Properties

Label 531.8.a.d.1.7
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(4.11298\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.11298 q^{2} -123.535 q^{4} +537.118 q^{5} +1259.15 q^{7} +531.490 q^{8} +O(q^{10})\) \(q-2.11298 q^{2} -123.535 q^{4} +537.118 q^{5} +1259.15 q^{7} +531.490 q^{8} -1134.92 q^{10} -7730.24 q^{11} -8860.73 q^{13} -2660.56 q^{14} +14689.5 q^{16} -22368.4 q^{17} -24886.1 q^{19} -66353.0 q^{20} +16333.9 q^{22} +26686.2 q^{23} +210371. q^{25} +18722.6 q^{26} -155549. q^{28} -10101.0 q^{29} -265180. q^{31} -99069.4 q^{32} +47264.0 q^{34} +676311. q^{35} +122638. q^{37} +52583.9 q^{38} +285473. q^{40} +741866. q^{41} -166554. q^{43} +954958. q^{44} -56387.4 q^{46} +545699. q^{47} +761910. q^{49} -444510. q^{50} +1.09461e6 q^{52} +1.54083e6 q^{53} -4.15205e6 q^{55} +669225. q^{56} +21343.2 q^{58} +205379. q^{59} +2.74259e6 q^{61} +560320. q^{62} -1.67092e6 q^{64} -4.75926e6 q^{65} -41604.4 q^{67} +2.76328e6 q^{68} -1.42903e6 q^{70} +1.57840e6 q^{71} +5.59479e6 q^{73} -259132. q^{74} +3.07431e6 q^{76} -9.73352e6 q^{77} -3.74825e6 q^{79} +7.88999e6 q^{80} -1.56755e6 q^{82} +3.37715e6 q^{83} -1.20144e7 q^{85} +351926. q^{86} -4.10855e6 q^{88} +5.43486e6 q^{89} -1.11570e7 q^{91} -3.29668e6 q^{92} -1.15305e6 q^{94} -1.33668e7 q^{95} -1.14400e7 q^{97} -1.60990e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.11298 −0.186763 −0.0933816 0.995630i \(-0.529768\pi\)
−0.0933816 + 0.995630i \(0.529768\pi\)
\(3\) 0 0
\(4\) −123.535 −0.965120
\(5\) 537.118 1.92165 0.960826 0.277153i \(-0.0893908\pi\)
0.960826 + 0.277153i \(0.0893908\pi\)
\(6\) 0 0
\(7\) 1259.15 1.38750 0.693751 0.720215i \(-0.255957\pi\)
0.693751 + 0.720215i \(0.255957\pi\)
\(8\) 531.490 0.367012
\(9\) 0 0
\(10\) −1134.92 −0.358894
\(11\) −7730.24 −1.75113 −0.875566 0.483099i \(-0.839511\pi\)
−0.875566 + 0.483099i \(0.839511\pi\)
\(12\) 0 0
\(13\) −8860.73 −1.11858 −0.559291 0.828971i \(-0.688927\pi\)
−0.559291 + 0.828971i \(0.688927\pi\)
\(14\) −2660.56 −0.259134
\(15\) 0 0
\(16\) 14689.5 0.896575
\(17\) −22368.4 −1.10424 −0.552119 0.833765i \(-0.686181\pi\)
−0.552119 + 0.833765i \(0.686181\pi\)
\(18\) 0 0
\(19\) −24886.1 −0.832375 −0.416187 0.909279i \(-0.636634\pi\)
−0.416187 + 0.909279i \(0.636634\pi\)
\(20\) −66353.0 −1.85462
\(21\) 0 0
\(22\) 16333.9 0.327047
\(23\) 26686.2 0.457339 0.228670 0.973504i \(-0.426562\pi\)
0.228670 + 0.973504i \(0.426562\pi\)
\(24\) 0 0
\(25\) 210371. 2.69275
\(26\) 18722.6 0.208910
\(27\) 0 0
\(28\) −155549. −1.33911
\(29\) −10101.0 −0.0769077 −0.0384539 0.999260i \(-0.512243\pi\)
−0.0384539 + 0.999260i \(0.512243\pi\)
\(30\) 0 0
\(31\) −265180. −1.59873 −0.799363 0.600848i \(-0.794830\pi\)
−0.799363 + 0.600848i \(0.794830\pi\)
\(32\) −99069.4 −0.534459
\(33\) 0 0
\(34\) 47264.0 0.206231
\(35\) 676311. 2.66630
\(36\) 0 0
\(37\) 122638. 0.398032 0.199016 0.979996i \(-0.436225\pi\)
0.199016 + 0.979996i \(0.436225\pi\)
\(38\) 52583.9 0.155457
\(39\) 0 0
\(40\) 285473. 0.705269
\(41\) 741866. 1.68105 0.840527 0.541769i \(-0.182245\pi\)
0.840527 + 0.541769i \(0.182245\pi\)
\(42\) 0 0
\(43\) −166554. −0.319459 −0.159730 0.987161i \(-0.551062\pi\)
−0.159730 + 0.987161i \(0.551062\pi\)
\(44\) 954958. 1.69005
\(45\) 0 0
\(46\) −56387.4 −0.0854142
\(47\) 545699. 0.766674 0.383337 0.923608i \(-0.374775\pi\)
0.383337 + 0.923608i \(0.374775\pi\)
\(48\) 0 0
\(49\) 761910. 0.925161
\(50\) −444510. −0.502906
\(51\) 0 0
\(52\) 1.09461e6 1.07957
\(53\) 1.54083e6 1.42164 0.710819 0.703375i \(-0.248324\pi\)
0.710819 + 0.703375i \(0.248324\pi\)
\(54\) 0 0
\(55\) −4.15205e6 −3.36506
\(56\) 669225. 0.509230
\(57\) 0 0
\(58\) 21343.2 0.0143635
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 2.74259e6 1.54706 0.773529 0.633761i \(-0.218490\pi\)
0.773529 + 0.633761i \(0.218490\pi\)
\(62\) 560320. 0.298583
\(63\) 0 0
\(64\) −1.67092e6 −0.796758
\(65\) −4.75926e6 −2.14952
\(66\) 0 0
\(67\) −41604.4 −0.0168996 −0.00844981 0.999964i \(-0.502690\pi\)
−0.00844981 + 0.999964i \(0.502690\pi\)
\(68\) 2.76328e6 1.06572
\(69\) 0 0
\(70\) −1.42903e6 −0.497966
\(71\) 1.57840e6 0.523376 0.261688 0.965152i \(-0.415721\pi\)
0.261688 + 0.965152i \(0.415721\pi\)
\(72\) 0 0
\(73\) 5.59479e6 1.68327 0.841635 0.540047i \(-0.181594\pi\)
0.841635 + 0.540047i \(0.181594\pi\)
\(74\) −259132. −0.0743378
\(75\) 0 0
\(76\) 3.07431e6 0.803341
\(77\) −9.73352e6 −2.42970
\(78\) 0 0
\(79\) −3.74825e6 −0.855330 −0.427665 0.903937i \(-0.640664\pi\)
−0.427665 + 0.903937i \(0.640664\pi\)
\(80\) 7.88999e6 1.72291
\(81\) 0 0
\(82\) −1.56755e6 −0.313959
