Properties

Label 1764.2.x.b.293.4
Level $1764$
Weight $2$
Character 1764.293
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(293,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.x (of order \(6\), 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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 293.4
Root \(1.08696 - 1.34852i\) of defining polynomial
Character \(\chi\) \(=\) 1764.293
Dual form 1764.2.x.b.1469.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.317569 + 1.70269i) q^{3} +(-0.0382122 + 0.0661855i) q^{5} +(-2.79830 - 1.08144i) q^{9} +O(q^{10})\) \(q+(-0.317569 + 1.70269i) q^{3} +(-0.0382122 + 0.0661855i) q^{5} +(-2.79830 - 1.08144i) q^{9} +(-4.66300 + 2.69219i) q^{11} +(-4.60313 - 2.65762i) q^{13} +(-0.100558 - 0.0860820i) q^{15} +3.78183 q^{17} -5.01070i q^{19} +(2.02463 + 1.16892i) q^{23} +(2.49708 + 4.32507i) q^{25} +(2.73001 - 4.42120i) q^{27} +(8.84430 - 5.10626i) q^{29} +(-4.97636 - 2.87310i) q^{31} +(-3.10313 - 8.79460i) q^{33} -0.708972 q^{37} +(5.98691 - 6.99372i) q^{39} +(3.29910 - 5.71422i) q^{41} +(0.716520 + 1.24105i) q^{43} +(0.178505 - 0.143883i) q^{45} +(-1.46192 - 2.53213i) q^{47} +(-1.20099 + 6.43928i) q^{51} -12.1053i q^{53} -0.411498i q^{55} +(8.53166 + 1.59124i) q^{57} +(-0.289951 + 0.502210i) q^{59} +(2.40641 - 1.38934i) q^{61} +(0.351792 - 0.203107i) q^{65} +(-2.63593 + 4.56556i) q^{67} +(-2.63327 + 3.07610i) q^{69} +3.32103i q^{71} +7.12826i q^{73} +(-8.15724 + 2.87824i) q^{75} +(-0.469123 - 0.812544i) q^{79} +(6.66096 + 6.05240i) q^{81} +(-6.49790 - 11.2547i) q^{83} +(-0.144512 + 0.250303i) q^{85} +(5.88570 + 16.6807i) q^{87} +3.03588 q^{89} +(6.47234 - 7.56078i) q^{93} +(0.331636 + 0.191470i) q^{95} +(6.18183 - 3.56908i) q^{97} +(15.9599 - 2.49077i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{9} + 6 q^{11} + 3 q^{13} - 3 q^{15} + 18 q^{17} - 21 q^{23} - 8 q^{25} - 9 q^{27} + 6 q^{29} - 6 q^{31} + 27 q^{33} - 2 q^{37} + 6 q^{39} + 6 q^{41} - 2 q^{43} - 15 q^{45} - 18 q^{47} + 18 q^{51} + 15 q^{57} - 15 q^{59} - 3 q^{61} + 39 q^{65} - 7 q^{67} - 21 q^{69} - 42 q^{75} - q^{79} - 18 q^{81} + 6 q^{85} + 51 q^{87} + 42 q^{89} + 48 q^{93} - 6 q^{95} + 3 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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
\(3\) −0.317569 + 1.70269i −0.183349 + 0.983048i
\(4\) 0 0
\(5\) −0.0382122 + 0.0661855i −0.0170890 + 0.0295991i −0.874443 0.485127i \(-0.838773\pi\)
0.857354 + 0.514727i \(0.172107\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.79830 1.08144i −0.932767 0.360481i
\(10\) 0 0
\(11\) −4.66300 + 2.69219i −1.40595 + 0.811725i −0.994994 0.0999316i \(-0.968138\pi\)
−0.410954 + 0.911656i \(0.634804\pi\)
\(12\) 0 0
\(13\) −4.60313 2.65762i −1.27668 0.737091i −0.300442 0.953800i \(-0.597134\pi\)
−0.976236 + 0.216709i \(0.930468\pi\)
\(14\) 0 0
\(15\) −0.100558 0.0860820i −0.0259641 0.0222263i
\(16\) 0 0
\(17\) 3.78183 0.917229 0.458615 0.888635i \(-0.348346\pi\)
0.458615 + 0.888635i \(0.348346\pi\)
\(18\) 0 0
\(19\) 5.01070i 1.14953i −0.818317 0.574767i \(-0.805093\pi\)
0.818317 0.574767i \(-0.194907\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.02463 + 1.16892i 0.422164 + 0.243737i 0.696003 0.718039i \(-0.254960\pi\)
−0.273839 + 0.961776i \(0.588293\pi\)
\(24\) 0 0
\(25\) 2.49708 + 4.32507i 0.499416 + 0.865014i
\(26\) 0 0
\(27\) 2.73001 4.42120i 0.525392 0.850861i
\(28\) 0 0
\(29\) 8.84430 5.10626i 1.64235 0.948209i 0.662349 0.749196i \(-0.269560\pi\)
0.979997 0.199013i \(-0.0637736\pi\)
\(30\) 0 0
\(31\) −4.97636 2.87310i −0.893780 0.516024i −0.0186031 0.999827i \(-0.505922\pi\)
−0.875177 + 0.483803i \(0.839255\pi\)
\(32\) 0 0
\(33\) −3.10313 8.79460i −0.540185 1.53094i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.708972 −0.116554 −0.0582771 0.998300i \(-0.518561\pi\)
−0.0582771 + 0.998300i \(0.518561\pi\)
\(38\) 0 0
\(39\) 5.98691 6.99372i 0.958673 1.11989i
\(40\) 0 0
\(41\) 3.29910 5.71422i 0.515234 0.892411i −0.484610 0.874730i \(-0.661039\pi\)
0.999844 0.0176805i \(-0.00562816\pi\)
\(42\) 0 0
\(43\) 0.716520 + 1.24105i 0.109268 + 0.189258i 0.915474 0.402377i \(-0.131816\pi\)
−0.806206 + 0.591635i \(0.798483\pi\)
\(44\) 0 0
\(45\) 0.178505 0.143883i 0.0266100 0.0214488i
\(46\) 0 0
\(47\) −1.46192 2.53213i −0.213244 0.369349i 0.739484 0.673174i \(-0.235069\pi\)
−0.952728 + 0.303825i \(0.901736\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1.20099 + 6.43928i −0.168173 + 0.901680i
\(52\) 0 0
\(53\) 12.1053i 1.66279i −0.555683 0.831394i \(-0.687543\pi\)
0.555683 0.831394i \(-0.312457\pi\)
\(54\) 0 0
\(55\) 0.411498i 0.0554863i
\(56\) 0 0
\(57\) 8.53166 + 1.59124i 1.13005 + 0.210765i
\(58\) 0 0
\(59\) −0.289951 + 0.502210i −0.0377484 + 0.0653822i −0.884282 0.466953i \(-0.845352\pi\)
0.846534 + 0.532335i \(0.178685\pi\)
\(60\) 0 0
\(61\) 2.40641 1.38934i 0.308109 0.177887i −0.337971 0.941156i \(-0.609741\pi\)
0.646080 + 0.763270i \(0.276407\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.351792 0.203107i 0.0436344 0.0251923i
\(66\) 0 0
\(67\) −2.63593 + 4.56556i −0.322030 + 0.557771i −0.980907 0.194479i \(-0.937698\pi\)
0.658877 + 0.752251i \(0.271032\pi\)
\(68\) 0 0
\(69\) −2.63327 + 3.07610i −0.317008 + 0.370319i
\(70\) 0 0
\(71\) 3.32103i 0.394134i 0.980390 + 0.197067i \(0.0631416\pi\)
−0.980390 + 0.197067i \(0.936858\pi\)
\(72\) 0 0
\(73\) 7.12826i 0.834300i 0.908838 + 0.417150i \(0.136971\pi\)
