Properties

Label 600.2.r.f.593.4
Level $600$
Weight $2$
Character 600.593
Analytic conductor $4.791$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,2,Mod(257,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.r (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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.6040479020157644046336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 593.4
Root \(1.47240 + 0.912166i\) of defining polynomial
Character \(\chi\) \(=\) 600.593
Dual form 600.2.r.f.257.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0537601 - 1.73122i) q^{3} +(-0.560232 + 0.560232i) q^{7} +(-2.99422 + 0.186141i) q^{9} +O(q^{10})\) \(q+(-0.0537601 - 1.73122i) q^{3} +(-0.560232 + 0.560232i) q^{7} +(-2.99422 + 0.186141i) q^{9} -0.939764i q^{11} +(-4.13018 - 4.13018i) q^{13} +(-4.50588 - 4.50588i) q^{17} +3.74456i q^{19} +(1.00000 + 0.939764i) q^{21} +(6.18334 - 6.18334i) q^{23} +(0.483219 + 5.17364i) q^{27} -3.16915 q^{29} -5.37228 q^{31} +(-1.62693 + 0.0505218i) q^{33} +(-3.56995 + 3.56995i) q^{37} +(-6.92820 + 7.37228i) q^{39} -9.15759i q^{41} +(4.33875 + 4.33875i) q^{43} +(-5.65685 - 5.65685i) q^{47} +6.37228i q^{49} +(-7.55842 + 8.04290i) q^{51} +(-0.526485 + 0.526485i) q^{53} +(6.48265 - 0.201308i) q^{57} +11.9769 q^{59} +1.37228 q^{61} +(1.57317 - 1.78174i) q^{63} +(1.22474 - 1.22474i) q^{67} +(-11.0371 - 10.3723i) q^{69} +3.16915i q^{71} +(-5.56349 - 5.56349i) q^{73} +(0.526485 + 0.526485i) q^{77} -8.74456i q^{79} +(8.93070 - 1.11469i) q^{81} +(-3.97940 + 3.97940i) q^{83} +(0.170374 + 5.48648i) q^{87} +16.0858 q^{89} +4.62772 q^{91} +(0.288814 + 9.30058i) q^{93} +(5.25065 - 5.25065i) q^{97} +(0.174928 + 2.81386i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{21} - 40 q^{31} - 52 q^{51} - 24 q^{61} + 28 q^{81} + 120 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −0.0537601 1.73122i −0.0310384 0.999518i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.560232 + 0.560232i −0.211748 + 0.211748i −0.805010 0.593262i \(-0.797840\pi\)
0.593262 + 0.805010i \(0.297840\pi\)
\(8\) 0 0
\(9\) −2.99422 + 0.186141i −0.998073 + 0.0620469i
\(10\) 0 0
\(11\) 0.939764i 0.283349i −0.989913 0.141675i \(-0.954751\pi\)
0.989913 0.141675i \(-0.0452487\pi\)
\(12\) 0 0
\(13\) −4.13018 4.13018i −1.14551 1.14551i −0.987426 0.158081i \(-0.949469\pi\)
−0.158081 0.987426i \(-0.550531\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.50588 4.50588i −1.09284 1.09284i −0.995224 0.0976128i \(-0.968879\pi\)
−0.0976128 0.995224i \(-0.531121\pi\)
\(18\) 0 0
\(19\) 3.74456i 0.859062i 0.903052 + 0.429531i \(0.141321\pi\)
−0.903052 + 0.429531i \(0.858679\pi\)
\(20\) 0 0
\(21\) 1.00000 + 0.939764i 0.218218 + 0.205073i
\(22\) 0 0
\(23\) 6.18334 6.18334i 1.28932 1.28932i 0.354113 0.935203i \(-0.384783\pi\)
0.935203 0.354113i \(-0.115217\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.483219 + 5.17364i 0.0929956 + 0.995667i
\(28\) 0 0
\(29\) −3.16915 −0.588496 −0.294248 0.955729i \(-0.595069\pi\)
−0.294248 + 0.955729i \(0.595069\pi\)
\(30\) 0 0
\(31\) −5.37228 −0.964890 −0.482445 0.875926i \(-0.660251\pi\)
−0.482445 + 0.875926i \(0.660251\pi\)
\(32\) 0 0
\(33\) −1.62693 + 0.0505218i −0.283213 + 0.00879471i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.56995 + 3.56995i −0.586897 + 0.586897i −0.936790 0.349893i \(-0.886218\pi\)
0.349893 + 0.936790i \(0.386218\pi\)
\(38\) 0 0
\(39\) −6.92820 + 7.37228i −1.10940 + 1.18051i
\(40\) 0 0
\(41\) 9.15759i 1.43017i −0.699035 0.715087i \(-0.746387\pi\)
0.699035 0.715087i \(-0.253613\pi\)
\(42\) 0 0
\(43\) 4.33875 + 4.33875i 0.661653 + 0.661653i 0.955770 0.294117i \(-0.0950254\pi\)
−0.294117 + 0.955770i \(0.595025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.65685 5.65685i −0.825137 0.825137i 0.161703 0.986840i \(-0.448301\pi\)
−0.986840 + 0.161703i \(0.948301\pi\)
\(48\) 0 0
\(49\) 6.37228i 0.910326i
\(50\) 0 0
\(51\) −7.55842 + 8.04290i −1.05839 + 1.12623i
\(52\) 0 0
\(53\) −0.526485 + 0.526485i −0.0723183 + 0.0723183i −0.742341 0.670022i \(-0.766284\pi\)
0.670022 + 0.742341i \(0.266284\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.48265 0.201308i 0.858648 0.0266639i
\(58\) 0 0
\(59\) 11.9769 1.55926 0.779628 0.626242i \(-0.215408\pi\)
0.779628 + 0.626242i \(0.215408\pi\)
\(60\) 0 0
\(61\) 1.37228 0.175703 0.0878513 0.996134i \(-0.472000\pi\)
0.0878513 + 0.996134i \(0.472000\pi\)
\(62\) 0 0
\(63\) 1.57317 1.78174i 0.198201 0.224478i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.22474 1.22474i 0.149626 0.149626i −0.628325 0.777951i \(-0.716259\pi\)
0.777951 + 0.628325i \(0.216259\pi\)
\(68\) 0 0
\(69\) −11.0371 10.3723i −1.32871 1.24868i
\(70\) 0 0
\(71\) 3.16915i 0.376109i 0.982159 + 0.188054i \(0.0602181\pi\)
−0.982159 + 0.188054i \(0.939782\pi\)
\(72\) 0 0
\(73\) −5.56349 5.56349i −0.651158 0.651158i 0.302114 0.953272i \(-0.402308\pi\)