\(83\) 3.37715e6 0.648302 0.324151 0.946005i \(-0.394921\pi\)
0.324151 + 0.946005i \(0.394921\pi\)
\(84\) 0 0
\(85\) −1.20144e7 −2.12196
\(86\) 351926. 0.0596632
\(87\) 0 0
\(88\) −4.10855e6 −0.642686
\(89\) 5.43486e6 0.817191 0.408595 0.912716i \(-0.366019\pi\)
0.408595 + 0.912716i \(0.366019\pi\)
\(90\) 0 0
\(91\) −1.11570e7 −1.55203
\(92\) −3.29668e6 −0.441387
\(93\) 0 0
\(94\) −1.15305e6 −0.143187
\(95\) −1.33668e7 −1.59953
\(96\) 0 0
\(97\) −1.14400e7 −1.27270 −0.636349 0.771401i \(-0.719556\pi\)
−0.636349 + 0.771401i \(0.719556\pi\)
\(98\) −1.60990e6 −0.172786
\(99\) 0 0
\(100\) −2.59882e7 −2.59882
\(101\) 1.39505e7 1.34730 0.673652 0.739049i \(-0.264725\pi\)
0.673652 + 0.739049i \(0.264725\pi\)
\(102\) 0 0
\(103\) −9.43458e6 −0.850731 −0.425366 0.905022i \(-0.639855\pi\)
−0.425366 + 0.905022i \(0.639855\pi\)
\(104\) −4.70939e6 −0.410533
\(105\) 0 0
\(106\) −3.25575e6 −0.265510
\(107\) −1.19603e7 −0.943844 −0.471922 0.881640i \(-0.656440\pi\)
−0.471922 + 0.881640i \(0.656440\pi\)
\(108\) 0 0
\(109\) 2.39265e6 0.176964 0.0884822 0.996078i \(-0.471798\pi\)
0.0884822 + 0.996078i \(0.471798\pi\)
\(110\) 8.77322e6 0.628470
\(111\) 0 0
\(112\) 1.84962e7 1.24400
\(113\) 4.09793e6 0.267171 0.133586 0.991037i \(-0.457351\pi\)
0.133586 + 0.991037i \(0.457351\pi\)
\(114\) 0 0
\(115\) 1.43336e7 0.878847
\(116\) 1.24783e6 0.0742251
\(117\) 0 0
\(118\) −433963. −0.0243145
\(119\) −2.81651e7 −1.53213
\(120\) 0 0
\(121\) 4.02695e7 2.06646
\(122\) −5.79505e6 −0.288933
\(123\) 0 0
\(124\) 3.27590e7 1.54296
\(125\) 7.10316e7 3.25287
\(126\) 0 0
\(127\) 1.95190e7 0.845560 0.422780 0.906232i \(-0.361054\pi\)
0.422780 + 0.906232i \(0.361054\pi\)
\(128\) 1.62115e7 0.683264
\(129\) 0 0
\(130\) 1.00562e7 0.401452
\(131\) 3.90620e6 0.151811 0.0759057 0.997115i \(-0.475815\pi\)
0.0759057 + 0.997115i \(0.475815\pi\)
\(132\) 0 0
\(133\) −3.13352e7 −1.15492
\(134\) 87909.4 0.00315623
\(135\) 0 0
\(136\) −1.18886e7 −0.405269
\(137\) 2.53665e7 0.842828 0.421414 0.906868i \(-0.361534\pi\)
0.421414 + 0.906868i \(0.361534\pi\)
\(138\) 0 0
\(139\) 4.42236e7 1.39670 0.698348 0.715758i \(-0.253919\pi\)
0.698348 + 0.715758i \(0.253919\pi\)
\(140\) −8.35483e7 −2.57329
\(141\) 0 0
\(142\) −3.33514e6 −0.0977474
\(143\) 6.84956e7 1.95878
\(144\) 0 0
\(145\) −5.42541e6 −0.147790
\(146\) −1.18217e7 −0.314373
\(147\) 0 0
\(148\) −1.51501e7 −0.384149
\(149\) −2.22130e6 −0.0550117 −0.0275059 0.999622i \(-0.508756\pi\)
−0.0275059 + 0.999622i \(0.508756\pi\)
\(150\) 0 0
\(151\) −4.02533e7 −0.951443 −0.475721 0.879596i \(-0.657813\pi\)
−0.475721 + 0.879596i \(0.657813\pi\)
\(152\) −1.32267e7 −0.305491
\(153\) 0 0
\(154\) 2.05668e7 0.453778
\(155\) −1.42433e8 −3.07220
\(156\) 0 0
\(157\) 3.47322e7 0.716281 0.358140 0.933668i \(-0.383411\pi\)
0.358140 + 0.933668i \(0.383411\pi\)
\(158\) 7.91999e6 0.159744
\(159\) 0 0
\(160\) −5.32120e7 −1.02704
\(161\) 3.36018e7 0.634559
\(162\) 0 0
\(163\) −1.15628e7 −0.209126 −0.104563 0.994518i \(-0.533344\pi\)
−0.104563 + 0.994518i \(0.533344\pi\)
\(164\) −9.16466e7 −1.62242
\(165\) 0 0
\(166\) −7.13587e6 −0.121079
\(167\) 1.06887e8 1.77590 0.887949 0.459941i \(-0.152130\pi\)
0.887949 + 0.459941i \(0.152130\pi\)
\(168\) 0 0
\(169\) 1.57640e7 0.251225
\(170\) 2.53863e7 0.396304
\(171\) 0 0
\(172\) 2.05753e7 0.308316
\(173\) 7.33625e7 1.07724 0.538621 0.842548i \(-0.318946\pi\)
0.538621 + 0.842548i \(0.318946\pi\)
\(174\) 0 0
\(175\) 2.64888e8 3.73619
\(176\) −1.13553e8 −1.57002
\(177\) 0 0
\(178\) −1.14838e7 −0.152621
\(179\) 7.93198e7 1.03370 0.516852 0.856075i \(-0.327104\pi\)
0.516852 + 0.856075i \(0.327104\pi\)
\(180\) 0 0
\(181\) 4.76795e7 0.597663 0.298831 0.954306i \(-0.403403\pi\)
0.298831 + 0.954306i \(0.403403\pi\)
\(182\) 2.35745e7 0.289863
\(183\) 0 0
\(184\) 1.41834e7 0.167849
\(185\) 6.58710e7 0.764880
\(186\) 0 0
\(187\) 1.72913e8 1.93367
\(188\) −6.74131e7 −0.739932
\(189\) 0 0
\(190\) 2.82437e7 0.298734
\(191\) 5.66760e7 0.588549 0.294274 0.955721i \(-0.404922\pi\)
0.294274 + 0.955721i \(0.404922\pi\)
\(192\) 0 0
\(193\) −1.76636e8 −1.76859 −0.884297 0.466924i \(-0.845362\pi\)
−0.884297 + 0.466924i \(0.845362\pi\)
\(194\) 2.41726e7 0.237693
\(195\) 0 0
\(196\) −9.41228e7 −0.892891
\(197\) −1.27300e8 −1.18630 −0.593151 0.805091i \(-0.702116\pi\)
−0.593151 + 0.805091i \(0.702116\pi\)
\(198\) 0 0
\(199\) 1.45827e8 1.31175 0.655877 0.754868i \(-0.272299\pi\)
0.655877 + 0.754868i \(0.272299\pi\)
\(200\) 1.11810e8 0.988270
\(201\) 0 0
\(202\) −2.94772e7 −0.251627
\(203\) −1.27186e7 −0.106710
\(204\) 0 0
\(205\) 3.98469e8 3.23040
\(206\) 1.99351e7 0.158885
\(207\) 0 0
