Properties

Label 60.7.b.a.29.9
Level $60$
Weight $7$
Character 60.29
Analytic conductor $13.803$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,7,Mod(29,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.29");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 60.b (of order \(2\), degree \(1\), 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: \(13.8032450172\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 1880 x^{10} + 1266870 x^{8} + 399545800 x^{6} + 62009694600 x^{4} + 4432082624000 x^{2} + 109931031040000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 5^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 29.9
Root \(-7.15535i\) of defining polynomial
Character \(\chi\) \(=\) 60.29
Dual form 60.7.b.a.29.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(22.8848 - 14.3278i) q^{3} +(-93.9389 + 82.4651i) q^{5} -279.744i q^{7} +(318.429 - 655.777i) q^{9} +O(q^{10})\) \(q+(22.8848 - 14.3278i) q^{3} +(-93.9389 + 82.4651i) q^{5} -279.744i q^{7} +(318.429 - 655.777i) q^{9} -772.624i q^{11} -2877.85i q^{13} +(-968.231 + 3233.13i) q^{15} +2794.21 q^{17} -13069.1 q^{19} +(-4008.12 - 6401.89i) q^{21} +10557.8 q^{23} +(2024.03 - 15493.4i) q^{25} +(-2108.66 - 19569.7i) q^{27} -37261.6i q^{29} -8711.04 q^{31} +(-11070.0 - 17681.4i) q^{33} +(23069.1 + 26278.9i) q^{35} -3555.10i q^{37} +(-41233.2 - 65859.0i) q^{39} +108887. i q^{41} +66697.5i q^{43} +(24165.9 + 87862.3i) q^{45} -4022.16 q^{47} +39392.1 q^{49} +(63944.9 - 40034.8i) q^{51} +79567.7 q^{53} +(63714.5 + 72579.5i) q^{55} +(-299084. + 187251. i) q^{57} +151116. i q^{59} +207998. q^{61} +(-183450. - 89078.6i) q^{63} +(237322. + 270342. i) q^{65} -363435. i q^{67} +(241613. - 151270. i) q^{69} +37954.2i q^{71} +461513. i q^{73} +(-175666. - 383562. i) q^{75} -216137. q^{77} -227372. q^{79} +(-328647. - 417637. i) q^{81} +998762. q^{83} +(-262485. + 230425. i) q^{85} +(-533877. - 852725. i) q^{87} +1.03916e6i q^{89} -805062. q^{91} +(-199350. + 124810. i) q^{93} +(1.22770e6 - 1.07774e6i) q^{95} -734512. i q^{97} +(-506670. - 246026. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 712 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 712 q^{9} + 2480 q^{15} - 192 q^{19} + 5348 q^{21} + 18660 q^{25} + 40848 q^{31} - 45312 q^{39} + 45340 q^{45} - 242940 q^{49} - 40720 q^{51} - 24240 q^{55} - 99312 q^{61} + 108460 q^{69} + 126640 q^{75} + 626544 q^{79} - 798268 q^{81} - 732720 q^{85} + 1996032 q^{91} + 1632080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 22.8848 14.3278i 0.847585 0.530659i
\(4\) 0 0
\(5\) −93.9389 + 82.4651i −0.751511 + 0.659720i
\(6\) 0 0
\(7\) 279.744i 0.815581i −0.913076 0.407791i \(-0.866299\pi\)
0.913076 0.407791i \(-0.133701\pi\)
\(8\) 0 0
\(9\) 318.429 655.777i 0.436802 0.899558i
\(10\) 0 0
\(11\) 772.624i 0.580484i −0.956953 0.290242i \(-0.906264\pi\)
0.956953 0.290242i \(-0.0937358\pi\)
\(12\) 0 0
\(13\) 2877.85i 1.30990i −0.755672 0.654950i \(-0.772690\pi\)
0.755672 0.654950i \(-0.227310\pi\)
\(14\) 0 0
\(15\) −968.231 + 3233.13i −0.286883 + 0.957966i
\(16\) 0 0
\(17\) 2794.21 0.568738 0.284369 0.958715i \(-0.408216\pi\)
0.284369 + 0.958715i \(0.408216\pi\)
\(18\) 0 0
\(19\) −13069.1 −1.90539 −0.952697 0.303922i \(-0.901704\pi\)
−0.952697 + 0.303922i \(0.901704\pi\)
\(20\) 0 0
\(21\) −4008.12 6401.89i −0.432795 0.691275i
\(22\) 0 0
\(23\) 10557.8 0.867740 0.433870 0.900975i \(-0.357148\pi\)
0.433870 + 0.900975i \(0.357148\pi\)
\(24\) 0 0
\(25\) 2024.03 15493.4i 0.129538 0.991574i
\(26\) 0 0
\(27\) −2108.66 19569.7i −0.107131 0.994245i
\(28\) 0 0
\(29\) 37261.6i 1.52780i −0.645332 0.763902i \(-0.723281\pi\)
0.645332 0.763902i \(-0.276719\pi\)
\(30\) 0 0
\(31\) −8711.04 −0.292405 −0.146203 0.989255i \(-0.546705\pi\)
−0.146203 + 0.989255i \(0.546705\pi\)
\(32\) 0 0
\(33\) −11070.0 17681.4i −0.308039 0.492010i
\(34\) 0 0
\(35\) 23069.1 + 26278.9i 0.538055 + 0.612918i
\(36\) 0 0
\(37\) 3555.10i 0.0701854i −0.999384 0.0350927i \(-0.988827\pi\)
0.999384 0.0350927i \(-0.0111726\pi\)
\(38\) 0 0
\(39\) −41233.2 65859.0i −0.695110 1.11025i
\(40\) 0 0
\(41\) 108887.i 1.57988i 0.613187 + 0.789938i \(0.289887\pi\)
−0.613187 + 0.789938i \(0.710113\pi\)
\(42\) 0 0
\(43\) 66697.5i 0.838889i 0.907781 + 0.419444i \(0.137775\pi\)
−0.907781 + 0.419444i \(0.862225\pi\)
\(44\) 0 0
\(45\) 24165.9 + 87862.3i 0.265195 + 0.964195i
\(46\) 0 0
\(47\) −4022.16 −0.0387405 −0.0193703 0.999812i \(-0.506166\pi\)
−0.0193703 + 0.999812i \(0.506166\pi\)
\(48\) 0 0
\(49\) 39392.1 0.334828
\(50\) 0 0
\(51\) 63944.9 40034.8i 0.482054 0.301806i
\(52\) 0 0
\(53\) 79567.7 0.534453 0.267226 0.963634i \(-0.413893\pi\)
0.267226 + 0.963634i \(0.413893\pi\)
\(54\) 0 0
\(55\) 63714.5 + 72579.5i 0.382957 + 0.436240i
\(56\) 0 0
\(57\) −299084. + 187251.i −1.61498 + 1.01111i
\(58\) 0 0
\(59\) 151116.i 0.735793i 0.929867 + 0.367897i \(0.119922\pi\)
−0.929867 + 0.367897i \(0.880078\pi\)
\(60\) 0 0
\(61\) 207998. 0.916369 0.458184 0.888857i \(-0.348500\pi\)
0.458184 + 0.888857i \(0.348500\pi\)
\(62\) 0 0
\(63\) −183450. 89078.6i −0.733662 0.356248i
\(64\) 0 0
\(65\) 237322. + 270342.i 0.864168 + 0.984404i
\(66\) 0 0
