Properties

Label 5070.2.b.g.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.g.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} +1.00000 q^{22} +3.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} -1.00000 q^{29} -1.00000 q^{30} -3.00000i q^{31} +1.00000i q^{32} +1.00000i q^{33} -2.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +5.00000i q^{37} -6.00000 q^{38} -1.00000 q^{40} +10.0000i q^{41} -2.00000 q^{42} -5.00000 q^{43} +1.00000i q^{44} -1.00000i q^{45} +3.00000i q^{46} -3.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} +2.00000 q^{51} +14.0000 q^{53} -1.00000i q^{54} -1.00000 q^{55} -2.00000 q^{56} -6.00000i q^{57} -1.00000i q^{58} +5.00000i q^{59} -1.00000i q^{60} -10.0000 q^{61} +3.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} +2.00000 q^{68} -3.00000 q^{69} -2.00000i q^{70} +4.00000i q^{71} -1.00000i q^{72} +2.00000i q^{73} -5.00000 q^{74} +1.00000 q^{75} -6.00000i q^{76} -2.00000 q^{77} +5.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} -6.00000i q^{83} -2.00000i q^{84} +2.00000i q^{85} -5.00000i q^{86} +1.00000 q^{87} -1.00000 q^{88} -10.0000i q^{89} +1.00000 q^{90} -3.00000 q^{92} +3.00000i q^{93} +3.00000 q^{94} +6.00000 q^{95} -1.00000i q^{96} -10.0000i q^{97} +3.00000i q^{98} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + 2q^{10} + 2q^{12} + 4q^{14} + 2q^{16} - 4q^{17} + 2q^{22} + 6q^{23} - 2q^{25} - 2q^{27} - 2q^{29} - 2q^{30} - 4q^{35} - 2q^{36} - 12q^{38} - 2q^{40} - 4q^{42} - 10q^{43} - 2q^{48} + 6q^{49} + 4q^{51} + 28q^{53} - 2q^{55} - 4q^{56} - 20q^{61} + 6q^{62} - 2q^{64} - 2q^{66} + 4q^{68} - 6q^{69} - 10q^{74} + 2q^{75} - 4q^{77} + 10q^{79} + 2q^{81} - 20q^{82} + 2q^{87} - 2q^{88} + 2q^{90} - 6q^{92} + 6q^{94} + 12q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) − 1.00000i − 0.408248i
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) − 1.00000i − 0.301511i −0.988571 0.150756i \(-0.951829\pi\)
0.988571 0.150756i \(-0.0481707\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 2.00000i 0.436436i
\(22\) 1.00000 0.213201
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000i 0.377964i
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) −1.00000 −0.182574
\(31\) − 3.00000i − 0.538816i −0.963026 0.269408i \(-0.913172\pi\)
0.963026 0.269408i \(-0.0868280\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 1.00000i 0.174078i
\(34\) − 2.00000i − 0.342997i
\(35\) −2.00000 −0.338062
\(36\) −1.00000 −0.166667
\(37\) 5.00000i 0.821995i 0.911636 + 0.410997i \(0.134819\pi\)
−0.911636 + 0.410997i \(0.865181\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) −2.00000 −0.308607
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 1.00000i 0.150756i
\(45\) − 1.00000i − 0.149071i
\(46\) 3.00000i 0.442326i
\(47\) − 3.00000i − 0.437595i −0.975770 0.218797i \(-0.929787\pi\)
0.975770 0.218797i \(-0.0702134\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.00000 0.428571
\(50\) − 1.00000i − 0.141421i
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −1.00000 −0.134840
\(56\) −2.00000 −0.267261
\(57\) − 6.00000i − 0.794719i
\(58\) − 1.00000i − 0.131306i
\(59\) 5.00000i 0.650945i 0.945552 + 0.325472i \(0.105523\pi\)
−0.945552 + 0.325472i \(0.894477\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 3.00000 0.381000
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 2.00000 0.242536
\(69\) −3.00000 −0.361158
\(70\) − 2.00000i − 0.239046i
\(71\) 4.00000i 0.474713i 0.971423 + 0.237356i \(0.0762809\pi\)
−0.971423 + 0.237356i \(0.923719\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) −5.00000 −0.581238
\(75\) 1.00000 0.115470
\(76\) − 6.00000i − 0.688247i
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) − 2.00000i − 0.218218i
\(85\) 2.00000i 0.216930i
\(86\) − 5.00000i − 0.539164i
\(87\) 1.00000 0.107211
\(88\) −1.00000 −0.106600
\(89\) − 10.0000i − 1.06000i −0.847998 0.529999i \(-0.822192\pi\)
0.847998 0.529999i \(-0.177808\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 3.00000i 0.311086i
\(94\) 3.00000 0.309426
\(95\) 6.00000 0.615587
\(96\) − 1.00000i − 0.102062i
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 3.00000i 0.303046i
\(99\) − 1.00000i − 0.100504i
\(100\) 1.00000 0.100000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 2.00000i 0.198030i
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 14.0000i 1.35980i
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 6.00000i − 0.574696i −0.957826 0.287348i \(-0.907226\pi\)
