Properties

Label 1850.2.d.i.1701.14
Level $1850$
Weight $2$
Character 1850.1701
Analytic conductor $14.772$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1850,2,Mod(1701,1850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1850.1701");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 48 x^{18} + 878 x^{16} + 8102 x^{14} + 41081 x^{12} + 115688 x^{10} + 175041 x^{8} + 134990 x^{6} + 49782 x^{4} + 6932 x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1701.14
Root \(-0.0120170i\) of defining polynomial
Character \(\chi\) \(=\) 1850.1701
Dual form 1850.2.d.i.1701.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -0.987983 q^{3} -1.00000 q^{4} -0.987983i q^{6} -4.78937 q^{7} -1.00000i q^{8} -2.02389 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -0.987983 q^{3} -1.00000 q^{4} -0.987983i q^{6} -4.78937 q^{7} -1.00000i q^{8} -2.02389 q^{9} -5.98732 q^{11} +0.987983 q^{12} -3.49410i q^{13} -4.78937i q^{14} +1.00000 q^{16} -4.96343i q^{17} -2.02389i q^{18} +7.33092i q^{19} +4.73182 q^{21} -5.98732i q^{22} -1.74873i q^{23} +0.987983i q^{24} +3.49410 q^{26} +4.96352 q^{27} +4.78937 q^{28} +7.85004i q^{29} -3.24097i q^{31} +1.00000i q^{32} +5.91537 q^{33} +4.96343 q^{34} +2.02389 q^{36} +(4.61424 + 3.96343i) q^{37} -7.33092 q^{38} +3.45211i q^{39} -0.530665 q^{41} +4.73182i q^{42} +1.76838i q^{43} +5.98732 q^{44} +1.74873 q^{46} -4.30638 q^{47} -0.987983 q^{48} +15.9381 q^{49} +4.90379i q^{51} +3.49410i q^{52} +3.66238 q^{53} +4.96352i q^{54} +4.78937i q^{56} -7.24283i q^{57} -7.85004 q^{58} -2.15110i q^{59} +3.06584i q^{61} +3.24097 q^{62} +9.69316 q^{63} -1.00000 q^{64} +5.91537i q^{66} -3.79622 q^{67} +4.96343i q^{68} +1.72772i q^{69} -8.47719 q^{71} +2.02389i q^{72} -9.05445 q^{73} +(-3.96343 + 4.61424i) q^{74} -7.33092i q^{76} +28.6755 q^{77} -3.45211 q^{78} -5.56622i q^{79} +1.16780 q^{81} -0.530665i q^{82} +3.77680 q^{83} -4.73182 q^{84} -1.76838 q^{86} -7.75571i q^{87} +5.98732i q^{88} -8.45791i q^{89} +16.7345i q^{91} +1.74873i q^{92} +3.20203i q^{93} -4.30638i q^{94} -0.987983i q^{96} -3.64747i q^{97} +15.9381i q^{98} +12.1177 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{4} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{4} + 16 q^{9} + 20 q^{16} - 24 q^{21} + 4 q^{26} + 36 q^{34} - 16 q^{36} - 8 q^{41} - 20 q^{46} + 16 q^{49} - 20 q^{64} - 40 q^{71} - 16 q^{74} + 116 q^{81} + 24 q^{84} + 20 q^{86} + 164 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −0.987983 −0.570412 −0.285206 0.958466i \(-0.592062\pi\)
−0.285206 + 0.958466i \(0.592062\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0.987983i 0.403342i
\(7\) −4.78937 −1.81021 −0.905106 0.425186i \(-0.860209\pi\)
−0.905106 + 0.425186i \(0.860209\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −2.02389 −0.674630
\(10\) 0 0
\(11\) −5.98732 −1.80525 −0.902623 0.430432i \(-0.858361\pi\)
−0.902623 + 0.430432i \(0.858361\pi\)
\(12\) 0.987983 0.285206
\(13\) 3.49410i 0.969088i −0.874767 0.484544i \(-0.838985\pi\)
0.874767 0.484544i \(-0.161015\pi\)
\(14\) 4.78937i 1.28001i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.96343i 1.20381i −0.798568 0.601905i \(-0.794409\pi\)
0.798568 0.601905i \(-0.205591\pi\)
\(18\) 2.02389i 0.477035i
\(19\) 7.33092i 1.68183i 0.541168 + 0.840915i \(0.317982\pi\)
−0.541168 + 0.840915i \(0.682018\pi\)
\(20\) 0 0
\(21\) 4.73182 1.03257
\(22\) 5.98732i 1.27650i
\(23\) 1.74873i 0.364635i −0.983240 0.182318i \(-0.941640\pi\)
0.983240 0.182318i \(-0.0583599\pi\)
\(24\) 0.987983i 0.201671i
\(25\) 0 0
\(26\) 3.49410 0.685249
\(27\) 4.96352 0.955229
\(28\) 4.78937 0.905106
\(29\) 7.85004i 1.45772i 0.684665 + 0.728858i \(0.259949\pi\)
−0.684665 + 0.728858i \(0.740051\pi\)
\(30\) 0 0
\(31\) 3.24097i 0.582096i −0.956708 0.291048i \(-0.905996\pi\)
0.956708 0.291048i \(-0.0940039\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 5.91537 1.02973
\(34\) 4.96343 0.851222
\(35\) 0 0
\(36\) 2.02389 0.337315
\(37\) 4.61424 + 3.96343i 0.758576 + 0.651584i
\(38\) −7.33092 −1.18923
\(39\) 3.45211i 0.552780i
\(40\) 0 0
\(41\) −0.530665 −0.0828760 −0.0414380 0.999141i \(-0.513194\pi\)
−0.0414380 + 0.999141i \(0.513194\pi\)
\(42\) 4.73182i 0.730135i
\(43\) 1.76838i 0.269676i 0.990868 + 0.134838i \(0.0430514\pi\)
−0.990868 + 0.134838i \(0.956949\pi\)
\(44\) 5.98732 0.902623
\(45\) 0 0
\(46\) 1.74873 0.257836
\(47\) −4.30638 −0.628150 −0.314075 0.949398i \(-0.601694\pi\)
−0.314075 + 0.949398i \(0.601694\pi\)
\(48\) −0.987983 −0.142603
\(49\) 15.9381 2.27687
\(50\) 0 0
\(51\) 4.90379i 0.686668i
\(52\) 3.49410i 0.484544i
\(53\) 3.66238 0.503067 0.251534 0.967849i \(-0.419065\pi\)
0.251534 + 0.967849i \(0.419065\pi\)
\(54\) 4.96352i 0.675449i
\(55\) 0 0
\(56\) 4.78937i 0.640007i
\(57\) 7.24283i 0.959336i
\(58\) −7.85004 −1.03076
\(59\) 2.15110i 0.280049i −0.990148 0.140025i \(-0.955282\pi\)
0.990148 0.140025i \(-0.0447182\pi\)
\(60\) 0 0
\(61\) 3.06584i 0.392541i 0.980550 + 0.196270i \(0.0628830\pi\)
−0.980550 + 0.196270i \(0.937117\pi\)
