Properties

Label 363.4.a.q.1.2
Level $363$
Weight $4$
Character 363.1
Self dual yes
Analytic conductor $21.418$
Analytic rank $0$
Dimension $4$
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] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, 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("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 28x^{2} - 2x + 72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.59490\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.59490 q^{2} +3.00000 q^{3} -5.45630 q^{4} +8.73607 q^{5} -4.78470 q^{6} -29.0283 q^{7} +21.4614 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.59490 q^{2} +3.00000 q^{3} -5.45630 q^{4} +8.73607 q^{5} -4.78470 q^{6} -29.0283 q^{7} +21.4614 q^{8} +9.00000 q^{9} -13.9331 q^{10} -16.3689 q^{12} -31.1208 q^{13} +46.2971 q^{14} +26.2082 q^{15} +9.42156 q^{16} +109.520 q^{17} -14.3541 q^{18} -81.2185 q^{19} -47.6666 q^{20} -87.0848 q^{21} +143.858 q^{23} +64.3843 q^{24} -48.6811 q^{25} +49.6345 q^{26} +27.0000 q^{27} +158.387 q^{28} +85.5321 q^{29} -41.7994 q^{30} +308.509 q^{31} -186.718 q^{32} -174.673 q^{34} -253.593 q^{35} -49.1067 q^{36} -120.043 q^{37} +129.535 q^{38} -93.3624 q^{39} +187.489 q^{40} +134.057 q^{41} +138.891 q^{42} -207.008 q^{43} +78.6246 q^{45} -229.438 q^{46} +328.179 q^{47} +28.2647 q^{48} +499.640 q^{49} +77.6414 q^{50} +328.560 q^{51} +169.804 q^{52} +338.452 q^{53} -43.0623 q^{54} -622.988 q^{56} -243.655 q^{57} -136.415 q^{58} +657.950 q^{59} -143.000 q^{60} +414.074 q^{61} -492.041 q^{62} -261.254 q^{63} +222.424 q^{64} -271.873 q^{65} +122.643 q^{67} -597.574 q^{68} +431.573 q^{69} +404.455 q^{70} -1030.29 q^{71} +193.153 q^{72} +772.054 q^{73} +191.457 q^{74} -146.043 q^{75} +443.152 q^{76} +148.904 q^{78} +477.645 q^{79} +82.3074 q^{80} +81.0000 q^{81} -213.808 q^{82} +1124.93 q^{83} +475.160 q^{84} +956.775 q^{85} +330.156 q^{86} +256.596 q^{87} -472.532 q^{89} -125.398 q^{90} +903.383 q^{91} -784.930 q^{92} +925.527 q^{93} -523.412 q^{94} -709.530 q^{95} -560.154 q^{96} +1373.79 q^{97} -796.875 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} - 3 q^{6} + 20 q^{7} - 75 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} - 3 q^{6} + 20 q^{7} - 75 q^{8} + 36 q^{9} + 3 q^{10} + 75 q^{12} + 32 q^{13} + 62 q^{14} + 42 q^{15} + 289 q^{16} + 92 q^{17} - 9 q^{18} - 34 q^{19} + 391 q^{20} + 60 q^{21} - 26 q^{23} - 225 q^{24} + 334 q^{25} - 181 q^{26} + 108 q^{27} + 692 q^{28} + 174 q^{29} + 9 q^{30} + 422 q^{31} - 1271 q^{32} - 477 q^{34} - 82 q^{35} + 225 q^{36} + 518 q^{37} - 798 q^{38} + 96 q^{39} + 423 q^{40} + 428 q^{41} + 186 q^{42} - 550 q^{43} + 126 q^{45} - 2004 q^{46} + 556 q^{47} + 867 q^{48} + 282 q^{49} + 2074 q^{50} + 276 q^{51} + 467 q^{52} + 882 q^{53} - 27 q^{54} - 2112 q^{56} - 102 q^{57} - 225 q^{58} - 158 q^{59} + 1173 q^{60} + 290 q^{61} + 1142 q^{62} + 180 q^{63} + 3097 q^{64} - 1636 q^{65} + 992 q^{67} - 2033 q^{68} - 78 q^{69} + 948 q^{70} - 42 q^{71} - 675 q^{72} + 1274 q^{73} + 1509 q^{74} + 1002 q^{75} + 632 q^{76} - 543 q^{78} + 362 q^{79} + 1423 q^{80} + 324 q^{81} + 2403 q^{82} + 1500 q^{83} + 2076 q^{84} - 2388 q^{85} - 1272 q^{86} + 522 q^{87} + 1428 q^{89} + 27 q^{90} + 1750 q^{91} + 896 q^{92} + 1266 q^{93} - 1728 q^{94} - 4452 q^{95} - 3813 q^{96} + 1052 q^{97} - 615 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.59490 −0.563882 −0.281941 0.959432i \(-0.590978\pi\)
−0.281941 + 0.959432i \(0.590978\pi\)
\(3\) 3.00000 0.577350
\(4\) −5.45630 −0.682037
\(5\) 8.73607 0.781378 0.390689 0.920523i \(-0.372237\pi\)
0.390689 + 0.920523i \(0.372237\pi\)
\(6\) −4.78470 −0.325557
\(7\) −29.0283 −1.56738 −0.783689 0.621153i \(-0.786665\pi\)
−0.783689 + 0.621153i \(0.786665\pi\)
\(8\) 21.4614 0.948470
\(9\) 9.00000 0.333333
\(10\) −13.9331 −0.440605
\(11\) 0 0
\(12\) −16.3689 −0.393774
\(13\) −31.1208 −0.663951 −0.331975 0.943288i \(-0.607715\pi\)
−0.331975 + 0.943288i \(0.607715\pi\)
\(14\) 46.2971 0.883817
\(15\) 26.2082 0.451129
\(16\) 9.42156 0.147212
\(17\) 109.520 1.56250 0.781251 0.624217i \(-0.214582\pi\)
0.781251 + 0.624217i \(0.214582\pi\)
\(18\) −14.3541 −0.187961
\(19\) −81.2185 −0.980674 −0.490337 0.871533i \(-0.663126\pi\)
−0.490337 + 0.871533i \(0.663126\pi\)
\(20\) −47.6666 −0.532929
\(21\) −87.0848 −0.904927
\(22\) 0 0
\(23\) 143.858 1.30419 0.652095 0.758137i \(-0.273890\pi\)
0.652095 + 0.758137i \(0.273890\pi\)
\(24\) 64.3843 0.547600
\(25\) −48.6811 −0.389449
\(26\) 49.6345 0.374390
\(27\) 27.0000 0.192450
\(28\) 158.387 1.06901
\(29\) 85.5321 0.547687 0.273843 0.961774i \(-0.411705\pi\)
0.273843 + 0.961774i \(0.411705\pi\)
\(30\) −41.7994 −0.254383
\(31\) 308.509 1.78741 0.893707 0.448651i \(-0.148095\pi\)
0.893707 + 0.448651i \(0.148095\pi\)
\(32\) −186.718 −1.03148
\(33\) 0 0
\(34\) −174.673 −0.881066
\(35\) −253.593 −1.22471
\(36\) −49.1067 −0.227346
\(37\) −120.043 −0.533379 −0.266690 0.963782i \(-0.585930\pi\)
−0.266690 + 0.963782i \(0.585930\pi\)
\(38\) 129.535 0.552984
\(39\) −93.3624 −0.383332
\(40\) 187.489 0.741114
\(41\) 134.057 0.510641 0.255320 0.966857i \(-0.417819\pi\)
