Properties

Label 825.6.a.y.1.7
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.317411\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.31741 q^{2} +9.00000 q^{3} -30.2644 q^{4} +11.8567 q^{6} -87.4786 q^{7} -82.0279 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+1.31741 q^{2} +9.00000 q^{3} -30.2644 q^{4} +11.8567 q^{6} -87.4786 q^{7} -82.0279 q^{8} +81.0000 q^{9} +121.000 q^{11} -272.380 q^{12} +521.292 q^{13} -115.245 q^{14} +860.397 q^{16} -68.1523 q^{17} +106.710 q^{18} -1588.76 q^{19} -787.308 q^{21} +159.407 q^{22} +191.904 q^{23} -738.251 q^{24} +686.756 q^{26} +729.000 q^{27} +2647.49 q^{28} -4252.98 q^{29} -83.6771 q^{31} +3758.39 q^{32} +1089.00 q^{33} -89.7846 q^{34} -2451.42 q^{36} -14445.4 q^{37} -2093.06 q^{38} +4691.63 q^{39} +6151.94 q^{41} -1037.21 q^{42} +1408.62 q^{43} -3662.00 q^{44} +252.816 q^{46} -14797.5 q^{47} +7743.57 q^{48} -9154.49 q^{49} -613.371 q^{51} -15776.6 q^{52} +10938.4 q^{53} +960.393 q^{54} +7175.69 q^{56} -14298.9 q^{57} -5602.92 q^{58} -16099.1 q^{59} +41095.0 q^{61} -110.237 q^{62} -7085.77 q^{63} -22581.4 q^{64} +1434.66 q^{66} -1254.02 q^{67} +2062.59 q^{68} +1727.14 q^{69} +47638.4 q^{71} -6644.26 q^{72} -7358.01 q^{73} -19030.5 q^{74} +48083.1 q^{76} -10584.9 q^{77} +6180.81 q^{78} +90184.5 q^{79} +6561.00 q^{81} +8104.64 q^{82} +10857.0 q^{83} +23827.4 q^{84} +1855.73 q^{86} -38276.8 q^{87} -9925.37 q^{88} +112869. q^{89} -45601.9 q^{91} -5807.86 q^{92} -753.094 q^{93} -19494.4 q^{94} +33825.5 q^{96} +96993.9 q^{97} -12060.2 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.31741 0.232888 0.116444 0.993197i \(-0.462851\pi\)
0.116444 + 0.993197i \(0.462851\pi\)
\(3\) 9.00000 0.577350
\(4\) −30.2644 −0.945763
\(5\) 0 0
\(6\) 11.8567 0.134458
\(7\) −87.4786 −0.674772 −0.337386 0.941366i \(-0.609543\pi\)
−0.337386 + 0.941366i \(0.609543\pi\)
\(8\) −82.0279 −0.453144
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −272.380 −0.546037
\(13\) 521.292 0.855506 0.427753 0.903896i \(-0.359305\pi\)
0.427753 + 0.903896i \(0.359305\pi\)
\(14\) −115.245 −0.157146
\(15\) 0 0
\(16\) 860.397 0.840232
\(17\) −68.1523 −0.0571950 −0.0285975 0.999591i \(-0.509104\pi\)
−0.0285975 + 0.999591i \(0.509104\pi\)
\(18\) 106.710 0.0776292
\(19\) −1588.76 −1.00966 −0.504830 0.863219i \(-0.668445\pi\)
−0.504830 + 0.863219i \(0.668445\pi\)
\(20\) 0 0
\(21\) −787.308 −0.389580
\(22\) 159.407 0.0702183
\(23\) 191.904 0.0756422 0.0378211 0.999285i \(-0.487958\pi\)
0.0378211 + 0.999285i \(0.487958\pi\)
\(24\) −738.251 −0.261623
\(25\) 0 0
\(26\) 686.756 0.199237
\(27\) 729.000 0.192450
\(28\) 2647.49 0.638174
\(29\) −4252.98 −0.939071 −0.469535 0.882914i \(-0.655579\pi\)
−0.469535 + 0.882914i \(0.655579\pi\)
\(30\) 0 0
\(31\) −83.6771 −0.0156388 −0.00781938 0.999969i \(-0.502489\pi\)
−0.00781938 + 0.999969i \(0.502489\pi\)
\(32\) 3758.39 0.648824
\(33\) 1089.00 0.174078
\(34\) −89.7846 −0.0133200
\(35\) 0 0
\(36\) −2451.42 −0.315254
\(37\) −14445.4 −1.73470 −0.867352 0.497695i \(-0.834180\pi\)
−0.867352 + 0.497695i \(0.834180\pi\)
\(38\) −2093.06 −0.235138
\(39\) 4691.63 0.493926
\(40\) 0 0
\(41\) 6151.94 0.571548 0.285774 0.958297i \(-0.407749\pi\)
0.285774 + 0.958297i \(0.407749\pi\)
\(42\) −1037.21 −0.0907283
\(43\) 1408.62 0.116177 0.0580886 0.998311i \(-0.481499\pi\)
0.0580886 + 0.998311i \(0.481499\pi\)
\(44\) −3662.00 −0.285158
\(45\) 0 0
\(46\) 252.816 0.0176161
\(47\) −14797.5 −0.977110 −0.488555 0.872533i \(-0.662476\pi\)
−0.488555 + 0.872533i \(0.662476\pi\)
\(48\) 7743.57 0.485108
\(49\) −9154.49 −0.544683
\(50\) 0 0
\(51\) −613.371 −0.0330216
\(52\) −15776.6 −0.809106
\(53\) 10938.4 0.534891 0.267445 0.963573i \(-0.413820\pi\)
0.267445 + 0.963573i \(0.413820\pi\)
\(54\) 960.393 0.0448192
\(55\) 0 0
\(56\) 7175.69 0.305769
\(57\) −14298.9 −0.582928
\(58\) −5602.92 −0.218698
\(59\) −16099.1 −0.602103 −0.301052 0.953608i \(-0.597338\pi\)
−0.301052 + 0.953608i \(0.597338\pi\)
\(60\) 0 0
\(61\) 41095.0 1.41405 0.707025 0.707189i \(-0.250037\pi\)
0.707025 + 0.707189i \(0.250037\pi\)
\(62\) −110.237 −0.00364208
\(63\) −7085.77 −0.224924
\(64\) −22581.4 −0.689129
\(65\) 0 0
\(66\) 1434.66 0.0405405
\(67\) −1254.02 −0.0341286 −0.0170643 0.999854i \(-0.505432\pi\)
−0.0170643 + 0.999854i \(0.505432\pi\)
\(68\) 2062.59 0.0540930
\(69\) 1727.14 0.0436721
\(70\) 0 0
\(71\) 47638.4 1.12153 0.560765 0.827975i \(-0.310507\pi\)
0.560765 + 0.827975i \(0.310507\pi\)
\(72\) −6644.26 −0.151048
\(73\) −7358.01 −0.161604 −0.0808022 0.996730i \(-0.525748\pi\)
−0.0808022 + 0.996730i \(0.525748\pi\)
\(74\) −19030.5 −0.403991
\(75\) 0 0
\(76\) 48083.1 0.954900
\(77\) −10584.9 −0.203451
\(78\) 6180.81 0.115029
\(79\) 90184.5 1.62579 0.812895 0.582411i \(-0.197890\pi\)
0.812895 + 0.582411i \(0.197890\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 8104.64 0.133106
