Properties

Label 825.6.a.o.1.4
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.60892\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.60892 q^{2} -9.00000 q^{3} -25.1935 q^{4} -23.4803 q^{6} -174.969 q^{7} -149.214 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+2.60892 q^{2} -9.00000 q^{3} -25.1935 q^{4} -23.4803 q^{6} -174.969 q^{7} -149.214 q^{8} +81.0000 q^{9} +121.000 q^{11} +226.742 q^{12} +815.527 q^{13} -456.481 q^{14} +416.906 q^{16} -462.991 q^{17} +211.323 q^{18} -1996.09 q^{19} +1574.72 q^{21} +315.680 q^{22} -1342.70 q^{23} +1342.92 q^{24} +2127.65 q^{26} -729.000 q^{27} +4408.08 q^{28} +6586.33 q^{29} -4492.74 q^{31} +5862.51 q^{32} -1089.00 q^{33} -1207.91 q^{34} -2040.67 q^{36} +9089.95 q^{37} -5207.64 q^{38} -7339.74 q^{39} -1108.16 q^{41} +4108.32 q^{42} +20231.4 q^{43} -3048.42 q^{44} -3503.01 q^{46} +9098.00 q^{47} -3752.15 q^{48} +13807.1 q^{49} +4166.92 q^{51} -20546.0 q^{52} +38340.2 q^{53} -1901.91 q^{54} +26107.7 q^{56} +17964.8 q^{57} +17183.2 q^{58} +29309.0 q^{59} +32390.1 q^{61} -11721.2 q^{62} -14172.5 q^{63} +1953.85 q^{64} -2841.12 q^{66} -64412.9 q^{67} +11664.4 q^{68} +12084.3 q^{69} -63314.4 q^{71} -12086.3 q^{72} +4121.83 q^{73} +23715.0 q^{74} +50288.4 q^{76} -21171.2 q^{77} -19148.8 q^{78} -53233.3 q^{79} +6561.00 q^{81} -2891.10 q^{82} -119847. q^{83} -39672.7 q^{84} +52782.1 q^{86} -59277.0 q^{87} -18054.8 q^{88} +66306.8 q^{89} -142692. q^{91} +33827.4 q^{92} +40434.6 q^{93} +23736.0 q^{94} -52762.6 q^{96} -180567. q^{97} +36021.7 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 9 q^{2} - 63 q^{3} + 95 q^{4} - 81 q^{6} + 65 q^{7} + 207 q^{8} + 567 q^{9} + 847 q^{11} - 855 q^{12} - 113 q^{13} - 1212 q^{14} + 499 q^{16} + 1030 q^{17} + 729 q^{18} - 3803 q^{19} - 585 q^{21} + 1089 q^{22} + 514 q^{23} - 1863 q^{24} - 12111 q^{26} - 5103 q^{27} - 342 q^{28} - 2698 q^{29} - 17233 q^{31} + 9943 q^{32} - 7623 q^{33} + 4090 q^{34} + 7695 q^{36} + 23182 q^{37} - 11943 q^{38} + 1017 q^{39} - 16158 q^{41} + 10908 q^{42} - 4249 q^{43} + 11495 q^{44} - 28769 q^{46} + 7580 q^{47} - 4491 q^{48} + 37140 q^{49} - 9270 q^{51} - 23887 q^{52} + 20574 q^{53} - 6561 q^{54} - 73276 q^{56} + 34227 q^{57} + 8733 q^{58} - 364 q^{59} - 28127 q^{61} - 71917 q^{62} + 5265 q^{63} + 43379 q^{64} - 9801 q^{66} - 21493 q^{67} + 160660 q^{68} - 4626 q^{69} - 177084 q^{71} + 16767 q^{72} + 78670 q^{73} + 196750 q^{74} - 32701 q^{76} + 7865 q^{77} + 108999 q^{78} - 187432 q^{79} + 45927 q^{81} + 179552 q^{82} - 44592 q^{83} + 3078 q^{84} - 110433 q^{86} + 24282 q^{87} + 25047 q^{88} - 151168 q^{89} - 230153 q^{91} - 44767 q^{92} + 155097 q^{93} + 54040 q^{94} - 89487 q^{96} - 55589 q^{97} + 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.60892 0.461197 0.230598 0.973049i \(-0.425932\pi\)
0.230598 + 0.973049i \(0.425932\pi\)
\(3\) −9.00000 −0.577350
\(4\) −25.1935 −0.787297
\(5\) 0 0
\(6\) −23.4803 −0.266272
\(7\) −174.969 −1.34963 −0.674817 0.737985i \(-0.735777\pi\)
−0.674817 + 0.737985i \(0.735777\pi\)
\(8\) −149.214 −0.824296
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 226.742 0.454546
\(13\) 815.527 1.33838 0.669191 0.743091i \(-0.266641\pi\)
0.669191 + 0.743091i \(0.266641\pi\)
\(14\) −456.481 −0.622447
\(15\) 0 0
\(16\) 416.906 0.407135
\(17\) −462.991 −0.388553 −0.194276 0.980947i \(-0.562236\pi\)
−0.194276 + 0.980947i \(0.562236\pi\)
\(18\) 211.323 0.153732
\(19\) −1996.09 −1.26851 −0.634257 0.773122i \(-0.718694\pi\)
−0.634257 + 0.773122i \(0.718694\pi\)
\(20\) 0 0
\(21\) 1574.72 0.779211
\(22\) 315.680 0.139056
\(23\) −1342.70 −0.529250 −0.264625 0.964351i \(-0.585248\pi\)
−0.264625 + 0.964351i \(0.585248\pi\)
\(24\) 1342.92 0.475908
\(25\) 0 0
\(26\) 2127.65 0.617257
\(27\) −729.000 −0.192450
\(28\) 4408.08 1.06256
\(29\) 6586.33 1.45428 0.727141 0.686488i \(-0.240849\pi\)
0.727141 + 0.686488i \(0.240849\pi\)
\(30\) 0 0
\(31\) −4492.74 −0.839667 −0.419833 0.907601i \(-0.637911\pi\)
−0.419833 + 0.907601i \(0.637911\pi\)
\(32\) 5862.51 1.01207
\(33\) −1089.00 −0.174078
\(34\) −1207.91 −0.179199
\(35\) 0 0
\(36\) −2040.67 −0.262432
\(37\) 9089.95 1.09158 0.545792 0.837921i \(-0.316229\pi\)
0.545792 + 0.837921i \(0.316229\pi\)
\(38\) −5207.64 −0.585035
\(39\) −7339.74 −0.772715
\(40\) 0 0
\(41\) −1108.16 −0.102954 −0.0514769 0.998674i \(-0.516393\pi\)
−0.0514769 + 0.998674i \(0.516393\pi\)
\(42\) 4108.32 0.359370
\(43\) 20231.4 1.66861 0.834304 0.551304i \(-0.185870\pi\)
0.834304 + 0.551304i \(0.185870\pi\)
\(44\) −3048.42 −0.237379
\(45\) 0 0
\(46\) −3503.01 −0.244088
\(47\) 9098.00 0.600761 0.300380 0.953820i \(-0.402886\pi\)
0.300380 + 0.953820i \(0.402886\pi\)
\(48\) −3752.15 −0.235059
\(49\) 13807.1 0.821510
\(50\) 0 0
\(51\) 4166.92 0.224331
\(52\) −20546.0 −1.05370
\(53\) 38340.2 1.87484 0.937422 0.348196i \(-0.113206\pi\)
0.937422 + 0.348196i \(0.113206\pi\)
\(54\) −1901.91 −0.0887574
\(55\) 0 0
\(56\) 26107.7 1.11250
\(57\) 17964.8 0.732377
