Properties

Label 504.4.s.j.289.4
Level $504$
Weight $4$
Character 504.289
Analytic conductor $29.737$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 173x^{6} + 9457x^{4} + 168048x^{2} + 746496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.4
Root \(-8.67551i\) of defining polynomial
Character \(\chi\) \(=\) 504.289
Dual form 504.4.s.j.361.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(9.47901 - 16.4181i) q^{5} +(12.8033 - 13.3819i) q^{7} +O(q^{10})\) \(q+(9.47901 - 16.4181i) q^{5} +(12.8033 - 13.3819i) q^{7} +(27.3654 + 47.3983i) q^{11} +62.0173 q^{13} +(61.2195 + 106.035i) q^{17} +(6.25288 - 10.8303i) q^{19} +(-37.2195 + 64.4661i) q^{23} +(-117.203 - 203.002i) q^{25} +232.572 q^{29} +(-5.18387 - 8.97872i) q^{31} +(-98.3438 - 337.053i) q^{35} +(122.993 - 213.031i) q^{37} -238.653 q^{41} -92.9718 q^{43} +(-242.822 + 420.580i) q^{47} +(-15.1521 - 342.665i) q^{49} +(-189.278 - 327.840i) q^{53} +1037.59 q^{55} +(-91.3918 - 158.295i) q^{59} +(-198.235 + 343.353i) q^{61} +(587.863 - 1018.21i) q^{65} +(130.620 + 226.240i) q^{67} +874.523 q^{71} +(-76.2032 - 131.988i) q^{73} +(984.647 + 240.651i) q^{77} +(-286.679 + 496.542i) q^{79} -317.754 q^{83} +2321.20 q^{85} +(-47.5080 + 82.2863i) q^{89} +(794.025 - 829.911i) q^{91} +(-118.542 - 205.321i) q^{95} -1608.78 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} + 18 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} + 18 q^{7} + 14 q^{11} + 44 q^{13} + 96 q^{17} + 26 q^{19} + 96 q^{23} - 110 q^{25} + 152 q^{29} - 238 q^{31} - 152 q^{35} - 562 q^{37} - 856 q^{41} - 516 q^{43} - 80 q^{47} + 156 q^{49} + 2952 q^{55} + 262 q^{59} + 276 q^{61} + 2196 q^{65} - 150 q^{67} + 1696 q^{71} + 218 q^{73} + 764 q^{77} - 1762 q^{79} - 6900 q^{83} + 2904 q^{85} - 344 q^{89} - 2806 q^{91} + 2004 q^{95} - 1244 q^{97}+O(q^{100}) \) 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
\(5\) 9.47901 16.4181i 0.847828 1.46848i −0.0353138 0.999376i \(-0.511243\pi\)
0.883142 0.469106i \(-0.155424\pi\)
\(6\) 0 0
\(7\) 12.8033 13.3819i 0.691312 0.722556i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 27.3654 + 47.3983i 0.750089 + 1.29919i 0.947779 + 0.318928i \(0.103323\pi\)
−0.197690 + 0.980265i \(0.563344\pi\)
\(12\) 0 0
\(13\) 62.0173 1.32312 0.661558 0.749894i \(-0.269896\pi\)
0.661558 + 0.749894i \(0.269896\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 61.2195 + 106.035i 0.873407 + 1.51279i 0.858450 + 0.512897i \(0.171428\pi\)
0.0149571 + 0.999888i \(0.495239\pi\)
\(18\) 0 0
\(19\) 6.25288 10.8303i 0.0755004 0.130771i −0.825803 0.563958i \(-0.809278\pi\)
0.901304 + 0.433188i \(0.142611\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −37.2195 + 64.4661i −0.337427 + 0.584440i −0.983948 0.178456i \(-0.942890\pi\)
0.646521 + 0.762896i \(0.276223\pi\)
\(24\) 0 0
\(25\) −117.203 203.002i −0.937626 1.62402i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 232.572 1.48923 0.744613 0.667496i \(-0.232634\pi\)
0.744613 + 0.667496i \(0.232634\pi\)
\(30\) 0 0
\(31\) −5.18387 8.97872i −0.0300339 0.0520202i 0.850618 0.525785i \(-0.176228\pi\)
−0.880652 + 0.473764i \(0.842895\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −98.3438 337.053i −0.474947 1.62778i
\(36\) 0 0
\(37\) 122.993 213.031i 0.546486 0.946541i −0.452026 0.892005i \(-0.649299\pi\)
0.998512 0.0545365i \(-0.0173681\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −238.653 −0.909056 −0.454528 0.890733i \(-0.650192\pi\)
−0.454528 + 0.890733i \(0.650192\pi\)
\(42\) 0 0
\(43\) −92.9718 −0.329722 −0.164861 0.986317i \(-0.552718\pi\)
−0.164861 + 0.986317i \(0.552718\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −242.822 + 420.580i −0.753601 + 1.30528i 0.192465 + 0.981304i \(0.438352\pi\)
−0.946067 + 0.323972i \(0.894982\pi\)
\(48\) 0 0
\(49\) −15.1521 342.665i −0.0441751 0.999024i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −189.278 327.840i −0.490555 0.849666i 0.509386 0.860538i \(-0.329872\pi\)
−0.999941 + 0.0108725i \(0.996539\pi\)
\(54\) 0 0
\(55\) 1037.59 2.54379
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −91.3918 158.295i −0.201664 0.349293i 0.747400 0.664374i \(-0.231302\pi\)
−0.949065 + 0.315081i \(0.897968\pi\)
\(60\) 0 0
\(61\) −198.235 + 343.353i −0.416088 + 0.720686i −0.995542 0.0943190i \(-0.969933\pi\)
0.579454 + 0.815005i \(0.303266\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 587.863 1018.21i 1.12178 1.94297i
\(66\) 0 0
\(67\) 130.620 + 226.240i 0.238175 + 0.412531i 0.960191 0.279346i \(-0.0901175\pi\)
−0.722016 + 0.691877i \(0.756784\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 874.523 1.46179 0.730893 0.682492i \(-0.239104\pi\)
0.730893 + 0.682492i \(0.239104\pi\)
\(72\) 0 0
