Properties

Label 225.4.b.e.199.2
Level $225$
Weight $4$
Character 225.199
Analytic conductor $13.275$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.4.b.e.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +7.00000 q^{4} +24.0000i q^{7} +15.0000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} +7.00000 q^{4} +24.0000i q^{7} +15.0000i q^{8} -52.0000 q^{11} +22.0000i q^{13} -24.0000 q^{14} +41.0000 q^{16} -14.0000i q^{17} +20.0000 q^{19} -52.0000i q^{22} +168.000i q^{23} -22.0000 q^{26} +168.000i q^{28} +230.000 q^{29} -288.000 q^{31} +161.000i q^{32} +14.0000 q^{34} +34.0000i q^{37} +20.0000i q^{38} -122.000 q^{41} -188.000i q^{43} -364.000 q^{44} -168.000 q^{46} +256.000i q^{47} -233.000 q^{49} +154.000i q^{52} +338.000i q^{53} -360.000 q^{56} +230.000i q^{58} +100.000 q^{59} +742.000 q^{61} -288.000i q^{62} +167.000 q^{64} +84.0000i q^{67} -98.0000i q^{68} +328.000 q^{71} -38.0000i q^{73} -34.0000 q^{74} +140.000 q^{76} -1248.00i q^{77} +240.000 q^{79} -122.000i q^{82} -1212.00i q^{83} +188.000 q^{86} -780.000i q^{88} +330.000 q^{89} -528.000 q^{91} +1176.00i q^{92} -256.000 q^{94} -866.000i q^{97} -233.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{4} - 104 q^{11} - 48 q^{14} + 82 q^{16} + 40 q^{19} - 44 q^{26} + 460 q^{29} - 576 q^{31} + 28 q^{34} - 244 q^{41} - 728 q^{44} - 336 q^{46} - 466 q^{49} - 720 q^{56} + 200 q^{59} + 1484 q^{61} + 334 q^{64} + 656 q^{71} - 68 q^{74} + 280 q^{76} + 480 q^{79} + 376 q^{86} + 660 q^{89} - 1056 q^{91} - 512 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.353553i 0.984251 + 0.176777i \(0.0565670\pi\)
−0.984251 + 0.176777i \(0.943433\pi\)
\(3\) 0 0
\(4\) 7.00000 0.875000
\(5\) 0 0
\(6\) 0 0
\(7\) 24.0000i 1.29588i 0.761692 + 0.647939i \(0.224369\pi\)
−0.761692 + 0.647939i \(0.775631\pi\)
\(8\) 15.0000i 0.662913i
\(9\) 0 0
\(10\) 0 0
\(11\) −52.0000 −1.42533 −0.712663 0.701506i \(-0.752511\pi\)
−0.712663 + 0.701506i \(0.752511\pi\)
\(12\) 0 0
\(13\) 22.0000i 0.469362i 0.972072 + 0.234681i \(0.0754045\pi\)
−0.972072 + 0.234681i \(0.924595\pi\)
\(14\) −24.0000 −0.458162
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) − 14.0000i − 0.199735i −0.995001 0.0998676i \(-0.968158\pi\)
0.995001 0.0998676i \(-0.0318419\pi\)
\(18\) 0 0
\(19\) 20.0000 0.241490 0.120745 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 52.0000i − 0.503929i
\(23\) 168.000i 1.52306i 0.648129 + 0.761531i \(0.275552\pi\)
−0.648129 + 0.761531i \(0.724448\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −22.0000 −0.165944
\(27\) 0 0
\(28\) 168.000i 1.13389i
\(29\) 230.000 1.47276 0.736378 0.676570i \(-0.236535\pi\)
0.736378 + 0.676570i \(0.236535\pi\)
\(30\) 0 0
\(31\) −288.000 −1.66859 −0.834296 0.551317i \(-0.814125\pi\)
−0.834296 + 0.551317i \(0.814125\pi\)
\(32\) 161.000i 0.889408i
\(33\) 0 0
\(34\) 14.0000 0.0706171
\(35\) 0 0
\(36\) 0 0
\(37\) 34.0000i 0.151069i 0.997143 + 0.0755347i \(0.0240664\pi\)
−0.997143 + 0.0755347i \(0.975934\pi\)
\(38\) 20.0000i 0.0853797i
\(39\) 0 0
\(40\) 0 0
\(41\) −122.000 −0.464712 −0.232356 0.972631i \(-0.574643\pi\)
−0.232356 + 0.972631i \(0.574643\pi\)
\(42\) 0 0
\(43\) − 188.000i − 0.666738i −0.942796 0.333369i \(-0.891815\pi\)
0.942796 0.333369i \(-0.108185\pi\)
\(44\) −364.000 −1.24716
\(45\) 0 0
\(46\) −168.000 −0.538484
\(47\) 256.000i 0.794499i 0.917711 + 0.397249i \(0.130035\pi\)
−0.917711 + 0.397249i \(0.869965\pi\)
\(48\) 0 0
\(49\) −233.000 −0.679300
\(50\) 0 0
\(51\) 0 0
\(52\) 154.000i 0.410691i
\(53\) 338.000i 0.875998i 0.898976 + 0.437999i \(0.144313\pi\)
−0.898976 + 0.437999i \(0.855687\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −360.000 −0.859054
\(57\) 0 0
\(58\) 230.000i 0.520698i
\(59\) 100.000 0.220659 0.110330 0.993895i \(-0.464809\pi\)
0.110330 + 0.993895i \(0.464809\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) − 288.000i − 0.589936i
\(63\) 0 0
\(64\) 167.000 0.326172
\(65\) 0 0
\(66\) 0 0
\(67\) 84.0000i 0.153168i 0.997063 + 0.0765838i \(0.0244013\pi\)
−0.997063 + 0.0765838i \(0.975599\pi\)
\(68\) − 98.0000i − 0.174768i
\(69\) 0 0
\(70\) 0 0
\(71\) 328.000 0.548260 0.274130 0.961693i \(-0.411610\pi\)
0.274130 + 0.961693i \(0.411610\pi\)
\(72\) 0 0
\(73\) − 38.0000i − 0.0609255i −0.999536 0.0304628i \(-0.990302\pi\)
0.999536 0.0304628i \(-0.00969810\pi\)
\(74\) −34.0000 −0.0534111
\(75\) 0 0
\(76\) 140.000 0.211304
\(77\) − 1248.00i − 1.84705i
\(78\) 0 0
\(79\) 240.000 0.341799 0.170899 0.985288i \(-0.445333\pi\)
0.170899 + 0.985288i \(0.445333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 122.000i − 0.164301i