\(83\) 10857.0 0.172987 0.0864935 0.996252i \(-0.472434\pi\)
0.0864935 + 0.996252i \(0.472434\pi\)
\(84\) 23827.4 0.368450
\(85\) 0 0
\(86\) 1855.73 0.0270563
\(87\) −38276.8 −0.542173
\(88\) −9925.37 −0.136628
\(89\) 112869. 1.51043 0.755216 0.655476i \(-0.227532\pi\)
0.755216 + 0.655476i \(0.227532\pi\)
\(90\) 0 0
\(91\) −45601.9 −0.577271
\(92\) −5807.86 −0.0715396
\(93\) −753.094 −0.00902905
\(94\) −19494.4 −0.227557
\(95\) 0 0
\(96\) 33825.5 0.374599
\(97\) 96993.9 1.04668 0.523341 0.852123i \(-0.324685\pi\)
0.523341 + 0.852123i \(0.324685\pi\)
\(98\) −12060.2 −0.126850
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −55750.8 −0.543810 −0.271905 0.962324i \(-0.587654\pi\)
−0.271905 + 0.962324i \(0.587654\pi\)
\(102\) −808.062 −0.00769031
\(103\) 107469. 0.998139 0.499069 0.866562i \(-0.333675\pi\)
0.499069 + 0.866562i \(0.333675\pi\)
\(104\) −42760.5 −0.387667
\(105\) 0 0
\(106\) 14410.4 0.124569
\(107\) 25673.1 0.216780 0.108390 0.994108i \(-0.465430\pi\)
0.108390 + 0.994108i \(0.465430\pi\)
\(108\) −22062.8 −0.182012
\(109\) 98829.6 0.796748 0.398374 0.917223i \(-0.369575\pi\)
0.398374 + 0.917223i \(0.369575\pi\)
\(110\) 0 0
\(111\) −130009. −1.00153
\(112\) −75266.4 −0.566965
\(113\) −12055.0 −0.0888120 −0.0444060 0.999014i \(-0.514140\pi\)
−0.0444060 + 0.999014i \(0.514140\pi\)
\(114\) −18837.5 −0.135757
\(115\) 0 0
\(116\) 128714. 0.888139
\(117\) 42224.7 0.285169
\(118\) −21209.1 −0.140222
\(119\) 5961.87 0.0385936
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 54139.1 0.329315
\(123\) 55367.5 0.329983
\(124\) 2532.44 0.0147906
\(125\) 0 0
\(126\) −9334.87 −0.0523820
\(127\) 57953.4 0.318838 0.159419 0.987211i \(-0.449038\pi\)
0.159419 + 0.987211i \(0.449038\pi\)
\(128\) −150017. −0.809313
\(129\) 12677.5 0.0670750
\(130\) 0 0
\(131\) −124350. −0.633095 −0.316548 0.948577i \(-0.602524\pi\)
−0.316548 + 0.948577i \(0.602524\pi\)
\(132\) −32958.0 −0.164636
\(133\) 138983. 0.681291
\(134\) −1652.06 −0.00794812
\(135\) 0 0
\(136\) 5590.39 0.0259176
\(137\) 408683. 1.86031 0.930154 0.367170i \(-0.119673\pi\)
0.930154 + 0.367170i \(0.119673\pi\)
\(138\) 2275.35 0.0101707
\(139\) 378463. 1.66145 0.830723 0.556686i \(-0.187927\pi\)
0.830723 + 0.556686i \(0.187927\pi\)
\(140\) 0 0
\(141\) −133177. −0.564135
\(142\) 62759.4 0.261191
\(143\) 63076.4 0.257945
\(144\) 69692.2 0.280077
\(145\) 0 0
\(146\) −9693.52 −0.0376357
\(147\) −82390.4 −0.314473
\(148\) 437182. 1.64062
\(149\) −271317. −1.00118 −0.500588 0.865686i \(-0.666883\pi\)
−0.500588 + 0.865686i \(0.666883\pi\)
\(150\) 0 0
\(151\) −146748. −0.523758 −0.261879 0.965101i \(-0.584342\pi\)
−0.261879 + 0.965101i \(0.584342\pi\)
\(152\) 130323. 0.457522
\(153\) −5520.34 −0.0190650
\(154\) −13944.7 −0.0473813
\(155\) 0 0
\(156\) −141990. −0.467138
\(157\) 365562. 1.18362 0.591810 0.806077i \(-0.298414\pi\)
0.591810 + 0.806077i \(0.298414\pi\)
\(158\) 118810. 0.378626
\(159\) 98445.8 0.308819
\(160\) 0 0
\(161\) −16787.5 −0.0510412
\(162\) 8643.54 0.0258764
\(163\) −16695.2 −0.0492180 −0.0246090 0.999697i \(-0.507834\pi\)
−0.0246090 + 0.999697i \(0.507834\pi\)
\(164\) −186185. −0.540549
\(165\) 0 0
\(166\) 14303.1 0.0402865
\(167\) 568590. 1.57764 0.788821 0.614623i \(-0.210692\pi\)
0.788821 + 0.614623i \(0.210692\pi\)
\(168\) 64581.2 0.176536
\(169\) −99547.4 −0.268110
\(170\) 0 0
\(171\) −128690. −0.336554
\(172\) −42630.9 −0.109876
\(173\) −552203. −1.40276 −0.701380 0.712788i \(-0.747432\pi\)
−0.701380 + 0.712788i \(0.747432\pi\)
\(174\) −50426.3 −0.126265
\(175\) 0 0
\(176\) 104108. 0.253339
\(177\) −144892. −0.347624
\(178\) 148695. 0.351761
\(179\) −179962. −0.419807 −0.209903 0.977722i \(-0.567315\pi\)
−0.209903 + 0.977722i \(0.567315\pi\)
\(180\) 0 0
\(181\) 612439. 1.38953 0.694763 0.719239i \(-0.255509\pi\)
0.694763 + 0.719239i \(0.255509\pi\)
\(182\) −60076.5 −0.134439
\(183\) 369855. 0.816402
\(184\) −15741.5 −0.0342768
\(185\) 0 0
\(186\) −992.135 −0.00210275
\(187\) −8246.43 −0.0172449
\(188\) 447838. 0.924115
\(189\) −63771.9 −0.129860
\(190\) 0 0
\(191\) 65407.8 0.129732 0.0648659 0.997894i \(-0.479338\pi\)
0.0648659 + 0.997894i \(0.479338\pi\)
\(192\) −203232. −0.397869
\(193\) 661050. 1.27744 0.638721 0.769439i \(-0.279464\pi\)
0.638721 + 0.769439i \(0.279464\pi\)
\(194\) 127781. 0.243759
\(195\) 0 0
\(196\) 277055. 0.515141
\(197\) −590186. −1.08349 −0.541743 0.840544i \(-0.682235\pi\)
−0.541743 + 0.840544i \(0.682235\pi\)
\(198\) 12911.9 0.0234061
\(199\) −342750. −0.613543 −0.306771 0.951783i \(-0.599249\pi\)
−0.306771 + 0.951783i \(0.599249\pi\)
\(200\) 0 0
\(201\) −11286.2 −0.0197041
\(202\) −73446.7 −0.126647
\(203\) 372045. 0.633658
\(204\) 18563.3 0.0312306
\(205\) 0 0
\(206\) 141581. 0.232454
\(207\) 15544.2 0.0252141
\(208\) 448518. 0.718823
\(209\) −192241. −0.304424
\(210\) 0 0
