Properties

Label 4020.2.q.m.841.7
Level $4020$
Weight $2$
Character 4020.841
Analytic conductor $32.100$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4020,2,Mod(841,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.841");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.q (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 841.7
Character \(\chi\) \(=\) 4020.841
Dual form 4020.2.q.m.3781.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +(0.134465 - 0.232901i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +(0.134465 - 0.232901i) q^{7} +1.00000 q^{9} +(-3.05941 + 5.29905i) q^{11} +(-1.14872 - 1.98965i) q^{13} +1.00000 q^{15} +(-3.76596 - 6.52284i) q^{17} +(-1.88181 - 3.25939i) q^{19} +(0.134465 - 0.232901i) q^{21} +(2.57707 + 4.46362i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(-3.06664 + 5.31157i) q^{29} +(-5.25390 + 9.10002i) q^{31} +(-3.05941 + 5.29905i) q^{33} +(0.134465 - 0.232901i) q^{35} +(2.62278 + 4.54279i) q^{37} +(-1.14872 - 1.98965i) q^{39} +(1.80796 - 3.13148i) q^{41} +10.9631 q^{43} +1.00000 q^{45} +(-6.55135 + 11.3473i) q^{47} +(3.46384 + 5.99954i) q^{49} +(-3.76596 - 6.52284i) q^{51} +12.0675 q^{53} +(-3.05941 + 5.29905i) q^{55} +(-1.88181 - 3.25939i) q^{57} -11.8215 q^{59} +(4.89473 + 8.47792i) q^{61} +(0.134465 - 0.232901i) q^{63} +(-1.14872 - 1.98965i) q^{65} +(4.96860 + 6.50484i) q^{67} +(2.57707 + 4.46362i) q^{69} +(-6.02544 + 10.4364i) q^{71} +(0.167288 + 0.289752i) q^{73} +1.00000 q^{75} +(0.822768 + 1.42508i) q^{77} +(4.90196 - 8.49044i) q^{79} +1.00000 q^{81} +(4.16136 + 7.20768i) q^{83} +(-3.76596 - 6.52284i) q^{85} +(-3.06664 + 5.31157i) q^{87} -1.47751 q^{89} -0.617854 q^{91} +(-5.25390 + 9.10002i) q^{93} +(-1.88181 - 3.25939i) q^{95} +(-2.64816 - 4.58675i) q^{97} +(-3.05941 + 5.29905i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{3} + 24 q^{5} - 3 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{3} + 24 q^{5} - 3 q^{7} + 24 q^{9} - 2 q^{13} + 24 q^{15} - 6 q^{19} - 3 q^{21} + 2 q^{23} + 24 q^{25} + 24 q^{27} + 7 q^{29} - 8 q^{31} - 3 q^{35} - 10 q^{37} - 2 q^{39} + 2 q^{41} + 28 q^{43} + 24 q^{45} - 3 q^{47} - 17 q^{49} + 36 q^{53} - 6 q^{57} - 10 q^{59} + 9 q^{61} - 3 q^{63} - 2 q^{65} - 46 q^{67} + 2 q^{69} - 12 q^{71} + 6 q^{73} + 24 q^{75} - 5 q^{77} + 2 q^{79} + 24 q^{81} + 11 q^{83} + 7 q^{87} + 52 q^{89} - 22 q^{91} - 8 q^{93} - 6 q^{95} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.134465 0.232901i 0.0508231 0.0880282i −0.839495 0.543368i \(-0.817149\pi\)
0.890318 + 0.455340i \(0.150482\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.05941 + 5.29905i −0.922446 + 1.59772i −0.126829 + 0.991925i \(0.540480\pi\)
−0.795617 + 0.605799i \(0.792853\pi\)
\(12\) 0 0
\(13\) −1.14872 1.98965i −0.318599 0.551829i 0.661597 0.749859i \(-0.269879\pi\)
−0.980196 + 0.198030i \(0.936546\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −3.76596 6.52284i −0.913380 1.58202i −0.809256 0.587456i \(-0.800130\pi\)
−0.104123 0.994564i \(-0.533204\pi\)
\(18\) 0 0
\(19\) −1.88181 3.25939i −0.431716 0.747754i 0.565305 0.824882i \(-0.308758\pi\)
−0.997021 + 0.0771275i \(0.975425\pi\)
\(20\) 0 0
\(21\) 0.134465 0.232901i 0.0293427 0.0508231i
\(22\) 0 0
\(23\) 2.57707 + 4.46362i 0.537356 + 0.930728i 0.999045 + 0.0436864i \(0.0139102\pi\)
−0.461689 + 0.887042i \(0.652756\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.06664 + 5.31157i −0.569461 + 0.986335i 0.427159 + 0.904177i \(0.359515\pi\)
−0.996619 + 0.0821580i \(0.973819\pi\)
\(30\) 0 0
\(31\) −5.25390 + 9.10002i −0.943629 + 1.63441i −0.185154 + 0.982709i \(0.559278\pi\)
−0.758474 + 0.651703i \(0.774055\pi\)
\(32\) 0 0
\(33\) −3.05941 + 5.29905i −0.532575 + 0.922446i
\(34\) 0 0
\(35\) 0.134465 0.232901i 0.0227288 0.0393674i
\(36\) 0 0
\(37\) 2.62278 + 4.54279i 0.431182 + 0.746830i 0.996975 0.0777176i \(-0.0247632\pi\)
−0.565793 + 0.824547i \(0.691430\pi\)
\(38\) 0 0
\(39\) −1.14872 1.98965i −0.183943 0.318599i
\(40\) 0 0
\(41\) 1.80796 3.13148i 0.282356 0.489054i −0.689609 0.724182i \(-0.742218\pi\)
0.971964 + 0.235128i \(0.0755508\pi\)
\(42\) 0 0
\(43\) 10.9631 1.67186 0.835931 0.548834i \(-0.184928\pi\)
0.835931 + 0.548834i \(0.184928\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −6.55135 + 11.3473i −0.955612 + 1.65517i −0.222650 + 0.974898i \(0.571471\pi\)
−0.732962 + 0.680270i \(0.761863\pi\)
\(48\) 0 0
\(49\) 3.46384 + 5.99954i 0.494834 + 0.857078i
\(50\) 0 0
\(51\) −3.76596 6.52284i −0.527340 0.913380i
\(52\) 0 0
\(53\) 12.0675 1.65760 0.828800 0.559545i \(-0.189024\pi\)
0.828800 + 0.559545i \(0.189024\pi\)
\(54\) 0 0
\(55\) −3.05941 + 5.29905i −0.412531 + 0.714524i
\(56\) 0 0
\(57\) −1.88181 3.25939i −0.249251 0.431716i
\(58\) 0 0
\(59\) −11.8215 −1.53903 −0.769515 0.638628i \(-0.779502\pi\)
−0.769515 + 0.638628i \(0.779502\pi\)
\(60\) 0 0
\(61\) 4.89473 + 8.47792i 0.626706 + 1.08549i 0.988208 + 0.153115i \(0.0489306\pi\)
−0.361502 + 0.932371i \(0.617736\pi\)
\(62\) 0 0
\(63\) 0.134465 0.232901i 0.0169410 0.0293427i
\(64\) 0 0
\(65\) −1.14872 1.98965i −0.142482 0.246786i
\(66\) 0 0
\(67\) 4.96860 + 6.50484i 0.607011 + 0.794693i
\(68\) 0 0
