Properties

Label 350.4.c.a.99.1
Level $350$
Weight $4$
Character 350.99
Analytic conductor $20.651$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [350,4,Mod(99,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.99"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-8,0,-40,0,0,-146,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 350.99
Dual form 350.4.c.a.99.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -10.0000i q^{3} -4.00000 q^{4} -20.0000 q^{6} +7.00000i q^{7} +8.00000i q^{8} -73.0000 q^{9} +9.00000 q^{11} +40.0000i q^{12} +52.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} +96.0000i q^{17} +146.000i q^{18} +10.0000 q^{19} +70.0000 q^{21} -18.0000i q^{22} -75.0000i q^{23} +80.0000 q^{24} +104.000 q^{26} +460.000i q^{27} -28.0000i q^{28} -189.000 q^{29} -232.000 q^{31} -32.0000i q^{32} -90.0000i q^{33} +192.000 q^{34} +292.000 q^{36} +305.000i q^{37} -20.0000i q^{38} +520.000 q^{39} -438.000 q^{41} -140.000i q^{42} -353.000i q^{43} -36.0000 q^{44} -150.000 q^{46} -486.000i q^{47} -160.000i q^{48} -49.0000 q^{49} +960.000 q^{51} -208.000i q^{52} +354.000i q^{53} +920.000 q^{54} -56.0000 q^{56} -100.000i q^{57} +378.000i q^{58} +672.000 q^{59} +206.000 q^{61} +464.000i q^{62} -511.000i q^{63} -64.0000 q^{64} -180.000 q^{66} +599.000i q^{67} -384.000i q^{68} -750.000 q^{69} -471.000 q^{71} -584.000i q^{72} -614.000i q^{73} +610.000 q^{74} -40.0000 q^{76} +63.0000i q^{77} -1040.00i q^{78} -743.000 q^{79} +2629.00 q^{81} +876.000i q^{82} -996.000i q^{83} -280.000 q^{84} -706.000 q^{86} +1890.00i q^{87} +72.0000i q^{88} -180.000 q^{89} -364.000 q^{91} +300.000i q^{92} +2320.00i q^{93} -972.000 q^{94} -320.000 q^{96} -184.000i q^{97} +98.0000i q^{98} -657.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 40 q^{6} - 146 q^{9} + 18 q^{11} + 28 q^{14} + 32 q^{16} + 20 q^{19} + 140 q^{21} + 160 q^{24} + 208 q^{26} - 378 q^{29} - 464 q^{31} + 384 q^{34} + 584 q^{36} + 1040 q^{39} - 876 q^{41}+ \cdots - 1314 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) − 10.0000i − 1.92450i −0.272166 0.962250i \(-0.587740\pi\)
0.272166 0.962250i \(-0.412260\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −20.0000 −1.36083
\(7\) 7.00000i 0.377964i
\(8\) 8.00000i 0.353553i
\(9\) −73.0000 −2.70370
\(10\) 0 0
\(11\) 9.00000 0.246691 0.123346 0.992364i \(-0.460638\pi\)
0.123346 + 0.992364i \(0.460638\pi\)
\(12\) 40.0000i 0.962250i
\(13\) 52.0000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 14.0000 0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 96.0000i 1.36961i 0.728725 + 0.684806i \(0.240113\pi\)
−0.728725 + 0.684806i \(0.759887\pi\)
\(18\) 146.000i 1.91181i
\(19\) 10.0000 0.120745 0.0603726 0.998176i \(-0.480771\pi\)
0.0603726 + 0.998176i \(0.480771\pi\)
\(20\) 0 0
\(21\) 70.0000 0.727393
\(22\) − 18.0000i − 0.174437i
\(23\) − 75.0000i − 0.679938i −0.940437 0.339969i \(-0.889583\pi\)
0.940437 0.339969i \(-0.110417\pi\)
\(24\) 80.0000 0.680414
\(25\) 0 0
\(26\) 104.000 0.784465
\(27\) 460.000i 3.27878i
\(28\) − 28.0000i − 0.188982i
\(29\) −189.000 −1.21022 −0.605111 0.796141i \(-0.706871\pi\)
−0.605111 + 0.796141i \(0.706871\pi\)
\(30\) 0 0
\(31\) −232.000 −1.34414 −0.672071 0.740486i \(-0.734595\pi\)
−0.672071 + 0.740486i \(0.734595\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) − 90.0000i − 0.474757i
\(34\) 192.000 0.968463
\(35\) 0 0
\(36\) 292.000 1.35185
\(37\) 305.000i 1.35518i 0.735439 + 0.677590i \(0.236976\pi\)
−0.735439 + 0.677590i \(0.763024\pi\)
\(38\) − 20.0000i − 0.0853797i
\(39\) 520.000 2.13504
\(40\) 0 0
\(41\) −438.000 −1.66839 −0.834196 0.551467i \(-0.814068\pi\)
−0.834196 + 0.551467i \(0.814068\pi\)
\(42\) − 140.000i − 0.514344i
\(43\) − 353.000i − 1.25191i −0.779860 0.625953i \(-0.784710\pi\)
0.779860 0.625953i \(-0.215290\pi\)
\(44\) −36.0000 −0.123346
\(45\) 0 0
\(46\) −150.000 −0.480789
\(47\) − 486.000i − 1.50831i −0.656699 0.754153i \(-0.728048\pi\)
0.656699 0.754153i \(-0.271952\pi\)
\(48\) − 160.000i − 0.481125i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 960.000 2.63582
\(52\) − 208.000i − 0.554700i
\(53\) 354.000i 0.917465i 0.888574 + 0.458732i \(0.151696\pi\)
−0.888574 + 0.458732i \(0.848304\pi\)
\(54\) 920.000 2.31845
\(55\) 0 0
\(56\) −56.0000 −0.133631
\(57\) − 100.000i − 0.232374i
\(58\) 378.000i 0.855756i
\(59\) 672.000 1.48283 0.741415 0.671047i \(-0.234155\pi\)
0.741415 + 0.671047i \(0.234155\pi\)
\(60\) 0 0
\(61\) 206.000 0.432387 0.216193 0.976351i \(-0.430636\pi\)
0.216193 + 0.976351i \(0.430636\pi\)
\(62\) 464.000i 0.950453i
\(63\) − 511.000i − 1.02190i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −180.000 −0.335704
\(67\) 599.000i 1.09223i 0.837710 + 0.546116i \(0.183894\pi\)
−0.837710 + 0.546116i \(0.816106\pi\)
\(68\) − 384.000i − 0.684806i
\(69\) −750.000 −1.30854
\(70\) 0 0
\(71\) −471.000 −0.787288 −0.393644 0.919263i \(-0.628786\pi\)
−0.393644 + 0.919263i \(0.628786\pi\)
