Properties

Label 725.4.a.c.1.3
Level $725$
Weight $4$
Character 725.1
Self dual yes
Analytic conductor $42.776$
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] = [725,4,Mod(1,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, 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("725.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 725.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: \(42.7763847542\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13458092.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) 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 29)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.27399\) of defining polynomial
Character \(\chi\) \(=\) 725.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.63099 q^{2} -9.87991 q^{3} -5.33986 q^{4} +16.1141 q^{6} -5.21997 q^{7} +21.7572 q^{8} +70.6126 q^{9} +O(q^{10})\) \(q-1.63099 q^{2} -9.87991 q^{3} -5.33986 q^{4} +16.1141 q^{6} -5.21997 q^{7} +21.7572 q^{8} +70.6126 q^{9} -8.55158 q^{11} +52.7573 q^{12} +11.3429 q^{13} +8.51374 q^{14} +7.23299 q^{16} -68.4740 q^{17} -115.169 q^{18} +6.93014 q^{19} +51.5728 q^{21} +13.9476 q^{22} +132.042 q^{23} -214.959 q^{24} -18.5001 q^{26} -430.889 q^{27} +27.8739 q^{28} -29.0000 q^{29} -0.419319 q^{31} -185.855 q^{32} +84.4888 q^{33} +111.681 q^{34} -377.061 q^{36} -395.483 q^{37} -11.3030 q^{38} -112.067 q^{39} -447.209 q^{41} -84.1150 q^{42} -184.132 q^{43} +45.6642 q^{44} -215.360 q^{46} +97.2612 q^{47} -71.4613 q^{48} -315.752 q^{49} +676.516 q^{51} -60.5693 q^{52} +209.547 q^{53} +702.776 q^{54} -113.572 q^{56} -68.4691 q^{57} +47.2988 q^{58} +45.9651 q^{59} +427.655 q^{61} +0.683906 q^{62} -368.596 q^{63} +245.264 q^{64} -137.801 q^{66} +405.055 q^{67} +365.641 q^{68} -1304.57 q^{69} -557.971 q^{71} +1536.33 q^{72} +381.988 q^{73} +645.031 q^{74} -37.0060 q^{76} +44.6390 q^{77} +182.780 q^{78} +577.208 q^{79} +2350.60 q^{81} +729.396 q^{82} +353.745 q^{83} -275.392 q^{84} +300.318 q^{86} +286.517 q^{87} -186.059 q^{88} -277.871 q^{89} -59.2095 q^{91} -705.088 q^{92} +4.14283 q^{93} -158.632 q^{94} +1836.23 q^{96} -677.917 q^{97} +514.989 q^{98} -603.849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} + 26 q^{4} + 34 q^{6} - 40 q^{7} + 84 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{3} + 26 q^{4} + 34 q^{6} - 40 q^{7} + 84 q^{8} + 33 q^{9} + 12 q^{11} + 224 q^{12} - 14 q^{13} - 192 q^{14} + 146 q^{16} - 66 q^{17} + 108 q^{18} + 214 q^{19} + 98 q^{22} - 164 q^{23} + 314 q^{24} + 56 q^{26} - 362 q^{27} - 540 q^{28} - 145 q^{29} + 420 q^{31} + 652 q^{32} + 576 q^{33} + 204 q^{34} - 260 q^{36} - 378 q^{37} + 496 q^{38} - 374 q^{39} - 1158 q^{41} - 348 q^{42} + 204 q^{43} + 784 q^{44} + 580 q^{46} - 248 q^{47} + 1880 q^{48} - 283 q^{49} + 228 q^{51} - 1482 q^{52} + 554 q^{53} + 918 q^{54} - 608 q^{56} - 44 q^{57} + 440 q^{59} + 618 q^{61} - 1250 q^{62} - 804 q^{63} + 2594 q^{64} + 2940 q^{66} - 1164 q^{67} - 356 q^{68} - 1968 q^{69} - 692 q^{71} + 2648 q^{72} + 1950 q^{73} - 1832 q^{74} + 1376 q^{76} + 1616 q^{77} + 1302 q^{78} + 272 q^{79} + 1801 q^{81} - 92 q^{82} - 512 q^{83} - 3208 q^{84} + 2446 q^{86} + 232 q^{87} + 6954 q^{88} + 866 q^{89} + 2580 q^{91} - 3468 q^{92} + 40 q^{93} - 5942 q^{94} + 7386 q^{96} - 1562 q^{97} + 3408 q^{98} - 238 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.63099 −0.576643 −0.288322 0.957534i \(-0.593097\pi\)
−0.288322 + 0.957534i \(0.593097\pi\)
\(3\) −9.87991 −1.90139 −0.950695 0.310128i \(-0.899628\pi\)
−0.950695 + 0.310128i \(0.899628\pi\)
\(4\) −5.33986 −0.667483
\(5\) 0 0
\(6\) 16.1141 1.09642
\(7\) −5.21997 −0.281852 −0.140926 0.990020i \(-0.545008\pi\)
−0.140926 + 0.990020i \(0.545008\pi\)
\(8\) 21.7572 0.961543
\(9\) 70.6126 2.61528
\(10\) 0 0
\(11\) −8.55158 −0.234400 −0.117200 0.993108i \(-0.537392\pi\)
−0.117200 + 0.993108i \(0.537392\pi\)
\(12\) 52.7573 1.26914
\(13\) 11.3429 0.241996 0.120998 0.992653i \(-0.461391\pi\)
0.120998 + 0.992653i \(0.461391\pi\)
\(14\) 8.51374 0.162528
\(15\) 0 0
\(16\) 7.23299 0.113016
\(17\) −68.4740 −0.976904 −0.488452 0.872591i \(-0.662438\pi\)
−0.488452 + 0.872591i \(0.662438\pi\)
\(18\) −115.169 −1.50808
\(19\) 6.93014 0.0836780 0.0418390 0.999124i \(-0.486678\pi\)
0.0418390 + 0.999124i \(0.486678\pi\)
\(20\) 0 0
\(21\) 51.5728 0.535910
\(22\) 13.9476 0.135165
\(23\) 132.042 1.19708 0.598538 0.801095i \(-0.295749\pi\)
0.598538 + 0.801095i \(0.295749\pi\)
\(24\) −214.959 −1.82827
\(25\) 0 0
\(26\) −18.5001 −0.139545
\(27\) −430.889 −3.07128
\(28\) 27.8739 0.188131
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −0.419319 −0.00242941 −0.00121471 0.999999i \(-0.500387\pi\)
−0.00121471 + 0.999999i \(0.500387\pi\)
\(32\) −185.855 −1.02671
\(33\) 84.4888 0.445685
\(34\) 111.681 0.563325
\(35\) 0 0
\(36\) −377.061 −1.74565
\(37\) −395.483 −1.75722 −0.878609 0.477542i \(-0.841528\pi\)
−0.878609 + 0.477542i \(0.841528\pi\)
\(38\) −11.3030 −0.0482524
\(39\) −112.067 −0.460128