\(73\) −76.2032 131.988i −0.122177 0.211617i 0.798449 0.602062i \(-0.205654\pi\)
−0.920626 + 0.390446i \(0.872321\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 984.647 + 240.651i 1.45729 + 0.356166i
\(78\) 0 0
\(79\) −286.679 + 496.542i −0.408277 + 0.707156i −0.994697 0.102851i \(-0.967204\pi\)
0.586420 + 0.810007i \(0.300537\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −317.754 −0.420217 −0.210108 0.977678i \(-0.567382\pi\)
−0.210108 + 0.977678i \(0.567382\pi\)
\(84\) 0 0
\(85\) 2321.20 2.96200
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −47.5080 + 82.2863i −0.0565824 + 0.0980037i −0.892929 0.450197i \(-0.851354\pi\)
0.836347 + 0.548201i \(0.184687\pi\)
\(90\) 0 0
\(91\) 794.025 829.911i 0.914686 0.956026i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −118.542 205.321i −0.128023 0.221742i
\(96\) 0 0
\(97\) −1608.78 −1.68398 −0.841992 0.539490i \(-0.818617\pi\)
−0.841992 + 0.539490i \(0.818617\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −391.521 678.135i −0.385721 0.668089i 0.606148 0.795352i \(-0.292714\pi\)
−0.991869 + 0.127263i \(0.959381\pi\)
\(102\) 0 0
\(103\) 744.805 1290.04i 0.712503 1.23409i −0.251412 0.967880i \(-0.580895\pi\)
0.963915 0.266211i \(-0.0857717\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 356.385 617.276i 0.321991 0.557704i −0.658908 0.752223i \(-0.728981\pi\)
0.980899 + 0.194519i \(0.0623147\pi\)
\(108\) 0 0
\(109\) −520.924 902.267i −0.457757 0.792858i 0.541085 0.840968i \(-0.318014\pi\)
−0.998842 + 0.0481100i \(0.984680\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −352.093 −0.293116 −0.146558 0.989202i \(-0.546819\pi\)
−0.146558 + 0.989202i \(0.546819\pi\)
\(114\) 0 0
\(115\) 705.609 + 1222.15i 0.572160 + 0.991010i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2202.77 + 538.365i 1.69687 + 0.414721i
\(120\) 0 0
\(121\) −832.231 + 1441.47i −0.625267 + 1.08299i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2074.13 −1.48413
\(126\) 0 0
\(127\) −1093.73 −0.764194 −0.382097 0.924122i \(-0.624798\pi\)
−0.382097 + 0.924122i \(0.624798\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1083.19 + 1876.13i −0.722430 + 1.25129i 0.237593 + 0.971365i \(0.423642\pi\)
−0.960023 + 0.279921i \(0.909692\pi\)
\(132\) 0 0
\(133\) −64.8730 222.339i −0.0422947 0.144957i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −982.461 1701.67i −0.612681 1.06119i −0.990787 0.135432i \(-0.956758\pi\)
0.378105 0.925763i \(-0.376576\pi\)
\(138\) 0 0
\(139\) −136.976 −0.0835840 −0.0417920 0.999126i \(-0.513307\pi\)
−0.0417920 + 0.999126i \(0.513307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1697.13 + 2939.51i 0.992455 + 1.71898i
\(144\) 0 0
\(145\) 2204.55 3818.40i 1.26261 2.18690i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −140.053 + 242.578i −0.0770037 + 0.133374i −0.901956 0.431828i \(-0.857869\pi\)
0.824952 + 0.565203i \(0.191202\pi\)
\(150\) 0 0
\(151\) −1097.07 1900.17i −0.591245 1.02407i −0.994065 0.108787i \(-0.965303\pi\)
0.402820 0.915279i \(-0.368030\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −196.552 −0.101854
\(156\) 0 0
\(157\) 110.409 + 191.234i 0.0561248 + 0.0972111i 0.892723 0.450606i \(-0.148792\pi\)
−0.836598 + 0.547817i \(0.815459\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 386.149 + 1323.45i 0.189024 + 0.647840i
\(162\) 0 0
\(163\) −738.431 + 1279.00i −0.354837 + 0.614596i −0.987090 0.160166i \(-0.948797\pi\)
0.632253 + 0.774762i \(0.282130\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2197.66 −1.01832 −0.509162 0.860671i \(-0.670045\pi\)
−0.509162 + 0.860671i \(0.670045\pi\)
\(168\) 0 0
\(169\) 1649.15 0.750635
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1045.80 1811.38i 0.459601 0.796052i −0.539339 0.842089i \(-0.681326\pi\)
0.998940 + 0.0460370i \(0.0146592\pi\)
\(174\) 0 0
\(175\) −4217.14 1030.69i −1.82163 0.445214i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1167.27 + 2021.77i 0.487406 + 0.844212i 0.999895 0.0144814i \(-0.00460974\pi\)
−0.512489 + 0.858694i \(0.671276\pi\)
\(180\) 0 0
\(181\) 1758.40 0.722105 0.361053 0.932545i \(-0.382417\pi\)
0.361053 + 0.932545i \(0.382417\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2331.71 4038.64i −0.926652 1.60501i
\(186\) 0 0
\(187\) −3350.60 + 5803.40i −1.31027 + 2.26945i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1850.34 + 3204.88i −0.700973 + 1.21412i 0.267152 + 0.963654i \(0.413917\pi\)
−0.968125 + 0.250467i \(0.919416\pi\)
\(192\) 0 0
\(193\) −1354.22 2345.58i −0.505073 0.874812i −0.999983 0.00586773i \(-0.998132\pi\)
0.494910 0.868944i \(-0.335201\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 160.686 0.0581138 0.0290569 0.999578i \(-0.490750\pi\)
0.0290569 + 0.999578i \(0.490750\pi\)
