Properties

Label 261.6.a.a.1.3
Level $261$
Weight $6$
Character 261.1
Self dual yes
Analytic conductor $41.860$
Analytic rank $1$
Dimension $4$
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] = [261,6,Mod(1,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 261.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: \(41.8601769712\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3257317.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 34x^{2} - 27x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(6.17343\) of defining polynomial
Character \(\chi\) \(=\) 261.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.863638 q^{2} -31.2541 q^{4} +44.4758 q^{5} -36.7447 q^{7} -54.6287 q^{8} +O(q^{10})\) \(q+0.863638 q^{2} -31.2541 q^{4} +44.4758 q^{5} -36.7447 q^{7} -54.6287 q^{8} +38.4110 q^{10} +302.283 q^{11} -373.472 q^{13} -31.7341 q^{14} +952.953 q^{16} -280.365 q^{17} +1371.41 q^{19} -1390.05 q^{20} +261.063 q^{22} -1861.10 q^{23} -1146.91 q^{25} -322.545 q^{26} +1148.42 q^{28} +841.000 q^{29} +1472.03 q^{31} +2571.12 q^{32} -242.134 q^{34} -1634.25 q^{35} -11730.4 q^{37} +1184.40 q^{38} -2429.65 q^{40} +2177.39 q^{41} -9679.03 q^{43} -9447.58 q^{44} -1607.32 q^{46} +15909.1 q^{47} -15456.8 q^{49} -990.512 q^{50} +11672.6 q^{52} -24359.3 q^{53} +13444.3 q^{55} +2007.32 q^{56} +726.320 q^{58} -36304.7 q^{59} -22316.1 q^{61} +1271.30 q^{62} -28274.0 q^{64} -16610.5 q^{65} -54808.6 q^{67} +8762.58 q^{68} -1411.40 q^{70} -27790.4 q^{71} +31685.5 q^{73} -10130.8 q^{74} -42862.2 q^{76} -11107.3 q^{77} -55328.4 q^{79} +42383.3 q^{80} +1880.47 q^{82} +46888.8 q^{83} -12469.5 q^{85} -8359.18 q^{86} -16513.3 q^{88} -2564.30 q^{89} +13723.1 q^{91} +58167.2 q^{92} +13739.7 q^{94} +60994.5 q^{95} +34940.3 q^{97} -13349.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8} - 788 q^{10} + 124 q^{11} - 460 q^{13} - 768 q^{14} - 414 q^{16} - 184 q^{17} - 2392 q^{19} - 2822 q^{20} + 5538 q^{22} + 1192 q^{23} + 1824 q^{25} - 4724 q^{26} + 44 q^{28} + 3364 q^{29} - 19212 q^{31} - 6552 q^{32} - 7612 q^{34} + 22944 q^{35} - 10928 q^{37} + 456 q^{38} - 20 q^{40} + 1120 q^{41} - 21420 q^{43} + 1932 q^{44} - 7588 q^{46} - 23772 q^{47} + 10452 q^{49} - 43240 q^{50} - 29062 q^{52} - 8860 q^{53} - 52652 q^{55} - 34304 q^{56} + 10840 q^{59} + 49448 q^{61} - 18518 q^{62} - 20734 q^{64} - 97836 q^{65} - 7840 q^{67} - 20724 q^{68} - 77496 q^{70} + 48744 q^{71} - 74992 q^{73} + 35920 q^{74} - 140792 q^{76} - 128656 q^{77} - 106076 q^{79} - 58638 q^{80} - 234132 q^{82} - 62888 q^{83} + 23848 q^{85} + 216014 q^{86} - 39426 q^{88} - 107568 q^{89} - 268896 q^{91} + 26268 q^{92} + 30542 q^{94} - 147352 q^{95} - 49520 q^{97} - 242304 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.863638 0.152671 0.0763356 0.997082i \(-0.475678\pi\)
0.0763356 + 0.997082i \(0.475678\pi\)
\(3\) 0 0
\(4\) −31.2541 −0.976692
\(5\) 44.4758 0.795607 0.397803 0.917471i \(-0.369773\pi\)
0.397803 + 0.917471i \(0.369773\pi\)
\(6\) 0 0
\(7\) −36.7447 −0.283433 −0.141716 0.989907i \(-0.545262\pi\)
−0.141716 + 0.989907i \(0.545262\pi\)
\(8\) −54.6287 −0.301784
\(9\) 0 0
\(10\) 38.4110 0.121466
\(11\) 302.283 0.753237 0.376619 0.926368i \(-0.377087\pi\)
0.376619 + 0.926368i \(0.377087\pi\)
\(12\) 0 0
\(13\) −373.472 −0.612915 −0.306457 0.951884i \(-0.599144\pi\)
−0.306457 + 0.951884i \(0.599144\pi\)
\(14\) −31.7341 −0.0432720
\(15\) 0 0
\(16\) 952.953 0.930618
\(17\) −280.365 −0.235289 −0.117645 0.993056i \(-0.537534\pi\)
−0.117645 + 0.993056i \(0.537534\pi\)
\(18\) 0 0
\(19\) 1371.41 0.871532 0.435766 0.900060i \(-0.356478\pi\)
0.435766 + 0.900060i \(0.356478\pi\)
\(20\) −1390.05 −0.777062
\(21\) 0 0
\(22\) 261.063 0.114998
\(23\) −1861.10 −0.733586 −0.366793 0.930303i \(-0.619544\pi\)
−0.366793 + 0.930303i \(0.619544\pi\)
\(24\) 0 0
\(25\) −1146.91 −0.367010
\(26\) −322.545 −0.0935744
\(27\) 0 0
\(28\) 1148.42 0.276826
\(29\) 841.000 0.185695
\(30\) 0 0
\(31\) 1472.03 0.275114 0.137557 0.990494i \(-0.456075\pi\)
0.137557 + 0.990494i \(0.456075\pi\)
\(32\) 2571.12 0.443862
\(33\) 0 0
\(34\) −242.134 −0.0359219
\(35\) −1634.25 −0.225501
\(36\) 0 0
\(37\) −11730.4 −1.40867 −0.704335 0.709868i \(-0.748755\pi\)
−0.704335 + 0.709868i \(0.748755\pi\)
\(38\) 1184.40 0.133058
\(39\) 0 0
\(40\) −2429.65 −0.240101
\(41\) 2177.39 0.202291 0.101145 0.994872i \(-0.467749\pi\)
0.101145 + 0.994872i \(0.467749\pi\)
\(42\) 0 0
\(43\) −9679.03 −0.798290 −0.399145 0.916888i \(-0.630693\pi\)
−0.399145 + 0.916888i \(0.630693\pi\)
\(44\) −9447.58 −0.735680
\(45\) 0 0
\(46\) −1607.32 −0.111997
\(47\) 15909.1 1.05051 0.525255 0.850945i \(-0.323970\pi\)
0.525255 + 0.850945i \(0.323970\pi\)
\(48\) 0 0
\(49\) −15456.8 −0.919666