\(62\) 3.24097 0.411604
\(63\) 9.69316 1.22122
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 5.91537i 0.728132i
\(67\) −3.79622 −0.463782 −0.231891 0.972742i \(-0.574491\pi\)
−0.231891 + 0.972742i \(0.574491\pi\)
\(68\) 4.96343i 0.601905i
\(69\) 1.72772i 0.207992i
\(70\) 0 0
\(71\) −8.47719 −1.00606 −0.503028 0.864270i \(-0.667781\pi\)
−0.503028 + 0.864270i \(0.667781\pi\)
\(72\) 2.02389i 0.238518i
\(73\) −9.05445 −1.05974 −0.529872 0.848078i \(-0.677760\pi\)
−0.529872 + 0.848078i \(0.677760\pi\)
\(74\) −3.96343 + 4.61424i −0.460740 + 0.536394i
\(75\) 0 0
\(76\) 7.33092i 0.840915i
\(77\) 28.6755 3.26788
\(78\) −3.45211 −0.390874
\(79\) 5.56622i 0.626248i −0.949712 0.313124i \(-0.898624\pi\)
0.949712 0.313124i \(-0.101376\pi\)
\(80\) 0 0
\(81\) 1.16780 0.129755
\(82\) 0.530665i 0.0586022i
\(83\) 3.77680 0.414558 0.207279 0.978282i \(-0.433539\pi\)
0.207279 + 0.978282i \(0.433539\pi\)
\(84\) −4.73182 −0.516284
\(85\) 0 0
\(86\) −1.76838 −0.190690
\(87\) 7.75571i 0.831499i
\(88\) 5.98732i 0.638251i
\(89\) 8.45791i 0.896537i −0.893899 0.448268i \(-0.852041\pi\)
0.893899 0.448268i \(-0.147959\pi\)
\(90\) 0 0
\(91\) 16.7345i 1.75426i
\(92\) 1.74873i 0.182318i
\(93\) 3.20203i 0.332035i
\(94\) 4.30638i 0.444169i
\(95\) 0 0
\(96\) 0.987983i 0.100836i
\(97\) 3.64747i 0.370345i −0.982706 0.185172i \(-0.940716\pi\)
0.982706 0.185172i \(-0.0592843\pi\)
\(98\) 15.9381i 1.60999i
\(99\) 12.1177 1.21787
\(100\) 0 0
\(101\) 2.30416 0.229272 0.114636 0.993408i \(-0.463430\pi\)
0.114636 + 0.993408i \(0.463430\pi\)
\(102\) −4.90379 −0.485547
\(103\) 11.1716i 1.10077i −0.834912 0.550383i \(-0.814482\pi\)
0.834912 0.550383i \(-0.185518\pi\)
\(104\) −3.49410 −0.342625
\(105\) 0 0
\(106\) 3.66238i 0.355722i
\(107\) 6.51807 0.630126 0.315063 0.949071i \(-0.397974\pi\)
0.315063 + 0.949071i \(0.397974\pi\)
\(108\) −4.96352 −0.477615
\(109\) 1.33812i 0.128169i −0.997944 0.0640846i \(-0.979587\pi\)
0.997944 0.0640846i \(-0.0204127\pi\)
\(110\) 0 0
\(111\) −4.55879 3.91580i −0.432701 0.371672i
\(112\) −4.78937 −0.452553
\(113\) 1.39109i 0.130863i −0.997857 0.0654315i \(-0.979158\pi\)
0.997857 0.0654315i \(-0.0208424\pi\)
\(114\) 7.24283 0.678353
\(115\) 0 0
\(116\) 7.85004i 0.728858i
\(117\) 7.07167i 0.653776i
\(118\) 2.15110 0.198025
\(119\) 23.7717i 2.17915i
\(120\) 0 0
\(121\) 24.8480 2.25891
\(122\) −3.06584 −0.277568
\(123\) 0.524288 0.0472735
\(124\) 3.24097i 0.291048i
\(125\) 0 0
\(126\) 9.69316i 0.863535i
\(127\) −7.51024 −0.666426 −0.333213 0.942852i \(-0.608133\pi\)
−0.333213 + 0.942852i \(0.608133\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 1.74713i 0.153827i
\(130\) 0 0
\(131\) 6.19975i 0.541675i −0.962625 0.270837i \(-0.912699\pi\)
0.962625 0.270837i \(-0.0873006\pi\)
\(132\) −5.91537 −0.514867
\(133\) 35.1105i 3.04447i
\(134\) 3.79622i 0.327943i
\(135\) 0 0
\(136\) −4.96343 −0.425611
\(137\) −8.97186 −0.766518 −0.383259 0.923641i \(-0.625198\pi\)
−0.383259 + 0.923641i \(0.625198\pi\)
\(138\) −1.72772 −0.147073
\(139\) −5.04168 −0.427629 −0.213815 0.976874i \(-0.568589\pi\)
−0.213815 + 0.976874i \(0.568589\pi\)
\(140\) 0 0
\(141\) 4.25463 0.358305
\(142\) 8.47719i 0.711390i
\(143\) 20.9203i 1.74944i
\(144\) −2.02389 −0.168657
\(145\) 0 0
\(146\) 9.05445i 0.749352i
\(147\) −15.7466 −1.29875
\(148\) −4.61424 3.96343i −0.379288 0.325792i
\(149\) 21.6451 1.77324 0.886619 0.462500i \(-0.153048\pi\)
0.886619 + 0.462500i \(0.153048\pi\)
\(150\) 0 0
\(151\) 6.47544 0.526964 0.263482 0.964664i \(-0.415129\pi\)
0.263482 + 0.964664i \(0.415129\pi\)
\(152\) 7.33092 0.594616
\(153\) 10.0454i 0.812126i
\(154\) 28.6755i 2.31074i
\(155\) 0 0
\(156\) 3.45211i 0.276390i
\(157\) −13.1689 −1.05099 −0.525495 0.850797i \(-0.676120\pi\)
−0.525495 + 0.850797i \(0.676120\pi\)
\(158\) 5.56622 0.442825
\(159\) −3.61837 −0.286956
\(160\) 0 0
\(161\) 8.37532i 0.660067i
\(162\) 1.16780i 0.0917509i
\(163\) 18.7651i 1.46979i 0.678180 + 0.734896i \(0.262769\pi\)
−0.678180 + 0.734896i \(0.737231\pi\)
\(164\) 0.530665 0.0414380
\(165\) 0 0
\(166\) 3.77680i 0.293136i
\(167\) 7.19692i 0.556914i −0.960449 0.278457i \(-0.910177\pi\)
0.960449 0.278457i \(-0.0898230\pi\)
\(168\) 4.73182i 0.365068i
\(169\) 0.791278 0.0608676
\(170\) 0 0
\(171\) 14.8370i 1.13461i
\(172\) 1.76838i 0.134838i
\(173\) 9.79406 0.744629 0.372314 0.928107i \(-0.378564\pi\)
0.372314 + 0.928107i \(0.378564\pi\)
\(174\) 7.75571 0.587959
\(175\) 0 0
\(176\) −5.98732 −0.451311
\(177\) 2.12525i 0.159743i
\(178\) 8.45791 0.633947
\(179\) 11.2829i 0.843320i 0.906754 + 0.421660i \(0.138552\pi\)
−0.906754 + 0.421660i \(0.861448\pi\)
\(180\) 0 0
\(181\) 1.82697 0.135798 0.0678989 0.997692i \(-0.478370\pi\)
0.0678989 + 0.997692i \(0.478370\pi\)
\(182\) −16.7345 −1.24045
\(183\) 3.02900i 0.223910i
\(184\) −1.74873 −0.128918
\(185\) 0 0
\(186\) −3.20203 −0.234784
\(187\) 29.7177i 2.17317i
\(188\) 4.30638 0.314075
\(189\) −23.7721 −1.72917
\(190\) 0 0
\(191\) 23.8204i 1.72358i 0.507264 + 0.861791i \(0.330657\pi\)
