Properties

Label 825.6.a.v.1.13
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 1.13
Root \(11.4306\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+10.4306 q^{2} -9.00000 q^{3} +76.7973 q^{4} -93.8753 q^{6} -21.9028 q^{7} +467.262 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+10.4306 q^{2} -9.00000 q^{3} +76.7973 q^{4} -93.8753 q^{6} -21.9028 q^{7} +467.262 q^{8} +81.0000 q^{9} +121.000 q^{11} -691.175 q^{12} -460.797 q^{13} -228.459 q^{14} +2416.31 q^{16} -2318.63 q^{17} +844.878 q^{18} +800.141 q^{19} +197.125 q^{21} +1262.10 q^{22} -4781.90 q^{23} -4205.36 q^{24} -4806.39 q^{26} -729.000 q^{27} -1682.08 q^{28} +911.582 q^{29} -944.328 q^{31} +10251.1 q^{32} -1089.00 q^{33} -24184.7 q^{34} +6220.58 q^{36} -12825.4 q^{37} +8345.95 q^{38} +4147.17 q^{39} +8712.22 q^{41} +2056.13 q^{42} +20504.8 q^{43} +9292.47 q^{44} -49878.1 q^{46} +9288.25 q^{47} -21746.8 q^{48} -16327.3 q^{49} +20867.7 q^{51} -35388.0 q^{52} -23263.1 q^{53} -7603.90 q^{54} -10234.4 q^{56} -7201.27 q^{57} +9508.34 q^{58} +20726.7 q^{59} -6625.93 q^{61} -9849.90 q^{62} -1774.13 q^{63} +29603.6 q^{64} -11358.9 q^{66} -57904.9 q^{67} -178064. q^{68} +43037.1 q^{69} +38362.5 q^{71} +37848.2 q^{72} -8431.57 q^{73} -133777. q^{74} +61448.7 q^{76} -2650.24 q^{77} +43257.5 q^{78} -1683.43 q^{79} +6561.00 q^{81} +90873.6 q^{82} -7798.26 q^{83} +15138.7 q^{84} +213877. q^{86} -8204.23 q^{87} +56538.7 q^{88} -83079.0 q^{89} +10092.8 q^{91} -367237. q^{92} +8498.95 q^{93} +96882.0 q^{94} -92260.2 q^{96} -14678.5 q^{97} -170303. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.4306 1.84389 0.921943 0.387326i \(-0.126601\pi\)
0.921943 + 0.387326i \(0.126601\pi\)
\(3\) −9.00000 −0.577350
\(4\) 76.7973 2.39991
\(5\) 0 0
\(6\) −93.8753 −1.06457
\(7\) −21.9028 −0.168949 −0.0844744 0.996426i \(-0.526921\pi\)
−0.0844744 + 0.996426i \(0.526921\pi\)
\(8\) 467.262 2.58128
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −691.175 −1.38559
\(13\) −460.797 −0.756225 −0.378113 0.925760i \(-0.623427\pi\)
−0.378113 + 0.925760i \(0.623427\pi\)
\(14\) −228.459 −0.311522
\(15\) 0 0
\(16\) 2416.31 2.35967
\(17\) −2318.63 −1.94585 −0.972924 0.231127i \(-0.925759\pi\)
−0.972924 + 0.231127i \(0.925759\pi\)
\(18\) 844.878 0.614629
\(19\) 800.141 0.508490 0.254245 0.967140i \(-0.418173\pi\)
0.254245 + 0.967140i \(0.418173\pi\)
\(20\) 0 0
\(21\) 197.125 0.0975426
\(22\) 1262.10 0.555952
\(23\) −4781.90 −1.88487 −0.942434 0.334391i \(-0.891469\pi\)
−0.942434 + 0.334391i \(0.891469\pi\)
\(24\) −4205.36 −1.49030
\(25\) 0 0
\(26\) −4806.39 −1.39439
\(27\) −729.000 −0.192450
\(28\) −1682.08 −0.405462
\(29\) 911.582 0.201280 0.100640 0.994923i \(-0.467911\pi\)
0.100640 + 0.994923i \(0.467911\pi\)
\(30\) 0 0
\(31\) −944.328 −0.176489 −0.0882447 0.996099i \(-0.528126\pi\)
−0.0882447 + 0.996099i \(0.528126\pi\)
\(32\) 10251.1 1.76969
\(33\) −1089.00 −0.174078
\(34\) −24184.7 −3.58792
\(35\) 0 0
\(36\) 6220.58 0.799971
\(37\) −12825.4 −1.54017 −0.770084 0.637942i \(-0.779786\pi\)
−0.770084 + 0.637942i \(0.779786\pi\)
\(38\) 8345.95 0.937598
\(39\) 4147.17 0.436607
\(40\) 0 0
\(41\) 8712.22 0.809411 0.404706 0.914447i \(-0.367374\pi\)
0.404706 + 0.914447i \(0.367374\pi\)
\(42\) 2056.13 0.179857
\(43\) 20504.8 1.69116 0.845578 0.533852i \(-0.179256\pi\)
0.845578 + 0.533852i \(0.179256\pi\)
\(44\) 9292.47 0.723601
\(45\) 0 0
\(46\) −49878.1 −3.47548
\(47\) 9288.25 0.613323 0.306662 0.951819i \(-0.400788\pi\)
0.306662 + 0.951819i \(0.400788\pi\)
\(48\) −21746.8 −1.36236
\(49\) −16327.3 −0.971456
\(50\) 0 0
\(51\) 20867.7 1.12344
\(52\) −35388.0 −1.81488
\(53\) −23263.1 −1.13757 −0.568785 0.822486i \(-0.692586\pi\)
−0.568785 + 0.822486i \(0.692586\pi\)
\(54\) −7603.90 −0.354856
\(55\) 0 0
\(56\) −10234.4 −0.436104
\(57\) −7201.27 −0.293577
\(58\) 9508.34 0.371137
\(59\) 20726.7 0.775177 0.387588 0.921833i \(-0.373308\pi\)
0.387588 + 0.921833i \(0.373308\pi\)
\(60\) 0 0
\(61\) −6625.93 −0.227993 −0.113997 0.993481i \(-0.536365\pi\)
−0.113997 + 0.993481i \(0.536365\pi\)
\(62\) −9849.90 −0.325426
\(63\) −1774.13 −0.0563162
\(64\) 29603.6 0.903429
\(65\) 0 0
\(66\) −11358.9 −0.320979
\(67\) −57904.9 −1.57590 −0.787950 0.615739i \(-0.788857\pi\)
−0.787950 + 0.615739i \(0.788857\pi\)
\(68\) −178064. −4.66987
\(69\) 43037.1 1.08823
\(70\) 0 0
\(71\) 38362.5 0.903153 0.451577 0.892232i \(-0.350862\pi\)
0.451577 + 0.892232i \(0.350862\pi\)
\(72\) 37848.2 0.860427
\(73\) −8431.57 −0.185183 −0.0925916 0.995704i \(-0.529515\pi\)
−0.0925916 + 0.995704i \(0.529515\pi\)
\(74\) −133777. −2.83989
\(75\) 0 0
\(76\) 61448.7 1.22033
\(77\) −2650.24 −0.0509400
\(78\) 43257.5 0.805053
\(79\) −1683.43 −0.0303478 −0.0151739 0.999885i \(-0.504830\pi\)
−0.0151739 + 0.999885i \(0.504830\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 90873.6 1.49246
