Properties

Label 504.4.s.f.289.2
Level $504$
Weight $4$
Character 504.289
Analytic conductor $29.737$
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] = [504,4,Mod(289,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.s (of order \(3\), degree \(2\), 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: \(29.7369626429\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{505})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.2
Root \(5.86805 - 10.1638i\) of defining polynomial
Character \(\chi\) \(=\) 504.289
Dual form 504.4.s.f.361.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(7.86805 - 13.6279i) q^{5} +(11.2361 + 14.7224i) q^{7} +O(q^{10})\) \(q+(7.86805 - 13.6279i) q^{5} +(11.2361 + 14.7224i) q^{7} +(29.3403 + 50.8188i) q^{11} -20.7361 q^{13} +(21.4722 + 37.1910i) q^{17} +(-68.5764 + 118.778i) q^{19} +(14.5278 - 25.1629i) q^{23} +(-61.3125 - 106.196i) q^{25} -8.68051 q^{29} +(101.236 + 175.346i) q^{31} +(289.042 - 37.2872i) q^{35} +(7.63195 - 13.2189i) q^{37} -117.250 q^{41} -101.875 q^{43} +(294.250 - 509.656i) q^{47} +(-90.5000 + 330.846i) q^{49} +(202.340 + 350.464i) q^{53} +923.403 q^{55} +(5.43738 + 9.41783i) q^{59} +(447.305 - 774.756i) q^{61} +(-163.153 + 282.589i) q^{65} +(351.868 + 609.453i) q^{67} -1161.94 q^{71} +(558.187 + 966.809i) q^{73} +(-418.506 + 1002.97i) q^{77} +(-69.1665 + 119.800i) q^{79} +894.319 q^{83} +675.778 q^{85} +(-340.903 + 590.461i) q^{89} +(-232.993 - 305.286i) q^{91} +(1079.12 + 1869.10i) q^{95} +246.514 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 9 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 9 q^{5} + 5 q^{11} - 38 q^{13} - 4 q^{17} - 117 q^{19} + 148 q^{23} - 43 q^{25} + 190 q^{29} + 360 q^{31} + 482 q^{35} + 53 q^{37} + 340 q^{41} + 806 q^{43} + 368 q^{47} - 362 q^{49} + 697 q^{53} + 2570 q^{55} - 585 q^{59} + 1160 q^{61} - 338 q^{65} - 233 q^{67} - 1232 q^{71} + 817 q^{73} + 1135 q^{77} + 802 q^{79} + 566 q^{83} + 1984 q^{85} - 1858 q^{89} - 505 q^{91} + 2294 q^{95} + 3458 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 7.86805 13.6279i 0.703740 1.21891i −0.263404 0.964685i \(-0.584845\pi\)
0.967144 0.254228i \(-0.0818213\pi\)
\(6\) 0 0
\(7\) 11.2361 + 14.7224i 0.606693 + 0.794937i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 29.3403 + 50.8188i 0.804220 + 1.39295i 0.916816 + 0.399309i \(0.130750\pi\)
−0.112596 + 0.993641i \(0.535917\pi\)
\(12\) 0 0
\(13\) −20.7361 −0.442397 −0.221198 0.975229i \(-0.570997\pi\)
−0.221198 + 0.975229i \(0.570997\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 21.4722 + 37.1910i 0.306340 + 0.530596i 0.977559 0.210663i \(-0.0675623\pi\)
−0.671219 + 0.741259i \(0.734229\pi\)
\(18\) 0 0
\(19\) −68.5764 + 118.778i −0.828026 + 1.43418i 0.0715584 + 0.997436i \(0.477203\pi\)
−0.899584 + 0.436747i \(0.856131\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 14.5278 25.1629i 0.131707 0.228123i −0.792628 0.609706i \(-0.791288\pi\)
0.924335 + 0.381583i \(0.124621\pi\)
\(24\) 0 0
\(25\) −61.3125 106.196i −0.490500 0.849570i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.68051 −0.0555838 −0.0277919 0.999614i \(-0.508848\pi\)
−0.0277919 + 0.999614i \(0.508848\pi\)
\(30\) 0 0
\(31\) 101.236 + 175.346i 0.586534 + 1.01591i 0.994682 + 0.102991i \(0.0328411\pi\)
−0.408149 + 0.912915i \(0.633826\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 289.042 37.2872i 1.39591 0.180077i
\(36\) 0 0
\(37\) 7.63195 13.2189i 0.0339104 0.0587345i −0.848572 0.529080i \(-0.822537\pi\)
0.882483 + 0.470345i \(0.155871\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −117.250 −0.446618 −0.223309 0.974748i \(-0.571686\pi\)
−0.223309 + 0.974748i \(0.571686\pi\)
\(42\) 0 0
\(43\) −101.875 −0.361297 −0.180648 0.983548i \(-0.557820\pi\)
−0.180648 + 0.983548i \(0.557820\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 294.250 509.656i 0.913207 1.58172i 0.103703 0.994608i \(-0.466931\pi\)
0.809505 0.587113i \(-0.199736\pi\)
\(48\) 0 0
\(49\) −90.5000 + 330.846i −0.263848 + 0.964564i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 202.340 + 350.464i 0.524407 + 0.908300i 0.999596 + 0.0284159i \(0.00904629\pi\)
−0.475189 + 0.879884i \(0.657620\pi\)
\(54\) 0 0
\(55\) 923.403 2.26385
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.43738 + 9.41783i 0.0119981 + 0.0207813i 0.871962 0.489573i \(-0.162847\pi\)
−0.859964 + 0.510355i \(0.829514\pi\)
\(60\) 0 0
\(61\) 447.305 774.756i 0.938879 1.62619i 0.171311 0.985217i \(-0.445200\pi\)
0.767567 0.640968i \(-0.221467\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −163.153 + 282.589i −0.311332 + 0.539243i
\(66\) 0 0
\(67\) 351.868 + 609.453i 0.641604 + 1.11129i 0.985075 + 0.172128i \(0.0550642\pi\)
