Properties

Label 2925.2.c.w.2224.2
Level $2925$
Weight $2$
Character 2925.2224
Analytic conductor $23.356$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.3562425912\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2224.2
Root \(-0.671462 + 1.24464i\) of defining polynomial
Character \(\chi\) \(=\) 2925.2224
Dual form 2925.2.c.w.2224.5

$q$-expansion

\(f(q)\) \(=\) \(q-2.48929i q^{2} -4.19656 q^{4} +1.19656i q^{7} +5.46787i q^{8} +O(q^{10})\) \(q-2.48929i q^{2} -4.19656 q^{4} +1.19656i q^{7} +5.46787i q^{8} +1.19656 q^{11} +1.00000i q^{13} +2.97858 q^{14} +5.21798 q^{16} +6.17513i q^{17} -6.97858 q^{19} -2.97858i q^{22} -4.17513i q^{23} +2.48929 q^{26} -5.02142i q^{28} +6.00000 q^{29} -2.97858 q^{31} -2.05333i q^{32} +15.3717 q^{34} -7.78202i q^{37} +17.3717i q^{38} +6.17513 q^{41} -9.95715i q^{43} -5.02142 q^{44} -10.3931 q^{46} -1.02142i q^{47} +5.56825 q^{49} -4.19656i q^{52} -10.1751i q^{53} -6.54262 q^{56} -14.9357i q^{58} +5.37169 q^{59} +12.5682 q^{61} +7.41454i q^{62} +5.32464 q^{64} -9.37169i q^{67} -25.9143i q^{68} +5.19656 q^{71} -11.9572i q^{73} -19.3717 q^{74} +29.2860 q^{76} +1.43175i q^{77} +1.78202 q^{79} -15.3717i q^{82} +5.37169i q^{83} -24.7862 q^{86} +6.54262i q^{88} +10.1751 q^{89} -1.19656 q^{91} +17.5212i q^{92} -2.54262 q^{94} +1.82487i q^{97} -13.8610i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} - 2 q^{11} - 12 q^{14} + 52 q^{16} - 12 q^{19} + 36 q^{29} + 12 q^{31} + 44 q^{34} - 2 q^{41} - 60 q^{44} - 44 q^{46} - 24 q^{49} + 32 q^{56} - 16 q^{59} + 18 q^{61} - 60 q^{64} + 22 q^{71} - 68 q^{74} + 8 q^{76} - 10 q^{79} - 112 q^{86} + 22 q^{89} + 2 q^{91} + 56 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(2251\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.48929i − 1.76019i −0.474795 0.880096i \(-0.657478\pi\)
0.474795 0.880096i \(-0.342522\pi\)
\(3\) 0 0
\(4\) −4.19656 −2.09828
\(5\) 0 0
\(6\) 0 0
\(7\) 1.19656i 0.452256i 0.974098 + 0.226128i \(0.0726068\pi\)
−0.974098 + 0.226128i \(0.927393\pi\)
\(8\) 5.46787i 1.93318i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.19656 0.360776 0.180388 0.983596i \(-0.442265\pi\)
0.180388 + 0.983596i \(0.442265\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 2.97858 0.796058
\(15\) 0 0
\(16\) 5.21798 1.30450
\(17\) 6.17513i 1.49769i 0.662745 + 0.748845i \(0.269391\pi\)
−0.662745 + 0.748845i \(0.730609\pi\)
\(18\) 0 0
\(19\) −6.97858 −1.60100 −0.800498 0.599336i \(-0.795431\pi\)
−0.800498 + 0.599336i \(0.795431\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 2.97858i − 0.635035i
\(23\) − 4.17513i − 0.870576i −0.900291 0.435288i \(-0.856647\pi\)
0.900291 0.435288i \(-0.143353\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.48929 0.488190
\(27\) 0 0
\(28\) − 5.02142i − 0.948960i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −2.97858 −0.534968 −0.267484 0.963562i \(-0.586192\pi\)
−0.267484 + 0.963562i \(0.586192\pi\)
\(32\) − 2.05333i − 0.362980i
\(33\) 0 0
\(34\) 15.3717 2.63622
\(35\) 0 0
\(36\) 0 0
\(37\) − 7.78202i − 1.27936i −0.768643 0.639678i \(-0.779068\pi\)
0.768643 0.639678i \(-0.220932\pi\)
\(38\) 17.3717i 2.81806i
\(39\) 0 0
\(40\) 0 0
\(41\) 6.17513 0.964394 0.482197 0.876063i \(-0.339839\pi\)
0.482197 + 0.876063i \(0.339839\pi\)
\(42\) 0 0
\(43\) − 9.95715i − 1.51845i −0.650827 0.759226i \(-0.725578\pi\)
0.650827 0.759226i \(-0.274422\pi\)
\(44\) −5.02142 −0.757008
\(45\) 0 0
\(46\) −10.3931 −1.53238
\(47\) − 1.02142i − 0.148990i −0.997221 0.0744949i \(-0.976266\pi\)
0.997221 0.0744949i \(-0.0237345\pi\)
\(48\) 0 0
\(49\) 5.56825 0.795464
\(50\) 0 0
\(51\) 0 0
\(52\) − 4.19656i − 0.581958i
\(53\) − 10.1751i − 1.39766i −0.715287 0.698831i \(-0.753704\pi\)
0.715287 0.698831i \(-0.246296\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.54262 −0.874294
\(57\) 0 0
\(58\) − 14.9357i − 1.96116i
\(59\) 5.37169 0.699335 0.349667 0.936874i \(-0.386295\pi\)
0.349667 + 0.936874i \(0.386295\pi\)
\(60\) 0 0
\(61\) 12.5682 1.60920 0.804600 0.593818i \(-0.202380\pi\)
0.804600 + 0.593818i \(0.202380\pi\)
\(62\) 7.41454i 0.941647i
\(63\) 0 0
\(64\) 5.32464 0.665579
\(65\) 0 0
\(66\) 0 0
\(67\) − 9.37169i − 1.14493i −0.819928 0.572467i \(-0.805986\pi\)
0.819928 0.572467i \(-0.194014\pi\)
\(68\) − 25.9143i − 3.14257i
\(69\) 0 0
\(70\) 0 0
\(71\) 5.19656 0.616718 0.308359 0.951270i \(-0.400220\pi\)
0.308359 + 0.951270i \(0.400220\pi\)
\(72\) 0 0
\(73\) − 11.9572i − 1.39948i −0.714398 0.699740i \(-0.753299\pi\)