0.957826 0.287348i \(-0.0927736\pi\)
\(110\) − 1.00000i − 0.0953463i
\(111\) − 5.00000i − 0.474579i
\(112\) − 2.00000i − 0.188982i
\(113\) 17.0000 1.59923 0.799613 0.600516i \(-0.205038\pi\)
0.799613 + 0.600516i \(0.205038\pi\)
\(114\) 6.00000 0.561951
\(115\) − 3.00000i − 0.279751i
\(116\) 1.00000 0.0928477
\(117\) 0 0
\(118\) −5.00000 −0.460287
\(119\) 4.00000i 0.366679i
\(120\) 1.00000 0.0912871
\(121\) 10.0000 0.909091
\(122\) − 10.0000i − 0.905357i
\(123\) − 10.0000i − 0.901670i
\(124\) 3.00000i 0.269408i
\(125\) 1.00000i 0.0894427i
\(126\) 2.00000 0.178174
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) 13.0000 1.13582 0.567908 0.823092i \(-0.307753\pi\)
0.567908 + 0.823092i \(0.307753\pi\)
\(132\) − 1.00000i − 0.0870388i
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 1.00000i 0.0860663i
\(136\) 2.00000i 0.171499i
\(137\) 9.00000i 0.768922i 0.923141 + 0.384461i \(0.125613\pi\)
−0.923141 + 0.384461i \(0.874387\pi\)
\(138\) − 3.00000i − 0.255377i
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 2.00000 0.169031
\(141\) 3.00000i 0.252646i
\(142\) −4.00000 −0.335673
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 1.00000i 0.0830455i
\(146\) −2.00000 −0.165521
\(147\) −3.00000 −0.247436
\(148\) − 5.00000i − 0.410997i
\(149\) 11.0000i 0.901155i 0.892737 + 0.450578i \(0.148782\pi\)
−0.892737 + 0.450578i \(0.851218\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) − 24.0000i − 1.95309i −0.215308 0.976546i \(-0.569076\pi\)
0.215308 0.976546i \(-0.430924\pi\)
\(152\) 6.00000 0.486664
\(153\) −2.00000 −0.161690
\(154\) − 2.00000i − 0.161165i
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) 25.0000 1.99522 0.997609 0.0691164i \(-0.0220180\pi\)
0.997609 + 0.0691164i \(0.0220180\pi\)
\(158\) 5.00000i 0.397779i
\(159\) −14.0000 −1.11027
\(160\) 1.00000 0.0790569
\(161\) − 6.00000i − 0.472866i
\(162\) 1.00000i 0.0785674i
\(163\) − 17.0000i − 1.33154i −0.746156 0.665771i \(-0.768103\pi\)
0.746156 0.665771i \(-0.231897\pi\)
\(164\) − 10.0000i − 0.780869i
\(165\) 1.00000 0.0778499
\(166\) 6.00000 0.465690
\(167\) 7.00000i 0.541676i 0.962625 + 0.270838i \(0.0873008\pi\)
−0.962625 + 0.270838i \(0.912699\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) −2.00000 −0.153393
\(171\) 6.00000i 0.458831i
\(172\) 5.00000 0.381246
\(173\) −4.00000 −0.304114 −0.152057 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 1.00000i 0.0758098i
\(175\) 2.00000i 0.151186i
\(176\) − 1.00000i − 0.0753778i
\(177\) − 5.00000i − 0.375823i
\(178\) 10.0000 0.749532
\(179\) 7.00000 0.523205 0.261602 0.965176i \(-0.415749\pi\)
0.261602 + 0.965176i \(0.415749\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) − 3.00000i − 0.221163i
\(185\) 5.00000 0.367607
\(186\) −3.00000 −0.219971
\(187\) 2.00000i 0.146254i
\(188\) 3.00000i 0.218797i
\(189\) 2.00000i 0.145479i
\(190\) 6.00000i 0.435286i
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 12.0000i − 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 1.00000 0.0710669
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 0 0
\(202\) − 14.0000i − 0.985037i
\(203\) 2.00000i 0.140372i
\(204\) −2.00000 −0.140028
\(205\) 10.0000 0.698430
\(206\) 6.00000i 0.418040i
\(207\) 3.00000 0.208514
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 2.00000i 0.138013i
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −14.0000 −0.961524
\(213\) − 4.00000i − 0.274075i
\(214\) − 6.00000i − 0.410152i
\(215\) 5.00000i 0.340997i
\(216\) 1.00000i 0.0680414i
\(217\) −6.00000 −0.407307
\(218\) 6.00000 0.406371
\(219\) − 2.00000i − 0.135147i
\(220\) 1.00000 0.0674200
\(221\) 0 0
\(222\) 5.00000 0.335578
\(223\) 10.0000i 0.669650i 0.942280 + 0.334825i \(0.108677\pi\)
−0.942280 + 0.334825i \(0.891323\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) 17.0000i 1.13082i
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 6.00000i 0.397360i
\(229\) − 22.0000i − 1.45380i −0.686743 0.726900i \(-0.740960\pi\)
0.686743 0.726900i \(-0.259040\pi\)
\(230\) 3.00000 0.197814
\(231\) 2.00000 0.131590
\(232\) 1.00000i 0.0656532i
\(233\) −3.00000 −0.196537 −0.0982683 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) − 5.00000i − 0.325472i
\(237\) −5.00000 −0.324785
\(238\) −4.00000 −0.259281
\(239\) − 8.00000i − 0.517477i −0.965947 0.258738i \(-0.916693\pi\)
0.965947 0.258738i \(-0.0833068\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 7.00000i 0.450910i 0.974254 + 0.225455i \(0.0723868\pi\)