−0.343470 + 0.939164i \(0.611602\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1161.94 −1.94222 −0.971108 0.238640i \(-0.923298\pi\)
−0.971108 + 0.238640i \(0.923298\pi\)
\(72\) 0 0
\(73\) 558.187 + 966.809i 0.894943 + 1.55009i 0.833875 + 0.551954i \(0.186118\pi\)
0.0610689 + 0.998134i \(0.480549\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −418.506 + 1002.97i −0.619393 + 1.48440i
\(78\) 0 0
\(79\) −69.1665 + 119.800i −0.0985042 + 0.170614i −0.911066 0.412261i \(-0.864739\pi\)
0.812561 + 0.582875i \(0.198072\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 894.319 1.18270 0.591351 0.806414i \(-0.298595\pi\)
0.591351 + 0.806414i \(0.298595\pi\)
\(84\) 0 0
\(85\) 675.778 0.862334
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −340.903 + 590.461i −0.406018 + 0.703244i −0.994439 0.105310i \(-0.966416\pi\)
0.588421 + 0.808555i \(0.299750\pi\)
\(90\) 0 0
\(91\) −232.993 305.286i −0.268399 0.351678i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1079.12 + 1869.10i 1.16543 + 2.01858i
\(96\) 0 0
\(97\) 246.514 0.258039 0.129019 0.991642i \(-0.458817\pi\)
0.129019 + 0.991642i \(0.458817\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −544.972 943.919i −0.536898 0.929935i −0.999069 0.0431441i \(-0.986263\pi\)
0.462171 0.886791i \(-0.347071\pi\)
\(102\) 0 0
\(103\) 15.1182 26.1855i 0.0144625 0.0250498i −0.858704 0.512473i \(-0.828730\pi\)
0.873166 + 0.487423i \(0.162063\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 943.812 1634.73i 0.852727 1.47697i −0.0260107 0.999662i \(-0.508280\pi\)
0.878738 0.477305i \(-0.158386\pi\)
\(108\) 0 0
\(109\) 482.298 + 835.365i 0.423815 + 0.734069i 0.996309 0.0858401i \(-0.0273574\pi\)
−0.572494 + 0.819909i \(0.694024\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −119.806 −0.0997378 −0.0498689 0.998756i \(-0.515880\pi\)
−0.0498689 + 0.998756i \(0.515880\pi\)
\(114\) 0 0
\(115\) −228.611 395.966i −0.185375 0.321078i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −306.277 + 734.004i −0.235936 + 0.565429i
\(120\) 0 0
\(121\) −1056.20 + 1829.39i −0.793540 + 1.37445i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 37.3745 0.0267430
\(126\) 0 0
\(127\) 683.722 0.477721 0.238860 0.971054i \(-0.423226\pi\)
0.238860 + 0.971054i \(0.423226\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 130.119 225.372i 0.0867825 0.150312i −0.819367 0.573269i \(-0.805675\pi\)
0.906149 + 0.422958i \(0.139008\pi\)
\(132\) 0 0
\(133\) −2519.23 + 324.988i −1.64244 + 0.211880i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −365.847 633.666i −0.228149 0.395166i 0.729110 0.684396i \(-0.239934\pi\)
−0.957260 + 0.289230i \(0.906601\pi\)
\(138\) 0 0
\(139\) 2287.74 1.39599 0.697997 0.716101i \(-0.254075\pi\)
0.697997 + 0.716101i \(0.254075\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −608.403 1053.78i −0.355784 0.616237i
\(144\) 0 0
\(145\) −68.2987 + 118.297i −0.0391166 + 0.0677519i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1719.60 2978.43i 0.945469 1.63760i 0.190659 0.981656i \(-0.438937\pi\)
0.754810 0.655944i \(-0.227729\pi\)
\(150\) 0 0
\(151\) −1425.88 2469.70i −0.768455 1.33100i −0.938401 0.345549i \(-0.887693\pi\)
0.169946 0.985453i \(-0.445641\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3186.12 1.65107
\(156\) 0 0
\(157\) −1220.12 2113.32i −0.620232 1.07427i −0.989442 0.144928i \(-0.953705\pi\)
0.369210 0.929346i \(-0.379628\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 533.695 68.8482i 0.261249 0.0337019i
\(162\) 0 0
\(163\) −120.695 + 209.050i −0.0579974 + 0.100454i −0.893566 0.448931i \(-0.851805\pi\)
0.835569 + 0.549386i \(0.185138\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −326.445 −0.151264 −0.0756319 0.997136i \(-0.524097\pi\)
−0.0756319 + 0.997136i \(0.524097\pi\)
\(168\) 0 0
\(169\) −1767.01 −0.804285
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −797.847 + 1381.91i −0.350631 + 0.607311i −0.986360 0.164601i \(-0.947366\pi\)
0.635729 + 0.771912i \(0.280700\pi\)
\(174\) 0 0
\(175\) 874.555 2095.90i 0.377772 0.905344i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1441.43 + 2496.63i 0.601886 + 1.04250i 0.992535 + 0.121957i \(0.0389171\pi\)
−0.390650 + 0.920539i \(0.627750\pi\)
\(180\) 0 0
\(181\) 1128.60 0.463470 0.231735 0.972779i \(-0.425560\pi\)
0.231735 + 0.972779i \(0.425560\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −120.097 208.014i −0.0477282 0.0826677i
\(186\) 0 0
\(187\) −1260.00 + 2182.38i −0.492729 + 0.853432i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1102.62 1909.80i 0.417713 0.723500i −0.577996 0.816039i \(-0.696165\pi\)
0.995709 + 0.0925397i \(0.0294985\pi\)
\(192\) 0 0
\(193\) −1897.68 3286.88i −0.707761 1.22588i −0.965686 0.259714i \(-0.916372\pi\)
