Properties

Label 245.4.a.m.1.2
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
Analytic rank $0$
Dimension $5$
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] = [245,4,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("245.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.79706\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.79706 q^{2} +6.21383 q^{3} -0.176445 q^{4} -5.00000 q^{5} -17.3805 q^{6} +22.8700 q^{8} +11.6117 q^{9} +O(q^{10})\) \(q-2.79706 q^{2} +6.21383 q^{3} -0.176445 q^{4} -5.00000 q^{5} -17.3805 q^{6} +22.8700 q^{8} +11.6117 q^{9} +13.9853 q^{10} -49.6023 q^{11} -1.09640 q^{12} +71.6575 q^{13} -31.0692 q^{15} -62.5573 q^{16} +46.2981 q^{17} -32.4787 q^{18} +54.1893 q^{19} +0.882227 q^{20} +138.741 q^{22} +152.906 q^{23} +142.111 q^{24} +25.0000 q^{25} -200.430 q^{26} -95.6202 q^{27} -38.5268 q^{29} +86.9024 q^{30} +103.405 q^{31} -7.98354 q^{32} -308.220 q^{33} -129.499 q^{34} -2.04883 q^{36} +53.4830 q^{37} -151.571 q^{38} +445.268 q^{39} -114.350 q^{40} +40.4953 q^{41} +377.195 q^{43} +8.75210 q^{44} -58.0586 q^{45} -427.689 q^{46} +389.568 q^{47} -388.721 q^{48} -69.9265 q^{50} +287.689 q^{51} -12.6436 q^{52} +328.820 q^{53} +267.456 q^{54} +248.012 q^{55} +336.723 q^{57} +107.762 q^{58} -39.6843 q^{59} +5.48201 q^{60} +197.460 q^{61} -289.231 q^{62} +522.789 q^{64} -358.287 q^{65} +862.112 q^{66} +972.226 q^{67} -8.16908 q^{68} +950.135 q^{69} +386.897 q^{71} +265.560 q^{72} -219.720 q^{73} -149.595 q^{74} +155.346 q^{75} -9.56145 q^{76} -1245.44 q^{78} -386.474 q^{79} +312.787 q^{80} -907.684 q^{81} -113.268 q^{82} -1376.83 q^{83} -231.490 q^{85} -1055.04 q^{86} -239.399 q^{87} -1134.41 q^{88} +750.054 q^{89} +162.393 q^{90} -26.9796 q^{92} +642.544 q^{93} -1089.64 q^{94} -270.947 q^{95} -49.6084 q^{96} +533.467 q^{97} -575.968 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} - 8 q^{3} + 35 q^{4} - 25 q^{5} + 16 q^{6} + 33 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{2} - 8 q^{3} + 35 q^{4} - 25 q^{5} + 16 q^{6} + 33 q^{8} + 81 q^{9} - 5 q^{10} + 47 q^{11} - 98 q^{12} + q^{13} + 40 q^{15} + 171 q^{16} - 2 q^{17} - 51 q^{18} - 21 q^{19} - 175 q^{20} + 523 q^{22} + 201 q^{23} + 848 q^{24} + 125 q^{25} - 47 q^{26} - 518 q^{27} + 190 q^{29} - 80 q^{30} + 388 q^{31} - 95 q^{32} - 262 q^{33} + 130 q^{34} + 1229 q^{36} - 145 q^{37} + 835 q^{38} + 14 q^{39} - 165 q^{40} - 281 q^{41} + 568 q^{43} + 1091 q^{44} - 405 q^{45} + 337 q^{46} - 473 q^{47} + 70 q^{48} + 25 q^{50} + 732 q^{51} - 379 q^{52} + 351 q^{53} - 774 q^{54} - 235 q^{55} + 954 q^{57} + 1818 q^{58} + 708 q^{59} + 490 q^{60} + 1944 q^{61} + 448 q^{62} - 125 q^{64} - 5 q^{65} + 1482 q^{66} + 1118 q^{67} - 3118 q^{68} + 374 q^{69} + 864 q^{71} - 2219 q^{72} - 1652 q^{73} - 3285 q^{74} - 200 q^{75} + 691 q^{76} - 5574 q^{78} + 218 q^{79} - 855 q^{80} - 455 q^{81} + 1027 q^{82} - 1502 q^{83} + 10 q^{85} - 4264 q^{86} + 390 q^{87} + 2131 q^{88} + 2322 q^{89} + 255 q^{90} - 2957 q^{92} - 2288 q^{93} - 2677 q^{94} + 105 q^{95} + 4592 q^{96} + 598 q^{97} + 35 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.79706 −0.988911 −0.494455 0.869203i \(-0.664632\pi\)
−0.494455 + 0.869203i \(0.664632\pi\)
\(3\) 6.21383 1.19585 0.597926 0.801551i \(-0.295992\pi\)
0.597926 + 0.801551i \(0.295992\pi\)
\(4\) −0.176445 −0.0220557
\(5\) −5.00000 −0.447214
\(6\) −17.3805 −1.18259
\(7\) 0 0
\(8\) 22.8700 1.01072
\(9\) 11.6117 0.430064
\(10\) 13.9853 0.442254
\(11\) −49.6023 −1.35961 −0.679803 0.733395i \(-0.737935\pi\)
−0.679803 + 0.733395i \(0.737935\pi\)
\(12\) −1.09640 −0.0263753
\(13\) 71.6575 1.52879 0.764393 0.644751i \(-0.223039\pi\)
0.764393 + 0.644751i \(0.223039\pi\)
\(14\) 0 0
\(15\) −31.0692 −0.534802
\(16\) −62.5573 −0.977458
\(17\) 46.2981 0.660526 0.330263 0.943889i \(-0.392863\pi\)
0.330263 + 0.943889i \(0.392863\pi\)
\(18\) −32.4787 −0.425295
\(19\) 54.1893 0.654309 0.327155 0.944971i \(-0.393910\pi\)
0.327155 + 0.944971i \(0.393910\pi\)
\(20\) 0.882227 0.00986359
\(21\) 0 0
\(22\) 138.741 1.34453
\(23\) 152.906 1.38623 0.693113 0.720829i \(-0.256239\pi\)
0.693113 + 0.720829i \(0.256239\pi\)
\(24\) 142.111 1.20867
\(25\) 25.0000 0.200000
\(26\) −200.430 −1.51183
\(27\) −95.6202 −0.681560
\(28\) 0 0
\(29\) −38.5268 −0.246698 −0.123349 0.992363i \(-0.539363\pi\)
−0.123349 + 0.992363i \(0.539363\pi\)
\(30\) 86.9024 0.528871
\(31\) 103.405 0.599102 0.299551 0.954080i \(-0.403163\pi\)
0.299551 + 0.954080i \(0.403163\pi\)
\(32\) −7.98354 −0.0441032
\(33\) −308.220 −1.62589
\(34\) −129.499 −0.653201
\(35\) 0 0
\(36\) −2.04883 −0.00948534
\(37\) 53.4830 0.237637 0.118818 0.992916i \(-0.462089\pi\)
0.118818 + 0.992916i \(0.462089\pi\)
\(38\) −151.571 −0.647054
\(39\) 445.268 1.82820
\(40\) −114.350 −0.452009
\(41\) 40.4953 0.154251 0.0771256 0.997021i \(-0.475426\pi\)
0.0771256 + 0.997021i \(0.475426\pi\)
\(42\) 0 0
\(43\) 377.195 1.33771 0.668856 0.743392i \(-0.266784\pi\)
0.668856 + 0.743392i \(0.266784\pi\)
\(44\) 8.75210 0.0299870
\(45\) −58.0586 −0.192330
\(46\) −427.689 −1.37085
\(47\) 389.568 1.20903 0.604514 0.796595i \(-0.293367\pi\)
0.604514 + 0.796595i \(0.293367\pi\)
