Properties

Label 1305.2.a.q.1.3
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.a (trivial)

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: yes
Analytic conductor: \(10.4204774638\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.16425\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.16425 q^{2} +2.68397 q^{4} -1.00000 q^{5} -4.84822 q^{7} +1.48028 q^{8} +O(q^{10})\) \(q+2.16425 q^{2} +2.68397 q^{4} -1.00000 q^{5} -4.84822 q^{7} +1.48028 q^{8} -2.16425 q^{10} -3.00000 q^{11} +0.519721 q^{13} -10.4927 q^{14} -2.16425 q^{16} +2.84822 q^{17} -3.36794 q^{19} -2.68397 q^{20} -6.49274 q^{22} -6.80877 q^{23} +1.00000 q^{25} +1.12481 q^{26} -13.0125 q^{28} +1.00000 q^{29} -7.64453 q^{32} +6.16425 q^{34} +4.84822 q^{35} +10.8088 q^{37} -7.28905 q^{38} -1.48028 q^{40} -7.84822 q^{41} +4.17671 q^{43} -8.05191 q^{44} -14.7359 q^{46} -6.21616 q^{47} +16.5052 q^{49} +2.16425 q^{50} +1.39492 q^{52} -3.44084 q^{53} +3.00000 q^{55} -7.17671 q^{56} +2.16425 q^{58} -12.3285 q^{59} +5.36794 q^{61} -12.2162 q^{64} -0.519721 q^{65} -15.8088 q^{67} +7.64453 q^{68} +10.4927 q^{70} +14.9855 q^{71} +8.58409 q^{73} +23.3929 q^{74} -9.03944 q^{76} +14.5447 q^{77} -1.59262 q^{79} +2.16425 q^{80} -16.9855 q^{82} +6.48028 q^{83} -2.84822 q^{85} +9.03944 q^{86} -4.44084 q^{88} -14.4408 q^{89} -2.51972 q^{91} -18.2745 q^{92} -13.4533 q^{94} +3.36794 q^{95} +11.1373 q^{97} +35.7214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} - 4 q^{7} + q^{10} - 9 q^{11} + 6 q^{13} - 9 q^{14} + q^{16} - 2 q^{17} - 4 q^{19} - 5 q^{20} + 3 q^{22} - q^{23} + 3 q^{25} - 13 q^{26} - 21 q^{28} + 3 q^{29} - 11 q^{32} + 11 q^{34} + 4 q^{35} + 13 q^{37} + 2 q^{38} - 13 q^{41} - 13 q^{43} - 15 q^{44} - 32 q^{46} - 2 q^{47} + 9 q^{49} - q^{50} + 25 q^{52} + 3 q^{53} + 9 q^{55} + 4 q^{56} - q^{58} - 22 q^{59} + 10 q^{61} - 20 q^{64} - 6 q^{65} - 28 q^{67} + 11 q^{68} + 9 q^{70} + 3 q^{73} + 28 q^{74} - 36 q^{76} + 12 q^{77} - 2 q^{79} - q^{80} - 6 q^{82} + 15 q^{83} + 2 q^{85} + 36 q^{86} - 30 q^{89} - 12 q^{91} + 14 q^{92} - 9 q^{94} + 4 q^{95} - q^{97} + 50 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.16425 1.53035 0.765177 0.643820i \(-0.222651\pi\)
0.765177 + 0.643820i \(0.222651\pi\)
\(3\) 0 0
\(4\) 2.68397 1.34198
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.84822 −1.83245 −0.916227 0.400660i \(-0.868781\pi\)
−0.916227 + 0.400660i \(0.868781\pi\)
\(8\) 1.48028 0.523358
\(9\) 0 0
\(10\) −2.16425 −0.684395
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 0.519721 0.144145 0.0720724 0.997399i \(-0.477039\pi\)
0.0720724 + 0.997399i \(0.477039\pi\)
\(14\) −10.4927 −2.80430
\(15\) 0 0
\(16\) −2.16425 −0.541062
\(17\) 2.84822 0.690794 0.345397 0.938457i \(-0.387744\pi\)
0.345397 + 0.938457i \(0.387744\pi\)
\(18\) 0 0
\(19\) −3.36794 −0.772658 −0.386329 0.922361i \(-0.626257\pi\)
−0.386329 + 0.922361i \(0.626257\pi\)
\(20\) −2.68397 −0.600154
\(21\) 0 0
\(22\) −6.49274 −1.38426
\(23\) −6.80877 −1.41973 −0.709864 0.704339i \(-0.751244\pi\)
−0.709864 + 0.704339i \(0.751244\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.12481 0.220593
\(27\) 0 0
\(28\) −13.0125 −2.45912
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −7.64453 −1.35137
\(33\) 0 0
\(34\) 6.16425 1.05716
\(35\) 4.84822 0.819498
\(36\) 0 0
\(37\) 10.8088 1.77695 0.888476 0.458923i \(-0.151765\pi\)
0.888476 + 0.458923i \(0.151765\pi\)
\(38\) −7.28905 −1.18244
\(39\) 0 0
\(40\) −1.48028 −0.234053
\(41\) −7.84822 −1.22569 −0.612843 0.790205i \(-0.709974\pi\)
−0.612843 + 0.790205i \(0.709974\pi\)
\(42\) 0 0
\(43\) 4.17671 0.636943 0.318471 0.947932i \(-0.396831\pi\)
0.318471 + 0.947932i \(0.396831\pi\)
\(44\) −8.05191 −1.21387
\(45\) 0 0
\(46\) −14.7359 −2.17269
\(47\) −6.21616 −0.906719 −0.453360 0.891328i \(-0.649775\pi\)
−0.453360 + 0.891328i \(0.649775\pi\)
\(48\) 0 0
\(49\) 16.5052 2.35789
\(50\) 2.16425 0.306071
\(51\) 0 0
\(52\) 1.39492 0.193440
\(53\) −3.44084 −0.472635 −0.236318 0.971676i \(-0.575941\pi\)
−0.236318 + 0.971676i \(0.575941\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) −7.17671 −0.959029
\(57\) 0 0
\(58\) 2.16425 0.284180
\(59\) −12.3285 −1.60503 −0.802517 0.596630i \(-0.796506\pi\)
−0.802517 + 0.596630i \(0.796506\pi\)
\(60\) 0 0
\(61\) 5.36794 0.687294 0.343647 0.939099i \(-0.388338\pi\)
0.343647 + 0.939099i \(0.388338\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −12.2162 −1.52702
\(65\) −0.519721 −0.0644635
\(66\) 0 0
\(67\) −15.8088 −1.93135 −0.965675 0.259755i \(-0.916358\pi\)
