Properties

Label 6042.2.a.be.1.1
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
Analytic rank $1$
Dimension $12$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 36 x^{10} + 105 x^{9} + 465 x^{8} - 1287 x^{7} - 2668 x^{6} + 6443 x^{5} + 7140 x^{4} - 10858 x^{3} - 10086 x^{2} + 2072 x + 1496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.97453\) of defining polynomial
Character \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.97453 q^{5} +1.00000 q^{6} +2.10909 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.97453 q^{5} +1.00000 q^{6} +2.10909 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.97453 q^{10} -2.46089 q^{11} -1.00000 q^{12} -3.30271 q^{13} -2.10909 q^{14} +3.97453 q^{15} +1.00000 q^{16} -1.57490 q^{17} -1.00000 q^{18} -1.00000 q^{19} -3.97453 q^{20} -2.10909 q^{21} +2.46089 q^{22} +5.45116 q^{23} +1.00000 q^{24} +10.7969 q^{25} +3.30271 q^{26} -1.00000 q^{27} +2.10909 q^{28} -10.0009 q^{29} -3.97453 q^{30} +5.53168 q^{31} -1.00000 q^{32} +2.46089 q^{33} +1.57490 q^{34} -8.38265 q^{35} +1.00000 q^{36} +9.63437 q^{37} +1.00000 q^{38} +3.30271 q^{39} +3.97453 q^{40} -1.80309 q^{41} +2.10909 q^{42} +2.92497 q^{43} -2.46089 q^{44} -3.97453 q^{45} -5.45116 q^{46} +3.51719 q^{47} -1.00000 q^{48} -2.55172 q^{49} -10.7969 q^{50} +1.57490 q^{51} -3.30271 q^{52} -1.00000 q^{53} +1.00000 q^{54} +9.78088 q^{55} -2.10909 q^{56} +1.00000 q^{57} +10.0009 q^{58} +2.26193 q^{59} +3.97453 q^{60} -0.865226 q^{61} -5.53168 q^{62} +2.10909 q^{63} +1.00000 q^{64} +13.1267 q^{65} -2.46089 q^{66} +5.26523 q^{67} -1.57490 q^{68} -5.45116 q^{69} +8.38265 q^{70} -0.404292 q^{71} -1.00000 q^{72} -11.0627 q^{73} -9.63437 q^{74} -10.7969 q^{75} -1.00000 q^{76} -5.19025 q^{77} -3.30271 q^{78} +8.91239 q^{79} -3.97453 q^{80} +1.00000 q^{81} +1.80309 q^{82} +2.47817 q^{83} -2.10909 q^{84} +6.25949 q^{85} -2.92497 q^{86} +10.0009 q^{87} +2.46089 q^{88} +13.5839 q^{89} +3.97453 q^{90} -6.96573 q^{91} +5.45116 q^{92} -5.53168 q^{93} -3.51719 q^{94} +3.97453 q^{95} +1.00000 q^{96} -7.25414 q^{97} +2.55172 q^{98} -2.46089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 7 q^{11} - 12 q^{12} + 4 q^{13} + q^{14} + 3 q^{15} + 12 q^{16} - 4 q^{17} - 12 q^{18} - 12 q^{19} - 3 q^{20} + q^{21} + 7 q^{22} - 18 q^{23} + 12 q^{24} + 21 q^{25} - 4 q^{26} - 12 q^{27} - q^{28} + 10 q^{29} - 3 q^{30} + 14 q^{31} - 12 q^{32} + 7 q^{33} + 4 q^{34} - 19 q^{35} + 12 q^{36} + 14 q^{37} + 12 q^{38} - 4 q^{39} + 3 q^{40} - 9 q^{41} - q^{42} - 4 q^{43} - 7 q^{44} - 3 q^{45} + 18 q^{46} - 14 q^{47} - 12 q^{48} + 37 q^{49} - 21 q^{50} + 4 q^{51} + 4 q^{52} - 12 q^{53} + 12 q^{54} - 4 q^{55} + q^{56} + 12 q^{57} - 10 q^{58} - 18 q^{59} + 3 q^{60} - 7 q^{61} - 14 q^{62} - q^{63} + 12 q^{64} + 3 q^{65} - 7 q^{66} + 8 q^{67} - 4 q^{68} + 18 q^{69} + 19 q^{70} - q^{71} - 12 q^{72} - 31 q^{73} - 14 q^{74} - 21 q^{75} - 12 q^{76} - 25 q^{77} + 4 q^{78} - 3 q^{80} + 12 q^{81} + 9 q^{82} - 48 q^{83} + q^{84} + 4 q^{85} + 4 q^{86} - 10 q^{87} + 7 q^{88} + 13 q^{89} + 3 q^{90} + 9 q^{91} - 18 q^{92} - 14 q^{93} + 14 q^{94} + 3 q^{95} + 12 q^{96} + 25 q^{97} - 37 q^{98} - 7 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.97453 −1.77746 −0.888731 0.458428i \(-0.848413\pi\)
−0.888731 + 0.458428i \(0.848413\pi\)
\(6\) 1.00000 0.408248
\(7\) 2.10909 0.797163 0.398581 0.917133i \(-0.369503\pi\)
0.398581 + 0.917133i \(0.369503\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.97453 1.25686
\(11\) −2.46089 −0.741986 −0.370993 0.928636i \(-0.620983\pi\)
−0.370993 + 0.928636i \(0.620983\pi\)
\(12\) −1.00000 −0.288675
\(13\) −3.30271 −0.916007 −0.458003 0.888950i \(-0.651435\pi\)
−0.458003 + 0.888950i \(0.651435\pi\)
\(14\) −2.10909 −0.563679
\(15\) 3.97453 1.02622
\(16\) 1.00000 0.250000
\(17\) −1.57490 −0.381970 −0.190985 0.981593i \(-0.561168\pi\)
−0.190985 + 0.981593i \(0.561168\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) −3.97453 −0.888731
\(21\) −2.10909 −0.460242
\(22\) 2.46089 0.524664
\(23\) 5.45116 1.13665 0.568323 0.822805i \(-0.307592\pi\)
0.568323 + 0.822805i \(0.307592\pi\)
\(24\) 1.00000 0.204124
\(25\) 10.7969 2.15937
\(26\) 3.30271 0.647715
\(27\) −1.00000 −0.192450
\(28\) 2.10909 0.398581
\(29\) −10.0009 −1.85712 −0.928558 0.371187i \(-0.878951\pi\)
−0.928558 + 0.371187i \(0.878951\pi\)
\(30\) −3.97453 −0.725646
\(31\) 5.53168 0.993519 0.496759 0.867888i \(-0.334523\pi\)
0.496759 + 0.867888i \(0.334523\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.46089 0.428386
\(34\) 1.57490 0.270093
\(35\) −8.38265 −1.41693
\(36\) 1.00000 0.166667
\(37\) 9.63437 1.58388 0.791941 0.610598i \(-0.209071\pi\)
0.791941 + 0.610598i \(0.209071\pi\)
\(38\) 1.00000 0.162221
\(39\) 3.30271 0.528857
\(40\) 3.97453 0.628428
\(41\) −1.80309 −0.281595 −0.140797 0.990038i \(-0.544967\pi\)
−0.140797 + 0.990038i \(0.544967\pi\)
\(42\) 2.10909 0.325440
\(43\) 2.92497 0.446054 0.223027 0.974812i \(-0.428406\pi\)
0.223027 + 0.974812i \(0.428406\pi\)
\(44\) −2.46089 −0.370993
\(45\) −3.97453 −0.592488
\(46\) −5.45116 −0.803730
\(47\) 3.51719 0.513035 0.256517 0.966540i \(-0.417425\pi\)
0.256517 + 0.966540i \(0.417425\pi\)
\(48\) −1.00000 −0.144338
\(49\) −2.55172 −0.364532
\(50\) −10.7969 −1.52691