0.257924 0.966165i \(-0.416962\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3701.25 −1.33859 −0.669297 0.742995i \(-0.733405\pi\)
−0.669297 + 0.742995i \(0.733405\pi\)
\(198\) 0 0
\(199\) 1800.19 + 3118.03i 0.641268 + 1.11071i 0.985150 + 0.171696i \(0.0549246\pi\)
−0.343882 + 0.939013i \(0.611742\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −97.5351 127.798i −0.0337223 0.0441856i
\(204\) 0 0
\(205\) −922.528 + 1597.87i −0.314303 + 0.544389i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8048.19 −2.66366
\(210\) 0 0
\(211\) −4137.64 −1.34998 −0.674992 0.737825i \(-0.735853\pi\)
−0.674992 + 0.737825i \(0.735853\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −801.556 + 1388.34i −0.254259 + 0.440389i
\(216\) 0 0
\(217\) −1444.02 + 3460.65i −0.451735 + 1.08260i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −445.250 771.195i −0.135524 0.234734i
\(222\) 0 0
\(223\) 4261.62 1.27973 0.639864 0.768488i \(-0.278991\pi\)
0.639864 + 0.768488i \(0.278991\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −767.063 1328.59i −0.224281 0.388466i 0.731823 0.681495i \(-0.238670\pi\)
−0.956103 + 0.293029i \(0.905337\pi\)
\(228\) 0 0
\(229\) 287.452 497.881i 0.0829491 0.143672i −0.821566 0.570113i \(-0.806899\pi\)
0.904515 + 0.426441i \(0.140233\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2076.07 + 3595.86i −0.583724 + 1.01104i 0.411309 + 0.911496i \(0.365072\pi\)
−0.995033 + 0.0995443i \(0.968261\pi\)
\(234\) 0 0
\(235\) −4630.35 8019.99i −1.28532 2.22624i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4548.36 −1.23100 −0.615500 0.788137i \(-0.711046\pi\)
−0.615500 + 0.788137i \(0.711046\pi\)
\(240\) 0 0
\(241\) −3504.39 6069.79i −0.936672 1.62236i −0.771625 0.636077i \(-0.780556\pi\)
−0.165046 0.986286i \(-0.552777\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3796.66 + 3836.43i 0.990039 + 1.00041i
\(246\) 0 0
\(247\) 1422.01 2462.99i 0.366316 0.634478i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −682.819 −0.171710 −0.0858548 0.996308i \(-0.527362\pi\)
−0.0858548 + 0.996308i \(0.527362\pi\)
\(252\) 0 0
\(253\) 1705.00 0.423685
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 851.276 1474.45i 0.206619 0.357875i −0.744028 0.668148i \(-0.767087\pi\)
0.950647 + 0.310273i \(0.100421\pi\)
\(258\) 0 0
\(259\) 280.368 36.1683i 0.0672634 0.00867718i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1462.12 2532.47i −0.342808 0.593760i 0.642145 0.766583i \(-0.278045\pi\)
−0.984953 + 0.172823i \(0.944711\pi\)
\(264\) 0 0
\(265\) 6368.09 1.47618
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 686.910 + 1189.76i 0.155694 + 0.269670i 0.933311 0.359068i \(-0.116905\pi\)
−0.777618 + 0.628738i \(0.783572\pi\)
\(270\) 0 0
\(271\) 390.756 676.809i 0.0875894 0.151709i −0.818902 0.573933i \(-0.805417\pi\)
0.906492 + 0.422224i \(0.138750\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3597.85 6231.65i 0.788939 1.36648i
\(276\) 0 0
\(277\) 1075.81 + 1863.36i 0.233355 + 0.404182i 0.958793 0.284105i \(-0.0916964\pi\)
−0.725439 + 0.688287i \(0.758363\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2388.25 0.507014 0.253507 0.967334i \(-0.418416\pi\)
0.253507 + 0.967334i \(0.418416\pi\)
\(282\) 0 0
\(283\) 1647.33 + 2853.25i 0.346019 + 0.599322i 0.985538 0.169453i \(-0.0542000\pi\)
−0.639519 + 0.768775i \(0.720867\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1317.43 1726.20i −0.270960 0.355033i
\(288\) 0 0
\(289\) 1534.39 2657.64i 0.312312 0.540940i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4355.37 0.868408 0.434204 0.900815i \(-0.357030\pi\)
0.434204 + 0.900815i \(0.357030\pi\)
\(294\) 0 0
\(295\) 171.126 0.0337741
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −301.250 + 521.780i −0.0582667 + 0.100921i
\(300\) 0 0
\(301\) −1144.68 1499.84i −0.219196 0.287208i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7038.84 12191.6i −1.32145 2.28882i
\(306\) 0 0
\(307\) −8189.87 −1.52254 −0.761271 0.648434i \(-0.775424\pi\)
−0.761271 + 0.648434i \(0.775424\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 582.651 + 1009.18i 0.106235 + 0.184004i 0.914242 0.405168i \(-0.132787\pi\)
−0.808007 + 0.589173i \(0.799454\pi\)
\(312\) 0 0
\(313\) 2892.26 5009.54i 0.522301 0.904652i −0.477362 0.878707i \(-0.658407\pi\)
0.999663 0.0259457i \(-0.00825969\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2283.69 + 3955.46i −0.404620 + 0.700822i −0.994277 0.106831i \(-0.965930\pi\)
0.589657 + 0.807654i \(0.299263\pi\)
\(318\) 0 0
\(319\) −254.688 441.133i −0.0447016 0.0774255i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5889.94 −1.01463
\(324\) 0 0
\(325\) 1271.38 + 2202.10i 0.216996 + 0.375847i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10809.6 1394.47i 1.81140 0.233677i
\(330\) 0 0