0.255320 + 0.966857i \(0.417819\pi\)
\(42\) 138.891 0.510272
\(43\) −207.008 −0.734149 −0.367074 0.930192i \(-0.619641\pi\)
−0.367074 + 0.930192i \(0.619641\pi\)
\(44\) 0 0
\(45\) 78.6246 0.260459
\(46\) −229.438 −0.735410
\(47\) 328.179 1.01851 0.509253 0.860617i \(-0.329922\pi\)
0.509253 + 0.860617i \(0.329922\pi\)
\(48\) 28.2647 0.0849928
\(49\) 499.640 1.45668
\(50\) 77.6414 0.219603
\(51\) 328.560 0.902111
\(52\) 169.804 0.452839
\(53\) 338.452 0.877170 0.438585 0.898690i \(-0.355480\pi\)
0.438585 + 0.898690i \(0.355480\pi\)
\(54\) −43.0623 −0.108519
\(55\) 0 0
\(56\) −622.988 −1.48661
\(57\) −243.655 −0.566192
\(58\) −136.415 −0.308831
\(59\) 657.950 1.45183 0.725913 0.687786i \(-0.241417\pi\)
0.725913 + 0.687786i \(0.241417\pi\)
\(60\) −143.000 −0.307687
\(61\) 414.074 0.869128 0.434564 0.900641i \(-0.356903\pi\)
0.434564 + 0.900641i \(0.356903\pi\)
\(62\) −492.041 −1.00789
\(63\) −261.254 −0.522460
\(64\) 222.424 0.434421
\(65\) −271.873 −0.518796
\(66\) 0 0
\(67\) 122.643 0.223629 0.111815 0.993729i \(-0.464334\pi\)
0.111815 + 0.993729i \(0.464334\pi\)
\(68\) −597.574 −1.06568
\(69\) 431.573 0.752975
\(70\) 404.455 0.690595
\(71\) −1030.29 −1.72215 −0.861077 0.508475i \(-0.830210\pi\)
−0.861077 + 0.508475i \(0.830210\pi\)
\(72\) 193.153 0.316157
\(73\) 772.054 1.23784 0.618918 0.785456i \(-0.287571\pi\)
0.618918 + 0.785456i \(0.287571\pi\)
\(74\) 191.457 0.300763
\(75\) −146.043 −0.224848
\(76\) 443.152 0.668856
\(77\) 0 0
\(78\) 148.904 0.216154
\(79\) 477.645 0.680244 0.340122 0.940381i \(-0.389532\pi\)
0.340122 + 0.940381i \(0.389532\pi\)
\(80\) 82.3074 0.115028
\(81\) 81.0000 0.111111
\(82\) −213.808 −0.287941
\(83\) 1124.93 1.48768 0.743840 0.668358i \(-0.233002\pi\)
0.743840 + 0.668358i \(0.233002\pi\)
\(84\) 475.160 0.617194
\(85\) 956.775 1.22090
\(86\) 330.156 0.413973
\(87\) 256.596 0.316207
\(88\) 0 0
\(89\) −472.532 −0.562790 −0.281395 0.959592i \(-0.590797\pi\)
−0.281395 + 0.959592i \(0.590797\pi\)
\(90\) −125.398 −0.146868
\(91\) 903.383 1.04066
\(92\) −784.930 −0.889507
\(93\) 925.527 1.03196
\(94\) −523.412 −0.574318
\(95\) −709.530 −0.766277
\(96\) −560.154 −0.595526
\(97\) 1373.79 1.43801 0.719007 0.695003i \(-0.244597\pi\)
0.719007 + 0.695003i \(0.244597\pi\)
\(98\) −796.875 −0.821393
\(99\) 0 0
\(100\) 265.619 0.265619
\(101\) 1305.70 1.28635 0.643176 0.765718i \(-0.277616\pi\)
0.643176 + 0.765718i \(0.277616\pi\)
\(102\) −524.020 −0.508684
\(103\) −1176.29 −1.12527 −0.562636 0.826705i \(-0.690213\pi\)
−0.562636 + 0.826705i \(0.690213\pi\)
\(104\) −667.897 −0.629737
\(105\) −760.779 −0.707089
\(106\) −539.798 −0.494620
\(107\) −217.293 −0.196322 −0.0981612 0.995171i \(-0.531296\pi\)
−0.0981612 + 0.995171i \(0.531296\pi\)
\(108\) −147.320 −0.131258
\(109\) 145.175 0.127571 0.0637856 0.997964i \(-0.479683\pi\)
0.0637856 + 0.997964i \(0.479683\pi\)
\(110\) 0 0
\(111\) −360.130 −0.307947
\(112\) −273.491 −0.230737
\(113\) 1139.37 0.948523 0.474262 0.880384i \(-0.342715\pi\)
0.474262 + 0.880384i \(0.342715\pi\)
\(114\) 388.606 0.319266
\(115\) 1256.75 1.01907
\(116\) −466.688 −0.373543
\(117\) −280.087 −0.221317
\(118\) −1049.36 −0.818659
\(119\) −3179.18 −2.44903
\(120\) 562.466 0.427882
\(121\) 0 0
\(122\) −660.407 −0.490085
\(123\) 402.172 0.294818
\(124\) −1683.32 −1.21908
\(125\) −1517.29 −1.08568
\(126\) 416.674 0.294606
\(127\) −1529.03 −1.06834 −0.534171 0.845377i \(-0.679376\pi\)
−0.534171 + 0.845377i \(0.679376\pi\)
\(128\) 1139.00 0.786518
\(129\) −621.023 −0.423861
\(130\) 433.611 0.292540
\(131\) 67.0849 0.0447423 0.0223711 0.999750i \(-0.492878\pi\)
0.0223711 + 0.999750i \(0.492878\pi\)
\(132\) 0 0
\(133\) 2357.63 1.53709
\(134\) −195.602 −0.126101
\(135\) 235.874 0.150376
\(136\) 2350.46 1.48199
\(137\) −2003.93 −1.24969 −0.624845 0.780749i \(-0.714838\pi\)
−0.624845 + 0.780749i \(0.714838\pi\)
\(138\) −688.315 −0.424589
\(139\) −1006.97 −0.614462 −0.307231 0.951635i \(-0.599402\pi\)
−0.307231 + 0.951635i \(0.599402\pi\)
\(140\) 1383.68 0.835301
\(141\) 984.537 0.588035
\(142\) 1643.21 0.971091
\(143\) 0 0
\(144\) 84.7940 0.0490706
\(145\) 747.214 0.427950
\(146\) −1231.35 −0.697993
\(147\) 1498.92 0.841012
\(148\) 654.993 0.363784
\(149\) 2059.79 1.13252 0.566258 0.824228i \(-0.308391\pi\)
0.566258 + 0.824228i \(0.308391\pi\)
\(150\) 232.924 0.126788
\(151\) −2247.33 −1.21116 −0.605579 0.795785i \(-0.707058\pi\)
−0.605579 + 0.795785i \(0.707058\pi\)
\(152\) −1743.07 −0.930140
\(153\) 985.681 0.520834
\(154\) 0 0
\(155\) 2695.16 1.39665
\(156\) 509.413 0.261447
\(157\) −2897.67 −1.47299 −0.736494 0.676445i \(-0.763520\pi\)
−0.736494 + 0.676445i \(0.763520\pi\)
\(158\) −761.795 −0.383577
\(159\) 1015.36 0.506435
\(160\) −1631.18 −0.805976
\(161\) −4175.94 −2.04416
\(162\) −129.187 −0.0626535
\(163\) −3919.05 −1.88321 −0.941607 0.336715i \(-0.890684\pi\)
−0.941607 + 0.336715i \(0.890684\pi\)
\(164\) −731.458 −0.348276
\(165\) 0 0
\(166\) −1794.15 −0.838876
\(167\) 213.772 0.0990549 0.0495274 0.998773i \(-0.484228\pi\)
0.0495274 + 0.998773i \(0.484228\pi\)
\(168\) −1868.96 −0.858296
\(169\) −1228.50 −0.559170
\(170\) −1525.96 −0.688446
