Properties

Label 1127.2.c.c.1126.28
Level $1127$
Weight $2$
Character 1127.1126
Analytic conductor $8.999$
Analytic rank $0$
Dimension $28$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1127,2,Mod(1126,1127)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1127, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1127.1126");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1127 = 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1127.c (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: \(8.99914030780\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 161)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1126.28
Character \(\chi\) \(=\) 1127.1126
Dual form 1127.2.c.c.1126.25

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.46597 q^{2} -0.986579i q^{3} +4.08101 q^{4} +2.48691 q^{5} -2.43287i q^{6} +5.13171 q^{8} +2.02666 q^{9} +6.13265 q^{10} +4.49657i q^{11} -4.02624i q^{12} -2.10932i q^{13} -2.45353i q^{15} +4.49263 q^{16} -4.01646 q^{17} +4.99769 q^{18} -5.83898 q^{19} +10.1491 q^{20} +11.0884i q^{22} +(-4.49263 + 1.67817i) q^{23} -5.06284i q^{24} +1.18473 q^{25} -5.20153i q^{26} -4.95920i q^{27} +2.64981 q^{29} -6.05034i q^{30} +5.60378i q^{31} +0.815274 q^{32} +4.43622 q^{33} -9.90448 q^{34} +8.27083 q^{36} +0.778402i q^{37} -14.3988 q^{38} -2.08101 q^{39} +12.7621 q^{40} -1.68118i q^{41} -8.94389i q^{43} +18.3506i q^{44} +5.04013 q^{45} +(-11.0787 + 4.13831i) q^{46} -8.06722i q^{47} -4.43234i q^{48} +2.92150 q^{50} +3.96256i q^{51} -8.60817i q^{52} -6.34746i q^{53} -12.2292i q^{54} +11.1826i q^{55} +5.76061i q^{57} +6.53434 q^{58} +12.1792i q^{59} -10.0129i q^{60} +5.87140 q^{61} +13.8188i q^{62} -6.97482 q^{64} -5.24569i q^{65} +10.9396 q^{66} -15.7279i q^{67} -16.3912 q^{68} +(1.65564 + 4.43234i) q^{69} +5.33546 q^{71} +10.4003 q^{72} -3.64596i q^{73} +1.91952i q^{74} -1.16883i q^{75} -23.8290 q^{76} -5.13171 q^{78} -5.04997i q^{79} +11.1728 q^{80} +1.18735 q^{81} -4.14575i q^{82} +0.851415 q^{83} -9.98859 q^{85} -22.0554i q^{86} -2.61424i q^{87} +23.0751i q^{88} +7.95517 q^{89} +12.4288 q^{90} +(-18.3345 + 6.84862i) q^{92} +5.52857 q^{93} -19.8935i q^{94} -14.5210 q^{95} -0.804332i q^{96} +2.72009 q^{97} +9.11304i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 8 q^{2} + 24 q^{4} + 12 q^{8} - 8 q^{9} - 44 q^{18} + 44 q^{25} - 44 q^{29} + 12 q^{32} + 32 q^{39} - 36 q^{46} + 84 q^{50} - 28 q^{58} - 68 q^{64} + 16 q^{71} + 8 q^{72} - 12 q^{78} - 44 q^{81}+ \cdots - 112 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1127\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.46597 1.74370 0.871852 0.489769i \(-0.162919\pi\)
0.871852 + 0.489769i \(0.162919\pi\)
\(3\) 0.986579i 0.569601i −0.958587 0.284801i \(-0.908073\pi\)
0.958587 0.284801i \(-0.0919274\pi\)
\(4\) 4.08101 2.04051
\(5\) 2.48691 1.11218 0.556090 0.831122i \(-0.312301\pi\)
0.556090 + 0.831122i \(0.312301\pi\)
\(6\) 2.43287i 0.993217i
\(7\) 0 0
\(8\) 5.13171 1.81433
\(9\) 2.02666 0.675554
\(10\) 6.13265 1.93931
\(11\) 4.49657i 1.35577i 0.735169 + 0.677884i \(0.237103\pi\)
−0.735169 + 0.677884i \(0.762897\pi\)
\(12\) 4.02624i 1.16228i
\(13\) 2.10932i 0.585021i −0.956262 0.292510i \(-0.905509\pi\)
0.956262 0.292510i \(-0.0944906\pi\)
\(14\) 0 0
\(15\) 2.45353i 0.633500i
\(16\) 4.49263 1.12316
\(17\) −4.01646 −0.974135 −0.487068 0.873364i \(-0.661934\pi\)
−0.487068 + 0.873364i \(0.661934\pi\)
\(18\) 4.99769 1.17797
\(19\) −5.83898 −1.33955 −0.669777 0.742562i \(-0.733610\pi\)
−0.669777 + 0.742562i \(0.733610\pi\)
\(20\) 10.1491 2.26941
\(21\) 0 0
\(22\) 11.0884i 2.36406i
\(23\) −4.49263 + 1.67817i −0.936779 + 0.349922i
\(24\) 5.06284i 1.03345i
\(25\) 1.18473 0.236945
\(26\) 5.20153i 1.02010i
\(27\) 4.95920i 0.954398i
\(28\) 0 0
\(29\) 2.64981 0.492057 0.246028 0.969263i \(-0.420874\pi\)
0.246028 + 0.969263i \(0.420874\pi\)
\(30\) 6.05034i 1.10464i
\(31\) 5.60378i 1.00647i 0.864150 + 0.503234i \(0.167857\pi\)
−0.864150 + 0.503234i \(0.832143\pi\)
\(32\) 0.815274 0.144121
\(33\) 4.43622 0.772247
\(34\) −9.90448 −1.69860
\(35\) 0 0
\(36\) 8.27083 1.37847
\(37\) 0.778402i 0.127968i 0.997951 + 0.0639842i \(0.0203807\pi\)
−0.997951 + 0.0639842i \(0.979619\pi\)
\(38\) −14.3988 −2.33579
\(39\) −2.08101 −0.333229
\(40\) 12.7621 2.01787
\(41\) 1.68118i 0.262557i −0.991346 0.131278i \(-0.958092\pi\)
0.991346 0.131278i \(-0.0419082\pi\)
\(42\) 0 0
\(43\) 8.94389i 1.36393i −0.731384 0.681965i \(-0.761125\pi\)
0.731384 0.681965i \(-0.238875\pi\)
\(44\) 18.3506i 2.76645i
\(45\) 5.04013 0.751338
\(46\) −11.0787 + 4.13831i −1.63347 + 0.610161i
\(47\) 8.06722i 1.17673i −0.808597 0.588363i \(-0.799773\pi\)
0.808597 0.588363i \(-0.200227\pi\)
\(48\) 4.43234i 0.639753i
\(49\) 0 0
\(50\) 2.92150 0.413162
\(51\) 3.96256i 0.554869i
\(52\) 8.60817i 1.19374i
\(53\) 6.34746i 0.871891i −0.899973 0.435945i \(-0.856414\pi\)
0.899973 0.435945i \(-0.143586\pi\)
\(54\) 12.2292i 1.66419i
\(55\) 11.1826i 1.50786i
\(56\) 0 0
\(57\) 5.76061i 0.763012i
\(58\) 6.53434 0.858001
\(59\) 12.1792i 1.58560i 0.609483 + 0.792799i \(0.291377\pi\)
−0.609483 + 0.792799i \(0.708623\pi\)