\(48\) −388.721 −1.16890
\(49\) 0 0
\(50\) −69.9265 −0.197782
\(51\) 287.689 0.789891
\(52\) −12.6436 −0.0337184
\(53\) 328.820 0.852205 0.426102 0.904675i \(-0.359886\pi\)
0.426102 + 0.904675i \(0.359886\pi\)
\(54\) 267.456 0.674002
\(55\) 248.012 0.608034
\(56\) 0 0
\(57\) 336.723 0.782458
\(58\) 107.762 0.243962
\(59\) −39.6843 −0.0875671 −0.0437836 0.999041i \(-0.513941\pi\)
−0.0437836 + 0.999041i \(0.513941\pi\)
\(60\) 5.48201 0.0117954
\(61\) 197.460 0.414462 0.207231 0.978292i \(-0.433555\pi\)
0.207231 + 0.978292i \(0.433555\pi\)
\(62\) −289.231 −0.592458
\(63\) 0 0
\(64\) 522.789 1.02107
\(65\) −358.287 −0.683694
\(66\) 862.112 1.60786
\(67\) 972.226 1.77278 0.886390 0.462939i \(-0.153205\pi\)
0.886390 + 0.462939i \(0.153205\pi\)
\(68\) −8.16908 −0.0145683
\(69\) 950.135 1.65772
\(70\) 0 0
\(71\) 386.897 0.646708 0.323354 0.946278i \(-0.395190\pi\)
0.323354 + 0.946278i \(0.395190\pi\)
\(72\) 265.560 0.434675
\(73\) −219.720 −0.352278 −0.176139 0.984365i \(-0.556361\pi\)
−0.176139 + 0.984365i \(0.556361\pi\)
\(74\) −149.595 −0.235001
\(75\) 155.346 0.239171
\(76\) −9.56145 −0.0144312
\(77\) 0 0
\(78\) −1245.44 −1.80793
\(79\) −386.474 −0.550402 −0.275201 0.961387i \(-0.588744\pi\)
−0.275201 + 0.961387i \(0.588744\pi\)
\(80\) 312.787 0.437132
\(81\) −907.684 −1.24511
\(82\) −113.268 −0.152541
\(83\) −1376.83 −1.82081 −0.910404 0.413721i \(-0.864229\pi\)
−0.910404 + 0.413721i \(0.864229\pi\)
\(84\) 0 0
\(85\) −231.490 −0.295396
\(86\) −1055.04 −1.32288
\(87\) −239.399 −0.295014
\(88\) −1134.41 −1.37418
\(89\) 750.054 0.893321 0.446661 0.894703i \(-0.352613\pi\)
0.446661 + 0.894703i \(0.352613\pi\)
\(90\) 162.393 0.190198
\(91\) 0 0
\(92\) −26.9796 −0.0305741
\(93\) 642.544 0.716437
\(94\) −1089.64 −1.19562
\(95\) −270.947 −0.292616
\(96\) −49.6084 −0.0527410
\(97\) 533.467 0.558406 0.279203 0.960232i \(-0.409930\pi\)
0.279203 + 0.960232i \(0.409930\pi\)
\(98\) 0 0
\(99\) −575.968 −0.584717
\(100\) −4.41113 −0.00441113
\(101\) −1321.49 −1.30191 −0.650956 0.759115i \(-0.725632\pi\)
−0.650956 + 0.759115i \(0.725632\pi\)
\(102\) −804.683 −0.781132
\(103\) −1491.29 −1.42662 −0.713308 0.700851i \(-0.752804\pi\)
−0.713308 + 0.700851i \(0.752804\pi\)
\(104\) 1638.81 1.54518
\(105\) 0 0
\(106\) −919.729 −0.842754
\(107\) −439.516 −0.397099 −0.198550 0.980091i \(-0.563623\pi\)
−0.198550 + 0.980091i \(0.563623\pi\)
\(108\) 16.8717 0.0150323
\(109\) 1463.10 1.28568 0.642840 0.766000i \(-0.277756\pi\)
0.642840 + 0.766000i \(0.277756\pi\)
\(110\) −693.704 −0.601291
\(111\) 332.334 0.284178
\(112\) 0 0
\(113\) −1270.15 −1.05740 −0.528699 0.848809i \(-0.677320\pi\)
−0.528699 + 0.848809i \(0.677320\pi\)
\(114\) −941.836 −0.773781
\(115\) −764.532 −0.619939
\(116\) 6.79787 0.00544109
\(117\) 832.067 0.657475
\(118\) 111.000 0.0865960
\(119\) 0 0
\(120\) −710.553 −0.540536
\(121\) 1129.39 0.848527
\(122\) −552.308 −0.409866
\(123\) 251.631 0.184462
\(124\) −18.2454 −0.0132136
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −675.261 −0.471809 −0.235904 0.971776i \(-0.575805\pi\)
−0.235904 + 0.971776i \(0.575805\pi\)
\(128\) −1398.40 −0.965646
\(129\) 2343.82 1.59971
\(130\) 1002.15 0.676112
\(131\) 694.929 0.463483 0.231741 0.972777i \(-0.425558\pi\)
0.231741 + 0.972777i \(0.425558\pi\)
\(132\) 54.3841 0.0358600
\(133\) 0 0
\(134\) −2719.38 −1.75312
\(135\) 478.101 0.304803
\(136\) 1058.84 0.667608
\(137\) −2519.10 −1.57096 −0.785478 0.618890i \(-0.787583\pi\)
−0.785478 + 0.618890i \(0.787583\pi\)
\(138\) −2657.59 −1.63934
\(139\) −1711.61 −1.04444 −0.522219 0.852812i \(-0.674896\pi\)
−0.522219 + 0.852812i \(0.674896\pi\)
\(140\) 0 0
\(141\) 2420.71 1.44582
\(142\) −1082.18 −0.639536
\(143\) −3554.38 −2.07854
\(144\) −726.398 −0.420369
\(145\) 192.634 0.110327
\(146\) 614.571 0.348372
\(147\) 0 0
\(148\) −9.43683 −0.00524123
\(149\) −721.377 −0.396628 −0.198314 0.980139i \(-0.563546\pi\)
−0.198314 + 0.980139i \(0.563546\pi\)
\(150\) −434.512 −0.236518
\(151\) 930.692 0.501580 0.250790 0.968041i \(-0.419310\pi\)
0.250790 + 0.968041i \(0.419310\pi\)
\(152\) 1239.31 0.661325
\(153\) 537.600 0.284068
\(154\) 0 0
\(155\) −517.027 −0.267926
\(156\) −78.5654 −0.0403222
\(157\) 2266.66 1.15222 0.576111 0.817371i \(-0.304570\pi\)
0.576111 + 0.817371i \(0.304570\pi\)
\(158\) 1080.99 0.544298
\(159\) 2043.23 1.01911
\(160\) 39.9177 0.0197236
\(161\) 0 0
\(162\) 2538.85 1.23130
\(163\) 3194.30 1.53495 0.767474 0.641080i \(-0.221513\pi\)
0.767474 + 0.641080i \(0.221513\pi\)
\(164\) −7.14521 −0.00340212
\(165\) 1541.10 0.727119
\(166\) 3851.09 1.80062
\(167\) −611.079 −0.283154 −0.141577 0.989927i \(-0.545217\pi\)
−0.141577 + 0.989927i \(0.545217\pi\)
\(168\) 0 0
\(169\) 2937.79 1.33718
\(170\) 647.493 0.292120
\(171\) 629.231 0.281395
\(172\) −66.5542 −0.0295041
\(173\) 8.81121 0.00387228 0.00193614 0.999998i \(-0.499384\pi\)
0.00193614 + 0.999998i \(0.499384\pi\)
\(174\) 669.613 0.291743
\(175\) 0 0
\(176\) 3102.99 1.32896
\(177\) −246.592 −0.104717
\(178\) −2097.95 −0.883415
\(179\) −1250.16 −0.522018 −0.261009 0.965336i \(-0.584055\pi\)
−0.261009 + 0.965336i \(0.584055\pi\)