\(198\) 0 0
\(199\) 1033.00 + 1789.20i 0.367976 + 0.637353i 0.989249 0.146241i \(-0.0467176\pi\)
−0.621273 + 0.783594i \(0.713384\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2977.69 3112.27i 1.02952 1.07605i
\(204\) 0 0
\(205\) −2262.19 + 3918.23i −0.770723 + 1.33493i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 684.450 0.226528
\(210\) 0 0
\(211\) −1007.12 −0.328592 −0.164296 0.986411i \(-0.552535\pi\)
−0.164296 + 0.986411i \(0.552535\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −881.280 + 1526.42i −0.279548 + 0.484191i
\(216\) 0 0
\(217\) −186.523 45.5869i −0.0583503 0.0142610i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3796.67 + 6576.03i 1.15562 + 2.00159i
\(222\) 0 0
\(223\) 1644.83 0.493929 0.246964 0.969025i \(-0.420567\pi\)
0.246964 + 0.969025i \(0.420567\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 159.437 + 276.153i 0.0466177 + 0.0807442i 0.888393 0.459084i \(-0.151822\pi\)
−0.841775 + 0.539829i \(0.818489\pi\)
\(228\) 0 0
\(229\) −1268.05 + 2196.33i −0.365918 + 0.633789i −0.988923 0.148429i \(-0.952578\pi\)
0.623005 + 0.782218i \(0.285912\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1166.41 2020.28i 0.327957 0.568038i −0.654150 0.756365i \(-0.726973\pi\)
0.982106 + 0.188328i \(0.0603067\pi\)
\(234\) 0 0
\(235\) 4603.43 + 7973.37i 1.27785 + 2.21330i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2713.85 −0.734495 −0.367248 0.930123i \(-0.619700\pi\)
−0.367248 + 0.930123i \(0.619700\pi\)
\(240\) 0 0
\(241\) 2087.29 + 3615.30i 0.557902 + 0.966315i 0.997671 + 0.0682042i \(0.0217269\pi\)
−0.439769 + 0.898111i \(0.644940\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5769.55 2999.36i −1.50450 0.782130i
\(246\) 0 0
\(247\) 387.786 671.666i 0.0998958 0.173025i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6123.58 −1.53991 −0.769954 0.638099i \(-0.779721\pi\)
−0.769954 + 0.638099i \(0.779721\pi\)
\(252\) 0 0
\(253\) −4074.11 −1.01240
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2316.96 4013.09i 0.562365 0.974045i −0.434925 0.900467i \(-0.643225\pi\)
0.997289 0.0735777i \(-0.0234417\pi\)
\(258\) 0 0
\(259\) −1276.04 4373.38i −0.306137 1.04922i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1641.61 + 2843.36i 0.384891 + 0.666650i 0.991754 0.128156i \(-0.0409058\pi\)
−0.606863 + 0.794806i \(0.707572\pi\)
\(264\) 0 0
\(265\) −7176.69 −1.66362
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −399.493 691.942i −0.0905483 0.156834i 0.817194 0.576363i \(-0.195529\pi\)
−0.907742 + 0.419529i \(0.862195\pi\)
\(270\) 0 0
\(271\) 3353.04 5807.63i 0.751596 1.30180i −0.195453 0.980713i \(-0.562618\pi\)
0.947049 0.321089i \(-0.104049\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6414.63 11110.5i 1.40661 2.43631i
\(276\) 0 0
\(277\) −581.135 1006.56i −0.126054 0.218332i 0.796090 0.605178i \(-0.206898\pi\)
−0.922145 + 0.386845i \(0.873565\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2718.17 −0.577055 −0.288527 0.957472i \(-0.593166\pi\)
−0.288527 + 0.957472i \(0.593166\pi\)
\(282\) 0 0
\(283\) 1504.74 + 2606.29i 0.316069 + 0.547448i 0.979664 0.200644i \(-0.0643035\pi\)
−0.663595 + 0.748092i \(0.730970\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3055.54 + 3193.63i −0.628441 + 0.656844i
\(288\) 0 0
\(289\) −5039.17 + 8728.09i −1.02568 + 1.77653i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4209.15 −0.839252 −0.419626 0.907697i \(-0.637839\pi\)
−0.419626 + 0.907697i \(0.637839\pi\)
\(294\) 0 0
\(295\) −3465.21 −0.683907
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2308.26 + 3998.02i −0.446454 + 0.773282i
\(300\) 0 0
\(301\) −1190.34 + 1244.14i −0.227941 + 0.238243i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3758.14 + 6509.29i 0.705543 + 1.22204i
\(306\) 0 0
\(307\) −3114.82 −0.579063 −0.289531 0.957169i \(-0.593499\pi\)
−0.289531 + 0.957169i \(0.593499\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4243.62 7350.16i −0.773741 1.34016i −0.935499 0.353329i \(-0.885050\pi\)
0.161758 0.986831i \(-0.448284\pi\)
\(312\) 0 0
\(313\) 2477.46 4291.08i 0.447393 0.774908i −0.550822 0.834623i \(-0.685686\pi\)
0.998215 + 0.0597148i \(0.0190191\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2804.40 + 4857.36i −0.496879 + 0.860619i −0.999994 0.00360042i \(-0.998854\pi\)
0.503115 + 0.864220i \(0.332187\pi\)
\(318\) 0 0
\(319\) 6364.43 + 11023.5i 1.11705 + 1.93479i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1531.19 0.263770
\(324\) 0 0
\(325\) −7268.63 12589.6i −1.24059 2.14876i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2519.26 + 8634.24i 0.422162 + 1.44687i
\(330\) 0 0
\(331\) 3304.75 5723.99i 0.548777 0.950510i −0.449582 0.893239i \(-0.648427\pi\)
