Properties

Label 825.2.a.l.1.3
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} +2.17009 q^{6} +3.70928 q^{7} +1.53919 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} +2.17009 q^{6} +3.70928 q^{7} +1.53919 q^{8} +1.00000 q^{9} +1.00000 q^{11} +2.70928 q^{12} -1.70928 q^{13} +8.04945 q^{14} -2.07838 q^{16} -6.04945 q^{17} +2.17009 q^{18} -3.07838 q^{19} +3.70928 q^{21} +2.17009 q^{22} +4.00000 q^{23} +1.53919 q^{24} -3.70928 q^{26} +1.00000 q^{27} +10.0494 q^{28} +5.26180 q^{29} -6.34017 q^{31} -7.58864 q^{32} +1.00000 q^{33} -13.1278 q^{34} +2.70928 q^{36} -3.41855 q^{37} -6.68035 q^{38} -1.70928 q^{39} +9.57531 q^{41} +8.04945 q^{42} -3.12783 q^{43} +2.70928 q^{44} +8.68035 q^{46} +2.73820 q^{47} -2.07838 q^{48} +6.75872 q^{49} -6.04945 q^{51} -4.63090 q^{52} +13.7587 q^{53} +2.17009 q^{54} +5.70928 q^{56} -3.07838 q^{57} +11.4186 q^{58} -3.60197 q^{59} -14.6803 q^{61} -13.7587 q^{62} +3.70928 q^{63} -12.3112 q^{64} +2.17009 q^{66} +1.84324 q^{67} -16.3896 q^{68} +4.00000 q^{69} -7.23513 q^{71} +1.53919 q^{72} +6.38962 q^{73} -7.41855 q^{74} -8.34017 q^{76} +3.70928 q^{77} -3.70928 q^{78} -7.44521 q^{79} +1.00000 q^{81} +20.7792 q^{82} -7.86603 q^{83} +10.0494 q^{84} -6.78765 q^{86} +5.26180 q^{87} +1.53919 q^{88} -5.02052 q^{89} -6.34017 q^{91} +10.8371 q^{92} -6.34017 q^{93} +5.94214 q^{94} -7.58864 q^{96} +16.9939 q^{97} +14.6670 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{11} + q^{12} + 2 q^{13} + 6 q^{14} - 3 q^{16} + q^{18} - 6 q^{19} + 4 q^{21} + q^{22} + 12 q^{23} + 3 q^{24} - 4 q^{26} + 3 q^{27} + 12 q^{28} + 8 q^{29} - 8 q^{31} - 3 q^{32} + 3 q^{33} - 18 q^{34} + q^{36} + 4 q^{37} + 2 q^{38} + 2 q^{39} + 8 q^{41} + 6 q^{42} + 12 q^{43} + q^{44} + 4 q^{46} + 16 q^{47} - 3 q^{48} - 5 q^{49} - 10 q^{52} + 16 q^{53} + q^{54} + 10 q^{56} - 6 q^{57} + 20 q^{58} + 8 q^{59} - 22 q^{61} - 16 q^{62} + 4 q^{63} - 11 q^{64} + q^{66} + 12 q^{67} - 20 q^{68} + 12 q^{69} - 12 q^{71} + 3 q^{72} - 10 q^{73} - 8 q^{74} - 14 q^{76} + 4 q^{77} - 4 q^{78} - 10 q^{79} + 3 q^{81} + 4 q^{82} - 10 q^{83} + 12 q^{84} - 10 q^{86} + 8 q^{87} + 3 q^{88} + 18 q^{89} - 8 q^{91} + 4 q^{92} - 8 q^{93} - 12 q^{94} - 3 q^{96} + 16 q^{97} + 21 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.17009 1.53448 0.767241 0.641358i \(-0.221629\pi\)
0.767241 + 0.641358i \(0.221629\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.70928 1.35464
\(5\) 0 0
\(6\) 2.17009 0.885934
\(7\) 3.70928 1.40197 0.700987 0.713174i \(-0.252743\pi\)
0.700987 + 0.713174i \(0.252743\pi\)
\(8\) 1.53919 0.544185
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.70928 0.782100
\(13\) −1.70928 −0.474068 −0.237034 0.971501i \(-0.576175\pi\)
−0.237034 + 0.971501i \(0.576175\pi\)
\(14\) 8.04945 2.15131
\(15\) 0 0
\(16\) −2.07838 −0.519594
\(17\) −6.04945 −1.46721 −0.733603 0.679578i \(-0.762163\pi\)
−0.733603 + 0.679578i \(0.762163\pi\)
\(18\) 2.17009 0.511494
\(19\) −3.07838 −0.706228 −0.353114 0.935580i \(-0.614877\pi\)
−0.353114 + 0.935580i \(0.614877\pi\)
\(20\) 0 0
\(21\) 3.70928 0.809430
\(22\) 2.17009 0.462664
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.53919 0.314186
\(25\) 0 0
\(26\) −3.70928 −0.727449
\(27\) 1.00000 0.192450
\(28\) 10.0494 1.89917
\(29\) 5.26180 0.977091 0.488545 0.872538i \(-0.337528\pi\)
0.488545 + 0.872538i \(0.337528\pi\)
\(30\) 0 0
\(31\) −6.34017 −1.13873 −0.569364 0.822085i \(-0.692811\pi\)
−0.569364 + 0.822085i \(0.692811\pi\)
\(32\) −7.58864 −1.34149
\(33\) 1.00000 0.174078
\(34\) −13.1278 −2.25140
\(35\) 0 0
\(36\) 2.70928 0.451546
\(37\) −3.41855 −0.562006 −0.281003 0.959707i \(-0.590667\pi\)
−0.281003 + 0.959707i \(0.590667\pi\)
\(38\) −6.68035 −1.08370
\(39\) −1.70928 −0.273703
\(40\) 0 0
\(41\) 9.57531 1.49541 0.747706 0.664030i \(-0.231155\pi\)
0.747706 + 0.664030i \(0.231155\pi\)
\(42\) 8.04945 1.24206
\(43\) −3.12783 −0.476989 −0.238495 0.971144i \(-0.576654\pi\)
−0.238495 + 0.971144i \(0.576654\pi\)
\(44\) 2.70928 0.408439
\(45\) 0 0
\(46\) 8.68035 1.27985
\(47\) 2.73820 0.399408 0.199704 0.979856i \(-0.436002\pi\)
0.199704 + 0.979856i \(0.436002\pi\)
\(48\) −2.07838 −0.299988
\(49\) 6.75872 0.965532
\(50\) 0 0
\(51\) −6.04945 −0.847092
\(52\) −4.63090 −0.642190
\(53\) 13.7587 1.88991 0.944953 0.327206i \(-0.106107\pi\)
0.944953 + 0.327206i \(0.106107\pi\)
\(54\) 2.17009 0.295311
\(55\) 0 0
\(56\) 5.70928 0.762934
\(57\) −3.07838 −0.407741
\(58\) 11.4186 1.49933
\(59\) −3.60197 −0.468936 −0.234468 0.972124i \(-0.575335\pi\)
−0.234468 + 0.972124i \(0.575335\pi\)
\(60\) 0 0
\(61\) −14.6803 −1.87963 −0.939813 0.341690i \(-0.889001\pi\)
−0.939813 + 0.341690i \(0.889001\pi\)
\(62\) −13.7587 −1.74736
\(63\) 3.70928 0.467325
\(64\) −12.3112 −1.53891
\(65\) 0 0
\(66\) 2.17009 0.267119
\(67\) 1.84324 0.225188 0.112594 0.993641i \(-0.464084\pi\)
