Properties

Label 2475.4.a.z.1.3
Level $2475$
Weight $4$
Character 2475.1
Self dual yes
Analytic conductor $146.030$
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] = [2475,4,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, 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("2475.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(146.029727264\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1957.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.12946\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.59486 q^{2} +13.1127 q^{4} -20.6383 q^{7} +23.4921 q^{8} +O(q^{10})\) \(q+4.59486 q^{2} +13.1127 q^{4} -20.6383 q^{7} +23.4921 q^{8} -11.0000 q^{11} +15.6584 q^{13} -94.8302 q^{14} +3.04132 q^{16} +72.9507 q^{17} +61.0513 q^{19} -50.5434 q^{22} -13.6605 q^{23} +71.9483 q^{26} -270.624 q^{28} +31.4663 q^{29} -243.008 q^{31} -173.963 q^{32} +335.198 q^{34} +65.4018 q^{37} +280.522 q^{38} +109.087 q^{41} +121.750 q^{43} -144.240 q^{44} -62.7678 q^{46} -519.530 q^{47} +82.9413 q^{49} +205.324 q^{52} -542.673 q^{53} -484.839 q^{56} +144.583 q^{58} -109.478 q^{59} -89.6156 q^{61} -1116.59 q^{62} -823.664 q^{64} -488.446 q^{67} +956.581 q^{68} -837.423 q^{71} -351.216 q^{73} +300.512 q^{74} +800.547 q^{76} +227.022 q^{77} -831.205 q^{79} +501.238 q^{82} +1389.13 q^{83} +559.423 q^{86} -258.413 q^{88} -1523.70 q^{89} -323.164 q^{91} -179.125 q^{92} -2387.17 q^{94} +426.612 q^{97} +381.103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 17 q^{4} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 17 q^{4} - 6 q^{7} - 3 q^{8} - 33 q^{11} + 20 q^{13} - 144 q^{14} + 25 q^{16} + 32 q^{17} + 116 q^{19} - 11 q^{22} + 240 q^{23} - 302 q^{26} - 160 q^{28} - 238 q^{29} + 92 q^{31} + 197 q^{32} + 354 q^{34} + 90 q^{37} + 324 q^{38} + 46 q^{41} + 134 q^{43} - 187 q^{44} - 240 q^{46} - 220 q^{47} - 457 q^{49} + 1530 q^{52} - 798 q^{53} - 688 q^{56} + 978 q^{58} - 1236 q^{59} + 342 q^{61} - 1792 q^{62} - 1919 q^{64} - 764 q^{67} + 1074 q^{68} - 1816 q^{71} - 100 q^{73} + 1874 q^{74} + 396 q^{76} + 66 q^{77} - 96 q^{79} + 910 q^{82} + 858 q^{83} - 188 q^{86} + 33 q^{88} - 838 q^{89} + 332 q^{91} - 688 q^{92} - 3112 q^{94} + 1322 q^{97} + 1017 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.59486 1.62453 0.812263 0.583291i \(-0.198235\pi\)
0.812263 + 0.583291i \(0.198235\pi\)
\(3\) 0 0
\(4\) 13.1127 1.63909
\(5\) 0 0
\(6\) 0 0
\(7\) −20.6383 −1.11437 −0.557183 0.830390i \(-0.688118\pi\)
−0.557183 + 0.830390i \(0.688118\pi\)
\(8\) 23.4921 1.03822
\(9\) 0 0
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 15.6584 0.334067 0.167033 0.985951i \(-0.446581\pi\)
0.167033 + 0.985951i \(0.446581\pi\)
\(14\) −94.8302 −1.81032
\(15\) 0 0
\(16\) 3.04132 0.0475206
\(17\) 72.9507 1.04077 0.520387 0.853931i \(-0.325788\pi\)
0.520387 + 0.853931i \(0.325788\pi\)
\(18\) 0 0
\(19\) 61.0513 0.737165 0.368582 0.929595i \(-0.379843\pi\)
0.368582 + 0.929595i \(0.379843\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −50.5434 −0.489813
\(23\) −13.6605 −0.123844 −0.0619218 0.998081i \(-0.519723\pi\)
−0.0619218 + 0.998081i \(0.519723\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 71.9483 0.542701
\(27\) 0 0
\(28\) −270.624 −1.82654
\(29\) 31.4663 0.201487 0.100744 0.994912i \(-0.467878\pi\)
0.100744 + 0.994912i \(0.467878\pi\)
\(30\) 0 0
\(31\) −243.008 −1.40792 −0.703961 0.710239i \(-0.748587\pi\)
−0.703961 + 0.710239i \(0.748587\pi\)
\(32\) −173.963 −0.961016
\(33\) 0 0
\(34\) 335.198 1.69076
\(35\) 0 0
\(36\) 0 0
\(37\) 65.4018 0.290594 0.145297 0.989388i \(-0.453586\pi\)
0.145297 + 0.989388i \(0.453586\pi\)
\(38\) 280.522 1.19754
\(39\) 0 0
\(40\) 0 0
\(41\) 109.087 0.415524 0.207762 0.978179i \(-0.433382\pi\)
0.207762 + 0.978179i \(0.433382\pi\)
\(42\) 0 0
\(43\) 121.750 0.431783 0.215891 0.976417i \(-0.430734\pi\)
0.215891 + 0.976417i \(0.430734\pi\)
\(44\) −144.240 −0.494204
\(45\) 0 0
\(46\) −62.7678 −0.201187
\(47\) −519.530 −1.61237 −0.806184 0.591665i \(-0.798471\pi\)
−0.806184 + 0.591665i \(0.798471\pi\)
\(48\) 0 0
\(49\) 82.9413 0.241811
\(50\) 0 0
\(51\) 0 0
\(52\) 205.324 0.547565
\(53\) −542.673 −1.40645 −0.703226 0.710967i \(-0.748258\pi\)
−0.703226 + 0.710967i \(0.748258\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −484.839 −1.15695
\(57\) 0 0
\(58\) 144.583 0.327322
\(59\) −109.478 −0.241574 −0.120787 0.992678i \(-0.538542\pi\)