0.998359 0.0572706i \(-0.0182398\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4952.57 0.807725
\(336\) 0 0
\(337\) 11455.5 1.85170 0.925850 0.377891i \(-0.123350\pi\)
0.925850 + 0.377891i \(0.123350\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 283.717 491.413i 0.0450562 0.0780395i
\(342\) 0 0
\(343\) −4779.52 4184.47i −0.752390 0.658718i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3022.08 5234.39i −0.467532 0.809789i 0.531780 0.846883i \(-0.321523\pi\)
−0.999312 + 0.0370937i \(0.988190\pi\)
\(348\) 0 0
\(349\) −5487.93 −0.841726 −0.420863 0.907124i \(-0.638273\pi\)
−0.420863 + 0.907124i \(0.638273\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3440.08 + 5958.39i 0.518688 + 0.898394i 0.999764 + 0.0217151i \(0.00691268\pi\)
−0.481076 + 0.876679i \(0.659754\pi\)
\(354\) 0 0
\(355\) 8289.61 14358.0i 1.23934 2.14660i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1969.17 + 3410.70i −0.289495 + 0.501420i −0.973689 0.227880i \(-0.926821\pi\)
0.684194 + 0.729300i \(0.260154\pi\)
\(360\) 0 0
\(361\) 3351.30 + 5804.63i 0.488599 + 0.846279i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2889.32 −0.414340
\(366\) 0 0
\(367\) −719.814 1246.75i −0.102381 0.177330i 0.810284 0.586038i \(-0.199313\pi\)
−0.912665 + 0.408708i \(0.865980\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6810.52 1664.51i −0.953058 0.232931i
\(372\) 0 0
\(373\) 1247.05 2159.96i 0.173110 0.299835i −0.766396 0.642369i \(-0.777952\pi\)
0.939506 + 0.342534i \(0.111285\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14423.5 1.97042
\(378\) 0 0
\(379\) −1309.25 −0.177445 −0.0887225 0.996056i \(-0.528278\pi\)
−0.0887225 + 0.996056i \(0.528278\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3480.90 6029.10i 0.464402 0.804367i −0.534773 0.844996i \(-0.679603\pi\)
0.999174 + 0.0406289i \(0.0129361\pi\)
\(384\) 0 0
\(385\) 13284.5 13884.9i 1.75855 1.83803i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3176.72 + 5502.25i 0.414052 + 0.717159i 0.995328 0.0965470i \(-0.0307798\pi\)
−0.581276 + 0.813706i \(0.697446\pi\)
\(390\) 0 0
\(391\) −9114.25 −1.17884
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5434.86 + 9413.45i 0.692297 + 1.19909i
\(396\) 0 0
\(397\) −5752.70 + 9963.97i −0.727254 + 1.25964i 0.230785 + 0.973005i \(0.425871\pi\)
−0.958039 + 0.286637i \(0.907463\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 826.664 1431.82i 0.102947 0.178309i −0.809951 0.586498i \(-0.800506\pi\)
0.912897 + 0.408189i \(0.133840\pi\)
\(402\) 0 0
\(403\) −321.489 556.836i −0.0397383 0.0688287i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13463.0 1.63965
\(408\) 0 0
\(409\) 2447.41 + 4239.04i 0.295885 + 0.512487i 0.975190 0.221368i \(-0.0710523\pi\)
−0.679306 + 0.733855i \(0.737719\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3288.41 803.699i −0.391797 0.0957566i
\(414\) 0 0
\(415\) −3011.99 + 5216.92i −0.356272 + 0.617081i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 265.504 0.0309563 0.0154782 0.999880i \(-0.495073\pi\)
0.0154782 + 0.999880i \(0.495073\pi\)
\(420\) 0 0
\(421\) 11136.8 1.28925 0.644623 0.764500i \(-0.277014\pi\)
0.644623 + 0.764500i \(0.277014\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 14350.3 24855.4i 1.63786 2.83685i
\(426\) 0 0
\(427\) 2056.67 + 7048.81i 0.233089 + 0.798866i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2596.71 + 4497.63i 0.290206 + 0.502652i 0.973858 0.227156i \(-0.0729427\pi\)
−0.683652 + 0.729808i \(0.739609\pi\)
\(432\) 0 0
\(433\) 6314.17 0.700785 0.350392 0.936603i \(-0.386048\pi\)
0.350392 + 0.936603i \(0.386048\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 465.458 + 806.198i 0.0509517 + 0.0882509i
\(438\) 0 0
\(439\) 7711.66 13357.0i 0.838399 1.45215i −0.0528329 0.998603i \(-0.516825\pi\)
0.891232 0.453547i \(-0.149842\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4353.06 + 7539.72i −0.466862 + 0.808629i −0.999283 0.0378504i \(-0.987949\pi\)
0.532421 + 0.846480i \(0.321282\pi\)
\(444\) 0 0
\(445\) 900.657 + 1559.98i 0.0959444 + 0.166181i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −5495.91 −0.577657 −0.288829 0.957381i \(-0.593266\pi\)
−0.288829 + 0.957381i \(0.593266\pi\)
\(450\) 0 0
\(451\) −6530.83 11311.7i −0.681873 1.18104i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6099.02 20903.1i −0.628409 2.15375i
\(456\) 0 0
\(457\) 5678.63 9835.67i 0.581258 1.00677i −0.414072 0.910244i \(-0.635894\pi\)
0.995331 0.0965247i \(-0.0307727\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −14514.2 −1.46637 −0.733184 0.680030i \(-0.761967\pi\)
−0.733184 + 0.680030i \(0.761967\pi\)
\(462\) 0 0
\(463\) 9971.00 1.00085 0.500423 0.865781i \(-0.333178\pi\)
