Properties

Label 115.4.a.e.1.5
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
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] = [115,4,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, 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("115.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(6.78521965066\)
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: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(4.60878\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.60878 q^{2} -1.89520 q^{3} +23.4584 q^{4} -5.00000 q^{5} -10.6297 q^{6} +11.4426 q^{7} +86.7031 q^{8} -23.4082 q^{9} +O(q^{10})\) \(q+5.60878 q^{2} -1.89520 q^{3} +23.4584 q^{4} -5.00000 q^{5} -10.6297 q^{6} +11.4426 q^{7} +86.7031 q^{8} -23.4082 q^{9} -28.0439 q^{10} +37.7245 q^{11} -44.4584 q^{12} -8.69346 q^{13} +64.1788 q^{14} +9.47598 q^{15} +298.631 q^{16} -105.687 q^{17} -131.292 q^{18} -128.279 q^{19} -117.292 q^{20} -21.6859 q^{21} +211.588 q^{22} -23.0000 q^{23} -164.319 q^{24} +25.0000 q^{25} -48.7597 q^{26} +95.5335 q^{27} +268.425 q^{28} -133.383 q^{29} +53.1487 q^{30} +106.008 q^{31} +981.333 q^{32} -71.4953 q^{33} -592.773 q^{34} -57.2128 q^{35} -549.121 q^{36} -248.835 q^{37} -719.491 q^{38} +16.4758 q^{39} -433.515 q^{40} +134.233 q^{41} -121.631 q^{42} +108.684 q^{43} +884.957 q^{44} +117.041 q^{45} -129.002 q^{46} -76.2000 q^{47} -565.965 q^{48} -212.068 q^{49} +140.220 q^{50} +200.297 q^{51} -203.935 q^{52} +476.207 q^{53} +535.827 q^{54} -188.622 q^{55} +992.105 q^{56} +243.114 q^{57} -748.118 q^{58} +608.000 q^{59} +222.292 q^{60} -366.273 q^{61} +594.575 q^{62} -267.850 q^{63} +3115.03 q^{64} +43.4673 q^{65} -401.001 q^{66} +136.041 q^{67} -2479.24 q^{68} +43.5895 q^{69} -320.894 q^{70} -152.874 q^{71} -2029.57 q^{72} +1228.16 q^{73} -1395.66 q^{74} -47.3799 q^{75} -3009.23 q^{76} +431.664 q^{77} +92.4092 q^{78} -364.637 q^{79} -1493.16 q^{80} +450.968 q^{81} +752.882 q^{82} -762.744 q^{83} -508.717 q^{84} +528.433 q^{85} +609.583 q^{86} +252.788 q^{87} +3270.83 q^{88} +271.222 q^{89} +656.459 q^{90} -99.4754 q^{91} -539.544 q^{92} -200.906 q^{93} -427.389 q^{94} +641.396 q^{95} -1859.82 q^{96} +574.510 q^{97} -1189.44 q^{98} -883.063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19} - 110 q^{20} - 60 q^{21} + 193 q^{22} - 115 q^{23} + 105 q^{24} + 125 q^{25} - 257 q^{26} + 577 q^{27} + 17 q^{28} + 401 q^{29} - 95 q^{30} + 32 q^{31} + 670 q^{32} + 189 q^{33} - 663 q^{34} + 15 q^{35} - 659 q^{36} - 38 q^{37} - 875 q^{38} + 335 q^{39} - 690 q^{40} - 12 q^{41} - 798 q^{42} - 566 q^{43} + 47 q^{44} - 385 q^{45} - 138 q^{46} + 919 q^{47} - 773 q^{48} - 738 q^{49} + 150 q^{50} - 993 q^{51} - 305 q^{52} + 1156 q^{53} - 8 q^{54} - 115 q^{55} + 343 q^{56} + 114 q^{57} - 1042 q^{58} + 1324 q^{59} - 235 q^{60} - 1673 q^{61} + 565 q^{62} + 270 q^{63} + 2466 q^{64} - 660 q^{65} - 2781 q^{66} + 558 q^{67} - 2267 q^{68} - 92 q^{69} - 465 q^{70} - 108 q^{71} - 789 q^{72} + 1173 q^{73} + 1458 q^{74} + 100 q^{75} - 3477 q^{76} + 2608 q^{77} + 704 q^{78} + 656 q^{79} - 1410 q^{80} - 319 q^{81} + 3505 q^{82} - 82 q^{83} - 718 q^{84} - 115 q^{85} + 112 q^{86} + 2389 q^{87} + 2397 q^{88} + 570 q^{89} + 75 q^{90} - 1589 q^{91} - 506 q^{92} + 911 q^{93} - 948 q^{94} + 805 q^{95} - 5991 q^{96} + 633 q^{97} - 2555 q^{98} + 2021 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.60878 1.98300 0.991502 0.130090i \(-0.0415267\pi\)
0.991502 + 0.130090i \(0.0415267\pi\)
\(3\) −1.89520 −0.364731 −0.182365 0.983231i \(-0.558375\pi\)
−0.182365 + 0.983231i \(0.558375\pi\)
\(4\) 23.4584 2.93231
\(5\) −5.00000 −0.447214
\(6\) −10.6297 −0.723262
\(7\) 11.4426 0.617840 0.308920 0.951088i \(-0.400032\pi\)
0.308920 + 0.951088i \(0.400032\pi\)
\(8\) 86.7031 3.83177
\(9\) −23.4082 −0.866972
\(10\) −28.0439 −0.886826
\(11\) 37.7245 1.03403 0.517016 0.855976i \(-0.327043\pi\)
0.517016 + 0.855976i \(0.327043\pi\)
\(12\) −44.4584 −1.06950
\(13\) −8.69346 −0.185472 −0.0927358 0.995691i \(-0.529561\pi\)
−0.0927358 + 0.995691i \(0.529561\pi\)
\(14\) 64.1788 1.22518
\(15\) 9.47598 0.163112
\(16\) 298.631 4.66611
\(17\) −105.687 −1.50781 −0.753905 0.656983i \(-0.771832\pi\)
−0.753905 + 0.656983i \(0.771832\pi\)
\(18\) −131.292 −1.71921
\(19\) −128.279 −1.54891 −0.774455 0.632629i \(-0.781976\pi\)
−0.774455 + 0.632629i \(0.781976\pi\)
\(20\) −117.292 −1.31137
\(21\) −21.6859 −0.225345
\(22\) 211.588 2.05049
\(23\) −23.0000 −0.208514
\(24\) −164.319 −1.39756
\(25\) 25.0000 0.200000
\(26\) −48.7597 −0.367791
\(27\) 95.5335 0.680942
\(28\) 268.425 1.81170
\(29\) −133.383 −0.854092 −0.427046 0.904230i \(-0.640446\pi\)
−0.427046 + 0.904230i \(0.640446\pi\)
\(30\) 53.1487 0.323453
\(31\) 106.008 0.614179 0.307090 0.951681i \(-0.400645\pi\)
0.307090 + 0.951681i \(0.400645\pi\)
\(32\) 981.333 5.42115
\(33\) −71.4953 −0.377143
\(34\) −592.773 −2.98999
\(35\) −57.2128 −0.276306
\(36\) −549.121 −2.54223
\(37\) −248.835 −1.10563 −0.552814 0.833305i \(-0.686446\pi\)
−0.552814 + 0.833305i \(0.686446\pi\)
\(38\) −719.491 −3.07150
\(39\) 16.4758 0.0676472
\(40\) −433.515 −1.71362
\(41\) 134.233 0.511308 0.255654 0.966768i \(-0.417709\pi\)
0.255654 + 0.966768i \(0.417709\pi\)
\(42\) −121.631 −0.446860
\(43\) 108.684 0.385444 0.192722 0.981253i \(-0.438268\pi\)