\(211\) −214655. −0.331921 −0.165960 0.986132i \(-0.553072\pi\)
−0.165960 + 0.986132i \(0.553072\pi\)
\(212\) −331045. −0.505880
\(213\) 428746. 0.647516
\(214\) 33822.1 0.0504854
\(215\) 0 0
\(216\) −59798.3 −0.0872076
\(217\) 7319.96 0.0105526
\(218\) 130199. 0.185553
\(219\) −66222.1 −0.0933023
\(220\) 0 0
\(221\) −35527.3 −0.0489307
\(222\) −171275. −0.233244
\(223\) −118747. −0.159904 −0.0799521 0.996799i \(-0.525477\pi\)
−0.0799521 + 0.996799i \(0.525477\pi\)
\(224\) −328779. −0.437808
\(225\) 0 0
\(226\) −15881.4 −0.0206832
\(227\) 2183.16 0.00281204 0.00140602 0.999999i \(-0.499552\pi\)
0.00140602 + 0.999999i \(0.499552\pi\)
\(228\) 432747. 0.551312
\(229\) 123704. 0.155881 0.0779407 0.996958i \(-0.475166\pi\)
0.0779407 + 0.996958i \(0.475166\pi\)
\(230\) 0 0
\(231\) −95264.2 −0.117463
\(232\) 348863. 0.425534
\(233\) 1.16587e6 1.40689 0.703446 0.710748i \(-0.251644\pi\)
0.703446 + 0.710748i \(0.251644\pi\)
\(234\) 55627.3 0.0664122
\(235\) 0 0
\(236\) 487229. 0.569447
\(237\) 811661. 0.938650
\(238\) 7854.24 0.00898797
\(239\) 770619. 0.872659 0.436330 0.899787i \(-0.356278\pi\)
0.436330 + 0.899787i \(0.356278\pi\)
\(240\) 0 0
\(241\) −1.42541e6 −1.58087 −0.790434 0.612547i \(-0.790145\pi\)
−0.790434 + 0.612547i \(0.790145\pi\)
\(242\) 19288.2 0.0211716
\(243\) 59049.0 0.0641500
\(244\) −1.24372e6 −1.33736
\(245\) 0 0
\(246\) 72941.7 0.0768490
\(247\) −828211. −0.863771
\(248\) 6863.86 0.00708662
\(249\) 97712.7 0.0998741
\(250\) 0 0
\(251\) −668745. −0.670002 −0.335001 0.942218i \(-0.608737\pi\)
−0.335001 + 0.942218i \(0.608737\pi\)
\(252\) 214447. 0.212725
\(253\) 23220.4 0.0228070
\(254\) 76348.5 0.0742534
\(255\) 0 0
\(256\) 524969. 0.500650
\(257\) 1.08696e6 1.02655 0.513276 0.858223i \(-0.328432\pi\)
0.513276 + 0.858223i \(0.328432\pi\)
\(258\) 16701.5 0.0156209
\(259\) 1.26366e6 1.17053
\(260\) 0 0
\(261\) −344491. −0.313024
\(262\) −163821. −0.147440
\(263\) 865881. 0.771914 0.385957 0.922517i \(-0.373871\pi\)
0.385957 + 0.922517i \(0.373871\pi\)
\(264\) −89328.3 −0.0788823
\(265\) 0 0
\(266\) 183098. 0.158664
\(267\) 1.01582e6 0.872049
\(268\) 37952.2 0.0322775
\(269\) 455847. 0.384094 0.192047 0.981386i \(-0.438487\pi\)
0.192047 + 0.981386i \(0.438487\pi\)
\(270\) 0 0
\(271\) 1.93670e6 1.60191 0.800955 0.598725i \(-0.204326\pi\)
0.800955 + 0.598725i \(0.204326\pi\)
\(272\) −58638.1 −0.0480571
\(273\) −410417. −0.333288
\(274\) 538403. 0.433243
\(275\) 0 0
\(276\) −52270.8 −0.0413034
\(277\) −1.67891e6 −1.31470 −0.657352 0.753584i \(-0.728323\pi\)
−0.657352 + 0.753584i \(0.728323\pi\)
\(278\) 498591. 0.386930
\(279\) −6777.85 −0.00521292
\(280\) 0 0
\(281\) −1.66433e6 −1.25740 −0.628700 0.777648i \(-0.716413\pi\)
−0.628700 + 0.777648i \(0.716413\pi\)
\(282\) −175449. −0.131380
\(283\) 529051. 0.392673 0.196337 0.980537i \(-0.437095\pi\)
0.196337 + 0.980537i \(0.437095\pi\)
\(284\) −1.44175e6 −1.06070
\(285\) 0 0
\(286\) 83097.5 0.0600721
\(287\) −538163. −0.385664
\(288\) 304430. 0.216275
\(289\) −1.41521e6 −0.996729
\(290\) 0 0
\(291\) 872945. 0.604302
\(292\) 222686. 0.152839
\(293\) −1.17586e6 −0.800178 −0.400089 0.916476i \(-0.631021\pi\)
−0.400089 + 0.916476i \(0.631021\pi\)
\(294\) −108542. −0.0732369
\(295\) 0 0
\(296\) 1.18493e6 0.786071
\(297\) 88209.0 0.0580259
\(298\) −357436. −0.233162
\(299\) 100038. 0.0647123
\(300\) 0 0
\(301\) −123224. −0.0783931
\(302\) −193328. −0.121977
\(303\) −501757. −0.313969
\(304\) −1.36697e6 −0.848349
\(305\) 0 0
\(306\) −7272.55 −0.00444000
\(307\) 1.53089e6 0.927037 0.463518 0.886087i \(-0.346587\pi\)
0.463518 + 0.886087i \(0.346587\pi\)
\(308\) 320346. 0.192417
\(309\) 967223. 0.576276
\(310\) 0 0
\(311\) −1.11621e6 −0.654400 −0.327200 0.944955i \(-0.606105\pi\)
−0.327200 + 0.944955i \(0.606105\pi\)
\(312\) −384844. −0.223820
\(313\) 864587. 0.498824 0.249412 0.968397i \(-0.419763\pi\)
0.249412 + 0.968397i \(0.419763\pi\)
\(314\) 481596. 0.275651
\(315\) 0 0
\(316\) −2.72938e6 −1.53761
\(317\) −1.80604e6 −1.00943 −0.504717 0.863285i \(-0.668403\pi\)
−0.504717 + 0.863285i \(0.668403\pi\)
\(318\) 129694. 0.0719202
\(319\) −514611. −0.283140
\(320\) 0 0
\(321\) 231058. 0.125158
\(322\) −22116.0 −0.0118869
\(323\) 108278. 0.0577476
\(324\) −198565. −0.105085
\(325\) 0 0
\(326\) −21994.5 −0.0114623
\(327\) 889466. 0.460003
\(328\) −504630. −0.258994
\(329\) 1.29446e6 0.659326
\(330\) 0 0
\(331\) 1.85701e6 0.931634 0.465817 0.884881i \(-0.345761\pi\)
0.465817 + 0.884881i \(0.345761\pi\)
\(332\) −328580. −0.163605
\(333\) −1.17008e6 −0.578235
\(334\) 749068. 0.367413
\(335\) 0 0
\(336\) −677397. −0.327337
\(337\) 2.18068e6 1.04597 0.522983 0.852343i \(-0.324819\pi\)
0.522983 + 0.852343i \(0.324819\pi\)
\(338\) −131145. −0.0624395
\(339\) −108495. −0.0512757
\(340\) 0 0
\(341\) −10124.9 −0.00471527
\(342\) −169538. −0.0783792
