Properties

Label 525.4.d.h.274.4
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,4,Mod(274,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.274");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{65})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 33x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.4
Root \(4.53113i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.h.274.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.53113i q^{2} -3.00000i q^{3} -12.5311 q^{4} +13.5934 q^{6} -7.00000i q^{7} -20.5311i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+4.53113i q^{2} -3.00000i q^{3} -12.5311 q^{4} +13.5934 q^{6} -7.00000i q^{7} -20.5311i q^{8} -9.00000 q^{9} -19.0623 q^{11} +37.5934i q^{12} +2.93774i q^{13} +31.7179 q^{14} -7.21984 q^{16} -6.49806i q^{17} -40.7802i q^{18} +5.43580 q^{19} -21.0000 q^{21} -86.3735i q^{22} -49.3774i q^{23} -61.5934 q^{24} -13.3113 q^{26} +27.0000i q^{27} +87.7179i q^{28} +291.494 q^{29} +244.307 q^{31} -196.963i q^{32} +57.1868i q^{33} +29.4436 q^{34} +112.780 q^{36} -193.121i q^{37} +24.6303i q^{38} +8.81323 q^{39} +315.113 q^{41} -95.1537i q^{42} +300.996i q^{43} +238.872 q^{44} +223.735 q^{46} +86.5058i q^{47} +21.6595i q^{48} -49.0000 q^{49} -19.4942 q^{51} -36.8132i q^{52} -509.677i q^{53} -122.340 q^{54} -143.718 q^{56} -16.3074i q^{57} +1320.80i q^{58} +83.3852 q^{59} -5.25291 q^{61} +1106.99i q^{62} +63.0000i q^{63} +834.706 q^{64} -259.121 q^{66} +205.992i q^{67} +81.4281i q^{68} -148.132 q^{69} +1004.31 q^{71} +184.780i q^{72} +1007.29i q^{73} +875.055 q^{74} -68.1168 q^{76} +133.436i q^{77} +39.9339i q^{78} +863.237 q^{79} +81.0000 q^{81} +1427.82i q^{82} -1334.72i q^{83} +263.154 q^{84} -1363.85 q^{86} -874.483i q^{87} +391.370i q^{88} -326.249 q^{89} +20.5642 q^{91} +618.755i q^{92} -732.922i q^{93} -391.969 q^{94} -590.889 q^{96} +1526.77i q^{97} -222.025i q^{98} +171.560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 34 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 34 q^{4} + 6 q^{6} - 36 q^{9} - 44 q^{11} + 14 q^{14} - 174 q^{16} - 204 q^{19} - 84 q^{21} - 198 q^{24} + 108 q^{26} + 392 q^{29} + 300 q^{31} + 924 q^{34} + 306 q^{36} + 132 q^{39} - 352 q^{41} + 504 q^{44} - 1040 q^{46} - 196 q^{49} + 696 q^{51} - 54 q^{54} - 462 q^{56} + 1688 q^{59} - 408 q^{61} + 1678 q^{64} - 456 q^{66} - 1560 q^{69} + 3340 q^{71} + 2436 q^{74} + 824 q^{76} + 1776 q^{79} + 324 q^{81} + 714 q^{84} - 2424 q^{86} - 1176 q^{89} + 308 q^{91} + 2560 q^{94} - 90 q^{96} + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.53113i 1.60200i 0.598667 + 0.800998i \(0.295697\pi\)
−0.598667 + 0.800998i \(0.704303\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −12.5311 −1.56639
\(5\) 0 0
\(6\) 13.5934 0.924913
\(7\) − 7.00000i − 0.377964i
\(8\) − 20.5311i − 0.907356i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −19.0623 −0.522499 −0.261249 0.965271i \(-0.584135\pi\)
−0.261249 + 0.965271i \(0.584135\pi\)
\(12\) 37.5934i 0.904356i
\(13\) 2.93774i 0.0626756i 0.999509 + 0.0313378i \(0.00997677\pi\)
−0.999509 + 0.0313378i \(0.990023\pi\)
\(14\) 31.7179 0.605498
\(15\) 0 0
\(16\) −7.21984 −0.112810
\(17\) − 6.49806i − 0.0927066i −0.998925 0.0463533i \(-0.985240\pi\)
0.998925 0.0463533i \(-0.0147600\pi\)
\(18\) − 40.7802i − 0.533999i
\(19\) 5.43580 0.0656347 0.0328173 0.999461i \(-0.489552\pi\)
0.0328173 + 0.999461i \(0.489552\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) − 86.3735i − 0.837041i
\(23\) − 49.3774i − 0.447648i −0.974630 0.223824i \(-0.928146\pi\)
0.974630 0.223824i \(-0.0718541\pi\)
\(24\) −61.5934 −0.523862
\(25\) 0 0
\(26\) −13.3113 −0.100406
\(27\) 27.0000i 0.192450i
\(28\) 87.7179i 0.592040i
\(29\) 291.494 1.86652 0.933261 0.359200i \(-0.116950\pi\)
0.933261 + 0.359200i \(0.116950\pi\)
\(30\) 0 0
\(31\) 244.307 1.41545 0.707724 0.706489i \(-0.249722\pi\)
0.707724 + 0.706489i \(0.249722\pi\)
\(32\) − 196.963i − 1.08808i
\(33\) 57.1868i 0.301665i
\(34\) 29.4436 0.148516
\(35\) 0 0
\(36\) 112.780 0.522130
\(37\) − 193.121i − 0.858077i −0.903286 0.429038i \(-0.858853\pi\)
0.903286 0.429038i \(-0.141147\pi\)
\(38\) 24.6303i 0.105147i
\(39\) 8.81323 0.0361858
\(40\) 0 0
\(41\) 315.113 1.20030 0.600151 0.799887i \(-0.295107\pi\)
0.600151 + 0.799887i \(0.295107\pi\)
\(42\) − 95.1537i − 0.349584i
\(43\) 300.996i 1.06748i 0.845650 + 0.533738i \(0.179213\pi\)
−0.845650 + 0.533738i \(0.820787\pi\)
\(44\) 238.872 0.818437
\(45\) 0 0
\(46\) 223.735 0.717130
\(47\) 86.5058i 0.268472i 0.990949 + 0.134236i \(0.0428580\pi\)
−0.990949 + 0.134236i \(0.957142\pi\)
\(48\) 21.6595i 0.0651309i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −19.4942 −0.0535242
\(52\) − 36.8132i − 0.0981745i
\(53\) − 509.677i − 1.32093i −0.750855 0.660467i \(-0.770358\pi\)
0.750855 0.660467i \(-0.229642\pi\)
\(54\) −122.340 −0.308304
\(55\) 0 0
\(56\) −143.718 −0.342948
\(57\) − 16.3074i − 0.0378942i
\(58\) 1320.80i 2.99016i
\(59\) 83.3852 0.183997 0.0919985 0.995759i \(-0.470674\pi\)
0.0919985 + 0.995759i \(0.470674\pi\)
