Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1682,2,Mod(1,1682)] 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("1682.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1682, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,8,1,8,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.4308376200\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.32836640625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 18x^{6} + 17x^{5} + 95x^{4} - 77x^{3} - 128x^{2} + 51x + 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.06263\) of defining polynomial
Character \(\chi\) \(=\) 1682.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.06263 q^{3} +1.00000 q^{4} +4.09274 q^{5} -1.06263 q^{6} +2.34135 q^{7} +1.00000 q^{8} -1.87081 q^{9} +4.09274 q^{10} -4.82606 q^{11} -1.06263 q^{12} +4.41399 q^{13} +2.34135 q^{14} -4.34908 q^{15} +1.00000 q^{16} +2.51636 q^{17} -1.87081 q^{18} +2.07073 q^{19} +4.09274 q^{20} -2.48799 q^{21} -4.82606 q^{22} -2.02806 q^{23} -1.06263 q^{24} +11.7505 q^{25} +4.41399 q^{26} +5.17588 q^{27} +2.34135 q^{28} -4.34908 q^{30} +0.814848 q^{31} +1.00000 q^{32} +5.12833 q^{33} +2.51636 q^{34} +9.58253 q^{35} -1.87081 q^{36} +2.59308 q^{37} +2.07073 q^{38} -4.69045 q^{39} +4.09274 q^{40} +0.104627 q^{41} -2.48799 q^{42} +3.15502 q^{43} -4.82606 q^{44} -7.65675 q^{45} -2.02806 q^{46} -9.52807 q^{47} -1.06263 q^{48} -1.51809 q^{49} +11.7505 q^{50} -2.67397 q^{51} +4.41399 q^{52} -10.5005 q^{53} +5.17588 q^{54} -19.7518 q^{55} +2.34135 q^{56} -2.20043 q^{57} -8.95633 q^{59} -4.34908 q^{60} +5.91917 q^{61} +0.814848 q^{62} -4.38022 q^{63} +1.00000 q^{64} +18.0653 q^{65} +5.12833 q^{66} +5.20120 q^{67} +2.51636 q^{68} +2.15508 q^{69} +9.58253 q^{70} +5.54194 q^{71} -1.87081 q^{72} +2.52861 q^{73} +2.59308 q^{74} -12.4865 q^{75} +2.07073 q^{76} -11.2995 q^{77} -4.69045 q^{78} +13.5999 q^{79} +4.09274 q^{80} +0.112368 q^{81} +0.104627 q^{82} -13.9937 q^{83} -2.48799 q^{84} +10.2988 q^{85} +3.15502 q^{86} -4.82606 q^{88} +8.95536 q^{89} -7.65675 q^{90} +10.3347 q^{91} -2.02806 q^{92} -0.865884 q^{93} -9.52807 q^{94} +8.47497 q^{95} -1.06263 q^{96} -2.97302 q^{97} -1.51809 q^{98} +9.02865 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + q^{3} + 8 q^{4} + 5 q^{5} + q^{6} + 7 q^{7} + 8 q^{8} + 13 q^{9} + 5 q^{10} - 7 q^{11} + q^{12} + 13 q^{13} + 7 q^{14} - 20 q^{15} + 8 q^{16} + 9 q^{17} + 13 q^{18} - 13 q^{19} + 5 q^{20}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(3\) −1.06263 −0.613511 −0.306756 − 0.951788i \(-0.599243\pi\)
−0.306756 + 0.951788i \(0.599243\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.09274 1.83033 0.915165 − 0.403080i \(-0.132060\pi\)
0.915165 + 0.403080i \(0.132060\pi\)
\(6\) −1.06263 −0.433818
\(7\) 2.34135 0.884947 0.442473 − 0.896782i \(-0.354101\pi\)
0.442473 + 0.896782i \(0.354101\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.87081 −0.623604
\(10\) 4.09274 1.29424
\(11\) −4.82606 −1.45511 −0.727556 − 0.686048i \(-0.759344\pi\)
−0.727556 + 0.686048i \(0.759344\pi\)
\(12\) −1.06263 −0.306756
\(13\) 4.41399 1.22422 0.612110 − 0.790773i \(-0.290321\pi\)
0.612110 + 0.790773i \(0.290321\pi\)
\(14\) 2.34135 0.625752
\(15\) −4.34908 −1.12293
\(16\) 1.00000 0.250000
\(17\) 2.51636 0.610307 0.305153 − 0.952303i \(-0.401292\pi\)
0.305153 + 0.952303i \(0.401292\pi\)
\(18\) −1.87081 −0.440954
\(19\) 2.07073 0.475058 0.237529 − 0.971380i \(-0.423662\pi\)
0.237529 + 0.971380i \(0.423662\pi\)
\(20\) 4.09274 0.915165
\(21\) −2.48799 −0.542925
