Properties

Label 336.4.h.b.239.9
Level $336$
Weight $4$
Character 336.239
Analytic conductor $19.825$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,4,Mod(239,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("336.239");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.h (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: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.9
Character \(\chi\) \(=\) 336.239
Dual form 336.4.h.b.239.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.99829 - 4.79655i) q^{3} +16.7287i q^{5} -7.00000i q^{7} +(-19.0137 + 19.1697i) q^{9} +O(q^{10})\) \(q+(-1.99829 - 4.79655i) q^{3} +16.7287i q^{5} -7.00000i q^{7} +(-19.0137 + 19.1697i) q^{9} -60.9218 q^{11} +73.9074 q^{13} +(80.2398 - 33.4286i) q^{15} -16.3004i q^{17} -153.871i q^{19} +(-33.5758 + 13.9880i) q^{21} +111.882 q^{23} -154.848 q^{25} +(129.943 + 52.8935i) q^{27} -155.121i q^{29} +53.7611i q^{31} +(121.739 + 292.214i) q^{33} +117.101 q^{35} -63.0884 q^{37} +(-147.688 - 354.500i) q^{39} -458.858i q^{41} -95.5852i q^{43} +(-320.684 - 318.074i) q^{45} +70.5132 q^{47} -49.0000 q^{49} +(-78.1857 + 32.5729i) q^{51} -417.021i q^{53} -1019.14i q^{55} +(-738.050 + 307.478i) q^{57} -328.024 q^{59} -637.654 q^{61} +(134.188 + 133.096i) q^{63} +1236.37i q^{65} -201.474i q^{67} +(-223.572 - 536.647i) q^{69} +394.426 q^{71} +977.889 q^{73} +(309.430 + 742.735i) q^{75} +426.453i q^{77} -514.984i q^{79} +(-5.95806 - 728.976i) q^{81} -1133.84 q^{83} +272.684 q^{85} +(-744.045 + 309.976i) q^{87} -1035.66i q^{89} -517.352i q^{91} +(257.867 - 107.430i) q^{93} +2574.06 q^{95} +995.638 q^{97} +(1158.35 - 1167.86i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.99829 4.79655i −0.384570 0.923096i
\(4\) 0 0
\(5\) 16.7287i 1.49626i 0.663554 + 0.748128i \(0.269047\pi\)
−0.663554 + 0.748128i \(0.730953\pi\)
\(6\) 0 0
\(7\) 7.00000i 0.377964i
\(8\) 0 0
\(9\) −19.0137 + 19.1697i −0.704211 + 0.709990i
\(10\) 0 0
\(11\) −60.9218 −1.66987 −0.834937 0.550345i \(-0.814496\pi\)
−0.834937 + 0.550345i \(0.814496\pi\)
\(12\) 0 0
\(13\) 73.9074 1.57679 0.788393 0.615171i \(-0.210913\pi\)
0.788393 + 0.615171i \(0.210913\pi\)
\(14\) 0 0
\(15\) 80.2398 33.4286i 1.38119 0.575416i
\(16\) 0 0
\(17\) 16.3004i 0.232555i −0.993217 0.116277i \(-0.962904\pi\)
0.993217 0.116277i \(-0.0370961\pi\)
\(18\) 0 0
\(19\) 153.871i 1.85792i −0.370182 0.928959i \(-0.620705\pi\)
0.370182 0.928959i \(-0.379295\pi\)
\(20\) 0 0
\(21\) −33.5758 + 13.9880i −0.348897 + 0.145354i
\(22\) 0 0
\(23\) 111.882 1.01430 0.507152 0.861856i \(-0.330698\pi\)
0.507152 + 0.861856i \(0.330698\pi\)
\(24\) 0 0
\(25\) −154.848 −1.23878
\(26\) 0 0
\(27\) 129.943 + 52.8935i 0.926208 + 0.377013i
\(28\) 0 0
\(29\) 155.121i 0.993284i −0.867955 0.496642i \(-0.834566\pi\)
0.867955 0.496642i \(-0.165434\pi\)
\(30\) 0 0
\(31\) 53.7611i 0.311477i 0.987798 + 0.155738i \(0.0497756\pi\)
−0.987798 + 0.155738i \(0.950224\pi\)
\(32\) 0 0
\(33\) 121.739 + 292.214i 0.642184 + 1.54145i
\(34\) 0 0
\(35\) 117.101 0.565532
\(36\) 0 0
\(37\) −63.0884 −0.280315 −0.140158 0.990129i \(-0.544761\pi\)
−0.140158 + 0.990129i \(0.544761\pi\)
\(38\) 0 0
\(39\) −147.688 354.500i −0.606385 1.45553i
\(40\) 0 0
\(41\) 458.858i 1.74784i −0.486067 0.873921i \(-0.661569\pi\)
0.486067 0.873921i \(-0.338431\pi\)
\(42\) 0 0
\(43\) 95.5852i 0.338991i −0.985531 0.169495i \(-0.945786\pi\)
0.985531 0.169495i \(-0.0542138\pi\)
\(44\) 0 0
\(45\) −320.684 318.074i −1.06233 1.05368i
\(46\) 0 0
\(47\) 70.5132 0.218839 0.109419 0.993996i \(-0.465101\pi\)
0.109419 + 0.993996i \(0.465101\pi\)
\(48\) 0 0
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −78.1857 + 32.5729i −0.214670 + 0.0894336i
\(52\) 0 0
\(53\) 417.021i 1.08080i −0.841410 0.540398i \(-0.818274\pi\)
0.841410 0.540398i \(-0.181726\pi\)
\(54\) 0 0
\(55\) 1019.14i 2.49856i
\(56\) 0 0
\(57\) −738.050 + 307.478i −1.71504 + 0.714500i
\(58\) 0 0
\(59\) −328.024 −0.723815 −0.361907 0.932214i \(-0.617874\pi\)
−0.361907 + 0.932214i \(0.617874\pi\)
\(60\) 0 0
\(61\) −637.654 −1.33841 −0.669207 0.743076i \(-0.733366\pi\)
−0.669207 + 0.743076i \(0.733366\pi\)
\(62\) 0 0
\(63\) 134.188 + 133.096i 0.268351 + 0.266167i
\(64\) 0 0
\(65\) 1236.37i 2.35928i
\(66\) 0 0
\(67\) 201.474i 0.367372i −0.982985 0.183686i \(-0.941197\pi\)
0.982985 0.183686i \(-0.0588030\pi\)
\(68\) 0 0
\(69\) −223.572 536.647i −0.390071 0.936300i
\(70\) 0 0
\(71\) 394.426 0.659293 0.329646 0.944105i \(-0.393071\pi\)
0.329646 + 0.944105i \(0.393071\pi\)
\(72\) 0 0
\(73\) 977.889 1.56785 0.783926 0.620854i \(-0.213214\pi\)
0.783926 + 0.620854i \(0.213214\pi\)
\(74\) 0 0
\(75\) 309.430 + 742.735i 0.476399 + 1.14352i