\(83\) − 1212.00i − 1.60282i −0.598114 0.801411i \(-0.704083\pi\)
0.598114 0.801411i \(-0.295917\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 188.000 0.235727
\(87\) 0 0
\(88\) − 780.000i − 0.944867i
\(89\) 330.000 0.393033 0.196516 0.980501i \(-0.437037\pi\)
0.196516 + 0.980501i \(0.437037\pi\)
\(90\) 0 0
\(91\) −528.000 −0.608236
\(92\) 1176.00i 1.33268i
\(93\) 0 0
\(94\) −256.000 −0.280898
\(95\) 0 0
\(96\) 0 0
\(97\) − 866.000i − 0.906484i −0.891387 0.453242i \(-0.850267\pi\)
0.891387 0.453242i \(-0.149733\pi\)
\(98\) − 233.000i − 0.240169i
\(99\) 0 0
\(100\) 0 0
\(101\) 1218.00 1.19996 0.599978 0.800017i \(-0.295176\pi\)
0.599978 + 0.800017i \(0.295176\pi\)
\(102\) 0 0
\(103\) − 88.0000i − 0.0841835i −0.999114 0.0420917i \(-0.986598\pi\)
0.999114 0.0420917i \(-0.0134022\pi\)
\(104\) −330.000 −0.311146
\(105\) 0 0
\(106\) −338.000 −0.309712
\(107\) 36.0000i 0.0325257i 0.999868 + 0.0162629i \(0.00517686\pi\)
−0.999868 + 0.0162629i \(0.994823\pi\)
\(108\) 0 0
\(109\) 970.000 0.852378 0.426189 0.904634i \(-0.359856\pi\)
0.426189 + 0.904634i \(0.359856\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 984.000i 0.830172i
\(113\) − 1042.00i − 0.867461i −0.901043 0.433731i \(-0.857197\pi\)
0.901043 0.433731i \(-0.142803\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1610.00 1.28866
\(117\) 0 0
\(118\) 100.000i 0.0780148i
\(119\) 336.000 0.258833
\(120\) 0 0
\(121\) 1373.00 1.03156
\(122\) 742.000i 0.550635i
\(123\) 0 0
\(124\) −2016.00 −1.46002
\(125\) 0 0
\(126\) 0 0
\(127\) − 1936.00i − 1.35269i −0.736583 0.676347i \(-0.763562\pi\)
0.736583 0.676347i \(-0.236438\pi\)
\(128\) 1455.00i 1.00473i
\(129\) 0 0
\(130\) 0 0
\(131\) −732.000 −0.488207 −0.244104 0.969749i \(-0.578494\pi\)
−0.244104 + 0.969749i \(0.578494\pi\)
\(132\) 0 0
\(133\) 480.000i 0.312942i
\(134\) −84.0000 −0.0541529
\(135\) 0 0
\(136\) 210.000 0.132407
\(137\) − 2214.00i − 1.38069i −0.723479 0.690346i \(-0.757458\pi\)
0.723479 0.690346i \(-0.242542\pi\)
\(138\) 0 0
\(139\) −20.0000 −0.0122042 −0.00610208 0.999981i \(-0.501942\pi\)
−0.00610208 + 0.999981i \(0.501942\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 328.000i 0.193839i
\(143\) − 1144.00i − 0.668994i
\(144\) 0 0
\(145\) 0 0
\(146\) 38.0000 0.0215404
\(147\) 0 0
\(148\) 238.000i 0.132186i
\(149\) −1330.00 −0.731261 −0.365630 0.930760i \(-0.619147\pi\)
−0.365630 + 0.930760i \(0.619147\pi\)
\(150\) 0 0
\(151\) −1208.00 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 300.000i 0.160087i
\(153\) 0 0
\(154\) 1248.00 0.653031
\(155\) 0 0
\(156\) 0 0
\(157\) 3514.00i 1.78629i 0.449768 + 0.893146i \(0.351507\pi\)
−0.449768 + 0.893146i \(0.648493\pi\)
\(158\) 240.000i 0.120844i
\(159\) 0 0
\(160\) 0 0
\(161\) −4032.00 −1.97370
\(162\) 0 0
\(163\) − 2068.00i − 0.993732i −0.867827 0.496866i \(-0.834484\pi\)
0.867827 0.496866i \(-0.165516\pi\)
\(164\) −854.000 −0.406623
\(165\) 0 0
\(166\) 1212.00 0.566683
\(167\) − 24.0000i − 0.0111208i −0.999985 0.00556041i \(-0.998230\pi\)
0.999985 0.00556041i \(-0.00176994\pi\)
\(168\) 0 0
\(169\) 1713.00 0.779700
\(170\) 0 0
\(171\) 0 0
\(172\) − 1316.00i − 0.583396i
\(173\) 618.000i 0.271593i 0.990737 + 0.135797i \(0.0433594\pi\)
−0.990737 + 0.135797i \(0.956641\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2132.00 −0.913100
\(177\) 0 0
\(178\) 330.000i 0.138958i
\(179\) 3340.00 1.39466 0.697328 0.716752i \(-0.254372\pi\)
0.697328 + 0.716752i \(0.254372\pi\)
\(180\) 0 0
\(181\) −178.000 −0.0730974 −0.0365487 0.999332i \(-0.511636\pi\)
−0.0365487 + 0.999332i \(0.511636\pi\)
\(182\) − 528.000i − 0.215044i
\(183\) 0 0
\(184\) −2520.00 −1.00966
\(185\) 0 0
\(186\) 0 0
\(187\) 728.000i 0.284688i
\(188\) 1792.00i 0.695186i
\(189\) 0 0
\(190\) 0 0
\(191\) 1888.00 0.715240 0.357620 0.933867i \(-0.383588\pi\)
0.357620 + 0.933867i \(0.383588\pi\)
\(192\) 0 0
\(193\) 1922.00i 0.716832i 0.933562 + 0.358416i \(0.116683\pi\)
−0.933562 + 0.358416i \(0.883317\pi\)
\(194\) 866.000 0.320491
\(195\) 0 0
\(196\) −1631.00 −0.594388
\(197\) 2526.00i 0.913554i 0.889581 + 0.456777i \(0.150996\pi\)
−0.889581 + 0.456777i \(0.849004\pi\)
\(198\) 0 0
\(199\) 1160.00 0.413217 0.206609 0.978424i \(-0.433757\pi\)
0.206609 + 0.978424i \(0.433757\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1218.00i 0.424248i
\(203\) 5520.00i 1.90851i
\(204\) 0 0
\(205\) 0 0
\(206\) 88.0000 0.0297634
\(207\) 0 0
\(208\) 902.000i 0.300685i
\(209\) −1040.00 −0.344202
\(210\) 0 0
