Properties

Label 495.4.a.o.1.3
Level $495$
Weight $4$
Character 495.1
Self dual yes
Analytic conductor $29.206$
Analytic rank $0$
Dimension $7$
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] = [495,4,Mod(1,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 495.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: \(29.2059454528\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 41x^{5} + 40x^{4} + 424x^{3} - 168x^{2} - 1042x - 388 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.30247\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.30247 q^{2} -2.69863 q^{4} -5.00000 q^{5} +33.7164 q^{7} +24.6333 q^{8} +O(q^{10})\) \(q-2.30247 q^{2} -2.69863 q^{4} -5.00000 q^{5} +33.7164 q^{7} +24.6333 q^{8} +11.5123 q^{10} -11.0000 q^{11} +37.9156 q^{13} -77.6309 q^{14} -35.1283 q^{16} +44.2817 q^{17} -42.3986 q^{19} +13.4932 q^{20} +25.3272 q^{22} -30.6793 q^{23} +25.0000 q^{25} -87.2995 q^{26} -90.9881 q^{28} -194.548 q^{29} -82.9601 q^{31} -116.184 q^{32} -101.957 q^{34} -168.582 q^{35} +445.236 q^{37} +97.6216 q^{38} -123.166 q^{40} -206.184 q^{41} +197.228 q^{43} +29.6850 q^{44} +70.6382 q^{46} +275.301 q^{47} +793.792 q^{49} -57.5617 q^{50} -102.320 q^{52} +382.718 q^{53} +55.0000 q^{55} +830.544 q^{56} +447.941 q^{58} -771.622 q^{59} -210.576 q^{61} +191.013 q^{62} +548.537 q^{64} -189.578 q^{65} +452.672 q^{67} -119.500 q^{68} +388.154 q^{70} +310.139 q^{71} -174.350 q^{73} -1025.14 q^{74} +114.418 q^{76} -370.880 q^{77} +1191.11 q^{79} +175.641 q^{80} +474.732 q^{82} +183.669 q^{83} -221.409 q^{85} -454.111 q^{86} -270.966 q^{88} +613.710 q^{89} +1278.38 q^{91} +82.7923 q^{92} -633.873 q^{94} +211.993 q^{95} -1204.77 q^{97} -1827.68 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 5 q^{2} + 33 q^{4} - 35 q^{5} + 30 q^{7} - 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 5 q^{2} + 33 q^{4} - 35 q^{5} + 30 q^{7} - 45 q^{8} + 25 q^{10} - 77 q^{11} + 38 q^{13} - 20 q^{14} + 309 q^{16} - 12 q^{17} + 226 q^{19} - 165 q^{20} + 55 q^{22} - 334 q^{23} + 175 q^{25} + 372 q^{26} + 812 q^{28} + 258 q^{29} + 336 q^{31} - 485 q^{32} + 78 q^{34} - 150 q^{35} + 466 q^{37} + 494 q^{38} + 225 q^{40} + 258 q^{41} + 308 q^{43} - 363 q^{44} + 98 q^{46} - 546 q^{47} + 735 q^{49} - 125 q^{50} + 512 q^{52} - 110 q^{53} + 385 q^{55} - 20 q^{56} + 1362 q^{58} + 68 q^{59} + 1096 q^{61} - 356 q^{62} + 2761 q^{64} - 190 q^{65} + 2268 q^{67} + 1186 q^{68} + 100 q^{70} + 166 q^{71} + 200 q^{73} + 1710 q^{74} + 3310 q^{76} - 330 q^{77} + 2152 q^{79} - 1545 q^{80} - 1006 q^{82} - 370 q^{83} + 60 q^{85} - 106 q^{86} + 495 q^{88} + 252 q^{89} + 2768 q^{91} - 3774 q^{92} + 2218 q^{94} - 1130 q^{95} + 3698 q^{97} - 697 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.30247 −0.814046 −0.407023 0.913418i \(-0.633433\pi\)
−0.407023 + 0.913418i \(0.633433\pi\)
\(3\) 0 0
\(4\) −2.69863 −0.337329
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 33.7164 1.82051 0.910256 0.414046i \(-0.135885\pi\)
0.910256 + 0.414046i \(0.135885\pi\)
\(8\) 24.6333 1.08865
\(9\) 0 0
\(10\) 11.5123 0.364052
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 37.9156 0.808915 0.404457 0.914557i \(-0.367460\pi\)
0.404457 + 0.914557i \(0.367460\pi\)
\(14\) −77.6309 −1.48198
\(15\) 0 0
\(16\) −35.1283 −0.548880
\(17\) 44.2817 0.631758 0.315879 0.948799i \(-0.397701\pi\)
0.315879 + 0.948799i \(0.397701\pi\)
\(18\) 0 0
\(19\) −42.3986 −0.511943 −0.255971 0.966684i \(-0.582395\pi\)
−0.255971 + 0.966684i \(0.582395\pi\)
\(20\) 13.4932 0.150858
\(21\) 0 0
\(22\) 25.3272 0.245444
\(23\) −30.6793 −0.278134 −0.139067 0.990283i \(-0.544410\pi\)
−0.139067 + 0.990283i \(0.544410\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −87.2995 −0.658494
\(27\) 0 0
\(28\) −90.9881 −0.614112
\(29\) −194.548 −1.24575 −0.622873 0.782323i \(-0.714035\pi\)
−0.622873 + 0.782323i \(0.714035\pi\)
\(30\) 0 0
\(31\) −82.9601 −0.480648 −0.240324 0.970693i \(-0.577254\pi\)
−0.240324 + 0.970693i \(0.577254\pi\)
\(32\) −116.184 −0.641834
\(33\) 0 0
\(34\) −101.957 −0.514280
\(35\) −168.582 −0.814158
\(36\) 0 0
\(37\) 445.236 1.97828 0.989140 0.146980i \(-0.0469552\pi\)
0.989140 + 0.146980i \(0.0469552\pi\)
\(38\) 97.6216 0.416745
\(39\) 0 0
\(40\) −123.166 −0.486858
\(41\) −206.184 −0.785378 −0.392689 0.919671i \(-0.628455\pi\)
−0.392689 + 0.919671i \(0.628455\pi\)
\(42\) 0 0
\(43\) 197.228 0.699464 0.349732 0.936850i \(-0.386273\pi\)
0.349732 + 0.936850i \(0.386273\pi\)
\(44\) 29.6850 0.101709
\(45\) 0 0
\(46\) 70.6382 0.226414
\(47\) 275.301 0.854401 0.427201 0.904157i \(-0.359500\pi\)
0.427201 + 0.904157i \(0.359500\pi\)