\(69\) 2.57707 + 4.46362i 0.310243 + 0.537356i
\(70\) 0 0
\(71\) −6.02544 + 10.4364i −0.715088 + 1.23857i 0.247837 + 0.968802i \(0.420280\pi\)
−0.962926 + 0.269767i \(0.913053\pi\)
\(72\) 0 0
\(73\) 0.167288 + 0.289752i 0.0195796 + 0.0339129i 0.875649 0.482948i \(-0.160434\pi\)
−0.856070 + 0.516861i \(0.827101\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0.822768 + 1.42508i 0.0937631 + 0.162403i
\(78\) 0 0
\(79\) 4.90196 8.49044i 0.551514 0.955250i −0.446652 0.894708i \(-0.647384\pi\)
0.998166 0.0605418i \(-0.0192829\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.16136 + 7.20768i 0.456768 + 0.791146i 0.998788 0.0492198i \(-0.0156735\pi\)
−0.542020 + 0.840366i \(0.682340\pi\)
\(84\) 0 0
\(85\) −3.76596 6.52284i −0.408476 0.707501i
\(86\) 0 0
\(87\) −3.06664 + 5.31157i −0.328778 + 0.569461i
\(88\) 0 0
\(89\) −1.47751 −0.156616 −0.0783081 0.996929i \(-0.524952\pi\)
−0.0783081 + 0.996929i \(0.524952\pi\)
\(90\) 0 0
\(91\) −0.617854 −0.0647687
\(92\) 0 0
\(93\) −5.25390 + 9.10002i −0.544804 + 0.943629i
\(94\) 0 0
\(95\) −1.88181 3.25939i −0.193069 0.334406i
\(96\) 0 0
\(97\) −2.64816 4.58675i −0.268880 0.465714i 0.699693 0.714444i \(-0.253320\pi\)
−0.968573 + 0.248730i \(0.919987\pi\)
\(98\) 0 0
\(99\) −3.05941 + 5.29905i −0.307482 + 0.532575i
\(100\) 0 0
\(101\) 5.55036 9.61350i 0.552281 0.956579i −0.445829 0.895118i \(-0.647091\pi\)
0.998110 0.0614604i \(-0.0195758\pi\)
\(102\) 0 0
\(103\) 1.61545 2.79804i 0.159175 0.275699i −0.775396 0.631475i \(-0.782450\pi\)
0.934571 + 0.355776i \(0.115783\pi\)
\(104\) 0 0
\(105\) 0.134465 0.232901i 0.0131225 0.0227288i
\(106\) 0 0
\(107\) −8.58118 −0.829574 −0.414787 0.909918i \(-0.636144\pi\)
−0.414787 + 0.909918i \(0.636144\pi\)
\(108\) 0 0
\(109\) −18.0243 −1.72642 −0.863208 0.504848i \(-0.831548\pi\)
−0.863208 + 0.504848i \(0.831548\pi\)
\(110\) 0 0
\(111\) 2.62278 + 4.54279i 0.248943 + 0.431182i
\(112\) 0 0
\(113\) −9.83260 + 17.0306i −0.924973 + 1.60210i −0.133368 + 0.991067i \(0.542579\pi\)
−0.791605 + 0.611033i \(0.790754\pi\)
\(114\) 0 0
\(115\) 2.57707 + 4.46362i 0.240313 + 0.416234i
\(116\) 0 0
\(117\) −1.14872 1.98965i −0.106200 0.183943i
\(118\) 0 0
\(119\) −2.02556 −0.185683
\(120\) 0 0
\(121\) −13.2200 22.8976i −1.20181 2.08160i
\(122\) 0 0
\(123\) 1.80796 3.13148i 0.163018 0.282356i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.27915 + 7.41171i −0.379713 + 0.657683i −0.991020 0.133711i \(-0.957311\pi\)
0.611307 + 0.791394i \(0.290644\pi\)
\(128\) 0 0
\(129\) 10.9631 0.965250
\(130\) 0 0
\(131\) −9.67842 −0.845607 −0.422804 0.906221i \(-0.638954\pi\)
−0.422804 + 0.906221i \(0.638954\pi\)
\(132\) 0 0
\(133\) −1.01215 −0.0877646
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −5.37712 −0.459398 −0.229699 0.973262i \(-0.573774\pi\)
−0.229699 + 0.973262i \(0.573774\pi\)
\(138\) 0 0
\(139\) −12.8831 −1.09273 −0.546365 0.837547i \(-0.683989\pi\)
−0.546365 + 0.837547i \(0.683989\pi\)
\(140\) 0 0
\(141\) −6.55135 + 11.3473i −0.551723 + 0.955612i
\(142\) 0 0
\(143\) 14.0577 1.17556
\(144\) 0 0
\(145\) −3.06664 + 5.31157i −0.254671 + 0.441102i
\(146\) 0 0
\(147\) 3.46384 + 5.99954i 0.285693 + 0.494834i
\(148\) 0 0
\(149\) 3.97030 0.325260 0.162630 0.986687i \(-0.448002\pi\)
0.162630 + 0.986687i \(0.448002\pi\)
\(150\) 0 0
\(151\) 2.43423 + 4.21621i 0.198095 + 0.343110i 0.947911 0.318536i \(-0.103191\pi\)
−0.749816 + 0.661647i \(0.769858\pi\)
\(152\) 0 0
\(153\) −3.76596 6.52284i −0.304460 0.527340i
\(154\) 0 0
\(155\) −5.25390 + 9.10002i −0.422003 + 0.730931i
\(156\) 0 0
\(157\) −0.387721 0.671552i −0.0309435 0.0535957i 0.850139 0.526558i \(-0.176518\pi\)
−0.881082 + 0.472963i \(0.843185\pi\)
\(158\) 0 0
\(159\) 12.0675 0.957015
\(160\) 0 0
\(161\) 1.38611 0.109240
\(162\) 0 0
\(163\) 2.56301 4.43926i 0.200750 0.347710i −0.748020 0.663676i \(-0.768995\pi\)
0.948770 + 0.315966i \(0.102329\pi\)
\(164\) 0 0
\(165\) −3.05941 + 5.29905i −0.238175 + 0.412531i
\(166\) 0 0
\(167\) 5.81380 10.0698i 0.449885 0.779225i −0.548493 0.836155i \(-0.684798\pi\)
0.998378 + 0.0569309i \(0.0181315\pi\)
\(168\) 0 0
\(169\) 3.86087 6.68721i 0.296990 0.514401i
\(170\) 0 0
\(171\) −1.88181 3.25939i −0.143905 0.249251i
\(172\) 0 0
\(173\) −6.10683 10.5773i −0.464294 0.804180i 0.534876 0.844931i \(-0.320358\pi\)
−0.999169 + 0.0407504i \(0.987025\pi\)
\(174\) 0 0
\(175\) 0.134465 0.232901i 0.0101646 0.0176056i
\(176\) 0 0
\(177\) −11.8215 −0.888560
\(178\) 0 0
\(179\) 20.1511 1.50617 0.753083 0.657926i \(-0.228566\pi\)
0.753083 + 0.657926i \(0.228566\pi\)
\(180\) 0 0
\(181\) −2.76240 + 4.78462i −0.205328 + 0.355638i −0.950237 0.311527i \(-0.899159\pi\)
0.744909 + 0.667166i \(0.232493\pi\)
\(182\) 0 0
\(183\) 4.89473 + 8.47792i 0.361829 + 0.626706i
\(184\) 0 0
\(185\) 2.62278 + 4.54279i 0.192831 + 0.333992i
\(186\) 0 0
\(187\) 46.0865 3.37018
\(188\) 0 0
\(189\) 0.134465 0.232901i 0.00978091 0.0169410i
\(190\) 0 0
\(191\) 2.25042 + 3.89784i 0.162835 + 0.282038i 0.935884 0.352308i \(-0.114603\pi\)
−0.773050 + 0.634346i \(0.781270\pi\)
\(192\) 0 0
\(193\) 0.522022 0.0375760 0.0187880 0.999823i \(-0.494019\pi\)
0.0187880 + 0.999823i \(0.494019\pi\)
\(194\) 0 0
