Properties

Label 441.4.p.d.80.8
Level $441$
Weight $4$
Character 441.80
Analytic conductor $26.020$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,4,Mod(80,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.80");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.p (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: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 80.8
Character \(\chi\) \(=\) 441.80
Dual form 441.4.p.d.215.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.10847 + 1.79468i) q^{2} +(2.44173 - 4.22919i) q^{4} +(-0.428649 - 0.742442i) q^{5} -11.1864i q^{8} +O(q^{10})\) \(q+(-3.10847 + 1.79468i) q^{2} +(2.44173 - 4.22919i) q^{4} +(-0.428649 - 0.742442i) q^{5} -11.1864i q^{8} +(2.66489 + 1.53857i) q^{10} +(0.321369 + 0.185543i) q^{11} +62.0038i q^{13} +(39.6098 + 68.6061i) q^{16} +(51.9931 - 90.0547i) q^{17} +(-61.9469 + 35.7650i) q^{19} -4.18657 q^{20} -1.33196 q^{22} +(-118.248 + 68.2704i) q^{23} +(62.1325 - 107.617i) q^{25} +(-111.277 - 192.737i) q^{26} -17.8708i q^{29} +(-93.2782 - 53.8542i) q^{31} +(-168.750 - 97.4278i) q^{32} +373.243i q^{34} +(190.857 + 330.575i) q^{37} +(128.373 - 222.349i) q^{38} +(-8.30525 + 4.79504i) q^{40} -101.733 q^{41} -326.400 q^{43} +(1.56939 - 0.906088i) q^{44} +(245.047 - 424.433i) q^{46} +(-261.065 - 452.178i) q^{47} +446.031i q^{50} +(262.226 + 151.396i) q^{52} +(-268.049 - 154.758i) q^{53} -0.318131i q^{55} +(32.0724 + 55.5510i) q^{58} +(174.386 - 302.045i) q^{59} +(580.478 - 335.139i) q^{61} +386.604 q^{62} +65.6495 q^{64} +(46.0342 - 26.5779i) q^{65} +(119.685 - 207.301i) q^{67} +(-253.906 - 439.778i) q^{68} -178.563i q^{71} +(117.414 + 67.7891i) q^{73} +(-1186.55 - 685.054i) q^{74} +349.314i q^{76} +(-428.404 - 742.018i) q^{79} +(33.9574 - 58.8159i) q^{80} +(316.233 - 182.577i) q^{82} +1241.10 q^{83} -89.1472 q^{85} +(1014.60 - 585.782i) q^{86} +(2.07555 - 3.59496i) q^{88} +(282.464 + 489.242i) q^{89} +666.790i q^{92} +(1623.03 + 937.055i) q^{94} +(53.1069 + 30.6613i) q^{95} -1730.44i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 96 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 96 q^{4} - 144 q^{16} + 1248 q^{22} - 312 q^{25} + 864 q^{37} + 2496 q^{43} + 3888 q^{46} + 7440 q^{58} - 6720 q^{64} + 2688 q^{67} - 480 q^{79} + 26496 q^{85} + 7248 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.10847 + 1.79468i −1.09901 + 0.634514i −0.935961 0.352104i \(-0.885466\pi\)
−0.163049 + 0.986618i \(0.552133\pi\)
\(3\) 0 0
\(4\) 2.44173 4.22919i 0.305216 0.528649i
\(5\) −0.428649 0.742442i −0.0383395 0.0664060i 0.846219 0.532835i \(-0.178874\pi\)
−0.884558 + 0.466429i \(0.845540\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 11.1864i 0.494373i
\(9\) 0 0
\(10\) 2.66489 + 1.53857i 0.0842711 + 0.0486539i
\(11\) 0.321369 + 0.185543i 0.00880877 + 0.00508574i 0.504398 0.863471i \(-0.331714\pi\)
−0.495589 + 0.868557i \(0.665048\pi\)
\(12\) 0 0
\(13\) 62.0038i 1.32283i 0.750021 + 0.661414i \(0.230043\pi\)
−0.750021 + 0.661414i \(0.769957\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 39.6098 + 68.6061i 0.618902 + 1.07197i
\(17\) 51.9931 90.0547i 0.741775 1.28479i −0.209911 0.977720i \(-0.567318\pi\)
0.951686 0.307072i \(-0.0993491\pi\)
\(18\) 0 0
\(19\) −61.9469 + 35.7650i −0.747978 + 0.431845i −0.824963 0.565187i \(-0.808804\pi\)
0.0769848 + 0.997032i \(0.475471\pi\)
\(20\) −4.18657 −0.0468073
\(21\) 0 0
\(22\) −1.33196 −0.0129079
\(23\) −118.248 + 68.2704i −1.07202 + 0.618929i −0.928732 0.370753i \(-0.879100\pi\)
−0.143284 + 0.989682i \(0.545766\pi\)
\(24\) 0 0
\(25\) 62.1325 107.617i 0.497060 0.860933i
\(26\) −111.277 192.737i −0.839353 1.45380i
\(27\) 0 0
\(28\) 0 0
\(29\) 17.8708i 0.114432i −0.998362 0.0572161i \(-0.981778\pi\)
0.998362 0.0572161i \(-0.0182224\pi\)
\(30\) 0 0
\(31\) −93.2782 53.8542i −0.540428 0.312016i 0.204824 0.978799i \(-0.434338\pi\)
−0.745252 + 0.666783i \(0.767671\pi\)
\(32\) −168.750 97.4278i −0.932221 0.538218i
\(33\) 0 0
\(34\) 373.243i 1.88267i
\(35\) 0 0
\(36\) 0 0
\(37\) 190.857 + 330.575i 0.848020 + 1.46881i 0.882972 + 0.469425i \(0.155539\pi\)
−0.0349518 + 0.999389i \(0.511128\pi\)
\(38\) 128.373 222.349i 0.548024 0.949205i
\(39\) 0 0
\(40\) −8.30525 + 4.79504i −0.0328294 + 0.0189540i
\(41\) −101.733 −0.387511 −0.193756 0.981050i \(-0.562067\pi\)
−0.193756 + 0.981050i \(0.562067\pi\)
\(42\) 0 0
\(43\) −326.400 −1.15757 −0.578785 0.815480i \(-0.696473\pi\)
−0.578785 + 0.815480i \(0.696473\pi\)
\(44\) 1.56939 0.906088i 0.00537715 0.00310450i
\(45\) 0 0
\(46\) 245.047 424.433i 0.785438 1.36042i
\(47\) −261.065 452.178i −0.810218 1.40334i −0.912711 0.408606i \(-0.866015\pi\)
0.102493 0.994734i \(-0.467318\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 446.031i 1.26157i
\(51\) 0 0
\(52\) 262.226 + 151.396i 0.699312 + 0.403748i
\(53\) −268.049 154.758i −0.694704 0.401088i 0.110668 0.993857i \(-0.464701\pi\)
−0.805372 + 0.592770i \(0.798034\pi\)
\(54\) 0 0
\(55\) 0.318131i 0.000779940i
\(56\) 0 0
\(57\) 0 0
\(58\) 32.0724 + 55.5510i 0.0726088 + 0.125762i
\(59\) 174.386 302.045i 0.384799 0.666491i −0.606943 0.794746i \(-0.707604\pi\)
0.991741 + 0.128255i \(0.0409376\pi\)
\(60\) 0 0
\(61\) 580.478 335.139i 1.21840 0.703445i 0.253826 0.967250i \(-0.418311\pi\)
0.964576 + 0.263805i \(0.0849775\pi\)
\(62\) 386.604 0.791914
\(63\) 0 0
\(64\) 65.6495 0.128222
\(65\) 46.0342 26.5779i 0.0878437 0.0507166i
\(66\) 0 0
\(67\) 119.685 207.301i 0.218237 0.377997i −0.736032 0.676947i \(-0.763303\pi\)
0.954269 + 0.298949i \(0.0966362\pi\)
\(68\) −253.906 439.778i −0.452803 0.784278i
\(69\) 0 0
\(70\) 0 0
