Properties

Label 825.2.c.f.199.4
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(-1.75233 + 1.75233i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.f.199.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.36333i q^{2} -1.00000i q^{3} +0.141336 q^{4} +1.36333 q^{6} +2.50466i q^{7} +2.91934i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.36333i q^{2} -1.00000i q^{3} +0.141336 q^{4} +1.36333 q^{6} +2.50466i q^{7} +2.91934i q^{8} -1.00000 q^{9} +1.00000 q^{11} -0.141336i q^{12} -1.14134i q^{13} -3.41468 q^{14} -3.69735 q^{16} +7.64600i q^{17} -1.36333i q^{18} -1.77801 q^{19} +2.50466 q^{21} +1.36333i q^{22} -1.41468i q^{23} +2.91934 q^{24} +1.55602 q^{26} +1.00000i q^{27} +0.354000i q^{28} +0.726656 q^{29} +2.85866 q^{31} +0.797984i q^{32} -1.00000i q^{33} -10.4240 q^{34} -0.141336 q^{36} -8.42401i q^{37} -2.42401i q^{38} -1.14134 q^{39} +0.636672 q^{41} +3.41468i q^{42} +12.6974i q^{43} +0.141336 q^{44} +1.92867 q^{46} +6.14134i q^{47} +3.69735i q^{48} +0.726656 q^{49} +7.64600 q^{51} -0.161312i q^{52} +12.0187i q^{53} -1.36333 q^{54} -7.31198 q^{56} +1.77801i q^{57} +0.990671i q^{58} +3.41468 q^{59} +4.59465 q^{61} +3.89730i q^{62} -2.50466i q^{63} -8.48262 q^{64} +1.36333 q^{66} -9.32131i q^{67} +1.08066i q^{68} -1.41468 q^{69} +5.85866 q^{71} -2.91934i q^{72} -7.55602i q^{73} +11.4847 q^{74} -0.251297 q^{76} +2.50466i q^{77} -1.55602i q^{78} -6.91934 q^{79} +1.00000 q^{81} +0.867993i q^{82} +6.17997i q^{83} +0.354000 q^{84} -17.3107 q^{86} -0.726656i q^{87} +2.91934i q^{88} -3.45331 q^{89} +2.85866 q^{91} -0.199945i q^{92} -2.85866i q^{93} -8.37266 q^{94} +0.797984 q^{96} -19.4626i q^{97} +0.990671i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46} - 4 q^{49} + 8 q^{51} - 4 q^{54} - 44 q^{56} + 12 q^{59} - 6 q^{61} - 68 q^{64} + 4 q^{66} + 52 q^{71} - 28 q^{74} - 48 q^{76} - 12 q^{79} + 6 q^{81} + 40 q^{84} + 56 q^{86} - 4 q^{89} + 34 q^{91} - 4 q^{94} + 68 q^{96} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36333i 0.964019i 0.876166 + 0.482009i \(0.160093\pi\)
−0.876166 + 0.482009i \(0.839907\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 0.141336 0.0706681
\(5\) 0 0
\(6\) 1.36333 0.556576
\(7\) 2.50466i 0.946674i 0.880881 + 0.473337i \(0.156951\pi\)
−0.880881 + 0.473337i \(0.843049\pi\)
\(8\) 2.91934i 1.03214i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) − 0.141336i − 0.0408002i
\(13\) − 1.14134i − 0.316550i −0.987395 0.158275i \(-0.949407\pi\)
0.987395 0.158275i \(-0.0505932\pi\)
\(14\) −3.41468 −0.912612
\(15\) 0 0
\(16\) −3.69735 −0.924338
\(17\) 7.64600i 1.85443i 0.374533 + 0.927214i \(0.377803\pi\)
−0.374533 + 0.927214i \(0.622197\pi\)
\(18\) − 1.36333i − 0.321340i
\(19\) −1.77801 −0.407903 −0.203951 0.978981i \(-0.565378\pi\)
−0.203951 + 0.978981i \(0.565378\pi\)
\(20\) 0 0
\(21\) 2.50466 0.546563
\(22\) 1.36333i 0.290663i
\(23\) − 1.41468i − 0.294981i −0.989063 0.147491i \(-0.952880\pi\)
0.989063 0.147491i \(-0.0471196\pi\)
\(24\) 2.91934 0.595909
\(25\) 0 0
\(26\) 1.55602 0.305160
\(27\) 1.00000i 0.192450i
\(28\) 0.354000i 0.0668996i
\(29\) 0.726656 0.134937 0.0674684 0.997721i \(-0.478508\pi\)
0.0674684 + 0.997721i \(0.478508\pi\)
\(30\) 0 0
\(31\) 2.85866 0.513431 0.256716 0.966487i \(-0.417360\pi\)
0.256716 + 0.966487i \(0.417360\pi\)
\(32\) 0.797984i 0.141065i
\(33\) − 1.00000i − 0.174078i
\(34\) −10.4240 −1.78770
\(35\) 0 0
\(36\) −0.141336 −0.0235560
\(37\) − 8.42401i − 1.38490i −0.721467 0.692449i \(-0.756532\pi\)
0.721467 0.692449i \(-0.243468\pi\)
\(38\) − 2.42401i − 0.393226i
\(39\) −1.14134 −0.182760
\(40\) 0 0
\(41\) 0.636672 0.0994314 0.0497157 0.998763i \(-0.484168\pi\)
0.0497157 + 0.998763i \(0.484168\pi\)
\(42\) 3.41468i 0.526897i
\(43\) 12.6974i 1.93633i 0.250317 + 0.968164i \(0.419465\pi\)
−0.250317 + 0.968164i \(0.580535\pi\)
\(44\) 0.141336 0.0213072
\(45\) 0 0
\(46\) 1.92867 0.284367
\(47\) 6.14134i 0.895806i 0.894082 + 0.447903i \(0.147829\pi\)
−0.894082 + 0.447903i \(0.852171\pi\)
\(48\) 3.69735i 0.533667i
\(49\) 0.726656 0.103808
\(50\) 0 0
\(51\) 7.64600 1.07065
\(52\) − 0.161312i − 0.0223700i
\(53\) 12.0187i 1.65089i 0.564483 + 0.825445i \(0.309076\pi\)
−0.564483 + 0.825445i \(0.690924\pi\)
\(54\) −1.36333 −0.185525
\(55\) 0 0
\(56\) −7.31198 −0.977104
\(57\) 1.77801i 0.235503i
\(58\) 0.990671i 0.130082i
\(59\) 3.41468 0.444553 0.222277 0.974984i \(-0.428651\pi\)
0.222277 + 0.974984i \(0.428651\pi\)
\(60\) 0 0
\(61\) 4.59465 0.588285 0.294142 0.955762i \(-0.404966\pi\)
0.294142 + 0.955762i \(0.404966\pi\)
\(62\) 3.89730i 0.494957i
\(63\) − 2.50466i − 0.315558i
\(64\) −8.48262 −1.06033
\(65\) 0 0
\(66\) 1.36333 0.167814
\(67\) − 9.32131i − 1.13878i −0.822068 0.569389i \(-0.807180\pi\)
0.822068 0.569389i \(-0.192820\pi\)
\(68\) 1.08066i 0.131049i
\(69\) −1.41468 −0.170307
\(70\) 0 0
\(71\) 5.85866 0.695295 0.347648 0.937625i \(-0.386981\pi\)
0.347648 + 0.937625i \(0.386981\pi\)
\(72\) − 2.91934i − 0.344048i
