Properties

Label 300.7.b.e.149.10
Level $300$
Weight $7$
Character 300.149
Analytic conductor $69.016$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,7,Mod(149,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, 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("300.149");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 300.b (of order \(2\), degree \(1\), not 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: \(69.0162250860\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 406 x^{14} + 67561 x^{12} + 5921226 x^{10} + 291565644 x^{8} + 7924637994 x^{6} + 107459649193 x^{4} + 554332708870 x^{2} + \cdots + 276002078881 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{18}\cdot 5^{26} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.10
Root \(-6.94307i\) of defining polynomial
Character \(\chi\) \(=\) 300.149
Dual form 300.7.b.e.149.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(9.99060 + 25.0836i) q^{3} -134.460i q^{7} +(-529.376 + 501.201i) q^{9} +O(q^{10})\) \(q+(9.99060 + 25.0836i) q^{3} -134.460i q^{7} +(-529.376 + 501.201i) q^{9} -1155.16i q^{11} -78.2272i q^{13} -54.6849 q^{17} -1984.42 q^{19} +(3372.73 - 1343.33i) q^{21} +16917.8 q^{23} +(-17860.7 - 8271.35i) q^{27} +18486.2i q^{29} +38367.7 q^{31} +(28975.6 - 11540.8i) q^{33} -21242.8i q^{37} +(1962.22 - 781.537i) q^{39} +88980.3i q^{41} -65565.7i q^{43} +163822. q^{47} +99569.6 q^{49} +(-546.335 - 1371.69i) q^{51} -97179.1 q^{53} +(-19825.5 - 49776.4i) q^{57} -286230. i q^{59} +119504. q^{61} +(67391.2 + 71179.6i) q^{63} +142653. i q^{67} +(169019. + 424360. i) q^{69} +363327. i q^{71} +576517. i q^{73} -155322. q^{77} +627119. q^{79} +(29036.1 - 530647. i) q^{81} -991316. q^{83} +(-463701. + 184688. i) q^{87} +827592. i q^{89} -10518.4 q^{91} +(383316. + 962400. i) q^{93} -848738. i q^{97} +(578968. + 611514. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2984 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2984 q^{9} + 30544 q^{19} - 1736 q^{21} + 70064 q^{31} - 79216 q^{39} - 405120 q^{49} + 858240 q^{51} - 271216 q^{61} + 509880 q^{69} - 1415408 q^{79} - 2396224 q^{81} - 4008224 q^{91} - 5300160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 9.99060 + 25.0836i 0.370022 + 0.929023i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 134.460i 0.392010i −0.980603 0.196005i \(-0.937203\pi\)
0.980603 0.196005i \(-0.0627969\pi\)
\(8\) 0 0
\(9\) −529.376 + 501.201i −0.726167 + 0.687519i
\(10\) 0 0
\(11\) 1155.16i 0.867890i −0.900939 0.433945i \(-0.857121\pi\)
0.900939 0.433945i \(-0.142879\pi\)
\(12\) 0 0
\(13\) 78.2272i 0.0356064i −0.999842 0.0178032i \(-0.994333\pi\)
0.999842 0.0178032i \(-0.00566723\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −54.6849 −0.0111306 −0.00556532 0.999985i \(-0.501772\pi\)
−0.00556532 + 0.999985i \(0.501772\pi\)
\(18\) 0 0
\(19\) −1984.42 −0.289316 −0.144658 0.989482i \(-0.546208\pi\)
−0.144658 + 0.989482i \(0.546208\pi\)
\(20\) 0 0
\(21\) 3372.73 1343.33i 0.364186 0.145053i
\(22\) 0 0
\(23\) 16917.8 1.39047 0.695234 0.718784i \(-0.255301\pi\)
0.695234 + 0.718784i \(0.255301\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −17860.7 8271.35i −0.907418 0.420228i
\(28\) 0 0
\(29\) 18486.2i 0.757973i 0.925402 + 0.378986i \(0.123727\pi\)
−0.925402 + 0.378986i \(0.876273\pi\)
\(30\) 0 0
\(31\) 38367.7 1.28789 0.643947 0.765070i \(-0.277296\pi\)
0.643947 + 0.765070i \(0.277296\pi\)
\(32\) 0 0
\(33\) 28975.6 11540.8i 0.806290 0.321139i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 21242.8i 0.419378i −0.977768 0.209689i \(-0.932755\pi\)
0.977768 0.209689i \(-0.0672452\pi\)
\(38\) 0 0
\(39\) 1962.22 781.537i 0.0330791 0.0131751i
\(40\) 0 0
\(41\) 88980.3i 1.29105i 0.763740 + 0.645524i \(0.223361\pi\)
−0.763740 + 0.645524i \(0.776639\pi\)
\(42\) 0 0
\(43\) 65565.7i 0.824654i −0.911036 0.412327i \(-0.864716\pi\)
0.911036 0.412327i \(-0.135284\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 163822. 1.57790 0.788950 0.614457i \(-0.210625\pi\)
0.788950 + 0.614457i \(0.210625\pi\)
\(48\) 0 0
\(49\) 99569.6 0.846328
\(50\) 0 0
\(51\) −546.335 1371.69i −0.00411859 0.0103406i
\(52\) 0 0
\(53\) −97179.1 −0.652747 −0.326374 0.945241i \(-0.605827\pi\)
−0.326374 + 0.945241i \(0.605827\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −19825.5 49776.4i −0.107053 0.268781i
\(58\) 0 0
\(59\) 286230.i 1.39367i −0.717233 0.696834i \(-0.754592\pi\)
0.717233 0.696834i \(-0.245408\pi\)
\(60\) 0 0
\(61\) 119504. 0.526496 0.263248 0.964728i \(-0.415206\pi\)
0.263248 + 0.964728i \(0.415206\pi\)
\(62\) 0 0
\(63\) 67391.2 + 71179.6i 0.269514 + 0.284665i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 142653.i 0.474304i 0.971473 + 0.237152i \(0.0762139\pi\)
