Properties

Label 684.3.y.g.145.3
Level $684$
Weight $3$
Character 684.145
Analytic conductor $18.638$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,3,Mod(145,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.y (of order \(6\), 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: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 34x^{6} + 921x^{4} - 7990x^{2} + 55225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.3
Root \(-2.69047 - 1.55335i\) of defining polynomial
Character \(\chi\) \(=\) 684.145
Dual form 684.3.y.g.217.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.19676 + 3.80490i) q^{5} +1.44949 q^{7} +O(q^{10})\) \(q+(2.19676 + 3.80490i) q^{5} +1.44949 q^{7} -15.1554 q^{11} +(-14.8485 - 8.57277i) q^{13} +(-9.77447 - 16.9299i) q^{17} +(-3.34847 - 18.7026i) q^{19} +(2.19676 - 3.80490i) q^{23} +(2.84847 - 4.93369i) q^{25} +(29.3234 + 16.9299i) q^{29} -5.62475i q^{31} +(3.18418 + 5.51517i) q^{35} -23.7238i q^{37} +(32.2857 - 18.6401i) q^{41} +(-15.9722 - 27.6647i) q^{43} +(-39.0979 + 67.7195i) q^{47} -46.8990 q^{49} +(9.55255 + 5.51517i) q^{53} +(-33.2929 - 57.6649i) q^{55} +(78.4177 - 45.2745i) q^{59} +(57.1413 - 98.9717i) q^{61} -75.3293i q^{65} +(-98.3105 - 56.7596i) q^{67} +(-87.9702 + 50.7896i) q^{71} +(10.0505 + 17.4080i) q^{73} -21.9676 q^{77} +(-45.8258 + 26.4575i) q^{79} +29.2237 q^{83} +(42.9444 - 74.3819i) q^{85} +(-126.846 - 73.2347i) q^{89} +(-21.5227 - 12.4261i) q^{91} +(63.8059 - 53.8258i) q^{95} +(27.0000 - 15.5885i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} - 60 q^{13} + 32 q^{19} - 36 q^{25} - 20 q^{43} - 336 q^{49} + 8 q^{55} + 124 q^{61} - 228 q^{67} + 100 q^{73} - 396 q^{79} + 128 q^{85} - 84 q^{91} + 216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.19676 + 3.80490i 0.439352 + 0.760981i 0.997640 0.0686670i \(-0.0218746\pi\)
−0.558287 + 0.829648i \(0.688541\pi\)
\(6\) 0 0
\(7\) 1.44949 0.207070 0.103535 0.994626i \(-0.466985\pi\)
0.103535 + 0.994626i \(0.466985\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −15.1554 −1.37777 −0.688883 0.724873i \(-0.741898\pi\)
−0.688883 + 0.724873i \(0.741898\pi\)
\(12\) 0 0
\(13\) −14.8485 8.57277i −1.14219 0.659444i −0.195218 0.980760i \(-0.562541\pi\)
−0.946972 + 0.321316i \(0.895875\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −9.77447 16.9299i −0.574969 0.995875i −0.996045 0.0888501i \(-0.971681\pi\)
0.421076 0.907025i \(-0.361653\pi\)
\(18\) 0 0
\(19\) −3.34847 18.7026i −0.176235 0.984348i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.19676 3.80490i 0.0955114 0.165431i −0.814311 0.580429i \(-0.802885\pi\)
0.909822 + 0.414999i \(0.136218\pi\)
\(24\) 0 0
\(25\) 2.84847 4.93369i 0.113939 0.197348i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 29.3234 + 16.9299i 1.01115 + 0.583789i 0.911529 0.411236i \(-0.134903\pi\)
0.0996235 + 0.995025i \(0.468236\pi\)
\(30\) 0 0
\(31\) 5.62475i 0.181443i −0.995876 0.0907217i \(-0.971083\pi\)
0.995876 0.0907217i \(-0.0289174\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.18418 + 5.51517i 0.0909767 + 0.157576i
\(36\) 0 0
\(37\) 23.7238i 0.641184i −0.947217 0.320592i \(-0.896118\pi\)
0.947217 0.320592i \(-0.103882\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 32.2857 18.6401i 0.787456 0.454638i −0.0516104 0.998667i \(-0.516435\pi\)
0.839066 + 0.544030i \(0.183102\pi\)
\(42\) 0 0
\(43\) −15.9722 27.6647i −0.371446 0.643364i 0.618342 0.785909i \(-0.287805\pi\)
−0.989788 + 0.142545i \(0.954471\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −39.0979 + 67.7195i −0.831870 + 1.44084i 0.0646836 + 0.997906i \(0.479396\pi\)
−0.896554 + 0.442935i \(0.853937\pi\)
\(48\) 0 0
\(49\) −46.8990 −0.957122
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.55255 + 5.51517i 0.180237 + 0.104060i 0.587404 0.809294i \(-0.300150\pi\)
−0.407167 + 0.913354i \(0.633483\pi\)
\(54\) 0 0
\(55\) −33.2929 57.6649i −0.605325 1.04845i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 78.4177 45.2745i 1.32911 0.767364i 0.343951 0.938988i \(-0.388235\pi\)
0.985163 + 0.171624i \(0.0549013\pi\)
\(60\) 0 0
\(61\) 57.1413 98.9717i 0.936743 1.62249i 0.165247 0.986252i \(-0.447158\pi\)
0.771496 0.636234i \(-0.219509\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 75.3293i 1.15891i
\(66\) 0 0
\(67\) −98.3105 56.7596i −1.46732 0.847158i −0.467989 0.883734i \(-0.655021\pi\)
−0.999331 + 0.0365763i \(0.988355\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −87.9702 + 50.7896i −1.23902 + 0.715347i −0.968893 0.247478i \(-0.920398\pi\)
−0.270124 + 0.962825i \(0.587065\pi\)
\(72\) 0 0
\(73\) 10.0505 + 17.4080i 0.137678 + 0.238466i 0.926617 0.376006i \(-0.122703\pi\)
