Properties

Label 25.6.b.b.24.3
Level $25$
Weight $6$
Character 25.24
Analytic conductor $4.010$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.00959549532\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Defining polynomial: \(x^{4} + 121 x^{2} + 3600\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.3
Root \(7.26209i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.6.b.b.24.2

$q$-expansion

\(f(q)\) \(=\) \(q+5.26209i q^{2} +25.5242i q^{3} +4.31044 q^{4} -134.310 q^{6} -131.048i q^{7} +191.069i q^{8} -408.483 q^{9} +O(q^{10})\) \(q+5.26209i q^{2} +25.5242i q^{3} +4.31044 q^{4} -134.310 q^{6} -131.048i q^{7} +191.069i q^{8} -408.483 q^{9} +290.104 q^{11} +110.020i q^{12} +68.3868i q^{13} +689.588 q^{14} -867.486 q^{16} +310.644i q^{17} -2149.48i q^{18} +2133.35 q^{19} +3344.90 q^{21} +1526.55i q^{22} +873.145i q^{23} -4876.87 q^{24} -359.857 q^{26} -4223.83i q^{27} -564.876i q^{28} +2580.97 q^{29} -9086.30 q^{31} +1549.41i q^{32} +7404.67i q^{33} -1634.64 q^{34} -1760.74 q^{36} -3990.64i q^{37} +11225.9i q^{38} -1745.52 q^{39} +16981.8 q^{41} +17601.2i q^{42} -18017.7i q^{43} +1250.48 q^{44} -4594.57 q^{46} -24864.7i q^{47} -22141.9i q^{48} -366.670 q^{49} -7928.93 q^{51} +294.777i q^{52} -7652.91i q^{53} +22226.2 q^{54} +25039.2 q^{56} +54451.9i q^{57} +13581.3i q^{58} +9233.69 q^{59} +3326.17 q^{61} -47812.9i q^{62} +53531.1i q^{63} -35912.7 q^{64} -38964.0 q^{66} +32340.7i q^{67} +1339.01i q^{68} -22286.3 q^{69} -35885.9 q^{71} -78048.4i q^{72} -26513.6i q^{73} +20999.1 q^{74} +9195.65 q^{76} -38017.7i q^{77} -9185.06i q^{78} -71705.7 q^{79} +8548.28 q^{81} +89359.9i q^{82} +39630.1i q^{83} +14418.0 q^{84} +94810.8 q^{86} +65877.1i q^{87} +55429.9i q^{88} +117441. q^{89} +8961.98 q^{91} +3763.64i q^{92} -231920. i q^{93} +130840. q^{94} -39547.4 q^{96} -21878.3i q^{97} -1929.45i q^{98} -118503. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 138q^{4} - 382q^{6} - 392q^{9} + O(q^{10}) \) \( 4q - 138q^{4} - 382q^{6} - 392q^{9} - 392q^{11} - 36q^{14} + 2274q^{16} + 6360q^{19} + 5928q^{21} - 5070q^{24} - 9512q^{26} + 7840q^{29} - 2192q^{31} - 40226q^{34} - 34676q^{36} - 8224q^{39} + 55508q^{41} + 73774q^{44} + 4908q^{46} + 23372q^{49} - 35752q^{51} - 7190q^{54} + 108540q^{56} - 23920q^{59} - 48792q^{61} - 87298q^{64} - 22814q^{66} - 36984q^{69} - 174592q^{71} + 82444q^{74} - 135070q^{76} - 130960q^{79} + 92564q^{81} + 84684q^{84} + 497848q^{86} + 145620q^{89} - 41152q^{91} + 243304q^{94} - 156482q^{96} - 443584q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.26209i 0.930214i 0.885254 + 0.465107i \(0.153984\pi\)
−0.885254 + 0.465107i \(0.846016\pi\)
\(3\) 25.5242i 1.63738i 0.574238 + 0.818688i \(0.305298\pi\)
−0.574238 + 0.818688i \(0.694702\pi\)
\(4\) 4.31044 0.134701
\(5\) 0 0
\(6\) −134.310 −1.52311
\(7\) − 131.048i − 1.01085i −0.862871 0.505425i \(-0.831336\pi\)
0.862871 0.505425i \(-0.168664\pi\)
\(8\) 191.069i 1.05552i
\(9\) −408.483 −1.68100
\(10\) 0 0
\(11\) 290.104 0.722891 0.361445 0.932393i \(-0.382283\pi\)
0.361445 + 0.932393i \(0.382283\pi\)
\(12\) 110.020i 0.220557i
\(13\) 68.3868i 0.112231i 0.998424 + 0.0561156i \(0.0178715\pi\)
−0.998424 + 0.0561156i \(0.982128\pi\)
\(14\) 689.588 0.940307
\(15\) 0 0
\(16\) −867.486 −0.847154
\(17\) 310.644i 0.260700i 0.991468 + 0.130350i \(0.0416100\pi\)
−0.991468 + 0.130350i \(0.958390\pi\)
\(18\) − 2149.48i − 1.56369i
\(19\) 2133.35 1.35574 0.677871 0.735180i \(-0.262903\pi\)
0.677871 + 0.735180i \(0.262903\pi\)
\(20\) 0 0
\(21\) 3344.90 1.65514
\(22\) 1526.55i 0.672443i
\(23\) 873.145i 0.344165i 0.985083 + 0.172083i \(0.0550496\pi\)
−0.985083 + 0.172083i \(0.944950\pi\)
\(24\) −4876.87 −1.72828
\(25\) 0 0
\(26\) −359.857 −0.104399
\(27\) − 4223.83i − 1.11506i
\(28\) − 564.876i − 0.136163i
\(29\) 2580.97 0.569885 0.284943 0.958545i \(-0.408025\pi\)
0.284943 + 0.958545i \(0.408025\pi\)
\(30\) 0 0
\(31\) −9086.30 −1.69818 −0.849088 0.528252i \(-0.822848\pi\)
−0.849088 + 0.528252i \(0.822848\pi\)
\(32\) 1549.41i 0.267480i
\(33\) 7404.67i 1.18364i
\(34\) −1634.64 −0.242507
\(35\) 0 0
\(36\) −1760.74 −0.226433
\(37\) − 3990.64i − 0.479224i −0.970869 0.239612i \(-0.922980\pi\)
0.970869 0.239612i \(-0.0770202\pi\)
\(38\) 11225.9i 1.26113i
\(39\) −1745.52 −0.183765
\(40\) 0 0
\(41\) 16981.8 1.57770 0.788851 0.614584i \(-0.210676\pi\)
0.788851 + 0.614584i \(0.210676\pi\)
\(42\) 17601.2i 1.53964i
\(43\) − 18017.7i − 1.48603i −0.669273 0.743017i \(-0.733394\pi\)
0.669273 0.743017i \(-0.266606\pi\)
\(44\) 1250.48 0.0973742
\(45\) 0 0
\(46\) −4594.57 −0.320147
\(47\) − 24864.7i − 1.64187i −0.571024 0.820933i \(-0.693454\pi\)
0.571024 0.820933i \(-0.306546\pi\)
\(48\) − 22141.9i − 1.38711i
\(49\) −366.670 −0.0218165
\(50\) 0 0
\(51\) −7928.93 −0.426864
\(52\) 294.777i 0.0151177i
\(53\) − 7652.91i − 0.374229i −0.982338 0.187114i \(-0.940087\pi\)
0.982338 0.187114i \(-0.0599135\pi\)
\(54\) 22226.2 1.03724
\(55\) 0 0
\(56\) 25039.2 1.06697
\(57\) 54451.9i 2.21986i
\(58\) 13581.3i 0.530116i
\(59\) 9233.69 0.345339 0.172669 0.984980i \(-0.444761\pi\)
0.172669 + 0.984980i \(0.444761\pi\)
\(60\) 0 0
\(61\) 3326.17 0.114451 0.0572256 0.998361i \(-0.481775\pi\)
