Properties

Label 6525.2.a.ca.1.3
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $9$
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] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, 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("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.77820\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.77820 q^{2} +1.16198 q^{4} +2.84862 q^{7} +1.49016 q^{8} +O(q^{10})\) \(q-1.77820 q^{2} +1.16198 q^{4} +2.84862 q^{7} +1.49016 q^{8} -5.35505 q^{11} +2.50643 q^{13} -5.06541 q^{14} -4.97376 q^{16} -6.50704 q^{17} +2.25433 q^{19} +9.52233 q^{22} +0.355939 q^{23} -4.45692 q^{26} +3.31005 q^{28} +1.00000 q^{29} +5.45828 q^{31} +5.86400 q^{32} +11.5708 q^{34} +2.00281 q^{37} -4.00863 q^{38} +5.37258 q^{41} -12.8773 q^{43} -6.22247 q^{44} -0.632929 q^{46} +4.01725 q^{47} +1.11466 q^{49} +2.91242 q^{52} +3.42758 q^{53} +4.24492 q^{56} -1.77820 q^{58} +4.76617 q^{59} -4.29494 q^{61} -9.70588 q^{62} -0.479810 q^{64} +1.09723 q^{67} -7.56105 q^{68} -5.21539 q^{71} +6.78643 q^{73} -3.56138 q^{74} +2.61948 q^{76} -15.2545 q^{77} -1.84943 q^{79} -9.55349 q^{82} -13.7379 q^{83} +22.8984 q^{86} -7.97990 q^{88} +2.56752 q^{89} +7.13987 q^{91} +0.413594 q^{92} -7.14346 q^{94} -2.73855 q^{97} -1.98209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8} + 2 q^{11} - q^{13} - 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} - 3 q^{22} - 16 q^{23} + 6 q^{26} + 4 q^{28} + 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38} - 10 q^{41} - 3 q^{43} - 13 q^{44} + 4 q^{46} - 26 q^{47} - 8 q^{49} + 9 q^{52} - 22 q^{53} + 22 q^{56} - 2 q^{58} + 4 q^{59} + 7 q^{61} - 28 q^{62} + 9 q^{64} - 5 q^{67} - 39 q^{68} + 10 q^{73} - 34 q^{74} - 2 q^{76} - 34 q^{77} + 10 q^{79} + 8 q^{82} - 46 q^{83} + 28 q^{86} - 2 q^{88} + 4 q^{89} - 21 q^{91} - 20 q^{92} + 5 q^{94} - 7 q^{97} - 51 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.77820 −1.25737 −0.628687 0.777658i \(-0.716407\pi\)
−0.628687 + 0.777658i \(0.716407\pi\)
\(3\) 0 0
\(4\) 1.16198 0.580990
\(5\) 0 0
\(6\) 0 0
\(7\) 2.84862 1.07668 0.538339 0.842728i \(-0.319052\pi\)
0.538339 + 0.842728i \(0.319052\pi\)
\(8\) 1.49016 0.526852
\(9\) 0 0
\(10\) 0 0
\(11\) −5.35505 −1.61461 −0.807305 0.590135i \(-0.799075\pi\)
−0.807305 + 0.590135i \(0.799075\pi\)
\(12\) 0 0
\(13\) 2.50643 0.695158 0.347579 0.937651i \(-0.387004\pi\)
0.347579 + 0.937651i \(0.387004\pi\)
\(14\) −5.06541 −1.35379
\(15\) 0 0
\(16\) −4.97376 −1.24344
\(17\) −6.50704 −1.57819 −0.789095 0.614271i \(-0.789450\pi\)
−0.789095 + 0.614271i \(0.789450\pi\)
\(18\) 0 0
\(19\) 2.25433 0.517178 0.258589 0.965987i \(-0.416742\pi\)
0.258589 + 0.965987i \(0.416742\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 9.52233 2.03017
\(23\) 0.355939 0.0742184 0.0371092 0.999311i \(-0.488185\pi\)
0.0371092 + 0.999311i \(0.488185\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.45692 −0.874074
\(27\) 0 0
\(28\) 3.31005 0.625540
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 5.45828 0.980335 0.490168 0.871628i \(-0.336936\pi\)
0.490168 + 0.871628i \(0.336936\pi\)
\(32\) 5.86400 1.03662
\(33\) 0 0
\(34\) 11.5708 1.98438
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00281 0.329259 0.164630 0.986355i \(-0.447357\pi\)
0.164630 + 0.986355i \(0.447357\pi\)
\(38\) −4.00863 −0.650286
\(39\) 0 0
\(40\) 0 0
\(41\) 5.37258 0.839055 0.419528 0.907743i \(-0.362196\pi\)
0.419528 + 0.907743i \(0.362196\pi\)
\(42\) 0 0
\(43\) −12.8773 −1.96377 −0.981886 0.189472i \(-0.939322\pi\)
−0.981886 + 0.189472i \(0.939322\pi\)
\(44\) −6.22247 −0.938072
\(45\) 0 0
\(46\) −0.632929 −0.0933203
\(47\) 4.01725 0.585976 0.292988 0.956116i \(-0.405350\pi\)
0.292988 + 0.956116i \(0.405350\pi\)
\(48\) 0 0
\(49\) 1.11466 0.159238
\(50\) 0 0
\(51\) 0 0
\(52\) 2.91242 0.403880
\(53\) 3.42758 0.470814 0.235407 0.971897i \(-0.424358\pi\)
0.235407 + 0.971897i \(0.424358\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.24492 0.567251
\(57\) 0 0
\(58\) −1.77820 −0.233489
\(59\) 4.76617 0.620503 0.310251 0.950655i \(-0.399587\pi\)
0.310251 + 0.950655i \(0.399587\pi\)
\(60\) 0 0
\(61\) −4.29494 −0.549911 −0.274955 0.961457i \(-0.588663\pi\)
−0.274955 + 0.961457i \(0.588663\pi\)
\(62\) −9.70588 −1.23265
\(63\) 0 0
\(64\) −0.479810 −0.0599762
\(65\) 0 0
\(66\) 0 0
\(67\) 1.09723 0.134048 0.0670239 0.997751i \(-0.478650\pi\)