−0.120787 + 0.992678i \(0.538542\pi\)
\(60\) 0 0
\(61\) −89.6156 −0.188100 −0.0940501 0.995567i \(-0.529981\pi\)
−0.0940501 + 0.995567i \(0.529981\pi\)
\(62\) −1116.59 −2.28721
\(63\) 0 0
\(64\) −823.664 −1.60872
\(65\) 0 0
\(66\) 0 0
\(67\) −488.446 −0.890644 −0.445322 0.895371i \(-0.646911\pi\)
−0.445322 + 0.895371i \(0.646911\pi\)
\(68\) 956.581 1.70592
\(69\) 0 0
\(70\) 0 0
\(71\) −837.423 −1.39977 −0.699887 0.714254i \(-0.746766\pi\)
−0.699887 + 0.714254i \(0.746766\pi\)
\(72\) 0 0
\(73\) −351.216 −0.563105 −0.281553 0.959546i \(-0.590849\pi\)
−0.281553 + 0.959546i \(0.590849\pi\)
\(74\) 300.512 0.472078
\(75\) 0 0
\(76\) 800.547 1.20828
\(77\) 227.022 0.335994
\(78\) 0 0
\(79\) −831.205 −1.18377 −0.591885 0.806022i \(-0.701616\pi\)
−0.591885 + 0.806022i \(0.701616\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 501.238 0.675031
\(83\) 1389.13 1.83707 0.918537 0.395335i \(-0.129371\pi\)
0.918537 + 0.395335i \(0.129371\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 559.423 0.701443
\(87\) 0 0
\(88\) −258.413 −0.313034
\(89\) −1523.70 −1.81474 −0.907369 0.420335i \(-0.861912\pi\)
−0.907369 + 0.420335i \(0.861912\pi\)
\(90\) 0 0
\(91\) −323.164 −0.372273
\(92\) −179.125 −0.202990
\(93\) 0 0
\(94\) −2387.17 −2.61933
\(95\) 0 0
\(96\) 0 0
\(97\) 426.612 0.446555 0.223278 0.974755i \(-0.428324\pi\)
0.223278 + 0.974755i \(0.428324\pi\)
\(98\) 381.103 0.392829
\(99\) 0 0
\(100\) 0 0
\(101\) −74.1387 −0.0730403 −0.0365202 0.999333i \(-0.511627\pi\)
−0.0365202 + 0.999333i \(0.511627\pi\)
\(102\) 0 0
\(103\) 69.3916 0.0663821 0.0331911 0.999449i \(-0.489433\pi\)
0.0331911 + 0.999449i \(0.489433\pi\)
\(104\) 367.850 0.346833
\(105\) 0 0
\(106\) −2493.51 −2.28482
\(107\) 1141.71 1.03152 0.515761 0.856733i \(-0.327509\pi\)
0.515761 + 0.856733i \(0.327509\pi\)
\(108\) 0 0
\(109\) −2226.85 −1.95682 −0.978409 0.206680i \(-0.933734\pi\)
−0.978409 + 0.206680i \(0.933734\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −62.7678 −0.0529554
\(113\) −1719.76 −1.43169 −0.715847 0.698257i \(-0.753959\pi\)
−0.715847 + 0.698257i \(0.753959\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 412.608 0.330256
\(117\) 0 0
\(118\) −503.038 −0.392444
\(119\) −1505.58 −1.15980
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −411.771 −0.305574
\(123\) 0 0
\(124\) −3186.49 −2.30771
\(125\) 0 0
\(126\) 0 0
\(127\) 1601.63 1.11907 0.559534 0.828807i \(-0.310980\pi\)
0.559534 + 0.828807i \(0.310980\pi\)
\(128\) −2392.91 −1.65239
\(129\) 0 0
\(130\) 0 0
\(131\) −2004.13 −1.33665 −0.668327 0.743868i \(-0.732989\pi\)
−0.668327 + 0.743868i \(0.732989\pi\)
\(132\) 0 0
\(133\) −1260.00 −0.821471
\(134\) −2244.34 −1.44687
\(135\) 0 0
\(136\) 1713.77 1.08055
\(137\) −1672.85 −1.04322 −0.521610 0.853184i \(-0.674669\pi\)
−0.521610 + 0.853184i \(0.674669\pi\)
\(138\) 0 0
\(139\) 2540.38 1.55016 0.775080 0.631863i \(-0.217709\pi\)
0.775080 + 0.631863i \(0.217709\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3847.84 −2.27397
\(143\) −172.243 −0.100725
\(144\) 0 0
\(145\) 0 0
\(146\) −1613.79 −0.914780
\(147\) 0 0
\(148\) 857.594 0.476310
\(149\) −3090.68 −1.69932 −0.849658 0.527334i \(-0.823192\pi\)
−0.849658 + 0.527334i \(0.823192\pi\)
\(150\) 0 0
\(151\) 1358.74 0.732267 0.366134 0.930562i \(-0.380681\pi\)
0.366134 + 0.930562i \(0.380681\pi\)
\(152\) 1434.22 0.765335
\(153\) 0 0
\(154\) 1043.13 0.545831
\(155\) 0 0
\(156\) 0 0
\(157\) 1011.95 0.514411 0.257205 0.966357i \(-0.417198\pi\)
0.257205 + 0.966357i \(0.417198\pi\)
\(158\) −3819.27 −1.92307
\(159\) 0 0
\(160\) 0 0
\(161\) 281.929 0.138007
\(162\) 0 0
\(163\) 2816.37 1.35334 0.676672 0.736285i \(-0.263422\pi\)
0.676672 + 0.736285i \(0.263422\pi\)
\(164\) 1430.42 0.681081
\(165\) 0 0
\(166\) 6382.87 2.98438
\(167\) 3448.89 1.59810 0.799052 0.601262i \(-0.205335\pi\)
0.799052 + 0.601262i \(0.205335\pi\)
\(168\) 0 0
\(169\) −1951.81 −0.888399
\(170\) 0 0
\(171\) 0 0
\(172\) 1596.47 0.707730
\(173\) −2287.85 −1.00545 −0.502723 0.864448i \(-0.667668\pi\)
−0.502723 + 0.864448i \(0.667668\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −33.4545 −0.0143280
\(177\) 0 0
\(178\) −7001.17 −2.94809
\(179\) −3249.06 −1.35668 −0.678340 0.734748i \(-0.737300\pi\)
−0.678340 + 0.734748i \(0.737300\pi\)
\(180\) 0 0
\(181\) 1170.45 0.480655 0.240328 0.970692i \(-0.422745\pi\)
0.240328 + 0.970692i \(0.422745\pi\)
\(182\) −1484.89 −0.604767