0.112594 + 0.993641i \(0.464084\pi\)
\(68\) −16.3896 −1.98753
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −7.23513 −0.858652 −0.429326 0.903150i \(-0.641249\pi\)
−0.429326 + 0.903150i \(0.641249\pi\)
\(72\) 1.53919 0.181395
\(73\) 6.38962 0.747849 0.373924 0.927459i \(-0.378012\pi\)
0.373924 + 0.927459i \(0.378012\pi\)
\(74\) −7.41855 −0.862389
\(75\) 0 0
\(76\) −8.34017 −0.956683
\(77\) 3.70928 0.422711
\(78\) −3.70928 −0.419993
\(79\) −7.44521 −0.837652 −0.418826 0.908067i \(-0.637558\pi\)
−0.418826 + 0.908067i \(0.637558\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 20.7792 2.29468
\(83\) −7.86603 −0.863409 −0.431705 0.902015i \(-0.642088\pi\)
−0.431705 + 0.902015i \(0.642088\pi\)
\(84\) 10.0494 1.09648
\(85\) 0 0
\(86\) −6.78765 −0.731931
\(87\) 5.26180 0.564124
\(88\) 1.53919 0.164078
\(89\) −5.02052 −0.532174 −0.266087 0.963949i \(-0.585731\pi\)
−0.266087 + 0.963949i \(0.585731\pi\)
\(90\) 0 0
\(91\) −6.34017 −0.664631
\(92\) 10.8371 1.12985
\(93\) −6.34017 −0.657445
\(94\) 5.94214 0.612885
\(95\) 0 0
\(96\) −7.58864 −0.774512
\(97\) 16.9939 1.72546 0.862732 0.505661i \(-0.168751\pi\)
0.862732 + 0.505661i \(0.168751\pi\)
\(98\) 14.6670 1.48159
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −18.9360 −1.88420 −0.942101 0.335329i \(-0.891153\pi\)
−0.942101 + 0.335329i \(0.891153\pi\)
\(102\) −13.1278 −1.29985
\(103\) −11.7854 −1.16125 −0.580624 0.814172i \(-0.697191\pi\)
−0.580624 + 0.814172i \(0.697191\pi\)
\(104\) −2.63090 −0.257981
\(105\) 0 0
\(106\) 29.8576 2.90003
\(107\) 11.2846 1.09092 0.545461 0.838136i \(-0.316355\pi\)
0.545461 + 0.838136i \(0.316355\pi\)
\(108\) 2.70928 0.260700
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −3.41855 −0.324474
\(112\) −7.70928 −0.728458
\(113\) −0.496928 −0.0467471 −0.0233735 0.999727i \(-0.507441\pi\)
−0.0233735 + 0.999727i \(0.507441\pi\)
\(114\) −6.68035 −0.625672
\(115\) 0 0
\(116\) 14.2557 1.32360
\(117\) −1.70928 −0.158023
\(118\) −7.81658 −0.719575
\(119\) −22.4391 −2.05699
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −31.8576 −2.88425
\(123\) 9.57531 0.863376
\(124\) −17.1773 −1.54256
\(125\) 0 0
\(126\) 8.04945 0.717102
\(127\) 2.81432 0.249730 0.124865 0.992174i \(-0.460150\pi\)
0.124865 + 0.992174i \(0.460150\pi\)
\(128\) −11.5392 −1.01993
\(129\) −3.12783 −0.275390
\(130\) 0 0
\(131\) 8.68035 0.758405 0.379203 0.925314i \(-0.376198\pi\)
0.379203 + 0.925314i \(0.376198\pi\)
\(132\) 2.70928 0.235812
\(133\) −11.4186 −0.990114
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −9.31124 −0.798433
\(137\) 1.07838 0.0921320 0.0460660 0.998938i \(-0.485332\pi\)
0.0460660 + 0.998938i \(0.485332\pi\)
\(138\) 8.68035 0.738920
\(139\) 10.2823 0.872135 0.436067 0.899914i \(-0.356371\pi\)
0.436067 + 0.899914i \(0.356371\pi\)
\(140\) 0 0
\(141\) 2.73820 0.230598
\(142\) −15.7009 −1.31759
\(143\) −1.70928 −0.142937
\(144\) −2.07838 −0.173198
\(145\) 0 0
\(146\) 13.8660 1.14756
\(147\) 6.75872 0.557450
\(148\) −9.26180 −0.761315
\(149\) 11.4186 0.935444 0.467722 0.883876i \(-0.345075\pi\)
0.467722 + 0.883876i \(0.345075\pi\)
\(150\) 0 0
\(151\) −4.92162 −0.400516 −0.200258 0.979743i \(-0.564178\pi\)
−0.200258 + 0.979743i \(0.564178\pi\)
\(152\) −4.73820 −0.384319
\(153\) −6.04945 −0.489069
\(154\) 8.04945 0.648643
\(155\) 0 0
\(156\) −4.63090 −0.370769
\(157\) −3.41855 −0.272830 −0.136415 0.990652i \(-0.543558\pi\)
−0.136415 + 0.990652i \(0.543558\pi\)
\(158\) −16.1568 −1.28536
\(159\) 13.7587 1.09114
\(160\) 0 0
\(161\) 14.8371 1.16933
\(162\) 2.17009 0.170498
\(163\) 9.26180 0.725440 0.362720 0.931898i \(-0.381848\pi\)
0.362720 + 0.931898i \(0.381848\pi\)
\(164\) 25.9421 2.02574
\(165\) 0 0
\(166\) −17.0700 −1.32489
\(167\) 7.55252 0.584432 0.292216 0.956352i \(-0.405607\pi\)
0.292216 + 0.956352i \(0.405607\pi\)
\(168\) 5.70928 0.440480
\(169\) −10.0784 −0.775260
\(170\) 0 0
\(171\) −3.07838 −0.235409
\(172\) −8.47414 −0.646147
\(173\) 6.14834 0.467450 0.233725 0.972303i \(-0.424908\pi\)
0.233725 + 0.972303i \(0.424908\pi\)
\(174\) 11.4186 0.865638
\(175\) 0 0
\(176\) −2.07838 −0.156664
\(177\) −3.60197 −0.270741
\(178\) −10.8950 −0.816612
\(179\) −6.15676 −0.460178 −0.230089 0.973170i \(-0.573902\pi\)
−0.230089 + 0.973170i \(0.573902\pi\)
\(180\) 0 0
\(181\) 14.5958 1.08490 0.542450 0.840088i \(-0.317497\pi\)
0.542450 + 0.840088i \(0.317497\pi\)
\(182\) −13.7587 −1.01986
\(183\) −14.6803 −1.08520
\(184\) 6.15676 0.453882
\(185\) 0 0
\(186\) −13.7587 −1.00884
\(187\) −6.04945 −0.442379
\(188\) 7.41855 0.541053
\(189\) 3.70928 0.269810
\(190\) 0 0
\(191\) 5.84324 0.422802 0.211401 0.977399i \(-0.432197\pi\)
0.211401 + 0.977399i \(0.432197\pi\)
\(192\) −12.3112 −0.888487
\(193\) 2.02279 0.145603 0.0728017 0.997346i \(-0.476806\pi\)
0.0728017 + 0.997346i \(0.476806\pi\)
\(194\) 36.8781 2.64770