\(72\) − 584.000i − 0.955904i
\(73\) − 614.000i − 0.984428i −0.870474 0.492214i \(-0.836188\pi\)
0.870474 0.492214i \(-0.163812\pi\)
\(74\) 610.000 0.958258
\(75\) 0 0
\(76\) −40.0000 −0.0603726
\(77\) 63.0000i 0.0932405i
\(78\) − 1040.00i − 1.50970i
\(79\) −743.000 −1.05815 −0.529076 0.848574i \(-0.677461\pi\)
−0.529076 + 0.848574i \(0.677461\pi\)
\(80\) 0 0
\(81\) 2629.00 3.60631
\(82\) 876.000i 1.17973i
\(83\) − 996.000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) −280.000 −0.363696
\(85\) 0 0
\(86\) −706.000 −0.885232
\(87\) 1890.00i 2.32907i
\(88\) 72.0000i 0.0872185i
\(89\) −180.000 −0.214382 −0.107191 0.994238i \(-0.534186\pi\)
−0.107191 + 0.994238i \(0.534186\pi\)
\(90\) 0 0
\(91\) −364.000 −0.419314
\(92\) 300.000i 0.339969i
\(93\) 2320.00i 2.58680i
\(94\) −972.000 −1.06653
\(95\) 0 0
\(96\) −320.000 −0.340207
\(97\) − 184.000i − 0.192602i −0.995352 0.0963009i \(-0.969299\pi\)
0.995352 0.0963009i \(-0.0307011\pi\)
\(98\) 98.0000i 0.101015i
\(99\) −657.000 −0.666980
\(100\) 0 0
\(101\) −726.000 −0.715245 −0.357622 0.933866i \(-0.616412\pi\)
−0.357622 + 0.933866i \(0.616412\pi\)
\(102\) − 1920.00i − 1.86381i
\(103\) 1798.00i 1.72002i 0.510276 + 0.860011i \(0.329543\pi\)
−0.510276 + 0.860011i \(0.670457\pi\)
\(104\) −416.000 −0.392232
\(105\) 0 0
\(106\) 708.000 0.648746
\(107\) 876.000i 0.791459i 0.918367 + 0.395730i \(0.129508\pi\)
−0.918367 + 0.395730i \(0.870492\pi\)
\(108\) − 1840.00i − 1.63939i
\(109\) 691.000 0.607209 0.303605 0.952798i \(-0.401810\pi\)
0.303605 + 0.952798i \(0.401810\pi\)
\(110\) 0 0
\(111\) 3050.00 2.60805
\(112\) 112.000i 0.0944911i
\(113\) 1521.00i 1.26623i 0.774059 + 0.633113i \(0.218223\pi\)
−0.774059 + 0.633113i \(0.781777\pi\)
\(114\) −200.000 −0.164313
\(115\) 0 0
\(116\) 756.000 0.605111
\(117\) − 3796.00i − 2.99949i
\(118\) − 1344.00i − 1.04852i
\(119\) −672.000 −0.517665
\(120\) 0 0
\(121\) −1250.00 −0.939144
\(122\) − 412.000i − 0.305744i
\(123\) 4380.00i 3.21082i
\(124\) 928.000 0.672071
\(125\) 0 0
\(126\) −1022.00 −0.722595
\(127\) 1031.00i 0.720366i 0.932882 + 0.360183i \(0.117286\pi\)
−0.932882 + 0.360183i \(0.882714\pi\)
\(128\) 128.000i 0.0883883i
\(129\) −3530.00 −2.40930
\(130\) 0 0
\(131\) −1116.00 −0.744316 −0.372158 0.928169i \(-0.621382\pi\)
−0.372158 + 0.928169i \(0.621382\pi\)
\(132\) 360.000i 0.237379i
\(133\) 70.0000i 0.0456374i
\(134\) 1198.00 0.772324
\(135\) 0 0
\(136\) −768.000 −0.484231
\(137\) − 1398.00i − 0.871819i −0.899991 0.435909i \(-0.856427\pi\)
0.899991 0.435909i \(-0.143573\pi\)
\(138\) 1500.00i 0.925279i
\(139\) −674.000 −0.411280 −0.205640 0.978628i \(-0.565928\pi\)
−0.205640 + 0.978628i \(0.565928\pi\)
\(140\) 0 0
\(141\) −4860.00 −2.90274
\(142\) 942.000i 0.556696i
\(143\) 468.000i 0.273679i
\(144\) −1168.00 −0.675926
\(145\) 0 0
\(146\) −1228.00 −0.696096
\(147\) 490.000i 0.274929i
\(148\) − 1220.00i − 0.677590i
\(149\) 1281.00 0.704320 0.352160 0.935940i \(-0.385447\pi\)
0.352160 + 0.935940i \(0.385447\pi\)
\(150\) 0 0
\(151\) 953.000 0.513603 0.256801 0.966464i \(-0.417331\pi\)
0.256801 + 0.966464i \(0.417331\pi\)
\(152\) 80.0000i 0.0426898i
\(153\) − 7008.00i − 3.70303i
\(154\) 126.000 0.0659310
\(155\) 0 0
\(156\) −2080.00 −1.06752
\(157\) 650.000i 0.330418i 0.986259 + 0.165209i \(0.0528299\pi\)
−0.986259 + 0.165209i \(0.947170\pi\)
\(158\) 1486.00i 0.748227i
\(159\) 3540.00 1.76566
\(160\) 0 0
\(161\) 525.000 0.256993
\(162\) − 5258.00i − 2.55005i
\(163\) − 932.000i − 0.447852i −0.974606 0.223926i \(-0.928113\pi\)
0.974606 0.223926i \(-0.0718874\pi\)
\(164\) 1752.00 0.834196
\(165\) 0 0
\(166\) −1992.00 −0.931381
\(167\) − 180.000i − 0.0834061i −0.999130 0.0417030i \(-0.986722\pi\)
0.999130 0.0417030i \(-0.0132783\pi\)
\(168\) 560.000i 0.257172i
\(169\) −507.000 −0.230769
\(170\) 0 0
\(171\) −730.000 −0.326459
\(172\) 1412.00i 0.625953i
\(173\) 834.000i 0.366519i 0.983065 + 0.183260i \(0.0586649\pi\)
−0.983065 + 0.183260i \(0.941335\pi\)
\(174\) 3780.00 1.64690
\(175\) 0 0
\(176\) 144.000 0.0616728
\(177\) − 6720.00i − 2.85371i
\(178\) 360.000i 0.151591i
\(179\) 648.000 0.270580 0.135290 0.990806i \(-0.456803\pi\)
0.135290 + 0.990806i \(0.456803\pi\)
\(180\) 0 0
\(181\) −2914.00 −1.19666 −0.598331 0.801249i \(-0.704169\pi\)
−0.598331 + 0.801249i \(0.704169\pi\)
\(182\) 728.000i 0.296500i
\(183\) − 2060.00i − 0.832129i
\(184\) 600.000 0.240394
\(185\) 0 0
\(186\) 4640.00 1.82915
\(187\) 864.000i 0.337871i
\(188\) 1944.00i 0.754153i
\(189\) −3220.00 −1.23926
\(190\) 0 0
\(191\) −876.000 −0.331859 −0.165930 0.986138i \(-0.553062\pi\)
−0.165930 + 0.986138i \(0.553062\pi\)
\(192\) 640.000i 0.240563i
\(193\) 601.000i 0.224150i 0.993700 + 0.112075i \(0.0357497\pi\)
−0.993700 + 0.112075i \(0.964250\pi\)
\(194\) −368.000 −0.136190
\(195\) 0 0
\(196\) 196.000 0.0714286
\(197\) − 2013.00i − 0.728022i −0.931395 0.364011i \(-0.881407\pi\)
0.931395 0.364011i \(-0.118593\pi\)
