Properties

Label 490.4.c.e.99.5
Level $490$
Weight $4$
Character 490.99
Analytic conductor $28.911$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,4,Mod(99,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("490.99");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 490.c (of order \(2\), degree \(1\), 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: \(28.9109359028\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 185x^{10} + 12748x^{8} + 405460x^{6} + 5908496x^{4} + 33016000x^{2} + 60840000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.5
Root \(5.49484i\) of defining polynomial
Character \(\chi\) \(=\) 490.99
Dual form 490.4.c.e.99.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +4.49484i q^{3} -4.00000 q^{4} +(3.34202 + 10.6692i) q^{5} +8.98967 q^{6} +8.00000i q^{8} +6.79645 q^{9} +O(q^{10})\) \(q-2.00000i q^{2} +4.49484i q^{3} -4.00000 q^{4} +(3.34202 + 10.6692i) q^{5} +8.98967 q^{6} +8.00000i q^{8} +6.79645 q^{9} +(21.3383 - 6.68404i) q^{10} +13.9329 q^{11} -17.9793i q^{12} -78.0096i q^{13} +(-47.9561 + 15.0218i) q^{15} +16.0000 q^{16} -105.132i q^{17} -13.5929i q^{18} +152.300 q^{19} +(-13.3681 - 42.6766i) q^{20} -27.8658i q^{22} +108.116i q^{23} -35.9587 q^{24} +(-102.662 + 71.3130i) q^{25} -156.019 q^{26} +151.909i q^{27} +109.323 q^{29} +(30.0436 + 95.9122i) q^{30} +197.196 q^{31} -32.0000i q^{32} +62.6261i q^{33} -210.265 q^{34} -27.1858 q^{36} +254.290i q^{37} -304.601i q^{38} +350.641 q^{39} +(-85.3533 + 26.7361i) q^{40} -193.905 q^{41} +41.5329i q^{43} -55.7316 q^{44} +(22.7139 + 72.5124i) q^{45} +216.232 q^{46} +109.886i q^{47} +71.9174i q^{48} +(142.626 + 205.324i) q^{50} +472.553 q^{51} +312.039i q^{52} -11.0054i q^{53} +303.819 q^{54} +(46.5640 + 148.652i) q^{55} +684.566i q^{57} -218.645i q^{58} +192.141 q^{59} +(191.824 - 60.0873i) q^{60} +34.9035 q^{61} -394.392i q^{62} -64.0000 q^{64} +(832.297 - 260.710i) q^{65} +125.252 q^{66} +374.830i q^{67} +420.530i q^{68} -485.963 q^{69} -575.722 q^{71} +54.3716i q^{72} +522.161i q^{73} +508.580 q^{74} +(-320.540 - 461.448i) q^{75} -609.202 q^{76} -701.281i q^{78} +456.182 q^{79} +(53.4723 + 170.707i) q^{80} -499.304 q^{81} +387.810i q^{82} -773.057i q^{83} +(1121.67 - 351.355i) q^{85} +83.0657 q^{86} +491.387i q^{87} +111.463i q^{88} +1329.07 q^{89} +(145.025 - 45.4277i) q^{90} -432.463i q^{92} +886.364i q^{93} +219.772 q^{94} +(508.991 + 1624.92i) q^{95} +143.835 q^{96} +1038.18i q^{97} +94.6942 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 48 q^{4} - 8 q^{5} - 28 q^{6} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 48 q^{4} - 8 q^{5} - 28 q^{6} - 62 q^{9} + 12 q^{10} + 62 q^{11} + 86 q^{15} + 192 q^{16} + 186 q^{19} + 32 q^{20} + 112 q^{24} - 126 q^{25} - 236 q^{26} - 338 q^{29} - 28 q^{30} - 652 q^{31} + 272 q^{34} + 248 q^{36} + 868 q^{39} - 48 q^{40} - 396 q^{41} - 248 q^{44} + 664 q^{45} - 376 q^{46} + 160 q^{50} + 1448 q^{51} + 1540 q^{54} - 298 q^{55} + 1336 q^{59} - 344 q^{60} - 314 q^{61} - 768 q^{64} - 1862 q^{65} - 1600 q^{66} - 90 q^{69} + 2216 q^{71} + 1012 q^{74} - 4550 q^{75} - 744 q^{76} + 1772 q^{79} - 128 q^{80} - 1228 q^{81} + 2282 q^{85} - 396 q^{86} + 6094 q^{89} - 100 q^{90} + 3604 q^{94} - 1166 q^{95} - 448 q^{96} - 8546 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) 4.49484i 0.865032i 0.901626 + 0.432516i \(0.142374\pi\)
−0.901626 + 0.432516i \(0.857626\pi\)
\(4\) −4.00000 −0.500000
\(5\) 3.34202 + 10.6692i 0.298919 + 0.954278i
\(6\) 8.98967 0.611670
\(7\) 0 0
\(8\) 8.00000i 0.353553i
\(9\) 6.79645 0.251720
\(10\) 21.3383 6.68404i 0.674777 0.211368i
\(11\) 13.9329 0.381902 0.190951 0.981600i \(-0.438843\pi\)
0.190951 + 0.981600i \(0.438843\pi\)
\(12\) 17.9793i 0.432516i
\(13\) 78.0096i 1.66431i −0.554546 0.832153i \(-0.687108\pi\)
0.554546 0.832153i \(-0.312892\pi\)
\(14\) 0 0
\(15\) −47.9561 + 15.0218i −0.825481 + 0.258575i
\(16\) 16.0000 0.250000
\(17\) 105.132i 1.49990i −0.661492 0.749952i \(-0.730077\pi\)
0.661492 0.749952i \(-0.269923\pi\)
\(18\) 13.5929i 0.177993i
\(19\) 152.300 1.83895 0.919477 0.393144i \(-0.128613\pi\)
0.919477 + 0.393144i \(0.128613\pi\)
\(20\) −13.3681 42.6766i −0.149460 0.477139i
\(21\) 0 0
\(22\) 27.8658i 0.270046i
\(23\) 108.116i 0.980161i 0.871677 + 0.490081i \(0.163033\pi\)
−0.871677 + 0.490081i \(0.836967\pi\)
\(24\) −35.9587 −0.305835
\(25\) −102.662 + 71.3130i −0.821295 + 0.570504i
\(26\) −156.019 −1.17684
\(27\) 151.909i 1.08278i
\(28\) 0 0
\(29\) 109.323 0.700023 0.350012 0.936745i \(-0.386178\pi\)
0.350012 + 0.936745i \(0.386178\pi\)
\(30\) 30.0436 + 95.9122i 0.182840 + 0.583703i
\(31\) 197.196 1.14250 0.571249 0.820777i \(-0.306459\pi\)
0.571249 + 0.820777i \(0.306459\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 62.6261i 0.330358i
\(34\) −210.265 −1.06059
\(35\) 0 0
\(36\) −27.1858 −0.125860
\(37\) 254.290i 1.12986i 0.825137 + 0.564932i \(0.191098\pi\)
−0.825137 + 0.564932i \(0.808902\pi\)
\(38\) 304.601i 1.30034i
\(39\) 350.641 1.43968
\(40\) −85.3533 + 26.7361i −0.337388 + 0.105684i
\(41\) −193.905 −0.738606 −0.369303 0.929309i \(-0.620404\pi\)
−0.369303 + 0.929309i \(0.620404\pi\)
\(42\) 0 0
\(43\) 41.5329i 0.147295i 0.997284 + 0.0736477i \(0.0234640\pi\)
−0.997284 + 0.0736477i \(0.976536\pi\)
\(44\) −55.7316 −0.190951
\(45\) 22.7139 + 72.5124i 0.0752440 + 0.240211i
\(46\) 216.232 0.693079
\(47\) 109.886i 0.341032i 0.985355 + 0.170516i \(0.0545434\pi\)
−0.985355 + 0.170516i \(0.945457\pi\)
\(48\) 71.9174i 0.216258i
\(49\) 0 0
\(50\) 142.626 + 205.324i 0.403407 + 0.580743i
\(51\) 472.553 1.29746
\(52\) 312.039i 0.832153i
\(53\) 11.0054i 0.0285228i −0.999898 0.0142614i \(-0.995460\pi\)
