Properties

Label 576.3.q.h.257.1
Level $576$
Weight $3$
Character 576.257
Analytic conductor $15.695$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,3,Mod(65,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.q (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: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 257.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 576.257
Dual form 576.3.q.h.65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +(-6.39898 - 3.69445i) q^{5} +(3.39898 + 5.88721i) q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +(-6.39898 - 3.69445i) q^{5} +(3.39898 + 5.88721i) q^{7} +9.00000 q^{9} +(-5.29796 + 3.05878i) q^{11} +(8.39898 - 14.5475i) q^{13} +(-19.1969 - 11.0834i) q^{15} -25.1701i q^{17} +17.5959 q^{19} +(10.1969 + 17.6616i) q^{21} +(12.3990 + 7.15855i) q^{23} +(14.7980 + 25.6308i) q^{25} +27.0000 q^{27} +(-16.1969 + 9.35131i) q^{29} +(23.3990 - 40.5282i) q^{31} +(-15.8939 + 9.17633i) q^{33} -50.2295i q^{35} +49.5959 q^{37} +(25.1969 - 43.6424i) q^{39} +(-34.5000 - 19.9186i) q^{41} +(-22.0959 - 38.2713i) q^{43} +(-57.5908 - 33.2501i) q^{45} +(28.8031 - 16.6295i) q^{47} +(1.39388 - 2.41427i) q^{49} -75.5103i q^{51} +10.1708i q^{53} +45.2020 q^{55} +52.7878 q^{57} +(14.2980 + 8.25493i) q^{59} +(10.6010 + 18.3615i) q^{61} +(30.5908 + 52.9848i) q^{63} +(-107.490 + 62.0593i) q^{65} +(-43.4898 + 75.3265i) q^{67} +(37.1969 + 21.4757i) q^{69} +30.2555i q^{71} -48.7878 q^{73} +(44.3939 + 76.8925i) q^{75} +(-36.0153 - 20.7934i) q^{77} +(-55.7929 - 96.6361i) q^{79} +81.0000 q^{81} +(85.0857 - 49.1243i) q^{83} +(-92.9898 + 161.063i) q^{85} +(-48.5908 + 28.0539i) q^{87} -75.5103i q^{89} +114.192 q^{91} +(70.1969 - 121.585i) q^{93} +(-112.596 - 65.0073i) q^{95} +(70.2980 + 121.760i) q^{97} +(-47.6816 + 27.5290i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} - 6 q^{5} - 6 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} - 6 q^{5} - 6 q^{7} + 36 q^{9} + 18 q^{11} + 14 q^{13} - 18 q^{15} - 8 q^{19} - 18 q^{21} + 30 q^{23} + 20 q^{25} + 108 q^{27} - 6 q^{29} + 74 q^{31} + 54 q^{33} + 120 q^{37} + 42 q^{39} - 138 q^{41} - 10 q^{43} - 54 q^{45} + 174 q^{47} - 112 q^{49} + 220 q^{55} - 24 q^{57} + 18 q^{59} + 62 q^{61} - 54 q^{63} - 234 q^{65} + 22 q^{67} + 90 q^{69} + 40 q^{73} + 60 q^{75} - 438 q^{77} - 86 q^{79} + 324 q^{81} + 66 q^{83} - 176 q^{85} - 18 q^{87} + 300 q^{91} + 222 q^{93} - 372 q^{95} + 242 q^{97} + 162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 1.00000
\(4\) 0 0
\(5\) −6.39898 3.69445i −1.27980 0.738891i −0.302985 0.952995i \(-0.597983\pi\)
−0.976811 + 0.214105i \(0.931317\pi\)
\(6\) 0 0
\(7\) 3.39898 + 5.88721i 0.485568 + 0.841029i 0.999862 0.0165847i \(-0.00527930\pi\)
−0.514294 + 0.857614i \(0.671946\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) −5.29796 + 3.05878i −0.481633 + 0.278071i −0.721097 0.692835i \(-0.756362\pi\)
0.239464 + 0.970905i \(0.423028\pi\)
\(12\) 0 0
\(13\) 8.39898 14.5475i 0.646075 1.11904i −0.337977 0.941154i \(-0.609743\pi\)
0.984052 0.177881i \(-0.0569242\pi\)
\(14\) 0 0
\(15\) −19.1969 11.0834i −1.27980 0.738891i
\(16\) 0 0
\(17\) 25.1701i 1.48059i −0.672279 0.740297i \(-0.734685\pi\)
0.672279 0.740297i \(-0.265315\pi\)
\(18\) 0 0
\(19\) 17.5959 0.926101 0.463050 0.886332i \(-0.346755\pi\)
0.463050 + 0.886332i \(0.346755\pi\)
\(20\) 0 0
\(21\) 10.1969 + 17.6616i 0.485568 + 0.841029i
\(22\) 0 0
\(23\) 12.3990 + 7.15855i 0.539086 + 0.311241i 0.744708 0.667390i \(-0.232589\pi\)
−0.205622 + 0.978631i \(0.565922\pi\)
\(24\) 0 0
\(25\) 14.7980 + 25.6308i 0.591918 + 1.02523i
\(26\) 0 0
\(27\) 27.0000 1.00000
\(28\) 0 0
\(29\) −16.1969 + 9.35131i −0.558515 + 0.322459i −0.752549 0.658536i \(-0.771176\pi\)
0.194034 + 0.980995i \(0.437843\pi\)
\(30\) 0 0
\(31\) 23.3990 40.5282i 0.754806 1.30736i −0.190665 0.981655i \(-0.561064\pi\)
0.945471 0.325707i \(-0.105602\pi\)
\(32\) 0 0
\(33\) −15.8939 + 9.17633i −0.481633 + 0.278071i
\(34\) 0 0
\(35\) 50.2295i 1.43513i
\(36\) 0 0
\(37\) 49.5959 1.34043 0.670215 0.742167i \(-0.266202\pi\)
0.670215 + 0.742167i \(0.266202\pi\)
\(38\) 0 0
\(39\) 25.1969 43.6424i 0.646075 1.11904i
\(40\) 0 0
\(41\) −34.5000 19.9186i −0.841463 0.485819i 0.0162980 0.999867i \(-0.494812\pi\)
−0.857761 + 0.514048i \(0.828145\pi\)
\(42\) 0 0
\(43\) −22.0959 38.2713i −0.513859 0.890029i −0.999871 0.0160771i \(-0.994882\pi\)
0.486012 0.873952i \(-0.338451\pi\)
\(44\) 0 0
\(45\) −57.5908 33.2501i −1.27980 0.738891i
\(46\) 0 0
\(47\) 28.8031 16.6295i 0.612831 0.353818i −0.161242 0.986915i \(-0.551550\pi\)
0.774073 + 0.633097i \(0.218216\pi\)
\(48\) 0 0
\(49\) 1.39388 2.41427i 0.0284465 0.0492707i
\(50\) 0 0
\(51\) 75.5103i 1.48059i
\(52\) 0 0
\(53\) 10.1708i 0.191902i 0.995386 + 0.0959509i \(0.0305892\pi\)
−0.995386 + 0.0959509i \(0.969411\pi\)
\(54\) 0 0
\(55\) 45.2020 0.821855
\(56\) 0 0
\(57\) 52.7878 0.926101
\(58\) 0 0
\(59\) 14.2980 + 8.25493i 0.242338 + 0.139914i 0.616251 0.787550i \(-0.288651\pi\)
−0.373913 + 0.927464i \(0.621984\pi\)
\(60\) 0 0
\(61\) 10.6010 + 18.3615i 0.173787 + 0.301008i 0.939741 0.341887i \(-0.111066\pi\)
−0.765954 + 0.642896i \(0.777733\pi\)
\(62\) 0 0
\(63\) 30.5908 + 52.9848i 0.485568 + 0.841029i
\(64\) 0 0
\(65\) −107.490 + 62.0593i −1.65369 + 0.954758i
\(66\) 0 0