0.0572256 + 0.998361i \(0.481775\pi\)
\(62\) − 47812.9i − 1.57967i
\(63\) 53531.1i 1.69924i
\(64\) −35912.7 −1.09597
\(65\) 0 0
\(66\) −38964.0 −1.10104
\(67\) 32340.7i 0.880161i 0.897958 + 0.440080i \(0.145050\pi\)
−0.897958 + 0.440080i \(0.854950\pi\)
\(68\) 1339.01i 0.0351165i
\(69\) −22286.3 −0.563528
\(70\) 0 0
\(71\) −35885.9 −0.844847 −0.422424 0.906399i \(-0.638821\pi\)
−0.422424 + 0.906399i \(0.638821\pi\)
\(72\) − 78048.4i − 1.77432i
\(73\) − 26513.6i − 0.582319i −0.956675 0.291159i \(-0.905959\pi\)
0.956675 0.291159i \(-0.0940410\pi\)
\(74\) 20999.1 0.445781
\(75\) 0 0
\(76\) 9195.65 0.182620
\(77\) − 38017.7i − 0.730733i
\(78\) − 9185.06i − 0.170941i
\(79\) −71705.7 −1.29266 −0.646332 0.763056i \(-0.723698\pi\)
−0.646332 + 0.763056i \(0.723698\pi\)
\(80\) 0 0
\(81\) 8548.28 0.144766
\(82\) 89359.9i 1.46760i
\(83\) 39630.1i 0.631437i 0.948853 + 0.315719i \(0.102246\pi\)
−0.948853 + 0.315719i \(0.897754\pi\)
\(84\) 14418.0 0.222949
\(85\) 0 0
\(86\) 94810.8 1.38233
\(87\) 65877.1i 0.933117i
\(88\) 55429.9i 0.763022i
\(89\) 117441. 1.57161 0.785806 0.618473i \(-0.212248\pi\)
0.785806 + 0.618473i \(0.212248\pi\)
\(90\) 0 0
\(91\) 8961.98 0.113449
\(92\) 3763.64i 0.0463594i
\(93\) − 231920.i − 2.78055i
\(94\) 130840. 1.52729
\(95\) 0 0
\(96\) −39547.4 −0.437966
\(97\) − 21878.3i − 0.236093i −0.993008 0.118047i \(-0.962337\pi\)
0.993008 0.118047i \(-0.0376633\pi\)
\(98\) − 1929.45i − 0.0202940i
\(99\) −118503. −1.21518
\(100\) 0 0
\(101\) −75072.1 −0.732276 −0.366138 0.930561i \(-0.619320\pi\)
−0.366138 + 0.930561i \(0.619320\pi\)
\(102\) − 41722.7i − 0.397075i
\(103\) 47928.6i 0.445145i 0.974916 + 0.222573i \(0.0714455\pi\)
−0.974916 + 0.222573i \(0.928555\pi\)
\(104\) −13066.6 −0.118462
\(105\) 0 0
\(106\) 40270.3 0.348113
\(107\) − 92012.3i − 0.776938i −0.921462 0.388469i \(-0.873004\pi\)
0.921462 0.388469i \(-0.126996\pi\)
\(108\) − 18206.6i − 0.150199i
\(109\) 10647.5 0.0858387 0.0429194 0.999079i \(-0.486334\pi\)
0.0429194 + 0.999079i \(0.486334\pi\)
\(110\) 0 0
\(111\) 101858. 0.784670
\(112\) 113683.i 0.856346i
\(113\) 87373.9i 0.643703i 0.946790 + 0.321852i \(0.104305\pi\)
−0.946790 + 0.321852i \(0.895695\pi\)
\(114\) −286531. −2.06495
\(115\) 0 0
\(116\) 11125.1 0.0767642
\(117\) − 27934.9i − 0.188661i
\(118\) 48588.5i 0.321239i
\(119\) 40709.4 0.263528
\(120\) 0 0
\(121\) −76890.5 −0.477429
\(122\) 17502.6i 0.106464i
\(123\) 433447.i 2.58329i
\(124\) −39165.9 −0.228746
\(125\) 0 0
\(126\) −281685. −1.58066
\(127\) − 197379.i − 1.08591i −0.839763 0.542953i \(-0.817306\pi\)
0.839763 0.542953i \(-0.182694\pi\)
\(128\) − 139395.i − 0.752005i
\(129\) 459887. 2.43320
\(130\) 0 0
\(131\) −118490. −0.603258 −0.301629 0.953425i \(-0.597530\pi\)
−0.301629 + 0.953425i \(0.597530\pi\)
\(132\) 31917.4i 0.159438i
\(133\) − 279571.i − 1.37045i
\(134\) −170179. −0.818738
\(135\) 0 0
\(136\) −59354.3 −0.275173
\(137\) 302570.i 1.37728i 0.725101 + 0.688642i \(0.241793\pi\)
−0.725101 + 0.688642i \(0.758207\pi\)
\(138\) − 117272.i − 0.524202i
\(139\) −157190. −0.690062 −0.345031 0.938591i \(-0.612132\pi\)
−0.345031 + 0.938591i \(0.612132\pi\)
\(140\) 0 0
\(141\) 634650. 2.68835
\(142\) − 188835.i − 0.785889i
\(143\) 19839.3i 0.0811309i
\(144\) 354354. 1.42407
\(145\) 0 0
\(146\) 139517. 0.541681
\(147\) − 9358.95i − 0.0357218i
\(148\) − 17201.4i − 0.0645520i
\(149\) −526340. −1.94223 −0.971115 0.238612i \(-0.923308\pi\)
−0.971115 + 0.238612i \(0.923308\pi\)
\(150\) 0 0
\(151\) 1849.08 0.00659954 0.00329977 0.999995i \(-0.498950\pi\)
0.00329977 + 0.999995i \(0.498950\pi\)
\(152\) 407616.i 1.43101i
\(153\) − 126893.i − 0.438237i
\(154\) 200052. 0.679739
\(155\) 0 0
\(156\) −7523.94 −0.0247533
\(157\) − 343342.i − 1.11167i −0.831292 0.555837i \(-0.812398\pi\)
0.831292 0.555837i \(-0.187602\pi\)
\(158\) − 377322.i − 1.20246i
\(159\) 195334. 0.612753
\(160\) 0 0
\(161\) 114424. 0.347899
\(162\) 44981.8i 0.134663i
\(163\) − 267463.i − 0.788487i −0.919006 0.394243i \(-0.871007\pi\)
0.919006 0.394243i \(-0.128993\pi\)
\(164\) 73199.1 0.212518
\(165\) 0 0
\(166\) −208537. −0.587372
\(167\) − 122968.i − 0.341193i −0.985341 0.170596i \(-0.945431\pi\)
0.985341 0.170596i \(-0.0545694\pi\)
\(168\) 639106.i 1.74703i
\(169\) 366616. 0.987404
\(170\) 0 0
\(171\) −871437. −2.27901
\(172\) − 77664.3i − 0.200170i
\(173\) 288020.i 0.731657i 0.930682 + 0.365829i \(0.119214\pi\)
−0.930682 + 0.365829i \(0.880786\pi\)
\(174\) −346651. −0.867999
\(175\) 0 0
\(176\) −251662. −0.612400
\(177\) 235682.i 0.565450i
\(178\) 617986.i 1.46194i
\(179\) −246177. −0.574268 −0.287134 0.957890i \(-0.592702\pi\)
−0.287134 + 0.957890i \(0.592702\pi\)
\(180\) 0 0
\(181\) 433120. 0.982678 0.491339 0.870968i \(-0.336508\pi\)
0.491339 + 0.870968i \(0.336508\pi\)
\(182\) 47158.7i 0.105532i
\(183\) 84897.9i 0.187400i
\(184\) −166831. −0.363272
\(185\) 0 0
\(186\) 1.22038e6 2.58651
\(187\) 90119.1i 0.188457i
\(188\) − 107178.i − 0.221161i
\(189\) −553526. −1.12715
\(190\) 0 0
\(191\) −701011. −1.39040 −0.695202 0.718814i \(-0.744685\pi\)
−0.695202 + 0.718814i \(0.744685\pi\)
\(192\) − 916642.i − 1.79451i
