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 = [2,2,1,2,-3,1,-4] 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: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) 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 1.1
Root \(-1.79129\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.79129 q^{3} +1.00000 q^{4} +0.791288 q^{5} -1.79129 q^{6} -2.00000 q^{7} +1.00000 q^{8} +0.208712 q^{9} +0.791288 q^{10} +0.791288 q^{11} -1.79129 q^{12} -3.79129 q^{13} -2.00000 q^{14} -1.41742 q^{15} +1.00000 q^{16} +7.58258 q^{17} +0.208712 q^{18} +1.58258 q^{19} +0.791288 q^{20} +3.58258 q^{21} +0.791288 q^{22} -3.79129 q^{23} -1.79129 q^{24} -4.37386 q^{25} -3.79129 q^{26} +5.00000 q^{27} -2.00000 q^{28} -3.79129 q^{29} -1.41742 q^{30} +8.37386 q^{31} +1.00000 q^{32} -1.41742 q^{33} +7.58258 q^{34} -1.58258 q^{35} +0.208712 q^{36} +1.58258 q^{38} +6.79129 q^{39} +0.791288 q^{40} -9.79129 q^{41} +3.58258 q^{42} -6.00000 q^{43} +0.791288 q^{44} +0.165151 q^{45} -3.79129 q^{46} -7.58258 q^{47} -1.79129 q^{48} -3.00000 q^{49} -4.37386 q^{50} -13.5826 q^{51} -3.79129 q^{52} -1.58258 q^{53} +5.00000 q^{54} +0.626136 q^{55} -2.00000 q^{56} -2.83485 q^{57} -3.79129 q^{58} +1.58258 q^{59} -1.41742 q^{60} -12.7913 q^{61} +8.37386 q^{62} -0.417424 q^{63} +1.00000 q^{64} -3.00000 q^{65} -1.41742 q^{66} -6.37386 q^{67} +7.58258 q^{68} +6.79129 q^{69} -1.58258 q^{70} -9.16515 q^{71} +0.208712 q^{72} +4.37386 q^{73} +7.83485 q^{75} +1.58258 q^{76} -1.58258 q^{77} +6.79129 q^{78} +8.20871 q^{79} +0.791288 q^{80} -9.58258 q^{81} -9.79129 q^{82} +15.1652 q^{83} +3.58258 q^{84} +6.00000 q^{85} -6.00000 q^{86} +6.79129 q^{87} +0.791288 q^{88} -6.00000 q^{89} +0.165151 q^{90} +7.58258 q^{91} -3.79129 q^{92} -15.0000 q^{93} -7.58258 q^{94} +1.25227 q^{95} -1.79129 q^{96} -13.5826 q^{97} -3.00000 q^{98} +0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} - 3 q^{5} + q^{6} - 4 q^{7} + 2 q^{8} + 5 q^{9} - 3 q^{10} - 3 q^{11} + q^{12} - 3 q^{13} - 4 q^{14} - 12 q^{15} + 2 q^{16} + 6 q^{17} + 5 q^{18} - 6 q^{19} - 3 q^{20}+ \cdots - 18 q^{99}+O(q^{100}) \) 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
\(3\) −1.79129 −1.03420 −0.517100 − 0.855925i \(-0.672989\pi\)
−0.517100 + 0.855925i \(0.672989\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.791288 0.353875 0.176937 − 0.984222i \(-0.443381\pi\)
0.176937 + 0.984222i \(0.443381\pi\)
\(6\) −1.79129 −0.731290
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.208712 0.0695707
\(10\) 0.791288 0.250227
\(11\) 0.791288 0.238582 0.119291 − 0.992859i \(-0.461938\pi\)
0.119291 + 0.992859i \(0.461938\pi\)
\(12\) −1.79129 −0.517100
\(13\) −3.79129 −1.05151 −0.525757 − 0.850635i \(-0.676218\pi\)
−0.525757 + 0.850635i \(0.676218\pi\)
\(14\) −2.00000 −0.534522
\(15\) −1.41742 −0.365977
\(16\) 1.00000 0.250000
\(17\) 7.58258 1.83904 0.919522 − 0.393038i \(-0.128576\pi\)
0.919522 + 0.393038i \(0.128576\pi\)
\(18\) 0.208712 0.0491939
\(19\) 1.58258 0.363068 0.181534 − 0.983385i \(-0.441894\pi\)
0.181534 + 0.983385i \(0.441894\pi\)
\(20\) 0.791288 0.176937
\(21\) 3.58258 0.781782
\(22\) 0.791288 0.168703
\(23\) −3.79129 −0.790538 −0.395269 − 0.918565i \(-0.629349\pi\)
−0.395269 + 0.918565i \(0.629349\pi\)
\(24\) −1.79129 −0.365645
\(25\) −4.37386 −0.874773
\(26\) −3.79129 −0.743533
\(27\) 5.00000 0.962250
\(28\) −2.00000 −0.377964
\(29\) −3.79129 −0.704024 −0.352012 − 0.935995i \(-0.614502\pi\)
−0.352012 + 0.935995i \(0.614502\pi\)
\(30\) −1.41742 −0.258785
\(31\) 8.37386 1.50399 0.751995 − 0.659169i \(-0.229092\pi\)
0.751995 + 0.659169i \(0.229092\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.41742 −0.246742
\(34\) 7.58258 1.30040
\(35\) −1.58258 −0.267504
\(36\) 0.208712 0.0347854
\(37\) 0 0
\(38\) 1.58258 0.256728
\(39\) 6.79129 1.08748
\(40\) 0.791288 0.125114