\(50\) −990.512 −0.0560318
\(51\) 0 0
\(52\) 11672.6 0.598629
\(53\) −24359.3 −1.19117 −0.595586 0.803291i \(-0.703080\pi\)
−0.595586 + 0.803291i \(0.703080\pi\)
\(54\) 0 0
\(55\) 13444.3 0.599280
\(56\) 2007.32 0.0855353
\(57\) 0 0
\(58\) 726.320 0.0283503
\(59\) −36304.7 −1.35779 −0.678894 0.734236i \(-0.737541\pi\)
−0.678894 + 0.734236i \(0.737541\pi\)
\(60\) 0 0
\(61\) −22316.1 −0.767880 −0.383940 0.923358i \(-0.625433\pi\)
−0.383940 + 0.923358i \(0.625433\pi\)
\(62\) 1271.30 0.0420019
\(63\) 0 0
\(64\) −28274.0 −0.862853
\(65\) −16610.5 −0.487639
\(66\) 0 0
\(67\) −54808.6 −1.49163 −0.745817 0.666151i \(-0.767940\pi\)
−0.745817 + 0.666151i \(0.767940\pi\)
\(68\) 8762.58 0.229805
\(69\) 0 0
\(70\) −1411.40 −0.0344275
\(71\) −27790.4 −0.654258 −0.327129 0.944980i \(-0.606081\pi\)
−0.327129 + 0.944980i \(0.606081\pi\)
\(72\) 0 0
\(73\) 31685.5 0.695910 0.347955 0.937511i \(-0.386876\pi\)
0.347955 + 0.937511i \(0.386876\pi\)
\(74\) −10130.8 −0.215063
\(75\) 0 0
\(76\) −42862.2 −0.851218
\(77\) −11107.3 −0.213492
\(78\) 0 0
\(79\) −55328.4 −0.997426 −0.498713 0.866767i \(-0.666194\pi\)
−0.498713 + 0.866767i \(0.666194\pi\)
\(80\) 42383.3 0.740406
\(81\) 0 0
\(82\) 1880.47 0.0308839
\(83\) 46888.8 0.747092 0.373546 0.927612i \(-0.378142\pi\)
0.373546 + 0.927612i \(0.378142\pi\)
\(84\) 0 0
\(85\) −12469.5 −0.187198
\(86\) −8359.18 −0.121876
\(87\) 0 0
\(88\) −16513.3 −0.227315
\(89\) −2564.30 −0.0343157 −0.0171579 0.999853i \(-0.505462\pi\)
−0.0171579 + 0.999853i \(0.505462\pi\)
\(90\) 0 0
\(91\) 13723.1 0.173720
\(92\) 58167.2 0.716487
\(93\) 0 0
\(94\) 13739.7 0.160382
\(95\) 60994.5 0.693396
\(96\) 0 0
\(97\) 34940.3 0.377048 0.188524 0.982069i \(-0.439630\pi\)
0.188524 + 0.982069i \(0.439630\pi\)
\(98\) −13349.1 −0.140406
\(99\) 0 0
\(100\) 35845.6 0.358456
\(101\) −170224. −1.66042 −0.830210 0.557451i \(-0.811779\pi\)
−0.830210 + 0.557451i \(0.811779\pi\)
\(102\) 0 0
\(103\) 83962.9 0.779820 0.389910 0.920853i \(-0.372506\pi\)
0.389910 + 0.920853i \(0.372506\pi\)
\(104\) 20402.3 0.184968
\(105\) 0 0
\(106\) −21037.6 −0.181858
\(107\) −16842.4 −0.142215 −0.0711074 0.997469i \(-0.522653\pi\)
−0.0711074 + 0.997469i \(0.522653\pi\)
\(108\) 0 0
\(109\) −3855.63 −0.0310834 −0.0155417 0.999879i \(-0.504947\pi\)
−0.0155417 + 0.999879i \(0.504947\pi\)
\(110\) 11611.0 0.0914928
\(111\) 0 0
\(112\) −35016.0 −0.263767
\(113\) −107319. −0.790644 −0.395322 0.918543i \(-0.629367\pi\)
−0.395322 + 0.918543i \(0.629367\pi\)
\(114\) 0 0
\(115\) −82774.0 −0.583646
\(116\) −26284.7 −0.181367
\(117\) 0 0
\(118\) −31354.1 −0.207295
\(119\) 10301.9 0.0666886
\(120\) 0 0
\(121\) −69676.1 −0.432634
\(122\) −19273.0 −0.117233
\(123\) 0 0
\(124\) −46007.0 −0.268701
\(125\) −189996. −1.08760
\(126\) 0 0
\(127\) −14823.9 −0.0815554 −0.0407777 0.999168i \(-0.512984\pi\)
−0.0407777 + 0.999168i \(0.512984\pi\)
\(128\) −106694. −0.575595
\(129\) 0 0
\(130\) −14345.4 −0.0744484
\(131\) 313171. 1.59442 0.797211 0.603701i \(-0.206308\pi\)
0.797211 + 0.603701i \(0.206308\pi\)
\(132\) 0 0
\(133\) −50392.1 −0.247020
\(134\) −47334.8 −0.227729
\(135\) 0 0
\(136\) 15316.0 0.0710065
\(137\) 254312. 1.15762 0.578810 0.815463i \(-0.303517\pi\)
0.578810 + 0.815463i \(0.303517\pi\)
\(138\) 0 0
\(139\) −390931. −1.71618 −0.858091 0.513498i \(-0.828349\pi\)
−0.858091 + 0.513498i \(0.828349\pi\)
\(140\) 51077.0 0.220245
\(141\) 0 0
\(142\) −24000.9 −0.0998863
\(143\) −112894. −0.461670
\(144\) 0 0
\(145\) 37404.1 0.147740
\(146\) 27364.8 0.106245
\(147\) 0 0
\(148\) 366624. 1.37584
\(149\) −72345.7 −0.266961 −0.133480 0.991051i \(-0.542615\pi\)
−0.133480 + 0.991051i \(0.542615\pi\)
\(150\) 0 0
\(151\) −227419. −0.811679 −0.405840 0.913944i \(-0.633021\pi\)
−0.405840 + 0.913944i \(0.633021\pi\)
\(152\) −74918.3 −0.263014
\(153\) 0 0
\(154\) −9592.68 −0.0325940
\(155\) 65469.6 0.218882
\(156\) 0 0
\(157\) 539281. 1.74609 0.873044 0.487641i \(-0.162143\pi\)
0.873044 + 0.487641i \(0.162143\pi\)
\(158\) −47783.7 −0.152278
\(159\) 0 0
\(160\) 114353. 0.353140
\(161\) 68385.7 0.207922
\(162\) 0 0
\(163\) 87996.5 0.259416 0.129708 0.991552i \(-0.458596\pi\)
0.129708 + 0.991552i \(0.458596\pi\)
\(164\) −68052.3 −0.197576
\(165\) 0 0
\(166\) 40495.0 0.114059
\(167\) −199897. −0.554644 −0.277322 0.960777i \(-0.589447\pi\)
−0.277322 + 0.960777i \(0.589447\pi\)
\(168\) 0 0
\(169\) −231811. −0.624335
\(170\) −10769.1 −0.0285797
\(171\) 0 0
\(172\) 302510. 0.779683
\(173\) 313797. 0.797139 0.398569 0.917138i \(-0.369507\pi\)
0.398569 + 0.917138i \(0.369507\pi\)
\(174\) 0 0
\(175\) 42142.7 0.104023
\(176\) 288061. 0.700976
\(177\) 0 0
\(178\) −2214.63 −0.00523902
\(179\) 59814.7 0.139533 0.0697663 0.997563i \(-0.477775\pi\)
0.0697663 + 0.997563i \(0.477775\pi\)