\(183\) 0 0
\(184\) −320.913 −0.128576
\(185\) 0 0
\(186\) 0 0
\(187\) −802.458 −0.313805
\(188\) −6812.44 −2.64281
\(189\) 0 0
\(190\) 0 0
\(191\) 2760.35 1.04572 0.522859 0.852419i \(-0.324866\pi\)
0.522859 + 0.852419i \(0.324866\pi\)
\(192\) 0 0
\(193\) 1250.61 0.466430 0.233215 0.972425i \(-0.425075\pi\)
0.233215 + 0.972425i \(0.425075\pi\)
\(194\) 1960.22 0.725441
\(195\) 0 0
\(196\) 1087.58 0.396350
\(197\) −143.991 −0.0520756 −0.0260378 0.999661i \(-0.508289\pi\)
−0.0260378 + 0.999661i \(0.508289\pi\)
\(198\) 0 0
\(199\) 761.249 0.271174 0.135587 0.990765i \(-0.456708\pi\)
0.135587 + 0.990765i \(0.456708\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −340.657 −0.118656
\(203\) −649.411 −0.224531
\(204\) 0 0
\(205\) 0 0
\(206\) 318.844 0.107840
\(207\) 0 0
\(208\) 47.6223 0.0158751
\(209\) −671.564 −0.222263
\(210\) 0 0
\(211\) 3976.58 1.29743 0.648717 0.761029i \(-0.275306\pi\)
0.648717 + 0.761029i \(0.275306\pi\)
\(212\) −7115.91 −2.30530
\(213\) 0 0
\(214\) 5245.97 1.67573
\(215\) 0 0
\(216\) 0 0
\(217\) 5015.29 1.56894
\(218\) −10232.0 −3.17890
\(219\) 0 0
\(220\) 0 0
\(221\) 1142.29 0.347688
\(222\) 0 0
\(223\) −908.084 −0.272690 −0.136345 0.990661i \(-0.543536\pi\)
−0.136345 + 0.990661i \(0.543536\pi\)
\(224\) 3590.30 1.07092
\(225\) 0 0
\(226\) −7902.05 −2.32583
\(227\) −2062.15 −0.602951 −0.301475 0.953474i \(-0.597479\pi\)
−0.301475 + 0.953474i \(0.597479\pi\)
\(228\) 0 0
\(229\) 4077.47 1.17662 0.588312 0.808634i \(-0.299793\pi\)
0.588312 + 0.808634i \(0.299793\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 739.209 0.209187
\(233\) 1682.76 0.473138 0.236569 0.971615i \(-0.423977\pi\)
0.236569 + 0.971615i \(0.423977\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1435.56 −0.395961
\(237\) 0 0
\(238\) −6917.93 −1.88413
\(239\) −4024.96 −1.08934 −0.544672 0.838649i \(-0.683346\pi\)
−0.544672 + 0.838649i \(0.683346\pi\)
\(240\) 0 0
\(241\) −2784.27 −0.744194 −0.372097 0.928194i \(-0.621361\pi\)
−0.372097 + 0.928194i \(0.621361\pi\)
\(242\) 555.978 0.147684
\(243\) 0 0
\(244\) −1175.10 −0.308313
\(245\) 0 0
\(246\) 0 0
\(247\) 955.968 0.246262
\(248\) −5708.78 −1.46173
\(249\) 0 0
\(250\) 0 0
\(251\) 1827.60 0.459591 0.229796 0.973239i \(-0.426194\pi\)
0.229796 + 0.973239i \(0.426194\pi\)
\(252\) 0 0
\(253\) 150.265 0.0373402
\(254\) 7359.26 1.81796
\(255\) 0 0
\(256\) −4405.79 −1.07563
\(257\) 585.171 0.142031 0.0710155 0.997475i \(-0.477376\pi\)
0.0710155 + 0.997475i \(0.477376\pi\)
\(258\) 0 0
\(259\) −1349.78 −0.323828
\(260\) 0 0
\(261\) 0 0
\(262\) −9208.69 −2.17143
\(263\) −238.098 −0.0558241 −0.0279120 0.999610i \(-0.508886\pi\)
−0.0279120 + 0.999610i \(0.508886\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −5789.51 −1.33450
\(267\) 0 0
\(268\) −6404.84 −1.45984
\(269\) −4618.46 −1.04681 −0.523406 0.852083i \(-0.675339\pi\)
−0.523406 + 0.852083i \(0.675339\pi\)
\(270\) 0 0
\(271\) −143.439 −0.0321525 −0.0160762 0.999871i \(-0.505117\pi\)
−0.0160762 + 0.999871i \(0.505117\pi\)
\(272\) 221.867 0.0494582
\(273\) 0 0
\(274\) −7686.51 −1.69474
\(275\) 0 0
\(276\) 0 0
\(277\) 8602.51 1.86597 0.932987 0.359911i \(-0.117193\pi\)
0.932987 + 0.359911i \(0.117193\pi\)
\(278\) 11672.7 2.51828
\(279\) 0 0
\(280\) 0 0
\(281\) 2992.81 0.635360 0.317680 0.948198i \(-0.397096\pi\)
0.317680 + 0.948198i \(0.397096\pi\)
\(282\) 0 0
\(283\) 6858.89 1.44070 0.720351 0.693610i \(-0.243981\pi\)
0.720351 + 0.693610i \(0.243981\pi\)
\(284\) −10980.9 −2.29435
\(285\) 0 0
\(286\) −791.431 −0.163630
\(287\) −2251.37 −0.463046
\(288\) 0 0
\(289\) 408.809 0.0832096
\(290\) 0 0
\(291\) 0 0
\(292\) −4605.39 −0.922979
\(293\) 4049.70 0.807461 0.403731 0.914878i \(-0.367713\pi\)
0.403731 + 0.914878i \(0.367713\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1536.43 0.301699
\(297\) 0 0
\(298\) −14201.2 −2.76059
\(299\) −213.901 −0.0413720
\(300\) 0 0
\(301\) −2512.71 −0.481164
\(302\) 6243.20 1.18959
\(303\) 0 0
\(304\) 185.677 0.0350305
\(305\) 0 0
\(306\) 0 0
\(307\) 9572.69 1.77962 0.889808 0.456335i \(-0.150838\pi\)
0.889808 + 0.456335i \(0.150838\pi\)
\(308\) 2976.87 0.550724
\(309\) 0 0
\(310\) 0 0
\(311\) −5396.42 −0.983932 −0.491966 0.870614i \(-0.663722\pi\)
−0.491966 + 0.870614i \(0.663722\pi\)
\(312\) 0 0
\(313\) −9755.04 −1.76162 −0.880811 0.473469i \(-0.843002\pi\)
−0.880811 + 0.473469i \(0.843002\pi\)
\(314\) 4649.77 0.835674
\(315\) 0 0