\(58\) 17183.2 0.670711
\(59\) 29309.0 1.09615 0.548076 0.836429i \(-0.315361\pi\)
0.548076 + 0.836429i \(0.315361\pi\)
\(60\) 0 0
\(61\) 32390.1 1.11452 0.557260 0.830338i \(-0.311853\pi\)
0.557260 + 0.830338i \(0.311853\pi\)
\(62\) −11721.2 −0.387252
\(63\) −14172.5 −0.449878
\(64\) 1953.85 0.0596267
\(65\) 0 0
\(66\) −2841.12 −0.0802841
\(67\) −64412.9 −1.75302 −0.876508 0.481387i \(-0.840133\pi\)
−0.876508 + 0.481387i \(0.840133\pi\)
\(68\) 11664.4 0.305907
\(69\) 12084.3 0.305562
\(70\) 0 0
\(71\) −63314.4 −1.49058 −0.745292 0.666738i \(-0.767690\pi\)
−0.745292 + 0.666738i \(0.767690\pi\)
\(72\) −12086.3 −0.274765
\(73\) 4121.83 0.0905279 0.0452640 0.998975i \(-0.485587\pi\)
0.0452640 + 0.998975i \(0.485587\pi\)
\(74\) 23715.0 0.503435
\(75\) 0 0
\(76\) 50288.4 0.998698
\(77\) −21171.2 −0.406930
\(78\) −19148.8 −0.356374
\(79\) −53233.3 −0.959656 −0.479828 0.877363i \(-0.659301\pi\)
−0.479828 + 0.877363i \(0.659301\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −2891.10 −0.0474820
\(83\) −119847. −1.90956 −0.954780 0.297315i \(-0.903909\pi\)
−0.954780 + 0.297315i \(0.903909\pi\)
\(84\) −39672.7 −0.613471
\(85\) 0 0
\(86\) 52782.1 0.769557
\(87\) −59277.0 −0.839630
\(88\) −18054.8 −0.248535
\(89\) 66306.8 0.887326 0.443663 0.896194i \(-0.353679\pi\)
0.443663 + 0.896194i \(0.353679\pi\)
\(90\) 0 0
\(91\) −142692. −1.80632
\(92\) 33827.4 0.416677
\(93\) 40434.6 0.484782
\(94\) 23736.0 0.277069
\(95\) 0 0
\(96\) −52762.6 −0.584316
\(97\) −180567. −1.94854 −0.974268 0.225394i \(-0.927633\pi\)
−0.974268 + 0.225394i \(0.927633\pi\)
\(98\) 36021.7 0.378878
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −113813. −1.11017 −0.555084 0.831794i \(-0.687314\pi\)
−0.555084 + 0.831794i \(0.687314\pi\)
\(102\) 10871.2 0.103461
\(103\) 192758. 1.79027 0.895135 0.445795i \(-0.147079\pi\)
0.895135 + 0.445795i \(0.147079\pi\)
\(104\) −121688. −1.10322
\(105\) 0 0
\(106\) 100027. 0.864672
\(107\) −139114. −1.17465 −0.587327 0.809350i \(-0.699820\pi\)
−0.587327 + 0.809350i \(0.699820\pi\)
\(108\) 18366.1 0.151515
\(109\) 97360.0 0.784900 0.392450 0.919773i \(-0.371628\pi\)
0.392450 + 0.919773i \(0.371628\pi\)
\(110\) 0 0
\(111\) −81809.6 −0.630227
\(112\) −72945.6 −0.549483
\(113\) −20952.6 −0.154362 −0.0771811 0.997017i \(-0.524592\pi\)
−0.0771811 + 0.997017i \(0.524592\pi\)
\(114\) 46868.7 0.337770
\(115\) 0 0
\(116\) −165933. −1.14495
\(117\) 66057.7 0.446127
\(118\) 76464.9 0.505542
\(119\) 81009.0 0.524404
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 84503.4 0.514013
\(123\) 9973.43 0.0594404
\(124\) 113188. 0.661067
\(125\) 0 0
\(126\) −36974.9 −0.207482
\(127\) 347758. 1.91323 0.956616 0.291351i \(-0.0941046\pi\)
0.956616 + 0.291351i \(0.0941046\pi\)
\(128\) −182503. −0.984566
\(129\) −182083. −0.963372
\(130\) 0 0
\(131\) −160486. −0.817070 −0.408535 0.912743i \(-0.633960\pi\)
−0.408535 + 0.912743i \(0.633960\pi\)
\(132\) 27435.7 0.137051
\(133\) 349253. 1.71203
\(134\) −168048. −0.808486
\(135\) 0 0
\(136\) 69084.5 0.320282
\(137\) −142596. −0.649090 −0.324545 0.945870i \(-0.605211\pi\)
−0.324545 + 0.945870i \(0.605211\pi\)
\(138\) 31527.1 0.140924
\(139\) −124192. −0.545201 −0.272600 0.962127i \(-0.587884\pi\)
−0.272600 + 0.962127i \(0.587884\pi\)
\(140\) 0 0
\(141\) −81882.0 −0.346849
\(142\) −165182. −0.687453
\(143\) 98678.7 0.403537
\(144\) 33769.4 0.135712
\(145\) 0 0
\(146\) 10753.5 0.0417512
\(147\) −124264. −0.474299
\(148\) −229008. −0.859402
\(149\) −332027. −1.22520 −0.612601 0.790393i \(-0.709877\pi\)
−0.612601 + 0.790393i \(0.709877\pi\)
\(150\) 0 0
\(151\) −221114. −0.789177 −0.394589 0.918858i \(-0.629113\pi\)
−0.394589 + 0.918858i \(0.629113\pi\)
\(152\) 297843. 1.04563
\(153\) −37502.2 −0.129518
\(154\) −55234.1 −0.187675
\(155\) 0 0
\(156\) 184914. 0.608356
\(157\) 178933. 0.579350 0.289675 0.957125i \(-0.406453\pi\)
0.289675 + 0.957125i \(0.406453\pi\)
\(158\) −138882. −0.442590
\(159\) −345062. −1.08244
\(160\) 0 0
\(161\) 234931. 0.714293
\(162\) 17117.1 0.0512441
\(163\) 222185. 0.655007 0.327504 0.944850i \(-0.393793\pi\)
0.327504 + 0.944850i \(0.393793\pi\)
\(164\) 27918.4 0.0810553
\(165\) 0 0
\(166\) −312673. −0.880683
\(167\) 286150. 0.793968 0.396984 0.917826i \(-0.370057\pi\)
0.396984 + 0.917826i \(0.370057\pi\)
\(168\) −234970. −0.642301
\(169\) 293791. 0.791265
\(170\) 0 0
\(171\) −161683. −0.422838
\(172\) −509700. −1.31369
\(173\) 585626. 1.48767 0.743833 0.668366i \(-0.233006\pi\)
0.743833 + 0.668366i \(0.233006\pi\)
\(174\) −154649. −0.387235
\(175\) 0 0
\(176\) 50445.6 0.122756
\(177\) −263781. −0.632863
\(178\) 172989. 0.409232
\(179\) −511386. −1.19293 −0.596467 0.802638i \(-0.703429\pi\)
−0.596467 + 0.802638i \(0.703429\pi\)
\(180\) 0 0
\(181\) −16277.3 −0.0369306 −0.0184653 0.999830i \(-0.505878\pi\)
−0.0184653 + 0.999830i \(0.505878\pi\)
\(182\) −372272. −0.833071
\(183\) −291511. −0.643469
\(184\) 200350. 0.436258
\(185\) 0 0
\(186\) 105491. 0.223580