\(180\) 0 0
\(181\) 402231. 0.912597 0.456298 0.889827i \(-0.349175\pi\)
0.456298 + 0.889827i \(0.349175\pi\)
\(182\) 11851.8 0.0265220
\(183\) 0 0
\(184\) 101670. 0.221384
\(185\) −521719. −1.12075
\(186\) 0 0
\(187\) −84749.6 −0.177229
\(188\) −497224. −1.02602
\(189\) 0 0
\(190\) 52677.2 0.105862
\(191\) 903529. 1.79209 0.896043 0.443967i \(-0.146429\pi\)
0.896043 + 0.443967i \(0.146429\pi\)
\(192\) 0 0
\(193\) 1.02870e6 1.98791 0.993956 0.109780i \(-0.0350148\pi\)
0.993956 + 0.109780i \(0.0350148\pi\)
\(194\) 30175.7 0.0575644
\(195\) 0 0
\(196\) 483090. 0.898230
\(197\) 866099. 1.59002 0.795009 0.606598i \(-0.207466\pi\)
0.795009 + 0.606598i \(0.207466\pi\)
\(198\) 0 0
\(199\) 285836. 0.511663 0.255832 0.966721i \(-0.417651\pi\)
0.255832 + 0.966721i \(0.417651\pi\)
\(200\) 62654.0 0.110758
\(201\) 0 0
\(202\) −147012. −0.253498
\(203\) −30902.3 −0.0526321
\(204\) 0 0
\(205\) 96840.9 0.160944
\(206\) 72513.6 0.119056
\(207\) 0 0
\(208\) −355901. −0.570389
\(209\) 414554. 0.656470
\(210\) 0 0
\(211\) −918930. −1.42094 −0.710471 0.703727i \(-0.751518\pi\)
−0.710471 + 0.703727i \(0.751518\pi\)
\(212\) 761328. 1.16341
\(213\) 0 0
\(214\) −14545.8 −0.0217121
\(215\) −430482. −0.635125
\(216\) 0 0
\(217\) −54089.3 −0.0779762
\(218\) −3329.87 −0.00474554
\(219\) 0 0
\(220\) −420189. −0.585312
\(221\) 104709. 0.144212
\(222\) 0 0
\(223\) 486134. 0.654626 0.327313 0.944916i \(-0.393857\pi\)
0.327313 + 0.944916i \(0.393857\pi\)
\(224\) −94475.2 −0.125805
\(225\) 0 0
\(226\) −92684.9 −0.120708
\(227\) 557639. 0.718271 0.359136 0.933285i \(-0.383072\pi\)
0.359136 + 0.933285i \(0.383072\pi\)
\(228\) 0 0
\(229\) 440289. 0.554816 0.277408 0.960752i \(-0.410525\pi\)
0.277408 + 0.960752i \(0.410525\pi\)
\(230\) −71486.8 −0.0891059
\(231\) 0 0
\(232\) −45942.7 −0.0560398
\(233\) −1.46183e6 −1.76403 −0.882015 0.471221i \(-0.843813\pi\)
−0.882015 + 0.471221i \(0.843813\pi\)
\(234\) 0 0
\(235\) 707568. 0.835792
\(236\) 1.13467e6 1.32614
\(237\) 0 0
\(238\) 8897.16 0.0101814
\(239\) −1.51292e6 −1.71325 −0.856627 0.515936i \(-0.827444\pi\)
−0.856627 + 0.515936i \(0.827444\pi\)
\(240\) 0 0
\(241\) −800679. −0.888006 −0.444003 0.896025i \(-0.646442\pi\)
−0.444003 + 0.896025i \(0.646442\pi\)
\(242\) −60175.0 −0.0660507
\(243\) 0 0
\(244\) 697470. 0.749982
\(245\) −687454. −0.731692
\(246\) 0 0
\(247\) −512184. −0.534175
\(248\) −80415.0 −0.0830249
\(249\) 0 0
\(250\) −164088. −0.166045
\(251\) −1.66023e6 −1.66335 −0.831676 0.555261i \(-0.812618\pi\)
−0.831676 + 0.555261i \(0.812618\pi\)
\(252\) 0 0
\(253\) −562580. −0.552564
\(254\) −12802.5 −0.0124512
\(255\) 0 0
\(256\) 812621. 0.774976
\(257\) 1.00723e6 0.951250 0.475625 0.879648i \(-0.342222\pi\)
0.475625 + 0.879648i \(0.342222\pi\)
\(258\) 0 0
\(259\) 431031. 0.399263
\(260\) 519146. 0.476273
\(261\) 0 0
\(262\) 270466. 0.243422
\(263\) −1.01326e6 −0.903298 −0.451649 0.892196i \(-0.649164\pi\)
−0.451649 + 0.892196i \(0.649164\pi\)
\(264\) 0 0
\(265\) −1.08340e6 −0.947705
\(266\) −43520.5 −0.0377129
\(267\) 0 0
\(268\) 1.71300e6 1.45687
\(269\) 1.45518e6 1.22613 0.613064 0.790034i \(-0.289937\pi\)
0.613064 + 0.790034i \(0.289937\pi\)
\(270\) 0 0
\(271\) −1.61146e6 −1.33290 −0.666450 0.745550i \(-0.732187\pi\)
−0.666450 + 0.745550i \(0.732187\pi\)
\(272\) −267175. −0.218964
\(273\) 0 0
\(274\) 219634. 0.176735
\(275\) −346690. −0.276446
\(276\) 0 0
\(277\) 2.06358e6 1.61593 0.807966 0.589230i \(-0.200569\pi\)
0.807966 + 0.589230i \(0.200569\pi\)
\(278\) −337623. −0.262011
\(279\) 0 0
\(280\) 89276.9 0.0680525
\(281\) −21664.7 −0.0163676 −0.00818382 0.999967i \(-0.502605\pi\)
−0.00818382 + 0.999967i \(0.502605\pi\)
\(282\) 0 0
\(283\) −552726. −0.410245 −0.205123 0.978736i \(-0.565759\pi\)
−0.205123 + 0.978736i \(0.565759\pi\)
\(284\) 868565. 0.639008
\(285\) 0 0
\(286\) −97499.8 −0.0704837
\(287\) −80007.4 −0.0573358
\(288\) 0 0
\(289\) −1.34125e6 −0.944639
\(290\) 32303.6 0.0225557
\(291\) 0 0
\(292\) −990301. −0.679689
\(293\) 1.53503e6 1.04459 0.522297 0.852764i \(-0.325075\pi\)
0.522297 + 0.852764i \(0.325075\pi\)
\(294\) 0 0
\(295\) −1.61468e6 −1.08027
\(296\) 640817. 0.425114
\(297\) 0 0
\(298\) −62480.5 −0.0407572
\(299\) 695071. 0.449626
\(300\) 0 0
\(301\) 355653. 0.226261
\(302\) −196408. −0.123920
\(303\) 0 0
\(304\) 1.30689e6 0.811063
\(305\) −992525. −0.610930
\(306\) 0 0
\(307\) −2.69140e6 −1.62979 −0.814895 0.579609i \(-0.803205\pi\)
−0.814895 + 0.579609i \(0.803205\pi\)
\(308\) 347149. 0.208516
\(309\) 0 0
\(310\) 56542.1 0.0334170
\(311\) 1.88553e6 1.10543 0.552717 0.833369i \(-0.313591\pi\)
0.552717 + 0.833369i \(0.313591\pi\)
\(312\) 0 0
\(313\) −2.48142e6 −1.43166 −0.715828 0.698276i \(-0.753951\pi\)
