Properties

Label 780.2.bm.a.697.6
Level $780$
Weight $2$
Character 780.697
Analytic conductor $6.228$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [780,2,Mod(697,780)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(780, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("780.697");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 780.bm (of order \(4\), 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: \(6.22833135766\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 697.6
Character \(\chi\) \(=\) 780.697
Dual form 780.2.bm.a.733.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{3} +(1.53289 + 1.62796i) q^{5} +2.13466 q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{3} +(1.53289 + 1.62796i) q^{5} +2.13466 q^{7} -1.00000i q^{9} +(1.89894 - 1.89894i) q^{11} +(3.33364 - 1.37361i) q^{13} +(-2.23506 - 0.0672190i) q^{15} +(1.22748 - 1.22748i) q^{17} +(-0.581270 + 0.581270i) q^{19} +(-1.50943 + 1.50943i) q^{21} +(-4.10913 - 4.10913i) q^{23} +(-0.300476 + 4.99096i) q^{25} +(0.707107 + 0.707107i) q^{27} +1.81712i q^{29} +(4.74497 + 4.74497i) q^{31} +2.68551i q^{33} +(3.27221 + 3.47513i) q^{35} +6.00961 q^{37} +(-1.38595 + 3.32853i) q^{39} +(3.85523 + 3.85523i) q^{41} +(-3.78653 - 3.78653i) q^{43} +(1.62796 - 1.53289i) q^{45} -2.75112 q^{47} -2.44322 q^{49} +1.73592i q^{51} +(-7.79380 + 7.79380i) q^{53} +(6.00227 + 0.180517i) q^{55} -0.822039i q^{57} +(-0.434208 - 0.434208i) q^{59} +13.1423 q^{61} -2.13466i q^{63} +(7.34630 + 3.32142i) q^{65} +1.27294i q^{67} +5.81118 q^{69} +(-1.66190 - 1.66190i) q^{71} +0.637019i q^{73} +(-3.31667 - 3.74161i) q^{75} +(4.05360 - 4.05360i) q^{77} +0.837401i q^{79} -1.00000 q^{81} -5.22354 q^{83} +(3.87988 + 0.116687i) q^{85} +(-1.28490 - 1.28490i) q^{87} +(3.92688 + 3.92688i) q^{89} +(7.11620 - 2.93220i) q^{91} -6.71041 q^{93} +(-1.83731 - 0.0552566i) q^{95} +6.15970i q^{97} +(-1.89894 - 1.89894i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 28 q - 4 q^{5} + 8 q^{11} + 4 q^{15} + 4 q^{17} + 16 q^{19} + 8 q^{21} + 8 q^{23} - 12 q^{25} + 24 q^{37} - 8 q^{39} + 12 q^{41} - 16 q^{43} - 24 q^{47} + 36 q^{49} + 36 q^{53} + 40 q^{55} - 16 q^{59} + 8 q^{61} + 24 q^{65} + 8 q^{69} + 8 q^{71} + 48 q^{77} - 28 q^{81} - 24 q^{83} + 68 q^{85} - 24 q^{87} + 36 q^{89} - 24 q^{91} - 72 q^{95} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/780\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(301\) \(391\) \(521\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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
\(3\) −0.707107 + 0.707107i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 1.53289 + 1.62796i 0.685531 + 0.728044i
\(6\) 0 0
\(7\) 2.13466 0.806826 0.403413 0.915018i \(-0.367824\pi\)
0.403413 + 0.915018i \(0.367824\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 1.89894 1.89894i 0.572553 0.572553i −0.360288 0.932841i \(-0.617322\pi\)
0.932841 + 0.360288i \(0.117322\pi\)
\(12\) 0 0
\(13\) 3.33364 1.37361i 0.924587 0.380972i
\(14\) 0 0
\(15\) −2.23506 0.0672190i −0.577089 0.0173559i
\(16\) 0 0
\(17\) 1.22748 1.22748i 0.297708 0.297708i −0.542408 0.840115i \(-0.682487\pi\)
0.840115 + 0.542408i \(0.182487\pi\)
\(18\) 0 0
\(19\) −0.581270 + 0.581270i −0.133352 + 0.133352i −0.770632 0.637280i \(-0.780059\pi\)
0.637280 + 0.770632i \(0.280059\pi\)
\(20\) 0 0
\(21\) −1.50943 + 1.50943i −0.329385 + 0.329385i
\(22\) 0 0
\(23\) −4.10913 4.10913i −0.856812 0.856812i 0.134149 0.990961i \(-0.457170\pi\)
−0.990961 + 0.134149i \(0.957170\pi\)
\(24\) 0 0
\(25\) −0.300476 + 4.99096i −0.0600953 + 0.998193i
\(26\) 0 0
\(27\) 0.707107 + 0.707107i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 1.81712i 0.337431i 0.985665 + 0.168715i \(0.0539619\pi\)
−0.985665 + 0.168715i \(0.946038\pi\)
\(30\) 0 0
\(31\) 4.74497 + 4.74497i 0.852223 + 0.852223i 0.990407 0.138184i \(-0.0441265\pi\)
−0.138184 + 0.990407i \(0.544127\pi\)
\(32\) 0 0
\(33\) 2.68551i 0.467487i
\(34\) 0 0
\(35\) 3.27221 + 3.47513i 0.553104 + 0.587405i
\(36\) 0 0
\(37\) 6.00961 0.987974 0.493987 0.869469i \(-0.335539\pi\)
0.493987 + 0.869469i \(0.335539\pi\)
\(38\) 0 0
\(39\) −1.38595 + 3.32853i −0.221930 + 0.532992i
\(40\) 0 0
\(41\) 3.85523 + 3.85523i 0.602086 + 0.602086i 0.940866 0.338780i \(-0.110014\pi\)
−0.338780 + 0.940866i \(0.610014\pi\)
\(42\) 0 0
\(43\) −3.78653 3.78653i −0.577440 0.577440i 0.356757 0.934197i \(-0.383882\pi\)
−0.934197 + 0.356757i \(0.883882\pi\)
\(44\) 0 0
\(45\) 1.62796 1.53289i 0.242681 0.228510i
\(46\) 0 0
\(47\) −2.75112 −0.401292 −0.200646 0.979664i \(-0.564304\pi\)
−0.200646 + 0.979664i \(0.564304\pi\)
\(48\) 0 0
\(49\) −2.44322 −0.349032
\(50\) 0 0
\(51\) 1.73592i 0.243077i
\(52\) 0 0
\(53\) −7.79380 + 7.79380i −1.07056 + 1.07056i −0.0732468 + 0.997314i \(0.523336\pi\)
−0.997314 + 0.0732468i \(0.976664\pi\)
\(54\) 0 0
\(55\) 6.00227 + 0.180517i 0.809346 + 0.0243409i
\(56\) 0 0
\(57\) 0.822039i 0.108882i
\(58\) 0 0
\(59\) −0.434208 0.434208i −0.0565290 0.0565290i 0.678277 0.734806i \(-0.262727\pi\)
−0.734806 + 0.678277i \(0.762727\pi\)
\(60\) 0 0
\(61\) 13.1423 1.68270 0.841352 0.540488i \(-0.181760\pi\)
0.841352 + 0.540488i \(0.181760\pi\)
\(62\) 0 0
\(63\) 2.13466i 0.268942i
\(64\) 0 0
\(65\) 7.34630 + 3.32142i 0.911197 + 0.411972i
