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Properties

Label 1682.2.a.q.1.5
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 1.5
Root \(2.29664\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.74168 q^{3} +1.00000 q^{4} -0.494698 q^{5} -1.74168 q^{6} +3.13840 q^{7} -1.00000 q^{8} +0.0334417 q^{9} +0.494698 q^{10} -1.39672 q^{11} +1.74168 q^{12} -5.71545 q^{13} -3.13840 q^{14} -0.861605 q^{15} +1.00000 q^{16} -5.31800 q^{17} -0.0334417 q^{18} -4.96656 q^{19} -0.494698 q^{20} +5.46607 q^{21} +1.39672 q^{22} -0.813609 q^{23} -1.74168 q^{24} -4.75527 q^{25} +5.71545 q^{26} -5.16679 q^{27} +3.13840 q^{28} +0.861605 q^{30} -6.36328 q^{31} -1.00000 q^{32} -2.43263 q^{33} +5.31800 q^{34} -1.55256 q^{35} +0.0334417 q^{36} -8.92766 q^{37} +4.96656 q^{38} -9.95448 q^{39} +0.494698 q^{40} +4.01226 q^{41} -5.46607 q^{42} +1.39672 q^{43} -1.39672 q^{44} -0.0165436 q^{45} +0.813609 q^{46} +10.2982 q^{47} +1.74168 q^{48} +2.84952 q^{49} +4.75527 q^{50} -9.26224 q^{51} -5.71545 q^{52} +4.24359 q^{53} +5.16679 q^{54} +0.690954 q^{55} -3.13840 q^{56} -8.65014 q^{57} +11.1598 q^{59} -0.861605 q^{60} -4.86648 q^{61} +6.36328 q^{62} +0.104953 q^{63} +1.00000 q^{64} +2.82742 q^{65} +2.43263 q^{66} +7.13840 q^{67} -5.31800 q^{68} -1.41705 q^{69} +1.55256 q^{70} +6.03736 q^{71} -0.0334417 q^{72} -12.2792 q^{73} +8.92766 q^{74} -8.28215 q^{75} -4.96656 q^{76} -4.38345 q^{77} +9.95448 q^{78} +9.40223 q^{79} -0.494698 q^{80} -9.09921 q^{81} -4.01226 q^{82} +12.7094 q^{83} +5.46607 q^{84} +2.63080 q^{85} -1.39672 q^{86} +1.39672 q^{88} -1.79217 q^{89} +0.0165436 q^{90} -17.9373 q^{91} -0.813609 q^{92} -11.0828 q^{93} -10.2982 q^{94} +2.45695 q^{95} -1.74168 q^{96} -3.79630 q^{97} -2.84952 q^{98} -0.0467086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9} + q^{11} - 2 q^{12} - 3 q^{13} + 3 q^{14} - 27 q^{15} + 6 q^{16} + 6 q^{17} - 12 q^{18} - 18 q^{19} + 10 q^{21} - q^{22} - 14 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) 1.74168 1.00556 0.502779 − 0.864415i \(-0.332311\pi\)
0.502779 + 0.864415i \(0.332311\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.494698 −0.221236 −0.110618 − 0.993863i \(-0.535283\pi\)
−0.110618 + 0.993863i \(0.535283\pi\)
\(6\) −1.74168 −0.711037
\(7\) 3.13840 1.18620 0.593101 − 0.805128i \(-0.297904\pi\)
0.593101 + 0.805128i \(0.297904\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0.0334417 0.0111472
\(10\) 0.494698 0.156437
\(11\) −1.39672 −0.421126 −0.210563 − 0.977580i \(-0.567530\pi\)
−0.210563 + 0.977580i \(0.567530\pi\)
\(12\) 1.74168 0.502779
\(13\) −5.71545 −1.58518 −0.792591 − 0.609754i \(-0.791268\pi\)
−0.792591 + 0.609754i \(0.791268\pi\)
\(14\) −3.13840 −0.838771
\(15\) −0.861605 −0.222465
\(16\) 1.00000 0.250000
\(17\) −5.31800 −1.28980 −0.644902 − 0.764265i \(-0.723102\pi\)
−0.644902 + 0.764265i \(0.723102\pi\)
\(18\) −0.0334417 −0.00788229
\(19\) −4.96656 −1.13941 −0.569703 − 0.821850i \(-0.692942\pi\)
−0.569703 + 0.821850i \(0.692942\pi\)
\(20\) −0.494698 −0.110618
\(21\) 5.46607 1.19279
\(22\) 1.39672 0.297781
\(23\) −0.813609 −0.169649 −0.0848246 − 0.996396i \(-0.527033\pi\)
−0.0848246 + 0.996396i \(0.527033\pi\)
\(24\) −1.74168 −0.355519
\(25\) −4.75527 −0.951055
\(26\) 5.71545 1.12089
\(27\) −5.16679 −0.994349
\(28\) 3.13840 0.593101
\(29\) 0 0
\(30\) 0.861605 0.157307
\(31\) −6.36328 −1.14288 −0.571439 − 0.820645i \(-0.693615\pi\)
−0.571439 + 0.820645i \(0.693615\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.43263 −0.423467