\(195\) −1.14872 1.98965i −0.0822619 0.142482i
\(196\) 0 0
\(197\) 10.7953 18.6980i 0.769131 1.33217i −0.168904 0.985633i \(-0.554023\pi\)
0.938035 0.346541i \(-0.112644\pi\)
\(198\) 0 0
\(199\) −1.46680 2.54057i −0.103979 0.180096i 0.809342 0.587338i \(-0.199824\pi\)
−0.913320 + 0.407242i \(0.866491\pi\)
\(200\) 0 0
\(201\) 4.96860 + 6.50484i 0.350458 + 0.458816i
\(202\) 0 0
\(203\) 0.824713 + 1.42844i 0.0578835 + 0.100257i
\(204\) 0 0
\(205\) 1.80796 3.13148i 0.126273 0.218712i
\(206\) 0 0
\(207\) 2.57707 + 4.46362i 0.179119 + 0.310243i
\(208\) 0 0
\(209\) 23.0289 1.59294
\(210\) 0 0
\(211\) −9.86509 17.0868i −0.679141 1.17631i −0.975240 0.221149i \(-0.929019\pi\)
0.296099 0.955157i \(-0.404314\pi\)
\(212\) 0 0
\(213\) −6.02544 + 10.4364i −0.412856 + 0.715088i
\(214\) 0 0
\(215\) 10.9631 0.747680
\(216\) 0 0
\(217\) 1.41293 + 2.44727i 0.0959162 + 0.166132i
\(218\) 0 0
\(219\) 0.167288 + 0.289752i 0.0113043 + 0.0195796i
\(220\) 0 0
\(221\) −8.65210 + 14.9859i −0.582003 + 1.00806i
\(222\) 0 0
\(223\) −1.29322 −0.0866002 −0.0433001 0.999062i \(-0.513787\pi\)
−0.0433001 + 0.999062i \(0.513787\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −0.117554 + 0.203609i −0.00780233 + 0.0135140i −0.869900 0.493228i \(-0.835817\pi\)
0.862098 + 0.506742i \(0.169150\pi\)
\(228\) 0 0
\(229\) 4.55100 + 7.88257i 0.300739 + 0.520895i 0.976303 0.216406i \(-0.0694334\pi\)
−0.675565 + 0.737301i \(0.736100\pi\)
\(230\) 0 0
\(231\) 0.822768 + 1.42508i 0.0541342 + 0.0937631i
\(232\) 0 0
\(233\) −9.73932 + 16.8690i −0.638044 + 1.10513i 0.347817 + 0.937562i \(0.386923\pi\)
−0.985861 + 0.167563i \(0.946410\pi\)
\(234\) 0 0
\(235\) −6.55135 + 11.3473i −0.427363 + 0.740214i
\(236\) 0 0
\(237\) 4.90196 8.49044i 0.318417 0.551514i
\(238\) 0 0
\(239\) 4.89980 8.48670i 0.316942 0.548959i −0.662907 0.748702i \(-0.730677\pi\)
0.979848 + 0.199743i \(0.0640107\pi\)
\(240\) 0 0
\(241\) 15.3003 0.985577 0.492788 0.870149i \(-0.335978\pi\)
0.492788 + 0.870149i \(0.335978\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.46384 + 5.99954i 0.221297 + 0.383297i
\(246\) 0 0
\(247\) −4.32336 + 7.48827i −0.275089 + 0.476467i
\(248\) 0 0
\(249\) 4.16136 + 7.20768i 0.263715 + 0.456768i
\(250\) 0 0
\(251\) 0.00388720 + 0.00673283i 0.000245358 + 0.000424972i 0.866148 0.499787i \(-0.166589\pi\)
−0.865903 + 0.500212i \(0.833255\pi\)
\(252\) 0 0
\(253\) −31.5372 −1.98273
\(254\) 0 0
\(255\) −3.76596 6.52284i −0.235834 0.408476i
\(256\) 0 0
\(257\) 0.440192 0.762436i 0.0274585 0.0475594i −0.851970 0.523591i \(-0.824592\pi\)
0.879428 + 0.476032i \(0.157925\pi\)
\(258\) 0 0
\(259\) 1.41069 0.0876561
\(260\) 0 0
\(261\) −3.06664 + 5.31157i −0.189820 + 0.328778i
\(262\) 0 0
\(263\) −14.2044 −0.875880 −0.437940 0.899004i \(-0.644292\pi\)
−0.437940 + 0.899004i \(0.644292\pi\)
\(264\) 0 0
\(265\) 12.0675 0.741301
\(266\) 0 0
\(267\) −1.47751 −0.0904224
\(268\) 0 0
\(269\) −9.28569 −0.566158 −0.283079 0.959097i \(-0.591356\pi\)
−0.283079 + 0.959097i \(0.591356\pi\)
\(270\) 0 0
\(271\) 25.0251 1.52017 0.760083 0.649826i \(-0.225158\pi\)
0.760083 + 0.649826i \(0.225158\pi\)
\(272\) 0 0
\(273\) −0.617854 −0.0373942
\(274\) 0 0
\(275\) −3.05941 + 5.29905i −0.184489 + 0.319545i
\(276\) 0 0
\(277\) 1.55977 0.0937176 0.0468588 0.998902i \(-0.485079\pi\)
0.0468588 + 0.998902i \(0.485079\pi\)
\(278\) 0 0
\(279\) −5.25390 + 9.10002i −0.314543 + 0.544804i
\(280\) 0 0
\(281\) 16.3222 + 28.2709i 0.973703 + 1.68650i 0.684151 + 0.729341i \(0.260173\pi\)
0.289552 + 0.957162i \(0.406494\pi\)
\(282\) 0 0
\(283\) −23.0663 −1.37115 −0.685575 0.728002i \(-0.740449\pi\)
−0.685575 + 0.728002i \(0.740449\pi\)
\(284\) 0 0
\(285\) −1.88181 3.25939i −0.111469 0.193069i
\(286\) 0 0
\(287\) −0.486215 0.842149i −0.0287004 0.0497105i
\(288\) 0 0
\(289\) −19.8649 + 34.4071i −1.16853 + 2.02394i
\(290\) 0 0
\(291\) −2.64816 4.58675i −0.155238 0.268880i
\(292\) 0 0
\(293\) −29.7425 −1.73758 −0.868789 0.495182i \(-0.835101\pi\)
−0.868789 + 0.495182i \(0.835101\pi\)
\(294\) 0 0
\(295\) −11.8215 −0.688275
\(296\) 0 0
\(297\) −3.05941 + 5.29905i −0.177525 + 0.307482i
\(298\) 0 0
\(299\) 5.92069 10.2549i 0.342402 0.593058i
\(300\) 0 0
\(301\) 1.47416 2.55332i 0.0849692 0.147171i
\(302\) 0 0
\(303\) 5.55036 9.61350i 0.318860 0.552281i
\(304\) 0 0
\(305\) 4.89473 + 8.47792i 0.280271 + 0.485444i
\(306\) 0 0
\(307\) −7.69252 13.3238i −0.439036 0.760432i 0.558580 0.829451i \(-0.311346\pi\)
−0.997615 + 0.0690189i \(0.978013\pi\)
\(308\) 0 0
\(309\) 1.61545 2.79804i 0.0918997 0.159175i
\(310\) 0 0
\(311\) 13.6420 0.773568 0.386784 0.922170i \(-0.373586\pi\)
0.386784 + 0.922170i \(0.373586\pi\)
\(312\) 0 0
\(313\) 24.2528 1.37085 0.685424 0.728144i \(-0.259617\pi\)
0.685424 + 0.728144i \(0.259617\pi\)
\(314\) 0 0
\(315\) 0.134465 0.232901i 0.00757626 0.0131225i
\(316\) 0 0
\(317\) −2.84265 4.92362i −0.159659 0.276538i 0.775086 0.631855i \(-0.217706\pi\)
−0.934746 + 0.355317i \(0.884373\pi\)
\(318\) 0 0
\(319\) −18.7642 32.5006i −1.05059 1.81968i
\(320\) 0 0
\(321\) −8.58118 −0.478955
\(322\) 0 0
\(323\) −14.1736 + 24.5494i −0.788642 + 1.36597i
\(324\) 0 0
\(325\) −1.14872 1.98965i −0.0637198 0.110366i
