Properties

Label 5225.2.a.y.1.5
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $15$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.527123\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.527123 q^{2} +1.24468 q^{3} -1.72214 q^{4} -0.656097 q^{6} +3.22241 q^{7} +1.96203 q^{8} -1.45078 q^{9} +O(q^{10})\) \(q-0.527123 q^{2} +1.24468 q^{3} -1.72214 q^{4} -0.656097 q^{6} +3.22241 q^{7} +1.96203 q^{8} -1.45078 q^{9} +1.00000 q^{11} -2.14351 q^{12} -4.87506 q^{13} -1.69861 q^{14} +2.41005 q^{16} +7.11649 q^{17} +0.764741 q^{18} +1.00000 q^{19} +4.01085 q^{21} -0.527123 q^{22} +1.12671 q^{23} +2.44209 q^{24} +2.56976 q^{26} -5.53978 q^{27} -5.54944 q^{28} +8.47231 q^{29} -0.926610 q^{31} -5.19445 q^{32} +1.24468 q^{33} -3.75127 q^{34} +2.49845 q^{36} -10.7386 q^{37} -0.527123 q^{38} -6.06787 q^{39} +7.49805 q^{41} -2.11421 q^{42} -7.22055 q^{43} -1.72214 q^{44} -0.593913 q^{46} +12.4483 q^{47} +2.99973 q^{48} +3.38391 q^{49} +8.85772 q^{51} +8.39555 q^{52} +7.92848 q^{53} +2.92015 q^{54} +6.32245 q^{56} +1.24468 q^{57} -4.46595 q^{58} +1.91691 q^{59} -11.5308 q^{61} +0.488438 q^{62} -4.67502 q^{63} -2.08199 q^{64} -0.656097 q^{66} +10.2103 q^{67} -12.2556 q^{68} +1.40239 q^{69} +0.979715 q^{71} -2.84647 q^{72} -15.3116 q^{73} +5.66057 q^{74} -1.72214 q^{76} +3.22241 q^{77} +3.19852 q^{78} -3.05834 q^{79} -2.54288 q^{81} -3.95240 q^{82} -2.60802 q^{83} -6.90725 q^{84} +3.80612 q^{86} +10.5453 q^{87} +1.96203 q^{88} +5.38303 q^{89} -15.7094 q^{91} -1.94035 q^{92} -1.15333 q^{93} -6.56179 q^{94} -6.46540 q^{96} +15.6399 q^{97} -1.78374 q^{98} -1.45078 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23} + q^{24} + 3 q^{26} + q^{27} + 46 q^{28} + 9 q^{29} + 14 q^{31} + 18 q^{32} + 4 q^{33} - 13 q^{34} + 12 q^{36} + 9 q^{37} + 5 q^{38} - 22 q^{39} + 4 q^{41} - 6 q^{42} + 28 q^{43} + 17 q^{44} + 27 q^{46} + 14 q^{47} - 4 q^{48} + 32 q^{49} - 40 q^{51} + 14 q^{52} + 3 q^{53} - 39 q^{54} + 34 q^{56} + 4 q^{57} + 26 q^{58} + q^{59} + 2 q^{61} - 3 q^{62} + 45 q^{63} + 5 q^{64} - q^{66} + 37 q^{67} + 26 q^{68} - 7 q^{69} - 7 q^{71} + 16 q^{72} + 42 q^{73} - 43 q^{74} + 17 q^{76} + 21 q^{77} - 64 q^{78} - 10 q^{79} + 31 q^{81} + 22 q^{82} + 14 q^{83} - 32 q^{84} + 37 q^{86} + 29 q^{87} + 9 q^{88} + 15 q^{89} - 22 q^{91} + 26 q^{92} - 18 q^{93} - 44 q^{94} + 71 q^{96} + 8 q^{97} - 10 q^{98} + 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.527123 −0.372732 −0.186366 0.982480i \(-0.559671\pi\)
−0.186366 + 0.982480i \(0.559671\pi\)
\(3\) 1.24468 0.718614 0.359307 0.933219i \(-0.383013\pi\)
0.359307 + 0.933219i \(0.383013\pi\)
\(4\) −1.72214 −0.861071
\(5\) 0 0
\(6\) −0.656097 −0.267850
\(7\) 3.22241 1.21796 0.608978 0.793187i \(-0.291580\pi\)
0.608978 + 0.793187i \(0.291580\pi\)
\(8\) 1.96203 0.693681
\(9\) −1.45078 −0.483594
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −2.14351 −0.618777
\(13\) −4.87506 −1.35210 −0.676050 0.736856i \(-0.736310\pi\)
−0.676050 + 0.736856i \(0.736310\pi\)
\(14\) −1.69861 −0.453971
\(15\) 0 0
\(16\) 2.41005 0.602513
\(17\) 7.11649 1.72600 0.863002 0.505201i \(-0.168582\pi\)
0.863002 + 0.505201i \(0.168582\pi\)
\(18\) 0.764741 0.180251
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.01085 0.875240
\(22\) −0.527123 −0.112383
\(23\) 1.12671 0.234935 0.117467 0.993077i \(-0.462522\pi\)
0.117467 + 0.993077i \(0.462522\pi\)
\(24\) 2.44209 0.498489
\(25\) 0 0
\(26\) 2.56976 0.503971
\(27\) −5.53978 −1.06613
\(28\) −5.54944 −1.04875
\(29\) 8.47231 1.57327 0.786634 0.617420i \(-0.211822\pi\)
0.786634 + 0.617420i \(0.211822\pi\)
\(30\) 0 0
\(31\) −0.926610 −0.166424 −0.0832121 0.996532i \(-0.526518\pi\)
−0.0832121 + 0.996532i \(0.526518\pi\)
\(32\) −5.19445 −0.918257
\(33\) 1.24468 0.216670
\(34\) −3.75127 −0.643337
\(35\) 0 0
\(36\) 2.49845 0.416409
\(37\) −10.7386 −1.76542 −0.882709 0.469920i \(-0.844283\pi\)
−0.882709 + 0.469920i \(0.844283\pi\)
\(38\) −0.527123 −0.0855106
\(39\) −6.06787 −0.971637
\(40\) 0 0
\(41\) 7.49805 1.17100 0.585499 0.810673i \(-0.300898\pi\)
0.585499 + 0.810673i \(0.300898\pi\)
\(42\) −2.11421 −0.326230
\(43\) −7.22055 −1.10112 −0.550562 0.834794i \(-0.685586\pi\)
−0.550562 + 0.834794i \(0.685586\pi\)
\(44\) −1.72214 −0.259623
\(45\) 0 0
\(46\) −0.593913 −0.0875678
\(47\) 12.4483 1.81577 0.907886 0.419217i \(-0.137695\pi\)
0.907886 + 0.419217i \(0.137695\pi\)
\(48\) 2.99973 0.432974
\(49\) 3.38391 0.483416
\(50\) 0 0
\(51\) 8.85772 1.24033
\(52\) 8.39555 1.16425
\(53\) 7.92848 1.08906 0.544530 0.838741i \(-0.316708\pi\)
0.544530 + 0.838741i \(0.316708\pi\)
\(54\) 2.92015 0.397381
\(55\) 0 0
\(56\) 6.32245 0.844873
\(57\) 1.24468 0.164861
\(58\) −4.46595 −0.586408
\(59\) 1.91691 0.249561 0.124780 0.992184i \(-0.460177\pi\)
0.124780 + 0.992184i \(0.460177\pi\)
\(60\) 0 0
