Properties

Label 300.4.m.a.61.1
Level $300$
Weight $4$
Character 300.61
Analytic conductor $17.701$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,4,Mod(61,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.61");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 300.m (of order \(5\), degree \(4\), 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: \(17.7005730017\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 61.1
Character \(\chi\) \(=\) 300.61
Dual form 300.4.m.a.241.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.927051 - 2.85317i) q^{3} +(-10.7679 - 3.00887i) q^{5} -17.8479 q^{7} +(-7.28115 - 5.29007i) q^{9} +(33.6508 - 24.4487i) q^{11} +(-9.86695 - 7.16876i) q^{13} +(-18.5672 + 27.9331i) q^{15} +(36.4109 + 112.061i) q^{17} +(4.03136 + 12.4072i) q^{19} +(-16.5460 + 50.9232i) q^{21} +(-151.224 + 109.871i) q^{23} +(106.893 + 64.7981i) q^{25} +(-21.8435 + 15.8702i) q^{27} +(-13.3131 + 40.9734i) q^{29} +(40.4857 + 124.602i) q^{31} +(-38.5604 - 118.677i) q^{33} +(192.184 + 53.7021i) q^{35} +(318.449 + 231.367i) q^{37} +(-29.6008 + 21.5063i) q^{39} +(-360.662 - 262.037i) q^{41} +180.383 q^{43} +(62.4853 + 78.8707i) q^{45} +(34.7074 - 106.818i) q^{47} -24.4510 q^{49} +353.485 q^{51} +(-194.546 + 598.752i) q^{53} +(-435.910 + 162.010i) q^{55} +39.1372 q^{57} +(-85.2783 - 61.9583i) q^{59} +(-153.314 + 111.389i) q^{61} +(129.954 + 94.4168i) q^{63} +(84.6761 + 106.880i) q^{65} +(109.821 + 337.993i) q^{67} +(173.287 + 533.324i) q^{69} +(-31.3661 + 96.5350i) q^{71} +(561.527 - 407.973i) q^{73} +(283.976 - 244.914i) q^{75} +(-600.597 + 436.360i) q^{77} +(-121.320 + 373.384i) q^{79} +(25.0304 + 77.0356i) q^{81} +(-380.474 - 1170.98i) q^{83} +(-54.8902 - 1316.21i) q^{85} +(104.562 + 75.9688i) q^{87} +(-816.875 + 593.494i) q^{89} +(176.105 + 127.948i) q^{91} +393.044 q^{93} +(-6.07735 - 145.729i) q^{95} +(-328.606 + 1011.34i) q^{97} -374.352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 24 q^{3} + 8 q^{5} + 46 q^{7} - 72 q^{9} + 7 q^{11} + 50 q^{13} - 21 q^{15} + 96 q^{17} + 160 q^{19} - 27 q^{21} - 26 q^{23} + 46 q^{25} - 216 q^{27} - 18 q^{29} - 267 q^{31} + 66 q^{33} + 454 q^{35}+ \cdots - 522 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.927051 2.85317i 0.178411 0.549093i
\(4\) 0 0
\(5\) −10.7679 3.00887i −0.963106 0.269121i
\(6\) 0 0
\(7\) −17.8479 −0.963698 −0.481849 0.876254i \(-0.660035\pi\)
−0.481849 + 0.876254i \(0.660035\pi\)
\(8\) 0 0
\(9\) −7.28115 5.29007i −0.269672 0.195928i
\(10\) 0 0
\(11\) 33.6508 24.4487i 0.922372 0.670143i −0.0217410 0.999764i \(-0.506921\pi\)
0.944113 + 0.329621i \(0.106921\pi\)
\(12\) 0 0
\(13\) −9.86695 7.16876i −0.210508 0.152943i 0.477536 0.878612i \(-0.341530\pi\)
−0.688043 + 0.725670i \(0.741530\pi\)
\(14\) 0 0
\(15\) −18.5672 + 27.9331i −0.319601 + 0.480821i
\(16\) 0 0
\(17\) 36.4109 + 112.061i 0.519467 + 1.59876i 0.775004 + 0.631956i \(0.217748\pi\)
−0.255537 + 0.966799i \(0.582252\pi\)
\(18\) 0 0
\(19\) 4.03136 + 12.4072i 0.0486766 + 0.149811i 0.972440 0.233151i \(-0.0749038\pi\)
−0.923764 + 0.382963i \(0.874904\pi\)
\(20\) 0 0
\(21\) −16.5460 + 50.9232i −0.171934 + 0.529160i
\(22\) 0 0
\(23\) −151.224 + 109.871i −1.37097 + 0.996072i −0.373315 + 0.927705i \(0.621779\pi\)
−0.997660 + 0.0683668i \(0.978221\pi\)
\(24\) 0 0
\(25\) 106.893 + 64.7981i 0.855148 + 0.518384i
\(26\) 0 0
\(27\) −21.8435 + 15.8702i −0.155695 + 0.113119i
\(28\) 0 0
\(29\) −13.3131 + 40.9734i −0.0852474 + 0.262364i −0.984590 0.174881i \(-0.944046\pi\)
0.899342 + 0.437245i \(0.144046\pi\)
\(30\) 0 0
\(31\) 40.4857 + 124.602i 0.234563 + 0.721910i 0.997179 + 0.0750591i \(0.0239146\pi\)
−0.762616 + 0.646851i \(0.776085\pi\)
\(32\) 0 0
\(33\) −38.5604 118.677i −0.203409 0.626029i
\(34\) 0 0
\(35\) 192.184 + 53.7021i 0.928144 + 0.259352i
\(36\) 0 0
\(37\) 318.449 + 231.367i 1.41494 + 1.02801i 0.992581 + 0.121584i \(0.0387975\pi\)
0.422358 + 0.906429i \(0.361202\pi\)
\(38\) 0 0
\(39\) −29.6008 + 21.5063i −0.121537 + 0.0883015i
\(40\) 0 0
\(41\) −360.662 262.037i −1.37380 0.998128i −0.997429 0.0716603i \(-0.977170\pi\)
−0.376376 0.926467i \(-0.622830\pi\)
\(42\) 0 0
\(43\) 180.383 0.639723 0.319861 0.947464i \(-0.396364\pi\)
0.319861 + 0.947464i \(0.396364\pi\)
\(44\) 0 0
\(45\) 62.4853 + 78.8707i 0.206995 + 0.261274i
\(46\) 0 0
\(47\) 34.7074 106.818i 0.107715 0.331512i −0.882643 0.470044i \(-0.844238\pi\)
0.990358 + 0.138531i \(0.0442382\pi\)
\(48\) 0 0
\(49\) −24.4510 −0.0712856
\(50\) 0 0
\(51\) 353.485 0.970544
\(52\) 0 0
\(53\) −194.546 + 598.752i −0.504207 + 1.55179i 0.297892 + 0.954599i \(0.403716\pi\)
−0.802100 + 0.597190i \(0.796284\pi\)
\(54\) 0 0
\(55\) −435.910 + 162.010i −1.06869 + 0.397189i
\(56\) 0 0
\(57\) 39.1372 0.0909448
\(58\) 0 0
\(59\) −85.2783 61.9583i −0.188174 0.136717i 0.489710 0.871886i \(-0.337103\pi\)
−0.677884 + 0.735169i \(0.737103\pi\)
\(60\) 0 0
\(61\) −153.314 + 111.389i −0.321801 + 0.233802i −0.736943 0.675954i \(-0.763732\pi\)
0.415143 + 0.909756i \(0.363732\pi\)
\(62\) 0 0
\(63\) 129.954 + 94.4168i 0.259883 + 0.188816i
\(64\) 0 0
\(65\) 84.6761 + 106.880i 0.161581 + 0.203952i
\(66\) 0 0
\(67\) 109.821 + 337.993i 0.200250 + 0.616305i 0.999875 + 0.0158060i \(0.00503143\pi\)
−0.799625 + 0.600499i \(0.794969\pi\)
\(68\) 0 0
\(69\) 173.287 + 533.324i 0.302339 + 0.930503i
\(70\) 0 0
\(71\) −31.3661 + 96.5350i −0.0524292 + 0.161361i −0.973843 0.227223i \(-0.927035\pi\)
