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Properties

Label 1682.2.a.q.1.1
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 1.1
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.41744 q^{3} +1.00000 q^{4} +1.61551 q^{5} +3.41744 q^{6} -1.52091 q^{7} -1.00000 q^{8} +8.67893 q^{9} -1.61551 q^{10} -1.89654 q^{11} -3.41744 q^{12} -3.18562 q^{13} +1.52091 q^{14} -5.52091 q^{15} +1.00000 q^{16} +1.69929 q^{17} -8.67893 q^{18} +3.67893 q^{19} +1.61551 q^{20} +5.19761 q^{21} +1.89654 q^{22} -2.69101 q^{23} +3.41744 q^{24} -2.39014 q^{25} +3.18562 q^{26} -19.4074 q^{27} -1.52091 q^{28} +5.52091 q^{30} +1.78239 q^{31} -1.00000 q^{32} +6.48132 q^{33} -1.69929 q^{34} -2.45703 q^{35} +8.67893 q^{36} +7.07024 q^{37} -3.67893 q^{38} +10.8867 q^{39} -1.61551 q^{40} +5.03259 q^{41} -5.19761 q^{42} +1.89654 q^{43} -1.89654 q^{44} +14.0209 q^{45} +2.69101 q^{46} -2.57063 q^{47} -3.41744 q^{48} -4.68684 q^{49} +2.39014 q^{50} -5.80722 q^{51} -3.18562 q^{52} +8.24904 q^{53} +19.4074 q^{54} -3.06387 q^{55} +1.52091 q^{56} -12.5725 q^{57} +2.95027 q^{59} -5.52091 q^{60} +7.74593 q^{61} -1.78239 q^{62} -13.1998 q^{63} +1.00000 q^{64} -5.14639 q^{65} -6.48132 q^{66} +2.47909 q^{67} +1.69929 q^{68} +9.19638 q^{69} +2.45703 q^{70} +6.06870 q^{71} -8.67893 q^{72} -3.74813 q^{73} -7.07024 q^{74} +8.16816 q^{75} +3.67893 q^{76} +2.88446 q^{77} -10.8867 q^{78} -9.22066 q^{79} +1.61551 q^{80} +40.2870 q^{81} -5.03259 q^{82} -5.52313 q^{83} +5.19761 q^{84} +2.74521 q^{85} -1.89654 q^{86} +1.89654 q^{88} -15.9704 q^{89} -14.0209 q^{90} +4.84503 q^{91} -2.69101 q^{92} -6.09122 q^{93} +2.57063 q^{94} +5.94334 q^{95} +3.41744 q^{96} -14.9078 q^{97} +4.68684 q^{98} -16.4599 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9} + q^{11} - 2 q^{12} - 3 q^{13} + 3 q^{14} - 27 q^{15} + 6 q^{16} + 6 q^{17} - 12 q^{18} - 18 q^{19} + 10 q^{21} - q^{22} - 14 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) −3.41744 −1.97306 −0.986531 − 0.163572i \(-0.947698\pi\)
−0.986531 + 0.163572i \(0.947698\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.61551 0.722477 0.361238 − 0.932474i \(-0.382354\pi\)
0.361238 + 0.932474i \(0.382354\pi\)
\(6\) 3.41744 1.39517
\(7\) −1.52091 −0.574848 −0.287424 − 0.957803i \(-0.592799\pi\)
−0.287424 + 0.957803i \(0.592799\pi\)
\(8\) −1.00000 −0.353553
\(9\) 8.67893 2.89298
\(10\) −1.61551 −0.510868
\(11\) −1.89654 −0.571828 −0.285914 − 0.958255i \(-0.592297\pi\)
−0.285914 + 0.958255i \(0.592297\pi\)
\(12\) −3.41744 −0.986531
\(13\) −3.18562 −0.883532 −0.441766 − 0.897130i \(-0.645648\pi\)
−0.441766 + 0.897130i \(0.645648\pi\)
\(14\) 1.52091 0.406479
\(15\) −5.52091 −1.42549
\(16\) 1.00000 0.250000
\(17\) 1.69929 0.412138 0.206069 − 0.978537i \(-0.433933\pi\)
0.206069 + 0.978537i \(0.433933\pi\)
\(18\) −8.67893 −2.04564
\(19\) 3.67893 0.844004 0.422002 − 0.906595i \(-0.361327\pi\)
0.422002 + 0.906595i \(0.361327\pi\)
\(20\) 1.61551 0.361238
\(21\) 5.19761 1.13421
\(22\) 1.89654 0.404343
\(23\) −2.69101 −0.561114 −0.280557 − 0.959837i \(-0.590519\pi\)
−0.280557 + 0.959837i \(0.590519\pi\)
\(24\) 3.41744 0.697583
\(25\) −2.39014 −0.478027
\(26\) 3.18562 0.624751
\(27\) −19.4074 −3.73496
\(28\) −1.52091 −0.287424
\(29\) 0 0
\(30\) 5.52091 1.00797
\(31\) 1.78239 0.320127 0.160063 − 0.987107i \(-0.448830\pi\)
0.160063 + 0.987107i \(0.448830\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.48132 1.12825