\(71\) 178.563i 0.298472i −0.988802 0.149236i \(-0.952319\pi\)
0.988802 0.149236i \(-0.0476814\pi\)
\(72\) 0 0
\(73\) 117.414 + 67.7891i 0.188251 + 0.108687i 0.591163 0.806552i \(-0.298669\pi\)
−0.402913 + 0.915238i \(0.632002\pi\)
\(74\) −1186.55 685.054i −1.86397 1.07616i
\(75\) 0 0
\(76\) 349.314i 0.527224i
\(77\) 0 0
\(78\) 0 0
\(79\) −428.404 742.018i −0.610117 1.05675i −0.991220 0.132221i \(-0.957789\pi\)
0.381103 0.924532i \(-0.375544\pi\)
\(80\) 33.9574 58.8159i 0.0474569 0.0821977i
\(81\) 0 0
\(82\) 316.233 182.577i 0.425879 0.245881i
\(83\) 1241.10 1.64130 0.820651 0.571430i \(-0.193611\pi\)
0.820651 + 0.571430i \(0.193611\pi\)
\(84\) 0 0
\(85\) −89.1472 −0.113757
\(86\) 1014.60 585.782i 1.27218 0.734494i
\(87\) 0 0
\(88\) 2.07555 3.59496i 0.00251426 0.00435482i
\(89\) 282.464 + 489.242i 0.336417 + 0.582691i 0.983756 0.179511i \(-0.0574515\pi\)
−0.647339 + 0.762202i \(0.724118\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 666.790i 0.755627i
\(93\) 0 0
\(94\) 1623.03 + 937.055i 1.78088 + 1.02819i
\(95\) 53.1069 + 30.6613i 0.0573543 + 0.0331135i
\(96\) 0 0
\(97\) 1730.44i 1.81134i −0.423985 0.905669i \(-0.639369\pi\)
0.423985 0.905669i \(-0.360631\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −303.421 525.541i −0.303421 0.525541i
\(101\) 627.920 1087.59i 0.618617 1.07148i −0.371121 0.928585i \(-0.621026\pi\)
0.989738 0.142892i \(-0.0456402\pi\)
\(102\) 0 0
\(103\) 1351.32 780.186i 1.29272 0.746350i 0.313581 0.949561i \(-0.398471\pi\)
0.979135 + 0.203211i \(0.0651378\pi\)
\(104\) 693.599 0.653971
\(105\) 0 0
\(106\) 1110.96 1.01798
\(107\) −145.139 + 83.7959i −0.131132 + 0.0757089i −0.564131 0.825685i \(-0.690789\pi\)
0.432999 + 0.901394i \(0.357455\pi\)
\(108\) 0 0
\(109\) −10.5542 + 18.2803i −0.00927435 + 0.0160636i −0.870625 0.491947i \(-0.836285\pi\)
0.861351 + 0.508010i \(0.169619\pi\)
\(110\) 0.570941 + 0.988899i 0.000494883 + 0.000857162i
\(111\) 0 0
\(112\) 0 0
\(113\) 1111.80i 0.925573i −0.886470 0.462787i \(-0.846850\pi\)
0.886470 0.462787i \(-0.153150\pi\)
\(114\) 0 0
\(115\) 101.374 + 58.5281i 0.0822012 + 0.0474589i
\(116\) −75.5793 43.6357i −0.0604945 0.0349265i
\(117\) 0 0
\(118\) 1251.87i 0.976640i
\(119\) 0 0
\(120\) 0 0
\(121\) −665.431 1152.56i −0.499948 0.865936i
\(122\) −1202.93 + 2083.54i −0.892691 + 1.54619i
\(123\) 0 0
\(124\) −455.520 + 262.994i −0.329894 + 0.190465i
\(125\) −213.694 −0.152907
\(126\) 0 0
\(127\) −1725.08 −1.20532 −0.602662 0.797997i \(-0.705893\pi\)
−0.602662 + 0.797997i \(0.705893\pi\)
\(128\) 1145.93 661.603i 0.791304 0.456859i
\(129\) 0 0
\(130\) −95.3974 + 165.233i −0.0643608 + 0.111476i
\(131\) 426.358 + 738.474i 0.284359 + 0.492525i 0.972454 0.233097i \(-0.0748858\pi\)
−0.688094 + 0.725621i \(0.741552\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 859.184i 0.553897i
\(135\) 0 0
\(136\) −1007.39 581.615i −0.635167 0.366714i
\(137\) −264.199 152.536i −0.164760 0.0951241i 0.415353 0.909660i \(-0.363658\pi\)
−0.580113 + 0.814536i \(0.696991\pi\)
\(138\) 0 0
\(139\) 1689.85i 1.03116i −0.856841 0.515581i \(-0.827576\pi\)
0.856841 0.515581i \(-0.172424\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 320.462 + 555.057i 0.189384 + 0.328024i
\(143\) −11.5043 + 19.9261i −0.00672756 + 0.0116525i
\(144\) 0 0
\(145\) −13.2681 + 7.66032i −0.00759899 + 0.00438728i
\(146\) −486.638 −0.275852
\(147\) 0 0
\(148\) 1864.09 1.03532
\(149\) −1927.91 + 1113.08i −1.06000 + 0.611993i −0.925433 0.378911i \(-0.876299\pi\)
−0.134570 + 0.990904i \(0.542965\pi\)
\(150\) 0 0
\(151\) −410.604 + 711.188i −0.221288 + 0.383282i −0.955199 0.295963i \(-0.904359\pi\)
0.733911 + 0.679245i \(0.237693\pi\)
\(152\) 400.082 + 692.962i 0.213493 + 0.369780i
\(153\) 0 0
\(154\) 0 0
\(155\) 92.3382i 0.0478502i
\(156\) 0 0
\(157\) −1384.34 799.250i −0.703710 0.406287i 0.105018 0.994470i \(-0.466510\pi\)
−0.808728 + 0.588183i \(0.799843\pi\)
\(158\) 2663.36 + 1537.69i 1.34105 + 0.774255i
\(159\) 0 0
\(160\) 167.049i 0.0825401i
\(161\) 0 0
\(162\) 0 0
\(163\) 1296.33 + 2245.31i 0.622923 + 1.07893i 0.988939 + 0.148326i \(0.0473883\pi\)
−0.366016 + 0.930609i \(0.619278\pi\)
\(164\) −248.403 + 430.247i −0.118275 + 0.204857i
\(165\) 0 0
\(166\) −3857.91 + 2227.37i −1.80381 + 1.04143i
\(167\) −4136.77 −1.91685 −0.958423 0.285352i \(-0.907889\pi\)
−0.958423 + 0.285352i \(0.907889\pi\)
\(168\) 0 0
\(169\) −1647.47 −0.749873
\(170\) 277.111 159.990i 0.125020 0.0721806i
\(171\) 0 0
\(172\) −796.979 + 1380.41i −0.353309 + 0.611948i
\(173\) −1583.86 2743.33i −0.696063 1.20562i −0.969821 0.243818i \(-0.921600\pi\)
0.273757 0.961799i \(-0.411733\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 29.3972i 0.0125903i
\(177\) 0 0
\(178\) −1756.06 1013.86i −0.739451 0.426922i
\(179\) 3110.97 + 1796.12i 1.29902 + 0.749991i 0.980235 0.197836i \(-0.0633913\pi\)
0.318787 + 0.947827i \(0.396725\pi\)
\(180\) 0 0
\(181\) 314.936i 0.129332i 0.997907 + 0.0646658i \(0.0205981\pi\)
−0.997907 + 0.0646658i \(0.979402\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 763.699 + 1322.77i 0.305982 + 0.529976i
\(185\) 163.622 283.401i 0.0650254 0.112627i
\(186\) 0 0
\(187\) 33.4179 19.2939i 0.0130682 0.00754495i
\(188\) −2549.80 −0.989166
\(189\) 0 0
\(190\) −220.108 −0.0840439
\(191\) −3051.20 + 1761.61i −1.15590 + 0.667359i −0.950318 0.311281i \(-0.899242\pi\)
−0.205582 + 0.978640i \(0.565909\pi\)
\(192\) 0 0
\(193\) −360.020 + 623.573i −0.134274 + 0.232569i −0.925320 0.379188i \(-0.876203\pi\)
0.791046 + 0.611757i \(0.209537\pi\)
\(194\) 3105.58 + 5379.03i 1.14932 + 1.99068i
\(195\) 0 0
\(196\) 0 0
\(197\) 3479.92i 1.25855i −0.777184 0.629274i \(-0.783353\pi\)