0.192722 + 0.981253i \(0.438268\pi\)
\(44\) 884.957 3.03210
\(45\) 117.041 0.387721
\(46\) −129.002 −0.413485
\(47\) −76.2000 −0.236488 −0.118244 0.992985i \(-0.537726\pi\)
−0.118244 + 0.992985i \(0.537726\pi\)
\(48\) −565.965 −1.70187
\(49\) −212.068 −0.618274
\(50\) 140.220 0.396601
\(51\) 200.297 0.549945
\(52\) −203.935 −0.543859
\(53\) 476.207 1.23419 0.617095 0.786889i \(-0.288309\pi\)
0.617095 + 0.786889i \(0.288309\pi\)
\(54\) 535.827 1.35031
\(55\) −188.622 −0.462433
\(56\) 992.105 2.36742
\(57\) 243.114 0.564935
\(58\) −748.118 −1.69367
\(59\) 608.000 1.34161 0.670804 0.741635i \(-0.265949\pi\)
0.670804 + 0.741635i \(0.265949\pi\)
\(60\) 222.292 0.478296
\(61\) −366.273 −0.768794 −0.384397 0.923168i \(-0.625591\pi\)
−0.384397 + 0.923168i \(0.625591\pi\)
\(62\) 594.575 1.21792
\(63\) −267.850 −0.535650
\(64\) 3115.03 6.08405
\(65\) 43.4673 0.0829454
\(66\) −401.001 −0.747877
\(67\) 136.041 0.248060 0.124030 0.992278i \(-0.460418\pi\)
0.124030 + 0.992278i \(0.460418\pi\)
\(68\) −2479.24 −4.42136
\(69\) 43.5895 0.0760516
\(70\) −320.894 −0.547917
\(71\) −152.874 −0.255533 −0.127766 0.991804i \(-0.540781\pi\)
−0.127766 + 0.991804i \(0.540781\pi\)
\(72\) −2029.57 −3.32204
\(73\) 1228.16 1.96911 0.984556 0.175070i \(-0.0560151\pi\)
0.984556 + 0.175070i \(0.0560151\pi\)
\(74\) −1395.66 −2.19246
\(75\) −47.3799 −0.0729461
\(76\) −3009.23 −4.54188
\(77\) 431.664 0.638867
\(78\) 92.4092 0.134145
\(79\) −364.637 −0.519302 −0.259651 0.965702i \(-0.583608\pi\)
−0.259651 + 0.965702i \(0.583608\pi\)
\(80\) −1493.16 −2.08675
\(81\) 450.968 0.618611
\(82\) 752.882 1.01393
\(83\) −762.744 −1.00870 −0.504350 0.863499i \(-0.668268\pi\)
−0.504350 + 0.863499i \(0.668268\pi\)
\(84\) −508.717 −0.660781
\(85\) 528.433 0.674313
\(86\) 609.583 0.764338
\(87\) 252.788 0.311514
\(88\) 3270.83 3.96217
\(89\) 271.222 0.323028 0.161514 0.986870i \(-0.448362\pi\)
0.161514 + 0.986870i \(0.448362\pi\)
\(90\) 656.459 0.768853
\(91\) −99.4754 −0.114592
\(92\) −539.544 −0.611428
\(93\) −200.906 −0.224010
\(94\) −427.389 −0.468956
\(95\) 641.396 0.692694
\(96\) −1859.82 −1.97726
\(97\) 574.510 0.601367 0.300684 0.953724i \(-0.402785\pi\)
0.300684 + 0.953724i \(0.402785\pi\)
\(98\) −1189.44 −1.22604
\(99\) −883.063 −0.896477
\(100\) 586.461 0.586461
\(101\) 1372.25 1.35192 0.675958 0.736940i \(-0.263730\pi\)
0.675958 + 0.736940i \(0.263730\pi\)
\(102\) 1123.42 1.09054
\(103\) −242.428 −0.231914 −0.115957 0.993254i \(-0.536993\pi\)
−0.115957 + 0.993254i \(0.536993\pi\)
\(104\) −753.749 −0.710685
\(105\) 108.429 0.100777
\(106\) 2670.94 2.44740
\(107\) 650.896 0.588079 0.294039 0.955793i \(-0.405000\pi\)
0.294039 + 0.955793i \(0.405000\pi\)
\(108\) 2241.07 1.99673
\(109\) −1230.43 −1.08123 −0.540613 0.841271i \(-0.681808\pi\)
−0.540613 + 0.841271i \(0.681808\pi\)
\(110\) −1057.94 −0.917007
\(111\) 471.591 0.403256
\(112\) 3417.10 2.88291
\(113\) 238.959 0.198932 0.0994662 0.995041i \(-0.468286\pi\)
0.0994662 + 0.995041i \(0.468286\pi\)
\(114\) 1363.58 1.12027
\(115\) 115.000 0.0932505
\(116\) −3128.97 −2.50446
\(117\) 203.498 0.160799
\(118\) 3410.14 2.66041
\(119\) −1209.33 −0.931586
\(120\) 821.597 0.625010
\(121\) 92.1354 0.0692227
\(122\) −2054.34 −1.52452
\(123\) −254.397 −0.186490
\(124\) 2486.78 1.80096
\(125\) −125.000 −0.0894427
\(126\) −1502.31 −1.06220
\(127\) 2608.48 1.82256 0.911280 0.411788i \(-0.135096\pi\)
0.911280 + 0.411788i \(0.135096\pi\)
\(128\) 9620.89 6.64355
\(129\) −205.977 −0.140583
\(130\) 243.799 0.164481
\(131\) −936.409 −0.624538 −0.312269 0.949994i \(-0.601089\pi\)
−0.312269 + 0.949994i \(0.601089\pi\)
\(132\) −1677.17 −1.10590
\(133\) −1467.84 −0.956979
\(134\) 763.023 0.491904
\(135\) −477.667 −0.304526
\(136\) −9163.36 −5.77758
\(137\) 415.511 0.259121 0.129560 0.991572i \(-0.458643\pi\)
0.129560 + 0.991572i \(0.458643\pi\)
\(138\) 244.484 0.150811
\(139\) −949.629 −0.579471 −0.289736 0.957107i \(-0.593567\pi\)
−0.289736 + 0.957107i \(0.593567\pi\)
\(140\) −1342.12 −0.810215
\(141\) 144.414 0.0862542
\(142\) −857.439 −0.506723
\(143\) −327.956 −0.191784
\(144\) −6990.43 −4.04539
\(145\) 666.917 0.381962
\(146\) 6888.48 3.90476
\(147\) 401.910 0.225503
\(148\) −5837.28 −3.24204
\(149\) 2209.84 1.21502 0.607508 0.794314i \(-0.292169\pi\)
0.607508 + 0.794314i \(0.292169\pi\)
\(150\) −265.744 −0.144652
\(151\) −1384.25 −0.746018 −0.373009 0.927828i \(-0.621674\pi\)
−0.373009 + 0.927828i \(0.621674\pi\)
\(152\) −11122.2 −5.93507
\(153\) 2473.94 1.30723
\(154\) 2421.11 1.26688
\(155\) −530.039 −0.274669
\(156\) 386.497 0.198362
\(157\) 561.399 0.285379 0.142690 0.989767i \(-0.454425\pi\)
0.142690 + 0.989767i \(0.454425\pi\)
\(158\) −2045.17 −1.02978
\(159\) −902.506 −0.450147
\(160\) −4906.66 −2.42441
\(161\) −263.179 −0.128829
\(162\) 2529.38 1.22671
\(163\) 2134.47 1.02567 0.512836 0.858487i \(-0.328595\pi\)
0.512836 + 0.858487i \(0.328595\pi\)
\(164\) 3148.89 1.49931
\(165\) 357.476 0.168664
\(166\) −4278.07 −2.00026
\(167\) −1315.64 −0.609623 −0.304812 0.952413i \(-0.598594\pi\)
−0.304812 + 0.952413i \(0.598594\pi\)
\(168\) −1880.23 −0.863471
\(169\) −2121.42 −0.965600
\(170\) 2963.87 1.33717
\(171\) 3002.79 1.34286
\(172\) 2549.55 1.13024
\(173\) 676.565 0.297331 0.148666 0.988888i \(-0.452502\pi\)