\(195\) 0 0
\(196\) 18.3112 1.30795
\(197\) 17.8348 1.27068 0.635340 0.772233i \(-0.280860\pi\)
0.635340 + 0.772233i \(0.280860\pi\)
\(198\) 2.17009 0.154221
\(199\) 25.6742 1.82000 0.909998 0.414613i \(-0.136083\pi\)
0.909998 + 0.414613i \(0.136083\pi\)
\(200\) 0 0
\(201\) 1.84324 0.130012
\(202\) −41.0928 −2.89128
\(203\) 19.5174 1.36986
\(204\) −16.3896 −1.14750
\(205\) 0 0
\(206\) −25.5753 −1.78192
\(207\) 4.00000 0.278019
\(208\) 3.55252 0.246323
\(209\) −3.07838 −0.212936
\(210\) 0 0
\(211\) 8.43907 0.580970 0.290485 0.956880i \(-0.406183\pi\)
0.290485 + 0.956880i \(0.406183\pi\)
\(212\) 37.2762 2.56014
\(213\) −7.23513 −0.495743
\(214\) 24.4885 1.67400
\(215\) 0 0
\(216\) 1.53919 0.104729
\(217\) −23.5174 −1.59647
\(218\) −21.7009 −1.46977
\(219\) 6.38962 0.431771
\(220\) 0 0
\(221\) 10.3402 0.695555
\(222\) −7.41855 −0.497901
\(223\) 12.5814 0.842516 0.421258 0.906941i \(-0.361589\pi\)
0.421258 + 0.906941i \(0.361589\pi\)
\(224\) −28.1483 −1.88074
\(225\) 0 0
\(226\) −1.07838 −0.0717326
\(227\) 4.23287 0.280945 0.140473 0.990085i \(-0.455138\pi\)
0.140473 + 0.990085i \(0.455138\pi\)
\(228\) −8.34017 −0.552341
\(229\) −26.1978 −1.73120 −0.865599 0.500737i \(-0.833062\pi\)
−0.865599 + 0.500737i \(0.833062\pi\)
\(230\) 0 0
\(231\) 3.70928 0.244052
\(232\) 8.09890 0.531719
\(233\) −18.6309 −1.22055 −0.610275 0.792189i \(-0.708941\pi\)
−0.610275 + 0.792189i \(0.708941\pi\)
\(234\) −3.70928 −0.242483
\(235\) 0 0
\(236\) −9.75872 −0.635239
\(237\) −7.44521 −0.483619
\(238\) −48.6947 −3.15641
\(239\) 22.3545 1.44600 0.722998 0.690850i \(-0.242764\pi\)
0.722998 + 0.690850i \(0.242764\pi\)
\(240\) 0 0
\(241\) 9.20394 0.592878 0.296439 0.955052i \(-0.404201\pi\)
0.296439 + 0.955052i \(0.404201\pi\)
\(242\) 2.17009 0.139498
\(243\) 1.00000 0.0641500
\(244\) −39.7731 −2.54621
\(245\) 0 0
\(246\) 20.7792 1.32484
\(247\) 5.26180 0.334800
\(248\) −9.75872 −0.619680
\(249\) −7.86603 −0.498489
\(250\) 0 0
\(251\) −22.1256 −1.39655 −0.698276 0.715828i \(-0.746049\pi\)
−0.698276 + 0.715828i \(0.746049\pi\)
\(252\) 10.0494 0.633056
\(253\) 4.00000 0.251478
\(254\) 6.10731 0.383207
\(255\) 0 0
\(256\) −0.418551 −0.0261594
\(257\) 3.02052 0.188415 0.0942074 0.995553i \(-0.469968\pi\)
0.0942074 + 0.995553i \(0.469968\pi\)
\(258\) −6.78765 −0.422581
\(259\) −12.6803 −0.787918
\(260\) 0 0
\(261\) 5.26180 0.325697
\(262\) 18.8371 1.16376
\(263\) 18.2907 1.12785 0.563927 0.825825i \(-0.309290\pi\)
0.563927 + 0.825825i \(0.309290\pi\)
\(264\) 1.53919 0.0947305
\(265\) 0 0
\(266\) −24.7792 −1.51931
\(267\) −5.02052 −0.307251
\(268\) 4.99386 0.305048
\(269\) 30.5646 1.86356 0.931779 0.363026i \(-0.118256\pi\)
0.931779 + 0.363026i \(0.118256\pi\)
\(270\) 0 0
\(271\) −20.0722 −1.21930 −0.609651 0.792670i \(-0.708690\pi\)
−0.609651 + 0.792670i \(0.708690\pi\)
\(272\) 12.5730 0.762352
\(273\) −6.34017 −0.383725
\(274\) 2.34017 0.141375
\(275\) 0 0
\(276\) 10.8371 0.652317
\(277\) 0.760991 0.0457235 0.0228618 0.999739i \(-0.492722\pi\)
0.0228618 + 0.999739i \(0.492722\pi\)
\(278\) 22.3135 1.33828
\(279\) −6.34017 −0.379576
\(280\) 0 0
\(281\) 0.581449 0.0346864 0.0173432 0.999850i \(-0.494479\pi\)
0.0173432 + 0.999850i \(0.494479\pi\)
\(282\) 5.94214 0.353849
\(283\) −10.8143 −0.642844 −0.321422 0.946936i \(-0.604161\pi\)
−0.321422 + 0.946936i \(0.604161\pi\)
\(284\) −19.6020 −1.16316
\(285\) 0 0
\(286\) −3.70928 −0.219334
\(287\) 35.5174 2.09653
\(288\) −7.58864 −0.447165
\(289\) 19.5958 1.15270
\(290\) 0 0
\(291\) 16.9939 0.996198
\(292\) 17.3112 1.01306
\(293\) 7.04331 0.411474 0.205737 0.978607i \(-0.434041\pi\)
0.205737 + 0.978607i \(0.434041\pi\)
\(294\) 14.6670 0.855398
\(295\) 0 0
\(296\) −5.26180 −0.305836
\(297\) 1.00000 0.0580259
\(298\) 24.7792 1.43542
\(299\) −6.83710 −0.395400
\(300\) 0 0
\(301\) −11.6020 −0.668726
\(302\) −10.6803 −0.614585
\(303\) −18.9360 −1.08784
\(304\) 6.39803 0.366952
\(305\) 0 0
\(306\) −13.1278 −0.750468
\(307\) −20.1750 −1.15145 −0.575724 0.817644i \(-0.695280\pi\)
−0.575724 + 0.817644i \(0.695280\pi\)
\(308\) 10.0494 0.572620
\(309\) −11.7854 −0.670447
\(310\) 0 0
\(311\) 21.2762 1.20646 0.603230 0.797567i \(-0.293880\pi\)
0.603230 + 0.797567i \(0.293880\pi\)
\(312\) −2.63090 −0.148945
\(313\) 16.4657 0.930698 0.465349 0.885127i \(-0.345929\pi\)
0.465349 + 0.885127i \(0.345929\pi\)
\(314\) −7.41855 −0.418653
\(315\) 0 0
\(316\) −20.1711 −1.13471
\(317\) −22.1711 −1.24525 −0.622627 0.782518i \(-0.713935\pi\)
−0.622627 + 0.782518i \(0.713935\pi\)
\(318\) 29.8576 1.67433
\(319\) 5.26180 0.294604
\(320\) 0 0
\(321\) 11.2846 0.629844
\(322\) 32.1978 1.79431
\(323\) 18.6225 1.03618
\(324\) 2.70928 0.150515
\(325\) 0 0
\(326\) 20.0989 1.11317
\(327\) −10.0000 −0.553001
\(328\) 14.7382 0.813781
\(329\) 10.1568 0.559960
\(330\) 0 0
\(331\) −6.34017 −0.348487 −0.174244 0.984703i \(-0.555748\pi\)
