Properties

Label 1323.4.a.bd.1.1
Level $1323$
Weight $4$
Character 1323.1
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,4,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1323.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 40x^{4} + 453x^{2} - 1278 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.70753\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.70753 q^{2} +14.1608 q^{4} -0.830453 q^{5} -29.0024 q^{8} +O(q^{10})\) \(q-4.70753 q^{2} +14.1608 q^{4} -0.830453 q^{5} -29.0024 q^{8} +3.90938 q^{10} +68.2503 q^{11} -35.7721 q^{13} +23.2429 q^{16} -14.6779 q^{17} +86.7693 q^{19} -11.7599 q^{20} -321.290 q^{22} -58.4087 q^{23} -124.310 q^{25} +168.398 q^{26} +67.1398 q^{29} -69.5002 q^{31} +122.603 q^{32} +69.0965 q^{34} -289.240 q^{37} -408.469 q^{38} +24.0851 q^{40} +357.750 q^{41} -235.801 q^{43} +966.482 q^{44} +274.961 q^{46} -450.067 q^{47} +585.195 q^{50} -506.563 q^{52} -172.464 q^{53} -56.6787 q^{55} -316.063 q^{58} +445.856 q^{59} +476.314 q^{61} +327.174 q^{62} -763.099 q^{64} +29.7070 q^{65} -804.320 q^{67} -207.851 q^{68} -353.412 q^{71} -778.045 q^{73} +1361.61 q^{74} +1228.73 q^{76} -74.1996 q^{79} -19.3021 q^{80} -1684.12 q^{82} +1168.49 q^{83} +12.1893 q^{85} +1110.04 q^{86} -1979.42 q^{88} -1514.72 q^{89} -827.116 q^{92} +2118.70 q^{94} -72.0579 q^{95} +1631.46 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 32 q^{4} - 20 q^{10} + 52 q^{13} - 148 q^{16} - 62 q^{19} - 356 q^{22} - 46 q^{25} - 82 q^{31} + 420 q^{34} - 1132 q^{37} - 444 q^{40} - 1566 q^{43} - 888 q^{46} - 72 q^{52} + 224 q^{55} + 4 q^{58} + 886 q^{61} - 924 q^{64} - 2084 q^{67} - 2398 q^{73} + 3204 q^{76} + 984 q^{79} - 3892 q^{82} - 3600 q^{85} - 5796 q^{88} + 2772 q^{94} - 682 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.70753 −1.66436 −0.832182 0.554503i \(-0.812908\pi\)
−0.832182 + 0.554503i \(0.812908\pi\)
\(3\) 0 0
\(4\) 14.1608 1.77011
\(5\) −0.830453 −0.0742780 −0.0371390 0.999310i \(-0.511824\pi\)
−0.0371390 + 0.999310i \(0.511824\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −29.0024 −1.28174
\(9\) 0 0
\(10\) 3.90938 0.123626
\(11\) 68.2503 1.87075 0.935374 0.353659i \(-0.115063\pi\)
0.935374 + 0.353659i \(0.115063\pi\)
\(12\) 0 0
\(13\) −35.7721 −0.763184 −0.381592 0.924331i \(-0.624624\pi\)
−0.381592 + 0.924331i \(0.624624\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 23.2429 0.363170
\(17\) −14.6779 −0.209406 −0.104703 0.994504i \(-0.533389\pi\)
−0.104703 + 0.994504i \(0.533389\pi\)
\(18\) 0 0
\(19\) 86.7693 1.04770 0.523849 0.851811i \(-0.324496\pi\)
0.523849 + 0.851811i \(0.324496\pi\)
\(20\) −11.7599 −0.131480
\(21\) 0 0
\(22\) −321.290 −3.11361
\(23\) −58.4087 −0.529524 −0.264762 0.964314i \(-0.585293\pi\)
−0.264762 + 0.964314i \(0.585293\pi\)
\(24\) 0 0
\(25\) −124.310 −0.994483
\(26\) 168.398 1.27022
\(27\) 0 0
\(28\) 0 0
\(29\) 67.1398 0.429915 0.214958 0.976623i \(-0.431039\pi\)
0.214958 + 0.976623i \(0.431039\pi\)
\(30\) 0 0
\(31\) −69.5002 −0.402665 −0.201332 0.979523i \(-0.564527\pi\)
−0.201332 + 0.979523i \(0.564527\pi\)
\(32\) 122.603 0.677290
\(33\) 0 0
\(34\) 69.0965 0.348528
\(35\) 0 0
\(36\) 0 0
\(37\) −289.240 −1.28516 −0.642578 0.766220i \(-0.722135\pi\)
−0.642578 + 0.766220i \(0.722135\pi\)
\(38\) −408.469 −1.74375
\(39\) 0 0
\(40\) 24.0851 0.0952048
\(41\) 357.750 1.36271 0.681355 0.731953i \(-0.261391\pi\)
0.681355 + 0.731953i \(0.261391\pi\)
\(42\) 0 0
\(43\) −235.801 −0.836263 −0.418132 0.908386i \(-0.637315\pi\)
−0.418132 + 0.908386i \(0.637315\pi\)
\(44\) 966.482 3.31142
\(45\) 0 0
\(46\) 274.961 0.881320
\(47\) −450.067 −1.39679 −0.698394 0.715714i \(-0.746102\pi\)
−0.698394 + 0.715714i \(0.746102\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 585.195 1.65518
\(51\) 0 0
\(52\) −506.563 −1.35092
\(53\) −172.464 −0.446977 −0.223489 0.974707i \(-0.571745\pi\)
−0.223489 + 0.974707i \(0.571745\pi\)
\(54\) 0 0
\(55\) −56.6787 −0.138955
\(56\) 0 0
\(57\) 0 0
\(58\) −316.063 −0.715535
\(59\) 445.856 0.983823 0.491912 0.870645i \(-0.336298\pi\)
0.491912 + 0.870645i \(0.336298\pi\)
\(60\) 0 0
\(61\) 476.314 0.999767 0.499883 0.866093i \(-0.333376\pi\)
0.499883 + 0.866093i \(0.333376\pi\)
\(62\) 327.174 0.670180
\(63\) 0 0
\(64\) −763.099 −1.49043
\(65\) 29.7070 0.0566878
\(66\) 0 0
\(67\) −804.320 −1.46662 −0.733309 0.679896i \(-0.762025\pi\)
−0.733309 + 0.679896i \(0.762025\pi\)
\(68\) −207.851 −0.370671
\(69\) 0 0