−0.953272 + 0.302114i \(0.902308\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.526485 + 0.526485i 0.0599986 + 0.0599986i
\(78\) 0 0
\(79\) 8.74456i 0.983840i −0.870640 0.491920i \(-0.836295\pi\)
0.870640 0.491920i \(-0.163705\pi\)
\(80\) 0 0
\(81\) 8.93070 1.11469i 0.992300 0.123855i
\(82\) 0 0
\(83\) −3.97940 + 3.97940i −0.436796 + 0.436796i −0.890932 0.454136i \(-0.849948\pi\)
0.454136 + 0.890932i \(0.349948\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.170374 + 5.48648i 0.0182660 + 0.588212i
\(88\) 0 0
\(89\) 16.0858 1.70509 0.852545 0.522654i \(-0.175058\pi\)
0.852545 + 0.522654i \(0.175058\pi\)
\(90\) 0 0
\(91\) 4.62772 0.485117
\(92\) 0 0
\(93\) 0.288814 + 9.30058i 0.0299486 + 0.964425i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.25065 5.25065i 0.533122 0.533122i −0.388378 0.921500i \(-0.626964\pi\)
0.921500 + 0.388378i \(0.126964\pi\)
\(98\) 0 0
\(99\) 0.174928 + 2.81386i 0.0175810 + 0.282804i
\(100\) 0 0
\(101\) 8.80773i 0.876402i −0.898877 0.438201i \(-0.855616\pi\)
0.898877 0.438201i \(-0.144384\pi\)
\(102\) 0 0
\(103\) 8.26037 + 8.26037i 0.813918 + 0.813918i 0.985219 0.171301i \(-0.0547969\pi\)
−0.171301 + 0.985219i \(0.554797\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.15097 1.15097i −0.111269 0.111269i 0.649280 0.760549i \(-0.275070\pi\)
−0.760549 + 0.649280i \(0.775070\pi\)
\(108\) 0 0
\(109\) 10.8614i 1.04033i −0.854065 0.520167i \(-0.825870\pi\)
0.854065 0.520167i \(-0.174130\pi\)
\(110\) 0 0
\(111\) 6.37228 + 5.98844i 0.604830 + 0.568398i
\(112\) 0 0
\(113\) −1.67746 + 1.67746i −0.157802 + 0.157802i −0.781592 0.623790i \(-0.785592\pi\)
0.623790 + 0.781592i \(0.285592\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 13.1355 + 11.5979i 1.21438 + 1.07222i
\(118\) 0 0
\(119\) 5.04868 0.462811
\(120\) 0 0
\(121\) 10.1168 0.919713
\(122\) 0 0
\(123\) −15.8538 + 0.492313i −1.42949 + 0.0443903i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 9.58940 9.58940i 0.850921 0.850921i −0.139325 0.990247i \(-0.544493\pi\)
0.990247 + 0.139325i \(0.0444934\pi\)
\(128\) 0 0
\(129\) 7.27806 7.74456i 0.640797 0.681871i
\(130\) 0 0
\(131\) 5.04868i 0.441105i 0.975375 + 0.220552i \(0.0707860\pi\)
−0.975375 + 0.220552i \(0.929214\pi\)
\(132\) 0 0
\(133\) −2.09782 2.09782i −0.181904 0.181904i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.03237 5.03237i −0.429944 0.429944i 0.458665 0.888609i \(-0.348328\pi\)
−0.888609 + 0.458665i \(0.848328\pi\)
\(138\) 0 0
\(139\) 15.1168i 1.28219i −0.767460 0.641097i \(-0.778480\pi\)
0.767460 0.641097i \(-0.221520\pi\)
\(140\) 0 0
\(141\) −9.48913 + 10.0974i −0.799129 + 0.850350i
\(142\) 0 0
\(143\) −3.88140 + 3.88140i −0.324579 + 0.324579i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 11.0318 0.342574i 0.909887 0.0282551i
\(148\) 0 0
\(149\) 3.75906 0.307954 0.153977 0.988074i \(-0.450792\pi\)
0.153977 + 0.988074i \(0.450792\pi\)
\(150\) 0 0
\(151\) −2.62772 −0.213841 −0.106920 0.994268i \(-0.534099\pi\)
−0.106920 + 0.994268i \(0.534099\pi\)
\(152\) 0 0
\(153\) 14.3303 + 12.6529i 1.15854 + 1.02292i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −9.23773 + 9.23773i −0.737251 + 0.737251i −0.972045 0.234794i \(-0.924558\pi\)
0.234794 + 0.972045i \(0.424558\pi\)
\(158\) 0 0
\(159\) 0.939764 + 0.883156i 0.0745281 + 0.0700388i
\(160\) 0 0
\(161\) 6.92820i 0.546019i
\(162\) 0 0
\(163\) 13.2636 + 13.2636i 1.03889 + 1.03889i 0.999213 + 0.0396750i \(0.0126323\pi\)
0.0396750 + 0.999213i \(0.487368\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.82843 + 2.82843i 0.218870 + 0.218870i 0.808022 0.589152i \(-0.200538\pi\)
−0.589152 + 0.808022i \(0.700538\pi\)
\(168\) 0 0
\(169\) 21.1168i 1.62437i
\(170\) 0 0
\(171\) −0.697015 11.2120i −0.0533021 0.857406i
\(172\) 0 0
\(173\) −8.48528 + 8.48528i −0.645124 + 0.645124i −0.951811 0.306687i \(-0.900780\pi\)
0.306687 + 0.951811i \(0.400780\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −0.643878 20.7346i −0.0483968 1.55851i
\(178\) 0 0
\(179\) 5.98844 0.447597 0.223798 0.974635i \(-0.428154\pi\)
0.223798 + 0.974635i \(0.428154\pi\)
\(180\) 0 0
\(181\) −20.1168 −1.49527 −0.747637 0.664108i \(-0.768811\pi\)
−0.747637 + 0.664108i \(0.768811\pi\)
\(182\) 0 0
\(183\) −0.0737740 2.37572i −0.00545353 0.175618i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.23447 + 4.23447i −0.309655 + 0.309655i
\(188\) 0 0
\(189\) −3.16915 2.62772i −0.230522 0.191138i
\(190\) 0 0
\(191\) 25.2434i 1.82655i −0.407347 0.913273i \(-0.633546\pi\)
0.407347 0.913273i \(-0.366454\pi\)
\(192\) 0 0
\(193\) 8.36465 + 8.36465i 0.602101 + 0.602101i 0.940870 0.338769i \(-0.110010\pi\)
−0.338769 + 0.940870i \(0.610010\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.0647 + 10.0647i 0.717083 + 0.717083i 0.968007 0.250924i \(-0.0807344\pi\)
−0.250924 + 0.968007i \(0.580734\pi\)
\(198\) 0 0
\(199\) 4.62772i 0.328050i −0.986456 0.164025i \(-0.947552\pi\)
0.986456 0.164025i \(-0.0524478\pi\)
\(200\) 0 0
\(201\) −2.18614 2.05446i −0.154198 0.144910i