−0.971473 + 0.237152i \(0.923786\pi\)
\(68\) 0 0
\(69\) 169019. + 424360.i 0.514504 + 1.29178i
\(70\) 0 0
\(71\) 363327.i 1.01513i 0.861613 + 0.507566i \(0.169455\pi\)
−0.861613 + 0.507566i \(0.830545\pi\)
\(72\) 0 0
\(73\) 576517.i 1.48199i 0.671513 + 0.740993i \(0.265645\pi\)
−0.671513 + 0.740993i \(0.734355\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −155322. −0.340222
\(78\) 0 0
\(79\) 627119. 1.27195 0.635973 0.771711i \(-0.280599\pi\)
0.635973 + 0.771711i \(0.280599\pi\)
\(80\) 0 0
\(81\) 29036.1 530647.i 0.0546366 0.998506i
\(82\) 0 0
\(83\) −991316. −1.73372 −0.866858 0.498556i \(-0.833864\pi\)
−0.866858 + 0.498556i \(0.833864\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −463701. + 184688.i −0.704174 + 0.280467i
\(88\) 0 0
\(89\) 827592.i 1.17394i 0.809608 + 0.586970i \(0.199679\pi\)
−0.809608 + 0.586970i \(0.800321\pi\)
\(90\) 0 0
\(91\) −10518.4 −0.0139581
\(92\) 0 0
\(93\) 383316. + 962400.i 0.476550 + 1.19648i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 848738.i 0.929948i −0.885324 0.464974i \(-0.846064\pi\)
0.885324 0.464974i \(-0.153936\pi\)
\(98\) 0 0
\(99\) 578968. + 611514.i 0.596690 + 0.630233i
\(100\) 0 0
\(101\) 1.11358e6i 1.08083i −0.841398 0.540417i \(-0.818267\pi\)
0.841398 0.540417i \(-0.181733\pi\)
\(102\) 0 0
\(103\) 2.09668e6i 1.91876i 0.282118 + 0.959380i \(0.408963\pi\)
−0.282118 + 0.959380i \(0.591037\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 242367. 0.197844 0.0989218 0.995095i \(-0.468461\pi\)
0.0989218 + 0.995095i \(0.468461\pi\)
\(108\) 0 0
\(109\) 2.23474e6 1.72563 0.862816 0.505519i \(-0.168699\pi\)
0.862816 + 0.505519i \(0.168699\pi\)
\(110\) 0 0
\(111\) 532846. 212228.i 0.389612 0.155179i
\(112\) 0 0
\(113\) −245486. −0.170134 −0.0850672 0.996375i \(-0.527110\pi\)
−0.0850672 + 0.996375i \(0.527110\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 39207.5 + 41411.6i 0.0244800 + 0.0258562i
\(118\) 0 0
\(119\) 7352.90i 0.00436333i
\(120\) 0 0
\(121\) 437163. 0.246767
\(122\) 0 0
\(123\) −2.23195e6 + 888967.i −1.19941 + 0.477717i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 484140.i 0.236352i 0.992993 + 0.118176i \(0.0377047\pi\)
−0.992993 + 0.118176i \(0.962295\pi\)
\(128\) 0 0
\(129\) 1.64463e6 655041.i 0.766122 0.305140i
\(130\) 0 0
\(131\) 3.46823e6i 1.54274i 0.636385 + 0.771372i \(0.280429\pi\)
−0.636385 + 0.771372i \(0.719571\pi\)
\(132\) 0 0
\(133\) 266824.i 0.113415i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.74492e6 0.678600 0.339300 0.940678i \(-0.389810\pi\)
0.339300 + 0.940678i \(0.389810\pi\)
\(138\) 0 0
\(139\) 3.54561e6 1.32022 0.660111 0.751168i \(-0.270509\pi\)
0.660111 + 0.751168i \(0.270509\pi\)
\(140\) 0 0
\(141\) 1.63668e6 + 4.10926e6i 0.583859 + 1.46591i
\(142\) 0 0
\(143\) −90365.0 −0.0309024
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 994761. + 2.49757e6i 0.313160 + 0.786258i
\(148\) 0 0
\(149\) 4.13539e6i 1.25014i 0.780570 + 0.625069i \(0.214929\pi\)
−0.780570 + 0.625069i \(0.785071\pi\)
\(150\) 0 0
\(151\) 3.42874e6 0.995873 0.497937 0.867213i \(-0.334091\pi\)
0.497937 + 0.867213i \(0.334091\pi\)
\(152\) 0 0
\(153\) 28948.8 27408.1i 0.00808270 0.00765252i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.35519e6i 1.38381i −0.721990 0.691904i \(-0.756772\pi\)
0.721990 0.691904i \(-0.243228\pi\)
\(158\) 0 0
\(159\) −970878. 2.43760e6i −0.241531 0.606417i
\(160\) 0 0
\(161\) 2.27476e6i 0.545078i
\(162\) 0 0
\(163\) 590537.i 0.136359i 0.997673 + 0.0681796i \(0.0217191\pi\)
−0.997673 + 0.0681796i \(0.978281\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.06892e6 −0.873634 −0.436817 0.899550i \(-0.643894\pi\)
−0.436817 + 0.899550i \(0.643894\pi\)
\(168\) 0 0
\(169\) 4.82069e6 0.998732
\(170\) 0 0
\(171\) 1.05050e6 994593.i 0.210092 0.198910i
\(172\) 0 0
\(173\) 4.80964e6 0.928912 0.464456 0.885596i \(-0.346250\pi\)
0.464456 + 0.885596i \(0.346250\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.17968e6 2.85961e6i 1.29475 0.515688i
\(178\) 0 0
\(179\) 616529.i 0.107496i 0.998555 + 0.0537482i \(0.0171168\pi\)
−0.998555 + 0.0537482i \(0.982883\pi\)
\(180\) 0 0
\(181\) 169854. 0.0286444 0.0143222 0.999897i \(-0.495441\pi\)
0.0143222 + 0.999897i \(0.495441\pi\)
\(182\) 0 0
\(183\) 1.19392e6 + 2.99760e6i 0.194815 + 0.489126i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 63169.8i 0.00966017i
\(188\) 0 0
\(189\) −1.11216e6 + 2.40154e6i −0.164734 + 0.355717i
\(190\) 0 0
\(191\) 5.63748e6i 0.809068i −0.914523 0.404534i \(-0.867434\pi\)
0.914523 0.404534i \(-0.132566\pi\)
\(192\) 0 0
\(193\) 1.93534e6i 0.269207i −0.990900 0.134603i \(-0.957024\pi\)