\(61\) −11.5308 −1.47637 −0.738185 0.674599i \(-0.764317\pi\)
−0.738185 + 0.674599i \(0.764317\pi\)
\(62\) 0.488438 0.0620316
\(63\) −4.67502 −0.588997
\(64\) −2.08199 −0.260249
\(65\) 0 0
\(66\) −0.656097 −0.0807600
\(67\) 10.2103 1.24739 0.623693 0.781669i \(-0.285631\pi\)
0.623693 + 0.781669i \(0.285631\pi\)
\(68\) −12.2556 −1.48621
\(69\) 1.40239 0.168827
\(70\) 0 0
\(71\) 0.979715 0.116271 0.0581354 0.998309i \(-0.481484\pi\)
0.0581354 + 0.998309i \(0.481484\pi\)
\(72\) −2.84647 −0.335460
\(73\) −15.3116 −1.79209 −0.896044 0.443965i \(-0.853571\pi\)
−0.896044 + 0.443965i \(0.853571\pi\)
\(74\) 5.66057 0.658028
\(75\) 0 0
\(76\) −1.72214 −0.197543
\(77\) 3.22241 0.367227
\(78\) 3.19852 0.362161
\(79\) −3.05834 −0.344091 −0.172045 0.985089i \(-0.555038\pi\)
−0.172045 + 0.985089i \(0.555038\pi\)
\(80\) 0 0
\(81\) −2.54288 −0.282542
\(82\) −3.95240 −0.436469
\(83\) −2.60802 −0.286268 −0.143134 0.989703i \(-0.545718\pi\)
−0.143134 + 0.989703i \(0.545718\pi\)
\(84\) −6.90725 −0.753643
\(85\) 0 0
\(86\) 3.80612 0.410424
\(87\) 10.5453 1.13057
\(88\) 1.96203 0.209153
\(89\) 5.38303 0.570600 0.285300 0.958438i \(-0.407907\pi\)
0.285300 + 0.958438i \(0.407907\pi\)
\(90\) 0 0
\(91\) −15.7094 −1.64680
\(92\) −1.94035 −0.202295
\(93\) −1.15333 −0.119595
\(94\) −6.56179 −0.676797
\(95\) 0 0
\(96\) −6.46540 −0.659872
\(97\) 15.6399 1.58799 0.793995 0.607924i \(-0.207997\pi\)
0.793995 + 0.607924i \(0.207997\pi\)
\(98\) −1.78374 −0.180185
\(99\) −1.45078 −0.145809
\(100\) 0 0
\(101\) 14.5602 1.44879 0.724396 0.689384i \(-0.242119\pi\)
0.724396 + 0.689384i \(0.242119\pi\)
\(102\) −4.66911 −0.462311
\(103\) 10.2341 1.00839 0.504197 0.863589i \(-0.331789\pi\)
0.504197 + 0.863589i \(0.331789\pi\)
\(104\) −9.56501 −0.937926
\(105\) 0 0
\(106\) −4.17928 −0.405928
\(107\) 5.30542 0.512895 0.256447 0.966558i \(-0.417448\pi\)
0.256447 + 0.966558i \(0.417448\pi\)
\(108\) 9.54028 0.918014
\(109\) −6.54304 −0.626709 −0.313355 0.949636i \(-0.601453\pi\)
−0.313355 + 0.949636i \(0.601453\pi\)
\(110\) 0 0
\(111\) −13.3661 −1.26865
\(112\) 7.76618 0.733835
\(113\) −5.52729 −0.519963 −0.259982 0.965614i \(-0.583717\pi\)
−0.259982 + 0.965614i \(0.583717\pi\)
\(114\) −0.656097 −0.0614491
\(115\) 0 0
\(116\) −14.5905 −1.35470
\(117\) 7.07266 0.653868
\(118\) −1.01045 −0.0930194
\(119\) 22.9322 2.10220
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 6.07816 0.550291
\(123\) 9.33264 0.841496
\(124\) 1.59575 0.143303
\(125\) 0 0
\(126\) 2.46431 0.219538
\(127\) 2.97437 0.263932 0.131966 0.991254i \(-0.457871\pi\)
0.131966 + 0.991254i \(0.457871\pi\)
\(128\) 11.4864 1.01526
\(129\) −8.98724 −0.791283
\(130\) 0 0
\(131\) 11.1638 0.975388 0.487694 0.873015i \(-0.337838\pi\)
0.487694 + 0.873015i \(0.337838\pi\)
\(132\) −2.14351 −0.186568
\(133\) 3.22241 0.279418
\(134\) −5.38208 −0.464941
\(135\) 0 0
\(136\) 13.9627 1.19730
\(137\) −13.3569 −1.14116 −0.570580 0.821242i \(-0.693282\pi\)
−0.570580 + 0.821242i \(0.693282\pi\)
\(138\) −0.739229 −0.0629274
\(139\) 17.5311 1.48697 0.743485 0.668753i \(-0.233171\pi\)
0.743485 + 0.668753i \(0.233171\pi\)
\(140\) 0 0
\(141\) 15.4941 1.30484
\(142\) −0.516430 −0.0433379
\(143\) −4.87506 −0.407673
\(144\) −3.49647 −0.291372
\(145\) 0 0
\(146\) 8.07110 0.667969
\(147\) 4.21187 0.347389
\(148\) 18.4934 1.52015
\(149\) −9.45886 −0.774900 −0.387450 0.921891i \(-0.626644\pi\)
−0.387450 + 0.921891i \(0.626644\pi\)
\(150\) 0 0
\(151\) −0.0675175 −0.00549449 −0.00274725 0.999996i \(-0.500874\pi\)
−0.00274725 + 0.999996i \(0.500874\pi\)
\(152\) 1.96203 0.159141
\(153\) −10.3245 −0.834685
\(154\) −1.69861 −0.136878
\(155\) 0 0
\(156\) 10.4497 0.836648
\(157\) 13.6996 1.09334 0.546672 0.837347i \(-0.315894\pi\)
0.546672 + 0.837347i \(0.315894\pi\)
\(158\) 1.61212 0.128254
\(159\) 9.86838 0.782613
\(160\) 0 0
\(161\) 3.63071 0.286140
\(162\) 1.34041 0.105313
\(163\) 9.50798 0.744722 0.372361 0.928088i \(-0.378548\pi\)
0.372361 + 0.928088i \(0.378548\pi\)
\(164\) −12.9127 −1.00831
\(165\) 0 0
\(166\) 1.37475 0.106701
\(167\) −3.51330 −0.271868 −0.135934 0.990718i \(-0.543403\pi\)
−0.135934 + 0.990718i \(0.543403\pi\)
\(168\) 7.86940 0.607137
\(169\) 10.7663 0.828174
\(170\) 0 0
\(171\) −1.45078 −0.110944
\(172\) 12.4348 0.948145
\(173\) −3.64800 −0.277352 −0.138676 0.990338i \(-0.544285\pi\)
−0.138676 + 0.990338i \(0.544285\pi\)
\(174\) −5.55866 −0.421401
\(175\) 0 0
\(176\) 2.41005 0.181665
\(177\) 2.38594 0.179338
\(178\) −2.83752 −0.212681
\(179\) −18.2698 −1.36555 −0.682773 0.730630i \(-0.739226\pi\)
−0.682773 + 0.730630i \(0.739226\pi\)
\(180\) 0 0
\(181\) 4.87683 0.362492 0.181246 0.983438i \(-0.441987\pi\)
0.181246 + 0.983438i \(0.441987\pi\)
\(182\) 8.28081 0.613815
\(183\) −14.3521 −1.06094
\(184\) 2.21063 0.162970
\(185\) 0 0
\(186\) 0.607946 0.0445768
\(187\) 7.11649 0.520409
