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,-6,0,0] 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: \(\Q(\zeta_{36})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1.28558\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.87939 q^{3} +1.00000 q^{4} -0.800038 q^{5} -1.87939 q^{6} +0.147334 q^{7} +1.00000 q^{8} +0.532089 q^{9} -0.800038 q^{10} -4.81140 q^{11} -1.87939 q^{12} +1.94818 q^{13} +0.147334 q^{14} +1.50358 q^{15} +1.00000 q^{16} +3.49082 q^{17} +0.532089 q^{18} +6.74601 q^{19} -0.800038 q^{20} -0.276898 q^{21} -4.81140 q^{22} +4.17551 q^{23} -1.87939 q^{24} -4.35994 q^{25} +1.94818 q^{26} +4.63816 q^{27} +0.147334 q^{28} -3.03841 q^{29} +1.50358 q^{30} -8.13740 q^{31} +1.00000 q^{32} +9.04247 q^{33} +3.49082 q^{34} -0.117873 q^{35} +0.532089 q^{36} +6.74601 q^{38} -3.66138 q^{39} -0.800038 q^{40} -3.89766 q^{41} -0.276898 q^{42} -8.10852 q^{43} -4.81140 q^{44} -0.425691 q^{45} +4.17551 q^{46} +8.33822 q^{47} -1.87939 q^{48} -6.97829 q^{49} -4.35994 q^{50} -6.56060 q^{51} +1.94818 q^{52} -10.8894 q^{53} +4.63816 q^{54} +3.84930 q^{55} +0.147334 q^{56} -12.6784 q^{57} -3.03841 q^{58} -8.67110 q^{59} +1.50358 q^{60} +0.351687 q^{61} -8.13740 q^{62} +0.0783950 q^{63} +1.00000 q^{64} -1.55862 q^{65} +9.04247 q^{66} -2.39941 q^{67} +3.49082 q^{68} -7.84740 q^{69} -0.117873 q^{70} +6.98332 q^{71} +0.532089 q^{72} -1.13399 q^{73} +8.19401 q^{75} +6.74601 q^{76} -0.708885 q^{77} -3.66138 q^{78} -1.89027 q^{79} -0.800038 q^{80} -10.3131 q^{81} -3.89766 q^{82} -9.90490 q^{83} -0.276898 q^{84} -2.79279 q^{85} -8.10852 q^{86} +5.71034 q^{87} -4.81140 q^{88} +10.1141 q^{89} -0.425691 q^{90} +0.287034 q^{91} +4.17551 q^{92} +15.2933 q^{93} +8.33822 q^{94} -5.39707 q^{95} -1.87939 q^{96} -5.87526 q^{97} -6.97829 q^{98} -2.56009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} - 6 q^{5} + 6 q^{8} - 6 q^{9} - 6 q^{10} - 6 q^{11} - 12 q^{13} + 6 q^{15} + 6 q^{16} - 12 q^{17} - 6 q^{18} - 12 q^{19} - 6 q^{20} - 12 q^{21} - 6 q^{22} + 6 q^{23} + 6 q^{25} - 12 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) −1.87939 −1.08506 −0.542532 − 0.840035i \(-0.682534\pi\)
−0.542532 + 0.840035i \(0.682534\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.800038 −0.357788 −0.178894 − 0.983868i \(-0.557252\pi\)
−0.178894 + 0.983868i \(0.557252\pi\)
\(6\) −1.87939 −0.767256
\(7\) 0.147334 0.0556872 0.0278436 − 0.999612i \(-0.491136\pi\)
0.0278436 + 0.999612i \(0.491136\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.532089 0.177363
\(10\) −0.800038 −0.252994
\(11\) −4.81140 −1.45069 −0.725346 − 0.688385i \(-0.758320\pi\)
−0.725346 + 0.688385i \(0.758320\pi\)
\(12\) −1.87939 −0.542532
\(13\) 1.94818 0.540328 0.270164 − 0.962814i \(-0.412922\pi\)
0.270164 + 0.962814i \(0.412922\pi\)
\(14\) 0.147334 0.0393768
\(15\) 1.50358 0.388223
\(16\) 1.00000 0.250000
\(17\) 3.49082 0.846649 0.423324 − 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(18\) 0.532089 0.125415
\(19\) 6.74601 1.54764 0.773821 − 0.633405i \(-0.218343\pi\)
0.773821 + 0.633405i \(0.218343\pi\)
\(20\) −0.800038 −0.178894
\(21\) −0.276898 −0.0604241
\(22\) −4.81140 −1.02579
\(23\) 4.17551 0.870655 0.435327 − 0.900272i \(-0.356633\pi\)
0.435327 + 0.900272i \(0.356633\pi\)
\(24\) −1.87939 −0.383628
\(25\) −4.35994 −0.871988
\(26\) 1.94818 0.382070
\(27\) 4.63816 0.892613
\(28\) 0.147334 0.0278436
\(29\) −3.03841 −0.564219 −0.282109 − 0.959382i \(-0.591034\pi\)
−0.282109 + 0.959382i \(0.591034\pi\)
