Properties

Label 165.6.a.a.1.2
Level $165$
Weight $6$
Character 165.1
Self dual yes
Analytic conductor $26.463$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.4633302691\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.34253.1
Defining polynomial: \( x^{3} - x^{2} - 52x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.921799\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.92180 q^{2} +9.00000 q^{3} -23.4631 q^{4} +25.0000 q^{5} -26.2962 q^{6} -85.0105 q^{7} +162.052 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-2.92180 q^{2} +9.00000 q^{3} -23.4631 q^{4} +25.0000 q^{5} -26.2962 q^{6} -85.0105 q^{7} +162.052 q^{8} +81.0000 q^{9} -73.0450 q^{10} -121.000 q^{11} -211.168 q^{12} +724.085 q^{13} +248.384 q^{14} +225.000 q^{15} +277.335 q^{16} -2098.75 q^{17} -236.666 q^{18} -6.40223 q^{19} -586.577 q^{20} -765.094 q^{21} +353.538 q^{22} +1569.80 q^{23} +1458.47 q^{24} +625.000 q^{25} -2115.63 q^{26} +729.000 q^{27} +1994.61 q^{28} -5145.94 q^{29} -657.405 q^{30} -1031.88 q^{31} -5995.98 q^{32} -1089.00 q^{33} +6132.14 q^{34} -2125.26 q^{35} -1900.51 q^{36} -12641.6 q^{37} +18.7060 q^{38} +6516.76 q^{39} +4051.30 q^{40} +13808.7 q^{41} +2235.45 q^{42} -17012.0 q^{43} +2839.03 q^{44} +2025.00 q^{45} -4586.64 q^{46} +8078.06 q^{47} +2496.02 q^{48} -9580.22 q^{49} -1826.12 q^{50} -18888.8 q^{51} -16989.3 q^{52} -22110.8 q^{53} -2129.99 q^{54} -3025.00 q^{55} -13776.1 q^{56} -57.6201 q^{57} +15035.4 q^{58} -16890.6 q^{59} -5279.20 q^{60} -34398.6 q^{61} +3014.94 q^{62} -6885.85 q^{63} +8644.32 q^{64} +18102.1 q^{65} +3181.84 q^{66} -37306.5 q^{67} +49243.3 q^{68} +14128.2 q^{69} +6209.59 q^{70} -56607.5 q^{71} +13126.2 q^{72} -27777.9 q^{73} +36936.2 q^{74} +5625.00 q^{75} +150.216 q^{76} +10286.3 q^{77} -19040.7 q^{78} -12759.9 q^{79} +6933.39 q^{80} +6561.00 q^{81} -40346.2 q^{82} -69258.6 q^{83} +17951.5 q^{84} -52468.9 q^{85} +49705.6 q^{86} -46313.5 q^{87} -19608.3 q^{88} +59029.2 q^{89} -5916.64 q^{90} -61554.8 q^{91} -36832.4 q^{92} -9286.89 q^{93} -23602.5 q^{94} -160.056 q^{95} -53963.8 q^{96} +104905. q^{97} +27991.5 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9} - 175 q^{10} - 363 q^{11} + 225 q^{12} - 654 q^{13} - 728 q^{14} + 675 q^{15} - 415 q^{16} - 2366 q^{17} - 567 q^{18} - 2872 q^{19} + 625 q^{20} - 1548 q^{21} + 847 q^{22} + 2272 q^{23} - 2079 q^{24} + 1875 q^{25} + 3422 q^{26} + 2187 q^{27} + 4592 q^{28} - 7738 q^{29} - 1575 q^{30} + 568 q^{31} + 1001 q^{32} - 3267 q^{33} + 2506 q^{34} - 4300 q^{35} + 2025 q^{36} - 9126 q^{37} + 13076 q^{38} - 5886 q^{39} - 5775 q^{40} - 8758 q^{41} - 6552 q^{42} - 14672 q^{43} - 3025 q^{44} + 6075 q^{45} - 28768 q^{46} - 19392 q^{47} - 3735 q^{48} - 26629 q^{49} - 4375 q^{50} - 21294 q^{51} - 61506 q^{52} - 4598 q^{53} - 5103 q^{54} - 9075 q^{55} + 2688 q^{56} - 25848 q^{57} + 8550 q^{58} - 9348 q^{59} + 5625 q^{60} - 60078 q^{61} - 14096 q^{62} - 13932 q^{63} - 7087 q^{64} - 16350 q^{65} + 7623 q^{66} - 38468 q^{67} + 59778 q^{68} + 20448 q^{69} - 18200 q^{70} - 74032 q^{71} - 18711 q^{72} - 44442 q^{73} + 82542 q^{74} + 16875 q^{75} - 98708 q^{76} + 20812 q^{77} + 30798 q^{78} - 108116 q^{79} - 10375 q^{80} + 19683 q^{81} - 92230 q^{82} - 81892 q^{83} + 41328 q^{84} - 59150 q^{85} + 126412 q^{86} - 69642 q^{87} + 27951 q^{88} + 167342 q^{89} - 14175 q^{90} - 31832 q^{91} + 72960 q^{92} + 5112 q^{93} + 12728 q^{94} - 71800 q^{95} + 9009 q^{96} + 159702 q^{97} + 163121 q^{98} - 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.92180 −0.516506 −0.258253 0.966077i \(-0.583147\pi\)
−0.258253 + 0.966077i \(0.583147\pi\)
\(3\) 9.00000 0.577350
\(4\) −23.4631 −0.733222
\(5\) 25.0000 0.447214
\(6\) −26.2962 −0.298205
\(7\) −85.0105 −0.655734 −0.327867 0.944724i \(-0.606330\pi\)
−0.327867 + 0.944724i \(0.606330\pi\)
\(8\) 162.052 0.895219
\(9\) 81.0000 0.333333
\(10\) −73.0450 −0.230988
\(11\) −121.000 −0.301511
\(12\) −211.168 −0.423326
\(13\) 724.085 1.18831 0.594157 0.804349i \(-0.297486\pi\)
0.594157 + 0.804349i \(0.297486\pi\)
\(14\) 248.384 0.338690
\(15\) 225.000 0.258199
\(16\) 277.335 0.270835
\(17\) −2098.75 −1.76132 −0.880662 0.473745i \(-0.842902\pi\)
−0.880662 + 0.473745i \(0.842902\pi\)
\(18\) −236.666 −0.172169
\(19\) −6.40223 −0.00406862 −0.00203431 0.999998i \(-0.500648\pi\)
−0.00203431 + 0.999998i \(0.500648\pi\)
\(20\) −586.577 −0.327907
\(21\) −765.094 −0.378588
\(22\) 353.538 0.155732
\(23\) 1569.80 0.618764 0.309382 0.950938i \(-0.399878\pi\)
0.309382 + 0.950938i \(0.399878\pi\)
\(24\) 1458.47 0.516855
\(25\) 625.000 0.200000
\(26\) −2115.63 −0.613771
\(27\) 729.000 0.192450
\(28\) 1994.61 0.480798
\(29\) −5145.94 −1.13624 −0.568119 0.822946i \(-0.692329\pi\)
−0.568119 + 0.822946i \(0.692329\pi\)
\(30\) −657.405 −0.133361
\(31\) −1031.88 −0.192852 −0.0964259 0.995340i \(-0.530741\pi\)
−0.0964259 + 0.995340i \(0.530741\pi\)
\(32\) −5995.98 −1.03511
\(33\) −1089.00 −0.174078
\(34\) 6132.14 0.909735
\(35\) −2125.26 −0.293253
\(36\) −1900.51 −0.244407
\(37\) −12641.6 −1.51809 −0.759045 0.651039i \(-0.774334\pi\)
−0.759045 + 0.651039i \(0.774334\pi\)
\(38\) 18.7060 0.00210147
\(39\) 6516.76 0.686073
\(40\) 4051.30 0.400354
\(41\) 13808.7 1.28290 0.641450 0.767165i \(-0.278333\pi\)
0.641450 + 0.767165i \(0.278333\pi\)
\(42\) 2235.45 0.195543
\(43\) −17012.0 −1.40308 −0.701542 0.712628i \(-0.747505\pi\)
−0.701542 + 0.712628i \(0.747505\pi\)
\(44\) 2839.03 0.221075
\(45\) 2025.00 0.149071
