Properties

Label 304.4.a.g.1.3
Level $304$
Weight $4$
Character 304.1
Self dual yes
Analytic conductor $17.937$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,4,Mod(1,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.9365806417\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7057.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 22x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.50686\) of defining polynomial
Character \(\chi\) \(=\) 304.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.61812 q^{3} -13.8269 q^{5} +9.22252 q^{7} +4.56325 q^{9} +O(q^{10})\) \(q+5.61812 q^{3} -13.8269 q^{5} +9.22252 q^{7} +4.56325 q^{9} -59.8818 q^{11} +38.2034 q^{13} -77.6813 q^{15} -47.0714 q^{17} +19.0000 q^{19} +51.8132 q^{21} -179.632 q^{23} +66.1837 q^{25} -126.052 q^{27} -96.6484 q^{29} -125.918 q^{31} -336.423 q^{33} -127.519 q^{35} +407.764 q^{37} +214.631 q^{39} -220.654 q^{41} -100.777 q^{43} -63.0957 q^{45} +213.749 q^{47} -257.945 q^{49} -264.453 q^{51} +19.8284 q^{53} +827.981 q^{55} +106.744 q^{57} +97.7827 q^{59} +266.365 q^{61} +42.0847 q^{63} -528.235 q^{65} -627.332 q^{67} -1009.19 q^{69} +644.105 q^{71} +807.966 q^{73} +371.828 q^{75} -552.261 q^{77} -785.715 q^{79} -831.385 q^{81} -1174.90 q^{83} +650.853 q^{85} -542.982 q^{87} -234.248 q^{89} +352.331 q^{91} -707.421 q^{93} -262.711 q^{95} -1348.51 q^{97} -273.256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} + 7 q^{5} - 7 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{3} + 7 q^{5} - 7 q^{7} + 21 q^{9} - 103 q^{11} + 32 q^{13} - 122 q^{15} + 11 q^{17} + 57 q^{19} + 114 q^{21} - 316 q^{23} + 162 q^{25} - 178 q^{27} - 138 q^{29} - 420 q^{31} - 330 q^{33} - 333 q^{35} + 102 q^{37} - 164 q^{39} - 370 q^{41} - 431 q^{43} - 429 q^{45} + 199 q^{47} - 802 q^{49} - 272 q^{51} - 308 q^{53} - 85 q^{55} - 76 q^{57} + 188 q^{59} - 609 q^{61} + 61 q^{63} - 1536 q^{65} + 246 q^{67} - 892 q^{69} + 954 q^{71} - 629 q^{73} + 1074 q^{75} - 71 q^{77} - 452 q^{79} - 361 q^{81} + 780 q^{83} + 1883 q^{85} + 96 q^{87} - 1356 q^{89} + 670 q^{91} + 1092 q^{93} + 133 q^{95} - 548 q^{97} + 1305 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 5.61812 1.08121 0.540604 0.841277i \(-0.318196\pi\)
0.540604 + 0.841277i \(0.318196\pi\)
\(4\) 0 0
\(5\) −13.8269 −1.23672 −0.618359 0.785896i \(-0.712202\pi\)
−0.618359 + 0.785896i \(0.712202\pi\)
\(6\) 0 0
\(7\) 9.22252 0.497969 0.248985 0.968507i \(-0.419903\pi\)
0.248985 + 0.968507i \(0.419903\pi\)
\(8\) 0 0
\(9\) 4.56325 0.169009
\(10\) 0 0
\(11\) −59.8818 −1.64137 −0.820684 0.571383i \(-0.806407\pi\)
−0.820684 + 0.571383i \(0.806407\pi\)
\(12\) 0 0
\(13\) 38.2034 0.815055 0.407527 0.913193i \(-0.366391\pi\)
0.407527 + 0.913193i \(0.366391\pi\)
\(14\) 0 0
\(15\) −77.6813 −1.33715
\(16\) 0 0
\(17\) −47.0714 −0.671559 −0.335779 0.941941i \(-0.609000\pi\)
−0.335779 + 0.941941i \(0.609000\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) 51.8132 0.538408
\(22\) 0 0
\(23\) −179.632 −1.62851 −0.814256 0.580506i \(-0.802855\pi\)
−0.814256 + 0.580506i \(0.802855\pi\)
\(24\) 0 0
\(25\) 66.1837 0.529470
\(26\) 0 0
\(27\) −126.052 −0.898473
\(28\) 0 0
\(29\) −96.6484 −0.618868 −0.309434 0.950921i \(-0.600140\pi\)
−0.309434 + 0.950921i \(0.600140\pi\)
\(30\) 0 0
\(31\) −125.918 −0.729532 −0.364766 0.931099i \(-0.618851\pi\)
−0.364766 + 0.931099i \(0.618851\pi\)
\(32\) 0 0
\(33\) −336.423 −1.77466
\(34\) 0 0
\(35\) −127.519 −0.615847
\(36\) 0 0
\(37\) 407.764 1.81178 0.905891 0.423510i \(-0.139202\pi\)
0.905891 + 0.423510i \(0.139202\pi\)
\(38\) 0 0
\(39\) 214.631 0.881243
\(40\) 0 0
\(41\) −220.654 −0.840495 −0.420248 0.907409i \(-0.638057\pi\)
−0.420248 + 0.907409i \(0.638057\pi\)
\(42\) 0 0
\(43\) −100.777 −0.357403 −0.178702 0.983903i \(-0.557190\pi\)
−0.178702 + 0.983903i \(0.557190\pi\)
\(44\) 0 0
\(45\) −63.0957 −0.209017
\(46\) 0 0
\(47\) 213.749 0.663372 0.331686 0.943390i \(-0.392383\pi\)
0.331686 + 0.943390i \(0.392383\pi\)
\(48\) 0 0
\(49\) −257.945 −0.752027
\(50\) 0 0
\(51\) −264.453 −0.726095
\(52\) 0 0
\(53\) 19.8284 0.0513895 0.0256948 0.999670i \(-0.491820\pi\)
0.0256948 + 0.999670i \(0.491820\pi\)
\(54\) 0 0
\(55\) 827.981 2.02991
\(56\) 0 0
\(57\) 106.744 0.248046
\(58\) 0 0