−0.788939 + 0.614471i \(0.789369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −21.9676 −0.285294
\(78\) 0 0
\(79\) −45.8258 + 26.4575i −0.580073 + 0.334905i −0.761162 0.648561i \(-0.775371\pi\)
0.181089 + 0.983467i \(0.442038\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 29.2237 0.352092 0.176046 0.984382i \(-0.443669\pi\)
0.176046 + 0.984382i \(0.443669\pi\)
\(84\) 0 0
\(85\) 42.9444 74.3819i 0.505228 0.875081i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −126.846 73.2347i −1.42524 0.822862i −0.428498 0.903543i \(-0.640957\pi\)
−0.996740 + 0.0806807i \(0.974291\pi\)
\(90\) 0 0
\(91\) −21.5227 12.4261i −0.236513 0.136551i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 63.8059 53.8258i 0.671641 0.566587i
\(96\) 0 0
\(97\) 27.0000 15.5885i 0.278351 0.160706i −0.354326 0.935122i \(-0.615290\pi\)
0.632676 + 0.774416i \(0.281956\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −66.0026 + 114.320i −0.653491 + 1.13188i 0.328778 + 0.944407i \(0.393363\pi\)
−0.982270 + 0.187473i \(0.939970\pi\)
\(102\) 0 0
\(103\) 41.3156i 0.401122i 0.979681 + 0.200561i \(0.0642765\pi\)
−0.979681 + 0.200561i \(0.935723\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 33.8598i 0.316446i 0.987403 + 0.158223i \(0.0505766\pi\)
−0.987403 + 0.158223i \(0.949423\pi\)
\(108\) 0 0
\(109\) −30.3031 + 17.4955i −0.278010 + 0.160509i −0.632522 0.774542i \(-0.717980\pi\)
0.354512 + 0.935051i \(0.384647\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 163.227i 1.44448i 0.691641 + 0.722241i \(0.256888\pi\)
−0.691641 + 0.722241i \(0.743112\pi\)
\(114\) 0 0
\(115\) 19.3031 0.167853
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −14.1680 24.5397i −0.119059 0.206216i
\(120\) 0 0
\(121\) 108.687 0.898237
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 134.868 1.07894
\(126\) 0 0
\(127\) −135.576 78.2746i −1.06752 0.616335i −0.140020 0.990149i \(-0.544717\pi\)
−0.927504 + 0.373814i \(0.878050\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −107.963 186.997i −0.824145 1.42746i −0.902571 0.430541i \(-0.858323\pi\)
0.0784255 0.996920i \(-0.475011\pi\)
\(132\) 0 0
\(133\) −4.85357 27.1092i −0.0364930 0.203829i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 29.7673 51.5584i 0.217279 0.376339i −0.736696 0.676224i \(-0.763615\pi\)
0.953975 + 0.299885i \(0.0969485\pi\)
\(138\) 0 0
\(139\) −1.82577 + 3.16232i −0.0131350 + 0.0227505i −0.872518 0.488582i \(-0.837514\pi\)
0.859383 + 0.511332i \(0.170848\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 225.035 + 129.924i 1.57367 + 0.908559i
\(144\) 0 0
\(145\) 148.764i 1.02596i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −29.0017 50.2325i −0.194643 0.337131i 0.752141 0.659003i \(-0.229021\pi\)
−0.946783 + 0.321872i \(0.895688\pi\)
\(150\) 0 0
\(151\) 8.64258i 0.0572356i 0.999590 + 0.0286178i \(0.00911058\pi\)
−0.999590 + 0.0286178i \(0.990889\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 21.4016 12.3562i 0.138075 0.0797176i
\(156\) 0 0
\(157\) 60.0301 + 103.975i 0.382357 + 0.662262i 0.991399 0.130876i \(-0.0417790\pi\)
−0.609041 + 0.793138i \(0.708446\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.18418 5.51517i 0.0197775 0.0342557i
\(162\) 0 0
\(163\) −51.8332 −0.317995 −0.158997 0.987279i \(-0.550826\pi\)
−0.158997 + 0.987279i \(0.550826\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −125.215 72.2932i −0.749793 0.432893i 0.0758260 0.997121i \(-0.475841\pi\)
−0.825619 + 0.564228i \(0.809174\pi\)
\(168\) 0 0
\(169\) 62.4847 + 108.227i 0.369732 + 0.640394i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 57.0161 32.9182i 0.329573 0.190279i −0.326079 0.945343i \(-0.605727\pi\)
0.655651 + 0.755064i \(0.272394\pi\)
\(174\) 0 0
\(175\) 4.12883 7.15134i 0.0235933 0.0408648i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 60.1097i 0.335808i 0.985803 + 0.167904i \(0.0536999\pi\)
−0.985803 + 0.167904i \(0.946300\pi\)
\(180\) 0 0
\(181\) 218.530 + 126.168i 1.20735 + 0.697063i 0.962179 0.272417i \(-0.0878231\pi\)
0.245169 + 0.969480i \(0.421156\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 90.2668 52.1155i 0.487928 0.281706i
\(186\) 0 0
\(187\) 148.136 + 256.579i 0.792172 + 1.37208i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −211.533 −1.10750 −0.553750 0.832683i \(-0.686804\pi\)
−0.553750 + 0.832683i \(0.686804\pi\)
\(192\) 0 0
\(193\) 48.4546 27.9753i 0.251060 0.144950i −0.369189 0.929354i \(-0.620365\pi\)
0.620250 + 0.784405i \(0.287031\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 95.1265 0.482876 0.241438 0.970416i \(-0.422381\pi\)
0.241438 + 0.970416i \(0.422381\pi\)
\(198\) 0 0