0.777184 0.629274i \(-0.216647\pi\)
\(198\) 0 0
\(199\) −3381.21 1952.14i −1.20446 0.695395i −0.242916 0.970047i \(-0.578104\pi\)
−0.961544 + 0.274652i \(0.911437\pi\)
\(200\) −1203.84 695.039i −0.425622 0.245733i
\(201\) 0 0
\(202\) 4507.65i 1.57009i
\(203\) 0 0
\(204\) 0 0
\(205\) 43.6076 + 75.5305i 0.0148570 + 0.0257331i
\(206\) −2800.36 + 4850.37i −0.947139 + 1.64049i
\(207\) 0 0
\(208\) −4253.84 + 2455.96i −1.41803 + 0.818701i
\(209\) −26.5437 −0.00878502
\(210\) 0 0
\(211\) 2913.22 0.950494 0.475247 0.879852i \(-0.342359\pi\)
0.475247 + 0.879852i \(0.342359\pi\)
\(212\) −1309.00 + 755.753i −0.424069 + 0.244837i
\(213\) 0 0
\(214\) 300.773 520.954i 0.0960767 0.166410i
\(215\) 139.911 + 242.333i 0.0443807 + 0.0768696i
\(216\) 0 0
\(217\) 0 0
\(218\) 75.7651i 0.0235388i
\(219\) 0 0
\(220\) −1.34544 0.776788i −0.000412315 0.000238050i
\(221\) 5583.73 + 3223.77i 1.69956 + 0.981241i
\(222\) 0 0
\(223\) 1202.30i 0.361040i −0.983571 0.180520i \(-0.942222\pi\)
0.983571 0.180520i \(-0.0577780\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1995.33 + 3456.01i 0.587289 + 1.01721i
\(227\) 2567.71 4447.40i 0.750770 1.30037i −0.196681 0.980468i \(-0.563016\pi\)
0.947450 0.319903i \(-0.103650\pi\)
\(228\) 0 0
\(229\) −1625.94 + 938.735i −0.469192 + 0.270888i −0.715901 0.698202i \(-0.753984\pi\)
0.246710 + 0.969089i \(0.420651\pi\)
\(230\) −420.156 −0.120453
\(231\) 0 0
\(232\) −199.910 −0.0565722
\(233\) −3016.28 + 1741.45i −0.848081 + 0.489640i −0.860003 0.510289i \(-0.829538\pi\)
0.0119216 + 0.999929i \(0.496205\pi\)
\(234\) 0 0
\(235\) −223.811 + 387.651i −0.0621268 + 0.107607i
\(236\) −851.605 1475.02i −0.234893 0.406847i
\(237\) 0 0
\(238\) 0 0
\(239\) 2466.36i 0.667514i −0.942659 0.333757i \(-0.891684\pi\)
0.942659 0.333757i \(-0.108316\pi\)
\(240\) 0 0
\(241\) 2112.76 + 1219.80i 0.564709 + 0.326035i 0.755033 0.655686i \(-0.227621\pi\)
−0.190324 + 0.981721i \(0.560954\pi\)
\(242\) 4136.95 + 2388.47i 1.09890 + 0.634448i
\(243\) 0 0
\(244\) 3273.27i 0.858810i
\(245\) 0 0
\(246\) 0 0
\(247\) −2217.57 3840.94i −0.571257 0.989446i
\(248\) −602.434 + 1043.45i −0.154252 + 0.267173i
\(249\) 0 0
\(250\) 664.263 383.512i 0.168047 0.0970218i
\(251\) −4329.06 −1.08864 −0.544319 0.838878i \(-0.683212\pi\)
−0.544319 + 0.838878i \(0.683212\pi\)
\(252\) 0 0
\(253\) −50.6682 −0.0125909
\(254\) 5362.36 3095.96i 1.32466 0.764795i
\(255\) 0 0
\(256\) −2637.32 + 4567.98i −0.643878 + 1.11523i
\(257\) 1570.68 + 2720.50i 0.381231 + 0.660312i 0.991239 0.132084i \(-0.0421669\pi\)
−0.610007 + 0.792396i \(0.708834\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 259.584i 0.0619180i
\(261\) 0 0
\(262\) −2650.64 1530.35i −0.625028 0.360860i
\(263\) −6663.55 3847.20i −1.56233 0.902010i −0.997021 0.0771283i \(-0.975425\pi\)
−0.565306 0.824882i \(-0.691242\pi\)
\(264\) 0 0
\(265\) 265.348i 0.0615101i
\(266\) 0 0
\(267\) 0 0
\(268\) −584.477 1012.34i −0.133219 0.230741i
\(269\) −698.723 + 1210.22i −0.158371 + 0.274307i −0.934281 0.356536i \(-0.883958\pi\)
0.775910 + 0.630843i \(0.217291\pi\)
\(270\) 0 0
\(271\) 2135.91 1233.17i 0.478772 0.276419i −0.241133 0.970492i \(-0.577519\pi\)
0.719905 + 0.694073i \(0.244186\pi\)
\(272\) 8237.73 1.83635
\(273\) 0 0
\(274\) 1095.01 0.241430
\(275\) 39.9349 23.0564i 0.00875697 0.00505584i
\(276\) 0 0
\(277\) −1086.42 + 1881.73i −0.235655 + 0.408167i −0.959463 0.281835i \(-0.909057\pi\)
0.723808 + 0.690002i \(0.242390\pi\)
\(278\) 3032.74 + 5252.86i 0.654287 + 1.13326i
\(279\) 0 0
\(280\) 0 0
\(281\) 4631.24i 0.983191i 0.870824 + 0.491596i \(0.163586\pi\)
−0.870824 + 0.491596i \(0.836414\pi\)
\(282\) 0 0
\(283\) 5999.60 + 3463.87i 1.26021 + 0.727582i 0.973115 0.230320i \(-0.0739772\pi\)
0.287095 + 0.957902i \(0.407311\pi\)
\(284\) −755.176 436.001i −0.157787 0.0910983i
\(285\) 0 0
\(286\) 82.5863i 0.0170749i
\(287\) 0 0
\(288\) 0 0
\(289\) −2950.06 5109.66i −0.600460 1.04003i
\(290\) 27.4956 47.6238i 0.00556758 0.00964332i
\(291\) 0 0
\(292\) 573.387 331.045i 0.114914 0.0663457i
\(293\) −667.366 −0.133065 −0.0665323 0.997784i \(-0.521194\pi\)
−0.0665323 + 0.997784i \(0.521194\pi\)
\(294\) 0 0
\(295\) −299.002 −0.0590120
\(296\) 3697.94 2135.00i 0.726143 0.419239i
\(297\) 0 0
\(298\) 3995.23 6919.94i 0.776636 1.34517i
\(299\) −4233.02 7331.81i −0.818736 1.41809i
\(300\) 0 0
\(301\) 0 0
\(302\) 2947.61i 0.561642i
\(303\) 0 0
\(304\) −4907.40 2833.29i −0.925851 0.534540i
\(305\) −497.642 287.314i −0.0934260 0.0539395i
\(306\) 0 0
\(307\) 10190.1i 1.89439i −0.320658 0.947195i \(-0.603904\pi\)
0.320658 0.947195i \(-0.396096\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −165.717 287.031i −0.0303616 0.0525879i
\(311\) −4800.99 + 8315.56i −0.875367 + 1.51618i −0.0189952 + 0.999820i \(0.506047\pi\)
−0.856372 + 0.516360i \(0.827287\pi\)
\(312\) 0 0
\(313\) 5683.70 3281.48i 1.02639 0.592589i 0.110445 0.993882i \(-0.464772\pi\)
0.915950 + 0.401293i \(0.131439\pi\)
\(314\) 5737.58 1.03118
\(315\) 0 0
\(316\) −4184.18 −0.744869
\(317\) 115.552 66.7137i 0.0204732 0.0118202i −0.489728 0.871875i \(-0.662904\pi\)
0.510202 + 0.860055i \(0.329571\pi\)
\(318\) 0 0
\(319\) 3.31580 5.74314i 0.000581973 0.00100801i
\(320\) −28.1406 48.7410i −0.00491596 0.00851470i
\(321\) 0 0
\(322\) 0 0
\(323\) 7438.14i 1.28133i
\(324\) 0 0
\(325\) 6672.64 + 3852.45i 1.13887 + 0.657525i
\(326\) −8059.21 4652.99i −1.36920 0.790507i
\(327\) 0 0
\(328\) 1138.02i 0.191575i
\(329\) 0 0
\(330\) 0 0
\(331\) −2393.76 4146.11i −0.397501 0.688492i 0.595916 0.803047i \(-0.296789\pi\)
−0.993417 + 0.114555i \(0.963456\pi\)