\(67\) 363435.i 1.20838i −0.796842 0.604188i \(-0.793498\pi\)
0.796842 0.604188i \(-0.206502\pi\)
\(68\) 0 0
\(69\) 241613. 151270.i 0.735484 0.460474i
\(70\) 0 0
\(71\) 37954.2i 0.106044i 0.998593 + 0.0530218i \(0.0168853\pi\)
−0.998593 + 0.0530218i \(0.983115\pi\)
\(72\) 0 0
\(73\) 461513.i 1.18636i 0.805071 + 0.593179i \(0.202127\pi\)
−0.805071 + 0.593179i \(0.797873\pi\)
\(74\) 0 0
\(75\) −175666. 383562.i −0.416393 0.909185i
\(76\) 0 0
\(77\) −216137. −0.473432
\(78\) 0 0
\(79\) −227372. −0.461164 −0.230582 0.973053i \(-0.574063\pi\)
−0.230582 + 0.973053i \(0.574063\pi\)
\(80\) 0 0
\(81\) −328647. 417637.i −0.618408 0.785857i
\(82\) 0 0
\(83\) 998762. 1.74674 0.873369 0.487060i \(-0.161931\pi\)
0.873369 + 0.487060i \(0.161931\pi\)
\(84\) 0 0
\(85\) −262485. + 230425.i −0.427413 + 0.375208i
\(86\) 0 0
\(87\) −533877. 852725.i −0.810743 1.29494i
\(88\) 0 0
\(89\) 1.03916e6i 1.47405i 0.675864 + 0.737026i \(0.263771\pi\)
−0.675864 + 0.737026i \(0.736229\pi\)
\(90\) 0 0
\(91\) −805062. −1.06833
\(92\) 0 0
\(93\) −199350. + 124810.i −0.247838 + 0.155167i
\(94\) 0 0
\(95\) 1.22770e6 1.07774e6i 1.43192 1.25703i
\(96\) 0 0
\(97\) 734512.i 0.804792i −0.915466 0.402396i \(-0.868178\pi\)
0.915466 0.402396i \(-0.131822\pi\)
\(98\) 0 0
\(99\) −506670. 246026.i −0.522179 0.253557i
\(100\) 0 0
\(101\) 184614.i 0.179184i −0.995979 0.0895922i \(-0.971444\pi\)
0.995979 0.0895922i \(-0.0285564\pi\)
\(102\) 0 0
\(103\) 1.69989e6i 1.55564i −0.628489 0.777818i \(-0.716326\pi\)
0.628489 0.777818i \(-0.283674\pi\)
\(104\) 0 0
\(105\) 904451. + 270857.i 0.781299 + 0.233977i
\(106\) 0 0
\(107\) −1.51250e6 −1.23465 −0.617325 0.786708i \(-0.711784\pi\)
−0.617325 + 0.786708i \(0.711784\pi\)
\(108\) 0 0
\(109\) 1.08425e6 0.837237 0.418618 0.908162i \(-0.362514\pi\)
0.418618 + 0.908162i \(0.362514\pi\)
\(110\) 0 0
\(111\) −50936.7 81357.8i −0.0372445 0.0594881i
\(112\) 0 0
\(113\) 639940. 0.443511 0.221755 0.975102i \(-0.428821\pi\)
0.221755 + 0.975102i \(0.428821\pi\)
\(114\) 0 0
\(115\) −991787. + 870649.i −0.652116 + 0.572466i
\(116\) 0 0
\(117\) −1.88723e6 916390.i −1.17833 0.572167i
\(118\) 0 0
\(119\) 781664.i 0.463852i
\(120\) 0 0
\(121\) 1.17461e6 0.663038
\(122\) 0 0
\(123\) 1.56010e6 + 2.49185e6i 0.838375 + 1.33908i
\(124\) 0 0
\(125\) 1.08753e6 + 1.62234e6i 0.556813 + 0.830638i
\(126\) 0 0
\(127\) 215399.i 0.105155i −0.998617 0.0525777i \(-0.983256\pi\)
0.998617 0.0525777i \(-0.0167437\pi\)
\(128\) 0 0
\(129\) 955628. + 1.52636e6i 0.445164 + 0.711030i
\(130\) 0 0
\(131\) 1.27781e6i 0.568399i 0.958765 + 0.284199i \(0.0917278\pi\)
−0.958765 + 0.284199i \(0.908272\pi\)
\(132\) 0 0
\(133\) 3.65601e6i 1.55400i
\(134\) 0 0
\(135\) 1.81190e6 + 1.66447e6i 0.736434 + 0.676509i
\(136\) 0 0
\(137\) −5.09410e6 −1.98110 −0.990548 0.137166i \(-0.956200\pi\)
−0.990548 + 0.137166i \(0.956200\pi\)
\(138\) 0 0
\(139\) −1.88792e6 −0.702975 −0.351488 0.936193i \(-0.614324\pi\)
−0.351488 + 0.936193i \(0.614324\pi\)
\(140\) 0 0
\(141\) −92046.3 + 57628.6i −0.0328359 + 0.0205580i
\(142\) 0 0
\(143\) −2.22350e6 −0.760376
\(144\) 0 0
\(145\) 3.07278e6 + 3.50032e6i 1.00792 + 1.14816i
\(146\) 0 0
\(147\) 901481. 564402.i 0.283795 0.177679i
\(148\) 0 0
\(149\) 2.41329e6i 0.729542i −0.931097 0.364771i \(-0.881147\pi\)
0.931097 0.364771i \(-0.118853\pi\)
\(150\) 0 0
\(151\) 5.39450e6 1.56682 0.783412 0.621502i \(-0.213477\pi\)
0.783412 + 0.621502i \(0.213477\pi\)
\(152\) 0 0
\(153\) 889756. 1.83238e6i 0.248426 0.511612i
\(154\) 0 0
\(155\) 818305. 718356.i 0.219746 0.192906i
\(156\) 0 0
\(157\) 4.66333e6i 1.20503i −0.798108 0.602514i \(-0.794166\pi\)
0.798108 0.602514i \(-0.205834\pi\)
\(158\) 0 0
\(159\) 1.82089e6 1.14003e6i 0.452994 0.283612i
\(160\) 0 0
\(161\) 2.95348e6i 0.707712i
\(162\) 0 0
\(163\) 172365.i 0.0398003i −0.999802 0.0199002i \(-0.993665\pi\)
0.999802 0.0199002i \(-0.00633484\pi\)
\(164\) 0 0
\(165\) 2.49800e6 + 748079.i 0.556084 + 0.166531i
\(166\) 0 0
\(167\) −6.56485e6 −1.40953 −0.704767 0.709439i \(-0.748948\pi\)
−0.704767 + 0.709439i \(0.748948\pi\)
\(168\) 0 0
\(169\) −3.45521e6 −0.715837
\(170\) 0 0
\(171\) −4.16158e6 + 8.57042e6i −0.832280 + 1.71401i
\(172\) 0 0
\(173\) 4.35829e6 0.841740 0.420870 0.907121i \(-0.361725\pi\)
0.420870 + 0.907121i \(0.361725\pi\)
\(174\) 0 0
\(175\) −4.33418e6 566211.i −0.808709 0.105649i
\(176\) 0 0
\(177\) 2.16517e6 + 3.45827e6i 0.390455 + 0.623648i
\(178\) 0 0
\(179\) 8.77899e6i 1.53068i −0.643624 0.765341i \(-0.722570\pi\)
0.643624 0.765341i \(-0.277430\pi\)
\(180\) 0 0
\(181\) −7.19153e6 −1.21279 −0.606395 0.795164i \(-0.707385\pi\)
−0.606395 + 0.795164i \(0.707385\pi\)
\(182\) 0 0
\(183\) 4.76000e6 2.98016e6i 0.776701 0.486279i
\(184\) 0 0
\(185\) 293171. + 333962.i 0.0463027 + 0.0527451i
\(186\) 0 0
\(187\) 2.15887e6i 0.330143i
\(188\) 0 0
\(189\) −5.47452e6 + 589886.i −0.810887 + 0.0873741i
\(190\) 0 0
\(191\) 2.04101e6i 0.292917i −0.989217 0.146459i \(-0.953212\pi\)
0.989217 0.146459i \(-0.0467875\pi\)
\(192\) 0 0
\(193\) 4.28380e6i 0.595878i −0.954585 0.297939i \(-0.903701\pi\)
0.954585 0.297939i \(-0.0962992\pi\)
\(194\) 0 0
\(195\) 9.30447e6 + 2.78642e6i 1.25484 + 0.375788i
\(196\) 0 0