\(331\) −1770.31 + 3066.27i −0.293973 + 0.509177i −0.974746 0.223319i \(-0.928311\pi\)
0.680772 + 0.732495i \(0.261644\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11074.1 1.80609
\(336\) 0 0
\(337\) −9183.50 −1.48444 −0.742221 0.670155i \(-0.766227\pi\)
−0.742221 + 0.670155i \(0.766227\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5940.59 + 10289.4i −0.943404 + 1.63402i
\(342\) 0 0
\(343\) −5887.72 + 2385.03i −0.926842 + 0.375451i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2977.50 5157.18i −0.460636 0.797844i 0.538357 0.842717i \(-0.319045\pi\)
−0.998993 + 0.0448726i \(0.985712\pi\)
\(348\) 0 0
\(349\) 2816.11 0.431929 0.215964 0.976401i \(-0.430711\pi\)
0.215964 + 0.976401i \(0.430711\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −190.653 330.221i −0.0287463 0.0497900i 0.851294 0.524688i \(-0.175818\pi\)
−0.880041 + 0.474898i \(0.842485\pi\)
\(354\) 0 0
\(355\) −9142.23 + 15834.8i −1.36682 + 2.36739i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1497.38 + 2593.53i −0.220135 + 0.381285i −0.954849 0.297092i \(-0.903983\pi\)
0.734714 + 0.678377i \(0.237316\pi\)
\(360\) 0 0
\(361\) −5975.93 10350.6i −0.871254 1.50906i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 17567.4 2.51923
\(366\) 0 0
\(367\) 2325.01 + 4027.04i 0.330694 + 0.572779i 0.982648 0.185479i \(-0.0593838\pi\)
−0.651954 + 0.758259i \(0.726050\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2886.16 + 6916.79i −0.403887 + 0.967929i
\(372\) 0 0
\(373\) −2414.17 + 4181.47i −0.335123 + 0.580451i −0.983509 0.180862i \(-0.942111\pi\)
0.648385 + 0.761312i \(0.275445\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 180.000 0.0245901
\(378\) 0 0
\(379\) 13230.5 1.79315 0.896577 0.442887i \(-0.146046\pi\)
0.896577 + 0.442887i \(0.146046\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2204.87 + 3818.95i −0.294161 + 0.509502i −0.974789 0.223127i \(-0.928374\pi\)
0.680628 + 0.732629i \(0.261707\pi\)
\(384\) 0 0
\(385\) 10375.4 + 13594.7i 1.37346 + 1.79961i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 287.691 + 498.296i 0.0374975 + 0.0649476i 0.884165 0.467175i \(-0.154728\pi\)
−0.846668 + 0.532122i \(0.821395\pi\)
\(390\) 0 0
\(391\) 1247.78 0.161388
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1088.41 + 1885.18i 0.138643 + 0.240136i
\(396\) 0 0
\(397\) −3620.81 + 6271.43i −0.457741 + 0.792831i −0.998841 0.0481274i \(-0.984675\pi\)
0.541100 + 0.840958i \(0.318008\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1123.50 1945.96i 0.139912 0.242335i −0.787551 0.616250i \(-0.788651\pi\)
0.927463 + 0.373914i \(0.121985\pi\)
\(402\) 0 0
\(403\) −2099.24 3635.99i −0.259481 0.449434i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 895.693 0.109086
\(408\) 0 0
\(409\) 825.318 + 1429.49i 0.0997784 + 0.172821i 0.911593 0.411094i \(-0.134853\pi\)
−0.811814 + 0.583915i \(0.801520\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −77.5583 + 185.871i −0.00924066 + 0.0221456i
\(414\) 0 0
\(415\) 7036.55 12187.7i 0.832314 1.44161i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3178.64 0.370613 0.185307 0.982681i \(-0.440672\pi\)
0.185307 + 0.982681i \(0.440672\pi\)
\(420\) 0 0
\(421\) 8781.12 1.01655 0.508273 0.861196i \(-0.330284\pi\)
0.508273 + 0.861196i \(0.330284\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2633.03 4560.54i 0.300519 0.520514i
\(426\) 0 0
\(427\) 16432.3 2119.81i 1.86232 0.240246i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4543.35 + 7869.31i 0.507762 + 0.879469i 0.999960 + 0.00898583i \(0.00286032\pi\)
−0.492198 + 0.870483i \(0.663806\pi\)
\(432\) 0 0
\(433\) 3150.88 0.349703 0.174852 0.984595i \(-0.444055\pi\)
0.174852 + 0.984595i \(0.444055\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1992.53 + 3451.16i 0.218113 + 0.377783i
\(438\) 0 0
\(439\) −5165.69 + 8947.23i −0.561605 + 0.972729i 0.435751 + 0.900067i \(0.356483\pi\)
−0.997357 + 0.0726619i \(0.976851\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4136.03 + 7163.82i −0.443587 + 0.768315i −0.997953 0.0639584i \(-0.979627\pi\)
0.554366 + 0.832273i \(0.312961\pi\)
\(444\) 0 0
\(445\) 5364.48 + 9291.56i 0.571463 + 0.989802i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10111.1 −1.06275 −0.531374 0.847137i \(-0.678324\pi\)
−0.531374 + 0.847137i \(0.678324\pi\)
\(450\) 0 0
\(451\) −3440.14 5958.50i −0.359179 0.622117i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5993.59 + 773.192i −0.617547 + 0.0796655i
\(456\) 0 0
\(457\) 8920.37 15450.5i 0.913080 1.58150i 0.103392 0.994641i \(-0.467031\pi\)
0.809688 0.586860i \(-0.199636\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 15787.7 1.59503 0.797515 0.603299i \(-0.206148\pi\)
0.797515 + 0.603299i \(0.206148\pi\)
\(462\) 0 0