\(187\) −56021.9 −0.117153
\(188\) −229211. −0.472977
\(189\) 127552. 0.259737
\(190\) 0 0
\(191\) −219087. −0.434543 −0.217271 0.976111i \(-0.569716\pi\)
−0.217271 + 0.976111i \(0.569716\pi\)
\(192\) −17584.6 −0.0344255
\(193\) −704505. −1.36142 −0.680708 0.732555i \(-0.738328\pi\)
−0.680708 + 0.732555i \(0.738328\pi\)
\(194\) −471085. −0.898658
\(195\) 0 0
\(196\) −347850. −0.646773
\(197\) −670027. −1.23006 −0.615031 0.788503i \(-0.710856\pi\)
−0.615031 + 0.788503i \(0.710856\pi\)
\(198\) 25570.1 0.0463520
\(199\) −591128. −1.05815 −0.529077 0.848574i \(-0.677462\pi\)
−0.529077 + 0.848574i \(0.677462\pi\)
\(200\) 0 0
\(201\) 579716. 1.01210
\(202\) −296930. −0.512006
\(203\) −1.15240e6 −1.96275
\(204\) −104979. −0.176615
\(205\) 0 0
\(206\) 502890. 0.825667
\(207\) −108759. −0.176417
\(208\) 339998. 0.544901
\(209\) −241526. −0.382471
\(210\) 0 0
\(211\) −1070.59 −0.00165545 −0.000827723 1.00000i \(-0.500263\pi\)
−0.000827723 1.00000i \(0.500263\pi\)
\(212\) −965925. −1.47606
\(213\) 569829. 0.860589
\(214\) −362937. −0.541747
\(215\) 0 0
\(216\) 108777. 0.158636
\(217\) 786090. 1.13324
\(218\) 254005. 0.361993
\(219\) −37096.4 −0.0522663
\(220\) 0 0
\(221\) −377581. −0.520032
\(222\) −213435. −0.290659
\(223\) 515555. 0.694246 0.347123 0.937820i \(-0.387159\pi\)
0.347123 + 0.937820i \(0.387159\pi\)
\(224\) −1.02576e6 −1.36592
\(225\) 0 0
\(226\) −54663.6 −0.0711914
\(227\) 69874.2 0.0900020 0.0450010 0.998987i \(-0.485671\pi\)
0.0450010 + 0.998987i \(0.485671\pi\)
\(228\) −452596. −0.576599
\(229\) 502115. 0.632725 0.316362 0.948638i \(-0.397538\pi\)
0.316362 + 0.948638i \(0.397538\pi\)
\(230\) 0 0
\(231\) 190541. 0.234941
\(232\) −982770. −1.19876
\(233\) 1.52257e6 1.83733 0.918664 0.395041i \(-0.129270\pi\)
0.918664 + 0.395041i \(0.129270\pi\)
\(234\) 172339. 0.205752
\(235\) 0 0
\(236\) −738396. −0.862997
\(237\) 479099. 0.554058
\(238\) 211346. 0.241853
\(239\) −601975. −0.681685 −0.340843 0.940120i \(-0.610712\pi\)
−0.340843 + 0.940120i \(0.610712\pi\)
\(240\) 0 0
\(241\) −960570. −1.06534 −0.532668 0.846324i \(-0.678811\pi\)
−0.532668 + 0.846324i \(0.678811\pi\)
\(242\) 38197.3 0.0419270
\(243\) −59049.0 −0.0641500
\(244\) −816021. −0.877459
\(245\) 0 0
\(246\) 26019.9 0.0274137
\(247\) −1.62786e6 −1.69776
\(248\) 670377. 0.692134
\(249\) 1.07863e6 1.10248
\(250\) 0 0
\(251\) −999437. −1.00132 −0.500658 0.865645i \(-0.666909\pi\)
−0.500658 + 0.865645i \(0.666909\pi\)
\(252\) 357055. 0.354188
\(253\) −162467. −0.159575
\(254\) 907274. 0.882377
\(255\) 0 0
\(256\) −538659. −0.513705
\(257\) 222038. 0.209698 0.104849 0.994488i \(-0.466564\pi\)
0.104849 + 0.994488i \(0.466564\pi\)
\(258\) −475039. −0.444304
\(259\) −1.59046e6 −1.47324
\(260\) 0 0
\(261\) 533493. 0.484761
\(262\) −418696. −0.376830
\(263\) −501628. −0.447190 −0.223595 0.974682i \(-0.571779\pi\)
−0.223595 + 0.974682i \(0.571779\pi\)
\(264\) 162494. 0.143492
\(265\) 0 0
\(266\) 911175. 0.789583
\(267\) −596761. −0.512298
\(268\) 1.62279e6 1.38015
\(269\) −1.50544e6 −1.26848 −0.634238 0.773138i \(-0.718686\pi\)
−0.634238 + 0.773138i \(0.718686\pi\)
\(270\) 0 0
\(271\) 935698. 0.773949 0.386975 0.922090i \(-0.373520\pi\)
0.386975 + 0.922090i \(0.373520\pi\)
\(272\) −193024. −0.158193
\(273\) 1.28423e6 1.04288
\(274\) −372021. −0.299358
\(275\) 0 0
\(276\) −304447. −0.240569
\(277\) 1.06811e6 0.836404 0.418202 0.908354i \(-0.362660\pi\)
0.418202 + 0.908354i \(0.362660\pi\)
\(278\) −324007. −0.251445
\(279\) −363912. −0.279889
\(280\) 0 0
\(281\) 683491. 0.516377 0.258189 0.966095i \(-0.416874\pi\)
0.258189 + 0.966095i \(0.416874\pi\)
\(282\) −213624. −0.159966
\(283\) 272388. 0.202172 0.101086 0.994878i \(-0.467768\pi\)
0.101086 + 0.994878i \(0.467768\pi\)
\(284\) 1.59511e6 1.17353
\(285\) 0 0
\(286\) 257445. 0.186110
\(287\) 193893. 0.138950
\(288\) 474863. 0.337355
\(289\) −1.20550e6 −0.849027
\(290\) 0 0
\(291\) 1.62510e6 1.12499
\(292\) −103843. −0.0712724
\(293\) 1.53914e6 1.04739 0.523697 0.851905i \(-0.324553\pi\)
0.523697 + 0.851905i \(0.324553\pi\)
\(294\) −324196. −0.218745
\(295\) 0 0
\(296\) −1.35634e6 −0.899789
\(297\) −88209.0 −0.0580259
\(298\) −866233. −0.565059
\(299\) −1.09501e6 −0.708338
\(300\) 0 0
\(301\) −3.53986e6 −2.25201
\(302\) −576870. −0.363966
\(303\) 1.02432e6 0.640956
\(304\) −832180. −0.516456
\(305\) 0 0
\(306\) −97840.5 −0.0597331
\(307\) −800678. −0.484855 −0.242428 0.970170i \(-0.577944\pi\)
−0.242428 + 0.970170i \(0.577944\pi\)
\(308\) 533378. 0.320375
\(309\) −1.73482e6 −1.03361
\(310\) 0 0
\(311\) 3.17233e6 1.85985 0.929925 0.367750i \(-0.119872\pi\)
0.929925 + 0.367750i \(0.119872\pi\)
\(312\) 1.09519e6 0.636946
\(313\) 996401. 0.574875 0.287437 0.957799i \(-0.407197\pi\)
0.287437 + 0.957799i \(0.407197\pi\)
\(314\) 466822. 0.267194
\(315\) 0 0
\(316\) 1.34113e6 0.755535
\(317\) −1.73853e6 −0.971702 −0.485851 0.874042i \(-0.661490\pi\)
−0.485851 + 0.874042i \(0.661490\pi\)
\(318\) −900241. −0.499219