\(197\) 6.19176e6 0.809870 0.404935 0.914345i \(-0.367294\pi\)
0.404935 + 0.914345i \(0.367294\pi\)
\(198\) 0 0
\(199\) 1.95373e6 0.247917 0.123958 0.992287i \(-0.460441\pi\)
0.123958 + 0.992287i \(0.460441\pi\)
\(200\) 0 0
\(201\) −5.20722e6 8.31713e6i −0.641235 1.02420i
\(202\) 0 0
\(203\) −1.04237e7 −1.24605
\(204\) 0 0
\(205\) −8.97934e6 1.02287e7i −1.04228 1.18729i
\(206\) 0 0
\(207\) 3.36191e6 6.92356e6i 0.379031 0.780582i
\(208\) 0 0
\(209\) 1.00975e7i 1.10605i
\(210\) 0 0
\(211\) 1.15179e7 1.22610 0.613051 0.790043i \(-0.289942\pi\)
0.613051 + 0.790043i \(0.289942\pi\)
\(212\) 0 0
\(213\) 543799. + 868574.i 0.0562730 + 0.0898810i
\(214\) 0 0
\(215\) −5.50021e6 6.26549e6i −0.553432 0.630434i
\(216\) 0 0
\(217\) 2.43686e6i 0.238480i
\(218\) 0 0
\(219\) 6.61247e6 + 1.05616e7i 0.629552 + 1.00554i
\(220\) 0 0
\(221\) 8.04131e6i 0.744989i
\(222\) 0 0
\(223\) 1.19993e7i 1.08204i 0.841010 + 0.541019i \(0.181961\pi\)
−0.841010 + 0.541019i \(0.818039\pi\)
\(224\) 0 0
\(225\) −9.51568e6 6.26084e6i −0.835396 0.549649i
\(226\) 0 0
\(227\) 7.97219e6 0.681553 0.340777 0.940144i \(-0.389310\pi\)
0.340777 + 0.940144i \(0.389310\pi\)
\(228\) 0 0
\(229\) −192826. −0.0160568 −0.00802840 0.999968i \(-0.502556\pi\)
−0.00802840 + 0.999968i \(0.502556\pi\)
\(230\) 0 0
\(231\) −4.94626e6 + 3.09677e6i −0.401274 + 0.251231i
\(232\) 0 0
\(233\) 5.89892e6 0.466343 0.233171 0.972436i \(-0.425090\pi\)
0.233171 + 0.972436i \(0.425090\pi\)
\(234\) 0 0
\(235\) 377837. 331687.i 0.0291139 0.0255579i
\(236\) 0 0
\(237\) −5.20336e6 + 3.25774e6i −0.390876 + 0.244721i
\(238\) 0 0
\(239\) 1.92581e7i 1.41065i 0.708884 + 0.705325i \(0.249199\pi\)
−0.708884 + 0.705325i \(0.750801\pi\)
\(240\) 0 0
\(241\) 1.83867e7 1.31357 0.656785 0.754078i \(-0.271916\pi\)
0.656785 + 0.754078i \(0.271916\pi\)
\(242\) 0 0
\(243\) −1.35048e7 4.84875e6i −0.941176 0.337918i
\(244\) 0 0
\(245\) −3.70045e6 + 3.24847e6i −0.251627 + 0.220893i
\(246\) 0 0
\(247\) 3.76109e7i 2.49587i
\(248\) 0 0
\(249\) 2.28565e7 1.43100e7i 1.48051 0.926922i
\(250\) 0 0
\(251\) 1.31637e7i 0.832449i 0.909262 + 0.416224i \(0.136647\pi\)
−0.909262 + 0.416224i \(0.863353\pi\)
\(252\) 0 0
\(253\) 8.15721e6i 0.503709i
\(254\) 0 0
\(255\) −2.70544e6 + 9.03405e6i −0.163161 + 0.544831i
\(256\) 0 0
\(257\) 2.19050e7 1.29046 0.645230 0.763988i \(-0.276762\pi\)
0.645230 + 0.763988i \(0.276762\pi\)
\(258\) 0 0
\(259\) −994519. −0.0572419
\(260\) 0 0
\(261\) −2.44353e7 1.18652e7i −1.37435 0.667348i
\(262\) 0 0
\(263\) −1.12229e7 −0.616930 −0.308465 0.951236i \(-0.599815\pi\)
−0.308465 + 0.951236i \(0.599815\pi\)
\(264\) 0 0
\(265\) −7.47450e6 + 6.56155e6i −0.401647 + 0.352589i
\(266\) 0 0
\(267\) 1.48889e7 + 2.37810e7i 0.782219 + 1.24939i
\(268\) 0 0
\(269\) 1.28826e6i 0.0661829i 0.999452 + 0.0330914i \(0.0105353\pi\)
−0.999452 + 0.0330914i \(0.989465\pi\)
\(270\) 0 0
\(271\) 1.45523e7 0.731178 0.365589 0.930776i \(-0.380868\pi\)
0.365589 + 0.930776i \(0.380868\pi\)
\(272\) 0 0
\(273\) −1.84237e7 + 1.15348e7i −0.905500 + 0.566919i
\(274\) 0 0
\(275\) −1.19705e7 1.56381e6i −0.575593 0.0751947i
\(276\) 0 0
\(277\) 1.52269e7i 0.716426i −0.933640 0.358213i \(-0.883386\pi\)
0.933640 0.358213i \(-0.116614\pi\)
\(278\) 0 0
\(279\) −2.77385e6 + 5.71250e6i −0.127723 + 0.263035i
\(280\) 0 0
\(281\) 4.16226e7i 1.87590i 0.346770 + 0.937950i \(0.387279\pi\)
−0.346770 + 0.937950i \(0.612721\pi\)
\(282\) 0 0
\(283\) 3.10241e7i 1.36880i −0.729107 0.684400i \(-0.760064\pi\)
0.729107 0.684400i \(-0.239936\pi\)
\(284\) 0 0
\(285\) 1.26539e7 4.22541e7i 0.546626 1.82530i
\(286\) 0 0
\(287\) 3.04604e7 1.28852
\(288\) 0 0
\(289\) −1.63300e7 −0.676537
\(290\) 0 0
\(291\) −1.05239e7 1.68092e7i −0.427070 0.682130i
\(292\) 0 0
\(293\) 2.84645e7 1.13162 0.565810 0.824536i \(-0.308564\pi\)
0.565810 + 0.824536i \(0.308564\pi\)
\(294\) 0 0
\(295\) −1.24618e7 1.41957e7i −0.485418 0.552957i
\(296\) 0 0
\(297\) −1.51200e7 + 1.62920e6i −0.577143 + 0.0621879i
\(298\) 0 0
\(299\) 3.03837e7i 1.13665i
\(300\) 0 0
\(301\) 1.86583e7 0.684182
\(302\) 0 0
\(303\) −2.64511e6 4.22485e6i −0.0950858 0.151874i
\(304\) 0 0
\(305\) −1.95391e7 + 1.71526e7i −0.688661 + 0.604547i
\(306\) 0 0
\(307\) 1.58731e7i 0.548587i −0.961646 0.274293i \(-0.911556\pi\)
0.961646 0.274293i \(-0.0884440\pi\)
\(308\) 0 0
\(309\) −2.43556e7 3.89016e7i −0.825512 1.31853i
\(310\) 0 0
\(311\) 4.36168e7i 1.45002i −0.688741 0.725008i \(-0.741836\pi\)
0.688741 0.725008i \(-0.258164\pi\)
\(312\) 0 0
\(313\) 1.86898e7i 0.609498i 0.952433 + 0.304749i \(0.0985726\pi\)
−0.952433 + 0.304749i \(0.901427\pi\)
\(314\) 0 0
\(315\) 2.45790e7 6.76027e6i 0.786379 0.216288i
\(316\) 0 0
\(317\) 6.05789e7 1.90171 0.950853 0.309642i \(-0.100209\pi\)
0.950853 + 0.309642i \(0.100209\pi\)
\(318\) 0 0
\(319\) −2.87892e7 −0.886866
\(320\) 0 0
\(321\) −3.46132e7 + 2.16708e7i −1.04647 + 0.655178i
\(322\) 0 0
\(323\) −3.65178e7 −1.08367
\(324\) 0 0
\(325\) −4.45875e7 5.82485e6i −1.29886 0.169682i
\(326\) 0 0
\(327\) 2.48128e7 1.55349e7i 0.709630 0.444287i
\(328\) 0 0
\(329\) 1.12518e6i 0.0315960i
\(330\) 0 0
\(331\) 889460. 0.0245269 0.0122634 0.999925i \(-0.496096\pi\)
0.0122634 + 0.999925i \(0.496096\pi\)