−0.974254 + 0.225455i \(0.927613\pi\)
\(242\) 10.0000i 0.642824i
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) − 3.00000i − 0.191663i
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) −3.00000 −0.190500
\(249\) 6.00000i 0.380235i
\(250\) −1.00000 −0.0632456
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) 2.00000i 0.125988i
\(253\) − 3.00000i − 0.188608i
\(254\) 14.0000i 0.878438i
\(255\) − 2.00000i − 0.125245i
\(256\) 1.00000 0.0625000
\(257\) −21.0000 −1.30994 −0.654972 0.755653i \(-0.727320\pi\)
−0.654972 + 0.755653i \(0.727320\pi\)
\(258\) 5.00000i 0.311286i
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 13.0000i 0.803143i
\(263\) 23.0000 1.41824 0.709120 0.705087i \(-0.249092\pi\)
0.709120 + 0.705087i \(0.249092\pi\)
\(264\) 1.00000 0.0615457
\(265\) − 14.0000i − 0.860013i
\(266\) 12.0000i 0.735767i
\(267\) 10.0000i 0.611990i
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 29.0000i 1.76162i 0.473466 + 0.880812i \(0.343003\pi\)
−0.473466 + 0.880812i \(0.656997\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) 1.00000i 0.0603023i
\(276\) 3.00000 0.180579
\(277\) 19.0000 1.14160 0.570800 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) 10.0000i 0.599760i
\(279\) − 3.00000i − 0.179605i
\(280\) 2.00000i 0.119523i
\(281\) 30.0000i 1.78965i 0.446417 + 0.894825i \(0.352700\pi\)
−0.446417 + 0.894825i \(0.647300\pi\)
\(282\) −3.00000 −0.178647
\(283\) 13.0000 0.772770 0.386385 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) − 4.00000i − 0.237356i
\(285\) −6.00000 −0.355409
\(286\) 0 0
\(287\) 20.0000 1.18056
\(288\) 1.00000i 0.0589256i
\(289\) −13.0000 −0.764706
\(290\) −1.00000 −0.0587220
\(291\) 10.0000i 0.586210i
\(292\) − 2.00000i − 0.117041i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) − 3.00000i − 0.174964i
\(295\) 5.00000 0.291111
\(296\) 5.00000 0.290619
\(297\) 1.00000i 0.0580259i
\(298\) −11.0000 −0.637213
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 10.0000i 0.576390i
\(302\) 24.0000 1.38104
\(303\) 14.0000 0.804279
\(304\) 6.00000i 0.344124i
\(305\) 10.0000i 0.572598i
\(306\) − 2.00000i − 0.114332i
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 2.00000 0.113961
\(309\) −6.00000 −0.341328
\(310\) − 3.00000i − 0.170389i
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −12.0000 −0.678280 −0.339140 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(314\) 25.0000i 1.41083i
\(315\) −2.00000 −0.112687
\(316\) −5.00000 −0.281272
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) − 14.0000i − 0.785081i
\(319\) 1.00000i 0.0559893i
\(320\) 1.00000i 0.0559017i
\(321\) 6.00000 0.334887
\(322\) 6.00000 0.334367
\(323\) − 12.0000i − 0.667698i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 17.0000 0.941543
\(327\) 6.00000i 0.331801i
\(328\) 10.0000 0.552158
\(329\) −6.00000 −0.330791
\(330\) 1.00000i 0.0550482i
\(331\) 4.00000i 0.219860i 0.993939 + 0.109930i \(0.0350627\pi\)
−0.993939 + 0.109930i \(0.964937\pi\)
\(332\) 6.00000i 0.329293i
\(333\) 5.00000i 0.273998i
\(334\) −7.00000 −0.383023
\(335\) 0 0
\(336\) 2.00000i 0.109109i
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) −17.0000 −0.923313
\(340\) − 2.00000i − 0.108465i
\(341\) −3.00000 −0.162459
\(342\) −6.00000 −0.324443
\(343\) − 20.0000i − 1.07990i
\(344\) 5.00000i 0.269582i
\(345\) 3.00000i 0.161515i
\(346\) − 4.00000i − 0.215041i
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) −1.00000 −0.0536056
\(349\) − 8.00000i − 0.428230i −0.976808 0.214115i \(-0.931313\pi\)
0.976808 0.214115i \(-0.0686868\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 5.00000 0.265747
\(355\) 4.00000 0.212298
\(356\) 10.0000i 0.529999i
\(357\) − 4.00000i − 0.211702i
\(358\) 7.00000i 0.369961i
\(359\) − 12.0000i − 0.633336i −0.948536 0.316668i \(-0.897436\pi\)
0.948536 0.316668i \(-0.102564\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −17.0000 −0.894737
\(362\) − 16.0000i − 0.840941i
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 10.0000i 0.522708i
\(367\) 36.0000 1.87918 0.939592 0.342296i \(-0.111204\pi\)
0.939592 + 0.342296i \(0.111204\pi\)
\(368\) 3.00000 0.156386
\(369\) 10.0000i 0.520579i
\(370\) 5.00000i 0.259938i
\(371\) − 28.0000i − 1.45369i
\(372\) − 3.00000i − 0.155543i
\(373\) 37.0000 1.91579 0.957894 0.287123i \(-0.0926989\pi\)
0.957894 + 0.287123i \(0.0926989\pi\)
\(374\) −2.00000 −0.103418
\(375\) − 1.00000i − 0.0516398i
\(376\) −3.00000 −0.154713