\(199\) −40.4115 + 69.9947i −0.203073 + 0.351732i −0.949517 0.313716i \(-0.898426\pi\)
0.746444 + 0.665448i \(0.231759\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 42.5040 + 24.5397i 0.209379 + 0.120885i
\(204\) 0 0
\(205\) 141.848 + 81.8960i 0.691941 + 0.399492i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 50.7475 + 283.446i 0.242811 + 1.35620i
\(210\) 0 0
\(211\) −46.0982 + 26.6148i −0.218475 + 0.126137i −0.605244 0.796040i \(-0.706924\pi\)
0.386769 + 0.922177i \(0.373591\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 70.1742 121.545i 0.326392 0.565327i
\(216\) 0 0
\(217\) 8.15301i 0.0375715i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 335.177i 1.51664i
\(222\) 0 0
\(223\) −171.356 + 98.9324i −0.768412 + 0.443643i −0.832308 0.554314i \(-0.812981\pi\)
0.0638959 + 0.997957i \(0.479647\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 125.946i 0.554829i 0.960750 + 0.277415i \(0.0894775\pi\)
−0.960750 + 0.277415i \(0.910522\pi\)
\(228\) 0 0
\(229\) 193.586 0.845352 0.422676 0.906281i \(-0.361091\pi\)
0.422676 + 0.906281i \(0.361091\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −148.592 257.369i −0.637734 1.10459i −0.985929 0.167165i \(-0.946539\pi\)
0.348195 0.937422i \(-0.386795\pi\)
\(234\) 0 0
\(235\) −343.555 −1.46194
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 104.801 0.438499 0.219250 0.975669i \(-0.429639\pi\)
0.219250 + 0.975669i \(0.429639\pi\)
\(240\) 0 0
\(241\) 356.787 + 205.991i 1.48044 + 0.854735i 0.999755 0.0221551i \(-0.00705275\pi\)
0.480690 + 0.876890i \(0.340386\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −103.026 178.446i −0.420514 0.728352i
\(246\) 0 0
\(247\) −110.614 + 306.411i −0.447828 + 1.24053i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 87.8705 152.196i 0.350082 0.606359i −0.636182 0.771539i \(-0.719487\pi\)
0.986263 + 0.165180i \(0.0528205\pi\)
\(252\) 0 0
\(253\) −33.2929 + 57.6649i −0.131592 + 0.227925i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −373.283 215.515i −1.45246 0.838579i −0.453841 0.891083i \(-0.649947\pi\)
−0.998621 + 0.0525037i \(0.983280\pi\)
\(258\) 0 0
\(259\) 34.3874i 0.132770i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −60.0781 104.058i −0.228434 0.395659i 0.728910 0.684609i \(-0.240027\pi\)
−0.957344 + 0.288950i \(0.906694\pi\)
\(264\) 0 0
\(265\) 48.4621i 0.182876i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −415.487 + 239.882i −1.54456 + 0.891754i −0.546021 + 0.837772i \(0.683858\pi\)
−0.998542 + 0.0539820i \(0.982809\pi\)
\(270\) 0 0
\(271\) −63.0908 109.276i −0.232807 0.403234i 0.725826 0.687879i \(-0.241458\pi\)
−0.958633 + 0.284644i \(0.908125\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −43.1697 + 74.7722i −0.156981 + 0.271899i
\(276\) 0 0
\(277\) 459.918 1.66036 0.830178 0.557499i \(-0.188239\pi\)
0.830178 + 0.557499i \(0.188239\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −65.2371 37.6647i −0.232161 0.134038i 0.379408 0.925230i \(-0.376128\pi\)
−0.611568 + 0.791192i \(0.709461\pi\)
\(282\) 0 0
\(283\) 178.753 + 309.609i 0.631634 + 1.09402i 0.987218 + 0.159378i \(0.0509490\pi\)
−0.355583 + 0.934645i \(0.615718\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 46.7978 27.0187i 0.163058 0.0941418i
\(288\) 0 0
\(289\) −46.5806 + 80.6800i −0.161179 + 0.279169i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 228.294i 0.779161i −0.920992 0.389581i \(-0.872620\pi\)
0.920992 0.389581i \(-0.127380\pi\)
\(294\) 0 0
\(295\) 344.530 + 198.915i 1.16790 + 0.674287i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −65.2371 + 37.6647i −0.218184 + 0.125969i
\(300\) 0 0
\(301\) −23.1515 40.0996i −0.0769154 0.133221i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 502.104 1.64624
\(306\) 0 0
\(307\) −83.6969 + 48.3224i −0.272628 + 0.157402i −0.630082 0.776529i \(-0.716979\pi\)
0.357453 + 0.933931i \(0.383645\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 414.722 1.33351 0.666755 0.745277i \(-0.267683\pi\)
0.666755 + 0.745277i \(0.267683\pi\)
\(312\) 0 0
\(313\) 164.000 284.056i 0.523962 0.907528i −0.475649 0.879635i \(-0.657787\pi\)
0.999611 0.0278932i \(-0.00887983\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 461.253 + 266.304i 1.45506 + 0.840077i 0.998762 0.0497515i \(-0.0158429\pi\)
0.456295 + 0.889829i \(0.349176\pi\)
\(318\) 0 0
\(319\) −444.409 256.579i −1.39313 0.804324i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −283.904 + 239.497i −0.878958 + 0.741478i
\(324\) 0 0
\(325\) −84.5908 + 48.8385i −0.260279 + 0.150272i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −56.6720 + 98.1588i −0.172255 + 0.298355i
\(330\) 0 0