\(60\) 0 0
\(61\) −5.25291 −0.0110257 −0.00551283 0.999985i \(-0.501755\pi\)
−0.00551283 + 0.999985i \(0.501755\pi\)
\(62\) 1106.99i 2.26754i
\(63\) 63.0000i 0.125988i
\(64\) 834.706 1.63029
\(65\) 0 0
\(66\) −259.121 −0.483266
\(67\) 205.992i 0.375611i 0.982206 + 0.187806i \(0.0601375\pi\)
−0.982206 + 0.187806i \(0.939862\pi\)
\(68\) 81.4281i 0.145215i
\(69\) −148.132 −0.258450
\(70\) 0 0
\(71\) 1004.31 1.67872 0.839362 0.543573i \(-0.182929\pi\)
0.839362 + 0.543573i \(0.182929\pi\)
\(72\) 184.780i 0.302452i
\(73\) 1007.29i 1.61499i 0.589876 + 0.807494i \(0.299177\pi\)
−0.589876 + 0.807494i \(0.700823\pi\)
\(74\) 875.055 1.37464
\(75\) 0 0
\(76\) −68.1168 −0.102810
\(77\) 133.436i 0.197486i
\(78\) 39.9339i 0.0579695i
\(79\) 863.237 1.22939 0.614695 0.788765i \(-0.289279\pi\)
0.614695 + 0.788765i \(0.289279\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1427.82i 1.92288i
\(83\) − 1334.72i − 1.76512i −0.470200 0.882560i \(-0.655818\pi\)
0.470200 0.882560i \(-0.344182\pi\)
\(84\) 263.154 0.341815
\(85\) 0 0
\(86\) −1363.85 −1.71009
\(87\) − 874.483i − 1.07764i
\(88\) 391.370i 0.474093i
\(89\) −326.249 −0.388565 −0.194283 0.980946i \(-0.562238\pi\)
−0.194283 + 0.980946i \(0.562238\pi\)
\(90\) 0 0
\(91\) 20.5642 0.0236892
\(92\) 618.755i 0.701192i
\(93\) − 732.922i − 0.817210i
\(94\) −391.969 −0.430091
\(95\) 0 0
\(96\) −590.889 −0.628202
\(97\) 1526.77i 1.59815i 0.601232 + 0.799075i \(0.294677\pi\)
−0.601232 + 0.799075i \(0.705323\pi\)
\(98\) − 222.025i − 0.228857i
\(99\) 171.560 0.174166
\(100\) 0 0
\(101\) 96.8716 0.0954365 0.0477182 0.998861i \(-0.484805\pi\)
0.0477182 + 0.998861i \(0.484805\pi\)
\(102\) − 88.3307i − 0.0857455i
\(103\) − 1321.99i − 1.26466i −0.774700 0.632329i \(-0.782099\pi\)
0.774700 0.632329i \(-0.217901\pi\)
\(104\) 60.3152 0.0568691
\(105\) 0 0
\(106\) 2309.41 2.11613
\(107\) 1745.71i 1.57724i 0.614883 + 0.788619i \(0.289203\pi\)
−0.614883 + 0.788619i \(0.710797\pi\)
\(108\) − 338.340i − 0.301452i
\(109\) −476.856 −0.419032 −0.209516 0.977805i \(-0.567189\pi\)
−0.209516 + 0.977805i \(0.567189\pi\)
\(110\) 0 0
\(111\) −579.362 −0.495411
\(112\) 50.5389i 0.0426382i
\(113\) − 1641.65i − 1.36666i −0.730108 0.683332i \(-0.760530\pi\)
0.730108 0.683332i \(-0.239470\pi\)
\(114\) 73.8910 0.0607064
\(115\) 0 0
\(116\) −3652.75 −2.92370
\(117\) − 26.4397i − 0.0208919i
\(118\) 377.829i 0.294763i
\(119\) −45.4864 −0.0350398
\(120\) 0 0
\(121\) −967.630 −0.726995
\(122\) − 23.8016i − 0.0176631i
\(123\) − 945.339i − 0.692994i
\(124\) −3061.45 −2.21715
\(125\) 0 0
\(126\) −285.461 −0.201833
\(127\) − 844.016i − 0.589719i −0.955541 0.294859i \(-0.904727\pi\)
0.955541 0.294859i \(-0.0952728\pi\)
\(128\) 2206.46i 1.52363i
\(129\) 902.988 0.616308
\(130\) 0 0
\(131\) −2796.20 −1.86493 −0.932463 0.361265i \(-0.882345\pi\)
−0.932463 + 0.361265i \(0.882345\pi\)
\(132\) − 716.615i − 0.472525i
\(133\) − 38.0506i − 0.0248076i
\(134\) −933.377 −0.601728
\(135\) 0 0
\(136\) −133.413 −0.0841179
\(137\) − 2057.13i − 1.28287i −0.767179 0.641433i \(-0.778340\pi\)
0.767179 0.641433i \(-0.221660\pi\)
\(138\) − 671.206i − 0.414035i
\(139\) 1745.12 1.06489 0.532444 0.846465i \(-0.321274\pi\)
0.532444 + 0.846465i \(0.321274\pi\)
\(140\) 0 0
\(141\) 259.517 0.155002
\(142\) 4550.65i 2.68931i
\(143\) − 56.0000i − 0.0327479i
\(144\) 64.9786 0.0376033
\(145\) 0 0
\(146\) −4564.15 −2.58720
\(147\) 147.000i 0.0824786i
\(148\) 2420.02i 1.34408i
\(149\) 1173.57 0.645254 0.322627 0.946526i \(-0.395434\pi\)
0.322627 + 0.946526i \(0.395434\pi\)
\(150\) 0 0
\(151\) 1540.07 0.829994 0.414997 0.909823i \(-0.363783\pi\)
0.414997 + 0.909823i \(0.363783\pi\)
\(152\) − 111.603i − 0.0595540i
\(153\) 58.4826i 0.0309022i
\(154\) −604.615 −0.316372
\(155\) 0 0
\(156\) −110.440 −0.0566811
\(157\) − 2544.53i − 1.29348i −0.762712 0.646738i \(-0.776133\pi\)
0.762712 0.646738i \(-0.223867\pi\)
\(158\) 3911.44i 1.96948i
\(159\) −1529.03 −0.762642
\(160\) 0 0
\(161\) −345.642 −0.169195
\(162\) 367.021i 0.178000i
\(163\) 594.708i 0.285774i 0.989739 + 0.142887i \(0.0456385\pi\)
−0.989739 + 0.142887i \(0.954361\pi\)
\(164\) −3948.72 −1.88014
\(165\) 0 0
\(166\) 6047.81 2.82772
\(167\) 928.498i 0.430236i 0.976588 + 0.215118i \(0.0690135\pi\)
−0.976588 + 0.215118i \(0.930986\pi\)
\(168\) 431.154i 0.198001i
\(169\) 2188.37 0.996072
\(170\) 0 0
\(171\) −48.9222 −0.0218782
\(172\) − 3771.82i − 1.67209i
\(173\) 315.642i 0.138716i 0.997592 + 0.0693578i \(0.0220950\pi\)
−0.997592 + 0.0693578i \(0.977905\pi\)
\(174\) 3962.39 1.72637
\(175\) 0 0
\(176\) 137.626 0.0589431
\(177\) − 250.156i − 0.106231i
\(178\) − 1478.28i − 0.622480i
\(179\) 1445.49 0.603581 0.301791 0.953374i \(-0.402416\pi\)
0.301791 + 0.953374i \(0.402416\pi\)
\(180\) 0 0
\(181\) −1843.81 −0.757180 −0.378590 0.925564i \(-0.623591\pi\)
−0.378590 + 0.925564i \(0.623591\pi\)
\(182\) 93.1790i 0.0379499i
\(183\) 15.7587i 0.00636567i
\(184\) −1013.77 −0.406176
\(185\) 0 0
\(186\) 3320.97 1.30917
\(187\) 123.868i 0.0484391i
\(188\) − 1084.02i − 0.420532i