\(70\) 0 0
\(71\) −353.412 −0.590737 −0.295368 0.955383i \(-0.595442\pi\)
−0.295368 + 0.955383i \(0.595442\pi\)
\(72\) 0 0
\(73\) −778.045 −1.24744 −0.623721 0.781647i \(-0.714380\pi\)
−0.623721 + 0.781647i \(0.714380\pi\)
\(74\) 1361.61 2.13897
\(75\) 0 0
\(76\) 1228.73 1.85454
\(77\) 0 0
\(78\) 0 0
\(79\) −74.1996 −0.105672 −0.0528361 0.998603i \(-0.516826\pi\)
−0.0528361 + 0.998603i \(0.516826\pi\)
\(80\) −19.3021 −0.0269755
\(81\) 0 0
\(82\) −1684.12 −2.26804
\(83\) 1168.49 1.54529 0.772643 0.634840i \(-0.218934\pi\)
0.772643 + 0.634840i \(0.218934\pi\)
\(84\) 0 0
\(85\) 12.1893 0.0155543
\(86\) 1110.04 1.39185
\(87\) 0 0
\(88\) −1979.42 −2.39781
\(89\) −1514.72 −1.80405 −0.902023 0.431687i \(-0.857918\pi\)
−0.902023 + 0.431687i \(0.857918\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −827.116 −0.937313
\(93\) 0 0
\(94\) 2118.70 2.32476
\(95\) −72.0579 −0.0778209
\(96\) 0 0
\(97\) 1631.46 1.70772 0.853862 0.520499i \(-0.174254\pi\)
0.853862 + 0.520499i \(0.174254\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1760.34 −1.76034
\(101\) −1205.73 −1.18787 −0.593934 0.804514i \(-0.702426\pi\)
−0.593934 + 0.804514i \(0.702426\pi\)
\(102\) 0 0
\(103\) −485.363 −0.464313 −0.232156 0.972679i \(-0.574578\pi\)
−0.232156 + 0.972679i \(0.574578\pi\)
\(104\) 1037.48 0.978201
\(105\) 0 0
\(106\) 811.881 0.743933
\(107\) 258.101 0.233192 0.116596 0.993179i \(-0.462802\pi\)
0.116596 + 0.993179i \(0.462802\pi\)
\(108\) 0 0
\(109\) 581.246 0.510764 0.255382 0.966840i \(-0.417799\pi\)
0.255382 + 0.966840i \(0.417799\pi\)
\(110\) 266.817 0.231272
\(111\) 0 0
\(112\) 0 0
\(113\) −708.244 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(114\) 0 0
\(115\) 48.5057 0.0393320
\(116\) 950.756 0.760996
\(117\) 0 0
\(118\) −2098.88 −1.63744
\(119\) 0 0
\(120\) 0 0
\(121\) 3327.10 2.49970
\(122\) −2242.26 −1.66398
\(123\) 0 0
\(124\) −984.182 −0.712759
\(125\) 207.041 0.148146
\(126\) 0 0
\(127\) 2348.04 1.64059 0.820296 0.571939i \(-0.193809\pi\)
0.820296 + 0.571939i \(0.193809\pi\)
\(128\) 2611.49 1.80332
\(129\) 0 0
\(130\) −139.847 −0.0943491
\(131\) 2405.82 1.60456 0.802279 0.596949i \(-0.203621\pi\)
0.802279 + 0.596949i \(0.203621\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3786.36 2.44098
\(135\) 0 0
\(136\) 425.693 0.268403
\(137\) 1187.55 0.740579 0.370289 0.928916i \(-0.379258\pi\)
0.370289 + 0.928916i \(0.379258\pi\)
\(138\) 0 0
\(139\) −712.439 −0.434736 −0.217368 0.976090i \(-0.569747\pi\)
−0.217368 + 0.976090i \(0.569747\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1663.70 0.983201
\(143\) −2441.46 −1.42773
\(144\) 0 0
\(145\) −55.7564 −0.0319332
\(146\) 3662.67 2.07620
\(147\) 0 0
\(148\) −4095.89 −2.27486
\(149\) 2209.24 1.21468 0.607342 0.794441i \(-0.292236\pi\)
0.607342 + 0.794441i \(0.292236\pi\)
\(150\) 0 0
\(151\) −546.983 −0.294787 −0.147394 0.989078i \(-0.547088\pi\)
−0.147394 + 0.989078i \(0.547088\pi\)
\(152\) −2516.52 −1.34287
\(153\) 0 0
\(154\) 0 0
\(155\) 57.7166 0.0299091
\(156\) 0 0
\(157\) 1870.19 0.950686 0.475343 0.879801i \(-0.342324\pi\)
0.475343 + 0.879801i \(0.342324\pi\)
\(158\) 349.297 0.175877
\(159\) 0 0
\(160\) −101.816 −0.0503078
\(161\) 0 0
\(162\) 0 0
\(163\) −3037.40 −1.45955 −0.729777 0.683685i \(-0.760376\pi\)
−0.729777 + 0.683685i \(0.760376\pi\)
\(164\) 5066.04 2.41214
\(165\) 0 0
\(166\) −5500.72 −2.57192
\(167\) −2967.09 −1.37485 −0.687426 0.726255i \(-0.741259\pi\)
−0.687426 + 0.726255i \(0.741259\pi\)
\(168\) 0 0
\(169\) −917.358 −0.417550
\(170\) −57.3814 −0.0258879
\(171\) 0 0
\(172\) −3339.14 −1.48028
\(173\) 3866.81 1.69935 0.849677 0.527303i \(-0.176797\pi\)
0.849677 + 0.527303i \(0.176797\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1586.33 0.679399
\(177\) 0 0
\(178\) 7130.60 3.00259
\(179\) −334.726 −0.139769 −0.0698844 0.997555i \(-0.522263\pi\)
−0.0698844 + 0.997555i \(0.522263\pi\)
\(180\) 0 0
\(181\) −1564.56 −0.642502 −0.321251 0.946994i \(-0.604103\pi\)
−0.321251 + 0.946994i \(0.604103\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1693.99 0.678710
\(185\) 240.200 0.0954588
\(186\) 0 0
\(187\) −1001.77 −0.391746
\(188\) −6373.33 −2.47246
\(189\) 0 0
\(190\) 339.215 0.129522
\(191\) −3067.14 −1.16194 −0.580971 0.813925i \(-0.697327\pi\)
−0.580971 + 0.813925i \(0.697327\pi\)
\(192\) 0 0
\(193\) −2223.60 −0.829316 −0.414658 0.909977i \(-0.636099\pi\)
−0.414658 + 0.909977i \(0.636099\pi\)
