Properties

Label 1925.2.b.n.1849.6
Level $1925$
Weight $2$
Character 1925.1849
Analytic conductor $15.371$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1925.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.3712023891\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 385)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.6
Root \(0.403032 + 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 1925.1849
Dual form 1925.2.b.n.1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.67513i q^{2} -2.48119i q^{3} -5.15633 q^{4} +6.63752 q^{6} +1.00000i q^{7} -8.44358i q^{8} -3.15633 q^{9} +O(q^{10})\) \(q+2.67513i q^{2} -2.48119i q^{3} -5.15633 q^{4} +6.63752 q^{6} +1.00000i q^{7} -8.44358i q^{8} -3.15633 q^{9} -1.00000 q^{11} +12.7938i q^{12} +5.83146i q^{13} -2.67513 q^{14} +12.2750 q^{16} -5.44358i q^{17} -8.44358i q^{18} +1.35026 q^{19} +2.48119 q^{21} -2.67513i q^{22} -3.19394i q^{23} -20.9502 q^{24} -15.5999 q^{26} +0.387873i q^{27} -5.15633i q^{28} +3.61213 q^{29} -5.28726 q^{31} +15.9502i q^{32} +2.48119i q^{33} +14.5623 q^{34} +16.2750 q^{36} +8.54420i q^{37} +3.61213i q^{38} +14.4690 q^{39} -5.02539 q^{41} +6.63752i q^{42} +5.89446i q^{43} +5.15633 q^{44} +8.54420 q^{46} +11.8315i q^{47} -30.4568i q^{48} -1.00000 q^{49} -13.5066 q^{51} -30.0689i q^{52} -0.231548i q^{53} -1.03761 q^{54} +8.44358 q^{56} -3.35026i q^{57} +9.66291i q^{58} -13.5999 q^{59} -1.41327 q^{61} -14.1441i q^{62} -3.15633i q^{63} -18.1187 q^{64} -6.63752 q^{66} +10.8568i q^{67} +28.0689i q^{68} -7.92478 q^{69} -15.5369 q^{71} +26.6507i q^{72} -11.3684i q^{73} -22.8568 q^{74} -6.96239 q^{76} -1.00000i q^{77} +38.7064i q^{78} -1.96968 q^{79} -8.50659 q^{81} -13.4436i q^{82} +10.6253i q^{83} -12.7938 q^{84} -15.7685 q^{86} -8.96239i q^{87} +8.44358i q^{88} -7.22425 q^{89} -5.83146 q^{91} +16.4690i q^{92} +13.1187i q^{93} -31.6507 q^{94} +39.5755 q^{96} +0.836381i q^{97} -2.67513i q^{98} +3.15633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} + 8 q^{6} + 2 q^{9} + O(q^{10}) \) \( 6 q - 10 q^{4} + 8 q^{6} + 2 q^{9} - 6 q^{11} - 6 q^{14} + 10 q^{16} - 12 q^{19} + 4 q^{21} - 52 q^{24} - 40 q^{26} + 20 q^{29} - 20 q^{31} + 12 q^{34} + 34 q^{36} + 24 q^{39} + 10 q^{44} + 32 q^{46} - 6 q^{49} - 40 q^{51} - 28 q^{54} + 18 q^{56} - 28 q^{59} + 20 q^{61} - 66 q^{64} - 8 q^{66} - 4 q^{69} - 48 q^{71} - 76 q^{74} - 20 q^{76} - 16 q^{79} - 10 q^{81} - 24 q^{84} - 72 q^{86} - 40 q^{89} - 4 q^{91} - 76 q^{94} + 80 q^{96} - 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.67513i 1.89160i 0.324745 + 0.945802i \(0.394721\pi\)
−0.324745 + 0.945802i \(0.605279\pi\)
\(3\) − 2.48119i − 1.43252i −0.697834 0.716259i \(-0.745853\pi\)
0.697834 0.716259i \(-0.254147\pi\)
\(4\) −5.15633 −2.57816
\(5\) 0 0
\(6\) 6.63752 2.70976
\(7\) 1.00000i 0.377964i
\(8\) − 8.44358i − 2.98526i
\(9\) −3.15633 −1.05211
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 12.7938i 3.69326i
\(13\) 5.83146i 1.61735i 0.588252 + 0.808677i \(0.299816\pi\)
−0.588252 + 0.808677i \(0.700184\pi\)
\(14\) −2.67513 −0.714959
\(15\) 0 0
\(16\) 12.2750 3.06876
\(17\) − 5.44358i − 1.32026i −0.751150 0.660131i \(-0.770501\pi\)
0.751150 0.660131i \(-0.229499\pi\)
\(18\) − 8.44358i − 1.99017i
\(19\) 1.35026 0.309771 0.154886 0.987932i \(-0.450499\pi\)
0.154886 + 0.987932i \(0.450499\pi\)
\(20\) 0 0
\(21\) 2.48119 0.541441
\(22\) − 2.67513i − 0.570340i
\(23\) − 3.19394i − 0.665982i −0.942930 0.332991i \(-0.891942\pi\)
0.942930 0.332991i \(-0.108058\pi\)
\(24\) −20.9502 −4.27644
\(25\) 0 0
\(26\) −15.5999 −3.05939
\(27\) 0.387873i 0.0746462i
\(28\) − 5.15633i − 0.974454i
\(29\) 3.61213 0.670755 0.335378 0.942084i \(-0.391136\pi\)
0.335378 + 0.942084i \(0.391136\pi\)
\(30\) 0 0
\(31\) −5.28726 −0.949620 −0.474810 0.880088i \(-0.657483\pi\)
−0.474810 + 0.880088i \(0.657483\pi\)
\(32\) 15.9502i 2.81962i
\(33\) 2.48119i 0.431920i
\(34\) 14.5623 2.49741
\(35\) 0 0
\(36\) 16.2750 2.71251
\(37\) 8.54420i 1.40466i 0.711853 + 0.702329i \(0.247856\pi\)
−0.711853 + 0.702329i \(0.752144\pi\)
\(38\) 3.61213i 0.585964i
\(39\) 14.4690 2.31689
\(40\) 0 0
\(41\) −5.02539 −0.784834 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(42\) 6.63752i 1.02419i
\(43\) 5.89446i 0.898897i 0.893306 + 0.449448i \(0.148379\pi\)
−0.893306 + 0.449448i \(0.851621\pi\)
\(44\) 5.15633 0.777345
\(45\) 0 0
\(46\) 8.54420 1.25977
\(47\) 11.8315i 1.72580i 0.505379 + 0.862898i \(0.331353\pi\)
−0.505379 + 0.862898i \(0.668647\pi\)
\(48\) − 30.4568i − 4.39605i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −13.5066 −1.89130
\(52\) − 30.0689i − 4.16980i
\(53\) − 0.231548i − 0.0318056i −0.999874 0.0159028i \(-0.994938\pi\)
0.999874 0.0159028i \(-0.00506223\pi\)
\(54\) −1.03761 −0.141201
\(55\) 0 0
\(56\) 8.44358 1.12832
\(57\) − 3.35026i − 0.443753i
\(58\) 9.66291i 1.26880i
\(59\) −13.5999 −1.77056 −0.885279 0.465061i \(-0.846032\pi\)
−0.885279 + 0.465061i \(0.846032\pi\)
\(60\) 0 0
\(61\) −1.41327 −0.180950 −0.0904751 0.995899i \(-0.528839\pi\)