\(188\) −21.4377 −1.56351
\(189\) −17.8514 −1.29850
\(190\) 0 0
\(191\) −3.01617 −0.218242 −0.109121 0.994028i \(-0.534804\pi\)
−0.109121 + 0.994028i \(0.534804\pi\)
\(192\) −2.59141 −0.187019
\(193\) 11.8494 0.852940 0.426470 0.904502i \(-0.359757\pi\)
0.426470 + 0.904502i \(0.359757\pi\)
\(194\) −8.24415 −0.591895
\(195\) 0 0
\(196\) −5.82758 −0.416256
\(197\) 22.9156 1.63267 0.816333 0.577581i \(-0.196003\pi\)
0.816333 + 0.577581i \(0.196003\pi\)
\(198\) 0.764741 0.0543478
\(199\) −16.6587 −1.18090 −0.590452 0.807072i \(-0.701051\pi\)
−0.590452 + 0.807072i \(0.701051\pi\)
\(200\) 0 0
\(201\) 12.7085 0.896389
\(202\) −7.67500 −0.540011
\(203\) 27.3012 1.91617
\(204\) −15.2543 −1.06801
\(205\) 0 0
\(206\) −5.39462 −0.375861
\(207\) −1.63461 −0.113613
\(208\) −11.7492 −0.814658
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −11.0707 −0.762138 −0.381069 0.924547i \(-0.624444\pi\)
−0.381069 + 0.924547i \(0.624444\pi\)
\(212\) −13.6540 −0.937757
\(213\) 1.21943 0.0835538
\(214\) −2.79661 −0.191172
\(215\) 0 0
\(216\) −10.8692 −0.739555
\(217\) −2.98592 −0.202697
\(218\) 3.44898 0.233595
\(219\) −19.0580 −1.28782
\(220\) 0 0
\(221\) −34.6934 −2.33373
\(222\) 7.04558 0.472868
\(223\) −2.30152 −0.154121 −0.0770606 0.997026i \(-0.524553\pi\)
−0.0770606 + 0.997026i \(0.524553\pi\)
\(224\) −16.7386 −1.11840
\(225\) 0 0
\(226\) 2.91356 0.193807
\(227\) −15.8244 −1.05030 −0.525152 0.851008i \(-0.675992\pi\)
−0.525152 + 0.851008i \(0.675992\pi\)
\(228\) −2.14351 −0.141957
\(229\) 21.0470 1.39082 0.695412 0.718612i \(-0.255222\pi\)
0.695412 + 0.718612i \(0.255222\pi\)
\(230\) 0 0
\(231\) 4.01085 0.263895
\(232\) 16.6229 1.09135
\(233\) −1.45166 −0.0951017 −0.0475508 0.998869i \(-0.515142\pi\)
−0.0475508 + 0.998869i \(0.515142\pi\)
\(234\) −3.72816 −0.243718
\(235\) 0 0
\(236\) −3.30120 −0.214890
\(237\) −3.80665 −0.247268
\(238\) −12.0881 −0.783556
\(239\) 3.11689 0.201615 0.100807 0.994906i \(-0.467857\pi\)
0.100807 + 0.994906i \(0.467857\pi\)
\(240\) 0 0
\(241\) −25.3469 −1.63274 −0.816369 0.577531i \(-0.804016\pi\)
−0.816369 + 0.577531i \(0.804016\pi\)
\(242\) −0.527123 −0.0338847
\(243\) 13.4543 0.863093
\(244\) 19.8577 1.27126
\(245\) 0 0
\(246\) −4.91945 −0.313653
\(247\) −4.87506 −0.310193
\(248\) −1.81803 −0.115445
\(249\) −3.24614 −0.205716
\(250\) 0 0
\(251\) 12.0511 0.760656 0.380328 0.924852i \(-0.375811\pi\)
0.380328 + 0.924852i \(0.375811\pi\)
\(252\) 8.05104 0.507168
\(253\) 1.12671 0.0708355
\(254\) −1.56786 −0.0983761
\(255\) 0 0
\(256\) −1.89074 −0.118171
\(257\) 17.1129 1.06747 0.533737 0.845651i \(-0.320787\pi\)
0.533737 + 0.845651i \(0.320787\pi\)
\(258\) 4.73738 0.294937
\(259\) −34.6042 −2.15020
\(260\) 0 0
\(261\) −12.2915 −0.760824
\(262\) −5.88471 −0.363559
\(263\) 23.8861 1.47288 0.736440 0.676503i \(-0.236505\pi\)
0.736440 + 0.676503i \(0.236505\pi\)
\(264\) 2.44209 0.150300
\(265\) 0 0
\(266\) −1.69861 −0.104148
\(267\) 6.70012 0.410041
\(268\) −17.5836 −1.07409
\(269\) −5.15565 −0.314346 −0.157173 0.987571i \(-0.550238\pi\)
−0.157173 + 0.987571i \(0.550238\pi\)
\(270\) 0 0
\(271\) 9.25556 0.562235 0.281118 0.959673i \(-0.409295\pi\)
0.281118 + 0.959673i \(0.409295\pi\)
\(272\) 17.1511 1.03994
\(273\) −19.5532 −1.18341
\(274\) 7.04075 0.425347
\(275\) 0 0
\(276\) −2.41511 −0.145372
\(277\) −7.65078 −0.459691 −0.229845 0.973227i \(-0.573822\pi\)
−0.229845 + 0.973227i \(0.573822\pi\)
\(278\) −9.24106 −0.554242
\(279\) 1.34431 0.0804818
\(280\) 0 0
\(281\) 16.4752 0.982829 0.491415 0.870926i \(-0.336480\pi\)
0.491415 + 0.870926i \(0.336480\pi\)
\(282\) −8.16730 −0.486355
\(283\) 10.5470 0.626953 0.313476 0.949596i \(-0.398506\pi\)
0.313476 + 0.949596i \(0.398506\pi\)
\(284\) −1.68721 −0.100117
\(285\) 0 0
\(286\) 2.56976 0.151953
\(287\) 24.1618 1.42622
\(288\) 7.53602 0.444064
\(289\) 33.6445 1.97909
\(290\) 0 0
\(291\) 19.4666 1.14115
\(292\) 26.3688 1.54311
\(293\) 27.1073 1.58362 0.791811 0.610766i \(-0.209138\pi\)
0.791811 + 0.610766i \(0.209138\pi\)
\(294\) −2.22018 −0.129483
\(295\) 0 0
\(296\) −21.0695 −1.22464
\(297\) −5.53978 −0.321451
\(298\) 4.98598 0.288830
\(299\) −5.49277 −0.317655
\(300\) 0 0
\(301\) −23.2676 −1.34112
\(302\) 0.0355900 0.00204798
\(303\) 18.1227 1.04112
\(304\) 2.41005 0.138226
\(305\) 0 0
\(306\) 5.44227 0.311114
\(307\) 15.9437 0.909957 0.454978 0.890503i \(-0.349647\pi\)
0.454978 + 0.890503i \(0.349647\pi\)
\(308\) −5.54944 −0.316209
\(309\) 12.7381 0.724645
\(310\) 0 0
\(311\) 23.0972 1.30972 0.654862 0.755749i \(-0.272727\pi\)
0.654862 + 0.755749i \(0.272727\pi\)
\(312\) −11.9053 −0.674006
\(313\) 19.7892 1.11855 0.559275 0.828982i \(-0.311079\pi\)
0.559275 + 0.828982i \(0.311079\pi\)
\(314\) −7.22135 −0.407525
\(315\) 0 0
\(316\) 5.26690 0.296286
\(317\) −6.50111 −0.365139 −0.182569 0.983193i \(-0.558441\pi\)
