Properties

Label 336.3.bn.d.65.1
Level $336$
Weight $3$
Character 336.65
Analytic conductor $9.155$
Analytic rank $0$
Dimension $4$
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] = [336,3,Mod(65,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.bn (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: \(9.15533688251\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-35})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} - 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 65.1
Root \(2.81174 + 1.04601i\) of defining polynomial
Character \(\chi\) \(=\) 336.65
Dual form 336.3.bn.d.305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.81174 + 1.04601i) q^{3} +(5.12348 + 2.95804i) q^{5} +7.00000 q^{7} +(6.81174 - 5.88220i) q^{9} +O(q^{10})\) \(q+(-2.81174 + 1.04601i) q^{3} +(5.12348 + 2.95804i) q^{5} +7.00000 q^{7} +(6.81174 - 5.88220i) q^{9} +(-5.12348 + 2.95804i) q^{11} -6.00000 q^{13} +(-17.5000 - 2.95804i) q^{15} +(5.12348 - 2.95804i) q^{17} +(11.5000 - 19.9186i) q^{19} +(-19.6822 + 7.32205i) q^{21} +(35.8643 + 20.7063i) q^{23} +(5.00000 + 8.66025i) q^{25} +(-13.0000 + 23.6643i) q^{27} +47.3286i q^{29} +(19.5000 + 33.7750i) q^{31} +(11.3117 - 13.6764i) q^{33} +(35.8643 + 20.7063i) q^{35} +(-23.5000 + 40.7032i) q^{37} +(16.8704 - 6.27604i) q^{39} +22.0000 q^{43} +(52.2995 - 9.98789i) q^{45} +(-46.1113 - 26.6224i) q^{47} +49.0000 q^{49} +(-11.3117 + 13.6764i) q^{51} +(46.1113 - 26.6224i) q^{53} -35.0000 q^{55} +(-11.5000 + 68.0349i) q^{57} +(-87.0991 + 50.2867i) q^{59} +(40.5000 - 70.1481i) q^{61} +(47.6822 - 41.1754i) q^{63} +(-30.7409 - 17.7482i) q^{65} +(15.5000 + 26.8468i) q^{67} +(-122.500 - 20.7063i) q^{69} -94.6573i q^{71} +(8.50000 + 14.7224i) q^{73} +(-23.1174 - 19.1203i) q^{75} +(-35.8643 + 20.7063i) q^{77} +(-4.50000 + 7.79423i) q^{79} +(11.7995 - 80.1360i) q^{81} +47.3286i q^{83} +35.0000 q^{85} +(-49.5061 - 133.076i) q^{87} +(-76.8521 - 44.3706i) q^{89} -42.0000 q^{91} +(-90.1578 - 74.5693i) q^{93} +(117.840 - 68.0349i) q^{95} +82.0000 q^{97} +(-17.5000 + 50.2867i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 28 q^{7} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 28 q^{7} + 17 q^{9} - 24 q^{13} - 70 q^{15} + 46 q^{19} - 7 q^{21} + 20 q^{25} - 52 q^{27} + 78 q^{31} + 35 q^{33} - 94 q^{37} + 6 q^{39} + 88 q^{43} + 35 q^{45} + 196 q^{49} - 35 q^{51} - 140 q^{55} - 46 q^{57} + 162 q^{61} + 119 q^{63} + 62 q^{67} - 490 q^{69} + 34 q^{73} + 10 q^{75} - 18 q^{79} - 127 q^{81} + 140 q^{85} - 280 q^{87} - 168 q^{91} + 39 q^{93} + 328 q^{97} - 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −2.81174 + 1.04601i −0.937246 + 0.348669i
\(4\) 0 0
\(5\) 5.12348 + 2.95804i 1.02470 + 0.591608i 0.915460 0.402408i \(-0.131827\pi\)
0.109235 + 0.994016i \(0.465160\pi\)
\(6\) 0 0
\(7\) 7.00000 1.00000
\(8\) 0 0
\(9\) 6.81174 5.88220i 0.756860 0.653577i
\(10\) 0 0
\(11\) −5.12348 + 2.95804i −0.465770 + 0.268913i −0.714468 0.699669i \(-0.753331\pi\)
0.248697 + 0.968581i \(0.419998\pi\)
\(12\) 0 0
\(13\) −6.00000 −0.461538 −0.230769 0.973009i \(-0.574124\pi\)
−0.230769 + 0.973009i \(0.574124\pi\)
\(14\) 0 0
\(15\) −17.5000 2.95804i −1.16667 0.197203i
\(16\) 0 0
\(17\) 5.12348 2.95804i 0.301381 0.174002i −0.341682 0.939816i \(-0.610997\pi\)
0.643063 + 0.765813i \(0.277663\pi\)
\(18\) 0 0
\(19\) 11.5000 19.9186i 0.605263 1.04835i −0.386747 0.922186i \(-0.626401\pi\)
0.992010 0.126161i \(-0.0402654\pi\)
\(20\) 0 0
\(21\) −19.6822 + 7.32205i −0.937246 + 0.348669i
\(22\) 0 0
\(23\) 35.8643 + 20.7063i 1.55932 + 0.900273i 0.997322 + 0.0731333i \(0.0232999\pi\)
0.561996 + 0.827140i \(0.310033\pi\)
\(24\) 0 0
\(25\) 5.00000 + 8.66025i 0.200000 + 0.346410i
\(26\) 0 0
\(27\) −13.0000 + 23.6643i −0.481481 + 0.876456i
\(28\) 0 0
\(29\) 47.3286i 1.63202i 0.578036 + 0.816011i \(0.303819\pi\)
−0.578036 + 0.816011i \(0.696181\pi\)
\(30\) 0 0
\(31\) 19.5000 + 33.7750i 0.629032 + 1.08952i 0.987746 + 0.156068i \(0.0498820\pi\)
−0.358714 + 0.933448i \(0.616785\pi\)
\(32\) 0 0
\(33\) 11.3117 13.6764i 0.342780 0.414437i
\(34\) 0 0
\(35\) 35.8643 + 20.7063i 1.02470 + 0.591608i
\(36\) 0 0
\(37\) −23.5000 + 40.7032i −0.635135 + 1.10009i 0.351351 + 0.936244i \(0.385722\pi\)
−0.986486 + 0.163843i \(0.947611\pi\)
\(38\) 0 0
\(39\) 16.8704 6.27604i 0.432575 0.160924i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 22.0000 0.511628 0.255814 0.966726i \(-0.417657\pi\)
0.255814 + 0.966726i \(0.417657\pi\)
\(44\) 0 0
\(45\) 52.2995 9.98789i 1.16221 0.221953i
\(46\) 0 0
\(47\) −46.1113 26.6224i −0.981091 0.566433i −0.0784917 0.996915i \(-0.525010\pi\)
−0.902599 + 0.430482i \(0.858344\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) −11.3117 + 13.6764i −0.221799 + 0.268165i
\(52\) 0 0
\(53\) 46.1113 26.6224i 0.870024 0.502309i 0.00266787 0.999996i \(-0.499151\pi\)
0.867356 + 0.497688i \(0.165817\pi\)
\(54\) 0 0
\(55\) −35.0000 −0.636364
\(56\) 0 0
\(57\) −11.5000 + 68.0349i −0.201754 + 1.19360i
\(58\) 0 0
\(59\) −87.0991 + 50.2867i −1.47626 + 0.852317i −0.999641 0.0267957i \(-0.991470\pi\)
−0.476615 + 0.879112i \(0.658136\pi\)
\(60\) 0 0
\(61\) 40.5000 70.1481i 0.663934 1.14997i −0.315639 0.948879i \(-0.602219\pi\)
0.979573 0.201089i \(-0.0644479\pi\)
\(62\) 0 0
\(63\) 47.6822 41.1754i 0.756860 0.653577i
\(64\) 0 0
\(65\) −30.7409 17.7482i −0.472936 0.273050i
\(66\) 0 0