−0.715828 + 0.698276i \(0.753951\pi\)
\(314\) 465744. 0.266577
\(315\) 0 0
\(316\) 1.72924e6 0.974177
\(317\) −204476. −0.114286 −0.0571432 0.998366i \(-0.518199\pi\)
−0.0571432 + 0.998366i \(0.518199\pi\)
\(318\) 0 0
\(319\) 254220. 0.139873
\(320\) −1.25751e6 −0.686492
\(321\) 0 0
\(322\) 59060.5 0.0317437
\(323\) −384496. −0.205062
\(324\) 0 0
\(325\) 428338. 0.224946
\(326\) 75997.1 0.0396053
\(327\) 0 0
\(328\) −118948. −0.0610480
\(329\) −584574. −0.297749
\(330\) 0 0
\(331\) 1.58800e6 0.796672 0.398336 0.917240i \(-0.369588\pi\)
0.398336 + 0.917240i \(0.369588\pi\)
\(332\) −1.46547e6 −0.729678
\(333\) 0 0
\(334\) −172638. −0.0846782
\(335\) −2.43766e6 −1.18675
\(336\) 0 0
\(337\) −813616. −0.390252 −0.195126 0.980778i \(-0.562512\pi\)
−0.195126 + 0.980778i \(0.562512\pi\)
\(338\) −200201. −0.0953180
\(339\) 0 0
\(340\) 389722. 0.182834
\(341\) 444969. 0.207226
\(342\) 0 0
\(343\) 1.18553e6 0.544096
\(344\) 528753. 0.240911
\(345\) 0 0
\(346\) 271007. 0.121700
\(347\) −516038. −0.230069 −0.115034 0.993362i \(-0.536698\pi\)
−0.115034 + 0.993362i \(0.536698\pi\)
\(348\) 0 0
\(349\) −2.65164e6 −1.16533 −0.582667 0.812711i \(-0.697991\pi\)
−0.582667 + 0.812711i \(0.697991\pi\)
\(350\) 36396.1 0.0158812
\(351\) 0 0
\(352\) 777207. 0.334333
\(353\) −391323. −0.167147 −0.0835734 0.996502i \(-0.526633\pi\)
−0.0835734 + 0.996502i \(0.526633\pi\)
\(354\) 0 0
\(355\) −1.23600e6 −0.520532
\(356\) 80144.9 0.0335159
\(357\) 0 0
\(358\) 51658.3 0.0213026
\(359\) −3.03988e6 −1.24486 −0.622430 0.782675i \(-0.713854\pi\)
−0.622430 + 0.782675i \(0.713854\pi\)
\(360\) 0 0
\(361\) −595334. −0.240432
\(362\) 347382. 0.139327
\(363\) 0 0
\(364\) −428905. −0.169671
\(365\) 1.40924e6 0.553670
\(366\) 0 0
\(367\) 1.02897e6 0.398785 0.199392 0.979920i \(-0.436103\pi\)
0.199392 + 0.979920i \(0.436103\pi\)
\(368\) −1.77354e6 −0.682688
\(369\) 0 0
\(370\) −450577. −0.171106
\(371\) 895074. 0.337617
\(372\) 0 0
\(373\) −4.42358e6 −1.64627 −0.823136 0.567844i \(-0.807778\pi\)
−0.823136 + 0.567844i \(0.807778\pi\)
\(374\) −73193.0 −0.0270577
\(375\) 0 0
\(376\) −869091. −0.317027
\(377\) −314090. −0.113815
\(378\) 0 0
\(379\) −4.23499e6 −1.51445 −0.757223 0.653156i \(-0.773445\pi\)
−0.757223 + 0.653156i \(0.773445\pi\)
\(380\) −1.90633e6 −0.677234
\(381\) 0 0
\(382\) 780323. 0.273600
\(383\) 894484. 0.311584 0.155792 0.987790i \(-0.450207\pi\)
0.155792 + 0.987790i \(0.450207\pi\)
\(384\) 0 0
\(385\) −494005. −0.169856
\(386\) 888428. 0.303497
\(387\) 0 0
\(388\) −1.09203e6 −0.368260
\(389\) −3.60641e6 −1.20837 −0.604186 0.796843i \(-0.706502\pi\)
−0.604186 + 0.796843i \(0.706502\pi\)
\(390\) 0 0
\(391\) 521789. 0.172605
\(392\) 844386. 0.277540
\(393\) 0 0
\(394\) 747996. 0.242750
\(395\) −2.46077e6 −0.793559
\(396\) 0 0
\(397\) −1.96195e6 −0.624757 −0.312379 0.949958i \(-0.601126\pi\)
−0.312379 + 0.949958i \(0.601126\pi\)
\(398\) 246859. 0.0781162
\(399\) 0 0
\(400\) −1.09295e6 −0.341546
\(401\) 721116. 0.223947 0.111973 0.993711i \(-0.464283\pi\)
0.111973 + 0.993711i \(0.464283\pi\)
\(402\) 0 0
\(403\) −549762. −0.168621
\(404\) 5.32021e6 1.62172
\(405\) 0 0
\(406\) −26688.4 −0.00803540
\(407\) −3.54590e6 −1.06106
\(408\) 0 0
\(409\) 1.33383e6 0.394269 0.197135 0.980376i \(-0.436836\pi\)
0.197135 + 0.980376i \(0.436836\pi\)
\(410\) 83635.5 0.0245715
\(411\) 0 0
\(412\) −2.62419e6 −0.761644
\(413\) 1.33400e6 0.384842
\(414\) 0 0
\(415\) 2.08542e6 0.594391
\(416\) −960244. −0.272050
\(417\) 0 0
\(418\) 358024. 0.100224
\(419\) 4.51294e6 1.25581 0.627905 0.778290i \(-0.283913\pi\)
0.627905 + 0.778290i \(0.283913\pi\)
\(420\) 0 0
\(421\) 982181. 0.270076 0.135038 0.990840i \(-0.456884\pi\)
0.135038 + 0.990840i \(0.456884\pi\)
\(422\) −793623. −0.216937
\(423\) 0 0
\(424\) 1.33071e6 0.359476
\(425\) 321553. 0.0863535
\(426\) 0 0
\(427\) 819998. 0.217642
\(428\) 526395. 0.138900
\(429\) 0 0
\(430\) −371781. −0.0969652
\(431\) −5.31784e6 −1.37893 −0.689465 0.724319i \(-0.742154\pi\)
−0.689465 + 0.724319i \(0.742154\pi\)
\(432\) 0 0
\(433\) 679278. 0.174112 0.0870558 0.996203i \(-0.472254\pi\)
0.0870558 + 0.996203i \(0.472254\pi\)
\(434\) −46713.6 −0.0119047
\(435\) 0 0
\(436\) 120504. 0.0303589
\(437\) −2.55234e6 −0.639344
\(438\) 0 0
\(439\) 3.53619e6 0.875739 0.437870 0.899039i \(-0.355733\pi\)
0.437870 + 0.899039i \(0.355733\pi\)
\(440\) −734442. −0.180853
\(441\) 0 0
\(442\) 90430.5 0.0220170
\(443\) 1.35298e6 0.327553 0.163776 0.986497i \(-0.447632\pi\)
0.163776 + 0.986497i \(0.447632\pi\)
\(444\) 0 0
\(445\) −114049. −0.0273018
\(446\) 419844. 0.0999425
\(447\) 0 0
\(448\) 1.03892e6 0.244561
\(449\) 2.04554e6 0.478843 0.239421 0.970916i \(-0.423042\pi\)
0.239421 + 0.970916i \(0.423042\pi\)
\(450\) 0 0