−0.182569 + 0.983193i \(0.558441\pi\)
\(318\) −5.20185 −0.291705
\(319\) 8.47231 0.474358
\(320\) 0 0
\(321\) 6.60353 0.368573
\(322\) −1.91383 −0.106654
\(323\) 7.11649 0.395972
\(324\) 4.37920 0.243289
\(325\) 0 0
\(326\) −5.01187 −0.277582
\(327\) −8.14396 −0.450362
\(328\) 14.7114 0.812300
\(329\) 40.1135 2.21153
\(330\) 0 0
\(331\) 0.931117 0.0511788 0.0255894 0.999673i \(-0.491854\pi\)
0.0255894 + 0.999673i \(0.491854\pi\)
\(332\) 4.49139 0.246497
\(333\) 15.5794 0.853746
\(334\) 1.85194 0.101334
\(335\) 0 0
\(336\) 9.66637 0.527344
\(337\) 13.9841 0.761761 0.380881 0.924624i \(-0.375621\pi\)
0.380881 + 0.924624i \(0.375621\pi\)
\(338\) −5.67514 −0.308687
\(339\) −6.87968 −0.373653
\(340\) 0 0
\(341\) −0.926610 −0.0501788
\(342\) 0.764741 0.0413525
\(343\) −11.6525 −0.629176
\(344\) −14.1669 −0.763829
\(345\) 0 0
\(346\) 1.92294 0.103378
\(347\) −30.7347 −1.64992 −0.824961 0.565189i \(-0.808803\pi\)
−0.824961 + 0.565189i \(0.808803\pi\)
\(348\) −18.1605 −0.973502
\(349\) −12.9948 −0.695595 −0.347798 0.937570i \(-0.613070\pi\)
−0.347798 + 0.937570i \(0.613070\pi\)
\(350\) 0 0
\(351\) 27.0068 1.44152
\(352\) −5.19445 −0.276865
\(353\) 18.3337 0.975803 0.487901 0.872899i \(-0.337763\pi\)
0.487901 + 0.872899i \(0.337763\pi\)
\(354\) −1.25768 −0.0668450
\(355\) 0 0
\(356\) −9.27033 −0.491327
\(357\) 28.5432 1.51067
\(358\) 9.63041 0.508983
\(359\) −7.55341 −0.398654 −0.199327 0.979933i \(-0.563876\pi\)
−0.199327 + 0.979933i \(0.563876\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −2.57069 −0.135113
\(363\) 1.24468 0.0653285
\(364\) 27.0539 1.41801
\(365\) 0 0
\(366\) 7.56533 0.395446
\(367\) 6.80846 0.355399 0.177699 0.984085i \(-0.443135\pi\)
0.177699 + 0.984085i \(0.443135\pi\)
\(368\) 2.71543 0.141551
\(369\) −10.8780 −0.566289
\(370\) 0 0
\(371\) 25.5488 1.32643
\(372\) 1.98620 0.102979
\(373\) −0.104052 −0.00538763 −0.00269381 0.999996i \(-0.500857\pi\)
−0.00269381 + 0.999996i \(0.500857\pi\)
\(374\) −3.75127 −0.193973
\(375\) 0 0
\(376\) 24.4239 1.25957
\(377\) −41.3031 −2.12722
\(378\) 9.40990 0.483993
\(379\) −29.4765 −1.51411 −0.757053 0.653353i \(-0.773362\pi\)
−0.757053 + 0.653353i \(0.773362\pi\)
\(380\) 0 0
\(381\) 3.70212 0.189665
\(382\) 1.58989 0.0813460
\(383\) −15.0758 −0.770337 −0.385169 0.922846i \(-0.625857\pi\)
−0.385169 + 0.922846i \(0.625857\pi\)
\(384\) 14.2968 0.729580
\(385\) 0 0
\(386\) −6.24610 −0.317918
\(387\) 10.4755 0.532497
\(388\) −26.9341 −1.36737
\(389\) −8.16895 −0.414182 −0.207091 0.978322i \(-0.566400\pi\)
−0.207091 + 0.978322i \(0.566400\pi\)
\(390\) 0 0
\(391\) 8.01821 0.405498
\(392\) 6.63933 0.335337
\(393\) 13.8953 0.700927
\(394\) −12.0793 −0.608548
\(395\) 0 0
\(396\) 2.49845 0.125552
\(397\) 14.6856 0.737050 0.368525 0.929618i \(-0.379863\pi\)
0.368525 + 0.929618i \(0.379863\pi\)
\(398\) 8.78119 0.440161
\(399\) 4.01085 0.200794
\(400\) 0 0
\(401\) 1.14341 0.0570994 0.0285497 0.999592i \(-0.490911\pi\)
0.0285497 + 0.999592i \(0.490911\pi\)
\(402\) −6.69895 −0.334113
\(403\) 4.51729 0.225022
\(404\) −25.0747 −1.24751
\(405\) 0 0
\(406\) −14.3911 −0.714219
\(407\) −10.7386 −0.532294
\(408\) 17.3791 0.860393
\(409\) 34.6619 1.71392 0.856961 0.515381i \(-0.172350\pi\)
0.856961 + 0.515381i \(0.172350\pi\)
\(410\) 0 0
\(411\) −16.6250 −0.820053
\(412\) −17.6245 −0.868298
\(413\) 6.17708 0.303954
\(414\) 0.861640 0.0423473
\(415\) 0 0
\(416\) 25.3233 1.24158
\(417\) 21.8206 1.06856
\(418\) −0.527123 −0.0257824
\(419\) −19.8862 −0.971505 −0.485752 0.874096i \(-0.661454\pi\)
−0.485752 + 0.874096i \(0.661454\pi\)
\(420\) 0 0
\(421\) 32.9388 1.60534 0.802669 0.596424i \(-0.203412\pi\)
0.802669 + 0.596424i \(0.203412\pi\)
\(422\) 5.83562 0.284073
\(423\) −18.0598 −0.878097
\(424\) 15.5559 0.755460
\(425\) 0 0
\(426\) −0.642788 −0.0311432
\(427\) −37.1570 −1.79815
\(428\) −9.13669 −0.441639
\(429\) −6.06787 −0.292960
\(430\) 0 0
\(431\) −21.4153 −1.03154 −0.515770 0.856727i \(-0.672494\pi\)
−0.515770 + 0.856727i \(0.672494\pi\)
\(432\) −13.3512 −0.642358
\(433\) −31.7437 −1.52551 −0.762753 0.646690i \(-0.776153\pi\)
−0.762753 + 0.646690i \(0.776153\pi\)
\(434\) 1.57395 0.0755518
\(435\) 0 0
\(436\) 11.2680 0.539641
\(437\) 1.12671 0.0538977
\(438\) 10.0459 0.480012
\(439\) 9.40014 0.448644 0.224322 0.974515i \(-0.427983\pi\)
0.224322 + 0.974515i \(0.427983\pi\)
\(440\) 0 0
\(441\) −4.90932 −0.233777
\(442\) 18.2877 0.869856
\(443\) 21.5223 1.02256 0.511278 0.859415i \(-0.329172\pi\)
0.511278 + 0.859415i \(0.329172\pi\)
\(444\) 23.0183 1.09240
\(445\) 0 0
\(446\) 1.21318 0.0574459
\(447\) −11.7732 −0.556854
\(448\) −6.70904 −0.316972
\(449\) 30.7497 1.45117 0.725583 0.688134i \(-0.241570\pi\)
0.725583 + 0.688134i \(0.241570\pi\)
\(450\) 0 0
\(451\) 7.49805 0.353069
\(452\) 9.51877 0.447725