\(66\) 0 0
\(67\) 1.27294i 0.155515i 0.996972 + 0.0777573i \(0.0247759\pi\)
−0.996972 + 0.0777573i \(0.975224\pi\)
\(68\) 0 0
\(69\) 5.81118 0.699584
\(70\) 0 0
\(71\) −1.66190 1.66190i −0.197231 0.197231i 0.601581 0.798812i \(-0.294538\pi\)
−0.798812 + 0.601581i \(0.794538\pi\)
\(72\) 0 0
\(73\) 0.637019i 0.0745575i 0.999305 + 0.0372787i \(0.0118689\pi\)
−0.999305 + 0.0372787i \(0.988131\pi\)
\(74\) 0 0
\(75\) −3.31667 3.74161i −0.382977 0.432044i
\(76\) 0 0
\(77\) 4.05360 4.05360i 0.461950 0.461950i
\(78\) 0 0
\(79\) 0.837401i 0.0942149i 0.998890 + 0.0471075i \(0.0150003\pi\)
−0.998890 + 0.0471075i \(0.985000\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) −5.22354 −0.573358 −0.286679 0.958027i \(-0.592551\pi\)
−0.286679 + 0.958027i \(0.592551\pi\)
\(84\) 0 0
\(85\) 3.87988 + 0.116687i 0.420832 + 0.0126564i
\(86\) 0 0
\(87\) −1.28490 1.28490i −0.137756 0.137756i
\(88\) 0 0
\(89\) 3.92688 + 3.92688i 0.416248 + 0.416248i 0.883908 0.467660i \(-0.154903\pi\)
−0.467660 + 0.883908i \(0.654903\pi\)
\(90\) 0 0
\(91\) 7.11620 2.93220i 0.745981 0.307378i
\(92\) 0 0
\(93\) −6.71041 −0.695837
\(94\) 0 0
\(95\) −1.83731 0.0552566i −0.188504 0.00566921i
\(96\) 0 0
\(97\) 6.15970i 0.625422i 0.949848 + 0.312711i \(0.101237\pi\)
−0.949848 + 0.312711i \(0.898763\pi\)
\(98\) 0 0
\(99\) −1.89894 1.89894i −0.190851 0.190851i
\(100\) 0 0
\(101\) 3.93730i 0.391776i 0.980626 + 0.195888i \(0.0627589\pi\)
−0.980626 + 0.195888i \(0.937241\pi\)
\(102\) 0 0
\(103\) −3.97517 3.97517i −0.391685 0.391685i 0.483603 0.875288i \(-0.339328\pi\)
−0.875288 + 0.483603i \(0.839328\pi\)
\(104\) 0 0
\(105\) −4.77109 0.143490i −0.465611 0.0140032i
\(106\) 0 0
\(107\) −11.2103 11.2103i −1.08375 1.08375i −0.996157 0.0875884i \(-0.972084\pi\)
−0.0875884 0.996157i \(-0.527916\pi\)
\(108\) 0 0
\(109\) 8.33542 8.33542i 0.798388 0.798388i −0.184453 0.982841i \(-0.559051\pi\)
0.982841 + 0.184453i \(0.0590514\pi\)
\(110\) 0 0
\(111\) −4.24944 + 4.24944i −0.403339 + 0.403339i
\(112\) 0 0
\(113\) −4.73416 + 4.73416i −0.445353 + 0.445353i −0.893806 0.448453i \(-0.851975\pi\)
0.448453 + 0.893806i \(0.351975\pi\)
\(114\) 0 0
\(115\) 0.390622 12.9883i 0.0364257 1.21117i
\(116\) 0 0
\(117\) −1.37361 3.33364i −0.126991 0.308196i
\(118\) 0 0
\(119\) 2.62025 2.62025i 0.240198 0.240198i
\(120\) 0 0
\(121\) 3.78804i 0.344367i
\(122\) 0 0
\(123\) −5.45212 −0.491601
\(124\) 0 0
\(125\) −8.58566 + 7.16145i −0.767925 + 0.640540i
\(126\) 0 0
\(127\) 5.26184 5.26184i 0.466913 0.466913i −0.434000 0.900913i \(-0.642898\pi\)
0.900913 + 0.434000i \(0.142898\pi\)
\(128\) 0 0
\(129\) 5.35496 0.471478
\(130\) 0 0
\(131\) −21.1234 −1.84556 −0.922779 0.385330i \(-0.874088\pi\)
−0.922779 + 0.385330i \(0.874088\pi\)
\(132\) 0 0
\(133\) −1.24081 + 1.24081i −0.107592 + 0.107592i
\(134\) 0 0
\(135\) −0.0672190 + 2.23506i −0.00578529 + 0.192363i
\(136\) 0 0
\(137\) −13.1203 −1.12094 −0.560470 0.828175i \(-0.689379\pi\)
−0.560470 + 0.828175i \(0.689379\pi\)
\(138\) 0 0
\(139\) 19.3191i 1.63862i −0.573349 0.819311i \(-0.694356\pi\)
0.573349 0.819311i \(-0.305644\pi\)
\(140\) 0 0
\(141\) 1.94533 1.94533i 0.163827 0.163827i
\(142\) 0 0
\(143\) 3.72199 8.93881i 0.311248 0.747501i
\(144\) 0 0
\(145\) −2.95819 + 2.78545i −0.245665 + 0.231319i
\(146\) 0 0
\(147\) 1.72762 1.72762i 0.142492 0.142492i
\(148\) 0 0
\(149\) 3.41550 3.41550i 0.279808 0.279808i −0.553224 0.833032i \(-0.686603\pi\)
0.833032 + 0.553224i \(0.186603\pi\)
\(150\) 0 0
\(151\) −9.07740 + 9.07740i −0.738709 + 0.738709i −0.972328 0.233619i \(-0.924943\pi\)
0.233619 + 0.972328i \(0.424943\pi\)
\(152\) 0 0
\(153\) −1.22748 1.22748i −0.0992358 0.0992358i
\(154\) 0 0
\(155\) −0.451067 + 14.9981i −0.0362305 + 1.20468i
\(156\) 0 0
\(157\) −16.7071 16.7071i −1.33337 1.33337i −0.902335 0.431035i \(-0.858149\pi\)
−0.431035 0.902335i \(-0.641851\pi\)
\(158\) 0 0
\(159\) 11.0221i 0.874109i
\(160\) 0 0
\(161\) −8.77160 8.77160i −0.691299 0.691299i
\(162\) 0 0
\(163\) 5.73933i 0.449539i 0.974412 + 0.224769i \(0.0721629\pi\)
−0.974412 + 0.224769i \(0.927837\pi\)
\(164\) 0 0
\(165\) −4.37189 + 4.11660i −0.340351 + 0.320477i
\(166\) 0 0
\(167\) 12.4964 0.967002 0.483501 0.875344i \(-0.339365\pi\)
0.483501 + 0.875344i \(0.339365\pi\)
\(168\) 0 0
\(169\) 9.22638 9.15827i 0.709721 0.704483i
\(170\) 0 0
\(171\) 0.581270 + 0.581270i 0.0444508 + 0.0444508i
\(172\) 0 0
\(173\) −12.1397 12.1397i −0.922966 0.922966i 0.0742720 0.997238i \(-0.476337\pi\)
−0.997238 + 0.0742720i \(0.976337\pi\)
\(174\) 0 0
\(175\) −0.641415 + 10.6540i −0.0484865 + 0.805368i
\(176\) 0 0
\(177\) 0.614063 0.0461558
\(178\) 0 0
\(179\) −3.88220 −0.290169 −0.145085 0.989419i \(-0.546345\pi\)
−0.145085 + 0.989419i \(0.546345\pi\)
\(180\) 0 0
\(181\) 15.9058i 1.18227i 0.806572 + 0.591136i \(0.201320\pi\)
−0.806572 + 0.591136i \(0.798680\pi\)
\(182\) 0 0
\(183\) −9.29304 + 9.29304i −0.686961 + 0.686961i
\(184\) 0 0
\(185\) 9.21209 + 9.78338i 0.677286 + 0.719288i
\(186\) 0 0
\(187\) 4.66183i 0.340906i
\(188\) 0 0
\(189\) 1.50943 + 1.50943i 0.109795 + 0.109795i
\(190\) 0 0
\(191\) −3.96501 −0.286898 −0.143449 0.989658i \(-0.545819\pi\)
−0.143449 + 0.989658i \(0.545819\pi\)
\(192\) 0 0
\(193\) 16.9136i 1.21747i −0.793374 0.608734i \(-0.791678\pi\)
