Properties

Label 5445.2.a.ca.1.4
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $8$
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] = [5445,2,Mod(1,5445)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5445, 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("5445.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.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: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 3x^{6} + 22x^{5} - 3x^{4} - 32x^{3} + 9x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.205878\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.20588 q^{2} -0.545859 q^{4} -1.00000 q^{5} +4.31539 q^{7} +3.06999 q^{8} +O(q^{10})\) \(q-1.20588 q^{2} -0.545859 q^{4} -1.00000 q^{5} +4.31539 q^{7} +3.06999 q^{8} +1.20588 q^{10} +6.18245 q^{13} -5.20383 q^{14} -2.61032 q^{16} +0.803988 q^{17} -1.87389 q^{19} +0.545859 q^{20} -8.08431 q^{23} +1.00000 q^{25} -7.45528 q^{26} -2.35559 q^{28} +1.05948 q^{29} -1.61509 q^{31} -2.99226 q^{32} -0.969511 q^{34} -4.31539 q^{35} -5.54134 q^{37} +2.25968 q^{38} -3.06999 q^{40} -5.05198 q^{41} -9.87101 q^{43} +9.74869 q^{46} -12.9857 q^{47} +11.6226 q^{49} -1.20588 q^{50} -3.37475 q^{52} +2.66437 q^{53} +13.2482 q^{56} -1.27760 q^{58} -6.60968 q^{59} -5.34760 q^{61} +1.94760 q^{62} +8.82894 q^{64} -6.18245 q^{65} -7.49018 q^{67} -0.438864 q^{68} +5.20383 q^{70} +5.82947 q^{71} +4.80148 q^{73} +6.68218 q^{74} +1.02288 q^{76} -13.9499 q^{79} +2.61032 q^{80} +6.09207 q^{82} -12.0568 q^{83} -0.803988 q^{85} +11.9032 q^{86} -12.7727 q^{89} +26.6797 q^{91} +4.41289 q^{92} +15.6592 q^{94} +1.87389 q^{95} +1.34761 q^{97} -14.0154 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8} + 4 q^{10} + 6 q^{13} - 14 q^{14} + 14 q^{16} - 8 q^{17} - 2 q^{19} - 6 q^{20} - 4 q^{23} + 8 q^{25} + 2 q^{26} + 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} + 20 q^{38} + 12 q^{40} - 22 q^{41} + 14 q^{43} - 2 q^{46} - 10 q^{47} - 4 q^{50} - 10 q^{52} + 18 q^{53} - 34 q^{56} + 12 q^{58} - 2 q^{59} - 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} + 2 q^{71} + 16 q^{73} - 24 q^{74} - 22 q^{76} - 16 q^{79} - 14 q^{80} + 10 q^{82} - 46 q^{83} + 8 q^{85} + 28 q^{86} - 38 q^{89} + 8 q^{91} + 24 q^{92} - 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.20588 −0.852684 −0.426342 0.904562i \(-0.640198\pi\)
−0.426342 + 0.904562i \(0.640198\pi\)
\(3\) 0 0
\(4\) −0.545859 −0.272929
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.31539 1.63106 0.815532 0.578712i \(-0.196445\pi\)
0.815532 + 0.578712i \(0.196445\pi\)
\(8\) 3.06999 1.08541
\(9\) 0 0
\(10\) 1.20588 0.381332
\(11\) 0 0
\(12\) 0 0
\(13\) 6.18245 1.71470 0.857352 0.514731i \(-0.172108\pi\)
0.857352 + 0.514731i \(0.172108\pi\)
\(14\) −5.20383 −1.39078
\(15\) 0 0
\(16\) −2.61032 −0.652580
\(17\) 0.803988 0.194996 0.0974978 0.995236i \(-0.468916\pi\)
0.0974978 + 0.995236i \(0.468916\pi\)
\(18\) 0 0
\(19\) −1.87389 −0.429899 −0.214949 0.976625i \(-0.568959\pi\)
−0.214949 + 0.976625i \(0.568959\pi\)
\(20\) 0.545859 0.122058
\(21\) 0 0
\(22\) 0 0
\(23\) −8.08431 −1.68570 −0.842848 0.538152i \(-0.819123\pi\)
−0.842848 + 0.538152i \(0.819123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −7.45528 −1.46210
\(27\) 0 0
\(28\) −2.35559 −0.445165
\(29\) 1.05948 0.196741 0.0983703 0.995150i \(-0.468637\pi\)
0.0983703 + 0.995150i \(0.468637\pi\)
\(30\) 0 0
\(31\) −1.61509 −0.290078 −0.145039 0.989426i \(-0.546331\pi\)
−0.145039 + 0.989426i \(0.546331\pi\)
\(32\) −2.99226 −0.528962
\(33\) 0 0
\(34\) −0.969511 −0.166270
\(35\) −4.31539 −0.729434
\(36\) 0 0
\(37\) −5.54134 −0.910991 −0.455496 0.890238i \(-0.650538\pi\)
−0.455496 + 0.890238i \(0.650538\pi\)
\(38\) 2.25968 0.366568
\(39\) 0 0
\(40\) −3.06999 −0.485409
\(41\) −5.05198 −0.788987 −0.394494 0.918899i \(-0.629080\pi\)
−0.394494 + 0.918899i \(0.629080\pi\)
\(42\) 0 0
\(43\) −9.87101 −1.50531 −0.752657 0.658412i \(-0.771228\pi\)
−0.752657 + 0.658412i \(0.771228\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 9.74869 1.43737
\(47\) −12.9857 −1.89416 −0.947079 0.321002i \(-0.895980\pi\)
−0.947079 + 0.321002i \(0.895980\pi\)
\(48\) 0 0
\(49\) 11.6226 1.66037
\(50\) −1.20588 −0.170537
\(51\) 0 0
\(52\) −3.37475 −0.467993
\(53\) 2.66437 0.365980 0.182990 0.983115i \(-0.441422\pi\)
0.182990 + 0.983115i \(0.441422\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 13.2482 1.77037
\(57\) 0 0
\(58\) −1.27760 −0.167758
\(59\) −6.60968 −0.860507 −0.430254 0.902708i \(-0.641576\pi\)
−0.430254 + 0.902708i \(0.641576\pi\)