\(51\) 1.57490 0.220530
\(52\) −3.30271 −0.458003
\(53\) −1.00000 −0.137361
\(54\) 1.00000 0.136083
\(55\) 9.78088 1.31885
\(56\) −2.10909 −0.281840
\(57\) 1.00000 0.132453
\(58\) 10.0009 1.31318
\(59\) 2.26193 0.294478 0.147239 0.989101i \(-0.452961\pi\)
0.147239 + 0.989101i \(0.452961\pi\)
\(60\) 3.97453 0.513109
\(61\) −0.865226 −0.110781 −0.0553904 0.998465i \(-0.517640\pi\)
−0.0553904 + 0.998465i \(0.517640\pi\)
\(62\) −5.53168 −0.702524
\(63\) 2.10909 0.265721
\(64\) 1.00000 0.125000
\(65\) 13.1267 1.62817
\(66\) −2.46089 −0.302915
\(67\) 5.26523 0.643251 0.321625 0.946867i \(-0.395771\pi\)
0.321625 + 0.946867i \(0.395771\pi\)
\(68\) −1.57490 −0.190985
\(69\) −5.45116 −0.656243
\(70\) 8.38265 1.00192
\(71\) −0.404292 −0.0479807 −0.0239903 0.999712i \(-0.507637\pi\)
−0.0239903 + 0.999712i \(0.507637\pi\)
\(72\) −1.00000 −0.117851
\(73\) −11.0627 −1.29479 −0.647396 0.762154i \(-0.724142\pi\)
−0.647396 + 0.762154i \(0.724142\pi\)
\(74\) −9.63437 −1.11997
\(75\) −10.7969 −1.24672
\(76\) −1.00000 −0.114708
\(77\) −5.19025 −0.591484
\(78\) −3.30271 −0.373958
\(79\) 8.91239 1.00272 0.501361 0.865238i \(-0.332833\pi\)
0.501361 + 0.865238i \(0.332833\pi\)
\(80\) −3.97453 −0.444366
\(81\) 1.00000 0.111111
\(82\) 1.80309 0.199118
\(83\) 2.47817 0.272014 0.136007 0.990708i \(-0.456573\pi\)
0.136007 + 0.990708i \(0.456573\pi\)
\(84\) −2.10909 −0.230121
\(85\) 6.25949 0.678937
\(86\) −2.92497 −0.315408
\(87\) 10.0009 1.07221
\(88\) 2.46089 0.262332
\(89\) 13.5839 1.43989 0.719943 0.694033i \(-0.244168\pi\)
0.719943 + 0.694033i \(0.244168\pi\)
\(90\) 3.97453 0.418952
\(91\) −6.96573 −0.730206
\(92\) 5.45116 0.568323
\(93\) −5.53168 −0.573608
\(94\) −3.51719 −0.362770
\(95\) 3.97453 0.407778
\(96\) 1.00000 0.102062
\(97\) −7.25414 −0.736546 −0.368273 0.929718i \(-0.620051\pi\)
−0.368273 + 0.929718i \(0.620051\pi\)
\(98\) 2.55172 0.257763
\(99\) −2.46089 −0.247329
\(100\) 10.7969 1.07969
\(101\) −3.18748 −0.317166 −0.158583 0.987346i \(-0.550693\pi\)
−0.158583 + 0.987346i \(0.550693\pi\)
\(102\) −1.57490 −0.155939
\(103\) −6.88698 −0.678594 −0.339297 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(104\) 3.30271 0.323857
\(105\) 8.38265 0.818063
\(106\) 1.00000 0.0971286
\(107\) 15.3991 1.48869 0.744345 0.667795i \(-0.232762\pi\)
0.744345 + 0.667795i \(0.232762\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −16.0466 −1.53698 −0.768491 0.639860i \(-0.778992\pi\)
−0.768491 + 0.639860i \(0.778992\pi\)
\(110\) −9.78088 −0.932570
\(111\) −9.63437 −0.914454
\(112\) 2.10909 0.199291
\(113\) 17.1265 1.61113 0.805565 0.592508i \(-0.201862\pi\)
0.805565 + 0.592508i \(0.201862\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −21.6658 −2.02035
\(116\) −10.0009 −0.928558
\(117\) −3.30271 −0.305336
\(118\) −2.26193 −0.208228
\(119\) −3.32162 −0.304492
\(120\) −3.97453 −0.362823
\(121\) −4.94402 −0.449456
\(122\) 0.865226 0.0783339
\(123\) 1.80309 0.162579
\(124\) 5.53168 0.496759
\(125\) −23.0398 −2.06075
\(126\) −2.10909 −0.187893
\(127\) 8.31054 0.737441 0.368720 0.929540i \(-0.379796\pi\)
0.368720 + 0.929540i \(0.379796\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.92497 −0.257529
\(130\) −13.1267 −1.15129
\(131\) −5.70281 −0.498256 −0.249128 0.968471i \(-0.580144\pi\)
−0.249128 + 0.968471i \(0.580144\pi\)
\(132\) 2.46089 0.214193
\(133\) −2.10909 −0.182882
\(134\) −5.26523 −0.454847
\(135\) 3.97453 0.342073
\(136\) 1.57490 0.135047
\(137\) −9.04968 −0.773166 −0.386583 0.922255i \(-0.626345\pi\)
−0.386583 + 0.922255i \(0.626345\pi\)
\(138\) 5.45116 0.464034
\(139\) 10.4617 0.887346 0.443673 0.896189i \(-0.353675\pi\)
0.443673 + 0.896189i \(0.353675\pi\)
\(140\) −8.38265 −0.708464
\(141\) −3.51719 −0.296201
\(142\) 0.404292 0.0339275
\(143\) 8.12761 0.679664
\(144\) 1.00000 0.0833333
\(145\) 39.7488 3.30096
\(146\) 11.0627 0.915556
\(147\) 2.55172 0.210462
\(148\) 9.63437 0.791941
\(149\) 9.83811 0.805970 0.402985 0.915207i \(-0.367973\pi\)
0.402985 + 0.915207i \(0.367973\pi\)
\(150\) 10.7969 0.881561
\(151\) 9.31710 0.758215 0.379107 0.925353i \(-0.376231\pi\)
0.379107 + 0.925353i \(0.376231\pi\)
\(152\) 1.00000 0.0811107
\(153\) −1.57490 −0.127323
\(154\) 5.19025 0.418242
\(155\) −21.9858 −1.76594
\(156\) 3.30271 0.264428
\(157\) 3.48510 0.278142 0.139071 0.990282i \(-0.455588\pi\)
0.139071 + 0.990282i \(0.455588\pi\)
\(158\) −8.91239 −0.709032
\(159\) 1.00000 0.0793052
\(160\) 3.97453 0.314214
\(161\) 11.4970 0.906092
\(162\) −1.00000 −0.0785674
\(163\) 14.0163 1.09784 0.548920 0.835875i \(-0.315039\pi\)
0.548920 + 0.835875i \(0.315039\pi\)
\(164\) −1.80309 −0.140797
\(165\) −9.78088 −0.761440
\(166\) −2.47817 −0.192343
\(167\) −14.3928 −1.11375 −0.556876 0.830596i \(-0.688000\pi\)
−0.556876 + 0.830596i \(0.688000\pi\)
\(168\) 2.10909 0.162720
\(169\) −2.09211 −0.160932
\(170\) −6.25949 −0.480081
\(171\) −1.00000 −0.0764719
\(172\) 2.92497 0.223027
\(173\) 14.2954 1.08686 0.543429 0.839455i \(-0.317126\pi\)
0.543429 + 0.839455i \(0.317126\pi\)
\(174\) −10.0009 −0.758165
\(175\) 22.7716 1.72137
\(176\) −2.46089 −0.185497
\(177\) −2.26193 −0.170017
\(178\) −13.5839 −1.01815
\(179\) −18.5238 −1.38453 −0.692265 0.721643i \(-0.743387\pi\)
