Properties

Label 83.4.a.b.1.10
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 1.10
Root \(4.08231\) 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)
 
\(f(q)\) \(=\) \(q+4.08231 q^{2} +9.18464 q^{3} +8.66522 q^{4} -18.8026 q^{5} +37.4945 q^{6} +8.58868 q^{7} +2.71565 q^{8} +57.3577 q^{9} -76.7578 q^{10} -28.5783 q^{11} +79.5870 q^{12} +43.6240 q^{13} +35.0616 q^{14} -172.695 q^{15} -58.2357 q^{16} -124.285 q^{17} +234.152 q^{18} +29.2304 q^{19} -162.928 q^{20} +78.8840 q^{21} -116.665 q^{22} +49.5706 q^{23} +24.9422 q^{24} +228.536 q^{25} +178.086 q^{26} +278.824 q^{27} +74.4229 q^{28} -59.9456 q^{29} -704.993 q^{30} +307.994 q^{31} -259.461 q^{32} -262.481 q^{33} -507.369 q^{34} -161.489 q^{35} +497.017 q^{36} +22.9305 q^{37} +119.327 q^{38} +400.671 q^{39} -51.0611 q^{40} -335.507 q^{41} +322.029 q^{42} +121.394 q^{43} -247.637 q^{44} -1078.47 q^{45} +202.362 q^{46} +408.319 q^{47} -534.874 q^{48} -269.235 q^{49} +932.955 q^{50} -1141.51 q^{51} +378.012 q^{52} +594.811 q^{53} +1138.25 q^{54} +537.345 q^{55} +23.3238 q^{56} +268.471 q^{57} -244.716 q^{58} +245.939 q^{59} -1496.44 q^{60} -551.689 q^{61} +1257.32 q^{62} +492.627 q^{63} -593.314 q^{64} -820.243 q^{65} -1071.53 q^{66} -176.298 q^{67} -1076.96 q^{68} +455.289 q^{69} -659.248 q^{70} +686.161 q^{71} +155.763 q^{72} -451.604 q^{73} +93.6095 q^{74} +2099.02 q^{75} +253.288 q^{76} -245.450 q^{77} +1635.66 q^{78} -37.8012 q^{79} +1094.98 q^{80} +1012.24 q^{81} -1369.64 q^{82} -83.0000 q^{83} +683.547 q^{84} +2336.87 q^{85} +495.565 q^{86} -550.579 q^{87} -77.6085 q^{88} +721.262 q^{89} -4402.65 q^{90} +374.673 q^{91} +429.541 q^{92} +2828.81 q^{93} +1666.88 q^{94} -549.606 q^{95} -2383.06 q^{96} -46.6418 q^{97} -1099.10 q^{98} -1639.18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 5 q^{2} + 7 q^{3} + 73 q^{4} + 33 q^{5} - 3 q^{6} + 52 q^{7} + 63 q^{8} + 178 q^{9} + 35 q^{10} + 7 q^{11} + 127 q^{12} + 237 q^{13} + 114 q^{14} + 104 q^{15} + 365 q^{16} - 12 q^{17} + 262 q^{18}+ \cdots - 8456 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.08231 1.44331 0.721657 − 0.692251i \(-0.243381\pi\)
0.721657 + 0.692251i \(0.243381\pi\)
\(3\) 9.18464 1.76759 0.883793 − 0.467879i \(-0.154982\pi\)
0.883793 + 0.467879i \(0.154982\pi\)
\(4\) 8.66522 1.08315
\(5\) −18.8026 −1.68175 −0.840876 − 0.541228i \(-0.817960\pi\)
−0.840876 + 0.541228i \(0.817960\pi\)
\(6\) 37.4945 2.55118
\(7\) 8.58868 0.463745 0.231873 − 0.972746i \(-0.425515\pi\)
0.231873 + 0.972746i \(0.425515\pi\)
\(8\) 2.71565 0.120016
\(9\) 57.3577 2.12436
\(10\) −76.7578 −2.42729
\(11\) −28.5783 −0.783334 −0.391667 − 0.920107i \(-0.628101\pi\)
−0.391667 + 0.920107i \(0.628101\pi\)
\(12\) 79.5870 1.91457
\(13\) 43.6240 0.930701 0.465351 − 0.885126i \(-0.345928\pi\)
0.465351 + 0.885126i \(0.345928\pi\)
\(14\) 35.0616 0.669330
\(15\) −172.695 −2.97264
\(16\) −58.2357 −0.909933
\(17\) −124.285 −1.77315 −0.886574 − 0.462587i \(-0.846921\pi\)
−0.886574 + 0.462587i \(0.846921\pi\)
\(18\) 234.152 3.06611
\(19\) 29.2304 0.352943 0.176471 − 0.984306i \(-0.443532\pi\)
0.176471 + 0.984306i \(0.443532\pi\)
\(20\) −162.928 −1.82159
\(21\) 78.8840 0.819709
\(22\) −116.665 −1.13060
\(23\) 49.5706 0.449400 0.224700 − 0.974428i \(-0.427860\pi\)
0.224700 + 0.974428i \(0.427860\pi\)
\(24\) 24.9422 0.212138
