Properties

Label 224.3.n.a.145.14
Level $224$
Weight $3$
Character 224.145
Analytic conductor $6.104$
Analytic rank $0$
Dimension $28$
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] = [224,3,Mod(17,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.n (of order \(6\), degree \(2\), 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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.14
Character \(\chi\) \(=\) 224.145
Dual form 224.3.n.a.17.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.78005 + 4.81519i) q^{3} +(1.52921 - 2.64866i) q^{5} +(-0.608243 + 6.97352i) q^{7} +(-10.9574 + 18.9787i) q^{9} +O(q^{10})\) \(q+(2.78005 + 4.81519i) q^{3} +(1.52921 - 2.64866i) q^{5} +(-0.608243 + 6.97352i) q^{7} +(-10.9574 + 18.9787i) q^{9} +(0.106038 - 0.0612210i) q^{11} -4.11412 q^{13} +17.0051 q^{15} +(17.8551 - 10.3087i) q^{17} +(4.46893 - 7.74042i) q^{19} +(-35.2698 + 16.4580i) q^{21} +(-7.51940 + 13.0240i) q^{23} +(7.82306 + 13.5499i) q^{25} -71.8074 q^{27} -31.6239i q^{29} +(-23.0318 + 13.2974i) q^{31} +(0.589582 + 0.340395i) q^{33} +(17.5404 + 12.2750i) q^{35} +(25.1405 + 14.5149i) q^{37} +(-11.4375 - 19.8103i) q^{39} -9.26915i q^{41} -45.3391i q^{43} +(33.5122 + 58.0448i) q^{45} +(68.6931 + 39.6600i) q^{47} +(-48.2601 - 8.48319i) q^{49} +(99.2764 + 57.3172i) q^{51} +(55.0507 - 31.7835i) q^{53} -0.374478i q^{55} +49.6955 q^{57} +(-14.2561 - 24.6923i) q^{59} +(12.6191 - 21.8569i) q^{61} +(-125.684 - 87.9552i) q^{63} +(-6.29133 + 10.8969i) q^{65} +(65.4798 - 37.8048i) q^{67} -83.6173 q^{69} +2.81874 q^{71} +(11.0878 - 6.40155i) q^{73} +(-43.4970 + 75.3391i) q^{75} +(0.362429 + 0.776695i) q^{77} +(35.6186 - 61.6932i) q^{79} +(-101.012 - 174.958i) q^{81} -30.0525 q^{83} -63.0563i q^{85} +(152.275 - 87.9160i) q^{87} +(15.3030 + 8.83521i) q^{89} +(2.50238 - 28.6899i) q^{91} +(-128.059 - 73.9351i) q^{93} +(-13.6678 - 23.6734i) q^{95} -26.1737i q^{97} +2.68329i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 4 q^{7} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 4 q^{7} - 32 q^{9} - 28 q^{15} - 6 q^{17} - 30 q^{23} - 32 q^{25} + 6 q^{31} - 6 q^{33} + 20 q^{39} + 294 q^{47} - 20 q^{49} + 124 q^{57} - 432 q^{63} - 52 q^{65} + 136 q^{71} + 234 q^{73} + 162 q^{79} - 18 q^{81} - 48 q^{87} - 150 q^{89} - 290 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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
\(3\) 2.78005 + 4.81519i 0.926684 + 1.60506i 0.788830 + 0.614611i \(0.210687\pi\)
0.137854 + 0.990453i \(0.455980\pi\)
\(4\) 0 0
\(5\) 1.52921 2.64866i 0.305841 0.529732i −0.671607 0.740907i \(-0.734396\pi\)
0.977448 + 0.211175i \(0.0677290\pi\)
\(6\) 0 0
\(7\) −0.608243 + 6.97352i −0.0868918 + 0.996218i
\(8\) 0 0
\(9\) −10.9574 + 18.9787i −1.21749 + 2.10875i
\(10\) 0 0
\(11\) 0.106038 0.0612210i 0.00963981 0.00556554i −0.495172 0.868795i \(-0.664895\pi\)
0.504812 + 0.863229i \(0.331562\pi\)
\(12\) 0 0
\(13\) −4.11412 −0.316471 −0.158235 0.987401i \(-0.550580\pi\)
−0.158235 + 0.987401i \(0.550580\pi\)
\(14\) 0 0
\(15\) 17.0051 1.13367
\(16\) 0 0
\(17\) 17.8551 10.3087i 1.05030 0.606392i 0.127568 0.991830i \(-0.459283\pi\)
0.922734 + 0.385438i \(0.125950\pi\)
\(18\) 0 0
\(19\) 4.46893 7.74042i 0.235207 0.407390i −0.724126 0.689668i \(-0.757757\pi\)
0.959333 + 0.282277i \(0.0910899\pi\)
\(20\) 0 0
\(21\) −35.2698 + 16.4580i −1.67951 + 0.783712i
\(22\) 0 0
\(23\) −7.51940 + 13.0240i −0.326930 + 0.566260i −0.981901 0.189394i \(-0.939348\pi\)
0.654971 + 0.755654i \(0.272681\pi\)
\(24\) 0 0
\(25\) 7.82306 + 13.5499i 0.312922 + 0.541998i
\(26\) 0 0
\(27\) −71.8074 −2.65953
\(28\) 0 0
\(29\) 31.6239i 1.09048i −0.838280 0.545239i \(-0.816439\pi\)
0.838280 0.545239i \(-0.183561\pi\)
\(30\) 0 0
\(31\) −23.0318 + 13.2974i −0.742962 + 0.428949i −0.823145 0.567831i \(-0.807783\pi\)
0.0801833 + 0.996780i \(0.474449\pi\)
\(32\) 0 0
\(33\) 0.589582 + 0.340395i 0.0178661 + 0.0103150i
\(34\) 0 0
\(35\) 17.5404 + 12.2750i 0.501154 + 0.350714i
\(36\) 0 0
\(37\) 25.1405 + 14.5149i 0.679474 + 0.392295i 0.799657 0.600457i \(-0.205015\pi\)
−0.120183 + 0.992752i \(0.538348\pi\)
\(38\) 0 0
\(39\) −11.4375 19.8103i −0.293268 0.507956i
\(40\) 0 0
\(41\) 9.26915i 0.226077i −0.993591 0.113038i \(-0.963942\pi\)
0.993591 0.113038i \(-0.0360583\pi\)
\(42\) 0 0
\(43\) 45.3391i 1.05440i −0.849742 0.527199i \(-0.823242\pi\)
0.849742 0.527199i \(-0.176758\pi\)
\(44\) 0 0
\(45\) 33.5122 + 58.0448i 0.744715 + 1.28988i
\(46\) 0 0
\(47\) 68.6931 + 39.6600i 1.46156 + 0.843830i 0.999083 0.0428039i \(-0.0136291\pi\)
0.462472 + 0.886634i \(0.346962\pi\)
\(48\) 0 0
\(49\) −48.2601 8.48319i −0.984900 0.173126i
\(50\) 0 0
\(51\) 99.2764 + 57.3172i 1.94660 + 1.12387i
\(52\) 0 0
\(53\) 55.0507 31.7835i 1.03869 0.599689i 0.119229 0.992867i \(-0.461958\pi\)
0.919462 + 0.393178i \(0.128624\pi\)
\(54\) 0 0
\(55\) 0.374478i 0.00680869i
\(56\) 0 0
\(57\) 49.6955 0.871850
\(58\) 0 0
\(59\) −14.2561 24.6923i −0.241629 0.418514i 0.719550 0.694441i \(-0.244348\pi\)
−0.961178 + 0.275928i \(0.911015\pi\)
\(60\) 0 0
\(61\) 12.6191 21.8569i 0.206871 0.358311i −0.743856 0.668339i \(-0.767005\pi\)
0.950727 + 0.310029i \(0.100339\pi\)
\(62\) 0 0
\(63\) −125.684 87.9552i −1.99498 1.39611i
\(64\) 0 0
\(65\) −6.29133 + 10.8969i −0.0967897 + 0.167645i
\(66\) 0 0
\(67\) 65.4798 37.8048i 0.977311 0.564251i 0.0758537 0.997119i \(-0.475832\pi\)
0.901457 + 0.432868i \(0.142498\pi\)
