Properties

Label 975.2.c.i.274.2
Level $975$
Weight $2$
Character 975.274
Analytic conductor $7.785$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
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: \(\Z[a_1, a_2, a_3]\)
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 274.2
Root \(-0.671462 + 1.24464i\) of defining polynomial
Character \(\chi\) \(=\) 975.274
Dual form 975.2.c.i.274.5

$q$-expansion

\(f(q)\) \(=\) \(q-2.48929i q^{2} +1.00000i q^{3} -4.19656 q^{4} +2.48929 q^{6} -1.19656i q^{7} +5.46787i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-2.48929i q^{2} +1.00000i q^{3} -4.19656 q^{4} +2.48929 q^{6} -1.19656i q^{7} +5.46787i q^{8} -1.00000 q^{9} -1.19656 q^{11} -4.19656i q^{12} -1.00000i q^{13} -2.97858 q^{14} +5.21798 q^{16} +6.17513i q^{17} +2.48929i q^{18} -6.97858 q^{19} +1.19656 q^{21} +2.97858i q^{22} -4.17513i q^{23} -5.46787 q^{24} -2.48929 q^{26} -1.00000i q^{27} +5.02142i q^{28} -6.00000 q^{29} -2.97858 q^{31} -2.05333i q^{32} -1.19656i q^{33} +15.3717 q^{34} +4.19656 q^{36} +7.78202i q^{37} +17.3717i q^{38} +1.00000 q^{39} -6.17513 q^{41} -2.97858i q^{42} +9.95715i q^{43} +5.02142 q^{44} -10.3931 q^{46} -1.02142i q^{47} +5.21798i q^{48} +5.56825 q^{49} -6.17513 q^{51} +4.19656i q^{52} -10.1751i q^{53} -2.48929 q^{54} +6.54262 q^{56} -6.97858i q^{57} +14.9357i q^{58} -5.37169 q^{59} +12.5682 q^{61} +7.41454i q^{62} +1.19656i q^{63} +5.32464 q^{64} -2.97858 q^{66} +9.37169i q^{67} -25.9143i q^{68} +4.17513 q^{69} -5.19656 q^{71} -5.46787i q^{72} +11.9572i q^{73} +19.3717 q^{74} +29.2860 q^{76} +1.43175i q^{77} -2.48929i q^{78} +1.78202 q^{79} +1.00000 q^{81} +15.3717i q^{82} +5.37169i q^{83} -5.02142 q^{84} +24.7862 q^{86} -6.00000i q^{87} -6.54262i q^{88} -10.1751 q^{89} -1.19656 q^{91} +17.5212i q^{92} -2.97858i q^{93} -2.54262 q^{94} +2.05333 q^{96} -1.82487i q^{97} -13.8610i q^{98} +1.19656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} - 6 q^{9} + 2 q^{11} + 12 q^{14} + 52 q^{16} - 12 q^{19} - 2 q^{21} + 12 q^{24} - 36 q^{29} + 12 q^{31} + 44 q^{34} + 16 q^{36} + 6 q^{39} + 2 q^{41} + 60 q^{44} - 44 q^{46} - 24 q^{49} + 2 q^{51} - 32 q^{56} + 16 q^{59} + 18 q^{61} - 60 q^{64} + 12 q^{66} - 14 q^{69} - 22 q^{71} + 68 q^{74} + 8 q^{76} - 10 q^{79} + 6 q^{81} - 60 q^{84} + 112 q^{86} - 22 q^{89} + 2 q^{91} + 56 q^{94} - 44 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(-0.342522\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −4.19656 −2.09828
\(5\) 0 0
\(6\) 2.48929 1.01625
\(7\) − 1.19656i − 0.452256i −0.974098 0.226128i \(-0.927393\pi\)
0.974098 0.226128i \(-0.0726068\pi\)
\(8\) 5.46787i 1.93318i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.19656 −0.360776 −0.180388 0.983596i \(-0.557735\pi\)
−0.180388 + 0.983596i \(0.557735\pi\)
\(12\) − 4.19656i − 1.21144i
\(13\) − 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 \(0.730609\pi\)
\(18\) 2.48929i 0.586731i
\(19\) −6.97858 −1.60100 −0.800498 0.599336i \(-0.795431\pi\)
−0.800498 + 0.599336i \(0.795431\pi\)
\(20\) 0 0
\(21\) 1.19656 0.261110
\(22\) 2.97858i 0.635035i
\(23\) − 4.17513i − 0.870576i −0.900291 0.435288i \(-0.856647\pi\)
0.900291 0.435288i \(-0.143353\pi\)
\(24\) −5.46787 −1.11612
\(25\) 0 0
\(26\) −2.48929 −0.488190
\(27\) − 1.00000i − 0.192450i
\(28\) 5.02142i 0.948960i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 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
\(33\) − 1.19656i − 0.208294i
\(34\) 15.3717 2.63622
\(35\) 0 0
\(36\) 4.19656 0.699426
\(37\) 7.78202i 1.27936i 0.768643 + 0.639678i \(0.220932\pi\)
−0.768643 + 0.639678i \(0.779068\pi\)
\(38\) 17.3717i 2.81806i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −6.17513 −0.964394 −0.482197 0.876063i \(-0.660161\pi\)
−0.482197 + 0.876063i \(0.660161\pi\)
\(42\) − 2.97858i − 0.459604i
\(43\) 9.95715i 1.51845i 0.650827 + 0.759226i \(0.274422\pi\)
−0.650827 + 0.759226i \(0.725578\pi\)
\(44\) 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 \(-0.0237345\pi\)
\(48\) 5.21798i 0.753151i
\(49\) 5.56825 0.795464
\(50\) 0 0
\(51\) −6.17513 −0.864692
\(52\) 4.19656i 0.581958i
\(53\) − 10.1751i − 1.39766i −0.715287 0.698831i \(-0.753704\pi\)
0.715287 0.698831i \(-0.246296\pi\)
\(54\) −2.48929 −0.338749
\(55\) 0 0
\(56\) 6.54262 0.874294
\(57\) − 6.97858i − 0.924335i
\(58\) 14.9357i 1.96116i
\(59\) −5.37169 −0.699335 −0.349667 0.936874i \(-0.613705\pi\)
−0.349667 + 0.936874i \(0.613705\pi\)
\(60\) 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