\(171\) −730.966 −0.326891
\(172\) 1129.50 0.500717
\(173\) 2773.19 1.21874 0.609368 0.792887i \(-0.291423\pi\)
0.609368 + 0.792887i \(0.291423\pi\)
\(174\) −409.245 −0.178303
\(175\) 1413.13 0.610414
\(176\) 0 0
\(177\) 1973.85 0.838213
\(178\) 753.641 0.317347
\(179\) −327.622 −0.136802 −0.0684011 0.997658i \(-0.521790\pi\)
−0.0684011 + 0.997658i \(0.521790\pi\)
\(180\) −428.999 −0.177643
\(181\) 124.409 0.0510897 0.0255449 0.999674i \(-0.491868\pi\)
0.0255449 + 0.999674i \(0.491868\pi\)
\(182\) −1440.80 −0.586811
\(183\) 1242.22 0.501791
\(184\) 3087.39 1.23699
\(185\) −1048.71 −0.416771
\(186\) −1476.12 −0.581906
\(187\) 0 0
\(188\) −1790.64 −0.694660
\(189\) −783.763 −0.301642
\(190\) 1131.63 0.432090
\(191\) −2837.72 −1.07503 −0.537513 0.843255i \(-0.680636\pi\)
−0.537513 + 0.843255i \(0.680636\pi\)
\(192\) 667.271 0.250813
\(193\) 302.516 0.112827 0.0564134 0.998407i \(-0.482034\pi\)
0.0564134 + 0.998407i \(0.482034\pi\)
\(194\) −2191.06 −0.810870
\(195\) −815.620 −0.299527
\(196\) −2726.18 −0.993507
\(197\) −126.259 −0.0456627 −0.0228314 0.999739i \(-0.507268\pi\)
−0.0228314 + 0.999739i \(0.507268\pi\)
\(198\) 0 0
\(199\) −1617.78 −0.576290 −0.288145 0.957587i \(-0.593038\pi\)
−0.288145 + 0.957587i \(0.593038\pi\)
\(200\) −1044.77 −0.369381
\(201\) 367.928 0.129113
\(202\) −2082.45 −0.725351
\(203\) −2482.85 −0.858432
\(204\) −1792.72 −0.615273
\(205\) 1171.14 0.399003
\(206\) 1876.06 0.634520
\(207\) 1294.72 0.434730
\(208\) −293.206 −0.0977414
\(209\) 0 0
\(210\) 1213.37 0.398715
\(211\) 944.206 0.308065 0.154033 0.988066i \(-0.450774\pi\)
0.154033 + 0.988066i \(0.450774\pi\)
\(212\) −1846.70 −0.598263
\(213\) −3090.87 −0.994286
\(214\) 346.560 0.110703
\(215\) −1808.43 −0.573647
\(216\) 579.459 0.182533
\(217\) −8955.48 −2.80156
\(218\) −231.540 −0.0719350
\(219\) 2316.16 0.714665
\(220\) 0 0
\(221\) −3408.35 −1.03742
\(222\) 574.372 0.173646
\(223\) 3669.63 1.10196 0.550979 0.834519i \(-0.314254\pi\)
0.550979 + 0.834519i \(0.314254\pi\)
\(224\) 5420.10 1.61672
\(225\) −438.130 −0.129816
\(226\) −1817.18 −0.534855
\(227\) −5747.30 −1.68045 −0.840224 0.542240i \(-0.817576\pi\)
−0.840224 + 0.542240i \(0.817576\pi\)
\(228\) 1329.46 0.386164
\(229\) 3493.05 1.00798 0.503989 0.863710i \(-0.331865\pi\)
0.503989 + 0.863710i \(0.331865\pi\)
\(230\) −2004.39 −0.574633
\(231\) 0 0
\(232\) 1835.64 0.519464
\(233\) 4719.69 1.32703 0.663513 0.748165i \(-0.269065\pi\)
0.663513 + 0.748165i \(0.269065\pi\)
\(234\) 446.711 0.124797
\(235\) 2866.99 0.795839
\(236\) −3589.97 −0.990200
\(237\) 1432.93 0.392739
\(238\) 5070.47 1.38096
\(239\) −1001.85 −0.271147 −0.135574 0.990767i \(-0.543288\pi\)
−0.135574 + 0.990767i \(0.543288\pi\)
\(240\) 246.922 0.0664115
\(241\) −2726.05 −0.728632 −0.364316 0.931275i \(-0.618697\pi\)
−0.364316 + 0.931275i \(0.618697\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) −2259.31 −0.592777
\(245\) 4364.89 1.13821
\(246\) −641.424 −0.166243
\(247\) 2527.58 0.651119
\(248\) 6621.04 1.69531
\(249\) 3374.80 0.858912
\(250\) 2419.92 0.612198
\(251\) 2594.22 0.652373 0.326187 0.945305i \(-0.394236\pi\)
0.326187 + 0.945305i \(0.394236\pi\)
\(252\) 1425.48 0.356337
\(253\) 0 0
\(254\) 2438.65 0.602418
\(255\) 2870.33 0.704889
\(256\) −3595.98 −0.877925
\(257\) 1998.17 0.484989 0.242494 0.970153i \(-0.422034\pi\)
0.242494 + 0.970153i \(0.422034\pi\)
\(258\) 990.469 0.239007
\(259\) 3484.65 0.836007
\(260\) 1483.42 0.353838
\(261\) 769.789 0.182562
\(262\) −106.994 −0.0252294
\(263\) 7575.55 1.77615 0.888077 0.459695i \(-0.152041\pi\)
0.888077 + 0.459695i \(0.152041\pi\)
\(264\) 0 0
\(265\) 2956.74 0.685401
\(266\) −3760.18 −0.866736
\(267\) −1417.60 −0.324927
\(268\) −669.174 −0.152524
\(269\) 5722.09 1.29696 0.648479 0.761233i \(-0.275405\pi\)
0.648479 + 0.761233i \(0.275405\pi\)
\(270\) −376.195 −0.0847944
\(271\) −1224.92 −0.274570 −0.137285 0.990532i \(-0.543838\pi\)
−0.137285 + 0.990532i \(0.543838\pi\)
\(272\) 1031.85 0.230019
\(273\) 2710.15 0.600826
\(274\) 3196.07 0.704677
\(275\) 0 0
\(276\) −2354.79 −0.513557
\(277\) 1110.45 0.240868 0.120434 0.992721i \(-0.461571\pi\)
0.120434 + 0.992721i \(0.461571\pi\)
\(278\) 1606.02 0.346484
\(279\) 2776.58 0.595805
\(280\) −5442.47 −1.16161
\(281\) −241.064 −0.0511769 −0.0255884 0.999673i \(-0.508146\pi\)
−0.0255884 + 0.999673i \(0.508146\pi\)
\(282\) −1570.24 −0.331582
\(283\) 894.466 0.187882 0.0939408 0.995578i \(-0.470054\pi\)
0.0939408 + 0.995578i \(0.470054\pi\)
\(284\) 5621.57 1.17457
\(285\) −2128.59 −0.442410
\(286\) 0 0
\(287\) −3891.46 −0.800367
\(288\) −1680.46 −0.343827
\(289\) 7081.65 1.44141
\(290\) −1191.73 −0.241313
\(291\) 4121.37 0.830238
\(292\) −4212.55 −0.844250
\(293\) 5091.74 1.01523 0.507616 0.861584i \(-0.330527\pi\)
0.507616 + 0.861584i \(0.330527\pi\)
\(294\) −2390.63 −0.474232
\(295\) 5747.90 1.13443
\(296\) −2576.31 −0.505894
\(297\) 0 0
\(298\) −3285.16 −0.638606
\(299\) −4476.96 −0.865918
\(300\) 796.856 0.153355
\(301\) 6009.08 1.15069
\(302\) 3584.26 0.682950
\(303\) 3917.09 0.742676
\(304\) −765.205 −0.144367
\(305\) 3617.38 0.679117
\(306\) −1572.06 −0.293689