0.999898 0.0142614i \(-0.00453969\pi\)
\(54\) 303.819 0.765639
\(55\) 46.5640 + 148.652i 0.114158 + 0.364441i
\(56\) 0 0
\(57\) 684.566i 1.59075i
\(58\) 218.645i 0.494991i
\(59\) 192.141 0.423978 0.211989 0.977272i \(-0.432006\pi\)
0.211989 + 0.977272i \(0.432006\pi\)
\(60\) 191.824 60.0873i 0.412741 0.129287i
\(61\) 34.9035 0.0732611 0.0366306 0.999329i \(-0.488338\pi\)
0.0366306 + 0.999329i \(0.488338\pi\)
\(62\) 394.392i 0.807868i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 832.297 260.710i 1.58821 0.497493i
\(66\) 125.252 0.233598
\(67\) 374.830i 0.683474i 0.939796 + 0.341737i \(0.111015\pi\)
−0.939796 + 0.341737i \(0.888985\pi\)
\(68\) 420.530i 0.749952i
\(69\) −485.963 −0.847870
\(70\) 0 0
\(71\) −575.722 −0.962333 −0.481167 0.876629i \(-0.659787\pi\)
−0.481167 + 0.876629i \(0.659787\pi\)
\(72\) 54.3716i 0.0889966i
\(73\) 522.161i 0.837182i 0.908175 + 0.418591i \(0.137476\pi\)
−0.908175 + 0.418591i \(0.862524\pi\)
\(74\) 508.580 0.798935
\(75\) −320.540 461.448i −0.493504 0.710446i
\(76\) −609.202 −0.919477
\(77\) 0 0
\(78\) 701.281i 1.01801i
\(79\) 456.182 0.649676 0.324838 0.945770i \(-0.394690\pi\)
0.324838 + 0.945770i \(0.394690\pi\)
\(80\) 53.4723 + 170.707i 0.0747298 + 0.238570i
\(81\) −499.304 −0.684917
\(82\) 387.810i 0.522273i
\(83\) 773.057i 1.02234i −0.859480 0.511169i \(-0.829213\pi\)
0.859480 0.511169i \(-0.170787\pi\)
\(84\) 0 0
\(85\) 1121.67 351.355i 1.43133 0.448350i
\(86\) 83.0657 0.104154
\(87\) 491.387i 0.605542i
\(88\) 111.463i 0.135023i
\(89\) 1329.07 1.58294 0.791470 0.611208i \(-0.209316\pi\)
0.791470 + 0.611208i \(0.209316\pi\)
\(90\) 145.025 45.4277i 0.169855 0.0532056i
\(91\) 0 0
\(92\) 432.463i 0.490081i
\(93\) 886.364i 0.988297i
\(94\) 219.772 0.241146
\(95\) 508.991 + 1624.92i 0.549699 + 1.75487i
\(96\) 143.835 0.152917
\(97\) 1038.18i 1.08671i 0.839503 + 0.543356i \(0.182847\pi\)
−0.839503 + 0.543356i \(0.817153\pi\)
\(98\) 0 0
\(99\) 94.6942 0.0961325
\(100\) 410.647 285.252i 0.410647 0.285252i
\(101\) 489.559 0.482307 0.241153 0.970487i \(-0.422474\pi\)
0.241153 + 0.970487i \(0.422474\pi\)
\(102\) 945.107i 0.917446i
\(103\) 1185.15i 1.13375i −0.823802 0.566877i \(-0.808151\pi\)
0.823802 0.566877i \(-0.191849\pi\)
\(104\) 624.077 0.588421
\(105\) 0 0
\(106\) −22.0108 −0.0201686
\(107\) 1192.49i 1.07741i −0.842496 0.538703i \(-0.818915\pi\)
0.842496 0.538703i \(-0.181085\pi\)
\(108\) 607.638i 0.541389i
\(109\) 285.431 0.250820 0.125410 0.992105i \(-0.459975\pi\)
0.125410 + 0.992105i \(0.459975\pi\)
\(110\) 297.304 93.1280i 0.257699 0.0807218i
\(111\) −1142.99 −0.977369
\(112\) 0 0
\(113\) 350.019i 0.291389i 0.989330 + 0.145695i \(0.0465417\pi\)
−0.989330 + 0.145695i \(0.953458\pi\)
\(114\) 1369.13 1.12483
\(115\) −1153.50 + 361.325i −0.935347 + 0.292989i
\(116\) −437.290 −0.350012
\(117\) 530.188i 0.418940i
\(118\) 384.283i 0.299798i
\(119\) 0 0
\(120\) −120.175 383.649i −0.0914199 0.291852i
\(121\) −1136.87 −0.854151
\(122\) 69.8069i 0.0518034i
\(123\) 871.570i 0.638917i
\(124\) −788.784 −0.571249
\(125\) −1103.95 856.986i −0.789921 0.613209i
\(126\) 0 0
\(127\) 1112.45i 0.777274i −0.921391 0.388637i \(-0.872946\pi\)
0.921391 0.388637i \(-0.127054\pi\)
\(128\) 128.000i 0.0883883i
\(129\) −186.683 −0.127415
\(130\) −521.419 1664.59i −0.351781 1.12304i
\(131\) −605.974 −0.404154 −0.202077 0.979370i \(-0.564769\pi\)
−0.202077 + 0.979370i \(0.564769\pi\)
\(132\) 250.504i 0.165179i
\(133\) 0 0
\(134\) 749.659 0.483289
\(135\) −1620.75 + 507.684i −1.03327 + 0.323663i
\(136\) 841.060 0.530296
\(137\) 403.891i 0.251874i 0.992038 + 0.125937i \(0.0401938\pi\)
−0.992038 + 0.125937i \(0.959806\pi\)
\(138\) 971.926i 0.599535i
\(139\) 525.532 0.320683 0.160342 0.987062i \(-0.448740\pi\)
0.160342 + 0.987062i \(0.448740\pi\)
\(140\) 0 0
\(141\) −493.918 −0.295003
\(142\) 1151.44i 0.680472i
\(143\) 1086.90i 0.635602i
\(144\) 108.743 0.0629301
\(145\) 365.358 + 1166.38i 0.209250 + 0.668017i
\(146\) 1044.32 0.591977
\(147\) 0 0
\(148\) 1017.16i 0.564932i
\(149\) 2034.87 1.11881 0.559407 0.828893i \(-0.311029\pi\)
0.559407 + 0.828893i \(0.311029\pi\)
\(150\) −922.896 + 641.081i −0.502361 + 0.348960i
\(151\) 1748.70 0.942430 0.471215 0.882018i \(-0.343816\pi\)
0.471215 + 0.882018i \(0.343816\pi\)
\(152\) 1218.40i 0.650168i
\(153\) 714.527i 0.377556i
\(154\) 0 0
\(155\) 659.033 + 2103.91i 0.341515 + 1.09026i
\(156\) −1402.56 −0.719839
\(157\) 448.507i 0.227992i 0.993481 + 0.113996i \(0.0363651\pi\)
−0.993481 + 0.113996i \(0.963635\pi\)
\(158\) 912.363i 0.459391i
\(159\) 49.4674 0.0246731
\(160\) 341.413 106.945i 0.168694 0.0528419i
\(161\) 0 0
\(162\) 998.608i 0.484309i
\(163\) 911.681i 0.438088i 0.975715 + 0.219044i \(0.0702939\pi\)
−0.975715 + 0.219044i \(0.929706\pi\)
\(164\) 775.619 0.369303
\(165\) −668.167 + 209.297i −0.315253 + 0.0987502i
\(166\) −1546.11 −0.722902
\(167\) 1131.12i 0.524125i 0.965051 + 0.262063i \(0.0844027\pi\)
−0.965051 + 0.262063i \(0.915597\pi\)
\(168\) 0 0
\(169\) −3888.51 −1.76992
\(170\) −702.709 2243.35i −0.317031 1.01210i
\(171\) 1035.10 0.462902
\(172\) 166.131i 0.0736477i
\(173\) 2138.58i 0.939844i −0.882708 0.469922i \(-0.844282\pi\)
0.882708 0.469922i \(-0.155718\pi\)
\(174\) 982.773 0.428183
\(175\) 0 0
\(176\) 222.926 0.0954756
\(177\) 863.644i 0.366754i
\(178\) 2658.15i 1.11931i
\(179\) −1739.07 −0.726170 −0.363085 0.931756i \(-0.618276\pi\)
−0.363085 + 0.931756i \(0.618276\pi\)
\(180\) −90.8554 290.049i −0.0376220 0.120106i
\(181\) 2304.17 0.946231 0.473115 0.881001i \(-0.343129\pi\)
0.473115 + 0.881001i \(0.343129\pi\)
\(182\) 0 0
\(183\) 156.885i 0.0633732i