\(76\) 0 0
\(77\) 426.453i 0.631153i
\(78\) 0 0
\(79\) 514.984i 0.733420i −0.930335 0.366710i \(-0.880484\pi\)
0.930335 0.366710i \(-0.119516\pi\)
\(80\) 0 0
\(81\) −5.95806 728.976i −0.00817293 0.999967i
\(82\) 0 0
\(83\) −1133.84 −1.49946 −0.749728 0.661746i \(-0.769816\pi\)
−0.749728 + 0.661746i \(0.769816\pi\)
\(84\) 0 0
\(85\) 272.684 0.347961
\(86\) 0 0
\(87\) −744.045 + 309.976i −0.916897 + 0.381988i
\(88\) 0 0
\(89\) 1035.66i 1.23348i −0.787167 0.616740i \(-0.788453\pi\)
0.787167 0.616740i \(-0.211547\pi\)
\(90\) 0 0
\(91\) 517.352i 0.595969i
\(92\) 0 0
\(93\) 257.867 107.430i 0.287523 0.119785i
\(94\) 0 0
\(95\) 2574.06 2.77992
\(96\) 0 0
\(97\) 995.638 1.04218 0.521092 0.853501i \(-0.325525\pi\)
0.521092 + 0.853501i \(0.325525\pi\)
\(98\) 0 0
\(99\) 1158.35 1167.86i 1.17594 1.18560i
\(100\) 0 0
\(101\) 1457.31i 1.43572i 0.696190 + 0.717858i \(0.254877\pi\)
−0.696190 + 0.717858i \(0.745123\pi\)
\(102\) 0 0
\(103\) 359.018i 0.343448i 0.985145 + 0.171724i \(0.0549337\pi\)
−0.985145 + 0.171724i \(0.945066\pi\)
\(104\) 0 0
\(105\) −234.000 561.678i −0.217487 0.522040i
\(106\) 0 0
\(107\) 765.314 0.691455 0.345728 0.938335i \(-0.387632\pi\)
0.345728 + 0.938335i \(0.387632\pi\)
\(108\) 0 0
\(109\) 736.101 0.646842 0.323421 0.946255i \(-0.395167\pi\)
0.323421 + 0.946255i \(0.395167\pi\)
\(110\) 0 0
\(111\) 126.069 + 302.606i 0.107801 + 0.258758i
\(112\) 0 0
\(113\) 1228.45i 1.02268i 0.859380 + 0.511338i \(0.170850\pi\)
−0.859380 + 0.511338i \(0.829150\pi\)
\(114\) 0 0
\(115\) 1871.63i 1.51766i
\(116\) 0 0
\(117\) −1405.25 + 1416.79i −1.11039 + 1.11950i
\(118\) 0 0
\(119\) −114.103 −0.0878974
\(120\) 0 0
\(121\) 2380.47 1.78848
\(122\) 0 0
\(123\) −2200.93 + 916.929i −1.61343 + 0.672168i
\(124\) 0 0
\(125\) 499.315i 0.357281i
\(126\) 0 0
\(127\) 127.914i 0.0893745i −0.999001 0.0446872i \(-0.985771\pi\)
0.999001 0.0446872i \(-0.0142291\pi\)
\(128\) 0 0
\(129\) −458.479 + 191.007i −0.312921 + 0.130366i
\(130\) 0 0
\(131\) −1142.47 −0.761968 −0.380984 0.924582i \(-0.624415\pi\)
−0.380984 + 0.924582i \(0.624415\pi\)
\(132\) 0 0
\(133\) −1077.10 −0.702227
\(134\) 0 0
\(135\) −884.837 + 2173.78i −0.564108 + 1.38584i
\(136\) 0 0
\(137\) 660.795i 0.412084i −0.978543 0.206042i \(-0.933942\pi\)
0.978543 0.206042i \(-0.0660583\pi\)
\(138\) 0 0
\(139\) 502.841i 0.306837i 0.988161 + 0.153419i \(0.0490283\pi\)
−0.988161 + 0.153419i \(0.950972\pi\)
\(140\) 0 0
\(141\) −140.906 338.220i −0.0841588 0.202009i
\(142\) 0 0
\(143\) −4502.57 −2.63304
\(144\) 0 0
\(145\) 2594.97 1.48621
\(146\) 0 0
\(147\) 97.9160 + 235.031i 0.0549386 + 0.131871i
\(148\) 0 0
\(149\) 470.137i 0.258491i −0.991613 0.129245i \(-0.958745\pi\)
0.991613 0.129245i \(-0.0412555\pi\)
\(150\) 0 0
\(151\) 833.149i 0.449011i −0.974473 0.224506i \(-0.927923\pi\)
0.974473 0.224506i \(-0.0720767\pi\)
\(152\) 0 0
\(153\) 312.475 + 309.931i 0.165112 + 0.163768i
\(154\) 0 0
\(155\) −899.350 −0.466049
\(156\) 0 0
\(157\) −1091.24 −0.554715 −0.277358 0.960767i \(-0.589459\pi\)
−0.277358 + 0.960767i \(0.589459\pi\)
\(158\) 0 0
\(159\) −2000.26 + 833.326i −0.997678 + 0.415642i
\(160\) 0 0
\(161\) 783.174i 0.383371i
\(162\) 0 0
\(163\) 26.3933i 0.0126827i 0.999980 + 0.00634135i \(0.00201853\pi\)
−0.999980 + 0.00634135i \(0.997981\pi\)
\(164\) 0 0
\(165\) −4888.35 + 2036.53i −2.30641 + 0.960872i
\(166\) 0 0
\(167\) 2067.92 0.958206 0.479103 0.877759i \(-0.340962\pi\)
0.479103 + 0.877759i \(0.340962\pi\)
\(168\) 0 0
\(169\) 3265.31 1.48626
\(170\) 0 0
\(171\) 2949.67 + 2925.66i 1.31910 + 1.30837i
\(172\) 0 0
\(173\) 2289.25i 1.00606i −0.864269 0.503030i \(-0.832219\pi\)
0.864269 0.503030i \(-0.167781\pi\)
\(174\) 0 0
\(175\) 1083.94i 0.468216i
\(176\) 0 0
\(177\) 655.485 + 1573.38i 0.278358 + 0.668150i
\(178\) 0 0
\(179\) −108.183 −0.0451731 −0.0225865 0.999745i \(-0.507190\pi\)
−0.0225865 + 0.999745i \(0.507190\pi\)
\(180\) 0 0
\(181\) 752.631 0.309075 0.154538 0.987987i \(-0.450611\pi\)
0.154538 + 0.987987i \(0.450611\pi\)
\(182\) 0 0
\(183\) 1274.22 + 3058.54i 0.514714 + 1.23548i
\(184\) 0 0
\(185\) 1055.38i 0.419423i
\(186\) 0 0
\(187\) 993.051i 0.388337i
\(188\) 0 0
\(189\) 370.254 909.604i 0.142498 0.350074i
\(190\) 0 0
\(191\) −2279.55 −0.863572 −0.431786 0.901976i \(-0.642116\pi\)
−0.431786 + 0.901976i \(0.642116\pi\)
\(192\) 0 0
\(193\) −766.149 −0.285744 −0.142872 0.989741i \(-0.545634\pi\)
−0.142872 + 0.989741i \(0.545634\pi\)
\(194\) 0 0
\(195\) 5930.31 2470.62i 2.17784 0.907308i
\(196\) 0 0
\(197\) 3890.03i 1.40687i −0.710760 0.703435i \(-0.751649\pi\)
0.710760 0.703435i \(-0.248351\pi\)
\(198\) 0 0
\(199\) 2577.77i 0.918257i 0.888370 + 0.459129i \(0.151838\pi\)