0.990900 0.134603i \(-0.0429760\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.63667e6 −0.737265 −0.368632 0.929575i \(-0.620174\pi\)
−0.368632 + 0.929575i \(0.620174\pi\)
\(198\) 0 0
\(199\) −661790. −0.0839772 −0.0419886 0.999118i \(-0.513369\pi\)
−0.0419886 + 0.999118i \(0.513369\pi\)
\(200\) 0 0
\(201\) −3.57825e6 + 1.42519e6i −0.440639 + 0.175503i
\(202\) 0 0
\(203\) 2.48565e6 0.297133
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −8.95588e6 + 8.47923e6i −1.00971 + 0.955972i
\(208\) 0 0
\(209\) 2.29232e6i 0.251095i
\(210\) 0 0
\(211\) 726505. 0.0773376 0.0386688 0.999252i \(-0.487688\pi\)
0.0386688 + 0.999252i \(0.487688\pi\)
\(212\) 0 0
\(213\) −9.11355e6 + 3.62985e6i −0.943080 + 0.375621i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.15890e6i 0.504868i
\(218\) 0 0
\(219\) −1.44611e7 + 5.75976e6i −1.37680 + 0.548368i
\(220\) 0 0
\(221\) 4277.84i 0.000396322i
\(222\) 0 0
\(223\) 19464.1i 0.00175517i 1.00000 0.000877585i \(0.000279344\pi\)
−1.00000 0.000877585i \(0.999721\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.74686e6 −0.149342 −0.0746708 0.997208i \(-0.523791\pi\)
−0.0746708 + 0.997208i \(0.523791\pi\)
\(228\) 0 0
\(229\) −803494. −0.0669077 −0.0334539 0.999440i \(-0.510651\pi\)
−0.0334539 + 0.999440i \(0.510651\pi\)
\(230\) 0 0
\(231\) −1.55177e6 3.89605e6i −0.125890 0.316074i
\(232\) 0 0
\(233\) −1.41647e7 −1.11980 −0.559900 0.828561i \(-0.689160\pi\)
−0.559900 + 0.828561i \(0.689160\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.26530e6 + 1.57304e7i 0.470648 + 1.18167i
\(238\) 0 0
\(239\) 1.51609e7i 1.11054i −0.831672 0.555268i \(-0.812616\pi\)
0.831672 0.555268i \(-0.187384\pi\)
\(240\) 0 0
\(241\) −2.44786e7 −1.74878 −0.874391 0.485222i \(-0.838739\pi\)
−0.874391 + 0.485222i \(0.838739\pi\)
\(242\) 0 0
\(243\) 1.36006e7 4.57316e6i 0.947852 0.318711i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 155235.i 0.0103015i
\(248\) 0 0
\(249\) −9.90384e6 2.48658e7i −0.641513 1.61066i
\(250\) 0 0
\(251\) 5.38840e6i 0.340752i 0.985379 + 0.170376i \(0.0544982\pi\)
−0.985379 + 0.170376i \(0.945502\pi\)
\(252\) 0 0
\(253\) 1.95428e7i 1.20677i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.81725e7 1.65969 0.829843 0.557998i \(-0.188430\pi\)
0.829843 + 0.557998i \(0.188430\pi\)
\(258\) 0 0
\(259\) −2.85629e6 −0.164401
\(260\) 0 0
\(261\) −9.26530e6 9.78614e6i −0.521120 0.550415i
\(262\) 0 0
\(263\) −1.14892e7 −0.631570 −0.315785 0.948831i \(-0.602268\pi\)
−0.315785 + 0.948831i \(0.602268\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −2.07590e7 + 8.26814e6i −1.09062 + 0.434384i
\(268\) 0 0
\(269\) 1.46873e7i 0.754546i −0.926102 0.377273i \(-0.876862\pi\)
0.926102 0.377273i \(-0.123138\pi\)
\(270\) 0 0
\(271\) −1.77493e6 −0.0891812 −0.0445906 0.999005i \(-0.514198\pi\)
−0.0445906 + 0.999005i \(0.514198\pi\)
\(272\) 0 0
\(273\) −105085. 263839.i −0.00516479 0.0129674i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.74584e7i 1.29192i −0.763371 0.645960i \(-0.776457\pi\)
0.763371 0.645960i \(-0.223543\pi\)
\(278\) 0 0
\(279\) −2.03109e7 + 1.92299e7i −0.935226 + 0.885452i
\(280\) 0 0
\(281\) 2.73409e6i 0.123224i −0.998100 0.0616118i \(-0.980376\pi\)
0.998100 0.0616118i \(-0.0196241\pi\)
\(282\) 0 0
\(283\) 3.51523e7i 1.55094i −0.631386 0.775469i \(-0.717513\pi\)
0.631386 0.775469i \(-0.282487\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.19642e7 0.506104
\(288\) 0 0
\(289\) −2.41346e7 −0.999876
\(290\) 0 0
\(291\) 2.12894e7 8.47941e6i 0.863943 0.344102i
\(292\) 0 0
\(293\) 4.66113e7 1.85306 0.926528 0.376227i \(-0.122779\pi\)
0.926528 + 0.376227i \(0.122779\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −9.55475e6 + 2.06320e7i −0.364712 + 0.787539i
\(298\) 0 0
\(299\) 1.32343e6i 0.0495095i
\(300\) 0 0
\(301\) −8.81594e6 −0.323273
\(302\) 0 0
\(303\) 2.79327e7 1.11254e7i 1.00412 0.399932i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.11960e7i 1.07816i −0.842254 0.539081i \(-0.818772\pi\)
0.842254 0.539081i \(-0.181228\pi\)
\(308\) 0 0
\(309\) −5.25923e7 + 2.09471e7i −1.78257 + 0.709984i
\(310\) 0 0
\(311\) 3.51987e7i 1.17016i 0.810975 + 0.585081i \(0.198937\pi\)
−0.810975 + 0.585081i \(0.801063\pi\)
\(312\) 0 0
\(313\) 3.60591e7i 1.17593i −0.808886 0.587965i \(-0.799929\pi\)
0.808886 0.587965i \(-0.200071\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.74502e7 −1.80349 −0.901745 0.432269i \(-0.857713\pi\)
−0.901745 + 0.432269i \(0.857713\pi\)
\(318\) 0 0
\(319\) 2.13545e7 0.657837
\(320\) 0 0
\(321\) 2.42139e6 + 6.07944e6i 0.0732065 + 0.183801i
\(322\) 0 0
\(323\) 108518. 0.00322027
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.23264e7 + 5.60554e7i 0.638522 + 1.60315i
\(328\) 0 0