\(319\) 796946. 0.438483
\(320\) 0 0
\(321\) 1.25202e6 0.678187
\(322\) 612918. 0.329430
\(323\) 924170. 0.492885
\(324\) −165295. −0.0874775
\(325\) 0 0
\(326\) 579664. 0.302087
\(327\) −876240. −0.453162
\(328\) 165352. 0.0848644
\(329\) −1.59187e6 −0.810807
\(330\) 0 0
\(331\) −530501. −0.266144 −0.133072 0.991106i \(-0.542484\pi\)
−0.133072 + 0.991106i \(0.542484\pi\)
\(332\) 3.01938e6 1.50339
\(333\) 736286. 0.363861
\(334\) 746544. 0.366176
\(335\) 0 0
\(336\) 656510. 0.317244
\(337\) −1110.26 −0.000532538 0 −0.000266269 1.00000i \(-0.500085\pi\)
−0.000266269 1.00000i \(0.500085\pi\)
\(338\) 766478. 0.364929
\(339\) 188573. 0.0891211
\(340\) 0 0
\(341\) −543621. −0.253169
\(342\) −421819. −0.195012
\(343\) 524885. 0.240896
\(344\) −3.01880e6 −1.37543
\(345\) 0 0
\(346\) 1.52785e6 0.686107
\(347\) 1.08849e6 0.485289 0.242644 0.970115i \(-0.421985\pi\)
0.242644 + 0.970115i \(0.421985\pi\)
\(348\) 1.49340e6 0.661039
\(349\) −591038. −0.259748 −0.129874 0.991531i \(-0.541457\pi\)
−0.129874 + 0.991531i \(0.541457\pi\)
\(350\) 0 0
\(351\) −594519. −0.257572
\(352\) 709363. 0.305149
\(353\) 980027. 0.418602 0.209301 0.977851i \(-0.432881\pi\)
0.209301 + 0.977851i \(0.432881\pi\)
\(354\) −688184. −0.291875
\(355\) 0 0
\(356\) −1.67050e6 −0.698589
\(357\) −729081. −0.302765
\(358\) −1.33417e6 −0.550177
\(359\) −2.87712e6 −1.17821 −0.589104 0.808057i \(-0.700519\pi\)
−0.589104 + 0.808057i \(0.700519\pi\)
\(360\) 0 0
\(361\) 1.50826e6 0.609129
\(362\) −42466.3 −0.0170323
\(363\) −131769. −0.0524864
\(364\) 3.59491e6 1.42211
\(365\) 0 0
\(366\) −760530. −0.296766
\(367\) 3.28496e6 1.27311 0.636554 0.771232i \(-0.280359\pi\)
0.636554 + 0.771232i \(0.280359\pi\)
\(368\) −559781. −0.215476
\(369\) −89760.9 −0.0343179
\(370\) 0 0
\(371\) −6.70835e6 −2.53035
\(372\) −1.01869e6 −0.381667
\(373\) 1.03382e6 0.384744 0.192372 0.981322i \(-0.438382\pi\)
0.192372 + 0.981322i \(0.438382\pi\)
\(374\) −146157. −0.0540306
\(375\) 0 0
\(376\) −1.35755e6 −0.495205
\(377\) 5.37133e6 1.94638
\(378\) 332774. 0.119790
\(379\) 3.41421e6 1.22094 0.610468 0.792041i \(-0.290982\pi\)
0.610468 + 0.792041i \(0.290982\pi\)
\(380\) 0 0
\(381\) −3.12982e6 −1.10461
\(382\) −571580. −0.200410
\(383\) −1.82511e6 −0.635758 −0.317879 0.948131i \(-0.602971\pi\)
−0.317879 + 0.948131i \(0.602971\pi\)
\(384\) 1.64253e6 0.568439
\(385\) 0 0
\(386\) −1.83800e6 −0.627881
\(387\) 1.63874e6 0.556203
\(388\) 4.54911e6 1.53408
\(389\) −1.31381e6 −0.440210 −0.220105 0.975476i \(-0.570640\pi\)
−0.220105 + 0.975476i \(0.570640\pi\)
\(390\) 0 0
\(391\) 621659. 0.205641
\(392\) −2.06021e6 −0.677168
\(393\) 1.44438e6 0.471736
\(394\) −1.74805e6 −0.567301
\(395\) 0 0
\(396\) −246922. −0.0791264
\(397\) −4.07499e6 −1.29763 −0.648814 0.760947i \(-0.724735\pi\)
−0.648814 + 0.760947i \(0.724735\pi\)
\(398\) −1.54221e6 −0.488017
\(399\) −3.14328e6 −0.988441
\(400\) 0 0
\(401\) 1.88854e6 0.586495 0.293247 0.956037i \(-0.405264\pi\)
0.293247 + 0.956037i \(0.405264\pi\)
\(402\) 1.51244e6 0.466779
\(403\) −3.66395e6 −1.12379
\(404\) 2.86735e6 0.874033
\(405\) 0 0
\(406\) −3.00653e6 −0.905213
\(407\) 1.09988e6 0.329125
\(408\) −621760. −0.184915
\(409\) −2.77474e6 −0.820188 −0.410094 0.912043i \(-0.634504\pi\)
−0.410094 + 0.912043i \(0.634504\pi\)
\(410\) 0 0
\(411\) 1.28336e6 0.374752
\(412\) −4.85624e6 −1.40948
\(413\) −5.12816e6 −1.47940
\(414\) −283744. −0.0813628
\(415\) 0 0
\(416\) 4.78103e6 1.35453
\(417\) 1.11773e6 0.314772
\(418\) −630124. −0.176395
\(419\) 1.74694e6 0.486119 0.243060 0.970011i \(-0.421849\pi\)
0.243060 + 0.970011i \(0.421849\pi\)
\(420\) 0 0
\(421\) 321171. 0.0883143 0.0441572 0.999025i \(-0.485940\pi\)
0.0441572 + 0.999025i \(0.485940\pi\)
\(422\) −2793.08 −0.000763487 0
\(423\) 736938. 0.200254
\(424\) −5.72088e6 −1.54543
\(425\) 0 0
\(426\) 1.48664e6 0.396901
\(427\) −5.66726e6 −1.50419
\(428\) 3.50476e6 0.924802
\(429\) −888109. −0.232982
\(430\) 0 0
\(431\) 1.16098e6 0.301046 0.150523 0.988607i \(-0.451904\pi\)
0.150523 + 0.988607i \(0.451904\pi\)
\(432\) −303924. −0.0783531
\(433\) −1.65899e6 −0.425229 −0.212615 0.977136i \(-0.568198\pi\)
−0.212615 + 0.977136i \(0.568198\pi\)
\(434\) 2.05085e6 0.522648
\(435\) 0 0
\(436\) −2.45284e6 −0.617950
\(437\) 2.68015e6 0.671361
\(438\) −96781.8 −0.0241051
\(439\) 7.30846e6 1.80994 0.904971 0.425473i \(-0.139892\pi\)
0.904971 + 0.425473i \(0.139892\pi\)
\(440\) 0 0
\(441\) 1.11838e6 0.273837
\(442\) −985081. −0.239837
\(443\) −3.34639e6 −0.810153 −0.405077 0.914283i \(-0.632755\pi\)
−0.405077 + 0.914283i \(0.632755\pi\)
\(444\) 2.06107e6 0.496176
\(445\) 0 0
\(446\) 1.34504e6 0.320184
\(447\) 2.98824e6 0.707370
\(448\) −341863. −0.0804742
\(449\) −2.28621e6 −0.535180 −0.267590 0.963533i \(-0.586227\pi\)
−0.267590 + 0.963533i \(0.586227\pi\)
\(450\) 0 0
\(451\) −134087. −0.0310417
\(452\) 527869. 0.121529
\(453\) 1.99003e6 0.455632
\(454\) 182297. 0.0415087
\(455\) 0 0
\(456\) −2.68059e6 −0.603696