0.921414 + 0.388583i \(0.127035\pi\)
\(72\) 0 0
\(73\) 561.527 407.973i 0.900298 0.654105i −0.0382444 0.999268i \(-0.512177\pi\)
0.938543 + 0.345163i \(0.112177\pi\)
\(74\) 0 0
\(75\) 283.976 244.914i 0.437209 0.377070i
\(76\) 0 0
\(77\) −600.597 + 436.360i −0.888889 + 0.645815i
\(78\) 0 0
\(79\) −121.320 + 373.384i −0.172779 + 0.531759i −0.999525 0.0308162i \(-0.990189\pi\)
0.826746 + 0.562575i \(0.190189\pi\)
\(80\) 0 0
\(81\) 25.0304 + 77.0356i 0.0343352 + 0.105673i
\(82\) 0 0
\(83\) −380.474 1170.98i −0.503162 1.54857i −0.803838 0.594848i \(-0.797212\pi\)
0.300676 0.953726i \(-0.402788\pi\)
\(84\) 0 0
\(85\) −54.8902 1316.21i −0.0700433 1.67957i
\(86\) 0 0
\(87\) 104.562 + 75.9688i 0.128853 + 0.0936174i
\(88\) 0 0
\(89\) −816.875 + 593.494i −0.972905 + 0.706857i −0.956112 0.293002i \(-0.905346\pi\)
−0.0167932 + 0.999859i \(0.505346\pi\)
\(90\) 0 0
\(91\) 176.105 + 127.948i 0.202866 + 0.147391i
\(92\) 0 0
\(93\) 393.044 0.438244
\(94\) 0 0
\(95\) −6.07735 145.729i −0.00656340 0.157384i
\(96\) 0 0
\(97\) −328.606 + 1011.34i −0.343967 + 1.05862i 0.618167 + 0.786047i \(0.287876\pi\)
−0.962134 + 0.272576i \(0.912124\pi\)
\(98\) 0 0
\(99\) −374.352 −0.380038
\(100\) 0 0
\(101\) 242.161 0.238573 0.119287 0.992860i \(-0.461939\pi\)
0.119287 + 0.992860i \(0.461939\pi\)
\(102\) 0 0
\(103\) −496.703 + 1528.69i −0.475161 + 1.46240i 0.370580 + 0.928801i \(0.379159\pi\)
−0.845741 + 0.533594i \(0.820841\pi\)
\(104\) 0 0
\(105\) 331.385 498.549i 0.307999 0.463366i
\(106\) 0 0
\(107\) 964.672 0.871574 0.435787 0.900050i \(-0.356470\pi\)
0.435787 + 0.900050i \(0.356470\pi\)
\(108\) 0 0
\(109\) −1617.38 1175.09i −1.42125 1.03260i −0.991563 0.129623i \(-0.958623\pi\)
−0.429688 0.902977i \(-0.641377\pi\)
\(110\) 0 0
\(111\) 955.348 694.101i 0.816916 0.593524i
\(112\) 0 0
\(113\) −728.564 529.333i −0.606527 0.440668i 0.241663 0.970360i \(-0.422307\pi\)
−0.848190 + 0.529693i \(0.822307\pi\)
\(114\) 0 0
\(115\) 1958.95 728.060i 1.58846 0.590365i
\(116\) 0 0
\(117\) 33.9195 + 104.394i 0.0268023 + 0.0824888i
\(118\) 0 0
\(119\) −649.860 2000.06i −0.500610 1.54072i
\(120\) 0 0
\(121\) 123.334 379.582i 0.0926625 0.285186i
\(122\) 0 0
\(123\) −1081.99 + 786.110i −0.793166 + 0.576269i
\(124\) 0 0
\(125\) −956.045 1019.36i −0.684090 0.729398i
\(126\) 0 0
\(127\) 924.753 671.872i 0.646130 0.469441i −0.215820 0.976433i \(-0.569243\pi\)
0.861951 + 0.506992i \(0.169243\pi\)
\(128\) 0 0
\(129\) 167.224 514.662i 0.114134 0.351267i
\(130\) 0 0
\(131\) 91.8522 + 282.692i 0.0612608 + 0.188541i 0.977003 0.213224i \(-0.0683966\pi\)
−0.915742 + 0.401766i \(0.868397\pi\)
\(132\) 0 0
\(133\) −71.9514 221.444i −0.0469096 0.144373i
\(134\) 0 0
\(135\) 282.959 105.164i 0.180394 0.0670450i
\(136\) 0 0
\(137\) −1236.69 898.511i −0.771226 0.560329i 0.131107 0.991368i \(-0.458147\pi\)
−0.902333 + 0.431040i \(0.858147\pi\)
\(138\) 0 0
\(139\) 1882.70 1367.86i 1.14884 0.834681i 0.160513 0.987034i \(-0.448685\pi\)
0.988326 + 0.152353i \(0.0486851\pi\)
\(140\) 0 0
\(141\) −272.596 198.052i −0.162813 0.118291i
\(142\) 0 0
\(143\) −507.298 −0.296660
\(144\) 0 0
\(145\) 266.637 401.138i 0.152710 0.229743i
\(146\) 0 0
\(147\) −22.6673 + 69.7627i −0.0127181 + 0.0391424i
\(148\) 0 0
\(149\) −3307.05 −1.81828 −0.909142 0.416486i \(-0.863262\pi\)
−0.909142 + 0.416486i \(0.863262\pi\)
\(150\) 0 0
\(151\) 1947.87 1.04977 0.524885 0.851173i \(-0.324108\pi\)
0.524885 + 0.851173i \(0.324108\pi\)
\(152\) 0 0
\(153\) 327.698 1008.55i 0.173156 0.532919i
\(154\) 0 0
\(155\) −61.0331 1463.52i −0.0316277 0.758402i
\(156\) 0 0
\(157\) 948.343 0.482077 0.241038 0.970516i \(-0.422512\pi\)
0.241038 + 0.970516i \(0.422512\pi\)
\(158\) 0 0
\(159\) 1527.99 + 1110.15i 0.762120 + 0.553713i
\(160\) 0 0
\(161\) 2699.04 1960.97i 1.32121 0.959912i
\(162\) 0 0
\(163\) 872.549 + 633.944i 0.419284 + 0.304628i 0.777350 0.629069i \(-0.216564\pi\)
−0.358065 + 0.933696i \(0.616564\pi\)
\(164\) 0 0
\(165\) 58.1306 + 1393.92i 0.0274270 + 0.657674i
\(166\) 0 0
\(167\) 483.632 + 1488.47i 0.224099 + 0.689706i 0.998382 + 0.0568662i \(0.0181109\pi\)
−0.774283 + 0.632840i \(0.781889\pi\)
\(168\) 0 0
\(169\) −632.945 1948.00i −0.288095 0.886665i
\(170\) 0 0
\(171\) 36.2822 111.665i 0.0162255 0.0499371i
\(172\) 0 0
\(173\) −3269.90 + 2375.72i −1.43703 + 1.04406i −0.448373 + 0.893846i \(0.647996\pi\)
−0.988653 + 0.150214i \(0.952004\pi\)
\(174\) 0 0
\(175\) −1907.83 1156.51i −0.824104 0.499566i
\(176\) 0 0
\(177\) −255.835 + 185.875i −0.108643 + 0.0789334i
\(178\) 0 0
\(179\) −624.016 + 1920.52i −0.260565 + 0.801936i 0.732117 + 0.681179i \(0.238532\pi\)
−0.992682 + 0.120758i \(0.961468\pi\)
\(180\) 0 0
\(181\) −330.129 1016.03i −0.135571 0.417244i 0.860108 0.510113i \(-0.170396\pi\)
−0.995678 + 0.0928686i \(0.970396\pi\)
\(182\) 0 0
\(183\) 175.682 + 540.694i 0.0709661 + 0.218411i
\(184\) 0 0
\(185\) −2732.86 3449.50i −1.08608 1.37088i
\(186\) 0 0
\(187\) 3965.01 + 2880.75i 1.55054 + 1.12653i
\(188\) 0 0
\(189\) 389.861 283.250i 0.150043 0.109013i
\(190\) 0 0
\(191\) 80.9997 + 58.8497i 0.0306855 + 0.0222943i 0.603022 0.797724i \(-0.293963\pi\)
−0.572337 + 0.820019i \(0.693963\pi\)
\(192\) 0 0
\(193\) −2144.93 −0.799976 −0.399988 0.916520i \(-0.630986\pi\)
−0.399988 + 0.916520i \(0.630986\pi\)
\(194\) 0 0
\(195\) 383.447 142.511i 0.140816 0.0523357i
\(196\) 0 0