\(307\) −5218.29 −0.970109 −0.485054 0.874484i \(-0.661200\pi\)
−0.485054 + 0.874484i \(0.661200\pi\)
\(308\) 0 0
\(309\) −3528.86 −0.649676
\(310\) −4298.50 −0.787543
\(311\) 3591.72 0.654881 0.327440 0.944872i \(-0.393814\pi\)
0.327440 + 0.944872i \(0.393814\pi\)
\(312\) −2003.69 −0.363579
\(313\) 250.784 0.0452881 0.0226440 0.999744i \(-0.492792\pi\)
0.0226440 + 0.999744i \(0.492792\pi\)
\(314\) 4621.49 0.830591
\(315\) −2282.34 −0.408238
\(316\) −2606.17 −0.463951
\(317\) −3289.37 −0.582806 −0.291403 0.956600i \(-0.594122\pi\)
−0.291403 + 0.956600i \(0.594122\pi\)
\(318\) −1619.39 −0.285569
\(319\) 0 0
\(320\) 1943.11 0.339447
\(321\) −651.878 −0.113347
\(322\) 6660.20 1.15267
\(323\) −8895.06 −1.53230
\(324\) −441.960 −0.0757819
\(325\) 1514.99 0.258575
\(326\) 6250.49 1.06191
\(327\) 435.525 0.0736532
\(328\) 2877.07 0.484327
\(329\) −9526.46 −1.59639
\(330\) 0 0
\(331\) 156.251 0.0259467 0.0129733 0.999916i \(-0.495870\pi\)
0.0129733 + 0.999916i \(0.495870\pi\)
\(332\) −6137.97 −1.01465
\(333\) −1080.39 −0.177793
\(334\) −340.945 −0.0558553
\(335\) 1071.41 0.174739
\(336\) −820.474 −0.133216
\(337\) −142.712 −0.0230683 −0.0115342 0.999933i \(-0.503672\pi\)
−0.0115342 + 0.999933i \(0.503672\pi\)
\(338\) 1959.33 0.315306
\(339\) 3418.12 0.547630
\(340\) −5220.45 −0.832702
\(341\) 0 0
\(342\) 1165.82 0.184328
\(343\) −4546.98 −0.715784
\(344\) −4442.68 −0.696318
\(345\) 3770.25 0.588358
\(346\) −4422.95 −0.687224
\(347\) 1129.82 0.174789 0.0873943 0.996174i \(-0.472146\pi\)
0.0873943 + 0.996174i \(0.472146\pi\)
\(348\) −1400.07 −0.215665
\(349\) 4583.25 0.702968 0.351484 0.936194i \(-0.385677\pi\)
0.351484 + 0.936194i \(0.385677\pi\)
\(350\) −2253.80 −0.344201
\(351\) −840.262 −0.127777
\(352\) 0 0
\(353\) 1337.03 0.201595 0.100798 0.994907i \(-0.467861\pi\)
0.100798 + 0.994907i \(0.467861\pi\)
\(354\) −3148.09 −0.472653
\(355\) −9000.68 −1.34565
\(356\) 2578.27 0.383843
\(357\) −9537.53 −1.41395
\(358\) 522.524 0.0771403
\(359\) 2387.20 0.350952 0.175476 0.984484i \(-0.443854\pi\)
0.175476 + 0.984484i \(0.443854\pi\)
\(360\) 1687.40 0.247038
\(361\) −262.558 −0.0382793
\(362\) −198.420 −0.0288086
\(363\) 0 0
\(364\) −4929.12 −0.709770
\(365\) 6744.71 0.967218
\(366\) −1981.22 −0.282951
\(367\) 3905.72 0.555522 0.277761 0.960650i \(-0.410408\pi\)
0.277761 + 0.960650i \(0.410408\pi\)
\(368\) 1355.36 0.191992
\(369\) 1206.52 0.170214
\(370\) 1672.58 0.235009
\(371\) −9824.69 −1.37486
\(372\) −5049.95 −0.703838
\(373\) −8949.69 −1.24235 −0.621176 0.783671i \(-0.713345\pi\)
−0.621176 + 0.783671i \(0.713345\pi\)
\(374\) 0 0
\(375\) −4551.87 −0.626820
\(376\) 7043.19 0.966024
\(377\) −2661.83 −0.363637
\(378\) 1250.02 0.170091
\(379\) 7119.15 0.964871 0.482436 0.875931i \(-0.339752\pi\)
0.482436 + 0.875931i \(0.339752\pi\)
\(380\) 3871.41 0.522629
\(381\) −4587.08 −0.616807
\(382\) 4525.87 0.606188
\(383\) 10665.8 1.42297 0.711485 0.702701i \(-0.248023\pi\)
0.711485 + 0.702701i \(0.248023\pi\)
\(384\) 3417.00 0.454096
\(385\) 0 0
\(386\) −482.482 −0.0636210
\(387\) −1863.07 −0.244716
\(388\) −7495.81 −0.980779
\(389\) 5535.98 0.721556 0.360778 0.932652i \(-0.382511\pi\)
0.360778 + 0.932652i \(0.382511\pi\)
\(390\) 1300.83 0.168898
\(391\) 15755.3 2.03780
\(392\) 10723.0 1.38161
\(393\) 201.255 0.0258320
\(394\) 201.370 0.0257484
\(395\) 4172.74 0.531527
\(396\) 0 0
\(397\) 2864.65 0.362147 0.181074 0.983470i \(-0.442043\pi\)
0.181074 + 0.983470i \(0.442043\pi\)
\(398\) 2580.20 0.324959
\(399\) 7072.89 0.887438
\(400\) −458.652 −0.0573315
\(401\) −5123.98 −0.638103 −0.319052 0.947737i \(-0.603364\pi\)
−0.319052 + 0.947737i \(0.603364\pi\)
\(402\) −586.807 −0.0728042
\(403\) −9601.05 −1.18675
\(404\) −7124.26 −0.877340
\(405\) 707.622 0.0868198
\(406\) 3959.89 0.484054
\(407\) 0 0
\(408\) 7051.38 0.855625
\(409\) −8582.03 −1.03754 −0.518770 0.854914i \(-0.673610\pi\)
−0.518770 + 0.854914i \(0.673610\pi\)
\(410\) −1867.84 −0.224991
\(411\) −6011.80 −0.721508
\(412\) 6418.17 0.767477
\(413\) −19099.1 −2.27556
\(414\) −2064.95 −0.245137
\(415\) 9827.49 1.16244
\(416\) 5810.81 0.684852
\(417\) −3020.91 −0.354760
\(418\) 0 0
\(419\) 9110.33 1.06222 0.531108 0.847304i \(-0.321776\pi\)
0.531108 + 0.847304i \(0.321776\pi\)
\(420\) 4151.03 0.482261
\(421\) 5300.27 0.613586 0.306793 0.951776i \(-0.400744\pi\)
0.306793 + 0.951776i \(0.400744\pi\)
\(422\) −1505.91 −0.173713
\(423\) 2953.61 0.339502
\(424\) 7263.68 0.831970
\(425\) −5331.56 −0.608514
\(426\) 4929.63 0.560660
\(427\) −12019.9 −1.36225
\(428\) 1185.61 0.133899
\(429\) 0 0
\(430\) 2884.27 0.323469
\(431\) −3998.67 −0.446889 −0.223444 0.974717i \(-0.571730\pi\)
−0.223444 + 0.974717i \(0.571730\pi\)
\(432\) 254.382 0.0283309
\(433\) −11054.3 −1.22687 −0.613435 0.789745i \(-0.710213\pi\)
−0.613435 + 0.789745i \(0.710213\pi\)
\(434\) 14283.1 1.57975
\(435\) 2241.64 0.247077
\(436\) −792.118 −0.0870082
\(437\) −11683.9 −1.27899
\(438\) −3694.04 −0.402987
\(439\) 10015.5 1.08887 0.544437 0.838802i \(-0.316743\pi\)
0.544437 + 0.838802i \(0.316743\pi\)
\(440\) 0 0
\(441\) 4496.76 0.485559
\(442\) 5435.98 0.584985