\(343\) 2.27108e6 1.04231
\(344\) −115546. −0.0526451
\(345\) 0 0
\(346\) −727478. −0.326685
\(347\) 2.59930e6 1.15886 0.579432 0.815021i \(-0.303274\pi\)
0.579432 + 0.815021i \(0.303274\pi\)
\(348\) 1.15843e6 0.512767
\(349\) 1.17216e6 0.515138 0.257569 0.966260i \(-0.417079\pi\)
0.257569 + 0.966260i \(0.417079\pi\)
\(350\) 0 0
\(351\) 380022. 0.164642
\(352\) 454765. 0.195628
\(353\) 1.57881e6 0.674364 0.337182 0.941440i \(-0.390526\pi\)
0.337182 + 0.941440i \(0.390526\pi\)
\(354\) −190882. −0.0809574
\(355\) 0 0
\(356\) −3.41593e6 −1.42851
\(357\) 53656.8 0.0222820
\(358\) −237085. −0.0977678
\(359\) 3.19520e6 1.30847 0.654233 0.756293i \(-0.272992\pi\)
0.654233 + 0.756293i \(0.272992\pi\)
\(360\) 0 0
\(361\) 48073.9 0.0194152
\(362\) 806834. 0.323603
\(363\) 131769. 0.0524864
\(364\) 1.38012e6 0.545962
\(365\) 0 0
\(366\) 487252. 0.190130
\(367\) 2.52046e6 0.976819 0.488409 0.872615i \(-0.337577\pi\)
0.488409 + 0.872615i \(0.337577\pi\)
\(368\) 165114. 0.0635570
\(369\) 498307. 0.190516
\(370\) 0 0
\(371\) −956879. −0.360929
\(372\) 22792.0 0.00853934
\(373\) 1.66879e6 0.621053 0.310527 0.950565i \(-0.399495\pi\)
0.310527 + 0.950565i \(0.399495\pi\)
\(374\) −10863.9 −0.00401614
\(375\) 0 0
\(376\) 1.21381e6 0.442772
\(377\) −2.21705e6 −0.803380
\(378\) −84013.9 −0.0302428
\(379\) −3.13030e6 −1.11941 −0.559704 0.828692i \(-0.689085\pi\)
−0.559704 + 0.828692i \(0.689085\pi\)
\(380\) 0 0
\(381\) 521581. 0.184081
\(382\) 86169.0 0.0302129
\(383\) 4.30969e6 1.50124 0.750618 0.660737i \(-0.229756\pi\)
0.750618 + 0.660737i \(0.229756\pi\)
\(384\) −1.35016e6 −0.467257
\(385\) 0 0
\(386\) 870875. 0.297500
\(387\) 114098. 0.0387258
\(388\) −2.93546e6 −0.989914
\(389\) −2.45755e6 −0.823432 −0.411716 0.911312i \(-0.635071\pi\)
−0.411716 + 0.911312i \(0.635071\pi\)
\(390\) 0 0
\(391\) −13078.7 −0.00432636
\(392\) 750923. 0.246820
\(393\) −1.11915e6 −0.365518
\(394\) −777518. −0.252330
\(395\) 0 0
\(396\) −296622. −0.0950528
\(397\) 5.93203e6 1.88898 0.944489 0.328544i \(-0.106558\pi\)
0.944489 + 0.328544i \(0.106558\pi\)
\(398\) −451543. −0.142887
\(399\) 1.25085e6 0.393343
\(400\) 0 0
\(401\) 4.56758e6 1.41849 0.709243 0.704964i \(-0.249037\pi\)
0.709243 + 0.704964i \(0.249037\pi\)
\(402\) −14868.6 −0.00458885
\(403\) −43620.2 −0.0133791
\(404\) 1.68727e6 0.514316
\(405\) 0 0
\(406\) 490136. 0.147571
\(407\) −1.74789e6 −0.523033
\(408\) 50313.5 0.0149635
\(409\) −5.87934e6 −1.73788 −0.868941 0.494915i \(-0.835199\pi\)
−0.868941 + 0.494915i \(0.835199\pi\)
\(410\) 0 0
\(411\) 3.67814e6 1.07405
\(412\) −3.25249e6 −0.944003
\(413\) 1.40833e6 0.406282
\(414\) 20478.1 0.00587205
\(415\) 0 0
\(416\) 1.95922e6 0.555072
\(417\) 3.40616e6 0.959236
\(418\) −253260. −0.0708966
\(419\) 6.12464e6 1.70430 0.852149 0.523299i \(-0.175299\pi\)
0.852149 + 0.523299i \(0.175299\pi\)
\(420\) 0 0
\(421\) −2.83114e6 −0.778495 −0.389248 0.921133i \(-0.627265\pi\)
−0.389248 + 0.921133i \(0.627265\pi\)
\(422\) −282789. −0.0773003
\(423\) −1.19860e6 −0.325703
\(424\) −897256. −0.242383
\(425\) 0 0
\(426\) 564834. 0.150799
\(427\) −3.59494e6 −0.954161
\(428\) −776983. −0.205023
\(429\) 567687. 0.148924
\(430\) 0 0
\(431\) 6.20538e6 1.60907 0.804536 0.593904i \(-0.202414\pi\)
0.804536 + 0.593904i \(0.202414\pi\)
\(432\) 627230. 0.161703
\(433\) 422736. 0.108355 0.0541775 0.998531i \(-0.482746\pi\)
0.0541775 + 0.998531i \(0.482746\pi\)
\(434\) 9643.40 0.00245757
\(435\) 0 0
\(436\) −2.99102e6 −0.753535
\(437\) −304890. −0.0763730
\(438\) −87241.7 −0.0217290
\(439\) −4.07012e6 −1.00797 −0.503983 0.863713i \(-0.668133\pi\)
−0.503983 + 0.863713i \(0.668133\pi\)
\(440\) 0 0
\(441\) −741514. −0.181561
\(442\) −46804.0 −0.0113953
\(443\) 4.95245e6 1.19898 0.599488 0.800384i \(-0.295371\pi\)
0.599488 + 0.800384i \(0.295371\pi\)
\(444\) 3.93464e6 0.947212
\(445\) 0 0
\(446\) −156438. −0.0372397
\(447\) −2.44185e6 −0.578029
\(448\) 1.97539e6 0.465004
\(449\) 7.34037e6 1.71831 0.859157 0.511713i \(-0.170989\pi\)
0.859157 + 0.511713i \(0.170989\pi\)
\(450\) 0 0
\(451\) 744385. 0.172328
\(452\) 364838. 0.0839952
\(453\) −1.32073e6 −0.302392
\(454\) 2876.12 0.000654888 0
\(455\) 0 0
\(456\) 1.17291e6 0.264150
\(457\) 2.07655e6 0.465106 0.232553 0.972584i \(-0.425292\pi\)
0.232553 + 0.972584i \(0.425292\pi\)
\(458\) 162969. 0.0363029
\(459\) −49683.0 −0.0110072
\(460\) 0 0
\(461\) −1.68917e6 −0.370186 −0.185093 0.982721i \(-0.559259\pi\)
−0.185093 + 0.982721i \(0.559259\pi\)
\(462\) −125502. −0.0273556
\(463\) −1.35953e6 −0.294739 −0.147369 0.989082i \(-0.547081\pi\)
−0.147369 + 0.989082i \(0.547081\pi\)
\(464\) −3.65925e6 −0.789037
\(465\) 0 0
\(466\) 1.53593e6 0.327648
\(467\) −8.47580e6 −1.79841 −0.899205 0.437528i \(-0.855854\pi\)
−0.899205 + 0.437528i \(0.855854\pi\)
\(468\) −1.27791e6 −0.269702
\(469\) 109700. 0.0230290
\(470\) 0 0