0.148666 + 0.988888i \(0.452502\pi\)
\(174\) 1417.83 0.617733
\(175\) 286.064 0.123568
\(176\) 11265.7 4.82491
\(177\) −1152.28 −0.489325
\(178\) 1521.23 0.640566
\(179\) −3737.96 −1.56083 −0.780414 0.625263i \(-0.784992\pi\)
−0.780414 + 0.625263i \(0.784992\pi\)
\(180\) 2745.60 1.13692
\(181\) −1873.40 −0.769330 −0.384665 0.923056i \(-0.625683\pi\)
−0.384665 + 0.923056i \(0.625683\pi\)
\(182\) −557.936 −0.227236
\(183\) 694.158 0.280403
\(184\) −1994.17 −0.798979
\(185\) 1244.18 0.494452
\(186\) −1126.84 −0.444213
\(187\) −3986.97 −1.55912
\(188\) −1787.53 −0.693454
\(189\) 1093.15 0.420713
\(190\) 3597.45 1.37361
\(191\) −5158.92 −1.95438 −0.977190 0.212366i \(-0.931883\pi\)
−0.977190 + 0.212366i \(0.931883\pi\)
\(192\) −5903.60 −2.21904
\(193\) −4806.41 −1.79261 −0.896303 0.443442i \(-0.853757\pi\)
−0.896303 + 0.443442i \(0.853757\pi\)
\(194\) 3222.30 1.19251
\(195\) −82.3790 −0.0302527
\(196\) −4974.78 −1.81297
\(197\) 3202.59 1.15825 0.579125 0.815239i \(-0.303394\pi\)
0.579125 + 0.815239i \(0.303394\pi\)
\(198\) −4952.91 −1.77772
\(199\) −2210.46 −0.787415 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(200\) 2167.58 0.766354
\(201\) −257.824 −0.0904751
\(202\) 7696.62 2.68085
\(203\) −1526.25 −0.527692
\(204\) 4698.65 1.61261
\(205\) −671.163 −0.228664
\(206\) −1359.73 −0.459886
\(207\) 538.389 0.180776
\(208\) −2596.14 −0.865431
\(209\) −4839.27 −1.60162
\(210\) 608.157 0.199842
\(211\) 153.164 0.0499726 0.0249863 0.999688i \(-0.492046\pi\)
0.0249863 + 0.999688i \(0.492046\pi\)
\(212\) 11171.1 3.61902
\(213\) 289.727 0.0932007
\(214\) 3650.73 1.16616
\(215\) −543.419 −0.172376
\(216\) 8283.05 2.60921
\(217\) 1213.00 0.379465
\(218\) −6901.21 −2.14408
\(219\) −2327.60 −0.718195
\(220\) −4424.79 −1.35600
\(221\) 918.782 0.279656
\(222\) 2645.05 0.799659
\(223\) −3068.41 −0.921416 −0.460708 0.887552i \(-0.652405\pi\)
−0.460708 + 0.887552i \(0.652405\pi\)
\(224\) 11229.0 3.34940
\(225\) −585.206 −0.173394
\(226\) 1340.27 0.394484
\(227\) 4540.20 1.32750 0.663752 0.747953i \(-0.268963\pi\)
0.663752 + 0.747953i \(0.268963\pi\)
\(228\) 5703.09 1.65656
\(229\) 1476.25 0.425996 0.212998 0.977053i \(-0.431677\pi\)
0.212998 + 0.977053i \(0.431677\pi\)
\(230\) 645.010 0.184916
\(231\) −818.089 −0.233014
\(232\) −11564.7 −3.27269
\(233\) −90.7306 −0.0255106 −0.0127553 0.999919i \(-0.504060\pi\)
−0.0127553 + 0.999919i \(0.504060\pi\)
\(234\) 1141.38 0.318864
\(235\) 381.000 0.105760
\(236\) 14262.7 3.93400
\(237\) 691.059 0.189405
\(238\) −6782.85 −1.84734
\(239\) −1619.16 −0.438221 −0.219111 0.975700i \(-0.570316\pi\)
−0.219111 + 0.975700i \(0.570316\pi\)
\(240\) 2829.82 0.761101
\(241\) −6447.48 −1.72331 −0.861657 0.507491i \(-0.830573\pi\)
−0.861657 + 0.507491i \(0.830573\pi\)
\(242\) 516.768 0.137269
\(243\) −3434.08 −0.906568
\(244\) −8592.19 −2.25434
\(245\) 1060.34 0.276500
\(246\) −1426.86 −0.369810
\(247\) 1115.19 0.287279
\(248\) 9191.20 2.35339
\(249\) 1445.55 0.367904
\(250\) −701.098 −0.177365
\(251\) 3428.17 0.862089 0.431044 0.902331i \(-0.358145\pi\)
0.431044 + 0.902331i \(0.358145\pi\)
\(252\) −6283.35 −1.57069
\(253\) −867.663 −0.215611
\(254\) 14630.4 3.61414
\(255\) −1001.48 −0.245943
\(256\) 29041.2 7.09014
\(257\) 2261.08 0.548803 0.274402 0.961615i \(-0.411520\pi\)
0.274402 + 0.961615i \(0.411520\pi\)
\(258\) −1155.28 −0.278777
\(259\) −2847.31 −0.683101
\(260\) 1019.67 0.243221
\(261\) 3122.27 0.740474
\(262\) −5252.11 −1.23846
\(263\) 5319.64 1.24724 0.623618 0.781729i \(-0.285662\pi\)
0.623618 + 0.781729i \(0.285662\pi\)
\(264\) −6198.86 −1.44513
\(265\) −2381.04 −0.551947
\(266\) −8232.81 −1.89769
\(267\) −514.019 −0.117818
\(268\) 3191.31 0.727388
\(269\) −1992.51 −0.451620 −0.225810 0.974171i \(-0.572503\pi\)
−0.225810 + 0.974171i \(0.572503\pi\)
\(270\) −2679.13 −0.603877
\(271\) 3950.62 0.885546 0.442773 0.896634i \(-0.353995\pi\)
0.442773 + 0.896634i \(0.353995\pi\)
\(272\) −31561.3 −7.03561
\(273\) 188.525 0.0417951
\(274\) 2330.51 0.513837
\(275\) 943.112 0.206806
\(276\) 1022.54 0.223007
\(277\) −178.126 −0.0386375 −0.0193187 0.999813i \(-0.506150\pi\)
−0.0193187 + 0.999813i \(0.506150\pi\)
\(278\) −5326.27 −1.14909
\(279\) −2481.46 −0.532476
\(280\) −4960.53 −1.05874
\(281\) 3523.49 0.748020 0.374010 0.927425i \(-0.377983\pi\)
0.374010 + 0.927425i \(0.377983\pi\)
\(282\) 809.986 0.171043
\(283\) −699.284 −0.146884 −0.0734419 0.997300i \(-0.523398\pi\)
−0.0734419 + 0.997300i \(0.523398\pi\)
\(284\) −3586.19 −0.749301
\(285\) −1215.57 −0.252647
\(286\) −1839.43 −0.380308
\(287\) 1535.96 0.315906
\(288\) −22971.3 −4.69998
\(289\) 6256.67 1.27349
\(290\) 3740.59 0.757431
\(291\) −1088.81 −0.219337
\(292\) 28810.7 5.77404
\(293\) 8552.97 1.70536 0.852678 0.522436i \(-0.174977\pi\)
0.852678 + 0.522436i \(0.174977\pi\)
\(294\) 2254.23 0.447174
\(295\) −3040.00 −0.599985
\(296\) −21574.8 −4.23651
\(297\) 3603.95 0.704116
\(298\) 12394.5 2.40938
\(299\) 199.949 0.0386735
\(300\) −1111.46 −0.213900
\(301\) 1243.62 0.238143
\(302\) −7763.96 −1.47936
\(303\) −2600.67 −0.493085
\(304\) −38308.2 −7.22739
\(305\) 1831.36 0.343815
\(306\) 13875.8 2.59224
\(307\) −5621.73 −1.04511 −0.522555 0.852606i \(-0.675021\pi\)
−0.522555 + 0.852606i \(0.675021\pi\)
\(308\) 10126.2 1.87335