\(59\) 97.7827 0.215767 0.107883 0.994164i \(-0.465593\pi\)
0.107883 + 0.994164i \(0.465593\pi\)
\(60\) 0 0
\(61\) 266.365 0.559091 0.279545 0.960132i \(-0.409816\pi\)
0.279545 + 0.960132i \(0.409816\pi\)
\(62\) 0 0
\(63\) 42.0847 0.0841614
\(64\) 0 0
\(65\) −528.235 −1.00799
\(66\) 0 0
\(67\) −627.332 −1.14389 −0.571946 0.820291i \(-0.693811\pi\)
−0.571946 + 0.820291i \(0.693811\pi\)
\(68\) 0 0
\(69\) −1009.19 −1.76076
\(70\) 0 0
\(71\) 644.105 1.07664 0.538318 0.842741i \(-0.319060\pi\)
0.538318 + 0.842741i \(0.319060\pi\)
\(72\) 0 0
\(73\) 807.966 1.29541 0.647707 0.761889i \(-0.275728\pi\)
0.647707 + 0.761889i \(0.275728\pi\)
\(74\) 0 0
\(75\) 371.828 0.572467
\(76\) 0 0
\(77\) −552.261 −0.817350
\(78\) 0 0
\(79\) −785.715 −1.11899 −0.559493 0.828835i \(-0.689004\pi\)
−0.559493 + 0.828835i \(0.689004\pi\)
\(80\) 0 0
\(81\) −831.385 −1.14045
\(82\) 0 0
\(83\) −1174.90 −1.55376 −0.776879 0.629650i \(-0.783198\pi\)
−0.776879 + 0.629650i \(0.783198\pi\)
\(84\) 0 0
\(85\) 650.853 0.830529
\(86\) 0 0
\(87\) −542.982 −0.669124
\(88\) 0 0
\(89\) −234.248 −0.278991 −0.139496 0.990223i \(-0.544548\pi\)
−0.139496 + 0.990223i \(0.544548\pi\)
\(90\) 0 0
\(91\) 352.331 0.405872
\(92\) 0 0
\(93\) −707.421 −0.788775
\(94\) 0 0
\(95\) −262.711 −0.283722
\(96\) 0 0
\(97\) −1348.51 −1.41155 −0.705774 0.708437i \(-0.749401\pi\)
−0.705774 + 0.708437i \(0.749401\pi\)
\(98\) 0 0
\(99\) −273.256 −0.277406
\(100\) 0 0
\(101\) 867.044 0.854199 0.427100 0.904205i \(-0.359535\pi\)
0.427100 + 0.904205i \(0.359535\pi\)
\(102\) 0 0
\(103\) −492.373 −0.471019 −0.235509 0.971872i \(-0.575676\pi\)
−0.235509 + 0.971872i \(0.575676\pi\)
\(104\) 0 0
\(105\) −716.417 −0.665859
\(106\) 0 0
\(107\) 1307.06 1.18092 0.590460 0.807067i \(-0.298946\pi\)
0.590460 + 0.807067i \(0.298946\pi\)
\(108\) 0 0
\(109\) 1435.58 1.26150 0.630749 0.775987i \(-0.282748\pi\)
0.630749 + 0.775987i \(0.282748\pi\)
\(110\) 0 0
\(111\) 2290.87 1.95891
\(112\) 0 0
\(113\) 1779.17 1.48116 0.740578 0.671970i \(-0.234552\pi\)
0.740578 + 0.671970i \(0.234552\pi\)
\(114\) 0 0
\(115\) 2483.75 2.01401
\(116\) 0 0
\(117\) 174.332 0.137752
\(118\) 0 0
\(119\) −434.117 −0.334416
\(120\) 0 0
\(121\) 2254.83 1.69409
\(122\) 0 0
\(123\) −1239.66 −0.908750
\(124\) 0 0
\(125\) 813.248 0.581913
\(126\) 0 0
\(127\) −1334.73 −0.932582 −0.466291 0.884631i \(-0.654410\pi\)
−0.466291 + 0.884631i \(0.654410\pi\)
\(128\) 0 0
\(129\) −566.177 −0.386427
\(130\) 0 0
\(131\) −52.5550 −0.0350516 −0.0175258 0.999846i \(-0.505579\pi\)
−0.0175258 + 0.999846i \(0.505579\pi\)
\(132\) 0 0
\(133\) 175.228 0.114242
\(134\) 0 0
\(135\) 1742.92 1.11116
\(136\) 0 0
\(137\) 852.662 0.531736 0.265868 0.964009i \(-0.414341\pi\)
0.265868 + 0.964009i \(0.414341\pi\)
\(138\) 0 0
\(139\) 812.834 0.495998 0.247999 0.968760i \(-0.420227\pi\)
0.247999 + 0.968760i \(0.420227\pi\)
\(140\) 0 0
\(141\) 1200.87 0.717243
\(142\) 0 0
\(143\) −2287.69 −1.33780
\(144\) 0 0
\(145\) 1336.35 0.765364
\(146\) 0 0
\(147\) −1449.17 −0.813097
\(148\) 0 0
\(149\) 3103.16 1.70618 0.853089 0.521765i \(-0.174726\pi\)
0.853089 + 0.521765i \(0.174726\pi\)
\(150\) 0 0
\(151\) −109.068 −0.0587803 −0.0293902 0.999568i \(-0.509357\pi\)
−0.0293902 + 0.999568i \(0.509357\pi\)
\(152\) 0 0
\(153\) −214.799 −0.113500
\(154\) 0 0
\(155\) 1741.05 0.902225
\(156\) 0 0
\(157\) 1949.74 0.991123 0.495561 0.868573i \(-0.334962\pi\)
0.495561 + 0.868573i \(0.334962\pi\)
\(158\) 0 0
\(159\) 111.398 0.0555627
\(160\) 0 0
\(161\) −1656.66 −0.810949
\(162\) 0 0
\(163\) −2089.53 −1.00408 −0.502040 0.864845i \(-0.667417\pi\)
−0.502040 + 0.864845i \(0.667417\pi\)
\(164\) 0 0
\(165\) 4651.69 2.19475
\(166\) 0 0
\(167\) 506.705 0.234790 0.117395 0.993085i \(-0.462546\pi\)
0.117395 + 0.993085i \(0.462546\pi\)
\(168\) 0 0
\(169\) −737.502 −0.335686
\(170\) 0 0
\(171\) 86.7018 0.0387734
\(172\) 0 0
\(173\) −3668.57 −1.61223 −0.806116 0.591758i \(-0.798434\pi\)
−0.806116 + 0.591758i \(0.798434\pi\)
\(174\) 0 0
\(175\) 610.381 0.263660
\(176\) 0 0
\(177\) 549.355 0.233288
\(178\) 0 0
\(179\) 3873.56 1.61745 0.808725 0.588187i \(-0.200158\pi\)
0.808725 + 0.588187i \(0.200158\pi\)
\(180\) 0 0
\(181\) 559.567 0.229791 0.114896 0.993378i \(-0.463347\pi\)
0.114896 + 0.993378i \(0.463347\pi\)
