Properties

Label 5225.2.a.u.1.3
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: \( x^{15} - x^{14} - 21 x^{13} + 21 x^{12} + 168 x^{11} - 165 x^{10} - 645 x^{9} + 606 x^{8} + 1239 x^{7} + \cdots - 5 \) 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 1.3
Root \(1.98766\) 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)
 
\(f(q)\) \(=\) \(q-1.98766 q^{2} -0.104882 q^{3} +1.95080 q^{4} +0.208469 q^{6} +1.24528 q^{7} +0.0977875 q^{8} -2.98900 q^{9} +O(q^{10})\) \(q-1.98766 q^{2} -0.104882 q^{3} +1.95080 q^{4} +0.208469 q^{6} +1.24528 q^{7} +0.0977875 q^{8} -2.98900 q^{9} -1.00000 q^{11} -0.204603 q^{12} -2.45314 q^{13} -2.47520 q^{14} -4.09597 q^{16} -8.07996 q^{17} +5.94112 q^{18} +1.00000 q^{19} -0.130607 q^{21} +1.98766 q^{22} -0.595102 q^{23} -0.0102561 q^{24} +4.87602 q^{26} +0.628136 q^{27} +2.42930 q^{28} -6.20473 q^{29} -9.51628 q^{31} +7.94584 q^{32} +0.104882 q^{33} +16.0602 q^{34} -5.83095 q^{36} +6.49334 q^{37} -1.98766 q^{38} +0.257290 q^{39} -8.06971 q^{41} +0.259603 q^{42} -2.33313 q^{43} -1.95080 q^{44} +1.18286 q^{46} -8.72660 q^{47} +0.429593 q^{48} -5.44927 q^{49} +0.847439 q^{51} -4.78560 q^{52} +10.8618 q^{53} -1.24852 q^{54} +0.121773 q^{56} -0.104882 q^{57} +12.3329 q^{58} +0.601714 q^{59} -1.23483 q^{61} +18.9152 q^{62} -3.72215 q^{63} -7.60170 q^{64} -0.208469 q^{66} -5.79923 q^{67} -15.7624 q^{68} +0.0624153 q^{69} +7.74139 q^{71} -0.292287 q^{72} +13.3744 q^{73} -12.9066 q^{74} +1.95080 q^{76} -1.24528 q^{77} -0.511405 q^{78} +1.79880 q^{79} +8.90112 q^{81} +16.0399 q^{82} +15.9421 q^{83} -0.254789 q^{84} +4.63748 q^{86} +0.650763 q^{87} -0.0977875 q^{88} -7.25863 q^{89} -3.05486 q^{91} -1.16093 q^{92} +0.998084 q^{93} +17.3455 q^{94} -0.833373 q^{96} +8.72541 q^{97} +10.8313 q^{98} +2.98900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 17 q^{7} + 3 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 17 q^{7} + 3 q^{8} + 15 q^{9} - 15 q^{11} + 9 q^{12} + 3 q^{13} - 11 q^{14} + 13 q^{16} + q^{17} - 10 q^{18} + 15 q^{19} + 16 q^{21} + q^{22} + 18 q^{23} - 27 q^{24} + 15 q^{26} + 31 q^{27} + 34 q^{28} + 11 q^{29} - 2 q^{31} + 10 q^{32} - 4 q^{33} - 13 q^{34} + 8 q^{36} + 27 q^{37} - q^{38} + 4 q^{39} + 6 q^{41} + 2 q^{42} + 38 q^{43} - 13 q^{44} - 9 q^{46} + 14 q^{47} + 32 q^{48} + 28 q^{49} - 32 q^{51} + 16 q^{52} + 11 q^{53} - 11 q^{54} + 2 q^{56} + 4 q^{57} - 6 q^{58} + 13 q^{59} + 8 q^{61} + 7 q^{62} + 49 q^{63} + 9 q^{64} + q^{66} + 31 q^{67} - 26 q^{68} + 39 q^{69} - 3 q^{71} - 18 q^{72} + 18 q^{73} + 7 q^{74} + 13 q^{76} - 17 q^{77} + 18 q^{78} + 10 q^{79} + 31 q^{81} + 58 q^{82} + 16 q^{83} + 112 q^{84} - 63 q^{86} + 67 q^{87} - 3 q^{88} - 7 q^{89} - 50 q^{91} + 98 q^{92} + 26 q^{93} + 22 q^{94} - 37 q^{96} + 24 q^{97} - 46 q^{98} - 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.98766 −1.40549 −0.702745 0.711442i \(-0.748042\pi\)
−0.702745 + 0.711442i \(0.748042\pi\)
\(3\) −0.104882 −0.0605534 −0.0302767 0.999542i \(-0.509639\pi\)
−0.0302767 + 0.999542i \(0.509639\pi\)
\(4\) 1.95080 0.975401
\(5\) 0 0
\(6\) 0.208469 0.0851073
\(7\) 1.24528 0.470673 0.235336 0.971914i \(-0.424381\pi\)
0.235336 + 0.971914i \(0.424381\pi\)
\(8\) 0.0977875 0.0345731
\(9\) −2.98900 −0.996333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −0.204603 −0.0590639
\(13\) −2.45314 −0.680379 −0.340190 0.940357i \(-0.610491\pi\)
−0.340190 + 0.940357i \(0.610491\pi\)
\(14\) −2.47520 −0.661526
\(15\) 0 0
\(16\) −4.09597 −1.02399
\(17\) −8.07996 −1.95968 −0.979839 0.199789i \(-0.935974\pi\)
−0.979839 + 0.199789i \(0.935974\pi\)
\(18\) 5.94112 1.40034
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.130607 −0.0285009
\(22\) 1.98766 0.423771
\(23\) −0.595102 −0.124087 −0.0620437 0.998073i \(-0.519762\pi\)
−0.0620437 + 0.998073i \(0.519762\pi\)
\(24\) −0.0102561 −0.00209352
\(25\) 0 0
\(26\) 4.87602 0.956266
\(27\) 0.628136 0.120885
\(28\) 2.42930 0.459095
\(29\) −6.20473 −1.15219 −0.576095 0.817383i \(-0.695424\pi\)
−0.576095 + 0.817383i \(0.695424\pi\)
\(30\) 0 0
\(31\) −9.51628 −1.70918 −0.854588 0.519307i \(-0.826190\pi\)
−0.854588 + 0.519307i \(0.826190\pi\)
\(32\) 7.94584 1.40464
\(33\) 0.104882 0.0182576
\(34\) 16.0602 2.75431
\(35\) 0 0
\(36\) −5.83095 −0.971825
\(37\) 6.49334 1.06750 0.533749 0.845643i \(-0.320783\pi\)
0.533749 + 0.845643i \(0.320783\pi\)
\(38\) −1.98766 −0.322441
\(39\) 0.257290 0.0411993
\(40\) 0 0
\(41\) −8.06971 −1.26028 −0.630139 0.776483i \(-0.717002\pi\)
−0.630139 + 0.776483i \(0.717002\pi\)
\(42\) 0.259603 0.0400577
\(43\) −2.33313 −0.355800 −0.177900 0.984049i \(-0.556930\pi\)
−0.177900 + 0.984049i \(0.556930\pi\)
\(44\) −1.95080 −0.294095
\(45\) 0 0
\(46\) 1.18286 0.174403
\(47\) −8.72660 −1.27291 −0.636453 0.771316i \(-0.719599\pi\)
−0.636453 + 0.771316i \(0.719599\pi\)
\(48\) 0.429593 0.0620063
\(49\) −5.44927 −0.778467
\(50\) 0 0
\(51\) 0.847439 0.118665