\(463\) −5960.87 −0.598326 −0.299163 0.954202i \(-0.596707\pi\)
−0.299163 + 0.954202i \(0.596707\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5529.15 + 9576.78i −0.547877 + 0.948951i 0.450542 + 0.892755i \(0.351231\pi\)
−0.998420 + 0.0561964i \(0.982103\pi\)
\(468\) 0 0
\(469\) −5019.01 + 12028.2i −0.494150 + 1.18425i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2989.03 5177.15i −0.290562 0.503268i
\(474\) 0 0
\(475\) 16818.3 1.62459
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5931.90 10274.3i −0.565836 0.980056i −0.996971 0.0777692i \(-0.975220\pi\)
0.431136 0.902287i \(-0.358113\pi\)
\(480\) 0 0
\(481\) −158.257 + 274.109i −0.0150019 + 0.0259840i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1939.59 3359.46i 0.181592 0.314527i
\(486\) 0 0
\(487\) 2962.48 + 5131.17i 0.275653 + 0.477445i 0.970300 0.241906i \(-0.0777727\pi\)
−0.694647 + 0.719351i \(0.744439\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9203.12 0.845888 0.422944 0.906156i \(-0.360997\pi\)
0.422944 + 0.906156i \(0.360997\pi\)
\(492\) 0 0
\(493\) −186.390 322.837i −0.0170275 0.0294925i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13055.7 17106.6i −1.17833 1.54394i
\(498\) 0 0
\(499\) 4903.80 8493.63i 0.439928 0.761978i −0.557755 0.830006i \(-0.688337\pi\)
0.997683 + 0.0680273i \(0.0216705\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10462.6 −0.927446 −0.463723 0.885980i \(-0.653487\pi\)
−0.463723 + 0.885980i \(0.653487\pi\)
\(504\) 0 0
\(505\) −17151.5 −1.51135
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2670.91 4626.16i 0.232586 0.402850i −0.725983 0.687713i \(-0.758615\pi\)
0.958568 + 0.284863i \(0.0919481\pi\)
\(510\) 0 0
\(511\) −7961.93 + 19081.0i −0.689266 + 1.65185i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −237.902 412.058i −0.0203557 0.0352572i
\(516\) 0 0
\(517\) 34533.5 2.93768
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5776.63 + 10005.4i 0.485755 + 0.841353i 0.999866 0.0163709i \(-0.00521125\pi\)
−0.514111 + 0.857724i \(0.671878\pi\)
\(522\) 0 0
\(523\) −2625.76 + 4547.94i −0.219534 + 0.380244i −0.954666 0.297680i \(-0.903787\pi\)
0.735132 + 0.677925i \(0.237120\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4347.52 + 7530.13i −0.359357 + 0.622425i
\(528\) 0 0
\(529\) 5661.39 + 9805.81i 0.465307 + 0.805935i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2431.30 0.197583
\(534\) 0 0
\(535\) −14851.9 25724.3i −1.20020 2.07880i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −19468.5 + 5107.99i −1.55578 + 0.408194i
\(540\) 0 0
\(541\) 10308.8 17855.4i 0.819241 1.41897i −0.0870006 0.996208i \(-0.527728\pi\)
0.906242 0.422759i \(-0.138938\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15179.0 1.19302
\(546\) 0 0
\(547\) 10669.4 0.833985 0.416993 0.908910i \(-0.363084\pi\)
0.416993 + 0.908910i \(0.363084\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 595.278 1031.05i 0.0460249 0.0797174i
\(552\) 0 0
\(553\) −2540.91 + 327.785i −0.195389 + 0.0252058i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2688.94 4657.38i −0.204549 0.354290i 0.745440 0.666573i \(-0.232240\pi\)
−0.949989 + 0.312283i \(0.898906\pi\)
\(558\) 0 0
\(559\) 2112.49 0.159837
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1759.44 3047.43i −0.131708 0.228124i 0.792627 0.609706i \(-0.208713\pi\)
−0.924335 + 0.381582i \(0.875379\pi\)
\(564\) 0 0
\(565\) −942.638 + 1632.70i −0.0701895 + 0.121572i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1275.15 2208.63i 0.0939492 0.162725i −0.815220 0.579151i \(-0.803384\pi\)
0.909170 + 0.416426i \(0.136718\pi\)
\(570\) 0 0
\(571\) −6827.67 11825.9i −0.500401 0.866721i −1.00000 0.000463542i \(-0.999852\pi\)
0.499599 0.866257i \(-0.333481\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3562.94 −0.258408
\(576\) 0 0
\(577\) 8347.94 + 14459.1i 0.602304 + 1.04322i 0.992471 + 0.122477i \(0.0390837\pi\)
−0.390168 + 0.920744i \(0.627583\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10048.7 + 13166.5i 0.717536 + 0.940173i
\(582\) 0 0
\(583\) −11873.4 + 20565.4i −0.843477 + 1.46095i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16341.4 1.14903 0.574517 0.818493i \(-0.305190\pi\)
0.574517 + 0.818493i \(0.305190\pi\)
\(588\) 0 0
\(589\) −27769.6 −1.94266
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9962.59 17255.7i 0.689906 1.19495i −0.281961 0.959426i \(-0.590985\pi\)
0.971868 0.235527i \(-0.0756817\pi\)
\(594\) 0 0
\(595\) 7593.11 + 9949.09i 0.523171 + 0.685501i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 7390.84 + 12801.3i 0.504143 + 0.873201i 0.999989 + 0.00479026i \(0.00152479\pi\)
−0.495846 + 0.868411i \(0.665142\pi\)
\(600\) 0 0
\(601\) −20487.7 −1.39054 −0.695268 0.718751i \(-0.744714\pi\)