\(60\) 10.0129i 1.29266i
\(61\) 5.87140 0.751755 0.375878 0.926669i \(-0.377341\pi\)
0.375878 + 0.926669i \(0.377341\pi\)
\(62\) 13.8188i 1.75498i
\(63\) 0 0
\(64\) −6.97482 −0.871853
\(65\) 5.24569i 0.650648i
\(66\) 10.9396 1.34657
\(67\) 15.7279i 1.92147i −0.277471 0.960734i \(-0.589496\pi\)
0.277471 0.960734i \(-0.410504\pi\)
\(68\) −16.3912 −1.98773
\(69\) 1.65564 + 4.43234i 0.199316 + 0.533591i
\(70\) 0 0
\(71\) 5.33546 0.633203 0.316601 0.948559i \(-0.397458\pi\)
0.316601 + 0.948559i \(0.397458\pi\)
\(72\) 10.4003 1.22568
\(73\) 3.64596i 0.426727i −0.976973 0.213364i \(-0.931558\pi\)
0.976973 0.213364i \(-0.0684419\pi\)
\(74\) 1.91952i 0.223139i
\(75\) 1.16883i 0.134964i
\(76\) −23.8290 −2.73337
\(77\) 0 0
\(78\) −5.13171 −0.581052
\(79\) 5.04997i 0.568166i −0.958800 0.284083i \(-0.908311\pi\)
0.958800 0.284083i \(-0.0916891\pi\)
\(80\) 11.1728 1.24915
\(81\) 1.18735 0.131928
\(82\) 4.14575i 0.457822i
\(83\) 0.851415 0.0934550 0.0467275 0.998908i \(-0.485121\pi\)
0.0467275 + 0.998908i \(0.485121\pi\)
\(84\) 0 0
\(85\) −9.98859 −1.08341
\(86\) 22.0554i 2.37829i
\(87\) 2.61424i 0.280276i
\(88\) 23.0751i 2.45982i
\(89\) 7.95517 0.843246 0.421623 0.906771i \(-0.361461\pi\)
0.421623 + 0.906771i \(0.361461\pi\)
\(90\) 12.4288 1.31011
\(91\) 0 0
\(92\) −18.3345 + 6.84862i −1.91150 + 0.714018i
\(93\) 5.52857 0.573286
\(94\) 19.8935i 2.05186i
\(95\) −14.5210 −1.48983
\(96\) 0.804332i 0.0820918i
\(97\) 2.72009 0.276184 0.138092 0.990419i \(-0.455903\pi\)
0.138092 + 0.990419i \(0.455903\pi\)
\(98\) 0 0
\(99\) 9.11304i 0.915895i
\(100\) 4.83488 0.483488
\(101\) 14.2377i 1.41670i 0.705861 + 0.708351i \(0.250560\pi\)
−0.705861 + 0.708351i \(0.749440\pi\)
\(102\) 9.77155i 0.967528i
\(103\) −9.42494 −0.928666 −0.464333 0.885661i \(-0.653706\pi\)
−0.464333 + 0.885661i \(0.653706\pi\)
\(104\) 10.8244i 1.06142i
\(105\) 0 0
\(106\) 15.6527i 1.52032i
\(107\) 17.7613i 1.71705i 0.512773 + 0.858524i \(0.328618\pi\)
−0.512773 + 0.858524i \(0.671382\pi\)
\(108\) 20.2385i 1.94745i
\(109\) 6.50372i 0.622944i 0.950255 + 0.311472i \(0.100822\pi\)
−0.950255 + 0.311472i \(0.899178\pi\)
\(110\) 27.5759i 2.62926i
\(111\) 0.767955 0.0728910
\(112\) 0 0
\(113\) 3.54545i 0.333528i 0.985997 + 0.166764i \(0.0533318\pi\)
−0.985997 + 0.166764i \(0.946668\pi\)
\(114\) 14.2055i 1.33047i
\(115\) −11.1728 + 4.17345i −1.04187 + 0.389177i
\(116\) 10.8139 1.00404
\(117\) 4.27488i 0.395213i
\(118\) 30.0336i 2.76482i
\(119\) 0 0
\(120\) 12.5908i 1.14938i
\(121\) −9.21917 −0.838107
\(122\) 14.4787 1.31084
\(123\) −1.65862 −0.149553
\(124\) 22.8691i 2.05371i
\(125\) −9.48825 −0.848655
\(126\) 0 0
\(127\) 0.227609 0.0201970 0.0100985 0.999949i \(-0.496785\pi\)
0.0100985 + 0.999949i \(0.496785\pi\)
\(128\) −18.8303 −1.66438
\(129\) −8.82385 −0.776897
\(130\) 12.9357i 1.13454i
\(131\) 10.2755i 0.897779i 0.893587 + 0.448889i \(0.148180\pi\)
−0.893587 + 0.448889i \(0.851820\pi\)
\(132\) 18.1043 1.57578
\(133\) 0 0
\(134\) 38.7845i 3.35047i
\(135\) 12.3331i 1.06146i
\(136\) −20.6113 −1.76741
\(137\) 10.4070i 0.889126i 0.895748 + 0.444563i \(0.146641\pi\)
−0.895748 + 0.444563i \(0.853359\pi\)
\(138\) 4.08277 + 10.9300i 0.347548 + 0.930424i
\(139\) 6.21923i 0.527508i 0.964590 + 0.263754i \(0.0849607\pi\)
−0.964590 + 0.263754i \(0.915039\pi\)
\(140\) 0 0
\(141\) −7.95895 −0.670265
\(142\) 13.1571 1.10412
\(143\) 9.48472 0.793152
\(144\) 9.10505 0.758754
\(145\) 6.58983 0.547256
\(146\) 8.99082i 0.744086i
\(147\) 0 0
\(148\) 3.17667i 0.261120i
\(149\) 17.7095i 1.45082i −0.688316 0.725411i \(-0.741650\pi\)
0.688316 0.725411i \(-0.258350\pi\)
\(150\) 2.88229i 0.235338i
\(151\) −3.58392 −0.291656 −0.145828 0.989310i \(-0.546585\pi\)
−0.145828 + 0.989310i \(0.546585\pi\)
\(152\) −29.9640 −2.43040
\(153\) −8.14002 −0.658081
\(154\) 0 0
\(155\) 13.9361i 1.11937i
\(156\) −8.49263 −0.679955
\(157\) 8.71438 0.695484 0.347742 0.937590i \(-0.386949\pi\)
0.347742 + 0.937590i \(0.386949\pi\)
\(158\) 12.4531i 0.990713i
\(159\) −6.26227 −0.496630
\(160\) 2.02751 0.160289
\(161\) 0 0
\(162\) 2.92797 0.230043
\(163\) 6.06005 0.474660 0.237330 0.971429i \(-0.423728\pi\)
0.237330 + 0.971429i \(0.423728\pi\)
\(164\) 6.86093i 0.535749i
\(165\) 11.0325 0.858878
\(166\) 2.09956 0.162958
\(167\) 4.39200i 0.339863i −0.985456 0.169931i \(-0.945645\pi\)
0.985456 0.169931i \(-0.0543546\pi\)
\(168\) 0 0
\(169\) 8.55076 0.657751
\(170\) −24.6316 −1.88915
\(171\) −11.8336 −0.904941
\(172\) 36.5001i 2.78311i
\(173\) 8.42394i 0.640460i 0.947340 + 0.320230i \(0.103760\pi\)
−0.947340 + 0.320230i \(0.896240\pi\)
\(174\) 6.44664i 0.488719i
\(175\) 0 0
\(176\) 20.2015i 1.52274i
\(177\) 12.0158 0.903159
\(178\) 19.6172 1.47037
\(179\) 7.62608 0.570000 0.285000 0.958528i \(-0.408006\pi\)
0.285000 + 0.958528i \(0.408006\pi\)
\(180\) 20.5688 1.53311
\(181\) 13.6135 1.01189 0.505943 0.862567i \(-0.331145\pi\)
0.505943 + 0.862567i \(0.331145\pi\)
\(182\) 0 0
\(183\) 5.79259i 0.428201i
\(184\) −23.0549 + 8.61188i −1.69963 + 0.634876i
\(185\) 1.93582i 0.142324i
\(186\) 13.6333 0.999642
\(187\) 18.0603i 1.32070i
\(188\) 32.9224i 2.40112i
\(189\) 0 0