\(52\) −4.78560 −0.663643
\(53\) 10.8618 1.49198 0.745990 0.665957i \(-0.231977\pi\)
0.745990 + 0.665957i \(0.231977\pi\)
\(54\) −1.24852 −0.169902
\(55\) 0 0
\(56\) 0.121773 0.0162726
\(57\) −0.104882 −0.0138919
\(58\) 12.3329 1.61939
\(59\) 0.601714 0.0783365 0.0391683 0.999233i \(-0.487529\pi\)
0.0391683 + 0.999233i \(0.487529\pi\)
\(60\) 0 0
\(61\) −1.23483 −0.158104 −0.0790522 0.996870i \(-0.525189\pi\)
−0.0790522 + 0.996870i \(0.525189\pi\)
\(62\) 18.9152 2.40223
\(63\) −3.72215 −0.468947
\(64\) −7.60170 −0.950213
\(65\) 0 0
\(66\) −0.208469 −0.0256608
\(67\) −5.79923 −0.708488 −0.354244 0.935153i \(-0.615262\pi\)
−0.354244 + 0.935153i \(0.615262\pi\)
\(68\) −15.7624 −1.91147
\(69\) 0.0624153 0.00751392
\(70\) 0 0
\(71\) 7.74139 0.918734 0.459367 0.888247i \(-0.348076\pi\)
0.459367 + 0.888247i \(0.348076\pi\)
\(72\) −0.292287 −0.0344463
\(73\) 13.3744 1.56536 0.782680 0.622424i \(-0.213852\pi\)
0.782680 + 0.622424i \(0.213852\pi\)
\(74\) −12.9066 −1.50036
\(75\) 0 0
\(76\) 1.95080 0.223772
\(77\) −1.24528 −0.141913
\(78\) −0.511405 −0.0579052
\(79\) 1.79880 0.202381 0.101190 0.994867i \(-0.467735\pi\)
0.101190 + 0.994867i \(0.467735\pi\)
\(80\) 0 0
\(81\) 8.90112 0.989013
\(82\) 16.0399 1.77131
\(83\) 15.9421 1.74987 0.874934 0.484242i \(-0.160905\pi\)
0.874934 + 0.484242i \(0.160905\pi\)
\(84\) −0.254789 −0.0277998
\(85\) 0 0
\(86\) 4.63748 0.500073
\(87\) 0.650763 0.0697691
\(88\) −0.0977875 −0.0104242
\(89\) −7.25863 −0.769414 −0.384707 0.923039i \(-0.625697\pi\)
−0.384707 + 0.923039i \(0.625697\pi\)
\(90\) 0 0
\(91\) −3.05486 −0.320236
\(92\) −1.16093 −0.121035
\(93\) 0.998084 0.103496
\(94\) 17.3455 1.78906
\(95\) 0 0
\(96\) −0.833373 −0.0850558
\(97\) 8.72541 0.885931 0.442966 0.896539i \(-0.353926\pi\)
0.442966 + 0.896539i \(0.353926\pi\)
\(98\) 10.8313 1.09413
\(99\) 2.98900 0.300406
\(100\) 0 0
\(101\) −5.79671 −0.576794 −0.288397 0.957511i \(-0.593122\pi\)
−0.288397 + 0.957511i \(0.593122\pi\)
\(102\) −1.68442 −0.166783
\(103\) 7.85458 0.773935 0.386967 0.922093i \(-0.373523\pi\)
0.386967 + 0.922093i \(0.373523\pi\)
\(104\) −0.239887 −0.0235228
\(105\) 0 0
\(106\) −21.5895 −2.09696
\(107\) −0.0994255 −0.00961183 −0.00480591 0.999988i \(-0.501530\pi\)
−0.00480591 + 0.999988i \(0.501530\pi\)
\(108\) 1.22537 0.117911
\(109\) 16.9318 1.62177 0.810885 0.585206i \(-0.198986\pi\)
0.810885 + 0.585206i \(0.198986\pi\)
\(110\) 0 0
\(111\) −0.681032 −0.0646407
\(112\) −5.10065 −0.481966
\(113\) 3.54408 0.333399 0.166699 0.986008i \(-0.446689\pi\)
0.166699 + 0.986008i \(0.446689\pi\)
\(114\) 0.208469 0.0195249
\(115\) 0 0
\(116\) −12.1042 −1.12385
\(117\) 7.33244 0.677885
\(118\) −1.19601 −0.110101
\(119\) −10.0618 −0.922367
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.45443 0.222214
\(123\) 0.846365 0.0763142
\(124\) −18.5644 −1.66713
\(125\) 0 0
\(126\) 7.39838 0.659100
\(127\) 8.94688 0.793907 0.396954 0.917839i \(-0.370067\pi\)
0.396954 + 0.917839i \(0.370067\pi\)
\(128\) −0.782063 −0.0691253
\(129\) 0.244703 0.0215449
\(130\) 0 0
\(131\) −16.5671 −1.44747 −0.723735 0.690078i \(-0.757576\pi\)
−0.723735 + 0.690078i \(0.757576\pi\)
\(132\) 0.204603 0.0178084
\(133\) 1.24528 0.107980
\(134\) 11.5269 0.995773
\(135\) 0 0
\(136\) −0.790119 −0.0677521
\(137\) −2.55929 −0.218655 −0.109327 0.994006i \(-0.534870\pi\)
−0.109327 + 0.994006i \(0.534870\pi\)
\(138\) −0.124061 −0.0105607
\(139\) 17.7342 1.50419 0.752096 0.659053i \(-0.229043\pi\)
0.752096 + 0.659053i \(0.229043\pi\)
\(140\) 0 0
\(141\) 0.915260 0.0770788
\(142\) −15.3873 −1.29127
\(143\) 2.45314 0.205142
\(144\) 12.2429 1.02024
\(145\) 0 0
\(146\) −26.5839 −2.20010
\(147\) 0.571529 0.0471389
\(148\) 12.6672 1.04124
\(149\) −9.38089 −0.768513 −0.384256 0.923226i \(-0.625542\pi\)
−0.384256 + 0.923226i \(0.625542\pi\)
\(150\) 0 0
\(151\) −12.7995 −1.04161 −0.520804 0.853676i \(-0.674368\pi\)
−0.520804 + 0.853676i \(0.674368\pi\)
\(152\) 0.0977875 0.00793161
\(153\) 24.1510 1.95249
\(154\) 2.47520 0.199457
\(155\) 0 0
\(156\) 0.501921 0.0401859
\(157\) −20.1863 −1.61104 −0.805520 0.592569i \(-0.798114\pi\)
−0.805520 + 0.592569i \(0.798114\pi\)
\(158\) −3.57540 −0.284444
\(159\) −1.13920 −0.0903445
\(160\) 0 0
\(161\) −0.741070 −0.0584045
\(162\) −17.6924 −1.39005
\(163\) 17.1940 1.34674 0.673368 0.739307i \(-0.264847\pi\)
0.673368 + 0.739307i \(0.264847\pi\)
\(164\) −15.7424 −1.22928
\(165\) 0 0
\(166\) −31.6874 −2.45942
\(167\) −10.6651 −0.825286 −0.412643 0.910893i \(-0.635394\pi\)
−0.412643 + 0.910893i \(0.635394\pi\)
\(168\) −0.0127718 −0.000985363 0
\(169\) −6.98209 −0.537084
\(170\) 0 0
\(171\) −2.98900 −0.228575
\(172\) −4.55148 −0.347047
\(173\) 9.39794 0.714512 0.357256 0.934007i \(-0.383712\pi\)
0.357256 + 0.934007i \(0.383712\pi\)
\(174\) −1.29350 −0.0980597
\(175\) 0 0
\(176\) 4.09597 0.308746
\(177\) −0.0631088 −0.00474355
\(178\) 14.4277 1.08140