−0.965675 + 0.259755i \(0.916358\pi\)
\(68\) 7.64453 0.927035
\(69\) 0 0
\(70\) 10.4927 1.25412
\(71\) 14.9855 1.77845 0.889225 0.457470i \(-0.151244\pi\)
0.889225 + 0.457470i \(0.151244\pi\)
\(72\) 0 0
\(73\) 8.58409 1.00469 0.502346 0.864667i \(-0.332470\pi\)
0.502346 + 0.864667i \(0.332470\pi\)
\(74\) 23.3929 2.71937
\(75\) 0 0
\(76\) −9.03944 −1.03690
\(77\) 14.5447 1.65752
\(78\) 0 0
\(79\) −1.59262 −0.179184 −0.0895918 0.995979i \(-0.528556\pi\)
−0.0895918 + 0.995979i \(0.528556\pi\)
\(80\) 2.16425 0.241970
\(81\) 0 0
\(82\) −16.9855 −1.87573
\(83\) 6.48028 0.711303 0.355652 0.934619i \(-0.384259\pi\)
0.355652 + 0.934619i \(0.384259\pi\)
\(84\) 0 0
\(85\) −2.84822 −0.308933
\(86\) 9.03944 0.974748
\(87\) 0 0
\(88\) −4.44084 −0.473395
\(89\) −14.4408 −1.53073 −0.765363 0.643599i \(-0.777440\pi\)
−0.765363 + 0.643599i \(0.777440\pi\)
\(90\) 0 0
\(91\) −2.51972 −0.264139
\(92\) −18.2745 −1.90525
\(93\) 0 0
\(94\) −13.4533 −1.38760
\(95\) 3.36794 0.345543
\(96\) 0 0
\(97\) 11.1373 1.13082 0.565409 0.824811i \(-0.308718\pi\)
0.565409 + 0.824811i \(0.308718\pi\)
\(98\) 35.7214 3.60840
\(99\) 0 0
\(100\) 2.68397 0.268397
\(101\) −5.63206 −0.560411 −0.280206 0.959940i \(-0.590403\pi\)
−0.280206 + 0.959940i \(0.590403\pi\)
\(102\) 0 0
\(103\) 12.3534 1.21722 0.608610 0.793470i \(-0.291728\pi\)
0.608610 + 0.793470i \(0.291728\pi\)
\(104\) 0.769332 0.0754392
\(105\) 0 0
\(106\) −7.44682 −0.723299
\(107\) 6.96056 0.672903 0.336451 0.941701i \(-0.390773\pi\)
0.336451 + 0.941701i \(0.390773\pi\)
\(108\) 0 0
\(109\) −7.73588 −0.740963 −0.370481 0.928840i \(-0.620807\pi\)
−0.370481 + 0.928840i \(0.620807\pi\)
\(110\) 6.49274 0.619059
\(111\) 0 0
\(112\) 10.4927 0.991471
\(113\) 7.50521 0.706031 0.353015 0.935618i \(-0.385156\pi\)
0.353015 + 0.935618i \(0.385156\pi\)
\(114\) 0 0
\(115\) 6.80877 0.634922
\(116\) 2.68397 0.249200
\(117\) 0 0
\(118\) −26.6819 −2.45627
\(119\) −13.8088 −1.26585
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 11.6175 1.05180
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.58409 −0.584244 −0.292122 0.956381i \(-0.594361\pi\)
−0.292122 + 0.956381i \(0.594361\pi\)
\(128\) −11.1497 −0.985507
\(129\) 0 0
\(130\) −1.12481 −0.0986520
\(131\) 7.50521 0.655733 0.327867 0.944724i \(-0.393670\pi\)
0.327867 + 0.944724i \(0.393670\pi\)
\(132\) 0 0
\(133\) 16.3285 1.41586
\(134\) −34.2141 −2.95565
\(135\) 0 0
\(136\) 4.21616 0.361532
\(137\) 0.224681 0.0191958 0.00959790 0.999954i \(-0.496945\pi\)
0.00959790 + 0.999954i \(0.496945\pi\)
\(138\) 0 0
\(139\) −4.36794 −0.370484 −0.185242 0.982693i \(-0.559307\pi\)
−0.185242 + 0.982693i \(0.559307\pi\)
\(140\) 13.0125 1.09975
\(141\) 0 0
\(142\) 32.4323 2.72166
\(143\) −1.55916 −0.130384
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 18.5781 1.53754
\(147\) 0 0
\(148\) 29.0104 2.38464
\(149\) −10.3285 −0.846143 −0.423072 0.906096i \(-0.639048\pi\)
−0.423072 + 0.906096i \(0.639048\pi\)
\(150\) 0 0
\(151\) −5.84822 −0.475921 −0.237961 0.971275i \(-0.576479\pi\)
−0.237961 + 0.971275i \(0.576479\pi\)
\(152\) −4.98549 −0.404376
\(153\) 0 0
\(154\) 31.4782 2.53659
\(155\) 0 0
\(156\) 0 0
\(157\) 7.36794 0.588025 0.294013 0.955801i \(-0.405009\pi\)
0.294013 + 0.955801i \(0.405009\pi\)
\(158\) −3.44682 −0.274215
\(159\) 0 0
\(160\) 7.64453 0.604353
\(161\) 33.0104 2.60159
\(162\) 0 0
\(163\) −4.55916 −0.357101 −0.178551 0.983931i \(-0.557141\pi\)
−0.178551 + 0.983931i \(0.557141\pi\)
\(164\) −21.0644 −1.64485
\(165\) 0 0
\(166\) 14.0249 1.08855
\(167\) −19.3929 −1.50067 −0.750333 0.661061i \(-0.770107\pi\)
−0.750333 + 0.661061i \(0.770107\pi\)
\(168\) 0 0
\(169\) −12.7299 −0.979222
\(170\) −6.16425 −0.472776
\(171\) 0 0
\(172\) 11.2102 0.854767
\(173\) −3.92710 −0.298572 −0.149286 0.988794i \(-0.547698\pi\)
−0.149286 + 0.988794i \(0.547698\pi\)
\(174\) 0 0
\(175\) −4.84822 −0.366491
\(176\) 6.49274 0.489409
\(177\) 0 0
\(178\) −31.2536 −2.34255
\(179\) −4.40738 −0.329423 −0.164712 0.986342i \(-0.552669\pi\)
−0.164712 + 0.986342i \(0.552669\pi\)
\(180\) 0 0
\(181\) −7.65699 −0.569140 −0.284570 0.958655i \(-0.591851\pi\)
−0.284570 + 0.958655i \(0.591851\pi\)
\(182\) −5.45330 −0.404226
\(183\) 0 0
\(184\) −10.0789 −0.743025
\(185\) −10.8088 −0.794677
\(186\) 0 0
\(187\) −8.54465 −0.624847
\(188\) −16.6840 −1.21680
\(189\) 0 0
\(190\) 7.28905 0.528804
\(191\) −18.5052 −1.33899 −0.669495 0.742817i \(-0.733489\pi\)
−0.669495 + 0.742817i \(0.733489\pi\)
\(192\) 0 0