\(63\) 1.19656i 0.150752i
\(64\) 5.32464 0.665579
\(65\) 0 0
\(66\) −2.97858 −0.366638
\(67\) 9.37169i 1.14493i 0.819928 + 0.572467i \(0.194014\pi\)
−0.819928 + 0.572467i \(0.805986\pi\)
\(68\) − 25.9143i − 3.14257i
\(69\) 4.17513 0.502627
\(70\) 0 0
\(71\) −5.19656 −0.616718 −0.308359 0.951270i \(-0.599780\pi\)
−0.308359 + 0.951270i \(0.599780\pi\)
\(72\) − 5.46787i − 0.644394i
\(73\) 11.9572i 1.39948i 0.714398 + 0.699740i \(0.246701\pi\)
−0.714398 + 0.699740i \(0.753299\pi\)
\(74\) 19.3717 2.25191
\(75\) 0 0
\(76\) 29.2860 3.35933
\(77\) 1.43175i 0.163163i
\(78\) − 2.48929i − 0.281856i
\(79\) 1.78202 0.200493 0.100246 0.994963i \(-0.468037\pi\)
0.100246 + 0.994963i \(0.468037\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 15.3717i 1.69752i
\(83\) 5.37169i 0.589620i 0.955556 + 0.294810i \(0.0952563\pi\)
−0.955556 + 0.294810i \(0.904744\pi\)
\(84\) −5.02142 −0.547882
\(85\) 0 0
\(86\) 24.7862 2.67277
\(87\) − 6.00000i − 0.643268i
\(88\) − 6.54262i − 0.697445i
\(89\) −10.1751 −1.07856 −0.539281 0.842126i \(-0.681304\pi\)
−0.539281 + 0.842126i \(0.681304\pi\)
\(90\) 0 0
\(91\) −1.19656 −0.125433
\(92\) 17.5212i 1.82671i
\(93\) − 2.97858i − 0.308864i
\(94\) −2.54262 −0.262251
\(95\) 0 0
\(96\) 2.05333 0.209567
\(97\) − 1.82487i − 0.185287i −0.995699 0.0926435i \(-0.970468\pi\)
0.995699 0.0926435i \(-0.0295317\pi\)
\(98\) − 13.8610i − 1.40017i
\(99\) 1.19656 0.120259
\(100\) 0 0
\(101\) −10.3503 −1.02989 −0.514945 0.857223i \(-0.672188\pi\)
−0.514945 + 0.857223i \(0.672188\pi\)
\(102\) 15.3717i 1.52202i
\(103\) − 18.7434i − 1.84684i −0.383790 0.923420i \(-0.625381\pi\)
0.383790 0.923420i \(-0.374619\pi\)
\(104\) 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 \(-0.354641\pi\)
\(108\) 4.19656i 0.403814i
\(109\) −8.39312 −0.803915 −0.401957 0.915658i \(-0.631670\pi\)
−0.401957 + 0.915658i \(0.631670\pi\)
\(110\) 0 0
\(111\) −7.78202 −0.738637
\(112\) − 6.24361i − 0.589966i
\(113\) − 7.95715i − 0.748546i −0.927319 0.374273i \(-0.877892\pi\)
0.927319 0.374273i \(-0.122108\pi\)
\(114\) −17.3717 −1.62701
\(115\) 0 0
\(116\) 25.1793 2.33784
\(117\) 1.00000i 0.0924500i
\(118\) 13.3717i 1.23096i
\(119\) 7.38890 0.677340
\(120\) 0 0
\(121\) −9.56825 −0.869841
\(122\) − 31.2860i − 2.83250i
\(123\) − 6.17513i − 0.556793i
\(124\) 12.4998 1.12251
\(125\) 0 0
\(126\) 2.97858 0.265353
\(127\) − 10.3931i − 0.922240i −0.887338 0.461120i \(-0.847448\pi\)
0.887338 0.461120i \(-0.152552\pi\)
\(128\) − 17.3612i − 1.53453i
\(129\) −9.95715 −0.876679
\(130\) 0 0
\(131\) −6.39312 −0.558569 −0.279285 0.960208i \(-0.590097\pi\)
−0.279285 + 0.960208i \(0.590097\pi\)
\(132\) 5.02142i 0.437059i
\(133\) 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 \(0.746316\pi\)
\(138\) − 10.3931i − 0.884721i
\(139\) −5.78202 −0.490424 −0.245212 0.969469i \(-0.578858\pi\)
−0.245212 + 0.969469i \(0.578858\pi\)
\(140\) 0 0
\(141\) 1.02142 0.0860193
\(142\) 12.9357i 1.08554i
\(143\) 1.19656i 0.100061i
\(144\) −5.21798 −0.434832
\(145\) 0 0
\(146\) 29.7648 2.46335
\(147\) 5.56825i 0.459262i
\(148\) − 32.6577i − 2.68445i
\(149\) −15.3461 −1.25720 −0.628599 0.777730i \(-0.716371\pi\)
−0.628599 + 0.777730i \(0.716371\pi\)
\(150\) 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
\(153\) − 6.17513i − 0.499230i
\(154\) 3.56404 0.287198
\(155\) 0 0
\(156\) −4.19656 −0.335994
\(157\) 2.78623i 0.222365i 0.993800 + 0.111183i \(0.0354639\pi\)
−0.993800 + 0.111183i \(0.964536\pi\)
\(158\) − 4.43596i − 0.352906i
\(159\) 10.1751 0.806941
\(160\) 0 0
\(161\) −4.99579 −0.393723
\(162\) − 2.48929i − 0.195577i
\(163\) − 8.76060i − 0.686183i −0.939302 0.343091i \(-0.888526\pi\)
0.939302 0.343091i \(-0.111474\pi\)
\(164\) 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 \(-0.234621\pi\)
\(168\) 6.54262i 0.504774i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 6.97858 0.533665
\(172\) − 41.7858i − 3.18614i
\(173\) 7.95715i 0.604971i 0.953154 + 0.302486i \(0.0978164\pi\)
−0.953154 + 0.302486i \(0.902184\pi\)
\(174\) −14.9357 −1.13227
\(175\) 0 0
\(176\) −6.24361 −0.470630
\(177\) − 5.37169i − 0.403761i
\(178\) 25.3288i 1.89848i
\(179\) 15.5640 1.16331 0.581655 0.813435i \(-0.302405\pi\)
0.581655 + 0.813435i \(0.302405\pi\)
\(180\) 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
\(183\) 12.5682i 0.929072i
\(184\) 22.8291 1.68298
\(185\) 0 0
\(186\) −7.41454 −0.543660
\(187\) − 7.38890i − 0.540330i
\(188\) 4.28646i 0.312622i
\(189\) −1.19656 −0.0870368
