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Properties

Label 1682.2.a.q.1.4
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 1.4
Root \(3.41744\) 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 \(-0.390289\pi\)
0.337884 + 0.941188i \(0.390289\pi\)
\(4\) 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 \(-0.468780\pi\)
0.0979247 + 0.995194i \(0.468780\pi\)
\(12\) 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 \(-0.143679\pi\)
0.899846 + 0.436208i \(0.143679\pi\)
\(18\) 1.63001 0.384197
\(19\) −6.63001 −1.52103 −0.760514 − 0.649321i \(-0.775053\pi\)
−0.760514 + 0.649321i \(0.775053\pi\)
\(20\) −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 \(-0.680465\pi\)
−0.537060 + 0.843544i \(0.680465\pi\)
\(32\) −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 \(-0.434591\pi\)
0.204046 + 0.978961i \(0.434591\pi\)
\(38\) 6.63001 1.07553
\(39\) 0.577740 0.0925124
\(40\) 2.97240 0.469978
\(41\) −7.82245 −1.22166 −0.610830 − 0.791761i \(-0.709164\pi\)
−0.610830 + 0.791761i \(0.709164\pi\)
\(42\) −0.609702 −0.0940791
\(43\) −0.649559 −0.0990568 −0.0495284 − 0.998773i \(-0.515772\pi\)
−0.0495284 + 0.998773i \(0.515772\pi\)
\(44\) 0.649559 0.0979247
\(45\) 4.84505 0.722257
\(46\) −7.61793 −1.12320
\(47\) −8.79595 −1.28302 −0.641511 − 0.767114i \(-0.721692\pi\)
−0.641511 + 0.767114i \(0.721692\pi\)
\(48\) 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 \(-0.713187\pi\)
−0.620788 + 0.783979i \(0.713187\pi\)
\(62\) 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 \(-0.670909\pi\)
−0.511496 + 0.859286i \(0.670909\pi\)
\(74\) −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 \(-0.574607\pi\)
−0.232244 + 0.972657i \(0.574607\pi\)
\(80\) −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 \(-0.479866\pi\)
0.0632107 + 0.998000i \(0.479866\pi\)
\(90\) −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 \(-0.703104\pi\)
−0.595647 + 0.803246i \(0.703104\pi\)
\(98\) 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.q.1.4 6
29.9 even 14 58.2.d.b.23.2 ✓ 12
29.12 odd 4 1682.2.b.i.1681.4 12
29.13 even 14 58.2.d.b.53.2 yes 12
29.17 odd 4 1682.2.b.i.1681.9 12
29.28 even 2 1682.2.a.t.1.3 6
87.38 odd 14 522.2.k.h.487.2 12
87.71 odd 14 522.2.k.h.343.2 12
116.67 odd 14 464.2.u.h.81.1 12
116.71 odd 14 464.2.u.h.401.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.23.2 ✓ 12 29.9 even 14
58.2.d.b.53.2 yes 12 29.13 even 14
464.2.u.h.81.1 12 116.67 odd 14
464.2.u.h.401.1 12 116.71 odd 14
522.2.k.h.343.2 12 87.71 odd 14
522.2.k.h.487.2 12 87.38 odd 14
1682.2.a.q.1.4 6 1.1 even 1 trivial
1682.2.a.t.1.3 6 29.28 even 2
1682.2.b.i.1681.4 12 29.12 odd 4
1682.2.b.i.1681.9 12 29.17 odd 4