\(457\) −3.57185e6 −0.800024 −0.400012 0.916510i \(-0.630994\pi\)
−0.400012 + 0.916510i \(0.630994\pi\)
\(458\) 1.30998e6 0.291811
\(459\) 337520. 0.0747770
\(460\) 0 0
\(461\) −7.95525e6 −1.74342 −0.871709 0.490024i \(-0.836988\pi\)
−0.871709 + 0.490024i \(0.836988\pi\)
\(462\) 497107. 0.108354
\(463\) −5.70155e6 −1.23606 −0.618032 0.786153i \(-0.712070\pi\)
−0.618032 + 0.786153i \(0.712070\pi\)
\(464\) 2.74588e6 0.592089
\(465\) 0 0
\(466\) 3.97226e6 0.847370
\(467\) −513186. −0.108889 −0.0544444 0.998517i \(-0.517339\pi\)
−0.0544444 + 0.998517i \(0.517339\pi\)
\(468\) −1.66423e6 −0.351235
\(469\) 1.12703e7 2.36593
\(470\) 0 0
\(471\) −1.61040e6 −0.334488
\(472\) −4.37329e6 −0.903553
\(473\) 2.44800e6 0.503105
\(474\) 1.24993e6 0.255530
\(475\) 0 0
\(476\) −2.04090e6 −0.412862
\(477\) 3.10556e6 0.624948
\(478\) −1.57051e6 −0.314391
\(479\) 5.59628e6 1.11445 0.557225 0.830361i \(-0.311866\pi\)
0.557225 + 0.830361i \(0.311866\pi\)
\(480\) 0 0
\(481\) 7.41310e6 1.46096
\(482\) −2.50605e6 −0.491330
\(483\) −2.11438e6 −0.412397
\(484\) −368858. −0.0715725
\(485\) 0 0
\(486\) −154054. −0.0295858
\(487\) −425345. −0.0812678 −0.0406339 0.999174i \(-0.512938\pi\)
−0.0406339 + 0.999174i \(0.512938\pi\)
\(488\) −4.83304e6 −0.918695
\(489\) −1.99967e6 −0.378169
\(490\) 0 0
\(491\) −5.56781e6 −1.04227 −0.521136 0.853474i \(-0.674492\pi\)
−0.521136 + 0.853474i \(0.674492\pi\)
\(492\) −251266. −0.0467973
\(493\) −3.04941e6 −0.565065
\(494\) −4.24697e6 −0.783000
\(495\) 0 0
\(496\) −1.87305e6 −0.341857
\(497\) 1.10780e7 2.01174
\(498\) 2.81405e6 0.508462
\(499\) −5.31330e6 −0.955241 −0.477621 0.878566i \(-0.658501\pi\)
−0.477621 + 0.878566i \(0.658501\pi\)
\(500\) 0 0
\(501\) −2.57535e6 −0.458398
\(502\) −2.60745e6 −0.461804
\(503\) 1.03261e7 1.81976 0.909882 0.414867i \(-0.136172\pi\)
0.909882 + 0.414867i \(0.136172\pi\)
\(504\) 2.11473e6 0.370832
\(505\) 0 0
\(506\) −423864. −0.0735954
\(507\) −2.64412e6 −0.456837
\(508\) −8.76125e6 −1.50628
\(509\) 1.37144e6 0.234629 0.117314 0.993095i \(-0.462571\pi\)
0.117314 + 0.993095i \(0.462571\pi\)
\(510\) 0 0
\(511\) −721191. −0.122179
\(512\) 4.43477e6 0.747646
\(513\) 1.45515e6 0.244126
\(514\) 579280. 0.0967121
\(515\) 0 0
\(516\) 4.58730e6 0.758460
\(517\) 1.10086e6 0.181136
\(518\) −4.14939e6 −0.679453
\(519\) −5.27064e6 −0.858904
\(520\) 0 0
\(521\) 524277. 0.0846187 0.0423094 0.999105i \(-0.486528\pi\)
0.0423094 + 0.999105i \(0.486528\pi\)
\(522\) 1.39184e6 0.223570
\(523\) −1.42764e6 −0.228225 −0.114113 0.993468i \(-0.536402\pi\)
−0.114113 + 0.993468i \(0.536402\pi\)
\(524\) 4.04321e6 0.643277
\(525\) 0 0
\(526\) −1.30871e6 −0.206243
\(527\) 2.08010e6 0.326255
\(528\) −454010. −0.0708730
\(529\) −4.63349e6 −0.719895
\(530\) 0 0
\(531\) 2.37403e6 0.365384
\(532\) −8.79892e6 −1.34788
\(533\) −903733. −0.137791
\(534\) −1.55690e6 −0.236270
\(535\) 0 0
\(536\) 9.61128e6 1.44500
\(537\) 4.60248e6 0.688741
\(538\) −3.92757e6 −0.585017
\(539\) 1.67066e6 0.247695
\(540\) 0 0
\(541\) 5.39626e6 0.792683 0.396341 0.918103i \(-0.370280\pi\)
0.396341 + 0.918103i \(0.370280\pi\)
\(542\) 2.44116e6 0.356943
\(543\) 146496. 0.0213219
\(544\) −2.71429e6 −0.393241
\(545\) 0 0
\(546\) 3.35045e6 0.480974
\(547\) −7.34437e6 −1.04951 −0.524754 0.851254i \(-0.675843\pi\)
−0.524754 + 0.851254i \(0.675843\pi\)
\(548\) 3.59249e6 0.511027
\(549\) 2.62360e6 0.371507
\(550\) 0 0
\(551\) −1.31469e7 −1.84478
\(552\) −1.80315e6 −0.251874
\(553\) 9.31417e6 1.29518
\(554\) 2.78662e6 0.385747
\(555\) 0 0
\(556\) 3.12883e6 0.429235
\(557\) 7.05449e6 0.963447 0.481723 0.876323i \(-0.340011\pi\)
0.481723 + 0.876323i \(0.340011\pi\)
\(558\) −949418. −0.129084
\(559\) 1.64992e7 2.23324
\(560\) 0 0
\(561\) 504197. 0.0676383
\(562\) 1.78318e6 0.238152
\(563\) 126964. 0.0168814 0.00844072 0.999964i \(-0.497313\pi\)
0.00844072 + 0.999964i \(0.497313\pi\)
\(564\) 2.06290e6 0.273074
\(565\) 0 0
\(566\) 710638. 0.0932411
\(567\) −1.14797e6 −0.149959
\(568\) 9.44736e6 1.22868
\(569\) −1.36950e7 −1.77330 −0.886648 0.462445i \(-0.846972\pi\)
−0.886648 + 0.462445i \(0.846972\pi\)
\(570\) 0 0
\(571\) −484553. −0.0621944 −0.0310972 0.999516i \(-0.509900\pi\)
−0.0310972 + 0.999516i \(0.509900\pi\)
\(572\) −2.48606e6 −0.317704
\(573\) 1.97178e6 0.250883
\(574\) 505853. 0.0640833
\(575\) 0 0
\(576\) 158262. 0.0198756
\(577\) −4.31027e6 −0.538970 −0.269485 0.963005i \(-0.586853\pi\)
−0.269485 + 0.963005i \(0.586853\pi\)
\(578\) −3.14505e6 −0.391569
\(579\) 6.34054e6 0.786014
\(580\) 0 0
\(581\) 2.09696e7 2.57720
\(582\) 4.23976e6 0.518841
\(583\) 4.63917e6 0.565287
\(584\) −615032. −0.0746218
\(585\) 0 0
\(586\) 4.01551e6 0.483055
\(587\) −4.35136e6 −0.521230 −0.260615 0.965443i \(-0.583925\pi\)
−0.260615 + 0.965443i \(0.583925\pi\)
\(588\) 3.13065e6 0.373415
\(589\) 8.96790e6 1.06513
\(590\) 0 0
\(591\) 6.03024e6 0.710176
\(592\) 3.78965e6 0.444422
\(593\) −2.85779e6 −0.333729 −0.166865 0.985980i \(-0.553364\pi\)
−0.166865 + 0.985980i \(0.553364\pi\)
