Properties

Label 25.6.b.b.24.1
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 24.1
Root \(-8.26209i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.6.b.b.24.4

$q$-expansion

\(f(q)\) \(=\) \(q-10.2621i q^{2} -5.52417i q^{3} -73.3104 q^{4} -56.6896 q^{6} -68.9517i q^{7} +423.931i q^{8} +212.483 q^{9} +O(q^{10})\) \(q-10.2621i q^{2} -5.52417i q^{3} -73.3104 q^{4} -56.6896 q^{6} -68.9517i q^{7} +423.931i q^{8} +212.483 q^{9} -486.104 q^{11} +404.980i q^{12} -428.387i q^{13} -707.588 q^{14} +2004.49 q^{16} -1800.64i q^{17} -2180.52i q^{18} +1046.65 q^{19} -380.901 q^{21} +4988.45i q^{22} +686.855i q^{23} +2341.87 q^{24} -4396.14 q^{26} -2516.17i q^{27} +5054.88i q^{28} +1339.03 q^{29} +7990.30 q^{31} -7004.41i q^{32} +2685.33i q^{33} -18478.4 q^{34} -15577.3 q^{36} +1970.64i q^{37} -10740.9i q^{38} -2366.48 q^{39} +10772.2 q^{41} +3908.84i q^{42} +15017.7i q^{43} +35636.5 q^{44} +7048.57 q^{46} -895.337i q^{47} -11073.1i q^{48} +12052.7 q^{49} -9947.07 q^{51} +31405.2i q^{52} -19327.1i q^{53} -25821.2 q^{54} +29230.8 q^{56} -5781.90i q^{57} -13741.3i q^{58} -21193.7 q^{59} -27722.2 q^{61} -81997.1i q^{62} -14651.1i q^{63} -7736.31 q^{64} +27557.0 q^{66} +7719.33i q^{67} +132006. i q^{68} +3794.31 q^{69} -51410.1 q^{71} +90078.4i q^{72} -43776.4i q^{73} +20222.9 q^{74} -76730.7 q^{76} +33517.7i q^{77} +24285.1i q^{78} +6225.68 q^{79} +37733.7 q^{81} -110545. i q^{82} +52949.9i q^{83} +27924.0 q^{84} +154113. q^{86} -7397.05i q^{87} -206075. i q^{88} -44631.2 q^{89} -29538.0 q^{91} -50353.6i q^{92} -44139.8i q^{93} -9188.03 q^{94} -38693.6 q^{96} +148018. i q^{97} -123686. i q^{98} -103289. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 138q^{4} - 382q^{6} - 392q^{9} + O(q^{10}) \) \( 4q - 138q^{4} - 382q^{6} - 392q^{9} - 392q^{11} - 36q^{14} + 2274q^{16} + 6360q^{19} + 5928q^{21} - 5070q^{24} - 9512q^{26} + 7840q^{29} - 2192q^{31} - 40226q^{34} - 34676q^{36} - 8224q^{39} + 55508q^{41} + 73774q^{44} + 4908q^{46} + 23372q^{49} - 35752q^{51} - 7190q^{54} + 108540q^{56} - 23920q^{59} - 48792q^{61} - 87298q^{64} - 22814q^{66} - 36984q^{69} - 174592q^{71} + 82444q^{74} - 135070q^{76} - 130960q^{79} + 92564q^{81} + 84684q^{84} + 497848q^{86} + 145620q^{89} - 41152q^{91} + 243304q^{94} - 156482q^{96} - 443584q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 10.2621i − 1.81410i −0.421025 0.907049i \(-0.638330\pi\)
0.421025 0.907049i \(-0.361670\pi\)
\(3\) − 5.52417i − 0.354376i −0.984177 0.177188i \(-0.943300\pi\)
0.984177 0.177188i \(-0.0567001\pi\)
\(4\) −73.3104 −2.29095
\(5\) 0 0
\(6\) −56.6896 −0.642873
\(7\) − 68.9517i − 0.531863i −0.963992 0.265931i \(-0.914321\pi\)
0.963992 0.265931i \(-0.0856795\pi\)
\(8\) 423.931i 2.34191i
\(9\) 212.483 0.874418
\(10\) 0 0
\(11\) −486.104 −1.21129 −0.605645 0.795735i \(-0.707085\pi\)
−0.605645 + 0.795735i \(0.707085\pi\)
\(12\) 404.980i 0.811858i
\(13\) − 428.387i − 0.703036i −0.936181 0.351518i \(-0.885666\pi\)
0.936181 0.351518i \(-0.114334\pi\)
\(14\) −707.588 −0.964851
\(15\) 0 0
\(16\) 2004.49 1.95751
\(17\) − 1800.64i − 1.51114i −0.655066 0.755571i \(-0.727359\pi\)
0.655066 0.755571i \(-0.272641\pi\)
\(18\) − 2180.52i − 1.58628i
\(19\) 1046.65 0.665149 0.332575 0.943077i \(-0.392083\pi\)
0.332575 + 0.943077i \(0.392083\pi\)
\(20\) 0 0
\(21\) −380.901 −0.188479
\(22\) 4988.45i 2.19740i
\(23\) 686.855i 0.270736i 0.990795 + 0.135368i \(0.0432216\pi\)
−0.990795 + 0.135368i \(0.956778\pi\)
\(24\) 2341.87 0.829917
\(25\) 0 0
\(26\) −4396.14 −1.27538
\(27\) − 2516.17i − 0.664249i
\(28\) 5054.88i 1.21847i
\(29\) 1339.03 0.295663 0.147831 0.989013i \(-0.452771\pi\)
0.147831 + 0.989013i \(0.452771\pi\)
\(30\) 0 0
\(31\) 7990.30 1.49334 0.746670 0.665195i \(-0.231651\pi\)
0.746670 + 0.665195i \(0.231651\pi\)
\(32\) − 7004.41i − 1.20920i
\(33\) 2685.33i 0.429252i
\(34\) −18478.4 −2.74136
\(35\) 0 0
\(36\) −15577.3 −2.00325
\(37\) 1970.64i 0.236648i 0.992975 + 0.118324i \(0.0377522\pi\)
−0.992975 + 0.118324i \(0.962248\pi\)
\(38\) − 10740.9i − 1.20665i
\(39\) −2366.48 −0.249139
\(40\) 0 0
\(41\) 10772.2 1.00079 0.500395 0.865797i \(-0.333188\pi\)
0.500395 + 0.865797i \(0.333188\pi\)
\(42\) 3908.84i 0.341920i
\(43\) 15017.7i 1.23861i 0.785152 + 0.619303i \(0.212585\pi\)
−0.785152 + 0.619303i \(0.787415\pi\)
\(44\) 35636.5 2.77500
\(45\) 0 0
\(46\) 7048.57 0.491141
\(47\) − 895.337i − 0.0591210i −0.999563 0.0295605i \(-0.990589\pi\)
0.999563 0.0295605i \(-0.00941077\pi\)
\(48\) − 11073.1i − 0.693693i
\(49\) 12052.7 0.717122
\(50\) 0 0
\(51\) −9947.07 −0.535513
\(52\) 31405.2i 1.61062i
\(53\) − 19327.1i − 0.945098i −0.881304 0.472549i \(-0.843334\pi\)
0.881304 0.472549i \(-0.156666\pi\)
\(54\) −25821.2 −1.20501
\(55\) 0 0
\(56\) 29230.8 1.24558
\(57\) − 5781.90i − 0.235713i
\(58\) − 13741.3i − 0.536361i
\(59\) −21193.7 −0.792641 −0.396321 0.918112i \(-0.629713\pi\)
−0.396321 + 0.918112i \(0.629713\pi\)
\(60\) 0 0
\(61\) −27722.2 −0.953900 −0.476950 0.878931i \(-0.658258\pi\)
−0.476950 + 0.878931i \(0.658258\pi\)
\(62\) − 81997.1i − 2.70906i
\(63\) − 14651.1i − 0.465070i
\(64\) −7736.31 −0.236093
\(65\) 0 0
\(66\) 27557.0 0.778705