\(316\) −10899.3 −1.94030
\(317\) −4353.75 −0.771391 −0.385695 0.922626i \(-0.626038\pi\)
−0.385695 + 0.922626i \(0.626038\pi\)
\(318\) 0 0
\(319\) −346.129 −0.0607508
\(320\) 0 0
\(321\) 0 0
\(322\) 1295.42 0.224196
\(323\) 4453.74 0.767221
\(324\) 0 0
\(325\) 0 0
\(326\) 12940.8 2.19854
\(327\) 0 0
\(328\) 2562.68 0.431404
\(329\) 10722.2 1.79677
\(330\) 0 0
\(331\) 5387.64 0.894656 0.447328 0.894370i \(-0.352376\pi\)
0.447328 + 0.894370i \(0.352376\pi\)
\(332\) 18215.3 3.01113
\(333\) 0 0
\(334\) 15847.2 2.59616
\(335\) 0 0
\(336\) 0 0
\(337\) 4500.27 0.727434 0.363717 0.931509i \(-0.381508\pi\)
0.363717 + 0.931509i \(0.381508\pi\)
\(338\) −8968.30 −1.44323
\(339\) 0 0
\(340\) 0 0
\(341\) 2673.09 0.424504
\(342\) 0 0
\(343\) 5367.18 0.844899
\(344\) 2860.16 0.448283
\(345\) 0 0
\(346\) −10512.4 −1.63337
\(347\) 5906.32 0.913740 0.456870 0.889533i \(-0.348970\pi\)
0.456870 + 0.889533i \(0.348970\pi\)
\(348\) 0 0
\(349\) 3636.26 0.557721 0.278860 0.960332i \(-0.410043\pi\)
0.278860 + 0.960332i \(0.410043\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1913.59 0.289757
\(353\) 210.408 0.0317248 0.0158624 0.999874i \(-0.494951\pi\)
0.0158624 + 0.999874i \(0.494951\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −19979.8 −2.97451
\(357\) 0 0
\(358\) −14928.9 −2.20396
\(359\) −2499.68 −0.367488 −0.183744 0.982974i \(-0.558822\pi\)
−0.183744 + 0.982974i \(0.558822\pi\)
\(360\) 0 0
\(361\) −3131.74 −0.456588
\(362\) 5378.03 0.780837
\(363\) 0 0
\(364\) −4237.56 −0.610188
\(365\) 0 0
\(366\) 0 0
\(367\) −5748.70 −0.817656 −0.408828 0.912612i \(-0.634062\pi\)
−0.408828 + 0.912612i \(0.634062\pi\)
\(368\) −41.5458 −0.00588512
\(369\) 0 0
\(370\) 0 0
\(371\) 11199.9 1.56730
\(372\) 0 0
\(373\) −4467.78 −0.620196 −0.310098 0.950705i \(-0.600362\pi\)
−0.310098 + 0.950705i \(0.600362\pi\)
\(374\) −3687.18 −0.509785
\(375\) 0 0
\(376\) −12204.9 −1.67398
\(377\) 492.712 0.0673103
\(378\) 0 0
\(379\) 7804.08 1.05770 0.528851 0.848715i \(-0.322623\pi\)
0.528851 + 0.848715i \(0.322623\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 12683.4 1.69880
\(383\) −11161.1 −1.48904 −0.744522 0.667597i \(-0.767323\pi\)
−0.744522 + 0.667597i \(0.767323\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5746.38 0.757728
\(387\) 0 0
\(388\) 5594.04 0.731944
\(389\) −8490.24 −1.10661 −0.553306 0.832978i \(-0.686634\pi\)
−0.553306 + 0.832978i \(0.686634\pi\)
\(390\) 0 0
\(391\) −996.540 −0.128893
\(392\) 1948.47 0.251052
\(393\) 0 0
\(394\) −661.616 −0.0845983
\(395\) 0 0
\(396\) 0 0
\(397\) −6019.74 −0.761013 −0.380507 0.924778i \(-0.624250\pi\)
−0.380507 + 0.924778i \(0.624250\pi\)
\(398\) 3497.83 0.440529
\(399\) 0 0
\(400\) 0 0
\(401\) 10398.8 1.29499 0.647495 0.762069i \(-0.275816\pi\)
0.647495 + 0.762069i \(0.275816\pi\)
\(402\) 0 0
\(403\) −3805.13 −0.470340
\(404\) −972.158 −0.119720
\(405\) 0 0
\(406\) −2983.95 −0.364756
\(407\) −719.420 −0.0876175
\(408\) 0 0
\(409\) −4733.68 −0.572287 −0.286144 0.958187i \(-0.592373\pi\)
−0.286144 + 0.958187i \(0.592373\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 909.911 0.108806
\(413\) 2259.45 0.269202
\(414\) 0 0
\(415\) 0 0
\(416\) −2723.98 −0.321044
\(417\) 0 0
\(418\) −3085.74 −0.361073
\(419\) 8117.57 0.946466 0.473233 0.880937i \(-0.343087\pi\)
0.473233 + 0.880937i \(0.343087\pi\)
\(420\) 0 0
\(421\) −9484.27 −1.09795 −0.548973 0.835840i \(-0.684981\pi\)
−0.548973 + 0.835840i \(0.684981\pi\)
\(422\) 18271.8 2.10772
\(423\) 0 0
\(424\) −12748.5 −1.46020
\(425\) 0 0
\(426\) 0 0
\(427\) 1849.52 0.209612
\(428\) 14970.8 1.69075
\(429\) 0 0
\(430\) 0 0
\(431\) −9335.16 −1.04329 −0.521646 0.853162i \(-0.674682\pi\)
−0.521646 + 0.853162i \(0.674682\pi\)
\(432\) 0 0
\(433\) 2983.02 0.331074 0.165537 0.986204i \(-0.447064\pi\)
0.165537 + 0.986204i \(0.447064\pi\)
\(434\) 23044.5 2.54878
\(435\) 0 0
\(436\) −29200.0 −3.20739
\(437\) −833.988 −0.0912931
\(438\) 0 0
\(439\) −5232.32 −0.568850 −0.284425 0.958698i \(-0.591803\pi\)
−0.284425 + 0.958698i \(0.591803\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 5248.68 0.564828
\(443\) 7517.71 0.806269 0.403135 0.915141i \(-0.367921\pi\)
0.403135 + 0.915141i \(0.367921\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4172.52 −0.442992
\(447\) 0 0
\(448\) 16999.1 1.79270
\(449\) −16070.9 −1.68916 −0.844581 0.535428i \(-0.820150\pi\)
−0.844581 + 0.535428i \(0.820150\pi\)