\(46\) −4586.64 −0.319595
\(47\) 8078.06 0.533412 0.266706 0.963778i \(-0.414065\pi\)
0.266706 + 0.963778i \(0.414065\pi\)
\(48\) 2496.02 0.156367
\(49\) −9580.22 −0.570013
\(50\) −1826.12 −0.103301
\(51\) −18888.8 −1.01690
\(52\) −16989.3 −0.871297
\(53\) −22110.8 −1.08122 −0.540612 0.841272i \(-0.681807\pi\)
−0.540612 + 0.841272i \(0.681807\pi\)
\(54\) −2129.99 −0.0994016
\(55\) −3025.00 −0.134840
\(56\) −13776.1 −0.587025
\(57\) −57.6201 −0.00234902
\(58\) 15035.4 0.586874
\(59\) −16890.6 −0.631706 −0.315853 0.948808i \(-0.602291\pi\)
−0.315853 + 0.948808i \(0.602291\pi\)
\(60\) −5279.20 −0.189317
\(61\) −34398.6 −1.18363 −0.591816 0.806073i \(-0.701589\pi\)
−0.591816 + 0.806073i \(0.701589\pi\)
\(62\) 3014.94 0.0996091
\(63\) −6885.85 −0.218578
\(64\) 8644.32 0.263804
\(65\) 18102.1 0.531430
\(66\) 3181.84 0.0899122
\(67\) −37306.5 −1.01531 −0.507654 0.861561i \(-0.669487\pi\)
−0.507654 + 0.861561i \(0.669487\pi\)
\(68\) 49243.3 1.29144
\(69\) 14128.2 0.357244
\(70\) 6209.59 0.151467
\(71\) −56607.5 −1.33269 −0.666344 0.745645i \(-0.732142\pi\)
−0.666344 + 0.745645i \(0.732142\pi\)
\(72\) 13126.2 0.298406
\(73\) −27777.9 −0.610088 −0.305044 0.952338i \(-0.598671\pi\)
−0.305044 + 0.952338i \(0.598671\pi\)
\(74\) 36936.2 0.784102
\(75\) 5625.00 0.115470
\(76\) 150.216 0.00298320
\(77\) 10286.3 0.197711
\(78\) −19040.7 −0.354361
\(79\) −12759.9 −0.230028 −0.115014 0.993364i \(-0.536691\pi\)
−0.115014 + 0.993364i \(0.536691\pi\)
\(80\) 6933.39 0.121121
\(81\) 6561.00 0.111111
\(82\) −40346.2 −0.662625
\(83\) −69258.6 −1.10352 −0.551758 0.834004i \(-0.686043\pi\)
−0.551758 + 0.834004i \(0.686043\pi\)
\(84\) 17951.5 0.277589
\(85\) −52468.9 −0.787688
\(86\) 49705.6 0.724702
\(87\) −46313.5 −0.656008
\(88\) −19608.3 −0.269919
\(89\) 59029.2 0.789936 0.394968 0.918695i \(-0.370756\pi\)
0.394968 + 0.918695i \(0.370756\pi\)
\(90\) −5916.64 −0.0769962
\(91\) −61554.8 −0.779217
\(92\) −36832.4 −0.453691
\(93\) −9286.89 −0.111343
\(94\) −23602.5 −0.275510
\(95\) −160.056 −0.00181954
\(96\) −53963.8 −0.597620
\(97\) 104905. 1.13205 0.566027 0.824387i \(-0.308480\pi\)
0.566027 + 0.824387i \(0.308480\pi\)
\(98\) 27991.5 0.294415
\(99\) −9801.00 −0.100504
\(100\) −14664.4 −0.146644
\(101\) 135417. 1.32090 0.660451 0.750869i \(-0.270365\pi\)
0.660451 + 0.750869i \(0.270365\pi\)
\(102\) 55189.3 0.525236
\(103\) 167505. 1.55573 0.777867 0.628428i \(-0.216302\pi\)
0.777867 + 0.628428i \(0.216302\pi\)
\(104\) 117339. 1.06380
\(105\) −19127.4 −0.169310
\(106\) 64603.4 0.558458
\(107\) 57989.6 0.489656 0.244828 0.969567i \(-0.421269\pi\)
0.244828 + 0.969567i \(0.421269\pi\)
\(108\) −17104.6 −0.141109
\(109\) −14956.1 −0.120574 −0.0602869 0.998181i \(-0.519202\pi\)
−0.0602869 + 0.998181i \(0.519202\pi\)
\(110\) 8838.44 0.0696457
\(111\) −113774. −0.876469
\(112\) −23576.4 −0.177596
\(113\) 198323. 1.46109 0.730544 0.682866i \(-0.239267\pi\)
0.730544 + 0.682866i \(0.239267\pi\)
\(114\) 168.354 0.00121328
\(115\) 39245.0 0.276720
\(116\) 120740. 0.833115
\(117\) 58650.9 0.396105
\(118\) 49351.0 0.326280
\(119\) 178416. 1.15496
\(120\) 36461.7 0.231145
\(121\) 14641.0 0.0909091
\(122\) 100506. 0.611353
\(123\) 124278. 0.740682
\(124\) 24211.0 0.141403
\(125\) 15625.0 0.0894427
\(126\) 20119.1 0.112897
\(127\) −40986.5 −0.225492 −0.112746 0.993624i \(-0.535965\pi\)
−0.112746 + 0.993624i \(0.535965\pi\)
\(128\) 166614. 0.898851
\(129\) −153108. −0.810071
\(130\) −52890.8 −0.274487
\(131\) −238406. −1.21378 −0.606889 0.794787i \(-0.707583\pi\)
−0.606889 + 0.794787i \(0.707583\pi\)
\(132\) 25551.3 0.127637
\(133\) 544.257 0.00266793
\(134\) 109002. 0.524413
\(135\) 18225.0 0.0860663
\(136\) −340107. −1.57677
\(137\) 149534. 0.680673 0.340337 0.940304i \(-0.389459\pi\)
0.340337 + 0.940304i \(0.389459\pi\)
\(138\) −41279.8 −0.184518
\(139\) 167700. 0.736202 0.368101 0.929786i \(-0.380008\pi\)
0.368101 + 0.929786i \(0.380008\pi\)
\(140\) 49865.2 0.215019
\(141\) 72702.5 0.307965
\(142\) 165396. 0.688341
\(143\) −87614.3 −0.358290
\(144\) 22464.2 0.0902785
\(145\) −128648. −0.508142
\(146\) 81161.5 0.315114
\(147\) −86221.9 −0.329097
\(148\) 296611. 1.11310
\(149\) −272698. −1.00627 −0.503136 0.864207i \(-0.667821\pi\)
−0.503136 + 0.864207i \(0.667821\pi\)
\(150\) −16435.1 −0.0596410
\(151\) −448320. −1.60009 −0.800047 0.599937i \(-0.795192\pi\)
−0.800047 + 0.599937i \(0.795192\pi\)
\(152\) −1037.49 −0.00364231
\(153\) −169999. −0.587108
\(154\) −30054.4 −0.102119
\(155\) −25796.9 −0.0862459
\(156\) −152903. −0.503044
\(157\) −366634. −1.18709 −0.593545 0.804801i \(-0.702272\pi\)
−0.593545 + 0.804801i \(0.702272\pi\)
\(158\) 37281.9 0.118811
\(159\) −198998. −0.624245
\(160\) −149900. −0.462914
\(161\) −133450. −0.405744
\(162\) −19169.9 −0.0573896
\(163\) 501806. 1.47934 0.739668 0.672972i \(-0.234982\pi\)
0.739668 + 0.672972i \(0.234982\pi\)
\(164\) −323994. −0.940650
\(165\) −27225.0 −0.0778499
\(166\) 202360. 0.569973
\(167\) −195231. −0.541700 −0.270850 0.962622i \(-0.587305\pi\)
−0.270850 + 0.962622i \(0.587305\pi\)
\(168\) −123985. −0.338919
\(169\) 153006. 0.412089
\(170\) 153303. 0.406846
\(171\) −518.581 −0.00135621
\(172\) 399154. 1.02877
\(173\) 618180. 1.57036 0.785181 0.619266i \(-0.212570\pi\)
0.785181 + 0.619266i \(0.212570\pi\)
\(174\) 135319. 0.338832
\(175\) −53131.6 −0.131147
\(176\) −33557.6 −0.0816600
\(177\) −152015. −0.364716
\(178\) −172471. −0.408007
\(179\) −88898.6 −0.207378 −0.103689 0.994610i \(-0.533065\pi\)