\(67\) 7719.33i 0.210084i 0.994468 + 0.105042i \(0.0334977\pi\)
−0.994468 + 0.105042i \(0.966502\pi\)
\(68\) 132006.i 3.46195i
\(69\) 3794.31 0.0959422
\(70\) 0 0
\(71\) −51410.1 −1.21033 −0.605163 0.796101i \(-0.706892\pi\)
−0.605163 + 0.796101i \(0.706892\pi\)
\(72\) 90078.4i 2.04781i
\(73\) − 43776.4i − 0.961465i −0.876867 0.480732i \(-0.840371\pi\)
0.876867 0.480732i \(-0.159629\pi\)
\(74\) 20222.9 0.429303
\(75\) 0 0
\(76\) −76730.7 −1.52382
\(77\) 33517.7i 0.644240i
\(78\) 24285.1i 0.451963i
\(79\) 6225.68 0.112233 0.0561163 0.998424i \(-0.482128\pi\)
0.0561163 + 0.998424i \(0.482128\pi\)
\(80\) 0 0
\(81\) 37733.7 0.639024
\(82\) − 110545.i − 1.81553i
\(83\) 52949.9i 0.843664i 0.906674 + 0.421832i \(0.138613\pi\)
−0.906674 + 0.421832i \(0.861387\pi\)
\(84\) 27924.0 0.431797
\(85\) 0 0
\(86\) 154113. 2.24695
\(87\) − 7397.05i − 0.104776i
\(88\) − 206075.i − 2.83673i
\(89\) −44631.2 −0.597260 −0.298630 0.954369i \(-0.596530\pi\)
−0.298630 + 0.954369i \(0.596530\pi\)
\(90\) 0 0
\(91\) −29538.0 −0.373919
\(92\) − 50353.6i − 0.620242i
\(93\) − 44139.8i − 0.529204i
\(94\) −9188.03 −0.107251
\(95\) 0 0
\(96\) −38693.6 −0.428510
\(97\) 148018.i 1.59730i 0.601797 + 0.798649i \(0.294452\pi\)
−0.601797 + 0.798649i \(0.705548\pi\)
\(98\) − 123686.i − 1.30093i
\(99\) −103289. −1.05917
\(100\) 0 0
\(101\) 148476. 1.44828 0.724141 0.689652i \(-0.242237\pi\)
0.724141 + 0.689652i \(0.242237\pi\)
\(102\) 102078.i 0.971472i
\(103\) 188391.i 1.74972i 0.484378 + 0.874859i \(0.339046\pi\)
−0.484378 + 0.874859i \(0.660954\pi\)
\(104\) 181607. 1.64645
\(105\) 0 0
\(106\) −198336. −1.71450
\(107\) − 67887.7i − 0.573234i −0.958045 0.286617i \(-0.907469\pi\)
0.958045 0.286617i \(-0.0925307\pi\)
\(108\) 184462.i 1.52176i
\(109\) 219292. 1.76790 0.883949 0.467582i \(-0.154875\pi\)
0.883949 + 0.467582i \(0.154875\pi\)
\(110\) 0 0
\(111\) 10886.2 0.0838625
\(112\) − 138213.i − 1.04112i
\(113\) − 80783.9i − 0.595153i −0.954698 0.297577i \(-0.903822\pi\)
0.954698 0.297577i \(-0.0961784\pi\)
\(114\) −59334.4 −0.427606
\(115\) 0 0
\(116\) −98165.1 −0.677348
\(117\) − 91025.1i − 0.614747i
\(118\) 217492.i 1.43793i
\(119\) −124157. −0.803721
\(120\) 0 0
\(121\) 75246.5 0.467221
\(122\) 284487.i 1.73047i
\(123\) − 59507.3i − 0.354656i
\(124\) −585772. −3.42117
\(125\) 0 0
\(126\) −150351. −0.843683
\(127\) − 161301.i − 0.887417i −0.896171 0.443708i \(-0.853663\pi\)
0.896171 0.443708i \(-0.146337\pi\)
\(128\) − 144750.i − 0.780899i
\(129\) 82960.5 0.438932
\(130\) 0 0
\(131\) −193006. −0.982636 −0.491318 0.870980i \(-0.663485\pi\)
−0.491318 + 0.870980i \(0.663485\pi\)
\(132\) − 196862.i − 0.983395i
\(133\) − 72168.5i − 0.353768i
\(134\) 79216.4 0.381113
\(135\) 0 0
\(136\) 763349. 3.53896
\(137\) − 250340.i − 1.13954i −0.821806 0.569768i \(-0.807033\pi\)
0.821806 0.569768i \(-0.192967\pi\)
\(138\) − 38937.5i − 0.174049i
\(139\) 218650. 0.959871 0.479935 0.877304i \(-0.340660\pi\)
0.479935 + 0.877304i \(0.340660\pi\)
\(140\) 0 0
\(141\) −4946.00 −0.0209511
\(142\) 527575.i 2.19565i
\(143\) 208241.i 0.851580i
\(144\) 425920. 1.71168
\(145\) 0 0
\(146\) −449238. −1.74419
\(147\) − 66581.1i − 0.254131i
\(148\) − 144469.i − 0.542150i
\(149\) 38740.0 0.142953 0.0714766 0.997442i \(-0.477229\pi\)
0.0714766 + 0.997442i \(0.477229\pi\)
\(150\) 0 0
\(151\) −154945. −0.553013 −0.276507 0.961012i \(-0.589177\pi\)
−0.276507 + 0.961012i \(0.589177\pi\)
\(152\) 443709.i 1.55772i
\(153\) − 382607.i − 1.32137i
\(154\) 343962. 1.16871
\(155\) 0 0
\(156\) 173488. 0.570766
\(157\) 344442.i 1.11523i 0.830098 + 0.557617i \(0.188284\pi\)
−0.830098 + 0.557617i \(0.811716\pi\)
\(158\) − 63888.5i − 0.203601i
\(159\) −106766. −0.334920
\(160\) 0 0
\(161\) 47359.8 0.143994
\(162\) − 387227.i − 1.15925i
\(163\) 366203.i 1.07957i 0.841801 + 0.539787i \(0.181495\pi\)
−0.841801 + 0.539787i \(0.818505\pi\)
\(164\) −789712. −2.29276
\(165\) 0 0
\(166\) 543376. 1.53049
\(167\) − 249272.i − 0.691644i −0.938300 0.345822i \(-0.887600\pi\)
0.938300 0.345822i \(-0.112400\pi\)
\(168\) − 161476.i − 0.441402i
\(169\) 187778. 0.505740
\(170\) 0 0
\(171\) 222397. 0.581618
\(172\) − 1.10096e6i − 2.83758i
\(173\) − 61460.1i − 0.156127i −0.996948 0.0780635i \(-0.975126\pi\)
0.996948 0.0780635i \(-0.0248737\pi\)
\(174\) −75909.2 −0.190073
\(175\) 0 0
\(176\) −974389. −2.37111
\(177\) 117078.i 0.280893i
\(178\) 458009.i 1.08349i
\(179\) −606803. −1.41552 −0.707759 0.706454i \(-0.750294\pi\)
−0.707759 + 0.706454i \(0.750294\pi\)
\(180\) 0 0
\(181\) 153684. 0.348685 0.174343 0.984685i \(-0.444220\pi\)
0.174343 + 0.984685i \(0.444220\pi\)
\(182\) 303121.i 0.678325i
\(183\) 153142.i 0.338039i
\(184\) −291179. −0.634039
\(185\) 0 0
\(186\) −452966. −0.960027
\(187\) 875301.i 1.83043i
\(188\) 65637.6i 0.135443i
\(189\) −173494. −0.353289
\(190\) 0 0
\(191\) 182315. 0.361608 0.180804 0.983519i \(-0.442130\pi\)
0.180804 + 0.983519i \(0.442130\pi\)
\(192\) 42736.7i 0.0836659i
\(193\) − 102080.i − 0.197265i −0.995124 0.0986323i \(-0.968553\pi\)
0.995124 0.0986323i \(-0.0314468\pi\)
\(194\) 1.51898e6 2.89766
\(195\) 0 0
\(196\) −883586. −1.64289
\(197\) 404656.i 0.742882i 0.928456 + 0.371441i \(0.121136\pi\)