−0.908838 + 0.417150i \(0.863029\pi\)
\(74\) 0 0
\(75\) −8.15724 + 2.87824i −0.941917 + 0.332351i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.469123 0.812544i −0.0527804 0.0914184i 0.838428 0.545012i \(-0.183475\pi\)
−0.891208 + 0.453594i \(0.850142\pi\)
\(80\) 0 0
\(81\) 6.66096 + 6.05240i 0.740107 + 0.672489i
\(82\) 0 0
\(83\) −6.49790 11.2547i −0.713238 1.23536i −0.963635 0.267221i \(-0.913895\pi\)
0.250398 0.968143i \(-0.419439\pi\)
\(84\) 0 0
\(85\) −0.144512 + 0.250303i −0.0156746 + 0.0271491i
\(86\) 0 0
\(87\) 5.88570 + 16.6807i 0.631013 + 1.78836i
\(88\) 0 0
\(89\) 3.03588 0.321802 0.160901 0.986971i \(-0.448560\pi\)
0.160901 + 0.986971i \(0.448560\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.47234 7.56078i 0.671150 0.784016i
\(94\) 0 0
\(95\) 0.331636 + 0.191470i 0.0340251 + 0.0196444i
\(96\) 0 0
\(97\) 6.18183 3.56908i 0.627670 0.362385i −0.152179 0.988353i \(-0.548629\pi\)
0.779849 + 0.625967i \(0.215296\pi\)
\(98\) 0 0
\(99\) 15.9599 2.49077i 1.60403 0.250332i
\(100\) 0 0
\(101\) −4.08628 7.07765i −0.406600 0.704252i 0.587906 0.808929i \(-0.299952\pi\)
−0.994506 + 0.104677i \(0.966619\pi\)
\(102\) 0 0
\(103\) −6.46599 3.73314i −0.637113 0.367837i 0.146389 0.989227i \(-0.453235\pi\)
−0.783502 + 0.621390i \(0.786568\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.61870i 0.446507i 0.974760 + 0.223253i \(0.0716677\pi\)
−0.974760 + 0.223253i \(0.928332\pi\)
\(108\) 0 0
\(109\) −10.4558 −1.00149 −0.500744 0.865595i \(-0.666940\pi\)
−0.500744 + 0.865595i \(0.666940\pi\)
\(110\) 0 0
\(111\) 0.225148 1.20716i 0.0213701 0.114578i
\(112\) 0 0
\(113\) −16.6379 9.60591i −1.56516 0.903648i −0.996720 0.0809270i \(-0.974212\pi\)
−0.568445 0.822721i \(-0.692455\pi\)
\(114\) 0 0
\(115\) −0.154731 + 0.0893340i −0.0144287 + 0.00833044i
\(116\) 0 0
\(117\) 10.0069 + 12.4148i 0.925136 + 1.14775i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8.99573 15.5811i 0.817793 1.41646i
\(122\) 0 0
\(123\) 8.68184 + 7.43201i 0.782815 + 0.670122i
\(124\) 0 0
\(125\) −0.763798 −0.0683162
\(126\) 0 0
\(127\) 1.26488 0.112240 0.0561198 0.998424i \(-0.482127\pi\)
0.0561198 + 0.998424i \(0.482127\pi\)
\(128\) 0 0
\(129\) −2.34067 + 0.825892i −0.206084 + 0.0727157i
\(130\) 0 0
\(131\) 7.24394 12.5469i 0.632906 1.09623i −0.354049 0.935227i \(-0.615195\pi\)
0.986955 0.160998i \(-0.0514714\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.188300 + 0.349631i 0.0162062 + 0.0300915i
\(136\) 0 0
\(137\) 13.3414 7.70264i 1.13983 0.658081i 0.193442 0.981112i \(-0.438035\pi\)
0.946389 + 0.323030i \(0.104702\pi\)
\(138\) 0 0
\(139\) −0.374701 0.216333i −0.0317817 0.0183492i 0.484025 0.875054i \(-0.339174\pi\)
−0.515807 + 0.856705i \(0.672508\pi\)
\(140\) 0 0
\(141\) 4.77569 1.68508i 0.402185 0.141909i
\(142\) 0 0
\(143\) 28.6192 2.39326
\(144\) 0 0
\(145\) 0.780486i 0.0648159i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.04535 + 2.33558i 0.331408 + 0.191338i 0.656466 0.754356i \(-0.272051\pi\)
−0.325058 + 0.945694i \(0.605384\pi\)
\(150\) 0 0
\(151\) 4.12276 + 7.14083i 0.335506 + 0.581113i 0.983582 0.180463i \(-0.0577595\pi\)
−0.648076 + 0.761575i \(0.724426\pi\)
\(152\) 0 0
\(153\) −10.5827 4.08984i −0.855561 0.330644i
\(154\) 0 0
\(155\) 0.380316 0.219575i 0.0305477 0.0176367i
\(156\) 0 0
\(157\) −15.2334 8.79500i −1.21576 0.701917i −0.251749 0.967793i \(-0.581006\pi\)
−0.964007 + 0.265875i \(0.914339\pi\)
\(158\) 0 0
\(159\) 20.6115 + 3.84426i 1.63460 + 0.304870i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.5419 0.825709 0.412854 0.910797i \(-0.364532\pi\)
0.412854 + 0.910797i \(0.364532\pi\)
\(164\) 0 0
\(165\) 0.700653 + 0.130679i 0.0545457 + 0.0101733i
\(166\) 0 0
\(167\) −4.59146 + 7.95265i −0.355298 + 0.615395i −0.987169 0.159679i \(-0.948954\pi\)
0.631871 + 0.775074i \(0.282287\pi\)
\(168\) 0 0
\(169\) 7.62587 + 13.2084i 0.586605 + 1.01603i
\(170\) 0 0
\(171\) −5.41879 + 14.0214i −0.414385 + 1.07225i
\(172\) 0 0
\(173\) −1.22358 2.11931i −0.0930274 0.161128i 0.815756 0.578396i \(-0.196321\pi\)
−0.908784 + 0.417268i \(0.862988\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −0.763028 0.653183i −0.0573527 0.0490963i
\(178\) 0 0
\(179\) 5.83712i 0.436287i −0.975917 0.218143i \(-0.930000\pi\)
0.975917 0.218143i \(-0.0700001\pi\)
\(180\) 0 0
\(181\) 16.0704i 1.19451i −0.802053 0.597253i \(-0.796259\pi\)
0.802053 0.597253i \(-0.203741\pi\)
\(182\) 0 0
\(183\) 1.60141 + 4.53857i 0.118380 + 0.335501i
\(184\) 0 0
\(185\) 0.0270914 0.0469237i 0.00199180 0.00344990i
\(186\) 0 0
\(187\) −17.6347 + 10.1814i −1.28958 + 0.744537i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.90415 3.98611i 0.499567 0.288425i −0.228968 0.973434i \(-0.573535\pi\)
0.728535 + 0.685009i \(0.240202\pi\)
\(192\) 0 0
\(193\) −0.359027 + 0.621853i −0.0258433 + 0.0447620i −0.878658 0.477452i \(-0.841560\pi\)
0.852814 + 0.522214i \(0.174894\pi\)
\(194\) 0 0
\(195\) 0.234110 + 0.663492i 0.0167650 + 0.0475137i
\(196\) 0 0
\(197\) 13.5035i 0.962083i −0.876698 0.481042i \(-0.840259\pi\)
0.876698 0.481042i \(-0.159741\pi\)
\(198\) 0 0
\(199\) 24.5452i 1.73997i −0.493082 0.869983i \(-0.664130\pi\)
0.493082 0.869983i \(-0.335870\pi\)
\(200\) 0 0
\(201\) −6.93663 5.93804i −0.489272 0.418837i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.252132 + 0.436706i 0.0176097 + 0.0305009i
\(206\) 0 0
\(207\) −4.40139 5.46051i −0.305918 0.379531i