0.714398 0.699740i \(-0.246701\pi\)
\(74\) −19.3717 −2.25191
\(75\) 0 0
\(76\) 29.2860 3.35933
\(77\) 1.43175i 0.163163i
\(78\) 0 0
\(79\) 1.78202 0.200493 0.100246 0.994963i \(-0.468037\pi\)
0.100246 + 0.994963i \(0.468037\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 15.3717i − 1.69752i
\(83\) 5.37169i 0.589620i 0.955556 + 0.294810i \(0.0952563\pi\)
−0.955556 + 0.294810i \(0.904744\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −24.7862 −2.67277
\(87\) 0 0
\(88\) 6.54262i 0.697445i
\(89\) 10.1751 1.07856 0.539281 0.842126i \(-0.318696\pi\)
0.539281 + 0.842126i \(0.318696\pi\)
\(90\) 0 0
\(91\) −1.19656 −0.125433
\(92\) 17.5212i 1.82671i
\(93\) 0 0
\(94\) −2.54262 −0.262251
\(95\) 0 0
\(96\) 0 0
\(97\) 1.82487i 0.185287i 0.995699 + 0.0926435i \(0.0295317\pi\)
−0.995699 + 0.0926435i \(0.970468\pi\)
\(98\) − 13.8610i − 1.40017i
\(99\) 0 0
\(100\) 0 0
\(101\) 10.3503 1.02989 0.514945 0.857223i \(-0.327812\pi\)
0.514945 + 0.857223i \(0.327812\pi\)
\(102\) 0 0
\(103\) 18.7434i 1.84684i 0.383790 + 0.923420i \(0.374619\pi\)
−0.383790 + 0.923420i \(0.625381\pi\)
\(104\) −5.46787 −0.536168
\(105\) 0 0
\(106\) −25.3288 −2.46016
\(107\) − 18.5682i − 1.79506i −0.440953 0.897530i \(-0.645359\pi\)
0.440953 0.897530i \(-0.354641\pi\)
\(108\) 0 0
\(109\) −8.39312 −0.803915 −0.401957 0.915658i \(-0.631670\pi\)
−0.401957 + 0.915658i \(0.631670\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.24361i 0.589966i
\(113\) − 7.95715i − 0.748546i −0.927319 0.374273i \(-0.877892\pi\)
0.927319 0.374273i \(-0.122108\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −25.1793 −2.33784
\(117\) 0 0
\(118\) − 13.3717i − 1.23096i
\(119\) −7.38890 −0.677340
\(120\) 0 0
\(121\) −9.56825 −0.869841
\(122\) − 31.2860i − 2.83250i
\(123\) 0 0
\(124\) 12.4998 1.12251
\(125\) 0 0
\(126\) 0 0
\(127\) 10.3931i 0.922240i 0.887338 + 0.461120i \(0.152552\pi\)
−0.887338 + 0.461120i \(0.847448\pi\)
\(128\) − 17.3612i − 1.53453i
\(129\) 0 0
\(130\) 0 0
\(131\) 6.39312 0.558569 0.279285 0.960208i \(-0.409903\pi\)
0.279285 + 0.960208i \(0.409903\pi\)
\(132\) 0 0
\(133\) − 8.35027i − 0.724060i
\(134\) −23.3288 −2.01531
\(135\) 0 0
\(136\) −33.7648 −2.89531
\(137\) 16.7434i 1.43048i 0.698877 + 0.715242i \(0.253684\pi\)
−0.698877 + 0.715242i \(0.746316\pi\)
\(138\) 0 0
\(139\) −5.78202 −0.490424 −0.245212 0.969469i \(-0.578858\pi\)
−0.245212 + 0.969469i \(0.578858\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 12.9357i − 1.08554i
\(143\) 1.19656i 0.100061i
\(144\) 0 0
\(145\) 0 0
\(146\) −29.7648 −2.46335
\(147\) 0 0
\(148\) 32.6577i 2.68445i
\(149\) 15.3461 1.25720 0.628599 0.777730i \(-0.283629\pi\)
0.628599 + 0.777730i \(0.283629\pi\)
\(150\) 0 0
\(151\) −8.58546 −0.698675 −0.349337 0.936997i \(-0.613593\pi\)
−0.349337 + 0.936997i \(0.613593\pi\)
\(152\) − 38.1579i − 3.09502i
\(153\) 0 0
\(154\) 3.56404 0.287198
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.78623i − 0.222365i −0.993800 0.111183i \(-0.964536\pi\)
0.993800 0.111183i \(-0.0354639\pi\)
\(158\) − 4.43596i − 0.352906i
\(159\) 0 0
\(160\) 0 0
\(161\) 4.99579 0.393723
\(162\) 0 0
\(163\) 8.76060i 0.686183i 0.939302 + 0.343091i \(0.111474\pi\)
−0.939302 + 0.343091i \(0.888526\pi\)
\(164\) −25.9143 −2.02357
\(165\) 0 0
\(166\) 13.3717 1.03784
\(167\) − 17.3717i − 1.34426i −0.740432 0.672131i \(-0.765379\pi\)
0.740432 0.672131i \(-0.234621\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 41.7858i 3.18614i
\(173\) 7.95715i 0.604971i 0.953154 + 0.302486i \(0.0978164\pi\)
−0.953154 + 0.302486i \(0.902184\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.24361 0.470630
\(177\) 0 0
\(178\) − 25.3288i − 1.89848i
\(179\) −15.5640 −1.16331 −0.581655 0.813435i \(-0.697595\pi\)
−0.581655 + 0.813435i \(0.697595\pi\)
\(180\) 0 0
\(181\) 15.7820 1.17307 0.586534 0.809925i \(-0.300492\pi\)
0.586534 + 0.809925i \(0.300492\pi\)
\(182\) 2.97858i 0.220787i
\(183\) 0 0
\(184\) 22.8291 1.68298
\(185\) 0 0
\(186\) 0 0
\(187\) 7.38890i 0.540330i
\(188\) 4.28646i 0.312622i
\(189\) 0 0
\(190\) 0 0
\(191\) −10.7434 −0.777364 −0.388682 0.921372i \(-0.627070\pi\)
−0.388682 + 0.921372i \(0.627070\pi\)
\(192\) 0 0
\(193\) 9.73917i 0.701041i 0.936555 + 0.350521i \(0.113995\pi\)
−0.936555 + 0.350521i \(0.886005\pi\)
\(194\) 4.54262 0.326141
\(195\) 0 0
\(196\) −23.3675 −1.66911
\(197\) − 9.56404i − 0.681410i −0.940170 0.340705i \(-0.889334\pi\)