\(190\) −35.8084 −2.59782
\(191\) 10.0258i 0.725440i 0.931898 + 0.362720i \(0.118152\pi\)
−0.931898 + 0.362720i \(0.881848\pi\)
\(192\) 6.88121i 0.496609i
\(193\) −13.5576 −0.975899 −0.487950 0.872872i \(-0.662255\pi\)
−0.487950 + 0.872872i \(0.662255\pi\)
\(194\) 6.70767 0.481582
\(195\) −5.17529 −0.370610
\(196\) 0 0
\(197\) 24.0652 1.71458 0.857288 0.514837i \(-0.172148\pi\)
0.857288 + 0.514837i \(0.172148\pi\)
\(198\) 22.4725i 1.59705i
\(199\) 24.1281 1.71040 0.855198 0.518301i \(-0.173435\pi\)
0.855198 + 0.518301i \(0.173435\pi\)
\(200\) 6.07967 0.429898
\(201\) −15.5168 −1.09447
\(202\) 35.1097i 2.47031i
\(203\) 0 0
\(204\) 16.1712i 1.13221i
\(205\) 4.18095i 0.292011i
\(206\) −23.2416 −1.61932
\(207\) −9.10505 + 3.40108i −0.632845 + 0.236391i
\(208\) 9.47641i 0.657071i
\(209\) 26.2554i 1.81612i
\(210\) 0 0
\(211\) 22.0361 1.51703 0.758515 0.651656i \(-0.225925\pi\)
0.758515 + 0.651656i \(0.225925\pi\)
\(212\) 25.9041i 1.77910i
\(213\) 5.26385i 0.360673i
\(214\) 43.7988i 2.99402i
\(215\) 22.2427i 1.51694i
\(216\) 25.4492i 1.73160i
\(217\) 0 0
\(218\) 16.0380i 1.08623i
\(219\) −3.59702 −0.243064
\(220\) 45.6362i 3.07679i
\(221\) 8.47201i 0.569889i
\(222\) 1.89375 0.127100
\(223\) 16.7338i 1.12058i 0.828297 + 0.560290i \(0.189310\pi\)
−0.828297 + 0.560290i \(0.810690\pi\)
\(224\) 0 0
\(225\) 2.40104 0.160069
\(226\) 8.74298i 0.581575i
\(227\) −29.2679 −1.94258 −0.971289 0.237904i \(-0.923540\pi\)
−0.971289 + 0.237904i \(0.923540\pi\)
\(228\) 23.5091i 1.55693i
\(229\) 5.06294 0.334569 0.167284 0.985909i \(-0.446500\pi\)
0.167284 + 0.985909i \(0.446500\pi\)
\(230\) −27.5517 + 10.2916i −1.81671 + 0.678609i
\(231\) 0 0
\(232\) 13.5980 0.892755
\(233\) −2.11926 −0.138837 −0.0694186 0.997588i \(-0.522114\pi\)
−0.0694186 + 0.997588i \(0.522114\pi\)
\(234\) 10.5417i 0.689135i
\(235\) 20.0625i 1.30873i
\(236\) 49.7035i 3.23542i
\(237\) −4.98219 −0.323628
\(238\) 0 0
\(239\) −11.1996 −0.724444 −0.362222 0.932092i \(-0.617982\pi\)
−0.362222 + 0.932092i \(0.617982\pi\)
\(240\) 11.0228i 0.711520i
\(241\) −6.65950 −0.428976 −0.214488 0.976727i \(-0.568808\pi\)
−0.214488 + 0.976727i \(0.568808\pi\)
\(242\) −22.7342 −1.46141
\(243\) 16.0490i 1.02954i
\(244\) 23.9612 1.53396
\(245\) 0 0
\(246\) −4.09011 −0.260776
\(247\) 12.3163i 0.783667i
\(248\) 28.7570i 1.82607i
\(249\) 0.839988i 0.0532321i
\(250\) −23.3977 −1.47980
\(251\) −2.15008 −0.135712 −0.0678559 0.997695i \(-0.521616\pi\)
−0.0678559 + 0.997695i \(0.521616\pi\)
\(252\) 0 0
\(253\) −7.54600 20.2015i −0.474413 1.27005i
\(254\) 0.561276 0.0352176
\(255\) 9.85453i 0.617114i
\(256\) −32.4852 −2.03033
\(257\) 28.4754i 1.77624i −0.459608 0.888122i \(-0.652010\pi\)
0.459608 0.888122i \(-0.347990\pi\)
\(258\) −21.7594 −1.35468
\(259\) 0 0
\(260\) 21.4077i 1.32765i
\(261\) 5.37026 0.332411
\(262\) 25.3392i 1.56546i
\(263\) 12.5676i 0.774949i 0.921880 + 0.387475i \(0.126653\pi\)
−0.921880 + 0.387475i \(0.873347\pi\)
\(264\) 22.7654 1.40112
\(265\) 15.7856i 0.969700i
\(266\) 0 0
\(267\) 7.84840i 0.480314i
\(268\) 64.1857i 3.92077i
\(269\) 15.6214i 0.952453i −0.879323 0.476226i \(-0.842004\pi\)
0.879323 0.476226i \(-0.157996\pi\)
\(270\) 30.4130i 1.85088i
\(271\) 15.5059i 0.941915i 0.882156 + 0.470957i \(0.156091\pi\)
−0.882156 + 0.470957i \(0.843909\pi\)
\(272\) −18.0445 −1.09411
\(273\) 0 0
\(274\) 25.6632i 1.55037i
\(275\) 5.32721i 0.321243i
\(276\) 6.75670 + 18.0884i 0.406706 + 1.08879i
\(277\) 1.81998 0.109352 0.0546759 0.998504i \(-0.482587\pi\)
0.0546759 + 0.998504i \(0.482587\pi\)
\(278\) 15.3364i 0.919818i
\(279\) 11.3570i 0.679924i
\(280\) 0 0
\(281\) 16.5334i 0.986301i 0.869944 + 0.493150i \(0.164155\pi\)
−0.869944 + 0.493150i \(0.835845\pi\)
\(282\) −19.6265 −1.16874
\(283\) −29.2823 −1.74065 −0.870327 0.492475i \(-0.836092\pi\)
−0.870327 + 0.492475i \(0.836092\pi\)
\(284\) 21.7741 1.29205
\(285\) 14.3261i 0.848607i
\(286\) 23.3890 1.38302
\(287\) 0 0
\(288\) 1.65229 0.0973619
\(289\) −0.868025 −0.0510603
\(290\) 16.2503 0.954252
\(291\) 2.68358i 0.157315i
\(292\) 14.8792i 0.870739i
\(293\) 19.5508 1.14217 0.571085 0.820891i \(-0.306523\pi\)
0.571085 + 0.820891i \(0.306523\pi\)
\(294\) 0 0
\(295\) 30.2886i 1.76347i
\(296\) 3.99454i 0.232178i
\(297\) 22.2994 1.29394
\(298\) 43.6712i 2.52981i
\(299\) 3.53980 + 9.47641i 0.204712 + 0.548035i
\(300\) 4.76999i 0.275395i
\(301\) 0 0
\(302\) −8.83785 −0.508561
\(303\) 14.0466 0.806955
\(304\) −26.2324 −1.50453
\(305\) 14.6016 0.836087
\(306\) −20.0730 −1.14750
\(307\) 0.893505i 0.0509950i 0.999675 + 0.0254975i \(0.00811699\pi\)
−0.999675 + 0.0254975i \(0.991883\pi\)
\(308\) 0 0
\(309\) 9.29844i 0.528970i
\(310\) 34.3660i 1.95186i
\(311\) 27.2117i 1.54303i 0.636210 + 0.771516i \(0.280501\pi\)
−0.636210 + 0.771516i \(0.719499\pi\)
\(312\) −10.6792 −0.604588
\(313\) 17.9576 1.01502 0.507511 0.861646i \(-0.330566\pi\)
0.507511 + 0.861646i \(0.330566\pi\)
\(314\) 21.4894 1.21272
\(315\) 0 0
\(316\) 20.6090i 1.15935i
\(317\) −11.3066 −0.635043 −0.317522 0.948251i \(-0.602851\pi\)
−0.317522 + 0.948251i \(0.602851\pi\)
\(318\) −15.4426 −0.865976
\(319\) 11.9150i 0.667114i
