Properties

Label 83.4.a.b.1.12
Level $83$
Weight $4$
Character 83.1
Self dual yes
Analytic conductor $4.897$
Analytic rank $0$
Dimension $13$
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] = [83,4,Mod(1,83)] 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("83.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(83, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 83.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.89715853048\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 76 x^{11} + 374 x^{10} + 2129 x^{9} - 9943 x^{8} - 28670 x^{7} + 114750 x^{6} + \cdots - 6976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(4.97334\) of defining polynomial
Character \(\chi\) \(=\) 83.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.97334 q^{2} +1.07884 q^{3} +16.7341 q^{4} -3.52903 q^{5} +5.36546 q^{6} +7.39925 q^{7} +43.4375 q^{8} -25.8361 q^{9} -17.5510 q^{10} +52.5993 q^{11} +18.0535 q^{12} -85.6487 q^{13} +36.7990 q^{14} -3.80727 q^{15} +82.1568 q^{16} -3.63411 q^{17} -128.492 q^{18} -19.1465 q^{19} -59.0550 q^{20} +7.98264 q^{21} +261.594 q^{22} +86.1471 q^{23} +46.8623 q^{24} -112.546 q^{25} -425.960 q^{26} -57.0019 q^{27} +123.820 q^{28} -102.300 q^{29} -18.9349 q^{30} +104.109 q^{31} +61.0932 q^{32} +56.7465 q^{33} -18.0736 q^{34} -26.1122 q^{35} -432.343 q^{36} +247.021 q^{37} -95.2219 q^{38} -92.4016 q^{39} -153.292 q^{40} +162.904 q^{41} +39.7004 q^{42} +322.165 q^{43} +880.201 q^{44} +91.1763 q^{45} +428.439 q^{46} -202.336 q^{47} +88.6344 q^{48} -288.251 q^{49} -559.729 q^{50} -3.92064 q^{51} -1433.25 q^{52} +247.708 q^{53} -283.490 q^{54} -185.624 q^{55} +321.405 q^{56} -20.6561 q^{57} -508.775 q^{58} -456.459 q^{59} -63.7112 q^{60} +625.381 q^{61} +517.767 q^{62} -191.168 q^{63} -353.417 q^{64} +302.257 q^{65} +282.219 q^{66} +132.512 q^{67} -60.8134 q^{68} +92.9394 q^{69} -129.865 q^{70} +903.211 q^{71} -1122.26 q^{72} +422.822 q^{73} +1228.52 q^{74} -121.420 q^{75} -320.399 q^{76} +389.196 q^{77} -459.544 q^{78} +378.316 q^{79} -289.934 q^{80} +636.078 q^{81} +810.179 q^{82} -83.0000 q^{83} +133.582 q^{84} +12.8249 q^{85} +1602.24 q^{86} -110.366 q^{87} +2284.78 q^{88} -139.995 q^{89} +453.451 q^{90} -633.736 q^{91} +1441.59 q^{92} +112.317 q^{93} -1006.28 q^{94} +67.5685 q^{95} +65.9101 q^{96} -47.8805 q^{97} -1433.57 q^{98} -1358.96 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 5 q^{2} + 7 q^{3} + 73 q^{4} + 33 q^{5} - 3 q^{6} + 52 q^{7} + 63 q^{8} + 178 q^{9} + 35 q^{10} + 7 q^{11} + 127 q^{12} + 237 q^{13} + 114 q^{14} + 104 q^{15} + 365 q^{16} - 12 q^{17} + 262 q^{18}+ \cdots - 8456 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.97334 1.75834 0.879170 − 0.476508i \(-0.158098\pi\)
0.879170 + 0.476508i \(0.158098\pi\)
\(3\) 1.07884 0.207624 0.103812 − 0.994597i \(-0.466896\pi\)
0.103812 + 0.994597i \(0.466896\pi\)
\(4\) 16.7341 2.09176
\(5\) −3.52903 −0.315646 −0.157823 − 0.987467i \(-0.550448\pi\)
−0.157823 + 0.987467i \(0.550448\pi\)
\(6\) 5.36546 0.365073
\(7\) 7.39925 0.399522 0.199761 − 0.979845i \(-0.435983\pi\)
0.199761 + 0.979845i \(0.435983\pi\)
\(8\) 43.4375 1.91969
\(9\) −25.8361 −0.956892
\(10\) −17.5510 −0.555013
\(11\) 52.5993 1.44175 0.720877 − 0.693063i \(-0.243739\pi\)
0.720877 + 0.693063i \(0.243739\pi\)
\(12\) 18.0535 0.434299
\(13\) −85.6487 −1.82728 −0.913641 − 0.406521i \(-0.866742\pi\)
−0.913641 + 0.406521i \(0.866742\pi\)
\(14\) 36.7990 0.702496
\(15\) −3.80727 −0.0655356
\(16\) 82.1568 1.28370
\(17\) −3.63411 −0.0518471 −0.0259235 − 0.999664i \(-0.508253\pi\)
−0.0259235 + 0.999664i \(0.508253\pi\)
\(18\) −128.492 −1.68254