\(202\) 0 0
\(203\) 1.77546 1.77546i 0.124613 0.124613i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −17.3633 + 19.6652i −1.20683 + 1.36683i
\(208\) 0 0
\(209\) 3.51900 0.243415
\(210\) 0 0
\(211\) 9.74456 0.670843 0.335422 0.942068i \(-0.391121\pi\)
0.335422 + 0.942068i \(0.391121\pi\)
\(212\) 0 0
\(213\) 5.48648 0.170374i 0.375927 0.0116738i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.00972 3.00972i 0.204313 0.204313i
\(218\) 0 0
\(219\) −9.33252 + 9.93070i −0.630633 + 0.671055i
\(220\) 0 0
\(221\) 37.2203i 2.50371i
\(222\) 0 0
\(223\) −12.3906 12.3906i −0.829733 0.829733i 0.157747 0.987480i \(-0.449577\pi\)
−0.987480 + 0.157747i \(0.949577\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.82843 + 2.82843i 0.187729 + 0.187729i 0.794714 0.606984i \(-0.207621\pi\)
−0.606984 + 0.794714i \(0.707621\pi\)
\(228\) 0 0
\(229\) 4.62772i 0.305808i 0.988241 + 0.152904i \(0.0488626\pi\)
−0.988241 + 0.152904i \(0.951137\pi\)
\(230\) 0 0
\(231\) 0.883156 0.939764i 0.0581074 0.0618319i
\(232\) 0 0
\(233\) 12.8932 12.8932i 0.844659 0.844659i −0.144801 0.989461i \(-0.546254\pi\)
0.989461 + 0.144801i \(0.0462543\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −15.1387 + 0.470108i −0.983366 + 0.0305368i
\(238\) 0 0
\(239\) −13.2665 −0.858138 −0.429069 0.903272i \(-0.641158\pi\)
−0.429069 + 0.903272i \(0.641158\pi\)
\(240\) 0 0
\(241\) −13.2337 −0.852457 −0.426228 0.904616i \(-0.640158\pi\)
−0.426228 + 0.904616i \(0.640158\pi\)
\(242\) 0 0
\(243\) −2.40989 15.4011i −0.154594 0.987978i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 15.4657 15.4657i 0.984061 0.984061i
\(248\) 0 0
\(249\) 7.10313 + 6.67527i 0.450143 + 0.423028i
\(250\) 0 0
\(251\) 10.4472i 0.659422i −0.944082 0.329711i \(-0.893049\pi\)
0.944082 0.329711i \(-0.106951\pi\)
\(252\) 0 0
\(253\) −5.81088 5.81088i −0.365327 0.365327i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.0647 10.0647i −0.627821 0.627821i 0.319698 0.947519i \(-0.396418\pi\)
−0.947519 + 0.319698i \(0.896418\pi\)
\(258\) 0 0
\(259\) 4.00000i 0.248548i
\(260\) 0 0
\(261\) 9.48913 0.589907i 0.587362 0.0365143i
\(262\) 0 0
\(263\) −10.7872 + 10.7872i −0.665169 + 0.665169i −0.956594 0.291425i \(-0.905871\pi\)
0.291425 + 0.956594i \(0.405871\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −0.864773 27.8480i −0.0529233 1.70427i
\(268\) 0 0
\(269\) −27.1229 −1.65371 −0.826856 0.562413i \(-0.809873\pi\)
−0.826856 + 0.562413i \(0.809873\pi\)
\(270\) 0 0
\(271\) 18.2337 1.10762 0.553809 0.832644i \(-0.313174\pi\)
0.553809 + 0.832644i \(0.313174\pi\)
\(272\) 0 0
\(273\) −0.248787 8.01158i −0.0150572 0.484883i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.6315 + 14.6315i −0.879120 + 0.879120i −0.993444 0.114323i \(-0.963530\pi\)
0.114323 + 0.993444i \(0.463530\pi\)
\(278\) 0 0
\(279\) 16.0858 1.00000i 0.963031 0.0598684i
\(280\) 0 0
\(281\) 20.7846i 1.23991i −0.784639 0.619953i \(-0.787152\pi\)
0.784639 0.619953i \(-0.212848\pi\)
\(282\) 0 0
\(283\) −17.7455 17.7455i −1.05486 1.05486i −0.998405 0.0564542i \(-0.982021\pi\)
−0.0564542 0.998405i \(-0.517979\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.13037 + 5.13037i 0.302836 + 0.302836i
\(288\) 0 0
\(289\) 23.6060i 1.38859i
\(290\) 0 0
\(291\) −9.37228 8.80773i −0.549413 0.516318i
\(292\) 0 0
\(293\) 21.3784 21.3784i 1.24894 1.24894i 0.292754 0.956188i \(-0.405428\pi\)
0.956188 0.292754i \(-0.0945719\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.86199 0.454112i 0.282122 0.0263502i
\(298\) 0 0
\(299\) −51.0767 −2.95384
\(300\) 0 0
\(301\) −4.86141 −0.280207
\(302\) 0 0
\(303\) −15.2481 + 0.473504i −0.875980 + 0.0272021i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −19.4916 + 19.4916i −1.11245 + 1.11245i −0.119628 + 0.992819i \(0.538170\pi\)
−0.992819 + 0.119628i \(0.961830\pi\)
\(308\) 0 0
\(309\) 13.8564 14.7446i 0.788263 0.838789i
\(310\) 0 0
\(311\) 25.8333i 1.46487i 0.680836 + 0.732436i \(0.261617\pi\)
−0.680836 + 0.732436i \(0.738383\pi\)
\(312\) 0 0
\(313\) −9.02916 9.02916i −0.510359 0.510359i 0.404278 0.914636i \(-0.367523\pi\)
−0.914636 + 0.404278i \(0.867523\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.70982 + 6.70982i 0.376861 + 0.376861i 0.869969 0.493107i \(-0.164139\pi\)
−0.493107 + 0.869969i \(0.664139\pi\)
\(318\) 0 0
\(319\) 2.97825i 0.166750i
\(320\) 0 0
\(321\) −1.93070 + 2.05446i −0.107761 + 0.114669i
\(322\) 0 0
\(323\) 16.8726 16.8726i 0.938814 0.938814i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −18.8034 + 0.583910i −1.03983 + 0.0322903i
\(328\) 0 0
\(329\) 6.33830 0.349442
\(330\) 0 0
\(331\) 26.3723 1.44955 0.724776 0.688985i \(-0.241943\pi\)
0.724776 + 0.688985i \(0.241943\pi\)
\(332\) 0 0
\(333\) 10.0247 11.3537i 0.549351 0.622181i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.67423 + 3.67423i −0.200148 + 0.200148i −0.800064 0.599915i \(-0.795201\pi\)
0.599915 + 0.800064i \(0.295201\pi\)
\(338\) 0 0