\(60\) 0 0
\(61\) −5.34760 −0.684689 −0.342345 0.939574i \(-0.611221\pi\)
−0.342345 + 0.939574i \(0.611221\pi\)
\(62\) 1.94760 0.247345
\(63\) 0 0
\(64\) 8.82894 1.10362
\(65\) −6.18245 −0.766839
\(66\) 0 0
\(67\) −7.49018 −0.915071 −0.457536 0.889191i \(-0.651268\pi\)
−0.457536 + 0.889191i \(0.651268\pi\)
\(68\) −0.438864 −0.0532201
\(69\) 0 0
\(70\) 5.20383 0.621977
\(71\) 5.82947 0.691830 0.345915 0.938266i \(-0.387568\pi\)
0.345915 + 0.938266i \(0.387568\pi\)
\(72\) 0 0
\(73\) 4.80148 0.561970 0.280985 0.959712i \(-0.409339\pi\)
0.280985 + 0.959712i \(0.409339\pi\)
\(74\) 6.68218 0.776788
\(75\) 0 0
\(76\) 1.02288 0.117332
\(77\) 0 0
\(78\) 0 0
\(79\) −13.9499 −1.56949 −0.784744 0.619820i \(-0.787206\pi\)
−0.784744 + 0.619820i \(0.787206\pi\)
\(80\) 2.61032 0.291843
\(81\) 0 0
\(82\) 6.09207 0.672757
\(83\) −12.0568 −1.32340 −0.661701 0.749768i \(-0.730165\pi\)
−0.661701 + 0.749768i \(0.730165\pi\)
\(84\) 0 0
\(85\) −0.803988 −0.0872047
\(86\) 11.9032 1.28356
\(87\) 0 0
\(88\) 0 0
\(89\) −12.7727 −1.35390 −0.676951 0.736028i \(-0.736699\pi\)
−0.676951 + 0.736028i \(0.736699\pi\)
\(90\) 0 0
\(91\) 26.6797 2.79679
\(92\) 4.41289 0.460076
\(93\) 0 0
\(94\) 15.6592 1.61512
\(95\) 1.87389 0.192257
\(96\) 0 0
\(97\) 1.34761 0.136829 0.0684145 0.997657i \(-0.478206\pi\)
0.0684145 + 0.997657i \(0.478206\pi\)
\(98\) −14.0154 −1.41577
\(99\) 0 0
\(100\) −0.545859 −0.0545859
\(101\) 8.42949 0.838766 0.419383 0.907810i \(-0.362247\pi\)
0.419383 + 0.907810i \(0.362247\pi\)
\(102\) 0 0
\(103\) −9.90039 −0.975515 −0.487757 0.872979i \(-0.662185\pi\)
−0.487757 + 0.872979i \(0.662185\pi\)
\(104\) 18.9801 1.86115
\(105\) 0 0
\(106\) −3.21291 −0.312065
\(107\) −8.16742 −0.789574 −0.394787 0.918773i \(-0.629182\pi\)
−0.394787 + 0.918773i \(0.629182\pi\)
\(108\) 0 0
\(109\) 13.4997 1.29303 0.646517 0.762900i \(-0.276225\pi\)
0.646517 + 0.762900i \(0.276225\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −11.2645 −1.06440
\(113\) 13.5106 1.27097 0.635485 0.772114i \(-0.280800\pi\)
0.635485 + 0.772114i \(0.280800\pi\)
\(114\) 0 0
\(115\) 8.08431 0.753866
\(116\) −0.578327 −0.0536963
\(117\) 0 0
\(118\) 7.97047 0.733741
\(119\) 3.46952 0.318050
\(120\) 0 0
\(121\) 0 0
\(122\) 6.44855 0.583824
\(123\) 0 0
\(124\) 0.881610 0.0791709
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.12012 −0.188130 −0.0940652 0.995566i \(-0.529986\pi\)
−0.0940652 + 0.995566i \(0.529986\pi\)
\(128\) −4.66210 −0.412075
\(129\) 0 0
\(130\) 7.45528 0.653871
\(131\) −8.72324 −0.762153 −0.381076 0.924544i \(-0.624447\pi\)
−0.381076 + 0.924544i \(0.624447\pi\)
\(132\) 0 0
\(133\) −8.08654 −0.701192
\(134\) 9.03224 0.780267
\(135\) 0 0
\(136\) 2.46824 0.211650
\(137\) 1.54382 0.131898 0.0659489 0.997823i \(-0.478993\pi\)
0.0659489 + 0.997823i \(0.478993\pi\)
\(138\) 0 0
\(139\) 16.1982 1.37391 0.686957 0.726698i \(-0.258946\pi\)
0.686957 + 0.726698i \(0.258946\pi\)
\(140\) 2.35559 0.199084
\(141\) 0 0
\(142\) −7.02962 −0.589913
\(143\) 0 0
\(144\) 0 0
\(145\) −1.05948 −0.0879851
\(146\) −5.78999 −0.479183
\(147\) 0 0
\(148\) 3.02479 0.248636
\(149\) −14.0183 −1.14842 −0.574212 0.818707i \(-0.694692\pi\)
−0.574212 + 0.818707i \(0.694692\pi\)
\(150\) 0 0
\(151\) −13.5852 −1.10554 −0.552772 0.833332i \(-0.686430\pi\)
−0.552772 + 0.833332i \(0.686430\pi\)
\(152\) −5.75282 −0.466615
\(153\) 0 0
\(154\) 0 0
\(155\) 1.61509 0.129727
\(156\) 0 0
\(157\) −4.38074 −0.349621 −0.174810 0.984602i \(-0.555931\pi\)
−0.174810 + 0.984602i \(0.555931\pi\)
\(158\) 16.8219 1.33828
\(159\) 0 0
\(160\) 2.99226 0.236559
\(161\) −34.8869 −2.74948
\(162\) 0 0
\(163\) −4.68667 −0.367089 −0.183544 0.983011i \(-0.558757\pi\)
−0.183544 + 0.983011i \(0.558757\pi\)
\(164\) 2.75767 0.215338
\(165\) 0 0
\(166\) 14.5390 1.12844
\(167\) −1.64917 −0.127617 −0.0638084 0.997962i \(-0.520325\pi\)
−0.0638084 + 0.997962i \(0.520325\pi\)
\(168\) 0 0
\(169\) 25.2227 1.94021
\(170\) 0.969511 0.0743581
\(171\) 0 0
\(172\) 5.38818 0.410845
\(173\) 16.5722 1.25996 0.629981 0.776611i \(-0.283063\pi\)
0.629981 + 0.776611i \(0.283063\pi\)
\(174\) 0 0
\(175\) 4.31539 0.326213
\(176\) 0 0
\(177\) 0 0
\(178\) 15.4023 1.15445
\(179\) −17.5995 −1.31545 −0.657724 0.753259i \(-0.728481\pi\)
−0.657724 + 0.753259i \(0.728481\pi\)
\(180\) 0 0
\(181\) 8.40672 0.624867 0.312433 0.949940i \(-0.398856\pi\)
0.312433 + 0.949940i \(0.398856\pi\)
\(182\) −32.1724 −2.38478
\(183\) 0 0
\(184\) −24.8188 −1.82967
\(185\) 5.54134 0.407408
\(186\) 0 0
\(187\) 0 0
\(188\) 7.08835 0.516971