−0.103689 + 0.994610i \(0.533065\pi\)
\(180\) −47512.8 −0.109302
\(181\) −378128. −0.857911 −0.428956 0.903326i \(-0.641118\pi\)
−0.428956 + 0.903326i \(0.641118\pi\)
\(182\) 179851. 0.402470
\(183\) −309588. −0.683370
\(184\) 254389. 0.553930
\(185\) −316040. −0.678910
\(186\) 27134.4 0.0575093
\(187\) 253949. 0.531059
\(188\) −189536. −0.391109
\(189\) −61972.7 −0.126196
\(190\) 467.651 0.000939805 0
\(191\) 125442. 0.248806 0.124403 0.992232i \(-0.460299\pi\)
0.124403 + 0.992232i \(0.460299\pi\)
\(192\) 77798.9 0.152307
\(193\) −928398. −1.79408 −0.897038 0.441953i \(-0.854286\pi\)
−0.897038 + 0.441953i \(0.854286\pi\)
\(194\) −306512. −0.584713
\(195\) 162919. 0.306821
\(196\) 224781. 0.417946
\(197\) 1.03233e6 1.89519 0.947595 0.319473i \(-0.103506\pi\)
0.947595 + 0.319473i \(0.103506\pi\)
\(198\) 28636.6 0.0519108
\(199\) −155017. −0.277489 −0.138744 0.990328i \(-0.544307\pi\)
−0.138744 + 0.990328i \(0.544307\pi\)
\(200\) 101283. 0.179044
\(201\) −335759. −0.586188
\(202\) −395662. −0.682254
\(203\) 437459. 0.745070
\(204\) 443189. 0.745614
\(205\) 345217. 0.573730
\(206\) −489417. −0.803546
\(207\) 127154. 0.206255
\(208\) 200814. 0.321837
\(209\) 774.670 0.00122674
\(210\) 55886.3 0.0874495
\(211\) −364263. −0.563259 −0.281630 0.959523i \(-0.590875\pi\)
−0.281630 + 0.959523i \(0.590875\pi\)
\(212\) 518789. 0.792777
\(213\) −509468. −0.769428
\(214\) −169434. −0.252910
\(215\) −425300. −0.627478
\(216\) 118136. 0.172285
\(217\) 87720.4 0.126459
\(218\) 43698.8 0.0622771
\(219\) −250001. −0.352234
\(220\) 70975.8 0.0988676
\(221\) −1.51968e6 −2.09301
\(222\) 332426. 0.452702
\(223\) −74806.0 −0.100734 −0.0503668 0.998731i \(-0.516039\pi\)
−0.0503668 + 0.998731i \(0.516039\pi\)
\(224\) 509721. 0.678755
\(225\) 50625.0 0.0666667
\(226\) −579459. −0.754661
\(227\) 1.22677e6 1.58015 0.790075 0.613010i \(-0.210042\pi\)
0.790075 + 0.613010i \(0.210042\pi\)
\(228\) 1351.94 0.00172235
\(229\) −440852. −0.555526 −0.277763 0.960650i \(-0.589593\pi\)
−0.277763 + 0.960650i \(0.589593\pi\)
\(230\) −114666. −0.142927
\(231\) 92576.4 0.114149
\(232\) −833910. −1.01718
\(233\) −514549. −0.620922 −0.310461 0.950586i \(-0.600483\pi\)
−0.310461 + 0.950586i \(0.600483\pi\)
\(234\) −171366. −0.204590
\(235\) 201952. 0.238549
\(236\) 396306. 0.463181
\(237\) −114839. −0.132807
\(238\) −521296. −0.596544
\(239\) −984315. −1.11465 −0.557326 0.830294i \(-0.688173\pi\)
−0.557326 + 0.830294i \(0.688173\pi\)
\(240\) 62400.5 0.0699294
\(241\) 284794. 0.315855 0.157927 0.987451i \(-0.449519\pi\)
0.157927 + 0.987451i \(0.449519\pi\)
\(242\) −42778.1 −0.0469551
\(243\) 59049.0 0.0641500
\(244\) 807098. 0.867865
\(245\) −239505. −0.254918
\(246\) −363116. −0.382567
\(247\) −4635.76 −0.00483480
\(248\) −167218. −0.172645
\(249\) −623328. −0.637115
\(250\) −45653.1 −0.0461977
\(251\) 134529. 0.134782 0.0673911 0.997727i \(-0.478532\pi\)
0.0673911 + 0.997727i \(0.478532\pi\)
\(252\) 161563. 0.160266
\(253\) −189946. −0.186564
\(254\) 119754. 0.116468
\(255\) −472220. −0.454772
\(256\) −763432. −0.728066
\(257\) 2.06732e6 1.95242 0.976212 0.216817i \(-0.0695674\pi\)
0.976212 + 0.216817i \(0.0695674\pi\)
\(258\) 447350. 0.418407
\(259\) 1.07467e6 0.995462
\(260\) −424732. −0.389656
\(261\) −416821. −0.378746
\(262\) 696575. 0.626924
\(263\) −661739. −0.589926 −0.294963 0.955509i \(-0.595307\pi\)
−0.294963 + 0.955509i \(0.595307\pi\)
\(264\) −176475. −0.155838
\(265\) −552771. −0.483538
\(266\) −1590.21 −0.00137800
\(267\) 531263. 0.456070
\(268\) 875327. 0.744446
\(269\) 703008. 0.592351 0.296176 0.955133i \(-0.404289\pi\)
0.296176 + 0.955133i \(0.404289\pi\)
\(270\) −53249.8 −0.0444538
\(271\) −1.33847e6 −1.10710 −0.553549 0.832817i \(-0.686727\pi\)
−0.553549 + 0.832817i \(0.686727\pi\)
\(272\) −582059. −0.477029
\(273\) −553993. −0.449881
\(274\) −436908. −0.351572
\(275\) −75625.0 −0.0603023
\(276\) −331492. −0.261939
\(277\) −33171.1 −0.0259753 −0.0129877 0.999916i \(-0.504134\pi\)
−0.0129877 + 0.999916i \(0.504134\pi\)
\(278\) −489987. −0.380253
\(279\) −83582.0 −0.0642839
\(280\) −344403. −0.262526
\(281\) −321114. −0.242601 −0.121301 0.992616i \(-0.538707\pi\)
−0.121301 + 0.992616i \(0.538707\pi\)
\(282\) −212422. −0.159066
\(283\) −1.90591e6 −1.41461 −0.707305 0.706908i \(-0.750089\pi\)
−0.707305 + 0.706908i \(0.750089\pi\)
\(284\) 1.32819e6 0.977155
\(285\) −1440.50 −0.00105051
\(286\) 255991. 0.185059
\(287\) −1.17388e6 −0.841240
\(288\) −485675. −0.345036
\(289\) 2.98491e6 2.10226
\(290\) 375885. 0.262458
\(291\) 944146. 0.653592
\(292\) 651756. 0.447330
\(293\) 272957. 0.185748 0.0928742 0.995678i \(-0.470395\pi\)
0.0928742 + 0.995678i \(0.470395\pi\)
\(294\) 251923. 0.169981
\(295\) −422265. −0.282508
\(296\) −2.04859e6 −1.35902
\(297\) −88209.0 −0.0580259
\(298\) 796768. 0.519746
\(299\) 1.13667e6 0.735286
\(300\) −131980. −0.0846651
\(301\) 1.44620e6 0.920050
\(302\) 1.30990e6 0.826459
\(303\) 1.21876e6 0.762624
\(304\) −1775.57 −0.00110193
\(305\) −859966. −0.529336
\(306\) 496703. 0.303245
\(307\) −843806. −0.510971 −0.255486 0.966813i \(-0.582235\pi\)
−0.255486 + 0.966813i \(0.582235\pi\)
\(308\) −241348. −0.144966
\(309\) 1.50755e6 0.898204
\(310\) 75373.4 0.0445465
\(311\) 2.06909e6 1.21305 0.606523 0.795066i \(-0.292564\pi\)
0.606523 + 0.795066i \(0.292564\pi\)
\(312\) 1.05605e6 0.614186
\(313\) 603113. 0.347967 0.173983 0.984749i \(-0.444336\pi\)
0.173983 + 0.984749i \(0.444336\pi\)