\(19\) −19.1465 −0.231184 −0.115592 − 0.993297i \(-0.536877\pi\)
−0.115592 + 0.993297i \(0.536877\pi\)
\(20\) −59.0550 −0.660255
\(21\) 7.98264 0.0829503
\(22\) 261.594 2.53509
\(23\) 86.1471 0.780997 0.390498 − 0.920604i \(-0.372303\pi\)
0.390498 + 0.920604i \(0.372303\pi\)
\(24\) 46.8623 0.398572
\(25\) −112.546 −0.900368
\(26\) −425.960 −3.21298
\(27\) −57.0019 −0.406297
\(28\) 123.820 0.835704
\(29\) −102.300 −0.655059 −0.327530 − 0.944841i \(-0.606216\pi\)
−0.327530 + 0.944841i \(0.606216\pi\)
\(30\) −18.9349 −0.115234
\(31\) 104.109 0.603176 0.301588 − 0.953438i \(-0.402483\pi\)
0.301588 + 0.953438i \(0.402483\pi\)
\(32\) 61.0932 0.337496
\(33\) 56.7465 0.299342
\(34\) −18.0736 −0.0911648
\(35\) −26.1122 −0.126107
\(36\) −432.343 −2.00159
\(37\) 247.021 1.09757 0.548784 − 0.835964i \(-0.315091\pi\)
0.548784 + 0.835964i \(0.315091\pi\)
\(38\) −95.2219 −0.406501
\(39\) −92.4016 −0.379387
\(40\) −153.292 −0.605941
\(41\) 162.904 0.620522 0.310261 − 0.950651i \(-0.399584\pi\)
0.310261 + 0.950651i \(0.399584\pi\)
\(42\) 39.7004 0.145855
\(43\) 322.165 1.14255 0.571276 − 0.820758i \(-0.306449\pi\)
0.571276 + 0.820758i \(0.306449\pi\)
\(44\) 880.201 3.01580
\(45\) 91.1763 0.302039
\(46\) 428.439 1.37326
\(47\) −202.336 −0.627951 −0.313975 − 0.949431i \(-0.601661\pi\)
−0.313975 + 0.949431i \(0.601661\pi\)
\(48\) 88.6344 0.266527
\(49\) −288.251 −0.840382
\(50\) −559.729 −1.58315
\(51\) −3.92064 −0.0107647
\(52\) −1433.25 −3.82224
\(53\) 247.708 0.641988 0.320994 − 0.947081i \(-0.395983\pi\)
0.320994 + 0.947081i \(0.395983\pi\)
\(54\) −283.490 −0.714409
\(55\) −185.624 −0.455084
\(56\) 321.405 0.766957
\(57\) −20.6561 −0.0479994
\(58\) −508.775 −1.15182
\(59\) −456.459 −1.00722 −0.503610 − 0.863931i \(-0.667995\pi\)
−0.503610 + 0.863931i \(0.667995\pi\)
\(60\) −63.7112 −0.137085
\(61\) 625.381 1.31265 0.656326 − 0.754477i \(-0.272110\pi\)
0.656326 + 0.754477i \(0.272110\pi\)
\(62\) 517.767 1.06059
\(63\) −191.168 −0.382300
\(64\) −353.417 −0.690268
\(65\) 302.257 0.576774
\(66\) 282.219 0.526346
\(67\) 132.512 0.241626 0.120813 − 0.992675i \(-0.461450\pi\)
0.120813 + 0.992675i \(0.461450\pi\)
\(68\) −60.8134 −0.108452
\(69\) 92.9394 0.162153
\(70\) −129.865 −0.221740
\(71\) 903.211 1.50974 0.754869 − 0.655875i \(-0.227700\pi\)
0.754869 + 0.655875i \(0.227700\pi\)
\(72\) −1122.26 −1.83693
\(73\) 422.822 0.677912 0.338956 − 0.940802i \(-0.389926\pi\)
0.338956 + 0.940802i \(0.389926\pi\)
\(74\) 1228.52 1.92990
\(75\) −121.420 −0.186938
\(76\) −320.399 −0.483582
\(77\) 389.196 0.576012
\(78\) −459.544 −0.667092
\(79\) 378.316 0.538783 0.269392 − 0.963031i \(-0.413177\pi\)
0.269392 + 0.963031i \(0.413177\pi\)
\(80\) −289.934 −0.405195
\(81\) 636.078 0.872535
\(82\) 810.179 1.09109
\(83\) −83.0000 −0.109764
\(84\) 133.582 0.173512
\(85\) 12.8249 0.0163653
\(86\) 1602.24 2.00900
\(87\) −110.366 −0.136006
\(88\) 2284.78 2.76771
\(89\) −139.995 −0.166736 −0.0833679 − 0.996519i \(-0.526568\pi\)
−0.0833679 + 0.996519i \(0.526568\pi\)
\(90\) 453.451 0.531088
\(91\) −633.736 −0.730040
\(92\) 1441.59 1.63366
\(93\) 112.317 0.125234
\(94\) −1006.28 −1.10415
\(95\) 67.5685 0.0729724
\(96\) 65.9101 0.0700721
\(97\) −47.8805 −0.0501189 −0.0250594 − 0.999686i \(-0.507978\pi\)
−0.0250594 + 0.999686i \(0.507978\pi\)
\(98\) −1433.57 −1.47768
\(99\) −1358.96 −1.37960
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 83.4.a.b.1.12 ✓ 13
3.2 odd 2 747.4.a.f.1.2 13
4.3 odd 2 1328.4.a.j.1.7 13
5.4 even 2 2075.4.a.d.1.2 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
83.4.a.b.1.12 ✓ 13 1.1 even 1 trivial
747.4.a.f.1.2 13 3.2 odd 2
1328.4.a.j.1.7 13 4.3 odd 2
2075.4.a.d.1.2 13 5.4 even 2