\(83\) −7798.26 −0.124252 −0.0621259 0.998068i \(-0.519788\pi\)
−0.0621259 + 0.998068i \(0.519788\pi\)
\(84\) 15138.7 0.234094
\(85\) 0 0
\(86\) 213877. 3.11830
\(87\) −8204.23 −0.116209
\(88\) 56538.7 0.778286
\(89\) −83079.0 −1.11177 −0.555887 0.831258i \(-0.687621\pi\)
−0.555887 + 0.831258i \(0.687621\pi\)
\(90\) 0 0
\(91\) 10092.8 0.127763
\(92\) −367237. −4.52352
\(93\) 8498.95 0.101896
\(94\) 96882.0 1.13090
\(95\) 0 0
\(96\) −92260.2 −1.02173
\(97\) −14678.5 −0.158399 −0.0791997 0.996859i \(-0.525236\pi\)
−0.0791997 + 0.996859i \(0.525236\pi\)
\(98\) −170303. −1.79125
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −19178.1 −0.187069 −0.0935345 0.995616i \(-0.529817\pi\)
−0.0935345 + 0.995616i \(0.529817\pi\)
\(102\) 217662. 2.07149
\(103\) 127841. 1.18735 0.593675 0.804705i \(-0.297677\pi\)
0.593675 + 0.804705i \(0.297677\pi\)
\(104\) −215313. −1.95203
\(105\) 0 0
\(106\) −242648. −2.09755
\(107\) −52736.6 −0.445300 −0.222650 0.974898i \(-0.571471\pi\)
−0.222650 + 0.974898i \(0.571471\pi\)
\(108\) −55985.2 −0.461864
\(109\) −233661. −1.88374 −0.941869 0.335979i \(-0.890933\pi\)
−0.941869 + 0.335979i \(0.890933\pi\)
\(110\) 0 0
\(111\) 115429. 0.889216
\(112\) −52923.9 −0.398664
\(113\) 36635.5 0.269902 0.134951 0.990852i \(-0.456912\pi\)
0.134951 + 0.990852i \(0.456912\pi\)
\(114\) −75113.5 −0.541323
\(115\) 0 0
\(116\) 70007.0 0.483055
\(117\) −37324.6 −0.252075
\(118\) 216192. 1.42934
\(119\) 50784.5 0.328748
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −69112.4 −0.420394
\(123\) −78410.0 −0.467314
\(124\) −72521.8 −0.423559
\(125\) 0 0
\(126\) −18505.2 −0.103841
\(127\) −256889. −1.41330 −0.706652 0.707561i \(-0.749795\pi\)
−0.706652 + 0.707561i \(0.749795\pi\)
\(128\) −19253.6 −0.103869
\(129\) −184543. −0.976390
\(130\) 0 0
\(131\) −109508. −0.557531 −0.278765 0.960359i \(-0.589925\pi\)
−0.278765 + 0.960359i \(0.589925\pi\)
\(132\) −83632.2 −0.417771
\(133\) −17525.4 −0.0859088
\(134\) −603983. −2.90578
\(135\) 0 0
\(136\) −1.08341e6 −5.02278
\(137\) −149903. −0.682355 −0.341177 0.939999i \(-0.610826\pi\)
−0.341177 + 0.939999i \(0.610826\pi\)
\(138\) 448903. 2.00657
\(139\) −113251. −0.497168 −0.248584 0.968610i \(-0.579965\pi\)
−0.248584 + 0.968610i \(0.579965\pi\)
\(140\) 0 0
\(141\) −83594.3 −0.354102
\(142\) 400144. 1.66531
\(143\) −55756.4 −0.228011
\(144\) 195721. 0.786558
\(145\) 0 0
\(146\) −87946.3 −0.341456
\(147\) 146945. 0.560871
\(148\) −984959. −3.69627
\(149\) −61807.9 −0.228075 −0.114038 0.993476i \(-0.536378\pi\)
−0.114038 + 0.993476i \(0.536378\pi\)
\(150\) 0 0
\(151\) −256778. −0.916465 −0.458233 0.888832i \(-0.651517\pi\)
−0.458233 + 0.888832i \(0.651517\pi\)
\(152\) 373876. 1.31256
\(153\) −187809. −0.648616
\(154\) −27643.6 −0.0939275
\(155\) 0 0
\(156\) 318492. 1.04782
\(157\) 116054. 0.375761 0.187880 0.982192i \(-0.439838\pi\)
0.187880 + 0.982192i \(0.439838\pi\)
\(158\) −17559.2 −0.0559579
\(159\) 209368. 0.656776
\(160\) 0 0
\(161\) 104737. 0.318446
\(162\) 68435.1 0.204876
\(163\) 322117. 0.949609 0.474805 0.880091i \(-0.342519\pi\)
0.474805 + 0.880091i \(0.342519\pi\)
\(164\) 669075. 1.94252
\(165\) 0 0
\(166\) −81340.5 −0.229106
\(167\) 435361. 1.20797 0.603987 0.796994i \(-0.293578\pi\)
0.603987 + 0.796994i \(0.293578\pi\)
\(168\) 92109.2 0.251785
\(169\) −158959. −0.428123
\(170\) 0 0
\(171\) 64811.5 0.169497
\(172\) 1.57471e6 4.05863
\(173\) 700319. 1.77902 0.889510 0.456916i \(-0.151046\pi\)
0.889510 + 0.456916i \(0.151046\pi\)
\(174\) −85575.0 −0.214276
\(175\) 0 0
\(176\) 292373. 0.711469
\(177\) −186541. −0.447548
\(178\) −866563. −2.04998
\(179\) 228192. 0.532314 0.266157 0.963930i \(-0.414246\pi\)
0.266157 + 0.963930i \(0.414246\pi\)
\(180\) 0 0
\(181\) 511598. 1.16073 0.580366 0.814356i \(-0.302909\pi\)
0.580366 + 0.814356i \(0.302909\pi\)
\(182\) 105273. 0.235581
\(183\) 59633.4 0.131632
\(184\) −2.23440e6 −4.86538
\(185\) 0 0
\(186\) 88649.1 0.187885
\(187\) −280554. −0.586695
\(188\) 713312. 1.47192
\(189\) 15967.2 0.0325142
\(190\) 0 0
\(191\) −803150. −1.59299 −0.796496 0.604644i \(-0.793315\pi\)
−0.796496 + 0.604644i \(0.793315\pi\)
\(192\) −266432. −0.521595
\(193\) 578708. 1.11832 0.559160 0.829059i \(-0.311124\pi\)
0.559160 + 0.829059i \(0.311124\pi\)
\(194\) −153106. −0.292070
\(195\) 0 0
\(196\) −1.25389e6 −2.33141
\(197\) −278554. −0.511381 −0.255690 0.966759i \(-0.582303\pi\)
−0.255690 + 0.966759i \(0.582303\pi\)
\(198\) 102230. 0.185317
\(199\) 843631. 1.51015 0.755074 0.655639i \(-0.227601\pi\)
0.755074 + 0.655639i \(0.227601\pi\)
\(200\) 0 0
\(201\) 521144. 0.909846
\(202\) −200039. −0.344934
\(203\) −19966.2 −0.0340060
\(204\) 1.60258e6 2.69615
\(205\) 0 0
\(206\) 1.33346e6 2.18934
\(207\) −387334. −0.628290
\(208\) −1.11343e6 −1.78445
\(209\) 96817.1 0.153316
\(210\) 0 0