\(314\) 1.07123e6 0.613139
\(315\) −172146. −0.0977510
\(316\) 299387. 0.168661
\(317\) −1.67334e6 −0.935268 −0.467634 0.883922i \(-0.654893\pi\)
−0.467634 + 0.883922i \(0.654893\pi\)
\(318\) 581431. 0.322426
\(319\) 622659. 0.342589
\(320\) 216108. 0.117977
\(321\) 521907. 0.282703
\(322\) 389913. 0.209569
\(323\) 13436.7 0.00716616
\(324\) −153941. −0.0814691
\(325\) 452553. 0.237663
\(326\) −1.46618e6 −0.764086
\(327\) −134605. −0.0696133
\(328\) 2.23772e6 1.14848
\(329\) −686720. −0.349776
\(330\) 79546.0 0.0402099
\(331\) −2.20421e6 −1.10582 −0.552909 0.833242i \(-0.686482\pi\)
−0.552909 + 0.833242i \(0.686482\pi\)
\(332\) 1.62502e6 0.809122
\(333\) −1.02397e6 −0.506030
\(334\) 570427. 0.279791
\(335\) −932663. −0.454060
\(336\) −212188. −0.102535
\(337\) 1.07377e6 0.515036 0.257518 0.966273i \(-0.417095\pi\)
0.257518 + 0.966273i \(0.417095\pi\)
\(338\) −447052. −0.212847
\(339\) 1.78490e6 0.843559
\(340\) 1.23108e6 0.577550
\(341\) 124857. 0.0581470
\(342\) 1515.19 0.000700489 0
\(343\) 2.24319e6 1.02951
\(344\) −2.75683e6 −1.25607
\(345\) 353205. 0.159764
\(346\) −1.80620e6 −0.811101
\(347\) −1.39783e6 −0.623203 −0.311601 0.950213i \(-0.600865\pi\)
−0.311601 + 0.950213i \(0.600865\pi\)
\(348\) 1.08666e6 0.480999
\(349\) −2.66674e6 −1.17197 −0.585985 0.810322i \(-0.699292\pi\)
−0.585985 + 0.810322i \(0.699292\pi\)
\(350\) 155240. 0.0677381
\(351\) 527858. 0.228691
\(352\) 725514. 0.312097
\(353\) −594048. −0.253738 −0.126869 0.991920i \(-0.540493\pi\)
−0.126869 + 0.991920i \(0.540493\pi\)
\(354\) 444159. 0.188378
\(355\) −1.41519e6 −0.595996
\(356\) −1.38501e6 −0.579198
\(357\) 1.60575e6 0.666816
\(358\) 259744. 0.107112
\(359\) −3.35774e6 −1.37503 −0.687513 0.726172i \(-0.741298\pi\)
−0.687513 + 0.726172i \(0.741298\pi\)
\(360\) 328155. 0.133451
\(361\) −2.47606e6 −0.999983
\(362\) 1.10481e6 0.443116
\(363\) 131769. 0.0524864
\(364\) 1.44427e6 0.571339
\(365\) −694448. −0.272840
\(366\) 904553. 0.352965
\(367\) 2.58775e6 1.00290 0.501450 0.865187i \(-0.332800\pi\)
0.501450 + 0.865187i \(0.332800\pi\)
\(368\) 435362. 0.167583
\(369\) 1.11850e6 0.427633
\(370\) 923404. 0.350661
\(371\) 1.87965e6 0.708995
\(372\) 217899. 0.0816391
\(373\) −1.07376e6 −0.399608 −0.199804 0.979836i \(-0.564031\pi\)
−0.199804 + 0.979836i \(0.564031\pi\)
\(374\) −741989. −0.274295
\(375\) 140625. 0.0516398
\(376\) 1.30907e6 0.477520
\(377\) −3.72610e6 −1.35021
\(378\) 181072. 0.0651810
\(379\) 5.16745e6 1.84790 0.923949 0.382516i \(-0.124942\pi\)
0.923949 + 0.382516i \(0.124942\pi\)
\(380\) 3755.40 0.00133413
\(381\) −368878. −0.130188
\(382\) −366517. −0.128510
\(383\) 2.42092e6 0.843302 0.421651 0.906758i \(-0.361451\pi\)
0.421651 + 0.906758i \(0.361451\pi\)
\(384\) 1.49953e6 0.518952
\(385\) 257157. 0.0884191
\(386\) 2.71259e6 0.926651
\(387\) −1.37797e6 −0.467695
\(388\) −2.46140e6 −0.830047
\(389\) 1.92154e6 0.643835 0.321918 0.946768i \(-0.395673\pi\)
0.321918 + 0.946768i \(0.395673\pi\)
\(390\) −476017. −0.158475
\(391\) −3.29463e6 −1.08984
\(392\) −1.55249e6 −0.510287
\(393\) −2.14566e6 −0.700775
\(394\) −3.01626e6 −0.978877
\(395\) −318998. −0.102871
\(396\) 229962. 0.0736915
\(397\) 4.09572e6 1.30423 0.652114 0.758121i \(-0.273882\pi\)
0.652114 + 0.758121i \(0.273882\pi\)
\(398\) 452928. 0.143325
\(399\) 4898.31 0.00154033
\(400\) 173335. 0.0541671
\(401\) 5.97525e6 1.85565 0.927823 0.373022i \(-0.121678\pi\)
0.927823 + 0.373022i \(0.121678\pi\)
\(402\) 981020. 0.302770
\(403\) −747167. −0.229168
\(404\) −3.17731e6 −0.968514
\(405\) 164025. 0.0496904
\(406\) −1.27817e6 −0.384833
\(407\) 1.52963e6 0.457721
\(408\) −3.06097e6 −0.910350
\(409\) −1.92665e6 −0.569500 −0.284750 0.958602i \(-0.591911\pi\)
−0.284750 + 0.958602i \(0.591911\pi\)
\(410\) −1.00865e6 −0.296335
\(411\) 1.34581e6 0.392987
\(412\) −3.93019e6 −1.14070
\(413\) 1.43588e6 0.414231
\(414\) −371518. −0.106532
\(415\) −1.73147e6 −0.493507
\(416\) −4.34160e6 −1.23003
\(417\) 1.50930e6 0.425047
\(418\) −2263.43 −0.000633616 0
\(419\) −1.47857e6 −0.411441 −0.205720 0.978611i \(-0.565954\pi\)
−0.205720 + 0.978611i \(0.565954\pi\)
\(420\) 448787. 0.124142
\(421\) 4.82644e6 1.32715 0.663577 0.748108i \(-0.269037\pi\)
0.663577 + 0.748108i \(0.269037\pi\)
\(422\) 1.06430e6 0.290927
\(423\) 654323. 0.177804
\(424\) −3.58311e6 −0.967932
\(425\) −1.31172e6 −0.352265
\(426\) 1.48856e6 0.397414
\(427\) 2.92425e6 0.776147
\(428\) −1.36062e6 −0.359026
\(429\) −788528. −0.206859
\(430\) 1.24264e6 0.324096
\(431\) 2.89229e6 0.749978 0.374989 0.927029i \(-0.377647\pi\)
0.374989 + 0.927029i \(0.377647\pi\)
\(432\) 202178. 0.0521223
\(433\) 2.36700e6 0.606706 0.303353 0.952878i \(-0.401894\pi\)
0.303353 + 0.952878i \(0.401894\pi\)
\(434\) −256301. −0.0653170
\(435\) −1.15784e6 −0.293376
\(436\) 350917. 0.0884073
\(437\) −10050.2 −0.00251752
\(438\) 730453. 0.181931
\(439\) 3.22037e6 0.797525 0.398762 0.917054i \(-0.369440\pi\)
0.398762 + 0.917054i \(0.369440\pi\)
\(440\) −490207. −0.120711
\(441\) −775997. −0.190004
\(442\) 4.44019e6 1.08105
\(443\) −3.80335e6 −0.920782 −0.460391 0.887716i \(-0.652291\pi\)
−0.460391 + 0.887716i \(0.652291\pi\)
\(444\) 2.66950e6 0.642646
\(445\) 1.47573e6 0.353270
\(446\) 218568. 0.0520295
\(447\) −2.45428e6 −0.580972
\(448\) −734858. −0.172985
\(449\) −6.55303e6 −1.53400 −0.767001 0.641646i \(-0.778252\pi\)
−0.767001 + 0.641646i \(0.778252\pi\)
\(450\) −147916. −0.0344337
\(451\) −1.67085e6 −0.386809
