Properties

Label 1680.2.k.h.209.7
Level $1680$
Weight $2$
Character 1680.209
Analytic conductor $13.415$
Analytic rank $0$
Dimension $24$
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] = [1680,2,Mod(209,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.k (of order \(2\), degree \(1\), not 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: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.7
Character \(\chi\) \(=\) 1680.209
Dual form 1680.2.k.h.209.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.773053 - 1.54996i) q^{3} +(0.194052 + 2.22763i) q^{5} +(-0.942148 + 2.47232i) q^{7} +(-1.80478 + 2.39641i) q^{9} +O(q^{10})\) \(q+(-0.773053 - 1.54996i) q^{3} +(0.194052 + 2.22763i) q^{5} +(-0.942148 + 2.47232i) q^{7} +(-1.80478 + 2.39641i) q^{9} -1.45449i q^{11} +4.42393 q^{13} +(3.30274 - 2.02285i) q^{15} -0.440293i q^{17} -6.54815i q^{19} +(4.56033 - 0.450938i) q^{21} -2.43747 q^{23} +(-4.92469 + 0.864552i) q^{25} +(5.10954 + 0.944786i) q^{27} +4.30871i q^{29} +2.86295i q^{31} +(-2.25441 + 1.12440i) q^{33} +(-5.69024 - 1.61900i) q^{35} +9.10153i q^{37} +(-3.41994 - 6.85694i) q^{39} -4.55874 q^{41} +8.57261i q^{43} +(-5.68854 - 3.55535i) q^{45} +6.31118i q^{47} +(-5.22471 - 4.65858i) q^{49} +(-0.682438 + 0.340370i) q^{51} -9.37352 q^{53} +(3.24008 - 0.282247i) q^{55} +(-10.1494 + 5.06207i) q^{57} -8.33478 q^{59} +7.14436i q^{61} +(-4.22432 - 6.71976i) q^{63} +(0.858472 + 9.85489i) q^{65} +11.3488i q^{67} +(1.88430 + 3.77800i) q^{69} +1.12511i q^{71} +11.2423 q^{73} +(5.14707 + 6.96474i) q^{75} +(3.59597 + 1.37035i) q^{77} +6.83987 q^{79} +(-2.48556 - 8.64997i) q^{81} -5.75536i q^{83} +(0.980811 - 0.0854397i) q^{85} +(6.67834 - 3.33086i) q^{87} -11.7153 q^{89} +(-4.16800 + 10.9374i) q^{91} +(4.43747 - 2.21322i) q^{93} +(14.5869 - 1.27068i) q^{95} +6.27856 q^{97} +(3.48556 + 2.62504i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{9} - 2 q^{15} - 2 q^{21} + 16 q^{23} + 8 q^{25} - 8 q^{35} + 2 q^{39} - 6 q^{51} + 24 q^{53} + 8 q^{57} - 16 q^{63} + 16 q^{65} + 8 q^{77} - 4 q^{79} + 18 q^{81} - 12 q^{85} - 12 q^{91} + 32 q^{93} + 24 q^{95} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.773053 1.54996i −0.446323 0.894872i
\(4\) 0 0
\(5\) 0.194052 + 2.22763i 0.0867826 + 0.996227i
\(6\) 0 0
\(7\) −0.942148 + 2.47232i −0.356099 + 0.934448i
\(8\) 0 0
\(9\) −1.80478 + 2.39641i −0.601592 + 0.798803i
\(10\) 0 0
\(11\) 1.45449i 0.438546i −0.975664 0.219273i \(-0.929631\pi\)
0.975664 0.219273i \(-0.0703686\pi\)
\(12\) 0 0
\(13\) 4.42393 1.22698 0.613489 0.789703i \(-0.289766\pi\)
0.613489 + 0.789703i \(0.289766\pi\)
\(14\) 0 0
\(15\) 3.30274 2.02285i 0.852763 0.522298i
\(16\) 0 0
\(17\) 0.440293i 0.106787i −0.998574 0.0533934i \(-0.982996\pi\)
0.998574 0.0533934i \(-0.0170037\pi\)
\(18\) 0 0
\(19\) 6.54815i 1.50225i −0.660160 0.751125i \(-0.729512\pi\)
0.660160 0.751125i \(-0.270488\pi\)
\(20\) 0 0
\(21\) 4.56033 0.450938i 0.995147 0.0984027i
\(22\) 0 0
\(23\) −2.43747 −0.508248 −0.254124 0.967172i \(-0.581787\pi\)
−0.254124 + 0.967172i \(0.581787\pi\)
\(24\) 0 0
\(25\) −4.92469 + 0.864552i −0.984938 + 0.172910i
\(26\) 0 0
\(27\) 5.10954 + 0.944786i 0.983331 + 0.181824i
\(28\) 0 0
\(29\) 4.30871i 0.800107i 0.916492 + 0.400054i \(0.131009\pi\)
−0.916492 + 0.400054i \(0.868991\pi\)
\(30\) 0 0
\(31\) 2.86295i 0.514201i 0.966385 + 0.257101i \(0.0827672\pi\)
−0.966385 + 0.257101i \(0.917233\pi\)
\(32\) 0 0
\(33\) −2.25441 + 1.12440i −0.392443 + 0.195733i
\(34\) 0 0
\(35\) −5.69024 1.61900i −0.961826 0.273661i
\(36\) 0 0
\(37\) 9.10153i 1.49628i 0.663540 + 0.748141i \(0.269053\pi\)
−0.663540 + 0.748141i \(0.730947\pi\)
\(38\) 0 0
\(39\) −3.41994 6.85694i −0.547628 1.09799i
\(40\) 0 0
\(41\) −4.55874 −0.711955 −0.355978 0.934494i \(-0.615852\pi\)
−0.355978 + 0.934494i \(0.615852\pi\)
\(42\) 0 0
\(43\) 8.57261i 1.30731i 0.756793 + 0.653655i \(0.226765\pi\)
−0.756793 + 0.653655i \(0.773235\pi\)
\(44\) 0 0
\(45\) −5.68854 3.55535i −0.847997 0.530000i
\(46\) 0 0
\(47\) 6.31118i 0.920580i 0.887769 + 0.460290i \(0.152255\pi\)
−0.887769 + 0.460290i \(0.847745\pi\)
\(48\) 0 0
\(49\) −5.22471 4.65858i −0.746388 0.665512i
\(50\) 0 0
\(51\) −0.682438 + 0.340370i −0.0955605 + 0.0476613i
\(52\) 0 0
\(53\) −9.37352 −1.28755 −0.643776 0.765214i \(-0.722633\pi\)
−0.643776 + 0.765214i \(0.722633\pi\)
\(54\) 0 0
\(55\) 3.24008 0.282247i 0.436892 0.0380582i
\(56\) 0 0
\(57\) −10.1494 + 5.06207i −1.34432 + 0.670488i
\(58\) 0 0
\(59\) −8.33478 −1.08510 −0.542548 0.840025i \(-0.682540\pi\)
−0.542548 + 0.840025i \(0.682540\pi\)
\(60\) 0 0
\(61\) 7.14436i 0.914741i 0.889276 + 0.457371i \(0.151209\pi\)
−0.889276 + 0.457371i \(0.848791\pi\)
\(62\) 0 0
\(63\) −4.22432 6.71976i −0.532214 0.846610i
\(64\) 0 0
\(65\) 0.858472 + 9.85489i 0.106480 + 1.22235i
\(66\) 0 0
\(67\) 11.3488i 1.38648i 0.720706 + 0.693241i \(0.243818\pi\)
−0.720706 + 0.693241i \(0.756182\pi\)
\(68\) 0 0
\(69\) 1.88430 + 3.77800i 0.226843 + 0.454817i
\(70\) 0 0
\(71\) 1.12511i 0.133526i 0.997769 + 0.0667631i \(0.0212672\pi\)
−0.997769 + 0.0667631i \(0.978733\pi\)
\(72\) 0 0