\(180\) 10.2442 0.00424197
\(181\) −2448.02 −1.00531 −0.502653 0.864488i \(-0.667643\pi\)
−0.502653 + 0.864488i \(0.667643\pi\)
\(182\) 0 0
\(183\) 1226.98 0.495635
\(184\) 3496.97 1.40109
\(185\) −267.415 −0.106274
\(186\) −1797.23 −0.708493
\(187\) −2296.49 −0.898054
\(188\) −68.7374 −0.0266659
\(189\) 0 0
\(190\) 757.854 0.289371
\(191\) −5068.33 −1.92006 −0.960030 0.279898i \(-0.909699\pi\)
−0.960030 + 0.279898i \(0.909699\pi\)
\(192\) 3248.52 1.22105
\(193\) 3444.56 1.28469 0.642345 0.766416i \(-0.277962\pi\)
0.642345 + 0.766416i \(0.277962\pi\)
\(194\) −1492.14 −0.552214
\(195\) −2226.34 −0.817597
\(196\) 0 0
\(197\) −1769.61 −0.639998 −0.319999 0.947418i \(-0.603683\pi\)
−0.319999 + 0.947418i \(0.603683\pi\)
\(198\) 1611.02 0.578233
\(199\) 205.971 0.0733715 0.0366858 0.999327i \(-0.488320\pi\)
0.0366858 + 0.999327i \(0.488320\pi\)
\(200\) 571.751 0.202144
\(201\) 6041.25 2.11998
\(202\) 3696.29 1.28748
\(203\) 0 0
\(204\) −50.7613 −0.0174216
\(205\) −202.476 −0.0689833
\(206\) 4171.24 1.41080
\(207\) 1775.51 0.596165
\(208\) −4482.70 −1.49432
\(209\) −2687.91 −0.889603
\(210\) 0 0
\(211\) 4973.82 1.62281 0.811403 0.584488i \(-0.198704\pi\)
0.811403 + 0.584488i \(0.198704\pi\)
\(212\) −58.0187 −0.0187959
\(213\) 2404.11 0.773367
\(214\) 1229.35 0.392696
\(215\) −1885.97 −0.598243
\(216\) −2186.84 −0.688867
\(217\) 0 0
\(218\) −4092.37 −1.27142
\(219\) −1365.31 −0.421273
\(220\) −43.7605 −0.0134106
\(221\) 3317.60 1.00980
\(222\) −929.560 −0.281027
\(223\) 3355.68 1.00768 0.503841 0.863796i \(-0.331920\pi\)
0.503841 + 0.863796i \(0.331920\pi\)
\(224\) 0 0
\(225\) 290.293 0.0860127
\(226\) 3552.70 1.04567
\(227\) −2083.10 −0.609074 −0.304537 0.952500i \(-0.598502\pi\)
−0.304537 + 0.952500i \(0.598502\pi\)
\(228\) −59.4133 −0.0172576
\(229\) −5011.75 −1.44623 −0.723114 0.690729i \(-0.757290\pi\)
−0.723114 + 0.690729i \(0.757290\pi\)
\(230\) 2138.44 0.613064
\(231\) 0 0
\(232\) −881.108 −0.249343
\(233\) 1150.64 0.323525 0.161762 0.986830i \(-0.448282\pi\)
0.161762 + 0.986830i \(0.448282\pi\)
\(234\) −2327.34 −0.650184
\(235\) −1947.84 −0.540693
\(236\) 7.00211 0.00193135
\(237\) −2401.49 −0.658199
\(238\) 0 0
\(239\) 1919.63 0.519543 0.259771 0.965670i \(-0.416353\pi\)
0.259771 + 0.965670i \(0.416353\pi\)
\(240\) 1943.60 0.522746
\(241\) −250.169 −0.0668664 −0.0334332 0.999441i \(-0.510644\pi\)
−0.0334332 + 0.999441i \(0.510644\pi\)
\(242\) −3158.97 −0.839117
\(243\) −3058.45 −0.807407
\(244\) −34.8409 −0.00914123
\(245\) 0 0
\(246\) −703.828 −0.182416
\(247\) 3883.07 1.00030
\(248\) 2364.88 0.605525
\(249\) −8555.41 −2.17742
\(250\) 349.633 0.0884509
\(251\) 2905.43 0.730634 0.365317 0.930883i \(-0.380961\pi\)
0.365317 + 0.930883i \(0.380961\pi\)
\(252\) 0 0
\(253\) −7584.51 −1.88472
\(254\) 1888.75 0.466577
\(255\) −1438.44 −0.353250
\(256\) −270.887 −0.0661346
\(257\) 1489.30 0.361478 0.180739 0.983531i \(-0.442151\pi\)
0.180739 + 0.983531i \(0.442151\pi\)
\(258\) −6555.82 −1.58197
\(259\) 0 0
\(260\) 63.2181 0.0150793
\(261\) −447.362 −0.106096
\(262\) −1943.76 −0.458343
\(263\) 3309.86 0.776025 0.388013 0.921654i \(-0.373162\pi\)
0.388013 + 0.921654i \(0.373162\pi\)
\(264\) −7049.01 −1.64332
\(265\) −1644.10 −0.381118
\(266\) 0 0
\(267\) 4660.71 1.06828
\(268\) −171.545 −0.0390999
\(269\) 4110.05 0.931577 0.465789 0.884896i \(-0.345771\pi\)
0.465789 + 0.884896i \(0.345771\pi\)
\(270\) −1337.28 −0.301423
\(271\) −1728.27 −0.387399 −0.193700 0.981061i \(-0.562049\pi\)
−0.193700 + 0.981061i \(0.562049\pi\)
\(272\) −2896.28 −0.645636
\(273\) 0 0
\(274\) 7046.07 1.55353
\(275\) −1240.06 −0.271921
\(276\) −167.647 −0.0365622
\(277\) −4548.14 −0.986540 −0.493270 0.869876i \(-0.664198\pi\)
−0.493270 + 0.869876i \(0.664198\pi\)
\(278\) 4787.48 1.03286
\(279\) 1200.71 0.257652
\(280\) 0 0
\(281\) −3781.62 −0.802821 −0.401410 0.915898i \(-0.631480\pi\)
−0.401410 + 0.915898i \(0.631480\pi\)
\(282\) −6770.87 −1.42979
\(283\) 953.988 0.200384 0.100192 0.994968i \(-0.468054\pi\)
0.100192 + 0.994968i \(0.468054\pi\)
\(284\) −68.2662 −0.0142636
\(285\) −1683.62 −0.349926
\(286\) 9941.81 2.05550
\(287\) 0 0
\(288\) −92.7026 −0.0189672
\(289\) −2769.49 −0.563706
\(290\) −538.809 −0.109103
\(291\) 3314.88 0.667772
\(292\) 38.7686 0.00776973
\(293\) 4463.04 0.889875 0.444938 0.895562i \(-0.353226\pi\)
0.444938 + 0.895562i \(0.353226\pi\)
\(294\) 0 0
\(295\) 198.422 0.0391612
\(296\) 1223.16 0.240184
\(297\) 4742.98 0.926653
\(298\) 2017.74 0.392229
\(299\) 10956.9 2.11924
\(300\) −27.4100 −0.00527507
\(301\) 0 0
\(302\) −2603.20 −0.496018
\(303\) −8211.52 −1.55690
\(304\) −3389.94 −0.639560
\(305\) −987.300 −0.185353
\(306\) −1503.70 −0.280918
\(307\) 6617.48 1.23023 0.615113 0.788439i \(-0.289110\pi\)
0.615113 + 0.788439i \(0.289110\pi\)
\(308\) 0 0
\(309\) −9266.64 −1.70602
\(310\) 1446.16 0.264955
\(311\) 168.563 0.0307342 0.0153671 0.999882i \(-0.495108\pi\)
0.0153671 + 0.999882i \(0.495108\pi\)
\(312\) 10183.3 1.84780
\(313\) 7501.86 1.35473 0.677364 0.735648i \(-0.263122\pi\)
0.677364 + 0.735648i \(0.263122\pi\)
\(314\) −6339.98 −1.13944
\(315\) 0 0
\(316\) 68.1915 0.0121395