0.793374 0.608734i \(-0.208322\pi\)
\(194\) 0 0
\(195\) −7.54322 + 2.84602i −0.540181 + 0.203808i
\(196\) 0 0
\(197\) 23.1980i 1.65279i −0.563089 0.826396i \(-0.690387\pi\)
0.563089 0.826396i \(-0.309613\pi\)
\(198\) 0 0
\(199\) 24.2812 1.72125 0.860625 0.509239i \(-0.170073\pi\)
0.860625 + 0.509239i \(0.170073\pi\)
\(200\) 0 0
\(201\) −0.900106 0.900106i −0.0634886 0.0634886i
\(202\) 0 0
\(203\) 3.87894i 0.272248i
\(204\) 0 0
\(205\) −0.366486 + 12.1858i −0.0255965 + 0.851094i
\(206\) 0 0
\(207\) −4.10913 + 4.10913i −0.285604 + 0.285604i
\(208\) 0 0
\(209\) 2.20760i 0.152703i
\(210\) 0 0
\(211\) 9.03724 0.622149 0.311075 0.950386i \(-0.399311\pi\)
0.311075 + 0.950386i \(0.399311\pi\)
\(212\) 0 0
\(213\) 2.35028 0.161039
\(214\) 0 0
\(215\) 0.359955 11.9686i 0.0245487 0.816255i
\(216\) 0 0
\(217\) 10.1289 + 10.1289i 0.687595 + 0.687595i
\(218\) 0 0
\(219\) −0.450441 0.450441i −0.0304380 0.0304380i
\(220\) 0 0
\(221\) 2.40590 5.77806i 0.161838 0.388675i
\(222\) 0 0
\(223\) −25.1082 −1.68137 −0.840684 0.541527i \(-0.817847\pi\)
−0.840684 + 0.541527i \(0.817847\pi\)
\(224\) 0 0
\(225\) 4.99096 + 0.300476i 0.332731 + 0.0200318i
\(226\) 0 0
\(227\) 10.2921i 0.683110i −0.939862 0.341555i \(-0.889046\pi\)
0.939862 0.341555i \(-0.110954\pi\)
\(228\) 0 0
\(229\) −1.77751 1.77751i −0.117461 0.117461i 0.645933 0.763394i \(-0.276468\pi\)
−0.763394 + 0.645933i \(0.776468\pi\)
\(230\) 0 0
\(231\) 5.73265i 0.377181i
\(232\) 0 0
\(233\) 14.0627 + 14.0627i 0.921280 + 0.921280i 0.997120 0.0758398i \(-0.0241638\pi\)
−0.0758398 + 0.997120i \(0.524164\pi\)
\(234\) 0 0
\(235\) −4.21717 4.47870i −0.275098 0.292158i
\(236\) 0 0
\(237\) −0.592132 0.592132i −0.0384631 0.0384631i
\(238\) 0 0
\(239\) −21.1880 + 21.1880i −1.37054 + 1.37054i −0.510901 + 0.859640i \(0.670688\pi\)
−0.859640 + 0.510901i \(0.829312\pi\)
\(240\) 0 0
\(241\) 1.79901 1.79901i 0.115885 0.115885i −0.646786 0.762671i \(-0.723888\pi\)
0.762671 + 0.646786i \(0.223888\pi\)
\(242\) 0 0
\(243\) 0.707107 0.707107i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) −3.74520 3.97746i −0.239272 0.254110i
\(246\) 0 0
\(247\) −1.13931 + 2.73619i −0.0724924 + 0.174099i
\(248\) 0 0
\(249\) 3.69360 3.69360i 0.234072 0.234072i
\(250\) 0 0
\(251\) 19.4291i 1.22635i −0.789946 0.613177i \(-0.789891\pi\)
0.789946 0.613177i \(-0.210109\pi\)
\(252\) 0 0
\(253\) −15.6060 −0.981140
\(254\) 0 0
\(255\) −2.82600 + 2.66098i −0.176971 + 0.166637i
\(256\) 0 0
\(257\) 7.19958 7.19958i 0.449097 0.449097i −0.445957 0.895054i \(-0.647137\pi\)
0.895054 + 0.445957i \(0.147137\pi\)
\(258\) 0 0
\(259\) 12.8285 0.797123
\(260\) 0 0
\(261\) 1.81712 0.112477
\(262\) 0 0
\(263\) 9.21874 9.21874i 0.568452 0.568452i −0.363243 0.931695i \(-0.618330\pi\)
0.931695 + 0.363243i \(0.118330\pi\)
\(264\) 0 0
\(265\) −24.6350 0.740894i −1.51332 0.0455127i
\(266\) 0 0
\(267\) −5.55344 −0.339865
\(268\) 0 0
\(269\) 1.62017i 0.0987834i −0.998779 0.0493917i \(-0.984272\pi\)
0.998779 0.0493917i \(-0.0157283\pi\)
\(270\) 0 0
\(271\) 6.91766 6.91766i 0.420218 0.420218i −0.465061 0.885279i \(-0.653968\pi\)
0.885279 + 0.465061i \(0.153968\pi\)
\(272\) 0 0
\(273\) −2.95854 + 7.10529i −0.179059 + 0.430032i
\(274\) 0 0
\(275\) 8.90696 + 10.0481i 0.537110 + 0.605926i
\(276\) 0 0
\(277\) −7.18506 + 7.18506i −0.431709 + 0.431709i −0.889209 0.457501i \(-0.848745\pi\)
0.457501 + 0.889209i \(0.348745\pi\)
\(278\) 0 0
\(279\) 4.74497 4.74497i 0.284074 0.284074i
\(280\) 0 0
\(281\) 10.6309 10.6309i 0.634185 0.634185i −0.314930 0.949115i \(-0.601981\pi\)
0.949115 + 0.314930i \(0.101981\pi\)
\(282\) 0 0
\(283\) 12.7919 + 12.7919i 0.760399 + 0.760399i 0.976394 0.215995i \(-0.0692995\pi\)
−0.215995 + 0.976394i \(0.569300\pi\)
\(284\) 0 0
\(285\) 1.33824 1.26010i 0.0792707 0.0746418i
\(286\) 0 0
\(287\) 8.22962 + 8.22962i 0.485779 + 0.485779i
\(288\) 0 0
\(289\) 13.9866i 0.822740i
\(290\) 0 0
\(291\) −4.35556 4.35556i −0.255328 0.255328i
\(292\) 0 0
\(293\) 16.8122i 0.982178i 0.871109 + 0.491089i \(0.163401\pi\)
−0.871109 + 0.491089i \(0.836599\pi\)
\(294\) 0 0
\(295\) 0.0412767 1.37247i 0.00240322 0.0799080i
\(296\) 0 0
\(297\) 2.68551 0.155829
\(298\) 0 0
\(299\) −19.3427 8.05402i −1.11862 0.465776i
\(300\) 0 0
\(301\) −8.08296 8.08296i −0.465894 0.465894i
\(302\) 0 0
\(303\) −2.78409 2.78409i −0.159942 0.159942i
\(304\) 0 0
\(305\) 20.1458 + 21.3951i 1.15355 + 1.22508i
\(306\) 0 0
\(307\) 14.6050 0.833551 0.416775 0.909009i \(-0.363160\pi\)
0.416775 + 0.909009i \(0.363160\pi\)
\(308\) 0 0
\(309\) 5.62174 0.319809
\(310\) 0 0
\(311\) 27.4910i 1.55887i 0.626481 + 0.779436i \(0.284494\pi\)
−0.626481 + 0.779436i \(0.715506\pi\)
\(312\) 0 0
\(313\) 4.82784 4.82784i 0.272886 0.272886i −0.557375 0.830261i \(-0.688191\pi\)
0.830261 + 0.557375i \(0.188191\pi\)
\(314\) 0 0
\(315\) 3.47513 3.27221i 0.195802 0.184368i
\(316\) 0 0
\(317\) 27.3999i 1.53893i −0.638690 0.769464i \(-0.720523\pi\)
0.638690 0.769464i \(-0.279477\pi\)
\(318\) 0 0
\(319\) 3.45061 + 3.45061i 0.193197 + 0.193197i
\(320\) 0 0
\(321\) 15.8538 0.884874
\(322\) 0 0
\(323\) 1.42699i 0.0794000i
\(324\) 0 0
\(325\) 5.85397 + 17.0508i 0.324720 + 0.945810i
\(326\) 0 0
\(327\) 11.7881i 0.651881i
\(328\) 0 0
\(329\) −5.87271 −0.323773