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) −244.074 −0.0924637 −0.0462318 0.998931i \(-0.514721\pi\)
−0.0462318 + 0.998931i \(0.514721\pi\)
\(192\) − 2504.12i − 0.941246i
\(193\) 1733.03i 0.646355i 0.946338 + 0.323178i \(0.104751\pi\)
−0.946338 + 0.323178i \(0.895249\pi\)
\(194\) −6918.01 −2.56023
\(195\) 0 0
\(196\) 614.025 0.223770
\(197\) 358.230i 0.129557i 0.997900 + 0.0647787i \(0.0206342\pi\)
−0.997900 + 0.0647787i \(0.979366\pi\)
\(198\) 777.362i 0.279014i
\(199\) 3203.63 1.14120 0.570601 0.821227i \(-0.306710\pi\)
0.570601 + 0.821227i \(0.306710\pi\)
\(200\) 0 0
\(201\) 617.977 0.216859
\(202\) 438.938i 0.152889i
\(203\) − 2040.46i − 0.705479i
\(204\) 244.284 0.0838398
\(205\) 0 0
\(206\) 5990.12 2.02598
\(207\) 444.397i 0.149216i
\(208\) − 21.2100i − 0.00707044i
\(209\) −103.619 −0.0342940
\(210\) 0 0
\(211\) 4943.16 1.61280 0.806401 0.591369i \(-0.201412\pi\)
0.806401 + 0.591369i \(0.201412\pi\)
\(212\) 6386.83i 2.06910i
\(213\) − 3012.92i − 0.969211i
\(214\) −7910.05 −2.52673
\(215\) 0 0
\(216\) 554.340 0.174621
\(217\) − 1710.15i − 0.534989i
\(218\) − 2160.70i − 0.671288i
\(219\) 3021.86 0.932414
\(220\) 0 0
\(221\) 19.0896 0.00581044
\(222\) − 2625.16i − 0.793646i
\(223\) − 3160.15i − 0.948965i −0.880265 0.474482i \(-0.842635\pi\)
0.880265 0.474482i \(-0.157365\pi\)
\(224\) −1378.74 −0.411255
\(225\) 0 0
\(226\) 7438.51 2.18939
\(227\) − 3651.11i − 1.06755i −0.845628 0.533773i \(-0.820774\pi\)
0.845628 0.533773i \(-0.179226\pi\)
\(228\) 204.350i 0.0593571i
\(229\) 4083.70 1.17842 0.589210 0.807980i \(-0.299439\pi\)
0.589210 + 0.807980i \(0.299439\pi\)
\(230\) 0 0
\(231\) 400.307 0.114019
\(232\) − 5984.70i − 1.69360i
\(233\) 3682.51i 1.03540i 0.855561 + 0.517702i \(0.173212\pi\)
−0.855561 + 0.517702i \(0.826788\pi\)
\(234\) 119.802 0.0334687
\(235\) 0 0
\(236\) −1044.91 −0.288211
\(237\) − 2589.71i − 0.709789i
\(238\) − 206.105i − 0.0561336i
\(239\) −2658.78 −0.719591 −0.359796 0.933031i \(-0.617154\pi\)
−0.359796 + 0.933031i \(0.617154\pi\)
\(240\) 0 0
\(241\) −4820.39 −1.28842 −0.644209 0.764850i \(-0.722813\pi\)
−0.644209 + 0.764850i \(0.722813\pi\)
\(242\) − 4384.46i − 1.16464i
\(243\) − 243.000i − 0.0641500i
\(244\) 65.8249 0.0172705
\(245\) 0 0
\(246\) 4283.45 1.11017
\(247\) 15.9690i 0.00411369i
\(248\) − 5015.91i − 1.28432i
\(249\) −4004.17 −1.01909
\(250\) 0 0
\(251\) 1672.27 0.420530 0.210265 0.977644i \(-0.432567\pi\)
0.210265 + 0.977644i \(0.432567\pi\)
\(252\) − 789.461i − 0.197347i
\(253\) 941.245i 0.233896i
\(254\) 3824.34 0.944727
\(255\) 0 0
\(256\) −3320.09 −0.810569
\(257\) − 3697.74i − 0.897506i −0.893656 0.448753i \(-0.851868\pi\)
0.893656 0.448753i \(-0.148132\pi\)
\(258\) 4091.56i 0.987322i
\(259\) −1351.84 −0.324323
\(260\) 0 0
\(261\) −2623.45 −0.622174
\(262\) − 12670.0i − 2.98760i
\(263\) − 7319.00i − 1.71600i −0.513646 0.858002i \(-0.671706\pi\)
0.513646 0.858002i \(-0.328294\pi\)
\(264\) 1174.11 0.273717
\(265\) 0 0
\(266\) 172.412 0.0397416
\(267\) 978.747i 0.224338i
\(268\) − 2581.32i − 0.588354i
\(269\) 815.097 0.184749 0.0923743 0.995724i \(-0.470554\pi\)
0.0923743 + 0.995724i \(0.470554\pi\)
\(270\) 0 0
\(271\) −5106.02 −1.14453 −0.572267 0.820068i \(-0.693936\pi\)
−0.572267 + 0.820068i \(0.693936\pi\)
\(272\) 46.9150i 0.0104582i
\(273\) − 61.6926i − 0.0136769i
\(274\) 9321.13 2.05515
\(275\) 0 0
\(276\) 1856.26 0.404833
\(277\) 1398.72i 0.303398i 0.988427 + 0.151699i \(0.0484744\pi\)
−0.988427 + 0.151699i \(0.951526\pi\)
\(278\) 7907.39i 1.70595i
\(279\) −2198.77 −0.471816
\(280\) 0 0
\(281\) −7102.38 −1.50780 −0.753901 0.656988i \(-0.771830\pi\)
−0.753901 + 0.656988i \(0.771830\pi\)
\(282\) 1175.91i 0.248313i
\(283\) − 4465.18i − 0.937907i −0.883223 0.468953i \(-0.844631\pi\)
0.883223 0.468953i \(-0.155369\pi\)
\(284\) −12585.1 −2.62954
\(285\) 0 0
\(286\) 253.743 0.0524621
\(287\) − 2205.79i − 0.453671i
\(288\) 1772.67i 0.362692i
\(289\) 4870.78 0.991405
\(290\) 0 0
\(291\) 4580.32 0.922692
\(292\) − 12622.5i − 2.52970i
\(293\) − 7590.61i − 1.51348i −0.653718 0.756738i \(-0.726792\pi\)
0.653718 0.756738i \(-0.273208\pi\)
\(294\) −666.076 −0.132130
\(295\) 0 0
\(296\) −3964.98 −0.778581
\(297\) − 514.681i − 0.100555i
\(298\) 5317.61i 1.03369i
\(299\) 145.058 0.0280566
\(300\) 0 0
\(301\) 2106.97 0.403468
\(302\) 6978.26i 1.32965i
\(303\) − 290.615i − 0.0551003i
\(304\) −39.2456 −0.00740425
\(305\) 0 0
\(306\) −264.992 −0.0495052
\(307\) 9480.12i 1.76241i 0.472737 + 0.881203i \(0.343266\pi\)
−0.472737 + 0.881203i \(0.656734\pi\)
\(308\) − 1672.10i − 0.309340i
\(309\) −3965.98 −0.730151
\(310\) 0 0
\(311\) 7078.01 1.29054 0.645268 0.763956i \(-0.276746\pi\)
0.645268 + 0.763956i \(0.276746\pi\)
\(312\) − 180.945i − 0.0328334i
\(313\) − 5593.84i − 1.01017i −0.863070 0.505084i \(-0.831461\pi\)
0.863070 0.505084i \(-0.168539\pi\)
\(314\) 11529.6 2.07214
\(315\) 0 0
\(316\) −10817.3 −1.92571
\(317\) 3567.81i 0.632139i 0.948736 + 0.316070i \(0.102363\pi\)
−0.948736 + 0.316070i \(0.897637\pi\)
\(318\) − 6928.24i − 1.22175i