\(179\) 12.3622 0.923997 0.461999 0.886881i \(-0.347132\pi\)
0.461999 + 0.886881i \(0.347132\pi\)
\(180\) 0 0
\(181\) 10.5514 0.784279 0.392139 0.919906i \(-0.371735\pi\)
0.392139 + 0.919906i \(0.371735\pi\)
\(182\) 6.07202 0.450088
\(183\) 0.129511 0.00957376
\(184\) −0.0581935 −0.00429008
\(185\) 0 0
\(186\) −1.98385 −0.145463
\(187\) 8.07996 0.590865
\(188\) −17.0239 −1.24159
\(189\) 0.782207 0.0568972
\(190\) 0 0
\(191\) 7.75577 0.561188 0.280594 0.959827i \(-0.409469\pi\)
0.280594 + 0.959827i \(0.409469\pi\)
\(192\) 0.797279 0.0575386
\(193\) −18.0868 −1.30192 −0.650958 0.759114i \(-0.725632\pi\)
−0.650958 + 0.759114i \(0.725632\pi\)
\(194\) −17.3432 −1.24517
\(195\) 0 0
\(196\) −10.6305 −0.759318
\(197\) 18.6144 1.32622 0.663110 0.748522i \(-0.269236\pi\)
0.663110 + 0.748522i \(0.269236\pi\)
\(198\) −5.94112 −0.422217
\(199\) −6.39608 −0.453406 −0.226703 0.973964i \(-0.572795\pi\)
−0.226703 + 0.973964i \(0.572795\pi\)
\(200\) 0 0
\(201\) 0.608232 0.0429014
\(202\) 11.5219 0.810678
\(203\) −7.72665 −0.542304
\(204\) 1.65319 0.115746
\(205\) 0 0
\(206\) −15.6123 −1.08776
\(207\) 1.77876 0.123632
\(208\) 10.0480 0.696704
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −18.2797 −1.25842 −0.629212 0.777234i \(-0.716622\pi\)
−0.629212 + 0.777234i \(0.716622\pi\)
\(212\) 21.1892 1.45528
\(213\) −0.811930 −0.0556325
\(214\) 0.197624 0.0135093
\(215\) 0 0
\(216\) 0.0614239 0.00417936
\(217\) −11.8505 −0.804462
\(218\) −33.6546 −2.27938
\(219\) −1.40273 −0.0947879
\(220\) 0 0
\(221\) 19.8213 1.33332
\(222\) 1.35366 0.0908519
\(223\) −10.9820 −0.735411 −0.367705 0.929942i \(-0.619856\pi\)
−0.367705 + 0.929942i \(0.619856\pi\)
\(224\) 9.89482 0.661125
\(225\) 0 0
\(226\) −7.04443 −0.468589
\(227\) −16.6800 −1.10709 −0.553545 0.832819i \(-0.686725\pi\)
−0.553545 + 0.832819i \(0.686725\pi\)
\(228\) −0.204603 −0.0135502
\(229\) 20.2154 1.33587 0.667935 0.744220i \(-0.267179\pi\)
0.667935 + 0.744220i \(0.267179\pi\)
\(230\) 0 0
\(231\) 0.130607 0.00859333
\(232\) −0.606745 −0.0398348
\(233\) −12.3865 −0.811468 −0.405734 0.913991i \(-0.632984\pi\)
−0.405734 + 0.913991i \(0.632984\pi\)
\(234\) −14.5744 −0.952760
\(235\) 0 0
\(236\) 1.17383 0.0764096
\(237\) −0.188661 −0.0122548
\(238\) 19.9995 1.29638
\(239\) −5.13989 −0.332472 −0.166236 0.986086i \(-0.553161\pi\)
−0.166236 + 0.986086i \(0.553161\pi\)
\(240\) 0 0
\(241\) 20.3113 1.30837 0.654183 0.756336i \(-0.273012\pi\)
0.654183 + 0.756336i \(0.273012\pi\)
\(242\) −1.98766 −0.127772
\(243\) −2.81797 −0.180773
\(244\) −2.40892 −0.154215
\(245\) 0 0
\(246\) −1.68229 −0.107259
\(247\) −2.45314 −0.156090
\(248\) −0.930573 −0.0590915
\(249\) −1.67203 −0.105961
\(250\) 0 0
\(251\) −20.0586 −1.26609 −0.633043 0.774117i \(-0.718194\pi\)
−0.633043 + 0.774117i \(0.718194\pi\)
\(252\) −7.26118 −0.457411
\(253\) 0.595102 0.0374137
\(254\) −17.7834 −1.11583
\(255\) 0 0
\(256\) 16.7579 1.04737
\(257\) −8.63146 −0.538416 −0.269208 0.963082i \(-0.586762\pi\)
−0.269208 + 0.963082i \(0.586762\pi\)
\(258\) −0.486387 −0.0302811
\(259\) 8.08605 0.502442
\(260\) 0 0
\(261\) 18.5460 1.14797
\(262\) 32.9297 2.03441
\(263\) 4.81636 0.296990 0.148495 0.988913i \(-0.452557\pi\)
0.148495 + 0.988913i \(0.452557\pi\)
\(264\) 0.0102561 0.000631220 0
\(265\) 0 0
\(266\) −2.47520 −0.151764
\(267\) 0.761297 0.0465906
\(268\) −11.3131 −0.691060
\(269\) 3.30933 0.201774 0.100887 0.994898i \(-0.467832\pi\)
0.100887 + 0.994898i \(0.467832\pi\)
\(270\) 0 0
\(271\) 0.697805 0.0423886 0.0211943 0.999775i \(-0.493253\pi\)
0.0211943 + 0.999775i \(0.493253\pi\)
\(272\) 33.0953 2.00670
\(273\) 0.320398 0.0193914
\(274\) 5.08700 0.307317
\(275\) 0 0
\(276\) 0.121760 0.00732908
\(277\) −4.85656 −0.291803 −0.145901 0.989299i \(-0.546608\pi\)
−0.145901 + 0.989299i \(0.546608\pi\)
\(278\) −35.2495 −2.11413
\(279\) 28.4442 1.70291
\(280\) 0 0
\(281\) 12.0878 0.721095 0.360548 0.932741i \(-0.382590\pi\)
0.360548 + 0.932741i \(0.382590\pi\)
\(282\) −1.81923 −0.108333
\(283\) 16.2537 0.966182 0.483091 0.875570i \(-0.339514\pi\)
0.483091 + 0.875570i \(0.339514\pi\)
\(284\) 15.1019 0.896134
\(285\) 0 0
\(286\) −4.87602 −0.288325
\(287\) −10.0491 −0.593178
\(288\) −23.7501 −1.39949
\(289\) 48.2857 2.84034
\(290\) 0 0
\(291\) −0.915136 −0.0536462
\(292\) 26.0909 1.52685
\(293\) 10.8945 0.636462 0.318231 0.948013i \(-0.396911\pi\)
0.318231 + 0.948013i \(0.396911\pi\)
\(294\) −1.13601 −0.0662532
\(295\) 0 0
\(296\) 0.634967 0.0369067
\(297\) −0.628136 −0.0364482
\(298\) 18.6460 1.08014
\(299\) 1.45987 0.0844265
\(300\) 0 0
\(301\) −2.90541 −0.167465
\(302\) 25.4411 1.46397
\(303\) 0.607968 0.0349269
\(304\) −4.09597 −0.234920
\(305\) 0 0
\(306\) −48.0040 −2.74421
\(307\) −1.11684 −0.0637414 −0.0318707 0.999492i \(-0.510146\pi\)
−0.0318707 + 0.999492i \(0.510146\pi\)
\(308\) −2.42930 −0.138422
\(309\) −0.823801 −0.0468644
\(310\) 0 0
\(311\) −31.2395 −1.77143 −0.885714 0.464231i \(-0.846331\pi\)