\(211\) 48614.4 0.0751724 0.0375862 0.999293i \(-0.488033\pi\)
0.0375862 + 0.999293i \(0.488033\pi\)
\(212\) −1.78654e6 −2.73007
\(213\) −345263. −0.521436
\(214\) −550074. −0.821082
\(215\) 0 0
\(216\) −340634. −0.496768
\(217\) 20683.4 0.0298177
\(218\) −2.43723e6 −3.47340
\(219\) 75884.1 0.106916
\(220\) 0 0
\(221\) 1.06842e6 1.47150
\(222\) 1.20399e6 1.63961
\(223\) −473625. −0.637782 −0.318891 0.947791i \(-0.603310\pi\)
−0.318891 + 0.947791i \(0.603310\pi\)
\(224\) −224529. −0.298987
\(225\) 0 0
\(226\) 382130. 0.497668
\(227\) −767793. −0.988962 −0.494481 0.869188i \(-0.664642\pi\)
−0.494481 + 0.869188i \(0.664642\pi\)
\(228\) −553038. −0.704560
\(229\) 580162. 0.731073 0.365536 0.930797i \(-0.380886\pi\)
0.365536 + 0.930797i \(0.380886\pi\)
\(230\) 0 0
\(231\) 23852.2 0.0294102
\(232\) 425947. 0.519560
\(233\) −298663. −0.360406 −0.180203 0.983629i \(-0.557675\pi\)
−0.180203 + 0.983629i \(0.557675\pi\)
\(234\) −389317. −0.464798
\(235\) 0 0
\(236\) 1.59176e6 1.86036
\(237\) 15150.9 0.0175213
\(238\) 529712. 0.606174
\(239\) 324060. 0.366970 0.183485 0.983023i \(-0.441262\pi\)
0.183485 + 0.983023i \(0.441262\pi\)
\(240\) 0 0
\(241\) −679917. −0.754073 −0.377036 0.926198i \(-0.623057\pi\)
−0.377036 + 0.926198i \(0.623057\pi\)
\(242\) 152714. 0.167626
\(243\) −59049.0 −0.0641500
\(244\) −508853. −0.547165
\(245\) 0 0
\(246\) −817863. −0.861673
\(247\) −368703. −0.384533
\(248\) −441249. −0.455569
\(249\) 70184.4 0.0717368
\(250\) 0 0
\(251\) 1.78432e6 1.78767 0.893836 0.448393i \(-0.148004\pi\)
0.893836 + 0.448393i \(0.148004\pi\)
\(252\) −136248. −0.135154
\(253\) −578610. −0.568309
\(254\) −2.67950e6 −2.60597
\(255\) 0 0
\(256\) −1.14814e6 −1.09495
\(257\) 891074. 0.841552 0.420776 0.907165i \(-0.361758\pi\)
0.420776 + 0.907165i \(0.361758\pi\)
\(258\) −1.92489e6 −1.80035
\(259\) 280913. 0.260209
\(260\) 0 0
\(261\) 73838.1 0.0670933
\(262\) −1.14224e6 −1.02802
\(263\) 1.27856e6 1.13980 0.569902 0.821713i \(-0.306981\pi\)
0.569902 + 0.821713i \(0.306981\pi\)
\(264\) −508848. −0.449344
\(265\) 0 0
\(266\) −182800. −0.158406
\(267\) 747711. 0.641883
\(268\) −4.44694e6 −3.78203
\(269\) 454281. 0.382775 0.191388 0.981515i \(-0.438701\pi\)
0.191388 + 0.981515i \(0.438701\pi\)
\(270\) 0 0
\(271\) −2.02780e6 −1.67726 −0.838632 0.544698i \(-0.816644\pi\)
−0.838632 + 0.544698i \(0.816644\pi\)
\(272\) −5.60252e6 −4.59157
\(273\) −90834.8 −0.0737642
\(274\) −1.56358e6 −1.25818
\(275\) 0 0
\(276\) 3.30513e6 2.61166
\(277\) −1.27638e6 −0.999496 −0.499748 0.866171i \(-0.666574\pi\)
−0.499748 + 0.866171i \(0.666574\pi\)
\(278\) −1.18127e6 −0.916722
\(279\) −76490.6 −0.0588298
\(280\) 0 0
\(281\) −1.02183e6 −0.771994 −0.385997 0.922500i \(-0.626143\pi\)
−0.385997 + 0.922500i \(0.626143\pi\)
\(282\) −871938. −0.652924
\(283\) 75339.2 0.0559184 0.0279592 0.999609i \(-0.491099\pi\)
0.0279592 + 0.999609i \(0.491099\pi\)
\(284\) 2.94614e6 2.16749
\(285\) 0 0
\(286\) −581573. −0.420425
\(287\) −190822. −0.136749
\(288\) 830342. 0.589896
\(289\) 3.95618e6 2.78632
\(290\) 0 0
\(291\) 132107. 0.0914519
\(292\) −647522. −0.444424
\(293\) −661394. −0.450082 −0.225041 0.974349i \(-0.572252\pi\)
−0.225041 + 0.974349i \(0.572252\pi\)
\(294\) 1.53273e6 1.03418
\(295\) 0 0
\(296\) −5.99284e6 −3.97561
\(297\) −88209.0 −0.0580259
\(298\) −644693. −0.420545
\(299\) 2.20349e6 1.42539
\(300\) 0 0
\(301\) −449112. −0.285719
\(302\) −2.67835e6 −1.68986
\(303\) 172603. 0.108004
\(304\) 1.93339e6 1.19987
\(305\) 0 0
\(306\) −1.95896e6 −1.19597
\(307\) −962329. −0.582744 −0.291372 0.956610i \(-0.594112\pi\)
−0.291372 + 0.956610i \(0.594112\pi\)
\(308\) −203531. −0.122252
\(309\) −1.15057e6 −0.685517
\(310\) 0 0
\(311\) 645247. 0.378290 0.189145 0.981949i \(-0.439428\pi\)
0.189145 + 0.981949i \(0.439428\pi\)
\(312\) 1.93782e6 1.12701
\(313\) 3.21424e6 1.85446 0.927229 0.374495i \(-0.122184\pi\)
0.927229 + 0.374495i \(0.122184\pi\)
\(314\) 1.21051e6 0.692860
\(315\) 0 0
\(316\) −129283. −0.0728321
\(317\) −1.59691e6 −0.892552 −0.446276 0.894895i \(-0.647250\pi\)
−0.446276 + 0.894895i \(0.647250\pi\)
\(318\) 2.18383e6 1.21102
\(319\) 110301. 0.0606882
\(320\) 0 0
\(321\) 474629. 0.257094
\(322\) 1.09247e6 0.587178
\(323\) −1.85523e6 −0.989445
\(324\) 503867. 0.266657
\(325\) 0 0
\(326\) 3.35987e6 1.75097
\(327\) 2.10295e6 1.08758
\(328\) 4.07089e6 2.08932
\(329\) −203439. −0.103620
\(330\) 0 0
\(331\) 1.89235e6 0.949363 0.474682 0.880158i \(-0.342563\pi\)
0.474682 + 0.880158i \(0.342563\pi\)
\(332\) −598885. −0.298194
\(333\) −1.03886e6 −0.513389
\(334\) 4.54107e6 2.22737
\(335\) 0 0
\(336\) 476315. 0.230169
\(337\) −1.58818e6 −0.761772 −0.380886 0.924622i \(-0.624381\pi\)
−0.380886 + 0.924622i \(0.624381\pi\)
\(338\) −1.65804e6 −0.789410
\(339\) −329719. −0.155828
\(340\) 0 0
\(341\) −114264. −0.0532136