\(471\) 3.29006e6 0.683363
\(472\) 1.32057e6 0.272840
\(473\) 170442. 0.0350288
\(474\) 1.06929e6 0.218600
\(475\) 0 0
\(476\) −180433. −0.0365004
\(477\) 886013. 0.178297
\(478\) 1.01522e6 0.203232
\(479\) −3.52896e6 −0.702762 −0.351381 0.936233i \(-0.614288\pi\)
−0.351381 + 0.936233i \(0.614288\pi\)
\(480\) 0 0
\(481\) −7.53028e6 −1.48405
\(482\) −1.87784e6 −0.368165
\(483\) −151087. −0.0294687
\(484\) −443101. −0.0859785
\(485\) 0 0
\(486\) 77791.8 0.0149397
\(487\) 4.66713e6 0.891719 0.445859 0.895103i \(-0.352898\pi\)
0.445859 + 0.895103i \(0.352898\pi\)
\(488\) −3.37094e6 −0.640768
\(489\) −150257. −0.0284160
\(490\) 0 0
\(491\) −7.22516e6 −1.35252 −0.676260 0.736663i \(-0.736400\pi\)
−0.676260 + 0.736663i \(0.736400\pi\)
\(492\) −1.67566e6 −0.312086
\(493\) 289850. 0.0537102
\(494\) −1.09109e6 −0.201162
\(495\) 0 0
\(496\) −71995.6 −0.0131402
\(497\) −4.16734e6 −0.756777
\(498\) 128728. 0.0232594
\(499\) −1.04815e7 −1.88440 −0.942202 0.335046i \(-0.891248\pi\)
−0.942202 + 0.335046i \(0.891248\pi\)
\(500\) 0 0
\(501\) 5.11731e6 0.910852
\(502\) −881012. −0.156035
\(503\) −8.37701e6 −1.47628 −0.738140 0.674647i \(-0.764296\pi\)
−0.738140 + 0.674647i \(0.764296\pi\)
\(504\) 581230. 0.101923
\(505\) 0 0
\(506\) 30590.8 0.00531146
\(507\) −895926. −0.154793
\(508\) −1.75393e6 −0.301545
\(509\) 3.17550e6 0.543272 0.271636 0.962400i \(-0.412435\pi\)
0.271636 + 0.962400i \(0.412435\pi\)
\(510\) 0 0
\(511\) 643668. 0.109046
\(512\) 5.49216e6 0.925908
\(513\) −1.15821e6 −0.194309
\(514\) 1.43198e6 0.239071
\(515\) 0 0
\(516\) −383678. −0.0634371
\(517\) −1.79050e6 −0.294610
\(518\) 1.66477e6 0.272602
\(519\) −4.96982e6 −0.809884
\(520\) 0 0
\(521\) 3.72507e6 0.601230 0.300615 0.953746i \(-0.402808\pi\)
0.300615 + 0.953746i \(0.402808\pi\)
\(522\) −453837. −0.0728993
\(523\) −3.77963e6 −0.604219 −0.302110 0.953273i \(-0.597691\pi\)
−0.302110 + 0.953273i \(0.597691\pi\)
\(524\) 3.76339e6 0.598758
\(525\) 0 0
\(526\) 1.14072e6 0.179769
\(527\) 5702.79 0.000894460 0
\(528\) 936973. 0.146266
\(529\) −6.39952e6 −0.994278
\(530\) 0 0
\(531\) −1.30403e6 −0.200701
\(532\) −4.20624e6 −0.644340
\(533\) 3.20696e6 0.488962
\(534\) 1.33826e6 0.203089
\(535\) 0 0
\(536\) 102865. 0.0154652
\(537\) −1.61966e6 −0.242375
\(538\) 600538. 0.0894509
\(539\) −1.10769e6 −0.164228
\(540\) 0 0
\(541\) −6.08511e6 −0.893871 −0.446936 0.894566i \(-0.647485\pi\)
−0.446936 + 0.894566i \(0.647485\pi\)
\(542\) 2.55142e6 0.373065
\(543\) 5.51195e6 0.802243
\(544\) −256143. −0.0371095
\(545\) 0 0
\(546\) −540689. −0.0776186
\(547\) −2.49034e6 −0.355870 −0.177935 0.984042i \(-0.556942\pi\)
−0.177935 + 0.984042i \(0.556942\pi\)
\(548\) −1.23685e7 −1.75941
\(549\) 3.32870e6 0.471350
\(550\) 0 0
\(551\) 6.75698e6 0.948143
\(552\) −141673. −0.0197897
\(553\) −7.88922e6 −1.09704
\(554\) −2.21181e6 −0.306178
\(555\) 0 0
\(556\) −1.14540e7 −1.57133
\(557\) 5.36235e6 0.732347 0.366173 0.930547i \(-0.380668\pi\)
0.366173 + 0.930547i \(0.380668\pi\)
\(558\) −8929.21 −0.00121403
\(559\) 734300. 0.0993903
\(560\) 0 0
\(561\) −74217.9 −0.00995638
\(562\) −2.19261e6 −0.292833
\(563\) −479316. −0.0637310 −0.0318655 0.999492i \(-0.510145\pi\)
−0.0318655 + 0.999492i \(0.510145\pi\)
\(564\) 4.03054e6 0.533538
\(565\) 0 0
\(566\) 696978. 0.0914488
\(567\) −573947. −0.0749746
\(568\) −3.90768e6 −0.508215
\(569\) 3.08333e6 0.399245 0.199623 0.979873i \(-0.436028\pi\)
0.199623 + 0.979873i \(0.436028\pi\)
\(570\) 0 0
\(571\) −5.85646e6 −0.751701 −0.375850 0.926680i \(-0.622649\pi\)
−0.375850 + 0.926680i \(0.622649\pi\)
\(572\) −1.90897e6 −0.243955
\(573\) 588670. 0.0749007
\(574\) −708982. −0.0898164
\(575\) 0 0
\(576\) −1.82909e6 −0.229710
\(577\) −5.66144e6 −0.707925 −0.353963 0.935260i \(-0.615166\pi\)
−0.353963 + 0.935260i \(0.615166\pi\)
\(578\) −1.86442e6 −0.232126
\(579\) 5.94945e6 0.737531
\(580\) 0 0
\(581\) −949753. −0.116727
\(582\) 1.15003e6 0.140735
\(583\) 1.32355e6 0.161276
\(584\) 603562. 0.0732301
\(585\) 0 0
\(586\) −1.54909e6 −0.186352
\(587\) −1.39978e7 −1.67674 −0.838368 0.545104i \(-0.816490\pi\)
−0.838368 + 0.545104i \(0.816490\pi\)
\(588\) 2.49350e6 0.297417
\(589\) 132943. 0.0157899
\(590\) 0 0
\(591\) −5.31167e6 −0.625551
\(592\) −1.24288e7 −1.45755
\(593\) −1.27936e7 −1.49402 −0.747012 0.664810i \(-0.768512\pi\)
−0.747012 + 0.664810i \(0.768512\pi\)
\(594\) 116208. 0.0135135
\(595\) 0 0
\(596\) 8.21124e6 0.946876
\(597\) −3.08475e6 −0.354229
\(598\) 131791. 0.0150707
\(599\) −1.21733e7 −1.38625 −0.693125 0.720818i \(-0.743766\pi\)
−0.693125 + 0.720818i \(0.743766\pi\)
\(600\) 0 0
\(601\) −6.74801e6 −0.762060 −0.381030 0.924563i \(-0.624431\pi\)
−0.381030 + 0.924563i \(0.624431\pi\)
\(602\) −162336. −0.0182568
\(603\) −101576. −0.0113762
\(604\) 4.44125e6 0.495351
\(605\) 0 0
\(606\) −661020. −0.0731195