\(184\) −864.927 −0.346539
\(185\) −2713.06 + 849.841i −1.07821 + 0.337738i
\(186\) 1772.73 0.698832
\(187\) 1464.80i 0.572817i
\(188\) 439.543i 0.170516i
\(189\) 0 0
\(190\) 3249.84 1017.98i 1.24088 0.388696i
\(191\) −3611.07 −1.36800 −0.684000 0.729482i \(-0.739761\pi\)
−0.684000 + 0.729482i \(0.739761\pi\)
\(192\) 287.670i 0.108129i
\(193\) 4340.90i 1.61899i 0.587128 + 0.809494i \(0.300259\pi\)
−0.587128 + 0.809494i \(0.699741\pi\)
\(194\) 2076.36 0.768421
\(195\) 1171.85 + 3741.04i 0.430347 + 1.37385i
\(196\) 0 0
\(197\) 380.943i 0.137772i 0.997625 + 0.0688859i \(0.0219444\pi\)
−0.997625 + 0.0688859i \(0.978056\pi\)
\(198\) 189.388i 0.0679760i
\(199\) 3476.70 1.23848 0.619238 0.785203i \(-0.287441\pi\)
0.619238 + 0.785203i \(0.287441\pi\)
\(200\) −570.504 821.295i −0.201704 0.290372i
\(201\) −1684.80 −0.591226
\(202\) 979.119i 0.341042i
\(203\) 0 0
\(204\) −1890.21 −0.648732
\(205\) −648.033 2068.80i −0.220783 0.704836i
\(206\) −2370.31 −0.801685
\(207\) 734.804i 0.246726i
\(208\) 1248.15i 0.416077i
\(209\) 2121.99 0.702301
\(210\) 0 0
\(211\) 1693.38 0.552497 0.276248 0.961086i \(-0.410909\pi\)
0.276248 + 0.961086i \(0.410909\pi\)
\(212\) 44.0216i 0.0142614i
\(213\) 2587.78i 0.832449i
\(214\) −2384.98 −0.761841
\(215\) −443.121 + 138.804i −0.140561 + 0.0440294i
\(216\) −1215.28 −0.382820
\(217\) 0 0
\(218\) 570.862i 0.177356i
\(219\) −2347.03 −0.724189
\(220\) −186.256 594.609i −0.0570790 0.182221i
\(221\) −8201.35 −2.49630
\(222\) 2285.98i 0.691104i
\(223\) 260.918i 0.0783514i 0.999232 + 0.0391757i \(0.0124732\pi\)
−0.999232 + 0.0391757i \(0.987527\pi\)
\(224\) 0 0
\(225\) −697.736 + 484.675i −0.206737 + 0.143607i
\(226\) 700.038 0.206043
\(227\) 279.657i 0.0817688i 0.999164 + 0.0408844i \(0.0130175\pi\)
−0.999164 + 0.0408844i \(0.986982\pi\)
\(228\) 2738.26i 0.795377i
\(229\) −1722.39 −0.497024 −0.248512 0.968629i \(-0.579942\pi\)
−0.248512 + 0.968629i \(0.579942\pi\)
\(230\) 722.650 + 2307.01i 0.207175 + 0.661390i
\(231\) 0 0
\(232\) 874.580i 0.247496i
\(233\) 6659.62i 1.87247i 0.351370 + 0.936237i \(0.385716\pi\)
−0.351370 + 0.936237i \(0.614284\pi\)
\(234\) −1060.38 −0.296235
\(235\) −1172.39 + 367.240i −0.325439 + 0.101941i
\(236\) −768.566 −0.211989
\(237\) 2050.46i 0.561991i
\(238\) 0 0
\(239\) −1018.63 −0.275688 −0.137844 0.990454i \(-0.544017\pi\)
−0.137844 + 0.990454i \(0.544017\pi\)
\(240\) −767.298 + 240.349i −0.206370 + 0.0646436i
\(241\) −5853.84 −1.56464 −0.782322 0.622875i \(-0.785965\pi\)
−0.782322 + 0.622875i \(0.785965\pi\)
\(242\) 2273.75i 0.603976i
\(243\) 1857.27i 0.490303i
\(244\) −139.614 −0.0366306
\(245\) 0 0
\(246\) −1743.14 −0.451783
\(247\) 11880.9i 3.06058i
\(248\) 1577.57i 0.403934i
\(249\) 3474.77 0.884355
\(250\) −1713.97 + 2207.90i −0.433604 + 0.558558i
\(251\) 1101.55 0.277008 0.138504 0.990362i \(-0.455771\pi\)
0.138504 + 0.990362i \(0.455771\pi\)
\(252\) 0 0
\(253\) 1506.37i 0.374326i
\(254\) −2224.90 −0.549616
\(255\) 1579.28 + 5041.74i 0.387837 + 1.23814i
\(256\) 256.000 0.0625000
\(257\) 1442.47i 0.350111i −0.984559 0.175056i \(-0.943989\pi\)
0.984559 0.175056i \(-0.0560105\pi\)
\(258\) 373.367i 0.0900961i
\(259\) 0 0
\(260\) −3329.19 + 1042.84i −0.794106 + 0.248747i
\(261\) 743.005 0.176210
\(262\) 1211.95i 0.285780i
\(263\) 5430.39i 1.27320i −0.771194 0.636601i \(-0.780340\pi\)
0.771194 0.636601i \(-0.219660\pi\)
\(264\) −501.009 −0.116799
\(265\) 117.418 36.7802i 0.0272187 0.00852600i
\(266\) 0 0
\(267\) 5973.97i 1.36929i
\(268\) 1499.32i 0.341737i
\(269\) −5011.89 −1.13599 −0.567993 0.823033i \(-0.692280\pi\)
−0.567993 + 0.823033i \(0.692280\pi\)
\(270\) 1015.37 + 3241.49i 0.228864 + 0.730633i
\(271\) 3317.40 0.743607 0.371803 0.928311i \(-0.378740\pi\)
0.371803 + 0.928311i \(0.378740\pi\)
\(272\) 1682.12i 0.374976i
\(273\) 0 0
\(274\) 807.783 0.178102
\(275\) −1430.38 + 993.597i −0.313654 + 0.217877i
\(276\) 1943.85 0.423935
\(277\) 4160.35i 0.902424i −0.892417 0.451212i \(-0.850992\pi\)
0.892417 0.451212i \(-0.149008\pi\)
\(278\) 1051.06i 0.226757i
\(279\) 1340.23 0.287590
\(280\) 0 0
\(281\) −2423.12 −0.514418 −0.257209 0.966356i \(-0.582803\pi\)
−0.257209 + 0.966356i \(0.582803\pi\)
\(282\) 987.837i 0.208599i
\(283\) 8370.14i 1.75814i −0.476695 0.879069i \(-0.658165\pi\)
0.476695 0.879069i \(-0.341835\pi\)
\(284\) 2302.89 0.481167
\(285\) −7303.74 + 2287.83i −1.51802 + 0.475507i
\(286\) −2173.80 −0.449439
\(287\) 0 0
\(288\) 217.486i 0.0444983i
\(289\) −6139.84 −1.24971
\(290\) 2332.76 730.716i 0.472360 0.147962i
\(291\) −4666.44 −0.940040
\(292\) 2088.64i 0.418591i
\(293\) 5292.21i 1.05520i −0.849492 0.527601i \(-0.823092\pi\)
0.849492 0.527601i \(-0.176908\pi\)
\(294\) 0 0
\(295\) 642.140 + 2049.99i 0.126735 + 0.404593i
\(296\) −2034.32 −0.399467
\(297\) 2116.54i 0.413515i
\(298\) 4069.75i 0.791121i
\(299\) 8434.08 1.63129
\(300\) 1282.16 + 1845.79i 0.246752 + 0.355223i
\(301\) 0 0
\(302\) 3497.39i 0.666398i
\(303\) 2200.49i 0.417211i
\(304\) 2436.81 0.459738
\(305\) 116.648 + 372.390i 0.0218992 + 0.0699115i
\(306\) −1429.05 −0.266973
\(307\) 3852.07i 0.716121i 0.933698 + 0.358061i \(0.116562\pi\)
−0.933698 + 0.358061i \(0.883438\pi\)
\(308\) 0 0
\(309\) 5327.07 0.980733
\(310\) 4207.83 1318.07i 0.770931 0.241487i
\(311\) 1092.48 0.199192 0.0995962 0.995028i \(-0.468245\pi\)
0.0995962 + 0.995028i \(0.468245\pi\)
\(312\) 2805.12i 0.509003i
\(313\) 1324.90i 0.239257i 0.992819 + 0.119629i \(0.0381704\pi\)
−0.992819 + 0.119629i \(0.961830\pi\)
\(314\) 897.014 0.161215
\(315\) 0 0
\(316\) −1824.73 −0.324838
\(317\) 4985.47i 0.883319i 0.897183 + 0.441659i \(0.145610\pi\)