\(197\) −415.322 + 1278.23i −0.150206 + 0.462285i −0.997644 0.0686091i \(-0.978144\pi\)
0.847438 + 0.530894i \(0.178144\pi\)
\(198\) 0 0
\(199\) −462.968 −0.164919 −0.0824595 0.996594i \(-0.526278\pi\)
−0.0824595 + 0.996594i \(0.526278\pi\)
\(200\) 0 0
\(201\) 1066.16 0.374135
\(202\) 0 0
\(203\) 237.611 731.291i 0.0821527 0.252840i
\(204\) 0 0
\(205\) 3095.13 + 3906.76i 1.05450 + 1.33102i
\(206\) 0 0
\(207\) 1682.31 0.564873
\(208\) 0 0
\(209\) 439.000 + 318.952i 0.145293 + 0.105562i
\(210\) 0 0
\(211\) −1479.82 + 1075.15i −0.482820 + 0.350789i −0.802416 0.596765i \(-0.796453\pi\)
0.319597 + 0.947554i \(0.396453\pi\)
\(212\) 0 0
\(213\) 246.353 + 178.986i 0.0792480 + 0.0575770i
\(214\) 0 0
\(215\) −1942.33 542.747i −0.616121 0.172163i
\(216\) 0 0
\(217\) −722.587 2223.89i −0.226048 0.695704i
\(218\) 0 0
\(219\) −643.453 1980.34i −0.198541 0.611047i
\(220\) 0 0
\(221\) 444.075 1366.72i 0.135166 0.415999i
\(222\) 0 0
\(223\) −3906.20 + 2838.02i −1.17300 + 0.852233i −0.991365 0.131133i \(-0.958139\pi\)
−0.181634 + 0.983366i \(0.558139\pi\)
\(224\) 0 0
\(225\) −435.522 1037.28i −0.129043 0.307342i
\(226\) 0 0
\(227\) −4464.64 + 3243.75i −1.30541 + 0.948438i −0.999993 0.00378020i \(-0.998797\pi\)
−0.305420 + 0.952218i \(0.598797\pi\)
\(228\) 0 0
\(229\) 135.094 415.778i 0.0389838 0.119980i −0.929671 0.368392i \(-0.879909\pi\)
0.968655 + 0.248412i \(0.0799086\pi\)
\(230\) 0 0
\(231\) 688.223 + 2118.13i 0.196025 + 0.603303i
\(232\) 0 0
\(233\) 1155.52 + 3556.33i 0.324896 + 0.999926i 0.971488 + 0.237090i \(0.0761936\pi\)
−0.646592 + 0.762836i \(0.723806\pi\)
\(234\) 0 0
\(235\) −695.127 + 1045.78i −0.192958 + 0.290293i
\(236\) 0 0
\(237\) 952.858 + 692.292i 0.261159 + 0.189743i
\(238\) 0 0
\(239\) 2833.90 2058.95i 0.766986 0.557248i −0.134059 0.990973i \(-0.542801\pi\)
0.901045 + 0.433726i \(0.142801\pi\)
\(240\) 0 0
\(241\) 4492.47 + 3263.97i 1.20077 + 0.872410i 0.994360 0.106055i \(-0.0338220\pi\)
0.206410 + 0.978466i \(0.433822\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 263.284 + 73.5696i 0.0686556 + 0.0191845i
\(246\) 0 0
\(247\) 49.1673 151.321i 0.0126657 0.0389812i
\(248\) 0 0
\(249\) −3693.72 −0.940081
\(250\) 0 0
\(251\) 3738.96 0.940242 0.470121 0.882602i \(-0.344210\pi\)
0.470121 + 0.882602i \(0.344210\pi\)
\(252\) 0 0
\(253\) −2402.61 + 7394.48i −0.597039 + 1.83750i
\(254\) 0 0
\(255\) −3806.27 1063.59i −0.934737 0.261194i
\(256\) 0 0
\(257\) −339.325 −0.0823600 −0.0411800 0.999152i \(-0.513112\pi\)
−0.0411800 + 0.999152i \(0.513112\pi\)
\(258\) 0 0
\(259\) −5683.67 4129.42i −1.36357 0.990695i
\(260\) 0 0
\(261\) 313.686 227.907i 0.0743935 0.0540500i
\(262\) 0 0
\(263\) 1512.18 + 1098.66i 0.354544 + 0.257591i 0.750773 0.660560i \(-0.229681\pi\)
−0.396229 + 0.918152i \(0.629681\pi\)
\(264\) 0 0
\(265\) 3896.41 5861.91i 0.903224 1.35885i
\(266\) 0 0
\(267\) 936.055 + 2880.88i 0.214553 + 0.660326i
\(268\) 0 0
\(269\) −298.297 918.065i −0.0676115 0.208087i 0.911543 0.411206i \(-0.134892\pi\)
−0.979154 + 0.203119i \(0.934892\pi\)
\(270\) 0 0
\(271\) 1980.08 6094.05i 0.443841 1.36600i −0.439908 0.898043i \(-0.644989\pi\)
0.883749 0.467961i \(-0.155011\pi\)
\(272\) 0 0
\(273\) 528.314 383.843i 0.117125 0.0850960i
\(274\) 0 0
\(275\) 5181.28 432.904i 1.13616 0.0949275i
\(276\) 0 0
\(277\) −71.9960 + 52.3082i −0.0156167 + 0.0113462i −0.595566 0.803306i \(-0.703072\pi\)
0.579950 + 0.814652i \(0.303072\pi\)
\(278\) 0 0
\(279\) 364.372 1121.42i 0.0781876 0.240637i
\(280\) 0 0
\(281\) 2525.11 + 7771.48i 0.536069 + 1.64985i 0.741329 + 0.671142i \(0.234196\pi\)
−0.205260 + 0.978707i \(0.565804\pi\)
\(282\) 0 0
\(283\) 2282.50 + 7024.82i 0.479437 + 1.47556i 0.839879 + 0.542774i \(0.182626\pi\)
−0.360442 + 0.932782i \(0.617374\pi\)
\(284\) 0 0
\(285\) −421.424 117.759i −0.0875895 0.0244752i
\(286\) 0 0
\(287\) 6437.08 + 4676.81i 1.32393 + 0.961894i
\(288\) 0 0
\(289\) −7257.27 + 5272.71i −1.47716 + 1.07322i
\(290\) 0 0
\(291\) 2580.90 + 1875.13i 0.519915 + 0.377740i
\(292\) 0 0
\(293\) −451.780 −0.0900794 −0.0450397 0.998985i \(-0.514341\pi\)
−0.0450397 + 0.998985i \(0.514341\pi\)
\(294\) 0 0
\(295\) 731.840 + 923.749i 0.144439 + 0.182314i
\(296\) 0 0
\(297\) −347.043 + 1068.09i −0.0678030 + 0.208676i
\(298\) 0 0
\(299\) 2279.76 0.440943
\(300\) 0 0
\(301\) −3219.46 −0.616500
\(302\) 0 0
\(303\) 224.495 690.925i 0.0425641 0.130999i
\(304\) 0 0
\(305\) 1986.02 738.121i 0.372849 0.138573i
\(306\) 0 0
\(307\) 622.914 0.115803 0.0579016 0.998322i \(-0.481559\pi\)
0.0579016 + 0.998322i \(0.481559\pi\)
\(308\) 0 0
\(309\) 3901.15 + 2834.35i 0.718217 + 0.521815i
\(310\) 0 0
\(311\) −554.429 + 402.816i −0.101089 + 0.0734457i −0.637182 0.770714i \(-0.719900\pi\)
0.536092 + 0.844159i \(0.319900\pi\)
\(312\) 0 0
\(313\) −8162.49 5930.40i −1.47403 1.07095i −0.979421 0.201830i \(-0.935311\pi\)
−0.494609 0.869115i \(-0.664689\pi\)
\(314\) 0 0
\(315\) −1115.23 1407.68i −0.199480 0.251790i
\(316\) 0 0
\(317\) 586.284 + 1804.40i 0.103877 + 0.319701i 0.989465 0.144771i \(-0.0462444\pi\)
−0.885588 + 0.464471i \(0.846244\pi\)
\(318\) 0 0
\(319\) 553.752 + 1704.27i 0.0971918 + 0.299126i
\(320\) 0 0
\(321\) 894.300 2752.37i 0.155498 0.478575i
\(322\) 0 0
\(323\) −1243.59 + 903.517i −0.214226 + 0.155644i
\(324\) 0 0
\(325\) −590.191 1405.65i −0.100732 0.239912i
\(326\) 0 0
\(327\) −4852.13 + 3525.28i −0.820560 + 0.596172i
\(328\) 0 0