\(182\) 0 0
\(183\) 1496.47 0.604493
\(184\) 0 0
\(185\) −5638.12 −2.24066
\(186\) 0 0
\(187\) 2818.72 1.10227
\(188\) 0 0
\(189\) −1162.52 −0.447412
\(190\) 0 0
\(191\) 1333.34 0.505115 0.252558 0.967582i \(-0.418728\pi\)
0.252558 + 0.967582i \(0.418728\pi\)
\(192\) 0 0
\(193\) −3723.10 −1.38857 −0.694287 0.719698i \(-0.744280\pi\)
−0.694287 + 0.719698i \(0.744280\pi\)
\(194\) 0 0
\(195\) −2967.69 −1.08985
\(196\) 0 0
\(197\) −4989.58 −1.80453 −0.902267 0.431178i \(-0.858098\pi\)
−0.902267 + 0.431178i \(0.858098\pi\)
\(198\) 0 0
\(199\) −5308.31 −1.89094 −0.945468 0.325714i \(-0.894395\pi\)
−0.945468 + 0.325714i \(0.894395\pi\)
\(200\) 0 0
\(201\) −3524.43 −1.23679
\(202\) 0 0
\(203\) −891.342 −0.308177
\(204\) 0 0
\(205\) 3050.96 1.03946
\(206\) 0 0
\(207\) −819.704 −0.275234
\(208\) 0 0
\(209\) −1137.75 −0.376555
\(210\) 0 0
\(211\) 1738.60 0.567251 0.283626 0.958935i \(-0.408463\pi\)
0.283626 + 0.958935i \(0.408463\pi\)
\(212\) 0 0
\(213\) 3618.66 1.16407
\(214\) 0 0
\(215\) 1393.44 0.442007
\(216\) 0 0
\(217\) −1161.28 −0.363284
\(218\) 0 0
\(219\) 4539.25 1.40061
\(220\) 0 0
\(221\) −1798.29 −0.547357
\(222\) 0 0
\(223\) −1278.02 −0.383777 −0.191889 0.981417i \(-0.561461\pi\)
−0.191889 + 0.981417i \(0.561461\pi\)
\(224\) 0 0
\(225\) 302.013 0.0894853
\(226\) 0 0
\(227\) −1022.70 −0.299027 −0.149514 0.988760i \(-0.547771\pi\)
−0.149514 + 0.988760i \(0.547771\pi\)
\(228\) 0 0
\(229\) 600.407 0.173258 0.0866289 0.996241i \(-0.472391\pi\)
0.0866289 + 0.996241i \(0.472391\pi\)
\(230\) 0 0
\(231\) −3102.67 −0.883725
\(232\) 0 0
\(233\) 6067.68 1.70604 0.853019 0.521879i \(-0.174769\pi\)
0.853019 + 0.521879i \(0.174769\pi\)
\(234\) 0 0
\(235\) −2955.49 −0.820404
\(236\) 0 0
\(237\) −4414.24 −1.20986
\(238\) 0 0
\(239\) −11.8896 −0.00321788 −0.00160894 0.999999i \(-0.500512\pi\)
−0.00160894 + 0.999999i \(0.500512\pi\)
\(240\) 0 0
\(241\) −2816.80 −0.752888 −0.376444 0.926439i \(-0.622853\pi\)
−0.376444 + 0.926439i \(0.622853\pi\)
\(242\) 0 0
\(243\) −1267.40 −0.334584
\(244\) 0 0
\(245\) 3566.59 0.930044
\(246\) 0 0
\(247\) 725.864 0.186986
\(248\) 0 0
\(249\) −6600.72 −1.67993
\(250\) 0 0
\(251\) 3973.52 0.999228 0.499614 0.866248i \(-0.333475\pi\)
0.499614 + 0.866248i \(0.333475\pi\)
\(252\) 0 0
\(253\) 10756.7 2.67299
\(254\) 0 0
\(255\) 3656.57 0.897974
\(256\) 0 0
\(257\) −7356.09 −1.78545 −0.892724 0.450603i \(-0.851209\pi\)
−0.892724 + 0.450603i \(0.851209\pi\)
\(258\) 0 0
\(259\) 3760.61 0.902212
\(260\) 0 0
\(261\) −441.031 −0.104594
\(262\) 0 0
\(263\) −4092.74 −0.959578 −0.479789 0.877384i \(-0.659287\pi\)
−0.479789 + 0.877384i \(0.659287\pi\)
\(264\) 0 0
\(265\) −274.166 −0.0635543
\(266\) 0 0
\(267\) −1316.03 −0.301647
\(268\) 0 0
\(269\) −3662.93 −0.830234 −0.415117 0.909768i \(-0.636259\pi\)
−0.415117 + 0.909768i \(0.636259\pi\)
\(270\) 0 0
\(271\) −5576.56 −1.25001 −0.625003 0.780622i \(-0.714902\pi\)
−0.625003 + 0.780622i \(0.714902\pi\)
\(272\) 0 0
\(273\) 1979.44 0.438832
\(274\) 0 0
\(275\) −3963.20 −0.869055
\(276\) 0 0
\(277\) 764.590 0.165848 0.0829238 0.996556i \(-0.473574\pi\)
0.0829238 + 0.996556i \(0.473574\pi\)
\(278\) 0 0
\(279\) −574.594 −0.123298
\(280\) 0 0
\(281\) −2805.51 −0.595597 −0.297798 0.954629i \(-0.596252\pi\)
−0.297798 + 0.954629i \(0.596252\pi\)
\(282\) 0 0
\(283\) 2564.76 0.538725 0.269362 0.963039i \(-0.413187\pi\)
0.269362 + 0.963039i \(0.413187\pi\)
\(284\) 0 0
\(285\) −1475.94 −0.306763
\(286\) 0 0
\(287\) −2034.98 −0.418541
\(288\) 0 0
\(289\) −2697.28 −0.549009
\(290\) 0 0
\(291\) −7576.07 −1.52618
\(292\) 0 0
\(293\) −6350.30 −1.26617 −0.633086 0.774081i \(-0.718212\pi\)
−0.633086 + 0.774081i \(0.718212\pi\)
\(294\) 0 0
\(295\) −1352.03 −0.266842
\(296\) 0 0
\(297\) 7548.24 1.47472
\(298\) 0 0
\(299\) −6862.54 −1.32733
\(300\) 0 0
\(301\) −929.418 −0.177976
\(302\) 0 0
\(303\) 4871.16 0.923567
\(304\) 0 0
\(305\) −3683.01 −0.691437
\(306\) 0 0
\(307\) −7556.96 −1.40488 −0.702441 0.711742i \(-0.747906\pi\)
−0.702441 + 0.711742i \(0.747906\pi\)
\(308\) 0 0
\(309\) −2766.21 −0.509269
\(310\) 0 0
\(311\) 7200.87 1.31294 0.656470 0.754352i \(-0.272049\pi\)
0.656470 + 0.754352i \(0.272049\pi\)
\(312\) 0 0
\(313\) 7971.84 1.43960 0.719800 0.694181i \(-0.244233\pi\)