\(332\) 0 0
\(333\) −2.33135e6 1.13205e6i −0.0631358 0.0306571i
\(334\) 0 0
\(335\) 2.99707e7 + 3.41406e7i 0.797190 + 0.908108i
\(336\) 0 0
\(337\) 6.44293e7i 1.68343i 0.539926 + 0.841713i \(0.318452\pi\)
−0.539926 + 0.841713i \(0.681548\pi\)
\(338\) 0 0
\(339\) 1.46449e7 9.16893e6i 0.375913 0.235353i
\(340\) 0 0
\(341\) 6.73036e6i 0.169736i
\(342\) 0 0
\(343\) 4.39314e7i 1.08866i
\(344\) 0 0
\(345\) −1.02224e7 + 3.41348e7i −0.248940 + 0.831265i
\(346\) 0 0
\(347\) −2.40907e7 −0.576582 −0.288291 0.957543i \(-0.593087\pi\)
−0.288291 + 0.957543i \(0.593087\pi\)
\(348\) 0 0
\(349\) −1.95136e7 −0.459051 −0.229525 0.973303i \(-0.573717\pi\)
−0.229525 + 0.973303i \(0.573717\pi\)
\(350\) 0 0
\(351\) −5.63187e7 + 6.06841e6i −1.30236 + 0.140331i
\(352\) 0 0
\(353\) 3.83518e7 0.871891 0.435945 0.899973i \(-0.356414\pi\)
0.435945 + 0.899973i \(0.356414\pi\)
\(354\) 0 0
\(355\) −3.12989e6 3.56537e6i −0.0699591 0.0796929i
\(356\) 0 0
\(357\) −1.11995e7 1.78882e7i −0.246147 0.393154i
\(358\) 0 0
\(359\) 1.82974e7i 0.395463i −0.980256 0.197732i \(-0.936643\pi\)
0.980256 0.197732i \(-0.0633574\pi\)
\(360\) 0 0
\(361\) 1.23755e8 2.63053
\(362\) 0 0
\(363\) 2.68808e7 1.68296e7i 0.561982 0.351847i
\(364\) 0 0
\(365\) −3.80587e7 4.33541e7i −0.782665 0.891561i
\(366\) 0 0
\(367\) 1.49800e6i 0.0303049i 0.999885 + 0.0151525i \(0.00482336\pi\)
−0.999885 + 0.0151525i \(0.995177\pi\)
\(368\) 0 0
\(369\) 7.14054e7 + 3.46726e7i 1.42119 + 0.690093i
\(370\) 0 0
\(371\) 2.22586e7i 0.435889i
\(372\) 0 0
\(373\) 4.34299e7i 0.836878i 0.908245 + 0.418439i \(0.137423\pi\)
−0.908245 + 0.418439i \(0.862577\pi\)
\(374\) 0 0
\(375\) 4.81323e7 + 2.15451e7i 0.912732 + 0.408559i
\(376\) 0 0
\(377\) −1.07233e8 −2.00127
\(378\) 0 0
\(379\) −2.39252e7 −0.439480 −0.219740 0.975559i \(-0.570521\pi\)
−0.219740 + 0.975559i \(0.570521\pi\)
\(380\) 0 0
\(381\) −3.08619e6 4.92936e6i −0.0558017 0.0891283i
\(382\) 0 0
\(383\) −9.46356e7 −1.68445 −0.842225 0.539126i \(-0.818754\pi\)
−0.842225 + 0.539126i \(0.818754\pi\)
\(384\) 0 0
\(385\) 2.03037e7 1.78238e7i 0.355789 0.312333i
\(386\) 0 0
\(387\) 4.37387e7 + 2.12384e7i 0.754629 + 0.366428i
\(388\) 0 0
\(389\) 6.73483e7i 1.14414i 0.820206 + 0.572068i \(0.193859\pi\)
−0.820206 + 0.572068i \(0.806141\pi\)
\(390\) 0 0
\(391\) 2.95007e7 0.493516
\(392\) 0 0
\(393\) 1.83082e7 + 2.92425e7i 0.301626 + 0.481767i
\(394\) 0 0
\(395\) 2.13591e7 1.87502e7i 0.346570 0.304239i
\(396\) 0 0
\(397\) 4.88724e7i 0.781073i −0.920587 0.390537i \(-0.872289\pi\)
0.920587 0.390537i \(-0.127711\pi\)
\(398\) 0 0
\(399\) 5.23825e7 + 8.36670e7i 0.824646 + 1.31715i
\(400\) 0 0
\(401\) 4.51763e7i 0.700612i 0.936635 + 0.350306i \(0.113922\pi\)
−0.936635 + 0.350306i \(0.886078\pi\)
\(402\) 0 0
\(403\) 2.50691e7i 0.383021i
\(404\) 0 0
\(405\) 6.53132e7 + 1.21304e7i 0.983186 + 0.182604i
\(406\) 0 0
\(407\) −2.74676e6 −0.0407415
\(408\) 0 0
\(409\) 6.79302e7 0.992871 0.496436 0.868074i \(-0.334642\pi\)
0.496436 + 0.868074i \(0.334642\pi\)
\(410\) 0 0
\(411\) −1.16577e8 + 7.29872e7i −1.67915 + 1.05129i
\(412\) 0 0
\(413\) 4.22740e7 0.600099
\(414\) 0 0
\(415\) −9.38225e7 + 8.23629e7i −1.31269 + 1.15236i
\(416\) 0 0
\(417\) −4.32048e7 + 2.70498e7i −0.595832 + 0.373040i
\(418\) 0 0
\(419\) 3.84851e7i 0.523179i −0.965179 0.261590i \(-0.915753\pi\)
0.965179 0.261590i \(-0.0842467\pi\)
\(420\) 0 0
\(421\) −9.01299e6 −0.120788 −0.0603939 0.998175i \(-0.519236\pi\)
−0.0603939 + 0.998175i \(0.519236\pi\)
\(422\) 0 0
\(423\) −1.28077e6 + 2.63764e6i −0.0169219 + 0.0348493i
\(424\) 0 0
\(425\) 5.65556e6 4.32916e7i 0.0736731 0.563946i
\(426\) 0 0
\(427\) 5.81863e7i 0.747373i
\(428\) 0 0
\(429\) −5.08843e7 + 3.18578e7i −0.644484 + 0.403500i
\(430\) 0 0
\(431\) 1.03193e8i 1.28889i −0.764650 0.644446i \(-0.777088\pi\)
0.764650 0.644446i \(-0.222912\pi\)
\(432\) 0 0
\(433\) 1.44510e8i 1.78006i 0.455900 + 0.890031i \(0.349318\pi\)
−0.455900 + 0.890031i \(0.650682\pi\)
\(434\) 0 0
\(435\) 1.20472e8 + 3.60779e7i 1.46358 + 0.438302i
\(436\) 0 0
\(437\) −1.37981e8 −1.65339
\(438\) 0 0
\(439\) 1.69066e7 0.199831 0.0999156 0.994996i \(-0.468143\pi\)
0.0999156 + 0.994996i \(0.468143\pi\)
\(440\) 0 0
\(441\) 1.25436e7 2.58325e7i 0.146253 0.301197i
\(442\) 0 0
\(443\) 2.78401e7 0.320228 0.160114 0.987099i \(-0.448814\pi\)
0.160114 + 0.987099i \(0.448814\pi\)
\(444\) 0 0
\(445\) −8.56945e7 9.76177e7i −0.972463 1.10777i
\(446\) 0 0
\(447\) −3.45771e7 5.52276e7i −0.387138 0.618349i
\(448\) 0 0
\(449\) 4.38092e7i 0.483979i −0.970279 0.241990i \(-0.922200\pi\)
0.970279 0.241990i \(-0.0778000\pi\)
\(450\) 0 0
\(451\) 8.41285e7 0.917093
\(452\) 0 0
\(453\) 1.23452e8 7.72913e7i 1.32802 0.831450i
\(454\) 0 0
\(455\) 7.56266e7 6.63895e7i 0.802861 0.704799i
\(456\) 0 0
\(457\) 1.14940e8i 1.20426i −0.798397 0.602131i \(-0.794318\pi\)
0.798397 0.602131i \(-0.205682\pi\)
\(458\) 0 0
\(459\) −5.89204e6 5.46819e7i −0.0609295 0.565465i
\(460\) 0 0
\(461\) 1.03087e7i 0.105221i −0.998615 0.0526104i \(-0.983246\pi\)
0.998615 0.0526104i \(-0.0167542\pi\)
\(462\) 0 0
\(463\) 1.09209e8i 1.10032i −0.835061 0.550158i \(-0.814568\pi\)
0.835061 0.550158i \(-0.185432\pi\)