\(329\) 2.20275e7i 0.618553i
\(330\) 0 0
\(331\) 3.26327e7 0.899849 0.449924 0.893067i \(-0.351451\pi\)
0.449924 + 0.893067i \(0.351451\pi\)
\(332\) 0 0
\(333\) 1.06469e7 + 1.12454e7i 0.288330 + 0.304539i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4.34020e7i 1.13402i −0.823712 0.567009i \(-0.808100\pi\)
0.823712 0.567009i \(-0.191900\pi\)
\(338\) 0 0
\(339\) −2.45256e6 6.15769e6i −0.0629535 0.158059i
\(340\) 0 0
\(341\) 4.43209e7i 1.11775i
\(342\) 0 0
\(343\) 2.92071e7i 0.723779i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.50726e7 0.839421 0.419710 0.907658i \(-0.362132\pi\)
0.419710 + 0.907658i \(0.362132\pi\)
\(348\) 0 0
\(349\) −1.41882e7 −0.333772 −0.166886 0.985976i \(-0.553371\pi\)
−0.166886 + 0.985976i \(0.553371\pi\)
\(350\) 0 0
\(351\) −647045. + 1.39719e6i −0.0149628 + 0.0323099i
\(352\) 0 0
\(353\) 1.86131e7 0.423149 0.211575 0.977362i \(-0.432141\pi\)
0.211575 + 0.977362i \(0.432141\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −184437. + 73459.9i −0.00405363 + 0.00161453i
\(358\) 0 0
\(359\) 5.28748e6i 0.114279i −0.998366 0.0571394i \(-0.981802\pi\)
0.998366 0.0571394i \(-0.0181979\pi\)
\(360\) 0 0
\(361\) −4.31080e7 −0.916296
\(362\) 0 0
\(363\) 4.36752e6 + 1.09656e7i 0.0913094 + 0.229252i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.00133e6i 0.0202571i −0.999949 0.0101286i \(-0.996776\pi\)
0.999949 0.0101286i \(-0.00322408\pi\)
\(368\) 0 0
\(369\) −4.45970e7 4.71040e7i −0.887619 0.937516i
\(370\) 0 0
\(371\) 1.30666e7i 0.255884i
\(372\) 0 0
\(373\) 4.27333e7i 0.823455i 0.911307 + 0.411728i \(0.135074\pi\)
−0.911307 + 0.411728i \(0.864926\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.44612e6 0.0269887
\(378\) 0 0
\(379\) 1.05217e8 1.93272 0.966358 0.257202i \(-0.0828006\pi\)
0.966358 + 0.257202i \(0.0828006\pi\)
\(380\) 0 0
\(381\) −1.21440e7 + 4.83685e6i −0.219577 + 0.0874556i
\(382\) 0 0
\(383\) −6.60163e7 −1.17505 −0.587523 0.809208i \(-0.699897\pi\)
−0.587523 + 0.809208i \(0.699897\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.28616e7 + 3.47089e7i 0.566965 + 0.598836i
\(388\) 0 0
\(389\) 6.54648e7i 1.11214i 0.831136 + 0.556070i \(0.187691\pi\)
−0.831136 + 0.556070i \(0.812309\pi\)
\(390\) 0 0
\(391\) −925149. −0.0154768
\(392\) 0 0
\(393\) −8.69957e7 + 3.46497e7i −1.43324 + 0.570850i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.36739e7i 0.857811i −0.903349 0.428905i \(-0.858899\pi\)
0.903349 0.428905i \(-0.141101\pi\)
\(398\) 0 0
\(399\) −6.69291e6 + 2.66573e6i −0.105365 + 0.0419660i
\(400\) 0 0
\(401\) 8.82767e6i 0.136903i −0.997654 0.0684515i \(-0.978194\pi\)
0.997654 0.0684515i \(-0.0218058\pi\)
\(402\) 0 0
\(403\) 3.00139e6i 0.0458572i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.45388e7 −0.363974
\(408\) 0 0
\(409\) 1.11796e8 1.63402 0.817011 0.576622i \(-0.195629\pi\)
0.817011 + 0.576622i \(0.195629\pi\)
\(410\) 0 0
\(411\) 1.74328e7 + 4.37689e7i 0.251097 + 0.630435i
\(412\) 0 0
\(413\) −3.84863e7 −0.546332
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 3.54228e7 + 8.89368e7i 0.488512 + 1.22652i
\(418\) 0 0
\(419\) 2.09574e7i 0.284902i 0.989802 + 0.142451i \(0.0454983\pi\)
−0.989802 + 0.142451i \(0.954502\pi\)
\(420\) 0 0
\(421\) −1.87381e7 −0.251118 −0.125559 0.992086i \(-0.540073\pi\)
−0.125559 + 0.992086i \(0.540073\pi\)
\(422\) 0 0
\(423\) −8.67236e7 + 8.21080e7i −1.14582 + 1.08484i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.60685e7i 0.206392i
\(428\) 0 0
\(429\) −902801. 2.26668e6i −0.0114346 0.0287090i
\(430\) 0 0
\(431\) 7.51443e6i 0.0938565i −0.998898 0.0469283i \(-0.985057\pi\)
0.998898 0.0469283i \(-0.0149432\pi\)
\(432\) 0 0
\(433\) 6.88652e7i 0.848274i 0.905598 + 0.424137i \(0.139422\pi\)
−0.905598 + 0.424137i \(0.860578\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.35721e7 −0.402285
\(438\) 0 0
\(439\) 1.30096e8 1.53769 0.768846 0.639434i \(-0.220831\pi\)
0.768846 + 0.639434i \(0.220831\pi\)
\(440\) 0 0
\(441\) −5.27097e7 + 4.99044e7i −0.614575 + 0.581866i
\(442\) 0 0
\(443\) 1.19372e8 1.37306 0.686531 0.727100i \(-0.259132\pi\)
0.686531 + 0.727100i \(0.259132\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −1.03731e8 + 4.13151e7i −1.16141 + 0.462579i
\(448\) 0 0
\(449\) 5.58450e7i 0.616943i 0.951234 + 0.308471i \(0.0998174\pi\)
−0.951234 + 0.308471i \(0.900183\pi\)
\(450\) 0 0
\(451\) 1.02787e8 1.12049
\(452\) 0 0
\(453\) 3.42552e7 + 8.60053e7i 0.368495 + 0.925189i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.17246e8i 1.22843i −0.789140 0.614213i \(-0.789473\pi\)
0.789140 0.614213i \(-0.210527\pi\)
\(458\) 0 0
\(459\) 976711. + 452318.i 0.0101002 + 0.00467741i
\(460\) 0 0
\(461\) 1.77582e8i 1.81258i −0.422657 0.906290i \(-0.638903\pi\)