\(73\) 11.2423 1.31581 0.657904 0.753102i \(-0.271443\pi\)
0.657904 + 0.753102i \(0.271443\pi\)
\(74\) 0 0
\(75\) 5.14707 + 6.96474i 0.594333 + 0.804219i
\(76\) 0 0
\(77\) 3.59597 + 1.37035i 0.409799 + 0.156166i
\(78\) 0 0
\(79\) 6.83987 0.769546 0.384773 0.923011i \(-0.374280\pi\)
0.384773 + 0.923011i \(0.374280\pi\)
\(80\) 0 0
\(81\) −2.48556 8.64997i −0.276174 0.961108i
\(82\) 0 0
\(83\) 5.75536i 0.631733i −0.948804 0.315867i \(-0.897705\pi\)
0.948804 0.315867i \(-0.102295\pi\)
\(84\) 0 0
\(85\) 0.980811 0.0854397i 0.106384 0.00926723i
\(86\) 0 0
\(87\) 6.67834 3.33086i 0.715994 0.357106i
\(88\) 0 0
\(89\) −11.7153 −1.24182 −0.620908 0.783883i \(-0.713236\pi\)
−0.620908 + 0.783883i \(0.713236\pi\)
\(90\) 0 0
\(91\) −4.16800 + 10.9374i −0.436925 + 1.14655i
\(92\) 0 0
\(93\) 4.43747 2.21322i 0.460145 0.229500i
\(94\) 0 0
\(95\) 14.5869 1.27068i 1.49658 0.130369i
\(96\) 0 0
\(97\) 6.27856 0.637491 0.318746 0.947840i \(-0.396738\pi\)
0.318746 + 0.947840i \(0.396738\pi\)
\(98\) 0 0
\(99\) 3.48556 + 2.62504i 0.350312 + 0.263826i
\(100\) 0 0
\(101\) 11.5520 1.14946 0.574732 0.818342i \(-0.305106\pi\)
0.574732 + 0.818342i \(0.305106\pi\)
\(102\) 0 0
\(103\) −17.9525 −1.76892 −0.884458 0.466619i \(-0.845472\pi\)
−0.884458 + 0.466619i \(0.845472\pi\)
\(104\) 0 0
\(105\) 1.88947 + 10.0712i 0.184393 + 0.982853i
\(106\) 0 0
\(107\) −18.5926 −1.79742 −0.898709 0.438546i \(-0.855494\pi\)
−0.898709 + 0.438546i \(0.855494\pi\)
\(108\) 0 0
\(109\) 4.20960 0.403207 0.201603 0.979467i \(-0.435385\pi\)
0.201603 + 0.979467i \(0.435385\pi\)
\(110\) 0 0
\(111\) 14.1070 7.03597i 1.33898 0.667825i
\(112\) 0 0
\(113\) 10.3252 0.971316 0.485658 0.874149i \(-0.338580\pi\)
0.485658 + 0.874149i \(0.338580\pi\)
\(114\) 0 0
\(115\) −0.472996 5.42979i −0.0441071 0.506331i
\(116\) 0 0
\(117\) −7.98421 + 10.6016i −0.738140 + 0.980114i
\(118\) 0 0
\(119\) 1.08854 + 0.414821i 0.0997867 + 0.0380266i
\(120\) 0 0
\(121\) 8.88445 0.807677
\(122\) 0 0
\(123\) 3.52415 + 7.06588i 0.317762 + 0.637109i
\(124\) 0 0
\(125\) −2.88155 10.8026i −0.257734 0.966216i
\(126\) 0 0
\(127\) 0.00291325i 0.000258509i 1.00000 0.000129255i \(4.11430e-5\pi\)
−1.00000 0.000129255i \(0.999959\pi\)
\(128\) 0 0
\(129\) 13.2872 6.62708i 1.16988 0.583482i
\(130\) 0 0
\(131\) 4.00085 0.349556 0.174778 0.984608i \(-0.444079\pi\)
0.174778 + 0.984608i \(0.444079\pi\)
\(132\) 0 0
\(133\) 16.1891 + 6.16933i 1.40377 + 0.534949i
\(134\) 0 0
\(135\) −1.11312 + 11.5655i −0.0958021 + 0.995400i
\(136\) 0 0
\(137\) −17.6484 −1.50781 −0.753903 0.656986i \(-0.771831\pi\)
−0.753903 + 0.656986i \(0.771831\pi\)
\(138\) 0 0
\(139\) 4.22249i 0.358147i 0.983836 + 0.179073i \(0.0573099\pi\)
−0.983836 + 0.179073i \(0.942690\pi\)
\(140\) 0 0
\(141\) 9.78210 4.87888i 0.823801 0.410876i
\(142\) 0 0
\(143\) 6.43458i 0.538087i
\(144\) 0 0
\(145\) −9.59822 + 0.836113i −0.797089 + 0.0694354i
\(146\) 0 0
\(147\) −3.18165 + 11.6994i −0.262418 + 0.964954i
\(148\) 0 0
\(149\) 17.7877i 1.45723i −0.684926 0.728613i \(-0.740165\pi\)
0.684926 0.728613i \(-0.259835\pi\)
\(150\) 0 0
\(151\) −8.17122 −0.664965 −0.332482 0.943109i \(-0.607886\pi\)
−0.332482 + 0.943109i \(0.607886\pi\)
\(152\) 0 0
\(153\) 1.05512 + 0.794630i 0.0853016 + 0.0642421i
\(154\) 0 0
\(155\) −6.37760 + 0.555561i −0.512261 + 0.0446237i
\(156\) 0 0
\(157\) 0.0848924 0.00677515 0.00338758 0.999994i \(-0.498922\pi\)
0.00338758 + 0.999994i \(0.498922\pi\)
\(158\) 0 0
\(159\) 7.24624 + 14.5286i 0.574664 + 1.15219i
\(160\) 0 0
\(161\) 2.29646 6.02621i 0.180986 0.474932i
\(162\) 0 0
\(163\) 12.9990i 1.01816i 0.860718 + 0.509082i \(0.170015\pi\)
−0.860718 + 0.509082i \(0.829985\pi\)
\(164\) 0 0
\(165\) −2.94222 4.80381i −0.229052 0.373976i
\(166\) 0 0
\(167\) 0.585272i 0.0452897i −0.999744 0.0226449i \(-0.992791\pi\)
0.999744 0.0226449i \(-0.00720870\pi\)
\(168\) 0 0
\(169\) 6.57117 0.505475
\(170\) 0 0
\(171\) 15.6921 + 11.8180i 1.20000 + 0.903741i
\(172\) 0 0
\(173\) 17.4963i 1.33022i 0.746745 + 0.665111i \(0.231616\pi\)
−0.746745 + 0.665111i \(0.768384\pi\)
\(174\) 0 0
\(175\) 2.50234 12.9899i 0.189159 0.981946i
\(176\) 0 0
\(177\) 6.44323 + 12.9186i 0.484303 + 0.971022i
\(178\) 0 0
\(179\) 20.2719i 1.51519i 0.652724 + 0.757596i \(0.273626\pi\)
−0.652724 + 0.757596i \(0.726374\pi\)
\(180\) 0 0
\(181\) 12.9003i 0.958871i 0.877577 + 0.479435i \(0.159159\pi\)
−0.877577 + 0.479435i \(0.840841\pi\)
\(182\) 0 0
\(183\) 11.0735 5.52297i 0.818577 0.408270i
\(184\) 0 0
\(185\) −20.2749 + 1.76617i −1.49064 + 0.129851i
\(186\) 0 0
\(187\) −0.640403 −0.0468309
\(188\) 0 0
\(189\) −7.14975 + 11.7423i −0.520068 + 0.854125i
\(190\) 0 0
\(191\) 1.96182i 0.141952i −0.997478 0.0709760i \(-0.977389\pi\)
0.997478 0.0709760i \(-0.0226114\pi\)
\(192\) 0 0
\(193\) 17.9119i 1.28933i −0.764466 0.644664i \(-0.776997\pi\)
0.764466 0.644664i \(-0.223003\pi\)
\(194\) 0 0
\(195\) 14.6111 8.94896i 1.04632 0.640848i
\(196\) 0 0
\(197\) −12.2566 −0.873249 −0.436624 0.899644i \(-0.643826\pi\)
−0.436624 + 0.899644i \(0.643826\pi\)
\(198\) 0 0
\(199\) 1.32283i 0.0937726i 0.998900 + 0.0468863i \(0.0149298\pi\)
−0.998900 + 0.0468863i \(0.985070\pi\)
