Properties

Label 35.4.a.b.1.2
Level $35$
Weight $4$
Character 35.1
Self dual yes
Analytic conductor $2.065$
Analytic rank $0$
Dimension $2$
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] = [35,4,Mod(1,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, 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("35.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 35.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: \(2.06506685020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 35.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.41421 q^{2} -4.65685 q^{3} +21.3137 q^{4} -5.00000 q^{5} -25.2132 q^{6} -7.00000 q^{7} +72.0833 q^{8} -5.31371 q^{9} +O(q^{10})\) \(q+5.41421 q^{2} -4.65685 q^{3} +21.3137 q^{4} -5.00000 q^{5} -25.2132 q^{6} -7.00000 q^{7} +72.0833 q^{8} -5.31371 q^{9} -27.0711 q^{10} -52.2548 q^{11} -99.2548 q^{12} +30.6569 q^{13} -37.8995 q^{14} +23.2843 q^{15} +219.765 q^{16} +37.2254 q^{17} -28.7696 q^{18} +80.2254 q^{19} -106.569 q^{20} +32.5980 q^{21} -282.919 q^{22} +25.8335 q^{23} -335.681 q^{24} +25.0000 q^{25} +165.983 q^{26} +150.480 q^{27} -149.196 q^{28} +20.9411 q^{29} +126.066 q^{30} -314.558 q^{31} +613.186 q^{32} +243.343 q^{33} +201.546 q^{34} +35.0000 q^{35} -113.255 q^{36} +197.147 q^{37} +434.357 q^{38} -142.765 q^{39} -360.416 q^{40} +11.3625 q^{41} +176.492 q^{42} -33.8335 q^{43} -1113.74 q^{44} +26.5685 q^{45} +139.868 q^{46} -361.676 q^{47} -1023.41 q^{48} +49.0000 q^{49} +135.355 q^{50} -173.353 q^{51} +653.411 q^{52} +153.019 q^{53} +814.732 q^{54} +261.274 q^{55} -504.583 q^{56} -373.598 q^{57} +113.380 q^{58} -616.000 q^{59} +496.274 q^{60} +15.2649 q^{61} -1703.09 q^{62} +37.1960 q^{63} +1561.80 q^{64} -153.284 q^{65} +1317.51 q^{66} -166.510 q^{67} +793.411 q^{68} -120.303 q^{69} +189.497 q^{70} -952.000 q^{71} -383.029 q^{72} -148.489 q^{73} +1067.40 q^{74} -116.421 q^{75} +1709.90 q^{76} +365.784 q^{77} -772.958 q^{78} +857.725 q^{79} -1098.82 q^{80} -557.294 q^{81} +61.5189 q^{82} +660.528 q^{83} +694.784 q^{84} -186.127 q^{85} -183.182 q^{86} -97.5198 q^{87} -3766.70 q^{88} -45.7746 q^{89} +143.848 q^{90} -214.598 q^{91} +550.607 q^{92} +1464.85 q^{93} -1958.19 q^{94} -401.127 q^{95} -2855.52 q^{96} +1682.13 q^{97} +265.296 q^{98} +277.667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 2 q^{3} + 20 q^{4} - 10 q^{5} - 8 q^{6} - 14 q^{7} + 48 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 2 q^{3} + 20 q^{4} - 10 q^{5} - 8 q^{6} - 14 q^{7} + 48 q^{8} + 12 q^{9} - 40 q^{10} - 14 q^{11} - 108 q^{12} + 50 q^{13} - 56 q^{14} - 10 q^{15} + 168 q^{16} - 50 q^{17} + 16 q^{18} + 36 q^{19} - 100 q^{20} - 14 q^{21} - 184 q^{22} + 244 q^{23} - 496 q^{24} + 50 q^{25} + 216 q^{26} + 86 q^{27} - 140 q^{28} - 26 q^{29} + 40 q^{30} - 120 q^{31} + 672 q^{32} + 498 q^{33} - 24 q^{34} + 70 q^{35} - 136 q^{36} + 564 q^{37} + 320 q^{38} - 14 q^{39} - 240 q^{40} - 328 q^{41} + 56 q^{42} - 260 q^{43} - 1164 q^{44} - 60 q^{45} + 704 q^{46} - 350 q^{47} - 1368 q^{48} + 98 q^{49} + 200 q^{50} - 754 q^{51} + 628 q^{52} - 56 q^{53} + 648 q^{54} + 70 q^{55} - 336 q^{56} - 668 q^{57} - 8 q^{58} - 1232 q^{59} + 540 q^{60} + 336 q^{61} - 1200 q^{62} - 84 q^{63} + 2128 q^{64} - 250 q^{65} + 1976 q^{66} - 152 q^{67} + 908 q^{68} + 1332 q^{69} + 280 q^{70} - 1904 q^{71} - 800 q^{72} + 676 q^{73} + 2016 q^{74} + 50 q^{75} + 1768 q^{76} + 98 q^{77} - 440 q^{78} + 1014 q^{79} - 840 q^{80} - 1454 q^{81} - 816 q^{82} - 376 q^{83} + 756 q^{84} + 250 q^{85} - 768 q^{86} - 410 q^{87} - 4688 q^{88} - 216 q^{89} - 80 q^{90} - 350 q^{91} + 264 q^{92} + 2760 q^{93} - 1928 q^{94} - 180 q^{95} - 2464 q^{96} + 2742 q^{97} + 392 q^{98} + 940 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.41421 1.91421 0.957107 0.289735i \(-0.0935673\pi\)
0.957107 + 0.289735i \(0.0935673\pi\)
\(3\) −4.65685 −0.896212 −0.448106 0.893980i \(-0.647901\pi\)
−0.448106 + 0.893980i \(0.647901\pi\)
\(4\) 21.3137 2.66421
\(5\) −5.00000 −0.447214
\(6\) −25.2132 −1.71554
\(7\) −7.00000 −0.377964
\(8\) 72.0833 3.18566
\(9\) −5.31371 −0.196804
\(10\) −27.0711 −0.856062
\(11\) −52.2548 −1.43231 −0.716156 0.697941i \(-0.754100\pi\)
−0.716156 + 0.697941i \(0.754100\pi\)
\(12\) −99.2548 −2.38770
\(13\) 30.6569 0.654052 0.327026 0.945015i \(-0.393953\pi\)
0.327026 + 0.945015i \(0.393953\pi\)
\(14\) −37.8995 −0.723505
\(15\) 23.2843 0.400798
\(16\) 219.765 3.43382
\(17\) 37.2254 0.531087 0.265544 0.964099i \(-0.414449\pi\)
0.265544 + 0.964099i \(0.414449\pi\)
\(18\) −28.7696 −0.376725
\(19\) 80.2254 0.968683 0.484341 0.874879i \(-0.339059\pi\)
0.484341 + 0.874879i \(0.339059\pi\)
\(20\) −106.569 −1.19147
\(21\) 32.5980 0.338736
\(22\) −282.919 −2.74175
\(23\) 25.8335 0.234202 0.117101 0.993120i \(-0.462640\pi\)
0.117101 + 0.993120i \(0.462640\pi\)
\(24\) −335.681 −2.85503
\(25\) 25.0000 0.200000
\(26\) 165.983 1.25200
\(27\) 150.480 1.07259
\(28\) −149.196 −1.00698
\(29\) 20.9411 0.134092 0.0670460 0.997750i \(-0.478643\pi\)
0.0670460 + 0.997750i \(0.478643\pi\)
\(30\) 126.066 0.767213
\(31\) −314.558 −1.82246 −0.911232 0.411894i \(-0.864867\pi\)
−0.911232 + 0.411894i \(0.864867\pi\)
\(32\) 613.186 3.38741
\(33\) 243.343 1.28365
\(34\) 201.546 1.01661
\(35\) 35.0000 0.169031
\(36\) −113.255 −0.524328
\(37\) 197.147 0.875968 0.437984 0.898983i \(-0.355693\pi\)
0.437984 + 0.898983i \(0.355693\pi\)
\(38\) 434.357 1.85427