\(443\) −12779.4 −1.37058 −0.685292 0.728269i \(-0.740325\pi\)
−0.685292 + 0.728269i \(0.740325\pi\)
\(444\) 1964.98 0.210031
\(445\) −4128.07 −0.439751
\(446\) −5852.69 −0.621374
\(447\) 6179.38 0.653859
\(448\) −6456.57 −0.680903
\(449\) −5141.24 −0.540379 −0.270189 0.962807i \(-0.587086\pi\)
−0.270189 + 0.962807i \(0.587086\pi\)
\(450\) 698.773 0.0732010
\(451\) 0 0
\(452\) −6216.75 −0.646928
\(453\) −6741.98 −0.699262
\(454\) 9166.36 0.947574
\(455\) 7892.01 0.813150
\(456\) −5229.20 −0.537016
\(457\) −17198.0 −1.76037 −0.880183 0.474635i \(-0.842580\pi\)
−0.880183 + 0.474635i \(0.842580\pi\)
\(458\) −5571.06 −0.568381
\(459\) 2957.04 0.300704
\(460\) −6857.20 −0.695041
\(461\) 2206.21 0.222893 0.111446 0.993770i \(-0.464452\pi\)
0.111446 + 0.993770i \(0.464452\pi\)
\(462\) 0 0
\(463\) −8916.54 −0.895004 −0.447502 0.894283i \(-0.647686\pi\)
−0.447502 + 0.894283i \(0.647686\pi\)
\(464\) 805.845 0.0806259
\(465\) 8085.47 0.806354
\(466\) −7527.42 −0.748286
\(467\) 5732.86 0.568063 0.284031 0.958815i \(-0.408328\pi\)
0.284031 + 0.958815i \(0.408328\pi\)
\(468\) 1528.24 0.150946
\(469\) −3560.10 −0.350512
\(470\) −4572.57 −0.448759
\(471\) −8693.00 −0.850430
\(472\) 14120.6 1.37701
\(473\) 0 0
\(474\) −2285.39 −0.221458
\(475\) 3953.80 0.381922
\(476\) 17346.5 1.67033
\(477\) 3046.07 0.292390
\(478\) 1597.85 0.152895
\(479\) −16797.1 −1.60226 −0.801129 0.598492i \(-0.795767\pi\)
−0.801129 + 0.598492i \(0.795767\pi\)
\(480\) −4893.54 −0.465330
\(481\) 3735.85 0.354137
\(482\) 4347.77 0.410862
\(483\) −12527.8 −1.18020
\(484\) 0 0
\(485\) 12001.5 1.12363
\(486\) −387.560 −0.0361730
\(487\) −6090.82 −0.566738 −0.283369 0.959011i \(-0.591452\pi\)
−0.283369 + 0.959011i \(0.591452\pi\)
\(488\) 8886.63 0.824342
\(489\) −11757.2 −1.08727
\(490\) −6961.56 −0.641818
\(491\) 6213.12 0.571068 0.285534 0.958369i \(-0.407829\pi\)
0.285534 + 0.958369i \(0.407829\pi\)
\(492\) −2194.37 −0.201077
\(493\) 9367.48 0.855761
\(494\) −4031.24 −0.367154
\(495\) 0 0
\(496\) 2906.64 0.263129
\(497\) 29907.5 2.69927
\(498\) −5382.46 −0.484325
\(499\) −7089.58 −0.636018 −0.318009 0.948088i \(-0.603014\pi\)
−0.318009 + 0.948088i \(0.603014\pi\)
\(500\) 8278.79 0.740477
\(501\) 641.316 0.0571894
\(502\) −4137.52 −0.367861
\(503\) −919.713 −0.0815268 −0.0407634 0.999169i \(-0.512979\pi\)
−0.0407634 + 0.999169i \(0.512979\pi\)
\(504\) −5606.89 −0.495537
\(505\) 11406.6 1.00513
\(506\) 0 0
\(507\) −3685.49 −0.322837
\(508\) 8342.83 0.728648
\(509\) −14712.7 −1.28119 −0.640597 0.767877i \(-0.721313\pi\)
−0.640597 + 0.767877i \(0.721313\pi\)
\(510\) −4577.88 −0.397474
\(511\) −22411.4 −1.94016
\(512\) −3376.77 −0.291472
\(513\) −2192.90 −0.188731
\(514\) −3186.87 −0.273476
\(515\) −10276.1 −0.879262
\(516\) 3388.49 0.289089
\(517\) 0 0
\(518\) −5557.67 −0.471409
\(519\) 8319.56 0.703638
\(520\) −5834.80 −0.492063
\(521\) −5922.04 −0.497984 −0.248992 0.968506i \(-0.580099\pi\)
−0.248992 + 0.968506i \(0.580099\pi\)
\(522\) −1227.74 −0.102944
\(523\) 19017.6 1.59002 0.795010 0.606596i \(-0.207465\pi\)
0.795010 + 0.606596i \(0.207465\pi\)
\(524\) −366.035 −0.0305159
\(525\) 4239.38 0.352422
\(526\) −12082.2 −1.00154
\(527\) 33787.9 2.79284
\(528\) 0 0
\(529\) 8528.01 0.700913
\(530\) −4715.71 −0.386485
\(531\) 5921.55 0.483942
\(532\) −12863.9 −1.04835
\(533\) −4171.98 −0.339040
\(534\) 2260.92 0.183220
\(535\) −1898.29 −0.153402
\(536\) 2632.09 0.212106
\(537\) −982.866 −0.0789828
\(538\) −9126.15 −0.731331
\(539\) 0 0
\(540\) −1287.00 −0.102562
\(541\) −1368.85 −0.108783 −0.0543914 0.998520i \(-0.517322\pi\)
−0.0543914 + 0.998520i \(0.517322\pi\)
\(542\) 1953.62 0.154825
\(543\) 373.227 0.0294967
\(544\) −20449.4 −1.61169
\(545\) 1268.26 0.0996812
\(546\) −4322.41 −0.338795
\(547\) 10102.6 0.789685 0.394843 0.918749i \(-0.370799\pi\)
0.394843 + 0.918749i \(0.370799\pi\)
\(548\) 10934.0 0.852335
\(549\) 3726.67 0.289709
\(550\) 0 0
\(551\) −6946.79 −0.537102
\(552\) 9262.17 0.714174
\(553\) −13865.2 −1.06620
\(554\) −1771.06 −0.135821
\(555\) −3146.13 −0.240623
\(556\) 5494.33 0.419086
\(557\) 22956.8 1.74634 0.873171 0.487414i \(-0.162060\pi\)
0.873171 + 0.487414i \(0.162060\pi\)
\(558\) −4428.37 −0.335964
\(559\) 6442.25 0.487438
\(560\) −2389.24 −0.180293
\(561\) 0 0
\(562\) 384.473 0.0288577
\(563\) −7101.89 −0.531632 −0.265816 0.964024i \(-0.585641\pi\)
−0.265816 + 0.964024i \(0.585641\pi\)
\(564\) −5371.93 −0.401062
\(565\) 9953.64 0.741155
\(566\) −1426.58 −0.105943
\(567\) −2351.29 −0.174153
\(568\) −22111.5 −1.63341
\(569\) 7073.70 0.521169 0.260584 0.965451i \(-0.416085\pi\)
0.260584 + 0.965451i \(0.416085\pi\)
\(570\) 3394.89 0.249467
\(571\) −20207.2 −1.48099 −0.740496 0.672061i \(-0.765409\pi\)
−0.740496 + 0.672061i \(0.765409\pi\)
\(572\) 0 0
\(573\) −8513.15 −0.620667
\(574\) 6206.48 0.451313
\(575\) −7003.15 −0.507915
\(576\) 2001.81 0.144807
\(577\) 11516.1 0.830887 0.415443 0.909619i \(-0.363626\pi\)
0.415443 + 0.909619i \(0.363626\pi\)
\(578\) −11294.5 −0.812786
\(579\) 907.547 0.0651406
\(580\) −4077.02 −0.291878
\(581\) −32654.8 −2.33176
\(582\) −6573.17 −0.468156