\(309\) 459.448 0.0845861
\(310\) −2972.87 −0.544671
\(311\) −6533.62 −1.19128 −0.595639 0.803252i \(-0.703101\pi\)
−0.595639 + 0.803252i \(0.703101\pi\)
\(312\) 1428.50 0.259208
\(313\) −2713.24 −0.489973 −0.244987 0.969526i \(-0.578784\pi\)
−0.244987 + 0.969526i \(0.578784\pi\)
\(314\) 3148.77 0.565908
\(315\) 1339.25 0.239550
\(316\) −8553.82 −1.52275
\(317\) −7544.32 −1.33669 −0.668346 0.743851i \(-0.732997\pi\)
−0.668346 + 0.743851i \(0.732997\pi\)
\(318\) −5061.96 −0.892643
\(319\) −5031.82 −0.883159
\(320\) −15575.2 −2.72087
\(321\) −1233.57 −0.214490
\(322\) −1476.11 −0.255468
\(323\) 13557.4 2.33546
\(324\) 10579.0 1.81396
\(325\) −217.336 −0.0370943
\(326\) 11971.8 2.03391
\(327\) 2331.90 0.394356
\(328\) 11638.4 1.95921
\(329\) −871.923 −0.146111
\(330\) 2005.01 0.334461
\(331\) 5991.64 0.994956 0.497478 0.867477i \(-0.334260\pi\)
0.497478 + 0.867477i \(0.334260\pi\)
\(332\) −17892.8 −2.95782
\(333\) 5824.79 0.958548
\(334\) −7379.13 −1.20889
\(335\) −680.204 −0.110936
\(336\) −6476.08 −1.05149
\(337\) 5665.46 0.915778 0.457889 0.889009i \(-0.348606\pi\)
0.457889 + 0.889009i \(0.348606\pi\)
\(338\) −11898.6 −1.91479
\(339\) −452.874 −0.0725567
\(340\) 12396.2 1.97729
\(341\) 3999.09 0.635081
\(342\) 16842.0 2.66290
\(343\) −6351.40 −0.999834
\(344\) 9423.21 1.47693
\(345\) −217.948 −0.0340113
\(346\) 3794.71 0.589609
\(347\) 5593.30 0.865315 0.432657 0.901558i \(-0.357576\pi\)
0.432657 + 0.901558i \(0.357576\pi\)
\(348\) 5930.00 0.913453
\(349\) 4304.61 0.660230 0.330115 0.943941i \(-0.392912\pi\)
0.330115 + 0.943941i \(0.392912\pi\)
\(350\) 1604.47 0.245036
\(351\) −830.516 −0.126295
\(352\) 37020.3 5.60564
\(353\) 1056.64 0.159318 0.0796592 0.996822i \(-0.474617\pi\)
0.0796592 + 0.996822i \(0.474617\pi\)
\(354\) −6462.88 −0.970334
\(355\) 764.371 0.114278
\(356\) 6362.45 0.947217
\(357\) 2291.91 0.339778
\(358\) −20965.4 −3.09513
\(359\) 5186.43 0.762477 0.381238 0.924477i \(-0.375498\pi\)
0.381238 + 0.924477i \(0.375498\pi\)
\(360\) 10147.8 1.48566
\(361\) 9596.58 1.39912
\(362\) −10507.5 −1.52558
\(363\) −174.615 −0.0252476
\(364\) −2333.54 −0.336018
\(365\) −6140.80 −0.880614
\(366\) 3893.38 0.556039
\(367\) 178.772 0.0254274 0.0127137 0.999919i \(-0.495953\pi\)
0.0127137 + 0.999919i \(0.495953\pi\)
\(368\) −6868.52 −0.972952
\(369\) −3142.15 −0.443289
\(370\) 6978.31 0.980500
\(371\) 5449.03 0.762532
\(372\) −4712.93 −0.656866
\(373\) 7463.86 1.03610 0.518049 0.855351i \(-0.326659\pi\)
0.518049 + 0.855351i \(0.326659\pi\)
\(374\) −22362.1 −3.09175
\(375\) 236.899 0.0326225
\(376\) −6606.78 −0.906166
\(377\) 1159.56 0.158410
\(378\) 6131.23 0.834276
\(379\) −6075.99 −0.823490 −0.411745 0.911299i \(-0.635081\pi\)
−0.411745 + 0.911299i \(0.635081\pi\)
\(380\) 15046.2 2.03119
\(381\) −4943.58 −0.664743
\(382\) −28935.3 −3.87554
\(383\) −580.709 −0.0774747 −0.0387374 0.999249i \(-0.512334\pi\)
−0.0387374 + 0.999249i \(0.512334\pi\)
\(384\) −18233.5 −2.42311
\(385\) −2158.32 −0.285710
\(386\) −26958.1 −3.55475
\(387\) −2544.09 −0.334169
\(388\) 13477.1 1.76339
\(389\) 11373.5 1.48241 0.741205 0.671278i \(-0.234255\pi\)
0.741205 + 0.671278i \(0.234255\pi\)
\(390\) −462.046 −0.0599913
\(391\) 2430.79 0.314400
\(392\) −18386.9 −2.36908
\(393\) 1774.68 0.227788
\(394\) 17962.6 2.29681
\(395\) 1823.19 0.232239
\(396\) −20715.3 −2.62874
\(397\) 3701.46 0.467937 0.233969 0.972244i \(-0.424829\pi\)
0.233969 + 0.972244i \(0.424829\pi\)
\(398\) −12398.0 −1.56145
\(399\) 2781.85 0.349039
\(400\) 7465.78 0.933222
\(401\) −7615.01 −0.948317 −0.474159 0.880439i \(-0.657248\pi\)
−0.474159 + 0.880439i \(0.657248\pi\)
\(402\) −1446.08 −0.179413
\(403\) −921.574 −0.113913
\(404\) 32190.7 3.96423
\(405\) −2254.84 −0.276651
\(406\) −8560.39 −1.04642
\(407\) −9387.17 −1.14325
\(408\) 17366.4 2.10726
\(409\) −15423.6 −1.86466 −0.932330 0.361608i \(-0.882228\pi\)
−0.932330 + 0.361608i \(0.882228\pi\)
\(410\) −3764.41 −0.453441
\(411\) −787.475 −0.0945092
\(412\) −5686.98 −0.680042
\(413\) 6957.07 0.828899
\(414\) 3019.71 0.358480
\(415\) 3813.72 0.451104
\(416\) −8531.17 −1.00547
\(417\) 1799.73 0.211351
\(418\) −27142.4 −3.17603
\(419\) 4273.20 0.498233 0.249116 0.968474i \(-0.419860\pi\)
0.249116 + 0.968474i \(0.419860\pi\)
\(420\) 2543.59 0.295510
\(421\) −6076.38 −0.703432 −0.351716 0.936107i \(-0.614402\pi\)
−0.351716 + 0.936107i \(0.614402\pi\)
\(422\) 859.061 0.0990958
\(423\) 1783.71 0.205028
\(424\) 41288.6 4.72913
\(425\) −2642.17 −0.301562
\(426\) 1625.01 0.184817
\(427\) −4191.10 −0.474991
\(428\) 15269.0 1.72443
\(429\) 621.541 0.0699494
\(430\) −3047.92 −0.341822
\(431\) −3386.04 −0.378422 −0.189211 0.981936i \(-0.560593\pi\)
−0.189211 + 0.981936i \(0.560593\pi\)
\(432\) 28529.3 3.17735
\(433\) 7924.63 0.879523 0.439762 0.898114i \(-0.355063\pi\)
0.439762 + 0.898114i \(0.355063\pi\)
\(434\) 6803.46 0.752480
\(435\) −1263.94 −0.139313
\(436\) −28863.9 −3.17049
\(437\) 2950.42 0.322970
\(438\) −13055.0 −1.42418
\(439\) 13530.9 1.47106 0.735530 0.677492i \(-0.236933\pi\)
0.735530 + 0.677492i \(0.236933\pi\)
\(440\) −16354.1 −1.77194
\(441\) 4964.13 0.536026
\(442\) 5153.25 0.554559
\(443\) 996.052 0.106826 0.0534129 0.998573i \(-0.482990\pi\)
0.0534129 + 0.998573i \(0.482990\pi\)
\(444\) 11062.8 1.18247