\(73\) − 7.55602i − 0.884365i −0.896925 0.442182i \(-0.854204\pi\)
0.896925 0.442182i \(-0.145796\pi\)
\(74\) 11.4847 1.33507
\(75\) 0 0
\(76\) −0.251297 −0.0288257
\(77\) 2.50466i 0.285433i
\(78\) − 1.55602i − 0.176184i
\(79\) −6.91934 −0.778487 −0.389244 0.921135i \(-0.627264\pi\)
−0.389244 + 0.921135i \(0.627264\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.867993i 0.0958537i
\(83\) 6.17997i 0.678340i 0.940725 + 0.339170i \(0.110146\pi\)
−0.940725 + 0.339170i \(0.889854\pi\)
\(84\) 0.354000 0.0386245
\(85\) 0 0
\(86\) −17.3107 −1.86666
\(87\) − 0.726656i − 0.0779058i
\(88\) 2.91934i 0.311203i
\(89\) −3.45331 −0.366050 −0.183025 0.983108i \(-0.558589\pi\)
−0.183025 + 0.983108i \(0.558589\pi\)
\(90\) 0 0
\(91\) 2.85866 0.299669
\(92\) − 0.199945i − 0.0208457i
\(93\) − 2.85866i − 0.296430i
\(94\) −8.37266 −0.863574
\(95\) 0 0
\(96\) 0.797984 0.0814439
\(97\) − 19.4626i − 1.97613i −0.154032 0.988066i \(-0.549226\pi\)
0.154032 0.988066i \(-0.450774\pi\)
\(98\) 0.990671i 0.100073i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −8.37266 −0.833111 −0.416555 0.909110i \(-0.636763\pi\)
−0.416555 + 0.909110i \(0.636763\pi\)
\(102\) 10.4240i 1.03213i
\(103\) − 2.54669i − 0.250933i −0.992098 0.125466i \(-0.959957\pi\)
0.992098 0.125466i \(-0.0400427\pi\)
\(104\) 3.33195 0.326725
\(105\) 0 0
\(106\) −16.3854 −1.59149
\(107\) − 14.5653i − 1.40808i −0.710158 0.704042i \(-0.751376\pi\)
0.710158 0.704042i \(-0.248624\pi\)
\(108\) 0.141336i 0.0136001i
\(109\) −2.41468 −0.231284 −0.115642 0.993291i \(-0.536893\pi\)
−0.115642 + 0.993291i \(0.536893\pi\)
\(110\) 0 0
\(111\) −8.42401 −0.799571
\(112\) − 9.26063i − 0.875047i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) −2.42401 −0.227029
\(115\) 0 0
\(116\) 0.102703 0.00953572
\(117\) 1.14134i 0.105517i
\(118\) 4.65533i 0.428558i
\(119\) −19.1507 −1.75554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 6.26401i 0.567117i
\(123\) − 0.636672i − 0.0574068i
\(124\) 0.404032 0.0362832
\(125\) 0 0
\(126\) 3.41468 0.304204
\(127\) − 15.2020i − 1.34896i −0.738293 0.674480i \(-0.764368\pi\)
0.738293 0.674480i \(-0.235632\pi\)
\(128\) − 9.96862i − 0.881110i
\(129\) 12.6974 1.11794
\(130\) 0 0
\(131\) 18.7453 1.63779 0.818893 0.573946i \(-0.194588\pi\)
0.818893 + 0.573946i \(0.194588\pi\)
\(132\) − 0.141336i − 0.0123017i
\(133\) − 4.45331i − 0.386151i
\(134\) 12.7080 1.09780
\(135\) 0 0
\(136\) −22.3213 −1.91404
\(137\) 8.74531i 0.747163i 0.927597 + 0.373581i \(0.121870\pi\)
−0.927597 + 0.373581i \(0.878130\pi\)
\(138\) − 1.92867i − 0.164180i
\(139\) 22.3013 1.89157 0.945787 0.324787i \(-0.105293\pi\)
0.945787 + 0.324787i \(0.105293\pi\)
\(140\) 0 0
\(141\) 6.14134 0.517194
\(142\) 7.98728i 0.670278i
\(143\) − 1.14134i − 0.0954433i
\(144\) 3.69735 0.308113
\(145\) 0 0
\(146\) 10.3013 0.852544
\(147\) − 0.726656i − 0.0599336i
\(148\) − 1.19062i − 0.0978681i
\(149\) 15.9287 1.30493 0.652464 0.757820i \(-0.273735\pi\)
0.652464 + 0.757820i \(0.273735\pi\)
\(150\) 0 0
\(151\) −9.88665 −0.804564 −0.402282 0.915516i \(-0.631783\pi\)
−0.402282 + 0.915516i \(0.631783\pi\)
\(152\) − 5.19062i − 0.421015i
\(153\) − 7.64600i − 0.618143i
\(154\) −3.41468 −0.275163
\(155\) 0 0
\(156\) −0.161312 −0.0129153
\(157\) − 0.132007i − 0.0105353i −0.999986 0.00526767i \(-0.998323\pi\)
0.999986 0.00526767i \(-0.00167676\pi\)
\(158\) − 9.43334i − 0.750476i
\(159\) 12.0187 0.953142
\(160\) 0 0
\(161\) 3.54330 0.279251
\(162\) 1.36333i 0.107113i
\(163\) 4.31198i 0.337740i 0.985638 + 0.168870i \(0.0540118\pi\)
−0.985638 + 0.168870i \(0.945988\pi\)
\(164\) 0.0899847 0.00702663
\(165\) 0 0
\(166\) −8.42533 −0.653932
\(167\) − 11.2920i − 0.873801i −0.899510 0.436901i \(-0.856076\pi\)
0.899510 0.436901i \(-0.143924\pi\)
\(168\) 7.31198i 0.564131i
\(169\) 11.6974 0.899796
\(170\) 0 0
\(171\) 1.77801 0.135968
\(172\) 1.79459i 0.136837i
\(173\) − 21.6460i − 1.64571i −0.568248 0.822857i \(-0.692379\pi\)
0.568248 0.822857i \(-0.307621\pi\)
\(174\) 0.990671 0.0751026
\(175\) 0 0
\(176\) −3.69735 −0.278698
\(177\) − 3.41468i − 0.256663i
\(178\) − 4.70800i − 0.352879i
\(179\) −16.1413 −1.20646 −0.603230 0.797567i \(-0.706120\pi\)
−0.603230 + 0.797567i \(0.706120\pi\)
\(180\) 0 0
\(181\) 20.7546 1.54268 0.771340 0.636423i \(-0.219587\pi\)
0.771340 + 0.636423i \(0.219587\pi\)
\(182\) 3.89730i 0.288887i
\(183\) − 4.59465i − 0.339646i
\(184\) 4.12994 0.304463
\(185\) 0 0
\(186\) 3.89730 0.285764
\(187\) 7.64600i 0.559131i
\(188\) 0.867993i 0.0633049i
\(189\) −2.50466 −0.182188
\(190\) 0 0
\(191\) 3.31198 0.239646 0.119823 0.992795i \(-0.461767\pi\)
0.119823 + 0.992795i \(0.461767\pi\)
\(192\) 8.48262i 0.612180i
\(193\) 2.13201i 0.153465i 0.997052 + 0.0767326i \(0.0244488\pi\)
−0.997052 + 0.0767326i \(0.975551\pi\)
\(194\) 26.5340 1.90503
\(195\) 0 0
\(196\) 0.102703 0.00733591
\(197\) 17.9287i 1.27737i 0.769470 + 0.638683i \(0.220520\pi\)
−0.769470 + 0.638683i \(0.779480\pi\)
\(198\) − 1.36333i − 0.0968875i
\(199\) −15.1600 −1.07466 −0.537332 0.843371i \(-0.680568\pi\)