\(211\) −4468.00 −1.45777 −0.728886 0.684635i \(-0.759961\pi\)
−0.728886 + 0.684635i \(0.759961\pi\)
\(212\) 2366.00i 0.766498i
\(213\) 0 0
\(214\) −36.0000 −0.0114996
\(215\) 0 0
\(216\) 0 0
\(217\) − 6912.00i − 2.16229i
\(218\) 970.000i 0.301361i
\(219\) 0 0
\(220\) 0 0
\(221\) 308.000 0.0937481
\(222\) 0 0
\(223\) 6032.00i 1.81136i 0.423965 + 0.905678i \(0.360638\pi\)
−0.423965 + 0.905678i \(0.639362\pi\)
\(224\) −3864.00 −1.15256
\(225\) 0 0
\(226\) 1042.00 0.306694
\(227\) 2636.00i 0.770738i 0.922763 + 0.385369i \(0.125926\pi\)
−0.922763 + 0.385369i \(0.874074\pi\)
\(228\) 0 0
\(229\) −4830.00 −1.39378 −0.696889 0.717179i \(-0.745433\pi\)
−0.696889 + 0.717179i \(0.745433\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3450.00i 0.976309i
\(233\) − 2682.00i − 0.754093i −0.926194 0.377046i \(-0.876940\pi\)
0.926194 0.377046i \(-0.123060\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 700.000 0.193077
\(237\) 0 0
\(238\) 336.000i 0.0915111i
\(239\) 2320.00 0.627901 0.313950 0.949439i \(-0.398347\pi\)
0.313950 + 0.949439i \(0.398347\pi\)
\(240\) 0 0
\(241\) 2002.00 0.535104 0.267552 0.963543i \(-0.413785\pi\)
0.267552 + 0.963543i \(0.413785\pi\)
\(242\) 1373.00i 0.364710i
\(243\) 0 0
\(244\) 5194.00 1.36275
\(245\) 0 0
\(246\) 0 0
\(247\) 440.000i 0.113346i
\(248\) − 4320.00i − 1.10613i
\(249\) 0 0
\(250\) 0 0
\(251\) −132.000 −0.0331943 −0.0165971 0.999862i \(-0.505283\pi\)
−0.0165971 + 0.999862i \(0.505283\pi\)
\(252\) 0 0
\(253\) − 8736.00i − 2.17086i
\(254\) 1936.00 0.478250
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) − 7614.00i − 1.84805i −0.382335 0.924024i \(-0.624880\pi\)
0.382335 0.924024i \(-0.375120\pi\)
\(258\) 0 0
\(259\) −816.000 −0.195767
\(260\) 0 0
\(261\) 0 0
\(262\) − 732.000i − 0.172607i
\(263\) 4888.00i 1.14603i 0.819543 + 0.573017i \(0.194227\pi\)
−0.819543 + 0.573017i \(0.805773\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −480.000 −0.110642
\(267\) 0 0
\(268\) 588.000i 0.134022i
\(269\) 1270.00 0.287856 0.143928 0.989588i \(-0.454027\pi\)
0.143928 + 0.989588i \(0.454027\pi\)
\(270\) 0 0
\(271\) 1072.00 0.240293 0.120146 0.992756i \(-0.461664\pi\)
0.120146 + 0.992756i \(0.461664\pi\)
\(272\) − 574.000i − 0.127955i
\(273\) 0 0
\(274\) 2214.00 0.488148
\(275\) 0 0
\(276\) 0 0
\(277\) 5394.00i 1.17001i 0.811028 + 0.585007i \(0.198908\pi\)
−0.811028 + 0.585007i \(0.801092\pi\)
\(278\) − 20.0000i − 0.00431482i
\(279\) 0 0
\(280\) 0 0
\(281\) −2442.00 −0.518425 −0.259213 0.965820i \(-0.583463\pi\)
−0.259213 + 0.965820i \(0.583463\pi\)
\(282\) 0 0
\(283\) 2772.00i 0.582255i 0.956684 + 0.291128i \(0.0940305\pi\)
−0.956684 + 0.291128i \(0.905970\pi\)
\(284\) 2296.00 0.479727
\(285\) 0 0
\(286\) 1144.00 0.236525
\(287\) − 2928.00i − 0.602210i
\(288\) 0 0
\(289\) 4717.00 0.960106
\(290\) 0 0
\(291\) 0 0
\(292\) − 266.000i − 0.0533098i
\(293\) − 4542.00i − 0.905619i −0.891607 0.452810i \(-0.850422\pi\)
0.891607 0.452810i \(-0.149578\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −510.000 −0.100146
\(297\) 0 0
\(298\) − 1330.00i − 0.258540i
\(299\) −3696.00 −0.714867
\(300\) 0 0
\(301\) 4512.00 0.864011
\(302\) − 1208.00i − 0.230174i
\(303\) 0 0
\(304\) 820.000 0.154705
\(305\) 0 0
\(306\) 0 0
\(307\) − 5116.00i − 0.951093i −0.879691 0.475546i \(-0.842250\pi\)
0.879691 0.475546i \(-0.157750\pi\)
\(308\) − 8736.00i − 1.61617i
\(309\) 0 0
\(310\) 0 0
\(311\) 2808.00 0.511984 0.255992 0.966679i \(-0.417598\pi\)
0.255992 + 0.966679i \(0.417598\pi\)
\(312\) 0 0
\(313\) − 7318.00i − 1.32153i −0.750594 0.660763i \(-0.770233\pi\)
0.750594 0.660763i \(-0.229767\pi\)
\(314\) −3514.00 −0.631549
\(315\) 0 0
\(316\) 1680.00 0.299074
\(317\) 2246.00i 0.397943i 0.980005 + 0.198971i \(0.0637601\pi\)
−0.980005 + 0.198971i \(0.936240\pi\)
\(318\) 0 0
\(319\) −11960.0 −2.09916
\(320\) 0 0
\(321\) 0 0
\(322\) − 4032.00i − 0.697809i
\(323\) − 280.000i − 0.0482341i
\(324\) 0 0
\(325\) 0 0
\(326\) 2068.00 0.351337
\(327\) 0 0
\(328\) − 1830.00i − 0.308064i
\(329\) −6144.00 −1.02957
\(330\) 0 0
\(331\) 1332.00 0.221188 0.110594 0.993866i \(-0.464725\pi\)
0.110594 + 0.993866i \(0.464725\pi\)
\(332\) − 8484.00i − 1.40247i
\(333\) 0 0
\(334\) 24.0000 0.00393180
\(335\) 0 0
\(336\) 0 0
\(337\) 11534.0i 1.86438i 0.361966 + 0.932191i \(0.382106\pi\)
−0.361966 + 0.932191i \(0.617894\pi\)
\(338\) 1713.00i 0.275665i
\(339\) 0 0
\(340\) 0 0
\(341\) 14976.0 2.37829
\(342\) 0 0
\(343\) 2640.00i 0.415588i
\(344\) 2820.00 0.441989
\(345\) 0 0
\(346\) −618.000 −0.0960228