\(39\) −142.765 −0.586170
\(40\) −360.416 −1.42467
\(41\) 11.3625 0.0432810 0.0216405 0.999766i \(-0.493111\pi\)
0.0216405 + 0.999766i \(0.493111\pi\)
\(42\) 176.492 0.648414
\(43\) −33.8335 −0.119990 −0.0599948 0.998199i \(-0.519108\pi\)
−0.0599948 + 0.998199i \(0.519108\pi\)
\(44\) −1113.74 −3.81598
\(45\) 26.5685 0.0880134
\(46\) 139.868 0.448313
\(47\) −361.676 −1.12247 −0.561233 0.827658i \(-0.689673\pi\)
−0.561233 + 0.827658i \(0.689673\pi\)
\(48\) −1023.41 −3.07743
\(49\) 49.0000 0.142857
\(50\) 135.355 0.382843
\(51\) −173.353 −0.475967
\(52\) 653.411 1.74254
\(53\) 153.019 0.396582 0.198291 0.980143i \(-0.436461\pi\)
0.198291 + 0.980143i \(0.436461\pi\)
\(54\) 814.732 2.05317
\(55\) 261.274 0.640549
\(56\) −504.583 −1.20407
\(57\) −373.598 −0.868145
\(58\) 113.380 0.256681
\(59\) −616.000 −1.35926 −0.679630 0.733555i \(-0.737860\pi\)
−0.679630 + 0.733555i \(0.737860\pi\)
\(60\) 496.274 1.06781
\(61\) 15.2649 0.0320406 0.0160203 0.999872i \(-0.494900\pi\)
0.0160203 + 0.999872i \(0.494900\pi\)
\(62\) −1703.09 −3.48858
\(63\) 37.1960 0.0743849
\(64\) 1561.80 3.05040
\(65\) −153.284 −0.292501
\(66\) 1317.51 2.45719
\(67\) −166.510 −0.303618 −0.151809 0.988410i \(-0.548510\pi\)
−0.151809 + 0.988410i \(0.548510\pi\)
\(68\) 793.411 1.41493
\(69\) −120.303 −0.209895
\(70\) 189.497 0.323561
\(71\) −952.000 −1.59129 −0.795645 0.605763i \(-0.792868\pi\)
−0.795645 + 0.605763i \(0.792868\pi\)
\(72\) −383.029 −0.626951
\(73\) −148.489 −0.238074 −0.119037 0.992890i \(-0.537981\pi\)
−0.119037 + 0.992890i \(0.537981\pi\)
\(74\) 1067.40 1.67679
\(75\) −116.421 −0.179242
\(76\) 1709.90 2.58078
\(77\) 365.784 0.541363
\(78\) −772.958 −1.12205
\(79\) 857.725 1.22154 0.610770 0.791808i \(-0.290860\pi\)
0.610770 + 0.791808i \(0.290860\pi\)
\(80\) −1098.82 −1.53565
\(81\) −557.294 −0.764464
\(82\) 61.5189 0.0828491
\(83\) 660.528 0.873523 0.436761 0.899577i \(-0.356125\pi\)
0.436761 + 0.899577i \(0.356125\pi\)
\(84\) 694.784 0.902466
\(85\) −186.127 −0.237509
\(86\) −183.182 −0.229686
\(87\) −97.5198 −0.120175
\(88\) −3766.70 −4.56286
\(89\) −45.7746 −0.0545180 −0.0272590 0.999628i \(-0.508678\pi\)
−0.0272590 + 0.999628i \(0.508678\pi\)
\(90\) 143.848 0.168477
\(91\) −214.598 −0.247209
\(92\) 550.607 0.623965
\(93\) 1464.85 1.63331
\(94\) −1958.19 −2.14864
\(95\) −401.127 −0.433208
\(96\) −2855.52 −3.03583
\(97\) 1682.13 1.76076 0.880382 0.474265i \(-0.157286\pi\)
0.880382 + 0.474265i \(0.157286\pi\)
\(98\) 265.296 0.273459
\(99\) 277.667 0.281885
\(100\) 532.843 0.532843
\(101\) −434.167 −0.427734 −0.213867 0.976863i \(-0.568606\pi\)
−0.213867 + 0.976863i \(0.568606\pi\)
\(102\) −938.572 −0.911102
\(103\) 345.577 0.330589 0.165295 0.986244i \(-0.447142\pi\)
0.165295 + 0.986244i \(0.447142\pi\)
\(104\) 2209.85 2.08359
\(105\) −162.990 −0.151487
\(106\) 828.479 0.759142
\(107\) 217.119 0.196165 0.0980825 0.995178i \(-0.468729\pi\)
0.0980825 + 0.995178i \(0.468729\pi\)
\(108\) 3207.29 2.85761
\(109\) 1734.41 1.52409 0.762047 0.647521i \(-0.224194\pi\)
0.762047 + 0.647521i \(0.224194\pi\)
\(110\) 1414.59 1.22615
\(111\) −918.086 −0.785053
\(112\) −1538.35 −1.29786
\(113\) −1854.20 −1.54362 −0.771809 0.635855i \(-0.780648\pi\)
−0.771809 + 0.635855i \(0.780648\pi\)
\(114\) −2022.74 −1.66181
\(115\) −129.167 −0.104738
\(116\) 446.333 0.357250
\(117\) −162.902 −0.128720
\(118\) −3335.16 −2.60191
\(119\) −260.578 −0.200732
\(120\) 1678.41 1.27681
\(121\) 1399.57 1.05152
\(122\) 82.6476 0.0613325
\(123\) −52.9134 −0.0387890
\(124\) −6704.41 −4.85543
\(125\) −125.000 −0.0894427
\(126\) 201.387 0.142389
\(127\) 1394.51 0.974352 0.487176 0.873304i \(-0.338027\pi\)
0.487176 + 0.873304i \(0.338027\pi\)
\(128\) 3550.45 2.45171
\(129\) 157.558 0.107536
\(130\) −829.914 −0.559910
\(131\) 1762.42 1.17544 0.587722 0.809063i \(-0.300025\pi\)
0.587722 + 0.809063i \(0.300025\pi\)
\(132\) 5186.54 3.41993
\(133\) −561.578 −0.366128
\(134\) −901.519 −0.581189
\(135\) −752.401 −0.479677
\(136\) 2683.33 1.69186
\(137\) −922.949 −0.575568 −0.287784 0.957695i \(-0.592919\pi\)
−0.287784 + 0.957695i \(0.592919\pi\)
\(138\) −651.345 −0.401784
\(139\) −196.039 −0.119624 −0.0598122 0.998210i \(-0.519050\pi\)
−0.0598122 + 0.998210i \(0.519050\pi\)
\(140\) 745.980 0.450334
\(141\) 1684.27 1.00597
\(142\) −5154.33 −3.04607
\(143\) −1601.97 −0.936807
\(144\) −1167.76 −0.675790
\(145\) −104.706 −0.0599678
\(146\) −803.954 −0.455724
\(147\) −228.186 −0.128030
\(148\) 4201.94 2.33376
\(149\) 780.372 0.429064 0.214532 0.976717i \(-0.431177\pi\)
0.214532 + 0.976717i \(0.431177\pi\)
\(150\) −630.330 −0.343108
\(151\) −2319.43 −1.25002 −0.625008 0.780618i \(-0.714904\pi\)
−0.625008 + 0.780618i \(0.714904\pi\)
\(152\) 5782.91 3.08589
\(153\) −197.805 −0.104520
\(154\) 1980.43 1.03628
\(155\) 1572.79 0.815030
\(156\) −3042.84 −1.56168
\(157\) 1022.90 0.519977 0.259989 0.965612i \(-0.416281\pi\)
0.259989 + 0.965612i \(0.416281\pi\)
\(158\) 4643.91 2.33829
\(159\) −712.589 −0.355421
\(160\) −3065.93 −1.51489
\(161\) −180.834 −0.0885201
\(162\) −3017.31 −1.46335
\(163\) −1350.63 −0.649013 −0.324507 0.945883i \(-0.605198\pi\)
−0.324507 + 0.945883i \(0.605198\pi\)
\(164\) 242.177 0.115310
\(165\) −1216.72 −0.574068
\(166\) 3576.24 1.67211
\(167\) −1230.58 −0.570209 −0.285105 0.958496i \(-0.592028\pi\)
−0.285105 + 0.958496i \(0.592028\pi\)
\(168\) 2349.77 1.07910
\(169\) −1257.16 −0.572215