\(193\) 215730.i 0.416887i 0.978034 + 0.208443i \(0.0668397\pi\)
−0.978034 + 0.208443i \(0.933160\pi\)
\(194\) 115125. 0.219618
\(195\) 0 0
\(196\) −1580.51 −0.00293871
\(197\) 700484.i 1.28598i 0.765876 + 0.642988i \(0.222305\pi\)
−0.765876 + 0.642988i \(0.777695\pi\)
\(198\) − 623572.i − 1.13038i
\(199\) −22097.5 −0.0395558 −0.0197779 0.999804i \(-0.506296\pi\)
−0.0197779 + 0.999804i \(0.506296\pi\)
\(200\) 0 0
\(201\) −825469. −1.44115
\(202\) − 395036.i − 0.681174i
\(203\) − 338231.i − 0.576068i
\(204\) −34177.1 −0.0574990
\(205\) 0 0
\(206\) −252205. −0.414081
\(207\) − 356665.i − 0.578542i
\(208\) − 59324.6i − 0.0950772i
\(209\) 618893. 0.980054
\(210\) 0 0
\(211\) 910782. 1.40834 0.704172 0.710030i \(-0.251319\pi\)
0.704172 + 0.710030i \(0.251319\pi\)
\(212\) − 32987.4i − 0.0504090i
\(213\) − 915958.i − 1.38333i
\(214\) 484177. 0.722719
\(215\) 0 0
\(216\) 807042. 1.17696
\(217\) 1.19074e6i 1.71660i
\(218\) 56028.3i 0.0798484i
\(219\) 676737. 0.953475
\(220\) 0 0
\(221\) −21243.9 −0.0292587
\(222\) 535985.i 0.729911i
\(223\) 132745.i 0.178754i 0.995998 + 0.0893768i \(0.0284875\pi\)
−0.995998 + 0.0893768i \(0.971512\pi\)
\(224\) 203048. 0.270382
\(225\) 0 0
\(226\) −459769. −0.598782
\(227\) 354321.i 0.456386i 0.973616 + 0.228193i \(0.0732817\pi\)
−0.973616 + 0.228193i \(0.926718\pi\)
\(228\) 234711.i 0.299018i
\(229\) −366643. −0.462013 −0.231007 0.972952i \(-0.574202\pi\)
−0.231007 + 0.972952i \(0.574202\pi\)
\(230\) 0 0
\(231\) 970370. 1.19649
\(232\) 493142.i 0.601523i
\(233\) 1.02388e6i 1.23555i 0.786355 + 0.617776i \(0.211966\pi\)
−0.786355 + 0.617776i \(0.788034\pi\)
\(234\) 146996. 0.175495
\(235\) 0 0
\(236\) 39801.2 0.0465175
\(237\) − 1.83023e6i − 2.11658i
\(238\) 214216.i 0.245138i
\(239\) −1.19966e6 −1.35852 −0.679258 0.733899i \(-0.737698\pi\)
−0.679258 + 0.733899i \(0.737698\pi\)
\(240\) 0 0
\(241\) −94967.5 −0.105325 −0.0526626 0.998612i \(-0.516771\pi\)
−0.0526626 + 0.998612i \(0.516771\pi\)
\(242\) − 404604.i − 0.444112i
\(243\) − 808203.i − 0.878021i
\(244\) 14337.3 0.0154167
\(245\) 0 0
\(246\) −2.28084e6 −2.40302
\(247\) 145893.i 0.152157i
\(248\) − 1.73611e6i − 1.79245i
\(249\) −1.01153e6 −1.03390
\(250\) 0 0
\(251\) 418053. 0.418839 0.209419 0.977826i \(-0.432843\pi\)
0.209419 + 0.977826i \(0.432843\pi\)
\(252\) 230742.i 0.228890i
\(253\) 253303.i 0.248794i
\(254\) 1.03863e6 1.01012
\(255\) 0 0
\(256\) −415700. −0.396442
\(257\) − 2.04586e6i − 1.93216i −0.258246 0.966079i \(-0.583144\pi\)
0.258246 0.966079i \(-0.416856\pi\)
\(258\) 2.41997e6i 2.26340i
\(259\) −522967. −0.484423
\(260\) 0 0
\(261\) −1.05428e6 −0.957978
\(262\) − 623505.i − 0.561160i
\(263\) 1.64024e6i 1.46224i 0.682250 + 0.731119i \(0.261002\pi\)
−0.682250 + 0.731119i \(0.738998\pi\)
\(264\) −1.41480e6 −1.24935
\(265\) 0 0
\(266\) 1.47113e6 1.27481
\(267\) 2.99759e6i 2.57332i
\(268\) 139402.i 0.118559i
\(269\) 720582. 0.607160 0.303580 0.952806i \(-0.401818\pi\)
0.303580 + 0.952806i \(0.401818\pi\)
\(270\) 0 0
\(271\) 1.14186e6 0.944477 0.472238 0.881471i \(-0.343446\pi\)
0.472238 + 0.881471i \(0.343446\pi\)
\(272\) − 269479.i − 0.220853i
\(273\) 228747.i 0.185759i
\(274\) −1.59215e6 −1.28117
\(275\) 0 0
\(276\) −96063.7 −0.0759078
\(277\) − 377028.i − 0.295239i −0.989044 0.147620i \(-0.952839\pi\)
0.989044 0.147620i \(-0.0471612\pi\)
\(278\) − 827148.i − 0.641906i
\(279\) 3.71160e6 2.85464
\(280\) 0 0
\(281\) −617249. −0.466331 −0.233166 0.972437i \(-0.574908\pi\)
−0.233166 + 0.972437i \(0.574908\pi\)
\(282\) 3.33958e6i 2.50075i
\(283\) − 1.25311e6i − 0.930087i −0.885288 0.465044i \(-0.846039\pi\)
0.885288 0.465044i \(-0.153961\pi\)
\(284\) −154684. −0.113802
\(285\) 0 0
\(286\) −104396. −0.0754692
\(287\) − 2.22544e6i − 1.59482i
\(288\) − 632908.i − 0.449635i
\(289\) 1.32336e6 0.932036
\(290\) 0 0
\(291\) 558425. 0.386574
\(292\) − 114285.i − 0.0784390i
\(293\) 818972.i 0.557314i 0.960391 + 0.278657i \(0.0898893\pi\)
−0.960391 + 0.278657i \(0.910111\pi\)
\(294\) 49247.6 0.0332290
\(295\) 0 0
\(296\) 762487. 0.505828
\(297\) − 1.22535e6i − 0.806064i
\(298\) − 2.76965e6i − 1.80669i
\(299\) −59711.6 −0.0386261
\(300\) 0 0
\(301\) −2.36119e6 −1.50216
\(302\) 9730.03i 0.00613899i
\(303\) − 1.91615e6i − 1.19901i
\(304\) −1.85065e6 −1.14852
\(305\) 0 0
\(306\) 667722. 0.407654
\(307\) 136224.i 0.0824915i 0.999149 + 0.0412458i \(0.0131327\pi\)
−0.999149 + 0.0412458i \(0.986867\pi\)
\(308\) − 163873.i − 0.0984306i
\(309\) −1.22334e6 −0.728871
\(310\) 0 0
\(311\) 2.62886e6 1.54122 0.770612 0.637304i \(-0.219950\pi\)
0.770612 + 0.637304i \(0.219950\pi\)
\(312\) − 333514.i − 0.193967i
\(313\) − 218161.i − 0.125868i −0.998018 0.0629341i \(-0.979954\pi\)
0.998018 0.0629341i \(-0.0200458\pi\)
\(314\) 1.80669e6 1.03409
\(315\) 0 0
\(316\) −309083. −0.174123
\(317\) 1.25865e6i 0.703491i 0.936096 + 0.351745i \(0.114412\pi\)
−0.936096 + 0.351745i \(0.885588\pi\)
\(318\) 1.02787e6i 0.569992i
\(319\) 748750. 0.411965
\(320\) 0 0
\(321\) 2.34854e6 1.27214
\(322\) 602110.i 0.323621i
\(323\) 662711.i 0.353442i
\(324\) 36846.8 0.0195001
\(325\) 0 0
\(326\) 1.40741e6 0.733462
\(327\) 271770.i 0.140550i
\(328\) 3.24470e6i 1.66529i
\(329\) −3.25847e6 −1.65968
\(330\) 0 0