−0.0904751 + 0.995899i \(0.528839\pi\)
\(62\) − 14.1441i − 1.79630i
\(63\) − 3.15633i − 0.397660i
\(64\) −18.1187 −2.26484
\(65\) 0 0
\(66\) −6.63752 −0.817022
\(67\) 10.8568i 1.32638i 0.748453 + 0.663188i \(0.230797\pi\)
−0.748453 + 0.663188i \(0.769203\pi\)
\(68\) 28.0689i 3.40385i
\(69\) −7.92478 −0.954031
\(70\) 0 0
\(71\) −15.5369 −1.84389 −0.921946 0.387319i \(-0.873401\pi\)
−0.921946 + 0.387319i \(0.873401\pi\)
\(72\) 26.6507i 3.14081i
\(73\) − 11.3684i − 1.33057i −0.746591 0.665283i \(-0.768311\pi\)
0.746591 0.665283i \(-0.231689\pi\)
\(74\) −22.8568 −2.65705
\(75\) 0 0
\(76\) −6.96239 −0.798641
\(77\) − 1.00000i − 0.113961i
\(78\) 38.7064i 4.38264i
\(79\) −1.96968 −0.221607 −0.110803 0.993842i \(-0.535342\pi\)
−0.110803 + 0.993842i \(0.535342\pi\)
\(80\) 0 0
\(81\) −8.50659 −0.945176
\(82\) − 13.4436i − 1.48460i
\(83\) 10.6253i 1.16628i 0.812372 + 0.583139i \(0.198176\pi\)
−0.812372 + 0.583139i \(0.801824\pi\)
\(84\) −12.7938 −1.39592
\(85\) 0 0
\(86\) −15.7685 −1.70036
\(87\) − 8.96239i − 0.960869i
\(88\) 8.44358i 0.900089i
\(89\) −7.22425 −0.765769 −0.382885 0.923796i \(-0.625069\pi\)
−0.382885 + 0.923796i \(0.625069\pi\)
\(90\) 0 0
\(91\) −5.83146 −0.611303
\(92\) 16.4690i 1.71701i
\(93\) 13.1187i 1.36035i
\(94\) −31.6507 −3.26452
\(95\) 0 0
\(96\) 39.5755 4.03915
\(97\) 0.836381i 0.0849216i 0.999098 + 0.0424608i \(0.0135198\pi\)
−0.999098 + 0.0424608i \(0.986480\pi\)
\(98\) − 2.67513i − 0.270229i
\(99\) 3.15633 0.317223
\(100\) 0 0
\(101\) 7.41327 0.737648 0.368824 0.929499i \(-0.379761\pi\)
0.368824 + 0.929499i \(0.379761\pi\)
\(102\) − 36.1319i − 3.57759i
\(103\) 4.21933i 0.415743i 0.978156 + 0.207871i \(0.0666536\pi\)
−0.978156 + 0.207871i \(0.933346\pi\)
\(104\) 49.2384 4.82822
\(105\) 0 0
\(106\) 0.619421 0.0601635
\(107\) 11.5369i 1.11531i 0.830071 + 0.557657i \(0.188300\pi\)
−0.830071 + 0.557657i \(0.811700\pi\)
\(108\) − 2.00000i − 0.192450i
\(109\) 2.18664 0.209442 0.104721 0.994502i \(-0.466605\pi\)
0.104721 + 0.994502i \(0.466605\pi\)
\(110\) 0 0
\(111\) 21.1998 2.01220
\(112\) 12.2750i 1.15988i
\(113\) − 9.35026i − 0.879599i −0.898096 0.439799i \(-0.855050\pi\)
0.898096 0.439799i \(-0.144950\pi\)
\(114\) 8.96239 0.839405
\(115\) 0 0
\(116\) −18.6253 −1.72932
\(117\) − 18.4060i − 1.70163i
\(118\) − 36.3815i − 3.34919i
\(119\) 5.44358 0.499012
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 3.78067i − 0.342286i
\(123\) 12.4690i 1.12429i
\(124\) 27.2628 2.44827
\(125\) 0 0
\(126\) 8.44358 0.752214
\(127\) 16.9624i 1.50517i 0.658496 + 0.752584i \(0.271193\pi\)
−0.658496 + 0.752584i \(0.728807\pi\)
\(128\) − 16.5696i − 1.46456i
\(129\) 14.6253 1.28769
\(130\) 0 0
\(131\) −9.92478 −0.867132 −0.433566 0.901122i \(-0.642745\pi\)
−0.433566 + 0.901122i \(0.642745\pi\)
\(132\) − 12.7938i − 1.11356i
\(133\) 1.35026i 0.117083i
\(134\) −29.0435 −2.50898
\(135\) 0 0
\(136\) −45.9633 −3.94132
\(137\) 10.9927i 0.939170i 0.882887 + 0.469585i \(0.155596\pi\)
−0.882887 + 0.469585i \(0.844404\pi\)
\(138\) − 21.1998i − 1.80465i
\(139\) −6.88717 −0.584162 −0.292081 0.956394i \(-0.594348\pi\)
−0.292081 + 0.956394i \(0.594348\pi\)
\(140\) 0 0
\(141\) 29.3561 2.47223
\(142\) − 41.5633i − 3.48791i
\(143\) − 5.83146i − 0.487651i
\(144\) −38.7440 −3.22867
\(145\) 0 0
\(146\) 30.4119 2.51690
\(147\) 2.48119i 0.204645i
\(148\) − 44.0567i − 3.62144i
\(149\) −22.8119 −1.86883 −0.934414 0.356190i \(-0.884076\pi\)
−0.934414 + 0.356190i \(0.884076\pi\)
\(150\) 0 0
\(151\) −3.24472 −0.264052 −0.132026 0.991246i \(-0.542148\pi\)
−0.132026 + 0.991246i \(0.542148\pi\)
\(152\) − 11.4010i − 0.924747i
\(153\) 17.1817i 1.38906i
\(154\) 2.67513 0.215568
\(155\) 0 0
\(156\) −74.6067 −5.97332
\(157\) 5.42548i 0.433001i 0.976283 + 0.216500i \(0.0694643\pi\)
−0.976283 + 0.216500i \(0.930536\pi\)
\(158\) − 5.26916i − 0.419192i
\(159\) −0.574515 −0.0455620
\(160\) 0 0
\(161\) 3.19394 0.251717
\(162\) − 22.7562i − 1.78790i
\(163\) 3.38058i 0.264787i 0.991197 + 0.132394i \(0.0422663\pi\)
−0.991197 + 0.132394i \(0.957734\pi\)
\(164\) 25.9126 2.02343
\(165\) 0 0
\(166\) −28.4241 −2.20614
\(167\) − 11.2750i − 0.872489i −0.899828 0.436244i \(-0.856308\pi\)
0.899828 0.436244i \(-0.143692\pi\)
\(168\) − 20.9502i − 1.61634i
\(169\) −21.0059 −1.61584
\(170\) 0 0
\(171\) −4.26187 −0.325913
\(172\) − 30.3938i − 2.31750i
\(173\) − 8.98049i − 0.682774i −0.939923 0.341387i \(-0.889103\pi\)
0.939923 0.341387i \(-0.110897\pi\)
\(174\) 23.9756 1.81758
\(175\) 0 0
\(176\) −12.2750 −0.925266
\(177\) 33.7440i 2.53636i
\(178\) − 19.3258i − 1.44853i
\(179\) 26.2374 1.96108 0.980539 0.196326i \(-0.0629010\pi\)
0.980539 + 0.196326i \(0.0629010\pi\)
\(180\) 0 0
\(181\) −11.1998 −0.832476 −0.416238 0.909256i \(-0.636652\pi\)
−0.416238 + 0.909256i \(0.636652\pi\)
\(182\) − 15.5999i − 1.15634i
\(183\) 3.50659i 0.259214i
\(184\) −26.9683 −1.98813
\(185\) 0 0
\(186\) −35.0943 −2.57324
\(187\) 5.44358i 0.398074i
\(188\) − 61.0068i − 4.44938i