0.0670239 + 0.997751i \(0.478650\pi\)
\(68\) −7.56105 −0.916913
\(69\) 0 0
\(70\) 0 0
\(71\) −5.21539 −0.618953 −0.309476 0.950907i \(-0.600154\pi\)
−0.309476 + 0.950907i \(0.600154\pi\)
\(72\) 0 0
\(73\) 6.78643 0.794291 0.397146 0.917756i \(-0.370001\pi\)
0.397146 + 0.917756i \(0.370001\pi\)
\(74\) −3.56138 −0.414002
\(75\) 0 0
\(76\) 2.61948 0.300475
\(77\) −15.2545 −1.73842
\(78\) 0 0
\(79\) −1.84943 −0.208077 −0.104038 0.994573i \(-0.533176\pi\)
−0.104038 + 0.994573i \(0.533176\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −9.55349 −1.05501
\(83\) −13.7379 −1.50793 −0.753963 0.656917i \(-0.771860\pi\)
−0.753963 + 0.656917i \(0.771860\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 22.8984 2.46920
\(87\) 0 0
\(88\) −7.97990 −0.850660
\(89\) 2.56752 0.272157 0.136078 0.990698i \(-0.456550\pi\)
0.136078 + 0.990698i \(0.456550\pi\)
\(90\) 0 0
\(91\) 7.13987 0.748462
\(92\) 0.413594 0.0431202
\(93\) 0 0
\(94\) −7.14346 −0.736792
\(95\) 0 0
\(96\) 0 0
\(97\) −2.73855 −0.278057 −0.139029 0.990288i \(-0.544398\pi\)
−0.139029 + 0.990288i \(0.544398\pi\)
\(98\) −1.98209 −0.200221
\(99\) 0 0
\(100\) 0 0
\(101\) 5.73985 0.571137 0.285568 0.958358i \(-0.407818\pi\)
0.285568 + 0.958358i \(0.407818\pi\)
\(102\) 0 0
\(103\) −15.7896 −1.55579 −0.777895 0.628394i \(-0.783713\pi\)
−0.777895 + 0.628394i \(0.783713\pi\)
\(104\) 3.73499 0.366246
\(105\) 0 0
\(106\) −6.09491 −0.591990
\(107\) −16.5642 −1.60132 −0.800661 0.599118i \(-0.795518\pi\)
−0.800661 + 0.599118i \(0.795518\pi\)
\(108\) 0 0
\(109\) 9.32556 0.893226 0.446613 0.894727i \(-0.352630\pi\)
0.446613 + 0.894727i \(0.352630\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −14.1684 −1.33879
\(113\) −14.1979 −1.33563 −0.667815 0.744327i \(-0.732770\pi\)
−0.667815 + 0.744327i \(0.732770\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.16198 0.107887
\(117\) 0 0
\(118\) −8.47518 −0.780204
\(119\) −18.5361 −1.69920
\(120\) 0 0
\(121\) 17.6766 1.60696
\(122\) 7.63724 0.691444
\(123\) 0 0
\(124\) 6.34241 0.569565
\(125\) 0 0
\(126\) 0 0
\(127\) 5.36218 0.475816 0.237908 0.971288i \(-0.423538\pi\)
0.237908 + 0.971288i \(0.423538\pi\)
\(128\) −10.8748 −0.961205
\(129\) 0 0
\(130\) 0 0
\(131\) −10.8431 −0.947370 −0.473685 0.880694i \(-0.657076\pi\)
−0.473685 + 0.880694i \(0.657076\pi\)
\(132\) 0 0
\(133\) 6.42173 0.556835
\(134\) −1.95109 −0.168548
\(135\) 0 0
\(136\) −9.69656 −0.831473
\(137\) −7.41637 −0.633623 −0.316812 0.948489i \(-0.602612\pi\)
−0.316812 + 0.948489i \(0.602612\pi\)
\(138\) 0 0
\(139\) 19.6639 1.66787 0.833937 0.551859i \(-0.186082\pi\)
0.833937 + 0.551859i \(0.186082\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 9.27398 0.778255
\(143\) −13.4221 −1.12241
\(144\) 0 0
\(145\) 0 0
\(146\) −12.0676 −0.998722
\(147\) 0 0
\(148\) 2.32722 0.191296
\(149\) −14.4485 −1.18367 −0.591834 0.806060i \(-0.701596\pi\)
−0.591834 + 0.806060i \(0.701596\pi\)
\(150\) 0 0
\(151\) 6.69220 0.544604 0.272302 0.962212i \(-0.412215\pi\)
0.272302 + 0.962212i \(0.412215\pi\)
\(152\) 3.35931 0.272476
\(153\) 0 0
\(154\) 27.1256 2.18584
\(155\) 0 0
\(156\) 0 0
\(157\) 17.5789 1.40295 0.701473 0.712696i \(-0.252526\pi\)
0.701473 + 0.712696i \(0.252526\pi\)
\(158\) 3.28864 0.261630
\(159\) 0 0
\(160\) 0 0
\(161\) 1.01394 0.0799094
\(162\) 0 0
\(163\) −19.6576 −1.53971 −0.769853 0.638222i \(-0.779670\pi\)
−0.769853 + 0.638222i \(0.779670\pi\)
\(164\) 6.24283 0.487483
\(165\) 0 0
\(166\) 24.4286 1.89603
\(167\) 12.2841 0.950569 0.475285 0.879832i \(-0.342345\pi\)
0.475285 + 0.879832i \(0.342345\pi\)
\(168\) 0 0
\(169\) −6.71782 −0.516755
\(170\) 0 0
\(171\) 0 0
\(172\) −14.9632 −1.14093
\(173\) 12.9440 0.984118 0.492059 0.870562i \(-0.336244\pi\)
0.492059 + 0.870562i \(0.336244\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 26.6348 2.00767
\(177\) 0 0
\(178\) −4.56555 −0.342203
\(179\) 6.01888 0.449872 0.224936 0.974374i \(-0.427783\pi\)
0.224936 + 0.974374i \(0.427783\pi\)
\(180\) 0 0
\(181\) −19.8734 −1.47718 −0.738591 0.674154i \(-0.764508\pi\)
−0.738591 + 0.674154i \(0.764508\pi\)
\(182\) −12.6961 −0.941097
\(183\) 0 0
\(184\) 0.530407 0.0391021
\(185\) 0 0
\(186\) 0 0
\(187\) 34.8456 2.54816
\(188\) 4.66797 0.340446
\(189\) 0 0
\(190\) 0 0
\(191\) −3.04577 −0.220384 −0.110192 0.993910i \(-0.535147\pi\)
−0.110192 + 0.993910i \(0.535147\pi\)