\(40\) 0 0
\(41\) −447.209 −1.70347 −0.851736 0.523971i \(-0.824450\pi\)
−0.851736 + 0.523971i \(0.824450\pi\)
\(42\) −84.1150 −0.309029
\(43\) −184.132 −0.653020 −0.326510 0.945194i \(-0.605873\pi\)
−0.326510 + 0.945194i \(0.605873\pi\)
\(44\) 45.6642 0.156458
\(45\) 0 0
\(46\) −215.360 −0.690285
\(47\) 97.2612 0.301851 0.150926 0.988545i \(-0.451775\pi\)
0.150926 + 0.988545i \(0.451775\pi\)
\(48\) −71.4613 −0.214886
\(49\) −315.752 −0.920559
\(50\) 0 0
\(51\) 676.516 1.85748
\(52\) −60.5693 −0.161528
\(53\) 209.547 0.543086 0.271543 0.962426i \(-0.412466\pi\)
0.271543 + 0.962426i \(0.412466\pi\)
\(54\) 702.776 1.77103
\(55\) 0 0
\(56\) −113.572 −0.271013
\(57\) −68.4691 −0.159105
\(58\) 47.2988 0.107080
\(59\) 45.9651 0.101426 0.0507131 0.998713i \(-0.483851\pi\)
0.0507131 + 0.998713i \(0.483851\pi\)
\(60\) 0 0
\(61\) 427.655 0.897634 0.448817 0.893624i \(-0.351846\pi\)
0.448817 + 0.893624i \(0.351846\pi\)
\(62\) 0.683906 0.00140091
\(63\) −368.596 −0.737122
\(64\) 245.264 0.479031
\(65\) 0 0
\(66\) −137.801 −0.257001
\(67\) 405.055 0.738588 0.369294 0.929313i \(-0.379600\pi\)
0.369294 + 0.929313i \(0.379600\pi\)
\(68\) 365.641 0.652067
\(69\) −1304.57 −2.27611
\(70\) 0 0
\(71\) −557.971 −0.932662 −0.466331 0.884610i \(-0.654424\pi\)
−0.466331 + 0.884610i \(0.654424\pi\)
\(72\) 1536.33 2.51470
\(73\) 381.988 0.612443 0.306222 0.951960i \(-0.400935\pi\)
0.306222 + 0.951960i \(0.400935\pi\)
\(74\) 645.031 1.01329
\(75\) 0 0
\(76\) −37.0060 −0.0558536
\(77\) 44.6390 0.0660660
\(78\) 182.780 0.265330
\(79\) 577.208 0.822038 0.411019 0.911627i \(-0.365173\pi\)
0.411019 + 0.911627i \(0.365173\pi\)
\(80\) 0 0
\(81\) 2350.60 3.22442
\(82\) 729.396 0.982296
\(83\) 353.745 0.467813 0.233907 0.972259i \(-0.424849\pi\)
0.233907 + 0.972259i \(0.424849\pi\)
\(84\) −275.392 −0.357711
\(85\) 0 0
\(86\) 300.318 0.376560
\(87\) 286.517 0.353079
\(88\) −186.059 −0.225385
\(89\) −277.871 −0.330947 −0.165473 0.986214i \(-0.552915\pi\)
−0.165473 + 0.986214i \(0.552915\pi\)
\(90\) 0 0
\(91\) −59.2095 −0.0682070
\(92\) −705.088 −0.799027
\(93\) 4.14283 0.00461926
\(94\) −158.632 −0.174060
\(95\) 0 0
\(96\) 1836.23 1.95218
\(97\) −677.917 −0.709609 −0.354804 0.934941i \(-0.615453\pi\)
−0.354804 + 0.934941i \(0.615453\pi\)
\(98\) 514.989 0.530834
\(99\) −603.849 −0.613022
\(100\) 0 0
\(101\) 567.816 0.559404 0.279702 0.960087i \(-0.409764\pi\)
0.279702 + 0.960087i \(0.409764\pi\)
\(102\) −1103.39 −1.07110
\(103\) −319.205 −0.305362 −0.152681 0.988276i \(-0.548791\pi\)
−0.152681 + 0.988276i \(0.548791\pi\)
\(104\) 246.789 0.232689
\(105\) 0 0
\(106\) −341.771 −0.313167
\(107\) 79.6547 0.0719674 0.0359837 0.999352i \(-0.488544\pi\)
0.0359837 + 0.999352i \(0.488544\pi\)
\(108\) 2300.88 2.05003
\(109\) −1708.43 −1.50126 −0.750632 0.660721i \(-0.770251\pi\)
−0.750632 + 0.660721i \(0.770251\pi\)
\(110\) 0 0
\(111\) 3907.34 3.34116
\(112\) −37.7560 −0.0318536
\(113\) −1050.31 −0.874383 −0.437191 0.899369i \(-0.644027\pi\)
−0.437191 + 0.899369i \(0.644027\pi\)
\(114\) 111.673 0.0917465
\(115\) 0 0
\(116\) 154.856 0.123948
\(117\) 800.950 0.632887
\(118\) −74.9687 −0.0584867
\(119\) 357.432 0.275342
\(120\) 0 0
\(121\) −1257.87 −0.945057
\(122\) −697.503 −0.517615
\(123\) 4418.39 3.23896
\(124\) 2.23910 0.00162159
\(125\) 0 0
\(126\) 601.177 0.425057
\(127\) −366.926 −0.256373 −0.128187 0.991750i \(-0.540916\pi\)
−0.128187 + 0.991750i \(0.540916\pi\)
\(128\) 1086.81 0.750482
\(129\) 1819.21 1.24165
\(130\) 0 0
\(131\) 2310.86 1.54123 0.770614 0.637302i \(-0.219950\pi\)
0.770614 + 0.637302i \(0.219950\pi\)
\(132\) −451.158 −0.297487
\(133\) −36.1751 −0.0235848
\(134\) −660.643 −0.425902
\(135\) 0 0
\(136\) −1489.80 −0.939335
\(137\) −1899.65 −1.18466 −0.592329 0.805696i \(-0.701791\pi\)
−0.592329 + 0.805696i \(0.701791\pi\)
\(138\) 2127.74 1.31250
\(139\) 1309.49 0.799061 0.399531 0.916720i \(-0.369173\pi\)
0.399531 + 0.916720i \(0.369173\pi\)
\(140\) 0 0
\(141\) −960.932 −0.573937
\(142\) 910.047 0.537813
\(143\) −96.9994 −0.0567238
\(144\) 510.740 0.295567
\(145\) 0 0
\(146\) −623.020 −0.353161
\(147\) 3119.60 1.75034
\(148\) 2111.83 1.17291
\(149\) 1782.98 0.980320 0.490160 0.871632i \(-0.336938\pi\)
0.490160 + 0.871632i \(0.336938\pi\)
\(150\) 0 0
\(151\) −1631.96 −0.879515 −0.439758 0.898116i \(-0.644936\pi\)
−0.439758 + 0.898116i \(0.644936\pi\)
\(152\) 150.781 0.0804600
\(153\) −4835.12 −2.55488
\(154\) −72.8059 −0.0380965
\(155\) 0 0
\(156\) 598.420 0.307128
\(157\) −852.817 −0.433517 −0.216759 0.976225i \(-0.569548\pi\)
−0.216759 + 0.976225i \(0.569548\pi\)
\(158\) −941.422 −0.474022
\(159\) −2070.31 −1.03262
\(160\) 0 0
\(161\) −689.257 −0.337398
\(162\) −3833.81 −1.85934
\(163\) −3280.24 −1.57625 −0.788123 0.615518i \(-0.788947\pi\)
−0.788123 + 0.615518i \(0.788947\pi\)
\(164\) 2388.04 1.13704
\(165\) 0 0
\(166\) −576.955 −0.269761
\(167\) 1682.26 0.779504 0.389752 0.920920i \(-0.372561\pi\)
0.389752 + 0.920920i \(0.372561\pi\)