\(189\) 0 0
\(190\) −2.25968 −0.163934
\(191\) 5.31771 0.384776 0.192388 0.981319i \(-0.438377\pi\)
0.192388 + 0.981319i \(0.438377\pi\)
\(192\) 0 0
\(193\) 15.2872 1.10040 0.550199 0.835033i \(-0.314552\pi\)
0.550199 + 0.835033i \(0.314552\pi\)
\(194\) −1.62505 −0.116672
\(195\) 0 0
\(196\) −6.34429 −0.453163
\(197\) 1.69118 0.120492 0.0602458 0.998184i \(-0.480812\pi\)
0.0602458 + 0.998184i \(0.480812\pi\)
\(198\) 0 0
\(199\) 3.44855 0.244461 0.122230 0.992502i \(-0.460995\pi\)
0.122230 + 0.992502i \(0.460995\pi\)
\(200\) 3.06999 0.217081
\(201\) 0 0
\(202\) −10.1649 −0.715202
\(203\) 4.57207 0.320896
\(204\) 0 0
\(205\) 5.05198 0.352846
\(206\) 11.9387 0.831806
\(207\) 0 0
\(208\) −16.1382 −1.11898
\(209\) 0 0
\(210\) 0 0
\(211\) −0.394446 −0.0271548 −0.0135774 0.999908i \(-0.504322\pi\)
−0.0135774 + 0.999908i \(0.504322\pi\)
\(212\) −1.45437 −0.0998866
\(213\) 0 0
\(214\) 9.84890 0.673257
\(215\) 9.87101 0.673197
\(216\) 0 0
\(217\) −6.96973 −0.473136
\(218\) −16.2790 −1.10255
\(219\) 0 0
\(220\) 0 0
\(221\) 4.97062 0.334360
\(222\) 0 0
\(223\) −7.77146 −0.520416 −0.260208 0.965553i \(-0.583791\pi\)
−0.260208 + 0.965553i \(0.583791\pi\)
\(224\) −12.9128 −0.862771
\(225\) 0 0
\(226\) −16.2921 −1.08374
\(227\) −27.3162 −1.81304 −0.906519 0.422164i \(-0.861271\pi\)
−0.906519 + 0.422164i \(0.861271\pi\)
\(228\) 0 0
\(229\) 15.5822 1.02970 0.514851 0.857280i \(-0.327847\pi\)
0.514851 + 0.857280i \(0.327847\pi\)
\(230\) −9.74869 −0.642810
\(231\) 0 0
\(232\) 3.25260 0.213544
\(233\) 27.8846 1.82678 0.913390 0.407086i \(-0.133455\pi\)
0.913390 + 0.407086i \(0.133455\pi\)
\(234\) 0 0
\(235\) 12.9857 0.847093
\(236\) 3.60795 0.234858
\(237\) 0 0
\(238\) −4.18382 −0.271197
\(239\) −18.7458 −1.21257 −0.606284 0.795248i \(-0.707340\pi\)
−0.606284 + 0.795248i \(0.707340\pi\)
\(240\) 0 0
\(241\) −11.0049 −0.708886 −0.354443 0.935078i \(-0.615329\pi\)
−0.354443 + 0.935078i \(0.615329\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.91903 0.186872
\(245\) −11.6226 −0.742539
\(246\) 0 0
\(247\) −11.5852 −0.737149
\(248\) −4.95831 −0.314853
\(249\) 0 0
\(250\) 1.20588 0.0762664
\(251\) 23.3136 1.47154 0.735769 0.677232i \(-0.236821\pi\)
0.735769 + 0.677232i \(0.236821\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.55661 0.160416
\(255\) 0 0
\(256\) −12.0360 −0.752248
\(257\) −17.4730 −1.08994 −0.544969 0.838456i \(-0.683459\pi\)
−0.544969 + 0.838456i \(0.683459\pi\)
\(258\) 0 0
\(259\) −23.9130 −1.48588
\(260\) 3.37475 0.209293
\(261\) 0 0
\(262\) 10.5192 0.649876
\(263\) −1.61187 −0.0993921 −0.0496961 0.998764i \(-0.515825\pi\)
−0.0496961 + 0.998764i \(0.515825\pi\)
\(264\) 0 0
\(265\) −2.66437 −0.163671
\(266\) 9.75138 0.597896
\(267\) 0 0
\(268\) 4.08858 0.249750
\(269\) −8.17266 −0.498296 −0.249148 0.968465i \(-0.580151\pi\)
−0.249148 + 0.968465i \(0.580151\pi\)
\(270\) 0 0
\(271\) 10.4995 0.637801 0.318900 0.947788i \(-0.396686\pi\)
0.318900 + 0.947788i \(0.396686\pi\)
\(272\) −2.09867 −0.127250
\(273\) 0 0
\(274\) −1.86166 −0.112467
\(275\) 0 0
\(276\) 0 0
\(277\) 8.92222 0.536084 0.268042 0.963407i \(-0.413623\pi\)
0.268042 + 0.963407i \(0.413623\pi\)
\(278\) −19.5331 −1.17151
\(279\) 0 0
\(280\) −13.2482 −0.791732
\(281\) −4.39358 −0.262099 −0.131049 0.991376i \(-0.541835\pi\)
−0.131049 + 0.991376i \(0.541835\pi\)
\(282\) 0 0
\(283\) 13.0048 0.773056 0.386528 0.922278i \(-0.373674\pi\)
0.386528 + 0.922278i \(0.373674\pi\)
\(284\) −3.18207 −0.188821
\(285\) 0 0
\(286\) 0 0
\(287\) −21.8013 −1.28689
\(288\) 0 0
\(289\) −16.3536 −0.961977
\(290\) 1.27760 0.0750235
\(291\) 0 0
\(292\) −2.62093 −0.153378
\(293\) −15.5444 −0.908115 −0.454058 0.890972i \(-0.650024\pi\)
−0.454058 + 0.890972i \(0.650024\pi\)
\(294\) 0 0
\(295\) 6.60968 0.384830
\(296\) −17.0119 −0.988796
\(297\) 0 0
\(298\) 16.9044 0.979243
\(299\) −49.9809 −2.89047
\(300\) 0 0
\(301\) −42.5972 −2.45526
\(302\) 16.3820 0.942680
\(303\) 0 0
\(304\) 4.89144 0.280543
\(305\) 5.34760 0.306202
\(306\) 0 0
\(307\) 8.76859 0.500450 0.250225 0.968188i \(-0.419495\pi\)
0.250225 + 0.968188i \(0.419495\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.94760 −0.110616
\(311\) 1.75121 0.0993020 0.0496510 0.998767i \(-0.484189\pi\)
0.0496510 + 0.998767i \(0.484189\pi\)
\(312\) 0 0
\(313\) −17.5272 −0.990697 −0.495348 0.868694i \(-0.664960\pi\)
−0.495348 + 0.868694i \(0.664960\pi\)
\(314\) 5.28263 0.298116
\(315\) 0 0
\(316\) 7.61468 0.428359
\(317\) 1.99187 0.111874 0.0559372 0.998434i \(-0.482185\pi\)