−0.928456 + 0.371441i \(0.878864\pi\)
\(198\) 1.05996e6i 1.92144i
\(199\) 167297. 0.299472 0.149736 0.988726i \(-0.452158\pi\)
0.149736 + 0.988726i \(0.452158\pi\)
\(200\) 0 0
\(201\) 42642.9 0.0744487
\(202\) − 1.52367e6i − 2.62732i
\(203\) − 92328.5i − 0.157252i
\(204\) 729224. 1.22683
\(205\) 0 0
\(206\) 1.93329e6 3.17416
\(207\) 145945.i 0.236736i
\(208\) − 858695.i − 1.37620i
\(209\) −508783. −0.805688
\(210\) 0 0
\(211\) −460778. −0.712502 −0.356251 0.934390i \(-0.615945\pi\)
−0.356251 + 0.934390i \(0.615945\pi\)
\(212\) 1.41688e6i 2.16517i
\(213\) 283998.i 0.428911i
\(214\) −696670. −1.03990
\(215\) 0 0
\(216\) 1.06668e6 1.55561
\(217\) − 550944.i − 0.794252i
\(218\) − 2.25040e6i − 3.20714i
\(219\) −241829. −0.340720
\(220\) 0 0
\(221\) −771372. −1.06239
\(222\) − 111715.i − 0.152135i
\(223\) − 1.08298e6i − 1.45834i −0.684330 0.729172i \(-0.739905\pi\)
0.684330 0.729172i \(-0.260095\pi\)
\(224\) −482966. −0.643126
\(225\) 0 0
\(226\) −829012. −1.07967
\(227\) − 412201.i − 0.530938i −0.964119 0.265469i \(-0.914473\pi\)
0.964119 0.265469i \(-0.0855269\pi\)
\(228\) 423874.i 0.540007i
\(229\) 433163. 0.545836 0.272918 0.962037i \(-0.412011\pi\)
0.272918 + 0.962037i \(0.412011\pi\)
\(230\) 0 0
\(231\) 185158. 0.228303
\(232\) 567658.i 0.692416i
\(233\) 760097.i 0.917232i 0.888635 + 0.458616i \(0.151655\pi\)
−0.888635 + 0.458616i \(0.848345\pi\)
\(234\) −934108. −1.11521
\(235\) 0 0
\(236\) 1.55372e6 1.81590
\(237\) − 34391.7i − 0.0397725i
\(238\) 1.27411e6i 1.45803i
\(239\) 988624. 1.11953 0.559766 0.828651i \(-0.310891\pi\)
0.559766 + 0.828651i \(0.310891\pi\)
\(240\) 0 0
\(241\) −358878. −0.398020 −0.199010 0.979997i \(-0.563773\pi\)
−0.199010 + 0.979997i \(0.563773\pi\)
\(242\) − 772186.i − 0.847585i
\(243\) − 819877.i − 0.890703i
\(244\) 2.03232e6 2.18534
\(245\) 0 0
\(246\) −610669. −0.643381
\(247\) − 448373.i − 0.467624i
\(248\) 3.38734e6i 3.49727i
\(249\) 292504. 0.298974
\(250\) 0 0
\(251\) −851049. −0.852649 −0.426324 0.904570i \(-0.640192\pi\)
−0.426324 + 0.904570i \(0.640192\pi\)
\(252\) 1.07408e6i 1.06545i
\(253\) − 333883.i − 0.327939i
\(254\) −1.65528e6 −1.60986
\(255\) 0 0
\(256\) −1.73300e6 −1.65272
\(257\) 76358.4i 0.0721147i 0.999350 + 0.0360574i \(0.0114799\pi\)
−0.999350 + 0.0360574i \(0.988520\pi\)
\(258\) − 851348.i − 0.796266i
\(259\) 135879. 0.125864
\(260\) 0 0
\(261\) 284522. 0.258533
\(262\) 1.98064e6i 1.78260i
\(263\) 1.19420e6i 1.06460i 0.846555 + 0.532301i \(0.178673\pi\)
−0.846555 + 0.532301i \(0.821327\pi\)
\(264\) −1.13839e6 −1.00527
\(265\) 0 0
\(266\) −740600. −0.641770
\(267\) 246551.i 0.211655i
\(268\) − 565907.i − 0.481292i
\(269\) −1.02930e6 −0.867286 −0.433643 0.901085i \(-0.642772\pi\)
−0.433643 + 0.901085i \(0.642772\pi\)
\(270\) 0 0
\(271\) 2.12144e6 1.75472 0.877359 0.479834i \(-0.159303\pi\)
0.877359 + 0.479834i \(0.159303\pi\)
\(272\) − 3.60937e6i − 2.95807i
\(273\) 163173.i 0.132508i
\(274\) −2.56901e6 −2.06723
\(275\) 0 0
\(276\) −278162. −0.219799
\(277\) − 1.85145e6i − 1.44982i −0.688845 0.724908i \(-0.741882\pi\)
0.688845 0.724908i \(-0.258118\pi\)
\(278\) − 2.24381e6i − 1.74130i
\(279\) 1.69781e6 1.30580
\(280\) 0 0
\(281\) 90653.2 0.0684884 0.0342442 0.999413i \(-0.489098\pi\)
0.0342442 + 0.999413i \(0.489098\pi\)
\(282\) 50756.3i 0.0380073i
\(283\) − 929308.i − 0.689753i −0.938648 0.344877i \(-0.887921\pi\)
0.938648 0.344877i \(-0.112079\pi\)
\(284\) 3.76890e6 2.77280
\(285\) 0 0
\(286\) 2.13698e6 1.54485
\(287\) − 742759.i − 0.532283i
\(288\) − 1.48832e6i − 1.05734i
\(289\) −1.82246e6 −1.28355
\(290\) 0 0
\(291\) 817679. 0.566044
\(292\) 3.20927e6i 2.20267i
\(293\) 2.72733e6i 1.85596i 0.372632 + 0.927979i \(0.378455\pi\)
−0.372632 + 0.927979i \(0.621545\pi\)
\(294\) −683261. −0.461018
\(295\) 0 0
\(296\) −835417. −0.554209
\(297\) 1.22312e6i 0.804597i
\(298\) − 397553.i − 0.259331i
\(299\) 294240. 0.190337
\(300\) 0 0
\(301\) 1.03550e6 0.658768
\(302\) 1.59006e6i 1.00322i
\(303\) − 820208.i − 0.513236i
\(304\) 2.09800e6 1.30203
\(305\) 0 0
\(306\) −3.92635e6 −2.39709
\(307\) 2.29648e6i 1.39064i 0.718698 + 0.695322i \(0.244738\pi\)
−0.718698 + 0.695322i \(0.755262\pi\)
\(308\) − 2.45720e6i − 1.47592i
\(309\) 1.04071e6 0.620058
\(310\) 0 0
\(311\) 984847. 0.577388 0.288694 0.957421i \(-0.406779\pi\)
0.288694 + 0.957421i \(0.406779\pi\)
\(312\) − 1.00323e6i − 0.583462i
\(313\) 2.06650e6i 1.19227i 0.802884 + 0.596135i \(0.203298\pi\)
−0.802884 + 0.596135i \(0.796702\pi\)
\(314\) 3.53469e6 2.02315
\(315\) 0 0
\(316\) −456407. −0.257119
\(317\) − 1.14349e6i − 0.639125i −0.947565 0.319563i \(-0.896464\pi\)
0.947565 0.319563i \(-0.103536\pi\)
\(318\) 1.09564e6i 0.607578i
\(319\) −650910. −0.358133
\(320\) 0 0
\(321\) −375024. −0.203140
\(322\) − 486010.i − 0.261220i
\(323\) − 1.88465e6i − 1.00514i
\(324\) −2.76628e6 −1.46397
\(325\) 0 0
\(326\) 3.75801e6 1.95845
\(327\) − 1.21141e6i − 0.626501i
\(328\) 4.56666e6i 2.34376i
\(329\) −61735.0 −0.0314443
\(330\) 0 0
\(331\) 205230. 0.102961 0.0514804 0.998674i \(-0.483606\pi\)
0.0514804 + 0.998674i \(0.483606\pi\)
\(332\) − 3.88178e6i − 1.93279i