\(208\) 0 0
\(209\) 13.4897 + 23.3649i 0.933105 + 1.61618i
\(210\) 0 0
\(211\) −11.7838 + 20.4101i −0.811227 + 1.40509i 0.100778 + 0.994909i \(0.467867\pi\)
−0.912005 + 0.410178i \(0.865467\pi\)
\(212\) 0 0
\(213\) −5.65468 1.05466i −0.387453 0.0722639i
\(214\) 0 0
\(215\) −0.109519 −0.00746916
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.1372 2.26372i −0.820157 0.152968i
\(220\) 0 0
\(221\) −17.4083 10.0507i −1.17101 0.676081i
\(222\) 0 0
\(223\) −6.47489 + 3.73828i −0.433590 + 0.250334i −0.700875 0.713284i \(-0.747207\pi\)
0.267285 + 0.963618i \(0.413874\pi\)
\(224\) 0 0
\(225\) −2.31026 14.8033i −0.154017 0.986886i
\(226\) 0 0
\(227\) −0.318701 0.552006i −0.0211529 0.0366379i 0.855255 0.518207i \(-0.173400\pi\)
−0.876408 + 0.481569i \(0.840067\pi\)
\(228\) 0 0
\(229\) −1.58351 0.914239i −0.104641 0.0604146i 0.446766 0.894651i \(-0.352576\pi\)
−0.551407 + 0.834236i \(0.685909\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.1186i 1.31801i 0.752136 + 0.659007i \(0.229023\pi\)
−0.752136 + 0.659007i \(0.770977\pi\)
\(234\) 0 0
\(235\) 0.223454 0.0145765
\(236\) 0 0
\(237\) 1.53249 0.540731i 0.0995458 0.0351242i
\(238\) 0 0
\(239\) −2.41455 1.39404i −0.156184 0.0901730i 0.419871 0.907584i \(-0.362075\pi\)
−0.576055 + 0.817411i \(0.695409\pi\)
\(240\) 0 0
\(241\) −20.0304 + 11.5645i −1.29027 + 0.744938i −0.978702 0.205286i \(-0.934187\pi\)
−0.311568 + 0.950224i \(0.600854\pi\)
\(242\) 0 0
\(243\) −12.4207 + 9.41949i −0.796787 + 0.604261i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −13.3165 + 23.0649i −0.847310 + 1.46758i
\(248\) 0 0
\(249\) 21.2268 7.48976i 1.34519 0.474645i
\(250\) 0 0
\(251\) 18.6541 1.17743 0.588717 0.808339i \(-0.299633\pi\)
0.588717 + 0.808339i \(0.299633\pi\)
\(252\) 0 0
\(253\) −12.5878 −0.791388
\(254\) 0 0
\(255\) −0.380295 0.325548i −0.0238150 0.0203866i
\(256\) 0 0
\(257\) −5.43687 + 9.41694i −0.339143 + 0.587413i −0.984272 0.176661i \(-0.943470\pi\)
0.645129 + 0.764074i \(0.276804\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −30.2711 + 4.72424i −1.87374 + 0.292423i
\(262\) 0 0
\(263\) −16.4519 + 9.49852i −1.01447 + 0.585704i −0.912497 0.409083i \(-0.865849\pi\)
−0.101972 + 0.994787i \(0.532515\pi\)
\(264\) 0 0
\(265\) 0.801194 + 0.462570i 0.0492170 + 0.0284154i
\(266\) 0 0
\(267\) −0.964101 + 5.16915i −0.0590020 + 0.316347i
\(268\) 0 0
\(269\) 8.59576 0.524093 0.262046 0.965055i \(-0.415603\pi\)
0.262046 + 0.965055i \(0.415603\pi\)
\(270\) 0 0
\(271\) 1.83258i 0.111322i 0.998450 + 0.0556608i \(0.0177265\pi\)
−0.998450 + 0.0556608i \(0.982273\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −23.2878 13.4452i −1.40431 0.810776i
\(276\) 0 0
\(277\) −7.90931 13.6993i −0.475224 0.823113i 0.524373 0.851489i \(-0.324300\pi\)
−0.999597 + 0.0283760i \(0.990966\pi\)
\(278\) 0 0
\(279\) 10.8182 + 13.4214i 0.647671 + 0.803521i
\(280\) 0 0
\(281\) −9.95916 + 5.74992i −0.594114 + 0.343012i −0.766722 0.641979i \(-0.778114\pi\)
0.172609 + 0.984990i \(0.444780\pi\)
\(282\) 0 0
\(283\) 8.59806 + 4.96409i 0.511101 + 0.295085i 0.733286 0.679920i \(-0.237986\pi\)
−0.222185 + 0.975005i \(0.571319\pi\)
\(284\) 0 0
\(285\) −0.431331 + 0.503868i −0.0255499 + 0.0298466i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.69774 −0.158691
\(290\) 0 0
\(291\) 4.11388 + 11.6592i 0.241160 + 0.683473i
\(292\) 0 0
\(293\) 8.63598 14.9580i 0.504520 0.873854i −0.495467 0.868627i \(-0.665003\pi\)
0.999986 0.00522664i \(-0.00166370\pi\)
\(294\) 0 0
\(295\) −0.0221594 0.0383812i −0.00129017 0.00223464i
\(296\) 0 0
\(297\) −0.827372 + 27.9658i −0.0480090 + 1.62274i
\(298\) 0 0
\(299\) −6.21308 10.7614i −0.359312 0.622346i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 13.3487 4.71003i 0.766863 0.270584i
\(304\) 0 0
\(305\) 0.212359i 0.0121596i
\(306\) 0 0
\(307\) 21.6425i 1.23520i 0.786490 + 0.617602i \(0.211896\pi\)
−0.786490 + 0.617602i \(0.788104\pi\)
\(308\) 0 0
\(309\) 8.40978 9.82404i 0.478416 0.558870i
\(310\) 0 0
\(311\) 10.1016 17.4964i 0.572808 0.992133i −0.423468 0.905911i \(-0.639187\pi\)
0.996276 0.0862215i \(-0.0274793\pi\)
\(312\) 0 0
\(313\) 18.9146 10.9203i 1.06911 0.617254i 0.141175 0.989985i \(-0.454912\pi\)
0.927939 + 0.372731i \(0.121579\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.5288 12.4297i 1.20918 0.698120i 0.246599 0.969117i \(-0.420687\pi\)
0.962580 + 0.270997i \(0.0873535\pi\)
\(318\) 0 0
\(319\) −27.4940 + 47.6210i −1.53937 + 2.66626i
\(320\) 0 0
\(321\) −7.86421 1.46676i −0.438938 0.0818664i
\(322\) 0 0
\(323\) 18.9496i 1.05439i
\(324\) 0 0
\(325\) 26.5451i 1.47246i
\(326\) 0 0
\(327\) 3.32045 17.8030i 0.183621 0.984511i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.07219 13.9814i −0.443688 0.768490i 0.554272 0.832336i \(-0.312997\pi\)
−0.997960 + 0.0638459i \(0.979663\pi\)
\(332\) 0 0
\(333\) 1.98392 + 0.766713i 0.108718 + 0.0420156i
\(334\) 0 0
\(335\) −0.201449 0.348920i −0.0110063 0.0190635i
\(336\) 0 0
\(337\) −7.81522 + 13.5364i −0.425722 + 0.737372i −0.996488 0.0837408i \(-0.973313\pi\)
0.570765 + 0.821113i \(0.306647\pi\)
\(338\) 0 0
\(339\) 21.6396 25.2787i 1.17530 1.37295i
\(340\) 0 0
\(341\) 30.9397 1.67548
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −0.102970 0.291829i −0.00554373 0.0157115i
\(346\) 0 0
\(347\) −28.0445 16.1915i −1.50551 0.869206i −0.999980 0.00639573i \(-0.997964\pi\)
−0.505529 0.862810i \(-0.668703\pi\)