\(317\) 3425.16 0.606864 0.303432 0.952853i \(-0.401867\pi\)
0.303432 + 0.952853i \(0.401867\pi\)
\(318\) −5715.04 −1.00781
\(319\) 1911.02 0.335412
\(320\) −2613.94 −0.456637
\(321\) −2731.08 −0.474872
\(322\) 0 0
\(323\) 2508.86 0.432188
\(324\) 160.157 0.0274617
\(325\) 1791.44 0.305757
\(326\) −8934.65 −1.51793
\(327\) 9091.43 1.53748
\(328\) 926.128 0.155905
\(329\) 0 0
\(330\) −4310.56 −0.719056
\(331\) 1095.47 0.181911 0.0909557 0.995855i \(-0.471008\pi\)
0.0909557 + 0.995855i \(0.471008\pi\)
\(332\) 242.936 0.0401591
\(333\) 621.030 0.102199
\(334\) 1709.23 0.280014
\(335\) −4861.13 −0.792812
\(336\) 0 0
\(337\) 4241.92 0.685673 0.342837 0.939395i \(-0.388612\pi\)
0.342837 + 0.939395i \(0.388612\pi\)
\(338\) −8217.19 −1.32236
\(339\) −7892.52 −1.26449
\(340\) 40.8454 0.00651516
\(341\) −5129.14 −0.814542
\(342\) −1760.00 −0.278274
\(343\) 0 0
\(344\) 8626.45 1.35206
\(345\) −4750.67 −0.741356
\(346\) −24.6455 −0.00382934
\(347\) 78.1669 0.0120929 0.00604643 0.999982i \(-0.498075\pi\)
0.00604643 + 0.999982i \(0.498075\pi\)
\(348\) 42.2408 0.00650674
\(349\) −167.845 −0.0257436 −0.0128718 0.999917i \(-0.504097\pi\)
−0.0128718 + 0.999917i \(0.504097\pi\)
\(350\) 0 0
\(351\) −6851.90 −1.04196
\(352\) 396.002 0.0599630
\(353\) −4872.81 −0.734713 −0.367357 0.930080i \(-0.619737\pi\)
−0.367357 + 0.930080i \(0.619737\pi\)
\(354\) 689.732 0.103556
\(355\) −1934.49 −0.289216
\(356\) −132.344 −0.0197028
\(357\) 0 0
\(358\) 3496.77 0.516230
\(359\) 8449.53 1.24220 0.621099 0.783732i \(-0.286687\pi\)
0.621099 + 0.783732i \(0.286687\pi\)
\(360\) −1327.80 −0.194392
\(361\) −3922.52 −0.571879
\(362\) 6847.28 0.994157
\(363\) 7017.84 1.01471
\(364\) 0 0
\(365\) 1098.60 0.157544
\(366\) −3431.95 −0.490139
\(367\) −4801.20 −0.682890 −0.341445 0.939902i \(-0.610916\pi\)
−0.341445 + 0.939902i \(0.610916\pi\)
\(368\) −9565.41 −1.35498
\(369\) 470.220 0.0663379
\(370\) 747.976 0.105096
\(371\) 0 0
\(372\) −113.374 −0.0158015
\(373\) 6498.62 0.902106 0.451053 0.892497i \(-0.351048\pi\)
0.451053 + 0.892497i \(0.351048\pi\)
\(374\) 6423.43 0.888095
\(375\) −776.729 −0.106960
\(376\) 8909.42 1.22199
\(377\) −2760.73 −0.377148
\(378\) 0 0
\(379\) −11049.4 −1.49754 −0.748770 0.662830i \(-0.769355\pi\)
−0.748770 + 0.662830i \(0.769355\pi\)
\(380\) 47.8073 0.00645384
\(381\) −4195.96 −0.564214
\(382\) 14176.4 1.89877
\(383\) −8098.52 −1.08046 −0.540229 0.841518i \(-0.681662\pi\)
−0.540229 + 0.841518i \(0.681662\pi\)
\(384\) −8689.45 −1.15477
\(385\) 0 0
\(386\) −9634.66 −1.27044
\(387\) 4379.88 0.575302
\(388\) −94.1278 −0.0123160
\(389\) 5897.42 0.768666 0.384333 0.923194i \(-0.374431\pi\)
0.384333 + 0.923194i \(0.374431\pi\)
\(390\) 6227.21 0.808530
\(391\) 7079.27 0.915637
\(392\) 0 0
\(393\) 4318.17 0.554257
\(394\) 4949.71 0.632901
\(395\) 1932.37 0.246147
\(396\) 101.627 0.0128963
\(397\) −11561.4 −1.46159 −0.730795 0.682597i \(-0.760850\pi\)
−0.730795 + 0.682597i \(0.760850\pi\)
\(398\) −576.115 −0.0725579
\(399\) 0 0
\(400\) −1563.93 −0.195492
\(401\) −13069.2 −1.62754 −0.813769 0.581188i \(-0.802588\pi\)
−0.813769 + 0.581188i \(0.802588\pi\)
\(402\) −16897.7 −2.09648
\(403\) 7409.77 0.915898
\(404\) 233.171 0.0287146
\(405\) 4538.42 0.556830
\(406\) 0 0
\(407\) −2652.88 −0.323092
\(408\) 6579.44 0.798360
\(409\) −439.733 −0.0531623 −0.0265812 0.999647i \(-0.508462\pi\)
−0.0265812 + 0.999647i \(0.508462\pi\)
\(410\) 566.339 0.0682183
\(411\) −15653.2 −1.87863
\(412\) 263.132 0.0314650
\(413\) 0 0
\(414\) −4966.20 −0.589554
\(415\) 6884.16 0.814290
\(416\) −572.080 −0.0674244
\(417\) −10635.7 −1.24899
\(418\) 7518.26 0.879738
\(419\) 1673.71 0.195146 0.0975730 0.995228i \(-0.468892\pi\)
0.0975730 + 0.995228i \(0.468892\pi\)
\(420\) 0 0
\(421\) 805.413 0.0932385 0.0466192 0.998913i \(-0.485155\pi\)
0.0466192 + 0.998913i \(0.485155\pi\)
\(422\) −13912.1 −1.60481
\(423\) 4523.55 0.519959
\(424\) 7520.11 0.861342
\(425\) 1157.45 0.132105
\(426\) −6724.46 −0.764791
\(427\) 0 0
\(428\) 77.5506 0.00875829
\(429\) −22086.3 −2.48563
\(430\) 5275.18 0.591609
\(431\) 908.315 0.101513 0.0507564 0.998711i \(-0.483837\pi\)
0.0507564 + 0.998711i \(0.483837\pi\)
\(432\) 5981.74 0.666196
\(433\) 1464.37 0.162524 0.0812621 0.996693i \(-0.474105\pi\)
0.0812621 + 0.996693i \(0.474105\pi\)
\(434\) 0 0
\(435\) 1196.99 0.131934
\(436\) −258.157 −0.0283565
\(437\) 8285.89 0.907020
\(438\) 3818.84 0.416601
\(439\) −7413.86 −0.806024 −0.403012 0.915195i \(-0.632037\pi\)
−0.403012 + 0.915195i \(0.632037\pi\)
\(440\) 5672.03 0.614553
\(441\) 0 0
\(442\) −9279.54 −0.998604
\(443\) 8656.90 0.928447 0.464223 0.885718i \(-0.346333\pi\)
0.464223 + 0.885718i \(0.346333\pi\)
\(444\) −58.6389 −0.00626774
\(445\) −3750.27 −0.399505
\(446\) −9386.05 −0.996507
\(447\) −4482.52 −0.474308
\(448\) 0 0
\(449\) 7138.71 0.750326 0.375163 0.926959i \(-0.377587\pi\)
0.375163 + 0.926959i \(0.377587\pi\)
\(450\) −811.967 −0.0850589
\(451\) −2008.66 −0.209721
\(452\) 224.113 0.0233216
\(453\) 5783.16 0.599816
\(454\) 5826.55 0.602320
\(455\) 0 0
\(456\) 7700.87 0.790847
\(457\) 9955.60 1.01904 0.509522 0.860457i \(-0.329822\pi\)
0.509522 + 0.860457i \(0.329822\pi\)