\(339\) 2.99422 + 2.81386i 0.162624 + 0.152828i
\(340\) 0 0
\(341\) 5.04868i 0.273401i
\(342\) 0 0
\(343\) −7.49157 7.49157i −0.404507 0.404507i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.38728 8.38728i −0.450253 0.450253i 0.445186 0.895438i \(-0.353138\pi\)
−0.895438 + 0.445186i \(0.853138\pi\)
\(348\) 0 0
\(349\) 4.51087i 0.241462i −0.992685 0.120731i \(-0.961476\pi\)
0.992685 0.120731i \(-0.0385238\pi\)
\(350\) 0 0
\(351\) 19.3723 23.3639i 1.03402 1.24707i
\(352\) 0 0
\(353\) −5.13037 + 5.13037i −0.273062 + 0.273062i −0.830332 0.557270i \(-0.811849\pi\)
0.557270 + 0.830332i \(0.311849\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −0.271417 8.74035i −0.0143649 0.462588i
\(358\) 0 0
\(359\) 5.04868 0.266459 0.133229 0.991085i \(-0.457465\pi\)
0.133229 + 0.991085i \(0.457465\pi\)
\(360\) 0 0
\(361\) 4.97825 0.262013
\(362\) 0 0
\(363\) −0.543882 17.5144i −0.0285464 0.919270i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9.02916 9.02916i 0.471319 0.471319i −0.431023 0.902341i \(-0.641847\pi\)
0.902341 + 0.431023i \(0.141847\pi\)
\(368\) 0 0
\(369\) 1.70460 + 27.4198i 0.0887379 + 1.42742i
\(370\) 0 0
\(371\) 0.589907i 0.0306265i
\(372\) 0 0
\(373\) −0.560232 0.560232i −0.0290077 0.0290077i 0.692454 0.721462i \(-0.256529\pi\)
−0.721462 + 0.692454i \(0.756529\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 13.0892 + 13.0892i 0.674126 + 0.674126i
\(378\) 0 0
\(379\) 3.51087i 0.180342i 0.995926 + 0.0901708i \(0.0287413\pi\)
−0.995926 + 0.0901708i \(0.971259\pi\)
\(380\) 0 0
\(381\) −17.1168 16.0858i −0.876922 0.824100i
\(382\) 0 0
\(383\) −10.0647 + 10.0647i −0.514284 + 0.514284i −0.915836 0.401552i \(-0.868471\pi\)
0.401552 + 0.915836i \(0.368471\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −13.7988 12.1835i −0.701432 0.619325i
\(388\) 0 0
\(389\) −29.0024 −1.47048 −0.735241 0.677806i \(-0.762931\pi\)
−0.735241 + 0.677806i \(0.762931\pi\)
\(390\) 0 0
\(391\) −55.7228 −2.81802
\(392\) 0 0
\(393\) 8.74035 0.271417i 0.440892 0.0136912i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.66777 5.66777i 0.284457 0.284457i −0.550426 0.834884i \(-0.685535\pi\)
0.834884 + 0.550426i \(0.185535\pi\)
\(398\) 0 0
\(399\) −3.51900 + 3.74456i −0.176171 + 0.187463i
\(400\) 0 0
\(401\) 16.6757i 0.832745i 0.909194 + 0.416372i \(0.136699\pi\)
−0.909194 + 0.416372i \(0.863301\pi\)
\(402\) 0 0
\(403\) 22.1885 + 22.1885i 1.10529 + 1.10529i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.35491 + 3.35491i 0.166297 + 0.166297i
\(408\) 0 0
\(409\) 2.48913i 0.123079i −0.998105 0.0615397i \(-0.980399\pi\)
0.998105 0.0615397i \(-0.0196011\pi\)
\(410\) 0 0
\(411\) −8.44158 + 8.98266i −0.416392 + 0.443082i
\(412\) 0 0
\(413\) −6.70982 + 6.70982i −0.330169 + 0.330169i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −26.1705 + 0.812683i −1.28158 + 0.0397972i
\(418\) 0 0
\(419\) −2.81929 −0.137731 −0.0688657 0.997626i \(-0.521938\pi\)
−0.0688657 + 0.997626i \(0.521938\pi\)
\(420\) 0 0
\(421\) −5.25544 −0.256134 −0.128067 0.991765i \(-0.540877\pi\)
−0.128067 + 0.991765i \(0.540877\pi\)
\(422\) 0 0
\(423\) 17.9908 + 15.8849i 0.874744 + 0.772350i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.768795 + 0.768795i −0.0372046 + 0.0372046i
\(428\) 0 0
\(429\) 6.92820 + 6.51087i 0.334497 + 0.314348i
\(430\) 0 0
\(431\) 22.6641i 1.09169i 0.837885 + 0.545847i \(0.183792\pi\)
−0.837885 + 0.545847i \(0.816208\pi\)
\(432\) 0 0
\(433\) 6.12372 + 6.12372i 0.294287 + 0.294287i 0.838771 0.544484i \(-0.183274\pi\)
−0.544484 + 0.838771i \(0.683274\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 23.1539 + 23.1539i 1.10760 + 1.10760i
\(438\) 0 0
\(439\) 8.62772i 0.411779i 0.978575 + 0.205889i \(0.0660087\pi\)
−0.978575 + 0.205889i \(0.933991\pi\)
\(440\) 0 0
\(441\) −1.18614 19.0800i −0.0564829 0.908572i
\(442\) 0 0
\(443\) −23.0559 + 23.0559i −1.09542 + 1.09542i −0.100480 + 0.994939i \(0.532038\pi\)
−0.994939 + 0.100480i \(0.967962\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.202087 6.50774i −0.00955839 0.307805i
\(448\) 0 0
\(449\) −2.22938 −0.105211 −0.0526056 0.998615i \(-0.516753\pi\)
−0.0526056 + 0.998615i \(0.516753\pi\)
\(450\) 0 0
\(451\) −8.60597 −0.405239
\(452\) 0 0
\(453\) 0.141266 + 4.54915i 0.00663727 + 0.213738i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.3744 11.3744i 0.532071 0.532071i −0.389117 0.921188i \(-0.627220\pi\)
0.921188 + 0.389117i \(0.127220\pi\)
\(458\) 0 0
\(459\) 21.1345 25.4891i 0.986472 1.18973i
\(460\) 0 0
\(461\) 17.0256i 0.792959i −0.918044 0.396480i \(-0.870232\pi\)
0.918044 0.396480i \(-0.129768\pi\)
\(462\) 0 0
\(463\) −6.43657 6.43657i −0.299133 0.299133i 0.541541 0.840674i \(-0.317841\pi\)
−0.840674 + 0.541541i \(0.817841\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.82843 + 2.82843i 0.130884 + 0.130884i 0.769514 0.638630i \(-0.220499\pi\)
−0.638630 + 0.769514i \(0.720499\pi\)
\(468\) 0 0
\(469\) 1.37228i 0.0633661i