−0.507264 + 0.861791i \(0.669343\pi\)
\(192\) 0.987983 0.0713015
\(193\) 8.40405i 0.604937i 0.953159 + 0.302468i \(0.0978107\pi\)
−0.953159 + 0.302468i \(0.902189\pi\)
\(194\) 3.64747 0.261873
\(195\) 0 0
\(196\) −15.9381 −1.13843
\(197\) 4.22797 0.301230 0.150615 0.988592i \(-0.451875\pi\)
0.150615 + 0.988592i \(0.451875\pi\)
\(198\) 12.1177i 0.861166i
\(199\) 17.7545i 1.25858i 0.777170 + 0.629290i \(0.216654\pi\)
−0.777170 + 0.629290i \(0.783346\pi\)
\(200\) 0 0
\(201\) 3.75060 0.264547
\(202\) 2.30416i 0.162120i
\(203\) 37.5968i 2.63878i
\(204\) 4.90379i 0.343334i
\(205\) 0 0
\(206\) 11.1716 0.778360
\(207\) 3.53924i 0.245994i
\(208\) 3.49410i 0.242272i
\(209\) 43.8926i 3.03611i
\(210\) 0 0
\(211\) 4.57321 0.314833 0.157417 0.987532i \(-0.449683\pi\)
0.157417 + 0.987532i \(0.449683\pi\)
\(212\) −3.66238 −0.251534
\(213\) 8.37532 0.573867
\(214\) 6.51807i 0.445566i
\(215\) 0 0
\(216\) 4.96352i 0.337725i
\(217\) 15.5222i 1.05372i
\(218\) 1.33812 0.0906293
\(219\) 8.94565 0.604491
\(220\) 0 0
\(221\) −17.3427 −1.16660
\(222\) 3.91580 4.55879i 0.262812 0.305966i
\(223\) 20.3205 1.36076 0.680380 0.732860i \(-0.261815\pi\)
0.680380 + 0.732860i \(0.261815\pi\)
\(224\) 4.78937i 0.320003i
\(225\) 0 0
\(226\) 1.39109 0.0925341
\(227\) 5.74303i 0.381178i 0.981670 + 0.190589i \(0.0610398\pi\)
−0.981670 + 0.190589i \(0.938960\pi\)
\(228\) 7.24283i 0.479668i
\(229\) −23.4383 −1.54885 −0.774423 0.632669i \(-0.781960\pi\)
−0.774423 + 0.632669i \(0.781960\pi\)
\(230\) 0 0
\(231\) −28.3309 −1.86404
\(232\) 7.85004 0.515380
\(233\) −17.2961 −1.13310 −0.566551 0.824027i \(-0.691723\pi\)
−0.566551 + 0.824027i \(0.691723\pi\)
\(234\) −7.07167 −0.462289
\(235\) 0 0
\(236\) 2.15110i 0.140025i
\(237\) 5.49933i 0.357220i
\(238\) −23.7717 −1.54089
\(239\) 3.68796i 0.238554i 0.992861 + 0.119277i \(0.0380577\pi\)
−0.992861 + 0.119277i \(0.961942\pi\)
\(240\) 0 0
\(241\) 11.3613i 0.731844i −0.930646 0.365922i \(-0.880754\pi\)
0.930646 0.365922i \(-0.119246\pi\)
\(242\) 24.8480i 1.59729i
\(243\) −16.0443 −1.02924
\(244\) 3.06584i 0.196270i
\(245\) 0 0
\(246\) 0.524288i 0.0334274i
\(247\) 25.6150 1.62984
\(248\) −3.24097 −0.205802
\(249\) −3.73141 −0.236469
\(250\) 0 0
\(251\) 12.8260i 0.809567i −0.914412 0.404784i \(-0.867347\pi\)
0.914412 0.404784i \(-0.132653\pi\)
\(252\) −9.69316 −0.610612
\(253\) 10.4702i 0.658256i
\(254\) 7.51024i 0.471234i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0268i 0.874965i −0.899227 0.437483i \(-0.855870\pi\)
0.899227 0.437483i \(-0.144130\pi\)
\(258\) 1.74713 0.108772
\(259\) −22.0993 18.9824i −1.37318 1.17951i
\(260\) 0 0
\(261\) 15.8876i 0.983419i
\(262\) 6.19975 0.383022
\(263\) 3.57182 0.220248 0.110124 0.993918i \(-0.464875\pi\)
0.110124 + 0.993918i \(0.464875\pi\)
\(264\) 5.91537i 0.364066i
\(265\) 0 0
\(266\) 35.1105 2.15276
\(267\) 8.35627i 0.511396i
\(268\) 3.79622 0.231891
\(269\) 0.458895 0.0279793 0.0139897 0.999902i \(-0.495547\pi\)
0.0139897 + 0.999902i \(0.495547\pi\)
\(270\) 0 0
\(271\) 28.8897 1.75492 0.877462 0.479645i \(-0.159235\pi\)
0.877462 + 0.479645i \(0.159235\pi\)
\(272\) 4.96343i 0.300952i
\(273\) 16.5334i 1.00065i
\(274\) 8.97186i 0.542010i
\(275\) 0 0
\(276\) 1.72772i 0.103996i
\(277\) 22.4482i 1.34878i −0.738374 0.674391i \(-0.764406\pi\)
0.738374 0.674391i \(-0.235594\pi\)
\(278\) 5.04168i 0.302380i
\(279\) 6.55937i 0.392699i
\(280\) 0 0
\(281\) 6.04908i 0.360858i −0.983588 0.180429i \(-0.942251\pi\)
0.983588 0.180429i \(-0.0577486\pi\)
\(282\) 4.25463i 0.253360i
\(283\) 10.8317i 0.643878i 0.946760 + 0.321939i \(0.104335\pi\)
−0.946760 + 0.321939i \(0.895665\pi\)
\(284\) 8.47719 0.503028
\(285\) 0 0
\(286\) −20.9203 −1.23704
\(287\) 2.54155 0.150023
\(288\) 2.02389i 0.119259i
\(289\) −7.63567 −0.449157
\(290\) 0 0
\(291\) 3.60364i 0.211249i
\(292\) 9.05445 0.529872
\(293\) 23.3144 1.36204 0.681021 0.732264i \(-0.261536\pi\)
0.681021 + 0.732264i \(0.261536\pi\)
\(294\) 15.7466i 0.918357i
\(295\) 0 0
\(296\) 3.96343 4.61424i 0.230370 0.268197i
\(297\) −29.7182 −1.72442
\(298\) 21.6451i 1.25387i
\(299\) −6.11023 −0.353364
\(300\) 0 0
\(301\) 8.46945i 0.488171i
\(302\) 6.47544i 0.372620i
\(303\) −2.27647 −0.130780
\(304\) 7.33092i 0.420457i
\(305\) 0 0
\(306\) −10.0454 −0.574260
\(307\) −14.2651 −0.814153 −0.407077 0.913394i \(-0.633452\pi\)
−0.407077 + 0.913394i \(0.633452\pi\)
\(308\) −28.6755 −1.63394
\(309\) 11.0373i 0.627891i
\(310\) 0 0
\(311\) 22.5322i 1.27768i −0.769339 0.638841i \(-0.779414\pi\)
0.769339 0.638841i \(-0.220586\pi\)
\(312\) 3.45211 0.195437
\(313\) 10.0000i 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 13.1689i 0.743162i
\(315\) 0 0
\(316\) 5.56622i 0.313124i
\(317\) 12.3877 0.695764 0.347882 0.937538i \(-0.386901\pi\)
0.347882 + 0.937538i \(0.386901\pi\)
\(318\) 3.61837i 0.202908i
\(319\) 47.0007i 2.63154i
\(320\) 0 0
\(321\) −6.43975 −0.359432
\(322\) −8.37532 −0.466738
\(323\) 36.3865 2.02460
\(324\) −1.16780 −0.0648777
\(325\) 0 0
\(326\) −18.7651 −1.03930
