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Properties

Label 1682.2.a.q.1.3
Level $1682$
Weight $2$
Character 1682.1
Self dual yes
Analytic conductor $13.431$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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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,-2,6,0] 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: 6.6.13716913.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 13x^{4} + 17x^{3} + 52x^{2} - 32x - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.44077\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.63883 q^{3} +1.00000 q^{4} +1.19379 q^{5} +1.63883 q^{6} +2.04359 q^{7} -1.00000 q^{8} -0.314239 q^{9} -1.19379 q^{10} -3.68242 q^{11} -1.63883 q^{12} +2.15646 q^{13} -2.04359 q^{14} -1.95641 q^{15} +1.00000 q^{16} +6.53517 q^{17} +0.314239 q^{18} -5.31424 q^{19} +1.19379 q^{20} -3.34909 q^{21} +3.68242 q^{22} -5.89351 q^{23} +1.63883 q^{24} -3.57487 q^{25} -2.15646 q^{26} +5.43147 q^{27} +2.04359 q^{28} +1.95641 q^{30} -8.99665 q^{31} -1.00000 q^{32} +6.03485 q^{33} -6.53517 q^{34} +2.43961 q^{35} -0.314239 q^{36} -1.82798 q^{37} +5.31424 q^{38} -3.53407 q^{39} -1.19379 q^{40} +8.32895 q^{41} +3.34909 q^{42} +3.68242 q^{43} -3.68242 q^{44} -0.375134 q^{45} +5.89351 q^{46} +0.992767 q^{47} -1.63883 q^{48} -2.82375 q^{49} +3.57487 q^{50} -10.7100 q^{51} +2.15646 q^{52} -5.66449 q^{53} -5.43147 q^{54} -4.39602 q^{55} -2.04359 q^{56} +8.70913 q^{57} +2.94918 q^{59} -1.95641 q^{60} +1.80894 q^{61} +8.99665 q^{62} -0.642174 q^{63} +1.00000 q^{64} +2.57436 q^{65} -6.03485 q^{66} +6.04359 q^{67} +6.53517 q^{68} +9.65846 q^{69} -2.43961 q^{70} +3.75696 q^{71} +0.314239 q^{72} -12.7248 q^{73} +1.82798 q^{74} +5.85860 q^{75} -5.31424 q^{76} -7.52534 q^{77} +3.53407 q^{78} -14.1329 q^{79} +1.19379 q^{80} -7.95854 q^{81} -8.32895 q^{82} +2.05232 q^{83} -3.34909 q^{84} +7.80161 q^{85} -3.68242 q^{86} +3.68242 q^{88} +16.5120 q^{89} +0.375134 q^{90} +4.40692 q^{91} -5.89351 q^{92} +14.7440 q^{93} -0.992767 q^{94} -6.34407 q^{95} +1.63883 q^{96} +4.59626 q^{97} +2.82375 q^{98} +1.15716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9} + q^{11} - 2 q^{12} - 3 q^{13} + 3 q^{14} - 27 q^{15} + 6 q^{16} + 6 q^{17} - 12 q^{18} - 18 q^{19} + 10 q^{21} - q^{22} - 14 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.63883 −0.946178 −0.473089 − 0.881015i \(-0.656861\pi\)
−0.473089 + 0.881015i \(0.656861\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.19379 0.533878 0.266939 − 0.963713i \(-0.413988\pi\)
0.266939 + 0.963713i \(0.413988\pi\)
\(6\) 1.63883 0.669049
\(7\) 2.04359 0.772403 0.386202 − 0.922414i \(-0.373787\pi\)
0.386202 + 0.922414i \(0.373787\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.314239 −0.104746
\(10\) −1.19379 −0.377509
\(11\) −3.68242 −1.11029 −0.555145 − 0.831754i \(-0.687337\pi\)
−0.555145 + 0.831754i \(0.687337\pi\)
\(12\) −1.63883 −0.473089
\(13\) 2.15646 0.598095 0.299047 − 0.954238i \(-0.403331\pi\)
0.299047 + 0.954238i \(0.403331\pi\)
\(14\) −2.04359 −0.546171
\(15\) −1.95641 −0.505144
\(16\) 1.00000 0.250000
\(17\) 6.53517 1.58501 0.792506 − 0.609864i \(-0.208776\pi\)
0.792506 + 0.609864i \(0.208776\pi\)
\(18\) 0.314239 0.0740668
\(19\) −5.31424 −1.21917 −0.609585 − 0.792721i \(-0.708664\pi\)
−0.609585 + 0.792721i \(0.708664\pi\)
\(20\) 1.19379 0.266939
\(21\) −3.34909 −0.730831
\(22\) 3.68242 0.785094
\(23\) −5.89351 −1.22888 −0.614441 − 0.788963i \(-0.710618\pi\)
−0.614441 + 0.788963i \(0.710618\pi\)
\(24\) 1.63883 0.334525
\(25\) −3.57487 −0.714974
\(26\) −2.15646 −0.422917
\(27\) 5.43147 1.04529
\(28\) 2.04359 0.386202