\(452\) −4.65326e6 −1.07130
\(453\) −4.03488e6 −0.923815
\(454\) −3.58437e6 −0.816157
\(455\) −1.53887e6 −0.348477
\(456\) −9337.45 −0.00210289
\(457\) −7.41470e6 −1.66074 −0.830372 0.557209i \(-0.811872\pi\)
−0.830372 + 0.557209i \(0.811872\pi\)
\(458\) 1.28808e6 0.286932
\(459\) −1.52999e6 −0.338967
\(460\) −920810. −0.202897
\(461\) 409525. 0.0897487 0.0448743 0.998993i \(-0.485711\pi\)
0.0448743 + 0.998993i \(0.485711\pi\)
\(462\) −270490. −0.0589584
\(463\) 7.07192e6 1.53315 0.766575 0.642154i \(-0.221959\pi\)
0.766575 + 0.642154i \(0.221959\pi\)
\(464\) −1.42715e6 −0.307734
\(465\) −232172. −0.0497941
\(466\) 1.50341e6 0.320710
\(467\) −6.35423e6 −1.34825 −0.674125 0.738617i \(-0.735479\pi\)
−0.674125 + 0.738617i \(0.735479\pi\)
\(468\) −1.37613e6 −0.290432
\(469\) 3.17145e6 0.665772
\(470\) −590062. −0.123212
\(471\) −3.29970e6 −0.685366
\(472\) −2.73716e6 −0.565516
\(473\) 2.05845e6 0.423046
\(474\) 335537. 0.0685954
\(475\) −4001.39 −0.000813724 0
\(476\) −4.18619e6 −0.846841
\(477\) −1.79098e6 −0.360408
\(478\) 2.87597e6 0.575725
\(479\) 6.36751e6 1.26803 0.634017 0.773319i \(-0.281405\pi\)
0.634017 + 0.773319i \(0.281405\pi\)
\(480\) −1.34910e6 −0.267264
\(481\) −9.15358e6 −1.80397
\(482\) −832109. −0.163141
\(483\) −1.20105e6 −0.234257
\(484\) −343523. −0.0666565
\(485\) 2.62263e6 0.506270
\(486\) −172529. −0.0331339
\(487\) −2.94280e6 −0.562261 −0.281130 0.959670i \(-0.590709\pi\)
−0.281130 + 0.959670i \(0.590709\pi\)
\(488\) −5.57437e6 −1.05961
\(489\) 4.51626e6 0.854095
\(490\) 699787. 0.131667
\(491\) −501628. −0.0939027 −0.0469513 0.998897i \(-0.514951\pi\)
−0.0469513 + 0.998897i \(0.514951\pi\)
\(492\) −2.91595e6 −0.543084
\(493\) 1.08001e7 2.00129
\(494\) 13544.8 0.00249720
\(495\) −245025. −0.0449467
\(496\) −286176. −0.0522311
\(497\) 4.81224e6 0.873888
\(498\) 1.82124e6 0.329074
\(499\) −9.19784e6 −1.65362 −0.826808 0.562484i \(-0.809846\pi\)
−0.826808 + 0.562484i \(0.809846\pi\)
\(500\) −366611. −0.0655813
\(501\) −1.75708e6 −0.312750
\(502\) −393068. −0.0696158
\(503\) −6.53811e6 −1.15221 −0.576106 0.817375i \(-0.695428\pi\)
−0.576106 + 0.817375i \(0.695428\pi\)
\(504\) −1.11587e6 −0.195675
\(505\) 3.38543e6 0.590726
\(506\) 554984. 0.0963616
\(507\) 1.37705e6 0.237920
\(508\) 961669. 0.165336
\(509\) −1.49017e6 −0.254942 −0.127471 0.991842i \(-0.540686\pi\)
−0.127471 + 0.991842i \(0.540686\pi\)
\(510\) 1.37973e6 0.234892
\(511\) 2.36141e6 0.400055
\(512\) −3.10107e6 −0.522801
\(513\) −4667.23 −0.000783007 0
\(514\) −6.04029e6 −1.00844
\(515\) 4.18763e6 0.695746
\(516\) 3.59238e6 0.593962
\(517\) −977445. −0.160830
\(518\) −3.13996e6 −0.514162
\(519\) 5.56362e6 0.906649
\(520\) 2.93349e6 0.475746
\(521\) 3.82163e6 0.616814 0.308407 0.951255i \(-0.400204\pi\)
0.308407 + 0.951255i \(0.400204\pi\)
\(522\) 1.21787e6 0.195625
\(523\) −4.18273e6 −0.668660 −0.334330 0.942456i \(-0.608510\pi\)
−0.334330 + 0.942456i \(0.608510\pi\)
\(524\) 5.59375e6 0.889968
\(525\) −478184. −0.0757176
\(526\) 1.93347e6 0.304700
\(527\) 2.16566e6 0.339675
\(528\) −302018. −0.0471464
\(529\) −3.97207e6 −0.617131
\(530\) 1.61509e6 0.249750
\(531\) −1.36814e6 −0.210569
\(532\) −12769.9 −0.00195619
\(533\) 9.99866e6 1.52449
\(534\) −1.55224e6 −0.235563
\(535\) 1.44974e6 0.218981
\(536\) −6.04560e6 −0.908923
\(537\) −800088. −0.119730
\(538\) −2.05405e6 −0.305953
\(539\) 1.15921e6 0.171866
\(540\) −427615. −0.0631057
\(541\) 2.80424e6 0.411928 0.205964 0.978560i \(-0.433967\pi\)
0.205964 + 0.978560i \(0.433967\pi\)
\(542\) 3.91075e6 0.571823
\(543\) −3.40315e6 −0.495315
\(544\) 1.25841e7 1.82316
\(545\) −373903. −0.0539222
\(546\) 1.61866e6 0.232366
\(547\) −8.42403e6 −1.20379 −0.601896 0.798574i \(-0.705588\pi\)
−0.601896 + 0.798574i \(0.705588\pi\)
\(548\) −3.50853e6 −0.499084
\(549\) −2.78629e6 −0.394544
\(550\) 220961. 0.0311465
\(551\) 32945.5 0.00462293
\(552\) 2.28951e6 0.319811
\(553\) 1.08473e6 0.150837
\(554\) 96919.4 0.0134164
\(555\) −2.84436e6 −0.391969
\(556\) −3.93477e6 −0.539799
\(557\) −3.56702e6 −0.487155 −0.243577 0.969881i \(-0.578321\pi\)
−0.243577 + 0.969881i \(0.578321\pi\)
\(558\) 244210. 0.0332030
\(559\) −1.23181e7 −1.66730
\(560\) −589411. −0.0794233
\(561\) 2.28554e6 0.306607
\(562\) 938231. 0.125305
\(563\) 4.70552e6 0.625658 0.312829 0.949810i \(-0.398723\pi\)
0.312829 + 0.949810i \(0.398723\pi\)
\(564\) −1.70583e6 −0.225807
\(565\) 4.95807e6 0.653418
\(566\) 5.56869e6 0.730655
\(567\) −557754. −0.0728593
\(568\) −9.17337e6 −1.19305
\(569\) 2.60879e6 0.337799 0.168900 0.985633i \(-0.445979\pi\)
0.168900 + 0.985633i \(0.445979\pi\)
\(570\) 4208.86 0.000542597 0
\(571\) 1.13036e6 0.145086 0.0725431 0.997365i \(-0.476889\pi\)
0.0725431 + 0.997365i \(0.476889\pi\)
\(572\) 2.05570e6 0.262706
\(573\) 1.12898e6 0.143648
\(574\) 3.42985e6 0.434506
\(575\) 981126. 0.123753
\(576\) 700190. 0.0879346
\(577\) 1.12075e7 1.40142 0.700710 0.713446i \(-0.252867\pi\)
0.700710 + 0.713446i \(0.252867\pi\)
\(578\) −8.72132e6 −1.08583
\(579\) −8.35558e6 −1.03581
\(580\) 3.01849e6 0.372580
\(581\) 5.88771e6 0.723613
\(582\) −2.75861e6 −0.337584
\(583\) 2.67541e6 0.326001
\(584\) −4.50147e6 −0.546162
\(585\) 1.46627e6 0.177143
\(586\) −797526. −0.0959402
\(587\) 9.05798e6 1.08502 0.542508 0.840051i \(-0.317475\pi\)
0.542508 + 0.840051i \(0.317475\pi\)
\(588\) 2.02303e6 0.241301
\(589\) 6606.31 0.000784641 0
\(590\) 1.23377e6 0.145917
\(591\) 9.29097e6 1.09419
\(592\) −3.50596e6 −0.411152