\(67\) −43.4898 + 75.3265i −0.649101 + 1.12428i 0.334236 + 0.942489i \(0.391522\pi\)
−0.983338 + 0.181787i \(0.941812\pi\)
\(68\) 0 0
\(69\) 37.1969 + 21.4757i 0.539086 + 0.311241i
\(70\) 0 0
\(71\) 30.2555i 0.426134i 0.977038 + 0.213067i \(0.0683453\pi\)
−0.977038 + 0.213067i \(0.931655\pi\)
\(72\) 0 0
\(73\) −48.7878 −0.668325 −0.334163 0.942515i \(-0.608454\pi\)
−0.334163 + 0.942515i \(0.608454\pi\)
\(74\) 0 0
\(75\) 44.3939 + 76.8925i 0.591918 + 1.02523i
\(76\) 0 0
\(77\) −36.0153 20.7934i −0.467731 0.270045i
\(78\) 0 0
\(79\) −55.7929 96.6361i −0.706239 1.22324i −0.966243 0.257634i \(-0.917057\pi\)
0.260004 0.965608i \(-0.416276\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 85.0857 49.1243i 1.02513 0.591859i 0.109544 0.993982i \(-0.465061\pi\)
0.915585 + 0.402123i \(0.131728\pi\)
\(84\) 0 0
\(85\) −92.9898 + 161.063i −1.09400 + 1.89486i
\(86\) 0 0
\(87\) −48.5908 + 28.0539i −0.558515 + 0.322459i
\(88\) 0 0
\(89\) 75.5103i 0.848431i −0.905561 0.424215i \(-0.860550\pi\)
0.905561 0.424215i \(-0.139450\pi\)
\(90\) 0 0
\(91\) 114.192 1.25486
\(92\) 0 0
\(93\) 70.1969 121.585i 0.754806 1.30736i
\(94\) 0 0
\(95\) −112.596 65.0073i −1.18522 0.684287i
\(96\) 0 0
\(97\) 70.2980 + 121.760i 0.724721 + 1.25525i 0.959089 + 0.283106i \(0.0913648\pi\)
−0.234367 + 0.972148i \(0.575302\pi\)
\(98\) 0 0
\(99\) −47.6816 + 27.5290i −0.481633 + 0.278071i
\(100\) 0 0
\(101\) −28.1969 + 16.2795i −0.279178 + 0.161183i −0.633051 0.774110i \(-0.718198\pi\)
0.353873 + 0.935293i \(0.384864\pi\)
\(102\) 0 0
\(103\) −67.7929 + 117.421i −0.658183 + 1.14001i 0.322903 + 0.946432i \(0.395341\pi\)
−0.981086 + 0.193574i \(0.937992\pi\)
\(104\) 0 0
\(105\) 150.688i 1.43513i
\(106\) 0 0
\(107\) 35.3409i 0.330289i 0.986269 + 0.165144i \(0.0528090\pi\)
−0.986269 + 0.165144i \(0.947191\pi\)
\(108\) 0 0
\(109\) −53.5959 −0.491706 −0.245853 0.969307i \(-0.579068\pi\)
−0.245853 + 0.969307i \(0.579068\pi\)
\(110\) 0 0
\(111\) 148.788 1.34043
\(112\) 0 0
\(113\) 143.076 + 82.6047i 1.26615 + 0.731015i 0.974258 0.225435i \(-0.0723803\pi\)
0.291897 + 0.956450i \(0.405714\pi\)
\(114\) 0 0
\(115\) −52.8939 91.6149i −0.459947 0.796651i
\(116\) 0 0
\(117\) 75.5908 130.927i 0.646075 1.11904i
\(118\) 0 0
\(119\) 148.182 85.5527i 1.24522 0.718930i
\(120\) 0 0
\(121\) −41.7878 + 72.3785i −0.345353 + 0.598170i
\(122\) 0 0
\(123\) −103.500 59.7558i −0.841463 0.485819i
\(124\) 0 0
\(125\) 33.9588i 0.271670i
\(126\) 0 0
\(127\) 11.9796 0.0943275 0.0471637 0.998887i \(-0.484982\pi\)
0.0471637 + 0.998887i \(0.484982\pi\)
\(128\) 0 0
\(129\) −66.2878 114.814i −0.513859 0.890029i
\(130\) 0 0
\(131\) 13.3082 + 7.68347i 0.101589 + 0.0586525i 0.549934 0.835208i \(-0.314653\pi\)
−0.448345 + 0.893861i \(0.647986\pi\)
\(132\) 0 0
\(133\) 59.8082 + 103.591i 0.449685 + 0.778878i
\(134\) 0 0
\(135\) −172.772 99.7502i −1.27980 0.738891i
\(136\) 0 0
\(137\) −47.7122 + 27.5467i −0.348265 + 0.201071i −0.663921 0.747803i \(-0.731109\pi\)
0.315656 + 0.948874i \(0.397775\pi\)
\(138\) 0 0
\(139\) 50.4898 87.4509i 0.363236 0.629143i −0.625255 0.780420i \(-0.715005\pi\)
0.988491 + 0.151277i \(0.0483386\pi\)
\(140\) 0 0
\(141\) 86.4092 49.8884i 0.612831 0.353818i
\(142\) 0 0
\(143\) 102.762i 0.718619i
\(144\) 0 0
\(145\) 138.192 0.953047
\(146\) 0 0
\(147\) 4.18163 7.24280i 0.0284465 0.0492707i
\(148\) 0 0
\(149\) −187.389 108.189i −1.25764 0.726100i −0.285027 0.958519i \(-0.592003\pi\)
−0.972616 + 0.232419i \(0.925336\pi\)
\(150\) 0 0
\(151\) 76.7929 + 133.009i 0.508562 + 0.880855i 0.999951 + 0.00991488i \(0.00315606\pi\)
−0.491389 + 0.870940i \(0.663511\pi\)
\(152\) 0 0
\(153\) 226.531i 1.48059i
\(154\) 0 0
\(155\) −299.459 + 172.893i −1.93199 + 1.11544i
\(156\) 0 0
\(157\) 40.9847 70.9876i 0.261049 0.452150i −0.705472 0.708738i \(-0.749265\pi\)
0.966521 + 0.256588i \(0.0825983\pi\)
\(158\) 0 0
\(159\) 30.5124i 0.191902i
\(160\) 0 0
\(161\) 97.3271i 0.604516i
\(162\) 0 0
\(163\) −55.2122 −0.338725 −0.169363 0.985554i \(-0.554171\pi\)
−0.169363 + 0.985554i \(0.554171\pi\)
\(164\) 0 0
\(165\) 135.606 0.821855
\(166\) 0 0
\(167\) −133.803 77.2512i −0.801216 0.462582i 0.0426802 0.999089i \(-0.486410\pi\)
−0.843896 + 0.536507i \(0.819744\pi\)
\(168\) 0 0
\(169\) −56.5857 98.0093i −0.334827 0.579937i
\(170\) 0 0
\(171\) 158.363 0.926101
\(172\) 0 0
\(173\) −22.8031 + 13.1654i −0.131810 + 0.0761003i −0.564455 0.825464i \(-0.690914\pi\)
0.432645 + 0.901564i \(0.357580\pi\)
\(174\) 0 0
\(175\) −100.596 + 174.237i −0.574834 + 0.995641i
\(176\) 0 0
\(177\) 42.8939 + 24.7648i 0.242338 + 0.139914i
\(178\) 0 0
\(179\) 266.700i 1.48995i 0.667094 + 0.744973i \(0.267538\pi\)
−0.667094 + 0.744973i \(0.732462\pi\)
\(180\) 0 0
\(181\) 58.4041 0.322674 0.161337 0.986899i \(-0.448419\pi\)
0.161337 + 0.986899i \(0.448419\pi\)
\(182\) 0 0
\(183\) 31.8031 + 55.0845i 0.173787 + 0.301008i
\(184\) 0 0
\(185\) −317.363 183.230i −1.71548 0.990431i
\(186\) 0 0
\(187\) 76.9898 + 133.350i 0.411710 + 0.713103i
\(188\) 0 0
\(189\) 91.7724 + 158.955i 0.485568 + 0.841029i
\(190\) 0 0
\(191\) −99.5602 + 57.4811i −0.521258 + 0.300948i −0.737449 0.675403i \(-0.763970\pi\)
0.216191 + 0.976351i \(0.430636\pi\)
\(192\) 0 0
\(193\) −108.490 + 187.910i −0.562123 + 0.973626i 0.435188 + 0.900340i \(0.356682\pi\)
−0.997311 + 0.0732863i \(0.976651\pi\)
\(194\) 0 0
