Properties

Label 900.3.u.b.149.2
Level $900$
Weight $3$
Character 900.149
Analytic conductor $24.523$
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] = [900,3,Mod(149,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.u (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 149.2
Root \(1.40126 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 900.149
Dual form 900.3.u.b.749.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.59808 + 1.50000i) q^{3} +(3.24410 - 1.87298i) q^{7} +(4.50000 - 7.79423i) q^{9} +O(q^{10})\) \(q+(-2.59808 + 1.50000i) q^{3} +(3.24410 - 1.87298i) q^{7} +(4.50000 - 7.79423i) q^{9} +(-10.1190 + 5.84218i) q^{11} +(-1.95205 - 1.12702i) q^{13} -11.6844 q^{17} +26.7460 q^{19} +(-5.61895 + 9.73231i) q^{21} +(-9.95231 + 17.2379i) q^{23} +27.0000i q^{27} +(38.2379 - 22.0767i) q^{29} +(26.1109 - 45.2254i) q^{31} +(17.5265 - 30.3569i) q^{33} +14.0000i q^{37} +6.76210 q^{39} +(-22.5000 - 12.9904i) q^{41} +(-36.3591 + 20.9919i) q^{43} +(22.7367 + 39.3810i) q^{47} +(-17.4839 + 30.2829i) q^{49} +(30.3569 - 17.5265i) q^{51} -10.8323 q^{53} +(-69.4881 + 40.1190i) q^{57} +(31.8810 + 18.4065i) q^{59} +(22.6190 + 39.1772i) q^{61} -33.7137i q^{63} +(86.5606 + 49.9758i) q^{67} -59.7138i q^{69} +102.603i q^{71} +13.7621i q^{73} +(-21.8846 + 37.9052i) q^{77} +(28.3810 + 49.1574i) q^{79} +(-40.5000 - 70.1481i) q^{81} +(45.0333 + 78.0000i) q^{83} +(-66.2300 + 114.714i) q^{87} +95.2349i q^{89} -8.44353 q^{91} +156.665i q^{93} +(-88.0727 + 50.8488i) q^{97} +105.159i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{9} + 12 q^{11} + 152 q^{19} + 48 q^{21} + 120 q^{29} - 8 q^{31} + 240 q^{39} - 180 q^{41} + 108 q^{49} - 36 q^{51} + 348 q^{59} + 88 q^{61} + 320 q^{79} - 324 q^{81} + 800 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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
\(3\) −2.59808 + 1.50000i −0.866025 + 0.500000i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.24410 1.87298i 0.463443 0.267569i −0.250048 0.968234i \(-0.580446\pi\)
0.713491 + 0.700664i \(0.247113\pi\)
\(8\) 0 0
\(9\) 4.50000 7.79423i 0.500000 0.866025i
\(10\) 0 0
\(11\) −10.1190 + 5.84218i −0.919905 + 0.531107i −0.883605 0.468234i \(-0.844891\pi\)
−0.0362999 + 0.999341i \(0.511557\pi\)
\(12\) 0 0
\(13\) −1.95205 1.12702i −0.150158 0.0866936i 0.423039 0.906112i \(-0.360963\pi\)
−0.573196 + 0.819418i \(0.694297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −11.6844 −0.687315 −0.343658 0.939095i \(-0.611666\pi\)
−0.343658 + 0.939095i \(0.611666\pi\)
\(18\) 0 0
\(19\) 26.7460 1.40768 0.703841 0.710357i \(-0.251467\pi\)
0.703841 + 0.710357i \(0.251467\pi\)
\(20\) 0 0
\(21\) −5.61895 + 9.73231i −0.267569 + 0.463443i
\(22\) 0 0
\(23\) −9.95231 + 17.2379i −0.432709 + 0.749474i −0.997106 0.0760299i \(-0.975776\pi\)
0.564397 + 0.825504i \(0.309109\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 1.00000i
\(28\) 0 0
\(29\) 38.2379 22.0767i 1.31855 0.761264i 0.335053 0.942199i \(-0.391246\pi\)
0.983495 + 0.180935i \(0.0579124\pi\)
\(30\) 0 0
\(31\) 26.1109 45.2254i 0.842287 1.45888i −0.0456704 0.998957i \(-0.514542\pi\)
0.887957 0.459927i \(-0.152124\pi\)
\(32\) 0 0
\(33\) 17.5265 30.3569i 0.531107 0.919905i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 14.0000i 0.378378i 0.981941 + 0.189189i \(0.0605859\pi\)
−0.981941 + 0.189189i \(0.939414\pi\)
\(38\) 0 0
\(39\) 6.76210 0.173387
\(40\) 0 0
\(41\) −22.5000 12.9904i −0.548780 0.316839i 0.199849 0.979827i \(-0.435955\pi\)
−0.748630 + 0.662988i \(0.769288\pi\)
\(42\) 0 0
\(43\) −36.3591 + 20.9919i −0.845560 + 0.488184i −0.859150 0.511723i \(-0.829007\pi\)
0.0135900 + 0.999908i \(0.495674\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 22.7367 + 39.3810i 0.483759 + 0.837895i 0.999826 0.0186534i \(-0.00593789\pi\)
−0.516067 + 0.856548i \(0.672605\pi\)
\(48\) 0 0
\(49\) −17.4839 + 30.2829i −0.356814 + 0.618019i
\(50\) 0 0
\(51\) 30.3569 17.5265i 0.595232 0.343658i
\(52\) 0 0
\(53\) −10.8323 −0.204383 −0.102192 0.994765i \(-0.532585\pi\)
−0.102192 + 0.994765i \(0.532585\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −69.4881 + 40.1190i −1.21909 + 0.703841i
\(58\) 0 0
\(59\) 31.8810 + 18.4065i 0.540357 + 0.311975i 0.745224 0.666815i \(-0.232343\pi\)
−0.204867 + 0.978790i \(0.565676\pi\)
\(60\) 0 0