−0.692265 + 0.721643i \(0.743387\pi\)
\(180\) −3.97453 −0.296244
\(181\) −12.3606 −0.918754 −0.459377 0.888241i \(-0.651927\pi\)
−0.459377 + 0.888241i \(0.651927\pi\)
\(182\) 6.96573 0.516334
\(183\) 0.865226 0.0639594
\(184\) −5.45116 −0.401865
\(185\) −38.2921 −2.81529
\(186\) 5.53168 0.405602
\(187\) 3.87566 0.283416
\(188\) 3.51719 0.256517
\(189\) −2.10909 −0.153414
\(190\) −3.97453 −0.288343
\(191\) 9.88051 0.714929 0.357464 0.933927i \(-0.383641\pi\)
0.357464 + 0.933927i \(0.383641\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 15.7028 1.13031 0.565155 0.824985i \(-0.308816\pi\)
0.565155 + 0.824985i \(0.308816\pi\)
\(194\) 7.25414 0.520817
\(195\) −13.1267 −0.940023
\(196\) −2.55172 −0.182266
\(197\) −9.57072 −0.681885 −0.340943 0.940084i \(-0.610746\pi\)
−0.340943 + 0.940084i \(0.610746\pi\)
\(198\) 2.46089 0.174888
\(199\) 22.1102 1.56735 0.783674 0.621173i \(-0.213343\pi\)
0.783674 + 0.621173i \(0.213343\pi\)
\(200\) −10.7969 −0.763454
\(201\) −5.26523 −0.371381
\(202\) 3.18748 0.224270
\(203\) −21.0928 −1.48042
\(204\) 1.57490 0.110265
\(205\) 7.16642 0.500524
\(206\) 6.88698 0.479838
\(207\) 5.45116 0.378882
\(208\) −3.30271 −0.229002
\(209\) 2.46089 0.170223
\(210\) −8.38265 −0.578458
\(211\) −0.942588 −0.0648904 −0.0324452 0.999474i \(-0.510329\pi\)
−0.0324452 + 0.999474i \(0.510329\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 0.404292 0.0277017
\(214\) −15.3991 −1.05266
\(215\) −11.6254 −0.792844
\(216\) 1.00000 0.0680414
\(217\) 11.6668 0.791996
\(218\) 16.0466 1.08681
\(219\) 11.0627 0.747548
\(220\) 9.78088 0.659427
\(221\) 5.20144 0.349887
\(222\) 9.63437 0.646617
\(223\) −23.1801 −1.55225 −0.776127 0.630577i \(-0.782818\pi\)
−0.776127 + 0.630577i \(0.782818\pi\)
\(224\) −2.10909 −0.140920
\(225\) 10.7969 0.719792
\(226\) −17.1265 −1.13924
\(227\) 10.0649 0.668029 0.334014 0.942568i \(-0.391597\pi\)
0.334014 + 0.942568i \(0.391597\pi\)
\(228\) 1.00000 0.0662266
\(229\) 2.12287 0.140283 0.0701417 0.997537i \(-0.477655\pi\)
0.0701417 + 0.997537i \(0.477655\pi\)
\(230\) 21.6658 1.42860
\(231\) 5.19025 0.341493
\(232\) 10.0009 0.656590
\(233\) −27.1529 −1.77884 −0.889422 0.457088i \(-0.848893\pi\)
−0.889422 + 0.457088i \(0.848893\pi\)
\(234\) 3.30271 0.215905
\(235\) −13.9792 −0.911901
\(236\) 2.26193 0.147239
\(237\) −8.91239 −0.578922
\(238\) 3.32162 0.215308
\(239\) 5.70151 0.368800 0.184400 0.982851i \(-0.440966\pi\)
0.184400 + 0.982851i \(0.440966\pi\)
\(240\) 3.97453 0.256555
\(241\) 16.6620 1.07330 0.536648 0.843807i \(-0.319690\pi\)
0.536648 + 0.843807i \(0.319690\pi\)
\(242\) 4.94402 0.317814
\(243\) −1.00000 −0.0641500
\(244\) −0.865226 −0.0553904
\(245\) 10.1419 0.647941
\(246\) −1.80309 −0.114961
\(247\) 3.30271 0.210146
\(248\) −5.53168 −0.351262
\(249\) −2.47817 −0.157048
\(250\) 23.0398 1.45717
\(251\) 14.9166 0.941525 0.470762 0.882260i \(-0.343979\pi\)
0.470762 + 0.882260i \(0.343979\pi\)
\(252\) 2.10909 0.132860
\(253\) −13.4147 −0.843376
\(254\) −8.31054 −0.521449
\(255\) −6.25949 −0.391985
\(256\) 1.00000 0.0625000
\(257\) −13.1002 −0.817169 −0.408585 0.912720i \(-0.633977\pi\)
−0.408585 + 0.912720i \(0.633977\pi\)
\(258\) 2.92497 0.182101
\(259\) 20.3198 1.26261
\(260\) 13.1267 0.814084
\(261\) −10.0009 −0.619039
\(262\) 5.70281 0.352321
\(263\) 14.5218 0.895452 0.447726 0.894171i \(-0.352234\pi\)
0.447726 + 0.894171i \(0.352234\pi\)
\(264\) −2.46089 −0.151457
\(265\) 3.97453 0.244153
\(266\) 2.10909 0.129317
\(267\) −13.5839 −0.831318
\(268\) 5.26523 0.321625
\(269\) 12.6564 0.771676 0.385838 0.922567i \(-0.373912\pi\)
0.385838 + 0.922567i \(0.373912\pi\)
\(270\) −3.97453 −0.241882
\(271\) −16.5217 −1.00362 −0.501810 0.864978i \(-0.667332\pi\)
−0.501810 + 0.864978i \(0.667332\pi\)
\(272\) −1.57490 −0.0954925
\(273\) 6.96573 0.421585
\(274\) 9.04968 0.546711
\(275\) −26.5699 −1.60223
\(276\) −5.45116 −0.328121
\(277\) −21.2934 −1.27939 −0.639697 0.768627i \(-0.720940\pi\)
−0.639697 + 0.768627i \(0.720940\pi\)
\(278\) −10.4617 −0.627448
\(279\) 5.53168 0.331173
\(280\) 8.38265 0.500959
\(281\) 10.6289 0.634065 0.317033 0.948415i \(-0.397314\pi\)
0.317033 + 0.948415i \(0.397314\pi\)
\(282\) 3.51719 0.209446
\(283\) −18.8882 −1.12278 −0.561392 0.827550i \(-0.689734\pi\)
−0.561392 + 0.827550i \(0.689734\pi\)
\(284\) −0.404292 −0.0239903
\(285\) −3.97453 −0.235431
\(286\) −8.12761 −0.480595
\(287\) −3.80288 −0.224477
\(288\) −1.00000 −0.0589256
\(289\) −14.5197 −0.854099
\(290\) −39.7488 −2.33413
\(291\) 7.25414 0.425245
\(292\) −11.0627 −0.647396
\(293\) −30.9567 −1.80851 −0.904254 0.426995i \(-0.859572\pi\)
−0.904254 + 0.426995i \(0.859572\pi\)
\(294\) −2.55172 −0.148819
\(295\) −8.99011 −0.523424
\(296\) −9.63437 −0.559987
\(297\) 2.46089 0.142795
\(298\) −9.83811 −0.569907
\(299\) −18.0036 −1.04118
\(300\) −10.7969 −0.623358
\(301\) 6.16904 0.355577
\(302\) −9.31710 −0.536139
\(303\) 3.18748 0.183116
\(304\) −1.00000 −0.0573539
\(305\) 3.43887 0.196909
\(306\) 1.57490 0.0900312
\(307\) 8.83648 0.504325 0.252162 0.967685i \(-0.418858\pi\)
0.252162 + 0.967685i \(0.418858\pi\)
\(308\) −5.19025 −0.295742
\(309\) 6.88698 0.391786
\(310\) 21.9858 1.24871
\(311\) −10.0178 −0.568057 −0.284028 0.958816i \(-0.591671\pi\)
−0.284028 + 0.958816i \(0.591671\pi\)