\(195\) −322.469 + 186.178i −1.65369 + 0.954758i
\(196\) 0 0
\(197\) 171.105i 0.868555i 0.900779 + 0.434278i \(0.142996\pi\)
−0.900779 + 0.434278i \(0.857004\pi\)
\(198\) 0 0
\(199\) −62.0000 −0.311558 −0.155779 0.987792i \(-0.549789\pi\)
−0.155779 + 0.987792i \(0.549789\pi\)
\(200\) 0 0
\(201\) −130.469 + 225.980i −0.649101 + 1.12428i
\(202\) 0 0
\(203\) −110.106 63.5698i −0.542395 0.313152i
\(204\) 0 0
\(205\) 147.177 + 254.917i 0.717934 + 1.24350i
\(206\) 0 0
\(207\) 111.591 + 64.4270i 0.539086 + 0.311241i
\(208\) 0 0
\(209\) −93.2225 + 53.8220i −0.446040 + 0.257522i
\(210\) 0 0
\(211\) −64.7020 + 112.067i −0.306645 + 0.531124i −0.977626 0.210350i \(-0.932540\pi\)
0.670981 + 0.741474i \(0.265873\pi\)
\(212\) 0 0
\(213\) 90.7665i 0.426134i
\(214\) 0 0
\(215\) 326.529i 1.51874i
\(216\) 0 0
\(217\) 318.131 1.46604
\(218\) 0 0
\(219\) −146.363 −0.668325
\(220\) 0 0
\(221\) −366.161 211.403i −1.65684 0.956576i
\(222\) 0 0
\(223\) −49.1867 85.1939i −0.220568 0.382036i 0.734412 0.678704i \(-0.237458\pi\)
−0.954981 + 0.296668i \(0.904125\pi\)
\(224\) 0 0
\(225\) 133.182 + 230.677i 0.591918 + 1.02523i
\(226\) 0 0
\(227\) 383.651 221.501i 1.69009 0.975775i 0.735659 0.677352i \(-0.236873\pi\)
0.954434 0.298423i \(-0.0964607\pi\)
\(228\) 0 0
\(229\) −56.0051 + 97.0037i −0.244564 + 0.423597i −0.962009 0.273018i \(-0.911978\pi\)
0.717445 + 0.696615i \(0.245311\pi\)
\(230\) 0 0
\(231\) −108.046 62.3803i −0.467731 0.270045i
\(232\) 0 0
\(233\) 80.8526i 0.347007i −0.984833 0.173503i \(-0.944491\pi\)
0.984833 0.173503i \(-0.0555088\pi\)
\(234\) 0 0
\(235\) −245.747 −1.04573
\(236\) 0 0
\(237\) −167.379 289.908i −0.706239 1.22324i
\(238\) 0 0
\(239\) 37.3888 + 21.5864i 0.156438 + 0.0903197i 0.576176 0.817326i \(-0.304544\pi\)
−0.419737 + 0.907646i \(0.637878\pi\)
\(240\) 0 0
\(241\) 140.904 + 244.053i 0.584664 + 1.01267i 0.994917 + 0.100696i \(0.0321071\pi\)
−0.410253 + 0.911972i \(0.634560\pi\)
\(242\) 0 0
\(243\) 243.000 1.00000
\(244\) 0 0
\(245\) −17.8388 + 10.2992i −0.0728113 + 0.0420376i
\(246\) 0 0
\(247\) 147.788 255.976i 0.598331 1.03634i
\(248\) 0 0
\(249\) 255.257 147.373i 1.02513 0.591859i
\(250\) 0 0
\(251\) 15.5131i 0.0618050i 0.999522 + 0.0309025i \(0.00983814\pi\)
−0.999522 + 0.0309025i \(0.990162\pi\)
\(252\) 0 0
\(253\) −87.5857 −0.346189
\(254\) 0 0
\(255\) −278.969 + 483.189i −1.09400 + 1.89486i
\(256\) 0 0
\(257\) −312.035 180.153i −1.21414 0.700986i −0.250484 0.968121i \(-0.580590\pi\)
−0.963659 + 0.267135i \(0.913923\pi\)
\(258\) 0 0
\(259\) 168.576 + 291.981i 0.650871 + 1.12734i
\(260\) 0 0
\(261\) −145.772 + 84.1618i −0.558515 + 0.322459i
\(262\) 0 0
\(263\) −42.4296 + 24.4967i −0.161329 + 0.0931435i −0.578491 0.815689i \(-0.696358\pi\)
0.417162 + 0.908832i \(0.363025\pi\)
\(264\) 0 0
\(265\) 37.5755 65.0827i 0.141794 0.245595i
\(266\) 0 0
\(267\) 226.531i 0.848431i
\(268\) 0 0
\(269\) 281.700i 1.04721i −0.851961 0.523606i \(-0.824587\pi\)
0.851961 0.523606i \(-0.175413\pi\)
\(270\) 0 0
\(271\) −89.5959 −0.330612 −0.165306 0.986242i \(-0.552861\pi\)
−0.165306 + 0.986242i \(0.552861\pi\)
\(272\) 0 0
\(273\) 342.576 1.25486
\(274\) 0 0
\(275\) −156.798 90.5273i −0.570174 0.329190i
\(276\) 0 0
\(277\) 42.1969 + 73.0872i 0.152336 + 0.263853i 0.932086 0.362238i \(-0.117987\pi\)
−0.779750 + 0.626091i \(0.784654\pi\)
\(278\) 0 0
\(279\) 210.591 364.754i 0.754806 1.30736i
\(280\) 0 0
\(281\) −23.2673 + 13.4334i −0.0828019 + 0.0478057i −0.540829 0.841132i \(-0.681890\pi\)
0.458027 + 0.888938i \(0.348556\pi\)
\(282\) 0 0
\(283\) 90.4898 156.733i 0.319752 0.553827i −0.660684 0.750664i \(-0.729734\pi\)
0.980436 + 0.196837i \(0.0630671\pi\)
\(284\) 0 0
\(285\) −337.788 195.022i −1.18522 0.684287i
\(286\) 0 0
\(287\) 270.811i 0.943594i
\(288\) 0 0
\(289\) −344.535 −1.19216
\(290\) 0 0
\(291\) 210.894 + 365.279i 0.724721 + 1.25525i
\(292\) 0 0
\(293\) 407.985 + 235.550i 1.39244 + 0.803925i 0.993585 0.113089i \(-0.0360746\pi\)
0.398854 + 0.917014i \(0.369408\pi\)
\(294\) 0 0
\(295\) −60.9949 105.646i −0.206762 0.358123i
\(296\) 0 0
\(297\) −143.045 + 82.5870i −0.481633 + 0.278071i
\(298\) 0 0
\(299\) 208.278 120.249i 0.696580 0.402171i
\(300\) 0 0
\(301\) 150.207 260.166i 0.499027 0.864340i
\(302\) 0 0
\(303\) −84.5908 + 48.8385i −0.279178 + 0.161183i
\(304\) 0 0
\(305\) 156.660i 0.513639i
\(306\) 0 0
\(307\) −464.747 −1.51383 −0.756917 0.653511i \(-0.773295\pi\)
−0.756917 + 0.653511i \(0.773295\pi\)
\(308\) 0 0
\(309\) −203.379 + 352.262i −0.658183 + 1.14001i
\(310\) 0 0
\(311\) −218.348 126.063i −0.702083 0.405348i 0.106039 0.994362i \(-0.466183\pi\)
−0.808123 + 0.589014i \(0.799516\pi\)
\(312\) 0 0
\(313\) −98.1061 169.925i −0.313438 0.542891i 0.665666 0.746250i \(-0.268147\pi\)
−0.979104 + 0.203359i \(0.934814\pi\)
\(314\) 0 0
\(315\) 452.065i 1.43513i
\(316\) 0 0
\(317\) −98.9847 + 57.1488i −0.312255 + 0.180280i −0.647935 0.761696i \(-0.724367\pi\)
0.335680 + 0.941976i \(0.391034\pi\)
\(318\) 0 0
\(319\) 57.2071 99.0857i 0.179333 0.310613i
\(320\) 0 0
\(321\) 106.023i 0.330289i
\(322\) 0 0
\(323\) 442.891i 1.37118i
\(324\) 0 0
\(325\) 497.151 1.52970
\(326\) 0 0
\(327\) −160.788 −0.491706
\(328\) 0 0
\(329\) 195.802 + 113.046i 0.595143 + 0.343606i
\(330\) 0 0
\(331\) 27.2980 + 47.2815i 0.0824712 + 0.142844i 0.904311 0.426875i \(-0.140385\pi\)