\(342\) 676022. 0.312533
\(343\) 725734. 0.333075
\(344\) 9.58110e6 4.36535
\(345\) 0 0
\(346\) 7.30474e6 3.28031
\(347\) 1.31246e6 0.585141 0.292571 0.956244i \(-0.405489\pi\)
0.292571 + 0.956244i \(0.405489\pi\)
\(348\) −630063. −0.278892
\(349\) −3.90078e6 −1.71430 −0.857151 0.515065i \(-0.827768\pi\)
−0.857151 + 0.515065i \(0.827768\pi\)
\(350\) 0 0
\(351\) 335921. 0.145536
\(352\) 1.24039e6 0.533581
\(353\) −3.32103e6 −1.41852 −0.709261 0.704946i \(-0.750971\pi\)
−0.709261 + 0.704946i \(0.750971\pi\)
\(354\) −1.94573e6 −0.825228
\(355\) 0 0
\(356\) −6.38024e6 −2.66816
\(357\) −457060. −0.189803
\(358\) 2.38018e6 0.981527
\(359\) 2.37759e6 0.973647 0.486823 0.873500i \(-0.338155\pi\)
0.486823 + 0.873500i \(0.338155\pi\)
\(360\) 0 0
\(361\) −1.83587e6 −0.741438
\(362\) 5.33627e6 2.14026
\(363\) −131769. −0.0524864
\(364\) 775096. 0.306621
\(365\) 0 0
\(366\) 622011. 0.242714
\(367\) 3.52479e6 1.36605 0.683027 0.730393i \(-0.260663\pi\)
0.683027 + 0.730393i \(0.260663\pi\)
\(368\) −1.15545e7 −4.44768
\(369\) 705690. 0.269804
\(370\) 0 0
\(371\) 509528. 0.192191
\(372\) 652696. 0.244542
\(373\) 3.82557e6 1.42372 0.711859 0.702322i \(-0.247853\pi\)
0.711859 + 0.702322i \(0.247853\pi\)
\(374\) −2.92634e6 −1.08180
\(375\) 0 0
\(376\) 4.34005e6 1.58316
\(377\) −420054. −0.152213
\(378\) 166547. 0.0599525
\(379\) 3.70497e6 1.32491 0.662456 0.749101i \(-0.269514\pi\)
0.662456 + 0.749101i \(0.269514\pi\)
\(380\) 0 0
\(381\) 2.31200e6 0.815972
\(382\) −8.37733e6 −2.93729
\(383\) 789569. 0.275038 0.137519 0.990499i \(-0.456087\pi\)
0.137519 + 0.990499i \(0.456087\pi\)
\(384\) 173283. 0.0599691
\(385\) 0 0
\(386\) 6.03627e6 2.06206
\(387\) 1.66089e6 0.563719
\(388\) −1.12727e6 −0.380145
\(389\) −4.73042e6 −1.58499 −0.792494 0.609879i \(-0.791218\pi\)
−0.792494 + 0.609879i \(0.791218\pi\)
\(390\) 0 0
\(391\) 1.10875e7 3.66767
\(392\) −7.62911e6 −2.50760
\(393\) 985575. 0.321891
\(394\) −2.90549e6 −0.942928
\(395\) 0 0
\(396\) 752690. 0.241200
\(397\) 2.42880e6 0.773421 0.386711 0.922201i \(-0.373611\pi\)
0.386711 + 0.922201i \(0.373611\pi\)
\(398\) 8.79957e6 2.78454
\(399\) 157728. 0.0495995
\(400\) 0 0
\(401\) 276251. 0.0857914 0.0428957 0.999080i \(-0.486342\pi\)
0.0428957 + 0.999080i \(0.486342\pi\)
\(402\) 5.43585e6 1.67765
\(403\) 435144. 0.133466
\(404\) −1.47282e6 −0.448949
\(405\) 0 0
\(406\) −208259. −0.0627032
\(407\) −1.55188e6 −0.464378
\(408\) 9.75066e6 2.89990
\(409\) −563175. −0.166470 −0.0832349 0.996530i \(-0.526525\pi\)
−0.0832349 + 0.996530i \(0.526525\pi\)
\(410\) 0 0
\(411\) 1.34913e6 0.393958
\(412\) 9.81787e6 2.84954
\(413\) −453974. −0.130965
\(414\) −4.04012e6 −1.15849
\(415\) 0 0
\(416\) −4.72369e6 −1.33828
\(417\) 1.01926e6 0.287040
\(418\) 1.00986e6 0.282696
\(419\) 707010. 0.196739 0.0983695 0.995150i \(-0.468637\pi\)
0.0983695 + 0.995150i \(0.468637\pi\)
\(420\) 0 0
\(421\) 4.56766e6 1.25600 0.627999 0.778214i \(-0.283874\pi\)
0.627999 + 0.778214i \(0.283874\pi\)
\(422\) 507077. 0.138609
\(423\) 752349. 0.204441
\(424\) −1.08700e7 −2.93639
\(425\) 0 0
\(426\) −3.60130e6 −0.961468
\(427\) 145127. 0.0385192
\(428\) −4.05003e6 −1.06868
\(429\) 501808. 0.131642
\(430\) 0 0
\(431\) 6.16454e6 1.59848 0.799241 0.601011i \(-0.205235\pi\)
0.799241 + 0.601011i \(0.205235\pi\)
\(432\) −1.76149e6 −0.454120
\(433\) 2.94520e6 0.754910 0.377455 0.926028i \(-0.376799\pi\)
0.377455 + 0.926028i \(0.376799\pi\)
\(434\) 215741. 0.0549804
\(435\) 0 0
\(436\) −1.79446e7 −4.52081
\(437\) −3.82620e6 −0.958438
\(438\) 791517. 0.197140
\(439\) −5.90123e6 −1.46144 −0.730720 0.682677i \(-0.760816\pi\)
−0.730720 + 0.682677i \(0.760816\pi\)
\(440\) 0 0
\(441\) −1.32251e6 −0.323819
\(442\) 1.11442e7 2.71328
\(443\) −4.58354e6 −1.10966 −0.554832 0.831963i \(-0.687217\pi\)
−0.554832 + 0.831963i \(0.687217\pi\)
\(444\) 8.86463e6 2.13404
\(445\) 0 0
\(446\) −4.94019e6 −1.17600
\(447\) 556271. 0.131679
\(448\) −648401. −0.152633
\(449\) 6.61995e6 1.54967 0.774834 0.632164i \(-0.217833\pi\)
0.774834 + 0.632164i \(0.217833\pi\)
\(450\) 0 0
\(451\) 1.05418e6 0.244047
\(452\) 2.81350e6 0.647741
\(453\) 2.31101e6 0.529122
\(454\) −8.00854e6 −1.82353
\(455\) 0 0
\(456\) −3.36488e6 −0.757805
\(457\) 6.37429e6 1.42771 0.713857 0.700292i \(-0.246947\pi\)
0.713857 + 0.700292i \(0.246947\pi\)
\(458\) 6.05143e6 1.34801
\(459\) 1.69028e6 0.374478
\(460\) 0 0
\(461\) −5.36939e6 −1.17672 −0.588359 0.808600i \(-0.700226\pi\)
−0.588359 + 0.808600i \(0.700226\pi\)
\(462\) 248792. 0.0542290
\(463\) −7.13889e6 −1.54767 −0.773835 0.633387i \(-0.781664\pi\)
−0.773835 + 0.633387i \(0.781664\pi\)
\(464\) 2.20266e6 0.474955
\(465\) 0 0
\(466\) −3.11523e6 −0.664547
\(467\) 5.11617e6 1.08556 0.542779 0.839876i \(-0.317372\pi\)
0.542779 + 0.839876i \(0.317372\pi\)
\(468\) −2.86642e6 −0.604959
\(469\) 1.26828e6 0.266246