\(594\) −230131. −0.0267614
\(595\) 0 0
\(596\) 8.36492e6 0.964598
\(597\) 5.32015e6 0.610926
\(598\) −2.85680e6 −0.326683
\(599\) 1.62076e7 1.84566 0.922831 0.385206i \(-0.125870\pi\)
0.922831 + 0.385206i \(0.125870\pi\)
\(600\) 0 0
\(601\) −6.74322e6 −0.761520 −0.380760 0.924674i \(-0.624338\pi\)
−0.380760 + 0.924674i \(0.624338\pi\)
\(602\) −9.23524e6 −1.03862
\(603\) −5.21745e6 −0.584339
\(604\) 5.57065e6 0.621317
\(605\) 0 0
\(606\) 2.67237e6 0.295607
\(607\) −1.76156e6 −0.194055 −0.0970276 0.995282i \(-0.530933\pi\)
−0.0970276 + 0.995282i \(0.530933\pi\)
\(608\) −1.17021e7 −1.28382
\(609\) 1.03716e7 1.13319
\(610\) 0 0
\(611\) 7.41967e6 0.804047
\(612\) 944813. 0.101969
\(613\) −1.71643e7 −1.84491 −0.922456 0.386102i \(-0.873821\pi\)
−0.922456 + 0.386102i \(0.873821\pi\)
\(614\) −2.08891e6 −0.223614
\(615\) 0 0
\(616\) 3.15904e6 0.335431
\(617\) 5.60639e6 0.592885 0.296443 0.955051i \(-0.404200\pi\)
0.296443 + 0.955051i \(0.404200\pi\)
\(618\) −4.52601e6 −0.476699
\(619\) 679689. 0.0712990 0.0356495 0.999364i \(-0.488650\pi\)
0.0356495 + 0.999364i \(0.488650\pi\)
\(620\) 0 0
\(621\) 978831. 0.101854
\(622\) 8.27637e6 0.857757
\(623\) −1.16016e7 −1.19756
\(624\) −3.05998e6 −0.314599
\(625\) 0 0
\(626\) 2.59953e6 0.265131
\(627\) 2.17374e6 0.220820
\(628\) −4.50795e6 −0.456121
\(629\) −4.20856e6 −0.424138
\(630\) 0 0
\(631\) −1.79404e6 −0.179374 −0.0896868 0.995970i \(-0.528587\pi\)
−0.0896868 + 0.995970i \(0.528587\pi\)
\(632\) 7.94312e6 0.791040
\(633\) 9635.27 0.000955773 0
\(634\) −4.53568e6 −0.448146
\(635\) 0 0
\(636\) 8.69333e6 0.852203
\(637\) 1.12601e7 1.09949
\(638\) 2.07917e6 0.202227
\(639\) −5.12846e6 −0.496861
\(640\) 0 0
\(641\) −1.87077e6 −0.179836 −0.0899179 0.995949i \(-0.528660\pi\)
−0.0899179 + 0.995949i \(0.528660\pi\)
\(642\) 3.26643e6 0.312778
\(643\) −1.17131e7 −1.11724 −0.558618 0.829425i \(-0.688668\pi\)
−0.558618 + 0.829425i \(0.688668\pi\)
\(644\) −5.91875e6 −0.562361
\(645\) 0 0
\(646\) 2.41109e6 0.227317
\(647\) −8.80680e6 −0.827099 −0.413550 0.910482i \(-0.635711\pi\)
−0.413550 + 0.910482i \(0.635711\pi\)
\(648\) −978990. −0.0915884
\(649\) 3.54638e6 0.330502
\(650\) 0 0
\(651\) −7.07481e6 −0.654278
\(652\) −5.59763e6 −0.515686
\(653\) 6.56212e6 0.602229 0.301114 0.953588i \(-0.402641\pi\)
0.301114 + 0.953588i \(0.402641\pi\)
\(654\) −2.28604e6 −0.208997
\(655\) 0 0
\(656\) −461998. −0.0419161
\(657\) 333868. 0.0301760
\(658\) −4.15306e6 −0.373941
\(659\) −1.31437e7 −1.17897 −0.589486 0.807778i \(-0.700670\pi\)
−0.589486 + 0.807778i \(0.700670\pi\)
\(660\) 0 0
\(661\) −1.73548e7 −1.54496 −0.772478 0.635041i \(-0.780983\pi\)
−0.772478 + 0.635041i \(0.780983\pi\)
\(662\) −1.38404e6 −0.122745
\(663\) 3.39823e6 0.300240
\(664\) 1.78828e7 1.57404
\(665\) 0 0
\(666\) 1.92091e6 0.167812
\(667\) −8.84350e6 −0.769678
\(668\) −7.20913e6 −0.625089
\(669\) −4.64000e6 −0.400823
\(670\) 0 0
\(671\) 3.91920e6 0.336041
\(672\) 9.23181e6 0.788613
\(673\) −1.02596e7 −0.873159 −0.436580 0.899666i \(-0.643810\pi\)
−0.436580 + 0.899666i \(0.643810\pi\)
\(674\) −2896.59 −0.000245605 0
\(675\) 0 0
\(676\) −7.40163e6 −0.622961
\(677\) 1.22085e6 0.102375 0.0511873 0.998689i \(-0.483699\pi\)
0.0511873 + 0.998689i \(0.483699\pi\)
\(678\) 491973. 0.0411024
\(679\) 3.15936e7 2.62981
\(680\) 0 0
\(681\) −628868. −0.0519627
\(682\) −1.41827e6 −0.116761
\(683\) 8.61333e6 0.706511 0.353256 0.935527i \(-0.385075\pi\)
0.353256 + 0.935527i \(0.385075\pi\)
\(684\) 4.07336e6 0.332899
\(685\) 0 0
\(686\) 1.36939e6 0.111100
\(687\) −4.51904e6 −0.365304
\(688\) 8.43458e6 0.679348
\(689\) 3.12675e7 2.50926
\(690\) 0 0
\(691\) −1.53208e7 −1.22064 −0.610319 0.792156i \(-0.708959\pi\)
−0.610319 + 0.792156i \(0.708959\pi\)
\(692\) −1.47540e7 −1.17124
\(693\) −1.71487e6 −0.135643
\(694\) 2.83978e6 0.223814
\(695\) 0 0
\(696\) 8.84493e6 0.692104
\(697\) 513067. 0.0400030
\(698\) −1.54197e6 −0.119795
\(699\) −1.37031e7 −1.06078
\(700\) 0 0
\(701\) 233349. 0.0179354 0.00896768 0.999960i \(-0.497145\pi\)
0.00896768 + 0.999960i \(0.497145\pi\)
\(702\) −1.55105e6 −0.118791
\(703\) −1.81443e7 −1.38469
\(704\) 236416. 0.0179781
\(705\) 0 0
\(706\) 2.55682e6 0.193058
\(707\) 1.99138e7 1.49832
\(708\) 6.64556e6 0.498252
\(709\) −3.15108e6 −0.235420 −0.117710 0.993048i \(-0.537555\pi\)
−0.117710 + 0.993048i \(0.537555\pi\)
\(710\) 0 0
\(711\) −4.31190e6 −0.319885
\(712\) −9.89387e6 −0.731419
\(713\) 6.03242e6 0.444393
\(714\) −1.90212e6 −0.139634
\(715\) 0 0
\(716\) 1.28836e7 0.939194
\(717\) 5.41778e6 0.393571
\(718\) −7.50619e6 −0.543386
\(719\) −2.38682e7 −1.72186 −0.860930 0.508723i \(-0.830118\pi\)
−0.860930 + 0.508723i \(0.830118\pi\)
\(720\) 0 0
\(721\) −3.37266e7 −2.41621
\(722\) 3.93494e6 0.280928
\(723\) 8.64513e6 0.615072
\(724\) 410083. 0.0290754
\(725\) 0 0
\(726\) −343775. −0.0242066
\(727\) 2.60780e7 1.82995 0.914974 0.403513i \(-0.132211\pi\)
0.914974 + 0.403513i \(0.132211\pi\)
\(728\) 2.12916e7 1.48895