\(198\) 1314.00i 0.471626i
\(199\) −326.000 −0.116128 −0.0580641 0.998313i \(-0.518493\pi\)
−0.0580641 + 0.998313i \(0.518493\pi\)
\(200\) 0 0
\(201\) 5990.00 2.10200
\(202\) 1452.00i 0.505754i
\(203\) − 1323.00i − 0.457421i
\(204\) −3840.00 −1.31791
\(205\) 0 0
\(206\) 3596.00 1.21624
\(207\) 5475.00i 1.83835i
\(208\) 832.000i 0.277350i
\(209\) 90.0000 0.0297867
\(210\) 0 0
\(211\) −5956.00 −1.94326 −0.971630 0.236505i \(-0.923998\pi\)
−0.971630 + 0.236505i \(0.923998\pi\)
\(212\) − 1416.00i − 0.458732i
\(213\) 4710.00i 1.51514i
\(214\) 1752.00 0.559646
\(215\) 0 0
\(216\) −3680.00 −1.15922
\(217\) − 1624.00i − 0.508038i
\(218\) − 1382.00i − 0.429362i
\(219\) −6140.00 −1.89453
\(220\) 0 0
\(221\) −4992.00 −1.51945
\(222\) − 6100.00i − 1.84417i
\(223\) 3118.00i 0.936308i 0.883647 + 0.468154i \(0.155081\pi\)
−0.883647 + 0.468154i \(0.844919\pi\)
\(224\) 224.000 0.0668153
\(225\) 0 0
\(226\) 3042.00 0.895358
\(227\) − 6.00000i − 0.00175433i −1.00000 0.000877167i \(-0.999721\pi\)
1.00000 0.000877167i \(-0.000279211\pi\)
\(228\) 400.000i 0.116187i
\(229\) 586.000 0.169100 0.0845502 0.996419i \(-0.473055\pi\)
0.0845502 + 0.996419i \(0.473055\pi\)
\(230\) 0 0
\(231\) 630.000 0.179441
\(232\) − 1512.00i − 0.427878i
\(233\) − 1293.00i − 0.363550i −0.983340 0.181775i \(-0.941816\pi\)
0.983340 0.181775i \(-0.0581843\pi\)
\(234\) −7592.00 −2.12096
\(235\) 0 0
\(236\) −2688.00 −0.741415
\(237\) 7430.00i 2.03642i
\(238\) 1344.00i 0.366044i
\(239\) 5376.00 1.45500 0.727499 0.686109i \(-0.240683\pi\)
0.727499 + 0.686109i \(0.240683\pi\)
\(240\) 0 0
\(241\) −670.000 −0.179081 −0.0895404 0.995983i \(-0.528540\pi\)
−0.0895404 + 0.995983i \(0.528540\pi\)
\(242\) 2500.00i 0.664075i
\(243\) − 13870.0i − 3.66157i
\(244\) −824.000 −0.216193
\(245\) 0 0
\(246\) 8760.00 2.27040
\(247\) 520.000i 0.133955i
\(248\) − 1856.00i − 0.475226i
\(249\) −9960.00 −2.53490
\(250\) 0 0
\(251\) 1380.00 0.347031 0.173516 0.984831i \(-0.444487\pi\)
0.173516 + 0.984831i \(0.444487\pi\)
\(252\) 2044.00i 0.510952i
\(253\) − 675.000i − 0.167735i
\(254\) 2062.00 0.509376
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 3576.00i − 0.867956i −0.900923 0.433978i \(-0.857110\pi\)
0.900923 0.433978i \(-0.142890\pi\)
\(258\) 7060.00i 1.70363i
\(259\) −2135.00 −0.512210
\(260\) 0 0
\(261\) 13797.0 3.27208
\(262\) 2232.00i 0.526311i
\(263\) 5919.00i 1.38776i 0.720090 + 0.693881i \(0.244100\pi\)
−0.720090 + 0.693881i \(0.755900\pi\)
\(264\) 720.000 0.167852
\(265\) 0 0
\(266\) 140.000 0.0322705
\(267\) 1800.00i 0.412578i
\(268\) − 2396.00i − 0.546116i
\(269\) 1764.00 0.399825 0.199913 0.979814i \(-0.435934\pi\)
0.199913 + 0.979814i \(0.435934\pi\)
\(270\) 0 0
\(271\) −340.000 −0.0762123 −0.0381061 0.999274i \(-0.512132\pi\)
−0.0381061 + 0.999274i \(0.512132\pi\)
\(272\) 1536.00i 0.342403i
\(273\) 3640.00i 0.806970i
\(274\) −2796.00 −0.616469
\(275\) 0 0
\(276\) 3000.00 0.654271
\(277\) 4706.00i 1.02078i 0.859943 + 0.510390i \(0.170499\pi\)
−0.859943 + 0.510390i \(0.829501\pi\)
\(278\) 1348.00i 0.290819i
\(279\) 16936.0 3.63416
\(280\) 0 0
\(281\) 6681.00 1.41835 0.709173 0.705035i \(-0.249069\pi\)
0.709173 + 0.705035i \(0.249069\pi\)
\(282\) 9720.00i 2.05254i
\(283\) 226.000i 0.0474710i 0.999718 + 0.0237355i \(0.00755596\pi\)
−0.999718 + 0.0237355i \(0.992444\pi\)
\(284\) 1884.00 0.393644
\(285\) 0 0
\(286\) 936.000 0.193520
\(287\) − 3066.00i − 0.630593i
\(288\) 2336.00i 0.477952i
\(289\) −4303.00 −0.875840
\(290\) 0 0
\(291\) −1840.00 −0.370662
\(292\) 2456.00i 0.492214i
\(293\) − 4320.00i − 0.861355i −0.902506 0.430678i \(-0.858275\pi\)
0.902506 0.430678i \(-0.141725\pi\)
\(294\) 980.000 0.194404
\(295\) 0 0
\(296\) −2440.00 −0.479129
\(297\) 4140.00i 0.808846i
\(298\) − 2562.00i − 0.498029i
\(299\) 3900.00 0.754324
\(300\) 0 0
\(301\) 2471.00 0.473176
\(302\) − 1906.00i − 0.363172i
\(303\) 7260.00i 1.37649i
\(304\) 160.000 0.0301863
\(305\) 0 0
\(306\) −14016.0 −2.61844
\(307\) − 6604.00i − 1.22772i −0.789415 0.613860i \(-0.789616\pi\)
0.789415 0.613860i \(-0.210384\pi\)
\(308\) − 252.000i − 0.0466202i
\(309\) 17980.0 3.31018
\(310\) 0 0
\(311\) 6036.00 1.10055 0.550274 0.834984i \(-0.314523\pi\)
0.550274 + 0.834984i \(0.314523\pi\)
\(312\) 4160.00i 0.754851i
\(313\) − 9146.00i − 1.65164i −0.563936 0.825819i \(-0.690713\pi\)
0.563936 0.825819i \(-0.309287\pi\)
\(314\) 1300.00 0.233641
\(315\) 0 0
\(316\) 2972.00 0.529076
\(317\) − 4449.00i − 0.788267i −0.919053 0.394134i \(-0.871045\pi\)
0.919053 0.394134i \(-0.128955\pi\)
\(318\) − 7080.00i − 1.24851i
\(319\) −1701.00 −0.298551
\(320\) 0 0
\(321\) 8760.00 1.52316
\(322\) − 1050.00i − 0.181721i
\(323\) 960.000i 0.165374i
\(324\) −10516.0 −1.80316
\(325\) 0 0
\(326\) −1864.00 −0.316679
\(327\) − 6910.00i − 1.16857i
\(328\) − 3504.00i − 0.589866i
\(329\) 3402.00 0.570086
\(330\) 0 0
\(331\) −10081.0 −1.67402 −0.837012 0.547185i \(-0.815700\pi\)