−0.885714 + 0.464231i \(0.846331\pi\)
\(312\) 0.0251597 0.00142439
\(313\) −15.9427 −0.901133 −0.450566 0.892743i \(-0.648778\pi\)
−0.450566 + 0.892743i \(0.648778\pi\)
\(314\) 40.1235 2.26430
\(315\) 0 0
\(316\) 3.50910 0.197402
\(317\) −10.2653 −0.576558 −0.288279 0.957546i \(-0.593083\pi\)
−0.288279 + 0.957546i \(0.593083\pi\)
\(318\) 2.26435 0.126978
\(319\) 6.20473 0.347398
\(320\) 0 0
\(321\) 0.0104279 0.000582029 0
\(322\) 1.47300 0.0820869
\(323\) −8.07996 −0.449581
\(324\) 17.3643 0.964685
\(325\) 0 0
\(326\) −34.1758 −1.89282
\(327\) −1.77583 −0.0982037
\(328\) −0.789117 −0.0435717
\(329\) −10.8671 −0.599122
\(330\) 0 0
\(331\) 22.2045 1.22047 0.610234 0.792221i \(-0.291075\pi\)
0.610234 + 0.792221i \(0.291075\pi\)
\(332\) 31.0998 1.70682
\(333\) −19.4086 −1.06358
\(334\) 21.1985 1.15993
\(335\) 0 0
\(336\) 0.534964 0.0291847
\(337\) −21.7331 −1.18388 −0.591938 0.805983i \(-0.701637\pi\)
−0.591938 + 0.805983i \(0.701637\pi\)
\(338\) 13.8780 0.754866
\(339\) −0.371709 −0.0201884
\(340\) 0 0
\(341\) 9.51628 0.515336
\(342\) 5.94112 0.321259
\(343\) −15.5029 −0.837076
\(344\) −0.228151 −0.0123011
\(345\) 0 0
\(346\) −18.6799 −1.00424
\(347\) 6.96031 0.373649 0.186824 0.982393i \(-0.440180\pi\)
0.186824 + 0.982393i \(0.440180\pi\)
\(348\) 1.26951 0.0680529
\(349\) −8.22249 −0.440140 −0.220070 0.975484i \(-0.570629\pi\)
−0.220070 + 0.975484i \(0.570629\pi\)
\(350\) 0 0
\(351\) −1.54091 −0.0822476
\(352\) −7.94584 −0.423515
\(353\) 22.9432 1.22114 0.610571 0.791962i \(-0.290940\pi\)
0.610571 + 0.791962i \(0.290940\pi\)
\(354\) 0.125439 0.00666701
\(355\) 0 0
\(356\) −14.1602 −0.750487
\(357\) 1.05530 0.0558525
\(358\) −24.5720 −1.29867
\(359\) −0.321493 −0.0169678 −0.00848388 0.999964i \(-0.502701\pi\)
−0.00848388 + 0.999964i \(0.502701\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −20.9726 −1.10230
\(363\) −0.104882 −0.00550486
\(364\) −5.95942 −0.312359
\(365\) 0 0
\(366\) −0.257425 −0.0134558
\(367\) 19.0521 0.994510 0.497255 0.867604i \(-0.334341\pi\)
0.497255 + 0.867604i \(0.334341\pi\)
\(368\) 2.43752 0.127065
\(369\) 24.1204 1.25566
\(370\) 0 0
\(371\) 13.5260 0.702234
\(372\) 1.94706 0.100951
\(373\) −5.23572 −0.271096 −0.135548 0.990771i \(-0.543279\pi\)
−0.135548 + 0.990771i \(0.543279\pi\)
\(374\) −16.0602 −0.830455
\(375\) 0 0
\(376\) −0.853352 −0.0440083
\(377\) 15.2211 0.783926
\(378\) −1.55476 −0.0799684
\(379\) −21.2726 −1.09270 −0.546349 0.837558i \(-0.683983\pi\)
−0.546349 + 0.837558i \(0.683983\pi\)
\(380\) 0 0
\(381\) −0.938364 −0.0480738
\(382\) −15.4159 −0.788744
\(383\) −17.1531 −0.876482 −0.438241 0.898857i \(-0.644398\pi\)
−0.438241 + 0.898857i \(0.644398\pi\)
\(384\) 0.0820241 0.00418577
\(385\) 0 0
\(386\) 35.9504 1.82983
\(387\) 6.97374 0.354495
\(388\) 17.0216 0.864139
\(389\) 16.3320 0.828065 0.414033 0.910262i \(-0.364120\pi\)
0.414033 + 0.910262i \(0.364120\pi\)
\(390\) 0 0
\(391\) 4.80840 0.243171
\(392\) −0.532870 −0.0269140
\(393\) 1.73758 0.0876494
\(394\) −36.9991 −1.86399
\(395\) 0 0
\(396\) 5.83095 0.293016
\(397\) 35.5061 1.78200 0.891000 0.454003i \(-0.150004\pi\)
0.891000 + 0.454003i \(0.150004\pi\)
\(398\) 12.7133 0.637258
\(399\) −0.130607 −0.00653854
\(400\) 0 0
\(401\) 30.9517 1.54565 0.772827 0.634617i \(-0.218842\pi\)
0.772827 + 0.634617i \(0.218842\pi\)
\(402\) −1.20896 −0.0602975
\(403\) 23.3448 1.16289
\(404\) −11.3082 −0.562606
\(405\) 0 0
\(406\) 15.3580 0.762203
\(407\) −6.49334 −0.321863
\(408\) 0.0828690 0.00410262
\(409\) −3.57490 −0.176767 −0.0883837 0.996087i \(-0.528170\pi\)
−0.0883837 + 0.996087i \(0.528170\pi\)
\(410\) 0 0
\(411\) 0.268423 0.0132403
\(412\) 15.3227 0.754897
\(413\) 0.749304 0.0368709
\(414\) −3.53557 −0.173764
\(415\) 0 0
\(416\) −19.4923 −0.955687
\(417\) −1.85999 −0.0910840
\(418\) 1.98766 0.0972198
\(419\) 21.8193 1.06594 0.532972 0.846133i \(-0.321075\pi\)
0.532972 + 0.846133i \(0.321075\pi\)
\(420\) 0 0
\(421\) 36.1289 1.76081 0.880407 0.474220i \(-0.157270\pi\)
0.880407 + 0.474220i \(0.157270\pi\)
\(422\) 36.3338 1.76870
\(423\) 26.0838 1.26824
\(424\) 1.06215 0.0515824
\(425\) 0 0
\(426\) 1.61384 0.0781909
\(427\) −1.53772 −0.0744154
\(428\) −0.193960 −0.00937539
\(429\) −0.257290 −0.0124221
\(430\) 0 0
\(431\) −26.0067 −1.25270 −0.626349 0.779542i \(-0.715452\pi\)
−0.626349 + 0.779542i \(0.715452\pi\)
\(432\) −2.57283 −0.123785
\(433\) −32.2109 −1.54796 −0.773978 0.633212i \(-0.781736\pi\)
−0.773978 + 0.633212i \(0.781736\pi\)
\(434\) 23.5547 1.13066
\(435\) 0 0
\(436\) 33.0305 1.58188
\(437\) −0.595102 −0.0284676
\(438\) 2.78816 0.133223
\(439\) −36.1699 −1.72630 −0.863149 0.504950i \(-0.831511\pi\)
−0.863149 + 0.504950i \(0.831511\pi\)
\(440\) 0 0
\(441\) 16.2879 0.775613
\(442\) −39.3980 −1.87397
\(443\) −6.35992 −0.302169 −0.151084 0.988521i \(-0.548277\pi\)
−0.151084 + 0.988521i \(0.548277\pi\)
\(444\) −1.32856 −0.0630507