0.0559372 + 0.998434i \(0.482185\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −8.82894 −0.493553
\(321\) 0 0
\(322\) 42.0694 2.34444
\(323\) −1.50658 −0.0838284
\(324\) 0 0
\(325\) 6.18245 0.342941
\(326\) 5.65156 0.313011
\(327\) 0 0
\(328\) −15.5096 −0.856372
\(329\) −56.0383 −3.08949
\(330\) 0 0
\(331\) −10.3437 −0.568541 −0.284271 0.958744i \(-0.591751\pi\)
−0.284271 + 0.958744i \(0.591751\pi\)
\(332\) 6.58130 0.361196
\(333\) 0 0
\(334\) 1.98870 0.108817
\(335\) 7.49018 0.409232
\(336\) 0 0
\(337\) −20.6187 −1.12317 −0.561587 0.827418i \(-0.689809\pi\)
−0.561587 + 0.827418i \(0.689809\pi\)
\(338\) −30.4155 −1.65439
\(339\) 0 0
\(340\) 0.438864 0.0238007
\(341\) 0 0
\(342\) 0 0
\(343\) 19.9482 1.07710
\(344\) −30.3039 −1.63388
\(345\) 0 0
\(346\) −19.9841 −1.07435
\(347\) −2.98375 −0.160176 −0.0800879 0.996788i \(-0.525520\pi\)
−0.0800879 + 0.996788i \(0.525520\pi\)
\(348\) 0 0
\(349\) 11.2897 0.604323 0.302161 0.953257i \(-0.402292\pi\)
0.302161 + 0.953257i \(0.402292\pi\)
\(350\) −5.20383 −0.278156
\(351\) 0 0
\(352\) 0 0
\(353\) 13.4952 0.718275 0.359137 0.933285i \(-0.383071\pi\)
0.359137 + 0.933285i \(0.383071\pi\)
\(354\) 0 0
\(355\) −5.82947 −0.309396
\(356\) 6.97209 0.369520
\(357\) 0 0
\(358\) 21.2228 1.12166
\(359\) 24.6989 1.30356 0.651778 0.758410i \(-0.274023\pi\)
0.651778 + 0.758410i \(0.274023\pi\)
\(360\) 0 0
\(361\) −15.4886 −0.815187
\(362\) −10.1375 −0.532814
\(363\) 0 0
\(364\) −14.5633 −0.763327
\(365\) −4.80148 −0.251321
\(366\) 0 0
\(367\) 6.34491 0.331201 0.165601 0.986193i \(-0.447044\pi\)
0.165601 + 0.986193i \(0.447044\pi\)
\(368\) 21.1026 1.10005
\(369\) 0 0
\(370\) −6.68218 −0.347390
\(371\) 11.4978 0.596936
\(372\) 0 0
\(373\) 13.1832 0.682601 0.341301 0.939954i \(-0.389133\pi\)
0.341301 + 0.939954i \(0.389133\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −39.8660 −2.05593
\(377\) 6.55019 0.337352
\(378\) 0 0
\(379\) −18.5363 −0.952144 −0.476072 0.879406i \(-0.657940\pi\)
−0.476072 + 0.879406i \(0.657940\pi\)
\(380\) −1.02288 −0.0524725
\(381\) 0 0
\(382\) −6.41251 −0.328093
\(383\) 28.3843 1.45037 0.725185 0.688554i \(-0.241754\pi\)
0.725185 + 0.688554i \(0.241754\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −18.4345 −0.938293
\(387\) 0 0
\(388\) −0.735604 −0.0373446
\(389\) −19.1714 −0.972029 −0.486014 0.873951i \(-0.661550\pi\)
−0.486014 + 0.873951i \(0.661550\pi\)
\(390\) 0 0
\(391\) −6.49969 −0.328703
\(392\) 35.6812 1.80217
\(393\) 0 0
\(394\) −2.03936 −0.102741
\(395\) 13.9499 0.701896
\(396\) 0 0
\(397\) 16.2455 0.815339 0.407670 0.913130i \(-0.366342\pi\)
0.407670 + 0.913130i \(0.366342\pi\)
\(398\) −4.15852 −0.208448
\(399\) 0 0
\(400\) −2.61032 −0.130516
\(401\) 12.3247 0.615465 0.307732 0.951473i \(-0.400430\pi\)
0.307732 + 0.951473i \(0.400430\pi\)
\(402\) 0 0
\(403\) −9.98520 −0.497398
\(404\) −4.60131 −0.228924
\(405\) 0 0
\(406\) −5.51336 −0.273623
\(407\) 0 0
\(408\) 0 0
\(409\) −1.95751 −0.0967926 −0.0483963 0.998828i \(-0.515411\pi\)
−0.0483963 + 0.998828i \(0.515411\pi\)
\(410\) −6.09207 −0.300866
\(411\) 0 0
\(412\) 5.40422 0.266247
\(413\) −28.5233 −1.40354
\(414\) 0 0
\(415\) 12.0568 0.591843
\(416\) −18.4995 −0.907013
\(417\) 0 0
\(418\) 0 0
\(419\) −20.5776 −1.00528 −0.502640 0.864496i \(-0.667638\pi\)
−0.502640 + 0.864496i \(0.667638\pi\)
\(420\) 0 0
\(421\) −17.1205 −0.834400 −0.417200 0.908815i \(-0.636989\pi\)
−0.417200 + 0.908815i \(0.636989\pi\)
\(422\) 0.475654 0.0231545
\(423\) 0 0
\(424\) 8.17960 0.397237
\(425\) 0.803988 0.0389991
\(426\) 0 0
\(427\) −23.0769 −1.11677
\(428\) 4.45826 0.215498
\(429\) 0 0
\(430\) −11.9032 −0.574025
\(431\) −10.6079 −0.510963 −0.255481 0.966814i \(-0.582234\pi\)
−0.255481 + 0.966814i \(0.582234\pi\)
\(432\) 0 0
\(433\) 24.8262 1.19307 0.596536 0.802586i \(-0.296543\pi\)
0.596536 + 0.802586i \(0.296543\pi\)
\(434\) 8.40464 0.403436
\(435\) 0 0
\(436\) −7.36892 −0.352907
\(437\) 15.1491 0.724678
\(438\) 0 0
\(439\) 6.78781 0.323965 0.161982 0.986794i \(-0.448211\pi\)
0.161982 + 0.986794i \(0.448211\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −5.99395 −0.285103
\(443\) 15.1500 0.719799 0.359899 0.932991i \(-0.382811\pi\)
0.359899 + 0.932991i \(0.382811\pi\)
\(444\) 0 0
\(445\) 12.7727 0.605484
\(446\) 9.37143 0.443750
\(447\) 0 0
\(448\) 38.1003 1.80007
\(449\) 17.9310 0.846217 0.423108 0.906079i \(-0.360939\pi\)
0.423108 + 0.906079i \(0.360939\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −7.37488 −0.346885
\(453\) 0 0
\(454\) 32.9400 1.54595
\(455\) −26.6797 −1.25076