\(48\) 0 0
\(49\) 793.792 2.31426
\(50\) −57.5617 −0.162809
\(51\) 0 0
\(52\) −102.320 −0.272871
\(53\) 382.718 0.991894 0.495947 0.868353i \(-0.334821\pi\)
0.495947 + 0.868353i \(0.334821\pi\)
\(54\) 0 0
\(55\) 55.0000 0.134840
\(56\) 830.544 1.98190
\(57\) 0 0
\(58\) 447.941 1.01409
\(59\) −771.622 −1.70265 −0.851327 0.524635i \(-0.824202\pi\)
−0.851327 + 0.524635i \(0.824202\pi\)
\(60\) 0 0
\(61\) −210.576 −0.441992 −0.220996 0.975275i \(-0.570931\pi\)
−0.220996 + 0.975275i \(0.570931\pi\)
\(62\) 191.013 0.391269
\(63\) 0 0
\(64\) 548.537 1.07136
\(65\) −189.578 −0.361758
\(66\) 0 0
\(67\) 452.672 0.825413 0.412707 0.910864i \(-0.364584\pi\)
0.412707 + 0.910864i \(0.364584\pi\)
\(68\) −119.500 −0.213111
\(69\) 0 0
\(70\) 388.154 0.662762
\(71\) 310.139 0.518405 0.259203 0.965823i \(-0.416540\pi\)
0.259203 + 0.965823i \(0.416540\pi\)
\(72\) 0 0
\(73\) −174.350 −0.279536 −0.139768 0.990184i \(-0.544636\pi\)
−0.139768 + 0.990184i \(0.544636\pi\)
\(74\) −1025.14 −1.61041
\(75\) 0 0
\(76\) 114.418 0.172693
\(77\) −370.880 −0.548905
\(78\) 0 0
\(79\) 1191.11 1.69634 0.848168 0.529727i \(-0.177706\pi\)
0.848168 + 0.529727i \(0.177706\pi\)
\(80\) 175.641 0.245466
\(81\) 0 0
\(82\) 474.732 0.639334
\(83\) 183.669 0.242896 0.121448 0.992598i \(-0.461246\pi\)
0.121448 + 0.992598i \(0.461246\pi\)
\(84\) 0 0
\(85\) −221.409 −0.282531
\(86\) −454.111 −0.569396
\(87\) 0 0
\(88\) −270.966 −0.328240
\(89\) 613.710 0.730934 0.365467 0.930824i \(-0.380909\pi\)
0.365467 + 0.930824i \(0.380909\pi\)
\(90\) 0 0
\(91\) 1278.38 1.47264
\(92\) 82.7923 0.0938227
\(93\) 0 0
\(94\) −633.873 −0.695522
\(95\) 211.993 0.228948
\(96\) 0 0
\(97\) −1204.77 −1.26109 −0.630545 0.776152i \(-0.717169\pi\)
−0.630545 + 0.776152i \(0.717169\pi\)
\(98\) −1827.68 −1.88392
\(99\) 0 0
\(100\) −67.4659 −0.0674659
\(101\) 1389.76 1.36917 0.684586 0.728932i \(-0.259983\pi\)
0.684586 + 0.728932i \(0.259983\pi\)
\(102\) 0 0
\(103\) −981.947 −0.939361 −0.469680 0.882837i \(-0.655631\pi\)
−0.469680 + 0.882837i \(0.655631\pi\)
\(104\) 933.985 0.880623
\(105\) 0 0
\(106\) −881.197 −0.807448
\(107\) 738.649 0.667364 0.333682 0.942686i \(-0.391709\pi\)
0.333682 + 0.942686i \(0.391709\pi\)
\(108\) 0 0
\(109\) 599.887 0.527144 0.263572 0.964640i \(-0.415099\pi\)
0.263572 + 0.964640i \(0.415099\pi\)
\(110\) −126.636 −0.109766
\(111\) 0 0
\(112\) −1184.40 −0.999242
\(113\) 817.177 0.680297 0.340149 0.940372i \(-0.389523\pi\)
0.340149 + 0.940372i \(0.389523\pi\)
\(114\) 0 0
\(115\) 153.397 0.124385
\(116\) 525.014 0.420227
\(117\) 0 0
\(118\) 1776.64 1.38604
\(119\) 1493.02 1.15012
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 484.845 0.359802
\(123\) 0 0
\(124\) 223.879 0.162137
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 916.072 0.640065 0.320033 0.947407i \(-0.396306\pi\)
0.320033 + 0.947407i \(0.396306\pi\)
\(128\) −333.515 −0.230304
\(129\) 0 0
\(130\) 436.497 0.294487
\(131\) −903.159 −0.602361 −0.301181 0.953567i \(-0.597381\pi\)
−0.301181 + 0.953567i \(0.597381\pi\)
\(132\) 0 0
\(133\) −1429.53 −0.931998
\(134\) −1042.26 −0.671924
\(135\) 0 0
\(136\) 1090.80 0.687762
\(137\) 2373.28 1.48002 0.740009 0.672597i \(-0.234821\pi\)
0.740009 + 0.672597i \(0.234821\pi\)
\(138\) 0 0
\(139\) 126.777 0.0773601 0.0386801 0.999252i \(-0.487685\pi\)
0.0386801 + 0.999252i \(0.487685\pi\)
\(140\) 454.941 0.274639
\(141\) 0 0
\(142\) −714.086 −0.422006
\(143\) −417.071 −0.243897
\(144\) 0 0
\(145\) 972.739 0.557115
\(146\) 401.436 0.227555
\(147\) 0 0
\(148\) −1201.53 −0.667332
\(149\) 1448.83 0.796596 0.398298 0.917256i \(-0.369601\pi\)
0.398298 + 0.917256i \(0.369601\pi\)
\(150\) 0 0
\(151\) 1536.43 0.828033 0.414016 0.910269i \(-0.364126\pi\)
0.414016 + 0.910269i \(0.364126\pi\)
\(152\) −1044.42 −0.557325
\(153\) 0 0
\(154\) 853.940 0.446834
\(155\) 414.800 0.214952
\(156\) 0 0
\(157\) 2928.34 1.48858 0.744289 0.667857i \(-0.232788\pi\)
0.744289 + 0.667857i \(0.232788\pi\)
\(158\) −2742.50 −1.38090
\(159\) 0 0
\(160\) 580.922 0.287037
\(161\) −1034.39 −0.506346
\(162\) 0 0
\(163\) −2064.31 −0.991957 −0.495979 0.868335i \(-0.665191\pi\)
−0.495979 + 0.868335i \(0.665191\pi\)
\(164\) 556.415 0.264931
\(165\) 0 0
\(166\) −422.893 −0.197728
\(167\) 2896.40 1.34210 0.671049 0.741413i \(-0.265844\pi\)
0.671049 + 0.741413i \(0.265844\pi\)
\(168\) 0 0
\(169\) −759.408 −0.345657
\(170\) 509.786 0.229993
\(171\) 0 0
\(172\) −532.245 −0.235950
\(173\) −2938.08 −1.29120 −0.645601 0.763675i \(-0.723393\pi\)
−0.645601 + 0.763675i \(0.723393\pi\)
\(174\) 0 0
\(175\) 842.909 0.364102
\(176\) 386.411 0.165493
\(177\) 0 0