\(319\) −5556.54 −0.975255
\(320\) 0 0
\(321\) 5237.14 0.910618
\(322\) − 1566.15i − 0.271050i
\(323\) − 35.3222i − 0.00608477i
\(324\) −1015.02 −0.174043
\(325\) 0 0
\(326\) −2694.70 −0.457809
\(327\) 1430.57i 0.241928i
\(328\) − 6469.62i − 1.08910i
\(329\) 605.541 0.101473
\(330\) 0 0
\(331\) −4389.67 −0.728936 −0.364468 0.931216i \(-0.618749\pi\)
−0.364468 + 0.931216i \(0.618749\pi\)
\(332\) 16725.6i 2.76487i
\(333\) 1738.09i 0.286026i
\(334\) −4207.14 −0.689236
\(335\) 0 0
\(336\) 151.617 0.0246172
\(337\) 2348.83i 0.379671i 0.981816 + 0.189835i \(0.0607955\pi\)
−0.981816 + 0.189835i \(0.939205\pi\)
\(338\) 9915.79i 1.59570i
\(339\) −4924.94 −0.789044
\(340\) 0 0
\(341\) −4657.05 −0.739570
\(342\) − 221.673i − 0.0350488i
\(343\) 343.000i 0.0539949i
\(344\) 6179.79 0.968581
\(345\) 0 0
\(346\) −1430.21 −0.222222
\(347\) 558.436i 0.0863931i 0.999067 + 0.0431965i \(0.0137542\pi\)
−0.999067 + 0.0431965i \(0.986246\pi\)
\(348\) 10958.3i 1.68800i
\(349\) −3233.89 −0.496006 −0.248003 0.968759i \(-0.579774\pi\)
−0.248003 + 0.968759i \(0.579774\pi\)
\(350\) 0 0
\(351\) −79.3190 −0.0120619
\(352\) 3754.56i 0.568519i
\(353\) 7516.35i 1.13330i 0.823959 + 0.566650i \(0.191761\pi\)
−0.823959 + 0.566650i \(0.808239\pi\)
\(354\) 1133.49 0.170181
\(355\) 0 0
\(356\) 4088.27 0.608646
\(357\) 136.459i 0.0202302i
\(358\) 6549.70i 0.966935i
\(359\) 6577.76 0.967021 0.483511 0.875338i \(-0.339361\pi\)
0.483511 + 0.875338i \(0.339361\pi\)
\(360\) 0 0
\(361\) −6829.45 −0.995692
\(362\) − 8354.56i − 1.21300i
\(363\) 2902.89i 0.419731i
\(364\) −257.693 −0.0371065
\(365\) 0 0
\(366\) −71.4048 −0.0101978
\(367\) 8307.17i 1.18155i 0.806835 + 0.590777i \(0.201179\pi\)
−0.806835 + 0.590777i \(0.798821\pi\)
\(368\) 356.497i 0.0504992i
\(369\) −2836.02 −0.400101
\(370\) 0 0
\(371\) −3567.74 −0.499266
\(372\) 9184.34i 1.28007i
\(373\) 4551.09i 0.631760i 0.948799 + 0.315880i \(0.102300\pi\)
−0.948799 + 0.315880i \(0.897700\pi\)
\(374\) −561.261 −0.0775992
\(375\) 0 0
\(376\) 1776.06 0.243599
\(377\) 856.335i 0.116985i
\(378\) 856.383i 0.116528i
\(379\) 1788.29 0.242370 0.121185 0.992630i \(-0.461331\pi\)
0.121185 + 0.992630i \(0.461331\pi\)
\(380\) 0 0
\(381\) −2532.05 −0.340474
\(382\) − 1105.93i − 0.148126i
\(383\) 1358.47i 0.181240i 0.995886 + 0.0906199i \(0.0288848\pi\)
−0.995886 + 0.0906199i \(0.971115\pi\)
\(384\) 6619.37 0.879670
\(385\) 0 0
\(386\) −7852.60 −1.03546
\(387\) − 2708.97i − 0.355825i
\(388\) − 19132.2i − 2.50333i
\(389\) −9722.54 −1.26723 −0.633615 0.773649i \(-0.718430\pi\)
−0.633615 + 0.773649i \(0.718430\pi\)
\(390\) 0 0
\(391\) −320.858 −0.0414999
\(392\) 1006.03i 0.129622i
\(393\) 8388.61i 1.07672i
\(394\) −1623.19 −0.207551
\(395\) 0 0
\(396\) −2149.84 −0.272812
\(397\) − 4788.04i − 0.605302i −0.953101 0.302651i \(-0.902128\pi\)
0.953101 0.302651i \(-0.0978717\pi\)
\(398\) 14516.1i 1.82820i
\(399\) −114.152 −0.0143227
\(400\) 0 0
\(401\) 9681.41 1.20565 0.602826 0.797873i \(-0.294041\pi\)
0.602826 + 0.797873i \(0.294041\pi\)
\(402\) 2800.13i 0.347408i
\(403\) 717.712i 0.0887141i
\(404\) −1213.91 −0.149491
\(405\) 0 0
\(406\) 9245.58 1.13017
\(407\) 3681.32i 0.448344i
\(408\) 400.238i 0.0485655i
\(409\) −11113.1 −1.34353 −0.671767 0.740763i \(-0.734464\pi\)
−0.671767 + 0.740763i \(0.734464\pi\)
\(410\) 0 0
\(411\) −6171.40 −0.740663
\(412\) 16566.1i 1.98095i
\(413\) − 583.696i − 0.0695443i
\(414\) −2013.62 −0.239043
\(415\) 0 0
\(416\) 578.627 0.0681959
\(417\) − 5235.37i − 0.614814i
\(418\) − 469.510i − 0.0549389i
\(419\) 1230.09 0.143421 0.0717107 0.997425i \(-0.477154\pi\)
0.0717107 + 0.997425i \(0.477154\pi\)
\(420\) 0 0
\(421\) −12356.5 −1.43044 −0.715222 0.698897i \(-0.753674\pi\)
−0.715222 + 0.698897i \(0.753674\pi\)
\(422\) 22398.1i 2.58370i
\(423\) − 778.552i − 0.0894906i
\(424\) −10464.2 −1.19856
\(425\) 0 0
\(426\) 13651.9 1.55267
\(427\) 36.7703i 0.00416731i
\(428\) − 21875.7i − 2.47057i
\(429\) −168.000 −0.0189070
\(430\) 0 0
\(431\) −7375.27 −0.824256 −0.412128 0.911126i \(-0.635214\pi\)
−0.412128 + 0.911126i \(0.635214\pi\)
\(432\) − 194.936i − 0.0217103i
\(433\) 690.067i 0.0765877i 0.999267 + 0.0382939i \(0.0121923\pi\)
−0.999267 + 0.0382939i \(0.987808\pi\)
\(434\) 7748.92 0.857051
\(435\) 0 0
\(436\) 5975.55 0.656369
\(437\) − 268.406i − 0.0293812i
\(438\) 13692.5i 1.49372i
\(439\) −8408.79 −0.914191 −0.457095 0.889418i \(-0.651110\pi\)
−0.457095 + 0.889418i \(0.651110\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 86.4976i 0.00930830i
\(443\) − 6568.55i − 0.704473i −0.935911 0.352236i \(-0.885421\pi\)
0.935911 0.352236i \(-0.114579\pi\)
\(444\) 7260.06 0.776007
\(445\) 0 0
\(446\) 14319.0 1.52024
\(447\) − 3520.72i − 0.372537i
\(448\) − 5842.94i − 0.616190i
\(449\) −2954.55 −0.310543 −0.155271 0.987872i \(-0.549625\pi\)
−0.155271 + 0.987872i \(0.549625\pi\)
\(450\) 0 0
\(451\) −6006.76 −0.627156
\(452\) 20571.7i 2.14073i
\(453\) − 4620.21i − 0.479197i
\(454\) 16543.7 1.71020
\(455\) 0 0
\(456\) −334.810 −0.0343835
\(457\) − 8144.84i − 0.833697i −0.908976 0.416849i \(-0.863135\pi\)