−0.174244 + 0.984703i \(0.555748\pi\)
\(332\) −21.3112 −1.16961
\(333\) −3.41855 −0.187335
\(334\) 16.3896 0.896800
\(335\) 0 0
\(336\) −7.70928 −0.420575
\(337\) 3.18568 0.173535 0.0867677 0.996229i \(-0.472346\pi\)
0.0867677 + 0.996229i \(0.472346\pi\)
\(338\) −21.8710 −1.18962
\(339\) −0.496928 −0.0269894
\(340\) 0 0
\(341\) −6.34017 −0.343340
\(342\) −6.68035 −0.361232
\(343\) −0.894960 −0.0483233
\(344\) −4.81432 −0.259570
\(345\) 0 0
\(346\) 13.3424 0.717294
\(347\) −5.39576 −0.289660 −0.144830 0.989457i \(-0.546263\pi\)
−0.144830 + 0.989457i \(0.546263\pi\)
\(348\) 14.2557 0.764183
\(349\) −15.6742 −0.839021 −0.419510 0.907751i \(-0.637798\pi\)
−0.419510 + 0.907751i \(0.637798\pi\)
\(350\) 0 0
\(351\) −1.70928 −0.0912344
\(352\) −7.58864 −0.404476
\(353\) −5.75872 −0.306506 −0.153253 0.988187i \(-0.548975\pi\)
−0.153253 + 0.988187i \(0.548975\pi\)
\(354\) −7.81658 −0.415447
\(355\) 0 0
\(356\) −13.6020 −0.720903
\(357\) −22.4391 −1.18760
\(358\) −13.3607 −0.706135
\(359\) −10.5236 −0.555414 −0.277707 0.960666i \(-0.589574\pi\)
−0.277707 + 0.960666i \(0.589574\pi\)
\(360\) 0 0
\(361\) −9.52359 −0.501242
\(362\) 31.6742 1.66476
\(363\) 1.00000 0.0524864
\(364\) −17.1773 −0.900334
\(365\) 0 0
\(366\) −31.8576 −1.66522
\(367\) 8.89496 0.464313 0.232157 0.972678i \(-0.425422\pi\)
0.232157 + 0.972678i \(0.425422\pi\)
\(368\) −8.31351 −0.433372
\(369\) 9.57531 0.498471
\(370\) 0 0
\(371\) 51.0349 2.64960
\(372\) −17.1773 −0.890600
\(373\) −23.1689 −1.19964 −0.599819 0.800136i \(-0.704761\pi\)
−0.599819 + 0.800136i \(0.704761\pi\)
\(374\) −13.1278 −0.678824
\(375\) 0 0
\(376\) 4.21461 0.217352
\(377\) −8.99386 −0.463207
\(378\) 8.04945 0.414019
\(379\) −11.8310 −0.607716 −0.303858 0.952717i \(-0.598275\pi\)
−0.303858 + 0.952717i \(0.598275\pi\)
\(380\) 0 0
\(381\) 2.81432 0.144182
\(382\) 12.6803 0.648783
\(383\) 25.9421 1.32558 0.662791 0.748805i \(-0.269372\pi\)
0.662791 + 0.748805i \(0.269372\pi\)
\(384\) −11.5392 −0.588857
\(385\) 0 0
\(386\) 4.38962 0.223426
\(387\) −3.12783 −0.158996
\(388\) 46.0410 2.33738
\(389\) 1.00614 0.0510135 0.0255067 0.999675i \(-0.491880\pi\)
0.0255067 + 0.999675i \(0.491880\pi\)
\(390\) 0 0
\(391\) −24.1978 −1.22374
\(392\) 10.4030 0.525428
\(393\) 8.68035 0.437866
\(394\) 38.7031 1.94984
\(395\) 0 0
\(396\) 2.70928 0.136146
\(397\) 34.7214 1.74262 0.871308 0.490736i \(-0.163272\pi\)
0.871308 + 0.490736i \(0.163272\pi\)
\(398\) 55.7152 2.79275
\(399\) −11.4186 −0.571643
\(400\) 0 0
\(401\) −13.0205 −0.650214 −0.325107 0.945677i \(-0.605400\pi\)
−0.325107 + 0.945677i \(0.605400\pi\)
\(402\) 4.00000 0.199502
\(403\) 10.8371 0.539834
\(404\) −51.3028 −2.55241
\(405\) 0 0
\(406\) 42.3545 2.10202
\(407\) −3.41855 −0.169451
\(408\) −9.31124 −0.460975
\(409\) −29.5174 −1.45954 −0.729772 0.683691i \(-0.760374\pi\)
−0.729772 + 0.683691i \(0.760374\pi\)
\(410\) 0 0
\(411\) 1.07838 0.0531925
\(412\) −31.9299 −1.57307
\(413\) −13.3607 −0.657437
\(414\) 8.68035 0.426616
\(415\) 0 0
\(416\) 12.9711 0.635959
\(417\) 10.2823 0.503527
\(418\) −6.68035 −0.326746
\(419\) −6.15676 −0.300777 −0.150389 0.988627i \(-0.548052\pi\)
−0.150389 + 0.988627i \(0.548052\pi\)
\(420\) 0 0
\(421\) 9.96880 0.485850 0.242925 0.970045i \(-0.421893\pi\)
0.242925 + 0.970045i \(0.421893\pi\)
\(422\) 18.3135 0.891488
\(423\) 2.73820 0.133136
\(424\) 21.1773 1.02846
\(425\) 0 0
\(426\) −15.7009 −0.760709
\(427\) −54.4534 −2.63519
\(428\) 30.5730 1.47780
\(429\) −1.70928 −0.0825246
\(430\) 0 0
\(431\) 8.68035 0.418118 0.209059 0.977903i \(-0.432960\pi\)
0.209059 + 0.977903i \(0.432960\pi\)
\(432\) −2.07838 −0.0999960
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) −51.0349 −2.44975
\(435\) 0 0
\(436\) −27.0928 −1.29751
\(437\) −12.3135 −0.589035
\(438\) 13.8660 0.662545
\(439\) 3.07838 0.146923 0.0734615 0.997298i \(-0.476595\pi\)
0.0734615 + 0.997298i \(0.476595\pi\)
\(440\) 0 0
\(441\) 6.75872 0.321844
\(442\) 22.4391 1.06732
\(443\) 29.2618 1.39027 0.695135 0.718879i \(-0.255345\pi\)
0.695135 + 0.718879i \(0.255345\pi\)
\(444\) −9.26180 −0.439545
\(445\) 0 0
\(446\) 27.3028 1.29283
\(447\) 11.4186 0.540079
\(448\) −45.6658 −2.15751
\(449\) 10.6947 0.504715 0.252358 0.967634i \(-0.418794\pi\)
0.252358 + 0.967634i \(0.418794\pi\)
\(450\) 0 0
\(451\) 9.57531 0.450884
\(452\) −1.34632 −0.0633254
\(453\) −4.92162 −0.231238
\(454\) 9.18568 0.431106
\(455\) 0 0
\(456\) −4.73820 −0.221887
\(457\) 22.8554 1.06913 0.534564 0.845128i \(-0.320476\pi\)
0.534564 + 0.845128i \(0.320476\pi\)
\(458\) −56.8515 −2.65650
\(459\) −6.04945 −0.282364
\(460\) 0 0
\(461\) −21.7731 −1.01407 −0.507037 0.861924i \(-0.669259\pi\)
−0.507037 + 0.861924i \(0.669259\pi\)
\(462\) 8.04945 0.374494
\(463\) 24.8950 1.15697 0.578483 0.815694i \(-0.303645\pi\)
0.578483 + 0.815694i \(0.303645\pi\)
