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Properties

Label 1682.2.a.m.1.2
Level $1682$
Weight $2$
Character 1682.1
Self dual yes
Analytic conductor $13.431$
Analytic rank $1$
Dimension $3$
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 = [3,-3,-2,3,5] 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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(-1.24698\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.554958 q^{3} +1.00000 q^{4} +0.198062 q^{5} +0.554958 q^{6} +0.109916 q^{7} -1.00000 q^{8} -2.69202 q^{9} -0.198062 q^{10} -1.33513 q^{11} -0.554958 q^{12} +3.93900 q^{13} -0.109916 q^{14} -0.109916 q^{15} +1.00000 q^{16} -2.91185 q^{17} +2.69202 q^{18} +1.29590 q^{19} +0.198062 q^{20} -0.0609989 q^{21} +1.33513 q^{22} +7.78986 q^{23} +0.554958 q^{24} -4.96077 q^{25} -3.93900 q^{26} +3.15883 q^{27} +0.109916 q^{28} +0.109916 q^{30} -9.34481 q^{31} -1.00000 q^{32} +0.740939 q^{33} +2.91185 q^{34} +0.0217703 q^{35} -2.69202 q^{36} -3.02715 q^{37} -1.29590 q^{38} -2.18598 q^{39} -0.198062 q^{40} +3.76271 q^{41} +0.0609989 q^{42} -6.66487 q^{43} -1.33513 q^{44} -0.533188 q^{45} -7.78986 q^{46} +0.801938 q^{47} -0.554958 q^{48} -6.98792 q^{49} +4.96077 q^{50} +1.61596 q^{51} +3.93900 q^{52} +8.33513 q^{53} -3.15883 q^{54} -0.264438 q^{55} -0.109916 q^{56} -0.719169 q^{57} -5.08815 q^{59} -0.109916 q^{60} -11.0586 q^{61} +9.34481 q^{62} -0.295897 q^{63} +1.00000 q^{64} +0.780167 q^{65} -0.740939 q^{66} -11.0000 q^{67} -2.91185 q^{68} -4.32304 q^{69} -0.0217703 q^{70} +10.9487 q^{71} +2.69202 q^{72} +7.94869 q^{73} +3.02715 q^{74} +2.75302 q^{75} +1.29590 q^{76} -0.146752 q^{77} +2.18598 q^{78} +4.89008 q^{79} +0.198062 q^{80} +6.32304 q^{81} -3.76271 q^{82} -11.6746 q^{83} -0.0609989 q^{84} -0.576728 q^{85} +6.66487 q^{86} +1.33513 q^{88} -11.4940 q^{89} +0.533188 q^{90} +0.432960 q^{91} +7.78986 q^{92} +5.18598 q^{93} -0.801938 q^{94} +0.256668 q^{95} +0.554958 q^{96} -10.5918 q^{97} +6.98792 q^{98} +3.59419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 5 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} - 3 q^{9} - 5 q^{10} - 3 q^{11} - 2 q^{12} + 2 q^{13} - q^{14} - q^{15} + 3 q^{16} - 5 q^{17} + 3 q^{18} - 10 q^{19} + 5 q^{20}+ \cdots + 24 q^{99}+O(q^{100}) \) 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
\(3\) −0.554958 −0.320405 −0.160203 − 0.987084i \(-0.551215\pi\)
−0.160203 + 0.987084i \(0.551215\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.198062 0.0885761 0.0442881 − 0.999019i \(-0.485898\pi\)
0.0442881 + 0.999019i \(0.485898\pi\)
\(6\) 0.554958 0.226561
\(7\) 0.109916 0.0415444 0.0207722 − 0.999784i \(-0.493388\pi\)
0.0207722 + 0.999784i \(0.493388\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.69202 −0.897340
\(10\) −0.198062 −0.0626328
\(11\) −1.33513 −0.402556 −0.201278 − 0.979534i \(-0.564509\pi\)
−0.201278 + 0.979534i \(0.564509\pi\)
\(12\) −0.554958 −0.160203
\(13\) 3.93900 1.09248 0.546241 − 0.837628i \(-0.316058\pi\)
0.546241 + 0.837628i \(0.316058\pi\)
\(14\) −0.109916 −0.0293764
\(15\) −0.109916 −0.0283803
\(16\) 1.00000 0.250000
\(17\) −2.91185 −0.706228 −0.353114 − 0.935580i \(-0.614877\pi\)
−0.353114 + 0.935580i \(0.614877\pi\)
\(18\) 2.69202 0.634516
\(19\) 1.29590 0.297299 0.148650 − 0.988890i \(-0.452507\pi\)
0.148650 + 0.988890i \(0.452507\pi\)
\(20\) 0.198062 0.0442881
\(21\) −0.0609989 −0.0133111
\(22\) 1.33513 0.284650
\(23\) 7.78986 1.62430 0.812149 − 0.583451i \(-0.198298\pi\)
0.812149 + 0.583451i \(0.198298\pi\)
\(24\) 0.554958 0.113280
\(25\) −4.96077 −0.992154
\(26\) −3.93900 −0.772502
\(27\) 3.15883 0.607918
\(28\) 0.109916 0.0207722
\(29\) 0 0
\(30\) 0.109916 0.0200679