−0.837012 + 0.547185i \(0.815700\pi\)
\(332\) 3984.00i 0.658586i
\(333\) − 22265.0i − 3.66401i
\(334\) −360.000 −0.0589770
\(335\) 0 0
\(336\) 1120.00 0.181848
\(337\) 7778.00i 1.25725i 0.777707 + 0.628627i \(0.216383\pi\)
−0.777707 + 0.628627i \(0.783617\pi\)
\(338\) 1014.00i 0.163178i
\(339\) 15210.0 2.43685
\(340\) 0 0
\(341\) −2088.00 −0.331588
\(342\) 1460.00i 0.230841i
\(343\) − 343.000i − 0.0539949i
\(344\) 2824.00 0.442616
\(345\) 0 0
\(346\) 1668.00 0.259168
\(347\) 1017.00i 0.157336i 0.996901 + 0.0786678i \(0.0250666\pi\)
−0.996901 + 0.0786678i \(0.974933\pi\)
\(348\) − 7560.00i − 1.16454i
\(349\) −10370.0 −1.59053 −0.795263 0.606265i \(-0.792667\pi\)
−0.795263 + 0.606265i \(0.792667\pi\)
\(350\) 0 0
\(351\) −23920.0 −3.63748
\(352\) − 288.000i − 0.0436092i
\(353\) 9432.00i 1.42214i 0.703122 + 0.711069i \(0.251789\pi\)
−0.703122 + 0.711069i \(0.748211\pi\)
\(354\) −13440.0 −2.01788
\(355\) 0 0
\(356\) 720.000 0.107191
\(357\) 6720.00i 0.996247i
\(358\) − 1296.00i − 0.191329i
\(359\) −7557.00 −1.11098 −0.555492 0.831522i \(-0.687470\pi\)
−0.555492 + 0.831522i \(0.687470\pi\)
\(360\) 0 0
\(361\) −6759.00 −0.985421
\(362\) 5828.00i 0.846168i
\(363\) 12500.0i 1.80738i
\(364\) 1456.00 0.209657
\(365\) 0 0
\(366\) −4120.00 −0.588404
\(367\) − 11662.0i − 1.65872i −0.558712 0.829362i \(-0.688704\pi\)
0.558712 0.829362i \(-0.311296\pi\)
\(368\) − 1200.00i − 0.169985i
\(369\) 31974.0 4.51084
\(370\) 0 0
\(371\) −2478.00 −0.346769
\(372\) − 9280.00i − 1.29340i
\(373\) 2377.00i 0.329964i 0.986297 + 0.164982i \(0.0527565\pi\)
−0.986297 + 0.164982i \(0.947243\pi\)
\(374\) 1728.00 0.238911
\(375\) 0 0
\(376\) 3888.00 0.533267
\(377\) − 9828.00i − 1.34262i
\(378\) 6440.00i 0.876291i
\(379\) −4427.00 −0.599999 −0.300000 0.953939i \(-0.596987\pi\)
−0.300000 + 0.953939i \(0.596987\pi\)
\(380\) 0 0
\(381\) 10310.0 1.38634
\(382\) 1752.00i 0.234660i
\(383\) − 4608.00i − 0.614772i −0.951585 0.307386i \(-0.900546\pi\)
0.951585 0.307386i \(-0.0994543\pi\)
\(384\) 1280.00 0.170103
\(385\) 0 0
\(386\) 1202.00 0.158498
\(387\) 25769.0i 3.38479i
\(388\) 736.000i 0.0963009i
\(389\) 699.000 0.0911072 0.0455536 0.998962i \(-0.485495\pi\)
0.0455536 + 0.998962i \(0.485495\pi\)
\(390\) 0 0
\(391\) 7200.00 0.931252
\(392\) − 392.000i − 0.0505076i
\(393\) 11160.0i 1.43244i
\(394\) −4026.00 −0.514789
\(395\) 0 0
\(396\) 2628.00 0.333490
\(397\) − 7630.00i − 0.964581i −0.876011 0.482291i \(-0.839805\pi\)
0.876011 0.482291i \(-0.160195\pi\)
\(398\) 652.000i 0.0821151i
\(399\) 700.000 0.0878292
\(400\) 0 0
\(401\) 5601.00 0.697508 0.348754 0.937214i \(-0.386605\pi\)
0.348754 + 0.937214i \(0.386605\pi\)
\(402\) − 11980.0i − 1.48634i
\(403\) − 12064.0i − 1.49119i
\(404\) 2904.00 0.357622
\(405\) 0 0
\(406\) −2646.00 −0.323445
\(407\) 2745.00i 0.334311i
\(408\) 7680.00i 0.931904i
\(409\) −4670.00 −0.564588 −0.282294 0.959328i \(-0.591095\pi\)
−0.282294 + 0.959328i \(0.591095\pi\)
\(410\) 0 0
\(411\) −13980.0 −1.67782
\(412\) − 7192.00i − 0.860011i
\(413\) 4704.00i 0.560457i
\(414\) 10950.0 1.29991
\(415\) 0 0
\(416\) 1664.00 0.196116
\(417\) 6740.00i 0.791509i
\(418\) − 180.000i − 0.0210624i
\(419\) −36.0000 −0.00419741 −0.00209871 0.999998i \(-0.500668\pi\)
−0.00209871 + 0.999998i \(0.500668\pi\)
\(420\) 0 0
\(421\) 5495.00 0.636128 0.318064 0.948069i \(-0.396967\pi\)
0.318064 + 0.948069i \(0.396967\pi\)
\(422\) 11912.0i 1.37409i
\(423\) 35478.0i 4.07801i
\(424\) −2832.00 −0.324373
\(425\) 0 0
\(426\) 9420.00 1.07136
\(427\) 1442.00i 0.163427i
\(428\) − 3504.00i − 0.395730i
\(429\) 4680.00 0.526696
\(430\) 0 0
\(431\) 2700.00 0.301750 0.150875 0.988553i \(-0.451791\pi\)
0.150875 + 0.988553i \(0.451791\pi\)
\(432\) 7360.00i 0.819695i
\(433\) − 15104.0i − 1.67633i −0.545415 0.838166i \(-0.683628\pi\)
0.545415 0.838166i \(-0.316372\pi\)
\(434\) −3248.00 −0.359237
\(435\) 0 0
\(436\) −2764.00 −0.303605
\(437\) − 750.000i − 0.0820992i
\(438\) 12280.0i 1.33964i
\(439\) −14948.0 −1.62512 −0.812562 0.582875i \(-0.801928\pi\)
−0.812562 + 0.582875i \(0.801928\pi\)
\(440\) 0 0
\(441\) 3577.00 0.386243
\(442\) 9984.00i 1.07441i
\(443\) − 4980.00i − 0.534101i −0.963682 0.267051i \(-0.913951\pi\)
0.963682 0.267051i \(-0.0860491\pi\)
\(444\) −12200.0 −1.30402
\(445\) 0 0
\(446\) 6236.00 0.662070
\(447\) − 12810.0i − 1.35546i
\(448\) − 448.000i − 0.0472456i
\(449\) −12375.0 −1.30070 −0.650348 0.759637i \(-0.725377\pi\)
−0.650348 + 0.759637i \(0.725377\pi\)
\(450\) 0 0
\(451\) −3942.00 −0.411578
\(452\) − 6084.00i − 0.633113i
\(453\) − 9530.00i − 0.988429i
\(454\) −12.0000 −0.00124050
\(455\) 0 0
\(456\) 800.000 0.0821567
\(457\) 10835.0i 1.10906i 0.832164 + 0.554529i \(0.187102\pi\)
−0.832164 + 0.554529i \(0.812898\pi\)
\(458\) − 1172.00i − 0.119572i
\(459\) −44160.0 −4.49066
\(460\) 0 0
\(461\) 5700.00 0.575869 0.287934 0.957650i \(-0.407032\pi\)
0.287934 + 0.957650i \(0.407032\pi\)
\(462\) − 1260.00i − 0.126884i