\(312\) −3.30271 −0.186979
\(313\) −33.7498 −1.90765 −0.953825 0.300363i \(-0.902892\pi\)
−0.953825 + 0.300363i \(0.902892\pi\)
\(314\) −3.48510 −0.196676
\(315\) −8.38265 −0.472309
\(316\) 8.91239 0.501361
\(317\) −22.3335 −1.25438 −0.627188 0.778868i \(-0.715794\pi\)
−0.627188 + 0.778868i \(0.715794\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 24.6111 1.37795
\(320\) −3.97453 −0.222183
\(321\) −15.3991 −0.859496
\(322\) −11.4970 −0.640704
\(323\) 1.57490 0.0876299
\(324\) 1.00000 0.0555556
\(325\) −35.6589 −1.97800
\(326\) −14.0163 −0.776289
\(327\) 16.0466 0.887377
\(328\) 1.80309 0.0995588
\(329\) 7.41809 0.408972
\(330\) 9.78088 0.538420
\(331\) 9.18190 0.504683 0.252341 0.967638i \(-0.418799\pi\)
0.252341 + 0.967638i \(0.418799\pi\)
\(332\) 2.47817 0.136007
\(333\) 9.63437 0.527960
\(334\) 14.3928 0.787541
\(335\) −20.9268 −1.14335
\(336\) −2.10909 −0.115061
\(337\) 6.07232 0.330781 0.165390 0.986228i \(-0.447112\pi\)
0.165390 + 0.986228i \(0.447112\pi\)
\(338\) 2.09211 0.113796
\(339\) −17.1265 −0.930186
\(340\) 6.25949 0.339469
\(341\) −13.6129 −0.737177
\(342\) 1.00000 0.0540738
\(343\) −20.1455 −1.08775
\(344\) −2.92497 −0.157704
\(345\) 21.6658 1.16645
\(346\) −14.2954 −0.768524
\(347\) −30.0166 −1.61138 −0.805688 0.592340i \(-0.798204\pi\)
−0.805688 + 0.592340i \(0.798204\pi\)
\(348\) 10.0009 0.536103
\(349\) −25.7498 −1.37835 −0.689177 0.724593i \(-0.742028\pi\)
−0.689177 + 0.724593i \(0.742028\pi\)
\(350\) −22.7716 −1.21719
\(351\) 3.30271 0.176286
\(352\) 2.46089 0.131166
\(353\) −25.4504 −1.35459 −0.677293 0.735713i \(-0.736847\pi\)
−0.677293 + 0.735713i \(0.736847\pi\)
\(354\) 2.26193 0.120220
\(355\) 1.60687 0.0852839
\(356\) 13.5839 0.719943
\(357\) 3.32162 0.175799
\(358\) 18.5238 0.979011
\(359\) −7.94351 −0.419242 −0.209621 0.977783i \(-0.567223\pi\)
−0.209621 + 0.977783i \(0.567223\pi\)
\(360\) 3.97453 0.209476
\(361\) 1.00000 0.0526316
\(362\) 12.3606 0.649657
\(363\) 4.94402 0.259494
\(364\) −6.96573 −0.365103
\(365\) 43.9690 2.30144
\(366\) −0.865226 −0.0452261
\(367\) 13.8681 0.723910 0.361955 0.932196i \(-0.382109\pi\)
0.361955 + 0.932196i \(0.382109\pi\)
\(368\) 5.45116 0.284162
\(369\) −1.80309 −0.0938649
\(370\) 38.2921 1.99071
\(371\) −2.10909 −0.109499
\(372\) −5.53168 −0.286804
\(373\) −19.3130 −0.999989 −0.499994 0.866029i \(-0.666665\pi\)
−0.499994 + 0.866029i \(0.666665\pi\)
\(374\) −3.87566 −0.200406
\(375\) 23.0398 1.18977
\(376\) −3.51719 −0.181385
\(377\) 33.0300 1.70113
\(378\) 2.10909 0.108480
\(379\) 23.0378 1.18337 0.591687 0.806168i \(-0.298462\pi\)
0.591687 + 0.806168i \(0.298462\pi\)
\(380\) 3.97453 0.203889
\(381\) −8.31054 −0.425762
\(382\) −9.88051 −0.505531
\(383\) −11.6881 −0.597234 −0.298617 0.954373i \(-0.596525\pi\)
−0.298617 + 0.954373i \(0.596525\pi\)
\(384\) 1.00000 0.0510310
\(385\) 20.6288 1.05134
\(386\) −15.7028 −0.799250
\(387\) 2.92497 0.148685
\(388\) −7.25414 −0.368273
\(389\) 0.549640 0.0278678 0.0139339 0.999903i \(-0.495565\pi\)
0.0139339 + 0.999903i \(0.495565\pi\)
\(390\) 13.1267 0.664697
\(391\) −8.58505 −0.434165
\(392\) 2.55172 0.128881
\(393\) 5.70281 0.287669
\(394\) 9.57072 0.482166
\(395\) −35.4225 −1.78230
\(396\) −2.46089 −0.123664
\(397\) −35.9803 −1.80580 −0.902899 0.429852i \(-0.858566\pi\)
−0.902899 + 0.429852i \(0.858566\pi\)
\(398\) −22.1102 −1.10828
\(399\) 2.10909 0.105587
\(400\) 10.7969 0.539844
\(401\) −18.5237 −0.925031 −0.462515 0.886611i \(-0.653053\pi\)
−0.462515 + 0.886611i \(0.653053\pi\)
\(402\) 5.26523 0.262606
\(403\) −18.2695 −0.910070
\(404\) −3.18748 −0.158583
\(405\) −3.97453 −0.197496
\(406\) 21.0928 1.04682
\(407\) −23.7091 −1.17522
\(408\) −1.57490 −0.0779693
\(409\) 15.3045 0.756759 0.378380 0.925651i \(-0.376481\pi\)
0.378380 + 0.925651i \(0.376481\pi\)
\(410\) −7.16642 −0.353924
\(411\) 9.04968 0.446388
\(412\) −6.88698 −0.339297
\(413\) 4.77063 0.234747
\(414\) −5.45116 −0.267910
\(415\) −9.84955 −0.483495
\(416\) 3.30271 0.161929
\(417\) −10.4617 −0.512310
\(418\) −2.46089 −0.120366
\(419\) 2.86981 0.140200 0.0700998 0.997540i \(-0.477668\pi\)
0.0700998 + 0.997540i \(0.477668\pi\)
\(420\) 8.38265 0.409032
\(421\) −6.89059 −0.335827 −0.167913 0.985802i \(-0.553703\pi\)
−0.167913 + 0.985802i \(0.553703\pi\)
\(422\) 0.942588 0.0458844
\(423\) 3.51719 0.171012
\(424\) 1.00000 0.0485643
\(425\) −17.0040 −0.824816
\(426\) −0.404292 −0.0195880
\(427\) −1.82484 −0.0883104
\(428\) 15.3991 0.744345
\(429\) −8.12761 −0.392404
\(430\) 11.6254 0.560625
\(431\) −11.3253 −0.545519 −0.272759 0.962082i \(-0.587936\pi\)
−0.272759 + 0.962082i \(0.587936\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 6.17753 0.296873 0.148437 0.988922i \(-0.452576\pi\)
0.148437 + 0.988922i \(0.452576\pi\)
\(434\) −11.6668 −0.560026
\(435\) −39.7488 −1.90581
\(436\) −16.0466 −0.768491
\(437\) −5.45116 −0.260765
\(438\) −11.0627 −0.528597
\(439\) −29.3486 −1.40073 −0.700366 0.713784i \(-0.746980\pi\)
−0.700366 + 0.713784i \(0.746980\pi\)
\(440\) −9.78088 −0.466285
\(441\) −2.55172 −0.121511
\(442\) −5.20144 −0.247407
\(443\) 23.2012 1.10232 0.551162 0.834398i \(-0.314185\pi\)
0.551162 + 0.834398i \(0.314185\pi\)
\(444\) −9.63437 −0.457227
\(445\) −53.9894 −2.55934
