Properties

Label 825.6.a.j.1.2
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $3$
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: \(1\)
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: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.921799\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.92180 q^{2} -9.00000 q^{3} -23.4631 q^{4} -26.2962 q^{6} +85.0105 q^{7} -162.052 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+2.92180 q^{2} -9.00000 q^{3} -23.4631 q^{4} -26.2962 q^{6} +85.0105 q^{7} -162.052 q^{8} +81.0000 q^{9} -121.000 q^{11} +211.168 q^{12} -724.085 q^{13} +248.384 q^{14} +277.335 q^{16} +2098.75 q^{17} +236.666 q^{18} -6.40223 q^{19} -765.094 q^{21} -353.538 q^{22} -1569.80 q^{23} +1458.47 q^{24} -2115.63 q^{26} -729.000 q^{27} -1994.61 q^{28} -5145.94 q^{29} -1031.88 q^{31} +5995.98 q^{32} +1089.00 q^{33} +6132.14 q^{34} -1900.51 q^{36} +12641.6 q^{37} -18.7060 q^{38} +6516.76 q^{39} +13808.7 q^{41} -2235.45 q^{42} +17012.0 q^{43} +2839.03 q^{44} -4586.64 q^{46} -8078.06 q^{47} -2496.02 q^{48} -9580.22 q^{49} -18888.8 q^{51} +16989.3 q^{52} +22110.8 q^{53} -2129.99 q^{54} -13776.1 q^{56} +57.6201 q^{57} -15035.4 q^{58} -16890.6 q^{59} -34398.6 q^{61} -3014.94 q^{62} +6885.85 q^{63} +8644.32 q^{64} +3181.84 q^{66} +37306.5 q^{67} -49243.3 q^{68} +14128.2 q^{69} -56607.5 q^{71} -13126.2 q^{72} +27777.9 q^{73} +36936.2 q^{74} +150.216 q^{76} -10286.3 q^{77} +19040.7 q^{78} -12759.9 q^{79} +6561.00 q^{81} +40346.2 q^{82} +69258.6 q^{83} +17951.5 q^{84} +49705.6 q^{86} +46313.5 q^{87} +19608.3 q^{88} +59029.2 q^{89} -61554.8 q^{91} +36832.4 q^{92} +9286.89 q^{93} -23602.5 q^{94} -53963.8 q^{96} -104905. q^{97} -27991.5 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9} - 363 q^{11} - 225 q^{12} + 654 q^{13} - 728 q^{14} - 415 q^{16} + 2366 q^{17} + 567 q^{18} - 2872 q^{19} - 1548 q^{21} - 847 q^{22} - 2272 q^{23} - 2079 q^{24} + 3422 q^{26} - 2187 q^{27} - 4592 q^{28} - 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 3267 q^{33} + 2506 q^{34} + 2025 q^{36} + 9126 q^{37} - 13076 q^{38} - 5886 q^{39} - 8758 q^{41} + 6552 q^{42} + 14672 q^{43} - 3025 q^{44} - 28768 q^{46} + 19392 q^{47} + 3735 q^{48} - 26629 q^{49} - 21294 q^{51} + 61506 q^{52} + 4598 q^{53} - 5103 q^{54} + 2688 q^{56} + 25848 q^{57} - 8550 q^{58} - 9348 q^{59} - 60078 q^{61} + 14096 q^{62} + 13932 q^{63} - 7087 q^{64} + 7623 q^{66} + 38468 q^{67} - 59778 q^{68} + 20448 q^{69} - 74032 q^{71} + 18711 q^{72} + 44442 q^{73} + 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 30798 q^{78} - 108116 q^{79} + 19683 q^{81} + 92230 q^{82} + 81892 q^{83} + 41328 q^{84} + 126412 q^{86} + 69642 q^{87} - 27951 q^{88} + 167342 q^{89} - 31832 q^{91} - 72960 q^{92} - 5112 q^{93} + 12728 q^{94} + 9009 q^{96} - 159702 q^{97} - 163121 q^{98} - 29403 q^{99}+O(q^{100}) \) 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 \(-0.416853\pi\)
0.258253 + 0.966077i \(0.416853\pi\)
\(3\) −9.00000 −0.577350
\(4\) −23.4631 −0.733222
\(5\) 0 0
\(6\) −26.2962 −0.298205
\(7\) 85.0105 0.655734 0.327867 0.944724i \(-0.393670\pi\)
0.327867 + 0.944724i \(0.393670\pi\)
\(8\) −162.052 −0.895219
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 211.168 0.423326
\(13\) −724.085 −1.18831 −0.594157 0.804349i \(-0.702514\pi\)
−0.594157 + 0.804349i \(0.702514\pi\)
\(14\) 248.384 0.338690
\(15\) 0 0
\(16\) 277.335 0.270835
\(17\) 2098.75 1.76132 0.880662 0.473745i \(-0.157098\pi\)
0.880662 + 0.473745i \(0.157098\pi\)
\(18\) 236.666 0.172169
\(19\) −6.40223 −0.00406862 −0.00203431 0.999998i \(-0.500648\pi\)
−0.00203431 + 0.999998i \(0.500648\pi\)
\(20\) 0 0
\(21\) −765.094 −0.378588
\(22\) −353.538 −0.155732
\(23\) −1569.80 −0.618764 −0.309382 0.950938i \(-0.600122\pi\)
−0.309382 + 0.950938i \(0.600122\pi\)
\(24\) 1458.47 0.516855
\(25\) 0 0
\(26\) −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 \(0.692329\pi\)
\(30\) 0 0
\(31\) −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