0.500423 + 0.865781i \(0.333178\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −198.739 + 344.226i −0.0196928 + 0.0341089i −0.875704 0.482849i \(-0.839602\pi\)
0.856011 + 0.516957i \(0.172935\pi\)
\(468\) 0 0
\(469\) 4699.88 + 1148.67i 0.462730 + 0.113093i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2544.21 4406.70i −0.247321 0.428373i
\(474\) 0 0
\(475\) −2931.43 −0.283165
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −7030.20 12176.7i −0.670602 1.16152i −0.977734 0.209849i \(-0.932703\pi\)
0.307132 0.951667i \(-0.400631\pi\)
\(480\) 0 0
\(481\) 7627.71 13211.6i 0.723064 1.25238i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15249.6 + 26413.1i −1.42773 + 2.47290i
\(486\) 0 0
\(487\) 6767.34 + 11721.4i 0.629687 + 1.09065i 0.987614 + 0.156900i \(0.0501502\pi\)
−0.357927 + 0.933749i \(0.616516\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8693.29 0.799028 0.399514 0.916727i \(-0.369179\pi\)
0.399514 + 0.916727i \(0.369179\pi\)
\(492\) 0 0
\(493\) 14238.0 + 24660.9i 1.30070 + 2.25288i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11196.8 11702.8i 1.01055 1.05622i
\(498\) 0 0
\(499\) −2008.97 + 3479.63i −0.180228 + 0.312164i −0.941958 0.335731i \(-0.891017\pi\)
0.761730 + 0.647894i \(0.224350\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −52.2455 −0.00463124 −0.00231562 0.999997i \(-0.500737\pi\)
−0.00231562 + 0.999997i \(0.500737\pi\)
\(504\) 0 0
\(505\) −14844.9 −1.30810
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4619.55 8001.30i 0.402275 0.696761i −0.591725 0.806140i \(-0.701553\pi\)
0.994000 + 0.109379i \(0.0348862\pi\)
\(510\) 0 0
\(511\) −2741.90 670.131i −0.237367 0.0580134i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −14120.0 24456.6i −1.20816 2.09259i
\(516\) 0 0
\(517\) −26579.7 −2.26107
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7486.70 + 12967.3i 0.629555 + 1.09042i 0.987641 + 0.156733i \(0.0500961\pi\)
−0.358086 + 0.933689i \(0.616571\pi\)
\(522\) 0 0
\(523\) −6801.76 + 11781.0i −0.568681 + 0.984985i 0.428015 + 0.903771i \(0.359213\pi\)
−0.996697 + 0.0812134i \(0.974120\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 634.708 1099.35i 0.0524636 0.0908696i
\(528\) 0 0
\(529\) 3312.91 + 5738.13i 0.272287 + 0.471614i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −14800.6 −1.20279
\(534\) 0 0
\(535\) −6756.35 11702.3i −0.545986 0.945675i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 15827.1 10095.4i 1.26479 0.806749i
\(540\) 0 0
\(541\) 8443.19 14624.0i 0.670981 1.16217i −0.306645 0.951824i \(-0.599206\pi\)
0.977626 0.210350i \(-0.0674603\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −19751.4 −1.55240
\(546\) 0 0
\(547\) −5987.25 −0.468001 −0.234000 0.972237i \(-0.575182\pi\)
−0.234000 + 0.972237i \(0.575182\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1454.25 2518.83i 0.112437 0.194747i
\(552\) 0 0
\(553\) 2974.26 + 10193.7i 0.228713 + 0.783869i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1119.35 + 1938.78i 0.0851499 + 0.147484i 0.905455 0.424442i \(-0.139530\pi\)
−0.820305 + 0.571926i \(0.806196\pi\)
\(558\) 0 0
\(559\) −5765.86 −0.436261
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4226.42 + 7320.38i 0.316381 + 0.547988i 0.979730 0.200322i \(-0.0641989\pi\)
−0.663349 + 0.748310i \(0.730866\pi\)
\(564\) 0 0
\(565\) −3337.49 + 5780.70i −0.248512 + 0.430435i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7476.78 12950.2i 0.550866 0.954128i −0.447346 0.894361i \(-0.647631\pi\)
0.998212 0.0597671i \(-0.0190358\pi\)
\(570\) 0 0
\(571\) −8010.92 13875.3i −0.587122 1.01693i −0.994607 0.103713i \(-0.966928\pi\)
0.407485 0.913212i \(-0.366406\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 17449.0 1.26552
\(576\) 0 0
\(577\) 8056.54 + 13954.3i 0.581279 + 1.00681i 0.995328 + 0.0965505i \(0.0307809\pi\)
−0.414049 + 0.910255i \(0.635886\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −4068.29 + 4252.16i −0.290501 + 0.303630i
\(582\) 0 0
\(583\) 10359.4 17942.9i 0.735919 1.27465i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9552.04 0.671644 0.335822 0.941925i \(-0.390986\pi\)
0.335822 + 0.941925i \(0.390986\pi\)
\(588\) 0 0
\(589\) −129.656 −0.00907028
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −12083.9 + 20929.9i −0.836807 + 1.44939i 0.0557443 + 0.998445i \(0.482247\pi\)
−0.892551 + 0.450947i \(0.851087\pi\)
\(594\) 0 0
\(595\) 29719.0 31062.2i 2.04766 2.14021i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 410.665 + 711.293i 0.0280122 + 0.0485186i 0.879692 0.475544i \(-0.157749\pi\)
−0.851679 + 0.524063i \(0.824416\pi\)
\(600\) 0 0
\(601\) −14674.7 −0.995996 −0.497998 0.867178i \(-0.665931\pi\)