\(67\) 15.5000 + 26.8468i 0.231343 + 0.400698i 0.958204 0.286087i \(-0.0923546\pi\)
−0.726860 + 0.686785i \(0.759021\pi\)
\(68\) 0 0
\(69\) −122.500 20.7063i −1.77536 0.300091i
\(70\) 0 0
\(71\) 94.6573i 1.33320i −0.745415 0.666601i \(-0.767749\pi\)
0.745415 0.666601i \(-0.232251\pi\)
\(72\) 0 0
\(73\) 8.50000 + 14.7224i 0.116438 + 0.201677i 0.918354 0.395760i \(-0.129519\pi\)
−0.801915 + 0.597438i \(0.796186\pi\)
\(74\) 0 0
\(75\) −23.1174 19.1203i −0.308232 0.254938i
\(76\) 0 0
\(77\) −35.8643 + 20.7063i −0.465770 + 0.268913i
\(78\) 0 0
\(79\) −4.50000 + 7.79423i −0.0569620 + 0.0986611i −0.893100 0.449858i \(-0.851475\pi\)
0.836138 + 0.548519i \(0.184808\pi\)
\(80\) 0 0
\(81\) 11.7995 80.1360i 0.145673 0.989333i
\(82\) 0 0
\(83\) 47.3286i 0.570225i 0.958494 + 0.285112i \(0.0920309\pi\)
−0.958494 + 0.285112i \(0.907969\pi\)
\(84\) 0 0
\(85\) 35.0000 0.411765
\(86\) 0 0
\(87\) −49.5061 133.076i −0.569036 1.52961i
\(88\) 0 0
\(89\) −76.8521 44.3706i −0.863507 0.498546i 0.00167806 0.999999i \(-0.499466\pi\)
−0.865185 + 0.501453i \(0.832799\pi\)
\(90\) 0 0
\(91\) −42.0000 −0.461538
\(92\) 0 0
\(93\) −90.1578 74.5693i −0.969438 0.801820i
\(94\) 0 0
\(95\) 117.840 68.0349i 1.24042 0.716157i
\(96\) 0 0
\(97\) 82.0000 0.845361 0.422680 0.906279i \(-0.361089\pi\)
0.422680 + 0.906279i \(0.361089\pi\)
\(98\) 0 0
\(99\) −17.5000 + 50.2867i −0.176768 + 0.507946i
\(100\) 0 0
\(101\) −35.8643 + 20.7063i −0.355092 + 0.205013i −0.666926 0.745124i \(-0.732390\pi\)
0.311833 + 0.950137i \(0.399057\pi\)
\(102\) 0 0
\(103\) 11.5000 19.9186i 0.111650 0.193384i −0.804785 0.593566i \(-0.797720\pi\)
0.916436 + 0.400182i \(0.131053\pi\)
\(104\) 0 0
\(105\) −122.500 20.7063i −1.16667 0.197203i
\(106\) 0 0
\(107\) −46.1113 26.6224i −0.430947 0.248807i 0.268803 0.963195i \(-0.413372\pi\)
−0.699750 + 0.714388i \(0.746705\pi\)
\(108\) 0 0
\(109\) 68.5000 + 118.645i 0.628440 + 1.08849i 0.987865 + 0.155316i \(0.0496397\pi\)
−0.359424 + 0.933174i \(0.617027\pi\)
\(110\) 0 0
\(111\) 23.5000 139.028i 0.211712 1.25250i
\(112\) 0 0
\(113\) 94.6573i 0.837675i −0.908061 0.418838i \(-0.862438\pi\)
0.908061 0.418838i \(-0.137562\pi\)
\(114\) 0 0
\(115\) 122.500 + 212.176i 1.06522 + 1.84501i
\(116\) 0 0
\(117\) −40.8704 + 35.2932i −0.349320 + 0.301651i
\(118\) 0 0
\(119\) 35.8643 20.7063i 0.301381 0.174002i
\(120\) 0 0
\(121\) −43.0000 + 74.4782i −0.355372 + 0.615522i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 88.7412i 0.709930i
\(126\) 0 0
\(127\) 78.0000 0.614173 0.307087 0.951682i \(-0.400646\pi\)
0.307087 + 0.951682i \(0.400646\pi\)
\(128\) 0 0
\(129\) −61.8582 + 23.0122i −0.479521 + 0.178389i
\(130\) 0 0
\(131\) −128.087 73.9510i −0.977762 0.564511i −0.0761686 0.997095i \(-0.524269\pi\)
−0.901594 + 0.432584i \(0.857602\pi\)
\(132\) 0 0
\(133\) 80.5000 139.430i 0.605263 1.04835i
\(134\) 0 0
\(135\) −136.605 + 82.7890i −1.01189 + 0.613252i
\(136\) 0 0
\(137\) 87.0991 50.2867i 0.635760 0.367056i −0.147220 0.989104i \(-0.547032\pi\)
0.782979 + 0.622048i \(0.213699\pi\)
\(138\) 0 0
\(139\) −106.000 −0.762590 −0.381295 0.924453i \(-0.624522\pi\)
−0.381295 + 0.924453i \(0.624522\pi\)
\(140\) 0 0
\(141\) 157.500 + 26.6224i 1.11702 + 0.188811i
\(142\) 0 0
\(143\) 30.7409 17.7482i 0.214971 0.124114i
\(144\) 0 0
\(145\) −140.000 + 242.487i −0.965517 + 1.67232i
\(146\) 0 0
\(147\) −137.775 + 51.2544i −0.937246 + 0.348669i
\(148\) 0 0
\(149\) 169.075 + 97.6153i 1.13473 + 0.655136i 0.945120 0.326723i \(-0.105945\pi\)
0.189609 + 0.981860i \(0.439278\pi\)
\(150\) 0 0
\(151\) −20.5000 35.5070i −0.135762 0.235146i 0.790127 0.612944i \(-0.210015\pi\)
−0.925888 + 0.377798i \(0.876681\pi\)
\(152\) 0 0
\(153\) 17.5000 50.2867i 0.114379 0.328671i
\(154\) 0 0
\(155\) 230.727i 1.48856i
\(156\) 0 0
\(157\) −83.5000 144.626i −0.531847 0.921186i −0.999309 0.0371729i \(-0.988165\pi\)
0.467462 0.884013i \(-0.345169\pi\)
\(158\) 0 0
\(159\) −101.806 + 123.088i −0.640287 + 0.774137i
\(160\) 0 0
\(161\) 251.050 + 144.944i 1.55932 + 0.900273i
\(162\) 0 0
\(163\) 131.500 227.765i 0.806748 1.39733i −0.108356 0.994112i \(-0.534559\pi\)
0.915104 0.403217i \(-0.132108\pi\)
\(164\) 0 0
\(165\) 98.4108 36.6103i 0.596429 0.221880i
\(166\) 0 0
\(167\) 189.315i 1.13362i −0.823849 0.566810i \(-0.808177\pi\)
0.823849 0.566810i \(-0.191823\pi\)
\(168\) 0 0
\(169\) −133.000 −0.786982
\(170\) 0 0
\(171\) −38.8300 203.325i −0.227076 1.18904i
\(172\) 0 0
\(173\) −240.803 139.028i −1.39193 0.803629i −0.398398 0.917212i \(-0.630434\pi\)
−0.993528 + 0.113583i \(0.963767\pi\)
\(174\) 0 0
\(175\) 35.0000 + 60.6218i 0.200000 + 0.346410i
\(176\) 0 0
\(177\) 192.300 232.499i 1.08644 1.31355i
\(178\) 0 0
\(179\) 76.8521 44.3706i 0.429342 0.247880i −0.269725 0.962938i \(-0.586933\pi\)
0.699066 + 0.715057i \(0.253599\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.0331492 −0.0165746 0.999863i \(-0.505276\pi\)
−0.0165746 + 0.999863i \(0.505276\pi\)
\(182\) 0 0
\(183\) −40.5000 + 239.601i −0.221311 + 1.30930i
\(184\) 0 0
\(185\) −240.803 + 139.028i −1.30164 + 0.751502i
\(186\) 0 0
\(187\) −17.5000 + 30.3109i −0.0935829 + 0.162090i
\(188\) 0 0
\(189\) −91.0000 + 165.650i −0.481481 + 0.876456i
\(190\) 0 0
\(191\) −46.1113 26.6224i −0.241420 0.139384i 0.374409 0.927264i \(-0.377846\pi\)
−0.615829 + 0.787880i \(0.711179\pi\)
\(192\) 0 0
\(193\) −71.5000 123.842i −0.370466 0.641666i 0.619171 0.785256i \(-0.287469\pi\)
−0.989637 + 0.143590i \(0.954135\pi\)