\(326\) 0 0
\(327\) −18.0243 −0.996747
\(328\) 0 0
\(329\) 1.76186 + 3.05162i 0.0971343 + 0.168242i
\(330\) 0 0
\(331\) 6.80666 11.7895i 0.374128 0.648008i −0.616068 0.787693i \(-0.711275\pi\)
0.990196 + 0.139684i \(0.0446088\pi\)
\(332\) 0 0
\(333\) 2.62278 + 4.54279i 0.143727 + 0.248943i
\(334\) 0 0
\(335\) 4.96860 + 6.50484i 0.271464 + 0.355398i
\(336\) 0 0
\(337\) −17.0893 29.5995i −0.930913 1.61239i −0.781764 0.623575i \(-0.785680\pi\)
−0.149150 0.988815i \(-0.547654\pi\)
\(338\) 0 0
\(339\) −9.83260 + 17.0306i −0.534033 + 0.924973i
\(340\) 0 0
\(341\) −32.1477 55.6814i −1.74089 3.01532i
\(342\) 0 0
\(343\) 3.74558 0.202242
\(344\) 0 0
\(345\) 2.57707 + 4.46362i 0.138745 + 0.240313i
\(346\) 0 0
\(347\) −12.8534 + 22.2628i −0.690007 + 1.19513i 0.281828 + 0.959465i \(0.409059\pi\)
−0.971835 + 0.235662i \(0.924274\pi\)
\(348\) 0 0
\(349\) −20.1244 −1.07724 −0.538618 0.842550i \(-0.681053\pi\)
−0.538618 + 0.842550i \(0.681053\pi\)
\(350\) 0 0
\(351\) −1.14872 1.98965i −0.0613144 0.106200i
\(352\) 0 0
\(353\) 2.48311 + 4.30088i 0.132163 + 0.228913i 0.924510 0.381158i \(-0.124474\pi\)
−0.792347 + 0.610070i \(0.791141\pi\)
\(354\) 0 0
\(355\) −6.02544 + 10.4364i −0.319797 + 0.553905i
\(356\) 0 0
\(357\) −2.02556 −0.107204
\(358\) 0 0
\(359\) 23.3539 1.23257 0.616286 0.787523i \(-0.288637\pi\)
0.616286 + 0.787523i \(0.288637\pi\)
\(360\) 0 0
\(361\) 2.41760 4.18741i 0.127242 0.220390i
\(362\) 0 0
\(363\) −13.2200 22.8976i −0.693868 1.20181i
\(364\) 0 0
\(365\) 0.167288 + 0.289752i 0.00875627 + 0.0151663i
\(366\) 0 0
\(367\) 10.9902 19.0356i 0.573685 0.993651i −0.422498 0.906364i \(-0.638847\pi\)
0.996183 0.0872874i \(-0.0278199\pi\)
\(368\) 0 0
\(369\) 1.80796 3.13148i 0.0941186 0.163018i
\(370\) 0 0
\(371\) 1.62266 2.81053i 0.0842443 0.145915i
\(372\) 0 0
\(373\) −2.01096 + 3.48308i −0.104124 + 0.180347i −0.913380 0.407109i \(-0.866537\pi\)
0.809256 + 0.587456i \(0.199870\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 14.0909 0.725718
\(378\) 0 0
\(379\) 6.03876 + 10.4594i 0.310190 + 0.537265i 0.978403 0.206705i \(-0.0662739\pi\)
−0.668213 + 0.743970i \(0.732941\pi\)
\(380\) 0 0
\(381\) −4.27915 + 7.41171i −0.219228 + 0.379713i
\(382\) 0 0
\(383\) 16.4594 + 28.5084i 0.841034 + 1.45671i 0.889022 + 0.457865i \(0.151386\pi\)
−0.0479881 + 0.998848i \(0.515281\pi\)
\(384\) 0 0
\(385\) 0.822768 + 1.42508i 0.0419321 + 0.0726286i
\(386\) 0 0
\(387\) 10.9631 0.557288
\(388\) 0 0
\(389\) −8.37688 14.5092i −0.424725 0.735645i 0.571670 0.820484i \(-0.306296\pi\)
−0.996395 + 0.0848387i \(0.972962\pi\)
\(390\) 0 0
\(391\) 19.4103 33.6196i 0.981621 1.70022i
\(392\) 0 0
\(393\) −9.67842 −0.488212
\(394\) 0 0
\(395\) 4.90196 8.49044i 0.246644 0.427201i
\(396\) 0 0
\(397\) 11.9253 0.598515 0.299257 0.954172i \(-0.403261\pi\)
0.299257 + 0.954172i \(0.403261\pi\)
\(398\) 0 0
\(399\) −1.01215 −0.0506709
\(400\) 0 0
\(401\) 9.48657 0.473737 0.236868 0.971542i \(-0.423879\pi\)
0.236868 + 0.971542i \(0.423879\pi\)
\(402\) 0 0
\(403\) 24.1411 1.20256
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −32.0966 −1.59097
\(408\) 0 0
\(409\) −6.42013 + 11.1200i −0.317455 + 0.549848i −0.979956 0.199213i \(-0.936162\pi\)
0.662501 + 0.749061i \(0.269495\pi\)
\(410\) 0 0
\(411\) −5.37712 −0.265234
\(412\) 0 0
\(413\) −1.58958 + 2.75324i −0.0782183 + 0.135478i
\(414\) 0 0
\(415\) 4.16136 + 7.20768i 0.204273 + 0.353811i
\(416\) 0 0
\(417\) −12.8831 −0.630888
\(418\) 0 0
\(419\) 3.34364 + 5.79136i 0.163348 + 0.282927i 0.936067 0.351821i \(-0.114437\pi\)
−0.772720 + 0.634748i \(0.781104\pi\)
\(420\) 0 0
\(421\) 1.21145 + 2.09829i 0.0590425 + 0.102265i 0.894036 0.447995i \(-0.147862\pi\)
−0.834993 + 0.550260i \(0.814529\pi\)
\(422\) 0 0
\(423\) −6.55135 + 11.3473i −0.318537 + 0.551723i
\(424\) 0 0
\(425\) −3.76596 6.52284i −0.182676 0.316404i
\(426\) 0 0
\(427\) 2.63268 0.127405
\(428\) 0 0
\(429\) 14.0577 0.678711
\(430\) 0 0
\(431\) −18.4364 + 31.9327i −0.888048 + 1.53815i −0.0458691 + 0.998947i \(0.514606\pi\)
−0.842179 + 0.539198i \(0.818728\pi\)
\(432\) 0 0
\(433\) 12.5764 21.7830i 0.604384 1.04682i −0.387765 0.921758i \(-0.626753\pi\)
0.992149 0.125065i \(-0.0399139\pi\)
\(434\) 0 0
\(435\) −3.06664 + 5.31157i −0.147034 + 0.254671i
\(436\) 0 0
\(437\) 9.69910 16.7993i 0.463971 0.803621i
\(438\) 0 0
\(439\) −15.2586 26.4287i −0.728255 1.26137i −0.957620 0.288034i \(-0.906999\pi\)
0.229366 0.973340i \(-0.426335\pi\)
\(440\) 0 0
\(441\) 3.46384 + 5.99954i 0.164945 + 0.285693i
\(442\) 0 0
\(443\) 2.85162 4.93916i 0.135485 0.234666i −0.790298 0.612723i \(-0.790074\pi\)
0.925783 + 0.378056i \(0.123407\pi\)
\(444\) 0 0
\(445\) −1.47751 −0.0700409
\(446\) 0 0
\(447\) 3.97030 0.187789
\(448\) 0 0
\(449\) −2.29779 + 3.97989i −0.108439 + 0.187823i −0.915138 0.403140i \(-0.867919\pi\)
0.806699 + 0.590963i \(0.201252\pi\)
\(450\) 0 0
\(451\) 11.0626 + 19.1609i 0.520916 + 0.902253i
\(452\) 0 0
\(453\) 2.43423 + 4.21621i 0.114370 + 0.198095i
\(454\) 0 0
\(455\) −0.617854 −0.0289654
\(456\) 0 0
\(457\) −1.40065 + 2.42600i −0.0655196 + 0.113483i −0.896924 0.442184i \(-0.854204\pi\)
0.831405 + 0.555667i \(0.187537\pi\)
\(458\) 0 0