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) − 30.0000i − 1.54100i −0.637442 0.770498i \(-0.720007\pi\)
0.637442 0.770498i \(-0.279993\pi\)
\(380\) −6.00000 −0.307794
\(381\) −14.0000 −0.717242
\(382\) 24.0000i 1.22795i
\(383\) − 27.0000i − 1.37964i −0.723983 0.689818i \(-0.757691\pi\)
0.723983 0.689818i \(-0.242309\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 2.00000i 0.101929i
\(386\) −4.00000 −0.203595
\(387\) −5.00000 −0.254164
\(388\) 10.0000i 0.507673i
\(389\) 1.00000 0.0507020 0.0253510 0.999679i \(-0.491930\pi\)
0.0253510 + 0.999679i \(0.491930\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) − 3.00000i − 0.151523i
\(393\) −13.0000 −0.655763
\(394\) 12.0000 0.604551
\(395\) − 5.00000i − 0.251577i
\(396\) 1.00000i 0.0502519i
\(397\) 13.0000i 0.652451i 0.945292 + 0.326226i \(0.105777\pi\)
−0.945292 + 0.326226i \(0.894223\pi\)
\(398\) 0 0
\(399\) −12.0000 −0.600751
\(400\) −1.00000 −0.0500000
\(401\) 12.0000i 0.599251i 0.954057 + 0.299626i \(0.0968618\pi\)
−0.954057 + 0.299626i \(0.903138\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) − 1.00000i − 0.0496904i
\(406\) −2.00000 −0.0992583
\(407\) 5.00000 0.247841
\(408\) − 2.00000i − 0.0990148i
\(409\) − 26.0000i − 1.28562i −0.766027 0.642809i \(-0.777769\pi\)
0.766027 0.642809i \(-0.222231\pi\)
\(410\) 10.0000i 0.493865i
\(411\) − 9.00000i − 0.443937i
\(412\) −6.00000 −0.295599
\(413\) 10.0000 0.492068
\(414\) 3.00000i 0.147442i
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) −10.0000 −0.489702
\(418\) 6.00000i 0.293470i
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 28.0000i 1.36464i 0.731055 + 0.682318i \(0.239028\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) − 8.00000i − 0.389434i
\(423\) − 3.00000i − 0.145865i
\(424\) − 14.0000i − 0.679900i
\(425\) 2.00000 0.0970143
\(426\) 4.00000 0.193801
\(427\) 20.0000i 0.967868i
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −5.00000 −0.241121
\(431\) 12.0000i 0.578020i 0.957326 + 0.289010i \(0.0933260\pi\)
−0.957326 + 0.289010i \(0.906674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) − 6.00000i − 0.288009i
\(435\) − 1.00000i − 0.0479463i
\(436\) 6.00000i 0.287348i
\(437\) 18.0000i 0.861057i
\(438\) 2.00000 0.0955637
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 1.00000i 0.0476731i
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 5.00000i 0.237289i
\(445\) −10.0000 −0.474045
\(446\) −10.0000 −0.473514
\(447\) − 11.0000i − 0.520282i
\(448\) 2.00000i 0.0944911i
\(449\) 36.0000i 1.69895i 0.527633 + 0.849473i \(0.323080\pi\)
−0.527633 + 0.849473i \(0.676920\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 10.0000 0.470882
\(452\) −17.0000 −0.799613
\(453\) 24.0000i 1.12762i
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) 14.0000i 0.654892i 0.944870 + 0.327446i \(0.106188\pi\)
−0.944870 + 0.327446i \(0.893812\pi\)
\(458\) 22.0000 1.02799
\(459\) 2.00000 0.0933520
\(460\) 3.00000i 0.139876i
\(461\) 27.0000i 1.25752i 0.777601 + 0.628758i \(0.216436\pi\)
−0.777601 + 0.628758i \(0.783564\pi\)
\(462\) 2.00000i 0.0930484i
\(463\) − 10.0000i − 0.464739i −0.972628 0.232370i \(-0.925352\pi\)
0.972628 0.232370i \(-0.0746479\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 3.00000 0.139122
\(466\) − 3.00000i − 0.138972i
\(467\) 2.00000 0.0925490 0.0462745 0.998929i \(-0.485265\pi\)
0.0462745 + 0.998929i \(0.485265\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 3.00000i − 0.138380i
\(471\) −25.0000 −1.15194
\(472\) 5.00000 0.230144
\(473\) 5.00000i 0.229900i
\(474\) − 5.00000i − 0.229658i
\(475\) − 6.00000i − 0.275299i
\(476\) − 4.00000i − 0.183340i
\(477\) 14.0000 0.641016
\(478\) 8.00000 0.365911
\(479\) 4.00000i 0.182765i 0.995816 + 0.0913823i \(0.0291285\pi\)
−0.995816 + 0.0913823i \(0.970871\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −7.00000 −0.318841
\(483\) 6.00000i 0.273009i
\(484\) −10.0000 −0.454545
\(485\) −10.0000 −0.454077
\(486\) − 1.00000i − 0.0453609i
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 10.0000i 0.452679i
\(489\) 17.0000i 0.768767i
\(490\) 3.00000 0.135526
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 10.0000i 0.450835i
\(493\) 2.00000 0.0900755
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) − 3.00000i − 0.134704i
\(497\) 8.00000 0.358849
\(498\) −6.00000 −0.268866
\(499\) − 40.0000i − 1.79065i −0.445418 0.895323i \(-0.646945\pi\)
0.445418 0.895323i \(-0.353055\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 7.00000i − 0.312737i
\(502\) − 3.00000i − 0.133897i