\(35\) 0 0
\(36\) −1900.51 −0.244407
\(37\) 12641.6 1.51809 0.759045 0.651039i \(-0.225666\pi\)
0.759045 + 0.651039i \(0.225666\pi\)
\(38\) −18.7060 −0.00210147
\(39\) 6516.76 0.686073
\(40\) 0 0
\(41\) 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 \(-0.252495\pi\)
0.701542 + 0.712628i \(0.252495\pi\)
\(44\) 2839.03 0.221075
\(45\) 0 0
\(46\) −4586.64 −0.319595
\(47\) −8078.06 −0.533412 −0.266706 0.963778i \(-0.585935\pi\)
−0.266706 + 0.963778i \(0.585935\pi\)
\(48\) −2496.02 −0.156367
\(49\) −9580.22 −0.570013
\(50\) 0 0
\(51\) −18888.8 −1.01690
\(52\) 16989.3 0.871297
\(53\) 22110.8 1.08122 0.540612 0.841272i \(-0.318193\pi\)
0.540612 + 0.841272i \(0.318193\pi\)
\(54\) −2129.99 −0.0994016
\(55\) 0 0
\(56\) −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 \(0.602291\pi\)
\(60\) 0 0
\(61\) −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
\(65\) 0 0
\(66\) 3181.84 0.0899122
\(67\) 37306.5 1.01531 0.507654 0.861561i \(-0.330513\pi\)
0.507654 + 0.861561i \(0.330513\pi\)
\(68\) −49243.3 −1.29144
\(69\) 14128.2 0.357244
\(70\) 0 0
\(71\) −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 \(-0.401329\pi\)
0.305044 + 0.952338i \(0.401329\pi\)
\(74\) 36936.2 0.784102
\(75\) 0 0
\(76\) 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 \(0.536691\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 40346.2 0.662625
\(83\) 69258.6 1.10352 0.551758 0.834004i \(-0.313957\pi\)
0.551758 + 0.834004i \(0.313957\pi\)
\(84\) 17951.5 0.277589
\(85\) 0 0
\(86\) 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 \(0.370756\pi\)
\(90\) 0 0
\(91\) −61554.8 −0.779217
\(92\) 36832.4 0.453691
\(93\) 9286.89 0.111343
\(94\) −23602.5 −0.275510
\(95\) 0 0
\(96\) −53963.8 −0.597620
\(97\) −104905. −1.13205 −0.566027 0.824387i \(-0.691520\pi\)
−0.566027 + 0.824387i \(0.691520\pi\)
\(98\) −27991.5 −0.294415
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 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 \(-0.783698\pi\)
−0.777867 + 0.628428i \(0.783698\pi\)
\(104\) 117339. 1.06380
\(105\) 0 0
\(106\) 64603.4 0.558458
\(107\) −57989.6 −0.489656 −0.244828 0.969567i \(-0.578731\pi\)
−0.244828 + 0.969567i \(0.578731\pi\)
\(108\) 17104.6 0.141109
\(109\) −14956.1 −0.120574 −0.0602869 0.998181i \(-0.519202\pi\)
−0.0602869 + 0.998181i \(0.519202\pi\)
\(110\) 0 0
\(111\) −113774. −0.876469
\(112\) 23576.4 0.177596
\(113\) −198323. −1.46109 −0.730544 0.682866i \(-0.760733\pi\)
−0.730544 + 0.682866i \(0.760733\pi\)
\(114\) 168.354 0.00121328
\(115\) 0 0
\(116\) 120740. 0.833115
\(117\) −58650.9 −0.396105
\(118\) −49351.0 −0.326280
\(119\) 178416. 1.15496
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −100506. −0.611353
\(123\) −124278. −0.740682
\(124\) 24211.0 0.141403
\(125\) 0 0
\(126\) 20119.1 0.112897
\(127\) 40986.5 0.225492 0.112746 0.993624i \(-0.464035\pi\)
0.112746 + 0.993624i \(0.464035\pi\)
\(128\) −166614. −0.898851
\(129\) −153108. −0.810071
\(130\) 0 0
\(131\) −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
\(135\) 0 0
\(136\) −340107. −1.57677
\(137\) −149534. −0.680673 −0.340337 0.940304i \(-0.610541\pi\)
−0.340337 + 0.940304i \(0.610541\pi\)
\(138\) 41279.8 0.184518
\(139\) 167700. 0.736202 0.368101 0.929786i \(-0.380008\pi\)
0.368101 + 0.929786i \(0.380008\pi\)
\(140\) 0 0
\(141\) 72702.5 0.307965
\(142\) −165396. −0.688341
\(143\) 87614.3 0.358290
\(144\) 22464.2 0.0902785
\(145\) 0 0
\(146\) 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 \(0.667821\pi\)
\(150\) 0 0
\(151\) −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
\(155\) 0 0
\(156\) −152903. −0.503044
\(157\) 366634. 1.18709 0.593545 0.804801i \(-0.297728\pi\)
0.593545 + 0.804801i \(0.297728\pi\)
\(158\) −37281.9 −0.118811
\(159\) −198998. −0.624245
\(160\) 0 0
\(161\) −133450. −0.405744
\(162\) 19169.9 0.0573896
\(163\) −501806. −1.47934 −0.739668 0.672972i \(-0.765018\pi\)
−0.739668 + 0.672972i \(0.765018\pi\)
\(164\) −323994. −0.940650
\(165\) 0 0
