Properties

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

$q$-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 \(-0.839907\pi\)
0.876166 0.482009i \(-0.160093\pi\)
\(3\) 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 \(-0.843049\pi\)
0.880881 0.473337i \(-0.156951\pi\)
\(8\) − 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 \(0.0505932\pi\)
−0.987395 + 0.158275i \(0.949407\pi\)
\(14\) −3.41468 −0.912612
\(15\) 0 0
\(16\) −3.69735 −0.924338
\(17\) − 7.64600i − 1.85443i −0.374533 0.927214i \(-0.622197\pi\)
0.374533 0.927214i \(-0.377803\pi\)
\(18\) 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 \(0.0471196\pi\)
−0.989063 + 0.147491i \(0.952880\pi\)
\(24\) 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 \(0.243468\pi\)
−0.721467 + 0.692449i \(0.756532\pi\)
\(38\) 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 \(-0.580535\pi\)
0.250317 0.968164i \(-0.419465\pi\)
\(44\) 0.141336 0.0213072
\(45\) 0 0
\(46\) 1.92867 0.284367
\(47\) − 6.14134i − 0.895806i −0.894082 0.447903i \(-0.852171\pi\)
0.894082 0.447903i \(-0.147829\pi\)
\(48\) − 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 \(-0.690924\pi\)
0.564483 0.825445i \(-0.309076\pi\)
\(54\) −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 \(0.192820\pi\)
−0.822068 + 0.569389i \(0.807180\pi\)
\(68\) − 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 \(0.145796\pi\)
−0.896925 + 0.442182i \(0.854204\pi\)
\(74\) 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 \(-0.889854\pi\)
0.940725 0.339170i \(-0.110146\pi\)
\(84\) 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 \(0.450774\pi\)
−0.154032 + 0.988066i \(0.549226\pi\)
\(98\) − 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 \(0.0400427\pi\)
−0.992098 + 0.125466i \(0.959957\pi\)
\(104\) 3.33195 0.326725
\(105\) 0 0
\(106\) −16.3854 −1.59149
\(107\) 14.5653i 1.40808i 0.710158 + 0.704042i \(0.248624\pi\)
−0.710158 + 0.704042i \(0.751376\pi\)
\(108\) − 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 \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) −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 \(0.235632\pi\)
−0.738293 + 0.674480i \(0.764368\pi\)
\(128\) 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 \(-0.878130\pi\)
0.927597 0.373581i \(-0.121870\pi\)
\(138\) 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 \(0.00167676\pi\)
−0.999986 + 0.00526767i \(0.998323\pi\)
\(158\) 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 \(-0.945988\pi\)
0.985638 0.168870i \(-0.0540118\pi\)
\(164\) 0.0899847 0.00702663
\(165\) 0 0
\(166\) −8.42533 −0.653932
\(167\) 11.2920i 0.873801i 0.899510 + 0.436901i \(0.143924\pi\)
−0.899510 + 0.436901i \(0.856076\pi\)
\(168\) − 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 \(0.307621\pi\)
−0.568248 + 0.822857i \(0.692379\pi\)
\(174\) 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 \(-0.975551\pi\)
0.997052 0.0767326i \(-0.0244488\pi\)
\(194\) 26.5340 1.90503
\(195\) 0 0
\(196\) 0.102703 0.00733591
\(197\) − 17.9287i − 1.27737i −0.769470 0.638683i \(-0.779480\pi\)
0.769470 0.638683i \(-0.220520\pi\)
\(198\) 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 \(-0.865433\pi\)
0.911963 0.410273i \(-0.134567\pi\)
\(224\) −1.99868 −0.133543
\(225\) 0 0
\(226\) 8.17997 0.544123
\(227\) − 8.74531i − 0.580447i −0.956959 0.290223i \(-0.906270\pi\)
0.956959 0.290223i \(-0.0937296\pi\)
\(228\) − 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 \(0.0788435\pi\)
−0.969480 + 0.245169i \(0.921157\pi\)
\(234\) −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 \(0.316245\pi\)
−0.545749 + 0.837949i \(0.683755\pi\)
\(258\) − 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 \(0.120831\pi\)
−0.928812 + 0.370551i \(0.879169\pi\)
\(264\) 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 \(0.339815\pi\)
−0.482262 + 0.876027i \(0.660185\pi\)
\(278\) − 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 \(-0.903524\pi\)