\(61\) 22.6190 + 39.1772i 0.370802 + 0.642249i 0.989689 0.143232i \(-0.0457494\pi\)
−0.618887 + 0.785480i \(0.712416\pi\)
\(62\) 0 0
\(63\) 33.7137i 0.535138i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 86.5606 + 49.9758i 1.29195 + 0.745907i 0.979000 0.203862i \(-0.0653494\pi\)
0.312950 + 0.949770i \(0.398683\pi\)
\(68\) 0 0
\(69\) 59.7138i 0.865418i
\(70\) 0 0
\(71\) 102.603i 1.44511i 0.691312 + 0.722557i \(0.257033\pi\)
−0.691312 + 0.722557i \(0.742967\pi\)
\(72\) 0 0
\(73\) 13.7621i 0.188522i 0.995548 + 0.0942610i \(0.0300488\pi\)
−0.995548 + 0.0942610i \(0.969951\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −21.8846 + 37.9052i −0.284216 + 0.492276i
\(78\) 0 0
\(79\) 28.3810 + 49.1574i 0.359254 + 0.622246i 0.987836 0.155497i \(-0.0496979\pi\)
−0.628583 + 0.777743i \(0.716365\pi\)
\(80\) 0 0
\(81\) −40.5000 70.1481i −0.500000 0.866025i
\(82\) 0 0
\(83\) 45.0333 + 78.0000i 0.542570 + 0.939759i 0.998756 + 0.0498743i \(0.0158821\pi\)
−0.456185 + 0.889885i \(0.650785\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −66.2300 + 114.714i −0.761264 + 1.31855i
\(88\) 0 0
\(89\) 95.2349i 1.07005i 0.844835 + 0.535027i \(0.179699\pi\)
−0.844835 + 0.535027i \(0.820301\pi\)
\(90\) 0 0
\(91\) −8.44353 −0.0927861
\(92\) 0 0
\(93\) 156.665i 1.68457i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −88.0727 + 50.8488i −0.907966 + 0.524214i −0.879776 0.475388i \(-0.842308\pi\)
−0.0281897 + 0.999603i \(0.508974\pi\)
\(98\) 0 0
\(99\) 105.159i 1.06221i
\(100\) 0 0
\(101\) −87.7621 + 50.6695i −0.868932 + 0.501678i −0.866993 0.498320i \(-0.833950\pi\)
−0.00193860 + 0.999998i \(0.500617\pi\)
\(102\) 0 0
\(103\) 14.1043 + 8.14315i 0.136935 + 0.0790597i 0.566903 0.823785i \(-0.308142\pi\)
−0.429967 + 0.902844i \(0.641475\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 84.8705 0.793182 0.396591 0.917995i \(-0.370193\pi\)
0.396591 + 0.917995i \(0.370193\pi\)
\(108\) 0 0
\(109\) 161.968 1.48594 0.742971 0.669323i \(-0.233416\pi\)
0.742971 + 0.669323i \(0.233416\pi\)
\(110\) 0 0
\(111\) −21.0000 36.3731i −0.189189 0.327685i
\(112\) 0 0
\(113\) 3.46410 6.00000i 0.0306558 0.0530973i −0.850290 0.526314i \(-0.823574\pi\)
0.880946 + 0.473216i \(0.156907\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −17.5685 + 10.1431i −0.150158 + 0.0866936i
\(118\) 0 0
\(119\) −37.9052 + 21.8846i −0.318532 + 0.183904i
\(120\) 0 0
\(121\) 7.76210 13.4444i 0.0641496 0.111110i
\(122\) 0 0
\(123\) 77.9423 0.633677
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 225.903i 1.77877i −0.457163 0.889383i \(-0.651135\pi\)
0.457163 0.889383i \(-0.348865\pi\)
\(128\) 0 0
\(129\) 62.9758 109.077i 0.488184 0.845560i
\(130\) 0 0
\(131\) −81.7137 47.1774i −0.623769 0.360133i 0.154566 0.987982i \(-0.450602\pi\)
−0.778335 + 0.627849i \(0.783935\pi\)
\(132\) 0 0
\(133\) 86.7667 50.0948i 0.652381 0.376652i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 103.689 + 179.595i 0.756855 + 1.31091i 0.944447 + 0.328664i \(0.106598\pi\)
−0.187592 + 0.982247i \(0.560068\pi\)
\(138\) 0 0
\(139\) 16.3569 28.3309i 0.117675 0.203819i −0.801171 0.598436i \(-0.795789\pi\)
0.918846 + 0.394616i \(0.129123\pi\)
\(140\) 0 0
\(141\) −118.143 68.2100i −0.837895 0.483759i
\(142\) 0 0
\(143\) 26.3369 0.184174
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 104.903i 0.713627i
\(148\) 0 0
\(149\) −18.3327 10.5844i −0.123038 0.0710360i 0.437218 0.899356i \(-0.355964\pi\)
−0.560256 + 0.828320i \(0.689297\pi\)
\(150\) 0 0
\(151\) 137.714 + 238.527i 0.912011 + 1.57965i 0.811219 + 0.584742i \(0.198804\pi\)
0.100792 + 0.994908i \(0.467862\pi\)
\(152\) 0 0
\(153\) −52.5796 + 91.0706i −0.343658 + 0.595232i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −155.829 89.9677i −0.992539 0.573043i −0.0865070 0.996251i \(-0.527570\pi\)
−0.906032 + 0.423208i \(0.860904\pi\)
\(158\) 0 0
\(159\) 28.1431 16.2485i 0.177001 0.102192i
\(160\) 0 0
\(161\) 74.5620i 0.463118i
\(162\) 0 0
\(163\) 264.411i 1.62216i 0.584939 + 0.811078i \(0.301119\pi\)
−0.584939 + 0.811078i \(0.698881\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −86.3267 + 149.522i −0.516926 + 0.895342i 0.482881 + 0.875686i \(0.339590\pi\)
−0.999807 + 0.0196561i \(0.993743\pi\)
\(168\) 0 0
\(169\) −81.9597 141.958i −0.484968 0.839990i
\(170\) 0 0
\(171\) 120.357 208.464i 0.703841 1.21909i
\(172\) 0 0