\(607\) 1.16658e7 1.28512 0.642561 0.766234i \(-0.277872\pi\)
0.642561 + 0.766234i \(0.277872\pi\)
\(608\) −5.97120e6 −0.655092
\(609\) 3.34840e6 0.365843
\(610\) 0 0
\(611\) −7.71382e6 −0.835923
\(612\) 167070. 0.0180310
\(613\) −1.36615e7 −1.46841 −0.734204 0.678928i \(-0.762445\pi\)
−0.734204 + 0.678928i \(0.762445\pi\)
\(614\) 2.01681e6 0.215895
\(615\) 0 0
\(616\) 868258. 0.0921928
\(617\) 1.15742e7 1.22399 0.611995 0.790862i \(-0.290367\pi\)
0.611995 + 0.790862i \(0.290367\pi\)
\(618\) 1.27423e6 0.134207
\(619\) 9.22658e6 0.967863 0.483932 0.875106i \(-0.339208\pi\)
0.483932 + 0.875106i \(0.339208\pi\)
\(620\) 0 0
\(621\) 139898. 0.0145574
\(622\) −1.47050e6 −0.152402
\(623\) −9.87366e6 −1.01920
\(624\) 4.03667e6 0.415013
\(625\) 0 0
\(626\) 1.13902e6 0.116170
\(627\) −1.73016e6 −0.175759
\(628\) −1.10635e7 −1.11942
\(629\) 984487. 0.0992164
\(630\) 0 0
\(631\) 1.03002e7 1.02985 0.514923 0.857237i \(-0.327821\pi\)
0.514923 + 0.857237i \(0.327821\pi\)
\(632\) −7.39764e6 −0.736717
\(633\) −1.93189e6 −0.191635
\(634\) −2.37929e6 −0.235085
\(635\) 0 0
\(636\) −2.97941e6 −0.292070
\(637\) −4.77217e6 −0.465980
\(638\) −677954. −0.0659399
\(639\) 3.85871e6 0.373844
\(640\) 0 0
\(641\) −3.06165e6 −0.294313 −0.147157 0.989113i \(-0.547012\pi\)
−0.147157 + 0.989113i \(0.547012\pi\)
\(642\) 304399. 0.0291478
\(643\) −1.62044e6 −0.154563 −0.0772813 0.997009i \(-0.524624\pi\)
−0.0772813 + 0.997009i \(0.524624\pi\)
\(644\) 508064. 0.0482729
\(645\) 0 0
\(646\) 142647. 0.0134487
\(647\) 1.57992e6 0.148380 0.0741900 0.997244i \(-0.476363\pi\)
0.0741900 + 0.997244i \(0.476363\pi\)
\(648\) −538185. −0.0503494
\(649\) −1.94799e6 −0.181541
\(650\) 0 0
\(651\) 65879.6 0.00609254
\(652\) 505272. 0.0465486
\(653\) 1.29080e6 0.118462 0.0592308 0.998244i \(-0.481135\pi\)
0.0592308 + 0.998244i \(0.481135\pi\)
\(654\) 1.17179e6 0.107129
\(655\) 0 0
\(656\) 5.29311e6 0.480232
\(657\) −595999. −0.0538681
\(658\) 1.70534e6 0.153549
\(659\) 1.44401e7 1.29525 0.647627 0.761957i \(-0.275761\pi\)
0.647627 + 0.761957i \(0.275761\pi\)
\(660\) 0 0
\(661\) 8.80403e6 0.783751 0.391875 0.920018i \(-0.371826\pi\)
0.391875 + 0.920018i \(0.371826\pi\)
\(662\) 2.44645e6 0.216966
\(663\) −319745. −0.0282501
\(664\) −890574. −0.0783880
\(665\) 0 0
\(666\) −1.54147e6 −0.134664
\(667\) −816163. −0.0710334
\(668\) −1.72081e7 −1.49208
\(669\) −1.06872e6 −0.0923207
\(670\) 0 0
\(671\) 4.97250e6 0.426352
\(672\) −2.95901e6 −0.252769
\(673\) 1.43891e7 1.22461 0.612303 0.790623i \(-0.290243\pi\)
0.612303 + 0.790623i \(0.290243\pi\)
\(674\) 2.87286e6 0.243593
\(675\) 0 0
\(676\) 3.01274e6 0.253569
\(677\) 1.21710e7 1.02060 0.510299 0.859997i \(-0.329535\pi\)
0.510299 + 0.859997i \(0.329535\pi\)
\(678\) −142933. −0.0119415
\(679\) −8.48489e6 −0.706272
\(680\) 0 0
\(681\) 19648.4 0.00162353
\(682\) −13338.7 −0.00109813
\(683\) −5.96383e6 −0.489186 −0.244593 0.969626i \(-0.578654\pi\)
−0.244593 + 0.969626i \(0.578654\pi\)
\(684\) 3.89473e6 0.318300
\(685\) 0 0
\(686\) 2.99194e6 0.242741
\(687\) 1.11333e6 0.0899982
\(688\) 1.21197e6 0.0976158
\(689\) 5.70212e6 0.457602
\(690\) 0 0
\(691\) −1.10524e7 −0.880565 −0.440283 0.897859i \(-0.645122\pi\)
−0.440283 + 0.897859i \(0.645122\pi\)
\(692\) 1.67121e7 1.32668
\(693\) −857378. −0.0678171
\(694\) 3.42434e6 0.269885
\(695\) 0 0
\(696\) 3.13977e6 0.245682
\(697\) −419269. −0.0326897
\(698\) 1.54422e6 0.119969
\(699\) 1.04928e7 0.812270
\(700\) 0 0
\(701\) −9.86637e6 −0.758337 −0.379168 0.925328i \(-0.623790\pi\)
−0.379168 + 0.925328i \(0.623790\pi\)
\(702\) 500645. 0.0383431
\(703\) 2.29503e7 1.75146
\(704\) −2.73235e6 −0.207780
\(705\) 0 0
\(706\) 2.07995e6 0.157051
\(707\) 4.87700e6 0.366948
\(708\) 4.38506e6 0.328770
\(709\) −1.65852e7 −1.23910 −0.619550 0.784957i \(-0.712685\pi\)
−0.619550 + 0.784957i \(0.712685\pi\)
\(710\) 0 0
\(711\) 7.30495e6 0.541930
\(712\) −9.25844e6 −0.684444
\(713\) −16058.0 −0.00118295
\(714\) 70688.1 0.00518921
\(715\) 0 0
\(716\) 5.44646e6 0.397038
\(717\) 6.93557e6 0.503830
\(718\) 4.20940e6 0.304725
\(719\) −1.70621e6 −0.123086 −0.0615430 0.998104i \(-0.519602\pi\)
−0.0615430 + 0.998104i \(0.519602\pi\)
\(720\) 0 0
\(721\) −9.40126e6 −0.673516
\(722\) 63333.1 0.00452156
\(723\) −1.28286e7 −0.912715
\(724\) −1.85351e7 −1.31416
\(725\) 0 0
\(726\) 173594. 0.0122234
\(727\) 4.79877e6 0.336739 0.168370 0.985724i \(-0.446150\pi\)
0.168370 + 0.985724i \(0.446150\pi\)
\(728\) 3.74063e6 0.261587
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −96000.4 −0.00664476
\(732\) −1.11935e7 −0.772123
\(733\) −2.10050e7 −1.44398 −0.721991 0.691902i \(-0.756773\pi\)
−0.721991 + 0.691902i \(0.756773\pi\)
\(734\) 3.32048e6 0.227489
\(735\) 0 0
\(736\) 721250. 0.0490785
\(737\) −151737. −0.0102901
\(738\) 656475. 0.0443688
\(739\) −1.45144e7 −0.977660 −0.488830 0.872379i \(-0.662576\pi\)