0.940170 0.340705i \(-0.110666\pi\)
\(198\) 0 0
\(199\) −5.95715 −0.422291 −0.211146 0.977455i \(-0.567719\pi\)
−0.211146 + 0.977455i \(0.567719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 25.7648i − 1.81281i
\(203\) 7.17935i 0.503891i
\(204\) 0 0
\(205\) 0 0
\(206\) 46.6577 3.25080
\(207\) 0 0
\(208\) 5.21798i 0.361802i
\(209\) −8.35027 −0.577600
\(210\) 0 0
\(211\) 23.9143 1.64633 0.823164 0.567803i \(-0.192206\pi\)
0.823164 + 0.567803i \(0.192206\pi\)
\(212\) 42.7005i 2.93269i
\(213\) 0 0
\(214\) −46.2217 −3.15965
\(215\) 0 0
\(216\) 0 0
\(217\) − 3.56404i − 0.241943i
\(218\) 20.8929i 1.41504i
\(219\) 0 0
\(220\) 0 0
\(221\) −6.17513 −0.415385
\(222\) 0 0
\(223\) 2.62831i 0.176004i 0.996120 + 0.0880022i \(0.0280483\pi\)
−0.996120 + 0.0880022i \(0.971952\pi\)
\(224\) 2.45692 0.164160
\(225\) 0 0
\(226\) −19.8077 −1.31759
\(227\) 15.7648i 1.04635i 0.852226 + 0.523174i \(0.175252\pi\)
−0.852226 + 0.523174i \(0.824748\pi\)
\(228\) 0 0
\(229\) −8.74338 −0.577779 −0.288890 0.957362i \(-0.593286\pi\)
−0.288890 + 0.957362i \(0.593286\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 32.8072i 2.15390i
\(233\) 2.17513i 0.142498i 0.997459 + 0.0712489i \(0.0226985\pi\)
−0.997459 + 0.0712489i \(0.977302\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −22.5426 −1.46740
\(237\) 0 0
\(238\) 18.3931i 1.19225i
\(239\) 2.80344 0.181340 0.0906698 0.995881i \(-0.471099\pi\)
0.0906698 + 0.995881i \(0.471099\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 23.8181i 1.53109i
\(243\) 0 0
\(244\) −52.7434 −3.37655
\(245\) 0 0
\(246\) 0 0
\(247\) − 6.97858i − 0.444036i
\(248\) − 16.2865i − 1.03419i
\(249\) 0 0
\(250\) 0 0
\(251\) 23.9143 1.50946 0.754729 0.656037i \(-0.227768\pi\)
0.754729 + 0.656037i \(0.227768\pi\)
\(252\) 0 0
\(253\) − 4.99579i − 0.314083i
\(254\) 25.8715 1.62332
\(255\) 0 0
\(256\) −32.5678 −2.03549
\(257\) − 19.9572i − 1.24489i −0.782662 0.622447i \(-0.786139\pi\)
0.782662 0.622447i \(-0.213861\pi\)
\(258\) 0 0
\(259\) 9.31163 0.578597
\(260\) 0 0
\(261\) 0 0
\(262\) − 15.9143i − 0.983189i
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −20.7862 −1.27449
\(267\) 0 0
\(268\) 39.3288i 2.40239i
\(269\) −2.35027 −0.143298 −0.0716492 0.997430i \(-0.522826\pi\)
−0.0716492 + 0.997430i \(0.522826\pi\)
\(270\) 0 0
\(271\) 10.9786 0.666901 0.333451 0.942768i \(-0.391787\pi\)
0.333451 + 0.942768i \(0.391787\pi\)
\(272\) 32.2217i 1.95373i
\(273\) 0 0
\(274\) 41.6791 2.51793
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.21377i − 0.0729283i −0.999335 0.0364642i \(-0.988391\pi\)
0.999335 0.0364642i \(-0.0116095\pi\)
\(278\) 14.3931i 0.863242i
\(279\) 0 0
\(280\) 0 0
\(281\) −11.9572 −0.713304 −0.356652 0.934237i \(-0.616082\pi\)
−0.356652 + 0.934237i \(0.616082\pi\)
\(282\) 0 0
\(283\) − 29.8715i − 1.77567i −0.460158 0.887837i \(-0.652207\pi\)
0.460158 0.887837i \(-0.347793\pi\)
\(284\) −21.8077 −1.29405
\(285\) 0 0
\(286\) 2.97858 0.176127
\(287\) 7.38890i 0.436153i
\(288\) 0 0
\(289\) −21.1323 −1.24308
\(290\) 0 0
\(291\) 0 0
\(292\) 50.1789i 2.93650i
\(293\) − 0.777809i − 0.0454401i −0.999742 0.0227200i \(-0.992767\pi\)
0.999742 0.0227200i \(-0.00723263\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 42.5510 2.47323
\(297\) 0 0
\(298\) − 38.2008i − 2.21291i
\(299\) 4.17513 0.241454
\(300\) 0 0
\(301\) 11.9143 0.686729
\(302\) 21.3717i 1.22980i
\(303\) 0 0
\(304\) −36.4141 −2.08849
\(305\) 0 0
\(306\) 0 0
\(307\) − 0.760597i − 0.0434095i −0.999764 0.0217048i \(-0.993091\pi\)
0.999764 0.0217048i \(-0.00690939\pi\)
\(308\) − 6.00842i − 0.342362i
\(309\) 0 0
\(310\) 0 0
\(311\) 23.1281 1.31147 0.655736 0.754990i \(-0.272358\pi\)
0.655736 + 0.754990i \(0.272358\pi\)
\(312\) 0 0
\(313\) − 33.9143i − 1.91695i −0.285176 0.958475i \(-0.592052\pi\)
0.285176 0.958475i \(-0.407948\pi\)
\(314\) −6.93573 −0.391406
\(315\) 0 0
\(316\) −7.47835 −0.420690
\(317\) 9.64973i 0.541983i 0.962582 + 0.270991i \(0.0873515\pi\)
−0.962582 + 0.270991i \(0.912648\pi\)
\(318\) 0 0
\(319\) 7.17935 0.401966
\(320\) 0 0
\(321\) 0 0
\(322\) − 12.4360i − 0.693029i
\(323\) − 43.0937i − 2.39780i
\(324\) 0 0
\(325\) 0 0
\(326\) 21.8077 1.20781
\(327\) 0 0
\(328\) 33.7648i 1.86435i
\(329\) 1.22219 0.0673816
\(330\) 0 0
\(331\) 15.3288 0.842550 0.421275 0.906933i \(-0.361583\pi\)
0.421275 + 0.906933i \(0.361583\pi\)
\(332\) − 22.5426i − 1.23719i
\(333\) 0 0
\(334\) −43.2432 −2.36616
\(335\) 0 0
\(336\) 0 0