−0.497998 + 0.867178i \(0.665931\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15777.4 + 27327.3i 1.06024 + 1.83639i
\(606\) 0 0
\(607\) −2541.56 + 4402.11i −0.169949 + 0.294359i −0.938402 0.345547i \(-0.887693\pi\)
0.768453 + 0.639906i \(0.221027\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −15059.2 + 26083.3i −0.997102 + 1.72703i
\(612\) 0 0
\(613\) 11101.0 + 19227.5i 0.731428 + 1.26687i 0.956273 + 0.292476i \(0.0944792\pi\)
−0.224845 + 0.974395i \(0.572187\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14990.6 0.978121 0.489060 0.872250i \(-0.337340\pi\)
0.489060 + 0.872250i \(0.337340\pi\)
\(618\) 0 0
\(619\) −1536.51 2661.32i −0.0997700 0.172807i 0.811819 0.583909i \(-0.198477\pi\)
−0.911589 + 0.411102i \(0.865144\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 492.891 + 1689.28i 0.0316970 + 0.108635i
\(624\) 0 0
\(625\) −5010.29 + 8678.08i −0.320659 + 0.555397i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 30118.4 1.90922
\(630\) 0 0
\(631\) 26012.7 1.64113 0.820563 0.571556i \(-0.193660\pi\)
0.820563 + 0.571556i \(0.193660\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −10367.5 + 17957.0i −0.647905 + 1.12220i
\(636\) 0 0
\(637\) −939.691 21251.2i −0.0584488 1.32182i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4016.12 + 6956.12i 0.247468 + 0.428627i 0.962823 0.270134i \(-0.0870681\pi\)
−0.715355 + 0.698762i \(0.753735\pi\)
\(642\) 0 0
\(643\) −24887.7 −1.52640 −0.763200 0.646162i \(-0.776373\pi\)
−0.763200 + 0.646162i \(0.776373\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10542.2 18259.6i −0.640580 1.10952i −0.985303 0.170813i \(-0.945360\pi\)
0.344723 0.938704i \(-0.387973\pi\)
\(648\) 0 0
\(649\) 5001.95 8663.62i 0.302532 0.524002i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6583.16 11402.4i 0.394516 0.683322i −0.598523 0.801106i \(-0.704246\pi\)
0.993039 + 0.117783i \(0.0375788\pi\)
\(654\) 0 0
\(655\) 20535.1 + 35567.8i 1.22499 + 2.12175i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13903.4 0.821851 0.410926 0.911669i \(-0.365206\pi\)
0.410926 + 0.911669i \(0.365206\pi\)
\(660\) 0 0
\(661\) −3153.30 5461.68i −0.185551 0.321384i 0.758211 0.652009i \(-0.226074\pi\)
−0.943762 + 0.330625i \(0.892740\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4265.32 1042.46i −0.248725 0.0607892i
\(666\) 0 0
\(667\) −8656.23 + 14993.0i −0.502505 + 0.870364i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −21699.1 −1.24841
\(672\) 0 0
\(673\) −24407.6 −1.39798 −0.698992 0.715129i \(-0.746368\pi\)
−0.698992 + 0.715129i \(0.746368\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −15540.1 + 26916.3i −0.882209 + 1.52803i −0.0333299 + 0.999444i \(0.510611\pi\)
−0.848879 + 0.528587i \(0.822722\pi\)
\(678\) 0 0
\(679\) −20597.6 + 21528.5i −1.16416 + 1.21677i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 14379.7 + 24906.4i 0.805601 + 1.39534i 0.915885 + 0.401442i \(0.131491\pi\)
−0.110283 + 0.993900i \(0.535176\pi\)
\(684\) 0 0
\(685\) −37251.0 −2.07779
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −11738.5 20331.7i −0.649061 1.12421i
\(690\) 0 0
\(691\) −12447.7 + 21560.1i −0.685287 + 1.18695i 0.288059 + 0.957613i \(0.406990\pi\)
−0.973346 + 0.229340i \(0.926343\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1298.40 + 2248.89i −0.0708649 + 0.122742i
\(696\) 0 0
\(697\) −14610.2 25305.6i −0.793976 1.37521i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1702.74 0.0917427 0.0458714 0.998947i \(-0.485394\pi\)
0.0458714 + 0.998947i \(0.485394\pi\)
\(702\) 0 0
\(703\) −1538.12 2664.11i −0.0825198 0.142929i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14087.5 3443.04i −0.749385 0.183153i
\(708\) 0 0
\(709\) −3261.60 + 5649.26i −0.172767 + 0.299242i −0.939386 0.342860i \(-0.888604\pi\)
0.766619 + 0.642102i \(0.221938\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 771.765 0.0405369
\(714\) 0 0
\(715\) 64348.4 3.36572
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12627.2 + 21871.0i −0.654959 + 1.13442i 0.326945 + 0.945044i \(0.393981\pi\)
−0.981904 + 0.189379i \(0.939352\pi\)
\(720\) 0 0
\(721\) −7727.27 26483.7i −0.399138 1.36797i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −27258.2 47212.6i −1.39634 2.41853i
\(726\) 0 0
\(727\) −27964.9 −1.42663 −0.713316 0.700843i \(-0.752808\pi\)
−0.713316 + 0.700843i \(0.752808\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −5691.69 9858.29i −0.287982 0.498799i
\(732\) 0 0
\(733\) 4647.19 8049.17i 0.234172 0.405597i −0.724860 0.688896i \(-0.758096\pi\)
0.959032 + 0.283299i \(0.0914288\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7148.91 + 12382.3i −0.357305 + 0.618870i
\(738\) 0 0
\(739\) −9145.60 15840.6i −0.455245 0.788508i 0.543457 0.839437i \(-0.317115\pi\)