−0.897183 + 0.441659i \(0.854390\pi\)
\(318\) 98.9349i 0.0174465i
\(319\) 1523.18 0.267341
\(320\) −213.889 682.826i −0.0373649 0.119285i
\(321\) 5360.05 0.931990
\(322\) 0 0
\(323\) 16011.7i 2.75825i
\(324\) 1997.22 0.342458
\(325\) 5563.10 + 8008.61i 0.949494 + 1.36689i
\(326\) 1823.36 0.309775
\(327\) 1282.97i 0.216967i
\(328\) 1551.24i 0.261137i
\(329\) 0 0
\(330\) 418.595 + 1336.33i 0.0698269 + 0.222918i
\(331\) 10041.0 1.66738 0.833691 0.552231i \(-0.186223\pi\)
0.833691 + 0.552231i \(0.186223\pi\)
\(332\) 3092.23i 0.511169i
\(333\) 1728.27i 0.284410i
\(334\) 2262.25 0.370613
\(335\) −3999.12 + 1252.69i −0.652224 + 0.204303i
\(336\) 0 0
\(337\) 7227.40i 1.16825i −0.811662 0.584127i \(-0.801437\pi\)
0.811662 0.584127i \(-0.198563\pi\)
\(338\) 7777.01i 1.25152i
\(339\) −1573.28 −0.252061
\(340\) −4486.70 + 1405.42i −0.715663 + 0.224175i
\(341\) 2747.51 0.436323
\(342\) 2070.20i 0.327321i
\(343\) 0 0
\(344\) −332.263 −0.0520768
\(345\) −1624.10 5184.81i −0.253445 0.809104i
\(346\) −4277.16 −0.664570
\(347\) 3425.67i 0.529970i 0.964252 + 0.264985i \(0.0853671\pi\)
−0.964252 + 0.264985i \(0.914633\pi\)
\(348\) 1965.55i 0.302771i
\(349\) −7092.03 −1.08776 −0.543879 0.839164i \(-0.683045\pi\)
−0.543879 + 0.839164i \(0.683045\pi\)
\(350\) 0 0
\(351\) 11850.4 1.80207
\(352\) 445.853i 0.0675114i
\(353\) 11964.8i 1.80402i −0.431714 0.902011i \(-0.642091\pi\)
0.431714 0.902011i \(-0.357909\pi\)
\(354\) 1727.29 0.259334
\(355\) −1924.07 6142.47i −0.287660 0.918334i
\(356\) −5316.30 −0.791470
\(357\) 0 0
\(358\) 3478.14i 0.513479i
\(359\) 1556.01 0.228755 0.114377 0.993437i \(-0.463513\pi\)
0.114377 + 0.993437i \(0.463513\pi\)
\(360\) −580.099 + 181.711i −0.0849275 + 0.0266028i
\(361\) 16336.4 2.38175
\(362\) 4608.34i 0.669086i
\(363\) 5110.06i 0.738867i
\(364\) 0 0
\(365\) −5571.02 + 1745.07i −0.798905 + 0.250250i
\(366\) 313.771 0.0448116
\(367\) 2418.56i 0.343999i −0.985097 0.172000i \(-0.944977\pi\)
0.985097 0.172000i \(-0.0550228\pi\)
\(368\) 1729.85i 0.245040i
\(369\) −1317.86 −0.185922
\(370\) 1699.68 + 5426.12i 0.238817 + 0.762406i
\(371\) 0 0
\(372\) 3545.45i 0.494149i
\(373\) 2737.30i 0.379979i 0.981786 + 0.189989i \(0.0608453\pi\)
−0.981786 + 0.189989i \(0.939155\pi\)
\(374\) −2929.60 −0.405043
\(375\) 3852.01 4962.06i 0.530445 0.683306i
\(376\) −879.086 −0.120573
\(377\) 8528.21i 1.16505i
\(378\) 0 0
\(379\) −3489.69 −0.472964 −0.236482 0.971636i \(-0.575994\pi\)
−0.236482 + 0.971636i \(0.575994\pi\)
\(380\) −2035.96 6499.67i −0.274849 0.877437i
\(381\) 5000.27 0.672367
\(382\) 7222.15i 0.967323i
\(383\) 180.481i 0.0240787i 0.999928 + 0.0120394i \(0.00383234\pi\)
−0.999928 + 0.0120394i \(0.996168\pi\)
\(384\) −575.339 −0.0764587
\(385\) 0 0
\(386\) 8681.80 1.14480
\(387\) 282.276i 0.0370772i
\(388\) 4152.71i 0.543356i
\(389\) −6220.77 −0.810811 −0.405406 0.914137i \(-0.632870\pi\)
−0.405406 + 0.914137i \(0.632870\pi\)
\(390\) 7482.08 2343.69i 0.971461 0.304301i
\(391\) 11366.5 1.47015
\(392\) 0 0
\(393\) 2723.75i 0.349606i
\(394\) 761.885 0.0974194
\(395\) 1524.57 + 4867.07i 0.194201 + 0.619972i
\(396\) −378.777 −0.0480663
\(397\) 651.379i 0.0823470i −0.999152 0.0411735i \(-0.986890\pi\)
0.999152 0.0411735i \(-0.0131096\pi\)
\(398\) 6953.40i 0.875735i
\(399\) 0 0
\(400\) −1642.59 + 1141.01i −0.205324 + 0.142626i
\(401\) 12791.2 1.59292 0.796460 0.604692i \(-0.206704\pi\)
0.796460 + 0.604692i \(0.206704\pi\)
\(402\) 3369.60i 0.418060i
\(403\) 15383.2i 1.90147i
\(404\) −1958.24 −0.241153
\(405\) −1668.68 5327.16i −0.204735 0.653601i
\(406\) 0 0
\(407\) 3542.99i 0.431498i
\(408\) 3780.43i 0.458723i
\(409\) 12347.5 1.49278 0.746389 0.665510i \(-0.231786\pi\)
0.746389 + 0.665510i \(0.231786\pi\)
\(410\) −4137.60 + 1296.07i −0.498394 + 0.156117i
\(411\) −1815.43 −0.217879
\(412\) 4740.61i 0.566877i
\(413\) 0 0
\(414\) 1469.61 0.174462
\(415\) 8247.87 2583.57i 0.975595 0.305596i
\(416\) −2496.31 −0.294211
\(417\) 2362.18i 0.277401i
\(418\) 4243.97i 0.496602i
\(419\) −9286.60 −1.08277 −0.541385 0.840775i \(-0.682100\pi\)
−0.541385 + 0.840775i \(0.682100\pi\)
\(420\) 0 0
\(421\) 9253.33 1.07121 0.535605 0.844469i \(-0.320084\pi\)
0.535605 + 0.844469i \(0.320084\pi\)
\(422\) 3386.75i 0.390674i
\(423\) 746.833i 0.0858446i
\(424\) 88.0432 0.0100843
\(425\) 7497.32 + 10793.1i 0.855702 + 1.23186i
\(426\) −5175.55 −0.588630
\(427\) 0 0
\(428\) 4769.96i 0.538703i
\(429\) 4885.44 0.549816
\(430\) 277.607 + 886.241i 0.0311335 + 0.0993915i
\(431\) −3385.57 −0.378369 −0.189185 0.981942i \(-0.560584\pi\)
−0.189185 + 0.981942i \(0.560584\pi\)
\(432\) 2430.55i 0.270694i
\(433\) 12073.8i 1.34002i −0.742352 0.670010i \(-0.766290\pi\)
0.742352 0.670010i \(-0.233710\pi\)
\(434\) 0 0
\(435\) −5242.68 + 1642.22i −0.577856 + 0.181008i
\(436\) −1141.72 −0.125410
\(437\) 16466.1i 1.80247i
\(438\) 4694.05i 0.512079i
\(439\) −13382.5 −1.45492 −0.727462 0.686148i \(-0.759300\pi\)
−0.727462 + 0.686148i \(0.759300\pi\)
\(440\) −1189.22 + 372.512i −0.128849 + 0.0403609i
\(441\) 0 0
\(442\) 16402.7i 1.76515i
\(443\) 1205.17i 0.129254i −0.997909 0.0646271i \(-0.979414\pi\)
0.997909 0.0646271i \(-0.0205858\pi\)
\(444\) 4571.96 0.488684
\(445\) 4441.79 + 14180.1i 0.473171 + 1.51056i
\(446\) 521.836 0.0554028
\(447\) 9146.42i 0.967810i
\(448\) 0 0
\(449\) −7957.88 −0.836427 −0.418214 0.908349i \(-0.637344\pi\)
−0.418214 + 0.908349i \(0.637344\pi\)
\(450\) 969.351 + 1395.47i 0.101546 + 0.146185i
\(451\) −2701.65 −0.282075
\(452\) 1400.08i 0.145695i
\(453\) 7860.10i 0.815231i
\(454\) 559.315 0.0578193
\(455\) 0 0
\(456\) −5476.53 −0.562416
\(457\) 1697.84i 0.173789i 0.996218 + 0.0868943i \(0.0276942\pi\)