\(348\) 0 0
\(349\) −26.0421 + 15.0354i −1.39400 + 0.804827i −0.993755 0.111581i \(-0.964409\pi\)
−0.400246 + 0.916408i \(0.631075\pi\)
\(350\) 0 0
\(351\) −24.3165 + 13.0960i −1.29792 + 0.699014i
\(352\) 0 0
\(353\) −8.50607 14.7329i −0.452733 0.784156i 0.545822 0.837901i \(-0.316217\pi\)
−0.998555 + 0.0537453i \(0.982884\pi\)
\(354\) 0 0
\(355\) −0.219804 0.126904i −0.0116660 0.00673537i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.1783i 1.53997i 0.638060 + 0.769987i \(0.279737\pi\)
−0.638060 + 0.769987i \(0.720263\pi\)
\(360\) 0 0
\(361\) −6.10712 −0.321427
\(362\) 0 0
\(363\) 23.6729 + 20.2650i 1.24251 + 1.06364i
\(364\) 0 0
\(365\) −0.471788 0.272387i −0.0246945 0.0142574i
\(366\) 0 0
\(367\) −15.6981 + 9.06329i −0.819433 + 0.473100i −0.850221 0.526426i \(-0.823532\pi\)
0.0307880 + 0.999526i \(0.490198\pi\)
\(368\) 0 0
\(369\) −15.4115 + 12.4223i −0.802290 + 0.646679i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.1823 17.6362i 0.527219 0.913170i −0.472278 0.881450i \(-0.656568\pi\)
0.999497 0.0317200i \(-0.0100985\pi\)
\(374\) 0 0
\(375\) 0.242559 1.30051i 0.0125257 0.0671581i
\(376\) 0 0
\(377\) −54.2820 −2.79566
\(378\) 0 0
\(379\) −21.9961 −1.12986 −0.564931 0.825138i \(-0.691097\pi\)
−0.564931 + 0.825138i \(0.691097\pi\)
\(380\) 0 0
\(381\) −0.401686 + 2.15369i −0.0205790 + 0.110337i
\(382\) 0 0
\(383\) −16.3127 + 28.2544i −0.833538 + 1.44373i 0.0616774 + 0.998096i \(0.480355\pi\)
−0.895215 + 0.445634i \(0.852978\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.662914 4.24770i −0.0336978 0.215923i
\(388\) 0 0
\(389\) 13.6400 7.87504i 0.691574 0.399280i −0.112628 0.993637i \(-0.535927\pi\)
0.804201 + 0.594357i \(0.202593\pi\)
\(390\) 0 0
\(391\) 7.65680 + 4.42066i 0.387221 + 0.223562i
\(392\) 0 0
\(393\) 19.0630 + 16.3187i 0.961600 + 0.823168i
\(394\) 0 0
\(395\) 0.0717049 0.00360786
\(396\) 0 0
\(397\) 3.41635i 0.171462i −0.996318 0.0857308i \(-0.972678\pi\)
0.996318 0.0857308i \(-0.0273225\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.851348 + 0.491526i 0.0425143 + 0.0245456i 0.521106 0.853492i \(-0.325519\pi\)
−0.478592 + 0.878037i \(0.658853\pi\)
\(402\) 0 0
\(403\) 15.2712 + 26.4505i 0.760713 + 1.31759i
\(404\) 0 0
\(405\) −0.655112 + 0.209583i −0.0325528 + 0.0104143i
\(406\) 0 0
\(407\) 3.30594 1.90868i 0.163869 0.0946099i
\(408\) 0 0
\(409\) −25.0195 14.4450i −1.23714 0.714260i −0.268627 0.963244i \(-0.586570\pi\)
−0.968508 + 0.248984i \(0.919903\pi\)
\(410\) 0 0
\(411\) 8.87840 + 25.1623i 0.437939 + 1.24117i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0.993198 0.0487542
\(416\) 0 0
\(417\) 0.487342 0.569298i 0.0238652 0.0278786i
\(418\) 0 0
\(419\) 6.28926 10.8933i 0.307251 0.532174i −0.670509 0.741901i \(-0.733924\pi\)
0.977760 + 0.209727i \(0.0672577\pi\)
\(420\) 0 0
\(421\) −13.0232 22.5568i −0.634710 1.09935i −0.986576 0.163300i \(-0.947786\pi\)
0.351866 0.936050i \(-0.385547\pi\)
\(422\) 0 0
\(423\) 1.35255 + 8.66664i 0.0657633 + 0.421386i
\(424\) 0 0
\(425\) 9.44354 + 16.3567i 0.458079 + 0.793416i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −9.08858 + 48.7296i −0.438801 + 2.35269i
\(430\) 0 0
\(431\) 7.25676i 0.349546i −0.984609 0.174773i \(-0.944081\pi\)
0.984609 0.174773i \(-0.0559192\pi\)
\(432\) 0 0
\(433\) 8.29113i 0.398446i 0.979954 + 0.199223i \(0.0638419\pi\)
−0.979954 + 0.199223i \(0.936158\pi\)
\(434\) 0 0
\(435\) −1.32893 0.247858i −0.0637171 0.0118839i
\(436\) 0 0
\(437\) 5.85710 10.1448i 0.280183 0.485292i
\(438\) 0 0
\(439\) 2.83357 1.63596i 0.135239 0.0780802i −0.430854 0.902422i \(-0.641788\pi\)
0.566093 + 0.824341i \(0.308454\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.46737 + 1.42454i −0.117228 + 0.0676817i −0.557468 0.830199i \(-0.688227\pi\)
0.440239 + 0.897880i \(0.354894\pi\)
\(444\) 0 0
\(445\) −0.116008 + 0.200931i −0.00549929 + 0.00952505i
\(446\) 0 0
\(447\) −5.26145 + 6.14626i −0.248858 + 0.290708i
\(448\) 0 0
\(449\) 19.9802i 0.942925i 0.881886 + 0.471463i \(0.156274\pi\)
−0.881886 + 0.471463i \(0.843726\pi\)
\(450\) 0 0
\(451\) 35.5272i 1.67291i
\(452\) 0 0
\(453\) −13.4679 + 4.75207i −0.632776 + 0.223272i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.15008 15.8484i −0.428023 0.741357i 0.568675 0.822563i \(-0.307456\pi\)
−0.996697 + 0.0812053i \(0.974123\pi\)
\(458\) 0 0
\(459\) 10.3245 16.7202i 0.481904 0.780434i
\(460\) 0 0
\(461\) 4.52954 + 7.84539i 0.210962 + 0.365396i 0.952016 0.306049i \(-0.0990071\pi\)
−0.741054 + 0.671445i \(0.765674\pi\)
\(462\) 0 0
\(463\) 10.8227 18.7455i 0.502974 0.871176i −0.497021 0.867739i \(-0.665573\pi\)
0.999994 0.00343694i \(-0.00109401\pi\)
\(464\) 0 0
\(465\) 0.253092 + 0.717289i 0.0117369 + 0.0332635i
\(466\) 0 0
\(467\) 27.5523 1.27497 0.637484 0.770464i \(-0.279975\pi\)
0.637484 + 0.770464i \(0.279975\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 19.8128 23.1447i 0.912926 1.06645i
\(472\) 0 0
\(473\) −6.68227 3.85801i −0.307251 0.177392i
\(474\) 0 0
\(475\) 21.6716 12.5121i 0.994362 0.574095i
\(476\) 0 0
\(477\) −13.0912 + 33.8742i −0.599403 + 1.55099i
\(478\) 0 0
\(479\) −2.47325 4.28380i −0.113006 0.195732i 0.803975 0.594663i \(-0.202715\pi\)
−0.916981 + 0.398931i \(0.869381\pi\)
\(480\) 0 0
\(481\) 3.26349 + 1.88418i 0.148802 + 0.0859110i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.545530i 0.0247713i
\(486\) 0 0
\(487\) 9.57146 0.433724 0.216862 0.976202i \(-0.430418\pi\)