\(170\) −1007.73 −0.454644
\(171\) −426.294 −0.190641
\(172\) −721.117 −0.319678
\(173\) −2487.65 −1.09325 −0.546626 0.837377i \(-0.684088\pi\)
−0.546626 + 0.837377i \(0.684088\pi\)
\(174\) −527.993 −0.230040
\(175\) −175.000 −0.0755929
\(176\) −11483.8 −4.91830
\(177\) 2868.62 1.21819
\(178\) −247.833 −0.104359
\(179\) 1621.18 0.676941 0.338471 0.940977i \(-0.390090\pi\)
0.338471 + 0.940977i \(0.390090\pi\)
\(180\) 566.274 0.234487
\(181\) 2593.69 1.06512 0.532561 0.846392i \(-0.321230\pi\)
0.532561 + 0.846392i \(0.321230\pi\)
\(182\) −1161.88 −0.473210
\(183\) −71.0866 −0.0287151
\(184\) 1862.16 0.746089
\(185\) −985.736 −0.391745
\(186\) 7931.03 3.12651
\(187\) −1945.21 −0.760682
\(188\) −7708.66 −2.99049
\(189\) −1053.36 −0.405401
\(190\) −2171.79 −0.829253
\(191\) −1823.08 −0.690645 −0.345323 0.938484i \(-0.612231\pi\)
−0.345323 + 0.938484i \(0.612231\pi\)
\(192\) −7273.09 −2.73380
\(193\) −1541.03 −0.574744 −0.287372 0.957819i \(-0.592782\pi\)
−0.287372 + 0.957819i \(0.592782\pi\)
\(194\) 9107.39 3.37048
\(195\) 713.823 0.262143
\(196\) 1044.37 0.380602
\(197\) 701.243 0.253612 0.126806 0.991928i \(-0.459527\pi\)
0.126806 + 0.991928i \(0.459527\pi\)
\(198\) 1503.35 0.539587
\(199\) 3294.96 1.17374 0.586868 0.809682i \(-0.300361\pi\)
0.586868 + 0.809682i \(0.300361\pi\)
\(200\) 1802.08 0.637132
\(201\) 775.411 0.272106
\(202\) −2350.67 −0.818775
\(203\) −146.588 −0.0506820
\(204\) −3694.80 −1.26808
\(205\) −56.8124 −0.0193559
\(206\) 1871.03 0.632819
\(207\) −137.272 −0.0460920
\(208\) 6737.29 2.24590
\(209\) −4192.16 −1.38746
\(210\) −882.462 −0.289979
\(211\) 4082.35 1.33195 0.665974 0.745975i \(-0.268016\pi\)
0.665974 + 0.745975i \(0.268016\pi\)
\(212\) 3261.41 1.05658
\(213\) 4433.33 1.42613
\(214\) 1175.53 0.375502
\(215\) 169.167 0.0536610
\(216\) 10847.1 3.41691
\(217\) 2201.91 0.688826
\(218\) 9390.46 2.91744
\(219\) 691.494 0.213364
\(220\) 5568.72 1.70656
\(221\) 1141.21 0.347359
\(222\) −4970.71 −1.50276
\(223\) 747.161 0.224366 0.112183 0.993688i \(-0.464216\pi\)
0.112183 + 0.993688i \(0.464216\pi\)
\(224\) −4292.30 −1.28032
\(225\) −132.843 −0.0393608
\(226\) −10039.1 −2.95481
\(227\) 1665.67 0.487025 0.243513 0.969898i \(-0.421700\pi\)
0.243513 + 0.969898i \(0.421700\pi\)
\(228\) −7962.76 −2.31292
\(229\) −6628.35 −1.91272 −0.956362 0.292183i \(-0.905618\pi\)
−0.956362 + 0.292183i \(0.905618\pi\)
\(230\) −699.340 −0.200492
\(231\) −1703.40 −0.485176
\(232\) 1509.50 0.427172
\(233\) −432.431 −0.121586 −0.0607929 0.998150i \(-0.519363\pi\)
−0.0607929 + 0.998150i \(0.519363\pi\)
\(234\) −881.984 −0.246398
\(235\) 1808.38 0.501982
\(236\) −13129.2 −3.62136
\(237\) −3994.30 −1.09476
\(238\) −1410.82 −0.384244
\(239\) 5580.44 1.51033 0.755165 0.655535i \(-0.227557\pi\)
0.755165 + 0.655535i \(0.227557\pi\)
\(240\) 5117.06 1.37627
\(241\) −6296.87 −1.68306 −0.841529 0.540212i \(-0.818344\pi\)
−0.841529 + 0.540212i \(0.818344\pi\)
\(242\) 7577.56 2.01283
\(243\) −1467.73 −0.387468
\(244\) 325.352 0.0853629
\(245\) −245.000 −0.0638877
\(246\) −286.485 −0.0742504
\(247\) 2459.46 0.633569
\(248\) −22674.4 −5.80575
\(249\) −3075.98 −0.782862
\(250\) −676.777 −0.171212
\(251\) 311.921 0.0784393 0.0392197 0.999231i \(-0.487513\pi\)
0.0392197 + 0.999231i \(0.487513\pi\)
\(252\) 792.784 0.198177
\(253\) −1349.92 −0.335451
\(254\) 7550.17 1.86512
\(255\) 866.766 0.212859
\(256\) 6728.46 1.64269
\(257\) −7861.39 −1.90809 −0.954046 0.299659i \(-0.903127\pi\)
−0.954046 + 0.299659i \(0.903127\pi\)
\(258\) 853.050 0.205847
\(259\) −1380.03 −0.331085
\(260\) −3267.06 −0.779285
\(261\) −111.275 −0.0263899
\(262\) 9542.11 2.25005
\(263\) 5227.09 1.22554 0.612769 0.790262i \(-0.290056\pi\)
0.612769 + 0.790262i \(0.290056\pi\)
\(264\) 17541.0 4.08929
\(265\) −765.097 −0.177357
\(266\) −3040.50 −0.700846
\(267\) 213.166 0.0488596
\(268\) −3548.94 −0.808903
\(269\) 1281.71 0.290510 0.145255 0.989394i \(-0.453600\pi\)
0.145255 + 0.989394i \(0.453600\pi\)
\(270\) −4073.66 −0.918204
\(271\) 4704.14 1.05445 0.527226 0.849725i \(-0.323232\pi\)
0.527226 + 0.849725i \(0.323232\pi\)
\(272\) 8180.82 1.82366
\(273\) 999.352 0.221551
\(274\) −4997.04 −1.10176
\(275\) −1306.37 −0.286462
\(276\) −2564.10 −0.559205
\(277\) 8958.56 1.94321 0.971603 0.236619i \(-0.0760393\pi\)
0.971603 + 0.236619i \(0.0760393\pi\)
\(278\) −1061.40 −0.228987
\(279\) 1671.47 0.358668
\(280\) 2522.91 0.538475
\(281\) −370.904 −0.0787412 −0.0393706 0.999225i \(-0.512535\pi\)
−0.0393706 + 0.999225i \(0.512535\pi\)
\(282\) 9119.02 1.92564
\(283\) −5822.26 −1.22296 −0.611479 0.791261i \(-0.709425\pi\)
−0.611479 + 0.791261i \(0.709425\pi\)
\(284\) −20290.7 −4.23954
\(285\) 1867.99 0.388246
\(286\) −8673.40 −1.79325
\(287\) −79.5374 −0.0163587
\(288\) −3258.29 −0.666655
\(289\) −3527.27 −0.717946
\(290\) −566.899 −0.114791
\(291\) −7833.42 −1.57802
\(292\) −3164.86 −0.634279
\(293\) 7443.79 1.48420 0.742100 0.670289i \(-0.233830\pi\)
0.742100 + 0.670289i \(0.233830\pi\)
\(294\) −1235.45 −0.245077
\(295\) 3080.00 0.607880
\(296\) 14211.0 2.79053
\(297\) −7863.32 −1.53628
\(298\) 4225.10 0.821320
\(299\) 791.973 0.153181
\(300\) −2481.37 −0.477540
\(301\) 236.834 0.0453518
\(302\) −12557.9 −2.39280
\(303\) 2021.85 0.383341
\(304\) 17630.7 3.32628
\(305\) −76.3247 −0.0143290
\(306\) −1070.96 −0.200074
\(307\) −761.674 −0.141600 −0.0707998 0.997491i \(-0.522555\pi\)