\(190\) 0 0
\(191\) 10.7434 0.777364 0.388682 0.921372i \(-0.372930\pi\)
0.388682 + 0.921372i \(0.372930\pi\)
\(192\) 5.32464i 0.384272i
\(193\) − 9.73917i − 0.701041i −0.936555 0.350521i \(-0.886005\pi\)
0.936555 0.350521i \(-0.113995\pi\)
\(194\) −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 \(-0.110666\pi\)
\(198\) − 2.97858i − 0.211678i
\(199\) −5.95715 −0.422291 −0.211146 0.977455i \(-0.567719\pi\)
−0.211146 + 0.977455i \(0.567719\pi\)
\(200\) 0 0
\(201\) −9.37169 −0.661028
\(202\) 25.7648i 1.81281i
\(203\) 7.17935i 0.503891i
\(204\) 25.9143 1.81436
\(205\) 0 0
\(206\) −46.6577 −3.25080
\(207\) 4.17513i 0.290192i
\(208\) − 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
\(213\) − 5.19656i − 0.356062i
\(214\) −46.2217 −3.15965
\(215\) 0 0
\(216\) 5.46787 0.372041
\(217\) 3.56404i 0.241943i
\(218\) 20.8929i 1.41504i
\(219\) −11.9572 −0.807990
\(220\) 0 0
\(221\) 6.17513 0.415385
\(222\) 19.3717i 1.30014i
\(223\) − 2.62831i − 0.176004i −0.996120 0.0880022i \(-0.971952\pi\)
0.996120 0.0880022i \(-0.0280483\pi\)
\(224\) −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 \(0.824748\pi\)
\(228\) 29.2860i 1.93951i
\(229\) −8.74338 −0.577779 −0.288890 0.957362i \(-0.593286\pi\)
−0.288890 + 0.957362i \(0.593286\pi\)
\(230\) 0 0
\(231\) −1.43175 −0.0942022
\(232\) − 32.8072i − 2.15390i
\(233\) 2.17513i 0.142498i 0.997459 + 0.0712489i \(0.0226985\pi\)
−0.997459 + 0.0712489i \(0.977302\pi\)
\(234\) 2.48929 0.162730
\(235\) 0 0
\(236\) 22.5426 1.46740
\(237\) 1.78202i 0.115755i
\(238\) − 18.3931i − 1.19225i
\(239\) −2.80344 −0.181340 −0.0906698 0.995881i \(-0.528901\pi\)
−0.0906698 + 0.995881i \(0.528901\pi\)
\(240\) 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
\(243\) 1.00000i 0.0641500i
\(244\) −52.7434 −3.37655
\(245\) 0 0
\(246\) −15.3717 −0.980063
\(247\) 6.97858i 0.444036i
\(248\) − 16.2865i − 1.03419i
\(249\) −5.37169 −0.340417
\(250\) 0 0
\(251\) −23.9143 −1.50946 −0.754729 0.656037i \(-0.772232\pi\)
−0.754729 + 0.656037i \(0.772232\pi\)
\(252\) − 5.02142i − 0.316320i
\(253\) 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 \(-0.213861\pi\)
\(258\) 24.7862i 1.54312i
\(259\) 9.31163 0.578597
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 15.9143i 0.983189i
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) 6.54262 0.402670
\(265\) 0 0
\(266\) 20.7862 1.27449
\(267\) − 10.1751i − 0.622708i
\(268\) − 39.3288i − 2.40239i
\(269\) 2.35027 0.143298 0.0716492 0.997430i \(-0.477174\pi\)
0.0716492 + 0.997430i \(0.477174\pi\)
\(270\) 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
\(273\) − 1.19656i − 0.0724190i
\(274\) 41.6791 2.51793
\(275\) 0 0
\(276\) −17.5212 −1.05465
\(277\) 1.21377i 0.0729283i 0.999335 + 0.0364642i \(0.0116095\pi\)
−0.999335 + 0.0364642i \(0.988391\pi\)
\(278\) 14.3931i 0.863242i
\(279\) 2.97858 0.178323
\(280\) 0 0
\(281\) 11.9572 0.713304 0.356652 0.934237i \(-0.383918\pi\)
0.356652 + 0.934237i \(0.383918\pi\)
\(282\) − 2.54262i − 0.151411i
\(283\) 29.8715i 1.77567i 0.460158 + 0.887837i \(0.347793\pi\)
−0.460158 + 0.887837i \(0.652207\pi\)
\(284\) 21.8077 1.29405
\(285\) 0 0
\(286\) 2.97858 0.176127
\(287\) 7.38890i 0.436153i
\(288\) 2.05333i 0.120993i
\(289\) −21.1323 −1.24308
\(290\) 0 0
\(291\) 1.82487 0.106975
\(292\) − 50.1789i − 2.93650i
\(293\) − 0.777809i − 0.0454401i −0.999742 0.0227200i \(-0.992767\pi\)
0.999742 0.0227200i \(-0.00723263\pi\)
\(294\) 13.8610 0.808389
\(295\) 0 0
\(296\) −42.5510 −2.47323
\(297\) 1.19656i 0.0694313i
\(298\) 38.2008i 2.21291i
\(299\) −4.17513 −0.241454
\(300\) 0 0
\(301\) 11.9143 0.686729
\(302\) 21.3717i 1.22980i
\(303\) − 10.3503i − 0.594607i
\(304\) −36.4141 −2.08849
\(305\) 0 0
\(306\) −15.3717 −0.878741
\(307\) 0.760597i 0.0434095i 0.999764 + 0.0217048i \(0.00690939\pi\)
−0.999764 + 0.0217048i \(0.993091\pi\)
\(308\) − 6.00842i − 0.342362i
\(309\) 18.7434 1.06627
\(310\) 0 0
\(311\) −23.1281 −1.31147 −0.655736 0.754990i \(-0.727642\pi\)
−0.655736 + 0.754990i \(0.727642\pi\)
\(312\) 5.46787i 0.309557i
\(313\) 33.9143i 1.91695i 0.285176 + 0.958475i \(0.407948\pi\)
−0.285176 + 0.958475i \(0.592052\pi\)
\(314\) 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 \(0.912648\pi\)
\(318\) − 25.3288i − 1.42037i
\(319\) 7.17935 0.401966
\(320\) 0 0
\(321\) 18.5682 1.03638
\(322\) 12.4360i 0.693029i
\(323\) − 43.0937i − 2.39780i
\(324\) −4.19656 −0.233142
\(325\) 0 0
\(326\) −21.8077 −1.20781
\(327\) − 8.39312i − 0.464140i