\(194\) −7680.13 −2.84227
\(195\) 0 0
\(196\) 0 0
\(197\) −1856.18 −0.671306 −0.335653 0.941986i \(-0.608957\pi\)
−0.335653 + 0.941986i \(0.608957\pi\)
\(198\) 0 0
\(199\) −993.602 −0.353943 −0.176971 0.984216i \(-0.556630\pi\)
−0.176971 + 0.984216i \(0.556630\pi\)
\(200\) 3605.30 1.27467
\(201\) 0 0
\(202\) 5676.02 1.97704
\(203\) 0 0
\(204\) 0 0
\(205\) −297.094 −0.101219
\(206\) 2284.86 0.772785
\(207\) 0 0
\(208\) −831.446 −0.277165
\(209\) 5922.03 1.95998
\(210\) 0 0
\(211\) −545.079 −0.177843 −0.0889213 0.996039i \(-0.528342\pi\)
−0.0889213 + 0.996039i \(0.528342\pi\)
\(212\) −2442.24 −0.791197
\(213\) 0 0
\(214\) −1215.02 −0.388116
\(215\) 195.822 0.0621160
\(216\) 0 0
\(217\) 0 0
\(218\) −2736.23 −0.850097
\(219\) 0 0
\(220\) −802.618 −0.245966
\(221\) 525.058 0.159815
\(222\) 0 0
\(223\) −3446.72 −1.03502 −0.517510 0.855677i \(-0.673141\pi\)
−0.517510 + 0.855677i \(0.673141\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 3334.08 0.981326
\(227\) 486.297 0.142188 0.0710940 0.997470i \(-0.477351\pi\)
0.0710940 + 0.997470i \(0.477351\pi\)
\(228\) 0 0
\(229\) 4214.68 1.21622 0.608109 0.793853i \(-0.291928\pi\)
0.608109 + 0.793853i \(0.291928\pi\)
\(230\) −228.342 −0.0654627
\(231\) 0 0
\(232\) −1947.21 −0.551038
\(233\) 1099.34 0.309100 0.154550 0.987985i \(-0.450607\pi\)
0.154550 + 0.987985i \(0.450607\pi\)
\(234\) 0 0
\(235\) 373.760 0.103751
\(236\) 6313.71 1.74147
\(237\) 0 0
\(238\) 0 0
\(239\) −4558.99 −1.23388 −0.616939 0.787011i \(-0.711627\pi\)
−0.616939 + 0.787011i \(0.711627\pi\)
\(240\) 0 0
\(241\) −1089.40 −0.291181 −0.145591 0.989345i \(-0.546508\pi\)
−0.145591 + 0.989345i \(0.546508\pi\)
\(242\) −15662.4 −4.16041
\(243\) 0 0
\(244\) 6745.01 1.76969
\(245\) 0 0
\(246\) 0 0
\(247\) −3103.92 −0.799586
\(248\) 2015.67 0.516110
\(249\) 0 0
\(250\) −974.650 −0.246569
\(251\) −6356.75 −1.59855 −0.799273 0.600969i \(-0.794782\pi\)
−0.799273 + 0.600969i \(0.794782\pi\)
\(252\) 0 0
\(253\) −3986.41 −0.990606
\(254\) −11053.5 −2.73054
\(255\) 0 0
\(256\) −6188.88 −1.51096
\(257\) 1063.43 0.258113 0.129056 0.991637i \(-0.458805\pi\)
0.129056 + 0.991637i \(0.458805\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 420.677 0.100343
\(261\) 0 0
\(262\) −11325.5 −2.67057
\(263\) −3674.59 −0.861539 −0.430770 0.902462i \(-0.641758\pi\)
−0.430770 + 0.902462i \(0.641758\pi\)
\(264\) 0 0
\(265\) 143.224 0.0332006
\(266\) 0 0
\(267\) 0 0
\(268\) −11389.9 −2.59607
\(269\) −6959.94 −1.57753 −0.788764 0.614696i \(-0.789279\pi\)
−0.788764 + 0.614696i \(0.789279\pi\)
\(270\) 0 0
\(271\) −6595.78 −1.47847 −0.739235 0.673448i \(-0.764813\pi\)
−0.739235 + 0.673448i \(0.764813\pi\)
\(272\) −341.155 −0.0760499
\(273\) 0 0
\(274\) −5590.43 −1.23259
\(275\) −8484.22 −1.86043
\(276\) 0 0
\(277\) −3121.06 −0.676990 −0.338495 0.940968i \(-0.609918\pi\)
−0.338495 + 0.940968i \(0.609918\pi\)
\(278\) 3353.83 0.723559
\(279\) 0 0
\(280\) 0 0
\(281\) −1662.38 −0.352915 −0.176457 0.984308i \(-0.556464\pi\)
−0.176457 + 0.984308i \(0.556464\pi\)
\(282\) 0 0
\(283\) 453.279 0.0952107 0.0476053 0.998866i \(-0.484841\pi\)
0.0476053 + 0.998866i \(0.484841\pi\)
\(284\) −5004.62 −1.04567
\(285\) 0 0
\(286\) 11493.2 2.37625
\(287\) 0 0
\(288\) 0 0
\(289\) −4697.56 −0.956149
\(290\) 262.475 0.0531485
\(291\) 0 0
\(292\) −11017.8 −2.20811
\(293\) 772.774 0.154082 0.0770408 0.997028i \(-0.475453\pi\)
0.0770408 + 0.997028i \(0.475453\pi\)
\(294\) 0 0
\(295\) −370.263 −0.0730764
\(296\) 8388.65 1.64723
\(297\) 0 0
\(298\) −10400.1 −2.02168
\(299\) 2089.40 0.404124
\(300\) 0 0
\(301\) 0 0
\(302\) 2574.94 0.490633
\(303\) 0 0
\(304\) 2016.77 0.380492
\(305\) −395.557 −0.0742607
\(306\) 0 0
\(307\) 4193.64 0.779621 0.389810 0.920895i \(-0.372541\pi\)
0.389810 + 0.920895i \(0.372541\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −271.703 −0.0497796
\(311\) 399.740 0.0728848 0.0364424 0.999336i \(-0.488397\pi\)
0.0364424 + 0.999336i \(0.488397\pi\)
\(312\) 0 0
\(313\) 3730.48 0.673672 0.336836 0.941563i \(-0.390643\pi\)
0.336836 + 0.941563i \(0.390643\pi\)
\(314\) −8803.99 −1.58229
\(315\) 0 0
\(316\) −1050.73 −0.187051
\(317\) −8373.47 −1.48360 −0.741800 0.670622i \(-0.766027\pi\)
−0.741800 + 0.670622i \(0.766027\pi\)
\(318\) 0 0
\(319\) 4582.31 0.804263
\(320\) 633.718 0.110706
\(321\) 0 0
\(322\) 0 0
\(323\) −1273.59 −0.219394
\(324\) 0 0
\(325\) 4446.84 0.758973
\(326\) 14298.6 2.42923
\(327\) 0 0