0.422657 0.906290i \(-0.361097\pi\)
\(462\) 0 0
\(463\) 1.25613e8i 1.26558i 0.774323 + 0.632791i \(0.218091\pi\)
−0.774323 + 0.632791i \(0.781909\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.66195e7 −0.261366 −0.130683 0.991424i \(-0.541717\pi\)
−0.130683 + 0.991424i \(0.541717\pi\)
\(468\) 0 0
\(469\) 1.91810e7 0.185932
\(470\) 0 0
\(471\) 1.34327e8 5.35016e7i 1.28559 0.512040i
\(472\) 0 0
\(473\) −7.57390e7 −0.715709
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 5.14442e7 4.87062e7i 0.474003 0.448776i
\(478\) 0 0
\(479\) 1.63522e8i 1.48789i 0.668242 + 0.743944i \(0.267047\pi\)
−0.668242 + 0.743944i \(0.732953\pi\)
\(480\) 0 0
\(481\) −1.66176e6 −0.0149325
\(482\) 0 0
\(483\) 5.70593e7 2.27262e7i 0.506390 0.201691i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.56192e6i 0.0568125i 0.999596 + 0.0284062i \(0.00904320\pi\)
−0.999596 + 0.0284062i \(0.990957\pi\)
\(488\) 0 0
\(489\) −1.48128e7 + 5.89982e6i −0.126681 + 0.0504559i
\(490\) 0 0
\(491\) 1.29952e8i 1.09784i 0.835874 + 0.548921i \(0.184961\pi\)
−0.835874 + 0.548921i \(0.815039\pi\)
\(492\) 0 0
\(493\) 1.01092e6i 0.00843673i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.88527e7 0.397942
\(498\) 0 0
\(499\) 1.09191e8 0.878788 0.439394 0.898295i \(-0.355193\pi\)
0.439394 + 0.898295i \(0.355193\pi\)
\(500\) 0 0
\(501\) −4.06509e7 1.02063e8i −0.323264 0.811626i
\(502\) 0 0
\(503\) −3.22774e7 −0.253626 −0.126813 0.991927i \(-0.540475\pi\)
−0.126813 + 0.991927i \(0.540475\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.81616e7 + 1.20920e8i 0.369553 + 0.927845i
\(508\) 0 0
\(509\) 1.11482e8i 0.845382i −0.906274 0.422691i \(-0.861086\pi\)
0.906274 0.422691i \(-0.138914\pi\)
\(510\) 0 0
\(511\) 7.75183e7 0.580953
\(512\) 0 0
\(513\) 3.54432e7 + 1.64138e7i 0.262531 + 0.121579i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1.89241e8i 1.36944i
\(518\) 0 0
\(519\) 4.80513e7 + 1.20643e8i 0.343718 + 0.862981i
\(520\) 0 0
\(521\) 2.70153e8i 1.91028i 0.296160 + 0.955138i \(0.404294\pi\)
−0.296160 + 0.955138i \(0.595706\pi\)
\(522\) 0 0
\(523\) 1.29085e8i 0.902341i −0.892438 0.451171i \(-0.851007\pi\)
0.892438 0.451171i \(-0.148993\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.09813e6 −0.0143351
\(528\) 0 0
\(529\) 1.38177e8 0.933401
\(530\) 0 0
\(531\) 1.43459e8 + 1.51523e8i 0.958172 + 1.01203i
\(532\) 0 0
\(533\) 6.96068e6 0.0459695
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.54648e7 + 6.15949e6i −0.0998667 + 0.0397761i
\(538\) 0 0
\(539\) 1.15019e8i 0.734519i
\(540\) 0 0
\(541\) −1.19668e8 −0.755765 −0.377883 0.925853i \(-0.623348\pi\)
−0.377883 + 0.925853i \(0.623348\pi\)
\(542\) 0 0
\(543\) 1.69694e6 + 4.26055e6i 0.0105991 + 0.0266113i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7.12552e7i 0.435366i 0.976020 + 0.217683i \(0.0698499\pi\)
−0.976020 + 0.217683i \(0.930150\pi\)
\(548\) 0 0
\(549\) −6.32628e7 + 5.98958e7i −0.382324 + 0.361975i
\(550\) 0 0
\(551\) 3.66844e7i 0.219294i
\(552\) 0 0
\(553\) 8.43221e7i 0.498616i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.53675e8 1.46795 0.733976 0.679176i \(-0.237663\pi\)
0.733976 + 0.679176i \(0.237663\pi\)
\(558\) 0 0
\(559\) −5.12902e6 −0.0293629
\(560\) 0 0
\(561\) −1.58453e6 + 631105.i −0.00897452 + 0.00357448i
\(562\) 0 0
\(563\) −4.15634e7 −0.232909 −0.116454 0.993196i \(-0.537153\pi\)
−0.116454 + 0.993196i \(0.537153\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7.13506e7 3.90418e6i −0.391425 0.0214181i
\(568\) 0 0
\(569\) 1.69480e8i 0.919986i −0.887923 0.459993i \(-0.847852\pi\)
0.887923 0.459993i \(-0.152148\pi\)
\(570\) 0 0
\(571\) 2.30661e8 1.23898 0.619492 0.785003i \(-0.287338\pi\)
0.619492 + 0.785003i \(0.287338\pi\)
\(572\) 0 0
\(573\) 1.41408e8 5.63218e7i 0.751642 0.299373i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 9.38558e7i 0.488578i −0.969703 0.244289i \(-0.921446\pi\)
0.969703 0.244289i \(-0.0785545\pi\)
\(578\) 0 0
\(579\) 4.85454e7 1.93352e7i 0.250099 0.0996125i
\(580\) 0 0
\(581\) 1.33292e8i 0.679634i
\(582\) 0 0
\(583\) 1.12258e8i 0.566513i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.28514e8 −1.12979 −0.564897 0.825162i \(-0.691084\pi\)
−0.564897 + 0.825162i \(0.691084\pi\)
\(588\) 0 0
\(589\) −7.61376e7 −0.372609
\(590\) 0 0
\(591\) −5.63137e7 1.41388e8i −0.272805 0.684936i
\(592\) 0 0
\(593\) −1.47213e8 −0.705966 −0.352983 0.935630i \(-0.614833\pi\)
−0.352983 + 0.935630i \(0.614833\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.61169e6 1.66001e7i −0.0310734 0.0780167i
\(598\) 0 0
\(599\) 1.04762e8i 0.487440i −0.969846 0.243720i \(-0.921632\pi\)
0.969846 0.243720i \(-0.0783678\pi\)
\(600\) 0 0