\(333\) 418729.i 0.206929i
\(334\) −2.55805e6 −1.25471
\(335\) 0 0
\(336\) −763511. −0.368950
\(337\) − 488213.i − 0.234172i −0.993122 0.117086i \(-0.962645\pi\)
0.993122 0.117086i \(-0.0373553\pi\)
\(338\) − 1.92699e6i − 0.917462i
\(339\) −446265. −0.210908
\(340\) 0 0
\(341\) −3.88412e6 −1.80887
\(342\) − 2.28225e6i − 1.05511i
\(343\) − 1.98992e6i − 0.913273i
\(344\) −6.36648e6 −2.90070
\(345\) 0 0
\(346\) −630709. −0.283230
\(347\) 3.82809e6i 1.70670i 0.521336 + 0.853351i \(0.325434\pi\)
−0.521336 + 0.853351i \(0.674566\pi\)
\(348\) 542281.i 0.240036i
\(349\) −1.45476e6 −0.639333 −0.319667 0.947530i \(-0.603571\pi\)
−0.319667 + 0.947530i \(0.603571\pi\)
\(350\) 0 0
\(351\) −1.07789e6 −0.466991
\(352\) 3.40487e6i 1.46469i
\(353\) − 778492.i − 0.332520i −0.986082 0.166260i \(-0.946831\pi\)
0.986082 0.166260i \(-0.0531690\pi\)
\(354\) 1.20146e6 0.509567
\(355\) 0 0
\(356\) 3.27193e6 1.36829
\(357\) 685867.i 0.284819i
\(358\) 6.22707e6i 2.56789i
\(359\) 2.12510e6 0.870247 0.435124 0.900371i \(-0.356705\pi\)
0.435124 + 0.900371i \(0.356705\pi\)
\(360\) 0 0
\(361\) −1.38061e6 −0.557577
\(362\) − 1.57712e6i − 0.632549i
\(363\) − 415675.i − 0.165572i
\(364\) 2.16544e6 0.856630
\(365\) 0 0
\(366\) 1.57156e6 0.613236
\(367\) − 4.10801e6i − 1.59208i −0.605242 0.796042i \(-0.706923\pi\)
0.605242 0.796042i \(-0.293077\pi\)
\(368\) 1.37679e6i 0.529967i
\(369\) 2.28891e6 0.875109
\(370\) 0 0
\(371\) −1.33263e6 −0.502662
\(372\) 3.23591e6i 1.21238i
\(373\) − 4.54570e6i − 1.69172i −0.533405 0.845860i \(-0.679088\pi\)
0.533405 0.845860i \(-0.320912\pi\)
\(374\) 8.98241e6 3.32058
\(375\) 0 0
\(376\) 379561. 0.138456
\(377\) − 573624.i − 0.207861i
\(378\) 1.78041e6i 0.640901i
\(379\) −1.40554e6 −0.502626 −0.251313 0.967906i \(-0.580862\pi\)
−0.251313 + 0.967906i \(0.580862\pi\)
\(380\) 0 0
\(381\) −891054. −0.314479
\(382\) − 1.87093e6i − 0.655993i
\(383\) 4.64417e6i 1.61775i 0.587982 + 0.808874i \(0.299922\pi\)
−0.587982 + 0.808874i \(0.700078\pi\)
\(384\) −799627. −0.276732
\(385\) 0 0
\(386\) −1.04756e6 −0.357857
\(387\) 3.19102e6i 1.08306i
\(388\) − 1.08513e7i − 3.65933i
\(389\) −3.53606e6 −1.18480 −0.592400 0.805644i \(-0.701819\pi\)
−0.592400 + 0.805644i \(0.701819\pi\)
\(390\) 0 0
\(391\) 1.23678e6 0.409120
\(392\) 5.10950e6i 1.67944i
\(393\) 1.06620e6i 0.348223i
\(394\) 4.15261e6 1.34766
\(395\) 0 0
\(396\) 7.57217e6 2.42651
\(397\) 2.95611e6i 0.941336i 0.882310 + 0.470668i \(0.155987\pi\)
−0.882310 + 0.470668i \(0.844013\pi\)
\(398\) − 1.71682e6i − 0.543272i
\(399\) −398671. −0.125367
\(400\) 0 0
\(401\) −799254. −0.248213 −0.124106 0.992269i \(-0.539606\pi\)
−0.124106 + 0.992269i \(0.539606\pi\)
\(402\) − 437605.i − 0.135057i
\(403\) − 3.42294e6i − 1.04987i
\(404\) −1.08848e7 −3.31794
\(405\) 0 0
\(406\) −947484. −0.285270
\(407\) − 957937.i − 0.286649i
\(408\) − 4.21687e6i − 1.25412i
\(409\) −898422. −0.265566 −0.132783 0.991145i \(-0.542391\pi\)
−0.132783 + 0.991145i \(0.542391\pi\)
\(410\) 0 0
\(411\) −1.38292e6 −0.403824
\(412\) − 1.38111e7i − 4.00852i
\(413\) 1.46134e6i 0.421576i
\(414\) 1.49770e6 0.429462
\(415\) 0 0
\(416\) −3.00060e6 −0.850108
\(417\) − 1.20786e6i − 0.340155i
\(418\) 5.22118e6i 1.46160i
\(419\) −2.31259e6 −0.643523 −0.321761 0.946821i \(-0.604275\pi\)
−0.321761 + 0.946821i \(0.604275\pi\)
\(420\) 0 0
\(421\) 4.43296e6 1.21896 0.609478 0.792803i \(-0.291379\pi\)
0.609478 + 0.792803i \(0.291379\pi\)
\(422\) 4.72855e6i 1.29255i
\(423\) − 190244.i − 0.0516965i
\(424\) 8.19336e6 2.21334
\(425\) 0 0
\(426\) 2.91442e6 0.778086
\(427\) 1.91149e6i 0.507344i
\(428\) 4.97688e6i 1.31325i
\(429\) 1.15036e6 0.301780
\(430\) 0 0
\(431\) 5.97999e6 1.55063 0.775314 0.631576i \(-0.217592\pi\)
0.775314 + 0.631576i \(0.217592\pi\)
\(432\) − 5.04363e6i − 1.30027i
\(433\) 2.06419e6i 0.529089i 0.964373 + 0.264545i \(0.0852217\pi\)
−0.964373 + 0.264545i \(0.914778\pi\)
\(434\) −5.65384e6 −1.44085
\(435\) 0 0
\(436\) −1.60764e7 −4.05017
\(437\) 718899.i 0.180080i
\(438\) 2.48167e6i 0.618099i
\(439\) 4.09148e6 1.01326 0.506628 0.862165i \(-0.330892\pi\)
0.506628 + 0.862165i \(0.330892\pi\)
\(440\) 0 0
\(441\) 2.56099e6 0.627064
\(442\) 7.91589e6i 1.92728i
\(443\) 2.75822e6i 0.667759i 0.942616 + 0.333879i \(0.108358\pi\)
−0.942616 + 0.333879i \(0.891642\pi\)
\(444\) −798070. −0.192125
\(445\) 0 0
\(446\) −1.11137e7 −2.64558
\(447\) − 214006.i − 0.0506592i
\(448\) 533431.i 0.125569i
\(449\) 3.76648e6 0.881698 0.440849 0.897581i \(-0.354677\pi\)
0.440849 + 0.897581i \(0.354677\pi\)
\(450\) 0 0
\(451\) −5.23640e6 −1.21225
\(452\) 5.92231e6i 1.36347i
\(453\) 855944.i 0.195975i
\(454\) −4.23004e6 −0.963174
\(455\) 0 0
\(456\) 2.45113e6 0.552019
\(457\) − 480604.i − 0.107646i −0.998550 0.0538229i \(-0.982859\pi\)
0.998550 0.0538229i \(-0.0171407\pi\)
\(458\) − 4.44515e6i − 0.990200i
\(459\) −4.53073e6 −1.00377
\(460\) 0 0
\(461\) 4.52514e6 0.991699 0.495849 0.868409i \(-0.334857\pi\)
0.495849 + 0.868409i \(0.334857\pi\)
\(462\) − 1.90010e6i − 0.414164i
\(463\) 7.39975e6i 1.60422i 0.597175 + 0.802111i \(0.296290\pi\)
−0.597175 + 0.802111i \(0.703710\pi\)
\(464\) 2.68407e6 0.578761
\(465\) 0 0
\(466\) 7.80018e6 1.66395
