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Properties

Label 1682.2.a.t.1.1
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 1.1
Root \(-2.44077\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.44077 q^{3} +1.00000 q^{4} -2.88581 q^{5} -2.44077 q^{6} -3.04359 q^{7} +1.00000 q^{8} +2.95734 q^{9} -2.88581 q^{10} -5.48435 q^{11} -2.44077 q^{12} -0.107544 q^{13} -3.04359 q^{14} +7.04359 q^{15} +1.00000 q^{16} +0.816005 q^{17} +2.95734 q^{18} +2.04266 q^{19} -2.88581 q^{20} +7.42869 q^{21} -5.48435 q^{22} -9.16509 q^{23} -2.44077 q^{24} +3.32789 q^{25} -0.107544 q^{26} +0.104116 q^{27} -3.04359 q^{28} +7.04359 q^{30} -3.44170 q^{31} +1.00000 q^{32} +13.3860 q^{33} +0.816005 q^{34} +8.78321 q^{35} +2.95734 q^{36} +5.90758 q^{37} +2.04266 q^{38} +0.262489 q^{39} -2.88581 q^{40} -2.43376 q^{41} +7.42869 q^{42} +5.48435 q^{43} -5.48435 q^{44} -8.53433 q^{45} -9.16509 q^{46} +5.91000 q^{47} -2.44077 q^{48} +2.26342 q^{49} +3.32789 q^{50} -1.99168 q^{51} -0.107544 q^{52} +3.95070 q^{53} +0.104116 q^{54} +15.8268 q^{55} -3.04359 q^{56} -4.98565 q^{57} +1.13359 q^{59} +7.04359 q^{60} +8.16584 q^{61} -3.44170 q^{62} -9.00093 q^{63} +1.00000 q^{64} +0.310351 q^{65} +13.3860 q^{66} +0.956413 q^{67} +0.816005 q^{68} +22.3699 q^{69} +8.78321 q^{70} +2.38979 q^{71} +2.95734 q^{72} +8.89410 q^{73} +5.90758 q^{74} -8.12261 q^{75} +2.04266 q^{76} +16.6921 q^{77} +0.262489 q^{78} -4.20062 q^{79} -2.88581 q^{80} -9.12615 q^{81} -2.43376 q^{82} -15.4732 q^{83} +7.42869 q^{84} -2.35483 q^{85} +5.48435 q^{86} -5.48435 q^{88} +5.54154 q^{89} -8.53433 q^{90} +0.327319 q^{91} -9.16509 q^{92} +8.40038 q^{93} +5.91000 q^{94} -5.89472 q^{95} -2.44077 q^{96} +11.9217 q^{97} +2.26342 q^{98} -16.2191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{6} - 3 q^{7} + 6 q^{8} + 12 q^{9} - q^{11} + 2 q^{12} - 3 q^{13} - 3 q^{14} + 27 q^{15} + 6 q^{16} - 6 q^{17} + 12 q^{18} + 18 q^{19} - 10 q^{21} - q^{22} - 14 q^{23}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
\(3\) −2.44077 −1.40918 −0.704589 − 0.709616i \(-0.748868\pi\)
−0.704589 + 0.709616i \(0.748868\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.88581 −1.29057 −0.645286 − 0.763941i \(-0.723262\pi\)
−0.645286 + 0.763941i \(0.723262\pi\)
\(6\) −2.44077 −0.996439
\(7\) −3.04359 −1.15037 −0.575184 − 0.818024i \(-0.695069\pi\)
−0.575184 + 0.818024i \(0.695069\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.95734 0.985781
\(10\) −2.88581 −0.912573
\(11\) −5.48435 −1.65359 −0.826797 − 0.562500i \(-0.809840\pi\)
−0.826797 + 0.562500i \(0.809840\pi\)
\(12\) −2.44077 −0.704589
\(13\) −0.107544 −0.0298273 −0.0149136 − 0.999889i \(-0.504747\pi\)
−0.0149136 + 0.999889i \(0.504747\pi\)
\(14\) −3.04359 −0.813433
\(15\) 7.04359 1.81865
\(16\) 1.00000 0.250000
\(17\) 0.816005 0.197910 0.0989551 − 0.995092i \(-0.468450\pi\)
0.0989551 + 0.995092i \(0.468450\pi\)
\(18\) 2.95734 0.697052
\(19\) 2.04266 0.468618 0.234309 − 0.972162i \(-0.424717\pi\)
0.234309 + 0.972162i \(0.424717\pi\)
\(20\) −2.88581 −0.645286
\(21\) 7.42869 1.62107
\(22\) −5.48435 −1.16927
\(23\) −9.16509 −1.91105 −0.955527 − 0.294903i \(-0.904713\pi\)
−0.955527 + 0.294903i \(0.904713\pi\)
\(24\) −2.44077 −0.498219
\(25\) 3.32789 0.665578
\(26\) −0.107544 −0.0210911
\(27\) 0.104116 0.0200371
\(28\) −3.04359 −0.575184
\(29\) 0 0
\(30\) 7.04359 1.28598
\(31\) −3.44170 −0.618147 −0.309073 − 0.951038i \(-0.600019\pi\)
−0.309073 + 0.951038i \(0.600019\pi\)