0.216862 + 0.976202i \(0.430418\pi\)
\(488\) 0 0
\(489\) −3.34780 + 17.9496i −0.151393 + 0.811711i
\(490\) 0 0
\(491\) −33.0010 19.0531i −1.48931 0.859855i −0.489387 0.872067i \(-0.662779\pi\)
−0.999925 + 0.0122119i \(0.996113\pi\)
\(492\) 0 0
\(493\) 33.4477 19.3110i 1.50641 0.869725i
\(494\) 0 0
\(495\) −0.445011 + 1.15149i −0.0200018 + 0.0517558i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.4192 21.5107i 0.555960 0.962951i −0.441868 0.897080i \(-0.645684\pi\)
0.997828 0.0658709i \(-0.0209825\pi\)
\(500\) 0 0
\(501\) −12.0828 10.3434i −0.539819 0.462107i
\(502\) 0 0
\(503\) 27.2820 1.21645 0.608223 0.793766i \(-0.291883\pi\)
0.608223 + 0.793766i \(0.291883\pi\)
\(504\) 0 0
\(505\) 0.624584 0.0277936
\(506\) 0 0
\(507\) −24.9115 + 8.78990i −1.10636 + 0.390373i
\(508\) 0 0
\(509\) −20.8860 + 36.1757i −0.925758 + 1.60346i −0.135420 + 0.990788i \(0.543238\pi\)
−0.790338 + 0.612671i \(0.790095\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −22.1533 13.6793i −0.978093 0.603955i
\(514\) 0 0
\(515\) 0.494160 0.285303i 0.0217753 0.0125720i
\(516\) 0 0
\(517\) 13.6339 + 7.87154i 0.599619 + 0.346190i
\(518\) 0 0
\(519\) 3.99710 1.41036i 0.175453 0.0619078i
\(520\) 0 0
\(521\) 4.05257 0.177546 0.0887732 0.996052i \(-0.471705\pi\)
0.0887732 + 0.996052i \(0.471705\pi\)
\(522\) 0 0
\(523\) 30.3027i 1.32505i 0.749042 + 0.662523i \(0.230514\pi\)
−0.749042 + 0.662523i \(0.769486\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18.8198 10.8656i −0.819801 0.473312i
\(528\) 0 0
\(529\) −8.76726 15.1853i −0.381185 0.660232i
\(530\) 0 0
\(531\) 1.35448 1.09177i 0.0587795 0.0473788i
\(532\) 0 0
\(533\) −30.3724 + 17.5355i −1.31558 + 0.759548i
\(534\) 0 0
\(535\) −0.305691 0.176491i −0.0132162 0.00763037i
\(536\) 0 0
\(537\) 9.93880 + 1.85369i 0.428891 + 0.0799926i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −17.6536 −0.758988 −0.379494 0.925194i \(-0.623902\pi\)
−0.379494 + 0.925194i \(0.623902\pi\)
\(542\) 0 0
\(543\) 27.3630 + 5.10348i 1.17426 + 0.219011i
\(544\) 0 0
\(545\) 0.399541 0.692026i 0.0171145 0.0296431i
\(546\) 0 0
\(547\) −2.18319 3.78140i −0.0933466 0.161681i 0.815571 0.578657i \(-0.196423\pi\)
−0.908917 + 0.416976i \(0.863090\pi\)
\(548\) 0 0
\(549\) −8.23634 + 1.28540i −0.351518 + 0.0548594i
\(550\) 0 0
\(551\) −25.5859 44.3161i −1.09000 1.88793i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.0712930 + 0.0610297i 0.00302622 + 0.00259057i
\(556\) 0 0
\(557\) 17.0350i 0.721796i 0.932605 + 0.360898i \(0.117530\pi\)
−0.932605 + 0.360898i \(0.882470\pi\)
\(558\) 0 0
\(559\) 7.61695i 0.322163i
\(560\) 0 0
\(561\) −11.7355 33.2597i −0.495474 1.40423i
\(562\) 0 0
\(563\) −6.45992 + 11.1889i −0.272253 + 0.471556i −0.969438 0.245335i \(-0.921102\pi\)
0.697185 + 0.716891i \(0.254436\pi\)
\(564\) 0 0
\(565\) 1.27155 0.734127i 0.0534943 0.0308850i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.8280 10.8704i 0.789313 0.455710i −0.0504079 0.998729i \(-0.516052\pi\)
0.839720 + 0.543019i \(0.182719\pi\)
\(570\) 0 0
\(571\) 16.8254 29.1425i 0.704122 1.21958i −0.262885 0.964827i \(-0.584674\pi\)
0.967007 0.254748i \(-0.0819925\pi\)
\(572\) 0 0
\(573\) 4.59457 + 13.0215i 0.191941 + 0.543980i
\(574\) 0 0
\(575\) 11.6755i 0.486904i
\(576\) 0 0
\(577\) 14.5028i 0.603760i −0.953346 0.301880i \(-0.902386\pi\)
0.953346 0.301880i \(-0.0976142\pi\)
\(578\) 0 0
\(579\) −0.944806 0.808793i −0.0392648 0.0336123i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 32.5897 + 56.4469i 1.34973 + 2.33779i
\(584\) 0 0
\(585\) −1.20407 + 0.187912i −0.0497821 + 0.00776919i
\(586\) 0 0
\(587\) 15.8417 + 27.4386i 0.653857 + 1.13251i 0.982179 + 0.187948i \(0.0601837\pi\)
−0.328322 + 0.944566i \(0.606483\pi\)
\(588\) 0 0
\(589\) −14.3963 + 24.9350i −0.593187 + 1.02743i
\(590\) 0 0
\(591\) 22.9922 + 4.28829i 0.945774 + 0.176397i
\(592\) 0 0
\(593\) 7.08201 0.290823 0.145412 0.989371i \(-0.453549\pi\)
0.145412 + 0.989371i \(0.453549\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 41.7929 + 7.79481i 1.71047 + 0.319020i
\(598\) 0 0
\(599\) 5.20178 + 3.00325i 0.212539 + 0.122709i 0.602491 0.798126i \(-0.294175\pi\)
−0.389952 + 0.920835i \(0.627508\pi\)
\(600\) 0 0
\(601\) −0.530083 + 0.306043i −0.0216225 + 0.0124838i −0.510772 0.859716i \(-0.670640\pi\)
0.489150 + 0.872200i \(0.337307\pi\)
\(602\) 0 0
\(603\) 12.3135 9.92519i 0.501444 0.404185i
\(604\) 0 0
\(605\) 0.687494 + 1.19077i 0.0279506 + 0.0484119i
\(606\) 0 0
\(607\) 1.77500 + 1.02480i 0.0720450 + 0.0415952i 0.535590 0.844478i \(-0.320089\pi\)
−0.463545 + 0.886073i \(0.653423\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15.5409i 0.628719i
\(612\) 0 0
\(613\) 9.86332 0.398376 0.199188 0.979961i \(-0.436170\pi\)
0.199188 + 0.979961i \(0.436170\pi\)
\(614\) 0 0
\(615\) −0.823644 + 0.290619i −0.0332125 + 0.0117189i
\(616\) 0 0
\(617\) 23.2143 + 13.4028i 0.934571 + 0.539575i 0.888254 0.459352i \(-0.151918\pi\)
0.0463170 + 0.998927i \(0.485252\pi\)
\(618\) 0 0
\(619\) 0.0603011 0.0348148i 0.00242370 0.00139933i −0.498788 0.866724i \(-0.666221\pi\)
0.501211 + 0.865325i \(0.332888\pi\)
\(620\) 0 0
\(621\) 10.6953 5.76012i 0.429187 0.231146i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.4562 + 21.5748i −0.498248 + 0.862992i
\(626\) 0 0
\(627\) −44.0671 + 15.5489i −1.75987 + 0.620961i
\(628\) 0 0
\(629\) −2.68121 −0.106907
\(630\) 0 0