\(446\) 23.1801 1.09761
\(447\) −9.83811 −0.465327
\(448\) 2.10909 0.0996453
\(449\) 12.4029 0.585330 0.292665 0.956215i \(-0.405458\pi\)
0.292665 + 0.956215i \(0.405458\pi\)
\(450\) −10.7969 −0.508969
\(451\) 4.43720 0.208939
\(452\) 17.1265 0.805565
\(453\) −9.31710 −0.437755
\(454\) −10.0649 −0.472368
\(455\) 27.6855 1.29791
\(456\) −1.00000 −0.0468293
\(457\) −31.5034 −1.47367 −0.736833 0.676075i \(-0.763680\pi\)
−0.736833 + 0.676075i \(0.763680\pi\)
\(458\) −2.12287 −0.0991954
\(459\) 1.57490 0.0735101
\(460\) −21.6658 −1.01017
\(461\) −29.1293 −1.35669 −0.678344 0.734745i \(-0.737302\pi\)
−0.678344 + 0.734745i \(0.737302\pi\)
\(462\) −5.19025 −0.241472
\(463\) 23.3457 1.08497 0.542484 0.840066i \(-0.317484\pi\)
0.542484 + 0.840066i \(0.317484\pi\)
\(464\) −10.0009 −0.464279
\(465\) 21.9858 1.01957
\(466\) 27.1529 1.25783
\(467\) 29.9377 1.38535 0.692677 0.721248i \(-0.256431\pi\)
0.692677 + 0.721248i \(0.256431\pi\)
\(468\) −3.30271 −0.152668
\(469\) 11.1049 0.512775
\(470\) 13.9792 0.644811
\(471\) −3.48510 −0.160585
\(472\) −2.26193 −0.104114
\(473\) −7.19803 −0.330966
\(474\) 8.91239 0.409360
\(475\) −10.7969 −0.495394
\(476\) −3.32162 −0.152246
\(477\) −1.00000 −0.0457869
\(478\) −5.70151 −0.260781
\(479\) 10.2080 0.466413 0.233207 0.972427i \(-0.425078\pi\)
0.233207 + 0.972427i \(0.425078\pi\)
\(480\) −3.97453 −0.181412
\(481\) −31.8195 −1.45085
\(482\) −16.6620 −0.758934
\(483\) −11.4970 −0.523132
\(484\) −4.94402 −0.224728
\(485\) 28.8318 1.30918
\(486\) 1.00000 0.0453609
\(487\) 30.8479 1.39785 0.698926 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(488\) 0.865226 0.0391670
\(489\) −14.0163 −0.633838
\(490\) −10.1419 −0.458164
\(491\) −33.0651 −1.49221 −0.746104 0.665830i \(-0.768078\pi\)
−0.746104 + 0.665830i \(0.768078\pi\)
\(492\) 1.80309 0.0812894
\(493\) 15.7504 0.709362
\(494\) −3.30271 −0.148596
\(495\) 9.78088 0.439618
\(496\) 5.53168 0.248380
\(497\) −0.852691 −0.0382484
\(498\) 2.47817 0.111049
\(499\) 15.9857 0.715620 0.357810 0.933794i \(-0.383523\pi\)
0.357810 + 0.933794i \(0.383523\pi\)
\(500\) −23.0398 −1.03037
\(501\) 14.3928 0.643025
\(502\) −14.9166 −0.665759
\(503\) 0.287024 0.0127978 0.00639889 0.999980i \(-0.497963\pi\)
0.00639889 + 0.999980i \(0.497963\pi\)
\(504\) −2.10909 −0.0939465
\(505\) 12.6687 0.563750
\(506\) 13.4147 0.596357
\(507\) 2.09211 0.0929139
\(508\) 8.31054 0.368720
\(509\) 9.07055 0.402045 0.201023 0.979587i \(-0.435574\pi\)
0.201023 + 0.979587i \(0.435574\pi\)
\(510\) 6.25949 0.277175
\(511\) −23.3323 −1.03216
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) 13.1002 0.577826
\(515\) 27.3725 1.20618
\(516\) −2.92497 −0.128765
\(517\) −8.65542 −0.380665
\(518\) −20.3198 −0.892801
\(519\) −14.2954 −0.627497
\(520\) −13.1267 −0.575644
\(521\) −14.6070 −0.639944 −0.319972 0.947427i \(-0.603674\pi\)
−0.319972 + 0.947427i \(0.603674\pi\)
\(522\) 10.0009 0.437727
\(523\) −31.8464 −1.39255 −0.696274 0.717776i \(-0.745160\pi\)
−0.696274 + 0.717776i \(0.745160\pi\)
\(524\) −5.70281 −0.249128
\(525\) −22.7716 −0.993835
\(526\) −14.5218 −0.633180
\(527\) −8.71185 −0.379494
\(528\) 2.46089 0.107097
\(529\) 6.71519 0.291965
\(530\) −3.97453 −0.172642
\(531\) 2.26193 0.0981595
\(532\) −2.10909 −0.0914408
\(533\) 5.95507 0.257943
\(534\) 13.5839 0.587831
\(535\) −61.2043 −2.64609
\(536\) −5.26523 −0.227423
\(537\) 18.5238 0.799359
\(538\) −12.6564 −0.545657
\(539\) 6.27951 0.270477
\(540\) 3.97453 0.171036
\(541\) 30.6230 1.31658 0.658292 0.752763i \(-0.271279\pi\)
0.658292 + 0.752763i \(0.271279\pi\)
\(542\) 16.5217 0.709667
\(543\) 12.3606 0.530443
\(544\) 1.57490 0.0675234
\(545\) 63.7775 2.73193
\(546\) −6.96573 −0.298106
\(547\) −22.5604 −0.964613 −0.482307 0.876002i \(-0.660201\pi\)
−0.482307 + 0.876002i \(0.660201\pi\)
\(548\) −9.04968 −0.386583
\(549\) −0.865226 −0.0369270
\(550\) 26.5699 1.13295
\(551\) 10.0009 0.426052
\(552\) 5.45116 0.232017
\(553\) 18.7971 0.799333
\(554\) 21.2934 0.904669
\(555\) 38.2921 1.62541
\(556\) 10.4617 0.443673
\(557\) 13.6803 0.579653 0.289826 0.957079i \(-0.406402\pi\)
0.289826 + 0.957079i \(0.406402\pi\)
\(558\) −5.53168 −0.234175
\(559\) −9.66033 −0.408588
\(560\) −8.38265 −0.354232
\(561\) −3.87566 −0.163631
\(562\) −10.6289 −0.448352
\(563\) −18.9214 −0.797443 −0.398722 0.917072i \(-0.630546\pi\)
−0.398722 + 0.917072i \(0.630546\pi\)
\(564\) −3.51719 −0.148100
\(565\) −68.0699 −2.86372
\(566\) 18.8882 0.793928
\(567\) 2.10909 0.0885736
\(568\) 0.404292 0.0169637
\(569\) −15.4115 −0.646085 −0.323042 0.946384i \(-0.604706\pi\)
−0.323042 + 0.946384i \(0.604706\pi\)
\(570\) 3.97453 0.166475
\(571\) −29.8519 −1.24926 −0.624631 0.780920i \(-0.714751\pi\)
−0.624631 + 0.780920i \(0.714751\pi\)
\(572\) 8.12761 0.339832
\(573\) −9.88051 −0.412764
\(574\) 3.80288 0.158729
\(575\) 58.8555 2.45444
\(576\) 1.00000 0.0416667
\(577\) 13.7365 0.571857 0.285928 0.958251i \(-0.407698\pi\)
0.285928 + 0.958251i \(0.407698\pi\)
\(578\) 14.5197 0.603939
\(579\) −15.7028 −0.652585
\(580\) 39.7488 1.65048
\(581\) 5.22669 0.216840
\(582\) −7.25414 −0.300694
\(583\) 2.46089 0.101920
\(584\) 11.0627 0.457778
\(585\) 13.1267 0.542723
\(586\) 30.9567 1.27881
\(587\) 29.3934 1.21320 0.606598 0.795008i \(-0.292534\pi\)