\(178\) −1413.05 −0.595013
\(179\) 3352.50 1.39987 0.699937 0.714204i \(-0.253211\pi\)
0.699937 + 0.714204i \(0.253211\pi\)
\(180\) 0 0
\(181\) −4167.19 −1.71130 −0.855649 0.517556i \(-0.826842\pi\)
−0.855649 + 0.517556i \(0.826842\pi\)
\(182\) −2943.42 −1.19880
\(183\) 0 0
\(184\) −755.732 −0.302790
\(185\) −2226.18 −0.884713
\(186\) 0 0
\(187\) −487.099 −0.190482
\(188\) −742.938 −0.288215
\(189\) 0 0
\(190\) −488.108 −0.186374
\(191\) −1663.88 −0.630337 −0.315169 0.949036i \(-0.602061\pi\)
−0.315169 + 0.949036i \(0.602061\pi\)
\(192\) 0 0
\(193\) −3148.25 −1.17418 −0.587088 0.809523i \(-0.699726\pi\)
−0.587088 + 0.809523i \(0.699726\pi\)
\(194\) 2773.94 1.02659
\(195\) 0 0
\(196\) −2142.16 −0.780669
\(197\) −5034.38 −1.82074 −0.910368 0.413800i \(-0.864201\pi\)
−0.910368 + 0.413800i \(0.864201\pi\)
\(198\) 0 0
\(199\) 1918.95 0.683571 0.341785 0.939778i \(-0.388968\pi\)
0.341785 + 0.939778i \(0.388968\pi\)
\(200\) 615.832 0.217729
\(201\) 0 0
\(202\) −3199.88 −1.11457
\(203\) −6559.44 −2.26790
\(204\) 0 0
\(205\) 1030.92 0.351232
\(206\) 2260.90 0.764683
\(207\) 0 0
\(208\) −1331.91 −0.443997
\(209\) 466.385 0.154357
\(210\) 0 0
\(211\) −1200.62 −0.391725 −0.195862 0.980631i \(-0.562751\pi\)
−0.195862 + 0.980631i \(0.562751\pi\)
\(212\) −1032.82 −0.334595
\(213\) 0 0
\(214\) −1700.72 −0.543265
\(215\) −986.138 −0.312810
\(216\) 0 0
\(217\) −2797.11 −0.875025
\(218\) −1381.22 −0.429120
\(219\) 0 0
\(220\) −148.425 −0.0454855
\(221\) 1678.97 0.511039
\(222\) 0 0
\(223\) 3956.12 1.18799 0.593994 0.804469i \(-0.297550\pi\)
0.593994 + 0.804469i \(0.297550\pi\)
\(224\) −3917.31 −1.16847
\(225\) 0 0
\(226\) −1881.53 −0.553793
\(227\) 5849.85 1.71043 0.855216 0.518272i \(-0.173424\pi\)
0.855216 + 0.518272i \(0.173424\pi\)
\(228\) 0 0
\(229\) 483.283 0.139459 0.0697297 0.997566i \(-0.477786\pi\)
0.0697297 + 0.997566i \(0.477786\pi\)
\(230\) −353.191 −0.101255
\(231\) 0 0
\(232\) −4792.35 −1.35618
\(233\) −656.670 −0.184635 −0.0923173 0.995730i \(-0.529427\pi\)
−0.0923173 + 0.995730i \(0.529427\pi\)
\(234\) 0 0
\(235\) −1376.51 −0.382100
\(236\) 2082.33 0.574355
\(237\) 0 0
\(238\) −3437.63 −0.936253
\(239\) −2394.73 −0.648125 −0.324063 0.946036i \(-0.605049\pi\)
−0.324063 + 0.946036i \(0.605049\pi\)
\(240\) 0 0
\(241\) −1308.22 −0.349668 −0.174834 0.984598i \(-0.555939\pi\)
−0.174834 + 0.984598i \(0.555939\pi\)
\(242\) −278.599 −0.0740042
\(243\) 0 0
\(244\) 568.268 0.149097
\(245\) −3968.96 −1.03497
\(246\) 0 0
\(247\) −1607.57 −0.414118
\(248\) −2043.58 −0.523256
\(249\) 0 0
\(250\) 287.809 0.0728105
\(251\) 5173.00 1.30086 0.650432 0.759565i \(-0.274588\pi\)
0.650432 + 0.759565i \(0.274588\pi\)
\(252\) 0 0
\(253\) 337.472 0.0838605
\(254\) −2109.23 −0.521042
\(255\) 0 0
\(256\) −3620.39 −0.883884
\(257\) 7589.68 1.84215 0.921073 0.389390i \(-0.127314\pi\)
0.921073 + 0.389390i \(0.127314\pi\)
\(258\) 0 0
\(259\) 15011.7 3.60148
\(260\) 511.602 0.122031
\(261\) 0 0
\(262\) 2079.50 0.490350
\(263\) 6237.14 1.46235 0.731176 0.682189i \(-0.238972\pi\)
0.731176 + 0.682189i \(0.238972\pi\)
\(264\) 0 0
\(265\) −1913.59 −0.443589
\(266\) 3291.44 0.758689
\(267\) 0 0
\(268\) −1221.60 −0.278436
\(269\) −2250.12 −0.510009 −0.255004 0.966940i \(-0.582077\pi\)
−0.255004 + 0.966940i \(0.582077\pi\)
\(270\) 0 0
\(271\) −2157.04 −0.483508 −0.241754 0.970338i \(-0.577723\pi\)
−0.241754 + 0.970338i \(0.577723\pi\)
\(272\) −1555.54 −0.346759
\(273\) 0 0
\(274\) −5464.39 −1.20480
\(275\) −275.000 −0.0603023
\(276\) 0 0
\(277\) 6886.28 1.49371 0.746853 0.664990i \(-0.231564\pi\)
0.746853 + 0.664990i \(0.231564\pi\)
\(278\) −291.899 −0.0629747
\(279\) 0 0
\(280\) −4152.72 −0.886331
\(281\) −2279.62 −0.483953 −0.241977 0.970282i \(-0.577796\pi\)
−0.241977 + 0.970282i \(0.577796\pi\)
\(282\) 0 0
\(283\) −5289.42 −1.11104 −0.555519 0.831504i \(-0.687480\pi\)
−0.555519 + 0.831504i \(0.687480\pi\)
\(284\) −836.953 −0.174873
\(285\) 0 0
\(286\) 960.294 0.198543
\(287\) −6951.77 −1.42979
\(288\) 0 0
\(289\) −2952.13 −0.600881
\(290\) −2239.70 −0.453517
\(291\) 0 0
\(292\) 470.508 0.0942958
\(293\) −6130.70 −1.22239 −0.611193 0.791482i \(-0.709310\pi\)
−0.611193 + 0.791482i \(0.709310\pi\)
\(294\) 0 0
\(295\) 3858.11 0.761450
\(296\) 10967.6 2.15365
\(297\) 0 0
\(298\) −3335.89 −0.648465
\(299\) −1163.22 −0.224987
\(300\) 0 0
\(301\) 6649.80 1.27338
\(302\) −3537.58 −0.674057
\(303\) 0 0
\(304\) 1489.39 0.280995
\(305\) 1052.88 0.197665
\(306\) 0 0
\(307\) −3627.26 −0.674328 −0.337164 0.941446i \(-0.609468\pi\)
−0.337164 + 0.941446i \(0.609468\pi\)
\(308\) 1000.87 0.185162
\(309\) 0 0
\(310\) −955.065 −0.174981