\(208\) −1.30160e8 −1.00289
\(209\) 1.92375e8 1.45760
\(210\) 0 0
\(211\) −1.57257e8 −1.15245 −0.576223 0.817292i \(-0.695474\pi\)
−0.576223 + 0.817292i \(0.695474\pi\)
\(212\) −1.90347e8 −1.37205
\(213\) 0 0
\(214\) 2.52720e7 0.176275
\(215\) −8.94591e7 −0.613889
\(216\) 0 0
\(217\) −3.33900e8 −2.21824
\(218\) −5.05563e6 −0.0330504
\(219\) 0 0
\(220\) 5.12925e8 3.24769
\(221\) 1.98200e8 1.23518
\(222\) 0 0
\(223\) −1.56191e8 −0.943169 −0.471584 0.881821i \(-0.656318\pi\)
−0.471584 + 0.881821i \(0.656318\pi\)
\(224\) −1.24743e8 −0.741563
\(225\) 0 0
\(226\) −8.65886e6 −0.0498978
\(227\) 1.43058e8 0.811748 0.405874 0.913929i \(-0.366967\pi\)
0.405874 + 0.913929i \(0.366967\pi\)
\(228\) 0 0
\(229\) −9.00963e7 −0.495773 −0.247886 0.968789i \(-0.579736\pi\)
−0.247886 + 0.968789i \(0.579736\pi\)
\(230\) −3.02867e7 −0.164136
\(231\) 0 0
\(232\) −5.36856e6 −0.0282261
\(233\) −1.58220e8 −0.819437 −0.409718 0.912212i \(-0.634373\pi\)
−0.409718 + 0.912212i \(0.634373\pi\)
\(234\) 0 0
\(235\) 2.93105e8 1.47328
\(236\) −2.53716e7 −0.125648
\(237\) 0 0
\(238\) 5.95123e7 0.286146
\(239\) 7.78834e7 0.369022 0.184511 0.982830i \(-0.440930\pi\)
0.184511 + 0.982830i \(0.440930\pi\)
\(240\) 0 0
\(241\) 7.51343e7 0.345763 0.172882 0.984943i \(-0.444692\pi\)
0.172882 + 0.984943i \(0.444692\pi\)
\(242\) −8.50887e7 −0.385939
\(243\) 0 0
\(244\) −3.38807e8 −1.49310
\(245\) 4.09236e8 1.77784
\(246\) 0 0
\(247\) 2.20509e8 0.931079
\(248\) −1.40940e8 −0.586752
\(249\) 0 0
\(250\) −1.50089e8 −0.607516
\(251\) −1.66455e8 −0.664416 −0.332208 0.943206i \(-0.607794\pi\)
−0.332208 + 0.943206i \(0.607794\pi\)
\(252\) 0 0
\(253\) −2.06290e8 −0.800861
\(254\) −4.12433e7 −0.157919
\(255\) 0 0
\(256\) 1.79623e8 0.669149
\(257\) 5.67936e7 0.208705 0.104353 0.994540i \(-0.466723\pi\)
0.104353 + 0.994540i \(0.466723\pi\)
\(258\) 0 0
\(259\) 1.54419e8 0.552271
\(260\) 5.87936e8 2.07455
\(261\) 0 0
\(262\) −8.25373e6 −0.0283528
\(263\) 1.48859e8 0.504581 0.252291 0.967651i \(-0.418816\pi\)
0.252291 + 0.967651i \(0.418816\pi\)
\(264\) 0 0
\(265\) 8.27608e8 2.73189
\(266\) 6.62109e7 0.215697
\(267\) 0 0
\(268\) 5.13961e6 0.0163102
\(269\) 3.17458e7 0.0994382 0.0497191 0.998763i \(-0.484167\pi\)
0.0497191 + 0.998763i \(0.484167\pi\)
\(270\) 0 0
\(271\) 470544. 0.00143618 0.000718089 1.00000i \(-0.499771\pi\)
0.000718089 1.00000i \(0.499771\pi\)
\(272\) −3.28580e8 −0.990033
\(273\) 0 0
\(274\) −5.35991e7 −0.157409
\(275\) −1.62622e9 −4.71535
\(276\) 0 0
\(277\) 4.15036e8 1.17329 0.586646 0.809843i \(-0.300448\pi\)
0.586646 + 0.809843i \(0.300448\pi\)
\(278\) −9.34437e7 −0.260851
\(279\) 0 0
\(280\) 3.59453e8 0.978562
\(281\) −4.91128e8 −1.32045 −0.660226 0.751067i \(-0.729539\pi\)
−0.660226 + 0.751067i \(0.729539\pi\)
\(282\) 0 0
\(283\) −3.10302e8 −0.813827 −0.406914 0.913467i \(-0.633395\pi\)
−0.406914 + 0.913467i \(0.633395\pi\)
\(284\) −1.94989e8 −0.505121
\(285\) 0 0
\(286\) −1.44730e8 −0.365829
\(287\) 9.34119e8 2.33247
\(288\) 0 0
\(289\) 9.00046e7 0.219342
\(290\) 1.14638e7 0.0276017
\(291\) 0 0
\(292\) −6.91154e8 −1.62456
\(293\) 2.94684e8 0.684415 0.342208 0.939624i \(-0.388825\pi\)
0.342208 + 0.939624i \(0.388825\pi\)
\(294\) 0 0
\(295\) 1.10313e8 0.250178
\(296\) 6.51808e7 0.146083
\(297\) 0 0
\(298\) 4.69357e6 0.0102742
\(299\) −2.36459e8 −0.511572
\(300\) 0 0
\(301\) −2.09716e8 −0.443250
\(302\) 8.50547e7 0.177694
\(303\) 0 0
\(304\) −3.65564e8 −0.746286
\(305\) 1.47309e9 2.97291
\(306\) 0 0
\(307\) 2.26044e8 0.445871 0.222936 0.974833i \(-0.428436\pi\)
0.222936 + 0.974833i \(0.428436\pi\)
\(308\) 1.20243e9 2.34495
\(309\) 0 0
\(310\) 3.00958e8 0.573773
\(311\) 4.63267e8 0.873313 0.436656 0.899628i \(-0.356163\pi\)
0.436656 + 0.899628i \(0.356163\pi\)
\(312\) 0 0
\(313\) 6.22167e8 1.14684 0.573418 0.819263i \(-0.305617\pi\)
0.573418 + 0.819263i \(0.305617\pi\)
\(314\) −7.33885e7 −0.133775
\(315\) 0 0
\(316\) 4.63041e8 0.825496
\(317\) −8.20133e8 −1.44603 −0.723014 0.690833i \(-0.757244\pi\)
−0.723014 + 0.690833i \(0.757244\pi\)
\(318\) 0 0
\(319\) 7.80829e7 0.134675
\(320\) −8.97483e8 −1.53109
\(321\) 0 0
\(322\) −7.10001e7 −0.118512
\(323\) 5.56661e8 0.919140
\(324\) 0 0
\(325\) −1.86404e9 −3.01206
\(326\) 2.44321e7 0.0390570
\(327\) 0 0
\(328\) 3.94294e8 0.616967
\(329\) 6.87116e8 1.06376
\(330\) 0 0
\(331\) −4.99025e8 −0.756352 −0.378176 0.925734i \(-0.623449\pi\)
−0.378176 + 0.925734i \(0.623449\pi\)
\(332\) −4.17198e8 −0.625689
\(333\) 0 0
\(334\) −2.25851e8 −0.331672
\(335\) −2.23465e7 −0.0324752
\(336\) 0 0
\(337\) −2.16438e8 −0.308056 −0.154028 0.988066i \(-0.549225\pi\)