\(331\) −3.21863e6 −1.61473 −0.807366 0.590051i \(-0.799108\pi\)
−0.807366 + 0.590051i \(0.799108\pi\)
\(332\) 170823.i 0.0850553i
\(333\) 1.63011e6i 0.805576i
\(334\) 647067. 0.317382
\(335\) 0 0
\(336\) −2.90166e6 −1.40216
\(337\) 1.63574e6i 0.784585i 0.919840 + 0.392293i \(0.128318\pi\)
−0.919840 + 0.392293i \(0.871682\pi\)
\(338\) 1.92917e6i 0.918498i
\(339\) −2.23015e6 −1.05398
\(340\) 0 0
\(341\) −2.63597e6 −1.22760
\(342\) − 4.58558e6i − 2.11996i
\(343\) − 2.15448e6i − 0.988796i
\(344\) 3.44262e6 1.56853
\(345\) 0 0
\(346\) −1.51559e6 −0.680598
\(347\) − 1.83815e6i − 0.819514i −0.912195 0.409757i \(-0.865614\pi\)
0.912195 0.409757i \(-0.134386\pi\)
\(348\) 283959.i 0.125692i
\(349\) 2.53806e6 1.11542 0.557710 0.830036i \(-0.311680\pi\)
0.557710 + 0.830036i \(0.311680\pi\)
\(350\) 0 0
\(351\) 288854. 0.125144
\(352\) 449491.i 0.193359i
\(353\) 1.88471e6i 0.805023i 0.915415 + 0.402511i \(0.131863\pi\)
−0.915415 + 0.402511i \(0.868137\pi\)
\(354\) −1.24018e6 −0.525989
\(355\) 0 0
\(356\) 506223. 0.211698
\(357\) 1.03907e6i 0.431495i
\(358\) − 1.29540e6i − 0.534192i
\(359\) −305057. −0.124924 −0.0624619 0.998047i \(-0.519895\pi\)
−0.0624619 + 0.998047i \(0.519895\pi\)
\(360\) 0 0
\(361\) 2.07507e6 0.838039
\(362\) 2.27911e6i 0.914102i
\(363\) − 1.96257e6i − 0.781731i
\(364\) 38630.0 0.0152817
\(365\) 0 0
\(366\) −446740. −0.174322
\(367\) − 727834.i − 0.282077i −0.990004 0.141038i \(-0.954956\pi\)
0.990004 0.141038i \(-0.0450441\pi\)
\(368\) − 757441.i − 0.291561i
\(369\) −6.93680e6 −2.65212
\(370\) 0 0
\(371\) −1.00290e6 −0.378289
\(372\) − 999677.i − 0.374544i
\(373\) 4.77676e6i 1.77771i 0.458188 + 0.888855i \(0.348499\pi\)
−0.458188 + 0.888855i \(0.651501\pi\)
\(374\) −474215. −0.175306
\(375\) 0 0
\(376\) 4.75086e6 1.73302
\(377\) 176504.i 0.0639590i
\(378\) − 2.91270e6i − 1.04850i
\(379\) 701558. 0.250880 0.125440 0.992101i \(-0.459966\pi\)
0.125440 + 0.992101i \(0.459966\pi\)
\(380\) 0 0
\(381\) 5.03794e6 1.77804
\(382\) − 3.68878e6i − 1.29337i
\(383\) − 4.01069e6i − 1.39708i −0.715570 0.698541i \(-0.753833\pi\)
0.715570 0.698541i \(-0.246167\pi\)
\(384\) 3.55793e6 1.23132
\(385\) 0 0
\(386\) −1.13519e6 −0.387794
\(387\) 7.35994e6i 2.49803i
\(388\) − 94305.0i − 0.0318021i
\(389\) 4.45952e6 1.49422 0.747108 0.664702i \(-0.231442\pi\)
0.747108 + 0.664702i \(0.231442\pi\)
\(390\) 0 0
\(391\) −271237. −0.0897237
\(392\) − 70059.1i − 0.0230276i
\(393\) − 3.02436e6i − 0.987761i
\(394\) −3.68601e6 −1.19623
\(395\) 0 0
\(396\) −510799. −0.163686
\(397\) 3.36993e6i 1.07311i 0.843865 + 0.536555i \(0.180275\pi\)
−0.843865 + 0.536555i \(0.819725\pi\)
\(398\) − 116279.i − 0.0367953i
\(399\) 7.13583e6 2.24395
\(400\) 0 0
\(401\) −3.00679e6 −0.933775 −0.466888 0.884317i \(-0.654625\pi\)
−0.466888 + 0.884317i \(0.654625\pi\)
\(402\) − 4.34369e6i − 1.34058i
\(403\) − 621383.i − 0.190588i
\(404\) −323593. −0.0986384
\(405\) 0 0
\(406\) 1.77980e6 0.535867
\(407\) − 1.15770e6i − 0.346426i
\(408\) − 1.51497e6i − 0.450561i
\(409\) 998012. 0.295004 0.147502 0.989062i \(-0.452877\pi\)
0.147502 + 0.989062i \(0.452877\pi\)
\(410\) 0 0
\(411\) −7.72284e6 −2.25513
\(412\) 206593.i 0.0599616i
\(413\) − 1.21006e6i − 0.349085i
\(414\) 1.87680e6 0.538168
\(415\) 0 0
\(416\) −105959. −0.0300196
\(417\) − 4.01215e6i − 1.12989i
\(418\) 3.25667e6i 0.911660i
\(419\) −5.53743e6 −1.54090 −0.770448 0.637503i \(-0.779967\pi\)
−0.770448 + 0.637503i \(0.779967\pi\)
\(420\) 0 0
\(421\) 1.98635e6 0.546198 0.273099 0.961986i \(-0.411951\pi\)
0.273099 + 0.961986i \(0.411951\pi\)
\(422\) 4.79262e6i 1.31006i
\(423\) 1.01568e7i 2.75998i
\(424\) 1.46223e6 0.395004
\(425\) 0 0
\(426\) 4.81985e6 1.28680
\(427\) − 435890.i − 0.115693i
\(428\) − 396613.i − 0.104654i
\(429\) −506382. −0.132842
\(430\) 0 0
\(431\) 116512. 0.0302118 0.0151059 0.999886i \(-0.495191\pi\)
0.0151059 + 0.999886i \(0.495191\pi\)
\(432\) 3.66411e6i 0.944625i
\(433\) − 4.56166e6i − 1.16924i −0.811308 0.584619i \(-0.801244\pi\)
0.811308 0.584619i \(-0.198756\pi\)
\(434\) −6.26580e6 −1.59681
\(435\) 0 0
\(436\) 45895.6 0.0115626
\(437\) 1.86272e6i 0.466599i
\(438\) 3.56105e6i 0.886936i
\(439\) −2.92172e6 −0.723565 −0.361782 0.932263i \(-0.617832\pi\)
−0.361782 + 0.932263i \(0.617832\pi\)
\(440\) 0 0
\(441\) 149779. 0.0366736
\(442\) − 111787.i − 0.0272168i
\(443\) − 1.59752e6i − 0.386756i −0.981124 0.193378i \(-0.938056\pi\)
0.981124 0.193378i \(-0.0619444\pi\)
\(444\) 439052. 0.105696
\(445\) 0 0
\(446\) −698514. −0.166279
\(447\) − 1.34344e7i − 3.18016i
\(448\) 4.70630e6i 1.10786i
\(449\) −3.11073e6 −0.728193 −0.364096 0.931361i \(-0.618622\pi\)
−0.364096 + 0.931361i \(0.618622\pi\)
\(450\) 0 0
\(451\) 4.92650e6 1.14051
\(452\) 376620.i 0.0867076i
\(453\) 47196.3i 0.0108059i
\(454\) −1.86447e6 −0.424537
\(455\) 0 0
\(456\) −1.04041e7 −2.34310
\(457\) − 6.47145e6i − 1.44948i −0.689025 0.724738i \(-0.741961\pi\)
0.689025 0.724738i \(-0.258039\pi\)
\(458\) − 1.92931e6i − 0.429771i
\(459\) 1.31211e6 0.290695
\(460\) 0 0
\(461\) −5.47864e6 −1.20066 −0.600330 0.799752i \(-0.704964\pi\)
−0.600330 + 0.799752i \(0.704964\pi\)
\(462\) 5.10617e6i 1.11299i