\(311\) 701.327 0.127873 0.0639367 0.997954i \(-0.479634\pi\)
0.0639367 + 0.997954i \(0.479634\pi\)
\(312\) 0 0
\(313\) 7338.97 1.32531 0.662657 0.748923i \(-0.269429\pi\)
0.662657 + 0.748923i \(0.269429\pi\)
\(314\) −6742.41 −1.21177
\(315\) 0 0
\(316\) −3214.38 −0.572224
\(317\) 7376.07 1.30688 0.653440 0.756978i \(-0.273325\pi\)
0.653440 + 0.756978i \(0.273325\pi\)
\(318\) 0 0
\(319\) 2140.03 0.375607
\(320\) −2742.69 −0.479128
\(321\) 0 0
\(322\) 2381.66 0.412189
\(323\) −1877.48 −0.323424
\(324\) 0 0
\(325\) 947.890 0.161783
\(326\) 4753.01 0.807499
\(327\) 0 0
\(328\) −5078.98 −0.855000
\(329\) 9282.16 1.55545
\(330\) 0 0
\(331\) −7801.26 −1.29546 −0.647729 0.761871i \(-0.724281\pi\)
−0.647729 + 0.761871i \(0.724281\pi\)
\(332\) −495.657 −0.0819358
\(333\) 0 0
\(334\) −6668.88 −1.09253
\(335\) −2263.36 −0.369136
\(336\) 0 0
\(337\) 2829.95 0.457439 0.228720 0.973492i \(-0.426546\pi\)
0.228720 + 0.973492i \(0.426546\pi\)
\(338\) 1748.51 0.281381
\(339\) 0 0
\(340\) 597.501 0.0953060
\(341\) 912.561 0.144921
\(342\) 0 0
\(343\) 15199.1 2.39263
\(344\) 4858.36 0.761469
\(345\) 0 0
\(346\) 6764.83 1.05110
\(347\) 1629.68 0.252120 0.126060 0.992023i \(-0.459767\pi\)
0.126060 + 0.992023i \(0.459767\pi\)
\(348\) 0 0
\(349\) −2409.67 −0.369590 −0.184795 0.982777i \(-0.559162\pi\)
−0.184795 + 0.982777i \(0.559162\pi\)
\(350\) −1940.77 −0.296396
\(351\) 0 0
\(352\) 1278.03 0.193520
\(353\) −720.287 −0.108603 −0.0543017 0.998525i \(-0.517293\pi\)
−0.0543017 + 0.998525i \(0.517293\pi\)
\(354\) 0 0
\(355\) −1550.70 −0.231838
\(356\) −1656.18 −0.246565
\(357\) 0 0
\(358\) −7719.03 −1.13956
\(359\) −3839.51 −0.564461 −0.282230 0.959347i \(-0.591074\pi\)
−0.282230 + 0.959347i \(0.591074\pi\)
\(360\) 0 0
\(361\) −5061.36 −0.737914
\(362\) 9594.84 1.39308
\(363\) 0 0
\(364\) −3449.87 −0.496764
\(365\) 871.751 0.125012
\(366\) 0 0
\(367\) 3599.11 0.511913 0.255957 0.966688i \(-0.417610\pi\)
0.255957 + 0.966688i \(0.417610\pi\)
\(368\) 1077.71 0.152662
\(369\) 0 0
\(370\) 5125.71 0.720197
\(371\) 12903.9 1.80576
\(372\) 0 0
\(373\) −2214.60 −0.307419 −0.153710 0.988116i \(-0.549122\pi\)
−0.153710 + 0.988116i \(0.549122\pi\)
\(374\) 1121.53 0.155061
\(375\) 0 0
\(376\) 6781.58 0.930142
\(377\) −7376.40 −1.00770
\(378\) 0 0
\(379\) −13631.2 −1.84747 −0.923733 0.383037i \(-0.874878\pi\)
−0.923733 + 0.383037i \(0.874878\pi\)
\(380\) −572.092 −0.0772308
\(381\) 0 0
\(382\) 3831.04 0.513124
\(383\) 7154.42 0.954501 0.477250 0.878767i \(-0.341634\pi\)
0.477250 + 0.878767i \(0.341634\pi\)
\(384\) 0 0
\(385\) 1854.40 0.245478
\(386\) 7248.75 0.955833
\(387\) 0 0
\(388\) 3251.23 0.425403
\(389\) 3300.42 0.430175 0.215088 0.976595i \(-0.430996\pi\)
0.215088 + 0.976595i \(0.430996\pi\)
\(390\) 0 0
\(391\) −1358.53 −0.175713
\(392\) 19553.7 2.51942
\(393\) 0 0
\(394\) 11591.5 1.48216
\(395\) −5955.56 −0.758625
\(396\) 0 0
\(397\) 3855.87 0.487457 0.243729 0.969843i \(-0.421629\pi\)
0.243729 + 0.969843i \(0.421629\pi\)
\(398\) −4418.32 −0.556458
\(399\) 0 0
\(400\) −878.207 −0.109776
\(401\) −9412.25 −1.17213 −0.586066 0.810263i \(-0.699324\pi\)
−0.586066 + 0.810263i \(0.699324\pi\)
\(402\) 0 0
\(403\) −3145.48 −0.388803
\(404\) −3750.46 −0.461862
\(405\) 0 0
\(406\) 15102.9 1.84617
\(407\) −4897.59 −0.596474
\(408\) 0 0
\(409\) 10135.1 1.22530 0.612649 0.790355i \(-0.290104\pi\)
0.612649 + 0.790355i \(0.290104\pi\)
\(410\) −2373.66 −0.285919
\(411\) 0 0
\(412\) 2649.92 0.316874
\(413\) −26016.3 −3.09970
\(414\) 0 0
\(415\) −918.347 −0.108626
\(416\) −4405.20 −0.519189
\(417\) 0 0
\(418\) −1073.84 −0.125653
\(419\) −6764.61 −0.788718 −0.394359 0.918956i \(-0.629033\pi\)
−0.394359 + 0.918956i \(0.629033\pi\)
\(420\) 0 0
\(421\) 12417.9 1.43756 0.718778 0.695240i \(-0.244702\pi\)
0.718778 + 0.695240i \(0.244702\pi\)
\(422\) 2764.38 0.318882
\(423\) 0 0
\(424\) 9427.61 1.07982
\(425\) 1107.04 0.126352
\(426\) 0 0
\(427\) −7099.86 −0.804652
\(428\) −1993.34 −0.225121
\(429\) 0 0
\(430\) 2270.55 0.254641
\(431\) 15819.9 1.76802 0.884010 0.467467i \(-0.154833\pi\)
0.884010 + 0.467467i \(0.154833\pi\)
\(432\) 0 0
\(433\) 15554.9 1.72637 0.863185 0.504887i \(-0.168466\pi\)
0.863185 + 0.504887i \(0.168466\pi\)
\(434\) 6440.26 0.712310
\(435\) 0 0
\(436\) −1618.87 −0.177821
\(437\) 1300.76 0.142389
\(438\) 0 0
\(439\) −3128.47 −0.340123 −0.170061 0.985433i \(-0.554397\pi\)
−0.170061 + 0.985433i \(0.554397\pi\)
\(440\) 1354.83 0.146793
\(441\) 0 0
\(442\) −3865.77 −0.416009
\(443\) −10581.2 −1.13482 −0.567411 0.823435i \(-0.692055\pi\)
−0.567411 + 0.823435i \(0.692055\pi\)
\(444\) 0 0