\(445\) 0 0
\(446\) 21.8286 1.03361
\(447\) 0.983883 0.0465361
\(448\) −9.46627 −0.447239
\(449\) 29.0551 1.37120 0.685598 0.727981i \(-0.259541\pi\)
0.685598 + 0.727981i \(0.259541\pi\)
\(450\) 0 0
\(451\) 8.06971 0.379988
\(452\) 6.91380 0.325198
\(453\) 1.34243 0.0630729
\(454\) 33.1542 1.55600
\(455\) 0 0
\(456\) −0.0102561 −0.000480286 0
\(457\) 18.2529 0.853832 0.426916 0.904291i \(-0.359600\pi\)
0.426916 + 0.904291i \(0.359600\pi\)
\(458\) −40.1813 −1.87755
\(459\) −5.07531 −0.236895
\(460\) 0 0
\(461\) 22.6967 1.05709 0.528545 0.848905i \(-0.322738\pi\)
0.528545 + 0.848905i \(0.322738\pi\)
\(462\) −0.259603 −0.0120778
\(463\) 25.3667 1.17889 0.589446 0.807808i \(-0.299346\pi\)
0.589446 + 0.807808i \(0.299346\pi\)
\(464\) 25.4144 1.17984
\(465\) 0 0
\(466\) 24.6202 1.14051
\(467\) −35.1641 −1.62720 −0.813601 0.581423i \(-0.802496\pi\)
−0.813601 + 0.581423i \(0.802496\pi\)
\(468\) 14.3041 0.661210
\(469\) −7.22168 −0.333466
\(470\) 0 0
\(471\) 2.11717 0.0975540
\(472\) 0.0588401 0.00270834
\(473\) 2.33313 0.107278
\(474\) 0.374994 0.0172241
\(475\) 0 0
\(476\) −19.6287 −0.899678
\(477\) −32.4658 −1.48651
\(478\) 10.2164 0.467286
\(479\) −19.4028 −0.886536 −0.443268 0.896389i \(-0.646181\pi\)
−0.443268 + 0.896389i \(0.646181\pi\)
\(480\) 0 0
\(481\) −15.9291 −0.726304
\(482\) −40.3720 −1.83890
\(483\) 0.0777247 0.00353659
\(484\) 1.95080 0.0886729
\(485\) 0 0
\(486\) 5.60118 0.254075
\(487\) 25.6957 1.16438 0.582191 0.813052i \(-0.302196\pi\)
0.582191 + 0.813052i \(0.302196\pi\)
\(488\) −0.120751 −0.00546616
\(489\) −1.80333 −0.0815496
\(490\) 0 0
\(491\) −4.76339 −0.214969 −0.107484 0.994207i \(-0.534280\pi\)
−0.107484 + 0.994207i \(0.534280\pi\)
\(492\) 1.65109 0.0744369
\(493\) 50.1340 2.25792
\(494\) 4.87602 0.219382
\(495\) 0 0
\(496\) 38.9785 1.75018
\(497\) 9.64022 0.432423
\(498\) 3.32343 0.148926
\(499\) 26.8553 1.20221 0.601104 0.799171i \(-0.294728\pi\)
0.601104 + 0.799171i \(0.294728\pi\)
\(500\) 0 0
\(501\) 1.11857 0.0499739
\(502\) 39.8697 1.77947
\(503\) −2.37364 −0.105836 −0.0529178 0.998599i \(-0.516852\pi\)
−0.0529178 + 0.998599i \(0.516852\pi\)
\(504\) −0.363980 −0.0162129
\(505\) 0 0
\(506\) −1.18286 −0.0525846
\(507\) 0.732293 0.0325223
\(508\) 17.4536 0.774378
\(509\) −13.3350 −0.591063 −0.295531 0.955333i \(-0.595497\pi\)
−0.295531 + 0.955333i \(0.595497\pi\)
\(510\) 0 0
\(511\) 16.6550 0.736772
\(512\) −31.7449 −1.40294
\(513\) 0.628136 0.0277329
\(514\) 17.1564 0.756738
\(515\) 0 0
\(516\) 0.477367 0.0210149
\(517\) 8.72660 0.383795
\(518\) −16.0723 −0.706178
\(519\) −0.985671 −0.0432662
\(520\) 0 0
\(521\) −30.9479 −1.35585 −0.677926 0.735131i \(-0.737121\pi\)
−0.677926 + 0.735131i \(0.737121\pi\)
\(522\) −36.8631 −1.61345
\(523\) 17.2379 0.753762 0.376881 0.926262i \(-0.376996\pi\)
0.376881 + 0.926262i \(0.376996\pi\)
\(524\) −32.3191 −1.41187
\(525\) 0 0
\(526\) −9.57330 −0.417416
\(527\) 76.8912 3.34943
\(528\) −0.429593 −0.0186956
\(529\) −22.6459 −0.984602
\(530\) 0 0
\(531\) −1.79852 −0.0780493
\(532\) 2.42930 0.105324
\(533\) 19.7962 0.857467
\(534\) −1.51320 −0.0654827
\(535\) 0 0
\(536\) −0.567092 −0.0244946
\(537\) −1.29657 −0.0559512
\(538\) −6.57784 −0.283591
\(539\) 5.44927 0.234717
\(540\) 0 0
\(541\) −28.2470 −1.21443 −0.607217 0.794536i \(-0.707714\pi\)
−0.607217 + 0.794536i \(0.707714\pi\)
\(542\) −1.38700 −0.0595768
\(543\) −1.10665 −0.0474908
\(544\) −64.2021 −2.75264
\(545\) 0 0
\(546\) −0.636844 −0.0272544
\(547\) −7.04078 −0.301042 −0.150521 0.988607i \(-0.548095\pi\)
−0.150521 + 0.988607i \(0.548095\pi\)
\(548\) −4.99267 −0.213276
\(549\) 3.69092 0.157525
\(550\) 0 0
\(551\) −6.20473 −0.264331
\(552\) 0.00610343 0.000259779 0
\(553\) 2.24001 0.0952550
\(554\) 9.65321 0.410126
\(555\) 0 0
\(556\) 34.5959 1.46719
\(557\) −26.5954 −1.12688 −0.563441 0.826156i \(-0.690523\pi\)
−0.563441 + 0.826156i \(0.690523\pi\)
\(558\) −56.5374 −2.39342
\(559\) 5.72351 0.242079
\(560\) 0 0
\(561\) −0.847439 −0.0357789
\(562\) −24.0264 −1.01349
\(563\) 3.77560 0.159122 0.0795612 0.996830i \(-0.474648\pi\)
0.0795612 + 0.996830i \(0.474648\pi\)
\(564\) 1.78549 0.0751828
\(565\) 0 0
\(566\) −32.3069 −1.35796
\(567\) 11.0844 0.465501
\(568\) 0.757011 0.0317635
\(569\) 33.4474 1.40219 0.701094 0.713069i \(-0.252695\pi\)
0.701094 + 0.713069i \(0.252695\pi\)
\(570\) 0 0
\(571\) −12.1608 −0.508914 −0.254457 0.967084i \(-0.581897\pi\)
−0.254457 + 0.967084i \(0.581897\pi\)
\(572\) 4.78560 0.200096
\(573\) −0.813438 −0.0339819
\(574\) 19.9742 0.833706
\(575\) 0 0
\(576\) 22.7215 0.946728
\(577\) 8.99197 0.374341 0.187170 0.982327i \(-0.440068\pi\)
0.187170 + 0.982327i \(0.440068\pi\)
\(578\) −95.9757 −3.99206
\(579\) 1.89697 0.0788355
\(580\) 0 0
\(581\) 19.8524 0.823615
\(582\) 1.81898 0.0753992
\(583\) −10.8618 −0.449849
\(584\) 1.30785 0.0541193
\(585\) 0 0
\(586\) −21.6545 −0.894541