\(25\) 228.536 1.82829
\(26\) 178.086 1.34329
\(27\) 278.824 1.98740
\(28\) 74.4229 0.502307
\(29\) −59.9456 −0.383849 −0.191924 − 0.981410i \(-0.561473\pi\)
−0.191924 + 0.981410i \(0.561473\pi\)
\(30\) −704.993 −4.29045
\(31\) 307.994 1.78443 0.892214 − 0.451612i \(-0.149151\pi\)
0.892214 + 0.451612i \(0.149151\pi\)
\(32\) −259.461 −1.43333
\(33\) −262.481 −1.38461
\(34\) −507.369 −2.55921
\(35\) −161.489 −0.779904
\(36\) 497.017 2.30100
\(37\) 22.9305 0.101885 0.0509427 − 0.998702i \(-0.483777\pi\)
0.0509427 + 0.998702i \(0.483777\pi\)
\(38\) 119.327 0.509407
\(39\) 400.671 1.64509
\(40\) −51.0611 −0.201837
\(41\) −335.507 −1.27799 −0.638993 − 0.769212i \(-0.720649\pi\)
−0.638993 + 0.769212i \(0.720649\pi\)
\(42\) 322.029 1.18310
\(43\) 121.394 0.430519 0.215260 − 0.976557i \(-0.430940\pi\)
0.215260 + 0.976557i \(0.430940\pi\)
\(44\) −247.637 −0.848471
\(45\) −1078.47 −3.57264
\(46\) 202.362 0.648624
\(47\) 408.319 1.26722 0.633611 − 0.773651i \(-0.281572\pi\)
0.633611 + 0.773651i \(0.281572\pi\)
\(48\) −534.874 −1.60838
\(49\) −269.235 −0.784940
\(50\) 932.955 2.63879
\(51\) −1141.51 −3.13419
\(52\) 378.012 1.00809
\(53\) 594.811 1.54158 0.770788 − 0.637092i \(-0.219863\pi\)
0.770788 + 0.637092i \(0.219863\pi\)
\(54\) 1138.25 2.86844
\(55\) 537.345 1.31737
\(56\) 23.3238 0.0556567
\(57\) 268.471 0.623856
\(58\) −244.716 −0.554014
\(59\) 245.939 0.542686 0.271343 − 0.962483i \(-0.412532\pi\)
0.271343 + 0.962483i \(0.412532\pi\)
\(60\) −1496.44 −3.21982
\(61\) −551.689 −1.15798 −0.578988 − 0.815336i \(-0.696552\pi\)
−0.578988 + 0.815336i \(0.696552\pi\)
\(62\) 1257.32 2.57549
\(63\) 492.627 0.985161
\(64\) −593.314 −1.15882
\(65\) −820.243 −1.56521
\(66\) −1071.53 −1.99843
\(67\) −176.298 −0.321466 −0.160733 − 0.986998i \(-0.551386\pi\)
−0.160733 + 0.986998i \(0.551386\pi\)
\(68\) −1076.96 −1.92059
\(69\) 455.289 0.794352
\(70\) −659.248 −1.12565
\(71\) 686.161 1.14693 0.573467 − 0.819229i \(-0.305598\pi\)
0.573467 + 0.819229i \(0.305598\pi\)
\(72\) 155.763 0.254956
\(73\) −451.604 −0.724058 −0.362029 − 0.932167i \(-0.617916\pi\)
−0.362029 + 0.932167i \(0.617916\pi\)
\(74\) 93.6095 0.147052
\(75\) 2099.02 3.23166
\(76\) 253.288 0.382291
\(77\) −245.450 −0.363267
\(78\) 1635.66 2.37439
\(79\) −37.8012 −0.0538351 −0.0269175 − 0.999638i \(-0.508569\pi\)
−0.0269175 + 0.999638i \(0.508569\pi\)
\(80\) 1094.98 1.53028
\(81\) 1012.24 1.38854
\(82\) −1369.64 −1.84453
\(83\) −83.0000 −0.109764
\(84\) 683.547 0.887870
\(85\) 2336.87 2.98199
\(86\) 495.565 0.621374
\(87\) −550.579 −0.678486
\(88\) −77.6085 −0.0940124
\(89\) 721.262 0.859030 0.429515 − 0.903060i \(-0.358685\pi\)
0.429515 + 0.903060i \(0.358685\pi\)
\(90\) −4402.65 −5.15644
\(91\) 374.673 0.431608
\(92\) 429.541 0.486769
\(93\) 2828.81 3.15413
\(94\) 1666.88 1.82900
\(95\) −549.606 −0.593562
\(96\) −2383.06 −2.53354
\(97\) −46.6418 −0.0488223 −0.0244111 − 0.999702i \(-0.507771\pi\)
−0.0244111 + 0.999702i \(0.507771\pi\)
\(98\) −1099.10 −1.13291
\(99\) −1639.18 −1.66408
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.1.10 ✓ 13
3.2 odd 2 747.4.a.f.1.4 13
4.3 odd 2 1328.4.a.j.1.1 13
5.4 even 2 2075.4.a.d.1.4 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
83.4.a.b.1.10 ✓ 13 1.1 even 1 trivial
747.4.a.f.1.4 13 3.2 odd 2
1328.4.a.j.1.1 13 4.3 odd 2
2075.4.a.d.1.4 13 5.4 even 2