\(327\) 1.32204i 0.0731092i
\(328\) 0.530665i 0.0293011i
\(329\) 20.6249 1.13709
\(330\) 0 0
\(331\) 19.0068i 1.04471i 0.852728 + 0.522355i \(0.174946\pi\)
−0.852728 + 0.522355i \(0.825054\pi\)
\(332\) −3.77680 −0.207279
\(333\) −9.33871 8.02155i −0.511758 0.439578i
\(334\) 7.19692 0.393798
\(335\) 0 0
\(336\) 4.73182 0.258142
\(337\) 25.1046 1.36753 0.683766 0.729701i \(-0.260341\pi\)
0.683766 + 0.729701i \(0.260341\pi\)
\(338\) 0.791278i 0.0430399i
\(339\) 1.37438i 0.0746459i
\(340\) 0 0
\(341\) 19.4047i 1.05083i
\(342\) 14.8370 0.802292
\(343\) −42.8078 −2.31140
\(344\) 1.76838 0.0953449
\(345\) 0 0
\(346\) 9.79406i 0.526532i
\(347\) 30.4282i 1.63347i −0.577011 0.816736i \(-0.695781\pi\)
0.577011 0.816736i \(-0.304219\pi\)
\(348\) 7.75571i 0.415750i
\(349\) −15.1341 −0.810111 −0.405056 0.914292i \(-0.632748\pi\)
−0.405056 + 0.914292i \(0.632748\pi\)
\(350\) 0 0
\(351\) 17.3430i 0.925702i
\(352\) 5.98732i 0.319125i
\(353\) 10.1331i 0.539332i −0.962954 0.269666i \(-0.913087\pi\)
0.962954 0.269666i \(-0.0869133\pi\)
\(354\) −2.12525 −0.112956
\(355\) 0 0
\(356\) 8.45791i 0.448268i
\(357\) 23.4861i 1.24301i
\(358\) −11.2829 −0.596317
\(359\) 26.3685 1.39168 0.695838 0.718199i \(-0.255033\pi\)
0.695838 + 0.718199i \(0.255033\pi\)
\(360\) 0 0
\(361\) −34.7424 −1.82855
\(362\) 1.82697i 0.0960235i
\(363\) −24.5494 −1.28851
\(364\) 16.7345i 0.877128i
\(365\) 0 0
\(366\) 3.02900 0.158328
\(367\) −2.21157 −0.115443 −0.0577215 0.998333i \(-0.518384\pi\)
−0.0577215 + 0.998333i \(0.518384\pi\)
\(368\) 1.74873i 0.0911588i
\(369\) 1.07401 0.0559106
\(370\) 0 0
\(371\) −17.5405 −0.910658
\(372\) 3.20203i 0.166017i
\(373\) 6.08275 0.314953 0.157476 0.987523i \(-0.449664\pi\)
0.157476 + 0.987523i \(0.449664\pi\)
\(374\) −29.7177 −1.53666
\(375\) 0 0
\(376\) 4.30638i 0.222085i
\(377\) 27.4288 1.41266
\(378\) 23.7721i 1.22271i
\(379\) 20.0919 1.03205 0.516025 0.856574i \(-0.327411\pi\)
0.516025 + 0.856574i \(0.327411\pi\)
\(380\) 0 0
\(381\) 7.41999 0.380138
\(382\) −23.8204 −1.21876
\(383\) 11.5368i 0.589501i 0.955574 + 0.294751i \(0.0952366\pi\)
−0.955574 + 0.294751i \(0.904763\pi\)
\(384\) 0.987983i 0.0504178i
\(385\) 0 0
\(386\) −8.40405 −0.427755
\(387\) 3.57901i 0.181932i
\(388\) 3.64747i 0.185172i
\(389\) 8.55057i 0.433531i 0.976224 + 0.216766i \(0.0695507\pi\)
−0.976224 + 0.216766i \(0.930449\pi\)
\(390\) 0 0
\(391\) −8.67970 −0.438951
\(392\) 15.9381i 0.804995i
\(393\) 6.12525i 0.308978i
\(394\) 4.22797i 0.213002i
\(395\) 0 0
\(396\) −12.1177 −0.608936
\(397\) −9.84444 −0.494078 −0.247039 0.969006i \(-0.579458\pi\)
−0.247039 + 0.969006i \(0.579458\pi\)
\(398\) −17.7545 −0.889951
\(399\) 34.6886i 1.73660i
\(400\) 0 0
\(401\) 31.0959i 1.55285i 0.630207 + 0.776427i \(0.282970\pi\)
−0.630207 + 0.776427i \(0.717030\pi\)
\(402\) 3.75060i 0.187063i
\(403\) −11.3243 −0.564102
\(404\) −2.30416 −0.114636
\(405\) 0 0
\(406\) 37.5968 1.86590
\(407\) −27.6269 23.7304i −1.36942 1.17627i
\(408\) 4.90379 0.242774
\(409\) 38.3150i 1.89455i 0.320417 + 0.947277i \(0.396177\pi\)
−0.320417 + 0.947277i \(0.603823\pi\)
\(410\) 0 0
\(411\) 8.86404 0.437231
\(412\) 11.1716i 0.550383i
\(413\) 10.3024i 0.506948i
\(414\) −3.53924 −0.173944
\(415\) 0 0
\(416\) 3.49410 0.171312
\(417\) 4.98109 0.243925
\(418\) 43.8926 2.14686
\(419\) −29.0183 −1.41764 −0.708819 0.705391i \(-0.750772\pi\)
−0.708819 + 0.705391i \(0.750772\pi\)
\(420\) 0 0
\(421\) 23.6443i 1.15235i 0.817326 + 0.576175i \(0.195455\pi\)
−0.817326 + 0.576175i \(0.804545\pi\)
\(422\) 4.57321i 0.222621i
\(423\) 8.71564 0.423769
\(424\) 3.66238i 0.177861i
\(425\) 0 0
\(426\) 8.37532i 0.405785i
\(427\) 14.6834i 0.710582i
\(428\) −6.51807 −0.315063
\(429\) 20.6689i 0.997904i
\(430\) 0 0
\(431\) 32.1126i 1.54681i −0.633912 0.773406i \(-0.718552\pi\)
0.633912 0.773406i \(-0.281448\pi\)
\(432\) 4.96352 0.238807
\(433\) 13.0575 0.627504 0.313752 0.949505i \(-0.398414\pi\)
0.313752 + 0.949505i \(0.398414\pi\)
\(434\) −15.5222 −0.745091
\(435\) 0 0
\(436\) 1.33812i 0.0640846i
\(437\) 12.8198 0.613254
\(438\) 8.94565i 0.427440i
\(439\) 28.0499i 1.33875i −0.742924 0.669375i \(-0.766562\pi\)
0.742924 0.669375i \(-0.233438\pi\)
\(440\) 0 0
\(441\) −32.2569 −1.53604
\(442\) 17.3427i 0.824909i
\(443\) −6.79629 −0.322902 −0.161451 0.986881i \(-0.551617\pi\)
−0.161451 + 0.986881i \(0.551617\pi\)
\(444\) 4.55879 + 3.91580i 0.216351 + 0.185836i
\(445\) 0 0
\(446\) 20.3205i 0.962203i
\(447\) −21.3850 −1.01148
\(448\) 4.78937 0.226277
\(449\) 17.3278i 0.817747i −0.912591 0.408874i \(-0.865922\pi\)
0.912591 0.408874i \(-0.134078\pi\)
\(450\) 0 0
\(451\) 3.17726 0.149611
\(452\) 1.39109i 0.0654315i
\(453\) −6.39762 −0.300587
\(454\) −5.74303 −0.269534
\(455\) 0 0
\(456\) −7.24283 −0.339176
\(457\) 3.20960i 0.150139i −0.997178 0.0750693i \(-0.976082\pi\)
0.997178 0.0750693i \(-0.0239178\pi\)
\(458\) 23.4383i 1.09520i
\(459\) 24.6361i 1.14991i
\(460\) 0 0
\(461\) 34.0347i 1.58515i −0.609773 0.792576i \(-0.708739\pi\)
0.609773 0.792576i \(-0.291261\pi\)
