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 1.3
Root \(1.37564\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.37564 q^{3} +1.00000 q^{4} -0.347296 q^{5} +1.37564 q^{6} -0.477756 q^{7} -1.00000 q^{8} -1.10761 q^{9} +0.347296 q^{10} +2.58536 q^{11} -1.37564 q^{12} +4.48325 q^{13} +0.477756 q^{14} +0.477756 q^{15} +1.00000 q^{16} -2.00984 q^{17} +1.10761 q^{18} -3.52224 q^{19} -0.347296 q^{20} +0.657221 q^{21} -2.58536 q^{22} -5.20972 q^{23} +1.37564 q^{24} -4.87939 q^{25} -4.48325 q^{26} +5.65060 q^{27} -0.477756 q^{28} -0.228222 q^{29} -0.477756 q^{30} +6.74658 q^{31} -1.00000 q^{32} -3.55654 q^{33} +2.00984 q^{34} +0.165923 q^{35} -1.10761 q^{36} +3.52224 q^{38} -6.16735 q^{39} +0.347296 q^{40} +10.7164 q^{41} -0.657221 q^{42} -8.85889 q^{43} +2.58536 q^{44} +0.384668 q^{45} +5.20972 q^{46} +7.45780 q^{47} -1.37564 q^{48} -6.77175 q^{49} +4.87939 q^{50} +2.76483 q^{51} +4.48325 q^{52} +4.37564 q^{53} -5.65060 q^{54} -0.897887 q^{55} +0.477756 q^{56} +4.84535 q^{57} +0.228222 q^{58} -4.47776 q^{59} +0.477756 q^{60} +1.43851 q^{61} -6.74658 q^{62} +0.529166 q^{63} +1.00000 q^{64} -1.55702 q^{65} +3.55654 q^{66} -10.5230 q^{67} -2.00984 q^{68} +7.16671 q^{69} -0.165923 q^{70} +8.87964 q^{71} +1.10761 q^{72} +7.18667 q^{73} +6.71229 q^{75} -3.52224 q^{76} -1.23517 q^{77} +6.16735 q^{78} -3.53579 q^{79} -0.347296 q^{80} -4.45039 q^{81} -10.7164 q^{82} +1.41464 q^{83} +0.657221 q^{84} +0.698012 q^{85} +8.85889 q^{86} +0.313952 q^{87} -2.58536 q^{88} -3.82425 q^{89} -0.384668 q^{90} -2.14190 q^{91} -5.20972 q^{92} -9.28089 q^{93} -7.45780 q^{94} +1.22326 q^{95} +1.37564 q^{96} -18.4835 q^{97} +6.77175 q^{98} -2.86357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} - 3 q^{7} - 6 q^{8} + 12 q^{9} - 3 q^{11} + 3 q^{14} + 3 q^{15} + 6 q^{16} - 3 q^{17} - 12 q^{18} - 21 q^{19} + 6 q^{21} + 3 q^{22} - 21 q^{23} - 18 q^{25} - 3 q^{27} - 3 q^{28}+ \cdots - 36 q^{99}+O(q^{100}) \) 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
\(3\) −1.37564 −0.794228 −0.397114 − 0.917769i \(-0.629988\pi\)
−0.397114 + 0.917769i \(0.629988\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.347296 −0.155316 −0.0776578 − 0.996980i \(-0.524744\pi\)
−0.0776578 + 0.996980i \(0.524744\pi\)
\(6\) 1.37564 0.561604
\(7\) −0.477756 −0.180575 −0.0902873 − 0.995916i \(-0.528779\pi\)
−0.0902873 + 0.995916i \(0.528779\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.10761 −0.369202
\(10\) 0.347296 0.109825
\(11\) 2.58536 0.779516 0.389758 − 0.920917i \(-0.372559\pi\)
0.389758 + 0.920917i \(0.372559\pi\)
\(12\) −1.37564 −0.397114
\(13\) 4.48325 1.24343 0.621715 − 0.783244i \(-0.286436\pi\)
0.621715 + 0.783244i \(0.286436\pi\)
\(14\) 0.477756 0.127686
\(15\) 0.477756 0.123356
\(16\) 1.00000 0.250000
\(17\) −2.00984 −0.487459 −0.243729 − 0.969843i \(-0.578371\pi\)
−0.243729 + 0.969843i \(0.578371\pi\)
\(18\) 1.10761 0.261065
\(19\) −3.52224 −0.808058 −0.404029 − 0.914746i \(-0.632391\pi\)
−0.404029 + 0.914746i \(0.632391\pi\)
\(20\) −0.347296 −0.0776578
\(21\) 0.657221 0.143417
\(22\) −2.58536 −0.551201
\(23\) −5.20972 −1.08630 −0.543151 − 0.839635i \(-0.682769\pi\)
−0.543151 + 0.839635i \(0.682769\pi\)
\(24\) 1.37564 0.280802
\(25\) −4.87939 −0.975877
\(26\) −4.48325 −0.879238
\(27\) 5.65060 1.08746
\(28\) −0.477756 −0.0902873
\(29\) −0.228222 −0.0423797 −0.0211899 − 0.999775i \(-0.506745\pi\)
−0.0211899 + 0.999775i \(0.506745\pi\)