\(193\) −17.3929 −1.25197 −0.625983 0.779837i \(-0.715302\pi\)
−0.625983 + 0.779837i \(0.715302\pi\)
\(194\) 24.1038 1.73055
\(195\) 0 0
\(196\) 44.2995 3.16425
\(197\) −12.8088 −0.912587 −0.456294 0.889829i \(-0.650823\pi\)
−0.456294 + 0.889829i \(0.650823\pi\)
\(198\) 0 0
\(199\) −7.63206 −0.541023 −0.270511 0.962717i \(-0.587193\pi\)
−0.270511 + 0.962717i \(0.587193\pi\)
\(200\) 1.48028 0.104672
\(201\) 0 0
\(202\) −12.1892 −0.857628
\(203\) −4.84822 −0.340278
\(204\) 0 0
\(205\) 7.84822 0.548143
\(206\) 26.7359 1.86278
\(207\) 0 0
\(208\) −1.12481 −0.0779912
\(209\) 10.1038 0.698895
\(210\) 0 0
\(211\) 7.47175 0.514377 0.257188 0.966361i \(-0.417204\pi\)
0.257188 + 0.966361i \(0.417204\pi\)
\(212\) −9.23510 −0.634269
\(213\) 0 0
\(214\) 15.0644 1.02978
\(215\) −4.17671 −0.284849
\(216\) 0 0
\(217\) 0 0
\(218\) −16.7424 −1.13394
\(219\) 0 0
\(220\) 8.05191 0.542859
\(221\) 1.48028 0.0995743
\(222\) 0 0
\(223\) 8.54465 0.572192 0.286096 0.958201i \(-0.407642\pi\)
0.286096 + 0.958201i \(0.407642\pi\)
\(224\) 37.0623 2.47633
\(225\) 0 0
\(226\) 16.2431 1.08048
\(227\) −3.13727 −0.208228 −0.104114 0.994565i \(-0.533201\pi\)
−0.104114 + 0.994565i \(0.533201\pi\)
\(228\) 0 0
\(229\) −18.0499 −1.19277 −0.596384 0.802699i \(-0.703396\pi\)
−0.596384 + 0.802699i \(0.703396\pi\)
\(230\) 14.7359 0.971655
\(231\) 0 0
\(232\) 1.48028 0.0971851
\(233\) −1.92710 −0.126249 −0.0631243 0.998006i \(-0.520106\pi\)
−0.0631243 + 0.998006i \(0.520106\pi\)
\(234\) 0 0
\(235\) 6.21616 0.405497
\(236\) −33.0893 −2.15393
\(237\) 0 0
\(238\) −29.8856 −1.93720
\(239\) −17.3929 −1.12505 −0.562526 0.826780i \(-0.690170\pi\)
−0.562526 + 0.826780i \(0.690170\pi\)
\(240\) 0 0
\(241\) 20.6175 1.32809 0.664047 0.747691i \(-0.268838\pi\)
0.664047 + 0.747691i \(0.268838\pi\)
\(242\) −4.32850 −0.278246
\(243\) 0 0
\(244\) 14.4074 0.922338
\(245\) −16.5052 −1.05448
\(246\) 0 0
\(247\) −1.75039 −0.111375
\(248\) 0 0
\(249\) 0 0
\(250\) −2.16425 −0.136879
\(251\) 16.1622 1.02015 0.510075 0.860130i \(-0.329618\pi\)
0.510075 + 0.860130i \(0.329618\pi\)
\(252\) 0 0
\(253\) 20.4263 1.28419
\(254\) −14.2496 −0.894100
\(255\) 0 0
\(256\) 0.301518 0.0188449
\(257\) 10.5592 0.658663 0.329331 0.944214i \(-0.393177\pi\)
0.329331 + 0.944214i \(0.393177\pi\)
\(258\) 0 0
\(259\) −52.4033 −3.25618
\(260\) −1.39492 −0.0865090
\(261\) 0 0
\(262\) 16.2431 1.00350
\(263\) 13.4718 0.830704 0.415352 0.909661i \(-0.363658\pi\)
0.415352 + 0.909661i \(0.363658\pi\)
\(264\) 0 0
\(265\) 3.44084 0.211369
\(266\) 35.3389 2.16677
\(267\) 0 0
\(268\) −42.4303 −2.59184
\(269\) 6.44084 0.392705 0.196352 0.980533i \(-0.437090\pi\)
0.196352 + 0.980533i \(0.437090\pi\)
\(270\) 0 0
\(271\) 26.0499 1.58242 0.791208 0.611547i \(-0.209452\pi\)
0.791208 + 0.611547i \(0.209452\pi\)
\(272\) −6.16425 −0.373762
\(273\) 0 0
\(274\) 0.486265 0.0293764
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −32.7154 −1.96568 −0.982838 0.184469i \(-0.940943\pi\)
−0.982838 + 0.184469i \(0.940943\pi\)
\(278\) −9.45330 −0.566971
\(279\) 0 0
\(280\) 7.17671 0.428891
\(281\) 17.6425 1.05246 0.526231 0.850342i \(-0.323605\pi\)
0.526231 + 0.850342i \(0.323605\pi\)
\(282\) 0 0
\(283\) 26.4323 1.57124 0.785619 0.618711i \(-0.212345\pi\)
0.785619 + 0.618711i \(0.212345\pi\)
\(284\) 40.2206 2.38665
\(285\) 0 0
\(286\) −3.37442 −0.199533
\(287\) 38.0499 2.24601
\(288\) 0 0
\(289\) −8.88766 −0.522803
\(290\) −2.16425 −0.127089
\(291\) 0 0
\(292\) 23.0394 1.34828
\(293\) −15.5052 −0.905824 −0.452912 0.891555i \(-0.649615\pi\)
−0.452912 + 0.891555i \(0.649615\pi\)
\(294\) 0 0
\(295\) 12.3285 0.717793
\(296\) 16.0000 0.929981
\(297\) 0 0
\(298\) −22.3534 −1.29490
\(299\) −3.53866 −0.204646
\(300\) 0 0
\(301\) −20.2496 −1.16717
\(302\) −12.6570 −0.728328
\(303\) 0 0
\(304\) 7.28905 0.418056
\(305\) −5.36794 −0.307367
\(306\) 0 0
\(307\) 11.1123 0.634215 0.317107 0.948390i \(-0.397288\pi\)
0.317107 + 0.948390i \(0.397288\pi\)
\(308\) 39.0374 2.22436
\(309\) 0 0
\(310\) 0 0
\(311\) −11.0000 −0.623753 −0.311876 0.950123i \(-0.600957\pi\)
−0.311876 + 0.950123i \(0.600957\pi\)
\(312\) 0 0
\(313\) 17.3225 0.979126 0.489563 0.871968i \(-0.337156\pi\)
0.489563 + 0.871968i \(0.337156\pi\)
\(314\) 15.9460 0.899887
\(315\) 0 0
\(316\) −4.27454 −0.240462
\(317\) −22.2411 −1.24918 −0.624592 0.780951i \(-0.714735\pi\)
−0.624592 + 0.780951i \(0.714735\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) 12.2162 0.682904
\(321\) 0 0
\(322\) 71.4427 3.98135
\(323\) −9.59262 −0.533748
\(324\) 0 0