0.719800 + 0.694181i \(0.244233\pi\)
\(314\) 0 0
\(315\) −581.901 −0.104084
\(316\) 0 0
\(317\) 6328.95 1.12135 0.560677 0.828035i \(-0.310541\pi\)
0.560677 + 0.828035i \(0.310541\pi\)
\(318\) 0 0
\(319\) 5787.48 1.01579
\(320\) 0 0
\(321\) 7343.23 1.27682
\(322\) 0 0
\(323\) −894.358 −0.154066
\(324\) 0 0
\(325\) 2528.44 0.431547
\(326\) 0 0
\(327\) 8065.24 1.36394
\(328\) 0 0
\(329\) 1971.30 0.330339
\(330\) 0 0
\(331\) −9467.70 −1.57218 −0.786090 0.618112i \(-0.787898\pi\)
−0.786090 + 0.618112i \(0.787898\pi\)
\(332\) 0 0
\(333\) 1860.73 0.306208
\(334\) 0 0
\(335\) 8674.07 1.41467
\(336\) 0 0
\(337\) −10979.0 −1.77466 −0.887332 0.461131i \(-0.847444\pi\)
−0.887332 + 0.461131i \(0.847444\pi\)
\(338\) 0 0
\(339\) 9995.61 1.60144
\(340\) 0 0
\(341\) 7540.18 1.19743
\(342\) 0 0
\(343\) −5542.23 −0.872455
\(344\) 0 0
\(345\) 13954.0 2.17756
\(346\) 0 0
\(347\) −8731.23 −1.35077 −0.675385 0.737465i \(-0.736023\pi\)
−0.675385 + 0.737465i \(0.736023\pi\)
\(348\) 0 0
\(349\) −7156.72 −1.09768 −0.548840 0.835927i \(-0.684930\pi\)
−0.548840 + 0.835927i \(0.684930\pi\)
\(350\) 0 0
\(351\) −4815.62 −0.732305
\(352\) 0 0
\(353\) 5017.02 0.756456 0.378228 0.925713i \(-0.376534\pi\)
0.378228 + 0.925713i \(0.376534\pi\)
\(354\) 0 0
\(355\) −8905.99 −1.33150
\(356\) 0 0
\(357\) −2438.92 −0.361573
\(358\) 0 0
\(359\) −13325.3 −1.95901 −0.979503 0.201428i \(-0.935442\pi\)
−0.979503 + 0.201428i \(0.935442\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 12667.9 1.83166
\(364\) 0 0
\(365\) −11171.7 −1.60206
\(366\) 0 0
\(367\) 4706.78 0.669460 0.334730 0.942314i \(-0.391355\pi\)
0.334730 + 0.942314i \(0.391355\pi\)
\(368\) 0 0
\(369\) −1006.90 −0.142052
\(370\) 0 0
\(371\) 182.868 0.0255904
\(372\) 0 0
\(373\) 2524.08 0.350381 0.175190 0.984535i \(-0.443946\pi\)
0.175190 + 0.984535i \(0.443946\pi\)
\(374\) 0 0
\(375\) 4568.92 0.629168
\(376\) 0 0
\(377\) −3692.30 −0.504411
\(378\) 0 0
\(379\) 2987.03 0.404838 0.202419 0.979299i \(-0.435120\pi\)
0.202419 + 0.979299i \(0.435120\pi\)
\(380\) 0 0
\(381\) −7498.66 −1.00831
\(382\) 0 0
\(383\) −8661.02 −1.15550 −0.577751 0.816213i \(-0.696070\pi\)
−0.577751 + 0.816213i \(0.696070\pi\)
\(384\) 0 0
\(385\) 7636.07 1.01083
\(386\) 0 0
\(387\) −459.871 −0.0604045
\(388\) 0 0
\(389\) −1880.29 −0.245076 −0.122538 0.992464i \(-0.539103\pi\)
−0.122538 + 0.992464i \(0.539103\pi\)
\(390\) 0 0
\(391\) 8455.52 1.09364
\(392\) 0 0
\(393\) −295.260 −0.0378980
\(394\) 0 0
\(395\) 10864.0 1.38387
\(396\) 0 0
\(397\) −4453.25 −0.562978 −0.281489 0.959564i \(-0.590828\pi\)
−0.281489 + 0.959564i \(0.590828\pi\)
\(398\) 0 0
\(399\) 984.451 0.123519
\(400\) 0 0
\(401\) 3404.95 0.424027 0.212014 0.977267i \(-0.431998\pi\)
0.212014 + 0.977267i \(0.431998\pi\)
\(402\) 0 0
\(403\) −4810.48 −0.594608
\(404\) 0 0
\(405\) 11495.5 1.41041
\(406\) 0 0
\(407\) −24417.6 −2.97380
\(408\) 0 0
\(409\) 3784.84 0.457576 0.228788 0.973476i \(-0.426524\pi\)
0.228788 + 0.973476i \(0.426524\pi\)
\(410\) 0 0
\(411\) 4790.36 0.574917
\(412\) 0 0
\(413\) 901.803 0.107445
\(414\) 0 0
\(415\) 16245.2 1.92156
\(416\) 0 0
\(417\) 4566.60 0.536276
\(418\) 0 0
\(419\) 5688.12 0.663205 0.331602 0.943419i \(-0.392411\pi\)
0.331602 + 0.943419i \(0.392411\pi\)
\(420\) 0 0
\(421\) 5133.25 0.594250 0.297125 0.954839i \(-0.403972\pi\)
0.297125 + 0.954839i \(0.403972\pi\)
\(422\) 0 0
\(423\) 975.390 0.112116
\(424\) 0 0
\(425\) −3115.36 −0.355570
\(426\) 0 0
\(427\) 2456.56 0.278410
\(428\) 0 0
\(429\) −12852.5 −1.44644
\(430\) 0 0
\(431\) −4240.52 −0.473918 −0.236959 0.971520i \(-0.576151\pi\)
−0.236959 + 0.971520i \(0.576151\pi\)
\(432\) 0 0
\(433\) 536.405 0.0595334 0.0297667 0.999557i \(-0.490524\pi\)
0.0297667 + 0.999557i \(0.490524\pi\)
\(434\) 0 0
\(435\) 7507.77 0.827518
\(436\) 0 0
\(437\) −3413.00 −0.373606
\(438\) 0 0
\(439\) 2099.31 0.228234 0.114117 0.993467i \(-0.463596\pi\)
0.114117 + 0.993467i \(0.463596\pi\)
\(440\) 0 0
\(441\) −1177.07 −0.127099
\(442\) 0 0
\(443\) 2468.83 0.264780 0.132390 0.991198i \(-0.457735\pi\)
0.132390 + 0.991198i \(0.457735\pi\)
\(444\) 0 0
\(445\) 3238.92 0.345033
\(446\) 0 0
\(447\) 17433.9 1.84473
\(448\) 0 0
\(449\) −18561.5 −1.95094 −0.975468 0.220141i \(-0.929348\pi\)