−0.821840 + 0.569719i \(0.807052\pi\)
\(332\) 0 0
\(333\) 446.363 1.34043
\(334\) 0 0
\(335\) 556.581 321.342i 1.66143 0.959230i
\(336\) 0 0
\(337\) 118.884 205.913i 0.352771 0.611016i −0.633963 0.773363i \(-0.718573\pi\)
0.986734 + 0.162347i \(0.0519063\pi\)
\(338\) 0 0
\(339\) 429.227 + 247.814i 1.26615 + 0.731015i
\(340\) 0 0
\(341\) 286.289i 0.839558i
\(342\) 0 0
\(343\) 352.051 1.02639
\(344\) 0 0
\(345\) −158.682 274.845i −0.459947 0.796651i
\(346\) 0 0
\(347\) −108.349 62.5553i −0.312245 0.180275i 0.335686 0.941974i \(-0.391032\pi\)
−0.647931 + 0.761699i \(0.724365\pi\)
\(348\) 0 0
\(349\) −269.985 467.627i −0.773595 1.33991i −0.935581 0.353113i \(-0.885123\pi\)
0.161986 0.986793i \(-0.448210\pi\)
\(350\) 0 0
\(351\) 226.772 392.781i 0.646075 1.11904i
\(352\) 0 0
\(353\) 254.490 146.930i 0.720934 0.416232i −0.0941622 0.995557i \(-0.530017\pi\)
0.815096 + 0.579325i \(0.196684\pi\)
\(354\) 0 0
\(355\) 111.778 193.604i 0.314866 0.545364i
\(356\) 0 0
\(357\) 444.545 256.658i 1.24522 0.718930i
\(358\) 0 0
\(359\) 422.550i 1.17702i 0.808490 + 0.588509i \(0.200285\pi\)
−0.808490 + 0.588509i \(0.799715\pi\)
\(360\) 0 0
\(361\) −51.3837 −0.142337
\(362\) 0 0
\(363\) −125.363 + 217.136i −0.345353 + 0.598170i
\(364\) 0 0
\(365\) 312.192 + 180.244i 0.855320 + 0.493819i
\(366\) 0 0
\(367\) 131.358 + 227.519i 0.357924 + 0.619943i 0.987614 0.156904i \(-0.0501513\pi\)
−0.629690 + 0.776847i \(0.716818\pi\)
\(368\) 0 0
\(369\) −310.500 179.267i −0.841463 0.485819i
\(370\) 0 0
\(371\) −59.8775 + 34.5703i −0.161395 + 0.0931814i
\(372\) 0 0
\(373\) 60.9847 105.629i 0.163498 0.283187i −0.772623 0.634865i \(-0.781056\pi\)
0.936121 + 0.351679i \(0.114389\pi\)
\(374\) 0 0
\(375\) 101.876i 0.271670i
\(376\) 0 0
\(377\) 314.166i 0.833331i
\(378\) 0 0
\(379\) 641.151 1.69169 0.845846 0.533428i \(-0.179096\pi\)
0.845846 + 0.533428i \(0.179096\pi\)
\(380\) 0 0
\(381\) 35.9388 0.0943275
\(382\) 0 0
\(383\) 640.681 + 369.897i 1.67280 + 0.965789i 0.966063 + 0.258308i \(0.0831650\pi\)
0.706733 + 0.707481i \(0.250168\pi\)
\(384\) 0 0
\(385\) 153.641 + 266.114i 0.399067 + 0.691204i
\(386\) 0 0
\(387\) −198.863 344.441i −0.513859 0.890029i
\(388\) 0 0
\(389\) −75.7520 + 43.7355i −0.194735 + 0.112430i −0.594197 0.804319i \(-0.702530\pi\)
0.399462 + 0.916750i \(0.369197\pi\)
\(390\) 0 0
\(391\) 180.182 312.084i 0.460823 0.798168i
\(392\) 0 0
\(393\) 39.9245 + 23.0504i 0.101589 + 0.0586525i
\(394\) 0 0
\(395\) 824.496i 2.08733i
\(396\) 0 0
\(397\) −483.090 −1.21685 −0.608425 0.793611i \(-0.708199\pi\)
−0.608425 + 0.793611i \(0.708199\pi\)
\(398\) 0 0
\(399\) 179.424 + 310.772i 0.449685 + 0.778878i
\(400\) 0 0
\(401\) 317.682 + 183.414i 0.792224 + 0.457390i 0.840745 0.541432i \(-0.182118\pi\)
−0.0485212 + 0.998822i \(0.515451\pi\)
\(402\) 0 0
\(403\) −393.055 680.791i −0.975323 1.68931i
\(404\) 0 0
\(405\) −518.317 299.251i −1.27980 0.738891i
\(406\) 0 0
\(407\) −262.757 + 151.703i −0.645595 + 0.372734i
\(408\) 0 0
\(409\) −267.641 + 463.567i −0.654379 + 1.13342i 0.327671 + 0.944792i \(0.393736\pi\)
−0.982049 + 0.188625i \(0.939597\pi\)
\(410\) 0 0
\(411\) −143.137 + 82.6400i −0.348265 + 0.201071i
\(412\) 0 0
\(413\) 112.233i 0.271751i
\(414\) 0 0
\(415\) −725.949 −1.74927
\(416\) 0 0
\(417\) 151.469 262.353i 0.363236 0.629143i
\(418\) 0 0
\(419\) −605.620 349.655i −1.44539 0.834499i −0.447193 0.894438i \(-0.647576\pi\)
−0.998202 + 0.0599386i \(0.980910\pi\)
\(420\) 0 0
\(421\) −180.772 313.107i −0.429388 0.743722i 0.567431 0.823421i \(-0.307937\pi\)
−0.996819 + 0.0796989i \(0.974604\pi\)
\(422\) 0 0
\(423\) 259.228 149.665i 0.612831 0.353818i
\(424\) 0 0
\(425\) 645.131 372.466i 1.51795 0.876391i
\(426\) 0 0
\(427\) −72.0653 + 124.821i −0.168771 + 0.292320i
\(428\) 0 0
\(429\) 308.287i 0.718619i
\(430\) 0 0
\(431\) 463.747i 1.07598i −0.842952 0.537989i \(-0.819184\pi\)
0.842952 0.537989i \(-0.180816\pi\)
\(432\) 0 0
\(433\) −689.514 −1.59241 −0.796206 0.605026i \(-0.793163\pi\)
−0.796206 + 0.605026i \(0.793163\pi\)
\(434\) 0 0
\(435\) 414.576 0.953047
\(436\) 0 0
\(437\) 218.171 + 125.961i 0.499248 + 0.288241i
\(438\) 0 0
\(439\) 310.772 + 538.274i 0.707910 + 1.22614i 0.965631 + 0.259917i \(0.0836951\pi\)
−0.257721 + 0.966219i \(0.582972\pi\)
\(440\) 0 0
\(441\) 12.5449 21.7284i 0.0284465 0.0492707i
\(442\) 0 0
\(443\) 698.843 403.477i 1.57752 0.910784i 0.582320 0.812960i \(-0.302145\pi\)
0.995204 0.0978236i \(-0.0311881\pi\)
\(444\) 0 0
\(445\) −278.969 + 483.189i −0.626897 + 1.08582i
\(446\) 0 0
\(447\) −562.166 324.567i −1.25764 0.726100i
\(448\) 0 0
\(449\) 317.554i 0.707248i 0.935388 + 0.353624i \(0.115051\pi\)
−0.935388 + 0.353624i \(0.884949\pi\)
\(450\) 0 0
\(451\) 243.706 0.540368
\(452\) 0 0
\(453\) 230.379 + 399.027i 0.508562 + 0.880855i
\(454\) 0 0
\(455\) −730.711 421.876i −1.60596 0.927201i
\(456\) 0 0
\(457\) −285.843 495.094i −0.625477 1.08336i −0.988448 0.151557i \(-0.951571\pi\)
0.362972 0.931800i \(-0.381762\pi\)
\(458\) 0 0
\(459\) 679.593i 1.48059i
\(460\) 0 0
\(461\) 478.550 276.291i 1.03807 0.599330i 0.118784 0.992920i \(-0.462101\pi\)
0.919286 + 0.393591i \(0.128767\pi\)
\(462\) 0 0
\(463\) 60.1663 104.211i 0.129949 0.225078i −0.793708 0.608299i \(-0.791852\pi\)
0.923657 + 0.383221i \(0.125185\pi\)
\(464\) 0 0