\(445\) −1356.11 −0.144463
\(446\) −17210.0 −1.82717
\(447\) −4188.08 −0.443153
\(448\) 35644.0 3.75897
\(449\) −5079.36 −0.533875 −0.266938 0.963714i \(-0.586012\pi\)
−0.266938 + 0.963714i \(0.586012\pi\)
\(450\) −3282.29 −0.343842
\(451\) 5063.85 0.528709
\(452\) 5605.60 0.583331
\(453\) 2623.43 0.272096
\(454\) 25465.0 2.63245
\(455\) 497.377 0.0512470
\(456\) 21078.8 2.16470
\(457\) −11190.9 −1.14549 −0.572747 0.819732i \(-0.694122\pi\)
−0.572747 + 0.819732i \(0.694122\pi\)
\(458\) 8279.94 0.844752
\(459\) −10096.6 −1.02673
\(460\) 2697.72 0.273439
\(461\) 2426.07 0.245105 0.122553 0.992462i \(-0.460892\pi\)
0.122553 + 0.992462i \(0.460892\pi\)
\(462\) −4588.48 −0.462068
\(463\) −16349.2 −1.64106 −0.820531 0.571602i \(-0.806322\pi\)
−0.820531 + 0.571602i \(0.806322\pi\)
\(464\) −39832.4 −3.98529
\(465\) 1004.53 0.100180
\(466\) −508.889 −0.0505876
\(467\) −2880.80 −0.285455 −0.142728 0.989762i \(-0.545587\pi\)
−0.142728 + 0.989762i \(0.545587\pi\)
\(468\) 4773.76 0.471511
\(469\) 1556.65 0.153261
\(470\) 2136.95 0.209723
\(471\) −1063.96 −0.104087
\(472\) 52715.4 5.14073
\(473\) 4100.03 0.398562
\(474\) 3876.00 0.375592
\(475\) −3206.98 −0.309782
\(476\) −28368.9 −2.73169
\(477\) −11147.2 −1.07001
\(478\) −9081.53 −0.868994
\(479\) 7850.31 0.748831 0.374415 0.927261i \(-0.377843\pi\)
0.374415 + 0.927261i \(0.377843\pi\)
\(480\) 9299.09 0.884257
\(481\) 2163.24 0.205063
\(482\) −36162.5 −3.41734
\(483\) 498.775 0.0469877
\(484\) 2161.35 0.202982
\(485\) −2872.55 −0.268940
\(486\) −19261.0 −1.79773
\(487\) −11262.4 −1.04794 −0.523971 0.851736i \(-0.675550\pi\)
−0.523971 + 0.851736i \(0.675550\pi\)
\(488\) −31757.0 −2.94584
\(489\) −4045.24 −0.374094
\(490\) 5947.21 0.548301
\(491\) 15810.7 1.45321 0.726607 0.687054i \(-0.241096\pi\)
0.726607 + 0.687054i \(0.241096\pi\)
\(492\) −5967.76 −0.546844
\(493\) 14096.8 1.28781
\(494\) 6254.86 0.569675
\(495\) 4415.32 0.400917
\(496\) 31657.2 2.86583
\(497\) −1749.27 −0.157878
\(498\) 8107.78 0.729555
\(499\) −10470.4 −0.939322 −0.469661 0.882847i \(-0.655624\pi\)
−0.469661 + 0.882847i \(0.655624\pi\)
\(500\) −2932.31 −0.262273
\(501\) 2493.39 0.222348
\(502\) 19227.9 1.70953
\(503\) −5425.35 −0.480923 −0.240462 0.970659i \(-0.577299\pi\)
−0.240462 + 0.970659i \(0.577299\pi\)
\(504\) −23223.4 −2.05249
\(505\) −6861.23 −0.604595
\(506\) −4866.53 −0.427557
\(507\) 4020.51 0.352184
\(508\) 61190.9 5.34430
\(509\) 8098.38 0.705215 0.352608 0.935771i \(-0.385295\pi\)
0.352608 + 0.935771i \(0.385295\pi\)
\(510\) −5617.11 −0.487705
\(511\) 14053.3 1.21660
\(512\) 85918.7 7.41622
\(513\) −12255.0 −1.05472
\(514\) 12681.9 1.08828
\(515\) 1212.14 0.103715
\(516\) −4831.90 −0.412233
\(517\) −2874.60 −0.244536
\(518\) −15969.9 −1.35459
\(519\) −1282.22 −0.108446
\(520\) 3768.75 0.317828
\(521\) 14691.8 1.23543 0.617717 0.786400i \(-0.288058\pi\)
0.617717 + 0.786400i \(0.288058\pi\)
\(522\) 17512.1 1.46836
\(523\) −19263.6 −1.61059 −0.805297 0.592872i \(-0.797994\pi\)
−0.805297 + 0.592872i \(0.797994\pi\)
\(524\) −21966.7 −1.83134
\(525\) −542.147 −0.0450690
\(526\) 29836.7 2.47327
\(527\) −11203.6 −0.926066
\(528\) −21350.7 −1.75979
\(529\) 529.000 0.0434783
\(530\) −13354.7 −1.09451
\(531\) −14232.2 −1.16314
\(532\) −34433.3 −2.80615
\(533\) −1166.95 −0.0948331
\(534\) −2883.02 −0.233634
\(535\) −3254.48 −0.262997
\(536\) 11795.2 0.950510
\(537\) 7084.17 0.569282
\(538\) −11175.6 −0.895564
\(539\) −8000.15 −0.639315
\(540\) −11205.3 −0.892965
\(541\) −18737.4 −1.48906 −0.744531 0.667588i \(-0.767327\pi\)
−0.744531 + 0.667588i \(0.767327\pi\)
\(542\) 22158.2 1.75604
\(543\) 3550.46 0.280598
\(544\) −103714. −8.17407
\(545\) 6152.14 0.483539
\(546\) 1057.40 0.0828799
\(547\) −14180.2 −1.10841 −0.554207 0.832379i \(-0.686978\pi\)
−0.554207 + 0.832379i \(0.686978\pi\)
\(548\) 9747.25 0.759821
\(549\) 8573.80 0.666522
\(550\) 5289.71 0.410098
\(551\) 17110.3 1.32291
\(552\) 3779.34 0.291412
\(553\) −4172.38 −0.320846
\(554\) −999.073 −0.0766183
\(555\) −2357.96 −0.180342
\(556\) −22276.8 −1.69919
\(557\) 15154.2 1.15279 0.576397 0.817170i \(-0.304458\pi\)
0.576397 + 0.817170i \(0.304458\pi\)
\(558\) −13917.9 −1.05590
\(559\) −944.837 −0.0714890
\(560\) −17085.5 −1.28928
\(561\) 7556.09 0.568660
\(562\) 19762.5 1.48333
\(563\) 14444.7 1.08130 0.540651 0.841247i \(-0.318178\pi\)
0.540651 + 0.841247i \(0.318178\pi\)
\(564\) 3387.73 0.252924
\(565\) −1194.79 −0.0889653
\(566\) −3922.13 −0.291271
\(567\) 5160.22 0.382203
\(568\) −13254.7 −0.979144
\(569\) −8851.15 −0.652125 −0.326063 0.945348i \(-0.605722\pi\)
−0.326063 + 0.945348i \(0.605722\pi\)
\(570\) −6817.88 −0.500999
\(571\) −22318.9 −1.63576 −0.817879 0.575391i \(-0.804850\pi\)
−0.817879 + 0.575391i \(0.804850\pi\)
\(572\) −7693.34 −0.562368
\(573\) 9777.17 0.712822
\(574\) 8614.89 0.626444
\(575\) −575.000 −0.0417029
\(576\) −72917.4 −5.27470
\(577\) −4530.99 −0.326911 −0.163455 0.986551i \(-0.552264\pi\)
−0.163455 + 0.986551i \(0.552264\pi\)
\(578\) 35092.3 2.52534
\(579\) 9109.09 0.653818
\(580\) 15644.8 1.12003
\(581\) −8727.75 −0.623215
\(582\) −6106.89 −0.434946
\(583\) 17964.7 1.27619
\(584\) 106485. 7.54519
\(585\) −1017.49 −0.0719113
\(586\) 47971.7 3.38173
\(587\) −5092.89 −0.358103 −0.179051 0.983840i \(-0.557303\pi\)