\(463\) 2.35489e6i 0.510526i 0.966872 + 0.255263i \(0.0821621\pi\)
−0.966872 + 0.255263i \(0.917838\pi\)
\(464\) −2.23895e6 −0.482781
\(465\) 0 0
\(466\) −5.38776e6 −1.14933
\(467\) 4.56027e6i 0.967606i 0.875177 + 0.483803i \(0.160745\pi\)
−0.875177 + 0.483803i \(0.839255\pi\)
\(468\) − 120412.i − 0.0254129i
\(469\) 4.23819e6 0.889710
\(470\) 0 0
\(471\) 8.76351e6 1.82023
\(472\) 1.76427e6i 0.364510i
\(473\) − 5.22702e6i − 1.07424i
\(474\) 9.63082e6 1.96887
\(475\) 0 0
\(476\) 175475. 0.0354975
\(477\) 3.12609e6i 0.629079i
\(478\) − 6.31274e6i − 1.26371i
\(479\) 1.88004e6 0.374394 0.187197 0.982322i \(-0.440060\pi\)
0.187197 + 0.982322i \(0.440060\pi\)
\(480\) 0 0
\(481\) 272907. 0.0537839
\(482\) − 499727.i − 0.0979751i
\(483\) 2.92058e6i 0.569642i
\(484\) −331431. −0.0643103
\(485\) 0 0
\(486\) 4.25283e6 0.816747
\(487\) − 1.69396e6i − 0.323654i −0.986819 0.161827i \(-0.948261\pi\)
0.986819 0.161827i \(-0.0517386\pi\)
\(488\) 635528.i 0.120805i
\(489\) 6.82677e6 1.29105
\(490\) 0 0
\(491\) 1.48645e6 0.278258 0.139129 0.990274i \(-0.455570\pi\)
0.139129 + 0.990274i \(0.455570\pi\)
\(492\) 1.86835e6i 0.347972i
\(493\) 801762.i 0.148569i
\(494\) −767700. −0.141538
\(495\) 0 0
\(496\) 7.88224e6 1.43862
\(497\) 4.70279e6i 0.854013i
\(498\) − 5.32274e6i − 0.961749i
\(499\) −7.09934e6 −1.27634 −0.638170 0.769896i \(-0.720308\pi\)
−0.638170 + 0.769896i \(0.720308\pi\)
\(500\) 0 0
\(501\) 3.13865e6 0.558661
\(502\) 2.19983e6i 0.389610i
\(503\) 9.24224e6i 1.62876i 0.580331 + 0.814381i \(0.302923\pi\)
−0.580331 + 0.814381i \(0.697077\pi\)
\(504\) −1.02281e7 −1.79357
\(505\) 0 0
\(506\) −1.33290e6 −0.231431
\(507\) 9.35758e6i 1.61675i
\(508\) − 850790.i − 0.146273i
\(509\) 8.12506e6 1.39006 0.695028 0.718983i \(-0.255392\pi\)
0.695028 + 0.718983i \(0.255392\pi\)
\(510\) 0 0
\(511\) −3.47456e6 −0.588637
\(512\) − 6.64807e6i − 1.12078i
\(513\) − 9.01089e6i − 1.51173i
\(514\) 1.07655e7 1.79732
\(515\) 0 0
\(516\) 1.98232e6 0.327754
\(517\) − 7.21335e6i − 1.18689i
\(518\) − 2.75190e6i − 0.450617i
\(519\) −7.35148e6 −1.19800
\(520\) 0 0
\(521\) 5.06245e6 0.817084 0.408542 0.912740i \(-0.366037\pi\)
0.408542 + 0.912740i \(0.366037\pi\)
\(522\) − 5.54773e6i − 0.891125i
\(523\) − 4.76222e6i − 0.761299i −0.924719 0.380649i \(-0.875700\pi\)
0.924719 0.380649i \(-0.124300\pi\)
\(524\) −510744. −0.0812596
\(525\) 0 0
\(526\) −8.63109e6 −1.36019
\(527\) − 2.82260e6i − 0.442714i
\(528\) − 6.42345e6i − 1.00273i
\(529\) 5.67396e6 0.881550
\(530\) 0 0
\(531\) −3.77181e6 −0.580515
\(532\) − 1.20508e6i − 0.184601i
\(533\) 1.16133e6i 0.177067i
\(534\) −1.57736e7 −2.39374
\(535\) 0 0
\(536\) −6.17929e6 −0.929023
\(537\) − 6.28346e6i − 0.940292i
\(538\) 3.79177e6i 0.564789i
\(539\) −106373. −0.0157709
\(540\) 0 0
\(541\) −2.89920e6 −0.425877 −0.212939 0.977066i \(-0.568303\pi\)
−0.212939 + 0.977066i \(0.568303\pi\)
\(542\) 6.00859e6i 0.878566i
\(543\) 1.10550e7i 1.60901i
\(544\) −481315. −0.0697320
\(545\) 0 0
\(546\) −1.20369e6 −0.172795
\(547\) 5.74434e6i 0.820866i 0.911891 + 0.410433i \(0.134622\pi\)
−0.911891 + 0.410433i \(0.865378\pi\)
\(548\) 1.30421e6i 0.185522i
\(549\) −1.35869e6 −0.192393
\(550\) 0 0
\(551\) 5.50610e6 0.772618
\(552\) − 4.25822e6i − 0.594812i
\(553\) 9.39691e6i 1.30669i
\(554\) 1.98395e6 0.274636
\(555\) 0 0
\(556\) −677558. −0.0929522
\(557\) 7.29174e6i 0.995848i 0.867221 + 0.497924i \(0.165904\pi\)
−0.867221 + 0.497924i \(0.834096\pi\)
\(558\) 1.95308e7i 2.65542i
\(559\) 1.23217e6 0.166779
\(560\) 0 0
\(561\) −2.30022e6 −0.308576
\(562\) − 3.24802e6i − 0.433788i
\(563\) 6.65348e6i 0.884663i 0.896852 + 0.442331i \(0.145849\pi\)
−0.896852 + 0.442331i \(0.854151\pi\)
\(564\) 2.73562e6 0.362124
\(565\) 0 0
\(566\) 6.59398e6 0.865181
\(567\) − 1.12024e6i − 0.146336i
\(568\) − 6.85667e6i − 0.891749i
\(569\) −5.78715e6 −0.749349 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(570\) 0 0
\(571\) −1.22059e7 −1.56667 −0.783336 0.621599i \(-0.786483\pi\)
−0.783336 + 0.621599i \(0.786483\pi\)
\(572\) 85516.1i 0.0109284i
\(573\) − 1.78927e7i − 2.27662i
\(574\) 1.17105e7 1.48352
\(575\) 0 0
\(576\) 1.46697e7 1.84232
\(577\) 1.02981e7i 1.28771i 0.765149 + 0.643853i \(0.222665\pi\)
−0.765149 + 0.643853i \(0.777335\pi\)
\(578\) 6.96362e6i 0.866993i
\(579\) −5.50634e6 −0.682601
\(580\) 0 0
\(581\) 5.19346e6 0.638288
\(582\) 2.93848e6i 0.359597i
\(583\) − 2.22014e6i − 0.270526i
\(584\) 5.06591e6 0.614647
\(585\) 0 0
\(586\) −4.30950e6 −0.518422
\(587\) − 1.30519e7i − 1.56343i −0.623636 0.781715i \(-0.714345\pi\)
0.623636 0.781715i \(-0.285655\pi\)
\(588\) − 40341.1i − 0.00481177i
\(589\) −1.93842e7 −2.30229
\(590\) 0 0
\(591\) −1.78793e7 −2.10563
\(592\) 3.46183e6i 0.405977i
\(593\) − 6.43920e6i − 0.751961i −0.926628 0.375980i \(-0.877306\pi\)
0.926628 0.375980i \(-0.122694\pi\)
\(594\) 6.44791e6 0.749812
\(595\) 0 0
\(596\) −2.26876e6 −0.261621
\(597\) − 564020.i − 0.0647677i
\(598\) − 314208.i − 0.0359305i
\(599\) 1.00760e7 1.14741 0.573707 0.819061i \(-0.305505\pi\)
0.573707 + 0.819061i \(0.305505\pi\)
\(600\) 0 0
\(601\) 1.57050e6 0.177358 0.0886791 0.996060i \(-0.471735\pi\)
0.0886791 + 0.996060i \(0.471735\pi\)
\(602\) − 1.24248e7i − 1.39733i