\(166\) 202360. 0.569973
\(167\) 195231. 0.541700 0.270850 0.962622i \(-0.412695\pi\)
0.270850 + 0.962622i \(0.412695\pi\)
\(168\) 123985. 0.338919
\(169\) 153006. 0.412089
\(170\) 0 0
\(171\) −518.581 −0.00135621
\(172\) −399154. −1.02877
\(173\) −618180. −1.57036 −0.785181 0.619266i \(-0.787430\pi\)
−0.785181 + 0.619266i \(0.787430\pi\)
\(174\) 135319. 0.338832
\(175\) 0 0
\(176\) −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 \(0.533065\pi\)
\(180\) 0 0
\(181\) −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
\(185\) 0 0
\(186\) 27134.4 0.0575093
\(187\) −253949. −0.531059
\(188\) 189536. 0.391109
\(189\) −61972.7 −0.126196
\(190\) 0 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 \(-0.145714\pi\)
0.897038 + 0.441953i \(0.145714\pi\)
\(194\) −306512. −0.584713
\(195\) 0 0
\(196\) 224781. 0.417946
\(197\) −1.03233e6 −1.89519 −0.947595 0.319473i \(-0.896494\pi\)
−0.947595 + 0.319473i \(0.896494\pi\)
\(198\) −28636.6 −0.0519108
\(199\) −155017. −0.277489 −0.138744 0.990328i \(-0.544307\pi\)
−0.138744 + 0.990328i \(0.544307\pi\)
\(200\) 0 0
\(201\) −335759. −0.586188
\(202\) 395662. 0.682254
\(203\) −437459. −0.745070
\(204\) 443189. 0.745614
\(205\) 0 0
\(206\) −489417. −0.803546
\(207\) −127154. −0.206255
\(208\) −200814. −0.321837
\(209\) 774.670 0.00122674
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) 118136. 0.172285
\(217\) −87720.4 −0.126459
\(218\) −43698.8 −0.0622771
\(219\) −250001. −0.352234
\(220\) 0 0
\(221\) −1.51968e6 −2.09301
\(222\) −332426. −0.452702
\(223\) 74806.0 0.100734 0.0503668 0.998731i \(-0.483961\pi\)
0.0503668 + 0.998731i \(0.483961\pi\)
\(224\) 509721. 0.678755
\(225\) 0 0
\(226\) −579459. −0.754661
\(227\) −1.22677e6 −1.58015 −0.790075 0.613010i \(-0.789958\pi\)
−0.790075 + 0.613010i \(0.789958\pi\)
\(228\) −1351.94 −0.00172235
\(229\) −440852. −0.555526 −0.277763 0.960650i \(-0.589593\pi\)
−0.277763 + 0.960650i \(0.589593\pi\)
\(230\) 0 0
\(231\) 92576.4 0.114149
\(232\) 833910. 1.01718
\(233\) 514549. 0.620922 0.310461 0.950586i \(-0.399517\pi\)
0.310461 + 0.950586i \(0.399517\pi\)
\(234\) −171366. −0.204590
\(235\) 0 0
\(236\) 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 \(0.688173\pi\)
\(240\) 0 0
\(241\) 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
\(245\) 0 0
\(246\) −363116. −0.382567
\(247\) 4635.76 0.00483480
\(248\) 167218. 0.172645
\(249\) −623328. −0.637115
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) −763432. −0.728066
\(257\) −2.06732e6 −1.95242 −0.976212 0.216817i \(-0.930433\pi\)
−0.976212 + 0.216817i \(0.930433\pi\)
\(258\) −447350. −0.418407
\(259\) 1.07467e6 0.995462
\(260\) 0 0
\(261\) −416821. −0.378746
\(262\) −696575. −0.626924
\(263\) 661739. 0.589926 0.294963 0.955509i \(-0.404693\pi\)
0.294963 + 0.955509i \(0.404693\pi\)
\(264\) −176475. −0.155838
\(265\) 0 0
\(266\) −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 \(0.404289\pi\)
\(270\) 0 0
\(271\) −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
\(275\) 0 0
\(276\) −331492. −0.261939
\(277\) 33171.1 0.0259753 0.0129877 0.999916i \(-0.495866\pi\)
0.0129877 + 0.999916i \(0.495866\pi\)
\(278\) 489987. 0.380253
\(279\) −83582.0 −0.0642839
\(280\) 0 0
\(281\) −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 \(-0.249911\pi\)
0.707305 + 0.706908i \(0.249911\pi\)
\(284\) 1.32819e6 0.977155
\(285\) 0 0
\(286\) 255991. 0.185059
\(287\) 1.17388e6 0.841240
\(288\) 485675. 0.345036
\(289\) 2.98491e6 2.10226
\(290\) 0 0
\(291\) 944146. 0.653592
\(292\) −651756. −0.447330
\(293\) −272957. −0.185748 −0.0928742 0.995678i \(-0.529605\pi\)
−0.0928742 + 0.995678i \(0.529605\pi\)
\(294\) 251923. 0.169981
\(295\) 0 0
\(296\) −2.04859e6 −1.35902
\(297\) 88209.0 0.0580259
\(298\) −796768. −0.519746
\(299\) 1.13667e6 0.735286
\(300\) 0 0
\(301\) 1.44620e6 0.920050
\(302\) −1.30990e6 −0.826459
\(303\) −1.21876e6 −0.762624
\(304\) −1775.57 −0.00110193
\(305\) 0 0
\(306\) 496703. 0.303245