0.954420 0.298468i \(-0.0964757\pi\)
\(284\) 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 \(0.202668\pi\)
−0.804061 + 0.594547i \(0.797332\pi\)
\(294\) 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 \(-0.738970\pi\)
0.682184 0.731181i \(-0.261030\pi\)
\(308\) − 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 \(0.161567\pi\)
−0.873924 + 0.486062i \(0.838433\pi\)
\(314\) 0.179969 0.0101563
\(315\) 0 0
\(316\) −0.977953 −0.0550142
\(317\) 22.7453i 1.27750i 0.769413 + 0.638752i \(0.220549\pi\)
−0.769413 + 0.638752i \(0.779451\pi\)
\(318\) − 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 \(0.160004\pi\)
−0.876301 + 0.481764i \(0.839996\pi\)
\(338\) − 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 \(-0.818581\pi\)
0.841931 0.539585i \(-0.181419\pi\)
\(348\) 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 \(0.146844\pi\)
−0.895463 + 0.445135i \(0.853156\pi\)
\(354\) 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 \(-0.726733\pi\)
0.653578 0.756859i \(-0.273267\pi\)
\(368\) − 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 \(-0.846749\pi\)
0.886323 0.463068i \(-0.153251\pi\)
\(374\) −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 \(-0.699023\pi\)
0.585300 0.810817i \(-0.300977\pi\)
\(384\) −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 \(0.0242954\pi\)
−0.997089 + 0.0762523i \(0.975705\pi\)
\(398\) 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 \(-0.745642\pi\)
0.697359 0.716722i \(-0.254358\pi\)
\(434\) −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 \(-0.822617\pi\)
0.848705 0.528866i \(-0.177383\pi\)
\(444\) −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 \(0.0786930\pi\)
−0.969596 + 0.244711i \(0.921307\pi\)
\(458\) − 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 \(0.223115\pi\)
−0.764238 + 0.644934i \(0.776885\pi\)
\(464\) −2.68670 −0.124727
\(465\) 0 0
\(466\) 10.2041 0.472695
\(467\) − 20.8294i − 0.963868i −0.876208 0.481934i \(-0.839935\pi\)
0.876208 0.481934i \(-0.160065\pi\)
\(468\) − 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 \(-0.629456\pi\)
0.395580 0.918432i \(-0.370544\pi\)
\(488\) − 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 \(0.00643419\pi\)
−0.999796 + 0.0202122i \(0.993566\pi\)
\(504\) 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 \(-0.909381\pi\)
0.959750 0.280857i \(-0.0906187\pi\)
\(524\) 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 \(-0.859466\pi\)
0.904111 0.427297i \(-0.140534\pi\)
\(548\) − 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 \(0.343252\pi\)
−0.472775 + 0.881183i \(0.656748\pi\)
\(558\) 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 \(0.0153171\pi\)
−0.998842 + 0.0481016i \(0.984683\pi\)
\(564\) 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 \(-0.978233\pi\)
0.997663 0.0683297i \(-0.0217670\pi\)
\(578\) 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 \(-0.817067\pi\)
0.839355 0.543583i \(-0.182933\pi\)
\(588\) 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 \(-0.763303\pi\)
0.736034 0.676945i \(-0.236697\pi\)
\(594\) −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 \(-0.988783\pi\)
0.999379 0.0352307i \(-0.0112166\pi\)
\(608\) 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 \(0.0780378\pi\)
−0.970098 + 0.242715i \(0.921962\pi\)
\(614\) −34.9321 −1.40974
\(615\) 0 0
\(616\) −7.31198 −0.294608
\(617\) − 42.1587i − 1.69724i −0.528999 0.848622i \(-0.677433\pi\)
0.528999 0.848622i \(-0.322567\pi\)
\(618\) 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 \(0.146908\pi\)
−0.895375 + 0.445313i \(0.853092\pi\)
\(644\) 0.500796 0.0197341
\(645\) 0 0
\(646\) 18.5340 0.729209
\(647\) 13.3561i 0.525081i 0.964921 + 0.262541i \(0.0845604\pi\)
−0.964921 + 0.262541i \(0.915440\pi\)
\(648\) − 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 \(0.263086\pi\)
−0.677448 + 0.735570i \(0.736914\pi\)