\(330\) 0 0
\(331\) −5.68185 5.68185i −0.312303 0.312303i 0.533498 0.845801i \(-0.320877\pi\)
−0.845801 + 0.533498i \(0.820877\pi\)
\(332\) 0 0
\(333\) 6.00961i 0.329325i
\(334\) 0 0
\(335\) −2.07229 + 1.95128i −0.113221 + 0.106610i
\(336\) 0 0
\(337\) −15.7975 + 15.7975i −0.860544 + 0.860544i −0.991401 0.130858i \(-0.958227\pi\)
0.130858 + 0.991401i \(0.458227\pi\)
\(338\) 0 0
\(339\) 6.69512i 0.363629i
\(340\) 0 0
\(341\) 18.0209 0.975885
\(342\) 0 0
\(343\) −20.1581 −1.08843
\(344\) 0 0
\(345\) 8.90793 + 9.46035i 0.479587 + 0.509328i
\(346\) 0 0
\(347\) 17.2806 + 17.2806i 0.927670 + 0.927670i 0.997555 0.0698847i \(-0.0222632\pi\)
−0.0698847 + 0.997555i \(0.522263\pi\)
\(348\) 0 0
\(349\) −19.0865 19.0865i −1.02168 1.02168i −0.999760 0.0219165i \(-0.993023\pi\)
−0.0219165 0.999760i \(-0.506977\pi\)
\(350\) 0 0
\(351\) 3.32853 + 1.38595i 0.177664 + 0.0739766i
\(352\) 0 0
\(353\) 33.0078 1.75683 0.878414 0.477901i \(-0.158602\pi\)
0.878414 + 0.477901i \(0.158602\pi\)
\(354\) 0 0
\(355\) 0.157983 5.25301i 0.00838489 0.278801i
\(356\) 0 0
\(357\) 3.70560i 0.196121i
\(358\) 0 0
\(359\) −15.0109 15.0109i −0.792247 0.792247i 0.189612 0.981859i \(-0.439277\pi\)
−0.981859 + 0.189612i \(0.939277\pi\)
\(360\) 0 0
\(361\) 18.3243i 0.964434i
\(362\) 0 0
\(363\) −2.67855 2.67855i −0.140587 0.140587i
\(364\) 0 0
\(365\) −1.03704 + 0.976483i −0.0542811 + 0.0511114i
\(366\) 0 0
\(367\) −2.95951 2.95951i −0.154485 0.154485i 0.625633 0.780118i \(-0.284841\pi\)
−0.780118 + 0.625633i \(0.784841\pi\)
\(368\) 0 0
\(369\) 3.85523 3.85523i 0.200695 0.200695i
\(370\) 0 0
\(371\) −16.6371 + 16.6371i −0.863756 + 0.863756i
\(372\) 0 0
\(373\) 3.69960 3.69960i 0.191558 0.191558i −0.604811 0.796369i \(-0.706751\pi\)
0.796369 + 0.604811i \(0.206751\pi\)
\(374\) 0 0
\(375\) 1.00707 11.1349i 0.0520048 0.575003i
\(376\) 0 0
\(377\) 2.49602 + 6.05764i 0.128552 + 0.311984i
\(378\) 0 0
\(379\) −6.73427 + 6.73427i −0.345916 + 0.345916i −0.858586 0.512670i \(-0.828657\pi\)
0.512670 + 0.858586i \(0.328657\pi\)
\(380\) 0 0
\(381\) 7.44137i 0.381233i
\(382\) 0 0
\(383\) −16.7639 −0.856597 −0.428299 0.903637i \(-0.640887\pi\)
−0.428299 + 0.903637i \(0.640887\pi\)
\(384\) 0 0
\(385\) 12.8128 + 0.385343i 0.653001 + 0.0196389i
\(386\) 0 0
\(387\) −3.78653 + 3.78653i −0.192480 + 0.192480i
\(388\) 0 0
\(389\) 20.1793 1.02313 0.511566 0.859244i \(-0.329066\pi\)
0.511566 + 0.859244i \(0.329066\pi\)
\(390\) 0 0
\(391\) −10.0877 −0.510159
\(392\) 0 0
\(393\) 14.9365 14.9365i 0.753446 0.753446i
\(394\) 0 0
\(395\) −1.36325 + 1.28365i −0.0685926 + 0.0645872i
\(396\) 0 0
\(397\) −26.6617 −1.33811 −0.669055 0.743213i \(-0.733301\pi\)
−0.669055 + 0.743213i \(0.733301\pi\)
\(398\) 0 0
\(399\) 1.75478i 0.0878487i
\(400\) 0 0
\(401\) 21.5935 21.5935i 1.07833 1.07833i 0.0816678 0.996660i \(-0.473975\pi\)
0.996660 0.0816678i \(-0.0260246\pi\)
\(402\) 0 0
\(403\) 22.3358 + 9.30030i 1.11263 + 0.463281i
\(404\) 0 0
\(405\) −1.53289 1.62796i −0.0761701 0.0808937i
\(406\) 0 0
\(407\) 11.4119 11.4119i 0.565667 0.565667i
\(408\) 0 0
\(409\) −2.53012 + 2.53012i −0.125107 + 0.125107i −0.766888 0.641781i \(-0.778196\pi\)
0.641781 + 0.766888i \(0.278196\pi\)
\(410\) 0 0
\(411\) 9.27742 9.27742i 0.457622 0.457622i
\(412\) 0 0
\(413\) −0.926887 0.926887i −0.0456091 0.0456091i
\(414\) 0 0
\(415\) −8.00713 8.50369i −0.393054 0.417430i
\(416\) 0 0
\(417\) 13.6606 + 13.6606i 0.668965 + 0.668965i
\(418\) 0 0
\(419\) 20.4939i 1.00119i 0.865681 + 0.500596i \(0.166886\pi\)
−0.865681 + 0.500596i \(0.833114\pi\)
\(420\) 0 0
\(421\) −22.7083 22.7083i −1.10674 1.10674i −0.993577 0.113160i \(-0.963903\pi\)
−0.113160 0.993577i \(-0.536097\pi\)
\(422\) 0 0
\(423\) 2.75112i 0.133764i
\(424\) 0 0
\(425\) 5.75748 + 6.49513i 0.279279 + 0.315060i
\(426\) 0 0
\(427\) 28.0544 1.35765
\(428\) 0 0
\(429\) 3.68885 + 8.95254i 0.178099 + 0.432233i
\(430\) 0 0
\(431\) −7.07318 7.07318i −0.340703 0.340703i 0.515929 0.856632i \(-0.327447\pi\)
−0.856632 + 0.515929i \(0.827447\pi\)
\(432\) 0 0
\(433\) −27.4138 27.4138i −1.31742 1.31742i −0.915809 0.401614i \(-0.868449\pi\)
−0.401614 0.915809i \(-0.631551\pi\)
\(434\) 0 0
\(435\) 0.122145 4.06137i 0.00585641 0.194728i
\(436\) 0 0
\(437\) 4.77702 0.228516
\(438\) 0 0
\(439\) 37.5366 1.79152 0.895762 0.444533i \(-0.146630\pi\)
0.895762 + 0.444533i \(0.146630\pi\)
\(440\) 0 0
\(441\) 2.44322i 0.116344i
\(442\) 0 0
\(443\) 6.57913 6.57913i 0.312584 0.312584i −0.533326 0.845910i \(-0.679058\pi\)
0.845910 + 0.533326i \(0.179058\pi\)
\(444\) 0 0
\(445\) −0.373296 + 12.4123i −0.0176959 + 0.588397i
\(446\) 0 0
\(447\) 4.83024i 0.228463i
\(448\) 0 0
\(449\) 12.5481 + 12.5481i 0.592183 + 0.592183i 0.938221 0.346038i \(-0.112473\pi\)
−0.346038 + 0.938221i \(0.612473\pi\)
\(450\) 0 0
\(451\) 14.6417 0.689452
\(452\) 0 0
\(453\) 12.8374i 0.603153i
\(454\) 0 0
\(455\) 15.6819 + 7.09011i 0.735177 + 0.332390i
\(456\) 0 0
\(457\) 21.0694i 0.985583i 0.870147 + 0.492791i \(0.164023\pi\)
−0.870147 + 0.492791i \(0.835977\pi\)
\(458\) 0 0
\(459\) 1.73592 0.0810257
\(460\) 0 0
\(461\) −11.1853 11.1853i −0.520949 0.520949i 0.396909 0.917858i \(-0.370083\pi\)
−0.917858 + 0.396909i \(0.870083\pi\)
\(462\) 0 0
\(463\) 38.6791i 1.79757i 0.438389 + 0.898786i \(0.355549\pi\)