\(458\) 14018.2 1.43019
\(459\) −4427.03 −0.450188
\(460\) 134.898 0.0136732
\(461\) −10380.2 −1.04870 −0.524351 0.851502i \(-0.675692\pi\)
−0.524351 + 0.851502i \(0.675692\pi\)
\(462\) 0 0
\(463\) −2851.77 −0.286249 −0.143124 0.989705i \(-0.545715\pi\)
−0.143124 + 0.989705i \(0.545715\pi\)
\(464\) 2410.13 0.241137
\(465\) −3212.72 −0.320401
\(466\) −3218.42 −0.319937
\(467\) −12568.5 −1.24540 −0.622698 0.782462i \(-0.713963\pi\)
−0.622698 + 0.782462i \(0.713963\pi\)
\(468\) −146.814 −0.0145011
\(469\) 0 0
\(470\) 5448.22 0.534697
\(471\) 14084.6 1.37789
\(472\) −907.581 −0.0885060
\(473\) −18709.7 −1.81876
\(474\) 6717.10 0.650900
\(475\) 1354.73 0.130862
\(476\) 0 0
\(477\) 3818.16 0.366502
\(478\) −5369.33 −0.513781
\(479\) 5539.53 0.528409 0.264204 0.964467i \(-0.414891\pi\)
0.264204 + 0.964467i \(0.414891\pi\)
\(480\) 248.042 0.0235865
\(481\) 3832.46 0.363295
\(482\) 699.738 0.0661249
\(483\) 0 0
\(484\) −199.276 −0.0187148
\(485\) −2667.34 −0.249727
\(486\) 8554.68 0.798453
\(487\) −10949.9 −1.01886 −0.509432 0.860511i \(-0.670144\pi\)
−0.509432 + 0.860511i \(0.670144\pi\)
\(488\) 4515.92 0.418906
\(489\) 19848.8 1.83557
\(490\) 0 0
\(491\) 5381.76 0.494654 0.247327 0.968932i \(-0.420448\pi\)
0.247327 + 0.968932i \(0.420448\pi\)
\(492\) −44.3991 −0.00406843
\(493\) −1783.72 −0.162950
\(494\) −10861.2 −0.989206
\(495\) 2879.84 0.261493
\(496\) −6468.76 −0.585597
\(497\) 0 0
\(498\) 23930.0 2.15327
\(499\) 13142.8 1.17906 0.589530 0.807747i \(-0.299313\pi\)
0.589530 + 0.807747i \(0.299313\pi\)
\(500\) 22.0557 0.00197272
\(501\) −3797.14 −0.338611
\(502\) −8126.67 −0.722532
\(503\) −5172.75 −0.458532 −0.229266 0.973364i \(-0.573633\pi\)
−0.229266 + 0.973364i \(0.573633\pi\)
\(504\) 0 0
\(505\) 6607.45 0.582233
\(506\) 21214.3 1.86382
\(507\) 18255.0 1.59908
\(508\) 119.147 0.0104061
\(509\) −5487.45 −0.477852 −0.238926 0.971038i \(-0.576795\pi\)
−0.238926 + 0.971038i \(0.576795\pi\)
\(510\) 4023.41 0.349333
\(511\) 0 0
\(512\) 11944.9 1.03105
\(513\) −5181.59 −0.445951
\(514\) −4165.65 −0.357469
\(515\) 7456.46 0.638002
\(516\) −413.557 −0.0352826
\(517\) −19323.5 −1.64380
\(518\) 0 0
\(519\) 54.7514 0.00463067
\(520\) −8194.04 −0.691024
\(521\) −9911.27 −0.833437 −0.416719 0.909036i \(-0.636820\pi\)
−0.416719 + 0.909036i \(0.636820\pi\)
\(522\) 1251.30 0.104919
\(523\) 328.966 0.0275042 0.0137521 0.999905i \(-0.495622\pi\)
0.0137521 + 0.999905i \(0.495622\pi\)
\(524\) −122.617 −0.0102224
\(525\) 0 0
\(526\) −9257.88 −0.767420
\(527\) 4787.47 0.395722
\(528\) 19281.4 1.58924
\(529\) 11213.4 0.921621
\(530\) 4598.64 0.376891
\(531\) −460.803 −0.0376594
\(532\) 0 0
\(533\) 2901.79 0.235817
\(534\) −13036.3 −1.05643
\(535\) 2197.58 0.177588
\(536\) 22234.8 1.79179
\(537\) −7768.28 −0.624257
\(538\) −11496.1 −0.921247
\(539\) 0 0
\(540\) −84.3587 −0.00672263
\(541\) −11235.5 −0.892884 −0.446442 0.894813i \(-0.647309\pi\)
−0.446442 + 0.894813i \(0.647309\pi\)
\(542\) 4834.09 0.383103
\(543\) −15211.6 −1.20220
\(544\) −369.623 −0.0291313
\(545\) −7315.48 −0.574974
\(546\) 0 0
\(547\) −19423.1 −1.51823 −0.759115 0.650957i \(-0.774368\pi\)
−0.759115 + 0.650957i \(0.774368\pi\)
\(548\) 444.483 0.0346485
\(549\) 2292.85 0.178245
\(550\) 3468.52 0.268906
\(551\) −2087.74 −0.161417
\(552\) 21729.6 1.67550
\(553\) 0 0
\(554\) 12721.4 0.975600
\(555\) −1661.67 −0.127088
\(556\) 302.006 0.0230358
\(557\) 12912.4 0.982254 0.491127 0.871088i \(-0.336585\pi\)
0.491127 + 0.871088i \(0.336585\pi\)
\(558\) −3358.47 −0.254795
\(559\) 27028.8 2.04508
\(560\) 0 0
\(561\) −14270.0 −1.07394
\(562\) 10577.4 0.793918
\(563\) 5883.84 0.440452 0.220226 0.975449i \(-0.429321\pi\)
0.220226 + 0.975449i \(0.429321\pi\)
\(564\) −427.123 −0.0318885
\(565\) 6350.77 0.472883
\(566\) −2668.36 −0.198162
\(567\) 0 0
\(568\) 8848.35 0.653642
\(569\) 3037.55 0.223797 0.111899 0.993720i \(-0.464307\pi\)
0.111899 + 0.993720i \(0.464307\pi\)
\(570\) 4709.18 0.346045
\(571\) 15684.7 1.14953 0.574766 0.818318i \(-0.305093\pi\)
0.574766 + 0.818318i \(0.305093\pi\)
\(572\) 627.153 0.0458437
\(573\) −31493.7 −2.29611
\(574\) 0 0
\(575\) 3822.66 0.277245
\(576\) 6070.48 0.439126
\(577\) −16184.0 −1.16767 −0.583837 0.811871i \(-0.698449\pi\)
−0.583837 + 0.811871i \(0.698449\pi\)
\(578\) 7746.43 0.557455
\(579\) 21403.9 1.53630
\(580\) −33.9893 −0.00243333
\(581\) 0 0
\(582\) −9271.92 −0.660366
\(583\) −16310.2 −1.15866
\(584\) −5025.01 −0.356055
\(585\) −4160.33 −0.294032
\(586\) −12483.4 −0.880007
\(587\) 9256.04 0.650831 0.325415 0.945571i \(-0.394496\pi\)
0.325415 + 0.945571i \(0.394496\pi\)
\(588\) 0 0
\(589\) 5603.46 0.391998
\(590\) −554.998 −0.0387269
\(591\) −10996.1 −0.765344
\(592\) −3345.75 −0.232280
\(593\) 19154.5 1.32645 0.663223 0.748422i \(-0.269188\pi\)
0.663223 + 0.748422i \(0.269188\pi\)
\(594\) −13266.4 −0.916377
\(595\) 0 0
\(596\) 127.284 0.00874788
\(597\) 1279.87 0.0877415
\(598\) −30647.1 −2.09574
\(599\) 20884.5 1.42457 0.712285 0.701891i \(-0.247660\pi\)
0.712285 + 0.701891i \(0.247660\pi\)
\(600\) 3552.76 0.241735
\(601\) −8569.93 −0.581655 −0.290828 0.956775i \(-0.593931\pi\)