\(173\) −147.168 254.903i −0.850685 1.47343i −0.880591 0.473876i \(-0.842854\pi\)
0.0299069 0.999553i \(-0.490479\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −110.439 −0.623950
\(178\) 0 0
\(179\) 64.1138i 0.358178i −0.983833 0.179089i \(-0.942685\pi\)
0.983833 0.179089i \(-0.0573150\pi\)
\(180\) 0 0
\(181\) 291.206 1.60887 0.804435 0.594040i \(-0.202468\pi\)
0.804435 + 0.594040i \(0.202468\pi\)
\(182\) 0 0
\(183\) −117.532 67.8569i −0.642249 0.370802i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 118.233 68.2621i 0.632264 0.365038i
\(188\) 0 0
\(189\) 50.5706 + 87.5908i 0.267569 + 0.463443i
\(190\) 0 0
\(191\) 13.8085 7.97231i 0.0722956 0.0417399i −0.463416 0.886141i \(-0.653377\pi\)
0.535712 + 0.844401i \(0.320043\pi\)
\(192\) 0 0
\(193\) −211.928 122.357i −1.09807 0.633973i −0.162360 0.986732i \(-0.551910\pi\)
−0.935714 + 0.352758i \(0.885244\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 210.374 1.06789 0.533945 0.845519i \(-0.320709\pi\)
0.533945 + 0.845519i \(0.320709\pi\)
\(198\) 0 0
\(199\) 102.730 0.516230 0.258115 0.966114i \(-0.416899\pi\)
0.258115 + 0.966114i \(0.416899\pi\)
\(200\) 0 0
\(201\) −299.855 −1.49181
\(202\) 0 0
\(203\) 82.6984 143.238i 0.407381 0.705605i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 89.5708 + 155.141i 0.432709 + 0.749474i
\(208\) 0 0
\(209\) −270.641 + 156.255i −1.29493 + 0.747630i
\(210\) 0 0
\(211\) 113.984 197.426i 0.540208 0.935668i −0.458684 0.888600i \(-0.651679\pi\)
0.998892 0.0470680i \(-0.0149877\pi\)
\(212\) 0 0
\(213\) −153.905 266.571i −0.722557 1.25150i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 195.621i 0.901479i
\(218\) 0 0
\(219\) −20.6431 35.7550i −0.0942610 0.163265i
\(220\) 0 0
\(221\) 22.8085 + 13.1685i 0.103206 + 0.0595858i
\(222\) 0 0
\(223\) 241.935 139.681i 1.08491 0.626374i 0.152695 0.988273i \(-0.451205\pi\)
0.932217 + 0.361899i \(0.117872\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 124.282 + 215.262i 0.547496 + 0.948291i 0.998445 + 0.0557415i \(0.0177523\pi\)
−0.450949 + 0.892550i \(0.648914\pi\)
\(228\) 0 0
\(229\) 78.8891 136.640i 0.344494 0.596681i −0.640768 0.767735i \(-0.721384\pi\)
0.985262 + 0.171054i \(0.0547172\pi\)
\(230\) 0 0
\(231\) 131.308i 0.568431i
\(232\) 0 0
\(233\) −332.854 −1.42856 −0.714279 0.699861i \(-0.753245\pi\)
−0.714279 + 0.699861i \(0.753245\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −147.472 85.1431i −0.622246 0.359254i
\(238\) 0 0
\(239\) −207.714 119.924i −0.869095 0.501772i −0.00204759 0.999998i \(-0.500652\pi\)
−0.867047 + 0.498226i \(0.833985\pi\)
\(240\) 0 0
\(241\) 94.7540 + 164.119i 0.393170 + 0.680991i 0.992866 0.119237i \(-0.0380450\pi\)
−0.599696 + 0.800228i \(0.704712\pi\)
\(242\) 0 0
\(243\) 210.444 + 121.500i 0.866025 + 0.500000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −52.2095 30.1431i −0.211374 0.122037i
\(248\) 0 0
\(249\) −234.000 135.100i −0.939759 0.542570i
\(250\) 0 0
\(251\) 447.114i 1.78133i 0.454661 + 0.890664i \(0.349760\pi\)
−0.454661 + 0.890664i \(0.650240\pi\)
\(252\) 0 0
\(253\) 232.573i 0.919259i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 223.613 387.308i 0.870088 1.50704i 0.00818336 0.999967i \(-0.497395\pi\)
0.861905 0.507070i \(-0.169272\pi\)
\(258\) 0 0
\(259\) 26.2218 + 45.4174i 0.101242 + 0.175357i
\(260\) 0 0
\(261\) 397.380i 1.52253i
\(262\) 0 0
\(263\) 177.273 + 307.046i 0.674043 + 1.16748i 0.976748 + 0.214392i \(0.0687771\pi\)
−0.302705 + 0.953084i \(0.597890\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −142.852 247.427i −0.535027 0.926694i
\(268\) 0 0
\(269\) 227.367i 0.845229i 0.906310 + 0.422614i \(0.138888\pi\)
−0.906310 + 0.422614i \(0.861112\pi\)
\(270\) 0 0
\(271\) 228.573 0.843441 0.421721 0.906726i \(-0.361426\pi\)
0.421721 + 0.906726i \(0.361426\pi\)
\(272\) 0 0
\(273\) 21.9369 12.6653i 0.0803551 0.0463930i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 381.435 220.222i 1.37702 0.795024i 0.385223 0.922824i \(-0.374125\pi\)
0.991800 + 0.127799i \(0.0407913\pi\)
\(278\) 0 0
\(279\) −234.998 407.028i −0.842287 1.45888i
\(280\) 0 0
\(281\) −272.190 + 157.149i −0.968646 + 0.559248i −0.898823 0.438311i \(-0.855577\pi\)
−0.0698227 + 0.997559i \(0.522243\pi\)
\(282\) 0 0
\(283\) 232.067 + 133.984i 0.820024 + 0.473441i 0.850425 0.526097i \(-0.176345\pi\)