\(325\) 0.519721 0.0288289
\(326\) −9.86716 −0.546491
\(327\) 0 0
\(328\) −11.6175 −0.641472
\(329\) 30.1373 1.66152
\(330\) 0 0
\(331\) 29.5387 1.62359 0.811796 0.583941i \(-0.198490\pi\)
0.811796 + 0.583941i \(0.198490\pi\)
\(332\) 17.3929 0.954558
\(333\) 0 0
\(334\) −41.9710 −2.29655
\(335\) 15.8088 0.863726
\(336\) 0 0
\(337\) −9.34301 −0.508946 −0.254473 0.967080i \(-0.581902\pi\)
−0.254473 + 0.967080i \(0.581902\pi\)
\(338\) −27.5506 −1.49856
\(339\) 0 0
\(340\) −7.64453 −0.414583
\(341\) 0 0
\(342\) 0 0
\(343\) −46.0833 −2.48827
\(344\) 6.18270 0.333349
\(345\) 0 0
\(346\) −8.49922 −0.456921
\(347\) 25.8482 1.38760 0.693802 0.720165i \(-0.255934\pi\)
0.693802 + 0.720165i \(0.255934\pi\)
\(348\) 0 0
\(349\) −33.9770 −1.81875 −0.909373 0.415983i \(-0.863438\pi\)
−0.909373 + 0.415983i \(0.863438\pi\)
\(350\) −10.4927 −0.560861
\(351\) 0 0
\(352\) 22.9336 1.22236
\(353\) 24.7359 1.31656 0.658279 0.752774i \(-0.271285\pi\)
0.658279 + 0.752774i \(0.271285\pi\)
\(354\) 0 0
\(355\) −14.9855 −0.795347
\(356\) −38.7588 −2.05421
\(357\) 0 0
\(358\) −9.53866 −0.504134
\(359\) 16.0729 0.848295 0.424148 0.905593i \(-0.360574\pi\)
0.424148 + 0.905593i \(0.360574\pi\)
\(360\) 0 0
\(361\) −7.65699 −0.403000
\(362\) −16.5716 −0.870985
\(363\) 0 0
\(364\) −6.76285 −0.354470
\(365\) −8.58409 −0.449312
\(366\) 0 0
\(367\) −7.46577 −0.389710 −0.194855 0.980832i \(-0.562424\pi\)
−0.194855 + 0.980832i \(0.562424\pi\)
\(368\) 14.7359 0.768161
\(369\) 0 0
\(370\) −23.3929 −1.21614
\(371\) 16.6819 0.866082
\(372\) 0 0
\(373\) 17.0893 0.884851 0.442425 0.896805i \(-0.354118\pi\)
0.442425 + 0.896805i \(0.354118\pi\)
\(374\) −18.4927 −0.956237
\(375\) 0 0
\(376\) −9.20164 −0.474539
\(377\) 0.519721 0.0267670
\(378\) 0 0
\(379\) 19.1682 0.984604 0.492302 0.870425i \(-0.336156\pi\)
0.492302 + 0.870425i \(0.336156\pi\)
\(380\) 9.03944 0.463714
\(381\) 0 0
\(382\) −40.0499 −2.04913
\(383\) −13.8731 −0.708885 −0.354442 0.935078i \(-0.615329\pi\)
−0.354442 + 0.935078i \(0.615329\pi\)
\(384\) 0 0
\(385\) −14.5447 −0.741264
\(386\) −37.6425 −1.91595
\(387\) 0 0
\(388\) 29.8921 1.51754
\(389\) −20.3389 −1.03122 −0.515612 0.856822i \(-0.672435\pi\)
−0.515612 + 0.856822i \(0.672435\pi\)
\(390\) 0 0
\(391\) −19.3929 −0.980740
\(392\) 24.4323 1.23402
\(393\) 0 0
\(394\) −27.7214 −1.39658
\(395\) 1.59262 0.0801334
\(396\) 0 0
\(397\) 32.1287 1.61250 0.806248 0.591578i \(-0.201495\pi\)
0.806248 + 0.591578i \(0.201495\pi\)
\(398\) −16.5177 −0.827956
\(399\) 0 0
\(400\) −2.16425 −0.108212
\(401\) −21.0353 −1.05046 −0.525228 0.850962i \(-0.676020\pi\)
−0.525228 + 0.850962i \(0.676020\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −15.1163 −0.752063
\(405\) 0 0
\(406\) −10.4927 −0.520746
\(407\) −32.4263 −1.60731
\(408\) 0 0
\(409\) −7.51373 −0.371530 −0.185765 0.982594i \(-0.559476\pi\)
−0.185765 + 0.982594i \(0.559476\pi\)
\(410\) 16.9855 0.838853
\(411\) 0 0
\(412\) 33.1562 1.63349
\(413\) 59.7712 2.94115
\(414\) 0 0
\(415\) −6.48028 −0.318104
\(416\) −3.97302 −0.194793
\(417\) 0 0
\(418\) 21.8672 1.06956
\(419\) 14.6570 0.716041 0.358020 0.933714i \(-0.383452\pi\)
0.358020 + 0.933714i \(0.383452\pi\)
\(420\) 0 0
\(421\) −25.4967 −1.24263 −0.621316 0.783560i \(-0.713402\pi\)
−0.621316 + 0.783560i \(0.713402\pi\)
\(422\) 16.1707 0.787179
\(423\) 0 0
\(424\) −5.09340 −0.247357
\(425\) 2.84822 0.138159
\(426\) 0 0
\(427\) −26.0249 −1.25943
\(428\) 18.6819 0.903025
\(429\) 0 0
\(430\) −9.03944 −0.435921
\(431\) −36.1957 −1.74348 −0.871742 0.489966i \(-0.837009\pi\)
−0.871742 + 0.489966i \(0.837009\pi\)
\(432\) 0 0
\(433\) −14.5301 −0.698274 −0.349137 0.937072i \(-0.613525\pi\)
−0.349137 + 0.937072i \(0.613525\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −20.7629 −0.994360
\(437\) 22.9315 1.09696
\(438\) 0 0
\(439\) 10.5696 0.504459 0.252229 0.967667i \(-0.418836\pi\)
0.252229 + 0.967667i \(0.418836\pi\)
\(440\) 4.44084 0.211709
\(441\) 0 0
\(442\) 3.20369 0.152384
\(443\) −39.0019 −1.85304 −0.926518 0.376251i \(-0.877213\pi\)
−0.926518 + 0.376251i \(0.877213\pi\)
\(444\) 0 0
\(445\) 14.4408 0.684561
\(446\) 18.4927 0.875657
\(447\) 0 0
\(448\) 59.2266 2.79819
\(449\) 20.9460 0.988505 0.494252 0.869318i \(-0.335442\pi\)
0.494252 + 0.869318i \(0.335442\pi\)
\(450\) 0 0
\(451\) 23.5447 1.10867
\(452\) 20.1437 0.947482
\(453\) 0 0
\(454\) −6.78983 −0.318663
\(455\) 2.51972 0.118126
\(456\) 0 0
\(457\) −25.9126 −1.21214 −0.606070 0.795411i \(-0.707255\pi\)
−0.606070 + 0.795411i \(0.707255\pi\)