0.908976 0.416849i \(-0.136865\pi\)
\(458\) 18503.8i 1.88782i
\(459\) 175.448 0.0178414
\(460\) 0 0
\(461\) 2495.26 0.252095 0.126048 0.992024i \(-0.459771\pi\)
0.126048 + 0.992024i \(0.459771\pi\)
\(462\) 1813.84i 0.182657i
\(463\) 5755.66i 0.577728i 0.957370 + 0.288864i \(0.0932776\pi\)
−0.957370 + 0.288864i \(0.906722\pi\)
\(464\) −2104.54 −0.210562
\(465\) 0 0
\(466\) −16685.9 −1.65871
\(467\) − 4143.73i − 0.410598i −0.978699 0.205299i \(-0.934183\pi\)
0.978699 0.205299i \(-0.0658166\pi\)
\(468\) 331.319i 0.0327248i
\(469\) 1441.95 0.141968
\(470\) 0 0
\(471\) −7633.60 −0.746789
\(472\) − 1711.99i − 0.166951i
\(473\) − 5737.67i − 0.557755i
\(474\) 11734.3 1.13708
\(475\) 0 0
\(476\) 569.996 0.0548860
\(477\) 4587.09i 0.440312i
\(478\) − 12047.3i − 1.15278i
\(479\) 6765.96 0.645396 0.322698 0.946502i \(-0.395410\pi\)
0.322698 + 0.946502i \(0.395410\pi\)
\(480\) 0 0
\(481\) 567.339 0.0537805
\(482\) − 21841.8i − 2.06404i
\(483\) 1036.93i 0.0976848i
\(484\) 12125.5 1.13876
\(485\) 0 0
\(486\) 1101.06 0.102768
\(487\) 6360.42i 0.591824i 0.955215 + 0.295912i \(0.0956235\pi\)
−0.955215 + 0.295912i \(0.904377\pi\)
\(488\) 107.848i 0.0100042i
\(489\) 1784.12 0.164992
\(490\) 0 0
\(491\) −7072.54 −0.650060 −0.325030 0.945704i \(-0.605374\pi\)
−0.325030 + 0.945704i \(0.605374\pi\)
\(492\) 11846.2i 1.08550i
\(493\) − 1894.15i − 0.173039i
\(494\) −72.3576 −0.00659012
\(495\) 0 0
\(496\) −1763.86 −0.159677
\(497\) − 7030.15i − 0.634498i
\(498\) − 18143.4i − 1.63258i
\(499\) 18473.9 1.65732 0.828661 0.559751i \(-0.189103\pi\)
0.828661 + 0.559751i \(0.189103\pi\)
\(500\) 0 0
\(501\) 2785.49 0.248397
\(502\) 7577.28i 0.673687i
\(503\) − 11379.2i − 1.00869i −0.863501 0.504347i \(-0.831733\pi\)
0.863501 0.504347i \(-0.168267\pi\)
\(504\) 1293.46 0.114316
\(505\) 0 0
\(506\) −4264.90 −0.374700
\(507\) − 6565.11i − 0.575082i
\(508\) 10576.5i 0.923730i
\(509\) −6064.48 −0.528101 −0.264051 0.964509i \(-0.585059\pi\)
−0.264051 + 0.964509i \(0.585059\pi\)
\(510\) 0 0
\(511\) 7051.02 0.610408
\(512\) 2607.89i 0.225105i
\(513\) 146.767i 0.0126314i
\(514\) 16755.0 1.43780
\(515\) 0 0
\(516\) −11315.5 −0.965379
\(517\) − 1649.00i − 0.140276i
\(518\) − 6125.38i − 0.519563i
\(519\) 946.926 0.0800875
\(520\) 0 0
\(521\) 2682.88 0.225603 0.112801 0.993618i \(-0.464018\pi\)
0.112801 + 0.993618i \(0.464018\pi\)
\(522\) − 11887.2i − 0.996720i
\(523\) 4309.02i 0.360268i 0.983642 + 0.180134i \(0.0576532\pi\)
−0.983642 + 0.180134i \(0.942347\pi\)
\(524\) 35039.6 2.92120
\(525\) 0 0
\(526\) 33163.3 2.74903
\(527\) − 1587.52i − 0.131221i
\(528\) − 412.879i − 0.0340308i
\(529\) 9728.87 0.799611
\(530\) 0 0
\(531\) −750.467 −0.0613323
\(532\) 476.817i 0.0388584i
\(533\) 925.720i 0.0752296i
\(534\) −4434.83 −0.359389
\(535\) 0 0
\(536\) 4229.25 0.340813
\(537\) − 4336.47i − 0.348478i
\(538\) 3693.31i 0.295966i
\(539\) 934.051 0.0746427
\(540\) 0 0
\(541\) 4081.47 0.324355 0.162178 0.986762i \(-0.448148\pi\)
0.162178 + 0.986762i \(0.448148\pi\)
\(542\) − 23136.0i − 1.83354i
\(543\) 5531.44i 0.437158i
\(544\) −1279.88 −0.100872
\(545\) 0 0
\(546\) 279.537 0.0219104
\(547\) 8844.82i 0.691366i 0.938351 + 0.345683i \(0.112353\pi\)
−0.938351 + 0.345683i \(0.887647\pi\)
\(548\) 25778.2i 2.00947i
\(549\) 47.2762 0.00367522
\(550\) 0 0
\(551\) 1584.51 0.122509
\(552\) 3041.32i 0.234506i
\(553\) − 6042.66i − 0.464666i
\(554\) −6337.80 −0.486042
\(555\) 0 0
\(556\) −21868.4 −1.66803
\(557\) − 11144.7i − 0.847787i −0.905712 0.423894i \(-0.860663\pi\)
0.905712 0.423894i \(-0.139337\pi\)
\(558\) − 9962.90i − 0.755848i
\(559\) −884.249 −0.0669047
\(560\) 0 0
\(561\) 371.603 0.0279663
\(562\) − 32181.8i − 2.41549i
\(563\) 21857.5i 1.63621i 0.575071 + 0.818104i \(0.304975\pi\)
−0.575071 + 0.818104i \(0.695025\pi\)
\(564\) −3252.05 −0.242794
\(565\) 0 0
\(566\) 20232.3 1.50252
\(567\) − 567.000i − 0.0419961i
\(568\) − 20619.6i − 1.52320i
\(569\) −23496.4 −1.73115 −0.865573 0.500783i \(-0.833046\pi\)
−0.865573 + 0.500783i \(0.833046\pi\)
\(570\) 0 0
\(571\) 11067.8 0.811164 0.405582 0.914059i \(-0.367069\pi\)
0.405582 + 0.914059i \(0.367069\pi\)
\(572\) 701.743i 0.0512961i
\(573\) 732.222i 0.0533839i
\(574\) 9994.72 0.726780
\(575\) 0 0
\(576\) −7512.36 −0.543429
\(577\) 20482.9i 1.47784i 0.673791 + 0.738922i \(0.264665\pi\)
−0.673791 + 0.738922i \(0.735335\pi\)
\(578\) 22070.1i 1.58823i
\(579\) 5199.10 0.373173
\(580\) 0 0
\(581\) −9343.07 −0.667153
\(582\) 20754.0i 1.47815i
\(583\) 9715.60i 0.690187i
\(584\) 20680.8 1.46537
\(585\) 0 0
\(586\) 34394.1 2.42458
\(587\) − 23444.3i − 1.64847i −0.566248 0.824235i \(-0.691606\pi\)
0.566248 0.824235i \(-0.308394\pi\)
\(588\) − 1842.08i − 0.129194i
\(589\) 1328.01 0.0929025
\(590\) 0 0
\(591\) 1074.69 0.0748000
\(592\) 1394.30i 0.0967996i
\(593\) − 4404.69i − 0.305024i −0.988302 0.152512i \(-0.951264\pi\)
0.988302 0.152512i \(-0.0487362\pi\)
\(594\) 2332.09 0.161089
\(595\) 0 0
\(596\) −14706.2 −1.01072
\(597\) − 9610.89i − 0.658874i
\(598\) 657.277i 0.0449466i