\(601\) −2.92674e8 −1.34822 −0.674110 0.738631i \(-0.735473\pi\)
−0.674110 + 0.738631i \(0.735473\pi\)
\(602\) 0 0
\(603\) −7.14978e7 7.55170e7i −0.326092 0.344424i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 2.94250e8i 1.31568i 0.753158 + 0.657840i \(0.228530\pi\)
−0.753158 + 0.657840i \(0.771470\pi\)
\(608\) 0 0
\(609\) 2.48331e7 + 6.23490e7i 0.109946 + 0.276043i
\(610\) 0 0
\(611\) 1.28154e7i 0.0561833i
\(612\) 0 0
\(613\) 1.10142e8i 0.478158i −0.971000 0.239079i \(-0.923155\pi\)
0.971000 0.239079i \(-0.0768454\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.60731e8 1.53578 0.767888 0.640584i \(-0.221308\pi\)
0.767888 + 0.640584i \(0.221308\pi\)
\(618\) 0 0
\(619\) −4.02744e8 −1.69808 −0.849038 0.528332i \(-0.822818\pi\)
−0.849038 + 0.528332i \(0.822818\pi\)
\(620\) 0 0
\(621\) −3.02164e8 1.39933e8i −1.26174 0.584314i
\(622\) 0 0
\(623\) 1.11278e8 0.460197
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −5.74998e7 + 2.29017e7i −0.233273 + 0.0929106i
\(628\) 0 0
\(629\) 1.16166e6i 0.00466795i
\(630\) 0 0
\(631\) −7.78200e7 −0.309744 −0.154872 0.987935i \(-0.549497\pi\)
−0.154872 + 0.987935i \(0.549497\pi\)
\(632\) 0 0
\(633\) 7.25822e6 + 1.82234e7i 0.0286167 + 0.0718484i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 7.78905e6i 0.0301347i
\(638\) 0 0
\(639\) −1.82100e8 1.92336e8i −0.697921 0.737155i
\(640\) 0 0
\(641\) 4.82128e8i 1.83058i −0.402797 0.915290i \(-0.631962\pi\)
0.402797 0.915290i \(-0.368038\pi\)
\(642\) 0 0
\(643\) 1.57608e7i 0.0592850i 0.999561 + 0.0296425i \(0.00943688\pi\)
−0.999561 + 0.0296425i \(0.990563\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.10892e8 −0.778660 −0.389330 0.921098i \(-0.627293\pi\)
−0.389330 + 0.921098i \(0.627293\pi\)
\(648\) 0 0
\(649\) −3.30642e8 −1.20955
\(650\) 0 0
\(651\) 1.29404e8 5.15405e7i 0.469034 0.186812i
\(652\) 0 0
\(653\) −2.03936e8 −0.732410 −0.366205 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −2.88951e8 3.05194e8i −1.01889 1.07617i
\(658\) 0 0
\(659\) 4.80405e8i 1.67862i 0.543656 + 0.839308i \(0.317040\pi\)
−0.543656 + 0.839308i \(0.682960\pi\)
\(660\) 0 0
\(661\) −1.13313e8 −0.392351 −0.196176 0.980569i \(-0.562852\pi\)
−0.196176 + 0.980569i \(0.562852\pi\)
\(662\) 0 0
\(663\) −107304. + 42738.2i −0.000368192 + 0.000146648i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.12746e8i 1.05394i
\(668\) 0 0
\(669\) −488229. + 194458.i −0.00163059 + 0.000649452i
\(670\) 0 0
\(671\) 1.38047e8i 0.456940i
\(672\) 0 0
\(673\) 6.78723e7i 0.222663i −0.993783 0.111331i \(-0.964489\pi\)
0.993783 0.111331i \(-0.0355115\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.22505e8 −1.03937 −0.519685 0.854358i \(-0.673951\pi\)
−0.519685 + 0.854358i \(0.673951\pi\)
\(678\) 0 0
\(679\) −1.14121e8 −0.364549
\(680\) 0 0
\(681\) −1.74522e7 4.38176e7i −0.0552598 0.138742i
\(682\) 0 0
\(683\) 4.17199e8 1.30943 0.654714 0.755877i \(-0.272789\pi\)
0.654714 + 0.755877i \(0.272789\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −8.02739e6 2.01545e7i −0.0247574 0.0621588i
\(688\) 0 0
\(689\) 7.60204e6i 0.0232420i
\(690\) 0 0
\(691\) −3.88809e7 −0.117842 −0.0589212 0.998263i \(-0.518766\pi\)
−0.0589212 + 0.998263i \(0.518766\pi\)
\(692\) 0 0
\(693\) 8.22239e7 7.78478e7i 0.247058 0.233909i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.86587e6i 0.0143702i
\(698\) 0 0
\(699\) −1.41514e8 3.55302e8i −0.414351 1.04032i
\(700\) 0 0
\(701\) 7.02379e7i 0.203900i 0.994790 + 0.101950i \(0.0325082\pi\)
−0.994790 + 0.101950i \(0.967492\pi\)
\(702\) 0 0
\(703\) 4.21546e7i 0.121333i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.49732e8 −0.423698
\(708\) 0 0
\(709\) −3.27456e8 −0.918785 −0.459393 0.888233i \(-0.651933\pi\)
−0.459393 + 0.888233i \(0.651933\pi\)
\(710\) 0 0
\(711\) −3.31981e8 + 3.14313e8i −0.923645 + 0.874486i
\(712\) 0 0
\(713\) 6.49097e8 1.79078
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.80291e8 1.51467e8i 1.03171 0.410923i
\(718\) 0 0
\(719\) 6.46847e8i 1.74026i −0.492819 0.870132i \(-0.664034\pi\)
0.492819 0.870132i \(-0.335966\pi\)
\(720\) 0 0
\(721\) 2.81919e8 0.752173
\(722\) 0 0
\(723\) −2.44556e8 6.14012e8i −0.647089 1.62466i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 4.36086e8i 1.13493i 0.823398 + 0.567465i \(0.192076\pi\)
−0.823398 + 0.567465i \(0.807924\pi\)
\(728\) 0 0
\(729\) 2.50590e8 + 2.95465e8i 0.646816 + 0.762646i
\(730\) 0 0
\(731\) 3.58545e6i 0.00917893i
\(732\) 0 0
\(733\) 4.27585e8i 1.08570i −0.839829 0.542851i \(-0.817345\pi\)
0.839829 0.542851i \(-0.182655\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.64787e8 0.411643
\(738\) 0 0
\(739\) −3.65022e8 −0.904452 −0.452226 0.891903i \(-0.649370\pi\)
−0.452226 + 0.891903i \(0.649370\pi\)