\(337\) 22.3503i 1.21750i 0.793363 + 0.608748i \(0.208328\pi\)
−0.793363 + 0.608748i \(0.791672\pi\)
\(338\) 2.48929i 0.135399i
\(339\) 0 0
\(340\) 0 0
\(341\) −3.56404 −0.193004
\(342\) 0 0
\(343\) 15.0386i 0.812010i
\(344\) 54.4444 2.93544
\(345\) 0 0
\(346\) 19.8077 1.06487
\(347\) − 5.78202i − 0.310395i −0.987883 0.155198i \(-0.950399\pi\)
0.987883 0.155198i \(-0.0496014\pi\)
\(348\) 0 0
\(349\) 27.5212 1.47318 0.736588 0.676342i \(-0.236436\pi\)
0.736588 + 0.676342i \(0.236436\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 2.45692i − 0.130955i
\(353\) 28.7434i 1.52986i 0.644116 + 0.764928i \(0.277225\pi\)
−0.644116 + 0.764928i \(0.722775\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −42.7005 −2.26312
\(357\) 0 0
\(358\) 38.7434i 2.04765i
\(359\) −12.5855 −0.664235 −0.332118 0.943238i \(-0.607763\pi\)
−0.332118 + 0.943238i \(0.607763\pi\)
\(360\) 0 0
\(361\) 29.7005 1.56319
\(362\) − 39.2860i − 2.06483i
\(363\) 0 0
\(364\) 5.02142 0.263194
\(365\) 0 0
\(366\) 0 0
\(367\) 27.9143i 1.45712i 0.684985 + 0.728558i \(0.259809\pi\)
−0.684985 + 0.728558i \(0.740191\pi\)
\(368\) − 21.7858i − 1.13566i
\(369\) 0 0
\(370\) 0 0
\(371\) 12.1751 0.632102
\(372\) 0 0
\(373\) − 2.35027i − 0.121692i −0.998147 0.0608462i \(-0.980620\pi\)
0.998147 0.0608462i \(-0.0193799\pi\)
\(374\) 18.3931 0.951085
\(375\) 0 0
\(376\) 5.58500 0.288025
\(377\) 6.00000i 0.309016i
\(378\) 0 0
\(379\) −24.5510 −1.26110 −0.630551 0.776148i \(-0.717171\pi\)
−0.630551 + 0.776148i \(0.717171\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 26.7434i 1.36831i
\(383\) − 5.80765i − 0.296757i −0.988931 0.148379i \(-0.952595\pi\)
0.988931 0.148379i \(-0.0474054\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 24.2436 1.23397
\(387\) 0 0
\(388\) − 7.65815i − 0.388784i
\(389\) 13.6497 0.692069 0.346034 0.938222i \(-0.387528\pi\)
0.346034 + 0.938222i \(0.387528\pi\)
\(390\) 0 0
\(391\) 25.7820 1.30385
\(392\) 30.4464i 1.53778i
\(393\) 0 0
\(394\) −23.8077 −1.19941
\(395\) 0 0
\(396\) 0 0
\(397\) 12.1323i 0.608902i 0.952528 + 0.304451i \(0.0984730\pi\)
−0.952528 + 0.304451i \(0.901527\pi\)
\(398\) 14.8291i 0.743314i
\(399\) 0 0
\(400\) 0 0
\(401\) 37.4439 1.86986 0.934930 0.354832i \(-0.115462\pi\)
0.934930 + 0.354832i \(0.115462\pi\)
\(402\) 0 0
\(403\) − 2.97858i − 0.148373i
\(404\) −43.4355 −2.16100
\(405\) 0 0
\(406\) 17.8715 0.886946
\(407\) − 9.31163i − 0.461561i
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 78.6577i − 3.87519i
\(413\) 6.42754i 0.316279i
\(414\) 0 0
\(415\) 0 0
\(416\) 2.05333 0.100673
\(417\) 0 0
\(418\) 20.7862i 1.01669i
\(419\) 3.17935 0.155321 0.0776606 0.996980i \(-0.475255\pi\)
0.0776606 + 0.996980i \(0.475255\pi\)
\(420\) 0 0
\(421\) −16.3074 −0.794775 −0.397388 0.917651i \(-0.630083\pi\)
−0.397388 + 0.917651i \(0.630083\pi\)
\(422\) − 59.5296i − 2.89786i
\(423\) 0 0
\(424\) 55.6363 2.70194
\(425\) 0 0
\(426\) 0 0
\(427\) 15.0386i 0.727771i
\(428\) 77.9227i 3.76654i
\(429\) 0 0
\(430\) 0 0
\(431\) −4.58546 −0.220874 −0.110437 0.993883i \(-0.535225\pi\)
−0.110437 + 0.993883i \(0.535225\pi\)
\(432\) 0 0
\(433\) − 38.3503i − 1.84300i −0.388383 0.921498i \(-0.626966\pi\)
0.388383 0.921498i \(-0.373034\pi\)
\(434\) −8.87192 −0.425866
\(435\) 0 0
\(436\) 35.2222 1.68684
\(437\) 29.1365i 1.39379i
\(438\) 0 0
\(439\) −7.73917 −0.369371 −0.184685 0.982798i \(-0.559127\pi\)
−0.184685 + 0.982798i \(0.559127\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 15.3717i 0.731157i
\(443\) − 34.9185i − 1.65903i −0.558485 0.829514i \(-0.688617\pi\)
0.558485 0.829514i \(-0.311383\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 6.54262 0.309802
\(447\) 0 0
\(448\) 6.37123i 0.301012i
\(449\) −6.17513 −0.291423 −0.145711 0.989327i \(-0.546547\pi\)
−0.145711 + 0.989327i \(0.546547\pi\)
\(450\) 0 0
\(451\) 7.38890 0.347930
\(452\) 33.3927i 1.57066i
\(453\) 0 0
\(454\) 39.2432 1.84177
\(455\) 0 0
\(456\) 0 0
\(457\) − 1.38890i − 0.0649702i −0.999472 0.0324851i \(-0.989658\pi\)
0.999472 0.0324851i \(-0.0103421\pi\)
\(458\) 21.7648i 1.01700i
\(459\) 0 0
\(460\) 0 0
\(461\) −28.4826 −1.32657 −0.663283 0.748369i \(-0.730837\pi\)
−0.663283 + 0.748369i \(0.730837\pi\)
\(462\) 0 0
\(463\) 16.3759i 0.761053i 0.924770 + 0.380526i \(0.124257\pi\)
−0.924770 + 0.380526i \(0.875743\pi\)
\(464\) 31.3079 1.45343
\(465\) 0 0
\(466\) 5.41454 0.250824
\(467\) 25.7476i 1.19146i 0.803186 + 0.595728i \(0.203136\pi\)
−0.803186 + 0.595728i \(0.796864\pi\)