\(189\) −0.387873 −0.0282136
\(190\) 0 0
\(191\) 11.1998 0.810390 0.405195 0.914230i \(-0.367204\pi\)
0.405195 + 0.914230i \(0.367204\pi\)
\(192\) 44.9560i 3.24442i
\(193\) 0.604833i 0.0435368i 0.999763 + 0.0217684i \(0.00692965\pi\)
−0.999763 + 0.0217684i \(0.993070\pi\)
\(194\) −2.23743 −0.160638
\(195\) 0 0
\(196\) 5.15633 0.368309
\(197\) − 15.3054i − 1.09046i −0.838286 0.545231i \(-0.816442\pi\)
0.838286 0.545231i \(-0.183558\pi\)
\(198\) 8.44358i 0.600059i
\(199\) 12.5623 0.890518 0.445259 0.895402i \(-0.353112\pi\)
0.445259 + 0.895402i \(0.353112\pi\)
\(200\) 0 0
\(201\) 26.9380 1.90006
\(202\) 19.8315i 1.39534i
\(203\) 3.61213i 0.253522i
\(204\) 69.6444 4.87608
\(205\) 0 0
\(206\) −11.2873 −0.786421
\(207\) 10.0811i 0.700685i
\(208\) 71.5814i 4.96327i
\(209\) −1.35026 −0.0933996
\(210\) 0 0
\(211\) −4.43866 −0.305570 −0.152785 0.988259i \(-0.548824\pi\)
−0.152785 + 0.988259i \(0.548824\pi\)
\(212\) 1.19394i 0.0819999i
\(213\) 38.5501i 2.64141i
\(214\) −30.8627 −2.10973
\(215\) 0 0
\(216\) 3.27504 0.222838
\(217\) − 5.28726i − 0.358922i
\(218\) 5.84955i 0.396182i
\(219\) −28.2071 −1.90606
\(220\) 0 0
\(221\) 31.7440 2.13533
\(222\) 56.7123i 3.80628i
\(223\) − 7.78067i − 0.521032i −0.965469 0.260516i \(-0.916107\pi\)
0.965469 0.260516i \(-0.0838927\pi\)
\(224\) −15.9502 −1.06572
\(225\) 0 0
\(226\) 25.0132 1.66385
\(227\) 10.4485i 0.693492i 0.937959 + 0.346746i \(0.112713\pi\)
−0.937959 + 0.346746i \(0.887287\pi\)
\(228\) 17.2750i 1.14407i
\(229\) 29.4518 1.94623 0.973116 0.230316i \(-0.0739759\pi\)
0.973116 + 0.230316i \(0.0739759\pi\)
\(230\) 0 0
\(231\) −2.48119 −0.163251
\(232\) − 30.4993i − 2.00238i
\(233\) − 8.73084i − 0.571976i −0.958233 0.285988i \(-0.907678\pi\)
0.958233 0.285988i \(-0.0923218\pi\)
\(234\) 49.2384 3.21881
\(235\) 0 0
\(236\) 70.1255 4.56478
\(237\) 4.88717i 0.317456i
\(238\) 14.5623i 0.943933i
\(239\) 21.2144 1.37225 0.686123 0.727486i \(-0.259311\pi\)
0.686123 + 0.727486i \(0.259311\pi\)
\(240\) 0 0
\(241\) −9.33804 −0.601516 −0.300758 0.953700i \(-0.597240\pi\)
−0.300758 + 0.953700i \(0.597240\pi\)
\(242\) 2.67513i 0.171964i
\(243\) 22.2701i 1.42863i
\(244\) 7.28726 0.466519
\(245\) 0 0
\(246\) −33.3561 −2.12671
\(247\) 7.87399i 0.501010i
\(248\) 44.6434i 2.83486i
\(249\) 26.3634 1.67071
\(250\) 0 0
\(251\) 1.87636 0.118435 0.0592174 0.998245i \(-0.481139\pi\)
0.0592174 + 0.998245i \(0.481139\pi\)
\(252\) 16.2750i 1.02523i
\(253\) 3.19394i 0.200801i
\(254\) −45.3766 −2.84718
\(255\) 0 0
\(256\) 8.08840 0.505525
\(257\) − 27.1392i − 1.69290i −0.532472 0.846448i \(-0.678737\pi\)
0.532472 0.846448i \(-0.321263\pi\)
\(258\) 39.1246i 2.43579i
\(259\) −8.54420 −0.530911
\(260\) 0 0
\(261\) −11.4010 −0.705707
\(262\) − 26.5501i − 1.64027i
\(263\) 12.8119i 0.790018i 0.918677 + 0.395009i \(0.129259\pi\)
−0.918677 + 0.395009i \(0.870741\pi\)
\(264\) 20.9502 1.28939
\(265\) 0 0
\(266\) −3.61213 −0.221474
\(267\) 17.9248i 1.09698i
\(268\) − 55.9814i − 3.41961i
\(269\) 6.26187 0.381793 0.190896 0.981610i \(-0.438861\pi\)
0.190896 + 0.981610i \(0.438861\pi\)
\(270\) 0 0
\(271\) −5.73813 −0.348567 −0.174283 0.984696i \(-0.555761\pi\)
−0.174283 + 0.984696i \(0.555761\pi\)
\(272\) − 66.8202i − 4.05157i
\(273\) 14.4690i 0.875702i
\(274\) −29.4069 −1.77654
\(275\) 0 0
\(276\) 40.8627 2.45965
\(277\) 8.35756i 0.502157i 0.967967 + 0.251078i \(0.0807852\pi\)
−0.967967 + 0.251078i \(0.919215\pi\)
\(278\) − 18.4241i − 1.10500i
\(279\) 16.6883 0.999103
\(280\) 0 0
\(281\) −8.44851 −0.503996 −0.251998 0.967728i \(-0.581088\pi\)
−0.251998 + 0.967728i \(0.581088\pi\)
\(282\) 78.5315i 4.67648i
\(283\) 0.836381i 0.0497177i 0.999691 + 0.0248588i \(0.00791363\pi\)
−0.999691 + 0.0248588i \(0.992086\pi\)
\(284\) 80.1133 4.75385
\(285\) 0 0
\(286\) 15.5999 0.922442
\(287\) − 5.02539i − 0.296640i
\(288\) − 50.3439i − 2.96654i
\(289\) −12.6326 −0.743094
\(290\) 0 0
\(291\) 2.07522 0.121652
\(292\) 58.6190i 3.43042i
\(293\) − 2.71862i − 0.158824i −0.996842 0.0794118i \(-0.974696\pi\)
0.996842 0.0794118i \(-0.0253042\pi\)
\(294\) −6.63752 −0.387108
\(295\) 0 0
\(296\) 72.1436 4.19326
\(297\) − 0.387873i − 0.0225067i
\(298\) − 61.0249i − 3.53508i
\(299\) 18.6253 1.07713
\(300\) 0 0
\(301\) −5.89446 −0.339751
\(302\) − 8.68006i − 0.499481i
\(303\) − 18.3938i − 1.05669i
\(304\) 16.5745 0.950614
\(305\) 0 0
\(306\) −45.9633 −2.62755
\(307\) 8.36344i 0.477326i 0.971102 + 0.238663i \(0.0767092\pi\)
−0.971102 + 0.238663i \(0.923291\pi\)
\(308\) 5.15633i 0.293809i
\(309\) 10.4690 0.595559
\(310\) 0 0
\(311\) 4.43629 0.251559 0.125779 0.992058i \(-0.459857\pi\)
0.125779 + 0.992058i \(0.459857\pi\)
\(312\) − 122.170i − 6.91651i
\(313\) 29.7889i 1.68377i 0.539658 + 0.841885i \(0.318554\pi\)
−0.539658 + 0.841885i \(0.681446\pi\)
\(314\) −14.5139 −0.819066
\(315\) 0 0
\(316\) 10.1563 0.571338
\(317\) − 15.4010i − 0.865009i −0.901632 0.432504i \(-0.857630\pi\)
0.901632 0.432504i \(-0.142370\pi\)
\(318\) − 1.53690i − 0.0861853i
\(319\) −3.61213 −0.202240