\(467\) − 1.84711e6i − 0.391923i −0.980612 0.195962i \(-0.937217\pi\)
0.980612 0.195962i \(-0.0627828\pi\)
\(468\) 6.67309e6i 1.40836i
\(469\) 532261. 0.111736
\(470\) 0 0
\(471\) 1.90276e6 0.395212
\(472\) − 8.98467e6i − 1.85630i
\(473\) − 7.30018e6i − 1.50031i
\(474\) −352931. −0.0721513
\(475\) 0 0
\(476\) 9.10203e6 1.84128
\(477\) − 4.10669e6i − 0.826410i
\(478\) − 1.01453e7i − 2.03094i
\(479\) 3.05088e6 0.607555 0.303778 0.952743i \(-0.401752\pi\)
0.303778 + 0.952743i \(0.401752\pi\)
\(480\) 0 0
\(481\) 844197. 0.166372
\(482\) 3.68284e6i 0.722047i
\(483\) − 261624.i − 0.0510281i
\(484\) −5.51635e6 −1.07038
\(485\) 0 0
\(486\) −8.41365e6 −1.61582
\(487\) − 7.28136e6i − 1.39120i −0.718429 0.695601i \(-0.755138\pi\)
0.718429 0.695601i \(-0.244862\pi\)
\(488\) − 1.17523e7i − 2.23395i
\(489\) 2.02297e6 0.382575
\(490\) 0 0
\(491\) −6.60475e6 −1.23638 −0.618191 0.786028i \(-0.712134\pi\)
−0.618191 + 0.786028i \(0.712134\pi\)
\(492\) 4.36251e6i 0.812500i
\(493\) − 2.41112e6i − 0.446788i
\(494\) −4.60124e6 −0.848316
\(495\) 0 0
\(496\) 1.60164e7 2.92322
\(497\) 3.54481e6i 0.643727i
\(498\) − 3.00171e6i − 0.542369i
\(499\) −4.87006e6 −0.875555 −0.437777 0.899083i \(-0.644234\pi\)
−0.437777 + 0.899083i \(0.644234\pi\)
\(500\) 0 0
\(501\) −1.37702e6 −0.245102
\(502\) 8.73354e6i 1.54679i
\(503\) − 1.16752e6i − 0.205753i −0.994694 0.102876i \(-0.967195\pi\)
0.994694 0.102876i \(-0.0328046\pi\)
\(504\) 6.21105e6 1.08915
\(505\) 0 0
\(506\) −3.42634e6 −0.594914
\(507\) − 1.03732e6i − 0.179222i
\(508\) 1.18250e7i 2.03303i
\(509\) 7.41468e6 1.26852 0.634261 0.773119i \(-0.281305\pi\)
0.634261 + 0.773119i \(0.281305\pi\)
\(510\) 0 0
\(511\) −3.01846e6 −0.511367
\(512\) 1.31522e7i 2.21730i
\(513\) − 2.63356e6i − 0.441824i
\(514\) 783596. 0.130823
\(515\) 0 0
\(516\) −6.08187e6 −1.00557
\(517\) 435227.i 0.0716127i
\(518\) − 1.39440e6i − 0.228330i
\(519\) −339516. −0.0553277
\(520\) 0 0
\(521\) −811897. −0.131041 −0.0655204 0.997851i \(-0.520871\pi\)
−0.0655204 + 0.997851i \(0.520871\pi\)
\(522\) − 2.91979e6i − 0.469003i
\(523\) 5.06828e6i 0.810226i 0.914267 + 0.405113i \(0.132768\pi\)
−0.914267 + 0.405113i \(0.867232\pi\)
\(524\) 1.41494e7 2.25117
\(525\) 0 0
\(526\) 1.22550e7 1.93129
\(527\) − 1.43877e7i − 2.25665i
\(528\) 5.38270e6i 0.840263i
\(529\) 5.96457e6 0.926702
\(530\) 0 0
\(531\) −4.50331e6 −0.693099
\(532\) 5.29071e6i 0.810465i
\(533\) − 4.61465e6i − 0.703592i
\(534\) 2.53012e6 0.383962
\(535\) 0 0
\(536\) −3.27247e6 −0.491998
\(537\) 3.35209e6i 0.501626i
\(538\) 1.05628e7i 1.57334i
\(539\) −5.85886e6 −0.868642
\(540\) 0 0
\(541\) 1.52830e6 0.224499 0.112250 0.993680i \(-0.464194\pi\)
0.112250 + 0.993680i \(0.464194\pi\)
\(542\) − 2.17704e7i − 3.18323i
\(543\) − 848980.i − 0.123566i
\(544\) −1.26124e7 −1.82727
\(545\) 0 0
\(546\) 1.67449e6 0.240382
\(547\) 1.23234e7i 1.76101i 0.474036 + 0.880506i \(0.342797\pi\)
−0.474036 + 0.880506i \(0.657203\pi\)
\(548\) 1.83525e7i 2.61062i
\(549\) −5.89050e6 −0.834107
\(550\) 0 0
\(551\) 1.40150e6 0.196660
\(552\) 1.60853e6i 0.224688i
\(553\) − 429271.i − 0.0596923i
\(554\) −1.89998e7 −2.63011
\(555\) 0 0
\(556\) −1.60293e7 −2.19902
\(557\) 4.08606e6i 0.558042i 0.960285 + 0.279021i \(0.0900099\pi\)
−0.960285 + 0.279021i \(0.909990\pi\)
\(558\) − 1.74230e7i − 2.36885i
\(559\) 6.43339e6 0.870784
\(560\) 0 0
\(561\) 4.83531e6 0.648661
\(562\) − 930291.i − 0.124245i
\(563\) 24160.3i 0.00321241i 0.999999 + 0.00160621i \(0.000511272\pi\)
−0.999999 + 0.00160621i \(0.999489\pi\)
\(564\) 362593. 0.0479979
\(565\) 0 0
\(566\) −9.53664e6 −1.25128
\(567\) − 2.60180e6i − 0.339873i
\(568\) − 2.17943e7i − 2.83448i
\(569\) −1.42000e7 −1.83869 −0.919344 0.393454i \(-0.871280\pi\)
−0.919344 + 0.393454i \(0.871280\pi\)
\(570\) 0 0
\(571\) −767642. −0.0985300 −0.0492650 0.998786i \(-0.515688\pi\)
−0.0492650 + 0.998786i \(0.515688\pi\)
\(572\) − 1.52662e7i − 1.95093i
\(573\) − 1.00714e6i − 0.128145i
\(574\) −7.62225e6 −0.965614
\(575\) 0 0
\(576\) −1.64384e6 −0.206444
\(577\) 1.51488e6i 0.189426i 0.995505 + 0.0947129i \(0.0301933\pi\)
−0.995505 + 0.0947129i \(0.969807\pi\)
\(578\) 1.87023e7i 2.32849i
\(579\) −563910. −0.0699059
\(580\) 0 0
\(581\) 3.65098e6 0.448714
\(582\) − 8.39109e6i − 1.02686i
\(583\) 9.39498e6i 1.14479i
\(584\) 1.85582e7 2.25167
\(585\) 0 0
\(586\) 2.79881e7 3.36689
\(587\) 1.28973e7i 1.54491i 0.635070 + 0.772455i \(0.280971\pi\)
−0.635070 + 0.772455i \(0.719029\pi\)
\(588\) 4.88109e6i 0.582201i
\(589\) 8.36307e6 0.993294
\(590\) 0 0
\(591\) 2.23539e6 0.263260
\(592\) 3.95012e6i 0.463240i
\(593\) − 5.43125e6i − 0.634254i −0.948383 0.317127i \(-0.897282\pi\)
0.948383 0.317127i \(-0.102718\pi\)
\(594\) 1.25518e7 1.45962
\(595\) 0 0
\(596\) −2.84004e6 −0.327499
\(597\) − 924180.i − 0.106126i
\(598\) − 3.01951e6i − 0.345290i
\(599\) −3.92217e6 −0.446642 −0.223321 0.974745i \(-0.571690\pi\)
−0.223321 + 0.974745i \(0.571690\pi\)
\(600\) 0 0
\(601\) −5.64824e6 −0.637863 −0.318931 0.947778i \(-0.603324\pi\)
−0.318931 + 0.947778i \(0.603324\pi\)
\(602\) − 1.06264e7i − 1.19507i
\(603\) 1.64023e6i 0.183701i
\(604\) 1.13591e7 1.26693
\(605\) 0 0