\(593\) 1.00907e7 1.17837 0.589186 0.807997i \(-0.299448\pi\)
0.589186 + 0.807997i \(0.299448\pi\)
\(594\) 257729. 0.0299707
\(595\) 4.46040e6 0.516514
\(596\) 6.39833e6 0.737821
\(597\) −1.39515e6 −0.160208
\(598\) −3.32112e6 −0.379780
\(599\) −1.29126e7 −1.47044 −0.735222 0.677827i \(-0.762922\pi\)
−0.735222 + 0.677827i \(0.762922\pi\)
\(600\) 911543. 0.103371
\(601\) 1.82591e6 0.206202 0.103101 0.994671i \(-0.467124\pi\)
0.103101 + 0.994671i \(0.467124\pi\)
\(602\) −4.22550e6 −0.475211
\(603\) −3.02183e6 −0.338436
\(604\) 1.05190e7 1.17322
\(605\) 366025. 0.0406558
\(606\) −3.56096e6 −0.393900
\(607\) −1.26409e6 −0.139253 −0.0696267 0.997573i \(-0.522181\pi\)
−0.0696267 + 0.997573i \(0.522181\pi\)
\(608\) 38387.7 0.00421146
\(609\) 3.93713e6 0.430166
\(610\) 2.51265e6 0.273405
\(611\) 5.84920e6 0.633860
\(612\) 3.98870e6 0.430480
\(613\) 1.59125e7 1.71036 0.855180 0.518332i \(-0.173447\pi\)
0.855180 + 0.518332i \(0.173447\pi\)
\(614\) 2.46543e6 0.263920
\(615\) 3.10695e6 0.331243
\(616\) 1.66691e6 0.176995
\(617\) 85225.6 0.00901274 0.00450637 0.999990i \(-0.498566\pi\)
0.00450637 + 0.999990i \(0.498566\pi\)
\(618\) −4.40475e6 −0.463928
\(619\) −1.20387e7 −1.26285 −0.631425 0.775437i \(-0.717530\pi\)
−0.631425 + 0.775437i \(0.717530\pi\)
\(620\) 605276. 0.0632374
\(621\) 1.14439e6 0.119081
\(622\) −6.04545e6 −0.626546
\(623\) −5.01810e6 −0.517987
\(624\) 1.80733e6 0.185813
\(625\) 390625. 0.0400000
\(626\) −1.76218e6 −0.179727
\(627\) 6972.03 0.000708256 0
\(628\) 8.60236e6 0.870400
\(629\) 2.65316e7 2.67385
\(630\) 502977. 0.0504890
\(631\) 121417. 0.0121396 0.00606981 0.999982i \(-0.498068\pi\)
0.00606981 + 0.999982i \(0.498068\pi\)
\(632\) −2.06777e6 −0.205925
\(633\) −3.27836e6 −0.325198
\(634\) 4.88916e6 0.483071
\(635\) −1.02466e6 −0.100843
\(636\) 4.66910e6 0.457710
\(637\) −6.93689e6 −0.677355
\(638\) −1.81928e6 −0.176949
\(639\) −4.58521e6 −0.444229
\(640\) 4.16536e6 0.401978
\(641\) 1.25328e7 1.20477 0.602385 0.798206i \(-0.294217\pi\)
0.602385 + 0.798206i \(0.294217\pi\)
\(642\) −1.52491e6 −0.146018
\(643\) −3.45380e6 −0.329435 −0.164717 0.986341i \(-0.552671\pi\)
−0.164717 + 0.986341i \(0.552671\pi\)
\(644\) 3.13114e6 0.297501
\(645\) −3.82770e6 −0.362275
\(646\) −39259.4 −0.00370137
\(647\) 1.71115e7 1.60704 0.803521 0.595276i \(-0.202957\pi\)
0.803521 + 0.595276i \(0.202957\pi\)
\(648\) 1.06322e6 0.0994688
\(649\) 2.04376e6 0.190467
\(650\) −1.32227e6 −0.122754
\(651\) 789483. 0.0730114
\(652\) −1.17739e7 −1.08468
\(653\) 1.61622e7 1.48326 0.741630 0.670809i \(-0.234053\pi\)
0.741630 + 0.670809i \(0.234053\pi\)
\(654\) 393289. 0.0359557
\(655\) −5.96015e6 −0.542818
\(656\) 3.82964e6 0.347455
\(657\) −2.25001e6 −0.203363
\(658\) 2.00646e6 0.180661
\(659\) 9.54805e6 0.856449 0.428224 0.903672i \(-0.359139\pi\)
0.428224 + 0.903672i \(0.359139\pi\)
\(660\) 638783. 0.0570812
\(661\) −1.24374e7 −1.10720 −0.553599 0.832784i \(-0.686746\pi\)
−0.553599 + 0.832784i \(0.686746\pi\)
\(662\) 6.44027e6 0.571162
\(663\) −1.36771e7 −1.20840
\(664\) −1.12235e7 −0.987889
\(665\) 13606.4 0.00119314
\(666\) 2.99183e6 0.261367
\(667\) −8.07810e6 −0.703064
\(668\) 4.58073e6 0.397186
\(669\) −673254. −0.0581586
\(670\) 2.72505e6 0.234525
\(671\) 4.16224e6 0.356878
\(672\) 4.58749e6 0.391879
\(673\) 1.37772e7 1.17253 0.586266 0.810119i \(-0.300597\pi\)
0.586266 + 0.810119i \(0.300597\pi\)
\(674\) −3.13735e6 −0.266019
\(675\) 455625. 0.0384900
\(676\) −3.58999e6 −0.302153
\(677\) −1.06456e6 −0.0892683 −0.0446341 0.999003i \(-0.514212\pi\)
−0.0446341 + 0.999003i \(0.514212\pi\)
\(678\) −5.21513e6 −0.435703
\(679\) −8.91804e6 −0.742326
\(680\) −8.50268e6 −0.705154
\(681\) 1.10409e7 0.912300
\(682\) −364807. −0.0300333
\(683\) −1.48070e7 −1.21455 −0.607275 0.794492i \(-0.707737\pi\)
−0.607275 + 0.794492i \(0.707737\pi\)
\(684\) 12167.5 0.000994400 0
\(685\) 3.73835e6 0.304406
\(686\) −6.55415e6 −0.531748
\(687\) −3.96767e6 −0.320733
\(688\) −4.71803e6 −0.380005
\(689\) −1.60101e7 −1.28483
\(690\) −1.03200e6 −0.0825192
\(691\) 1.56554e7 1.24729 0.623646 0.781707i \(-0.285651\pi\)
0.623646 + 0.781707i \(0.285651\pi\)
\(692\) −1.45044e7 −1.15142
\(693\) 833188. 0.0659037
\(694\) 4.08417e6 0.321888
\(695\) 4.19251e6 0.329240
\(696\) −7.50519e6 −0.587271
\(697\) −2.89810e7 −2.25960
\(698\) 7.79167e6 0.605330
\(699\) −4.63094e6 −0.358490
\(700\) 1.24663e6 0.0961596
\(701\) 1.56474e7 1.20267 0.601334 0.798998i \(-0.294636\pi\)
0.601334 + 0.798998i \(0.294636\pi\)
\(702\) −1.54229e6 −0.118120
\(703\) 80934.4 0.00617653
\(704\) −1.04596e6 −0.0795398
\(705\) 1.81756e6 0.137726
\(706\) 1.73569e6 0.131057
\(707\) −1.15119e7 −0.866160
\(708\) 3.56675e6 0.267417
\(709\) −635875. −0.0475068 −0.0237534 0.999718i \(-0.507562\pi\)
−0.0237534 + 0.999718i \(0.507562\pi\)
\(710\) 4.13490e6 0.307836
\(711\) −1.03355e6 −0.0766759
\(712\) 9.56580e6 0.707166
\(713\) −1.61984e6 −0.119330
\(714\) −4.69167e6 −0.344415
\(715\) −2.19036e6 −0.160232
\(716\) 2.08584e6 0.152054
\(717\) −8.85884e6 −0.643545
\(718\) 9.81064e6 0.710209
\(719\) −2.50359e6 −0.180610 −0.0903048 0.995914i \(-0.528784\pi\)
−0.0903048 + 0.995914i \(0.528784\pi\)
\(720\) 561604. 0.0403738
\(721\) −1.42397e7 −1.02015
\(722\) 7.23454e6 0.516497
\(723\) 2.56314e6 0.182359
\(724\) 8.87205e6 0.629039
\(725\) −3.21621e6 −0.227248
\(726\) −385003. −0.0271095
\(727\) 8.82888e6 0.619540 0.309770 0.950811i \(-0.399748\pi\)
0.309770 + 0.950811i \(0.399748\pi\)