\(320\) 0 0
\(321\) 28.6253 1.59771
\(322\) 8.54420i 0.476150i
\(323\) − 7.35026i − 0.408980i
\(324\) 43.8627 2.43682
\(325\) 0 0
\(326\) −9.04349 −0.500873
\(327\) − 5.42548i − 0.300030i
\(328\) 42.4323i 2.34293i
\(329\) −11.8315 −0.652289
\(330\) 0 0
\(331\) 6.26187 0.344183 0.172092 0.985081i \(-0.444947\pi\)
0.172092 + 0.985081i \(0.444947\pi\)
\(332\) − 54.7875i − 3.00685i
\(333\) − 26.9683i − 1.47785i
\(334\) 30.1622 1.65040
\(335\) 0 0
\(336\) 30.4568 1.66155
\(337\) 15.8700i 0.864495i 0.901755 + 0.432248i \(0.142279\pi\)
−0.901755 + 0.432248i \(0.857721\pi\)
\(338\) − 56.1935i − 3.05652i
\(339\) −23.1998 −1.26004
\(340\) 0 0
\(341\) 5.28726 0.286321
\(342\) − 11.4010i − 0.616498i
\(343\) − 1.00000i − 0.0539949i
\(344\) 49.7704 2.68344
\(345\) 0 0
\(346\) 24.0240 1.29154
\(347\) − 6.79147i − 0.364585i −0.983244 0.182293i \(-0.941648\pi\)
0.983244 0.182293i \(-0.0583519\pi\)
\(348\) 46.2130i 2.47728i
\(349\) −26.7489 −1.43184 −0.715919 0.698183i \(-0.753992\pi\)
−0.715919 + 0.698183i \(0.753992\pi\)
\(350\) 0 0
\(351\) −2.26187 −0.120729
\(352\) − 15.9502i − 0.850147i
\(353\) − 16.8627i − 0.897512i −0.893654 0.448756i \(-0.851867\pi\)
0.893654 0.448756i \(-0.148133\pi\)
\(354\) −90.2697 −4.79778
\(355\) 0 0
\(356\) 37.2506 1.97428
\(357\) − 13.5066i − 0.714844i
\(358\) 70.1886i 3.70958i
\(359\) 3.79289 0.200181 0.100091 0.994978i \(-0.468087\pi\)
0.100091 + 0.994978i \(0.468087\pi\)
\(360\) 0 0
\(361\) −17.1768 −0.904042
\(362\) − 29.9610i − 1.57471i
\(363\) − 2.48119i − 0.130229i
\(364\) 30.0689 1.57604
\(365\) 0 0
\(366\) −9.38058 −0.490331
\(367\) 6.36977i 0.332500i 0.986084 + 0.166250i \(0.0531658\pi\)
−0.986084 + 0.166250i \(0.946834\pi\)
\(368\) − 39.2057i − 2.04374i
\(369\) 15.8618 0.825731
\(370\) 0 0
\(371\) 0.231548 0.0120214
\(372\) − 67.6444i − 3.50720i
\(373\) − 21.3317i − 1.10451i −0.833674 0.552257i \(-0.813767\pi\)
0.833674 0.552257i \(-0.186233\pi\)
\(374\) −14.5623 −0.752998
\(375\) 0 0
\(376\) 99.8999 5.15194
\(377\) 21.0640i 1.08485i
\(378\) − 1.03761i − 0.0533690i
\(379\) −24.7875 −1.27325 −0.636624 0.771174i \(-0.719670\pi\)
−0.636624 + 0.771174i \(0.719670\pi\)
\(380\) 0 0
\(381\) 42.0870 2.15618
\(382\) 29.9610i 1.53294i
\(383\) − 5.45817i − 0.278900i −0.990229 0.139450i \(-0.955467\pi\)
0.990229 0.139450i \(-0.0445334\pi\)
\(384\) −41.1124 −2.09801
\(385\) 0 0
\(386\) −1.61801 −0.0823544
\(387\) − 18.6048i − 0.945737i
\(388\) − 4.31265i − 0.218942i
\(389\) 13.7235 0.695811 0.347906 0.937530i \(-0.386893\pi\)
0.347906 + 0.937530i \(0.386893\pi\)
\(390\) 0 0
\(391\) −17.3865 −0.879271
\(392\) 8.44358i 0.426465i
\(393\) 24.6253i 1.24218i
\(394\) 40.9438 2.06272
\(395\) 0 0
\(396\) −16.2750 −0.817851
\(397\) − 2.11142i − 0.105969i −0.998595 0.0529846i \(-0.983127\pi\)
0.998595 0.0529846i \(-0.0168734\pi\)
\(398\) 33.6058i 1.68451i
\(399\) 3.35026 0.167723
\(400\) 0 0
\(401\) 19.1490 0.956257 0.478128 0.878290i \(-0.341315\pi\)
0.478128 + 0.878290i \(0.341315\pi\)
\(402\) 72.0625i 3.59415i
\(403\) − 30.8324i − 1.53587i
\(404\) −38.2252 −1.90178
\(405\) 0 0
\(406\) −9.66291 −0.479562
\(407\) − 8.54420i − 0.423520i
\(408\) 114.044i 5.64602i
\(409\) 18.6883 0.924077 0.462039 0.886860i \(-0.347118\pi\)
0.462039 + 0.886860i \(0.347118\pi\)
\(410\) 0 0
\(411\) 27.2750 1.34538
\(412\) − 21.7562i − 1.07185i
\(413\) − 13.5999i − 0.669208i
\(414\) −26.9683 −1.32542
\(415\) 0 0
\(416\) −93.0127 −4.56032
\(417\) 17.0884i 0.836822i
\(418\) − 3.61213i − 0.176675i
\(419\) 0.773377 0.0377819 0.0188910 0.999822i \(-0.493986\pi\)
0.0188910 + 0.999822i \(0.493986\pi\)
\(420\) 0 0
\(421\) −10.5198 −0.512702 −0.256351 0.966584i \(-0.582520\pi\)
−0.256351 + 0.966584i \(0.582520\pi\)
\(422\) − 11.8740i − 0.578017i
\(423\) − 37.3439i − 1.81572i
\(424\) −1.95509 −0.0949478
\(425\) 0 0
\(426\) −103.127 −4.99650
\(427\) − 1.41327i − 0.0683927i
\(428\) − 59.4880i − 2.87546i
\(429\) −14.4690 −0.698569
\(430\) 0 0
\(431\) −24.7308 −1.19124 −0.595621 0.803265i \(-0.703094\pi\)
−0.595621 + 0.803265i \(0.703094\pi\)
\(432\) 4.76116i 0.229071i
\(433\) − 18.5599i − 0.891933i −0.895050 0.445967i \(-0.852860\pi\)
0.895050 0.445967i \(-0.147140\pi\)
\(434\) 14.1441 0.678939
\(435\) 0 0
\(436\) −11.2750 −0.539976
\(437\) − 4.31265i − 0.206302i
\(438\) − 75.4577i − 3.60551i
\(439\) 1.42548 0.0680347 0.0340173 0.999421i \(-0.489170\pi\)
0.0340173 + 0.999421i \(0.489170\pi\)
\(440\) 0 0
\(441\) 3.15633 0.150301
\(442\) 84.9194i 4.03920i
\(443\) 40.1925i 1.90960i 0.297242 + 0.954802i \(0.403933\pi\)
−0.297242 + 0.954802i \(0.596067\pi\)
\(444\) −109.313 −5.18777
\(445\) 0 0
\(446\) 20.8143 0.985586
\(447\) 56.6009i 2.67713i
\(448\) − 18.1187i − 0.856029i
\(449\) −12.6556 −0.597256 −0.298628 0.954370i \(-0.596529\pi\)
−0.298628 + 0.954370i \(0.596529\pi\)
\(450\) 0 0
\(451\) 5.02539 0.236636
\(452\) 48.2130i 2.26775i
\(453\) 8.05079i 0.378259i
\(454\) −27.9511 −1.31181
\(455\) 0 0
\(456\) −28.2882 −1.32472
\(457\) 0.544198i 0.0254565i 0.999919 + 0.0127283i \(0.00405164\pi\)