\(450\) 0 0
\(451\) −1199.96 −0.125285
\(452\) −22550.7 −2.34667
\(453\) 0 0
\(454\) −9475.29 −0.979510
\(455\) 0 0
\(456\) 0 0
\(457\) −9718.51 −0.994776 −0.497388 0.867528i \(-0.665707\pi\)
−0.497388 + 0.867528i \(0.665707\pi\)
\(458\) 18735.4 1.91146
\(459\) 0 0
\(460\) 0 0
\(461\) 14538.0 1.46877 0.734385 0.678733i \(-0.237471\pi\)
0.734385 + 0.678733i \(0.237471\pi\)
\(462\) 0 0
\(463\) 9978.17 1.00157 0.500783 0.865573i \(-0.333045\pi\)
0.500783 + 0.865573i \(0.333045\pi\)
\(464\) 95.6990 0.00957481
\(465\) 0 0
\(466\) 7732.03 0.768625
\(467\) 15188.2 1.50498 0.752489 0.658605i \(-0.228853\pi\)
0.752489 + 0.658605i \(0.228853\pi\)
\(468\) 0 0
\(469\) 10080.7 0.992503
\(470\) 0 0
\(471\) 0 0
\(472\) −2571.88 −0.250806
\(473\) −1339.25 −0.130187
\(474\) 0 0
\(475\) 0 0
\(476\) −19742.3 −1.90102
\(477\) 0 0
\(478\) −18494.1 −1.76967
\(479\) −11330.8 −1.08083 −0.540415 0.841399i \(-0.681733\pi\)
−0.540415 + 0.841399i \(0.681733\pi\)
\(480\) 0 0
\(481\) 1024.09 0.0970779
\(482\) −12793.3 −1.20896
\(483\) 0 0
\(484\) 1586.64 0.149008
\(485\) 0 0
\(486\) 0 0
\(487\) −19086.9 −1.77599 −0.887997 0.459850i \(-0.847903\pi\)
−0.887997 + 0.459850i \(0.847903\pi\)
\(488\) −2105.26 −0.195288
\(489\) 0 0
\(490\) 0 0
\(491\) 8112.85 0.745677 0.372839 0.927896i \(-0.378384\pi\)
0.372839 + 0.927896i \(0.378384\pi\)
\(492\) 0 0
\(493\) 2295.49 0.209703
\(494\) 4392.53 0.400060
\(495\) 0 0
\(496\) −739.066 −0.0669053
\(497\) 17283.0 1.55986
\(498\) 0 0
\(499\) 18329.1 1.64433 0.822167 0.569246i \(-0.192765\pi\)
0.822167 + 0.569246i \(0.192765\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 8397.58 0.746618
\(503\) 7739.57 0.686064 0.343032 0.939324i \(-0.388546\pi\)
0.343032 + 0.939324i \(0.388546\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 690.446 0.0606602
\(507\) 0 0
\(508\) 21001.7 1.83425
\(509\) −15914.9 −1.38589 −0.692943 0.720993i \(-0.743686\pi\)
−0.692943 + 0.720993i \(0.743686\pi\)
\(510\) 0 0
\(511\) 7248.51 0.627505
\(512\) −1100.65 −0.0950048
\(513\) 0 0
\(514\) 2688.78 0.230733
\(515\) 0 0
\(516\) 0 0
\(517\) 5714.83 0.486147
\(518\) −6202.07 −0.526068
\(519\) 0 0
\(520\) 0 0
\(521\) −2274.50 −0.191262 −0.0956312 0.995417i \(-0.530487\pi\)
−0.0956312 + 0.995417i \(0.530487\pi\)
\(522\) 0 0
\(523\) −10971.1 −0.917274 −0.458637 0.888624i \(-0.651662\pi\)
−0.458637 + 0.888624i \(0.651662\pi\)
\(524\) −26279.6 −2.19089
\(525\) 0 0
\(526\) −1094.02 −0.0906877
\(527\) −17727.6 −1.46533
\(528\) 0 0
\(529\) −11980.4 −0.984663
\(530\) 0 0
\(531\) 0 0
\(532\) −16522.0 −1.34646
\(533\) 1708.13 0.138813
\(534\) 0 0
\(535\) 0 0
\(536\) −11474.6 −0.924680
\(537\) 0 0
\(538\) −21221.2 −1.70058
\(539\) −912.355 −0.0729089
\(540\) 0 0
\(541\) 5313.05 0.422229 0.211115 0.977461i \(-0.432291\pi\)
0.211115 + 0.977461i \(0.432291\pi\)
\(542\) −659.084 −0.0522326
\(543\) 0 0
\(544\) −12690.7 −1.00020
\(545\) 0 0
\(546\) 0 0
\(547\) −20685.1 −1.61688 −0.808439 0.588581i \(-0.799687\pi\)
−0.808439 + 0.588581i \(0.799687\pi\)
\(548\) −21935.6 −1.70993
\(549\) 0 0
\(550\) 0 0
\(551\) 1921.06 0.148529
\(552\) 0 0
\(553\) 17154.7 1.31915
\(554\) 39527.3 3.03132
\(555\) 0 0
\(556\) 33311.2 2.54085
\(557\) −10853.8 −0.825659 −0.412830 0.910808i \(-0.635460\pi\)
−0.412830 + 0.910808i \(0.635460\pi\)
\(558\) 0 0
\(559\) 1906.41 0.144244
\(560\) 0 0
\(561\) 0 0
\(562\) 13751.5 1.03216
\(563\) −15381.2 −1.15141 −0.575704 0.817658i \(-0.695272\pi\)
−0.575704 + 0.817658i \(0.695272\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 31515.6 2.34046
\(567\) 0 0
\(568\) −19672.9 −1.45327
\(569\) 1348.88 0.0993814 0.0496907 0.998765i \(-0.484176\pi\)
0.0496907 + 0.998765i \(0.484176\pi\)
\(570\) 0 0
\(571\) 3463.51 0.253841 0.126920 0.991913i \(-0.459491\pi\)
0.126920 + 0.991913i \(0.459491\pi\)
\(572\) −2258.57 −0.165097
\(573\) 0 0
\(574\) −10344.7 −0.752231
\(575\) 0 0
\(576\) 0 0
\(577\) −12052.6 −0.869598 −0.434799 0.900528i \(-0.643181\pi\)
−0.434799 + 0.900528i \(0.643181\pi\)
\(578\) 1878.42 0.135176
\(579\) 0 0
\(580\) 0 0
\(581\) −28669.4 −2.04717
\(582\) 0 0
\(583\) 5969.41 0.424061
\(584\) −8250.80 −0.584624
\(585\) 0 0
\(586\) 18607.8 1.31174
\(587\) −11133.1 −0.782813 −0.391407 0.920218i \(-0.628011\pi\)
−0.391407 + 0.920218i \(0.628011\pi\)
\(588\) 0 0
\(589\) −14836.0 −1.03787
\(590\) 0 0
\(591\) 0 0
\(592\) 198.908 0.0138092
\(593\) −7939.69 −0.549821 −0.274911 0.961470i \(-0.588648\pi\)