\(464\) −10.9360 −0.507691
\(465\) 0 0
\(466\) −40.4307 −1.87291
\(467\) −19.2039 −0.888652 −0.444326 0.895865i \(-0.646557\pi\)
−0.444326 + 0.895865i \(0.646557\pi\)
\(468\) −4.63090 −0.214063
\(469\) 6.83710 0.315708
\(470\) 0 0
\(471\) −3.41855 −0.157519
\(472\) −5.54411 −0.255188
\(473\) −3.12783 −0.143818
\(474\) −16.1568 −0.742104
\(475\) 0 0
\(476\) −60.7936 −2.78647
\(477\) 13.7587 0.629969
\(478\) 48.5113 2.21886
\(479\) 9.47641 0.432988 0.216494 0.976284i \(-0.430538\pi\)
0.216494 + 0.976284i \(0.430538\pi\)
\(480\) 0 0
\(481\) 5.84324 0.266429
\(482\) 19.9733 0.909761
\(483\) 14.8371 0.675111
\(484\) 2.70928 0.123149
\(485\) 0 0
\(486\) 2.17009 0.0984371
\(487\) 35.2039 1.59524 0.797621 0.603159i \(-0.206091\pi\)
0.797621 + 0.603159i \(0.206091\pi\)
\(488\) −22.5958 −1.02286
\(489\) 9.26180 0.418833
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 25.9421 1.16956
\(493\) −31.8310 −1.43359
\(494\) 11.4186 0.513745
\(495\) 0 0
\(496\) 13.1773 0.591677
\(497\) −26.8371 −1.20381
\(498\) −17.0700 −0.764924
\(499\) 26.1568 1.17094 0.585469 0.810695i \(-0.300911\pi\)
0.585469 + 0.810695i \(0.300911\pi\)
\(500\) 0 0
\(501\) 7.55252 0.337422
\(502\) −48.0144 −2.14299
\(503\) −28.2784 −1.26087 −0.630437 0.776241i \(-0.717124\pi\)
−0.630437 + 0.776241i \(0.717124\pi\)
\(504\) 5.70928 0.254311
\(505\) 0 0
\(506\) 8.68035 0.385888
\(507\) −10.0784 −0.447596
\(508\) 7.62475 0.338294
\(509\) 8.47027 0.375438 0.187719 0.982223i \(-0.439891\pi\)
0.187719 + 0.982223i \(0.439891\pi\)
\(510\) 0 0
\(511\) 23.7009 1.04846
\(512\) 22.1701 0.979789
\(513\) −3.07838 −0.135914
\(514\) 6.55479 0.289119
\(515\) 0 0
\(516\) −8.47414 −0.373053
\(517\) 2.73820 0.120426
\(518\) −27.5174 −1.20905
\(519\) 6.14834 0.269882
\(520\) 0 0
\(521\) 13.7009 0.600246 0.300123 0.953901i \(-0.402972\pi\)
0.300123 + 0.953901i \(0.402972\pi\)
\(522\) 11.4186 0.499776
\(523\) 24.4885 1.07081 0.535404 0.844596i \(-0.320159\pi\)
0.535404 + 0.844596i \(0.320159\pi\)
\(524\) 23.5174 1.02736
\(525\) 0 0
\(526\) 39.6925 1.73067
\(527\) 38.3545 1.67075
\(528\) −2.07838 −0.0904498
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −3.60197 −0.156312
\(532\) −30.9360 −1.34125
\(533\) −16.3668 −0.708926
\(534\) −10.8950 −0.471471
\(535\) 0 0
\(536\) 2.83710 0.122544
\(537\) −6.15676 −0.265684
\(538\) 66.3279 2.85960
\(539\) 6.75872 0.291119
\(540\) 0 0
\(541\) 4.83710 0.207963 0.103982 0.994579i \(-0.466842\pi\)
0.103982 + 0.994579i \(0.466842\pi\)
\(542\) −43.5585 −1.87100
\(543\) 14.5958 0.626367
\(544\) 45.9071 1.96825
\(545\) 0 0
\(546\) −13.7587 −0.588819
\(547\) −30.5464 −1.30607 −0.653034 0.757328i \(-0.726504\pi\)
−0.653034 + 0.757328i \(0.726504\pi\)
\(548\) 2.92162 0.124806
\(549\) −14.6803 −0.626542
\(550\) 0 0
\(551\) −16.1978 −0.690049
\(552\) 6.15676 0.262049
\(553\) −27.6163 −1.17437
\(554\) 1.65142 0.0701620
\(555\) 0 0
\(556\) 27.8576 1.18143
\(557\) −7.57918 −0.321140 −0.160570 0.987024i \(-0.551333\pi\)
−0.160570 + 0.987024i \(0.551333\pi\)
\(558\) −13.7587 −0.582453
\(559\) 5.34632 0.226125
\(560\) 0 0
\(561\) −6.04945 −0.255408
\(562\) 1.26180 0.0532256
\(563\) −14.1750 −0.597405 −0.298703 0.954346i \(-0.596554\pi\)
−0.298703 + 0.954346i \(0.596554\pi\)
\(564\) 7.41855 0.312377
\(565\) 0 0
\(566\) −23.4680 −0.986434
\(567\) 3.70928 0.155775
\(568\) −11.1362 −0.467266
\(569\) −14.7382 −0.617858 −0.308929 0.951085i \(-0.599970\pi\)
−0.308929 + 0.951085i \(0.599970\pi\)
\(570\) 0 0
\(571\) 1.23513 0.0516887 0.0258444 0.999666i \(-0.491773\pi\)
0.0258444 + 0.999666i \(0.491773\pi\)
\(572\) −4.63090 −0.193628
\(573\) 5.84324 0.244105
\(574\) 77.0759 3.21709
\(575\) 0 0
\(576\) −12.3112 −0.512968
\(577\) 28.9770 1.20633 0.603165 0.797617i \(-0.293906\pi\)
0.603165 + 0.797617i \(0.293906\pi\)
\(578\) 42.5246 1.76879
\(579\) 2.02279 0.0840641
\(580\) 0 0
\(581\) −29.1773 −1.21048
\(582\) 36.8781 1.52865
\(583\) 13.7587 0.569828
\(584\) 9.83483 0.406968
\(585\) 0 0
\(586\) 15.2846 0.631400
\(587\) −20.9939 −0.866509 −0.433255 0.901272i \(-0.642635\pi\)
−0.433255 + 0.901272i \(0.642635\pi\)
\(588\) 18.3112 0.755143
\(589\) 19.5174 0.804202
\(590\) 0 0
\(591\) 17.8348 0.733627
\(592\) 7.10504 0.292015
\(593\) −23.8927 −0.981155 −0.490578 0.871397i \(-0.663214\pi\)
−0.490578 + 0.871397i \(0.663214\pi\)
\(594\) 2.17009 0.0890397
\(595\) 0 0
\(596\) 30.9360 1.26719
\(597\) 25.6742 1.05078
\(598\) −14.8371 −0.606734
\(599\) −45.6742 −1.86620 −0.933099 0.359620i \(-0.882906\pi\)
−0.933099 + 0.359620i \(0.882906\pi\)
\(600\) 0 0
\(601\) −16.2101 −0.661223 −0.330611 0.943767i \(-0.607255\pi\)
−0.330611 + 0.943767i \(0.607255\pi\)
\(602\) −25.1773 −1.02615
\(603\) 1.84324 0.0750627
\(604\) −13.3340 −0.542554
\(605\) 0 0
\(606\) −41.0928 −1.66928
\(607\) 20.8020 0.844328 0.422164 0.906519i \(-0.361271\pi\)
