) −10.4012 −1.14862
\(83\) 3.55891 0.390641 0.195321 − 0.980739i \(-0.437425\pi\)
0.195321 + 0.980739i \(0.437425\pi\)
\(84\) 2.47135 0.269647
\(85\) −5.86060 −0.635671
\(86\) 3.84681 0.414812
\(87\) −3.47265 −0.372307
\(88\) −0.441088 −0.0470202
\(89\) 11.8701 1.25823 0.629115 − 0.777312i \(-0.283417\pi\)
0.629115 + 0.777312i \(0.283417\pi\)
\(90\) −2.12491 −0.223986
\(91\) 4.11912 0.431801
\(92\) −5.71115 −0.595429
\(93\) −12.5421 −1.30055
\(94\) −7.68632 −0.792783
\(95\) 9.10957 0.934622
\(96\) −1.27006 −0.129625
\(97\) 0.593497 0.0602604 0.0301302 − 0.999546i \(-0.490408\pi\)
0.0301302 + 0.999546i \(0.490408\pi\)
\(98\) 3.21367 0.324629
\(99\) −0.611763 −0.0614845
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.1.4 6
37.21 even 18 74.2.f.b.71.2 yes 12
37.30 even 18 74.2.f.b.49.2 ✓ 12
37.36 even 2 2738.2.a.t.1.4 6
111.95 odd 18 666.2.x.g.145.1 12
111.104 odd 18 666.2.x.g.271.1 12
148.67 odd 18 592.2.bc.d.49.1 12
148.95 odd 18 592.2.bc.d.145.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.49.2 ✓ 12 37.30 even 18
74.2.f.b.71.2 yes 12 37.21 even 18
592.2.bc.d.49.1 12 148.67 odd 18
592.2.bc.d.145.1 12 148.95 odd 18
666.2.x.g.145.1 12 111.95 odd 18
666.2.x.g.271.1 12 111.104 odd 18
2738.2.a.q.1.4 6 1.1 even 1 trivial
2738.2.a.t.1.4 6 37.36 even 2