−0.154028 + 0.988066i \(0.549225\pi\)
\(338\) −3.33091e7 −0.0469196
\(339\) 0 0
\(340\) 1.48421e9 2.04795
\(341\) 2.04990e9 2.79958
\(342\) 0 0
\(343\) −7.76048e7 −0.103839
\(344\) −8.85218e7 −0.117245
\(345\) 0 0
\(346\) −1.55014e8 −0.201189
\(347\) −2.49019e8 −0.319948 −0.159974 0.987121i \(-0.551141\pi\)
−0.159974 + 0.987121i \(0.551141\pi\)
\(348\) 0 0
\(349\) −1.28064e9 −1.61264 −0.806319 0.591481i \(-0.798543\pi\)
−0.806319 + 0.591481i \(0.798543\pi\)
\(350\) −5.59704e8 −0.697783
\(351\) 0 0
\(352\) 7.65830e8 0.935908
\(353\) −1.01386e8 −0.122678 −0.0613390 0.998117i \(-0.519537\pi\)
−0.0613390 + 0.998117i \(0.519537\pi\)
\(354\) 0 0
\(355\) 8.47790e8 1.00575
\(356\) −6.71397e8 −0.788687
\(357\) 0 0
\(358\) −1.67601e8 −0.193058
\(359\) 9.38042e8 1.07002 0.535010 0.844846i \(-0.320308\pi\)
0.535010 + 0.844846i \(0.320308\pi\)
\(360\) 0 0
\(361\) −2.74555e8 −0.307153
\(362\) −1.00746e8 −0.111621
\(363\) 0 0
\(364\) 1.37828e9 1.49790
\(365\) 3.00506e9 3.23466
\(366\) 0 0
\(367\) 7.40350e8 0.781818 0.390909 0.920429i \(-0.372161\pi\)
0.390909 + 0.920429i \(0.372161\pi\)
\(368\) 3.92006e8 0.410039
\(369\) 0 0
\(370\) −1.39184e8 −0.142851
\(371\) 1.94013e9 1.97253
\(372\) 0 0
\(373\) 1.05390e9 1.05152 0.525761 0.850632i \(-0.323781\pi\)
0.525761 + 0.850632i \(0.323781\pi\)
\(374\) −3.65362e8 −0.361138
\(375\) 0 0
\(376\) 2.90034e8 0.281379
\(377\) 8.95019e7 0.0860276
\(378\) 0 0
\(379\) 1.16570e9 1.09989 0.549947 0.835199i \(-0.314648\pi\)
0.549947 + 0.835199i \(0.314648\pi\)
\(380\) 1.65127e9 1.54374
\(381\) 0 0
\(382\) −1.19755e8 −0.109919
\(383\) 1.70280e7 0.0154870 0.00774349 0.999970i \(-0.497535\pi\)
0.00774349 + 0.999970i \(0.497535\pi\)
\(384\) 0 0
\(385\) −5.22805e9 −4.66903
\(386\) 3.73229e8 0.330308
\(387\) 0 0
\(388\) 1.41325e9 1.22831
\(389\) −3.59169e8 −0.309368 −0.154684 0.987964i \(-0.549436\pi\)
−0.154684 + 0.987964i \(0.549436\pi\)
\(390\) 0 0
\(391\) −5.96925e8 −0.505012
\(392\) 4.04948e8 0.339545
\(393\) 0 0
\(394\) 2.68982e8 0.221558
\(395\) −2.01325e9 −1.64365
\(396\) 0 0
\(397\) −1.33556e9 −1.07127 −0.535634 0.844450i \(-0.679927\pi\)
−0.535634 + 0.844450i \(0.679927\pi\)
\(398\) −3.08130e8 −0.244987
\(399\) 0 0
\(400\) 3.09024e9 2.41425
\(401\) −2.22573e9 −1.72372 −0.861862 0.507143i \(-0.830702\pi\)
−0.861862 + 0.507143i \(0.830702\pi\)
\(402\) 0 0
\(403\) 2.34968e9 1.78831
\(404\) −1.72338e9 −1.30031
\(405\) 0 0
\(406\) 2.68742e7 0.0199294
\(407\) −9.48020e8 −0.697007
\(408\) 0 0
\(409\) 1.44669e9 1.04555 0.522774 0.852471i \(-0.324897\pi\)
0.522774 + 0.852471i \(0.324897\pi\)
\(410\) −8.41960e8 −0.603320
\(411\) 0 0
\(412\) 1.16550e9 0.821057
\(413\) 2.58603e8 0.180637
\(414\) 0 0
\(415\) 1.81393e9 1.24581
\(416\) 8.77827e8 0.597836
\(417\) 0 0
\(418\) −4.06486e8 −0.272225
\(419\) 1.91705e9 1.27316 0.636581 0.771210i \(-0.280348\pi\)
0.636581 + 0.771210i \(0.280348\pi\)
\(420\) 0 0
\(421\) −1.78027e9 −1.16278 −0.581392 0.813623i \(-0.697492\pi\)
−0.581392 + 0.813623i \(0.697492\pi\)
\(422\) 3.32281e8 0.215235
\(423\) 0 0
\(424\) 8.18936e8 0.521758
\(425\) −4.70565e9 −2.97343
\(426\) 0 0
\(427\) 3.45333e9 2.14655
\(428\) 1.47752e9 0.910923
\(429\) 0 0
\(430\) 1.89026e8 0.114652
\(431\) −3.29923e8 −0.198492 −0.0992458 0.995063i \(-0.531643\pi\)
−0.0992458 + 0.995063i \(0.531643\pi\)
\(432\) 0 0
\(433\) −4.75888e7 −0.0281707 −0.0140853 0.999901i \(-0.504484\pi\)
−0.0140853 + 0.999901i \(0.504484\pi\)
\(434\) 7.05526e8 0.414285
\(435\) 0 0
\(436\) −2.95576e8 −0.170792
\(437\) −6.64114e8 −0.380678
\(438\) 0 0
\(439\) −3.04664e8 −0.171868 −0.0859342 0.996301i \(-0.527387\pi\)
−0.0859342 + 0.996301i \(0.527387\pi\)
\(440\) −2.20677e9 −1.23502
\(441\) 0 0
\(442\) −4.18793e8 −0.230686
\(443\) −2.92263e9 −1.59721 −0.798603 0.601858i \(-0.794427\pi\)
−0.798603 + 0.601858i \(0.794427\pi\)
\(444\) 0 0
\(445\) 2.91916e9 1.57036
\(446\) 3.30029e8 0.176149
\(447\) 0 0
\(448\) −2.10394e9 −1.10550
\(449\) 3.10711e9 1.61993 0.809963 0.586481i \(-0.199487\pi\)
0.809963 + 0.586481i \(0.199487\pi\)
\(450\) 0 0
\(451\) −5.73480e9 −2.94375
\(452\) −5.06239e8 −0.257852
\(453\) 0 0
\(454\) −3.02279e8 −0.151605
\(455\) −5.99261e9 −2.98247
\(456\) 0 0
\(457\) −1.02063e8 −0.0500221 −0.0250110 0.999687i \(-0.507962\pi\)
−0.0250110 + 0.999687i \(0.507962\pi\)
\(458\) 1.90372e8 0.0925921
\(459\) 0 0
\(460\) −1.77071e9 −0.848192
\(461\) 1.96777e9 0.935452 0.467726 0.883873i \(-0.345073\pi\)
0.467726 + 0.883873i \(0.345073\pi\)
\(462\) 0 0
\(463\) 1.67779e9 0.785604 0.392802 0.919623i \(-0.371506\pi\)
0.392802 + 0.919623i \(0.371506\pi\)
\(464\) −1.48378e8 −0.0689536
