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 1.6
Root \(-1.96962\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.53209 q^{3} +1.00000 q^{4} +0.384754 q^{5} +1.53209 q^{6} -3.26414 q^{7} +1.00000 q^{8} -0.652704 q^{9} +0.384754 q^{10} -2.58472 q^{11} +1.53209 q^{12} +1.36492 q^{13} -3.26414 q^{14} +0.589478 q^{15} +1.00000 q^{16} -3.33213 q^{17} -0.652704 q^{18} -5.98578 q^{19} +0.384754 q^{20} -5.00095 q^{21} -2.58472 q^{22} -7.54605 q^{23} +1.53209 q^{24} -4.85196 q^{25} +1.36492 q^{26} -5.59627 q^{27} -3.26414 q^{28} -3.21297 q^{29} +0.589478 q^{30} +2.53737 q^{31} +1.00000 q^{32} -3.96002 q^{33} -3.33213 q^{34} -1.25589 q^{35} -0.652704 q^{36} -5.98578 q^{38} +2.09118 q^{39} +0.384754 q^{40} +8.26915 q^{41} -5.00095 q^{42} +4.33920 q^{43} -2.58472 q^{44} -0.251131 q^{45} -7.54605 q^{46} +5.22909 q^{47} +1.53209 q^{48} +3.65461 q^{49} -4.85196 q^{50} -5.10511 q^{51} +1.36492 q^{52} -8.67122 q^{53} -5.59627 q^{54} -0.994481 q^{55} -3.26414 q^{56} -9.17075 q^{57} -3.21297 q^{58} +12.9674 q^{59} +0.589478 q^{60} -7.01264 q^{61} +2.53737 q^{62} +2.13052 q^{63} +1.00000 q^{64} +0.525160 q^{65} -3.96002 q^{66} +11.2380 q^{67} -3.33213 q^{68} -11.5612 q^{69} -1.25589 q^{70} +14.9995 q^{71} -0.652704 q^{72} +15.0792 q^{73} -7.43364 q^{75} -5.98578 q^{76} +8.43688 q^{77} +2.09118 q^{78} -1.46329 q^{79} +0.384754 q^{80} -6.61587 q^{81} +8.26915 q^{82} -0.0111178 q^{83} -5.00095 q^{84} -1.28205 q^{85} +4.33920 q^{86} -4.92256 q^{87} -2.58472 q^{88} +0.953262 q^{89} -0.251131 q^{90} -4.45530 q^{91} -7.54605 q^{92} +3.88748 q^{93} +5.22909 q^{94} -2.30306 q^{95} +1.53209 q^{96} -14.8185 q^{97} +3.65461 q^{98} +1.68705 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} - 6 q^{5} + 6 q^{8} - 6 q^{9} - 6 q^{10} - 6 q^{11} - 12 q^{13} + 6 q^{15} + 6 q^{16} - 12 q^{17} - 6 q^{18} - 12 q^{19} - 6 q^{20} - 12 q^{21} - 6 q^{22} + 6 q^{23} + 6 q^{25} - 12 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) 1.53209 0.884552 0.442276 − 0.896879i \(-0.354171\pi\)
0.442276 + 0.896879i \(0.354171\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.384754 0.172067 0.0860337 − 0.996292i \(-0.472581\pi\)
0.0860337 + 0.996292i \(0.472581\pi\)
\(6\) 1.53209 0.625473
\(7\) −3.26414 −1.23373 −0.616864 − 0.787069i \(-0.711597\pi\)
−0.616864 + 0.787069i \(0.711597\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.652704 −0.217568
\(10\) 0.384754 0.121670
\(11\) −2.58472 −0.779321 −0.389661 − 0.920959i \(-0.627408\pi\)
−0.389661 + 0.920959i \(0.627408\pi\)
\(12\) 1.53209 0.442276
\(13\) 1.36492 0.378561 0.189281 − 0.981923i \(-0.439384\pi\)
0.189281 + 0.981923i \(0.439384\pi\)
\(14\) −3.26414 −0.872378
\(15\) 0.589478 0.152203
\(16\) 1.00000 0.250000
\(17\) −3.33213 −0.808159 −0.404080 − 0.914724i \(-0.632408\pi\)
−0.404080 + 0.914724i \(0.632408\pi\)
\(18\) −0.652704 −0.153844
\(19\) −5.98578 −1.37323 −0.686616 − 0.727020i \(-0.740905\pi\)
−0.686616 + 0.727020i \(0.740905\pi\)
\(20\) 0.384754 0.0860337
\(21\) −5.00095 −1.09130
\(22\) −2.58472 −0.551063
\(23\) −7.54605 −1.57346 −0.786730 − 0.617297i \(-0.788228\pi\)
−0.786730 + 0.617297i \(0.788228\pi\)
\(24\) 1.53209 0.312736
\(25\) −4.85196 −0.970393
\(26\) 1.36492 0.267683
\(27\) −5.59627 −1.07700
\(28\) −3.26414 −0.616864
\(29\) −3.21297 −0.596634 −0.298317 − 0.954467i \(-0.596425\pi\)
−0.298317 + 0.954467i \(0.596425\pi\)
\(30\) 0.589478 0.107623