\(603\) − 1.32106e7i − 1.47955i
\(604\) 7970.35 0.000888966 0
\(605\) 0 0
\(606\) 1.00830e7 1.11534
\(607\) − 7.31039e6i − 0.805321i −0.915349 0.402660i \(-0.868086\pi\)
0.915349 0.402660i \(-0.131914\pi\)
\(608\) 3.30543e6i 0.362634i
\(609\) 8.63308e6 0.943241
\(610\) 0 0
\(611\) 1.70041e6 0.184269
\(612\) − 546964.i − 0.0590310i
\(613\) − 1.31997e7i − 1.41878i −0.704817 0.709389i \(-0.748971\pi\)
0.704817 0.709389i \(-0.251029\pi\)
\(614\) −716825. −0.0767348
\(615\) 0 0
\(616\) 7.26399e6 0.771300
\(617\) 1.02423e7i 1.08314i 0.840655 + 0.541570i \(0.182170\pi\)
−0.840655 + 0.541570i \(0.817830\pi\)
\(618\) − 6.43732e6i − 0.678006i
\(619\) −1.05614e7 −1.10788 −0.553942 0.832555i \(-0.686877\pi\)
−0.553942 + 0.832555i \(0.686877\pi\)
\(620\) 0 0
\(621\) 3.68802e6 0.383764
\(622\) 1.38333e7i 1.43367i
\(623\) − 1.53905e7i − 1.58866i
\(624\) 1.51421e6 0.155677
\(625\) 0 0
\(626\) 1.14798e6 0.117084
\(627\) 1.57967e7i 1.60472i
\(628\) − 1.47995e6i − 0.149744i
\(629\) 1.23967e6 0.124933
\(630\) 0 0
\(631\) 1.90535e7 1.90503 0.952513 0.304497i \(-0.0984883\pi\)
0.952513 + 0.304497i \(0.0984883\pi\)
\(632\) − 1.37007e7i − 1.36443i
\(633\) 2.32470e7i 2.30599i
\(634\) −6.62315e6 −0.654397
\(635\) 0 0
\(636\) 841976. 0.0825385
\(637\) − 25075.4i − 0.00244849i
\(638\) 3.93999e6i 0.383216i
\(639\) 1.46588e7 1.42019
\(640\) 0 0
\(641\) 8.56937e6 0.823766 0.411883 0.911237i \(-0.364871\pi\)
0.411883 + 0.911237i \(0.364871\pi\)
\(642\) 1.23582e7i 1.18336i
\(643\) 1.79513e7i 1.71226i 0.516761 + 0.856130i \(0.327138\pi\)
−0.516761 + 0.856130i \(0.672862\pi\)
\(644\) 493218. 0.0468624
\(645\) 0 0
\(646\) −3.48724e6 −0.328777
\(647\) − 1.05470e7i − 0.990534i −0.868741 0.495267i \(-0.835070\pi\)
0.868741 0.495267i \(-0.164930\pi\)
\(648\) 1.63331e6i 0.152803i
\(649\) 2.67873e6 0.249642
\(650\) 0 0
\(651\) −3.03928e7 −2.81072
\(652\) − 1.15288e6i − 0.106210i
\(653\) 1.00324e7i 0.920712i 0.887734 + 0.460356i \(0.152278\pi\)
−0.887734 + 0.460356i \(0.847722\pi\)
\(654\) −1.43008e6 −0.130742
\(655\) 0 0
\(656\) −1.47315e7 −1.33656
\(657\) 1.08304e7i 0.978879i
\(658\) − 1.71464e7i − 1.54386i
\(659\) 8.99161e6 0.806536 0.403268 0.915082i \(-0.367874\pi\)
0.403268 + 0.915082i \(0.367874\pi\)
\(660\) 0 0
\(661\) 2.39297e6 0.213027 0.106513 0.994311i \(-0.466031\pi\)
0.106513 + 0.994311i \(0.466031\pi\)
\(662\) − 1.69367e7i − 1.50205i
\(663\) − 542234.i − 0.0479074i
\(664\) −7.57208e6 −0.666492
\(665\) 0 0
\(666\) −8.57779e6 −0.749359
\(667\) 2.25356e6i 0.196135i
\(668\) − 530044.i − 0.0459590i
\(669\) −3.38820e6 −0.292687
\(670\) 0 0
\(671\) 964938. 0.0827357
\(672\) 5.18262e6i 0.442717i
\(673\) 1.53612e7i 1.30733i 0.756783 + 0.653666i \(0.226770\pi\)
−0.756783 + 0.653666i \(0.773230\pi\)
\(674\) −8.60742e6 −0.729833
\(675\) 0 0
\(676\) 1.58028e6 0.133004
\(677\) − 1.16026e7i − 0.972934i −0.873699 0.486467i \(-0.838285\pi\)
0.873699 0.486467i \(-0.161715\pi\)
\(678\) − 1.17352e7i − 0.980432i
\(679\) −2.86711e6 −0.238655
\(680\) 0 0
\(681\) −9.04375e6 −0.747275
\(682\) − 1.38707e7i − 1.14193i
\(683\) − 1.20315e7i − 0.986890i −0.869777 0.493445i \(-0.835737\pi\)
0.869777 0.493445i \(-0.164263\pi\)
\(684\) −3.75627e6 −0.306985
\(685\) 0 0
\(686\) 1.13371e7 0.919792
\(687\) − 9.35825e6i − 0.756490i
\(688\) 1.56301e7i 1.25890i
\(689\) 523358. 0.0420001
\(690\) 0 0
\(691\) 5.18616e6 0.413191 0.206595 0.978426i \(-0.433762\pi\)
0.206595 + 0.978426i \(0.433762\pi\)
\(692\) 1.24149e6i 0.0985550i
\(693\) 1.55296e7i 1.22836i
\(694\) 9.67248e6 0.762323
\(695\) 0 0
\(696\) −1.25870e7 −0.984919
\(697\) 5.27530e6i 0.411306i
\(698\) 1.33555e7i 1.03758i
\(699\) −2.61338e7 −2.02306
\(700\) 0 0
\(701\) 6.00859e6 0.461825 0.230913 0.972974i \(-0.425829\pi\)
0.230913 + 0.972974i \(0.425829\pi\)
\(702\) 1.51998e6i 0.116411i
\(703\) − 8.51342e6i − 0.649704i
\(704\) −1.04184e7 −0.792265
\(705\) 0 0
\(706\) −9.91752e6 −0.748844
\(707\) 9.83807e6i 0.740221i
\(708\) 1.01589e6i 0.0761667i
\(709\) −5.90083e6 −0.440857 −0.220429 0.975403i \(-0.570746\pi\)
−0.220429 + 0.975403i \(0.570746\pi\)
\(710\) 0 0
\(711\) 2.92906e7 2.17297
\(712\) 2.24393e7i 1.65886i
\(713\) − 7.93365e6i − 0.584453i
\(714\) −5.46769e6 −0.401383
\(715\) 0 0
\(716\) −1.06113e6 −0.0773545
\(717\) − 3.06204e7i − 2.22440i
\(718\) − 1.60524e6i − 0.116206i
\(719\) 1.36592e7 0.985382 0.492691 0.870204i \(-0.336013\pi\)
0.492691 + 0.870204i \(0.336013\pi\)
\(720\) 0 0
\(721\) 6.28097e6 0.449975
\(722\) 1.09192e7i 0.779556i
\(723\) − 2.42397e6i − 0.172457i
\(724\) 1.86693e6 0.132368
\(725\) 0 0
\(726\) 1.03272e7 0.727178
\(727\) 1.11594e7i 0.783079i 0.920161 + 0.391539i \(0.128057\pi\)
−0.920161 + 0.391539i \(0.871943\pi\)
\(728\) 1.71235e6i 0.119747i
\(729\) 2.27059e7 1.58242
\(730\) 0 0
\(731\) 5.59710e6 0.387409
\(732\) 365947.i 0.0252430i
\(733\) − 1.52510e7i − 1.04843i −0.851586 0.524215i \(-0.824359\pi\)
0.851586 0.524215i \(-0.175641\pi\)
\(734\) 3.82993e6 0.262392
\(735\) 0 0
\(736\) −1.35286e6 −0.0920573
\(737\) 9.38217e6i 0.636260i
\(738\) − 3.65020e7i − 2.46704i
\(739\) 1.11820e7 0.753196 0.376598 0.926377i \(-0.377094\pi\)
0.376598 + 0.926377i \(0.377094\pi\)
\(740\) 0 0