\(463\) − 6128.00i − 0.615102i −0.951532 0.307551i \(-0.900491\pi\)
0.951532 0.307551i \(-0.0995095\pi\)
\(464\) −3024.00 −0.302555
\(465\) 0 0
\(466\) −2586.00 −0.257069
\(467\) − 9810.00i − 0.972061i −0.873942 0.486031i \(-0.838444\pi\)
0.873942 0.486031i \(-0.161556\pi\)
\(468\) 15184.0i 1.49974i
\(469\) −4193.00 −0.412825
\(470\) 0 0
\(471\) 6500.00 0.635890
\(472\) 5376.00i 0.524259i
\(473\) − 3177.00i − 0.308834i
\(474\) 14860.0 1.43996
\(475\) 0 0
\(476\) 2688.00 0.258833
\(477\) − 25842.0i − 2.48055i
\(478\) − 10752.0i − 1.02884i
\(479\) 204.000 0.0194593 0.00972964 0.999953i \(-0.496903\pi\)
0.00972964 + 0.999953i \(0.496903\pi\)
\(480\) 0 0
\(481\) −15860.0 −1.50344
\(482\) 1340.00i 0.126629i
\(483\) − 5250.00i − 0.494582i
\(484\) 5000.00 0.469572
\(485\) 0 0
\(486\) −27740.0 −2.58912
\(487\) 15401.0i 1.43303i 0.697571 + 0.716515i \(0.254264\pi\)
−0.697571 + 0.716515i \(0.745736\pi\)
\(488\) 1648.00i 0.152872i
\(489\) −9320.00 −0.861892
\(490\) 0 0
\(491\) 3897.00 0.358186 0.179093 0.983832i \(-0.442684\pi\)
0.179093 + 0.983832i \(0.442684\pi\)
\(492\) − 17520.0i − 1.60541i
\(493\) − 18144.0i − 1.65753i
\(494\) 1040.00 0.0947203
\(495\) 0 0
\(496\) −3712.00 −0.336036
\(497\) − 3297.00i − 0.297567i
\(498\) 19920.0i 1.79244i
\(499\) −8132.00 −0.729536 −0.364768 0.931098i \(-0.618852\pi\)
−0.364768 + 0.931098i \(0.618852\pi\)
\(500\) 0 0
\(501\) −1800.00 −0.160515
\(502\) − 2760.00i − 0.245388i
\(503\) 10998.0i 0.974904i 0.873150 + 0.487452i \(0.162074\pi\)
−0.873150 + 0.487452i \(0.837926\pi\)
\(504\) 4088.00 0.361298
\(505\) 0 0
\(506\) −1350.00 −0.118606
\(507\) 5070.00i 0.444116i
\(508\) − 4124.00i − 0.360183i
\(509\) −5940.00 −0.517261 −0.258631 0.965976i \(-0.583271\pi\)
−0.258631 + 0.965976i \(0.583271\pi\)
\(510\) 0 0
\(511\) 4298.00 0.372079
\(512\) − 512.000i − 0.0441942i
\(513\) 4600.00i 0.395897i
\(514\) −7152.00 −0.613738
\(515\) 0 0
\(516\) 14120.0 1.20465
\(517\) − 4374.00i − 0.372086i
\(518\) 4270.00i 0.362187i
\(519\) 8340.00 0.705367
\(520\) 0 0
\(521\) 17022.0 1.43138 0.715688 0.698420i \(-0.246113\pi\)
0.715688 + 0.698420i \(0.246113\pi\)
\(522\) − 27594.0i − 2.31371i
\(523\) 15748.0i 1.31666i 0.752730 + 0.658329i \(0.228736\pi\)
−0.752730 + 0.658329i \(0.771264\pi\)
\(524\) 4464.00 0.372158
\(525\) 0 0
\(526\) 11838.0 0.981295
\(527\) − 22272.0i − 1.84096i
\(528\) − 1440.00i − 0.118689i
\(529\) 6542.00 0.537684
\(530\) 0 0
\(531\) −49056.0 −4.00913
\(532\) − 280.000i − 0.0228187i
\(533\) − 22776.0i − 1.85092i
\(534\) 3600.00 0.291736
\(535\) 0 0
\(536\) −4792.00 −0.386162
\(537\) − 6480.00i − 0.520731i
\(538\) − 3528.00i − 0.282719i
\(539\) −441.000 −0.0352416
\(540\) 0 0
\(541\) −3373.00 −0.268053 −0.134026 0.990978i \(-0.542791\pi\)
−0.134026 + 0.990978i \(0.542791\pi\)
\(542\) 680.000i 0.0538902i
\(543\) 29140.0i 2.30298i
\(544\) 3072.00 0.242116
\(545\) 0 0
\(546\) 7280.00 0.570614
\(547\) − 14389.0i − 1.12473i −0.826888 0.562367i \(-0.809891\pi\)
0.826888 0.562367i \(-0.190109\pi\)
\(548\) 5592.00i 0.435909i
\(549\) −15038.0 −1.16905
\(550\) 0 0
\(551\) −1890.00 −0.146128
\(552\) − 6000.00i − 0.462639i
\(553\) − 5201.00i − 0.399944i
\(554\) 9412.00 0.721801
\(555\) 0 0
\(556\) 2696.00 0.205640
\(557\) − 4929.00i − 0.374952i −0.982269 0.187476i \(-0.939969\pi\)
0.982269 0.187476i \(-0.0600307\pi\)
\(558\) − 33872.0i − 2.56974i
\(559\) 18356.0 1.38887
\(560\) 0 0
\(561\) 8640.00 0.650234
\(562\) − 13362.0i − 1.00292i
\(563\) − 15678.0i − 1.17362i −0.809724 0.586811i \(-0.800383\pi\)
0.809724 0.586811i \(-0.199617\pi\)
\(564\) 19440.0 1.45137
\(565\) 0 0
\(566\) 452.000 0.0335671
\(567\) 18403.0i 1.36306i
\(568\) − 3768.00i − 0.278348i
\(569\) 14499.0 1.06824 0.534121 0.845408i \(-0.320643\pi\)
0.534121 + 0.845408i \(0.320643\pi\)
\(570\) 0 0
\(571\) −9457.00 −0.693105 −0.346553 0.938031i \(-0.612648\pi\)
−0.346553 + 0.938031i \(0.612648\pi\)
\(572\) − 1872.00i − 0.136840i
\(573\) 8760.00i 0.638664i
\(574\) −6132.00 −0.445897
\(575\) 0 0
\(576\) 4672.00 0.337963
\(577\) 10370.0i 0.748195i 0.927389 + 0.374098i \(0.122048\pi\)
−0.927389 + 0.374098i \(0.877952\pi\)
\(578\) 8606.00i 0.619312i
\(579\) 6010.00 0.431377
\(580\) 0 0
\(581\) 6972.00 0.497844
\(582\) 3680.00i 0.262098i
\(583\) 3186.00i 0.226330i
\(584\) 4912.00 0.348048
\(585\) 0 0
\(586\) −8640.00 −0.609070
\(587\) − 12126.0i − 0.852630i −0.904575 0.426315i \(-0.859812\pi\)
0.904575 0.426315i \(-0.140188\pi\)
\(588\) − 1960.00i − 0.137464i
\(589\) −2320.00 −0.162299
\(590\) 0 0
\(591\) −20130.0 −1.40108
\(592\) 4880.00i 0.338795i
\(593\) − 1068.00i − 0.0739587i −0.999316 0.0369793i \(-0.988226\pi\)
0.999316 0.0369793i \(-0.0117736\pi\)
\(594\) 8280.00 0.571940
\(595\) 0 0
\(596\) −5124.00 −0.352160
\(597\) 3260.00i 0.223489i
\(598\) − 7800.00i − 0.533387i
\(599\) −3375.00 −0.230215 −0.115107 0.993353i \(-0.536721\pi\)
−0.115107 + 0.993353i \(0.536721\pi\)
\(600\) 0 0