\(599\) −3327.05 −0.226945 −0.113472 0.993541i \(-0.536197\pi\)
−0.113472 + 0.993541i \(0.536197\pi\)
\(600\) 0 0
\(601\) −14244.8 −0.966818 −0.483409 0.875395i \(-0.660602\pi\)
−0.483409 + 0.875395i \(0.660602\pi\)
\(602\) 9546.97i 0.646354i
\(603\) − 1853.93i − 0.125204i
\(604\) −19298.8 −1.30010
\(605\) 0 0
\(606\) 1316.81 0.0882704
\(607\) 11446.5i 0.765402i 0.923872 + 0.382701i \(0.125006\pi\)
−0.923872 + 0.382701i \(0.874994\pi\)
\(608\) − 1070.65i − 0.0714156i
\(609\) −6121.38 −0.407308
\(610\) 0 0
\(611\) −254.132 −0.0168266
\(612\) − 732.852i − 0.0484049i
\(613\) 19436.4i 1.28063i 0.768111 + 0.640316i \(0.221197\pi\)
−0.768111 + 0.640316i \(0.778803\pi\)
\(614\) −42955.6 −2.82337
\(615\) 0 0
\(616\) 2739.59 0.179190
\(617\) − 20530.1i − 1.33956i −0.742558 0.669781i \(-0.766388\pi\)
0.742558 0.669781i \(-0.233612\pi\)
\(618\) − 17970.4i − 1.16970i
\(619\) −5833.35 −0.378776 −0.189388 0.981902i \(-0.560650\pi\)
−0.189388 + 0.981902i \(0.560650\pi\)
\(620\) 0 0
\(621\) 1333.19 0.0861499
\(622\) 32071.4i 2.06744i
\(623\) 2283.74i 0.146864i
\(624\) −63.6301 −0.00408212
\(625\) 0 0
\(626\) 25346.4 1.61829
\(627\) 310.856i 0.0197997i
\(628\) 31885.9i 2.02609i
\(629\) −1254.91 −0.0795493
\(630\) 0 0
\(631\) 24776.6 1.56314 0.781568 0.623820i \(-0.214420\pi\)
0.781568 + 0.623820i \(0.214420\pi\)
\(632\) − 17723.2i − 1.11549i
\(633\) − 14829.5i − 0.931152i
\(634\) −16166.2 −1.01268
\(635\) 0 0
\(636\) 19160.5 1.19460
\(637\) − 143.949i − 0.00895366i
\(638\) − 25177.4i − 1.56235i
\(639\) −9038.77 −0.559574
\(640\) 0 0
\(641\) 27219.4 1.67723 0.838613 0.544728i \(-0.183367\pi\)
0.838613 + 0.544728i \(0.183367\pi\)
\(642\) 23730.1i 1.45881i
\(643\) − 7091.79i − 0.434950i −0.976066 0.217475i \(-0.930218\pi\)
0.976066 0.217475i \(-0.0697821\pi\)
\(644\) 4331.28 0.265026
\(645\) 0 0
\(646\) 160.049 0.00974777
\(647\) 27773.0i 1.68758i 0.536670 + 0.843792i \(0.319682\pi\)
−0.536670 + 0.843792i \(0.680318\pi\)
\(648\) − 1663.02i − 0.100817i
\(649\) −1589.51 −0.0961382
\(650\) 0 0
\(651\) −5130.46 −0.308876
\(652\) − 7452.37i − 0.447634i
\(653\) − 21380.4i − 1.28129i −0.767839 0.640643i \(-0.778668\pi\)
0.767839 0.640643i \(-0.221332\pi\)
\(654\) −6482.09 −0.387568
\(655\) 0 0
\(656\) −2275.06 −0.135406
\(657\) − 9065.59i − 0.538329i
\(658\) 2743.78i 0.162559i
\(659\) 17232.3 1.01863 0.509315 0.860580i \(-0.329899\pi\)
0.509315 + 0.860580i \(0.329899\pi\)
\(660\) 0 0
\(661\) 26577.7 1.56392 0.781962 0.623326i \(-0.214219\pi\)
0.781962 + 0.623326i \(0.214219\pi\)
\(662\) − 19890.1i − 1.16775i
\(663\) − 57.2689i − 0.00335466i
\(664\) −27403.4 −1.60159
\(665\) 0 0
\(666\) −7875.49 −0.458212
\(667\) − 14393.2i − 0.835544i
\(668\) − 11635.1i − 0.673917i
\(669\) −9480.44 −0.547885
\(670\) 0 0
\(671\) 100.132 0.00576090
\(672\) 4136.22i 0.237438i
\(673\) 31695.2i 1.81540i 0.419624 + 0.907698i \(0.362162\pi\)
−0.419624 + 0.907698i \(0.637838\pi\)
\(674\) −10642.9 −0.608231
\(675\) 0 0
\(676\) −27422.7 −1.56024
\(677\) 20440.3i 1.16039i 0.814477 + 0.580195i \(0.197024\pi\)
−0.814477 + 0.580195i \(0.802976\pi\)
\(678\) − 22315.5i − 1.26405i
\(679\) 10687.4 0.604044
\(680\) 0 0
\(681\) −10953.3 −0.616348
\(682\) − 21101.7i − 1.18479i
\(683\) 22896.9i 1.28276i 0.767223 + 0.641381i \(0.221638\pi\)
−0.767223 + 0.641381i \(0.778362\pi\)
\(684\) 613.051 0.0342699
\(685\) 0 0
\(686\) −1554.18 −0.0864997
\(687\) − 12251.1i − 0.680361i
\(688\) − 2173.14i − 0.120422i
\(689\) 1497.30 0.0827904
\(690\) 0 0
\(691\) −23764.0 −1.30829 −0.654143 0.756371i \(-0.726971\pi\)
−0.654143 + 0.756371i \(0.726971\pi\)
\(692\) − 3955.35i − 0.217283i
\(693\) − 1200.92i − 0.0658287i
\(694\) −2530.34 −0.138401
\(695\) 0 0
\(696\) −17954.1 −0.977800
\(697\) − 2047.62i − 0.111276i
\(698\) − 14653.2i − 0.794600i
\(699\) 11047.5 0.597791
\(700\) 0 0
\(701\) 26259.5 1.41485 0.707423 0.706791i \(-0.249858\pi\)
0.707423 + 0.706791i \(0.249858\pi\)
\(702\) − 359.405i − 0.0193232i
\(703\) − 1049.77i − 0.0563196i
\(704\) −15911.4 −0.851822
\(705\) 0 0
\(706\) −34057.6 −1.81554
\(707\) − 678.101i − 0.0360716i
\(708\) 3134.73i 0.166399i
\(709\) −12783.0 −0.677116 −0.338558 0.940945i \(-0.609939\pi\)
−0.338558 + 0.940945i \(0.609939\pi\)
\(710\) 0 0
\(711\) −7769.14 −0.409797
\(712\) 6698.26i 0.352567i
\(713\) − 12063.3i − 0.633623i
\(714\) −618.315 −0.0324087
\(715\) 0 0
\(716\) −18113.6 −0.945444
\(717\) 7976.35i 0.415456i
\(718\) 29804.7i 1.54916i
\(719\) 27609.0 1.43205 0.716025 0.698075i \(-0.245960\pi\)
0.716025 + 0.698075i \(0.245960\pi\)
\(720\) 0 0
\(721\) −9253.95 −0.477996
\(722\) − 30945.1i − 1.59509i
\(723\) 14461.2i 0.743868i
\(724\) 23105.1 1.18604
\(725\) 0 0
\(726\) −13153.4 −0.672407
\(727\) 31306.2i 1.59709i 0.601937 + 0.798544i \(0.294396\pi\)
−0.601937 + 0.798544i \(0.705604\pi\)
\(728\) − 422.206i − 0.0214945i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 1955.89 0.0989621
\(732\) − 197.475i − 0.00997113i
\(733\) 15765.8i 0.794441i 0.917723 + 0.397220i \(0.130025\pi\)
−0.917723 + 0.397220i \(0.869975\pi\)