\(328\) − 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
\(333\) − 7.78202i − 0.426452i
\(334\) −43.2432 −2.36616
\(335\) 0 0
\(336\) 6.24361 0.340617
\(337\) − 22.3503i − 1.21750i −0.793363 0.608748i \(-0.791672\pi\)
0.793363 0.608748i \(-0.208328\pi\)
\(338\) 2.48929i 0.135399i
\(339\) 7.95715 0.432173
\(340\) 0 0
\(341\) 3.56404 0.193004
\(342\) − 17.3717i − 0.939354i
\(343\) − 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 \(-0.0496014\pi\)
\(348\) 25.1793i 1.34975i
\(349\) 27.5212 1.47318 0.736588 0.676342i \(-0.236436\pi\)
0.736588 + 0.676342i \(0.236436\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 2.45692i 0.130955i
\(353\) 28.7434i 1.52986i 0.644116 + 0.764928i \(0.277225\pi\)
−0.644116 + 0.764928i \(0.722775\pi\)
\(354\) −13.3717 −0.710697
\(355\) 0 0
\(356\) 42.7005 2.26312
\(357\) 7.38890i 0.391062i
\(358\) − 38.7434i − 2.04765i
\(359\) 12.5855 0.664235 0.332118 0.943238i \(-0.392237\pi\)
0.332118 + 0.943238i \(0.392237\pi\)
\(360\) 0 0
\(361\) 29.7005 1.56319
\(362\) − 39.2860i − 2.06483i
\(363\) − 9.56825i − 0.502203i
\(364\) 5.02142 0.263194
\(365\) 0 0
\(366\) 31.2860 1.63535
\(367\) − 27.9143i − 1.45712i −0.684985 0.728558i \(-0.740191\pi\)
0.684985 0.728558i \(-0.259809\pi\)
\(368\) − 21.7858i − 1.13566i
\(369\) 6.17513 0.321465
\(370\) 0 0
\(371\) −12.1751 −0.632102
\(372\) 12.4998i 0.648083i
\(373\) 2.35027i 0.121692i 0.998147 + 0.0608462i \(0.0193799\pi\)
−0.998147 + 0.0608462i \(0.980620\pi\)
\(374\) −18.3931 −0.951085
\(375\) 0 0
\(376\) 5.58500 0.288025
\(377\) 6.00000i 0.309016i
\(378\) 2.97858i 0.153201i
\(379\) −24.5510 −1.26110 −0.630551 0.776148i \(-0.717171\pi\)
−0.630551 + 0.776148i \(0.717171\pi\)
\(380\) 0 0
\(381\) 10.3931 0.532455
\(382\) − 26.7434i − 1.36831i
\(383\) − 5.80765i − 0.296757i −0.988931 0.148379i \(-0.952595\pi\)
0.988931 0.148379i \(-0.0474054\pi\)
\(384\) 17.3612 0.885961
\(385\) 0 0
\(386\) −24.2436 −1.23397
\(387\) − 9.95715i − 0.506151i
\(388\) 7.65815i 0.388784i
\(389\) −13.6497 −0.692069 −0.346034 0.938222i \(-0.612472\pi\)
−0.346034 + 0.938222i \(0.612472\pi\)
\(390\) 0 0
\(391\) 25.7820 1.30385
\(392\) 30.4464i 1.53778i
\(393\) − 6.39312i − 0.322490i
\(394\) −23.8077 −1.19941
\(395\) 0 0
\(396\) −5.02142 −0.252336
\(397\) − 12.1323i − 0.608902i −0.952528 0.304451i \(-0.901527\pi\)
0.952528 0.304451i \(-0.0984730\pi\)
\(398\) 14.8291i 0.743314i
\(399\) −8.35027 −0.418036
\(400\) 0 0
\(401\) −37.4439 −1.86986 −0.934930 0.354832i \(-0.884538\pi\)
−0.934930 + 0.354832i \(0.884538\pi\)
\(402\) 23.3288i 1.16354i
\(403\) 2.97858i 0.148373i
\(404\) 43.4355 2.16100
\(405\) 0 0
\(406\) 17.8715 0.886946
\(407\) − 9.31163i − 0.461561i
\(408\) − 33.7648i − 1.67161i
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −16.7434 −0.825890
\(412\) 78.6577i 3.87519i
\(413\) 6.42754i 0.316279i
\(414\) 10.3931 0.510794
\(415\) 0 0
\(416\) −2.05333 −0.100673
\(417\) − 5.78202i − 0.283147i
\(418\) − 20.7862i − 1.01669i
\(419\) −3.17935 −0.155321 −0.0776606 0.996980i \(-0.524745\pi\)
−0.0776606 + 0.996980i \(0.524745\pi\)
\(420\) 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
\(423\) 1.02142i 0.0496633i
\(424\) 55.6363 2.70194
\(425\) 0 0
\(426\) −12.9357 −0.626738
\(427\) − 15.0386i − 0.727771i
\(428\) 77.9227i 3.76654i
\(429\) −1.19656 −0.0577703
\(430\) 0 0
\(431\) 4.58546 0.220874 0.110437 0.993883i \(-0.464775\pi\)
0.110437 + 0.993883i \(0.464775\pi\)
\(432\) − 5.21798i − 0.251050i
\(433\) 38.3503i 1.84300i 0.388383 + 0.921498i \(0.373034\pi\)
−0.388383 + 0.921498i \(0.626966\pi\)
\(434\) 8.87192 0.425866
\(435\) 0 0
\(436\) 35.2222 1.68684
\(437\) 29.1365i 1.39379i
\(438\) 29.7648i 1.42222i
\(439\) −7.73917 −0.369371 −0.184685 0.982798i \(-0.559127\pi\)
−0.184685 + 0.982798i \(0.559127\pi\)
\(440\) 0 0
\(441\) −5.56825 −0.265155
\(442\) − 15.3717i − 0.731157i
\(443\) − 34.9185i − 1.65903i −0.558485 0.829514i \(-0.688617\pi\)
0.558485 0.829514i \(-0.311383\pi\)
\(444\) 32.6577 1.54987
\(445\) 0 0
\(446\) −6.54262 −0.309802
\(447\) − 15.3461i − 0.725844i
\(448\) − 6.37123i − 0.301012i
\(449\) 6.17513 0.291423 0.145711 0.989327i \(-0.453453\pi\)
0.145711 + 0.989327i \(0.453453\pi\)
\(450\) 0 0
\(451\) 7.38890 0.347930
\(452\) 33.3927i 1.57066i
\(453\) − 8.58546i − 0.403380i
\(454\) 39.2432 1.84177
\(455\) 0 0
\(456\) 38.1579 1.78691
\(457\) 1.38890i 0.0649702i 0.999472 + 0.0324851i \(0.0103421\pi\)
−0.999472 + 0.0324851i \(0.989658\pi\)
\(458\) 21.7648i 1.01700i
\(459\) 6.17513 0.288231