\(34\) 5.31800 0.912029
\(35\) −1.55256 −0.262430
\(36\) 0.0334417 0.00557362
\(37\) −8.92766 −1.46770 −0.733849 − 0.679313i \(-0.762278\pi\)
−0.733849 + 0.679313i \(0.762278\pi\)
\(38\) 4.96656 0.805682
\(39\) −9.95448 −1.59399
\(40\) 0.494698 0.0782187
\(41\) 4.01226 0.626610 0.313305 − 0.949652i \(-0.398564\pi\)
0.313305 + 0.949652i \(0.398564\pi\)
\(42\) −5.46607 −0.843433
\(43\) 1.39672 0.212997 0.106499 − 0.994313i \(-0.466036\pi\)
0.106499 + 0.994313i \(0.466036\pi\)
\(44\) −1.39672 −0.210563
\(45\) −0.0165436 −0.00246617
\(46\) 0.813609 0.119960
\(47\) 10.2982 1.50214 0.751070 − 0.660223i \(-0.229538\pi\)
0.751070 + 0.660223i \(0.229538\pi\)
\(48\) 1.74168 0.251390
\(49\) 2.84952 0.407075
\(50\) 4.75527 0.672497
\(51\) −9.26224 −1.29697
\(52\) −5.71545 −0.792591
\(53\) 4.24359 0.582902 0.291451 − 0.956586i \(-0.405862\pi\)
0.291451 + 0.956586i \(0.405862\pi\)
\(54\) 5.16679 0.703111
\(55\) 0.690954 0.0931682
\(56\) −3.13840 −0.419386
\(57\) −8.65014 −1.14574
\(58\) 0 0
\(59\) 11.1598 1.45288 0.726438 − 0.687232i \(-0.241174\pi\)
0.726438 + 0.687232i \(0.241174\pi\)
\(60\) −0.861605 −0.111233
\(61\) −4.86648 −0.623089 −0.311544 − 0.950232i \(-0.600846\pi\)
−0.311544 + 0.950232i \(0.600846\pi\)
\(62\) 6.36328 0.808137
\(63\) 0.104953 0.0132229
\(64\) 1.00000 0.125000
\(65\) 2.82742 0.350699
\(66\) 2.43263 0.299436
\(67\) 7.13840 0.872094 0.436047 − 0.899924i \(-0.356378\pi\)
0.436047 + 0.899924i \(0.356378\pi\)
\(68\) −5.31800 −0.644902
\(69\) −1.41705 −0.170592
\(70\) 1.55256 0.185566
\(71\) 6.03736 0.716502 0.358251 − 0.933625i \(-0.383373\pi\)
0.358251 + 0.933625i \(0.383373\pi\)
\(72\) −0.0334417 −0.00394114
\(73\) −12.2792 −1.43717 −0.718585 − 0.695439i \(-0.755210\pi\)
−0.718585 + 0.695439i \(0.755210\pi\)
\(74\) 8.92766 1.03782
\(75\) −8.28215 −0.956341
\(76\) −4.96656 −0.569703
\(77\) −4.38345 −0.499541
\(78\) 9.95448 1.12712
\(79\) 9.40223 1.05783 0.528917 − 0.848674i \(-0.322598\pi\)
0.528917 + 0.848674i \(0.322598\pi\)
\(80\) −0.494698 −0.0553089
\(81\) −9.09921 −1.01102
\(82\) −4.01226 −0.443080
\(83\) 12.7094 1.39504 0.697520 − 0.716565i \(-0.254287\pi\)
0.697520 + 0.716565i \(0.254287\pi\)
\(84\) 5.46607 0.596397
\(85\) 2.63080 0.285351
\(86\) −1.39672 −0.150612
\(87\) 0 0
\(88\) 1.39672 0.148891
\(89\) −1.79217 −0.189970 −0.0949849 − 0.995479i \(-0.530280\pi\)
−0.0949849 + 0.995479i \(0.530280\pi\)
\(90\) 0.0165436 0.00174384
\(91\) −17.9373 −1.88034
\(92\) −0.813609 −0.0848246
\(93\) −11.0828 −1.14923
\(94\) −10.2982 −1.06217
\(95\) 2.45695 0.252078
\(96\) −1.74168 −0.177759
\(97\) −3.79630 −0.385456 −0.192728 − 0.981252i \(-0.561733\pi\)
−0.192728 + 0.981252i \(0.561733\pi\)
\(98\) −2.84952 −0.287845
\(99\) −0.0467086 −0.00469439
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.1.5 6
29.5 even 14 58.2.d.b.25.2 yes 12
29.6 even 14 58.2.d.b.7.2 ✓ 12
29.12 odd 4 1682.2.b.i.1681.5 12
29.17 odd 4 1682.2.b.i.1681.8 12
29.28 even 2 1682.2.a.t.1.2 6
87.5 odd 14 522.2.k.h.199.1 12
87.35 odd 14 522.2.k.h.181.1 12
116.35 odd 14 464.2.u.h.65.1 12
116.63 odd 14 464.2.u.h.257.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.7.2 ✓ 12 29.6 even 14
58.2.d.b.25.2 yes 12 29.5 even 14
464.2.u.h.65.1 12 116.35 odd 14
464.2.u.h.257.1 12 116.63 odd 14
522.2.k.h.181.1 12 87.35 odd 14
522.2.k.h.199.1 12 87.5 odd 14
1682.2.a.q.1.5 6 1.1 even 1 trivial
1682.2.a.t.1.2 6 29.28 even 2
1682.2.b.i.1681.5 12 29.12 odd 4
1682.2.b.i.1681.8 12 29.17 odd 4