−0.537332 + 0.843371i \(0.680568\pi\)
\(200\) 0 0
\(201\) −9.32131 −0.657474
\(202\) − 11.4147i − 0.803134i
\(203\) 1.82003i 0.127741i
\(204\) 1.08066 0.0756611
\(205\) 0 0
\(206\) 3.47197 0.241904
\(207\) 1.41468i 0.0983270i
\(208\) 4.21992i 0.292599i
\(209\) −1.77801 −0.122987
\(210\) 0 0
\(211\) −13.3213 −0.917076 −0.458538 0.888675i \(-0.651627\pi\)
−0.458538 + 0.888675i \(0.651627\pi\)
\(212\) 1.69867i 0.116665i
\(213\) − 5.85866i − 0.401429i
\(214\) 19.8573 1.35742
\(215\) 0 0
\(216\) −2.91934 −0.198636
\(217\) 7.15999i 0.486052i
\(218\) − 3.29200i − 0.222962i
\(219\) −7.55602 −0.510588
\(220\) 0 0
\(221\) 8.72666 0.587018
\(222\) − 11.4847i − 0.770802i
\(223\) 12.2534i 0.820546i 0.911963 + 0.410273i \(0.134567\pi\)
−0.911963 + 0.410273i \(0.865433\pi\)
\(224\) −1.99868 −0.133543
\(225\) 0 0
\(226\) 8.17997 0.544123
\(227\) 8.74531i 0.580447i 0.956959 + 0.290223i \(0.0937296\pi\)
−0.956959 + 0.290223i \(0.906270\pi\)
\(228\) 0.251297i 0.0166425i
\(229\) 11.4626 0.757473 0.378736 0.925505i \(-0.376359\pi\)
0.378736 + 0.925505i \(0.376359\pi\)
\(230\) 0 0
\(231\) 2.50466 0.164795
\(232\) 2.12136i 0.139274i
\(233\) − 7.48469i − 0.490338i −0.969480 0.245169i \(-0.921157\pi\)
0.969480 0.245169i \(-0.0788435\pi\)
\(234\) −1.55602 −0.101720
\(235\) 0 0
\(236\) 0.482618 0.0314157
\(237\) 6.91934i 0.449460i
\(238\) − 26.1086i − 1.69237i
\(239\) −12.9066 −0.834860 −0.417430 0.908709i \(-0.637069\pi\)
−0.417430 + 0.908709i \(0.637069\pi\)
\(240\) 0 0
\(241\) −16.8773 −1.08716 −0.543582 0.839356i \(-0.682932\pi\)
−0.543582 + 0.839356i \(0.682932\pi\)
\(242\) 1.36333i 0.0876381i
\(243\) − 1.00000i − 0.0641500i
\(244\) 0.649390 0.0415729
\(245\) 0 0
\(246\) 0.867993 0.0553412
\(247\) 2.02930i 0.129122i
\(248\) 8.34542i 0.529935i
\(249\) 6.17997 0.391640
\(250\) 0 0
\(251\) 2.28267 0.144081 0.0720405 0.997402i \(-0.477049\pi\)
0.0720405 + 0.997402i \(0.477049\pi\)
\(252\) − 0.354000i − 0.0222999i
\(253\) − 1.41468i − 0.0889401i
\(254\) 20.7253 1.30042
\(255\) 0 0
\(256\) −3.37473 −0.210920
\(257\) − 26.8667i − 1.67590i −0.545749 0.837949i \(-0.683755\pi\)
0.545749 0.837949i \(-0.316245\pi\)
\(258\) 17.3107i 1.07771i
\(259\) 21.0993 1.31105
\(260\) 0 0
\(261\) −0.726656 −0.0449789
\(262\) 25.5560i 1.57886i
\(263\) − 12.0187i − 0.741102i −0.928812 0.370551i \(-0.879169\pi\)
0.928812 0.370551i \(-0.120831\pi\)
\(264\) 2.91934 0.179673
\(265\) 0 0
\(266\) 6.07133 0.372257
\(267\) 3.45331i 0.211339i
\(268\) − 1.31744i − 0.0804753i
\(269\) 14.3854 0.877092 0.438546 0.898709i \(-0.355494\pi\)
0.438546 + 0.898709i \(0.355494\pi\)
\(270\) 0 0
\(271\) 28.6553 1.74069 0.870344 0.492445i \(-0.163897\pi\)
0.870344 + 0.492445i \(0.163897\pi\)
\(272\) − 28.2700i − 1.71412i
\(273\) − 2.85866i − 0.173014i
\(274\) −11.9227 −0.720279
\(275\) 0 0
\(276\) −0.199945 −0.0120353
\(277\) − 29.1600i − 1.75205i −0.482262 0.876027i \(-0.660185\pi\)
0.482262 0.876027i \(-0.339815\pi\)
\(278\) 30.4040i 1.82351i
\(279\) −2.85866 −0.171144
\(280\) 0 0
\(281\) −5.36333 −0.319949 −0.159975 0.987121i \(-0.551141\pi\)
−0.159975 + 0.987121i \(0.551141\pi\)
\(282\) 8.37266i 0.498584i
\(283\) 10.0420i 0.596936i 0.954420 + 0.298468i \(0.0964757\pi\)
−0.954420 + 0.298468i \(0.903524\pi\)
\(284\) 0.828041 0.0491352
\(285\) 0 0
\(286\) 1.55602 0.0920091
\(287\) 1.59465i 0.0941292i
\(288\) − 0.797984i − 0.0470216i
\(289\) −41.4613 −2.43890
\(290\) 0 0
\(291\) −19.4626 −1.14092
\(292\) − 1.06794i − 0.0624963i
\(293\) − 20.3540i − 1.18909i −0.804061 0.594547i \(-0.797332\pi\)
0.804061 0.594547i \(-0.202668\pi\)
\(294\) 0.990671 0.0577771
\(295\) 0 0
\(296\) 24.5926 1.42941
\(297\) 1.00000i 0.0580259i
\(298\) 21.7160i 1.25797i
\(299\) −1.61462 −0.0933762
\(300\) 0 0
\(301\) −31.8026 −1.83307
\(302\) − 13.4787i − 0.775615i
\(303\) 8.37266i 0.480997i
\(304\) 6.57392 0.377040
\(305\) 0 0
\(306\) 10.4240 0.595901
\(307\) 25.6226i 1.46236i 0.682184 + 0.731181i \(0.261030\pi\)
−0.682184 + 0.731181i \(0.738970\pi\)
\(308\) 0.354000i 0.0201710i
\(309\) −2.54669 −0.144876
\(310\) 0 0
\(311\) 11.9414 0.677134 0.338567 0.940942i \(-0.390058\pi\)
0.338567 + 0.940942i \(0.390058\pi\)
\(312\) − 3.33195i − 0.188635i
\(313\) − 17.1986i − 0.972124i −0.873924 0.486062i \(-0.838433\pi\)
0.873924 0.486062i \(-0.161567\pi\)
\(314\) 0.179969 0.0101563
\(315\) 0 0
\(316\) −0.977953 −0.0550142
\(317\) − 22.7453i − 1.27750i −0.769413 0.638752i \(-0.779451\pi\)
0.769413 0.638752i \(-0.220549\pi\)
\(318\) 16.3854i 0.918846i
\(319\) 0.726656 0.0406850
\(320\) 0 0
\(321\) −14.5653 −0.812958
\(322\) 4.83068i 0.269203i
\(323\) − 13.5946i − 0.756427i
\(324\) 0.141336 0.00785201
\(325\) 0 0
\(326\) −5.87864 −0.325588
\(327\) 2.41468i 0.133532i
\(328\) 1.85866i 0.102628i
\(329\) −15.3820 −0.848036
\(330\) 0 0
\(331\) 25.2733 1.38915 0.694574 0.719421i \(-0.255593\pi\)
0.694574 + 0.719421i \(0.255593\pi\)
\(332\) 0.873453i 0.0479370i
\(333\) 8.42401i 0.461633i
\(334\) 15.3947 0.842361
\(335\) 0 0