\(470\) 0 0
\(471\) −1.04449e6 −0.216946
\(472\) 9.68481e6 2.00095
\(473\) 2.48108e6 0.509903
\(474\) 158033. 0.0323073
\(475\) 0 0
\(476\) 3.90011e6 0.788968
\(477\) −1.88431e6 −0.379190
\(478\) 3.38013e6 0.676650
\(479\) −715144. −0.142415 −0.0712074 0.997462i \(-0.522685\pi\)
−0.0712074 + 0.997462i \(0.522685\pi\)
\(480\) 0 0
\(481\) 5.90993e6 1.16471
\(482\) −7.09194e6 −1.39042
\(483\) −942634. −0.183855
\(484\) 1.12439e6 0.218174
\(485\) 0 0
\(486\) −615916. −0.118285
\(487\) −9.41305e6 −1.79849 −0.899245 0.437445i \(-0.855883\pi\)
−0.899245 + 0.437445i \(0.855883\pi\)
\(488\) −3.09605e6 −0.588515
\(489\) −2.89906e6 −0.548257
\(490\) 0 0
\(491\) −2.03825e6 −0.381551 −0.190776 0.981634i \(-0.561100\pi\)
−0.190776 + 0.981634i \(0.561100\pi\)
\(492\) −6.02167e6 −1.12151
\(493\) −2.11362e6 −0.391660
\(494\) −3.84579e6 −0.709036
\(495\) 0 0
\(496\) −2.28179e6 −0.416458
\(497\) −840248. −0.152587
\(498\) 732065. 0.132274
\(499\) −2.29272e6 −0.412192 −0.206096 0.978532i \(-0.566076\pi\)
−0.206096 + 0.978532i \(0.566076\pi\)
\(500\) 0 0
\(501\) −3.91825e6 −0.697425
\(502\) 1.86115e7 3.29626
\(503\) −6.36926e6 −1.12246 −0.561228 0.827661i \(-0.689671\pi\)
−0.561228 + 0.827661i \(0.689671\pi\)
\(504\) −828983. −0.145368
\(505\) 0 0
\(506\) −6.03525e6 −1.04790
\(507\) 1.43063e6 0.247177
\(508\) −1.97284e7 −3.39181
\(509\) −5.66660e6 −0.969456 −0.484728 0.874665i \(-0.661081\pi\)
−0.484728 + 0.874665i \(0.661081\pi\)
\(510\) 0 0
\(511\) 184675. 0.0312864
\(512\) −1.13597e7 −1.91510
\(513\) −583303. −0.0978590
\(514\) 9.29442e6 1.55172
\(515\) 0 0
\(516\) −1.41724e7 −2.34325
\(517\) 1.12388e6 0.184924
\(518\) 2.93009e6 0.479796
\(519\) −6.30287e6 −1.02712
\(520\) 0 0
\(521\) 6.07157e6 0.979956 0.489978 0.871735i \(-0.337005\pi\)
0.489978 + 0.871735i \(0.337005\pi\)
\(522\) 770175. 0.123712
\(523\) 2.50083e6 0.399789 0.199894 0.979817i \(-0.435940\pi\)
0.199894 + 0.979817i \(0.435940\pi\)
\(524\) −8.40994e6 −1.33803
\(525\) 0 0
\(526\) 1.33361e7 2.10167
\(527\) 2.18954e6 0.343421
\(528\) −2.63136e6 −0.410767
\(529\) 1.64302e7 2.55273
\(530\) 0 0
\(531\) 1.67886e6 0.258392
\(532\) −1.34590e6 −0.206174
\(533\) −4.01457e6 −0.612098
\(534\) 7.79907e6 1.18356
\(535\) 0 0
\(536\) −2.70568e7 −4.06784
\(537\) −2.05373e6 −0.307332
\(538\) 4.73842e6 0.705794
\(539\) −1.97560e6 −0.292905
\(540\) 0 0
\(541\) 2.70948e6 0.398009 0.199004 0.979999i \(-0.436229\pi\)
0.199004 + 0.979999i \(0.436229\pi\)
\(542\) −2.11511e7 −3.09268
\(543\) −4.60438e6 −0.670149
\(544\) −2.37686e7 −3.44354
\(545\) 0 0
\(546\) −947461. −0.136013
\(547\) −9.99302e6 −1.42800 −0.714000 0.700145i \(-0.753119\pi\)
−0.714000 + 0.700145i \(0.753119\pi\)
\(548\) −1.15122e7 −1.63759
\(549\) −536700. −0.0759978
\(550\) 0 0
\(551\) 729394. 0.102349
\(552\) 2.01096e7 2.80903
\(553\) 36871.9 0.00512722
\(554\) −1.33134e7 −1.84296
\(555\) 0 0
\(556\) −8.69734e6 −1.19316
\(557\) 2.85519e6 0.389940 0.194970 0.980809i \(-0.437539\pi\)
0.194970 + 0.980809i \(0.437539\pi\)
\(558\) −797842. −0.108475
\(559\) −9.44854e6 −1.27890
\(560\) 0 0
\(561\) 2.52499e6 0.338729
\(562\) −1.06583e7 −1.42347
\(563\) 8.74359e6 1.16257 0.581285 0.813700i \(-0.302550\pi\)
0.581285 + 0.813700i \(0.302550\pi\)
\(564\) −6.41981e6 −0.849815
\(565\) 0 0
\(566\) 785832. 0.103107
\(567\) −143704. −0.0187721
\(568\) 1.79254e7 2.33129
\(569\) 4.89251e6 0.633507 0.316753 0.948508i \(-0.397407\pi\)
0.316753 + 0.948508i \(0.397407\pi\)
\(570\) 0 0
\(571\) 1.65650e6 0.212619 0.106310 0.994333i \(-0.466097\pi\)
0.106310 + 0.994333i \(0.466097\pi\)
\(572\) −4.28194e6 −0.547206
\(573\) 7.22835e6 0.919714
\(574\) −1.99039e6 −0.252150
\(575\) 0 0
\(576\) 2.39789e6 0.301143
\(577\) −1.33417e7 −1.66829 −0.834144 0.551547i \(-0.814038\pi\)
−0.834144 + 0.551547i \(0.814038\pi\)
\(578\) 4.12653e7 5.13766
\(579\) −5.20837e6 −0.645663
\(580\) 0 0
\(581\) 170804. 0.0209922
\(582\) 1.37795e6 0.168627
\(583\) −2.81484e6 −0.342990
\(584\) −3.93975e6 −0.478010
\(585\) 0 0
\(586\) −6.89874e6 −0.829900
\(587\) −3.82733e6 −0.458460 −0.229230 0.973372i \(-0.573621\pi\)
−0.229230 + 0.973372i \(0.573621\pi\)
\(588\) 1.12850e7 1.34604
\(589\) −755596. −0.0897432
\(590\) 0 0
\(591\) 2.50699e6 0.295246
\(592\) −3.09902e7 −3.63430
\(593\) −3.31931e6 −0.387625 −0.193812 0.981039i \(-0.562085\pi\)
−0.193812 + 0.981039i \(0.562085\pi\)
\(594\) −920072. −0.106993
\(595\) 0 0
\(596\) −4.74668e6 −0.547362
\(597\) −7.59268e6 −0.871885
\(598\) 2.29837e7 2.62825
\(599\) −7.82165e6 −0.890699 −0.445350 0.895357i \(-0.646921\pi\)
−0.445350 + 0.895357i \(0.646921\pi\)
\(600\) 0 0
\(601\) 1.52108e7 1.71778 0.858889 0.512162i \(-0.171155\pi\)
0.858889 + 0.512162i \(0.171155\pi\)
\(602\) −4.68451e6 −0.526833
\(603\) −4.69030e6 −0.525300
\(604\) −1.97199e7 −2.19944
\(605\) 0 0
\(606\) 1.80035e6 0.199148