\(200\) 0 0
\(201\) 17.5903 8.77326i 1.24072 0.618818i
\(202\) 0 0
\(203\) −10.6525 4.05944i −0.747659 0.284917i
\(204\) 0 0
\(205\) −0.884632 10.1552i −0.0617854 0.709269i
\(206\) 0 0
\(207\) 4.39909 5.84118i 0.305758 0.405990i
\(208\) 0 0
\(209\) −9.52424 −0.658806
\(210\) 0 0
\(211\) 15.1148 1.04054 0.520272 0.854001i \(-0.325831\pi\)
0.520272 + 0.854001i \(0.325831\pi\)
\(212\) 0 0
\(213\) 1.74388 0.869771i 0.119489 0.0595958i
\(214\) 0 0
\(215\) −19.0966 + 1.66353i −1.30238 + 0.113452i
\(216\) 0 0
\(217\) −7.07813 2.69733i −0.480495 0.183106i
\(218\) 0 0
\(219\) −8.69088 17.4251i −0.587275 1.17748i
\(220\) 0 0
\(221\) 1.94783i 0.131025i
\(222\) 0 0
\(223\) −22.8929 −1.53302 −0.766511 0.642231i \(-0.778009\pi\)
−0.766511 + 0.642231i \(0.778009\pi\)
\(224\) 0 0
\(225\) 6.81614 13.3619i 0.454409 0.890793i
\(226\) 0 0
\(227\) 12.5749i 0.834629i 0.908762 + 0.417314i \(0.137029\pi\)
−0.908762 + 0.417314i \(0.862971\pi\)
\(228\) 0 0
\(229\) 0.500577i 0.0330791i −0.999863 0.0165395i \(-0.994735\pi\)
0.999863 0.0165395i \(-0.00526494\pi\)
\(230\) 0 0
\(231\) −0.655886 6.63298i −0.0431541 0.436418i
\(232\) 0 0
\(233\) 12.9171 0.846228 0.423114 0.906076i \(-0.360937\pi\)
0.423114 + 0.906076i \(0.360937\pi\)
\(234\) 0 0
\(235\) −14.0590 + 1.22470i −0.917107 + 0.0798903i
\(236\) 0 0
\(237\) −5.28759 10.6016i −0.343466 0.688645i
\(238\) 0 0
\(239\) 27.1112i 1.75368i 0.480783 + 0.876840i \(0.340353\pi\)
−0.480783 + 0.876840i \(0.659647\pi\)
\(240\) 0 0
\(241\) 4.75478i 0.306282i −0.988204 0.153141i \(-0.951061\pi\)
0.988204 0.153141i \(-0.0489389\pi\)
\(242\) 0 0
\(243\) −11.4857 + 10.5394i −0.736806 + 0.676104i
\(244\) 0 0
\(245\) 9.36374 12.5427i 0.598227 0.801326i
\(246\) 0 0
\(247\) 28.9686i 1.84323i
\(248\) 0 0
\(249\) −8.92061 + 4.44920i −0.565321 + 0.281957i
\(250\) 0 0
\(251\) 10.8100 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(252\) 0 0
\(253\) 3.54529i 0.222890i
\(254\) 0 0
\(255\) −0.890647 1.45417i −0.0557745 0.0910638i
\(256\) 0 0
\(257\) 22.4592i 1.40097i −0.713669 0.700483i \(-0.752968\pi\)
0.713669 0.700483i \(-0.247032\pi\)
\(258\) 0 0
\(259\) −22.5019 8.57499i −1.39820 0.532824i
\(260\) 0 0
\(261\) −10.3254 7.77626i −0.639128 0.481338i
\(262\) 0 0
\(263\) 17.7846 1.09664 0.548322 0.836267i \(-0.315267\pi\)
0.548322 + 0.836267i \(0.315267\pi\)
\(264\) 0 0
\(265\) −1.81895 20.8808i −0.111737 1.28269i
\(266\) 0 0
\(267\) 9.05653 + 18.1582i 0.554251 + 1.11127i
\(268\) 0 0
\(269\) −16.0968 −0.981438 −0.490719 0.871318i \(-0.663266\pi\)
−0.490719 + 0.871318i \(0.663266\pi\)
\(270\) 0 0
\(271\) 9.44781i 0.573914i −0.957943 0.286957i \(-0.907356\pi\)
0.957943 0.286957i \(-0.0926437\pi\)
\(272\) 0 0
\(273\) 20.1746 1.99492i 1.22102 0.120738i
\(274\) 0 0
\(275\) 1.25749 + 7.16293i 0.0758292 + 0.431941i
\(276\) 0 0
\(277\) 4.67510i 0.280899i −0.990088 0.140450i \(-0.955145\pi\)
0.990088 0.140450i \(-0.0448548\pi\)
\(278\) 0 0
\(279\) −6.86081 5.16699i −0.410746 0.309340i
\(280\) 0 0
\(281\) 11.8520i 0.707032i 0.935429 + 0.353516i \(0.115014\pi\)
−0.935429 + 0.353516i \(0.884986\pi\)
\(282\) 0 0
\(283\) −5.84917 −0.347697 −0.173849 0.984772i \(-0.555620\pi\)
−0.173849 + 0.984772i \(0.555620\pi\)
\(284\) 0 0
\(285\) −13.2459 21.6268i −0.784622 1.28106i
\(286\) 0 0
\(287\) 4.29501 11.2707i 0.253526 0.665285i
\(288\) 0 0
\(289\) 16.8061 0.988597
\(290\) 0 0
\(291\) −4.85366 9.73154i −0.284527 0.570473i
\(292\) 0 0
\(293\) 0.346350i 0.0202340i −0.999949 0.0101170i \(-0.996780\pi\)
0.999949 0.0101170i \(-0.00322040\pi\)
\(294\) 0 0
\(295\) −1.61738 18.5668i −0.0941675 1.08100i
\(296\) 0 0
\(297\) 1.37418 7.43179i 0.0797383 0.431236i
\(298\) 0 0
\(299\) −10.7832 −0.623609
\(300\) 0 0
\(301\) −21.1942 8.07667i −1.22161 0.465531i
\(302\) 0 0
\(303\) −8.93029 17.9051i −0.513032 1.02862i
\(304\) 0 0
\(305\) −15.9150 + 1.38638i −0.911290 + 0.0793837i
\(306\) 0 0
\(307\) 24.7866 1.41465 0.707323 0.706890i \(-0.249902\pi\)
0.707323 + 0.706890i \(0.249902\pi\)
\(308\) 0 0
\(309\) 13.8783 + 27.8258i 0.789508 + 1.58295i
\(310\) 0 0
\(311\) −13.0969 −0.742659 −0.371330 0.928501i \(-0.621098\pi\)
−0.371330 + 0.928501i \(0.621098\pi\)
\(312\) 0 0
\(313\) 1.69327 0.0957092 0.0478546 0.998854i \(-0.484762\pi\)
0.0478546 + 0.998854i \(0.484762\pi\)
\(314\) 0 0
\(315\) 14.1494 10.7142i 0.797229 0.603677i
\(316\) 0 0
\(317\) 32.6867 1.83587 0.917934 0.396734i \(-0.129857\pi\)
0.917934 + 0.396734i \(0.129857\pi\)
\(318\) 0 0
\(319\) 6.26699 0.350884
\(320\) 0 0
\(321\) 14.3731 + 28.8179i 0.802228 + 1.60846i
\(322\) 0 0
\(323\) −2.88311 −0.160420
\(324\) 0 0
\(325\) −21.7865 + 3.82472i −1.20850 + 0.212157i
\(326\) 0 0
\(327\) −3.25425 6.52473i −0.179960 0.360819i
\(328\) 0 0
\(329\) −15.6032 5.94607i −0.860234 0.327817i
\(330\) 0 0
\(331\) 22.5819 1.24121 0.620606 0.784123i \(-0.286887\pi\)
0.620606 + 0.784123i \(0.286887\pi\)
\(332\) 0 0
\(333\) −21.8110 16.4262i −1.19524 0.900152i
\(334\) 0 0
\(335\) −25.2810 + 2.20226i −1.38125 + 0.120322i
\(336\) 0 0
\(337\) 26.8892i 1.46475i −0.680901 0.732375i \(-0.738412\pi\)
0.680901 0.732375i \(-0.261588\pi\)