\(328\) −10375.6 −1.74664
\(329\) 0 0
\(330\) 0 0
\(331\) −1349.72 −0.224130 −0.112065 0.993701i \(-0.535747\pi\)
−0.112065 + 0.993701i \(0.535747\pi\)
\(332\) 16546.9 2.73532
\(333\) 0 0
\(334\) 13967.7 2.28825
\(335\) 667.950 0.108937
\(336\) 0 0
\(337\) −6629.72 −1.07164 −0.535822 0.844331i \(-0.679998\pi\)
−0.535822 + 0.844331i \(0.679998\pi\)
\(338\) 4318.49 0.694955
\(339\) 0 0
\(340\) 172.610 0.0275327
\(341\) −4743.41 −0.753284
\(342\) 0 0
\(343\) 0 0
\(344\) 6838.79 1.07187
\(345\) 0 0
\(346\) −18203.1 −2.82834
\(347\) −4952.94 −0.766246 −0.383123 0.923697i \(-0.625152\pi\)
−0.383123 + 0.923697i \(0.625152\pi\)
\(348\) 0 0
\(349\) −4498.12 −0.689911 −0.344955 0.938619i \(-0.612106\pi\)
−0.344955 + 0.938619i \(0.612106\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 8367.67 1.26704
\(353\) 1613.35 0.243258 0.121629 0.992576i \(-0.461188\pi\)
0.121629 + 0.992576i \(0.461188\pi\)
\(354\) 0 0
\(355\) 293.492 0.0438787
\(356\) −21449.7 −3.19335
\(357\) 0 0
\(358\) 1575.73 0.232626
\(359\) −13186.9 −1.93866 −0.969328 0.245770i \(-0.920959\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(360\) 0 0
\(361\) 669.919 0.0976700
\(362\) 7365.22 1.06936
\(363\) 0 0
\(364\) 0 0
\(365\) 646.130 0.0926575
\(366\) 0 0
\(367\) 5103.39 0.725871 0.362936 0.931814i \(-0.381774\pi\)
0.362936 + 0.931814i \(0.381774\pi\)
\(368\) −1357.58 −0.192307
\(369\) 0 0
\(370\) −1130.75 −0.158878
\(371\) 0 0
\(372\) 0 0
\(373\) −1459.19 −0.202557 −0.101279 0.994858i \(-0.532293\pi\)
−0.101279 + 0.994858i \(0.532293\pi\)
\(374\) 4715.85 0.652008
\(375\) 0 0
\(376\) 13053.0 1.79031
\(377\) −2401.73 −0.328104
\(378\) 0 0
\(379\) 2332.75 0.316162 0.158081 0.987426i \(-0.449469\pi\)
0.158081 + 0.987426i \(0.449469\pi\)
\(380\) −1020.40 −0.137751
\(381\) 0 0
\(382\) 14438.7 1.93389
\(383\) 5651.11 0.753938 0.376969 0.926226i \(-0.376966\pi\)
0.376969 + 0.926226i \(0.376966\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10467.7 1.38028
\(387\) 0 0
\(388\) 23102.8 3.02285
\(389\) 9040.82 1.17838 0.589188 0.807996i \(-0.299448\pi\)
0.589188 + 0.807996i \(0.299448\pi\)
\(390\) 0 0
\(391\) 857.314 0.110886
\(392\) 0 0
\(393\) 0 0
\(394\) 8738.02 1.11730
\(395\) 61.6193 0.00784912
\(396\) 0 0
\(397\) 3524.35 0.445547 0.222774 0.974870i \(-0.428489\pi\)
0.222774 + 0.974870i \(0.428489\pi\)
\(398\) 4677.41 0.589089
\(399\) 0 0
\(400\) −2889.33 −0.361166
\(401\) 4469.81 0.556638 0.278319 0.960489i \(-0.410223\pi\)
0.278319 + 0.960489i \(0.410223\pi\)
\(402\) 0 0
\(403\) 2486.17 0.307307
\(404\) −17074.2 −2.10265
\(405\) 0 0
\(406\) 0 0
\(407\) −19740.7 −2.40420
\(408\) 0 0
\(409\) −7532.36 −0.910639 −0.455320 0.890328i \(-0.650475\pi\)
−0.455320 + 0.890328i \(0.650475\pi\)
\(410\) 1398.58 0.168466
\(411\) 0 0
\(412\) −6873.15 −0.821882
\(413\) 0 0
\(414\) 0 0
\(415\) −970.379 −0.114781
\(416\) −4385.75 −0.516897
\(417\) 0 0
\(418\) −27878.2 −3.26212
\(419\) 15486.4 1.80563 0.902814 0.430031i \(-0.141498\pi\)
0.902814 + 0.430031i \(0.141498\pi\)
\(420\) 0 0
\(421\) 4408.87 0.510393 0.255196 0.966889i \(-0.417860\pi\)
0.255196 + 0.966889i \(0.417860\pi\)
\(422\) 2565.98 0.295995
\(423\) 0 0
\(424\) 5001.88 0.572907
\(425\) 1824.61 0.208251
\(426\) 0 0
\(427\) 0 0
\(428\) 3654.93 0.412774
\(429\) 0 0
\(430\) −921.837 −0.103384
\(431\) −10356.6 −1.15745 −0.578726 0.815522i \(-0.696450\pi\)
−0.578726 + 0.815522i \(0.696450\pi\)
\(432\) 0 0
\(433\) −1398.15 −0.155175 −0.0775875 0.996986i \(-0.524722\pi\)
−0.0775875 + 0.996986i \(0.524722\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 8230.94 0.904107
\(437\) −5068.08 −0.554781
\(438\) 0 0
\(439\) −3449.32 −0.375005 −0.187502 0.982264i \(-0.560039\pi\)
−0.187502 + 0.982264i \(0.560039\pi\)
\(440\) 1643.82 0.178104
\(441\) 0 0
\(442\) −2471.73 −0.265991
\(443\) 2244.65 0.240737 0.120369 0.992729i \(-0.461592\pi\)
0.120369 + 0.992729i \(0.461592\pi\)
\(444\) 0 0
\(445\) 1257.91 0.134001
\(446\) 16225.6 1.72265
\(447\) 0 0
\(448\) 0 0
\(449\) −7639.18 −0.802929 −0.401465 0.915875i \(-0.631499\pi\)
−0.401465 + 0.915875i \(0.631499\pi\)
\(450\) 0 0
\(451\) 24416.5 2.54929
\(452\) −10029.3 −1.04367
\(453\) 0 0
\(454\) −2289.26 −0.236652
\(455\) 0 0
\(456\) 0 0
\(457\) 540.823 0.0553581 0.0276790 0.999617i \(-0.491188\pi\)
0.0276790 + 0.999617i \(0.491188\pi\)
\(458\) −19840.8 −2.02423
\(459\) 0 0
\(460\) 686.881 0.0696217
\(461\) 8627.25 0.871608 0.435804 0.900042i \(-0.356464\pi\)