\(631\) 11.8214 0.470603 0.235301 0.971922i \(-0.424392\pi\)
0.235301 + 0.971922i \(0.424392\pi\)
\(632\) 0 0
\(633\) −31.0098 26.5457i −1.23253 1.05510i
\(634\) 0 0
\(635\) −0.0483338 + 0.0837165i −0.00191807 + 0.00332219i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.59151 9.29324i 0.142078 0.367635i
\(640\) 0 0
\(641\) 17.7673 10.2580i 0.701766 0.405165i −0.106239 0.994341i \(-0.533881\pi\)
0.808005 + 0.589176i \(0.200547\pi\)
\(642\) 0 0
\(643\) −15.6081 9.01132i −0.615522 0.355372i 0.159602 0.987182i \(-0.448979\pi\)
−0.775123 + 0.631810i \(0.782312\pi\)
\(644\) 0 0
\(645\) 0.0347800 0.186477i 0.00136946 0.00734254i
\(646\) 0 0
\(647\) 18.2365 0.716952 0.358476 0.933539i \(-0.383296\pi\)
0.358476 + 0.933539i \(0.383296\pi\)
\(648\) 0 0
\(649\) 3.12241i 0.122565i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.79559 + 4.50079i 0.305065 + 0.176129i 0.644716 0.764422i \(-0.276976\pi\)
−0.339651 + 0.940552i \(0.610309\pi\)
\(654\) 0 0
\(655\) 0.553614 + 0.958888i 0.0216315 + 0.0374669i
\(656\) 0 0
\(657\) 7.70881 19.9470i 0.300749 0.778207i
\(658\) 0 0
\(659\) 30.4806 17.5980i 1.18735 0.685519i 0.229650 0.973273i \(-0.426242\pi\)
0.957704 + 0.287754i \(0.0929086\pi\)
\(660\) 0 0
\(661\) 10.8797 + 6.28141i 0.423172 + 0.244318i 0.696433 0.717621i \(-0.254769\pi\)
−0.273262 + 0.961940i \(0.588102\pi\)
\(662\) 0 0
\(663\) 22.6415 26.4491i 0.879322 1.02720i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 23.8752 0.924452
\(668\) 0 0
\(669\) −4.30890 12.2119i −0.166592 0.472139i
\(670\) 0 0
\(671\) −7.48072 + 12.9570i −0.288790 + 0.500199i
\(672\) 0 0
\(673\) 23.8913 + 41.3810i 0.920942 + 1.59512i 0.797960 + 0.602710i \(0.205913\pi\)
0.122982 + 0.992409i \(0.460754\pi\)
\(674\) 0 0
\(675\) 25.9391 + 0.767411i 0.998395 + 0.0295377i
\(676\) 0 0
\(677\) −18.5235 32.0837i −0.711918 1.23308i −0.964136 0.265407i \(-0.914494\pi\)
0.252219 0.967670i \(-0.418840\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1.04110 0.367348i 0.0398952 0.0140768i
\(682\) 0 0
\(683\) 25.0390i 0.958092i 0.877790 + 0.479046i \(0.159017\pi\)
−0.877790 + 0.479046i \(0.840983\pi\)
\(684\) 0 0
\(685\) 1.17734i 0.0449839i
\(686\) 0 0
\(687\) 2.05954 2.40589i 0.0785763 0.0917904i
\(688\) 0 0
\(689\) −32.1712 + 55.7222i −1.22563 + 2.12285i
\(690\) 0 0
\(691\) −40.2655 + 23.2473i −1.53177 + 0.884370i −0.532493 + 0.846434i \(0.678745\pi\)
−0.999280 + 0.0379352i \(0.987922\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.0286363 0.0165332i 0.00108624 0.000627139i
\(696\) 0 0
\(697\) 12.4767 21.6102i 0.472587 0.818545i
\(698\) 0 0
\(699\) −34.2558 6.38905i −1.29567 0.241656i
\(700\) 0 0
\(701\) 36.0041i 1.35986i −0.733279 0.679928i \(-0.762011\pi\)
0.733279 0.679928i \(-0.237989\pi\)
\(702\) 0 0
\(703\) 3.55244i 0.133983i
\(704\) 0 0
\(705\) −0.0709620 + 0.380472i −0.00267258 + 0.0143294i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15.9158 27.5670i −0.597731 1.03530i −0.993155 0.116802i \(-0.962736\pi\)
0.395424 0.918499i \(-0.370598\pi\)
\(710\) 0 0
\(711\) 0.434025 + 2.78107i 0.0162772 + 0.104298i
\(712\) 0 0
\(713\) −6.71685 11.6339i −0.251548 0.435694i
\(714\) 0 0
\(715\) −1.09360 + 1.89418i −0.0408985 + 0.0708382i
\(716\) 0 0
\(717\) 3.14041 3.66852i 0.117281 0.137004i
\(718\) 0 0
\(719\) 40.0541 1.49377 0.746883 0.664955i \(-0.231550\pi\)
0.746883 + 0.664955i \(0.231550\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −13.3298 37.7780i −0.495740 1.40498i
\(724\) 0 0
\(725\) 44.1699 + 25.5015i 1.64043 + 0.947101i
\(726\) 0 0
\(727\) −3.39242 + 1.95862i −0.125818 + 0.0726411i −0.561588 0.827417i \(-0.689809\pi\)
0.435770 + 0.900058i \(0.356476\pi\)
\(728\) 0 0
\(729\) −12.0940 24.1399i −0.447927 0.894070i
\(730\) 0 0
\(731\) 2.70976 + 4.69344i 0.100224 + 0.173593i
\(732\) 0 0
\(733\) 20.4239 + 11.7918i 0.754376 + 0.435539i 0.827273 0.561800i \(-0.189891\pi\)
−0.0728971 + 0.997339i \(0.523224\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 28.3856i 1.04560i
\(738\) 0 0
\(739\) −33.7282 −1.24071 −0.620355 0.784321i \(-0.713011\pi\)
−0.620355 + 0.784321i \(0.713011\pi\)
\(740\) 0 0
\(741\) −35.0434 29.9986i −1.28735 1.10203i
\(742\) 0 0
\(743\) 29.4003 + 16.9743i 1.07859 + 0.622725i 0.930516 0.366251i \(-0.119359\pi\)
0.148076 + 0.988976i \(0.452692\pi\)
\(744\) 0 0
\(745\) −0.309164 + 0.178496i −0.0113269 + 0.00653958i
\(746\) 0 0
\(747\) 6.01177 + 38.5211i 0.219959 + 1.40941i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.69831 + 2.94157i −0.0619724 + 0.107339i −0.895347 0.445369i \(-0.853072\pi\)
0.833375 + 0.552709i \(0.186406\pi\)
\(752\) 0 0
\(753\) −5.92395 + 31.7621i −0.215881 + 1.15747i
\(754\) 0 0
\(755\) −0.630160 −0.0229339
\(756\) 0 0
\(757\) 29.1344 1.05891 0.529454 0.848339i \(-0.322397\pi\)
0.529454 + 0.848339i \(0.322397\pi\)
\(758\) 0 0
\(759\) 3.99749 21.4331i 0.145100 0.777972i
\(760\) 0 0
\(761\) −8.36288 + 14.4849i −0.303154 + 0.525079i −0.976849 0.213931i \(-0.931373\pi\)
0.673694 + 0.739010i \(0.264706\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.675077 0.544140i 0.0244074 0.0196734i
\(766\) 0 0
\(767\) 2.66937 1.54116i 0.0963852 0.0556480i
\(768\) 0 0
\(769\) −24.0816 13.9035i −0.868404 0.501373i −0.00158643 0.999999i \(-0.500505\pi\)
−0.866818 + 0.498625i \(0.833838\pi\)
\(770\) 0 0
\(771\) −14.3075 12.2478i −0.515273 0.441095i
\(772\) 0 0
\(773\) 12.8448 0.461994 0.230997 0.972954i \(-0.425801\pi\)