\(320\) −17.3458 −0.969658
\(321\) 17.5229 0.978033
\(322\) 0 0
\(323\) 23.4521 1.30491
\(324\) 4.84558 0.269199
\(325\) 2.49897i 0.138618i
\(326\) 14.9439 0.827667
\(327\) 6.41643 0.354830
\(328\) 8.62736i 0.476366i
\(329\) 0 0
\(330\) 27.2058 1.49763
\(331\) −35.6885 −1.96162 −0.980808 0.194975i \(-0.937537\pi\)
−0.980808 + 0.194975i \(0.937537\pi\)
\(332\) 3.47464 0.190695
\(333\) 1.57756i 0.0864497i
\(334\) 10.8305i 0.592621i
\(335\) 39.1139i 2.13702i
\(336\) 0 0
\(337\) 17.0125i 0.926730i −0.886167 0.463365i \(-0.846642\pi\)
0.886167 0.463365i \(-0.153358\pi\)
\(338\) 21.0859 1.14692
\(339\) 3.49787 0.189978
\(340\) −40.7635 −2.21071
\(341\) −25.1978 −1.36454
\(342\) −29.1814 −1.57795
\(343\) 0 0
\(344\) 45.8975i 2.47463i
\(345\) 4.11744 + 11.0228i 0.221676 + 0.593449i
\(346\) 20.7732i 1.11677i
\(347\) 1.88896 0.101405 0.0507024 0.998714i \(-0.483854\pi\)
0.0507024 + 0.998714i \(0.483854\pi\)
\(348\) 10.6687i 0.571905i
\(349\) 29.0312i 1.55401i −0.629497 0.777003i \(-0.716739\pi\)
0.629497 0.777003i \(-0.283261\pi\)
\(350\) 0 0
\(351\) −10.4605 −0.558343
\(352\) 3.66594i 0.195395i
\(353\) 18.2642i 0.972103i −0.873930 0.486052i \(-0.838437\pi\)
0.873930 0.486052i \(-0.161563\pi\)
\(354\) 29.6305 1.57484
\(355\) 13.2688 0.704236
\(356\) 32.4651 1.72065
\(357\) 0 0
\(358\) 18.8057 0.993911
\(359\) 5.13956i 0.271256i −0.990760 0.135628i \(-0.956695\pi\)
0.990760 0.135628i \(-0.0433052\pi\)
\(360\) 25.8645 1.36318
\(361\) 15.0937 0.794405
\(362\) 33.5706 1.76443
\(363\) 9.09544i 0.477387i
\(364\) 0 0
\(365\) 9.06717i 0.474597i
\(366\) 14.2844i 0.746656i
\(367\) 9.65616 0.504047 0.252024 0.967721i \(-0.418904\pi\)
0.252024 + 0.967721i \(0.418904\pi\)
\(368\) −20.1838 + 7.53939i −1.05215 + 0.393018i
\(369\) 3.40719i 0.177371i
\(370\) 4.77367i 0.248171i
\(371\) 0 0
\(372\) 22.5622 1.16979
\(373\) 6.18433i 0.320212i 0.987100 + 0.160106i \(0.0511837\pi\)
−0.987100 + 0.160106i \(0.948816\pi\)
\(374\) 44.5362i 2.30291i
\(375\) 9.36090i 0.483395i
\(376\) 41.3987i 2.13497i
\(377\) 5.58929i 0.287863i
\(378\) 0 0
\(379\) 13.0402i 0.669830i −0.942248 0.334915i \(-0.891292\pi\)
0.942248 0.334915i \(-0.108708\pi\)
\(380\) −59.2605 −3.04000
\(381\) 0.224554i 0.0115042i
\(382\) 24.7233i 1.26495i
\(383\) 24.4317 1.24840 0.624201 0.781264i \(-0.285425\pi\)
0.624201 + 0.781264i \(0.285425\pi\)
\(384\) 18.5775i 0.948031i
\(385\) 0 0
\(386\) −33.4327 −1.70168
\(387\) 18.1263i 0.921409i
\(388\) 11.1007 0.563554
\(389\) 15.9172i 0.807034i −0.914972 0.403517i \(-0.867788\pi\)
0.914972 0.403517i \(-0.132212\pi\)
\(390\) −12.7621 −0.646235
\(391\) 18.0445 6.74030i 0.912549 0.340872i
\(392\) 0 0
\(393\) 10.1376 0.511376
\(394\) 59.3441 2.98971
\(395\) 12.5588i 0.631903i
\(396\) 37.1904i 1.86889i
\(397\) 15.8323i 0.794599i 0.917689 + 0.397299i \(0.130053\pi\)
−0.917689 + 0.397299i \(0.869947\pi\)
\(398\) 59.4992 2.98243
\(399\) 0 0
\(400\) 5.32254 0.266127
\(401\) 0.706831i 0.0352975i −0.999844 0.0176487i \(-0.994382\pi\)
0.999844 0.0176487i \(-0.00561806\pi\)
\(402\) −38.2640 −1.90843
\(403\) 11.8202 0.588805
\(404\) 58.1041i 2.89079i
\(405\) 2.95283 0.146727
\(406\) 0 0
\(407\) −3.50014 −0.173496
\(408\) 20.3347i 1.00672i
\(409\) 22.9311i 1.13387i −0.823763 0.566935i \(-0.808129\pi\)
0.823763 0.566935i \(-0.191871\pi\)
\(410\) 10.3101i 0.509180i
\(411\) 10.2673 0.506448
\(412\) −38.4633 −1.89495
\(413\) 0 0
\(414\) −22.4528 + 8.38696i −1.10349 + 0.412197i
\(415\) 2.11739 0.103939
\(416\) 1.71968i 0.0843140i
\(417\) 6.13576 0.300469
\(418\) 64.7451i 3.16678i
\(419\) 8.27756 0.404385 0.202193 0.979346i \(-0.435193\pi\)
0.202193 + 0.979346i \(0.435193\pi\)
\(420\) 0 0
\(421\) 2.27003i 0.110635i −0.998469 0.0553173i \(-0.982383\pi\)
0.998469 0.0553173i \(-0.0176170\pi\)
\(422\) 54.3404 2.64525
\(423\) 16.3495i 0.794942i
\(424\) 32.5734i 1.58190i
\(425\) −4.75841 −0.230817
\(426\) 12.9805i 0.628908i
\(427\) 0 0
\(428\) 72.4840i 3.50365i
\(429\) 9.35742i 0.451781i
\(430\) 54.8498i 2.64509i
\(431\) 0.603523i 0.0290707i −0.999894 0.0145353i \(-0.995373\pi\)
0.999894 0.0145353i \(-0.00462690\pi\)
\(432\) 22.2799i 1.07194i
\(433\) −27.0009 −1.29758 −0.648790 0.760968i \(-0.724725\pi\)
−0.648790 + 0.760968i \(0.724725\pi\)
\(434\) 0 0
\(435\) 6.50139i 0.311718i
\(436\) 26.5418i 1.27112i
\(437\) 26.2324 9.79879i 1.25487 0.468740i
\(438\) −8.87016 −0.423832
\(439\) 7.81671i 0.373071i −0.982448 0.186536i \(-0.940274\pi\)
0.982448 0.186536i \(-0.0597260\pi\)
\(440\) 57.3858i 2.73576i
\(441\) 0 0
\(442\) 20.8917i 0.993718i
\(443\) 6.19844 0.294497 0.147248 0.989100i \(-0.452958\pi\)
0.147248 + 0.989100i \(0.452958\pi\)
\(444\) 3.13403 0.148735
\(445\) 19.7838 0.937842
\(446\) 41.2651i 1.95396i
\(447\) −17.4719 −0.826391
\(448\) 0 0
\(449\) −12.0380 −0.568109 −0.284055 0.958808i \(-0.591680\pi\)
−0.284055 + 0.958808i \(0.591680\pi\)
\(450\) 5.92089 0.279114
\(451\) 7.55957 0.355966
\(452\) 14.4690i 0.680566i
\(453\) 3.53582i 0.166127i
\(454\) −72.1737 −3.38728
\(455\) 0 0
\(456\) 29.5618i 1.38436i
\(457\) 17.9498i 0.839656i 0.907604 + 0.419828i \(0.137910\pi\)