−0.975468 + 0.220141i \(0.929348\pi\)
\(450\) 0 0
\(451\) 13213.1 1.37956
\(452\) 0 0
\(453\) −612.757 −0.0635537
\(454\) 0 0
\(455\) −4871.66 −0.501949
\(456\) 0 0
\(457\) −4128.97 −0.422637 −0.211319 0.977417i \(-0.567776\pi\)
−0.211319 + 0.977417i \(0.567776\pi\)
\(458\) 0 0
\(459\) 5933.46 0.603378
\(460\) 0 0
\(461\) −15795.0 −1.59576 −0.797879 0.602818i \(-0.794045\pi\)
−0.797879 + 0.602818i \(0.794045\pi\)
\(462\) 0 0
\(463\) 16293.3 1.63545 0.817725 0.575610i \(-0.195235\pi\)
0.817725 + 0.575610i \(0.195235\pi\)
\(464\) 0 0
\(465\) 9781.45 0.975492
\(466\) 0 0
\(467\) −13787.7 −1.36621 −0.683106 0.730320i \(-0.739371\pi\)
−0.683106 + 0.730320i \(0.739371\pi\)
\(468\) 0 0
\(469\) −5785.58 −0.569623
\(470\) 0 0
\(471\) 10953.9 1.07161
\(472\) 0 0
\(473\) 6034.71 0.586630
\(474\) 0 0
\(475\) 1257.49 0.121469
\(476\) 0 0
\(477\) 90.4821 0.00868531
\(478\) 0 0
\(479\) 8123.95 0.774932 0.387466 0.921884i \(-0.373350\pi\)
0.387466 + 0.921884i \(0.373350\pi\)
\(480\) 0 0
\(481\) 15578.0 1.47670
\(482\) 0 0
\(483\) −9307.29 −0.876804
\(484\) 0 0
\(485\) 18645.7 1.74569
\(486\) 0 0
\(487\) 1100.55 0.102404 0.0512018 0.998688i \(-0.483695\pi\)
0.0512018 + 0.998688i \(0.483695\pi\)
\(488\) 0 0
\(489\) −11739.3 −1.08562
\(490\) 0 0
\(491\) −18720.2 −1.72063 −0.860315 0.509762i \(-0.829733\pi\)
−0.860315 + 0.509762i \(0.829733\pi\)
\(492\) 0 0
\(493\) 4549.38 0.415606
\(494\) 0 0
\(495\) 3778.28 0.343073
\(496\) 0 0
\(497\) 5940.27 0.536132
\(498\) 0 0
\(499\) 6309.11 0.566001 0.283001 0.959120i \(-0.408670\pi\)
0.283001 + 0.959120i \(0.408670\pi\)
\(500\) 0 0
\(501\) 2846.73 0.253857
\(502\) 0 0
\(503\) −8091.31 −0.717244 −0.358622 0.933483i \(-0.616753\pi\)
−0.358622 + 0.933483i \(0.616753\pi\)
\(504\) 0 0
\(505\) −11988.6 −1.05640
\(506\) 0 0
\(507\) −4143.37 −0.362946
\(508\) 0 0
\(509\) 11338.8 0.987398 0.493699 0.869633i \(-0.335644\pi\)
0.493699 + 0.869633i \(0.335644\pi\)
\(510\) 0 0
\(511\) 7451.48 0.645077
\(512\) 0 0
\(513\) −2394.99 −0.206124
\(514\) 0 0
\(515\) 6808.00 0.582517
\(516\) 0 0
\(517\) −12799.7 −1.08884
\(518\) 0 0
\(519\) −20610.4 −1.74316
\(520\) 0 0
\(521\) 12848.2 1.08041 0.540203 0.841535i \(-0.318348\pi\)
0.540203 + 0.841535i \(0.318348\pi\)
\(522\) 0 0
\(523\) 14224.9 1.18932 0.594658 0.803978i \(-0.297287\pi\)
0.594658 + 0.803978i \(0.297287\pi\)
\(524\) 0 0
\(525\) 3429.19 0.285071
\(526\) 0 0
\(527\) 5927.13 0.489924
\(528\) 0 0
\(529\) 20100.5 1.65205
\(530\) 0 0
\(531\) 446.207 0.0364666
\(532\) 0 0
\(533\) −8429.71 −0.685050
\(534\) 0 0
\(535\) −18072.6 −1.46046
\(536\) 0 0
\(537\) 21762.1 1.74880
\(538\) 0 0
\(539\) 15446.2 1.23435
\(540\) 0 0
\(541\) 24551.3 1.95109 0.975547 0.219789i \(-0.0705368\pi\)
0.975547 + 0.219789i \(0.0705368\pi\)
\(542\) 0 0
\(543\) 3143.71 0.248452
\(544\) 0 0
\(545\) −19849.6 −1.56012
\(546\) 0 0
\(547\) −2290.42 −0.179034 −0.0895169 0.995985i \(-0.528532\pi\)
−0.0895169 + 0.995985i \(0.528532\pi\)
\(548\) 0 0
\(549\) 1215.49 0.0944915
\(550\) 0 0
\(551\) −1836.32 −0.141978
\(552\) 0 0
\(553\) −7246.27 −0.557220
\(554\) 0 0
\(555\) −31675.6 −2.42262
\(556\) 0 0
\(557\) 11590.3 0.881683 0.440841 0.897585i \(-0.354680\pi\)
0.440841 + 0.897585i \(0.354680\pi\)
\(558\) 0 0
\(559\) −3850.02 −0.291303
\(560\) 0 0
\(561\) 15835.9 1.19179
\(562\) 0 0
\(563\) 15650.3 1.17155 0.585773 0.810475i \(-0.300791\pi\)
0.585773 + 0.810475i \(0.300791\pi\)
\(564\) 0 0
\(565\) −24600.5 −1.83177
\(566\) 0 0
\(567\) −7667.46 −0.567907
\(568\) 0 0
\(569\) 745.409 0.0549194 0.0274597 0.999623i \(-0.491258\pi\)
0.0274597 + 0.999623i \(0.491258\pi\)
\(570\) 0 0
\(571\) 13663.3 1.00138 0.500691 0.865626i \(-0.333079\pi\)
0.500691 + 0.865626i \(0.333079\pi\)
\(572\) 0 0
\(573\) 7490.86 0.546134
\(574\) 0 0
\(575\) −11888.7 −0.862248
\(576\) 0 0
\(577\) 12816.7 0.924721 0.462361 0.886692i \(-0.347003\pi\)
0.462361 + 0.886692i \(0.347003\pi\)
\(578\) 0 0
\(579\) −20916.8 −1.50134
\(580\) 0 0
\(581\) −10835.5 −0.773724
\(582\) 0 0
\(583\) −1187.36 −0.0843491
\(584\) 0 0
\(585\) −2410.47 −0.170360
\(586\) 0 0
\(587\) −23688.4 −1.66563 −0.832817 0.553549i \(-0.813273\pi\)
−0.832817 + 0.553549i \(0.813273\pi\)
\(588\) 0 0
\(589\) −2392.44 −0.167366
\(590\) 0 0
\(591\) −28032.1 −1.95108
\(592\) 0 0