\(31\) −9.34481 −1.67838 −0.839189 − 0.543840i \(-0.816970\pi\)
−0.839189 + 0.543840i \(0.816970\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.740939 0.128981
\(34\) 2.91185 0.499379
\(35\) 0.0217703 0.00367985
\(36\) −2.69202 −0.448670
\(37\) −3.02715 −0.497660 −0.248830 − 0.968547i \(-0.580046\pi\)
−0.248830 + 0.968547i \(0.580046\pi\)
\(38\) −1.29590 −0.210222
\(39\) −2.18598 −0.350037
\(40\) −0.198062 −0.0313164
\(41\) 3.76271 0.587636 0.293818 − 0.955861i \(-0.405074\pi\)
0.293818 + 0.955861i \(0.405074\pi\)
\(42\) 0.0609989 0.00941234
\(43\) −6.66487 −1.01638 −0.508192 − 0.861244i \(-0.669686\pi\)
−0.508192 + 0.861244i \(0.669686\pi\)
\(44\) −1.33513 −0.201278
\(45\) −0.533188 −0.0794830
\(46\) −7.78986 −1.14855
\(47\) 0.801938 0.116975 0.0584873 − 0.998288i \(-0.481372\pi\)
0.0584873 + 0.998288i \(0.481372\pi\)
\(48\) −0.554958 −0.0801013
\(49\) −6.98792 −0.998274
\(50\) 4.96077 0.701559
\(51\) 1.61596 0.226279
\(52\) 3.93900 0.546241
\(53\) 8.33513 1.14492 0.572459 − 0.819934i \(-0.305990\pi\)
0.572459 + 0.819934i \(0.305990\pi\)
\(54\) −3.15883 −0.429863
\(55\) −0.264438 −0.0356568
\(56\) −0.109916 −0.0146882
\(57\) −0.719169 −0.0952562
\(58\) 0 0
\(59\) −5.08815 −0.662420 −0.331210 − 0.943557i \(-0.607457\pi\)
−0.331210 + 0.943557i \(0.607457\pi\)
\(60\) −0.109916 −0.0141901
\(61\) −11.0586 −1.41591 −0.707955 − 0.706258i \(-0.750382\pi\)
−0.707955 + 0.706258i \(0.750382\pi\)
\(62\) 9.34481 1.18679
\(63\) −0.295897 −0.0372795
\(64\) 1.00000 0.125000
\(65\) 0.780167 0.0967679
\(66\) −0.740939 −0.0912033
\(67\) −11.0000 −1.34386 −0.671932 − 0.740613i \(-0.734535\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) −2.91185 −0.353114
\(69\) −4.32304 −0.520433
\(70\) −0.0217703 −0.00260204
\(71\) 10.9487 1.29937 0.649685 − 0.760203i \(-0.274901\pi\)
0.649685 + 0.760203i \(0.274901\pi\)
\(72\) 2.69202 0.317258
\(73\) 7.94869 0.930324 0.465162 − 0.885226i \(-0.345996\pi\)
0.465162 + 0.885226i \(0.345996\pi\)
\(74\) 3.02715 0.351899
\(75\) 2.75302 0.317891
\(76\) 1.29590 0.148650
\(77\) −0.146752 −0.0167239
\(78\) 2.18598 0.247514
\(79\) 4.89008 0.550177 0.275089 − 0.961419i \(-0.411293\pi\)
0.275089 + 0.961419i \(0.411293\pi\)
\(80\) 0.198062 0.0221440
\(81\) 6.32304 0.702560
\(82\) −3.76271 −0.415522
\(83\) −11.6746 −1.28145 −0.640725 − 0.767771i \(-0.721366\pi\)
−0.640725 + 0.767771i \(0.721366\pi\)
\(84\) −0.0609989 −0.00665553
\(85\) −0.576728 −0.0625550
\(86\) 6.66487 0.718692
\(87\) 0 0
\(88\) 1.33513 0.142325
\(89\) −11.4940 −1.21836 −0.609179 − 0.793033i \(-0.708501\pi\)
−0.609179 + 0.793033i \(0.708501\pi\)
\(90\) 0.533188 0.0562029
\(91\) 0.432960 0.0453866
\(92\) 7.78986 0.812149
\(93\) 5.18598 0.537761
\(94\) −0.801938 −0.0827136
\(95\) 0.256668 0.0263336
\(96\) 0.554958 0.0566402
\(97\) −10.5918 −1.07543 −0.537717 − 0.843125i \(-0.680713\pi\)
−0.537717 + 0.843125i \(0.680713\pi\)
\(98\) 6.98792 0.705886
\(99\) 3.59419 0.361229
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.m.1.2 3
29.12 odd 4 1682.2.b.g.1681.2 6
29.17 odd 4 1682.2.b.g.1681.5 6
29.23 even 7 58.2.d.a.7.1 ✓ 6
29.24 even 7 58.2.d.a.25.1 yes 6
29.28 even 2 1682.2.a.n.1.2 3
87.23 odd 14 522.2.k.c.181.1 6
87.53 odd 14 522.2.k.c.199.1 6
116.23 odd 14 464.2.u.b.65.1 6
116.111 odd 14 464.2.u.b.257.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.a.7.1 ✓ 6 29.23 even 7
58.2.d.a.25.1 yes 6 29.24 even 7
464.2.u.b.65.1 6 116.23 odd 14
464.2.u.b.257.1 6 116.111 odd 14
522.2.k.c.181.1 6 87.23 odd 14
522.2.k.c.199.1 6 87.53 odd 14
1682.2.a.m.1.2 3 1.1 even 1 trivial
1682.2.a.n.1.2 3 29.28 even 2
1682.2.b.g.1681.2 6 29.12 odd 4
1682.2.b.g.1681.5 6 29.17 odd 4