−0.290828 + 0.956775i \(0.593931\pi\)
\(602\) 0 0
\(603\) 11289.2 0.762409
\(604\) −164.216 −0.0110627
\(605\) −5646.95 −0.379473
\(606\) 22968.1 1.53963
\(607\) −19361.0 −1.29463 −0.647313 0.762225i \(-0.724107\pi\)
−0.647313 + 0.762225i \(0.724107\pi\)
\(608\) −432.622 −0.0288572
\(609\) 0 0
\(610\) 2761.54 0.183298
\(611\) 27915.4 1.84834
\(612\) −94.8571 −0.00626531
\(613\) −15907.2 −1.04810 −0.524050 0.851688i \(-0.675579\pi\)
−0.524050 + 0.851688i \(0.675579\pi\)
\(614\) −18509.5 −1.21658
\(615\) −1258.16 −0.0824938
\(616\) 0 0
\(617\) 17496.4 1.14162 0.570810 0.821082i \(-0.306629\pi\)
0.570810 + 0.821082i \(0.306629\pi\)
\(618\) 25919.4 1.68710
\(619\) 22908.6 1.48752 0.743758 0.668449i \(-0.233041\pi\)
0.743758 + 0.668449i \(0.233041\pi\)
\(620\) 91.2270 0.00590930
\(621\) −14620.9 −0.944796
\(622\) −471.482 −0.0303934
\(623\) 0 0
\(624\) −27854.7 −1.78699
\(625\) 625.000 0.0400000
\(626\) −20983.2 −1.33971
\(627\) −16702.3 −1.06383
\(628\) −399.941 −0.0254130
\(629\) 2476.16 0.156965
\(630\) 0 0
\(631\) 27379.1 1.72733 0.863665 0.504066i \(-0.168163\pi\)
0.863665 + 0.504066i \(0.168163\pi\)
\(632\) −8838.67 −0.556303
\(633\) 30906.5 1.94064
\(634\) −9580.37 −0.600134
\(635\) 3376.30 0.210999
\(636\) −360.519 −0.0224772
\(637\) 0 0
\(638\) −5345.23 −0.331692
\(639\) 4492.54 0.278125
\(640\) 6992.02 0.431850
\(641\) −8816.10 −0.543237 −0.271619 0.962405i \(-0.587559\pi\)
−0.271619 + 0.962405i \(0.587559\pi\)
\(642\) 7639.00 0.469606
\(643\) 25760.8 1.57995 0.789975 0.613139i \(-0.210094\pi\)
0.789975 + 0.613139i \(0.210094\pi\)
\(644\) 0 0
\(645\) −11719.1 −0.715411
\(646\) −7017.44 −0.427395
\(647\) −21618.1 −1.31359 −0.656797 0.754067i \(-0.728089\pi\)
−0.656797 + 0.754067i \(0.728089\pi\)
\(648\) −20758.8 −1.25846
\(649\) 1968.43 0.119057
\(650\) −5010.76 −0.302366
\(651\) 0 0
\(652\) −563.619 −0.0338543
\(653\) −25008.3 −1.49870 −0.749349 0.662175i \(-0.769633\pi\)
−0.749349 + 0.662175i \(0.769633\pi\)
\(654\) −25429.3 −1.52044
\(655\) −3474.64 −0.207276
\(656\) −2533.28 −0.150774
\(657\) −2551.33 −0.151502
\(658\) 0 0
\(659\) −29424.3 −1.73932 −0.869658 0.493655i \(-0.835660\pi\)
−0.869658 + 0.493655i \(0.835660\pi\)
\(660\) −271.920 −0.0160371
\(661\) −21022.4 −1.23703 −0.618513 0.785774i \(-0.712265\pi\)
−0.618513 + 0.785774i \(0.712265\pi\)
\(662\) −3064.11 −0.179894
\(663\) 20615.0 1.20757
\(664\) −31488.2 −1.84033
\(665\) 0 0
\(666\) −1737.06 −0.101066
\(667\) −5890.99 −0.341979
\(668\) 107.822 0.00624515
\(669\) 20851.6 1.20504
\(670\) 13596.9 0.784020
\(671\) −9794.48 −0.563504
\(672\) 0 0
\(673\) 5241.08 0.300192 0.150096 0.988671i \(-0.452042\pi\)
0.150096 + 0.988671i \(0.452042\pi\)
\(674\) −11864.9 −0.678069
\(675\) −2390.51 −0.136312
\(676\) −518.360 −0.0294925
\(677\) 5348.26 0.303620 0.151810 0.988410i \(-0.451490\pi\)
0.151810 + 0.988410i \(0.451490\pi\)
\(678\) 22075.9 1.25047
\(679\) 0 0
\(680\) −5294.19 −0.298563
\(681\) −12944.0 −0.728363
\(682\) 14346.5 0.805509
\(683\) 17974.4 1.00699 0.503494 0.863999i \(-0.332048\pi\)
0.503494 + 0.863999i \(0.332048\pi\)
\(684\) −111.025 −0.00620635
\(685\) 12595.5 0.702553
\(686\) 0 0
\(687\) −31142.2 −1.72947
\(688\) −23596.3 −1.30756
\(689\) 23562.4 1.30284
\(690\) 13287.9 0.733134
\(691\) 23734.1 1.30664 0.653319 0.757083i \(-0.273376\pi\)
0.653319 + 0.757083i \(0.273376\pi\)
\(692\) −1.55470 −8.54056e−5 0
\(693\) 0 0
\(694\) −218.638 −0.0119587
\(695\) 8558.05 0.467087
\(696\) −5475.06 −0.298177
\(697\) 1874.85 0.101887
\(698\) 469.472 0.0254581
\(699\) 7149.91 0.386888
\(700\) 0 0
\(701\) −31094.0 −1.67533 −0.837664 0.546186i \(-0.816079\pi\)
−0.837664 + 0.546186i \(0.816079\pi\)
\(702\) 19165.2 1.03040
\(703\) 2898.21 0.155488
\(704\) −25931.5 −1.38826
\(705\) −12103.5 −0.646590
\(706\) 13629.6 0.726566
\(707\) 0 0
\(708\) 43.5100 0.00230961
\(709\) −9053.94 −0.479588 −0.239794 0.970824i \(-0.577080\pi\)
−0.239794 + 0.970824i \(0.577080\pi\)
\(710\) 5410.88 0.286009
\(711\) −4487.63 −0.236708
\(712\) 17153.8 0.902899
\(713\) 15811.3 0.830490
\(714\) 0 0
\(715\) 17771.9 0.929554
\(716\) 220.585 0.0115135
\(717\) 11928.3 0.621296
\(718\) −23633.8 −1.22842
\(719\) −1101.81 −0.0571498 −0.0285749 0.999592i \(-0.509097\pi\)
−0.0285749 + 0.999592i \(0.509097\pi\)
\(720\) 3631.99 0.187995
\(721\) 0 0
\(722\) 10971.5 0.565537
\(723\) −1554.51 −0.0799624
\(724\) 431.943 0.0221727
\(725\) −963.169 −0.0493396
\(726\) −19629.3 −1.00346
\(727\) −6476.47 −0.330397 −0.165199 0.986260i \(-0.552827\pi\)
−0.165199 + 0.986260i \(0.552827\pi\)
\(728\) 0 0
\(729\) 5502.76 0.279569
\(730\) −3072.86 −0.155797
\(731\) 17463.4 0.883593
\(732\) −216.496 −0.0109316
\(733\) −16111.9 −0.811880 −0.405940 0.913900i \(-0.633056\pi\)
−0.405940 + 0.913900i \(0.633056\pi\)
\(734\) 13429.3 0.675317
\(735\) 0 0
\(736\) −1220.73 −0.0611370
\(737\) −48224.7 −2.41028
\(738\) −1315.23 −0.0656022
\(739\) −9172.27 −0.456573 −0.228286 0.973594i \(-0.573312\pi\)
−0.228286 + 0.973594i \(0.573312\pi\)
\(740\) 47.1841 0.00234395
\(741\) 24128.7 1.19621
\(742\) 0 0
\(743\) 3249.07 0.160426 0.0802131 0.996778i \(-0.474440\pi\)
0.0802131 + 0.996778i \(0.474440\pi\)