\(194\) 0 0
\(195\) 105.000 + 17.7482i 0.538462 + 0.0910166i
\(196\) 0 0
\(197\) 47.3286i 0.240247i −0.992759 0.120123i \(-0.961671\pi\)
0.992759 0.120123i \(-0.0383290\pi\)
\(198\) 0 0
\(199\) −84.5000 146.358i −0.424623 0.735469i 0.571762 0.820420i \(-0.306260\pi\)
−0.996385 + 0.0849507i \(0.972927\pi\)
\(200\) 0 0
\(201\) −71.6639 59.2730i −0.356537 0.294891i
\(202\) 0 0
\(203\) 331.300i 1.63202i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 366.097 69.9153i 1.76858 0.337755i
\(208\) 0 0
\(209\) 136.070i 0.651052i
\(210\) 0 0
\(211\) 166.000 0.786730 0.393365 0.919382i \(-0.371311\pi\)
0.393365 + 0.919382i \(0.371311\pi\)
\(212\) 0 0
\(213\) 99.0122 + 266.151i 0.464846 + 1.24954i
\(214\) 0 0
\(215\) 112.716 + 65.0769i 0.524263 + 0.302683i
\(216\) 0 0
\(217\) 136.500 + 236.425i 0.629032 + 1.08952i
\(218\) 0 0
\(219\) −39.2995 32.5046i −0.179450 0.148423i
\(220\) 0 0
\(221\) −30.7409 + 17.7482i −0.139099 + 0.0803088i
\(222\) 0 0
\(223\) 142.000 0.636771 0.318386 0.947961i \(-0.396859\pi\)
0.318386 + 0.947961i \(0.396859\pi\)
\(224\) 0 0
\(225\) 85.0000 + 29.5804i 0.377778 + 0.131468i
\(226\) 0 0
\(227\) 76.8521 44.3706i 0.338556 0.195465i −0.321077 0.947053i \(-0.604045\pi\)
0.659633 + 0.751588i \(0.270712\pi\)
\(228\) 0 0
\(229\) 128.500 222.569i 0.561135 0.971915i −0.436262 0.899820i \(-0.643698\pi\)
0.997398 0.0720955i \(-0.0229686\pi\)
\(230\) 0 0
\(231\) 79.1822 95.7350i 0.342780 0.414437i
\(232\) 0 0
\(233\) −158.828 91.6992i −0.681664 0.393559i 0.118818 0.992916i \(-0.462090\pi\)
−0.800482 + 0.599357i \(0.795423\pi\)
\(234\) 0 0
\(235\) −157.500 272.798i −0.670213 1.16084i
\(236\) 0 0
\(237\) 4.50000 26.6224i 0.0189873 0.112331i
\(238\) 0 0
\(239\) 283.972i 1.18817i 0.804404 + 0.594083i \(0.202485\pi\)
−0.804404 + 0.594083i \(0.797515\pi\)
\(240\) 0 0
\(241\) 0.500000 + 0.866025i 0.00207469 + 0.00359347i 0.867061 0.498202i \(-0.166006\pi\)
−0.864986 + 0.501796i \(0.832673\pi\)
\(242\) 0 0
\(243\) 50.6456 + 237.664i 0.208418 + 0.978040i
\(244\) 0 0
\(245\) 251.050 + 144.944i 1.02470 + 0.591608i
\(246\) 0 0
\(247\) −69.0000 + 119.512i −0.279352 + 0.483852i
\(248\) 0 0
\(249\) −49.5061 133.076i −0.198820 0.534441i
\(250\) 0 0
\(251\) 141.986i 0.565681i −0.959167 0.282840i \(-0.908723\pi\)
0.959167 0.282840i \(-0.0912767\pi\)
\(252\) 0 0
\(253\) −245.000 −0.968379
\(254\) 0 0
\(255\) −98.4108 + 36.6103i −0.385925 + 0.143570i
\(256\) 0 0
\(257\) 169.075 + 97.6153i 0.657878 + 0.379826i 0.791468 0.611211i \(-0.209317\pi\)
−0.133590 + 0.991037i \(0.542650\pi\)
\(258\) 0 0
\(259\) −164.500 + 284.922i −0.635135 + 1.10009i
\(260\) 0 0
\(261\) 278.396 + 322.390i 1.06665 + 1.23521i
\(262\) 0 0
\(263\) −46.1113 + 26.6224i −0.175328 + 0.101226i −0.585096 0.810964i \(-0.698943\pi\)
0.409768 + 0.912190i \(0.365610\pi\)
\(264\) 0 0
\(265\) 315.000 1.18868
\(266\) 0 0
\(267\) 262.500 + 44.3706i 0.983146 + 0.166182i
\(268\) 0 0
\(269\) −35.8643 + 20.7063i −0.133325 + 0.0769750i −0.565179 0.824968i \(-0.691193\pi\)
0.431854 + 0.901943i \(0.357860\pi\)
\(270\) 0 0
\(271\) 43.5000 75.3442i 0.160517 0.278023i −0.774537 0.632528i \(-0.782017\pi\)
0.935054 + 0.354505i \(0.115351\pi\)
\(272\) 0 0
\(273\) 118.093 43.9323i 0.432575 0.160924i
\(274\) 0 0
\(275\) −51.2348 29.5804i −0.186308 0.107565i
\(276\) 0 0
\(277\) −155.500 269.334i −0.561372 0.972325i −0.997377 0.0723804i \(-0.976940\pi\)
0.436005 0.899944i \(-0.356393\pi\)
\(278\) 0 0
\(279\) 331.500 + 115.364i 1.18817 + 0.413489i
\(280\) 0 0
\(281\) 378.629i 1.34743i 0.738989 + 0.673717i \(0.235303\pi\)
−0.738989 + 0.673717i \(0.764697\pi\)
\(282\) 0 0
\(283\) 159.500 + 276.262i 0.563604 + 0.976191i 0.997178 + 0.0750731i \(0.0239190\pi\)
−0.433574 + 0.901118i \(0.642748\pi\)
\(284\) 0 0
\(285\) −260.170 + 314.558i −0.912877 + 1.10371i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −127.000 + 219.970i −0.439446 + 0.761143i
\(290\) 0 0
\(291\) −230.562 + 85.7726i −0.792311 + 0.294751i
\(292\) 0 0
\(293\) 141.986i 0.484594i −0.970202 0.242297i \(-0.922099\pi\)
0.970202 0.242297i \(-0.0779008\pi\)
\(294\) 0 0
\(295\) −595.000 −2.01695
\(296\) 0 0
\(297\) −3.39482 159.698i −0.0114304 0.537704i
\(298\) 0 0
\(299\) −215.186 124.238i −0.719686 0.415511i
\(300\) 0 0
\(301\) 154.000 0.511628
\(302\) 0 0
\(303\) 79.1822 95.7350i 0.261327 0.315957i
\(304\) 0 0
\(305\) 415.002 239.601i 1.36066 0.785578i
\(306\) 0 0
\(307\) −442.000 −1.43974 −0.719870 0.694109i \(-0.755798\pi\)
−0.719870 + 0.694109i \(0.755798\pi\)
\(308\) 0 0
\(309\) −11.5000 + 68.0349i −0.0372168 + 0.220178i
\(310\) 0 0
\(311\) −210.062 + 121.280i −0.675442 + 0.389967i −0.798136 0.602478i \(-0.794180\pi\)
0.122693 + 0.992445i \(0.460847\pi\)
\(312\) 0 0
\(313\) 128.500 222.569i 0.410543 0.711082i −0.584406 0.811461i \(-0.698672\pi\)
0.994949 + 0.100380i \(0.0320058\pi\)
\(314\) 0 0
\(315\) 366.097 69.9153i 1.16221 0.221953i
\(316\) 0 0
\(317\) −76.8521 44.3706i −0.242436 0.139970i 0.373860 0.927485i \(-0.378034\pi\)
−0.616296 + 0.787515i \(0.711367\pi\)
\(318\) 0 0
\(319\) −140.000 242.487i −0.438871 0.760148i
\(320\) 0 0
\(321\) 157.500 + 26.6224i 0.490654 + 0.0829357i
\(322\) 0 0
\(323\) 136.070i 0.421269i
\(324\) 0 0
\(325\) −30.0000 51.9615i −0.0923077 0.159882i
\(326\) 0 0
\(327\) −316.708 261.948i −0.968526 0.801066i
\(328\) 0 0
\(329\) −322.779 186.357i −0.981091 0.566433i
\(330\) 0 0