−0.999919 + 0.0127283i \(0.995948\pi\)
\(458\) 78.7875i 3.68150i
\(459\) 2.11142 0.0985526
\(460\) 0 0
\(461\) −11.5755 −0.539123 −0.269562 0.962983i \(-0.586879\pi\)
−0.269562 + 0.962983i \(0.586879\pi\)
\(462\) − 6.63752i − 0.308805i
\(463\) 23.7948i 1.10584i 0.833235 + 0.552919i \(0.186486\pi\)
−0.833235 + 0.552919i \(0.813514\pi\)
\(464\) 44.3390 2.05839
\(465\) 0 0
\(466\) 23.3561 1.08195
\(467\) 2.66784i 0.123453i 0.998093 + 0.0617264i \(0.0196606\pi\)
−0.998093 + 0.0617264i \(0.980339\pi\)
\(468\) 94.9072i 4.38709i
\(469\) −10.8568 −0.501323
\(470\) 0 0
\(471\) 13.4617 0.620282
\(472\) 114.832i 5.28557i
\(473\) − 5.89446i − 0.271028i
\(474\) −13.0738 −0.600500
\(475\) 0 0
\(476\) −28.0689 −1.28654
\(477\) 0.730841i 0.0334629i
\(478\) 56.7513i 2.59574i
\(479\) 10.7104 0.489369 0.244685 0.969603i \(-0.421316\pi\)
0.244685 + 0.969603i \(0.421316\pi\)
\(480\) 0 0
\(481\) −49.8251 −2.27183
\(482\) − 24.9805i − 1.13783i
\(483\) − 7.92478i − 0.360590i
\(484\) −5.15633 −0.234378
\(485\) 0 0
\(486\) −59.5755 −2.70240
\(487\) − 17.4314i − 0.789891i −0.918705 0.394945i \(-0.870764\pi\)
0.918705 0.394945i \(-0.129236\pi\)
\(488\) 11.9330i 0.540183i
\(489\) 8.38787 0.379313
\(490\) 0 0
\(491\) −28.3693 −1.28029 −0.640145 0.768254i \(-0.721126\pi\)
−0.640145 + 0.768254i \(0.721126\pi\)
\(492\) − 64.2941i − 2.89860i
\(493\) − 19.6629i − 0.885573i
\(494\) −21.0640 −0.947712
\(495\) 0 0
\(496\) −64.9013 −2.91415
\(497\) − 15.5369i − 0.696925i
\(498\) 70.5256i 3.16033i
\(499\) −27.4763 −1.23001 −0.615003 0.788524i \(-0.710845\pi\)
−0.615003 + 0.788524i \(0.710845\pi\)
\(500\) 0 0
\(501\) −27.9756 −1.24986
\(502\) 5.01951i 0.224032i
\(503\) 20.2981i 0.905046i 0.891753 + 0.452523i \(0.149476\pi\)
−0.891753 + 0.452523i \(0.850524\pi\)
\(504\) −26.6507 −1.18712
\(505\) 0 0
\(506\) −8.54420 −0.379836
\(507\) 52.1197i 2.31472i
\(508\) − 87.4636i − 3.88057i
\(509\) 24.2619 1.07539 0.537694 0.843140i \(-0.319296\pi\)
0.537694 + 0.843140i \(0.319296\pi\)
\(510\) 0 0
\(511\) 11.3684 0.502907
\(512\) − 11.5017i − 0.508306i
\(513\) 0.523730i 0.0231233i
\(514\) 72.6009 3.20229
\(515\) 0 0
\(516\) −75.4128 −3.31986
\(517\) − 11.8315i − 0.520347i
\(518\) − 22.8568i − 1.00427i
\(519\) −22.2823 −0.978086
\(520\) 0 0
\(521\) 2.20123 0.0964377 0.0482188 0.998837i \(-0.484646\pi\)
0.0482188 + 0.998837i \(0.484646\pi\)
\(522\) − 30.4993i − 1.33492i
\(523\) − 22.1378i − 0.968017i −0.875063 0.484008i \(-0.839180\pi\)
0.875063 0.484008i \(-0.160820\pi\)
\(524\) 51.1754 2.23561
\(525\) 0 0
\(526\) −34.2736 −1.49440
\(527\) 28.7816i 1.25375i
\(528\) 30.4568i 1.32546i
\(529\) 12.7988 0.556468
\(530\) 0 0
\(531\) 42.9257 1.86282
\(532\) − 6.96239i − 0.301858i
\(533\) − 29.3054i − 1.26936i
\(534\) −47.9511 −2.07505
\(535\) 0 0
\(536\) 91.6707 3.95957
\(537\) − 65.1002i − 2.80928i
\(538\) 16.7513i 0.722200i
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 23.0640 0.991597 0.495799 0.868438i \(-0.334875\pi\)
0.495799 + 0.868438i \(0.334875\pi\)
\(542\) − 15.3503i − 0.659350i
\(543\) 27.7889i 1.19254i
\(544\) 86.8261 3.72264
\(545\) 0 0
\(546\) −38.7064 −1.65648
\(547\) − 21.3766i − 0.913998i −0.889467 0.456999i \(-0.848924\pi\)
0.889467 0.456999i \(-0.151076\pi\)
\(548\) − 56.6820i − 2.42133i
\(549\) 4.46073 0.190379
\(550\) 0 0
\(551\) 4.87732 0.207781
\(552\) 66.9135i 2.84803i
\(553\) − 1.96968i − 0.0837594i
\(554\) −22.3576 −0.949882
\(555\) 0 0
\(556\) 35.5125 1.50606
\(557\) 9.19394i 0.389560i 0.980847 + 0.194780i \(0.0623992\pi\)
−0.980847 + 0.194780i \(0.937601\pi\)
\(558\) 44.6434i 1.88991i
\(559\) −34.3733 −1.45384
\(560\) 0 0
\(561\) 13.5066 0.570249
\(562\) − 22.6009i − 0.953360i
\(563\) − 9.79877i − 0.412969i −0.978450 0.206484i \(-0.933798\pi\)
0.978450 0.206484i \(-0.0662023\pi\)
\(564\) −151.370 −6.37382
\(565\) 0 0
\(566\) −2.23743 −0.0940461
\(567\) − 8.50659i − 0.357243i
\(568\) 131.187i 5.50449i
\(569\) 33.5125 1.40492 0.702458 0.711725i \(-0.252086\pi\)
0.702458 + 0.711725i \(0.252086\pi\)
\(570\) 0 0
\(571\) 43.1392 1.80532 0.902659 0.430356i \(-0.141612\pi\)
0.902659 + 0.430356i \(0.141612\pi\)
\(572\) 30.0689i 1.25724i
\(573\) − 27.7889i − 1.16090i
\(574\) 13.4436 0.561124
\(575\) 0 0
\(576\) 57.1886 2.38286
\(577\) 14.8510i 0.618254i 0.951021 + 0.309127i \(0.100037\pi\)
−0.951021 + 0.309127i \(0.899963\pi\)
\(578\) − 33.7938i − 1.40564i
\(579\) 1.50071 0.0623673
\(580\) 0 0
\(581\) −10.6253 −0.440812
\(582\) 5.55149i 0.230117i
\(583\) 0.231548i 0.00958974i
\(584\) −95.9897 −3.97208
\(585\) 0 0
\(586\) 7.27267 0.300431
\(587\) − 14.7938i − 0.610607i −0.952255 0.305304i \(-0.901242\pi\)
0.952255 0.305304i \(-0.0987580\pi\)
\(588\) − 12.7938i − 0.527609i
\(589\) −7.13918 −0.294165
\(590\) 0 0
\(591\) −37.9756 −1.56211
\(592\) 104.880i 4.31056i
\(593\) − 27.4191i − 1.12597i −0.826467 0.562985i \(-0.809653\pi\)
0.826467 0.562985i \(-0.190347\pi\)
\(594\) 1.03761 0.0425737
\(595\) 0 0
\(596\) 117.626 4.81814