\(468\) 0 0
\(469\) 11.2138 0.517804
\(470\) 0 0
\(471\) 0 0
\(472\) 29.3717i 1.35194i
\(473\) − 11.9143i − 0.547820i
\(474\) 0 0
\(475\) 0 0
\(476\) 31.0080 1.42125
\(477\) 0 0
\(478\) − 6.97858i − 0.319193i
\(479\) −7.58967 −0.346781 −0.173391 0.984853i \(-0.555472\pi\)
−0.173391 + 0.984853i \(0.555472\pi\)
\(480\) 0 0
\(481\) 7.78202 0.354830
\(482\) 14.9357i 0.680304i
\(483\) 0 0
\(484\) 40.1537 1.82517
\(485\) 0 0
\(486\) 0 0
\(487\) − 4.41033i − 0.199851i −0.994995 0.0999255i \(-0.968140\pi\)
0.994995 0.0999255i \(-0.0318604\pi\)
\(488\) 68.7215i 3.11088i
\(489\) 0 0
\(490\) 0 0
\(491\) 0.0856914 0.00386720 0.00193360 0.999998i \(-0.499385\pi\)
0.00193360 + 0.999998i \(0.499385\pi\)
\(492\) 0 0
\(493\) 37.0508i 1.66868i
\(494\) −17.3717 −0.781589
\(495\) 0 0
\(496\) −15.5422 −0.697863
\(497\) 6.21798i 0.278915i
\(498\) 0 0
\(499\) 17.7220 0.793344 0.396672 0.917960i \(-0.370165\pi\)
0.396672 + 0.917960i \(0.370165\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 59.5296i − 2.65694i
\(503\) 8.70054i 0.387938i 0.981008 + 0.193969i \(0.0621361\pi\)
−0.981008 + 0.193969i \(0.937864\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −12.4360 −0.552846
\(507\) 0 0
\(508\) − 43.6153i − 1.93512i
\(509\) −33.3545 −1.47841 −0.739206 0.673480i \(-0.764799\pi\)
−0.739206 + 0.673480i \(0.764799\pi\)
\(510\) 0 0
\(511\) 14.3074 0.632923
\(512\) 46.3482i 2.04832i
\(513\) 0 0
\(514\) −49.6791 −2.19125
\(515\) 0 0
\(516\) 0 0
\(517\) − 1.22219i − 0.0537519i
\(518\) − 23.1793i − 1.01844i
\(519\) 0 0
\(520\) 0 0
\(521\) −18.7005 −0.819285 −0.409643 0.912246i \(-0.634347\pi\)
−0.409643 + 0.912246i \(0.634347\pi\)
\(522\) 0 0
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) −26.8291 −1.17203
\(525\) 0 0
\(526\) −19.9143 −0.868305
\(527\) − 18.3931i − 0.801217i
\(528\) 0 0
\(529\) 5.56825 0.242098
\(530\) 0 0
\(531\) 0 0
\(532\) 35.0424i 1.51928i
\(533\) 6.17513i 0.267475i
\(534\) 0 0
\(535\) 0 0
\(536\) 51.2432 2.21337
\(537\) 0 0
\(538\) 5.85050i 0.252233i
\(539\) 6.66273 0.286984
\(540\) 0 0
\(541\) −41.5296 −1.78550 −0.892749 0.450555i \(-0.851226\pi\)
−0.892749 + 0.450555i \(0.851226\pi\)
\(542\) − 27.3288i − 1.17387i
\(543\) 0 0
\(544\) 12.6796 0.543632
\(545\) 0 0
\(546\) 0 0
\(547\) 7.91431i 0.338391i 0.985582 + 0.169196i \(0.0541170\pi\)
−0.985582 + 0.169196i \(0.945883\pi\)
\(548\) − 70.2646i − 3.00155i
\(549\) 0 0
\(550\) 0 0
\(551\) −41.8715 −1.78378
\(552\) 0 0
\(553\) 2.13229i 0.0906742i
\(554\) −3.02142 −0.128368
\(555\) 0 0
\(556\) 24.2646 1.02905
\(557\) 42.7005i 1.80928i 0.426177 + 0.904640i \(0.359860\pi\)
−0.426177 + 0.904640i \(0.640140\pi\)
\(558\) 0 0
\(559\) 9.95715 0.421143
\(560\) 0 0
\(561\) 0 0
\(562\) 29.7648i 1.25555i
\(563\) − 1.04706i − 0.0441282i −0.999757 0.0220641i \(-0.992976\pi\)
0.999757 0.0220641i \(-0.00702379\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −74.3587 −3.12553
\(567\) 0 0
\(568\) 28.4141i 1.19223i
\(569\) 16.7778 0.703362 0.351681 0.936120i \(-0.385610\pi\)
0.351681 + 0.936120i \(0.385610\pi\)
\(570\) 0 0
\(571\) −20.6111 −0.862548 −0.431274 0.902221i \(-0.641936\pi\)
−0.431274 + 0.902221i \(0.641936\pi\)
\(572\) − 5.02142i − 0.209956i
\(573\) 0 0
\(574\) 18.3931 0.767714
\(575\) 0 0
\(576\) 0 0
\(577\) − 1.38890i − 0.0578208i −0.999582 0.0289104i \(-0.990796\pi\)
0.999582 0.0289104i \(-0.00920376\pi\)
\(578\) 52.6044i 2.18805i
\(579\) 0 0
\(580\) 0 0
\(581\) −6.42754 −0.266659
\(582\) 0 0
\(583\) − 12.1751i − 0.504243i
\(584\) 65.3801 2.70545
\(585\) 0 0
\(586\) −1.93619 −0.0799833
\(587\) − 0.935731i − 0.0386218i −0.999814 0.0193109i \(-0.993853\pi\)
0.999814 0.0193109i \(-0.00614723\pi\)
\(588\) 0 0
\(589\) 20.7862 0.856482
\(590\) 0 0
\(591\) 0 0
\(592\) − 40.6064i − 1.66891i
\(593\) 0.478807i 0.0196622i 0.999952 + 0.00983112i \(0.00312939\pi\)
−0.999952 + 0.00983112i \(0.996871\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −64.4006 −2.63795
\(597\) 0 0
\(598\) − 10.3931i − 0.425006i
\(599\) 29.0852 1.18839 0.594195 0.804321i \(-0.297471\pi\)
0.594195 + 0.804321i \(0.297471\pi\)
\(600\) 0 0
\(601\) 11.4318 0.466311 0.233155 0.972439i \(-0.425095\pi\)
0.233155 + 0.972439i \(0.425095\pi\)
\(602\) − 29.6582i − 1.20878i
\(603\) 0 0
\(604\) 36.0294 1.46601
\(605\) 0 0
\(606\) 0 0
\(607\) − 27.9143i − 1.13301i −0.824059 0.566503i \(-0.808296\pi\)
0.824059 0.566503i \(-0.191704\pi\)
\(608\) 14.3293i 0.581130i
\(609\) 0 0
\(610\) 0 0