\(168\) 1122.08 0.515301
\(169\) −2068.34 −0.941438
\(170\) 0 0
\(171\) 489.355 0.218842
\(172\) 983.239 0.435879
\(173\) 1590.22 0.698854 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(174\) −467.308 −0.203601
\(175\) 0 0
\(176\) −61.8535 −0.0264908
\(177\) −454.131 −0.192851
\(178\) 453.206 0.190838
\(179\) −1794.60 −0.749356 −0.374678 0.927155i \(-0.622247\pi\)
−0.374678 + 0.927155i \(0.622247\pi\)
\(180\) 0 0
\(181\) 2353.41 0.966450 0.483225 0.875496i \(-0.339465\pi\)
0.483225 + 0.875496i \(0.339465\pi\)
\(182\) 96.5702 0.0393311
\(183\) −4225.20 −1.70675
\(184\) 2872.88 1.15104
\(185\) 0 0
\(186\) −6.75693 −0.00266367
\(187\) 585.560 0.228986
\(188\) −519.361 −0.201480
\(189\) 2249.23 0.865646
\(190\) 0 0
\(191\) −2184.35 −0.827508 −0.413754 0.910389i \(-0.635783\pi\)
−0.413754 + 0.910389i \(0.635783\pi\)
\(192\) −2423.19 −0.910825
\(193\) 3109.71 1.15980 0.579901 0.814687i \(-0.303091\pi\)
0.579901 + 0.814687i \(0.303091\pi\)
\(194\) 1105.68 0.409191
\(195\) 0 0
\(196\) 1686.07 0.614457
\(197\) −923.756 −0.334086 −0.167043 0.985950i \(-0.553422\pi\)
−0.167043 + 0.985950i \(0.553422\pi\)
\(198\) 984.874 0.353495
\(199\) 4548.71 1.62035 0.810174 0.586189i \(-0.199372\pi\)
0.810174 + 0.586189i \(0.199372\pi\)
\(200\) 0 0
\(201\) −4001.91 −1.40434
\(202\) −926.103 −0.322576
\(203\) 151.379 0.0523386
\(204\) −3612.50 −1.23983
\(205\) 0 0
\(206\) 520.622 0.176085
\(207\) 9323.85 3.13069
\(208\) 82.0429 0.0273493
\(209\) −59.2636 −0.0196141
\(210\) 0 0
\(211\) −318.664 −0.103970 −0.0519851 0.998648i \(-0.516555\pi\)
−0.0519851 + 0.998648i \(0.516555\pi\)
\(212\) −1118.95 −0.362500
\(213\) 5512.70 1.77335
\(214\) −129.916 −0.0414995
\(215\) 0 0
\(216\) −9374.94 −2.95317
\(217\) 2.18883 0.000684735 0
\(218\) 2786.44 0.865694
\(219\) −3774.01 −1.16449
\(220\) 0 0
\(221\) −776.691 −0.236407
\(222\) −6372.85 −1.92666
\(223\) −1706.70 −0.512509 −0.256254 0.966609i \(-0.582488\pi\)
−0.256254 + 0.966609i \(0.582488\pi\)
\(224\) 970.157 0.289381
\(225\) 0 0
\(226\) 1713.06 0.504207
\(227\) −3043.63 −0.889925 −0.444962 0.895549i \(-0.646783\pi\)
−0.444962 + 0.895549i \(0.646783\pi\)
\(228\) 365.616 0.106199
\(229\) −2621.57 −0.756500 −0.378250 0.925704i \(-0.623474\pi\)
−0.378250 + 0.925704i \(0.623474\pi\)
\(230\) 0 0
\(231\) −441.029 −0.125617
\(232\) −630.960 −0.178554
\(233\) 2778.73 0.781291 0.390646 0.920541i \(-0.372252\pi\)
0.390646 + 0.920541i \(0.372252\pi\)
\(234\) −1306.34 −0.364950
\(235\) 0 0
\(236\) −245.447 −0.0677002
\(237\) −5702.76 −1.56301
\(238\) −582.969 −0.158774
\(239\) 4722.67 1.27818 0.639088 0.769133i \(-0.279312\pi\)
0.639088 + 0.769133i \(0.279312\pi\)
\(240\) 0 0
\(241\) −4704.13 −1.25734 −0.628672 0.777671i \(-0.716401\pi\)
−0.628672 + 0.777671i \(0.716401\pi\)
\(242\) 2051.58 0.544961
\(243\) −11589.7 −3.05959
\(244\) −2283.62 −0.599155
\(245\) 0 0
\(246\) −7206.36 −1.86773
\(247\) 78.6076 0.0202497
\(248\) −9.12321 −0.00233599
\(249\) −3494.96 −0.889496
\(250\) 0 0
\(251\) 4449.07 1.11882 0.559408 0.828893i \(-0.311029\pi\)
0.559408 + 0.828893i \(0.311029\pi\)
\(252\) 1968.25 0.492016
\(253\) −1129.17 −0.280594
\(254\) 598.454 0.147836
\(255\) 0 0
\(256\) −3734.70 −0.911792
\(257\) −7226.68 −1.75404 −0.877019 0.480456i \(-0.840471\pi\)
−0.877019 + 0.480456i \(0.840471\pi\)
\(258\) −2967.11 −0.715986
\(259\) 2064.41 0.495275
\(260\) 0 0
\(261\) −2047.77 −0.485646
\(262\) −3769.00 −0.888739
\(263\) 4789.59 1.12296 0.561481 0.827490i \(-0.310232\pi\)
0.561481 + 0.827490i \(0.310232\pi\)
\(264\) 1838.24 0.428545
\(265\) 0 0
\(266\) 59.0014 0.0136000
\(267\) 2745.34 0.629259
\(268\) −2162.94 −0.492995
\(269\) −47.0448 −0.0106631 −0.00533154 0.999986i \(-0.501697\pi\)
−0.00533154 + 0.999986i \(0.501697\pi\)
\(270\) 0 0
\(271\) 7721.78 1.73087 0.865433 0.501025i \(-0.167044\pi\)
0.865433 + 0.501025i \(0.167044\pi\)
\(272\) −495.272 −0.110405
\(273\) 584.984 0.129688
\(274\) 3098.32 0.683125
\(275\) 0 0
\(276\) 6966.20 1.51926
\(277\) 7554.66 1.63868 0.819342 0.573305i \(-0.194339\pi\)
0.819342 + 0.573305i \(0.194339\pi\)
\(278\) −2135.77 −0.460773
\(279\) −29.6092 −0.00635360
\(280\) 0 0
\(281\) −2094.83 −0.444723 −0.222361 0.974964i \(-0.571377\pi\)
−0.222361 + 0.974964i \(0.571377\pi\)
\(282\) 1567.27 0.330957
\(283\) 8200.83 1.72257 0.861287 0.508119i \(-0.169659\pi\)
0.861287 + 0.508119i \(0.169659\pi\)
\(284\) 2979.49 0.622536
\(285\) 0 0
\(286\) 158.205 0.0327094
\(287\) 2334.42 0.480127
\(288\) −13123.7 −2.68514
\(289\) −224.318 −0.0456580
\(290\) 0 0
\(291\) 6697.76 1.34924
\(292\) −2039.76 −0.408795
\(293\) −3518.87 −0.701620 −0.350810 0.936447i \(-0.614094\pi\)
−0.350810 + 0.936447i \(0.614094\pi\)
\(294\) −5088.05 −1.00932
\(295\) 0 0
\(296\) −8604.62 −1.68964
\(297\) 3684.78 0.719907
\(298\) −2908.03 −0.565295
\(299\) 1497.74 0.289687
\(300\) 0 0
\(301\) 961.163 0.184055
\(302\) 2661.71 0.507167
\(303\) −5609.97 −1.06364
\(304\) 50.1256 0.00945691
\(305\) 0 0