−0.888370 + 0.459129i \(0.848162\pi\)
\(200\) 0 0
\(201\) −966.378 + 402.602i −0.339119 + 0.141280i
\(202\) 0 0
\(203\) −1085.85 −0.375426
\(204\) 0 0
\(205\) 7676.07 2.61522
\(206\) 0 0
\(207\) −2127.29 + 2144.75i −0.714285 + 0.720147i
\(208\) 0 0
\(209\) 9374.11i 3.10249i
\(210\) 0 0
\(211\) 4839.87i 1.57910i −0.613685 0.789551i \(-0.710313\pi\)
0.613685 0.789551i \(-0.289687\pi\)
\(212\) 0 0
\(213\) −788.176 1891.88i −0.253544 0.608590i
\(214\) 0 0
\(215\) 1599.01 0.507217
\(216\) 0 0
\(217\) 376.327 0.117727
\(218\) 0 0
\(219\) −1954.10 4690.49i −0.602950 1.44728i
\(220\) 0 0
\(221\) 1204.72i 0.366689i
\(222\) 0 0
\(223\) 1258.94i 0.378049i 0.981972 + 0.189024i \(0.0605325\pi\)
−0.981972 + 0.189024i \(0.939467\pi\)
\(224\) 0 0
\(225\) 2944.23 2968.39i 0.872365 0.879524i
\(226\) 0 0
\(227\) 3028.13 0.885393 0.442696 0.896672i \(-0.354022\pi\)
0.442696 + 0.896672i \(0.354022\pi\)
\(228\) 0 0
\(229\) −4162.54 −1.20117 −0.600585 0.799561i \(-0.705066\pi\)
−0.600585 + 0.799561i \(0.705066\pi\)
\(230\) 0 0
\(231\) 2045.50 852.175i 0.582615 0.242723i
\(232\) 0 0
\(233\) 631.397i 0.177529i −0.996053 0.0887644i \(-0.971708\pi\)
0.996053 0.0887644i \(-0.0282918\pi\)
\(234\) 0 0
\(235\) 1179.59i 0.327439i
\(236\) 0 0
\(237\) −2470.14 + 1029.08i −0.677017 + 0.282052i
\(238\) 0 0
\(239\) −1634.45 −0.442358 −0.221179 0.975233i \(-0.570990\pi\)
−0.221179 + 0.975233i \(0.570990\pi\)
\(240\) 0 0
\(241\) −3537.63 −0.945555 −0.472777 0.881182i \(-0.656748\pi\)
−0.472777 + 0.881182i \(0.656748\pi\)
\(242\) 0 0
\(243\) −3484.66 + 1485.28i −0.919922 + 0.392102i
\(244\) 0 0
\(245\) 819.704i 0.213751i
\(246\) 0 0
\(247\) 11372.2i 2.92954i
\(248\) 0 0
\(249\) 2265.73 + 5438.50i 0.576646 + 1.38414i
\(250\) 0 0
\(251\) −3866.21 −0.972244 −0.486122 0.873891i \(-0.661589\pi\)
−0.486122 + 0.873891i \(0.661589\pi\)
\(252\) 0 0
\(253\) −6816.05 −1.69376
\(254\) 0 0
\(255\) −544.901 1307.94i −0.133816 0.321202i
\(256\) 0 0
\(257\) 2615.37i 0.634794i 0.948293 + 0.317397i \(0.102809\pi\)
−0.948293 + 0.317397i \(0.897191\pi\)
\(258\) 0 0
\(259\) 441.619i 0.105949i
\(260\) 0 0
\(261\) 2973.63 + 2949.43i 0.705223 + 0.699482i
\(262\) 0 0
\(263\) 7004.30 1.64222 0.821110 0.570770i \(-0.193355\pi\)
0.821110 + 0.570770i \(0.193355\pi\)
\(264\) 0 0
\(265\) 6976.19 1.61715
\(266\) 0 0
\(267\) −4967.59 + 2069.54i −1.13862 + 0.474360i
\(268\) 0 0
\(269\) 404.298i 0.0916374i 0.998950 + 0.0458187i \(0.0145897\pi\)
−0.998950 + 0.0458187i \(0.985410\pi\)
\(270\) 0 0
\(271\) 758.555i 0.170033i −0.996380 0.0850165i \(-0.972906\pi\)
0.996380 0.0850165i \(-0.0270943\pi\)
\(272\) 0 0
\(273\) −2481.50 + 1033.82i −0.550137 + 0.229192i
\(274\) 0 0
\(275\) 9433.62 2.06861
\(276\) 0 0
\(277\) −1084.77 −0.235298 −0.117649 0.993055i \(-0.537536\pi\)
−0.117649 + 0.993055i \(0.537536\pi\)
\(278\) 0 0
\(279\) −1030.59 1022.20i −0.221145 0.219345i
\(280\) 0 0
\(281\) 4094.10i 0.869159i 0.900633 + 0.434580i \(0.143103\pi\)
−0.900633 + 0.434580i \(0.856897\pi\)
\(282\) 0 0
\(283\) 5030.95i 1.05675i −0.849013 0.528373i \(-0.822802\pi\)
0.849013 0.528373i \(-0.177198\pi\)
\(284\) 0 0
\(285\) −5143.70 12346.6i −1.06908 2.56613i
\(286\) 0 0
\(287\) −3212.00 −0.660622
\(288\) 0 0
\(289\) 4647.30 0.945918
\(290\) 0 0
\(291\) −1989.57 4775.62i −0.400793 0.962035i
\(292\) 0 0
\(293\) 2585.51i 0.515519i 0.966209 + 0.257759i \(0.0829842\pi\)
−0.966209 + 0.257759i \(0.917016\pi\)
\(294\) 0 0
\(295\) 5487.40i 1.08301i
\(296\) 0 0
\(297\) −7916.39 3222.37i −1.54665 0.629565i
\(298\) 0 0
\(299\) 8268.91 1.59934
\(300\) 0 0
\(301\) −669.097 −0.128127
\(302\) 0 0
\(303\) 6990.03 2912.11i 1.32530 0.552134i
\(304\) 0 0
\(305\) 10667.1i 2.00261i
\(306\) 0 0
\(307\) 766.680i 0.142530i −0.997457 0.0712651i \(-0.977296\pi\)
0.997457 0.0712651i \(-0.0227036\pi\)
\(308\) 0 0
\(309\) 1722.05 717.421i 0.317035 0.132080i
\(310\) 0 0
\(311\) −5681.41 −1.03589 −0.517947 0.855412i \(-0.673304\pi\)
−0.517947 + 0.855412i \(0.673304\pi\)
\(312\) 0 0
\(313\) −9636.05 −1.74013 −0.870067 0.492934i \(-0.835924\pi\)
−0.870067 + 0.492934i \(0.835924\pi\)
\(314\) 0 0
\(315\) −2226.52 + 2244.79i −0.398254 + 0.401522i
\(316\) 0 0
\(317\) 1585.82i 0.280972i 0.990083 + 0.140486i \(0.0448666\pi\)
−0.990083 + 0.140486i \(0.955133\pi\)
\(318\) 0 0
\(319\) 9450.26i 1.65866i
\(320\) 0 0
\(321\) −1529.32 3670.86i −0.265913 0.638279i
\(322\) 0 0
\(323\) −2508.16 −0.432068
\(324\) 0 0
\(325\) −11444.4 −1.95330
\(326\) 0 0
\(327\) −1470.94 3530.74i −0.248756 0.597097i
\(328\) 0 0
\(329\) 493.593i 0.0827132i
\(330\) 0 0
\(331\) 3330.99i 0.553135i 0.960995 + 0.276567i \(0.0891969\pi\)
−0.960995 + 0.276567i \(0.910803\pi\)