\(587\) 34.4915 1.42362 0.711809 0.702373i \(-0.247876\pi\)
0.711809 + 0.702373i \(0.247876\pi\)
\(588\) 1.11494 0.0459793
\(589\) −9.51628 −0.392112
\(590\) 0 0
\(591\) −1.95231 −0.0803072
\(592\) −26.5966 −1.09311
\(593\) −13.9176 −0.571527 −0.285763 0.958300i \(-0.592247\pi\)
−0.285763 + 0.958300i \(0.592247\pi\)
\(594\) 1.24852 0.0512275
\(595\) 0 0
\(596\) −18.3003 −0.749608
\(597\) 0.670832 0.0274553
\(598\) −2.90173 −0.118661
\(599\) −19.8981 −0.813014 −0.406507 0.913648i \(-0.633253\pi\)
−0.406507 + 0.913648i \(0.633253\pi\)
\(600\) 0 0
\(601\) −35.9397 −1.46601 −0.733005 0.680224i \(-0.761883\pi\)
−0.733005 + 0.680224i \(0.761883\pi\)
\(602\) 5.77498 0.235371
\(603\) 17.3339 0.705890
\(604\) −24.9693 −1.01599
\(605\) 0 0
\(606\) −1.20844 −0.0490894
\(607\) 3.78275 0.153537 0.0767686 0.997049i \(-0.475540\pi\)
0.0767686 + 0.997049i \(0.475540\pi\)
\(608\) 7.94584 0.322246
\(609\) 0.810384 0.0328384
\(610\) 0 0
\(611\) 21.4076 0.866058
\(612\) 47.1138 1.90446
\(613\) −1.87040 −0.0755446 −0.0377723 0.999286i \(-0.512026\pi\)
−0.0377723 + 0.999286i \(0.512026\pi\)
\(614\) 2.21990 0.0895879
\(615\) 0 0
\(616\) −0.121773 −0.00490638
\(617\) 48.3584 1.94683 0.973417 0.229040i \(-0.0735585\pi\)
0.973417 + 0.229040i \(0.0735585\pi\)
\(618\) 1.63744 0.0658674
\(619\) 24.5105 0.985162 0.492581 0.870267i \(-0.336054\pi\)
0.492581 + 0.870267i \(0.336054\pi\)
\(620\) 0 0
\(621\) −0.373805 −0.0150003
\(622\) 62.0935 2.48972
\(623\) −9.03905 −0.362142
\(624\) −1.05385 −0.0421878
\(625\) 0 0
\(626\) 31.6886 1.26653
\(627\) 0.104882 0.00418857
\(628\) −39.3794 −1.57141
\(629\) −52.4659 −2.09195
\(630\) 0 0
\(631\) 30.7812 1.22538 0.612691 0.790323i \(-0.290087\pi\)
0.612691 + 0.790323i \(0.290087\pi\)
\(632\) 0.175900 0.00699692
\(633\) 1.91720 0.0762019
\(634\) 20.4040 0.810346
\(635\) 0 0
\(636\) −2.22236 −0.0881222
\(637\) 13.3678 0.529653
\(638\) −12.3329 −0.488265
\(639\) −23.1390 −0.915365
\(640\) 0 0
\(641\) 18.2583 0.721159 0.360579 0.932729i \(-0.382579\pi\)
0.360579 + 0.932729i \(0.382579\pi\)
\(642\) −0.0207272 −0.000818036 0
\(643\) 24.9299 0.983140 0.491570 0.870838i \(-0.336423\pi\)
0.491570 + 0.870838i \(0.336423\pi\)
\(644\) −1.44568 −0.0569678
\(645\) 0 0
\(646\) 16.0602 0.631881
\(647\) −10.0526 −0.395209 −0.197604 0.980282i \(-0.563316\pi\)
−0.197604 + 0.980282i \(0.563316\pi\)
\(648\) 0.870418 0.0341932
\(649\) −0.601714 −0.0236194
\(650\) 0 0
\(651\) 1.24290 0.0487129
\(652\) 33.5421 1.31361
\(653\) 3.99482 0.156330 0.0781648 0.996940i \(-0.475094\pi\)
0.0781648 + 0.996940i \(0.475094\pi\)
\(654\) 3.52975 0.138024
\(655\) 0 0
\(656\) 33.0533 1.29052
\(657\) −39.9762 −1.55962
\(658\) 21.6001 0.842059
\(659\) −16.1838 −0.630430 −0.315215 0.949020i \(-0.602077\pi\)
−0.315215 + 0.949020i \(0.602077\pi\)
\(660\) 0 0
\(661\) 27.2908 1.06149 0.530744 0.847532i \(-0.321912\pi\)
0.530744 + 0.847532i \(0.321912\pi\)
\(662\) −44.1350 −1.71536
\(663\) −2.07889 −0.0807374
\(664\) 1.55893 0.0604983
\(665\) 0 0
\(666\) 38.5777 1.49486
\(667\) 3.69245 0.142972
\(668\) −20.8054 −0.804986
\(669\) 1.15181 0.0445317
\(670\) 0 0
\(671\) 1.23483 0.0476703
\(672\) −1.03778 −0.0400334
\(673\) 23.9335 0.922568 0.461284 0.887252i \(-0.347389\pi\)
0.461284 + 0.887252i \(0.347389\pi\)
\(674\) 43.1980 1.66393
\(675\) 0 0
\(676\) −13.6207 −0.523873
\(677\) −4.23402 −0.162727 −0.0813633 0.996685i \(-0.525927\pi\)
−0.0813633 + 0.996685i \(0.525927\pi\)
\(678\) 0.738832 0.0283747
\(679\) 10.8656 0.416984
\(680\) 0 0
\(681\) 1.74942 0.0670381
\(682\) −18.9152 −0.724299
\(683\) 14.4350 0.552341 0.276170 0.961109i \(-0.410935\pi\)
0.276170 + 0.961109i \(0.410935\pi\)
\(684\) −5.83095 −0.222952
\(685\) 0 0
\(686\) 30.8145 1.17650
\(687\) −2.12022 −0.0808915
\(688\) 9.55646 0.364336
\(689\) −26.6455 −1.01511
\(690\) 0 0
\(691\) 48.4203 1.84200 0.920998 0.389568i \(-0.127376\pi\)
0.920998 + 0.389568i \(0.127376\pi\)
\(692\) 18.3335 0.696936
\(693\) 3.72215 0.141393
\(694\) −13.8347 −0.525159
\(695\) 0 0
\(696\) 0.0636365 0.00241213
\(697\) 65.2030 2.46974
\(698\) 16.3435 0.618612
\(699\) 1.29912 0.0491372
\(700\) 0 0
\(701\) 22.7724 0.860103 0.430052 0.902804i \(-0.358495\pi\)
0.430052 + 0.902804i \(0.358495\pi\)
\(702\) 3.06280 0.115598
\(703\) 6.49334 0.244901
\(704\) 7.60170 0.286500
\(705\) 0 0
\(706\) −45.6033 −1.71630
\(707\) −7.21854 −0.271481
\(708\) −0.123113 −0.00462686
\(709\) 32.9979 1.23926 0.619631 0.784893i \(-0.287282\pi\)
0.619631 + 0.784893i \(0.287282\pi\)
\(710\) 0 0
\(711\) −5.37661 −0.201639
\(712\) −0.709803 −0.0266010
\(713\) 5.66316 0.212087
\(714\) −2.09758 −0.0785001
\(715\) 0 0
\(716\) 24.1163 0.901268
\(717\) 0.539080 0.0201323
\(718\) 0.639020 0.0238480
\(719\) −25.3585 −0.945712 −0.472856 0.881140i \(-0.656777\pi\)
−0.472856 + 0.881140i \(0.656777\pi\)
\(720\) 0 0
\(721\) 9.78117 0.364270
\(722\) −1.98766 −0.0739731
\(723\) −2.13028 −0.0792261