\(68\) 0 0
\(69\) −83.6173 −1.21184
\(70\) 0 0
\(71\) 2.81874 0.0397006 0.0198503 0.999803i \(-0.493681\pi\)
0.0198503 + 0.999803i \(0.493681\pi\)
\(72\) 0 0
\(73\) 11.0878 6.40155i 0.151888 0.0876925i −0.422130 0.906535i \(-0.638717\pi\)
0.574018 + 0.818843i \(0.305384\pi\)
\(74\) 0 0
\(75\) −43.4970 + 75.3391i −0.579960 + 1.00452i
\(76\) 0 0
\(77\) 0.362429 + 0.776695i 0.00470687 + 0.0100869i
\(78\) 0 0
\(79\) 35.6186 61.6932i 0.450868 0.780926i −0.547572 0.836758i \(-0.684448\pi\)
0.998440 + 0.0558321i \(0.0177812\pi\)
\(80\) 0 0
\(81\) −101.012 174.958i −1.24706 2.15997i
\(82\) 0 0
\(83\) −30.0525 −0.362078 −0.181039 0.983476i \(-0.557946\pi\)
−0.181039 + 0.983476i \(0.557946\pi\)
\(84\) 0 0
\(85\) 63.0563i 0.741839i
\(86\) 0 0
\(87\) 152.275 87.9160i 1.75029 1.01053i
\(88\) 0 0
\(89\) 15.3030 + 8.83521i 0.171944 + 0.0992720i 0.583502 0.812112i \(-0.301682\pi\)
−0.411558 + 0.911384i \(0.635015\pi\)
\(90\) 0 0
\(91\) 2.50238 28.6899i 0.0274987 0.315274i
\(92\) 0 0
\(93\) −128.059 73.9351i −1.37698 0.795001i
\(94\) 0 0
\(95\) −13.6678 23.6734i −0.143872 0.249193i
\(96\) 0 0
\(97\) 26.1737i 0.269832i −0.990857 0.134916i \(-0.956923\pi\)
0.990857 0.134916i \(-0.0430765\pi\)
\(98\) 0 0
\(99\) 2.68329i 0.0271039i
\(100\) 0 0
\(101\) 67.8445 + 117.510i 0.671727 + 1.16347i 0.977414 + 0.211334i \(0.0677807\pi\)
−0.305686 + 0.952132i \(0.598886\pi\)
\(102\) 0 0
\(103\) −110.258 63.6577i −1.07047 0.618036i −0.142160 0.989844i \(-0.545405\pi\)
−0.928310 + 0.371807i \(0.878738\pi\)
\(104\) 0 0
\(105\) −10.3432 + 118.585i −0.0985068 + 1.12938i
\(106\) 0 0
\(107\) −69.1003 39.8951i −0.645797 0.372851i 0.141047 0.990003i \(-0.454953\pi\)
−0.786844 + 0.617152i \(0.788287\pi\)
\(108\) 0 0
\(109\) 27.3608 15.7968i 0.251017 0.144925i −0.369213 0.929345i \(-0.620373\pi\)
0.620230 + 0.784420i \(0.287039\pi\)
\(110\) 0 0
\(111\) 161.409i 1.45413i
\(112\) 0 0
\(113\) 57.7985 0.511491 0.255745 0.966744i \(-0.417679\pi\)
0.255745 + 0.966744i \(0.417679\pi\)
\(114\) 0 0
\(115\) 22.9974 + 39.8327i 0.199978 + 0.346371i
\(116\) 0 0
\(117\) 45.0799 78.0808i 0.385299 0.667357i
\(118\) 0 0
\(119\) 61.0275 + 130.783i 0.512836 + 1.09902i
\(120\) 0 0
\(121\) −60.4925 + 104.776i −0.499938 + 0.865918i
\(122\) 0 0
\(123\) 44.6327 25.7687i 0.362868 0.209502i
\(124\) 0 0
\(125\) 124.313 0.994500
\(126\) 0 0
\(127\) −67.8062 −0.533907 −0.266954 0.963709i \(-0.586017\pi\)
−0.266954 + 0.963709i \(0.586017\pi\)
\(128\) 0 0
\(129\) 218.316 126.045i 1.69238 0.977093i
\(130\) 0 0
\(131\) −56.0784 + 97.1307i −0.428080 + 0.741456i −0.996702 0.0811427i \(-0.974143\pi\)
0.568623 + 0.822598i \(0.307476\pi\)
\(132\) 0 0
\(133\) 51.2598 + 35.8723i 0.385412 + 0.269716i
\(134\) 0 0
\(135\) −109.808 + 190.193i −0.813394 + 1.40884i
\(136\) 0 0
\(137\) 29.3413 + 50.8207i 0.214170 + 0.370954i 0.953016 0.302921i \(-0.0979619\pi\)
−0.738845 + 0.673875i \(0.764629\pi\)
\(138\) 0 0
\(139\) 175.260 1.26086 0.630430 0.776246i \(-0.282879\pi\)
0.630430 + 0.776246i \(0.282879\pi\)
\(140\) 0 0
\(141\) 441.027i 3.12785i
\(142\) 0 0
\(143\) −0.436252 + 0.251870i −0.00305072 + 0.00176133i
\(144\) 0 0
\(145\) −83.7610 48.3594i −0.577662 0.333513i
\(146\) 0 0
\(147\) −93.3174 255.965i −0.634812 1.74126i
\(148\) 0 0
\(149\) −61.6922 35.6180i −0.414041 0.239047i 0.278483 0.960441i \(-0.410168\pi\)
−0.692525 + 0.721394i \(0.743502\pi\)
\(150\) 0 0
\(151\) −86.3801 149.615i −0.572053 0.990825i −0.996355 0.0853045i \(-0.972814\pi\)
0.424302 0.905521i \(-0.360520\pi\)
\(152\) 0 0
\(153\) 451.824i 2.95310i
\(154\) 0 0
\(155\) 81.3380i 0.524761i
\(156\) 0 0
\(157\) −134.922 233.692i −0.859378 1.48849i −0.872524 0.488572i \(-0.837518\pi\)
0.0131460 0.999914i \(-0.495815\pi\)
\(158\) 0 0
\(159\) 306.087 + 176.720i 1.92508 + 1.11144i
\(160\) 0 0
\(161\) −86.2494 60.3584i −0.535711 0.374897i
\(162\) 0 0
\(163\) −236.230 136.387i −1.44926 0.836733i −0.450826 0.892612i \(-0.648871\pi\)
−0.998438 + 0.0558788i \(0.982204\pi\)
\(164\) 0 0
\(165\) 1.80318 1.04107i 0.0109284 0.00630950i
\(166\) 0 0
\(167\) 82.5676i 0.494417i 0.968962 + 0.247208i \(0.0795132\pi\)
−0.968962 + 0.247208i \(0.920487\pi\)
\(168\) 0 0
\(169\) −152.074 −0.899846
\(170\) 0 0
\(171\) 97.9355 + 169.629i 0.572722 + 0.991984i
\(172\) 0 0
\(173\) −115.129 + 199.410i −0.665488 + 1.15266i 0.313665 + 0.949534i \(0.398443\pi\)
−0.979153 + 0.203125i \(0.934890\pi\)
\(174\) 0 0
\(175\) −99.2491 + 46.3127i −0.567138 + 0.264644i
\(176\) 0 0
\(177\) 79.2654 137.292i 0.447827 0.775660i
\(178\) 0 0
\(179\) −228.664 + 132.019i −1.27745 + 0.737538i −0.976379 0.216063i \(-0.930678\pi\)
−0.301074 + 0.953601i \(0.597345\pi\)
\(180\) 0 0
\(181\) −183.991 −1.01653 −0.508263 0.861202i \(-0.669712\pi\)
−0.508263 + 0.861202i \(0.669712\pi\)
\(182\) 0 0
\(183\) 140.327 0.766815
\(184\) 0 0
\(185\) 76.8901 44.3925i 0.415622 0.239960i
\(186\) 0 0
\(187\) 1.26221 2.18622i 0.00674980 0.0116910i
\(188\) 0 0
\(189\) 43.6763 500.751i 0.231092 2.64947i
\(190\) 0 0
\(191\) 148.189 256.671i 0.775860 1.34383i −0.158450 0.987367i \(-0.550650\pi\)
0.934310 0.356462i \(-0.116017\pi\)
\(192\) 0 0
\(193\) −47.8173 82.8220i −0.247758 0.429129i 0.715145 0.698976i \(-0.246360\pi\)
−0.962903 + 0.269846i \(0.913027\pi\)
\(194\) 0 0
\(195\) −69.9609 −0.358774
\(196\) 0 0
\(197\) 161.104i 0.817786i −0.912582 0.408893i \(-0.865915\pi\)
0.912582 0.408893i \(-0.134085\pi\)
