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Properties

Label 1682.2.a.t.1.3
Level $1682$
Weight $2$
Character 1682.1
Self dual yes
Analytic conductor $13.431$
Analytic rank $0$
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: \(0\)
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 \(-1.17047\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.17047 q^{3} +1.00000 q^{4} -2.97240 q^{5} -1.17047 q^{6} +0.520906 q^{7} +1.00000 q^{8} -1.63001 q^{9} -2.97240 q^{10} -0.649559 q^{11} -1.17047 q^{12} +0.493598 q^{13} +0.520906 q^{14} +3.47909 q^{15} +1.00000 q^{16} -7.42032 q^{17} -1.63001 q^{18} +6.63001 q^{19} -2.97240 q^{20} -0.609702 q^{21} -0.649559 q^{22} +7.61793 q^{23} -1.17047 q^{24} +3.83518 q^{25} +0.493598 q^{26} +5.41927 q^{27} +0.520906 q^{28} +3.47909 q^{30} +5.98045 q^{31} +1.00000 q^{32} +0.760286 q^{33} -7.42032 q^{34} -1.54834 q^{35} -1.63001 q^{36} -2.48233 q^{37} +6.63001 q^{38} -0.577740 q^{39} -2.97240 q^{40} +7.82245 q^{41} -0.609702 q^{42} +0.649559 q^{43} -0.649559 q^{44} +4.84505 q^{45} +7.61793 q^{46} +8.79595 q^{47} -1.17047 q^{48} -6.72866 q^{49} +3.83518 q^{50} +8.68522 q^{51} +0.493598 q^{52} -1.15121 q^{53} +5.41927 q^{54} +1.93075 q^{55} +0.520906 q^{56} -7.76020 q^{57} -5.31686 q^{59} +3.47909 q^{60} +9.69702 q^{61} +5.98045 q^{62} -0.849083 q^{63} +1.00000 q^{64} -1.46717 q^{65} +0.760286 q^{66} +4.52091 q^{67} -7.42032 q^{68} -8.91652 q^{69} -1.54834 q^{70} -14.1448 q^{71} -1.63001 q^{72} +8.74045 q^{73} -2.48233 q^{74} -4.48894 q^{75} +6.63001 q^{76} -0.338359 q^{77} -0.577740 q^{78} +4.12847 q^{79} -2.97240 q^{80} -1.45303 q^{81} +7.82245 q^{82} +4.28153 q^{83} -0.609702 q^{84} +22.0562 q^{85} +0.649559 q^{86} -0.649559 q^{88} -1.19266 q^{89} +4.84505 q^{90} +0.257118 q^{91} +7.61793 q^{92} -6.99991 q^{93} +8.79595 q^{94} -19.7071 q^{95} -1.17047 q^{96} +11.7329 q^{97} -6.72866 q^{98} +1.05879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{6} - 3 q^{7} + 6 q^{8} + 12 q^{9} - q^{11} + 2 q^{12} - 3 q^{13} - 3 q^{14} + 27 q^{15} + 6 q^{16} - 6 q^{17} + 12 q^{18} + 18 q^{19} - 10 q^{21} - q^{22} - 14 q^{23}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.17047 −0.675768 −0.337884 − 0.941188i \(-0.609711\pi\)
−0.337884 + 0.941188i \(0.609711\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.97240 −1.32930 −0.664649 − 0.747155i \(-0.731419\pi\)
−0.664649 + 0.747155i \(0.731419\pi\)
\(6\) −1.17047 −0.477840
\(7\) 0.520906 0.196884 0.0984420 − 0.995143i \(-0.468614\pi\)
0.0984420 + 0.995143i \(0.468614\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.63001 −0.543337
\(10\) −2.97240 −0.939956
\(11\) −0.649559 −0.195849 −0.0979247 − 0.995194i \(-0.531220\pi\)
−0.0979247 + 0.995194i \(0.531220\pi\)
\(12\) −1.17047 −0.337884
\(13\) 0.493598 0.136900 0.0684498 − 0.997655i \(-0.478195\pi\)
0.0684498 + 0.997655i \(0.478195\pi\)
\(14\) 0.520906 0.139218
\(15\) 3.47909 0.898298
\(16\) 1.00000 0.250000
\(17\) −7.42032 −1.79969 −0.899846 − 0.436208i \(-0.856321\pi\)
−0.899846 + 0.436208i \(0.856321\pi\)
\(18\) −1.63001 −0.384197
\(19\) 6.63001 1.52103 0.760514 − 0.649321i \(-0.224947\pi\)
0.760514 + 0.649321i \(0.224947\pi\)
\(20\) −2.97240 −0.664649
\(21\) −0.609702 −0.133048
\(22\) −0.649559 −0.138486
\(23\) 7.61793 1.58845 0.794224 − 0.607625i \(-0.207878\pi\)
0.794224 + 0.607625i \(0.207878\pi\)
\(24\) −1.17047 −0.238920
\(25\) 3.83518 0.767036
\(26\) 0.493598 0.0968026
\(27\) 5.41927 1.04294
\(28\) 0.520906 0.0984420
\(29\) 0 0
\(30\) 3.47909 0.635193
\(31\) 5.98045 1.07412 0.537060 − 0.843544i \(-0.319535\pi\)