\(460\) 0 0
\(461\) 28.4826 1.32657 0.663283 0.748369i \(-0.269163\pi\)
0.663283 + 0.748369i \(0.269163\pi\)
\(462\) 3.56404i 0.165814i
\(463\) − 16.3759i − 0.761053i −0.924770 0.380526i \(-0.875743\pi\)
0.924770 0.380526i \(-0.124257\pi\)
\(464\) −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 \(0.796864\pi\)
\(468\) − 4.19656i − 0.193986i
\(469\) 11.2138 0.517804
\(470\) 0 0
\(471\) −2.78623 −0.128383
\(472\) − 29.3717i − 1.35194i
\(473\) − 11.9143i − 0.547820i
\(474\) 4.43596 0.203750
\(475\) 0 0
\(476\) −31.0080 −1.42125
\(477\) 10.1751i 0.465887i
\(478\) 6.97858i 0.319193i
\(479\) 7.58967 0.346781 0.173391 0.984853i \(-0.444528\pi\)
0.173391 + 0.984853i \(0.444528\pi\)
\(480\) 0 0
\(481\) 7.78202 0.354830
\(482\) 14.9357i 0.680304i
\(483\) − 4.99579i − 0.227316i
\(484\) 40.1537 1.82517
\(485\) 0 0
\(486\) 2.48929 0.112916
\(487\) 4.41033i 0.199851i 0.994995 + 0.0999255i \(0.0318604\pi\)
−0.994995 + 0.0999255i \(0.968140\pi\)
\(488\) 68.7215i 3.11088i
\(489\) 8.76060 0.396168
\(490\) 0 0
\(491\) −0.0856914 −0.00386720 −0.00193360 0.999998i \(-0.500615\pi\)
−0.00193360 + 0.999998i \(0.500615\pi\)
\(492\) 25.9143i 1.16831i
\(493\) − 37.0508i − 1.66868i
\(494\) 17.3717 0.781589
\(495\) 0 0
\(496\) −15.5422 −0.697863
\(497\) 6.21798i 0.278915i
\(498\) 13.3717i 0.599200i
\(499\) 17.7220 0.793344 0.396672 0.917960i \(-0.370165\pi\)
0.396672 + 0.917960i \(0.370165\pi\)
\(500\) 0 0
\(501\) 17.3717 0.776110
\(502\) 59.5296i 2.65694i
\(503\) 8.70054i 0.387938i 0.981008 + 0.193969i \(0.0621361\pi\)
−0.981008 + 0.193969i \(0.937864\pi\)
\(504\) −6.54262 −0.291431
\(505\) 0 0
\(506\) 12.4360 0.552846
\(507\) − 1.00000i − 0.0444116i
\(508\) 43.6153i 1.93512i
\(509\) 33.3545 1.47841 0.739206 0.673480i \(-0.235201\pi\)
0.739206 + 0.673480i \(0.235201\pi\)
\(510\) 0 0
\(511\) 14.3074 0.632923
\(512\) 46.3482i 2.04832i
\(513\) 6.97858i 0.308112i
\(514\) −49.6791 −2.19125
\(515\) 0 0
\(516\) 41.7858 1.83952
\(517\) 1.22219i 0.0537519i
\(518\) − 23.1793i − 1.01844i
\(519\) −7.95715 −0.349280
\(520\) 0 0
\(521\) 18.7005 0.819285 0.409643 0.912246i \(-0.365653\pi\)
0.409643 + 0.912246i \(0.365653\pi\)
\(522\) − 14.9357i − 0.653719i
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 26.8291 1.17203
\(525\) 0 0
\(526\) −19.9143 −0.868305
\(527\) − 18.3931i − 0.801217i
\(528\) − 6.24361i − 0.271718i
\(529\) 5.56825 0.242098
\(530\) 0 0
\(531\) 5.37169 0.233112
\(532\) − 35.0424i − 1.51928i
\(533\) 6.17513i 0.267475i
\(534\) −25.3288 −1.09609
\(535\) 0 0
\(536\) −51.2432 −2.21337
\(537\) 15.5640i 0.671638i
\(538\) − 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
\(543\) 15.7820i 0.677271i
\(544\) 12.6796 0.543632
\(545\) 0 0
\(546\) −2.97858 −0.127471
\(547\) − 7.91431i − 0.338391i −0.985582 0.169196i \(-0.945883\pi\)
0.985582 0.169196i \(-0.0541170\pi\)
\(548\) − 70.2646i − 3.00155i
\(549\) −12.5682 −0.536400
\(550\) 0 0
\(551\) 41.8715 1.78378
\(552\) 22.8291i 0.971670i
\(553\) − 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 \(0.640140\pi\)
\(558\) − 7.41454i − 0.313882i
\(559\) 9.95715 0.421143
\(560\) 0 0
\(561\) 7.38890 0.311960
\(562\) − 29.7648i − 1.25555i
\(563\) − 1.04706i − 0.0441282i −0.999757 0.0220641i \(-0.992976\pi\)
0.999757 0.0220641i \(-0.00702379\pi\)
\(564\) −4.28646 −0.180493
\(565\) 0 0
\(566\) 74.3587 3.12553
\(567\) − 1.19656i − 0.0502507i
\(568\) − 28.4141i − 1.19223i
\(569\) −16.7778 −0.703362 −0.351681 0.936120i \(-0.614390\pi\)
−0.351681 + 0.936120i \(0.614390\pi\)
\(570\) 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
\(573\) 10.7434i 0.448811i
\(574\) 18.3931 0.767714
\(575\) 0 0
\(576\) −5.32464 −0.221860
\(577\) 1.38890i 0.0578208i 0.999582 + 0.0289104i \(0.00920376\pi\)
−0.999582 + 0.0289104i \(0.990796\pi\)
\(578\) 52.6044i 2.18805i
\(579\) 9.73917 0.404746
\(580\) 0 0
\(581\) 6.42754 0.266659
\(582\) − 4.54262i − 0.188298i
\(583\) 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 \(-0.00614723\pi\)
\(588\) − 23.3675i − 0.963659i
\(589\) 20.7862 0.856482
\(590\) 0 0
\(591\) 9.56404 0.393412
\(592\) 40.6064i 1.66891i
\(593\) 0.478807i 0.0196622i 0.999952 + 0.00983112i \(0.00312939\pi\)
−0.999952 + 0.00983112i \(0.996871\pi\)
\(594\) 2.97858 0.122213
\(595\) 0 0
\(596\) 64.4006 2.63795
\(597\) − 5.95715i − 0.243810i
\(598\) 10.3931i 0.425006i
\(599\) −29.0852 −1.18839 −0.594195 0.804321i \(-0.702529\pi\)
−0.594195 + 0.804321i \(0.702529\pi\)
\(600\) 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