\(465\) 0 0
\(466\) 3.34316e8 0.153041
\(467\) 1.19888e9 0.544709 0.272355 0.962197i \(-0.412198\pi\)
0.272355 + 0.962197i \(0.412198\pi\)
\(468\) 0 0
\(469\) −5.23860e7 −0.0234483
\(470\) −6.19326e8 −0.275155
\(471\) 0 0
\(472\) 1.09157e8 0.0477809
\(473\) 1.28750e9 0.559415
\(474\) 0 0
\(475\) −5.23530e9 −2.24137
\(476\) 3.47938e9 1.47869
\(477\) 0 0
\(478\) −1.64566e8 −0.0689197
\(479\) 3.41930e9 1.42155 0.710777 0.703418i \(-0.248344\pi\)
0.710777 + 0.703418i \(0.248344\pi\)
\(480\) 0 0
\(481\) −1.08666e9 −0.445232
\(482\) −1.58758e8 −0.0645758
\(483\) 0 0
\(484\) −4.97470e9 −1.99438
\(485\) −6.14464e9 −2.44568
\(486\) 0 0
\(487\) −1.02116e9 −0.400628 −0.200314 0.979732i \(-0.564196\pi\)
−0.200314 + 0.979732i \(0.564196\pi\)
\(488\) 1.45766e9 0.567789
\(489\) 0 0
\(490\) −8.64709e8 −0.332035
\(491\) −3.59695e9 −1.37135 −0.685677 0.727906i \(-0.740494\pi\)
−0.685677 + 0.727906i \(0.740494\pi\)
\(492\) 0 0
\(493\) 2.25942e8 0.0849245
\(494\) −4.65932e8 −0.173891
\(495\) 0 0
\(496\) −3.89535e9 −1.43338
\(497\) 1.98744e9 0.726186
\(498\) 0 0
\(499\) −1.17064e9 −0.421765 −0.210883 0.977511i \(-0.567634\pi\)
−0.210883 + 0.977511i \(0.567634\pi\)
\(500\) −8.77491e9 −3.13941
\(501\) 0 0
\(502\) 3.51718e8 0.124088
\(503\) 1.37007e9 0.480014 0.240007 0.970771i \(-0.422850\pi\)
0.240007 + 0.970771i \(0.422850\pi\)
\(504\) 0 0
\(505\) 7.49308e9 2.58905
\(506\) 4.35888e8 0.149571
\(507\) 0 0
\(508\) −2.41129e9 −0.816067
\(509\) 6.94286e8 0.233360 0.116680 0.993170i \(-0.462775\pi\)
0.116680 + 0.993170i \(0.462775\pi\)
\(510\) 0 0
\(511\) 7.04466e9 2.33554
\(512\) −2.45462e9 −0.808237
\(513\) 0 0
\(514\) −1.20004e8 −0.0389785
\(515\) −5.06748e9 −1.63481
\(516\) 0 0
\(517\) −4.21839e9 −1.34255
\(518\) −3.26285e8 −0.103144
\(519\) 0 0
\(520\) −2.52950e9 −0.788901
\(521\) 3.26480e9 1.01140 0.505702 0.862708i \(-0.331233\pi\)
0.505702 + 0.862708i \(0.331233\pi\)
\(522\) 0 0
\(523\) 1.44867e9 0.442806 0.221403 0.975182i \(-0.428936\pi\)
0.221403 + 0.975182i \(0.428936\pi\)
\(524\) −4.82553e8 −0.146516
\(525\) 0 0
\(526\) −3.14538e8 −0.0942372
\(527\) 5.93163e9 1.76537
\(528\) 0 0
\(529\) −2.69267e9 −0.790841
\(530\) −1.74872e9 −0.510217
\(531\) 0 0
\(532\) 3.87101e9 1.11464
\(533\) −6.57347e9 −1.88040
\(534\) 0 0
\(535\) −6.42412e9 −1.81374
\(536\) −2.21123e7 −0.00620237
\(537\) 0 0
\(538\) −6.70784e7 −0.0185714
\(539\) −5.88975e9 −1.62008
\(540\) 0 0
\(541\) 5.30499e9 1.44044 0.720218 0.693748i \(-0.244042\pi\)
0.720218 + 0.693748i \(0.244042\pi\)
\(542\) −994253. −0.000268225 0
\(543\) 0 0
\(544\) 2.21602e9 0.590170
\(545\) 1.28513e9 0.340064
\(546\) 0 0
\(547\) 2.77315e9 0.724465 0.362232 0.932088i \(-0.382015\pi\)
0.362232 + 0.932088i \(0.382015\pi\)
\(548\) −3.13366e9 −0.813430
\(549\) 0 0
\(550\) 3.43617e9 0.880654
\(551\) 2.51373e8 0.0640160
\(552\) 0 0
\(553\) −4.71960e9 −1.18677
\(554\) −8.76964e8 −0.219128
\(555\) 0 0
\(556\) −5.46317e9 −1.34798
\(557\) 2.33572e9 0.572701 0.286350 0.958125i \(-0.407558\pi\)
0.286350 + 0.958125i \(0.407558\pi\)
\(558\) 0 0
\(559\) 1.47579e9 0.357341
\(560\) 9.93466e9 2.39053
\(561\) 0 0
\(562\) 1.03775e9 0.246612
\(563\) −5.53231e9 −1.30655 −0.653276 0.757120i \(-0.726606\pi\)
−0.653276 + 0.757120i \(0.726606\pi\)
\(564\) 0 0
\(565\) 2.20107e9 0.513411
\(566\) 6.55663e8 0.151993
\(567\) 0 0
\(568\) 8.38906e8 0.192085
\(569\) −5.01743e8 −0.114180 −0.0570898 0.998369i \(-0.518182\pi\)
−0.0570898 + 0.998369i \(0.518182\pi\)
\(570\) 0 0
\(571\) −1.45684e9 −0.327481 −0.163741 0.986503i \(-0.552356\pi\)
−0.163741 + 0.986503i \(0.552356\pi\)
\(572\) −8.46162e9 −1.89046
\(573\) 0 0
\(574\) −1.97378e9 −0.435619
\(575\) 5.61399e9 1.23150
\(576\) 0 0
\(577\) −5.26247e9 −1.14044 −0.570222 0.821490i \(-0.693143\pi\)
−0.570222 + 0.821490i \(0.693143\pi\)
\(578\) −1.90178e8 −0.0409650
\(579\) 0 0
\(580\) 6.70230e8 0.142635
\(581\) 4.25234e9 0.899521
\(582\) 0 0
\(583\) −1.19110e10 −2.48948
\(584\) 2.97357e9 0.617780
\(585\) 0 0
\(586\) −6.22662e8 −0.127824
\(587\) 3.91337e9 0.798579 0.399290 0.916825i \(-0.369257\pi\)
0.399290 + 0.916825i \(0.369257\pi\)
\(588\) 0 0
\(589\) 6.59928e9 1.33074
\(590\) −2.33089e8 −0.0467240
\(591\) 0 0
\(592\) 1.80149e9 0.356866
\(593\) −6.52849e9 −1.28564 −0.642822 0.766015i \(-0.722237\pi\)
−0.642822 + 0.766015i \(0.722237\pi\)
\(594\) 0 0
\(595\) −1.51280e10 −2.94423
\(596\) 2.74409e8 0.0530929
\(597\) 0 0
\(598\) 4.99634e8 0.0955427
\(599\) −8.24579e9 −1.56761 −0.783805 0.621007i \(-0.786724\pi\)
−0.783805 + 0.621007i \(0.786724\pi\)
\(600\) 0 0
\(601\) −6.15461e9 −1.15648 −0.578242 0.815865i \(-0.696261\pi\)