\(445\) −3068.55 −0.326883
\(446\) −9108.84 −0.967077
\(447\) 0 0
\(448\) 18494.7 1.95043
\(449\) 8549.76 0.898637 0.449318 0.893372i \(-0.351667\pi\)
0.449318 + 0.893372i \(0.351667\pi\)
\(450\) 0 0
\(451\) 2268.02 0.236800
\(452\) −2205.26 −0.229484
\(453\) 0 0
\(454\) −13469.1 −1.39237
\(455\) −6391.88 −0.658584
\(456\) 0 0
\(457\) 10920.2 1.11778 0.558888 0.829243i \(-0.311228\pi\)
0.558888 + 0.829243i \(0.311228\pi\)
\(458\) −1112.74 −0.113526
\(459\) 0 0
\(460\) −413.961 −0.0419588
\(461\) 10700.4 1.08106 0.540528 0.841326i \(-0.318225\pi\)
0.540528 + 0.841326i \(0.318225\pi\)
\(462\) 0 0
\(463\) 713.866 0.0716548 0.0358274 0.999358i \(-0.488593\pi\)
0.0358274 + 0.999358i \(0.488593\pi\)
\(464\) 6834.14 0.683765
\(465\) 0 0
\(466\) 1511.96 0.150301
\(467\) −11686.7 −1.15802 −0.579010 0.815320i \(-0.696561\pi\)
−0.579010 + 0.815320i \(0.696561\pi\)
\(468\) 0 0
\(469\) 15262.4 1.50267
\(470\) 3169.37 0.311047
\(471\) 0 0
\(472\) −19007.6 −1.85359
\(473\) −2169.50 −0.210896
\(474\) 0 0
\(475\) −1059.97 −0.102389
\(476\) −4029.11 −0.387970
\(477\) 0 0
\(478\) 5513.78 0.527604
\(479\) −1443.85 −0.137727 −0.0688633 0.997626i \(-0.521937\pi\)
−0.0688633 + 0.997626i \(0.521937\pi\)
\(480\) 0 0
\(481\) 16881.4 1.60026
\(482\) 3012.15 0.284646
\(483\) 0 0
\(484\) −326.535 −0.0306663
\(485\) 6023.85 0.563977
\(486\) 0 0
\(487\) 7658.37 0.712595 0.356297 0.934373i \(-0.384039\pi\)
0.356297 + 0.934373i \(0.384039\pi\)
\(488\) −5187.18 −0.481174
\(489\) 0 0
\(490\) 9138.41 0.842513
\(491\) −3394.27 −0.311978 −0.155989 0.987759i \(-0.549856\pi\)
−0.155989 + 0.987759i \(0.549856\pi\)
\(492\) 0 0
\(493\) −8614.91 −0.787010
\(494\) 3701.38 0.337111
\(495\) 0 0
\(496\) 2914.25 0.263818
\(497\) 10456.8 0.943763
\(498\) 0 0
\(499\) −11997.3 −1.07630 −0.538149 0.842850i \(-0.680876\pi\)
−0.538149 + 0.842850i \(0.680876\pi\)
\(500\) 337.329 0.0301717
\(501\) 0 0
\(502\) −11910.7 −1.05896
\(503\) −11969.4 −1.06101 −0.530506 0.847681i \(-0.677998\pi\)
−0.530506 + 0.847681i \(0.677998\pi\)
\(504\) 0 0
\(505\) −6948.81 −0.612313
\(506\) −777.020 −0.0682663
\(507\) 0 0
\(508\) −2472.14 −0.215913
\(509\) 18724.8 1.63057 0.815286 0.579059i \(-0.196580\pi\)
0.815286 + 0.579059i \(0.196580\pi\)
\(510\) 0 0
\(511\) −5878.45 −0.508899
\(512\) 11004.0 0.949826
\(513\) 0 0
\(514\) −17475.0 −1.49959
\(515\) 4909.74 0.420095
\(516\) 0 0
\(517\) −3028.32 −0.257612
\(518\) −34564.0 −2.93177
\(519\) 0 0
\(520\) −4669.93 −0.393827
\(521\) −18285.5 −1.53762 −0.768812 0.639475i \(-0.779152\pi\)
−0.768812 + 0.639475i \(0.779152\pi\)
\(522\) 0 0
\(523\) −19025.4 −1.59068 −0.795338 0.606167i \(-0.792706\pi\)
−0.795338 + 0.606167i \(0.792706\pi\)
\(524\) 2437.30 0.203194
\(525\) 0 0
\(526\) −14360.8 −1.19042
\(527\) −3673.61 −0.303653
\(528\) 0 0
\(529\) −11225.8 −0.922642
\(530\) 4405.99 0.361102
\(531\) 0 0
\(532\) 3857.77 0.314390
\(533\) −7817.58 −0.635304
\(534\) 0 0
\(535\) −3693.25 −0.298454
\(536\) 11150.8 0.898584
\(537\) 0 0
\(538\) 5180.84 0.415171
\(539\) −8731.72 −0.697777
\(540\) 0 0
\(541\) 4886.50 0.388331 0.194166 0.980969i \(-0.437800\pi\)
0.194166 + 0.980969i \(0.437800\pi\)
\(542\) 4966.51 0.393598
\(543\) 0 0
\(544\) −5144.84 −0.405484
\(545\) −2999.43 −0.235746
\(546\) 0 0
\(547\) 12672.6 0.990570 0.495285 0.868731i \(-0.335064\pi\)
0.495285 + 0.868731i \(0.335064\pi\)
\(548\) −6404.60 −0.499254
\(549\) 0 0
\(550\) 633.179 0.0490888
\(551\) 8248.57 0.637751
\(552\) 0 0
\(553\) 40159.9 3.08820
\(554\) −15855.4 −1.21594
\(555\) 0 0
\(556\) −342.124 −0.0260958
\(557\) −8274.65 −0.629458 −0.314729 0.949182i \(-0.601914\pi\)
−0.314729 + 0.949182i \(0.601914\pi\)
\(558\) 0 0
\(559\) 7478.00 0.565807
\(560\) 5921.99 0.446875
\(561\) 0 0
\(562\) 5248.76 0.393960
\(563\) 9174.35 0.686772 0.343386 0.939194i \(-0.388426\pi\)
0.343386 + 0.939194i \(0.388426\pi\)
\(564\) 0 0
\(565\) −4085.89 −0.304238
\(566\) 12178.7 0.904435
\(567\) 0 0
\(568\) 7639.75 0.564360
\(569\) −4686.05 −0.345254 −0.172627 0.984987i \(-0.555226\pi\)
−0.172627 + 0.984987i \(0.555226\pi\)
\(570\) 0 0
\(571\) −19892.5 −1.45792 −0.728962 0.684554i \(-0.759997\pi\)
−0.728962 + 0.684554i \(0.759997\pi\)
\(572\) 1125.52 0.0822736
\(573\) 0 0
\(574\) 16006.2 1.16391
\(575\) −766.983 −0.0556268
\(576\) 0 0
\(577\) −2566.17 −0.185149 −0.0925746 0.995706i \(-0.529510\pi\)
−0.0925746 + 0.995706i \(0.529510\pi\)
\(578\) 6797.19 0.489145
\(579\) 0 0
\(580\) −2625.07 −0.187931
\(581\) 6192.66 0.442194
\(582\) 0 0
\(583\) −4209.90 −0.299067
\(584\) −4294.82 −0.304316
\(585\) 0 0
\(586\) 14115.7 0.995078
\(587\) −19097.4 −1.34282 −0.671409 0.741087i \(-0.734310\pi\)