\(347\) 11956.0i 1.84966i 0.380382 + 0.924830i \(0.375793\pi\)
−0.380382 + 0.924830i \(0.624207\pi\)
\(348\) 0 0
\(349\) −4870.00 −0.746949 −0.373474 0.927640i \(-0.621834\pi\)
−0.373474 + 0.927640i \(0.621834\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 8372.00i − 1.26770i
\(353\) − 10722.0i − 1.61664i −0.588742 0.808321i \(-0.700377\pi\)
0.588742 0.808321i \(-0.299623\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2310.00 0.343904
\(357\) 0 0
\(358\) 3340.00i 0.493085i
\(359\) 120.000 0.0176417 0.00882083 0.999961i \(-0.497192\pi\)
0.00882083 + 0.999961i \(0.497192\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) − 178.000i − 0.0258438i
\(363\) 0 0
\(364\) −3696.00 −0.532206
\(365\) 0 0
\(366\) 0 0
\(367\) − 3936.00i − 0.559830i −0.960025 0.279915i \(-0.909694\pi\)
0.960025 0.279915i \(-0.0903063\pi\)
\(368\) 6888.00i 0.975711i
\(369\) 0 0
\(370\) 0 0
\(371\) −8112.00 −1.13519
\(372\) 0 0
\(373\) 3022.00i 0.419499i 0.977755 + 0.209750i \(0.0672649\pi\)
−0.977755 + 0.209750i \(0.932735\pi\)
\(374\) −728.000 −0.100652
\(375\) 0 0
\(376\) −3840.00 −0.526683
\(377\) 5060.00i 0.691255i
\(378\) 0 0
\(379\) 13340.0 1.80799 0.903997 0.427539i \(-0.140619\pi\)
0.903997 + 0.427539i \(0.140619\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1888.00i 0.252876i
\(383\) 1008.00i 0.134481i 0.997737 + 0.0672407i \(0.0214195\pi\)
−0.997737 + 0.0672407i \(0.978580\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1922.00 −0.253438
\(387\) 0 0
\(388\) − 6062.00i − 0.793174i
\(389\) 9630.00 1.25517 0.627584 0.778549i \(-0.284044\pi\)
0.627584 + 0.778549i \(0.284044\pi\)
\(390\) 0 0
\(391\) 2352.00 0.304209
\(392\) − 3495.00i − 0.450317i
\(393\) 0 0
\(394\) −2526.00 −0.322990
\(395\) 0 0
\(396\) 0 0
\(397\) − 7126.00i − 0.900866i −0.892810 0.450433i \(-0.851270\pi\)
0.892810 0.450433i \(-0.148730\pi\)
\(398\) 1160.00i 0.146094i
\(399\) 0 0
\(400\) 0 0
\(401\) 8718.00 1.08568 0.542838 0.839837i \(-0.317350\pi\)
0.542838 + 0.839837i \(0.317350\pi\)
\(402\) 0 0
\(403\) − 6336.00i − 0.783173i
\(404\) 8526.00 1.04996
\(405\) 0 0
\(406\) −5520.00 −0.674761
\(407\) − 1768.00i − 0.215323i
\(408\) 0 0
\(409\) 10870.0 1.31415 0.657074 0.753826i \(-0.271794\pi\)
0.657074 + 0.753826i \(0.271794\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 616.000i − 0.0736605i
\(413\) 2400.00i 0.285947i
\(414\) 0 0
\(415\) 0 0
\(416\) −3542.00 −0.417454
\(417\) 0 0
\(418\) − 1040.00i − 0.121694i
\(419\) −9700.00 −1.13097 −0.565484 0.824759i \(-0.691311\pi\)
−0.565484 + 0.824759i \(0.691311\pi\)
\(420\) 0 0
\(421\) 862.000 0.0997893 0.0498947 0.998754i \(-0.484111\pi\)
0.0498947 + 0.998754i \(0.484111\pi\)
\(422\) − 4468.00i − 0.515400i
\(423\) 0 0
\(424\) −5070.00 −0.580710
\(425\) 0 0
\(426\) 0 0
\(427\) 17808.0i 2.01824i
\(428\) 252.000i 0.0284600i
\(429\) 0 0
\(430\) 0 0
\(431\) −15792.0 −1.76490 −0.882452 0.470402i \(-0.844109\pi\)
−0.882452 + 0.470402i \(0.844109\pi\)
\(432\) 0 0
\(433\) 11602.0i 1.28766i 0.765169 + 0.643830i \(0.222655\pi\)
−0.765169 + 0.643830i \(0.777345\pi\)
\(434\) 6912.00 0.764485
\(435\) 0 0
\(436\) 6790.00 0.745830
\(437\) 3360.00i 0.367805i
\(438\) 0 0
\(439\) 440.000 0.0478361 0.0239181 0.999714i \(-0.492386\pi\)
0.0239181 + 0.999714i \(0.492386\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 308.000i 0.0331449i
\(443\) 10188.0i 1.09266i 0.837571 + 0.546328i \(0.183975\pi\)
−0.837571 + 0.546328i \(0.816025\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −6032.00 −0.640411
\(447\) 0 0
\(448\) 4008.00i 0.422679i
\(449\) −13310.0 −1.39897 −0.699485 0.714647i \(-0.746587\pi\)
−0.699485 + 0.714647i \(0.746587\pi\)
\(450\) 0 0
\(451\) 6344.00 0.662367
\(452\) − 7294.00i − 0.759029i
\(453\) 0 0
\(454\) −2636.00 −0.272497
\(455\) 0 0
\(456\) 0 0
\(457\) − 3226.00i − 0.330210i −0.986276 0.165105i \(-0.947204\pi\)
0.986276 0.165105i \(-0.0527963\pi\)
\(458\) − 4830.00i − 0.492775i
\(459\) 0 0
\(460\) 0 0
\(461\) −6582.00 −0.664977 −0.332488 0.943107i \(-0.607888\pi\)
−0.332488 + 0.943107i \(0.607888\pi\)
\(462\) 0 0
\(463\) 15072.0i 1.51286i 0.654073 + 0.756431i \(0.273059\pi\)
−0.654073 + 0.756431i \(0.726941\pi\)
\(464\) 9430.00 0.943484
\(465\) 0 0
\(466\) 2682.00 0.266612
\(467\) 476.000i 0.0471663i 0.999722 + 0.0235831i \(0.00750744\pi\)
−0.999722 + 0.0235831i \(0.992493\pi\)
\(468\) 0 0
\(469\) −2016.00 −0.198487
\(470\) 0 0
\(471\) 0 0
\(472\) 1500.00i 0.146278i
\(473\) 9776.00i 0.950319i
\(474\) 0 0
\(475\) 0 0
\(476\) 2352.00 0.226478
\(477\) 0 0
\(478\) 2320.00i 0.221997i