\(470\) 0 0
\(471\) 16.4891 + 15.4959i 0.759779 + 0.714013i
\(472\) 0 0
\(473\) 4.07740 4.07740i 0.187479 0.187479i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.47841 1.67441i 0.0676919 0.0766661i
\(478\) 0 0
\(479\) 4.45877 0.203726 0.101863 0.994798i \(-0.467520\pi\)
0.101863 + 0.994798i \(0.467520\pi\)
\(480\) 0 0
\(481\) 29.4891 1.34459
\(482\) 0 0
\(483\) 11.9942 0.372461i 0.545756 0.0169476i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −9.94106 + 9.94106i −0.450473 + 0.450473i −0.895511 0.445039i \(-0.853190\pi\)
0.445039 + 0.895511i \(0.353190\pi\)
\(488\) 0 0
\(489\) 22.2492 23.6753i 1.00614 1.07063i
\(490\) 0 0
\(491\) 4.45877i 0.201221i −0.994926 0.100611i \(-0.967920\pi\)
0.994926 0.100611i \(-0.0320797\pi\)
\(492\) 0 0
\(493\) 14.2798 + 14.2798i 0.643130 + 0.643130i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.77546 1.77546i −0.0796401 0.0796401i
\(498\) 0 0
\(499\) 23.3723i 1.04629i 0.852245 + 0.523144i \(0.175241\pi\)
−0.852245 + 0.523144i \(0.824759\pi\)
\(500\) 0 0
\(501\) 4.74456 5.04868i 0.211971 0.225558i
\(502\) 0 0
\(503\) 16.2481 16.2481i 0.724466 0.724466i −0.245046 0.969512i \(-0.578803\pi\)
0.969512 + 0.245046i \(0.0788029\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 36.5578 1.13524i 1.62359 0.0504179i
\(508\) 0 0
\(509\) 22.0742 0.978423 0.489212 0.872165i \(-0.337285\pi\)
0.489212 + 0.872165i \(0.337285\pi\)
\(510\) 0 0
\(511\) 6.23369 0.275762
\(512\) 0 0
\(513\) −19.3730 + 1.80944i −0.855339 + 0.0798889i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5.31611 + 5.31611i −0.233802 + 0.233802i
\(518\) 0 0
\(519\) 15.1460 + 14.2337i 0.664837 + 0.624790i
\(520\) 0 0
\(521\) 16.6757i 0.730576i 0.930895 + 0.365288i \(0.119029\pi\)
−0.930895 + 0.365288i \(0.880971\pi\)
\(522\) 0 0
\(523\) 12.8465 + 12.8465i 0.561738 + 0.561738i 0.929801 0.368063i \(-0.119979\pi\)
−0.368063 + 0.929801i \(0.619979\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.2069 + 24.2069i 1.05447 + 1.05447i
\(528\) 0 0
\(529\) 53.4674i 2.32467i
\(530\) 0 0
\(531\) −35.8614 + 2.22938i −1.55625 + 0.0967470i
\(532\) 0 0
\(533\) −37.8225 + 37.8225i −1.63828 + 1.63828i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.321939 10.3673i −0.0138927 0.447381i
\(538\) 0 0
\(539\) 5.98844 0.257940
\(540\) 0 0
\(541\) 5.88316 0.252937 0.126468 0.991971i \(-0.459636\pi\)
0.126468 + 0.991971i \(0.459636\pi\)
\(542\) 0 0
\(543\) 1.08148 + 34.8266i 0.0464109 + 1.49455i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.53113 + 3.53113i −0.150980 + 0.150980i −0.778556 0.627575i \(-0.784047\pi\)
0.627575 + 0.778556i \(0.284047\pi\)
\(548\) 0 0
\(549\) −4.10891 + 0.255437i −0.175364 + 0.0109018i
\(550\) 0 0
\(551\) 11.8671i 0.505554i
\(552\) 0 0
\(553\) 4.89898 + 4.89898i 0.208326 + 0.208326i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24.7334 24.7334i −1.04799 1.04799i −0.998789 0.0491969i \(-0.984334\pi\)
−0.0491969 0.998789i \(-0.515666\pi\)
\(558\) 0 0
\(559\) 35.8397i 1.51586i
\(560\) 0 0
\(561\) 7.55842 + 7.10313i 0.319117 + 0.299894i
\(562\) 0 0
\(563\) 22.9579 22.9579i 0.967560 0.967560i −0.0319299 0.999490i \(-0.510165\pi\)
0.999490 + 0.0319299i \(0.0101653\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −4.37878 + 5.62775i −0.183891 + 0.236343i
\(568\) 0 0
\(569\) −34.9909 −1.46689 −0.733447 0.679747i \(-0.762090\pi\)
−0.733447 + 0.679747i \(0.762090\pi\)
\(570\) 0 0
\(571\) 31.8397 1.33245 0.666224 0.745752i \(-0.267909\pi\)
0.666224 + 0.745752i \(0.267909\pi\)
\(572\) 0 0
\(573\) −43.7017 + 1.35709i −1.82567 + 0.0566931i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 26.2144 26.2144i 1.09132 1.09132i 0.0959325 0.995388i \(-0.469417\pi\)
0.995388 0.0959325i \(-0.0305833\pi\)
\(578\) 0 0
\(579\) 14.0313 14.9307i 0.583122 0.620499i
\(580\) 0 0
\(581\) 4.45877i 0.184981i
\(582\) 0 0
\(583\) 0.494772 + 0.494772i 0.0204914 + 0.0204914i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.8726 + 16.8726i 0.696405 + 0.696405i 0.963633 0.267228i \(-0.0861077\pi\)
−0.267228 + 0.963633i \(0.586108\pi\)
\(588\) 0 0
\(589\) 20.1168i 0.828900i
\(590\) 0 0
\(591\) 16.8832 17.9653i 0.694480 0.738994i
\(592\) 0 0
\(593\) −13.5177 + 13.5177i −0.555103 + 0.555103i −0.927909 0.372806i \(-0.878396\pi\)
0.372806 + 0.927909i \(0.378396\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.01158 + 0.248787i −0.327892 + 0.0101822i
\(598\) 0 0
\(599\) 28.4125 1.16090 0.580452 0.814294i \(-0.302876\pi\)
0.580452 + 0.814294i \(0.302876\pi\)
\(600\) 0 0
\(601\) −3.74456 −0.152744 −0.0763719 0.997079i \(-0.524334\pi\)
−0.0763719 + 0.997079i \(0.524334\pi\)
\(602\) 0 0
\(603\) −3.43918 + 3.89513i −0.140054 + 0.158622i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 18.4755 18.4755i 0.749896 0.749896i −0.224564 0.974459i \(-0.572096\pi\)
0.974459 + 0.224564i \(0.0720957\pi\)
\(608\) 0 0
\(609\) −3.16915 2.97825i −0.128420 0.120685i
\(610\) 0 0