\(603\) − 9.37169i − 0.381645i
\(604\) 36.0294 1.46601
\(605\) 0 0
\(606\) −25.7648 −1.04662
\(607\) 27.9143i 1.13301i 0.824059 + 0.566503i \(0.191704\pi\)
−0.824059 + 0.566503i \(0.808296\pi\)
\(608\) 14.3293i 0.581130i
\(609\) −7.17935 −0.290922
\(610\) 0 0
\(611\) −1.02142 −0.0413223
\(612\) 25.9143i 1.04752i
\(613\) 4.65394i 0.187971i 0.995574 + 0.0939855i \(0.0299607\pi\)
−0.995574 + 0.0939855i \(0.970039\pi\)
\(614\) 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 \(-0.104088\pi\)
\(618\) − 46.6577i − 1.87685i
\(619\) −1.02142 −0.0410545 −0.0205272 0.999789i \(-0.506534\pi\)
−0.0205272 + 0.999789i \(0.506534\pi\)
\(620\) 0 0
\(621\) −4.17513 −0.167542
\(622\) 57.5725i 2.30845i
\(623\) 12.1751i 0.487786i
\(624\) 5.21798 0.208886
\(625\) 0 0
\(626\) 84.4225 3.37420
\(627\) 8.35027i 0.333478i
\(628\) − 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
\(633\) 23.9143i 0.950508i
\(634\) 24.0210 0.953994
\(635\) 0 0
\(636\) −42.7005 −1.69319
\(637\) − 5.56825i − 0.220622i
\(638\) − 17.8715i − 0.707538i
\(639\) 5.19656 0.205573
\(640\) 0 0
\(641\) 38.2646 1.51136 0.755680 0.654941i \(-0.227307\pi\)
0.755680 + 0.654941i \(0.227307\pi\)
\(642\) − 46.2217i − 1.82423i
\(643\) 33.1109i 1.30577i 0.757459 + 0.652883i \(0.226440\pi\)
−0.757459 + 0.652883i \(0.773560\pi\)
\(644\) 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 \(-0.108199\pi\)
\(648\) 5.46787i 0.214798i
\(649\) 6.42754 0.252303
\(650\) 0 0
\(651\) −3.56404 −0.139686
\(652\) 36.7643i 1.43980i
\(653\) 19.1709i 0.750216i 0.926981 + 0.375108i \(0.122394\pi\)
−0.926981 + 0.375108i \(0.877606\pi\)
\(654\) −20.8929 −0.816976
\(655\) 0 0
\(656\) −32.2217 −1.25805
\(657\) − 11.9572i − 0.466493i
\(658\) 3.04239i 0.118605i
\(659\) −37.8715 −1.47526 −0.737631 0.675204i \(-0.764056\pi\)
−0.737631 + 0.675204i \(0.764056\pi\)
\(660\) 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
\(663\) 6.17513i 0.239822i
\(664\) −29.3717 −1.13984
\(665\) 0 0
\(666\) −19.3717 −0.750638
\(667\) 25.0508i 0.969971i
\(668\) 72.9013i 2.82064i
\(669\) 2.62831 0.101616
\(670\) 0 0
\(671\) −15.0386 −0.580560
\(672\) − 2.45692i − 0.0947779i
\(673\) 21.1281i 0.814428i 0.913333 + 0.407214i \(0.133500\pi\)
−0.913333 + 0.407214i \(0.866500\pi\)
\(674\) −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 \(-0.0956141\pi\)
\(678\) − 19.8077i − 0.760708i
\(679\) −2.18356 −0.0837972
\(680\) 0 0
\(681\) −15.7648 −0.604109
\(682\) − 8.87192i − 0.339723i
\(683\) − 30.0722i − 1.15068i −0.817914 0.575341i \(-0.804869\pi\)
0.817914 0.575341i \(-0.195131\pi\)
\(684\) −29.2860 −1.11978
\(685\) 0 0
\(686\) −37.4355 −1.42929
\(687\) − 8.74338i − 0.333581i
\(688\) 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
\(693\) − 1.43175i − 0.0543877i
\(694\) −14.3931 −0.546355
\(695\) 0 0
\(696\) 32.8072 1.24355
\(697\) − 38.1323i − 1.44436i
\(698\) − 68.5082i − 2.59307i
\(699\) −2.17513 −0.0822712
\(700\) 0 0
\(701\) −28.6921 −1.08369 −0.541843 0.840480i \(-0.682273\pi\)
−0.541843 + 0.840480i \(0.682273\pi\)
\(702\) 2.48929i 0.0939521i
\(703\) − 54.3074i − 2.04824i
\(704\) −6.37123 −0.240125
\(705\) 0 0
\(706\) 71.5506 2.69284
\(707\) 12.3847i 0.465774i
\(708\) 22.5426i 0.847203i
\(709\) −12.3074 −0.462215 −0.231108 0.972928i \(-0.574235\pi\)
−0.231108 + 0.972928i \(0.574235\pi\)
\(710\) 0 0
\(711\) −1.78202 −0.0668310
\(712\) − 55.6363i − 2.08506i
\(713\) 12.4360i 0.465730i
\(714\) 18.3931 0.688345
\(715\) 0 0
\(716\) −65.3154 −2.44095
\(717\) − 2.80344i − 0.104696i
\(718\) − 31.3288i − 1.16918i
\(719\) 28.7862 1.07355 0.536773 0.843727i \(-0.319643\pi\)
0.536773 + 0.843727i \(0.319643\pi\)
\(720\) 0 0
\(721\) −22.4275 −0.835245
\(722\) − 73.9332i − 2.75151i
\(723\) − 6.00000i − 0.223142i
\(724\) −66.2302 −2.46142
\(725\) 0 0
\(726\) −23.8181 −0.883974
\(727\) 34.3931i 1.27557i 0.770214 + 0.637785i \(0.220149\pi\)
−0.770214 + 0.637785i \(0.779851\pi\)
\(728\) − 6.54262i − 0.242485i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −61.4868 −2.27417
\(732\) − 52.7434i − 1.94945i
\(733\) 29.0042i 1.07129i 0.844442 + 0.535647i \(0.179932\pi\)
−0.844442 + 0.535647i \(0.820068\pi\)
\(734\) −69.4868 −2.56480
\(735\) 0 0
\(736\) −8.57292 −0.316002
\(737\) − 11.2138i − 0.413065i
\(738\) − 15.3717i − 0.565840i
\(739\) 6.27804 0.230941 0.115471 0.993311i \(-0.463162\pi\)
0.115471 + 0.993311i \(0.463162\pi\)
\(740\) 0 0
\(741\) −6.97858 −0.256364
\(742\) 30.3074i 1.11262i
\(743\) − 12.2352i − 0.448866i −0.974490 0.224433i \(-0.927947\pi\)