−0.695268 + 0.718751i \(0.744714\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 16620.5 + 28787.5i 1.11689 + 1.93451i
\(606\) 0 0
\(607\) −3762.02 + 6516.01i −0.251558 + 0.435711i −0.963955 0.266066i \(-0.914276\pi\)
0.712397 + 0.701777i \(0.247610\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6101.59 + 10568.3i −0.404000 + 0.699749i
\(612\) 0 0
\(613\) −1707.32 2957.16i −0.112493 0.194843i 0.804282 0.594248i \(-0.202550\pi\)
−0.916775 + 0.399405i \(0.869217\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22219.4 −1.44979 −0.724895 0.688859i \(-0.758112\pi\)
−0.724895 + 0.688859i \(0.758112\pi\)
\(618\) 0 0
\(619\) −10931.7 18934.3i −0.709828 1.22946i −0.964921 0.262542i \(-0.915439\pi\)
0.255092 0.966917i \(-0.417894\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12523.4 + 1615.56i −0.805363 + 0.103894i
\(624\) 0 0
\(625\) 7958.12 13783.9i 0.509320 0.882168i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 655.499 0.0415524
\(630\) 0 0
\(631\) −16080.8 −1.01452 −0.507262 0.861792i \(-0.669342\pi\)
−0.507262 + 0.861792i \(0.669342\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 5379.56 9317.67i 0.336191 0.582300i
\(636\) 0 0
\(637\) 1876.62 6860.45i 0.116726 0.426720i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9231.90 15990.1i −0.568858 0.985292i −0.996679 0.0814280i \(-0.974052\pi\)
0.427821 0.903864i \(-0.359281\pi\)
\(642\) 0 0
\(643\) −3110.87 −0.190794 −0.0953971 0.995439i \(-0.530412\pi\)
−0.0953971 + 0.995439i \(0.530412\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12364.5 21416.0i −0.751313 1.30131i −0.947186 0.320683i \(-0.896087\pi\)
0.195873 0.980629i \(-0.437246\pi\)
\(648\) 0 0
\(649\) −319.068 + 552.643i −0.0192982 + 0.0334255i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3526.32 6107.76i 0.211325 0.366026i −0.740804 0.671721i \(-0.765555\pi\)
0.952130 + 0.305695i \(0.0988887\pi\)
\(654\) 0 0
\(655\) −2047.56 3546.47i −0.122145 0.211561i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −7895.55 −0.466718 −0.233359 0.972391i \(-0.574972\pi\)
−0.233359 + 0.972391i \(0.574972\pi\)
\(660\) 0 0
\(661\) 6550.56 + 11345.9i 0.385457 + 0.667631i 0.991833 0.127547i \(-0.0407104\pi\)
−0.606375 + 0.795179i \(0.707377\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −15392.5 + 36888.7i −0.897589 + 2.15110i
\(666\) 0 0
\(667\) −126.109 + 218.427i −0.00732076 + 0.0126799i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 52496.2 3.02026
\(672\) 0 0
\(673\) −9128.25 −0.522836 −0.261418 0.965226i \(-0.584190\pi\)
−0.261418 + 0.965226i \(0.584190\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8014.93 13882.3i 0.455006 0.788093i −0.543683 0.839291i \(-0.682971\pi\)
0.998689 + 0.0511978i \(0.0163039\pi\)
\(678\) 0 0
\(679\) 2769.86 + 3629.29i 0.156550 + 0.205124i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −11088.2 19205.3i −0.621197 1.07594i −0.989263 0.146145i \(-0.953313\pi\)
0.368066 0.929800i \(-0.380020\pi\)
\(684\) 0 0
\(685\) −11514.0 −0.642231
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4195.75 7267.25i −0.231996 0.401829i
\(690\) 0 0
\(691\) 592.072 1025.50i 0.0325955 0.0564570i −0.849267 0.527963i \(-0.822956\pi\)
0.881863 + 0.471506i \(0.156289\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18000.0 31176.9i 0.982417 1.70160i
\(696\) 0 0
\(697\) −2517.61 4360.63i −0.136817 0.236974i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −31064.6 −1.67374 −0.836870 0.547401i \(-0.815617\pi\)
−0.836870 + 0.547401i \(0.815617\pi\)
\(702\) 0 0
\(703\) 1046.74 + 1813.01i 0.0561574 + 0.0972674i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7773.42 18629.3i 0.413507 0.990985i
\(708\) 0 0
\(709\) −287.946 + 498.736i −0.0152525 + 0.0264181i −0.873551 0.486733i \(-0.838189\pi\)
0.858298 + 0.513151i \(0.171522\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5882.95 0.309002
\(714\) 0 0
\(715\) −19147.8 −1.00152
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2417.77 4187.70i 0.125407 0.217211i −0.796485 0.604658i \(-0.793310\pi\)
0.921892 + 0.387447i \(0.126643\pi\)
\(720\) 0 0
\(721\) 555.384 71.6462i 0.0286874 0.00370076i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 532.224 + 921.838i 0.0272638 + 0.0472224i
\(726\) 0 0
\(727\) −17999.8 −0.918259 −0.459129 0.888369i \(-0.651839\pi\)
−0.459129 + 0.888369i \(0.651839\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2187.48 3788.82i −0.110680 0.191703i
\(732\) 0 0
\(733\) −2556.70 + 4428.33i −0.128832 + 0.223144i −0.923224 0.384261i \(-0.874456\pi\)
0.794392 + 0.607405i \(0.207789\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −20647.8 + 35763.0i −1.03198 + 1.78745i
\(738\) 0 0
\(739\) −6528.06 11306.9i −0.324951 0.562831i 0.656552 0.754281i \(-0.272014\pi\)