\(741\) −3.72379e6 −0.249138
\(742\) − 5.27735e6i − 0.351890i
\(743\) 7.71450e6i 0.512667i 0.966588 + 0.256334i \(0.0825146\pi\)
−0.966588 + 0.256334i \(0.917485\pi\)
\(744\) 4.43127e7 2.93492
\(745\) 0 0
\(746\) −2.51357e7 −1.65365
\(747\) − 1.61883e7i − 1.06145i
\(748\) 388453.i 0.0253854i
\(749\) −1.20581e7 −0.785367
\(750\) 0 0
\(751\) −2.23973e7 −1.44909 −0.724545 0.689228i \(-0.757950\pi\)
−0.724545 + 0.689228i \(0.757950\pi\)
\(752\) 2.15698e7i 1.39091i
\(753\) 1.06704e7i 0.685796i
\(754\) −928780. −0.0594955
\(755\) 0 0
\(756\) −2.38594e6 −0.151829
\(757\) − 2.57267e7i − 1.63171i −0.578254 0.815857i \(-0.696266\pi\)
0.578254 0.815857i \(-0.303734\pi\)
\(758\) 3.69166e6i 0.233372i
\(759\) −6.46535e6 −0.407369
\(760\) 0 0
\(761\) −1.48340e7 −0.928533 −0.464267 0.885696i \(-0.653682\pi\)
−0.464267 + 0.885696i \(0.653682\pi\)
\(762\) 2.65101e7i 1.65395i
\(763\) − 1.39534e6i − 0.0867700i
\(764\) −3.02166e6 −0.187289
\(765\) 0 0
\(766\) 2.11046e7 1.29959
\(767\) 631463.i 0.0387578i
\(768\) − 1.06104e7i − 0.649125i
\(769\) 5.57112e6 0.339724 0.169862 0.985468i \(-0.445668\pi\)
0.169862 + 0.985468i \(0.445668\pi\)
\(770\) 0 0
\(771\) 5.22188e7 3.16367
\(772\) 929892.i 0.0561551i
\(773\) − 1.58230e7i − 0.952447i −0.879324 0.476224i \(-0.842005\pi\)
0.879324 0.476224i \(-0.157995\pi\)
\(774\) −3.87287e7 −2.32370
\(775\) 0 0
\(776\) 4.18026e6 0.249200
\(777\) − 1.33483e7i − 0.793183i
\(778\) 2.34664e7i 1.38994i
\(779\) 3.62281e7 2.13896
\(780\) 0 0
\(781\) −1.04107e7 −0.610732
\(782\) − 1.42727e6i − 0.0834623i
\(783\) − 1.09016e7i − 0.635454i
\(784\) 318081. 0.0184819
\(785\) 0 0
\(786\) 1.59144e7 0.918830
\(787\) 1.31529e7i 0.756981i 0.925605 + 0.378491i \(0.123557\pi\)
−0.925605 + 0.378491i \(0.876443\pi\)
\(788\) 3.01939e6i 0.173222i
\(789\) −4.18658e7 −2.39423
\(790\) 0 0
\(791\) 1.14502e7 0.650687
\(792\) − 2.26422e7i − 1.28264i
\(793\) 227466.i 0.0128450i
\(794\) −1.77328e7 −0.998222
\(795\) 0 0
\(796\) −95249.7 −0.00532821
\(797\) 2.58443e7i 1.44118i 0.693361 + 0.720590i \(0.256129\pi\)
−0.693361 + 0.720590i \(0.743871\pi\)
\(798\) 3.75494e7i 2.08735i
\(799\) 7.72406e6 0.428034
\(800\) 0 0
\(801\) −4.79728e7 −2.64188
\(802\) − 1.58220e7i − 0.868611i
\(803\) − 7.69170e6i − 0.420953i
\(804\) −3.55813e6 −0.194125
\(805\) 0 0
\(806\) 3.26977e6 0.177288
\(807\) 1.83923e7i 0.994149i
\(808\) − 1.43439e7i − 0.772929i
\(809\) −1.78857e7 −0.960804 −0.480402 0.877048i \(-0.659509\pi\)
−0.480402 + 0.877048i \(0.659509\pi\)
\(810\) 0 0
\(811\) −1.41608e7 −0.756026 −0.378013 0.925800i \(-0.623393\pi\)
−0.378013 + 0.925800i \(0.623393\pi\)
\(812\) − 1.45793e6i − 0.0775970i
\(813\) 2.91451e7i 1.54646i
\(814\) 6.09193e6 0.322251
\(815\) 0 0
\(816\) 6.87824e6 0.361619
\(817\) − 3.84380e7i − 2.01468i
\(818\) 5.25162e6i 0.274417i
\(819\) −3.66082e6 −0.190708
\(820\) 0 0
\(821\) −3.46248e7 −1.79279 −0.896394 0.443258i \(-0.853823\pi\)
−0.896394 + 0.443258i \(0.853823\pi\)
\(822\) − 4.06382e7i − 2.09776i
\(823\) 2.13360e7i 1.09803i 0.835813 + 0.549015i \(0.184997\pi\)
−0.835813 + 0.549015i \(0.815003\pi\)
\(824\) −9.15766e6 −0.469858
\(825\) 0 0
\(826\) 6.36744e6 0.324724
\(827\) 1.59813e6i 0.0812548i 0.999174 + 0.0406274i \(0.0129357\pi\)
−0.999174 + 0.0406274i \(0.987064\pi\)
\(828\) − 1.53738e6i − 0.0779303i
\(829\) 2.53923e7 1.28327 0.641633 0.767012i \(-0.278257\pi\)
0.641633 + 0.767012i \(0.278257\pi\)
\(830\) 0 0
\(831\) 9.62333e6 0.483418
\(832\) − 2.45595e6i − 0.123002i
\(833\) − 113904.i − 0.00568755i
\(834\) 2.11123e7 1.05104
\(835\) 0 0
\(836\) 2.66770e6 0.132014
\(837\) 3.83790e7i 1.89356i
\(838\) − 2.91384e7i − 1.43336i
\(839\) −1.98528e7 −0.973681 −0.486841 0.873491i \(-0.661851\pi\)
−0.486841 + 0.873491i \(0.661851\pi\)
\(840\) 0 0
\(841\) −1.38498e7 −0.675231
\(842\) 1.04523e7i 0.508081i
\(843\) − 1.57548e7i − 0.763560i
\(844\) 3.92587e6 0.189705
\(845\) 0 0
\(846\) −5.34460e7 −2.56737
\(847\) 1.00764e7i 0.482609i
\(848\) 6.63879e6i 0.317029i
\(849\) 3.19846e7 1.52290
\(850\) 0 0
\(851\) 3.48441e6 0.164932
\(852\) − 3.94818e6i − 0.186337i
\(853\) 1.59794e7i 0.751948i 0.926630 + 0.375974i \(0.122692\pi\)
−0.926630 + 0.375974i \(0.877308\pi\)
\(854\) 2.29369e6 0.107619
\(855\) 0 0
\(856\) 1.75807e7 0.820070
\(857\) 7.00157e6i 0.325644i 0.986655 + 0.162822i \(0.0520597\pi\)
−0.986655 + 0.162822i \(0.947940\pi\)
\(858\) − 2.66463e6i − 0.123571i
\(859\) 7.28414e6 0.336818 0.168409 0.985717i \(-0.446137\pi\)
0.168409 + 0.985717i \(0.446137\pi\)
\(860\) 0 0
\(861\) 5.68026e7 2.61132
\(862\) 613094.i 0.0281034i
\(863\) 1.76361e7i 0.806075i 0.915184 + 0.403037i \(0.132046\pi\)
−0.915184 + 0.403037i \(0.867954\pi\)
\(864\) 6.54444e6 0.298255
\(865\) 0 0
\(866\) 2.40038e7 1.08764
\(867\) 3.37776e7i 1.52609i
\(868\) 5.13263e6i 0.231228i
\(869\) −2.08021e7 −0.934455
\(870\) 0 0
\(871\) −2.21167e6 −0.0987816
\(872\) 2.03441e6i 0.0906041i
\(873\) 8.93692e6i 0.396874i
\(874\) −9.80180e6 −0.434037
\(875\) 0 0
\(876\) 2.91703e6 0.128434
\(877\) − 2.69004e7i − 1.18102i −0.807029 0.590512i \(-0.798926\pi\)
0.807029 0.590512i \(-0.201074\pi\)
\(878\) − 1.53743e7i − 0.673070i
\(879\) −2.09036e7 −0.912533
\(880\) 0 0