−0.0304006 + 0.999538i \(0.509678\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −97.3231 −0.339105
\(288\) 0 0
\(289\) −152.476 −0.527598
\(290\) 0 0
\(291\) 152.546 264.218i 0.524214 0.907966i
\(292\) 0 0
\(293\) 155.033 268.524i 0.529121 0.916465i −0.470302 0.882506i \(-0.655855\pi\)
0.999423 0.0339593i \(-0.0108117\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −157.739 273.212i −0.531107 0.919905i
\(298\) 0 0
\(299\) 38.8548 22.4328i 0.129949 0.0750262i
\(300\) 0 0
\(301\) −78.6351 + 136.200i −0.261246 + 0.452492i
\(302\) 0 0
\(303\) 152.008 263.286i 0.501678 0.868932i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 495.952i 1.61548i −0.589541 0.807739i \(-0.700691\pi\)
0.589541 0.807739i \(-0.299309\pi\)
\(308\) 0 0
\(309\) −48.8589 −0.158119
\(310\) 0 0
\(311\) −195.762 113.023i −0.629460 0.363419i 0.151083 0.988521i \(-0.451724\pi\)
−0.780543 + 0.625102i \(0.785057\pi\)
\(312\) 0 0
\(313\) −283.326 + 163.579i −0.905196 + 0.522615i −0.878882 0.477038i \(-0.841710\pi\)
−0.0263139 + 0.999654i \(0.508377\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.09616 7.09475i −0.0129216 0.0223809i 0.859492 0.511149i \(-0.170780\pi\)
−0.872414 + 0.488768i \(0.837447\pi\)
\(318\) 0 0
\(319\) −257.952 + 446.785i −0.808626 + 1.40058i
\(320\) 0 0
\(321\) −220.500 + 127.306i −0.686916 + 0.396591i
\(322\) 0 0
\(323\) −312.509 −0.967521
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −420.805 + 242.952i −1.28686 + 0.742971i
\(328\) 0 0
\(329\) 147.520 + 85.1708i 0.448389 + 0.258878i
\(330\) 0 0
\(331\) −138.903 240.587i −0.419647 0.726850i 0.576257 0.817269i \(-0.304513\pi\)
−0.995904 + 0.0904186i \(0.971180\pi\)
\(332\) 0 0
\(333\) 109.119 + 63.0000i 0.327685 + 0.189189i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −48.3473 27.9133i −0.143464 0.0828288i 0.426550 0.904464i \(-0.359729\pi\)
−0.570014 + 0.821635i \(0.693062\pi\)
\(338\) 0 0
\(339\) 20.7846i 0.0613115i
\(340\) 0 0
\(341\) 610.178i 1.78938i
\(342\) 0 0
\(343\) 314.540i 0.917027i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 254.102 440.117i 0.732281 1.26835i −0.223625 0.974675i \(-0.571789\pi\)
0.955906 0.293673i \(-0.0948777\pi\)
\(348\) 0 0
\(349\) 69.2379 + 119.924i 0.198389 + 0.343621i 0.948006 0.318251i \(-0.103096\pi\)
−0.749617 + 0.661872i \(0.769762\pi\)
\(350\) 0 0
\(351\) 30.4294 52.7054i 0.0866936 0.150158i
\(352\) 0 0
\(353\) 288.027 + 498.877i 0.815940 + 1.41325i 0.908651 + 0.417556i \(0.137113\pi\)
−0.0927114 + 0.995693i \(0.529553\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 65.6538 113.716i 0.183904 0.318532i
\(358\) 0 0
\(359\) 267.888i 0.746207i −0.927790 0.373103i \(-0.878294\pi\)
0.927790 0.373103i \(-0.121706\pi\)
\(360\) 0 0
\(361\) 354.347 0.981570
\(362\) 0 0
\(363\) 46.5726i 0.128299i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 52.3736 30.2379i 0.142707 0.0823921i −0.426946 0.904277i \(-0.640411\pi\)
0.569654 + 0.821885i \(0.307077\pi\)
\(368\) 0 0
\(369\) −202.500 + 116.913i −0.548780 + 0.316839i
\(370\) 0 0
\(371\) −35.1411 + 20.2887i −0.0947199 + 0.0546866i
\(372\) 0 0
\(373\) −187.501 108.254i −0.502685 0.290225i 0.227137 0.973863i \(-0.427064\pi\)
−0.729822 + 0.683638i \(0.760397\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −99.5231 −0.263987
\(378\) 0 0
\(379\) −285.254 −0.752649 −0.376325 0.926488i \(-0.622812\pi\)
−0.376325 + 0.926488i \(0.622812\pi\)
\(380\) 0 0
\(381\) 338.855 + 586.914i 0.889383 + 1.54046i
\(382\) 0 0
\(383\) 81.7905 141.665i 0.213552 0.369883i −0.739272 0.673407i \(-0.764830\pi\)
0.952824 + 0.303524i \(0.0981633\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 377.855i 0.976369i
\(388\) 0 0
\(389\) −250.760 + 144.776i −0.644627 + 0.372176i −0.786395 0.617724i \(-0.788055\pi\)
0.141767 + 0.989900i \(0.454721\pi\)
\(390\) 0 0
\(391\) 116.286 201.414i 0.297407 0.515125i
\(392\) 0 0
\(393\) 283.065 0.720266
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 66.5081i 0.167527i −0.996486 0.0837633i \(-0.973306\pi\)
0.996486 0.0837633i \(-0.0266940\pi\)
\(398\) 0 0
\(399\) −150.284 + 260.300i −0.376652 + 0.652381i
\(400\) 0 0
\(401\) −392.165 226.417i −0.977968 0.564630i −0.0763122 0.997084i \(-0.524315\pi\)
−0.901656 + 0.432454i \(0.857648\pi\)
\(402\) 0 0
\(403\) −101.940 + 58.8548i −0.252952 + 0.146042i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −81.7905 141.665i −0.200959 0.348072i