\(329\) −619.456 + 1906.49i −0.103805 + 0.319478i
\(330\) 0 0
\(331\) 3181.22 + 9790.79i 0.528265 + 1.62583i 0.757768 + 0.652524i \(0.226290\pi\)
−0.229504 + 0.973308i \(0.573710\pi\)
\(332\) 0 0
\(333\) −1094.73 3369.24i −0.180153 0.554454i
\(334\) 0 0
\(335\) −165.557 3969.90i −0.0270010 0.647459i
\(336\) 0 0
\(337\) 817.018 + 593.598i 0.132065 + 0.0959506i 0.651856 0.758342i \(-0.273990\pi\)
−0.519792 + 0.854293i \(0.673990\pi\)
\(338\) 0 0
\(339\) −2185.69 + 1588.00i −0.350178 + 0.254420i
\(340\) 0 0
\(341\) 4408.74 + 3203.14i 0.700137 + 0.508680i
\(342\) 0 0
\(343\) 6558.24 1.03240
\(344\) 0 0
\(345\) −261.235 6264.16i −0.0407664 0.977539i
\(346\) 0 0
\(347\) −64.3581 + 198.074i −0.00995655 + 0.0306431i −0.955911 0.293655i \(-0.905128\pi\)
0.945955 + 0.324298i \(0.105128\pi\)
\(348\) 0 0
\(349\) −1067.46 −0.163725 −0.0818624 0.996644i \(-0.526087\pi\)
−0.0818624 + 0.996644i \(0.526087\pi\)
\(350\) 0 0
\(351\) 329.298 0.0500758
\(352\) 0 0
\(353\) 1333.79 4104.98i 0.201106 0.618941i −0.798745 0.601670i \(-0.794502\pi\)
0.999851 0.0172709i \(-0.00549779\pi\)
\(354\) 0 0
\(355\) 628.207 945.099i 0.0939205 0.141298i
\(356\) 0 0
\(357\) −6308.97 −0.935311
\(358\) 0 0
\(359\) −4899.88 3559.97i −0.720350 0.523365i 0.166146 0.986101i \(-0.446868\pi\)
−0.886496 + 0.462736i \(0.846868\pi\)
\(360\) 0 0
\(361\) 5411.36 3931.58i 0.788943 0.573201i
\(362\) 0 0
\(363\) −968.676 703.785i −0.140061 0.101761i
\(364\) 0 0
\(365\) −7273.98 + 2703.44i −1.04312 + 0.387683i
\(366\) 0 0
\(367\) −2254.77 6939.48i −0.320704 0.987024i −0.973343 0.229355i \(-0.926338\pi\)
0.652639 0.757669i \(-0.273662\pi\)
\(368\) 0 0
\(369\) 1239.85 + 3815.86i 0.174916 + 0.538335i
\(370\) 0 0
\(371\) 3472.25 10686.5i 0.485903 1.49546i
\(372\) 0 0
\(373\) −2908.38 + 2113.06i −0.403727 + 0.293325i −0.771057 0.636766i \(-0.780272\pi\)
0.367330 + 0.930091i \(0.380272\pi\)
\(374\) 0 0
\(375\) −3794.72 + 1782.76i −0.522556 + 0.245496i
\(376\) 0 0
\(377\) 425.088 308.844i 0.0580720 0.0421917i
\(378\) 0 0
\(379\) 3122.89 9611.25i 0.423250 1.30263i −0.481409 0.876496i \(-0.659875\pi\)
0.904660 0.426135i \(-0.140125\pi\)
\(380\) 0 0
\(381\) −1059.67 3261.34i −0.142490 0.438539i
\(382\) 0 0
\(383\) −1828.18 5626.56i −0.243905 0.750662i −0.995815 0.0913961i \(-0.970867\pi\)
0.751910 0.659266i \(-0.229133\pi\)
\(384\) 0 0
\(385\) 7780.09 2891.54i 1.02990 0.382770i
\(386\) 0 0
\(387\) −1313.39 954.236i −0.172516 0.125340i
\(388\) 0 0
\(389\) −4528.90 + 3290.44i −0.590294 + 0.428874i −0.842421 0.538820i \(-0.818870\pi\)
0.252126 + 0.967694i \(0.418870\pi\)
\(390\) 0 0
\(391\) −17818.5 12945.9i −2.30465 1.67443i
\(392\) 0 0
\(393\) 891.720 0.114456
\(394\) 0 0
\(395\) 2429.82 3655.51i 0.309512 0.465642i
\(396\) 0 0
\(397\) 1149.34 3537.29i 0.145298 0.447183i −0.851751 0.523947i \(-0.824459\pi\)
0.997049 + 0.0767645i \(0.0244590\pi\)
\(398\) 0 0
\(399\) −698.519 −0.0876433
\(400\) 0 0
\(401\) −3376.07 −0.420431 −0.210216 0.977655i \(-0.567417\pi\)
−0.210216 + 0.977655i \(0.567417\pi\)
\(402\) 0 0
\(403\) 493.773 1519.68i 0.0610337 0.187842i
\(404\) 0 0
\(405\) −37.7338 904.821i −0.00462965 0.111015i
\(406\) 0 0
\(407\) 16372.7 1.99402
\(408\) 0 0
\(409\) −2593.28 1884.13i −0.313519 0.227785i 0.419886 0.907577i \(-0.362070\pi\)
−0.733405 + 0.679792i \(0.762070\pi\)
\(410\) 0 0
\(411\) −3710.08 + 2695.53i −0.445268 + 0.323506i
\(412\) 0 0
\(413\) 1522.04 + 1105.83i 0.181343 + 0.131754i
\(414\) 0 0
\(415\) 573.573 + 13753.7i 0.0678448 + 1.62685i
\(416\) 0 0
\(417\) −2157.38 6639.75i −0.253351 0.779736i
\(418\) 0 0
\(419\) −948.726 2919.88i −0.110616 0.340442i 0.880391 0.474248i \(-0.157280\pi\)
−0.991008 + 0.133806i \(0.957280\pi\)
\(420\) 0 0
\(421\) −1381.37 + 4251.41i −0.159914 + 0.492164i −0.998626 0.0524116i \(-0.983309\pi\)
0.838712 + 0.544576i \(0.183309\pi\)
\(422\) 0 0
\(423\) −817.787 + 594.157i −0.0940003 + 0.0682952i
\(424\) 0 0
\(425\) −3369.26 + 14338.0i −0.384549 + 1.63646i
\(426\) 0 0
\(427\) 2736.34 1988.07i 0.310119 0.225314i
\(428\) 0 0
\(429\) −470.291 + 1447.41i −0.0529274 + 0.162894i
\(430\) 0 0
\(431\) −1133.94 3489.90i −0.126728 0.390030i 0.867484 0.497466i \(-0.165736\pi\)
−0.994212 + 0.107436i \(0.965736\pi\)
\(432\) 0 0
\(433\) 4041.88 + 12439.6i 0.448592 + 1.38062i 0.878496 + 0.477750i \(0.158547\pi\)
−0.429904 + 0.902875i \(0.641453\pi\)
\(434\) 0 0
\(435\) −897.330 1132.64i −0.0989050 0.124841i
\(436\) 0 0
\(437\) −1972.83 1433.35i −0.215957 0.156902i
\(438\) 0 0
\(439\) 2360.27 1714.84i 0.256605 0.186434i −0.452044 0.891996i \(-0.649305\pi\)
0.708649 + 0.705561i \(0.249305\pi\)
\(440\) 0 0
\(441\) 178.031 + 129.347i 0.0192238 + 0.0139669i
\(442\) 0 0
\(443\) 3937.86 0.422332 0.211166 0.977450i \(-0.432274\pi\)
0.211166 + 0.977450i \(0.432274\pi\)
\(444\) 0 0
\(445\) 10581.7 3932.79i 1.12724 0.418949i
\(446\) 0 0
\(447\) −3065.81 + 9435.58i −0.324402 + 0.998407i
\(448\) 0 0
\(449\) 17103.5 1.79770 0.898848 0.438260i \(-0.144405\pi\)
0.898848 + 0.438260i \(0.144405\pi\)
\(450\) 0 0
\(451\) −18543.0 −1.93605
\(452\) 0 0
\(453\) 1805.77 5557.60i 0.187291 0.576422i
\(454\) 0 0
\(455\) −1511.29 1907.60i −0.155715 0.196548i
\(456\) 0 0
\(457\) 6842.22 0.700362 0.350181 0.936682i \(-0.386120\pi\)
0.350181 + 0.936682i \(0.386120\pi\)
\(458\) 0 0
\(459\) −2573.77 1869.96i −0.261729 0.190157i
\(460\) 0 0