\(597\) − 31.1695i − 1.27568i
\(598\) 49.8251i 2.03750i
\(599\) −11.3258 −0.462761 −0.231380 0.972863i \(-0.574324\pi\)
−0.231380 + 0.972863i \(0.574324\pi\)
\(600\) 0 0
\(601\) 15.5393 0.633860 0.316930 0.948449i \(-0.397348\pi\)
0.316930 + 0.948449i \(0.397348\pi\)
\(602\) − 15.7685i − 0.642674i
\(603\) − 34.2677i − 1.39549i
\(604\) 16.7308 0.680768
\(605\) 0 0
\(606\) 49.2057 1.99884
\(607\) − 17.7235i − 0.719377i −0.933073 0.359688i \(-0.882883\pi\)
0.933073 0.359688i \(-0.117117\pi\)
\(608\) 21.5369i 0.873437i
\(609\) 8.96239 0.363174
\(610\) 0 0
\(611\) −68.9946 −2.79122
\(612\) − 88.5945i − 3.58122i
\(613\) 22.2941i 0.900450i 0.892915 + 0.450225i \(0.148656\pi\)
−0.892915 + 0.450225i \(0.851344\pi\)
\(614\) −22.3733 −0.902912
\(615\) 0 0
\(616\) −8.44358 −0.340202
\(617\) 30.9438i 1.24575i 0.782321 + 0.622876i \(0.214036\pi\)
−0.782321 + 0.622876i \(0.785964\pi\)
\(618\) 28.0059i 1.12656i
\(619\) −32.4119 −1.30274 −0.651371 0.758759i \(-0.725806\pi\)
−0.651371 + 0.758759i \(0.725806\pi\)
\(620\) 0 0
\(621\) 1.23884 0.0497130
\(622\) 11.8677i 0.475850i
\(623\) − 7.22425i − 0.289434i
\(624\) 177.607 7.10998
\(625\) 0 0
\(626\) −79.6893 −3.18502
\(627\) 3.35026i 0.133797i
\(628\) − 27.9756i − 1.11635i
\(629\) 46.5111 1.85452
\(630\) 0 0
\(631\) −27.3258 −1.08782 −0.543912 0.839142i \(-0.683057\pi\)
−0.543912 + 0.839142i \(0.683057\pi\)
\(632\) 16.6312i 0.661553i
\(633\) 11.0132i 0.437734i
\(634\) 41.1998 1.63625
\(635\) 0 0
\(636\) 2.96239 0.117466
\(637\) − 5.83146i − 0.231051i
\(638\) − 9.66291i − 0.382558i
\(639\) 49.0395 1.93997
\(640\) 0 0
\(641\) 19.4460 0.768069 0.384034 0.923319i \(-0.374534\pi\)
0.384034 + 0.923319i \(0.374534\pi\)
\(642\) 76.5764i 3.02223i
\(643\) 5.29314i 0.208741i 0.994538 + 0.104370i \(0.0332828\pi\)
−0.994538 + 0.104370i \(0.966717\pi\)
\(644\) −16.4690 −0.648969
\(645\) 0 0
\(646\) 19.6629 0.773627
\(647\) − 35.0966i − 1.37979i −0.723909 0.689896i \(-0.757656\pi\)
0.723909 0.689896i \(-0.242344\pi\)
\(648\) 71.8261i 2.82159i
\(649\) 13.5999 0.533843
\(650\) 0 0
\(651\) −13.1187 −0.514163
\(652\) − 17.4314i − 0.682665i
\(653\) 27.7988i 1.08785i 0.839134 + 0.543925i \(0.183062\pi\)
−0.839134 + 0.543925i \(0.816938\pi\)
\(654\) 14.5139 0.567538
\(655\) 0 0
\(656\) −61.6869 −2.40847
\(657\) 35.8822i 1.39990i
\(658\) − 31.6507i − 1.23387i
\(659\) −19.6180 −0.764209 −0.382105 0.924119i \(-0.624801\pi\)
−0.382105 + 0.924119i \(0.624801\pi\)
\(660\) 0 0
\(661\) 21.5633 0.838713 0.419357 0.907822i \(-0.362256\pi\)
0.419357 + 0.907822i \(0.362256\pi\)
\(662\) 16.7513i 0.651058i
\(663\) − 78.7631i − 3.05890i
\(664\) 89.7156 3.48164
\(665\) 0 0
\(666\) 72.1436 2.79551
\(667\) − 11.5369i − 0.446711i
\(668\) 58.1378i 2.24942i
\(669\) −19.3054 −0.746388
\(670\) 0 0
\(671\) 1.41327 0.0545585
\(672\) 39.5755i 1.52666i
\(673\) − 21.0679i − 0.812109i −0.913849 0.406054i \(-0.866904\pi\)
0.913849 0.406054i \(-0.133096\pi\)
\(674\) −42.4544 −1.63528
\(675\) 0 0
\(676\) 108.313 4.16589
\(677\) − 34.5174i − 1.32661i −0.748349 0.663306i \(-0.769153\pi\)
0.748349 0.663306i \(-0.230847\pi\)
\(678\) − 62.0625i − 2.38350i
\(679\) −0.836381 −0.0320973
\(680\) 0 0
\(681\) 25.9248 0.993440
\(682\) 14.1441i 0.541606i
\(683\) 33.7802i 1.29256i 0.763099 + 0.646282i \(0.223677\pi\)
−0.763099 + 0.646282i \(0.776323\pi\)
\(684\) 21.9756 0.840257
\(685\) 0 0
\(686\) 2.67513 0.102137
\(687\) − 73.0757i − 2.78801i
\(688\) 72.3547i 2.75850i
\(689\) 1.35026 0.0514409
\(690\) 0 0
\(691\) −13.8618 −0.527327 −0.263663 0.964615i \(-0.584931\pi\)
−0.263663 + 0.964615i \(0.584931\pi\)
\(692\) 46.3063i 1.76030i
\(693\) 3.15633i 0.119899i
\(694\) 18.1681 0.689651
\(695\) 0 0
\(696\) −75.6747 −2.86844
\(697\) 27.3561i 1.03619i
\(698\) − 71.5569i − 2.70847i
\(699\) −21.6629 −0.819367
\(700\) 0 0
\(701\) 40.5256 1.53063 0.765316 0.643655i \(-0.222583\pi\)
0.765316 + 0.643655i \(0.222583\pi\)
\(702\) − 6.05079i − 0.228372i
\(703\) 11.5369i 0.435123i
\(704\) 18.1187 0.682875
\(705\) 0 0
\(706\) 45.1100 1.69774
\(707\) 7.41327i 0.278805i
\(708\) − 173.995i − 6.53914i
\(709\) 0.850969 0.0319588 0.0159794 0.999872i \(-0.494913\pi\)
0.0159794 + 0.999872i \(0.494913\pi\)
\(710\) 0 0
\(711\) 6.21696 0.233154
\(712\) 60.9986i 2.28602i
\(713\) 16.8872i 0.632429i
\(714\) 36.1319 1.35220
\(715\) 0 0
\(716\) −135.289 −5.05598
\(717\) − 52.6371i − 1.96577i
\(718\) 10.1465i 0.378663i
\(719\) −22.5769 −0.841976 −0.420988 0.907066i \(-0.638317\pi\)
−0.420988 + 0.907066i \(0.638317\pi\)
\(720\) 0 0
\(721\) −4.21933 −0.157136
\(722\) − 45.9502i − 1.71009i
\(723\) 23.1695i 0.861683i
\(724\) 57.7499 2.14626
\(725\) 0 0
\(726\) 6.63752 0.246341
\(727\) − 12.5174i − 0.464244i −0.972687 0.232122i \(-0.925433\pi\)
0.972687 0.232122i \(-0.0745669\pi\)
\(728\) 49.2384i 1.82490i
\(729\) 29.7367 1.10136
\(730\) 0 0
\(731\) 32.0870 1.18678
\(732\) − 18.0811i − 0.668297i
\(733\) 16.6678i 0.615641i 0.951445 + 0.307820i \(0.0995996\pi\)
−0.951445 + 0.307820i \(0.900400\pi\)