\(459\) −3.76596 6.52284i −0.175780 0.304460i
\(460\) 0 0
\(461\) 39.4011 1.83509 0.917545 0.397631i \(-0.130168\pi\)
0.917545 + 0.397631i \(0.130168\pi\)
\(462\) 0 0
\(463\) −2.78868 4.83014i −0.129601 0.224476i 0.793921 0.608021i \(-0.208036\pi\)
−0.923522 + 0.383545i \(0.874703\pi\)
\(464\) 0 0
\(465\) −5.25390 + 9.10002i −0.243644 + 0.422003i
\(466\) 0 0
\(467\) −3.61786 6.26632i −0.167415 0.289971i 0.770096 0.637929i \(-0.220208\pi\)
−0.937510 + 0.347958i \(0.886875\pi\)
\(468\) 0 0
\(469\) 2.18309 0.282515i 0.100806 0.0130453i
\(470\) 0 0
\(471\) −0.387721 0.671552i −0.0178652 0.0309435i
\(472\) 0 0
\(473\) −33.5407 + 58.0942i −1.54220 + 2.67117i
\(474\) 0 0
\(475\) −1.88181 3.25939i −0.0863432 0.149551i
\(476\) 0 0
\(477\) 12.0675 0.552533
\(478\) 0 0
\(479\) 9.76713 + 16.9172i 0.446272 + 0.772965i 0.998140 0.0609662i \(-0.0194182\pi\)
−0.551868 + 0.833931i \(0.686085\pi\)
\(480\) 0 0
\(481\) 6.02570 10.4368i 0.274748 0.475878i
\(482\) 0 0
\(483\) 1.38611 0.0630700
\(484\) 0 0
\(485\) −2.64816 4.58675i −0.120247 0.208274i
\(486\) 0 0
\(487\) 1.15817 + 2.00601i 0.0524818 + 0.0909012i 0.891073 0.453860i \(-0.149954\pi\)
−0.838591 + 0.544762i \(0.816620\pi\)
\(488\) 0 0
\(489\) 2.56301 4.43926i 0.115903 0.200750i
\(490\) 0 0
\(491\) 5.42241 0.244710 0.122355 0.992486i \(-0.460955\pi\)
0.122355 + 0.992486i \(0.460955\pi\)
\(492\) 0 0
\(493\) 46.1954 2.08054
\(494\) 0 0
\(495\) −3.05941 + 5.29905i −0.137510 + 0.238175i
\(496\) 0 0
\(497\) 1.62042 + 2.80666i 0.0726860 + 0.125896i
\(498\) 0 0
\(499\) 14.0470 + 24.3301i 0.628830 + 1.08917i 0.987787 + 0.155811i \(0.0497992\pi\)
−0.358957 + 0.933354i \(0.616867\pi\)
\(500\) 0 0
\(501\) 5.81380 10.0698i 0.259742 0.449885i
\(502\) 0 0
\(503\) −19.8880 + 34.4471i −0.886764 + 1.53592i −0.0430860 + 0.999071i \(0.513719\pi\)
−0.843678 + 0.536849i \(0.819614\pi\)
\(504\) 0 0
\(505\) 5.55036 9.61350i 0.246988 0.427795i
\(506\) 0 0
\(507\) 3.86087 6.68721i 0.171467 0.296990i
\(508\) 0 0
\(509\) −19.9647 −0.884920 −0.442460 0.896788i \(-0.645894\pi\)
−0.442460 + 0.896788i \(0.645894\pi\)
\(510\) 0 0
\(511\) 0.0899779 0.00398039
\(512\) 0 0
\(513\) −1.88181 3.25939i −0.0830838 0.143905i
\(514\) 0 0
\(515\) 1.61545 2.79804i 0.0711852 0.123296i
\(516\) 0 0
\(517\) −40.0865 69.4318i −1.76300 3.05361i
\(518\) 0 0
\(519\) −6.10683 10.5773i −0.268060 0.464294i
\(520\) 0 0
\(521\) −27.7518 −1.21583 −0.607913 0.794003i \(-0.707993\pi\)
−0.607913 + 0.794003i \(0.707993\pi\)
\(522\) 0 0
\(523\) 1.57105 + 2.72113i 0.0686971 + 0.118987i 0.898328 0.439325i \(-0.144782\pi\)
−0.829631 + 0.558312i \(0.811449\pi\)
\(524\) 0 0
\(525\) 0.134465 0.232901i 0.00586854 0.0101646i
\(526\) 0 0
\(527\) 79.1440 3.44756
\(528\) 0 0
\(529\) −1.78258 + 3.08751i −0.0775034 + 0.134240i
\(530\) 0 0
\(531\) −11.8215 −0.513010
\(532\) 0 0
\(533\) −8.30738 −0.359833
\(534\) 0 0
\(535\) −8.58118 −0.370997
\(536\) 0 0
\(537\) 20.1511 0.869585
\(538\) 0 0
\(539\) −42.3892 −1.82583
\(540\) 0 0
\(541\) 2.81725 0.121123 0.0605614 0.998164i \(-0.480711\pi\)
0.0605614 + 0.998164i \(0.480711\pi\)
\(542\) 0 0
\(543\) −2.76240 + 4.78462i −0.118546 + 0.205328i
\(544\) 0 0
\(545\) −18.0243 −0.772077
\(546\) 0 0
\(547\) 1.77829 3.08009i 0.0760342 0.131695i −0.825501 0.564400i \(-0.809108\pi\)
0.901536 + 0.432705i \(0.142441\pi\)
\(548\) 0 0
\(549\) 4.89473 + 8.47792i 0.208902 + 0.361829i
\(550\) 0 0
\(551\) 23.0833 0.983381
\(552\) 0 0
\(553\) −1.31829 2.28334i −0.0560592 0.0970975i
\(554\) 0 0
\(555\) 2.62278 + 4.54279i 0.111331 + 0.192831i
\(556\) 0 0
\(557\) −13.9232 + 24.1156i −0.589943 + 1.02181i 0.404296 + 0.914628i \(0.367517\pi\)
−0.994239 + 0.107184i \(0.965817\pi\)
\(558\) 0 0
\(559\) −12.5936 21.8128i −0.532653 0.922583i
\(560\) 0 0
\(561\) 46.0865 1.94577
\(562\) 0 0
\(563\) −2.71161 −0.114281 −0.0571403 0.998366i \(-0.518198\pi\)
−0.0571403 + 0.998366i \(0.518198\pi\)
\(564\) 0 0
\(565\) −9.83260 + 17.0306i −0.413660 + 0.716481i
\(566\) 0 0
\(567\) 0.134465 0.232901i 0.00564701 0.00978091i
\(568\) 0 0
\(569\) 7.92018 13.7182i 0.332031 0.575095i −0.650879 0.759182i \(-0.725599\pi\)
0.982910 + 0.184087i \(0.0589327\pi\)
\(570\) 0 0
\(571\) 12.0111 20.8038i 0.502649 0.870614i −0.497346 0.867552i \(-0.665692\pi\)
0.999995 0.00306177i \(-0.000974595\pi\)
\(572\) 0 0
\(573\) 2.25042 + 3.89784i 0.0940127 + 0.162835i
\(574\) 0 0
\(575\) 2.57707 + 4.46362i 0.107471 + 0.186146i
\(576\) 0 0
\(577\) 4.00330 6.93392i 0.166660 0.288663i −0.770584 0.637339i \(-0.780035\pi\)
0.937243 + 0.348676i \(0.113369\pi\)
\(578\) 0 0
\(579\) 0.522022 0.0216945
\(580\) 0 0
\(581\) 2.23823 0.0928575
\(582\) 0 0
\(583\) −36.9194 + 63.9463i −1.52905 + 2.64839i
\(584\) 0 0
\(585\) −1.14872 1.98965i −0.0474939 0.0822619i
\(586\) 0 0
\(587\) 4.39318 + 7.60921i 0.181326 + 0.314066i 0.942332 0.334679i \(-0.108628\pi\)
−0.761006 + 0.648744i \(0.775294\pi\)
\(588\) 0 0
\(589\) 39.5473 1.62952
\(590\) 0 0
\(591\) 10.7953 18.6980i 0.444058 0.769131i
\(592\) 0 0
\(593\) 5.56163 + 9.63303i 0.228389 + 0.395581i 0.957331 0.288994i \(-0.0933209\pi\)
−0.728942 + 0.684576i \(0.759988\pi\)
\(594\) 0 0
\(595\) −2.02556 −0.0830400
\(596\) 0 0