0.537060 + 0.843544i \(0.319535\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.760286 0.132349
\(34\) −7.42032 −1.27257
\(35\) −1.54834 −0.261718
\(36\) −1.63001 −0.271669
\(37\) −2.48233 −0.408092 −0.204046 − 0.978961i \(-0.565409\pi\)
−0.204046 + 0.978961i \(0.565409\pi\)
\(38\) 6.63001 1.07553
\(39\) −0.577740 −0.0925124
\(40\) −2.97240 −0.469978
\(41\) 7.82245 1.22166 0.610830 − 0.791761i \(-0.290836\pi\)
0.610830 + 0.791761i \(0.290836\pi\)
\(42\) −0.609702 −0.0940791
\(43\) 0.649559 0.0990568 0.0495284 − 0.998773i \(-0.484228\pi\)
0.0495284 + 0.998773i \(0.484228\pi\)
\(44\) −0.649559 −0.0979247
\(45\) 4.84505 0.722257
\(46\) 7.61793 1.12320
\(47\) 8.79595 1.28302 0.641511 − 0.767114i \(-0.278308\pi\)
0.641511 + 0.767114i \(0.278308\pi\)
\(48\) −1.17047 −0.168942
\(49\) −6.72866 −0.961237
\(50\) 3.83518 0.542376
\(51\) 8.68522 1.21617
\(52\) 0.493598 0.0684498
\(53\) −1.15121 −0.158130 −0.0790652 − 0.996869i \(-0.525194\pi\)
−0.0790652 + 0.996869i \(0.525194\pi\)
\(54\) 5.41927 0.737469
\(55\) 1.93075 0.260342
\(56\) 0.520906 0.0696090
\(57\) −7.76020 −1.02786
\(58\) 0 0
\(59\) −5.31686 −0.692196 −0.346098 − 0.938198i \(-0.612493\pi\)
−0.346098 + 0.938198i \(0.612493\pi\)
\(60\) 3.47909 0.449149
\(61\) 9.69702 1.24158 0.620788 − 0.783979i \(-0.286813\pi\)
0.620788 + 0.783979i \(0.286813\pi\)
\(62\) 5.98045 0.759518
\(63\) −0.849083 −0.106974
\(64\) 1.00000 0.125000
\(65\) −1.46717 −0.181980
\(66\) 0.760286 0.0935848
\(67\) 4.52091 0.552317 0.276158 − 0.961112i \(-0.410939\pi\)
0.276158 + 0.961112i \(0.410939\pi\)
\(68\) −7.42032 −0.899846
\(69\) −8.91652 −1.07342
\(70\) −1.54834 −0.185062
\(71\) −14.1448 −1.67868 −0.839338 − 0.543610i \(-0.817057\pi\)
−0.839338 + 0.543610i \(0.817057\pi\)
\(72\) −1.63001 −0.192099
\(73\) 8.74045 1.02299 0.511496 − 0.859286i \(-0.329091\pi\)
0.511496 + 0.859286i \(0.329091\pi\)
\(74\) −2.48233 −0.288565
\(75\) −4.48894 −0.518339
\(76\) 6.63001 0.760514
\(77\) −0.338359 −0.0385596
\(78\) −0.577740 −0.0654161
\(79\) 4.12847 0.464489 0.232244 − 0.972657i \(-0.425393\pi\)
0.232244 + 0.972657i \(0.425393\pi\)
\(80\) −2.97240 −0.332325
\(81\) −1.45303 −0.161448
\(82\) 7.82245 0.863845
\(83\) 4.28153 0.469958 0.234979 − 0.972000i \(-0.424498\pi\)
0.234979 + 0.972000i \(0.424498\pi\)
\(84\) −0.609702 −0.0665240
\(85\) 22.0562 2.39233
\(86\) 0.649559 0.0700438
\(87\) 0 0
\(88\) −0.649559 −0.0692432
\(89\) −1.19266 −0.126421 −0.0632107 − 0.998000i \(-0.520134\pi\)
−0.0632107 + 0.998000i \(0.520134\pi\)
\(90\) 4.84505 0.510713
\(91\) 0.257118 0.0269533
\(92\) 7.61793 0.794224
\(93\) −6.99991 −0.725857
\(94\) 8.79595 0.907233
\(95\) −19.7071 −2.02190
\(96\) −1.17047 −0.119460
\(97\) 11.7329 1.19129 0.595647 − 0.803246i \(-0.296896\pi\)
0.595647 + 0.803246i \(0.296896\pi\)
\(98\) −6.72866 −0.679697
\(99\) 1.05879 0.106412
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.t.1.3 6
29.12 odd 4 1682.2.b.i.1681.9 12
29.16 even 7 58.2.d.b.53.2 yes 12
29.17 odd 4 1682.2.b.i.1681.4 12
29.20 even 7 58.2.d.b.23.2 ✓ 12
29.28 even 2 1682.2.a.q.1.4 6
87.20 odd 14 522.2.k.h.487.2 12
87.74 odd 14 522.2.k.h.343.2 12
116.103 odd 14 464.2.u.h.401.1 12
116.107 odd 14 464.2.u.h.81.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.23.2 ✓ 12 29.20 even 7
58.2.d.b.53.2 yes 12 29.16 even 7
464.2.u.h.81.1 12 116.107 odd 14
464.2.u.h.401.1 12 116.103 odd 14
522.2.k.h.343.2 12 87.74 odd 14
522.2.k.h.487.2 12 87.20 odd 14
1682.2.a.q.1.4 6 29.28 even 2
1682.2.a.t.1.3 6 1.1 even 1 trivial
1682.2.b.i.1681.4 12 29.17 odd 4
1682.2.b.i.1681.9 12 29.12 odd 4