−0.438389 + 0.898786i \(0.644451\pi\)
\(464\) 0 0
\(465\) −10.2863 10.9242i −0.477017 0.506600i
\(466\) 0 0
\(467\) −13.3530 + 13.3530i −0.617904 + 0.617904i −0.944993 0.327090i \(-0.893932\pi\)
0.327090 + 0.944993i \(0.393932\pi\)
\(468\) 0 0
\(469\) 2.71730i 0.125473i
\(470\) 0 0
\(471\) 23.6274 1.08869
\(472\) 0 0
\(473\) −14.3808 −0.661230
\(474\) 0 0
\(475\) −2.72644 3.07575i −0.125098 0.141125i
\(476\) 0 0
\(477\) 7.79380 + 7.79380i 0.356854 + 0.356854i
\(478\) 0 0
\(479\) 1.95863 + 1.95863i 0.0894919 + 0.0894919i 0.750436 0.660944i \(-0.229844\pi\)
−0.660944 + 0.750436i \(0.729844\pi\)
\(480\) 0 0
\(481\) 20.0339 8.25488i 0.913467 0.376390i
\(482\) 0 0
\(483\) 12.4049 0.564443
\(484\) 0 0
\(485\) −10.0277 + 9.44216i −0.455335 + 0.428746i
\(486\) 0 0
\(487\) 17.7479i 0.804234i −0.915588 0.402117i \(-0.868275\pi\)
0.915588 0.402117i \(-0.131725\pi\)
\(488\) 0 0
\(489\) −4.05832 4.05832i −0.183524 0.183524i
\(490\) 0 0
\(491\) 24.6538i 1.11261i 0.830978 + 0.556306i \(0.187782\pi\)
−0.830978 + 0.556306i \(0.812218\pi\)
\(492\) 0 0
\(493\) 2.23048 + 2.23048i 0.100456 + 0.100456i
\(494\) 0 0
\(495\) 0.180517 6.00227i 0.00811364 0.269782i
\(496\) 0 0
\(497\) −3.54759 3.54759i −0.159131 0.159131i
\(498\) 0 0
\(499\) −0.763548 + 0.763548i −0.0341811 + 0.0341811i −0.723991 0.689810i \(-0.757694\pi\)
0.689810 + 0.723991i \(0.257694\pi\)
\(500\) 0 0
\(501\) −8.83631 + 8.83631i −0.394777 + 0.394777i
\(502\) 0 0
\(503\) 17.2923 17.2923i 0.771026 0.771026i −0.207260 0.978286i \(-0.566454\pi\)
0.978286 + 0.207260i \(0.0664545\pi\)
\(504\) 0 0
\(505\) −6.40975 + 6.03546i −0.285230 + 0.268574i
\(506\) 0 0
\(507\) −0.0481545 + 12.9999i −0.00213862 + 0.577346i
\(508\) 0 0
\(509\) −22.2401 + 22.2401i −0.985776 + 0.985776i −0.999900 0.0141241i \(-0.995504\pi\)
0.0141241 + 0.999900i \(0.495504\pi\)
\(510\) 0 0
\(511\) 1.35982i 0.0601549i
\(512\) 0 0
\(513\) −0.822039 −0.0362939
\(514\) 0 0
\(515\) 0.377887 12.5649i 0.0166517 0.553676i
\(516\) 0 0
\(517\) −5.22421 + 5.22421i −0.229761 + 0.229761i
\(518\) 0 0
\(519\) 17.1682 0.753599
\(520\) 0 0
\(521\) 3.21481 0.140843 0.0704217 0.997517i \(-0.477565\pi\)
0.0704217 + 0.997517i \(0.477565\pi\)
\(522\) 0 0
\(523\) 7.33192 7.33192i 0.320603 0.320603i −0.528396 0.848998i \(-0.677206\pi\)
0.848998 + 0.528396i \(0.177206\pi\)
\(524\) 0 0
\(525\) −7.07998 7.98708i −0.308996 0.348585i
\(526\) 0 0
\(527\) 11.6487 0.507426
\(528\) 0 0
\(529\) 10.7699i 0.468255i
\(530\) 0 0
\(531\) −0.434208 + 0.434208i −0.0188430 + 0.0188430i
\(532\) 0 0
\(533\) 18.1476 + 7.55638i 0.786059 + 0.327303i
\(534\) 0 0
\(535\) 1.06568 35.4342i 0.0460733 1.53195i
\(536\) 0 0
\(537\) 2.74513 2.74513i 0.118461 0.118461i
\(538\) 0 0
\(539\) −4.63954 + 4.63954i −0.199839 + 0.199839i
\(540\) 0 0
\(541\) −0.865604 + 0.865604i −0.0372152 + 0.0372152i −0.725470 0.688254i \(-0.758377\pi\)
0.688254 + 0.725470i \(0.258377\pi\)
\(542\) 0 0
\(543\) −11.2471 11.2471i −0.482661 0.482661i
\(544\) 0 0
\(545\) 26.3470 + 0.792381i 1.12858 + 0.0339419i
\(546\) 0 0
\(547\) 14.4611 + 14.4611i 0.618311 + 0.618311i 0.945098 0.326787i \(-0.105966\pi\)
−0.326787 + 0.945098i \(0.605966\pi\)
\(548\) 0 0
\(549\) 13.1423i 0.560901i
\(550\) 0 0
\(551\) −1.05624 1.05624i −0.0449972 0.0449972i
\(552\) 0 0
\(553\) 1.78757i 0.0760151i
\(554\) 0 0
\(555\) −13.4318 0.403960i −0.570149 0.0171471i
\(556\) 0 0
\(557\) −20.5392 −0.870272 −0.435136 0.900365i \(-0.643300\pi\)
−0.435136 + 0.900365i \(0.643300\pi\)
\(558\) 0 0
\(559\) −17.8242 7.42172i −0.753882 0.313905i
\(560\) 0 0
\(561\) 3.29641 + 3.29641i 0.139174 + 0.139174i
\(562\) 0 0
\(563\) 3.91042 + 3.91042i 0.164804 + 0.164804i 0.784691 0.619887i \(-0.212821\pi\)
−0.619887 + 0.784691i \(0.712821\pi\)
\(564\) 0 0
\(565\) −14.9640 0.450039i −0.629539 0.0189333i
\(566\) 0 0
\(567\) −2.13466 −0.0896473
\(568\) 0 0
\(569\) −29.0908 −1.21955 −0.609775 0.792574i \(-0.708740\pi\)
−0.609775 + 0.792574i \(0.708740\pi\)
\(570\) 0 0
\(571\) 27.8608i 1.16594i 0.812494 + 0.582969i \(0.198109\pi\)
−0.812494 + 0.582969i \(0.801891\pi\)
\(572\) 0 0
\(573\) 2.80368 2.80368i 0.117126 0.117126i
\(574\) 0 0
\(575\) 21.7432 19.2738i 0.906754 0.803773i
\(576\) 0 0
\(577\) 11.1273i 0.463235i 0.972807 + 0.231618i \(0.0744018\pi\)
−0.972807 + 0.231618i \(0.925598\pi\)
\(578\) 0 0
\(579\) 11.9597 + 11.9597i 0.497029 + 0.497029i
\(580\) 0 0
\(581\) −11.1505 −0.462600
\(582\) 0 0
\(583\) 29.5999i 1.22590i
\(584\) 0 0
\(585\) 3.32142 7.34630i 0.137324 0.303732i
\(586\) 0 0
\(587\) 27.0251i 1.11545i −0.830027 0.557723i \(-0.811675\pi\)
0.830027 0.557723i \(-0.188325\pi\)
\(588\) 0 0
\(589\) −5.51622 −0.227292
\(590\) 0 0
\(591\) 16.4035 + 16.4035i 0.674750 + 0.674750i
\(592\) 0 0
\(593\) 23.2668i 0.955454i −0.878508 0.477727i \(-0.841461\pi\)
0.878508 0.477727i \(-0.158539\pi\)
\(594\) 0 0
\(595\) 8.28222 + 0.249086i 0.339538 + 0.0102115i
\(596\) 0 0
\(597\) −17.1694 + 17.1694i −0.702698 + 0.702698i
\(598\) 0 0
\(599\) 35.1115i 1.43462i 0.696756 + 0.717308i \(0.254626\pi\)
−0.696756 + 0.717308i \(0.745374\pi\)
\(600\) 0 0
\(601\) −11.3510 −0.463015 −0.231508 0.972833i \(-0.574366\pi\)
−0.231508 + 0.972833i \(0.574366\pi\)
\(602\) 0 0
\(603\) 1.27294 0.0518382