\(332\) 3030.42 5248.84i 0.500951 0.867673i
\(333\) 0 0
\(334\) 12859.0 7424.17i 2.10663 1.21627i
\(335\) −205.212 −0.0334684
\(336\) 0 0
\(337\) 8273.70 1.33738 0.668690 0.743541i \(-0.266855\pi\)
0.668690 + 0.743541i \(0.266855\pi\)
\(338\) 5121.12 2956.68i 0.824118 0.475805i
\(339\) 0 0
\(340\) −217.673 + 377.021i −0.0347205 + 0.0601377i
\(341\) −19.9845 34.6142i −0.00317367 0.00549696i
\(342\) 0 0
\(343\) 0 0
\(344\) 3651.24i 0.572272i
\(345\) 0 0
\(346\) 9846.79 + 5685.05i 1.52996 + 0.883324i
\(347\) 3203.79 + 1849.71i 0.495644 + 0.286160i 0.726913 0.686730i \(-0.240954\pi\)
−0.231269 + 0.972890i \(0.574288\pi\)
\(348\) 0 0
\(349\) 903.273i 0.138542i 0.997598 + 0.0692709i \(0.0220673\pi\)
−0.997598 + 0.0692709i \(0.977933\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −36.1540 62.6206i −0.00547447 0.00948207i
\(353\) 5080.80 8800.20i 0.766073 1.32688i −0.173605 0.984815i \(-0.555542\pi\)
0.939678 0.342061i \(-0.111125\pi\)
\(354\) 0 0
\(355\) −132.572 + 76.5408i −0.0198203 + 0.0114433i
\(356\) 2758.80 0.410719
\(357\) 0 0
\(358\) −12893.8 −1.90352
\(359\) 3651.25 2108.05i 0.536784 0.309912i −0.206991 0.978343i \(-0.566367\pi\)
0.743775 + 0.668431i \(0.233034\pi\)
\(360\) 0 0
\(361\) −871.225 + 1509.01i −0.127019 + 0.220004i
\(362\) −565.209 978.970i −0.0820627 0.142137i
\(363\) 0 0
\(364\) 0 0
\(365\) 116.231i 0.0166680i
\(366\) 0 0
\(367\) −468.678 270.591i −0.0666615 0.0384871i 0.466299 0.884627i \(-0.345587\pi\)
−0.532960 + 0.846140i \(0.678921\pi\)
\(368\) −9367.53 5408.35i −1.32695 0.766113i
\(369\) 0 0
\(370\) 1174.59i 0.165038i
\(371\) 0 0
\(372\) 0 0
\(373\) −1120.17 1940.19i −0.155496 0.269328i 0.777743 0.628582i \(-0.216364\pi\)
−0.933240 + 0.359254i \(0.883031\pi\)
\(374\) −69.2525 + 119.949i −0.00957476 + 0.0165840i
\(375\) 0 0
\(376\) −5058.24 + 2920.38i −0.693773 + 0.400550i
\(377\) 1108.06 0.151374
\(378\) 0 0
\(379\) −2922.72 −0.396121 −0.198060 0.980190i \(-0.563464\pi\)
−0.198060 + 0.980190i \(0.563464\pi\)
\(380\) 259.345 149.733i 0.0350109 0.0202135i
\(381\) 0 0
\(382\) 6323.04 10951.8i 0.846897 1.46687i
\(383\) −3526.59 6108.23i −0.470497 0.814924i 0.528934 0.848663i \(-0.322592\pi\)
−0.999431 + 0.0337386i \(0.989259\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2584.48i 0.340794i
\(387\) 0 0
\(388\) −7318.38 4225.27i −0.957562 0.552849i
\(389\) −8598.98 4964.62i −1.12079 0.647086i −0.179184 0.983816i \(-0.557346\pi\)
−0.941601 + 0.336730i \(0.890679\pi\)
\(390\) 0 0
\(391\) 14198.4i 1.83642i
\(392\) 0 0
\(393\) 0 0
\(394\) 6245.32 + 10817.2i 0.798566 + 1.38316i
\(395\) −367.270 + 636.131i −0.0467832 + 0.0810309i
\(396\) 0 0
\(397\) 2207.29 1274.38i 0.279044 0.161106i −0.353946 0.935266i \(-0.615160\pi\)
0.632991 + 0.774159i \(0.281827\pi\)
\(398\) 14013.9 1.76495
\(399\) 0 0
\(400\) 9844.22 1.23053
\(401\) −4468.23 + 2579.74i −0.556441 + 0.321261i −0.751716 0.659487i \(-0.770774\pi\)
0.195275 + 0.980749i \(0.437440\pi\)
\(402\) 0 0
\(403\) 3339.17 5783.61i 0.412744 0.714893i
\(404\) −3066.42 5311.19i −0.377624 0.654063i
\(405\) 0 0
\(406\) 0 0
\(407\) 141.649i 0.0172513i
\(408\) 0 0
\(409\) 6922.74 + 3996.85i 0.836938 + 0.483206i 0.856222 0.516608i \(-0.172805\pi\)
−0.0192844 + 0.999814i \(0.506139\pi\)
\(410\) −271.106 156.523i −0.0326560 0.0188539i
\(411\) 0 0
\(412\) 7620.01i 0.911191i
\(413\) 0 0
\(414\) 0 0
\(415\) −531.995 921.443i −0.0629268 0.108992i
\(416\) 6040.90 10463.1i 0.711969 1.23317i
\(417\) 0 0
\(418\) 82.5104 47.6374i 0.00965483 0.00557422i
\(419\) −5972.30 −0.696338 −0.348169 0.937432i \(-0.613197\pi\)
−0.348169 + 0.937432i \(0.613197\pi\)
\(420\) 0 0
\(421\) 12647.3 1.46412 0.732058 0.681242i \(-0.238560\pi\)
0.732058 + 0.681242i \(0.238560\pi\)
\(422\) −9055.66 + 5228.29i −1.04460 + 0.603102i
\(423\) 0 0
\(424\) −1731.18 + 2998.50i −0.198287 + 0.343443i
\(425\) −6460.92 11190.6i −0.737414 1.27724i
\(426\) 0 0
\(427\) 0 0
\(428\) 818.426i 0.0924302i
\(429\) 0 0
\(430\) −869.818 502.190i −0.0975497 0.0563203i
\(431\) 2947.84 + 1701.94i 0.329449 + 0.190207i 0.655596 0.755111i \(-0.272417\pi\)
−0.326148 + 0.945319i \(0.605751\pi\)
\(432\) 0 0
\(433\) 7413.29i 0.822772i 0.911461 + 0.411386i \(0.134955\pi\)
−0.911461 + 0.411386i \(0.865045\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 51.5407 + 89.2711i 0.00566136 + 0.00980576i
\(437\) 4883.39 8458.27i 0.534563 0.925890i
\(438\) 0 0
\(439\) −4020.87 + 2321.45i −0.437142 + 0.252384i −0.702385 0.711797i \(-0.747881\pi\)
0.265242 + 0.964182i \(0.414548\pi\)
\(440\) −3.55873 −0.000385582
\(441\) 0 0
\(442\) −23142.5 −2.49044
\(443\) −2570.90 + 1484.31i −0.275727 + 0.159191i −0.631487 0.775386i \(-0.717555\pi\)
0.355761 + 0.934577i \(0.384222\pi\)
\(444\) 0 0
\(445\) 242.156 419.426i 0.0257961 0.0446802i
\(446\) 2157.74 + 3737.31i 0.229085 + 0.396786i
\(447\) 0 0
\(448\) 0 0
\(449\) 8732.52i 0.917847i −0.888476 0.458923i \(-0.848235\pi\)
0.888476 0.458923i \(-0.151765\pi\)
\(450\) 0 0
\(451\) −32.6937 18.8757i −0.00341350 0.00197078i
\(452\) −4702.04 2714.72i −0.489304 0.282500i
\(453\) 0 0
\(454\) 18432.8i 1.90549i
\(455\) 0 0
\(456\) 0 0
\(457\) 6049.30 + 10477.7i 0.619200 + 1.07249i 0.989632 + 0.143626i \(0.0458762\pi\)
−0.370432 + 0.928859i \(0.620790\pi\)
\(458\) 3369.45 5836.06i 0.343764 0.595417i
\(459\) 0 0
\(460\) 495.053 285.819i 0.0501782 0.0289704i
\(461\) −7746.32 −0.782608 −0.391304 0.920261i \(-0.627976\pi\)
−0.391304 + 0.920261i \(0.627976\pi\)
\(462\) 0 0
\(463\) 2573.36 0.258303 0.129152 0.991625i \(-0.458775\pi\)
0.129152 + 0.991625i \(0.458775\pi\)
\(464\) 1226.05 707.860i 0.122668 0.0708223i
\(465\) 0 0