\(744\) 14695.0 0.724119
\(745\) 3606.88 0.177377
\(746\) −18177.0 −0.892103
\(747\) −15987.4 −0.783063
\(748\) 405.205 0.0198072
\(749\) 0 0
\(750\) 2172.56 0.105774
\(751\) −24191.6 −1.17545 −0.587726 0.809060i \(-0.699977\pi\)
−0.587726 + 0.809060i \(0.699977\pi\)
\(752\) −24370.3 −1.18177
\(753\) 18053.9 0.873731
\(754\) 7721.94 0.372966
\(755\) −4653.46 −0.224313
\(756\) 0 0
\(757\) −21282.7 −1.02184 −0.510919 0.859629i \(-0.670695\pi\)
−0.510919 + 0.859629i \(0.670695\pi\)
\(758\) 30905.8 1.48093
\(759\) −47128.9 −2.25385
\(760\) −6196.55 −0.295753
\(761\) 32224.5 1.53500 0.767501 0.641047i \(-0.221500\pi\)
0.767501 + 0.641047i \(0.221500\pi\)
\(762\) 11736.4 0.557957
\(763\) 0 0
\(764\) 894.283 0.0423482
\(765\) −2688.00 −0.127039
\(766\) 22652.1 1.06848
\(767\) −2843.68 −0.133871
\(768\) −1683.25 −0.0790872
\(769\) −25116.5 −1.17780 −0.588898 0.808208i \(-0.700438\pi\)
−0.588898 + 0.808208i \(0.700438\pi\)
\(770\) 0 0
\(771\) 9254.24 0.432274
\(772\) −607.777 −0.0283347
\(773\) −11824.7 −0.550202 −0.275101 0.961415i \(-0.588711\pi\)
−0.275101 + 0.961415i \(0.588711\pi\)
\(774\) −12250.8 −0.568922
\(775\) 2585.13 0.119820
\(776\) 12200.4 0.564393
\(777\) 0 0
\(778\) −16495.5 −0.760142
\(779\) 2194.41 0.100928
\(780\) 392.827 0.0180326
\(781\) −19191.0 −0.879267
\(782\) −19801.2 −0.905484
\(783\) 3683.94 0.168139
\(784\) 0 0
\(785\) −11333.3 −0.515289
\(786\) −12078.2 −0.548111
\(787\) −20291.0 −0.919053 −0.459527 0.888164i \(-0.651981\pi\)
−0.459527 + 0.888164i \(0.651981\pi\)
\(788\) 312.240 0.0141156
\(789\) 20566.9 0.928012
\(790\) −5404.96 −0.243417
\(791\) 0 0
\(792\) −13172.4 −0.590986
\(793\) 14149.5 0.633623
\(794\) 32338.0 1.44538
\(795\) −10216.2 −0.455760
\(796\) −36.3427 −0.00161826
\(797\) −37140.0 −1.65065 −0.825325 0.564658i \(-0.809008\pi\)
−0.825325 + 0.564658i \(0.809008\pi\)
\(798\) 0 0
\(799\) 18036.2 0.798594
\(800\) −199.588 −0.00882065
\(801\) 8709.42 0.384185
\(802\) 36555.3 1.60949
\(803\) 10898.6 0.478959
\(804\) −1065.95 −0.0467577
\(805\) 0 0
\(806\) −20725.6 −0.905741
\(807\) 25539.2 1.11403
\(808\) −30222.5 −1.31587
\(809\) −3834.95 −0.166662 −0.0833311 0.996522i \(-0.526556\pi\)
−0.0833311 + 0.996522i \(0.526556\pi\)
\(810\) −12694.2 −0.550655
\(811\) −19628.9 −0.849894 −0.424947 0.905218i \(-0.639707\pi\)
−0.424947 + 0.905218i \(0.639707\pi\)
\(812\) 0 0
\(813\) −10739.2 −0.463272
\(814\) 7420.27 0.319509
\(815\) −15971.5 −0.686450
\(816\) −17997.0 −0.772086
\(817\) 20439.9 0.875278
\(818\) 1229.96 0.0525728
\(819\) 0 0
\(820\) 35.7260 0.00152147
\(821\) −12225.4 −0.519696 −0.259848 0.965650i \(-0.583672\pi\)
−0.259848 + 0.965650i \(0.583672\pi\)
\(822\) 43783.1 1.85780
\(823\) 25094.4 1.06286 0.531431 0.847102i \(-0.321655\pi\)
0.531431 + 0.847102i \(0.321655\pi\)
\(824\) −34105.9 −1.44191
\(825\) −7705.51 −0.325178
\(826\) 0 0
\(827\) 36846.0 1.54929 0.774644 0.632397i \(-0.217929\pi\)
0.774644 + 0.632397i \(0.217929\pi\)
\(828\) −313.280 −0.0131488
\(829\) −35126.1 −1.47163 −0.735814 0.677184i \(-0.763200\pi\)
−0.735814 + 0.677184i \(0.763200\pi\)
\(830\) −19255.4 −0.805260
\(831\) −28261.4 −1.17976
\(832\) 37461.7 1.56100
\(833\) 0 0
\(834\) 29748.6 1.23514
\(835\) 3055.40 0.126630
\(836\) 474.270 0.0196208
\(837\) −9887.64 −0.408324
\(838\) −4681.48 −0.192982
\(839\) 20946.7 0.861931 0.430965 0.902368i \(-0.358173\pi\)
0.430965 + 0.902368i \(0.358173\pi\)
\(840\) 0 0
\(841\) −22904.7 −0.939140
\(842\) −2252.79 −0.0922045
\(843\) −23498.4 −0.960055
\(844\) −877.607 −0.0357921
\(845\) −14689.0 −0.598007
\(846\) −12652.6 −0.514193
\(847\) 0 0
\(848\) −20570.1 −0.832994
\(849\) 5927.92 0.239630
\(850\) −3237.47 −0.130640
\(851\) 8177.89 0.329418
\(852\) −424.195 −0.0170571
\(853\) −30591.5 −1.22794 −0.613969 0.789330i \(-0.710428\pi\)
−0.613969 + 0.789330i \(0.710428\pi\)
\(854\) 0 0
\(855\) −3146.16 −0.125844
\(856\) −10051.7 −0.401357
\(857\) −6484.50 −0.258467 −0.129233 0.991614i \(-0.541252\pi\)
−0.129233 + 0.991614i \(0.541252\pi\)
\(858\) 61776.8 2.45807
\(859\) −43597.9 −1.73171 −0.865857 0.500292i \(-0.833226\pi\)
−0.865857 + 0.500292i \(0.833226\pi\)
\(860\) 332.771 0.0131947
\(861\) 0 0
\(862\) −2540.61 −0.100387
\(863\) −5402.94 −0.213115 −0.106558 0.994307i \(-0.533983\pi\)
−0.106558 + 0.994307i \(0.533983\pi\)
\(864\) 763.387 0.0300590
\(865\) −44.0560 −0.00173173
\(866\) −4095.93 −0.160722
\(867\) −17209.1 −0.674109
\(868\) 0 0
\(869\) 19170.0 0.748329
\(870\) −3348.07 −0.130471
\(871\) 69667.3 2.71020
\(872\) 33461.0 1.29947
\(873\) 6194.47 0.240150
\(874\) −23176.1 −0.896962
\(875\) 0 0
\(876\) 240.902 0.00929146
\(877\) 48981.8 1.88597 0.942987 0.332831i \(-0.108004\pi\)
0.942987 + 0.332831i \(0.108004\pi\)
\(878\) 20737.0 0.797085
\(879\) 27732.6 1.06416
\(880\) −15514.9 −0.594328
\(881\) 13125.8 0.501952 0.250976 0.967993i \(-0.419248\pi\)
0.250976 + 0.967993i \(0.419248\pi\)
\(882\) 0 0
\(883\) −7496.22 −0.285694 −0.142847 0.989745i \(-0.545626\pi\)
−0.142847 + 0.989745i \(0.545626\pi\)
\(884\) −585.376 −0.0222719
\(885\) 1232.96 0.0468310
\(886\) −24213.9 −0.918151
\(887\) 46027.7 1.74235 0.871173 0.490977i \(-0.163360\pi\)