\(462\) 28.3309i 1.31807i
\(463\) 26.0543i 1.21085i 0.795904 + 0.605423i \(0.206996\pi\)
−0.795904 + 0.605423i \(0.793004\pi\)
\(464\) 7.85004i 0.364429i
\(465\) 0 0
\(466\) 17.2961i 0.801224i
\(467\) 20.3289i 0.940710i 0.882477 + 0.470355i \(0.155874\pi\)
−0.882477 + 0.470355i \(0.844126\pi\)
\(468\) 7.07167i 0.326888i
\(469\) 18.1815 0.839544
\(470\) 0 0
\(471\) 13.0106 0.599498
\(472\) −2.15110 −0.0990123
\(473\) 10.5879i 0.486832i
\(474\) −5.49933 −0.252593
\(475\) 0 0
\(476\) 23.7717i 1.08958i
\(477\) −7.41226 −0.339384
\(478\) −3.68796 −0.168683
\(479\) 6.22972i 0.284643i −0.989820 0.142322i \(-0.954543\pi\)
0.989820 0.142322i \(-0.0454567\pi\)
\(480\) 0 0
\(481\) 13.8486 16.1226i 0.631443 0.735127i
\(482\) 11.3613 0.517492
\(483\) 8.27467i 0.376511i
\(484\) −24.8480 −1.12946
\(485\) 0 0
\(486\) 16.0443i 0.727785i
\(487\) 30.5352i 1.38368i 0.722050 + 0.691841i \(0.243200\pi\)
−0.722050 + 0.691841i \(0.756800\pi\)
\(488\) 3.06584 0.138784
\(489\) 18.5396i 0.838387i
\(490\) 0 0
\(491\) 24.5944 1.10993 0.554965 0.831874i \(-0.312732\pi\)
0.554965 + 0.831874i \(0.312732\pi\)
\(492\) −0.524288 −0.0236367
\(493\) 38.9632 1.75481
\(494\) 25.6150i 1.15247i
\(495\) 0 0
\(496\) 3.24097i 0.145524i
\(497\) 40.6004 1.82118
\(498\) 3.73141i 0.167209i
\(499\) 33.5739i 1.50297i −0.659747 0.751487i \(-0.729337\pi\)
0.659747 0.751487i \(-0.270663\pi\)
\(500\) 0 0
\(501\) 7.11043i 0.317671i
\(502\) 12.8260 0.572451
\(503\) 1.77931i 0.0793357i −0.999213 0.0396679i \(-0.987370\pi\)
0.999213 0.0396679i \(-0.0126300\pi\)
\(504\) 9.69316i 0.431768i
\(505\) 0 0
\(506\) −10.4702 −0.465458
\(507\) −0.781770 −0.0347196
\(508\) 7.51024 0.333213
\(509\) −29.5114 −1.30807 −0.654035 0.756464i \(-0.726925\pi\)
−0.654035 + 0.756464i \(0.726925\pi\)
\(510\) 0 0
\(511\) 43.3651 1.91836
\(512\) 1.00000i 0.0441942i
\(513\) 36.3872i 1.60653i
\(514\) 14.0268 0.618694
\(515\) 0 0
\(516\) 1.74713i 0.0769133i
\(517\) 25.7837 1.13397
\(518\) 18.9824 22.0993i 0.834037 0.970988i
\(519\) −9.67637 −0.424745
\(520\) 0 0
\(521\) 29.2841 1.28296 0.641481 0.767139i \(-0.278320\pi\)
0.641481 + 0.767139i \(0.278320\pi\)
\(522\) 15.8876 0.695382
\(523\) 29.9729i 1.31062i −0.755359 0.655312i \(-0.772537\pi\)
0.755359 0.655312i \(-0.227463\pi\)
\(524\) 6.19975i 0.270837i
\(525\) 0 0
\(526\) 3.57182i 0.155739i
\(527\) −16.0864 −0.700732
\(528\) 5.91537 0.257434
\(529\) 19.9419 0.867041
\(530\) 0 0
\(531\) 4.35359i 0.188930i
\(532\) 35.1105i 1.52223i
\(533\) 1.85420i 0.0803141i
\(534\) −8.35627 −0.361611
\(535\) 0 0
\(536\) 3.79622i 0.163972i
\(537\) 11.1473i 0.481040i
\(538\) 0.458895i 0.0197844i
\(539\) −95.4264 −4.11031
\(540\) 0 0
\(541\) 18.7155i 0.804644i 0.915498 + 0.402322i \(0.131797\pi\)
−0.915498 + 0.402322i \(0.868203\pi\)
\(542\) 28.8897i 1.24092i
\(543\) −1.80502 −0.0774607
\(544\) 4.96343 0.212805
\(545\) 0 0
\(546\) 16.5334 0.707566
\(547\) 32.4792i 1.38871i 0.719633 + 0.694355i \(0.244310\pi\)
−0.719633 + 0.694355i \(0.755690\pi\)
\(548\) 8.97186 0.383259
\(549\) 6.20492i 0.264820i
\(550\) 0 0
\(551\) −57.5480 −2.45163
\(552\) 1.72772 0.0735364
\(553\) 26.6587i 1.13364i
\(554\) 22.4482 0.953733
\(555\) 0 0
\(556\) 5.04168 0.213815
\(557\) 16.0158i 0.678612i 0.940676 + 0.339306i \(0.110192\pi\)
−0.940676 + 0.339306i \(0.889808\pi\)
\(558\) −6.55937 −0.277680
\(559\) 6.17891 0.261340
\(560\) 0 0
\(561\) 29.3606i 1.23960i
\(562\) 6.04908 0.255165
\(563\) 15.3069i 0.645109i −0.946551 0.322554i \(-0.895458\pi\)
0.946551 0.322554i \(-0.104542\pi\)
\(564\) −4.25463 −0.179152
\(565\) 0 0
\(566\) −10.8317 −0.455291
\(567\) −5.59302 −0.234885
\(568\) 8.47719i 0.355695i
\(569\) 26.6494i 1.11720i 0.829437 + 0.558601i \(0.188662\pi\)
−0.829437 + 0.558601i \(0.811338\pi\)
\(570\) 0 0
\(571\) 23.1683 0.969564 0.484782 0.874635i \(-0.338899\pi\)
0.484782 + 0.874635i \(0.338899\pi\)
\(572\) 20.9203i 0.874721i
\(573\) 23.5341i 0.983152i
\(574\) 2.54155i 0.106082i
\(575\) 0 0
\(576\) 2.02389 0.0843287
\(577\) 3.54110i 0.147418i 0.997280 + 0.0737091i \(0.0234836\pi\)
−0.997280 + 0.0737091i \(0.976516\pi\)
\(578\) 7.63567i 0.317602i
\(579\) 8.30306i 0.345063i
\(580\) 0 0
\(581\) −18.0885 −0.750437
\(582\) −3.60364 −0.149376
\(583\) −21.9279 −0.908160
\(584\) 9.05445i 0.374676i
\(585\) 0 0
\(586\) 23.3144i 0.963109i
\(587\) 12.0976i 0.499320i −0.968334 0.249660i \(-0.919681\pi\)
0.968334 0.249660i \(-0.0803188\pi\)
\(588\) 15.7466 0.649377
\(589\) 23.7593 0.978986
\(590\) 0 0
\(591\) −4.17716 −0.171825
\(592\) 4.61424 + 3.96343i 0.189644 + 0.162896i
\(593\) 17.2007 0.706348 0.353174 0.935558i \(-0.385102\pi\)
0.353174 + 0.935558i \(0.385102\pi\)
\(594\) 29.7182i 1.21935i
\(595\) 0 0
\(596\) −21.6451 −0.886619
\(597\) 17.5411i 0.717910i
\(598\) 6.11023i 0.249866i
\(599\) −2.41919 −0.0988454 −0.0494227 0.998778i \(-0.515738\pi\)
−0.0494227 + 0.998778i \(0.515738\pi\)
\(600\) 0 0
\(601\) −39.0483 −1.59281 −0.796407 0.604761i \(-0.793269\pi\)
−0.796407 + 0.604761i \(0.793269\pi\)
\(602\) 8.46945 0.345189