\(466\) 6250.68 10826.5i 0.621367 1.07624i
\(467\) −1193.94 2067.96i −0.118306 0.204911i 0.800791 0.598944i \(-0.204413\pi\)
−0.919096 + 0.394033i \(0.871080\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1606.67i 0.157681i
\(471\) 0 0
\(472\) −3378.80 1950.75i −0.329495 0.190234i
\(473\) −104.895 60.5610i −0.0101968 0.00588710i
\(474\) 0 0
\(475\) 8888.69i 0.858612i
\(476\) 0 0
\(477\) 0 0
\(478\) 4426.32 + 7666.62i 0.423547 + 0.733605i
\(479\) −1453.00 + 2516.67i −0.138600 + 0.240062i −0.926967 0.375143i \(-0.877593\pi\)
0.788367 + 0.615205i \(0.210927\pi\)
\(480\) 0 0
\(481\) −20496.9 + 11833.9i −1.94299 + 1.12178i
\(482\) −8756.60 −0.827495
\(483\) 0 0
\(484\) −6499.20 −0.610368
\(485\) −1284.75 + 741.753i −0.120284 + 0.0694459i
\(486\) 0 0
\(487\) −132.268 + 229.095i −0.0123072 + 0.0213168i −0.872113 0.489304i \(-0.837251\pi\)
0.859806 + 0.510621i \(0.170584\pi\)
\(488\) −3748.99 6493.45i −0.347764 0.602346i
\(489\) 0 0
\(490\) 0 0
\(491\) 19788.0i 1.81878i 0.415943 + 0.909391i \(0.363452\pi\)
−0.415943 + 0.909391i \(0.636548\pi\)
\(492\) 0 0
\(493\) −1609.35 929.160i −0.147022 0.0848829i
\(494\) 13786.5 + 7959.63i 1.25563 + 0.724941i
\(495\) 0 0
\(496\) 8532.61i 0.772430i
\(497\) 0 0
\(498\) 0 0
\(499\) 5735.87 + 9934.82i 0.514575 + 0.891270i 0.999857 + 0.0169124i \(0.00538363\pi\)
−0.485282 + 0.874358i \(0.661283\pi\)
\(500\) −521.783 + 903.755i −0.0466697 + 0.0808343i
\(501\) 0 0
\(502\) 13456.8 7769.27i 1.19642 0.690756i
\(503\) −1261.66 −0.111839 −0.0559193 0.998435i \(-0.517809\pi\)
−0.0559193 + 0.998435i \(0.517809\pi\)
\(504\) 0 0
\(505\) −1076.63 −0.0948700
\(506\) 157.501 90.9331i 0.0138375 0.00798907i
\(507\) 0 0
\(508\) −4212.17 + 7295.70i −0.367884 + 0.637194i
\(509\) 4261.39 + 7380.94i 0.371086 + 0.642740i 0.989733 0.142929i \(-0.0456522\pi\)
−0.618647 + 0.785669i \(0.712319\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 8346.93i 0.720480i
\(513\) 0 0
\(514\) −9764.83 5637.73i −0.837954 0.483793i
\(515\) −1158.49 668.853i −0.0991243 0.0572294i
\(516\) 0 0
\(517\) 193.755i 0.0164823i
\(518\) 0 0
\(519\) 0 0
\(520\) −297.310 514.957i −0.0250729 0.0434276i
\(521\) −6608.58 + 11446.4i −0.555714 + 0.962525i 0.442133 + 0.896949i \(0.354222\pi\)
−0.997848 + 0.0655760i \(0.979112\pi\)
\(522\) 0 0
\(523\) 7783.83 4493.99i 0.650790 0.375734i −0.137969 0.990437i \(-0.544057\pi\)
0.788759 + 0.614703i \(0.210724\pi\)
\(524\) 4164.20 0.347164
\(525\) 0 0
\(526\) 27617.9 2.28935
\(527\) −9699.65 + 5600.09i −0.801752 + 0.462892i
\(528\) 0 0
\(529\) 3238.19 5608.72i 0.266146 0.460978i
\(530\) −476.213 824.825i −0.0390290 0.0676002i
\(531\) 0 0
\(532\) 0 0
\(533\) 6307.81i 0.512611i
\(534\) 0 0
\(535\) 124.427 + 71.8381i 0.0100551 + 0.00580529i
\(536\) −2318.95 1338.84i −0.186872 0.107890i
\(537\) 0 0
\(538\) 5015.92i 0.401955i
\(539\) 0 0
\(540\) 0 0
\(541\) −5815.74 10073.2i −0.462178 0.800516i 0.536891 0.843652i \(-0.319599\pi\)
−0.999069 + 0.0431356i \(0.986265\pi\)
\(542\) −4426.27 + 7666.53i −0.350784 + 0.607575i
\(543\) 0 0
\(544\) −17547.7 + 10131.1i −1.38300 + 0.798473i
\(545\) 18.0961 0.00142230
\(546\) 0 0
\(547\) −23271.9 −1.81908 −0.909538 0.415622i \(-0.863564\pi\)
−0.909538 + 0.415622i \(0.863564\pi\)
\(548\) −1290.21 + 744.900i −0.100575 + 0.0580667i
\(549\) 0 0
\(550\) −82.7577 + 143.341i −0.00641600 + 0.0111128i
\(551\) 639.151 + 1107.04i 0.0494170 + 0.0855927i
\(552\) 0 0
\(553\) 0 0
\(554\) 7799.07i 0.598106i
\(555\) 0 0
\(556\) −7146.72 4126.16i −0.545123 0.314727i
\(557\) 211.015 + 121.830i 0.0160521 + 0.00926767i 0.508005 0.861354i \(-0.330383\pi\)
−0.491952 + 0.870622i \(0.663717\pi\)
\(558\) 0 0
\(559\) 20238.0i 1.53127i
\(560\) 0 0
\(561\) 0 0
\(562\) −8311.58 14396.1i −0.623848 1.08054i
\(563\) −4606.13 + 7978.05i −0.344805 + 0.597220i −0.985318 0.170727i \(-0.945388\pi\)
0.640513 + 0.767947i \(0.278722\pi\)
\(564\) 0 0
\(565\) −825.450 + 476.574i −0.0614637 + 0.0354861i
\(566\) −24866.1 −1.84664
\(567\) 0 0
\(568\) −1997.47 −0.147556
\(569\) 6941.28 4007.55i 0.511413 0.295264i −0.222001 0.975046i \(-0.571259\pi\)
0.733414 + 0.679782i \(0.237926\pi\)
\(570\) 0 0
\(571\) −10058.3 + 17421.5i −0.737173 + 1.27682i 0.216590 + 0.976263i \(0.430507\pi\)
−0.953763 + 0.300559i \(0.902827\pi\)
\(572\) 56.1809 + 97.3082i 0.00410672 + 0.00711304i
\(573\) 0 0
\(574\) 0 0
\(575\) 16967.2i 1.23058i
\(576\) 0 0
\(577\) −10355.4 5978.68i −0.747141 0.431362i 0.0775191 0.996991i \(-0.475300\pi\)
−0.824660 + 0.565629i \(0.808633\pi\)
\(578\) 18340.4 + 10588.8i 1.31982 + 0.762001i
\(579\) 0 0
\(580\) 74.8176i 0.00535626i
\(581\) 0 0
\(582\) 0 0
\(583\) −57.4284 99.4689i −0.00407966 0.00706617i
\(584\) 758.316 1313.44i 0.0537317 0.0930661i
\(585\) 0 0
\(586\) 2074.49 1197.71i 0.146239 0.0844314i
\(587\) −5400.12 −0.379705 −0.189852 0.981813i \(-0.560801\pi\)
−0.189852 + 0.981813i \(0.560801\pi\)
\(588\) 0 0
\(589\) 7704.39 0.538971
\(590\) 929.437 536.611i 0.0648548 0.0374439i
\(591\) 0 0
\(592\) −15119.6 + 26188.0i −1.04968 + 1.81811i
\(593\) 11877.2 + 20571.9i 0.822491 + 1.42460i 0.903822 + 0.427909i \(0.140750\pi\)
−0.0813305 + 0.996687i \(0.525917\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10871.3i 0.747160i
\(597\) 0 0
\(598\) 26316.5 + 15193.8i 1.79960 + 1.03900i
\(599\) 1263.64 + 729.564i 0.0861953 + 0.0497649i 0.542478 0.840070i \(-0.317486\pi\)
−0.456283 + 0.889835i \(0.650819\pi\)
\(600\) 0 0
\(601\) 5357.92i 0.363651i 0.983331 + 0.181825i \(0.0582006\pi\)
−0.983331 + 0.181825i \(0.941799\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2005.17 + 3473.05i 0.135081 + 0.233968i