\(408\) 0 0
\(409\) 150.435 260.562i 0.367813 0.637071i −0.621410 0.783485i \(-0.713440\pi\)
0.989223 + 0.146415i \(0.0467734\pi\)
\(410\) 0 0
\(411\) −538.784 311.067i −1.31091 0.756855i
\(412\) 0 0
\(413\) 137.901 0.333900
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 98.1411i 0.235350i
\(418\) 0 0
\(419\) −103.524 59.7697i −0.247074 0.142649i 0.371349 0.928493i \(-0.378895\pi\)
−0.618424 + 0.785845i \(0.712229\pi\)
\(420\) 0 0
\(421\) 81.1431 + 140.544i 0.192739 + 0.333834i 0.946157 0.323708i \(-0.104930\pi\)
−0.753418 + 0.657542i \(0.771596\pi\)
\(422\) 0 0
\(423\) 409.260 0.967517
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 146.756 + 84.7298i 0.343692 + 0.198431i
\(428\) 0 0
\(429\) −68.4254 + 39.5054i −0.159500 + 0.0920872i
\(430\) 0 0
\(431\) 499.815i 1.15966i −0.814736 0.579832i \(-0.803118\pi\)
0.814736 0.579832i \(-0.196882\pi\)
\(432\) 0 0
\(433\) 739.883i 1.70874i −0.519668 0.854368i \(-0.673944\pi\)
0.519668 0.854368i \(-0.326056\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −266.184 + 461.044i −0.609117 + 1.05502i
\(438\) 0 0
\(439\) 124.333 + 215.350i 0.283218 + 0.490548i 0.972175 0.234254i \(-0.0752647\pi\)
−0.688958 + 0.724802i \(0.741931\pi\)
\(440\) 0 0
\(441\) 157.355 + 272.547i 0.356814 + 0.618019i
\(442\) 0 0
\(443\) 45.5432 + 78.8831i 0.102806 + 0.178066i 0.912840 0.408318i \(-0.133884\pi\)
−0.810034 + 0.586384i \(0.800551\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 63.5062 0.142072
\(448\) 0 0
\(449\) 93.6705i 0.208620i 0.994545 + 0.104310i \(0.0332634\pi\)
−0.994545 + 0.104310i \(0.966737\pi\)
\(450\) 0 0
\(451\) 303.569 0.673101
\(452\) 0 0
\(453\) −715.581 413.141i −1.57965 0.912011i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 349.005 201.498i 0.763686 0.440915i −0.0669313 0.997758i \(-0.521321\pi\)
0.830618 + 0.556843i \(0.187988\pi\)
\(458\) 0 0
\(459\) 315.478i 0.687315i
\(460\) 0 0
\(461\) −57.9496 + 33.4572i −0.125704 + 0.0725752i −0.561533 0.827454i \(-0.689788\pi\)
0.435829 + 0.900029i \(0.356455\pi\)
\(462\) 0 0
\(463\) 153.824 + 88.8105i 0.332234 + 0.191815i 0.656832 0.754037i \(-0.271896\pi\)
−0.324599 + 0.945852i \(0.605229\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 620.647 1.32901 0.664504 0.747285i \(-0.268643\pi\)
0.664504 + 0.747285i \(0.268643\pi\)
\(468\) 0 0
\(469\) 374.415 0.798327
\(470\) 0 0
\(471\) 539.806 1.14609
\(472\) 0 0
\(473\) 245.277 424.833i 0.518557 0.898166i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −48.7454 + 84.4294i −0.102192 + 0.177001i
\(478\) 0 0
\(479\) −146.044 + 84.3187i −0.304894 + 0.176031i −0.644639 0.764487i \(-0.722992\pi\)
0.339745 + 0.940517i \(0.389659\pi\)
\(480\) 0 0
\(481\) 15.7782 27.3287i 0.0328030 0.0568164i
\(482\) 0 0
\(483\) −111.843 193.718i −0.231559 0.401072i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 401.935i 0.825330i 0.910883 + 0.412665i \(0.135402\pi\)
−0.910883 + 0.412665i \(0.864598\pi\)
\(488\) 0 0
\(489\) −396.617 686.961i −0.811078 1.40483i
\(490\) 0 0
\(491\) −307.833 177.727i −0.626950 0.361970i 0.152620 0.988285i \(-0.451229\pi\)
−0.779570 + 0.626315i \(0.784562\pi\)
\(492\) 0 0
\(493\) −446.785 + 257.952i −0.906258 + 0.523228i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 192.174 + 332.855i 0.386668 + 0.669728i
\(498\) 0 0
\(499\) −8.30845 + 14.3907i −0.0166502 + 0.0288390i −0.874230 0.485511i \(-0.838634\pi\)
0.857580 + 0.514350i \(0.171967\pi\)
\(500\) 0 0
\(501\) 517.960i 1.03385i
\(502\) 0 0
\(503\) −362.522 −0.720721 −0.360360 0.932813i \(-0.617346\pi\)
−0.360360 + 0.932813i \(0.617346\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 425.875 + 245.879i 0.839990 + 0.484968i
\(508\) 0 0
\(509\) 578.286 + 333.874i 1.13612 + 0.655941i 0.945467 0.325717i \(-0.105606\pi\)
0.190655 + 0.981657i \(0.438939\pi\)
\(510\) 0 0
\(511\) 25.7762 + 44.6457i 0.0504426 + 0.0873692i
\(512\) 0 0
\(513\) 722.141i 1.40768i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −460.142 265.663i −0.890024 0.513855i
\(518\) 0 0
\(519\) 764.710 + 441.505i 1.47343 + 0.850685i
\(520\) 0 0
\(521\) 676.352i 1.29818i 0.760712 + 0.649090i \(0.224850\pi\)
−0.760712 + 0.649090i \(0.775150\pi\)
\(522\) 0 0
\(523\) 189.427i 0.362194i −0.983465 0.181097i \(-0.942035\pi\)