\(465\) −898.378 + 518.679i −1.93199 + 1.11544i
\(466\) 0 0
\(467\) 880.440i 1.88531i 0.333767 + 0.942656i \(0.391680\pi\)
−0.333767 + 0.942656i \(0.608320\pi\)
\(468\) 0 0
\(469\) −591.284 −1.26073
\(470\) 0 0
\(471\) 122.954 212.963i 0.261049 0.452150i
\(472\) 0 0
\(473\) 234.127 + 135.173i 0.494982 + 0.285778i
\(474\) 0 0
\(475\) 260.384 + 450.998i 0.548176 + 0.949469i
\(476\) 0 0
\(477\) 91.5371i 0.191902i
\(478\) 0 0
\(479\) 593.793 342.826i 1.23965 0.715713i 0.270628 0.962684i \(-0.412769\pi\)
0.969023 + 0.246971i \(0.0794353\pi\)
\(480\) 0 0
\(481\) 416.555 721.495i 0.866019 1.49999i
\(482\) 0 0
\(483\) 291.981i 0.604516i
\(484\) 0 0
\(485\) 1038.85i 2.14196i
\(486\) 0 0
\(487\) 391.131 0.803143 0.401571 0.915828i \(-0.368464\pi\)
0.401571 + 0.915828i \(0.368464\pi\)
\(488\) 0 0
\(489\) −165.637 −0.338725
\(490\) 0 0
\(491\) −166.469 96.1111i −0.339042 0.195746i 0.320807 0.947145i \(-0.396046\pi\)
−0.659848 + 0.751399i \(0.729379\pi\)
\(492\) 0 0
\(493\) 235.373 + 407.679i 0.477431 + 0.826935i
\(494\) 0 0
\(495\) 406.818 0.821855
\(496\) 0 0
\(497\) −178.120 + 102.838i −0.358391 + 0.206917i
\(498\) 0 0
\(499\) 304.692 527.742i 0.610605 1.05760i −0.380534 0.924767i \(-0.624260\pi\)
0.991139 0.132832i \(-0.0424070\pi\)
\(500\) 0 0
\(501\) −401.409 231.754i −0.801216 0.462582i
\(502\) 0 0
\(503\) 232.130i 0.461491i 0.973014 + 0.230746i \(0.0741165\pi\)
−0.973014 + 0.230746i \(0.925883\pi\)
\(504\) 0 0
\(505\) 240.576 0.476387
\(506\) 0 0
\(507\) −169.757 294.028i −0.334827 0.579937i
\(508\) 0 0
\(509\) 223.136 + 128.827i 0.438381 + 0.253099i 0.702910 0.711278i \(-0.251883\pi\)
−0.264530 + 0.964377i \(0.585217\pi\)
\(510\) 0 0
\(511\) −165.829 287.224i −0.324518 0.562081i
\(512\) 0 0
\(513\) 475.090 0.926101
\(514\) 0 0
\(515\) 867.610 500.915i 1.68468 0.972650i
\(516\) 0 0
\(517\) −101.732 + 176.204i −0.196773 + 0.340821i
\(518\) 0 0
\(519\) −68.4092 + 39.4961i −0.131810 + 0.0761003i
\(520\) 0 0
\(521\) 484.088i 0.929152i 0.885533 + 0.464576i \(0.153793\pi\)
−0.885533 + 0.464576i \(0.846207\pi\)
\(522\) 0 0
\(523\) 644.384 1.23209 0.616046 0.787711i \(-0.288734\pi\)
0.616046 + 0.787711i \(0.288734\pi\)
\(524\) 0 0
\(525\) −301.788 + 522.712i −0.574834 + 0.995641i
\(526\) 0 0
\(527\) −1020.10 588.955i −1.93567 1.11756i
\(528\) 0 0
\(529\) −162.010 280.610i −0.306257 0.530454i
\(530\) 0 0
\(531\) 128.682 + 74.2944i 0.242338 + 0.139914i
\(532\) 0 0
\(533\) −579.530 + 334.592i −1.08730 + 0.627752i
\(534\) 0 0
\(535\) 130.565 226.146i 0.244047 0.422702i
\(536\) 0 0
\(537\) 800.101i 1.48995i
\(538\) 0 0
\(539\) 17.0542i 0.0316405i
\(540\) 0 0
\(541\) 332.302 0.614237 0.307118 0.951671i \(-0.400635\pi\)
0.307118 + 0.951671i \(0.400635\pi\)
\(542\) 0 0
\(543\) 175.212 0.322674
\(544\) 0 0
\(545\) 342.959 + 198.008i 0.629283 + 0.363317i
\(546\) 0 0
\(547\) −157.329 272.501i −0.287621 0.498174i 0.685621 0.727959i \(-0.259531\pi\)
−0.973241 + 0.229785i \(0.926198\pi\)
\(548\) 0 0
\(549\) 95.4092 + 165.254i 0.173787 + 0.301008i
\(550\) 0 0
\(551\) −285.000 + 164.545i −0.517241 + 0.298629i
\(552\) 0 0
\(553\) 379.278 656.928i 0.685855 1.18793i
\(554\) 0 0
\(555\) −952.090 549.689i −1.71548 0.990431i
\(556\) 0 0
\(557\) 664.080i 1.19224i 0.802894 + 0.596122i \(0.203293\pi\)
−0.802894 + 0.596122i \(0.796707\pi\)
\(558\) 0 0
\(559\) −742.333 −1.32797
\(560\) 0 0
\(561\) 230.969 + 400.051i 0.411710 + 0.713103i
\(562\) 0 0
\(563\) 211.024 + 121.835i 0.374821 + 0.216403i 0.675563 0.737302i \(-0.263901\pi\)
−0.300741 + 0.953706i \(0.597234\pi\)
\(564\) 0 0
\(565\) −610.358 1057.17i −1.08028 1.87110i
\(566\) 0 0
\(567\) 275.317 + 476.864i 0.485568 + 0.841029i
\(568\) 0 0
\(569\) −502.155 + 289.919i −0.882522 + 0.509524i −0.871489 0.490415i \(-0.836845\pi\)
−0.0110330 + 0.999939i \(0.503512\pi\)
\(570\) 0 0
\(571\) −356.843 + 618.070i −0.624944 + 1.08243i 0.363608 + 0.931552i \(0.381545\pi\)
−0.988552 + 0.150882i \(0.951789\pi\)
\(572\) 0 0
\(573\) −298.681 + 172.443i −0.521258 + 0.300948i
\(574\) 0 0
\(575\) 423.728i 0.736918i
\(576\) 0 0
\(577\) 829.433 1.43749 0.718746 0.695273i \(-0.244717\pi\)
0.718746 + 0.695273i \(0.244717\pi\)
\(578\) 0 0
\(579\) −325.469 + 563.730i −0.562123 + 0.973626i
\(580\) 0 0
\(581\) 578.409 + 333.945i 0.995541 + 0.574776i
\(582\) 0 0
\(583\) −31.1102 53.8844i −0.0533623 0.0924261i
\(584\) 0 0
\(585\) −967.408 + 558.533i −1.65369 + 0.954758i
\(586\) 0 0
\(587\) −777.480 + 448.878i −1.32450 + 0.764699i −0.984442 0.175707i \(-0.943779\pi\)
−0.340054 + 0.940406i \(0.610445\pi\)
\(588\) 0 0
\(589\) 411.727 713.131i 0.699026 1.21075i
\(590\) 0 0
\(591\) 513.316i 0.868555i
\(592\) 0 0
\(593\) 378.065i 0.637547i −0.947831 0.318774i \(-0.896729\pi\)
0.947831 0.318774i \(-0.103271\pi\)
\(594\) 0 0
\(595\) −1264.28 −2.12484
\(596\) 0 0
\(597\) −186.000 −0.311558
\(598\) 0 0
\(599\) 822.438 + 474.835i 1.37302 + 0.792712i 0.991307 0.131569i \(-0.0420016\pi\)
0.381711 + 0.924282i \(0.375335\pi\)
\(600\) 0 0
\(601\) −252.308 437.011i −0.419814 0.727139i 0.576107 0.817375i \(-0.304571\pi\)
−0.995920 + 0.0902356i \(0.971238\pi\)
\(602\) 0 0
\(603\) −391.408 + 677.939i −0.649101 + 1.12428i
\(604\) 0 0
\(605\) 534.798 308.766i 0.883964 0.510357i
\(606\) 0 0
\(607\) −429.954 + 744.702i −0.708326 + 1.22686i 0.257151 + 0.966371i \(0.417216\pi\)