\(605\) −570.473 + 988.088i −0.0383356 + 0.0663992i
\(606\) 0 0
\(607\) 8773.63 5065.46i 0.586673 0.338716i −0.177108 0.984191i \(-0.556674\pi\)
0.763781 + 0.645475i \(0.223341\pi\)
\(608\) 13938.0 0.929707
\(609\) 0 0
\(610\) 2062.54 0.136901
\(611\) 28036.8 16187.0i 1.85638 1.07178i
\(612\) 0 0
\(613\) 1217.62 2108.99i 0.0802274 0.138958i −0.823120 0.567867i \(-0.807769\pi\)
0.903348 + 0.428909i \(0.141102\pi\)
\(614\) 18287.9 + 31675.5i 1.20202 + 2.08195i
\(615\) 0 0
\(616\) 0 0
\(617\) 28047.4i 1.83006i −0.403389 0.915029i \(-0.632168\pi\)
0.403389 0.915029i \(-0.367832\pi\)
\(618\) 0 0
\(619\) −14086.2 8132.66i −0.914655 0.528076i −0.0327292 0.999464i \(-0.510420\pi\)
−0.881926 + 0.471388i \(0.843753\pi\)
\(620\) 390.516 + 225.465i 0.0252960 + 0.0146046i
\(621\) 0 0
\(622\) 34464.9i 2.22173i
\(623\) 0 0
\(624\) 0 0
\(625\) −7674.97 13293.4i −0.491198 0.850779i
\(626\) −11778.4 + 20400.8i −0.752012 + 1.30252i
\(627\) 0 0
\(628\) −6760.36 + 3903.10i −0.429567 + 0.248010i
\(629\) 39693.0 2.51616
\(630\) 0 0
\(631\) 13662.1 0.861931 0.430966 0.902368i \(-0.358173\pi\)
0.430966 + 0.902368i \(0.358173\pi\)
\(632\) −8300.50 + 4792.30i −0.522431 + 0.301625i
\(633\) 0 0
\(634\) −239.459 + 414.755i −0.0150002 + 0.0259811i
\(635\) 739.454 + 1280.77i 0.0462116 + 0.0800408i
\(636\) 0 0
\(637\) 0 0
\(638\) 23.8032i 0.00147708i
\(639\) 0 0
\(640\) −982.404 567.191i −0.0606764 0.0350316i
\(641\) 23643.1 + 13650.3i 1.45686 + 0.841117i 0.998855 0.0478346i \(-0.0152320\pi\)
0.458002 + 0.888951i \(0.348565\pi\)
\(642\) 0 0
\(643\) 6857.85i 0.420602i −0.977637 0.210301i \(-0.932556\pi\)
0.977637 0.210301i \(-0.0674445\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −13349.1 23121.2i −0.813021 1.40819i
\(647\) −4134.32 + 7160.85i −0.251216 + 0.435119i −0.963861 0.266406i \(-0.914164\pi\)
0.712645 + 0.701525i \(0.247497\pi\)
\(648\) 0 0
\(649\) 112.084 64.7120i 0.00677920 0.00391397i
\(650\) −27655.6 −1.66883
\(651\) 0 0
\(652\) 12661.1 0.760504
\(653\) 10492.9 6058.07i 0.628818 0.363048i −0.151476 0.988461i \(-0.548403\pi\)
0.780294 + 0.625413i \(0.215069\pi\)
\(654\) 0 0
\(655\) 365.516 633.092i 0.0218044 0.0377663i
\(656\) −4029.60 6979.48i −0.239832 0.415401i
\(657\) 0 0
\(658\) 0 0
\(659\) 11814.0i 0.698343i −0.937059 0.349171i \(-0.886463\pi\)
0.937059 0.349171i \(-0.113537\pi\)
\(660\) 0 0
\(661\) 23198.2 + 13393.5i 1.36506 + 0.788119i 0.990292 0.138999i \(-0.0443886\pi\)
0.374769 + 0.927118i \(0.377722\pi\)
\(662\) 14881.9 + 8592.05i 0.873716 + 0.504440i
\(663\) 0 0
\(664\) 13883.4i 0.811416i
\(665\) 0 0
\(666\) 0 0
\(667\) 1220.05 + 2113.19i 0.0708253 + 0.122673i
\(668\) −10100.9 + 17495.2i −0.585051 + 1.01334i
\(669\) 0 0
\(670\) 637.895 368.289i 0.0367821 0.0212362i
\(671\) 248.730 0.0143102
\(672\) 0 0
\(673\) 3317.25 0.190001 0.0950004 0.995477i \(-0.469715\pi\)
0.0950004 + 0.995477i \(0.469715\pi\)
\(674\) −25718.6 + 14848.6i −1.46980 + 0.848587i
\(675\) 0 0
\(676\) −4022.67 + 6967.48i −0.228873 + 0.396420i
\(677\) 14873.9 + 25762.4i 0.844388 + 1.46252i 0.886152 + 0.463395i \(0.153369\pi\)
−0.0417637 + 0.999128i \(0.513298\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 997.235i 0.0562385i
\(681\) 0 0
\(682\) 124.242 + 71.7314i 0.00697579 + 0.00402747i
\(683\) −16830.2 9716.94i −0.942886 0.544375i −0.0520221 0.998646i \(-0.516567\pi\)
−0.890864 + 0.454270i \(0.849900\pi\)
\(684\) 0 0
\(685\) 261.537i 0.0145881i
\(686\) 0 0
\(687\) 0 0
\(688\) −12928.6 22393.0i −0.716423 1.24088i
\(689\) 9595.58 16620.0i 0.530570 0.918974i
\(690\) 0 0
\(691\) 2798.73 1615.85i 0.154079 0.0889578i −0.420978 0.907071i \(-0.638313\pi\)
0.575057 + 0.818113i \(0.304980\pi\)
\(692\) −15469.5 −0.849798
\(693\) 0 0
\(694\) −13278.5 −0.726290
\(695\) −1254.62 + 724.354i −0.0684754 + 0.0395343i
\(696\) 0 0
\(697\) −5289.39 + 9161.49i −0.287446 + 0.497871i
\(698\) −1621.08 2807.80i −0.0879067 0.152259i
\(699\) 0 0
\(700\) 0 0
\(701\) 6802.40i 0.366509i −0.983065 0.183255i \(-0.941337\pi\)
0.983065 0.183255i \(-0.0586633\pi\)
\(702\) 0 0
\(703\) −23646.0 13652.0i −1.26860 0.732427i
\(704\) 21.0977 + 12.1808i 0.00112948 + 0.000652103i
\(705\) 0 0
\(706\) 36473.6i 1.94433i
\(707\) 0 0
\(708\) 0 0
\(709\) 3508.04 + 6076.11i 0.185821 + 0.321852i 0.943853 0.330366i \(-0.107172\pi\)
−0.758032 + 0.652218i \(0.773839\pi\)
\(710\) 274.732 475.849i 0.0145218 0.0251525i
\(711\) 0 0
\(712\) 5472.85 3159.75i 0.288067 0.166316i
\(713\) 14706.6 0.772463
\(714\) 0 0
\(715\) 19.7253 0.00103173
\(716\) 15192.3 8771.27i 0.792964 0.457818i
\(717\) 0 0
\(718\) −7566.53 + 13105.6i −0.393287 + 0.681194i
\(719\) 9045.55 + 15667.3i 0.469182 + 0.812647i 0.999379 0.0352269i \(-0.0112154\pi\)
−0.530197 + 0.847874i \(0.677882\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6254.26i 0.322382i
\(723\) 0 0
\(724\) 1331.93 + 768.988i 0.0683711 + 0.0394741i
\(725\) −1923.20 1110.36i −0.0985185 0.0568797i
\(726\) 0 0
\(727\) 35637.5i 1.81805i −0.416745 0.909024i \(-0.636829\pi\)
0.416745 0.909024i \(-0.363171\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 208.597 + 361.301i 0.0105761 + 0.0183183i
\(731\) −16970.5 + 29393.8i −0.858656 + 1.48724i
\(732\) 0 0
\(733\) −10723.2 + 6191.06i −0.540343 + 0.311967i −0.745218 0.666821i \(-0.767655\pi\)
0.204875 + 0.978788i \(0.434321\pi\)
\(734\) 1942.50 0.0976823
\(735\) 0 0
\(736\) 26605.7 1.33247
\(737\) 76.9262 44.4134i 0.00384479 0.00221979i
\(738\) 0 0
\(739\) 5933.76 10277.6i 0.295368 0.511592i −0.679703 0.733488i \(-0.737891\pi\)
0.975070 + 0.221896i \(0.0712245\pi\)
\(740\) −799.039 1383.98i −0.0396936 0.0687513i
\(741\) 0 0
\(742\) 0 0