\(464\) 0 0
\(465\) 8.43430e6 2.81640e7i 0.0838861 0.280114i
\(466\) 0 0
\(467\) −9.33692e7 −0.916755 −0.458377 0.888758i \(-0.651569\pi\)
−0.458377 + 0.888758i \(0.651569\pi\)
\(468\) 0 0
\(469\) −1.01669e8 −0.985528
\(470\) 0 0
\(471\) −6.68153e7 1.06719e8i −0.639459 1.02137i
\(472\) 0 0
\(473\) 5.15321e7 0.486962
\(474\) 0 0
\(475\) −2.64522e7 + 2.02484e8i −0.246821 + 1.88934i
\(476\) 0 0
\(477\) 2.53366e7 5.21787e7i 0.233450 0.480771i
\(478\) 0 0
\(479\) 1.21460e8i 1.10516i 0.833459 + 0.552581i \(0.186357\pi\)
−0.833459 + 0.552581i \(0.813643\pi\)
\(480\) 0 0
\(481\) −1.02310e7 −0.0919358
\(482\) 0 0
\(483\) −4.23169e7 6.75899e7i −0.375554 0.599847i
\(484\) 0 0
\(485\) 6.05715e7 + 6.89992e7i 0.530937 + 0.604810i
\(486\) 0 0
\(487\) 7.28588e7i 0.630805i −0.948958 0.315402i \(-0.897861\pi\)
0.948958 0.315402i \(-0.102139\pi\)
\(488\) 0 0
\(489\) −2.46961e6 3.94455e6i −0.0211204 0.0337342i
\(490\) 0 0
\(491\) 1.10666e7i 0.0934911i −0.998907 0.0467456i \(-0.985115\pi\)
0.998907 0.0467456i \(-0.0148850\pi\)
\(492\) 0 0
\(493\) 1.04117e8i 0.868920i
\(494\) 0 0
\(495\) 6.78845e7 1.86711e7i 0.559700 0.153941i
\(496\) 0 0
\(497\) 1.06175e7 0.0864872
\(498\) 0 0
\(499\) −1.65048e8 −1.32834 −0.664168 0.747584i \(-0.731214\pi\)
−0.664168 + 0.747584i \(0.731214\pi\)
\(500\) 0 0
\(501\) −1.50235e8 + 9.40598e7i −1.19470 + 0.747982i
\(502\) 0 0
\(503\) −5.48649e7 −0.431112 −0.215556 0.976491i \(-0.569156\pi\)
−0.215556 + 0.976491i \(0.569156\pi\)
\(504\) 0 0
\(505\) 1.52242e7 + 1.73424e7i 0.118212 + 0.134659i
\(506\) 0 0
\(507\) −7.90718e7 + 4.95055e7i −0.606733 + 0.379865i
\(508\) 0 0
\(509\) 2.47827e8i 1.87929i −0.342147 0.939647i \(-0.611154\pi\)
0.342147 0.939647i \(-0.388846\pi\)
\(510\) 0 0
\(511\) 1.29106e8 0.967571
\(512\) 0 0
\(513\) 2.75583e7 + 2.55759e8i 0.204127 + 1.89443i
\(514\) 0 0
\(515\) 1.40181e8 + 1.59685e8i 1.02629 + 1.16908i
\(516\) 0 0
\(517\) 3.10762e6i 0.0224883i
\(518\) 0 0
\(519\) 9.97386e7 6.24447e7i 0.713446 0.446677i
\(520\) 0 0
\(521\) 4.86823e6i 0.0344237i −0.999852 0.0172118i \(-0.994521\pi\)
0.999852 0.0172118i \(-0.00547897\pi\)
\(522\) 0 0
\(523\) 5.30392e7i 0.370759i 0.982667 + 0.185379i \(0.0593514\pi\)
−0.982667 + 0.185379i \(0.940649\pi\)
\(524\) 0 0
\(525\) −1.07299e8 + 4.91416e7i −0.741514 + 0.339603i
\(526\) 0 0
\(527\) −2.43405e7 −0.166302
\(528\) 0 0
\(529\) −3.65689e7 −0.247027
\(530\) 0 0
\(531\) 9.90988e7 + 4.81198e7i 0.661888 + 0.321396i
\(532\) 0 0
\(533\) 3.13359e8 2.06948
\(534\) 0 0
\(535\) 1.42082e8 1.24728e8i 0.927853 0.814523i
\(536\) 0 0
\(537\) −1.25783e8 2.00905e8i −0.812271 1.29738i
\(538\) 0 0
\(539\) 3.04353e7i 0.194362i
\(540\) 0 0
\(541\) −2.43394e7 −0.153716 −0.0768578 0.997042i \(-0.524489\pi\)
−0.0768578 + 0.997042i \(0.524489\pi\)
\(542\) 0 0
\(543\) −1.64577e8 + 1.03039e8i −1.02794 + 0.643578i
\(544\) 0 0
\(545\) −1.01853e8 + 8.94124e7i −0.629193 + 0.552342i
\(546\) 0 0
\(547\) 1.94157e8i 1.18629i 0.805094 + 0.593147i \(0.202115\pi\)
−0.805094 + 0.593147i \(0.797885\pi\)
\(548\) 0 0
\(549\) 6.62326e7 1.36401e8i 0.400272 0.824326i
\(550\) 0 0
\(551\) 4.86976e8i 2.91107i
\(552\) 0 0
\(553\) 6.36060e7i 0.376117i
\(554\) 0 0
\(555\) 1.14941e7 + 3.44216e6i 0.0672352 + 0.0201350i
\(556\) 0 0
\(557\) −3.28160e7 −0.189898 −0.0949490 0.995482i \(-0.530269\pi\)
−0.0949490 + 0.995482i \(0.530269\pi\)
\(558\) 0 0
\(559\) 1.91945e8 1.09886
\(560\) 0 0
\(561\) −3.09319e7 4.94054e7i −0.175193 0.279825i
\(562\) 0 0
\(563\) −2.44281e8 −1.36888 −0.684439 0.729070i \(-0.739953\pi\)
−0.684439 + 0.729070i \(0.739953\pi\)
\(564\) 0 0
\(565\) −6.01153e7 + 5.27727e7i −0.333303 + 0.292593i
\(566\) 0 0
\(567\) −1.16832e8 + 9.19372e7i −0.640930 + 0.504362i
\(568\) 0 0
\(569\) 2.53339e8i 1.37520i 0.726090 + 0.687600i \(0.241336\pi\)
−0.726090 + 0.687600i \(0.758664\pi\)
\(570\) 0 0
\(571\) 1.40185e8 0.752999 0.376499 0.926417i \(-0.377128\pi\)
0.376499 + 0.926417i \(0.377128\pi\)
\(572\) 0 0
\(573\) −2.92432e7 4.67081e7i −0.155439 0.248272i
\(574\) 0 0
\(575\) 2.13693e7 1.63576e8i 0.112405 0.860429i
\(576\) 0 0
\(577\) 5.11319e7i 0.266173i 0.991104 + 0.133087i \(0.0424888\pi\)
−0.991104 + 0.133087i \(0.957511\pi\)
\(578\) 0 0
\(579\) −6.13774e7 9.80339e7i −0.316208 0.505057i
\(580\) 0 0
\(581\) 2.79398e8i 1.42461i
\(582\) 0 0
\(583\) 6.14759e7i 0.310241i
\(584\) 0 0
\(585\) 2.52854e8 6.95458e7i 1.26300 0.347379i
\(586\) 0 0
\(587\) 3.47081e8 1.71599 0.857997 0.513654i \(-0.171709\pi\)
0.857997 + 0.513654i \(0.171709\pi\)
\(588\) 0 0
\(589\) 1.13845e8 0.557147
\(590\) 0 0
\(591\) 1.41697e8 8.87142e7i 0.686434 0.429765i
\(592\) 0 0
\(593\) 2.55958e7 0.122745 0.0613725 0.998115i \(-0.480452\pi\)
0.0613725 + 0.998115i \(0.480452\pi\)
\(594\) 0 0
\(595\) 6.44599e7 + 7.34286e7i 0.306012 + 0.348590i
\(596\) 0 0
\(597\) 4.47108e7 2.79927e7i 0.210131 0.131559i
\(598\) 0 0
\(599\) 2.40177e8i 1.11751i 0.829333 + 0.558755i \(0.188721\pi\)
−0.829333 + 0.558755i \(0.811279\pi\)
\(600\) 0 0
\(601\) 103123. 0.000475041 0.000237520 1.00000i \(-0.499924\pi\)
0.000237520 1.00000i \(0.499924\pi\)
\(602\) 0 0
\(603\) −2.38332e8 1.15728e8i −1.08700 0.527821i
\(604\) 0 0