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −9.36694e6 −0.648343
\(732\) 7.34419e6 0.506601
\(733\) −822931. −0.0565723 −0.0282861 0.999600i \(-0.509005\pi\)
−0.0282861 + 0.999600i \(0.509005\pi\)
\(734\) 8.57021e6 0.587153
\(735\) 0 0
\(736\) −7.87161e6 −0.535635
\(737\) −7.79396e6 −0.528554
\(738\) −234179. −0.0158273
\(739\) 1.58916e7 1.07043 0.535215 0.844716i \(-0.320231\pi\)
0.535215 + 0.844716i \(0.320231\pi\)
\(740\) 0 0
\(741\) 1.46508e7 0.980200
\(742\) −1.75016e7 −1.16699
\(743\) −8.78449e6 −0.583774 −0.291887 0.956453i \(-0.594283\pi\)
−0.291887 + 0.956453i \(0.594283\pi\)
\(744\) −6.03340e6 −0.399604
\(745\) 0 0
\(746\) 2.69715e6 0.177443
\(747\) −9.70763e6 −0.636520
\(748\) 1.41139e6 0.0922343
\(749\) 2.43406e7 1.58535
\(750\) 0 0
\(751\) −1.46203e7 −0.945926 −0.472963 0.881082i \(-0.656816\pi\)
−0.472963 + 0.881082i \(0.656816\pi\)
\(752\) 3.79301e6 0.244590
\(753\) 8.99493e6 0.578110
\(754\) 1.40134e7 0.897666
\(755\) 0 0
\(756\) −3.21349e6 −0.204490
\(757\) 7.20669e6 0.457084 0.228542 0.973534i \(-0.426604\pi\)
0.228542 + 0.973534i \(0.426604\pi\)
\(758\) 8.90742e6 0.563092
\(759\) 1.46220e6 0.0921305
\(760\) 0 0
\(761\) −2.92986e7 −1.83394 −0.916969 0.398958i \(-0.869372\pi\)
−0.916969 + 0.398958i \(0.869372\pi\)
\(762\) −8.16547e6 −0.509441
\(763\) −1.70350e7 −1.05933
\(764\) 5.51956e6 0.342114
\(765\) 0 0
\(766\) −4.76157e6 −0.293209
\(767\) 2.39022e7 1.46707
\(768\) 4.84793e6 0.296588
\(769\) −9.38411e6 −0.572239 −0.286119 0.958194i \(-0.592365\pi\)
−0.286119 + 0.958194i \(0.592365\pi\)
\(770\) 0 0
\(771\) −1.99834e6 −0.121069
\(772\) 1.77490e7 1.07184
\(773\) 3.43323e6 0.206659 0.103329 0.994647i \(-0.467050\pi\)
0.103329 + 0.994647i \(0.467050\pi\)
\(774\) 4.27535e6 0.256519
\(775\) 0 0
\(776\) 2.69430e7 1.60617
\(777\) 1.43141e7 0.850575
\(778\) −3.42764e6 −0.203024
\(779\) 2.21198e6 0.130598
\(780\) 0 0
\(781\) −7.66104e6 −0.449428
\(782\) 1.62186e6 0.0948412
\(783\) −4.80144e6 −0.279877
\(784\) 5.75627e6 0.334465
\(785\) 0 0
\(786\) 3.76827e6 0.217563
\(787\) −9.21027e6 −0.530073 −0.265036 0.964238i \(-0.585384\pi\)
−0.265036 + 0.964238i \(0.585384\pi\)
\(788\) 1.68803e7 0.968424
\(789\) 4.51465e6 0.258185
\(790\) 0 0
\(791\) 3.66605e6 0.208332
\(792\) −1.46244e6 −0.0828449
\(793\) 2.64150e7 1.49165
\(794\) −1.06313e7 −0.598462
\(795\) 0 0
\(796\) 1.48926e7 0.833082
\(797\) 1.76659e7 0.985120 0.492560 0.870279i \(-0.336061\pi\)
0.492560 + 0.870279i \(0.336061\pi\)
\(798\) −8.20057e6 −0.455866
\(799\) −4.21229e6 −0.233427
\(800\) 0 0
\(801\) 5.37085e6 0.295775
\(802\) 4.92705e6 0.270490
\(803\) 498741. 0.0272952
\(804\) −1.46051e7 −0.796827
\(805\) 0 0
\(806\) −9.55896e6 −0.518290
\(807\) 1.35489e7 0.732355
\(808\) 1.69825e7 0.915108
\(809\) 5.29802e6 0.284605 0.142302 0.989823i \(-0.454549\pi\)
0.142302 + 0.989823i \(0.454549\pi\)
\(810\) 0 0
\(811\) 1.81272e7 0.967786 0.483893 0.875127i \(-0.339222\pi\)
0.483893 + 0.875127i \(0.339222\pi\)
\(812\) 2.90331e7 1.54527
\(813\) −8.42128e6 −0.446840
\(814\) 2.86951e6 0.151791
\(815\) 0 0
\(816\) 1.73721e6 0.0913329
\(817\) −4.03836e7 −2.11665
\(818\) −7.23908e6 −0.378268
\(819\) −1.15580e7 −0.602108
\(820\) 0 0
\(821\) 3.68262e7 1.90677 0.953387 0.301751i \(-0.0975710\pi\)
0.953387 + 0.301751i \(0.0975710\pi\)
\(822\) 3.34819e6 0.172835
\(823\) 3.25146e7 1.67332 0.836660 0.547723i \(-0.184505\pi\)
0.836660 + 0.547723i \(0.184505\pi\)
\(824\) −2.87621e7 −1.47571
\(825\) 0 0
\(826\) −1.33790e7 −0.682296
\(827\) −4.14035e6 −0.210510 −0.105255 0.994445i \(-0.533566\pi\)
−0.105255 + 0.994445i \(0.533566\pi\)
\(828\) 2.74002e6 0.138892
\(829\) −2.81888e7 −1.42459 −0.712296 0.701879i \(-0.752345\pi\)
−0.712296 + 0.701879i \(0.752345\pi\)
\(830\) 0 0
\(831\) −9.61298e6 −0.482898
\(832\) 1.59342e6 0.0798033
\(833\) −6.39257e6 −0.319200
\(834\) 2.91606e6 0.145172
\(835\) 0 0
\(836\) 6.08490e6 0.301119
\(837\) 3.27521e6 0.161594
\(838\) 4.55763e6 0.224197
\(839\) −470091. −0.0230556 −0.0115278 0.999934i \(-0.503669\pi\)
−0.0115278 + 0.999934i \(0.503669\pi\)
\(840\) 0 0
\(841\) 2.28686e7 1.11494
\(842\) 837911. 0.0407303
\(843\) −6.15142e6 −0.298130
\(844\) 26971.8 0.00130333
\(845\) 0 0
\(846\) 1.92262e6 0.0923563
\(847\) −2.56172e6 −0.122694
\(848\) 1.59843e7 0.763314
\(849\) −2.45149e6 −0.116724
\(850\) 0 0
\(851\) −1.22051e7 −0.577721
\(852\) −1.43560e7 −0.677540
\(853\) −3.80179e7 −1.78902 −0.894510 0.447047i \(-0.852476\pi\)
−0.894510 + 0.447047i \(0.852476\pi\)
\(854\) −1.47855e7 −0.693730
\(855\) 0 0
\(856\) 2.07576e7 0.968263
\(857\) 2.72661e7 1.26815 0.634076 0.773271i \(-0.281381\pi\)
0.634076 + 0.773271i \(0.281381\pi\)
\(858\) −2.31701e6 −0.107451
\(859\) 3.07007e7 1.41960 0.709799 0.704404i \(-0.248786\pi\)
0.709799 + 0.704404i \(0.248786\pi\)
\(860\) 0 0
\(861\) −1.74504e6 −0.0802228
\(862\) 3.02892e6 0.138841
\(863\) −2.20422e7 −1.00746 −0.503731 0.863861i \(-0.668040\pi\)
−0.503731 + 0.863861i \(0.668040\pi\)