\(734\) −37640.8 −1.89285
\(735\) 0 0
\(736\) −9725.53 −0.487076
\(737\) − 3926.68i − 0.196256i
\(738\) − 12850.4i − 0.640959i
\(739\) −3966.51 −0.197443 −0.0987216 0.995115i \(-0.531475\pi\)
−0.0987216 + 0.995115i \(0.531475\pi\)
\(740\) 0 0
\(741\) 47.9070 0.00237504
\(742\) − 16165.9i − 0.799823i
\(743\) − 8224.50i − 0.406094i −0.979169 0.203047i \(-0.934916\pi\)
0.979169 0.203047i \(-0.0650844\pi\)
\(744\) −15047.7 −0.741500
\(745\) 0 0
\(746\) −20621.6 −1.01208
\(747\) 12012.5i 0.588373i
\(748\) − 1552.20i − 0.0758745i
\(749\) 12220.0 0.596140
\(750\) 0 0
\(751\) 18929.2 0.919754 0.459877 0.887983i \(-0.347894\pi\)
0.459877 + 0.887983i \(0.347894\pi\)
\(752\) − 624.558i − 0.0302863i
\(753\) − 5016.82i − 0.242793i
\(754\) −3880.16 −0.187410
\(755\) 0 0
\(756\) −2368.38 −0.113938
\(757\) − 34906.8i − 1.67597i −0.545695 0.837984i \(-0.683734\pi\)
0.545695 0.837984i \(-0.316266\pi\)
\(758\) 8102.96i 0.388276i
\(759\) 2823.74 0.135040
\(760\) 0 0
\(761\) −13683.4 −0.651803 −0.325902 0.945404i \(-0.605668\pi\)
−0.325902 + 0.945404i \(0.605668\pi\)
\(762\) − 11473.0i − 0.545438i
\(763\) 3337.99i 0.158379i
\(764\) 3058.52 0.144834
\(765\) 0 0
\(766\) −6155.42 −0.290345
\(767\) 244.964i 0.0115321i
\(768\) 9960.28i 0.467982i
\(769\) −41837.3 −1.96189 −0.980943 0.194294i \(-0.937758\pi\)
−0.980943 + 0.194294i \(0.937758\pi\)
\(770\) 0 0
\(771\) −11093.2 −0.518175
\(772\) − 21716.9i − 1.01245i
\(773\) − 19640.0i − 0.913843i −0.889507 0.456921i \(-0.848952\pi\)
0.889507 0.456921i \(-0.151048\pi\)
\(774\) 12274.7 0.570031
\(775\) 0 0
\(776\) 31346.4 1.45009
\(777\) 4055.53i 0.187248i
\(778\) − 44054.1i − 2.03010i
\(779\) 1712.89 0.0787814
\(780\) 0 0
\(781\) −19144.4 −0.877131
\(782\) − 1453.85i − 0.0664827i
\(783\) 7870.34i 0.359212i
\(784\) 353.772 0.0161157
\(785\) 0 0
\(786\) −38009.9 −1.72489
\(787\) 24935.3i 1.12941i 0.825293 + 0.564705i \(0.191010\pi\)
−0.825293 + 0.564705i \(0.808990\pi\)
\(788\) − 4489.02i − 0.202938i
\(789\) −21957.0 −0.990736
\(790\) 0 0
\(791\) −11491.5 −0.516551
\(792\) − 3522.33i − 0.158031i
\(793\) − 15.4317i 0 0.000691041i
\(794\) 21695.2 0.969692
\(795\) 0 0
\(796\) −40145.1 −1.78757
\(797\) − 1168.33i − 0.0519251i −0.999663 0.0259625i \(-0.991735\pi\)
0.999663 0.0259625i \(-0.00826506\pi\)
\(798\) − 517.237i − 0.0229448i
\(799\) 562.120 0.0248891
\(800\) 0 0
\(801\) 2936.24 0.129522
\(802\) 43867.7i 1.93145i
\(803\) − 19201.2i − 0.843829i
\(804\) −7743.95 −0.339686
\(805\) 0 0
\(806\) −3252.05 −0.142120
\(807\) − 2445.29i − 0.106665i
\(808\) − 1988.88i − 0.0865949i
\(809\) 35175.7 1.52869 0.764345 0.644807i \(-0.223062\pi\)
0.764345 + 0.644807i \(0.223062\pi\)
\(810\) 0 0
\(811\) −15256.5 −0.660577 −0.330288 0.943880i \(-0.607146\pi\)
−0.330288 + 0.943880i \(0.607146\pi\)
\(812\) 25569.3i 1.10506i
\(813\) 15318.1i 0.660797i
\(814\) −16680.5 −0.718245
\(815\) 0 0
\(816\) 140.745 0.00603806
\(817\) 1636.16i 0.0700635i
\(818\) − 50354.7i − 2.15233i
\(819\) −185.078 −0.00789639
\(820\) 0 0
\(821\) −15971.9 −0.678956 −0.339478 0.940614i \(-0.610250\pi\)
−0.339478 + 0.940614i \(0.610250\pi\)
\(822\) − 27963.4i − 1.18654i
\(823\) − 2312.41i − 0.0979409i −0.998800 0.0489705i \(-0.984406\pi\)
0.998800 0.0489705i \(-0.0155940\pi\)
\(824\) −27142.0 −1.14750
\(825\) 0 0
\(826\) 2644.80 0.111410
\(827\) 10422.4i 0.438238i 0.975698 + 0.219119i \(0.0703183\pi\)
−0.975698 + 0.219119i \(0.929682\pi\)
\(828\) − 5568.79i − 0.233731i
\(829\) 13213.4 0.553584 0.276792 0.960930i \(-0.410729\pi\)
0.276792 + 0.960930i \(0.410729\pi\)
\(830\) 0 0
\(831\) 4196.17 0.175167
\(832\) 2452.15i 0.102179i
\(833\) 318.405i 0.0132438i
\(834\) 23722.2 0.984929
\(835\) 0 0
\(836\) 1298.46 0.0537179
\(837\) 6596.30i 0.272403i
\(838\) 5573.67i 0.229761i
\(839\) 10119.6 0.416409 0.208205 0.978085i \(-0.433238\pi\)
0.208205 + 0.978085i \(0.433238\pi\)
\(840\) 0 0
\(841\) 60579.9 2.48390
\(842\) − 55988.7i − 2.29157i
\(843\) 21307.1i 0.870530i
\(844\) −61943.4 −2.52628
\(845\) 0 0
\(846\) 3527.72 0.143364
\(847\) 6773.41i 0.274778i
\(848\) 3679.79i 0.149015i
\(849\) −13395.5 −0.541501
\(850\) 0 0
\(851\) −9535.80 −0.384116
\(852\) 37755.3i 1.51816i
\(853\) − 35378.1i − 1.42007i −0.704165 0.710037i \(-0.748678\pi\)
0.704165 0.710037i \(-0.251322\pi\)
\(854\) −166.611 −0.00667602
\(855\) 0 0
\(856\) 35841.4 1.43112
\(857\) 6697.57i 0.266960i 0.991052 + 0.133480i \(0.0426152\pi\)
−0.991052 + 0.133480i \(0.957385\pi\)
\(858\) − 761.230i − 0.0302890i
\(859\) 24298.4 0.965135 0.482568 0.875859i \(-0.339704\pi\)
0.482568 + 0.875859i \(0.339704\pi\)
\(860\) 0 0
\(861\) −6617.37 −0.261927
\(862\) − 33418.3i − 1.32045i
\(863\) − 24942.9i − 0.983853i −0.870637 0.491926i \(-0.836293\pi\)
0.870637 0.491926i \(-0.163707\pi\)
\(864\) 5318.00 0.209401
\(865\) 0 0
\(866\) −3126.78 −0.122693
\(867\) − 14612.3i − 0.572388i
\(868\) 21430.1i 0.838002i
\(869\) −16455.3 −0.642355
\(870\) 0 0
\(871\) −605.152 −0.0235417
\(872\) 9790.39i 0.380212i
\(873\) − 13741.0i − 0.532716i
\(874\) 1216.18 0.0470686
\(875\) 0 0