\(724\) 20.5837 0.764986
\(725\) 0 0
\(726\) 0.208469 0.00773702
\(727\) −37.2296 −1.38077 −0.690385 0.723442i \(-0.742559\pi\)
−0.690385 + 0.723442i \(0.742559\pi\)
\(728\) −0.298727 −0.0110715
\(729\) −26.4078 −0.978067
\(730\) 0 0
\(731\) 18.8516 0.697253
\(732\) 0.252651 0.00933826
\(733\) 48.0618 1.77520 0.887601 0.460614i \(-0.152371\pi\)
0.887601 + 0.460614i \(0.152371\pi\)
\(734\) −37.8691 −1.39777
\(735\) 0 0
\(736\) −4.72859 −0.174298
\(737\) 5.79923 0.213617
\(738\) −47.9432 −1.76481
\(739\) 39.7257 1.46133 0.730667 0.682734i \(-0.239209\pi\)
0.730667 + 0.682734i \(0.239209\pi\)
\(740\) 0 0
\(741\) 0.257290 0.00945177
\(742\) −26.8851 −0.986983
\(743\) −35.6645 −1.30840 −0.654202 0.756320i \(-0.726995\pi\)
−0.654202 + 0.756320i \(0.726995\pi\)
\(744\) 0.0976001 0.00357819
\(745\) 0 0
\(746\) 10.4069 0.381022
\(747\) −47.6508 −1.74345
\(748\) 15.7624 0.576331
\(749\) −0.123813 −0.00452402
\(750\) 0 0
\(751\) −35.0581 −1.27929 −0.639645 0.768671i \(-0.720918\pi\)
−0.639645 + 0.768671i \(0.720918\pi\)
\(752\) 35.7439 1.30345
\(753\) 2.10378 0.0766658
\(754\) −30.2544 −1.10180
\(755\) 0 0
\(756\) 1.52593 0.0554976
\(757\) 42.8016 1.55565 0.777826 0.628480i \(-0.216323\pi\)
0.777826 + 0.628480i \(0.216323\pi\)
\(758\) 42.2827 1.53578
\(759\) −0.0624153 −0.00226553
\(760\) 0 0
\(761\) −12.6482 −0.458497 −0.229249 0.973368i \(-0.573627\pi\)
−0.229249 + 0.973368i \(0.573627\pi\)
\(762\) 1.86515 0.0675673
\(763\) 21.0848 0.763322
\(764\) 15.1300 0.547384
\(765\) 0 0
\(766\) 34.0946 1.23189
\(767\) −1.47609 −0.0532986
\(768\) −1.75759 −0.0634217
\(769\) −9.67462 −0.348876 −0.174438 0.984668i \(-0.555811\pi\)
−0.174438 + 0.984668i \(0.555811\pi\)
\(770\) 0 0
\(771\) 0.905282 0.0326029
\(772\) −35.2838 −1.26989
\(773\) −31.1687 −1.12106 −0.560531 0.828133i \(-0.689403\pi\)
−0.560531 + 0.828133i \(0.689403\pi\)
\(774\) −13.8614 −0.498239
\(775\) 0 0
\(776\) 0.853236 0.0306294
\(777\) −0.848078 −0.0304246
\(778\) −32.4625 −1.16384
\(779\) −8.06971 −0.289128
\(780\) 0 0
\(781\) −7.74139 −0.277009
\(782\) −9.55748 −0.341775
\(783\) −3.89742 −0.139282
\(784\) 22.3201 0.797145
\(785\) 0 0
\(786\) −3.45372 −0.123190
\(787\) 44.4253 1.58359 0.791795 0.610787i \(-0.209147\pi\)
0.791795 + 0.610787i \(0.209147\pi\)
\(788\) 36.3130 1.29360
\(789\) −0.505148 −0.0179837
\(790\) 0 0
\(791\) 4.41338 0.156922
\(792\) 0.292287 0.0103860
\(793\) 3.02923 0.107571
\(794\) −70.5742 −2.50458
\(795\) 0 0
\(796\) −12.4775 −0.442253
\(797\) 20.1652 0.714288 0.357144 0.934049i \(-0.383750\pi\)
0.357144 + 0.934049i \(0.383750\pi\)
\(798\) 0.259603 0.00918986
\(799\) 70.5106 2.49448
\(800\) 0 0
\(801\) 21.6961 0.766592
\(802\) −61.5215 −2.17240
\(803\) −13.3744 −0.471974
\(804\) 1.18654 0.0418461
\(805\) 0 0
\(806\) −46.4016 −1.63443
\(807\) −0.347088 −0.0122181
\(808\) −0.566846 −0.0199416
\(809\) −32.6594 −1.14824 −0.574121 0.818770i \(-0.694656\pi\)
−0.574121 + 0.818770i \(0.694656\pi\)
\(810\) 0 0
\(811\) 45.7373 1.60606 0.803028 0.595942i \(-0.203221\pi\)
0.803028 + 0.595942i \(0.203221\pi\)
\(812\) −15.0732 −0.528964
\(813\) −0.0731870 −0.00256678
\(814\) 12.9066 0.452375
\(815\) 0 0
\(816\) −3.47109 −0.121512
\(817\) −2.33313 −0.0816260
\(818\) 7.10569 0.248445
\(819\) 9.13096 0.319062
\(820\) 0 0
\(821\) 13.5002 0.471159 0.235580 0.971855i \(-0.424301\pi\)
0.235580 + 0.971855i \(0.424301\pi\)
\(822\) −0.533533 −0.0186091
\(823\) 45.8337 1.59766 0.798831 0.601556i \(-0.205452\pi\)
0.798831 + 0.601556i \(0.205452\pi\)
\(824\) 0.768079 0.0267573
\(825\) 0 0
\(826\) −1.48936 −0.0518216
\(827\) −48.5796 −1.68928 −0.844639 0.535337i \(-0.820185\pi\)
−0.844639 + 0.535337i \(0.820185\pi\)
\(828\) 3.47001 0.120591
\(829\) −23.5476 −0.817842 −0.408921 0.912570i \(-0.634095\pi\)
−0.408921 + 0.912570i \(0.634095\pi\)
\(830\) 0 0
\(831\) 0.509364 0.0176697
\(832\) 18.6481 0.646505
\(833\) 44.0299 1.52555
\(834\) 3.69703 0.128018
\(835\) 0 0
\(836\) −1.95080 −0.0674699
\(837\) −5.97752 −0.206613
\(838\) −43.3695 −1.49817
\(839\) −30.7980 −1.06327 −0.531633 0.846975i \(-0.678421\pi\)
−0.531633 + 0.846975i \(0.678421\pi\)
\(840\) 0 0
\(841\) 9.49873 0.327543
\(842\) −71.8120 −2.47480
\(843\) −1.26778 −0.0436648
\(844\) −35.6600 −1.22747
\(845\) 0 0
\(846\) −51.8458 −1.78250
\(847\) 1.24528 0.0427884
\(848\) −44.4896 −1.52778
\(849\) −1.70471 −0.0585056
\(850\) 0 0
\(851\) −3.86420 −0.132463
\(852\) −1.58391 −0.0542640
\(853\) −8.62823 −0.295425 −0.147712 0.989030i \(-0.547191\pi\)
−0.147712 + 0.989030i \(0.547191\pi\)
\(854\) 3.05646 0.104590
\(855\) 0 0
\(856\) −0.00972257 −0.000332311 0
\(857\) −47.7829 −1.63223 −0.816117 0.577887i \(-0.803877\pi\)
−0.816117 + 0.577887i \(0.803877\pi\)
\(858\) 0.511405 0.0174591
\(859\) −38.9923 −1.33040 −0.665200 0.746665i \(-0.731654\pi\)
−0.665200 + 0.746665i \(0.731654\pi\)
\(860\) 0 0
\(861\) 1.05396 0.0359190
\(862\) 51.6926 1.76066