0.422164 + 0.906519i \(0.361271\pi\)
\(608\) 23.3607 0.947401
\(609\) 19.5174 0.790887
\(610\) 0 0
\(611\) −4.68035 −0.189347
\(612\) −16.3896 −0.662511
\(613\) 15.3835 0.621333 0.310666 0.950519i \(-0.399448\pi\)
0.310666 + 0.950519i \(0.399448\pi\)
\(614\) −43.7815 −1.76688
\(615\) 0 0
\(616\) 5.70928 0.230033
\(617\) 21.9733 0.884613 0.442307 0.896864i \(-0.354160\pi\)
0.442307 + 0.896864i \(0.354160\pi\)
\(618\) −25.5753 −1.02879
\(619\) 10.3935 0.417750 0.208875 0.977942i \(-0.433020\pi\)
0.208875 + 0.977942i \(0.433020\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 46.1711 1.85129
\(623\) −18.6225 −0.746094
\(624\) 3.55252 0.142215
\(625\) 0 0
\(626\) 35.7321 1.42814
\(627\) −3.07838 −0.122939
\(628\) −9.26180 −0.369586
\(629\) 20.6803 0.824579
\(630\) 0 0
\(631\) 38.7214 1.54147 0.770737 0.637153i \(-0.219888\pi\)
0.770737 + 0.637153i \(0.219888\pi\)
\(632\) −11.4596 −0.455838
\(633\) 8.43907 0.335423
\(634\) −48.1133 −1.91082
\(635\) 0 0
\(636\) 37.2762 1.47810
\(637\) −11.5525 −0.457728
\(638\) 11.4186 0.452065
\(639\) −7.23513 −0.286217
\(640\) 0 0
\(641\) 24.2245 0.956808 0.478404 0.878140i \(-0.341215\pi\)
0.478404 + 0.878140i \(0.341215\pi\)
\(642\) 24.4885 0.966485
\(643\) −25.8888 −1.02096 −0.510478 0.859891i \(-0.670531\pi\)
−0.510478 + 0.859891i \(0.670531\pi\)
\(644\) 40.1978 1.58401
\(645\) 0 0
\(646\) 40.4124 1.59000
\(647\) −0.581449 −0.0228591 −0.0114296 0.999935i \(-0.503638\pi\)
−0.0114296 + 0.999935i \(0.503638\pi\)
\(648\) 1.53919 0.0604650
\(649\) −3.60197 −0.141390
\(650\) 0 0
\(651\) −23.5174 −0.921721
\(652\) 25.0928 0.982708
\(653\) 37.0082 1.44824 0.724122 0.689672i \(-0.242245\pi\)
0.724122 + 0.689672i \(0.242245\pi\)
\(654\) −21.7009 −0.848571
\(655\) 0 0
\(656\) −19.9011 −0.777008
\(657\) 6.38962 0.249283
\(658\) 22.0410 0.859249
\(659\) 30.5236 1.18903 0.594515 0.804084i \(-0.297344\pi\)
0.594515 + 0.804084i \(0.297344\pi\)
\(660\) 0 0
\(661\) 5.88428 0.228872 0.114436 0.993431i \(-0.463494\pi\)
0.114436 + 0.993431i \(0.463494\pi\)
\(662\) −13.7587 −0.534748
\(663\) 10.3402 0.401579
\(664\) −12.1073 −0.469855
\(665\) 0 0
\(666\) −7.41855 −0.287463
\(667\) 21.0472 0.814950
\(668\) 20.4619 0.791693
\(669\) 12.5814 0.486427
\(670\) 0 0
\(671\) −14.6803 −0.566728
\(672\) −28.1483 −1.08585
\(673\) −26.9711 −1.03966 −0.519829 0.854270i \(-0.674004\pi\)
−0.519829 + 0.854270i \(0.674004\pi\)
\(674\) 6.91321 0.266287
\(675\) 0 0
\(676\) −27.3051 −1.05020
\(677\) −17.9506 −0.689896 −0.344948 0.938622i \(-0.612103\pi\)
−0.344948 + 0.938622i \(0.612103\pi\)
\(678\) −1.07838 −0.0414148
\(679\) 63.0349 2.41906
\(680\) 0 0
\(681\) 4.23287 0.162204
\(682\) −13.7587 −0.526849
\(683\) 8.77924 0.335928 0.167964 0.985793i \(-0.446281\pi\)
0.167964 + 0.985793i \(0.446281\pi\)
\(684\) −8.34017 −0.318894
\(685\) 0 0
\(686\) −1.94214 −0.0741513
\(687\) −26.1978 −0.999508
\(688\) 6.50080 0.247841
\(689\) −23.5174 −0.895943
\(690\) 0 0
\(691\) −14.7214 −0.560028 −0.280014 0.959996i \(-0.590339\pi\)
−0.280014 + 0.959996i \(0.590339\pi\)
\(692\) 16.6576 0.633225
\(693\) 3.70928 0.140904
\(694\) −11.7093 −0.444478
\(695\) 0 0
\(696\) 8.09890 0.306988
\(697\) −57.9253 −2.19408
\(698\) −34.0144 −1.28746
\(699\) −18.6309 −0.704685
\(700\) 0 0
\(701\) −7.10504 −0.268354 −0.134177 0.990957i \(-0.542839\pi\)
−0.134177 + 0.990957i \(0.542839\pi\)
\(702\) −3.70928 −0.139998
\(703\) 10.5236 0.396905
\(704\) −12.3112 −0.463997
\(705\) 0 0
\(706\) −12.4969 −0.470328
\(707\) −70.2388 −2.64160
\(708\) −9.75872 −0.366755
\(709\) −34.1666 −1.28315 −0.641577 0.767059i \(-0.721719\pi\)
−0.641577 + 0.767059i \(0.721719\pi\)
\(710\) 0 0
\(711\) −7.44521 −0.279217
\(712\) −7.72753 −0.289601
\(713\) −25.3607 −0.949765
\(714\) −48.6947 −1.82235
\(715\) 0 0
\(716\) −16.6803 −0.623374
\(717\) 22.3545 0.834846
\(718\) −22.8371 −0.852273
\(719\) 31.8310 1.18709 0.593547 0.804799i \(-0.297727\pi\)
0.593547 + 0.804799i \(0.297727\pi\)
\(720\) 0 0
\(721\) −43.7152 −1.62804
\(722\) −20.6670 −0.769147
\(723\) 9.20394 0.342298
\(724\) 39.5441 1.46965
\(725\) 0 0
\(726\) 2.17009 0.0805395
\(727\) 5.16290 0.191481 0.0957407 0.995406i \(-0.469478\pi\)
0.0957407 + 0.995406i \(0.469478\pi\)
\(728\) −9.75872 −0.361682
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 18.9216 0.699841
\(732\) −39.7731 −1.47006
\(733\) −36.4475 −1.34622 −0.673109 0.739543i \(-0.735042\pi\)
−0.673109 + 0.739543i \(0.735042\pi\)
\(734\) 19.3028 0.712481
\(735\) 0 0
\(736\) −30.3545 −1.11888
\(737\) 1.84324 0.0678968
\(738\) 20.7792 0.764894
\(739\) 25.4329 0.935565 0.467783 0.883844i \(-0.345053\pi\)
0.467783 + 0.883844i \(0.345053\pi\)
\(740\) 0 0
\(741\) 5.26180 0.193297
\(742\) 110.750 4.06577
\(743\) 10.1217 0.371329 0.185664 0.982613i \(-0.440556\pi\)
0.185664 + 0.982613i \(0.440556\pi\)
\(744\) −9.75872 −0.357772