−0.578242 + 0.815865i \(0.696261\pi\)
\(602\) 4.43127e8 0.0827828
\(603\) 0 0
\(604\) 4.97271e9 0.918256
\(605\) 2.16295e10 3.97102
\(606\) 0 0
\(607\) 7.75348e6 0.00140714 0.000703569 1.00000i \(-0.499776\pi\)
0.000703569 1.00000i \(0.499776\pi\)
\(608\) 2.46545e9 0.444870
\(609\) 0 0
\(610\) −3.11263e9 −0.555230
\(611\) −4.83529e9 −0.857588
\(612\) 0 0
\(613\) 5.41748e9 0.949917 0.474958 0.880008i \(-0.342463\pi\)
0.474958 + 0.880008i \(0.342463\pi\)
\(614\) −4.77628e8 −0.0832723
\(615\) 0 0
\(616\) −5.17327e9 −0.891728
\(617\) 1.53588e9 0.263245 0.131622 0.991300i \(-0.457981\pi\)
0.131622 + 0.991300i \(0.457981\pi\)
\(618\) 0 0
\(619\) −1.24887e9 −0.211641 −0.105821 0.994385i \(-0.533747\pi\)
−0.105821 + 0.994385i \(0.533747\pi\)
\(620\) 1.75955e10 2.96504
\(621\) 0 0
\(622\) −9.78875e8 −0.163103
\(623\) 6.84329e9 1.13385
\(624\) 0 0
\(625\) 2.17171e10 3.55813
\(626\) −1.31463e9 −0.214187
\(627\) 0 0
\(628\) −4.29065e9 −0.691296
\(629\) −2.74321e9 −0.439523
\(630\) 0 0
\(631\) −8.97235e9 −1.42169 −0.710843 0.703351i \(-0.751686\pi\)
−0.710843 + 0.703351i \(0.751686\pi\)
\(632\) −1.99216e9 −0.313916
\(633\) 0 0
\(634\) 1.73293e9 0.270065
\(635\) 1.04840e10 1.62487
\(636\) 0 0
\(637\) −6.75108e9 −1.03487
\(638\) −1.64988e8 −0.0251524
\(639\) 0 0
\(640\) 8.70750e9 1.31300
\(641\) 5.94956e9 0.892240 0.446120 0.894973i \(-0.352805\pi\)
0.446120 + 0.894973i \(0.352805\pi\)
\(642\) 0 0
\(643\) 8.03210e9 1.19149 0.595745 0.803173i \(-0.296857\pi\)
0.595745 + 0.803173i \(0.296857\pi\)
\(644\) −4.15101e9 −0.612426
\(645\) 0 0
\(646\) −1.17621e9 −0.171661
\(647\) 8.88936e9 1.29034 0.645172 0.764038i \(-0.276786\pi\)
0.645172 + 0.764038i \(0.276786\pi\)
\(648\) 0 0
\(649\) −1.58763e9 −0.227978
\(650\) 3.93868e9 0.562541
\(651\) 0 0
\(652\) 1.42842e9 0.201831
\(653\) 1.14448e10 1.60846 0.804231 0.594316i \(-0.202577\pi\)
0.804231 + 0.594316i \(0.202577\pi\)
\(654\) 0 0
\(655\) 2.09809e9 0.291729
\(656\) 1.08976e10 1.50719
\(657\) 0 0
\(658\) −1.45187e9 −0.198672
\(659\) −9.40014e9 −1.27949 −0.639743 0.768589i \(-0.720959\pi\)
−0.639743 + 0.768589i \(0.720959\pi\)
\(660\) 0 0
\(661\) 1.37618e9 0.185341 0.0926704 0.995697i \(-0.470460\pi\)
0.0926704 + 0.995697i \(0.470460\pi\)
\(662\) 1.05443e9 0.141259
\(663\) 0 0
\(664\) 1.79492e9 0.237935
\(665\) −1.68307e10 −2.21936
\(666\) 0 0
\(667\) −2.69556e8 −0.0351729
\(668\) −1.32043e10 −1.71395
\(669\) 0 0
\(670\) 4.72177e7 0.00606517
\(671\) −2.12009e10 −2.70910
\(672\) 0 0
\(673\) 9.13119e9 1.15471 0.577357 0.816492i \(-0.304084\pi\)
0.577357 + 0.816492i \(0.304084\pi\)
\(674\) 4.57331e8 0.0575335
\(675\) 0 0
\(676\) −1.94741e9 −0.242462
\(677\) −6.14968e9 −0.761715 −0.380857 0.924634i \(-0.624371\pi\)
−0.380857 + 0.924634i \(0.624371\pi\)
\(678\) 0 0
\(679\) −1.44047e10 −1.76587
\(680\) −6.38556e9 −0.778785
\(681\) 0 0
\(682\) −4.33141e9 −0.522858
\(683\) 1.15096e10 1.38225 0.691127 0.722734i \(-0.257115\pi\)
0.691127 + 0.722734i \(0.257115\pi\)
\(684\) 0 0
\(685\) 1.36248e10 1.61962
\(686\) 1.63978e8 0.0193932
\(687\) 0 0
\(688\) −2.44659e9 −0.286419
\(689\) −1.36529e10 −1.59022
\(690\) 0 0
\(691\) −6.76616e9 −0.780134 −0.390067 0.920786i \(-0.627548\pi\)
−0.390067 + 0.920786i \(0.627548\pi\)
\(692\) −9.06286e9 −1.03967
\(693\) 0 0
\(694\) 5.26173e8 0.0597545
\(695\) 2.37533e10 2.68396
\(696\) 0 0
\(697\) −1.65943e10 −1.85628
\(698\) 2.70596e9 0.301181
\(699\) 0 0
\(700\) −3.27230e10 −3.60587
\(701\) −2.85225e9 −0.312734 −0.156367 0.987699i \(-0.549978\pi\)
−0.156367 + 0.987699i \(0.549978\pi\)
\(702\) 0 0
\(703\) −3.05197e9 −0.331312
\(704\) 1.29166e10 1.39523
\(705\) 0 0
\(706\) 2.14227e8 0.0229117
\(707\) 1.75658e10 1.86939
\(708\) 0 0
\(709\) 1.72710e10 1.81994 0.909969 0.414676i \(-0.136105\pi\)
0.909969 + 0.414676i \(0.136105\pi\)
\(710\) −1.79137e9 −0.187837
\(711\) 0 0
\(712\) 2.88858e9 0.299919
\(713\) −7.07662e9 −0.731161
\(714\) 0 0
\(715\) 3.67902e10 3.76410
\(716\) −9.79879e9 −0.997647
\(717\) 0 0
\(718\) −1.98207e9 −0.199840
\(719\) −6.37828e9 −0.639959 −0.319980 0.947424i \(-0.603676\pi\)
−0.319980 + 0.947424i \(0.603676\pi\)
\(720\) 0 0
\(721\) −1.18795e10 −1.18039
\(722\) 5.80130e8 0.0573648
\(723\) 0 0
\(724\) −5.89010e9 −0.576816
\(725\) −2.12495e9 −0.207093
\(726\) 0 0
\(727\) 1.75274e10 1.69179 0.845895 0.533350i \(-0.179067\pi\)
0.845895 + 0.533350i \(0.179067\pi\)
\(728\) −5.92982e9 −0.569615
\(729\) 0 0
\(730\) −6.34965e9 −0.604115
\(731\) 3.72554e9 0.352759
\(732\) 0 0
\(733\) 6.90854e9 0.647922 0.323961 0.946070i \(-0.394985\pi\)
0.323961 + 0.946070i \(0.394985\pi\)
\(734\) −1.56435e9 −0.146015