\(451\) 658186. 0.152373
\(452\) 3.35416e6 0.772215
\(453\) 0 0
\(454\) 481598. 0.109659
\(455\) 610347. 0.138213
\(456\) 0 0
\(457\) 8.48379e6 1.90020 0.950100 0.311944i \(-0.100980\pi\)
0.950100 + 0.311944i \(0.100980\pi\)
\(458\) 380250. 0.0847044
\(459\) 0 0
\(460\) 2.58703e6 0.570042
\(461\) 2.06795e6 0.453198 0.226599 0.973988i \(-0.427239\pi\)
0.226599 + 0.973988i \(0.427239\pi\)
\(462\) 0 0
\(463\) −368290. −0.0798430 −0.0399215 0.999203i \(-0.512711\pi\)
−0.0399215 + 0.999203i \(0.512711\pi\)
\(464\) 801433. 0.172811
\(465\) 0 0
\(466\) −1.26249e6 −0.269317
\(467\) 2.78365e6 0.590640 0.295320 0.955398i \(-0.404574\pi\)
0.295320 + 0.955398i \(0.404574\pi\)
\(468\) 0 0
\(469\) 2.01393e6 0.422777
\(470\) 611082. 0.127601
\(471\) 0 0
\(472\) 1.98328e6 0.409759
\(473\) −2.92580e6 −0.601302
\(474\) 0 0
\(475\) −1.57288e6 −0.319861
\(476\) −321978. −0.0651342
\(477\) 0 0
\(478\) −1.30662e6 −0.261564
\(479\) −1.49820e6 −0.298352 −0.149176 0.988811i \(-0.547662\pi\)
−0.149176 + 0.988811i \(0.547662\pi\)
\(480\) 0 0
\(481\) 4.38099e6 0.863394
\(482\) −691497. −0.135573
\(483\) 0 0
\(484\) 2.17767e6 0.422550
\(485\) 1.55399e6 0.299982
\(486\) 0 0
\(487\) 7.95675e6 1.52024 0.760122 0.649781i \(-0.225139\pi\)
0.760122 + 0.649781i \(0.225139\pi\)
\(488\) 1.21910e6 0.231734
\(489\) 0 0
\(490\) −593712. −0.111708
\(491\) 1.87342e6 0.350697 0.175348 0.984506i \(-0.443895\pi\)
0.175348 + 0.984506i \(0.443895\pi\)
\(492\) 0 0
\(493\) −235787. −0.0436921
\(494\) −442341. −0.0815530
\(495\) 0 0
\(496\) 1.40277e6 0.256026
\(497\) 1.02115e6 0.185438
\(498\) 0 0
\(499\) −352286. −0.0633350 −0.0316675 0.999498i \(-0.510082\pi\)
−0.0316675 + 0.999498i \(0.510082\pi\)
\(500\) 5.93817e6 1.06225
\(501\) 0 0
\(502\) −1.43384e6 −0.253946
\(503\) 3.57266e6 0.629611 0.314805 0.949156i \(-0.398061\pi\)
0.314805 + 0.949156i \(0.398061\pi\)
\(504\) 0 0
\(505\) −7.57085e6 −1.32104
\(506\) −485865. −0.0843606
\(507\) 0 0
\(508\) 463308. 0.0796545
\(509\) 1.05719e7 1.80867 0.904337 0.426819i \(-0.140366\pi\)
0.904337 + 0.426819i \(0.140366\pi\)
\(510\) 0 0
\(511\) −1.16427e6 −0.197243
\(512\) 4.11603e6 0.693911
\(513\) 0 0
\(514\) 869880. 0.145228
\(515\) 3.73431e6 0.620430
\(516\) 0 0
\(517\) 4.80903e6 0.791282
\(518\) 372255. 0.0609559
\(519\) 0 0
\(520\) 907408. 0.147162
\(521\) −7.42701e6 −1.19872 −0.599362 0.800478i \(-0.704579\pi\)
−0.599362 + 0.800478i \(0.704579\pi\)
\(522\) 0 0
\(523\) −1.03292e7 −1.65125 −0.825623 0.564222i \(-0.809176\pi\)
−0.825623 + 0.564222i \(0.809176\pi\)
\(524\) −9.78788e6 −1.55726
\(525\) 0 0
\(526\) −875089. −0.137908
\(527\) −412706. −0.0647313
\(528\) 0 0
\(529\) −2.97263e6 −0.461851
\(530\) −935663. −0.144687
\(531\) 0 0
\(532\) 1.57496e6 0.241263
\(533\) −813193. −0.123987
\(534\) 0 0
\(535\) −749079. −0.113147
\(536\) 2.99412e6 0.450151
\(537\) 0 0
\(538\) 1.25675e6 0.187194
\(539\) −4.67233e6 −0.692726
\(540\) 0 0
\(541\) 4.60833e6 0.676941 0.338470 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(542\) −1.39172e6 −0.203495
\(543\) 0 0
\(544\) −720854. −0.104436
\(545\) −171482. −0.0247302
\(546\) 0 0
\(547\) −1.79146e6 −0.255999 −0.127999 0.991774i \(-0.540856\pi\)
−0.127999 + 0.991774i \(0.540856\pi\)
\(548\) −7.94831e6 −1.13064
\(549\) 0 0
\(550\) −299415. −0.0422052
\(551\) 1.15336e6 0.161839
\(552\) 0 0
\(553\) 2.03303e6 0.282703
\(554\) 1.78219e6 0.246706
\(555\) 0 0
\(556\) 1.22182e7 1.67618
\(557\) −1.07840e7 −1.47280 −0.736399 0.676548i \(-0.763475\pi\)
−0.736399 + 0.676548i \(0.763475\pi\)
\(558\) 0 0
\(559\) 3.61485e6 0.489284
\(560\) −1.55736e6 −0.209855
\(561\) 0 0
\(562\) −18710.4 −0.00249887
\(563\) 1.40598e7 1.86942 0.934710 0.355411i \(-0.115659\pi\)
0.934710 + 0.355411i \(0.115659\pi\)
\(564\) 0 0
\(565\) −4.77310e6 −0.629041
\(566\) −477355. −0.0626326
\(567\) 0 0
\(568\) 1.51815e6 0.197444
\(569\) −4.09890e6 −0.530746 −0.265373 0.964146i \(-0.585495\pi\)
−0.265373 + 0.964146i \(0.585495\pi\)
\(570\) 0 0
\(571\) −2.46148e6 −0.315941 −0.157971 0.987444i \(-0.550495\pi\)
−0.157971 + 0.987444i \(0.550495\pi\)
\(572\) 3.52841e6 0.450909
\(573\) 0 0
\(574\) −69097.5 −0.00875351
\(575\) 2.13451e6 0.269233
\(576\) 0 0
\(577\) 7.09785e6 0.887539 0.443770 0.896141i \(-0.353641\pi\)
0.443770 + 0.896141i \(0.353641\pi\)
\(578\) −1.15836e6 −0.144219
\(579\) 0 0
\(580\) −1.16903e6 −0.144297
\(581\) −1.72292e6 −0.211750
\(582\) 0 0
\(583\) −7.36339e6 −0.897235
\(584\) −1.73094e6 −0.210014
\(585\) 0 0
\(586\) 1.32571e6 0.159479
\(587\) 1.43549e7 1.71951 0.859753 0.510710i \(-0.170617\pi\)
0.859753 + 0.510710i \(0.170617\pi\)
\(588\) 0 0
\(589\) 2.01876e6 0.239770
\(590\) −1.39450e6 −0.164925
\(591\) 0 0
\(592\) −1.11785e7 −1.31093
\(593\) −1.01492e7 −1.18521 −0.592606 0.805493i \(-0.701901\pi\)