0.606598 + 0.795008i \(0.292534\pi\)
\(588\) 2.55172 0.105231
\(589\) −5.53168 −0.227929
\(590\) 8.99011 0.370117
\(591\) 9.57072 0.393687
\(592\) 9.63437 0.395970
\(593\) 16.7364 0.687283 0.343642 0.939101i \(-0.388339\pi\)
0.343642 + 0.939101i \(0.388339\pi\)
\(594\) −2.46089 −0.100972
\(595\) 13.2019 0.541223
\(596\) 9.83811 0.402985
\(597\) −22.1102 −0.904908
\(598\) 18.0036 0.736222
\(599\) −14.0789 −0.575249 −0.287624 0.957743i \(-0.592865\pi\)
−0.287624 + 0.957743i \(0.592865\pi\)
\(600\) 10.7969 0.440780
\(601\) −5.54555 −0.226208 −0.113104 0.993583i \(-0.536079\pi\)
−0.113104 + 0.993583i \(0.536079\pi\)
\(602\) −6.16904 −0.251431
\(603\) 5.26523 0.214417
\(604\) 9.31710 0.379107
\(605\) 19.6501 0.798892
\(606\) −3.18748 −0.129482
\(607\) −13.1971 −0.535653 −0.267826 0.963467i \(-0.586305\pi\)
−0.267826 + 0.963467i \(0.586305\pi\)
\(608\) 1.00000 0.0405554
\(609\) 21.0928 0.854723
\(610\) −3.43887 −0.139236
\(611\) −11.6163 −0.469943
\(612\) −1.57490 −0.0636616
\(613\) 5.52185 0.223025 0.111513 0.993763i \(-0.464430\pi\)
0.111513 + 0.993763i \(0.464430\pi\)
\(614\) −8.83648 −0.356611
\(615\) −7.16642 −0.288978
\(616\) 5.19025 0.209121
\(617\) −47.9540 −1.93056 −0.965279 0.261223i \(-0.915874\pi\)
−0.965279 + 0.261223i \(0.915874\pi\)
\(618\) −6.88698 −0.277035
\(619\) 27.8577 1.11970 0.559848 0.828595i \(-0.310860\pi\)
0.559848 + 0.828595i \(0.310860\pi\)
\(620\) −21.9858 −0.882971
\(621\) −5.45116 −0.218748
\(622\) 10.0178 0.401677
\(623\) 28.6496 1.14782
\(624\) 3.30271 0.132214
\(625\) 37.5881 1.50352
\(626\) 33.7498 1.34891
\(627\) −2.46089 −0.0982785
\(628\) 3.48510 0.139071
\(629\) −15.1732 −0.604995
\(630\) 8.38265 0.333973
\(631\) 42.5805 1.69510 0.847552 0.530712i \(-0.178075\pi\)
0.847552 + 0.530712i \(0.178075\pi\)
\(632\) −8.91239 −0.354516
\(633\) 0.942588 0.0374645
\(634\) 22.3335 0.886978
\(635\) −33.0305 −1.31077
\(636\) 1.00000 0.0396526
\(637\) 8.42759 0.333913
\(638\) −24.6111 −0.974361
\(639\) −0.404292 −0.0159936
\(640\) 3.97453 0.157107
\(641\) 5.47068 0.216079 0.108039 0.994147i \(-0.465543\pi\)
0.108039 + 0.994147i \(0.465543\pi\)
\(642\) 15.3991 0.607756
\(643\) −18.4437 −0.727348 −0.363674 0.931526i \(-0.618478\pi\)
−0.363674 + 0.931526i \(0.618478\pi\)
\(644\) 11.4970 0.453046
\(645\) 11.6254 0.457749
\(646\) −1.57490 −0.0619637
\(647\) 6.20660 0.244006 0.122003 0.992530i \(-0.461068\pi\)
0.122003 + 0.992530i \(0.461068\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −5.56636 −0.218499
\(650\) 35.6589 1.39866
\(651\) −11.6668 −0.457259
\(652\) 14.0163 0.548920
\(653\) 8.02181 0.313918 0.156959 0.987605i \(-0.449831\pi\)
0.156959 + 0.987605i \(0.449831\pi\)
\(654\) −16.0466 −0.627470
\(655\) 22.6660 0.885632
\(656\) −1.80309 −0.0703987
\(657\) −11.0627 −0.431597
\(658\) −7.41809 −0.289187
\(659\) −25.8944 −1.00870 −0.504352 0.863498i \(-0.668269\pi\)
−0.504352 + 0.863498i \(0.668269\pi\)
\(660\) −9.78088 −0.380720
\(661\) 5.62074 0.218622 0.109311 0.994008i \(-0.465136\pi\)
0.109311 + 0.994008i \(0.465136\pi\)
\(662\) −9.18190 −0.356865
\(663\) −5.20144 −0.202007
\(664\) −2.47817 −0.0961716
\(665\) 8.38265 0.325065
\(666\) −9.63437 −0.373324
\(667\) −54.5164 −2.11088
\(668\) −14.3928 −0.556876
\(669\) 23.1801 0.896194
\(670\) 20.9268 0.808473
\(671\) 2.12923 0.0821979
\(672\) 2.10909 0.0813601
\(673\) 5.07757 0.195726 0.0978629 0.995200i \(-0.468799\pi\)
0.0978629 + 0.995200i \(0.468799\pi\)
\(674\) −6.07232 −0.233897
\(675\) −10.7969 −0.415572
\(676\) −2.09211 −0.0804658
\(677\) 32.7041 1.25692 0.628460 0.777842i \(-0.283686\pi\)
0.628460 + 0.777842i \(0.283686\pi\)
\(678\) 17.1265 0.657741
\(679\) −15.2997 −0.587147
\(680\) −6.25949 −0.240041
\(681\) −10.0649 −0.385686
\(682\) 13.6129 0.521263
\(683\) −30.4393 −1.16473 −0.582364 0.812928i \(-0.697872\pi\)
−0.582364 + 0.812928i \(0.697872\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 35.9682 1.37427
\(686\) 20.1455 0.769158
\(687\) −2.12287 −0.0809927
\(688\) 2.92497 0.111513
\(689\) 3.30271 0.125823
\(690\) −21.6658 −0.824803
\(691\) −22.4429 −0.853769 −0.426884 0.904306i \(-0.640389\pi\)
−0.426884 + 0.904306i \(0.640389\pi\)
\(692\) 14.2954 0.543429
\(693\) −5.19025 −0.197161
\(694\) 30.0166 1.13942
\(695\) −41.5801 −1.57722
\(696\) −10.0009 −0.379082
\(697\) 2.83968 0.107561
\(698\) 25.7498 0.974643
\(699\) 27.1529 1.02702
\(700\) 22.7716 0.860686
\(701\) 30.1518 1.13882 0.569408 0.822055i \(-0.307172\pi\)
0.569408 + 0.822055i \(0.307172\pi\)
\(702\) −3.30271 −0.124653
\(703\) −9.63437 −0.363367
\(704\) −2.46089 −0.0927483
\(705\) 13.9792 0.526486
\(706\) 25.4504 0.957838
\(707\) −6.72269 −0.252833
\(708\) −2.26193 −0.0850086
\(709\) −41.1902 −1.54693 −0.773465 0.633839i \(-0.781478\pi\)
−0.773465 + 0.633839i \(0.781478\pi\)
\(710\) −1.60687 −0.0603048
\(711\) 8.91239 0.334241
\(712\) −13.5839 −0.509076
\(713\) 30.1541 1.12928
\(714\) −3.32162 −0.124308
\(715\) −32.3034 −1.20808
\(716\) −18.5238 −0.692265
\(717\) −5.70151 −0.212927
\(718\) 7.94351 0.296449
\(719\) 2.64266 0.0985547 0.0492773 0.998785i \(-0.484308\pi\)
0.0492773 + 0.998785i \(0.484308\pi\)
\(720\) −3.97453 −0.148122
\(721\) −14.5253 −0.540950
\(722\) −1.00000 −0.0372161