\(331\) −60.5000 + 104.789i −0.182779 + 0.316583i −0.942826 0.333285i \(-0.891843\pi\)
0.760047 + 0.649869i \(0.225176\pi\)
\(332\) 0 0
\(333\) 79.3483 + 415.491i 0.238283 + 1.24772i
\(334\) 0 0
\(335\) 183.398i 0.547458i
\(336\) 0 0
\(337\) −78.0000 −0.231454 −0.115727 0.993281i \(-0.536920\pi\)
−0.115727 + 0.993281i \(0.536920\pi\)
\(338\) 0 0
\(339\) 99.0122 + 266.151i 0.292071 + 0.785107i
\(340\) 0 0
\(341\) −199.816 115.364i −0.585969 0.338310i
\(342\) 0 0
\(343\) 343.000 1.00000
\(344\) 0 0
\(345\) −566.376 468.448i −1.64167 1.35782i
\(346\) 0 0
\(347\) 568.706 328.342i 1.63892 0.946232i 0.657716 0.753266i \(-0.271523\pi\)
0.981205 0.192966i \(-0.0618106\pi\)
\(348\) 0 0
\(349\) −422.000 −1.20917 −0.604585 0.796541i \(-0.706661\pi\)
−0.604585 + 0.796541i \(0.706661\pi\)
\(350\) 0 0
\(351\) 78.0000 141.986i 0.222222 0.404518i
\(352\) 0 0
\(353\) 169.075 97.6153i 0.478965 0.276531i −0.241020 0.970520i \(-0.577482\pi\)
0.719985 + 0.693990i \(0.244149\pi\)
\(354\) 0 0
\(355\) 280.000 484.974i 0.788732 1.36612i
\(356\) 0 0
\(357\) −79.1822 + 95.7350i −0.221799 + 0.268165i
\(358\) 0 0
\(359\) 117.840 + 68.0349i 0.328245 + 0.189512i 0.655062 0.755575i \(-0.272643\pi\)
−0.326817 + 0.945088i \(0.605976\pi\)
\(360\) 0 0
\(361\) −84.0000 145.492i −0.232687 0.403026i
\(362\) 0 0
\(363\) 43.0000 254.391i 0.118457 0.700803i
\(364\) 0 0
\(365\) 100.573i 0.275543i
\(366\) 0 0
\(367\) −36.5000 63.2199i −0.0994550 0.172261i 0.812004 0.583652i \(-0.198377\pi\)
−0.911459 + 0.411390i \(0.865043\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 322.779 186.357i 0.870024 0.502309i
\(372\) 0 0
\(373\) 200.500 347.276i 0.537534 0.931035i −0.461503 0.887139i \(-0.652689\pi\)
0.999036 0.0438965i \(-0.0139772\pi\)
\(374\) 0 0
\(375\) 92.8239 + 249.517i 0.247530 + 0.665379i
\(376\) 0 0
\(377\) 283.972i 0.753241i
\(378\) 0 0
\(379\) −538.000 −1.41953 −0.709763 0.704441i \(-0.751198\pi\)
−0.709763 + 0.704441i \(0.751198\pi\)
\(380\) 0 0
\(381\) −219.316 + 81.5886i −0.575631 + 0.214143i
\(382\) 0 0
\(383\) −210.062 121.280i −0.548466 0.316657i 0.200037 0.979788i \(-0.435894\pi\)
−0.748503 + 0.663131i \(0.769227\pi\)
\(384\) 0 0
\(385\) −245.000 −0.636364
\(386\) 0 0
\(387\) 149.858 129.408i 0.387231 0.334388i
\(388\) 0 0
\(389\) −35.8643 + 20.7063i −0.0921962 + 0.0532295i −0.545389 0.838183i \(-0.683618\pi\)
0.453193 + 0.891412i \(0.350285\pi\)
\(390\) 0 0
\(391\) 245.000 0.626598
\(392\) 0 0
\(393\) 437.500 + 73.9510i 1.11323 + 0.188170i
\(394\) 0 0
\(395\) −46.1113 + 26.6224i −0.116737 + 0.0673984i
\(396\) 0 0
\(397\) 16.5000 28.5788i 0.0415617 0.0719870i −0.844496 0.535561i \(-0.820100\pi\)
0.886058 + 0.463574i \(0.153433\pi\)
\(398\) 0 0
\(399\) −80.5000 + 476.244i −0.201754 + 1.19360i
\(400\) 0 0
\(401\) −322.779 186.357i −0.804935 0.464729i 0.0402588 0.999189i \(-0.487182\pi\)
−0.845194 + 0.534460i \(0.820515\pi\)
\(402\) 0 0
\(403\) −117.000 202.650i −0.290323 0.502853i
\(404\) 0 0
\(405\) 297.500 375.671i 0.734568 0.927583i
\(406\) 0 0
\(407\) 278.056i 0.683184i
\(408\) 0 0
\(409\) 288.500 + 499.697i 0.705379 + 1.22175i 0.966555 + 0.256461i \(0.0825565\pi\)
−0.261176 + 0.965291i \(0.584110\pi\)
\(410\) 0 0
\(411\) −192.300 + 232.499i −0.467882 + 0.565692i
\(412\) 0 0
\(413\) −609.694 + 352.007i −1.47626 + 0.852317i
\(414\) 0 0
\(415\) −140.000 + 242.487i −0.337349 + 0.584306i
\(416\) 0 0
\(417\) 298.044 110.877i 0.714734 0.265892i
\(418\) 0 0
\(419\) 141.986i 0.338869i −0.985541 0.169434i \(-0.945806\pi\)
0.985541 0.169434i \(-0.0541940\pi\)
\(420\) 0 0
\(421\) −246.000 −0.584323 −0.292162 0.956369i \(-0.594374\pi\)
−0.292162 + 0.956369i \(0.594374\pi\)
\(422\) 0 0
\(423\) −470.696 + 89.8911i −1.11276 + 0.212508i
\(424\) 0 0
\(425\) 51.2348 + 29.5804i 0.120552 + 0.0696009i
\(426\) 0 0
\(427\) 283.500 491.036i 0.663934 1.14997i
\(428\) 0 0
\(429\) −67.8704 + 82.0585i −0.158206 + 0.191279i
\(430\) 0 0
\(431\) −537.965 + 310.594i −1.24818 + 0.720636i −0.970746 0.240109i \(-0.922817\pi\)
−0.277433 + 0.960745i \(0.589483\pi\)
\(432\) 0 0
\(433\) −622.000 −1.43649 −0.718245 0.695790i \(-0.755054\pi\)
−0.718245 + 0.695790i \(0.755054\pi\)
\(434\) 0 0
\(435\) 140.000 828.251i 0.321839 1.90403i
\(436\) 0 0
\(437\) 824.880 476.244i 1.88760 1.08980i
\(438\) 0 0
\(439\) −124.500 + 215.640i −0.283599 + 0.491208i −0.972268 0.233867i \(-0.924862\pi\)
0.688669 + 0.725075i \(0.258195\pi\)
\(440\) 0 0
\(441\) 333.775 288.228i 0.756860 0.653577i
\(442\) 0 0
\(443\) −210.062 121.280i −0.474182 0.273769i 0.243807 0.969824i \(-0.421604\pi\)
−0.717989 + 0.696055i \(0.754937\pi\)
\(444\) 0 0
\(445\) −262.500 454.663i −0.589888 1.02172i
\(446\) 0 0
\(447\) −577.500 97.6153i −1.29195 0.218379i
\(448\) 0 0
\(449\) 473.286i 1.05409i 0.849837 + 0.527045i \(0.176700\pi\)
−0.849837 + 0.527045i \(0.823300\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 94.7812 + 78.3933i 0.209230 + 0.173054i
\(454\) 0 0
\(455\) −215.186 124.238i −0.472936 0.273050i
\(456\) 0 0
\(457\) −303.500 + 525.677i −0.664114 + 1.15028i 0.315411 + 0.948955i \(0.397858\pi\)
−0.979525 + 0.201324i \(0.935476\pi\)
\(458\) 0 0
\(459\) 3.39482 + 159.698i 0.00739612 + 0.347926i
\(460\) 0 0
\(461\) 520.615i 1.12932i −0.825325 0.564658i \(-0.809008\pi\)
0.825325 0.564658i \(-0.190992\pi\)
\(462\) 0 0
\(463\) 302.000 0.652268 0.326134 0.945324i \(-0.394254\pi\)
0.326134 + 0.945324i \(0.394254\pi\)