−0.996218 + 0.0868943i \(0.972306\pi\)
\(458\) 3444.77i 0.351449i
\(459\) 15970.6 1.62406
\(460\) 4614.02 1445.30i 0.467673 0.146494i
\(461\) −14516.2 −1.46657 −0.733283 0.679924i \(-0.762013\pi\)
−0.733283 + 0.679924i \(0.762013\pi\)
\(462\) 0 0
\(463\) 6570.80i 0.659549i −0.944060 0.329774i \(-0.893027\pi\)
0.944060 0.329774i \(-0.106973\pi\)
\(464\) 1749.16 0.175006
\(465\) −9456.75 + 2962.24i −0.943111 + 0.295421i
\(466\) 13319.2 1.32404
\(467\) 18664.4i 1.84943i 0.380659 + 0.924715i \(0.375697\pi\)
−0.380659 + 0.924715i \(0.624303\pi\)
\(468\) 2120.75i 0.209470i
\(469\) 0 0
\(470\) 734.480 + 2344.78i 0.0720831 + 0.230120i
\(471\) −2015.97 −0.197220
\(472\) 1537.13i 0.149899i
\(473\) 578.673i 0.0562524i
\(474\) 4100.92 0.397387
\(475\) −15635.4 + 10861.0i −1.51032 + 1.04913i
\(476\) 0 0
\(477\) 74.7976i 0.00717976i
\(478\) 2037.25i 0.194941i
\(479\) −5296.28 −0.505205 −0.252602 0.967570i \(-0.581286\pi\)
−0.252602 + 0.967570i \(0.581286\pi\)
\(480\) 480.698 + 1534.60i 0.0457100 + 0.145926i
\(481\) 19837.1 1.88044
\(482\) 11707.7i 1.10637i
\(483\) 0 0
\(484\) 4547.50 0.427075
\(485\) −11076.5 + 3469.61i −1.03703 + 0.324839i
\(486\) 3714.53 0.346697
\(487\) 13257.2i 1.23355i −0.787138 0.616777i \(-0.788438\pi\)
0.787138 0.616777i \(-0.211562\pi\)
\(488\) 279.228i 0.0259017i
\(489\) −4097.86 −0.378960
\(490\) 0 0
\(491\) −11404.4 −1.04821 −0.524107 0.851653i \(-0.675601\pi\)
−0.524107 + 0.851653i \(0.675601\pi\)
\(492\) 3486.28i 0.319459i
\(493\) 11493.3i 1.04997i
\(494\) −23761.8 −2.16416
\(495\) 316.470 + 1010.31i 0.0287359 + 0.0917372i
\(496\) 3155.14 0.285625
\(497\) 0 0
\(498\) 6949.53i 0.625333i
\(499\) 18988.6 1.70350 0.851751 0.523947i \(-0.175541\pi\)
0.851751 + 0.523947i \(0.175541\pi\)
\(500\) 4415.79 + 3427.94i 0.394960 + 0.306605i
\(501\) −5084.21 −0.453385
\(502\) 2203.09i 0.195874i
\(503\) 5173.78i 0.458623i 0.973353 + 0.229311i \(0.0736474\pi\)
−0.973353 + 0.229311i \(0.926353\pi\)
\(504\) 0 0
\(505\) 1636.12 + 5223.19i 0.144171 + 0.460255i
\(506\) 3012.73 0.264688
\(507\) 17478.2i 1.53103i
\(508\) 4449.79i 0.388637i
\(509\) −15987.0 −1.39216 −0.696080 0.717964i \(-0.745074\pi\)
−0.696080 + 0.717964i \(0.745074\pi\)
\(510\) 10083.5 3158.56i 0.875499 0.274242i
\(511\) 0 0
\(512\) 512.000i 0.0441942i
\(513\) 23135.9i 1.99118i
\(514\) −2884.93 −0.247566
\(515\) 12644.6 3960.80i 1.08192 0.338901i
\(516\) 746.734 0.0637076
\(517\) 1531.03i 0.130241i
\(518\) 0 0
\(519\) 9612.56 0.812995
\(520\) 2085.68 + 6658.38i 0.175890 + 0.561518i
\(521\) 6278.45 0.527954 0.263977 0.964529i \(-0.414966\pi\)
0.263977 + 0.964529i \(0.414966\pi\)
\(522\) 1486.01i 0.124599i
\(523\) 16580.9i 1.38629i −0.720797 0.693147i \(-0.756224\pi\)
0.720797 0.693147i \(-0.243776\pi\)
\(524\) 2423.90 0.202077
\(525\) 0 0
\(526\) −10860.8 −0.900289
\(527\) 20731.7i 1.71364i
\(528\) 1002.02i 0.0825894i
\(529\) 477.968 0.0392840
\(530\) −73.5605 234.837i −0.00602880 0.0192465i
\(531\) 1305.88 0.106724
\(532\) 0 0
\(533\) 15126.4i 1.22927i
\(534\) 11947.9 0.968236
\(535\) 12722.9 3985.32i 1.02815 0.322057i
\(536\) −2998.64 −0.241644
\(537\) 7816.84i 0.628160i
\(538\) 10023.8i 0.803263i
\(539\) 0 0
\(540\) 6482.99 2030.74i 0.516636 0.161832i
\(541\) −10593.3 −0.841851 −0.420925 0.907095i \(-0.638295\pi\)
−0.420925 + 0.907095i \(0.638295\pi\)
\(542\) 6634.79i 0.525809i
\(543\) 10356.9i 0.818519i
\(544\) −3364.24 −0.265148
\(545\) 953.915 + 3045.31i 0.0749748 + 0.239352i
\(546\) 0 0
\(547\) 163.918i 0.0128129i 0.999979 + 0.00640644i \(0.00203925\pi\)
−0.999979 + 0.00640644i \(0.997961\pi\)
\(548\) 1615.57i 0.125937i
\(549\) 237.219 0.0184413
\(550\) 1987.19 + 2860.75i 0.154062 + 0.221787i
\(551\) 16649.9 1.28731
\(552\) 3887.70i 0.299767i
\(553\) 0 0
\(554\) −8320.71 −0.638110
\(555\) −3819.90 12194.8i −0.292154 0.932682i
\(556\) −2102.13 −0.160342
\(557\) 3673.20i 0.279423i −0.990192 0.139712i \(-0.955382\pi\)
0.990192 0.139712i \(-0.0446175\pi\)
\(558\) 2680.46i 0.203357i
\(559\) 3239.96 0.245145
\(560\) 0 0
\(561\) 6584.03 0.495505
\(562\) 4846.24i 0.363748i
\(563\) 10985.5i 0.822353i −0.911556 0.411176i \(-0.865118\pi\)
0.911556 0.411176i \(-0.134882\pi\)
\(564\) 1975.67 0.147502
\(565\) −3734.41 + 1169.77i −0.278067 + 0.0871019i
\(566\) −16740.3 −1.24319
\(567\) 0 0
\(568\) 4605.78i 0.340236i
\(569\) −26436.8 −1.94778 −0.973891 0.227016i \(-0.927103\pi\)
−0.973891 + 0.227016i \(0.927103\pi\)
\(570\) 4575.66 + 14607.5i 0.336234 + 1.07340i
\(571\) −4688.41 −0.343614 −0.171807 0.985131i \(-0.554961\pi\)
−0.171807 + 0.985131i \(0.554961\pi\)
\(572\) 4347.60i 0.317801i
\(573\) 16231.2i 1.18336i
\(574\) 0 0
\(575\) −7710.07 11099.4i −0.559186 0.805001i
\(576\) −434.973 −0.0314650
\(577\) 21915.1i 1.58118i 0.612349 + 0.790588i \(0.290225\pi\)
−0.612349 + 0.790588i \(0.709775\pi\)
\(578\) 12279.7i 0.883680i
\(579\) −19511.6 −1.40048
\(580\) −1461.43 4665.52i −0.104625 0.334009i
\(581\) 0 0
\(582\) 9332.88i 0.664709i
\(583\) 153.337i 0.0108929i
\(584\) −4177.29 −0.295989
\(585\) 5656.66 1771.90i 0.399785 0.125229i
\(586\) −10584.4 −0.746141
\(587\) 10705.4i 0.752742i −0.926469 0.376371i \(-0.877172\pi\)
0.926469 0.376371i \(-0.122828\pi\)
\(588\) 0 0
\(589\) 30033.0 2.10100
\(590\) 4099.97 1284.28i 0.286090 0.0896152i
\(591\) −1712.28 −0.119177
\(592\) 4068.64i 0.282466i
\(593\) 17638.7i 1.22148i 0.791832 + 0.610739i \(0.209128\pi\)
−0.791832 + 0.610739i \(0.790872\pi\)
\(594\) 4233.08 0.292399
\(595\) 0 0
\(596\) −8139.49 −0.559407
\(597\) 15627.2i 1.07132i
\(598\) 16868.2i 1.15350i
\(599\) −12921.6 −0.881407 −0.440703 0.897653i \(-0.645271\pi\)
−0.440703 + 0.897653i \(0.645271\pi\)