\(479\) −19680.0 −1.87725 −0.938624 0.344941i \(-0.887899\pi\)
−0.938624 + 0.344941i \(0.887899\pi\)
\(480\) 0 0
\(481\) −748.000 −0.0709062
\(482\) 2002.00i 0.189188i
\(483\) 0 0
\(484\) 9611.00 0.902611
\(485\) 0 0
\(486\) 0 0
\(487\) 5944.00i 0.553077i 0.961003 + 0.276538i \(0.0891873\pi\)
−0.961003 + 0.276538i \(0.910813\pi\)
\(488\) 11130.0i 1.03244i
\(489\) 0 0
\(490\) 0 0
\(491\) −10772.0 −0.990089 −0.495044 0.868868i \(-0.664848\pi\)
−0.495044 + 0.868868i \(0.664848\pi\)
\(492\) 0 0
\(493\) − 3220.00i − 0.294161i
\(494\) −440.000 −0.0400740
\(495\) 0 0
\(496\) −11808.0 −1.06894
\(497\) 7872.00i 0.710478i
\(498\) 0 0
\(499\) −8140.00 −0.730253 −0.365127 0.930958i \(-0.618974\pi\)
−0.365127 + 0.930958i \(0.618974\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 132.000i − 0.0117360i
\(503\) 13768.0i 1.22045i 0.792229 + 0.610223i \(0.208920\pi\)
−0.792229 + 0.610223i \(0.791080\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8736.00 0.767515
\(507\) 0 0
\(508\) − 13552.0i − 1.18361i
\(509\) 22150.0 1.92884 0.964422 0.264368i \(-0.0851633\pi\)
0.964422 + 0.264368i \(0.0851633\pi\)
\(510\) 0 0
\(511\) 912.000 0.0789521
\(512\) 11521.0i 0.994455i
\(513\) 0 0
\(514\) 7614.00 0.653384
\(515\) 0 0
\(516\) 0 0
\(517\) − 13312.0i − 1.13242i
\(518\) − 816.000i − 0.0692143i
\(519\) 0 0
\(520\) 0 0
\(521\) −1562.00 −0.131348 −0.0656741 0.997841i \(-0.520920\pi\)
−0.0656741 + 0.997841i \(0.520920\pi\)
\(522\) 0 0
\(523\) − 668.000i − 0.0558501i −0.999610 0.0279250i \(-0.991110\pi\)
0.999610 0.0279250i \(-0.00888997\pi\)
\(524\) −5124.00 −0.427181
\(525\) 0 0
\(526\) −4888.00 −0.405184
\(527\) 4032.00i 0.333276i
\(528\) 0 0
\(529\) −16057.0 −1.31972
\(530\) 0 0
\(531\) 0 0
\(532\) 3360.00i 0.273824i
\(533\) − 2684.00i − 0.218118i
\(534\) 0 0
\(535\) 0 0
\(536\) −1260.00 −0.101537
\(537\) 0 0
\(538\) 1270.00i 0.101772i
\(539\) 12116.0 0.968225
\(540\) 0 0
\(541\) −6138.00 −0.487788 −0.243894 0.969802i \(-0.578425\pi\)
−0.243894 + 0.969802i \(0.578425\pi\)
\(542\) 1072.00i 0.0849564i
\(543\) 0 0
\(544\) 2254.00 0.177646
\(545\) 0 0
\(546\) 0 0
\(547\) 10484.0i 0.819494i 0.912199 + 0.409747i \(0.134383\pi\)
−0.912199 + 0.409747i \(0.865617\pi\)
\(548\) − 15498.0i − 1.20811i
\(549\) 0 0
\(550\) 0 0
\(551\) 4600.00 0.355656
\(552\) 0 0
\(553\) 5760.00i 0.442930i
\(554\) −5394.00 −0.413663
\(555\) 0 0
\(556\) −140.000 −0.0106786
\(557\) 3606.00i 0.274311i 0.990550 + 0.137155i \(0.0437960\pi\)
−0.990550 + 0.137155i \(0.956204\pi\)
\(558\) 0 0
\(559\) 4136.00 0.312941
\(560\) 0 0
\(561\) 0 0
\(562\) − 2442.00i − 0.183291i
\(563\) − 12252.0i − 0.917159i −0.888654 0.458579i \(-0.848359\pi\)
0.888654 0.458579i \(-0.151641\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −2772.00 −0.205858
\(567\) 0 0
\(568\) 4920.00i 0.363448i
\(569\) −14550.0 −1.07200 −0.536000 0.844218i \(-0.680065\pi\)
−0.536000 + 0.844218i \(0.680065\pi\)
\(570\) 0 0
\(571\) −25468.0 −1.86655 −0.933277 0.359157i \(-0.883064\pi\)
−0.933277 + 0.359157i \(0.883064\pi\)
\(572\) − 8008.00i − 0.585369i
\(573\) 0 0
\(574\) 2928.00 0.212914
\(575\) 0 0
\(576\) 0 0
\(577\) − 12866.0i − 0.928282i −0.885761 0.464141i \(-0.846363\pi\)
0.885761 0.464141i \(-0.153637\pi\)
\(578\) 4717.00i 0.339449i
\(579\) 0 0
\(580\) 0 0
\(581\) 29088.0 2.07706
\(582\) 0 0
\(583\) − 17576.0i − 1.24858i
\(584\) 570.000 0.0403883
\(585\) 0 0
\(586\) 4542.00 0.320185
\(587\) − 14844.0i − 1.04374i −0.853024 0.521872i \(-0.825234\pi\)
0.853024 0.521872i \(-0.174766\pi\)
\(588\) 0 0
\(589\) −5760.00 −0.402948
\(590\) 0 0
\(591\) 0 0
\(592\) 1394.00i 0.0967788i
\(593\) − 20402.0i − 1.41283i −0.707797 0.706416i \(-0.750311\pi\)
0.707797 0.706416i \(-0.249689\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −9310.00 −0.639853
\(597\) 0 0
\(598\) − 3696.00i − 0.252744i
\(599\) 10760.0 0.733959 0.366980 0.930229i \(-0.380392\pi\)
0.366980 + 0.930229i \(0.380392\pi\)
\(600\) 0 0
\(601\) 14282.0 0.969343 0.484671 0.874696i \(-0.338939\pi\)
0.484671 + 0.874696i \(0.338939\pi\)
\(602\) 4512.00i 0.305474i
\(603\) 0 0
\(604\) −8456.00 −0.569652
\(605\) 0 0
\(606\) 0 0
\(607\) − 11056.0i − 0.739290i −0.929173 0.369645i \(-0.879479\pi\)
0.929173 0.369645i \(-0.120521\pi\)
\(608\) 3220.00i 0.214783i
\(609\) 0 0
\(610\) 0 0
\(611\) −5632.00 −0.372907
\(612\) 0 0
\(613\) − 16418.0i − 1.08176i −0.841101 0.540878i \(-0.818092\pi\)
0.841101 0.540878i \(-0.181908\pi\)
\(614\) 5116.00 0.336262
\(615\) 0 0
\(616\) 18720.0 1.22443
\(617\) − 10374.0i − 0.676891i −0.940986 0.338445i \(-0.890099\pi\)
