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Properties

Label 87.4.g.a.16.2
Level $87$
Weight $4$
Character 87.16
Analytic conductor $5.133$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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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] = [87,4,Mod(7,87)] 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("87.7"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 87.g (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [42,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.13316617050\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 16.2
Character \(\chi\) \(=\) 87.16
Dual form 87.4.g.a.49.2

$q$-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+(-2.78928 - 1.34325i) q^{2} +(-1.87047 + 2.34549i) q^{3} +(0.987842 + 1.23871i) q^{4} +(-13.3238 - 6.41638i) q^{5} +(8.36783 - 4.02974i) q^{6} +(12.4583 - 15.6222i) q^{7} +(4.41969 + 19.3639i) q^{8} +(-2.00269 - 8.77435i) q^{9} +(28.5449 + 35.7941i) q^{10} +(-12.7093 + 55.6832i) q^{11} -4.75313 q^{12} +(-3.59831 + 15.7652i) q^{13} +(-55.7340 + 26.8401i) q^{14} +(39.9713 - 19.2491i) q^{15} +(16.5032 - 72.3053i) q^{16} +101.424 q^{17} +(-6.20005 + 27.1642i) q^{18} +(79.8093 + 100.078i) q^{19} +(-5.21370 - 22.8427i) q^{20} +(13.3389 + 58.4416i) q^{21} +(110.246 - 138.244i) q^{22} +(-132.994 + 64.0464i) q^{23} +(-53.6849 - 25.8533i) q^{24} +(58.4163 + 73.2517i) q^{25} +(31.2132 - 39.1402i) q^{26} +(24.3262 + 11.7149i) q^{27} +31.6582 q^{28} +(61.8646 - 143.394i) q^{29} -137.347 q^{30} +(74.1050 + 35.6871i) q^{31} +(-44.0862 + 55.2824i) q^{32} +(-106.832 - 133.963i) q^{33} +(-282.901 - 136.238i) q^{34} +(-266.229 + 128.209i) q^{35} +(8.89058 - 11.1484i) q^{36} +(33.9725 + 148.843i) q^{37} +(-88.1814 - 386.348i) q^{38} +(-30.2467 - 37.9282i) q^{39} +(65.3595 - 286.359i) q^{40} -399.179 q^{41} +(41.2955 - 180.927i) q^{42} +(-158.921 + 76.5326i) q^{43} +(-81.5305 + 39.2630i) q^{44} +(-29.6163 + 129.757i) q^{45} +456.986 q^{46} +(-63.0515 + 276.247i) q^{47} +(138.723 + 173.953i) q^{48} +(-12.5193 - 54.8508i) q^{49} +(-64.5442 - 282.787i) q^{50} +(-189.711 + 237.891i) q^{51} +(-23.0832 + 11.1163i) q^{52} +(524.280 + 252.480i) q^{53} +(-52.1165 - 65.3520i) q^{54} +(526.621 - 660.362i) q^{55} +(357.569 + 172.196i) q^{56} -384.013 q^{57} +(-365.171 + 316.866i) q^{58} +362.895 q^{59} +(63.3295 + 30.4979i) q^{60} +(42.2361 - 52.9624i) q^{61} +(-158.763 - 199.082i) q^{62} +(-162.025 - 78.0269i) q^{63} +(-337.335 + 162.452i) q^{64} +(149.099 - 186.964i) q^{65} +(118.039 + 517.163i) q^{66} +(156.987 + 687.806i) q^{67} +(100.191 + 125.636i) q^{68} +(98.5402 - 431.733i) q^{69} +914.803 q^{70} +(234.680 - 1028.20i) q^{71} +(161.055 - 77.5598i) q^{72} +(-707.661 + 340.792i) q^{73} +(105.174 - 460.799i) q^{74} -281.077 q^{75} +(-45.1288 + 197.722i) q^{76} +(711.558 + 892.265i) q^{77} +(33.4196 + 146.421i) q^{78} +(93.7234 + 410.629i) q^{79} +(-683.823 + 857.487i) q^{80} +(-72.9785 + 35.1446i) q^{81} +(1113.42 + 536.195i) q^{82} +(464.666 + 582.673i) q^{83} +(-59.2158 + 74.2542i) q^{84} +(-1351.35 - 650.778i) q^{85} +546.078 q^{86} +(220.613 + 413.317i) q^{87} -1134.42 q^{88} +(-2.76641 - 1.33223i) q^{89} +(256.904 - 322.147i) q^{90} +(201.458 + 252.621i) q^{91} +(-210.712 - 101.474i) q^{92} +(-222.315 + 107.061i) q^{93} +(546.936 - 685.835i) q^{94} +(-421.223 - 1845.50i) q^{95} +(-47.2026 - 206.808i) q^{96} +(-70.5467 - 88.4628i) q^{97} +(-38.7582 + 169.811i) q^{98} +514.037 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q - 2 q^{2} + 21 q^{3} - 34 q^{4} - 47 q^{5} + 6 q^{6} + 20 q^{7} - 81 q^{8} - 63 q^{9} - 108 q^{10} + 85 q^{11} - 402 q^{12} - 96 q^{13} - 197 q^{14} + 141 q^{15} + 150 q^{16} + 488 q^{17} - 18 q^{18}+ \cdots + 1206 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(e\left(\frac{1}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.78928 − 1.34325i −0.986158 − 0.474909i −0.129939 − 0.991522i \(-0.541478\pi\)
−0.856219 + 0.516613i \(0.827193\pi\)
\(3\) −1.87047 + 2.34549i −0.359972 + 0.451391i
\(4\) 0.987842 + 1.23871i 0.123480 + 0.154839i
\(5\) −13.3238 − 6.41638i −1.19171 − 0.573899i −0.270410 − 0.962745i \(-0.587159\pi\)
−0.921303 + 0.388846i \(0.872874\pi\)
\(6\) 8.36783 − 4.02974i 0.569359 − 0.274189i
\(7\) 12.4583 − 15.6222i 0.672684 − 0.843519i −0.321974 − 0.946749i \(-0.604346\pi\)
0.994658 + 0.103230i \(0.0329177\pi\)
\(8\) 4.41969 + 19.3639i 0.195325 + 0.855773i
\(9\) −2.00269 − 8.77435i −0.0741736 − 0.324976i
\(10\) 28.5449 + 35.7941i 0.902668 + 1.13191i
\(11\) −12.7093 + 55.6832i −0.348364 + 1.52628i 0.432529 + 0.901620i \(0.357621\pi\)
−0.780893 + 0.624664i \(0.785236\pi\)
\(12\) −4.75313 −0.114342
\(13\) −3.59831 + 15.7652i −0.0767686 + 0.336345i −0.998698 − 0.0510128i \(-0.983755\pi\)
0.921929 + 0.387358i \(0.126612\pi\)
\(14\) −55.7340 + 26.8401i −1.06397 + 0.512380i
\(15\) 39.9713 − 19.2491i 0.688036 − 0.331341i
\(16\) 16.5032 − 72.3053i 0.257863 − 1.12977i
\(17\) 101.424 1.44700 0.723501 − 0.690323i \(-0.242532\pi\)
0.723501 + 0.690323i \(0.242532\pi\)
\(18\) −6.20005 + 27.1642i −0.0811870 + 0.355704i
\(19\) 79.8093 + 100.078i 0.963659 + 1.20839i 0.978024 + 0.208493i \(0.0668558\pi\)