\(338\) 0 0
\(339\) −7.98195 16.0037i −0.433520 0.869203i
\(340\) 0 0
\(341\) 4.16414 0.225501
\(342\) 0 0
\(343\) 16.4399 8.52808i 0.887674 0.460473i
\(344\) 0 0
\(345\) −8.05033 + 4.93065i −0.433415 + 0.265457i
\(346\) 0 0
\(347\) 29.0533 1.55966 0.779831 0.625991i \(-0.215305\pi\)
0.779831 + 0.625991i \(0.215305\pi\)
\(348\) 0 0
\(349\) 27.1001i 1.45063i −0.688416 0.725317i \(-0.741693\pi\)
0.688416 0.725317i \(-0.258307\pi\)
\(350\) 0 0
\(351\) 22.6042 + 4.17967i 1.20653 + 0.223094i
\(352\) 0 0
\(353\) 11.7715i 0.626536i 0.949665 + 0.313268i \(0.101424\pi\)
−0.949665 + 0.313268i \(0.898576\pi\)
\(354\) 0 0
\(355\) −2.50633 + 0.218330i −0.133022 + 0.0115878i
\(356\) 0 0
\(357\) −0.198545 2.00788i −0.0105081 0.106268i
\(358\) 0 0
\(359\) 8.39177i 0.442901i −0.975172 0.221450i \(-0.928921\pi\)
0.975172 0.221450i \(-0.0710791\pi\)
\(360\) 0 0
\(361\) −23.8783 −1.25675
\(362\) 0 0
\(363\) −6.86815 13.7706i −0.360485 0.722768i
\(364\) 0 0
\(365\) 2.18158 + 25.0436i 0.114189 + 1.31084i
\(366\) 0 0
\(367\) −3.13112 −0.163443 −0.0817216 0.996655i \(-0.526042\pi\)
−0.0817216 + 0.996655i \(0.526042\pi\)
\(368\) 0 0
\(369\) 8.22750 10.9246i 0.428307 0.568712i
\(370\) 0 0
\(371\) 8.83125 23.1743i 0.458496 1.20315i
\(372\) 0 0
\(373\) 9.27241i 0.480107i 0.970760 + 0.240054i \(0.0771650\pi\)
−0.970760 + 0.240054i \(0.922835\pi\)
\(374\) 0 0
\(375\) −14.5161 + 12.8173i −0.749607 + 0.661883i
\(376\) 0 0
\(377\) 19.0614i 0.981714i
\(378\) 0 0
\(379\) 19.4784 1.00054 0.500268 0.865871i \(-0.333235\pi\)
0.500268 + 0.865871i \(0.333235\pi\)
\(380\) 0 0
\(381\) 0.00451543 0.00225210i 0.000231333 0.000115378i
\(382\) 0 0
\(383\) 1.95349i 0.0998186i −0.998754 0.0499093i \(-0.984107\pi\)
0.998754 0.0499093i \(-0.0158932\pi\)
\(384\) 0 0
\(385\) −2.35483 + 8.27642i −0.120013 + 0.421805i
\(386\) 0 0
\(387\) −20.5435 15.4716i −1.04428 0.786468i
\(388\) 0 0
\(389\) 1.48098i 0.0750884i −0.999295 0.0375442i \(-0.988046\pi\)
0.999295 0.0375442i \(-0.0119535\pi\)
\(390\) 0 0
\(391\) 1.07320i 0.0542742i
\(392\) 0 0
\(393\) −3.09287 6.20117i −0.156015 0.312808i
\(394\) 0 0
\(395\) 1.32729 + 15.2367i 0.0667832 + 0.766642i
\(396\) 0 0
\(397\) −12.9737 −0.651132 −0.325566 0.945519i \(-0.605555\pi\)
−0.325566 + 0.945519i \(0.605555\pi\)
\(398\) 0 0
\(399\) −2.95281 29.8618i −0.147825 1.49496i
\(400\) 0 0
\(401\) 31.1453i 1.55532i 0.628683 + 0.777662i \(0.283594\pi\)
−0.628683 + 0.777662i \(0.716406\pi\)
\(402\) 0 0
\(403\) 12.6655i 0.630914i
\(404\) 0 0
\(405\) 18.7866 7.21546i 0.933515 0.358539i
\(406\) 0 0
\(407\) 13.2381 0.656189
\(408\) 0 0
\(409\) 27.3261i 1.35119i 0.737273 + 0.675594i \(0.236113\pi\)
−0.737273 + 0.675594i \(0.763887\pi\)
\(410\) 0 0
\(411\) 13.6432 + 27.3544i 0.672968 + 1.34929i
\(412\) 0 0
\(413\) 7.85260 20.6062i 0.386401 1.01397i
\(414\) 0 0
\(415\) 12.8208 1.11684i 0.629350 0.0548235i
\(416\) 0 0
\(417\) 6.54470 3.26421i 0.320495 0.159849i
\(418\) 0 0
\(419\) −24.6777 −1.20559 −0.602793 0.797897i \(-0.705946\pi\)
−0.602793 + 0.797897i \(0.705946\pi\)
\(420\) 0 0
\(421\) 31.5258 1.53647 0.768237 0.640165i \(-0.221134\pi\)
0.768237 + 0.640165i \(0.221134\pi\)
\(422\) 0 0
\(423\) −15.1242 11.3903i −0.735362 0.553814i
\(424\) 0 0
\(425\) 0.380656 + 2.16831i 0.0184645 + 0.105178i
\(426\) 0 0
\(427\) −17.6631 6.73105i −0.854779 0.325738i
\(428\) 0 0
\(429\) −9.97337 + 4.97427i −0.481519 + 0.240160i
\(430\) 0 0
\(431\) 24.6470i 1.18720i −0.804759 0.593602i \(-0.797705\pi\)
0.804759 0.593602i \(-0.202295\pi\)
\(432\) 0 0
\(433\) −15.3862 −0.739412 −0.369706 0.929149i \(-0.620542\pi\)
−0.369706 + 0.929149i \(0.620542\pi\)
\(434\) 0 0
\(435\) 8.71588 + 14.2305i 0.417895 + 0.682302i
\(436\) 0 0
\(437\) 15.9609i 0.763515i
\(438\) 0 0
\(439\) 23.8686i 1.13919i −0.821926 0.569594i \(-0.807101\pi\)
0.821926 0.569594i \(-0.192899\pi\)
\(440\) 0 0
\(441\) 20.5933 4.11286i 0.980634 0.195850i
\(442\) 0 0
\(443\) 3.84558 0.182709 0.0913546 0.995818i \(-0.470880\pi\)
0.0913546 + 0.995818i \(0.470880\pi\)
\(444\) 0 0
\(445\) −2.27337 26.0973i −0.107768 1.23713i
\(446\) 0 0
\(447\) −27.5703 + 13.7508i −1.30403 + 0.650393i
\(448\) 0 0
\(449\) 2.50182i 0.118068i −0.998256 0.0590340i \(-0.981198\pi\)
0.998256 0.0590340i \(-0.0188020\pi\)
\(450\) 0 0
\(451\) 6.63065i 0.312225i
\(452\) 0 0
\(453\) 6.31679 + 12.6651i 0.296789 + 0.595058i
\(454\) 0 0
\(455\) −25.1732 7.16235i −1.18014 0.335776i
\(456\) 0 0
\(457\) 10.4851i 0.490472i 0.969463 + 0.245236i \(0.0788655\pi\)
−0.969463 + 0.245236i \(0.921134\pi\)
\(458\) 0 0
\(459\) 0.415982 2.24969i 0.0194164 0.105007i
\(460\) 0 0
\(461\) 24.6007 1.14577 0.572885 0.819636i \(-0.305824\pi\)
0.572885 + 0.819636i \(0.305824\pi\)
\(462\) 0 0
\(463\) 1.28981i 0.0599424i −0.999551 0.0299712i \(-0.990458\pi\)
0.999551 0.0299712i \(-0.00954156\pi\)
\(464\) 0 0
\(465\) 5.79133 + 9.45558i 0.268566 + 0.438492i
\(466\) 0 0
\(467\) 40.9656i 1.89566i −0.318771 0.947832i \(-0.603270\pi\)
0.318771 0.947832i \(-0.396730\pi\)
\(468\) 0 0
\(469\) −28.0579 10.6923i −1.29560 0.493724i
\(470\) 0 0
\(471\) −0.0656264 0.131580i −0.00302390 0.00606290i