\(307\) 843806. 0.510971 0.255486 0.966813i \(-0.417765\pi\)
0.255486 + 0.966813i \(0.417765\pi\)
\(308\) 241348. 0.144966
\(309\) 1.50755e6 0.898204
\(310\) 0 0
\(311\) 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 \(-0.555664\pi\)
−0.173983 + 0.984749i \(0.555664\pi\)
\(314\) 1.07123e6 0.613139
\(315\) 0 0
\(316\) 299387. 0.168661
\(317\) 1.67334e6 0.935268 0.467634 0.883922i \(-0.345107\pi\)
0.467634 + 0.883922i \(0.345107\pi\)
\(318\) −581431. −0.322426
\(319\) 622659. 0.342589
\(320\) 0 0
\(321\) 521907. 0.282703
\(322\) −389913. −0.209569
\(323\) −13436.7 −0.00716616
\(324\) −153941. −0.0814691
\(325\) 0 0
\(326\) −1.46618e6 −0.764086
\(327\) 134605. 0.0696133
\(328\) −2.23772e6 −1.14848
\(329\) −686720. −0.349776
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) −212188. −0.102535
\(337\) −1.07377e6 −0.515036 −0.257518 0.966273i \(-0.582905\pi\)
−0.257518 + 0.966273i \(0.582905\pi\)
\(338\) 447052. 0.212847
\(339\) 1.78490e6 0.843559
\(340\) 0 0
\(341\) 124857. 0.0581470
\(342\) −1515.19 −0.000700489 0
\(343\) −2.24319e6 −1.02951
\(344\) −2.75683e6 −1.25607
\(345\) 0 0
\(346\) −1.80620e6 −0.811101
\(347\) 1.39783e6 0.623203 0.311601 0.950213i \(-0.399135\pi\)
0.311601 + 0.950213i \(0.399135\pi\)
\(348\) −1.08666e6 −0.480999
\(349\) −2.66674e6 −1.17197 −0.585985 0.810322i \(-0.699292\pi\)
−0.585985 + 0.810322i \(0.699292\pi\)
\(350\) 0 0
\(351\) 527858. 0.228691
\(352\) −725514. −0.312097
\(353\) 594048. 0.253738 0.126869 0.991920i \(-0.459507\pi\)
0.126869 + 0.991920i \(0.459507\pi\)
\(354\) 444159. 0.188378
\(355\) 0 0
\(356\) −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 \(0.741298\pi\)
\(360\) 0 0
\(361\) −2.47606e6 −0.999983
\(362\) −1.10481e6 −0.443116
\(363\) −131769. −0.0524864
\(364\) 1.44427e6 0.571339
\(365\) 0 0
\(366\) 904553. 0.352965
\(367\) −2.58775e6 −1.00290 −0.501450 0.865187i \(-0.667200\pi\)
−0.501450 + 0.865187i \(0.667200\pi\)
\(368\) −435362. −0.167583
\(369\) 1.11850e6 0.427633
\(370\) 0 0
\(371\) 1.87965e6 0.708995
\(372\) −217899. −0.0816391
\(373\) 1.07376e6 0.399608 0.199804 0.979836i \(-0.435969\pi\)
0.199804 + 0.979836i \(0.435969\pi\)
\(374\) −741989. −0.274295
\(375\) 0 0
\(376\) 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 \(0.124942\pi\)
\(380\) 0 0
\(381\) −368878. −0.130188
\(382\) 366517. 0.128510
\(383\) −2.42092e6 −0.843302 −0.421651 0.906758i \(-0.638549\pi\)
−0.421651 + 0.906758i \(0.638549\pi\)
\(384\) 1.49953e6 0.518952
\(385\) 0 0
\(386\) 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 \(0.395673\pi\)
\(390\) 0 0
\(391\) −3.29463e6 −1.08984
\(392\) 1.55249e6 0.510287
\(393\) 2.14566e6 0.700775
\(394\) −3.01626e6 −0.978877
\(395\) 0 0
\(396\) 229962. 0.0736915
\(397\) −4.09572e6 −1.30423 −0.652114 0.758121i \(-0.726118\pi\)
−0.652114 + 0.758121i \(0.726118\pi\)
\(398\) −452928. −0.143325
\(399\) 4898.31 0.00154033
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) −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 \(0.591911\pi\)
\(410\) 0 0
\(411\) 1.34581e6 0.392987
\(412\) 3.93019e6 1.14070
\(413\) −1.43588e6 −0.414231
\(414\) −371518. −0.106532
\(415\) 0 0
\(416\) −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 \(0.565954\pi\)
\(420\) 0 0
\(421\) 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
\(425\) 0 0
\(426\) 1.48856e6 0.397414
\(427\) −2.92425e6 −0.776147
\(428\) 1.36062e6 0.359026
\(429\) −788528. −0.206859
\(430\) 0 0
\(431\) 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 \(-0.598106\pi\)
−0.303353 + 0.952878i \(0.598106\pi\)
\(434\) −256301. −0.0653170
\(435\) 0 0
\(436\) 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 \(0.369440\pi\)
\(440\) 0 0
\(441\) −775997. −0.190004
\(442\) −4.44019e6 −1.08105
\(443\) 3.80335e6 0.920782 0.460391 0.887716i \(-0.347709\pi\)
0.460391 + 0.887716i \(0.347709\pi\)
\(444\) 2.66950e6 0.642646
\(445\) 0 0
\(446\) 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 \(0.778252\pi\)
\(450\) 0 0
\(451\) −1.67085e6 −0.386809