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 14.0000i 0.622992i
\(506\) 3.00000 0.133366
\(507\) 0 0
\(508\) −14.0000 −0.621150
\(509\) − 9.00000i − 0.398918i −0.979906 0.199459i \(-0.936082\pi\)
0.979906 0.199459i \(-0.0639185\pi\)
\(510\) 2.00000 0.0885615
\(511\) 4.00000 0.176950
\(512\) 1.00000i 0.0441942i
\(513\) − 6.00000i − 0.264906i
\(514\) − 21.0000i − 0.926270i
\(515\) − 6.00000i − 0.264392i
\(516\) −5.00000 −0.220113
\(517\) −3.00000 −0.131940
\(518\) 10.0000i 0.439375i
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) − 1.00000i − 0.0437688i
\(523\) 33.0000 1.44299 0.721495 0.692420i \(-0.243455\pi\)
0.721495 + 0.692420i \(0.243455\pi\)
\(524\) −13.0000 −0.567908
\(525\) − 2.00000i − 0.0872872i
\(526\) 23.0000i 1.00285i
\(527\) 6.00000i 0.261364i
\(528\) 1.00000i 0.0435194i
\(529\) −14.0000 −0.608696
\(530\) 14.0000 0.608121
\(531\) 5.00000i 0.216982i
\(532\) −12.0000 −0.520266
\(533\) 0 0
\(534\) −10.0000 −0.432742
\(535\) 6.00000i 0.259403i
\(536\) 0 0
\(537\) −7.00000 −0.302072
\(538\) 18.0000i 0.776035i
\(539\) − 3.00000i − 0.129219i
\(540\) − 1.00000i − 0.0430331i
\(541\) 8.00000i 0.343947i 0.985102 + 0.171973i \(0.0550143\pi\)
−0.985102 + 0.171973i \(0.944986\pi\)
\(542\) −29.0000 −1.24566
\(543\) 16.0000 0.686626
\(544\) − 2.00000i − 0.0857493i
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) − 9.00000i − 0.384461i
\(549\) −10.0000 −0.426790
\(550\) −1.00000 −0.0426401
\(551\) − 6.00000i − 0.255609i
\(552\) 3.00000i 0.127688i
\(553\) − 10.0000i − 0.425243i
\(554\) 19.0000i 0.807233i
\(555\) −5.00000 −0.212238
\(556\) −10.0000 −0.424094
\(557\) 34.0000i 1.44063i 0.693649 + 0.720313i \(0.256002\pi\)
−0.693649 + 0.720313i \(0.743998\pi\)
\(558\) 3.00000 0.127000
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) − 2.00000i − 0.0844401i
\(562\) −30.0000 −1.26547
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) − 3.00000i − 0.126323i
\(565\) − 17.0000i − 0.715195i
\(566\) 13.0000i 0.546431i
\(567\) − 2.00000i − 0.0839921i
\(568\) 4.00000 0.167836
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) − 6.00000i − 0.251312i
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 20.0000i 0.834784i
\(575\) −3.00000 −0.125109
\(576\) −1.00000 −0.0416667
\(577\) − 18.0000i − 0.749350i −0.927156 0.374675i \(-0.877754\pi\)
0.927156 0.374675i \(-0.122246\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) − 4.00000i − 0.166234i
\(580\) − 1.00000i − 0.0415227i
\(581\) −12.0000 −0.497844
\(582\) −10.0000 −0.414513
\(583\) − 14.0000i − 0.579821i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) − 18.0000i − 0.742940i −0.928445 0.371470i \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) 3.00000 0.123718
\(589\) 18.0000 0.741677
\(590\) 5.00000i 0.205847i
\(591\) 12.0000i 0.493614i
\(592\) 5.00000i 0.205499i
\(593\) 35.0000i 1.43728i 0.695383 + 0.718639i \(0.255235\pi\)
−0.695383 + 0.718639i \(0.744765\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 4.00000 0.163984
\(596\) − 11.0000i − 0.450578i
\(597\) 0 0
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −11.0000 −0.448699 −0.224350 0.974509i \(-0.572026\pi\)
−0.224350 + 0.974509i \(0.572026\pi\)
\(602\) −10.0000 −0.407570
\(603\) 0 0
\(604\) 24.0000i 0.976546i
\(605\) − 10.0000i − 0.406558i
\(606\) 14.0000i 0.568711i
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) −6.00000 −0.243332
\(609\) − 2.00000i − 0.0810441i
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) − 37.0000i − 1.49442i −0.664590 0.747208i \(-0.731394\pi\)
0.664590 0.747208i \(-0.268606\pi\)
\(614\) 0 0
\(615\) −10.0000 −0.403239
\(616\) 2.00000i 0.0805823i
\(617\) 3.00000i 0.120775i 0.998175 + 0.0603877i \(0.0192337\pi\)
−0.998175 + 0.0603877i \(0.980766\pi\)
\(618\) − 6.00000i − 0.241355i
\(619\) 10.0000i 0.401934i 0.979598 + 0.200967i \(0.0644084\pi\)
−0.979598 + 0.200967i \(0.935592\pi\)
\(620\) 3.00000 0.120483
\(621\) −3.00000 −0.120386
\(622\) 0 0
\(623\) −20.0000 −0.801283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 12.0000i − 0.479616i
\(627\) −6.00000 −0.239617
\(628\) −25.0000 −0.997609
\(629\) − 10.0000i − 0.398726i
\(630\) − 2.00000i − 0.0796819i
\(631\) − 48.0000i − 1.91085i −0.295234 0.955425i \(-0.595398\pi\)
0.295234 0.955425i \(-0.404602\pi\)
\(632\) − 5.00000i − 0.198889i
\(633\) 8.00000 0.317971
\(634\) −12.0000 −0.476581
\(635\) − 14.0000i − 0.555573i
\(636\) 14.0000 0.555136
\(637\) 0 0
\(638\) −1.00000 −0.0395904