\(723\) −16.6620 −0.619667
\(724\) −12.3606 −0.459377
\(725\) −107.978 −4.01021
\(726\) −4.94402 −0.183490
\(727\) 32.7498 1.21462 0.607311 0.794465i \(-0.292248\pi\)
0.607311 + 0.794465i \(0.292248\pi\)
\(728\) 6.96573 0.258167
\(729\) 1.00000 0.0370370
\(730\) −43.9690 −1.62737
\(731\) −4.60654 −0.170379
\(732\) 0.865226 0.0319797
\(733\) −50.7532 −1.87461 −0.937305 0.348510i \(-0.886688\pi\)
−0.937305 + 0.348510i \(0.886688\pi\)
\(734\) −13.8681 −0.511882
\(735\) −10.1419 −0.374089
\(736\) −5.45116 −0.200933
\(737\) −12.9572 −0.477283
\(738\) 1.80309 0.0663725
\(739\) −40.6894 −1.49678 −0.748392 0.663257i \(-0.769174\pi\)
−0.748392 + 0.663257i \(0.769174\pi\)
\(740\) −38.2921 −1.40765
\(741\) −3.30271 −0.121328
\(742\) 2.10909 0.0774273
\(743\) −10.8906 −0.399536 −0.199768 0.979843i \(-0.564019\pi\)
−0.199768 + 0.979843i \(0.564019\pi\)
\(744\) 5.53168 0.202801
\(745\) −39.1019 −1.43258
\(746\) 19.3130 0.707099
\(747\) 2.47817 0.0906714
\(748\) 3.87566 0.141708
\(749\) 32.4782 1.18673
\(750\) −23.0398 −0.841296
\(751\) −10.8828 −0.397120 −0.198560 0.980089i \(-0.563626\pi\)
−0.198560 + 0.980089i \(0.563626\pi\)
\(752\) 3.51719 0.128259
\(753\) −14.9166 −0.543590
\(754\) −33.0300 −1.20288
\(755\) −37.0311 −1.34770
\(756\) −2.10909 −0.0767070
\(757\) 41.3886 1.50429 0.752147 0.658996i \(-0.229019\pi\)
0.752147 + 0.658996i \(0.229019\pi\)
\(758\) −23.0378 −0.836772
\(759\) 13.4147 0.486923
\(760\) −3.97453 −0.144171
\(761\) −18.7619 −0.680117 −0.340059 0.940404i \(-0.610447\pi\)
−0.340059 + 0.940404i \(0.610447\pi\)
\(762\) 8.31054 0.301059
\(763\) −33.8437 −1.22522
\(764\) 9.88051 0.357464
\(765\) 6.25949 0.226312
\(766\) 11.6881 0.422308
\(767\) −7.47050 −0.269744
\(768\) −1.00000 −0.0360844
\(769\) 20.6982 0.746395 0.373197 0.927752i \(-0.378261\pi\)
0.373197 + 0.927752i \(0.378261\pi\)
\(770\) −20.6288 −0.743410
\(771\) 13.1002 0.471793
\(772\) 15.7028 0.565155
\(773\) 42.3504 1.52324 0.761619 0.648026i \(-0.224405\pi\)
0.761619 + 0.648026i \(0.224405\pi\)
\(774\) −2.92497 −0.105136
\(775\) 59.7248 2.14538
\(776\) 7.25414 0.260408
\(777\) −20.3198 −0.728969
\(778\) −0.549640 −0.0197055
\(779\) 1.80309 0.0646023
\(780\) −13.1267 −0.470012
\(781\) 0.994919 0.0356010
\(782\) 8.58505 0.307001
\(783\) 10.0009 0.357402
\(784\) −2.55172 −0.0911329
\(785\) −13.8516 −0.494386
\(786\) −5.70281 −0.203412
\(787\) 2.72995 0.0973121 0.0486560 0.998816i \(-0.484506\pi\)
0.0486560 + 0.998816i \(0.484506\pi\)
\(788\) −9.57072 −0.340943
\(789\) −14.5218 −0.516989
\(790\) 35.4225 1.26028
\(791\) 36.1215 1.28433
\(792\) 2.46089 0.0874439
\(793\) 2.85759 0.101476
\(794\) 35.9803 1.27689
\(795\) −3.97453 −0.140962
\(796\) 22.1102 0.783674
\(797\) −22.1632 −0.785061 −0.392530 0.919739i \(-0.628400\pi\)
−0.392530 + 0.919739i \(0.628400\pi\)
\(798\) −2.10909 −0.0746611
\(799\) −5.53923 −0.195964
\(800\) −10.7969 −0.381727
\(801\) 13.5839 0.479962
\(802\) 18.5237 0.654096
\(803\) 27.2241 0.960718
\(804\) −5.26523 −0.185690
\(805\) −45.6952 −1.61054
\(806\) 18.2695 0.643517
\(807\) −12.6564 −0.445527
\(808\) 3.18748 0.112135
\(809\) −20.1693 −0.709113 −0.354557 0.935035i \(-0.615368\pi\)
−0.354557 + 0.935035i \(0.615368\pi\)
\(810\) 3.97453 0.139651
\(811\) 25.1648 0.883656 0.441828 0.897100i \(-0.354330\pi\)
0.441828 + 0.897100i \(0.354330\pi\)
\(812\) −21.0928 −0.740212
\(813\) 16.5217 0.579441
\(814\) 23.7091 0.831005
\(815\) −55.7081 −1.95137
\(816\) 1.57490 0.0551326
\(817\) −2.92497 −0.102332
\(818\) −15.3045 −0.535109
\(819\) −6.96573 −0.243402
\(820\) 7.16642 0.250262
\(821\) 35.0158 1.22206 0.611029 0.791608i \(-0.290756\pi\)
0.611029 + 0.791608i \(0.290756\pi\)
\(822\) −9.04968 −0.315644
\(823\) −37.6598 −1.31274 −0.656369 0.754440i \(-0.727908\pi\)
−0.656369 + 0.754440i \(0.727908\pi\)
\(824\) 6.88698 0.239919
\(825\) 26.5699 0.925046
\(826\) −4.77063 −0.165991
\(827\) −3.14803 −0.109468 −0.0547339 0.998501i \(-0.517431\pi\)
−0.0547339 + 0.998501i \(0.517431\pi\)
\(828\) 5.45116 0.189441
\(829\) 15.3048 0.531556 0.265778 0.964034i \(-0.414371\pi\)
0.265778 + 0.964034i \(0.414371\pi\)
\(830\) 9.84955 0.341883
\(831\) 21.2934 0.738659
\(832\) −3.30271 −0.114501
\(833\) 4.01871 0.139240
\(834\) 10.4617 0.362258
\(835\) 57.2047 1.97965
\(836\) 2.46089 0.0851117
\(837\) −5.53168 −0.191203
\(838\) −2.86981 −0.0991361
\(839\) −25.8995 −0.894151 −0.447075 0.894496i \(-0.647534\pi\)
−0.447075 + 0.894496i \(0.647534\pi\)
\(840\) −8.38265 −0.289229
\(841\) 71.0176 2.44888
\(842\) 6.89059 0.237465
\(843\) −10.6289 −0.366078
\(844\) −0.942588 −0.0324452
\(845\) 8.31515 0.286050
\(846\) −3.51719 −0.120923
\(847\) −10.4274 −0.358290
\(848\) −1.00000 −0.0343401
\(849\) 18.8882 0.648240
\(850\) 17.0040 0.583233
\(851\) 52.5186 1.80031
\(852\) 0.404292 0.0138508
\(853\) −48.3913 −1.65689 −0.828443 0.560074i \(-0.810773\pi\)
−0.828443 + 0.560074i \(0.810773\pi\)
\(854\) 1.82484 0.0624449
\(855\) 3.97453 0.135926
\(856\) −15.3991 −0.526332
\(857\) 19.2617 0.657968 0.328984 0.944336i \(-0.393294\pi\)
0.328984 + 0.944336i \(0.393294\pi\)
\(858\) 8.12761 0.277472
\(859\) −16.4311 −0.560621 −0.280311 0.959909i \(-0.590437\pi\)
−0.280311 + 0.959909i \(0.590437\pi\)
\(860\) −11.6254 −0.396422
\(861\) 3.80288 0.129602
