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Properties

Label 1682.2.a.q.1.2
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 1.2
Root \(-1.74168\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.29664 q^{3} +1.00000 q^{4} +3.54362 q^{5} +2.29664 q^{6} -4.13840 q^{7} -1.00000 q^{8} +2.27454 q^{9} -3.54362 q^{10} +1.84176 q^{11} -2.29664 q^{12} +3.35856 q^{13} +4.13840 q^{14} -8.13840 q^{15} +1.00000 q^{16} -3.52078 q^{17} -2.27454 q^{18} -2.72546 q^{19} +3.54362 q^{20} +9.50439 q^{21} -1.84176 q^{22} -3.05470 q^{23} +2.29664 q^{24} +7.55721 q^{25} -3.35856 q^{26} +1.66612 q^{27} -4.13840 q^{28} +8.13840 q^{30} -0.883704 q^{31} -1.00000 q^{32} -4.22985 q^{33} +3.52078 q^{34} -14.6649 q^{35} +2.27454 q^{36} -4.88934 q^{37} +2.72546 q^{38} -7.71338 q^{39} -3.54362 q^{40} +3.01488 q^{41} -9.50439 q^{42} -1.84176 q^{43} +1.84176 q^{44} +8.06008 q^{45} +3.05470 q^{46} -2.01433 q^{47} -2.29664 q^{48} +10.1263 q^{49} -7.55721 q^{50} +8.08594 q^{51} +3.35856 q^{52} -6.62764 q^{53} -1.66612 q^{54} +6.52649 q^{55} +4.13840 q^{56} +6.25940 q^{57} +6.12406 q^{59} -8.13840 q^{60} -1.82554 q^{61} +0.883704 q^{62} -9.41293 q^{63} +1.00000 q^{64} +11.9014 q^{65} +4.22985 q^{66} -0.138395 q^{67} -3.52078 q^{68} +7.01554 q^{69} +14.6649 q^{70} -13.1080 q^{71} -2.27454 q^{72} +15.3867 q^{73} +4.88934 q^{74} -17.3562 q^{75} -2.72546 q^{76} -7.62193 q^{77} +7.71338 q^{78} +15.8792 q^{79} +3.54362 q^{80} -10.6501 q^{81} -3.01488 q^{82} -0.0469400 q^{83} +9.50439 q^{84} -12.4763 q^{85} +1.84176 q^{86} -1.84176 q^{88} +3.59949 q^{89} -8.06008 q^{90} -13.8990 q^{91} -3.05470 q^{92} +2.02955 q^{93} +2.01433 q^{94} -9.65799 q^{95} +2.29664 q^{96} -5.23755 q^{97} -10.1263 q^{98} +4.18915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9} + q^{11} - 2 q^{12} - 3 q^{13} + 3 q^{14} - 27 q^{15} + 6 q^{16} + 6 q^{17} - 12 q^{18} - 18 q^{19} + 10 q^{21} - q^{22} - 14 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) −2.29664 −1.32596 −0.662982 − 0.748636i \(-0.730709\pi\)
−0.662982 + 0.748636i \(0.730709\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.54362 1.58475 0.792377 − 0.610032i \(-0.208844\pi\)
0.792377 + 0.610032i \(0.208844\pi\)
\(6\) 2.29664 0.937598
\(7\) −4.13840 −1.56417 −0.782083 − 0.623174i \(-0.785843\pi\)
−0.782083 + 0.623174i \(0.785843\pi\)
\(8\) −1.00000 −0.353553
\(9\) 2.27454 0.758179
\(10\) −3.54362 −1.12059
\(11\) 1.84176 0.555311 0.277656 − 0.960681i \(-0.410443\pi\)
0.277656 + 0.960681i \(0.410443\pi\)
\(12\) −2.29664 −0.662982
\(13\) 3.35856 0.931496 0.465748 − 0.884917i \(-0.345785\pi\)
0.465748 + 0.884917i \(0.345785\pi\)
\(14\) 4.13840 1.10603
\(15\) −8.13840 −2.10132
\(16\) 1.00000 0.250000
\(17\) −3.52078 −0.853914 −0.426957 − 0.904272i \(-0.640414\pi\)
−0.426957 + 0.904272i \(0.640414\pi\)
\(18\) −2.27454 −0.536113
\(19\) −2.72546 −0.625264 −0.312632 − 0.949874i \(-0.601211\pi\)
−0.312632 + 0.949874i \(0.601211\pi\)
\(20\) 3.54362 0.792377
\(21\) 9.50439 2.07403
\(22\) −1.84176 −0.392664
\(23\) −3.05470 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(24\) 2.29664 0.468799
\(25\) 7.55721 1.51144
\(26\) −3.35856 −0.658667
\(27\) 1.66612 0.320646
\(28\) −4.13840 −0.782083
\(29\) 0 0
\(30\) 8.13840 1.48586
\(31\) −0.883704 −0.158718 −0.0793590 − 0.996846i \(-0.525287\pi\)
−0.0793590 + 0.996846i \(0.525287\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.22985 −0.736322
\(34\) 3.52078 0.603808