\(22\) −4.82606 −1.02892
\(23\) −2.02806 −0.422879 −0.211439 − 0.977391i \(-0.567815\pi\)
−0.211439 + 0.977391i \(0.567815\pi\)
\(24\) −1.06263 −0.216909
\(25\) 11.7505 2.35011
\(26\) 4.41399 0.865654
\(27\) 5.17588 0.996099
\(28\) 2.34135 0.442473
\(29\) 0 0
\(30\) −4.34908 −0.794030
\(31\) 0.814848 0.146351 0.0731755 − 0.997319i \(-0.476687\pi\)
0.0731755 + 0.997319i \(0.476687\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.12833 0.892728
\(34\) 2.51636 0.431552
\(35\) 9.58253 1.61974
\(36\) −1.87081 −0.311802
\(37\) 2.59308 0.426300 0.213150 − 0.977019i \(-0.431628\pi\)
0.213150 + 0.977019i \(0.431628\pi\)
\(38\) 2.07073 0.335917
\(39\) −4.69045 −0.751073
\(40\) 4.09274 0.647119
\(41\) 0.104627 0.0163400 0.00817002 − 0.999967i \(-0.497399\pi\)
0.00817002 + 0.999967i \(0.497399\pi\)
\(42\) −2.48799 −0.383906
\(43\) 3.15502 0.481137 0.240568 − 0.970632i \(-0.422666\pi\)
0.240568 + 0.970632i \(0.422666\pi\)
\(44\) −4.82606 −0.727556
\(45\) −7.65675 −1.14140
\(46\) −2.02806 −0.299020
\(47\) −9.52807 −1.38981 −0.694906 − 0.719100i \(-0.744554\pi\)
−0.694906 + 0.719100i \(0.744554\pi\)
\(48\) −1.06263 −0.153378
\(49\) −1.51809 −0.216870
\(50\) 11.7505 1.66178
\(51\) −2.67397 −0.374430
\(52\) 4.41399 0.612110
\(53\) −10.5005 −1.44236 −0.721181 − 0.692747i \(-0.756400\pi\)
−0.721181 + 0.692747i \(0.756400\pi\)
\(54\) 5.17588 0.704349
\(55\) −19.7518 −2.66334
\(56\) 2.34135 0.312876
\(57\) −2.20043 −0.291454
\(58\) 0 0
\(59\) −8.95633 −1.16601 −0.583007 − 0.812467i \(-0.698124\pi\)
−0.583007 + 0.812467i \(0.698124\pi\)
\(60\) −4.34908 −0.561464
\(61\) 5.91917 0.757872 0.378936 − 0.925423i \(-0.376290\pi\)
0.378936 + 0.925423i \(0.376290\pi\)
\(62\) 0.814848 0.103486
\(63\) −4.38022 −0.551856
\(64\) 1.00000 0.125000
\(65\) 18.0653 2.24073
\(66\) 5.12833 0.631254
\(67\) 5.20120 0.635428 0.317714 − 0.948187i \(-0.397085\pi\)
0.317714 + 0.948187i \(0.397085\pi\)
\(68\) 2.51636 0.305153
\(69\) 2.15508 0.259441
\(70\) 9.58253 1.14533
\(71\) 5.54194 0.657707 0.328854 − 0.944381i \(-0.393338\pi\)
0.328854 + 0.944381i \(0.393338\pi\)
\(72\) −1.87081 −0.220477
\(73\) 2.52861 0.295952 0.147976 − 0.988991i \(-0.452724\pi\)
0.147976 + 0.988991i \(0.452724\pi\)
\(74\) 2.59308 0.301440
\(75\) −12.4865 −1.44182
\(76\) 2.07073 0.237529
\(77\) −11.2995 −1.28770
\(78\) −4.69045 −0.531089
\(79\) 13.5999 1.53011 0.765053 − 0.643967i \(-0.222713\pi\)
0.765053 + 0.643967i \(0.222713\pi\)
\(80\) 4.09274 0.457582
\(81\) 0.112368 0.0124853
\(82\) 0.104627 0.0115541
\(83\) −13.9937 −1.53601 −0.768003 − 0.640446i \(-0.778750\pi\)
−0.768003 + 0.640446i \(0.778750\pi\)
\(84\) −2.48799 −0.271462
\(85\) 10.2988 1.11706
\(86\) 3.15502 0.340215
\(87\) 0 0
\(88\) −4.82606 −0.514460
\(89\) 8.95536 0.949267 0.474633 − 0.880184i \(-0.342581\pi\)
0.474633 + 0.880184i \(0.342581\pi\)
\(90\) −7.65675 −0.807092
\(91\) 10.3347 1.08337
\(92\) −2.02806 −0.211439
\(93\) −0.865884 −0.0897880
\(94\) −9.52807 −0.982746
\(95\) 8.47497 0.869513
\(96\) −1.06263 −0.108455
\(97\) −2.97302 −0.301865 −0.150932 − 0.988544i \(-0.548228\pi\)
−0.150932 + 0.988544i \(0.548228\pi\)
\(98\) −1.51809 −0.153350
\(99\) 9.02865 0.907414
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 1682.2.a.v.1.3 yes 8
29.12 odd 4 1682.2.b.k.1681.11 16
29.17 odd 4 1682.2.b.k.1681.6 16
29.28 even 2 1682.2.a.u.1.6 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1682.2.a.u.1.6 ✓ 8 29.28 even 2
1682.2.a.v.1.3 yes 8 1.1 even 1 trivial
1682.2.b.k.1681.6 16 29.17 odd 4
1682.2.b.k.1681.11 16 29.12 odd 4