\(458\) −39.0644 −1.82536
\(459\) 0 0
\(460\) 18.2745 0.852055
\(461\) −25.7693 −1.20020 −0.600099 0.799926i \(-0.704872\pi\)
−0.600099 + 0.799926i \(0.704872\pi\)
\(462\) 0 0
\(463\) −33.2805 −1.54668 −0.773339 0.633993i \(-0.781415\pi\)
−0.773339 + 0.633993i \(0.781415\pi\)
\(464\) −2.16425 −0.100473
\(465\) 0 0
\(466\) −4.17073 −0.193205
\(467\) 34.4822 1.59564 0.797822 0.602893i \(-0.205985\pi\)
0.797822 + 0.602893i \(0.205985\pi\)
\(468\) 0 0
\(469\) 76.6444 3.53911
\(470\) 13.4533 0.620555
\(471\) 0 0
\(472\) −18.2496 −0.840006
\(473\) −12.5301 −0.576136
\(474\) 0 0
\(475\) −3.36794 −0.154532
\(476\) −37.0623 −1.69875
\(477\) 0 0
\(478\) −37.6425 −1.72173
\(479\) 13.2641 0.606053 0.303027 0.952982i \(-0.402003\pi\)
0.303027 + 0.952982i \(0.402003\pi\)
\(480\) 0 0
\(481\) 5.61755 0.256138
\(482\) 44.6215 2.03245
\(483\) 0 0
\(484\) −5.36794 −0.243997
\(485\) −11.1373 −0.505717
\(486\) 0 0
\(487\) −23.2351 −1.05288 −0.526441 0.850211i \(-0.676474\pi\)
−0.526441 + 0.850211i \(0.676474\pi\)
\(488\) 7.94605 0.359701
\(489\) 0 0
\(490\) −35.7214 −1.61373
\(491\) −17.9211 −0.808769 −0.404384 0.914589i \(-0.632514\pi\)
−0.404384 + 0.914589i \(0.632514\pi\)
\(492\) 0 0
\(493\) 2.84822 0.128277
\(494\) −3.78828 −0.170443
\(495\) 0 0
\(496\) 0 0
\(497\) −72.6529 −3.25893
\(498\) 0 0
\(499\) −2.46986 −0.110566 −0.0552831 0.998471i \(-0.517606\pi\)
−0.0552831 + 0.998471i \(0.517606\pi\)
\(500\) −2.68397 −0.120031
\(501\) 0 0
\(502\) 34.9790 1.56119
\(503\) 21.2266 0.946446 0.473223 0.880943i \(-0.343090\pi\)
0.473223 + 0.880943i \(0.343090\pi\)
\(504\) 0 0
\(505\) 5.63206 0.250623
\(506\) 44.2076 1.96527
\(507\) 0 0
\(508\) −17.6715 −0.784046
\(509\) −37.9460 −1.68193 −0.840964 0.541090i \(-0.818012\pi\)
−0.840964 + 0.541090i \(0.818012\pi\)
\(510\) 0 0
\(511\) −41.6175 −1.84105
\(512\) 22.9520 1.01435
\(513\) 0 0
\(514\) 22.8526 1.00799
\(515\) −12.3534 −0.544357
\(516\) 0 0
\(517\) 18.6485 0.820159
\(518\) −113.414 −4.98311
\(519\) 0 0
\(520\) −0.769332 −0.0337375
\(521\) −8.40738 −0.368334 −0.184167 0.982895i \(-0.558959\pi\)
−0.184167 + 0.982895i \(0.558959\pi\)
\(522\) 0 0
\(523\) 3.45535 0.151092 0.0755459 0.997142i \(-0.475930\pi\)
0.0755459 + 0.997142i \(0.475930\pi\)
\(524\) 20.1437 0.879984
\(525\) 0 0
\(526\) 29.1562 1.27127
\(527\) 0 0
\(528\) 0 0
\(529\) 23.3594 1.01563
\(530\) 7.44682 0.323469
\(531\) 0 0
\(532\) 43.8252 1.90006
\(533\) −4.07888 −0.176676
\(534\) 0 0
\(535\) −6.96056 −0.300931
\(536\) −23.4014 −1.01079
\(537\) 0 0
\(538\) 13.9396 0.600977
\(539\) −49.5156 −2.13279
\(540\) 0 0
\(541\) 16.7359 0.719532 0.359766 0.933043i \(-0.382857\pi\)
0.359766 + 0.933043i \(0.382857\pi\)
\(542\) 56.3784 2.42166
\(543\) 0 0
\(544\) −21.7733 −0.933521
\(545\) 7.73588 0.331369
\(546\) 0 0
\(547\) 12.1123 0.517886 0.258943 0.965893i \(-0.416626\pi\)
0.258943 + 0.965893i \(0.416626\pi\)
\(548\) 0.603037 0.0257605
\(549\) 0 0
\(550\) −6.49274 −0.276852
\(551\) −3.36794 −0.143479
\(552\) 0 0
\(553\) 7.72136 0.328346
\(554\) −70.8042 −3.00818
\(555\) 0 0
\(556\) −11.7234 −0.497183
\(557\) −2.33448 −0.0989152 −0.0494576 0.998776i \(-0.515749\pi\)
−0.0494576 + 0.998776i \(0.515749\pi\)
\(558\) 0 0
\(559\) 2.17073 0.0918119
\(560\) −10.4927 −0.443399
\(561\) 0 0
\(562\) 38.1827 1.61064
\(563\) 4.13727 0.174365 0.0871826 0.996192i \(-0.472214\pi\)
0.0871826 + 0.996192i \(0.472214\pi\)
\(564\) 0 0
\(565\) −7.50521 −0.315747
\(566\) 57.2061 2.40455
\(567\) 0 0
\(568\) 22.1827 0.930765
\(569\) −30.6485 −1.28485 −0.642425 0.766348i \(-0.722072\pi\)
−0.642425 + 0.766348i \(0.722072\pi\)
\(570\) 0 0
\(571\) −35.2121 −1.47358 −0.736789 0.676122i \(-0.763659\pi\)
−0.736789 + 0.676122i \(0.763659\pi\)
\(572\) −4.18475 −0.174973
\(573\) 0 0
\(574\) 82.3493 3.43719
\(575\) −6.80877 −0.283946
\(576\) 0 0
\(577\) −3.03944 −0.126534 −0.0632668 0.997997i \(-0.520152\pi\)
−0.0632668 + 0.997997i \(0.520152\pi\)
\(578\) −19.2351 −0.800075
\(579\) 0 0
\(580\) −2.68397 −0.111446
\(581\) −31.4178 −1.30343
\(582\) 0 0
\(583\) 10.3225 0.427515
\(584\) 12.7069 0.525813
\(585\) 0 0
\(586\) −33.5571 −1.38623
\(587\) 3.51373 0.145027 0.0725137 0.997367i \(-0.476898\pi\)
0.0725137 + 0.997367i \(0.476898\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 26.6819 1.09848
\(591\) 0 0
\(592\) −23.3929 −0.961441
\(593\) 16.8397 0.691523 0.345762 0.938322i \(-0.387621\pi\)
0.345762 + 0.938322i \(0.387621\pi\)
\(594\) 0 0
\(595\) 13.8088 0.566105
\(596\) −27.7214 −1.13551
\(597\) 0 0