0.230997 + 0.972954i \(0.425801\pi\)
\(774\) 0 0
\(775\) 28.6975i 1.03084i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −28.6322 16.5308i −1.02586 0.592278i
\(780\) 0 0
\(781\) −8.94083 15.4860i −0.319928 0.554132i
\(782\) 0 0
\(783\) 1.56927 53.0426i 0.0560812 1.89559i
\(784\) 0 0
\(785\) 1.16420 0.672153i 0.0415522 0.0239902i
\(786\) 0 0
\(787\) −6.55243 3.78305i −0.233569 0.134851i 0.378648 0.925541i \(-0.376389\pi\)
−0.612217 + 0.790689i \(0.709722\pi\)
\(788\) 0 0
\(789\) −10.9484 31.0289i −0.389774 1.10466i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −14.7693 −0.524474
\(794\) 0 0
\(795\) −1.04205 + 1.21729i −0.0369576 + 0.0431727i
\(796\) 0 0
\(797\) −4.03362 + 6.98643i −0.142878 + 0.247472i −0.928579 0.371134i \(-0.878969\pi\)
0.785701 + 0.618606i \(0.212302\pi\)
\(798\) 0 0
\(799\) −5.52875 9.57608i −0.195593 0.338777i
\(800\) 0 0
\(801\) −8.49529 3.28313i −0.300166 0.116004i
\(802\) 0 0
\(803\) −19.1906 33.2391i −0.677222 1.17298i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −2.72975 + 14.6359i −0.0960917 + 0.515208i
\(808\) 0 0
\(809\) 0.0980908i 0.00344869i 0.999999 + 0.00172435i \(0.000548876\pi\)
−0.999999 + 0.00172435i \(0.999451\pi\)
\(810\) 0 0
\(811\) 30.3085i 1.06428i −0.846658 0.532138i \(-0.821389\pi\)
0.846658 0.532138i \(-0.178611\pi\)
\(812\) 0 0
\(813\) −3.12032 0.581972i −0.109434 0.0204107i
\(814\) 0 0
\(815\) −0.402831 + 0.697724i −0.0141106 + 0.0244402i
\(816\) 0 0
\(817\) 6.21853 3.59027i 0.217559 0.125608i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.5499 11.2871i 0.682295 0.393923i −0.118424 0.992963i \(-0.537784\pi\)
0.800719 + 0.599040i \(0.204451\pi\)
\(822\) 0 0
\(823\) 12.2655 21.2445i 0.427549 0.740536i −0.569106 0.822264i \(-0.692711\pi\)
0.996655 + 0.0817282i \(0.0260439\pi\)
\(824\) 0 0
\(825\) 30.2885 35.3821i 1.05451 1.23185i
\(826\) 0 0
\(827\) 40.3057i 1.40157i 0.713375 + 0.700783i \(0.247166\pi\)
−0.713375 + 0.700783i \(0.752834\pi\)
\(828\) 0 0
\(829\) 54.0493i 1.87721i 0.344993 + 0.938605i \(0.387881\pi\)
−0.344993 + 0.938605i \(0.612119\pi\)
\(830\) 0 0
\(831\) 25.8374 9.11661i 0.896291 0.316252i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −0.350900 0.607777i −0.0121434 0.0210330i
\(836\) 0 0
\(837\) −26.2881 + 14.1579i −0.908649 + 0.489368i
\(838\) 0 0
\(839\) −11.8650 20.5507i −0.409624 0.709489i 0.585224 0.810872i \(-0.301007\pi\)
−0.994847 + 0.101383i \(0.967673\pi\)
\(840\) 0 0
\(841\) 37.6478 65.2079i 1.29820 2.24855i
\(842\) 0 0
\(843\) −6.62761 18.7833i −0.228267 0.646933i
\(844\) 0 0
\(845\) −1.16561 −0.0400980
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −11.1828 + 13.0634i −0.383792 + 0.448334i
\(850\) 0 0
\(851\) −1.43540 0.828731i −0.0492050 0.0284085i
\(852\) 0 0
\(853\) 48.0748 27.7560i 1.64605 0.950347i 0.667429 0.744673i \(-0.267395\pi\)
0.978621 0.205674i \(-0.0659387\pi\)
\(854\) 0 0
\(855\) −0.720953 0.894436i −0.0246561 0.0305891i
\(856\) 0 0
\(857\) −15.3048 26.5088i −0.522803 0.905522i −0.999648 0.0265343i \(-0.991553\pi\)
0.476845 0.878988i \(-0.341780\pi\)
\(858\) 0 0
\(859\) −36.4944 21.0700i −1.24517 0.718900i −0.275030 0.961436i \(-0.588688\pi\)
−0.970143 + 0.242535i \(0.922021\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 26.3020i 0.895330i 0.894201 + 0.447665i \(0.147744\pi\)
−0.894201 + 0.447665i \(0.852256\pi\)
\(864\) 0 0
\(865\) 0.187024 0.00635899
\(866\) 0 0
\(867\) 0.856720 4.59342i 0.0290957 0.156001i
\(868\) 0 0
\(869\) 4.37504 + 2.52593i 0.148413 + 0.0856863i
\(870\) 0 0
\(871\) 24.2670 14.0106i 0.822256 0.474730i
\(872\) 0 0
\(873\) −21.1584 + 3.30206i −0.716103 + 0.111758i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −17.8533 + 30.9228i −0.602863 + 1.04419i 0.389522 + 0.921017i \(0.372640\pi\)
−0.992385 + 0.123172i \(0.960693\pi\)
\(878\) 0 0
\(879\) 22.7262 + 19.4546i 0.766537 + 0.656187i
\(880\) 0 0
\(881\) −12.4482 −0.419392 −0.209696 0.977767i \(-0.567247\pi\)
−0.209696 + 0.977767i \(0.567247\pi\)
\(882\) 0 0
\(883\) 2.35637 0.0792982 0.0396491 0.999214i \(-0.487376\pi\)
0.0396491 + 0.999214i \(0.487376\pi\)
\(884\) 0 0
\(885\) 0.0723883 0.0255418i 0.00243331 0.000858580i
\(886\) 0 0
\(887\) 16.7299 28.9770i 0.561734 0.972952i −0.435611 0.900135i \(-0.643468\pi\)
0.997345 0.0728170i \(-0.0231989\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −47.3543 10.2898i −1.58643 0.344722i
\(892\) 0 0
\(893\) −12.6877 + 7.32526i −0.424579 + 0.245131i
\(894\) 0 0
\(895\) 0.386333 + 0.223049i 0.0129137 + 0.00745572i
\(896\) 0 0
\(897\) 20.2964 7.16147i 0.677676 0.239114i
\(898\) 0 0
\(899\) −58.6832 −1.95719
\(900\) 0 0
\(901\) 45.7801i 1.52516i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.06363 + 0.614087i 0.0353563 + 0.0204130i
\(906\) 0 0
\(907\) 0.467962 + 0.810535i 0.0155384 + 0.0269134i 0.873690 0.486483i \(-0.161720\pi\)
−0.858152 + 0.513396i \(0.828387\pi\)
\(908\) 0 0
\(909\) 3.78057 + 24.2245i 0.125394 + 0.803475i
\(910\) 0 0
\(911\) 28.8739 16.6703i 0.956634 0.552313i 0.0614988 0.998107i \(-0.480412\pi\)
0.895136 + 0.445794i \(0.147079\pi\)
\(912\) 0 0
\(913\) 60.5995 + 34.9871i 2.00555 + 1.15791i
\(914\) 0 0
\(915\) −0.361581 0.0674387i −0.0119535 0.00222945i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 3.46967 0.114454 0.0572270 0.998361i \(-0.481774\pi\)
0.0572270 + 0.998361i \(0.481774\pi\)
\(920\) 0 0
\(921\) −36.8505 6.87300i −1.21427 0.226473i