\(456\) 0 0
\(457\) −41.1984 −1.92718 −0.963590 0.267385i \(-0.913841\pi\)
−0.963590 + 0.267385i \(0.913841\pi\)
\(458\) −18.7902 −0.878010
\(459\) 0 0
\(460\) −4.41289 −0.205752
\(461\) −16.8145 −0.783128 −0.391564 0.920151i \(-0.628066\pi\)
−0.391564 + 0.920151i \(0.628066\pi\)
\(462\) 0 0
\(463\) −35.7050 −1.65935 −0.829676 0.558245i \(-0.811475\pi\)
−0.829676 + 0.558245i \(0.811475\pi\)
\(464\) −2.76558 −0.128389
\(465\) 0 0
\(466\) −33.6254 −1.55767
\(467\) 35.9512 1.66362 0.831811 0.555059i \(-0.187304\pi\)
0.831811 + 0.555059i \(0.187304\pi\)
\(468\) 0 0
\(469\) −32.3230 −1.49254
\(470\) −15.6592 −0.722303
\(471\) 0 0
\(472\) −20.2917 −0.934000
\(473\) 0 0
\(474\) 0 0
\(475\) −1.87389 −0.0859798
\(476\) −1.89387 −0.0868053
\(477\) 0 0
\(478\) 22.6052 1.03394
\(479\) 13.8915 0.634721 0.317360 0.948305i \(-0.397204\pi\)
0.317360 + 0.948305i \(0.397204\pi\)
\(480\) 0 0
\(481\) −34.2591 −1.56208
\(482\) 13.2705 0.604456
\(483\) 0 0
\(484\) 0 0
\(485\) −1.34761 −0.0611918
\(486\) 0 0
\(487\) −14.0003 −0.634413 −0.317207 0.948356i \(-0.602745\pi\)
−0.317207 + 0.948356i \(0.602745\pi\)
\(488\) −16.4171 −0.743167
\(489\) 0 0
\(490\) 14.0154 0.633151
\(491\) 25.2477 1.13941 0.569707 0.821848i \(-0.307057\pi\)
0.569707 + 0.821848i \(0.307057\pi\)
\(492\) 0 0
\(493\) 0.851809 0.0383636
\(494\) 13.9703 0.628555
\(495\) 0 0
\(496\) 4.21590 0.189299
\(497\) 25.1564 1.12842
\(498\) 0 0
\(499\) −8.86257 −0.396743 −0.198371 0.980127i \(-0.563565\pi\)
−0.198371 + 0.980127i \(0.563565\pi\)
\(500\) 0.545859 0.0244116
\(501\) 0 0
\(502\) −28.1133 −1.25476
\(503\) −0.661184 −0.0294807 −0.0147404 0.999891i \(-0.504692\pi\)
−0.0147404 + 0.999891i \(0.504692\pi\)
\(504\) 0 0
\(505\) −8.42949 −0.375107
\(506\) 0 0
\(507\) 0 0
\(508\) 1.15729 0.0513463
\(509\) −38.7166 −1.71608 −0.858042 0.513580i \(-0.828319\pi\)
−0.858042 + 0.513580i \(0.828319\pi\)
\(510\) 0 0
\(511\) 20.7202 0.916609
\(512\) 23.8381 1.05351
\(513\) 0 0
\(514\) 21.0704 0.929373
\(515\) 9.90039 0.436263
\(516\) 0 0
\(517\) 0 0
\(518\) 28.8362 1.26699
\(519\) 0 0
\(520\) −18.9801 −0.832332
\(521\) 24.5734 1.07658 0.538290 0.842760i \(-0.319071\pi\)
0.538290 + 0.842760i \(0.319071\pi\)
\(522\) 0 0
\(523\) 33.6944 1.47335 0.736677 0.676245i \(-0.236394\pi\)
0.736677 + 0.676245i \(0.236394\pi\)
\(524\) 4.76166 0.208014
\(525\) 0 0
\(526\) 1.94372 0.0847501
\(527\) −1.29851 −0.0565640
\(528\) 0 0
\(529\) 42.3561 1.84157
\(530\) 3.21291 0.139560
\(531\) 0 0
\(532\) 4.41411 0.191376
\(533\) −31.2336 −1.35288
\(534\) 0 0
\(535\) 8.16742 0.353108
\(536\) −22.9948 −0.993225
\(537\) 0 0
\(538\) 9.85523 0.424889
\(539\) 0 0
\(540\) 0 0
\(541\) −8.20840 −0.352906 −0.176453 0.984309i \(-0.556462\pi\)
−0.176453 + 0.984309i \(0.556462\pi\)
\(542\) −12.6611 −0.543843
\(543\) 0 0
\(544\) −2.40574 −0.103145
\(545\) −13.4997 −0.578262
\(546\) 0 0
\(547\) 35.0759 1.49974 0.749868 0.661587i \(-0.230117\pi\)
0.749868 + 0.661587i \(0.230117\pi\)
\(548\) −0.842711 −0.0359988
\(549\) 0 0
\(550\) 0 0
\(551\) −1.98534 −0.0845785
\(552\) 0 0
\(553\) −60.1993 −2.55993
\(554\) −10.7591 −0.457111
\(555\) 0 0
\(556\) −8.84194 −0.374982
\(557\) −39.6130 −1.67846 −0.839229 0.543779i \(-0.816993\pi\)
−0.839229 + 0.543779i \(0.816993\pi\)
\(558\) 0 0
\(559\) −61.0270 −2.58117
\(560\) 11.2645 0.476014
\(561\) 0 0
\(562\) 5.29812 0.223488
\(563\) −21.6803 −0.913717 −0.456858 0.889539i \(-0.651025\pi\)
−0.456858 + 0.889539i \(0.651025\pi\)
\(564\) 0 0
\(565\) −13.5106 −0.568395
\(566\) −15.6822 −0.659173
\(567\) 0 0
\(568\) 17.8964 0.750918
\(569\) 27.4638 1.15134 0.575672 0.817681i \(-0.304741\pi\)
0.575672 + 0.817681i \(0.304741\pi\)
\(570\) 0 0
\(571\) 23.8306 0.997281 0.498641 0.866809i \(-0.333833\pi\)
0.498641 + 0.866809i \(0.333833\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 26.2897 1.09731
\(575\) −8.08431 −0.337139
\(576\) 0 0
\(577\) −16.0731 −0.669130 −0.334565 0.942373i \(-0.608589\pi\)
−0.334565 + 0.942373i \(0.608589\pi\)
\(578\) 19.7204 0.820262
\(579\) 0 0
\(580\) 0.578327 0.0240137
\(581\) −52.0296 −2.15855
\(582\) 0 0
\(583\) 0 0
\(584\) 14.7405 0.609966
\(585\) 0 0
\(586\) 18.7447 0.774336
\(587\) 17.7793 0.733829 0.366915 0.930255i \(-0.380414\pi\)
0.366915 + 0.930255i \(0.380414\pi\)
\(588\) 0 0
\(589\) 3.02649 0.124704
\(590\) −7.97047 −0.328139
\(591\) 0 0
\(592\) 14.4647 0.594495
\(593\) 20.2944 0.833389 0.416695 0.909046i \(-0.363188\pi\)
0.416695 + 0.909046i \(0.363188\pi\)
\(594\) 0 0
\(595\) −3.46952 −0.142236
\(596\) 7.65202 0.313439