−0.981502 + 0.191450i \(0.938681\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28402.9 1.40242 0.701212 0.712953i \(-0.252643\pi\)
0.701212 + 0.712953i \(0.252643\pi\)
\(744\) 0 0
\(745\) −27059.8 46868.9i −1.33073 2.30489i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 34672.0 4472.79i 1.69144 0.218201i
\(750\) 0 0
\(751\) 16635.1 28812.9i 0.808288 1.40000i −0.105760 0.994392i \(-0.533728\pi\)
0.914049 0.405605i \(-0.132939\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −44875.7 −2.16317
\(756\) 0 0
\(757\) 18331.2 0.880129 0.440064 0.897966i \(-0.354956\pi\)
0.440064 + 0.897966i \(0.354956\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2498.50 4327.53i 0.119015 0.206141i −0.800362 0.599517i \(-0.795360\pi\)
0.919378 + 0.393376i \(0.128693\pi\)
\(762\) 0 0
\(763\) −6879.46 + 16486.9i −0.326413 + 0.782260i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −112.750 195.289i −0.00530792 0.00919358i
\(768\) 0 0
\(769\) −9834.47 −0.461171 −0.230585 0.973052i \(-0.574064\pi\)
−0.230585 + 0.973052i \(0.574064\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6991.49 12109.6i −0.325312 0.563458i 0.656263 0.754532i \(-0.272136\pi\)
−0.981576 + 0.191074i \(0.938803\pi\)
\(774\) 0 0
\(775\) 12414.1 21501.8i 0.575389 0.996603i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 8040.57 13926.7i 0.369812 0.640532i
\(780\) 0 0
\(781\) −34091.7 59048.6i −1.56197 2.70541i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −38400.0 −1.74593
\(786\) 0 0
\(787\) 15639.3 + 27088.1i 0.708362 + 1.22692i 0.965464 + 0.260535i \(0.0838990\pi\)
−0.257102 + 0.966384i \(0.582768\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1346.15 1763.83i −0.0605102 0.0792853i
\(792\) 0 0
\(793\) −9275.37 + 16065.4i −0.415357 + 0.719419i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30546.9 −1.35762 −0.678811 0.734313i \(-0.737505\pi\)
−0.678811 + 0.734313i \(0.737505\pi\)
\(798\) 0 0
\(799\) 25272.8 1.11901
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −32754.7 + 56732.8i −1.43946 + 2.49322i
\(804\) 0 0
\(805\) 3260.88 7814.82i 0.142771 0.342157i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −14786.3 25610.7i −0.642596 1.11301i −0.984851 0.173401i \(-0.944524\pi\)
0.342256 0.939607i \(-0.388809\pi\)
\(810\) 0 0
\(811\) 45069.0 1.95140 0.975701 0.219107i \(-0.0703144\pi\)
0.975701 + 0.219107i \(0.0703144\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1899.27 + 3289.64i 0.0816302 + 0.141388i
\(816\) 0 0
\(817\) 6986.20 12100.5i 0.299163 0.518166i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4700.47 8141.45i 0.199814 0.346088i −0.748654 0.662961i \(-0.769299\pi\)
0.948468 + 0.316873i \(0.102633\pi\)
\(822\) 0 0
\(823\) 14004.4 + 24256.4i 0.593152 + 1.02737i 0.993805 + 0.111139i \(0.0354499\pi\)
−0.400653 + 0.916230i \(0.631217\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28650.8 1.20470 0.602349 0.798232i \(-0.294231\pi\)
0.602349 + 0.798232i \(0.294231\pi\)
\(828\) 0 0
\(829\) 2124.82 + 3680.29i 0.0890205 + 0.154188i 0.907097 0.420921i \(-0.138293\pi\)
−0.818077 + 0.575109i \(0.804960\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −14247.7 + 3738.20i −0.592621 + 0.155487i
\(834\) 0 0
\(835\) −2568.48 + 4448.74i −0.106450 + 0.184377i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −9265.50 −0.381264 −0.190632 0.981662i \(-0.561054\pi\)
−0.190632 + 0.981662i \(0.561054\pi\)
\(840\) 0 0
\(841\) −24313.6 −0.996910
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −13903.0 + 24080.6i −0.566007 + 0.980354i
\(846\) 0 0
\(847\) −38800.7 + 5005.41i −1.57404 + 0.203055i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −221.751 384.084i −0.00893245 0.0154715i
\(852\) 0 0
\(853\) 36321.1 1.45793 0.728964 0.684552i \(-0.240002\pi\)
0.728964 + 0.684552i \(0.240002\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −11838.1 20504.2i −0.471858 0.817282i 0.527623 0.849478i \(-0.323083\pi\)
−0.999482 + 0.0321962i \(0.989750\pi\)
\(858\) 0 0
\(859\) −13706.2 + 23739.8i −0.544410 + 0.942946i 0.454233 + 0.890883i \(0.349913\pi\)
−0.998644 + 0.0520636i \(0.983420\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11362.7 + 19680.9i −0.448195 + 0.776297i −0.998269 0.0588195i \(-0.981266\pi\)
0.550074 + 0.835116i \(0.314600\pi\)
\(864\) 0 0
\(865\) 12555.0 + 21745.9i 0.493506 + 0.854778i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −8117.45 −0.316876
\(870\) 0 0
\(871\) −7296.37 12637.7i −0.283844 0.491632i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 419.943 + 550.243i 0.0162248 + 0.0212590i
\(876\) 0 0
\(877\) −5990.93 + 10376.6i −0.230672 + 0.399536i −0.958006 0.286748i \(-0.907426\pi\)