\(453\) −0.0840373 −0.00394842
\(454\) 8.34142 0.391482
\(455\) 0 0
\(456\) 2.44209 0.114361
\(457\) −0.507061 −0.0237193 −0.0118596 0.999930i \(-0.503775\pi\)
−0.0118596 + 0.999930i \(0.503775\pi\)
\(458\) −11.0943 −0.518405
\(459\) −39.4238 −1.84015
\(460\) 0 0
\(461\) 5.60128 0.260878 0.130439 0.991456i \(-0.458361\pi\)
0.130439 + 0.991456i \(0.458361\pi\)
\(462\) −2.11421 −0.0983621
\(463\) 22.5309 1.04710 0.523550 0.851995i \(-0.324607\pi\)
0.523550 + 0.851995i \(0.324607\pi\)
\(464\) 20.4187 0.947915
\(465\) 0 0
\(466\) 0.765205 0.0354475
\(467\) −15.1056 −0.699003 −0.349502 0.936936i \(-0.613649\pi\)
−0.349502 + 0.936936i \(0.613649\pi\)
\(468\) −12.1801 −0.563026
\(469\) 32.9018 1.51926
\(470\) 0 0
\(471\) 17.0515 0.785692
\(472\) 3.76103 0.173116
\(473\) −7.22055 −0.332001
\(474\) 2.00657 0.0921648
\(475\) 0 0
\(476\) −39.4926 −1.81014
\(477\) −11.5025 −0.526663
\(478\) −1.64298 −0.0751482
\(479\) −1.97446 −0.0902153 −0.0451076 0.998982i \(-0.514363\pi\)
−0.0451076 + 0.998982i \(0.514363\pi\)
\(480\) 0 0
\(481\) 52.3515 2.38702
\(482\) 13.3609 0.608574
\(483\) 4.51906 0.205624
\(484\) −1.72214 −0.0782792
\(485\) 0 0
\(486\) −7.09206 −0.321702
\(487\) −6.23748 −0.282647 −0.141324 0.989963i \(-0.545136\pi\)
−0.141324 + 0.989963i \(0.545136\pi\)
\(488\) −22.6238 −1.02413
\(489\) 11.8343 0.535168
\(490\) 0 0
\(491\) 29.4073 1.32713 0.663566 0.748117i \(-0.269042\pi\)
0.663566 + 0.748117i \(0.269042\pi\)
\(492\) −16.0721 −0.724587
\(493\) 60.2931 2.71547
\(494\) 2.56976 0.115619
\(495\) 0 0
\(496\) −2.23318 −0.100273
\(497\) 3.15704 0.141613
\(498\) 1.71112 0.0766770
\(499\) −8.33612 −0.373176 −0.186588 0.982438i \(-0.559743\pi\)
−0.186588 + 0.982438i \(0.559743\pi\)
\(500\) 0 0
\(501\) −4.37292 −0.195368
\(502\) −6.35239 −0.283521
\(503\) 36.9242 1.64637 0.823184 0.567774i \(-0.192195\pi\)
0.823184 + 0.567774i \(0.192195\pi\)
\(504\) −9.17250 −0.408576
\(505\) 0 0
\(506\) −0.593913 −0.0264027
\(507\) 13.4005 0.595137
\(508\) −5.12228 −0.227265
\(509\) −14.0280 −0.621782 −0.310891 0.950446i \(-0.600627\pi\)
−0.310891 + 0.950446i \(0.600627\pi\)
\(510\) 0 0
\(511\) −49.3402 −2.18268
\(512\) −21.9761 −0.971214
\(513\) −5.53978 −0.244587
\(514\) −9.02061 −0.397882
\(515\) 0 0
\(516\) 15.4773 0.681350
\(517\) 12.4483 0.547476
\(518\) 18.2407 0.801449
\(519\) −4.54057 −0.199309
\(520\) 0 0
\(521\) 6.56794 0.287747 0.143873 0.989596i \(-0.454044\pi\)
0.143873 + 0.989596i \(0.454044\pi\)
\(522\) 6.47912 0.283583
\(523\) 6.72747 0.294172 0.147086 0.989124i \(-0.453011\pi\)
0.147086 + 0.989124i \(0.453011\pi\)
\(524\) −19.2257 −0.839878
\(525\) 0 0
\(526\) −12.5909 −0.548990
\(527\) −6.59422 −0.287249
\(528\) 2.99973 0.130547
\(529\) −21.7305 −0.944806
\(530\) 0 0
\(531\) −2.78103 −0.120686
\(532\) −5.54944 −0.240599
\(533\) −36.5535 −1.58331
\(534\) −3.53179 −0.152835
\(535\) 0 0
\(536\) 20.0329 0.865288
\(537\) −22.7399 −0.981300
\(538\) 2.71766 0.117167
\(539\) 3.38391 0.145755
\(540\) 0 0
\(541\) −5.50043 −0.236482 −0.118241 0.992985i \(-0.537726\pi\)
−0.118241 + 0.992985i \(0.537726\pi\)
\(542\) −4.87882 −0.209563
\(543\) 6.07008 0.260492
\(544\) −36.9662 −1.58491
\(545\) 0 0
\(546\) 10.3069 0.441096
\(547\) 30.6112 1.30884 0.654420 0.756131i \(-0.272913\pi\)
0.654420 + 0.756131i \(0.272913\pi\)
\(548\) 23.0025 0.982619
\(549\) 16.7287 0.713964
\(550\) 0 0
\(551\) 8.47231 0.360932
\(552\) 2.75152 0.117112
\(553\) −9.85523 −0.419087
\(554\) 4.03290 0.171342
\(555\) 0 0
\(556\) −30.1911 −1.28039
\(557\) −14.5219 −0.615313 −0.307656 0.951498i \(-0.599545\pi\)
−0.307656 + 0.951498i \(0.599545\pi\)
\(558\) −0.708617 −0.0299982
\(559\) 35.2007 1.48883
\(560\) 0 0
\(561\) 8.85772 0.373973
\(562\) −8.68447 −0.366332
\(563\) 18.6917 0.787762 0.393881 0.919161i \(-0.371132\pi\)
0.393881 + 0.919161i \(0.371132\pi\)
\(564\) −26.6830 −1.12356
\(565\) 0 0
\(566\) −5.55955 −0.233685
\(567\) −8.19419 −0.344124
\(568\) 1.92223 0.0806549
\(569\) 4.12594 0.172968 0.0864842 0.996253i \(-0.472437\pi\)
0.0864842 + 0.996253i \(0.472437\pi\)
\(570\) 0 0
\(571\) −20.6081 −0.862422 −0.431211 0.902251i \(-0.641913\pi\)
−0.431211 + 0.902251i \(0.641913\pi\)
\(572\) 8.39555 0.351036
\(573\) −3.75415 −0.156832
\(574\) −12.7362 −0.531600
\(575\) 0 0
\(576\) 3.02052 0.125855
\(577\) 18.0882 0.753020 0.376510 0.926413i \(-0.377124\pi\)
0.376510 + 0.926413i \(0.377124\pi\)
\(578\) −17.7348 −0.737669
\(579\) 14.7487 0.612934
\(580\) 0 0
\(581\) −8.40412 −0.348662
\(582\) −10.2613 −0.425344
\(583\) 7.92848 0.328364
\(584\) −30.0418 −1.24314
\(585\) 0 0
\(586\) −14.2889 −0.590267
\(587\) −31.2026 −1.28787 −0.643934 0.765081i \(-0.722699\pi\)
−0.643934 + 0.765081i \(0.722699\pi\)
\(588\) −7.25344 −0.299127
\(589\) −0.926610 −0.0381803
\(590\) 0 0
\(591\) 28.5224 1.17326
\(592\) −25.8806 −1.06369