\(603\) 7.68313 0.312881
\(604\) −6.47544 −0.263482
\(605\) 0 0
\(606\) 2.27647i 0.0924752i
\(607\) 38.0508i 1.54443i 0.635359 + 0.772217i \(0.280852\pi\)
−0.635359 + 0.772217i \(0.719148\pi\)
\(608\) −7.33092 −0.297308
\(609\) 37.1450i 1.50519i
\(610\) 0 0
\(611\) 15.0469i 0.608733i
\(612\) 10.0454i 0.406063i
\(613\) 7.12176 0.287645 0.143823 0.989603i \(-0.454061\pi\)
0.143823 + 0.989603i \(0.454061\pi\)
\(614\) 14.2651i 0.575693i
\(615\) 0 0
\(616\) 28.6755i 1.15537i
\(617\) −42.6678 −1.71774 −0.858871 0.512192i \(-0.828834\pi\)
−0.858871 + 0.512192i \(0.828834\pi\)
\(618\) −11.0373 −0.443986
\(619\) 6.30429 0.253391 0.126695 0.991942i \(-0.459563\pi\)
0.126695 + 0.991942i \(0.459563\pi\)
\(620\) 0 0
\(621\) 8.67985i 0.348310i
\(622\) 22.5322 0.903458
\(623\) 40.5081i 1.62292i
\(624\) 3.45211i 0.138195i
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 43.3651i 1.73184i
\(628\) 13.1689 0.525495
\(629\) 19.6722 22.9025i 0.784383 0.913181i
\(630\) 0 0
\(631\) 32.6385i 1.29932i 0.760225 + 0.649660i \(0.225089\pi\)
−0.760225 + 0.649660i \(0.774911\pi\)
\(632\) −5.56622 −0.221412
\(633\) −4.51826 −0.179585
\(634\) 12.3877i 0.491979i
\(635\) 0 0
\(636\) 3.61837 0.143478
\(637\) 55.6892i 2.20649i
\(638\) 47.0007 1.86078
\(639\) 17.1569 0.678716
\(640\) 0 0
\(641\) −12.1721 −0.480767 −0.240384 0.970678i \(-0.577273\pi\)
−0.240384 + 0.970678i \(0.577273\pi\)
\(642\) 6.43975i 0.254156i
\(643\) 10.0853i 0.397727i 0.980027 + 0.198864i \(0.0637251\pi\)
−0.980027 + 0.198864i \(0.936275\pi\)
\(644\) 8.37532i 0.330034i
\(645\) 0 0
\(646\) 36.3865i 1.43161i
\(647\) 38.5061i 1.51383i 0.653512 + 0.756916i \(0.273295\pi\)
−0.653512 + 0.756916i \(0.726705\pi\)
\(648\) 1.16780i 0.0458754i
\(649\) 12.8793i 0.505558i
\(650\) 0 0
\(651\) 15.3357i 0.601053i
\(652\) 18.7651i 0.734896i
\(653\) 2.20349i 0.0862293i −0.999070 0.0431146i \(-0.986272\pi\)
0.999070 0.0431146i \(-0.0137281\pi\)
\(654\) −1.32204 −0.0516960
\(655\) 0 0
\(656\) −0.530665 −0.0207190
\(657\) 18.3252 0.714935
\(658\) 20.6249i 0.804041i
\(659\) 20.4048 0.794859 0.397430 0.917633i \(-0.369902\pi\)
0.397430 + 0.917633i \(0.369902\pi\)
\(660\) 0 0
\(661\) 27.0306i 1.05137i 0.850680 + 0.525684i \(0.176191\pi\)
−0.850680 + 0.525684i \(0.823809\pi\)
\(662\) −19.0068 −0.738721
\(663\) 17.1343 0.665442
\(664\) 3.77680i 0.146568i
\(665\) 0 0
\(666\) 8.02155 9.33871i 0.310829 0.361868i
\(667\) 13.7276 0.531535
\(668\) 7.19692i 0.278457i
\(669\) −20.0763 −0.776194
\(670\) 0 0
\(671\) 18.3562i 0.708632i
\(672\) 4.73182i 0.182534i
\(673\) 11.6776 0.450139 0.225070 0.974343i \(-0.427739\pi\)
0.225070 + 0.974343i \(0.427739\pi\)
\(674\) 25.1046i 0.966991i
\(675\) 0 0
\(676\) −0.791278 −0.0304338
\(677\) −37.2578 −1.43193 −0.715966 0.698135i \(-0.754014\pi\)
−0.715966 + 0.698135i \(0.754014\pi\)
\(678\) −1.37438 −0.0527826
\(679\) 17.4691i 0.670402i
\(680\) 0 0
\(681\) 5.67402i 0.217429i
\(682\) −19.4047 −0.743046
\(683\) 46.3018i 1.77169i −0.463981 0.885845i \(-0.653580\pi\)
0.463981 0.885845i \(-0.346420\pi\)
\(684\) 14.8370i 0.567306i
\(685\) 0 0
\(686\) 42.8078i 1.63441i
\(687\) 23.1566 0.883480
\(688\) 1.76838i 0.0674190i
\(689\) 12.7967i 0.487516i
\(690\) 0 0
\(691\) 29.7037 1.12998 0.564991 0.825097i \(-0.308879\pi\)
0.564991 + 0.825097i \(0.308879\pi\)
\(692\) −9.79406 −0.372314
\(693\) −58.0361 −2.20461
\(694\) 30.4282 1.15504
\(695\) 0 0
\(696\) −7.75571 −0.293979
\(697\) 2.63392i 0.0997669i
\(698\) 15.1341i 0.572835i
\(699\) 17.0882 0.646336
\(700\) 0 0
\(701\) 45.9214i 1.73443i −0.497936 0.867214i \(-0.665909\pi\)
0.497936 0.867214i \(-0.334091\pi\)
\(702\) 17.3430 0.654570
\(703\) −29.0556 + 33.8266i −1.09585 + 1.27580i
\(704\) 5.98732 0.225656
\(705\) 0 0
\(706\) 10.1331 0.381365
\(707\) −11.0355 −0.415031
\(708\) 2.12525i 0.0798717i
\(709\) 52.6575i 1.97759i 0.149266 + 0.988797i \(0.452309\pi\)
−0.149266 + 0.988797i \(0.547691\pi\)
\(710\) 0 0
\(711\) 11.2654i 0.422486i
\(712\) −8.45791 −0.316974
\(713\) −5.66759 −0.212253
\(714\) 23.4861 0.878944
\(715\) 0 0
\(716\) 11.2829i 0.421660i
\(717\) 3.64364i 0.136074i
\(718\) 26.3685i 0.984063i
\(719\) 13.3494 0.497850 0.248925 0.968523i \(-0.419923\pi\)
0.248925 + 0.968523i \(0.419923\pi\)
\(720\) 0 0
\(721\) 53.5048i 1.99262i
\(722\) 34.7424i 1.29298i
\(723\) 11.2247i 0.417453i
\(724\) −1.82697 −0.0678989
\(725\) 0 0
\(726\) 24.5494i 0.911115i
\(727\) 13.1733i 0.488571i −0.969703 0.244286i \(-0.921447\pi\)
0.969703 0.244286i \(-0.0785534\pi\)
\(728\) 16.7345 0.620223
\(729\) 12.3481 0.457338
\(730\) 0 0
\(731\) 8.77726 0.324639
\(732\) 3.02900i 0.111955i
\(733\) −32.3264 −1.19400 −0.597001 0.802241i \(-0.703641\pi\)
−0.597001 + 0.802241i \(0.703641\pi\)
\(734\) 2.21157i 0.0816305i
\(735\) 0 0
\(736\) 1.74873 0.0644590
\(737\) 22.7292 0.837240
\(738\) 1.07401i 0.0395348i
\(739\) −6.94169 −0.255354 −0.127677 0.991816i \(-0.540752\pi\)
−0.127677 + 0.991816i \(0.540752\pi\)
\(740\) 0 0
\(741\) −25.3072 −0.929681
\(742\) 17.5405i 0.643933i