\(192\) 0 0
\(193\) −21.3408 −1.53615 −0.768073 0.640362i \(-0.778784\pi\)
−0.768073 + 0.640362i \(0.778784\pi\)
\(194\) 4.86967 0.349622
\(195\) 0 0
\(196\) 1.29522 0.0925154
\(197\) 10.0730 0.717670 0.358835 0.933401i \(-0.383174\pi\)
0.358835 + 0.933401i \(0.383174\pi\)
\(198\) 0 0
\(199\) −4.22842 −0.299744 −0.149872 0.988705i \(-0.547886\pi\)
−0.149872 + 0.988705i \(0.547886\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −10.2066 −0.718133
\(203\) 2.84862 0.199934
\(204\) 0 0
\(205\) 0 0
\(206\) 28.0769 1.95621
\(207\) 0 0
\(208\) −12.4664 −0.864388
\(209\) −12.0720 −0.835040
\(210\) 0 0
\(211\) 28.2252 1.94310 0.971550 0.236834i \(-0.0761098\pi\)
0.971550 + 0.236834i \(0.0761098\pi\)
\(212\) 3.98278 0.273539
\(213\) 0 0
\(214\) 29.4544 2.01346
\(215\) 0 0
\(216\) 0 0
\(217\) 15.5486 1.05551
\(218\) −16.5827 −1.12312
\(219\) 0 0
\(220\) 0 0
\(221\) −16.3094 −1.09709
\(222\) 0 0
\(223\) 8.48452 0.568165 0.284083 0.958800i \(-0.408311\pi\)
0.284083 + 0.958800i \(0.408311\pi\)
\(224\) 16.7043 1.11610
\(225\) 0 0
\(226\) 25.2467 1.67939
\(227\) 14.7872 0.981460 0.490730 0.871312i \(-0.336730\pi\)
0.490730 + 0.871312i \(0.336730\pi\)
\(228\) 0 0
\(229\) −21.0993 −1.39428 −0.697141 0.716934i \(-0.745545\pi\)
−0.697141 + 0.716934i \(0.745545\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.49016 0.0978340
\(233\) −26.0762 −1.70831 −0.854155 0.520018i \(-0.825925\pi\)
−0.854155 + 0.520018i \(0.825925\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 5.53820 0.360506
\(237\) 0 0
\(238\) 32.9609 2.13653
\(239\) 8.17070 0.528519 0.264259 0.964452i \(-0.414872\pi\)
0.264259 + 0.964452i \(0.414872\pi\)
\(240\) 0 0
\(241\) −10.5741 −0.681135 −0.340568 0.940220i \(-0.610619\pi\)
−0.340568 + 0.940220i \(0.610619\pi\)
\(242\) −31.4324 −2.02055
\(243\) 0 0
\(244\) −4.99064 −0.319493
\(245\) 0 0
\(246\) 0 0
\(247\) 5.65031 0.359520
\(248\) 8.13372 0.516492
\(249\) 0 0
\(250\) 0 0
\(251\) 21.3791 1.34944 0.674720 0.738074i \(-0.264264\pi\)
0.674720 + 0.738074i \(0.264264\pi\)
\(252\) 0 0
\(253\) −1.90607 −0.119834
\(254\) −9.53500 −0.598279
\(255\) 0 0
\(256\) 20.2971 1.26857
\(257\) 6.90806 0.430913 0.215456 0.976513i \(-0.430876\pi\)
0.215456 + 0.976513i \(0.430876\pi\)
\(258\) 0 0
\(259\) 5.70524 0.354506
\(260\) 0 0
\(261\) 0 0
\(262\) 19.2812 1.19120
\(263\) −2.36575 −0.145878 −0.0729391 0.997336i \(-0.523238\pi\)
−0.0729391 + 0.997336i \(0.523238\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −11.4191 −0.700150
\(267\) 0 0
\(268\) 1.27496 0.0778805
\(269\) 13.0500 0.795670 0.397835 0.917457i \(-0.369762\pi\)
0.397835 + 0.917457i \(0.369762\pi\)
\(270\) 0 0
\(271\) −24.2733 −1.47450 −0.737250 0.675620i \(-0.763876\pi\)
−0.737250 + 0.675620i \(0.763876\pi\)
\(272\) 32.3645 1.96239
\(273\) 0 0
\(274\) 13.1878 0.796701
\(275\) 0 0
\(276\) 0 0
\(277\) −0.0846766 −0.00508773 −0.00254386 0.999997i \(-0.500810\pi\)
−0.00254386 + 0.999997i \(0.500810\pi\)
\(278\) −34.9663 −2.09714
\(279\) 0 0
\(280\) 0 0
\(281\) 8.18399 0.488216 0.244108 0.969748i \(-0.421505\pi\)
0.244108 + 0.969748i \(0.421505\pi\)
\(282\) 0 0
\(283\) −10.6021 −0.630227 −0.315114 0.949054i \(-0.602043\pi\)
−0.315114 + 0.949054i \(0.602043\pi\)
\(284\) −6.06018 −0.359605
\(285\) 0 0
\(286\) 23.8670 1.41129
\(287\) 15.3045 0.903393
\(288\) 0 0
\(289\) 25.3416 1.49068
\(290\) 0 0
\(291\) 0 0
\(292\) 7.88570 0.461475
\(293\) −26.2617 −1.53422 −0.767112 0.641513i \(-0.778307\pi\)
−0.767112 + 0.641513i \(0.778307\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.98451 0.173471
\(297\) 0 0
\(298\) 25.6923 1.48831
\(299\) 0.892136 0.0515935
\(300\) 0 0
\(301\) −36.6826 −2.11435
\(302\) −11.9000 −0.684771
\(303\) 0 0
\(304\) −11.2125 −0.643080
\(305\) 0 0
\(306\) 0 0
\(307\) −20.8923 −1.19239 −0.596193 0.802841i \(-0.703321\pi\)
−0.596193 + 0.802841i \(0.703321\pi\)
\(308\) −17.7255 −1.01000
\(309\) 0 0
\(310\) 0 0
\(311\) −5.25221 −0.297826 −0.148913 0.988850i \(-0.547577\pi\)
−0.148913 + 0.988850i \(0.547577\pi\)
\(312\) 0 0
\(313\) 32.0146 1.80957 0.904787 0.425865i \(-0.140030\pi\)
0.904787 + 0.425865i \(0.140030\pi\)
\(314\) −31.2587 −1.76403
\(315\) 0 0
\(316\) −2.14900 −0.120891
\(317\) −22.1715 −1.24527 −0.622637 0.782511i \(-0.713939\pi\)
−0.622637 + 0.782511i \(0.713939\pi\)
\(318\) 0 0
\(319\) −5.35505 −0.299825
\(320\) 0 0
\(321\) 0 0