\(605\) −1.10342e8 + 9.68645e7i −0.498281 + 0.437420i
\(606\) 0 0
\(607\) 1.50975e8i 0.675053i 0.941316 + 0.337526i \(0.109590\pi\)
−0.941316 + 0.337526i \(0.890410\pi\)
\(608\) 0 0
\(609\) −2.38545e8 + 1.49349e8i −1.05613 + 0.661227i
\(610\) 0 0
\(611\) 1.15752e7i 0.0507462i
\(612\) 0 0
\(613\) 4.49042e7i 0.194942i 0.995238 + 0.0974711i \(0.0310754\pi\)
−0.995238 + 0.0974711i \(0.968925\pi\)
\(614\) 0 0
\(615\) −3.52045e8 1.05427e8i −1.51347 0.453240i
\(616\) 0 0
\(617\) −4.74818e7 −0.202149 −0.101074 0.994879i \(-0.532228\pi\)
−0.101074 + 0.994879i \(0.532228\pi\)
\(618\) 0 0
\(619\) −5.46225e7 −0.230303 −0.115152 0.993348i \(-0.536735\pi\)
−0.115152 + 0.993348i \(0.536735\pi\)
\(620\) 0 0
\(621\) −2.22628e7 2.06613e8i −0.0929619 0.862746i
\(622\) 0 0
\(623\) 2.90699e8 1.20221
\(624\) 0 0
\(625\) −2.35947e8 6.27180e7i −0.966440 0.256893i
\(626\) 0 0
\(627\) 1.44675e8 + 2.31079e8i 0.586936 + 0.937473i
\(628\) 0 0
\(629\) 9.93369e6i 0.0399171i
\(630\) 0 0
\(631\) −4.24138e8 −1.68818 −0.844090 0.536201i \(-0.819859\pi\)
−0.844090 + 0.536201i \(0.819859\pi\)
\(632\) 0 0
\(633\) 2.63585e8 1.65026e8i 1.03923 0.650642i
\(634\) 0 0
\(635\) 1.77629e7 + 2.02343e7i 0.0693732 + 0.0790255i
\(636\) 0 0
\(637\) 1.13365e8i 0.438591i
\(638\) 0 0
\(639\) 2.48895e7 + 1.20857e7i 0.0953923 + 0.0463201i
\(640\) 0 0
\(641\) 3.86487e7i 0.146744i −0.997305 0.0733721i \(-0.976624\pi\)
0.997305 0.0733721i \(-0.0233761\pi\)
\(642\) 0 0
\(643\) 5.25945e7i 0.197837i −0.995096 0.0989185i \(-0.968462\pi\)
0.995096 0.0989185i \(-0.0315383\pi\)
\(644\) 0 0
\(645\) −2.15642e8 6.45786e7i −0.803626 0.240663i
\(646\) 0 0
\(647\) 3.78974e8 1.39925 0.699626 0.714509i \(-0.253350\pi\)
0.699626 + 0.714509i \(0.253350\pi\)
\(648\) 0 0
\(649\) 1.16756e8 0.427116
\(650\) 0 0
\(651\) 3.49149e7 + 5.57671e7i 0.126552 + 0.202132i
\(652\) 0 0
\(653\) −2.47799e8 −0.889940 −0.444970 0.895546i \(-0.646786\pi\)
−0.444970 + 0.895546i \(0.646786\pi\)
\(654\) 0 0
\(655\) −1.05375e8 1.20036e8i −0.374984 0.427158i
\(656\) 0 0
\(657\) 3.02650e8 + 1.46959e8i 1.06720 + 0.518204i
\(658\) 0 0
\(659\) 1.29762e8i 0.453410i −0.973963 0.226705i \(-0.927205\pi\)
0.973963 0.226705i \(-0.0727954\pi\)
\(660\) 0 0
\(661\) −1.47342e8 −0.510180 −0.255090 0.966917i \(-0.582105\pi\)
−0.255090 + 0.966917i \(0.582105\pi\)
\(662\) 0 0
\(663\) −1.15214e8 1.84024e8i −0.395335 0.631442i
\(664\) 0 0
\(665\) −3.01493e8 3.43441e8i −1.02521 1.16785i
\(666\) 0 0
\(667\) 3.93400e8i 1.32574i
\(668\) 0 0
\(669\) 1.71924e8 + 2.74603e8i 0.574194 + 0.917120i
\(670\) 0 0
\(671\) 1.60705e8i 0.531937i
\(672\) 0 0
\(673\) 2.34333e8i 0.768756i −0.923176 0.384378i \(-0.874416\pi\)
0.923176 0.384378i \(-0.125584\pi\)
\(674\) 0 0
\(675\) −3.07469e8 6.93945e6i −0.999745 0.0225639i
\(676\) 0 0
\(677\) 3.68160e8 1.18651 0.593254 0.805015i \(-0.297843\pi\)
0.593254 + 0.805015i \(0.297843\pi\)
\(678\) 0 0
\(679\) −2.05475e8 −0.656373
\(680\) 0 0
\(681\) 1.82442e8 1.14224e8i 0.577675 0.361672i
\(682\) 0 0
\(683\) −3.78221e8 −1.18709 −0.593545 0.804801i \(-0.702272\pi\)
−0.593545 + 0.804801i \(0.702272\pi\)
\(684\) 0 0
\(685\) 4.78534e8 4.20085e8i 1.48882 1.30697i
\(686\) 0 0
\(687\) −4.41278e6 + 2.76277e6i −0.0136095 + 0.00852068i
\(688\) 0 0
\(689\) 2.28984e8i 0.700079i
\(690\) 0 0
\(691\) −2.94037e6 −0.00891186 −0.00445593 0.999990i \(-0.501418\pi\)
−0.00445593 + 0.999990i \(0.501418\pi\)
\(692\) 0 0
\(693\) −6.88243e7 + 1.41738e8i −0.206796 + 0.425879i
\(694\) 0 0
\(695\) 1.77349e8 1.55688e8i 0.528294 0.463767i
\(696\) 0 0
\(697\) 3.04252e8i 0.898535i
\(698\) 0 0
\(699\) 1.34996e8 8.45186e7i 0.395265 0.247469i
\(700\) 0 0
\(701\) 8.90944e7i 0.258640i −0.991603 0.129320i \(-0.958720\pi\)
0.991603 0.129320i \(-0.0412795\pi\)
\(702\) 0 0
\(703\) 4.64619e7i 0.133731i
\(704\) 0 0
\(705\) 3.89438e6 1.30042e7i 0.0111140 0.0371121i
\(706\) 0 0
\(707\) −5.16447e7 −0.146139
\(708\) 0 0
\(709\) 5.18589e7 0.145507 0.0727536 0.997350i \(-0.476821\pi\)
0.0727536 + 0.997350i \(0.476821\pi\)
\(710\) 0 0
\(711\) −7.24018e7 + 1.49105e8i −0.201438 + 0.414844i
\(712\) 0 0
\(713\) −9.19693e7 −0.253732
\(714\) 0 0
\(715\) 2.08873e8 1.83361e8i 0.571431 0.501636i
\(716\) 0 0
\(717\) 2.75926e8 + 4.40717e8i 0.748574 + 1.19565i
\(718\) 0 0
\(719\) 2.07056e8i 0.557059i 0.960428 + 0.278530i \(0.0898471\pi\)
−0.960428 + 0.278530i \(0.910153\pi\)
\(720\) 0 0
\(721\) −4.75533e8 −1.26875
\(722\) 0 0
\(723\) 4.20776e8 2.63441e8i 1.11336 0.697057i
\(724\) 0 0
\(725\) −5.77307e8 7.54186e7i −1.51493 0.197909i
\(726\) 0 0
\(727\) 4.83279e8i 1.25775i 0.777506 + 0.628875i \(0.216484\pi\)
−0.777506 + 0.628875i \(0.783516\pi\)
\(728\) 0 0
\(729\) −3.78528e8 + 8.25318e7i −0.977046 + 0.213029i
\(730\) 0 0
\(731\) 1.86367e8i 0.477108i
\(732\) 0 0
\(733\) 6.32007e8i 1.60476i 0.596813 + 0.802380i \(0.296433\pi\)
−0.596813 + 0.802380i \(0.703567\pi\)
\(734\) 0 0
\(735\) −3.81407e7 + 1.27360e8i −0.0960565 + 0.320753i
\(736\) 0 0
\(737\) −2.80798e8 −0.701443
\(738\) 0 0
\(739\) 6.40814e7 0.158781 0.0793905 0.996844i \(-0.474703\pi\)
0.0793905 + 0.996844i \(0.474703\pi\)
\(740\) 0 0
\(741\) 5.38881e8 + 8.60718e8i 1.32446 + 2.11547i
\(742\) 0 0