−0.907604 + 0.419828i \(0.862090\pi\)
\(458\) 12.4851 0.583389
\(459\) 19.9184i 0.929713i
\(460\) −45.5962 + 17.0319i −2.12594 + 0.794117i
\(461\) 12.3887i 0.577001i −0.957480 0.288501i \(-0.906843\pi\)
0.957480 0.288501i \(-0.0931567\pi\)
\(462\) 0 0
\(463\) 12.7244 0.591354 0.295677 0.955288i \(-0.404455\pi\)
0.295677 + 0.955288i \(0.404455\pi\)
\(464\) 11.9046 0.552657
\(465\) 13.7491 0.637598
\(466\) −5.22603 −0.242091
\(467\) −4.10530 −0.189971 −0.0949853 0.995479i \(-0.530280\pi\)
−0.0949853 + 0.995479i \(0.530280\pi\)
\(468\) 17.4458i 0.806435i
\(469\) 0 0
\(470\) 49.4735i 2.28204i
\(471\) 8.59742i 0.396148i
\(472\) 62.5003i 2.87681i
\(473\) 40.2169 1.84917
\(474\) −12.2859 −0.564312
\(475\) −6.91759 −0.317401
\(476\) 0 0
\(477\) 12.8642i 0.589009i
\(478\) −27.6180 −1.26322
\(479\) 20.0488 0.916051 0.458025 0.888939i \(-0.348557\pi\)
0.458025 + 0.888939i \(0.348557\pi\)
\(480\) 2.00030i 0.0913009i
\(481\) 1.64190 0.0748642
\(482\) −16.4221 −0.748008
\(483\) 0 0
\(484\) −37.6236 −1.71016
\(485\) 6.76463 0.307166
\(486\) 39.5764i 1.79522i
\(487\) −27.0461 −1.22557 −0.612787 0.790248i \(-0.709952\pi\)
−0.612787 + 0.790248i \(0.709952\pi\)
\(488\) 30.1303 1.36394
\(489\) 5.97872i 0.270367i
\(490\) 0 0
\(491\) 10.5974 0.478256 0.239128 0.970988i \(-0.423138\pi\)
0.239128 + 0.970988i \(0.423138\pi\)
\(492\) −6.76885 −0.305163
\(493\) −10.6428 −0.479330
\(494\) 30.3716i 1.36648i
\(495\) 22.6633i 1.01864i
\(496\) 25.1757i 1.13042i
\(497\) 0 0
\(498\) 2.07139i 0.0928210i
\(499\) 27.7025 1.24013 0.620067 0.784549i \(-0.287105\pi\)
0.620067 + 0.784549i \(0.287105\pi\)
\(500\) −38.7216 −1.73168
\(501\) −4.33305 −0.193586
\(502\) −5.30203 −0.236641
\(503\) −14.7700 −0.658561 −0.329280 0.944232i \(-0.606806\pi\)
−0.329280 + 0.944232i \(0.606806\pi\)
\(504\) 0 0
\(505\) 35.4078i 1.57563i
\(506\) −18.6082 49.8162i −0.827236 2.21460i
\(507\) 8.43600i 0.374656i
\(508\) 0.928874 0.0412121
\(509\) 12.6691i 0.561550i 0.959774 + 0.280775i \(0.0905914\pi\)
−0.959774 + 0.280775i \(0.909409\pi\)
\(510\) 24.3010i 1.07607i
\(511\) 0 0
\(512\) −42.4471 −1.87591
\(513\) 28.9567i 1.27847i
\(514\) 70.2194i 3.09724i
\(515\) −23.4390 −1.03284
\(516\) −36.0103 −1.58526
\(517\) 36.2749 1.59537
\(518\) 0 0
\(519\) 8.31088 0.364807
\(520\) 26.9194i 1.18049i
\(521\) −7.79378 −0.341452 −0.170726 0.985319i \(-0.554611\pi\)
−0.170726 + 0.985319i \(0.554611\pi\)
\(522\) 13.2429 0.579626
\(523\) −3.76428 −0.164600 −0.0823002 0.996608i \(-0.526227\pi\)
−0.0823002 + 0.996608i \(0.526227\pi\)
\(524\) 41.9346i 1.83192i
\(525\) 0 0
\(526\) 30.9912i 1.35128i
\(527\) 22.5074i 0.980437i
\(528\) 19.9303 0.867356
\(529\) 17.3675 15.0788i 0.755109 0.655599i
\(530\) 38.9267i 1.69087i
\(531\) 24.6832i 1.07116i
\(532\) 0 0
\(533\) −3.54616 −0.153601
\(534\) 19.3539i 0.837526i
\(535\) 44.1707i 1.90967i
\(536\) 80.7110i 3.48619i
\(537\) 7.52373i 0.324673i
\(538\) 38.5219i 1.66080i
\(539\) 0 0
\(540\) 50.3315i 2.16592i
\(541\) −13.2377 −0.569133 −0.284567 0.958656i \(-0.591850\pi\)
−0.284567 + 0.958656i \(0.591850\pi\)
\(542\) 38.2370i 1.64242i
\(543\) 13.4308i 0.576372i
\(544\) −3.27452 −0.140394
\(545\) 16.1742i 0.692826i
\(546\) 0 0
\(547\) −40.0668 −1.71313 −0.856567 0.516036i \(-0.827407\pi\)
−0.856567 + 0.516036i \(0.827407\pi\)
\(548\) 42.4709i 1.81427i
\(549\) 11.8993 0.507851
\(550\) 13.1367i 0.560152i
\(551\) −15.4722 −0.659136
\(552\) 8.49629 + 22.7455i 0.361626 + 0.968112i
\(553\) 0 0
\(554\) 4.48801 0.190677
\(555\) 1.90984 0.0810680
\(556\) 25.3807i 1.07638i
\(557\) 10.4022i 0.440755i −0.975415 0.220377i \(-0.929271\pi\)
0.975415 0.220377i \(-0.0707289\pi\)
\(558\) 28.0060i 1.18559i
\(559\) −18.8655 −0.797928
\(560\) 0 0
\(561\) −17.8179 −0.752274
\(562\) 40.7709i 1.71982i
\(563\) 1.95284 0.0823024 0.0411512 0.999153i \(-0.486897\pi\)
0.0411512 + 0.999153i \(0.486897\pi\)
\(564\) −32.4806 −1.36768
\(565\) 8.81722i 0.370943i
\(566\) −72.2093 −3.03518
\(567\) 0 0
\(568\) 27.3801 1.14884
\(569\) 13.0841i 0.548513i −0.961657 0.274256i \(-0.911568\pi\)
0.961657 0.274256i \(-0.0884317\pi\)
\(570\) 35.3278i 1.47972i
\(571\) 17.6990i 0.740680i −0.928896 0.370340i \(-0.879241\pi\)
0.928896 0.370340i \(-0.120759\pi\)
\(572\) 38.7073 1.61843
\(573\) 9.89122 0.413212
\(574\) 0 0
\(575\) −5.32254 + 1.98817i −0.221965 + 0.0829124i
\(576\) −14.1356 −0.588984
\(577\) 6.79439i 0.282854i 0.989949 + 0.141427i \(0.0451691\pi\)
−0.989949 + 0.141427i \(0.954831\pi\)
\(578\) −2.14052 −0.0890340
\(579\) 13.3757i 0.555874i
\(580\) 26.8932 1.11668
\(581\) 0 0
\(582\) 6.61764i 0.274310i
\(583\) 28.5418 1.18208
\(584\) 18.7100i 0.774226i
\(585\) 10.6313i 0.439548i
\(586\) 48.2117 1.99161
\(587\) 28.6736i 1.18349i 0.806126 + 0.591744i \(0.201560\pi\)
−0.806126 + 0.591744i \(0.798440\pi\)
\(588\) 0 0
\(589\) 32.7204i 1.34822i
\(590\) 74.6909i 3.07497i
\(591\) 23.7422i 0.976625i
\(592\) 3.49707i 0.143729i
\(593\) 15.5881i 0.640126i −0.947396 0.320063i \(-0.896296\pi\)
0.947396 0.320063i \(-0.103704\pi\)
\(594\) 54.9897 2.25625
\(595\) 0 0
\(596\) 72.2729i 2.96041i
\(597\) 23.8043i 0.974244i
\(598\) 8.72903 + 23.3685i 0.356957 + 0.955611i