\(331\) 138.310i 0.417856i −0.977931 0.208928i \(-0.933003\pi\)
0.977931 0.208928i \(-0.0669975\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 498.749i 1.48880i
\(336\) 0 0
\(337\) −200.409 + 115.706i −0.594686 + 0.343342i −0.766948 0.641709i \(-0.778226\pi\)
0.172262 + 0.985051i \(0.444892\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 85.2454i 0.249986i
\(342\) 0 0
\(343\) −139.005 −0.405261
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 291.137 + 504.264i 0.839012 + 1.45321i 0.890722 + 0.454549i \(0.150200\pi\)
−0.0517097 + 0.998662i \(0.516467\pi\)
\(348\) 0 0
\(349\) 160.444 0.459725 0.229862 0.973223i \(-0.426172\pi\)
0.229862 + 0.973223i \(0.426172\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 162.959 0.461641 0.230821 0.972996i \(-0.425859\pi\)
0.230821 + 0.972996i \(0.425859\pi\)
\(354\) 0 0
\(355\) −386.499 223.146i −1.08873 0.628579i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 130.674 + 226.334i 0.363994 + 0.630456i 0.988614 0.150473i \(-0.0480796\pi\)
−0.624620 + 0.780928i \(0.714746\pi\)
\(360\) 0 0
\(361\) −338.576 + 125.250i −0.937882 + 0.346954i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −44.1572 + 76.4825i −0.120979 + 0.209541i
\(366\) 0 0
\(367\) 229.750 397.938i 0.626021 1.08430i −0.362321 0.932053i \(-0.618016\pi\)
0.988342 0.152247i \(-0.0486510\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 13.8463 + 7.99418i 0.0373216 + 0.0215477i
\(372\) 0 0
\(373\) 48.6194i 0.130347i −0.997874 0.0651734i \(-0.979240\pi\)
0.997874 0.0651734i \(-0.0207601\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −290.272 502.766i −0.769952 1.33360i
\(378\) 0 0
\(379\) 523.962i 1.38249i −0.722622 0.691243i \(-0.757063\pi\)
0.722622 0.691243i \(-0.242937\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −339.665 + 196.106i −0.886855 + 0.512026i −0.872912 0.487877i \(-0.837771\pi\)
−0.0139424 + 0.999903i \(0.504438\pi\)
\(384\) 0 0
\(385\) −48.2577 83.5847i −0.125345 0.217103i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 266.108 460.912i 0.684081 1.18486i −0.289644 0.957135i \(-0.593537\pi\)
0.973725 0.227729i \(-0.0731299\pi\)
\(390\) 0 0
\(391\) −85.8888 −0.219664
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −201.337 116.242i −0.509713 0.294283i
\(396\) 0 0
\(397\) −62.3332 107.964i −0.157010 0.271950i 0.776779 0.629774i \(-0.216852\pi\)
−0.933789 + 0.357823i \(0.883519\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 174.975 101.022i 0.436348 0.251926i −0.265699 0.964056i \(-0.585603\pi\)
0.702047 + 0.712130i \(0.252270\pi\)
\(402\) 0 0
\(403\) −48.2196 + 83.5189i −0.119652 + 0.207243i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 359.544i 0.883401i
\(408\) 0 0
\(409\) 473.091 + 273.139i 1.15670 + 0.667822i 0.950511 0.310690i \(-0.100560\pi\)
0.206190 + 0.978512i \(0.433893\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 113.666 65.6249i 0.275219 0.158898i
\(414\) 0 0
\(415\) 64.1975 + 111.193i 0.154693 + 0.267936i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −529.552 −1.26385 −0.631924 0.775031i \(-0.717734\pi\)
−0.631924 + 0.775031i \(0.717734\pi\)
\(420\) 0 0
\(421\) 225.681 130.297i 0.536060 0.309494i −0.207421 0.978252i \(-0.566507\pi\)
0.743480 + 0.668758i \(0.233174\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −111.369 −0.262045
\(426\) 0 0
\(427\) 82.8258 143.458i 0.193971 0.335968i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −208.226 120.219i −0.483123 0.278931i 0.238594 0.971119i \(-0.423314\pi\)
−0.721717 + 0.692188i \(0.756647\pi\)
\(432\) 0 0
\(433\) −715.847 413.295i −1.65323 0.954491i −0.975733 0.218963i \(-0.929733\pi\)
−0.677494 0.735528i \(-0.736934\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −78.5174 28.3446i −0.179674 0.0648618i
\(438\) 0 0
\(439\) 516.901 298.433i 1.17745 0.679801i 0.222027 0.975040i \(-0.428733\pi\)
0.955423 + 0.295239i \(0.0953993\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 259.517 449.497i 0.585818 1.01467i −0.408955 0.912554i \(-0.634107\pi\)
0.994773 0.102112i \(-0.0325599\pi\)
\(444\) 0 0
\(445\) 643.517i 1.44611i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 50.6169i 0.112732i 0.998410 + 0.0563662i \(0.0179514\pi\)
−0.998410 + 0.0563662i \(0.982049\pi\)
\(450\) 0 0
\(451\) −489.303 + 282.499i −1.08493 + 0.626384i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 109.189i 0.239976i
\(456\) 0 0
\(457\) 219.788 0.480936 0.240468 0.970657i \(-0.422699\pi\)
0.240468 + 0.970657i \(0.422699\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 23.1770 + 40.1437i 0.0502754 + 0.0870796i 0.890068 0.455828i \(-0.150657\pi\)
−0.839793 + 0.542907i \(0.817323\pi\)