\(862\) 11.3253 0.385740
\(863\) 7.24931 0.246769 0.123385 0.992359i \(-0.460625\pi\)
0.123385 + 0.992359i \(0.460625\pi\)
\(864\) 1.00000 0.0340207
\(865\) −56.8174 −1.93185
\(866\) −6.17753 −0.209921
\(867\) 14.5197 0.493114
\(868\) 11.6668 0.395998
\(869\) −21.9324 −0.744006
\(870\) 39.7488 1.34761
\(871\) −17.3895 −0.589222
\(872\) 16.0466 0.543405
\(873\) −7.25414 −0.245515
\(874\) 5.45116 0.184388
\(875\) −48.5932 −1.64275
\(876\) 11.0627 0.373774
\(877\) 14.4556 0.488130 0.244065 0.969759i \(-0.421519\pi\)
0.244065 + 0.969759i \(0.421519\pi\)
\(878\) 29.3486 0.990467
\(879\) 30.9567 1.04414
\(880\) 9.78088 0.329713
\(881\) 34.4101 1.15930 0.579652 0.814864i \(-0.303188\pi\)
0.579652 + 0.814864i \(0.303188\pi\)
\(882\) 2.55172 0.0859209
\(883\) −39.8251 −1.34022 −0.670111 0.742261i \(-0.733754\pi\)
−0.670111 + 0.742261i \(0.733754\pi\)
\(884\) 5.20144 0.174943
\(885\) 8.99011 0.302199
\(886\) −23.2012 −0.779461
\(887\) −19.4413 −0.652774 −0.326387 0.945236i \(-0.605831\pi\)
−0.326387 + 0.945236i \(0.605831\pi\)
\(888\) 9.63437 0.323308
\(889\) 17.5277 0.587860
\(890\) 53.9894 1.80973
\(891\) −2.46089 −0.0824429
\(892\) −23.1801 −0.776127
\(893\) −3.51719 −0.117698
\(894\) 9.83811 0.329036
\(895\) 73.6232 2.46095
\(896\) −2.10909 −0.0704599
\(897\) 18.0036 0.601123
\(898\) −12.4029 −0.413891
\(899\) −55.3216 −1.84508
\(900\) 10.7969 0.359896
\(901\) 1.57490 0.0524676
\(902\) −4.43720 −0.147743
\(903\) −6.16904 −0.205293
\(904\) −17.1265 −0.569620
\(905\) 49.1274 1.63305
\(906\) 9.31710 0.309540
\(907\) −26.5232 −0.880690 −0.440345 0.897829i \(-0.645144\pi\)
−0.440345 + 0.897829i \(0.645144\pi\)
\(908\) 10.0649 0.334014
\(909\) −3.18748 −0.105722
\(910\) −27.6855 −0.917764
\(911\) 15.2301 0.504596 0.252298 0.967650i \(-0.418814\pi\)
0.252298 + 0.967650i \(0.418814\pi\)
\(912\) 1.00000 0.0331133
\(913\) −6.09850 −0.201831
\(914\) 31.5034 1.04204
\(915\) −3.43887 −0.113685
\(916\) 2.12287 0.0701417
\(917\) −12.0278 −0.397191
\(918\) −1.57490 −0.0519795
\(919\) 54.5711 1.80014 0.900068 0.435749i \(-0.143517\pi\)
0.900068 + 0.435749i \(0.143517\pi\)
\(920\) 21.6658 0.714300
\(921\) −8.83648 −0.291172
\(922\) 29.1293 0.959323
\(923\) 1.33526 0.0439506
\(924\) 5.19025 0.170747
\(925\) 104.021 3.42019
\(926\) −23.3457 −0.767188
\(927\) −6.88698 −0.226198
\(928\) 10.0009 0.328295
\(929\) −23.1136 −0.758333 −0.379166 0.925328i \(-0.623789\pi\)
−0.379166 + 0.925328i \(0.623789\pi\)
\(930\) −21.9858 −0.720943
\(931\) 2.55172 0.0836293
\(932\) −27.1529 −0.889422
\(933\) 10.0178 0.327968
\(934\) −29.9377 −0.979593
\(935\) −15.4039 −0.503762
\(936\) 3.30271 0.107952
\(937\) 5.88785 0.192348 0.0961738 0.995365i \(-0.469340\pi\)
0.0961738 + 0.995365i \(0.469340\pi\)
\(938\) −11.1049 −0.362587
\(939\) 33.7498 1.10138
\(940\) −13.9792 −0.455950
\(941\) 53.9887 1.75998 0.879990 0.474992i \(-0.157549\pi\)
0.879990 + 0.474992i \(0.157549\pi\)
\(942\) 3.48510 0.113551
\(943\) −9.82892 −0.320074
\(944\) 2.26193 0.0736196
\(945\) 8.38265 0.272688
\(946\) 7.19803 0.234028
\(947\) −1.72250 −0.0559737 −0.0279869 0.999608i \(-0.508910\pi\)
−0.0279869 + 0.999608i \(0.508910\pi\)
\(948\) −8.91239 −0.289461
\(949\) 36.5369 1.18604
\(950\) 10.7969 0.350297
\(951\) 22.3335 0.724214
\(952\) 3.32162 0.107654
\(953\) 7.23123 0.234243 0.117121 0.993118i \(-0.462633\pi\)
0.117121 + 0.993118i \(0.462633\pi\)
\(954\) 1.00000 0.0323762
\(955\) −39.2704 −1.27076
\(956\) 5.70151 0.184400
\(957\) −24.6111 −0.795563
\(958\) −10.2080 −0.329804
\(959\) −19.0866 −0.616339
\(960\) 3.97453 0.128277
\(961\) −0.400529 −0.0129203
\(962\) 31.8195 1.02590
\(963\) 15.3991 0.496230
\(964\) 16.6620 0.536648
\(965\) −62.4111 −2.00908
\(966\) 11.4970 0.369910
\(967\) 6.51292 0.209441 0.104721 0.994502i \(-0.466605\pi\)
0.104721 + 0.994502i \(0.466605\pi\)
\(968\) 4.94402 0.158907
\(969\) −1.57490 −0.0505931
\(970\) −28.8318 −0.925733
\(971\) −43.8563 −1.40741 −0.703707 0.710490i \(-0.748473\pi\)
−0.703707 + 0.710490i \(0.748473\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 22.0646 0.707359
\(974\) −30.8479 −0.988431
\(975\) 35.6589 1.14200
\(976\) −0.865226 −0.0276952
\(977\) −17.1241 −0.547847 −0.273924 0.961751i \(-0.588322\pi\)
−0.273924 + 0.961751i \(0.588322\pi\)
\(978\) 14.0163 0.448191
\(979\) −33.4284 −1.06838
\(980\) 10.1419 0.323971
\(981\) −16.0466 −0.512327
\(982\) 33.0651 1.05515
\(983\) −9.31816 −0.297203 −0.148602 0.988897i \(-0.547477\pi\)
−0.148602 + 0.988897i \(0.547477\pi\)
\(984\) −1.80309 −0.0574803
\(985\) 38.0391 1.21203
\(986\) −15.7504 −0.501595
\(987\) −7.41809 −0.236120
\(988\) 3.30271 0.105073
\(989\) 15.9445 0.507005
\(990\) −9.78088 −0.310857
\(991\) −9.09884 −0.289034 −0.144517 0.989502i \(-0.546163\pi\)
−0.144517 + 0.989502i \(0.546163\pi\)
\(992\) −5.53168 −0.175631
\(993\) −9.18190 −0.291379
\(994\) 0.852691 0.0270457
\(995\) −87.8774 −2.78590
\(996\) −2.47817 −0.0785238
\(997\) 2.24827 0.0712034 0.0356017 0.999366i \(-0.488665\pi\)
0.0356017 + 0.999366i \(0.488665\pi\)
\(998\) −15.9857 −0.506020
\(999\) −9.63437 −0.304818
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 6042.2.a.be.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.be.1.1 12 1.1 even 1 trivial