\(598\) −7.65855 −0.313181
\(599\) 8.16220 0.333498 0.166749 0.985999i \(-0.446673\pi\)
0.166749 + 0.985999i \(0.446673\pi\)
\(600\) 0 0
\(601\) −6.30357 −0.257128 −0.128564 0.991701i \(-0.541037\pi\)
−0.128564 + 0.991701i \(0.541037\pi\)
\(602\) −43.8252 −1.78618
\(603\) 0 0
\(604\) −15.6964 −0.638679
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 16.1707 0.656350 0.328175 0.944617i \(-0.393567\pi\)
0.328175 + 0.944617i \(0.393567\pi\)
\(608\) 25.7463 1.04415
\(609\) 0 0
\(610\) −11.6175 −0.470381
\(611\) −3.23067 −0.130699
\(612\) 0 0
\(613\) 0.469861 0.0189775 0.00948876 0.999955i \(-0.496980\pi\)
0.00948876 + 0.999955i \(0.496980\pi\)
\(614\) 24.0499 0.970573
\(615\) 0 0
\(616\) 21.5301 0.867474
\(617\) 22.2076 0.894046 0.447023 0.894523i \(-0.352484\pi\)
0.447023 + 0.894523i \(0.352484\pi\)
\(618\) 0 0
\(619\) −40.5781 −1.63097 −0.815486 0.578777i \(-0.803530\pi\)
−0.815486 + 0.578777i \(0.803530\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −23.8067 −0.954563
\(623\) 70.0123 2.80498
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 37.4902 1.49841
\(627\) 0 0
\(628\) 19.7753 0.789121
\(629\) 30.7857 1.22751
\(630\) 0 0
\(631\) 6.07036 0.241657 0.120829 0.992673i \(-0.461445\pi\)
0.120829 + 0.992673i \(0.461445\pi\)
\(632\) −2.35752 −0.0937771
\(633\) 0 0
\(634\) −48.1352 −1.91169
\(635\) 6.58409 0.261282
\(636\) 0 0
\(637\) 8.57811 0.339877
\(638\) −6.49274 −0.257050
\(639\) 0 0
\(640\) 11.1497 0.440732
\(641\) 40.7712 1.61037 0.805183 0.593026i \(-0.202067\pi\)
0.805183 + 0.593026i \(0.202067\pi\)
\(642\) 0 0
\(643\) 48.5447 1.91441 0.957207 0.289404i \(-0.0934571\pi\)
0.957207 + 0.289404i \(0.0934571\pi\)
\(644\) 88.5989 3.49129
\(645\) 0 0
\(646\) −20.7608 −0.816823
\(647\) −24.6879 −0.970582 −0.485291 0.874353i \(-0.661286\pi\)
−0.485291 + 0.874353i \(0.661286\pi\)
\(648\) 0 0
\(649\) 36.9855 1.45181
\(650\) 1.12481 0.0441185
\(651\) 0 0
\(652\) −12.2367 −0.479224
\(653\) 24.1912 0.946676 0.473338 0.880881i \(-0.343049\pi\)
0.473338 + 0.880881i \(0.343049\pi\)
\(654\) 0 0
\(655\) −7.50521 −0.293253
\(656\) 16.9855 0.663172
\(657\) 0 0
\(658\) 65.2245 2.54272
\(659\) 26.9041 1.04803 0.524017 0.851708i \(-0.324433\pi\)
0.524017 + 0.851708i \(0.324433\pi\)
\(660\) 0 0
\(661\) −6.24707 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(662\) 63.9290 2.48467
\(663\) 0 0
\(664\) 9.59262 0.372266
\(665\) −16.3285 −0.633192
\(666\) 0 0
\(667\) −6.80877 −0.263637
\(668\) −52.0499 −2.01387
\(669\) 0 0
\(670\) 34.2141 1.32181
\(671\) −16.1038 −0.621681
\(672\) 0 0
\(673\) −17.4014 −0.670774 −0.335387 0.942080i \(-0.608867\pi\)
−0.335387 + 0.942080i \(0.608867\pi\)
\(674\) −20.2206 −0.778868
\(675\) 0 0
\(676\) −34.1666 −1.31410
\(677\) 32.2121 1.23801 0.619005 0.785387i \(-0.287536\pi\)
0.619005 + 0.785387i \(0.287536\pi\)
\(678\) 0 0
\(679\) −53.9959 −2.07217
\(680\) −4.21616 −0.161682
\(681\) 0 0
\(682\) 0 0
\(683\) −48.9375 −1.87254 −0.936271 0.351278i \(-0.885747\pi\)
−0.936271 + 0.351278i \(0.885747\pi\)
\(684\) 0 0
\(685\) −0.224681 −0.00858462
\(686\) −99.7357 −3.80793
\(687\) 0 0
\(688\) −9.03944 −0.344626
\(689\) −1.78828 −0.0681279
\(690\) 0 0
\(691\) 31.6759 1.20501 0.602505 0.798115i \(-0.294169\pi\)
0.602505 + 0.798115i \(0.294169\pi\)
\(692\) −10.5402 −0.400679
\(693\) 0 0
\(694\) 55.9420 2.12353
\(695\) 4.36794 0.165685
\(696\) 0 0
\(697\) −22.3534 −0.846696
\(698\) −73.5346 −2.78332
\(699\) 0 0
\(700\) −13.0125 −0.491825
\(701\) −46.2206 −1.74573 −0.872864 0.487964i \(-0.837740\pi\)
−0.872864 + 0.487964i \(0.837740\pi\)
\(702\) 0 0
\(703\) −36.4033 −1.37298
\(704\) 36.6485 1.38124
\(705\) 0 0
\(706\) 53.5346 2.01480
\(707\) 27.3055 1.02693
\(708\) 0 0
\(709\) −1.97696 −0.0742464 −0.0371232 0.999311i \(-0.511819\pi\)
−0.0371232 + 0.999311i \(0.511819\pi\)
\(710\) −32.4323 −1.21716
\(711\) 0 0
\(712\) −21.3765 −0.801117
\(713\) 0 0
\(714\) 0 0
\(715\) 1.55916 0.0583094
\(716\) −11.8293 −0.442081
\(717\) 0 0
\(718\) 34.7857 1.29819
\(719\) −19.2891 −0.719360 −0.359680 0.933076i \(-0.617114\pi\)
−0.359680 + 0.933076i \(0.617114\pi\)
\(720\) 0 0
\(721\) −59.8921 −2.23050
\(722\) −16.5716 −0.616732
\(723\) 0 0
\(724\) −20.5511 −0.763777
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 1.92112 0.0712502 0.0356251 0.999365i \(-0.488658\pi\)
0.0356251 + 0.999365i \(0.488658\pi\)
\(728\) −3.72989 −0.138239
\(729\) 0 0
\(730\) −18.5781 −0.687607
\(731\) 11.8962 0.439996
\(732\) 0 0
\(733\) 31.8002 1.17457 0.587284 0.809381i \(-0.300197\pi\)