\(462\) 0 0
\(463\) −678.308 −1.46503 −0.732514 0.680752i \(-0.761653\pi\)
−0.732514 + 0.680752i \(0.761653\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −687.475 −1.47211 −0.736054 0.676923i \(-0.763313\pi\)
−0.736054 + 0.676923i \(0.763313\pi\)
\(468\) 0 0
\(469\) −142.500 82.2724i −0.303838 0.175421i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 242.065 + 419.269i 0.511766 + 0.886405i
\(474\) 0 0
\(475\) −101.811 36.7535i −0.214339 0.0773758i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 194.946 337.656i 0.406985 0.704919i −0.587565 0.809177i \(-0.699913\pi\)
0.994550 + 0.104258i \(0.0332468\pi\)
\(480\) 0 0
\(481\) −203.379 + 352.262i −0.422824 + 0.732353i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 118.625 + 68.4883i 0.244588 + 0.141213i
\(486\) 0 0
\(487\) 179.520i 0.368624i −0.982868 0.184312i \(-0.940994\pi\)
0.982868 0.184312i \(-0.0590057\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 253.593 + 439.235i 0.516482 + 0.894573i 0.999817 + 0.0191376i \(0.00609205\pi\)
−0.483335 + 0.875436i \(0.660575\pi\)
\(492\) 0 0
\(493\) 661.923i 1.34264i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −127.512 + 73.6191i −0.256563 + 0.148127i
\(498\) 0 0
\(499\) −359.729 623.069i −0.720900 1.24864i −0.960639 0.277799i \(-0.910395\pi\)
0.239739 0.970837i \(-0.422938\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −315.102 + 545.773i −0.626445 + 1.08504i 0.361814 + 0.932250i \(0.382158\pi\)
−0.988259 + 0.152785i \(0.951176\pi\)
\(504\) 0 0
\(505\) −579.968 −1.14845
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 83.0780 + 47.9651i 0.163218 + 0.0942340i 0.579384 0.815055i \(-0.303293\pi\)
−0.416166 + 0.909289i \(0.636626\pi\)
\(510\) 0 0
\(511\) 14.5681 + 25.2327i 0.0285090 + 0.0493791i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −157.202 + 90.7606i −0.305246 + 0.176234i
\(516\) 0 0
\(517\) 592.545 1026.32i 1.14612 1.98514i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 253.352i 0.486281i 0.969991 + 0.243140i \(0.0781776\pi\)
−0.969991 + 0.243140i \(0.921822\pi\)
\(522\) 0 0
\(523\) −406.309 234.583i −0.776882 0.448533i 0.0584420 0.998291i \(-0.481387\pi\)
−0.835324 + 0.549758i \(0.814720\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −95.2263 + 54.9789i −0.180695 + 0.104324i
\(528\) 0 0
\(529\) 254.848 + 441.410i 0.481755 + 0.834424i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −639.191 −1.19923
\(534\) 0 0
\(535\) −128.833 + 74.3819i −0.240810 + 0.139032i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 710.774 1.31869
\(540\) 0 0
\(541\) 461.257 798.921i 0.852601 1.47675i −0.0262519 0.999655i \(-0.508357\pi\)
0.878853 0.477093i \(-0.158309\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −133.137 76.8668i −0.244289 0.141040i
\(546\) 0 0
\(547\) 159.901 + 92.3187i 0.292323 + 0.168773i 0.638989 0.769216i \(-0.279353\pi\)
−0.346666 + 0.937989i \(0.612686\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 218.444 605.114i 0.396451 1.09821i
\(552\) 0 0
\(553\) −66.4240 + 38.3499i −0.120116 + 0.0693488i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 392.410 679.674i 0.704507 1.22024i −0.262363 0.964969i \(-0.584502\pi\)
0.966869 0.255272i \(-0.0821650\pi\)
\(558\) 0 0
\(559\) 547.704i 0.979792i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 700.448i 1.24413i −0.782964 0.622067i \(-0.786293\pi\)
0.782964 0.622067i \(-0.213707\pi\)
\(564\) 0 0
\(565\) −621.061 + 358.570i −1.09922 + 0.634637i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 251.046i 0.441206i 0.975364 + 0.220603i \(0.0708025\pi\)
−0.975364 + 0.220603i \(0.929198\pi\)
\(570\) 0 0
\(571\) 728.378 1.27562 0.637809 0.770194i \(-0.279841\pi\)
0.637809 + 0.770194i \(0.279841\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.5148 21.6763i −0.0217649 0.0376979i
\(576\) 0 0
\(577\) −200.504 −0.347494 −0.173747 0.984790i \(-0.555588\pi\)
−0.173747 + 0.984790i \(0.555588\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 42.3594 0.0729078
\(582\) 0 0
\(583\) −144.773 83.5847i −0.248324 0.143370i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 257.221 + 445.519i 0.438195 + 0.758977i 0.997550 0.0699515i \(-0.0222845\pi\)
−0.559355 + 0.828928i \(0.688951\pi\)
\(588\) 0 0
\(589\) −105.197 + 18.8343i −0.178603 + 0.0319767i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −399.444 + 691.858i −0.673599 + 1.16671i 0.303277 + 0.952902i \(0.401919\pi\)
−0.976876 + 0.213805i \(0.931414\pi\)
\(594\) 0 0
\(595\) 62.2474 107.816i 0.104618 0.181203i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 147.881 + 85.3793i 0.246880 + 0.142536i 0.618335 0.785915i \(-0.287808\pi\)