\(30\) 1.50358 0.274515
\(31\) −8.13740 −1.46152 −0.730760 − 0.682634i \(-0.760834\pi\)
−0.730760 + 0.682634i \(0.760834\pi\)
\(32\) 1.00000 0.176777
\(33\) 9.04247 1.57409
\(34\) 3.49082 0.598671
\(35\) −0.117873 −0.0199242
\(36\) 0.532089 0.0886815
\(37\) 0 0
\(38\) 6.74601 1.09435
\(39\) −3.66138 −0.586290
\(40\) −0.800038 −0.126497
\(41\) −3.89766 −0.608713 −0.304356 − 0.952558i \(-0.598441\pi\)
−0.304356 + 0.952558i \(0.598441\pi\)
\(42\) −0.276898 −0.0427263
\(43\) −8.10852 −1.23654 −0.618269 − 0.785966i \(-0.712166\pi\)
−0.618269 + 0.785966i \(0.712166\pi\)
\(44\) −4.81140 −0.725346
\(45\) −0.425691 −0.0634583
\(46\) 4.17551 0.615646
\(47\) 8.33822 1.21625 0.608127 − 0.793840i \(-0.291921\pi\)
0.608127 + 0.793840i \(0.291921\pi\)
\(48\) −1.87939 −0.271266
\(49\) −6.97829 −0.996899
\(50\) −4.35994 −0.616588
\(51\) −6.56060 −0.918667
\(52\) 1.94818 0.270164
\(53\) −10.8894 −1.49578 −0.747889 − 0.663824i \(-0.768932\pi\)
−0.747889 + 0.663824i \(0.768932\pi\)
\(54\) 4.63816 0.631173
\(55\) 3.84930 0.519040
\(56\) 0.147334 0.0196884
\(57\) −12.6784 −1.67929
\(58\) −3.03841 −0.398963
\(59\) −8.67110 −1.12888 −0.564440 − 0.825474i \(-0.690908\pi\)
−0.564440 + 0.825474i \(0.690908\pi\)
\(60\) 1.50358 0.194111
\(61\) 0.351687 0.0450289 0.0225145 − 0.999747i \(-0.492833\pi\)
0.0225145 + 0.999747i \(0.492833\pi\)
\(62\) −8.13740 −1.03345
\(63\) 0.0783950 0.00987684
\(64\) 1.00000 0.125000
\(65\) −1.55862 −0.193323
\(66\) 9.04247 1.11305
\(67\) −2.39941 −0.293135 −0.146567 − 0.989201i \(-0.546823\pi\)
−0.146567 + 0.989201i \(0.546823\pi\)
\(68\) 3.49082 0.423324
\(69\) −7.84740 −0.944716
\(70\) −0.117873 −0.0140885
\(71\) 6.98332 0.828767 0.414384 − 0.910102i \(-0.363997\pi\)
0.414384 + 0.910102i \(0.363997\pi\)
\(72\) 0.532089 0.0627073
\(73\) −1.13399 −0.132724 −0.0663618 − 0.997796i \(-0.521139\pi\)
−0.0663618 + 0.997796i \(0.521139\pi\)
\(74\) 0 0
\(75\) 8.19401 0.946162
\(76\) 6.74601 0.773821
\(77\) −0.708885 −0.0807849
\(78\) −3.66138 −0.414570
\(79\) −1.89027 −0.212672 −0.106336 − 0.994330i \(-0.533912\pi\)
−0.106336 + 0.994330i \(0.533912\pi\)
\(80\) −0.800038 −0.0894470
\(81\) −10.3131 −1.14591
\(82\) −3.89766 −0.430425
\(83\) −9.90490 −1.08720 −0.543602 − 0.839343i \(-0.682940\pi\)
−0.543602 + 0.839343i \(0.682940\pi\)
\(84\) −0.276898 −0.0302121
\(85\) −2.79279 −0.302921
\(86\) −8.10852 −0.874365
\(87\) 5.71034 0.612213
\(88\) −4.81140 −0.512897
\(89\) 10.1141 1.07210 0.536048 − 0.844187i \(-0.319917\pi\)
0.536048 + 0.844187i \(0.319917\pi\)
\(90\) −0.425691 −0.0448718
\(91\) 0.287034 0.0300893
\(92\) 4.17551 0.435327
\(93\) 15.2933 1.58584
\(94\) 8.33822 0.860022
\(95\) −5.39707 −0.553727
\(96\) −1.87939 −0.191814
\(97\) −5.87526 −0.596542 −0.298271 − 0.954481i \(-0.596410\pi\)
−0.298271 + 0.954481i \(0.596410\pi\)
\(98\) −6.97829 −0.704914
\(99\) −2.56009 −0.257299
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.s.1.2 6
37.5 odd 36 74.2.h.a.25.1 yes 12
37.15 odd 36 74.2.h.a.3.1 ✓ 12
37.36 even 2 2738.2.a.r.1.1 6
111.5 even 36 666.2.bj.c.469.2 12
111.89 even 36 666.2.bj.c.595.2 12
148.15 even 36 592.2.bq.b.225.2 12
148.79 even 36 592.2.bq.b.321.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.3.1 ✓ 12 37.15 odd 36
74.2.h.a.25.1 yes 12 37.5 odd 36
592.2.bq.b.225.2 12 148.15 even 36
592.2.bq.b.321.2 12 148.79 even 36
666.2.bj.c.469.2 12 111.5 even 36
666.2.bj.c.595.2 12 111.89 even 36
2738.2.a.r.1.1 6 37.36 even 2
2738.2.a.s.1.2 6 1.1 even 1 trivial