\(639\) 4.00000i 0.158238i
\(640\) −1.00000 −0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 6.00000i 0.236801i
\(643\) − 40.0000i − 1.57745i −0.614749 0.788723i \(-0.710743\pi\)
0.614749 0.788723i \(-0.289257\pi\)
\(644\) 6.00000i 0.236433i
\(645\) − 5.00000i − 0.196875i
\(646\) 12.0000 0.472134
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 5.00000 0.196267
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 17.0000i 0.665771i
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −6.00000 −0.234619
\(655\) − 13.0000i − 0.507952i
\(656\) 10.0000i 0.390434i
\(657\) 2.00000i 0.0780274i
\(658\) − 6.00000i − 0.233904i
\(659\) −13.0000 −0.506408 −0.253204 0.967413i \(-0.581484\pi\)
−0.253204 + 0.967413i \(0.581484\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 12.0000i 0.466746i 0.972387 + 0.233373i \(0.0749763\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) − 12.0000i − 0.465340i
\(666\) −5.00000 −0.193746
\(667\) −3.00000 −0.116160
\(668\) − 7.00000i − 0.270838i
\(669\) − 10.0000i − 0.386622i
\(670\) 0 0
\(671\) 10.0000i 0.386046i
\(672\) −2.00000 −0.0771517
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) 22.0000i 0.847408i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) − 17.0000i − 0.652881i
\(679\) −20.0000 −0.767530
\(680\) 2.00000 0.0766965
\(681\) − 20.0000i − 0.766402i
\(682\) − 3.00000i − 0.114876i
\(683\) 44.0000i 1.68361i 0.539779 + 0.841807i \(0.318508\pi\)
−0.539779 + 0.841807i \(0.681492\pi\)
\(684\) − 6.00000i − 0.229416i
\(685\) 9.00000 0.343872
\(686\) 20.0000 0.763604
\(687\) 22.0000i 0.839352i
\(688\) −5.00000 −0.190623
\(689\) 0 0
\(690\) −3.00000 −0.114208
\(691\) − 14.0000i − 0.532585i −0.963892 0.266293i \(-0.914201\pi\)
0.963892 0.266293i \(-0.0857987\pi\)
\(692\) 4.00000 0.152057
\(693\) −2.00000 −0.0759737
\(694\) − 18.0000i − 0.683271i
\(695\) − 10.0000i − 0.379322i
\(696\) − 1.00000i − 0.0379049i
\(697\) − 20.0000i − 0.757554i
\(698\) 8.00000 0.302804
\(699\) 3.00000 0.113470
\(700\) − 2.00000i − 0.0755929i
\(701\) −23.0000 −0.868698 −0.434349 0.900745i \(-0.643022\pi\)
−0.434349 + 0.900745i \(0.643022\pi\)
\(702\) 0 0
\(703\) −30.0000 −1.13147
\(704\) 1.00000i 0.0376889i
\(705\) 3.00000 0.112987
\(706\) 14.0000 0.526897
\(707\) 28.0000i 1.05305i
\(708\) 5.00000i 0.187912i
\(709\) 16.0000i 0.600893i 0.953799 + 0.300446i \(0.0971356\pi\)
−0.953799 + 0.300446i \(0.902864\pi\)
\(710\) 4.00000i 0.150117i
\(711\) 5.00000 0.187515
\(712\) −10.0000 −0.374766
\(713\) − 9.00000i − 0.337053i
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) −7.00000 −0.261602
\(717\) 8.00000i 0.298765i
\(718\) 12.0000 0.447836
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) − 12.0000i − 0.446903i
\(722\) − 17.0000i − 0.632674i
\(723\) − 7.00000i − 0.260333i
\(724\) 16.0000 0.594635
\(725\) 1.00000 0.0371391
\(726\) − 10.0000i − 0.371135i
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.00000i 0.0740233i
\(731\) 10.0000 0.369863
\(732\) −10.0000 −0.369611
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) 36.0000i 1.32878i
\(735\) 3.00000i 0.110657i
\(736\) 3.00000i 0.110581i
\(737\) 0 0
\(738\) −10.0000 −0.368105
\(739\) − 24.0000i − 0.882854i −0.897297 0.441427i \(-0.854472\pi\)
0.897297 0.441427i \(-0.145528\pi\)
\(740\) −5.00000 −0.183804
\(741\) 0 0
\(742\) 28.0000 1.02791
\(743\) − 15.0000i − 0.550297i −0.961402 0.275148i \(-0.911273\pi\)
0.961402 0.275148i \(-0.0887270\pi\)
\(744\) 3.00000 0.109985
\(745\) 11.0000 0.403009
\(746\) 37.0000i 1.35467i
\(747\) − 6.00000i − 0.219529i
\(748\) − 2.00000i − 0.0731272i
\(749\) 12.0000i 0.438470i
\(750\) 1.00000 0.0365148
\(751\) −41.0000 −1.49611 −0.748056 0.663636i \(-0.769012\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(752\) − 3.00000i − 0.109399i
\(753\) 3.00000 0.109326
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) − 2.00000i − 0.0727393i
\(757\) −54.0000 −1.96266 −0.981332 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) 30.0000 1.08965
\(759\) 3.00000i 0.108893i
\(760\) − 6.00000i − 0.217643i
\(761\) 20.0000i 0.724999i 0.931984 + 0.362500i \(0.118077\pi\)
−0.931984 + 0.362500i \(0.881923\pi\)
\(762\) − 14.0000i − 0.507166i
\(763\) −12.0000 −0.434429
\(764\) −24.0000 −0.868290
\(765\) 2.00000i 0.0723102i
\(766\) 27.0000 0.975550
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) − 29.0000i − 1.04577i −0.852404 0.522883i \(-0.824856\pi\)
0.852404 0.522883i \(-0.175144\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 21.0000 0.756297