\(472\) 0 0
\(473\) 12.4688 0.573316
\(474\) 0 0
\(475\) 5.66122 + 32.2476i 0.259755 + 1.47962i
\(476\) 0 0
\(477\) 16.9171 22.4628i 0.774582 1.02850i
\(478\) 0 0
\(479\) 28.8388 1.31768 0.658838 0.752285i \(-0.271048\pi\)
0.658838 + 0.752285i \(0.271048\pi\)
\(480\) 0 0
\(481\) 40.2645i 1.83591i
\(482\) 0 0
\(483\) −11.1157 + 1.09915i −0.505782 + 0.0500130i
\(484\) 0 0
\(485\) 1.21837 + 13.9863i 0.0553232 + 0.635086i
\(486\) 0 0
\(487\) 30.1405i 1.36580i −0.730514 0.682898i \(-0.760720\pi\)
0.730514 0.682898i \(-0.239280\pi\)
\(488\) 0 0
\(489\) 20.1480 10.0489i 0.911126 0.454429i
\(490\) 0 0
\(491\) 15.9029i 0.717690i −0.933397 0.358845i \(-0.883171\pi\)
0.933397 0.358845i \(-0.116829\pi\)
\(492\) 0 0
\(493\) 1.89709 0.0854408
\(494\) 0 0
\(495\) −5.17123 + 8.27394i −0.232430 + 0.371886i
\(496\) 0 0
\(497\) −2.78163 1.06002i −0.124773 0.0475485i
\(498\) 0 0
\(499\) −1.93099 −0.0864429 −0.0432214 0.999066i \(-0.513762\pi\)
−0.0432214 + 0.999066i \(0.513762\pi\)
\(500\) 0 0
\(501\) −0.907151 + 0.452447i −0.0405285 + 0.0202138i
\(502\) 0 0
\(503\) 10.3348i 0.460807i 0.973095 + 0.230404i \(0.0740046\pi\)
−0.973095 + 0.230404i \(0.925995\pi\)
\(504\) 0 0
\(505\) 2.24168 + 25.7335i 0.0997535 + 1.14513i
\(506\) 0 0
\(507\) −5.07987 10.1851i −0.225605 0.452335i
\(508\) 0 0
\(509\) 9.90113 0.438860 0.219430 0.975628i \(-0.429580\pi\)
0.219430 + 0.975628i \(0.429580\pi\)
\(510\) 0 0
\(511\) −10.5919 + 27.7945i −0.468558 + 1.22956i
\(512\) 0 0
\(513\) 6.18660 33.4580i 0.273145 1.47721i
\(514\) 0 0
\(515\) −3.48372 39.9917i −0.153511 1.76224i
\(516\) 0 0
\(517\) 9.17956 0.403717
\(518\) 0 0
\(519\) 27.1187 13.5256i 1.19038 0.593708i
\(520\) 0 0
\(521\) 22.3619 0.979694 0.489847 0.871808i \(-0.337053\pi\)
0.489847 + 0.871808i \(0.337053\pi\)
\(522\) 0 0
\(523\) 39.1612 1.71240 0.856201 0.516643i \(-0.172819\pi\)
0.856201 + 0.516643i \(0.172819\pi\)
\(524\) 0 0
\(525\) −22.0684 + 6.16338i −0.963142 + 0.268992i
\(526\) 0 0
\(527\) 1.26054 0.0549099
\(528\) 0 0
\(529\) −17.0587 −0.741684
\(530\) 0 0
\(531\) 15.0424 19.9735i 0.652785 0.866778i
\(532\) 0 0
\(533\) −20.1675 −0.873553
\(534\) 0 0
\(535\) −3.60793 41.4175i −0.155985 1.79064i
\(536\) 0 0
\(537\) 31.4207 15.6712i 1.35590 0.676264i
\(538\) 0 0
\(539\) −6.77587 + 7.59931i −0.291858 + 0.327325i
\(540\) 0 0
\(541\) 12.7660 0.548853 0.274427 0.961608i \(-0.411512\pi\)
0.274427 + 0.961608i \(0.411512\pi\)
\(542\) 0 0
\(543\) 19.9950 9.97261i 0.858067 0.427966i
\(544\) 0 0
\(545\) 0.816881 + 9.37745i 0.0349913 + 0.401686i
\(546\) 0 0
\(547\) 18.1815i 0.777383i 0.921368 + 0.388691i \(0.127073\pi\)
−0.921368 + 0.388691i \(0.872927\pi\)
\(548\) 0 0
\(549\) −17.1208 12.8940i −0.730699 0.550301i
\(550\) 0 0
\(551\) 28.2141 1.20196
\(552\) 0 0
\(553\) −6.44417 + 16.9103i −0.274034 + 0.719101i
\(554\) 0 0
\(555\) 18.4110 + 30.0600i 0.781505 + 1.27597i
\(556\) 0 0
\(557\) 2.18158 0.0924367 0.0462183 0.998931i \(-0.485283\pi\)
0.0462183 + 0.998931i \(0.485283\pi\)
\(558\) 0 0
\(559\) 37.9246i 1.60404i
\(560\) 0 0
\(561\) 0.495066 + 0.992602i 0.0209017 + 0.0419077i
\(562\) 0 0
\(563\) 36.6727i 1.54557i 0.634667 + 0.772786i \(0.281137\pi\)
−0.634667 + 0.772786i \(0.718863\pi\)
\(564\) 0 0
\(565\) 2.00363 + 23.0008i 0.0842933 + 0.967651i
\(566\) 0 0
\(567\) 23.7272 + 2.00445i 0.996451 + 0.0841792i
\(568\) 0 0
\(569\) 6.87817i 0.288348i −0.989552 0.144174i \(-0.953948\pi\)
0.989552 0.144174i \(-0.0460525\pi\)
\(570\) 0 0
\(571\) −21.8721 −0.915319 −0.457659 0.889128i \(-0.651312\pi\)
−0.457659 + 0.889128i \(0.651312\pi\)
\(572\) 0 0
\(573\) −3.04074 + 1.51659i −0.127029 + 0.0633564i
\(574\) 0 0
\(575\) 12.0038 2.10732i 0.500593 0.0878814i
\(576\) 0 0
\(577\) −40.0096 −1.66562 −0.832810 0.553559i \(-0.813269\pi\)
−0.832810 + 0.553559i \(0.813269\pi\)
\(578\) 0 0
\(579\) −27.7628 + 13.8469i −1.15378 + 0.575456i
\(580\) 0 0
\(581\) 14.2291 + 5.42241i 0.590322 + 0.224959i
\(582\) 0 0
\(583\) 13.6337i 0.564651i
\(584\) 0 0
\(585\) −25.1657 15.7286i −1.04047 0.650299i
\(586\) 0 0
\(587\) 15.0493i 0.621151i 0.950549 + 0.310576i \(0.100522\pi\)
−0.950549 + 0.310576i \(0.899478\pi\)
\(588\) 0 0
\(589\) 18.7470 0.772459
\(590\) 0 0
\(591\) 9.47503 + 18.9973i 0.389751 + 0.781446i
\(592\) 0 0
\(593\) 39.0931i 1.60536i −0.596408 0.802681i \(-0.703406\pi\)
0.596408 0.802681i \(-0.296594\pi\)
\(594\) 0 0
\(595\) −0.712835 + 2.50537i −0.0292234 + 0.102710i
\(596\) 0 0
\(597\) 2.05033 1.02261i 0.0839145 0.0418528i
\(598\) 0 0
\(599\) 14.1576i 0.578463i 0.957259 + 0.289231i \(0.0933997\pi\)
−0.957259 + 0.289231i \(0.906600\pi\)
\(600\) 0 0
\(601\) 4.57952i 0.186803i 0.995629 + 0.0934013i \(0.0297740\pi\)
−0.995629 + 0.0934013i \(0.970226\pi\)
\(602\) 0 0
\(603\) −27.1965 20.4821i −1.10753 0.834096i
\(604\) 0 0
\(605\) 1.72404 + 19.7913i 0.0700923 + 0.804630i
\(606\) 0 0
\(607\) 22.3014 0.905188 0.452594 0.891717i \(-0.350499\pi\)
0.452594 + 0.891717i \(0.350499\pi\)
\(608\) 0 0
\(609\) 1.94296 + 19.6492i 0.0787327 + 0.796224i
\(610\) 0 0
\(611\) 27.9202i 1.12953i
\(612\) 0 0
\(613\) 44.5311i 1.79859i 0.437340 + 0.899296i \(0.355921\pi\)