−0.965478 + 0.260486i \(0.916117\pi\)
\(608\) 0 0
\(609\) −330.318 190.709i −0.542395 0.313152i
\(610\) 0 0
\(611\) 558.682i 0.914373i
\(612\) 0 0
\(613\) 655.253 1.06893 0.534464 0.845191i \(-0.320513\pi\)
0.534464 + 0.845191i \(0.320513\pi\)
\(614\) 0 0
\(615\) 441.530 + 764.752i 0.717934 + 1.24350i
\(616\) 0 0
\(617\) −147.227 85.0013i −0.238617 0.137765i 0.375924 0.926650i \(-0.377325\pi\)
−0.614541 + 0.788885i \(0.710659\pi\)
\(618\) 0 0
\(619\) 270.531 + 468.573i 0.437045 + 0.756983i 0.997460 0.0712282i \(-0.0226918\pi\)
−0.560415 + 0.828212i \(0.689359\pi\)
\(620\) 0 0
\(621\) 334.772 + 193.281i 0.539086 + 0.311241i
\(622\) 0 0
\(623\) 444.545 256.658i 0.713555 0.411971i
\(624\) 0 0
\(625\) 244.490 423.469i 0.391184 0.677550i
\(626\) 0 0
\(627\) −279.667 + 161.466i −0.446040 + 0.257522i
\(628\) 0 0
\(629\) 1248.33i 1.98463i
\(630\) 0 0
\(631\) 260.788 0.413293 0.206646 0.978416i \(-0.433745\pi\)
0.206646 + 0.978416i \(0.433745\pi\)
\(632\) 0 0
\(633\) −194.106 + 336.202i −0.306645 + 0.531124i
\(634\) 0 0
\(635\) −76.6571 44.2580i −0.120720 0.0696977i
\(636\) 0 0
\(637\) −23.4143 40.5547i −0.0367571 0.0636652i
\(638\) 0 0
\(639\) 272.300i 0.426134i
\(640\) 0 0
\(641\) 585.418 337.991i 0.913289 0.527288i 0.0318012 0.999494i \(-0.489876\pi\)
0.881488 + 0.472206i \(0.156542\pi\)
\(642\) 0 0
\(643\) −378.318 + 655.267i −0.588364 + 1.01908i 0.406082 + 0.913837i \(0.366895\pi\)
−0.994447 + 0.105241i \(0.966439\pi\)
\(644\) 0 0
\(645\) 979.588i 1.51874i
\(646\) 0 0
\(647\) 554.770i 0.857450i 0.903435 + 0.428725i \(0.141037\pi\)
−0.903435 + 0.428725i \(0.858963\pi\)
\(648\) 0 0
\(649\) −101.000 −0.155624
\(650\) 0 0
\(651\) 954.392 1.46604
\(652\) 0 0
\(653\) −174.499 100.747i −0.267227 0.154283i 0.360400 0.932798i \(-0.382640\pi\)
−0.627627 + 0.778514i \(0.715974\pi\)
\(654\) 0 0
\(655\) −56.7724 98.3328i −0.0866755 0.150126i
\(656\) 0 0
\(657\) −439.090 −0.668325
\(658\) 0 0
\(659\) 89.7122 51.7954i 0.136134 0.0785969i −0.430386 0.902645i \(-0.641623\pi\)
0.566520 + 0.824048i \(0.308289\pi\)
\(660\) 0 0
\(661\) 109.207 189.152i 0.165215 0.286161i −0.771517 0.636209i \(-0.780502\pi\)
0.936732 + 0.350048i \(0.113835\pi\)
\(662\) 0 0
\(663\) −1098.48 634.210i −1.65684 0.956576i
\(664\) 0 0
\(665\) 883.834i 1.32907i
\(666\) 0 0
\(667\) −267.767 −0.401450
\(668\) 0 0
\(669\) −147.560 255.582i −0.220568 0.382036i
\(670\) 0 0
\(671\) −112.328 64.8523i −0.167403 0.0966503i
\(672\) 0 0
\(673\) −394.429 683.170i −0.586075 1.01511i −0.994740 0.102428i \(-0.967339\pi\)
0.408665 0.912684i \(-0.365994\pi\)
\(674\) 0 0
\(675\) 399.545 + 692.032i 0.591918 + 1.02523i
\(676\) 0 0
\(677\) 634.550 366.358i 0.937297 0.541149i 0.0481850 0.998838i \(-0.484656\pi\)
0.889112 + 0.457690i \(0.151323\pi\)
\(678\) 0 0
\(679\) −477.883 + 827.717i −0.703804 + 1.21902i
\(680\) 0 0
\(681\) 1150.95 664.503i 1.69009 0.975775i
\(682\) 0 0
\(683\) 1353.33i 1.98145i 0.135883 + 0.990725i \(0.456613\pi\)
−0.135883 + 0.990725i \(0.543387\pi\)
\(684\) 0 0
\(685\) 407.080 0.594277
\(686\) 0 0
\(687\) −168.015 + 291.011i −0.244564 + 0.423597i
\(688\) 0 0
\(689\) 147.959 + 85.4243i 0.214745 + 0.123983i
\(690\) 0 0
\(691\) −368.257 637.840i −0.532934 0.923068i −0.999260 0.0384555i \(-0.987756\pi\)
0.466327 0.884613i \(-0.345577\pi\)
\(692\) 0 0
\(693\) −324.138 187.141i −0.467731 0.270045i
\(694\) 0 0
\(695\) −646.166 + 373.064i −0.929736 + 0.536783i
\(696\) 0 0
\(697\) −501.353 + 868.369i −0.719301 + 1.24587i
\(698\) 0 0
\(699\) 242.558i 0.347007i
\(700\) 0 0
\(701\) 1068.34i 1.52403i 0.647561 + 0.762014i \(0.275789\pi\)
−0.647561 + 0.762014i \(0.724211\pi\)
\(702\) 0 0
\(703\) 872.686 1.24137
\(704\) 0 0
\(705\) −737.241 −1.04573
\(706\) 0 0
\(707\) −191.682 110.667i −0.271120 0.156531i
\(708\) 0 0
\(709\) 136.944 + 237.194i 0.193151 + 0.334547i 0.946293 0.323311i \(-0.104796\pi\)
−0.753142 + 0.657858i \(0.771463\pi\)
\(710\) 0 0
\(711\) −502.136 869.725i −0.706239 1.22324i
\(712\) 0 0
\(713\) 580.247 335.006i 0.813811 0.469854i
\(714\) 0 0
\(715\) 379.651 657.575i 0.530980 0.919685i
\(716\) 0 0
\(717\) 112.166 + 64.7593i 0.156438 + 0.0903197i
\(718\) 0 0
\(719\) 654.423i 0.910185i −0.890444 0.455092i \(-0.849606\pi\)
0.890444 0.455092i \(-0.150394\pi\)
\(720\) 0 0
\(721\) −921.706 −1.27837
\(722\) 0 0
\(723\) 422.712 + 732.159i 0.584664 + 1.01267i
\(724\) 0 0
\(725\) −479.363 276.761i −0.661191 0.381739i
\(726\) 0 0
\(727\) −583.166 1010.07i −0.802155 1.38937i −0.918195 0.396128i \(-0.870354\pi\)
0.116041 0.993244i \(-0.462980\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) −963.292 + 556.157i −1.31777 + 0.760816i
\(732\) 0 0
\(733\) 439.146 760.623i 0.599108 1.03768i −0.393845 0.919177i \(-0.628855\pi\)
0.992953 0.118508i \(-0.0378112\pi\)
\(734\) 0 0
\(735\) −53.5163 + 30.8977i −0.0728113 + 0.0420376i
\(736\) 0 0
\(737\) 532.103i 0.721984i
\(738\) 0 0
\(739\) 593.151 0.802640 0.401320 0.915938i \(-0.368552\pi\)
0.401320 + 0.915938i \(0.368552\pi\)
\(740\) 0 0
\(741\) 443.363 767.928i 0.598331 1.03634i
\(742\) 0 0
\(743\) −47.0561 27.1679i −0.0633326 0.0365651i 0.467999 0.883729i \(-0.344975\pi\)
−0.531332 + 0.847164i \(0.678308\pi\)
\(744\) 0 0
\(745\) 799.398 + 1384.60i 1.07302 + 1.85852i
\(746\) 0 0
\(747\) 765.771 442.118i 1.02513 0.591859i
\(748\) 0 0