−0.998702 + 0.0509292i \(0.983782\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −14742.4 −0.727921 −0.363960 0.931414i \(-0.618576\pi\)
−0.363960 + 0.931414i \(0.618576\pi\)
\(744\) 0 0
\(745\) 2655.12 + 4598.80i 0.130572 + 0.226157i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3697.46 12672.3i −0.180377 0.618204i
\(750\) 0 0
\(751\) −14231.7 + 24650.0i −0.691507 + 1.19773i 0.279837 + 0.960048i \(0.409720\pi\)
−0.971344 + 0.237678i \(0.923614\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −41596.4 −2.00510
\(756\) 0 0
\(757\) −20336.7 −0.976422 −0.488211 0.872726i \(-0.662350\pi\)
−0.488211 + 0.872726i \(0.662350\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19790.8 34278.7i 0.942728 1.63285i 0.182489 0.983208i \(-0.441585\pi\)
0.760238 0.649644i \(-0.225082\pi\)
\(762\) 0 0
\(763\) −18743.6 4581.01i −0.889337 0.217357i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5667.87 9817.04i −0.266825 0.462155i
\(768\) 0 0
\(769\) 10580.3 0.496146 0.248073 0.968741i \(-0.420203\pi\)
0.248073 + 0.968741i \(0.420203\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12961.1 + 22449.3i 0.603077 + 1.04456i 0.992352 + 0.123439i \(0.0393923\pi\)
−0.389275 + 0.921122i \(0.627274\pi\)
\(774\) 0 0
\(775\) −1215.13 + 2104.67i −0.0563211 + 0.0975509i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1492.27 + 2584.68i −0.0686341 + 0.118878i
\(780\) 0 0
\(781\) 23931.7 + 41450.9i 1.09647 + 1.89914i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4186.27 0.190337
\(786\) 0 0
\(787\) −8610.00 14913.0i −0.389979 0.675463i 0.602467 0.798143i \(-0.294184\pi\)
−0.992446 + 0.122680i \(0.960851\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4507.94 + 4711.68i −0.202635 + 0.211793i
\(792\) 0 0
\(793\) −12294.0 + 21293.8i −0.550533 + 0.953551i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6275.52 0.278909 0.139454 0.990228i \(-0.455465\pi\)
0.139454 + 0.990228i \(0.455465\pi\)
\(798\) 0 0
\(799\) −59461.9 −2.63280
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4170.66 7223.80i 0.183287 0.317463i
\(804\) 0 0
\(805\) 25388.8 + 6205.12i 1.11160 + 0.271679i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2302.01 3987.20i −0.100043 0.173279i 0.811659 0.584131i \(-0.198565\pi\)
−0.911702 + 0.410852i \(0.865231\pi\)
\(810\) 0 0
\(811\) 5104.36 0.221009 0.110505 0.993876i \(-0.464753\pi\)
0.110505 + 0.993876i \(0.464753\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 13999.2 + 24247.3i 0.601682 + 1.04214i
\(816\) 0 0
\(817\) −581.341 + 1006.91i −0.0248942 + 0.0431180i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −22817.7 + 39521.4i −0.969967 + 1.68003i −0.274334 + 0.961634i \(0.588458\pi\)
−0.695633 + 0.718398i \(0.744876\pi\)
\(822\) 0 0
\(823\) 19688.6 + 34101.6i 0.833901 + 1.44436i 0.894922 + 0.446222i \(0.147231\pi\)
−0.0610219 + 0.998136i \(0.519436\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 30916.3 1.29996 0.649978 0.759953i \(-0.274778\pi\)
0.649978 + 0.759953i \(0.274778\pi\)
\(828\) 0 0
\(829\) 1271.66 + 2202.59i 0.0532771 + 0.0922786i 0.891434 0.453151i \(-0.149700\pi\)
−0.838157 + 0.545429i \(0.816367\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 35407.0 22584.5i 1.47273 0.939382i
\(834\) 0 0
\(835\) −20831.6 + 36081.5i −0.863364 + 1.49539i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7930.82 0.326343 0.163172 0.986598i \(-0.447828\pi\)
0.163172 + 0.986598i \(0.447828\pi\)
\(840\) 0 0
\(841\) 29700.8 1.21780
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 15632.3 27075.9i 0.636410 1.10229i
\(846\) 0 0
\(847\) 8634.31 + 29592.3i 0.350270 + 1.20048i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9155.51 + 15857.8i 0.368798 + 0.638776i
\(852\) 0 0
\(853\) 41983.2 1.68520 0.842601 0.538538i \(-0.181023\pi\)
0.842601 + 0.538538i \(0.181023\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5927.16 10266.1i −0.236252 0.409200i 0.723384 0.690446i \(-0.242586\pi\)
−0.959636 + 0.281246i \(0.909252\pi\)
\(858\) 0 0
\(859\) 4056.94 7026.83i 0.161142 0.279106i −0.774136 0.633019i \(-0.781816\pi\)
0.935279 + 0.353912i \(0.115149\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 22908.5 39678.7i 0.903609 1.56510i 0.0808353 0.996727i \(-0.474241\pi\)
0.822774 0.568369i \(-0.192425\pi\)
\(864\) 0 0
\(865\) −19826.3 34340.2i −0.779325 1.34983i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −31380.3 −1.22498
\(870\) 0 0
\(871\) 8100.67 + 14030.8i 0.315133 + 0.545826i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −26555.7 + 27755.9i −1.02599 + 1.07236i
\(876\) 0 0
\(877\) −15906.5 + 27550.9i −0.612456 + 1.06081i 0.378369 + 0.925655i \(0.376485\pi\)