\(600\) 3691.58 2564.32i 0.251181 0.174480i
\(601\) 25860.7 1.75521 0.877603 0.479388i \(-0.159141\pi\)
0.877603 + 0.479388i \(0.159141\pi\)
\(602\) 0 0
\(603\) 2547.51i 0.172044i
\(604\) −6994.78 −0.471215
\(605\) −3799.46 12129.5i −0.255322 0.815098i
\(606\) 4400.98 0.295012
\(607\) 6462.68i 0.432145i 0.976377 + 0.216073i \(0.0693248\pi\)
−0.976377 + 0.216073i \(0.930675\pi\)
\(608\) 4873.62i 0.325084i
\(609\) 0 0
\(610\) 744.781 233.296i 0.0494349 0.0154850i
\(611\) 8572.15 0.567581
\(612\) 2858.11i 0.188778i
\(613\) 13103.7i 0.863382i −0.902022 0.431691i \(-0.857917\pi\)
0.902022 0.431691i \(-0.142083\pi\)
\(614\) 7704.14 0.506374
\(615\) 9298.92 2912.80i 0.609705 0.190985i
\(616\) 0 0
\(617\) 1840.69i 0.120103i −0.998195 0.0600514i \(-0.980874\pi\)
0.998195 0.0600514i \(-0.0191265\pi\)
\(618\) 10654.1i 0.693483i
\(619\) −20894.4 −1.35673 −0.678365 0.734725i \(-0.737311\pi\)
−0.678365 + 0.734725i \(0.737311\pi\)
\(620\) −2636.13 8415.66i −0.170757 0.545131i
\(621\) −16423.8 −1.06130
\(622\) 2184.96i 0.140850i
\(623\) 0 0
\(624\) 5610.25 0.359919
\(625\) 5453.90 14642.3i 0.349050 0.937104i
\(626\) 2649.79 0.169181
\(627\) 9537.98i 0.607512i
\(628\) 1794.03i 0.113996i
\(629\) 26734.1 1.69469
\(630\) 0 0
\(631\) −11128.4 −0.702086 −0.351043 0.936359i \(-0.614173\pi\)
−0.351043 + 0.936359i \(0.614173\pi\)
\(632\) 3649.45i 0.229695i
\(633\) 7611.44i 0.477927i
\(634\) 9970.95 0.624601
\(635\) 11868.9 3717.82i 0.741736 0.232342i
\(636\) −197.870 −0.0123366
\(637\) 0 0
\(638\) 3046.36i 0.189038i
\(639\) −3912.87 −0.242239
\(640\) −1365.65 + 427.778i −0.0843471 + 0.0264210i
\(641\) −31784.4 −1.95851 −0.979257 0.202620i \(-0.935054\pi\)
−0.979257 + 0.202620i \(0.935054\pi\)
\(642\) 10720.1i 0.659016i
\(643\) 1781.74i 0.109277i −0.998506 0.0546384i \(-0.982599\pi\)
0.998506 0.0546384i \(-0.0174006\pi\)
\(644\) 0 0
\(645\) −623.899 1991.75i −0.0380868 0.121590i
\(646\) −32023.5 −1.95038
\(647\) 13936.7i 0.846843i −0.905933 0.423422i \(-0.860829\pi\)
0.905933 0.423422i \(-0.139171\pi\)
\(648\) 3994.43i 0.242155i
\(649\) 2677.09 0.161918
\(650\) 16017.2 11126.2i 0.966534 0.671394i
\(651\) 0 0
\(652\) 3646.72i 0.219044i
\(653\) 856.900i 0.0513523i 0.999670 + 0.0256762i \(0.00817388\pi\)
−0.999670 + 0.0256762i \(0.991826\pi\)
\(654\) 2565.93 0.153419
\(655\) −2025.18 6465.23i −0.120809 0.385676i
\(656\) −3102.48 −0.184651
\(657\) 3548.84i 0.210736i
\(658\) 0 0
\(659\) −11560.8 −0.683374 −0.341687 0.939814i \(-0.610998\pi\)
−0.341687 + 0.939814i \(0.610998\pi\)
\(660\) 2672.67 837.190i 0.157627 0.0493751i
\(661\) 9094.31 0.535140 0.267570 0.963538i \(-0.413779\pi\)
0.267570 + 0.963538i \(0.413779\pi\)
\(662\) 20082.0i 1.17902i
\(663\) 36863.7i 2.15938i
\(664\) 6184.46 0.361451
\(665\) 0 0
\(666\) 3456.54 0.201108
\(667\) 11819.5i 0.686136i
\(668\) 4524.49i 0.262063i
\(669\) −1172.78 −0.0677765
\(670\) 2505.37 + 7998.23i 0.144464 + 0.461192i
\(671\) 486.306 0.0279786
\(672\) 0 0
\(673\) 5216.21i 0.298767i −0.988779 0.149384i \(-0.952271\pi\)
0.988779 0.149384i \(-0.0477289\pi\)
\(674\) −14454.8 −0.826080
\(675\) −10833.1 15595.3i −0.617729 0.889279i
\(676\) 15554.0 0.884958
\(677\) 5631.52i 0.319700i −0.987141 0.159850i \(-0.948899\pi\)
0.987141 0.159850i \(-0.0511010\pi\)
\(678\) 3146.56i 0.178234i
\(679\) 0 0
\(680\) 2810.84 + 8973.40i 0.158516 + 0.506050i
\(681\) −1257.01 −0.0707326
\(682\) 5495.02i 0.308527i
\(683\) 20993.9i 1.17615i −0.808807 0.588074i \(-0.799886\pi\)
0.808807 0.588074i \(-0.200114\pi\)
\(684\) −4140.41 −0.231451
\(685\) −4309.18 + 1349.81i −0.240358 + 0.0752901i
\(686\) 0 0
\(687\) 7741.84i 0.429941i
\(688\) 664.526i 0.0368238i
\(689\) −858.527 −0.0474706
\(690\) −10369.6 + 3248.19i −0.572123 + 0.179212i
\(691\) −20881.2 −1.14958 −0.574790 0.818301i \(-0.694916\pi\)
−0.574790 + 0.818301i \(0.694916\pi\)
\(692\) 8554.31i 0.469922i
\(693\) 0 0
\(694\) 6851.34 0.374746
\(695\) 1756.34 + 5606.98i 0.0958584 + 0.306021i
\(696\) −3931.09 −0.214092
\(697\) 20385.7i 1.10784i
\(698\) 14184.1i 0.769161i
\(699\) −29933.9 −1.61975
\(700\) 0 0
\(701\) −14493.3 −0.780889 −0.390444 0.920626i \(-0.627679\pi\)
−0.390444 + 0.920626i \(0.627679\pi\)
\(702\) 23700.8i 1.27426i
\(703\) 38728.5i 2.07777i
\(704\) −891.705 −0.0477378
\(705\) −1650.68 5269.69i −0.0881821 0.281515i
\(706\) −23929.5 −1.27564
\(707\) 0 0
\(708\) 3454.58i 0.183377i
\(709\) −19466.9 −1.03116 −0.515580 0.856841i \(-0.672424\pi\)
−0.515580 + 0.856841i \(0.672424\pi\)
\(710\) −12284.9 + 3848.15i −0.649360 + 0.203406i
\(711\) 3100.41 0.163537
\(712\) 10632.6i 0.559654i
\(713\) 21320.0i 1.11983i
\(714\) 0 0
\(715\) 11596.3 3632.44i 0.606542 0.189994i
\(716\) 6956.29 0.363085
\(717\) 4578.56i 0.238479i
\(718\) 3112.01i 0.161754i
\(719\) 31341.8 1.62566 0.812831 0.582499i \(-0.197925\pi\)
0.812831 + 0.582499i \(0.197925\pi\)
\(720\) 363.422 + 1160.20i 0.0188110 + 0.0600528i
\(721\) 0 0
\(722\) 32672.9i 1.68415i
\(723\) 26312.1i 1.35347i
\(724\) −9216.69 −0.473115
\(725\) −11223.2 + 7796.12i −0.574925 + 0.399366i
\(726\) −10220.1 −0.522458
\(727\) 21225.4i 1.08282i 0.840760 + 0.541408i \(0.182108\pi\)
−0.840760 + 0.541408i \(0.817892\pi\)
\(728\) 0 0
\(729\) −21829.3 −1.10904
\(730\) 3490.14 + 11142.0i 0.176953 + 0.564911i
\(731\) 4366.45 0.220929
\(732\) 627.541i 0.0316866i
\(733\) 10542.1i 0.531217i 0.964081 + 0.265609i \(0.0855729\pi\)
−0.964081 + 0.265609i \(0.914427\pi\)
\(734\) −4837.12 −0.243244
\(735\) 0 0
\(736\) 3459.71 0.173270
\(737\) 5222.46i 0.261020i
\(738\) 2635.73i 0.131467i
\(739\) −13623.8 −0.678158 −0.339079 0.940758i \(-0.610115\pi\)
−0.339079 + 0.940758i \(0.610115\pi\)