−0.179051 + 0.983840i \(0.557303\pi\)
\(588\) 9428.19 0.661245
\(589\) −13598.6 −0.951309
\(590\) −17050.7 −1.18977
\(591\) −6069.54 −0.422449
\(592\) −74309.9 −5.15898
\(593\) 4193.92 0.290427 0.145214 0.989400i \(-0.453613\pi\)
0.145214 + 0.989400i \(0.453613\pi\)
\(594\) 20213.8 1.39626
\(595\) 6046.63 0.416618
\(596\) 51839.5 3.56280
\(597\) 4189.26 0.287194
\(598\) 1121.47 0.0766897
\(599\) 17663.0 1.20483 0.602414 0.798184i \(-0.294206\pi\)
0.602414 + 0.798184i \(0.294206\pi\)
\(600\) −4107.98 −0.279513
\(601\) −15817.7 −1.07357 −0.536785 0.843719i \(-0.680361\pi\)
−0.536785 + 0.843719i \(0.680361\pi\)
\(602\) 6975.19 0.472239
\(603\) −3184.47 −0.215061
\(604\) −32472.4 −2.18755
\(605\) −460.677 −0.0309573
\(606\) −14586.6 −0.977790
\(607\) −740.284 −0.0495011 −0.0247506 0.999694i \(-0.507879\pi\)
−0.0247506 + 0.999694i \(0.507879\pi\)
\(608\) −125885. −8.39687
\(609\) 2892.54 0.192466
\(610\) 10271.7 0.681786
\(611\) 662.441 0.0438617
\(612\) 58034.7 3.83319
\(613\) −12412.1 −0.817815 −0.408908 0.912576i \(-0.634090\pi\)
−0.408908 + 0.912576i \(0.634090\pi\)
\(614\) −31531.0 −2.07246
\(615\) 1271.99 0.0834007
\(616\) 37426.6 2.44799
\(617\) −27047.9 −1.76484 −0.882422 0.470459i \(-0.844089\pi\)
−0.882422 + 0.470459i \(0.844089\pi\)
\(618\) 2576.95 0.167735
\(619\) 193.644 0.0125738 0.00628691 0.999980i \(-0.497999\pi\)
0.00628691 + 0.999980i \(0.497999\pi\)
\(620\) −12433.9 −0.805415
\(621\) −2197.27 −0.141986
\(622\) −36645.6 −2.36231
\(623\) 3103.48 0.199580
\(624\) 4920.19 0.315649
\(625\) 625.000 0.0400000
\(626\) −15218.0 −0.971619
\(627\) 9171.36 0.584161
\(628\) 13169.6 0.836819
\(629\) 26298.5 1.66708
\(630\) 7511.56 0.475028
\(631\) 11459.7 0.722982 0.361491 0.932376i \(-0.382268\pi\)
0.361491 + 0.932376i \(0.382268\pi\)
\(632\) −31615.2 −1.98985
\(633\) −290.275 −0.0182265
\(634\) −42314.4 −2.65066
\(635\) −13042.4 −0.815073
\(636\) −21171.4 −1.31997
\(637\) 1843.60 0.114672
\(638\) −28222.4 −1.75131
\(639\) 3578.52 0.221540
\(640\) −48104.4 −2.97109
\(641\) 2261.29 0.139338 0.0696689 0.997570i \(-0.477806\pi\)
0.0696689 + 0.997570i \(0.477806\pi\)
\(642\) −6918.85 −0.425335
\(643\) 10224.8 0.627101 0.313551 0.949572i \(-0.398481\pi\)
0.313551 + 0.949572i \(0.398481\pi\)
\(644\) −6173.77 −0.377765
\(645\) 1029.88 0.0628708
\(646\) 76040.6 4.63123
\(647\) −16733.4 −1.01678 −0.508390 0.861127i \(-0.669759\pi\)
−0.508390 + 0.861127i \(0.669759\pi\)
\(648\) 39100.3 2.37038
\(649\) 22936.5 1.38727
\(650\) −1218.99 −0.0735582
\(651\) −2298.87 −0.138402
\(652\) 50071.3 3.00758
\(653\) −9106.08 −0.545710 −0.272855 0.962055i \(-0.587968\pi\)
−0.272855 + 0.962055i \(0.587968\pi\)
\(654\) 13079.1 0.782010
\(655\) 4682.04 0.279302
\(656\) 40086.0 2.38582
\(657\) −28749.0 −1.70716
\(658\) −4890.43 −0.289740
\(659\) −19951.5 −1.17936 −0.589680 0.807637i \(-0.700746\pi\)
−0.589680 + 0.807637i \(0.700746\pi\)
\(660\) 8385.84 0.494573
\(661\) 16531.2 0.972750 0.486375 0.873750i \(-0.338319\pi\)
0.486375 + 0.873750i \(0.338319\pi\)
\(662\) 33605.8 1.97300
\(663\) −1741.27 −0.101999
\(664\) −66132.3 −3.86511
\(665\) 7339.22 0.427974
\(666\) 32670.0 1.90080
\(667\) 3067.82 0.178091
\(668\) −30862.8 −1.78760
\(669\) 5815.24 0.336069
\(670\) −3815.12 −0.219986
\(671\) −13817.4 −0.794957
\(672\) −21281.1 −1.22163
\(673\) −10268.9 −0.588169 −0.294084 0.955779i \(-0.595015\pi\)
−0.294084 + 0.955779i \(0.595015\pi\)
\(674\) 31776.3 1.81599
\(675\) 2388.34 0.136188
\(676\) −49765.3 −2.83144
\(677\) 4729.79 0.268509 0.134254 0.990947i \(-0.457136\pi\)
0.134254 + 0.990947i \(0.457136\pi\)
\(678\) −2540.07 −0.143880
\(679\) 6573.86 0.371549
\(680\) 45816.8 2.58381
\(681\) −8604.56 −0.484182
\(682\) 22430.0 1.25937
\(683\) 11551.0 0.647123 0.323561 0.946207i \(-0.395120\pi\)
0.323561 + 0.946207i \(0.395120\pi\)
\(684\) 70440.8 3.93768
\(685\) −2077.56 −0.115882
\(686\) −35623.6 −1.98268
\(687\) −2797.77 −0.155374
\(688\) 32456.3 1.79853
\(689\) −4139.89 −0.228907
\(690\) −1222.42 −0.0674446
\(691\) 26575.2 1.46305 0.731526 0.681813i \(-0.238808\pi\)
0.731526 + 0.681813i \(0.238808\pi\)
\(692\) 15871.2 0.871866
\(693\) −10104.5 −0.553879
\(694\) 31371.6 1.71592
\(695\) 4748.15 0.259147
\(696\) 21917.5 1.19365
\(697\) −14186.6 −0.770955
\(698\) 24143.6 1.30924
\(699\) 171.952 0.00930448
\(700\) 6710.62 0.362339
\(701\) 5886.27 0.317149 0.158574 0.987347i \(-0.449310\pi\)
0.158574 + 0.987347i \(0.449310\pi\)
\(702\) −4658.18 −0.250444
\(703\) 31920.4 1.71252
\(704\) 117513. 6.29110
\(705\) −722.070 −0.0385741
\(706\) 5926.48 0.315929
\(707\) 15702.0 0.835268
\(708\) −27030.7 −1.43485
\(709\) −2588.26 −0.137100 −0.0685501 0.997648i \(-0.521837\pi\)
−0.0685501 + 0.997648i \(0.521837\pi\)
\(710\) 4287.19 0.226613
\(711\) 8535.51 0.450220
\(712\) 23515.8 1.23777
\(713\) −2438.18 −0.128065
\(714\) 12854.8 0.673781
\(715\) 1639.78 0.0857682
\(716\) −87686.8 −4.57683
\(717\) 3068.63 0.159833
\(718\) 29089.5 1.51199
\(719\) 1170.46 0.0607106 0.0303553 0.999539i \(-0.490336\pi\)
0.0303553 + 0.999539i \(0.490336\pi\)
\(720\) 34952.1 1.80915
\(721\) −2774.00 −0.143286
\(722\) 53825.1 2.77447
\(723\) 12219.2 0.628545
\(724\) −43947.0 −2.25591
\(725\) −3334.58 −0.170818
\(726\) −979.376 −0.0500662
\(727\) −3135.70 −0.159968 −0.0799838 0.996796i \(-0.525487\pi\)