\(606\) −8.41704e6 −0.931061
\(607\) 1.07148e7i 1.18035i 0.807274 + 0.590177i \(0.200942\pi\)
−0.807274 + 0.590177i \(0.799058\pi\)
\(608\) − 7.33119e6i − 0.804296i
\(609\) −510039. −0.0557263
\(610\) 0 0
\(611\) −383551. −0.0415642
\(612\) 2.80491e7i 3.02719i
\(613\) 4.08748e6i 0.439344i 0.975574 + 0.219672i \(0.0704987\pi\)
−0.975574 + 0.219672i \(0.929501\pi\)
\(614\) 2.35666e7 2.52276
\(615\) 0 0
\(616\) −1.42092e7 −1.50875
\(617\) 7.83395e6i 0.828453i 0.910174 + 0.414227i \(0.135948\pi\)
−0.910174 + 0.414227i \(0.864052\pi\)
\(618\) − 1.06798e7i − 1.12485i
\(619\) −1.23423e7 −1.29470 −0.647352 0.762191i \(-0.724124\pi\)
−0.647352 + 0.762191i \(0.724124\pi\)
\(620\) 0 0
\(621\) 1.72824e6 0.179836
\(622\) − 1.01066e7i − 1.04744i
\(623\) 3.07739e6i 0.317660i
\(624\) −4.74358e6 −0.487691
\(625\) 0 0
\(626\) 2.12066e7 2.16290
\(627\) 2.81061e6i 0.285516i
\(628\) − 2.52512e7i − 2.55495i
\(629\) 3.54842e6 0.357609
\(630\) 0 0
\(631\) −1.31578e6 −0.131556 −0.0657780 0.997834i \(-0.520953\pi\)
−0.0657780 + 0.997834i \(0.520953\pi\)
\(632\) 2.63926e6i 0.262839i
\(633\) 2.54542e6i 0.252494i
\(634\) −1.17346e7 −1.15944
\(635\) 0 0
\(636\) 7.82708e6 0.767285
\(637\) − 5.16320e6i − 0.504163i
\(638\) 6.67969e6i 0.649688i
\(639\) −1.09238e7 −1.05833
\(640\) 0 0
\(641\) 6.55744e6 0.630360 0.315180 0.949032i \(-0.397935\pi\)
0.315180 + 0.949032i \(0.397935\pi\)
\(642\) 3.84852e6i 0.368516i
\(643\) − 4.69954e6i − 0.448258i −0.974559 0.224129i \(-0.928046\pi\)
0.974559 0.224129i \(-0.0719537\pi\)
\(644\) −3.47197e6 −0.329884
\(645\) 0 0
\(646\) −1.93405e7 −1.82341
\(647\) − 2.05827e7i − 1.93305i −0.256580 0.966523i \(-0.582596\pi\)
0.256580 0.966523i \(-0.417404\pi\)
\(648\) 1.59965e7i 1.49654i
\(649\) 1.03023e7 0.960117
\(650\) 0 0
\(651\) −3.04351e6 −0.281464
\(652\) − 2.68465e7i − 2.47325i
\(653\) − 1.42466e7i − 1.30746i −0.756727 0.653731i \(-0.773203\pi\)
0.756727 0.653731i \(-0.226797\pi\)
\(654\) −1.24316e7 −1.13653
\(655\) 0 0
\(656\) 2.15927e7 1.95905
\(657\) − 9.30177e6i − 0.840722i
\(658\) 633530.i 0.0570430i
\(659\) 1.35369e7 1.21425 0.607123 0.794608i \(-0.292324\pi\)
0.607123 + 0.794608i \(0.292324\pi\)
\(660\) 0 0
\(661\) −1.30443e7 −1.16122 −0.580612 0.814180i \(-0.697187\pi\)
−0.580612 + 0.814180i \(0.697187\pi\)
\(662\) − 2.10609e6i − 0.186781i
\(663\) 4.26119e6i 0.376485i
\(664\) −2.24471e7 −1.97579
\(665\) 0 0
\(666\) 4.29703e6 0.375390
\(667\) 919721.i 0.0800464i
\(668\) 1.82743e7i 1.58452i
\(669\) −5.98260e6 −0.516802
\(670\) 0 0
\(671\) 1.34759e7 1.15545
\(672\) 2.66799e6i 0.227908i
\(673\) 4.75951e6i 0.405065i 0.979276 + 0.202532i \(0.0649171\pi\)
−0.979276 + 0.202532i \(0.935083\pi\)
\(674\) −5.01008e6 −0.424810
\(675\) 0 0
\(676\) −1.37661e7 −1.15863
\(677\) 1.51397e7i 1.26954i 0.772701 + 0.634770i \(0.218905\pi\)
−0.772701 + 0.634770i \(0.781095\pi\)
\(678\) 4.57961e6i 0.382608i
\(679\) 1.02061e7 0.849543
\(680\) 0 0
\(681\) −2.27707e6 −0.188152
\(682\) 3.98592e7i 3.28146i
\(683\) − 2.34145e7i − 1.92058i −0.278998 0.960292i \(-0.590002\pi\)
0.278998 0.960292i \(-0.409998\pi\)
\(684\) −1.63040e7 −1.33246
\(685\) 0 0
\(686\) −2.04208e7 −1.65677
\(687\) − 2.39287e6i − 0.193431i
\(688\) 3.01028e7i 2.42458i
\(689\) −8.27947e6 −0.664438
\(690\) 0 0
\(691\) −1.62194e7 −1.29223 −0.646113 0.763242i \(-0.723606\pi\)
−0.646113 + 0.763242i \(0.723606\pi\)
\(692\) 4.50567e6i 0.357679i
\(693\) 7.12196e6i 0.563334i
\(694\) 3.92841e7 3.09613
\(695\) 0 0
\(696\) 3.13584e6 0.245375
\(697\) − 1.93968e7i − 1.51234i
\(698\) 1.49289e7i 1.15981i
\(699\) 4.19891e6 0.325045
\(700\) 0 0
\(701\) −1.89605e7 −1.45732 −0.728659 0.684876i \(-0.759856\pi\)
−0.728659 + 0.684876i \(0.759856\pi\)
\(702\) 1.10614e7i 0.847167i
\(703\) 2.06258e6i 0.157406i
\(704\) 3.76065e6 0.285977
\(705\) 0 0
\(706\) −7.98895e6 −0.603223
\(707\) − 1.02377e7i − 0.770287i
\(708\) − 8.58301e6i − 0.643512i
\(709\) −128325. −0.00958732 −0.00479366 0.999989i \(-0.501526\pi\)
−0.00479366 + 0.999989i \(0.501526\pi\)
\(710\) 0 0
\(711\) 1.32285e6 0.0981382
\(712\) − 1.89206e7i − 1.39873i
\(713\) 5.48817e6i 0.404300i
\(714\) 7.03843e6 0.516690
\(715\) 0 0
\(716\) 4.44850e7 3.24288
\(717\) − 5.46133e6i − 0.396735i
\(718\) − 2.18079e7i − 1.57871i
\(719\) 2.41874e7 1.74489 0.872444 0.488714i \(-0.162534\pi\)
0.872444 + 0.488714i \(0.162534\pi\)
\(720\) 0 0
\(721\) 1.29899e7 0.930610
\(722\) 1.41680e7i 1.01150i
\(723\) 1.98251e6i 0.141049i
\(724\) −1.12667e7 −0.798821
\(725\) 0 0
\(726\) −4.26569e6 −0.300364
\(727\) 513307.i 0.0360198i 0.999838 + 0.0180099i \(0.00573303\pi\)
−0.999838 + 0.0180099i \(0.994267\pi\)
\(728\) − 1.25221e7i − 0.875685i
\(729\) 4.64015e6 0.323380
\(730\) 0 0
\(731\) 2.70416e7 1.87171
\(732\) − 1.12269e7i − 0.774431i
\(733\) 1.64153e7i 1.12847i 0.825615 + 0.564234i \(0.190828\pi\)
−0.825615 + 0.564234i \(0.809172\pi\)
\(734\) −4.21567e7 −2.88820
\(735\) 0 0
\(736\) 4.81101e6 0.327372
\(737\) − 3.75240e6i − 0.254472i
\(738\) − 2.34890e7i − 1.58753i
\(739\) −1.16112e7 −0.782109 −0.391054 0.920368i \(-0.627890\pi\)
−0.391054 + 0.920368i \(0.627890\pi\)
\(740\) 0 0
\(741\) −2.47689e6 −0.165715
\(742\) 1.36756e7i 0.911879i