\(734\) −17.0400 −0.628957
\(735\) 0 0
\(736\) 50.9438 1.87781
\(737\) − 10.8568i − 0.399917i
\(738\) 42.4323i 1.56196i
\(739\) −42.7005 −1.57076 −0.785382 0.619011i \(-0.787533\pi\)
−0.785382 + 0.619011i \(0.787533\pi\)
\(740\) 0 0
\(741\) 19.5369 0.717706
\(742\) 0.619421i 0.0227397i
\(743\) − 19.6873i − 0.722259i −0.932516 0.361129i \(-0.882391\pi\)
0.932516 0.361129i \(-0.117609\pi\)
\(744\) 110.769 4.06099
\(745\) 0 0
\(746\) 57.0651 2.08930
\(747\) − 33.5369i − 1.22705i
\(748\) − 28.0689i − 1.02630i
\(749\) −11.5369 −0.421549
\(750\) 0 0
\(751\) −5.85940 −0.213813 −0.106906 0.994269i \(-0.534094\pi\)
−0.106906 + 0.994269i \(0.534094\pi\)
\(752\) 145.232i 5.29605i
\(753\) − 4.65562i − 0.169660i
\(754\) −56.3488 −2.05210
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 40.5863i 1.47513i 0.675274 + 0.737567i \(0.264025\pi\)
−0.675274 + 0.737567i \(0.735975\pi\)
\(758\) − 66.3098i − 2.40848i
\(759\) 7.92478 0.287651
\(760\) 0 0
\(761\) 21.8472 0.791960 0.395980 0.918259i \(-0.370405\pi\)
0.395980 + 0.918259i \(0.370405\pi\)
\(762\) 112.588i 4.07864i
\(763\) 2.18664i 0.0791618i
\(764\) −57.7499 −2.08932
\(765\) 0 0
\(766\) 14.6013 0.527567
\(767\) − 79.3073i − 2.86362i
\(768\) − 20.0689i − 0.724173i
\(769\) −45.2892 −1.63317 −0.816585 0.577226i \(-0.804135\pi\)
−0.816585 + 0.577226i \(0.804135\pi\)
\(770\) 0 0
\(771\) −67.3376 −2.42510
\(772\) − 3.11871i − 0.112245i
\(773\) − 33.8153i − 1.21625i −0.793841 0.608125i \(-0.791922\pi\)
0.793841 0.608125i \(-0.208078\pi\)
\(774\) 49.7704 1.78896
\(775\) 0 0
\(776\) 7.06205 0.253513
\(777\) 21.1998i 0.760539i
\(778\) 36.7123i 1.31620i
\(779\) −6.78560 −0.243119
\(780\) 0 0
\(781\) 15.5369 0.555954
\(782\) − 46.5111i − 1.66323i
\(783\) 1.40105i 0.0500693i
\(784\) −12.2750 −0.438394
\(785\) 0 0
\(786\) −65.8759 −2.34972
\(787\) − 1.27504i − 0.0454502i −0.999742 0.0227251i \(-0.992766\pi\)
0.999742 0.0227251i \(-0.00723425\pi\)
\(788\) 78.9194i 2.81139i
\(789\) 31.7889 1.13172
\(790\) 0 0
\(791\) 9.35026 0.332457
\(792\) − 26.6507i − 0.946991i
\(793\) − 8.24140i − 0.292661i
\(794\) 5.64832 0.200452
\(795\) 0 0
\(796\) −64.7753 −2.29590
\(797\) 42.5256i 1.50634i 0.657829 + 0.753168i \(0.271475\pi\)
−0.657829 + 0.753168i \(0.728525\pi\)
\(798\) 8.96239i 0.317265i
\(799\) 64.4055 2.27850
\(800\) 0 0
\(801\) 22.8021 0.805672
\(802\) 51.2262i 1.80886i
\(803\) 11.3684i 0.401181i
\(804\) −138.901 −4.89865
\(805\) 0 0
\(806\) 82.4807 2.90526
\(807\) − 15.5369i − 0.546925i
\(808\) − 62.5945i − 2.20207i
\(809\) 14.7151 0.517356 0.258678 0.965964i \(-0.416713\pi\)
0.258678 + 0.965964i \(0.416713\pi\)
\(810\) 0 0
\(811\) 51.7743 1.81804 0.909021 0.416750i \(-0.136831\pi\)
0.909021 + 0.416750i \(0.136831\pi\)
\(812\) − 18.6253i − 0.653620i
\(813\) 14.2374i 0.499328i
\(814\) 22.8568 0.801132
\(815\) 0 0
\(816\) −165.794 −5.80395
\(817\) 7.95906i 0.278452i
\(818\) 49.9937i 1.74799i
\(819\) 18.4060 0.643157
\(820\) 0 0
\(821\) 2.64974 0.0924765 0.0462383 0.998930i \(-0.485277\pi\)
0.0462383 + 0.998930i \(0.485277\pi\)
\(822\) 72.9643i 2.54492i
\(823\) 5.76845i 0.201076i 0.994933 + 0.100538i \(0.0320563\pi\)
−0.994933 + 0.100538i \(0.967944\pi\)
\(824\) 35.6263 1.24110
\(825\) 0 0
\(826\) 36.3815 1.26588
\(827\) 13.4920i 0.469163i 0.972096 + 0.234581i \(0.0753719\pi\)
−0.972096 + 0.234581i \(0.924628\pi\)
\(828\) − 51.9814i − 1.80648i
\(829\) 4.70052 0.163256 0.0816280 0.996663i \(-0.473988\pi\)
0.0816280 + 0.996663i \(0.473988\pi\)
\(830\) 0 0
\(831\) 20.7367 0.719349
\(832\) − 105.658i − 3.66305i
\(833\) 5.44358i 0.188609i
\(834\) −45.7137 −1.58294
\(835\) 0 0
\(836\) 6.96239 0.240799
\(837\) − 2.05079i − 0.0708855i
\(838\) 2.06888i 0.0714684i
\(839\) 38.8045 1.33968 0.669839 0.742506i \(-0.266363\pi\)
0.669839 + 0.742506i \(0.266363\pi\)
\(840\) 0 0
\(841\) −15.9525 −0.550088
\(842\) − 28.1417i − 0.969828i
\(843\) 20.9624i 0.721983i
\(844\) 22.8872 0.787809
\(845\) 0 0
\(846\) 99.8999 3.43463
\(847\) 1.00000i 0.0343604i
\(848\) − 2.84226i − 0.0976036i
\(849\) 2.07522 0.0712215
\(850\) 0 0
\(851\) 27.2896 0.935476
\(852\) − 198.777i − 6.80998i
\(853\) 20.6824i 0.708153i 0.935217 + 0.354076i \(0.115205\pi\)
−0.935217 + 0.354076i \(0.884795\pi\)
\(854\) 3.78067 0.129372
\(855\) 0 0
\(856\) 97.4128 3.32950
\(857\) − 26.3453i − 0.899940i −0.893044 0.449970i \(-0.851435\pi\)
0.893044 0.449970i \(-0.148565\pi\)
\(858\) − 38.7064i − 1.32141i
\(859\) 8.51151 0.290409 0.145205 0.989402i \(-0.453616\pi\)
0.145205 + 0.989402i \(0.453616\pi\)
\(860\) 0 0
\(861\) −12.4690 −0.424942
\(862\) − 66.1582i − 2.25336i
\(863\) 7.56722i 0.257591i 0.991671 + 0.128796i \(0.0411111\pi\)
−0.991671 + 0.128796i \(0.958889\pi\)
\(864\) −6.18664 −0.210474
\(865\) 0 0
\(866\) 49.6502 1.68718
\(867\) 31.3439i 1.06450i
\(868\) 27.2628i 0.925360i
\(869\) 1.96968 0.0668169
\(870\) 0 0
\(871\) −63.3112 −2.14522
\(872\) − 18.4631i − 0.625239i
\(873\) − 2.63989i − 0.0893467i
\(874\) 11.5369 0.390242
\(875\) 0 0
\(876\) 145.445 4.91413