\(922\) 0 0
\(923\) 8.82603 15.2871i 0.290512 0.503182i
\(924\) 0 0
\(925\) −1.77036 3.06635i −0.0582090 0.100821i
\(926\) 0 0
\(927\) 14.0566 + 17.4390i 0.461679 + 0.572774i
\(928\) 0 0
\(929\) 7.57680 + 13.1234i 0.248587 + 0.430565i 0.963134 0.269022i \(-0.0867005\pi\)
−0.714547 + 0.699587i \(0.753367\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 26.5831 + 22.7562i 0.870290 + 0.745004i
\(934\) 0 0
\(935\) 1.55622i 0.0508937i
\(936\) 0 0
\(937\) 33.6651i 1.09979i 0.835233 + 0.549896i \(0.185333\pi\)
−0.835233 + 0.549896i \(0.814667\pi\)
\(938\) 0 0
\(939\) 12.5872 + 35.6736i 0.410769 + 1.16416i
\(940\) 0 0
\(941\) −18.8980 + 32.7323i −0.616058 + 1.06704i 0.374140 + 0.927372i \(0.377938\pi\)
−0.990198 + 0.139671i \(0.955395\pi\)
\(942\) 0 0
\(943\) 13.3589 7.71277i 0.435026 0.251162i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9.47426 + 5.46997i −0.307872 + 0.177750i −0.645974 0.763360i \(-0.723548\pi\)
0.338102 + 0.941110i \(0.390215\pi\)
\(948\) 0 0
\(949\) 18.9442 32.8123i 0.614955 1.06513i
\(950\) 0 0
\(951\) 14.3270 + 40.6042i 0.464584 + 1.31668i
\(952\) 0 0
\(953\) 11.0914i 0.359284i 0.983732 + 0.179642i \(0.0574939\pi\)
−0.983732 + 0.179642i \(0.942506\pi\)
\(954\) 0 0
\(955\) 0.609273i 0.0197156i
\(956\) 0 0
\(957\) −72.3525 61.9367i −2.33882 2.00213i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00942 + 1.74838i 0.0325621 + 0.0563992i
\(962\) 0 0
\(963\) 4.99486 12.9245i 0.160957 0.416487i
\(964\) 0 0
\(965\) −0.0274384 0.0475248i −0.000883275 0.00152988i
\(966\) 0 0
\(967\) 20.1446 34.8915i 0.647807 1.12203i −0.335839 0.941920i \(-0.609020\pi\)
0.983646 0.180115i \(-0.0576470\pi\)
\(968\) 0 0
\(969\) 32.2653 + 6.01782i 1.03651 + 0.193320i
\(970\) 0 0
\(971\) −47.6916 −1.53050 −0.765248 0.643736i \(-0.777384\pi\)
−0.765248 + 0.643736i \(0.777384\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 45.1981 + 8.42992i 1.44750 + 0.269973i
\(976\) 0 0
\(977\) 14.4540 + 8.34504i 0.462426 + 0.266982i 0.713064 0.701099i \(-0.247307\pi\)
−0.250638 + 0.968081i \(0.580640\pi\)
\(978\) 0 0
\(979\) −14.1563 + 8.17314i −0.452437 + 0.261215i
\(980\) 0 0
\(981\) 29.2586 + 11.3074i 0.934155 + 0.361017i
\(982\) 0 0
\(983\) −16.9255 29.3157i −0.539838 0.935027i −0.998912 0.0466291i \(-0.985152\pi\)
0.459074 0.888398i \(-0.348181\pi\)
\(984\) 0 0
\(985\) 0.893735 + 0.515998i 0.0284768 + 0.0164411i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.35022i 0.106531i
\(990\) 0 0
\(991\) 8.19550 0.260339 0.130169 0.991492i \(-0.458448\pi\)
0.130169 + 0.991492i \(0.458448\pi\)
\(992\) 0 0
\(993\) 26.3695 9.30435i 0.836812 0.295265i
\(994\) 0 0
\(995\) 1.62454 + 0.937928i 0.0515014 + 0.0297343i
\(996\) 0 0
\(997\) −18.7391 + 10.8190i −0.593472 + 0.342641i −0.766469 0.642281i \(-0.777988\pi\)
0.172997 + 0.984922i \(0.444655\pi\)
\(998\) 0 0
\(999\) −1.93550 + 3.13451i −0.0612366 + 0.0991714i
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 1764.2.x.b.293.4 16
3.2 odd 2 5292.2.x.b.881.4 16
7.2 even 3 252.2.bm.a.185.3 yes 16
7.3 odd 6 252.2.w.a.5.1 16
7.4 even 3 1764.2.w.b.509.8 16
7.5 odd 6 1764.2.bm.a.1697.6 16
7.6 odd 2 1764.2.x.a.293.5 16
9.2 odd 6 1764.2.x.a.1469.5 16
9.7 even 3 5292.2.x.a.4409.5 16
21.2 odd 6 756.2.bm.a.17.5 16
21.5 even 6 5292.2.bm.a.2285.4 16
21.11 odd 6 5292.2.w.b.1097.4 16
21.17 even 6 756.2.w.a.341.5 16
21.20 even 2 5292.2.x.a.881.5 16
28.3 even 6 1008.2.ca.d.257.8 16
28.23 odd 6 1008.2.df.d.689.6 16
63.2 odd 6 252.2.w.a.101.1 yes 16
63.11 odd 6 1764.2.bm.a.1685.6 16
63.16 even 3 756.2.w.a.521.5 16
63.20 even 6 inner 1764.2.x.b.1469.4 16
63.23 odd 6 2268.2.t.b.1781.4 16
63.25 even 3 5292.2.bm.a.4625.4 16
63.31 odd 6 2268.2.t.b.2105.4 16
63.34 odd 6 5292.2.x.b.4409.4 16
63.38 even 6 252.2.bm.a.173.3 yes 16
63.47 even 6 1764.2.w.b.1109.8 16
63.52 odd 6 756.2.bm.a.89.5 16
63.58 even 3 2268.2.t.a.1781.5 16
63.59 even 6 2268.2.t.a.2105.5 16
63.61 odd 6 5292.2.w.b.521.4 16
84.23 even 6 3024.2.df.d.17.5 16
84.59 odd 6 3024.2.ca.d.2609.5 16
252.79 odd 6 3024.2.ca.d.2033.5 16
252.115 even 6 3024.2.df.d.1601.5 16
252.191 even 6 1008.2.ca.d.353.8 16
252.227 odd 6 1008.2.df.d.929.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.1 16 7.3 odd 6
252.2.w.a.101.1 yes 16 63.2 odd 6
252.2.bm.a.173.3 yes 16 63.38 even 6
252.2.bm.a.185.3 yes 16 7.2 even 3
756.2.w.a.341.5 16 21.17 even 6
756.2.w.a.521.5 16 63.16 even 3
756.2.bm.a.17.5 16 21.2 odd 6
756.2.bm.a.89.5 16 63.52 odd 6
1008.2.ca.d.257.8 16 28.3 even 6
1008.2.ca.d.353.8 16 252.191 even 6
1008.2.df.d.689.6 16 28.23 odd 6
1008.2.df.d.929.6 16 252.227 odd 6
1764.2.w.b.509.8 16 7.4 even 3
1764.2.w.b.1109.8 16 63.47 even 6
1764.2.x.a.293.5 16 7.6 odd 2
1764.2.x.a.1469.5 16 9.2 odd 6
1764.2.x.b.293.4 16 1.1 even 1 trivial
1764.2.x.b.1469.4 16 63.20 even 6 inner
1764.2.bm.a.1685.6 16 63.11 odd 6
1764.2.bm.a.1697.6 16 7.5 odd 6
2268.2.t.a.1781.5 16 63.58 even 3
2268.2.t.a.2105.5 16 63.59 even 6
2268.2.t.b.1781.4 16 63.23 odd 6
2268.2.t.b.2105.4 16 63.31 odd 6
3024.2.ca.d.2033.5 16 252.79 odd 6
3024.2.ca.d.2609.5 16 84.59 odd 6
3024.2.df.d.17.5 16 84.23 even 6
3024.2.df.d.1601.5 16 252.115 even 6
5292.2.w.b.521.4 16 63.61 odd 6
5292.2.w.b.1097.4 16 21.11 odd 6
5292.2.x.a.881.5 16 21.20 even 2
5292.2.x.a.4409.5 16 9.7 even 3
5292.2.x.b.881.4 16 3.2 odd 2
5292.2.x.b.4409.4 16 63.34 odd 6
5292.2.bm.a.2285.4 16 21.5 even 6
5292.2.bm.a.4625.4 16 63.25 even 3