\(452\) 4.65326e6 1.07130
\(453\) 4.03488e6 0.923815
\(454\) −3.58437e6 −0.816157
\(455\) 0 0
\(456\) −9337.45 −0.00210289
\(457\) 7.41470e6 1.66074 0.830372 0.557209i \(-0.188128\pi\)
0.830372 + 0.557209i \(0.188128\pi\)
\(458\) −1.28808e6 −0.286932
\(459\) −1.52999e6 −0.338967
\(460\) 0 0
\(461\) 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 \(-0.778041\pi\)
−0.766575 + 0.642154i \(0.778041\pi\)
\(464\) −1.42715e6 −0.307734
\(465\) 0 0
\(466\) 1.50341e6 0.320710
\(467\) 6.35423e6 1.34825 0.674125 0.738617i \(-0.264521\pi\)
0.674125 + 0.738617i \(0.264521\pi\)
\(468\) 1.37613e6 0.290432
\(469\) 3.17145e6 0.665772
\(470\) 0 0
\(471\) −3.29970e6 −0.685366
\(472\) 2.73716e6 0.565516
\(473\) −2.05845e6 −0.423046
\(474\) 335537. 0.0685954
\(475\) 0 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 \(0.281405\pi\)
\(480\) 0 0
\(481\) −9.15358e6 −1.80397
\(482\) 832109. 0.163141
\(483\) 1.20105e6 0.234257
\(484\) −343523. −0.0666565
\(485\) 0 0
\(486\) −172529. −0.0331339
\(487\) 2.94280e6 0.562261 0.281130 0.959670i \(-0.409291\pi\)
0.281130 + 0.959670i \(0.409291\pi\)
\(488\) 5.57437e6 1.05961
\(489\) 4.51626e6 0.854095
\(490\) 0 0
\(491\) −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
\(495\) 0 0
\(496\) −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 \(0.809846\pi\)
\(500\) 0 0
\(501\) −1.75708e6 −0.312750
\(502\) 393068. 0.0696158
\(503\) 6.53811e6 1.15221 0.576106 0.817375i \(-0.304572\pi\)
0.576106 + 0.817375i \(0.304572\pi\)
\(504\) −1.11587e6 −0.195675
\(505\) 0 0
\(506\) 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 \(0.540686\pi\)
\(510\) 0 0
\(511\) 2.36141e6 0.400055
\(512\) 3.10107e6 0.522801
\(513\) 4667.23 0.000783007 0
\(514\) −6.04029e6 −1.00844
\(515\) 0 0
\(516\) 3.59238e6 0.593962
\(517\) 977445. 0.160830
\(518\) 3.13996e6 0.514162
\(519\) 5.56362e6 0.906649
\(520\) 0 0
\(521\) 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 \(-0.391490\pi\)
0.334330 + 0.942456i \(0.391490\pi\)
\(524\) 5.59375e6 0.889968
\(525\) 0 0
\(526\) 1.93347e6 0.304700
\(527\) −2.16566e6 −0.339675
\(528\) 302018. 0.0471464
\(529\) −3.97207e6 −0.617131
\(530\) 0 0
\(531\) −1.36814e6 −0.210569
\(532\) 12769.9 0.00195619
\(533\) −9.99866e6 −1.52449
\(534\) −1.55224e6 −0.235563
\(535\) 0 0
\(536\) −6.04560e6 −0.908923
\(537\) 800088. 0.119730
\(538\) 2.05405e6 0.305953
\(539\) 1.15921e6 0.171866
\(540\) 0 0
\(541\) 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
\(545\) 0 0
\(546\) 1.61866e6 0.232366
\(547\) 8.42403e6 1.20379 0.601896 0.798574i \(-0.294412\pi\)
0.601896 + 0.798574i \(0.294412\pi\)
\(548\) 3.50853e6 0.499084
\(549\) −2.78629e6 −0.394544
\(550\) 0 0
\(551\) 32945.5 0.00462293
\(552\) −2.28951e6 −0.319811
\(553\) −1.08473e6 −0.150837
\(554\) 96919.4 0.0134164
\(555\) 0 0
\(556\) −3.93477e6 −0.539799
\(557\) 3.56702e6 0.487155 0.243577 0.969881i \(-0.421679\pi\)
0.243577 + 0.969881i \(0.421679\pi\)
\(558\) −244210. −0.0332030
\(559\) −1.23181e7 −1.66730
\(560\) 0 0
\(561\) 2.28554e6 0.306607
\(562\) −938231. −0.125305
\(563\) −4.70552e6 −0.625658 −0.312829 0.949810i \(-0.601277\pi\)
−0.312829 + 0.949810i \(0.601277\pi\)
\(564\) −1.70583e6 −0.225807
\(565\) 0 0
\(566\) 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 \(0.445979\pi\)
\(570\) 0 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
\(575\) 0 0
\(576\) 700190. 0.0879346
\(577\) −1.12075e7 −1.40142 −0.700710 0.713446i \(-0.747133\pi\)
−0.700710 + 0.713446i \(0.747133\pi\)
\(578\) 8.72132e6 1.08583
\(579\) −8.35558e6 −1.03581
\(580\) 0 0
\(581\) 5.88771e6 0.723613
\(582\) 2.75861e6 0.337584
\(583\) −2.67541e6 −0.326001
\(584\) −4.50147e6 −0.546162
\(585\) 0 0
\(586\) −797526. −0.0959402
\(587\) −9.05798e6 −1.08502 −0.542508 0.840051i \(-0.682525\pi\)
−0.542508 + 0.840051i \(0.682525\pi\)
\(588\) −2.02303e6 −0.241301
\(589\) 6606.31 0.000784641 0
\(590\) 0 0
\(591\) 9.29097e6 1.09419
\(592\) 3.50596e6 0.411152
\(593\) −1.00907e7 −1.17837 −0.589186 0.807997i \(-0.700552\pi\)