\(864\) −4.27377e6 −0.194772
\(865\) 0 0
\(866\) −4.32817e6 −0.196114
\(867\) 1.08495e7 0.490186
\(868\) −1.98044e7 −0.892199
\(869\) −6.44123e6 −0.289347
\(870\) 0 0
\(871\) −5.25305e7 −2.34620
\(872\) −1.45274e7 −0.646990
\(873\) −1.46259e7 −0.649512
\(874\) 6.99231e6 0.309630
\(875\) 0 0
\(876\) 934590. 0.0411491
\(877\) 6.53105e6 0.286737 0.143369 0.989669i \(-0.454207\pi\)
0.143369 + 0.989669i \(0.454207\pi\)
\(878\) 1.90672e7 0.834740
\(879\) −1.38523e7 −0.604713
\(880\) 0 0
\(881\) 2.27023e7 0.985439 0.492719 0.870188i \(-0.336003\pi\)
0.492719 + 0.870188i \(0.336003\pi\)
\(882\) 2.91776e6 0.126293
\(883\) 2.91594e7 1.25857 0.629285 0.777174i \(-0.283348\pi\)
0.629285 + 0.777174i \(0.283348\pi\)
\(884\) 9.51260e6 0.409420
\(885\) 0 0
\(886\) −8.73048e6 −0.373640
\(887\) −1.47884e7 −0.631120 −0.315560 0.948906i \(-0.602192\pi\)
−0.315560 + 0.948906i \(0.602192\pi\)
\(888\) 1.22071e7 0.519493
\(889\) −6.08468e7 −2.58216
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −1.29887e7 −0.546578
\(893\) −1.81604e7 −0.762073
\(894\) 7.79609e6 0.326237
\(895\) 0 0
\(896\) 3.19323e7 1.32880
\(897\) 9.85510e6 0.408959
\(898\) −5.96454e6 −0.246823
\(899\) −2.95907e7 −1.22111
\(900\) 0 0
\(901\) −1.77512e7 −0.728475
\(902\) −349823. −0.0143164
\(903\) 3.18588e7 1.30020
\(904\) 3.12641e6 0.127240
\(905\) 0 0
\(906\) 5.19183e6 0.210136
\(907\) −6.96226e6 −0.281017 −0.140508 0.990080i \(-0.544874\pi\)
−0.140508 + 0.990080i \(0.544874\pi\)
\(908\) −1.76038e6 −0.0708584
\(909\) −9.21887e6 −0.370056
\(910\) 0 0
\(911\) 3.40287e7 1.35847 0.679234 0.733922i \(-0.262312\pi\)
0.679234 + 0.733922i \(0.262312\pi\)
\(912\) 7.48962e6 0.298176
\(913\) −1.45015e7 −0.575754
\(914\) −9.31869e6 −0.368968
\(915\) 0 0
\(916\) −1.26501e7 −0.498143
\(917\) 2.80801e7 1.10275
\(918\) 880564. 0.0344869
\(919\) −2.25426e7 −0.880470 −0.440235 0.897883i \(-0.645105\pi\)
−0.440235 + 0.897883i \(0.645105\pi\)
\(920\) 0 0
\(921\) 7.20610e6 0.279931
\(922\) −2.07546e7 −0.804059
\(923\) −5.16346e7 −1.99497
\(924\) −4.80040e6 −0.184968
\(925\) 0 0
\(926\) −1.48749e7 −0.570069
\(927\) 1.56134e7 0.596757
\(928\) 3.86124e7 1.47183
\(929\) 3.13230e6 0.119076 0.0595381 0.998226i \(-0.481037\pi\)
0.0595381 + 0.998226i \(0.481037\pi\)
\(930\) 0 0
\(931\) −2.75602e7 −1.04210
\(932\) −3.83588e7 −1.44652
\(933\) −2.85510e7 −1.07378
\(934\) −1.33886e6 −0.0502191
\(935\) 0 0
\(936\) −9.85670e6 −0.367741
\(937\) −1.11644e7 −0.415417 −0.207709 0.978191i \(-0.566601\pi\)
−0.207709 + 0.978191i \(0.566601\pi\)
\(938\) 2.94032e7 1.09116
\(939\) −8.96761e6 −0.331904
\(940\) 0 0
\(941\) 2.20914e7 0.813296 0.406648 0.913585i \(-0.366698\pi\)
0.406648 + 0.913585i \(0.366698\pi\)
\(942\) −4.20140e6 −0.154265
\(943\) 1.48793e6 0.0544883
\(944\) 1.22191e7 0.446281
\(945\) 0 0
\(946\) 6.38664e6 0.232030
\(947\) 3.88629e7 1.40819 0.704093 0.710107i \(-0.251354\pi\)
0.704093 + 0.710107i \(0.251354\pi\)
\(948\) −1.20702e7 −0.436208
\(949\) 3.36146e6 0.121161
\(950\) 0 0
\(951\) 1.56467e7 0.561012
\(952\) −1.20876e7 −0.432264
\(953\) −1.65779e6 −0.0591287 −0.0295643 0.999563i \(-0.509412\pi\)
−0.0295643 + 0.999563i \(0.509412\pi\)
\(954\) 8.10216e6 0.288224
\(955\) 0 0
\(956\) 1.51659e7 0.536689
\(957\) −7.17252e6 −0.253158
\(958\) 1.46003e7 0.513981
\(959\) 2.49498e7 0.876033
\(960\) 0 0
\(961\) −8.44445e6 −0.294960
\(962\) 1.93402e7 0.673789
\(963\) −1.12682e7 −0.391551
\(964\) 2.42001e7 0.838736
\(965\) 0 0
\(966\) −5.51626e6 −0.190196
\(967\) −4.00243e7 −1.37644 −0.688221 0.725501i \(-0.741608\pi\)
−0.688221 + 0.725501i \(0.741608\pi\)
\(968\) −2.18464e6 −0.0749360
\(969\) −8.31753e6 −0.284567
\(970\) 0 0
\(971\) −3.50158e7 −1.19183 −0.595917 0.803046i \(-0.703211\pi\)
−0.595917 + 0.803046i \(0.703211\pi\)
\(972\) 1.48765e6 0.0505052
\(973\) 2.17297e7 0.735821
\(974\) −1.10969e6 −0.0374805
\(975\) 0 0
\(976\) 1.35036e7 0.453760
\(977\) −3.66007e7 −1.22674 −0.613370 0.789795i \(-0.710187\pi\)
−0.613370 + 0.789795i \(0.710187\pi\)
\(978\) −5.21698e6 −0.174410
\(979\) 8.02313e6 0.267539
\(980\) 0 0
\(981\) 7.88616e6 0.261633
\(982\) −1.45260e7 −0.480692
\(983\) 4.10117e7 1.35371 0.676853 0.736119i \(-0.263343\pi\)
0.676853 + 0.736119i \(0.263343\pi\)
\(984\) −1.48817e6 −0.0489965
\(985\) 0 0
\(986\) −7.95568e6 −0.260606
\(987\) 1.43268e7 0.468119
\(988\) 4.10116e7 1.33664
\(989\) −2.71648e7 −0.883111
\(990\) 0 0
\(991\) 3.49847e7 1.13160 0.565802 0.824541i \(-0.308567\pi\)
0.565802 + 0.824541i \(0.308567\pi\)
\(992\) −2.63387e7 −0.849797
\(993\) 4.77451e6 0.153658
\(994\) 2.89018e7 0.927809
\(995\) 0 0
\(996\) −2.71744e7 −0.867983
\(997\) −5.02983e7 −1.60256 −0.801282 0.598287i \(-0.795848\pi\)
−0.801282 + 0.598287i \(0.795848\pi\)
\(998\) −1.38620e7 −0.440554
\(999\) −6.62658e6 −0.210076
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.o.1.4 yes 7
5.4 even 2 825.6.a.m.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.m.1.4 7 5.4 even 2
825.6.a.o.1.4 yes 7 1.1 even 1 trivial