\(332\) 0 0
\(333\) 1199.54 1209.39i 0.197401 0.199021i
\(334\) 0 0
\(335\) 3370.38 0.549683
\(336\) 0 0
\(337\) 10234.2 1.65428 0.827140 0.561996i \(-0.189967\pi\)
0.827140 + 0.561996i \(0.189967\pi\)
\(338\) 0 0
\(339\) 5892.30 2454.79i 0.944028 0.393291i
\(340\) 0 0
\(341\) 3275.22i 0.520127i
\(342\) 0 0
\(343\) 343.000i 0.0539949i
\(344\) 0 0
\(345\) 8977.38 3740.06i 1.40095 0.583647i
\(346\) 0 0
\(347\) −4657.40 −0.720525 −0.360263 0.932851i \(-0.617313\pi\)
−0.360263 + 0.932851i \(0.617313\pi\)
\(348\) 0 0
\(349\) 3280.13 0.503098 0.251549 0.967845i \(-0.419060\pi\)
0.251549 + 0.967845i \(0.419060\pi\)
\(350\) 0 0
\(351\) 9603.78 + 3909.22i 1.46043 + 0.594469i
\(352\) 0 0
\(353\) 6977.11i 1.05200i −0.850486 0.525998i \(-0.823692\pi\)
0.850486 0.525998i \(-0.176308\pi\)
\(354\) 0 0
\(355\) 6598.22i 0.986471i
\(356\) 0 0
\(357\) 228.010 + 547.300i 0.0338027 + 0.0811377i
\(358\) 0 0
\(359\) −457.102 −0.0672004 −0.0336002 0.999435i \(-0.510697\pi\)
−0.0336002 + 0.999435i \(0.510697\pi\)
\(360\) 0 0
\(361\) −16817.3 −2.45186
\(362\) 0 0
\(363\) −4756.86 11418.0i −0.687797 1.65094i
\(364\) 0 0
\(365\) 16358.8i 2.34591i
\(366\) 0 0
\(367\) 6117.95i 0.870175i −0.900388 0.435088i \(-0.856717\pi\)
0.900388 0.435088i \(-0.143283\pi\)
\(368\) 0 0
\(369\) 8796.19 + 8724.59i 1.24095 + 1.23085i
\(370\) 0 0
\(371\) −2919.14 −0.408502
\(372\) 0 0
\(373\) −2466.70 −0.342415 −0.171208 0.985235i \(-0.554767\pi\)
−0.171208 + 0.985235i \(0.554767\pi\)
\(374\) 0 0
\(375\) −2394.99 + 997.775i −0.329804 + 0.137400i
\(376\) 0 0
\(377\) 11464.6i 1.56620i
\(378\) 0 0
\(379\) 6021.87i 0.816155i 0.912947 + 0.408078i \(0.133801\pi\)
−0.912947 + 0.408078i \(0.866199\pi\)
\(380\) 0 0
\(381\) −613.547 + 255.609i −0.0825012 + 0.0343708i
\(382\) 0 0
\(383\) −4212.17 −0.561963 −0.280982 0.959713i \(-0.590660\pi\)
−0.280982 + 0.959713i \(0.590660\pi\)
\(384\) 0 0
\(385\) −7133.98 −0.944367
\(386\) 0 0
\(387\) 1832.34 + 1817.43i 0.240680 + 0.238721i
\(388\) 0 0
\(389\) 2689.11i 0.350497i 0.984524 + 0.175248i \(0.0560729\pi\)
−0.984524 + 0.175248i \(0.943927\pi\)
\(390\) 0 0
\(391\) 1823.72i 0.235881i
\(392\) 0 0
\(393\) 2282.98 + 5479.90i 0.293030 + 0.703369i
\(394\) 0 0
\(395\) 8614.98 1.09738
\(396\) 0 0
\(397\) −8370.78 −1.05823 −0.529115 0.848550i \(-0.677476\pi\)
−0.529115 + 0.848550i \(0.677476\pi\)
\(398\) 0 0
\(399\) 2152.35 + 5166.35i 0.270056 + 0.648223i
\(400\) 0 0
\(401\) 4771.81i 0.594247i 0.954839 + 0.297123i \(0.0960272\pi\)
−0.954839 + 0.297123i \(0.903973\pi\)
\(402\) 0 0
\(403\) 3973.34i 0.491132i
\(404\) 0 0
\(405\) 12194.8 99.6704i 1.49621 0.0122288i
\(406\) 0 0
\(407\) 3843.46 0.468091
\(408\) 0 0
\(409\) −3505.84 −0.423846 −0.211923 0.977286i \(-0.567973\pi\)
−0.211923 + 0.977286i \(0.567973\pi\)
\(410\) 0 0
\(411\) −3169.53 + 1320.46i −0.380393 + 0.158475i
\(412\) 0 0
\(413\) 2296.17i 0.273576i
\(414\) 0 0
\(415\) 18967.6i 2.24357i
\(416\) 0 0
\(417\) 2411.90 1004.82i 0.283240 0.118001i
\(418\) 0 0
\(419\) −13302.4 −1.55099 −0.775496 0.631352i \(-0.782500\pi\)
−0.775496 + 0.631352i \(0.782500\pi\)
\(420\) 0 0
\(421\) 5120.87 0.592816 0.296408 0.955061i \(-0.404211\pi\)
0.296408 + 0.955061i \(0.404211\pi\)
\(422\) 0 0
\(423\) −1340.72 + 1351.72i −0.154109 + 0.155373i
\(424\) 0 0
\(425\) 2524.08i 0.288085i
\(426\) 0 0
\(427\) 4463.58i 0.505873i
\(428\) 0 0
\(429\) 8997.43 + 21596.8i 1.01259 + 2.43054i
\(430\) 0 0
\(431\) −17758.5 −1.98468 −0.992339 0.123544i \(-0.960574\pi\)
−0.992339 + 0.123544i \(0.960574\pi\)
\(432\) 0 0
\(433\) 17278.3 1.91765 0.958826 0.283994i \(-0.0916595\pi\)
0.958826 + 0.283994i \(0.0916595\pi\)
\(434\) 0 0
\(435\) −5185.48 12446.9i −0.571552 1.37191i
\(436\) 0 0
\(437\) 17215.4i 1.88449i
\(438\) 0 0
\(439\) 4721.22i 0.513284i −0.966507 0.256642i \(-0.917384\pi\)
0.966507 0.256642i \(-0.0826162\pi\)
\(440\) 0 0
\(441\) 931.672 939.317i 0.100602 0.101427i
\(442\) 0 0
\(443\) 2733.59 0.293176 0.146588 0.989198i \(-0.453171\pi\)
0.146588 + 0.989198i \(0.453171\pi\)
\(444\) 0 0
\(445\) 17325.2 1.84560
\(446\) 0 0
\(447\) −2255.03 + 939.467i −0.238611 + 0.0994078i
\(448\) 0 0
\(449\) 2204.08i 0.231663i 0.993269 + 0.115832i \(0.0369533\pi\)
−0.993269 + 0.115832i \(0.963047\pi\)
\(450\) 0 0
\(451\) 27954.5i 2.91868i
\(452\) 0 0
\(453\) −3996.24 + 1664.87i −0.414480 + 0.172676i
\(454\) 0 0
\(455\) 8654.60 0.891723
\(456\) 0 0
\(457\) −1401.18 −0.143423 −0.0717116 0.997425i \(-0.522846\pi\)
−0.0717116 + 0.997425i \(0.522846\pi\)
\(458\) 0 0
\(459\) 862.185 2118.13i 0.0876762 0.215394i
\(460\) 0 0
\(461\) 5775.00i 0.583446i −0.956503 0.291723i \(-0.905772\pi\)
0.956503 0.291723i \(-0.0942285\pi\)
\(462\) 0 0