\(745\) 0 0
\(746\) −50.2784 −1.84082
\(747\) −7.86603 −0.287803
\(748\) −16.3896 −0.599264
\(749\) 41.8576 1.52944
\(750\) 0 0
\(751\) 10.4703 0.382065 0.191033 0.981584i \(-0.438816\pi\)
0.191033 + 0.981584i \(0.438816\pi\)
\(752\) −5.69102 −0.207530
\(753\) −22.1256 −0.806300
\(754\) −19.5174 −0.710784
\(755\) 0 0
\(756\) 10.0494 0.365495
\(757\) −6.73820 −0.244904 −0.122452 0.992474i \(-0.539076\pi\)
−0.122452 + 0.992474i \(0.539076\pi\)
\(758\) −25.6742 −0.932529
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −5.57531 −0.202105 −0.101052 0.994881i \(-0.532221\pi\)
−0.101052 + 0.994881i \(0.532221\pi\)
\(762\) 6.10731 0.221244
\(763\) −37.0928 −1.34285
\(764\) 15.8310 0.572744
\(765\) 0 0
\(766\) 56.2967 2.03408
\(767\) 6.15676 0.222308
\(768\) −0.418551 −0.0151031
\(769\) −11.5297 −0.415773 −0.207886 0.978153i \(-0.566658\pi\)
−0.207886 + 0.978153i \(0.566658\pi\)
\(770\) 0 0
\(771\) 3.02052 0.108781
\(772\) 5.48029 0.197240
\(773\) −28.7480 −1.03400 −0.516998 0.855987i \(-0.672950\pi\)
−0.516998 + 0.855987i \(0.672950\pi\)
\(774\) −6.78765 −0.243977
\(775\) 0 0
\(776\) 26.1568 0.938973
\(777\) −12.6803 −0.454905
\(778\) 2.18342 0.0782793
\(779\) −29.4764 −1.05610
\(780\) 0 0
\(781\) −7.23513 −0.258893
\(782\) −52.5113 −1.87780
\(783\) 5.26180 0.188041
\(784\) −14.0472 −0.501685
\(785\) 0 0
\(786\) 18.8371 0.671897
\(787\) −13.4536 −0.479570 −0.239785 0.970826i \(-0.577077\pi\)
−0.239785 + 0.970826i \(0.577077\pi\)
\(788\) 48.3195 1.72131
\(789\) 18.2907 0.651167
\(790\) 0 0
\(791\) −1.84324 −0.0655382
\(792\) 1.53919 0.0546927
\(793\) 25.0928 0.891070
\(794\) 75.3484 2.67401
\(795\) 0 0
\(796\) 69.5585 2.46544
\(797\) −48.3279 −1.71186 −0.855931 0.517090i \(-0.827015\pi\)
−0.855931 + 0.517090i \(0.827015\pi\)
\(798\) −24.7792 −0.877176
\(799\) −16.5646 −0.586014
\(800\) 0 0
\(801\) −5.02052 −0.177391
\(802\) −28.2557 −0.997742
\(803\) 6.38962 0.225485
\(804\) 4.99386 0.176120
\(805\) 0 0
\(806\) 23.5174 0.828367
\(807\) 30.5646 1.07593
\(808\) −29.1461 −1.02536
\(809\) −43.8141 −1.54042 −0.770212 0.637789i \(-0.779849\pi\)
−0.770212 + 0.637789i \(0.779849\pi\)
\(810\) 0 0
\(811\) −47.2762 −1.66009 −0.830045 0.557696i \(-0.811686\pi\)
−0.830045 + 0.557696i \(0.811686\pi\)
\(812\) 52.8781 1.85566
\(813\) −20.0722 −0.703964
\(814\) −7.41855 −0.260020
\(815\) 0 0
\(816\) 12.5730 0.440144
\(817\) 9.62863 0.336863
\(818\) −64.0554 −2.23965
\(819\) −6.34017 −0.221544
\(820\) 0 0
\(821\) −3.30283 −0.115270 −0.0576348 0.998338i \(-0.518356\pi\)
−0.0576348 + 0.998338i \(0.518356\pi\)
\(822\) 2.34017 0.0816229
\(823\) 11.0517 0.385239 0.192619 0.981274i \(-0.438302\pi\)
0.192619 + 0.981274i \(0.438302\pi\)
\(824\) −18.1399 −0.631935
\(825\) 0 0
\(826\) −28.9939 −1.00883
\(827\) 5.12783 0.178312 0.0891560 0.996018i \(-0.471583\pi\)
0.0891560 + 0.996018i \(0.471583\pi\)
\(828\) 10.8371 0.376615
\(829\) −6.39803 −0.222213 −0.111106 0.993809i \(-0.535439\pi\)
−0.111106 + 0.993809i \(0.535439\pi\)
\(830\) 0 0
\(831\) 0.760991 0.0263985
\(832\) 21.0433 0.729545
\(833\) −40.8865 −1.41663
\(834\) 22.3135 0.772654
\(835\) 0 0
\(836\) −8.34017 −0.288451
\(837\) −6.34017 −0.219148
\(838\) −13.3607 −0.461537
\(839\) −22.0722 −0.762018 −0.381009 0.924571i \(-0.624423\pi\)
−0.381009 + 0.924571i \(0.624423\pi\)
\(840\) 0 0
\(841\) −1.31351 −0.0452935
\(842\) 21.6332 0.745528
\(843\) 0.581449 0.0200262
\(844\) 22.8638 0.787003
\(845\) 0 0
\(846\) 5.94214 0.204295
\(847\) 3.70928 0.127452
\(848\) −28.5958 −0.981985
\(849\) −10.8143 −0.371146
\(850\) 0 0
\(851\) −13.6742 −0.468746
\(852\) −19.6020 −0.671552
\(853\) −8.76099 −0.299971 −0.149985 0.988688i \(-0.547923\pi\)
−0.149985 + 0.988688i \(0.547923\pi\)
\(854\) −118.169 −4.04365
\(855\) 0 0
\(856\) 17.3691 0.593664
\(857\) −36.5730 −1.24931 −0.624656 0.780900i \(-0.714761\pi\)
−0.624656 + 0.780900i \(0.714761\pi\)
\(858\) −3.70928 −0.126633
\(859\) −49.6886 −1.69535 −0.847676 0.530514i \(-0.821999\pi\)
−0.847676 + 0.530514i \(0.821999\pi\)
\(860\) 0 0
\(861\) 35.5174 1.21043
\(862\) 18.8371 0.641594
\(863\) −34.1399 −1.16214 −0.581068 0.813855i \(-0.697365\pi\)
−0.581068 + 0.813855i \(0.697365\pi\)
\(864\) −7.58864 −0.258171
\(865\) 0 0
\(866\) −34.7214 −1.17988
\(867\) 19.5958 0.665509
\(868\) −63.7152 −2.16264
\(869\) −7.44521 −0.252562
\(870\) 0 0
\(871\) −3.15061 −0.106754
\(872\) −15.3919 −0.521235
\(873\) 16.9939 0.575155
\(874\) −26.7214 −0.903864
\(875\) 0 0
\(876\) 17.3112 0.584893
\(877\) −43.5357 −1.47010 −0.735048 0.678015i \(-0.762840\pi\)
−0.735048 + 0.678015i \(0.762840\pi\)
\(878\) 6.68035 0.225451
\(879\) 7.04331 0.237565
\(880\) 0 0
\(881\) −8.52359 −0.287167 −0.143584 0.989638i \(-0.545863\pi\)
−0.143584 + 0.989638i \(0.545863\pi\)
\(882\) 14.6670 0.493864
\(883\) 43.0349 1.44824 0.724120 0.689674i \(-0.242246\pi\)