0.435804 + 0.900042i \(0.356464\pi\)
\(462\) 0 0
\(463\) 4497.71 0.451461 0.225730 0.974190i \(-0.427523\pi\)
0.225730 + 0.974190i \(0.427523\pi\)
\(464\) 1560.52 0.156132
\(465\) 0 0
\(466\) −5175.19 −0.514455
\(467\) −2493.07 −0.247035 −0.123518 0.992342i \(-0.539418\pi\)
−0.123518 + 0.992342i \(0.539418\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1759.48 −0.172679
\(471\) 0 0
\(472\) −12930.9 −1.26100
\(473\) −16093.5 −1.56444
\(474\) 0 0
\(475\) −10786.3 −1.04192
\(476\) 0 0
\(477\) 0 0
\(478\) 21461.6 2.05362
\(479\) −8339.27 −0.795472 −0.397736 0.917500i \(-0.630204\pi\)
−0.397736 + 0.917500i \(0.630204\pi\)
\(480\) 0 0
\(481\) 10346.7 0.980811
\(482\) 5128.41 0.484632
\(483\) 0 0
\(484\) 47114.6 4.42474
\(485\) −1354.85 −0.126846
\(486\) 0 0
\(487\) −3104.85 −0.288899 −0.144450 0.989512i \(-0.546141\pi\)
−0.144450 + 0.989512i \(0.546141\pi\)
\(488\) −13814.2 −1.28144
\(489\) 0 0
\(490\) 0 0
\(491\) −496.579 −0.0456421 −0.0228211 0.999740i \(-0.507265\pi\)
−0.0228211 + 0.999740i \(0.507265\pi\)
\(492\) 0 0
\(493\) −985.468 −0.0900269
\(494\) 14611.8 1.33080
\(495\) 0 0
\(496\) −1615.38 −0.146236
\(497\) 0 0
\(498\) 0 0
\(499\) −13253.6 −1.18900 −0.594501 0.804095i \(-0.702650\pi\)
−0.594501 + 0.804095i \(0.702650\pi\)
\(500\) 2931.87 0.262234
\(501\) 0 0
\(502\) 29924.6 2.66056
\(503\) −1625.59 −0.144099 −0.0720493 0.997401i \(-0.522954\pi\)
−0.0720493 + 0.997401i \(0.522954\pi\)
\(504\) 0 0
\(505\) 1001.30 0.0882325
\(506\) 18766.1 1.64873
\(507\) 0 0
\(508\) 33250.3 2.90402
\(509\) −7083.05 −0.616799 −0.308399 0.951257i \(-0.599793\pi\)
−0.308399 + 0.951257i \(0.599793\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 8242.42 0.711459
\(513\) 0 0
\(514\) −5006.14 −0.429594
\(515\) 403.071 0.0344882
\(516\) 0 0
\(517\) −30717.2 −2.61304
\(518\) 0 0
\(519\) 0 0
\(520\) −861.575 −0.0726588
\(521\) −16565.3 −1.39298 −0.696489 0.717568i \(-0.745255\pi\)
−0.696489 + 0.717568i \(0.745255\pi\)
\(522\) 0 0
\(523\) −6472.69 −0.541168 −0.270584 0.962696i \(-0.587217\pi\)
−0.270584 + 0.962696i \(0.587217\pi\)
\(524\) 34068.4 2.84024
\(525\) 0 0
\(526\) 17298.2 1.43391
\(527\) 1020.11 0.0843204
\(528\) 0 0
\(529\) −8755.43 −0.719605
\(530\) −674.229 −0.0552578
\(531\) 0 0
\(532\) 0 0
\(533\) −12797.5 −1.04000
\(534\) 0 0
\(535\) −214.341 −0.0173210
\(536\) 23327.2 1.87982
\(537\) 0 0
\(538\) 32764.1 2.62558
\(539\) 0 0
\(540\) 0 0
\(541\) −3388.92 −0.269318 −0.134659 0.990892i \(-0.542994\pi\)
−0.134659 + 0.990892i \(0.542994\pi\)
\(542\) 31049.8 2.46071
\(543\) 0 0
\(544\) −1799.54 −0.141829
\(545\) −482.698 −0.0379385
\(546\) 0 0
\(547\) 10055.2 0.785980 0.392990 0.919543i \(-0.371441\pi\)
0.392990 + 0.919543i \(0.371441\pi\)
\(548\) 16816.7 1.31090
\(549\) 0 0
\(550\) 39939.7 3.09643
\(551\) 5825.67 0.450421
\(552\) 0 0
\(553\) 0 0
\(554\) 14692.5 1.12676
\(555\) 0 0
\(556\) −10088.7 −0.769529
\(557\) 5632.51 0.428469 0.214234 0.976782i \(-0.431274\pi\)
0.214234 + 0.976782i \(0.431274\pi\)
\(558\) 0 0
\(559\) 8435.10 0.638223
\(560\) 0 0
\(561\) 0 0
\(562\) 7825.68 0.587378
\(563\) −1450.41 −0.108575 −0.0542874 0.998525i \(-0.517289\pi\)
−0.0542874 + 0.998525i \(0.517289\pi\)
\(564\) 0 0
\(565\) 588.163 0.0437951
\(566\) −2133.82 −0.158465
\(567\) 0 0
\(568\) 10249.8 0.757169
\(569\) 16030.4 1.18107 0.590536 0.807012i \(-0.298916\pi\)
0.590536 + 0.807012i \(0.298916\pi\)
\(570\) 0 0
\(571\) −16540.8 −1.21228 −0.606139 0.795359i \(-0.707282\pi\)
−0.606139 + 0.795359i \(0.707282\pi\)
\(572\) −34573.1 −2.52723
\(573\) 0 0
\(574\) 0 0
\(575\) 7260.80 0.526602
\(576\) 0 0
\(577\) −2039.27 −0.147133 −0.0735666 0.997290i \(-0.523438\pi\)
−0.0735666 + 0.997290i \(0.523438\pi\)
\(578\) 22113.9 1.59138
\(579\) 0 0
\(580\) −789.558 −0.0565252
\(581\) 0 0
\(582\) 0 0
\(583\) −11770.7 −0.836182
\(584\) 22565.2 1.59889
\(585\) 0 0
\(586\) −3637.86 −0.256448
\(587\) −20392.1 −1.43385 −0.716927 0.697148i \(-0.754452\pi\)
−0.716927 + 0.697148i \(0.754452\pi\)
\(588\) 0 0
\(589\) −6030.49 −0.421871
\(590\) 1743.02 0.121626
\(591\) 0 0
\(592\) −6722.77 −0.466730
\(593\) 20269.0 1.40362 0.701810 0.712364i \(-0.252375\pi\)
0.701810 + 0.712364i \(0.252375\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 31284.7 2.15012
\(597\) 0 0
\(598\) −9835.91 −0.672609
\(599\) 22317.8 1.52234 0.761169 0.648554i \(-0.224626\pi\)
0.761169 + 0.648554i \(0.224626\pi\)
\(600\) 0 0