−0.437340 + 0.899296i \(0.644079\pi\)
\(614\) 0 0
\(615\) −15.0563 + 9.22165i −0.607129 + 0.371853i
\(616\) 0 0
\(617\) −4.11299 −0.165583 −0.0827913 0.996567i \(-0.526383\pi\)
−0.0827913 + 0.996567i \(0.526383\pi\)
\(618\) 0 0
\(619\) 8.36511i 0.336222i −0.985768 0.168111i \(-0.946233\pi\)
0.985768 0.168111i \(-0.0537667\pi\)
\(620\) 0 0
\(621\) −12.4544 2.30289i −0.499776 0.0924118i
\(622\) 0 0
\(623\) 11.0375 28.9639i 0.442209 1.16041i
\(624\) 0 0
\(625\) 23.5051 8.51530i 0.940204 0.340612i
\(626\) 0 0
\(627\) 7.36275 + 14.7622i 0.294040 + 0.589547i
\(628\) 0 0
\(629\) 4.00734 0.159783
\(630\) 0 0
\(631\) 16.3025 0.648991 0.324496 0.945887i \(-0.394805\pi\)
0.324496 + 0.945887i \(0.394805\pi\)
\(632\) 0 0
\(633\) −11.6845 23.4273i −0.464418 0.931153i
\(634\) 0 0
\(635\) −0.00648965 0.000565321i −0.000257534 2.24341e-5i
\(636\) 0 0
\(637\) −23.1138 20.6092i −0.915801 0.816568i
\(638\) 0 0
\(639\) −2.69623 2.03058i −0.106661 0.0803283i
\(640\) 0 0
\(641\) 45.8483i 1.81090i −0.424455 0.905449i \(-0.639534\pi\)
0.424455 0.905449i \(-0.360466\pi\)
\(642\) 0 0
\(643\) 26.3467 1.03901 0.519505 0.854467i \(-0.326116\pi\)
0.519505 + 0.854467i \(0.326116\pi\)
\(644\) 0 0
\(645\) 17.3411 + 28.3131i 0.682806 + 1.11483i
\(646\) 0 0
\(647\) 4.41576i 0.173601i 0.996226 + 0.0868007i \(0.0276643\pi\)
−0.996226 + 0.0868007i \(0.972336\pi\)
\(648\) 0 0
\(649\) 12.1229i 0.475865i
\(650\) 0 0
\(651\) 1.29101 + 13.0560i 0.0505988 + 0.511706i
\(652\) 0 0
\(653\) −46.8989 −1.83530 −0.917648 0.397393i \(-0.869915\pi\)
−0.917648 + 0.397393i \(0.869915\pi\)
\(654\) 0 0
\(655\) 0.776372 + 8.91242i 0.0303354 + 0.348237i
\(656\) 0 0
\(657\) −20.2898 + 26.9411i −0.791580 + 1.05107i
\(658\) 0 0
\(659\) 26.9930i 1.05150i 0.850640 + 0.525748i \(0.176215\pi\)
−0.850640 + 0.525748i \(0.823785\pi\)
\(660\) 0 0
\(661\) 22.4053i 0.871467i −0.900076 0.435733i \(-0.856489\pi\)
0.900076 0.435733i \(-0.143511\pi\)
\(662\) 0 0
\(663\) −3.01906 + 1.50577i −0.117251 + 0.0584794i
\(664\) 0 0
\(665\) −10.6015 + 37.2606i −0.411107 + 1.44490i
\(666\) 0 0
\(667\) 10.5024i 0.406653i
\(668\) 0 0
\(669\) 17.6974 + 35.4832i 0.684222 + 1.37186i
\(670\) 0 0
\(671\) 10.3914 0.401156
\(672\) 0 0
\(673\) 13.7897i 0.531554i 0.964035 + 0.265777i \(0.0856284\pi\)
−0.964035 + 0.265777i \(0.914372\pi\)
\(674\) 0 0
\(675\) −25.9797 0.235312i −0.999959 0.00905717i
\(676\) 0 0
\(677\) 36.5086i 1.40314i −0.712600 0.701570i \(-0.752483\pi\)
0.712600 0.701570i \(-0.247517\pi\)
\(678\) 0 0
\(679\) −5.91534 + 15.5226i −0.227010 + 0.595703i
\(680\) 0 0
\(681\) 19.4907 9.72111i 0.746886 0.372514i
\(682\) 0 0
\(683\) −31.7095 −1.21333 −0.606666 0.794957i \(-0.707493\pi\)
−0.606666 + 0.794957i \(0.707493\pi\)
\(684\) 0 0
\(685\) −3.42471 39.3142i −0.130851 1.50212i
\(686\) 0 0
\(687\) −0.775877 + 0.386973i −0.0296015 + 0.0147639i
\(688\) 0 0
\(689\) −41.4678 −1.57980
\(690\) 0 0
\(691\) 30.2906i 1.15231i −0.817340 0.576155i \(-0.804552\pi\)
0.817340 0.576155i \(-0.195448\pi\)
\(692\) 0 0
\(693\) −9.77384 + 6.14425i −0.371277 + 0.233401i
\(694\) 0 0
\(695\) −9.40615 + 0.819381i −0.356795 + 0.0310809i
\(696\) 0 0
\(697\) 2.00718i 0.0760274i
\(698\) 0 0
\(699\) −9.98562 20.0211i −0.377691 0.757266i
\(700\) 0 0
\(701\) 23.8010i 0.898952i −0.893293 0.449476i \(-0.851611\pi\)
0.893293 0.449476i \(-0.148389\pi\)
\(702\) 0 0
\(703\) 59.5982 2.24779
\(704\) 0 0
\(705\) 12.7666 + 20.8442i 0.480817 + 0.785036i
\(706\) 0 0
\(707\) −10.8837 + 28.5602i −0.409323 + 1.07412i
\(708\) 0 0
\(709\) −39.8188 −1.49543 −0.747714 0.664021i \(-0.768849\pi\)
−0.747714 + 0.664021i \(0.768849\pi\)
\(710\) 0 0
\(711\) −12.3444 + 16.3911i −0.462953 + 0.614716i
\(712\) 0 0
\(713\) 6.97837i 0.261342i
\(714\) 0 0
\(715\) 14.3339 1.24864i 0.536057 0.0466966i
\(716\) 0 0
\(717\) 42.0214 20.9584i 1.56932 0.782707i
\(718\) 0 0
\(719\) 9.97913 0.372159 0.186079 0.982535i \(-0.440422\pi\)
0.186079 + 0.982535i \(0.440422\pi\)
\(720\) 0 0
\(721\) 16.9140 44.3844i 0.629909 1.65296i
\(722\) 0 0
\(723\) −7.36973 + 3.67570i −0.274083 + 0.136701i
\(724\) 0 0
\(725\) −3.72510 21.2190i −0.138347 0.788056i
\(726\) 0 0
\(727\) −31.2440 −1.15878 −0.579388 0.815052i \(-0.696709\pi\)
−0.579388 + 0.815052i \(0.696709\pi\)
\(728\) 0 0
\(729\) 25.2148 + 9.65484i 0.933880 + 0.357587i
\(730\) 0 0
\(731\) 3.77446 0.139603
\(732\) 0 0
\(733\) −10.4386 −0.385558 −0.192779 0.981242i \(-0.561750\pi\)
−0.192779 + 0.981242i \(0.561750\pi\)
\(734\) 0 0
\(735\) −26.6795 4.81725i −0.984087 0.177687i
\(736\) 0 0
\(737\) 16.5068 0.608036
\(738\) 0 0
\(739\) −23.7381 −0.873218 −0.436609 0.899651i \(-0.643821\pi\)
−0.436609 + 0.899651i \(0.643821\pi\)
\(740\) 0 0
\(741\) −44.9003 + 22.3943i −1.64945 + 0.822674i
\(742\) 0 0
\(743\) −16.6347 −0.610269 −0.305135 0.952309i \(-0.598701\pi\)
−0.305135 + 0.952309i \(0.598701\pi\)
\(744\) 0 0
\(745\) 39.6245 3.45174i 1.45173 0.126462i
\(746\) 0 0
\(747\) 13.7922 + 10.3871i 0.504631 + 0.380046i
\(748\) 0 0
\(749\) 17.5170 45.9669i 0.640058 1.67959i
\(750\) 0 0
\(751\) −7.46709 −0.272478 −0.136239 0.990676i \(-0.543501\pi\)
−0.136239 + 0.990676i \(0.543501\pi\)