0.724120 + 0.689674i \(0.242246\pi\)
\(884\) 28.0144 0.942225
\(885\) 0 0
\(886\) 63.5006 2.13335
\(887\) 17.0289 0.571775 0.285888 0.958263i \(-0.407712\pi\)
0.285888 + 0.958263i \(0.407712\pi\)
\(888\) −5.26180 −0.176574
\(889\) 10.4391 0.350115
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 34.0866 1.14130
\(893\) −8.42923 −0.282073
\(894\) 24.7792 0.828742
\(895\) 0 0
\(896\) −42.8020 −1.42992
\(897\) −6.83710 −0.228284
\(898\) 23.2085 0.774477
\(899\) −33.3607 −1.11264
\(900\) 0 0
\(901\) −83.2327 −2.77288
\(902\) 20.7792 0.691873
\(903\) −11.6020 −0.386089
\(904\) −0.764867 −0.0254391
\(905\) 0 0
\(906\) −10.6803 −0.354831
\(907\) 6.13993 0.203873 0.101937 0.994791i \(-0.467496\pi\)
0.101937 + 0.994791i \(0.467496\pi\)
\(908\) 11.4680 0.380579
\(909\) −18.9360 −0.628067
\(910\) 0 0
\(911\) 53.8720 1.78486 0.892429 0.451187i \(-0.148999\pi\)
0.892429 + 0.451187i \(0.148999\pi\)
\(912\) 6.39803 0.211860
\(913\) −7.86603 −0.260328
\(914\) 49.5981 1.64056
\(915\) 0 0
\(916\) −70.9770 −2.34515
\(917\) 32.1978 1.06326
\(918\) −13.1278 −0.433283
\(919\) −48.2700 −1.59228 −0.796141 0.605112i \(-0.793128\pi\)
−0.796141 + 0.605112i \(0.793128\pi\)
\(920\) 0 0
\(921\) −20.1750 −0.664789
\(922\) −47.2495 −1.55608
\(923\) 12.3668 0.407059
\(924\) 10.0494 0.330603
\(925\) 0 0
\(926\) 54.0242 1.77535
\(927\) −11.7854 −0.387083
\(928\) −39.9299 −1.31076
\(929\) 4.10343 0.134629 0.0673146 0.997732i \(-0.478557\pi\)
0.0673146 + 0.997732i \(0.478557\pi\)
\(930\) 0 0
\(931\) −20.8059 −0.681886
\(932\) −50.4762 −1.65340
\(933\) 21.2762 0.696551
\(934\) −41.6742 −1.36362
\(935\) 0 0
\(936\) −2.63090 −0.0859936
\(937\) −38.5341 −1.25885 −0.629427 0.777060i \(-0.716710\pi\)
−0.629427 + 0.777060i \(0.716710\pi\)
\(938\) 14.8371 0.484449
\(939\) 16.4657 0.537339
\(940\) 0 0
\(941\) 14.6849 0.478713 0.239357 0.970932i \(-0.423063\pi\)
0.239357 + 0.970932i \(0.423063\pi\)
\(942\) −7.41855 −0.241709
\(943\) 38.3012 1.24726
\(944\) 7.48625 0.243657
\(945\) 0 0
\(946\) −6.78765 −0.220686
\(947\) 6.05786 0.196854 0.0984270 0.995144i \(-0.468619\pi\)
0.0984270 + 0.995144i \(0.468619\pi\)
\(948\) −20.1711 −0.655128
\(949\) −10.9216 −0.354531
\(950\) 0 0
\(951\) −22.1711 −0.718948
\(952\) −34.5380 −1.11938
\(953\) 40.1438 1.30039 0.650193 0.759769i \(-0.274688\pi\)
0.650193 + 0.759769i \(0.274688\pi\)
\(954\) 29.8576 0.966676
\(955\) 0 0
\(956\) 60.5646 1.95880
\(957\) 5.26180 0.170090
\(958\) 20.5646 0.664413
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) 9.19779 0.296703
\(962\) 12.6803 0.408831
\(963\) 11.2846 0.363641
\(964\) 24.9360 0.803134
\(965\) 0 0
\(966\) 32.1978 1.03595
\(967\) −50.6285 −1.62810 −0.814051 0.580794i \(-0.802742\pi\)
−0.814051 + 0.580794i \(0.802742\pi\)
\(968\) 1.53919 0.0494714
\(969\) 18.6225 0.598240
\(970\) 0 0
\(971\) −6.69263 −0.214777 −0.107388 0.994217i \(-0.534249\pi\)
−0.107388 + 0.994217i \(0.534249\pi\)
\(972\) 2.70928 0.0869000
\(973\) 38.1399 1.22271
\(974\) 76.3956 2.44787
\(975\) 0 0
\(976\) 30.5113 0.976643
\(977\) 38.5835 1.23440 0.617198 0.786808i \(-0.288268\pi\)
0.617198 + 0.786808i \(0.288268\pi\)
\(978\) 20.0989 0.642692
\(979\) −5.02052 −0.160456
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −17.3607 −0.554002
\(983\) −28.4657 −0.907916 −0.453958 0.891023i \(-0.649988\pi\)
−0.453958 + 0.891023i \(0.649988\pi\)
\(984\) 14.7382 0.469837
\(985\) 0 0
\(986\) −69.0759 −2.19983
\(987\) 10.1568 0.323293
\(988\) 14.2557 0.453533
\(989\) −12.5113 −0.397836
\(990\) 0 0
\(991\) 2.65368 0.0842970 0.0421485 0.999111i \(-0.486580\pi\)
0.0421485 + 0.999111i \(0.486580\pi\)
\(992\) 48.1133 1.52760
\(993\) −6.34017 −0.201199
\(994\) −58.2388 −1.84722
\(995\) 0 0
\(996\) −21.3112 −0.675273
\(997\) −8.08065 −0.255917 −0.127958 0.991780i \(-0.540842\pi\)
−0.127958 + 0.991780i \(0.540842\pi\)
\(998\) 56.7624 1.79678
\(999\) −3.41855 −0.108158
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 825.2.a.l.1.3 3
3.2 odd 2 2475.2.a.ba.1.1 3
5.2 odd 4 165.2.c.b.34.6 yes 6
5.3 odd 4 165.2.c.b.34.1 6
5.4 even 2 825.2.a.j.1.1 3
11.10 odd 2 9075.2.a.cg.1.1 3
15.2 even 4 495.2.c.e.199.1 6
15.8 even 4 495.2.c.e.199.6 6
15.14 odd 2 2475.2.a.bc.1.3 3
20.3 even 4 2640.2.d.h.529.3 6
20.7 even 4 2640.2.d.h.529.6 6
55.32 even 4 1815.2.c.e.364.1 6
55.43 even 4 1815.2.c.e.364.6 6
55.54 odd 2 9075.2.a.ch.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.b.34.1 6 5.3 odd 4
165.2.c.b.34.6 yes 6 5.2 odd 4
495.2.c.e.199.1 6 15.2 even 4
495.2.c.e.199.6 6 15.8 even 4
825.2.a.j.1.1 3 5.4 even 2
825.2.a.l.1.3 3 1.1 even 1 trivial
1815.2.c.e.364.1 6 55.32 even 4
1815.2.c.e.364.6 6 55.43 even 4
2475.2.a.ba.1.1 3 3.2 odd 2
2475.2.a.bc.1.3 3 15.14 odd 2
2640.2.d.h.529.3 6 20.3 even 4
2640.2.d.h.529.6 6 20.7 even 4
9075.2.a.cg.1.1 3 11.10 odd 2
9075.2.a.ch.1.3 3 55.54 odd 2