\(593\) −4251.84 −0.294439 −0.147219 0.989104i \(-0.547032\pi\)
−0.147219 + 0.989104i \(0.547032\pi\)
\(594\) 0 0
\(595\) 6002.51 0.413578
\(596\) 0 0
\(597\) −29822.7 −2.04449
\(598\) 0 0
\(599\) 3852.14 0.262761 0.131381 0.991332i \(-0.458059\pi\)
0.131381 + 0.991332i \(0.458059\pi\)
\(600\) 0 0
\(601\) 4596.36 0.311963 0.155981 0.987760i \(-0.450146\pi\)
0.155981 + 0.987760i \(0.450146\pi\)
\(602\) 0 0
\(603\) −2862.67 −0.193328
\(604\) 0 0
\(605\) −31177.3 −2.09511
\(606\) 0 0
\(607\) −7814.06 −0.522509 −0.261255 0.965270i \(-0.584136\pi\)
−0.261255 + 0.965270i \(0.584136\pi\)
\(608\) 0 0
\(609\) −5007.66 −0.333203
\(610\) 0 0
\(611\) 8165.93 0.540685
\(612\) 0 0
\(613\) −10013.8 −0.659795 −0.329897 0.944017i \(-0.607014\pi\)
−0.329897 + 0.944017i \(0.607014\pi\)
\(614\) 0 0
\(615\) 17140.7 1.12387
\(616\) 0 0
\(617\) −18316.8 −1.19515 −0.597576 0.801813i \(-0.703869\pi\)
−0.597576 + 0.801813i \(0.703869\pi\)
\(618\) 0 0
\(619\) 16755.3 1.08797 0.543985 0.839095i \(-0.316915\pi\)
0.543985 + 0.839095i \(0.316915\pi\)
\(620\) 0 0
\(621\) 22643.0 1.46317
\(622\) 0 0
\(623\) −2160.35 −0.138929
\(624\) 0 0
\(625\) −19517.7 −1.24913
\(626\) 0 0
\(627\) −6392.04 −0.407135
\(628\) 0 0
\(629\) −19194.0 −1.21672
\(630\) 0 0
\(631\) −27586.7 −1.74042 −0.870212 0.492677i \(-0.836018\pi\)
−0.870212 + 0.492677i \(0.836018\pi\)
\(632\) 0 0
\(633\) 9767.64 0.613316
\(634\) 0 0
\(635\) 18455.2 1.15334
\(636\) 0 0
\(637\) −9854.38 −0.612943
\(638\) 0 0
\(639\) 2939.21 0.181962
\(640\) 0 0
\(641\) 17669.0 1.08874 0.544371 0.838845i \(-0.316769\pi\)
0.544371 + 0.838845i \(0.316769\pi\)
\(642\) 0 0
\(643\) −15999.3 −0.981259 −0.490630 0.871368i \(-0.663233\pi\)
−0.490630 + 0.871368i \(0.663233\pi\)
\(644\) 0 0
\(645\) 7828.49 0.477901
\(646\) 0 0
\(647\) −9922.65 −0.602936 −0.301468 0.953476i \(-0.597477\pi\)
−0.301468 + 0.953476i \(0.597477\pi\)
\(648\) 0 0
\(649\) −5855.40 −0.354152
\(650\) 0 0
\(651\) −6524.20 −0.392786
\(652\) 0 0
\(653\) 3000.87 0.179836 0.0899181 0.995949i \(-0.471339\pi\)
0.0899181 + 0.995949i \(0.471339\pi\)
\(654\) 0 0
\(655\) 726.674 0.0433489
\(656\) 0 0
\(657\) 3686.95 0.218937
\(658\) 0 0
\(659\) −23808.3 −1.40734 −0.703672 0.710525i \(-0.748457\pi\)
−0.703672 + 0.710525i \(0.748457\pi\)
\(660\) 0 0
\(661\) −19729.7 −1.16096 −0.580482 0.814273i \(-0.697136\pi\)
−0.580482 + 0.814273i \(0.697136\pi\)
\(662\) 0 0
\(663\) −10103.0 −0.591807
\(664\) 0 0
\(665\) −2422.86 −0.141285
\(666\) 0 0
\(667\) 17361.1 1.00783
\(668\) 0 0
\(669\) −7180.05 −0.414943
\(670\) 0 0
\(671\) −15950.4 −0.917673
\(672\) 0 0
\(673\) 20300.0 1.16272 0.581358 0.813648i \(-0.302522\pi\)
0.581358 + 0.813648i \(0.302522\pi\)
\(674\) 0 0
\(675\) −8342.61 −0.475715
\(676\) 0 0
\(677\) 16523.6 0.938040 0.469020 0.883188i \(-0.344607\pi\)
0.469020 + 0.883188i \(0.344607\pi\)
\(678\) 0 0
\(679\) −12436.6 −0.702908
\(680\) 0 0
\(681\) −5745.66 −0.323310
\(682\) 0 0
\(683\) 5359.32 0.300247 0.150123 0.988667i \(-0.452033\pi\)
0.150123 + 0.988667i \(0.452033\pi\)
\(684\) 0 0
\(685\) −11789.7 −0.657607
\(686\) 0 0
\(687\) 3373.16 0.187328
\(688\) 0 0
\(689\) 757.513 0.0418853
\(690\) 0 0
\(691\) −19231.8 −1.05877 −0.529385 0.848382i \(-0.677577\pi\)
−0.529385 + 0.848382i \(0.677577\pi\)
\(692\) 0 0
\(693\) −2520.11 −0.138140
\(694\) 0 0
\(695\) −11239.0 −0.613409
\(696\) 0 0
\(697\) 10386.5 0.564442
\(698\) 0 0
\(699\) 34089.0 1.84458
\(700\) 0 0
\(701\) 6552.10 0.353023 0.176512 0.984299i \(-0.443519\pi\)
0.176512 + 0.984299i \(0.443519\pi\)
\(702\) 0 0
\(703\) 7747.51 0.415651
\(704\) 0 0
\(705\) −16604.3 −0.887027
\(706\) 0 0
\(707\) 7996.33 0.425365
\(708\) 0 0
\(709\) −27985.5 −1.48240 −0.741198 0.671287i \(-0.765742\pi\)
−0.741198 + 0.671287i \(0.765742\pi\)
\(710\) 0 0
\(711\) −3585.42 −0.189119
\(712\) 0 0
\(713\) 22618.8 1.18805
\(714\) 0 0
\(715\) 31631.7 1.65449
\(716\) 0 0
\(717\) −66.7971 −0.00347920
\(718\) 0 0
\(719\) 20268.3 1.05129 0.525646 0.850703i \(-0.323823\pi\)
0.525646 + 0.850703i \(0.323823\pi\)
\(720\) 0 0
\(721\) −4540.92 −0.234553
\(722\) 0 0
\(723\) −15825.1 −0.814029
\(724\) 0 0
\(725\) −6396.55 −0.327672
\(726\) 0 0
\(727\) −2068.98 −0.105549 −0.0527745 0.998606i \(-0.516806\pi\)
−0.0527745 + 0.998606i \(0.516806\pi\)