\(604\) 0 0
\(605\) −6.16675 + 5.80666i −0.250714 + 0.236074i
\(606\) 0 0
\(607\) −25.0768 25.0768i −1.01784 1.01784i −0.999838 0.0179998i \(-0.994270\pi\)
−0.0179998 0.999838i \(-0.505730\pi\)
\(608\) 0 0
\(609\) −2.74282 2.74282i −0.111145 0.111145i
\(610\) 0 0
\(611\) −9.17125 + 3.77897i −0.371029 + 0.152881i
\(612\) 0 0
\(613\) −32.7471 −1.32264 −0.661322 0.750103i \(-0.730004\pi\)
−0.661322 + 0.750103i \(0.730004\pi\)
\(614\) 0 0
\(615\) −8.35752 8.87581i −0.337008 0.357907i
\(616\) 0 0
\(617\) 3.04921i 0.122757i 0.998115 + 0.0613784i \(0.0195496\pi\)
−0.998115 + 0.0613784i \(0.980450\pi\)
\(618\) 0 0
\(619\) 2.15135 + 2.15135i 0.0864701 + 0.0864701i 0.749019 0.662549i \(-0.230525\pi\)
−0.662549 + 0.749019i \(0.730525\pi\)
\(620\) 0 0
\(621\) 5.81118i 0.233195i
\(622\) 0 0
\(623\) 8.38255 + 8.38255i 0.335840 + 0.335840i
\(624\) 0 0
\(625\) −24.8194 2.99933i −0.992777 0.119973i
\(626\) 0 0
\(627\) −1.56101 1.56101i −0.0623406 0.0623406i
\(628\) 0 0
\(629\) 7.37667 7.37667i 0.294127 0.294127i
\(630\) 0 0
\(631\) −26.1168 + 26.1168i −1.03969 + 1.03969i −0.0405145 + 0.999179i \(0.512900\pi\)
−0.999179 + 0.0405145i \(0.987100\pi\)
\(632\) 0 0
\(633\) −6.39029 + 6.39029i −0.253991 + 0.253991i
\(634\) 0 0
\(635\) 16.6319 + 0.500201i 0.660016 + 0.0198499i
\(636\) 0 0
\(637\) −8.14483 + 3.35604i −0.322710 + 0.132971i
\(638\) 0 0
\(639\) −1.66190 + 1.66190i −0.0657437 + 0.0657437i
\(640\) 0 0
\(641\) 14.0903i 0.556534i −0.960504 0.278267i \(-0.910240\pi\)
0.960504 0.278267i \(-0.0897599\pi\)
\(642\) 0 0
\(643\) −9.89537 −0.390235 −0.195118 0.980780i \(-0.562509\pi\)
−0.195118 + 0.980780i \(0.562509\pi\)
\(644\) 0 0
\(645\) 8.20858 + 8.71764i 0.323213 + 0.343257i
\(646\) 0 0
\(647\) 7.48001 7.48001i 0.294069 0.294069i −0.544616 0.838685i \(-0.683325\pi\)
0.838685 + 0.544616i \(0.183325\pi\)
\(648\) 0 0
\(649\) −1.64907 −0.0647317
\(650\) 0 0
\(651\) −14.3244 −0.561419
\(652\) 0 0
\(653\) 10.0579 10.0579i 0.393595 0.393595i −0.482372 0.875967i \(-0.660225\pi\)
0.875967 + 0.482372i \(0.160225\pi\)
\(654\) 0 0
\(655\) −32.3799 34.3879i −1.26519 1.34365i
\(656\) 0 0
\(657\) 0.637019 0.0248525
\(658\) 0 0
\(659\) 36.4548i 1.42008i −0.704164 0.710038i \(-0.748678\pi\)
0.704164 0.710038i \(-0.251322\pi\)
\(660\) 0 0
\(661\) −6.58265 + 6.58265i −0.256035 + 0.256035i −0.823439 0.567404i \(-0.807948\pi\)
0.567404 + 0.823439i \(0.307948\pi\)
\(662\) 0 0
\(663\) 2.38448 + 5.78693i 0.0926055 + 0.224746i
\(664\) 0 0
\(665\) −3.92202 0.117954i −0.152090 0.00457407i
\(666\) 0 0
\(667\) 7.46678 7.46678i 0.289115 0.289115i
\(668\) 0 0
\(669\) 17.7542 17.7542i 0.686415 0.686415i
\(670\) 0 0
\(671\) 24.9565 24.9565i 0.963437 0.963437i
\(672\) 0 0
\(673\) 10.3834 + 10.3834i 0.400252 + 0.400252i 0.878322 0.478070i \(-0.158663\pi\)
−0.478070 + 0.878322i \(0.658663\pi\)
\(674\) 0 0
\(675\) −3.74161 + 3.31667i −0.144015 + 0.127659i
\(676\) 0 0
\(677\) −22.0470 22.0470i −0.847334 0.847334i 0.142466 0.989800i \(-0.454497\pi\)
−0.989800 + 0.142466i \(0.954497\pi\)
\(678\) 0 0
\(679\) 13.1489i 0.504607i
\(680\) 0 0
\(681\) 7.27761 + 7.27761i 0.278879 + 0.278879i
\(682\) 0 0
\(683\) 44.6066i 1.70682i 0.521237 + 0.853412i \(0.325471\pi\)
−0.521237 + 0.853412i \(0.674529\pi\)
\(684\) 0 0
\(685\) −20.1120 21.3592i −0.768438 0.816093i
\(686\) 0 0
\(687\) 2.51377 0.0959064
\(688\) 0 0
\(689\) −15.2761 + 36.6874i −0.581973 + 1.39768i
\(690\) 0 0
\(691\) −33.1306 33.1306i −1.26035 1.26035i −0.950924 0.309424i \(-0.899864\pi\)
−0.309424 0.950924i \(-0.600136\pi\)
\(692\) 0 0
\(693\) −4.05360 4.05360i −0.153983 0.153983i
\(694\) 0 0
\(695\) 31.4506 29.6141i 1.19299 1.12333i
\(696\) 0 0
\(697\) 9.46444 0.358491
\(698\) 0 0
\(699\) −19.8877 −0.752222
\(700\) 0 0
\(701\) 5.40260i 0.204053i 0.994782 + 0.102027i \(0.0325327\pi\)
−0.994782 + 0.102027i \(0.967467\pi\)
\(702\) 0 0
\(703\) −3.49320 + 3.49320i −0.131749 + 0.131749i
\(704\) 0 0
\(705\) 6.14891 + 0.184927i 0.231581 + 0.00696477i
\(706\) 0 0
\(707\) 8.40480i 0.316095i
\(708\) 0 0
\(709\) −34.9371 34.9371i −1.31209 1.31209i −0.919873 0.392216i \(-0.871708\pi\)
−0.392216 0.919873i \(-0.628292\pi\)
\(710\) 0 0
\(711\) 0.837401 0.0314050
\(712\) 0 0
\(713\) 38.9954i 1.46039i
\(714\) 0 0
\(715\) 20.2574 7.64301i 0.757584 0.285832i
\(716\) 0 0
\(717\) 29.9644i 1.11904i
\(718\) 0 0
\(719\) −23.4590 −0.874872 −0.437436 0.899249i \(-0.644113\pi\)
−0.437436 + 0.899249i \(0.644113\pi\)
\(720\) 0 0
\(721\) −8.48564 8.48564i −0.316022 0.316022i
\(722\) 0 0
\(723\) 2.54419i 0.0946194i
\(724\) 0 0
\(725\) −9.06919 0.546002i −0.336821 0.0202780i
\(726\) 0 0
\(727\) −8.12835 + 8.12835i −0.301464 + 0.301464i −0.841586 0.540123i \(-0.818378\pi\)
0.540123 + 0.841586i \(0.318378\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) −9.29577 −0.343817
\(732\) 0 0
\(733\) −15.4133 −0.569303 −0.284652 0.958631i \(-0.591878\pi\)
−0.284652 + 0.958631i \(0.591878\pi\)
\(734\) 0 0
\(735\) 5.46074 + 0.164231i 0.201422 + 0.00605774i
\(736\) 0 0
\(737\) 2.41724 + 2.41724i 0.0890403 + 0.0890403i
\(738\) 0 0
\(739\) 6.33555 + 6.33555i 0.233057 + 0.233057i 0.813967 0.580910i \(-0.197303\pi\)
−0.580910 + 0.813967i \(0.697303\pi\)
\(740\) 0 0
\(741\) −1.12916 2.74039i −0.0414809 0.100671i
\(742\) 0 0