\(198\) 0 0
\(199\) −0.961074 + 0.554877i −0.00482952 + 0.00278832i −0.502413 0.864628i \(-0.667554\pi\)
0.497583 + 0.867416i \(0.334221\pi\)
\(200\) 0 0
\(201\) 364.075 + 210.199i 1.81132 + 1.04576i
\(202\) 0 0
\(203\) 220.530 + 19.2350i 1.08635 + 0.0947536i
\(204\) 0 0
\(205\) −24.5508 14.1744i −0.119760 0.0691436i
\(206\) 0 0
\(207\) −164.786 285.417i −0.796067 1.37883i
\(208\) 0 0
\(209\) 1.09437i 0.00523622i
\(210\) 0 0
\(211\) 214.045i 1.01443i 0.861819 + 0.507216i \(0.169325\pi\)
−0.861819 + 0.507216i \(0.830675\pi\)
\(212\) 0 0
\(213\) 7.83624 + 13.5728i 0.0367899 + 0.0637219i
\(214\) 0 0
\(215\) −120.088 69.3328i −0.558549 0.322478i
\(216\) 0 0
\(217\) −78.7210 168.701i −0.362770 0.777424i
\(218\) 0 0
\(219\) 61.6494 + 35.5933i 0.281504 + 0.162526i
\(220\) 0 0
\(221\) −73.4581 + 42.4111i −0.332390 + 0.191905i
\(222\) 0 0
\(223\) 290.270i 1.30166i 0.759224 + 0.650829i \(0.225579\pi\)
−0.759224 + 0.650829i \(0.774421\pi\)
\(224\) 0 0
\(225\) −342.881 −1.52392
\(226\) 0 0
\(227\) −40.7118 70.5149i −0.179347 0.310638i 0.762310 0.647212i \(-0.224065\pi\)
−0.941657 + 0.336574i \(0.890732\pi\)
\(228\) 0 0
\(229\) −117.111 + 202.842i −0.511400 + 0.885771i 0.488512 + 0.872557i \(0.337540\pi\)
−0.999913 + 0.0132145i \(0.995794\pi\)
\(230\) 0 0
\(231\) −2.73236 + 3.90442i −0.0118284 + 0.0169022i
\(232\) 0 0
\(233\) 30.9903 53.6768i 0.133006 0.230372i −0.791828 0.610744i \(-0.790871\pi\)
0.924834 + 0.380371i \(0.124204\pi\)
\(234\) 0 0
\(235\) 210.092 121.297i 0.894008 0.516156i
\(236\) 0 0
\(237\) 396.086 1.67125
\(238\) 0 0
\(239\) −97.0822 −0.406202 −0.203101 0.979158i \(-0.565102\pi\)
−0.203101 + 0.979158i \(0.565102\pi\)
\(240\) 0 0
\(241\) −207.622 + 119.871i −0.861502 + 0.497388i −0.864515 0.502607i \(-0.832374\pi\)
0.00301303 + 0.999995i \(0.499041\pi\)
\(242\) 0 0
\(243\) 238.503 413.100i 0.981494 1.70000i
\(244\) 0 0
\(245\) −96.2687 + 114.852i −0.392933 + 0.468784i
\(246\) 0 0
\(247\) −18.3857 + 31.8450i −0.0744361 + 0.128927i
\(248\) 0 0
\(249\) −83.5475 144.709i −0.335532 0.581159i
\(250\) 0 0
\(251\) 136.078 0.542144 0.271072 0.962559i \(-0.412622\pi\)
0.271072 + 0.962559i \(0.412622\pi\)
\(252\) 0 0
\(253\) 1.84138i 0.00727818i
\(254\) 0 0
\(255\) 303.628 175.300i 1.19070 0.687450i
\(256\) 0 0
\(257\) −16.4497 9.49721i −0.0640064 0.0369541i 0.467655 0.883911i \(-0.345099\pi\)
−0.531662 + 0.846957i \(0.678432\pi\)
\(258\) 0 0
\(259\) −116.512 + 166.490i −0.449852 + 0.642817i
\(260\) 0 0
\(261\) 600.181 + 346.515i 2.29954 + 1.32764i
\(262\) 0 0
\(263\) 123.286 + 213.537i 0.468767 + 0.811928i 0.999363 0.0356971i \(-0.0113652\pi\)
−0.530596 + 0.847625i \(0.678032\pi\)
\(264\) 0 0
\(265\) 194.414i 0.733638i
\(266\) 0 0
\(267\) 98.2494i 0.367975i
\(268\) 0 0
\(269\) 147.121 + 254.821i 0.546918 + 0.947290i 0.998483 + 0.0550522i \(0.0175325\pi\)
−0.451565 + 0.892238i \(0.649134\pi\)
\(270\) 0 0
\(271\) −392.032 226.340i −1.44661 0.835202i −0.448335 0.893866i \(-0.647983\pi\)
−0.998278 + 0.0586635i \(0.981316\pi\)
\(272\) 0 0
\(273\) 145.104 67.7100i 0.531517 0.248022i
\(274\) 0 0
\(275\) 1.65908 + 0.957871i 0.00603302 + 0.00348317i
\(276\) 0 0
\(277\) 252.424 145.737i 0.911277 0.526126i 0.0304353 0.999537i \(-0.490311\pi\)
0.880842 + 0.473411i \(0.156977\pi\)
\(278\) 0 0
\(279\) 582.820i 2.08896i
\(280\) 0 0
\(281\) 495.433 1.76311 0.881553 0.472086i \(-0.156499\pi\)
0.881553 + 0.472086i \(0.156499\pi\)
\(282\) 0 0
\(283\) −18.3685 31.8151i −0.0649062 0.112421i 0.831746 0.555156i \(-0.187341\pi\)
−0.896652 + 0.442735i \(0.854008\pi\)
\(284\) 0 0
\(285\) 75.9946 131.626i 0.266648 0.461847i
\(286\) 0 0
\(287\) 64.6386 + 5.63789i 0.225222 + 0.0196442i
\(288\) 0 0
\(289\) 68.0371 117.844i 0.235423 0.407764i
\(290\) 0 0
\(291\) 126.032 72.7644i 0.433098 0.250049i
\(292\) 0 0
\(293\) −527.984 −1.80199 −0.900996 0.433828i \(-0.857163\pi\)
−0.900996 + 0.433828i \(0.857163\pi\)
\(294\) 0 0
\(295\) −87.2021 −0.295600
\(296\) 0 0
\(297\) −7.61430 + 4.39612i −0.0256374 + 0.0148017i
\(298\) 0 0
\(299\) 30.9357 53.5822i 0.103464 0.179205i
\(300\) 0 0
\(301\) 316.173 + 27.5772i 1.05041 + 0.0916185i
\(302\) 0 0
\(303\) −377.222 + 653.368i −1.24496 + 2.15633i
\(304\) 0 0
\(305\) −38.5944 66.8475i −0.126539 0.219172i
\(306\) 0 0
\(307\) −174.486 −0.568359 −0.284179 0.958771i \(-0.591721\pi\)
−0.284179 + 0.958771i \(0.591721\pi\)
\(308\) 0 0
\(309\) 707.887i 2.29090i
\(310\) 0 0
\(311\) −11.9119 + 6.87736i −0.0383020 + 0.0221137i −0.519029 0.854757i \(-0.673706\pi\)
0.480727 + 0.876870i \(0.340373\pi\)
\(312\) 0 0
\(313\) −365.368 210.945i −1.16731 0.673947i −0.214265 0.976776i \(-0.568736\pi\)
−0.953045 + 0.302829i \(0.902069\pi\)
\(314\) 0 0
\(315\) −425.160 + 198.393i −1.34971 + 0.629818i
\(316\) 0 0
\(317\) −408.352 235.762i −1.28818 0.743730i −0.309848 0.950786i \(-0.600278\pi\)
−0.978329 + 0.207056i \(0.933612\pi\)
\(318\) 0 0
\(319\) −1.93605 3.35333i −0.00606911 0.0105120i
\(320\) 0 0
\(321\) 443.642i 1.38206i
\(322\) 0 0
\(323\) 184.275i 0.570510i
\(324\) 0 0
\(325\) −32.1850 55.7460i −0.0990308 0.171526i
\(326\) 0 0
\(327\) 152.129 + 87.8317i 0.465226 + 0.268598i
\(328\) 0 0
\(329\) −318.352 + 454.910i −0.967635 + 1.38271i
\(330\) 0 0
\(331\) −383.707 221.533i −1.15923 0.669284i −0.208114 0.978105i \(-0.566732\pi\)
−0.951120 + 0.308820i \(0.900066\pi\)
\(332\) 0 0
\(333\) −550.949 + 318.090i −1.65450 + 0.955227i