\(611\) 1.02142 0.0413223
\(612\) 0 0
\(613\) − 4.65394i − 0.187971i −0.995574 0.0939855i \(-0.970039\pi\)
0.995574 0.0939855i \(-0.0299607\pi\)
\(614\) −1.89334 −0.0764092
\(615\) 0 0
\(616\) −7.82862 −0.315424
\(617\) − 15.9572i − 0.642411i −0.947010 0.321205i \(-0.895912\pi\)
0.947010 0.321205i \(-0.104088\pi\)
\(618\) 0 0
\(619\) −1.02142 −0.0410545 −0.0205272 0.999789i \(-0.506534\pi\)
−0.0205272 + 0.999789i \(0.506534\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 57.5725i − 2.30845i
\(623\) 12.1751i 0.487786i
\(624\) 0 0
\(625\) 0 0
\(626\) −84.4225 −3.37420
\(627\) 0 0
\(628\) 11.6926i 0.466585i
\(629\) 48.0550 1.91608
\(630\) 0 0
\(631\) −20.4998 −0.816083 −0.408041 0.912963i \(-0.633788\pi\)
−0.408041 + 0.912963i \(0.633788\pi\)
\(632\) 9.74384i 0.387589i
\(633\) 0 0
\(634\) 24.0210 0.953994
\(635\) 0 0
\(636\) 0 0
\(637\) 5.56825i 0.220622i
\(638\) − 17.8715i − 0.707538i
\(639\) 0 0
\(640\) 0 0
\(641\) −38.2646 −1.51136 −0.755680 0.654941i \(-0.772693\pi\)
−0.755680 + 0.654941i \(0.772693\pi\)
\(642\) 0 0
\(643\) − 33.1109i − 1.30577i −0.757459 0.652883i \(-0.773560\pi\)
0.757459 0.652883i \(-0.226440\pi\)
\(644\) −20.9651 −0.826141
\(645\) 0 0
\(646\) −107.273 −4.22058
\(647\) − 16.9614i − 0.666820i −0.942782 0.333410i \(-0.891801\pi\)
0.942782 0.333410i \(-0.108199\pi\)
\(648\) 0 0
\(649\) 6.42754 0.252303
\(650\) 0 0
\(651\) 0 0
\(652\) − 36.7643i − 1.43980i
\(653\) 19.1709i 0.750216i 0.926981 + 0.375108i \(0.122394\pi\)
−0.926981 + 0.375108i \(0.877606\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 32.2217 1.25805
\(657\) 0 0
\(658\) − 3.04239i − 0.118605i
\(659\) 37.8715 1.47526 0.737631 0.675204i \(-0.235944\pi\)
0.737631 + 0.675204i \(0.235944\pi\)
\(660\) 0 0
\(661\) 24.3931 0.948782 0.474391 0.880314i \(-0.342668\pi\)
0.474391 + 0.880314i \(0.342668\pi\)
\(662\) − 38.1579i − 1.48305i
\(663\) 0 0
\(664\) −29.3717 −1.13984
\(665\) 0 0
\(666\) 0 0
\(667\) − 25.0508i − 0.969971i
\(668\) 72.9013i 2.82064i
\(669\) 0 0
\(670\) 0 0
\(671\) 15.0386 0.580560
\(672\) 0 0
\(673\) − 21.1281i − 0.814428i −0.913333 0.407214i \(-0.866500\pi\)
0.913333 0.407214i \(-0.133500\pi\)
\(674\) 55.6363 2.14303
\(675\) 0 0
\(676\) 4.19656 0.161406
\(677\) − 15.3973i − 0.591767i −0.955224 0.295884i \(-0.904386\pi\)
0.955224 0.295884i \(-0.0956141\pi\)
\(678\) 0 0
\(679\) −2.18356 −0.0837972
\(680\) 0 0
\(681\) 0 0
\(682\) 8.87192i 0.339723i
\(683\) − 30.0722i − 1.15068i −0.817914 0.575341i \(-0.804869\pi\)
0.817914 0.575341i \(-0.195131\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 37.4355 1.42929
\(687\) 0 0
\(688\) − 51.9562i − 1.98081i
\(689\) 10.1751 0.387642
\(690\) 0 0
\(691\) −8.14950 −0.310022 −0.155011 0.987913i \(-0.549541\pi\)
−0.155011 + 0.987913i \(0.549541\pi\)
\(692\) − 33.3927i − 1.26940i
\(693\) 0 0
\(694\) −14.3931 −0.546355
\(695\) 0 0
\(696\) 0 0
\(697\) 38.1323i 1.44436i
\(698\) − 68.5082i − 2.59307i
\(699\) 0 0
\(700\) 0 0
\(701\) 28.6921 1.08369 0.541843 0.840480i \(-0.317727\pi\)
0.541843 + 0.840480i \(0.317727\pi\)
\(702\) 0 0
\(703\) 54.3074i 2.04824i
\(704\) 6.37123 0.240125
\(705\) 0 0
\(706\) 71.5506 2.69284
\(707\) 12.3847i 0.465774i
\(708\) 0 0
\(709\) −12.3074 −0.462215 −0.231108 0.972928i \(-0.574235\pi\)
−0.231108 + 0.972928i \(0.574235\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 55.6363i 2.08506i
\(713\) 12.4360i 0.465730i
\(714\) 0 0
\(715\) 0 0
\(716\) 65.3154 2.44095
\(717\) 0 0
\(718\) 31.3288i 1.16918i
\(719\) −28.7862 −1.07355 −0.536773 0.843727i \(-0.680357\pi\)
−0.536773 + 0.843727i \(0.680357\pi\)
\(720\) 0 0
\(721\) −22.4275 −0.835245
\(722\) − 73.9332i − 2.75151i
\(723\) 0 0
\(724\) −66.2302 −2.46142
\(725\) 0 0
\(726\) 0 0
\(727\) − 34.3931i − 1.27557i −0.770214 0.637785i \(-0.779851\pi\)
0.770214 0.637785i \(-0.220149\pi\)
\(728\) − 6.54262i − 0.242485i
\(729\) 0 0
\(730\) 0 0
\(731\) 61.4868 2.27417
\(732\) 0 0
\(733\) − 29.0042i − 1.07129i −0.844442 0.535647i \(-0.820068\pi\)
0.844442 0.535647i \(-0.179932\pi\)
\(734\) 69.4868 2.56480
\(735\) 0 0
\(736\) −8.57292 −0.316002
\(737\) − 11.2138i − 0.413065i
\(738\) 0 0
\(739\) 6.27804 0.230941 0.115471 0.993311i \(-0.463162\pi\)
0.115471 + 0.993311i \(0.463162\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 30.3074i − 1.11262i
\(743\) − 12.2352i − 0.448866i −0.974490 0.224433i \(-0.927947\pi\)
0.974490 0.224433i \(-0.0720529\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −5.85050 −0.214202