−0.671409 + 0.741087i \(0.734310\pi\)
\(588\) 0 0
\(589\) 3517.40 0.246064
\(590\) −8883.18 −0.619855
\(591\) 0 0
\(592\) −15640.4 −1.08584
\(593\) 598.368 0.0414368 0.0207184 0.999785i \(-0.493405\pi\)
0.0207184 + 0.999785i \(0.493405\pi\)
\(594\) 0 0
\(595\) −7465.09 −0.514351
\(596\) −3909.86 −0.268715
\(597\) 0 0
\(598\) 2678.29 0.183149
\(599\) −1927.27 −0.131463 −0.0657315 0.997837i \(-0.520938\pi\)
−0.0657315 + 0.997837i \(0.520938\pi\)
\(600\) 0 0
\(601\) 11578.8 0.785869 0.392935 0.919566i \(-0.371460\pi\)
0.392935 + 0.919566i \(0.371460\pi\)
\(602\) −15311.0 −1.03659
\(603\) 0 0
\(604\) −4146.27 −0.279320
\(605\) −605.000 −0.0406558
\(606\) 0 0
\(607\) 1284.90 0.0859182 0.0429591 0.999077i \(-0.486321\pi\)
0.0429591 + 0.999077i \(0.486321\pi\)
\(608\) 4926.06 0.328582
\(609\) 0 0
\(610\) −2424.23 −0.160908
\(611\) 10438.2 0.691138
\(612\) 0 0
\(613\) 24325.7 1.60278 0.801391 0.598141i \(-0.204094\pi\)
0.801391 + 0.598141i \(0.204094\pi\)
\(614\) 8351.65 0.548934
\(615\) 0 0
\(616\) −9135.99 −0.597564
\(617\) −18016.9 −1.17558 −0.587789 0.809014i \(-0.700002\pi\)
−0.587789 + 0.809014i \(0.700002\pi\)
\(618\) 0 0
\(619\) −20542.5 −1.33388 −0.666941 0.745110i \(-0.732397\pi\)
−0.666941 + 0.745110i \(0.732397\pi\)
\(620\) −1119.39 −0.0725097
\(621\) 0 0
\(622\) −1614.79 −0.104095
\(623\) 20692.0 1.33067
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −16897.8 −1.07887
\(627\) 0 0
\(628\) −7902.52 −0.502141
\(629\) 19715.8 1.24979
\(630\) 0 0
\(631\) 5441.98 0.343331 0.171665 0.985155i \(-0.445085\pi\)
0.171665 + 0.985155i \(0.445085\pi\)
\(632\) 29341.0 1.84671
\(633\) 0 0
\(634\) −16983.2 −1.06386
\(635\) −4580.36 −0.286246
\(636\) 0 0
\(637\) 30097.1 1.87204
\(638\) −4927.35 −0.305761
\(639\) 0 0
\(640\) 1667.58 0.102995
\(641\) −4499.67 −0.277264 −0.138632 0.990344i \(-0.544271\pi\)
−0.138632 + 0.990344i \(0.544271\pi\)
\(642\) 0 0
\(643\) 2929.30 0.179659 0.0898293 0.995957i \(-0.471368\pi\)
0.0898293 + 0.995957i \(0.471368\pi\)
\(644\) 2791.45 0.170805
\(645\) 0 0
\(646\) 4322.85 0.263282
\(647\) 20128.9 1.22311 0.611554 0.791203i \(-0.290545\pi\)
0.611554 + 0.791203i \(0.290545\pi\)
\(648\) 0 0
\(649\) 8487.84 0.513370
\(650\) −2182.49 −0.131699
\(651\) 0 0
\(652\) 5570.81 0.334616
\(653\) −13802.2 −0.827141 −0.413571 0.910472i \(-0.635719\pi\)
−0.413571 + 0.910472i \(0.635719\pi\)
\(654\) 0 0
\(655\) 4515.79 0.269384
\(656\) 7242.88 0.431078
\(657\) 0 0
\(658\) −21371.9 −1.26621
\(659\) −29377.3 −1.73653 −0.868267 0.496097i \(-0.834766\pi\)
−0.868267 + 0.496097i \(0.834766\pi\)
\(660\) 0 0
\(661\) −22438.5 −1.32036 −0.660179 0.751108i \(-0.729520\pi\)
−0.660179 + 0.751108i \(0.729520\pi\)
\(662\) 17962.2 1.05456
\(663\) 0 0
\(664\) 4524.38 0.264428
\(665\) 7147.64 0.416802
\(666\) 0 0
\(667\) 5968.60 0.346484
\(668\) −7816.33 −0.452729
\(669\) 0 0
\(670\) 5211.32 0.300494
\(671\) 2316.34 0.133266
\(672\) 0 0
\(673\) −31739.0 −1.81790 −0.908951 0.416903i \(-0.863116\pi\)
−0.908951 + 0.416903i \(0.863116\pi\)
\(674\) −6515.87 −0.372377
\(675\) 0 0
\(676\) 2049.36 0.116600
\(677\) −28959.9 −1.64404 −0.822022 0.569456i \(-0.807154\pi\)
−0.822022 + 0.569456i \(0.807154\pi\)
\(678\) 0 0
\(679\) −40620.4 −2.29583
\(680\) −5454.02 −0.307577
\(681\) 0 0
\(682\) −2101.14 −0.117972
\(683\) 20978.4 1.17528 0.587640 0.809122i \(-0.300057\pi\)
0.587640 + 0.809122i \(0.300057\pi\)
\(684\) 0 0
\(685\) −11866.4 −0.661885
\(686\) −34995.4 −1.94771
\(687\) 0 0
\(688\) −6928.27 −0.383921
\(689\) 14511.0 0.802358
\(690\) 0 0
\(691\) −14713.7 −0.810037 −0.405018 0.914309i \(-0.632735\pi\)
−0.405018 + 0.914309i \(0.632735\pi\)
\(692\) 7928.80 0.435560
\(693\) 0 0
\(694\) −3752.29 −0.205238
\(695\) −633.883 −0.0345965
\(696\) 0 0
\(697\) −9130.17 −0.496169
\(698\) 5548.20 0.300863
\(699\) 0 0
\(700\) −2274.70 −0.122822
\(701\) 30033.8 1.61821 0.809103 0.587667i \(-0.199954\pi\)
0.809103 + 0.587667i \(0.199954\pi\)
\(702\) 0 0
\(703\) −18877.4 −1.01277
\(704\) −6033.91 −0.323028
\(705\) 0 0
\(706\) 1658.44 0.0884082
\(707\) 46857.7 2.49260
\(708\) 0 0
\(709\) 6202.27 0.328535 0.164267 0.986416i \(-0.447474\pi\)
0.164267 + 0.986416i \(0.447474\pi\)
\(710\) 3570.43 0.188727
\(711\) 0 0
\(712\) 15117.7 0.795729
\(713\) 2545.16 0.133684
\(714\) 0 0
\(715\) 2085.36 0.109074
\(716\) −9047.17 −0.472219
\(717\) 0 0
\(718\) 8840.35 0.459497
\(719\) −28650.6 −1.48608 −0.743038 0.669250i \(-0.766616\pi\)
−0.743038 + 0.669250i \(0.766616\pi\)
\(720\) 0 0
\(721\) −33107.7 −1.71012
\(722\) 11653.6 0.600696
\(723\) 0 0
\(724\) 11245.7 0.577271
\(725\) −4863.70 −0.249149