\(611\) 46.7277i 1.89040i
\(612\) 0 0
\(613\) −8.88606 8.88606i −0.358905 0.358905i 0.504504 0.863409i \(-0.331675\pi\)
−0.863409 + 0.504504i \(0.831675\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.9461 + 13.9461i 0.561450 + 0.561450i 0.929719 0.368269i \(-0.120049\pi\)
−0.368269 + 0.929719i \(0.620049\pi\)
\(618\) 0 0
\(619\) 12.8614i 0.516944i 0.966019 + 0.258472i \(0.0832189\pi\)
−0.966019 + 0.258472i \(0.916781\pi\)
\(620\) 0 0
\(621\) 34.9783 + 29.0024i 1.40363 + 1.16383i
\(622\) 0 0
\(623\) −9.01177 + 9.01177i −0.361049 + 0.361049i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −0.189182 6.09216i −0.00755520 0.243297i
\(628\) 0 0
\(629\) 32.1716 1.28276
\(630\) 0 0
\(631\) −25.6060 −1.01936 −0.509679 0.860365i \(-0.670236\pi\)
−0.509679 + 0.860365i \(0.670236\pi\)
\(632\) 0 0
\(633\) −0.523868 16.8699i −0.0208219 0.670520i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 26.3187 26.3187i 1.04278 1.04278i
\(638\) 0 0
\(639\) −0.589907 9.48913i −0.0233364 0.375384i
\(640\) 0 0
\(641\) 23.9538i 0.946117i −0.881031 0.473058i \(-0.843150\pi\)
0.881031 0.473058i \(-0.156850\pi\)
\(642\) 0 0
\(643\) −5.10754 5.10754i −0.201422 0.201422i 0.599187 0.800609i \(-0.295491\pi\)
−0.800609 + 0.599187i \(0.795491\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) 11.2554i 0.441815i
\(650\) 0 0
\(651\) −5.37228 5.04868i −0.210556 0.197873i
\(652\) 0 0
\(653\) 6.90583 6.90583i 0.270246 0.270246i −0.558953 0.829199i \(-0.688797\pi\)
0.829199 + 0.558953i \(0.188797\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 17.6939 + 15.6227i 0.690305 + 0.609501i
\(658\) 0 0
\(659\) 35.6906 1.39031 0.695154 0.718861i \(-0.255336\pi\)
0.695154 + 0.718861i \(0.255336\pi\)
\(660\) 0 0
\(661\) 40.2337 1.56491 0.782455 0.622708i \(-0.213967\pi\)
0.782455 + 0.622708i \(0.213967\pi\)
\(662\) 0 0
\(663\) 64.4363 2.00096i 2.50250 0.0777110i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −19.5959 + 19.5959i −0.758757 + 0.758757i
\(668\) 0 0
\(669\) −20.7846 + 22.1168i −0.803579 + 0.855087i
\(670\) 0 0
\(671\) 1.28962i 0.0497852i
\(672\) 0 0
\(673\) −1.82380 1.82380i −0.0703023 0.0703023i 0.671081 0.741384i \(-0.265830\pi\)
−0.741384 + 0.671081i \(0.765830\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −16.2481 16.2481i −0.624464 0.624464i 0.322205 0.946670i \(-0.395576\pi\)
−0.946670 + 0.322205i \(0.895576\pi\)
\(678\) 0 0
\(679\) 5.88316i 0.225775i
\(680\) 0 0
\(681\) 4.74456 5.04868i 0.181812 0.193466i
\(682\) 0 0
\(683\) −17.3990 + 17.3990i −0.665756 + 0.665756i −0.956731 0.290975i \(-0.906020\pi\)
0.290975 + 0.956731i \(0.406020\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 8.01158 0.248787i 0.305661 0.00949180i
\(688\) 0 0
\(689\) 4.34896 0.165682
\(690\) 0 0
\(691\) −21.3505 −0.812213 −0.406106 0.913826i \(-0.633114\pi\)
−0.406106 + 0.913826i \(0.633114\pi\)
\(692\) 0 0
\(693\) −1.67441 1.47841i −0.0636057 0.0561602i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −41.2630 + 41.2630i −1.56295 + 1.56295i
\(698\) 0 0
\(699\) −23.0140 21.6277i −0.870469 0.818035i
\(700\) 0 0
\(701\) 13.2665i 0.501069i −0.968108 0.250534i \(-0.919394\pi\)
0.968108 0.250534i \(-0.0806063\pi\)
\(702\) 0 0
\(703\) −13.3679 13.3679i −0.504180 0.504180i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.93437 + 4.93437i 0.185576 + 0.185576i
\(708\) 0 0
\(709\) 0.861407i 0.0323508i −0.999869 0.0161754i \(-0.994851\pi\)
0.999869 0.0161754i \(-0.00514902\pi\)
\(710\) 0 0
\(711\) 1.62772 + 26.1831i 0.0610442 + 0.981945i
\(712\) 0 0
\(713\) −33.2186 + 33.2186i −1.24405 + 1.24405i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0.713208 + 22.9672i 0.0266352 + 0.857725i
\(718\) 0 0
\(719\) −27.8226 −1.03761 −0.518804 0.854893i \(-0.673623\pi\)
−0.518804 + 0.854893i \(0.673623\pi\)
\(720\) 0 0
\(721\) −9.25544 −0.344690
\(722\) 0 0
\(723\) 0.711444 + 22.9104i 0.0264589 + 0.852046i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −10.5668 + 10.5668i −0.391899 + 0.391899i −0.875364 0.483465i \(-0.839378\pi\)
0.483465 + 0.875364i \(0.339378\pi\)
\(728\) 0 0
\(729\) −26.5330 + 5.00000i −0.982704 + 0.185185i
\(730\) 0 0
\(731\) 39.0998i 1.44616i
\(732\) 0 0
\(733\) 34.2929 + 34.2929i 1.26664 + 1.26664i 0.947815 + 0.318820i \(0.103287\pi\)
0.318820 + 0.947815i \(0.396713\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.15097 1.15097i −0.0423966 0.0423966i
\(738\) 0 0
\(739\) 44.4674i 1.63576i 0.575390 + 0.817879i \(0.304850\pi\)
−0.575390 + 0.817879i \(0.695150\pi\)
\(740\) 0 0
\(741\) −27.6060 25.9431i −1.01413 0.953043i
\(742\) 0 0
\(743\) 32.6922 32.6922i 1.19936 1.19936i 0.225000 0.974359i \(-0.427762\pi\)
0.974359 0.225000i \(-0.0722382\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 11.1745 12.6559i 0.408852 0.463056i
\(748\) 0 0
\(749\) 1.28962 0.0471217
\(750\) 0 0
\(751\) −44.0000 −1.60558 −0.802791 0.596260i \(-0.796653\pi\)
−0.802791 + 0.596260i \(0.796653\pi\)
\(752\) 0 0