\(743\) −1.46058e8 −0.356089 −0.178045 0.984022i \(-0.556977\pi\)
−0.178045 + 0.984022i \(0.556977\pi\)
\(744\) 0 0
\(745\) 1.99012e8 + 2.26701e8i 0.481294 + 0.548259i
\(746\) 0 0
\(747\) 3.18034e8 6.54965e8i 0.762979 1.57129i
\(748\) 0 0
\(749\) 4.23113e8i 1.00696i
\(750\) 0 0
\(751\) 7.06104e7 0.166705 0.0833526 0.996520i \(-0.473437\pi\)
0.0833526 + 0.996520i \(0.473437\pi\)
\(752\) 0 0
\(753\) 1.88607e8 + 3.01249e8i 0.441746 + 0.705571i
\(754\) 0 0
\(755\) −5.06753e8 + 4.44858e8i −1.17749 + 1.03367i
\(756\) 0 0
\(757\) 6.20517e8i 1.43043i 0.698905 + 0.715215i \(0.253671\pi\)
−0.698905 + 0.715215i \(0.746329\pi\)
\(758\) 0 0
\(759\) −1.16875e8 1.86676e8i −0.267298 0.426937i
\(760\) 0 0
\(761\) 2.01803e8i 0.457904i −0.973438 0.228952i \(-0.926470\pi\)
0.973438 0.228952i \(-0.0735298\pi\)
\(762\) 0 0
\(763\) 3.03312e8i 0.682835i
\(764\) 0 0
\(765\) 6.75245e7 + 2.45505e8i 0.150826 + 0.548374i
\(766\) 0 0
\(767\) 4.34891e8 0.963815
\(768\) 0 0
\(769\) 7.56998e8 1.66462 0.832311 0.554308i \(-0.187017\pi\)
0.832311 + 0.554308i \(0.187017\pi\)
\(770\) 0 0
\(771\) 5.01293e8 3.13851e8i 1.09378 0.684794i
\(772\) 0 0
\(773\) −5.74970e8 −1.24482 −0.622410 0.782691i \(-0.713846\pi\)
−0.622410 + 0.782691i \(0.713846\pi\)
\(774\) 0 0
\(775\) −1.76314e7 + 1.34963e8i −0.0378775 + 0.289941i
\(776\) 0 0
\(777\) −2.27594e7 + 1.42493e7i −0.0485174 + 0.0303759i
\(778\) 0 0
\(779\) 1.42305e9i 3.01029i
\(780\) 0 0
\(781\) 2.93243e7 0.0615566
\(782\) 0 0
\(783\) −7.29200e8 + 7.85721e7i −1.51901 + 0.163675i
\(784\) 0 0
\(785\) 3.84562e8 + 4.38068e8i 0.794982 + 0.905593i
\(786\) 0 0
\(787\) 3.48390e8i 0.714729i 0.933965 + 0.357365i \(0.116325\pi\)
−0.933965 + 0.357365i \(0.883675\pi\)
\(788\) 0 0
\(789\) −2.56833e8 + 1.60799e8i −0.522901 + 0.327380i
\(790\) 0 0
\(791\) 1.79020e8i 0.361719i
\(792\) 0 0
\(793\) 5.98588e8i 1.20035i
\(794\) 0 0
\(795\) −7.70399e7 + 2.57253e8i −0.153326 + 0.511987i
\(796\) 0 0
\(797\) −5.25438e8 −1.03788 −0.518939 0.854811i \(-0.673673\pi\)
−0.518939 + 0.854811i \(0.673673\pi\)
\(798\) 0 0
\(799\) −1.12387e7 −0.0220332
\(800\) 0 0
\(801\) 6.81459e8 + 3.30899e8i 1.32600 + 0.643869i
\(802\) 0 0
\(803\) 3.56577e8 0.688662
\(804\) 0 0
\(805\) 2.43559e8 + 2.77447e8i 0.466892 + 0.531854i
\(806\) 0 0
\(807\) 1.84579e7 + 2.94815e7i 0.0351205 + 0.0560956i
\(808\) 0 0
\(809\) 5.86771e8i 1.10821i −0.832446 0.554106i \(-0.813060\pi\)
0.832446 0.554106i \(-0.186940\pi\)
\(810\) 0 0
\(811\) −2.92475e8 −0.548309 −0.274155 0.961686i \(-0.588398\pi\)
−0.274155 + 0.961686i \(0.588398\pi\)
\(812\) 0 0
\(813\) 3.33026e8 2.08502e8i 0.619736 0.388006i
\(814\) 0 0
\(815\) 1.42141e7 + 1.61918e7i 0.0262571 + 0.0299104i
\(816\) 0 0
\(817\) 8.71676e8i 1.59841i
\(818\) 0 0
\(819\) −2.56355e8 + 5.27942e8i −0.466649 + 0.961024i
\(820\) 0 0
\(821\) 1.82491e8i 0.329770i 0.986313 + 0.164885i \(0.0527254\pi\)
−0.986313 + 0.164885i \(0.947275\pi\)
\(822\) 0 0
\(823\) 2.87227e8i 0.515259i −0.966244 0.257629i \(-0.917059\pi\)
0.966244 0.257629i \(-0.0829413\pi\)
\(824\) 0 0
\(825\) −2.96350e8 + 1.35724e8i −0.527767 + 0.241710i
\(826\) 0 0
\(827\) −4.52810e8 −0.800570 −0.400285 0.916391i \(-0.631089\pi\)
−0.400285 + 0.916391i \(0.631089\pi\)
\(828\) 0 0
\(829\) 1.05808e8 0.185719 0.0928594 0.995679i \(-0.470399\pi\)
0.0928594 + 0.995679i \(0.470399\pi\)
\(830\) 0 0
\(831\) −2.18167e8 3.48464e8i −0.380178 0.607232i
\(832\) 0 0
\(833\) 1.10070e8 0.190429
\(834\) 0 0
\(835\) 6.16695e8 5.41371e8i 1.05928 0.929898i
\(836\) 0 0
\(837\) 1.83686e7 + 1.70473e8i 0.0313257 + 0.290722i
\(838\) 0 0
\(839\) 1.09165e9i 1.84841i 0.381896 + 0.924205i \(0.375271\pi\)
−0.381896 + 0.924205i \(0.624729\pi\)
\(840\) 0 0
\(841\) −7.93605e8 −1.33419
\(842\) 0 0
\(843\) 5.96359e8 + 9.52524e8i 0.995463 + 1.58999i
\(844\) 0 0
\(845\) 3.24579e8 2.84934e8i 0.537960 0.472253i
\(846\) 0 0
\(847\) 3.28591e8i 0.540761i
\(848\) 0 0
\(849\) −4.44507e8 7.09981e8i −0.726366 1.16017i
\(850\) 0 0
\(851\) 3.75340e7i 0.0609027i
\(852\) 0 0
\(853\) 7.70719e8i 1.24179i 0.783893 + 0.620896i \(0.213231\pi\)
−0.783893 + 0.620896i \(0.786769\pi\)
\(854\) 0 0
\(855\) −3.15826e8 1.14828e9i −0.505301 1.83717i
\(856\) 0 0
\(857\) 1.95426e8 0.310485 0.155242 0.987876i \(-0.450384\pi\)
0.155242 + 0.987876i \(0.450384\pi\)
\(858\) 0 0
\(859\) −9.88095e8 −1.55890 −0.779451 0.626463i \(-0.784502\pi\)
−0.779451 + 0.626463i \(0.784502\pi\)
\(860\) 0 0
\(861\) 6.97081e8 4.36430e8i 1.09213 0.683763i
\(862\) 0 0
\(863\) 6.59608e8 1.02625 0.513125 0.858314i \(-0.328488\pi\)
0.513125 + 0.858314i \(0.328488\pi\)
\(864\) 0 0
\(865\) −4.09413e8 + 3.59407e8i −0.632577 + 0.555313i
\(866\) 0 0
\(867\) −3.73708e8 + 2.33972e8i −0.573423 + 0.359011i
\(868\) 0 0
\(869\) 1.75673e8i 0.267698i
\(870\) 0 0
\(871\) −1.04591e9 −1.58285
\(872\) 0 0
\(873\) −4.81676e8 2.33890e8i −0.723956 0.351535i
\(874\) 0 0
\(875\) 4.53840e8 3.04229e8i 0.677453 0.454126i
\(876\) 0 0
\(877\) 9.54769e8i 1.41547i −0.706480 0.707733i \(-0.749718\pi\)
0.706480 0.707733i \(-0.250282\pi\)
\(878\) 0 0
\(879\) 6.51404e8 4.07833e8i 0.959145 0.600504i
\(880\) 0 0
\(881\) 2.27849e7i 0.0333211i −0.999861 0.0166606i \(-0.994697\pi\)
0.999861 0.0166606i \(-0.00530346\pi\)