\(597\) −1.46680 2.54057i −0.0600322 0.103979i
\(598\) 0 0
\(599\) 24.0600 41.6731i 0.983065 1.70272i 0.332825 0.942989i \(-0.391998\pi\)
0.650240 0.759729i \(-0.274668\pi\)
\(600\) 0 0
\(601\) −4.46941 7.74125i −0.182311 0.315772i 0.760356 0.649507i \(-0.225025\pi\)
−0.942667 + 0.333734i \(0.891691\pi\)
\(602\) 0 0
\(603\) 4.96860 + 6.50484i 0.202337 + 0.264898i
\(604\) 0 0
\(605\) −13.2200 22.8976i −0.537468 0.930922i
\(606\) 0 0
\(607\) 19.8664 34.4097i 0.806354 1.39665i −0.109019 0.994040i \(-0.534771\pi\)
0.915373 0.402607i \(-0.131896\pi\)
\(608\) 0 0
\(609\) 0.824713 + 1.42844i 0.0334190 + 0.0578835i
\(610\) 0 0
\(611\) 30.1028 1.21783
\(612\) 0 0
\(613\) −0.726735 1.25874i −0.0293525 0.0508401i 0.850976 0.525205i \(-0.176011\pi\)
−0.880329 + 0.474365i \(0.842678\pi\)
\(614\) 0 0
\(615\) 1.80796 3.13148i 0.0729039 0.126273i
\(616\) 0 0
\(617\) 17.4071 0.700782 0.350391 0.936604i \(-0.386049\pi\)
0.350391 + 0.936604i \(0.386049\pi\)
\(618\) 0 0
\(619\) 3.15801 + 5.46983i 0.126931 + 0.219851i 0.922486 0.386030i \(-0.126154\pi\)
−0.795555 + 0.605881i \(0.792821\pi\)
\(620\) 0 0
\(621\) 2.57707 + 4.46362i 0.103414 + 0.179119i
\(622\) 0 0
\(623\) −0.198674 + 0.344114i −0.00795972 + 0.0137866i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 23.0289 0.919684
\(628\) 0 0
\(629\) 19.7546 34.2159i 0.787666 1.36428i
\(630\) 0 0
\(631\) −3.45825 5.98987i −0.137671 0.238453i 0.788944 0.614465i \(-0.210628\pi\)
−0.926615 + 0.376013i \(0.877295\pi\)
\(632\) 0 0
\(633\) −9.86509 17.0868i −0.392102 0.679141i
\(634\) 0 0
\(635\) −4.27915 + 7.41171i −0.169813 + 0.294125i
\(636\) 0 0
\(637\) 7.95799 13.7836i 0.315307 0.546128i
\(638\) 0 0
\(639\) −6.02544 + 10.4364i −0.238363 + 0.412856i
\(640\) 0 0
\(641\) −5.94521 + 10.2974i −0.234822 + 0.406723i −0.959221 0.282658i \(-0.908784\pi\)
0.724399 + 0.689381i \(0.242117\pi\)
\(642\) 0 0
\(643\) 18.3036 0.721822 0.360911 0.932600i \(-0.382466\pi\)
0.360911 + 0.932600i \(0.382466\pi\)
\(644\) 0 0
\(645\) 10.9631 0.431673
\(646\) 0 0
\(647\) −14.5162 25.1429i −0.570692 0.988468i −0.996495 0.0836518i \(-0.973342\pi\)
0.425803 0.904816i \(-0.359992\pi\)
\(648\) 0 0
\(649\) 36.1669 62.6428i 1.41967 2.45895i
\(650\) 0 0
\(651\) 1.41293 + 2.44727i 0.0553773 + 0.0959162i
\(652\) 0 0
\(653\) 14.5471 + 25.1963i 0.569273 + 0.986009i 0.996638 + 0.0819302i \(0.0261085\pi\)
−0.427365 + 0.904079i \(0.640558\pi\)
\(654\) 0 0
\(655\) −9.67842 −0.378167
\(656\) 0 0
\(657\) 0.167288 + 0.289752i 0.00652654 + 0.0113043i
\(658\) 0 0
\(659\) 3.33052 5.76863i 0.129739 0.224714i −0.793837 0.608131i \(-0.791919\pi\)
0.923575 + 0.383417i \(0.125253\pi\)
\(660\) 0 0
\(661\) 24.1141 0.937931 0.468965 0.883217i \(-0.344627\pi\)
0.468965 + 0.883217i \(0.344627\pi\)
\(662\) 0 0
\(663\) −8.65210 + 14.9859i −0.336020 + 0.582003i
\(664\) 0 0
\(665\) −1.01215 −0.0392495
\(666\) 0 0
\(667\) −31.6118 −1.22401
\(668\) 0 0
\(669\) −1.29322 −0.0499986
\(670\) 0 0
\(671\) −59.8999 −2.31241
\(672\) 0 0
\(673\) −1.74705 −0.0673437 −0.0336719 0.999433i \(-0.510720\pi\)
−0.0336719 + 0.999433i \(0.510720\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 2.43731 4.22154i 0.0936733 0.162247i −0.815381 0.578925i \(-0.803472\pi\)
0.909054 + 0.416678i \(0.136806\pi\)
\(678\) 0 0
\(679\) −1.42434 −0.0546613
\(680\) 0 0
\(681\) −0.117554 + 0.203609i −0.00450468 + 0.00780233i
\(682\) 0 0
\(683\) −7.05378 12.2175i −0.269905 0.467490i 0.698932 0.715188i \(-0.253659\pi\)
−0.968837 + 0.247699i \(0.920326\pi\)
\(684\) 0 0
\(685\) −5.37712 −0.205449
\(686\) 0 0
\(687\) 4.55100 + 7.88257i 0.173632 + 0.300739i
\(688\) 0 0
\(689\) −13.8622 24.0101i −0.528109 0.914712i
\(690\) 0 0
\(691\) −13.7011 + 23.7311i −0.521215 + 0.902772i 0.478480 + 0.878098i \(0.341188\pi\)
−0.999696 + 0.0246733i \(0.992145\pi\)
\(692\) 0 0
\(693\) 0.822768 + 1.42508i 0.0312544 + 0.0541342i
\(694\) 0 0
\(695\) −12.8831 −0.488684
\(696\) 0 0
\(697\) −27.2348 −1.03159
\(698\) 0 0
\(699\) −9.73932 + 16.8690i −0.368375 + 0.638044i
\(700\) 0 0
\(701\) −5.74457 + 9.94989i −0.216969 + 0.375802i −0.953880 0.300188i \(-0.902951\pi\)
0.736911 + 0.675990i \(0.236284\pi\)
\(702\) 0 0
\(703\) 9.87113 17.0973i 0.372297 0.644837i
\(704\) 0 0
\(705\) −6.55135 + 11.3473i −0.246738 + 0.427363i
\(706\) 0 0
\(707\) −1.49266 2.58536i −0.0561372 0.0972326i
\(708\) 0 0
\(709\) −11.6373 20.1564i −0.437048 0.756989i 0.560412 0.828214i \(-0.310643\pi\)
−0.997460 + 0.0712246i \(0.977309\pi\)
\(710\) 0 0
\(711\) 4.90196 8.49044i 0.183838 0.318417i
\(712\) 0 0
\(713\) −54.1587 −2.02826
\(714\) 0 0
\(715\) 14.0577 0.525727
\(716\) 0 0
\(717\) 4.89980 8.48670i 0.182986 0.316942i
\(718\) 0 0
\(719\) 11.1752 + 19.3560i 0.416764 + 0.721857i 0.995612 0.0935790i \(-0.0298308\pi\)
−0.578848 + 0.815436i \(0.696497\pi\)
\(720\) 0 0
\(721\) −0.434444 0.752478i −0.0161795 0.0280238i
\(722\) 0 0
\(723\) 15.3003 0.569023
\(724\) 0 0
\(725\) −3.06664 + 5.31157i −0.113892 + 0.197267i
\(726\) 0 0
\(727\) 0.885689 + 1.53406i 0.0328484 + 0.0568951i 0.881982 0.471283i \(-0.156209\pi\)
−0.849134 + 0.528178i \(0.822875\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −41.2867 71.5107i −1.52705 2.64492i