−0.488830 + 0.872379i \(0.662576\pi\)
\(740\) 0 0
\(741\) −7.45390e6 −0.498698
\(742\) −1.26060e6 −0.0840560
\(743\) 1.33510e7 0.887245 0.443622 0.896214i \(-0.353693\pi\)
0.443622 + 0.896214i \(0.353693\pi\)
\(744\) 61774.7 0.00409146
\(745\) 0 0
\(746\) 2.19848e6 0.144636
\(747\) 879414. 0.0576623
\(748\) 249573. 0.0163096
\(749\) −2.24585e6 −0.146277
\(750\) 0 0
\(751\) 1.03902e7 0.672242 0.336121 0.941819i \(-0.390885\pi\)
0.336121 + 0.941819i \(0.390885\pi\)
\(752\) −1.27317e7 −0.820999
\(753\) −6.01870e6 −0.386826
\(754\) −2.92076e6 −0.187097
\(755\) 0 0
\(756\) 1.93002e6 0.122817
\(757\) −1.09141e7 −0.692227 −0.346113 0.938193i \(-0.612499\pi\)
−0.346113 + 0.938193i \(0.612499\pi\)
\(758\) −4.12390e6 −0.260696
\(759\) 208983. 0.0131676
\(760\) 0 0
\(761\) 1.04145e7 0.651894 0.325947 0.945388i \(-0.394317\pi\)
0.325947 + 0.945388i \(0.394317\pi\)
\(762\) 687136. 0.0428702
\(763\) −8.64548e6 −0.537623
\(764\) −1.97953e6 −0.122696
\(765\) 0 0
\(766\) 5.67763e6 0.349619
\(767\) −8.39232e6 −0.515103
\(768\) 4.72472e6 0.289050
\(769\) −9.42825e6 −0.574930 −0.287465 0.957791i \(-0.592813\pi\)
−0.287465 + 0.957791i \(0.592813\pi\)
\(770\) 0 0
\(771\) 9.78265e6 0.592681
\(772\) −2.00063e7 −1.20816
\(773\) −1.18529e7 −0.713469 −0.356734 0.934206i \(-0.616110\pi\)
−0.356734 + 0.934206i \(0.616110\pi\)
\(774\) 150314. 0.00901875
\(775\) 0 0
\(776\) −7.95620e6 −0.474298
\(777\) 1.13730e7 0.675805
\(778\) −3.23760e6 −0.191767
\(779\) −9.77398e6 −0.577069
\(780\) 0 0
\(781\) 5.76425e6 0.338154
\(782\) −17230.0 −0.00100756
\(783\) −3.10042e6 −0.180724
\(784\) −7.87650e6 −0.457660
\(785\) 0 0
\(786\) −1.47439e6 −0.0851246
\(787\) 1.84213e7 1.06019 0.530094 0.847939i \(-0.322157\pi\)
0.530094 + 0.847939i \(0.322157\pi\)
\(788\) 1.78616e7 1.02472
\(789\) 7.79293e6 0.445664
\(790\) 0 0
\(791\) 1.05456e6 0.0599279
\(792\) −803955. −0.0455427
\(793\) 2.14225e7 1.20973
\(794\) 7.81492e6 0.439919
\(795\) 0 0
\(796\) 1.03731e7 0.580266
\(797\) 8.83478e6 0.492663 0.246331 0.969186i \(-0.420775\pi\)
0.246331 + 0.969186i \(0.420775\pi\)
\(798\) 1.64788e6 0.0916048
\(799\) 1.00848e6 0.0558858
\(800\) 0 0
\(801\) 9.14242e6 0.503478
\(802\) 6.01738e6 0.330348
\(803\) −890319. −0.0487255
\(804\) 341570. 0.0186354
\(805\) 0 0
\(806\) −57465.8 −0.00311582
\(807\) 4.10262e6 0.221757
\(808\) 4.57312e6 0.246425
\(809\) 1.51577e7 0.814259 0.407129 0.913370i \(-0.366530\pi\)
0.407129 + 0.913370i \(0.366530\pi\)
\(810\) 0 0
\(811\) −1.02018e7 −0.544658 −0.272329 0.962204i \(-0.587794\pi\)
−0.272329 + 0.962204i \(0.587794\pi\)
\(812\) −1.12597e7 −0.599291
\(813\) 1.74303e7 0.924863
\(814\) −2.30270e6 −0.121808
\(815\) 0 0
\(816\) −527742. −0.0277458
\(817\) −2.23796e6 −0.117300
\(818\) −7.74551e6 −0.404731
\(819\) −3.69376e6 −0.192424
\(820\) 0 0
\(821\) 2.60152e7 1.34700 0.673501 0.739186i \(-0.264790\pi\)
0.673501 + 0.739186i \(0.264790\pi\)
\(822\) 4.84563e6 0.250133
\(823\) −1.36038e7 −0.700103 −0.350052 0.936730i \(-0.613836\pi\)
−0.350052 + 0.936730i \(0.613836\pi\)
\(824\) −8.81547e6 −0.452301
\(825\) 0 0
\(826\) 1.85534e6 0.0946181
\(827\) 6.94229e6 0.352971 0.176486 0.984303i \(-0.443527\pi\)
0.176486 + 0.984303i \(0.443527\pi\)
\(828\) −470437. −0.0238465
\(829\) −6.04757e6 −0.305629 −0.152814 0.988255i \(-0.548834\pi\)
−0.152814 + 0.988255i \(0.548834\pi\)
\(830\) 0 0
\(831\) −1.51102e7 −0.759044
\(832\) −1.17715e7 −0.589553
\(833\) 623900. 0.0311532
\(834\) 4.48732e6 0.223394
\(835\) 0 0
\(836\) 5.81805e6 0.287913
\(837\) −61000.6 −0.00300968
\(838\) 8.06867e6 0.396910
\(839\) 4.32449e6 0.212095 0.106047 0.994361i \(-0.466180\pi\)
0.106047 + 0.994361i \(0.466180\pi\)
\(840\) 0 0
\(841\) −2.42331e6 −0.118146
\(842\) −3.72977e6 −0.181302
\(843\) −1.49790e7 −0.725961
\(844\) 6.49641e6 0.313919
\(845\) 0 0
\(846\) −1.57904e6 −0.0758523
\(847\) −1.28077e6 −0.0613429
\(848\) 9.41139e6 0.449432
\(849\) 4.76146e6 0.226710
\(850\) 0 0
\(851\) −2.77213e6 −0.131217
\(852\) −1.29757e7 −0.612397
\(853\) −4.01062e7 −1.88729 −0.943645 0.330959i \(-0.892628\pi\)
−0.943645 + 0.330959i \(0.892628\pi\)
\(854\) −4.73601e6 −0.222212
\(855\) 0 0
\(856\) −2.10591e6 −0.0982327
\(857\) 1.21434e7 0.564792 0.282396 0.959298i \(-0.408871\pi\)
0.282396 + 0.959298i \(0.408871\pi\)
\(858\) 747878. 0.0346827
\(859\) −1.69726e7 −0.784810 −0.392405 0.919792i \(-0.628357\pi\)
−0.392405 + 0.919792i \(0.628357\pi\)
\(860\) 0 0
\(861\) −4.84347e6 −0.222663
\(862\) 8.17504e6 0.374733
\(863\) 3.51856e7 1.60819 0.804095 0.594501i \(-0.202650\pi\)
0.804095 + 0.594501i \(0.202650\pi\)
\(864\) 2.73987e6 0.124866
\(865\) 0 0
\(866\) 556917. 0.0252346
\(867\) −1.27369e7 −0.575462
\(868\) −221534. −0.00998026
\(869\) 1.09123e7 0.490194
\(870\) 0 0
\(871\) −653712. −0.0291972
\(872\) −8.10678e6 −0.361042
\(873\) 7.85650e6 0.348894
\(874\) −401666. −0.0177863