\(306\) 7886.06 1.47325
\(307\) 8725.36 1.62209 0.811046 0.584982i \(-0.198898\pi\)
0.811046 + 0.584982i \(0.198898\pi\)
\(308\) −238.366 −0.0440979
\(309\) 3153.72 0.580611
\(310\) 0 0
\(311\) 2640.09 0.481370 0.240685 0.970603i \(-0.422628\pi\)
0.240685 + 0.970603i \(0.422628\pi\)
\(312\) −2438.26 −0.442433
\(313\) 1938.00 0.349976 0.174988 0.984571i \(-0.444011\pi\)
0.174988 + 0.984571i \(0.444011\pi\)
\(314\) 1390.94 0.249985
\(315\) 0 0
\(316\) −3082.21 −0.548696
\(317\) 5546.11 0.982651 0.491326 0.870976i \(-0.336512\pi\)
0.491326 + 0.870976i \(0.336512\pi\)
\(318\) 3376.66 0.595452
\(319\) 247.996 0.0435270
\(320\) 0 0
\(321\) −786.981 −0.136838
\(322\) 1124.17 0.194558
\(323\) −474.534 −0.0817454
\(324\) −12551.9 −2.15224
\(325\) 0 0
\(326\) 5350.04 0.908931
\(327\) 16879.1 2.85449
\(328\) −9730.04 −1.63796
\(329\) −507.701 −0.0850773
\(330\) 0 0
\(331\) 185.492 0.0308023 0.0154012 0.999881i \(-0.495097\pi\)
0.0154012 + 0.999881i \(0.495097\pi\)
\(332\) −1888.95 −0.312257
\(333\) −27926.1 −4.59562
\(334\) −2743.76 −0.449496
\(335\) 0 0
\(336\) 373.026 0.0605662
\(337\) 8508.32 1.37530 0.687652 0.726040i \(-0.258641\pi\)
0.687652 + 0.726040i \(0.258641\pi\)
\(338\) 3373.45 0.542874
\(339\) 10377.0 1.66254
\(340\) 0 0
\(341\) 3.58584 0.000569454 0
\(342\) −798.135 −0.126194
\(343\) 3438.67 0.541313
\(344\) −4006.20 −0.627906
\(345\) 0 0
\(346\) −2593.63 −0.402990
\(347\) −7853.53 −1.21498 −0.607492 0.794326i \(-0.707824\pi\)
−0.607492 + 0.794326i \(0.707824\pi\)
\(348\) −1529.96 −0.235674
\(349\) 6328.33 0.970624 0.485312 0.874341i \(-0.338706\pi\)
0.485312 + 0.874341i \(0.338706\pi\)
\(350\) 0 0
\(351\) −4887.51 −0.743237
\(352\) 1589.35 0.240661
\(353\) 7973.45 1.20222 0.601111 0.799166i \(-0.294725\pi\)
0.601111 + 0.799166i \(0.294725\pi\)
\(354\) 740.684 0.111206
\(355\) 0 0
\(356\) 1483.79 0.220901
\(357\) −3531.40 −0.523533
\(358\) 2926.98 0.432111
\(359\) −10059.4 −1.47887 −0.739435 0.673228i \(-0.764907\pi\)
−0.739435 + 0.673228i \(0.764907\pi\)
\(360\) 0 0
\(361\) −6810.97 −0.992998
\(362\) −3838.39 −0.557297
\(363\) 12427.6 1.79692
\(364\) 316.170 0.0455270
\(365\) 0 0
\(366\) 6891.27 0.984187
\(367\) 3032.67 0.431346 0.215673 0.976466i \(-0.430805\pi\)
0.215673 + 0.976466i \(0.430805\pi\)
\(368\) 955.061 0.135288
\(369\) −31578.6 −4.45506
\(370\) 0 0
\(371\) −1093.83 −0.153070
\(372\) −22.1221 −0.00308328
\(373\) 11109.4 1.54215 0.771077 0.636742i \(-0.219718\pi\)
0.771077 + 0.636742i \(0.219718\pi\)
\(374\) −955.045 −0.132043
\(375\) 0 0
\(376\) 2116.13 0.290243
\(377\) −328.943 −0.0449375
\(378\) −3668.47 −0.499169
\(379\) 4510.20 0.611275 0.305638 0.952148i \(-0.401130\pi\)
0.305638 + 0.952148i \(0.401130\pi\)
\(380\) 0 0
\(381\) 3625.20 0.487466
\(382\) 3562.66 0.477177
\(383\) −7810.96 −1.04209 −0.521047 0.853528i \(-0.674458\pi\)
−0.521047 + 0.853528i \(0.674458\pi\)
\(384\) −10737.6 −1.42696
\(385\) 0 0
\(386\) −5071.92 −0.668792
\(387\) −13002.0 −1.70783
\(388\) 3619.98 0.473652
\(389\) 9221.90 1.20198 0.600989 0.799258i \(-0.294774\pi\)
0.600989 + 0.799258i \(0.294774\pi\)
\(390\) 0 0
\(391\) −9041.46 −1.16943
\(392\) −6869.88 −0.885157
\(393\) −22831.1 −2.93047
\(394\) 1506.64 0.192648
\(395\) 0 0
\(396\) 3224.47 0.409181
\(397\) 10034.3 1.26854 0.634268 0.773113i \(-0.281301\pi\)
0.634268 + 0.773113i \(0.281301\pi\)
\(398\) −7418.91 −0.934363
\(399\) 357.407 0.0448439
\(400\) 0 0
\(401\) −8193.64 −1.02038 −0.510188 0.860063i \(-0.670424\pi\)
−0.510188 + 0.860063i \(0.670424\pi\)
\(402\) 6527.09 0.809805
\(403\) −4.75628 −0.000587908 0
\(404\) −3032.06 −0.373392
\(405\) 0 0
\(406\) −246.898 −0.0301807
\(407\) 3382.01 0.411892
\(408\) 14719.1 1.78604
\(409\) 3921.69 0.474120 0.237060 0.971495i \(-0.423816\pi\)
0.237060 + 0.971495i \(0.423816\pi\)
\(410\) 0 0
\(411\) 18768.4 2.25250
\(412\) 1704.51 0.203824
\(413\) −239.936 −0.0285872
\(414\) −15207.1 −1.80529
\(415\) 0 0
\(416\) −2108.13 −0.248460
\(417\) −12937.6 −1.51933
\(418\) 96.6586 0.0113103
\(419\) 3026.66 0.352892 0.176446 0.984310i \(-0.443540\pi\)
0.176446 + 0.984310i \(0.443540\pi\)
\(420\) 0 0
\(421\) 4823.43 0.558384 0.279192 0.960235i \(-0.409933\pi\)
0.279192 + 0.960235i \(0.409933\pi\)
\(422\) 519.739 0.0599538
\(423\) 6867.87 0.789426
\(424\) 4559.17 0.522200
\(425\) 0 0
\(426\) −8991.19 −1.02259
\(427\) −2232.35 −0.253000
\(428\) −425.345 −0.0480370
\(429\) 958.346 0.107854
\(430\) 0 0
\(431\) −2029.51 −0.226816 −0.113408 0.993548i \(-0.536177\pi\)
−0.113408 + 0.993548i \(0.536177\pi\)
\(432\) −3116.61 −0.347102
\(433\) 535.808 0.0594672 0.0297336 0.999558i \(-0.490534\pi\)
0.0297336 + 0.999558i \(0.490534\pi\)
\(434\) −3.56997 −0.000394848 0
\(435\) 0 0
\(436\) 9122.77 1.00207
\(437\) 915.072 0.100169
\(438\) 6155.38 0.671497
\(439\) 8791.74 0.955824 0.477912 0.878408i \(-0.341394\pi\)
0.477912 + 0.878408i \(0.341394\pi\)
\(440\) 0 0
\(441\) −22296.1 −2.40752
\(442\) 1266.78 0.136322
\(443\) −13916.9 −1.49258 −0.746291 0.665619i \(-0.768167\pi\)