\(31\) 2.53737 0.455726 0.227863 − 0.973693i \(-0.426826\pi\)
0.227863 + 0.973693i \(0.426826\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.96002 −0.689350
\(34\) −3.33213 −0.571455
\(35\) −1.25589 −0.212285
\(36\) −0.652704 −0.108784
\(37\) 0 0
\(38\) −5.98578 −0.971022
\(39\) 2.09118 0.334857
\(40\) 0.384754 0.0608350
\(41\) 8.26915 1.29142 0.645712 − 0.763581i \(-0.276561\pi\)
0.645712 + 0.763581i \(0.276561\pi\)
\(42\) −5.00095 −0.771664
\(43\) 4.33920 0.661722 0.330861 − 0.943680i \(-0.392661\pi\)
0.330861 + 0.943680i \(0.392661\pi\)
\(44\) −2.58472 −0.389661
\(45\) −0.251131 −0.0374363
\(46\) −7.54605 −1.11260
\(47\) 5.22909 0.762742 0.381371 − 0.924422i \(-0.375452\pi\)
0.381371 + 0.924422i \(0.375452\pi\)
\(48\) 1.53209 0.221138
\(49\) 3.65461 0.522087
\(50\) −4.85196 −0.686171
\(51\) −5.10511 −0.714859
\(52\) 1.36492 0.189281
\(53\) −8.67122 −1.19108 −0.595542 − 0.803324i \(-0.703063\pi\)
−0.595542 + 0.803324i \(0.703063\pi\)
\(54\) −5.59627 −0.761555
\(55\) −0.994481 −0.134096
\(56\) −3.26414 −0.436189
\(57\) −9.17075 −1.21470
\(58\) −3.21297 −0.421884
\(59\) 12.9674 1.68821 0.844107 − 0.536174i \(-0.180131\pi\)
0.844107 + 0.536174i \(0.180131\pi\)
\(60\) 0.589478 0.0761013
\(61\) −7.01264 −0.897877 −0.448938 − 0.893563i \(-0.648198\pi\)
−0.448938 + 0.893563i \(0.648198\pi\)
\(62\) 2.53737 0.322247
\(63\) 2.13052 0.268420
\(64\) 1.00000 0.125000
\(65\) 0.525160 0.0651381
\(66\) −3.96002 −0.487444
\(67\) 11.2380 1.37294 0.686470 − 0.727158i \(-0.259159\pi\)
0.686470 + 0.727158i \(0.259159\pi\)
\(68\) −3.33213 −0.404080
\(69\) −11.5612 −1.39181
\(70\) −1.25589 −0.150108
\(71\) 14.9995 1.78012 0.890059 − 0.455845i \(-0.150663\pi\)
0.890059 + 0.455845i \(0.150663\pi\)
\(72\) −0.652704 −0.0769219
\(73\) 15.0792 1.76489 0.882445 − 0.470416i \(-0.155896\pi\)
0.882445 + 0.470416i \(0.155896\pi\)
\(74\) 0 0
\(75\) −7.43364 −0.858363
\(76\) −5.98578 −0.686616
\(77\) 8.43688 0.961471
\(78\) 2.09118 0.236780
\(79\) −1.46329 −0.164633 −0.0823167 − 0.996606i \(-0.526232\pi\)
−0.0823167 + 0.996606i \(0.526232\pi\)
\(80\) 0.384754 0.0430169
\(81\) −6.61587 −0.735096
\(82\) 8.26915 0.913175
\(83\) −0.0111178 −0.00122034 −0.000610170 − 1.00000i \(-0.500194\pi\)
−0.000610170 1.00000i \(0.500194\pi\)
\(84\) −5.00095 −0.545649
\(85\) −1.28205 −0.139058
\(86\) 4.33920 0.467908
\(87\) −4.92256 −0.527754
\(88\) −2.58472 −0.275532
\(89\) 0.953262 0.101046 0.0505228 − 0.998723i \(-0.483911\pi\)
0.0505228 + 0.998723i \(0.483911\pi\)
\(90\) −0.251131 −0.0264715
\(91\) −4.45530 −0.467042
\(92\) −7.54605 −0.786730
\(93\) 3.88748 0.403113
\(94\) 5.22909 0.539340
\(95\) −2.30306 −0.236289
\(96\) 1.53209 0.156368
\(97\) −14.8185 −1.50459 −0.752297 − 0.658824i \(-0.771054\pi\)
−0.752297 + 0.658824i \(0.771054\pi\)
\(98\) 3.65461 0.369171
\(99\) 1.68705 0.169555
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.1.6 6
37.2 odd 36 74.2.h.a.41.2 ✓ 12
37.19 odd 36 74.2.h.a.65.2 yes 12
37.36 even 2 2738.2.a.r.1.5 6
111.2 even 36 666.2.bj.c.559.1 12
111.56 even 36 666.2.bj.c.361.1 12
148.19 even 36 592.2.bq.b.65.2 12
148.39 even 36 592.2.bq.b.337.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.41.2 ✓ 12 37.2 odd 36
74.2.h.a.65.2 yes 12 37.19 odd 36
592.2.bq.b.65.2 12 148.19 even 36
592.2.bq.b.337.2 12 148.39 even 36
666.2.bj.c.361.1 12 111.56 even 36
666.2.bj.c.559.1 12 111.2 even 36
2738.2.a.r.1.5 6 37.36 even 2
2738.2.a.s.1.6 6 1.1 even 1 trivial