\(743\) 44.7202 1.64063 0.820313 0.571915i \(-0.193799\pi\)
0.820313 + 0.571915i \(0.193799\pi\)
\(744\) 0 0
\(745\) 10.7959 + 0.324684i 0.395530 + 0.0118955i
\(746\) 0 0
\(747\) 5.22354i 0.191119i
\(748\) 0 0
\(749\) −23.9303 23.9303i −0.874394 0.874394i
\(750\) 0 0
\(751\) 4.64023i 0.169324i −0.996410 0.0846621i \(-0.973019\pi\)
0.996410 0.0846621i \(-0.0269811\pi\)
\(752\) 0 0
\(753\) 13.7384 + 13.7384i 0.500657 + 0.500657i
\(754\) 0 0
\(755\) −28.6923 0.862916i −1.04422 0.0314047i
\(756\) 0 0
\(757\) −20.4954 20.4954i −0.744918 0.744918i 0.228602 0.973520i \(-0.426584\pi\)
−0.973520 + 0.228602i \(0.926584\pi\)
\(758\) 0 0
\(759\) 11.0351 11.0351i 0.400549 0.400549i
\(760\) 0 0
\(761\) 17.6249 17.6249i 0.638902 0.638902i −0.311383 0.950285i \(-0.600792\pi\)
0.950285 + 0.311383i \(0.100792\pi\)
\(762\) 0 0
\(763\) 17.7933 17.7933i 0.644160 0.644160i
\(764\) 0 0
\(765\) 0.116687 3.87988i 0.00421881 0.140277i
\(766\) 0 0
\(767\) −2.04393 0.851061i −0.0738020 0.0307300i
\(768\) 0 0
\(769\) −35.6848 + 35.6848i −1.28683 + 1.28683i −0.350123 + 0.936704i \(0.613860\pi\)
−0.936704 + 0.350123i \(0.886140\pi\)
\(770\) 0 0
\(771\) 10.1817i 0.366687i
\(772\) 0 0
\(773\) 10.4900 0.377301 0.188650 0.982044i \(-0.439589\pi\)
0.188650 + 0.982044i \(0.439589\pi\)
\(774\) 0 0
\(775\) −25.1077 + 22.2562i −0.901897 + 0.799468i
\(776\) 0 0
\(777\) −9.07111 + 9.07111i −0.325424 + 0.325424i
\(778\) 0 0
\(779\) −4.48186 −0.160579
\(780\) 0 0
\(781\) −6.31170 −0.225850
\(782\) 0 0
\(783\) −1.28490 + 1.28490i −0.0459185 + 0.0459185i
\(784\) 0 0
\(785\) 1.58821 52.8085i 0.0566856 1.88482i
\(786\) 0 0
\(787\) 16.0980 0.573832 0.286916 0.957956i \(-0.407370\pi\)
0.286916 + 0.957956i \(0.407370\pi\)
\(788\) 0 0
\(789\) 13.0373i 0.464139i
\(790\) 0 0
\(791\) −10.1058 + 10.1058i −0.359322 + 0.359322i
\(792\) 0 0
\(793\) 43.8119 18.0525i 1.55581 0.641062i
\(794\) 0 0
\(795\) 17.9435 16.8957i 0.636390 0.599229i
\(796\) 0 0
\(797\) −17.8833 + 17.8833i −0.633460 + 0.633460i −0.948934 0.315474i \(-0.897836\pi\)
0.315474 + 0.948934i \(0.397836\pi\)
\(798\) 0 0
\(799\) −3.37694 + 3.37694i −0.119468 + 0.119468i
\(800\) 0 0
\(801\) 3.92688 3.92688i 0.138749 0.138749i
\(802\) 0 0
\(803\) 1.20966 + 1.20966i 0.0426881 + 0.0426881i
\(804\) 0 0
\(805\) 0.833845 27.7257i 0.0293892 0.977202i
\(806\) 0 0
\(807\) 1.14563 + 1.14563i 0.0403282 + 0.0403282i
\(808\) 0 0
\(809\) 20.4265i 0.718157i 0.933307 + 0.359079i \(0.116909\pi\)
−0.933307 + 0.359079i \(0.883091\pi\)
\(810\) 0 0
\(811\) 4.25247 + 4.25247i 0.149325 + 0.149325i 0.777816 0.628492i \(-0.216327\pi\)
−0.628492 + 0.777816i \(0.716327\pi\)
\(812\) 0 0
\(813\) 9.78305i 0.343106i
\(814\) 0 0
\(815\) −9.34337 + 8.79778i −0.327284 + 0.308173i
\(816\) 0 0
\(817\) 4.40199 0.154006
\(818\) 0 0
\(819\) −2.93220 7.11620i −0.102459 0.248660i
\(820\) 0 0
\(821\) −26.9521 26.9521i −0.940636 0.940636i 0.0576979 0.998334i \(-0.481624\pi\)
−0.998334 + 0.0576979i \(0.981624\pi\)
\(822\) 0 0
\(823\) 25.9491 + 25.9491i 0.904528 + 0.904528i 0.995824 0.0912955i \(-0.0291008\pi\)
−0.0912955 + 0.995824i \(0.529101\pi\)
\(824\) 0 0
\(825\) −13.4033 0.806933i −0.466642 0.0280938i
\(826\) 0 0
\(827\) 45.6260 1.58657 0.793287 0.608848i \(-0.208368\pi\)
0.793287 + 0.608848i \(0.208368\pi\)
\(828\) 0 0
\(829\) 42.7307 1.48410 0.742049 0.670346i \(-0.233854\pi\)
0.742049 + 0.670346i \(0.233854\pi\)
\(830\) 0 0
\(831\) 10.1612i 0.352489i
\(832\) 0 0
\(833\) −2.99900 + 2.99900i −0.103909 + 0.103909i
\(834\) 0 0
\(835\) 19.1557 + 20.3436i 0.662910 + 0.704020i
\(836\) 0 0
\(837\) 6.71041i 0.231946i
\(838\) 0 0
\(839\) −3.57427 3.57427i −0.123398 0.123398i 0.642711 0.766109i \(-0.277810\pi\)
−0.766109 + 0.642711i \(0.777810\pi\)
\(840\) 0 0
\(841\) 25.6981 0.886140
\(842\) 0 0
\(843\) 15.0343i 0.517810i
\(844\) 0 0
\(845\) 29.0523 + 0.981468i 0.999430 + 0.0337635i
\(846\) 0 0
\(847\) 8.08617i 0.277844i
\(848\) 0 0
\(849\) −18.0905 −0.620863
\(850\) 0 0
\(851\) −24.6943 24.6943i −0.846508 0.846508i
\(852\) 0 0
\(853\) 10.6644i 0.365141i 0.983193 + 0.182571i \(0.0584418\pi\)
−0.983193 + 0.182571i \(0.941558\pi\)
\(854\) 0 0
\(855\) −0.0552566 + 1.83731i −0.00188974 + 0.0628345i
\(856\) 0 0
\(857\) −17.7615 + 17.7615i −0.606720 + 0.606720i −0.942087 0.335367i \(-0.891140\pi\)
0.335367 + 0.942087i \(0.391140\pi\)
\(858\) 0 0
\(859\) 0.529028i 0.0180502i 0.999959 + 0.00902510i \(0.00287282\pi\)
−0.999959 + 0.00902510i \(0.997127\pi\)
\(860\) 0 0
\(861\) −11.6384 −0.396637
\(862\) 0 0
\(863\) 16.5919 0.564795 0.282397 0.959298i \(-0.408870\pi\)
0.282397 + 0.959298i \(0.408870\pi\)
\(864\) 0 0
\(865\) 1.15403 38.3718i 0.0392381 1.30468i
\(866\) 0 0
\(867\) −9.89001 9.89001i −0.335882 0.335882i
\(868\) 0 0
\(869\) 1.59018 + 1.59018i 0.0539430 + 0.0539430i
\(870\) 0 0
\(871\) 1.74853 + 4.24353i 0.0592466 + 0.143787i
\(872\) 0 0
\(873\) 6.15970 0.208474
\(874\) 0 0
\(875\) −18.3275 + 15.2873i −0.619582 + 0.516804i
\(876\) 0 0
\(877\) 31.7095i 1.07075i −0.844613 0.535377i \(-0.820170\pi\)
0.844613 0.535377i \(-0.179830\pi\)
\(878\) 0 0
\(879\) −11.8880 11.8880i −0.400973 0.400973i
\(880\) 0 0
\(881\) 45.8764i 1.54561i −0.634641 0.772807i \(-0.718852\pi\)
0.634641 0.772807i \(-0.281148\pi\)
\(882\) 0 0