−0.592606 + 0.805493i \(0.701901\pi\)
\(594\) 0 0
\(595\) 458187. 0.0530579
\(596\) 2.26110e6 0.260738
\(597\) 0 0
\(598\) 600290. 0.0686449
\(599\) 8.76777e6 0.998440 0.499220 0.866475i \(-0.333620\pi\)
0.499220 + 0.866475i \(0.333620\pi\)
\(600\) 0 0
\(601\) 1.49619e7 1.68966 0.844831 0.535034i \(-0.179701\pi\)
0.844831 + 0.535034i \(0.179701\pi\)
\(602\) 307156. 0.0345436
\(603\) 0 0
\(604\) 7.10779e6 0.792760
\(605\) −3.09890e6 −0.344206
\(606\) 0 0
\(607\) 7.03290e6 0.774752 0.387376 0.921922i \(-0.373382\pi\)
0.387376 + 0.921922i \(0.373382\pi\)
\(608\) 3.52606e6 0.386840
\(609\) 0 0
\(610\) −857182. −0.0932714
\(611\) −5.94159e6 −0.643873
\(612\) 0 0
\(613\) −806845. −0.0867239 −0.0433620 0.999059i \(-0.513807\pi\)
−0.0433620 + 0.999059i \(0.513807\pi\)
\(614\) −2.32439e6 −0.248822
\(615\) 0 0
\(616\) 606777. 0.0644284
\(617\) −7.36483e6 −0.778843 −0.389421 0.921060i \(-0.627325\pi\)
−0.389421 + 0.921060i \(0.627325\pi\)
\(618\) 0 0
\(619\) −1.30528e7 −1.36923 −0.684617 0.728903i \(-0.740031\pi\)
−0.684617 + 0.728903i \(0.740031\pi\)
\(620\) −2.04620e6 −0.213781
\(621\) 0 0
\(622\) 1.62842e6 0.168768
\(623\) 94224.4 0.00972620
\(624\) 0 0
\(625\) −4.86615e6 −0.498294
\(626\) −2.14305e6 −0.218573
\(627\) 0 0
\(628\) −1.68548e7 −1.70539
\(629\) 3.28880e6 0.331445
\(630\) 0 0
\(631\) 2.89105e6 0.289056 0.144528 0.989501i \(-0.453834\pi\)
0.144528 + 0.989501i \(0.453834\pi\)
\(632\) 3.02252e6 0.301007
\(633\) 0 0
\(634\) −176594. −0.0174482
\(635\) −659304. −0.0648860
\(636\) 0 0
\(637\) 5.77270e6 0.563677
\(638\) 219554. 0.0213545
\(639\) 0 0
\(640\) −4.74532e6 −0.457947
\(641\) 1.31397e7 1.26311 0.631555 0.775331i \(-0.282417\pi\)
0.631555 + 0.775331i \(0.282417\pi\)
\(642\) 0 0
\(643\) −1.17266e7 −1.11852 −0.559259 0.828993i \(-0.688914\pi\)
−0.559259 + 0.828993i \(0.688914\pi\)
\(644\) −2.13734e6 −0.203076
\(645\) 0 0
\(646\) −332065. −0.0313071
\(647\) −1.10662e7 −1.03929 −0.519646 0.854382i \(-0.673936\pi\)
−0.519646 + 0.854382i \(0.673936\pi\)
\(648\) 0 0
\(649\) −1.09743e7 −1.02274
\(650\) 369929. 0.0343427
\(651\) 0 0
\(652\) −2.75025e6 −0.253369
\(653\) 3.44597e6 0.316249 0.158124 0.987419i \(-0.449455\pi\)
0.158124 + 0.987419i \(0.449455\pi\)
\(654\) 0 0
\(655\) 1.39285e7 1.26853
\(656\) 2.07495e6 0.188255
\(657\) 0 0
\(658\) −504860. −0.0454576
\(659\) 9.12181e6 0.818215 0.409108 0.912486i \(-0.365840\pi\)
0.409108 + 0.912486i \(0.365840\pi\)
\(660\) 0 0
\(661\) 1.68113e7 1.49657 0.748287 0.663375i \(-0.230877\pi\)
0.748287 + 0.663375i \(0.230877\pi\)
\(662\) 1.37145e6 0.121629
\(663\) 0 0
\(664\) −2.56147e6 −0.225460
\(665\) −2.24123e6 −0.196531
\(666\) 0 0
\(667\) −1.56519e6 −0.136224
\(668\) 6.24760e6 0.541716
\(669\) 0 0
\(670\) −2.10525e6 −0.181183
\(671\) −6.74577e6 −0.578396
\(672\) 0 0
\(673\) 8.53511e6 0.726393 0.363196 0.931713i \(-0.381685\pi\)
0.363196 + 0.931713i \(0.381685\pi\)
\(674\) −702670. −0.0595802
\(675\) 0 0
\(676\) 7.24506e6 0.609783
\(677\) 912690. 0.0765335 0.0382667 0.999268i \(-0.487816\pi\)
0.0382667 + 0.999268i \(0.487816\pi\)
\(678\) 0 0
\(679\) −1.28387e6 −0.106868
\(680\) 681191. 0.0564932
\(681\) 0 0
\(682\) 384292. 0.0316374
\(683\) 4.20567e6 0.344972 0.172486 0.985012i \(-0.444820\pi\)
0.172486 + 0.985012i \(0.444820\pi\)
\(684\) 0 0
\(685\) 1.13107e7 0.921010
\(686\) 1.02386e6 0.0830677
\(687\) 0 0
\(688\) −9.22366e6 −0.742903
\(689\) 9.09751e6 0.730087
\(690\) 0 0
\(691\) 6.23030e6 0.496379 0.248190 0.968711i \(-0.420164\pi\)
0.248190 + 0.968711i \(0.420164\pi\)
\(692\) −9.80746e6 −0.778559
\(693\) 0 0
\(694\) −445670. −0.0351249
\(695\) −1.73870e7 −1.36541
\(696\) 0 0
\(697\) −610464. −0.0475968
\(698\) −2.29005e6 −0.177913
\(699\) 0 0
\(700\) −1.31713e6 −0.101598
\(701\) −1.69453e7 −1.30243 −0.651215 0.758893i \(-0.725741\pi\)
−0.651215 + 0.758893i \(0.725741\pi\)
\(702\) 0 0
\(703\) −1.60872e7 −1.22770
\(704\) −8.54673e6 −0.649933
\(705\) 0 0
\(706\) −337961. −0.0255185
\(707\) 6.25484e6 0.470617
\(708\) 0 0
\(709\) −1.87326e7 −1.39953 −0.699766 0.714372i \(-0.746712\pi\)
−0.699766 + 0.714372i \(0.746712\pi\)
\(710\) −1.06746e6 −0.0794702
\(711\) 0 0
\(712\) 140084. 0.0103559
\(713\) −2.73960e6 −0.201820
\(714\) 0 0
\(715\) −5.02106e6 −0.367308
\(716\) −1.86946e6 −0.136280
\(717\) 0 0
\(718\) −2.62536e6 −0.190054
\(719\) −2.15611e6 −0.155542 −0.0777711 0.996971i \(-0.524780\pi\)
−0.0777711 + 0.996971i \(0.524780\pi\)
\(720\) 0 0
\(721\) −3.08519e6 −0.221026
\(722\) −514154. −0.0367071
\(723\) 0 0
\(724\) −1.25714e7 −0.891326
\(725\) −964548. −0.0681520
\(726\) 0 0
\(727\) 4.87466e6 0.342065 0.171032 0.985265i \(-0.445290\pi\)
0.171032 + 0.985265i \(0.445290\pi\)
\(728\) −749677. −0.0524259