\(599\) 1.11974 0.0457515 0.0228757 0.999738i \(-0.492718\pi\)
0.0228757 + 0.999738i \(0.492718\pi\)
\(600\) 5.99808i 0.244870i
\(601\) 3.71483i 0.151531i −0.997126 0.0757656i \(-0.975860\pi\)
0.997126 0.0757656i \(-0.0241401\pi\)
\(602\) 0 0
\(603\) 31.8751i 1.29806i
\(604\) −14.6260 −0.595125
\(605\) −22.9273 −0.932126
\(606\) 34.6385 1.40709
\(607\) 10.2553i 0.416251i 0.978102 + 0.208126i \(0.0667363\pi\)
−0.978102 + 0.208126i \(0.933264\pi\)
\(608\) −4.76037 −0.193059
\(609\) 0 0
\(610\) 36.0072 1.45789
\(611\) −17.0164 −0.688409
\(612\) −33.2195 −1.34282
\(613\) 39.4514i 1.59343i 0.604357 + 0.796714i \(0.293430\pi\)
−0.604357 + 0.796714i \(0.706570\pi\)
\(614\) 2.20336i 0.0889202i
\(615\) −4.12484 −0.166330
\(616\) 0 0
\(617\) 32.7292i 1.31763i 0.752305 + 0.658815i \(0.228942\pi\)
−0.752305 + 0.658815i \(0.771058\pi\)
\(618\) 22.9297i 0.922367i
\(619\) 26.7689 1.07593 0.537965 0.842967i \(-0.319193\pi\)
0.537965 + 0.842967i \(0.319193\pi\)
\(620\) 56.8734i 2.28409i
\(621\) 8.32237 + 22.2799i 0.333965 + 0.894060i
\(622\) 67.1032i 2.69059i
\(623\) 0 0
\(624\) −9.34922 −0.374268
\(625\) −29.5201 −1.18080
\(626\) 44.2828 1.76990
\(627\) −25.9030 −1.03447
\(628\) 35.5635 1.41914
\(629\) 3.12642i 0.124659i
\(630\) 0 0
\(631\) 3.62895i 0.144466i 0.997388 + 0.0722331i \(0.0230125\pi\)
−0.997388 + 0.0722331i \(0.976987\pi\)
\(632\) 25.9150i 1.03084i
\(633\) 21.7404i 0.864102i
\(634\) −27.8818 −1.10733
\(635\) 0.566043 0.0224627
\(636\) −25.5564 −1.01338
\(637\) 0 0
\(638\) 29.3822i 1.16325i
\(639\) 10.8132 0.427763
\(640\) −46.8292 −1.85109
\(641\) 18.7443i 0.740355i −0.928961 0.370177i \(-0.879297\pi\)
0.928961 0.370177i \(-0.120703\pi\)
\(642\) 43.2110 1.70540
\(643\) 21.3880 0.843460 0.421730 0.906722i \(-0.361423\pi\)
0.421730 + 0.906722i \(0.361423\pi\)
\(644\) 0 0
\(645\) −21.9441 −0.864050
\(646\) 57.8321 2.27537
\(647\) 11.1153i 0.436988i −0.975838 0.218494i \(-0.929886\pi\)
0.975838 0.218494i \(-0.0701145\pi\)
\(648\) 6.09313 0.239361
\(649\) −54.7647 −2.14970
\(650\) 6.16238i 0.241708i
\(651\) 0 0
\(652\) 24.7312 0.968547
\(653\) 2.94991 0.115439 0.0577194 0.998333i \(-0.481617\pi\)
0.0577194 + 0.998333i \(0.481617\pi\)
\(654\) 15.8227 0.618718
\(655\) 25.5544i 0.998492i
\(656\) 7.55294i 0.294893i
\(657\) 7.38913i 0.288277i
\(658\) 0 0
\(659\) 13.6891i 0.533252i 0.963800 + 0.266626i \(0.0859089\pi\)
−0.963800 + 0.266626i \(0.914091\pi\)
\(660\) 45.0237 1.75255
\(661\) −20.2120 −0.786155 −0.393078 0.919505i \(-0.628590\pi\)
−0.393078 + 0.919505i \(0.628590\pi\)
\(662\) −88.0068 −3.42048
\(663\) 8.35831 0.324610
\(664\) 4.36922 0.169559
\(665\) 0 0
\(666\) 3.89021i 0.150743i
\(667\) −11.9046 + 4.44682i −0.460948 + 0.172181i
\(668\) 17.9238i 0.693492i
\(669\) 16.5092 0.638283
\(670\) 96.4537i 3.72633i
\(671\) 26.4012i 1.01921i
\(672\) 0 0
\(673\) 35.7251 1.37710 0.688551 0.725188i \(-0.258247\pi\)
0.688551 + 0.725188i \(0.258247\pi\)
\(674\) 41.9523i 1.61594i
\(675\) 5.87529i 0.226140i
\(676\) 34.8958 1.34214
\(677\) −32.2210 −1.23835 −0.619177 0.785252i \(-0.712533\pi\)
−0.619177 + 0.785252i \(0.712533\pi\)
\(678\) 8.62564 0.331266
\(679\) 0 0
\(680\) −51.2586 −1.96568
\(681\) 28.8751i 1.10649i
\(682\) −62.1371 −2.37935
\(683\) 5.63137 0.215478 0.107739 0.994179i \(-0.465639\pi\)
0.107739 + 0.994179i \(0.465639\pi\)
\(684\) −48.2932 −1.84654
\(685\) 25.8812i 0.988869i
\(686\) 0 0
\(687\) 4.99499i 0.190571i
\(688\) 40.1816i 1.53191i
\(689\) −13.3888 −0.510074
\(690\) 10.1535 + 27.1820i 0.386537 + 1.03480i
\(691\) 10.1159i 0.384827i −0.981314 0.192414i \(-0.938368\pi\)
0.981314 0.192414i \(-0.0616315\pi\)
\(692\) 34.3782i 1.30686i
\(693\) 0 0
\(694\) 4.65812 0.176820
\(695\) 15.4667i 0.586684i
\(696\) 13.4155i 0.508515i
\(697\) 6.75241i 0.255766i
\(698\) 71.5901i 2.70973i
\(699\) 2.09081i 0.0790819i
\(700\) 0 0
\(701\) 33.1633i 1.25256i 0.779598 + 0.626280i \(0.215423\pi\)
−0.779598 + 0.626280i \(0.784577\pi\)
\(702\) −25.7954 −0.973584
\(703\) 4.54507i 0.171421i
\(704\) 31.3628i 1.18203i
\(705\) −19.7932 −0.745455
\(706\) 45.0389i 1.69506i
\(707\) 0 0
\(708\) 49.0364 1.84290
\(709\) 29.7999i 1.11916i 0.828776 + 0.559580i \(0.189038\pi\)
−0.828776 + 0.559580i \(0.810962\pi\)
\(710\) 32.7205 1.22798
\(711\) 10.2346i 0.383827i
\(712\) 40.8236 1.52993
\(713\) −9.40408 25.1757i −0.352186 0.942839i
\(714\) 0 0
\(715\) 23.5877 0.882128
\(716\) 31.1221 1.16309
\(717\) 11.0493i 0.412644i
\(718\) 12.6740i 0.472990i
\(719\) 46.4035i 1.73056i −0.501291 0.865279i \(-0.667141\pi\)
0.501291 0.865279i \(-0.332859\pi\)
\(720\) 22.6435 0.843872
\(721\) 0 0
\(722\) 37.2206 1.38521
\(723\) 6.57012i 0.244345i
\(724\) 55.5570 2.06476
\(725\) 3.13929 0.116590
\(726\) 22.4291i 0.832422i
\(727\) 2.32964 0.0864016 0.0432008 0.999066i \(-0.486244\pi\)
0.0432008 + 0.999066i \(0.486244\pi\)
\(728\) 0 0
\(729\) −12.2716 −0.454502
\(730\) 22.3594i 0.827558i
\(731\) 35.9228i 1.32865i
\(732\) 23.6396i 0.873746i
\(733\) −32.3300 −1.19413 −0.597067 0.802191i \(-0.703668\pi\)
−0.597067 + 0.802191i \(0.703668\pi\)
\(734\) 23.8118 0.878909
\(735\) 0 0