\(882\) 0 0
\(883\) 1.64348e8i 0.238717i −0.992851 0.119359i \(-0.961916\pi\)
0.992851 0.119359i \(-0.0380838\pi\)
\(884\) 0 0
\(885\) −4.88580e8 1.46316e8i −0.704865 0.211087i
\(886\) 0 0
\(887\) −1.76046e8 −0.252265 −0.126132 0.992013i \(-0.540256\pi\)
−0.126132 + 0.992013i \(0.540256\pi\)
\(888\) 0 0
\(889\) −6.02566e7 −0.0857628
\(890\) 0 0
\(891\) −3.22676e8 + 2.53921e8i −0.456178 + 0.358976i
\(892\) 0 0
\(893\) 5.25660e7 0.0738159
\(894\) 0 0
\(895\) 7.23960e8 + 8.24688e8i 1.00982 + 1.15033i
\(896\) 0 0
\(897\) −4.35332e8 6.95326e8i −0.603175 0.963410i
\(898\) 0 0
\(899\) 3.24587e8i 0.446738i
\(900\) 0 0
\(901\) 2.22329e8 0.303963
\(902\) 0 0
\(903\) 4.26990e8 2.67332e8i 0.579902 0.363067i
\(904\) 0 0
\(905\) 6.75564e8 5.93050e8i 0.911425 0.800102i
\(906\) 0 0
\(907\) 1.07506e8i 0.144083i −0.997402 0.0720413i \(-0.977049\pi\)
0.997402 0.0720413i \(-0.0229513\pi\)
\(908\) 0 0
\(909\) −1.21066e8 5.87864e7i −0.161187 0.0782681i
\(910\) 0 0
\(911\) 1.74267e8i 0.230494i 0.993337 + 0.115247i \(0.0367659\pi\)
−0.993337 + 0.115247i \(0.963234\pi\)
\(912\) 0 0
\(913\) 7.71667e8i 1.01395i
\(914\) 0 0
\(915\) −2.01390e8 + 6.72486e8i −0.262891 + 0.877850i
\(916\) 0 0
\(917\) 3.57461e8 0.463575
\(918\) 0 0
\(919\) 6.89452e8 0.888296 0.444148 0.895953i \(-0.353506\pi\)
0.444148 + 0.895953i \(0.353506\pi\)
\(920\) 0 0
\(921\) −2.27426e8 3.63252e8i −0.291113 0.464974i
\(922\) 0 0
\(923\) 1.09226e8 0.138906
\(924\) 0 0
\(925\) −5.50804e7 7.19562e6i −0.0695940 0.00909166i
\(926\) 0 0
\(927\) −1.11475e9 5.41293e8i −1.39938 0.679505i
\(928\) 0 0
\(929\) 3.86093e8i 0.481554i −0.970580 0.240777i \(-0.922598\pi\)
0.970580 0.240777i \(-0.0774022\pi\)
\(930\) 0 0
\(931\) −5.14820e8 −0.637978
\(932\) 0 0
\(933\) −6.24932e8 9.98162e8i −0.769464 1.22901i
\(934\) 0 0
\(935\) 1.78032e8 + 2.02802e8i 0.217802 + 0.248106i
\(936\) 0 0
\(937\) 2.56424e7i 0.0311702i 0.999879 + 0.0155851i \(0.00496109\pi\)
−0.999879 + 0.0155851i \(0.995039\pi\)
\(938\) 0 0
\(939\) 2.67784e8 + 4.27713e8i 0.323436 + 0.516602i
\(940\) 0 0
\(941\) 1.32023e9i 1.58446i 0.610224 + 0.792229i \(0.291080\pi\)
−0.610224 + 0.792229i \(0.708920\pi\)
\(942\) 0 0
\(943\) 1.14960e9i 1.37092i
\(944\) 0 0
\(945\) 4.65625e8 5.06870e8i 0.551748 0.600622i
\(946\) 0 0
\(947\) −9.59807e8 −1.13014 −0.565072 0.825041i \(-0.691152\pi\)
−0.565072 + 0.825041i \(0.691152\pi\)
\(948\) 0 0
\(949\) 1.32817e9 1.55401
\(950\) 0 0
\(951\) 1.38634e9 8.67962e8i 1.61186 1.00916i
\(952\) 0 0
\(953\) 9.86881e8 1.14021 0.570106 0.821571i \(-0.306902\pi\)
0.570106 + 0.821571i \(0.306902\pi\)
\(954\) 0 0
\(955\) 1.68312e8 + 1.91730e8i 0.193244 + 0.220131i
\(956\) 0 0
\(957\) −6.58836e8 + 4.12486e8i −0.751695 + 0.470623i
\(958\) 0 0
\(959\) 1.42504e9i 1.61574i
\(960\) 0 0
\(961\) −8.11621e8 −0.914499
\(962\) 0 0
\(963\) −4.81623e8 + 9.91862e8i −0.539298 + 1.11064i
\(964\) 0 0
\(965\) 3.53264e8 + 4.02415e8i 0.393113 + 0.447809i
\(966\) 0 0
\(967\) 8.20291e8i 0.907170i −0.891213 0.453585i \(-0.850145\pi\)
0.891213 0.453585i \(-0.149855\pi\)
\(968\) 0 0
\(969\) −8.35702e8 + 5.23219e8i −0.918502 + 0.575059i
\(970\) 0 0
\(971\) 1.18742e9i 1.29702i −0.761205 0.648512i \(-0.775392\pi\)
0.761205 0.648512i \(-0.224608\pi\)
\(972\) 0 0
\(973\) 5.28136e8i 0.573333i
\(974\) 0 0
\(975\) −1.10383e9 + 5.05540e8i −1.19094 + 0.545434i
\(976\) 0 0
\(977\) −7.04362e8 −0.755287 −0.377644 0.925951i \(-0.623266\pi\)
−0.377644 + 0.925951i \(0.623266\pi\)
\(978\) 0 0
\(979\) 8.02881e8 0.855664
\(980\) 0 0
\(981\) 3.45255e8 7.11024e8i 0.365707 0.753143i
\(982\) 0 0
\(983\) 1.07712e9 1.13398 0.566988 0.823726i \(-0.308108\pi\)
0.566988 + 0.823726i \(0.308108\pi\)
\(984\) 0 0
\(985\) −5.81647e8 + 5.10604e8i −0.608626 + 0.534288i
\(986\) 0 0
\(987\) 1.61213e7 + 2.57494e7i 0.0167667 + 0.0267803i
\(988\) 0 0
\(989\) 7.04179e8i 0.727937i
\(990\) 0 0
\(991\) 1.19667e9 1.22957 0.614786 0.788694i \(-0.289242\pi\)
0.614786 + 0.788694i \(0.289242\pi\)
\(992\) 0 0
\(993\) 2.03551e7 1.27440e7i 0.0207886 0.0130154i
\(994\) 0 0
\(995\) −1.83531e8 + 1.61115e8i −0.186312 + 0.163556i
\(996\) 0 0
\(997\) 1.14696e9i 1.15735i 0.815559 + 0.578674i \(0.196430\pi\)
−0.815559 + 0.578674i \(0.803570\pi\)
\(998\) 0 0
\(999\) −6.95723e7 + 7.49650e6i −0.0697814 + 0.00751904i
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 60.7.b.a.29.9 yes 12
3.2 odd 2 inner 60.7.b.a.29.3 12
4.3 odd 2 240.7.c.e.209.4 12
5.2 odd 4 300.7.g.i.101.3 12
5.3 odd 4 300.7.g.i.101.10 12
5.4 even 2 inner 60.7.b.a.29.4 yes 12
12.11 even 2 240.7.c.e.209.10 12
15.2 even 4 300.7.g.i.101.4 12
15.8 even 4 300.7.g.i.101.9 12
15.14 odd 2 inner 60.7.b.a.29.10 yes 12
20.19 odd 2 240.7.c.e.209.9 12
60.59 even 2 240.7.c.e.209.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.7.b.a.29.3 12 3.2 odd 2 inner
60.7.b.a.29.4 yes 12 5.4 even 2 inner
60.7.b.a.29.9 yes 12 1.1 even 1 trivial
60.7.b.a.29.10 yes 12 15.14 odd 2 inner
240.7.c.e.209.3 12 60.59 even 2
240.7.c.e.209.4 12 4.3 odd 2
240.7.c.e.209.9 12 20.19 odd 2
240.7.c.e.209.10 12 12.11 even 2
300.7.g.i.101.3 12 5.2 odd 4
300.7.g.i.101.4 12 15.2 even 4
300.7.g.i.101.9 12 15.8 even 4
300.7.g.i.101.10 12 5.3 odd 4