\(35\) −14.6649 −2.47882
\(36\) 2.27454 0.379089
\(37\) −4.88934 −0.803803 −0.401902 − 0.915683i \(-0.631651\pi\)
−0.401902 + 0.915683i \(0.631651\pi\)
\(38\) 2.72546 0.442129
\(39\) −7.71338 −1.23513
\(40\) −3.54362 −0.560295
\(41\) 3.01488 0.470846 0.235423 − 0.971893i \(-0.424353\pi\)
0.235423 + 0.971893i \(0.424353\pi\)
\(42\) −9.50439 −1.46656
\(43\) −1.84176 −0.280866 −0.140433 − 0.990090i \(-0.544849\pi\)
−0.140433 + 0.990090i \(0.544849\pi\)
\(44\) 1.84176 0.277656
\(45\) 8.06008 1.20153
\(46\) 3.05470 0.450392
\(47\) −2.01433 −0.293821 −0.146910 − 0.989150i \(-0.546933\pi\)
−0.146910 + 0.989150i \(0.546933\pi\)
\(48\) −2.29664 −0.331491
\(49\) 10.1263 1.44662
\(50\) −7.55721 −1.06875
\(51\) 8.08594 1.13226
\(52\) 3.35856 0.465748
\(53\) −6.62764 −0.910376 −0.455188 − 0.890395i \(-0.650428\pi\)
−0.455188 + 0.890395i \(0.650428\pi\)
\(54\) −1.66612 −0.226731
\(55\) 6.52649 0.880031
\(56\) 4.13840 0.553016
\(57\) 6.25940 0.829077
\(58\) 0 0
\(59\) 6.12406 0.797285 0.398642 − 0.917106i \(-0.369481\pi\)
0.398642 + 0.917106i \(0.369481\pi\)
\(60\) −8.13840 −1.05066
\(61\) −1.82554 −0.233737 −0.116869 − 0.993147i \(-0.537286\pi\)
−0.116869 + 0.993147i \(0.537286\pi\)
\(62\) 0.883704 0.112231
\(63\) −9.41293 −1.18592
\(64\) 1.00000 0.125000
\(65\) 11.9014 1.47619
\(66\) 4.22985 0.520659
\(67\) −0.138395 −0.0169076 −0.00845382 − 0.999964i \(-0.502691\pi\)
−0.00845382 + 0.999964i \(0.502691\pi\)
\(68\) −3.52078 −0.426957
\(69\) 7.01554 0.844572
\(70\) 14.6649 1.75279
\(71\) −13.1080 −1.55564 −0.777819 − 0.628488i \(-0.783674\pi\)
−0.777819 + 0.628488i \(0.783674\pi\)
\(72\) −2.27454 −0.268057
\(73\) 15.3867 1.80088 0.900439 − 0.434982i \(-0.143245\pi\)
0.900439 + 0.434982i \(0.143245\pi\)
\(74\) 4.88934 0.568375
\(75\) −17.3562 −2.00412
\(76\) −2.72546 −0.312632
\(77\) −7.62193 −0.868599
\(78\) 7.71338 0.873368
\(79\) 15.8792 1.78655 0.893274 − 0.449512i \(-0.148402\pi\)
0.893274 + 0.449512i \(0.148402\pi\)
\(80\) 3.54362 0.396188
\(81\) −10.6501 −1.18334
\(82\) −3.01488 −0.332938
\(83\) −0.0469400 −0.00515233 −0.00257616 − 0.999997i \(-0.500820\pi\)
−0.00257616 + 0.999997i \(0.500820\pi\)
\(84\) 9.50439 1.03701
\(85\) −12.4763 −1.35324
\(86\) 1.84176 0.198602
\(87\) 0 0
\(88\) −1.84176 −0.196332
\(89\) 3.59949 0.381545 0.190772 − 0.981634i \(-0.438901\pi\)
0.190772 + 0.981634i \(0.438901\pi\)
\(90\) −8.06008 −0.849607
\(91\) −13.8990 −1.45701
\(92\) −3.05470 −0.318475
\(93\) 2.02955 0.210454
\(94\) 2.01433 0.207763
\(95\) −9.65799 −0.990889
\(96\) 2.29664 0.234399
\(97\) −5.23755 −0.531793 −0.265897 − 0.964002i \(-0.585668\pi\)
−0.265897 + 0.964002i \(0.585668\pi\)
\(98\) −10.1263 −1.02291
\(99\) 4.18915 0.421025
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.1.2 6
29.5 even 14 58.2.d.b.25.1 yes 12
29.6 even 14 58.2.d.b.7.1 ✓ 12
29.12 odd 4 1682.2.b.i.1681.2 12
29.17 odd 4 1682.2.b.i.1681.11 12
29.28 even 2 1682.2.a.t.1.5 6
87.5 odd 14 522.2.k.h.199.2 12
87.35 odd 14 522.2.k.h.181.2 12
116.35 odd 14 464.2.u.h.65.2 12
116.63 odd 14 464.2.u.h.257.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.7.1 ✓ 12 29.6 even 14
58.2.d.b.25.1 yes 12 29.5 even 14
464.2.u.h.65.2 12 116.35 odd 14
464.2.u.h.257.2 12 116.63 odd 14
522.2.k.h.181.2 12 87.35 odd 14
522.2.k.h.199.2 12 87.5 odd 14
1682.2.a.q.1.2 6 1.1 even 1 trivial
1682.2.a.t.1.5 6 29.28 even 2
1682.2.b.i.1681.2 12 29.12 odd 4
1682.2.b.i.1681.11 12 29.17 odd 4