\(743\) − 5.72590e6i − 0.380515i −0.981734 0.190257i \(-0.939068\pi\)
0.981734 0.190257i \(-0.0609323\pi\)
\(744\) 1.87122e7 1.23935
\(745\) 0 0
\(746\) −4.66483e7 −3.06894
\(747\) 1.12510e7i 0.737715i
\(748\) − 6.41687e7i − 4.19343i
\(749\) −4.68097e6 −0.304882
\(750\) 0 0
\(751\) 1.15324e7 0.746137 0.373069 0.927804i \(-0.378306\pi\)
0.373069 + 0.927804i \(0.378306\pi\)
\(752\) − 1.79469e6i − 0.115730i
\(753\) 4.70134e6i 0.302158i
\(754\) −5.88658e6 −0.377081
\(755\) 0 0
\(756\) 1.27189e7 0.809368
\(757\) − 8.63293e6i − 0.547544i −0.961795 0.273772i \(-0.911729\pi\)
0.961795 0.273772i \(-0.0882713\pi\)
\(758\) 1.44238e7i 0.911812i
\(759\) −1.84443e6 −0.116214
\(760\) 0 0
\(761\) −3.52622e6 −0.220723 −0.110361 0.993892i \(-0.535201\pi\)
−0.110361 + 0.993892i \(0.535201\pi\)
\(762\) 9.14408e6i 0.570496i
\(763\) − 1.51206e7i − 0.940280i
\(764\) −1.33656e7 −0.828427
\(765\) 0 0
\(766\) 4.76588e7 2.93475
\(767\) 9.07910e6i 0.557255i
\(768\) 9.57341e6i 0.585685i
\(769\) −1.40471e7 −0.856585 −0.428293 0.903640i \(-0.640885\pi\)
−0.428293 + 0.903640i \(0.640885\pi\)
\(770\) 0 0
\(771\) 421817. 0.0255557
\(772\) 7.48356e6i 0.451924i
\(773\) 2.44760e7i 1.47330i 0.676274 + 0.736651i \(0.263594\pi\)
−0.676274 + 0.736651i \(0.736406\pi\)
\(774\) 3.27465e7 1.96477
\(775\) 0 0
\(776\) −6.27496e7 −3.74073
\(777\) − 750619.i − 0.0446033i
\(778\) 3.62873e7i 2.14934i
\(779\) 1.12747e7 0.665675
\(780\) 0 0
\(781\) 2.49907e7 1.46606
\(782\) − 1.26920e7i − 0.742184i
\(783\) − 3.36924e6i − 0.196393i
\(784\) 2.41594e7 1.40377
\(785\) 0 0
\(786\) 1.09414e7 0.631710
\(787\) 4.35977e6i 0.250915i 0.992099 + 0.125458i \(0.0400399\pi\)
−0.992099 + 0.125458i \(0.959960\pi\)
\(788\) − 2.96655e7i − 1.70191i
\(789\) 6.59697e6 0.377270
\(790\) 0 0
\(791\) −5.57019e6 −0.316540
\(792\) − 4.37875e7i − 2.48049i
\(793\) 1.18758e7i 0.670626i
\(794\) 3.03359e7 1.70768
\(795\) 0 0
\(796\) −1.22647e7 −0.686076
\(797\) 1.06887e7i 0.596044i 0.954559 + 0.298022i \(0.0963269\pi\)
−0.954559 + 0.298022i \(0.903673\pi\)
\(798\) 4.09120e6i 0.227428i
\(799\) −1.61218e6 −0.0893403
\(800\) 0 0
\(801\) −9.48339e6 −0.522255
\(802\) 8.20202e6i 0.450282i
\(803\) 2.12799e7i 1.16461i
\(804\) −3.12617e6 −0.170558
\(805\) 0 0
\(806\) −3.51265e7 −1.90457
\(807\) 5.68605e6i 0.307345i
\(808\) 6.29436e7i 3.39175i
\(809\) −9.12014e6 −0.489926 −0.244963 0.969532i \(-0.578776\pi\)
−0.244963 + 0.969532i \(0.578776\pi\)
\(810\) 0 0
\(811\) 5.22575e6 0.278995 0.139497 0.990222i \(-0.455451\pi\)
0.139497 + 0.990222i \(0.455451\pi\)
\(812\) 6.76865e6i 0.360256i
\(813\) − 1.17192e7i − 0.621830i
\(814\) −9.83044e6 −0.520010
\(815\) 0 0
\(816\) −1.99388e7 −1.04827
\(817\) 1.57184e7i 0.823857i
\(818\) 9.21968e6i 0.481762i
\(819\) −6.27633e6 −0.326961
\(820\) 0 0
\(821\) 9.00437e6 0.466225 0.233112 0.972450i \(-0.425109\pi\)
0.233112 + 0.972450i \(0.425109\pi\)
\(822\) 1.41916e7i 0.732576i
\(823\) − 2.78867e7i − 1.43515i −0.696482 0.717574i \(-0.745252\pi\)
0.696482 0.717574i \(-0.254748\pi\)
\(824\) −7.98650e7 −4.09768
\(825\) 0 0
\(826\) 1.49964e7 0.764781
\(827\) 6.64309e6i 0.337758i 0.985637 + 0.168879i \(0.0540148\pi\)
−0.985637 + 0.168879i \(0.945985\pi\)
\(828\) − 1.06993e7i − 0.542351i
\(829\) −2.17030e7 −1.09682 −0.548408 0.836211i \(-0.684766\pi\)
−0.548408 + 0.836211i \(0.684766\pi\)
\(830\) 0 0
\(831\) −1.02277e7 −0.513780
\(832\) 3.31413e6i 0.165982i
\(833\) − 2.17026e7i − 1.08367i
\(834\) −1.23952e7 −0.617075
\(835\) 0 0
\(836\) 3.72991e7 1.84579
\(837\) − 2.01049e7i − 0.991949i
\(838\) 2.37320e7i 1.16741i
\(839\) 1.01238e7 0.496520 0.248260 0.968693i \(-0.420141\pi\)
0.248260 + 0.968693i \(0.420141\pi\)
\(840\) 0 0
\(841\) −1.87181e7 −0.912584
\(842\) − 4.54914e7i − 2.21131i
\(843\) − 500784.i − 0.0242707i
\(844\) 3.37799e7 1.63231
\(845\) 0 0
\(846\) −1.95230e6 −0.0937825
\(847\) − 5.18837e6i − 0.248498i
\(848\) − 3.87409e7i − 1.85003i
\(849\) −5.13366e6 −0.244432
\(850\) 0 0
\(851\) −1.35354e6 −0.0640691
\(852\) − 2.08200e7i − 0.982613i
\(853\) 1.46326e7i 0.688573i 0.938865 + 0.344286i \(0.111879\pi\)
−0.938865 + 0.344286i \(0.888121\pi\)
\(854\) 1.96159e7 0.920371
\(855\) 0 0
\(856\) 2.87797e7 1.34246
\(857\) 5.52218e6i 0.256838i 0.991720 + 0.128419i \(0.0409902\pi\)
−0.991720 + 0.128419i \(0.959010\pi\)
\(858\) − 1.18051e7i − 0.547458i
\(859\) 3.02260e6 0.139765 0.0698824 0.997555i \(-0.477738\pi\)
0.0698824 + 0.997555i \(0.477738\pi\)
\(860\) 0 0
\(861\) −4.10313e6 −0.188628
\(862\) − 6.13672e7i − 2.81299i
\(863\) − 3.06818e7i − 1.40234i −0.712992 0.701172i \(-0.752661\pi\)
0.712992 0.701172i \(-0.247339\pi\)
\(864\) −1.76243e7 −0.803207
\(865\) 0 0
\(866\) 2.11829e7 0.959820
\(867\) 1.00676e7i 0.454860i
\(868\) 4.03900e7i 1.81959i
\(869\) −3.02633e6 −0.135946
\(870\) 0 0
\(871\) 3.30686e6 0.147697
\(872\) 9.29649e7i 4.14026i
\(873\) 3.14514e7i 1.39671i
\(874\) 7.37741e6 0.326682
\(875\) 0 0
\(876\) 1.77286e7 0.780573
\(877\) 5.17607e6i 0.227249i 0.993524 + 0.113624i \(0.0362460\pi\)
−0.993524 + 0.113624i \(0.963754\pi\)
\(878\) − 4.19871e7i − 1.83815i
\(879\) 1.50662e7 0.657707
\(880\) 0 0
\(881\) −4.25937e7 −1.84887 −0.924433 0.381345i \(-0.875461\pi\)