\(607\) 1.34079e7 1.47703 0.738516 0.674236i \(-0.235527\pi\)
0.738516 + 0.674236i \(0.235527\pi\)
\(608\) 8.20235e6 0.899870
\(609\) 179696. 0.0196334
\(610\) 0 0
\(611\) −4.28000e6 −0.463811
\(612\) −1.44232e7 −1.55662
\(613\) −1.15459e7 −1.24102 −0.620509 0.784199i \(-0.713074\pi\)
−0.620509 + 0.784199i \(0.713074\pi\)
\(614\) −1.00377e7 −1.07451
\(615\) 0 0
\(616\) −1.23836e6 −0.131490
\(617\) 1.07188e7 1.13353 0.566763 0.823881i \(-0.308195\pi\)
0.566763 + 0.823881i \(0.308195\pi\)
\(618\) −1.20012e7 −1.26401
\(619\) 8.11579e6 0.851342 0.425671 0.904878i \(-0.360038\pi\)
0.425671 + 0.904878i \(0.360038\pi\)
\(620\) 0 0
\(621\) 3.48601e6 0.362743
\(622\) 6.73031e6 0.697524
\(623\) 1.81966e6 0.187833
\(624\) 1.00208e7 1.03025
\(625\) 0 0
\(626\) 3.35264e7 3.41941
\(627\) −871354. −0.0885168
\(628\) 8.91264e6 0.901793
\(629\) 2.97374e7 2.99693
\(630\) 0 0
\(631\) 1.00476e7 1.00459 0.502295 0.864696i \(-0.332489\pi\)
0.502295 + 0.864696i \(0.332489\pi\)
\(632\) −786603. −0.0783362
\(633\) −437529. −0.0434008
\(634\) −1.66568e7 −1.64576
\(635\) 0 0
\(636\) 1.60789e7 1.57621
\(637\) 7.52356e6 0.734640
\(638\) 1.15051e6 0.111902
\(639\) 3.10737e6 0.301051
\(640\) 0 0
\(641\) −1.25041e6 −0.120200 −0.0601002 0.998192i \(-0.519142\pi\)
−0.0601002 + 0.998192i \(0.519142\pi\)
\(642\) 4.95067e6 0.474052
\(643\) −1.68108e7 −1.60347 −0.801734 0.597681i \(-0.796089\pi\)
−0.801734 + 0.597681i \(0.796089\pi\)
\(644\) 8.04353e6 0.764244
\(645\) 0 0
\(646\) −1.93511e7 −1.82442
\(647\) −1.96602e6 −0.184640 −0.0923202 0.995729i \(-0.529428\pi\)
−0.0923202 + 0.995729i \(0.529428\pi\)
\(648\) 3.06571e6 0.286809
\(649\) 2.50793e6 0.233725
\(650\) 0 0
\(651\) −186151. −0.0172152
\(652\) 2.47377e7 2.27898
\(653\) 2.26355e6 0.207734 0.103867 0.994591i \(-0.466878\pi\)
0.103867 + 0.994591i \(0.466878\pi\)
\(654\) 2.19350e7 2.00537
\(655\) 0 0
\(656\) 2.10514e7 1.90995
\(657\) −682957. −0.0617277
\(658\) −2.12199e6 −0.191064
\(659\) −4.02181e6 −0.360752 −0.180376 0.983598i \(-0.557731\pi\)
−0.180376 + 0.983598i \(0.557731\pi\)
\(660\) 0 0
\(661\) −1.60875e7 −1.43214 −0.716071 0.698027i \(-0.754061\pi\)
−0.716071 + 0.698027i \(0.754061\pi\)
\(662\) 1.97384e7 1.75052
\(663\) −9.61575e6 −0.849570
\(664\) −3.64383e6 −0.320729
\(665\) 0 0
\(666\) −1.08359e7 −0.946631
\(667\) −4.35909e6 −0.379386
\(668\) 3.34345e7 2.89904
\(669\) 4.26262e6 0.368224
\(670\) 0 0
\(671\) −801738. −0.0687426
\(672\) 2.02076e6 0.172620
\(673\) −5.69741e6 −0.484886 −0.242443 0.970166i \(-0.577949\pi\)
−0.242443 + 0.970166i \(0.577949\pi\)
\(674\) −1.65657e7 −1.40462
\(675\) 0 0
\(676\) −1.22076e7 −1.02746
\(677\) 2.57893e6 0.216256 0.108128 0.994137i \(-0.465514\pi\)
0.108128 + 0.994137i \(0.465514\pi\)
\(678\) −3.43917e6 −0.287329
\(679\) 321501. 0.0267614
\(680\) 0 0
\(681\) 6.91014e6 0.570978
\(682\) −1.19184e6 −0.0981197
\(683\) 1.28680e7 1.05550 0.527750 0.849400i \(-0.323036\pi\)
0.527750 + 0.849400i \(0.323036\pi\)
\(684\) 4.97734e6 0.406778
\(685\) 0 0
\(686\) 7.56983e6 0.614152
\(687\) −5.22146e6 −0.422085
\(688\) 4.95458e7 3.99058
\(689\) 1.07196e7 0.860259
\(690\) 0 0
\(691\) −3.61306e6 −0.287859 −0.143930 0.989588i \(-0.545974\pi\)
−0.143930 + 0.989588i \(0.545974\pi\)
\(692\) 5.37826e7 4.26949
\(693\) −214670. −0.0169800
\(694\) 1.36897e7 1.07893
\(695\) 0 0
\(696\) −3.83353e6 −0.299968
\(697\) −2.02004e7 −1.57499
\(698\) −4.06874e7 −3.16098
\(699\) 2.68797e6 0.208080
\(700\) 0 0
\(701\) 1.60588e7 1.23429 0.617147 0.786848i \(-0.288288\pi\)
0.617147 + 0.786848i \(0.288288\pi\)
\(702\) 3.50386e6 0.268351
\(703\) −1.02622e7 −0.783161
\(704\) 3.58203e6 0.272394
\(705\) 0 0
\(706\) −3.46403e7 −2.61559
\(707\) 420054. 0.0316051
\(708\) −1.43258e7 −1.07408
\(709\) −6.79151e6 −0.507400 −0.253700 0.967283i \(-0.581648\pi\)
−0.253700 + 0.967283i \(0.581648\pi\)
\(710\) 0 0
\(711\) −136358. −0.0101159
\(712\) −3.88197e7 −2.86980
\(713\) 4.51568e6 0.332659
\(714\) −4.76741e6 −0.349975
\(715\) 0 0
\(716\) 1.75245e7 1.27751
\(717\) −2.91654e6 −0.211870
\(718\) 2.47997e7 1.79529
\(719\) 9.17382e6 0.661802 0.330901 0.943666i \(-0.392647\pi\)
0.330901 + 0.943666i \(0.392647\pi\)
\(720\) 0 0
\(721\) −2.80009e6 −0.200601
\(722\) −1.91492e7 −1.36713
\(723\) 6.11925e6 0.435364
\(724\) 3.92893e7 2.78566
\(725\) 0 0
\(726\) −1.37443e6 −0.0967789
\(727\) 1.19986e7 0.841966 0.420983 0.907069i \(-0.361685\pi\)
0.420983 + 0.907069i \(0.361685\pi\)
\(728\) 4.71596e6 0.329793
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −4.75429e7 −3.29073
\(732\) 4.57968e6 0.315906
\(733\) 3.87027e6 0.266061 0.133031 0.991112i \(-0.457529\pi\)
0.133031 + 0.991112i \(0.457529\pi\)
\(734\) 3.67657e7 2.51885
\(735\) 0 0
\(736\) −4.90199e7 −3.33563
\(737\) −7.00650e6 −0.475152
\(738\) 7.36077e6 0.497487
\(739\) −1.26699e7 −0.853416 −0.426708 0.904390i \(-0.640327\pi\)