−0.990825 + 0.135150i \(0.956848\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −43551.6 −1.66548 −0.832742 0.553661i \(-0.813230\pi\)
−0.832742 + 0.553661i \(0.813230\pi\)
\(882\) 0 0
\(883\) 40645.1 1.54906 0.774528 0.632540i \(-0.217988\pi\)
0.774528 + 0.632540i \(0.217988\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 17014.1 29469.3i 0.644056 1.11554i −0.340462 0.940258i \(-0.610584\pi\)
0.984519 0.175280i \(-0.0560831\pi\)
\(888\) 0 0
\(889\) −14003.3 + 14636.2i −0.528297 + 0.552173i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3036.67 + 5259.68i 0.113794 + 0.197098i
\(894\) 0 0
\(895\) 44258.2 1.65295
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1205.62 2088.20i −0.0447272 0.0774699i
\(900\) 0 0
\(901\) 23175.1 40140.4i 0.856908 1.48421i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16667.9 28869.7i 0.612221 1.06040i
\(906\) 0 0
\(907\) −25402.4 43998.2i −0.929958 1.61073i −0.783387 0.621534i \(-0.786510\pi\)
−0.146571 0.989200i \(-0.546824\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −28738.3 −1.04516 −0.522582 0.852589i \(-0.675031\pi\)
−0.522582 + 0.852589i \(0.675031\pi\)
\(912\) 0 0
\(913\) −8695.46 15061.0i −0.315200 0.545943i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11237.9 + 38515.8i 0.404700 + 1.38703i
\(918\) 0 0
\(919\) 27371.1 47408.2i 0.982470 1.70169i 0.329790 0.944054i \(-0.393022\pi\)
0.652680 0.757634i \(-0.273645\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 54235.5 1.93411
\(924\) 0 0
\(925\) −57660.9 −2.04960
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 21887.9 37911.0i 0.773002 1.33888i −0.162908 0.986641i \(-0.552088\pi\)
0.935911 0.352238i \(-0.114579\pi\)
\(930\) 0 0
\(931\) −3805.91 1978.54i −0.133978 0.0696499i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 63520.6 + 110021.i 2.22176 + 3.84820i
\(936\) 0 0
\(937\) 35090.9 1.22345 0.611724 0.791071i \(-0.290476\pi\)
0.611724 + 0.791071i \(0.290476\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −15892.4 27526.4i −0.550560 0.953599i −0.998234 0.0594016i \(-0.981081\pi\)
0.447674 0.894197i \(-0.352253\pi\)
\(942\) 0 0
\(943\) 8882.54 15385.0i 0.306740 0.531288i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28250.4 48931.2i 0.969394 1.67904i 0.272079 0.962275i \(-0.412289\pi\)
0.697315 0.716765i \(-0.254378\pi\)
\(948\) 0 0
\(949\) −4725.92 8185.53i −0.161654 0.279993i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −36669.1 −1.24641 −0.623204 0.782059i \(-0.714169\pi\)
−0.623204 + 0.782059i \(0.714169\pi\)
\(954\) 0 0
\(955\) 35078.8 + 60758.2i 1.18861 + 2.05873i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −35350.4 8639.76i −1.19033 0.290920i
\(960\) 0 0
\(961\) 14841.8 25706.7i 0.498196 0.862901i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −51346.8 −1.71286
\(966\) 0 0
\(967\) −12012.4 −0.399474 −0.199737 0.979850i \(-0.564009\pi\)
−0.199737 + 0.979850i \(0.564009\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −18648.1 + 32299.5i −0.616320 + 1.06750i 0.373832 + 0.927497i \(0.378044\pi\)
−0.990151 + 0.140000i \(0.955290\pi\)
\(972\) 0 0
\(973\) −1753.75 + 1833.01i −0.0577826 + 0.0603942i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4894.22 + 8477.04i 0.160266 + 0.277589i 0.934964 0.354742i \(-0.115431\pi\)
−0.774698 + 0.632331i \(0.782098\pi\)
\(978\) 0 0
\(979\) −5200.30 −0.169767
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2310.20 4001.39i −0.0749583 0.129832i 0.826110 0.563509i \(-0.190549\pi\)
−0.901068 + 0.433678i \(0.857216\pi\)
\(984\) 0 0
\(985\) 1523.15 2638.17i 0.0492705 0.0853390i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3460.37 5993.53i 0.111257 0.192703i
\(990\) 0 0
\(991\) −20182.7 34957.4i −0.646946 1.12054i −0.983848 0.179004i \(-0.942713\pi\)
0.336903 0.941540i \(-0.390621\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 39167.1 1.24792
\(996\) 0 0
\(997\) 1497.18 + 2593.19i 0.0475587 + 0.0823742i 0.888825 0.458247i \(-0.151523\pi\)
−0.841266 + 0.540621i \(0.818189\pi\)
\(998\) 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.j.289.4 8
3.2 odd 2 168.4.q.f.121.1 yes 8
7.4 even 3 inner 504.4.s.j.361.4 8
12.11 even 2 336.4.q.m.289.1 8
21.2 odd 6 1176.4.a.bd.1.4 4
21.5 even 6 1176.4.a.ba.1.1 4
21.11 odd 6 168.4.q.f.25.1 8
84.11 even 6 336.4.q.m.193.1 8
84.23 even 6 2352.4.a.cm.1.4 4
84.47 odd 6 2352.4.a.cp.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.f.25.1 8 21.11 odd 6
168.4.q.f.121.1 yes 8 3.2 odd 2
336.4.q.m.193.1 8 84.11 even 6
336.4.q.m.289.1 8 12.11 even 2
504.4.s.j.289.4 8 1.1 even 1 trivial
504.4.s.j.361.4 8 7.4 even 3 inner
1176.4.a.ba.1.1 4 21.5 even 6
1176.4.a.bd.1.4 4 21.2 odd 6
2352.4.a.cm.1.4 4 84.23 even 6
2352.4.a.cp.1.1 4 84.47 odd 6