\(743\) 6713.61i 0.331492i 0.986168 + 0.165746i \(0.0530032\pi\)
−0.986168 + 0.165746i \(0.946997\pi\)
\(744\) 0 0
\(745\) 1652.79 + 954.240i 0.0812801 + 0.0469271i
\(746\) 6964.03 + 4020.68i 0.341784 + 0.197329i
\(747\) 0 0
\(748\) 188.441i 0.00921136i
\(749\) 0 0
\(750\) 0 0
\(751\) 13144.8 + 22767.5i 0.638696 + 1.10625i 0.985719 + 0.168397i \(0.0538592\pi\)
−0.347023 + 0.937857i \(0.612807\pi\)
\(752\) 20681.5 35821.3i 1.00289 1.73706i
\(753\) 0 0
\(754\) −3444.37 + 1988.61i −0.166362 + 0.0960489i
\(755\) 704.021 0.0339363
\(756\) 0 0
\(757\) −37236.6 −1.78783 −0.893915 0.448237i \(-0.852052\pi\)
−0.893915 + 0.448237i \(0.852052\pi\)
\(758\) 9085.18 5245.33i 0.435341 0.251344i
\(759\) 0 0
\(760\) 342.989 594.075i 0.0163704 0.0283544i
\(761\) −6028.60 10441.8i −0.287170 0.497393i 0.685963 0.727636i \(-0.259381\pi\)
−0.973133 + 0.230243i \(0.926048\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 17205.5i 0.814754i
\(765\) 0 0
\(766\) 21924.6 + 12658.2i 1.03416 + 0.597074i
\(767\) 18728.0 + 10812.6i 0.881652 + 0.509022i
\(768\) 0 0
\(769\) 17232.1i 0.808070i 0.914744 + 0.404035i \(0.132393\pi\)
−0.914744 + 0.404035i \(0.867607\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1758.14 + 3045.19i 0.0819649 + 0.141967i
\(773\) 5556.40 9623.98i 0.258538 0.447801i −0.707312 0.706901i \(-0.750093\pi\)
0.965851 + 0.259100i \(0.0834259\pi\)
\(774\) 0 0
\(775\) −11591.2 + 6692.20i −0.537250 + 0.310182i
\(776\) −19357.4 −0.895477
\(777\) 0 0
\(778\) 35639.6 1.64234
\(779\) 6302.01 3638.47i 0.289850 0.167345i
\(780\) 0 0
\(781\) 33.1310 57.3845i 0.00151795 0.00262917i
\(782\) −25481.5 44135.2i −1.16524 2.01825i
\(783\) 0 0
\(784\) 0 0
\(785\) 1370.39i 0.0623074i
\(786\) 0 0
\(787\) −28578.1 16499.6i −1.29441 0.747328i −0.314978 0.949099i \(-0.601997\pi\)
−0.979433 + 0.201771i \(0.935330\pi\)
\(788\) −14717.2 8497.00i −0.665330 0.384128i
\(789\) 0 0
\(790\) 2636.52i 0.118738i
\(791\) 0 0
\(792\) 0 0
\(793\) 20779.9 + 35991.8i 0.930537 + 1.61174i
\(794\) −4574.19 + 7922.73i −0.204448 + 0.354115i
\(795\) 0 0
\(796\) −16512.0 + 9533.19i −0.735240 + 0.424491i
\(797\) 17247.7 0.766554 0.383277 0.923634i \(-0.374796\pi\)
0.383277 + 0.923634i \(0.374796\pi\)
\(798\) 0 0
\(799\) −54294.3 −2.40400
\(800\) −20969.7 + 12106.9i −0.926739 + 0.535053i
\(801\) 0 0
\(802\) 9259.58 16038.1i 0.407690 0.706139i
\(803\) 25.1555 + 43.5707i 0.00110550 + 0.00191479i
\(804\) 0 0
\(805\) 0 0
\(806\) 23970.9i 1.04757i
\(807\) 0 0
\(808\) −12166.2 7024.15i −0.529709 0.305828i
\(809\) −19029.1 10986.5i −0.826981 0.477458i 0.0258371 0.999666i \(-0.491775\pi\)
−0.852818 + 0.522209i \(0.825108\pi\)
\(810\) 0 0
\(811\) 30078.0i 1.30232i 0.758941 + 0.651159i \(0.225717\pi\)
−0.758941 + 0.651159i \(0.774283\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −254.213 440.311i −0.0109462 0.0189593i
\(815\) 1111.34 1924.90i 0.0477652 0.0827317i
\(816\) 0 0
\(817\) 20219.4 11673.7i 0.865837 0.499891i
\(818\) −28692.2 −1.22640
\(819\) 0 0
\(820\) 425.911 0.0181384
\(821\) −33411.0 + 19289.9i −1.42028 + 0.820001i −0.996323 0.0856788i \(-0.972694\pi\)
−0.423961 + 0.905680i \(0.639361\pi\)
\(822\) 0 0
\(823\) 14026.1 24293.9i 0.594068 1.02896i −0.399610 0.916685i \(-0.630854\pi\)
0.993678 0.112271i \(-0.0358124\pi\)
\(824\) −8727.47 15116.4i −0.368976 0.639084i
\(825\) 0 0
\(826\) 0 0
\(827\) 2038.04i 0.0856948i −0.999082 0.0428474i \(-0.986357\pi\)
0.999082 0.0428474i \(-0.0136429\pi\)
\(828\) 0 0
\(829\) −34603.1 19978.1i −1.44972 0.836995i −0.451254 0.892396i \(-0.649023\pi\)
−0.998464 + 0.0554004i \(0.982356\pi\)
\(830\) 3307.38 + 1909.52i 0.138314 + 0.0798558i
\(831\) 0 0
\(832\) 4070.52i 0.169615i
\(833\) 0 0
\(834\) 0 0
\(835\) 1773.22 + 3071.31i 0.0734910 + 0.127290i
\(836\) −64.8125 + 112.259i −0.00268133 + 0.00464419i
\(837\) 0 0
\(838\) 18564.7 10718.3i 0.765283 0.441836i
\(839\) −24277.8 −0.999004 −0.499502 0.866313i \(-0.666484\pi\)
−0.499502 + 0.866313i \(0.666484\pi\)
\(840\) 0 0
\(841\) 24069.6 0.986905
\(842\) −39313.9 + 22697.9i −1.60908 + 0.929002i
\(843\) 0 0
\(844\) 7113.28 12320.6i 0.290106 0.502478i
\(845\) 706.187 + 1223.15i 0.0287498 + 0.0497961i
\(846\) 0 0
\(847\) 0 0
\(848\) 24519.7i 0.992937i
\(849\) 0 0
\(850\) 40167.2 + 23190.5i 1.62085 + 0.935798i
\(851\) −45136.9 26059.8i −1.81818 1.04973i
\(852\) 0 0
\(853\) 28395.8i 1.13981i 0.821712 + 0.569903i \(0.193019\pi\)
−0.821712 + 0.569903i \(0.806981\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 937.373 + 1623.58i 0.0374285 + 0.0648280i
\(857\) 17496.3 30304.5i 0.697388 1.20791i −0.271981 0.962303i \(-0.587679\pi\)
0.969369 0.245609i \(-0.0789879\pi\)
\(858\) 0 0
\(859\) −8360.94 + 4827.19i −0.332097 + 0.191737i −0.656772 0.754089i \(-0.728079\pi\)
0.324675 + 0.945826i \(0.394745\pi\)
\(860\) 1366.50 0.0541828
\(861\) 0 0
\(862\) −12217.7 −0.482757
\(863\) 36502.8 21074.9i 1.43983 0.831284i 0.441989 0.897021i \(-0.354273\pi\)
0.997837 + 0.0657370i \(0.0209398\pi\)
\(864\) 0 0
\(865\) −1357.84 + 2351.85i −0.0533735 + 0.0924456i
\(866\) −13304.5 23044.0i −0.522060 0.904235i
\(867\) 0 0
\(868\) 0 0
\(869\) 317.949i 0.0124116i
\(870\) 0 0
\(871\) 12853.4 + 7420.93i 0.500025 + 0.288690i
\(872\) 204.491 + 118.063i 0.00794144 + 0.00458499i
\(873\) 0 0
\(874\) 35056.4i 1.35675i
\(875\) 0 0
\(876\) 0 0
\(877\) 365.254 + 632.639i 0.0140636 + 0.0243588i 0.872972 0.487771i \(-0.162190\pi\)
−0.858908 + 0.512130i \(0.828857\pi\)
\(878\) 8332.50 14432.3i 0.320283 0.554746i
\(879\) 0 0
\(880\) 21.8257 12.6011i 0.000836073 0.000482707i
\(881\) 36187.9 1.38388 0.691942 0.721953i \(-0.256755\pi\)
0.691942 + 0.721953i \(0.256755\pi\)