\(336\) −9.26063 −0.505209
\(337\) − 17.6880i − 0.963528i −0.876301 0.481764i \(-0.839996\pi\)
0.876301 0.481764i \(-0.160004\pi\)
\(338\) 15.9473i 0.867420i
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 2.85866 0.154805
\(342\) 2.42401i 0.131075i
\(343\) 19.3527i 1.04495i
\(344\) −37.0679 −1.99857
\(345\) 0 0
\(346\) 29.5106 1.58650
\(347\) 20.1027i 1.07917i 0.841931 + 0.539585i \(0.181419\pi\)
−0.841931 + 0.539585i \(0.818581\pi\)
\(348\) − 0.102703i − 0.00550545i
\(349\) 10.9907 0.588317 0.294159 0.955757i \(-0.404961\pi\)
0.294159 + 0.955757i \(0.404961\pi\)
\(350\) 0 0
\(351\) 1.14134 0.0609200
\(352\) 0.797984i 0.0425327i
\(353\) − 16.7267i − 0.890270i −0.895463 0.445135i \(-0.853156\pi\)
0.895463 0.445135i \(-0.146844\pi\)
\(354\) 4.65533 0.247428
\(355\) 0 0
\(356\) −0.488078 −0.0258681
\(357\) 19.1507i 1.01356i
\(358\) − 22.0059i − 1.16305i
\(359\) 13.5933 0.717429 0.358714 0.933447i \(-0.383215\pi\)
0.358714 + 0.933447i \(0.383215\pi\)
\(360\) 0 0
\(361\) −15.8387 −0.833615
\(362\) 28.2954i 1.48717i
\(363\) − 1.00000i − 0.0524864i
\(364\) 0.404032 0.0211771
\(365\) 0 0
\(366\) 6.26401 0.327425
\(367\) 28.9987i 1.51372i 0.653578 + 0.756859i \(0.273267\pi\)
−0.653578 + 0.756859i \(0.726733\pi\)
\(368\) 5.23057i 0.272662i
\(369\) −0.636672 −0.0331438
\(370\) 0 0
\(371\) −30.1027 −1.56285
\(372\) − 0.404032i − 0.0209481i
\(373\) 17.8867i 0.926136i 0.886323 + 0.463068i \(0.153251\pi\)
−0.886323 + 0.463068i \(0.846749\pi\)
\(374\) −10.4240 −0.539013
\(375\) 0 0
\(376\) −17.9287 −0.924601
\(377\) − 0.829359i − 0.0427142i
\(378\) − 3.41468i − 0.175632i
\(379\) −23.7839 −1.22170 −0.610850 0.791747i \(-0.709172\pi\)
−0.610850 + 0.791747i \(0.709172\pi\)
\(380\) 0 0
\(381\) −15.2020 −0.778823
\(382\) 4.51531i 0.231023i
\(383\) 31.7360i 1.62163i 0.585300 + 0.810817i \(0.300977\pi\)
−0.585300 + 0.810817i \(0.699023\pi\)
\(384\) −9.96862 −0.508709
\(385\) 0 0
\(386\) −2.90663 −0.147943
\(387\) − 12.6974i − 0.645443i
\(388\) − 2.75077i − 0.139649i
\(389\) −25.7546 −1.30581 −0.652906 0.757439i \(-0.726450\pi\)
−0.652906 + 0.757439i \(0.726450\pi\)
\(390\) 0 0
\(391\) 10.8166 0.547021
\(392\) 2.12136i 0.107145i
\(393\) − 18.7453i − 0.945576i
\(394\) −24.4427 −1.23140
\(395\) 0 0
\(396\) −0.141336 −0.00710241
\(397\) − 3.03863i − 0.152505i −0.997089 0.0762523i \(-0.975705\pi\)
0.997089 0.0762523i \(-0.0242954\pi\)
\(398\) − 20.6680i − 1.03600i
\(399\) −4.45331 −0.222945
\(400\) 0 0
\(401\) −18.1800 −0.907864 −0.453932 0.891036i \(-0.649979\pi\)
−0.453932 + 0.891036i \(0.649979\pi\)
\(402\) − 12.7080i − 0.633817i
\(403\) − 3.26270i − 0.162526i
\(404\) −1.18336 −0.0588743
\(405\) 0 0
\(406\) −2.48130 −0.123145
\(407\) − 8.42401i − 0.417563i
\(408\) 22.3213i 1.10507i
\(409\) 20.5946 1.01834 0.509170 0.860666i \(-0.329952\pi\)
0.509170 + 0.860666i \(0.329952\pi\)
\(410\) 0 0
\(411\) 8.74531 0.431375
\(412\) − 0.359939i − 0.0177329i
\(413\) 8.55263i 0.420847i
\(414\) −1.92867 −0.0947891
\(415\) 0 0
\(416\) 0.910768 0.0446541
\(417\) − 22.3013i − 1.09210i
\(418\) − 2.42401i − 0.118562i
\(419\) −32.7253 −1.59874 −0.799369 0.600841i \(-0.794833\pi\)
−0.799369 + 0.600841i \(0.794833\pi\)
\(420\) 0 0
\(421\) 32.5640 1.58707 0.793537 0.608522i \(-0.208237\pi\)
0.793537 + 0.608522i \(0.208237\pi\)
\(422\) − 18.1613i − 0.884079i
\(423\) − 6.14134i − 0.298602i
\(424\) −35.0866 −1.70396
\(425\) 0 0
\(426\) 7.98728 0.386985
\(427\) 11.5081i 0.556914i
\(428\) − 2.05861i − 0.0995066i
\(429\) −1.14134 −0.0551042
\(430\) 0 0
\(431\) 1.82003 0.0876678 0.0438339 0.999039i \(-0.486043\pi\)
0.0438339 + 0.999039i \(0.486043\pi\)
\(432\) − 3.69735i − 0.177889i
\(433\) 29.8280i 1.43344i 0.697359 + 0.716722i \(0.254358\pi\)
−0.697359 + 0.716722i \(0.745642\pi\)
\(434\) −9.76142 −0.468563
\(435\) 0 0
\(436\) −0.341281 −0.0163444
\(437\) 2.51531i 0.120324i
\(438\) − 10.3013i − 0.492217i
\(439\) −18.2220 −0.869688 −0.434844 0.900506i \(-0.643197\pi\)
−0.434844 + 0.900506i \(0.643197\pi\)
\(440\) 0 0
\(441\) −0.726656 −0.0346027
\(442\) 11.8973i 0.565897i
\(443\) 22.2627i 1.05773i 0.848705 + 0.528866i \(0.177383\pi\)
−0.848705 + 0.528866i \(0.822617\pi\)
\(444\) −1.19062 −0.0565042
\(445\) 0 0
\(446\) −16.7054 −0.791022
\(447\) − 15.9287i − 0.753400i
\(448\) − 21.2461i − 1.00378i
\(449\) −25.8760 −1.22116 −0.610582 0.791953i \(-0.709064\pi\)
−0.610582 + 0.791953i \(0.709064\pi\)
\(450\) 0 0
\(451\) 0.636672 0.0299797
\(452\) − 0.848017i − 0.0398874i
\(453\) 9.88665i 0.464515i
\(454\) −11.9227 −0.559562
\(455\) 0 0
\(456\) −5.19062 −0.243073
\(457\) − 10.4626i − 0.489422i −0.969596 0.244711i \(-0.921307\pi\)
0.969596 0.244711i \(-0.0786930\pi\)
\(458\) 15.6273i 0.730218i
\(459\) −7.64600 −0.356885
\(460\) 0 0
\(461\) 11.8387 0.551383 0.275691 0.961246i \(-0.411093\pi\)
0.275691 + 0.961246i \(0.411093\pi\)
\(462\) 3.41468i 0.158865i
\(463\) − 27.7546i − 1.28987i −0.764238 0.644934i \(-0.776885\pi\)
0.764238 0.644934i \(-0.223115\pi\)
\(464\) −2.68670 −0.124727