\(726\) 0 0
\(727\) −11144.9 −0.568558 −0.284279 0.958742i \(-0.591754\pi\)
−0.284279 + 0.958742i \(0.591754\pi\)
\(728\) 31490.6 1.60318
\(729\) 0 0
\(730\) −2007.18 −0.101766
\(731\) 8733.58 0.441892
\(732\) 0 0
\(733\) −28546.2 −1.43844 −0.719221 0.694781i \(-0.755501\pi\)
−0.719221 + 0.694781i \(0.755501\pi\)
\(734\) −8286.84 −0.416721
\(735\) 0 0
\(736\) 3564.46 0.178516
\(737\) −4979.39 −0.248871
\(738\) 0 0
\(739\) −19992.4 −0.995175 −0.497587 0.867414i \(-0.665781\pi\)
−0.497587 + 0.867414i \(0.665781\pi\)
\(740\) 6007.64 0.298440
\(741\) 0 0
\(742\) −29710.8 −1.46997
\(743\) 10763.2 0.531446 0.265723 0.964049i \(-0.414389\pi\)
0.265723 + 0.964049i \(0.414389\pi\)
\(744\) 0 0
\(745\) −7244.15 −0.356248
\(746\) 5099.04 0.250253
\(747\) 0 0
\(748\) 1314.50 0.0642553
\(749\) 24904.6 1.21494
\(750\) 0 0
\(751\) 3243.24 0.157586 0.0787932 0.996891i \(-0.474893\pi\)
0.0787932 + 0.996891i \(0.474893\pi\)
\(752\) −9670.87 −0.468963
\(753\) 0 0
\(754\) 16983.9 0.820316
\(755\) −7682.15 −0.370308
\(756\) 0 0
\(757\) 11503.0 0.552288 0.276144 0.961116i \(-0.410943\pi\)
0.276144 + 0.961116i \(0.410943\pi\)
\(758\) 31385.5 1.50392
\(759\) 0 0
\(760\) 5222.09 0.249243
\(761\) 36993.2 1.76216 0.881080 0.472968i \(-0.156817\pi\)
0.881080 + 0.472968i \(0.156817\pi\)
\(762\) 0 0
\(763\) 20226.0 0.959672
\(764\) 4490.22 0.212631
\(765\) 0 0
\(766\) −16472.8 −0.777007
\(767\) −29256.5 −1.37730
\(768\) 0 0
\(769\) −21702.1 −1.01768 −0.508841 0.860861i \(-0.669926\pi\)
−0.508841 + 0.860861i \(0.669926\pi\)
\(770\) −4269.70 −0.199830
\(771\) 0 0
\(772\) 8495.98 0.396084
\(773\) 14968.0 0.696459 0.348230 0.937409i \(-0.386783\pi\)
0.348230 + 0.937409i \(0.386783\pi\)
\(774\) 0 0
\(775\) −2074.00 −0.0961295
\(776\) −29677.4 −1.37288
\(777\) 0 0
\(778\) −7599.13 −0.350182
\(779\) 8741.91 0.402069
\(780\) 0 0
\(781\) −3411.53 −0.156305
\(782\) 3127.98 0.143039
\(783\) 0 0
\(784\) −27884.6 −1.27025
\(785\) −14641.7 −0.665713
\(786\) 0 0
\(787\) −43780.7 −1.98299 −0.991494 0.130149i \(-0.958455\pi\)
−0.991494 + 0.130149i \(0.958455\pi\)
\(788\) 13586.0 0.614187
\(789\) 0 0
\(790\) 13712.5 0.617555
\(791\) 27552.2 1.23849
\(792\) 0 0
\(793\) −7984.12 −0.357534
\(794\) −8878.02 −0.396813
\(795\) 0 0
\(796\) −5178.54 −0.230588
\(797\) −4398.73 −0.195497 −0.0977484 0.995211i \(-0.531164\pi\)
−0.0977484 + 0.995211i \(0.531164\pi\)
\(798\) 0 0
\(799\) 12190.8 0.539775
\(800\) −2904.61 −0.128367
\(801\) 0 0
\(802\) 21671.4 0.954170
\(803\) 1917.85 0.0842834
\(804\) 0 0
\(805\) 5171.97 0.226445
\(806\) 7242.37 0.316503
\(807\) 0 0
\(808\) 34234.4 1.49055
\(809\) −164.162 −0.00713427 −0.00356714 0.999994i \(-0.501135\pi\)
−0.00356714 + 0.999994i \(0.501135\pi\)
\(810\) 0 0
\(811\) −35517.8 −1.53785 −0.768926 0.639338i \(-0.779209\pi\)
−0.768926 + 0.639338i \(0.779209\pi\)
\(812\) 17701.5 0.765028
\(813\) 0 0
\(814\) 11276.6 0.485557
\(815\) 10321.5 0.443617
\(816\) 0 0
\(817\) −8362.18 −0.358085
\(818\) −23335.7 −0.997449
\(819\) 0 0
\(820\) −2782.07 −0.118481
\(821\) −8770.33 −0.372822 −0.186411 0.982472i \(-0.559686\pi\)
−0.186411 + 0.982472i \(0.559686\pi\)
\(822\) 0 0
\(823\) 9344.99 0.395803 0.197902 0.980222i \(-0.436587\pi\)
0.197902 + 0.980222i \(0.436587\pi\)
\(824\) −24188.6 −1.02263
\(825\) 0 0
\(826\) 59901.7 2.52330
\(827\) −37820.4 −1.59026 −0.795130 0.606439i \(-0.792597\pi\)
−0.795130 + 0.606439i \(0.792597\pi\)
\(828\) 0 0
\(829\) 8605.31 0.360525 0.180262 0.983619i \(-0.442305\pi\)
0.180262 + 0.983619i \(0.442305\pi\)
\(830\) 2114.47 0.0884267
\(831\) 0 0
\(832\) 20798.1 0.866641
\(833\) 35150.5 1.46206
\(834\) 0 0
\(835\) −14482.0 −0.600205
\(836\) −1258.60 −0.0520690
\(837\) 0 0
\(838\) 15575.3 0.642053
\(839\) −46477.5 −1.91249 −0.956247 0.292560i \(-0.905493\pi\)
−0.956247 + 0.292560i \(0.905493\pi\)
\(840\) 0 0
\(841\) 13459.9 0.551883
\(842\) −28591.8 −1.17024
\(843\) 0 0
\(844\) 3240.03 0.132140
\(845\) 3797.04 0.154582
\(846\) 0 0
\(847\) 4079.68 0.165501
\(848\) −13444.2 −0.544431
\(849\) 0 0
\(850\) −2548.93 −0.102856
\(851\) −13659.5 −0.550226
\(852\) 0 0
\(853\) 3867.32 0.155234 0.0776169 0.996983i \(-0.475269\pi\)
0.0776169 + 0.996983i \(0.475269\pi\)
\(854\) 16347.2 0.655024
\(855\) 0 0
\(856\) 18195.4 0.726524
\(857\) 37944.2 1.51243 0.756213 0.654325i \(-0.227047\pi\)
0.756213 + 0.654325i \(0.227047\pi\)
\(858\) 0 0
\(859\) 9438.58 0.374901 0.187451 0.982274i \(-0.439978\pi\)
0.187451 + 0.982274i \(0.439978\pi\)
\(860\) 2661.23 0.105520
\(861\) 0 0
\(862\) −36424.8 −1.43925
\(863\) −2626.80 −0.103612 −0.0518061 0.998657i \(-0.516498\pi\)
−0.0518061 + 0.998657i \(0.516498\pi\)