−0.0143654 + 0.999897i \(0.504573\pi\)
\(20\) −5.21370 − 22.8427i −0.0582909 − 0.255389i
\(21\) 13.3389 + 58.4416i 0.138609 + 0.607286i
\(22\) 110.246 − 138.244i 1.06839 − 1.33972i
\(23\) −132.994 + 64.0464i −1.20570 + 0.580635i −0.925295 − 0.379247i \(-0.876183\pi\)
−0.280405 + 0.959882i \(0.590469\pi\)
\(24\) −53.6849 − 25.8533i −0.456599 − 0.219887i
\(25\) 58.4163 + 73.2517i 0.467330 + 0.586014i
\(26\) 31.2132 − 39.1402i 0.235439 − 0.295231i
\(27\) 24.3262 + 11.7149i 0.173392 + 0.0835010i
\(28\) 31.6582 0.213673
\(29\) 61.8646 − 143.394i 0.396137 − 0.918191i
\(30\) −137.347 −0.835869
\(31\) 74.1050 + 35.6871i 0.429344 + 0.206761i 0.636060 − 0.771640i \(-0.280563\pi\)
−0.206716 + 0.978401i \(0.566278\pi\)
\(32\) −44.0862 + 55.2824i −0.243544 + 0.305395i
\(33\) −106.832 − 133.963i −0.563549 − 0.706668i
\(34\) −282.901 − 136.238i −1.42697 − 0.687194i
\(35\) −266.229 + 128.209i −1.28574 + 0.619180i
\(36\) 8.89058 − 11.1484i 0.0411601 − 0.0516131i
\(37\) 33.9725 + 148.843i 0.150947 + 0.661343i 0.992611 + 0.121341i \(0.0387194\pi\)
−0.841664 + 0.540002i \(0.818423\pi\)
\(38\) −88.1814 − 386.348i −0.376445 − 1.64931i
\(39\) −30.2467 − 37.9282i −0.124188 − 0.155727i
\(40\) 65.3595 − 286.359i 0.258356 − 1.13193i
\(41\) −399.179 −1.52052 −0.760260 − 0.649619i \(-0.774928\pi\)
−0.760260 + 0.649619i \(0.774928\pi\)
\(42\) 41.2955 − 180.927i 0.151715 − 0.664707i
\(43\) −158.921 + 76.5326i −0.563612 + 0.271421i −0.693920 − 0.720053i \(-0.744118\pi\)
0.130308 + 0.991474i \(0.458403\pi\)
\(44\) −81.5305 + 39.2630i −0.279345 + 0.134525i
\(45\) −29.6163 + 129.757i −0.0981096 + 0.429846i
\(46\) 456.986 1.46476
\(47\) −63.0515 + 276.247i −0.195681 + 0.857335i 0.777790 + 0.628524i \(0.216341\pi\)
−0.973471 + 0.228811i \(0.926516\pi\)
\(48\) 138.723 + 173.953i 0.417144 + 0.523082i
\(49\) −12.5193 − 54.8508i −0.0364996 − 0.159915i
\(50\) −64.5442 − 282.787i −0.182559 − 0.799841i
\(51\) −189.711 + 237.891i −0.520880 + 0.653163i
\(52\) −23.0832 + 11.1163i −0.0615588 + 0.0296452i
\(53\) 524.280 + 252.480i 1.35878 + 0.654355i 0.964365 − 0.264577i \(-0.0852324\pi\)
0.394417 + 0.918932i \(0.370947\pi\)
\(54\) −52.1165 − 65.3520i −0.131336 − 0.164690i
\(55\) 526.621 − 660.362i 1.29108 − 1.61897i
\(56\) 357.569 + 172.196i 0.853252 + 0.410904i
\(57\) −384.013 −0.892346
\(58\) −365.171 + 316.866i −0.826711 + 0.717353i
\(59\) 362.895 0.800761 0.400381 − 0.916349i \(-0.368878\pi\)
0.400381 + 0.916349i \(0.368878\pi\)
\(60\) 63.3295 + 30.4979i 0.136263 + 0.0656210i
\(61\) 42.2361 − 52.9624i 0.0886522 − 0.111166i −0.735527 − 0.677495i \(-0.763065\pi\)