\(876\) −37867.4 −1.46052
\(877\) − 16276.6i − 0.626705i −0.949637 0.313353i \(-0.898548\pi\)
0.949637 0.313353i \(-0.101452\pi\)
\(878\) − 38101.3i − 1.46453i
\(879\) −22771.8 −0.873806
\(880\) 0 0
\(881\) 26636.5 1.01862 0.509311 0.860582i \(-0.329900\pi\)
0.509311 + 0.860582i \(0.329900\pi\)
\(882\) 1998.23i 0.0762855i
\(883\) 21788.3i 0.830392i 0.909732 + 0.415196i \(0.136287\pi\)
−0.909732 + 0.415196i \(0.863713\pi\)
\(884\) −239.215 −0.00910142
\(885\) 0 0
\(886\) 29763.0 1.12856
\(887\) − 26813.2i − 1.01499i −0.861654 0.507496i \(-0.830571\pi\)
0.861654 0.507496i \(-0.169429\pi\)
\(888\) 11895.0i 0.449514i
\(889\) −5908.11 −0.222893
\(890\) 0 0
\(891\) −1544.04 −0.0580554
\(892\) 39600.2i 1.48645i
\(893\) 470.229i 0.0176211i
\(894\) 15952.8 0.596803
\(895\) 0 0
\(896\) 15445.2 0.575879
\(897\) − 435.174i − 0.0161985i
\(898\) − 13387.4i − 0.497488i
\(899\) 71214.2 2.64196
\(900\) 0 0
\(901\) −3311.91 −0.122459
\(902\) − 27217.4i − 1.00470i
\(903\) − 6320.92i − 0.232942i
\(904\) −33704.8 −1.24005
\(905\) 0 0
\(906\) 20934.8 0.767672
\(907\) 15543.0i 0.569014i 0.958674 + 0.284507i \(0.0918300\pi\)
−0.958674 + 0.284507i \(0.908170\pi\)
\(908\) 45752.6i 1.67219i
\(909\) −871.844 −0.0318122
\(910\) 0 0
\(911\) 48711.1 1.77154 0.885768 0.464128i \(-0.153632\pi\)
0.885768 + 0.464128i \(0.153632\pi\)
\(912\) 117.737i 0.00427485i
\(913\) 25442.8i 0.922273i
\(914\) 36905.3 1.33558
\(915\) 0 0
\(916\) −51173.3 −1.84587
\(917\) 19573.4i 0.704876i
\(918\) 794.976i 0.0285818i
\(919\) −1030.47 −0.0369883 −0.0184941 0.999829i \(-0.505887\pi\)
−0.0184941 + 0.999829i \(0.505887\pi\)
\(920\) 0 0
\(921\) 28440.4 1.01753
\(922\) 11306.3i 0.403855i
\(923\) 2950.40i 0.105215i
\(924\) −5016.30 −0.178598
\(925\) 0 0
\(926\) −26079.6 −0.925518
\(927\) 11897.9i 0.421553i
\(928\) − 57413.6i − 2.03092i
\(929\) −879.756 −0.0310698 −0.0155349 0.999879i \(-0.504945\pi\)
−0.0155349 + 0.999879i \(0.504945\pi\)
\(930\) 0 0
\(931\) −266.354 −0.00937638
\(932\) − 46146.0i − 1.62185i
\(933\) − 21234.0i − 0.745092i
\(934\) 18775.8 0.657776
\(935\) 0 0
\(936\) −542.836 −0.0189564
\(937\) − 18668.1i − 0.650864i −0.945565 0.325432i \(-0.894490\pi\)
0.945565 0.325432i \(-0.105510\pi\)
\(938\) 6533.64i 0.227432i
\(939\) −16781.5 −0.583221
\(940\) 0 0
\(941\) 29613.4 1.02590 0.512948 0.858420i \(-0.328553\pi\)
0.512948 + 0.858420i \(0.328553\pi\)
\(942\) − 34588.8i − 1.19635i
\(943\) − 15559.5i − 0.537313i
\(944\) −602.028 −0.0207567
\(945\) 0 0
\(946\) 25998.1 0.893521
\(947\) 20738.9i 0.711640i 0.934554 + 0.355820i \(0.115798\pi\)
−0.934554 + 0.355820i \(0.884202\pi\)
\(948\) 32452.0i 1.11181i
\(949\) −2959.15 −0.101220
\(950\) 0 0
\(951\) 10703.4 0.364966
\(952\) 933.888i 0.0317936i
\(953\) 45776.5i 1.55598i 0.628279 + 0.777988i \(0.283760\pi\)
−0.628279 + 0.777988i \(0.716240\pi\)
\(954\) −20784.7 −0.705377
\(955\) 0 0
\(956\) 33317.5 1.12716
\(957\) 16669.6i 0.563064i
\(958\) 30657.4i 1.03392i
\(959\) −14399.9 −0.484878
\(960\) 0 0
\(961\) 29895.1 1.00349
\(962\) 2570.68i 0.0861561i
\(963\) − 15711.4i − 0.525746i
\(964\) 60404.9 2.01817
\(965\) 0 0
\(966\) −4698.44 −0.156491
\(967\) 34461.0i 1.14601i 0.819552 + 0.573005i \(0.194222\pi\)
−0.819552 + 0.573005i \(0.805778\pi\)
\(968\) 19866.5i 0.659643i
\(969\) −105.967 −0.00351304
\(970\) 0 0
\(971\) −22762.8 −0.752309 −0.376154 0.926557i \(-0.622754\pi\)
−0.376154 + 0.926557i \(0.622754\pi\)
\(972\) 3045.06i 0.100484i
\(973\) − 12215.9i − 0.402490i
\(974\) −28819.9 −0.948099
\(975\) 0 0
\(976\) 37.9251 0.00124381
\(977\) 4809.57i 0.157494i 0.996895 + 0.0787470i \(0.0250919\pi\)
−0.996895 + 0.0787470i \(0.974908\pi\)
\(978\) 8084.10i 0.264316i
\(979\) 6219.04 0.203025
\(980\) 0 0
\(981\) 4291.70 0.139677
\(982\) − 32046.6i − 1.04139i
\(983\) 27591.6i 0.895256i 0.894220 + 0.447628i \(0.147731\pi\)
−0.894220 + 0.447628i \(0.852269\pi\)
\(984\) −19408.9 −0.628793
\(985\) 0 0
\(986\) 8582.63 0.277207
\(987\) − 1816.62i − 0.0585853i
\(988\) − 200.109i − 0.00644365i
\(989\) 14862.4 0.477854
\(990\) 0 0
\(991\) −22263.4 −0.713643 −0.356822 0.934173i \(-0.616140\pi\)
−0.356822 + 0.934173i \(0.616140\pi\)
\(992\) − 48119.5i − 1.54012i
\(993\) 13169.0i 0.420851i
\(994\) 31854.5 1.01646
\(995\) 0 0
\(996\) 50176.8 1.59630
\(997\) 30378.2i 0.964983i 0.875901 + 0.482491i \(0.160268\pi\)
−0.875901 + 0.482491i \(0.839732\pi\)
\(998\) 83707.4i 2.65502i
\(999\) 5214.26 0.165137
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 525.4.d.h.274.4 4
5.2 odd 4 525.4.a.k.1.1 2
5.3 odd 4 105.4.a.f.1.2 2
5.4 even 2 inner 525.4.d.h.274.1 4
15.2 even 4 1575.4.a.w.1.2 2
15.8 even 4 315.4.a.i.1.1 2
20.3 even 4 1680.4.a.bg.1.2 2
35.13 even 4 735.4.a.p.1.2 2
105.83 odd 4 2205.4.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.2 2 5.3 odd 4
315.4.a.i.1.1 2 15.8 even 4
525.4.a.k.1.1 2 5.2 odd 4
525.4.d.h.274.1 4 5.4 even 2 inner
525.4.d.h.274.4 4 1.1 even 1 trivial
735.4.a.p.1.2 2 35.13 even 4
1575.4.a.w.1.2 2 15.2 even 4
1680.4.a.bg.1.2 2 20.3 even 4
2205.4.a.z.1.1 2 105.83 odd 4