\(864\) 0 0
\(865\) 14690.4 0.577443
\(866\) −35814.6 −1.40534
\(867\) 0 0
\(868\) 7548.38 0.295171
\(869\) −13102.2 −0.511465
\(870\) 0 0
\(871\) 17163.3 0.667689
\(872\) 14777.2 0.573874
\(873\) 0 0
\(874\) −2994.96 −0.115911
\(875\) −4214.54 −0.162832
\(876\) 0 0
\(877\) −39926.8 −1.53732 −0.768662 0.639655i \(-0.779077\pi\)
−0.768662 + 0.639655i \(0.779077\pi\)
\(878\) 7203.22 0.276876
\(879\) 0 0
\(880\) −1932.06 −0.0740109
\(881\) −48387.5 −1.85041 −0.925207 0.379462i \(-0.876109\pi\)
−0.925207 + 0.379462i \(0.876109\pi\)
\(882\) 0 0
\(883\) −24690.2 −0.940986 −0.470493 0.882404i \(-0.655924\pi\)
−0.470493 + 0.882404i \(0.655924\pi\)
\(884\) −4530.92 −0.172388
\(885\) 0 0
\(886\) 24362.8 0.923797
\(887\) −25504.5 −0.965453 −0.482726 0.875771i \(-0.660353\pi\)
−0.482726 + 0.875771i \(0.660353\pi\)
\(888\) 0 0
\(889\) 30886.6 1.16525
\(890\) 7065.24 0.266098
\(891\) 0 0
\(892\) −10676.1 −0.400743
\(893\) −11672.4 −0.437405
\(894\) 0 0
\(895\) −16762.5 −0.626043
\(896\) −11244.9 −0.419271
\(897\) 0 0
\(898\) −19685.6 −0.731532
\(899\) 16139.7 0.598765
\(900\) 0 0
\(901\) 16947.4 0.626638
\(902\) −5222.05 −0.192766
\(903\) 0 0
\(904\) 20129.8 0.740604
\(905\) 20836.0 0.765316
\(906\) 0 0
\(907\) 4050.02 0.148268 0.0741338 0.997248i \(-0.476381\pi\)
0.0741338 + 0.997248i \(0.476381\pi\)
\(908\) −15786.6 −0.576979
\(909\) 0 0
\(910\) 14717.1 0.536118
\(911\) −10405.9 −0.378445 −0.189223 0.981934i \(-0.560597\pi\)
−0.189223 + 0.981934i \(0.560597\pi\)
\(912\) 0 0
\(913\) −2020.36 −0.0732358
\(914\) −25143.4 −0.909922
\(915\) 0 0
\(916\) −1304.20 −0.0470438
\(917\) −30451.2 −1.09661
\(918\) 0 0
\(919\) 31719.1 1.13854 0.569268 0.822152i \(-0.307227\pi\)
0.569268 + 0.822152i \(0.307227\pi\)
\(920\) 3778.66 0.135412
\(921\) 0 0
\(922\) −24637.3 −0.880029
\(923\) 11759.1 0.419346
\(924\) 0 0
\(925\) 11130.9 0.395656
\(926\) −1643.65 −0.0583303
\(927\) 0 0
\(928\) 22603.4 0.799562
\(929\) 36958.3 1.30523 0.652617 0.757688i \(-0.273671\pi\)
0.652617 + 0.757688i \(0.273671\pi\)
\(930\) 0 0
\(931\) −33655.7 −1.18477
\(932\) 1772.11 0.0622827
\(933\) 0 0
\(934\) 26908.2 0.942681
\(935\) 2435.49 0.0851863
\(936\) 0 0
\(937\) −10365.2 −0.361383 −0.180691 0.983540i \(-0.557833\pi\)
−0.180691 + 0.983540i \(0.557833\pi\)
\(938\) −35141.3 −1.22325
\(939\) 0 0
\(940\) 3714.69 0.128893
\(941\) −10350.4 −0.358570 −0.179285 0.983797i \(-0.557378\pi\)
−0.179285 + 0.983797i \(0.557378\pi\)
\(942\) 0 0
\(943\) 6325.58 0.218440
\(944\) 27105.8 0.934552
\(945\) 0 0
\(946\) 4995.22 0.171679
\(947\) −24145.1 −0.828521 −0.414260 0.910158i \(-0.635960\pi\)
−0.414260 + 0.910158i \(0.635960\pi\)
\(948\) 0 0
\(949\) −6610.59 −0.226121
\(950\) 2440.54 0.0833490
\(951\) 0 0
\(952\) 36777.9 1.25208
\(953\) 53268.6 1.81064 0.905319 0.424732i \(-0.139632\pi\)
0.905319 + 0.424732i \(0.139632\pi\)
\(954\) 0 0
\(955\) 8319.42 0.281895
\(956\) 6462.49 0.218632
\(957\) 0 0
\(958\) 3324.41 0.112116
\(959\) 80018.2 2.69439
\(960\) 0 0
\(961\) −22908.6 −0.768978
\(962\) −38868.9 −1.30268
\(963\) 0 0
\(964\) 3530.42 0.117953
\(965\) 15741.2 0.525107
\(966\) 0 0
\(967\) −10029.7 −0.333541 −0.166771 0.985996i \(-0.553334\pi\)
−0.166771 + 0.985996i \(0.553334\pi\)
\(968\) 2980.63 0.0989679
\(969\) 0 0
\(970\) −13869.7 −0.459103
\(971\) −42152.5 −1.39314 −0.696569 0.717489i \(-0.745291\pi\)
−0.696569 + 0.717489i \(0.745291\pi\)
\(972\) 0 0
\(973\) 4274.45 0.140835
\(974\) −17633.2 −0.580085
\(975\) 0 0
\(976\) 7397.18 0.242601
\(977\) −40437.3 −1.32416 −0.662079 0.749434i \(-0.730326\pi\)
−0.662079 + 0.749434i \(0.730326\pi\)
\(978\) 0 0
\(979\) −6750.81 −0.220385
\(980\) 10710.8 0.349126
\(981\) 0 0
\(982\) 7815.21 0.253965
\(983\) 40951.5 1.32874 0.664369 0.747405i \(-0.268701\pi\)
0.664369 + 0.747405i \(0.268701\pi\)
\(984\) 0 0
\(985\) 25171.9 0.814258
\(986\) 19835.6 0.640663
\(987\) 0 0
\(988\) 4338.24 0.139694
\(989\) −6050.81 −0.194545
\(990\) 0 0
\(991\) 255.910 0.00820306 0.00410153 0.999992i \(-0.498694\pi\)
0.00410153 + 0.999992i \(0.498694\pi\)
\(992\) 9638.67 0.308496
\(993\) 0 0
\(994\) −24076.4 −0.768266
\(995\) −9594.73 −0.305702
\(996\) 0 0
\(997\) 14729.2 0.467883 0.233941 0.972251i \(-0.424838\pi\)
0.233941 + 0.972251i \(0.424838\pi\)
\(998\) 27623.4 0.876156
\(999\) 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 495.4.a.o.1.3 7
3.2 odd 2 495.4.a.p.1.5 yes 7
5.4 even 2 2475.4.a.bt.1.5 7
15.14 odd 2 2475.4.a.bp.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.4.a.o.1.3 7 1.1 even 1 trivial
495.4.a.p.1.5 yes 7 3.2 odd 2
2475.4.a.bp.1.3 7 15.14 odd 2
2475.4.a.bt.1.5 7 5.4 even 2