\(743\) −36.0784 −1.32359 −0.661795 0.749685i \(-0.730205\pi\)
−0.661795 + 0.749685i \(0.730205\pi\)
\(744\) 3.20203 0.117392
\(745\) 0 0
\(746\) 6.08275i 0.222705i
\(747\) −7.64382 −0.279673
\(748\) 29.7177i 1.08659i
\(749\) −31.2175 −1.14066
\(750\) 0 0
\(751\) −29.9052 −1.09126 −0.545629 0.838027i \(-0.683709\pi\)
−0.545629 + 0.838027i \(0.683709\pi\)
\(752\) −4.30638 −0.157038
\(753\) 12.6718i 0.461787i
\(754\) 27.4288i 0.998898i
\(755\) 0 0
\(756\) 23.7721 0.864584
\(757\) 24.9464i 0.906693i −0.891334 0.453347i \(-0.850230\pi\)
0.891334 0.453347i \(-0.149770\pi\)
\(758\) 20.0919i 0.729769i
\(759\) 10.3444i 0.375477i
\(760\) 0 0
\(761\) 11.7647 0.426469 0.213235 0.977001i \(-0.431600\pi\)
0.213235 + 0.977001i \(0.431600\pi\)
\(762\) 7.41999i 0.268798i
\(763\) 6.40878i 0.232013i
\(764\) 23.8204i 0.861791i
\(765\) 0 0
\(766\) −11.5368 −0.416840
\(767\) −7.51615 −0.271392
\(768\) −0.987983 −0.0356508
\(769\) 26.7939i 0.966214i −0.875561 0.483107i \(-0.839508\pi\)
0.875561 0.483107i \(-0.160492\pi\)
\(770\) 0 0
\(771\) 13.8582i 0.499091i
\(772\) 8.40405i 0.302468i
\(773\) 2.77358 0.0997589 0.0498794 0.998755i \(-0.484116\pi\)
0.0498794 + 0.998755i \(0.484116\pi\)
\(774\) 3.57901 0.128645
\(775\) 0 0
\(776\) −3.64747 −0.130937
\(777\) 21.8337 + 18.7542i 0.783281 + 0.672805i
\(778\) −8.55057 −0.306553
\(779\) 3.89026i 0.139383i
\(780\) 0 0
\(781\) 50.7556 1.81618
\(782\) 8.67970i 0.310386i
\(783\) 38.9638i 1.39245i
\(784\) 15.9381 0.569217
\(785\) 0 0
\(786\) −6.12525 −0.218480
\(787\) 28.0910 1.00134 0.500669 0.865639i \(-0.333088\pi\)
0.500669 + 0.865639i \(0.333088\pi\)
\(788\) −4.22797 −0.150615
\(789\) −3.52890 −0.125632
\(790\) 0 0
\(791\) 6.66246i 0.236890i
\(792\) 12.1177i 0.430583i
\(793\) 10.7123 0.380407
\(794\) 9.84444i 0.349366i
\(795\) 0 0
\(796\) 17.7545i 0.629290i
\(797\) 1.36472i 0.0483407i −0.999708 0.0241704i \(-0.992306\pi\)
0.999708 0.0241704i \(-0.00769441\pi\)
\(798\) −34.6886 −1.22796
\(799\) 21.3744i 0.756173i
\(800\) 0 0
\(801\) 17.1179i 0.604831i
\(802\) −31.0959 −1.09803
\(803\) 54.2119 1.91310
\(804\) −3.75060 −0.132273
\(805\) 0 0
\(806\) 11.3243i 0.398881i
\(807\) −0.453381 −0.0159598
\(808\) 2.30416i 0.0810600i
\(809\) 0.636644i 0.0223832i −0.999937 0.0111916i \(-0.996438\pi\)
0.999937 0.0111916i \(-0.00356247\pi\)
\(810\) 0 0
\(811\) 11.3776 0.399521 0.199760 0.979845i \(-0.435984\pi\)
0.199760 + 0.979845i \(0.435984\pi\)
\(812\) 37.5968i 1.31939i
\(813\) −28.5425 −1.00103
\(814\) 23.7304 27.6269i 0.831748 0.968324i
\(815\) 0 0
\(816\) 4.90379i 0.171667i
\(817\) −12.9639 −0.453549
\(818\) −38.3150 −1.33965
\(819\) 33.8688i 1.18347i
\(820\) 0 0
\(821\) −26.1807 −0.913711 −0.456856 0.889541i \(-0.651024\pi\)
−0.456856 + 0.889541i \(0.651024\pi\)
\(822\) 8.86404i 0.309169i
\(823\) −21.7020 −0.756485 −0.378242 0.925707i \(-0.623471\pi\)
−0.378242 + 0.925707i \(0.623471\pi\)
\(824\) −11.1716 −0.389180
\(825\) 0 0
\(826\) −10.3024 −0.358467
\(827\) 7.90798i 0.274988i −0.990503 0.137494i \(-0.956095\pi\)
0.990503 0.137494i \(-0.0439047\pi\)
\(828\) 3.53924i 0.122997i
\(829\) 21.7391i 0.755030i 0.926003 + 0.377515i \(0.123221\pi\)
−0.926003 + 0.377515i \(0.876779\pi\)
\(830\) 0 0
\(831\) 22.1785i 0.769362i
\(832\) 3.49410i 0.121136i
\(833\) 79.1076i 2.74092i
\(834\) 4.98109i 0.172481i
\(835\) 0 0
\(836\) 43.8926i 1.51806i
\(837\) 16.0866i 0.556035i
\(838\) 29.0183i 1.00242i
\(839\) −11.3258 −0.391011 −0.195506 0.980703i \(-0.562635\pi\)
−0.195506 + 0.980703i \(0.562635\pi\)
\(840\) 0 0
\(841\) −32.6231 −1.12494
\(842\) −23.6443 −0.814835
\(843\) 5.97639i 0.205838i
\(844\) −4.57321 −0.157417
\(845\) 0 0
\(846\) 8.71564i 0.299650i
\(847\) −119.006 −4.08911
\(848\) 3.66238 0.125767
\(849\) 10.7015i 0.367276i
\(850\) 0 0
\(851\) 6.93097 8.06906i 0.237591 0.276604i
\(852\) −8.37532 −0.286934
\(853\) 48.1277i 1.64786i 0.566691 + 0.823930i \(0.308223\pi\)
−0.566691 + 0.823930i \(0.691777\pi\)
\(854\) 14.6834 0.502457
\(855\) 0 0
\(856\) 6.51807i 0.222783i
\(857\) 40.0486i 1.36803i −0.729466 0.684017i \(-0.760231\pi\)
0.729466 0.684017i \(-0.239769\pi\)
\(858\) 20.6689 0.705624
\(859\) 7.66509i 0.261530i 0.991413 + 0.130765i \(0.0417433\pi\)
−0.991413 + 0.130765i \(0.958257\pi\)
\(860\) 0 0
\(861\) −2.51101 −0.0855750
\(862\) 32.1126 1.09376
\(863\) −50.9196 −1.73332 −0.866662 0.498896i \(-0.833739\pi\)
−0.866662 + 0.498896i \(0.833739\pi\)
\(864\) 4.96352i 0.168862i
\(865\) 0 0
\(866\) 13.0575i 0.443712i
\(867\) 7.54391 0.256205
\(868\) 15.5222i 0.526859i
\(869\) 33.3267i 1.13053i
\(870\) 0 0
\(871\) 13.2644i 0.449446i
\(872\) −1.33812 −0.0453146
\(873\) 7.38208i 0.249846i
\(874\) 12.8198i 0.433636i
\(875\) 0 0
\(876\) −8.94565 −0.302245
\(877\) 2.50813 0.0846937 0.0423468 0.999103i \(-0.486517\pi\)
0.0423468 + 0.999103i \(0.486517\pi\)
\(878\) 28.0499 0.946640
\(879\) −23.0342 −0.776925
\(880\) 0 0
\(881\) 24.0074 0.808829 0.404414 0.914576i \(-0.367475\pi\)
0.404414 + 0.914576i \(0.367475\pi\)
\(882\) 32.2569i 1.08615i