0.974490 0.224433i \(-0.0720529\pi\)
\(744\) 16.2865 0.597091
\(745\) 0 0
\(746\) 5.85050 0.214202
\(747\) − 5.37169i − 0.196540i
\(748\) 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
\(753\) − 23.9143i − 0.871486i
\(754\) 14.9357 0.543927
\(755\) 0 0
\(756\) 5.02142 0.182627
\(757\) 30.3503i 1.10310i 0.834142 + 0.551550i \(0.185963\pi\)
−0.834142 + 0.551550i \(0.814037\pi\)
\(758\) 61.1146i 2.21978i
\(759\) −4.99579 −0.181336
\(760\) 0 0
\(761\) −27.1709 −0.984945 −0.492473 0.870328i \(-0.663907\pi\)
−0.492473 + 0.870328i \(0.663907\pi\)
\(762\) − 25.8715i − 0.937224i
\(763\) 10.0428i 0.363575i
\(764\) −45.0852 −1.63113
\(765\) 0 0
\(766\) −14.4569 −0.522350
\(767\) 5.37169i 0.193961i
\(768\) − 32.5678i − 1.17519i
\(769\) 38.3503 1.38295 0.691473 0.722402i \(-0.256962\pi\)
0.691473 + 0.722402i \(0.256962\pi\)
\(770\) 0 0
\(771\) 19.9572 0.718739
\(772\) 40.8710i 1.47098i
\(773\) 50.2646i 1.80789i 0.427647 + 0.903946i \(0.359342\pi\)
−0.427647 + 0.903946i \(0.640658\pi\)
\(774\) −24.7862 −0.890923
\(775\) 0 0
\(776\) 9.97812 0.358194
\(777\) 9.31163i 0.334053i
\(778\) 33.9781i 1.21817i
\(779\) 43.0937 1.54399
\(780\) 0 0
\(781\) 6.21798 0.222497
\(782\) − 64.1789i − 2.29503i
\(783\) 6.00000i 0.214423i
\(784\) 29.0550 1.03768
\(785\) 0 0
\(786\) −15.9143 −0.567645
\(787\) − 13.2860i − 0.473595i −0.971559 0.236797i \(-0.923902\pi\)
0.971559 0.236797i \(-0.0760978\pi\)
\(788\) 40.1360i 1.42979i
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) −9.52119 −0.338535
\(792\) 6.54262i 0.232482i
\(793\) − 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 \(0.944328\pi\)
\(798\) 20.7862i 0.735825i
\(799\) 6.30742 0.223141
\(800\) 0 0
\(801\) 10.1751 0.359521
\(802\) 93.2087i 3.29131i
\(803\) − 14.3074i − 0.504898i
\(804\) 39.3288 1.38702
\(805\) 0 0
\(806\) 7.41454 0.261166
\(807\) 2.35027i 0.0827334i
\(808\) − 56.5939i − 1.99097i
\(809\) 9.91431 0.348569 0.174284 0.984695i \(-0.444239\pi\)
0.174284 + 0.984695i \(0.444239\pi\)
\(810\) 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
\(813\) 10.9786i 0.385036i
\(814\) −23.1793 −0.812436
\(815\) 0 0
\(816\) −32.2217 −1.12799
\(817\) − 69.4868i − 2.43103i
\(818\) − 34.8500i − 1.21850i
\(819\) 1.19656 0.0418111
\(820\) 0 0
\(821\) −34.4741 −1.20316 −0.601578 0.798814i \(-0.705461\pi\)
−0.601578 + 0.798814i \(0.705461\pi\)
\(822\) 41.6791i 1.45373i
\(823\) − 13.2566i − 0.462097i −0.972942 0.231048i \(-0.925784\pi\)
0.972942 0.231048i \(-0.0742155\pi\)
\(824\) 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 \(0.837206\pi\)
\(828\) − 17.5212i − 0.608904i
\(829\) −16.3418 −0.567576 −0.283788 0.958887i \(-0.591591\pi\)
−0.283788 + 0.958887i \(0.591591\pi\)
\(830\) 0 0
\(831\) −1.21377 −0.0421052
\(832\) − 5.32464i − 0.184599i
\(833\) 34.3847i 1.19136i
\(834\) −14.3931 −0.498393
\(835\) 0 0
\(836\) −35.0424 −1.21197
\(837\) 2.97858i 0.102955i
\(838\) 7.91431i 0.273395i
\(839\) 30.3675 1.04840 0.524201 0.851595i \(-0.324364\pi\)
0.524201 + 0.851595i \(0.324364\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 40.5939i 1.39896i
\(843\) 11.9572i 0.411826i
\(844\) −100.358 −3.45446
\(845\) 0 0
\(846\) 2.54262 0.0874169
\(847\) 11.4490i 0.393391i
\(848\) − 53.0937i − 1.82324i
\(849\) −29.8715 −1.02519
\(850\) 0 0
\(851\) 32.4910 1.11378
\(852\) 21.8077i 0.747118i
\(853\) 42.1407i 1.44287i 0.692482 + 0.721435i \(0.256517\pi\)
−0.692482 + 0.721435i \(0.743483\pi\)
\(854\) −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 \(-0.0118281\pi\)
\(858\) 2.97858i 0.101687i
\(859\) −18.5682 −0.633541 −0.316770 0.948502i \(-0.602598\pi\)
−0.316770 + 0.948502i \(0.602598\pi\)
\(860\) 0 0
\(861\) −7.38890 −0.251813
\(862\) − 11.4145i − 0.388781i
\(863\) − 33.7220i − 1.14791i −0.818887 0.573954i \(-0.805409\pi\)
0.818887 0.573954i \(-0.194591\pi\)
\(864\) −2.05333 −0.0698556
\(865\) 0 0
\(866\) 95.4649 3.24403
\(867\) − 21.1323i − 0.717690i
\(868\) − 14.9567i − 0.507663i
\(869\) −2.13229 −0.0723330
\(870\) 0 0
\(871\) 9.37169 0.317548
\(872\) − 45.8924i − 1.55411i
\(873\) 1.82487i 0.0617623i
\(874\) 72.5292 2.45334
\(875\) 0 0
\(876\) 50.1789 1.69539
\(877\) 43.4868i 1.46844i 0.678910 + 0.734222i \(0.262453\pi\)
−0.678910 + 0.734222i \(0.737547\pi\)
\(878\) 19.2650i 0.650164i
\(879\) 0.777809 0.0262348
\(880\) 0 0
\(881\) 26.7005 0.899564 0.449782 0.893138i \(-0.351502\pi\)
0.449782 + 0.893138i \(0.351502\pi\)
\(882\) 13.8610i 0.466724i
\(883\) − 20.2990i − 0.683116i −0.939861 0.341558i \(-0.889045\pi\)