0.983465 0.181097i \(-0.0579648\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −305.089 + 528.429i −0.578916 + 1.00271i
\(528\) 0 0
\(529\) 66.4032 + 115.014i 0.125526 + 0.217417i
\(530\) 0 0
\(531\) 286.929 165.659i 0.540357 0.311975i
\(532\) 0 0
\(533\) 29.2808 + 50.7157i 0.0549357 + 0.0951515i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 96.1707 + 166.573i 0.179089 + 0.310191i
\(538\) 0 0
\(539\) 408.575i 0.758025i
\(540\) 0 0
\(541\) −590.629 −1.09174 −0.545868 0.837871i \(-0.683800\pi\)
−0.545868 + 0.837871i \(0.683800\pi\)
\(542\) 0 0
\(543\) −756.574 + 436.808i −1.39332 + 0.804435i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −683.783 + 394.782i −1.25006 + 0.721722i −0.971121 0.238586i \(-0.923316\pi\)
−0.278939 + 0.960309i \(0.589983\pi\)
\(548\) 0 0
\(549\) 407.141 0.741605
\(550\) 0 0
\(551\) 1022.71 590.462i 1.85610 1.07162i
\(552\) 0 0
\(553\) 184.142 + 106.314i 0.332987 + 0.192250i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1004.69 −1.80376 −0.901880 0.431987i \(-0.857813\pi\)
−0.901880 + 0.431987i \(0.857813\pi\)
\(558\) 0 0
\(559\) 94.6330 0.169290
\(560\) 0 0
\(561\) −204.786 + 354.700i −0.365038 + 0.632264i
\(562\) 0 0
\(563\) −157.603 + 272.976i −0.279934 + 0.484859i −0.971368 0.237580i \(-0.923646\pi\)
0.691434 + 0.722439i \(0.256979\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −262.772 151.712i −0.463443 0.267569i
\(568\) 0 0
\(569\) 823.923 475.692i 1.44802 0.836015i 0.449656 0.893202i \(-0.351546\pi\)
0.998363 + 0.0571870i \(0.0182131\pi\)
\(570\) 0 0
\(571\) −229.183 + 396.957i −0.401372 + 0.695197i −0.993892 0.110359i \(-0.964800\pi\)
0.592520 + 0.805556i \(0.298133\pi\)
\(572\) 0 0
\(573\) −23.9169 + 41.4254i −0.0417399 + 0.0722956i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 991.190i 1.71783i 0.512116 + 0.858916i \(0.328862\pi\)
−0.512116 + 0.858916i \(0.671138\pi\)
\(578\) 0 0
\(579\) 734.141 1.26795
\(580\) 0 0
\(581\) 292.185 + 168.693i 0.502901 + 0.290350i
\(582\) 0 0
\(583\) 109.612 63.2843i 0.188013 0.108549i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 83.6483 + 144.883i 0.142501 + 0.246820i 0.928438 0.371488i \(-0.121152\pi\)
−0.785937 + 0.618307i \(0.787819\pi\)
\(588\) 0 0
\(589\) 698.361 1209.60i 1.18567 2.05364i
\(590\) 0 0
\(591\) −546.569 + 315.561i −0.924820 + 0.533945i
\(592\) 0 0
\(593\) −868.330 −1.46430 −0.732150 0.681143i \(-0.761483\pi\)
−0.732150 + 0.681143i \(0.761483\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −266.900 + 154.095i −0.447069 + 0.258115i
\(598\) 0 0
\(599\) 170.335 + 98.3428i 0.284365 + 0.164178i 0.635398 0.772185i \(-0.280836\pi\)
−0.351033 + 0.936363i \(0.614169\pi\)
\(600\) 0 0
\(601\) −332.181 575.355i −0.552715 0.957330i −0.998077 0.0619795i \(-0.980259\pi\)
0.445363 0.895350i \(-0.353075\pi\)
\(602\) 0 0
\(603\) 779.046 449.782i 1.29195 0.745907i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 64.4979 + 37.2379i 0.106257 + 0.0613474i 0.552187 0.833720i \(-0.313794\pi\)
−0.445930 + 0.895068i \(0.647127\pi\)
\(608\) 0 0
\(609\) 496.191i 0.814763i
\(610\) 0 0
\(611\) 102.498i 0.167755i
\(612\) 0 0
\(613\) 304.569i 0.496849i 0.968651 + 0.248425i \(0.0799128\pi\)
−0.968651 + 0.248425i \(0.920087\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 470.744 815.353i 0.762956 1.32148i −0.178364 0.983965i \(-0.557080\pi\)
0.941320 0.337515i \(-0.109586\pi\)
\(618\) 0 0
\(619\) −117.736 203.924i −0.190203 0.329442i 0.755114 0.655593i \(-0.227581\pi\)
−0.945318 + 0.326151i \(0.894248\pi\)
\(620\) 0 0
\(621\) −465.423 268.712i −0.749474 0.432709i
\(622\) 0 0
\(623\) 178.373 + 308.952i 0.286313 + 0.495909i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 468.764 811.923i 0.747630 1.29493i
\(628\) 0 0
\(629\) 163.581i 0.260065i
\(630\) 0 0
\(631\) −832.125 −1.31874 −0.659370 0.751819i \(-0.729177\pi\)
−0.659370 + 0.751819i \(0.729177\pi\)
\(632\) 0 0
\(633\) 683.903i 1.08042i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 68.2588 39.4092i 0.107157 0.0618669i
\(638\) 0 0
\(639\) 799.712 + 461.714i 1.25150 + 0.722557i
\(640\) 0 0
\(641\) 913.355 527.326i 1.42489 0.822661i 0.428179 0.903694i \(-0.359155\pi\)
0.996711 + 0.0810330i \(0.0258219\pi\)
\(642\) 0 0
\(643\) 854.928 + 493.593i 1.32959 + 0.767640i 0.985236 0.171199i \(-0.0547641\pi\)