\(322\) −1.80298 −0.100476
\(323\) −14.6690 −0.816205
\(324\) 0 0
\(325\) 0 0
\(326\) 34.9551 1.93599
\(327\) 0 0
\(328\) 8.00602 0.442058
\(329\) 11.4436 0.630908
\(330\) 0 0
\(331\) −31.8652 −1.75147 −0.875734 0.482794i \(-0.839622\pi\)
−0.875734 + 0.482794i \(0.839622\pi\)
\(332\) −15.9631 −0.876090
\(333\) 0 0
\(334\) −21.8435 −1.19522
\(335\) 0 0
\(336\) 0 0
\(337\) 4.68105 0.254993 0.127496 0.991839i \(-0.459306\pi\)
0.127496 + 0.991839i \(0.459306\pi\)
\(338\) 11.9456 0.649755
\(339\) 0 0
\(340\) 0 0
\(341\) −29.2294 −1.58286
\(342\) 0 0
\(343\) −16.7651 −0.905231
\(344\) −19.1893 −1.03462
\(345\) 0 0
\(346\) −23.0170 −1.23740
\(347\) 10.0153 0.537650 0.268825 0.963189i \(-0.413365\pi\)
0.268825 + 0.963189i \(0.413365\pi\)
\(348\) 0 0
\(349\) 14.9709 0.801376 0.400688 0.916215i \(-0.368771\pi\)
0.400688 + 0.916215i \(0.368771\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −31.4020 −1.67373
\(353\) 12.0450 0.641092 0.320546 0.947233i \(-0.396134\pi\)
0.320546 + 0.947233i \(0.396134\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.98341 0.158120
\(357\) 0 0
\(358\) −10.7027 −0.565658
\(359\) −4.72615 −0.249436 −0.124718 0.992192i \(-0.539803\pi\)
−0.124718 + 0.992192i \(0.539803\pi\)
\(360\) 0 0
\(361\) −13.9180 −0.732527
\(362\) 35.3389 1.85737
\(363\) 0 0
\(364\) 8.29639 0.434849
\(365\) 0 0
\(366\) 0 0
\(367\) 11.9598 0.624295 0.312148 0.950034i \(-0.398952\pi\)
0.312148 + 0.950034i \(0.398952\pi\)
\(368\) −1.77036 −0.0922862
\(369\) 0 0
\(370\) 0 0
\(371\) 9.76389 0.506916
\(372\) 0 0
\(373\) 8.31719 0.430648 0.215324 0.976543i \(-0.430919\pi\)
0.215324 + 0.976543i \(0.430919\pi\)
\(374\) −61.9622 −3.20399
\(375\) 0 0
\(376\) 5.98636 0.308723
\(377\) 2.50643 0.129088
\(378\) 0 0
\(379\) −24.5715 −1.26215 −0.631075 0.775721i \(-0.717386\pi\)
−0.631075 + 0.775721i \(0.717386\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 5.41598 0.277106
\(383\) −34.0260 −1.73865 −0.869325 0.494241i \(-0.835446\pi\)
−0.869325 + 0.494241i \(0.835446\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 37.9482 1.93151
\(387\) 0 0
\(388\) −3.18214 −0.161549
\(389\) −25.3691 −1.28626 −0.643131 0.765756i \(-0.722365\pi\)
−0.643131 + 0.765756i \(0.722365\pi\)
\(390\) 0 0
\(391\) −2.31611 −0.117131
\(392\) 1.66103 0.0838947
\(393\) 0 0
\(394\) −17.9117 −0.902379
\(395\) 0 0
\(396\) 0 0
\(397\) −16.7222 −0.839265 −0.419632 0.907694i \(-0.637841\pi\)
−0.419632 + 0.907694i \(0.637841\pi\)
\(398\) 7.51895 0.376891
\(399\) 0 0
\(400\) 0 0
\(401\) −8.01416 −0.400208 −0.200104 0.979775i \(-0.564128\pi\)
−0.200104 + 0.979775i \(0.564128\pi\)
\(402\) 0 0
\(403\) 13.6808 0.681488
\(404\) 6.66960 0.331825
\(405\) 0 0
\(406\) −5.06541 −0.251392
\(407\) −10.7251 −0.531625
\(408\) 0 0
\(409\) −19.4335 −0.960924 −0.480462 0.877016i \(-0.659531\pi\)
−0.480462 + 0.877016i \(0.659531\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −18.3471 −0.903899
\(413\) 13.5770 0.668082
\(414\) 0 0
\(415\) 0 0
\(416\) 14.6977 0.720613
\(417\) 0 0
\(418\) 21.4664 1.04996
\(419\) 6.92370 0.338245 0.169122 0.985595i \(-0.445907\pi\)
0.169122 + 0.985595i \(0.445907\pi\)
\(420\) 0 0
\(421\) −12.9872 −0.632956 −0.316478 0.948600i \(-0.602500\pi\)
−0.316478 + 0.948600i \(0.602500\pi\)
\(422\) −50.1899 −2.44320
\(423\) 0 0
\(424\) 5.10766 0.248050
\(425\) 0 0
\(426\) 0 0
\(427\) −12.2347 −0.592077
\(428\) −19.2473 −0.930352
\(429\) 0 0
\(430\) 0 0
\(431\) 39.6752 1.91109 0.955543 0.294851i \(-0.0952701\pi\)
0.955543 + 0.294851i \(0.0952701\pi\)
\(432\) 0 0
\(433\) −4.93087 −0.236962 −0.118481 0.992956i \(-0.537803\pi\)
−0.118481 + 0.992956i \(0.537803\pi\)
\(434\) −27.6484 −1.32717
\(435\) 0 0
\(436\) 10.8361 0.518956
\(437\) 0.802403 0.0383841
\(438\) 0 0
\(439\) 22.1458 1.05696 0.528481 0.848945i \(-0.322762\pi\)
0.528481 + 0.848945i \(0.322762\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 29.0014 1.37945
\(443\) −1.21156 −0.0575631 −0.0287815 0.999586i \(-0.509163\pi\)
−0.0287815 + 0.999586i \(0.509163\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −15.0871 −0.714396
\(447\) 0 0
\(448\) −1.36680 −0.0645751
\(449\) −5.22771 −0.246711 −0.123355 0.992363i \(-0.539365\pi\)
−0.123355 + 0.992363i \(0.539365\pi\)
\(450\) 0 0
\(451\) −28.7704 −1.35475
\(452\) −16.4977 −0.775988
\(453\) 0 0
\(454\) −26.2945 −1.23406
\(455\) 0 0
\(456\) 0 0