\(334\) 0 0
\(335\) 231.245i 0.690284i
\(336\) 0 0
\(337\) 556.978 1.65276 0.826378 0.563117i \(-0.190398\pi\)
0.826378 + 0.563117i \(0.190398\pi\)
\(338\) 0 0
\(339\) 160.683 + 278.311i 0.473990 + 0.820975i
\(340\) 0 0
\(341\) −1.62816 + 2.82006i −0.00477467 + 0.00826998i
\(342\) 0 0
\(343\) 88.5116 331.383i 0.258051 0.966131i
\(344\) 0 0
\(345\) −127.868 + 221.474i −0.370632 + 0.641953i
\(346\) 0 0
\(347\) 277.806 160.392i 0.800595 0.462223i −0.0430845 0.999071i \(-0.513718\pi\)
0.843679 + 0.536848i \(0.180385\pi\)
\(348\) 0 0
\(349\) 222.198 0.636670 0.318335 0.947978i \(-0.396876\pi\)
0.318335 + 0.947978i \(0.396876\pi\)
\(350\) 0 0
\(351\) 295.424 0.841664
\(352\) 0 0
\(353\) −118.142 + 68.2096i −0.334681 + 0.193228i −0.657918 0.753090i \(-0.728562\pi\)
0.323236 + 0.946318i \(0.395229\pi\)
\(354\) 0 0
\(355\) 4.31043 7.46589i 0.0121421 0.0210307i
\(356\) 0 0
\(357\) −460.087 + 657.443i −1.28876 + 1.84158i
\(358\) 0 0
\(359\) 124.441 215.538i 0.346632 0.600384i −0.639017 0.769193i \(-0.720659\pi\)
0.985649 + 0.168809i \(0.0539920\pi\)
\(360\) 0 0
\(361\) 140.557 + 243.452i 0.389355 + 0.674383i
\(362\) 0 0
\(363\) −672.689 −1.85314
\(364\) 0 0
\(365\) 39.1572i 0.107280i
\(366\) 0 0
\(367\) 225.916 130.432i 0.615574 0.355402i −0.159570 0.987187i \(-0.551011\pi\)
0.775144 + 0.631785i \(0.217677\pi\)
\(368\) 0 0
\(369\) 175.917 + 101.566i 0.476739 + 0.275245i
\(370\) 0 0
\(371\) 188.159 + 403.229i 0.507167 + 1.08687i
\(372\) 0 0
\(373\) 381.464 + 220.239i 1.02269 + 0.590452i 0.914883 0.403720i \(-0.132283\pi\)
0.107810 + 0.994172i \(0.465616\pi\)
\(374\) 0 0
\(375\) 345.595 + 598.589i 0.921588 + 1.59624i
\(376\) 0 0
\(377\) 130.104i 0.345104i
\(378\) 0 0
\(379\) 283.715i 0.748587i −0.927310 0.374294i \(-0.877885\pi\)
0.927310 0.374294i \(-0.122115\pi\)
\(380\) 0 0
\(381\) −188.505 326.500i −0.494763 0.856955i
\(382\) 0 0
\(383\) −138.511 79.9691i −0.361646 0.208797i 0.308156 0.951336i \(-0.400288\pi\)
−0.669803 + 0.742539i \(0.733621\pi\)
\(384\) 0 0
\(385\) 2.61143 + 0.227773i 0.00678294 + 0.000591619i
\(386\) 0 0
\(387\) 860.479 + 496.798i 2.22346 + 1.28371i
\(388\) 0 0
\(389\) 430.295 248.431i 1.10616 0.638640i 0.168326 0.985731i \(-0.446164\pi\)
0.937831 + 0.347091i \(0.112830\pi\)
\(390\) 0 0
\(391\) 310.060i 0.792992i
\(392\) 0 0
\(393\) −623.604 −1.58678
\(394\) 0 0
\(395\) −108.936 188.683i −0.275788 0.477679i
\(396\) 0 0
\(397\) −142.186 + 246.273i −0.358150 + 0.620334i −0.987652 0.156665i \(-0.949926\pi\)
0.629502 + 0.776999i \(0.283259\pi\)
\(398\) 0 0
\(399\) −30.2269 + 346.552i −0.0757566 + 0.868552i
\(400\) 0 0
\(401\) −70.8759 + 122.761i −0.176748 + 0.306136i −0.940765 0.339060i \(-0.889891\pi\)
0.764017 + 0.645196i \(0.223224\pi\)
\(402\) 0 0
\(403\) 94.7556 54.7072i 0.235126 0.135750i
\(404\) 0 0
\(405\) −617.872 −1.52561
\(406\) 0 0
\(407\) 3.55447 0.00873333
\(408\) 0 0
\(409\) −323.318 + 186.668i −0.790508 + 0.456400i −0.840141 0.542368i \(-0.817528\pi\)
0.0496336 + 0.998767i \(0.484195\pi\)
\(410\) 0 0
\(411\) −163.141 + 282.568i −0.396937 + 0.687514i
\(412\) 0 0
\(413\) 180.864 84.3964i 0.437926 0.204350i
\(414\) 0 0
\(415\) −45.9565 + 79.5989i −0.110738 + 0.191805i
\(416\) 0 0
\(417\) 487.231 + 843.908i 1.16842 + 2.02376i
\(418\) 0 0
\(419\) 418.864 0.999676 0.499838 0.866119i \(-0.333393\pi\)
0.499838 + 0.866119i \(0.333393\pi\)
\(420\) 0 0
\(421\) 315.112i 0.748485i −0.927331 0.374243i \(-0.877903\pi\)
0.927331 0.374243i \(-0.122097\pi\)
\(422\) 0 0
\(423\) −1505.39 + 869.139i −3.55885 + 2.05470i
\(424\) 0 0
\(425\) 279.364 + 161.291i 0.657326 + 0.379507i
\(426\) 0 0
\(427\) 144.744 + 101.294i 0.338980 + 0.237222i
\(428\) 0 0
\(429\) −2.42561 1.40043i −0.00565410 0.00326440i
\(430\) 0 0
\(431\) 111.663 + 193.405i 0.259078 + 0.448736i 0.965995 0.258560i \(-0.0832481\pi\)
−0.706917 + 0.707296i \(0.749915\pi\)
\(432\) 0 0
\(433\) 591.725i 1.36657i 0.730151 + 0.683286i \(0.239450\pi\)
−0.730151 + 0.683286i \(0.760550\pi\)
\(434\) 0 0
\(435\) 537.767i 1.23625i
\(436\) 0 0
\(437\) 67.2074 + 116.407i 0.153793 + 0.266377i
\(438\) 0 0
\(439\) 443.687 + 256.163i 1.01068 + 0.583515i 0.911390 0.411544i \(-0.135010\pi\)
0.0992873 + 0.995059i \(0.468344\pi\)
\(440\) 0 0
\(441\) 689.804 822.962i 1.56418 1.86613i
\(442\) 0 0
\(443\) 134.591 + 77.7063i 0.303818 + 0.175409i 0.644157 0.764894i \(-0.277208\pi\)
−0.340339 + 0.940303i \(0.610542\pi\)
\(444\) 0 0
\(445\) 46.8030 27.0217i 0.105175 0.0607229i
\(446\) 0 0
\(447\) 396.079i 0.886084i
\(448\) 0 0
\(449\) −369.139 −0.822136 −0.411068 0.911605i \(-0.634844\pi\)
−0.411068 + 0.911605i \(0.634844\pi\)
\(450\) 0 0
\(451\) −0.567467 0.982881i −0.00125824 0.00217934i
\(452\) 0 0
\(453\) 480.282 831.873i 1.06023 1.83636i
\(454\) 0 0
\(455\) −72.1632 50.5007i −0.158600 0.110991i
\(456\) 0 0
\(457\) 214.079 370.795i 0.468444 0.811369i −0.530906 0.847431i \(-0.678148\pi\)
0.999350 + 0.0360623i \(0.0114815\pi\)
\(458\) 0 0
\(459\) −1282.13 + 740.238i −2.79331 + 1.61272i
\(460\) 0 0
\(461\) 165.578 0.359171 0.179586 0.983742i \(-0.442524\pi\)
0.179586 + 0.983742i \(0.442524\pi\)
\(462\) 0 0
\(463\) 605.376 1.30751 0.653754 0.756708i \(-0.273193\pi\)
0.653754 + 0.756708i \(0.273193\pi\)
\(464\) 0 0
\(465\) −391.658 + 226.124i −0.842275 + 0.486288i
\(466\) 0 0
\(467\) −286.063 + 495.476i −0.612555 + 1.06098i 0.378253 + 0.925702i \(0.376525\pi\)
−0.990808 + 0.135275i \(0.956808\pi\)