0.587284 + 0.809381i \(0.300197\pi\)
\(734\) −16.1578 −0.596394
\(735\) 0 0
\(736\) 52.0499 1.91858
\(737\) 47.4263 1.74697
\(738\) 0 0
\(739\) 11.4967 0.422912 0.211456 0.977387i \(-0.432179\pi\)
0.211456 + 0.977387i \(0.432179\pi\)
\(740\) −29.0104 −1.06644
\(741\) 0 0
\(742\) 36.1038 1.32541
\(743\) 25.3843 0.931261 0.465631 0.884979i \(-0.345828\pi\)
0.465631 + 0.884979i \(0.345828\pi\)
\(744\) 0 0
\(745\) 10.3285 0.378407
\(746\) 36.9855 1.35413
\(747\) 0 0
\(748\) −22.9336 −0.838535
\(749\) −33.7463 −1.23306
\(750\) 0 0
\(751\) 50.8356 1.85502 0.927509 0.373802i \(-0.121946\pi\)
0.927509 + 0.373802i \(0.121946\pi\)
\(752\) 13.4533 0.490591
\(753\) 0 0
\(754\) 1.12481 0.0409630
\(755\) 5.84822 0.212838
\(756\) 0 0
\(757\) 34.9546 1.27045 0.635223 0.772329i \(-0.280908\pi\)
0.635223 + 0.772329i \(0.280908\pi\)
\(758\) 41.4847 1.50679
\(759\) 0 0
\(760\) 4.98549 0.180843
\(761\) −17.1931 −0.623250 −0.311625 0.950205i \(-0.600873\pi\)
−0.311625 + 0.950205i \(0.600873\pi\)
\(762\) 0 0
\(763\) 37.5052 1.35778
\(764\) −49.6674 −1.79690
\(765\) 0 0
\(766\) −30.0249 −1.08484
\(767\) −6.40738 −0.231357
\(768\) 0 0
\(769\) −13.2102 −0.476371 −0.238185 0.971220i \(-0.576553\pi\)
−0.238185 + 0.971220i \(0.576553\pi\)
\(770\) −31.4782 −1.13440
\(771\) 0 0
\(772\) −46.6819 −1.68012
\(773\) 18.0499 0.649208 0.324604 0.945850i \(-0.394769\pi\)
0.324604 + 0.945850i \(0.394769\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 16.4863 0.591822
\(777\) 0 0
\(778\) −44.0185 −1.57814
\(779\) 26.4323 0.947036
\(780\) 0 0
\(781\) −44.9565 −1.60867
\(782\) −41.9710 −1.50088
\(783\) 0 0
\(784\) −35.7214 −1.27576
\(785\) −7.36794 −0.262973
\(786\) 0 0
\(787\) 9.31398 0.332008 0.166004 0.986125i \(-0.446914\pi\)
0.166004 + 0.986125i \(0.446914\pi\)
\(788\) −34.3784 −1.22468
\(789\) 0 0
\(790\) 3.44682 0.122632
\(791\) −36.3869 −1.29377
\(792\) 0 0
\(793\) 2.78983 0.0990698
\(794\) 69.5346 2.46769
\(795\) 0 0
\(796\) −20.4842 −0.726044
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −17.7050 −0.626356
\(800\) −7.64453 −0.270275
\(801\) 0 0
\(802\) −45.5257 −1.60757
\(803\) −25.7523 −0.908778
\(804\) 0 0
\(805\) −33.0104 −1.16346
\(806\) 0 0
\(807\) 0 0
\(808\) −8.33702 −0.293295
\(809\) −3.85674 −0.135596 −0.0677979 0.997699i \(-0.521597\pi\)
−0.0677979 + 0.997699i \(0.521597\pi\)
\(810\) 0 0
\(811\) 42.7883 1.50250 0.751250 0.660018i \(-0.229451\pi\)
0.751250 + 0.660018i \(0.229451\pi\)
\(812\) −13.0125 −0.456648
\(813\) 0 0
\(814\) −70.1786 −2.45976
\(815\) 4.55916 0.159701
\(816\) 0 0
\(817\) −14.0669 −0.492139
\(818\) −16.2616 −0.568573
\(819\) 0 0
\(820\) 21.0644 0.735600
\(821\) −17.7463 −0.619350 −0.309675 0.950842i \(-0.600220\pi\)
−0.309675 + 0.950842i \(0.600220\pi\)
\(822\) 0 0
\(823\) −49.2600 −1.71710 −0.858548 0.512733i \(-0.828633\pi\)
−0.858548 + 0.512733i \(0.828633\pi\)
\(824\) 18.2865 0.637041
\(825\) 0 0
\(826\) 129.360 4.50100
\(827\) −19.7463 −0.686646 −0.343323 0.939217i \(-0.611553\pi\)
−0.343323 + 0.939217i \(0.611553\pi\)
\(828\) 0 0
\(829\) 47.5387 1.65109 0.825543 0.564339i \(-0.190869\pi\)
0.825543 + 0.564339i \(0.190869\pi\)
\(830\) −14.0249 −0.486812
\(831\) 0 0
\(832\) −6.34899 −0.220112
\(833\) 47.0104 1.62881
\(834\) 0 0
\(835\) 19.3929 0.671118
\(836\) 27.1183 0.937907
\(837\) 0 0
\(838\) 31.7214 1.09580
\(839\) −30.4658 −1.05180 −0.525898 0.850548i \(-0.676270\pi\)
−0.525898 + 0.850548i \(0.676270\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −55.1811 −1.90167
\(843\) 0 0
\(844\) 20.0540 0.690286
\(845\) 12.7299 0.437922
\(846\) 0 0
\(847\) 9.69643 0.333173
\(848\) 7.44682 0.255725
\(849\) 0 0
\(850\) 6.16425 0.211432
\(851\) −73.5945 −2.52279
\(852\) 0 0
\(853\) −41.5406 −1.42232 −0.711161 0.703029i \(-0.751830\pi\)
−0.711161 + 0.703029i \(0.751830\pi\)
\(854\) −56.3244 −1.92738
\(855\) 0 0
\(856\) 10.3036 0.352169
\(857\) 8.05585 0.275182 0.137591 0.990489i \(-0.456064\pi\)
0.137591 + 0.990489i \(0.456064\pi\)
\(858\) 0 0
\(859\) −50.6529 −1.72825 −0.864127 0.503273i \(-0.832129\pi\)
−0.864127 + 0.503273i \(0.832129\pi\)
\(860\) −11.2102 −0.382264
\(861\) 0 0
\(862\) −78.3364 −2.66815
\(863\) 25.2600 0.859861 0.429931 0.902862i \(-0.358538\pi\)
0.429931 + 0.902862i \(0.358538\pi\)
\(864\) 0 0
\(865\) 3.92710 0.133525
\(866\) −31.4468 −1.06861
\(867\) 0 0
\(868\) 0 0
\(869\) 4.77786 0.162078
\(870\) 0 0
\(871\) −8.21616 −0.278394
\(872\) −11.4513 −0.387788
\(873\) 0 0
\(874\) 49.6295 1.67874
\(875\) 4.84822 0.163900