\(601\) −27448.0 −1.86294 −0.931470 0.363817i \(-0.881473\pi\)
−0.931470 + 0.363817i \(0.881473\pi\)
\(602\) − 4942.00i − 0.334586i
\(603\) − 43727.0i − 2.95307i
\(604\) −3812.00 −0.256801
\(605\) 0 0
\(606\) 14520.0 0.973325
\(607\) 3884.00i 0.259714i 0.991533 + 0.129857i \(0.0414519\pi\)
−0.991533 + 0.129857i \(0.958548\pi\)
\(608\) − 320.000i − 0.0213449i
\(609\) −13230.0 −0.880306
\(610\) 0 0
\(611\) 25272.0 1.67332
\(612\) 28032.0i 1.85151i
\(613\) 3643.00i 0.240032i 0.992772 + 0.120016i \(0.0382945\pi\)
−0.992772 + 0.120016i \(0.961705\pi\)
\(614\) −13208.0 −0.868129
\(615\) 0 0
\(616\) −504.000 −0.0329655
\(617\) 30369.0i 1.98154i 0.135555 + 0.990770i \(0.456718\pi\)
−0.135555 + 0.990770i \(0.543282\pi\)
\(618\) − 35960.0i − 2.34065i
\(619\) 10888.0 0.706988 0.353494 0.935437i \(-0.384993\pi\)
0.353494 + 0.935437i \(0.384993\pi\)
\(620\) 0 0
\(621\) 34500.0 2.22937
\(622\) − 12072.0i − 0.778204i
\(623\) − 1260.00i − 0.0810286i
\(624\) 8320.00 0.533761
\(625\) 0 0
\(626\) −18292.0 −1.16788
\(627\) − 900.000i − 0.0573246i
\(628\) − 2600.00i − 0.165209i
\(629\) −29280.0 −1.85607
\(630\) 0 0
\(631\) 28499.0 1.79798 0.898992 0.437966i \(-0.144301\pi\)
0.898992 + 0.437966i \(0.144301\pi\)
\(632\) − 5944.00i − 0.374113i
\(633\) 59560.0i 3.73981i
\(634\) −8898.00 −0.557389
\(635\) 0 0
\(636\) −14160.0 −0.882831
\(637\) − 2548.00i − 0.158486i
\(638\) 3402.00i 0.211107i
\(639\) 34383.0 2.12859
\(640\) 0 0
\(641\) 5817.00 0.358436 0.179218 0.983809i \(-0.442643\pi\)
0.179218 + 0.983809i \(0.442643\pi\)
\(642\) − 17520.0i − 1.07704i
\(643\) − 22376.0i − 1.37235i −0.727435 0.686177i \(-0.759288\pi\)
0.727435 0.686177i \(-0.240712\pi\)
\(644\) −2100.00 −0.128496
\(645\) 0 0
\(646\) 1920.00 0.116937
\(647\) 3018.00i 0.183385i 0.995787 + 0.0916923i \(0.0292276\pi\)
−0.995787 + 0.0916923i \(0.970772\pi\)
\(648\) 21032.0i 1.27502i
\(649\) 6048.00 0.365801
\(650\) 0 0
\(651\) −16240.0 −0.977720
\(652\) 3728.00i 0.223926i
\(653\) 29682.0i 1.77878i 0.457145 + 0.889392i \(0.348872\pi\)
−0.457145 + 0.889392i \(0.651128\pi\)
\(654\) −13820.0 −0.826307
\(655\) 0 0
\(656\) −7008.00 −0.417098
\(657\) 44822.0i 2.66160i
\(658\) − 6804.00i − 0.403112i
\(659\) −2052.00 −0.121297 −0.0606484 0.998159i \(-0.519317\pi\)
−0.0606484 + 0.998159i \(0.519317\pi\)
\(660\) 0 0
\(661\) 14222.0 0.836871 0.418435 0.908247i \(-0.362579\pi\)
0.418435 + 0.908247i \(0.362579\pi\)
\(662\) 20162.0i 1.18371i
\(663\) 49920.0i 2.92418i
\(664\) 7968.00 0.465690
\(665\) 0 0
\(666\) −44530.0 −2.59084
\(667\) 14175.0i 0.822876i
\(668\) 720.000i 0.0417030i
\(669\) 31180.0 1.80193
\(670\) 0 0
\(671\) 1854.00 0.106666
\(672\) − 2240.00i − 0.128586i
\(673\) − 20942.0i − 1.19949i −0.800192 0.599744i \(-0.795269\pi\)
0.800192 0.599744i \(-0.204731\pi\)
\(674\) 15556.0 0.889013
\(675\) 0 0
\(676\) 2028.00 0.115385
\(677\) 13074.0i 0.742208i 0.928591 + 0.371104i \(0.121021\pi\)
−0.928591 + 0.371104i \(0.878979\pi\)
\(678\) − 30420.0i − 1.72312i
\(679\) 1288.00 0.0727966
\(680\) 0 0
\(681\) −60.0000 −0.00337622
\(682\) 4176.00i 0.234468i
\(683\) 31383.0i 1.75818i 0.476656 + 0.879090i \(0.341849\pi\)
−0.476656 + 0.879090i \(0.658151\pi\)
\(684\) 2920.00 0.163230
\(685\) 0 0
\(686\) −686.000 −0.0381802
\(687\) − 5860.00i − 0.325434i
\(688\) − 5648.00i − 0.312977i
\(689\) −18408.0 −1.01784
\(690\) 0 0
\(691\) −622.000 −0.0342431 −0.0171216 0.999853i \(-0.505450\pi\)
−0.0171216 + 0.999853i \(0.505450\pi\)
\(692\) − 3336.00i − 0.183260i
\(693\) − 4599.00i − 0.252095i
\(694\) 2034.00 0.111253
\(695\) 0 0
\(696\) −15120.0 −0.823451
\(697\) − 42048.0i − 2.28505i
\(698\) 20740.0i 1.12467i
\(699\) −12930.0 −0.699653
\(700\) 0 0
\(701\) 7782.00 0.419290 0.209645 0.977778i \(-0.432769\pi\)
0.209645 + 0.977778i \(0.432769\pi\)
\(702\) 47840.0i 2.57209i
\(703\) 3050.00i 0.163631i
\(704\) −576.000 −0.0308364
\(705\) 0 0
\(706\) 18864.0 1.00560
\(707\) − 5082.00i − 0.270337i
\(708\) 26880.0i 1.42685i
\(709\) −7502.00 −0.397382 −0.198691 0.980062i \(-0.563669\pi\)
−0.198691 + 0.980062i \(0.563669\pi\)
\(710\) 0 0
\(711\) 54239.0 2.86093
\(712\) − 1440.00i − 0.0757953i
\(713\) 17400.0i 0.913934i
\(714\) 13440.0 0.704453
\(715\) 0 0
\(716\) −2592.00 −0.135290
\(717\) − 53760.0i − 2.80015i
\(718\) 15114.0i 0.785584i
\(719\) 20814.0 1.07960 0.539799 0.841794i \(-0.318500\pi\)
0.539799 + 0.841794i \(0.318500\pi\)
\(720\) 0 0
\(721\) −12586.0 −0.650107
\(722\) 13518.0i 0.696798i
\(723\) 6700.00i 0.344641i
\(724\) 11656.0 0.598331
\(725\) 0 0
\(726\) 25000.0 1.27801
\(727\) 14360.0i 0.732576i 0.930501 + 0.366288i \(0.119372\pi\)
−0.930501 + 0.366288i \(0.880628\pi\)
\(728\) − 2912.00i − 0.148250i
\(729\) −67717.0 −3.44038
\(730\) 0 0
\(731\) 33888.0 1.71463
\(732\) 8240.00i 0.416064i
\(733\) 1588.00i 0.0800193i 0.999199 + 0.0400096i \(0.0127389\pi\)
−0.999199 + 0.0400096i \(0.987261\pi\)
\(734\) −23324.0 −1.17289
\(735\) 0 0
\(736\) −2400.00 −0.120197