\(736\) −3.66273 + 1.36817i −0.135010 + 0.0504313i
\(737\) 70.7216 2.60506
\(738\) 8.40204i 0.309283i
\(739\) 20.5438 0.755716 0.377858 0.925864i \(-0.376661\pi\)
0.377858 + 0.925864i \(0.376661\pi\)
\(740\) 7.90009i 0.290413i
\(741\) 12.1510 0.446378
\(742\) 0 0
\(743\) 7.41754i 0.272123i 0.990700 + 0.136062i \(0.0434445\pi\)
−0.990700 + 0.136062i \(0.956556\pi\)
\(744\) 28.3710 1.04013
\(745\) 44.0421i 1.61358i
\(746\) 15.2504i 0.558356i
\(747\) 1.72553 0.0631339
\(748\) 73.7044i 2.69490i
\(749\) 0 0
\(750\) 23.0837i 0.842898i
\(751\) 31.5847i 1.15254i −0.817258 0.576272i \(-0.804507\pi\)
0.817258 0.576272i \(-0.195493\pi\)
\(752\) 36.2431i 1.32165i
\(753\) 2.12122i 0.0773016i
\(754\) 13.7830i 0.501948i
\(755\) −8.91290 −0.324374
\(756\) 0 0
\(757\) 31.9227i 1.16025i −0.814528 0.580125i \(-0.803004\pi\)
0.814528 0.580125i \(-0.196996\pi\)
\(758\) 32.1567i 1.16799i
\(759\) −19.9303 + 7.44473i −0.723425 + 0.270226i
\(760\) −74.5178 −2.70304
\(761\) 4.34317i 0.157440i −0.996897 0.0787199i \(-0.974917\pi\)
0.996897 0.0787199i \(-0.0250833\pi\)
\(762\) 0.553743i 0.0200600i
\(763\) 0 0
\(764\) 40.9153i 1.48026i
\(765\) −20.2435 −0.731905
\(766\) 60.2479 2.17684
\(767\) 25.6899 0.927608
\(768\) 32.0492i 1.15648i
\(769\) −30.9232 −1.11512 −0.557559 0.830137i \(-0.688262\pi\)
−0.557559 + 0.830137i \(0.688262\pi\)
\(770\) 0 0
\(771\) −28.0932 −1.01175
\(772\) −55.3288 −1.99133
\(773\) 9.67418 0.347956 0.173978 0.984750i \(-0.444338\pi\)
0.173978 + 0.984750i \(0.444338\pi\)
\(774\) 44.6988i 1.60667i
\(775\) 6.63894i 0.238478i
\(776\) 13.9587 0.501089
\(777\) 0 0
\(778\) 39.2514i 1.40723i
\(779\) 9.81640i 0.351709i
\(780\) −21.1204 −0.756232
\(781\) 23.9913i 0.858476i
\(782\) 44.4972 16.6214i 1.59122 0.594379i
\(783\) 13.1409i 0.469618i
\(784\) 0 0
\(785\) 21.6719 0.773503
\(786\) 24.9991 0.891689
\(787\) −21.8488 −0.778827 −0.389413 0.921063i \(-0.627322\pi\)
−0.389413 + 0.921063i \(0.627322\pi\)
\(788\) 98.2104 3.49860
\(789\) 12.3989 0.441412
\(790\) 30.9697i 1.10185i
\(791\) 0 0
\(792\) 46.7655i 1.66174i
\(793\) 12.3847i 0.439792i
\(794\) 39.0419i 1.38555i
\(795\) −15.5737 −0.552342
\(796\) 98.4671 3.49007
\(797\) 21.0566 0.745864 0.372932 0.927859i \(-0.378352\pi\)
0.372932 + 0.927859i \(0.378352\pi\)
\(798\) 0 0
\(799\) 32.4017i 1.14629i
\(800\) 0.965876 0.0341489
\(801\) 16.1224 0.569658
\(802\) 1.74302i 0.0615483i
\(803\) 16.3943 0.578543
\(804\) −63.3243 −2.23327
\(805\) 0 0
\(806\) 29.1482 1.02670
\(807\) −15.4117 −0.542519
\(808\) 73.0637i 2.57037i
\(809\) −21.4849 −0.755369 −0.377684 0.925934i \(-0.623280\pi\)
−0.377684 + 0.925934i \(0.623280\pi\)
\(810\) 7.28159 0.255849
\(811\) 20.3429i 0.714337i −0.934040 0.357169i \(-0.883742\pi\)
0.934040 0.357169i \(-0.116258\pi\)
\(812\) 0 0
\(813\) 15.2978 0.536516
\(814\) −8.63125 −0.302525
\(815\) 15.0708 0.527908
\(816\) 17.8023i 0.623206i
\(817\) 52.2232i 1.82706i
\(818\) 56.5474i 1.97713i
\(819\) 0 0
\(820\) 17.0625i 0.595849i
\(821\) −28.1375 −0.982005 −0.491003 0.871158i \(-0.663369\pi\)
−0.491003 + 0.871158i \(0.663369\pi\)
\(822\) 25.3188 0.883095
\(823\) −33.5526 −1.16957 −0.584786 0.811188i \(-0.698821\pi\)
−0.584786 + 0.811188i \(0.698821\pi\)
\(824\) −48.3661 −1.68491
\(825\) 5.25571 0.182980
\(826\) 0 0
\(827\) 27.0254i 0.939764i −0.882729 0.469882i \(-0.844297\pi\)
0.882729 0.469882i \(-0.155703\pi\)
\(828\) −37.1578 + 13.8798i −1.29132 + 0.482358i
\(829\) 16.1942i 0.562448i −0.959642 0.281224i \(-0.909260\pi\)
0.959642 0.281224i \(-0.0907403\pi\)
\(830\) 5.22143 0.181239
\(831\) 1.79555i 0.0622869i
\(832\) 14.7121i 0.510052i
\(833\) 0 0
\(834\) 15.1306 0.523930
\(835\) 10.9225i 0.377989i
\(836\) 107.149i 3.70581i
\(837\) 27.7903 0.960572
\(838\) 20.4122 0.705128
\(839\) −43.1817 −1.49080 −0.745399 0.666618i \(-0.767741\pi\)
−0.745399 + 0.666618i \(0.767741\pi\)
\(840\) 0 0
\(841\) −21.9785 −0.757880
\(842\) 5.59784i 0.192914i
\(843\) 16.3115 0.561798
\(844\) 89.9297 3.09551
\(845\) 21.2650 0.731538
\(846\) 40.3175i 1.38614i
\(847\) 0 0
\(848\) 28.5168i 0.979271i
\(849\) 28.8893i 0.991479i
\(850\) −11.7341 −0.402476
\(851\) −1.30629 3.49707i −0.0447790 0.119878i
\(852\) 21.4818i 0.735956i
\(853\) 18.9128i 0.647562i −0.946132 0.323781i \(-0.895046\pi\)
0.946132 0.323781i \(-0.104954\pi\)
\(854\) 0 0
\(855\) −29.4292 −1.00646
\(856\) 91.1458i 3.11530i
\(857\) 0.263693i 0.00900758i 0.999990 + 0.00450379i \(0.00143361\pi\)
−0.999990 + 0.00450379i \(0.998566\pi\)
\(858\) 23.0751i 0.787772i
\(859\) 40.7829i 1.39150i 0.718286 + 0.695748i \(0.244927\pi\)
−0.718286 + 0.695748i \(0.755073\pi\)
\(860\) 90.7726i 3.09532i
\(861\) 0 0
\(862\) 1.48827i 0.0506907i
\(863\) 12.6099 0.429248 0.214624 0.976697i \(-0.431147\pi\)
0.214624 + 0.976697i \(0.431147\pi\)
\(864\) 4.04311i 0.137549i
\(865\) 20.9496i 0.712307i
\(866\) −66.5834 −2.26260
\(867\) 0.856375i 0.0290840i
\(868\) 0 0
\(869\) 22.7075 0.770301
\(870\) 16.0322i 0.543543i
\(871\) −33.1752 −1.12410
\(872\) 33.3752i 1.13023i
\(873\) 5.51271 0.186577
\(874\) 64.6883 24.1635i 2.18812 0.817343i
\(875\) 0 0
\(876\) −14.6795 −0.495974