\(753\) −18.0864 + 0.561643i −0.659104 + 0.0204674i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 28.6251 28.6251i 1.04040 1.04040i 0.0412473 0.999149i \(-0.486867\pi\)
0.999149 0.0412473i \(-0.0131332\pi\)
\(758\) 0 0
\(759\) −9.74749 + 10.3723i −0.353812 + 0.376490i
\(760\) 0 0
\(761\) 14.2063i 0.514977i 0.966281 + 0.257488i \(0.0828949\pi\)
−0.966281 + 0.257488i \(0.917105\pi\)
\(762\) 0 0
\(763\) 6.08490 + 6.08490i 0.220288 + 0.220288i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −49.4667 49.4667i −1.78614 1.78614i
\(768\) 0 0
\(769\) 7.51087i 0.270849i −0.990788 0.135425i \(-0.956760\pi\)
0.990788 0.135425i \(-0.0432398\pi\)
\(770\) 0 0
\(771\) −16.8832 + 17.9653i −0.608032 + 0.647005i
\(772\) 0 0
\(773\) 36.7696 36.7696i 1.32251 1.32251i 0.410770 0.911739i \(-0.365260\pi\)
0.911739 0.410770i \(-0.134740\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −6.92487 + 0.215040i −0.248428 + 0.00771453i
\(778\) 0 0
\(779\) 34.2912 1.22861
\(780\) 0 0
\(781\) 2.97825 0.106570
\(782\) 0 0
\(783\) −1.53139 16.3960i −0.0547275 0.585946i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0.560232 0.560232i 0.0199701 0.0199701i −0.697051 0.717021i \(-0.745505\pi\)
0.717021 + 0.697051i \(0.245505\pi\)
\(788\) 0 0
\(789\) 19.2549 + 18.0951i 0.685494 + 0.644202i
\(790\) 0 0
\(791\) 1.87953i 0.0668283i
\(792\) 0 0
\(793\) −5.66777 5.66777i −0.201269 0.201269i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10.0647 10.0647i −0.356511 0.356511i 0.506014 0.862525i \(-0.331118\pi\)
−0.862525 + 0.506014i \(0.831118\pi\)
\(798\) 0 0
\(799\) 50.9783i 1.80348i
\(800\) 0 0
\(801\) −48.1644 + 2.99422i −1.70181 + 0.105796i
\(802\) 0 0
\(803\) −5.22837 + 5.22837i −0.184505 + 0.184505i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.45813 + 46.9556i 0.0513286 + 1.65292i
\(808\) 0 0
\(809\) −5.74839 −0.202103 −0.101051 0.994881i \(-0.532221\pi\)
−0.101051 + 0.994881i \(0.532221\pi\)
\(810\) 0 0
\(811\) 32.6277 1.14571 0.572857 0.819655i \(-0.305835\pi\)
0.572857 + 0.819655i \(0.305835\pi\)
\(812\) 0 0
\(813\) −0.980245 31.5665i −0.0343787 1.10708i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −16.2467 + 16.2467i −0.568401 + 0.568401i
\(818\) 0 0
\(819\) −13.8564 + 0.861407i −0.484182 + 0.0301000i
\(820\) 0 0
\(821\) 25.8333i 0.901588i −0.892628 0.450794i \(-0.851141\pi\)
0.892628 0.450794i \(-0.148859\pi\)
\(822\) 0 0
\(823\) 2.59259 + 2.59259i 0.0903721 + 0.0903721i 0.750848 0.660475i \(-0.229645\pi\)
−0.660475 + 0.750848i \(0.729645\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.2157 11.2157i −0.390008 0.390008i 0.484682 0.874690i \(-0.338935\pi\)
−0.874690 + 0.484682i \(0.838935\pi\)
\(828\) 0 0
\(829\) 13.2554i 0.460380i −0.973146 0.230190i \(-0.926065\pi\)
0.973146 0.230190i \(-0.0739348\pi\)
\(830\) 0 0
\(831\) 26.1168 + 24.5437i 0.905983 + 0.851410i
\(832\) 0 0
\(833\) 28.7128 28.7128i 0.994838 0.994838i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.59599 27.7942i −0.0897305 0.960709i
\(838\) 0 0
\(839\) −40.9793 −1.41476 −0.707381 0.706832i \(-0.750124\pi\)
−0.707381 + 0.706832i \(0.750124\pi\)
\(840\) 0 0
\(841\) −18.9565 −0.653672
\(842\) 0 0
\(843\) −35.9827 + 1.11738i −1.23931 + 0.0384847i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −5.66777 + 5.66777i −0.194747 + 0.194747i
\(848\) 0 0
\(849\) −29.7673 + 31.6753i −1.02161 + 1.08709i
\(850\) 0 0
\(851\) 44.1485i 1.51339i
\(852\) 0 0
\(853\) −27.5823 27.5823i −0.944399 0.944399i 0.0541349 0.998534i \(-0.482760\pi\)
−0.998534 + 0.0541349i \(0.982760\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14.2401 14.2401i −0.486434 0.486434i 0.420745 0.907179i \(-0.361769\pi\)
−0.907179 + 0.420745i \(0.861769\pi\)
\(858\) 0 0
\(859\) 11.8614i 0.404706i −0.979313 0.202353i \(-0.935141\pi\)
0.979313 0.202353i \(-0.0648588\pi\)
\(860\) 0 0
\(861\) 8.60597 9.15759i 0.293291 0.312090i
\(862\) 0 0
\(863\) 5.65685 5.65685i 0.192562 0.192562i −0.604240 0.796802i \(-0.706523\pi\)
0.796802 + 0.604240i \(0.206523\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 40.8670 1.26906i 1.38792 0.0430995i
\(868\) 0 0
\(869\) −8.21782 −0.278771
\(870\) 0 0
\(871\) −10.1168 −0.342796
\(872\) 0 0
\(873\) −14.7442 + 16.6990i −0.499017 + 0.565174i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.768795 0.768795i 0.0259604 0.0259604i −0.694007 0.719968i \(-0.744157\pi\)
0.719968 + 0.694007i \(0.244157\pi\)
\(878\) 0 0
\(879\) −38.1600 35.8614i −1.28711 1.20958i
\(880\) 0 0
\(881\) 17.6155i 0.593480i −0.954958 0.296740i \(-0.904101\pi\)
0.954958 0.296740i \(-0.0958995\pi\)
\(882\) 0 0
\(883\) −5.28947 5.28947i −0.178005 0.178005i 0.612481 0.790486i \(-0.290172\pi\)
−0.790486 + 0.612481i \(0.790172\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22.6274 22.6274i −0.759754 0.759754i 0.216523 0.976277i \(-0.430528\pi\)
−0.976277 + 0.216523i \(0.930528\pi\)
\(888\) 0 0
\(889\) 10.7446i 0.360361i
\(890\) 0 0