\(457\) 16.9064 0.790849 0.395425 0.918498i \(-0.370597\pi\)
0.395425 + 0.918498i \(0.370597\pi\)
\(458\) 37.5187 1.75313
\(459\) 0 0
\(460\) 0 0
\(461\) 7.79634 0.363112 0.181556 0.983381i \(-0.441887\pi\)
0.181556 + 0.983381i \(0.441887\pi\)
\(462\) 0 0
\(463\) −22.3232 −1.03745 −0.518724 0.854942i \(-0.673593\pi\)
−0.518724 + 0.854942i \(0.673593\pi\)
\(464\) −4.97376 −0.230901
\(465\) 0 0
\(466\) 46.3686 2.14799
\(467\) −8.19178 −0.379070 −0.189535 0.981874i \(-0.560698\pi\)
−0.189535 + 0.981874i \(0.560698\pi\)
\(468\) 0 0
\(469\) 3.12559 0.144327
\(470\) 0 0
\(471\) 0 0
\(472\) 7.10237 0.326913
\(473\) 68.9587 3.17072
\(474\) 0 0
\(475\) 0 0
\(476\) −21.5386 −0.987220
\(477\) 0 0
\(478\) −14.5291 −0.664546
\(479\) −17.4830 −0.798817 −0.399409 0.916773i \(-0.630785\pi\)
−0.399409 + 0.916773i \(0.630785\pi\)
\(480\) 0 0
\(481\) 5.01989 0.228887
\(482\) 18.8028 0.856442
\(483\) 0 0
\(484\) 20.5398 0.933630
\(485\) 0 0
\(486\) 0 0
\(487\) −23.2620 −1.05410 −0.527051 0.849833i \(-0.676702\pi\)
−0.527051 + 0.849833i \(0.676702\pi\)
\(488\) −6.40016 −0.289722
\(489\) 0 0
\(490\) 0 0
\(491\) −34.4941 −1.55670 −0.778348 0.627833i \(-0.783942\pi\)
−0.778348 + 0.627833i \(0.783942\pi\)
\(492\) 0 0
\(493\) −6.50704 −0.293062
\(494\) −10.0474 −0.452052
\(495\) 0 0
\(496\) −27.1482 −1.21899
\(497\) −14.8567 −0.666413
\(498\) 0 0
\(499\) 12.8752 0.576372 0.288186 0.957574i \(-0.406948\pi\)
0.288186 + 0.957574i \(0.406948\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −38.0163 −1.69675
\(503\) −6.19856 −0.276380 −0.138190 0.990406i \(-0.544128\pi\)
−0.138190 + 0.990406i \(0.544128\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3.38937 0.150676
\(507\) 0 0
\(508\) 6.23074 0.276444
\(509\) 29.6558 1.31447 0.657235 0.753686i \(-0.271726\pi\)
0.657235 + 0.753686i \(0.271726\pi\)
\(510\) 0 0
\(511\) 19.3320 0.855197
\(512\) −14.3427 −0.633863
\(513\) 0 0
\(514\) −12.2839 −0.541819
\(515\) 0 0
\(516\) 0 0
\(517\) −21.5126 −0.946123
\(518\) −10.1450 −0.445747
\(519\) 0 0
\(520\) 0 0
\(521\) 31.2676 1.36986 0.684929 0.728610i \(-0.259833\pi\)
0.684929 + 0.728610i \(0.259833\pi\)
\(522\) 0 0
\(523\) −18.7657 −0.820567 −0.410284 0.911958i \(-0.634570\pi\)
−0.410284 + 0.911958i \(0.634570\pi\)
\(524\) −12.5995 −0.550413
\(525\) 0 0
\(526\) 4.20676 0.183423
\(527\) −35.5172 −1.54716
\(528\) 0 0
\(529\) −22.8733 −0.994492
\(530\) 0 0
\(531\) 0 0
\(532\) 7.46192 0.323515
\(533\) 13.4660 0.583276
\(534\) 0 0
\(535\) 0 0
\(536\) 1.63505 0.0706234
\(537\) 0 0
\(538\) −23.2054 −1.00045
\(539\) −5.96908 −0.257106
\(540\) 0 0
\(541\) −32.7471 −1.40791 −0.703955 0.710245i \(-0.748584\pi\)
−0.703955 + 0.710245i \(0.748584\pi\)
\(542\) 43.1627 1.85400
\(543\) 0 0
\(544\) −38.1573 −1.63598
\(545\) 0 0
\(546\) 0 0
\(547\) −41.0014 −1.75309 −0.876547 0.481316i \(-0.840159\pi\)
−0.876547 + 0.481316i \(0.840159\pi\)
\(548\) −8.61767 −0.368129
\(549\) 0 0
\(550\) 0 0
\(551\) 2.25433 0.0960375
\(552\) 0 0
\(553\) −5.26832 −0.224032
\(554\) 0.150572 0.00639718
\(555\) 0 0
\(556\) 22.8491 0.969018
\(557\) −32.8805 −1.39319 −0.696596 0.717463i \(-0.745303\pi\)
−0.696596 + 0.717463i \(0.745303\pi\)
\(558\) 0 0
\(559\) −32.2761 −1.36513
\(560\) 0 0
\(561\) 0 0
\(562\) −14.5527 −0.613870
\(563\) −19.8934 −0.838407 −0.419204 0.907892i \(-0.637691\pi\)
−0.419204 + 0.907892i \(0.637691\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 18.8525 0.792432
\(567\) 0 0
\(568\) −7.77178 −0.326097
\(569\) 1.72158 0.0721724 0.0360862 0.999349i \(-0.488511\pi\)
0.0360862 + 0.999349i \(0.488511\pi\)
\(570\) 0 0
\(571\) 26.4954 1.10880 0.554399 0.832251i \(-0.312948\pi\)
0.554399 + 0.832251i \(0.312948\pi\)
\(572\) −15.5962 −0.652108
\(573\) 0 0
\(574\) −27.2143 −1.13590
\(575\) 0 0
\(576\) 0 0
\(577\) 40.8791 1.70182 0.850909 0.525314i \(-0.176052\pi\)
0.850909 + 0.525314i \(0.176052\pi\)
\(578\) −45.0623 −1.87435
\(579\) 0 0
\(580\) 0 0
\(581\) −39.1340 −1.62355
\(582\) 0 0
\(583\) −18.3549 −0.760181
\(584\) 10.1129 0.418474
\(585\) 0 0
\(586\) 46.6984 1.92909
\(587\) −40.0943 −1.65487 −0.827434 0.561563i \(-0.810200\pi\)
−0.827434 + 0.561563i \(0.810200\pi\)
\(588\) 0 0
\(589\) 12.3047 0.507008
\(590\) 0 0
\(591\) 0 0
\(592\) −9.96148 −0.409414
\(593\) −7.94061 −0.326082 −0.163041 0.986619i \(-0.552130\pi\)
−0.163041 + 0.986619i \(0.552130\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −16.7889 −0.687700