\(747\) 0 0
\(748\) − 31.0080i − 1.13376i
\(749\) 22.2180 0.811827
\(750\) 0 0
\(751\) −28.8757 −1.05369 −0.526844 0.849962i \(-0.676625\pi\)
−0.526844 + 0.849962i \(0.676625\pi\)
\(752\) − 5.32976i − 0.194357i
\(753\) 0 0
\(754\) 14.9357 0.543927
\(755\) 0 0
\(756\) 0 0
\(757\) − 30.3503i − 1.10310i −0.834142 0.551550i \(-0.814037\pi\)
0.834142 0.551550i \(-0.185963\pi\)
\(758\) 61.1146i 2.21978i
\(759\) 0 0
\(760\) 0 0
\(761\) 27.1709 0.984945 0.492473 0.870328i \(-0.336093\pi\)
0.492473 + 0.870328i \(0.336093\pi\)
\(762\) 0 0
\(763\) − 10.0428i − 0.363575i
\(764\) 45.0852 1.63113
\(765\) 0 0
\(766\) −14.4569 −0.522350
\(767\) 5.37169i 0.193961i
\(768\) 0 0
\(769\) 38.3503 1.38295 0.691473 0.722402i \(-0.256962\pi\)
0.691473 + 0.722402i \(0.256962\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 40.8710i − 1.47098i
\(773\) 50.2646i 1.80789i 0.427647 + 0.903946i \(0.359342\pi\)
−0.427647 + 0.903946i \(0.640658\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −9.97812 −0.358194
\(777\) 0 0
\(778\) − 33.9781i − 1.21817i
\(779\) −43.0937 −1.54399
\(780\) 0 0
\(781\) 6.21798 0.222497
\(782\) − 64.1789i − 2.29503i
\(783\) 0 0
\(784\) 29.0550 1.03768
\(785\) 0 0
\(786\) 0 0
\(787\) 13.2860i 0.473595i 0.971559 + 0.236797i \(0.0760978\pi\)
−0.971559 + 0.236797i \(0.923902\pi\)
\(788\) 40.1360i 1.42979i
\(789\) 0 0
\(790\) 0 0
\(791\) 9.52119 0.338535
\(792\) 0 0
\(793\) 12.5682i 0.446312i
\(794\) 30.2008 1.07179
\(795\) 0 0
\(796\) 24.9995 0.886085
\(797\) 9.82487i 0.348015i 0.984744 + 0.174007i \(0.0556716\pi\)
−0.984744 + 0.174007i \(0.944328\pi\)
\(798\) 0 0
\(799\) 6.30742 0.223141
\(800\) 0 0
\(801\) 0 0
\(802\) − 93.2087i − 3.29131i
\(803\) − 14.3074i − 0.504898i
\(804\) 0 0
\(805\) 0 0
\(806\) −7.41454 −0.261166
\(807\) 0 0
\(808\) 56.5939i 1.99097i
\(809\) −9.91431 −0.348569 −0.174284 0.984695i \(-0.555761\pi\)
−0.174284 + 0.984695i \(0.555761\pi\)
\(810\) 0 0
\(811\) −36.5855 −1.28469 −0.642345 0.766416i \(-0.722038\pi\)
−0.642345 + 0.766416i \(0.722038\pi\)
\(812\) − 30.1285i − 1.05730i
\(813\) 0 0
\(814\) −23.1793 −0.812436
\(815\) 0 0
\(816\) 0 0
\(817\) 69.4868i 2.43103i
\(818\) − 34.8500i − 1.21850i
\(819\) 0 0
\(820\) 0 0
\(821\) 34.4741 1.20316 0.601578 0.798814i \(-0.294539\pi\)
0.601578 + 0.798814i \(0.294539\pi\)
\(822\) 0 0
\(823\) 13.2566i 0.462097i 0.972942 + 0.231048i \(0.0742155\pi\)
−0.972942 + 0.231048i \(0.925784\pi\)
\(824\) −102.486 −3.57028
\(825\) 0 0
\(826\) 16.0000 0.556711
\(827\) 28.1495i 0.978854i 0.872044 + 0.489427i \(0.162794\pi\)
−0.872044 + 0.489427i \(0.837206\pi\)
\(828\) 0 0
\(829\) −16.3418 −0.567576 −0.283788 0.958887i \(-0.591591\pi\)
−0.283788 + 0.958887i \(0.591591\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.32464i 0.184599i
\(833\) 34.3847i 1.19136i
\(834\) 0 0
\(835\) 0 0
\(836\) 35.0424 1.21197
\(837\) 0 0
\(838\) − 7.91431i − 0.273395i
\(839\) −30.3675 −1.04840 −0.524201 0.851595i \(-0.675636\pi\)
−0.524201 + 0.851595i \(0.675636\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 40.5939i 1.39896i
\(843\) 0 0
\(844\) −100.358 −3.45446
\(845\) 0 0
\(846\) 0 0
\(847\) − 11.4490i − 0.393391i
\(848\) − 53.0937i − 1.82324i
\(849\) 0 0
\(850\) 0 0
\(851\) −32.4910 −1.11378
\(852\) 0 0
\(853\) − 42.1407i − 1.44287i −0.692482 0.721435i \(-0.743483\pi\)
0.692482 0.721435i \(-0.256517\pi\)
\(854\) 37.4355 1.28102
\(855\) 0 0
\(856\) 101.529 3.47018
\(857\) − 2.17513i − 0.0743012i −0.999310 0.0371506i \(-0.988172\pi\)
0.999310 0.0371506i \(-0.0118281\pi\)
\(858\) 0 0
\(859\) −18.5682 −0.633541 −0.316770 0.948502i \(-0.602598\pi\)
−0.316770 + 0.948502i \(0.602598\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 11.4145i 0.388781i
\(863\) − 33.7220i − 1.14791i −0.818887 0.573954i \(-0.805409\pi\)
0.818887 0.573954i \(-0.194591\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −95.4649 −3.24403
\(867\) 0 0
\(868\) 14.9567i 0.507663i
\(869\) 2.13229 0.0723330
\(870\) 0 0
\(871\) 9.37169 0.317548
\(872\) − 45.8924i − 1.55411i
\(873\) 0 0
\(874\) 72.5292 2.45334
\(875\) 0 0
\(876\) 0 0
\(877\) − 43.4868i − 1.46844i −0.678910 0.734222i \(-0.737547\pi\)
0.678910 0.734222i \(-0.262453\pi\)
\(878\) 19.2650i 0.650164i
\(879\) 0 0
\(880\) 0 0
\(881\) −26.7005 −0.899564 −0.449782 0.893138i \(-0.648498\pi\)
−0.449782 + 0.893138i \(0.648498\pi\)
\(882\) 0 0
\(883\) 20.2990i 0.683116i 0.939861 + 0.341558i \(0.110955\pi\)