\(737\) 5391.00i 0.269444i
\(738\) − 63948.0i − 3.18965i
\(739\) −2957.00 −0.147192 −0.0735961 0.997288i \(-0.523448\pi\)
−0.0735961 + 0.997288i \(0.523448\pi\)
\(740\) 0 0
\(741\) 5200.00 0.257796
\(742\) 4956.00i 0.245203i
\(743\) 12384.0i 0.611474i 0.952116 + 0.305737i \(0.0989028\pi\)
−0.952116 + 0.305737i \(0.901097\pi\)
\(744\) −18560.0 −0.914573
\(745\) 0 0
\(746\) 4754.00 0.233319
\(747\) 72708.0i 3.56124i
\(748\) − 3456.00i − 0.168936i
\(749\) −6132.00 −0.299143
\(750\) 0 0
\(751\) −20236.0 −0.983252 −0.491626 0.870806i \(-0.663597\pi\)
−0.491626 + 0.870806i \(0.663597\pi\)
\(752\) − 7776.00i − 0.377077i
\(753\) − 13800.0i − 0.667862i
\(754\) −19656.0 −0.949376
\(755\) 0 0
\(756\) 12880.0 0.619631
\(757\) 37601.0i 1.80533i 0.430348 + 0.902663i \(0.358391\pi\)
−0.430348 + 0.902663i \(0.641609\pi\)
\(758\) 8854.00i 0.424264i
\(759\) −6750.00 −0.322806
\(760\) 0 0
\(761\) −13392.0 −0.637923 −0.318962 0.947768i \(-0.603334\pi\)
−0.318962 + 0.947768i \(0.603334\pi\)
\(762\) − 20620.0i − 0.980294i
\(763\) 4837.00i 0.229503i
\(764\) 3504.00 0.165930
\(765\) 0 0
\(766\) −9216.00 −0.434710
\(767\) 34944.0i 1.64505i
\(768\) − 2560.00i − 0.120281i
\(769\) −22430.0 −1.05182 −0.525908 0.850541i \(-0.676274\pi\)
−0.525908 + 0.850541i \(0.676274\pi\)
\(770\) 0 0
\(771\) −35760.0 −1.67038
\(772\) − 2404.00i − 0.112075i
\(773\) − 34704.0i − 1.61477i −0.590026 0.807384i \(-0.700882\pi\)
0.590026 0.807384i \(-0.299118\pi\)
\(774\) 51538.0 2.39340
\(775\) 0 0
\(776\) 1472.00 0.0680950
\(777\) 21350.0i 0.985749i
\(778\) − 1398.00i − 0.0644225i
\(779\) −4380.00 −0.201450
\(780\) 0 0
\(781\) −4239.00 −0.194217
\(782\) − 14400.0i − 0.658495i
\(783\) − 86940.0i − 3.96805i
\(784\) −784.000 −0.0357143
\(785\) 0 0
\(786\) 22320.0 1.01289
\(787\) 37646.0i 1.70513i 0.522624 + 0.852564i \(0.324953\pi\)
−0.522624 + 0.852564i \(0.675047\pi\)
\(788\) 8052.00i 0.364011i
\(789\) 59190.0 2.67075
\(790\) 0 0
\(791\) −10647.0 −0.478589
\(792\) − 5256.00i − 0.235813i
\(793\) 10712.0i 0.479690i
\(794\) −15260.0 −0.682062
\(795\) 0 0
\(796\) 1304.00 0.0580641
\(797\) − 8988.00i − 0.399462i −0.979851 0.199731i \(-0.935993\pi\)
0.979851 0.199731i \(-0.0640068\pi\)
\(798\) − 1400.00i − 0.0621046i
\(799\) 46656.0 2.06580
\(800\) 0 0
\(801\) 13140.0 0.579624
\(802\) − 11202.0i − 0.493212i
\(803\) − 5526.00i − 0.242850i
\(804\) −23960.0 −1.05100
\(805\) 0 0
\(806\) −24128.0 −1.05443
\(807\) − 17640.0i − 0.769464i
\(808\) − 5808.00i − 0.252877i
\(809\) 28029.0 1.21811 0.609053 0.793130i \(-0.291550\pi\)
0.609053 + 0.793130i \(0.291550\pi\)
\(810\) 0 0
\(811\) 8078.00 0.349762 0.174881 0.984590i \(-0.444046\pi\)
0.174881 + 0.984590i \(0.444046\pi\)
\(812\) 5292.00i 0.228710i
\(813\) 3400.00i 0.146671i
\(814\) 5490.00 0.236394
\(815\) 0 0
\(816\) 15360.0 0.658955
\(817\) − 3530.00i − 0.151162i
\(818\) 9340.00i 0.399224i
\(819\) 26572.0 1.13370
\(820\) 0 0
\(821\) −35574.0 −1.51223 −0.756115 0.654439i \(-0.772905\pi\)
−0.756115 + 0.654439i \(0.772905\pi\)
\(822\) 27960.0i 1.18640i
\(823\) − 6599.00i − 0.279498i −0.990187 0.139749i \(-0.955370\pi\)
0.990187 0.139749i \(-0.0446295\pi\)
\(824\) −14384.0 −0.608119
\(825\) 0 0
\(826\) 9408.00 0.396303
\(827\) − 663.000i − 0.0278776i −0.999903 0.0139388i \(-0.995563\pi\)
0.999903 0.0139388i \(-0.00443700\pi\)
\(828\) − 21900.0i − 0.919176i
\(829\) 22564.0 0.945332 0.472666 0.881242i \(-0.343292\pi\)
0.472666 + 0.881242i \(0.343292\pi\)
\(830\) 0 0
\(831\) 47060.0 1.96449
\(832\) − 3328.00i − 0.138675i
\(833\) − 4704.00i − 0.195659i
\(834\) 13480.0 0.559681
\(835\) 0 0
\(836\) −360.000 −0.0148934
\(837\) − 106720.i − 4.40715i
\(838\) 72.0000i 0.00296802i
\(839\) 294.000 0.0120977 0.00604887 0.999982i \(-0.498075\pi\)
0.00604887 + 0.999982i \(0.498075\pi\)
\(840\) 0 0
\(841\) 11332.0 0.464636
\(842\) − 10990.0i − 0.449810i
\(843\) − 66810.0i − 2.72961i
\(844\) 23824.0 0.971630
\(845\) 0 0
\(846\) 70956.0 2.88359
\(847\) − 8750.00i − 0.354963i
\(848\) 5664.00i 0.229366i
\(849\) 2260.00 0.0913581
\(850\) 0 0
\(851\) 22875.0 0.921439
\(852\) − 18840.0i − 0.757568i
\(853\) 28852.0i 1.15812i 0.815286 + 0.579058i \(0.196580\pi\)
−0.815286 + 0.579058i \(0.803420\pi\)
\(854\) 2884.00 0.115560
\(855\) 0 0
\(856\) −7008.00 −0.279823
\(857\) − 7422.00i − 0.295835i −0.989000 0.147918i \(-0.952743\pi\)
0.989000 0.147918i \(-0.0472570\pi\)
\(858\) − 9360.00i − 0.372430i
\(859\) −8138.00 −0.323242 −0.161621 0.986853i \(-0.551672\pi\)
−0.161621 + 0.986853i \(0.551672\pi\)
\(860\) 0 0
\(861\) −30660.0 −1.21358
\(862\) − 5400.00i − 0.213370i
\(863\) 32199.0i 1.27007i 0.772485 + 0.635033i \(0.219013\pi\)
−0.772485 + 0.635033i \(0.780987\pi\)
\(864\) 14720.0 0.579612
\(865\) 0 0
\(866\) −30208.0 −1.18535
\(867\) 43030.0i 1.68555i
\(868\) 6496.00i 0.254019i
\(869\) −6687.00 −0.261037
\(870\) 0 0
\(871\) −31148.0 −1.21172
\(872\) 5528.00i 0.214681i
\(873\) 13432.0i 0.520738i
\(874\) −1500.00 −0.0580529