\(597\) 0 0
\(598\) −1.58639 −0.0648724
\(599\) 37.2143 1.52054 0.760268 0.649609i \(-0.225068\pi\)
0.760268 + 0.649609i \(0.225068\pi\)
\(600\) 0 0
\(601\) −9.87927 −0.402984 −0.201492 0.979490i \(-0.564579\pi\)
−0.201492 + 0.979490i \(0.564579\pi\)
\(602\) 65.2289 2.65853
\(603\) 0 0
\(604\) 7.77621 0.316409
\(605\) 0 0
\(606\) 0 0
\(607\) −3.95850 −0.160670 −0.0803352 0.996768i \(-0.525599\pi\)
−0.0803352 + 0.996768i \(0.525599\pi\)
\(608\) 13.2194 0.536116
\(609\) 0 0
\(610\) 0 0
\(611\) 10.0690 0.407346
\(612\) 0 0
\(613\) 19.8179 0.800436 0.400218 0.916420i \(-0.368934\pi\)
0.400218 + 0.916420i \(0.368934\pi\)
\(614\) 37.1506 1.49927
\(615\) 0 0
\(616\) −22.7317 −0.915888
\(617\) 8.70648 0.350510 0.175255 0.984523i \(-0.443925\pi\)
0.175255 + 0.984523i \(0.443925\pi\)
\(618\) 0 0
\(619\) −18.3012 −0.735589 −0.367794 0.929907i \(-0.619887\pi\)
−0.367794 + 0.929907i \(0.619887\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 9.33946 0.374478
\(623\) 7.31390 0.293025
\(624\) 0 0
\(625\) 0 0
\(626\) −56.9283 −2.27531
\(627\) 0 0
\(628\) 20.4263 0.815098
\(629\) −13.0323 −0.519633
\(630\) 0 0
\(631\) −48.9315 −1.94793 −0.973966 0.226693i \(-0.927209\pi\)
−0.973966 + 0.226693i \(0.927209\pi\)
\(632\) −2.75595 −0.109626
\(633\) 0 0
\(634\) 39.4252 1.56578
\(635\) 0 0
\(636\) 0 0
\(637\) 2.79382 0.110695
\(638\) 9.52233 0.376993
\(639\) 0 0
\(640\) 0 0
\(641\) 5.22083 0.206210 0.103105 0.994670i \(-0.467122\pi\)
0.103105 + 0.994670i \(0.467122\pi\)
\(642\) 0 0
\(643\) 25.8165 1.01810 0.509051 0.860736i \(-0.329996\pi\)
0.509051 + 0.860736i \(0.329996\pi\)
\(644\) 1.17817 0.0464266
\(645\) 0 0
\(646\) 26.0844 1.02628
\(647\) −1.68358 −0.0661885 −0.0330943 0.999452i \(-0.510536\pi\)
−0.0330943 + 0.999452i \(0.510536\pi\)
\(648\) 0 0
\(649\) −25.5231 −1.00187
\(650\) 0 0
\(651\) 0 0
\(652\) −22.8418 −0.894553
\(653\) −26.5807 −1.04018 −0.520091 0.854111i \(-0.674102\pi\)
−0.520091 + 0.854111i \(0.674102\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −26.7219 −1.04332
\(657\) 0 0
\(658\) −20.3490 −0.793288
\(659\) 5.88644 0.229303 0.114652 0.993406i \(-0.463425\pi\)
0.114652 + 0.993406i \(0.463425\pi\)
\(660\) 0 0
\(661\) 21.3474 0.830319 0.415160 0.909749i \(-0.363726\pi\)
0.415160 + 0.909749i \(0.363726\pi\)
\(662\) 56.6625 2.20225
\(663\) 0 0
\(664\) −20.4717 −0.794454
\(665\) 0 0
\(666\) 0 0
\(667\) 0.355939 0.0137820
\(668\) 14.2738 0.552271
\(669\) 0 0
\(670\) 0 0
\(671\) 22.9996 0.887891
\(672\) 0 0
\(673\) 20.3306 0.783688 0.391844 0.920032i \(-0.371837\pi\)
0.391844 + 0.920032i \(0.371837\pi\)
\(674\) −8.32382 −0.320622
\(675\) 0 0
\(676\) −7.80597 −0.300230
\(677\) −27.0573 −1.03990 −0.519949 0.854197i \(-0.674049\pi\)
−0.519949 + 0.854197i \(0.674049\pi\)
\(678\) 0 0
\(679\) −7.80110 −0.299379
\(680\) 0 0
\(681\) 0 0
\(682\) 51.9755 1.99025
\(683\) −14.8635 −0.568737 −0.284369 0.958715i \(-0.591784\pi\)
−0.284369 + 0.958715i \(0.591784\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 29.8117 1.13821
\(687\) 0 0
\(688\) 64.0487 2.44183
\(689\) 8.59099 0.327291
\(690\) 0 0
\(691\) −3.66787 −0.139532 −0.0697662 0.997563i \(-0.522225\pi\)
−0.0697662 + 0.997563i \(0.522225\pi\)
\(692\) 15.0407 0.571763
\(693\) 0 0
\(694\) −17.8092 −0.676028
\(695\) 0 0
\(696\) 0 0
\(697\) −34.9596 −1.32419
\(698\) −26.6212 −1.00763
\(699\) 0 0
\(700\) 0 0
\(701\) −39.0313 −1.47419 −0.737097 0.675787i \(-0.763804\pi\)
−0.737097 + 0.675787i \(0.763804\pi\)
\(702\) 0 0
\(703\) 4.51498 0.170286
\(704\) 2.56941 0.0968382
\(705\) 0 0
\(706\) −21.4184 −0.806092
\(707\) 16.3507 0.614931
\(708\) 0 0
\(709\) −36.9135 −1.38632 −0.693158 0.720785i \(-0.743781\pi\)
−0.693158 + 0.720785i \(0.743781\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3.82602 0.143386
\(713\) 1.94281 0.0727589
\(714\) 0 0
\(715\) 0 0
\(716\) 6.99382 0.261371
\(717\) 0 0
\(718\) 8.40401 0.313635
\(719\) 41.9646 1.56502 0.782508 0.622640i \(-0.213940\pi\)
0.782508 + 0.622640i \(0.213940\pi\)
\(720\) 0 0
\(721\) −44.9785 −1.67509
\(722\) 24.7489 0.921060
\(723\) 0 0
\(724\) −23.0926 −0.858228
\(725\) 0 0
\(726\) 0 0
\(727\) 37.0774 1.37513 0.687563 0.726125i \(-0.258680\pi\)
0.687563 + 0.726125i \(0.258680\pi\)
\(728\) 10.6396 0.394329
\(729\) 0 0
\(730\) 0 0
\(731\) 83.7932 3.09920
\(732\) 0 0