−0.589186 + 0.807997i \(0.700552\pi\)
\(594\) 257729. 0.0299707
\(595\) 0 0
\(596\) 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 \(0.762922\pi\)
\(600\) 0 0
\(601\) 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
\(605\) 0 0
\(606\) −3.56096e6 −0.393900
\(607\) 1.26409e6 0.139253 0.0696267 0.997573i \(-0.477819\pi\)
0.0696267 + 0.997573i \(0.477819\pi\)
\(608\) −38387.7 −0.00421146
\(609\) 3.93713e6 0.430166
\(610\) 0 0
\(611\) 5.84920e6 0.633860
\(612\) −3.98870e6 −0.430480
\(613\) −1.59125e7 −1.71036 −0.855180 0.518332i \(-0.826553\pi\)
−0.855180 + 0.518332i \(0.826553\pi\)
\(614\) 2.46543e6 0.263920
\(615\) 0 0
\(616\) 1.66691e6 0.176995
\(617\) −85225.6 −0.00901274 −0.00450637 0.999990i \(-0.501434\pi\)
−0.00450637 + 0.999990i \(0.501434\pi\)
\(618\) 4.40475e6 0.463928
\(619\) −1.20387e7 −1.26285 −0.631425 0.775437i \(-0.717530\pi\)
−0.631425 + 0.775437i \(0.717530\pi\)
\(620\) 0 0
\(621\) 1.14439e6 0.119081
\(622\) 6.04545e6 0.626546
\(623\) 5.01810e6 0.517987
\(624\) 1.80733e6 0.185813
\(625\) 0 0
\(626\) −1.76218e6 −0.179727
\(627\) −6972.03 −0.000708256 0
\(628\) −8.60236e6 −0.870400
\(629\) 2.65316e7 2.67385
\(630\) 0 0
\(631\) 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
\(635\) 0 0
\(636\) 4.66910e6 0.457710
\(637\) 6.93689e6 0.677355
\(638\) 1.81928e6 0.176949
\(639\) −4.58521e6 −0.444229
\(640\) 0 0
\(641\) 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 \(-0.447329\pi\)
0.164717 + 0.986341i \(0.447329\pi\)
\(644\) 3.13114e6 0.297501
\(645\) 0 0
\(646\) −39259.4 −0.00370137
\(647\) −1.71115e7 −1.60704 −0.803521 0.595276i \(-0.797043\pi\)
−0.803521 + 0.595276i \(0.797043\pi\)
\(648\) −1.06322e6 −0.0994688
\(649\) 2.04376e6 0.190467
\(650\) 0 0
\(651\) 789483. 0.0730114
\(652\) 1.17739e7 1.08468
\(653\) −1.61622e7 −1.48326 −0.741630 0.670809i \(-0.765947\pi\)
−0.741630 + 0.670809i \(0.765947\pi\)
\(654\) 393289. 0.0359557
\(655\) 0 0
\(656\) 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 \(0.359139\pi\)
\(660\) 0 0
\(661\) −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
\(665\) 0 0
\(666\) 2.99183e6 0.261367
\(667\) 8.07810e6 0.703064
\(668\) −4.58073e6 −0.397186
\(669\) −673254. −0.0581586
\(670\) 0 0
\(671\) 4.16224e6 0.356878
\(672\) −4.58749e6 −0.391879
\(673\) −1.37772e7 −1.17253 −0.586266 0.810119i \(-0.699403\pi\)
−0.586266 + 0.810119i \(0.699403\pi\)
\(674\) −3.13735e6 −0.266019
\(675\) 0 0
\(676\) −3.58999e6 −0.302153
\(677\) 1.06456e6 0.0892683 0.0446341 0.999003i \(-0.485788\pi\)
0.0446341 + 0.999003i \(0.485788\pi\)
\(678\) 5.21513e6 0.435703
\(679\) −8.91804e6 −0.742326
\(680\) 0 0
\(681\) 1.10409e7 0.912300
\(682\) 364807. 0.0300333
\(683\) 1.48070e7 1.21455 0.607275 0.794492i \(-0.292263\pi\)
0.607275 + 0.794492i \(0.292263\pi\)
\(684\) 12167.5 0.000994400 0
\(685\) 0 0
\(686\) −6.55415e6 −0.531748
\(687\) 3.96767e6 0.320733
\(688\) 4.71803e6 0.380005
\(689\) −1.60101e7 −1.28483
\(690\) 0 0
\(691\) 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
\(695\) 0 0
\(696\) −7.50519e6 −0.587271
\(697\) 2.89810e7 2.25960
\(698\) −7.79167e6 −0.605330
\(699\) −4.63094e6 −0.358490
\(700\) 0 0
\(701\) 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
\(705\) 0 0
\(706\) 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 \(0.507562\pi\)
\(710\) 0 0
\(711\) −1.03355e6 −0.0766759
\(712\) −9.56580e6 −0.707166
\(713\) 1.61984e6 0.119330
\(714\) −4.69167e6 −0.344415
\(715\) 0 0
\(716\) 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 \(0.528784\pi\)
\(720\) 0 0
\(721\) −1.42397e7 −1.02015
\(722\) −7.23454e6 −0.516497
\(723\) −2.56314e6 −0.182359
\(724\) 8.87205e6 0.629039
\(725\) 0 0
\(726\) −385003. −0.0271095
\(727\) −8.82888e6 −0.619540 −0.309770 0.950811i \(-0.600252\pi\)
−0.309770 + 0.950811i \(0.600252\pi\)
\(728\) 9.97508e6 0.697570
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 3.57040e7 2.47129
\(732\) −7.26389e6 −0.501062
\(733\) 2.93960e6 0.202082 0.101041 0.994882i \(-0.467783\pi\)