\(601\) 1880.78 0.127652 0.0638259 0.997961i \(-0.479670\pi\)
0.0638259 + 0.997961i \(0.479670\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −7745.74 −0.521804
\(605\) −2763.00 −0.185673
\(606\) 0 0
\(607\) −16361.6 −1.09407 −0.547033 0.837111i \(-0.684243\pi\)
−0.547033 + 0.837111i \(0.684243\pi\)
\(608\) 10638.2 0.709595
\(609\) 0 0
\(610\) 1862.09 0.123597
\(611\) 16099.8 1.06601
\(612\) 0 0
\(613\) −15921.4 −1.04904 −0.524518 0.851399i \(-0.675755\pi\)
−0.524518 + 0.851399i \(0.675755\pi\)
\(614\) −19741.7 −1.29757
\(615\) 0 0
\(616\) 0 0
\(617\) −7690.80 −0.501815 −0.250908 0.968011i \(-0.580729\pi\)
−0.250908 + 0.968011i \(0.580729\pi\)
\(618\) 0 0
\(619\) −604.012 −0.0392202 −0.0196101 0.999808i \(-0.506242\pi\)
−0.0196101 + 0.999808i \(0.506242\pi\)
\(620\) 817.317 0.0529423
\(621\) 0 0
\(622\) −1881.79 −0.121307
\(623\) 0 0
\(624\) 0 0
\(625\) 15366.9 0.983479
\(626\) −17561.3 −1.12123
\(627\) 0 0
\(628\) 26483.5 1.68281
\(629\) 4245.43 0.269120
\(630\) 0 0
\(631\) −10806.1 −0.681752 −0.340876 0.940108i \(-0.610724\pi\)
−0.340876 + 0.940108i \(0.610724\pi\)
\(632\) 2151.97 0.135444
\(633\) 0 0
\(634\) 39418.4 2.46925
\(635\) −1949.94 −0.121860
\(636\) 0 0
\(637\) 0 0
\(638\) −21571.4 −1.33859
\(639\) 0 0
\(640\) −2168.72 −0.133947
\(641\) 4447.62 0.274057 0.137029 0.990567i \(-0.456245\pi\)
0.137029 + 0.990567i \(0.456245\pi\)
\(642\) 0 0
\(643\) 13531.3 0.829897 0.414949 0.909845i \(-0.363800\pi\)
0.414949 + 0.909845i \(0.363800\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 5995.46 0.365152
\(647\) −18976.9 −1.15311 −0.576553 0.817059i \(-0.695603\pi\)
−0.576553 + 0.817059i \(0.695603\pi\)
\(648\) 0 0
\(649\) 30429.8 1.84049
\(650\) −20933.6 −1.26321
\(651\) 0 0
\(652\) −43012.1 −2.58357
\(653\) 8796.72 0.527171 0.263585 0.964636i \(-0.415095\pi\)
0.263585 + 0.964636i \(0.415095\pi\)
\(654\) 0 0
\(655\) −1997.92 −0.119183
\(656\) 8315.13 0.494895
\(657\) 0 0
\(658\) 0 0
\(659\) 24279.6 1.43520 0.717601 0.696455i \(-0.245240\pi\)
0.717601 + 0.696455i \(0.245240\pi\)
\(660\) 0 0
\(661\) −25635.3 −1.50847 −0.754234 0.656605i \(-0.771992\pi\)
−0.754234 + 0.656605i \(0.771992\pi\)
\(662\) 6353.83 0.373034
\(663\) 0 0
\(664\) −33889.1 −1.98065
\(665\) 0 0
\(666\) 0 0
\(667\) −3921.54 −0.227650
\(668\) −42016.5 −2.43363
\(669\) 0 0
\(670\) −3144.40 −0.181311
\(671\) 32508.6 1.87031
\(672\) 0 0
\(673\) −23355.1 −1.33770 −0.668852 0.743396i \(-0.733214\pi\)
−0.668852 + 0.743396i \(0.733214\pi\)
\(674\) 31209.6 1.78361
\(675\) 0 0
\(676\) −12990.6 −0.739108
\(677\) 17993.7 1.02150 0.510750 0.859730i \(-0.329368\pi\)
0.510750 + 0.859730i \(0.329368\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −353.518 −0.0199365
\(681\) 0 0
\(682\) 22329.7 1.25374
\(683\) 17930.9 1.00455 0.502275 0.864708i \(-0.332496\pi\)
0.502275 + 0.864708i \(0.332496\pi\)
\(684\) 0 0
\(685\) −986.205 −0.0550087
\(686\) 0 0
\(687\) 0 0
\(688\) −5480.69 −0.303706
\(689\) 6169.41 0.341126
\(690\) 0 0
\(691\) 25818.0 1.42137 0.710683 0.703512i \(-0.248386\pi\)
0.710683 + 0.703512i \(0.248386\pi\)
\(692\) 54757.4 3.00804
\(693\) 0 0
\(694\) 23316.1 1.27531
\(695\) 591.647 0.0322913
\(696\) 0 0
\(697\) −5251.00 −0.285360
\(698\) 21175.0 1.14826
\(699\) 0 0
\(700\) 0 0
\(701\) 16727.2 0.901250 0.450625 0.892713i \(-0.351201\pi\)
0.450625 + 0.892713i \(0.351201\pi\)
\(702\) 0 0
\(703\) −25097.2 −1.34646
\(704\) −52081.7 −2.78821
\(705\) 0 0
\(706\) −7594.92 −0.404870
\(707\) 0 0
\(708\) 0 0
\(709\) −6891.41 −0.365039 −0.182519 0.983202i \(-0.558425\pi\)
−0.182519 + 0.983202i \(0.558425\pi\)
\(710\) −1381.62 −0.0730302
\(711\) 0 0
\(712\) 43930.5 2.31231
\(713\) 4059.41 0.213220
\(714\) 0 0
\(715\) 2027.51 0.106049
\(716\) −4740.01 −0.247406
\(717\) 0 0
\(718\) 62077.7 3.22663
\(719\) 10406.4 0.539767 0.269884 0.962893i \(-0.413015\pi\)
0.269884 + 0.962893i \(0.413015\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3153.66 −0.162558
\(723\) 0 0
\(724\) −22155.5 −1.13730
\(725\) −8346.17 −0.427543
\(726\) 0 0
\(727\) 2203.46 0.112410 0.0562049 0.998419i \(-0.482100\pi\)
0.0562049 + 0.998419i \(0.482100\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −3041.68 −0.154216
\(731\) 3461.05 0.175119
\(732\) 0 0
\(733\) −8773.70 −0.442106 −0.221053 0.975262i \(-0.570949\pi\)
−0.221053 + 0.975262i \(0.570949\pi\)
\(734\) −24024.4 −1.20811
\(735\) 0 0
\(736\) −7161.06 −0.358641
\(737\) −54895.1 −2.74367
\(738\) 0 0