\(32\) 1.00000 0.176777
\(33\) 13.3860 2.33021
\(34\) 0.816005 0.139944
\(35\) 8.78321 1.48463
\(36\) 2.95734 0.492891
\(37\) 5.90758 0.971200 0.485600 − 0.874181i \(-0.338601\pi\)
0.485600 + 0.874181i \(0.338601\pi\)
\(38\) 2.04266 0.331363
\(39\) 0.262489 0.0420319
\(40\) −2.88581 −0.456286
\(41\) −2.43376 −0.380090 −0.190045 − 0.981775i \(-0.560863\pi\)
−0.190045 + 0.981775i \(0.560863\pi\)
\(42\) 7.42869 1.14627
\(43\) 5.48435 0.836356 0.418178 − 0.908365i \(-0.362669\pi\)
0.418178 + 0.908365i \(0.362669\pi\)
\(44\) −5.48435 −0.826797
\(45\) −8.53433 −1.27222
\(46\) −9.16509 −1.35132
\(47\) 5.91000 0.862062 0.431031 − 0.902337i \(-0.358150\pi\)
0.431031 + 0.902337i \(0.358150\pi\)
\(48\) −2.44077 −0.352294
\(49\) 2.26342 0.323346
\(50\) 3.32789 0.470635
\(51\) −1.99168 −0.278891
\(52\) −0.107544 −0.0149136
\(53\) 3.95070 0.542670 0.271335 − 0.962485i \(-0.412535\pi\)
0.271335 + 0.962485i \(0.412535\pi\)
\(54\) 0.104116 0.0141683
\(55\) 15.8268 2.13408
\(56\) −3.04359 −0.406716
\(57\) −4.98565 −0.660365
\(58\) 0 0
\(59\) 1.13359 0.147581 0.0737904 − 0.997274i \(-0.476490\pi\)
0.0737904 + 0.997274i \(0.476490\pi\)
\(60\) 7.04359 0.909323
\(61\) 8.16584 1.04553 0.522764 − 0.852477i \(-0.324901\pi\)
0.522764 + 0.852477i \(0.324901\pi\)
\(62\) −3.44170 −0.437096
\(63\) −9.00093 −1.13401
\(64\) 1.00000 0.125000
\(65\) 0.310351 0.0384943
\(66\) 13.3860 1.64771
\(67\) 0.956413 0.116844 0.0584222 − 0.998292i \(-0.481393\pi\)
0.0584222 + 0.998292i \(0.481393\pi\)
\(68\) 0.816005 0.0989551
\(69\) 22.3699 2.69301
\(70\) 8.78321 1.04979
\(71\) 2.38979 0.283616 0.141808 − 0.989894i \(-0.454709\pi\)
0.141808 + 0.989894i \(0.454709\pi\)
\(72\) 2.95734 0.348526
\(73\) 8.89410 1.04098 0.520488 − 0.853869i \(-0.325750\pi\)
0.520488 + 0.853869i \(0.325750\pi\)
\(74\) 5.90758 0.686742
\(75\) −8.12261 −0.937918
\(76\) 2.04266 0.234309
\(77\) 16.6921 1.90224
\(78\) 0.262489 0.0297211
\(79\) −4.20062 −0.472607 −0.236303 − 0.971679i \(-0.575936\pi\)
−0.236303 + 0.971679i \(0.575936\pi\)
\(80\) −2.88581 −0.322643
\(81\) −9.12615 −1.01402
\(82\) −2.43376 −0.268764
\(83\) −15.4732 −1.69840 −0.849202 − 0.528068i \(-0.822917\pi\)
−0.849202 + 0.528068i \(0.822917\pi\)
\(84\) 7.42869 0.810536
\(85\) −2.35483 −0.255418
\(86\) 5.48435 0.591393
\(87\) 0 0
\(88\) −5.48435 −0.584634
\(89\) 5.54154 0.587402 0.293701 − 0.955897i \(-0.405113\pi\)
0.293701 + 0.955897i \(0.405113\pi\)
\(90\) −8.53433 −0.899597
\(91\) 0.327319 0.0343124
\(92\) −9.16509 −0.955527
\(93\) 8.40038 0.871079
\(94\) 5.91000 0.609570
\(95\) −5.89472 −0.604785
\(96\) −2.44077 −0.249110
\(97\) 11.9217 1.21046 0.605232 − 0.796049i \(-0.293080\pi\)
0.605232 + 0.796049i \(0.293080\pi\)
\(98\) 2.26342 0.228640
\(99\) −16.2191 −1.63008
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.1.1 6
29.7 even 7 58.2.d.b.49.1 yes 12
29.12 odd 4 1682.2.b.i.1681.7 12
29.17 odd 4 1682.2.b.i.1681.6 12
29.25 even 7 58.2.d.b.45.1 ✓ 12
29.28 even 2 1682.2.a.q.1.6 6
87.65 odd 14 522.2.k.h.397.1 12
87.83 odd 14 522.2.k.h.451.1 12
116.7 odd 14 464.2.u.h.49.2 12
116.83 odd 14 464.2.u.h.161.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.45.1 ✓ 12 29.25 even 7
58.2.d.b.49.1 yes 12 29.7 even 7
464.2.u.h.49.2 12 116.7 odd 14
464.2.u.h.161.2 12 116.83 odd 14
522.2.k.h.397.1 12 87.65 odd 14
522.2.k.h.451.1 12 87.83 odd 14
1682.2.a.q.1.6 6 29.28 even 2
1682.2.a.t.1.1 6 1.1 even 1 trivial
1682.2.b.i.1681.6 12 29.17 odd 4
1682.2.b.i.1681.7 12 29.12 odd 4