\(883\) 7.59797 + 7.59797i 0.255692 + 0.255692i 0.823299 0.567607i \(-0.192131\pi\)
−0.567607 + 0.823299i \(0.692131\pi\)
\(884\) 0 0
\(885\) 0.941292 + 0.999666i 0.0316412 + 0.0336034i
\(886\) 0 0
\(887\) −2.01004 2.01004i −0.0674905 0.0674905i 0.672556 0.740046i \(-0.265196\pi\)
−0.740046 + 0.672556i \(0.765196\pi\)
\(888\) 0 0
\(889\) 11.2323 11.2323i 0.376718 0.376718i
\(890\) 0 0
\(891\) −1.89894 + 1.89894i −0.0636170 + 0.0636170i
\(892\) 0 0
\(893\) 1.59914 1.59914i 0.0535132 0.0535132i
\(894\) 0 0
\(895\) −5.95099 6.32004i −0.198920 0.211256i
\(896\) 0 0
\(897\) 19.3724 7.98232i 0.646826 0.266522i
\(898\) 0 0
\(899\) −8.62219 + 8.62219i −0.287566 + 0.287566i
\(900\) 0 0
\(901\) 19.1335i 0.637428i
\(902\) 0 0
\(903\) 11.4310 0.380401
\(904\) 0 0
\(905\) −25.8940 + 24.3820i −0.860746 + 0.810484i
\(906\) 0 0
\(907\) −24.3483 + 24.3483i −0.808472 + 0.808472i −0.984403 0.175931i \(-0.943707\pi\)
0.175931 + 0.984403i \(0.443707\pi\)
\(908\) 0 0
\(909\) 3.93730 0.130592
\(910\) 0 0
\(911\) −41.9116 −1.38859 −0.694297 0.719689i \(-0.744284\pi\)
−0.694297 + 0.719689i \(0.744284\pi\)
\(912\) 0 0
\(913\) −9.91920 + 9.91920i −0.328278 + 0.328278i
\(914\) 0 0
\(915\) −29.3739 0.883414i −0.971070 0.0292048i
\(916\) 0 0
\(917\) −45.0912 −1.48904
\(918\) 0 0
\(919\) 31.6303i 1.04339i −0.853133 0.521693i \(-0.825301\pi\)
0.853133 0.521693i \(-0.174699\pi\)
\(920\) 0 0
\(921\) −10.3273 + 10.3273i −0.340296 + 0.340296i
\(922\) 0 0
\(923\) −7.82299 3.25737i −0.257497 0.107218i
\(924\) 0 0
\(925\) −1.80575 + 29.9937i −0.0593726 + 0.986188i
\(926\) 0 0
\(927\) −3.97517 + 3.97517i −0.130562 + 0.130562i
\(928\) 0 0
\(929\) −19.6715 + 19.6715i −0.645400 + 0.645400i −0.951878 0.306477i \(-0.900850\pi\)
0.306477 + 0.951878i \(0.400850\pi\)
\(930\) 0 0
\(931\) 1.42017 1.42017i 0.0465442 0.0465442i
\(932\) 0 0
\(933\) −19.4391 19.4391i −0.636407 0.636407i
\(934\) 0 0
\(935\) 7.58924 7.14608i 0.248195 0.233702i
\(936\) 0 0
\(937\) 9.54342 + 9.54342i 0.311770 + 0.311770i 0.845595 0.533825i \(-0.179246\pi\)
−0.533825 + 0.845595i \(0.679246\pi\)
\(938\) 0 0
\(939\) 6.82759i 0.222810i
\(940\) 0 0
\(941\) 10.4194 + 10.4194i 0.339662 + 0.339662i 0.856240 0.516578i \(-0.172794\pi\)
−0.516578 + 0.856240i \(0.672794\pi\)
\(942\) 0 0
\(943\) 31.6833i 1.03175i
\(944\) 0 0
\(945\) −0.143490 + 4.77109i −0.00466772 + 0.155204i
\(946\) 0 0
\(947\) −3.83373 −0.124580 −0.0622898 0.998058i \(-0.519840\pi\)
−0.0622898 + 0.998058i \(0.519840\pi\)
\(948\) 0 0
\(949\) 0.875018 + 2.12360i 0.0284043 + 0.0689349i
\(950\) 0 0
\(951\) 19.3746 + 19.3746i 0.628265 + 0.628265i
\(952\) 0 0
\(953\) 27.0921 + 27.0921i 0.877598 + 0.877598i 0.993286 0.115687i \(-0.0369071\pi\)
−0.115687 + 0.993286i \(0.536907\pi\)
\(954\) 0 0
\(955\) −6.07794 6.45486i −0.196677 0.208874i
\(956\) 0 0
\(957\) −4.87990 −0.157745
\(958\) 0 0
\(959\) −28.0073 −0.904403
\(960\) 0 0
\(961\) 14.0296i 0.452566i
\(962\) 0 0
\(963\) −11.2103 + 11.2103i −0.361248 + 0.361248i
\(964\) 0 0
\(965\) 27.5346 25.9268i 0.886370 0.834612i
\(966\) 0 0
\(967\) 60.6472i 1.95028i 0.221586 + 0.975141i \(0.428877\pi\)
−0.221586 + 0.975141i \(0.571123\pi\)
\(968\) 0 0
\(969\) −1.00904 1.00904i −0.0324149 0.0324149i
\(970\) 0 0
\(971\) −21.1997 −0.680331 −0.340165 0.940366i \(-0.610483\pi\)
−0.340165 + 0.940366i \(0.610483\pi\)
\(972\) 0 0
\(973\) 41.2397i 1.32208i
\(974\) 0 0
\(975\) −16.1961 7.91738i −0.518692 0.253559i
\(976\) 0 0
\(977\) 24.6849i 0.789739i 0.918737 + 0.394869i \(0.129210\pi\)
−0.918737 + 0.394869i \(0.870790\pi\)
\(978\) 0 0
\(979\) 14.9138 0.476648
\(980\) 0 0
\(981\) −8.33542 8.33542i −0.266129 0.266129i
\(982\) 0 0
\(983\) 35.5810i 1.13486i 0.823423 + 0.567428i \(0.192062\pi\)
−0.823423 + 0.567428i \(0.807938\pi\)
\(984\) 0 0
\(985\) 37.7654 35.5601i 1.20330 1.13304i
\(986\) 0 0
\(987\) 4.15263 4.15263i 0.132180 0.132180i
\(988\) 0 0
\(989\) 31.1187i 0.989516i
\(990\) 0 0
\(991\) −5.07037 −0.161065 −0.0805327 0.996752i \(-0.525662\pi\)
−0.0805327 + 0.996752i \(0.525662\pi\)
\(992\) 0 0
\(993\) 8.03535 0.254994
\(994\) 0 0
\(995\) 37.2205 + 39.5287i 1.17997 + 1.25315i
\(996\) 0 0
\(997\) 21.7495 + 21.7495i 0.688814 + 0.688814i 0.961970 0.273156i \(-0.0880674\pi\)
−0.273156 + 0.961970i \(0.588067\pi\)
\(998\) 0 0
\(999\) 4.24944 + 4.24944i 0.134446 + 0.134446i
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 780.2.bm.a.697.6 yes 28
3.2 odd 2 2340.2.bp.i.1477.3 28
5.2 odd 4 3900.2.r.b.3193.12 28
5.3 odd 4 780.2.r.a.73.6 28
5.4 even 2 3900.2.bm.b.2257.12 28
13.5 odd 4 780.2.r.a.577.6 yes 28
15.8 even 4 2340.2.u.i.73.5 28
39.5 even 4 2340.2.u.i.577.5 28
65.18 even 4 inner 780.2.bm.a.733.6 yes 28
65.44 odd 4 3900.2.r.b.1357.12 28
65.57 even 4 3900.2.bm.b.2293.12 28
195.83 odd 4 2340.2.bp.i.1513.3 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
780.2.r.a.73.6 28 5.3 odd 4
780.2.r.a.577.6 yes 28 13.5 odd 4
780.2.bm.a.697.6 yes 28 1.1 even 1 trivial
780.2.bm.a.733.6 yes 28 65.18 even 4 inner
2340.2.u.i.73.5 28 15.8 even 4
2340.2.u.i.577.5 28 39.5 even 4
2340.2.bp.i.1477.3 28 3.2 odd 2
2340.2.bp.i.1513.3 28 195.83 odd 4
3900.2.r.b.1357.12 28 65.44 odd 4
3900.2.r.b.3193.12 28 5.2 odd 4
3900.2.bm.b.2257.12 28 5.4 even 2
3900.2.bm.b.2293.12 28 65.57 even 4