\(740\) 0 0
\(741\) −3.89387e6 + 1.55090e6i −0.00957032 + 0.00381178i
\(742\) 0 0
\(743\) −6.75024e8 −1.64571 −0.822854 0.568253i \(-0.807620\pi\)
−0.822854 + 0.568253i \(0.807620\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.24778e8 4.96848e8i 1.25897 1.19196i
\(748\) 0 0
\(749\) 3.25885e7i 0.0775567i
\(750\) 0 0
\(751\) −5.14564e8 −1.21484 −0.607421 0.794380i \(-0.707796\pi\)
−0.607421 + 0.794380i \(0.707796\pi\)
\(752\) 0 0
\(753\) −1.35161e8 + 5.38334e7i −0.316567 + 0.126086i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.32695e8i 0.536413i 0.963361 + 0.268207i \(0.0864310\pi\)
−0.963361 + 0.268207i \(0.913569\pi\)
\(758\) 0 0
\(759\) 4.90204e8 1.95245e8i 1.12112 0.446533i
\(760\) 0 0
\(761\) 3.18044e7i 0.0721661i −0.999349 0.0360831i \(-0.988512\pi\)
0.999349 0.0360831i \(-0.0114881\pi\)
\(762\) 0 0
\(763\) 3.00482e8i 0.676465i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.23910e7 −0.0496234
\(768\) 0 0
\(769\) −6.46145e8 −1.42086 −0.710430 0.703768i \(-0.751499\pi\)
−0.710430 + 0.703768i \(0.751499\pi\)
\(770\) 0 0
\(771\) 2.81460e8 + 7.06668e8i 0.614121 + 1.54189i
\(772\) 0 0
\(773\) 1.82196e8 0.394457 0.197228 0.980358i \(-0.436806\pi\)
0.197228 + 0.980358i \(0.436806\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −2.85361e7 7.16462e7i −0.0608319 0.152732i
\(778\) 0 0
\(779\) 1.76574e8i 0.373521i
\(780\) 0 0
\(781\) 4.19701e8 0.881022
\(782\) 0 0
\(783\) 1.52906e8 3.30177e8i 0.318522 0.687799i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.57411e8i 0.322932i 0.986878 + 0.161466i \(0.0516222\pi\)
−0.986878 + 0.161466i \(0.948378\pi\)
\(788\) 0 0
\(789\) −1.14784e8 2.88190e8i −0.233695 0.586743i
\(790\) 0 0
\(791\) 3.30080e7i 0.0666944i
\(792\) 0 0
\(793\) 9.34850e6i 0.0187466i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.11553e7 0.101045 0.0505225 0.998723i \(-0.483911\pi\)
0.0505225 + 0.998723i \(0.483911\pi\)
\(798\) 0 0
\(799\) −8.95860e6 −0.0175631
\(800\) 0 0
\(801\) −4.14790e8 4.38107e8i −0.807106 0.852477i
\(802\) 0 0
\(803\) 6.65971e8 1.28620
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.68411e8 1.46735e8i 0.700990 0.279199i
\(808\) 0 0
\(809\) 4.30177e8i 0.812459i 0.913771 + 0.406229i \(0.133157\pi\)
−0.913771 + 0.406229i \(0.866843\pi\)
\(810\) 0 0
\(811\) 1.35268e8 0.253590 0.126795 0.991929i \(-0.459531\pi\)
0.126795 + 0.991929i \(0.459531\pi\)
\(812\) 0 0
\(813\) −1.77326e7 4.45216e7i −0.0329990 0.0828513i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.30110e8i 0.238586i
\(818\) 0 0
\(819\) 5.56818e6 5.27183e6i 0.0101359 0.00959642i
\(820\) 0 0
\(821\) 6.01637e8i 1.08719i 0.839348 + 0.543595i \(0.182937\pi\)
−0.839348 + 0.543595i \(0.817063\pi\)
\(822\) 0 0
\(823\) 5.45245e8i 0.978119i 0.872250 + 0.489060i \(0.162660\pi\)
−0.872250 + 0.489060i \(0.837340\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.83751e8 0.501673 0.250836 0.968030i \(-0.419294\pi\)
0.250836 + 0.968030i \(0.419294\pi\)
\(828\) 0 0
\(829\) −9.17319e7 −0.161011 −0.0805057 0.996754i \(-0.525654\pi\)
−0.0805057 + 0.996754i \(0.525654\pi\)
\(830\) 0 0
\(831\) 6.88755e8 2.74326e8i 1.20022 0.478039i
\(832\) 0 0
\(833\) −5.44495e6 −0.00942018
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −6.85274e8 3.17353e8i −1.16866 0.541210i
\(838\) 0 0
\(839\) 7.01084e8i 1.18709i 0.804800 + 0.593545i \(0.202272\pi\)
−0.804800 + 0.593545i \(0.797728\pi\)
\(840\) 0 0
\(841\) 2.53084e8 0.425477
\(842\) 0 0
\(843\) 6.85808e7 2.73152e7i 0.114477 0.0455955i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.87807e7i 0.0967352i
\(848\) 0 0
\(849\) 8.81747e8 3.51193e8i 1.44086 0.573882i
\(850\) 0 0
\(851\) 3.59382e8i 0.583132i
\(852\) 0 0
\(853\) 1.36003e8i 0.219130i 0.993980 + 0.109565i \(0.0349457\pi\)
−0.993980 + 0.109565i \(0.965054\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.07337e9 −1.70532 −0.852661 0.522465i \(-0.825013\pi\)
−0.852661 + 0.522465i \(0.825013\pi\)
\(858\) 0 0
\(859\) 4.00149e8 0.631309 0.315654 0.948874i \(-0.397776\pi\)
0.315654 + 0.948874i \(0.397776\pi\)
\(860\) 0 0
\(861\) 1.19530e8 + 3.00107e8i 0.187270 + 0.470182i
\(862\) 0 0
\(863\) −4.98364e8 −0.775379 −0.387690 0.921790i \(-0.626727\pi\)
−0.387690 + 0.921790i \(0.626727\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −2.41119e8 6.05383e8i −0.369977 0.928908i
\(868\) 0 0
\(869\) 7.24424e8i 1.10391i
\(870\) 0 0
\(871\) 1.11593e7 0.0168882
\(872\) 0 0
\(873\) 4.25389e8 + 4.49301e8i 0.639356 + 0.675297i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.06775e9i 1.58296i 0.611194 + 0.791481i \(0.290689\pi\)
−0.611194 + 0.791481i \(0.709311\pi\)
\(878\) 0 0
\(879\) 4.65675e8 + 1.16918e9i 0.685672 + 1.72153i
\(880\) 0 0