\(597\) 0 0
\(598\) 60.2708 2.46466
\(599\) −9.00707 −0.368019 −0.184009 0.982924i \(-0.558908\pi\)
−0.184009 + 0.982924i \(0.558908\pi\)
\(600\) 0 0
\(601\) 41.1081 1.67683 0.838417 0.545029i \(-0.183481\pi\)
0.838417 + 0.545029i \(0.183481\pi\)
\(602\) 51.3671 2.09356
\(603\) 0 0
\(604\) 7.41558 0.301736
\(605\) 0 0
\(606\) 0 0
\(607\) 1.19057 0.0483236 0.0241618 0.999708i \(-0.492308\pi\)
0.0241618 + 0.999708i \(0.492308\pi\)
\(608\) 5.60716 0.227400
\(609\) 0 0
\(610\) −6.44855 −0.261094
\(611\) −80.2834 −3.24792
\(612\) 0 0
\(613\) 0.136596 0.00551706 0.00275853 0.999996i \(-0.499122\pi\)
0.00275853 + 0.999996i \(0.499122\pi\)
\(614\) −10.5738 −0.426726
\(615\) 0 0
\(616\) 0 0
\(617\) 17.7011 0.712620 0.356310 0.934368i \(-0.384035\pi\)
0.356310 + 0.934368i \(0.384035\pi\)
\(618\) 0 0
\(619\) −7.08920 −0.284939 −0.142470 0.989799i \(-0.545504\pi\)
−0.142470 + 0.989799i \(0.545504\pi\)
\(620\) −0.881610 −0.0354063
\(621\) 0 0
\(622\) −2.11174 −0.0846733
\(623\) −55.1191 −2.20830
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 21.1357 0.844752
\(627\) 0 0
\(628\) 2.39127 0.0954219
\(629\) −4.45517 −0.177639
\(630\) 0 0
\(631\) 24.1403 0.961010 0.480505 0.876992i \(-0.340453\pi\)
0.480505 + 0.876992i \(0.340453\pi\)
\(632\) −42.8262 −1.70353
\(633\) 0 0
\(634\) −2.40195 −0.0953935
\(635\) 2.12012 0.0841344
\(636\) 0 0
\(637\) 71.8560 2.84704
\(638\) 0 0
\(639\) 0 0
\(640\) 4.66210 0.184286
\(641\) 42.8201 1.69129 0.845646 0.533745i \(-0.179216\pi\)
0.845646 + 0.533745i \(0.179216\pi\)
\(642\) 0 0
\(643\) −34.2410 −1.35033 −0.675166 0.737666i \(-0.735928\pi\)
−0.675166 + 0.737666i \(0.735928\pi\)
\(644\) 19.0433 0.750413
\(645\) 0 0
\(646\) 1.81675 0.0714792
\(647\) 10.8930 0.428247 0.214124 0.976807i \(-0.431310\pi\)
0.214124 + 0.976807i \(0.431310\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −7.45528 −0.292420
\(651\) 0 0
\(652\) 2.55826 0.100189
\(653\) 42.0505 1.64556 0.822781 0.568359i \(-0.192421\pi\)
0.822781 + 0.568359i \(0.192421\pi\)
\(654\) 0 0
\(655\) 8.72324 0.340845
\(656\) 13.1873 0.514877
\(657\) 0 0
\(658\) 67.5753 2.63436
\(659\) −0.425249 −0.0165654 −0.00828268 0.999966i \(-0.502636\pi\)
−0.00828268 + 0.999966i \(0.502636\pi\)
\(660\) 0 0
\(661\) −7.08477 −0.275565 −0.137783 0.990462i \(-0.543998\pi\)
−0.137783 + 0.990462i \(0.543998\pi\)
\(662\) 12.4732 0.484786
\(663\) 0 0
\(664\) −37.0142 −1.43643
\(665\) 8.08654 0.313583
\(666\) 0 0
\(667\) −8.56517 −0.331645
\(668\) 0.900216 0.0348304
\(669\) 0 0
\(670\) −9.03224 −0.348946
\(671\) 0 0
\(672\) 0 0
\(673\) −12.2833 −0.473487 −0.236743 0.971572i \(-0.576080\pi\)
−0.236743 + 0.971572i \(0.576080\pi\)
\(674\) 24.8637 0.957712
\(675\) 0 0
\(676\) −13.7680 −0.529540
\(677\) 17.2760 0.663972 0.331986 0.943284i \(-0.392281\pi\)
0.331986 + 0.943284i \(0.392281\pi\)
\(678\) 0 0
\(679\) 5.81545 0.223177
\(680\) −2.46824 −0.0946526
\(681\) 0 0
\(682\) 0 0
\(683\) 26.8858 1.02876 0.514379 0.857563i \(-0.328022\pi\)
0.514379 + 0.857563i \(0.328022\pi\)
\(684\) 0 0
\(685\) −1.54382 −0.0589865
\(686\) −24.0551 −0.918428
\(687\) 0 0
\(688\) 25.7665 0.982338
\(689\) 16.4723 0.627546
\(690\) 0 0
\(691\) 24.9330 0.948496 0.474248 0.880391i \(-0.342720\pi\)
0.474248 + 0.880391i \(0.342720\pi\)
\(692\) −9.04609 −0.343881
\(693\) 0 0
\(694\) 3.59803 0.136579
\(695\) −16.1982 −0.614433
\(696\) 0 0
\(697\) −4.06173 −0.153849
\(698\) −13.6140 −0.515297
\(699\) 0 0
\(700\) −2.35559 −0.0890331
\(701\) 20.0721 0.758112 0.379056 0.925374i \(-0.376249\pi\)
0.379056 + 0.925374i \(0.376249\pi\)
\(702\) 0 0
\(703\) 10.3838 0.391634
\(704\) 0 0
\(705\) 0 0
\(706\) −16.2735 −0.612461
\(707\) 36.3765 1.36808
\(708\) 0 0
\(709\) −11.3356 −0.425717 −0.212858 0.977083i \(-0.568277\pi\)
−0.212858 + 0.977083i \(0.568277\pi\)
\(710\) 7.02962 0.263817
\(711\) 0 0
\(712\) −39.2121 −1.46954
\(713\) 13.0569 0.488984
\(714\) 0 0
\(715\) 0 0
\(716\) 9.60684 0.359025
\(717\) 0 0
\(718\) −29.7838 −1.11152
\(719\) 4.63537 0.172870 0.0864351 0.996257i \(-0.472452\pi\)
0.0864351 + 0.996257i \(0.472452\pi\)
\(720\) 0 0
\(721\) −42.7240 −1.59113
\(722\) 18.6773 0.695097
\(723\) 0 0
\(724\) −4.58889 −0.170545
\(725\) 1.05948 0.0393481
\(726\) 0 0
\(727\) −2.22651 −0.0825766 −0.0412883 0.999147i \(-0.513146\pi\)
−0.0412883 + 0.999147i \(0.513146\pi\)
\(728\) 81.9065 3.03566
\(729\) 0 0
\(730\) 5.78999 0.214297
\(731\) −7.93617 −0.293530
\(732\) 0 0
\(733\) −30.4780 −1.12573 −0.562866 0.826548i \(-0.690301\pi\)
−0.562866 + 0.826548i \(0.690301\pi\)