0.940986 0.338445i \(-0.109901\pi\)
\(618\) 0 0
\(619\) 5260.00 0.341546 0.170773 0.985310i \(-0.445373\pi\)
0.170773 + 0.985310i \(0.445373\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 2808.00i 0.181014i
\(623\) 7920.00i 0.509323i
\(624\) 0 0
\(625\) 0 0
\(626\) 7318.00 0.467230
\(627\) 0 0
\(628\) 24598.0i 1.56300i
\(629\) 476.000 0.0301739
\(630\) 0 0
\(631\) 21352.0 1.34708 0.673542 0.739149i \(-0.264772\pi\)
0.673542 + 0.739149i \(0.264772\pi\)
\(632\) 3600.00i 0.226583i
\(633\) 0 0
\(634\) −2246.00 −0.140694
\(635\) 0 0
\(636\) 0 0
\(637\) − 5126.00i − 0.318838i
\(638\) − 11960.0i − 0.742164i
\(639\) 0 0
\(640\) 0 0
\(641\) 29118.0 1.79422 0.897108 0.441812i \(-0.145664\pi\)
0.897108 + 0.441812i \(0.145664\pi\)
\(642\) 0 0
\(643\) 5772.00i 0.354005i 0.984210 + 0.177003i \(0.0566401\pi\)
−0.984210 + 0.177003i \(0.943360\pi\)
\(644\) −28224.0 −1.72699
\(645\) 0 0
\(646\) 280.000 0.0170533
\(647\) − 14264.0i − 0.866732i −0.901218 0.433366i \(-0.857326\pi\)
0.901218 0.433366i \(-0.142674\pi\)
\(648\) 0 0
\(649\) −5200.00 −0.314511
\(650\) 0 0
\(651\) 0 0
\(652\) − 14476.0i − 0.869515i
\(653\) − 6902.00i − 0.413623i −0.978381 0.206812i \(-0.933691\pi\)
0.978381 0.206812i \(-0.0663088\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −5002.00 −0.297706
\(657\) 0 0
\(658\) − 6144.00i − 0.364009i
\(659\) 20140.0 1.19051 0.595253 0.803539i \(-0.297052\pi\)
0.595253 + 0.803539i \(0.297052\pi\)
\(660\) 0 0
\(661\) −3218.00 −0.189358 −0.0946790 0.995508i \(-0.530182\pi\)
−0.0946790 + 0.995508i \(0.530182\pi\)
\(662\) 1332.00i 0.0782019i
\(663\) 0 0
\(664\) 18180.0 1.06253
\(665\) 0 0
\(666\) 0 0
\(667\) 38640.0i 2.24310i
\(668\) − 168.000i − 0.00973071i
\(669\) 0 0
\(670\) 0 0
\(671\) −38584.0 −2.21985
\(672\) 0 0
\(673\) − 7518.00i − 0.430606i −0.976547 0.215303i \(-0.930926\pi\)
0.976547 0.215303i \(-0.0690739\pi\)
\(674\) −11534.0 −0.659159
\(675\) 0 0
\(676\) 11991.0 0.682237
\(677\) − 18114.0i − 1.02833i −0.857692 0.514164i \(-0.828102\pi\)
0.857692 0.514164i \(-0.171898\pi\)
\(678\) 0 0
\(679\) 20784.0 1.17469
\(680\) 0 0
\(681\) 0 0
\(682\) 14976.0i 0.840851i
\(683\) 23868.0i 1.33716i 0.743638 + 0.668582i \(0.233099\pi\)
−0.743638 + 0.668582i \(0.766901\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2640.00 −0.146932
\(687\) 0 0
\(688\) − 7708.00i − 0.427129i
\(689\) −7436.00 −0.411160
\(690\) 0 0
\(691\) 172.000 0.00946916 0.00473458 0.999989i \(-0.498493\pi\)
0.00473458 + 0.999989i \(0.498493\pi\)
\(692\) 4326.00i 0.237644i
\(693\) 0 0
\(694\) −11956.0 −0.653953
\(695\) 0 0
\(696\) 0 0
\(697\) 1708.00i 0.0928194i
\(698\) − 4870.00i − 0.264086i
\(699\) 0 0
\(700\) 0 0
\(701\) 22138.0 1.19278 0.596391 0.802694i \(-0.296601\pi\)
0.596391 + 0.802694i \(0.296601\pi\)
\(702\) 0 0
\(703\) 680.000i 0.0364818i
\(704\) −8684.00 −0.464901
\(705\) 0 0
\(706\) 10722.0 0.571569
\(707\) 29232.0i 1.55500i
\(708\) 0 0
\(709\) −3070.00 −0.162618 −0.0813091 0.996689i \(-0.525910\pi\)
−0.0813091 + 0.996689i \(0.525910\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4950.00i 0.260546i
\(713\) − 48384.0i − 2.54137i
\(714\) 0 0
\(715\) 0 0
\(716\) 23380.0 1.22032
\(717\) 0 0
\(718\) 120.000i 0.00623727i
\(719\) 15600.0 0.809154 0.404577 0.914504i \(-0.367419\pi\)
0.404577 + 0.914504i \(0.367419\pi\)
\(720\) 0 0
\(721\) 2112.00 0.109092
\(722\) − 6459.00i − 0.332935i
\(723\) 0 0
\(724\) −1246.00 −0.0639603
\(725\) 0 0
\(726\) 0 0
\(727\) − 20696.0i − 1.05581i −0.849304 0.527904i \(-0.822978\pi\)
0.849304 0.527904i \(-0.177022\pi\)
\(728\) − 7920.00i − 0.403207i
\(729\) 0 0
\(730\) 0 0
\(731\) −2632.00 −0.133171
\(732\) 0 0
\(733\) − 30778.0i − 1.55090i −0.631408 0.775451i \(-0.717522\pi\)
0.631408 0.775451i \(-0.282478\pi\)
\(734\) 3936.00 0.197930
\(735\) 0 0
\(736\) −27048.0 −1.35462
\(737\) − 4368.00i − 0.218314i
\(738\) 0 0
\(739\) −11740.0 −0.584388 −0.292194 0.956359i \(-0.594385\pi\)
−0.292194 + 0.956359i \(0.594385\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 8112.00i − 0.401349i
\(743\) − 2632.00i − 0.129958i −0.997887 0.0649789i \(-0.979302\pi\)
0.997887 0.0649789i \(-0.0206980\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3022.00 −0.148315
\(747\) 0 0
\(748\) 5096.00i 0.249102i
\(749\) −864.000 −0.0421494
\(750\) 0 0
\(751\) −20528.0 −0.997440 −0.498720 0.866763i \(-0.666196\pi\)
−0.498720 + 0.866763i \(0.666196\pi\)
\(752\) 10496.0i 0.508976i
\(753\) 0 0
\(754\) −5060.00 −0.244396
\(755\) 0 0
\(756\) 0 0
\(757\) − 21646.0i − 1.03928i −0.854384 0.519642i \(-0.826066\pi\)