\(465\) 0 0
\(466\) 10.2041 0.472695
\(467\) 20.8294i 0.963868i 0.876208 + 0.481934i \(0.160065\pi\)
−0.876208 + 0.481934i \(0.839935\pi\)
\(468\) 0.161312i 0.00745665i
\(469\) 23.3467 1.07805
\(470\) 0 0
\(471\) −0.132007 −0.00608258
\(472\) 9.96862i 0.458843i
\(473\) 12.6974i 0.583825i
\(474\) −9.43334 −0.433288
\(475\) 0 0
\(476\) −2.70668 −0.124061
\(477\) − 12.0187i − 0.550297i
\(478\) − 17.5960i − 0.804821i
\(479\) −2.64939 −0.121054 −0.0605269 0.998167i \(-0.519278\pi\)
−0.0605269 + 0.998167i \(0.519278\pi\)
\(480\) 0 0
\(481\) −9.61462 −0.438389
\(482\) − 23.0093i − 1.04805i
\(483\) − 3.54330i − 0.161226i
\(484\) 0.141336 0.00642437
\(485\) 0 0
\(486\) 1.36333 0.0618418
\(487\) 40.5360i 1.83686i 0.395580 + 0.918432i \(0.370544\pi\)
−0.395580 + 0.918432i \(0.629456\pi\)
\(488\) 13.4134i 0.607194i
\(489\) 4.31198 0.194994
\(490\) 0 0
\(491\) −16.1214 −0.727547 −0.363773 0.931487i \(-0.618512\pi\)
−0.363773 + 0.931487i \(0.618512\pi\)
\(492\) − 0.0899847i − 0.00405682i
\(493\) 5.55602i 0.250230i
\(494\) −2.76661 −0.124476
\(495\) 0 0
\(496\) −10.5695 −0.474584
\(497\) 14.6740i 0.658218i
\(498\) 8.42533i 0.377548i
\(499\) 28.0666 1.25643 0.628217 0.778038i \(-0.283785\pi\)
0.628217 + 0.778038i \(0.283785\pi\)
\(500\) 0 0
\(501\) −11.2920 −0.504489
\(502\) 3.11203i 0.138897i
\(503\) − 0.906626i − 0.0404245i −0.999796 0.0202122i \(-0.993566\pi\)
0.999796 0.0202122i \(-0.00643419\pi\)
\(504\) 7.31198 0.325701
\(505\) 0 0
\(506\) 1.92867 0.0857400
\(507\) − 11.6974i − 0.519498i
\(508\) − 2.14859i − 0.0953284i
\(509\) 14.3013 0.633895 0.316948 0.948443i \(-0.397342\pi\)
0.316948 + 0.948443i \(0.397342\pi\)
\(510\) 0 0
\(511\) 18.9253 0.837205
\(512\) − 24.5381i − 1.08444i
\(513\) − 1.77801i − 0.0785010i
\(514\) 36.6281 1.61560
\(515\) 0 0
\(516\) 1.79459 0.0790026
\(517\) 6.14134i 0.270096i
\(518\) 28.7653i 1.26387i
\(519\) −21.6460 −0.950154
\(520\) 0 0
\(521\) 20.1214 0.881533 0.440766 0.897622i \(-0.354707\pi\)
0.440766 + 0.897622i \(0.354707\pi\)
\(522\) − 0.990671i − 0.0433605i
\(523\) 12.8459i 0.561714i 0.959750 + 0.280857i \(0.0906187\pi\)
−0.959750 + 0.280857i \(0.909381\pi\)
\(524\) 2.64939 0.115739
\(525\) 0 0
\(526\) 16.3854 0.714436
\(527\) 21.8573i 0.952121i
\(528\) 3.69735i 0.160907i
\(529\) 20.9987 0.912986
\(530\) 0 0
\(531\) −3.41468 −0.148184
\(532\) − 0.629414i − 0.0272886i
\(533\) − 0.726656i − 0.0314750i
\(534\) −4.70800 −0.203735
\(535\) 0 0
\(536\) 27.2121 1.17538
\(537\) 16.1413i 0.696550i
\(538\) 19.6120i 0.845533i
\(539\) 0.726656 0.0312993
\(540\) 0 0
\(541\) 17.6040 0.756854 0.378427 0.925631i \(-0.376465\pi\)
0.378427 + 0.925631i \(0.376465\pi\)
\(542\) 39.0666i 1.67805i
\(543\) − 20.7546i − 0.890667i
\(544\) −6.10138 −0.261595
\(545\) 0 0
\(546\) 3.89730 0.166789
\(547\) 19.9873i 0.854594i 0.904111 + 0.427297i \(0.140534\pi\)
−0.904111 + 0.427297i \(0.859466\pi\)
\(548\) 1.23603i 0.0528005i
\(549\) −4.59465 −0.196095
\(550\) 0 0
\(551\) −1.29200 −0.0550411
\(552\) − 4.12994i − 0.175782i
\(553\) − 17.3306i − 0.736974i
\(554\) 39.7546 1.68901
\(555\) 0 0
\(556\) 3.15198 0.133674
\(557\) − 41.5933i − 1.76237i −0.472775 0.881183i \(-0.656748\pi\)
0.472775 0.881183i \(-0.343252\pi\)
\(558\) − 3.89730i − 0.164986i
\(559\) 14.4919 0.612944
\(560\) 0 0
\(561\) 7.64600 0.322814
\(562\) − 7.31198i − 0.308437i
\(563\) − 2.28267i − 0.0962032i −0.998842 0.0481016i \(-0.984683\pi\)
0.998842 0.0481016i \(-0.0153171\pi\)
\(564\) 0.867993 0.0365491
\(565\) 0 0
\(566\) −13.6906 −0.575458
\(567\) 2.50466i 0.105186i
\(568\) 17.1035i 0.717645i
\(569\) 2.37266 0.0994670 0.0497335 0.998763i \(-0.484163\pi\)
0.0497335 + 0.998763i \(0.484163\pi\)
\(570\) 0 0
\(571\) −31.9053 −1.33520 −0.667598 0.744522i \(-0.732677\pi\)
−0.667598 + 0.744522i \(0.732677\pi\)
\(572\) − 0.161312i − 0.00674479i
\(573\) − 3.31198i − 0.138360i
\(574\) −2.17403 −0.0907423
\(575\) 0 0
\(576\) 8.48262 0.353442
\(577\) 3.28267i 0.136659i 0.997663 + 0.0683297i \(0.0217670\pi\)
−0.997663 + 0.0683297i \(0.978233\pi\)
\(578\) − 56.5254i − 2.35115i
\(579\) 2.13201 0.0886032
\(580\) 0 0
\(581\) −15.4787 −0.642167
\(582\) − 26.5340i − 1.09987i
\(583\) 12.0187i 0.497762i
\(584\) 22.0586 0.912792
\(585\) 0 0
\(586\) 27.7492 1.14631
\(587\) 26.3400i 1.08717i 0.839355 + 0.543583i \(0.182933\pi\)
−0.839355 + 0.543583i \(0.817067\pi\)
\(588\) − 0.102703i − 0.00423539i
\(589\) −5.08273 −0.209430
\(590\) 0 0
\(591\) 17.9287 0.737487
\(592\) 31.1465i 1.28011i
\(593\) 32.9694i 1.35389i 0.736034 + 0.676945i \(0.236697\pi\)
−0.736034 + 0.676945i \(0.763303\pi\)
\(594\) −1.36333 −0.0559380
\(595\) 0 0
\(596\) 2.25130 0.0922167
\(597\) 15.1600i 0.620457i
\(598\) − 2.20126i − 0.0900164i
\(599\) −24.5454 −1.00290 −0.501448 0.865188i \(-0.667199\pi\)
−0.501448 + 0.865188i \(0.667199\pi\)
\(600\) 0 0
\(601\) −9.70668 −0.395944 −0.197972 0.980208i \(-0.563435\pi\)
−0.197972 + 0.980208i \(0.563435\pi\)
\(602\) − 43.3574i − 1.76712i
\(603\) 9.32131i 0.379593i