\(734\) −7.65118 −0.282410
\(735\) 0 0
\(736\) 24.1904 0.891669
\(737\) 0 0
\(738\) 0 0
\(739\) −9.20178 −0.338493 −0.169246 0.985574i \(-0.554133\pi\)
−0.169246 + 0.985574i \(0.554133\pi\)
\(740\) −3.02479 −0.111194
\(741\) 0 0
\(742\) −13.8649 −0.508998
\(743\) 1.72902 0.0634316 0.0317158 0.999497i \(-0.489903\pi\)
0.0317158 + 0.999497i \(0.489903\pi\)
\(744\) 0 0
\(745\) 14.0183 0.513591
\(746\) −15.8974 −0.582043
\(747\) 0 0
\(748\) 0 0
\(749\) −35.2456 −1.28785
\(750\) 0 0
\(751\) 33.8778 1.23622 0.618109 0.786092i \(-0.287899\pi\)
0.618109 + 0.786092i \(0.287899\pi\)
\(752\) 33.8968 1.23609
\(753\) 0 0
\(754\) −7.89872 −0.287655
\(755\) 13.5852 0.494415
\(756\) 0 0
\(757\) −26.8677 −0.976522 −0.488261 0.872698i \(-0.662369\pi\)
−0.488261 + 0.872698i \(0.662369\pi\)
\(758\) 22.3525 0.811879
\(759\) 0 0
\(760\) 5.75282 0.208677
\(761\) −17.4413 −0.632247 −0.316124 0.948718i \(-0.602381\pi\)
−0.316124 + 0.948718i \(0.602381\pi\)
\(762\) 0 0
\(763\) 58.2563 2.10902
\(764\) −2.90272 −0.105017
\(765\) 0 0
\(766\) −34.2280 −1.23671
\(767\) −40.8640 −1.47551
\(768\) 0 0
\(769\) −41.6485 −1.50188 −0.750942 0.660368i \(-0.770400\pi\)
−0.750942 + 0.660368i \(0.770400\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.34467 −0.300331
\(773\) 22.1515 0.796733 0.398366 0.917226i \(-0.369577\pi\)
0.398366 + 0.917226i \(0.369577\pi\)
\(774\) 0 0
\(775\) −1.61509 −0.0580157
\(776\) 4.13715 0.148515
\(777\) 0 0
\(778\) 23.1184 0.828834
\(779\) 9.46684 0.339185
\(780\) 0 0
\(781\) 0 0
\(782\) 7.83783 0.280280
\(783\) 0 0
\(784\) −30.3386 −1.08352
\(785\) 4.38074 0.156355
\(786\) 0 0
\(787\) −8.41235 −0.299868 −0.149934 0.988696i \(-0.547906\pi\)
−0.149934 + 0.988696i \(0.547906\pi\)
\(788\) −0.923145 −0.0328857
\(789\) 0 0
\(790\) −16.8219 −0.598496
\(791\) 58.3034 2.07303
\(792\) 0 0
\(793\) −33.0612 −1.17404
\(794\) −19.5901 −0.695227
\(795\) 0 0
\(796\) −1.88242 −0.0667206
\(797\) −40.9234 −1.44958 −0.724791 0.688969i \(-0.758063\pi\)
−0.724791 + 0.688969i \(0.758063\pi\)
\(798\) 0 0
\(799\) −10.4403 −0.369352
\(800\) −2.99226 −0.105792
\(801\) 0 0
\(802\) −14.8621 −0.524797
\(803\) 0 0
\(804\) 0 0
\(805\) 34.8869 1.22960
\(806\) 12.0409 0.424124
\(807\) 0 0
\(808\) 25.8785 0.910402
\(809\) 21.6716 0.761931 0.380966 0.924589i \(-0.375592\pi\)
0.380966 + 0.924589i \(0.375592\pi\)
\(810\) 0 0
\(811\) −54.5031 −1.91386 −0.956931 0.290314i \(-0.906240\pi\)
−0.956931 + 0.290314i \(0.906240\pi\)
\(812\) −2.49571 −0.0875821
\(813\) 0 0
\(814\) 0 0
\(815\) 4.68667 0.164167
\(816\) 0 0
\(817\) 18.4971 0.647133
\(818\) 2.36052 0.0825335
\(819\) 0 0
\(820\) −2.75767 −0.0963020
\(821\) −17.9510 −0.626495 −0.313247 0.949672i \(-0.601417\pi\)
−0.313247 + 0.949672i \(0.601417\pi\)
\(822\) 0 0
\(823\) 35.2352 1.22822 0.614110 0.789220i \(-0.289515\pi\)
0.614110 + 0.789220i \(0.289515\pi\)
\(824\) −30.3942 −1.05883
\(825\) 0 0
\(826\) 34.3957 1.19678
\(827\) −39.9373 −1.38876 −0.694378 0.719610i \(-0.744320\pi\)
−0.694378 + 0.719610i \(0.744320\pi\)
\(828\) 0 0
\(829\) −45.5317 −1.58138 −0.790690 0.612216i \(-0.790278\pi\)
−0.790690 + 0.612216i \(0.790278\pi\)
\(830\) −14.5390 −0.504656
\(831\) 0 0
\(832\) 54.5845 1.89238
\(833\) 9.34441 0.323765
\(834\) 0 0
\(835\) 1.64917 0.0570720
\(836\) 0 0
\(837\) 0 0
\(838\) 24.8140 0.857186
\(839\) 18.0373 0.622718 0.311359 0.950292i \(-0.399216\pi\)
0.311359 + 0.950292i \(0.399216\pi\)
\(840\) 0 0
\(841\) −27.8775 −0.961293
\(842\) 20.6452 0.711480
\(843\) 0 0
\(844\) 0.215312 0.00741134
\(845\) −25.2227 −0.867688
\(846\) 0 0
\(847\) 0 0
\(848\) −6.95486 −0.238831
\(849\) 0 0
\(850\) −0.969511 −0.0332540
\(851\) 44.7979 1.53565
\(852\) 0 0
\(853\) −1.18704 −0.0406434 −0.0203217 0.999793i \(-0.506469\pi\)
−0.0203217 + 0.999793i \(0.506469\pi\)
\(854\) 27.8280 0.952254
\(855\) 0 0
\(856\) −25.0739 −0.857009
\(857\) 5.59493 0.191119 0.0955597 0.995424i \(-0.469536\pi\)
0.0955597 + 0.995424i \(0.469536\pi\)
\(858\) 0 0
\(859\) −18.5783 −0.633882 −0.316941 0.948445i \(-0.602656\pi\)
−0.316941 + 0.948445i \(0.602656\pi\)
\(860\) −5.38818 −0.183735
\(861\) 0 0
\(862\) 12.7918 0.435690
\(863\) 14.2249 0.484220 0.242110 0.970249i \(-0.422161\pi\)
0.242110 + 0.970249i \(0.422161\pi\)
\(864\) 0 0
\(865\) −16.5722 −0.563472
\(866\) −29.9374 −1.01731
\(867\) 0 0
\(868\) 3.80449 0.129133
\(869\) 0 0
\(870\) 0 0
\(871\) −46.3077 −1.56908
\(872\) 41.4439 1.40347
\(873\) 0 0
\(874\) −18.2679 −0.617922
\(875\) −4.31539 −0.145887
\(876\) 0 0
\(877\) −18.6019 −0.628141 −0.314070 0.949400i \(-0.601693\pi\)