−0.426708 + 0.904390i \(0.640327\pi\)
\(740\) 0 0
\(741\) 3.31832e6 0.222010
\(742\) 5.31468e6 0.354378
\(743\) −1.07430e7 −0.713929 −0.356965 0.934118i \(-0.616188\pi\)
−0.356965 + 0.934118i \(0.616188\pi\)
\(744\) 3.97124e6 0.263023
\(745\) 0 0
\(746\) 3.99030e7 2.62517
\(747\) −631659. −0.0414173
\(748\) −2.15458e7 −1.40802
\(749\) 1.15508e6 0.0752329
\(750\) 0 0
\(751\) −1.60278e7 −1.03699 −0.518493 0.855082i \(-0.673507\pi\)
−0.518493 + 0.855082i \(0.673507\pi\)
\(752\) 2.24433e7 1.44724
\(753\) −1.60589e7 −1.03211
\(754\) −4.38141e6 −0.280663
\(755\) 0 0
\(756\) 1.22623e6 0.0780313
\(757\) −2.27246e7 −1.44131 −0.720654 0.693294i \(-0.756159\pi\)
−0.720654 + 0.693294i \(0.756159\pi\)
\(758\) 3.86451e7 2.44299
\(759\) 5.20749e6 0.328114
\(760\) 0 0
\(761\) 1.62869e7 1.01948 0.509738 0.860330i \(-0.329742\pi\)
0.509738 + 0.860330i \(0.329742\pi\)
\(762\) 2.41155e7 1.50456
\(763\) 5.11784e6 0.318255
\(764\) −6.16797e7 −3.82304
\(765\) 0 0
\(766\) 8.23568e6 0.507139
\(767\) −9.55081e6 −0.586208
\(768\) 1.03333e7 0.632171
\(769\) 8.82491e6 0.538139 0.269069 0.963121i \(-0.413284\pi\)
0.269069 + 0.963121i \(0.413284\pi\)
\(770\) 0 0
\(771\) −8.01966e6 −0.485870
\(772\) 4.44432e7 2.68387
\(773\) 2.36592e7 1.42413 0.712067 0.702111i \(-0.247759\pi\)
0.712067 + 0.702111i \(0.247759\pi\)
\(774\) 1.73240e7 1.03943
\(775\) 0 0
\(776\) −6.85872e6 −0.408873
\(777\) −2.52822e6 −0.150232
\(778\) −4.93411e7 −2.92254
\(779\) 6.97101e6 0.411578
\(780\) 0 0
\(781\) 4.64187e6 0.272311
\(782\) 1.15649e8 6.76276
\(783\) −664543. −0.0387363
\(784\) −3.94517e7 −2.29232
\(785\) 0 0
\(786\) 1.02801e7 0.593529
\(787\) 1.96332e7 1.12994 0.564969 0.825112i \(-0.308888\pi\)
0.564969 + 0.825112i \(0.308888\pi\)
\(788\) −2.13922e7 −1.22727
\(789\) −1.15070e7 −0.658066
\(790\) 0 0
\(791\) −802420. −0.0455995
\(792\) 4.57963e6 0.259429
\(793\) 3.05321e6 0.172414
\(794\) 2.53339e7 1.42610
\(795\) 0 0
\(796\) 6.47885e7 3.62423
\(797\) −2.52639e7 −1.40882 −0.704408 0.709795i \(-0.748788\pi\)
−0.704408 + 0.709795i \(0.748788\pi\)
\(798\) 1.64520e6 0.0914557
\(799\) −2.15360e7 −1.19343
\(800\) 0 0
\(801\) −6.72940e6 −0.370591
\(802\) 2.88147e6 0.158189
\(803\) −1.02022e6 −0.0558348
\(804\) 4.00225e7 2.18355
\(805\) 0 0
\(806\) 4.53880e6 0.246096
\(807\) −4.08853e6 −0.220995
\(808\) −8.96119e6 −0.482878
\(809\) −1.60088e6 −0.0859978 −0.0429989 0.999075i \(-0.513691\pi\)
−0.0429989 + 0.999075i \(0.513691\pi\)
\(810\) 0 0
\(811\) −1.70463e7 −0.910079 −0.455040 0.890471i \(-0.650375\pi\)
−0.455040 + 0.890471i \(0.650375\pi\)
\(812\) −1.53335e6 −0.0816115
\(813\) 1.82502e7 0.968369
\(814\) −1.61870e7 −0.856260
\(815\) 0 0
\(816\) 5.04227e7 2.65094
\(817\) 1.64067e7 0.859937
\(818\) −5.87425e6 −0.306951
\(819\) 817513. 0.0425878
\(820\) 0 0
\(821\) 1.11036e7 0.574919 0.287459 0.957793i \(-0.407189\pi\)
0.287459 + 0.957793i \(0.407189\pi\)
\(822\) 1.40722e7 0.726413
\(823\) 1.15634e7 0.595097 0.297548 0.954707i \(-0.403831\pi\)
0.297548 + 0.954707i \(0.403831\pi\)
\(824\) 5.97355e7 3.06488
\(825\) 0 0
\(826\) −4.73522e6 −0.241485
\(827\) −3.02353e7 −1.53727 −0.768636 0.639686i \(-0.779064\pi\)
−0.768636 + 0.639686i \(0.779064\pi\)
\(828\) −2.97462e7 −1.50784
\(829\) −1.68978e7 −0.853974 −0.426987 0.904258i \(-0.640425\pi\)
−0.426987 + 0.904258i \(0.640425\pi\)
\(830\) 0 0
\(831\) 1.14874e7 0.577060
\(832\) −1.36412e7 −0.683196
\(833\) 3.78569e7 1.89031
\(834\) 1.06314e7 0.529270
\(835\) 0 0
\(836\) 7.43529e6 0.367944
\(837\) 688415. 0.0339654
\(838\) 7.37454e6 0.362764
\(839\) −6.03413e6 −0.295944 −0.147972 0.988992i \(-0.547275\pi\)
−0.147972 + 0.988992i \(0.547275\pi\)
\(840\) 0 0
\(841\) −1.96802e7 −0.959486
\(842\) 4.76434e7 2.31592
\(843\) 9.19650e6 0.445711
\(844\) 3.73345e6 0.180407
\(845\) 0 0
\(846\) 7.84744e6 0.376966
\(847\) −320679. −0.0153590
\(848\) −5.62108e7 −2.68429
\(849\) −678053. −0.0322845
\(850\) 0 0
\(851\) 6.13300e7 2.90301
\(852\) −2.65152e7 −1.25140
\(853\) −2.44547e7 −1.15077 −0.575386 0.817882i \(-0.695148\pi\)
−0.575386 + 0.817882i \(0.695148\pi\)
\(854\) 1.51376e6 0.0710250
\(855\) 0 0
\(856\) −2.46418e7 −1.14945
\(857\) 9.19253e6 0.427546 0.213773 0.976883i \(-0.431425\pi\)
0.213773 + 0.976883i \(0.431425\pi\)
\(858\) 5.23415e6 0.242733
\(859\) −1.84087e7 −0.851219 −0.425610 0.904907i \(-0.639940\pi\)
−0.425610 + 0.904907i \(0.639940\pi\)
\(860\) 0 0
\(861\) 1.71740e6 0.0789521
\(862\) 6.42999e7 2.94742
\(863\) −8.55693e6 −0.391103 −0.195551 0.980693i \(-0.562650\pi\)
−0.195551 + 0.980693i \(0.562650\pi\)
\(864\) −7.47308e6 −0.340577
\(865\) 0 0
\(866\) 3.07202e7 1.39197
\(867\) −3.56056e7 −1.60868
\(868\) 1.58843e6 0.0715598
\(869\) −203695. −0.00915021
\(870\) 0 0
\(871\) 2.66824e7 1.19174
\(872\) −1.09181e8 −4.86246
\(873\) −1.18896e6 −0.0527998