\(863\) −34.7056 −1.18139 −0.590697 0.806894i \(-0.701147\pi\)
−0.590697 + 0.806894i \(0.701147\pi\)
\(864\) 4.99107 0.169800
\(865\) 0 0
\(866\) 64.0244 2.17564
\(867\) −5.06429 −0.171992
\(868\) −23.1179 −0.784673
\(869\) −1.79880 −0.0610201
\(870\) 0 0
\(871\) 14.2263 0.482041
\(872\) 1.65571 0.0560696
\(873\) −26.0803 −0.882683
\(874\) 1.18286 0.0400109
\(875\) 0 0
\(876\) −2.73646 −0.0924563
\(877\) −14.3162 −0.483422 −0.241711 0.970348i \(-0.577709\pi\)
−0.241711 + 0.970348i \(0.577709\pi\)
\(878\) 71.8936 2.42629
\(879\) −1.14263 −0.0385400
\(880\) 0 0
\(881\) 0.516699 0.0174080 0.00870402 0.999962i \(-0.497229\pi\)
0.00870402 + 0.999962i \(0.497229\pi\)
\(882\) −32.3748 −1.09012
\(883\) 51.8049 1.74337 0.871687 0.490063i \(-0.163026\pi\)
0.871687 + 0.490063i \(0.163026\pi\)
\(884\) 38.6674 1.30053
\(885\) 0 0
\(886\) 12.6414 0.424695
\(887\) 14.6407 0.491585 0.245793 0.969322i \(-0.420952\pi\)
0.245793 + 0.969322i \(0.420952\pi\)
\(888\) −0.0665964 −0.00223483
\(889\) 11.1414 0.373670
\(890\) 0 0
\(891\) −8.90112 −0.298199
\(892\) −21.4238 −0.717321
\(893\) −8.72660 −0.292025
\(894\) −1.95563 −0.0654060
\(895\) 0 0
\(896\) −0.973890 −0.0325354
\(897\) −0.153114 −0.00511231
\(898\) −57.7518 −1.92720
\(899\) 59.0460 1.96930
\(900\) 0 0
\(901\) −87.7627 −2.92380
\(902\) −16.0399 −0.534069
\(903\) 0.304724 0.0101406
\(904\) 0.346566 0.0115266
\(905\) 0 0
\(906\) −2.66830 −0.0886484
\(907\) 13.4655 0.447114 0.223557 0.974691i \(-0.428233\pi\)
0.223557 + 0.974691i \(0.428233\pi\)
\(908\) −32.5394 −1.07986
\(909\) 17.3264 0.574679
\(910\) 0 0
\(911\) −3.43920 −0.113946 −0.0569729 0.998376i \(-0.518145\pi\)
−0.0569729 + 0.998376i \(0.518145\pi\)
\(912\) 0.429593 0.0142252
\(913\) −15.9421 −0.527605
\(914\) −36.2805 −1.20005
\(915\) 0 0
\(916\) 39.4362 1.30301
\(917\) −20.6307 −0.681285
\(918\) 10.0880 0.332954
\(919\) 46.3892 1.53024 0.765120 0.643888i \(-0.222680\pi\)
0.765120 + 0.643888i \(0.222680\pi\)
\(920\) 0 0
\(921\) 0.117136 0.00385976
\(922\) −45.1134 −1.48573
\(923\) −18.9907 −0.625087
\(924\) 0.254789 0.00838195
\(925\) 0 0
\(926\) −50.4205 −1.65692
\(927\) −23.4773 −0.771097
\(928\) −49.3018 −1.61841
\(929\) −29.7987 −0.977665 −0.488832 0.872378i \(-0.662577\pi\)
−0.488832 + 0.872378i \(0.662577\pi\)
\(930\) 0 0
\(931\) −5.44927 −0.178593
\(932\) −24.1636 −0.791507
\(933\) 3.27645 0.107266
\(934\) 69.8945 2.28702
\(935\) 0 0
\(936\) 0.717021 0.0234366
\(937\) −43.2661 −1.41344 −0.706720 0.707493i \(-0.749826\pi\)
−0.706720 + 0.707493i \(0.749826\pi\)
\(938\) 14.3543 0.468683
\(939\) 1.67209 0.0545667
\(940\) 0 0
\(941\) 25.8405 0.842377 0.421189 0.906973i \(-0.361613\pi\)
0.421189 + 0.906973i \(0.361613\pi\)
\(942\) −4.20822 −0.137111
\(943\) 4.80230 0.156385
\(944\) −2.46461 −0.0802161
\(945\) 0 0
\(946\) −4.63748 −0.150778
\(947\) 42.8107 1.39116 0.695580 0.718449i \(-0.255147\pi\)
0.695580 + 0.718449i \(0.255147\pi\)
\(948\) −0.368040 −0.0119534
\(949\) −32.8094 −1.06504
\(950\) 0 0
\(951\) 1.07664 0.0349126
\(952\) −0.983921 −0.0318891
\(953\) 35.0067 1.13398 0.566990 0.823725i \(-0.308108\pi\)
0.566990 + 0.823725i \(0.308108\pi\)
\(954\) 64.5312 2.08927
\(955\) 0 0
\(956\) −10.0269 −0.324293
\(957\) −0.650763 −0.0210362
\(958\) 38.5662 1.24602
\(959\) −3.18704 −0.102915
\(960\) 0 0
\(961\) 59.5597 1.92128
\(962\) 31.6617 1.02081
\(963\) 0.297183 0.00957658
\(964\) 39.6234 1.27618
\(965\) 0 0
\(966\) −0.154490 −0.00497065
\(967\) −2.63095 −0.0846057 −0.0423028 0.999105i \(-0.513469\pi\)
−0.0423028 + 0.999105i \(0.513469\pi\)
\(968\) 0.0977875 0.00314301
\(969\) 0.847439 0.0272237
\(970\) 0 0
\(971\) 26.6425 0.854999 0.427499 0.904016i \(-0.359395\pi\)
0.427499 + 0.904016i \(0.359395\pi\)
\(972\) −5.49731 −0.176326
\(973\) 22.0840 0.707982
\(974\) −51.0743 −1.63653
\(975\) 0 0
\(976\) 5.05785 0.161898
\(977\) −33.7219 −1.07886 −0.539429 0.842031i \(-0.681360\pi\)
−0.539429 + 0.842031i \(0.681360\pi\)
\(978\) 3.58442 0.114617
\(979\) 7.25863 0.231987
\(980\) 0 0
\(981\) −50.6090 −1.61582
\(982\) 9.46802 0.302137
\(983\) −6.93604 −0.221225 −0.110613 0.993864i \(-0.535281\pi\)
−0.110613 + 0.993864i \(0.535281\pi\)
\(984\) 0.0827639 0.00263842
\(985\) 0 0
\(986\) −99.6495 −3.17349
\(987\) 1.13976 0.0362789
\(988\) −4.78560 −0.152250
\(989\) 1.38845 0.0441502
\(990\) 0 0
\(991\) 15.8403 0.503183 0.251591 0.967834i \(-0.419046\pi\)
0.251591 + 0.967834i \(0.419046\pi\)
\(992\) −75.6149 −2.40077
\(993\) −2.32884 −0.0739036
\(994\) −19.1615 −0.607766
\(995\) 0 0
\(996\) −3.26180 −0.103354
\(997\) −53.3663 −1.69013 −0.845064 0.534665i \(-0.820438\pi\)
−0.845064 + 0.534665i \(0.820438\pi\)
\(998\) −53.3792 −1.68969
\(999\) 4.07870 0.129044
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.u.1.3 15
5.4 even 2 5225.2.a.v.1.13 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.u.1.3 15 1.1 even 1 trivial
5225.2.a.v.1.13 yes 15 5.4 even 2