0.854384 0.519642i \(-0.173934\pi\)
\(758\) 13340.0i 0.639222i
\(759\) 0 0
\(760\) 0 0
\(761\) −18282.0 −0.870857 −0.435428 0.900223i \(-0.643403\pi\)
−0.435428 + 0.900223i \(0.643403\pi\)
\(762\) 0 0
\(763\) 23280.0i 1.10458i
\(764\) 13216.0 0.625835
\(765\) 0 0
\(766\) −1008.00 −0.0475464
\(767\) 2200.00i 0.103569i
\(768\) 0 0
\(769\) 24190.0 1.13435 0.567174 0.823598i \(-0.308037\pi\)
0.567174 + 0.823598i \(0.308037\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 13454.0i 0.627228i
\(773\) 25698.0i 1.19572i 0.801600 + 0.597861i \(0.203982\pi\)
−0.801600 + 0.597861i \(0.796018\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12990.0 0.600920
\(777\) 0 0
\(778\) 9630.00i 0.443769i
\(779\) −2440.00 −0.112223
\(780\) 0 0
\(781\) −17056.0 −0.781449
\(782\) 2352.00i 0.107554i
\(783\) 0 0
\(784\) −9553.00 −0.435177
\(785\) 0 0
\(786\) 0 0
\(787\) − 33436.0i − 1.51444i −0.653160 0.757220i \(-0.726557\pi\)
0.653160 0.757220i \(-0.273443\pi\)
\(788\) 17682.0i 0.799359i
\(789\) 0 0
\(790\) 0 0
\(791\) 25008.0 1.12412
\(792\) 0 0
\(793\) 16324.0i 0.730999i
\(794\) 7126.00 0.318504
\(795\) 0 0
\(796\) 8120.00 0.361565
\(797\) − 37594.0i − 1.67083i −0.549623 0.835413i \(-0.685229\pi\)
0.549623 0.835413i \(-0.314771\pi\)
\(798\) 0 0
\(799\) 3584.00 0.158689
\(800\) 0 0
\(801\) 0 0
\(802\) 8718.00i 0.383844i
\(803\) 1976.00i 0.0868388i
\(804\) 0 0
\(805\) 0 0
\(806\) 6336.00 0.276893
\(807\) 0 0
\(808\) 18270.0i 0.795466i
\(809\) 4730.00 0.205560 0.102780 0.994704i \(-0.467226\pi\)
0.102780 + 0.994704i \(0.467226\pi\)
\(810\) 0 0
\(811\) −8748.00 −0.378772 −0.189386 0.981903i \(-0.560650\pi\)
−0.189386 + 0.981903i \(0.560650\pi\)
\(812\) 38640.0i 1.66995i
\(813\) 0 0
\(814\) 1768.00 0.0761282
\(815\) 0 0
\(816\) 0 0
\(817\) − 3760.00i − 0.161011i
\(818\) 10870.0i 0.464622i
\(819\) 0 0
\(820\) 0 0
\(821\) −44142.0 −1.87645 −0.938226 0.346024i \(-0.887532\pi\)
−0.938226 + 0.346024i \(0.887532\pi\)
\(822\) 0 0
\(823\) 3992.00i 0.169079i 0.996420 + 0.0845397i \(0.0269420\pi\)
−0.996420 + 0.0845397i \(0.973058\pi\)
\(824\) 1320.00 0.0558063
\(825\) 0 0
\(826\) −2400.00 −0.101098
\(827\) − 14444.0i − 0.607336i −0.952778 0.303668i \(-0.901789\pi\)
0.952778 0.303668i \(-0.0982114\pi\)
\(828\) 0 0
\(829\) −42150.0 −1.76590 −0.882949 0.469468i \(-0.844446\pi\)
−0.882949 + 0.469468i \(0.844446\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3674.00i 0.153093i
\(833\) 3262.00i 0.135680i
\(834\) 0 0
\(835\) 0 0
\(836\) −7280.00 −0.301177
\(837\) 0 0
\(838\) − 9700.00i − 0.399858i
\(839\) 13400.0 0.551394 0.275697 0.961245i \(-0.411091\pi\)
0.275697 + 0.961245i \(0.411091\pi\)
\(840\) 0 0
\(841\) 28511.0 1.16901
\(842\) 862.000i 0.0352809i
\(843\) 0 0
\(844\) −31276.0 −1.27555
\(845\) 0 0
\(846\) 0 0
\(847\) 32952.0i 1.33677i
\(848\) 13858.0i 0.561186i
\(849\) 0 0
\(850\) 0 0
\(851\) −5712.00 −0.230088
\(852\) 0 0
\(853\) − 8658.00i − 0.347531i −0.984787 0.173766i \(-0.944406\pi\)
0.984787 0.173766i \(-0.0555935\pi\)
\(854\) −17808.0 −0.713556
\(855\) 0 0
\(856\) −540.000 −0.0215617
\(857\) 42826.0i 1.70701i 0.521084 + 0.853505i \(0.325528\pi\)
−0.521084 + 0.853505i \(0.674472\pi\)
\(858\) 0 0
\(859\) 35900.0 1.42595 0.712976 0.701189i \(-0.247347\pi\)
0.712976 + 0.701189i \(0.247347\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 15792.0i − 0.623988i
\(863\) 3088.00i 0.121804i 0.998144 + 0.0609019i \(0.0193977\pi\)
−0.998144 + 0.0609019i \(0.980602\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −11602.0 −0.455256
\(867\) 0 0
\(868\) − 48384.0i − 1.89200i
\(869\) −12480.0 −0.487175
\(870\) 0 0
\(871\) −1848.00 −0.0718910
\(872\) 14550.0i 0.565052i
\(873\) 0 0
\(874\) −3360.00 −0.130039
\(875\) 0 0
\(876\) 0 0
\(877\) 35274.0i 1.35817i 0.734058 + 0.679087i \(0.237624\pi\)
−0.734058 + 0.679087i \(0.762376\pi\)
\(878\) 440.000i 0.0169126i
\(879\) 0 0
\(880\) 0 0
\(881\) −25042.0 −0.957646 −0.478823 0.877911i \(-0.658936\pi\)
−0.478823 + 0.877911i \(0.658936\pi\)
\(882\) 0 0
\(883\) 12572.0i 0.479141i 0.970879 + 0.239570i \(0.0770066\pi\)
−0.970879 + 0.239570i \(0.922993\pi\)
\(884\) 2156.00 0.0820296
\(885\) 0 0
\(886\) −10188.0 −0.386312
\(887\) − 21864.0i − 0.827645i −0.910358 0.413823i \(-0.864193\pi\)
0.910358 0.413823i \(-0.135807\pi\)
\(888\) 0 0
\(889\) 46464.0 1.75293
\(890\) 0 0
\(891\) 0 0
\(892\) 42224.0i 1.58494i
\(893\) 5120.00i 0.191864i
\(894\) 0 0
\(895\) 0 0
\(896\) −34920.0 −1.30200
\(897\) 0 0
\(898\) − 13310.0i − 0.494611i
\(899\) −66240.0 −2.45743