0.344356 + 0.938839i \(0.388097\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −75.6097 −0.116862 −0.0584310 0.998291i \(-0.518610\pi\)
−0.0584310 + 0.998291i \(0.518610\pi\)
\(648\) 0 0
\(649\) −430.137 −0.662769
\(650\) 0 0
\(651\) 293.431 + 508.238i 0.450740 + 0.780704i
\(652\) 0 0
\(653\) 623.510 1079.95i 0.954840 1.65383i 0.220105 0.975476i \(-0.429360\pi\)
0.734734 0.678355i \(-0.237307\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 107.265 + 61.9294i 0.163265 + 0.0942610i
\(658\) 0 0
\(659\) −544.851 + 314.570i −0.826784 + 0.477344i −0.852750 0.522319i \(-0.825067\pi\)
0.0259662 + 0.999663i \(0.491734\pi\)
\(660\) 0 0
\(661\) −323.282 + 559.941i −0.489080 + 0.847112i −0.999921 0.0125632i \(-0.996001\pi\)
0.510841 + 0.859675i \(0.329334\pi\)
\(662\) 0 0
\(663\) −79.0108 −0.119172
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 878.855i 1.31762i
\(668\) 0 0
\(669\) −419.044 + 725.806i −0.626374 + 1.08491i
\(670\) 0 0
\(671\) −457.760 264.288i −0.682206 0.393872i
\(672\) 0 0
\(673\) −79.5905 + 45.9516i −0.118262 + 0.0682788i −0.557964 0.829865i \(-0.688417\pi\)
0.439702 + 0.898144i \(0.355084\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 365.763 + 633.520i 0.540270 + 0.935776i 0.998888 + 0.0471421i \(0.0150113\pi\)
−0.458618 + 0.888634i \(0.651655\pi\)
\(678\) 0 0
\(679\) −190.478 + 329.917i −0.280527 + 0.485887i
\(680\) 0 0
\(681\) −645.786 372.845i −0.948291 0.547496i
\(682\) 0 0
\(683\) −682.177 −0.998794 −0.499397 0.866373i \(-0.666445\pi\)
−0.499397 + 0.866373i \(0.666445\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 473.335i 0.688988i
\(688\) 0 0
\(689\) 21.1452 + 12.2082i 0.0306897 + 0.0177187i
\(690\) 0 0
\(691\) 210.331 + 364.303i 0.304386 + 0.527212i 0.977124 0.212669i \(-0.0682155\pi\)
−0.672739 + 0.739880i \(0.734882\pi\)
\(692\) 0 0
\(693\) 196.961 + 341.147i 0.284216 + 0.492276i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 262.898 + 151.784i 0.377185 + 0.217768i
\(698\) 0 0
\(699\) 864.780 499.281i 1.23717 0.714279i
\(700\) 0 0
\(701\) 70.8185i 0.101025i 0.998723 + 0.0505125i \(0.0160855\pi\)
−0.998723 + 0.0505125i \(0.983915\pi\)
\(702\) 0 0
\(703\) 374.444i 0.532637i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −189.806 + 328.754i −0.268467 + 0.464998i
\(708\) 0 0
\(709\) 399.696 + 692.293i 0.563745 + 0.976436i 0.997165 + 0.0752436i \(0.0239734\pi\)
−0.433420 + 0.901192i \(0.642693\pi\)
\(710\) 0 0
\(711\) 510.859 0.718508
\(712\) 0 0
\(713\) 519.727 + 900.194i 0.728930 + 1.26254i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 719.541 1.00354
\(718\) 0 0
\(719\) 368.675i 0.512761i 0.966576 + 0.256381i \(0.0825300\pi\)
−0.966576 + 0.256381i \(0.917470\pi\)
\(720\) 0 0
\(721\) 61.0079 0.0846157
\(722\) 0 0
\(723\) −492.356 284.262i −0.680991 0.393170i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −784.783 + 453.095i −1.07948 + 0.623239i −0.930757 0.365640i \(-0.880850\pi\)
−0.148725 + 0.988879i \(0.547517\pi\)
\(728\) 0 0
\(729\) −729.000 −1.00000
\(730\) 0 0
\(731\) 424.833 245.277i 0.581166 0.335537i
\(732\) 0 0
\(733\) 87.9190 + 50.7601i 0.119944 + 0.0692497i 0.558772 0.829322i \(-0.311273\pi\)
−0.438828 + 0.898571i \(0.644606\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1167.87 −1.58463
\(738\) 0 0
\(739\) −557.665 −0.754622 −0.377311 0.926087i \(-0.623151\pi\)
−0.377311 + 0.926087i \(0.623151\pi\)
\(740\) 0 0
\(741\) 180.859 0.244074
\(742\) 0 0
\(743\) −178.289 + 308.806i −0.239959 + 0.415621i −0.960702 0.277581i \(-0.910467\pi\)
0.720743 + 0.693202i \(0.243801\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 810.600 1.08514
\(748\) 0 0
\(749\) 275.329 158.961i 0.367595 0.212231i
\(750\) 0 0
\(751\) −244.810 + 424.024i −0.325979 + 0.564613i −0.981710 0.190382i \(-0.939027\pi\)
0.655731 + 0.754995i \(0.272361\pi\)
\(752\) 0 0
\(753\) −670.670 1161.63i −0.890664 1.54268i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 198.379i 0.262059i 0.991378 + 0.131030i \(0.0418283\pi\)
−0.991378 + 0.131030i \(0.958172\pi\)
\(758\) 0 0
\(759\) 348.859 + 604.241i 0.459630 + 0.796102i
\(760\) 0 0
\(761\) −30.6653 17.7046i −0.0402961 0.0232649i 0.479717 0.877423i \(-0.340739\pi\)
−0.520013 + 0.854159i \(0.674073\pi\)
\(762\) 0 0
\(763\) 525.440 303.363i 0.688650 0.397592i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −41.4889 71.8609i −0.0540925 0.0936909i