0.939861 0.341558i \(-0.110955\pi\)
\(884\) −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 \(0.793051\pi\)
\(888\) − 42.5510i − 1.42792i
\(889\) −12.4360 −0.417089
\(890\) 0 0
\(891\) −1.19656 −0.0400862
\(892\) 11.0298i 0.369307i
\(893\) 7.12808i 0.238532i
\(894\) −38.2008 −1.27762
\(895\) 0 0
\(896\) −20.7737 −0.694000
\(897\) − 4.17513i − 0.139404i
\(898\) − 15.3717i − 0.512960i
\(899\) 17.8715 0.596047
\(900\) 0 0
\(901\) 62.8328 2.09326
\(902\) − 18.3931i − 0.612424i
\(903\) 11.9143i 0.396483i
\(904\) 43.5087 1.44708
\(905\) 0 0
\(906\) −21.3717 −0.710027
\(907\) − 7.26504i − 0.241232i −0.992699 0.120616i \(-0.961513\pi\)
0.992699 0.120616i \(-0.0384869\pi\)
\(908\) − 66.1579i − 2.19553i
\(909\) 10.3503 0.343297
\(910\) 0 0
\(911\) −6.65769 −0.220579 −0.110290 0.993899i \(-0.535178\pi\)
−0.110290 + 0.993899i \(0.535178\pi\)
\(912\) − 36.4141i − 1.20579i
\(913\) − 6.42754i − 0.212721i
\(914\) 3.45738 0.114360
\(915\) 0 0
\(916\) 36.6921 1.21234
\(917\) 7.64973i 0.252616i
\(918\) − 15.3717i − 0.507341i
\(919\) 27.1831 0.896688 0.448344 0.893861i \(-0.352014\pi\)
0.448344 + 0.893861i \(0.352014\pi\)
\(920\) 0 0
\(921\) −0.760597 −0.0250625
\(922\) − 70.9013i − 2.33501i
\(923\) 5.19656i 0.171047i
\(924\) 6.00842 0.197663
\(925\) 0 0
\(926\) −40.7643 −1.33960
\(927\) 18.7434i 0.615614i
\(928\) 12.3200i 0.404423i
\(929\) 15.3973 0.505170 0.252585 0.967575i \(-0.418719\pi\)
0.252585 + 0.967575i \(0.418719\pi\)
\(930\) 0 0
\(931\) −38.8585 −1.27353
\(932\) − 9.12808i − 0.299000i
\(933\) − 23.1281i − 0.757179i
\(934\) 64.0932 2.09719
\(935\) 0 0
\(936\) −5.46787 −0.178723
\(937\) 1.12808i 0.0368527i 0.999830 + 0.0184264i \(0.00586562\pi\)
−0.999830 + 0.0184264i \(0.994134\pi\)
\(938\) − 27.9143i − 0.911434i
\(939\) −33.9143 −1.10675
\(940\) 0 0
\(941\) −30.1407 −0.982559 −0.491280 0.871002i \(-0.663471\pi\)
−0.491280 + 0.871002i \(0.663471\pi\)
\(942\) 6.93573i 0.225978i
\(943\) 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 \(-0.105510\pi\)
\(948\) − 7.47835i − 0.242885i
\(949\) 11.9572 0.388146
\(950\) 0 0
\(951\) −9.64973 −0.312914
\(952\) 40.4015i 1.30942i
\(953\) 43.2259i 1.40023i 0.714032 + 0.700113i \(0.246867\pi\)
−0.714032 + 0.700113i \(0.753133\pi\)
\(954\) 25.3288 0.820052
\(955\) 0 0
\(956\) 11.7648 0.380501
\(957\) 7.17935i 0.232075i
\(958\) − 18.8929i − 0.610401i
\(959\) 20.0344 0.646945
\(960\) 0 0
\(961\) −22.1281 −0.713809
\(962\) − 19.3717i − 0.624568i
\(963\) 18.5682i 0.598353i
\(964\) 25.1793 0.810972
\(965\) 0 0
\(966\) −12.4360 −0.400120
\(967\) 57.6875i 1.85511i 0.373691 + 0.927553i \(0.378092\pi\)
−0.373691 + 0.927553i \(0.621908\pi\)
\(968\) − 52.3179i − 1.68156i
\(969\) 43.0937 1.38437
\(970\) 0 0
\(971\) −19.5296 −0.626735 −0.313368 0.949632i \(-0.601457\pi\)
−0.313368 + 0.949632i \(0.601457\pi\)
\(972\) − 4.19656i − 0.134605i
\(973\) 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 \(-0.223049\pi\)
\(978\) − 21.8077i − 0.697332i
\(979\) 12.1751 0.389119
\(980\) 0 0
\(981\) 8.39312 0.267972
\(982\) 0.213311i 0.00680702i
\(983\) − 32.2008i − 1.02705i −0.858076 0.513523i \(-0.828340\pi\)
0.858076 0.513523i \(-0.171660\pi\)
\(984\) 33.7648 1.07638
\(985\) 0 0
\(986\) −92.2302 −2.93721
\(987\) − 1.22219i − 0.0389028i
\(988\) − 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
\(993\) 15.3288i 0.486446i
\(994\) 15.4783 0.490943
\(995\) 0 0
\(996\) 22.5426 0.714290
\(997\) 35.1365i 1.11278i 0.830920 + 0.556392i \(0.187815\pi\)
−0.830920 + 0.556392i \(0.812185\pi\)
\(998\) − 44.1151i − 1.39644i
\(999\) 7.78202 0.246212
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 975.2.c.i.274.2 6
3.2 odd 2 2925.2.c.w.2224.5 6
5.2 odd 4 975.2.a.o.1.3 3
5.3 odd 4 195.2.a.e.1.1 3
5.4 even 2 inner 975.2.c.i.274.5 6
15.2 even 4 2925.2.a.bh.1.1 3
15.8 even 4 585.2.a.n.1.3 3
15.14 odd 2 2925.2.c.w.2224.2 6
20.3 even 4 3120.2.a.bj.1.2 3
35.13 even 4 9555.2.a.bq.1.1 3
60.23 odd 4 9360.2.a.dd.1.2 3
65.38 odd 4 2535.2.a.bc.1.3 3
195.38 even 4 7605.2.a.bx.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.1 3 5.3 odd 4
585.2.a.n.1.3 3 15.8 even 4
975.2.a.o.1.3 3 5.2 odd 4
975.2.c.i.274.2 6 1.1 even 1 trivial
975.2.c.i.274.5 6 5.4 even 2 inner
2535.2.a.bc.1.3 3 65.38 odd 4
2925.2.a.bh.1.1 3 15.2 even 4
2925.2.c.w.2224.2 6 15.14 odd 2
2925.2.c.w.2224.5 6 3.2 odd 2
3120.2.a.bj.1.2 3 20.3 even 4
7605.2.a.bx.1.1 3 195.38 even 4
9360.2.a.dd.1.2 3 60.23 odd 4
9555.2.a.bq.1.1 3 35.13 even 4