0.871173 + 0.490977i \(0.163360\pi\)
\(888\) 7600.50 0.287225
\(889\) 0 0
\(890\) 10489.7 0.395075
\(891\) 45023.2 1.69286
\(892\) −592.094 −0.0222251
\(893\) 21110.4 0.791078
\(894\) 12537.9 0.469048
\(895\) 6250.80 0.233454
\(896\) 0 0
\(897\) 68084.3 2.53430
\(898\) −19967.4 −0.742006
\(899\) −3983.87 −0.147797
\(900\) −51.2208 −0.00189707
\(901\) 15223.7 0.562903
\(902\) 5618.35 0.207395
\(903\) 0 0
\(904\) −29048.4 −1.06874
\(905\) 12240.1 0.449586
\(906\) −16175.9 −0.593164
\(907\) 11769.4 0.430866 0.215433 0.976519i \(-0.430884\pi\)
0.215433 + 0.976519i \(0.430884\pi\)
\(908\) 367.552 0.0134335
\(909\) −15344.8 −0.559905
\(910\) 0 0
\(911\) 4591.84 0.166997 0.0834986 0.996508i \(-0.473391\pi\)
0.0834986 + 0.996508i \(0.473391\pi\)
\(912\) −21064.5 −0.764819
\(913\) 68294.1 2.47558
\(914\) −27846.4 −1.00774
\(915\) −6134.92 −0.221655
\(916\) 884.301 0.0318975
\(917\) 0 0
\(918\) 12382.7 0.445196
\(919\) 5444.40 0.195424 0.0977118 0.995215i \(-0.468848\pi\)
0.0977118 + 0.995215i \(0.468848\pi\)
\(920\) −17484.9 −0.626586
\(921\) 41119.9 1.47117
\(922\) 29033.9 1.03707
\(923\) 27724.1 0.988677
\(924\) 0 0
\(925\) 1337.08 0.0475273
\(926\) 7976.58 0.283074
\(927\) −17316.5 −0.613536
\(928\) 307.580 0.0108802
\(929\) 22114.5 0.781004 0.390502 0.920602i \(-0.372302\pi\)
0.390502 + 0.920602i \(0.372302\pi\)
\(930\) 8986.17 0.316847
\(931\) 0 0
\(932\) −203.026 −0.00713555
\(933\) 1047.42 0.0367536
\(934\) 35154.8 1.23159
\(935\) 11482.5 0.401622
\(936\) 19029.4 0.664524
\(937\) −38332.7 −1.33647 −0.668236 0.743949i \(-0.732950\pi\)
−0.668236 + 0.743949i \(0.732950\pi\)
\(938\) 0 0
\(939\) 46615.3 1.62006
\(940\) 343.687 0.0119254
\(941\) −41340.7 −1.43216 −0.716082 0.698016i \(-0.754066\pi\)
−0.716082 + 0.698016i \(0.754066\pi\)
\(942\) −39395.6 −1.36261
\(943\) 6191.99 0.213827
\(944\) 2482.54 0.0855932
\(945\) 0 0
\(946\) 52332.3 1.79859
\(947\) −4417.34 −0.151578 −0.0757889 0.997124i \(-0.524148\pi\)
−0.0757889 + 0.997124i \(0.524148\pi\)
\(948\) 423.731 0.0145170
\(949\) −15744.6 −0.538558
\(950\) −3789.27 −0.129411
\(951\) 21283.3 0.725720
\(952\) 0 0
\(953\) −1981.88 −0.0673656 −0.0336828 0.999433i \(-0.510724\pi\)
−0.0336828 + 0.999433i \(0.510724\pi\)
\(954\) −10679.6 −0.362438
\(955\) 25341.6 0.858677
\(956\) −338.710 −0.0114589
\(957\) 11874.7 0.401103
\(958\) −15494.4 −0.522549
\(959\) 0 0
\(960\) −16242.6 −0.546071
\(961\) −19098.3 −0.641077
\(962\) −10719.6 −0.359267
\(963\) −5103.54 −0.170778
\(964\) 44.1412 0.00147478
\(965\) −17222.8 −0.574531
\(966\) 0 0
\(967\) 12406.6 0.412586 0.206293 0.978490i \(-0.433860\pi\)
0.206293 + 0.978490i \(0.433860\pi\)
\(968\) 25829.2 0.857625
\(969\) 15589.6 0.516833
\(970\) 7460.71 0.246958
\(971\) −22191.2 −0.733418 −0.366709 0.930336i \(-0.619516\pi\)
−0.366709 + 0.930336i \(0.619516\pi\)
\(972\) 539.650 0.0178079
\(973\) 0 0
\(974\) 30627.5 1.00756
\(975\) 11131.7 0.365640
\(976\) −12352.6 −0.405119
\(977\) −1300.82 −0.0425966 −0.0212983 0.999773i \(-0.506780\pi\)
−0.0212983 + 0.999773i \(0.506780\pi\)
\(978\) −55518.4 −1.81522
\(979\) −37204.4 −1.21456
\(980\) 0 0
\(981\) 16989.1 0.552925
\(982\) −15053.1 −0.489169
\(983\) −38899.7 −1.26216 −0.631082 0.775716i \(-0.717389\pi\)
−0.631082 + 0.775716i \(0.717389\pi\)
\(984\) 5754.81 0.186440
\(985\) 8848.06 0.286216
\(986\) 4989.16 0.161143
\(987\) 0 0
\(988\) −685.149 −0.0220623
\(989\) 57675.5 1.85437
\(990\) −8055.09 −0.258594
\(991\) −2713.22 −0.0869711 −0.0434856 0.999054i \(-0.513846\pi\)
−0.0434856 + 0.999054i \(0.513846\pi\)
\(992\) −825.541 −0.0264223
\(993\) 6807.09 0.217539
\(994\) 0 0
\(995\) −1029.86 −0.0328127
\(996\) 1509.56 0.0480244
\(997\) 51048.1 1.62158 0.810788 0.585340i \(-0.199039\pi\)
0.810788 + 0.585340i \(0.199039\pi\)
\(998\) −36761.1 −1.16598
\(999\) −5114.06 −0.161964
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 245.4.a.m.1.2 5
3.2 odd 2 2205.4.a.bu.1.4 5
5.4 even 2 1225.4.a.bg.1.4 5
7.2 even 3 35.4.e.c.11.4 10
7.3 odd 6 245.4.e.o.226.4 10
7.4 even 3 35.4.e.c.16.4 yes 10
7.5 odd 6 245.4.e.o.116.4 10
7.6 odd 2 245.4.a.n.1.2 5
21.2 odd 6 315.4.j.g.46.2 10
21.11 odd 6 315.4.j.g.226.2 10
21.20 even 2 2205.4.a.bt.1.4 5
28.11 odd 6 560.4.q.n.401.5 10
28.23 odd 6 560.4.q.n.81.5 10
35.2 odd 12 175.4.k.d.74.4 20
35.4 even 6 175.4.e.d.51.2 10
35.9 even 6 175.4.e.d.151.2 10
35.18 odd 12 175.4.k.d.149.4 20
35.23 odd 12 175.4.k.d.74.7 20
35.32 odd 12 175.4.k.d.149.7 20
35.34 odd 2 1225.4.a.bf.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.e.c.11.4 10 7.2 even 3
35.4.e.c.16.4 yes 10 7.4 even 3
175.4.e.d.51.2 10 35.4 even 6
175.4.e.d.151.2 10 35.9 even 6
175.4.k.d.74.4 20 35.2 odd 12
175.4.k.d.74.7 20 35.23 odd 12
175.4.k.d.149.4 20 35.18 odd 12
175.4.k.d.149.7 20 35.32 odd 12
245.4.a.m.1.2 5 1.1 even 1 trivial
245.4.a.n.1.2 5 7.6 odd 2
245.4.e.o.116.4 10 7.5 odd 6
245.4.e.o.226.4 10 7.3 odd 6
315.4.j.g.46.2 10 21.2 odd 6
315.4.j.g.226.2 10 21.11 odd 6
560.4.q.n.81.5 10 28.23 odd 6
560.4.q.n.401.5 10 28.11 odd 6
1225.4.a.bf.1.4 5 35.34 odd 2
1225.4.a.bg.1.4 5 5.4 even 2
2205.4.a.bt.1.4 5 21.20 even 2
2205.4.a.bu.1.4 5 3.2 odd 2