\(583\) 0 0
\(584\) 16569.4 1.17405
\(585\) −2446.86 −0.172932
\(586\) −8120.82 −0.572471
\(587\) 3747.76 0.263520 0.131760 0.991282i \(-0.457937\pi\)
0.131760 + 0.991282i \(0.457937\pi\)
\(588\) −8178.55 −0.573602
\(589\) −25056.6 −1.75287
\(590\) −9167.31 −0.639682
\(591\) −378.776 −0.0263634
\(592\) −1131.00 −0.0785197
\(593\) 19378.3 1.34194 0.670972 0.741483i \(-0.265877\pi\)
0.670972 + 0.741483i \(0.265877\pi\)
\(594\) 0 0
\(595\) −27773.5 −1.91362
\(596\) −11238.9 −0.772418
\(597\) −4853.35 −0.332721
\(598\) 7140.31 0.488276
\(599\) 6307.26 0.430230 0.215115 0.976589i \(-0.430987\pi\)
0.215115 + 0.976589i \(0.430987\pi\)
\(600\) −3134.30 −0.213262
\(601\) 8722.09 0.591983 0.295991 0.955191i \(-0.404350\pi\)
0.295991 + 0.955191i \(0.404350\pi\)
\(602\) −9583.87 −0.648853
\(603\) 1103.78 0.0745431
\(604\) 12262.1 0.826054
\(605\) 0 0
\(606\) −6247.36 −0.418781
\(607\) −10407.5 −0.695926 −0.347963 0.937508i \(-0.613126\pi\)
−0.347963 + 0.937508i \(0.613126\pi\)
\(608\) 15164.9 1.01155
\(609\) −7448.54 −0.495616
\(610\) −5769.36 −0.382942
\(611\) −10213.2 −0.676238
\(612\) −5378.17 −0.355228
\(613\) 21401.3 1.41010 0.705048 0.709159i \(-0.250925\pi\)
0.705048 + 0.709159i \(0.250925\pi\)
\(614\) 8322.64 0.547027
\(615\) 3513.41 0.230365
\(616\) 0 0
\(617\) 20892.3 1.36320 0.681599 0.731726i \(-0.261285\pi\)
0.681599 + 0.731726i \(0.261285\pi\)
\(618\) 5628.18 0.366340
\(619\) 17588.1 1.14204 0.571022 0.820935i \(-0.306547\pi\)
0.571022 + 0.820935i \(0.306547\pi\)
\(620\) −14705.6 −0.952564
\(621\) 3884.16 0.250992
\(622\) −5728.44 −0.369276
\(623\) 13716.8 0.882105
\(624\) −879.619 −0.0564310
\(625\) −7170.01 −0.458881
\(626\) −399.976 −0.0255371
\(627\) 0 0
\(628\) 15810.5 1.00463
\(629\) −13147.2 −0.833406
\(630\) 3640.10 0.230198
\(631\) −11923.7 −0.752259 −0.376129 0.926567i \(-0.622745\pi\)
−0.376129 + 0.926567i \(0.622745\pi\)
\(632\) 10250.9 0.645191
\(633\) 2832.62 0.177862
\(634\) 5246.22 0.328634
\(635\) −13357.7 −0.834778
\(636\) −5540.09 −0.345407
\(637\) −15549.2 −0.967161
\(638\) 0 0
\(639\) −9272.61 −0.574051
\(640\) 9950.38 0.614568
\(641\) 592.419 0.0365041 0.0182521 0.999833i \(-0.494190\pi\)
0.0182521 + 0.999833i \(0.494190\pi\)
\(642\) 1039.68 0.0639142
\(643\) 17511.6 1.07401 0.537007 0.843578i \(-0.319555\pi\)
0.537007 + 0.843578i \(0.319555\pi\)
\(644\) 22785.1 1.39419
\(645\) −5425.30 −0.331195
\(646\) 14186.7 0.864039
\(647\) −13456.4 −0.817660 −0.408830 0.912611i \(-0.634063\pi\)
−0.408830 + 0.912611i \(0.634063\pi\)
\(648\) 1738.38 0.105386
\(649\) 0 0
\(650\) −2416.26 −0.145806
\(651\) −26866.4 −1.61748
\(652\) 21383.5 1.28442
\(653\) 19046.3 1.14141 0.570703 0.821157i \(-0.306671\pi\)
0.570703 + 0.821157i \(0.306671\pi\)
\(654\) −694.619 −0.0415317
\(655\) 586.059 0.0349606
\(656\) 1263.03 0.0751723
\(657\) 6948.48 0.412612
\(658\) 15193.7 0.900173
\(659\) −12897.9 −0.762415 −0.381208 0.924489i \(-0.624492\pi\)
−0.381208 + 0.924489i \(0.624492\pi\)
\(660\) 0 0
\(661\) 7772.66 0.457370 0.228685 0.973501i \(-0.426557\pi\)
0.228685 + 0.973501i \(0.426557\pi\)
\(662\) −249.205 −0.0146309
\(663\) −10225.1 −0.598957
\(664\) 24142.7 1.41102
\(665\) 20596.4 1.20105
\(666\) 1723.12 0.100254
\(667\) 12304.4 0.714288
\(668\) −1166.40 −0.0675591
\(669\) 11008.9 0.636216
\(670\) −1708.80 −0.0985322
\(671\) 0 0
\(672\) 16260.3 0.933414
\(673\) 10948.8 0.627108 0.313554 0.949570i \(-0.398480\pi\)
0.313554 + 0.949570i \(0.398480\pi\)
\(674\) 227.612 0.0130078
\(675\) −1314.39 −0.0749494
\(676\) 6703.04 0.381374
\(677\) 6557.64 0.372276 0.186138 0.982524i \(-0.440403\pi\)
0.186138 + 0.982524i \(0.440403\pi\)
\(678\) −5451.55 −0.308799
\(679\) −39878.8 −2.25391
\(680\) 20533.8 1.15799
\(681\) −17241.9 −0.970207
\(682\) 0 0
\(683\) −23613.3 −1.32290 −0.661449 0.749990i \(-0.730058\pi\)
−0.661449 + 0.749990i \(0.730058\pi\)
\(684\) 3988.37 0.222952
\(685\) −17506.5 −0.976480
\(686\) 7251.98 0.403618
\(687\) 10479.1 0.581957
\(688\) −1950.34 −0.108075
\(689\) −10532.9 −0.582398
\(690\) −6013.17 −0.331764
\(691\) −13535.7 −0.745186 −0.372593 0.927995i \(-0.621531\pi\)
−0.372593 + 0.927995i \(0.621531\pi\)
\(692\) −15131.3 −0.831224
\(693\) 0 0
\(694\) −1801.94 −0.0985601
\(695\) −8796.97 −0.480127
\(696\) 5506.92 0.299913
\(697\) 14682.0 0.797877
\(698\) −7309.83 −0.396391
\(699\) 14159.1 0.766159
\(700\) −7710.44 −0.416325
\(701\) 23699.2 1.27690 0.638450 0.769663i \(-0.279576\pi\)
0.638450 + 0.769663i \(0.279576\pi\)
\(702\) 1340.13 0.0720513
\(703\) 9749.75 0.523071
\(704\) 0 0
\(705\) 8600.98 0.459478
\(706\) −2132.43 −0.113676
\(707\) −37902.1 −2.01620
\(708\) −10769.9 −0.571692
\(709\) −9518.98 −0.504221 −0.252111 0.967698i \(-0.581125\pi\)
−0.252111 + 0.967698i \(0.581125\pi\)
\(710\) 14355.2 0.758789
\(711\) 4298.80 0.226748
\(712\) −10141.2 −0.533789
\(713\) 44381.4 2.33113
\(714\) 15211.4 0.797300
\(715\) 0 0
\(716\) 1787.60 0.0933042
\(717\) −3005.55 −0.156547
\(718\) −3807.35 −0.197896
\(719\) −13051.3 −0.676956 −0.338478 0.940974i \(-0.609912\pi\)
−0.338478 + 0.940974i \(0.609912\pi\)
\(720\) 740.766 0.0383427
\(721\) 34145.6 1.76373