−0.274911 + 0.961470i \(0.588648\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −40527.1 −2.78533
\(597\) 0 0
\(598\) −982.846 −0.0672100
\(599\) 19474.7 1.32840 0.664202 0.747553i \(-0.268771\pi\)
0.664202 + 0.747553i \(0.268771\pi\)
\(600\) 0 0
\(601\) −19946.1 −1.35377 −0.676887 0.736087i \(-0.736671\pi\)
−0.676887 + 0.736087i \(0.736671\pi\)
\(602\) −11545.6 −0.781664
\(603\) 0 0
\(604\) 17816.7 1.20025
\(605\) 0 0
\(606\) 0 0
\(607\) −1427.44 −0.0954496 −0.0477248 0.998861i \(-0.515197\pi\)
−0.0477248 + 0.998861i \(0.515197\pi\)
\(608\) −10620.6 −0.708427
\(609\) 0 0
\(610\) 0 0
\(611\) −8135.03 −0.538638
\(612\) 0 0
\(613\) 8029.40 0.529045 0.264522 0.964380i \(-0.414786\pi\)
0.264522 + 0.964380i \(0.414786\pi\)
\(614\) 43985.1 2.89103
\(615\) 0 0
\(616\) 5333.22 0.348834
\(617\) 20795.5 1.35688 0.678440 0.734655i \(-0.262656\pi\)
0.678440 + 0.734655i \(0.262656\pi\)
\(618\) 0 0
\(619\) 1677.43 0.108920 0.0544602 0.998516i \(-0.482656\pi\)
0.0544602 + 0.998516i \(0.482656\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24795.8 −1.59842
\(623\) 31446.6 2.02228
\(624\) 0 0
\(625\) 0 0
\(626\) −44823.0 −2.86180
\(627\) 0 0
\(628\) 13269.4 0.843165
\(629\) 4771.11 0.302443
\(630\) 0 0
\(631\) −25225.2 −1.59144 −0.795719 0.605666i \(-0.792907\pi\)
−0.795719 + 0.605666i \(0.792907\pi\)
\(632\) −19526.8 −1.22901
\(633\) 0 0
\(634\) −20004.8 −1.25315
\(635\) 0 0
\(636\) 0 0
\(637\) 1298.73 0.0807812
\(638\) −1590.41 −0.0986912
\(639\) 0 0
\(640\) 0 0
\(641\) −15165.3 −0.934468 −0.467234 0.884134i \(-0.654749\pi\)
−0.467234 + 0.884134i \(0.654749\pi\)
\(642\) 0 0
\(643\) −27156.1 −1.66553 −0.832763 0.553630i \(-0.813242\pi\)
−0.832763 + 0.553630i \(0.813242\pi\)
\(644\) 3696.85 0.226206
\(645\) 0 0
\(646\) 20464.3 1.24637
\(647\) 29154.9 1.77156 0.885778 0.464110i \(-0.153626\pi\)
0.885778 + 0.464110i \(0.153626\pi\)
\(648\) 0 0
\(649\) 1204.26 0.0728374
\(650\) 0 0
\(651\) 0 0
\(652\) 36930.2 2.21825
\(653\) −19141.7 −1.14713 −0.573564 0.819161i \(-0.694440\pi\)
−0.573564 + 0.819161i \(0.694440\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 331.768 0.0197460
\(657\) 0 0
\(658\) 49267.2 2.91890
\(659\) 24939.6 1.47422 0.737110 0.675773i \(-0.236190\pi\)
0.737110 + 0.675773i \(0.236190\pi\)
\(660\) 0 0
\(661\) 22617.7 1.33090 0.665452 0.746440i \(-0.268239\pi\)
0.665452 + 0.746440i \(0.268239\pi\)
\(662\) 24755.4 1.45339
\(663\) 0 0
\(664\) 32633.7 1.90728
\(665\) 0 0
\(666\) 0 0
\(667\) −429.843 −0.0249529
\(668\) 45224.3 2.61943
\(669\) 0 0
\(670\) 0 0
\(671\) 985.772 0.0567143
\(672\) 0 0
\(673\) 13855.8 0.793615 0.396807 0.917902i \(-0.370118\pi\)
0.396807 + 0.917902i \(0.370118\pi\)
\(674\) 20678.1 1.18174
\(675\) 0 0
\(676\) −25593.5 −1.45616
\(677\) −24992.8 −1.41884 −0.709419 0.704787i \(-0.751043\pi\)
−0.709419 + 0.704787i \(0.751043\pi\)
\(678\) 0 0
\(679\) −8804.57 −0.497626
\(680\) 0 0
\(681\) 0 0
\(682\) 12282.5 0.689619
\(683\) −14420.5 −0.807887 −0.403943 0.914784i \(-0.632361\pi\)
−0.403943 + 0.914784i \(0.632361\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 24661.4 1.37256
\(687\) 0 0
\(688\) 370.280 0.0205186
\(689\) −8497.42 −0.469849
\(690\) 0 0
\(691\) 30552.4 1.68201 0.841005 0.541027i \(-0.181964\pi\)
0.841005 + 0.541027i \(0.181964\pi\)
\(692\) −29999.9 −1.64801
\(693\) 0 0
\(694\) 27138.7 1.48440
\(695\) 0 0
\(696\) 0 0
\(697\) 7957.96 0.432467
\(698\) 16708.1 0.906032
\(699\) 0 0
\(700\) 0 0
\(701\) 9151.47 0.493076 0.246538 0.969133i \(-0.420707\pi\)
0.246538 + 0.969133i \(0.420707\pi\)
\(702\) 0 0
\(703\) 3992.86 0.214216
\(704\) 9060.30 0.485047
\(705\) 0 0
\(706\) 966.793 0.0515378
\(707\) 1530.10 0.0813937
\(708\) 0 0
\(709\) −6261.96 −0.331697 −0.165848 0.986151i \(-0.553036\pi\)
−0.165848 + 0.986151i \(0.553036\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −35794.9 −1.88409
\(713\) 3319.60 0.174362
\(714\) 0 0
\(715\) 0 0
\(716\) −42603.9 −2.22372
\(717\) 0 0
\(718\) −11485.7 −0.596995
\(719\) 18228.7 0.945500 0.472750 0.881197i \(-0.343261\pi\)
0.472750 + 0.881197i \(0.343261\pi\)
\(720\) 0 0
\(721\) −1432.13 −0.0739740
\(722\) −14389.9 −0.741740
\(723\) 0 0
\(724\) 15347.7 0.787836
\(725\) 0 0
\(726\) 0 0
\(727\) 7233.66 0.369026 0.184513 0.982830i \(-0.440929\pi\)
0.184513 + 0.982830i \(0.440929\pi\)
\(728\) −7591.81 −0.386499
\(729\) 0 0
\(730\) 0 0
\(731\) 8881.73 0.449388