\(463\) 15866.2i 1.59258i 0.604916 + 0.796289i \(0.293207\pi\)
−0.604916 + 0.796289i \(0.706793\pi\)
\(464\) 0 0
\(465\) 1797.16 + 4313.77i 0.179229 + 0.430208i
\(466\) 0 0
\(467\) −4802.77 −0.475900 −0.237950 0.971277i \(-0.576476\pi\)
−0.237950 + 0.971277i \(0.576476\pi\)
\(468\) 0 0
\(469\) −1410.32 −0.138854
\(470\) 0 0
\(471\) 2180.61 + 5234.17i 0.213327 + 0.512055i
\(472\) 0 0
\(473\) 5823.23i 0.566072i
\(474\) 0 0
\(475\) 23826.6i 2.30156i
\(476\) 0 0
\(477\) 7994.18 + 7929.11i 0.767355 + 0.761109i
\(478\) 0 0
\(479\) 5778.37 0.551191 0.275595 0.961274i \(-0.411125\pi\)
0.275595 + 0.961274i \(0.411125\pi\)
\(480\) 0 0
\(481\) −4662.70 −0.441997
\(482\) 0 0
\(483\) −3756.53 + 1565.01i −0.353888 + 0.147433i
\(484\) 0 0
\(485\) 16655.7i 1.55937i
\(486\) 0 0
\(487\) 8916.11i 0.829625i 0.909907 + 0.414813i \(0.136153\pi\)
−0.909907 + 0.414813i \(0.863847\pi\)
\(488\) 0 0
\(489\) 126.597 52.7413i 0.0117074 0.00487739i
\(490\) 0 0
\(491\) 9826.22 0.903159 0.451579 0.892231i \(-0.350861\pi\)
0.451579 + 0.892231i \(0.350861\pi\)
\(492\) 0 0
\(493\) −2528.54 −0.230993
\(494\) 0 0
\(495\) 19536.7 + 19377.6i 1.77395 + 1.75951i
\(496\) 0 0
\(497\) 2760.98i 0.249189i
\(498\) 0 0
\(499\) 14608.3i 1.31053i −0.755398 0.655266i \(-0.772557\pi\)
0.755398 0.655266i \(-0.227443\pi\)
\(500\) 0 0
\(501\) −4132.29 9918.87i −0.368497 0.884515i
\(502\) 0 0
\(503\) 10453.8 0.926662 0.463331 0.886185i \(-0.346654\pi\)
0.463331 + 0.886185i \(0.346654\pi\)
\(504\) 0 0
\(505\) −24378.8 −2.14820
\(506\) 0 0
\(507\) −6525.02 15662.2i −0.571570 1.37196i
\(508\) 0 0
\(509\) 9286.13i 0.808645i −0.914616 0.404323i \(-0.867507\pi\)
0.914616 0.404323i \(-0.132493\pi\)
\(510\) 0 0
\(511\) 6845.22i 0.592593i
\(512\) 0 0
\(513\) 8138.78 19994.5i 0.700460 1.72082i
\(514\) 0 0
\(515\) −6005.89 −0.513886
\(516\) 0 0
\(517\) −4295.80 −0.365433
\(518\) 0 0
\(519\) −10980.5 + 4574.57i −0.928689 + 0.386901i
\(520\) 0 0
\(521\) 2764.21i 0.232442i 0.993223 + 0.116221i \(0.0370781\pi\)
−0.993223 + 0.116221i \(0.962922\pi\)
\(522\) 0 0
\(523\) 11042.1i 0.923209i 0.887086 + 0.461605i \(0.152726\pi\)
−0.887086 + 0.461605i \(0.847274\pi\)
\(524\) 0 0
\(525\) 5199.15 2166.01i 0.432208 0.180062i
\(526\) 0 0
\(527\) 876.327 0.0724353
\(528\) 0 0
\(529\) 350.576 0.0288137
\(530\) 0 0
\(531\) 6236.95 6288.13i 0.509718 0.513902i
\(532\) 0 0
\(533\) 33913.0i 2.75598i
\(534\) 0 0
\(535\) 12802.7i 1.03459i
\(536\) 0 0
\(537\) 216.181 + 518.905i 0.0173722 + 0.0416991i
\(538\) 0 0
\(539\) 2985.17 0.238554
\(540\) 0 0
\(541\) 23666.7 1.88080 0.940398 0.340075i \(-0.110452\pi\)
0.940398 + 0.340075i \(0.110452\pi\)
\(542\) 0 0
\(543\) −1503.97 3610.03i −0.118861 0.285306i
\(544\) 0 0
\(545\) 12314.0i 0.967841i
\(546\) 0 0
\(547\) 6867.33i 0.536793i −0.963309 0.268397i \(-0.913506\pi\)
0.963309 0.268397i \(-0.0864938\pi\)
\(548\) 0 0
\(549\) 12124.2 12223.7i 0.942526 0.950261i
\(550\) 0 0
\(551\) −23868.6 −1.84544
\(552\) 0 0
\(553\) −3604.88 −0.277207
\(554\) 0 0
\(555\) −5062.19 + 2108.96i −0.387168 + 0.161298i
\(556\) 0 0
\(557\) 17973.1i 1.36722i 0.729845 + 0.683612i \(0.239592\pi\)
−0.729845 + 0.683612i \(0.760408\pi\)
\(558\) 0 0
\(559\) 7064.46i 0.534516i
\(560\) 0 0
\(561\) 4763.21 1984.40i 0.358472 0.149343i
\(562\) 0 0
\(563\) −19148.6 −1.43342 −0.716711 0.697370i \(-0.754353\pi\)
−0.716711 + 0.697370i \(0.754353\pi\)
\(564\) 0 0
\(565\) −20550.2 −1.53019
\(566\) 0 0
\(567\) −5102.83 + 41.7064i −0.377952 + 0.00308908i
\(568\) 0 0
\(569\) 13679.2i 1.00784i 0.863750 + 0.503921i \(0.168110\pi\)
−0.863750 + 0.503921i \(0.831890\pi\)
\(570\) 0 0
\(571\) 10288.3i 0.754033i 0.926207 + 0.377017i \(0.123050\pi\)
−0.926207 + 0.377017i \(0.876950\pi\)
\(572\) 0 0
\(573\) 4555.19 + 10934.0i 0.332104 + 0.797160i
\(574\) 0 0
\(575\) −17324.7 −1.25650
\(576\) 0 0
\(577\) −13939.4 −1.00573 −0.502864 0.864365i \(-0.667720\pi\)
−0.502864 + 0.864365i \(0.667720\pi\)
\(578\) 0 0
\(579\) 1530.99 + 3674.87i 0.109889 + 0.263769i
\(580\) 0 0
\(581\) 7936.86i 0.566741i
\(582\) 0 0
\(583\) 25405.7i 1.80479i
\(584\) 0 0
\(585\) −23700.9 23508.0i −1.67506 1.66143i
\(586\) 0 0
\(587\) 8490.79 0.597023 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(588\) 0 0
\(589\) 8272.27 0.578698
\(590\) 0 0
\(591\) −18658.7 + 7773.40i −1.29868 + 0.541040i
\(592\) 0 0
\(593\) 3335.30i 0.230968i −0.993309 0.115484i \(-0.963158\pi\)
0.993309 0.115484i \(-0.0368420\pi\)
\(594\) 0 0
\(595\) 1908.79i 0.131517i
\(596\) 0 0
\(597\) 12364.4 5151.12i 0.847639 0.353134i
\(598\) 0 0
\(599\) 14950.3 1.01978 0.509892 0.860238i \(-0.329685\pi\)
0.509892 + 0.860238i \(0.329685\pi\)
\(600\) 0 0