\(30\) −0.477756 −0.0872259
\(31\) 6.74658 1.21172 0.605861 − 0.795571i \(-0.292829\pi\)
0.605861 + 0.795571i \(0.292829\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.55654 −0.619113
\(34\) 2.00984 0.344686
\(35\) 0.165923 0.0280461
\(36\) −1.10761 −0.184601
\(37\) 0 0
\(38\) 3.52224 0.571383
\(39\) −6.16735 −0.987566
\(40\) 0.347296 0.0549124
\(41\) 10.7164 1.67361 0.836807 − 0.547498i \(-0.184420\pi\)
0.836807 + 0.547498i \(0.184420\pi\)
\(42\) −0.657221 −0.101411
\(43\) −8.85889 −1.35097 −0.675484 − 0.737374i \(-0.736065\pi\)
−0.675484 + 0.737374i \(0.736065\pi\)
\(44\) 2.58536 0.389758
\(45\) 0.384668 0.0573429
\(46\) 5.20972 0.768131
\(47\) 7.45780 1.08783 0.543916 − 0.839140i \(-0.316941\pi\)
0.543916 + 0.839140i \(0.316941\pi\)
\(48\) −1.37564 −0.198557
\(49\) −6.77175 −0.967393
\(50\) 4.87939 0.690049
\(51\) 2.76483 0.387153
\(52\) 4.48325 0.621715
\(53\) 4.37564 0.601041 0.300520 − 0.953775i \(-0.402840\pi\)
0.300520 + 0.953775i \(0.402840\pi\)
\(54\) −5.65060 −0.768949
\(55\) −0.897887 −0.121071
\(56\) 0.477756 0.0638428
\(57\) 4.84535 0.641782
\(58\) 0.228222 0.0299670
\(59\) −4.47776 −0.582954 −0.291477 − 0.956578i \(-0.594147\pi\)
−0.291477 + 0.956578i \(0.594147\pi\)
\(60\) 0.477756 0.0616780
\(61\) 1.43851 0.184182 0.0920909 − 0.995751i \(-0.470645\pi\)
0.0920909 + 0.995751i \(0.470645\pi\)
\(62\) −6.74658 −0.856817
\(63\) 0.529166 0.0666686
\(64\) 1.00000 0.125000
\(65\) −1.55702 −0.193124
\(66\) 3.55654 0.437779
\(67\) −10.5230 −1.28559 −0.642797 − 0.766037i \(-0.722226\pi\)
−0.642797 + 0.766037i \(0.722226\pi\)
\(68\) −2.00984 −0.243729
\(69\) 7.16671 0.862771
\(70\) −0.165923 −0.0198316
\(71\) 8.87964 1.05382 0.526910 − 0.849921i \(-0.323351\pi\)
0.526910 + 0.849921i \(0.323351\pi\)
\(72\) 1.10761 0.130533
\(73\) 7.18667 0.841136 0.420568 − 0.907261i \(-0.361831\pi\)
0.420568 + 0.907261i \(0.361831\pi\)
\(74\) 0 0
\(75\) 6.71229 0.775069
\(76\) −3.52224 −0.404029
\(77\) −1.23517 −0.140761
\(78\) 6.16735 0.698315
\(79\) −3.53579 −0.397807 −0.198904 − 0.980019i \(-0.563738\pi\)
−0.198904 + 0.980019i \(0.563738\pi\)
\(80\) −0.347296 −0.0388289
\(81\) −4.45039 −0.494487
\(82\) −10.7164 −1.18342
\(83\) 1.41464 0.155277 0.0776383 − 0.996982i \(-0.475262\pi\)
0.0776383 + 0.996982i \(0.475262\pi\)
\(84\) 0.657221 0.0717087
\(85\) 0.698012 0.0757100
\(86\) 8.85889 0.955279
\(87\) 0.313952 0.0336591
\(88\) −2.58536 −0.275601
\(89\) −3.82425 −0.405370 −0.202685 − 0.979244i \(-0.564967\pi\)
−0.202685 + 0.979244i \(0.564967\pi\)
\(90\) −0.384668 −0.0405476
\(91\) −2.14190 −0.224532
\(92\) −5.20972 −0.543151
\(93\) −9.28089 −0.962383
\(94\) −7.45780 −0.769213
\(95\) 1.22326 0.125504
\(96\) 1.37564 0.140401
\(97\) −18.4835 −1.87672 −0.938358 − 0.345665i \(-0.887653\pi\)
−0.938358 + 0.345665i \(0.887653\pi\)
\(98\) 6.77175 0.684050
\(99\) −2.86357 −0.287799
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.1.3 6
37.3 even 18 74.2.f.b.9.1 ✓ 12
37.25 even 18 74.2.f.b.33.1 yes 12
37.36 even 2 2738.2.a.t.1.3 6
111.62 odd 18 666.2.x.g.181.1 12
111.77 odd 18 666.2.x.g.379.1 12
148.3 odd 18 592.2.bc.d.305.2 12
148.99 odd 18 592.2.bc.d.33.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.9.1 ✓ 12 37.3 even 18
74.2.f.b.33.1 yes 12 37.25 even 18
592.2.bc.d.33.2 12 148.99 odd 18
592.2.bc.d.305.2 12 148.3 odd 18
666.2.x.g.181.1 12 111.62 odd 18
666.2.x.g.379.1 12 111.77 odd 18
2738.2.a.q.1.3 6 1.1 even 1 trivial
2738.2.a.t.1.3 6 37.36 even 2