\(875\) 0 0
\(876\) 24560.0 0.947267
\(877\) − 11158.0i − 0.429622i −0.976656 0.214811i \(-0.931086\pi\)
0.976656 0.214811i \(-0.0689136\pi\)
\(878\) 29896.0i 1.14914i
\(879\) −43200.0 −1.65768
\(880\) 0 0
\(881\) 16272.0 0.622267 0.311134 0.950366i \(-0.399291\pi\)
0.311134 + 0.950366i \(0.399291\pi\)
\(882\) − 7154.00i − 0.273115i
\(883\) − 9071.00i − 0.345712i −0.984947 0.172856i \(-0.944701\pi\)
0.984947 0.172856i \(-0.0552995\pi\)
\(884\) 19968.0 0.759725
\(885\) 0 0
\(886\) −9960.00 −0.377667
\(887\) − 30138.0i − 1.14085i −0.821349 0.570426i \(-0.806778\pi\)
0.821349 0.570426i \(-0.193222\pi\)
\(888\) 24400.0i 0.922084i
\(889\) −7217.00 −0.272273
\(890\) 0 0
\(891\) 23661.0 0.889645
\(892\) − 12472.0i − 0.468154i
\(893\) − 4860.00i − 0.182121i
\(894\) −25620.0 −0.958457
\(895\) 0 0
\(896\) −896.000 −0.0334077
\(897\) − 39000.0i − 1.45170i
\(898\) 24750.0i 0.919731i
\(899\) 43848.0 1.62671
\(900\) 0 0
\(901\) −33984.0 −1.25657
\(902\) 7884.00i 0.291029i
\(903\) − 24710.0i − 0.910628i
\(904\) −12168.0 −0.447679
\(905\) 0 0
\(906\) −19060.0 −0.698925
\(907\) 25844.0i 0.946126i 0.881029 + 0.473063i \(0.156852\pi\)
−0.881029 + 0.473063i \(0.843148\pi\)
\(908\) 24.0000i 0 0.000877167i
\(909\) 52998.0 1.93381
\(910\) 0 0
\(911\) 40815.0 1.48437 0.742185 0.670195i \(-0.233790\pi\)
0.742185 + 0.670195i \(0.233790\pi\)
\(912\) − 1600.00i − 0.0580935i
\(913\) − 8964.00i − 0.324934i
\(914\) 21670.0 0.784223
\(915\) 0 0
\(916\) −2344.00 −0.0845502
\(917\) − 7812.00i − 0.281325i
\(918\) 88320.0i 3.17538i
\(919\) 5389.00 0.193435 0.0967175 0.995312i \(-0.469166\pi\)
0.0967175 + 0.995312i \(0.469166\pi\)
\(920\) 0 0
\(921\) −66040.0 −2.36275
\(922\) − 11400.0i − 0.407201i
\(923\) − 24492.0i − 0.873417i
\(924\) −2520.00 −0.0897207
\(925\) 0 0
\(926\) −12256.0 −0.434943
\(927\) − 131254.i − 4.65043i
\(928\) 6048.00i 0.213939i
\(929\) −43662.0 −1.54198 −0.770992 0.636844i \(-0.780239\pi\)
−0.770992 + 0.636844i \(0.780239\pi\)
\(930\) 0 0
\(931\) −490.000 −0.0172493
\(932\) 5172.00i 0.181775i
\(933\) − 60360.0i − 2.11800i
\(934\) −19620.0 −0.687351
\(935\) 0 0
\(936\) 30368.0 1.06048
\(937\) − 11950.0i − 0.416638i −0.978061 0.208319i \(-0.933201\pi\)
0.978061 0.208319i \(-0.0667992\pi\)
\(938\) 8386.00i 0.291911i
\(939\) −91460.0 −3.17858
\(940\) 0 0
\(941\) −8448.00 −0.292664 −0.146332 0.989236i \(-0.546747\pi\)
−0.146332 + 0.989236i \(0.546747\pi\)
\(942\) − 13000.0i − 0.449642i
\(943\) 32850.0i 1.13440i
\(944\) 10752.0 0.370707
\(945\) 0 0
\(946\) −6354.00 −0.218379
\(947\) 25692.0i 0.881603i 0.897605 + 0.440801i \(0.145306\pi\)
−0.897605 + 0.440801i \(0.854694\pi\)
\(948\) − 29720.0i − 1.01821i
\(949\) 31928.0 1.09213
\(950\) 0 0
\(951\) −44490.0 −1.51702
\(952\) − 5376.00i − 0.183022i
\(953\) − 47547.0i − 1.61616i −0.589074 0.808079i \(-0.700507\pi\)
0.589074 0.808079i \(-0.299493\pi\)
\(954\) −51684.0 −1.75402
\(955\) 0 0
\(956\) −21504.0 −0.727499
\(957\) 17010.0i 0.574561i
\(958\) − 408.000i − 0.0137598i
\(959\) 9786.00 0.329517
\(960\) 0 0
\(961\) 24033.0 0.806720
\(962\) 31720.0i 1.06309i
\(963\) − 63948.0i − 2.13987i
\(964\) 2680.00 0.0895404
\(965\) 0 0
\(966\) −10500.0 −0.349723
\(967\) 51608.0i 1.71624i 0.513452 + 0.858119i \(0.328367\pi\)
−0.513452 + 0.858119i \(0.671633\pi\)
\(968\) − 10000.0i − 0.332037i
\(969\) 9600.00 0.318263
\(970\) 0 0
\(971\) −11754.0 −0.388469 −0.194235 0.980955i \(-0.562222\pi\)
−0.194235 + 0.980955i \(0.562222\pi\)
\(972\) 55480.0i 1.83078i
\(973\) − 4718.00i − 0.155449i
\(974\) 30802.0 1.01331
\(975\) 0 0
\(976\) 3296.00 0.108097
\(977\) 12765.0i 0.418003i 0.977915 + 0.209001i \(0.0670213\pi\)
−0.977915 + 0.209001i \(0.932979\pi\)
\(978\) 18640.0i 0.609449i
\(979\) −1620.00 −0.0528860
\(980\) 0 0
\(981\) −50443.0 −1.64171
\(982\) − 7794.00i − 0.253275i
\(983\) 32112.0i 1.04193i 0.853579 + 0.520963i \(0.174427\pi\)
−0.853579 + 0.520963i \(0.825573\pi\)
\(984\) −35040.0 −1.13520
\(985\) 0 0
\(986\) −36288.0 −1.17205
\(987\) − 34020.0i − 1.09713i
\(988\) − 2080.00i − 0.0669773i
\(989\) −26475.0 −0.851219
\(990\) 0 0
\(991\) −42505.0 −1.36248 −0.681239 0.732061i \(-0.738559\pi\)
−0.681239 + 0.732061i \(0.738559\pi\)
\(992\) 7424.00i 0.237613i
\(993\) 100810.i 3.22166i
\(994\) −6594.00 −0.210411
\(995\) 0 0
\(996\) 39840.0 1.26745
\(997\) 59654.0i 1.89495i 0.319836 + 0.947473i \(0.396372\pi\)
−0.319836 + 0.947473i \(0.603628\pi\)
\(998\) 16264.0i 0.515860i
\(999\) −140300. −4.44334
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 350.4.c.a.99.1 2
5.2 odd 4 350.4.a.k.1.1 yes 1
5.3 odd 4 350.4.a.j.1.1 1
5.4 even 2 inner 350.4.c.a.99.2 2
35.13 even 4 2450.4.a.a.1.1 1
35.27 even 4 2450.4.a.bp.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.4.a.j.1.1 1 5.3 odd 4
350.4.a.k.1.1 yes 1 5.2 odd 4
350.4.c.a.99.1 2 1.1 even 1 trivial
350.4.c.a.99.2 2 5.4 even 2 inner
2450.4.a.a.1.1 1 35.13 even 4
2450.4.a.bp.1.1 1 35.27 even 4