\(881\) 7.83809e7i 0.114626i −0.998356 0.0573129i \(-0.981747\pi\)
0.998356 0.0573129i \(-0.0182533\pi\)
\(882\) 0 0
\(883\) 8.34186e8i 1.21166i 0.795594 + 0.605830i \(0.207159\pi\)
−0.795594 + 0.605830i \(0.792841\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.77272e8 0.970493 0.485247 0.874377i \(-0.338730\pi\)
0.485247 + 0.874377i \(0.338730\pi\)
\(888\) 0 0
\(889\) 6.50972e7 0.0926525
\(890\) 0 0
\(891\) −6.12983e8 3.35414e7i −0.866594 0.0474185i
\(892\) 0 0
\(893\) −3.25092e8 −0.456512
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.31965e7 1.32219e7i 0.0459954 0.0183196i
\(898\) 0 0
\(899\) 7.09273e8i 0.976189i
\(900\) 0 0
\(901\) 5.31422e6 0.00726550
\(902\) 0 0
\(903\) −8.80765e7 2.21136e8i −0.119618 0.300328i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.70732e8i 0.496865i −0.968649 0.248432i \(-0.920085\pi\)
0.968649 0.248432i \(-0.0799154\pi\)
\(908\) 0 0
\(909\) 5.58129e8 + 5.89504e8i 0.743093 + 0.784865i
\(910\) 0 0
\(911\) 5.03944e8i 0.666541i −0.942831 0.333271i \(-0.891848\pi\)
0.942831 0.333271i \(-0.108152\pi\)
\(912\) 0 0
\(913\) 1.14513e9i 1.50467i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.66336e8 0.604771
\(918\) 0 0
\(919\) 3.59392e7 0.0463043 0.0231522 0.999732i \(-0.492630\pi\)
0.0231522 + 0.999732i \(0.492630\pi\)
\(920\) 0 0
\(921\) 7.82509e8 3.11667e8i 1.00164 0.398944i
\(922\) 0 0
\(923\) 2.84220e7 0.0361451
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.05086e9 1.10993e9i −1.31918 1.39334i
\(928\) 0 0
\(929\) 4.60827e8i 0.574766i 0.957816 + 0.287383i \(0.0927853\pi\)
−0.957816 + 0.287383i \(0.907215\pi\)
\(930\) 0 0
\(931\) −1.97588e8 −0.244856
\(932\) 0 0
\(933\) −8.82911e8 + 3.51656e8i −1.08711 + 0.432986i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.28388e8i 0.642295i 0.947029 + 0.321147i \(0.104068\pi\)
−0.947029 + 0.321147i \(0.895932\pi\)
\(938\) 0 0
\(939\) 9.04492e8 3.60252e8i 1.09247 0.435120i
\(940\) 0 0
\(941\) 2.96965e8i 0.356399i 0.983994 + 0.178200i \(0.0570273\pi\)
−0.983994 + 0.178200i \(0.942973\pi\)
\(942\) 0 0
\(943\) 1.50535e9i 1.79516i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.43742e8 0.640240 0.320120 0.947377i \(-0.396277\pi\)
0.320120 + 0.947377i \(0.396277\pi\)
\(948\) 0 0
\(949\) 4.50993e7 0.0527681
\(950\) 0 0
\(951\) −5.73962e8 1.44106e9i −0.667332 1.67548i
\(952\) 0 0
\(953\) −1.43357e9 −1.65630 −0.828150 0.560506i \(-0.810607\pi\)
−0.828150 + 0.560506i \(0.810607\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.13345e8 + 5.35649e8i 0.243414 + 0.611146i
\(958\) 0 0
\(959\) 2.34621e8i 0.266018i
\(960\) 0 0
\(961\) 5.84575e8 0.658673
\(962\) 0 0
\(963\) −1.28303e8 + 1.21475e8i −0.143667 + 0.136021i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.86723e8i 0.206500i −0.994655 0.103250i \(-0.967076\pi\)
0.994655 0.103250i \(-0.0329241\pi\)
\(968\) 0 0
\(969\) 1.08416e6 + 2.72202e6i 0.00119157 + 0.00299171i
\(970\) 0 0
\(971\) 7.68364e8i 0.839284i 0.907690 + 0.419642i \(0.137844\pi\)
−0.907690 + 0.419642i \(0.862156\pi\)
\(972\) 0 0
\(973\) 4.76741e8i 0.517540i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.60128e7 0.0922316 0.0461158 0.998936i \(-0.485316\pi\)
0.0461158 + 0.998936i \(0.485316\pi\)
\(978\) 0 0
\(979\) 9.56002e8 1.01885
\(980\) 0 0
\(981\) −1.18302e9 + 1.12006e9i −1.25310 + 1.18640i
\(982\) 0 0
\(983\) −9.61067e8 −1.01180 −0.505898 0.862593i \(-0.668839\pi\)
−0.505898 + 0.862593i \(0.668839\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 5.52529e8 2.20068e8i 0.574650 0.228879i
\(988\) 0 0
\(989\) 1.10923e9i 1.14665i
\(990\) 0 0
\(991\) −6.24596e8 −0.641768 −0.320884 0.947118i \(-0.603980\pi\)
−0.320884 + 0.947118i \(0.603980\pi\)
\(992\) 0 0
\(993\) 3.26021e8 + 8.18547e8i 0.332964 + 0.835980i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.53088e9i 1.54474i 0.635171 + 0.772371i \(0.280929\pi\)
−0.635171 + 0.772371i \(0.719071\pi\)
\(998\) 0 0
\(999\) −1.75707e8 + 3.79411e8i −0.176235 + 0.380552i
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 300.7.b.e.149.10 16
3.2 odd 2 inner 300.7.b.e.149.8 16
5.2 odd 4 60.7.g.a.41.7 8
5.3 odd 4 300.7.g.h.101.2 8
5.4 even 2 inner 300.7.b.e.149.7 16
15.2 even 4 60.7.g.a.41.8 yes 8
15.8 even 4 300.7.g.h.101.1 8
15.14 odd 2 inner 300.7.b.e.149.9 16
20.7 even 4 240.7.l.c.161.2 8
60.47 odd 4 240.7.l.c.161.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.7.g.a.41.7 8 5.2 odd 4
60.7.g.a.41.8 yes 8 15.2 even 4
240.7.l.c.161.1 8 60.47 odd 4
240.7.l.c.161.2 8 20.7 even 4
300.7.b.e.149.7 16 5.4 even 2 inner
300.7.b.e.149.8 16 3.2 odd 2 inner
300.7.b.e.149.9 16 15.14 odd 2 inner
300.7.b.e.149.10 16 1.1 even 1 trivial
300.7.g.h.101.1 8 15.8 even 4
300.7.g.h.101.2 8 5.3 odd 4