−0.746291 + 0.665619i \(0.768167\pi\)
\(444\) −20864.7 −2.23016
\(445\) 0 0
\(446\) 2783.62 0.295535
\(447\) −17615.7 −1.86397
\(448\) −1280.27 −0.135016
\(449\) 2129.54 0.223829 0.111915 0.993718i \(-0.464302\pi\)
0.111915 + 0.993718i \(0.464302\pi\)
\(450\) 0 0
\(451\) 3824.35 0.399294
\(452\) 5608.53 0.583635
\(453\) 16123.6 1.67230
\(454\) 4964.14 0.513169
\(455\) 0 0
\(456\) −1489.70 −0.152986
\(457\) −1932.25 −0.197783 −0.0988915 0.995098i \(-0.531530\pi\)
−0.0988915 + 0.995098i \(0.531530\pi\)
\(458\) 4275.77 0.436231
\(459\) 29504.6 3.00035
\(460\) 0 0
\(461\) −16518.3 −1.66884 −0.834418 0.551132i \(-0.814196\pi\)
−0.834418 + 0.551132i \(0.814196\pi\)
\(462\) 719.316 0.0724364
\(463\) −11535.2 −1.15785 −0.578926 0.815380i \(-0.696528\pi\)
−0.578926 + 0.815380i \(0.696528\pi\)
\(464\) −209.757 −0.0209865
\(465\) 0 0
\(466\) −4532.10 −0.450526
\(467\) −15667.8 −1.55250 −0.776250 0.630425i \(-0.782880\pi\)
−0.776250 + 0.630425i \(0.782880\pi\)
\(468\) −4276.96 −0.422441
\(469\) −2114.38 −0.208172
\(470\) 0 0
\(471\) 8425.75 0.824285
\(472\) 1000.07 0.0975255
\(473\) 1574.62 0.153068
\(474\) 9301.17 0.901301
\(475\) 0 0
\(476\) −1908.64 −0.183786
\(477\) 14796.7 1.42032
\(478\) −7702.65 −0.737052
\(479\) 4672.28 0.445682 0.222841 0.974855i \(-0.428467\pi\)
0.222841 + 0.974855i \(0.428467\pi\)
\(480\) 0 0
\(481\) −4485.92 −0.425239
\(482\) 7672.41 0.725038
\(483\) 6809.80 0.641525
\(484\) 6716.85 0.630809
\(485\) 0 0
\(486\) 18902.8 1.76429
\(487\) 8465.76 0.787722 0.393861 0.919170i \(-0.371139\pi\)
0.393861 + 0.919170i \(0.371139\pi\)
\(488\) 9304.60 0.863113
\(489\) 32408.4 2.99706
\(490\) 0 0
\(491\) −12302.6 −1.13077 −0.565385 0.824827i \(-0.691272\pi\)
−0.565385 + 0.824827i \(0.691272\pi\)
\(492\) −23593.6 −2.16195
\(493\) 1985.74 0.181407
\(494\) −128.209 −0.0116769
\(495\) 0 0
\(496\) −3.03293 −0.000274562 0
\(497\) 2912.59 0.262873
\(498\) 5700.26 0.512922
\(499\) −2781.31 −0.249516 −0.124758 0.992187i \(-0.539815\pi\)
−0.124758 + 0.992187i \(0.539815\pi\)
\(500\) 0 0
\(501\) −16620.6 −1.48214
\(502\) −7256.40 −0.645158
\(503\) 13673.9 1.21211 0.606054 0.795424i \(-0.292752\pi\)
0.606054 + 0.795424i \(0.292752\pi\)
\(504\) −8019.62 −0.708774
\(505\) 0 0
\(506\) 1841.67 0.161803
\(507\) 20435.0 1.79004
\(508\) 1959.33 0.171125
\(509\) 7431.68 0.647158 0.323579 0.946201i \(-0.395114\pi\)
0.323579 + 0.946201i \(0.395114\pi\)
\(510\) 0 0
\(511\) −1993.97 −0.172618
\(512\) −2603.24 −0.224704
\(513\) −2986.12 −0.256999
\(514\) 11786.7 1.01145
\(515\) 0 0
\(516\) −9714.31 −0.828776
\(517\) −831.737 −0.0707538
\(518\) −3367.04 −0.285597
\(519\) −15711.2 −1.32879
\(520\) 0 0
\(521\) 7797.86 0.655721 0.327860 0.944726i \(-0.393672\pi\)
0.327860 + 0.944726i \(0.393672\pi\)
\(522\) 3339.89 0.280044
\(523\) 16670.9 1.39382 0.696908 0.717160i \(-0.254558\pi\)
0.696908 + 0.717160i \(0.254558\pi\)
\(524\) −12339.7 −1.02874
\(525\) 0 0
\(526\) −7811.79 −0.647548
\(527\) 28.7124 0.00237331
\(528\) 611.107 0.0503693
\(529\) 5268.18 0.432990
\(530\) 0 0
\(531\) 3245.71 0.265258
\(532\) 193.170 0.0157425
\(533\) −5072.64 −0.412233
\(534\) −4477.63 −0.362858
\(535\) 0 0
\(536\) 8812.88 0.710184
\(537\) 17730.5 1.42482
\(538\) 76.7297 0.00614880
\(539\) 2700.18 0.215779
\(540\) 0 0
\(541\) 13400.1 1.06491 0.532455 0.846458i \(-0.321270\pi\)
0.532455 + 0.846458i \(0.321270\pi\)
\(542\) −12594.2 −0.998092
\(543\) −23251.5 −1.83760
\(544\) 12726.2 1.00300
\(545\) 0 0
\(546\) −954.105 −0.0747838
\(547\) 17339.4 1.35535 0.677677 0.735360i \(-0.262987\pi\)
0.677677 + 0.735360i \(0.262987\pi\)
\(548\) 10143.9 0.790739
\(549\) 30197.9 2.34757
\(550\) 0 0
\(551\) −200.974 −0.0155386
\(552\) −28383.7 −2.18857
\(553\) −3013.01 −0.231693
\(554\) −12321.6 −0.944936
\(555\) 0 0
\(556\) −6992.50 −0.533360
\(557\) 7790.90 0.592659 0.296330 0.955086i \(-0.404237\pi\)
0.296330 + 0.955086i \(0.404237\pi\)
\(558\) 48.2924 0.00366376
\(559\) −2088.58 −0.158028
\(560\) 0 0
\(561\) −5785.28 −0.435392
\(562\) 3416.66 0.256447
\(563\) 21514.2 1.61051 0.805253 0.592932i \(-0.202030\pi\)
0.805253 + 0.592932i \(0.202030\pi\)
\(564\) 5131.24 0.383093
\(565\) 0 0
\(566\) −13375.5 −0.993311
\(567\) −12270.1 −0.908808
\(568\) −12139.9 −0.896794
\(569\) 16993.5 1.25203 0.626015 0.779811i \(-0.284685\pi\)
0.626015 + 0.779811i \(0.284685\pi\)
\(570\) 0 0
\(571\) 16791.4 1.23064 0.615321 0.788277i \(-0.289027\pi\)
0.615321 + 0.788277i \(0.289027\pi\)
\(572\) 517.963 0.0378621
\(573\) 21581.2 1.57342
\(574\) −3807.42 −0.276862
\(575\) 0 0
\(576\) 17318.7 1.25280
\(577\) −8108.87 −0.585055 −0.292527 0.956257i \(-0.594496\pi\)
−0.292527 + 0.956257i \(0.594496\pi\)
\(578\) 365.860 0.0263284
\(579\) −30723.7 −2.20524
\(580\) 0 0
\(581\) −1846.54 −0.131854
\(582\) −10924.0 −0.778032
\(583\) −1791.96 −0.127299
\(584\) 8311.00 0.588890
\(585\) 0 0
\(586\) 5739.25 0.404584
\(587\) −16076.3 −1.13039 −0.565197 0.824956i \(-0.691200\pi\)