\(461\) 6184.42 4493.24i 0.624809 0.453950i −0.229789 0.973240i \(-0.573804\pi\)
0.854598 + 0.519290i \(0.173804\pi\)
\(462\) 0 0
\(463\) −11520.6 8370.17i −1.15638 0.840162i −0.167067 0.985946i \(-0.553430\pi\)
−0.989316 + 0.145784i \(0.953430\pi\)
\(464\) 0 0
\(465\) −4232.24 1182.62i −0.422076 0.117941i
\(466\) 0 0
\(467\) 1735.26 + 5340.59i 0.171945 + 0.529193i 0.999481 0.0322203i \(-0.0102578\pi\)
−0.827536 + 0.561413i \(0.810258\pi\)
\(468\) 0 0
\(469\) −1960.07 6032.48i −0.192980 0.593932i
\(470\) 0 0
\(471\) 879.163 2705.78i 0.0860078 0.264705i
\(472\) 0 0
\(473\) 6070.02 4410.12i 0.590063 0.428706i
\(474\) 0 0
\(475\) −373.039 + 1587.48i −0.0360341 + 0.153344i
\(476\) 0 0
\(477\) 4583.96 3330.44i 0.440010 0.319686i
\(478\) 0 0
\(479\) −1641.63 + 5052.41i −0.156593 + 0.481943i −0.998319 0.0579619i \(-0.981540\pi\)
0.841726 + 0.539905i \(0.181540\pi\)
\(480\) 0 0
\(481\) −1483.51 4565.77i −0.140628 0.432809i
\(482\) 0 0
\(483\) −3092.83 9518.74i −0.291363 0.896724i
\(484\) 0 0
\(485\) 6581.38 9901.28i 0.616175 0.926998i
\(486\) 0 0
\(487\) 647.849 + 470.690i 0.0602810 + 0.0437967i 0.617518 0.786557i \(-0.288138\pi\)
−0.557237 + 0.830354i \(0.688138\pi\)
\(488\) 0 0
\(489\) 2617.65 1901.83i 0.242074 0.175877i
\(490\) 0 0
\(491\) 3979.58 + 2891.34i 0.365776 + 0.265752i 0.755457 0.655198i \(-0.227415\pi\)
−0.389681 + 0.920950i \(0.627415\pi\)
\(492\) 0 0
\(493\) −5076.27 −0.463740
\(494\) 0 0
\(495\) 4030.97 + 1126.37i 0.366017 + 0.102276i
\(496\) 0 0
\(497\) 559.821 1722.95i 0.0505260 0.155503i
\(498\) 0 0
\(499\) −13306.5 −1.19375 −0.596875 0.802334i \(-0.703591\pi\)
−0.596875 + 0.802334i \(0.703591\pi\)
\(500\) 0 0
\(501\) 4695.20 0.418694
\(502\) 0 0
\(503\) 4829.02 14862.2i 0.428062 1.31744i −0.471969 0.881615i \(-0.656456\pi\)
0.900031 0.435826i \(-0.143544\pi\)
\(504\) 0 0
\(505\) −2607.55 728.629i −0.229771 0.0642050i
\(506\) 0 0
\(507\) −6144.76 −0.538261
\(508\) 0 0
\(509\) 12310.3 + 8943.93i 1.07199 + 0.778847i 0.976269 0.216562i \(-0.0694842\pi\)
0.0957215 + 0.995408i \(0.469484\pi\)
\(510\) 0 0
\(511\) −10022.1 + 7281.48i −0.867616 + 0.630360i
\(512\) 0 0
\(513\) −284.964 207.039i −0.0245253 0.0178187i
\(514\) 0 0
\(515\) 9948.06 14966.2i 0.851192 1.28057i
\(516\) 0 0
\(517\) −1443.64 4443.08i −0.122807 0.377962i
\(518\) 0 0
\(519\) 3746.97 + 11532.0i 0.316905 + 0.975333i
\(520\) 0 0
\(521\) 1552.47 4778.02i 0.130547 0.401783i −0.864324 0.502936i \(-0.832253\pi\)
0.994871 + 0.101153i \(0.0322531\pi\)
\(522\) 0 0
\(523\) 5448.42 3958.51i 0.455531 0.330963i −0.336245 0.941775i \(-0.609157\pi\)
0.791776 + 0.610812i \(0.209157\pi\)
\(524\) 0 0
\(525\) −5068.38 + 4371.21i −0.421338 + 0.363382i
\(526\) 0 0
\(527\) −12489.0 + 9073.76i −1.03231 + 0.750018i
\(528\) 0 0
\(529\) 7037.35 21658.7i 0.578397 1.78012i
\(530\) 0 0
\(531\) 293.161 + 902.256i 0.0239587 + 0.0737374i
\(532\) 0 0
\(533\) 1680.16 + 5171.00i 0.136540 + 0.420227i
\(534\) 0 0
\(535\) −10387.5 2902.57i −0.839418 0.234559i
\(536\) 0 0
\(537\) 4901.08 + 3560.85i 0.393850 + 0.286149i
\(538\) 0 0
\(539\) −822.794 + 597.795i −0.0657519 + 0.0477715i
\(540\) 0 0
\(541\) −4991.42 3626.48i −0.396669 0.288197i 0.371514 0.928427i \(-0.378839\pi\)
−0.768183 + 0.640231i \(0.778839\pi\)
\(542\) 0 0
\(543\) −3204.96 −0.253293
\(544\) 0 0
\(545\) 13880.0 + 17519.7i 1.09092 + 1.37699i
\(546\) 0 0
\(547\) 262.911 809.158i 0.0205508 0.0632488i −0.940255 0.340471i \(-0.889414\pi\)
0.960806 + 0.277222i \(0.0894136\pi\)
\(548\) 0 0
\(549\) 1705.56 0.132589
\(550\) 0 0
\(551\) −562.036 −0.0434547
\(552\) 0 0
\(553\) 2165.31 6664.13i 0.166507 0.512455i
\(554\) 0 0
\(555\) −12375.5 + 4599.47i −0.946506 + 0.351777i
\(556\) 0 0
\(557\) −22738.9 −1.72976 −0.864882 0.501975i \(-0.832607\pi\)
−0.864882 + 0.501975i \(0.832607\pi\)
\(558\) 0 0
\(559\) −1779.82 1293.12i −0.134666 0.0978409i
\(560\) 0 0
\(561\) 11895.0 8642.25i 0.895203 0.650403i
\(562\) 0 0
\(563\) −1053.67 765.535i −0.0788754 0.0573063i 0.547649 0.836708i \(-0.315523\pi\)
−0.626524 + 0.779402i \(0.715523\pi\)
\(564\) 0 0
\(565\) 6252.38 + 7891.93i 0.465557 + 0.587639i
\(566\) 0 0
\(567\) −446.741 1374.93i −0.0330888 0.101837i
\(568\) 0 0
\(569\) 5638.24 + 17352.7i 0.415408 + 1.27850i 0.911885 + 0.410445i \(0.134627\pi\)
−0.496477 + 0.868050i \(0.665373\pi\)
\(570\) 0 0
\(571\) −6852.65 + 21090.3i −0.502232 + 1.54571i 0.303143 + 0.952945i \(0.401964\pi\)
−0.805375 + 0.592766i \(0.798036\pi\)
\(572\) 0 0
\(573\) 242.999 176.549i 0.0177163 0.0128716i
\(574\) 0 0
\(575\) −23284.3 + 1945.44i −1.68873 + 0.141096i
\(576\) 0 0
\(577\) 10327.0 7502.99i 0.745092 0.541341i −0.149210 0.988806i \(-0.547673\pi\)
0.894302 + 0.447465i \(0.147673\pi\)
\(578\) 0 0
\(579\) −1988.46 + 6119.84i −0.142725 + 0.439261i
\(580\) 0 0
\(581\) 6790.68 + 20899.6i 0.484897 + 1.49236i
\(582\) 0 0
\(583\) 8092.08 + 24904.9i 0.574854 + 1.76922i
\(584\) 0 0
\(585\) −51.1344 1226.16i −0.00361393 0.0866586i
\(586\) 0 0
\(587\) −20463.9 14867.9i −1.43890 1.04543i −0.988270 0.152717i \(-0.951198\pi\)
−0.450634 0.892709i \(-0.648802\pi\)
\(588\) 0 0
\(589\) −1382.76 + 1004.63i −0.0967326 + 0.0702804i
\(590\) 0 0
\(591\) 3261.98 + 2369.97i 0.227039 + 0.164954i
\(592\) 0 0
\(593\) −6370.90 −0.441183 −0.220591 0.975366i \(-0.570799\pi\)
−0.220591 + 0.975366i \(0.570799\pi\)
\(594\) 0 0
\(595\) 979.677 + 23491.7i 0.0675006 + 1.61860i
\(596\) 0 0
