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 = [6,6,-8,6,-6] 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: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{28})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 7x^{4} + 14x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.867767\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.44504 q^{3} +1.00000 q^{4} -0.613807 q^{5} -1.44504 q^{6} +2.87647 q^{7} +1.00000 q^{8} -0.911854 q^{9} -0.613807 q^{10} -4.34475 q^{11} -1.44504 q^{12} +0.478274 q^{13} +2.87647 q^{14} +0.886977 q^{15} +1.00000 q^{16} +0.259558 q^{17} -0.911854 q^{18} -8.42322 q^{19} -0.613807 q^{20} -4.15662 q^{21} -4.34475 q^{22} -0.142575 q^{23} -1.44504 q^{24} -4.62324 q^{25} +0.478274 q^{26} +5.65279 q^{27} +2.87647 q^{28} +0.886977 q^{30} +6.62704 q^{31} +1.00000 q^{32} +6.27835 q^{33} +0.259558 q^{34} -1.76560 q^{35} -0.911854 q^{36} -9.60431 q^{37} -8.42322 q^{38} -0.691125 q^{39} -0.613807 q^{40} -4.28236 q^{41} -4.15662 q^{42} -3.25912 q^{43} -4.34475 q^{44} +0.559702 q^{45} -0.142575 q^{46} +5.08365 q^{47} -1.44504 q^{48} +1.27409 q^{49} -4.62324 q^{50} -0.375073 q^{51} +0.478274 q^{52} +9.26256 q^{53} +5.65279 q^{54} +2.66684 q^{55} +2.87647 q^{56} +12.1719 q^{57} -10.2463 q^{59} +0.886977 q^{60} -9.93996 q^{61} +6.62704 q^{62} -2.62292 q^{63} +1.00000 q^{64} -0.293568 q^{65} +6.27835 q^{66} +1.86332 q^{67} +0.259558 q^{68} +0.206027 q^{69} -1.76560 q^{70} -9.58445 q^{71} -0.911854 q^{72} +2.98180 q^{73} -9.60431 q^{74} +6.68078 q^{75} -8.42322 q^{76} -12.4976 q^{77} -0.691125 q^{78} -13.5549 q^{79} -0.613807 q^{80} -5.43296 q^{81} -4.28236 q^{82} +1.05487 q^{83} -4.15662 q^{84} -0.159319 q^{85} -3.25912 q^{86} -4.34475 q^{88} +2.57596 q^{89} +0.559702 q^{90} +1.37574 q^{91} -0.142575 q^{92} -9.57635 q^{93} +5.08365 q^{94} +5.17023 q^{95} -1.44504 q^{96} -16.6449 q^{97} +1.27409 q^{98} +3.96178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - 8 q^{3} + 6 q^{4} - 6 q^{5} - 8 q^{6} + 2 q^{7} + 6 q^{8} + 2 q^{9} - 6 q^{10} - 2 q^{11} - 8 q^{12} - 8 q^{13} + 2 q^{14} + 8 q^{15} + 6 q^{16} - 12 q^{17} + 2 q^{18} - 4 q^{19} - 6 q^{20}+ \cdots - 24 q^{99}+O(q^{100}) \) 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
\(3\) −1.44504 −0.834295 −0.417148 − 0.908839i \(-0.636970\pi\)
−0.417148 + 0.908839i \(0.636970\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.613807 −0.274503 −0.137251 − 0.990536i \(-0.543827\pi\)
−0.137251 + 0.990536i \(0.543827\pi\)
\(6\) −1.44504 −0.589936
\(7\) 2.87647 1.08720 0.543602 − 0.839343i \(-0.317060\pi\)
0.543602 + 0.839343i \(0.317060\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.911854 −0.303951
\(10\) −0.613807 −0.194103
\(11\) −4.34475 −1.30999 −0.654996 − 0.755632i \(-0.727330\pi\)
−0.654996 + 0.755632i \(0.727330\pi\)
\(12\) −1.44504 −0.417148
\(13\) 0.478274 0.132649 0.0663246 − 0.997798i \(-0.478873\pi\)
0.0663246 + 0.997798i \(0.478873\pi\)
\(14\) 2.87647 0.768770
\(15\) 0.886977 0.229016
\(16\) 1.00000 0.250000
\(17\) 0.259558 0.0629522 0.0314761 − 0.999505i \(-0.489979\pi\)
0.0314761 + 0.999505i \(0.489979\pi\)
\(18\) −0.911854 −0.214926
\(19\) −8.42322 −1.93242 −0.966210 − 0.257756i \(-0.917017\pi\)
−0.966210 + 0.257756i \(0.917017\pi\)
\(20\) −0.613807 −0.137251
\(21\) −4.15662 −0.907049
\(22\) −4.34475 −0.926305
\(23\) −0.142575 −0.0297289 −0.0148645 − 0.999890i \(-0.504732\pi\)
−0.0148645 + 0.999890i \(0.504732\pi\)
\(24\) −1.44504 −0.294968
\(25\) −4.62324 −0.924648
\(26\) 0.478274 0.0937972
\(27\) 5.65279 1.08788
\(28\) 2.87647 0.543602
\(29\) 0 0
\(30\) 0.886977 0.161939