\(752\) 0 0
\(753\) −8.35667 16.7550i −0.304534 0.610587i
\(754\) 0 0
\(755\) −1.58564 18.2025i −0.0577074 0.662456i
\(756\) 0 0
\(757\) 6.54945i 0.238044i 0.992892 + 0.119022i \(0.0379759\pi\)
−0.992892 + 0.119022i \(0.962024\pi\)
\(758\) 0 0
\(759\) 5.49507 2.74070i 0.199458 0.0994810i
\(760\) 0 0
\(761\) −12.5194 −0.453826 −0.226913 0.973915i \(-0.572863\pi\)
−0.226913 + 0.973915i \(0.572863\pi\)
\(762\) 0 0
\(763\) −3.96607 + 10.4075i −0.143581 + 0.376776i
\(764\) 0 0
\(765\) −1.56540 + 2.50462i −0.0565970 + 0.0905549i
\(766\) 0 0
\(767\) −36.8725 −1.33139
\(768\) 0 0
\(769\) 15.0198i 0.541629i −0.962632 0.270815i \(-0.912707\pi\)
0.962632 0.270815i \(-0.0872931\pi\)
\(770\) 0 0
\(771\) −34.8110 + 17.3622i −1.25369 + 0.625283i
\(772\) 0 0
\(773\) 31.2676i 1.12462i 0.826927 + 0.562309i \(0.190087\pi\)
−0.826927 + 0.562309i \(0.809913\pi\)
\(774\) 0 0
\(775\) −2.47517 14.0991i −0.0889108 0.506456i
\(776\) 0 0
\(777\) 4.10422 + 41.5060i 0.147238 + 1.48902i
\(778\) 0 0
\(779\) 29.8513i 1.06953i
\(780\) 0 0
\(781\) 1.63647 0.0585574
\(782\) 0 0
\(783\) −4.07081 + 22.0155i −0.145479 + 0.786770i
\(784\) 0 0
\(785\) 0.0164735 + 0.189109i 0.000587966 + 0.00674959i
\(786\) 0 0
\(787\) −22.0233 −0.785048 −0.392524 0.919742i \(-0.628398\pi\)
−0.392524 + 0.919742i \(0.628398\pi\)
\(788\) 0 0
\(789\) −13.7484 27.5654i −0.489457 0.981356i
\(790\) 0 0
\(791\) −9.72790 + 25.5272i −0.345884 + 0.907644i
\(792\) 0 0
\(793\) 31.6062i 1.12237i
\(794\) 0 0
\(795\) −30.9583 + 18.9613i −1.09798 + 0.672486i
\(796\) 0 0
\(797\) 8.33314i 0.295175i 0.989049 + 0.147587i \(0.0471508\pi\)
−0.989049 + 0.147587i \(0.952849\pi\)
\(798\) 0 0
\(799\) 2.77877 0.0983057
\(800\) 0 0
\(801\) 21.1434 28.0746i 0.747067 0.991967i
\(802\) 0 0
\(803\) 16.3518i 0.577043i
\(804\) 0 0
\(805\) 13.8698 + 3.94627i 0.488846 + 0.139088i
\(806\) 0 0
\(807\) 12.4437 + 24.9494i 0.438038 + 0.878261i
\(808\) 0 0
\(809\) 5.19293i 0.182574i −0.995825 0.0912868i \(-0.970902\pi\)
0.995825 0.0912868i \(-0.0290980\pi\)
\(810\) 0 0
\(811\) 30.8998i 1.08504i −0.840044 0.542519i \(-0.817471\pi\)
0.840044 0.542519i \(-0.182529\pi\)
\(812\) 0 0
\(813\) −14.6438 + 7.30367i −0.513579 + 0.256151i
\(814\) 0 0
\(815\) −28.9571 + 2.52249i −1.01432 + 0.0883589i
\(816\) 0 0
\(817\) 56.1347 1.96391
\(818\) 0 0
\(819\) −18.6881 29.7277i −0.653015 1.03877i
\(820\) 0 0
\(821\) 16.6956i 0.582679i −0.956620 0.291340i \(-0.905899\pi\)
0.956620 0.291340i \(-0.0941010\pi\)
\(822\) 0 0
\(823\) 30.9576i 1.07911i 0.841949 + 0.539557i \(0.181408\pi\)
−0.841949 + 0.539557i \(0.818592\pi\)
\(824\) 0 0
\(825\) 10.1302 7.48638i 0.352687 0.260642i
\(826\) 0 0
\(827\) 24.6612 0.857555 0.428777 0.903410i \(-0.358944\pi\)
0.428777 + 0.903410i \(0.358944\pi\)
\(828\) 0 0
\(829\) 26.6977i 0.927249i 0.886032 + 0.463625i \(0.153451\pi\)
−0.886032 + 0.463625i \(0.846549\pi\)
\(830\) 0 0
\(831\) −7.24624 + 3.61410i −0.251369 + 0.125372i
\(832\) 0 0
\(833\) −2.05114 + 2.30040i −0.0710678 + 0.0797043i
\(834\) 0 0
\(835\) 1.30377 0.113573i 0.0451189 0.00393036i
\(836\) 0 0
\(837\) −2.70488 + 14.6284i −0.0934942 + 0.505630i
\(838\) 0 0
\(839\) 51.2367 1.76889 0.884443 0.466647i \(-0.154538\pi\)
0.884443 + 0.466647i \(0.154538\pi\)
\(840\) 0 0
\(841\) 10.4350 0.359828
\(842\) 0 0
\(843\) 18.3702 9.16224i 0.632703 0.315564i
\(844\) 0 0
\(845\) 1.27515 + 14.6382i 0.0438664 + 0.503568i
\(846\) 0 0
\(847\) −8.37047 + 21.9652i −0.287613 + 0.754733i
\(848\) 0 0
\(849\) 4.52172 + 9.06600i 0.155185 + 0.311144i
\(850\) 0 0
\(851\) 22.1847i 0.760483i
\(852\) 0 0
\(853\) −15.3303 −0.524901 −0.262450 0.964946i \(-0.584531\pi\)
−0.262450 + 0.964946i \(0.584531\pi\)
\(854\) 0 0
\(855\) −23.2810 + 37.2494i −0.796193 + 1.27390i
\(856\) 0 0
\(857\) 10.2905i 0.351518i 0.984433 + 0.175759i \(0.0562379\pi\)
−0.984433 + 0.175759i \(0.943762\pi\)
\(858\) 0 0
\(859\) 19.8890i 0.678603i −0.940678 0.339301i \(-0.889809\pi\)
0.940678 0.339301i \(-0.110191\pi\)
\(860\) 0 0
\(861\) −20.7894 + 2.05571i −0.708500 + 0.0700583i
\(862\) 0 0
\(863\) 17.8374 0.607193 0.303596 0.952801i \(-0.401813\pi\)
0.303596 + 0.952801i \(0.401813\pi\)
\(864\) 0 0
\(865\) −38.9754 + 3.39520i −1.32520 + 0.115440i
\(866\) 0 0
\(867\) −12.9920 26.0489i −0.441233 0.884668i
\(868\) 0 0
\(869\) 9.94855i 0.337481i
\(870\) 0 0
\(871\) 50.2065i 1.70118i
\(872\) 0 0
\(873\) −11.3314 + 15.0460i −0.383510 + 0.509230i
\(874\) 0 0
\(875\) 29.4224 + 3.05357i 0.994658 + 0.103230i
\(876\) 0 0
\(877\) 8.09995i 0.273516i 0.990604 + 0.136758i \(0.0436682\pi\)
−0.990604 + 0.136758i \(0.956332\pi\)
\(878\) 0 0
\(879\) −0.536831 + 0.267747i −0.0181068 + 0.00903089i
\(880\) 0 0
\(881\) −46.3728 −1.56234 −0.781169 0.624319i \(-0.785376\pi\)
−0.781169 + 0.624319i \(0.785376\pi\)
\(882\) 0 0
\(883\) 8.51933i 0.286698i 0.989672 + 0.143349i \(0.0457872\pi\)
−0.989672 + 0.143349i \(0.954213\pi\)
\(884\) 0 0
\(885\) −27.5276 + 16.8600i −0.925330 + 0.566744i
\(886\) 0 0
\(887\) 54.7644i 1.83881i −0.393312 0.919405i \(-0.628671\pi\)
0.393312 0.919405i \(-0.371329\pi\)
\(888\) 0 0
\(889\) −0.00720248 0.00274471i −0.000241563 9.20547e-5i