\(597\) −429.195 + 1320.92i −0.0294234 + 0.0905559i
\(598\) 0 0
\(599\) 22943.4 1.56501 0.782507 0.622642i \(-0.213941\pi\)
0.782507 + 0.622642i \(0.213941\pi\)
\(600\) 0 0
\(601\) −17845.5 −1.21121 −0.605603 0.795767i \(-0.707068\pi\)
−0.605603 + 0.795767i \(0.707068\pi\)
\(602\) 0 0
\(603\) 988.386 3041.94i 0.0667499 0.205435i
\(604\) 0 0
\(605\) −2470.15 + 3716.19i −0.165993 + 0.249727i
\(606\) 0 0
\(607\) 15935.9 1.06560 0.532798 0.846243i \(-0.321141\pi\)
0.532798 + 0.846243i \(0.321141\pi\)
\(608\) 0 0
\(609\) −1866.22 1355.89i −0.124176 0.0902189i
\(610\) 0 0
\(611\) −1108.21 + 805.163i −0.0733771 + 0.0533116i
\(612\) 0 0
\(613\) 22328.9 + 16222.9i 1.47122 + 1.06890i 0.980258 + 0.197720i \(0.0633538\pi\)
0.490960 + 0.871182i \(0.336646\pi\)
\(614\) 0 0
\(615\) 14016.0 5209.16i 0.918990 0.341551i
\(616\) 0 0
\(617\) −5461.72 16809.5i −0.356371 1.09680i −0.955211 0.295927i \(-0.904371\pi\)
0.598840 0.800869i \(-0.295629\pi\)
\(618\) 0 0
\(619\) 392.316 + 1207.42i 0.0254741 + 0.0784013i 0.962985 0.269554i \(-0.0868762\pi\)
−0.937511 + 0.347955i \(0.886876\pi\)
\(620\) 0 0
\(621\) 1559.59 4799.92i 0.100780 0.310168i
\(622\) 0 0
\(623\) 14579.5 10592.7i 0.937587 0.681197i
\(624\) 0 0
\(625\) 7227.42 + 13853.0i 0.462555 + 0.886591i
\(626\) 0 0
\(627\) 1317.00 956.855i 0.0838849 0.0609460i
\(628\) 0 0
\(629\) −14332.2 + 44110.1i −0.908528 + 2.79616i
\(630\) 0 0
\(631\) 8404.62 + 25866.8i 0.530242 + 1.63192i 0.753711 + 0.657206i \(0.228262\pi\)
−0.223468 + 0.974711i \(0.571738\pi\)
\(632\) 0 0
\(633\) 1695.72 + 5218.90i 0.106475 + 0.327697i
\(634\) 0 0
\(635\) −11979.2 + 4452.17i −0.748629 + 0.278235i
\(636\) 0 0
\(637\) 241.256 + 175.283i 0.0150062 + 0.0109026i
\(638\) 0 0
\(639\) 739.058 536.957i 0.0457538 0.0332421i
\(640\) 0 0
\(641\) −21841.7 15868.9i −1.34586 0.977823i −0.999207 0.0398293i \(-0.987319\pi\)
−0.346652 0.937994i \(-0.612681\pi\)
\(642\) 0 0
\(643\) 12216.8 0.749275 0.374637 0.927171i \(-0.377767\pi\)
0.374637 + 0.927171i \(0.377767\pi\)
\(644\) 0 0
\(645\) −3349.19 + 5038.65i −0.204456 + 0.307592i
\(646\) 0 0
\(647\) 1903.36 5857.93i 0.115655 0.355949i −0.876428 0.481532i \(-0.840080\pi\)
0.992083 + 0.125584i \(0.0400804\pi\)
\(648\) 0 0
\(649\) −4384.48 −0.265187
\(650\) 0 0
\(651\) −7015.02 −0.422335
\(652\) 0 0
\(653\) 6155.05 18943.3i 0.368860 1.13524i −0.578668 0.815563i \(-0.696427\pi\)
0.947528 0.319672i \(-0.103573\pi\)
\(654\) 0 0
\(655\) −138.469 3320.36i −0.00826021 0.198072i
\(656\) 0 0
\(657\) −6246.77 −0.370943
\(658\) 0 0
\(659\) 4382.17 + 3183.83i 0.259037 + 0.188201i 0.709722 0.704482i \(-0.248820\pi\)
−0.450686 + 0.892683i \(0.648820\pi\)
\(660\) 0 0
\(661\) −14170.3 + 10295.3i −0.833830 + 0.605813i −0.920640 0.390412i \(-0.872333\pi\)
0.0868103 + 0.996225i \(0.472333\pi\)
\(662\) 0 0
\(663\) −3487.81 2534.04i −0.204307 0.148438i
\(664\) 0 0
\(665\) 108.468 + 2600.97i 0.00632514 + 0.151671i
\(666\) 0 0
\(667\) −2488.52 7658.89i −0.144462 0.444608i
\(668\) 0 0
\(669\) 4476.11 + 13776.0i 0.258679 + 0.796133i
\(670\) 0 0
\(671\) −2435.81 + 7496.66i −0.140139 + 0.431305i
\(672\) 0 0
\(673\) −8199.83 + 5957.52i −0.469659 + 0.341227i −0.797308 0.603572i \(-0.793743\pi\)
0.327650 + 0.944799i \(0.393743\pi\)
\(674\) 0 0
\(675\) −3363.28 + 281.007i −0.191782 + 0.0160237i
\(676\) 0 0
\(677\) −13791.8 + 10020.4i −0.782959 + 0.568853i −0.905866 0.423565i \(-0.860778\pi\)
0.122907 + 0.992418i \(0.460778\pi\)
\(678\) 0 0
\(679\) 5864.93 18050.4i 0.331481 1.02019i
\(680\) 0 0
\(681\) 5116.02 + 15745.5i 0.287880 + 0.886004i
\(682\) 0 0
\(683\) 3758.08 + 11566.2i 0.210540 + 0.647976i 0.999440 + 0.0334546i \(0.0106509\pi\)
−0.788900 + 0.614522i \(0.789349\pi\)
\(684\) 0 0
\(685\) 10613.1 + 13396.1i 0.591977 + 0.747209i
\(686\) 0 0
\(687\) −1061.05 770.895i −0.0589249 0.0428115i
\(688\) 0 0
\(689\) 6211.88 4513.20i 0.343474 0.249549i
\(690\) 0 0
\(691\) −21187.2 15393.4i −1.16643 0.847458i −0.175849 0.984417i \(-0.556267\pi\)
−0.990577 + 0.136959i \(0.956267\pi\)
\(692\) 0 0
\(693\) 6681.41 0.366242
\(694\) 0 0
\(695\) −24388.4 + 9064.15i −1.33108 + 0.494709i
\(696\) 0 0
\(697\) 16232.1 49957.3i 0.882116 2.71487i
\(698\) 0 0
\(699\) 11218.0 0.607017
\(700\) 0 0
\(701\) −4648.66 −0.250467 −0.125234 0.992127i \(-0.539968\pi\)
−0.125234 + 0.992127i \(0.539968\pi\)
\(702\) 0 0
\(703\) −1586.84 + 4883.80i −0.0851336 + 0.262014i
\(704\) 0 0
\(705\) 2339.36 + 2952.80i 0.124972 + 0.157743i
\(706\) 0 0
\(707\) −4322.07 −0.229912
\(708\) 0 0
\(709\) 29078.7 + 21126.9i 1.54030 + 1.11910i 0.950145 + 0.311808i \(0.100935\pi\)
0.590158 + 0.807288i \(0.299065\pi\)
\(710\) 0 0
\(711\) 2858.57 2076.88i 0.150780 0.109548i
\(712\) 0 0
\(713\) −19812.6 14394.7i −1.04065 0.756080i
\(714\) 0 0
\(715\) 5462.51 + 1526.39i 0.285715 + 0.0798374i
\(716\) 0 0
\(717\) −3247.36 9994.34i −0.169142 0.520565i
\(718\) 0 0
\(719\) −7738.74 23817.4i −0.401399 1.23538i −0.923865 0.382720i \(-0.874988\pi\)
0.522465 0.852661i \(-0.325012\pi\)
\(720\) 0 0
\(721\) 8865.12 27284.0i 0.457912 1.40931i
\(722\) 0 0
\(723\) 13477.4 9791.92i 0.693265 0.503686i
\(724\) 0 0
\(725\) −4078.08 + 3517.13i −0.208905 + 0.180169i
\(726\) 0 0
\(727\) −2409.43 + 1750.55i −0.122917 + 0.0893046i −0.647545 0.762027i \(-0.724204\pi\)
0.524628 + 0.851331i \(0.324204\pi\)
\(728\) 0 0
\(729\) 225.273 693.320i 0.0114451 0.0352243i
\(730\) 0 0
\(731\) 6567.89 + 20213.9i 0.332315 + 1.02276i