−0.924433 + 0.381345i \(0.875461\pi\)
\(882\) − 2.62811e7i − 1.13756i
\(883\) − 1.72076e7i − 0.742709i −0.928491 0.371354i \(-0.878893\pi\)
0.928491 0.371354i \(-0.121107\pi\)
\(884\) 5.65496e7 2.43388
\(885\) 0 0
\(886\) 2.83051e7 1.21138
\(887\) − 2.53773e6i − 0.108302i −0.998533 0.0541510i \(-0.982755\pi\)
0.998533 0.0541510i \(-0.0172452\pi\)
\(888\) 4.61499e6i 0.196398i
\(889\) −1.11220e7 −0.471984
\(890\) 0 0
\(891\) −1.83425e7 −0.774043
\(892\) 7.93941e7i 3.34100i
\(893\) − 937108.i − 0.0393243i
\(894\) −2.19615e6 −0.0919007
\(895\) 0 0
\(896\) −9.98078e6 −0.415331
\(897\) − 1.62543e6i − 0.0674508i
\(898\) − 3.86519e7i − 1.59949i
\(899\) 1.06993e7 0.441525
\(900\) 0 0
\(901\) −3.48012e7 −1.42818
\(902\) 5.37364e7i 2.19913i
\(903\) − 5.72026e6i − 0.233452i
\(904\) 3.42468e7 1.39380
\(905\) 0 0
\(906\) 8.78377e6 0.355517
\(907\) − 2.60899e7i − 1.05306i −0.850156 0.526531i \(-0.823492\pi\)
0.850156 0.526531i \(-0.176508\pi\)
\(908\) 3.02186e7i 1.21635i
\(909\) 3.15487e7 1.26640
\(910\) 0 0
\(911\) 1.44818e7 0.578130 0.289065 0.957309i \(-0.406656\pi\)
0.289065 + 0.957309i \(0.406656\pi\)
\(912\) − 1.15897e7i − 0.461409i
\(913\) − 2.57392e7i − 1.02192i
\(914\) −4.93200e6 −0.195280
\(915\) 0 0
\(916\) −3.17553e7 −1.25048
\(917\) 1.33081e7i 0.522627i
\(918\) 4.64947e7i 1.82095i
\(919\) −4.61041e7 −1.80074 −0.900369 0.435127i \(-0.856704\pi\)
−0.900369 + 0.435127i \(0.856704\pi\)
\(920\) 0 0
\(921\) 1.26861e7 0.492811
\(922\) − 4.64374e7i − 1.79904i
\(923\) 2.20234e7i 0.850903i
\(924\) −1.35740e7 −0.523031
\(925\) 0 0
\(926\) 7.59369e7 2.91022
\(927\) 4.00301e7i 1.52998i
\(928\) − 9.37914e6i − 0.357514i
\(929\) 1.81557e7 0.690197 0.345098 0.938567i \(-0.387846\pi\)
0.345098 + 0.938567i \(0.387846\pi\)
\(930\) 0 0
\(931\) 1.26150e7 0.476993
\(932\) − 5.57230e7i − 2.10133i
\(933\) − 5.44047e6i − 0.204612i
\(934\) −1.89552e7 −0.710987
\(935\) 0 0
\(936\) 3.85884e7 1.43968
\(937\) − 1.78946e7i − 0.665844i −0.942954 0.332922i \(-0.891965\pi\)
0.942954 0.332922i \(-0.108035\pi\)
\(938\) − 5.46210e6i − 0.202700i
\(939\) 1.14157e7 0.422512
\(940\) 0 0
\(941\) −3.04463e6 −0.112088 −0.0560441 0.998428i \(-0.517849\pi\)
−0.0560441 + 0.998428i \(0.517849\pi\)
\(942\) − 1.95262e7i − 0.716954i
\(943\) 7.39891e6i 0.270950i
\(944\) −4.24825e7 −1.55160
\(945\) 0 0
\(946\) −7.49151e7 −2.72171
\(947\) − 3.17110e7i − 1.14904i −0.818491 0.574519i \(-0.805189\pi\)
0.818491 0.574519i \(-0.194811\pi\)
\(948\) 2.52127e6i 0.0911169i
\(949\) −1.87532e7 −0.675944
\(950\) 0 0
\(951\) −6.31686e6 −0.226491
\(952\) − 5.26342e7i − 1.88224i
\(953\) 1.01913e7i 0.363494i 0.983345 + 0.181747i \(0.0581752\pi\)
−0.983345 + 0.181747i \(0.941825\pi\)
\(954\) −4.21432e7 −1.49919
\(955\) 0 0
\(956\) −7.24764e7 −2.56479
\(957\) 3.59574e6i 0.126914i
\(958\) − 3.13084e7i − 1.10216i
\(959\) −1.72613e7 −0.606077
\(960\) 0 0
\(961\) 3.52157e7 1.23006
\(962\) − 8.66322e6i − 0.301816i
\(963\) − 1.44250e7i − 0.501246i
\(964\) 2.63095e7 0.911844
\(965\) 0 0
\(966\) −2.68481e6 −0.0925699
\(967\) 3.21125e7i 1.10435i 0.833727 + 0.552177i \(0.186203\pi\)
−0.833727 + 0.552177i \(0.813797\pi\)
\(968\) 3.18993e7i 1.09419i
\(969\) −1.04111e7 −0.356196
\(970\) 0 0
\(971\) 2.29867e7 0.782399 0.391200 0.920306i \(-0.372060\pi\)
0.391200 + 0.920306i \(0.372060\pi\)
\(972\) 6.01055e7i 2.04056i
\(973\) − 1.50763e7i − 0.510519i
\(974\) −7.47219e7 −2.52378
\(975\) 0 0
\(976\) −5.55687e7 −1.86726
\(977\) 2.47331e7i 0.828978i 0.910054 + 0.414489i \(0.136040\pi\)
−0.910054 + 0.414489i \(0.863960\pi\)
\(978\) − 2.07599e7i − 0.694029i
\(979\) 2.16954e7 0.723455
\(980\) 0 0
\(981\) 4.65960e7 1.54588
\(982\) 6.77785e7i 2.24292i
\(983\) 5.57031e7i 1.83863i 0.393518 + 0.919317i \(0.371258\pi\)
−0.393518 + 0.919317i \(0.628742\pi\)
\(984\) 2.52270e7 0.830574
\(985\) 0 0
\(986\) −2.47431e7 −0.810518
\(987\) 341035.i 0.0111431i
\(988\) 3.28704e7i 1.07130i
\(989\) −1.03150e7 −0.335335
\(990\) 0 0
\(991\) −6.86029e6 −0.221901 −0.110950 0.993826i \(-0.535389\pi\)
−0.110950 + 0.993826i \(0.535389\pi\)
\(992\) − 5.59673e7i − 1.80574i
\(993\) − 1.13373e6i − 0.0364868i
\(994\) 3.63772e7 1.16778
\(995\) 0 0
\(996\) −2.14436e7 −0.684936
\(997\) − 6.03725e7i − 1.92354i −0.273856 0.961771i \(-0.588299\pi\)
0.273856 0.961771i \(-0.411701\pi\)
\(998\) 4.99770e7i 1.58834i
\(999\) 4.95847e6 0.157193
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.24.1 4
3.2 odd 2 225.6.b.i.199.4 4
4.3 odd 2 400.6.c.n.49.3 4
5.2 odd 4 25.6.a.d.1.2 yes 2
5.3 odd 4 25.6.a.b.1.1 2
5.4 even 2 inner 25.6.b.b.24.4 4
15.2 even 4 225.6.a.l.1.1 2
15.8 even 4 225.6.a.s.1.2 2
15.14 odd 2 225.6.b.i.199.1 4
20.3 even 4 400.6.a.w.1.1 2
20.7 even 4 400.6.a.o.1.2 2
20.19 odd 2 400.6.c.n.49.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.a.b.1.1 2 5.3 odd 4
25.6.a.d.1.2 yes 2 5.2 odd 4
25.6.b.b.24.1 4 1.1 even 1 trivial
25.6.b.b.24.4 4 5.4 even 2 inner
225.6.a.l.1.1 2 15.2 even 4
225.6.a.s.1.2 2 15.8 even 4
225.6.b.i.199.1 4 15.14 odd 2
225.6.b.i.199.4 4 3.2 odd 2
400.6.a.o.1.2 2 20.7 even 4
400.6.a.w.1.1 2 20.3 even 4
400.6.c.n.49.2 4 20.19 odd 2
400.6.c.n.49.3 4 4.3 odd 2