\(464\) 0 0
\(465\) −241.342 648.744i −0.519016 1.39515i
\(466\) 0 0
\(467\) −128.087 73.9510i −0.274276 0.158353i 0.356553 0.934275i \(-0.383952\pi\)
−0.630829 + 0.775922i \(0.717285\pi\)
\(468\) 0 0
\(469\) 108.500 + 187.928i 0.231343 + 0.400698i
\(470\) 0 0
\(471\) 386.060 + 319.309i 0.819661 + 0.677939i
\(472\) 0 0
\(473\) −112.716 + 65.0769i −0.238301 + 0.137583i
\(474\) 0 0
\(475\) 230.000 0.484211
\(476\) 0 0
\(477\) 157.500 452.580i 0.330189 0.948805i
\(478\) 0 0
\(479\) −128.087 + 73.9510i −0.267405 + 0.154386i −0.627708 0.778449i \(-0.716007\pi\)
0.360303 + 0.932835i \(0.382673\pi\)
\(480\) 0 0
\(481\) 141.000 244.219i 0.293139 0.507732i
\(482\) 0 0
\(483\) −857.500 144.944i −1.77536 0.300091i
\(484\) 0 0
\(485\) 420.125 + 242.559i 0.866237 + 0.500122i
\(486\) 0 0
\(487\) 323.500 + 560.318i 0.664271 + 1.15055i 0.979482 + 0.201530i \(0.0645913\pi\)
−0.315211 + 0.949021i \(0.602075\pi\)
\(488\) 0 0
\(489\) −131.500 + 777.964i −0.268916 + 1.59093i
\(490\) 0 0
\(491\) 141.986i 0.289177i −0.989492 0.144589i \(-0.953814\pi\)
0.989492 0.144589i \(-0.0461858\pi\)
\(492\) 0 0
\(493\) 140.000 + 242.487i 0.283976 + 0.491860i
\(494\) 0 0
\(495\) −238.411 + 205.877i −0.481638 + 0.415913i
\(496\) 0 0
\(497\) 662.601i 1.33320i
\(498\) 0 0
\(499\) 51.5000 89.2006i 0.103206 0.178759i −0.809798 0.586709i \(-0.800423\pi\)
0.913004 + 0.407951i \(0.133756\pi\)
\(500\) 0 0
\(501\) 198.024 + 532.303i 0.395258 + 1.06248i
\(502\) 0 0
\(503\) 283.972i 0.564556i 0.959333 + 0.282278i \(0.0910901\pi\)
−0.959333 + 0.282278i \(0.908910\pi\)
\(504\) 0 0
\(505\) −245.000 −0.485149
\(506\) 0 0
\(507\) 373.961 139.119i 0.737596 0.274396i
\(508\) 0 0
\(509\) 169.075 + 97.6153i 0.332170 + 0.191779i 0.656804 0.754061i \(-0.271908\pi\)
−0.324634 + 0.945840i \(0.605241\pi\)
\(510\) 0 0
\(511\) 59.5000 + 103.057i 0.116438 + 0.201677i
\(512\) 0 0
\(513\) 321.860 + 531.081i 0.627407 + 1.03525i
\(514\) 0 0
\(515\) 117.840 68.0349i 0.228815 0.132107i
\(516\) 0 0
\(517\) 315.000 0.609284
\(518\) 0 0
\(519\) 822.500 + 139.028i 1.58478 + 0.267876i
\(520\) 0 0
\(521\) −322.779 + 186.357i −0.619537 + 0.357690i −0.776689 0.629884i \(-0.783102\pi\)
0.157152 + 0.987575i \(0.449769\pi\)
\(522\) 0 0
\(523\) 11.5000 19.9186i 0.0219885 0.0380852i −0.854822 0.518922i \(-0.826334\pi\)
0.876810 + 0.480836i \(0.159667\pi\)
\(524\) 0 0
\(525\) −161.822 133.842i −0.308232 0.254938i
\(526\) 0 0
\(527\) 199.816 + 115.364i 0.379157 + 0.218906i
\(528\) 0 0
\(529\) 593.000 + 1027.11i 1.12098 + 1.94160i
\(530\) 0 0
\(531\) −297.500 + 854.874i −0.560264 + 1.60993i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −157.500 272.798i −0.294393 0.509903i
\(536\) 0 0
\(537\) −169.676 + 205.146i −0.315970 + 0.382023i
\(538\) 0 0
\(539\) −251.050 + 144.944i −0.465770 + 0.268913i
\(540\) 0 0
\(541\) 256.500 444.271i 0.474122 0.821203i −0.525439 0.850831i \(-0.676099\pi\)
0.999561 + 0.0296279i \(0.00943223\pi\)
\(542\) 0 0
\(543\) 16.8704 6.27604i 0.0310689 0.0115581i
\(544\) 0 0
\(545\) 810.503i 1.48716i
\(546\) 0 0
\(547\) 54.0000 0.0987203 0.0493601 0.998781i \(-0.484282\pi\)
0.0493601 + 0.998781i \(0.484282\pi\)
\(548\) 0 0
\(549\) −136.749 716.059i −0.249088 1.30430i
\(550\) 0 0
\(551\) 942.719 + 544.279i 1.71092 + 0.987803i
\(552\) 0 0
\(553\) −31.5000 + 54.5596i −0.0569620 + 0.0986611i
\(554\) 0 0
\(555\) 531.652 642.792i 0.957931 1.15818i
\(556\) 0 0
\(557\) −773.645 + 446.664i −1.38895 + 0.801910i −0.993197 0.116447i \(-0.962849\pi\)
−0.395752 + 0.918357i \(0.629516\pi\)
\(558\) 0 0
\(559\) −132.000 −0.236136
\(560\) 0 0
\(561\) 17.5000 103.531i 0.0311943 0.184548i
\(562\) 0 0
\(563\) 486.730 281.014i 0.864530 0.499136i −0.000996920 1.00000i \(-0.500317\pi\)
0.865527 + 0.500863i \(0.166984\pi\)
\(564\) 0 0
\(565\) 280.000 484.974i 0.495575 0.858361i
\(566\) 0 0
\(567\) 82.5968 560.952i 0.145673 0.989333i
\(568\) 0 0
\(569\) −568.706 328.342i −0.999483 0.577052i −0.0913876 0.995815i \(-0.529130\pi\)
−0.908095 + 0.418764i \(0.862464\pi\)
\(570\) 0 0
\(571\) 463.500 + 802.806i 0.811734 + 1.40596i 0.911650 + 0.410968i \(0.134809\pi\)
−0.0999158 + 0.994996i \(0.531857\pi\)
\(572\) 0 0
\(573\) 157.500 + 26.6224i 0.274869 + 0.0464614i
\(574\) 0 0
\(575\) 414.126i 0.720218i
\(576\) 0 0
\(577\) 176.500 + 305.707i 0.305893 + 0.529821i 0.977460 0.211122i \(-0.0677118\pi\)
−0.671567 + 0.740944i \(0.734379\pi\)
\(578\) 0 0
\(579\) 330.578 + 273.421i 0.570947 + 0.472229i
\(580\) 0 0
\(581\) 331.300i 0.570225i
\(582\) 0 0
\(583\) −157.500 + 272.798i −0.270154 + 0.467921i
\(584\) 0 0
\(585\) −313.797 + 59.9274i −0.536406 + 0.102440i
\(586\) 0 0
\(587\) 236.643i 0.403140i −0.979474 0.201570i \(-0.935396\pi\)
0.979474 0.201570i \(-0.0646044\pi\)
\(588\) 0 0
\(589\) 897.000 1.52292
\(590\) 0 0
\(591\) 49.5061 + 133.076i 0.0837667 + 0.225170i
\(592\) 0 0
\(593\) 660.928 + 381.587i 1.11455 + 0.643486i 0.940004 0.341163i \(-0.110821\pi\)
0.174546 + 0.984649i \(0.444154\pi\)
\(594\) 0 0
\(595\) 245.000 0.411765
\(596\) 0 0
\(597\) 390.684 + 323.134i 0.654412 + 0.541262i
\(598\) 0 0
\(599\) −537.965 + 310.594i −0.898105 + 0.518521i −0.876585 0.481247i \(-0.840184\pi\)
−0.0215201 + 0.999768i \(0.506851\pi\)
\(600\) 0 0
\(601\) −958.000 −1.59401 −0.797005 0.603973i \(-0.793584\pi\)
−0.797005 + 0.603973i \(0.793584\pi\)
\(602\) 0 0
\(603\) 263.500 + 91.6992i 0.436982 + 0.152072i
\(604\) 0 0