\(877\) −16.8956 −0.570524 −0.285262 0.958450i \(-0.592081\pi\)
−0.285262 + 0.958450i \(0.592081\pi\)
\(878\) 19.2758i 0.650526i
\(879\) 19.2884i 0.650581i
\(880\) 50.2392i 1.69356i
\(881\) 12.0628 0.406406 0.203203 0.979137i \(-0.434865\pi\)
0.203203 + 0.979137i \(0.434865\pi\)
\(882\) 0 0
\(883\) −49.2906 −1.65876 −0.829381 0.558684i \(-0.811307\pi\)
−0.829381 + 0.558684i \(0.811307\pi\)
\(884\) 34.5744i 1.16286i
\(885\) 29.8821 1.00448
\(886\) 15.2852 0.513516
\(887\) 6.00364i 0.201583i −0.994908 0.100791i \(-0.967863\pi\)
0.994908 0.100791i \(-0.0321374\pi\)
\(888\) 3.94092 0.132249
\(889\) 0 0
\(890\) 48.7863 1.63532
\(891\) 5.33900i 0.178863i
\(892\) 68.2909i 2.28655i
\(893\) 47.1044i 1.57629i
\(894\) −43.0851 −1.44098
\(895\) 18.9654 0.633943
\(896\) 0 0
\(897\) 9.34922 3.49229i 0.312161 0.116604i
\(898\) −29.6854 −0.990615
\(899\) 14.8489i 0.495240i
\(900\) 9.79867 0.326622
\(901\) 25.4943i 0.849340i
\(902\) 18.6417 0.620700
\(903\) 0 0
\(904\) 18.1942i 0.605132i
\(905\) 33.8556 1.12540
\(906\) 8.71923i 0.289677i
\(907\) 19.4967i 0.647377i −0.946164 0.323688i \(-0.895077\pi\)
0.946164 0.323688i \(-0.104923\pi\)
\(908\) −119.443 −3.96384
\(909\) 28.8550i 0.957058i
\(910\) 0 0
\(911\) 39.2121i 1.29916i −0.760295 0.649578i \(-0.774946\pi\)
0.760295 0.649578i \(-0.225054\pi\)
\(912\) 25.8803i 0.856983i
\(913\) 3.82845i 0.126703i
\(914\) 44.2637i 1.46411i
\(915\) 14.4057i 0.476237i
\(916\) 20.6619 0.682689
\(917\) 0 0
\(918\) 49.1183i 1.62114i
\(919\) 40.4966i 1.33586i −0.744225 0.667929i \(-0.767181\pi\)
0.744225 0.667929i \(-0.232819\pi\)
\(920\) −57.3355 + 21.4170i −1.89030 + 0.706097i
\(921\) 0.881513 0.0290468
\(922\) 30.5503i 1.00612i
\(923\) 11.2542i 0.370437i
\(924\) 0 0
\(925\) 0.922193i 0.0303215i
\(926\) 31.3781 1.03115
\(927\) −19.1012 −0.627364
\(928\) 2.16032 0.0709159
\(929\) 22.1070i 0.725306i 0.931924 + 0.362653i \(0.118129\pi\)
−0.931924 + 0.362653i \(0.881871\pi\)
\(930\) 33.9048 1.11178
\(931\) 0 0
\(932\) −8.64872 −0.283298
\(933\) 26.8464 0.878913
\(934\) −10.1236 −0.331253
\(935\) 44.9144i 1.46886i
\(936\) 21.9375i 0.717049i
\(937\) −18.7417 −0.612266 −0.306133 0.951989i \(-0.599035\pi\)
−0.306133 + 0.951989i \(0.599035\pi\)
\(938\) 0 0
\(939\) 17.7165i 0.578157i
\(940\) 81.8752i 2.67047i
\(941\) −22.5284 −0.734406 −0.367203 0.930141i \(-0.619685\pi\)
−0.367203 + 0.930141i \(0.619685\pi\)
\(942\) 21.2010i 0.690766i
\(943\) 2.82131 + 7.55294i 0.0918744 + 0.245958i
\(944\) 54.7167i 1.78088i
\(945\) 0 0
\(946\) 99.1736 3.22441
\(947\) −21.2167 −0.689449 −0.344724 0.938704i \(-0.612028\pi\)
−0.344724 + 0.938704i \(0.612028\pi\)
\(948\) −20.3324 −0.660365
\(949\) −7.69050 −0.249644
\(950\) −17.0586 −0.553453
\(951\) 11.1549i 0.361722i
\(952\) 0 0
\(953\) 37.6405i 1.21930i 0.792673 + 0.609648i \(0.208689\pi\)
−0.792673 + 0.609648i \(0.791311\pi\)
\(954\) 31.7226i 1.02706i
\(955\) 24.9332i 0.806820i
\(956\) −45.7058 −1.47823
\(957\) 11.7551 0.379989
\(958\) 49.4396 1.59732
\(959\) 0 0
\(960\) 17.1130i 0.552319i
\(961\) −0.402367 −0.0129796
\(962\) 4.04888 0.130541
\(963\) 35.9961i 1.15996i
\(964\) −27.1775 −0.875328
\(965\) −33.7166 −1.08538
\(966\) 0 0
\(967\) −22.5027 −0.723636 −0.361818 0.932249i \(-0.617844\pi\)
−0.361818 + 0.932249i \(0.617844\pi\)
\(968\) −47.3102 −1.52061
\(969\) 23.1373i 0.743277i
\(970\) 16.6814 0.535607
\(971\) 47.1721 1.51383 0.756913 0.653516i \(-0.226707\pi\)
0.756913 + 0.653516i \(0.226707\pi\)
\(972\) 65.4962i 2.10079i
\(973\) 0 0
\(974\) −66.6948 −2.13704
\(975\) −2.46543 −0.0789569
\(976\) 26.3780 0.844340
\(977\) 2.43944i 0.0780445i −0.999238 0.0390223i \(-0.987576\pi\)
0.999238 0.0390223i \(-0.0124243\pi\)
\(978\) 14.7433i 0.471440i
\(979\) 35.7710i 1.14325i
\(980\) 0 0
\(981\) 13.1808i 0.420832i
\(982\) 26.1330 0.833937
\(983\) 24.8899 0.793864 0.396932 0.917848i \(-0.370075\pi\)
0.396932 + 0.917848i \(0.370075\pi\)
\(984\) −8.51156 −0.271339
\(985\) 59.8481 1.90692
\(986\) −26.2449 −0.835809
\(987\) 0 0
\(988\) 50.2629i 1.59908i
\(989\) 15.0094 + 40.1816i 0.477270 + 1.27770i
\(990\) 55.8871i 1.77621i
\(991\) −24.2765 −0.771170 −0.385585 0.922672i \(-0.626000\pi\)
−0.385585 + 0.922672i \(0.626000\pi\)
\(992\) 4.56862i 0.145054i
\(993\) 35.2095i 1.11734i
\(994\) 0 0
\(995\) 60.0045 1.90227
\(996\) 3.42800i 0.108620i
\(997\) 3.32387i 0.105268i −0.998614 0.0526340i \(-0.983238\pi\)
0.998614 0.0526340i \(-0.0167617\pi\)
\(998\) 68.3135 2.16243
\(999\) 3.86025 0.122133
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 1127.2.c.c.1126.28 28
7.2 even 3 161.2.g.a.45.1 28
7.3 odd 6 161.2.g.a.68.2 yes 28
7.6 odd 2 inner 1127.2.c.c.1126.27 28
23.22 odd 2 inner 1127.2.c.c.1126.26 28
161.45 even 6 161.2.g.a.68.1 yes 28
161.114 odd 6 161.2.g.a.45.2 yes 28
161.160 even 2 inner 1127.2.c.c.1126.25 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.g.a.45.1 28 7.2 even 3
161.2.g.a.45.2 yes 28 161.114 odd 6
161.2.g.a.68.1 yes 28 161.45 even 6
161.2.g.a.68.2 yes 28 7.3 odd 6
1127.2.c.c.1126.25 28 161.160 even 2 inner
1127.2.c.c.1126.26 28 23.22 odd 2 inner
1127.2.c.c.1126.27 28 7.6 odd 2 inner
1127.2.c.c.1126.28 28 1.1 even 1 trivial