\(468\) 0 0
\(469\) 223.805 + 479.620i 0.477196 + 1.02264i
\(470\) 0 0
\(471\) 750.182 1299.35i 1.59274 2.75871i
\(472\) 0 0
\(473\) −2.77570 4.80766i −0.00586830 0.0101642i
\(474\) 0 0
\(475\) 139.843 0.294406
\(476\) 0 0
\(477\) 1393.06i 2.92045i
\(478\) 0 0
\(479\) −32.2540 + 18.6218i −0.0673361 + 0.0388765i −0.533290 0.845932i \(-0.679045\pi\)
0.465954 + 0.884809i \(0.345711\pi\)
\(480\) 0 0
\(481\) −103.431 59.7160i −0.215034 0.124150i
\(482\) 0 0
\(483\) 50.8596 583.107i 0.105299 1.20726i
\(484\) 0 0
\(485\) −69.3254 40.0250i −0.142939 0.0825259i
\(486\) 0 0
\(487\) 137.172 + 237.589i 0.281668 + 0.487863i 0.971796 0.235824i \(-0.0757790\pi\)
−0.690128 + 0.723688i \(0.742446\pi\)
\(488\) 0 0
\(489\) 1516.66i 3.10155i
\(490\) 0 0
\(491\) 881.994i 1.79632i −0.439667 0.898161i \(-0.644903\pi\)
0.439667 0.898161i \(-0.355097\pi\)
\(492\) 0 0
\(493\) −326.000 564.648i −0.661257 1.14533i
\(494\) 0 0
\(495\) 7.10712 + 4.10330i 0.0143578 + 0.00828949i
\(496\) 0 0
\(497\) −1.71448 + 19.6566i −0.00344965 + 0.0395504i
\(498\) 0 0
\(499\) −305.733 176.515i −0.612692 0.353738i 0.161327 0.986901i \(-0.448423\pi\)
−0.774018 + 0.633163i \(0.781756\pi\)
\(500\) 0 0
\(501\) −397.579 + 229.542i −0.793570 + 0.458168i
\(502\) 0 0
\(503\) 291.993i 0.580502i −0.956951 0.290251i \(-0.906261\pi\)
0.956951 0.290251i \(-0.0937388\pi\)
\(504\) 0 0
\(505\) 414.993 0.821768
\(506\) 0 0
\(507\) −422.774 732.266i −0.833873 1.44431i
\(508\) 0 0
\(509\) −41.5606 + 71.9851i −0.0816515 + 0.141425i −0.903959 0.427618i \(-0.859353\pi\)
0.822308 + 0.569043i \(0.192686\pi\)
\(510\) 0 0
\(511\) 37.8973 + 81.2148i 0.0741630 + 0.158933i
\(512\) 0 0
\(513\) −320.902 + 555.819i −0.625541 + 1.08347i
\(514\) 0 0
\(515\) −337.216 + 194.692i −0.654788 + 0.378042i
\(516\) 0 0
\(517\) 9.71210 0.0187855
\(518\) 0 0
\(519\) −1280.26 −2.46679
\(520\) 0 0
\(521\) 513.150 296.267i 0.984933 0.568651i 0.0811772 0.996700i \(-0.474132\pi\)
0.903756 + 0.428048i \(0.140799\pi\)
\(522\) 0 0
\(523\) 151.233 261.943i 0.289164 0.500847i −0.684446 0.729063i \(-0.739956\pi\)
0.973610 + 0.228217i \(0.0732894\pi\)
\(524\) 0 0
\(525\) −498.922 349.152i −0.950328 0.665051i
\(526\) 0 0
\(527\) −274.157 + 474.855i −0.520223 + 0.901052i
\(528\) 0 0
\(529\) 151.417 + 262.262i 0.286233 + 0.495770i
\(530\) 0 0
\(531\) 624.838 1.17672
\(532\) 0 0
\(533\) 38.1344i 0.0715467i
\(534\) 0 0
\(535\) −211.337 + 122.016i −0.395023 + 0.228067i
\(536\) 0 0
\(537\) −1271.40 734.041i −2.36759 1.36693i
\(538\) 0 0
\(539\) −5.63675 + 2.05499i −0.0104578 + 0.00381260i
\(540\) 0 0
\(541\) −630.140 363.811i −1.16477 0.672480i −0.212327 0.977199i \(-0.568104\pi\)
−0.952442 + 0.304719i \(0.901437\pi\)
\(542\) 0 0
\(543\) −511.505 885.952i −0.941997 1.63159i
\(544\) 0 0
\(545\) 96.6260i 0.177296i
\(546\) 0 0
\(547\) 1033.51i 1.88941i 0.327921 + 0.944705i \(0.393652\pi\)
−0.327921 + 0.944705i \(0.606348\pi\)
\(548\) 0 0
\(549\) 276.545 + 478.990i 0.503724 + 0.872476i
\(550\) 0 0
\(551\) −244.782 141.325i −0.444250 0.256488i
\(552\) 0 0
\(553\) 408.554 + 285.911i 0.738796 + 0.517019i
\(554\) 0 0
\(555\) 427.517 + 246.827i 0.770301 + 0.444734i
\(556\) 0 0
\(557\) 625.736 361.269i 1.12340 0.648597i 0.181136 0.983458i \(-0.442023\pi\)
0.942268 + 0.334861i \(0.108689\pi\)
\(558\) 0 0
\(559\) 186.530i 0.333686i
\(560\) 0 0
\(561\) 14.0361 0.0250197
\(562\) 0 0
\(563\) −206.897 358.355i −0.367489 0.636510i 0.621683 0.783269i \(-0.286449\pi\)
−0.989172 + 0.146759i \(0.953116\pi\)
\(564\) 0 0
\(565\) 88.3857 153.089i 0.156435 0.270953i
\(566\) 0 0
\(567\) 1281.51 597.992i 2.26016 1.05466i
\(568\) 0 0
\(569\) 258.602 447.911i 0.454485 0.787190i −0.544174 0.838972i \(-0.683157\pi\)
0.998658 + 0.0517822i \(0.0164901\pi\)
\(570\) 0 0
\(571\) 615.938 355.612i 1.07870 0.622788i 0.148155 0.988964i \(-0.452667\pi\)
0.930545 + 0.366177i \(0.119333\pi\)
\(572\) 0 0
\(573\) 1647.89 2.87591
\(574\) 0 0
\(575\) −235.299 −0.409215
\(576\) 0 0
\(577\) 527.662 304.646i 0.914491 0.527982i 0.0326179 0.999468i \(-0.489616\pi\)
0.881874 + 0.471486i \(0.156282\pi\)
\(578\) 0 0
\(579\) 265.869 460.499i 0.459187 0.795335i
\(580\) 0 0
\(581\) 18.2792 209.572i 0.0314616 0.360709i
\(582\) 0 0
\(583\) 3.89164 6.74051i 0.00667519 0.0115618i
\(584\) 0 0
\(585\) −137.873 238.803i −0.235680 0.408210i
\(586\) 0 0
\(587\) 972.801 1.65724 0.828621 0.559810i \(-0.189126\pi\)
0.828621 + 0.559810i \(0.189126\pi\)
\(588\) 0 0
\(589\) 237.701i 0.403567i
\(590\) 0 0
\(591\) 775.745 447.877i 1.31260 0.757829i
\(592\) 0 0
\(593\) −281.520 162.536i −0.474739 0.274091i 0.243482 0.969905i \(-0.421710\pi\)
−0.718222 + 0.695814i \(0.755044\pi\)
\(594\) 0 0
\(595\) 439.724 + 38.3535i 0.739033 + 0.0644597i
\(596\) 0 0
\(597\) −5.34367 3.08517i −0.00895088 0.00516779i
\(598\) 0 0
\(599\) 231.570 + 401.091i 0.386595 + 0.669602i 0.991989 0.126324i \(-0.0403179\pi\)
−0.605394 + 0.795926i \(0.706985\pi\)
\(600\) 0 0
\(601\) 325.247i 0.541176i 0.962695 + 0.270588i \(0.0872182\pi\)
−0.962695 + 0.270588i \(0.912782\pi\)
\(602\) 0 0
\(603\) 1656.97i 2.74787i
\(604\) 0 0
\(605\) 185.011 + 320.448i 0.305803 + 0.529667i
\(606\) 0 0
\(607\) −346.450 200.023i −0.570758 0.329527i 0.186694 0.982418i \(-0.440223\pi\)
−0.757452 + 0.652891i \(0.773556\pi\)
\(608\) 0 0
\(609\) 520.464 + 1115.37i 0.854621 + 1.83147i
\(610\) 0 0
\(611\) −282.612 163.166i −0.462540 0.267047i