−0.371455 + 0.928451i \(0.621141\pi\)
\(600\) 0 0
\(601\) 836.789i 1.39233i 0.717883 + 0.696164i \(0.245111\pi\)
−0.717883 + 0.696164i \(0.754889\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 238.759 + 413.543i 0.394643 + 0.683542i
\(606\) 0 0
\(607\) 285.238i 0.469914i −0.972006 0.234957i \(-0.924505\pi\)
0.972006 0.234957i \(-0.0754949\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1161.09 670.354i 1.90031 1.09714i
\(612\) 0 0
\(613\) −330.388 572.249i −0.538969 0.933522i −0.998960 0.0455986i \(-0.985480\pi\)
0.459990 0.887924i \(-0.347853\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 372.340 644.912i 0.603468 1.04524i −0.388823 0.921312i \(-0.627118\pi\)
0.992292 0.123926i \(-0.0395484\pi\)
\(618\) 0 0
\(619\) 94.1964 0.152175 0.0760876 0.997101i \(-0.475757\pi\)
0.0760876 + 0.997101i \(0.475757\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −183.862 106.153i −0.295124 0.170390i
\(624\) 0 0
\(625\) 225.061 + 389.817i 0.360097 + 0.623707i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −401.641 + 231.888i −0.638539 + 0.368661i
\(630\) 0 0
\(631\) 284.195 492.240i 0.450388 0.780094i −0.548022 0.836464i \(-0.684619\pi\)
0.998410 + 0.0563693i \(0.0179524\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 687.802i 1.08315i
\(636\) 0 0
\(637\) 696.378 + 402.054i 1.09322 + 0.631168i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −233.256 + 134.670i −0.363894 + 0.210094i −0.670787 0.741650i \(-0.734044\pi\)
0.306894 + 0.951744i \(0.400710\pi\)
\(642\) 0 0
\(643\) 577.725 + 1000.65i 0.898483 + 1.55622i 0.829434 + 0.558605i \(0.188663\pi\)
0.0690498 + 0.997613i \(0.478003\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1039.16 1.60611 0.803057 0.595902i \(-0.203205\pi\)
0.803057 + 0.595902i \(0.203205\pi\)
\(648\) 0 0
\(649\) −1188.45 + 686.154i −1.83121 + 1.05725i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1115.67 −1.70852 −0.854262 0.519843i \(-0.825991\pi\)
−0.854262 + 0.519843i \(0.825991\pi\)
\(654\) 0 0
\(655\) 474.338 821.578i 0.724181 1.25432i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 86.7060 + 50.0597i 0.131572 + 0.0759632i 0.564341 0.825542i \(-0.309130\pi\)
−0.432769 + 0.901505i \(0.642464\pi\)
\(660\) 0 0
\(661\) −759.303 438.384i −1.14872 0.663213i −0.200144 0.979766i \(-0.564141\pi\)
−0.948575 + 0.316553i \(0.897474\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 92.4859 78.0200i 0.139077 0.117323i
\(666\) 0 0
\(667\) 128.833 74.3819i 0.193153 0.111517i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −866.001 + 1499.96i −1.29061 + 2.23541i
\(672\) 0 0
\(673\) 838.032i 1.24522i 0.782533 + 0.622609i \(0.213927\pi\)
−0.782533 + 0.622609i \(0.786073\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 572.195i 0.845193i 0.906318 + 0.422596i \(0.138881\pi\)
−0.906318 + 0.422596i \(0.861119\pi\)
\(678\) 0 0
\(679\) 39.1362 22.5953i 0.0576380 0.0332773i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 191.437i 0.280289i −0.990131 0.140144i \(-0.955243\pi\)
0.990131 0.140144i \(-0.0447567\pi\)
\(684\) 0 0
\(685\) 261.566 0.381849
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −94.5605 163.784i −0.137243 0.237712i
\(690\) 0 0
\(691\) 1090.79 1.57856 0.789281 0.614033i \(-0.210454\pi\)
0.789281 + 0.614033i \(0.210454\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.0431 −0.0230836
\(696\) 0 0
\(697\) −631.151 364.395i −0.905525 0.522805i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −404.337 700.331i −0.576800 0.999046i −0.995844 0.0910803i \(-0.970968\pi\)
0.419044 0.907966i \(-0.362365\pi\)
\(702\) 0 0
\(703\) −443.697 + 79.4384i −0.631148 + 0.112999i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −95.6701 + 165.706i −0.135318 + 0.234378i
\(708\) 0 0
\(709\) 153.803 266.395i 0.216930 0.375733i −0.736938 0.675960i \(-0.763729\pi\)
0.953868 + 0.300227i \(0.0970625\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −21.4016 12.3562i −0.0300163 0.0173299i
\(714\) 0 0
\(715\) 1141.65i 1.59671i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −156.569 271.185i −0.217759 0.377169i 0.736364 0.676586i \(-0.236541\pi\)
−0.954122 + 0.299417i \(0.903208\pi\)
\(720\) 0 0
\(721\) 59.8865i 0.0830604i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 167.054 96.4485i 0.230419 0.133032i
\(726\) 0 0
\(727\) 295.082 + 511.098i 0.405890 + 0.703023i 0.994425 0.105449i \(-0.0336281\pi\)
−0.588534 + 0.808472i \(0.700295\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −312.240 + 540.815i −0.427140 + 0.739829i
\(732\) 0 0
\(733\) −775.514 −1.05800 −0.529000 0.848622i \(-0.677433\pi\)