\(900\) 0 0
\(901\) 4732.00 0.174968
\(902\) 6344.00i 0.234182i
\(903\) 0 0
\(904\) 15630.0 0.575051
\(905\) 0 0
\(906\) 0 0
\(907\) − 31236.0i − 1.14352i −0.820420 0.571761i \(-0.806260\pi\)
0.820420 0.571761i \(-0.193740\pi\)
\(908\) 18452.0i 0.674396i
\(909\) 0 0
\(910\) 0 0
\(911\) −8272.00 −0.300838 −0.150419 0.988622i \(-0.548062\pi\)
−0.150419 + 0.988622i \(0.548062\pi\)
\(912\) 0 0
\(913\) 63024.0i 2.28455i
\(914\) 3226.00 0.116747
\(915\) 0 0
\(916\) −33810.0 −1.21956
\(917\) − 17568.0i − 0.632657i
\(918\) 0 0
\(919\) −20200.0 −0.725067 −0.362533 0.931971i \(-0.618088\pi\)
−0.362533 + 0.931971i \(0.618088\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 6582.00i − 0.235105i
\(923\) 7216.00i 0.257332i
\(924\) 0 0
\(925\) 0 0
\(926\) −15072.0 −0.534878
\(927\) 0 0
\(928\) 37030.0i 1.30988i
\(929\) 31010.0 1.09516 0.547581 0.836753i \(-0.315549\pi\)
0.547581 + 0.836753i \(0.315549\pi\)
\(930\) 0 0
\(931\) −4660.00 −0.164044
\(932\) − 18774.0i − 0.659831i
\(933\) 0 0
\(934\) −476.000 −0.0166758
\(935\) 0 0
\(936\) 0 0
\(937\) 39174.0i 1.36580i 0.730510 + 0.682902i \(0.239283\pi\)
−0.730510 + 0.682902i \(0.760717\pi\)
\(938\) − 2016.00i − 0.0701756i
\(939\) 0 0
\(940\) 0 0
\(941\) 4138.00 0.143353 0.0716764 0.997428i \(-0.477165\pi\)
0.0716764 + 0.997428i \(0.477165\pi\)
\(942\) 0 0
\(943\) − 20496.0i − 0.707785i
\(944\) 4100.00 0.141360
\(945\) 0 0
\(946\) −9776.00 −0.335989
\(947\) 23676.0i 0.812425i 0.913779 + 0.406213i \(0.133151\pi\)
−0.913779 + 0.406213i \(0.866849\pi\)
\(948\) 0 0
\(949\) 836.000 0.0285961
\(950\) 0 0
\(951\) 0 0
\(952\) 5040.00i 0.171583i
\(953\) − 18922.0i − 0.643173i −0.946880 0.321586i \(-0.895784\pi\)
0.946880 0.321586i \(-0.104216\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 16240.0 0.549413
\(957\) 0 0
\(958\) − 19680.0i − 0.663708i
\(959\) 53136.0 1.78921
\(960\) 0 0
\(961\) 53153.0 1.78420
\(962\) − 748.000i − 0.0250691i
\(963\) 0 0
\(964\) 14014.0 0.468216
\(965\) 0 0
\(966\) 0 0
\(967\) − 39656.0i − 1.31877i −0.751805 0.659385i \(-0.770817\pi\)
0.751805 0.659385i \(-0.229183\pi\)
\(968\) 20595.0i 0.683831i
\(969\) 0 0
\(970\) 0 0
\(971\) 33228.0 1.09818 0.549092 0.835762i \(-0.314974\pi\)
0.549092 + 0.835762i \(0.314974\pi\)
\(972\) 0 0
\(973\) − 480.000i − 0.0158151i
\(974\) −5944.00 −0.195542
\(975\) 0 0
\(976\) 30422.0 0.997730
\(977\) − 974.000i − 0.0318946i −0.999873 0.0159473i \(-0.994924\pi\)
0.999873 0.0159473i \(-0.00507640\pi\)
\(978\) 0 0
\(979\) −17160.0 −0.560200
\(980\) 0 0
\(981\) 0 0
\(982\) − 10772.0i − 0.350049i
\(983\) 13608.0i 0.441534i 0.975327 + 0.220767i \(0.0708560\pi\)
−0.975327 + 0.220767i \(0.929144\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 3220.00 0.104002
\(987\) 0 0
\(988\) 3080.00i 0.0991780i
\(989\) 31584.0 1.01548
\(990\) 0 0
\(991\) 13472.0 0.431839 0.215919 0.976411i \(-0.430725\pi\)
0.215919 + 0.976411i \(0.430725\pi\)
\(992\) − 46368.0i − 1.48406i
\(993\) 0 0
\(994\) −7872.00 −0.251192
\(995\) 0 0
\(996\) 0 0
\(997\) 3234.00i 0.102730i 0.998680 + 0.0513650i \(0.0163572\pi\)
−0.998680 + 0.0513650i \(0.983643\pi\)
\(998\) − 8140.00i − 0.258184i
\(999\) 0 0
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 225.4.b.e.199.2 2
3.2 odd 2 75.4.b.b.49.1 2
5.2 odd 4 45.4.a.c.1.1 1
5.3 odd 4 225.4.a.f.1.1 1
5.4 even 2 inner 225.4.b.e.199.1 2
12.11 even 2 1200.4.f.b.49.1 2
15.2 even 4 15.4.a.a.1.1 1
15.8 even 4 75.4.a.b.1.1 1
15.14 odd 2 75.4.b.b.49.2 2
20.7 even 4 720.4.a.n.1.1 1
35.27 even 4 2205.4.a.l.1.1 1
45.2 even 12 405.4.e.g.271.1 2
45.7 odd 12 405.4.e.i.271.1 2
45.22 odd 12 405.4.e.i.136.1 2
45.32 even 12 405.4.e.g.136.1 2
60.23 odd 4 1200.4.a.t.1.1 1
60.47 odd 4 240.4.a.e.1.1 1
60.59 even 2 1200.4.f.b.49.2 2
105.62 odd 4 735.4.a.e.1.1 1
120.77 even 4 960.4.a.b.1.1 1
120.107 odd 4 960.4.a.ba.1.1 1
165.32 odd 4 1815.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 15.2 even 4
45.4.a.c.1.1 1 5.2 odd 4
75.4.a.b.1.1 1 15.8 even 4
75.4.b.b.49.1 2 3.2 odd 2
75.4.b.b.49.2 2 15.14 odd 2
225.4.a.f.1.1 1 5.3 odd 4
225.4.b.e.199.1 2 5.4 even 2 inner
225.4.b.e.199.2 2 1.1 even 1 trivial
240.4.a.e.1.1 1 60.47 odd 4
405.4.e.g.136.1 2 45.32 even 12
405.4.e.g.271.1 2 45.2 even 12
405.4.e.i.136.1 2 45.22 odd 12
405.4.e.i.271.1 2 45.7 odd 12
720.4.a.n.1.1 1 20.7 even 4
735.4.a.e.1.1 1 105.62 odd 4
960.4.a.b.1.1 1 120.77 even 4
960.4.a.ba.1.1 1 120.107 odd 4
1200.4.a.t.1.1 1 60.23 odd 4
1200.4.f.b.49.1 2 12.11 even 2
1200.4.f.b.49.2 2 60.59 even 2
1815.4.a.e.1.1 1 165.32 odd 4
2205.4.a.l.1.1 1 35.27 even 4