\(654\) −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 \(0.241507\pi\)
−0.725720 + 0.687990i \(0.758493\pi\)
\(674\) 24.1146 0.928859
\(675\) 0 0
\(676\) 1.65326 0.0635869
\(677\) − 33.2920i − 1.27952i −0.768577 0.639758i \(-0.779035\pi\)
0.768577 0.639758i \(-0.220965\pi\)
\(678\) 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 \(-0.756655\pi\)
0.721736 0.692169i \(-0.243345\pi\)
\(684\) 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 \(0.305705\pi\)
−0.573193 + 0.819421i \(0.694295\pi\)
\(728\) − 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 \(-0.660758\pi\)
0.483840 0.875156i \(-0.339242\pi\)
\(734\) −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 \(-0.916242\pi\)
0.965580 0.260106i \(-0.0837576\pi\)
\(744\) 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 \(-0.851603\pi\)
0.893281 0.449498i \(-0.148397\pi\)
\(758\) 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 \(0.0307838\pi\)
−0.995327 + 0.0965596i \(0.969216\pi\)
\(774\) 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 \(-0.911043\pi\)
0.961203 0.275842i \(-0.0889567\pi\)
\(788\) − 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 \(0.281529\pi\)
−0.633716 + 0.773566i \(0.718471\pi\)
\(798\) 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 \(-0.989738\pi\)
0.999480 0.0322319i \(-0.0102615\pi\)
\(824\) 7.43466 0.258998
\(825\) 0 0
\(826\) −11.6600 −0.405705
\(827\) − 15.6987i − 0.545896i −0.962029 0.272948i \(-0.912001\pi\)
0.962029 0.272948i \(-0.0879987\pi\)
\(828\) − 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 \(-0.790248\pi\)
0.790633 0.612290i \(-0.209752\pi\)
\(854\) −15.6893 −0.536875
\(855\) 0 0
\(856\) 42.5213 1.45335
\(857\) 24.7513i 0.845487i 0.906249 + 0.422743i \(0.138933\pi\)
−0.906249 + 0.422743i \(0.861067\pi\)
\(858\) 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 \(-0.908810\pi\)
0.959244 0.282581i \(-0.0911904\pi\)
\(864\) −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 \(0.0603376\pi\)
−0.982088 + 0.188423i \(0.939662\pi\)
\(878\) 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 \(0.0689589\pi\)
−0.976625 + 0.214950i \(0.931041\pi\)
\(884\) 1.23339 0.0414835
\(885\) 0 0
\(886\) −30.3514 −1.01967
\(887\) 47.1053i 1.58164i 0.612049 + 0.790820i \(0.290345\pi\)
−0.612049 + 0.790820i \(0.709655\pi\)
\(888\) 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 \(0.134976\pi\)
−0.911434 + 0.411447i \(0.865024\pi\)
\(908\) − 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 \(0.172551\pi\)
−0.856635 + 0.515924i \(0.827449\pi\)
\(938\) − 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 \(-0.986094\pi\)
0.999046 0.0436745i \(-0.0139065\pi\)
\(948\) − 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 \(0.0742405\pi\)
−0.972924 + 0.231125i \(0.925760\pi\)
\(954\) 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 \(-0.948292\pi\)
0.986835 0.161732i \(-0.0517079\pi\)
\(968\) − 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 \(0.175992\pi\)
−0.851008 + 0.525152i \(0.824008\pi\)
\(978\) − 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 \(0.245969\pi\)
−0.716003 + 0.698097i \(0.754031\pi\)
\(984\) 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 \(0.0890357\pi\)
−0.961135 + 0.276081i \(0.910964\pi\)
\(998\) − 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.199.3 6
3.2 odd 2 2475.2.c.q.199.4 6
5.2 odd 4 825.2.a.m.1.2 yes 3
5.3 odd 4 825.2.a.i.1.2 3
5.4 even 2 inner 825.2.c.f.199.4 6
15.2 even 4 2475.2.a.z.1.2 3
15.8 even 4 2475.2.a.bd.1.2 3
15.14 odd 2 2475.2.c.q.199.3 6
55.32 even 4 9075.2.a.cd.1.2 3
55.43 even 4 9075.2.a.cj.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.2 3 5.3 odd 4
825.2.a.m.1.2 yes 3 5.2 odd 4
825.2.c.f.199.3 6 1.1 even 1 trivial
825.2.c.f.199.4 6 5.4 even 2 inner
2475.2.a.z.1.2 3 15.2 even 4
2475.2.a.bd.1.2 3 15.8 even 4
2475.2.c.q.199.3 6 15.14 odd 2
2475.2.c.q.199.4 6 3.2 odd 2
9075.2.a.cd.1.2 3 55.32 even 4
9075.2.a.cj.1.2 3 55.43 even 4