\(604\) −1.39734 −0.0568570
\(605\) 0 0
\(606\) −11.4147 −0.463690
\(607\) 1.73599i 0.0704615i 0.999379 + 0.0352307i \(0.0112166\pi\)
−0.999379 + 0.0352307i \(0.988783\pi\)
\(608\) − 1.41882i − 0.0575408i
\(609\) 1.82003 0.0737514
\(610\) 0 0
\(611\) 7.00933 0.283567
\(612\) − 1.08066i − 0.0436829i
\(613\) − 12.0187i − 0.485429i −0.970098 0.242715i \(-0.921962\pi\)
0.970098 0.242715i \(-0.0780378\pi\)
\(614\) −34.9321 −1.40974
\(615\) 0 0
\(616\) −7.31198 −0.294608
\(617\) 42.1587i 1.69724i 0.528999 + 0.848622i \(0.322567\pi\)
−0.528999 + 0.848622i \(0.677433\pi\)
\(618\) − 3.47197i − 0.139663i
\(619\) −9.03863 −0.363293 −0.181647 0.983364i \(-0.558143\pi\)
−0.181647 + 0.983364i \(0.558143\pi\)
\(620\) 0 0
\(621\) 1.41468 0.0567691
\(622\) 16.2800i 0.652770i
\(623\) − 8.64939i − 0.346530i
\(624\) 4.21992 0.168932
\(625\) 0 0
\(626\) 23.4474 0.937146
\(627\) 1.77801i 0.0710068i
\(628\) − 0.0186574i 0 0.000744512i
\(629\) 64.4100 2.56819
\(630\) 0 0
\(631\) 10.9614 0.436365 0.218183 0.975908i \(-0.429987\pi\)
0.218183 + 0.975908i \(0.429987\pi\)
\(632\) − 20.1999i − 0.803511i
\(633\) 13.3213i 0.529474i
\(634\) 31.0093 1.23154
\(635\) 0 0
\(636\) 1.69867 0.0673567
\(637\) − 0.829359i − 0.0328604i
\(638\) 0.990671i 0.0392211i
\(639\) −5.85866 −0.231765
\(640\) 0 0
\(641\) −44.1587 −1.74416 −0.872081 0.489361i \(-0.837230\pi\)
−0.872081 + 0.489361i \(0.837230\pi\)
\(642\) − 19.8573i − 0.783707i
\(643\) − 22.5840i − 0.890626i −0.895375 0.445313i \(-0.853092\pi\)
0.895375 0.445313i \(-0.146908\pi\)
\(644\) 0.500796 0.0197341
\(645\) 0 0
\(646\) 18.5340 0.729209
\(647\) − 13.3561i − 0.525081i −0.964921 0.262541i \(-0.915440\pi\)
0.964921 0.262541i \(-0.0845604\pi\)
\(648\) 2.91934i 0.114683i
\(649\) 3.41468 0.134038
\(650\) 0 0
\(651\) 7.15999 0.280622
\(652\) 0.609438i 0.0238674i
\(653\) − 37.5933i − 1.47114i −0.677448 0.735570i \(-0.736914\pi\)
0.677448 0.735570i \(-0.263086\pi\)
\(654\) −3.29200 −0.128727
\(655\) 0 0
\(656\) −2.35400 −0.0919082
\(657\) 7.55602i 0.294788i
\(658\) − 20.9707i − 0.817523i
\(659\) −7.00933 −0.273045 −0.136522 0.990637i \(-0.543593\pi\)
−0.136522 + 0.990637i \(0.543593\pi\)
\(660\) 0 0
\(661\) 26.6226 1.03550 0.517750 0.855532i \(-0.326770\pi\)
0.517750 + 0.855532i \(0.326770\pi\)
\(662\) 34.4559i 1.33917i
\(663\) − 8.72666i − 0.338915i
\(664\) −18.0415 −0.700144
\(665\) 0 0
\(666\) −11.4847 −0.445023
\(667\) − 1.02799i − 0.0398038i
\(668\) − 1.59597i − 0.0617498i
\(669\) 12.2534 0.473743
\(670\) 0 0
\(671\) 4.59465 0.177374
\(672\) 1.99868i 0.0771008i
\(673\) − 35.6960i − 1.37598i −0.725720 0.687990i \(-0.758493\pi\)
0.725720 0.687990i \(-0.241507\pi\)
\(674\) 24.1146 0.928859
\(675\) 0 0
\(676\) 1.65326 0.0635869
\(677\) 33.2920i 1.27952i 0.768577 + 0.639758i \(0.220965\pi\)
−0.768577 + 0.639758i \(0.779035\pi\)
\(678\) − 8.17997i − 0.314150i
\(679\) 48.7474 1.87075
\(680\) 0 0
\(681\) 8.74531 0.335121
\(682\) 3.89730i 0.149235i
\(683\) 36.1787i 1.38434i 0.721736 + 0.692169i \(0.243345\pi\)
−0.721736 + 0.692169i \(0.756655\pi\)
\(684\) 0.251297 0.00960857
\(685\) 0 0
\(686\) −26.3841 −1.00735
\(687\) − 11.4626i − 0.437327i
\(688\) − 46.9466i − 1.78982i
\(689\) 13.7173 0.522589
\(690\) 0 0
\(691\) 35.3293 1.34399 0.671995 0.740555i \(-0.265438\pi\)
0.671995 + 0.740555i \(0.265438\pi\)
\(692\) − 3.05936i − 0.116299i
\(693\) − 2.50466i − 0.0951443i
\(694\) −27.4066 −1.04034
\(695\) 0 0
\(696\) 2.12136 0.0804100
\(697\) 4.86799i 0.184388i
\(698\) 14.9839i 0.567149i
\(699\) −7.48469 −0.283097
\(700\) 0 0
\(701\) −49.4006 −1.86584 −0.932918 0.360088i \(-0.882747\pi\)
−0.932918 + 0.360088i \(0.882747\pi\)
\(702\) 1.55602i 0.0587280i
\(703\) 14.9780i 0.564904i
\(704\) −8.48262 −0.319701
\(705\) 0 0
\(706\) 22.8039 0.858237
\(707\) − 20.9707i − 0.788684i
\(708\) − 0.482618i − 0.0181379i
\(709\) 11.1027 0.416971 0.208485 0.978025i \(-0.433147\pi\)
0.208485 + 0.978025i \(0.433147\pi\)
\(710\) 0 0
\(711\) 6.91934 0.259496
\(712\) − 10.0814i − 0.377817i
\(713\) − 4.04409i − 0.151452i
\(714\) −26.1086 −0.977091
\(715\) 0 0
\(716\) −2.28135 −0.0852582
\(717\) 12.9066i 0.482007i
\(718\) 18.5322i 0.691614i
\(719\) 43.9787 1.64013 0.820064 0.572271i \(-0.193938\pi\)
0.820064 + 0.572271i \(0.193938\pi\)
\(720\) 0 0
\(721\) 6.37860 0.237551
\(722\) − 21.5933i − 0.803621i
\(723\) 16.8773i 0.627674i
\(724\) 2.93338 0.109018
\(725\) 0 0
\(726\) 1.36333 0.0505979
\(727\) − 44.1880i − 1.63884i −0.573193 0.819421i \(-0.694295\pi\)
0.573193 0.819421i \(-0.305705\pi\)
\(728\) 8.34542i 0.309302i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −97.0840 −3.59078
\(732\) − 0.649390i − 0.0240021i
\(733\) 47.3879i 1.75031i 0.483840 + 0.875156i \(0.339242\pi\)
−0.483840 + 0.875156i \(0.660758\pi\)
\(734\) −39.5347 −1.45925
\(735\) 0 0
\(736\) 1.12889 0.0416115
\(737\) − 9.32131i − 0.343355i
\(738\) − 0.867993i − 0.0319512i
\(739\) 13.0407 0.479710 0.239855 0.970809i \(-0.422900\pi\)
0.239855 + 0.970809i \(0.422900\pi\)
\(740\) 0 0