\(732\) 0 0
\(733\) 3.15295 5.46108i 0.116457 0.201709i −0.801904 0.597453i \(-0.796180\pi\)
0.918361 + 0.395743i \(0.129513\pi\)
\(734\) 0 0
\(735\) 3.46384 + 5.99954i 0.127766 + 0.221297i
\(736\) 0 0
\(737\) −49.6705 + 6.42789i −1.82964 + 0.236774i
\(738\) 0 0
\(739\) 8.04082 + 13.9271i 0.295786 + 0.512317i 0.975167 0.221469i \(-0.0710851\pi\)
−0.679381 + 0.733785i \(0.737752\pi\)
\(740\) 0 0
\(741\) −4.32336 + 7.48827i −0.158822 + 0.275089i
\(742\) 0 0
\(743\) −13.6661 23.6704i −0.501361 0.868383i −0.999999 0.00157266i \(-0.999499\pi\)
0.498637 0.866811i \(-0.333834\pi\)
\(744\) 0 0
\(745\) 3.97030 0.145461
\(746\) 0 0
\(747\) 4.16136 + 7.20768i 0.152256 + 0.263715i
\(748\) 0 0
\(749\) −1.15387 + 1.99856i −0.0421615 + 0.0730259i
\(750\) 0 0
\(751\) 21.9798 0.802054 0.401027 0.916066i \(-0.368653\pi\)
0.401027 + 0.916066i \(0.368653\pi\)
\(752\) 0 0
\(753\) 0.00388720 + 0.00673283i 0.000141657 + 0.000245358i
\(754\) 0 0
\(755\) 2.43423 + 4.21621i 0.0885907 + 0.153444i
\(756\) 0 0
\(757\) −17.1659 + 29.7321i −0.623904 + 1.08063i 0.364848 + 0.931067i \(0.381121\pi\)
−0.988752 + 0.149566i \(0.952212\pi\)
\(758\) 0 0
\(759\) −31.5372 −1.14473
\(760\) 0 0
\(761\) −21.9275 −0.794871 −0.397436 0.917630i \(-0.630100\pi\)
−0.397436 + 0.917630i \(0.630100\pi\)
\(762\) 0 0
\(763\) −2.42364 + 4.19787i −0.0877418 + 0.151973i
\(764\) 0 0
\(765\) −3.76596 6.52284i −0.136159 0.235834i
\(766\) 0 0
\(767\) 13.5797 + 23.5207i 0.490333 + 0.849282i
\(768\) 0 0
\(769\) −7.91490 + 13.7090i −0.285419 + 0.494360i −0.972711 0.232022i \(-0.925466\pi\)
0.687292 + 0.726381i \(0.258799\pi\)
\(770\) 0 0
\(771\) 0.440192 0.762436i 0.0158531 0.0274585i
\(772\) 0 0
\(773\) −7.02348 + 12.1650i −0.252617 + 0.437546i −0.964246 0.265010i \(-0.914625\pi\)
0.711628 + 0.702556i \(0.247958\pi\)
\(774\) 0 0
\(775\) −5.25390 + 9.10002i −0.188726 + 0.326883i
\(776\) 0 0
\(777\) 1.41069 0.0506082
\(778\) 0 0
\(779\) −13.6089 −0.487590
\(780\) 0 0
\(781\) −36.8686 63.8582i −1.31926 2.28503i
\(782\) 0 0
\(783\) −3.06664 + 5.31157i −0.109593 + 0.189820i
\(784\) 0 0
\(785\) −0.387721 0.671552i −0.0138384 0.0239687i
\(786\) 0 0
\(787\) −1.15172 1.99484i −0.0410544 0.0711083i 0.844768 0.535133i \(-0.179738\pi\)
−0.885822 + 0.464024i \(0.846405\pi\)
\(788\) 0 0
\(789\) −14.2044 −0.505690
\(790\) 0 0
\(791\) 2.64428 + 4.58003i 0.0940199 + 0.162847i
\(792\) 0 0
\(793\) 11.2454 19.4776i 0.399335 0.691669i
\(794\) 0 0
\(795\) 12.0675 0.427990
\(796\) 0 0
\(797\) −4.64793 + 8.05046i −0.164638 + 0.285162i −0.936527 0.350596i \(-0.885979\pi\)
0.771889 + 0.635758i \(0.219312\pi\)
\(798\) 0 0
\(799\) 98.6884 3.49135
\(800\) 0 0
\(801\) −1.47751 −0.0522054
\(802\) 0 0
\(803\) −2.04721 −0.0722446
\(804\) 0 0
\(805\) 1.38611 0.0488538
\(806\) 0 0
\(807\) −9.28569 −0.326872
\(808\) 0 0
\(809\) −37.1029 −1.30447 −0.652235 0.758017i \(-0.726168\pi\)
−0.652235 + 0.758017i \(0.726168\pi\)
\(810\) 0 0
\(811\) 4.31942 7.48145i 0.151675 0.262709i −0.780168 0.625570i \(-0.784867\pi\)
0.931843 + 0.362861i \(0.118200\pi\)
\(812\) 0 0
\(813\) 25.0251 0.877669
\(814\) 0 0
\(815\) 2.56301 4.43926i 0.0897783 0.155501i
\(816\) 0 0
\(817\) −20.6305 35.7331i −0.721770 1.25014i
\(818\) 0 0
\(819\) −0.617854 −0.0215896
\(820\) 0 0
\(821\) −8.11182 14.0501i −0.283105 0.490351i 0.689043 0.724720i \(-0.258031\pi\)
−0.972148 + 0.234369i \(0.924698\pi\)
\(822\) 0 0
\(823\) 16.5546 + 28.6733i 0.577055 + 0.999489i 0.995815 + 0.0913921i \(0.0291317\pi\)
−0.418760 + 0.908097i \(0.637535\pi\)
\(824\) 0 0
\(825\) −3.05941 + 5.29905i −0.106515 + 0.184489i
\(826\) 0 0
\(827\) 20.0811 + 34.7816i 0.698290 + 1.20947i 0.969059 + 0.246829i \(0.0793886\pi\)
−0.270769 + 0.962644i \(0.587278\pi\)
\(828\) 0 0
\(829\) 43.5191 1.51148 0.755741 0.654871i \(-0.227277\pi\)
0.755741 + 0.654871i \(0.227277\pi\)
\(830\) 0 0
\(831\) 1.55977 0.0541079
\(832\) 0 0
\(833\) 26.0894 45.1881i 0.903943 1.56567i
\(834\) 0 0
\(835\) 5.81380 10.0698i 0.201195 0.348480i
\(836\) 0 0
\(837\) −5.25390 + 9.10002i −0.181601 + 0.314543i
\(838\) 0 0
\(839\) −9.12578 + 15.8063i −0.315057 + 0.545695i −0.979450 0.201689i \(-0.935357\pi\)
0.664393 + 0.747384i \(0.268690\pi\)
\(840\) 0 0
\(841\) −4.30855 7.46263i −0.148571 0.257332i
\(842\) 0 0
\(843\) 16.3222 + 28.2709i 0.562168 + 0.973703i
\(844\) 0 0
\(845\) 3.86087 6.68721i 0.132818 0.230047i
\(846\) 0 0
\(847\) −7.11050 −0.244320
\(848\) 0 0
\(849\) −23.0663 −0.791633
\(850\) 0 0
\(851\) −13.5182 + 23.4142i −0.463397 + 0.802627i
\(852\) 0 0
\(853\) −11.1112 19.2451i −0.380440 0.658941i 0.610685 0.791873i \(-0.290894\pi\)
−0.991125 + 0.132932i \(0.957561\pi\)
\(854\) 0 0
\(855\) −1.88181 3.25939i −0.0643565 0.111469i
\(856\) 0 0
\(857\) 23.7509 0.811315 0.405657 0.914025i \(-0.367043\pi\)
0.405657 + 0.914025i \(0.367043\pi\)
\(858\) 0 0
\(859\) −15.1911 + 26.3118i −0.518315 + 0.897748i 0.481459 + 0.876469i \(0.340107\pi\)
−0.999774 + 0.0212788i \(0.993226\pi\)
\(860\) 0 0
\(861\) −0.486215 0.842149i −0.0165702 0.0287004i
\(862\) 0 0
\(863\) 45.1045 1.53537 0.767687 0.640825i \(-0.221407\pi\)
0.767687 + 0.640825i \(0.221407\pi\)
\(864\) 0 0