\(733\) −6.29483 −0.232505 −0.116252 0.993220i \(-0.537088\pi\)
−0.116252 + 0.993220i \(0.537088\pi\)
\(734\) −21.2668 −0.784973
\(735\) 0 0
\(736\) 2.08723 0.0769361
\(737\) −5.87572 −0.216435
\(738\) 0 0
\(739\) −34.6989 −1.27642 −0.638210 0.769862i \(-0.720325\pi\)
−0.638210 + 0.769862i \(0.720325\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −17.3621 −0.637383
\(743\) −10.8648 −0.398590 −0.199295 0.979940i \(-0.563865\pi\)
−0.199295 + 0.979940i \(0.563865\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −14.7896 −0.541485
\(747\) 0 0
\(748\) 40.4898 1.48046
\(749\) −47.1852 −1.72411
\(750\) 0 0
\(751\) 6.02497 0.219854 0.109927 0.993940i \(-0.464938\pi\)
0.109927 + 0.993940i \(0.464938\pi\)
\(752\) −19.9809 −0.728627
\(753\) 0 0
\(754\) −4.45692 −0.162311
\(755\) 0 0
\(756\) 0 0
\(757\) −34.1212 −1.24016 −0.620078 0.784540i \(-0.712899\pi\)
−0.620078 + 0.784540i \(0.712899\pi\)
\(758\) 43.6929 1.58700
\(759\) 0 0
\(760\) 0 0
\(761\) 14.0126 0.507956 0.253978 0.967210i \(-0.418261\pi\)
0.253978 + 0.967210i \(0.418261\pi\)
\(762\) 0 0
\(763\) 26.5650 0.961718
\(764\) −3.53913 −0.128041
\(765\) 0 0
\(766\) 60.5050 2.18613
\(767\) 11.9461 0.431347
\(768\) 0 0
\(769\) 34.0766 1.22883 0.614417 0.788982i \(-0.289391\pi\)
0.614417 + 0.788982i \(0.289391\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −24.7976 −0.892486
\(773\) 14.3678 0.516773 0.258386 0.966042i \(-0.416809\pi\)
0.258386 + 0.966042i \(0.416809\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4.08088 −0.146495
\(777\) 0 0
\(778\) 45.1112 1.61731
\(779\) 12.1115 0.433941
\(780\) 0 0
\(781\) 27.9287 0.999367
\(782\) 4.11850 0.147277
\(783\) 0 0
\(784\) −5.54407 −0.198002
\(785\) 0 0
\(786\) 0 0
\(787\) 1.65976 0.0591642 0.0295821 0.999562i \(-0.490582\pi\)
0.0295821 + 0.999562i \(0.490582\pi\)
\(788\) 11.7046 0.416959
\(789\) 0 0
\(790\) 0 0
\(791\) −40.4446 −1.43804
\(792\) 0 0
\(793\) −10.7650 −0.382275
\(794\) 29.7354 1.05527
\(795\) 0 0
\(796\) −4.91334 −0.174149
\(797\) 8.57389 0.303703 0.151851 0.988403i \(-0.451477\pi\)
0.151851 + 0.988403i \(0.451477\pi\)
\(798\) 0 0
\(799\) −26.1404 −0.924782
\(800\) 0 0
\(801\) 0 0
\(802\) 14.2507 0.503211
\(803\) −36.3417 −1.28247
\(804\) 0 0
\(805\) 0 0
\(806\) −24.3271 −0.856886
\(807\) 0 0
\(808\) 8.55332 0.300905
\(809\) 44.3986 1.56097 0.780485 0.625175i \(-0.214972\pi\)
0.780485 + 0.625175i \(0.214972\pi\)
\(810\) 0 0
\(811\) −13.2677 −0.465892 −0.232946 0.972490i \(-0.574837\pi\)
−0.232946 + 0.972490i \(0.574837\pi\)
\(812\) 3.31005 0.116160
\(813\) 0 0
\(814\) 19.0714 0.668451
\(815\) 0 0
\(816\) 0 0
\(817\) −29.0297 −1.01562
\(818\) 34.5565 1.20824
\(819\) 0 0
\(820\) 0 0
\(821\) 18.5405 0.647069 0.323534 0.946216i \(-0.395129\pi\)
0.323534 + 0.946216i \(0.395129\pi\)
\(822\) 0 0
\(823\) −4.42801 −0.154351 −0.0771753 0.997018i \(-0.524590\pi\)
−0.0771753 + 0.997018i \(0.524590\pi\)
\(824\) −23.5290 −0.819672
\(825\) 0 0
\(826\) −24.1426 −0.840029
\(827\) −26.1067 −0.907819 −0.453910 0.891048i \(-0.649971\pi\)
−0.453910 + 0.891048i \(0.649971\pi\)
\(828\) 0 0
\(829\) −25.1458 −0.873350 −0.436675 0.899619i \(-0.643844\pi\)
−0.436675 + 0.899619i \(0.643844\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.20261 −0.0416930
\(833\) −7.25316 −0.251307
\(834\) 0 0
\(835\) 0 0
\(836\) −14.0275 −0.485150
\(837\) 0 0
\(838\) −12.3117 −0.425300
\(839\) 31.3784 1.08330 0.541651 0.840603i \(-0.317799\pi\)
0.541651 + 0.840603i \(0.317799\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 23.0937 0.795863
\(843\) 0 0
\(844\) 32.7971 1.12892
\(845\) 0 0
\(846\) 0 0
\(847\) 50.3540 1.73018
\(848\) −17.0480 −0.585430
\(849\) 0 0
\(850\) 0 0
\(851\) 0.712877 0.0244371
\(852\) 0 0
\(853\) −36.1547 −1.23791 −0.618956 0.785425i \(-0.712444\pi\)
−0.618956 + 0.785425i \(0.712444\pi\)
\(854\) 21.7556 0.744463
\(855\) 0 0
\(856\) −24.6834 −0.843660
\(857\) −20.2270 −0.690943 −0.345471 0.938429i \(-0.612281\pi\)
−0.345471 + 0.938429i \(0.612281\pi\)
\(858\) 0 0
\(859\) 29.8005 1.01678 0.508390 0.861127i \(-0.330241\pi\)
0.508390 + 0.861127i \(0.330241\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −70.5502 −2.40295
\(863\) 35.7369 1.21650 0.608248 0.793747i \(-0.291872\pi\)
0.608248 + 0.793747i \(0.291872\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 8.76805 0.297951
\(867\) 0 0
\(868\) 18.0671 0.613239