0.101041 + 0.994882i \(0.467783\pi\)
\(734\) −7.56089e6 −0.518004
\(735\) 0 0
\(736\) −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 \(0.516277\pi\)
\(740\) 0 0
\(741\) −41721.8 −0.00279137
\(742\) 5.49197e6 0.366200
\(743\) 1.07165e7 0.712168 0.356084 0.934454i \(-0.384112\pi\)
0.356084 + 0.934454i \(0.384112\pi\)
\(744\) −1.50496e6 −0.0996764
\(745\) 0 0
\(746\) 3.13731e6 0.206400
\(747\) 5.60995e6 0.367839
\(748\) 5.95844e6 0.389384
\(749\) −4.92973e6 −0.321084
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) 1.45407e6 0.0925296
\(757\) 2.81995e7 1.78855 0.894275 0.447518i \(-0.147692\pi\)
0.894275 + 0.447518i \(0.147692\pi\)
\(758\) 1.50982e7 0.954450
\(759\) −1.70951e6 −0.107713
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) −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 \(0.440494\pi\)
\(770\) 0 0
\(771\) 1.86059e7 1.12723
\(772\) −2.17831e7 −1.31546
\(773\) −1.02486e7 −0.616901 −0.308451 0.951240i \(-0.599810\pi\)
−0.308451 + 0.951240i \(0.599810\pi\)
\(774\) 4.02615e6 0.241567
\(775\) 0 0
\(776\) 1.70001e7 1.01344
\(777\) −9.67201e6 −0.574730
\(778\) 5.61434e6 0.332545
\(779\) −88406.4 −0.00521963
\(780\) 0 0
\(781\) 6.84951e6 0.401820
\(782\) −9.62624e6 −0.562911
\(783\) 3.75139e6 0.218669
\(784\) −2.65693e6 −0.154380
\(785\) 0 0
\(786\) 6.26917e6 0.361954
\(787\) 1.62333e7 0.934267 0.467133 0.884187i \(-0.345287\pi\)
0.467133 + 0.884187i \(0.345287\pi\)
\(788\) 2.42217e7 1.38959
\(789\) −5.95565e6 −0.340594
\(790\) 0 0
\(791\) −1.68595e7 −0.958084
\(792\) 1.58827e6 0.0899729
\(793\) 2.49075e7 1.40653
\(794\) −1.19669e7 −0.673642
\(795\) 0 0
\(796\) 3.63717e6 0.203461
\(797\) −1.29153e7 −0.720211 −0.360105 0.932912i \(-0.617259\pi\)
−0.360105 + 0.932912i \(0.617259\pi\)
\(798\) 14311.9 0.000795590 0
\(799\) −1.69539e7 −0.939511
\(800\) 0 0
\(801\) 4.78136e6 0.263312
\(802\) 1.74585e7 0.958452
\(803\) −3.36113e6 −0.183948
\(804\) 7.87794e6 0.429806
\(805\) 0 0
\(806\) 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 \(0.487997\pi\)
\(810\) 0 0
\(811\) −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
\(815\) 0 0
\(816\) −5.23853e6 −0.275413
\(817\) −108915. −0.00570862
\(818\) −5.62928e6 −0.294150
\(819\) −4.98594e6 −0.259739
\(820\) 0 0
\(821\) −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 \(-0.924357\pi\)
−0.971896 + 0.235409i \(0.924357\pi\)
\(824\) 2.71446e7 1.39272
\(825\) 0 0
\(826\) −4.19535e6 −0.213953
\(827\) 2.54890e7 1.29595 0.647976 0.761661i \(-0.275616\pi\)
0.647976 + 0.761661i \(0.275616\pi\)
\(828\) 2.98342e6 0.151230
\(829\) 1.71410e7 0.866262 0.433131 0.901331i \(-0.357409\pi\)
0.433131 + 0.901331i \(0.357409\pi\)
\(830\) 0 0
\(831\) −298540. −0.0149969
\(832\) −6.25922e6 −0.313482
\(833\) −2.01065e7 −1.00398
\(834\) −4.40988e6 −0.219539
\(835\) 0 0
\(836\) −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 \(0.579743\pi\)
\(840\) 0 0
\(841\) 5.96954e6 0.291039
\(842\) 1.41019e7 0.685483
\(843\) 2.89003e6 0.140066
\(844\) 8.54672e6 0.412994
\(845\) 0 0
\(846\) −1.91180e6 −0.0918368
\(847\) 1.24464e6 0.0596121
\(848\) 6.13212e6 0.292834
\(849\) −1.71532e7 −0.816726
\(850\) 0 0
\(851\) −1.98448e7 −0.939339
\(852\) −1.19537e7 −0.564161
\(853\) −3.29141e7 −1.54885 −0.774425 0.632666i \(-0.781961\pi\)
−0.774425 + 0.632666i \(0.781961\pi\)
\(854\) −8.54406e6 −0.400885
\(855\) 0 0
\(856\) 9.39734e6 0.438349
\(857\) −6.94241e6 −0.322893 −0.161446 0.986881i \(-0.551616\pi\)
−0.161446 + 0.986881i \(0.551616\pi\)
\(858\) −2.30392e6 −0.106844
\(859\) −1.84275e7 −0.852085 −0.426042 0.904703i \(-0.640093\pi\)
−0.426042 + 0.904703i \(0.640093\pi\)
\(860\) 0 0
\(861\) −1.05649e7 −0.485690
\(862\) 8.45068e6 0.387368
\(863\) 2.60648e7 1.19131 0.595657 0.803239i \(-0.296892\pi\)
0.595657 + 0.803239i \(0.296892\pi\)
\(864\) −4.37107e6 −0.199207
\(865\) 0 0
\(866\) −6.91589e6 −0.313367
\(867\) −2.68642e7 −1.21374
\(868\) 2.05819e6 0.0927228
\(869\) 1.54395e6 0.0693559
\(870\) 0 0