\(728\) −9.97508e6 −0.697570
\(729\) 531441. 0.0370370
\(730\) 2.02904e6 0.140923
\(731\) 3.57040e7 2.47129
\(732\) 7.26389e6 0.501062
\(733\) −2.93960e6 −0.202082 −0.101041 0.994882i \(-0.532217\pi\)
−0.101041 + 0.994882i \(0.532217\pi\)
\(734\) −7.56089e6 −0.518004
\(735\) −2.15555e6 −0.147177
\(736\) −9.41250e6 −0.640487
\(737\) 4.51409e6 0.306127
\(738\) −3.26804e6 −0.220875
\(739\) −1.51762e6 −0.102224 −0.0511120 0.998693i \(-0.516277\pi\)
−0.0511120 + 0.998693i \(0.516277\pi\)
\(740\) 7.41527e6 0.497792
\(741\) −41721.8 −0.00279137
\(742\) −5.49197e6 −0.366200
\(743\) −1.07165e7 −0.712168 −0.356084 0.934454i \(-0.615888\pi\)
−0.356084 + 0.934454i \(0.615888\pi\)
\(744\) −1.50496e6 −0.0996764
\(745\) −6.81744e6 −0.450019
\(746\) 3.13731e6 0.206400
\(747\) −5.60995e6 −0.367839
\(748\) −5.95844e6 −0.389384
\(749\) −4.92973e6 −0.321084
\(750\) −410878. −0.0266723
\(751\) −2.69694e7 −1.74490 −0.872452 0.488700i \(-0.837471\pi\)
−0.872452 + 0.488700i \(0.837471\pi\)
\(752\) 2.24033e6 0.144467
\(753\) 1.21076e6 0.0778165
\(754\) 1.08869e7 0.697391
\(755\) −1.12080e7 −0.715584
\(756\) 1.45407e6 0.0925296
\(757\) −2.81995e7 −1.78855 −0.894275 0.447518i \(-0.852308\pi\)
−0.894275 + 0.447518i \(0.852308\pi\)
\(758\) −1.50982e7 −0.954450
\(759\) −1.70951e6 −0.107713
\(760\) −25937.4 −0.00162889
\(761\) 5.29643e6 0.331529 0.165764 0.986165i \(-0.446991\pi\)
0.165764 + 0.986165i \(0.446991\pi\)
\(762\) 1.07779e6 0.0672428
\(763\) 1.27143e6 0.0790643
\(764\) −2.94326e6 −0.182430
\(765\) −4.24998e6 −0.262563
\(766\) −7.07344e6 −0.435571
\(767\) −1.22302e7 −0.750665
\(768\) −6.87089e6 −0.420349
\(769\) 6.09571e6 0.371714 0.185857 0.982577i \(-0.440494\pi\)
0.185857 + 0.982577i \(0.440494\pi\)
\(770\) −751360. −0.0456690
\(771\) 1.86059e7 1.12723
\(772\) 2.17831e7 1.31546
\(773\) 1.02486e7 0.616901 0.308451 0.951240i \(-0.400190\pi\)
0.308451 + 0.951240i \(0.400190\pi\)
\(774\) 4.02615e6 0.241567
\(775\) −644923. −0.0385704
\(776\) 1.70001e7 1.01344
\(777\) 9.67201e6 0.574730
\(778\) −5.61434e6 −0.332545
\(779\) −88406.4 −0.00521963
\(780\) −3.82259e6 −0.224968
\(781\) 6.84951e6 0.401820
\(782\) 9.62624e6 0.562911
\(783\) −3.75139e6 −0.218669
\(784\) −2.65693e6 −0.154380
\(785\) −9.16585e6 −0.530883
\(786\) 6.26917e6 0.361954
\(787\) −1.62333e7 −0.934267 −0.467133 0.884187i \(-0.654713\pi\)
−0.467133 + 0.884187i \(0.654713\pi\)
\(788\) −2.42217e7 −1.38959
\(789\) −5.95565e6 −0.340594
\(790\) 932048. 0.0531337
\(791\) −1.68595e7 −0.958084
\(792\) −1.58827e6 −0.0899729
\(793\) −2.49075e7 −1.40653
\(794\) −1.19669e7 −0.673642
\(795\) −4.97494e6 −0.279171
\(796\) 3.63717e6 0.203461
\(797\) 1.29153e7 0.720211 0.360105 0.932912i \(-0.382741\pi\)
0.360105 + 0.932912i \(0.382741\pi\)
\(798\) −14311.9 −0.000795590 0
\(799\) −1.69539e7 −0.939511
\(800\) −3.74749e6 −0.207021
\(801\) 4.78136e6 0.263312
\(802\) −1.74585e7 −0.958452
\(803\) 3.36113e6 0.183948
\(804\) 7.87794e6 0.429806
\(805\) −3.33624e6 −0.181454
\(806\) 2.18307e6 0.118367
\(807\) 6.32707e6 0.341994
\(808\) 2.19447e7 1.18250
\(809\) 1.40361e6 0.0754007 0.0377004 0.999289i \(-0.487997\pi\)
0.0377004 + 0.999289i \(0.487997\pi\)
\(810\) −479248. −0.0256654
\(811\) −1.27909e7 −0.682887 −0.341444 0.939902i \(-0.610916\pi\)
−0.341444 + 0.939902i \(0.610916\pi\)
\(812\) −1.02641e7 −0.546301
\(813\) −1.20462e7 −0.639183
\(814\) −4.46928e6 −0.236416
\(815\) 1.25452e7 0.661580
\(816\) −5.23853e6 −0.275413
\(817\) 108915. 0.00570862
\(818\) 5.62928e6 0.294150
\(819\) −4.98594e6 −0.259739
\(820\) −8.09986e6 −0.420671
\(821\) −5.20603e6 −0.269556 −0.134778 0.990876i \(-0.543032\pi\)
−0.134778 + 0.990876i \(0.543032\pi\)
\(822\) −3.93217e6 −0.202980
\(823\) 3.77702e7 1.94379 0.971896 0.235409i \(-0.0756429\pi\)
0.971896 + 0.235409i \(0.0756429\pi\)
\(824\) 2.71446e7 1.39272
\(825\) −680625. −0.0348155
\(826\) −4.19535e6 −0.213953
\(827\) −2.54890e7 −1.29595 −0.647976 0.761661i \(-0.724384\pi\)
−0.647976 + 0.761661i \(0.724384\pi\)
\(828\) −2.98342e6 −0.151230
\(829\) 1.71410e7 0.866262 0.433131 0.901331i \(-0.357409\pi\)
0.433131 + 0.901331i \(0.357409\pi\)
\(830\) 5.05900e6 0.254900
\(831\) −298540. −0.0149969
\(832\) 6.25922e6 0.313482
\(833\) 2.01065e7 1.00398
\(834\) −4.40988e6 −0.219539
\(835\) −4.88079e6 −0.242255
\(836\) −18176.1 −0.000899469 0
\(837\) −752238. −0.0371143
\(838\) 4.32009e6 0.212512
\(839\) −1.01094e7 −0.495818 −0.247909 0.968783i \(-0.579743\pi\)
−0.247909 + 0.968783i \(0.579743\pi\)
\(840\) −3.09963e6 −0.151569
\(841\) 5.96954e6 0.291039
\(842\) −1.41019e7 −0.685483
\(843\) −2.89003e6 −0.140066
\(844\) 8.54672e6 0.412994
\(845\) 3.82515e6 0.184292
\(846\) −1.91180e6 −0.0918368
\(847\) −1.24464e6 −0.0596121
\(848\) −6.13212e6 −0.292834
\(849\) −1.71532e7 −0.816726
\(850\) 3.83259e6 0.181947
\(851\) −1.98448e7 −0.939339
\(852\) 1.19537e7 0.564161
\(853\) 3.29141e7 1.54885 0.774425 0.632666i \(-0.218039\pi\)
0.774425 + 0.632666i \(0.218039\pi\)
\(854\) −8.54406e6 −0.400885
\(855\) −12964.5 −0.000606514 0
\(856\) 9.39734e6 0.438349
\(857\) 6.94241e6 0.322893 0.161446 0.986881i \(-0.448384\pi\)
0.161446 + 0.986881i \(0.448384\pi\)
\(858\) 2.30392e6 0.106844
\(859\) −1.84275e7 −0.852085 −0.426042 0.904703i \(-0.640093\pi\)
−0.426042 + 0.904703i \(0.640093\pi\)
\(860\) 9.97884e6 0.460081
\(861\) −1.05649e7 −0.485690
\(862\) −8.45068e6 −0.387368
\(863\) −2.60648e7 −1.19131 −0.595657 0.803239i \(-0.703108\pi\)
−0.595657 + 0.803239i \(0.703108\pi\)