\(732\) 0 0
\(733\) 13444.8 0.677485 0.338743 0.940879i \(-0.389998\pi\)
0.338743 + 0.940879i \(0.389998\pi\)
\(734\) −26414.4 −1.32830
\(735\) 0 0
\(736\) 2376.41 0.119016
\(737\) 5372.90 0.268539
\(738\) 0 0
\(739\) 18490.9 0.920432 0.460216 0.887807i \(-0.347772\pi\)
0.460216 + 0.887807i \(0.347772\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 51461.8 2.54612
\(743\) −25160.9 −1.24235 −0.621173 0.783674i \(-0.713343\pi\)
−0.621173 + 0.783674i \(0.713343\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −20528.8 −1.00752
\(747\) 0 0
\(748\) −10522.4 −0.514354
\(749\) −23562.9 −1.14949
\(750\) 0 0
\(751\) −13419.5 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(752\) −1580.06 −0.0766207
\(753\) 0 0
\(754\) 2263.94 0.109347
\(755\) 0 0
\(756\) 0 0
\(757\) 7014.90 0.336804 0.168402 0.985718i \(-0.446139\pi\)
0.168402 + 0.985718i \(0.446139\pi\)
\(758\) 35858.6 1.71826
\(759\) 0 0
\(760\) 0 0
\(761\) 30156.9 1.43651 0.718256 0.695779i \(-0.244941\pi\)
0.718256 + 0.695779i \(0.244941\pi\)
\(762\) 0 0
\(763\) 45958.4 2.18061
\(764\) 36195.7 1.71402
\(765\) 0 0
\(766\) −51283.5 −2.41899
\(767\) −1714.26 −0.0807019
\(768\) 0 0
\(769\) 11292.2 0.529530 0.264765 0.964313i \(-0.414706\pi\)
0.264765 + 0.964313i \(0.414706\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 16398.9 0.764519
\(773\) 8524.10 0.396624 0.198312 0.980139i \(-0.436454\pi\)
0.198312 + 0.980139i \(0.436454\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10022.0 0.463621
\(777\) 0 0
\(778\) −39011.4 −1.79772
\(779\) 6659.89 0.306310
\(780\) 0 0
\(781\) 9211.66 0.422047
\(782\) −4578.96 −0.209390
\(783\) 0 0
\(784\) 252.251 0.0114910
\(785\) 0 0
\(786\) 0 0
\(787\) 14983.9 0.678676 0.339338 0.940665i \(-0.389797\pi\)
0.339338 + 0.940665i \(0.389797\pi\)
\(788\) −1888.10 −0.0853565
\(789\) 0 0
\(790\) 0 0
\(791\) 35493.0 1.59543
\(792\) 0 0
\(793\) −1403.24 −0.0628380
\(794\) −27659.9 −1.23629
\(795\) 0 0
\(796\) 9982.04 0.444477
\(797\) 37172.3 1.65208 0.826041 0.563610i \(-0.190588\pi\)
0.826041 + 0.563610i \(0.190588\pi\)
\(798\) 0 0
\(799\) −37900.1 −1.67811
\(800\) 0 0
\(801\) 0 0
\(802\) 47781.0 2.10375
\(803\) 3863.37 0.169783
\(804\) 0 0
\(805\) 0 0
\(806\) −17484.0 −0.764080
\(807\) 0 0
\(808\) −1741.68 −0.0758316
\(809\) −23797.1 −1.03419 −0.517096 0.855928i \(-0.672987\pi\)
−0.517096 + 0.855928i \(0.672987\pi\)
\(810\) 0 0
\(811\) 8988.35 0.389178 0.194589 0.980885i \(-0.437663\pi\)
0.194589 + 0.980885i \(0.437663\pi\)
\(812\) −8515.54 −0.368026
\(813\) 0 0
\(814\) −3305.63 −0.142337
\(815\) 0 0
\(816\) 0 0
\(817\) 7432.98 0.318295
\(818\) −21750.6 −0.929696
\(819\) 0 0
\(820\) 0 0
\(821\) 25156.8 1.06940 0.534702 0.845041i \(-0.320424\pi\)
0.534702 + 0.845041i \(0.320424\pi\)
\(822\) 0 0
\(823\) −1318.51 −0.0558447 −0.0279224 0.999610i \(-0.508889\pi\)
−0.0279224 + 0.999610i \(0.508889\pi\)
\(824\) 1630.16 0.0689189
\(825\) 0 0
\(826\) 10381.9 0.437326
\(827\) 124.982 0.00525519 0.00262760 0.999997i \(-0.499164\pi\)
0.00262760 + 0.999997i \(0.499164\pi\)
\(828\) 0 0
\(829\) −8886.80 −0.372318 −0.186159 0.982520i \(-0.559604\pi\)
−0.186159 + 0.982520i \(0.559604\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −12897.3 −0.537419
\(833\) 6050.63 0.251671
\(834\) 0 0
\(835\) 0 0
\(836\) −8806.02 −0.364309
\(837\) 0 0
\(838\) 37299.1 1.53756
\(839\) −2995.21 −0.123249 −0.0616247 0.998099i \(-0.519628\pi\)
−0.0616247 + 0.998099i \(0.519628\pi\)
\(840\) 0 0
\(841\) −23398.9 −0.959403
\(842\) −43578.8 −1.78364
\(843\) 0 0
\(844\) 52143.6 2.12661
\(845\) 0 0
\(846\) 0 0
\(847\) −2497.24 −0.101306
\(848\) −1650.44 −0.0668355
\(849\) 0 0
\(850\) 0 0
\(851\) −893.418 −0.0359882
\(852\) 0 0
\(853\) −18130.5 −0.727757 −0.363878 0.931446i \(-0.618548\pi\)
−0.363878 + 0.931446i \(0.618548\pi\)
\(854\) 8498.27 0.340521
\(855\) 0 0
\(856\) 26821.1 1.07094
\(857\) 26394.1 1.05205 0.526024 0.850470i \(-0.323682\pi\)
0.526024 + 0.850470i \(0.323682\pi\)
\(858\) 0 0
\(859\) −29456.2 −1.17000 −0.585002 0.811032i \(-0.698906\pi\)
−0.585002 + 0.811032i \(0.698906\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −42893.7 −1.69486
\(863\) −762.616 −0.0300808 −0.0150404 0.999887i \(-0.504788\pi\)
−0.0150404 + 0.999887i \(0.504788\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 13706.6 0.537838
\(867\) 0 0
\(868\) 65764.0 2.57163
\(869\) 9143.26 0.356920
\(870\) 0 0
\(871\) −7648.29 −0.297535