\(740\) 10852.2 3399.36i 0.539103 0.168869i
\(741\) 53402.7 2.64750
\(742\) 0 0
\(743\) 8366.07i 0.413084i −0.978438 0.206542i \(-0.933779\pi\)
0.978438 0.206542i \(-0.0662210\pi\)
\(744\) −7090.91 −0.349416
\(745\) 6800.58 + 21710.4i 0.334435 + 1.06766i
\(746\) 5474.60 0.268685
\(747\) 5254.04i 0.257343i
\(748\) 5859.20i 0.286408i
\(749\) 0 0
\(750\) −9924.13 7704.02i −0.483171 0.375081i
\(751\) −4026.96 −0.195667 −0.0978334 0.995203i \(-0.531191\pi\)
−0.0978334 + 0.995203i \(0.531191\pi\)
\(752\) 1758.17i 0.0852579i
\(753\) 4951.27i 0.239621i
\(754\) −17056.4 −0.823817
\(755\) 5844.17 + 18657.1i 0.281710 + 0.899340i
\(756\) 0 0
\(757\) 7909.33i 0.379748i −0.981808 0.189874i \(-0.939192\pi\)
0.981808 0.189874i \(-0.0608080\pi\)
\(758\) 6979.38i 0.334436i
\(759\) −6770.87 −0.323804
\(760\) −12999.3 + 4071.93i −0.620442 + 0.194348i
\(761\) 12265.2 0.584247 0.292124 0.956381i \(-0.405638\pi\)
0.292124 + 0.956381i \(0.405638\pi\)
\(762\) 10000.5i 0.475435i
\(763\) 0 0
\(764\) 14444.3 0.684000
\(765\) 7623.40 2387.96i 0.360294 0.112859i
\(766\) 360.962 0.0170262
\(767\) 14988.9i 0.705629i
\(768\) 1150.68i 0.0540645i
\(769\) 7111.92 0.333501 0.166750 0.985999i \(-0.446673\pi\)
0.166750 + 0.985999i \(0.446673\pi\)
\(770\) 0 0
\(771\) 6483.65 0.302857
\(772\) 17363.6i 0.809494i
\(773\) 7131.22i 0.331814i 0.986141 + 0.165907i \(0.0530551\pi\)
−0.986141 + 0.165907i \(0.946945\pi\)
\(774\) 564.552 0.0262176
\(775\) −20244.5 + 14062.6i −0.938328 + 0.651800i
\(776\) −8305.43 −0.384211
\(777\) 0 0
\(778\) 12441.5i 0.573330i
\(779\) −29531.8 −1.35826
\(780\) −4687.39 14964.2i −0.215174 0.686927i
\(781\) −8021.48 −0.367517
\(782\) 22733.0i 1.03955i
\(783\) 16607.1i 0.757970i
\(784\) 0 0
\(785\) −4785.19 + 1498.92i −0.217568 + 0.0681512i
\(786\) −5447.51 −0.247209
\(787\) 12062.3i 0.546345i 0.961965 + 0.273172i \(0.0880729\pi\)
−0.961965 + 0.273172i \(0.911927\pi\)
\(788\) 1523.77i 0.0688859i
\(789\) 24408.7 1.10136
\(790\) 9734.14 3049.13i 0.438386 0.137321i
\(791\) 0 0
\(792\) 757.553i 0.0339880i
\(793\) 2722.81i 0.121929i
\(794\) −1302.76 −0.0582281
\(795\) 165.321 + 527.776i 0.00737526 + 0.0235450i
\(796\) −13906.8 −0.619238
\(797\) 15849.5i 0.704415i −0.935922 0.352207i \(-0.885431\pi\)
0.935922 0.352207i \(-0.114569\pi\)
\(798\) 0 0
\(799\) 11552.6 0.511515
\(800\) 2282.02 + 3285.18i 0.100852 + 0.145186i
\(801\) 9032.98 0.398458
\(802\) 25582.3i 1.12636i
\(803\) 7275.21i 0.319722i
\(804\) 6739.19 0.295613
\(805\) 0 0
\(806\) −30766.4 −1.34454
\(807\) 22527.6i 0.982664i
\(808\) 3916.47i 0.170521i
\(809\) 4367.34 0.189799 0.0948995 0.995487i \(-0.469747\pi\)
0.0948995 + 0.995487i \(0.469747\pi\)
\(810\) −10654.3 + 3337.37i −0.462166 + 0.144769i
\(811\) −284.034 −0.0122981 −0.00614907 0.999981i \(-0.501957\pi\)
−0.00614907 + 0.999981i \(0.501957\pi\)
\(812\) 0 0
\(813\) 14911.2i 0.643243i
\(814\) 7085.99 0.305115
\(815\) −9726.87 + 3046.85i −0.418058 + 0.130953i
\(816\) 7560.85 0.324366
\(817\) 6325.47i 0.270869i
\(818\) 24695.1i 1.05555i
\(819\) 0 0
\(820\) 2592.13 + 8275.20i 0.110392 + 0.352418i
\(821\) −11862.3 −0.504261 −0.252131 0.967693i \(-0.581131\pi\)
−0.252131 + 0.967693i \(0.581131\pi\)
\(822\) 3630.85i 0.154064i
\(823\) 33092.7i 1.40163i 0.713344 + 0.700814i \(0.247180\pi\)
−0.713344 + 0.700814i \(0.752820\pi\)
\(824\) 9481.23 0.400843
\(825\) −4466.05 6429.31i −0.188470 0.271321i
\(826\) 0 0
\(827\) 38358.2i 1.61287i 0.591322 + 0.806436i \(0.298606\pi\)
−0.591322 + 0.806436i \(0.701394\pi\)
\(828\) 2939.21i 0.123363i
\(829\) −44688.0 −1.87223 −0.936114 0.351697i \(-0.885605\pi\)
−0.936114 + 0.351697i \(0.885605\pi\)
\(830\) −5167.14 16495.7i −0.216089 0.689850i
\(831\) 18700.1 0.780625
\(832\) 4992.62i 0.208038i
\(833\) 0 0
\(834\) 4724.36 0.196152
\(835\) −12068.1 + 3780.23i −0.500161 + 0.156671i
\(836\) −8487.94 −0.351150
\(837\) 29955.9i 1.23707i
\(838\) 18573.2i 0.765633i
\(839\) −25673.2 −1.05642 −0.528211 0.849113i \(-0.677137\pi\)
−0.528211 + 0.849113i \(0.677137\pi\)
\(840\) 0 0
\(841\) −12437.6 −0.509967
\(842\) 18506.7i 0.757460i
\(843\) 10891.5i 0.444987i
\(844\) −6773.50 −0.276248
\(845\) −12995.5 41487.1i −0.529062 1.68899i
\(846\) 1493.67 0.0607013
\(847\) 0 0
\(848\) 176.086i 0.00713069i
\(849\) 37622.4 1.52084
\(850\) 21586.2 14994.6i 0.871059 0.605072i
\(851\) −27492.8 −1.10745
\(852\) 10351.1i 0.416224i
\(853\) 1652.74i 0.0663408i −0.999450 0.0331704i \(-0.989440\pi\)
0.999450 0.0331704i \(-0.0105604\pi\)
\(854\) 0 0
\(855\) 3459.33 + 11043.7i 0.138370 + 0.441737i
\(856\) 9539.92 0.380920
\(857\) 1566.46i 0.0624379i −0.999513 0.0312190i \(-0.990061\pi\)
0.999513 0.0312190i \(-0.00993892\pi\)
\(858\) 9770.87i 0.388779i
\(859\) −17466.8 −0.693783 −0.346892 0.937905i \(-0.612763\pi\)
−0.346892 + 0.937905i \(0.612763\pi\)
\(860\) 1772.48 555.214i 0.0702804 0.0220147i
\(861\) 0 0
\(862\) 6771.14i 0.267547i
\(863\) 42901.9i 1.69224i −0.532996 0.846118i \(-0.678934\pi\)
0.532996 0.846118i \(-0.321066\pi\)
\(864\) 4861.10 0.191410
\(865\) 22816.8 7147.17i 0.896873 0.280937i
\(866\) −24147.6 −0.947537
\(867\) 27597.6i 1.08104i
\(868\) 0 0
\(869\) 6355.93 0.248113
\(870\) 3284.45 + 10485.4i 0.127992 + 0.408606i
\(871\) 29240.3 1.13751
\(872\) 2283.45i 0.0886781i
\(873\) 7055.92i 0.273547i
\(874\) 32932.2 1.27454
\(875\) 0 0
\(876\) 9388.11 0.362095
\(877\) 35156.0i 1.35363i 0.736153 + 0.676815i \(0.236640\pi\)
−0.736153 + 0.676815i \(0.763360\pi\)
\(878\) 26765.0i 1.02879i
\(879\) 23787.6 0.912783
\(880\) 745.024 + 2378.44i 0.0285395 + 0.0911103i
\(881\) −26203.5 −1.00206 −0.501032 0.865429i \(-0.667046\pi\)
−0.501032 + 0.865429i \(0.667046\pi\)