\(605\) −440.619 + 254.391i −0.728296 + 0.420482i
\(606\) 0 0
\(607\) 403.500 698.883i 0.664745 1.15137i −0.314610 0.949221i \(-0.601874\pi\)
0.979354 0.202150i \(-0.0647930\pi\)
\(608\) 0 0
\(609\) −346.543 931.530i −0.569036 1.52961i
\(610\) 0 0
\(611\) 276.668 + 159.734i 0.452811 + 0.261431i
\(612\) 0 0
\(613\) −259.500 449.467i −0.423328 0.733225i 0.572935 0.819601i \(-0.305805\pi\)
−0.996263 + 0.0863756i \(0.972472\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 757.258i 1.22732i −0.789569 0.613661i \(-0.789696\pi\)
0.789569 0.613661i \(-0.210304\pi\)
\(618\) 0 0
\(619\) −344.500 596.692i −0.556543 0.963960i −0.997782 0.0665707i \(-0.978794\pi\)
0.441239 0.897390i \(-0.354539\pi\)
\(620\) 0 0
\(621\) −956.236 + 579.523i −1.53983 + 0.933210i
\(622\) 0 0
\(623\) −537.965 310.594i −0.863507 0.498546i
\(624\) 0 0
\(625\) 387.500 671.170i 0.620000 1.07387i
\(626\) 0 0
\(627\) −142.330 382.593i −0.227002 0.610196i
\(628\) 0 0
\(629\) 278.056i 0.442060i
\(630\) 0 0
\(631\) −674.000 −1.06815 −0.534073 0.845438i \(-0.679339\pi\)
−0.534073 + 0.845438i \(0.679339\pi\)
\(632\) 0 0
\(633\) −466.748 + 173.637i −0.737359 + 0.274308i
\(634\) 0 0
\(635\) 399.631 + 230.727i 0.629340 + 0.363350i
\(636\) 0 0
\(637\) −294.000 −0.461538
\(638\) 0 0
\(639\) −556.793 644.781i −0.871350 1.00905i
\(640\) 0 0
\(641\) 742.904 428.916i 1.15898 0.669135i 0.207918 0.978146i \(-0.433331\pi\)
0.951059 + 0.309011i \(0.0999980\pi\)
\(642\) 0 0
\(643\) −218.000 −0.339036 −0.169518 0.985527i \(-0.554221\pi\)
−0.169518 + 0.985527i \(0.554221\pi\)
\(644\) 0 0
\(645\) −385.000 65.0769i −0.596899 0.100894i
\(646\) 0 0
\(647\) 445.742 257.349i 0.688937 0.397758i −0.114277 0.993449i \(-0.536455\pi\)
0.803214 + 0.595691i \(0.203122\pi\)
\(648\) 0 0
\(649\) 297.500 515.285i 0.458398 0.793968i
\(650\) 0 0
\(651\) −631.104 521.985i −0.969438 0.801820i
\(652\) 0 0
\(653\) 906.855 + 523.573i 1.38875 + 0.801796i 0.993175 0.116636i \(-0.0372112\pi\)
0.395577 + 0.918433i \(0.370544\pi\)
\(654\) 0 0
\(655\) −437.500 757.772i −0.667939 1.15690i
\(656\) 0 0
\(657\) 144.500 + 50.2867i 0.219939 + 0.0765398i
\(658\) 0 0
\(659\) 615.272i 0.933645i 0.884351 + 0.466823i \(0.154601\pi\)
−0.884351 + 0.466823i \(0.845399\pi\)
\(660\) 0 0
\(661\) 260.500 + 451.199i 0.394100 + 0.682601i 0.992986 0.118233i \(-0.0377231\pi\)
−0.598886 + 0.800834i \(0.704390\pi\)
\(662\) 0 0
\(663\) 67.8704 82.0585i 0.102369 0.123769i
\(664\) 0 0
\(665\) 824.880 476.244i 1.24042 0.716157i
\(666\) 0 0
\(667\) −980.000 + 1697.41i −1.46927 + 2.54484i
\(668\) 0 0
\(669\) −399.267 + 148.533i −0.596811 + 0.222022i
\(670\) 0 0
\(671\) 479.202i 0.714162i
\(672\) 0 0
\(673\) 818.000 1.21545 0.607727 0.794146i \(-0.292082\pi\)
0.607727 + 0.794146i \(0.292082\pi\)
\(674\) 0 0
\(675\) −269.939 + 5.73829i −0.399910 + 0.00850118i
\(676\) 0 0
\(677\) 660.928 + 381.587i 0.976260 + 0.563644i 0.901139 0.433530i \(-0.142732\pi\)
0.0751214 + 0.997174i \(0.476066\pi\)
\(678\) 0 0
\(679\) 574.000 0.845361
\(680\) 0 0
\(681\) −169.676 + 205.146i −0.249157 + 0.301243i
\(682\) 0 0
\(683\) −169.075 + 97.6153i −0.247547 + 0.142921i −0.618641 0.785674i \(-0.712316\pi\)
0.371093 + 0.928596i \(0.378983\pi\)
\(684\) 0 0
\(685\) 595.000 0.868613
\(686\) 0 0
\(687\) −128.500 + 760.216i −0.187045 + 1.10657i
\(688\) 0 0
\(689\) −276.668 + 159.734i −0.401550 + 0.231835i
\(690\) 0 0
\(691\) 179.500 310.903i 0.259768 0.449932i −0.706411 0.707802i \(-0.749687\pi\)
0.966180 + 0.257869i \(0.0830204\pi\)
\(692\) 0 0
\(693\) −122.500 + 352.007i −0.176768 + 0.507946i
\(694\) 0 0
\(695\) −543.088 313.552i −0.781422 0.451154i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 542.500 + 91.6992i 0.776109 + 0.131186i
\(700\) 0 0
\(701\) 804.587i 1.14777i −0.818936 0.573885i \(-0.805436\pi\)
0.818936 0.573885i \(-0.194564\pi\)
\(702\) 0 0
\(703\) 540.500 + 936.173i 0.768848 + 1.33168i
\(704\) 0 0
\(705\) 728.197 + 602.290i 1.03290 + 0.854312i
\(706\) 0 0
\(707\) −251.050 + 144.944i −0.355092 + 0.205013i
\(708\) 0 0
\(709\) −215.500 + 373.257i −0.303949 + 0.526455i −0.977027 0.213116i \(-0.931639\pi\)
0.673078 + 0.739572i \(0.264972\pi\)
\(710\) 0 0
\(711\) 15.1944 + 79.5621i 0.0213704 + 0.111902i
\(712\) 0 0
\(713\) 1615.09i 2.26520i
\(714\) 0 0
\(715\) 210.000 0.293706
\(716\) 0 0
\(717\) −297.037 798.454i −0.414277 1.11360i
\(718\) 0 0
\(719\) 199.816 + 115.364i 0.277908 + 0.160450i 0.632476 0.774580i \(-0.282039\pi\)
−0.354568 + 0.935030i \(0.615372\pi\)
\(720\) 0 0
\(721\) 80.5000 139.430i 0.111650 0.193384i
\(722\) 0 0
\(723\) −2.31174 1.91203i −0.00319742 0.00264458i
\(724\) 0 0
\(725\) −409.878 + 236.643i −0.565349 + 0.326404i
\(726\) 0 0
\(727\) 734.000 1.00963 0.504814 0.863228i \(-0.331561\pi\)
0.504814 + 0.863228i \(0.331561\pi\)
\(728\) 0 0
\(729\) −391.000 615.272i −0.536351 0.843995i
\(730\) 0 0
\(731\) 112.716 65.0769i 0.154195 0.0890245i
\(732\) 0 0
\(733\) −151.500 + 262.406i −0.206685 + 0.357989i −0.950668 0.310209i \(-0.899601\pi\)
0.743983 + 0.668198i \(0.232934\pi\)
\(734\) 0 0
\(735\) −857.500 144.944i −1.16667 0.197203i
\(736\) 0 0
\(737\) −158.828 91.6992i −0.215506 0.124422i
\(738\) 0 0
\(739\) 295.500 + 511.821i 0.399865 + 0.692586i 0.993709 0.111994i \(-0.0357238\pi\)
−0.593844 + 0.804580i \(0.702390\pi\)
\(740\) 0 0
\(741\) 69.0000 408.210i 0.0931174 0.550890i
\(742\) 0 0
\(743\) 851.915i 1.14659i −0.819349 0.573294i \(-0.805665\pi\)