\(875\) 0 0
\(876\) 2.00417e6 0.0882419
\(877\) 3.15265e7 1.38413 0.692066 0.721834i \(-0.256701\pi\)
0.692066 + 0.721834i \(0.256701\pi\)
\(878\) −5.36203e6 −0.234743
\(879\) −1.05827e7 −0.461983
\(880\) 0 0
\(881\) −2.96796e7 −1.28831 −0.644153 0.764897i \(-0.722790\pi\)
−0.644153 + 0.764897i \(0.722790\pi\)
\(882\) −976879. −0.0422833
\(883\) 1.56983e7 0.677564 0.338782 0.940865i \(-0.389985\pi\)
0.338782 + 0.940865i \(0.389985\pi\)
\(884\) 1.07521e6 0.0462768
\(885\) 0 0
\(886\) 6.52441e6 0.279227
\(887\) 532251. 0.0227147 0.0113574 0.999936i \(-0.496385\pi\)
0.0113574 + 0.999936i \(0.496385\pi\)
\(888\) 1.06643e7 0.453838
\(889\) −5.06969e6 −0.215143
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 3.59381e6 0.151232
\(893\) 2.35097e7 0.986550
\(894\) −3.21692e6 −0.134616
\(895\) 0 0
\(896\) 1.31233e7 0.546102
\(897\) 900342. 0.0373617
\(898\) 9.67029e6 0.400174
\(899\) 355877. 0.0146859
\(900\) 0 0
\(901\) −745479. −0.0305931
\(902\) 980661. 0.0401331
\(903\) −1.10901e6 −0.0452603
\(904\) 988848. 0.0402447
\(905\) 0 0
\(906\) −1.73995e6 −0.0704233
\(907\) −7.46788e6 −0.301425 −0.150713 0.988578i \(-0.548157\pi\)
−0.150713 + 0.988578i \(0.548157\pi\)
\(908\) −66072.1 −0.00265952
\(909\) −4.51581e6 −0.181270
\(910\) 0 0
\(911\) 4.14937e6 0.165648 0.0828240 0.996564i \(-0.473606\pi\)
0.0828240 + 0.996564i \(0.473606\pi\)
\(912\) −1.23027e7 −0.489795
\(913\) 1.31369e6 0.0521575
\(914\) 2.73567e6 0.108318
\(915\) 0 0
\(916\) −3.74383e6 −0.147427
\(917\) 1.08780e7 0.427195
\(918\) −65453.0 −0.00256344
\(919\) −2.25977e7 −0.882625 −0.441313 0.897353i \(-0.645487\pi\)
−0.441313 + 0.897353i \(0.645487\pi\)
\(920\) 0 0
\(921\) 1.37780e7 0.535225
\(922\) −2.22533e6 −0.0862118
\(923\) 2.48335e7 0.959476
\(924\) 2.88312e6 0.111092
\(925\) 0 0
\(926\) −1.79106e6 −0.0686410
\(927\) 8.70500e6 0.332713
\(928\) −1.59844e7 −0.609291
\(929\) 4.69291e7 1.78403 0.892017 0.452002i \(-0.149290\pi\)
0.892017 + 0.452002i \(0.149290\pi\)
\(930\) 0 0
\(931\) 1.45443e7 0.549945
\(932\) −3.52844e7 −1.33059
\(933\) −1.00458e7 −0.377818
\(934\) −1.11661e7 −0.418827
\(935\) 0 0
\(936\) −3.46360e6 −0.129222
\(937\) −5.63886e6 −0.209818 −0.104909 0.994482i \(-0.533455\pi\)
−0.104909 + 0.994482i \(0.533455\pi\)
\(938\) 144520. 0.00536317
\(939\) 7.78128e6 0.287996
\(940\) 0 0
\(941\) −2.03249e7 −0.748261 −0.374131 0.927376i \(-0.622059\pi\)
−0.374131 + 0.927376i \(0.622059\pi\)
\(942\) 4.33436e6 0.159147
\(943\) 1.18058e6 0.0432331
\(944\) −1.38516e7 −0.505906
\(945\) 0 0
\(946\) 224543. 0.00815777
\(947\) 1.36521e7 0.494681 0.247340 0.968929i \(-0.420443\pi\)
0.247340 + 0.968929i \(0.420443\pi\)
\(948\) −2.45644e7 −0.887741
\(949\) −3.83567e6 −0.138253
\(950\) 0 0
\(951\) −1.62543e7 −0.582797
\(952\) −489039. −0.0174885
\(953\) −5.93702e6 −0.211756 −0.105878 0.994379i \(-0.533765\pi\)
−0.105878 + 0.994379i \(0.533765\pi\)
\(954\) 1.16724e6 0.0415232
\(955\) 0 0
\(956\) −2.33223e7 −0.825329
\(957\) −4.63149e6 −0.163471
\(958\) −4.64909e6 −0.163664
\(959\) −3.57510e7 −1.25528
\(960\) 0 0
\(961\) −2.86221e7 −0.999755
\(962\) −9.92047e6 −0.345617
\(963\) 2.07952e6 0.0722601
\(964\) 4.31391e7 1.49513
\(965\) 0 0
\(966\) −199044. −0.00686289
\(967\) 4.40436e7 1.51467 0.757333 0.653029i \(-0.226502\pi\)
0.757333 + 0.653029i \(0.226502\pi\)
\(968\) −1.20097e6 −0.0411949
\(969\) 974502. 0.0333406
\(970\) 0 0
\(971\) −1.01824e7 −0.346577 −0.173289 0.984871i \(-0.555439\pi\)
−0.173289 + 0.984871i \(0.555439\pi\)
\(972\) −1.78708e6 −0.0606707
\(973\) −3.31074e7 −1.12110
\(974\) 6.14854e6 0.207670
\(975\) 0 0
\(976\) 3.53580e7 1.18813
\(977\) 4.43840e7 1.48761 0.743806 0.668395i \(-0.233018\pi\)
0.743806 + 0.668395i \(0.233018\pi\)
\(978\) −197951. −0.00661774
\(979\) 1.36572e7 0.455413
\(980\) 0 0
\(981\) 8.00520e6 0.265583
\(982\) −9.51850e6 −0.314985
\(983\) 2.94574e7 0.972324 0.486162 0.873869i \(-0.338396\pi\)
0.486162 + 0.873869i \(0.338396\pi\)
\(984\) −4.54167e6 −0.149530
\(985\) 0 0
\(986\) 381852. 0.0125084
\(987\) 1.16502e7 0.380662
\(988\) 2.50653e7 0.816923
\(989\) 270319. 0.00878791
\(990\) 0 0
\(991\) −1.97622e7 −0.639221 −0.319611 0.947549i \(-0.603552\pi\)
−0.319611 + 0.947549i \(0.603552\pi\)
\(992\) −314491. −0.0101468
\(993\) 1.67131e7 0.537879
\(994\) −5.49010e6 −0.176244
\(995\) 0 0
\(996\) −2.95722e6 −0.0944572
\(997\) −2.39291e7 −0.762408 −0.381204 0.924491i \(-0.624491\pi\)
−0.381204 + 0.924491i \(0.624491\pi\)
\(998\) −1.38085e7 −0.438854
\(999\) −1.05307e7 −0.333844
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 825.6.a.y.1.7 13
5.2 odd 4 165.6.c.b.34.15 yes 26
5.3 odd 4 165.6.c.b.34.12 26
5.4 even 2 825.6.a.v.1.7 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.12 26 5.3 odd 4
165.6.c.b.34.15 yes 26 5.2 odd 4
825.6.a.v.1.7 13 5.4 even 2
825.6.a.y.1.7 13 1.1 even 1 trivial