\(729\) 0 0
\(730\) 1.21707e6 0.0845295
\(731\) 2.71367e6 0.187829
\(732\) 0 0
\(733\) 5.15920e6 0.354669 0.177334 0.984151i \(-0.443253\pi\)
0.177334 + 0.984151i \(0.443253\pi\)
\(734\) 888660. 0.0608829
\(735\) 0 0
\(736\) −4.78513e6 −0.325611
\(737\) −1.65677e7 −1.12355
\(738\) 0 0
\(739\) −4.60821e6 −0.310400 −0.155200 0.987883i \(-0.549602\pi\)
−0.155200 + 0.987883i \(0.549602\pi\)
\(740\) 1.63059e7 1.09462
\(741\) 0 0
\(742\) 773020. 0.0515444
\(743\) −2.50473e7 −1.66452 −0.832261 0.554384i \(-0.812954\pi\)
−0.832261 + 0.554384i \(0.812954\pi\)
\(744\) 0 0
\(745\) −3.21763e6 −0.212396
\(746\) −3.82037e6 −0.251338
\(747\) 0 0
\(748\) 2.64878e6 0.173098
\(749\) 618870. 0.0403083
\(750\) 0 0
\(751\) −1.87279e7 −1.21168 −0.605841 0.795586i \(-0.707163\pi\)
−0.605841 + 0.795586i \(0.707163\pi\)
\(752\) 1.51606e7 0.977623
\(753\) 0 0
\(754\) −271260. −0.0173763
\(755\) −1.01146e7 −0.645778
\(756\) 0 0
\(757\) −5.50814e6 −0.349354 −0.174677 0.984626i \(-0.555888\pi\)
−0.174677 + 0.984626i \(0.555888\pi\)
\(758\) −3.65750e6 −0.231212
\(759\) 0 0
\(760\) −3.33205e6 −0.209256
\(761\) 2.67826e7 1.67645 0.838226 0.545323i \(-0.183593\pi\)
0.838226 + 0.545323i \(0.183593\pi\)
\(762\) 0 0
\(763\) 141674. 0.00881005
\(764\) −2.82390e7 −1.75032
\(765\) 0 0
\(766\) 772511. 0.0475699
\(767\) 1.35588e7 0.832209
\(768\) 0 0
\(769\) 1.76374e7 1.07552 0.537761 0.843097i \(-0.319270\pi\)
0.537761 + 0.843097i \(0.319270\pi\)
\(770\) −426642. −0.0259320
\(771\) 0 0
\(772\) −3.21512e7 −1.94158
\(773\) 6.59079e6 0.396724 0.198362 0.980129i \(-0.436438\pi\)
0.198362 + 0.980129i \(0.436438\pi\)
\(774\) 0 0
\(775\) −1.68828e6 −0.100970
\(776\) −1.90874e6 −0.113787
\(777\) 0 0
\(778\) −3.11463e6 −0.184484
\(779\) 2.98609e6 0.176303
\(780\) 0 0
\(781\) −8.40056e6 −0.492811
\(782\) 450637. 0.0263518
\(783\) 0 0
\(784\) −1.47296e7 −0.855858
\(785\) 2.39849e7 1.38920
\(786\) 0 0
\(787\) 1.83730e7 1.05741 0.528704 0.848806i \(-0.322678\pi\)
0.528704 + 0.848806i \(0.322678\pi\)
\(788\) −2.70692e7 −1.55296
\(789\) 0 0
\(790\) −2.12522e6 −0.121153
\(791\) 3.94341e6 0.224094
\(792\) 0 0
\(793\) 8.33444e6 0.470645
\(794\) −1.69441e6 −0.0953824
\(795\) 0 0
\(796\) −8.93356e6 −0.499737
\(797\) 3.06769e7 1.71067 0.855334 0.518078i \(-0.173352\pi\)
0.855334 + 0.518078i \(0.173352\pi\)
\(798\) 0 0
\(799\) −4.46035e6 −0.247174
\(800\) −2.94884e6 −0.162902
\(801\) 0 0
\(802\) 622784. 0.0341902
\(803\) 9.57797e6 0.524185
\(804\) 0 0
\(805\) 3.04151e6 0.165424
\(806\) −474796. −0.0257436
\(807\) 0 0
\(808\) 9.29912e6 0.501088
\(809\) 2.81720e6 0.151338 0.0756688 0.997133i \(-0.475891\pi\)
0.0756688 + 0.997133i \(0.475891\pi\)
\(810\) 0 0
\(811\) −1.57819e7 −0.842574 −0.421287 0.906927i \(-0.638421\pi\)
−0.421287 + 0.906927i \(0.638421\pi\)
\(812\) 965825. 0.0514053
\(813\) 0 0
\(814\) −3.06238e6 −0.161994
\(815\) 3.91371e6 0.206393
\(816\) 0 0
\(817\) −1.32739e7 −0.695735
\(818\) 1.15195e6 0.0601935
\(819\) 0 0
\(820\) −3.02668e6 −0.157192
\(821\) 2.40563e6 0.124558 0.0622789 0.998059i \(-0.480163\pi\)
0.0622789 + 0.998059i \(0.480163\pi\)
\(822\) 0 0
\(823\) 4.47397e6 0.230247 0.115123 0.993351i \(-0.463274\pi\)
0.115123 + 0.993351i \(0.463274\pi\)
\(824\) −4.58678e6 −0.235337
\(825\) 0 0
\(826\) 1.15210e6 0.0587542
\(827\) −2.36054e7 −1.20018 −0.600091 0.799932i \(-0.704869\pi\)
−0.600091 + 0.799932i \(0.704869\pi\)
\(828\) 0 0
\(829\) −7.52668e6 −0.380380 −0.190190 0.981747i \(-0.560910\pi\)
−0.190190 + 0.981747i \(0.560910\pi\)
\(830\) 1.80105e6 0.0907464
\(831\) 0 0
\(832\) 1.05595e7 0.528855
\(833\) 4.33356e6 0.216388
\(834\) 0 0
\(835\) −8.89056e6 −0.441279
\(836\) −1.29565e7 −0.641169
\(837\) 0 0
\(838\) 3.89754e6 0.191726
\(839\) 7.28878e6 0.357478 0.178739 0.983896i \(-0.442798\pi\)
0.178739 + 0.983896i \(0.442798\pi\)
\(840\) 0 0
\(841\) 707281. 0.0344828
\(842\) 848249. 0.0412328
\(843\) 0 0
\(844\) 2.87204e7 1.38782
\(845\) −1.03100e7 −0.496725
\(846\) 0 0
\(847\) 2.56023e6 0.122623
\(848\) −2.32132e7 −1.10853
\(849\) 0 0
\(850\) 277705. 0.0131837
\(851\) 2.18315e7 1.03338
\(852\) 0 0
\(853\) 2.18767e7 1.02946 0.514729 0.857353i \(-0.327893\pi\)
0.514729 + 0.857353i \(0.327893\pi\)
\(854\) 708182. 0.0332277
\(855\) 0 0
\(856\) 920079. 0.0429181
\(857\) 1.62738e7 0.756897 0.378448 0.925622i \(-0.376458\pi\)
0.378448 + 0.925622i \(0.376458\pi\)
\(858\) 0 0
\(859\) 1.50220e7 0.694615 0.347308 0.937751i \(-0.387096\pi\)
0.347308 + 0.937751i \(0.387096\pi\)
\(860\) 1.34544e7 0.620321
\(861\) 0 0
\(862\) −4.59269e6 −0.210523
\(863\) −2.24278e7 −1.02508 −0.512542 0.858662i \(-0.671296\pi\)
−0.512542 + 0.858662i \(0.671296\pi\)
\(864\) 0 0
\(865\) 1.39564e7 0.634209
\(866\) 586650. 0.0265818
\(867\) 0 0