−0.529000 + 0.848622i \(0.677433\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1489.94 + 860.215i 2.02162 + 1.16718i
\(738\) 0 0
\(739\) 473.628 + 820.348i 0.640904 + 1.11008i 0.985231 + 0.171229i \(0.0547738\pi\)
−0.344327 + 0.938850i \(0.611893\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 487.913 281.697i 0.656680 0.379134i −0.134331 0.990937i \(-0.542888\pi\)
0.791011 + 0.611802i \(0.209555\pi\)
\(744\) 0 0
\(745\) 127.420 220.698i 0.171033 0.296239i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 49.0794i 0.0655265i
\(750\) 0 0
\(751\) 863.689 + 498.651i 1.15005 + 0.663982i 0.948900 0.315578i \(-0.102198\pi\)
0.201152 + 0.979560i \(0.435532\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −32.8842 + 18.9857i −0.0435552 + 0.0251466i
\(756\) 0 0
\(757\) 595.954 + 1032.22i 0.787258 + 1.36357i 0.927641 + 0.373473i \(0.121833\pi\)
−0.140383 + 0.990097i \(0.544833\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 516.905 0.679244 0.339622 0.940562i \(-0.389701\pi\)
0.339622 + 0.940562i \(0.389701\pi\)
\(762\) 0 0
\(763\) −43.9240 + 25.3595i −0.0575675 + 0.0332366i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1552.51 −2.02413
\(768\) 0 0
\(769\) −142.307 + 246.483i −0.185055 + 0.320524i −0.943595 0.331102i \(-0.892580\pi\)
0.758540 + 0.651626i \(0.225913\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −792.031 457.280i −1.02462 0.591565i −0.109181 0.994022i \(-0.534823\pi\)
−0.915439 + 0.402457i \(0.868156\pi\)
\(774\) 0 0
\(775\) −27.7508 16.0219i −0.0358074 0.0206734i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −456.727 541.411i −0.586299 0.695007i
\(780\) 0 0
\(781\) 1333.23 769.738i 1.70708 0.985581i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −263.744 + 456.818i −0.335979 + 0.581933i
\(786\) 0 0
\(787\) 199.823i 0.253905i 0.991909 + 0.126952i \(0.0405195\pi\)
−0.991909 + 0.126952i \(0.959480\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 236.595i 0.299109i
\(792\) 0 0
\(793\) −1696.92 + 979.719i −2.13988 + 1.23546i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1490.18i 1.86973i −0.355003 0.934865i \(-0.615520\pi\)
0.355003 0.934865i \(-0.384480\pi\)
\(798\) 0 0
\(799\) 1528.64 1.91320
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −152.320 263.825i −0.189688 0.328550i
\(804\) 0 0
\(805\) 27.9796 0.0347573
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −235.121 −0.290631 −0.145316 0.989385i \(-0.546420\pi\)
−0.145316 + 0.989385i \(0.546420\pi\)
\(810\) 0 0
\(811\) 406.665 + 234.788i 0.501437 + 0.289505i 0.729307 0.684187i \(-0.239843\pi\)
−0.227870 + 0.973692i \(0.573176\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −113.865 197.220i −0.139712 0.241988i
\(816\) 0 0
\(817\) −463.919 + 391.356i −0.567832 + 0.479016i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 370.642 641.971i 0.451452 0.781938i −0.547025 0.837117i \(-0.684240\pi\)
0.998476 + 0.0551789i \(0.0175729\pi\)
\(822\) 0 0
\(823\) −514.242 + 890.693i −0.624838 + 1.08225i 0.363734 + 0.931503i \(0.381502\pi\)
−0.988572 + 0.150749i \(0.951832\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −613.862 354.413i −0.742275 0.428553i 0.0806207 0.996745i \(-0.474310\pi\)
−0.822896 + 0.568192i \(0.807643\pi\)
\(828\) 0 0
\(829\) 695.252i 0.838664i 0.907833 + 0.419332i \(0.137736\pi\)
−0.907833 + 0.419332i \(0.862264\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 458.413 + 793.994i 0.550315 + 0.953174i
\(834\) 0 0
\(835\) 635.244i 0.760771i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 747.665 431.664i 0.891138 0.514499i 0.0168234 0.999858i \(-0.494645\pi\)
0.874315 + 0.485360i \(0.161311\pi\)
\(840\) 0 0
\(841\) 152.742 + 264.557i 0.181619 + 0.314574i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −274.528 + 475.497i −0.324885 + 0.562718i
\(846\) 0 0
\(847\) 157.540 0.185998
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −90.2668 52.1155i −0.106071 0.0612404i
\(852\) 0 0
\(853\) −92.1663 159.637i −0.108050 0.187147i 0.806930 0.590646i \(-0.201127\pi\)
−0.914980 + 0.403499i \(0.867794\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −684.058 + 394.941i −0.798201 + 0.460842i −0.842842 0.538161i \(-0.819119\pi\)
0.0446406 + 0.999003i \(0.485786\pi\)
\(858\) 0 0
\(859\) 627.043 1086.07i 0.729968 1.26434i −0.226928 0.973912i \(-0.572868\pi\)
0.956896 0.290430i \(-0.0937984\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1094.97i 1.26879i 0.773009 + 0.634395i \(0.218751\pi\)
−0.773009 + 0.634395i \(0.781249\pi\)
\(864\) 0 0
\(865\) 250.502 + 144.627i 0.289597 + 0.167199i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 694.509 400.975i 0.799204 0.461421i
\(870\) 0 0