\(739\) 27385.9 1.36320 0.681601 0.731724i \(-0.261284\pi\)
0.681601 + 0.731724i \(0.261284\pi\)
\(740\) 3401.44 0.168972
\(741\) 0 0
\(742\) 0 0
\(743\) −28890.0 −1.42648 −0.713239 0.700921i \(-0.752772\pi\)
−0.713239 + 0.700921i \(0.752772\pi\)
\(744\) 0 0
\(745\) −1834.67 −0.0902243
\(746\) 6869.17 0.337129
\(747\) 0 0
\(748\) −14185.9 −0.693432
\(749\) 0 0
\(750\) 0 0
\(751\) −20807.7 −1.01103 −0.505515 0.862818i \(-0.668697\pi\)
−0.505515 + 0.862818i \(0.668697\pi\)
\(752\) −10460.8 −0.507271
\(753\) 0 0
\(754\) 11306.2 0.546085
\(755\) 454.244 0.0218962
\(756\) 0 0
\(757\) 14372.8 0.690078 0.345039 0.938588i \(-0.387866\pi\)
0.345039 + 0.938588i \(0.387866\pi\)
\(758\) −10981.5 −0.526208
\(759\) 0 0
\(760\) 2089.85 0.0997459
\(761\) −26973.6 −1.28488 −0.642439 0.766337i \(-0.722077\pi\)
−0.642439 + 0.766337i \(0.722077\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −43433.4 −2.05676
\(765\) 0 0
\(766\) −26602.8 −1.25483
\(767\) −15949.2 −0.750838
\(768\) 0 0
\(769\) −3264.38 −0.153077 −0.0765387 0.997067i \(-0.524387\pi\)
−0.0765387 + 0.997067i \(0.524387\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −31488.0 −1.46798
\(773\) 22135.4 1.02995 0.514977 0.857204i \(-0.327800\pi\)
0.514977 + 0.857204i \(0.327800\pi\)
\(774\) 0 0
\(775\) 8639.59 0.400443
\(776\) −47316.1 −2.18885
\(777\) 0 0
\(778\) −42560.0 −1.96124
\(779\) 31041.7 1.42771
\(780\) 0 0
\(781\) −24120.5 −1.10512
\(782\) −4035.83 −0.184554
\(783\) 0 0
\(784\) 0 0
\(785\) −1553.11 −0.0706150
\(786\) 0 0
\(787\) 29347.2 1.32924 0.664622 0.747180i \(-0.268593\pi\)
0.664622 + 0.747180i \(0.268593\pi\)
\(788\) −26285.1 −1.18828
\(789\) 0 0
\(790\) −290.075 −0.0130638
\(791\) 0 0
\(792\) 0 0
\(793\) −17038.7 −0.763006
\(794\) −16591.0 −0.741553
\(795\) 0 0
\(796\) −14070.3 −0.626516
\(797\) −26622.6 −1.18321 −0.591606 0.806227i \(-0.701506\pi\)
−0.591606 + 0.806227i \(0.701506\pi\)
\(798\) 0 0
\(799\) 6606.02 0.292496
\(800\) −15240.8 −0.673554
\(801\) 0 0
\(802\) −21041.8 −0.926448
\(803\) −53101.8 −2.33365
\(804\) 0 0
\(805\) 0 0
\(806\) −11703.7 −0.511471
\(807\) 0 0
\(808\) 34969.1 1.52253
\(809\) 21385.7 0.929395 0.464697 0.885470i \(-0.346163\pi\)
0.464697 + 0.885470i \(0.346163\pi\)
\(810\) 0 0
\(811\) −26846.2 −1.16239 −0.581195 0.813764i \(-0.697415\pi\)
−0.581195 + 0.813764i \(0.697415\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 92930.1 4.00147
\(815\) 2522.42 0.108413
\(816\) 0 0
\(817\) −20460.3 −0.876151
\(818\) 35458.8 1.51563
\(819\) 0 0
\(820\) −4207.11 −0.179169
\(821\) −23679.3 −1.00659 −0.503296 0.864114i \(-0.667880\pi\)
−0.503296 + 0.864114i \(0.667880\pi\)
\(822\) 0 0
\(823\) −20546.6 −0.870242 −0.435121 0.900372i \(-0.643294\pi\)
−0.435121 + 0.900372i \(0.643294\pi\)
\(824\) 14076.7 0.595126
\(825\) 0 0
\(826\) 0 0
\(827\) 24006.6 1.00942 0.504712 0.863288i \(-0.331599\pi\)
0.504712 + 0.863288i \(0.331599\pi\)
\(828\) 0 0
\(829\) −20544.1 −0.860706 −0.430353 0.902661i \(-0.641611\pi\)
−0.430353 + 0.902661i \(0.641611\pi\)
\(830\) 4568.09 0.191037
\(831\) 0 0
\(832\) 27297.6 1.13747
\(833\) 0 0
\(834\) 0 0
\(835\) 2464.03 0.102121
\(836\) 83861.0 3.46937
\(837\) 0 0
\(838\) −72902.5 −3.00522
\(839\) −8213.78 −0.337987 −0.168993 0.985617i \(-0.554052\pi\)
−0.168993 + 0.985617i \(0.554052\pi\)
\(840\) 0 0
\(841\) −19881.3 −0.815173
\(842\) −20754.9 −0.849479
\(843\) 0 0
\(844\) −7718.79 −0.314800
\(845\) 761.823 0.0310148
\(846\) 0 0
\(847\) 0 0
\(848\) −4008.56 −0.162329
\(849\) 0 0
\(850\) −8589.41 −0.346605
\(851\) 16894.1 0.680521
\(852\) 0 0
\(853\) 9158.34 0.367615 0.183808 0.982962i \(-0.441158\pi\)
0.183808 + 0.982962i \(0.441158\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7485.54 −0.298891
\(857\) 25763.8 1.02692 0.513462 0.858112i \(-0.328363\pi\)
0.513462 + 0.858112i \(0.328363\pi\)
\(858\) 0 0
\(859\) 41844.0 1.66205 0.831024 0.556236i \(-0.187755\pi\)
0.831024 + 0.556236i \(0.187755\pi\)
\(860\) 2773.00 0.109952
\(861\) 0 0
\(862\) 48754.2 1.92642
\(863\) −43821.4 −1.72850 −0.864250 0.503062i \(-0.832207\pi\)
−0.864250 + 0.503062i \(0.832207\pi\)
\(864\) 0 0
\(865\) −3211.21 −0.126225
\(866\) 6581.83 0.258268
\(867\) 0 0
\(868\) 0 0
\(869\) −5064.15 −0.197686
\(870\) 0 0
\(871\) 28772.2 1.11930
\(872\) −16857.5 −0.654665
\(873\) 0 0
\(874\) 23858.1 0.923357
\(875\) 0 0
\(876\) 0 0
\(877\) −6690.44 −0.257606 −0.128803 0.991670i \(-0.541113\pi\)
−0.128803 + 0.991670i \(0.541113\pi\)