\(722\) 418.753 0.0215850
\(723\) −8178.15 −0.420676
\(724\) −678.812 −0.0348451
\(725\) −4163.80 −0.213296
\(726\) 0 0
\(727\) 16042.6 0.818414 0.409207 0.912442i \(-0.365805\pi\)
0.409207 + 0.912442i \(0.365805\pi\)
\(728\) 19387.9 0.987037
\(729\) 729.000 0.0370370
\(730\) −10757.1 −0.545397
\(731\) −22671.5 −1.14711
\(732\) −6777.94 −0.342240
\(733\) 18575.8 0.936035 0.468018 0.883719i \(-0.344968\pi\)
0.468018 + 0.883719i \(0.344968\pi\)
\(734\) −6229.22 −0.313249
\(735\) 13094.7 0.657148
\(736\) −26860.8 −1.34525
\(737\) 0 0
\(738\) −1924.27 −0.0959803
\(739\) −29628.7 −1.47484 −0.737421 0.675434i \(-0.763956\pi\)
−0.737421 + 0.675434i \(0.763956\pi\)
\(740\) 5722.06 0.284253
\(741\) 7582.75 0.375924
\(742\) 15669.4 0.775258
\(743\) 3203.46 0.158174 0.0790871 0.996868i \(-0.474799\pi\)
0.0790871 + 0.996868i \(0.474799\pi\)
\(744\) 19863.1 0.978787
\(745\) 17994.5 0.884923
\(746\) 14273.9 0.700540
\(747\) 10124.4 0.495893
\(748\) 0 0
\(749\) 6307.63 0.307711
\(750\) 7259.77 0.353453
\(751\) 26419.7 1.28371 0.641856 0.766825i \(-0.278165\pi\)
0.641856 + 0.766825i \(0.278165\pi\)
\(752\) 3091.96 0.149936
\(753\) 7782.66 0.376648
\(754\) 4245.35 0.205048
\(755\) −19632.8 −0.946371
\(756\) 4276.44 0.205731
\(757\) 16095.5 0.772789 0.386395 0.922334i \(-0.373720\pi\)
0.386395 + 0.922334i \(0.373720\pi\)
\(758\) −11354.3 −0.544074
\(759\) 0 0
\(760\) −15227.5 −0.726791
\(761\) 18290.7 0.871273 0.435636 0.900123i \(-0.356523\pi\)
0.435636 + 0.900123i \(0.356523\pi\)
\(762\) 7315.94 0.347806
\(763\) −4214.18 −0.199952
\(764\) 15483.4 0.733208
\(765\) 8610.98 0.406968
\(766\) −17010.9 −0.802387
\(767\) −20475.9 −0.963941
\(768\) −10787.9 −0.506870
\(769\) −23870.0 −1.11934 −0.559672 0.828714i \(-0.689073\pi\)
−0.559672 + 0.828714i \(0.689073\pi\)
\(770\) 0 0
\(771\) 5994.50 0.280008
\(772\) −1650.62 −0.0769520
\(773\) 30486.5 1.41853 0.709264 0.704943i \(-0.249028\pi\)
0.709264 + 0.704943i \(0.249028\pi\)
\(774\) 2971.41 0.137991
\(775\) −15018.6 −0.696106
\(776\) 29483.5 1.36391
\(777\) 10454.0 0.482669
\(778\) −8829.33 −0.406872
\(779\) −10887.9 −0.500772
\(780\) 4450.27 0.204289
\(781\) 0 0
\(782\) −25128.1 −1.14908
\(783\) 2309.37 0.105402
\(784\) 4707.39 0.214440
\(785\) −25314.2 −1.15096
\(786\) −320.981 −0.0145662
\(787\) 17185.1 0.778378 0.389189 0.921158i \(-0.372755\pi\)
0.389189 + 0.921158i \(0.372755\pi\)
\(788\) 688.904 0.0311437
\(789\) 22726.6 1.02546
\(790\) −6655.10 −0.299719
\(791\) −33074.0 −1.48670
\(792\) 0 0
\(793\) −12886.3 −0.577058
\(794\) −4568.82 −0.204208
\(795\) 8870.23 0.395717
\(796\) 8827.11 0.393051
\(797\) −27085.7 −1.20379 −0.601897 0.798574i \(-0.705588\pi\)
−0.601897 + 0.798574i \(0.705588\pi\)
\(798\) −11280.6 −0.500410
\(799\) 35942.2 1.59142
\(800\) 9089.63 0.401709
\(801\) −4252.79 −0.187597
\(802\) 8172.23 0.359815
\(803\) 0 0
\(804\) −2007.52 −0.0880595
\(805\) −36481.3 −1.59726
\(806\) 15312.7 0.669190
\(807\) 17166.3 0.748799
\(808\) 28022.1 1.22007
\(809\) −31395.6 −1.36441 −0.682207 0.731159i \(-0.738980\pi\)
−0.682207 + 0.731159i \(0.738980\pi\)
\(810\) −1128.59 −0.0489561
\(811\) −31205.7 −1.35115 −0.675575 0.737292i \(-0.736104\pi\)
−0.675575 + 0.737292i \(0.736104\pi\)
\(812\) 13547.2 0.585483
\(813\) −3674.76 −0.158523
\(814\) 0 0
\(815\) −34237.1 −1.47150
\(816\) 3095.55 0.132801
\(817\) 16812.9 0.719960
\(818\) 13687.5 0.585050
\(819\) 8130.44 0.346887
\(820\) −6390.06 −0.272135
\(821\) −34624.8 −1.47188 −0.735940 0.677046i \(-0.763260\pi\)
−0.735940 + 0.677046i \(0.763260\pi\)
\(822\) 9588.21 0.406846
\(823\) 7913.71 0.335182 0.167591 0.985857i \(-0.446401\pi\)
0.167591 + 0.985857i \(0.446401\pi\)
\(824\) −25244.8 −1.06729
\(825\) 0 0
\(826\) 30461.2 1.28315
\(827\) −25482.6 −1.07148 −0.535742 0.844382i \(-0.679968\pi\)
−0.535742 + 0.844382i \(0.679968\pi\)
\(828\) −7064.37 −0.296502
\(829\) 13174.9 0.551972 0.275986 0.961162i \(-0.410996\pi\)
0.275986 + 0.961162i \(0.410996\pi\)
\(830\) −15673.9 −0.655479
\(831\) 3331.35 0.139065
\(832\) −6922.01 −0.288434
\(833\) 54720.6 2.27606
\(834\) 4818.05 0.200043
\(835\) 1867.53 0.0773993
\(836\) 0 0
\(837\) 8329.74 0.343988
\(838\) −14530.1 −0.598965
\(839\) −2995.99 −0.123281 −0.0616407 0.998098i \(-0.519633\pi\)
−0.0616407 + 0.998098i \(0.519633\pi\)
\(840\) −16327.4 −0.670653
\(841\) −17073.3 −0.700039
\(842\) −8453.40 −0.345990
\(843\) −723.193 −0.0295470
\(844\) −5151.87 −0.210112
\(845\) −10732.2 −0.436923
\(846\) −4710.71 −0.191439
\(847\) 0 0
\(848\) 3188.75 0.129130
\(849\) 2683.40 0.108474
\(850\) 8503.30 0.343130
\(851\) −17269.2 −0.695628
\(852\) 16864.7 0.678140
\(853\) −27612.6 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(854\) 19170.5 0.768149
\(855\) −6385.77 −0.255426
\(856\) −4663.42 −0.186206
\(857\) 8310.71 0.331258 0.165629 0.986188i \(-0.447035\pi\)
0.165629 + 0.986188i \(0.447035\pi\)
\(858\) 0 0
\(859\) −28790.9 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(860\) 9867.35 0.391249
\(861\) −11674.4 −0.462092
\(862\) 6377.47 0.251992
\(863\) −1188.11 −0.0468643 −0.0234322 0.999725i \(-0.507459\pi\)
−0.0234322 + 0.999725i \(0.507459\pi\)