\(601\) −15554.3 −1.05570 −0.527849 0.849338i \(-0.677001\pi\)
−0.527849 + 0.849338i \(0.677001\pi\)
\(602\) 0 0
\(603\) 3862.20 + 3830.76i 0.260831 + 0.258707i
\(604\) 0 0
\(605\) 39822.0i 2.67603i
\(606\) 0 0
\(607\) 23162.9i 1.54885i −0.632663 0.774427i \(-0.718038\pi\)
0.632663 0.774427i \(-0.281962\pi\)
\(608\) 0 0
\(609\) 2169.83 + 5208.32i 0.144378 + 0.346554i
\(610\) 0 0
\(611\) 5211.45 0.345062
\(612\) 0 0
\(613\) −517.112 −0.0340717 −0.0170359 0.999855i \(-0.505423\pi\)
−0.0170359 + 0.999855i \(0.505423\pi\)
\(614\) 0 0
\(615\) −15339.0 36818.6i −1.00574 2.41410i
\(616\) 0 0
\(617\) 3318.03i 0.216497i 0.994124 + 0.108249i \(0.0345242\pi\)
−0.994124 + 0.108249i \(0.965476\pi\)
\(618\) 0 0
\(619\) 6295.88i 0.408809i −0.978886 0.204405i \(-0.934474\pi\)
0.978886 0.204405i \(-0.0655258\pi\)
\(620\) 0 0
\(621\) 14538.3 + 5917.83i 0.939457 + 0.382406i
\(622\) 0 0
\(623\) −7249.61 −0.466211
\(624\) 0 0
\(625\) −11003.1 −0.704199
\(626\) 0 0
\(627\) 44963.3 18732.1i 2.86390 1.19313i
\(628\) 0 0
\(629\) 1028.37i 0.0651886i
\(630\) 0 0
\(631\) 16754.4i 1.05702i −0.848926 0.528511i \(-0.822750\pi\)
0.848926 0.528511i \(-0.177250\pi\)
\(632\) 0 0
\(633\) −23214.7 + 9671.45i −1.45766 + 0.607276i
\(634\) 0 0
\(635\) 2139.83 0.133727
\(636\) 0 0
\(637\) −3621.46 −0.225255
\(638\) 0 0
\(639\) −7499.50 + 7561.05i −0.464281 + 0.468091i
\(640\) 0 0
\(641\) 5871.80i 0.361813i −0.983500 0.180907i \(-0.942097\pi\)
0.983500 0.180907i \(-0.0579032\pi\)
\(642\) 0 0
\(643\) 12954.8i 0.794539i 0.917702 + 0.397269i \(0.130042\pi\)
−0.917702 + 0.397269i \(0.869958\pi\)
\(644\) 0 0
\(645\) −3195.28 7669.74i −0.195061 0.468210i
\(646\) 0 0
\(647\) 20113.4 1.22216 0.611080 0.791569i \(-0.290735\pi\)
0.611080 + 0.791569i \(0.290735\pi\)
\(648\) 0 0
\(649\) 19983.8 1.20868
\(650\) 0 0
\(651\) −752.010 1805.07i −0.0452743 0.108673i
\(652\) 0 0
\(653\) 31517.3i 1.88877i −0.328844 0.944384i \(-0.606659\pi\)
0.328844 0.944384i \(-0.393341\pi\)
\(654\) 0 0
\(655\) 19111.9i 1.14010i
\(656\) 0 0
\(657\) −18593.3 + 18745.9i −1.10410 + 1.11316i
\(658\) 0 0
\(659\) 19556.1 1.15599 0.577994 0.816041i \(-0.303836\pi\)
0.577994 + 0.816041i \(0.303836\pi\)
\(660\) 0 0
\(661\) 26582.3 1.56419 0.782097 0.623157i \(-0.214150\pi\)
0.782097 + 0.623157i \(0.214150\pi\)
\(662\) 0 0
\(663\) −5778.50 + 2407.38i −0.338489 + 0.141018i
\(664\) 0 0
\(665\) 18018.4i 1.05071i
\(666\) 0 0
\(667\) 17355.2i 1.00749i
\(668\) 0 0
\(669\) 6038.57 2515.72i 0.348975 0.145386i
\(670\) 0 0
\(671\) 38847.1 2.23498
\(672\) 0 0
\(673\) 201.721 0.0115539 0.00577695 0.999983i \(-0.498161\pi\)
0.00577695 + 0.999983i \(0.498161\pi\)
\(674\) 0 0
\(675\) −20121.5 8190.44i −1.14737 0.467038i
\(676\) 0 0
\(677\) 4804.24i 0.272736i 0.990658 + 0.136368i \(0.0435429\pi\)
−0.990658 + 0.136368i \(0.956457\pi\)
\(678\) 0 0
\(679\) 6969.47i 0.393908i
\(680\) 0 0
\(681\) −6051.07 14524.6i −0.340496 0.817302i
\(682\) 0 0
\(683\) 3043.76 0.170521 0.0852607 0.996359i \(-0.472828\pi\)
0.0852607 + 0.996359i \(0.472828\pi\)
\(684\) 0 0
\(685\) 11054.2 0.616583
\(686\) 0 0
\(687\) 8317.94 + 19965.8i 0.461935 + 1.10880i
\(688\) 0 0
\(689\) 30820.9i 1.70418i
\(690\) 0 0
\(691\) 8025.16i 0.441811i −0.975295 0.220906i \(-0.929099\pi\)
0.975295 0.220906i \(-0.0709013\pi\)
\(692\) 0 0
\(693\) −8174.99 8108.45i −0.448113 0.444465i
\(694\) 0 0
\(695\) −8411.85 −0.459108
\(696\) 0 0
\(697\) −7479.57 −0.406469
\(698\) 0 0
\(699\) −3028.52 + 1261.71i −0.163876 + 0.0682723i
\(700\) 0 0
\(701\) 9256.11i 0.498714i −0.968412 0.249357i \(-0.919781\pi\)
0.968412 0.249357i \(-0.0802192\pi\)
\(702\) 0 0
\(703\) 9707.47i 0.520803i
\(704\) 0 0
\(705\) 5657.97 2357.16i 0.302257 0.125923i
\(706\) 0 0
\(707\) 10201.1 0.542650
\(708\) 0 0
\(709\) −10786.1 −0.571342 −0.285671 0.958328i \(-0.592217\pi\)
−0.285671 + 0.958328i \(0.592217\pi\)
\(710\) 0 0
\(711\) 9872.10 + 9791.74i 0.520721 + 0.516483i
\(712\) 0 0
\(713\) 6014.89i 0.315932i
\(714\) 0 0
\(715\) 75322.0i 3.93970i
\(716\) 0 0
\(717\) 3266.09 + 7839.69i 0.170118 + 0.408338i
\(718\) 0 0
\(719\) 16903.4 0.876759 0.438379 0.898790i \(-0.355553\pi\)
0.438379 + 0.898790i \(0.355553\pi\)
\(720\) 0 0
\(721\) 2513.13 0.129811
\(722\) 0 0
\(723\) 7069.19 + 16968.4i 0.363632 + 0.872837i
\(724\) 0 0
\(725\) 24020.2i 1.23046i
\(726\) 0 0
\(727\) 17968.4i 0.916661i 0.888782 + 0.458330i \(0.151552\pi\)
−0.888782 + 0.458330i \(0.848448\pi\)
\(728\) 0 0
\(729\) 14087.6 + 13746.3i 0.715722 + 0.698385i
\(730\) 0 0
\(731\) −1558.08 −0.0788339
\(732\) 0 0
\(733\) 26141.6 1.31728 0.658638 0.752460i \(-0.271133\pi\)
0.658638 + 0.752460i \(0.271133\pi\)
\(734\) 0 0
\(735\) −3931.75 + 1638.00i −0.197313 + 0.0822023i