\(749\) −208.059 + 120.123i −0.277783 + 0.160378i
\(750\) 0 0
\(751\) −455.570 + 789.071i −0.606618 + 1.05069i 0.385175 + 0.922844i \(0.374141\pi\)
−0.991793 + 0.127850i \(0.959192\pi\)
\(752\) 0 0
\(753\) 46.5392i 0.0618050i
\(754\) 0 0
\(755\) 1134.83i 1.50309i
\(756\) 0 0
\(757\) −1272.22 −1.68061 −0.840304 0.542115i \(-0.817624\pi\)
−0.840304 + 0.542115i \(0.817624\pi\)
\(758\) 0 0
\(759\) −262.757 −0.346189
\(760\) 0 0
\(761\) 399.480 + 230.640i 0.524940 + 0.303074i 0.738954 0.673756i \(-0.235320\pi\)
−0.214013 + 0.976831i \(0.568654\pi\)
\(762\) 0 0
\(763\) −182.171 315.530i −0.238757 0.413539i
\(764\) 0 0
\(765\) −836.908 + 1449.57i −1.09400 + 1.89486i
\(766\) 0 0
\(767\) 240.177 138.666i 0.313138 0.180790i
\(768\) 0 0
\(769\) −269.439 + 466.682i −0.350376 + 0.606868i −0.986315 0.164871i \(-0.947279\pi\)
0.635940 + 0.771739i \(0.280613\pi\)
\(770\) 0 0
\(771\) −936.104 540.460i −1.21414 0.700986i
\(772\) 0 0
\(773\) 1036.29i 1.34061i 0.742087 + 0.670304i \(0.233836\pi\)
−0.742087 + 0.670304i \(0.766164\pi\)
\(774\) 0 0
\(775\) 1385.03 1.78713
\(776\) 0 0
\(777\) 505.727 + 875.944i 0.650871 + 1.12734i
\(778\) 0 0
\(779\) −607.059 350.486i −0.779280 0.449918i
\(780\) 0 0
\(781\) −92.5449 160.292i −0.118495 0.205240i
\(782\) 0 0
\(783\) −437.317 + 252.485i −0.558515 + 0.322459i
\(784\) 0 0
\(785\) −524.520 + 302.832i −0.668179 + 0.385773i
\(786\) 0 0
\(787\) −706.096 + 1222.99i −0.897199 + 1.55399i −0.0661406 + 0.997810i \(0.521069\pi\)
−0.831059 + 0.556185i \(0.812265\pi\)
\(788\) 0 0
\(789\) −127.289 + 73.4902i −0.161329 + 0.0931435i
\(790\) 0 0
\(791\) 1123.09i 1.41983i
\(792\) 0 0
\(793\) 356.151 0.449119
\(794\) 0 0
\(795\) 112.727 195.248i 0.141794 0.245595i
\(796\) 0 0
\(797\) −60.4602 34.9067i −0.0758597 0.0437976i 0.461590 0.887093i \(-0.347279\pi\)
−0.537450 + 0.843296i \(0.680612\pi\)
\(798\) 0 0
\(799\) −418.565 724.976i −0.523861 0.907355i
\(800\) 0 0
\(801\) 679.593i 0.848431i
\(802\) 0 0
\(803\) 258.476 149.231i 0.321887 0.185842i
\(804\) 0 0
\(805\) 359.570 622.794i 0.446671 0.773657i
\(806\) 0 0
\(807\) 845.099i 1.04721i
\(808\) 0 0
\(809\) 1263.33i 1.56160i 0.624781 + 0.780800i \(0.285188\pi\)
−0.624781 + 0.780800i \(0.714812\pi\)
\(810\) 0 0
\(811\) 442.241 0.545303 0.272652 0.962113i \(-0.412099\pi\)
0.272652 + 0.962113i \(0.412099\pi\)
\(812\) 0 0
\(813\) −268.788 −0.330612
\(814\) 0 0
\(815\) 353.302 + 203.979i 0.433499 + 0.250281i
\(816\) 0 0
\(817\) −388.798 673.418i −0.475885 0.824257i
\(818\) 0 0
\(819\) 1027.73 1.25486
\(820\) 0 0
\(821\) −498.077 + 287.565i −0.606671 + 0.350261i −0.771661 0.636034i \(-0.780574\pi\)
0.164991 + 0.986295i \(0.447241\pi\)
\(822\) 0 0
\(823\) 11.6214 20.1289i 0.0141208 0.0244580i −0.858879 0.512179i \(-0.828838\pi\)
0.872999 + 0.487721i \(0.162172\pi\)
\(824\) 0 0
\(825\) −470.394 271.582i −0.570174 0.329190i
\(826\) 0 0
\(827\) 790.958i 0.956418i −0.878246 0.478209i \(-0.841286\pi\)
0.878246 0.478209i \(-0.158714\pi\)
\(828\) 0 0
\(829\) −1159.78 −1.39901 −0.699503 0.714630i \(-0.746595\pi\)
−0.699503 + 0.714630i \(0.746595\pi\)
\(830\) 0 0
\(831\) 126.591 + 219.262i 0.152336 + 0.263853i
\(832\) 0 0
\(833\) −60.7673 35.0840i −0.0729500 0.0421177i
\(834\) 0 0
\(835\) 570.802 + 988.658i 0.683595 + 1.18402i
\(836\) 0 0
\(837\) 631.772 1094.26i 0.754806 1.30736i
\(838\) 0 0
\(839\) 473.470 273.358i 0.564327 0.325814i −0.190553 0.981677i \(-0.561028\pi\)
0.754880 + 0.655862i \(0.227695\pi\)
\(840\) 0 0
\(841\) −245.606 + 425.402i −0.292041 + 0.505829i
\(842\) 0 0
\(843\) −69.8020 + 40.3002i −0.0828019 + 0.0478057i
\(844\) 0 0
\(845\) 836.213i 0.989601i
\(846\) 0 0
\(847\) −568.143 −0.670771
\(848\) 0 0
\(849\) 271.469 470.199i 0.319752 0.553827i
\(850\) 0 0
\(851\) 614.939 + 355.035i 0.722607 + 0.417197i
\(852\) 0 0
\(853\) 108.317 + 187.611i 0.126984 + 0.219943i 0.922507 0.385981i \(-0.126137\pi\)
−0.795523 + 0.605924i \(0.792804\pi\)
\(854\) 0 0
\(855\) −1013.36 585.066i −1.18522 0.684287i
\(856\) 0 0
\(857\) 489.741 282.752i 0.571460 0.329932i −0.186273 0.982498i \(-0.559641\pi\)
0.757732 + 0.652566i \(0.226307\pi\)
\(858\) 0 0
\(859\) 187.884 325.424i 0.218724 0.378841i −0.735694 0.677314i \(-0.763144\pi\)
0.954418 + 0.298473i \(0.0964773\pi\)
\(860\) 0 0
\(861\) 812.434i 0.943594i
\(862\) 0 0
\(863\) 1429.35i 1.65626i 0.560534 + 0.828131i \(0.310596\pi\)
−0.560534 + 0.828131i \(0.689404\pi\)
\(864\) 0 0
\(865\) 194.555 0.224919
\(866\) 0 0
\(867\) −1033.60 −1.19216
\(868\) 0 0
\(869\) 591.177 + 341.316i 0.680295 + 0.392769i
\(870\) 0 0
\(871\) 730.540 + 1265.33i 0.838737 + 1.45273i
\(872\) 0 0
\(873\) 632.682 + 1095.84i 0.724721 + 1.25525i
\(874\) 0 0
\(875\) 199.922 115.425i 0.228483 0.131915i
\(876\) 0 0
\(877\) −420.813 + 728.870i −0.479833 + 0.831095i −0.999732 0.0231327i \(-0.992636\pi\)
0.519900 + 0.854227i \(0.325969\pi\)
\(878\) 0 0
\(879\) 1223.95 + 706.650i 1.39244 + 0.803925i
\(880\) 0 0
\(881\) 449.261i 0.509944i −0.966948 0.254972i \(-0.917934\pi\)
0.966948 0.254972i \(-0.0820663\pi\)
\(882\) 0 0
\(883\) −122.445 −0.138669 −0.0693346 0.997593i \(-0.522088\pi\)
−0.0693346 + 0.997593i \(0.522088\pi\)
\(884\) 0 0
\(885\) −182.985 316.939i −0.206762 0.358123i
\(886\) 0 0
\(887\) −1361.09 785.828i −1.53449 0.885940i −0.999147 0.0413069i \(-0.986848\pi\)
−0.535346 0.844633i \(-0.679819\pi\)