\(878\) 16237.8 0.624144
\(879\) 0 0
\(880\) −1317.37 −0.0504644
\(881\) −11946.0 −0.456835 −0.228417 0.973563i \(-0.573355\pi\)
−0.228417 + 0.973563i \(0.573355\pi\)
\(882\) 0 0
\(883\) −4878.25 −0.185919 −0.0929593 0.995670i \(-0.529633\pi\)
−0.0929593 + 0.995670i \(0.529633\pi\)
\(884\) 7435.26 0.282890
\(885\) 0 0
\(886\) −10566.8 −0.400674
\(887\) 30111.7 1.13986 0.569928 0.821695i \(-0.306971\pi\)
0.569928 + 0.821695i \(0.306971\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −5921.63 −0.223026
\(891\) 0 0
\(892\) −48808.5 −1.83210
\(893\) −39052.0 −1.46341
\(894\) 0 0
\(895\) 277.974 0.0103817
\(896\) 0 0
\(897\) 0 0
\(898\) 35961.7 1.33637
\(899\) −4666.23 −0.173112
\(900\) 0 0
\(901\) 2531.41 0.0935998
\(902\) −114942. −4.24294
\(903\) 0 0
\(904\) 20540.8 0.755725
\(905\) 1299.30 0.0477238
\(906\) 0 0
\(907\) −1986.84 −0.0727364 −0.0363682 0.999338i \(-0.511579\pi\)
−0.0363682 + 0.999338i \(0.511579\pi\)
\(908\) 6886.38 0.251688
\(909\) 0 0
\(910\) 0 0
\(911\) 31604.7 1.14941 0.574703 0.818362i \(-0.305118\pi\)
0.574703 + 0.818362i \(0.305118\pi\)
\(912\) 0 0
\(913\) 79750.0 2.89084
\(914\) −2545.94 −0.0921360
\(915\) 0 0
\(916\) 59683.5 2.15284
\(917\) 0 0
\(918\) 0 0
\(919\) −26999.8 −0.969140 −0.484570 0.874752i \(-0.661024\pi\)
−0.484570 + 0.874752i \(0.661024\pi\)
\(920\) −1406.78 −0.0504132
\(921\) 0 0
\(922\) −40613.1 −1.45067
\(923\) 12642.3 0.450841
\(924\) 0 0
\(925\) 35955.5 1.27807
\(926\) −21173.1 −0.751395
\(927\) 0 0
\(928\) 8231.51 0.291177
\(929\) −17312.9 −0.611431 −0.305715 0.952123i \(-0.598896\pi\)
−0.305715 + 0.952123i \(0.598896\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 15567.6 0.547140
\(933\) 0 0
\(934\) 11736.2 0.411156
\(935\) 831.922 0.0290981
\(936\) 0 0
\(937\) −31294.1 −1.09107 −0.545535 0.838088i \(-0.683673\pi\)
−0.545535 + 0.838088i \(0.683673\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 5292.75 0.183650
\(941\) 44549.5 1.54333 0.771665 0.636029i \(-0.219424\pi\)
0.771665 + 0.636029i \(0.219424\pi\)
\(942\) 0 0
\(943\) −20895.7 −0.721587
\(944\) 10363.0 0.357295
\(945\) 0 0
\(946\) 75760.6 2.60380
\(947\) 54541.4 1.87155 0.935775 0.352596i \(-0.114701\pi\)
0.935775 + 0.352596i \(0.114701\pi\)
\(948\) 0 0
\(949\) 27832.3 0.952028
\(950\) 50777.0 1.73413
\(951\) 0 0
\(952\) 0 0
\(953\) 19349.7 0.657710 0.328855 0.944380i \(-0.393337\pi\)
0.328855 + 0.944380i \(0.393337\pi\)
\(954\) 0 0
\(955\) 2547.12 0.0863066
\(956\) −64559.2 −2.18409
\(957\) 0 0
\(958\) 39257.4 1.32395
\(959\) 0 0
\(960\) 0 0
\(961\) −24960.7 −0.837861
\(962\) −48707.5 −1.63243
\(963\) 0 0
\(964\) −15426.9 −0.515422
\(965\) 1846.59 0.0615999
\(966\) 0 0
\(967\) 52450.4 1.74425 0.872125 0.489283i \(-0.162742\pi\)
0.872125 + 0.489283i \(0.162742\pi\)
\(968\) −96493.9 −3.20396
\(969\) 0 0
\(970\) 6377.99 0.211118
\(971\) −13635.8 −0.450662 −0.225331 0.974282i \(-0.572346\pi\)
−0.225331 + 0.974282i \(0.572346\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 14616.2 0.480834
\(975\) 0 0
\(976\) 11070.9 0.363085
\(977\) −30018.6 −0.982989 −0.491495 0.870881i \(-0.663549\pi\)
−0.491495 + 0.870881i \(0.663549\pi\)
\(978\) 0 0
\(979\) −103380. −3.37492
\(980\) 0 0
\(981\) 0 0
\(982\) 2337.66 0.0759651
\(983\) −6470.42 −0.209943 −0.104972 0.994475i \(-0.533475\pi\)
−0.104972 + 0.994475i \(0.533475\pi\)
\(984\) 0 0
\(985\) 1541.47 0.0498633
\(986\) 4639.12 0.149837
\(987\) 0 0
\(988\) −43954.2 −1.41535
\(989\) 13772.8 0.442821
\(990\) 0 0
\(991\) −23884.4 −0.765605 −0.382802 0.923830i \(-0.625041\pi\)
−0.382802 + 0.923830i \(0.625041\pi\)
\(992\) −8520.91 −0.272721
\(993\) 0 0
\(994\) 0 0
\(995\) 825.140 0.0262902
\(996\) 0 0
\(997\) 7790.90 0.247483 0.123741 0.992315i \(-0.460511\pi\)
0.123741 + 0.992315i \(0.460511\pi\)
\(998\) 62391.6 1.97893
\(999\) 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 1323.4.a.bd.1.1 6
3.2 odd 2 inner 1323.4.a.bd.1.6 6
7.2 even 3 189.4.e.e.109.6 yes 12
7.4 even 3 189.4.e.e.163.6 yes 12
7.6 odd 2 1323.4.a.be.1.1 6
21.2 odd 6 189.4.e.e.109.1 12
21.11 odd 6 189.4.e.e.163.1 yes 12
21.20 even 2 1323.4.a.be.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.e.e.109.1 12 21.2 odd 6
189.4.e.e.109.6 yes 12 7.2 even 3
189.4.e.e.163.1 yes 12 21.11 odd 6
189.4.e.e.163.6 yes 12 7.4 even 3
1323.4.a.bd.1.1 6 1.1 even 1 trivial
1323.4.a.bd.1.6 6 3.2 odd 2 inner
1323.4.a.be.1.1 6 7.6 odd 2
1323.4.a.be.1.6 6 21.20 even 2