0.819349 0.573294i \(-0.194335\pi\)
\(744\) 0 0
\(745\) 577.500 + 1000.26i 0.775168 + 1.34263i
\(746\) 0 0
\(747\) 278.396 + 322.390i 0.372686 + 0.431580i
\(748\) 0 0
\(749\) −322.779 186.357i −0.430947 0.248807i
\(750\) 0 0
\(751\) 499.500 865.159i 0.665113 1.15201i −0.314141 0.949376i \(-0.601717\pi\)
0.979255 0.202634i \(-0.0649501\pi\)
\(752\) 0 0
\(753\) 148.518 + 399.227i 0.197235 + 0.530182i
\(754\) 0 0
\(755\) 242.559i 0.321271i
\(756\) 0 0
\(757\) −1398.00 −1.84676 −0.923382 0.383883i \(-0.874587\pi\)
−0.923382 + 0.383883i \(0.874587\pi\)
\(758\) 0 0
\(759\) 688.876 256.272i 0.907610 0.337644i
\(760\) 0 0
\(761\) −1142.54 659.643i −1.50136 0.866811i −0.999999 0.00157261i \(-0.999499\pi\)
−0.501361 0.865238i \(-0.667167\pi\)
\(762\) 0 0
\(763\) 479.500 + 830.518i 0.628440 + 1.08849i
\(764\) 0 0
\(765\) 238.411 205.877i 0.311648 0.269120i
\(766\) 0 0
\(767\) 522.594 301.720i 0.681349 0.393377i
\(768\) 0 0
\(769\) 946.000 1.23017 0.615085 0.788461i \(-0.289122\pi\)
0.615085 + 0.788461i \(0.289122\pi\)
\(770\) 0 0
\(771\) −577.500 97.6153i −0.749027 0.126609i
\(772\) 0 0
\(773\) 455.989 263.266i 0.589896 0.340576i −0.175161 0.984540i \(-0.556044\pi\)
0.765056 + 0.643963i \(0.222711\pi\)
\(774\) 0 0
\(775\) −195.000 + 337.750i −0.251613 + 0.435806i
\(776\) 0 0
\(777\) 164.500 973.195i 0.211712 1.25250i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 280.000 + 484.974i 0.358515 + 0.620966i
\(782\) 0 0
\(783\) −1120.00 615.272i −1.43040 0.785788i
\(784\) 0 0
\(785\) 987.985i 1.25858i
\(786\) 0 0
\(787\) 223.500 + 387.113i 0.283990 + 0.491885i 0.972364 0.233471i \(-0.0750085\pi\)
−0.688374 + 0.725356i \(0.741675\pi\)
\(788\) 0 0
\(789\) 101.806 123.088i 0.129031 0.156005i
\(790\) 0 0
\(791\) 662.601i 0.837675i
\(792\) 0 0
\(793\) −243.000 + 420.888i −0.306431 + 0.530755i
\(794\) 0 0
\(795\) −885.697 + 329.492i −1.11408 + 0.414456i
\(796\) 0 0
\(797\) 1277.87i 1.60335i −0.597757 0.801677i \(-0.703941\pi\)
0.597757 0.801677i \(-0.296059\pi\)
\(798\) 0 0
\(799\) −315.000 −0.394243
\(800\) 0 0
\(801\) −784.493 + 149.818i −0.979392 + 0.187039i
\(802\) 0 0
\(803\) −87.0991 50.2867i −0.108467 0.0626235i
\(804\) 0 0
\(805\) 857.500 + 1485.23i 1.06522 + 1.84501i
\(806\) 0 0
\(807\) 79.1822 95.7350i 0.0981192 0.118631i
\(808\) 0 0
\(809\) −1142.54 + 659.643i −1.41228 + 0.815381i −0.995603 0.0936737i \(-0.970139\pi\)
−0.416678 + 0.909054i \(0.636806\pi\)
\(810\) 0 0
\(811\) 86.0000 0.106042 0.0530210 0.998593i \(-0.483115\pi\)
0.0530210 + 0.998593i \(0.483115\pi\)
\(812\) 0 0
\(813\) −43.5000 + 257.349i −0.0535055 + 0.316543i
\(814\) 0 0
\(815\) 1347.47 777.964i 1.65334 0.954558i
\(816\) 0 0
\(817\) 253.000 438.209i 0.309670 0.536363i
\(818\) 0 0
\(819\) −286.093 + 247.052i −0.349320 + 0.301651i
\(820\) 0 0
\(821\) 333.026 + 192.273i 0.405634 + 0.234193i 0.688912 0.724845i \(-0.258089\pi\)
−0.283278 + 0.959038i \(0.591422\pi\)
\(822\) 0 0
\(823\) −188.500 326.492i −0.229040 0.396709i 0.728484 0.685063i \(-0.240225\pi\)
−0.957524 + 0.288354i \(0.906892\pi\)
\(824\) 0 0
\(825\) 175.000 + 29.5804i 0.212121 + 0.0358550i
\(826\) 0 0
\(827\) 141.986i 0.171688i 0.996309 + 0.0858440i \(0.0273587\pi\)
−0.996309 + 0.0858440i \(0.972641\pi\)
\(828\) 0 0
\(829\) −75.5000 130.770i −0.0910736 0.157744i 0.816890 0.576794i \(-0.195697\pi\)
−0.907963 + 0.419050i \(0.862363\pi\)
\(830\) 0 0
\(831\) 718.950 + 594.642i 0.865163 + 0.715574i
\(832\) 0 0
\(833\) 251.050 144.944i 0.301381 0.174002i
\(834\) 0 0
\(835\) 560.000 969.948i 0.670659 1.16161i
\(836\) 0 0
\(837\) −1052.76 + 22.3793i −1.25778 + 0.0267376i
\(838\) 0 0
\(839\) 473.286i 0.564108i 0.959399 + 0.282054i \(0.0910157\pi\)
−0.959399 + 0.282054i \(0.908984\pi\)
\(840\) 0 0
\(841\) −1399.00 −1.66350
\(842\) 0 0
\(843\) −396.049 1064.61i −0.469809 1.26288i
\(844\) 0 0
\(845\) −681.422 393.419i −0.806417 0.465585i
\(846\) 0 0
\(847\) −301.000 + 521.347i −0.355372 + 0.615522i
\(848\) 0 0
\(849\) −737.444 609.938i −0.868603 0.718420i
\(850\) 0 0
\(851\) −1685.62 + 973.195i −1.98076 + 1.14359i
\(852\) 0 0
\(853\) −1462.00 −1.71395 −0.856975 0.515357i \(-0.827659\pi\)
−0.856975 + 0.515357i \(0.827659\pi\)
\(854\) 0 0
\(855\) 402.500 1156.59i 0.470760 1.35274i
\(856\) 0 0
\(857\) 169.075 97.6153i 0.197287 0.113904i −0.398103 0.917341i \(-0.630331\pi\)
0.595389 + 0.803437i \(0.296998\pi\)
\(858\) 0 0
\(859\) 491.500 851.303i 0.572177 0.991040i −0.424165 0.905585i \(-0.639432\pi\)
0.996342 0.0854547i \(-0.0272343\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −128.087 73.9510i −0.148420 0.0856906i 0.423951 0.905685i \(-0.360643\pi\)
−0.572371 + 0.819995i \(0.693976\pi\)
\(864\) 0 0
\(865\) −822.500 1424.61i −0.950867 1.64695i
\(866\) 0 0
\(867\) 127.000 751.342i 0.146482 0.866600i
\(868\) 0 0
\(869\) 53.2447i 0.0612713i
\(870\) 0 0
\(871\) −93.0000 161.081i −0.106774 0.184938i
\(872\) 0 0
\(873\) 558.562 482.340i 0.639820 0.552509i
\(874\) 0 0
\(875\) 621.188i 0.709930i
\(876\) 0 0
\(877\) 176.500 305.707i 0.201254 0.348583i −0.747679 0.664061i \(-0.768832\pi\)
0.948933 + 0.315478i \(0.102165\pi\)
\(878\) 0 0
\(879\) 148.518 + 399.227i 0.168963 + 0.454183i
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −314.000 −0.355606 −0.177803 0.984066i \(-0.556899\pi\)
−0.177803 + 0.984066i \(0.556899\pi\)
\(884\) 0 0
\(885\) 1672.98 622.374i 1.89038 0.703248i