−0.314070 + 0.949400i \(0.601693\pi\)
\(878\) −8.18527 −0.276240
\(879\) 0 0
\(880\) 0 0
\(881\) −30.5201 −1.02825 −0.514124 0.857716i \(-0.671883\pi\)
−0.514124 + 0.857716i \(0.671883\pi\)
\(882\) 0 0
\(883\) −16.2877 −0.548125 −0.274062 0.961712i \(-0.588367\pi\)
−0.274062 + 0.961712i \(0.588367\pi\)
\(884\) −2.71326 −0.0912566
\(885\) 0 0
\(886\) −18.2691 −0.613761
\(887\) −5.29250 −0.177705 −0.0888524 0.996045i \(-0.528320\pi\)
−0.0888524 + 0.996045i \(0.528320\pi\)
\(888\) 0 0
\(889\) −9.14915 −0.306853
\(890\) −15.4023 −0.516286
\(891\) 0 0
\(892\) 4.24212 0.142037
\(893\) 24.3337 0.814296
\(894\) 0 0
\(895\) 17.5995 0.588286
\(896\) −20.1188 −0.672121
\(897\) 0 0
\(898\) −21.6226 −0.721556
\(899\) −1.71115 −0.0570702
\(900\) 0 0
\(901\) 2.14212 0.0713644
\(902\) 0 0
\(903\) 0 0
\(904\) 41.4774 1.37952
\(905\) −8.40672 −0.279449
\(906\) 0 0
\(907\) 1.15084 0.0382132 0.0191066 0.999817i \(-0.493918\pi\)
0.0191066 + 0.999817i \(0.493918\pi\)
\(908\) 14.9108 0.494832
\(909\) 0 0
\(910\) 32.1724 1.06651
\(911\) 24.4697 0.810717 0.405359 0.914158i \(-0.367147\pi\)
0.405359 + 0.914158i \(0.367147\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 49.6802 1.64328
\(915\) 0 0
\(916\) −8.50569 −0.281036
\(917\) −37.6442 −1.24312
\(918\) 0 0
\(919\) −27.1167 −0.894499 −0.447249 0.894409i \(-0.647596\pi\)
−0.447249 + 0.894409i \(0.647596\pi\)
\(920\) 24.8188 0.818251
\(921\) 0 0
\(922\) 20.2762 0.667761
\(923\) 36.0404 1.18628
\(924\) 0 0
\(925\) −5.54134 −0.182198
\(926\) 43.0559 1.41490
\(927\) 0 0
\(928\) −3.17024 −0.104068
\(929\) 29.0648 0.953587 0.476793 0.879015i \(-0.341799\pi\)
0.476793 + 0.879015i \(0.341799\pi\)
\(930\) 0 0
\(931\) −21.7794 −0.713790
\(932\) −15.2211 −0.498582
\(933\) 0 0
\(934\) −43.3527 −1.41854
\(935\) 0 0
\(936\) 0 0
\(937\) −59.8539 −1.95534 −0.977671 0.210143i \(-0.932607\pi\)
−0.977671 + 0.210143i \(0.932607\pi\)
\(938\) 38.9776 1.27266
\(939\) 0 0
\(940\) −7.08835 −0.231197
\(941\) 30.2191 0.985115 0.492558 0.870280i \(-0.336062\pi\)
0.492558 + 0.870280i \(0.336062\pi\)
\(942\) 0 0
\(943\) 40.8418 1.32999
\(944\) 17.2534 0.561550
\(945\) 0 0
\(946\) 0 0
\(947\) 38.6489 1.25592 0.627960 0.778245i \(-0.283890\pi\)
0.627960 + 0.778245i \(0.283890\pi\)
\(948\) 0 0
\(949\) 29.6849 0.963612
\(950\) 2.25968 0.0733136
\(951\) 0 0
\(952\) 10.6514 0.345214
\(953\) 44.7169 1.44852 0.724261 0.689526i \(-0.242181\pi\)
0.724261 + 0.689526i \(0.242181\pi\)
\(954\) 0 0
\(955\) −5.31771 −0.172077
\(956\) 10.2326 0.330945
\(957\) 0 0
\(958\) −16.7515 −0.541216
\(959\) 6.66220 0.215134
\(960\) 0 0
\(961\) −28.3915 −0.915855
\(962\) 41.3123 1.33196
\(963\) 0 0
\(964\) 6.00711 0.193476
\(965\) −15.2872 −0.492113
\(966\) 0 0
\(967\) −34.5876 −1.11226 −0.556131 0.831095i \(-0.687715\pi\)
−0.556131 + 0.831095i \(0.687715\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 1.62505 0.0521773
\(971\) −6.91193 −0.221814 −0.110907 0.993831i \(-0.535376\pi\)
−0.110907 + 0.993831i \(0.535376\pi\)
\(972\) 0 0
\(973\) 69.9016 2.24094
\(974\) 16.8826 0.540954
\(975\) 0 0
\(976\) 13.9589 0.446815
\(977\) 12.2661 0.392427 0.196214 0.980561i \(-0.437135\pi\)
0.196214 + 0.980561i \(0.437135\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.34429 0.202661
\(981\) 0 0
\(982\) −30.4457 −0.971561
\(983\) −0.161755 −0.00515918 −0.00257959 0.999997i \(-0.500821\pi\)
−0.00257959 + 0.999997i \(0.500821\pi\)
\(984\) 0 0
\(985\) −1.69118 −0.0538855
\(986\) −1.02718 −0.0327120
\(987\) 0 0
\(988\) 6.32389 0.201190
\(989\) 79.8003 2.53750
\(990\) 0 0
\(991\) 43.1252 1.36992 0.684958 0.728582i \(-0.259820\pi\)
0.684958 + 0.728582i \(0.259820\pi\)
\(992\) 4.83277 0.153440
\(993\) 0 0
\(994\) −30.3356 −0.962185
\(995\) −3.44855 −0.109326
\(996\) 0 0
\(997\) −1.47228 −0.0466276 −0.0233138 0.999728i \(-0.507422\pi\)
−0.0233138 + 0.999728i \(0.507422\pi\)
\(998\) 10.6872 0.338296
\(999\) 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 5445.2.a.ca.1.4 8
3.2 odd 2 5445.2.a.cd.1.5 8
11.3 even 5 495.2.n.h.361.2 yes 16
11.4 even 5 495.2.n.h.181.2 yes 16
11.10 odd 2 5445.2.a.cc.1.5 8
33.14 odd 10 495.2.n.g.361.3 yes 16
33.26 odd 10 495.2.n.g.181.3 16
33.32 even 2 5445.2.a.cb.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.n.g.181.3 16 33.26 odd 10
495.2.n.g.361.3 yes 16 33.14 odd 10
495.2.n.h.181.2 yes 16 11.4 even 5
495.2.n.h.361.2 yes 16 11.3 even 5
5445.2.a.ca.1.4 8 1.1 even 1 trivial
5445.2.a.cb.1.4 8 33.32 even 2
5445.2.a.cc.1.5 8 11.10 odd 2
5445.2.a.cd.1.5 8 3.2 odd 2