\(732\) 0 0
\(733\) −7178.34 22092.7i −0.361716 1.11325i −0.952012 0.306061i \(-0.900989\pi\)
0.590296 0.807187i \(-0.299011\pi\)
\(734\) 0 0
\(735\) 453.985 682.992i 0.0227830 0.0342756i
\(736\) 0 0
\(737\) 11959.1 + 8688.77i 0.597717 + 0.434267i
\(738\) 0 0
\(739\) 20298.9 14748.0i 1.01043 0.734119i 0.0461291 0.998935i \(-0.485311\pi\)
0.964299 + 0.264817i \(0.0853114\pi\)
\(740\) 0 0
\(741\) −386.165 280.565i −0.0191446 0.0139093i
\(742\) 0 0
\(743\) 21482.7 1.06073 0.530367 0.847768i \(-0.322054\pi\)
0.530367 + 0.847768i \(0.322054\pi\)
\(744\) 0 0
\(745\) 35609.9 + 9950.48i 1.75120 + 0.489339i
\(746\) 0 0
\(747\) −3424.27 + 10538.8i −0.167721 + 0.516191i
\(748\) 0 0
\(749\) −17217.4 −0.839934
\(750\) 0 0
\(751\) −24836.3 −1.20678 −0.603389 0.797447i \(-0.706183\pi\)
−0.603389 + 0.797447i \(0.706183\pi\)
\(752\) 0 0
\(753\) 3466.20 10667.9i 0.167750 0.516280i
\(754\) 0 0
\(755\) −20974.4 5860.88i −1.01104 0.282516i
\(756\) 0 0
\(757\) −13579.1 −0.651969 −0.325985 0.945375i \(-0.605696\pi\)
−0.325985 + 0.945375i \(0.605696\pi\)
\(758\) 0 0
\(759\) 18870.4 + 13710.1i 0.902438 + 0.655660i
\(760\) 0 0
\(761\) 23520.2 17088.4i 1.12038 0.814001i 0.136110 0.990694i \(-0.456540\pi\)
0.984266 + 0.176693i \(0.0565401\pi\)
\(762\) 0 0
\(763\) 28866.8 + 20973.0i 1.36966 + 0.995115i
\(764\) 0 0
\(765\) −6563.20 + 9873.94i −0.310187 + 0.466657i
\(766\) 0 0
\(767\) 397.272 + 1222.68i 0.0187023 + 0.0575598i
\(768\) 0 0
\(769\) 412.241 + 1268.75i 0.0193313 + 0.0594957i 0.960257 0.279118i \(-0.0900421\pi\)
−0.940926 + 0.338614i \(0.890042\pi\)
\(770\) 0 0
\(771\) −314.572 + 968.152i −0.0146939 + 0.0452233i
\(772\) 0 0
\(773\) −21454.6 + 15587.7i −0.998277 + 0.725291i −0.961718 0.274041i \(-0.911640\pi\)
−0.0365590 + 0.999331i \(0.511640\pi\)
\(774\) 0 0
\(775\) −3746.32 + 15942.6i −0.173641 + 0.738934i
\(776\) 0 0
\(777\) −17051.0 + 12388.3i −0.787260 + 0.571978i
\(778\) 0 0
\(779\) 1797.19 5531.18i 0.0826586 0.254397i
\(780\) 0 0
\(781\) 1304.66 + 4015.34i 0.0597754 + 0.183970i
\(782\) 0 0
\(783\) −359.453 1106.28i −0.0164059 0.0504921i
\(784\) 0 0
\(785\) −10211.6 2853.44i −0.464291 0.129737i
\(786\) 0 0
\(787\) −18118.7 13164.0i −0.820662 0.596246i 0.0962401 0.995358i \(-0.469318\pi\)
−0.916902 + 0.399112i \(0.869318\pi\)
\(788\) 0 0
\(789\) 4536.55 3295.99i 0.204696 0.148720i
\(790\) 0 0
\(791\) 13003.4 + 9447.50i 0.584509 + 0.424671i
\(792\) 0 0
\(793\) 2311.26 0.103500
\(794\) 0 0
\(795\) −13112.8 16551.4i −0.584987 0.738387i
\(796\) 0 0
\(797\) 1777.37 5470.18i 0.0789933 0.243116i −0.903759 0.428041i \(-0.859204\pi\)
0.982753 + 0.184924i \(0.0592040\pi\)
\(798\) 0 0
\(799\) 13233.9 0.585961
\(800\) 0 0
\(801\) 9087.41 0.400859
\(802\) 0 0
\(803\) 8921.40 27457.3i 0.392067 1.20666i
\(804\) 0 0
\(805\) −34963.2 + 12994.4i −1.53079 + 0.568933i
\(806\) 0 0
\(807\) −2895.93 −0.126322
\(808\) 0 0
\(809\) −15685.1 11395.9i −0.681655 0.495251i 0.192251 0.981346i \(-0.438421\pi\)
−0.873906 + 0.486094i \(0.838421\pi\)
\(810\) 0 0
\(811\) 10271.5 7462.65i 0.444734 0.323118i −0.342779 0.939416i \(-0.611368\pi\)
0.787513 + 0.616298i \(0.211368\pi\)
\(812\) 0 0
\(813\) −15551.7 11299.0i −0.670876 0.487420i
\(814\) 0 0
\(815\) −7488.03 9451.60i −0.321834 0.406227i
\(816\) 0 0
\(817\) 727.186 + 2238.05i 0.0311396 + 0.0958377i
\(818\) 0 0
\(819\) −605.394 1863.21i −0.0258293 0.0794944i
\(820\) 0 0
\(821\) 2046.46 6298.34i 0.0869937 0.267739i −0.898091 0.439810i \(-0.855046\pi\)
0.985085 + 0.172071i \(0.0550458\pi\)
\(822\) 0 0
\(823\) −35068.1 + 25478.4i −1.48529 + 1.07913i −0.509489 + 0.860477i \(0.670166\pi\)
−0.975803 + 0.218651i \(0.929834\pi\)
\(824\) 0 0
\(825\) 3568.16 15184.4i 0.150579 0.640791i
\(826\) 0 0
\(827\) 23468.4 17050.8i 0.986793 0.716947i 0.0275762 0.999620i \(-0.491221\pi\)
0.959216 + 0.282673i \(0.0912211\pi\)
\(828\) 0 0
\(829\) −937.054 + 2883.96i −0.0392584 + 0.120825i −0.968765 0.247980i \(-0.920233\pi\)
0.929507 + 0.368805i \(0.120233\pi\)
\(830\) 0 0
\(831\) 82.5001 + 253.909i 0.00344392 + 0.0105993i
\(832\) 0 0
\(833\) −890.282 2740.01i −0.0370305 0.113968i
\(834\) 0 0
\(835\) −729.085 17482.8i −0.0302168 0.724570i
\(836\) 0 0
\(837\) −2861.81 2079.23i −0.118182 0.0858645i
\(838\) 0 0
\(839\) 34179.2 24832.7i 1.40643 1.02183i 0.412606 0.910910i \(-0.364619\pi\)
0.993829 0.110925i \(-0.0353812\pi\)
\(840\) 0 0
\(841\) 18229.5 + 13244.5i 0.747449 + 0.543053i
\(842\) 0 0
\(843\) 24514.3 1.00156
\(844\) 0 0
\(845\) 954.178 + 22880.3i 0.0388458 + 0.931485i
\(846\) 0 0
\(847\) −2201.25 + 6774.77i −0.0892987 + 0.274833i
\(848\) 0 0
\(849\) 22159.0 0.895754
\(850\) 0 0
\(851\) −73577.7 −2.96382
\(852\) 0 0
\(853\) −3891.06 + 11975.5i −0.156187 + 0.480694i −0.998279 0.0586395i \(-0.981324\pi\)
0.842092 + 0.539333i \(0.181324\pi\)
\(854\) 0 0
\(855\) −726.667 + 1093.23i −0.0290661 + 0.0437281i
\(856\) 0 0
\(857\) −10465.5 −0.417146 −0.208573 0.978007i \(-0.566882\pi\)
−0.208573 + 0.978007i \(0.566882\pi\)
\(858\) 0 0
\(859\) −26753.8 19437.8i −1.06266 0.772069i −0.0880833 0.996113i \(-0.528074\pi\)
−0.974579 + 0.224044i \(0.928074\pi\)
\(860\) 0 0
\(861\) 19311.2 14030.4i 0.764373 0.555350i
\(862\) 0 0
\(863\) −27516.9 19992.2i −1.08538 0.788577i −0.106770 0.994284i \(-0.534051\pi\)
−0.978614 + 0.205706i \(0.934051\pi\)
\(864\) 0 0
\(865\) 42358.0 15742.7i 1.66499 0.618808i
\(866\) 0 0