\(877\) − 17.2955i − 0.584028i −0.956414 0.292014i \(-0.905675\pi\)
0.956414 0.292014i \(-0.0943254\pi\)
\(878\) 3.81336i 0.128695i
\(879\) −6.74543 −0.227518
\(880\) 0 0
\(881\) 20.4504 0.688992 0.344496 0.938788i \(-0.388050\pi\)
0.344496 + 0.938788i \(0.388050\pi\)
\(882\) 8.44358i 0.284310i
\(883\) − 49.6589i − 1.67116i −0.549371 0.835578i \(-0.685133\pi\)
0.549371 0.835578i \(-0.314867\pi\)
\(884\) −163.682 −5.50524
\(885\) 0 0
\(886\) −107.520 −3.61221
\(887\) 47.1100i 1.58180i 0.611946 + 0.790900i \(0.290387\pi\)
−0.611946 + 0.790900i \(0.709613\pi\)
\(888\) − 179.002i − 6.00693i
\(889\) −16.9624 −0.568900
\(890\) 0 0
\(891\) 8.50659 0.284981
\(892\) 40.1197i 1.34331i
\(893\) 15.9756i 0.534602i
\(894\) −151.415 −5.06407
\(895\) 0 0
\(896\) 16.5696 0.553551
\(897\) − 46.2130i − 1.54301i
\(898\) − 33.8554i − 1.12977i
\(899\) −19.0982 −0.636962
\(900\) 0 0
\(901\) −1.26045 −0.0419917
\(902\) 13.4436i 0.447622i
\(903\) 14.6253i 0.486700i
\(904\) −78.9497 −2.62583
\(905\) 0 0
\(906\) −21.5369 −0.715516
\(907\) − 14.4591i − 0.480107i −0.970760 0.240054i \(-0.922835\pi\)
0.970760 0.240054i \(-0.0771651\pi\)
\(908\) − 53.8759i − 1.78793i
\(909\) −23.3987 −0.776085
\(910\) 0 0
\(911\) −31.5369 −1.04486 −0.522432 0.852681i \(-0.674975\pi\)
−0.522432 + 0.852681i \(0.674975\pi\)
\(912\) − 41.1246i − 1.36177i
\(913\) − 10.6253i − 0.351646i
\(914\) −1.45580 −0.0481536
\(915\) 0 0
\(916\) −151.863 −5.01770
\(917\) − 9.92478i − 0.327745i
\(918\) 5.64832i 0.186422i
\(919\) −5.26328 −0.173620 −0.0868098 0.996225i \(-0.527667\pi\)
−0.0868098 + 0.996225i \(0.527667\pi\)
\(920\) 0 0
\(921\) 20.7513 0.683779
\(922\) − 30.9659i − 1.01981i
\(923\) − 90.6028i − 2.98223i
\(924\) 12.7938 0.420887
\(925\) 0 0
\(926\) −63.6542 −2.09181
\(927\) − 13.3176i − 0.437407i
\(928\) 57.6140i 1.89127i
\(929\) −26.0508 −0.854699 −0.427349 0.904087i \(-0.640553\pi\)
−0.427349 + 0.904087i \(0.640553\pi\)
\(930\) 0 0
\(931\) −1.35026 −0.0442530
\(932\) 45.0191i 1.47465i
\(933\) − 11.0073i − 0.360363i
\(934\) −7.13681 −0.233524
\(935\) 0 0
\(936\) −155.412 −5.07981
\(937\) 29.3439i 0.958624i 0.877645 + 0.479312i \(0.159114\pi\)
−0.877645 + 0.479312i \(0.840886\pi\)
\(938\) − 29.0435i − 0.948304i
\(939\) 73.9121 2.41203
\(940\) 0 0
\(941\) −28.6375 −0.933556 −0.466778 0.884374i \(-0.654585\pi\)
−0.466778 + 0.884374i \(0.654585\pi\)
\(942\) 36.0118i 1.17333i
\(943\) 16.0508i 0.522685i
\(944\) −166.939 −5.43341
\(945\) 0 0
\(946\) 15.7685 0.512677
\(947\) 52.8178i 1.71635i 0.513358 + 0.858174i \(0.328401\pi\)
−0.513358 + 0.858174i \(0.671599\pi\)
\(948\) − 25.1998i − 0.818452i
\(949\) 66.2941 2.15200
\(950\) 0 0
\(951\) −38.2130 −1.23914
\(952\) − 45.9633i − 1.48968i
\(953\) 37.1939i 1.20483i 0.798183 + 0.602415i \(0.205795\pi\)
−0.798183 + 0.602415i \(0.794205\pi\)
\(954\) −1.95509 −0.0632985
\(955\) 0 0
\(956\) −109.388 −3.53787
\(957\) 8.96239i 0.289713i
\(958\) 28.6516i 0.925693i
\(959\) −10.9927 −0.354973
\(960\) 0 0
\(961\) −3.04491 −0.0982228
\(962\) − 133.289i − 4.29740i
\(963\) − 36.4142i − 1.17343i
\(964\) 48.1500 1.55081
\(965\) 0 0
\(966\) 21.1998 0.682093
\(967\) − 4.07125i − 0.130923i −0.997855 0.0654613i \(-0.979148\pi\)
0.997855 0.0654613i \(-0.0208519\pi\)
\(968\) − 8.44358i − 0.271387i
\(969\) −18.2374 −0.585871
\(970\) 0 0
\(971\) 0.773377 0.0248188 0.0124094 0.999923i \(-0.496050\pi\)
0.0124094 + 0.999923i \(0.496050\pi\)
\(972\) − 114.832i − 3.68324i
\(973\) − 6.88717i − 0.220792i
\(974\) 46.6312 1.49416
\(975\) 0 0
\(976\) −17.3479 −0.555292
\(977\) 37.8740i 1.21170i 0.795580 + 0.605848i \(0.207166\pi\)
−0.795580 + 0.605848i \(0.792834\pi\)
\(978\) 22.4387i 0.717509i
\(979\) 7.22425 0.230888
\(980\) 0 0
\(981\) −6.90175 −0.220356
\(982\) − 75.8916i − 2.42180i
\(983\) − 15.5794i − 0.496907i −0.968644 0.248453i \(-0.920078\pi\)
0.968644 0.248453i \(-0.0799223\pi\)
\(984\) 105.283 3.35629
\(985\) 0 0
\(986\) 52.6009 1.67515
\(987\) 29.3561i 0.934416i
\(988\) − 40.6009i − 1.29169i
\(989\) 18.8265 0.598649
\(990\) 0 0
\(991\) 27.0982 0.860804 0.430402 0.902637i \(-0.358372\pi\)
0.430402 + 0.902637i \(0.358372\pi\)
\(992\) − 84.3327i − 2.67756i
\(993\) − 15.5369i − 0.493049i
\(994\) 41.5633 1.31831
\(995\) 0 0
\(996\) −135.938 −4.30737
\(997\) − 50.4060i − 1.59637i −0.602410 0.798187i \(-0.705793\pi\)
0.602410 0.798187i \(-0.294207\pi\)
\(998\) − 73.5026i − 2.32668i
\(999\) −3.31406 −0.104852
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 1925.2.b.n.1849.6 6
5.2 odd 4 385.2.a.f.1.1 3
5.3 odd 4 1925.2.a.v.1.3 3
5.4 even 2 inner 1925.2.b.n.1849.1 6
15.2 even 4 3465.2.a.bh.1.3 3
20.7 even 4 6160.2.a.bn.1.3 3
35.27 even 4 2695.2.a.g.1.1 3
55.32 even 4 4235.2.a.q.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.f.1.1 3 5.2 odd 4
1925.2.a.v.1.3 3 5.3 odd 4
1925.2.b.n.1849.1 6 5.4 even 2 inner
1925.2.b.n.1849.6 6 1.1 even 1 trivial
2695.2.a.g.1.1 3 35.27 even 4
3465.2.a.bh.1.3 3 15.2 even 4
4235.2.a.q.1.3 3 55.32 even 4
6160.2.a.bn.1.3 3 20.7 even 4