\(864\) −4.37107e6 −0.199207
\(865\) 1.54545e7 0.702287
\(866\) −6.91589e6 −0.313367
\(867\) 2.68642e7 1.21374
\(868\) −2.05819e6 −0.0927228
\(869\) 1.54395e6 0.0693559
\(870\) 3.38297e6 0.151530
\(871\) −2.70131e7 −1.20650
\(872\) −2.42367e6 −0.107940
\(873\) 8.49731e6 0.377352
\(874\) 29364.8 0.00130031
\(875\) −1.32829e6 −0.0586506
\(876\) 5.86580e6 0.258266
\(877\) 1.86288e6 0.0817875 0.0408937 0.999164i \(-0.486979\pi\)
0.0408937 + 0.999164i \(0.486979\pi\)
\(878\) −9.40927e6 −0.411926
\(879\) 2.45661e6 0.107242
\(880\) −838940. −0.0365194
\(881\) 1.01838e7 0.442051 0.221025 0.975268i \(-0.429060\pi\)
0.221025 + 0.975268i \(0.429060\pi\)
\(882\) 2.26731e6 0.0981384
\(883\) 6.25954e6 0.270172 0.135086 0.990834i \(-0.456869\pi\)
0.135086 + 0.990834i \(0.456869\pi\)
\(884\) 3.56563e7 1.53464
\(885\) −3.80039e6 −0.163106
\(886\) 1.11126e7 0.475589
\(887\) 1.80319e7 0.769540 0.384770 0.923012i \(-0.374281\pi\)
0.384770 + 0.923012i \(0.374281\pi\)
\(888\) −1.84374e7 −0.784632
\(889\) 3.48428e6 0.147863
\(890\) −4.31179e6 −0.182466
\(891\) −793881. −0.0335013
\(892\) 1.75518e6 0.0738600
\(893\) −51717.6 −0.00217025
\(894\) 7.17091e6 0.300075
\(895\) −2.22247e6 −0.0927422
\(896\) −1.41640e7 −0.589407
\(897\) 1.02300e7 0.424518
\(898\) 1.91466e7 0.792321
\(899\) 5.30998e6 0.219126
\(900\) −1.18782e6 −0.0488814
\(901\) 4.64052e7 1.90439
\(902\) 4.88189e6 0.199789
\(903\) 1.30158e7 0.531191
\(904\) 3.21386e7 1.30799
\(905\) −9.45320e6 −0.383670
\(906\) 1.17891e7 0.477156
\(907\) 2.69733e7 1.08872 0.544360 0.838852i \(-0.316772\pi\)
0.544360 + 0.838852i \(0.316772\pi\)
\(908\) −2.87838e7 −1.15860
\(909\) 1.09688e7 0.440301
\(910\) 4.49627e6 0.179990
\(911\) −4.31660e7 −1.72324 −0.861621 0.507553i \(-0.830550\pi\)
−0.861621 + 0.507553i \(0.830550\pi\)
\(912\) −15980.1 −0.000636198 0
\(913\) 8.38029e6 0.332723
\(914\) 2.16643e7 0.857784
\(915\) −7.73969e6 −0.305612
\(916\) 1.03438e7 0.407323
\(917\) 2.02670e7 0.795915
\(918\) 4.47033e6 0.175079
\(919\) −3.69944e7 −1.44493 −0.722465 0.691407i \(-0.756991\pi\)
−0.722465 + 0.691407i \(0.756991\pi\)
\(920\) 6.35974e6 0.247725
\(921\) −7.59425e6 −0.295010
\(922\) −1.19655e6 −0.0463557
\(923\) −4.09887e7 −1.58365
\(924\) −2.17213e6 −0.0836962
\(925\) −7.90099e6 −0.303618
\(926\) −2.06627e7 −0.791882
\(927\) 1.35679e7 0.518578
\(928\) 3.08550e7 1.17613
\(929\) 1.63853e6 0.0622895 0.0311447 0.999515i \(-0.490085\pi\)
0.0311447 + 0.999515i \(0.490085\pi\)
\(930\) 678361. 0.0257190
\(931\) 61334.7 0.00231917
\(932\) 1.20729e7 0.455273
\(933\) 1.86218e7 0.700353
\(934\) 1.85658e7 0.696380
\(935\) 6.34873e6 0.237497
\(936\) 9.50449e6 0.354600
\(937\) −2.02831e7 −0.754719 −0.377360 0.926067i \(-0.623168\pi\)
−0.377360 + 0.926067i \(0.623168\pi\)
\(938\) −9.26633e6 −0.343875
\(939\) 5.42802e6 0.200899
\(940\) −4.73841e6 −0.174909
\(941\) −1.83342e7 −0.674975 −0.337487 0.941330i \(-0.609577\pi\)
−0.337487 + 0.941330i \(0.609577\pi\)
\(942\) 9.64107e6 0.353996
\(943\) 2.16769e7 0.793812
\(944\) −4.68436e6 −0.171088
\(945\) −1.54932e6 −0.0564366
\(946\) −6.01438e6 −0.218506
\(947\) −6.09095e6 −0.220704 −0.110352 0.993893i \(-0.535198\pi\)
−0.110352 + 0.993893i \(0.535198\pi\)
\(948\) 2.69448e6 0.0973766
\(949\) −2.01136e7 −0.724976
\(950\) 11691.3 0.000420293 0
\(951\) −1.50601e7 −0.539977
\(952\) 2.89127e7 1.03394
\(953\) −5.31211e7 −1.89467 −0.947337 0.320238i \(-0.896237\pi\)
−0.947337 + 0.320238i \(0.896237\pi\)
\(954\) 5.23288e6 0.186153
\(955\) 3.13605e6 0.111269
\(956\) 2.30951e7 0.817287
\(957\) 5.60393e6 0.197794
\(958\) −1.86046e7 −0.654947
\(959\) −1.27120e7 −0.446340
\(960\) 1.94497e6 0.0681138
\(961\) −2.75644e7 −0.962808
\(962\) 2.67449e7 0.931759
\(963\) 4.69716e6 0.163219
\(964\) −6.68214e6 −0.231592
\(965\) −2.32099e7 −0.802335
\(966\) 3.50922e6 0.120995
\(967\) −2.17670e6 −0.0748570 −0.0374285 0.999299i \(-0.511917\pi\)
−0.0374285 + 0.999299i \(0.511917\pi\)
\(968\) 2.37260e6 0.0813836
\(969\) 120930. 0.00413739
\(970\) −7.66279e6 −0.261492
\(971\) 4.83305e7 1.64503 0.822514 0.568745i \(-0.192571\pi\)
0.822514 + 0.568745i \(0.192571\pi\)
\(972\) −1.38547e6 −0.0470362
\(973\) −1.42563e7 −0.482753
\(974\) 8.59826e6 0.290411
\(975\) 4.07298e6 0.137215
\(976\) −9.53996e6 −0.320569
\(977\) −3.84816e7 −1.28978 −0.644892 0.764274i \(-0.723098\pi\)
−0.644892 + 0.764274i \(0.723098\pi\)
\(978\) −1.31956e7 −0.441145
\(979\) −7.14253e6 −0.238175
\(980\) 5.61954e6 0.186911
\(981\) −1.21145e6 −0.0401913
\(982\) 1.46566e6 0.0485013
\(983\) −1.53385e7 −0.506288 −0.253144 0.967429i \(-0.581465\pi\)
−0.253144 + 0.967429i \(0.581465\pi\)
\(984\) 2.01395e7 0.663073
\(985\) 2.58082e7 0.847555
\(986\) −3.15556e7 −1.03368
\(987\) −6.18048e6 −0.201943
\(988\) 108769. 0.00354498
\(989\) −2.67054e7 −0.868178
\(990\) 715914. 0.0232152
\(991\) −4.29143e7 −1.38809 −0.694046 0.719931i \(-0.744174\pi\)
−0.694046 + 0.719931i \(0.744174\pi\)
\(992\) 6.18712e6 0.199622
\(993\) −1.98379e7 −0.638444
\(994\) −1.40604e7 −0.451368
\(995\) −3.87542e6 −0.124097
\(996\) 1.46252e7 0.467147
\(997\) 6.10887e7 1.94636 0.973180 0.230044i \(-0.0738869\pi\)
0.973180 + 0.230044i \(0.0738869\pi\)
\(998\) 2.68742e7 0.854102
\(999\) −9.21572e6 −0.292156
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 165.6.a.a.1.2 3
3.2 odd 2 495.6.a.e.1.2 3
5.4 even 2 825.6.a.j.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.a.1.2 3 1.1 even 1 trivial
495.6.a.e.1.2 3 3.2 odd 2
825.6.a.j.1.2 3 5.4 even 2