\(876\) 0 0
\(877\) 24.9315 0.841878 0.420939 0.907089i \(-0.361701\pi\)
0.420939 + 0.907089i \(0.361701\pi\)
\(878\) 22.8752 0.772000
\(879\) 0 0
\(880\) −6.49274 −0.218870
\(881\) 21.9356 0.739030 0.369515 0.929225i \(-0.379524\pi\)
0.369515 + 0.929225i \(0.379524\pi\)
\(882\) 0 0
\(883\) 7.39287 0.248790 0.124395 0.992233i \(-0.460301\pi\)
0.124395 + 0.992233i \(0.460301\pi\)
\(884\) 3.97302 0.133627
\(885\) 0 0
\(886\) −84.4098 −2.83580
\(887\) −19.5472 −0.656330 −0.328165 0.944620i \(-0.606430\pi\)
−0.328165 + 0.944620i \(0.606430\pi\)
\(888\) 0 0
\(889\) 31.9211 1.07060
\(890\) 31.2536 1.04762
\(891\) 0 0
\(892\) 22.9336 0.767873
\(893\) 20.9356 0.700584
\(894\) 0 0
\(895\) 4.40738 0.147322
\(896\) 54.0563 1.80590
\(897\) 0 0
\(898\) 45.3324 1.51276
\(899\) 0 0
\(900\) 0 0
\(901\) −9.80025 −0.326494
\(902\) 50.9565 1.69666
\(903\) 0 0
\(904\) 11.1098 0.369507
\(905\) 7.65699 0.254527
\(906\) 0 0
\(907\) 26.2266 0.870839 0.435420 0.900228i \(-0.356600\pi\)
0.435420 + 0.900228i \(0.356600\pi\)
\(908\) −8.42034 −0.279439
\(909\) 0 0
\(910\) 5.45330 0.180775
\(911\) 28.1392 0.932292 0.466146 0.884708i \(-0.345642\pi\)
0.466146 + 0.884708i \(0.345642\pi\)
\(912\) 0 0
\(913\) −19.4408 −0.643398
\(914\) −56.0813 −1.85500
\(915\) 0 0
\(916\) −48.4453 −1.60068
\(917\) −36.3869 −1.20160
\(918\) 0 0
\(919\) −42.9520 −1.41686 −0.708428 0.705783i \(-0.750595\pi\)
−0.708428 + 0.705783i \(0.750595\pi\)
\(920\) 10.0789 0.332291
\(921\) 0 0
\(922\) −55.7712 −1.83673
\(923\) 7.78828 0.256354
\(924\) 0 0
\(925\) 10.8088 0.355390
\(926\) −72.0273 −2.36696
\(927\) 0 0
\(928\) −7.64453 −0.250944
\(929\) 56.4822 1.85312 0.926560 0.376147i \(-0.122751\pi\)
0.926560 + 0.376147i \(0.122751\pi\)
\(930\) 0 0
\(931\) −55.5885 −1.82184
\(932\) −5.17228 −0.169424
\(933\) 0 0
\(934\) 74.6280 2.44190
\(935\) 8.54465 0.279440
\(936\) 0 0
\(937\) −58.3449 −1.90604 −0.953022 0.302900i \(-0.902045\pi\)
−0.953022 + 0.302900i \(0.902045\pi\)
\(938\) 165.877 5.41609
\(939\) 0 0
\(940\) 16.6840 0.544171
\(941\) −0.457241 −0.0149056 −0.00745281 0.999972i \(-0.502372\pi\)
−0.00745281 + 0.999972i \(0.502372\pi\)
\(942\) 0 0
\(943\) 53.4367 1.74014
\(944\) 26.6819 0.868423
\(945\) 0 0
\(946\) −27.1183 −0.881693
\(947\) −15.0309 −0.488439 −0.244220 0.969720i \(-0.578532\pi\)
−0.244220 + 0.969720i \(0.578532\pi\)
\(948\) 0 0
\(949\) 4.46134 0.144821
\(950\) −7.28905 −0.236488
\(951\) 0 0
\(952\) −20.4408 −0.662491
\(953\) 4.19975 0.136043 0.0680216 0.997684i \(-0.478331\pi\)
0.0680216 + 0.997684i \(0.478331\pi\)
\(954\) 0 0
\(955\) 18.5052 0.598814
\(956\) −46.6819 −1.50980
\(957\) 0 0
\(958\) 28.7069 0.927476
\(959\) −1.08930 −0.0351754
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 12.1578 0.391982
\(963\) 0 0
\(964\) 55.3369 1.78228
\(965\) 17.3929 0.559896
\(966\) 0 0
\(967\) 24.3055 0.781611 0.390805 0.920473i \(-0.372197\pi\)
0.390805 + 0.920473i \(0.372197\pi\)
\(968\) −2.96056 −0.0951559
\(969\) 0 0
\(970\) −24.1038 −0.773927
\(971\) 30.8088 0.988701 0.494350 0.869263i \(-0.335406\pi\)
0.494350 + 0.869263i \(0.335406\pi\)
\(972\) 0 0
\(973\) 21.1767 0.678894
\(974\) −50.2865 −1.61128
\(975\) 0 0
\(976\) −11.6175 −0.371869
\(977\) 32.0729 1.02610 0.513051 0.858358i \(-0.328515\pi\)
0.513051 + 0.858358i \(0.328515\pi\)
\(978\) 0 0
\(979\) 43.3225 1.38459
\(980\) −44.2995 −1.41509
\(981\) 0 0
\(982\) −38.7857 −1.23770
\(983\) −28.3534 −0.904334 −0.452167 0.891933i \(-0.649349\pi\)
−0.452167 + 0.891933i \(0.649349\pi\)
\(984\) 0 0
\(985\) 12.8088 0.408121
\(986\) 6.16425 0.196310
\(987\) 0 0
\(988\) −4.69799 −0.149463
\(989\) −28.4383 −0.904285
\(990\) 0 0
\(991\) −14.6425 −0.465134 −0.232567 0.972580i \(-0.574712\pi\)
−0.232567 + 0.972580i \(0.574712\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −157.239 −4.98731
\(995\) 7.63206 0.241953
\(996\) 0 0
\(997\) −34.7128 −1.09937 −0.549683 0.835373i \(-0.685252\pi\)
−0.549683 + 0.835373i \(0.685252\pi\)
\(998\) −5.34539 −0.169205
\(999\) 0 0
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 1305.2.a.q.1.3 3
3.2 odd 2 435.2.a.i.1.1 3
5.4 even 2 6525.2.a.bf.1.1 3
12.11 even 2 6960.2.a.cl.1.3 3
15.2 even 4 2175.2.c.m.349.2 6
15.8 even 4 2175.2.c.m.349.5 6
15.14 odd 2 2175.2.a.u.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.i.1.1 3 3.2 odd 2
1305.2.a.q.1.3 3 1.1 even 1 trivial
2175.2.a.u.1.3 3 15.14 odd 2
2175.2.c.m.349.2 6 15.2 even 4
2175.2.c.m.349.5 6 15.8 even 4
6525.2.a.bf.1.1 3 5.4 even 2
6960.2.a.cl.1.3 3 12.11 even 2