\(728\) 0 0
\(729\) 15327.0 0.778690
\(730\) 0 0
\(731\) 4743.72 0.240017
\(732\) 0 0
\(733\) 28074.9 1.41469 0.707347 0.706866i \(-0.249892\pi\)
0.707347 + 0.706866i \(0.249892\pi\)
\(734\) 0 0
\(735\) 20037.5 1.00557
\(736\) 0 0
\(737\) 37565.8 1.87755
\(738\) 0 0
\(739\) 17218.0 0.857068 0.428534 0.903526i \(-0.359030\pi\)
0.428534 + 0.903526i \(0.359030\pi\)
\(740\) 0 0
\(741\) 4077.99 0.202171
\(742\) 0 0
\(743\) 6618.16 0.326779 0.163390 0.986562i \(-0.447757\pi\)
0.163390 + 0.986562i \(0.447757\pi\)
\(744\) 0 0
\(745\) −42907.1 −2.11006
\(746\) 0 0
\(747\) −5361.36 −0.262600
\(748\) 0 0
\(749\) 12054.4 0.588062
\(750\) 0 0
\(751\) 22945.4 1.11490 0.557450 0.830210i \(-0.311780\pi\)
0.557450 + 0.830210i \(0.311780\pi\)
\(752\) 0 0
\(753\) 22323.7 1.08037
\(754\) 0 0
\(755\) 1508.07 0.0726947
\(756\) 0 0
\(757\) −30316.2 −1.45556 −0.727782 0.685808i \(-0.759449\pi\)
−0.727782 + 0.685808i \(0.759449\pi\)
\(758\) 0 0
\(759\) 60432.2 2.89005
\(760\) 0 0
\(761\) −22856.6 −1.08877 −0.544384 0.838836i \(-0.683236\pi\)
−0.544384 + 0.838836i \(0.683236\pi\)
\(762\) 0 0
\(763\) 13239.6 0.628187
\(764\) 0 0
\(765\) 2970.01 0.140367
\(766\) 0 0
\(767\) 3735.63 0.175862
\(768\) 0 0
\(769\) −28279.6 −1.32612 −0.663061 0.748565i \(-0.730743\pi\)
−0.663061 + 0.748565i \(0.730743\pi\)
\(770\) 0 0
\(771\) −41327.4 −1.93044
\(772\) 0 0
\(773\) 7705.64 0.358542 0.179271 0.983800i \(-0.442626\pi\)
0.179271 + 0.983800i \(0.442626\pi\)
\(774\) 0 0
\(775\) −8333.70 −0.386265
\(776\) 0 0
\(777\) 21127.5 0.975478
\(778\) 0 0
\(779\) −4192.42 −0.192823
\(780\) 0 0
\(781\) −38570.2 −1.76716
\(782\) 0 0
\(783\) 12182.8 0.556036
\(784\) 0 0
\(785\) −26958.9 −1.22574
\(786\) 0 0
\(787\) 42608.4 1.92989 0.964946 0.262448i \(-0.0845300\pi\)
0.964946 + 0.262448i \(0.0845300\pi\)
\(788\) 0 0
\(789\) −22993.5 −1.03750
\(790\) 0 0
\(791\) 16408.5 0.737570
\(792\) 0 0
\(793\) 10176.0 0.455690
\(794\) 0 0
\(795\) −1540.30 −0.0687154
\(796\) 0 0
\(797\) 42744.6 1.89974 0.949869 0.312647i \(-0.101216\pi\)
0.949869 + 0.312647i \(0.101216\pi\)
\(798\) 0 0
\(799\) −10061.5 −0.445494
\(800\) 0 0
\(801\) −1068.93 −0.0471521
\(802\) 0 0
\(803\) −48382.4 −2.12625
\(804\) 0 0
\(805\) 22906.5 1.00291
\(806\) 0 0
\(807\) −20578.8 −0.897656
\(808\) 0 0
\(809\) 17770.3 0.772277 0.386139 0.922441i \(-0.373809\pi\)
0.386139 + 0.922441i \(0.373809\pi\)
\(810\) 0 0
\(811\) −17635.6 −0.763588 −0.381794 0.924247i \(-0.624694\pi\)
−0.381794 + 0.924247i \(0.624694\pi\)
\(812\) 0 0
\(813\) −31329.8 −1.35152
\(814\) 0 0
\(815\) 28891.8 1.24176
\(816\) 0 0
\(817\) −1914.76 −0.0819940
\(818\) 0 0
\(819\) 1607.78 0.0685962
\(820\) 0 0
\(821\) 6447.65 0.274086 0.137043 0.990565i \(-0.456240\pi\)
0.137043 + 0.990565i \(0.456240\pi\)
\(822\) 0 0
\(823\) 21513.1 0.911179 0.455590 0.890190i \(-0.349428\pi\)
0.455590 + 0.890190i \(0.349428\pi\)
\(824\) 0 0
\(825\) −22265.7 −0.939628
\(826\) 0 0
\(827\) 42712.8 1.79597 0.897987 0.440022i \(-0.145030\pi\)
0.897987 + 0.440022i \(0.145030\pi\)
\(828\) 0 0
\(829\) 28421.8 1.19075 0.595374 0.803449i \(-0.297004\pi\)
0.595374 + 0.803449i \(0.297004\pi\)
\(830\) 0 0
\(831\) 4295.56 0.179316
\(832\) 0 0
\(833\) 12141.9 0.505030
\(834\) 0 0
\(835\) −7006.17 −0.290369
\(836\) 0 0
\(837\) 15872.2 0.655465
\(838\) 0 0
\(839\) 12288.2 0.505645 0.252823 0.967513i \(-0.418641\pi\)
0.252823 + 0.967513i \(0.418641\pi\)
\(840\) 0 0
\(841\) −15048.1 −0.617003
\(842\) 0 0
\(843\) −15761.7 −0.643963
\(844\) 0 0
\(845\) 10197.4 0.415149
\(846\) 0 0
\(847\) 20795.2 0.843603
\(848\) 0 0
\(849\) 14409.1 0.582473
\(850\) 0 0
\(851\) −73247.3 −2.95051
\(852\) 0 0
\(853\) 23447.1 0.941166 0.470583 0.882356i \(-0.344044\pi\)
0.470583 + 0.882356i \(0.344044\pi\)
\(854\) 0 0
\(855\) −1198.82 −0.0479517
\(856\) 0 0
\(857\) −34878.3 −1.39022 −0.695111 0.718902i \(-0.744645\pi\)
−0.695111 + 0.718902i \(0.744645\pi\)
\(858\) 0 0
\(859\) −31553.6 −1.25331 −0.626656 0.779296i \(-0.715577\pi\)
−0.626656 + 0.779296i \(0.715577\pi\)
\(860\) 0 0
\(861\) −11432.8 −0.452529
\(862\) 0 0
\(863\) 3137.99 0.123776 0.0618879 0.998083i \(-0.480288\pi\)
0.0618879 + 0.998083i \(0.480288\pi\)
\(864\) 0 0
\(865\) 50725.0 1.99387
\(866\) 0 0
\(867\) −15153.6 −0.593592