−0.565197 + 0.824956i \(0.691200\pi\)
\(588\) −16658.2 −1.16832
\(589\) −2.90594 −0.000203289 0
\(590\) 0 0
\(591\) 9126.63 0.635227
\(592\) −2860.53 −0.198593
\(593\) 4341.83 0.300671 0.150335 0.988635i \(-0.451965\pi\)
0.150335 + 0.988635i \(0.451965\pi\)
\(594\) −6009.85 −0.415130
\(595\) 0 0
\(596\) −9520.88 −0.654346
\(597\) −44940.8 −3.08091
\(598\) −2442.80 −0.167046
\(599\) −10540.1 −0.718960 −0.359480 0.933153i \(-0.617046\pi\)
−0.359480 + 0.933153i \(0.617046\pi\)
\(600\) 0 0
\(601\) 16485.6 1.11890 0.559451 0.828863i \(-0.311012\pi\)
0.559451 + 0.828863i \(0.311012\pi\)
\(602\) −1567.65 −0.106134
\(603\) 28602.0 1.93162
\(604\) 8714.43 0.587061
\(605\) 0 0
\(606\) 9149.82 0.613343
\(607\) −13326.5 −0.891112 −0.445556 0.895254i \(-0.646994\pi\)
−0.445556 + 0.895254i \(0.646994\pi\)
\(608\) −1288.00 −0.0859132
\(609\) −1495.61 −0.0995161
\(610\) 0 0
\(611\) 1103.22 0.0730467
\(612\) 25818.9 1.70534
\(613\) −20459.4 −1.34804 −0.674019 0.738714i \(-0.735433\pi\)
−0.674019 + 0.738714i \(0.735433\pi\)
\(614\) −14231.0 −0.935369
\(615\) 0 0
\(616\) 971.221 0.0635253
\(617\) 3108.50 0.202826 0.101413 0.994844i \(-0.467664\pi\)
0.101413 + 0.994844i \(0.467664\pi\)
\(618\) −5143.70 −0.334806
\(619\) 10426.3 0.677012 0.338506 0.940964i \(-0.390078\pi\)
0.338506 + 0.940964i \(0.390078\pi\)
\(620\) 0 0
\(621\) −56895.5 −3.67655
\(622\) −4305.97 −0.277579
\(623\) 1450.48 0.0932780
\(624\) −810.576 −0.0520016
\(625\) 0 0
\(626\) −3160.87 −0.201811
\(627\) 585.519 0.0372941
\(628\) 4553.92 0.289365
\(629\) 27080.3 1.71663
\(630\) 0 0
\(631\) 18018.2 1.13675 0.568377 0.822768i \(-0.307572\pi\)
0.568377 + 0.822768i \(0.307572\pi\)
\(632\) 12558.4 0.790424
\(633\) 3148.37 0.197688
\(634\) −9045.67 −0.566639
\(635\) 0 0
\(636\) 11055.2 0.689255
\(637\) −3581.53 −0.222772
\(638\) −404.479 −0.0250995
\(639\) −39399.8 −2.43917
\(640\) 0 0
\(641\) 15178.7 0.935295 0.467648 0.883915i \(-0.345102\pi\)
0.467648 + 0.883915i \(0.345102\pi\)
\(642\) 1283.56 0.0789067
\(643\) −23957.1 −1.46932 −0.734662 0.678433i \(-0.762659\pi\)
−0.734662 + 0.678433i \(0.762659\pi\)
\(644\) 3680.54 0.225207
\(645\) 0 0
\(646\) 773.962 0.0471379
\(647\) 12835.2 0.779915 0.389958 0.920833i \(-0.372490\pi\)
0.389958 + 0.920833i \(0.372490\pi\)
\(648\) 51142.5 3.10041
\(649\) −393.074 −0.0237743
\(650\) 0 0
\(651\) −21.6255 −0.00130195
\(652\) 17516.0 1.05212
\(653\) 4355.97 0.261045 0.130523 0.991445i \(-0.458335\pi\)
0.130523 + 0.991445i \(0.458335\pi\)
\(654\) −27529.7 −1.64602
\(655\) 0 0
\(656\) −3234.66 −0.192519
\(657\) 26973.2 1.60171
\(658\) 828.056 0.0490593
\(659\) −32704.1 −1.93319 −0.966594 0.256313i \(-0.917492\pi\)
−0.966594 + 0.256313i \(0.917492\pi\)
\(660\) 0 0
\(661\) −2839.66 −0.167095 −0.0835475 0.996504i \(-0.526625\pi\)
−0.0835475 + 0.996504i \(0.526625\pi\)
\(662\) −302.536 −0.0177620
\(663\) 7673.64 0.449501
\(664\) 7696.50 0.449823
\(665\) 0 0
\(666\) 45547.3 2.65003
\(667\) −3829.23 −0.222291
\(668\) −8983.04 −0.520306
\(669\) 16862.1 0.974478
\(670\) 0 0
\(671\) −3657.13 −0.210405
\(672\) −9585.06 −0.550226
\(673\) 18569.9 1.06362 0.531810 0.846864i \(-0.321512\pi\)
0.531810 + 0.846864i \(0.321512\pi\)
\(674\) −13877.0 −0.793060
\(675\) 0 0
\(676\) 11044.6 0.628393
\(677\) −18106.5 −1.02790 −0.513952 0.857819i \(-0.671819\pi\)
−0.513952 + 0.857819i \(0.671819\pi\)
\(678\) −16924.8 −0.958693
\(679\) 3538.71 0.200005
\(680\) 0 0
\(681\) 30070.8 1.69209
\(682\) −5.84848 −0.000328372 0
\(683\) 5510.97 0.308743 0.154371 0.988013i \(-0.450665\pi\)
0.154371 + 0.988013i \(0.450665\pi\)
\(684\) −2613.09 −0.146073
\(685\) 0 0
\(686\) −5608.44 −0.312145
\(687\) 25900.9 1.43840
\(688\) −1331.82 −0.0738014
\(689\) 2376.87 0.131425
\(690\) 0 0
\(691\) 24652.0 1.35717 0.678587 0.734520i \(-0.262592\pi\)
0.678587 + 0.734520i \(0.262592\pi\)
\(692\) −8491.53 −0.466473
\(693\) 3152.08 0.172781
\(694\) 12809.1 0.700613
\(695\) 0 0
\(696\) 6233.82 0.339501
\(697\) 30622.2 1.66413
\(698\) −10321.5 −0.559704
\(699\) −27453.6 −1.48554
\(700\) 0 0
\(701\) −3399.10 −0.183142 −0.0915708 0.995799i \(-0.529189\pi\)
−0.0915708 + 0.995799i \(0.529189\pi\)
\(702\) 7971.50 0.428582
\(703\) −2740.75 −0.147041
\(704\) −2097.39 −0.112285
\(705\) 0 0
\(706\) −13004.7 −0.693253
\(707\) −2963.98 −0.157669
\(708\) 2424.99 0.128724
\(709\) 26329.1 1.39465 0.697327 0.716753i \(-0.254373\pi\)
0.697327 + 0.716753i \(0.254373\pi\)
\(710\) 0 0
\(711\) 40758.2 2.14986
\(712\) −6045.70 −0.318219
\(713\) −55.3678 −0.00290819
\(714\) 5759.68 0.301892
\(715\) 0 0
\(716\) 9582.91 0.500182
\(717\) −46659.6 −2.43031
\(718\) 16406.8 0.852781
\(719\) 22345.6 1.15904 0.579521 0.814957i \(-0.303240\pi\)
0.579521 + 0.814957i \(0.303240\pi\)
\(720\) 0 0
\(721\) 1666.24 0.0860668
\(722\) 11108.7 0.572606
\(723\) 46476.4 2.39070
\(724\) −12566.9 −0.645089
\(725\) 0 0
\(726\) −20269.4 −1.03618
\(727\) 29862.8 1.52345 0.761727 0.647898i \(-0.224352\pi\)
0.761727 + 0.647898i \(0.224352\pi\)