−0.0707998 + 0.997491i \(0.522555\pi\)
\(308\) 7796.21 1.44231
\(309\) −1609.30 −0.296278
\(310\) 8515.43 1.56014
\(311\) 7718.69 1.40735 0.703677 0.710520i \(-0.251540\pi\)
0.703677 + 0.710520i \(0.251540\pi\)
\(312\) −10290.9 −1.86734
\(313\) 8556.00 1.54509 0.772546 0.634959i \(-0.218983\pi\)
0.772546 + 0.634959i \(0.218983\pi\)
\(314\) 5538.21 0.995348
\(315\) −185.980 −0.0332660
\(316\) 18281.3 3.25444
\(317\) −7780.95 −1.37862 −0.689309 0.724468i \(-0.742086\pi\)
−0.689309 + 0.724468i \(0.742086\pi\)
\(318\) −3858.11 −0.680352
\(319\) −1094.28 −0.192062
\(320\) −7809.02 −1.36418
\(321\) −1011.09 −0.175805
\(322\) −979.076 −0.169446
\(323\) 2986.42 0.514455
\(324\) −11878.0 −2.03670
\(325\) 766.421 0.130810
\(326\) −7312.58 −1.24235
\(327\) −8076.89 −1.36591
\(328\) 819.045 0.137879
\(329\) 2531.73 0.424252
\(330\) −6587.56 −1.09889
\(331\) −4932.12 −0.819015 −0.409507 0.912307i \(-0.634299\pi\)
−0.409507 + 0.912307i \(0.634299\pi\)
\(332\) 14078.3 2.32725
\(333\) −1047.58 −0.172394
\(334\) −6662.61 −1.09150
\(335\) 832.548 0.135782
\(336\) 7163.88 1.16316
\(337\) −7121.13 −1.15108 −0.575538 0.817775i \(-0.695207\pi\)
−0.575538 + 0.817775i \(0.695207\pi\)
\(338\) −6806.52 −1.09534
\(339\) 8634.76 1.38341
\(340\) −3967.06 −0.632776
\(341\) 16437.2 2.61034
\(342\) −2308.05 −0.364927
\(343\) −343.000 −0.0539949
\(344\) −2438.83 −0.382246
\(345\) 601.514 0.0938679
\(346\) −13468.7 −2.09272
\(347\) 9540.58 1.47598 0.737991 0.674811i \(-0.235775\pi\)
0.737991 + 0.674811i \(0.235775\pi\)
\(348\) −2078.51 −0.320172
\(349\) 1281.65 0.196576 0.0982880 0.995158i \(-0.468663\pi\)
0.0982880 + 0.995158i \(0.468663\pi\)
\(350\) −947.487 −0.144701
\(351\) 4613.25 0.701530
\(352\) −32041.9 −4.85182
\(353\) 5798.07 0.874221 0.437110 0.899408i \(-0.356002\pi\)
0.437110 + 0.899408i \(0.356002\pi\)
\(354\) 15531.3 2.33187
\(355\) 4760.00 0.711647
\(356\) −975.627 −0.145247
\(357\) 1213.47 0.179899
\(358\) 8777.40 1.29581
\(359\) 2267.29 0.333323 0.166662 0.986014i \(-0.446701\pi\)
0.166662 + 0.986014i \(0.446701\pi\)
\(360\) 1915.15 0.280381
\(361\) −422.886 −0.0616541
\(362\) 14042.8 2.03887
\(363\) −6517.58 −0.942381
\(364\) −4573.88 −0.658616
\(365\) 742.447 0.106470
\(366\) −384.878 −0.0549669
\(367\) −7372.85 −1.04866 −0.524332 0.851514i \(-0.675685\pi\)
−0.524332 + 0.851514i \(0.675685\pi\)
\(368\) 5677.28 0.804209
\(369\) −60.3769 −0.00851788
\(370\) −5336.98 −0.749883
\(371\) −1071.14 −0.149894
\(372\) 31221.4 4.35150
\(373\) 6447.14 0.894961 0.447480 0.894294i \(-0.352321\pi\)
0.447480 + 0.894294i \(0.352321\pi\)
\(374\) −10531.8 −1.45611
\(375\) 582.107 0.0801596
\(376\) −26070.8 −3.57579
\(377\) 641.989 0.0877032
\(378\) −5703.12 −0.776024
\(379\) −4247.57 −0.575680 −0.287840 0.957678i \(-0.592937\pi\)
−0.287840 + 0.957678i \(0.592937\pi\)
\(380\) −8549.50 −1.15416
\(381\) −6494.03 −0.873226
\(382\) −9870.53 −1.32204
\(383\) −6681.86 −0.891454 −0.445727 0.895169i \(-0.647055\pi\)
−0.445727 + 0.895169i \(0.647055\pi\)
\(384\) −16533.9 −2.19725
\(385\) −1828.92 −0.242105
\(386\) −8343.45 −1.10018
\(387\) 179.781 0.0236145
\(388\) 35852.4 4.69105
\(389\) −6371.78 −0.830494 −0.415247 0.909709i \(-0.636305\pi\)
−0.415247 + 0.909709i \(0.636305\pi\)
\(390\) 3864.79 0.501798
\(391\) 961.661 0.124382
\(392\) 3532.08 0.455094
\(393\) −8207.33 −1.05345
\(394\) 3796.68 0.485467
\(395\) −4288.62 −0.546289
\(396\) 5918.11 0.751001
\(397\) 4247.93 0.537021 0.268510 0.963277i \(-0.413469\pi\)
0.268510 + 0.963277i \(0.413469\pi\)
\(398\) 17839.6 2.24678
\(399\) 2615.19 0.328128
\(400\) 5494.11 0.686764
\(401\) −8833.62 −1.10008 −0.550038 0.835140i \(-0.685387\pi\)
−0.550038 + 0.835140i \(0.685387\pi\)
\(402\) 4198.24 0.520869
\(403\) −9643.37 −1.19199
\(404\) −9253.70 −1.13958
\(405\) 2786.47 0.341879
\(406\) −793.658 −0.0970162
\(407\) −10301.9 −1.25466
\(408\) −12495.9 −1.51627
\(409\) −319.205 −0.0385908 −0.0192954 0.999814i \(-0.506142\pi\)
−0.0192954 + 0.999814i \(0.506142\pi\)
\(410\) −307.595 −0.0370512
\(411\) 4298.04 0.515831
\(412\) 7365.53 0.880761
\(413\) 4312.00 0.513752
\(414\) −743.218 −0.0882298
\(415\) −3302.64 −0.390651
\(416\) 18798.3 2.21554
\(417\) 912.924 0.107209
\(418\) −22697.3 −2.65589
\(419\) −12789.2 −1.49115 −0.745577 0.666420i \(-0.767826\pi\)
−0.745577 + 0.666420i \(0.767826\pi\)
\(420\) −3473.92 −0.403595
\(421\) −6747.40 −0.781112 −0.390556 0.920579i \(-0.627717\pi\)
−0.390556 + 0.920579i \(0.627717\pi\)
\(422\) 22102.7 2.54963
\(423\) 1921.84 0.220906
\(424\) 11030.1 1.26337
\(425\) 930.635 0.106217
\(426\) 24003.0 2.72992
\(427\) −106.855 −0.0121102
\(428\) 4627.60 0.522625
\(429\) 7460.14 0.839577
\(430\) 915.908 0.102719
\(431\) −5184.75 −0.579444 −0.289722 0.957111i \(-0.593563\pi\)
−0.289722 + 0.957111i \(0.593563\pi\)
\(432\) 33070.2 3.68308
\(433\) −4242.03 −0.470806 −0.235403 0.971898i \(-0.575641\pi\)
−0.235403 + 0.971898i \(0.575641\pi\)
\(434\) 11921.6 1.31856
\(435\) 487.599 0.0537439
\(436\) 36966.7 4.06051
\(437\) 2072.50 0.226868
\(438\) 3743.90 0.408425
\(439\) −5434.12 −0.590789 −0.295394 0.955375i \(-0.595451\pi\)
−0.295394 + 0.955375i \(0.595451\pi\)
\(440\) 18833.5 2.04057
\(441\) −260.372 −0.0281149
\(442\) 6178.77 0.664919
\(443\) −11493.8 −1.23270 −0.616350 0.787472i \(-0.711389\pi\)
−0.616350 + 0.787472i \(0.711389\pi\)
\(444\) −19567.8 −2.09155
\(445\) 228.873 0.0243812
\(446\) 4045.29 0.429484