\(612\) 0 0
\(613\) −821.365 + 474.215i −1.33991 + 0.773597i −0.986794 0.161982i \(-0.948211\pi\)
−0.353116 + 0.935579i \(0.614878\pi\)
\(614\) 0 0
\(615\) 157.623i 0.256297i
\(616\) 0 0
\(617\) −1066.14 −1.72793 −0.863967 0.503548i \(-0.832028\pi\)
−0.863967 + 0.503548i \(0.832028\pi\)
\(618\) 0 0
\(619\) 471.501 + 816.664i 0.761715 + 1.31933i 0.941966 + 0.335708i \(0.108976\pi\)
−0.180251 + 0.983621i \(0.557691\pi\)
\(620\) 0 0
\(621\) 539.948 935.218i 0.869482 1.50599i
\(622\) 0 0
\(623\) −70.9205 + 101.342i −0.113837 + 0.162668i
\(624\) 0 0
\(625\) −5.47705 + 9.48652i −0.00876328 + 0.0151784i
\(626\) 0 0
\(627\) 5.26960 3.04240i 0.00840446 0.00485232i
\(628\) 0 0
\(629\) 598.517 0.951537
\(630\) 0 0
\(631\) 575.646 0.912276 0.456138 0.889909i \(-0.349232\pi\)
0.456138 + 0.889909i \(0.349232\pi\)
\(632\) 0 0
\(633\) −1030.67 + 595.056i −1.62823 + 0.940057i
\(634\) 0 0
\(635\) −103.690 + 179.596i −0.163291 + 0.282828i
\(636\) 0 0
\(637\) 198.548 + 34.9008i 0.311692 + 0.0547894i
\(638\) 0 0
\(639\) −30.8860 + 53.4961i −0.0483349 + 0.0837185i
\(640\) 0 0
\(641\) −396.899 687.449i −0.619187 1.07246i −0.989634 0.143610i \(-0.954129\pi\)
0.370447 0.928854i \(-0.379204\pi\)
\(642\) 0 0
\(643\) −841.343 −1.30847 −0.654233 0.756293i \(-0.727008\pi\)
−0.654233 + 0.756293i \(0.727008\pi\)
\(644\) 0 0
\(645\) 770.995i 1.19534i
\(646\) 0 0
\(647\) −476.604 + 275.167i −0.736637 + 0.425297i −0.820845 0.571151i \(-0.806497\pi\)
0.0842084 + 0.996448i \(0.473164\pi\)
\(648\) 0 0
\(649\) −3.02337 1.74555i −0.00465851 0.00268959i
\(650\) 0 0
\(651\) 593.479 848.054i 0.911642 1.30269i
\(652\) 0 0
\(653\) −713.506 411.943i −1.09266 0.630847i −0.158375 0.987379i \(-0.550626\pi\)
−0.934283 + 0.356532i \(0.883959\pi\)
\(654\) 0 0
\(655\) 171.511 + 297.066i 0.261849 + 0.453535i
\(656\) 0 0
\(657\) 280.577i 0.427058i
\(658\) 0 0
\(659\) 354.257i 0.537567i −0.963201 0.268784i \(-0.913378\pi\)
0.963201 0.268784i \(-0.0866217\pi\)
\(660\) 0 0
\(661\) −84.3031 146.017i −0.127539 0.220904i 0.795184 0.606369i \(-0.207374\pi\)
−0.922722 + 0.385465i \(0.874041\pi\)
\(662\) 0 0
\(663\) −408.435 235.810i −0.616040 0.355671i
\(664\) 0 0
\(665\) 173.400 80.9138i 0.260752 0.121675i
\(666\) 0 0
\(667\) 411.869 + 237.793i 0.617494 + 0.356511i
\(668\) 0 0
\(669\) −1397.70 + 806.965i −2.08924 + 1.20623i
\(670\) 0 0
\(671\) 3.09022i 0.00460539i
\(672\) 0 0
\(673\) 514.054 0.763824 0.381912 0.924199i \(-0.375266\pi\)
0.381912 + 0.924199i \(0.375266\pi\)
\(674\) 0 0
\(675\) −561.753 972.986i −0.832227 1.44146i
\(676\) 0 0
\(677\) −260.650 + 451.460i −0.385008 + 0.666853i −0.991770 0.128030i \(-0.959135\pi\)
0.606762 + 0.794883i \(0.292468\pi\)
\(678\) 0 0
\(679\) 182.523 + 15.9200i 0.268812 + 0.0234462i
\(680\) 0 0
\(681\) 226.362 392.070i 0.332396 0.575727i
\(682\) 0 0
\(683\) −441.591 + 254.953i −0.646547 + 0.373284i −0.787132 0.616785i \(-0.788435\pi\)
0.140585 + 0.990069i \(0.455102\pi\)
\(684\) 0 0
\(685\) 179.476 0.262009
\(686\) 0 0
\(687\) −1302.30 −1.89563
\(688\) 0 0
\(689\) −226.485 + 130.761i −0.328715 + 0.189784i
\(690\) 0 0
\(691\) −467.402 + 809.565i −0.676415 + 1.17158i 0.299639 + 0.954053i \(0.403134\pi\)
−0.976053 + 0.217532i \(0.930199\pi\)
\(692\) 0 0
\(693\) −18.7120 1.63209i −0.0270014 0.00235511i
\(694\) 0 0
\(695\) 268.008 464.203i 0.385623 0.667919i
\(696\) 0 0
\(697\) −95.5526 165.502i −0.137091 0.237449i
\(698\) 0 0
\(699\) 344.619 0.493017
\(700\) 0 0
\(701\) 364.276i 0.519651i 0.965656 + 0.259826i \(0.0836651\pi\)
−0.965656 + 0.259826i \(0.916335\pi\)
\(702\) 0 0
\(703\) 224.703 129.732i 0.319634 0.184541i
\(704\) 0 0
\(705\) 1168.13 + 674.422i 1.65693 + 0.956626i
\(706\) 0 0
\(707\) −860.725 + 401.640i −1.21743 + 0.568091i
\(708\) 0 0
\(709\) −429.168 247.780i −0.605315 0.349479i 0.165815 0.986157i \(-0.446975\pi\)
−0.771130 + 0.636678i \(0.780308\pi\)
\(710\) 0 0
\(711\) 780.572 + 1351.99i 1.09785 + 1.90153i
\(712\) 0 0
\(713\) 399.955i 0.560946i
\(714\) 0 0
\(715\) 1.54065i 0.00215475i
\(716\) 0 0
\(717\) −269.894 467.470i −0.376421 0.651980i
\(718\) 0 0
\(719\) 582.836 + 336.500i 0.810620 + 0.468012i 0.847171 0.531320i \(-0.178304\pi\)
−0.0365511 + 0.999332i \(0.511637\pi\)
\(720\) 0 0
\(721\) 510.983 730.171i 0.708714 1.01272i
\(722\) 0 0
\(723\) −1154.40 666.493i −1.59668 0.921844i
\(724\) 0 0
\(725\) 428.502 247.395i 0.591037 0.341235i
\(726\) 0 0
\(727\) 165.434i 0.227557i −0.993506 0.113778i \(-0.963705\pi\)
0.993506 0.113778i \(-0.0362954\pi\)
\(728\) 0 0
\(729\) 833.991 1.14402
\(730\) 0 0
\(731\) −467.386 809.536i −0.639378 1.10744i
\(732\) 0 0
\(733\) −474.614 + 822.055i −0.647495 + 1.12149i 0.336225 + 0.941782i \(0.390850\pi\)
−0.983719 + 0.179712i \(0.942483\pi\)
\(734\) 0 0
\(735\) −820.667 144.257i −1.11655 0.196268i
\(736\) 0 0
\(737\) 4.62889 8.01748i 0.00628073 0.0108785i
\(738\) 0 0
\(739\) 62.4587 36.0605i 0.0845178 0.0487964i −0.457145 0.889392i \(-0.651128\pi\)
0.541663 + 0.840596i \(0.317795\pi\)
\(740\) 0 0
\(741\) −204.453 −0.275915
\(742\) 0 0
\(743\) −159.310 −0.214415 −0.107208 0.994237i \(-0.534191\pi\)
−0.107208 + 0.994237i \(0.534191\pi\)
\(744\) 0 0
\(745\) −188.680 + 108.934i −0.253262 + 0.146221i
\(746\) 0 0
\(747\) 329.297 570.358i 0.440825 0.763532i
\(748\) 0 0
\(749\) 320.239 457.607i 0.427556 0.610957i
\(750\) 0 0
\(751\) 382.562 662.616i 0.509403 0.882312i −0.490538 0.871420i \(-0.663200\pi\)