\(864\) −5041.38 −0.198509
\(865\) 24226.7 0.952294
\(866\) 17630.4 0.691809
\(867\) 21245.0 0.832199
\(868\) 48863.8 1.91076
\(869\) 0 0
\(870\) −3575.19 −0.139322
\(871\) −3816.73 −0.148479
\(872\) 3115.67 0.120997
\(873\) 12364.1 0.479338
\(874\) 18634.6 0.721197
\(875\) 44044.3 1.70168
\(876\) −12637.7 −0.487428
\(877\) −3253.31 −0.125264 −0.0626319 0.998037i \(-0.519949\pi\)
−0.0626319 + 0.998037i \(0.519949\pi\)
\(878\) −15973.8 −0.613996
\(879\) 15275.2 0.586144
\(880\) 0 0
\(881\) −1996.30 −0.0763417 −0.0381709 0.999271i \(-0.512153\pi\)
−0.0381709 + 0.999271i \(0.512153\pi\)
\(882\) −7171.88 −0.273798
\(883\) −12259.6 −0.467236 −0.233618 0.972328i \(-0.575057\pi\)
−0.233618 + 0.972328i \(0.575057\pi\)
\(884\) 18597.0 0.707562
\(885\) 17243.7 0.654961
\(886\) 20381.9 0.772847
\(887\) 17185.3 0.650536 0.325268 0.945622i \(-0.394546\pi\)
0.325268 + 0.945622i \(0.394546\pi\)
\(888\) −7728.92 −0.292078
\(889\) 44385.0 1.67450
\(890\) 6583.86 0.247968
\(891\) 0 0
\(892\) −20022.6 −0.751576
\(893\) −26654.2 −0.998823
\(894\) −9855.49 −0.368699
\(895\) −2862.13 −0.106894
\(896\) −33063.2 −1.23277
\(897\) −13430.9 −0.499938
\(898\) 8199.76 0.304710
\(899\) 26387.4 0.978943
\(900\) 2390.57 0.0885395
\(901\) 37067.4 1.37058
\(902\) 0 0
\(903\) 18027.2 0.664351
\(904\) 24452.6 0.899646
\(905\) 1086.85 0.0399204
\(906\) 10752.8 0.394301
\(907\) −11506.1 −0.421229 −0.210614 0.977569i \(-0.567546\pi\)
−0.210614 + 0.977569i \(0.567546\pi\)
\(908\) 31359.0 1.14613
\(909\) 11751.3 0.428784
\(910\) −12587.0 −0.458521
\(911\) −42758.6 −1.55506 −0.777529 0.628848i \(-0.783527\pi\)
−0.777529 + 0.628848i \(0.783527\pi\)
\(912\) −2295.61 −0.0833502
\(913\) 0 0
\(914\) 27429.0 0.992638
\(915\) 10852.1 0.392088
\(916\) −19059.1 −0.687479
\(917\) −1947.36 −0.0701281
\(918\) −4716.18 −0.169561
\(919\) −16491.2 −0.591940 −0.295970 0.955197i \(-0.595643\pi\)
−0.295970 + 0.955197i \(0.595643\pi\)
\(920\) 26971.7 0.966554
\(921\) −15654.9 −0.560093
\(922\) −3518.68 −0.125685
\(923\) 32063.5 1.14343
\(924\) 0 0
\(925\) 5843.85 0.207724
\(926\) 14221.0 0.504677
\(927\) −10586.6 −0.375091
\(928\) −15970.4 −0.564928
\(929\) −8099.05 −0.286029 −0.143015 0.989721i \(-0.545680\pi\)
−0.143015 + 0.989721i \(0.545680\pi\)
\(930\) −12895.5 −0.454688
\(931\) −40580.0 −1.42852
\(932\) −25752.0 −0.905081
\(933\) 10775.2 0.378096
\(934\) −9143.34 −0.320320
\(935\) 0 0
\(936\) −6011.07 −0.209912
\(937\) 34725.7 1.21071 0.605357 0.795954i \(-0.293030\pi\)
0.605357 + 0.795954i \(0.293030\pi\)
\(938\) 5678.00 0.197647
\(939\) 752.353 0.0261471
\(940\) −15643.2 −0.542792
\(941\) −3044.16 −0.105459 −0.0527294 0.998609i \(-0.516792\pi\)
−0.0527294 + 0.998609i \(0.516792\pi\)
\(942\) 13864.5 0.479542
\(943\) 19285.2 0.665973
\(944\) 6198.91 0.213726
\(945\) −6847.01 −0.235696
\(946\) 0 0
\(947\) 16063.2 0.551199 0.275599 0.961273i \(-0.411124\pi\)
0.275599 + 0.961273i \(0.411124\pi\)
\(948\) −7818.52 −0.267863
\(949\) −24026.9 −0.821862
\(950\) −6305.92 −0.215359
\(951\) −9868.12 −0.336483
\(952\) −68229.7 −2.32283
\(953\) −47423.8 −1.61197 −0.805985 0.591936i \(-0.798364\pi\)
−0.805985 + 0.591936i \(0.798364\pi\)
\(954\) −4858.18 −0.164873
\(955\) −24790.5 −0.840002
\(956\) 5466.38 0.184933
\(957\) 0 0
\(958\) 26789.8 0.903484
\(959\) 58170.7 1.95874
\(960\) 5829.33 0.195980
\(961\) 65386.8 2.19485
\(962\) −5958.30 −0.199692
\(963\) −1955.64 −0.0654408
\(964\) 14874.1 0.496954
\(965\) 2642.80 0.0881603
\(966\) 19980.6 0.665492
\(967\) 3912.18 0.130100 0.0650502 0.997882i \(-0.479279\pi\)
0.0650502 + 0.997882i \(0.479279\pi\)
\(968\) 0 0
\(969\) −26685.2 −0.884676
\(970\) −19141.2 −0.633596
\(971\) −165.480 −0.00546910 −0.00273455 0.999996i \(-0.500870\pi\)
−0.00273455 + 0.999996i \(0.500870\pi\)
\(972\) −1325.88 −0.0437527
\(973\) 29230.6 0.963094
\(974\) 9714.25 0.319574
\(975\) 4544.98 0.149288
\(976\) 3901.23 0.127946
\(977\) −5420.60 −0.177503 −0.0887515 0.996054i \(-0.528288\pi\)
−0.0887515 + 0.996054i \(0.528288\pi\)
\(978\) 18751.5 0.613094
\(979\) 0 0
\(980\) −23816.1 −0.776304
\(981\) 1306.58 0.0425237
\(982\) −9909.30 −0.322015
\(983\) −23990.9 −0.778423 −0.389212 0.921148i \(-0.627253\pi\)
−0.389212 + 0.921148i \(0.627253\pi\)
\(984\) 8631.20 0.279627
\(985\) −1103.00 −0.0356798
\(986\) −14940.2 −0.482548
\(987\) −28579.4 −0.921674
\(988\) −13791.3 −0.444087
\(989\) −29779.6 −0.957470
\(990\) 0 0
\(991\) −49090.3 −1.57357 −0.786784 0.617228i \(-0.788256\pi\)
−0.786784 + 0.617228i \(0.788256\pi\)
\(992\) −57604.1 −1.84368
\(993\) 468.754 0.0149803
\(994\) −47699.5 −1.52207
\(995\) −14133.1 −0.450300
\(996\) −18413.9 −0.585810
\(997\) 15827.9 0.502784 0.251392 0.967885i \(-0.419112\pi\)
0.251392 + 0.967885i \(0.419112\pi\)
\(998\) 11307.2 0.358639
\(999\) −3241.17 −0.102649
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 363.4.a.q.1.2 4
3.2 odd 2 1089.4.a.bf.1.3 4
11.10 odd 2 363.4.a.s.1.3 yes 4
33.32 even 2 1089.4.a.ba.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.q.1.2 4 1.1 even 1 trivial
363.4.a.s.1.3 yes 4 11.10 odd 2
1089.4.a.ba.1.2 4 33.32 even 2
1089.4.a.bf.1.3 4 3.2 odd 2