\(736\) 0 0
\(737\) 12274.1i 0.613465i
\(738\) 0 0
\(739\) 20364.7i 1.01370i 0.862033 + 0.506851i \(0.169191\pi\)
−0.862033 + 0.506851i \(0.830809\pi\)
\(740\) 0 0
\(741\) −54547.4 + 22724.9i −2.70425 + 1.12661i
\(742\) 0 0
\(743\) −5300.06 −0.261696 −0.130848 0.991402i \(-0.541770\pi\)
−0.130848 + 0.991402i \(0.541770\pi\)
\(744\) 0 0
\(745\) 7864.75 0.386768
\(746\) 0 0
\(747\) 21558.5 21735.4i 1.05593 1.06460i
\(748\) 0 0
\(749\) 5357.20i 0.261346i
\(750\) 0 0
\(751\) 38359.6i 1.86386i −0.362634 0.931932i \(-0.618123\pi\)
0.362634 0.931932i \(-0.381877\pi\)
\(752\) 0 0
\(753\) 7725.80 + 18544.5i 0.373896 + 0.897474i
\(754\) 0 0
\(755\) 13937.5 0.671836
\(756\) 0 0
\(757\) −29775.4 −1.42960 −0.714799 0.699330i \(-0.753482\pi\)
−0.714799 + 0.699330i \(0.753482\pi\)
\(758\) 0 0
\(759\) 13620.4 + 32693.5i 0.651370 + 1.56350i
\(760\) 0 0
\(761\) 24451.7i 1.16475i −0.812921 0.582373i \(-0.802124\pi\)
0.812921 0.582373i \(-0.197876\pi\)
\(762\) 0 0
\(763\) 5152.71i 0.244483i
\(764\) 0 0
\(765\) −5184.73 + 5227.28i −0.245038 + 0.247049i
\(766\) 0 0
\(767\) −24243.4 −1.14130
\(768\) 0 0
\(769\) 20070.8 0.941187 0.470593 0.882350i \(-0.344040\pi\)
0.470593 + 0.882350i \(0.344040\pi\)
\(770\) 0 0
\(771\) 12544.7 5226.25i 0.585976 0.244123i
\(772\) 0 0
\(773\) 39134.5i 1.82092i 0.413597 + 0.910460i \(0.364272\pi\)
−0.413597 + 0.910460i \(0.635728\pi\)
\(774\) 0 0
\(775\) 8324.79i 0.385852i
\(776\) 0 0
\(777\) 2118.24 882.480i 0.0978012 0.0407449i
\(778\) 0 0
\(779\) −70605.0 −3.24735
\(780\) 0 0
\(781\) −24029.2 −1.10094
\(782\) 0 0
\(783\) 8204.89 20156.9i 0.374481 0.919988i
\(784\) 0 0
\(785\) 18254.9i 0.829996i
\(786\) 0 0
\(787\) 9041.56i 0.409526i −0.978812 0.204763i \(-0.934358\pi\)
0.978812 0.204763i \(-0.0656423\pi\)
\(788\) 0 0
\(789\) −13996.6 33596.5i −0.631549 1.51593i
\(790\) 0 0
\(791\) 8599.12 0.386535
\(792\) 0 0
\(793\) −47127.4 −2.11039
\(794\) 0 0
\(795\) −13940.4 33461.6i −0.621907 1.49278i
\(796\) 0 0
\(797\) 36193.9i 1.60860i −0.594222 0.804301i \(-0.702540\pi\)
0.594222 0.804301i \(-0.297460\pi\)
\(798\) 0 0
\(799\) 1149.39i 0.0508919i
\(800\) 0 0
\(801\) 19853.3 + 19691.7i 0.875758 + 0.868630i
\(802\) 0 0
\(803\) −59574.8 −2.61812
\(804\) 0 0
\(805\) 13101.4 0.573621
\(806\) 0 0
\(807\) 1939.23 807.902i 0.0845901 0.0352410i
\(808\) 0 0
\(809\) 13745.8i 0.597377i −0.954351 0.298689i \(-0.903451\pi\)
0.954351 0.298689i \(-0.0965492\pi\)
\(810\) 0 0
\(811\) 38555.3i 1.66937i 0.550728 + 0.834685i \(0.314350\pi\)
−0.550728 + 0.834685i \(0.685650\pi\)
\(812\) 0 0
\(813\) −3638.44 + 1515.81i −0.156957 + 0.0653896i
\(814\) 0 0
\(815\) −441.524 −0.0189766
\(816\) 0 0
\(817\) −14707.8 −0.629817
\(818\) 0 0
\(819\) 9917.50 + 9836.78i 0.423133 + 0.419688i
\(820\) 0 0
\(821\) 36235.3i 1.54034i −0.637838 0.770171i \(-0.720171\pi\)
0.637838 0.770171i \(-0.279829\pi\)
\(822\) 0 0
\(823\) 23707.3i 1.00411i −0.864835 0.502056i \(-0.832577\pi\)
0.864835 0.502056i \(-0.167423\pi\)
\(824\) 0 0
\(825\) −18851.1 45248.8i −0.795527 1.90953i
\(826\) 0 0
\(827\) 35174.4 1.47900 0.739500 0.673156i \(-0.235062\pi\)
0.739500 + 0.673156i \(0.235062\pi\)
\(828\) 0 0
\(829\) 24029.0 1.00671 0.503355 0.864080i \(-0.332099\pi\)
0.503355 + 0.864080i \(0.332099\pi\)
\(830\) 0 0
\(831\) 2167.68 + 5203.15i 0.0904887 + 0.217203i
\(832\) 0 0
\(833\) 798.720i 0.0332221i
\(834\) 0 0
\(835\) 34593.5i 1.43372i
\(836\) 0 0
\(837\) −2843.61 + 6985.89i −0.117431 + 0.288492i
\(838\) 0 0
\(839\) −46306.1 −1.90544 −0.952720 0.303850i \(-0.901728\pi\)
−0.952720 + 0.303850i \(0.901728\pi\)
\(840\) 0 0
\(841\) 326.470 0.0133860
\(842\) 0 0
\(843\) 19637.6 8181.19i 0.802317 0.334253i
\(844\) 0 0
\(845\) 54624.2i 2.22382i
\(846\) 0 0
\(847\) 16663.3i 0.675982i
\(848\) 0 0
\(849\) −24131.2 + 10053.3i −0.975477 + 0.406393i
\(850\) 0 0
\(851\) −7058.45 −0.284325
\(852\) 0 0
\(853\) 44352.0 1.78028 0.890142 0.455684i \(-0.150605\pi\)
0.890142 + 0.455684i \(0.150605\pi\)
\(854\) 0 0
\(855\) −48942.3 + 49344.0i −1.95765 + 1.97372i
\(856\) 0 0
\(857\) 42349.5i 1.68802i −0.536328 0.844009i \(-0.680189\pi\)
0.536328 0.844009i \(-0.319811\pi\)
\(858\) 0 0
\(859\) 41927.4i 1.66536i 0.553755 + 0.832680i \(0.313194\pi\)
−0.553755 + 0.832680i \(0.686806\pi\)
\(860\) 0 0
\(861\) 6418.50 + 15406.5i 0.254056 + 0.609818i
\(862\) 0 0
\(863\) 28670.6 1.13089 0.565445 0.824786i \(-0.308704\pi\)
0.565445 + 0.824786i \(0.308704\pi\)
\(864\) 0 0
\(865\) 38296.0 1.50532
\(866\) 0 0
\(867\) −9286.63 22291.0i −0.363772 0.873173i
\(868\) 0 0
\(869\) 31373.7i 1.22472i
\(870\) 0 0
\(871\) 14890.4i 0.579267i
\(872\) 0 0
\(873\) −18930.8 + 19086.1i −0.733917 + 0.739940i