\(868\) 0 0
\(869\) 47050.0 1.83667
\(870\) 0 0
\(871\) −23966.2 −0.932335
\(872\) 0 0
\(873\) −6153.58 −0.238565
\(874\) 0 0
\(875\) 7500.19 0.289775
\(876\) 0 0
\(877\) −26872.9 −1.03470 −0.517351 0.855773i \(-0.673082\pi\)
−0.517351 + 0.855773i \(0.673082\pi\)
\(878\) 0 0
\(879\) −35676.8 −1.36900
\(880\) 0 0
\(881\) 10220.7 0.390856 0.195428 0.980718i \(-0.437390\pi\)
0.195428 + 0.980718i \(0.437390\pi\)
\(882\) 0 0
\(883\) 18978.1 0.723290 0.361645 0.932316i \(-0.382215\pi\)
0.361645 + 0.932316i \(0.382215\pi\)
\(884\) 0 0
\(885\) −7595.89 −0.288512
\(886\) 0 0
\(887\) 25106.7 0.950395 0.475197 0.879879i \(-0.342377\pi\)
0.475197 + 0.879879i \(0.342377\pi\)
\(888\) 0 0
\(889\) −12309.5 −0.464397
\(890\) 0 0
\(891\) 49784.8 1.87189
\(892\) 0 0
\(893\) 4061.23 0.152188
\(894\) 0 0
\(895\) −53559.4 −2.00033
\(896\) 0 0
\(897\) −38554.5 −1.43512
\(898\) 0 0
\(899\) 12169.7 0.451484
\(900\) 0 0
\(901\) −933.353 −0.0345111
\(902\) 0 0
\(903\) −5221.58 −0.192429
\(904\) 0 0
\(905\) −7737.08 −0.284187
\(906\) 0 0
\(907\) −40789.2 −1.49325 −0.746627 0.665243i \(-0.768328\pi\)
−0.746627 + 0.665243i \(0.768328\pi\)
\(908\) 0 0
\(909\) 3956.54 0.144368
\(910\) 0 0
\(911\) −8653.99 −0.314731 −0.157365 0.987540i \(-0.550300\pi\)
−0.157365 + 0.987540i \(0.550300\pi\)
\(912\) 0 0
\(913\) 70355.1 2.55029
\(914\) 0 0
\(915\) −20691.6 −0.747587
\(916\) 0 0
\(917\) −484.690 −0.0174546
\(918\) 0 0
\(919\) 9025.35 0.323960 0.161980 0.986794i \(-0.448212\pi\)
0.161980 + 0.986794i \(0.448212\pi\)
\(920\) 0 0
\(921\) −42455.9 −1.51897
\(922\) 0 0
\(923\) 24607.0 0.877518
\(924\) 0 0
\(925\) 26987.3 0.959284
\(926\) 0 0
\(927\) −2246.82 −0.0796066
\(928\) 0 0
\(929\) −30560.6 −1.07929 −0.539646 0.841892i \(-0.681442\pi\)
−0.539646 + 0.841892i \(0.681442\pi\)
\(930\) 0 0
\(931\) −4900.96 −0.172527
\(932\) 0 0
\(933\) 40455.4 1.41956
\(934\) 0 0
\(935\) −38974.3 −1.36320
\(936\) 0 0
\(937\) 15242.6 0.531436 0.265718 0.964051i \(-0.414391\pi\)
0.265718 + 0.964051i \(0.414391\pi\)
\(938\) 0 0
\(939\) 44786.7 1.55651
\(940\) 0 0
\(941\) −6241.91 −0.216238 −0.108119 0.994138i \(-0.534483\pi\)
−0.108119 + 0.994138i \(0.534483\pi\)
\(942\) 0 0
\(943\) 39636.4 1.36876
\(944\) 0 0
\(945\) 16074.1 0.553322
\(946\) 0 0
\(947\) −19299.8 −0.662259 −0.331130 0.943585i \(-0.607430\pi\)
−0.331130 + 0.943585i \(0.607430\pi\)
\(948\) 0 0
\(949\) 30867.0 1.05583
\(950\) 0 0
\(951\) 35556.8 1.21242
\(952\) 0 0
\(953\) 1598.11 0.0543209 0.0271605 0.999631i \(-0.491353\pi\)
0.0271605 + 0.999631i \(0.491353\pi\)
\(954\) 0 0
\(955\) −18436.0 −0.624685
\(956\) 0 0
\(957\) 32514.7 1.09828
\(958\) 0 0
\(959\) 7863.70 0.264788
\(960\) 0 0
\(961\) −13935.7 −0.467783
\(962\) 0 0
\(963\) 5964.45 0.199586
\(964\) 0 0
\(965\) 51479.1 1.71727
\(966\) 0 0
\(967\) −31860.4 −1.05952 −0.529762 0.848146i \(-0.677719\pi\)
−0.529762 + 0.848146i \(0.677719\pi\)
\(968\) 0 0
\(969\) −5024.61 −0.166578
\(970\) 0 0
\(971\) 3196.13 0.105632 0.0528160 0.998604i \(-0.483180\pi\)
0.0528160 + 0.998604i \(0.483180\pi\)
\(972\) 0 0
\(973\) 7496.38 0.246992
\(974\) 0 0
\(975\) 14205.1 0.466592
\(976\) 0 0
\(977\) −36786.9 −1.20462 −0.602312 0.798261i \(-0.705754\pi\)
−0.602312 + 0.798261i \(0.705754\pi\)
\(978\) 0 0
\(979\) 14027.2 0.457927
\(980\) 0 0
\(981\) 6550.90 0.213205
\(982\) 0 0
\(983\) 25120.5 0.815076 0.407538 0.913188i \(-0.366387\pi\)
0.407538 + 0.913188i \(0.366387\pi\)
\(984\) 0 0
\(985\) 68990.6 2.23170
\(986\) 0 0
\(987\) 11075.0 0.357165
\(988\) 0 0
\(989\) 18102.7 0.582036
\(990\) 0 0
\(991\) 9526.01 0.305352 0.152676 0.988276i \(-0.451211\pi\)
0.152676 + 0.988276i \(0.451211\pi\)
\(992\) 0 0
\(993\) −53190.6 −1.69985
\(994\) 0 0
\(995\) 73397.6 2.33855
\(996\) 0 0
\(997\) 13162.4 0.418111 0.209056 0.977904i \(-0.432961\pi\)
0.209056 + 0.977904i \(0.432961\pi\)
\(998\) 0 0
\(999\) −51399.6 −1.62784
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 304.4.a.g.1.3 3
4.3 odd 2 152.4.a.c.1.1 3
8.3 odd 2 1216.4.a.r.1.3 3
8.5 even 2 1216.4.a.w.1.1 3
12.11 even 2 1368.4.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.4.a.c.1.1 3 4.3 odd 2
304.4.a.g.1.3 3 1.1 even 1 trivial
1216.4.a.r.1.3 3 8.3 odd 2
1216.4.a.w.1.1 3 8.5 even 2
1368.4.a.d.1.3 3 12.11 even 2