\(867\) 8316.09 + 25594.3i 0.325755 + 1.00257i
\(868\) 0 0
\(869\) 5046.26 + 15530.8i 0.196988 + 0.606266i
\(870\) 0 0
\(871\) 1339.40 4122.24i 0.0521053 0.160364i
\(872\) 0 0
\(873\) 7742.70 5625.40i 0.300173 0.218088i
\(874\) 0 0
\(875\) 17063.4 + 18193.6i 0.659256 + 0.702919i
\(876\) 0 0
\(877\) 32744.6 23790.3i 1.26078 0.916012i 0.261987 0.965071i \(-0.415622\pi\)
0.998796 + 0.0490593i \(0.0156223\pi\)
\(878\) 0 0
\(879\) −418.823 + 1289.00i −0.0160712 + 0.0494619i
\(880\) 0 0
\(881\) −9724.64 29929.4i −0.371886 1.14455i −0.945556 0.325461i \(-0.894481\pi\)
0.573670 0.819087i \(-0.305519\pi\)
\(882\) 0 0
\(883\) −5452.62 16781.4i −0.207809 0.639570i −0.999586 0.0287605i \(-0.990844\pi\)
0.791777 0.610810i \(-0.209156\pi\)
\(884\) 0 0
\(885\) 3314.07 1231.70i 0.125877 0.0467833i
\(886\) 0 0
\(887\) −19087.1 13867.6i −0.722529 0.524948i 0.164663 0.986350i \(-0.447347\pi\)
−0.887191 + 0.461402i \(0.847347\pi\)
\(888\) 0 0
\(889\) −16504.9 + 11991.5i −0.622675 + 0.452400i
\(890\) 0 0
\(891\) 2725.71 + 1980.35i 0.102486 + 0.0744603i
\(892\) 0 0
\(893\) 1465.24 0.0549075
\(894\) 0 0
\(895\) 12497.9 18802.3i 0.466770 0.702227i
\(896\) 0 0
\(897\) 2113.45 6504.54i 0.0786690 0.242118i
\(898\) 0 0
\(899\) −5644.37 −0.209399
\(900\) 0 0
\(901\) −74180.5 −2.74285
\(902\) 0 0
\(903\) −2984.60 + 9185.66i −0.109990 + 0.338515i
\(904\) 0 0
\(905\) 497.677 + 11933.8i 0.0182799 + 0.438336i
\(906\) 0 0
\(907\) 33213.9 1.21593 0.607966 0.793963i \(-0.291986\pi\)
0.607966 + 0.793963i \(0.291986\pi\)
\(908\) 0 0
\(909\) −1763.21 1281.05i −0.0643366 0.0467432i
\(910\) 0 0
\(911\) −13321.6 + 9678.70i −0.484483 + 0.351997i −0.803059 0.595900i \(-0.796795\pi\)
0.318576 + 0.947897i \(0.396795\pi\)
\(912\) 0 0
\(913\) −41432.2 30102.3i −1.50187 1.09117i
\(914\) 0 0
\(915\) −264.845 6350.72i −0.00956884 0.229452i
\(916\) 0 0
\(917\) −1639.37 5045.47i −0.0590369 0.181697i
\(918\) 0 0
\(919\) −12005.6 36949.3i −0.430932 1.32627i −0.897198 0.441628i \(-0.854401\pi\)
0.466266 0.884645i \(-0.345599\pi\)
\(920\) 0 0
\(921\) 577.473 1777.28i 0.0206606 0.0635867i
\(922\) 0 0
\(923\) 1001.52 727.650i 0.0357157 0.0259490i
\(924\) 0 0
\(925\) 19048.0 + 45366.5i 0.677076 + 1.61259i
\(926\) 0 0
\(927\) 11703.5 8503.06i 0.414663 0.301270i
\(928\) 0 0
\(929\) 10093.7 31065.3i 0.356474 1.09711i −0.598676 0.800991i \(-0.704306\pi\)
0.955150 0.296123i \(-0.0956937\pi\)
\(930\) 0 0
\(931\) −98.5705 303.369i −0.00346994 0.0106794i
\(932\) 0 0
\(933\) 635.319 + 1955.31i 0.0222930 + 0.0686109i
\(934\) 0 0
\(935\) −34026.9 42949.7i −1.19016 1.50225i
\(936\) 0 0
\(937\) −13700.2 9953.76i −0.477658 0.347039i 0.322760 0.946481i \(-0.395389\pi\)
−0.800418 + 0.599442i \(0.795389\pi\)
\(938\) 0 0
\(939\) −24487.5 + 17791.2i −0.851032 + 0.618311i
\(940\) 0 0
\(941\) −8939.88 6495.20i −0.309704 0.225013i 0.422065 0.906565i \(-0.361305\pi\)
−0.731770 + 0.681552i \(0.761305\pi\)
\(942\) 0 0
\(943\) 83331.0 2.87766
\(944\) 0 0
\(945\) −5050.23 + 1876.96i −0.173845 + 0.0646112i
\(946\) 0 0
\(947\) −10750.0 + 33085.0i −0.368878 + 1.13529i 0.578639 + 0.815584i \(0.303584\pi\)
−0.947517 + 0.319706i \(0.896416\pi\)
\(948\) 0 0
\(949\) −8465.22 −0.289560
\(950\) 0 0
\(951\) 5691.77 0.194078
\(952\) 0 0
\(953\) 94.7665 291.661i 0.00322118 0.00991378i −0.949433 0.313970i \(-0.898341\pi\)
0.952654 + 0.304056i \(0.0983410\pi\)
\(954\) 0 0
\(955\) −695.122 877.403i −0.0235535 0.0297299i
\(956\) 0 0
\(957\) 5375.94 0.181588
\(958\) 0 0
\(959\) 22072.5 + 16036.6i 0.743229 + 0.539988i
\(960\) 0 0
\(961\) 10214.8 7421.49i 0.342882 0.249118i
\(962\) 0 0
\(963\) −7023.93 5103.18i −0.235039 0.170766i
\(964\) 0 0
\(965\) 23096.3 + 6453.80i 0.770462 + 0.215290i
\(966\) 0 0
\(967\) −9517.31 29291.3i −0.316500 0.974088i −0.975132 0.221623i \(-0.928865\pi\)
0.658632 0.752465i \(-0.271135\pi\)
\(968\) 0 0
\(969\) 1425.02 + 4385.77i 0.0472428 + 0.145398i
\(970\) 0 0
\(971\) 9174.90 28237.4i 0.303230 0.933247i −0.677102 0.735890i \(-0.736764\pi\)
0.980332 0.197357i \(-0.0632358\pi\)
\(972\) 0 0
\(973\) −33602.3 + 24413.5i −1.10713 + 0.804380i
\(974\) 0 0
\(975\) −4557.70 + 380.803i −0.149706 + 0.0125081i
\(976\) 0 0
\(977\) −11685.1 + 8489.69i −0.382639 + 0.278003i −0.762432 0.647068i \(-0.775995\pi\)
0.379794 + 0.925071i \(0.375995\pi\)
\(978\) 0 0
\(979\) −12978.3 + 39943.1i −0.423686 + 1.30397i
\(980\) 0 0
\(981\) 5560.04 + 17112.0i 0.180957 + 0.556927i
\(982\) 0 0
\(983\) 483.034 + 1486.63i 0.0156728 + 0.0482360i 0.958587 0.284800i \(-0.0919271\pi\)
−0.942914 + 0.333036i \(0.891927\pi\)
\(984\) 0 0
\(985\) 8318.16 12514.2i 0.269075 0.404806i
\(986\) 0 0
\(987\) 4865.27 + 3534.82i 0.156903 + 0.113997i
\(988\) 0 0
\(989\) −27278.2 + 19818.8i −0.877044 + 0.637210i
\(990\) 0 0
\(991\) 37144.7 + 26987.2i 1.19066 + 0.865063i 0.993334 0.115275i \(-0.0367750\pi\)
0.197324 + 0.980338i \(0.436775\pi\)
\(992\) 0 0
\(993\) 30883.9 0.986981
\(994\) 0 0
\(995\) 4985.17 + 1393.01i 0.158835 + 0.0443832i
\(996\) 0 0
\(997\) 1872.45 5762.82i 0.0594796 0.183059i −0.916902 0.399112i \(-0.869318\pi\)
0.976382 + 0.216053i \(0.0693183\pi\)
\(998\) 0 0
\(999\) −10627.9 −0.336588
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 300.4.m.a.61.1 32
25.16 even 5 inner 300.4.m.a.241.1 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.4.m.a.61.1 32 1.1 even 1 trivial
300.4.m.a.241.1 yes 32 25.16 even 5 inner