\(741\) 2.02930 0.0745484
\(742\) − 41.0399i − 1.50662i
\(743\) 14.1800i 0.520213i 0.965580 + 0.260106i \(0.0837576\pi\)
−0.965580 + 0.260106i \(0.916242\pi\)
\(744\) 8.34542 0.305958
\(745\) 0 0
\(746\) −24.3854 −0.892812
\(747\) − 6.17997i − 0.226113i
\(748\) 1.08066i 0.0395127i
\(749\) 36.4813 1.33300
\(750\) 0 0
\(751\) 19.1120 0.697408 0.348704 0.937233i \(-0.386622\pi\)
0.348704 + 0.937233i \(0.386622\pi\)
\(752\) − 22.7067i − 0.828027i
\(753\) − 2.28267i − 0.0831852i
\(754\) 1.13069 0.0411773
\(755\) 0 0
\(756\) −0.354000 −0.0128748
\(757\) 24.7347i 0.898997i 0.893281 + 0.449498i \(0.148397\pi\)
−0.893281 + 0.449498i \(0.851603\pi\)
\(758\) − 32.4253i − 1.17774i
\(759\) −1.41468 −0.0513496
\(760\) 0 0
\(761\) −9.82003 −0.355976 −0.177988 0.984033i \(-0.556959\pi\)
−0.177988 + 0.984033i \(0.556959\pi\)
\(762\) − 20.7253i − 0.750800i
\(763\) − 6.04796i − 0.218951i
\(764\) 0.468102 0.0169353
\(765\) 0 0
\(766\) −43.2666 −1.56328
\(767\) − 3.89730i − 0.140723i
\(768\) 3.37473i 0.121775i
\(769\) 15.3026 0.551828 0.275914 0.961182i \(-0.411020\pi\)
0.275914 + 0.961182i \(0.411020\pi\)
\(770\) 0 0
\(771\) −26.8667 −0.967580
\(772\) 0.301330i 0.0108451i
\(773\) − 5.36927i − 0.193119i −0.995327 0.0965596i \(-0.969216\pi\)
0.995327 0.0965596i \(-0.0307838\pi\)
\(774\) 17.3107 0.622219
\(775\) 0 0
\(776\) 56.8181 2.03965
\(777\) − 21.0993i − 0.756934i
\(778\) − 35.1120i − 1.25883i
\(779\) −1.13201 −0.0405584
\(780\) 0 0
\(781\) 5.85866 0.209639
\(782\) 14.7466i 0.527338i
\(783\) 0.726656i 0.0259686i
\(784\) −2.68670 −0.0959537
\(785\) 0 0
\(786\) 25.5560 0.911553
\(787\) 15.4767i 0.551684i 0.961203 + 0.275842i \(0.0889567\pi\)
−0.961203 + 0.275842i \(0.911043\pi\)
\(788\) 2.53397i 0.0902689i
\(789\) −12.0187 −0.427876
\(790\) 0 0
\(791\) 15.0280 0.534334
\(792\) − 2.91934i − 0.103734i
\(793\) − 5.24404i − 0.186221i
\(794\) 4.14265 0.147017
\(795\) 0 0
\(796\) −2.14265 −0.0759444
\(797\) − 43.6774i − 1.54713i −0.633716 0.773566i \(-0.718471\pi\)
0.633716 0.773566i \(-0.281529\pi\)
\(798\) − 6.07133i − 0.214923i
\(799\) −46.9567 −1.66121
\(800\) 0 0
\(801\) 3.45331 0.122017
\(802\) − 24.7853i − 0.875198i
\(803\) − 7.55602i − 0.266646i
\(804\) −1.31744 −0.0464624
\(805\) 0 0
\(806\) 4.44813 0.156679
\(807\) − 14.3854i − 0.506389i
\(808\) − 24.4427i − 0.859890i
\(809\) 26.9966 0.949150 0.474575 0.880215i \(-0.342602\pi\)
0.474575 + 0.880215i \(0.342602\pi\)
\(810\) 0 0
\(811\) −14.2220 −0.499402 −0.249701 0.968323i \(-0.580332\pi\)
−0.249701 + 0.968323i \(0.580332\pi\)
\(812\) 0.257236i 0.00902722i
\(813\) − 28.6553i − 1.00499i
\(814\) 11.4847 0.402538
\(815\) 0 0
\(816\) −28.2700 −0.989646
\(817\) − 22.5760i − 0.789834i
\(818\) 28.0773i 0.981699i
\(819\) −2.85866 −0.0998898
\(820\) 0 0
\(821\) 23.7801 0.829930 0.414965 0.909837i \(-0.363794\pi\)
0.414965 + 0.909837i \(0.363794\pi\)
\(822\) 11.9227i 0.415853i
\(823\) 1.84934i 0.0644638i 0.999480 + 0.0322319i \(0.0102615\pi\)
−0.999480 + 0.0322319i \(0.989738\pi\)
\(824\) 7.43466 0.258998
\(825\) 0 0
\(826\) −11.6600 −0.405705
\(827\) 15.6987i 0.545896i 0.962029 + 0.272948i \(0.0879987\pi\)
−0.962029 + 0.272948i \(0.912001\pi\)
\(828\) 0.199945i 0.00694858i
\(829\) 4.34128 0.150779 0.0753895 0.997154i \(-0.475980\pi\)
0.0753895 + 0.997154i \(0.475980\pi\)
\(830\) 0 0
\(831\) −29.1600 −1.01155
\(832\) 9.68152i 0.335646i
\(833\) 5.55602i 0.192505i
\(834\) 30.4040 1.05281
\(835\) 0 0
\(836\) −0.251297 −0.00869128
\(837\) 2.85866i 0.0988099i
\(838\) − 44.6154i − 1.54121i
\(839\) −28.3013 −0.977070 −0.488535 0.872544i \(-0.662469\pi\)
−0.488535 + 0.872544i \(0.662469\pi\)
\(840\) 0 0
\(841\) −28.4720 −0.981792
\(842\) 44.3955i 1.52997i
\(843\) 5.36333i 0.184723i
\(844\) −1.88278 −0.0648080
\(845\) 0 0
\(846\) 8.37266 0.287858
\(847\) 2.50466i 0.0860613i
\(848\) − 44.4372i − 1.52598i
\(849\) 10.0420 0.344641
\(850\) 0 0
\(851\) −11.9173 −0.408519
\(852\) − 0.828041i − 0.0283682i
\(853\) 35.7653i 1.22458i 0.790633 + 0.612290i \(0.209752\pi\)
−0.790633 + 0.612290i \(0.790248\pi\)
\(854\) −15.6893 −0.536875
\(855\) 0 0
\(856\) 42.5213 1.45335
\(857\) − 24.7513i − 0.845487i −0.906249 0.422743i \(-0.861067\pi\)
0.906249 0.422743i \(-0.138933\pi\)
\(858\) − 1.55602i − 0.0531215i
\(859\) 10.8039 0.368625 0.184313 0.982868i \(-0.440994\pi\)
0.184313 + 0.982868i \(0.440994\pi\)
\(860\) 0 0
\(861\) 1.59465 0.0543455
\(862\) 2.48130i 0.0845134i
\(863\) 16.6027i 0.565161i 0.959244 + 0.282581i \(0.0911904\pi\)
−0.959244 + 0.282581i \(0.908810\pi\)
\(864\) −0.797984 −0.0271480
\(865\) 0 0
\(866\) −40.6654 −1.38187
\(867\) 41.4613i 1.40810i
\(868\) 1.01197i 0.0343484i
\(869\) −6.91934 −0.234723
\(870\) 0 0
\(871\) −10.6387 −0.360480
\(872\) − 7.04928i − 0.238719i
\(873\) 19.4626i 0.658711i
\(874\) −3.42920 −0.115994
\(875\) 0 0
\(876\) −1.06794 −0.0360823
\(877\) − 11.1600i − 0.376846i −0.982088 0.188423i \(-0.939662\pi\)
0.982088 0.188423i \(-0.0603376\pi\)
\(878\) − 24.8426i − 0.838396i
\(879\) −20.3540 −0.686523