\(868\) 1.69051e6 0.0761587
\(869\) −1.67248e7 −0.751298
\(870\) 0 0
\(871\) 2.04695e7 0.914244
\(872\) 210628. 0.00938047
\(873\) 0 0
\(874\) −2.20430e6 −0.0976093
\(875\) 6.98136e6 0.308262
\(876\) 0 0
\(877\) 1.25450e7 0.550771 0.275385 0.961334i \(-0.411195\pi\)
0.275385 + 0.961334i \(0.411195\pi\)
\(878\) 3.05399e6 0.133700
\(879\) 0 0
\(880\) 1.28117e7 0.557701
\(881\) −7.60341e6 −0.330042 −0.165021 0.986290i \(-0.552769\pi\)
−0.165021 + 0.986290i \(0.552769\pi\)
\(882\) 0 0
\(883\) 2.15220e7 0.928923 0.464462 0.885593i \(-0.346248\pi\)
0.464462 + 0.885593i \(0.346248\pi\)
\(884\) −3.27258e6 −0.140851
\(885\) 0 0
\(886\) 1.16848e6 0.0500079
\(887\) 333279. 0.0142232 0.00711162 0.999975i \(-0.497736\pi\)
0.00711162 + 0.999975i \(0.497736\pi\)
\(888\) 0 0
\(889\) 544700. 0.0231155
\(890\) −98497.2 −0.00416820
\(891\) 0 0
\(892\) −1.51937e7 −0.639368
\(893\) 2.18178e7 0.915552
\(894\) 0 0
\(895\) 2.66031e6 0.111013
\(896\) 3.92046e6 0.163142
\(897\) 0 0
\(898\) 1.76661e6 0.0731055
\(899\) 1.23798e6 0.0510873
\(900\) 0 0
\(901\) 6.82950e6 0.280270
\(902\) 568435. 0.0232629
\(903\) 0 0
\(904\) 5.86270e6 0.238603
\(905\) 1.78895e7 0.726068
\(906\) 0 0
\(907\) 1.65687e7 0.668759 0.334380 0.942439i \(-0.391473\pi\)
0.334380 + 0.942439i \(0.391473\pi\)
\(908\) −1.74285e7 −0.701529
\(909\) 0 0
\(910\) 527119. 0.0211011
\(911\) −1.30648e7 −0.521561 −0.260781 0.965398i \(-0.583980\pi\)
−0.260781 + 0.965398i \(0.583980\pi\)
\(912\) 0 0
\(913\) 1.41737e7 0.562737
\(914\) 7.32693e6 0.290106
\(915\) 0 0
\(916\) −1.37608e7 −0.541884
\(917\) −1.15074e7 −0.451911
\(918\) 0 0
\(919\) −7.30805e6 −0.285439 −0.142719 0.989763i \(-0.545585\pi\)
−0.142719 + 0.989763i \(0.545585\pi\)
\(920\) 4.52184e6 0.176135
\(921\) 0 0
\(922\) 1.78596e6 0.0691902
\(923\) 1.03789e7 0.401004
\(924\) 0 0
\(925\) 1.34537e7 0.516996
\(926\) −318069. −0.0121897
\(927\) 0 0
\(928\) 2.16232e6 0.0824231
\(929\) −2.69866e7 −1.02591 −0.512955 0.858416i \(-0.671449\pi\)
−0.512955 + 0.858416i \(0.671449\pi\)
\(930\) 0 0
\(931\) −2.11976e7 −0.801518
\(932\) 4.56881e7 1.72291
\(933\) 0 0
\(934\) 2.40407e6 0.0901737
\(935\) −3.76930e6 −0.141004
\(936\) 0 0
\(937\) −2.15536e7 −0.801993 −0.400997 0.916080i \(-0.631336\pi\)
−0.400997 + 0.916080i \(0.631336\pi\)
\(938\) 1.73930e6 0.0645459
\(939\) 0 0
\(940\) −2.21144e7 −0.816311
\(941\) 1.22668e7 0.451602 0.225801 0.974173i \(-0.427500\pi\)
0.225801 + 0.974173i \(0.427500\pi\)
\(942\) 0 0
\(943\) −4.05234e6 −0.148398
\(944\) −3.45966e7 −1.26358
\(945\) 0 0
\(946\) −2.52684e6 −0.0918014
\(947\) −5.13280e7 −1.85986 −0.929928 0.367741i \(-0.880131\pi\)
−0.929928 + 0.367741i \(0.880131\pi\)
\(948\) 0 0
\(949\) −1.18336e7 −0.426533
\(950\) −1.35840e6 −0.0488335
\(951\) 0 0
\(952\) −562782. −0.0201255
\(953\) 3.30301e7 1.17809 0.589044 0.808101i \(-0.299504\pi\)
0.589044 + 0.808101i \(0.299504\pi\)
\(954\) 0 0
\(955\) 4.01852e7 1.42580
\(956\) 4.72851e7 1.67332
\(957\) 0 0
\(958\) −1.29390e6 −0.0455498
\(959\) −9.34463e6 −0.328107
\(960\) 0 0
\(961\) −2.64623e7 −0.924312
\(962\) 3.78359e6 0.131815
\(963\) 0 0
\(964\) 2.50245e7 0.867307
\(965\) 4.57524e7 1.58160
\(966\) 0 0
\(967\) −7.77281e6 −0.267308 −0.133654 0.991028i \(-0.542671\pi\)
−0.133654 + 0.991028i \(0.542671\pi\)
\(968\) 3.80632e6 0.130562
\(969\) 0 0
\(970\) 1.34209e6 0.0457986
\(971\) −2.31111e7 −0.786635 −0.393317 0.919403i \(-0.628673\pi\)
−0.393317 + 0.919403i \(0.628673\pi\)
\(972\) 0 0
\(973\) 1.43647e7 0.486422
\(974\) 6.87175e6 0.232097
\(975\) 0 0
\(976\) −2.12662e7 −0.714603
\(977\) −1.04792e7 −0.351231 −0.175616 0.984459i \(-0.556192\pi\)
−0.175616 + 0.984459i \(0.556192\pi\)
\(978\) 0 0
\(979\) −775143. −0.0258479
\(980\) 2.14858e7 0.714638
\(981\) 0 0
\(982\) 1.61796e6 0.0535413
\(983\) −3.94189e7 −1.30113 −0.650565 0.759450i \(-0.725468\pi\)
−0.650565 + 0.759450i \(0.725468\pi\)
\(984\) 0 0
\(985\) 3.85204e7 1.26503
\(986\) −203635. −0.00667052
\(987\) 0 0
\(988\) 1.60079e7 0.521724
\(989\) 1.80137e7 0.585615
\(990\) 0 0
\(991\) −4.53294e7 −1.46621 −0.733104 0.680117i \(-0.761929\pi\)
−0.733104 + 0.680117i \(0.761929\pi\)
\(992\) 3.78477e6 0.122113
\(993\) 0 0
\(994\) 881905. 0.0283110
\(995\) 1.27128e7 0.407083
\(996\) 0 0
\(997\) 4.74723e7 1.51253 0.756263 0.654268i \(-0.227023\pi\)
0.756263 + 0.654268i \(0.227023\pi\)
\(998\) −304247. −0.00966942
\(999\) 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 261.6.a.a.1.3 4
3.2 odd 2 29.6.a.a.1.2 4
12.11 even 2 464.6.a.i.1.1 4
15.14 odd 2 725.6.a.a.1.3 4
87.86 odd 2 841.6.a.a.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.6.a.a.1.2 4 3.2 odd 2
261.6.a.a.1.3 4 1.1 even 1 trivial
464.6.a.i.1.1 4 12.11 even 2
725.6.a.a.1.3 4 15.14 odd 2
841.6.a.a.1.3 4 87.86 odd 2