\(447\) −3634.08 −0.384532
\(448\) −10932.6 −1.15294
\(449\) −16849.3 −1.77098 −0.885489 0.464661i \(-0.846176\pi\)
−0.885489 + 0.464661i \(0.846176\pi\)
\(450\) −719.239 −0.0753450
\(451\) −593.745 −0.0619919
\(452\) −39520.0 −4.11253
\(453\) 10801.2 1.12028
\(454\) 9018.32 0.932270
\(455\) 1072.99 0.110555
\(456\) −26930.2 −2.76561
\(457\) 15348.5 1.57106 0.785528 0.618826i \(-0.212391\pi\)
0.785528 + 0.618826i \(0.212391\pi\)
\(458\) −35887.3 −3.66136
\(459\) 5601.69 0.569639
\(460\) −2753.04 −0.279046
\(461\) 14038.4 1.41830 0.709148 0.705059i \(-0.249080\pi\)
0.709148 + 0.705059i \(0.249080\pi\)
\(462\) −9222.58 −0.928730
\(463\) −8661.23 −0.869377 −0.434689 0.900581i \(-0.643142\pi\)
−0.434689 + 0.900581i \(0.643142\pi\)
\(464\) 4602.12 0.460448
\(465\) −7324.26 −0.730440
\(466\) −2341.27 −0.232741
\(467\) 7014.71 0.695079 0.347539 0.937665i \(-0.387017\pi\)
0.347539 + 0.937665i \(0.387017\pi\)
\(468\) −3472.04 −0.342938
\(469\) 1165.57 0.114757
\(470\) 9790.96 0.960901
\(471\) −4763.50 −0.466010
\(472\) −44403.3 −4.33014
\(473\) 1767.96 0.171863
\(474\) −21626.0 −2.09560
\(475\) 2005.63 0.193737
\(476\) −5553.88 −0.534793
\(477\) −813.100 −0.0780488
\(478\) 30213.7 2.89109
\(479\) 18134.7 1.72984 0.864922 0.501907i \(-0.167368\pi\)
0.864922 + 0.501907i \(0.167368\pi\)
\(480\) 14277.6 1.35767
\(481\) 6043.91 0.572929
\(482\) −34092.6 −3.22173
\(483\) 842.119 0.0793328
\(484\) 29830.0 2.80146
\(485\) −8410.63 −0.787438
\(486\) −7946.59 −0.741697
\(487\) 16537.8 1.53881 0.769405 0.638761i \(-0.220553\pi\)
0.769405 + 0.638761i \(0.220553\pi\)
\(488\) 1100.35 0.102070
\(489\) 6289.67 0.581654
\(490\) −1326.48 −0.122295
\(491\) 220.608 0.0202768 0.0101384 0.999949i \(-0.496773\pi\)
0.0101384 + 0.999949i \(0.496773\pi\)
\(492\) −1127.78 −0.103342
\(493\) 779.542 0.0712146
\(494\) 13316.0 1.21279
\(495\) −1388.33 −0.126063
\(496\) −69128.8 −6.25801
\(497\) 6664.00 0.601451
\(498\) −16654.0 −1.49856
\(499\) 5939.04 0.532801 0.266401 0.963862i \(-0.414166\pi\)
0.266401 + 0.963862i \(0.414166\pi\)
\(500\) −2664.21 −0.238295
\(501\) 5730.62 0.511029
\(502\) 1688.81 0.150150
\(503\) −11604.8 −1.02869 −0.514345 0.857584i \(-0.671965\pi\)
−0.514345 + 0.857584i \(0.671965\pi\)
\(504\) 2681.21 0.236965
\(505\) 2170.83 0.191289
\(506\) −7308.78 −0.642124
\(507\) 5854.40 0.512826
\(508\) 29722.2 2.59588
\(509\) −1867.67 −0.162639 −0.0813193 0.996688i \(-0.525913\pi\)
−0.0813193 + 0.996688i \(0.525913\pi\)
\(510\) 4692.86 0.407457
\(511\) 1039.43 0.0899834
\(512\) 8025.75 0.692757
\(513\) 12072.3 1.03900
\(514\) −42563.2 −3.65250
\(515\) −1727.88 −0.147844
\(516\) 3358.14 0.286499
\(517\) 18899.3 1.60772
\(518\) −7471.78 −0.633767
\(519\) 11584.6 0.979786
\(520\) −11049.2 −0.931809
\(521\) 6117.21 0.514395 0.257197 0.966359i \(-0.417201\pi\)
0.257197 + 0.966359i \(0.417201\pi\)
\(522\) −602.467 −0.0505158
\(523\) −16685.6 −1.39505 −0.697524 0.716561i \(-0.745715\pi\)
−0.697524 + 0.716561i \(0.745715\pi\)
\(524\) 37563.7 3.13164
\(525\) 814.949 0.0677473
\(526\) 28300.6 2.34594
\(527\) −11709.6 −0.967887
\(528\) 53478.2 4.40784
\(529\) −11499.6 −0.945149
\(530\) −4142.40 −0.339499
\(531\) 3273.24 0.267508
\(532\) −11969.3 −0.975442
\(533\) 348.338 0.0283081
\(534\) 1154.12 0.0935278
\(535\) −1085.59 −0.0877276
\(536\) −12002.6 −0.967223
\(537\) −7549.59 −0.606683
\(538\) 6939.44 0.556097
\(539\) −2560.49 −0.204616
\(540\) −16036.5 −1.27796
\(541\) 9309.03 0.739790 0.369895 0.929074i \(-0.379394\pi\)
0.369895 + 0.929074i \(0.379394\pi\)
\(542\) 25469.2 2.01845
\(543\) −12078.4 −0.954575
\(544\) 22826.1 1.79901
\(545\) −8672.05 −0.681596
\(546\) 5410.70 0.424097
\(547\) 10894.7 0.851598 0.425799 0.904818i \(-0.359993\pi\)
0.425799 + 0.904818i \(0.359993\pi\)
\(548\) −19671.5 −1.53344
\(549\) −81.1134 −0.00630571
\(550\) −7072.97 −0.548350
\(551\) 1680.01 0.129893
\(552\) −8671.81 −0.668654
\(553\) −6004.07 −0.461698
\(554\) 48503.6 3.71971
\(555\) 4590.43 0.351086
\(556\) −4178.31 −0.318705
\(557\) −7873.90 −0.598973 −0.299486 0.954101i \(-0.596815\pi\)
−0.299486 + 0.954101i \(0.596815\pi\)
\(558\) 9049.71 0.686567
\(559\) −1037.23 −0.0784796
\(560\) 7691.76 0.580422
\(561\) 9058.55 0.681733
\(562\) −2008.15 −0.150728
\(563\) 21770.7 1.62971 0.814854 0.579666i \(-0.196817\pi\)
0.814854 + 0.579666i \(0.196817\pi\)
\(564\) 35898.1 2.68011
\(565\) 9271.02 0.690327
\(566\) −31522.9 −2.34100
\(567\) 3901.06 0.288940
\(568\) −68623.3 −5.06931
\(569\) −12381.3 −0.912213 −0.456106 0.889925i \(-0.650756\pi\)
−0.456106 + 0.889925i \(0.650756\pi\)
\(570\) 10113.7 0.743186
\(571\) −5768.38 −0.422765 −0.211383 0.977403i \(-0.567797\pi\)
−0.211383 + 0.977403i \(0.567797\pi\)
\(572\) −34143.9 −2.49585
\(573\) 8489.81 0.618965
\(574\) −430.632 −0.0313140
\(575\) 645.837 0.0468405
\(576\) −8298.97 −0.600330
\(577\) 4733.38 0.341513 0.170757 0.985313i \(-0.445379\pi\)
0.170757 + 0.985313i \(0.445379\pi\)
\(578\) −19097.4 −1.37430
\(579\) 7176.34 0.515093
\(580\) −2231.67 −0.159767
\(581\) −4623.70 −0.330161
\(582\) −42411.8 −3.02066
\(583\) −7996.00 −0.568028
\(584\) −10703.6 −0.758422
\(585\) 814.508 0.0575654
\(586\) 40302.3 2.84108
\(587\) −8441.67 −0.593569 −0.296785 0.954944i \(-0.595914\pi\)
−0.296785 + 0.954944i \(0.595914\pi\)
\(588\) −4863.49 −0.341100
\(589\) −25235.6 −1.76539
\(590\) 16675.8 1.16361
\(591\) −3265.59 −0.227290
\(592\) 43326.0 3.00792