−0.0799838 + 0.996796i \(0.525487\pi\)
\(728\) −8624.82 −0.439089
\(729\) −5667.88 −0.287958
\(730\) −34442.4 −1.74626
\(731\) −11486.4 −0.581177
\(732\) 16283.9 0.822226
\(733\) −25792.4 −1.29968 −0.649839 0.760072i \(-0.725164\pi\)
−0.649839 + 0.760072i \(0.725164\pi\)
\(734\) 1002.70 0.0504225
\(735\) −2009.55 −0.100848
\(736\) −22570.7 −1.13039
\(737\) 5132.07 0.256502
\(738\) −17623.6 −0.879044
\(739\) −809.931 −0.0403164 −0.0201582 0.999797i \(-0.506417\pi\)
−0.0201582 + 0.999797i \(0.506417\pi\)
\(740\) 29186.4 1.44988
\(741\) −2113.50 −0.104779
\(742\) 30562.4 1.51210
\(743\) 1825.32 0.0901270 0.0450635 0.998984i \(-0.485651\pi\)
0.0450635 + 0.998984i \(0.485651\pi\)
\(744\) −17419.1 −0.858355
\(745\) −11049.2 −0.543371
\(746\) 41863.2 2.05459
\(747\) 17854.5 0.874514
\(748\) −93528.2 −4.57183
\(749\) 7447.91 0.363339
\(750\) 1328.72 0.0646906
\(751\) 4310.27 0.209433 0.104716 0.994502i \(-0.466606\pi\)
0.104716 + 0.994502i \(0.466606\pi\)
\(752\) −22755.7 −1.10348
\(753\) −6497.06 −0.314430
\(754\) 6503.73 0.314127
\(755\) 6921.25 0.333629
\(756\) 25643.5 1.23366
\(757\) 19302.2 0.926750 0.463375 0.886162i \(-0.346638\pi\)
0.463375 + 0.886162i \(0.346638\pi\)
\(758\) −34078.9 −1.63298
\(759\) 1644.39 0.0786398
\(760\) 55611.1 2.65424
\(761\) −30144.8 −1.43594 −0.717968 0.696076i \(-0.754927\pi\)
−0.717968 + 0.696076i \(0.754927\pi\)
\(762\) −27727.5 −1.31819
\(763\) −14079.3 −0.668025
\(764\) −121020. −5.73084
\(765\) −12369.7 −0.584610
\(766\) −3257.07 −0.153633
\(767\) −5285.62 −0.248830
\(768\) −55038.8 −2.58599
\(769\) 37297.7 1.74901 0.874506 0.485015i \(-0.161186\pi\)
0.874506 + 0.485015i \(0.161186\pi\)
\(770\) −12105.6 −0.566564
\(771\) −4285.19 −0.200165
\(772\) −112751. −5.25647
\(773\) −13602.8 −0.632933 −0.316467 0.948604i \(-0.602497\pi\)
−0.316467 + 0.948604i \(0.602497\pi\)
\(774\) −14269.3 −0.662659
\(775\) 2650.19 0.122836
\(776\) 49811.8 2.30430
\(777\) 5396.21 0.249148
\(778\) 63791.3 2.93963
\(779\) −17219.3 −0.791970
\(780\) −1932.48 −0.0887103
\(781\) −5767.10 −0.264229
\(782\) 13633.8 0.623457
\(783\) −12742.6 −0.581587
\(784\) −63330.1 −2.88493
\(785\) −2807.00 −0.127625
\(786\) 9953.79 0.451705
\(787\) 14129.9 0.639997 0.319998 0.947418i \(-0.396318\pi\)
0.319998 + 0.947418i \(0.396318\pi\)
\(788\) 75127.8 3.39634
\(789\) −10081.8 −0.454905
\(790\) 10225.9 0.460531
\(791\) 2734.30 0.122908
\(792\) −76564.3 −3.43509
\(793\) 3184.18 0.142589
\(794\) 20760.7 0.927922
\(795\) 4512.53 0.201312
\(796\) −51854.0 −2.30894
\(797\) 16171.3 0.718715 0.359357 0.933200i \(-0.382996\pi\)
0.359357 + 0.933200i \(0.382996\pi\)
\(798\) 15602.8 0.692147
\(799\) 8053.32 0.356578
\(800\) 24533.3 1.08423
\(801\) −6348.83 −0.280056
\(802\) −42710.9 −1.88052
\(803\) 46331.7 2.03613
\(804\) −6048.15 −0.265301
\(805\) 1315.89 0.0576139
\(806\) −5168.91 −0.225890
\(807\) 3776.20 0.164720
\(808\) 118978. 5.18023
\(809\) −41744.9 −1.81418 −0.907090 0.420937i \(-0.861701\pi\)
−0.907090 + 0.420937i \(0.861701\pi\)
\(810\) −12646.9 −0.548601
\(811\) −12210.2 −0.528679 −0.264340 0.964430i \(-0.585154\pi\)
−0.264340 + 0.964430i \(0.585154\pi\)
\(812\) −35803.4 −1.54736
\(813\) −7487.19 −0.322986
\(814\) −52650.6 −2.26708
\(815\) −10672.3 −0.458694
\(816\) 59814.9 2.56610
\(817\) −13941.9 −0.597019
\(818\) −86507.4 −3.69763
\(819\) 2328.54 0.0993478
\(820\) −15744.4 −0.670512
\(821\) −22326.2 −0.949074 −0.474537 0.880236i \(-0.657385\pi\)
−0.474537 + 0.880236i \(0.657385\pi\)
\(822\) −4416.78 −0.187412
\(823\) −29128.6 −1.23373 −0.616864 0.787069i \(-0.711597\pi\)
−0.616864 + 0.787069i \(0.711597\pi\)
\(824\) −21019.2 −0.888641
\(825\) −1787.38 −0.0754286
\(826\) 39020.7 1.64371
\(827\) −13540.8 −0.569357 −0.284679 0.958623i \(-0.591887\pi\)
−0.284679 + 0.958623i \(0.591887\pi\)
\(828\) 12629.8 0.530091
\(829\) 93.8298 0.00393105 0.00196553 0.999998i \(-0.499374\pi\)
0.00196553 + 0.999998i \(0.499374\pi\)
\(830\) 21390.3 0.894542
\(831\) 337.585 0.0140923
\(832\) −27080.4 −1.12842
\(833\) 22412.7 0.932239
\(834\) 10094.3 0.419110
\(835\) 6578.19 0.272632
\(836\) −113522. −4.69645
\(837\) 10127.3 0.418220
\(838\) 23967.4 0.987997
\(839\) −33750.6 −1.38880 −0.694398 0.719591i \(-0.744329\pi\)
−0.694398 + 0.719591i \(0.744329\pi\)
\(840\) 9401.17 0.386156
\(841\) −6597.88 −0.270527
\(842\) −34081.1 −1.39491
\(843\) −6677.70 −0.272826
\(844\) 3592.98 0.146535
\(845\) 10607.1 0.431830
\(846\) 10004.4 0.406571
\(847\) 1054.26 0.0427686
\(848\) 142210. 5.75887
\(849\) 1325.28 0.0535730
\(850\) −14819.3 −0.597999
\(851\) 5723.21 0.230539
\(852\) 6796.54 0.273293
\(853\) −2680.73 −0.107604 −0.0538022 0.998552i \(-0.517134\pi\)
−0.0538022 + 0.998552i \(0.517134\pi\)
\(854\) −23506.9 −0.941910
\(855\) −15014.0 −0.600546
\(856\) 56434.7 2.25338
\(857\) 29267.7 1.16659 0.583295 0.812261i \(-0.301763\pi\)
0.583295 + 0.812261i \(0.301763\pi\)
\(858\) 3486.09 0.138710
\(859\) 34842.9 1.38396 0.691982 0.721914i \(-0.256738\pi\)
0.691982 + 0.721914i \(0.256738\pi\)
\(860\) −12747.8 −0.505459
\(861\) −2910.95 −0.115221
\(862\) −18991.6 −0.750412
\(863\) −4041.96 −0.159432 −0.0797161 0.996818i \(-0.525401\pi\)
−0.0797161 + 0.996818i \(0.525401\pi\)
\(864\) 93750.1 3.69149
\(865\) −3382.83 −0.132971