−0.939861 + 0.341558i \(0.889045\pi\)
\(884\) 25.9143 0.871593
\(885\) 0 0
\(886\) −86.9223 −2.92021
\(887\) 36.0550i 1.21061i 0.795994 + 0.605305i \(0.206949\pi\)
−0.795994 + 0.605305i \(0.793051\pi\)
\(888\) 0 0
\(889\) −12.4360 −0.417089
\(890\) 0 0
\(891\) 0 0
\(892\) − 11.0298i − 0.369307i
\(893\) 7.12808i 0.238532i
\(894\) 0 0
\(895\) 0 0
\(896\) 20.7737 0.694000
\(897\) 0 0
\(898\) 15.3717i 0.512960i
\(899\) −17.8715 −0.596047
\(900\) 0 0
\(901\) 62.8328 2.09326
\(902\) − 18.3931i − 0.612424i
\(903\) 0 0
\(904\) 43.5087 1.44708
\(905\) 0 0
\(906\) 0 0
\(907\) 7.26504i 0.241232i 0.992699 + 0.120616i \(0.0384869\pi\)
−0.992699 + 0.120616i \(0.961513\pi\)
\(908\) − 66.1579i − 2.19553i
\(909\) 0 0
\(910\) 0 0
\(911\) 6.65769 0.220579 0.110290 0.993899i \(-0.464822\pi\)
0.110290 + 0.993899i \(0.464822\pi\)
\(912\) 0 0
\(913\) 6.42754i 0.212721i
\(914\) −3.45738 −0.114360
\(915\) 0 0
\(916\) 36.6921 1.21234
\(917\) 7.64973i 0.252616i
\(918\) 0 0
\(919\) 27.1831 0.896688 0.448344 0.893861i \(-0.352014\pi\)
0.448344 + 0.893861i \(0.352014\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 70.9013i 2.33501i
\(923\) 5.19656i 0.171047i
\(924\) 0 0
\(925\) 0 0
\(926\) 40.7643 1.33960
\(927\) 0 0
\(928\) − 12.3200i − 0.404423i
\(929\) −15.3973 −0.505170 −0.252585 0.967575i \(-0.581281\pi\)
−0.252585 + 0.967575i \(0.581281\pi\)
\(930\) 0 0
\(931\) −38.8585 −1.27353
\(932\) − 9.12808i − 0.299000i
\(933\) 0 0
\(934\) 64.0932 2.09719
\(935\) 0 0
\(936\) 0 0
\(937\) − 1.12808i − 0.0368527i −0.999830 0.0184264i \(-0.994134\pi\)
0.999830 0.0184264i \(-0.00586562\pi\)
\(938\) − 27.9143i − 0.911434i
\(939\) 0 0
\(940\) 0 0
\(941\) 30.1407 0.982559 0.491280 0.871002i \(-0.336529\pi\)
0.491280 + 0.871002i \(0.336529\pi\)
\(942\) 0 0
\(943\) − 25.7820i − 0.839578i
\(944\) 28.0294 0.912279
\(945\) 0 0
\(946\) −29.6582 −0.964270
\(947\) − 20.0294i − 0.650868i −0.945565 0.325434i \(-0.894490\pi\)
0.945565 0.325434i \(-0.105510\pi\)
\(948\) 0 0
\(949\) 11.9572 0.388146
\(950\) 0 0
\(951\) 0 0
\(952\) − 40.4015i − 1.30942i
\(953\) 43.2259i 1.40023i 0.714032 + 0.700113i \(0.246867\pi\)
−0.714032 + 0.700113i \(0.753133\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −11.7648 −0.380501
\(957\) 0 0
\(958\) 18.8929i 0.610401i
\(959\) −20.0344 −0.646945
\(960\) 0 0
\(961\) −22.1281 −0.713809
\(962\) − 19.3717i − 0.624568i
\(963\) 0 0
\(964\) 25.1793 0.810972
\(965\) 0 0
\(966\) 0 0
\(967\) − 57.6875i − 1.85511i −0.373691 0.927553i \(-0.621908\pi\)
0.373691 0.927553i \(-0.378092\pi\)
\(968\) − 52.3179i − 1.68156i
\(969\) 0 0
\(970\) 0 0
\(971\) 19.5296 0.626735 0.313368 0.949632i \(-0.398543\pi\)
0.313368 + 0.949632i \(0.398543\pi\)
\(972\) 0 0
\(973\) − 6.91852i − 0.221798i
\(974\) −10.9786 −0.351776
\(975\) 0 0
\(976\) 65.5809 2.09919
\(977\) − 40.3074i − 1.28955i −0.764373 0.644774i \(-0.776951\pi\)
0.764373 0.644774i \(-0.223049\pi\)
\(978\) 0 0
\(979\) 12.1751 0.389119
\(980\) 0 0
\(981\) 0 0
\(982\) − 0.213311i − 0.00680702i
\(983\) − 32.2008i − 1.02705i −0.858076 0.513523i \(-0.828340\pi\)
0.858076 0.513523i \(-0.171660\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 92.2302 2.93721
\(987\) 0 0
\(988\) 29.2860i 0.931712i
\(989\) −41.5725 −1.32193
\(990\) 0 0
\(991\) 26.4826 0.841246 0.420623 0.907235i \(-0.361811\pi\)
0.420623 + 0.907235i \(0.361811\pi\)
\(992\) 6.11599i 0.194183i
\(993\) 0 0
\(994\) 15.4783 0.490943
\(995\) 0 0
\(996\) 0 0
\(997\) − 35.1365i − 1.11278i −0.830920 0.556392i \(-0.812185\pi\)
0.830920 0.556392i \(-0.187815\pi\)
\(998\) − 44.1151i − 1.39644i
\(999\) 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 2925.2.c.w.2224.2 6
3.2 odd 2 975.2.c.i.274.5 6
5.2 odd 4 585.2.a.n.1.3 3
5.3 odd 4 2925.2.a.bh.1.1 3
5.4 even 2 inner 2925.2.c.w.2224.5 6
15.2 even 4 195.2.a.e.1.1 3
15.8 even 4 975.2.a.o.1.3 3
15.14 odd 2 975.2.c.i.274.2 6
20.7 even 4 9360.2.a.dd.1.2 3
60.47 odd 4 3120.2.a.bj.1.2 3
65.12 odd 4 7605.2.a.bx.1.1 3
105.62 odd 4 9555.2.a.bq.1.1 3
195.77 even 4 2535.2.a.bc.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.1 3 15.2 even 4
585.2.a.n.1.3 3 5.2 odd 4
975.2.a.o.1.3 3 15.8 even 4
975.2.c.i.274.2 6 15.14 odd 2
975.2.c.i.274.5 6 3.2 odd 2
2535.2.a.bc.1.3 3 195.77 even 4
2925.2.a.bh.1.1 3 5.3 odd 4
2925.2.c.w.2224.2 6 1.1 even 1 trivial
2925.2.c.w.2224.5 6 5.4 even 2 inner
3120.2.a.bj.1.2 3 60.47 odd 4
7605.2.a.bx.1.1 3 65.12 odd 4
9360.2.a.dd.1.2 3 20.7 even 4
9555.2.a.bq.1.1 3 105.62 odd 4