\(593\) 18939.9 1.31158 0.655791 0.754943i \(-0.272335\pi\)
0.655791 + 0.754943i \(0.272335\pi\)
\(594\) −42573.7 −2.94077
\(595\) 1302.89 0.0897701
\(596\) 16632.6 1.14312
\(597\) −15344.2 −1.05192
\(598\) 4287.91 0.293220
\(599\) 22655.3 1.54536 0.772681 0.634794i \(-0.218915\pi\)
0.772681 + 0.634794i \(0.218915\pi\)
\(600\) −8392.03 −0.571005
\(601\) −15947.4 −1.08237 −0.541187 0.840902i \(-0.682025\pi\)
−0.541187 + 0.840902i \(0.682025\pi\)
\(602\) 1282.27 0.0868131
\(603\) 884.784 0.0597532
\(604\) −49435.6 −3.33031
\(605\) −6997.84 −0.470252
\(606\) 10946.7 0.733796
\(607\) −25993.2 −1.73811 −0.869053 0.494719i \(-0.835271\pi\)
−0.869053 + 0.494719i \(0.835271\pi\)
\(608\) 49193.1 3.28132
\(609\) 682.638 0.0454218
\(610\) −413.238 −0.0274287
\(611\) −11087.9 −0.734152
\(612\) −4215.96 −0.278464
\(613\) 665.408 0.0438427 0.0219213 0.999760i \(-0.493022\pi\)
0.0219213 + 0.999760i \(0.493022\pi\)
\(614\) −4123.87 −0.271052
\(615\) 264.567 0.0173470
\(616\) 26366.9 1.72460
\(617\) 18401.3 1.20066 0.600330 0.799752i \(-0.295036\pi\)
0.600330 + 0.799752i \(0.295036\pi\)
\(618\) −8713.10 −0.567140
\(619\) −11150.6 −0.724040 −0.362020 0.932170i \(-0.617913\pi\)
−0.362020 + 0.932170i \(0.617913\pi\)
\(620\) 33522.0 2.17142
\(621\) 3887.43 0.251203
\(622\) 41790.6 2.69397
\(623\) 320.422 0.0206059
\(624\) −31374.6 −2.01280
\(625\) 625.000 0.0400000
\(626\) 46324.0 2.95764
\(627\) 19522.3 1.24345
\(628\) 21801.8 1.38533
\(629\) 7338.88 0.465215
\(630\) −1006.93 −0.0636781
\(631\) 5381.79 0.339534 0.169767 0.985484i \(-0.445699\pi\)
0.169767 + 0.985484i \(0.445699\pi\)
\(632\) 61827.6 3.89141
\(633\) −19010.9 −1.19371
\(634\) −42127.7 −2.63897
\(635\) −6972.55 −0.435744
\(636\) −15187.9 −0.946918
\(637\) 1502.19 0.0934361
\(638\) −5924.64 −0.367647
\(639\) 5058.65 0.313172
\(640\) −17752.2 −1.09644
\(641\) −19455.1 −1.19880 −0.599398 0.800451i \(-0.704593\pi\)
−0.599398 + 0.800451i \(0.704593\pi\)
\(642\) −5474.26 −0.336529
\(643\) −14695.8 −0.901317 −0.450658 0.892696i \(-0.648811\pi\)
−0.450658 + 0.892696i \(0.648811\pi\)
\(644\) −3854.25 −0.235837
\(645\) −787.788 −0.0480917
\(646\) 16169.1 0.984777
\(647\) −12694.8 −0.771383 −0.385691 0.922628i \(-0.626037\pi\)
−0.385691 + 0.922628i \(0.626037\pi\)
\(648\) −40171.6 −2.43532
\(649\) 32189.0 1.94688
\(650\) 4149.57 0.250399
\(651\) −10254.0 −0.617334
\(652\) −28786.8 −1.72911
\(653\) −12385.6 −0.742247 −0.371124 0.928583i \(-0.621027\pi\)
−0.371124 + 0.928583i \(0.621027\pi\)
\(654\) −43730.0 −2.61465
\(655\) −8812.09 −0.525675
\(656\) 2497.07 0.148619
\(657\) 789.030 0.0468539
\(658\) 13707.3 0.812109
\(659\) −2072.18 −0.122489 −0.0612447 0.998123i \(-0.519507\pi\)
−0.0612447 + 0.998123i \(0.519507\pi\)
\(660\) −25932.7 −1.52944
\(661\) 1074.36 0.0632193 0.0316096 0.999500i \(-0.489937\pi\)
0.0316096 + 0.999500i \(0.489937\pi\)
\(662\) −26703.6 −1.56777
\(663\) −5314.47 −0.311307
\(664\) 47613.0 2.78275
\(665\) 2807.89 0.163737
\(666\) −5671.84 −0.329999
\(667\) 540.982 0.0314047
\(668\) −26228.2 −1.51916
\(669\) −3479.42 −0.201080
\(670\) 4507.59 0.259916
\(671\) −797.667 −0.0458921
\(672\) 19988.6 1.14744
\(673\) 26195.2 1.50037 0.750186 0.661226i \(-0.229964\pi\)
0.750186 + 0.661226i \(0.229964\pi\)
\(674\) −38555.3 −2.20341
\(675\) 3762.01 0.214518
\(676\) −26794.7 −1.52450
\(677\) −4228.44 −0.240047 −0.120024 0.992771i \(-0.538297\pi\)
−0.120024 + 0.992771i \(0.538297\pi\)
\(678\) 46750.4 2.64814
\(679\) −11774.9 −0.665506
\(680\) −13416.6 −0.756624
\(681\) −7756.80 −0.436478
\(682\) 88994.5 4.99674
\(683\) 27525.5 1.54207 0.771036 0.636792i \(-0.219739\pi\)
0.771036 + 0.636792i \(0.219739\pi\)
\(684\) −9085.91 −0.507907
\(685\) 4614.74 0.257402
\(686\) −1857.08 −0.103358
\(687\) 30867.3 1.71421
\(688\) −7435.40 −0.412023
\(689\) 4691.09 0.259385
\(690\) 3256.72 0.179683
\(691\) −33324.4 −1.83462 −0.917309 0.398177i \(-0.869643\pi\)
−0.917309 + 0.398177i \(0.869643\pi\)
\(692\) −53021.1 −2.91266
\(693\) −1943.67 −0.106542
\(694\) 51654.8 2.82534
\(695\) 980.193 0.0534976
\(696\) −7029.54 −0.382836
\(697\) 422.973 0.0229860
\(698\) 6939.12 0.376289
\(699\) 2013.77 0.108967
\(700\) −3729.90 −0.201396
\(701\) −33262.9 −1.79219 −0.896094 0.443864i \(-0.853607\pi\)
−0.896094 + 0.443864i \(0.853607\pi\)
\(702\) 24977.1 1.34288
\(703\) 15816.2 0.848534
\(704\) −81611.8 −4.36912
\(705\) −8421.37 −0.449882
\(706\) 31392.0 1.67345
\(707\) 3039.17 0.161668
\(708\) 61141.0 3.24551
\(709\) 13703.0 0.725851 0.362926 0.931818i \(-0.381778\pi\)
0.362926 + 0.931818i \(0.381778\pi\)
\(710\) 25771.7 1.36224
\(711\) −4557.70 −0.240404
\(712\) −3299.58 −0.173676
\(713\) −8126.14 −0.426825
\(714\) 6570.00 0.344364
\(715\) 8009.84 0.418953
\(716\) 34553.3 1.80352
\(717\) −25987.3 −1.35358
\(718\) 12275.6 0.638052
\(719\) −8074.93 −0.418838 −0.209419 0.977826i \(-0.567157\pi\)
−0.209419 + 0.977826i \(0.567157\pi\)
\(720\) 5838.82 0.302222
\(721\) −2419.04 −0.124951
\(722\) −2289.59 −0.118019
\(723\) 29323.6 1.50838
\(724\) 55281.1 2.83771
\(725\) 523.528 0.0268184
\(726\) −35287.6 −1.80392
\(727\) −3668.70 −0.187159 −0.0935794 0.995612i \(-0.529831\pi\)
−0.0935794 + 0.995612i \(0.529831\pi\)
\(728\) −15468.9 −0.787523
\(729\) 21881.9 1.11172
\(730\) 4019.77 0.203806
\(731\) −1259.46 −0.0637250
\(732\) −1515.12 −0.0765033
\(733\) −14980.3 −0.754857 −0.377428 0.926039i \(-0.623192\pi\)