\(772\) − 4.00000i − 0.143963i
\(773\) − 16.0000i − 0.575480i −0.957709 0.287740i \(-0.907096\pi\)
0.957709 0.287740i \(-0.0929039\pi\)
\(774\) − 5.00000i − 0.179721i
\(775\) 3.00000i 0.107763i
\(776\) −10.0000 −0.358979
\(777\) −10.0000 −0.358748
\(778\) 1.00000i 0.0358517i
\(779\) −60.0000 −2.14972
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) − 6.00000i − 0.214560i
\(783\) 1.00000 0.0357371
\(784\) 3.00000 0.107143
\(785\) − 25.0000i − 0.892288i
\(786\) − 13.0000i − 0.463695i
\(787\) − 25.0000i − 0.891154i −0.895244 0.445577i \(-0.852999\pi\)
0.895244 0.445577i \(-0.147001\pi\)
\(788\) 12.0000i 0.427482i
\(789\) −23.0000 −0.818822
\(790\) 5.00000 0.177892
\(791\) − 34.0000i − 1.20890i
\(792\) −1.00000 −0.0355335
\(793\) 0 0
\(794\) −13.0000 −0.461353
\(795\) 14.0000i 0.496529i
\(796\) 0 0
\(797\) 52.0000 1.84193 0.920967 0.389640i \(-0.127401\pi\)
0.920967 + 0.389640i \(0.127401\pi\)
\(798\) − 12.0000i − 0.424795i
\(799\) 6.00000i 0.212265i
\(800\) − 1.00000i − 0.0353553i
\(801\) − 10.0000i − 0.353333i
\(802\) −12.0000 −0.423735
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) −6.00000 −0.211472
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) 14.0000i 0.492518i
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 1.00000 0.0351364
\(811\) 52.0000i 1.82597i 0.407997 + 0.912983i \(0.366228\pi\)
−0.407997 + 0.912983i \(0.633772\pi\)
\(812\) − 2.00000i − 0.0701862i
\(813\) − 29.0000i − 1.01707i
\(814\) 5.00000i 0.175250i
\(815\) −17.0000 −0.595484
\(816\) 2.00000 0.0700140
\(817\) − 30.0000i − 1.04957i
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) 23.0000i 0.802706i 0.915924 + 0.401353i \(0.131460\pi\)
−0.915924 + 0.401353i \(0.868540\pi\)
\(822\) 9.00000 0.313911
\(823\) 22.0000 0.766872 0.383436 0.923567i \(-0.374741\pi\)
0.383436 + 0.923567i \(0.374741\pi\)
\(824\) − 6.00000i − 0.209020i
\(825\) − 1.00000i − 0.0348155i
\(826\) 10.0000i 0.347945i
\(827\) 18.0000i 0.625921i 0.949766 + 0.312961i \(0.101321\pi\)
−0.949766 + 0.312961i \(0.898679\pi\)
\(828\) −3.00000 −0.104257
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) − 6.00000i − 0.208263i
\(831\) −19.0000 −0.659103
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) − 10.0000i − 0.346272i
\(835\) 7.00000 0.242245
\(836\) −6.00000 −0.207514
\(837\) 3.00000i 0.103695i
\(838\) 4.00000i 0.138178i
\(839\) − 38.0000i − 1.31191i −0.754802 0.655953i \(-0.772267\pi\)
0.754802 0.655953i \(-0.227733\pi\)
\(840\) − 2.00000i − 0.0690066i
\(841\) −28.0000 −0.965517
\(842\) −28.0000 −0.964944
\(843\) − 30.0000i − 1.03325i
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) − 20.0000i − 0.687208i
\(848\) 14.0000 0.480762
\(849\) −13.0000 −0.446159
\(850\) 2.00000i 0.0685994i
\(851\) 15.0000i 0.514193i
\(852\) 4.00000i 0.137038i
\(853\) − 7.00000i − 0.239675i −0.992793 0.119838i \(-0.961763\pi\)
0.992793 0.119838i \(-0.0382374\pi\)
\(854\) −20.0000 −0.684386
\(855\) 6.00000 0.205196
\(856\) 6.00000i 0.205076i
\(857\) 35.0000 1.19558 0.597789 0.801654i \(-0.296046\pi\)
0.597789 + 0.801654i \(0.296046\pi\)
\(858\) 0 0
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) − 5.00000i − 0.170499i
\(861\) −20.0000 −0.681598
\(862\) −12.0000 −0.408722
\(863\) − 51.0000i − 1.73606i −0.496512 0.868030i \(-0.665386\pi\)
0.496512 0.868030i \(-0.334614\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 4.00000i 0.136004i
\(866\) − 16.0000i − 0.543702i
\(867\) 13.0000 0.441503
\(868\) 6.00000 0.203653
\(869\) − 5.00000i − 0.169613i
\(870\) 1.00000 0.0339032
\(871\) 0 0
\(872\) −6.00000 −0.203186
\(873\) − 10.0000i − 0.338449i
\(874\) −18.0000 −0.608859
\(875\) 2.00000 0.0676123
\(876\) 2.00000i 0.0675737i
\(877\) 13.0000i 0.438979i 0.975615 + 0.219489i \(0.0704391\pi\)
−0.975615 + 0.219489i \(0.929561\pi\)
\(878\) − 28.0000i − 0.944954i
\(879\) − 14.0000i − 0.472208i
\(880\) −1.00000 −0.0337100
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 1.00000 0.0336527 0.0168263 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) 0 0
\(885\) −5.00000 −0.168073
\(886\) 16.0000i 0.537531i
\(887\) 1.00000 0.0335767 0.0167884 0.999859i \(-0.494656\pi\)
0.0167884 + 0.999859i \(0.494656\pi\)
\(888\) −5.00000 −0.167789
\(889\) − 28.0000i − 0.939090i
\(890\) − 10.0000i − 0.335201i
\(891\) − 1.00000i − 0.0335013i
\(892\) − 10.0000i − 0.334825i
\(893\) 18.0000 0.602347
\(894\) 11.0000 0.367895
\(895\) − 7.00000i − 0.233984i
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) 3.00000i 0.100056i