\(880\) 0 0
\(881\) 54.5254 1.83701 0.918504 0.395413i \(-0.129398\pi\)
0.918504 + 0.395413i \(0.129398\pi\)
\(882\) − 0.990671i − 0.0333576i
\(883\) − 12.7746i − 0.429900i −0.976625 0.214950i \(-0.931041\pi\)
0.976625 0.214950i \(-0.0689589\pi\)
\(884\) 1.23339 0.0414835
\(885\) 0 0
\(886\) −30.3514 −1.01967
\(887\) − 47.1053i − 1.58164i −0.612049 0.790820i \(-0.709655\pi\)
0.612049 0.790820i \(-0.290345\pi\)
\(888\) − 24.5926i − 0.825273i
\(889\) 38.0759 1.27703
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 1.73184i 0.0579864i
\(893\) − 10.9193i − 0.365402i
\(894\) 21.7160 0.726292
\(895\) 0 0
\(896\) 24.9681 0.834124
\(897\) 1.61462i 0.0539108i
\(898\) − 35.2775i − 1.17722i
\(899\) 2.07727 0.0692807
\(900\) 0 0
\(901\) −91.8947 −3.06146
\(902\) 0.867993i 0.0289010i
\(903\) 31.8026i 1.05832i
\(904\) 17.5161 0.582576
\(905\) 0 0
\(906\) −13.4787 −0.447801
\(907\) − 24.7826i − 0.822894i −0.911434 0.411447i \(-0.865024\pi\)
0.911434 0.411447i \(-0.134976\pi\)
\(908\) 1.23603i 0.0410191i
\(909\) 8.37266 0.277704
\(910\) 0 0
\(911\) −6.08273 −0.201530 −0.100765 0.994910i \(-0.532129\pi\)
−0.100765 + 0.994910i \(0.532129\pi\)
\(912\) − 6.57392i − 0.217684i
\(913\) 6.17997i 0.204527i
\(914\) 14.2640 0.471812
\(915\) 0 0
\(916\) 1.62009 0.0535291
\(917\) 46.9507i 1.55045i
\(918\) − 10.4240i − 0.344044i
\(919\) −17.9953 −0.593610 −0.296805 0.954938i \(-0.595921\pi\)
−0.296805 + 0.954938i \(0.595921\pi\)
\(920\) 0 0
\(921\) 25.6226 0.844295
\(922\) 16.1400i 0.531543i
\(923\) − 6.68670i − 0.220096i
\(924\) 0.354000 0.0116457
\(925\) 0 0
\(926\) 37.8387 1.24346
\(927\) 2.54669i 0.0836442i
\(928\) 0.579860i 0.0190348i
\(929\) 16.7453 0.549396 0.274698 0.961531i \(-0.411422\pi\)
0.274698 + 0.961531i \(0.411422\pi\)
\(930\) 0 0
\(931\) −1.29200 −0.0423436
\(932\) − 1.05786i − 0.0346513i
\(933\) − 11.9414i − 0.390944i
\(934\) −28.3973 −0.929187
\(935\) 0 0
\(936\) −3.33195 −0.108908
\(937\) − 31.5853i − 1.03185i −0.856635 0.515924i \(-0.827449\pi\)
0.856635 0.515924i \(-0.172551\pi\)
\(938\) 31.8293i 1.03926i
\(939\) −17.1986 −0.561256
\(940\) 0 0
\(941\) −14.8421 −0.483838 −0.241919 0.970296i \(-0.577777\pi\)
−0.241919 + 0.970296i \(0.577777\pi\)
\(942\) − 0.179969i − 0.00586372i
\(943\) − 0.900687i − 0.0293304i
\(944\) −12.6253 −0.410918
\(945\) 0 0
\(946\) −17.3107 −0.562818
\(947\) 2.68802i 0.0873490i 0.999046 + 0.0436745i \(0.0139065\pi\)
−0.999046 + 0.0436745i \(0.986094\pi\)
\(948\) 0.977953i 0.0317624i
\(949\) −8.62395 −0.279945
\(950\) 0 0
\(951\) −22.7453 −0.737567
\(952\) − 55.9074i − 1.81197i
\(953\) − 14.2700i − 0.462249i −0.972924 0.231125i \(-0.925760\pi\)
0.972924 0.231125i \(-0.0742405\pi\)
\(954\) 16.3854 0.530496
\(955\) 0 0
\(956\) −1.82417 −0.0589980
\(957\) − 0.726656i − 0.0234895i
\(958\) − 3.61199i − 0.116698i
\(959\) −21.9041 −0.707320
\(960\) 0 0
\(961\) −22.8280 −0.736388
\(962\) − 13.1079i − 0.422615i
\(963\) 14.5653i 0.469362i
\(964\) −2.38538 −0.0768278
\(965\) 0 0
\(966\) 4.83068 0.155425
\(967\) 10.0586i 0.323463i 0.986835 + 0.161732i \(0.0517079\pi\)
−0.986835 + 0.161732i \(0.948292\pi\)
\(968\) 2.91934i 0.0938313i
\(969\) −13.5946 −0.436723
\(970\) 0 0
\(971\) −13.4520 −0.431695 −0.215848 0.976427i \(-0.569251\pi\)
−0.215848 + 0.976427i \(0.569251\pi\)
\(972\) − 0.141336i − 0.00453336i
\(973\) 55.8573i 1.79070i
\(974\) −55.2639 −1.77077
\(975\) 0 0
\(976\) −16.9880 −0.543774
\(977\) − 32.8294i − 1.05030i −0.851008 0.525152i \(-0.824008\pi\)
0.851008 0.525152i \(-0.175992\pi\)
\(978\) 5.87864i 0.187978i
\(979\) −3.45331 −0.110368
\(980\) 0 0
\(981\) 2.41468 0.0770948
\(982\) − 21.9787i − 0.701369i
\(983\) − 43.7746i − 1.39619i −0.716003 0.698097i \(-0.754031\pi\)
0.716003 0.698097i \(-0.245969\pi\)
\(984\) 1.85866 0.0592520
\(985\) 0 0
\(986\) −7.57467 −0.241227
\(987\) 15.3820i 0.489614i
\(988\) 0.286814i 0.00912477i
\(989\) 17.9627 0.571180
\(990\) 0 0
\(991\) 42.1507 1.33896 0.669480 0.742830i \(-0.266517\pi\)
0.669480 + 0.742830i \(0.266517\pi\)
\(992\) 2.28117i 0.0724271i
\(993\) − 25.2733i − 0.802025i
\(994\) −20.0055 −0.634535
\(995\) 0 0
\(996\) 0.873453 0.0276764
\(997\) − 17.4347i − 0.552161i −0.961135 0.276081i \(-0.910964\pi\)
0.961135 0.276081i \(-0.0890357\pi\)
\(998\) 38.2640i 1.21123i
\(999\) 8.42401 0.266524
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.c.f.199.4 6
3.2 odd 2 2475.2.c.q.199.3 6
5.2 odd 4 825.2.a.i.1.2 3
5.3 odd 4 825.2.a.m.1.2 yes 3
5.4 even 2 inner 825.2.c.f.199.3 6
15.2 even 4 2475.2.a.bd.1.2 3
15.8 even 4 2475.2.a.z.1.2 3
15.14 odd 2 2475.2.c.q.199.4 6
55.32 even 4 9075.2.a.cj.1.2 3
55.43 even 4 9075.2.a.cd.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.2 3 5.2 odd 4
825.2.a.m.1.2 yes 3 5.3 odd 4
825.2.c.f.199.3 6 5.4 even 2 inner
825.2.c.f.199.4 6 1.1 even 1 trivial
2475.2.a.z.1.2 3 15.8 even 4
2475.2.a.bd.1.2 3 15.2 even 4
2475.2.c.q.199.3 6 3.2 odd 2
2475.2.c.q.199.4 6 15.14 odd 2
9075.2.a.cd.1.2 3 55.43 even 4
9075.2.a.cj.1.2 3 55.32 even 4