0.999941 0.0108919i \(-0.00346706\pi\)
\(752\) 0 0
\(753\) 378.305 + 655.243i 0.502397 + 0.870176i
\(754\) 0 0
\(755\) −528.371 −0.699830
\(756\) 0 0
\(757\) 950.822i 1.25604i 0.778197 + 0.628020i \(0.216134\pi\)
−0.778197 + 0.628020i \(0.783866\pi\)
\(758\) 0 0
\(759\) −8.86660 + 5.11913i −0.0116819 + 0.00674457i
\(760\) 0 0
\(761\) 529.627 + 305.781i 0.695962 + 0.401814i 0.805842 0.592131i \(-0.201713\pi\)
−0.109879 + 0.993945i \(0.535046\pi\)
\(762\) 0 0
\(763\) 93.5172 + 200.410i 0.122565 + 0.262660i
\(764\) 0 0
\(765\) 1196.73 + 690.931i 1.56435 + 0.903178i
\(766\) 0 0
\(767\) 58.6513 + 101.587i 0.0764685 + 0.132447i
\(768\) 0 0
\(769\) 979.152i 1.27328i 0.771161 + 0.636640i \(0.219676\pi\)
−0.771161 + 0.636640i \(0.780324\pi\)
\(770\) 0 0
\(771\) 105.611i 0.136979i
\(772\) 0 0
\(773\) −228.660 396.051i −0.295809 0.512356i 0.679364 0.733802i \(-0.262256\pi\)
−0.975173 + 0.221445i \(0.928923\pi\)
\(774\) 0 0
\(775\) −360.359 208.053i −0.464979 0.268456i
\(776\) 0 0
\(777\) −1125.59 98.1756i −1.44863 0.126352i
\(778\) 0 0
\(779\) −71.7471 41.4232i −0.0921015 0.0531748i
\(780\) 0 0
\(781\) 0.298893 0.172566i 0.000382706 0.000220955i
\(782\) 0 0
\(783\) 2270.83i 2.90016i
\(784\) 0 0
\(785\) −825.296 −1.05133
\(786\) 0 0
\(787\) −91.5206 158.518i −0.116290 0.201421i 0.802004 0.597318i \(-0.203767\pi\)
−0.918295 + 0.395897i \(0.870434\pi\)
\(788\) 0 0
\(789\) −685.481 + 1187.29i −0.868797 + 1.50480i
\(790\) 0 0
\(791\) −35.1555 + 403.059i −0.0444444 + 0.509556i
\(792\) 0 0
\(793\) −51.9165 + 89.9220i −0.0654685 + 0.113395i
\(794\) 0 0
\(795\) 936.141 540.481i 1.17754 0.679851i
\(796\) 0 0
\(797\) −619.727 −0.777575 −0.388787 0.921328i \(-0.627106\pi\)
−0.388787 + 0.921328i \(0.627106\pi\)
\(798\) 0 0
\(799\) 1635.37 2.04677
\(800\) 0 0
\(801\) −335.362 + 193.621i −0.418679 + 0.241725i
\(802\) 0 0
\(803\) 0.783819 1.35761i 0.000976113 0.00169068i
\(804\) 0 0
\(805\) −291.762 + 136.145i −0.362438 + 0.169124i
\(806\) 0 0
\(807\) −818.008 + 1416.83i −1.01364 + 1.75568i
\(808\) 0 0
\(809\) 335.874 + 581.750i 0.415171 + 0.719098i 0.995446 0.0953227i \(-0.0303883\pi\)
−0.580275 + 0.814421i \(0.697055\pi\)
\(810\) 0 0
\(811\) −1133.22 −1.39732 −0.698658 0.715456i \(-0.746219\pi\)
−0.698658 + 0.715456i \(0.746219\pi\)
\(812\) 0 0
\(813\) 2516.95i 3.09587i
\(814\) 0 0
\(815\) −722.489 + 417.129i −0.886489 + 0.511815i
\(816\) 0 0
\(817\) −350.944 202.617i −0.429551 0.248002i
\(818\) 0 0
\(819\) 517.079 + 361.858i 0.631354 + 0.441829i
\(820\) 0 0
\(821\) 620.954 + 358.508i 0.756338 + 0.436672i 0.827979 0.560758i \(-0.189490\pi\)
−0.0716414 + 0.997430i \(0.522824\pi\)
\(822\) 0 0
\(823\) −362.895 628.553i −0.440942 0.763734i 0.556818 0.830635i \(-0.312022\pi\)
−0.997760 + 0.0669009i \(0.978689\pi\)
\(824\) 0 0
\(825\) 10.6517i 0.0129112i
\(826\) 0 0
\(827\) 436.858i 0.528244i 0.964489 + 0.264122i \(0.0850822\pi\)
−0.964489 + 0.264122i \(0.914918\pi\)
\(828\) 0 0
\(829\) 588.361 + 1019.07i 0.709724 + 1.22928i 0.964959 + 0.262399i \(0.0845137\pi\)
−0.255235 + 0.966879i \(0.582153\pi\)
\(830\) 0 0
\(831\) 1403.50 + 810.313i 1.68893 + 0.975105i
\(832\) 0 0
\(833\) −949.140 + 346.029i −1.13942 + 0.415400i
\(834\) 0 0
\(835\) 218.694 + 126.263i 0.261909 + 0.151213i
\(836\) 0 0
\(837\) 1653.85 954.853i 1.97593 1.14080i
\(838\) 0 0
\(839\) 1551.16i 1.84881i −0.381407 0.924407i \(-0.624560\pi\)
0.381407 0.924407i \(-0.375440\pi\)
\(840\) 0 0
\(841\) −159.070 −0.189143
\(842\) 0 0
\(843\) 1377.33 + 2385.60i 1.63384 + 2.82990i
\(844\) 0 0
\(845\) −232.552 + 402.793i −0.275210 + 0.476678i
\(846\) 0 0
\(847\) −693.864 485.575i −0.819202 0.573288i
\(848\) 0 0
\(849\) 102.131 176.895i 0.120295 0.208357i
\(850\) 0 0
\(851\) −378.084 + 218.287i −0.444282 + 0.256506i
\(852\) 0 0
\(853\) −138.736 −0.162645 −0.0813225 0.996688i \(-0.525914\pi\)
−0.0813225 + 0.996688i \(0.525914\pi\)
\(854\) 0 0
\(855\) 599.054 0.700648
\(856\) 0 0
\(857\) 1475.23 851.725i 1.72139 0.993844i 0.805302 0.592865i \(-0.202003\pi\)
0.916087 0.400980i \(-0.131330\pi\)
\(858\) 0 0
\(859\) 256.796 444.784i 0.298948 0.517793i −0.676948 0.736031i \(-0.736698\pi\)
0.975896 + 0.218238i \(0.0700310\pi\)
\(860\) 0 0
\(861\) 152.551 + 326.921i 0.177179 + 0.379699i
\(862\) 0 0
\(863\) 117.938 204.275i 0.136661 0.236703i −0.789570 0.613660i \(-0.789696\pi\)
0.926231 + 0.376958i \(0.123030\pi\)
\(864\) 0 0
\(865\) 352.113 + 609.878i 0.407067 + 0.705061i
\(866\) 0 0
\(867\) 756.587 0.872649
\(868\) 0 0
\(869\) 8.72242i 0.0100373i
\(870\) 0 0
\(871\) −269.392 + 155.533i −0.309290 + 0.178569i
\(872\) 0 0
\(873\) 496.745 + 286.796i 0.569009 + 0.328517i
\(874\) 0 0
\(875\) −75.6122 + 866.897i −0.0864139 + 0.990739i
\(876\) 0 0
\(877\) 384.546 + 222.017i 0.438478 + 0.253156i 0.702952 0.711237i \(-0.251865\pi\)
−0.264474 + 0.964393i \(0.585198\pi\)
\(878\) 0 0
\(879\) −1467.82 2542.34i −1.66988 2.89231i
\(880\) 0 0
\(881\) 71.7252i 0.0814134i 0.999171 + 0.0407067i \(0.0129609\pi\)
−0.999171 + 0.0407067i \(0.987039\pi\)
\(882\) 0 0
\(883\) 973.840i 1.10288i −0.834216 0.551438i \(-0.814079\pi\)
0.834216 0.551438i \(-0.185921\pi\)
\(884\) 0 0
\(885\) −242.426 419.895i −0.273928 0.474457i
\(886\) 0 0
\(887\) −715.042 412.830i −0.806135 0.465422i 0.0394767 0.999220i \(-0.487431\pi\)
−0.845612 + 0.533798i \(0.820764\pi\)
\(888\) 0 0
\(889\) 41.2426 472.848i 0.0463922 0.531888i
\(890\) 0 0