0.824179 + 0.566329i \(0.191637\pi\)
\(62\) −158.763 − 199.082i −0.325208 − 0.407798i
\(63\) −162.025 − 78.0269i −0.324019 − 0.156039i
\(64\) −337.335 + 162.452i −0.658857 + 0.317289i
\(65\) 149.099 − 186.964i 0.284514 − 0.356769i
\(66\) 118.039 + 517.163i 0.220146 + 0.964521i
\(67\) 156.987 + 687.806i 0.286254 + 1.25416i 0.889622 + 0.456698i \(0.150968\pi\)
−0.603367 + 0.797463i \(0.706175\pi\)
\(68\) 100.191 + 125.636i 0.178676 + 0.224053i
\(69\) 98.5402 − 431.733i 0.171925 − 0.753254i
\(70\) 914.803 1.56200
\(71\) 234.680 − 1028.20i 0.392274 − 1.71866i −0.264333 − 0.964431i \(-0.585152\pi\)
0.656607 − 0.754233i \(-0.271991\pi\)
\(72\) 161.055 − 77.5598i 0.263618 − 0.126952i
\(73\) −707.661 + 340.792i −1.13460 + 0.546392i −0.904372 − 0.426745i \(-0.859660\pi\)
−0.230224 + 0.973138i \(0.573946\pi\)
\(74\) 105.174 − 460.799i 0.165220 − 0.723875i
\(75\) −281.077 −0.432747
\(76\) −45.1288 + 197.722i −0.0681135 + 0.298425i
\(77\) 711.558 + 892.265i 1.05311 + 1.32056i
\(78\) 33.4196 + 146.421i 0.0485132 + 0.212550i
\(79\) 93.7234 + 410.629i 0.133477 + 0.584802i 0.996785 + 0.0801242i \(0.0255317\pi\)
−0.863308 + 0.504678i \(0.831611\pi\)
\(80\) −683.823 + 857.487i −0.955672 + 1.19837i
\(81\) −72.9785 + 35.1446i −0.100108 + 0.0482093i
\(82\) 1113.42 + 536.195i 1.49947 + 0.722108i
\(83\) 464.666 + 582.673i 0.614503 + 0.770563i 0.987559 − 0.157246i \(-0.0502616\pi\)
−0.373056 + 0.927809i \(0.621690\pi\)
\(84\) −59.2158 + 74.2542i −0.0769163 + 0.0964500i
\(85\) −1351.35 − 650.778i −1.72441 − 0.830433i
\(86\) 546.078 0.684711
\(87\) 220.613 + 413.317i 0.271865 + 0.509336i
\(88\) −1134.42 −1.37420
\(89\) −2.76641 − 1.33223i −0.00329481 − 0.00158670i 0.432236 − 0.901761i \(-0.357725\pi\)
−0.435530 + 0.900174i \(0.643439\pi\)
\(90\) 256.904 − 322.147i 0.300889 − 0.377303i
\(91\) 201.458 + 252.621i 0.232072 + 0.291010i
\(92\) −210.712 − 101.474i −0.238785 − 0.114993i
\(93\) −222.315 + 107.061i −0.247882 + 0.119374i
\(94\) 546.936 − 685.835i 0.600129 − 0.752537i
\(95\) −421.223 − 1845.50i −0.454911 − 1.99310i
\(96\) −47.2026 − 206.808i −0.0501833 − 0.219867i
\(97\) −70.5467 − 88.4628i −0.0738447 − 0.0925983i 0.743537 − 0.668695i \(-0.233147\pi\)
−0.817382 + 0.576096i \(0.804575\pi\)
\(98\) −38.7582 + 169.811i −0.0399507 + 0.175035i
\(99\) 514.037 0.521845
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 87.4.g.a.16.2 ✓ 42
29.20 even 7 inner 87.4.g.a.49.2 yes 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.4.g.a.16.2 ✓ 42 1.1 even 1 trivial
87.4.g.a.49.2 yes 42 29.20 even 7 inner