\(871\) 973.173 + 1685.59i 1.11731 + 1.93523i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 195.489 0.223417
\(876\) 0 0
\(877\) −15.6520 + 9.03671i −0.0178473 + 0.0103041i −0.508897 0.860827i \(-0.669947\pi\)
0.491050 + 0.871131i \(0.336613\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −362.089 −0.410998 −0.205499 0.978657i \(-0.565882\pi\)
−0.205499 + 0.978657i \(0.565882\pi\)
\(882\) 0 0
\(883\) 517.836 896.918i 0.586451 1.01576i −0.408242 0.912874i \(-0.633858\pi\)
0.994693 0.102889i \(-0.0328086\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 374.913 + 216.456i 0.422676 + 0.244032i 0.696221 0.717827i \(-0.254863\pi\)
−0.273546 + 0.961859i \(0.588196\pi\)
\(888\) 0 0
\(889\) −196.515 113.458i −0.221052 0.127624i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1397.45 + 504.476i 1.56489 + 0.564923i
\(894\) 0 0
\(895\) −228.712 + 132.047i −0.255544 + 0.147538i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 95.2263 164.937i 0.105925 0.183467i
\(900\) 0 0
\(901\) 215.631i 0.239325i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1108.65i 1.22503i
\(906\) 0 0
\(907\) −1446.74 + 835.277i −1.59509 + 0.920923i −0.602670 + 0.797990i \(0.705897\pi\)
−0.992415 + 0.122933i \(0.960770\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1087.43i 1.19367i −0.802364 0.596835i \(-0.796425\pi\)
0.802364 0.596835i \(-0.203575\pi\)
\(912\) 0 0
\(913\) −442.897 −0.485101
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −156.491 271.051i −0.170656 0.295584i
\(918\) 0 0
\(919\) −984.258 −1.07101 −0.535505 0.844532i \(-0.679879\pi\)
−0.535505 + 0.844532i \(0.679879\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1741.63 1.88692
\(924\) 0 0
\(925\) −117.046 67.5765i −0.126536 0.0730557i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −223.803 387.638i −0.240907 0.417264i 0.720066 0.693906i \(-0.244112\pi\)
−0.960973 + 0.276642i \(0.910778\pi\)
\(930\) 0 0
\(931\) 157.040 + 877.133i 0.168679 + 0.942141i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −650.840 + 1127.29i −0.696086 + 1.20566i
\(936\) 0 0
\(937\) −153.045 + 265.082i −0.163335 + 0.282905i −0.936063 0.351833i \(-0.885558\pi\)
0.772728 + 0.634738i \(0.218892\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 739.908 + 427.186i 0.786299 + 0.453970i 0.838658 0.544658i \(-0.183341\pi\)
−0.0523588 + 0.998628i \(0.516674\pi\)
\(942\) 0 0
\(943\) 163.792i 0.173692i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 573.188 + 992.791i 0.605267 + 1.04835i 0.992009 + 0.126166i \(0.0402671\pi\)
−0.386742 + 0.922188i \(0.626400\pi\)
\(948\) 0 0
\(949\) 344.643i 0.363164i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 675.404 389.944i 0.708713 0.409176i −0.101871 0.994798i \(-0.532483\pi\)
0.810584 + 0.585622i \(0.199150\pi\)
\(954\) 0 0
\(955\) −464.687 804.861i −0.486583 0.842786i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 43.1473 74.7334i 0.0449920 0.0779284i
\(960\) 0 0
\(961\) 929.362 0.967078
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 212.886 + 122.910i 0.220608 + 0.127368i
\(966\) 0 0
\(967\) 687.674 + 1191.09i 0.711141 + 1.23173i 0.964429 + 0.264342i \(0.0851548\pi\)
−0.253288 + 0.967391i \(0.581512\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1484.68 + 857.179i −1.52902 + 0.882780i −0.529616 + 0.848238i \(0.677664\pi\)
−0.999403 + 0.0345418i \(0.989003\pi\)
\(972\) 0 0
\(973\) −2.64643 + 4.58375i −0.00271986 + 0.00471094i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1792.68i 1.83489i −0.397866 0.917443i \(-0.630249\pi\)
0.397866 0.917443i \(-0.369751\pi\)
\(978\) 0 0
\(979\) 1922.41 + 1109.90i 1.96364 + 1.13371i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 487.315 281.351i 0.495742 0.286217i −0.231211 0.972904i \(-0.574269\pi\)
0.726954 + 0.686687i \(0.240936\pi\)
\(984\) 0 0
\(985\) 208.970 + 361.947i 0.212153 + 0.367459i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −140.348 −0.141909
\(990\) 0 0
\(991\) 1085.70 626.830i 1.09556 0.632523i 0.160510 0.987034i \(-0.448686\pi\)
0.935052 + 0.354511i \(0.115353\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −355.098 −0.356882
\(996\) 0 0
\(997\) 502.212 869.856i 0.503723 0.872474i −0.496268 0.868169i \(-0.665297\pi\)
0.999991 0.00430418i \(-0.00137007\pi\)
\(998\) 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 684.3.y.g.145.3 yes 8
3.2 odd 2 inner 684.3.y.g.145.2 8
19.8 odd 6 inner 684.3.y.g.217.3 yes 8
57.8 even 6 inner 684.3.y.g.217.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.3.y.g.145.2 8 3.2 odd 2 inner
684.3.y.g.145.3 yes 8 1.1 even 1 trivial
684.3.y.g.217.2 yes 8 57.8 even 6 inner
684.3.y.g.217.3 yes 8 19.8 odd 6 inner