\(735\) 0 0
\(736\) −2.64378e9 −0.244429
\(737\) 3.21612e8 0.0295935
\(738\) 0 0
\(739\) 1.05720e10 0.963610 0.481805 0.876278i \(-0.339981\pi\)
0.481805 + 0.876278i \(0.339981\pi\)
\(740\) −8.13739e9 −0.738200
\(741\) 0 0
\(742\) −4.09947e9 −0.368395
\(743\) −1.14806e10 −1.02685 −0.513423 0.858136i \(-0.671623\pi\)
−0.513423 + 0.858136i \(0.671623\pi\)
\(744\) 0 0
\(745\) −1.19310e9 −0.105713
\(746\) −2.22687e9 −0.196386
\(747\) 0 0
\(748\) −2.13608e10 −1.86622
\(749\) −1.50598e10 −1.30959
\(750\) 0 0
\(751\) −6.77468e9 −0.583645 −0.291823 0.956472i \(-0.594262\pi\)
−0.291823 + 0.956472i \(0.594262\pi\)
\(752\) 8.01604e9 0.687381
\(753\) 0 0
\(754\) −1.89116e8 −0.0160668
\(755\) −2.16208e10 −1.82834
\(756\) 0 0
\(757\) −1.20655e10 −1.01090 −0.505451 0.862855i \(-0.668674\pi\)
−0.505451 + 0.862855i \(0.668674\pi\)
\(758\) −2.46311e9 −0.205420
\(759\) 0 0
\(760\) −7.10430e9 −0.587048
\(761\) 6.97304e9 0.573556 0.286778 0.957997i \(-0.407416\pi\)
0.286778 + 0.957997i \(0.407416\pi\)
\(762\) 0 0
\(763\) 3.01270e9 0.245539
\(764\) −7.00149e9 −0.568020
\(765\) 0 0
\(766\) −3.59798e7 −0.00289240
\(767\) −1.81981e9 −0.145627
\(768\) 0 0
\(769\) −6.18686e9 −0.490600 −0.245300 0.969447i \(-0.578887\pi\)
−0.245300 + 0.969447i \(0.578887\pi\)
\(770\) 1.10468e10 0.872003
\(771\) 0 0
\(772\) 2.18208e10 1.70691
\(773\) −9.77041e9 −0.760825 −0.380412 0.924817i \(-0.624218\pi\)
−0.380412 + 0.924817i \(0.624218\pi\)
\(774\) 0 0
\(775\) −5.57860e10 −4.30496
\(776\) −6.08025e9 −0.467096
\(777\) 0 0
\(778\) 7.58918e8 0.0577785
\(779\) −1.84621e10 −1.39927
\(780\) 0 0
\(781\) −1.22014e10 −0.916501
\(782\) 1.26129e9 0.0943176
\(783\) 0 0
\(784\) 1.11921e10 0.829477
\(785\) 1.86553e10 1.37644
\(786\) 0 0
\(787\) −3.17612e9 −0.232266 −0.116133 0.993234i \(-0.537050\pi\)
−0.116133 + 0.993234i \(0.537050\pi\)
\(788\) 1.57260e10 1.14492
\(789\) 0 0
\(790\) 4.25397e9 0.306973
\(791\) 5.15990e9 0.370701
\(792\) 0 0
\(793\) −2.43014e10 −1.73051
\(794\) 2.82203e9 0.200073
\(795\) 0 0
\(796\) −1.80148e10 −1.26600
\(797\) 1.64367e10 1.15003 0.575016 0.818142i \(-0.304996\pi\)
0.575016 + 0.818142i \(0.304996\pi\)
\(798\) 0 0
\(799\) −1.22064e10 −0.846591
\(800\) −2.08413e10 −1.43916
\(801\) 0 0
\(802\) 4.70294e9 0.321928
\(803\) −4.32491e10 −2.94762
\(804\) 0 0
\(805\) 1.80481e10 1.21940
\(806\) −4.96485e9 −0.333990
\(807\) 0 0
\(808\) 7.41457e9 0.494477
\(809\) −6.28475e9 −0.417319 −0.208659 0.977988i \(-0.566910\pi\)
−0.208659 + 0.977988i \(0.566910\pi\)
\(810\) 0 0
\(811\) 2.31027e10 1.52086 0.760432 0.649418i \(-0.224987\pi\)
0.760432 + 0.649418i \(0.224987\pi\)
\(812\) 1.57120e9 0.102988
\(813\) 0 0
\(814\) 2.00315e9 0.130175
\(815\) −6.21060e9 −0.401866
\(816\) 0 0
\(817\) 4.14487e9 0.265910
\(818\) −3.05683e9 −0.195270
\(819\) 0 0
\(820\) −4.92250e10 −3.11772
\(821\) 2.91799e10 1.84027 0.920137 0.391597i \(-0.128077\pi\)
0.920137 + 0.391597i \(0.128077\pi\)
\(822\) 0 0
\(823\) 2.76539e10 1.72925 0.864624 0.502419i \(-0.167557\pi\)
0.864624 + 0.502419i \(0.167557\pi\)
\(824\) −5.01439e9 −0.312229
\(825\) 0 0
\(826\) −5.46423e8 −0.0337364
\(827\) 5.03285e9 0.309417 0.154709 0.987960i \(-0.450556\pi\)
0.154709 + 0.987960i \(0.450556\pi\)
\(828\) 0 0
\(829\) 1.94317e10 1.18459 0.592296 0.805720i \(-0.298222\pi\)
0.592296 + 0.805720i \(0.298222\pi\)
\(830\) −3.83281e9 −0.232672
\(831\) 0 0
\(832\) 1.48056e10 0.891239
\(833\) −1.70427e10 −1.02160
\(834\) 0 0
\(835\) 5.74111e10 3.41266
\(836\) −2.37651e10 −1.40676
\(837\) 0 0
\(838\) −4.05069e9 −0.237780
\(839\) −1.03412e10 −0.604509 −0.302255 0.953227i \(-0.597739\pi\)
−0.302255 + 0.953227i \(0.597739\pi\)
\(840\) 0 0
\(841\) −1.71478e10 −0.994085
\(842\) 3.76169e9 0.217165
\(843\) 0 0
\(844\) 1.94268e10 1.11225
\(845\) 8.46713e9 0.482767
\(846\) 0 0
\(847\) 5.07052e10 2.86722
\(848\) 2.26340e10 1.27461
\(849\) 0 0
\(850\) 9.94296e9 0.555328
\(851\) 3.27273e9 0.182036
\(852\) 0 0
\(853\) 2.35292e10 1.29803 0.649016 0.760775i \(-0.275181\pi\)
0.649016 + 0.760775i \(0.275181\pi\)
\(854\) −7.29683e9 −0.400896
\(855\) 0 0
\(856\) −6.35680e9 −0.346402
\(857\) 2.62808e10 1.42628 0.713142 0.701020i \(-0.247272\pi\)
0.713142 + 0.701020i \(0.247272\pi\)
\(858\) 0 0
\(859\) 1.11154e9 0.0598339 0.0299170 0.999552i \(-0.490476\pi\)
0.0299170 + 0.999552i \(0.490476\pi\)
\(860\) 1.10514e10 0.592476
\(861\) 0 0
\(862\) 6.97122e8 0.0370709
\(863\) 1.39838e10 0.740604 0.370302 0.928911i \(-0.379254\pi\)
0.370302 + 0.928911i \(0.379254\pi\)
\(864\) 0 0
\(865\) 3.94043e10 2.07008
\(866\) 1.00554e8 0.00526125
\(867\) 0 0
\(868\) 4.12485e10 2.14086