\(900\) 1.00000 0.0333333
\(901\) −28.0000 −0.932815
\(902\) 10.0000i 0.332964i
\(903\) − 10.0000i − 0.332779i
\(904\) − 17.0000i − 0.565412i
\(905\) 16.0000i 0.531858i
\(906\) −24.0000 −0.797347
\(907\) 53.0000 1.75984 0.879918 0.475125i \(-0.157597\pi\)
0.879918 + 0.475125i \(0.157597\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 34.0000 1.12647 0.563235 0.826297i \(-0.309557\pi\)
0.563235 + 0.826297i \(0.309557\pi\)
\(912\) − 6.00000i − 0.198680i
\(913\) −6.00000 −0.198571
\(914\) −14.0000 −0.463079
\(915\) − 10.0000i − 0.330590i
\(916\) 22.0000i 0.726900i
\(917\) − 26.0000i − 0.858596i
\(918\) 2.00000i 0.0660098i
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 0 0
\(922\) −27.0000 −0.889198
\(923\) 0 0
\(924\) −2.00000 −0.0657952
\(925\) − 5.00000i − 0.164399i
\(926\) 10.0000 0.328620
\(927\) 6.00000 0.197066
\(928\) − 1.00000i − 0.0328266i
\(929\) 24.0000i 0.787414i 0.919236 + 0.393707i \(0.128808\pi\)
−0.919236 + 0.393707i \(0.871192\pi\)
\(930\) 3.00000i 0.0983739i
\(931\) 18.0000i 0.589926i
\(932\) 3.00000 0.0982683
\(933\) 0 0
\(934\) 2.00000i 0.0654420i
\(935\) 2.00000 0.0654070
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) 12.0000 0.391605
\(940\) 3.00000 0.0978492
\(941\) 26.0000i 0.847576i 0.905761 + 0.423788i \(0.139300\pi\)
−0.905761 + 0.423788i \(0.860700\pi\)
\(942\) − 25.0000i − 0.814544i
\(943\) 30.0000i 0.976934i
\(944\) 5.00000i 0.162736i
\(945\) 2.00000 0.0650600
\(946\) −5.00000 −0.162564
\(947\) − 52.0000i − 1.68977i −0.534946 0.844886i \(-0.679668\pi\)
0.534946 0.844886i \(-0.320332\pi\)
\(948\) 5.00000 0.162392
\(949\) 0 0
\(950\) 6.00000 0.194666
\(951\) − 12.0000i − 0.389127i
\(952\) 4.00000 0.129641
\(953\) −15.0000 −0.485898 −0.242949 0.970039i \(-0.578115\pi\)
−0.242949 + 0.970039i \(0.578115\pi\)
\(954\) 14.0000i 0.453267i
\(955\) − 24.0000i − 0.776622i
\(956\) 8.00000i 0.258738i
\(957\) − 1.00000i − 0.0323254i
\(958\) −4.00000 −0.129234
\(959\) 18.0000 0.581250
\(960\) − 1.00000i − 0.0322749i
\(961\) 22.0000 0.709677
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) − 7.00000i − 0.225455i
\(965\) 4.00000 0.128765
\(966\) −6.00000 −0.193047
\(967\) 16.0000i 0.514525i 0.966342 + 0.257263i \(0.0828206\pi\)
−0.966342 + 0.257263i \(0.917179\pi\)
\(968\) − 10.0000i − 0.321412i
\(969\) 12.0000i 0.385496i
\(970\) − 10.0000i − 0.321081i
\(971\) −40.0000 −1.28366 −0.641831 0.766846i \(-0.721825\pi\)
−0.641831 + 0.766846i \(0.721825\pi\)
\(972\) 1.00000 0.0320750
\(973\) − 20.0000i − 0.641171i
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) − 39.0000i − 1.24772i −0.781536 0.623860i \(-0.785563\pi\)
0.781536 0.623860i \(-0.214437\pi\)
\(978\) −17.0000 −0.543600
\(979\) −10.0000 −0.319601
\(980\) 3.00000i 0.0958315i
\(981\) − 6.00000i − 0.191565i
\(982\) − 24.0000i − 0.765871i
\(983\) 25.0000i 0.797376i 0.917087 + 0.398688i \(0.130534\pi\)
−0.917087 + 0.398688i \(0.869466\pi\)
\(984\) −10.0000 −0.318788
\(985\) −12.0000 −0.382352
\(986\) 2.00000i 0.0636930i
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) −15.0000 −0.476972
\(990\) − 1.00000i − 0.0317821i
\(991\) 39.0000 1.23888 0.619438 0.785046i \(-0.287361\pi\)
0.619438 + 0.785046i \(0.287361\pi\)
\(992\) 3.00000 0.0952501
\(993\) − 4.00000i − 0.126936i
\(994\) 8.00000i 0.253745i
\(995\) 0 0
\(996\) − 6.00000i − 0.190117i
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 40.0000 1.26618
\(999\) − 5.00000i − 0.158193i
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 5070.2.b.g.1351.2 2
13.2 odd 12 390.2.i.a.61.1 2
13.5 odd 4 5070.2.a.o.1.1 1
13.6 odd 12 390.2.i.a.211.1 yes 2
13.8 odd 4 5070.2.a.f.1.1 1
13.12 even 2 inner 5070.2.b.g.1351.1 2
39.2 even 12 1170.2.i.k.451.1 2
39.32 even 12 1170.2.i.k.991.1 2
65.2 even 12 1950.2.z.h.1699.1 4
65.19 odd 12 1950.2.i.s.601.1 2
65.28 even 12 1950.2.z.h.1699.2 4
65.32 even 12 1950.2.z.h.1849.2 4
65.54 odd 12 1950.2.i.s.451.1 2
65.58 even 12 1950.2.z.h.1849.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.a.61.1 2 13.2 odd 12
390.2.i.a.211.1 yes 2 13.6 odd 12
1170.2.i.k.451.1 2 39.2 even 12
1170.2.i.k.991.1 2 39.32 even 12
1950.2.i.s.451.1 2 65.54 odd 12
1950.2.i.s.601.1 2 65.19 odd 12
1950.2.z.h.1699.1 4 65.2 even 12
1950.2.z.h.1699.2 4 65.28 even 12
1950.2.z.h.1849.1 4 65.58 even 12
1950.2.z.h.1849.2 4 65.32 even 12
5070.2.a.f.1.1 1 13.8 odd 4
5070.2.a.o.1.1 1 13.5 odd 4
5070.2.b.g.1351.1 2 13.12 even 2 inner
5070.2.b.g.1351.2 2 1.1 even 1 trivial