\(891\) −1.04755 8.39275i −0.0350942 0.281168i
\(892\) 0 0
\(893\) 21.1824 21.1824i 0.708843 0.708843i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.74589 + 88.4248i 0.0916825 + 2.95242i
\(898\) 0 0
\(899\) 17.0256 0.567834
\(900\) 0 0
\(901\) 4.74456 0.158064
\(902\) 0 0
\(903\) 0.261350 + 8.41615i 0.00869717 + 0.280072i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −35.4910 + 35.4910i −1.17846 + 1.17846i −0.198321 + 0.980137i \(0.563549\pi\)
−0.980137 + 0.198321i \(0.936451\pi\)
\(908\) 0 0
\(909\) 1.63948 + 26.3723i 0.0543780 + 0.874713i
\(910\) 0 0
\(911\) 54.2458i 1.79724i −0.438724 0.898622i \(-0.644569\pi\)
0.438724 0.898622i \(-0.355431\pi\)
\(912\) 0 0
\(913\) 3.73969 + 3.73969i 0.123766 + 0.123766i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.82843 2.82843i −0.0934029 0.0934029i
\(918\) 0 0
\(919\) 33.8397i 1.11627i 0.829751 + 0.558134i \(0.188482\pi\)
−0.829751 + 0.558134i \(0.811518\pi\)
\(920\) 0 0
\(921\) 34.7921 + 32.6964i 1.14644 + 1.07738i
\(922\) 0 0
\(923\) 13.0892 13.0892i 0.430835 0.430835i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −26.2709 23.1958i −0.862851 0.761849i
\(928\) 0 0
\(929\) −3.16915 −0.103976 −0.0519882 0.998648i \(-0.516556\pi\)
−0.0519882 + 0.998648i \(0.516556\pi\)
\(930\) 0 0
\(931\) −23.8614 −0.782026
\(932\) 0 0
\(933\) 44.7230 1.38880i 1.46417 0.0454673i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.67423 3.67423i 0.120032 0.120032i −0.644539 0.764571i \(-0.722951\pi\)
0.764571 + 0.644539i \(0.222951\pi\)
\(938\) 0 0
\(939\) −15.1460 + 16.1168i −0.494272 + 0.525953i
\(940\) 0 0
\(941\) 30.2921i 0.987493i −0.869606 0.493746i \(-0.835627\pi\)
0.869606 0.493746i \(-0.164373\pi\)
\(942\) 0 0
\(943\) −56.6245 56.6245i −1.84395 1.84395i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20.8520 20.8520i −0.677598 0.677598i 0.281858 0.959456i \(-0.409049\pi\)
−0.959456 + 0.281858i \(0.909049\pi\)
\(948\) 0 0
\(949\) 45.9565i 1.49181i
\(950\) 0 0
\(951\) 11.2554 11.9769i 0.364983 0.388377i
\(952\) 0 0
\(953\) 17.3990 17.3990i 0.563610 0.563610i −0.366721 0.930331i \(-0.619520\pi\)
0.930331 + 0.366721i \(0.119520\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5.15600 0.160111i 0.166670 0.00517565i
\(958\) 0 0
\(959\) 5.63858 0.182079
\(960\) 0 0
\(961\) −2.13859 −0.0689869
\(962\) 0 0
\(963\) 3.66050 + 3.23202i 0.117958 + 0.104150i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 42.1361 42.1361i 1.35501 1.35501i 0.475043 0.879963i \(-0.342433\pi\)
0.879963 0.475043i \(-0.157567\pi\)
\(968\) 0 0
\(969\) −30.1171 28.3030i −0.967501 0.909223i
\(970\) 0 0
\(971\) 22.4241i 0.719623i −0.933025 0.359812i \(-0.882841\pi\)
0.933025 0.359812i \(-0.117159\pi\)
\(972\) 0 0
\(973\) 8.46893 + 8.46893i 0.271502 + 0.271502i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.33431 + 7.33431i 0.234645 + 0.234645i 0.814628 0.579983i \(-0.196941\pi\)
−0.579983 + 0.814628i \(0.696941\pi\)
\(978\) 0 0
\(979\) 15.1168i 0.483136i
\(980\) 0 0
\(981\) 2.02175 + 32.5214i 0.0645495 + 1.03833i
\(982\) 0 0
\(983\) −7.95880 + 7.95880i −0.253846 + 0.253846i −0.822545 0.568699i \(-0.807447\pi\)
0.568699 + 0.822545i \(0.307447\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −0.340747 10.9730i −0.0108461 0.349273i
\(988\) 0 0
\(989\) 53.6559 1.70616
\(990\) 0 0
\(991\) 57.8397 1.83734 0.918669 0.395029i \(-0.129265\pi\)
0.918669 + 0.395029i \(0.129265\pi\)
\(992\) 0 0
\(993\) −1.41778 45.6561i −0.0449918 1.44885i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4.06472 4.06472i 0.128731 0.128731i −0.639806 0.768537i \(-0.720985\pi\)
0.768537 + 0.639806i \(0.220985\pi\)
\(998\) 0 0
\(999\) −20.1947 16.7446i −0.638932 0.529775i
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 600.2.r.f.593.4 yes 16
3.2 odd 2 inner 600.2.r.f.593.1 yes 16
4.3 odd 2 1200.2.v.m.593.5 16
5.2 odd 4 inner 600.2.r.f.257.1 16
5.3 odd 4 inner 600.2.r.f.257.8 yes 16
5.4 even 2 inner 600.2.r.f.593.5 yes 16
12.11 even 2 1200.2.v.m.593.8 16
15.2 even 4 inner 600.2.r.f.257.4 yes 16
15.8 even 4 inner 600.2.r.f.257.5 yes 16
15.14 odd 2 inner 600.2.r.f.593.8 yes 16
20.3 even 4 1200.2.v.m.257.1 16
20.7 even 4 1200.2.v.m.257.8 16
20.19 odd 2 1200.2.v.m.593.4 16
60.23 odd 4 1200.2.v.m.257.4 16
60.47 odd 4 1200.2.v.m.257.5 16
60.59 even 2 1200.2.v.m.593.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.r.f.257.1 16 5.2 odd 4 inner
600.2.r.f.257.4 yes 16 15.2 even 4 inner
600.2.r.f.257.5 yes 16 15.8 even 4 inner
600.2.r.f.257.8 yes 16 5.3 odd 4 inner
600.2.r.f.593.1 yes 16 3.2 odd 2 inner
600.2.r.f.593.4 yes 16 1.1 even 1 trivial
600.2.r.f.593.5 yes 16 5.4 even 2 inner
600.2.r.f.593.8 yes 16 15.14 odd 2 inner
1200.2.v.m.257.1 16 20.3 even 4
1200.2.v.m.257.4 16 60.23 odd 4
1200.2.v.m.257.5 16 60.47 odd 4
1200.2.v.m.257.8 16 20.7 even 4
1200.2.v.m.593.1 16 60.59 even 2
1200.2.v.m.593.4 16 20.19 odd 2
1200.2.v.m.593.5 16 4.3 odd 2
1200.2.v.m.593.8 16 12.11 even 2