\(883\) 34.7959i 1.17098i −0.810681 0.585488i \(-0.800903\pi\)
0.810681 0.585488i \(-0.199097\pi\)
\(884\) 17.3427 0.583299
\(885\) 0 0
\(886\) 6.79629i 0.228326i
\(887\) −0.672248 −0.0225719 −0.0112859 0.999936i \(-0.503592\pi\)
−0.0112859 + 0.999936i \(0.503592\pi\)
\(888\) −3.91580 + 4.55879i −0.131406 + 0.152983i
\(889\) 35.9693 1.20637
\(890\) 0 0
\(891\) −6.99198 −0.234240
\(892\) −20.3205 −0.680380
\(893\) 31.5698i 1.05644i
\(894\) 21.3850i 0.715222i
\(895\) 0 0
\(896\) 4.78937i 0.160002i
\(897\) 6.03681 0.201563
\(898\) 17.3278 0.578235
\(899\) 25.4418 0.848531
\(900\) 0 0
\(901\) 18.1780i 0.605597i
\(902\) 3.17726i 0.105791i
\(903\) 8.36767i 0.278459i
\(904\) −1.39109 −0.0462671
\(905\) 0 0
\(906\) 6.39762i 0.212547i
\(907\) 36.6618i 1.21734i 0.793425 + 0.608668i \(0.208296\pi\)
−0.793425 + 0.608668i \(0.791704\pi\)
\(908\) 5.74303i 0.190589i
\(909\) −4.66336 −0.154674
\(910\) 0 0
\(911\) 20.5358i 0.680381i −0.940356 0.340191i \(-0.889508\pi\)
0.940356 0.340191i \(-0.110492\pi\)
\(912\) 7.24283i 0.239834i
\(913\) −22.6129 −0.748378
\(914\) 3.20960 0.106164
\(915\) 0 0
\(916\) 23.4383 0.774423
\(917\) 29.6929i 0.980546i
\(918\) 24.6361 0.813112
\(919\) 42.3926i 1.39840i 0.714925 + 0.699202i \(0.246461\pi\)
−0.714925 + 0.699202i \(0.753539\pi\)
\(920\) 0 0
\(921\) 14.0937 0.464403
\(922\) 34.0347 1.12087
\(923\) 29.6201i 0.974958i
\(924\) 28.3309 0.932019
\(925\) 0 0
\(926\) −26.0543 −0.856197
\(927\) 22.6100i 0.742610i
\(928\) −7.85004 −0.257690
\(929\) −1.22700 −0.0402565 −0.0201282 0.999797i \(-0.506407\pi\)
−0.0201282 + 0.999797i \(0.506407\pi\)
\(930\) 0 0
\(931\) 116.841i 3.82930i
\(932\) 17.2961 0.566551
\(933\) 22.2614i 0.728806i
\(934\) −20.3289 −0.665183
\(935\) 0 0
\(936\) 7.07167 0.231145
\(937\) 31.9456 1.04362 0.521808 0.853063i \(-0.325258\pi\)
0.521808 + 0.853063i \(0.325258\pi\)
\(938\) 18.1815i 0.593647i
\(939\) 9.87983i 0.322416i
\(940\) 0 0
\(941\) 6.72808 0.219329 0.109665 0.993969i \(-0.465022\pi\)
0.109665 + 0.993969i \(0.465022\pi\)
\(942\) 13.0106i 0.423909i
\(943\) 0.927990i 0.0302195i
\(944\) 2.15110i 0.0700123i
\(945\) 0 0
\(946\) 10.5879 0.344242
\(947\) 3.77013i 0.122513i 0.998122 + 0.0612564i \(0.0195107\pi\)
−0.998122 + 0.0612564i \(0.980489\pi\)
\(948\) 5.49933i 0.178610i
\(949\) 31.6372i 1.02699i
\(950\) 0 0
\(951\) −12.2389 −0.396872
\(952\) 23.7717 0.770446
\(953\) 13.7237 0.444553 0.222276 0.974984i \(-0.428651\pi\)
0.222276 + 0.974984i \(0.428651\pi\)
\(954\) 7.41226i 0.239981i
\(955\) 0 0
\(956\) 3.68796i 0.119277i
\(957\) 46.4359i 1.50106i
\(958\) 6.22972 0.201273
\(959\) 42.9696 1.38756
\(960\) 0 0
\(961\) 20.4961 0.661164
\(962\) 16.1226 + 13.8486i 0.519814 + 0.446498i
\(963\) −13.1919 −0.425102
\(964\) 11.3613i 0.365922i
\(965\) 0 0
\(966\) 8.27467 0.266233
\(967\) 43.7304i 1.40627i −0.711055 0.703137i \(-0.751782\pi\)
0.711055 0.703137i \(-0.248218\pi\)
\(968\) 24.8480i 0.798646i
\(969\) −35.9493 −1.15486
\(970\) 0 0
\(971\) 36.2807 1.16430 0.582151 0.813081i \(-0.302211\pi\)
0.582151 + 0.813081i \(0.302211\pi\)
\(972\) 16.0443 0.514622
\(973\) 24.1465 0.774100
\(974\) −30.5352 −0.978410
\(975\) 0 0
\(976\) 3.06584i 0.0981351i
\(977\) 29.6513i 0.948628i −0.880356 0.474314i \(-0.842696\pi\)
0.880356 0.474314i \(-0.157304\pi\)
\(978\) 18.5396 0.592829
\(979\) 50.6402i 1.61847i
\(980\) 0 0
\(981\) 2.70822i 0.0864667i
\(982\) 24.5944i 0.784839i
\(983\) 42.0082 1.33985 0.669927 0.742427i \(-0.266325\pi\)
0.669927 + 0.742427i \(0.266325\pi\)
\(984\) 0.524288i 0.0167137i
\(985\) 0 0
\(986\) 38.9632i 1.24084i
\(987\) −20.3770 −0.648607
\(988\) −25.6150 −0.814921
\(989\) 3.09243 0.0983334
\(990\) 0 0
\(991\) 20.7399i 0.658824i −0.944186 0.329412i \(-0.893149\pi\)
0.944186 0.329412i \(-0.106851\pi\)
\(992\) 3.24097 0.102901
\(993\) 18.7784i 0.595915i
\(994\) 40.6004i 1.28777i
\(995\) 0 0
\(996\) 3.73141 0.118234
\(997\) 16.0224i 0.507435i −0.967278 0.253718i \(-0.918347\pi\)
0.967278 0.253718i \(-0.0816534\pi\)
\(998\) 33.5739 1.06276
\(999\) 22.9029 + 19.6726i 0.724614 + 0.622413i
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 1850.2.d.i.1701.14 20
5.2 odd 4 370.2.c.a.369.7 yes 10
5.3 odd 4 370.2.c.b.369.4 yes 10
5.4 even 2 inner 1850.2.d.i.1701.7 20
15.2 even 4 3330.2.e.d.739.9 10
15.8 even 4 3330.2.e.c.739.1 10
37.36 even 2 inner 1850.2.d.i.1701.4 20
185.73 odd 4 370.2.c.a.369.4 10
185.147 odd 4 370.2.c.b.369.7 yes 10
185.184 even 2 inner 1850.2.d.i.1701.17 20
555.332 even 4 3330.2.e.c.739.2 10
555.443 even 4 3330.2.e.d.739.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.c.a.369.4 10 185.73 odd 4
370.2.c.a.369.7 yes 10 5.2 odd 4
370.2.c.b.369.4 yes 10 5.3 odd 4
370.2.c.b.369.7 yes 10 185.147 odd 4
1850.2.d.i.1701.4 20 37.36 even 2 inner
1850.2.d.i.1701.7 20 5.4 even 2 inner
1850.2.d.i.1701.14 20 1.1 even 1 trivial
1850.2.d.i.1701.17 20 185.184 even 2 inner
3330.2.e.c.739.1 10 15.8 even 4
3330.2.e.c.739.2 10 555.332 even 4
3330.2.e.d.739.9 10 15.2 even 4
3330.2.e.d.739.10 10 555.443 even 4