\(866\) 44447.6 1.74410
\(867\) −11857.6 −0.464482
\(868\) 28455.1 1.11271
\(869\) −13755.7 −0.536975
\(870\) −7089.15 −0.276258
\(871\) −1182.66 −0.0460081
\(872\) −106682. −4.14301
\(873\) −13448.3 −0.521368
\(874\) 16548.3 0.640451
\(875\) −1430.32 −0.0552613
\(876\) −54601.9 −2.10597
\(877\) 34918.5 1.34448 0.672242 0.740331i \(-0.265331\pi\)
0.672242 + 0.740331i \(0.265331\pi\)
\(878\) 75891.9 2.91712
\(879\) −16209.5 −0.621996
\(880\) −56328.5 −2.15777
\(881\) 2473.77 0.0946008 0.0473004 0.998881i \(-0.484938\pi\)
0.0473004 + 0.998881i \(0.484938\pi\)
\(882\) 27842.8 1.06294
\(883\) 16956.8 0.646252 0.323126 0.946356i \(-0.395266\pi\)
0.323126 + 0.946356i \(0.395266\pi\)
\(884\) 21553.2 0.820037
\(885\) 5761.39 0.218833
\(886\) 5586.64 0.211836
\(887\) 582.058 0.0220334 0.0110167 0.999939i \(-0.496493\pi\)
0.0110167 + 0.999939i \(0.496493\pi\)
\(888\) 40888.4 1.54519
\(889\) 29847.7 1.12605
\(890\) −7606.13 −0.286470
\(891\) 17012.5 0.639664
\(892\) −71980.1 −2.70188
\(893\) 9774.88 0.366298
\(894\) −23490.1 −0.878775
\(895\) 18689.8 0.698024
\(896\) 110088. 4.10465
\(897\) −378.943 −0.0141054
\(898\) −28489.0 −1.05868
\(899\) −14139.7 −0.524566
\(900\) −13728.0 −0.508445
\(901\) −50328.7 −1.86092
\(902\) 28402.1 1.04843
\(903\) −2356.90 −0.0868580
\(904\) 20718.5 0.762263
\(905\) 9367.00 0.344055
\(906\) 14714.2 0.539567
\(907\) 26224.8 0.960065 0.480033 0.877251i \(-0.340625\pi\)
0.480033 + 0.877251i \(0.340625\pi\)
\(908\) 106506. 3.89265
\(909\) −32121.8 −1.17207
\(910\) 2789.68 0.101623
\(911\) −12266.6 −0.446117 −0.223058 0.974805i \(-0.571604\pi\)
−0.223058 + 0.974805i \(0.571604\pi\)
\(912\) 72601.5 2.63605
\(913\) −28774.1 −1.04303
\(914\) −62767.6 −2.27152
\(915\) −3470.79 −0.125400
\(916\) 34630.4 1.24915
\(917\) −10714.9 −0.385864
\(918\) −56629.7 −2.03601
\(919\) 12114.7 0.434850 0.217425 0.976077i \(-0.430234\pi\)
0.217425 + 0.976077i \(0.430234\pi\)
\(920\) 9970.85 0.357314
\(921\) 10654.3 0.381184
\(922\) 13607.3 0.486045
\(923\) 1329.01 0.0473941
\(924\) −19191.1 −0.683269
\(925\) −6220.88 −0.221126
\(926\) −91699.1 −3.25423
\(927\) 5674.81 0.201063
\(928\) −130893. −4.63016
\(929\) −2418.31 −0.0854059 −0.0427029 0.999088i \(-0.513597\pi\)
−0.0427029 + 0.999088i \(0.513597\pi\)
\(930\) 5634.18 0.198658
\(931\) 27203.9 0.957650
\(932\) −2128.40 −0.0748048
\(933\) 12382.5 0.434496
\(934\) −16157.8 −0.566059
\(935\) 19934.9 0.697262
\(936\) 17643.9 0.616143
\(937\) 16448.7 0.573486 0.286743 0.958008i \(-0.407427\pi\)
0.286743 + 0.958008i \(0.407427\pi\)
\(938\) 8730.94 0.303918
\(939\) 5142.13 0.178708
\(940\) 8937.67 0.310122
\(941\) −17373.2 −0.601859 −0.300929 0.953646i \(-0.597297\pi\)
−0.300929 + 0.953646i \(0.597297\pi\)
\(942\) −5967.53 −0.206404
\(943\) −3087.35 −0.106615
\(944\) 181568. 6.26009
\(945\) −5465.74 −0.188149
\(946\) 22996.2 0.790350
\(947\) 18638.9 0.639579 0.319790 0.947489i \(-0.396388\pi\)
0.319790 + 0.947489i \(0.396388\pi\)
\(948\) 16211.2 0.555395
\(949\) −10676.9 −0.365214
\(950\) −17987.3 −0.614299
\(951\) 14298.0 0.487532
\(952\) −104852. −3.56962
\(953\) −31188.1 −1.06011 −0.530053 0.847965i \(-0.677828\pi\)
−0.530053 + 0.847965i \(0.677828\pi\)
\(954\) −62522.0 −2.12183
\(955\) 25794.6 0.874025
\(956\) −37983.0 −1.28500
\(957\) 9536.28 0.322115
\(958\) 44030.7 1.48493
\(959\) 4754.51 0.160095
\(960\) 29518.0 0.992385
\(961\) −18553.3 −0.622784
\(962\) 12133.1 0.406640
\(963\) −15236.3 −0.509848
\(964\) −151248. −5.05328
\(965\) 24032.1 0.801678
\(966\) 2797.52 0.0931768
\(967\) 43197.8 1.43655 0.718277 0.695757i \(-0.244931\pi\)
0.718277 + 0.695757i \(0.244931\pi\)
\(968\) 7988.42 0.265246
\(969\) −25693.9 −0.851815
\(970\) −16111.5 −0.533308
\(971\) 13497.0 0.446074 0.223037 0.974810i \(-0.428403\pi\)
0.223037 + 0.974810i \(0.428403\pi\)
\(972\) −80558.1 −2.65834
\(973\) −10866.2 −0.358021
\(974\) −63168.4 −2.07808
\(975\) 411.895 0.0135294
\(976\) −109380. −3.58728
\(977\) −34955.4 −1.14465 −0.572325 0.820027i \(-0.693958\pi\)
−0.572325 + 0.820027i \(0.693958\pi\)
\(978\) −22688.8 −0.741830
\(979\) 10231.7 0.334021
\(980\) 24873.9 0.810784
\(981\) 28802.2 0.937393
\(982\) 88678.9 2.88173
\(983\) −27521.6 −0.892984 −0.446492 0.894788i \(-0.647327\pi\)
−0.446492 + 0.894788i \(0.647327\pi\)
\(984\) −22057.0 −0.714585
\(985\) −16013.0 −0.517985
\(986\) 79066.1 2.55373
\(987\) 1652.46 0.0532913
\(988\) 26160.6 0.842389
\(989\) −2499.73 −0.0803707
\(990\) 24764.5 0.795019
\(991\) 24567.9 0.787513 0.393756 0.919215i \(-0.371175\pi\)
0.393756 + 0.919215i \(0.371175\pi\)
\(992\) 104029. 3.32956
\(993\) −11355.3 −0.362891
\(994\) −9811.29 −0.313074
\(995\) 11052.3 0.352143
\(996\) 33910.4 1.07881
\(997\) 2293.84 0.0728652 0.0364326 0.999336i \(-0.488401\pi\)
0.0364326 + 0.999336i \(0.488401\pi\)
\(998\) −58726.5 −1.86268
\(999\) −23772.1 −0.752868
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 115.4.a.e.1.5 5
3.2 odd 2 1035.4.a.k.1.1 5
4.3 odd 2 1840.4.a.n.1.3 5
5.2 odd 4 575.4.b.i.24.10 10
5.3 odd 4 575.4.b.i.24.1 10
5.4 even 2 575.4.a.j.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.5 5 1.1 even 1 trivial
575.4.a.j.1.1 5 5.4 even 2
575.4.b.i.24.1 10 5.3 odd 4
575.4.b.i.24.10 10 5.2 odd 4
1035.4.a.k.1.1 5 3.2 odd 2
1840.4.a.n.1.3 5 4.3 odd 2