\(881\) 2.51911e7 1.09347 0.546735 0.837306i \(-0.315870\pi\)
0.546735 + 0.837306i \(0.315870\pi\)
\(882\) 788148.i 0.0341143i
\(883\) − 3.22126e7i − 1.39035i −0.718840 0.695175i \(-0.755327\pi\)
0.718840 0.695175i \(-0.244673\pi\)
\(884\) −91570.7 −0.00394117
\(885\) 0 0
\(886\) 8.40629e6 0.359766
\(887\) − 8.96139e6i − 0.382443i −0.981547 0.191221i \(-0.938755\pi\)
0.981547 0.191221i \(-0.0612448\pi\)
\(888\) 1.94618e7i 0.828231i
\(889\) −2.58662e7 −1.09769
\(890\) 0 0
\(891\) 2.47989e6 0.104650
\(892\) 572187.i 0.0240783i
\(893\) − 5.30449e7i − 2.22595i
\(894\) 7.06930e7 2.95823
\(895\) 0 0
\(896\) −1.82674e7 −0.760164
\(897\) − 1.52409e6i − 0.0632454i
\(898\) − 1.63689e7i − 0.677376i
\(899\) −2.34514e7 −0.967765
\(900\) 0 0
\(901\) 2.37733e6 0.0975613
\(902\) 2.59237e7i 1.06092i
\(903\) − 6.02675e7i − 2.45960i
\(904\) −1.66944e7 −0.679439
\(905\) 0 0
\(906\) −248351. −0.0100518
\(907\) 5.81689e6i 0.234786i 0.993086 + 0.117393i \(0.0374537\pi\)
−0.993086 + 0.117393i \(0.962546\pi\)
\(908\) 1.52728e6i 0.0614757i
\(909\) 3.06657e7 1.23096
\(910\) 0 0
\(911\) −1.96435e7 −0.784192 −0.392096 0.919924i \(-0.628250\pi\)
−0.392096 + 0.919924i \(0.628250\pi\)
\(912\) − 4.72363e7i − 1.88057i
\(913\) 1.14969e7i 0.456460i
\(914\) 3.40533e7 1.34832
\(915\) 0 0
\(916\) −1.58039e6 −0.0622337
\(917\) 1.55279e7i 0.609803i
\(918\) 6.90442e6i 0.270409i
\(919\) 89962.4 0.00351376 0.00175688 0.999998i \(-0.499441\pi\)
0.00175688 + 0.999998i \(0.499441\pi\)
\(920\) 0 0
\(921\) −3.47702e6 −0.135070
\(922\) − 2.88291e7i − 1.11687i
\(923\) − 2.45412e6i − 0.0948183i
\(924\) 4.18272e6 0.161168
\(925\) 0 0
\(926\) −1.23916e7 −0.474899
\(927\) − 1.95781e7i − 0.748290i
\(928\) 3.99898e6i 0.152433i
\(929\) −3.65192e7 −1.38830 −0.694149 0.719832i \(-0.744219\pi\)
−0.694149 + 0.719832i \(0.744219\pi\)
\(930\) 0 0
\(931\) −782234. −0.0295776
\(932\) 4.41338e6i 0.166430i
\(933\) 6.70994e7i 2.52357i
\(934\) −2.39966e7 −0.900081
\(935\) 0 0
\(936\) 5.33748e6 0.199135
\(937\) 3.58659e7i 1.33454i 0.744814 + 0.667272i \(0.232538\pi\)
−0.744814 + 0.667272i \(0.767462\pi\)
\(938\) 2.23017e7i 0.827621i
\(939\) 5.56838e6 0.206094
\(940\) 0 0
\(941\) −3.19693e7 −1.17695 −0.588476 0.808515i \(-0.700272\pi\)
−0.588476 + 0.808515i \(0.700272\pi\)
\(942\) 4.61143e7i 1.69320i
\(943\) 1.48276e7i 0.542990i
\(944\) −8.01010e6 −0.292555
\(945\) 0 0
\(946\) 2.75050e7 0.999273
\(947\) − 4.71846e7i − 1.70972i −0.518858 0.854861i \(-0.673643\pi\)
0.518858 0.854861i \(-0.326357\pi\)
\(948\) − 7.88908e6i − 0.285106i
\(949\) 1.81318e6 0.0653544
\(950\) 0 0
\(951\) −3.21261e7 −1.15188
\(952\) 7.77829e6i 0.278158i
\(953\) 1.65226e6i 0.0589315i 0.999566 + 0.0294657i \(0.00938059\pi\)
−0.999566 + 0.0294657i \(0.990619\pi\)
\(954\) −1.64497e7 −0.585178
\(955\) 0 0
\(956\) −5.17108e6 −0.182994
\(957\) 1.91112e7i 0.674541i
\(958\) 9.89295e6i 0.348267i
\(959\) 3.96512e7 1.39223
\(960\) 0 0
\(961\) 5.39316e7 1.88380
\(962\) 1.43606e6i 0.0500306i
\(963\) 3.75855e7i 1.30603i
\(964\) −409351. −0.0141874
\(965\) 0 0
\(966\) −1.53684e7 −0.529889
\(967\) 3.23040e7i 1.11094i 0.831537 + 0.555470i \(0.187462\pi\)
−0.831537 + 0.555470i \(0.812538\pi\)
\(968\) − 1.46914e7i − 0.503934i
\(969\) −1.69151e7 −0.578717
\(970\) 0 0
\(971\) 1.15927e7 0.394582 0.197291 0.980345i \(-0.436786\pi\)
0.197291 + 0.980345i \(0.436786\pi\)
\(972\) − 3.48371e6i − 0.118270i
\(973\) 2.05995e7i 0.697549i
\(974\) 8.91377e6 0.301068
\(975\) 0 0
\(976\) −2.88541e6 −0.0969579
\(977\) − 2.58947e7i − 0.867909i −0.900935 0.433954i \(-0.857118\pi\)
0.900935 0.433954i \(-0.142882\pi\)
\(978\) 3.59231e7i 1.20095i
\(979\) 3.40702e7 1.13610
\(980\) 0 0
\(981\) −4.34935e6 −0.144295
\(982\) 7.82184e6i 0.258839i
\(983\) 3.46040e7i 1.14220i 0.820880 + 0.571101i \(0.193483\pi\)
−0.820880 + 0.571101i \(0.806517\pi\)
\(984\) −8.28182e7 −2.72670
\(985\) 0 0
\(986\) −4.21894e6 −0.138201
\(987\) − 8.31698e7i − 2.71752i
\(988\) 628861.i 0.0204957i
\(989\) 1.57321e7 0.511441
\(990\) 0 0
\(991\) −3.71464e7 −1.20152 −0.600762 0.799428i \(-0.705136\pi\)
−0.600762 + 0.799428i \(0.705136\pi\)
\(992\) − 1.40784e7i − 0.454228i
\(993\) − 8.21528e7i − 2.64393i
\(994\) −2.47465e7 −0.794415
\(995\) 0 0
\(996\) −4.36012e6 −0.139268
\(997\) 2.47350e7i 0.788086i 0.919092 + 0.394043i \(0.128924\pi\)
−0.919092 + 0.394043i \(0.871076\pi\)
\(998\) − 3.73573e7i − 1.18727i
\(999\) −1.68558e7 −0.534362
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 25.6.b.b.24.3 4
3.2 odd 2 225.6.b.i.199.2 4
4.3 odd 2 400.6.c.n.49.1 4
5.2 odd 4 25.6.a.d.1.1 yes 2
5.3 odd 4 25.6.a.b.1.2 2
5.4 even 2 inner 25.6.b.b.24.2 4
15.2 even 4 225.6.a.l.1.2 2
15.8 even 4 225.6.a.s.1.1 2
15.14 odd 2 225.6.b.i.199.3 4
20.3 even 4 400.6.a.w.1.2 2
20.7 even 4 400.6.a.o.1.1 2
20.19 odd 2 400.6.c.n.49.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.a.b.1.2 2 5.3 odd 4
25.6.a.d.1.1 yes 2 5.2 odd 4
25.6.b.b.24.2 4 5.4 even 2 inner
25.6.b.b.24.3 4 1.1 even 1 trivial
225.6.a.l.1.2 2 15.2 even 4
225.6.a.s.1.1 2 15.8 even 4
225.6.b.i.199.2 4 3.2 odd 2
225.6.b.i.199.3 4 15.14 odd 2
400.6.a.o.1.1 2 20.7 even 4
400.6.a.w.1.2 2 20.3 even 4
400.6.c.n.49.1 4 4.3 odd 2
400.6.c.n.49.4 4 20.19 odd 2