\(891\) −21.4222 12.3681i −0.0240428 0.0138811i
\(892\) 0 0
\(893\) 613.970 354.476i 0.687536 0.396949i
\(894\) 0 0
\(895\) 807.539i 0.902278i
\(896\) 0 0
\(897\) 344.011 0.383513
\(898\) 0 0
\(899\) 420.516 + 728.355i 0.467760 + 0.810184i
\(900\) 0 0
\(901\) 655.291 1135.00i 0.727293 1.25971i
\(902\) 0 0
\(903\) 746.189 + 1599.10i 0.826344 + 1.77088i
\(904\) 0 0
\(905\) −281.360 + 487.330i −0.310895 + 0.538486i
\(906\) 0 0
\(907\) −637.046 + 367.799i −0.702366 + 0.405511i −0.808228 0.588870i \(-0.799573\pi\)
0.105862 + 0.994381i \(0.466240\pi\)
\(908\) 0 0
\(909\) −2973.59 −3.27128
\(910\) 0 0
\(911\) −235.528 −0.258538 −0.129269 0.991610i \(-0.541263\pi\)
−0.129269 + 0.991610i \(0.541263\pi\)
\(912\) 0 0
\(913\) −3.18670 + 1.83984i −0.00349036 + 0.00201516i
\(914\) 0 0
\(915\) 214.589 371.679i 0.234524 0.406207i
\(916\) 0 0
\(917\) −643.234 450.143i −0.701455 0.490887i
\(918\) 0 0
\(919\) −717.115 + 1242.08i −0.780321 + 1.35156i 0.151434 + 0.988467i \(0.451611\pi\)
−0.931755 + 0.363088i \(0.881722\pi\)
\(920\) 0 0
\(921\) −485.080 840.184i −0.526689 0.912252i
\(922\) 0 0
\(923\) −11.5966 −0.0125641
\(924\) 0 0
\(925\) 454.204i 0.491031i
\(926\) 0 0
\(927\) 2416.29 1395.04i 2.60657 1.50490i
\(928\) 0 0
\(929\) −1194.57 689.687i −1.28587 0.742397i −0.307955 0.951401i \(-0.599645\pi\)
−0.977915 + 0.209003i \(0.932978\pi\)
\(930\) 0 0
\(931\) −281.334 + 335.642i −0.302185 + 0.360518i
\(932\) 0 0
\(933\) −66.2316 38.2388i −0.0709878 0.0409848i
\(934\) 0 0
\(935\) −3.86037 6.68635i −0.00412874 0.00715118i
\(936\) 0 0
\(937\) 541.545i 0.577956i 0.957336 + 0.288978i \(0.0933154\pi\)
−0.957336 + 0.288978i \(0.906685\pi\)
\(938\) 0 0
\(939\) 2345.76i 2.49814i
\(940\) 0 0
\(941\) −48.9285 84.7467i −0.0519963 0.0900603i 0.838856 0.544354i \(-0.183225\pi\)
−0.890852 + 0.454294i \(0.849892\pi\)
\(942\) 0 0
\(943\) 120.721 + 69.6984i 0.128018 + 0.0739114i
\(944\) 0 0
\(945\) −1259.53 881.434i −1.33283 0.932735i
\(946\) 0 0
\(947\) 674.176 + 389.236i 0.711907 + 0.411020i 0.811767 0.583982i \(-0.198506\pi\)
−0.0998594 + 0.995002i \(0.531839\pi\)
\(948\) 0 0
\(949\) −45.6166 + 26.3367i −0.0480680 + 0.0277521i
\(950\) 0 0
\(951\) 2621.73i 2.75681i
\(952\) 0 0
\(953\) 348.435 0.365620 0.182810 0.983148i \(-0.441481\pi\)
0.182810 + 0.983148i \(0.441481\pi\)
\(954\) 0 0
\(955\) −453.224 785.006i −0.474580 0.821996i
\(956\) 0 0
\(957\) 10.7646 18.6449i 0.0112483 0.0194826i
\(958\) 0 0
\(959\) −372.246 + 173.701i −0.388161 + 0.181128i
\(960\) 0 0
\(961\) −126.857 + 219.722i −0.132005 + 0.228639i
\(962\) 0 0
\(963\) 1514.32 874.291i 1.57250 0.907883i
\(964\) 0 0
\(965\) −292.490 −0.303098
\(966\) 0 0
\(967\) 1055.75 1.09177 0.545887 0.837859i \(-0.316193\pi\)
0.545887 + 0.837859i \(0.316193\pi\)
\(968\) 0 0
\(969\) 887.319 512.294i 0.915706 0.528683i
\(970\) 0 0
\(971\) 426.302 738.377i 0.439034 0.760429i −0.558581 0.829450i \(-0.688654\pi\)
0.997615 + 0.0690206i \(0.0219874\pi\)
\(972\) 0 0
\(973\) −106.600 + 1222.18i −0.109558 + 1.25609i
\(974\) 0 0
\(975\) 178.952 309.954i 0.183540 0.317901i
\(976\) 0 0
\(977\) −169.835 294.163i −0.173834 0.301089i 0.765923 0.642932i \(-0.222282\pi\)
−0.939757 + 0.341843i \(0.888949\pi\)
\(978\) 0 0
\(979\) 2.16360 0.00221001
\(980\) 0 0
\(981\) 692.365i 0.705774i
\(982\) 0 0
\(983\) 905.444 522.759i 0.921103 0.531799i 0.0371164 0.999311i \(-0.488183\pi\)
0.883987 + 0.467512i \(0.154849\pi\)
\(984\) 0 0
\(985\) −426.709 246.361i −0.433208 0.250112i
\(986\) 0 0
\(987\) −3075.52 268.252i −3.11602 0.271785i
\(988\) 0 0
\(989\) 590.496 + 340.923i 0.597063 + 0.344715i
\(990\) 0 0
\(991\) −409.911 709.987i −0.413634 0.716435i 0.581650 0.813439i \(-0.302407\pi\)
−0.995284 + 0.0970040i \(0.969074\pi\)
\(992\) 0 0
\(993\) 2463.49i 2.48086i
\(994\) 0 0
\(995\) 3.39408i 0.00341114i
\(996\) 0 0
\(997\) −380.211 658.545i −0.381355 0.660526i 0.609901 0.792478i \(-0.291209\pi\)
−0.991256 + 0.131951i \(0.957876\pi\)
\(998\) 0 0
\(999\) −1805.28 1042.28i −1.80708 1.04332i
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 224.3.n.a.145.14 28
4.3 odd 2 56.3.j.a.5.8 yes 28
7.2 even 3 1568.3.h.a.881.1 28
7.3 odd 6 inner 224.3.n.a.17.1 28
7.5 odd 6 1568.3.h.a.881.27 28
8.3 odd 2 56.3.j.a.5.3 28
8.5 even 2 inner 224.3.n.a.145.1 28
28.3 even 6 56.3.j.a.45.3 yes 28
28.11 odd 6 392.3.j.e.325.3 28
28.19 even 6 392.3.h.a.293.21 28
28.23 odd 6 392.3.h.a.293.22 28
28.27 even 2 392.3.j.e.117.8 28
56.3 even 6 56.3.j.a.45.8 yes 28
56.5 odd 6 1568.3.h.a.881.2 28
56.11 odd 6 392.3.j.e.325.8 28
56.19 even 6 392.3.h.a.293.24 28
56.27 even 2 392.3.j.e.117.3 28
56.37 even 6 1568.3.h.a.881.28 28
56.45 odd 6 inner 224.3.n.a.17.14 28
56.51 odd 6 392.3.h.a.293.23 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.j.a.5.3 28 8.3 odd 2
56.3.j.a.5.8 yes 28 4.3 odd 2
56.3.j.a.45.3 yes 28 28.3 even 6
56.3.j.a.45.8 yes 28 56.3 even 6
224.3.n.a.17.1 28 7.3 odd 6 inner
224.3.n.a.17.14 28 56.45 odd 6 inner
224.3.n.a.145.1 28 8.5 even 2 inner
224.3.n.a.145.14 28 1.1 even 1 trivial
392.3.h.a.293.21 28 28.19 even 6
392.3.h.a.293.22 28 28.23 odd 6
392.3.h.a.293.23 28 56.51 odd 6
392.3.h.a.293.24 28 56.19 even 6
392.3.j.e.117.3 28 56.27 even 2
392.3.j.e.117.8 28 28.27 even 2
392.3.j.e.325.3 28 28.11 odd 6
392.3.j.e.325.8 28 56.11 odd 6
1568.3.h.a.881.1 28 7.2 even 3
1568.3.h.a.881.2 28 56.5 odd 6
1568.3.h.a.881.27 28 7.5 odd 6
1568.3.h.a.881.28 28 56.37 even 6