−0.377428 + 0.926039i \(0.623192\pi\)
\(734\) −39918.2 −2.00737
\(735\) 1140.93 0.0572569
\(736\) 15840.7 0.793338
\(737\) 8700.94 0.434875
\(738\) −326.894 −0.0163050
\(739\) 6530.59 0.325077 0.162538 0.986702i \(-0.448032\pi\)
0.162538 + 0.986702i \(0.448032\pi\)
\(740\) −21009.7 −1.04369
\(741\) −11453.3 −0.567812
\(742\) −5799.36 −0.286929
\(743\) 25952.0 1.28141 0.640704 0.767788i \(-0.278643\pi\)
0.640704 + 0.767788i \(0.278643\pi\)
\(744\) 105591. 5.20318
\(745\) −3901.86 −0.191883
\(746\) 34906.2 1.71315
\(747\) −3509.85 −0.171913
\(748\) −41459.6 −2.02662
\(749\) −1519.83 −0.0741434
\(750\) 3151.65 0.153443
\(751\) −14093.9 −0.684813 −0.342407 0.939552i \(-0.611242\pi\)
−0.342407 + 0.939552i \(0.611242\pi\)
\(752\) −79483.6 −3.85435
\(753\) −1452.57 −0.0702983
\(754\) 3475.87 0.167883
\(755\) 11597.1 0.559024
\(756\) −22451.0 −1.08007
\(757\) −2554.41 −0.122644 −0.0613220 0.998118i \(-0.519532\pi\)
−0.0613220 + 0.998118i \(0.519532\pi\)
\(758\) −22997.2 −1.10197
\(759\) 6286.40 0.300635
\(760\) −28914.5 −1.38005
\(761\) 2219.08 0.105705 0.0528527 0.998602i \(-0.483169\pi\)
0.0528527 + 0.998602i \(0.483169\pi\)
\(762\) −35160.1 −1.67154
\(763\) −12140.9 −0.576054
\(764\) −38856.5 −1.84003
\(765\) 989.025 0.0467428
\(766\) −36177.0 −1.70643
\(767\) −18884.6 −0.889028
\(768\) −31333.5 −1.47220
\(769\) −22466.2 −1.05352 −0.526758 0.850015i \(-0.676592\pi\)
−0.526758 + 0.850015i \(0.676592\pi\)
\(770\) −9902.16 −0.463440
\(771\) 36609.3 1.71006
\(772\) −32845.0 −1.53124
\(773\) 9674.79 0.450165 0.225083 0.974340i \(-0.427735\pi\)
0.225083 + 0.974340i \(0.427735\pi\)
\(774\) 973.374 0.0452031
\(775\) −7863.96 −0.364493
\(776\) 121253. 5.60920
\(777\) 6426.60 0.296722
\(778\) −34498.2 −1.58974
\(779\) 911.560 0.0419256
\(780\) 15214.2 0.698405
\(781\) 49746.6 2.27922
\(782\) 5206.64 0.238093
\(783\) 3151.23 0.143826
\(784\) 10768.5 0.490546
\(785\) −5114.51 −0.232541
\(786\) −44436.2 −2.01652
\(787\) −20942.8 −0.948577 −0.474288 0.880370i \(-0.657295\pi\)
−0.474288 + 0.880370i \(0.657295\pi\)
\(788\) 14946.1 0.675676
\(789\) −24341.8 −1.09834
\(790\) −23219.5 −1.04571
\(791\) 12979.4 0.583433
\(792\) 20015.1 0.897989
\(793\) 467.975 0.0209562
\(794\) 22999.2 1.02797
\(795\) 3562.94 0.158949
\(796\) 70227.8 3.12708
\(797\) 23526.6 1.04561 0.522807 0.852451i \(-0.324885\pi\)
0.522807 + 0.852451i \(0.324885\pi\)
\(798\) 14159.2 0.628107
\(799\) −13463.5 −0.596127
\(800\) 15329.6 0.677481
\(801\) 243.233 0.0107294
\(802\) −47827.1 −2.10578
\(803\) 7759.29 0.340996
\(804\) 16526.9 0.724948
\(805\) 904.172 0.0395874
\(806\) −52211.3 −2.28172
\(807\) −5968.72 −0.260358
\(808\) −31296.1 −1.36262
\(809\) −18202.2 −0.791047 −0.395523 0.918456i \(-0.629437\pi\)
−0.395523 + 0.918456i \(0.629437\pi\)
\(810\) 15086.6 0.654429
\(811\) −2510.24 −0.108689 −0.0543443 0.998522i \(-0.517307\pi\)
−0.0543443 + 0.998522i \(0.517307\pi\)
\(812\) −3124.33 −0.135028
\(813\) −21906.5 −0.945012
\(814\) −55776.7 −2.40168
\(815\) 6753.13 0.290248
\(816\) −38096.9 −1.63438
\(817\) −2714.30 −0.116232
\(818\) −1728.24 −0.0738711
\(819\) 1140.31 0.0486516
\(820\) −1210.88 −0.0515681
\(821\) 17899.6 0.760903 0.380451 0.924801i \(-0.375769\pi\)
0.380451 + 0.924801i \(0.375769\pi\)
\(822\) 23270.5 0.987411
\(823\) 14039.5 0.594637 0.297318 0.954778i \(-0.403908\pi\)
0.297318 + 0.954778i \(0.403908\pi\)
\(824\) 24910.3 1.05315
\(825\) 6083.58 0.256731
\(826\) 23346.1 0.983431
\(827\) 15127.4 0.636073 0.318036 0.948079i \(-0.396977\pi\)
0.318036 + 0.948079i \(0.396977\pi\)
\(828\) −2925.77 −0.122799
\(829\) 21986.5 0.921136 0.460568 0.887624i \(-0.347646\pi\)
0.460568 + 0.887624i \(0.347646\pi\)
\(830\) −17881.2 −0.747790
\(831\) −41718.7 −1.74152
\(832\) 47880.0 1.99512
\(833\) 1824.04 0.0758696
\(834\) 4942.76 0.205220
\(835\) 6152.89 0.255005
\(836\) −89350.6 −3.69648
\(837\) −47334.8 −1.95476
\(838\) −69243.5 −2.85439
\(839\) −2276.89 −0.0936914 −0.0468457 0.998902i \(-0.514917\pi\)
−0.0468457 + 0.998902i \(0.514917\pi\)
\(840\) −11748.8 −0.482588
\(841\) −23950.5 −0.982019
\(842\) −36531.8 −1.49521
\(843\) 1727.25 0.0705688
\(844\) 87010.1 3.54859
\(845\) 6285.79 0.255903
\(846\) 10405.3 0.422861
\(847\) −9796.97 −0.397436
\(848\) 33628.2 1.36179
\(849\) 27113.4 1.09603
\(850\) 5038.66 0.203323
\(851\) 5093.00 0.205154
\(852\) 94490.6 3.79952
\(853\) 13342.6 0.535570 0.267785 0.963479i \(-0.413708\pi\)
0.267785 + 0.963479i \(0.413708\pi\)
\(854\) −578.533 −0.0231815
\(855\) 2131.47 0.0852571
\(856\) 15650.6 0.624915
\(857\) 18690.9 0.745003 0.372502 0.928032i \(-0.378500\pi\)
0.372502 + 0.928032i \(0.378500\pi\)
\(858\) 40390.8 1.60713
\(859\) 18318.9 0.727628 0.363814 0.931472i \(-0.381474\pi\)
0.363814 + 0.931472i \(0.381474\pi\)
\(860\) 3605.58 0.142964
\(861\) 370.394 0.0146609
\(862\) −28071.3 −1.10918
\(863\) 38133.1 1.50413 0.752067 0.659087i \(-0.229057\pi\)
0.752067 + 0.659087i \(0.229057\pi\)
\(864\) 92272.3 3.63330
\(865\) 12438.3 0.488917
\(866\) −22967.3 −0.901223
\(867\) 16426.0 0.643432
\(868\) 46930.8 1.83518
\(869\) −44820.3 −1.74962
\(870\) 2639.96 0.102877
\(871\) −5104.66 −0.198582
\(872\) 125022. 4.85525
\(873\) −8938.33 −0.346525
\(874\) 11221.0 0.434273
\(875\) 875.000 0.0338062
\(876\) 14738.3 0.568449
\(877\) −19707.5 −0.758807 −0.379404 0.925231i \(-0.623871\pi\)
−0.379404 + 0.925231i \(0.623871\pi\)
\(878\) −29421.5 −1.13090