\(869\) 9.90378 0.335963
\(870\) 0 0
\(871\) 2.75013 0.0931845
\(872\) 13.8966 0.470598
\(873\) 0 0
\(874\) −1.42683 −0.0482632
\(875\) 0 0
\(876\) 0 0
\(877\) −0.626314 −0.0211491 −0.0105746 0.999944i \(-0.503366\pi\)
−0.0105746 + 0.999944i \(0.503366\pi\)
\(878\) −39.3796 −1.32900
\(879\) 0 0
\(880\) 0 0
\(881\) −6.65299 −0.224145 −0.112072 0.993700i \(-0.535749\pi\)
−0.112072 + 0.993700i \(0.535749\pi\)
\(882\) 0 0
\(883\) 1.03215 0.0347345 0.0173673 0.999849i \(-0.494472\pi\)
0.0173673 + 0.999849i \(0.494472\pi\)
\(884\) −18.9512 −0.637399
\(885\) 0 0
\(886\) 2.15440 0.0723783
\(887\) −55.8755 −1.87611 −0.938057 0.346480i \(-0.887377\pi\)
−0.938057 + 0.346480i \(0.887377\pi\)
\(888\) 0 0
\(889\) 15.2748 0.512301
\(890\) 0 0
\(891\) 0 0
\(892\) 9.85884 0.330098
\(893\) 9.05620 0.303054
\(894\) 0 0
\(895\) 0 0
\(896\) −30.9782 −1.03491
\(897\) 0 0
\(898\) 9.29588 0.310208
\(899\) 5.45828 0.182044
\(900\) 0 0
\(901\) −22.3034 −0.743035
\(902\) 51.1595 1.70342
\(903\) 0 0
\(904\) −21.1572 −0.703679
\(905\) 0 0
\(906\) 0 0
\(907\) −23.0687 −0.765982 −0.382991 0.923752i \(-0.625106\pi\)
−0.382991 + 0.923752i \(0.625106\pi\)
\(908\) 17.1824 0.570219
\(909\) 0 0
\(910\) 0 0
\(911\) −52.8451 −1.75084 −0.875418 0.483367i \(-0.839414\pi\)
−0.875418 + 0.483367i \(0.839414\pi\)
\(912\) 0 0
\(913\) 73.5670 2.43471
\(914\) −30.0629 −0.994393
\(915\) 0 0
\(916\) −24.5170 −0.810064
\(917\) −30.8881 −1.02001
\(918\) 0 0
\(919\) −49.6417 −1.63753 −0.818765 0.574129i \(-0.805341\pi\)
−0.818765 + 0.574129i \(0.805341\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −13.8634 −0.456567
\(923\) −13.0720 −0.430270
\(924\) 0 0
\(925\) 0 0
\(926\) 39.6951 1.30446
\(927\) 0 0
\(928\) 5.86400 0.192495
\(929\) −10.7335 −0.352153 −0.176077 0.984376i \(-0.556341\pi\)
−0.176077 + 0.984376i \(0.556341\pi\)
\(930\) 0 0
\(931\) 2.51281 0.0823542
\(932\) −30.3001 −0.992512
\(933\) 0 0
\(934\) 14.5666 0.476633
\(935\) 0 0
\(936\) 0 0
\(937\) 11.4028 0.372513 0.186256 0.982501i \(-0.440365\pi\)
0.186256 + 0.982501i \(0.440365\pi\)
\(938\) −5.55792 −0.181472
\(939\) 0 0
\(940\) 0 0
\(941\) −26.5670 −0.866060 −0.433030 0.901380i \(-0.642556\pi\)
−0.433030 + 0.901380i \(0.642556\pi\)
\(942\) 0 0
\(943\) 1.91231 0.0622734
\(944\) −23.7058 −0.771558
\(945\) 0 0
\(946\) −122.622 −3.98679
\(947\) −15.3658 −0.499322 −0.249661 0.968333i \(-0.580319\pi\)
−0.249661 + 0.968333i \(0.580319\pi\)
\(948\) 0 0
\(949\) 17.0097 0.552158
\(950\) 0 0
\(951\) 0 0
\(952\) −27.6218 −0.895229
\(953\) −12.7062 −0.411596 −0.205798 0.978595i \(-0.565979\pi\)
−0.205798 + 0.978595i \(0.565979\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 9.49419 0.307064
\(957\) 0 0
\(958\) 31.0881 1.00441
\(959\) −21.1264 −0.682209
\(960\) 0 0
\(961\) −1.20722 −0.0389426
\(962\) −8.92634 −0.287797
\(963\) 0 0
\(964\) −12.2869 −0.395733
\(965\) 0 0
\(966\) 0 0
\(967\) −15.8218 −0.508795 −0.254398 0.967100i \(-0.581877\pi\)
−0.254398 + 0.967100i \(0.581877\pi\)
\(968\) 26.3410 0.846632
\(969\) 0 0
\(970\) 0 0
\(971\) −47.8287 −1.53490 −0.767448 0.641111i \(-0.778474\pi\)
−0.767448 + 0.641111i \(0.778474\pi\)
\(972\) 0 0
\(973\) 56.0152 1.79577
\(974\) 41.3644 1.32540
\(975\) 0 0
\(976\) 21.3620 0.683781
\(977\) −46.2147 −1.47854 −0.739270 0.673409i \(-0.764829\pi\)
−0.739270 + 0.673409i \(0.764829\pi\)
\(978\) 0 0
\(979\) −13.7492 −0.439426
\(980\) 0 0
\(981\) 0 0
\(982\) 61.3372 1.95735
\(983\) −40.2132 −1.28260 −0.641301 0.767289i \(-0.721605\pi\)
−0.641301 + 0.767289i \(0.721605\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 11.5708 0.368489
\(987\) 0 0
\(988\) 6.56555 0.208878
\(989\) −4.58354 −0.145748
\(990\) 0 0
\(991\) −8.54051 −0.271298 −0.135649 0.990757i \(-0.543312\pi\)
−0.135649 + 0.990757i \(0.543312\pi\)
\(992\) 32.0073 1.01623
\(993\) 0 0
\(994\) 26.4181 0.837931
\(995\) 0 0
\(996\) 0 0
\(997\) −26.7017 −0.845651 −0.422825 0.906211i \(-0.638962\pi\)
−0.422825 + 0.906211i \(0.638962\pi\)
\(998\) −22.8946 −0.724715
\(999\) 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 6525.2.a.ca.1.3 9
3.2 odd 2 6525.2.a.cc.1.7 yes 9
5.4 even 2 6525.2.a.cd.1.7 yes 9
15.14 odd 2 6525.2.a.cb.1.3 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6525.2.a.ca.1.3 9 1.1 even 1 trivial
6525.2.a.cb.1.3 yes 9 15.14 odd 2
6525.2.a.cc.1.7 yes 9 3.2 odd 2
6525.2.a.cd.1.7 yes 9 5.4 even 2