\(886\) 0 0
\(887\) −537.965 310.594i −0.606499 0.350163i 0.165095 0.986278i \(-0.447207\pi\)
−0.771594 + 0.636115i \(0.780540\pi\)
\(888\) 0 0
\(889\) 546.000 0.614173
\(890\) 0 0
\(891\) 176.591 + 445.478i 0.198194 + 0.499975i
\(892\) 0 0
\(893\) −1060.56 + 612.314i −1.18764 + 0.685682i
\(894\) 0 0
\(895\) 525.000 0.586592
\(896\) 0 0
\(897\) 735.000 + 124.238i 0.819398 + 0.138504i
\(898\) 0 0
\(899\) −1598.52 + 922.908i −1.77811 + 1.02659i
\(900\) 0 0
\(901\) 157.500 272.798i 0.174806 0.302772i
\(902\) 0 0
\(903\) −433.008 + 161.085i −0.479521 + 0.178389i
\(904\) 0 0
\(905\) −30.7409 17.7482i −0.0339678 0.0196113i
\(906\) 0 0
\(907\) −376.500 652.117i −0.415105 0.718983i 0.580335 0.814378i \(-0.302922\pi\)
−0.995439 + 0.0953956i \(0.969588\pi\)
\(908\) 0 0
\(909\) −122.500 + 352.007i −0.134763 + 0.387246i
\(910\) 0 0
\(911\) 1703.83i 1.87029i −0.354270 0.935143i \(-0.615271\pi\)
0.354270 0.935143i \(-0.384729\pi\)
\(912\) 0 0
\(913\) −140.000 242.487i −0.153341 0.265594i
\(914\) 0 0
\(915\) −916.251 + 1107.79i −1.00137 + 1.21070i
\(916\) 0 0
\(917\) −896.608 517.657i −0.977762 0.564511i
\(918\) 0 0
\(919\) −284.500 + 492.768i −0.309576 + 0.536201i −0.978270 0.207337i \(-0.933520\pi\)
0.668694 + 0.743538i \(0.266854\pi\)
\(920\) 0 0
\(921\) 1242.79 462.335i 1.34939 0.501993i
\(922\) 0 0
\(923\) 567.944i 0.615324i
\(924\) 0 0
\(925\) −470.000 −0.508108
\(926\) 0 0
\(927\) −38.8300 203.325i −0.0418878 0.219337i
\(928\) 0 0
\(929\) 1398.71 + 807.545i 1.50561 + 0.869263i 0.999979 + 0.00651097i \(0.00207252\pi\)
0.505628 + 0.862752i \(0.331261\pi\)
\(930\) 0 0
\(931\) 563.500 976.011i 0.605263 1.04835i
\(932\) 0 0
\(933\) 463.781 560.733i 0.497086 0.601000i
\(934\) 0 0
\(935\) −179.322 + 103.531i −0.191788 + 0.110729i
\(936\) 0 0
\(937\) 722.000 0.770544 0.385272 0.922803i \(-0.374108\pi\)
0.385272 + 0.922803i \(0.374108\pi\)
\(938\) 0 0
\(939\) −128.500 + 760.216i −0.136848 + 0.809602i
\(940\) 0 0
\(941\) 455.989 263.266i 0.484579 0.279772i −0.237743 0.971328i \(-0.576408\pi\)
0.722323 + 0.691556i \(0.243074\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −956.236 + 579.523i −1.01189 + 0.613252i
\(946\) 0 0
\(947\) −128.087 73.9510i −0.135255 0.0780898i 0.430845 0.902426i \(-0.358215\pi\)
−0.566101 + 0.824336i \(0.691549\pi\)
\(948\) 0 0
\(949\) −51.0000 88.3346i −0.0537408 0.0930818i
\(950\) 0 0
\(951\) 262.500 + 44.3706i 0.276025 + 0.0466568i
\(952\) 0 0
\(953\) 378.629i 0.397302i 0.980070 + 0.198651i \(0.0636561\pi\)
−0.980070 + 0.198651i \(0.936344\pi\)
\(954\) 0 0
\(955\) −157.500 272.798i −0.164921 0.285652i
\(956\) 0 0
\(957\) 647.287 + 535.369i 0.676370 + 0.559424i
\(958\) 0 0
\(959\) 609.694 352.007i 0.635760 0.367056i
\(960\) 0 0
\(961\) −280.000 + 484.974i −0.291363 + 0.504656i
\(962\) 0 0
\(963\) −470.696 + 89.8911i −0.488781 + 0.0933448i
\(964\) 0 0
\(965\) 845.999i 0.876683i
\(966\) 0 0
\(967\) −482.000 −0.498449 −0.249224 0.968446i \(-0.580176\pi\)
−0.249224 + 0.968446i \(0.580176\pi\)
\(968\) 0 0
\(969\) 142.330 + 382.593i 0.146883 + 0.394832i
\(970\) 0 0
\(971\) 199.816 + 115.364i 0.205783 + 0.118809i 0.599350 0.800487i \(-0.295426\pi\)
−0.393567 + 0.919296i \(0.628759\pi\)
\(972\) 0 0
\(973\) −742.000 −0.762590
\(974\) 0 0
\(975\) 138.704 + 114.722i 0.142261 + 0.117664i
\(976\) 0 0
\(977\) 169.075 97.6153i 0.173055 0.0999133i −0.410971 0.911649i \(-0.634810\pi\)
0.584026 + 0.811735i \(0.301477\pi\)
\(978\) 0 0
\(979\) 525.000 0.536261
\(980\) 0 0
\(981\) 1164.50 + 405.251i 1.18705 + 0.413100i
\(982\) 0 0
\(983\) −1275.75 + 736.552i −1.29781 + 0.749290i −0.980025 0.198874i \(-0.936272\pi\)
−0.317783 + 0.948163i \(0.602938\pi\)
\(984\) 0 0
\(985\) 140.000 242.487i 0.142132 0.246180i
\(986\) 0 0
\(987\) 1102.50 + 186.357i 1.11702 + 0.188811i
\(988\) 0 0
\(989\) 789.015 + 455.538i 0.797791 + 0.460605i
\(990\) 0 0
\(991\) −180.500 312.635i −0.182139 0.315474i 0.760470 0.649374i \(-0.224969\pi\)
−0.942609 + 0.333899i \(0.891636\pi\)
\(992\) 0 0
\(993\) 60.5000 357.923i 0.0609265 0.360446i
\(994\) 0 0
\(995\) 999.817i 1.00484i
\(996\) 0 0
\(997\) 36.5000 + 63.2199i 0.0366098 + 0.0634101i 0.883750 0.467960i \(-0.155011\pi\)
−0.847140 + 0.531370i \(0.821677\pi\)
\(998\) 0 0
\(999\) −657.713 1085.25i −0.658372 1.08634i
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 336.3.bn.d.65.1 4
3.2 odd 2 inner 336.3.bn.d.65.2 4
4.3 odd 2 84.3.p.c.65.2 yes 4
7.4 even 3 inner 336.3.bn.d.305.2 4
12.11 even 2 84.3.p.c.65.1 yes 4
21.11 odd 6 inner 336.3.bn.d.305.1 4
28.3 even 6 588.3.p.d.557.2 4
28.11 odd 6 84.3.p.c.53.1 4
28.19 even 6 588.3.c.f.197.1 2
28.23 odd 6 588.3.c.e.197.2 2
28.27 even 2 588.3.p.d.569.1 4
84.11 even 6 84.3.p.c.53.2 yes 4
84.23 even 6 588.3.c.e.197.1 2
84.47 odd 6 588.3.c.f.197.2 2
84.59 odd 6 588.3.p.d.557.1 4
84.83 odd 2 588.3.p.d.569.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.3.p.c.53.1 4 28.11 odd 6
84.3.p.c.53.2 yes 4 84.11 even 6
84.3.p.c.65.1 yes 4 12.11 even 2
84.3.p.c.65.2 yes 4 4.3 odd 2
336.3.bn.d.65.1 4 1.1 even 1 trivial
336.3.bn.d.65.2 4 3.2 odd 2 inner
336.3.bn.d.305.1 4 21.11 odd 6 inner
336.3.bn.d.305.2 4 7.4 even 3 inner
588.3.c.e.197.1 2 84.23 even 6
588.3.c.e.197.2 2 28.23 odd 6
588.3.c.f.197.1 2 28.19 even 6
588.3.c.f.197.2 2 84.47 odd 6
588.3.p.d.557.1 4 84.59 odd 6
588.3.p.d.557.2 4 28.3 even 6
588.3.p.d.569.1 4 28.27 even 2
588.3.p.d.569.2 4 84.83 odd 2