\(874\) 0 0
\(875\) −3495.21 −0.135039
\(876\) 0 0
\(877\) −1035.04 −0.0398529 −0.0199264 0.999801i \(-0.506343\pi\)
−0.0199264 + 0.999801i \(0.506343\pi\)
\(878\) 0 0
\(879\) 12401.5 5166.58i 0.475873 0.198253i
\(880\) 0 0
\(881\) 7978.04i 0.305093i 0.988296 + 0.152547i \(0.0487474\pi\)
−0.988296 + 0.152547i \(0.951253\pi\)
\(882\) 0 0
\(883\) 22682.9i 0.864484i 0.901758 + 0.432242i \(0.142277\pi\)
−0.901758 + 0.432242i \(0.857723\pi\)
\(884\) 0 0
\(885\) −26320.6 + 10965.4i −0.999724 + 0.416494i
\(886\) 0 0
\(887\) 41695.5 1.57835 0.789175 0.614168i \(-0.210508\pi\)
0.789175 + 0.614168i \(0.210508\pi\)
\(888\) 0 0
\(889\) −895.400 −0.0337804
\(890\) 0 0
\(891\) 362.976 + 44410.5i 0.0136478 + 1.66982i
\(892\) 0 0
\(893\) 10849.9i 0.406584i
\(894\) 0 0
\(895\) 1809.76i 0.0675905i
\(896\) 0 0
\(897\) −16523.6 39662.2i −0.615060 1.47635i
\(898\) 0 0
\(899\) 8339.47 0.309385
\(900\) 0 0
\(901\) −6797.61 −0.251344
\(902\) 0 0
\(903\) 1337.05 + 3209.35i 0.0492737 + 0.118273i
\(904\) 0 0
\(905\) 12590.5i 0.462456i
\(906\) 0 0
\(907\) 5918.76i 0.216680i −0.994114 0.108340i \(-0.965446\pi\)
0.994114 0.108340i \(-0.0345536\pi\)
\(908\) 0 0
\(909\) −27936.2 27708.8i −1.01934 1.01105i
\(910\) 0 0
\(911\) 16275.8 0.591921 0.295961 0.955200i \(-0.404360\pi\)
0.295961 + 0.955200i \(0.404360\pi\)
\(912\) 0 0
\(913\) 69075.5 2.50390
\(914\) 0 0
\(915\) −51165.2 + 21315.9i −1.84860 + 0.770145i
\(916\) 0 0
\(917\) 7997.27i 0.287997i
\(918\) 0 0
\(919\) 10367.7i 0.372141i 0.982536 + 0.186070i \(0.0595752\pi\)
−0.982536 + 0.186070i \(0.940425\pi\)
\(920\) 0 0
\(921\) −3677.42 + 1532.05i −0.131569 + 0.0548129i
\(922\) 0 0
\(923\) 29151.0 1.03956
\(924\) 0 0
\(925\) 9769.10 0.347250
\(926\) 0 0
\(927\) −6882.29 6826.27i −0.243845 0.241860i
\(928\) 0 0
\(929\) 38112.2i 1.34598i 0.739649 + 0.672992i \(0.234991\pi\)
−0.739649 + 0.672992i \(0.765009\pi\)
\(930\) 0 0
\(931\) 7539.68i 0.265417i
\(932\) 0 0
\(933\) 11353.1 + 27251.2i 0.398375 + 0.956230i
\(934\) 0 0
\(935\) −16612.4 −0.581052
\(936\) 0 0
\(937\) −36739.1 −1.28091 −0.640456 0.767995i \(-0.721255\pi\)
−0.640456 + 0.767995i \(0.721255\pi\)
\(938\) 0 0
\(939\) 19255.6 + 46219.8i 0.669204 + 1.60631i
\(940\) 0 0
\(941\) 31135.3i 1.07862i −0.842107 0.539310i \(-0.818685\pi\)
0.842107 0.539310i \(-0.181315\pi\)
\(942\) 0 0
\(943\) 51337.9i 1.77284i
\(944\) 0 0
\(945\) 15216.4 + 6193.86i 0.523800 + 0.213213i
\(946\) 0 0
\(947\) 16651.0 0.571368 0.285684 0.958324i \(-0.407779\pi\)
0.285684 + 0.958324i \(0.407779\pi\)
\(948\) 0 0
\(949\) 72273.2 2.47217
\(950\) 0 0
\(951\) 7606.44 3168.91i 0.259364 0.108054i
\(952\) 0 0
\(953\) 21802.6i 0.741086i 0.928815 + 0.370543i \(0.120828\pi\)
−0.928815 + 0.370543i \(0.879172\pi\)
\(954\) 0 0
\(955\) 38133.8i 1.29213i
\(956\) 0 0
\(957\) 45328.6 18884.3i 1.53110 0.637872i
\(958\) 0 0
\(959\) −4625.56 −0.155753
\(960\) 0 0
\(961\) 26900.7 0.902982
\(962\) 0 0
\(963\) −14551.5 + 14670.9i −0.486931 + 0.490927i
\(964\) 0 0
\(965\) 12816.6i 0.427546i
\(966\) 0 0
\(967\) 2820.93i 0.0938106i 0.998899 + 0.0469053i \(0.0149359\pi\)
−0.998899 + 0.0469053i \(0.985064\pi\)
\(968\) 0 0
\(969\) 5012.02 + 12030.5i 0.166160 + 0.398840i
\(970\) 0 0
\(971\) 40165.3 1.32746 0.663730 0.747972i \(-0.268972\pi\)
0.663730 + 0.747972i \(0.268972\pi\)
\(972\) 0 0
\(973\) 3519.89 0.115974
\(974\) 0 0
\(975\) 22869.2 + 54893.6i 0.751180 + 1.80308i
\(976\) 0 0
\(977\) 18264.9i 0.598102i 0.954237 + 0.299051i \(0.0966701\pi\)
−0.954237 + 0.299051i \(0.903330\pi\)
\(978\) 0 0
\(979\) 63094.2i 2.05976i
\(980\) 0 0
\(981\) −13996.0 + 14110.9i −0.455513 + 0.459251i
\(982\) 0 0
\(983\) −1405.50 −0.0456038 −0.0228019 0.999740i \(-0.507259\pi\)
−0.0228019 + 0.999740i \(0.507259\pi\)
\(984\) 0 0
\(985\) 65075.0 2.10504
\(986\) 0 0
\(987\) −2367.54 + 986.339i −0.0763522 + 0.0318090i
\(988\) 0 0
\(989\) 10694.3i 0.343840i
\(990\) 0 0
\(991\) 32951.8i 1.05625i 0.849165 + 0.528127i \(0.177105\pi\)
−0.849165 + 0.528127i \(0.822895\pi\)
\(992\) 0 0
\(993\) 15977.2 6656.26i 0.510596 0.212719i
\(994\) 0 0
\(995\) −43122.6 −1.37395
\(996\) 0 0
\(997\) −19008.7 −0.603822 −0.301911 0.953336i \(-0.597625\pi\)
−0.301911 + 0.953336i \(0.597625\pi\)
\(998\) 0 0
\(999\) −8197.91 3336.96i −0.259630 0.105683i
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 336.4.h.b.239.9 24
3.2 odd 2 inner 336.4.h.b.239.15 yes 24
4.3 odd 2 inner 336.4.h.b.239.16 yes 24
12.11 even 2 inner 336.4.h.b.239.10 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.4.h.b.239.9 24 1.1 even 1 trivial
336.4.h.b.239.10 yes 24 12.11 even 2 inner
336.4.h.b.239.15 yes 24 3.2 odd 2 inner
336.4.h.b.239.16 yes 24 4.3 odd 2 inner