\(34\) −1.69929 −0.291425
\(35\) −2.45703 −0.415315
\(36\) 8.67893 1.44649
\(37\) 7.07024 1.16234 0.581170 − 0.813782i \(-0.302595\pi\)
0.581170 + 0.813782i \(0.302595\pi\)
\(38\) −3.67893 −0.596801
\(39\) 10.8867 1.74326
\(40\) −1.61551 −0.255434
\(41\) 5.03259 0.785959 0.392979 − 0.919547i \(-0.371444\pi\)
0.392979 + 0.919547i \(0.371444\pi\)
\(42\) −5.19761 −0.802009
\(43\) 1.89654 0.289219 0.144610 − 0.989489i \(-0.453807\pi\)
0.144610 + 0.989489i \(0.453807\pi\)
\(44\) −1.89654 −0.285914
\(45\) 14.0209 2.09011
\(46\) 2.69101 0.396768
\(47\) −2.57063 −0.374966 −0.187483 − 0.982268i \(-0.560033\pi\)
−0.187483 + 0.982268i \(0.560033\pi\)
\(48\) −3.41744 −0.493266
\(49\) −4.68684 −0.669549
\(50\) 2.39014 0.338016
\(51\) −5.80722 −0.813174
\(52\) −3.18562 −0.441766
\(53\) 8.24904 1.13309 0.566546 − 0.824030i \(-0.308279\pi\)
0.566546 + 0.824030i \(0.308279\pi\)
\(54\) 19.4074 2.64102
\(55\) −3.06387 −0.413132
\(56\) 1.52091 0.203240
\(57\) −12.5725 −1.66527
\(58\) 0 0
\(59\) 2.95027 0.384093 0.192046 − 0.981386i \(-0.438488\pi\)
0.192046 + 0.981386i \(0.438488\pi\)
\(60\) −5.52091 −0.712746
\(61\) 7.74593 0.991765 0.495883 − 0.868390i \(-0.334845\pi\)
0.495883 + 0.868390i \(0.334845\pi\)
\(62\) −1.78239 −0.226364
\(63\) −13.1998 −1.66302
\(64\) 1.00000 0.125000
\(65\) −5.14639 −0.638331
\(66\) −6.48132 −0.797795
\(67\) 2.47909 0.302870 0.151435 − 0.988467i \(-0.451611\pi\)
0.151435 + 0.988467i \(0.451611\pi\)
\(68\) 1.69929 0.206069
\(69\) 9.19638 1.10711
\(70\) 2.45703 0.293672
\(71\) 6.06870 0.720223 0.360111 − 0.932909i \(-0.382739\pi\)
0.360111 + 0.932909i \(0.382739\pi\)
\(72\) −8.67893 −1.02282
\(73\) −3.74813 −0.438686 −0.219343 − 0.975648i \(-0.570391\pi\)
−0.219343 + 0.975648i \(0.570391\pi\)
\(74\) −7.07024 −0.821898
\(75\) 8.16816 0.943178
\(76\) 3.67893 0.422002
\(77\) 2.88446 0.328714
\(78\) −10.8867 −1.23267
\(79\) −9.22066 −1.03741 −0.518703 − 0.854955i \(-0.673585\pi\)
−0.518703 + 0.854955i \(0.673585\pi\)
\(80\) 1.61551 0.180619
\(81\) 40.2870 4.47634
\(82\) −5.03259 −0.555757
\(83\) −5.52313 −0.606242 −0.303121 − 0.952952i \(-0.598029\pi\)
−0.303121 + 0.952952i \(0.598029\pi\)
\(84\) 5.19761 0.567106
\(85\) 2.74521 0.297760
\(86\) −1.89654 −0.204509
\(87\) 0 0
\(88\) 1.89654 0.202172
\(89\) −15.9704 −1.69286 −0.846431 − 0.532498i \(-0.821253\pi\)
−0.846431 + 0.532498i \(0.821253\pi\)
\(90\) −14.0209 −1.47793
\(91\) 4.84503 0.507897
\(92\) −2.69101 −0.280557
\(93\) −6.09122 −0.631630
\(94\) 2.57063 0.265141
\(95\) 5.94334 0.609773
\(96\) 3.41744 0.348791
\(97\) −14.9078 −1.51366 −0.756830 − 0.653612i \(-0.773253\pi\)
−0.756830 + 0.653612i \(0.773253\pi\)
\(98\) 4.68684 0.473443
\(99\) −16.4599 −1.65428
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.1.1 6
29.9 even 14 58.2.d.b.23.1 ✓ 12
29.12 odd 4 1682.2.b.i.1681.1 12
29.13 even 14 58.2.d.b.53.1 yes 12
29.17 odd 4 1682.2.b.i.1681.12 12
29.28 even 2 1682.2.a.t.1.6 6
87.38 odd 14 522.2.k.h.487.1 12
87.71 odd 14 522.2.k.h.343.1 12
116.67 odd 14 464.2.u.h.81.2 12
116.71 odd 14 464.2.u.h.401.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.23.1 ✓ 12 29.9 even 14
58.2.d.b.53.1 yes 12 29.13 even 14
464.2.u.h.81.2 12 116.67 odd 14
464.2.u.h.401.2 12 116.71 odd 14
522.2.k.h.343.1 12 87.71 odd 14
522.2.k.h.487.1 12 87.38 odd 14
1682.2.a.q.1.1 6 1.1 even 1 trivial
1682.2.a.t.1.6 6 29.28 even 2
1682.2.b.i.1681.1 12 29.12 odd 4
1682.2.b.i.1681.12 12 29.17 odd 4