\(31\) 6.62704 1.19025 0.595126 − 0.803633i \(-0.297102\pi\)
0.595126 + 0.803633i \(0.297102\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.27835 1.09292
\(34\) 0.259558 0.0445139
\(35\) −1.76560 −0.298441
\(36\) −0.911854 −0.151976
\(37\) −9.60431 −1.57894 −0.789470 − 0.613790i \(-0.789644\pi\)
−0.789470 + 0.613790i \(0.789644\pi\)
\(38\) −8.42322 −1.36643
\(39\) −0.691125 −0.110669
\(40\) −0.613807 −0.0970514
\(41\) −4.28236 −0.668792 −0.334396 − 0.942433i \(-0.608532\pi\)
−0.334396 + 0.942433i \(0.608532\pi\)
\(42\) −4.15662 −0.641381
\(43\) −3.25912 −0.497011 −0.248506 − 0.968630i \(-0.579939\pi\)
−0.248506 + 0.968630i \(0.579939\pi\)
\(44\) −4.34475 −0.654996
\(45\) 0.559702 0.0834355
\(46\) −0.142575 −0.0210215
\(47\) 5.08365 0.741527 0.370763 − 0.928727i \(-0.379096\pi\)
0.370763 + 0.928727i \(0.379096\pi\)
\(48\) −1.44504 −0.208574
\(49\) 1.27409 0.182013
\(50\) −4.62324 −0.653825
\(51\) −0.375073 −0.0525207
\(52\) 0.478274 0.0663246
\(53\) 9.26256 1.27231 0.636155 − 0.771561i \(-0.280524\pi\)
0.636155 + 0.771561i \(0.280524\pi\)
\(54\) 5.65279 0.769248
\(55\) 2.66684 0.359597
\(56\) 2.87647 0.384385
\(57\) 12.1719 1.61221
\(58\) 0 0
\(59\) −10.2463 −1.33395 −0.666977 − 0.745078i \(-0.732412\pi\)
−0.666977 + 0.745078i \(0.732412\pi\)
\(60\) 0.886977 0.114508
\(61\) −9.93996 −1.27268 −0.636341 − 0.771408i \(-0.719553\pi\)
−0.636341 + 0.771408i \(0.719553\pi\)
\(62\) 6.62704 0.841635
\(63\) −2.62292 −0.330457
\(64\) 1.00000 0.125000
\(65\) −0.293568 −0.0364126
\(66\) 6.27835 0.772812
\(67\) 1.86332 0.227640 0.113820 − 0.993501i \(-0.463691\pi\)
0.113820 + 0.993501i \(0.463691\pi\)
\(68\) 0.259558 0.0314761
\(69\) 0.206027 0.0248027
\(70\) −1.76560 −0.211029
\(71\) −9.58445 −1.13746 −0.568732 − 0.822523i \(-0.692566\pi\)
−0.568732 + 0.822523i \(0.692566\pi\)
\(72\) −0.911854 −0.107463
\(73\) 2.98180 0.348993 0.174497 − 0.984658i \(-0.444170\pi\)
0.174497 + 0.984658i \(0.444170\pi\)
\(74\) −9.60431 −1.11648
\(75\) 6.68078 0.771430
\(76\) −8.42322 −0.966210
\(77\) −12.4976 −1.42423
\(78\) −0.691125 −0.0782545
\(79\) −13.5549 −1.52505 −0.762525 − 0.646959i \(-0.776041\pi\)
−0.762525 + 0.646959i \(0.776041\pi\)
\(80\) −0.613807 −0.0686257
\(81\) −5.43296 −0.603662
\(82\) −4.28236 −0.472908
\(83\) 1.05487 0.115787 0.0578933 − 0.998323i \(-0.481562\pi\)
0.0578933 + 0.998323i \(0.481562\pi\)
\(84\) −4.15662 −0.453525
\(85\) −0.159319 −0.0172806
\(86\) −3.25912 −0.351440
\(87\) 0 0
\(88\) −4.34475 −0.463152
\(89\) 2.57596 0.273051 0.136526 − 0.990637i \(-0.456406\pi\)
0.136526 + 0.990637i \(0.456406\pi\)
\(90\) 0.559702 0.0589978
\(91\) 1.37574 0.144217
\(92\) −0.142575 −0.0148645
\(93\) −9.57635 −0.993021
\(94\) 5.08365 0.524338
\(95\) 5.17023 0.530455
\(96\) −1.44504 −0.147484
\(97\) −16.6449 −1.69003 −0.845015 − 0.534742i \(-0.820409\pi\)
−0.845015 + 0.534742i \(0.820409\pi\)
\(98\) 1.27409 0.128703
\(99\) 3.96178 0.398174
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.s.1.4 6
29.2 odd 28 58.2.e.a.33.2 ✓ 12
29.12 odd 4 1682.2.b.j.1681.9 12
29.15 odd 28 58.2.e.a.51.2 yes 12
29.17 odd 4 1682.2.b.j.1681.3 12
29.28 even 2 1682.2.a.r.1.4 6
87.2 even 28 522.2.n.a.91.1 12
87.44 even 28 522.2.n.a.109.1 12
116.15 even 28 464.2.y.c.225.2 12
116.31 even 28 464.2.y.c.33.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.33.2 ✓ 12 29.2 odd 28
58.2.e.a.51.2 yes 12 29.15 odd 28
464.2.y.c.33.2 12 116.31 even 28
464.2.y.c.225.2 12 116.15 even 28
522.2.n.a.91.1 12 87.2 even 28
522.2.n.a.109.1 12 87.44 even 28
1682.2.a.r.1.4 6 29.28 even 2
1682.2.a.s.1.4 6 1.1 even 1 trivial
1682.2.b.j.1681.3 12 29.17 odd 4
1682.2.b.j.1681.9 12 29.12 odd 4