\(872\) −52313.3 −2.03160
\(873\) 0 0
\(874\) −3832.06 −0.148308
\(875\) 0 0
\(876\) 0 0
\(877\) −44767.2 −1.72369 −0.861847 0.507168i \(-0.830692\pi\)
−0.861847 + 0.507168i \(0.830692\pi\)
\(878\) −24041.8 −0.924112
\(879\) 0 0
\(880\) 0 0
\(881\) 32057.9 1.22595 0.612973 0.790104i \(-0.289973\pi\)
0.612973 + 0.790104i \(0.289973\pi\)
\(882\) 0 0
\(883\) −7078.95 −0.269791 −0.134896 0.990860i \(-0.543070\pi\)
−0.134896 + 0.990860i \(0.543070\pi\)
\(884\) 14978.6 0.569891
\(885\) 0 0
\(886\) 34542.8 1.30981
\(887\) −25148.1 −0.951964 −0.475982 0.879455i \(-0.657907\pi\)
−0.475982 + 0.879455i \(0.657907\pi\)
\(888\) 0 0
\(889\) −33055.0 −1.24705
\(890\) 0 0
\(891\) 0 0
\(892\) −11907.4 −0.446963
\(893\) −31718.0 −1.18858
\(894\) 0 0
\(895\) 0 0
\(896\) 49385.8 1.84137
\(897\) 0 0
\(898\) −73843.6 −2.74409
\(899\) −7646.56 −0.283679
\(900\) 0 0
\(901\) −39588.4 −1.46380
\(902\) −5513.62 −0.203529
\(903\) 0 0
\(904\) −40400.8 −1.48641
\(905\) 0 0
\(906\) 0 0
\(907\) −1269.76 −0.0464848 −0.0232424 0.999730i \(-0.507399\pi\)
−0.0232424 + 0.999730i \(0.507399\pi\)
\(908\) −27040.4 −0.988289
\(909\) 0 0
\(910\) 0 0
\(911\) 33783.1 1.22863 0.614316 0.789060i \(-0.289432\pi\)
0.614316 + 0.789060i \(0.289432\pi\)
\(912\) 0 0
\(913\) −15280.5 −0.553899
\(914\) −44655.1 −1.61604
\(915\) 0 0
\(916\) 53466.6 1.92859
\(917\) 41361.9 1.48952
\(918\) 0 0
\(919\) 39262.5 1.40930 0.704652 0.709553i \(-0.251103\pi\)
0.704652 + 0.709553i \(0.251103\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 66800.1 2.38606
\(923\) −13112.7 −0.467618
\(924\) 0 0
\(925\) 0 0
\(926\) 45848.3 1.62707
\(927\) 0 0
\(928\) −5473.95 −0.193633
\(929\) 21175.0 0.747825 0.373913 0.927464i \(-0.378016\pi\)
0.373913 + 0.927464i \(0.378016\pi\)
\(930\) 0 0
\(931\) 5063.68 0.178255
\(932\) 22065.5 0.775515
\(933\) 0 0
\(934\) 69787.5 2.44488
\(935\) 0 0
\(936\) 0 0
\(937\) 5135.11 0.179036 0.0895180 0.995985i \(-0.471467\pi\)
0.0895180 + 0.995985i \(0.471467\pi\)
\(938\) 46319.4 1.61235
\(939\) 0 0
\(940\) 0 0
\(941\) 9702.77 0.336133 0.168067 0.985776i \(-0.446248\pi\)
0.168067 + 0.985776i \(0.446248\pi\)
\(942\) 0 0
\(943\) −1490.18 −0.0514600
\(944\) −332.959 −0.0114798
\(945\) 0 0
\(946\) −6153.65 −0.211493
\(947\) −699.579 −0.0240055 −0.0120028 0.999928i \(-0.503821\pi\)
−0.0120028 + 0.999928i \(0.503821\pi\)
\(948\) 0 0
\(949\) −5499.49 −0.188115
\(950\) 0 0
\(951\) 0 0
\(952\) −35369.3 −1.20412
\(953\) 42039.3 1.42895 0.714473 0.699663i \(-0.246667\pi\)
0.714473 + 0.699663i \(0.246667\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −52778.1 −1.78553
\(957\) 0 0
\(958\) −52063.4 −1.75584
\(959\) 34524.9 1.16253
\(960\) 0 0
\(961\) 29262.0 0.982243
\(962\) 4705.55 0.157706
\(963\) 0 0
\(964\) −36509.3 −1.21980
\(965\) 0 0
\(966\) 0 0
\(967\) 32794.8 1.09060 0.545299 0.838242i \(-0.316416\pi\)
0.545299 + 0.838242i \(0.316416\pi\)
\(968\) 2842.55 0.0943832
\(969\) 0 0
\(970\) 0 0
\(971\) 3322.53 0.109810 0.0549048 0.998492i \(-0.482514\pi\)
0.0549048 + 0.998492i \(0.482514\pi\)
\(972\) 0 0
\(973\) −52429.3 −1.72745
\(974\) −87701.4 −2.88515
\(975\) 0 0
\(976\) −272.550 −0.00893864
\(977\) 22192.5 0.726716 0.363358 0.931650i \(-0.381630\pi\)
0.363358 + 0.931650i \(0.381630\pi\)
\(978\) 0 0
\(979\) 16760.7 0.547164
\(980\) 0 0
\(981\) 0 0
\(982\) 37277.4 1.21137
\(983\) 7383.09 0.239556 0.119778 0.992801i \(-0.461782\pi\)
0.119778 + 0.992801i \(0.461782\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 10547.4 0.340668
\(987\) 0 0
\(988\) 12535.3 0.403645
\(989\) −1663.16 −0.0534735
\(990\) 0 0
\(991\) 46260.8 1.48287 0.741434 0.671026i \(-0.234146\pi\)
0.741434 + 0.671026i \(0.234146\pi\)
\(992\) 42274.3 1.35304
\(993\) 0 0
\(994\) 79413.1 2.53403
\(995\) 0 0
\(996\) 0 0
\(997\) −41196.8 −1.30864 −0.654320 0.756217i \(-0.727045\pi\)
−0.654320 + 0.756217i \(0.727045\pi\)
\(998\) 84219.5 2.67127
\(999\) 0 0
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 2475.4.a.z.1.3 3
3.2 odd 2 825.4.a.p.1.1 3
5.4 even 2 495.4.a.i.1.1 3
15.2 even 4 825.4.c.m.199.1 6
15.8 even 4 825.4.c.m.199.6 6
15.14 odd 2 165.4.a.g.1.3 3
165.164 even 2 1815.4.a.q.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.g.1.3 3 15.14 odd 2
495.4.a.i.1.1 3 5.4 even 2
825.4.a.p.1.1 3 3.2 odd 2
825.4.c.m.199.1 6 15.2 even 4
825.4.c.m.199.6 6 15.8 even 4
1815.4.a.q.1.1 3 165.164 even 2
2475.4.a.z.1.3 3 1.1 even 1 trivial