\(865\) −6.10683 10.5773i −0.207638 0.359640i
\(866\) 0 0
\(867\) −19.8649 + 34.4071i −0.674648 + 1.16853i
\(868\) 0 0
\(869\) 29.9942 + 51.9515i 1.01748 + 1.76233i
\(870\) 0 0
\(871\) 7.23480 17.3580i 0.245142 0.588155i
\(872\) 0 0
\(873\) −2.64816 4.58675i −0.0896268 0.155238i
\(874\) 0 0
\(875\) 0.134465 0.232901i 0.00454575 0.00787348i
\(876\) 0 0
\(877\) 3.54078 + 6.13281i 0.119564 + 0.207090i 0.919595 0.392868i \(-0.128517\pi\)
−0.800031 + 0.599958i \(0.795184\pi\)
\(878\) 0 0
\(879\) −29.7425 −1.00319
\(880\) 0 0
\(881\) 4.57111 + 7.91740i 0.154005 + 0.266744i 0.932696 0.360663i \(-0.117450\pi\)
−0.778691 + 0.627407i \(0.784116\pi\)
\(882\) 0 0
\(883\) −21.2989 + 36.8907i −0.716764 + 1.24147i 0.245512 + 0.969394i \(0.421044\pi\)
−0.962275 + 0.272077i \(0.912289\pi\)
\(884\) 0 0
\(885\) −11.8215 −0.397376
\(886\) 0 0
\(887\) −10.3149 17.8660i −0.346342 0.599881i 0.639255 0.768995i \(-0.279243\pi\)
−0.985597 + 0.169114i \(0.945910\pi\)
\(888\) 0 0
\(889\) 1.15079 + 1.99323i 0.0385964 + 0.0668510i
\(890\) 0 0
\(891\) −3.05941 + 5.29905i −0.102494 + 0.177525i
\(892\) 0 0
\(893\) 49.3135 1.65021
\(894\) 0 0
\(895\) 20.1511 0.673578
\(896\) 0 0
\(897\) 5.92069 10.2549i 0.197686 0.342402i
\(898\) 0 0
\(899\) −32.2236 55.8130i −1.07472 1.86147i
\(900\) 0 0
\(901\) −45.4458 78.7144i −1.51402 2.62236i
\(902\) 0 0
\(903\) 1.47416 2.55332i 0.0490570 0.0849692i
\(904\) 0 0
\(905\) −2.76240 + 4.78462i −0.0918254 + 0.159046i
\(906\) 0 0
\(907\) 1.45033 2.51205i 0.0481575 0.0834112i −0.840942 0.541125i \(-0.817998\pi\)
0.889099 + 0.457714i \(0.151332\pi\)
\(908\) 0 0
\(909\) 5.55036 9.61350i 0.184094 0.318860i
\(910\) 0 0
\(911\) 47.0652 1.55934 0.779669 0.626192i \(-0.215387\pi\)
0.779669 + 0.626192i \(0.215387\pi\)
\(912\) 0 0
\(913\) −50.9252 −1.68538
\(914\) 0 0
\(915\) 4.89473 + 8.47792i 0.161815 + 0.280271i
\(916\) 0 0
\(917\) −1.30141 + 2.25411i −0.0429764 + 0.0744373i
\(918\) 0 0
\(919\) −10.1149 17.5195i −0.333658 0.577913i 0.649568 0.760304i \(-0.274950\pi\)
−0.983226 + 0.182390i \(0.941617\pi\)
\(920\) 0 0
\(921\) −7.69252 13.3238i −0.253477 0.439036i
\(922\) 0 0
\(923\) 27.6863 0.911305
\(924\) 0 0
\(925\) 2.62278 + 4.54279i 0.0862365 + 0.149366i
\(926\) 0 0
\(927\) 1.61545 2.79804i 0.0530583 0.0918997i
\(928\) 0 0
\(929\) −5.78842 −0.189912 −0.0949560 0.995481i \(-0.530271\pi\)
−0.0949560 + 0.995481i \(0.530271\pi\)
\(930\) 0 0
\(931\) 13.0366 22.5800i 0.427256 0.740029i
\(932\) 0 0
\(933\) 13.6420 0.446620
\(934\) 0 0
\(935\) 46.0865 1.50719
\(936\) 0 0
\(937\) 7.62353 0.249050 0.124525 0.992216i \(-0.460259\pi\)
0.124525 + 0.992216i \(0.460259\pi\)
\(938\) 0 0
\(939\) 24.2528 0.791460
\(940\) 0 0
\(941\) 8.02812 0.261709 0.130855 0.991402i \(-0.458228\pi\)
0.130855 + 0.991402i \(0.458228\pi\)
\(942\) 0 0
\(943\) 18.6369 0.606902
\(944\) 0 0
\(945\) 0.134465 0.232901i 0.00437415 0.00757626i
\(946\) 0 0
\(947\) 39.8835 1.29604 0.648021 0.761623i \(-0.275597\pi\)
0.648021 + 0.761623i \(0.275597\pi\)
\(948\) 0 0
\(949\) 0.384336 0.665690i 0.0124761 0.0216092i
\(950\) 0 0
\(951\) −2.84265 4.92362i −0.0921794 0.159659i
\(952\) 0 0
\(953\) 22.8924 0.741557 0.370778 0.928721i \(-0.379091\pi\)
0.370778 + 0.928721i \(0.379091\pi\)
\(954\) 0 0
\(955\) 2.25042 + 3.89784i 0.0728219 + 0.126131i
\(956\) 0 0
\(957\) −18.7642 32.5006i −0.606561 1.05059i
\(958\) 0 0
\(959\) −0.723036 + 1.25233i −0.0233480 + 0.0404400i
\(960\) 0 0
\(961\) −39.7070 68.7745i −1.28087 2.21853i
\(962\) 0 0
\(963\) −8.58118 −0.276525
\(964\) 0 0
\(965\) 0.522022 0.0168045
\(966\) 0 0
\(967\) 11.2004 19.3997i 0.360182 0.623854i −0.627808 0.778368i \(-0.716048\pi\)
0.987991 + 0.154514i \(0.0493812\pi\)
\(968\) 0 0
\(969\) −14.1736 + 24.5494i −0.455322 + 0.788642i
\(970\) 0 0
\(971\) 22.5266 39.0173i 0.722914 1.25212i −0.236913 0.971531i \(-0.576136\pi\)
0.959827 0.280593i \(-0.0905310\pi\)
\(972\) 0 0
\(973\) −1.73233 + 3.00048i −0.0555359 + 0.0961910i
\(974\) 0 0
\(975\) −1.14872 1.98965i −0.0367886 0.0637198i
\(976\) 0 0
\(977\) 19.2937 + 33.4178i 0.617262 + 1.06913i 0.989983 + 0.141185i \(0.0450913\pi\)
−0.372722 + 0.927943i \(0.621575\pi\)
\(978\) 0 0
\(979\) 4.52032 7.82942i 0.144470 0.250229i
\(980\) 0 0
\(981\) −18.0243 −0.575472
\(982\) 0 0
\(983\) 29.0071 0.925182 0.462591 0.886572i \(-0.346920\pi\)
0.462591 + 0.886572i \(0.346920\pi\)
\(984\) 0 0
\(985\) 10.7953 18.6980i 0.343966 0.595766i
\(986\) 0 0
\(987\) 1.76186 + 3.05162i 0.0560805 + 0.0971343i
\(988\) 0 0
\(989\) 28.2528 + 48.9352i 0.898386 + 1.55605i
\(990\) 0 0
\(991\) −8.56394 −0.272043 −0.136021 0.990706i \(-0.543432\pi\)
−0.136021 + 0.990706i \(0.543432\pi\)
\(992\) 0 0
\(993\) 6.80666 11.7895i 0.216003 0.374128i
\(994\) 0 0
\(995\) −1.46680 2.54057i −0.0465007 0.0805416i
\(996\) 0 0
\(997\) 49.4074 1.56475 0.782375 0.622808i \(-0.214008\pi\)
0.782375 + 0.622808i \(0.214008\pi\)
\(998\) 0 0
\(999\) 2.62278 + 4.54279i 0.0829811 + 0.143727i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4020.2.q.m.841.7 24
67.29 even 3 inner 4020.2.q.m.3781.7 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.q.m.841.7 24 1.1 even 1 trivial
4020.2.q.m.3781.7 yes 24 67.29 even 3 inner