\(874\) −3.99095e7 −1.76725
\(875\) 0 0
\(876\) 5.82769e6 0.256588
\(877\) −3.62307e7 −1.59066 −0.795330 0.606177i \(-0.792702\pi\)
−0.795330 + 0.606177i \(0.792702\pi\)
\(878\) −6.15533e7 −2.69473
\(879\) 5.95255e6 0.259855
\(880\) 0 0
\(881\) −2.54635e7 −1.10530 −0.552649 0.833414i \(-0.686383\pi\)
−0.552649 + 0.833414i \(0.686383\pi\)
\(882\) −1.37945e7 −0.597085
\(883\) 1.25573e6 0.0541995 0.0270998 0.999633i \(-0.491373\pi\)
0.0270998 + 0.999633i \(0.491373\pi\)
\(884\) 8.20515e7 3.53147
\(885\) 0 0
\(886\) −4.78090e7 −2.04609
\(887\) 2.38712e7 1.01874 0.509371 0.860547i \(-0.329878\pi\)
0.509371 + 0.860547i \(0.329878\pi\)
\(888\) 5.39356e7 2.29532
\(889\) 5.62659e6 0.238776
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −3.63731e7 −1.53062
\(893\) 7.43192e6 0.311869
\(894\) 5.80224e6 0.242802
\(895\) 0 0
\(896\) 421709. 0.0175486
\(897\) −1.98314e7 −0.822947
\(898\) 6.90500e7 2.85741
\(899\) −860832. −0.0355238
\(900\) 0 0
\(901\) 5.39385e7 2.21354
\(902\) 1.09957e7 0.449994
\(903\) 4.04201e6 0.164960
\(904\) 1.71184e7 0.696692
\(905\) 0 0
\(906\) 2.41052e7 0.975640
\(907\) 1.51874e7 0.613007 0.306504 0.951870i \(-0.400841\pi\)
0.306504 + 0.951870i \(0.400841\pi\)
\(908\) −5.89644e7 −2.37342
\(909\) −1.55342e6 −0.0623563
\(910\) 0 0
\(911\) 2.87742e6 0.114870 0.0574352 0.998349i \(-0.481708\pi\)
0.0574352 + 0.998349i \(0.481708\pi\)
\(912\) −1.74005e7 −0.692746
\(913\) −943590. −0.0374633
\(914\) 6.64876e7 2.63254
\(915\) 0 0
\(916\) 4.45549e7 1.75451
\(917\) 2.39854e6 0.0941941
\(918\) 1.76306e7 0.690495
\(919\) 3.83454e7 1.49770 0.748849 0.662741i \(-0.230607\pi\)
0.748849 + 0.662741i \(0.230607\pi\)
\(920\) 0 0
\(921\) 8.66096e6 0.336447
\(922\) −5.60059e7 −2.16973
\(923\) −1.76773e7 −0.682987
\(924\) 1.83178e6 0.0705820
\(925\) 0 0
\(926\) −7.44629e7 −2.85373
\(927\) 1.03552e7 0.395783
\(928\) 9.34474e6 0.356203
\(929\) 3.31484e7 1.26015 0.630077 0.776533i \(-0.283023\pi\)
0.630077 + 0.776533i \(0.283023\pi\)
\(930\) 0 0
\(931\) −1.30641e7 −0.493976
\(932\) −2.29365e7 −0.864943
\(933\) −5.80722e6 −0.218406
\(934\) 5.33647e7 2.00164
\(935\) 0 0
\(936\) −1.74403e7 −0.650677
\(937\) −2.93312e7 −1.09139 −0.545695 0.837984i \(-0.683734\pi\)
−0.545695 + 0.837984i \(0.683734\pi\)
\(938\) 1.32289e7 0.490928
\(939\) −2.89281e7 −1.07067
\(940\) 0 0
\(941\) −9.43943e6 −0.347513 −0.173757 0.984789i \(-0.555591\pi\)
−0.173757 + 0.984789i \(0.555591\pi\)
\(942\) −1.08946e7 −0.400023
\(943\) −4.16610e7 −1.52563
\(944\) 5.00821e7 1.82917
\(945\) 0 0
\(946\) 2.58791e7 0.940202
\(947\) −2.37783e7 −0.861601 −0.430801 0.902447i \(-0.641769\pi\)
−0.430801 + 0.902447i \(0.641769\pi\)
\(948\) 1.16355e6 0.0420496
\(949\) 3.88524e6 0.140040
\(950\) 0 0
\(951\) 1.43722e7 0.515315
\(952\) 2.37297e7 0.848592
\(953\) 2.31442e7 0.825488 0.412744 0.910847i \(-0.364571\pi\)
0.412744 + 0.910847i \(0.364571\pi\)
\(954\) −1.96545e7 −0.699183
\(955\) 0 0
\(956\) 2.48869e7 0.880696
\(957\) −992712. −0.0350383
\(958\) −7.45938e6 −0.262597
\(959\) 3.28331e6 0.115283
\(960\) 0 0
\(961\) −2.77374e7 −0.968851
\(962\) 6.16440e7 2.14760
\(963\) −4.27167e6 −0.148433
\(964\) −5.22158e7 −1.80971
\(965\) 0 0
\(966\) −9.83223e6 −0.339008
\(967\) −7.71333e6 −0.265262 −0.132631 0.991165i \(-0.542343\pi\)
−0.132631 + 0.991165i \(0.542343\pi\)
\(968\) 6.84118e6 0.234662
\(969\) 1.66971e7 0.571256
\(970\) 0 0
\(971\) −4.66559e7 −1.58803 −0.794014 0.607899i \(-0.792012\pi\)
−0.794014 + 0.607899i \(0.792012\pi\)
\(972\) −4.53480e6 −0.153955
\(973\) 2.48051e6 0.0839960
\(974\) −9.81837e7 −3.31621
\(975\) 0 0
\(976\) −1.60103e7 −0.537990
\(977\) −1.28308e7 −0.430050 −0.215025 0.976609i \(-0.568983\pi\)
−0.215025 + 0.976609i \(0.568983\pi\)
\(978\) −3.02389e7 −1.01092
\(979\) −1.00526e7 −0.335212
\(980\) 0 0
\(981\) −1.89266e7 −0.627913
\(982\) −2.12601e7 −0.703537
\(983\) −7.06277e6 −0.233126 −0.116563 0.993183i \(-0.537188\pi\)
−0.116563 + 0.993183i \(0.537188\pi\)
\(984\) −3.66380e7 −1.20627
\(985\) 0 0
\(986\) −2.20463e7 −0.722176
\(987\) 1.83095e6 0.0598251
\(988\) −2.83154e7 −0.922847
\(989\) −9.80518e7 −3.18761
\(990\) 0 0
\(991\) 1.51956e7 0.491512 0.245756 0.969332i \(-0.420964\pi\)
0.245756 + 0.969332i \(0.420964\pi\)
\(992\) −9.68043e6 −0.312331
\(993\) −1.70312e7 −0.548115
\(994\) −8.76428e6 −0.281352
\(995\) 0 0
\(996\) 5.38997e6 0.172162
\(997\) −3.19469e7 −1.01787 −0.508934 0.860806i \(-0.669960\pi\)
−0.508934 + 0.860806i \(0.669960\pi\)
\(998\) −2.39144e7 −0.760035
\(999\) 9.34975e6 0.296405
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.v.1.13 13
5.2 odd 4 165.6.c.b.34.25 yes 26
5.3 odd 4 165.6.c.b.34.2 26
5.4 even 2 825.6.a.y.1.1 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.2 26 5.3 odd 4
165.6.c.b.34.25 yes 26 5.2 odd 4
825.6.a.v.1.13 13 1.1 even 1 trivial
825.6.a.y.1.1 13 5.4 even 2