\(593\) 31.0847 1.27649 0.638247 0.769831i \(-0.279660\pi\)
0.638247 + 0.769831i \(0.279660\pi\)
\(594\) 2.92015 0.119815
\(595\) 0 0
\(596\) 16.2895 0.667244
\(597\) −20.7347 −0.848614
\(598\) 2.89537 0.118400
\(599\) −3.46676 −0.141648 −0.0708240 0.997489i \(-0.522563\pi\)
−0.0708240 + 0.997489i \(0.522563\pi\)
\(600\) 0 0
\(601\) −22.6643 −0.924498 −0.462249 0.886750i \(-0.652957\pi\)
−0.462249 + 0.886750i \(0.652957\pi\)
\(602\) 12.2649 0.499879
\(603\) −14.8129 −0.603229
\(604\) 0.116275 0.00473115
\(605\) 0 0
\(606\) −9.55289 −0.388060
\(607\) −45.2747 −1.83764 −0.918821 0.394675i \(-0.870857\pi\)
−0.918821 + 0.394675i \(0.870857\pi\)
\(608\) −5.19445 −0.210663
\(609\) 33.9812 1.37699
\(610\) 0 0
\(611\) −60.6863 −2.45511
\(612\) 17.7802 0.718723
\(613\) 32.5470 1.31456 0.657280 0.753646i \(-0.271707\pi\)
0.657280 + 0.753646i \(0.271707\pi\)
\(614\) −8.40431 −0.339170
\(615\) 0 0
\(616\) 6.32245 0.254739
\(617\) 0.114512 0.00461007 0.00230503 0.999997i \(-0.499266\pi\)
0.00230503 + 0.999997i \(0.499266\pi\)
\(618\) −6.71455 −0.270099
\(619\) −21.7981 −0.876139 −0.438069 0.898941i \(-0.644338\pi\)
−0.438069 + 0.898941i \(0.644338\pi\)
\(620\) 0 0
\(621\) −6.24171 −0.250471
\(622\) −12.1751 −0.488176
\(623\) 17.3463 0.694965
\(624\) −14.6239 −0.585425
\(625\) 0 0
\(626\) −10.4313 −0.416920
\(627\) 1.24468 0.0497075
\(628\) −23.5926 −0.941447
\(629\) −76.4213 −3.04712
\(630\) 0 0
\(631\) −7.64209 −0.304227 −0.152113 0.988363i \(-0.548608\pi\)
−0.152113 + 0.988363i \(0.548608\pi\)
\(632\) −6.00055 −0.238689
\(633\) −13.7794 −0.547683
\(634\) 3.42688 0.136099
\(635\) 0 0
\(636\) −16.9947 −0.673885
\(637\) −16.4968 −0.653627
\(638\) −4.46595 −0.176809
\(639\) −1.42135 −0.0562279
\(640\) 0 0
\(641\) −15.6286 −0.617291 −0.308646 0.951177i \(-0.599876\pi\)
−0.308646 + 0.951177i \(0.599876\pi\)
\(642\) −3.48087 −0.137379
\(643\) −21.8689 −0.862425 −0.431213 0.902250i \(-0.641914\pi\)
−0.431213 + 0.902250i \(0.641914\pi\)
\(644\) −6.25260 −0.246387
\(645\) 0 0
\(646\) −3.75127 −0.147592
\(647\) 44.3165 1.74226 0.871131 0.491051i \(-0.163387\pi\)
0.871131 + 0.491051i \(0.163387\pi\)
\(648\) −4.98919 −0.195994
\(649\) 1.91691 0.0752454
\(650\) 0 0
\(651\) −3.71650 −0.145661
\(652\) −16.3741 −0.641259
\(653\) −23.2868 −0.911284 −0.455642 0.890163i \(-0.650590\pi\)
−0.455642 + 0.890163i \(0.650590\pi\)
\(654\) 4.29287 0.167864
\(655\) 0 0
\(656\) 18.0707 0.705543
\(657\) 22.2138 0.866644
\(658\) −21.1448 −0.824309
\(659\) −33.4706 −1.30383 −0.651914 0.758293i \(-0.726034\pi\)
−0.651914 + 0.758293i \(0.726034\pi\)
\(660\) 0 0
\(661\) 4.30725 0.167533 0.0837663 0.996485i \(-0.473305\pi\)
0.0837663 + 0.996485i \(0.473305\pi\)
\(662\) −0.490813 −0.0190760
\(663\) −43.1820 −1.67705
\(664\) −5.11701 −0.198579
\(665\) 0 0
\(666\) −8.21226 −0.318219
\(667\) 9.54581 0.369615
\(668\) 6.05041 0.234097
\(669\) −2.86464 −0.110754
\(670\) 0 0
\(671\) −11.5308 −0.445142
\(672\) −20.8342 −0.803695
\(673\) −43.9621 −1.69461 −0.847307 0.531103i \(-0.821778\pi\)
−0.847307 + 0.531103i \(0.821778\pi\)
\(674\) −7.37133 −0.283933
\(675\) 0 0
\(676\) −18.5410 −0.713116
\(677\) 19.9483 0.766676 0.383338 0.923608i \(-0.374774\pi\)
0.383338 + 0.923608i \(0.374774\pi\)
\(678\) 3.62644 0.139272
\(679\) 50.3981 1.93410
\(680\) 0 0
\(681\) −19.6963 −0.754763
\(682\) 0.488438 0.0187032
\(683\) −41.1720 −1.57540 −0.787701 0.616058i \(-0.788729\pi\)
−0.787701 + 0.616058i \(0.788729\pi\)
\(684\) 2.49845 0.0955308
\(685\) 0 0
\(686\) 6.14230 0.234514
\(687\) 26.1967 0.999465
\(688\) −17.4019 −0.663442
\(689\) −38.6518 −1.47252
\(690\) 0 0
\(691\) −7.48838 −0.284872 −0.142436 0.989804i \(-0.545493\pi\)
−0.142436 + 0.989804i \(0.545493\pi\)
\(692\) 6.28237 0.238820
\(693\) −4.67502 −0.177589
\(694\) 16.2009 0.614979
\(695\) 0 0
\(696\) 20.6901 0.784256
\(697\) 53.3598 2.02115
\(698\) 6.84985 0.259271
\(699\) −1.80685 −0.0683414
\(700\) 0 0
\(701\) 17.9421 0.677666 0.338833 0.940847i \(-0.389968\pi\)
0.338833 + 0.940847i \(0.389968\pi\)
\(702\) −14.2359 −0.537299
\(703\) −10.7386 −0.405015
\(704\) −2.08199 −0.0784681
\(705\) 0 0
\(706\) −9.66410 −0.363713
\(707\) 46.9188 1.76456
\(708\) −4.10892 −0.154423
\(709\) 12.2738 0.460954 0.230477 0.973078i \(-0.425971\pi\)
0.230477 + 0.973078i \(0.425971\pi\)
\(710\) 0 0
\(711\) 4.43699 0.166400
\(712\) 10.5616 0.395814
\(713\) −1.04402 −0.0390988
\(714\) −15.0458 −0.563074
\(715\) 0 0
\(716\) 31.4631 1.17583
\(717\) 3.87951 0.144883
\(718\) 3.98158 0.148591
\(719\) 34.2879 1.27872 0.639361 0.768907i \(-0.279199\pi\)
0.639361 + 0.768907i \(0.279199\pi\)
\(720\) 0 0
\(721\) 32.9784 1.22818
\(722\) −0.527123 −0.0196175
\(723\) −31.5487 −1.17331
\(724\) −8.39860 −0.312131
\(725\) 0 0
\(726\) −0.656097 −0.0243500
\(727\) 15.7447 0.583939 0.291970 0.956428i \(-0.405689\pi\)