\(871\) −2.70131e7 −1.20650
\(872\) 2.42367e6 0.107940
\(873\) −8.49731e6 −0.377352
\(874\) 29364.8 0.00130031
\(875\) 0 0
\(876\) 5.86580e6 0.258266
\(877\) −1.86288e6 −0.0817875 −0.0408937 0.999164i \(-0.513021\pi\)
−0.0408937 + 0.999164i \(0.513021\pi\)
\(878\) 9.40927e6 0.411926
\(879\) 2.45661e6 0.107242
\(880\) 0 0
\(881\) 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 \(-0.543131\pi\)
−0.135086 + 0.990834i \(0.543131\pi\)
\(884\) 3.56563e7 1.53464
\(885\) 0 0
\(886\) 1.11126e7 0.475589
\(887\) −1.80319e7 −0.769540 −0.384770 0.923012i \(-0.625719\pi\)
−0.384770 + 0.923012i \(0.625719\pi\)
\(888\) 1.84374e7 0.784632
\(889\) 3.48428e6 0.147863
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −1.75518e6 −0.0738600
\(893\) 51717.6 0.00217025
\(894\) 7.17091e6 0.300075
\(895\) 0 0
\(896\) −1.41640e7 −0.589407
\(897\) −1.02300e7 −0.424518
\(898\) −1.91466e7 −0.792321
\(899\) 5.30998e6 0.219126
\(900\) 0 0
\(901\) 4.64052e7 1.90439
\(902\) −4.88189e6 −0.199789
\(903\) −1.30158e7 −0.531191
\(904\) 3.21386e7 1.30799
\(905\) 0 0
\(906\) 1.17891e7 0.477156
\(907\) −2.69733e7 −1.08872 −0.544360 0.838852i \(-0.683228\pi\)
−0.544360 + 0.838852i \(0.683228\pi\)
\(908\) 2.87838e7 1.15860
\(909\) 1.09688e7 0.440301
\(910\) 0 0
\(911\) −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
\(915\) 0 0
\(916\) 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 \(0.756991\pi\)
\(920\) 0 0
\(921\) −7.59425e6 −0.295010
\(922\) 1.19655e6 0.0463557
\(923\) 4.09887e7 1.58365
\(924\) −2.17213e6 −0.0836962
\(925\) 0 0
\(926\) −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 \(0.490085\pi\)
\(930\) 0 0
\(931\) 61334.7 0.00231917
\(932\) −1.20729e7 −0.455273
\(933\) −1.86218e7 −0.700353
\(934\) 1.85658e7 0.696380
\(935\) 0 0
\(936\) 9.50449e6 0.354600
\(937\) 2.02831e7 0.754719 0.377360 0.926067i \(-0.376832\pi\)
0.377360 + 0.926067i \(0.376832\pi\)
\(938\) 9.26633e6 0.343875
\(939\) 5.42802e6 0.200899
\(940\) 0 0
\(941\) −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
\(945\) 0 0
\(946\) −6.01438e6 −0.218506
\(947\) 6.09095e6 0.220704 0.110352 0.993893i \(-0.464802\pi\)
0.110352 + 0.993893i \(0.464802\pi\)
\(948\) −2.69448e6 −0.0973766
\(949\) −2.01136e7 −0.724976
\(950\) 0 0
\(951\) −1.50601e7 −0.539977
\(952\) −2.89127e7 −1.03394
\(953\) 5.31211e7 1.89467 0.947337 0.320238i \(-0.103763\pi\)
0.947337 + 0.320238i \(0.103763\pi\)
\(954\) 5.23288e6 0.186153
\(955\) 0 0
\(956\) 2.30951e7 0.817287
\(957\) −5.60393e6 −0.197794
\(958\) 1.86046e7 0.654947
\(959\) −1.27120e7 −0.446340
\(960\) 0 0
\(961\) −2.75644e7 −0.962808
\(962\) −2.67449e7 −0.931759
\(963\) −4.69716e6 −0.163219
\(964\) −6.68214e6 −0.231592
\(965\) 0 0
\(966\) 3.50922e6 0.120995
\(967\) 2.17670e6 0.0748570 0.0374285 0.999299i \(-0.488083\pi\)
0.0374285 + 0.999299i \(0.488083\pi\)
\(968\) −2.37260e6 −0.0813836
\(969\) 120930. 0.00413739
\(970\) 0 0
\(971\) 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
\(975\) 0 0
\(976\) −9.53996e6 −0.320569
\(977\) 3.84816e7 1.28978 0.644892 0.764274i \(-0.276902\pi\)
0.644892 + 0.764274i \(0.276902\pi\)
\(978\) 1.31956e7 0.441145
\(979\) −7.14253e6 −0.238175
\(980\) 0 0
\(981\) −1.21145e6 −0.0401913
\(982\) −1.46566e6 −0.0485013
\(983\) 1.53385e7 0.506288 0.253144 0.967429i \(-0.418535\pi\)
0.253144 + 0.967429i \(0.418535\pi\)
\(984\) 2.01395e7 0.663073
\(985\) 0 0
\(986\) −3.15556e7 −1.03368
\(987\) 6.18048e6 0.201943
\(988\) −108769. −0.00354498
\(989\) −2.67054e7 −0.868178
\(990\) 0 0
\(991\) −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
\(995\) 0 0
\(996\) 1.46252e7 0.467147
\(997\) −6.10887e7 −1.94636 −0.973180 0.230044i \(-0.926113\pi\)
−0.973180 + 0.230044i \(0.926113\pi\)
\(998\) −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 825.6.a.j.1.2 3
5.4 even 2 165.6.a.a.1.2 3
15.14 odd 2 495.6.a.e.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.a.1.2 3 5.4 even 2
495.6.a.e.1.2 3 15.14 odd 2
825.6.a.j.1.2 3 1.1 even 1 trivial