\(888\) 0 0
\(889\) 40.7184 + 70.5263i 0.0458025 + 0.0793322i
\(890\) 0 0
\(891\) −429.135 + 247.761i −0.481633 + 0.278071i
\(892\) 0 0
\(893\) 506.816 292.611i 0.567543 0.327671i
\(894\) 0 0
\(895\) 985.312 1706.61i 1.10091 1.90683i
\(896\) 0 0
\(897\) 624.833 360.747i 0.696580 0.402171i
\(898\) 0 0
\(899\) 875.244i 0.973575i
\(900\) 0 0
\(901\) 256.000 0.284129
\(902\) 0 0
\(903\) 450.621 780.499i 0.499027 0.864340i
\(904\) 0 0
\(905\) −373.727 215.771i −0.412957 0.238421i
\(906\) 0 0
\(907\) −349.288 604.984i −0.385102 0.667017i 0.606681 0.794945i \(-0.292500\pi\)
−0.991783 + 0.127929i \(0.959167\pi\)
\(908\) 0 0
\(909\) −253.772 + 146.516i −0.279178 + 0.161183i
\(910\) 0 0
\(911\) −70.4888 + 40.6967i −0.0773752 + 0.0446726i −0.538188 0.842825i \(-0.680891\pi\)
0.460813 + 0.887497i \(0.347558\pi\)
\(912\) 0 0
\(913\) −300.520 + 520.517i −0.329157 + 0.570117i
\(914\) 0 0
\(915\) 469.980i 0.513639i
\(916\) 0 0
\(917\) 104.464i 0.113919i
\(918\) 0 0
\(919\) −348.665 −0.379396 −0.189698 0.981842i \(-0.560751\pi\)
−0.189698 + 0.981842i \(0.560751\pi\)
\(920\) 0 0
\(921\) −1394.24 −1.51383
\(922\) 0 0
\(923\) 440.141 + 254.115i 0.476859 + 0.275315i
\(924\) 0 0
\(925\) 733.918 + 1271.18i 0.793425 + 1.37425i
\(926\) 0 0
\(927\) −610.136 + 1056.79i −0.658183 + 1.14001i
\(928\) 0 0
\(929\) −932.298 + 538.262i −1.00355 + 0.579400i −0.909297 0.416148i \(-0.863380\pi\)
−0.0942533 + 0.995548i \(0.530046\pi\)
\(930\) 0 0
\(931\) 24.5265 42.4812i 0.0263443 0.0456297i
\(932\) 0 0
\(933\) −655.044 378.190i −0.702083 0.405348i
\(934\) 0 0
\(935\) 1137.74i 1.21683i
\(936\) 0 0
\(937\) −1437.39 −1.53404 −0.767018 0.641626i \(-0.778260\pi\)
−0.767018 + 0.641626i \(0.778260\pi\)
\(938\) 0 0
\(939\) −294.318 509.774i −0.313438 0.542891i
\(940\) 0 0
\(941\) 186.591 + 107.728i 0.198290 + 0.114483i 0.595858 0.803090i \(-0.296812\pi\)
−0.397568 + 0.917573i \(0.630146\pi\)
\(942\) 0 0
\(943\) −285.177 493.940i −0.302414 0.523797i
\(944\) 0 0
\(945\) 1356.20i 1.43513i
\(946\) 0 0
\(947\) 407.651 235.357i 0.430466 0.248529i −0.269079 0.963118i \(-0.586719\pi\)
0.699545 + 0.714589i \(0.253386\pi\)
\(948\) 0 0
\(949\) −409.767 + 709.738i −0.431789 + 0.747880i
\(950\) 0 0
\(951\) −296.954 + 171.447i −0.312255 + 0.180280i
\(952\) 0 0
\(953\) 1192.14i 1.25093i 0.780251 + 0.625466i \(0.215091\pi\)
−0.780251 + 0.625466i \(0.784909\pi\)
\(954\) 0 0
\(955\) 849.445 0.889471
\(956\) 0 0
\(957\) 171.621 297.257i 0.179333 0.310613i
\(958\) 0 0
\(959\) −324.346 187.261i −0.338213 0.195267i
\(960\) 0 0
\(961\) −614.524 1064.39i −0.639464 1.10758i
\(962\) 0 0
\(963\) 318.068i 0.330289i
\(964\) 0 0
\(965\) 1388.45 801.621i 1.43881 0.830695i
\(966\) 0 0
\(967\) 96.3888 166.950i 0.0996782 0.172648i −0.811873 0.583834i \(-0.801552\pi\)
0.911551 + 0.411186i \(0.134885\pi\)
\(968\) 0 0
\(969\) 1328.67i 1.37118i
\(970\) 0 0
\(971\) 202.388i 0.208433i −0.994555 0.104216i \(-0.966767\pi\)
0.994555 0.104216i \(-0.0332335\pi\)
\(972\) 0 0
\(973\) 686.455 0.705504
\(974\) 0 0
\(975\) 1491.45 1.52970
\(976\) 0 0
\(977\) 35.1980 + 20.3216i 0.0360266 + 0.0208000i 0.517905 0.855438i \(-0.326712\pi\)
−0.481879 + 0.876238i \(0.660045\pi\)
\(978\) 0 0
\(979\) 230.969 + 400.051i 0.235924 + 0.408632i
\(980\) 0 0
\(981\) −482.363 −0.491706
\(982\) 0 0
\(983\) 631.105 364.369i 0.642019 0.370670i −0.143373 0.989669i \(-0.545795\pi\)
0.785392 + 0.618999i \(0.212461\pi\)
\(984\) 0 0
\(985\) 632.141 1094.90i 0.641767 1.11157i
\(986\) 0 0
\(987\) 587.406 + 339.139i 0.595143 + 0.343606i
\(988\) 0 0
\(989\) 632.699i 0.639736i
\(990\) 0 0
\(991\) 746.527 0.753306 0.376653 0.926354i \(-0.377075\pi\)
0.376653 + 0.926354i \(0.377075\pi\)
\(992\) 0 0
\(993\) 81.8939 + 141.844i 0.0824712 + 0.142844i
\(994\) 0 0
\(995\) 396.737 + 229.056i 0.398730 + 0.230207i
\(996\) 0 0
\(997\) 119.046 + 206.194i 0.119404 + 0.206814i 0.919532 0.393016i \(-0.128568\pi\)
−0.800128 + 0.599830i \(0.795235\pi\)
\(998\) 0 0
\(999\) 1339.09 1.34043
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 576.3.q.h.257.1 4
3.2 odd 2 1728.3.q.e.449.2 4
4.3 odd 2 576.3.q.c.257.1 4
8.3 odd 2 144.3.q.d.113.2 4
8.5 even 2 72.3.m.a.41.2 4
9.2 odd 6 inner 576.3.q.h.65.1 4
9.7 even 3 1728.3.q.e.1601.2 4
12.11 even 2 1728.3.q.f.449.2 4
24.5 odd 2 216.3.m.a.17.1 4
24.11 even 2 432.3.q.c.17.1 4
36.7 odd 6 1728.3.q.f.1601.2 4
36.11 even 6 576.3.q.c.65.1 4
72.5 odd 6 648.3.e.b.161.4 4
72.11 even 6 144.3.q.d.65.2 4
72.13 even 6 648.3.e.b.161.1 4
72.29 odd 6 72.3.m.a.65.2 yes 4
72.43 odd 6 432.3.q.c.305.1 4
72.59 even 6 1296.3.e.c.161.4 4
72.61 even 6 216.3.m.a.89.1 4
72.67 odd 6 1296.3.e.c.161.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.m.a.41.2 4 8.5 even 2
72.3.m.a.65.2 yes 4 72.29 odd 6
144.3.q.d.65.2 4 72.11 even 6
144.3.q.d.113.2 4 8.3 odd 2
216.3.m.a.17.1 4 24.5 odd 2
216.3.m.a.89.1 4 72.61 even 6
432.3.q.c.17.1 4 24.11 even 2
432.3.q.c.305.1 4 72.43 odd 6
576.3.q.c.65.1 4 36.11 even 6
576.3.q.c.257.1 4 4.3 odd 2
576.3.q.h.65.1 4 9.2 odd 6 inner
576.3.q.h.257.1 4 1.1 even 1 trivial
648.3.e.b.161.1 4 72.13 even 6
648.3.e.b.161.4 4 72.5 odd 6
1296.3.e.c.161.1 4 72.67 odd 6
1296.3.e.c.161.4 4 72.59 even 6
1728.3.q.e.449.2 4 3.2 odd 2
1728.3.q.e.1601.2 4 9.7 even 3
1728.3.q.f.449.2 4 12.11 even 2
1728.3.q.f.1601.2 4 36.7 odd 6