\(879\) −34664.6 −1.33016
\(880\) 57418.8 2.19953
\(881\) −14091.5 −0.538883 −0.269441 0.963017i \(-0.586839\pi\)
−0.269441 + 0.963017i \(0.586839\pi\)
\(882\) −1409.71 −0.0538178
\(883\) 3115.87 0.118751 0.0593757 0.998236i \(-0.481089\pi\)
0.0593757 + 0.998236i \(0.481089\pi\)
\(884\) 24323.5 0.925438
\(885\) −14343.1 −0.544789
\(886\) −62229.8 −2.35965
\(887\) 38734.6 1.46627 0.733134 0.680084i \(-0.238057\pi\)
0.733134 + 0.680084i \(0.238057\pi\)
\(888\) −66178.6 −2.50091
\(889\) −9761.57 −0.368270
\(890\) 1239.17 0.0466708
\(891\) 29121.3 1.09495
\(892\) 15924.8 0.597759
\(893\) −29015.6 −1.08731
\(894\) −19675.7 −0.736077
\(895\) −8105.89 −0.302737
\(896\) −24853.1 −0.926658
\(897\) −3688.10 −0.137282
\(898\) −91225.8 −3.39003
\(899\) −6587.21 −0.244378
\(900\) −2831.37 −0.104866
\(901\) 5696.21 0.210619
\(902\) −3214.66 −0.118666
\(903\) −1102.90 −0.0406449
\(904\) −133657. −4.91744
\(905\) −12968.4 −0.476337
\(906\) 58480.2 2.14445
\(907\) 19242.9 0.704464 0.352232 0.935913i \(-0.385423\pi\)
0.352232 + 0.935913i \(0.385423\pi\)
\(908\) 35501.7 1.29754
\(909\) 2307.03 0.0841799
\(910\) 5809.40 0.211626
\(911\) 34613.3 1.25882 0.629412 0.777072i \(-0.283296\pi\)
0.629412 + 0.777072i \(0.283296\pi\)
\(912\) −82103.6 −2.98105
\(913\) −34515.8 −1.25116
\(914\) 83100.1 3.00734
\(915\) 355.433 0.0128418
\(916\) −141275. −5.09591
\(917\) −12336.9 −0.444276
\(918\) 30328.7 1.09041
\(919\) 25826.4 0.927022 0.463511 0.886091i \(-0.346589\pi\)
0.463511 + 0.886091i \(0.346589\pi\)
\(920\) −9310.81 −0.333661
\(921\) 3547.01 0.126903
\(922\) 76007.0 2.71492
\(923\) −29185.3 −1.04079
\(924\) −36305.8 −1.29261
\(925\) 4928.68 0.175194
\(926\) −46893.8 −1.66417
\(927\) −1836.29 −0.0650613
\(928\) 12840.8 0.454224
\(929\) 19451.6 0.686960 0.343480 0.939160i \(-0.388394\pi\)
0.343480 + 0.939160i \(0.388394\pi\)
\(930\) −39655.1 −1.39822
\(931\) 3931.04 0.138383
\(932\) −9216.70 −0.323930
\(933\) −35944.8 −1.26129
\(934\) 37979.1 1.33053
\(935\) 9726.03 0.340188
\(936\) −11742.5 −0.410059
\(937\) 34469.1 1.20177 0.600884 0.799336i \(-0.294815\pi\)
0.600884 + 0.799336i \(0.294815\pi\)
\(938\) 6310.63 0.219669
\(939\) −39844.1 −1.38473
\(940\) 38543.3 1.33739
\(941\) 14156.4 0.490419 0.245209 0.969470i \(-0.421143\pi\)
0.245209 + 0.969470i \(0.421143\pi\)
\(942\) −25790.6 −0.892042
\(943\) 293.532 0.0101365
\(944\) −135375. −4.66746
\(945\) 5266.81 0.181301
\(946\) 9572.13 0.328982
\(947\) −38092.4 −1.30711 −0.653557 0.756877i \(-0.726724\pi\)
−0.653557 + 0.756877i \(0.726724\pi\)
\(948\) −85133.3 −2.91667
\(949\) −4552.22 −0.155713
\(950\) 10858.9 0.370853
\(951\) 36234.8 1.23553
\(952\) −18783.3 −0.639464
\(953\) 5037.40 0.171225 0.0856126 0.996329i \(-0.472715\pi\)
0.0856126 + 0.996329i \(0.472715\pi\)
\(954\) −4402.30 −0.149402
\(955\) 9115.39 0.308866
\(956\) 118940. 4.02384
\(957\) 5095.88 0.172128
\(958\) 98185.1 3.31129
\(959\) 6460.64 0.217544
\(960\) 36365.4 1.22259
\(961\) 69156.0 2.32137
\(962\) 32723.0 1.09671
\(963\) −1153.71 −0.0386060
\(964\) −134210. −4.48403
\(965\) 7705.14 0.257033
\(966\) 4559.41 0.151860
\(967\) 11495.3 0.382278 0.191139 0.981563i \(-0.438782\pi\)
0.191139 + 0.981563i \(0.438782\pi\)
\(968\) 100885. 3.34977
\(969\) −13907.3 −0.461061
\(970\) −45537.0 −1.50732
\(971\) 22352.7 0.738757 0.369379 0.929279i \(-0.379571\pi\)
0.369379 + 0.929279i \(0.379571\pi\)
\(972\) −31282.7 −1.03230
\(973\) 1372.27 0.0452138
\(974\) 89539.4 2.94561
\(975\) −3569.11 −0.117234
\(976\) 3354.69 0.110022
\(977\) 14345.7 0.469765 0.234882 0.972024i \(-0.424530\pi\)
0.234882 + 0.972024i \(0.424530\pi\)
\(978\) 34053.6 1.11341
\(979\) 2391.94 0.0780867
\(980\) −5221.86 −0.170210
\(981\) −9216.15 −0.299948
\(982\) 1194.42 0.0388141
\(983\) 34460.9 1.11814 0.559070 0.829120i \(-0.311158\pi\)
0.559070 + 0.829120i \(0.311158\pi\)
\(984\) −3814.17 −0.123568
\(985\) −3506.21 −0.113419
\(986\) 4220.61 0.136320
\(987\) −11789.9 −0.380220
\(988\) 52420.2 1.68796
\(989\) −874.036 −0.0281019
\(990\) −7516.74 −0.241311
\(991\) −35189.6 −1.12799 −0.563993 0.825780i \(-0.690735\pi\)
−0.563993 + 0.825780i \(0.690735\pi\)
\(992\) −192883. −6.17342
\(993\) 22968.2 0.734011
\(994\) 36080.3 1.15131
\(995\) −16474.8 −0.524911
\(996\) −65560.6 −2.08571
\(997\) −50730.0 −1.61147 −0.805734 0.592277i \(-0.798229\pi\)
−0.805734 + 0.592277i \(0.798229\pi\)
\(998\) 32155.2 1.01990
\(999\) 29666.8 0.939554
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 35.4.a.b.1.2 2
3.2 odd 2 315.4.a.f.1.1 2
4.3 odd 2 560.4.a.r.1.2 2
5.2 odd 4 175.4.b.c.99.4 4
5.3 odd 4 175.4.b.c.99.1 4
5.4 even 2 175.4.a.c.1.1 2
7.2 even 3 245.4.e.h.116.1 4
7.3 odd 6 245.4.e.i.226.1 4
7.4 even 3 245.4.e.h.226.1 4
7.5 odd 6 245.4.e.i.116.1 4
7.6 odd 2 245.4.a.k.1.2 2
8.3 odd 2 2240.4.a.bo.1.1 2
8.5 even 2 2240.4.a.bn.1.2 2
15.14 odd 2 1575.4.a.z.1.2 2
21.20 even 2 2205.4.a.u.1.1 2
35.34 odd 2 1225.4.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 2 1.1 even 1 trivial
175.4.a.c.1.1 2 5.4 even 2
175.4.b.c.99.1 4 5.3 odd 4
175.4.b.c.99.4 4 5.2 odd 4
245.4.a.k.1.2 2 7.6 odd 2
245.4.e.h.116.1 4 7.2 even 3
245.4.e.h.226.1 4 7.4 even 3
245.4.e.i.116.1 4 7.5 odd 6
245.4.e.i.226.1 4 7.3 odd 6
315.4.a.f.1.1 2 3.2 odd 2
560.4.a.r.1.2 2 4.3 odd 2
1225.4.a.m.1.1 2 35.34 odd 2
1575.4.a.z.1.2 2 15.14 odd 2
2205.4.a.u.1.1 2 21.20 even 2
2240.4.a.bn.1.2 2 8.5 even 2
2240.4.a.bo.1.1 2 8.3 odd 2