\(41\) −9.79129 −1.52914 −0.764571 − 0.644539i \(-0.777049\pi\)
−0.764571 + 0.644539i \(0.777049\pi\)
\(42\) 3.58258 0.552803
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0.791288 0.119291
\(45\) 0.165151 0.0246193
\(46\) −3.79129 −0.558995
\(47\) −7.58258 −1.10603 −0.553016 − 0.833171i \(-0.686523\pi\)
−0.553016 + 0.833171i \(0.686523\pi\)
\(48\) −1.79129 −0.258550
\(49\) −3.00000 −0.428571
\(50\) −4.37386 −0.618558
\(51\) −13.5826 −1.90194
\(52\) −3.79129 −0.525757
\(53\) −1.58258 −0.217383 −0.108692 − 0.994076i \(-0.534666\pi\)
−0.108692 + 0.994076i \(0.534666\pi\)
\(54\) 5.00000 0.680414
\(55\) 0.626136 0.0844282
\(56\) −2.00000 −0.267261
\(57\) −2.83485 −0.375485
\(58\) −3.79129 −0.497820
\(59\) 1.58258 0.206034 0.103017 − 0.994680i \(-0.467150\pi\)
0.103017 + 0.994680i \(0.467150\pi\)
\(60\) −1.41742 −0.182989
\(61\) −12.7913 −1.63776 −0.818878 − 0.573967i \(-0.805404\pi\)
−0.818878 + 0.573967i \(0.805404\pi\)
\(62\) 8.37386 1.06348
\(63\) −0.417424 −0.0525905
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) −1.41742 −0.174473
\(67\) −6.37386 −0.778691 −0.389346 − 0.921092i \(-0.627299\pi\)
−0.389346 + 0.921092i \(0.627299\pi\)
\(68\) 7.58258 0.919522
\(69\) 6.79129 0.817575
\(70\) −1.58258 −0.189154
\(71\) −9.16515 −1.08770 −0.543852 − 0.839181i \(-0.683035\pi\)
−0.543852 + 0.839181i \(0.683035\pi\)
\(72\) 0.208712 0.0245970
\(73\) 4.37386 0.511922 0.255961 − 0.966687i \(-0.417608\pi\)
0.255961 + 0.966687i \(0.417608\pi\)
\(74\) 0 0
\(75\) 7.83485 0.904690
\(76\) 1.58258 0.181534
\(77\) −1.58258 −0.180351
\(78\) 6.79129 0.768962
\(79\) 8.20871 0.923552 0.461776 − 0.886997i \(-0.347212\pi\)
0.461776 + 0.886997i \(0.347212\pi\)
\(80\) 0.791288 0.0884687
\(81\) −9.58258 −1.06473
\(82\) −9.79129 −1.08127
\(83\) 15.1652 1.66459 0.832296 − 0.554332i \(-0.187026\pi\)
0.832296 + 0.554332i \(0.187026\pi\)
\(84\) 3.58258 0.390891
\(85\) 6.00000 0.650791
\(86\) −6.00000 −0.646997
\(87\) 6.79129 0.728102
\(88\) 0.791288 0.0843516
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0.165151 0.0174085
\(91\) 7.58258 0.794870
\(92\) −3.79129 −0.395269
\(93\) −15.0000 −1.55543
\(94\) −7.58258 −0.782083
\(95\) 1.25227 0.128480
\(96\) −1.79129 −0.182823
\(97\) −13.5826 −1.37910 −0.689551 − 0.724237i \(-0.742192\pi\)
−0.689551 + 0.724237i \(0.742192\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0.165151 0.0165983
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.k.1.1 2
37.6 odd 4 74.2.b.a.73.2 ✓ 4
37.31 odd 4 74.2.b.a.73.4 yes 4
37.36 even 2 2738.2.a.h.1.1 2
111.68 even 4 666.2.c.b.73.2 4
111.80 even 4 666.2.c.b.73.3 4
148.31 even 4 592.2.g.c.369.1 4
148.43 even 4 592.2.g.c.369.2 4
185.43 even 4 1850.2.c.g.1849.3 4
185.68 even 4 1850.2.c.h.1849.3 4
185.117 even 4 1850.2.c.h.1849.2 4
185.142 even 4 1850.2.c.g.1849.2 4
185.154 odd 4 1850.2.d.e.1701.3 4
185.179 odd 4 1850.2.d.e.1701.1 4
296.43 even 4 2368.2.g.h.961.3 4
296.117 odd 4 2368.2.g.j.961.1 4
296.179 even 4 2368.2.g.h.961.4 4
296.253 odd 4 2368.2.g.j.961.2 4
444.179 odd 4 5328.2.h.m.2737.3 4
444.191 odd 4 5328.2.h.m.2737.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.b.a.73.2 ✓ 4 37.6 odd 4
74.2.b.a.73.4 yes 4 37.31 odd 4
592.2.g.c.369.1 4 148.31 even 4
592.2.g.c.369.2 4 148.43 even 4
666.2.c.b.73.2 4 111.68 even 4
666.2.c.b.73.3 4 111.80 even 4
1850.2.c.g.1849.2 4 185.142 even 4
1850.2.c.g.1849.3 4 185.43 even 4
1850.2.c.h.1849.2 4 185.117 even 4
1850.2.c.h.1849.3 4 185.68 even 4
1850.2.d.e.1701.1 4 185.179 odd 4
1850.2.d.e.1701.3 4 185.154 odd 4
2368.2.g.h.961.3 4 296.43 even 4
2368.2.g.h.961.4 4 296.179 even 4
2368.2.g.j.961.1 4 296.117 odd 4
2368.2.g.j.961.2 4 296.253 odd 4
2738.2.a.h.1.1 2 37.36 even 2
2738.2.a.k.1.1 2 1.1 even 1 trivial
5328.2.h.m.2737.2 4 444.191 odd 4
5328.2.h.m.2737.3 4 444.179 odd 4