0.291970 + 0.956428i \(0.405689\pi\)
\(728\) −30.8223 −1.14235
\(729\) 24.3748 0.902772
\(730\) 0 0
\(731\) −51.3850 −1.90054
\(732\) 24.7164 0.913544
\(733\) 51.1191 1.88813 0.944063 0.329765i \(-0.106970\pi\)
0.944063 + 0.329765i \(0.106970\pi\)
\(734\) −3.58890 −0.132469
\(735\) 0 0
\(736\) −5.85262 −0.215731
\(737\) 10.2103 0.376101
\(738\) 5.73407 0.211074
\(739\) −50.3516 −1.85221 −0.926107 0.377260i \(-0.876866\pi\)
−0.926107 + 0.377260i \(0.876866\pi\)
\(740\) 0 0
\(741\) −6.06787 −0.222909
\(742\) −13.4673 −0.494402
\(743\) 2.11671 0.0776544 0.0388272 0.999246i \(-0.487638\pi\)
0.0388272 + 0.999246i \(0.487638\pi\)
\(744\) −2.26286 −0.0829606
\(745\) 0 0
\(746\) 0.0548484 0.00200814
\(747\) 3.78368 0.138438
\(748\) −12.2556 −0.448109
\(749\) 17.0962 0.624683
\(750\) 0 0
\(751\) −7.61971 −0.278047 −0.139024 0.990289i \(-0.544396\pi\)
−0.139024 + 0.990289i \(0.544396\pi\)
\(752\) 30.0011 1.09403
\(753\) 14.9997 0.546618
\(754\) 21.7718 0.792882
\(755\) 0 0
\(756\) 30.7427 1.11810
\(757\) −45.4268 −1.65107 −0.825533 0.564354i \(-0.809125\pi\)
−0.825533 + 0.564354i \(0.809125\pi\)
\(758\) 15.5377 0.564356
\(759\) 1.40239 0.0509034
\(760\) 0 0
\(761\) 3.78459 0.137191 0.0685957 0.997645i \(-0.478148\pi\)
0.0685957 + 0.997645i \(0.478148\pi\)
\(762\) −1.95147 −0.0706944
\(763\) −21.0843 −0.763304
\(764\) 5.19427 0.187922
\(765\) 0 0
\(766\) 7.94680 0.287130
\(767\) −9.34508 −0.337431
\(768\) −2.35335 −0.0849193
\(769\) 12.0500 0.434534 0.217267 0.976112i \(-0.430286\pi\)
0.217267 + 0.976112i \(0.430286\pi\)
\(770\) 0 0
\(771\) 21.3000 0.767101
\(772\) −20.4064 −0.734441
\(773\) 5.54555 0.199460 0.0997298 0.995015i \(-0.468202\pi\)
0.0997298 + 0.995015i \(0.468202\pi\)
\(774\) −5.52185 −0.198479
\(775\) 0 0
\(776\) 30.6859 1.10156
\(777\) −43.0710 −1.54516
\(778\) 4.30604 0.154379
\(779\) 7.49805 0.268646
\(780\) 0 0
\(781\) 0.979715 0.0350570
\(782\) −4.22658 −0.151142
\(783\) −46.9347 −1.67731
\(784\) 8.15541 0.291265
\(785\) 0 0
\(786\) −7.32456 −0.261258
\(787\) −46.5114 −1.65795 −0.828976 0.559284i \(-0.811076\pi\)
−0.828976 + 0.559284i \(0.811076\pi\)
\(788\) −39.4639 −1.40584
\(789\) 29.7304 1.05843
\(790\) 0 0
\(791\) −17.8112 −0.633292
\(792\) −2.84647 −0.101145
\(793\) 56.2135 1.99620
\(794\) −7.74113 −0.274722
\(795\) 0 0
\(796\) 28.6887 1.01684
\(797\) 3.60512 0.127700 0.0638500 0.997960i \(-0.479662\pi\)
0.0638500 + 0.997960i \(0.479662\pi\)
\(798\) −2.11421 −0.0748423
\(799\) 88.5883 3.13403
\(800\) 0 0
\(801\) −7.80960 −0.275939
\(802\) −0.602720 −0.0212828
\(803\) −15.3116 −0.540335
\(804\) −21.8858 −0.771854
\(805\) 0 0
\(806\) −2.38117 −0.0838730
\(807\) −6.41712 −0.225893
\(808\) 28.5674 1.00500
\(809\) 32.1749 1.13121 0.565605 0.824676i \(-0.308643\pi\)
0.565605 + 0.824676i \(0.308643\pi\)
\(810\) 0 0
\(811\) −43.8016 −1.53808 −0.769041 0.639200i \(-0.779266\pi\)
−0.769041 + 0.639200i \(0.779266\pi\)
\(812\) −47.0166 −1.64996
\(813\) 11.5202 0.404030
\(814\) 5.66057 0.198403
\(815\) 0 0
\(816\) 21.3476 0.747315
\(817\) −7.22055 −0.252615
\(818\) −18.2711 −0.638834
\(819\) 22.7910 0.796382
\(820\) 0 0
\(821\) −37.6149 −1.31277 −0.656385 0.754426i \(-0.727915\pi\)
−0.656385 + 0.754426i \(0.727915\pi\)
\(822\) 8.76345 0.305660
\(823\) 8.52852 0.297285 0.148643 0.988891i \(-0.452510\pi\)
0.148643 + 0.988891i \(0.452510\pi\)
\(824\) 20.0795 0.699503
\(825\) 0 0
\(826\) −3.25608 −0.113293
\(827\) 43.5444 1.51419 0.757093 0.653307i \(-0.226619\pi\)
0.757093 + 0.653307i \(0.226619\pi\)
\(828\) 2.81503 0.0978289
\(829\) −1.16979 −0.0406286 −0.0203143 0.999794i \(-0.506467\pi\)
−0.0203143 + 0.999794i \(0.506467\pi\)
\(830\) 0 0
\(831\) −9.52273 −0.330340
\(832\) 10.1499 0.351883
\(833\) 24.0816 0.834378
\(834\) −11.5021 −0.398286
\(835\) 0 0
\(836\) −1.72214 −0.0595615
\(837\) 5.13322 0.177430
\(838\) 10.4825 0.362111
\(839\) −45.7360 −1.57898 −0.789491 0.613762i \(-0.789656\pi\)
−0.789491 + 0.613762i \(0.789656\pi\)
\(840\) 0 0
\(841\) 42.7800 1.47517
\(842\) −17.3628 −0.598362
\(843\) 20.5063 0.706275
\(844\) 19.0653 0.656255
\(845\) 0 0
\(846\) 9.51973 0.327295
\(847\) 3.22241 0.110723
\(848\) 19.1081 0.656173
\(849\) 13.1276 0.450537
\(850\) 0 0
\(851\) −12.0993 −0.414758
\(852\) −2.10003 −0.0719457
\(853\) −56.7867 −1.94434 −0.972170 0.234277i \(-0.924728\pi\)
−0.972170 + 0.234277i \(0.924728\pi\)
\(854\) 19.5863 0.670230
\(855\) 0 0
\(856\) 10.4094 0.355785
\(857\) −24.0415 −0.821241 −0.410620 0.911806i \(-0.634688\pi\)
−0.410620 + 0.911806i \(0.634688\pi\)
\(858\) 3.19852 0.109196
\(859\) −21.4533 −0.731978 −0.365989 0.930619i \(-0.619269\pi\)
−0.365989 + 0.930619i \(0.619269\pi\)
\(860\) 0 0
\(861\) 30.0736 1.02490
\(862\) 11.2885 0.384488
\(863\) −25.3546 −0.863082 −0.431541 0.902093i \(-0.642030\pi\)