\(890\) 0 0
\(891\) −12.5813 + 3.61523i −0.421490 + 0.121115i
\(892\) 0 0
\(893\) 41.3265 1.38294
\(894\) 0 0
\(895\) −45.1583 + 3.93380i −1.50947 + 0.131492i
\(896\) 0 0
\(897\) 8.33600 + 16.7136i 0.278331 + 0.558051i
\(898\) 0 0
\(899\) −12.3356 −0.411416
\(900\) 0 0
\(901\) 4.12710i 0.137494i
\(902\) 0 0
\(903\) 3.86571 + 39.0940i 0.128643 + 1.30097i
\(904\) 0 0
\(905\) −28.7371 + 2.50332i −0.955253 + 0.0832133i
\(906\) 0 0
\(907\) 20.6308i 0.685036i −0.939511 0.342518i \(-0.888720\pi\)
0.939511 0.342518i \(-0.111280\pi\)
\(908\) 0 0
\(909\) −20.8487 + 27.6833i −0.691509 + 0.918196i
\(910\) 0 0
\(911\) 25.4837i 0.844313i 0.906523 + 0.422156i \(0.138727\pi\)
−0.906523 + 0.422156i \(0.861273\pi\)
\(912\) 0 0
\(913\) −8.37114 −0.277044
\(914\) 0 0
\(915\) 14.4520 + 23.5959i 0.477768 + 0.780058i
\(916\) 0 0
\(917\) −3.76939 + 9.89137i −0.124476 + 0.326642i
\(918\) 0 0
\(919\) −17.9121 −0.590865 −0.295432 0.955364i \(-0.595464\pi\)
−0.295432 + 0.955364i \(0.595464\pi\)
\(920\) 0 0
\(921\) −19.1614 38.4184i −0.631389 1.26593i
\(922\) 0 0
\(923\) 4.97742i 0.163834i
\(924\) 0 0
\(925\) −7.86875 44.8222i −0.258723 1.47374i
\(926\) 0 0
\(927\) 32.4003 43.0217i 1.06417 1.41302i
\(928\) 0 0
\(929\) 10.9000 0.357616 0.178808 0.983884i \(-0.442776\pi\)
0.178808 + 0.983884i \(0.442776\pi\)
\(930\) 0 0
\(931\) −30.5051 + 34.2122i −0.999764 + 1.12126i
\(932\) 0 0
\(933\) 10.1246 + 20.2998i 0.331466 + 0.664585i
\(934\) 0 0
\(935\) −0.124271 1.42658i −0.00406411 0.0466542i
\(936\) 0 0
\(937\) −26.9791 −0.881368 −0.440684 0.897662i \(-0.645264\pi\)
−0.440684 + 0.897662i \(0.645264\pi\)
\(938\) 0 0
\(939\) −1.30899 2.62451i −0.0427172 0.0856475i
\(940\) 0 0
\(941\) 37.6275 1.22662 0.613310 0.789842i \(-0.289838\pi\)
0.613310 + 0.789842i \(0.289838\pi\)
\(942\) 0 0
\(943\) 11.1118 0.361850
\(944\) 0 0
\(945\) −27.5449 13.6484i −0.896035 0.443983i
\(946\) 0 0
\(947\) −2.38298 −0.0774364 −0.0387182 0.999250i \(-0.512327\pi\)
−0.0387182 + 0.999250i \(0.512327\pi\)
\(948\) 0 0
\(949\) 49.7351 1.61447
\(950\) 0 0
\(951\) −25.2686 50.6632i −0.819389 1.64287i
\(952\) 0 0
\(953\) −10.3582 −0.335534 −0.167767 0.985827i \(-0.553656\pi\)
−0.167767 + 0.985827i \(0.553656\pi\)
\(954\) 0 0
\(955\) 4.37020 0.380694i 0.141416 0.0123190i
\(956\) 0 0
\(957\) −4.84472 9.71361i −0.156607 0.313996i
\(958\) 0 0
\(959\) 16.6274 43.6325i 0.536928 1.40897i
\(960\) 0 0
\(961\) 22.8035 0.735597
\(962\) 0 0
\(963\) 33.5555 44.5556i 1.08131 1.43578i
\(964\) 0 0
\(965\) 39.9012 3.47584i 1.28446 0.111891i
\(966\) 0 0
\(967\) 10.3006i 0.331245i 0.986189 + 0.165622i \(0.0529633\pi\)
−0.986189 + 0.165622i \(0.947037\pi\)
\(968\) 0 0
\(969\) 2.22879 + 4.46871i 0.0715992 + 0.143556i
\(970\) 0 0
\(971\) 18.2487 0.585629 0.292814 0.956169i \(-0.405408\pi\)
0.292814 + 0.956169i \(0.405408\pi\)
\(972\) 0 0
\(973\) −10.4393 3.97821i −0.334670 0.127536i
\(974\) 0 0
\(975\) 22.7703 + 30.8116i 0.729233 + 0.986759i
\(976\) 0 0
\(977\) 34.9393 1.11781 0.558904 0.829233i \(-0.311222\pi\)
0.558904 + 0.829233i \(0.311222\pi\)
\(978\) 0 0
\(979\) 17.0398i 0.544594i
\(980\) 0 0
\(981\) −7.59739 + 10.0879i −0.242566 + 0.322083i
\(982\) 0 0
\(983\) 37.9493i 1.21039i 0.796076 + 0.605197i \(0.206906\pi\)
−0.796076 + 0.605197i \(0.793094\pi\)
\(984\) 0 0
\(985\) −2.37842 27.3033i −0.0757828 0.869954i
\(986\) 0 0
\(987\) 2.84595 + 28.7811i 0.0905875 + 0.916112i
\(988\) 0 0
\(989\) 20.8955i 0.664438i
\(990\) 0 0
\(991\) −19.9068 −0.632359 −0.316180 0.948699i \(-0.602400\pi\)
−0.316180 + 0.948699i \(0.602400\pi\)
\(992\) 0 0
\(993\) −17.4570 35.0011i −0.553981 1.11073i
\(994\) 0 0
\(995\) −2.94677 + 0.256697i −0.0934188 + 0.00813783i
\(996\) 0 0
\(997\) 43.8198 1.38779 0.693893 0.720078i \(-0.255894\pi\)
0.693893 + 0.720078i \(0.255894\pi\)
\(998\) 0 0
\(999\) −8.59899 + 46.5046i −0.272060 + 1.47134i
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 1680.2.k.h.209.7 24
3.2 odd 2 1680.2.k.i.209.8 24
4.3 odd 2 840.2.k.b.209.18 yes 24
5.4 even 2 1680.2.k.i.209.18 24
7.6 odd 2 inner 1680.2.k.h.209.18 24
12.11 even 2 840.2.k.a.209.17 yes 24
15.14 odd 2 inner 1680.2.k.h.209.17 24
20.19 odd 2 840.2.k.a.209.7 24
21.20 even 2 1680.2.k.i.209.17 24
28.27 even 2 840.2.k.b.209.7 yes 24
35.34 odd 2 1680.2.k.i.209.7 24
60.59 even 2 840.2.k.b.209.8 yes 24
84.83 odd 2 840.2.k.a.209.8 yes 24
105.104 even 2 inner 1680.2.k.h.209.8 24
140.139 even 2 840.2.k.a.209.18 yes 24
420.419 odd 2 840.2.k.b.209.17 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.k.a.209.7 24 20.19 odd 2
840.2.k.a.209.8 yes 24 84.83 odd 2
840.2.k.a.209.17 yes 24 12.11 even 2
840.2.k.a.209.18 yes 24 140.139 even 2
840.2.k.b.209.7 yes 24 28.27 even 2
840.2.k.b.209.8 yes 24 60.59 even 2
840.2.k.b.209.17 yes 24 420.419 odd 2
840.2.k.b.209.18 yes 24 4.3 odd 2
1680.2.k.h.209.7 24 1.1 even 1 trivial
1680.2.k.h.209.8 24 105.104 even 2 inner
1680.2.k.h.209.17 24 15.14 odd 2 inner
1680.2.k.h.209.18 24 7.6 odd 2 inner
1680.2.k.i.209.7 24 35.34 odd 2
1680.2.k.i.209.8 24 3.2 odd 2
1680.2.k.i.209.17 24 21.20 even 2
1680.2.k.i.209.18 24 5.4 even 2