\(882\) 0 0
\(883\) −38996.6 −1.48623 −0.743115 0.669164i \(-0.766652\pi\)
−0.743115 + 0.669164i \(0.766652\pi\)
\(884\) 27267.9 15743.1i 1.03746 0.598980i
\(885\) 0 0
\(886\) 5327.70 9227.85i 0.202018 0.349905i
\(887\) −15147.0 26235.4i −0.573380 0.993123i −0.996216 0.0869165i \(-0.972299\pi\)
0.422836 0.906206i \(-0.361035\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1738.36i 0.0654720i
\(891\) 0 0
\(892\) −5084.75 2935.68i −0.190863 0.110195i
\(893\) 32344.3 + 18674.0i 1.21205 + 0.699778i
\(894\) 0 0
\(895\) 3079.62i 0.115017i
\(896\) 0 0
\(897\) 0 0
\(898\) 15672.1 + 27144.8i 0.582387 + 1.00872i
\(899\) −962.420 + 1666.96i −0.0357047 + 0.0618423i
\(900\) 0 0
\(901\) −27873.4 + 16092.7i −1.03063 + 0.595034i
\(902\) 135.503 0.00500196
\(903\) 0 0
\(904\) −12437.1 −0.457579
\(905\) 233.822 134.997i 0.00858840 0.00495852i
\(906\) 0 0
\(907\) 26811.7 46439.3i 0.981554 1.70010i 0.325205 0.945644i \(-0.394567\pi\)
0.656349 0.754458i \(-0.272100\pi\)
\(908\) −12539.3 21718.7i −0.458293 0.793787i
\(909\) 0 0
\(910\) 0 0
\(911\) 29609.2i 1.07683i 0.842678 + 0.538417i \(0.180977\pi\)
−0.842678 + 0.538417i \(0.819023\pi\)
\(912\) 0 0
\(913\) 398.850 + 230.276i 0.0144578 + 0.00834724i
\(914\) −37608.1 21713.1i −1.36101 0.785782i
\(915\) 0 0
\(916\) 9168.53i 0.330717i
\(917\) 0 0
\(918\) 0 0
\(919\) 22783.3 + 39461.8i 0.817792 + 1.41646i 0.907306 + 0.420472i \(0.138135\pi\)
−0.0895134 + 0.995986i \(0.528531\pi\)
\(920\) 654.718 1134.00i 0.0234624 0.0406381i
\(921\) 0 0
\(922\) 24079.2 13902.1i 0.860094 0.496576i
\(923\) 11071.6 0.394827
\(924\) 0 0
\(925\) 47433.8 1.68607
\(926\) −7999.23 + 4618.35i −0.283878 + 0.163897i
\(927\) 0 0
\(928\) −1741.12 + 3015.70i −0.0615894 + 0.106676i
\(929\) 19829.7 + 34346.0i 0.700313 + 1.21298i 0.968357 + 0.249571i \(0.0802895\pi\)
−0.268044 + 0.963407i \(0.586377\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 17008.6i 0.597783i
\(933\) 0 0
\(934\) 7422.62 + 4285.45i 0.260038 + 0.150133i
\(935\) −28.6491 16.5406i −0.00100206 0.000578540i
\(936\) 0 0
\(937\) 1645.64i 0.0573753i −0.999588 0.0286877i \(-0.990867\pi\)
0.999588 0.0286877i \(-0.00913282\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1092.97 + 1893.08i 0.0379242 + 0.0656866i
\(941\) 16471.6 28529.6i 0.570626 0.988352i −0.425876 0.904781i \(-0.640034\pi\)
0.996502 0.0835710i \(-0.0266325\pi\)
\(942\) 0 0
\(943\) 12029.7 6945.32i 0.415418 0.239842i
\(944\) 27629.5 0.952611
\(945\) 0 0
\(946\) 434.750 0.0149418
\(947\) −19826.4 + 11446.8i −0.680328 + 0.392788i −0.799979 0.600028i \(-0.795156\pi\)
0.119650 + 0.992816i \(0.461823\pi\)
\(948\) 0 0
\(949\) −4203.18 + 7280.13i −0.143774 + 0.249023i
\(950\) −15952.3 27630.2i −0.544802 0.943624i
\(951\) 0 0
\(952\) 0 0
\(953\) 31350.3i 1.06562i −0.846234 0.532811i \(-0.821136\pi\)
0.846234 0.532811i \(-0.178864\pi\)
\(954\) 0 0
\(955\) 2615.79 + 1510.22i 0.0886333 + 0.0511725i
\(956\) −10430.7 6022.18i −0.352881 0.203736i
\(957\) 0 0
\(958\) 10430.6i 0.351773i
\(959\) 0 0
\(960\) 0 0
\(961\) −9094.95 15752.9i −0.305292 0.528781i
\(962\) 42476.0 73570.5i 1.42358 2.46571i
\(963\) 0 0
\(964\) 10317.6 5956.85i 0.344716 0.199022i
\(965\) 617.289 0.0205920
\(966\) 0 0
\(967\) −1272.04 −0.0423021 −0.0211511 0.999776i \(-0.506733\pi\)
−0.0211511 + 0.999776i \(0.506733\pi\)
\(968\) −12893.0 + 7443.77i −0.428095 + 0.247161i
\(969\) 0 0
\(970\) 2662.41 4611.43i 0.0881287 0.152643i
\(971\) −11659.6 20195.0i −0.385350 0.667445i 0.606468 0.795108i \(-0.292586\pi\)
−0.991818 + 0.127663i \(0.959253\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 949.512i 0.0312365i
\(975\) 0 0
\(976\) 45985.2 + 26549.5i 1.50814 + 0.870728i
\(977\) −25918.0 14963.7i −0.848710 0.490003i 0.0115056 0.999934i \(-0.496338\pi\)
−0.860215 + 0.509931i \(0.829671\pi\)
\(978\) 0 0
\(979\) 209.636i 0.00684372i
\(980\) 0 0
\(981\) 0 0
\(982\) −35513.1 61510.5i −1.15404 1.99886i
\(983\) −17281.3 + 29932.1i −0.560720 + 0.971195i 0.436714 + 0.899600i \(0.356142\pi\)
−0.997434 + 0.0715947i \(0.977191\pi\)
\(984\) 0 0
\(985\) −2583.64 + 1491.66i −0.0835751 + 0.0482521i
\(986\) 6670.17 0.215438
\(987\) 0 0
\(988\) −21658.8 −0.697427
\(989\) 38596.1 22283.4i 1.24093 0.716453i
\(990\) 0 0
\(991\) −11458.6 + 19846.9i −0.367300 + 0.636183i −0.989142 0.146960i \(-0.953051\pi\)
0.621842 + 0.783143i \(0.286385\pi\)
\(992\) 10493.8 + 18175.8i 0.335865 + 0.581736i
\(993\) 0 0
\(994\) 0 0
\(995\) 3347.14i 0.106645i
\(996\) 0 0
\(997\) 28270.4 + 16321.9i 0.898027 + 0.518476i 0.876560 0.481294i \(-0.159833\pi\)
0.0214673 + 0.999770i \(0.493166\pi\)
\(998\) −35659.6 20588.1i −1.13105 0.653010i
\(999\) 0 0
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 441.4.p.d.80.8 48
3.2 odd 2 inner 441.4.p.d.80.17 48
7.2 even 3 inner 441.4.p.d.215.17 48
7.3 odd 6 441.4.c.b.440.11 24
7.4 even 3 441.4.c.b.440.13 yes 24
7.5 odd 6 inner 441.4.p.d.215.18 48
7.6 odd 2 inner 441.4.p.d.80.7 48
21.2 odd 6 inner 441.4.p.d.215.8 48
21.5 even 6 inner 441.4.p.d.215.7 48
21.11 odd 6 441.4.c.b.440.12 yes 24
21.17 even 6 441.4.c.b.440.14 yes 24
21.20 even 2 inner 441.4.p.d.80.18 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.4.c.b.440.11 24 7.3 odd 6
441.4.c.b.440.12 yes 24 21.11 odd 6
441.4.c.b.440.13 yes 24 7.4 even 3
441.4.c.b.440.14 yes 24 21.17 even 6
441.4.p.d.80.7 48 7.6 odd 2 inner
441.4.p.d.80.8 48 1.1 even 1 trivial
441.4.p.d.80.17 48 3.2 odd 2 inner
441.4.p.d.80.18 48 21.20 even 2 inner
441.4.p.d.215.7 48 21.5 even 6 inner
441.4.p.d.215.8 48 21.2 odd 6 inner
441.4.p.d.215.17 48 7.2 even 3 inner
441.4.p.d.215.18 48 7.5 odd 6 inner