\(869\) 2.89749e10 1.49780
\(870\) 0 0
\(871\) 3.68645e8 0.0189036
\(872\) 1.27167e9 0.0649481
\(873\) 0 0
\(874\) 1.40326e9 0.0710966
\(875\) 8.94392e10 4.51336
\(876\) 0 0
\(877\) 3.43482e10 1.71951 0.859756 0.510705i \(-0.170615\pi\)
0.859756 + 0.510705i \(0.170615\pi\)
\(878\) 6.43751e8 0.0320987
\(879\) 0 0
\(880\) −6.09915e10 −3.01703
\(881\) −3.32281e10 −1.63715 −0.818576 0.574398i \(-0.805236\pi\)
−0.818576 + 0.574398i \(0.805236\pi\)
\(882\) 0 0
\(883\) −2.10543e10 −1.02915 −0.514576 0.857445i \(-0.672051\pi\)
−0.514576 + 0.857445i \(0.672051\pi\)
\(884\) −2.44847e10 −1.19210
\(885\) 0 0
\(886\) 6.17547e9 0.298299
\(887\) −1.23601e10 −0.594688 −0.297344 0.954770i \(-0.596101\pi\)
−0.297344 + 0.954770i \(0.596101\pi\)
\(888\) 0 0
\(889\) 2.45773e10 1.17322
\(890\) −6.16814e9 −0.293285
\(891\) 0 0
\(892\) 1.92951e10 0.910271
\(893\) −1.35803e10 −0.638160
\(894\) 0 0
\(895\) 4.26041e10 1.98642
\(896\) 2.04127e10 0.948030
\(897\) 0 0
\(898\) −6.56528e9 −0.302542
\(899\) 2.67857e9 0.122954
\(900\) 0 0
\(901\) −3.44658e10 −1.56983
\(902\) 1.21175e10 0.549784
\(903\) 0 0
\(904\) 2.17801e9 0.0980551
\(905\) 2.56095e10 1.14850
\(906\) 0 0
\(907\) 1.40907e10 0.627058 0.313529 0.949579i \(-0.398489\pi\)
0.313529 + 0.949579i \(0.398489\pi\)
\(908\) −1.76727e10 −0.783433
\(909\) 0 0
\(910\) 1.26623e10 0.557015
\(911\) −2.85261e10 −1.25005 −0.625027 0.780603i \(-0.714912\pi\)
−0.625027 + 0.780603i \(0.714912\pi\)
\(912\) 0 0
\(913\) −2.61062e10 −1.13526
\(914\) 2.15657e8 0.00934228
\(915\) 0 0
\(916\) 1.11301e10 0.478480
\(917\) 4.91848e9 0.210639
\(918\) 0 0
\(919\) −1.21759e10 −0.517485 −0.258742 0.965946i \(-0.583308\pi\)
−0.258742 + 0.965946i \(0.583308\pi\)
\(920\) 7.61817e9 0.322547
\(921\) 0 0
\(922\) −4.15787e9 −0.174708
\(923\) −1.39858e10 −0.585439
\(924\) 0 0
\(925\) 2.57994e10 1.07180
\(926\) −3.54514e9 −0.146722
\(927\) 0 0
\(928\) 1.00070e9 0.0411040
\(929\) 4.26915e9 0.174697 0.0873487 0.996178i \(-0.472161\pi\)
0.0873487 + 0.996178i \(0.472161\pi\)
\(930\) 0 0
\(931\) −1.89610e10 −0.770081
\(932\) 1.95457e10 0.790854
\(933\) 0 0
\(934\) −2.53320e9 −0.101732
\(935\) 9.28746e10 3.71583
\(936\) 0 0
\(937\) 1.52673e10 0.606282 0.303141 0.952946i \(-0.401965\pi\)
0.303141 + 0.952946i \(0.401965\pi\)
\(938\) 1.10691e8 0.00437927
\(939\) 0 0
\(940\) −3.62088e10 −1.42189
\(941\) −3.60621e10 −1.41087 −0.705435 0.708775i \(-0.749248\pi\)
−0.705435 + 0.708775i \(0.749248\pi\)
\(942\) 0 0
\(943\) 1.97975e10 0.768813
\(944\) 3.01691e9 0.116724
\(945\) 0 0
\(946\) −2.72047e9 −0.104478
\(947\) −1.21747e10 −0.465836 −0.232918 0.972496i \(-0.574827\pi\)
−0.232918 + 0.972496i \(0.574827\pi\)
\(948\) 0 0
\(949\) −4.95739e10 −1.88287
\(950\) 1.10621e10 0.418606
\(951\) 0 0
\(952\) −1.49695e10 −0.562311
\(953\) 2.94721e10 1.10303 0.551514 0.834166i \(-0.314050\pi\)
0.551514 + 0.834166i \(0.314050\pi\)
\(954\) 0 0
\(955\) 3.04417e10 1.13099
\(956\) −9.62135e9 −0.356150
\(957\) 0 0
\(958\) −7.22493e9 −0.265494
\(959\) 3.19402e10 1.16943
\(960\) 0 0
\(961\) 4.28076e10 1.55593
\(962\) 2.29610e9 0.0831529
\(963\) 0 0
\(964\) −9.28174e9 −0.333703
\(965\) −9.48743e10 −3.39862
\(966\) 0 0
\(967\) −3.39812e10 −1.20850 −0.604250 0.796795i \(-0.706527\pi\)
−0.604250 + 0.796795i \(0.706527\pi\)
\(968\) 2.14028e10 0.758415
\(969\) 0 0
\(970\) 1.29835e10 0.456764
\(971\) 1.42757e9 0.0500415 0.0250208 0.999687i \(-0.492035\pi\)
0.0250208 + 0.999687i \(0.492035\pi\)
\(972\) 0 0
\(973\) 5.56840e10 1.93792
\(974\) 2.15769e9 0.0748226
\(975\) 0 0
\(976\) 4.02873e10 1.38705
\(977\) −2.69567e10 −0.924775 −0.462388 0.886678i \(-0.653007\pi\)
−0.462388 + 0.886678i \(0.653007\pi\)
\(978\) 0 0
\(979\) −4.20128e10 −1.43101
\(980\) −5.05551e10 −1.71583
\(981\) 0 0
\(982\) 7.60030e9 0.256118
\(983\) 1.23730e10 0.415469 0.207735 0.978185i \(-0.433391\pi\)
0.207735 + 0.978185i \(0.433391\pi\)
\(984\) 0 0
\(985\) −6.83749e10 −2.27966
\(986\) −4.77412e8 −0.0158608
\(987\) 0 0
\(988\) −2.72406e10 −0.898602
\(989\) −4.44468e9 −0.146101
\(990\) 0 0
\(991\) −6.08067e10 −1.98470 −0.992348 0.123474i \(-0.960597\pi\)
−0.992348 + 0.123474i \(0.960597\pi\)
\(992\) 2.62712e10 0.854454
\(993\) 0 0
\(994\) −4.19944e9 −0.135625
\(995\) 7.83264e10 2.52074
\(996\) 0 0
\(997\) 4.42391e10 1.41375 0.706876 0.707338i \(-0.250104\pi\)
0.706876 + 0.707338i \(0.250104\pi\)
\(998\) 2.47354e9 0.0787702
\(999\) 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 531.8.a.d.1.7 17
3.2 odd 2 177.8.a.b.1.11 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.11 17 3.2 odd 2
531.8.a.d.1.7 17 1.1 even 1 trivial