0.727334 + 0.686284i \(0.240759\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 22891.8 0.875419 0.437709 0.899117i \(-0.355790\pi\)
0.437709 + 0.899117i \(0.355790\pi\)
\(882\) 0 0
\(883\) 36489.6 1.39068 0.695341 0.718680i \(-0.255253\pi\)
0.695341 + 0.718680i \(0.255253\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12052.9 20876.2i 0.456253 0.790254i −0.542506 0.840052i \(-0.682524\pi\)
0.998759 + 0.0497981i \(0.0158578\pi\)
\(888\) 0 0
\(889\) 7682.37 + 10066.1i 0.289830 + 0.379758i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 40357.2 + 69900.7i 1.51232 + 2.61941i
\(894\) 0 0
\(895\) 45365.0 1.69428
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −878.781 1522.09i −0.0326018 0.0564679i
\(900\) 0 0
\(901\) −8689.38 + 15050.5i −0.321293 + 0.556496i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8879.87 15380.4i 0.326162 0.564929i
\(906\) 0 0
\(907\) −11181.6 19367.2i −0.409350 0.709014i 0.585467 0.810696i \(-0.300911\pi\)
−0.994817 + 0.101682i \(0.967578\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 21426.3 0.779238 0.389619 0.920976i \(-0.372607\pi\)
0.389619 + 0.920976i \(0.372607\pi\)
\(912\) 0 0
\(913\) 26239.5 + 45448.2i 0.951152 + 1.64744i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4780.05 616.641i 0.172139 0.0222064i
\(918\) 0 0
\(919\) 15263.0 26436.4i 0.547858 0.948917i −0.450563 0.892744i \(-0.648777\pi\)
0.998421 0.0561730i \(-0.0178898\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 24094.2 0.859231
\(924\) 0 0
\(925\) −1871.73 −0.0665322
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2319.45 4017.40i 0.0819146 0.141880i −0.822158 0.569260i \(-0.807230\pi\)
0.904072 + 0.427380i \(0.140563\pi\)
\(930\) 0 0
\(931\) −33090.9 33437.6i −1.16489 1.17709i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 19827.5 + 34342.2i 0.693506 + 1.20119i
\(936\) 0 0
\(937\) −7713.38 −0.268928 −0.134464 0.990919i \(-0.542931\pi\)
−0.134464 + 0.990919i \(0.542931\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −3745.31 6487.07i −0.129749 0.224732i 0.793830 0.608139i \(-0.208084\pi\)
−0.923579 + 0.383408i \(0.874750\pi\)
\(942\) 0 0
\(943\) −1703.38 + 2950.34i −0.0588226 + 0.101884i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12569.6 21771.1i 0.431316 0.747061i −0.565671 0.824631i \(-0.691383\pi\)
0.996987 + 0.0775698i \(0.0247161\pi\)
\(948\) 0 0
\(949\) −11574.6 20047.8i −0.395920 0.685754i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2505.29 0.0851568 0.0425784 0.999093i \(-0.486443\pi\)
0.0425784 + 0.999093i \(0.486443\pi\)
\(954\) 0 0
\(955\) −17351.0 30052.8i −0.587922 1.01831i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5218.41 12506.1i 0.175716 0.421108i
\(960\) 0 0
\(961\) −5602.00 + 9702.94i −0.188043 + 0.325700i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −59724.2 −1.99232
\(966\) 0 0
\(967\) −9331.61 −0.310325 −0.155162 0.987889i \(-0.549590\pi\)
−0.155162 + 0.987889i \(0.549590\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −20993.0 + 36360.9i −0.693817 + 1.20173i 0.276761 + 0.960939i \(0.410739\pi\)
−0.970578 + 0.240787i \(0.922594\pi\)
\(972\) 0 0
\(973\) 25705.2 + 33681.0i 0.846939 + 1.10973i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3975.47 6885.72i −0.130181 0.225480i 0.793565 0.608485i \(-0.208222\pi\)
−0.923746 + 0.383005i \(0.874889\pi\)
\(978\) 0 0
\(979\) −40008.7 −1.30611
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5269.74 + 9127.47i 0.170986 + 0.296156i 0.938765 0.344559i \(-0.111972\pi\)
−0.767779 + 0.640715i \(0.778638\pi\)
\(984\) 0 0
\(985\) −29121.6 + 50440.1i −0.942022 + 1.63163i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1480.02 + 2563.46i −0.0475852 + 0.0824200i
\(990\) 0 0
\(991\) 2215.93 + 3838.10i 0.0710306 + 0.123029i 0.899353 0.437223i \(-0.144038\pi\)
−0.828323 + 0.560251i \(0.810705\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 56656.0 1.80514
\(996\) 0 0
\(997\) 19707.8 + 34134.9i 0.626031 + 1.08432i 0.988341 + 0.152259i \(0.0486548\pi\)
−0.362310 + 0.932058i \(0.618012\pi\)
\(998\) 0 0
\(999\) 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 504.4.s.f.289.2 4
3.2 odd 2 168.4.q.d.121.1 yes 4
7.4 even 3 inner 504.4.s.f.361.2 4
12.11 even 2 336.4.q.g.289.1 4
21.2 odd 6 1176.4.a.r.1.2 2
21.5 even 6 1176.4.a.u.1.1 2
21.11 odd 6 168.4.q.d.25.1 4
84.11 even 6 336.4.q.g.193.1 4
84.23 even 6 2352.4.a.cc.1.2 2
84.47 odd 6 2352.4.a.bo.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.d.25.1 4 21.11 odd 6
168.4.q.d.121.1 yes 4 3.2 odd 2
336.4.q.g.193.1 4 84.11 even 6
336.4.q.g.289.1 4 12.11 even 2
504.4.s.f.289.2 4 1.1 even 1 trivial
504.4.s.f.361.2 4 7.4 even 3 inner
1176.4.a.r.1.2 2 21.2 odd 6
1176.4.a.u.1.1 2 21.5 even 6
2352.4.a.bo.1.1 2 84.47 odd 6
2352.4.a.cc.1.2 2 84.23 even 6