\(728\) −1288.23 −0.0655839
\(729\) 51039.2 2.59306
\(730\) 0 0
\(731\) 12608.2 0.637938
\(732\) 22562.0 1.13923
\(733\) 987.919 0.0497812 0.0248906 0.999690i \(-0.492076\pi\)
0.0248906 + 0.999690i \(0.492076\pi\)
\(734\) −4946.27 −0.248733
\(735\) 0 0
\(736\) −24540.7 −1.22905
\(737\) −3463.86 −0.173125
\(738\) 51504.5 2.56898
\(739\) −35870.9 −1.78556 −0.892782 0.450489i \(-0.851250\pi\)
−0.892782 + 0.450489i \(0.851250\pi\)
\(740\) 0 0
\(741\) −776.636 −0.0385026
\(742\) 1784.03 0.0882667
\(743\) −8322.20 −0.410918 −0.205459 0.978666i \(-0.565869\pi\)
−0.205459 + 0.978666i \(0.565869\pi\)
\(744\) 90.1365 0.00444162
\(745\) 0 0
\(746\) −18119.4 −0.889273
\(747\) 24978.8 1.22346
\(748\) −3126.81 −0.152844
\(749\) −415.795 −0.0202841
\(750\) 0 0
\(751\) 20781.2 1.00974 0.504872 0.863194i \(-0.331540\pi\)
0.504872 + 0.863194i \(0.331540\pi\)
\(752\) 703.489 0.0341139
\(753\) −43956.4 −2.12730
\(754\) 536.504 0.0259129
\(755\) 0 0
\(756\) −12010.6 −0.577804
\(757\) 16424.7 0.788596 0.394298 0.918983i \(-0.370988\pi\)
0.394298 + 0.918983i \(0.370988\pi\)
\(758\) −7356.10 −0.352488
\(759\) 11156.1 0.533519
\(760\) 0 0
\(761\) 416.312 0.0198309 0.00991543 0.999951i \(-0.496844\pi\)
0.00991543 + 0.999951i \(0.496844\pi\)
\(762\) −5912.67 −0.281094
\(763\) 8917.95 0.423134
\(764\) 11664.1 0.552347
\(765\) 0 0
\(766\) 12739.6 0.600916
\(767\) 521.376 0.0245447
\(768\) 36898.5 1.73367
\(769\) −37129.2 −1.74111 −0.870554 0.492073i \(-0.836239\pi\)
−0.870554 + 0.492073i \(0.836239\pi\)
\(770\) 0 0
\(771\) 71398.9 3.33511
\(772\) −16605.4 −0.774148
\(773\) 2182.83 0.101567 0.0507833 0.998710i \(-0.483828\pi\)
0.0507833 + 0.998710i \(0.483828\pi\)
\(774\) 21206.2 0.984809
\(775\) 0 0
\(776\) −14749.6 −0.682319
\(777\) −20396.2 −0.941711
\(778\) −15040.9 −0.693112
\(779\) −3099.22 −0.142543
\(780\) 0 0
\(781\) 4771.53 0.218616
\(782\) 14746.6 0.674343
\(783\) 12495.8 0.570322
\(784\) −2283.83 −0.104037
\(785\) 0 0
\(786\) 37237.4 1.68984
\(787\) −7068.15 −0.320143 −0.160071 0.987105i \(-0.551172\pi\)
−0.160071 + 0.987105i \(0.551172\pi\)
\(788\) 4932.73 0.222996
\(789\) −47320.7 −2.13519
\(790\) 0 0
\(791\) 5482.61 0.246446
\(792\) −13138.1 −0.589446
\(793\) 4850.84 0.217224
\(794\) −16365.9 −0.731493
\(795\) 0 0
\(796\) −24289.5 −1.08155
\(797\) −15086.0 −0.670482 −0.335241 0.942132i \(-0.608818\pi\)
−0.335241 + 0.942132i \(0.608818\pi\)
\(798\) −582.928 −0.0258589
\(799\) −6659.86 −0.294880
\(800\) 0 0
\(801\) −19621.2 −0.865519
\(802\) 13363.8 0.588393
\(803\) −3266.60 −0.143557
\(804\) 21369.6 0.937375
\(805\) 0 0
\(806\) 7.75746 0.000339013 0
\(807\) 464.798 0.0202747
\(808\) 12354.1 0.537890
\(809\) −21798.3 −0.947326 −0.473663 0.880706i \(-0.657069\pi\)
−0.473663 + 0.880706i \(0.657069\pi\)
\(810\) 0 0
\(811\) 5569.61 0.241153 0.120577 0.992704i \(-0.461526\pi\)
0.120577 + 0.992704i \(0.461526\pi\)
\(812\) −808.344 −0.0349351
\(813\) −76290.4 −3.29105
\(814\) −5516.03 −0.237515
\(815\) 0 0
\(816\) 4893.24 0.209924
\(817\) −1276.06 −0.0546434
\(818\) −6396.25 −0.273398
\(819\) −4180.93 −0.178381
\(820\) 0 0
\(821\) 21281.8 0.904676 0.452338 0.891847i \(-0.350590\pi\)
0.452338 + 0.891847i \(0.350590\pi\)
\(822\) −30611.1 −1.29889
\(823\) −16799.0 −0.711514 −0.355757 0.934578i \(-0.615777\pi\)
−0.355757 + 0.934578i \(0.615777\pi\)
\(824\) −6945.02 −0.293618
\(825\) 0 0
\(826\) 391.335 0.0164846
\(827\) −7360.62 −0.309497 −0.154748 0.987954i \(-0.549457\pi\)
−0.154748 + 0.987954i \(0.549457\pi\)
\(828\) −49788.1 −2.08968
\(829\) −11634.6 −0.487438 −0.243719 0.969846i \(-0.578367\pi\)
−0.243719 + 0.969846i \(0.578367\pi\)
\(830\) 0 0
\(831\) −74639.4 −3.11578
\(832\) 2782.00 0.115924
\(833\) 21620.8 0.899299
\(834\) 21101.2 0.876110
\(835\) 0 0
\(836\) 316.459 0.0130921
\(837\) 180.680 0.00746141
\(838\) −4936.46 −0.203493
\(839\) −32151.0 −1.32297 −0.661486 0.749957i \(-0.730074\pi\)
−0.661486 + 0.749957i \(0.730074\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) −7866.99 −0.321988
\(843\) 20696.7 0.845592
\(844\) 1701.62 0.0693984
\(845\) 0 0
\(846\) −11201.4 −0.455217
\(847\) 6566.05 0.266366
\(848\) 1515.66 0.0613771
\(849\) −81023.4 −3.27528
\(850\) 0 0
\(851\) −52220.6 −2.10352
\(852\) −29437.1 −1.18368
\(853\) −35044.9 −1.40670 −0.703349 0.710844i \(-0.748313\pi\)
−0.703349 + 0.710844i \(0.748313\pi\)
\(854\) 3640.95 0.145891
\(855\) 0 0
\(856\) 1733.07 0.0691997
\(857\) −7143.05 −0.284716 −0.142358 0.989815i \(-0.545468\pi\)
−0.142358 + 0.989815i \(0.545468\pi\)
\(858\) −1563.06 −0.0621933
\(859\) 33960.3 1.34891 0.674454 0.738317i \(-0.264379\pi\)
0.674454 + 0.738317i \(0.264379\pi\)
\(860\) 0 0
\(861\) −23063.9 −0.912909
\(862\) 3310.11 0.130792
\(863\) −45823.4 −1.80747 −0.903736 0.428090i \(-0.859186\pi\)
−0.903736 + 0.428090i \(0.859186\pi\)
\(864\) 80082.7 3.15332
\(865\) 0 0
\(866\) −873.900 −0.0342914
\(867\) 2216.24 0.0868136
\(868\) −11.6881 −0.000457049 0
\(869\) −4936.04 −0.192685