−0.431541 + 0.902093i \(0.642030\pi\)
\(864\) 28.7761 0.978983
\(865\) 0 0
\(866\) 16.7329 0.568606
\(867\) 41.8764 1.42220
\(868\) 5.14217 0.174537
\(869\) −3.05834 −0.103747
\(870\) 0 0
\(871\) −49.7759 −1.68659
\(872\) −12.8376 −0.434736
\(873\) −22.6901 −0.767944
\(874\) −0.593913 −0.0200894
\(875\) 0 0
\(876\) 32.8205 1.10890
\(877\) −39.1208 −1.32102 −0.660508 0.750819i \(-0.729659\pi\)
−0.660508 + 0.750819i \(0.729659\pi\)
\(878\) −4.95503 −0.167224
\(879\) 33.7397 1.13801
\(880\) 0 0
\(881\) −18.4388 −0.621219 −0.310610 0.950538i \(-0.600533\pi\)
−0.310610 + 0.950538i \(0.600533\pi\)
\(882\) 2.58782 0.0871364
\(883\) −4.68809 −0.157767 −0.0788834 0.996884i \(-0.525135\pi\)
−0.0788834 + 0.996884i \(0.525135\pi\)
\(884\) 59.7469 2.00950
\(885\) 0 0
\(886\) −11.3449 −0.381140
\(887\) −29.0079 −0.973990 −0.486995 0.873405i \(-0.661907\pi\)
−0.486995 + 0.873405i \(0.661907\pi\)
\(888\) −26.2246 −0.880041
\(889\) 9.58463 0.321458
\(890\) 0 0
\(891\) −2.54288 −0.0851896
\(892\) 3.96354 0.132709
\(893\) 12.4483 0.416567
\(894\) 6.20593 0.207557
\(895\) 0 0
\(896\) 37.0137 1.23654
\(897\) −6.83672 −0.228271
\(898\) −16.2089 −0.540897
\(899\) −7.85053 −0.261830
\(900\) 0 0
\(901\) 56.4229 1.87972
\(902\) −3.95240 −0.131600
\(903\) −28.9606 −0.963747
\(904\) −10.8447 −0.360689
\(905\) 0 0
\(906\) 0.0442980 0.00147170
\(907\) −31.2319 −1.03704 −0.518519 0.855066i \(-0.673516\pi\)
−0.518519 + 0.855066i \(0.673516\pi\)
\(908\) 27.2519 0.904386
\(909\) −21.1237 −0.700628
\(910\) 0 0
\(911\) −12.0740 −0.400031 −0.200015 0.979793i \(-0.564099\pi\)
−0.200015 + 0.979793i \(0.564099\pi\)
\(912\) 2.99973 0.0993311
\(913\) −2.60802 −0.0863130
\(914\) 0.267283 0.00884095
\(915\) 0 0
\(916\) −36.2459 −1.19760
\(917\) 35.9744 1.18798
\(918\) 20.7812 0.685882
\(919\) −28.9331 −0.954416 −0.477208 0.878790i \(-0.658351\pi\)
−0.477208 + 0.878790i \(0.658351\pi\)
\(920\) 0 0
\(921\) 19.8448 0.653907
\(922\) −2.95257 −0.0972376
\(923\) −4.77618 −0.157210
\(924\) −6.90725 −0.227232
\(925\) 0 0
\(926\) −11.8766 −0.390288
\(927\) −14.8474 −0.487653
\(928\) −44.0090 −1.44466
\(929\) −45.6501 −1.49773 −0.748866 0.662722i \(-0.769401\pi\)
−0.748866 + 0.662722i \(0.769401\pi\)
\(930\) 0 0
\(931\) 3.38391 0.110903
\(932\) 2.49997 0.0818893
\(933\) 28.7485 0.941185
\(934\) 7.96250 0.260541
\(935\) 0 0
\(936\) 13.8767 0.453576
\(937\) −16.6566 −0.544146 −0.272073 0.962277i \(-0.587709\pi\)
−0.272073 + 0.962277i \(0.587709\pi\)
\(938\) −17.3433 −0.566278
\(939\) 24.6311 0.803806
\(940\) 0 0
\(941\) 17.6743 0.576165 0.288082 0.957606i \(-0.406982\pi\)
0.288082 + 0.957606i \(0.406982\pi\)
\(942\) −8.98824 −0.292853
\(943\) 8.44811 0.275108
\(944\) 4.61986 0.150364
\(945\) 0 0
\(946\) 3.80612 0.123748
\(947\) 18.3019 0.594731 0.297366 0.954764i \(-0.403892\pi\)
0.297366 + 0.954764i \(0.403892\pi\)
\(948\) 6.55558 0.212915
\(949\) 74.6451 2.42308
\(950\) 0 0
\(951\) −8.09177 −0.262394
\(952\) 44.9937 1.45825
\(953\) 11.9242 0.386261 0.193131 0.981173i \(-0.438136\pi\)
0.193131 + 0.981173i \(0.438136\pi\)
\(954\) 6.06323 0.196304
\(955\) 0 0
\(956\) −5.36772 −0.173604
\(957\) 10.5453 0.340880
\(958\) 1.04078 0.0336261
\(959\) −43.0415 −1.38988
\(960\) 0 0
\(961\) −30.1414 −0.972303
\(962\) −27.5957 −0.889720
\(963\) −7.69702 −0.248033
\(964\) 43.6510 1.40590
\(965\) 0 0
\(966\) −2.38210 −0.0766428
\(967\) −9.97275 −0.320702 −0.160351 0.987060i \(-0.551263\pi\)
−0.160351 + 0.987060i \(0.551263\pi\)
\(968\) 1.96203 0.0630619
\(969\) 8.85772 0.284551
\(970\) 0 0
\(971\) −20.4714 −0.656960 −0.328480 0.944511i \(-0.606536\pi\)
−0.328480 + 0.944511i \(0.606536\pi\)
\(972\) −23.1702 −0.743184
\(973\) 56.4924 1.81106
\(974\) 3.28792 0.105352
\(975\) 0 0
\(976\) −27.7899 −0.889533
\(977\) −15.9786 −0.511202 −0.255601 0.966782i \(-0.582273\pi\)
−0.255601 + 0.966782i \(0.582273\pi\)
\(978\) −6.23816 −0.199474
\(979\) 5.38303 0.172042
\(980\) 0 0
\(981\) 9.49253 0.303073
\(982\) −15.5013 −0.494665
\(983\) 10.3802 0.331077 0.165538 0.986203i \(-0.447064\pi\)
0.165538 + 0.986203i \(0.447064\pi\)
\(984\) 18.3109 0.583730
\(985\) 0 0
\(986\) −31.7819 −1.01214
\(987\) 49.9283 1.58924
\(988\) 8.39555 0.267098
\(989\) −8.13545 −0.258692
\(990\) 0 0
\(991\) 47.3252 1.50334 0.751668 0.659542i \(-0.229250\pi\)
0.751668 + 0.659542i \(0.229250\pi\)
\(992\) 4.81323 0.152820
\(993\) 1.15894 0.0367778
\(994\) −1.66415 −0.0527836
\(995\) 0 0
\(996\) 5.59032 0.177136
\(997\) 1.92478 0.0609584 0.0304792 0.999535i \(-0.490297\pi\)
0.0304792 + 0.999535i \(0.490297\pi\)
\(998\) 4.39416 0.139095
\(999\) 59.4896 1.88217
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 5225.2.a.y.1.5 yes 15
5.4 even 2 5225.2.a.r.1.11 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.r.1.11 15 5.4 even 2
5225.2.a.y.1.5 yes 15 1.1 even 1 trivial