\(882\) 0 0
\(883\) 12182.9i 0.464312i 0.972679 + 0.232156i \(0.0745780\pi\)
−0.972679 + 0.232156i \(0.925422\pi\)
\(884\) 32805.4 1.24815
\(885\) −9214.36 + 2886.31i −0.349986 + 0.109630i
\(886\) −2410.35 −0.0913965
\(887\) 22769.7i 0.861931i −0.902368 0.430965i \(-0.858173\pi\)
0.902368 0.430965i \(-0.141827\pi\)
\(888\) 9143.93i 0.345552i
\(889\) 0 0
\(890\) 28360.2 8883.58i 1.06813 0.334582i
\(891\) −6956.75 −0.261571
\(892\) 1043.67i 0.0391757i
\(893\) 16735.7i 0.627141i
\(894\) 18292.8 0.684345
\(895\) −5812.01 18554.4i −0.217066 0.692968i
\(896\) 0 0
\(897\) 37909.8i 1.41112i
\(898\) 15915.8i 0.591443i
\(899\) 21558.0 0.799775
\(900\) 2790.94 1938.70i 0.103368 0.0718037i
\(901\) −1157.02 −0.0427814
\(902\) 5403.31i 0.199457i
\(903\) 0 0
\(904\) −2800.15 −0.103022
\(905\) 7700.58 + 24583.6i 0.282846 + 0.902967i
\(906\) 15720.2 0.576456
\(907\) 28419.0i 1.04039i 0.854046 + 0.520197i \(0.174142\pi\)
−0.854046 + 0.520197i \(0.825858\pi\)
\(908\) 1118.63i 0.0408844i
\(909\) 3327.26 0.121406
\(910\) 0 0
\(911\) 21021.6 0.764519 0.382259 0.924055i \(-0.375146\pi\)
0.382259 + 0.924055i \(0.375146\pi\)
\(912\) 10953.1i 0.397688i
\(913\) 10770.9i 0.390433i
\(914\) 3395.67 0.122887
\(915\) −1673.83 + 524.314i −0.0604757 + 0.0189435i
\(916\) 6889.54 0.248512
\(917\) 0 0
\(918\) 31941.2i 1.14839i
\(919\) 24704.5 0.886752 0.443376 0.896336i \(-0.353781\pi\)
0.443376 + 0.896336i \(0.353781\pi\)
\(920\) −2890.60 9228.04i −0.103587 0.330695i
\(921\) −17314.4 −0.619467
\(922\) 29032.4i 1.03702i
\(923\) 44911.9i 1.60162i
\(924\) 0 0
\(925\) −18134.2 26105.9i −0.644593 0.927952i
\(926\) −13141.6 −0.466371
\(927\) 8054.83i 0.285389i
\(928\) 3498.32i 0.123748i
\(929\) −17304.1 −0.611119 −0.305560 0.952173i \(-0.598844\pi\)
−0.305560 + 0.952173i \(0.598844\pi\)
\(930\) 5924.49 + 18913.5i 0.208894 + 0.666880i
\(931\) 0 0
\(932\) 26638.5i 0.936237i
\(933\) 4910.52i 0.172308i
\(934\) 37328.7 1.30775
\(935\) 15628.2 4895.39i 0.546627 0.171226i
\(936\) 4241.51 0.148118
\(937\) 31154.0i 1.08619i −0.839672 0.543094i \(-0.817253\pi\)
0.839672 0.543094i \(-0.182747\pi\)
\(938\) 0 0
\(939\) −5955.19 −0.206965
\(940\) 4689.55 1468.96i 0.162720 0.0509704i
\(941\) −53006.5 −1.83631 −0.918153 0.396227i \(-0.870319\pi\)
−0.918153 + 0.396227i \(0.870319\pi\)
\(942\) 4031.93i 0.139456i
\(943\) 20964.2i 0.723953i
\(944\) 3074.26 0.105994
\(945\) 0 0
\(946\) 1157.35 0.0397765
\(947\) 54176.8i 1.85904i 0.368772 + 0.929520i \(0.379778\pi\)
−0.368772 + 0.929520i \(0.620222\pi\)
\(948\) 8201.85i 0.280995i
\(949\) 40733.6 1.39333
\(950\) 21722.0 + 31270.9i 0.741848 + 1.06796i
\(951\) −22408.9 −0.764099
\(952\) 0 0
\(953\) 6214.74i 0.211244i 0.994406 + 0.105622i \(0.0336833\pi\)
−0.994406 + 0.105622i \(0.966317\pi\)
\(954\) −149.595 −0.00507686
\(955\) −12068.3 38527.1i −0.408922 1.30545i
\(956\) 4074.51 0.137844
\(957\) 6846.44i 0.231258i
\(958\) 10592.6i 0.357234i
\(959\) 0 0
\(960\) 3069.19 961.397i 0.103185 0.0323218i
\(961\) 9095.25 0.305302
\(962\) 39674.1i 1.32967i
\(963\) 8104.70i 0.271205i
\(964\) 23415.4 0.782322
\(965\) −46313.7 + 14507.4i −1.54497 + 0.483947i
\(966\) 0 0
\(967\) 45681.3i 1.51914i −0.650424 0.759571i \(-0.725409\pi\)
0.650424 0.759571i \(-0.274591\pi\)
\(968\) 9095.00i 0.301988i
\(969\) 71970.1 2.38598
\(970\) 6939.22 + 22153.0i 0.229696 + 0.733288i
\(971\) 52107.6 1.72215 0.861077 0.508474i \(-0.169790\pi\)
0.861077 + 0.508474i \(0.169790\pi\)
\(972\) 7429.06i 0.245152i
\(973\) 0 0
\(974\) −26514.4 −0.872255
\(975\) −35997.4 + 25005.2i −1.18240 + 0.821342i
\(976\) 558.455 0.0183153
\(977\) 4174.15i 0.136687i 0.997662 + 0.0683433i \(0.0217713\pi\)
−0.997662 + 0.0683433i \(0.978229\pi\)
\(978\) 8195.71i 0.267965i
\(979\) 18517.8 0.604528
\(980\) 0 0
\(981\) 1939.92 0.0631364
\(982\) 22808.8i 0.741199i
\(983\) 19717.9i 0.639781i 0.947455 + 0.319890i \(0.103646\pi\)
−0.947455 + 0.319890i \(0.896354\pi\)
\(984\) 6972.56 0.225891
\(985\) −4064.34 + 1273.12i −0.131473 + 0.0411826i
\(986\) −22986.7 −0.742440
\(987\) 0 0
\(988\) 47523.6i 1.53029i
\(989\) −4490.36 −0.144373
\(990\) 2020.61 632.939i 0.0648680 0.0203193i
\(991\) 24502.6 0.785419 0.392710 0.919663i \(-0.371538\pi\)
0.392710 + 0.919663i \(0.371538\pi\)
\(992\) 6310.27i 0.201967i
\(993\) 45132.7i 1.44234i
\(994\) 0 0
\(995\) 11619.2 + 37093.5i 0.370204 + 1.18185i
\(996\) −13899.1 −0.442177
\(997\) 22942.8i 0.728791i −0.931244 0.364395i \(-0.881276\pi\)
0.931244 0.364395i \(-0.118724\pi\)
\(998\) 37977.2i 1.20456i
\(999\) −38629.0 −1.22339
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 490.4.c.e.99.5 12
5.2 odd 4 2450.4.a.cx.1.5 6
5.3 odd 4 2450.4.a.cw.1.2 6
5.4 even 2 inner 490.4.c.e.99.8 12
7.3 odd 6 70.4.i.a.9.8 yes 24
7.5 odd 6 70.4.i.a.39.5 yes 24
7.6 odd 2 490.4.c.f.99.2 12
35.3 even 12 350.4.e.o.51.2 12
35.12 even 12 350.4.e.n.151.5 12
35.13 even 4 2450.4.a.cv.1.5 6
35.17 even 12 350.4.e.n.51.5 12
35.19 odd 6 70.4.i.a.39.8 yes 24
35.24 odd 6 70.4.i.a.9.5 24
35.27 even 4 2450.4.a.cy.1.2 6
35.33 even 12 350.4.e.o.151.2 12
35.34 odd 2 490.4.c.f.99.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.4.i.a.9.5 24 35.24 odd 6
70.4.i.a.9.8 yes 24 7.3 odd 6
70.4.i.a.39.5 yes 24 7.5 odd 6
70.4.i.a.39.8 yes 24 35.19 odd 6
350.4.e.n.51.5 12 35.17 even 12
350.4.e.n.151.5 12 35.12 even 12
350.4.e.o.51.2 12 35.3 even 12
350.4.e.o.151.2 12 35.33 even 12
490.4.c.e.99.5 12 1.1 even 1 trivial
490.4.c.e.99.8 12 5.4 even 2 inner
490.4.c.f.99.2 12 7.6 odd 2
490.4.c.f.99.11 12 35.34 odd 2
2450.4.a.cv.1.5 6 35.13 even 4
2450.4.a.cw.1.2 6 5.3 odd 4
2450.4.a.cx.1.5 6 5.2 odd 4
2450.4.a.cy.1.2 6 35.27 even 4