\(29\) 0 0
\(30\) 1.95641 0.357191
\(31\) −8.99665 −1.61585 −0.807923 − 0.589287i \(-0.799409\pi\)
−0.807923 + 0.589287i \(0.799409\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.03485 1.05053
\(34\) −6.53517 −1.12077
\(35\) 2.43961 0.412369
\(36\) −0.314239 −0.0523732
\(37\) −1.82798 −0.300519 −0.150259 − 0.988647i \(-0.548011\pi\)
−0.150259 + 0.988647i \(0.548011\pi\)
\(38\) 5.31424 0.862083
\(39\) −3.53407 −0.565904
\(40\) −1.19379 −0.188754
\(41\) 8.32895 1.30076 0.650382 − 0.759608i \(-0.274609\pi\)
0.650382 + 0.759608i \(0.274609\pi\)
\(42\) 3.34909 0.516776
\(43\) 3.68242 0.561563 0.280782 − 0.959772i \(-0.409406\pi\)
0.280782 + 0.959772i \(0.409406\pi\)
\(44\) −3.68242 −0.555145
\(45\) −0.375134 −0.0559217
\(46\) 5.89351 0.868951
\(47\) 0.992767 0.144810 0.0724050 − 0.997375i \(-0.476933\pi\)
0.0724050 + 0.997375i \(0.476933\pi\)
\(48\) −1.63883 −0.236545
\(49\) −2.82375 −0.403393
\(50\) 3.57487 0.505563
\(51\) −10.7100 −1.49970
\(52\) 2.15646 0.299047
\(53\) −5.66449 −0.778077 −0.389039 − 0.921221i \(-0.627193\pi\)
−0.389039 + 0.921221i \(0.627193\pi\)
\(54\) −5.43147 −0.739130
\(55\) −4.39602 −0.592759
\(56\) −2.04359 −0.273086
\(57\) 8.70913 1.15355
\(58\) 0 0
\(59\) 2.94918 0.383951 0.191975 − 0.981400i \(-0.438511\pi\)
0.191975 + 0.981400i \(0.438511\pi\)
\(60\) −1.95641 −0.252572
\(61\) 1.80894 0.231612 0.115806 − 0.993272i \(-0.463055\pi\)
0.115806 + 0.993272i \(0.463055\pi\)
\(62\) 8.99665 1.14258
\(63\) −0.642174 −0.0809064
\(64\) 1.00000 0.125000
\(65\) 2.57436 0.319310
\(66\) −6.03485 −0.742839
\(67\) 6.04359 0.738342 0.369171 − 0.929362i \(-0.379642\pi\)
0.369171 + 0.929362i \(0.379642\pi\)
\(68\) 6.53517 0.792506
\(69\) 9.65846 1.16274
\(70\) −2.43961 −0.291589
\(71\) 3.75696 0.445870 0.222935 − 0.974833i \(-0.428436\pi\)
0.222935 + 0.974833i \(0.428436\pi\)
\(72\) 0.314239 0.0370334
\(73\) −12.7248 −1.48933 −0.744665 − 0.667438i \(-0.767391\pi\)
−0.744665 + 0.667438i \(0.767391\pi\)
\(74\) 1.82798 0.212499
\(75\) 5.85860 0.676493
\(76\) −5.31424 −0.609585
\(77\) −7.52534 −0.857592
\(78\) 3.53407 0.400155
\(79\) −14.1329 −1.59008 −0.795039 − 0.606559i \(-0.792550\pi\)
−0.795039 + 0.606559i \(0.792550\pi\)
\(80\) 1.19379 0.133469
\(81\) −7.95854 −0.884282
\(82\) −8.32895 −0.919778
\(83\) 2.05232 0.225272 0.112636 − 0.993636i \(-0.464071\pi\)
0.112636 + 0.993636i \(0.464071\pi\)
\(84\) −3.34909 −0.365416
\(85\) 7.80161 0.846203
\(86\) −3.68242 −0.397085
\(87\) 0 0
\(88\) 3.68242 0.392547
\(89\) 16.5120 1.75027 0.875134 − 0.483880i \(-0.160773\pi\)
0.875134 + 0.483880i \(0.160773\pi\)
\(90\) 0.375134 0.0395426
\(91\) 4.40692 0.461970
\(92\) −5.89351 −0.614441
\(93\) 14.7440 1.52888
\(94\) −0.992767 −0.102396
\(95\) −6.34407 −0.650888
\(96\) 1.63883 0.167262
\(97\) 4.59626 0.466679 0.233340 − 0.972395i \(-0.425035\pi\)
0.233340 + 0.972395i \(0.425035\pi\)
\(98\) 2.82375 0.285242
\(99\) 1.15716 0.116299
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.q.1.3 6
29.4 even 14 58.2.d.b.45.2 ✓ 12
29.12 odd 4 1682.2.b.i.1681.3 12
29.17 odd 4 1682.2.b.i.1681.10 12
29.22 even 14 58.2.d.b.49.2 yes 12
29.28 even 2 1682.2.a.t.1.4 6
87.62 odd 14 522.2.k.h.451.2 12
87.80 odd 14 522.2.k.h.397.2 12
116.51 odd 14 464.2.u.h.49.1 12
116.91 odd 14 464.2.u.h.161.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.45.2 ✓ 12 29.4 even 14
58.2.d.b.49.2 yes 12 29.22 even 14
464.2.u.h.49.1 12 116.51 odd 14
464.2.u.h.161.1 12 116.91 odd 14
522.2.k.h.397.2 12 87.80 odd 14
522.2.k.h.451.2 12 87.62 odd 14
1682.2.a.q.1.3 6 1.1 even 1 trivial
1682.2.a.t.1.4 6 29.28 even 2
1682.2.b.i.1681.3 12 29.12 odd 4
1682.2.b.i.1681.10 12 29.17 odd 4