\(870\) 0 0
\(871\) 4594.49 0.178735
\(872\) −37170.7 −1.44353
\(873\) −47869.5 −1.85583
\(874\) −1492.48 −0.0577617
\(875\) 0 0
\(876\) 20152.7 0.777279
\(877\) 21281.3 0.819406 0.409703 0.912219i \(-0.365632\pi\)
0.409703 + 0.912219i \(0.365632\pi\)
\(878\) −14339.3 −0.551169
\(879\) 34766.1 1.33405
\(880\) 0 0
\(881\) 1359.32 0.0519826 0.0259913 0.999662i \(-0.491726\pi\)
0.0259913 + 0.999662i \(0.491726\pi\)
\(882\) 36364.7 1.38828
\(883\) 47928.2 1.82663 0.913313 0.407257i \(-0.133515\pi\)
0.913313 + 0.407257i \(0.133515\pi\)
\(884\) 4147.42 0.157797
\(885\) 0 0
\(886\) 22698.5 0.860688
\(887\) −3100.58 −0.117370 −0.0586851 0.998277i \(-0.518691\pi\)
−0.0586851 + 0.998277i \(0.518691\pi\)
\(888\) 85012.9 3.21266
\(889\) 1915.34 0.0722593
\(890\) 0 0
\(891\) −20101.3 −0.755803
\(892\) 9113.56 0.342090
\(893\) 674.033 0.0252583
\(894\) 28731.1 1.07485
\(895\) 0 0
\(896\) −5673.14 −0.211525
\(897\) −14797.5 −0.550808
\(898\) −3473.27 −0.129070
\(899\) 12.1602 0.000451131 0
\(900\) 0 0
\(901\) −14348.5 −0.530543
\(902\) −6237.48 −0.230250
\(903\) −9496.21 −0.349960
\(904\) −22851.9 −0.840756
\(905\) 0 0
\(906\) −26297.5 −0.964321
\(907\) −23428.1 −0.857680 −0.428840 0.903380i \(-0.641078\pi\)
−0.428840 + 0.903380i \(0.641078\pi\)
\(908\) 16252.6 0.594009
\(909\) 40094.9 1.46300
\(910\) 0 0
\(911\) −12868.3 −0.467999 −0.234000 0.972237i \(-0.575181\pi\)
−0.234000 + 0.972237i \(0.575181\pi\)
\(912\) −495.237 −0.0179813
\(913\) −3025.07 −0.109655
\(914\) 3151.49 0.114050
\(915\) 0 0
\(916\) 13998.8 0.504950
\(917\) −12062.6 −0.434398
\(918\) −48121.9 −1.73013
\(919\) 3938.41 0.141367 0.0706834 0.997499i \(-0.477482\pi\)
0.0706834 + 0.997499i \(0.477482\pi\)
\(920\) 0 0
\(921\) −86205.7 −3.08423
\(922\) 26941.2 0.962323
\(923\) −6328.99 −0.225700
\(924\) 2355.03 0.0838473
\(925\) 0 0
\(926\) 18813.8 0.667667
\(927\) −22539.9 −0.798607
\(928\) 5389.79 0.190656
\(929\) −15856.7 −0.560001 −0.280001 0.960000i \(-0.590335\pi\)
−0.280001 + 0.960000i \(0.590335\pi\)
\(930\) 0 0
\(931\) −2188.20 −0.0770306
\(932\) −14838.0 −0.521498
\(933\) −26083.9 −0.915271
\(934\) 25554.0 0.895238
\(935\) 0 0
\(936\) 17426.4 0.608548
\(937\) 18559.7 0.647085 0.323542 0.946214i \(-0.395126\pi\)
0.323542 + 0.946214i \(0.395126\pi\)
\(938\) 3448.54 0.120041
\(939\) −19147.3 −0.665441
\(940\) 0 0
\(941\) 24125.6 0.835785 0.417892 0.908497i \(-0.362769\pi\)
0.417892 + 0.908497i \(0.362769\pi\)
\(942\) −13742.4 −0.475318
\(943\) −59050.6 −2.03919
\(944\) 332.465 0.0114627
\(945\) 0 0
\(946\) −2568.19 −0.0882655
\(947\) 6883.31 0.236196 0.118098 0.993002i \(-0.462320\pi\)
0.118098 + 0.993002i \(0.462320\pi\)
\(948\) 30452.0 1.04328
\(949\) 4332.84 0.148209
\(950\) 0 0
\(951\) −54795.0 −1.86840
\(952\) 7776.73 0.264753
\(953\) −37573.8 −1.27716 −0.638580 0.769555i \(-0.720478\pi\)
−0.638580 + 0.769555i \(0.720478\pi\)
\(954\) −24133.3 −0.819020
\(955\) 0 0
\(956\) −25218.4 −0.853161
\(957\) −2450.18 −0.0827617
\(958\) −7620.45 −0.257000
\(959\) 9916.13 0.333898
\(960\) 0 0
\(961\) −29790.8 −0.999994
\(962\) 7316.50 0.245211
\(963\) 5624.63 0.188215
\(964\) 25119.4 0.839255
\(965\) 0 0
\(966\) −11106.7 −0.369931
\(967\) 1584.28 0.0526856 0.0263428 0.999653i \(-0.491614\pi\)
0.0263428 + 0.999653i \(0.491614\pi\)
\(968\) −27367.8 −0.908712
\(969\) 4688.35 0.155430
\(970\) 0 0
\(971\) 36569.1 1.20861 0.604304 0.796754i \(-0.293451\pi\)
0.604304 + 0.796754i \(0.293451\pi\)
\(972\) 61887.5 2.04222
\(973\) −6835.50 −0.225217
\(974\) −13807.6 −0.454234
\(975\) 0 0
\(976\) 3093.23 0.101447
\(977\) 33583.0 1.09971 0.549854 0.835261i \(-0.314683\pi\)
0.549854 + 0.835261i \(0.314683\pi\)
\(978\) −52858.0 −1.72823
\(979\) 2376.23 0.0775738
\(980\) 0 0
\(981\) −120637. −3.92623
\(982\) 20065.4 0.652051
\(983\) 25900.6 0.840387 0.420193 0.907435i \(-0.361962\pi\)
0.420193 + 0.907435i \(0.361962\pi\)
\(984\) 96131.9 3.11440
\(985\) 0 0
\(986\) −3238.74 −0.104607
\(987\) 5016.04 0.161765
\(988\) −419.754 −0.0135163
\(989\) −24313.2 −0.781714
\(990\) 0 0
\(991\) 29604.3 0.948951 0.474475 0.880269i \(-0.342638\pi\)
0.474475 + 0.880269i \(0.342638\pi\)
\(992\) 77.9324 0.00249431
\(993\) −1832.65 −0.0585672
\(994\) −4750.42 −0.151584
\(995\) 0 0
\(996\) 18662.6 0.593723
\(997\) −39267.2 −1.24735 −0.623673 0.781685i \(-0.714360\pi\)
−0.623673 + 0.781685i \(0.714360\pi\)
\(998\) 4536.30 0.143882
\(999\) 170409. 5.39691
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 725.4.a.c.1.3 5
5.4 even 2 29.4.a.b.1.3 5
15.14 odd 2 261.4.a.f.1.3 5
20.19 odd 2 464.4.a.l.1.1 5
35.34 odd 2 1421.4.a.e.1.3 5
40.19 odd 2 1856.4.a.bb.1.5 5
40.29 even 2 1856.4.a.y.1.1 5
145.144 even 2 841.4.a.b.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.b.1.3 5 5.4 even 2
261.4.a.f.1.3 5 15.14 odd 2
464.4.a.l.1.1 5 20.19 odd 2
725.4.a.c.1.3 5 1.1 even 1 trivial
841.4.a.b.1.3 5 145.144 even 2
1421.4.a.e.1.3 5 35.34 odd 2
1856.4.a.y.1.1 5 40.29 even 2
1856.4.a.bb.1.5 5 40.19 odd 2