\(768\) 0 0
\(769\) −114.552 + 198.411i −0.148963 + 0.258011i −0.930844 0.365416i \(-0.880927\pi\)
0.781882 + 0.623427i \(0.214260\pi\)
\(770\) 0 0
\(771\) 1341.68i 1.74018i
\(772\) 0 0
\(773\) 905.388 1.17126 0.585632 0.810577i \(-0.300846\pi\)
0.585632 + 0.810577i \(0.300846\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −136.252 78.6653i −0.175357 0.101242i
\(778\) 0 0
\(779\) −601.784 347.440i −0.772509 0.446008i
\(780\) 0 0
\(781\) −599.425 1038.24i −0.767510 1.32937i
\(782\) 0 0
\(783\) 596.070 + 1032.42i 0.761264 + 1.31855i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 128.423 + 74.1452i 0.163181 + 0.0942125i 0.579366 0.815067i \(-0.303300\pi\)
−0.416186 + 0.909280i \(0.636633\pi\)
\(788\) 0 0
\(789\) −921.139 531.820i −1.16748 0.674043i
\(790\) 0 0
\(791\) 25.9528i 0.0328101i
\(792\) 0 0
\(793\) 101.968i 0.128585i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −122.864 + 212.806i −0.154158 + 0.267009i −0.932752 0.360518i \(-0.882600\pi\)
0.778594 + 0.627528i \(0.215933\pi\)
\(798\) 0 0
\(799\) −265.663 460.142i −0.332495 0.575898i
\(800\) 0 0
\(801\) 742.282 + 428.557i 0.926694 + 0.535027i
\(802\) 0 0
\(803\) −80.4006 139.258i −0.100125 0.173422i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −341.050 590.716i −0.422614 0.731990i
\(808\) 0 0
\(809\) 1551.44i 1.91773i 0.283865 + 0.958864i \(0.408383\pi\)
−0.283865 + 0.958864i \(0.591617\pi\)
\(810\) 0 0
\(811\) −439.512 −0.541939 −0.270969 0.962588i \(-0.587344\pi\)
−0.270969 + 0.962588i \(0.587344\pi\)
\(812\) 0 0
\(813\) −593.849 + 342.859i −0.730442 + 0.421721i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −972.459 + 561.450i −1.19028 + 0.687209i
\(818\) 0 0
\(819\) −37.9959 + 65.8108i −0.0463930 + 0.0803551i
\(820\) 0 0
\(821\) 942.187 543.972i 1.14761 0.662573i 0.199306 0.979937i \(-0.436131\pi\)
0.948304 + 0.317365i \(0.102798\pi\)
\(822\) 0 0
\(823\) −812.709 469.218i −0.987496 0.570131i −0.0829708 0.996552i \(-0.526441\pi\)
−0.904525 + 0.426421i \(0.859774\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −615.395 −0.744129 −0.372065 0.928207i \(-0.621350\pi\)
−0.372065 + 0.928207i \(0.621350\pi\)
\(828\) 0 0
\(829\) 559.230 0.674583 0.337292 0.941400i \(-0.390489\pi\)
0.337292 + 0.941400i \(0.390489\pi\)
\(830\) 0 0
\(831\) −660.665 + 1144.31i −0.795024 + 1.37702i
\(832\) 0 0
\(833\) 204.288 353.837i 0.245243 0.424774i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1221.09 + 704.994i 1.45888 + 0.842287i
\(838\) 0 0
\(839\) 914.135 527.776i 1.08955 0.629054i 0.156096 0.987742i \(-0.450109\pi\)
0.933457 + 0.358688i \(0.116776\pi\)
\(840\) 0 0
\(841\) 554.258 960.003i 0.659046 1.14150i
\(842\) 0 0
\(843\) 471.446 816.569i 0.559248 0.968646i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 58.1531i 0.0686578i
\(848\) 0 0
\(849\) −803.903 −0.946882
\(850\) 0 0
\(851\) −241.331 139.332i −0.283585 0.163728i
\(852\) 0 0
\(853\) 79.5941 45.9536i 0.0933107 0.0538730i −0.452619 0.891704i \(-0.649510\pi\)
0.545929 + 0.837831i \(0.316177\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −589.498 1021.04i −0.687862 1.19141i −0.972528 0.232785i \(-0.925216\pi\)
0.284666 0.958627i \(-0.408117\pi\)
\(858\) 0 0
\(859\) −669.784 + 1160.10i −0.779726 + 1.35052i 0.152374 + 0.988323i \(0.451308\pi\)
−0.932100 + 0.362201i \(0.882025\pi\)
\(860\) 0 0
\(861\) 252.853 145.985i 0.293673 0.169552i
\(862\) 0 0
\(863\) −1332.42 −1.54393 −0.771967 0.635663i \(-0.780727\pi\)
−0.771967 + 0.635663i \(0.780727\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 396.144 228.714i 0.456913 0.263799i
\(868\) 0 0
\(869\) −574.373 331.614i −0.660958 0.381605i
\(870\) 0 0
\(871\) −112.647 195.111i −0.129331 0.224007i
\(872\) 0 0
\(873\) 915.278i 1.04843i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 556.537 + 321.317i 0.634591 + 0.366381i 0.782528 0.622615i \(-0.213930\pi\)
−0.147937 + 0.988997i \(0.547263\pi\)
\(878\) 0 0
\(879\) 930.195i 1.05824i
\(880\) 0 0
\(881\) 983.693i 1.11656i −0.829651 0.558282i \(-0.811461\pi\)
0.829651 0.558282i \(-0.188539\pi\)
\(882\) 0 0
\(883\) 197.540i 0.223715i −0.993724 0.111857i \(-0.964320\pi\)
0.993724 0.111857i \(-0.0356800\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 655.920 1136.09i 0.739481 1.28082i −0.213248 0.976998i \(-0.568404\pi\)
0.952729 0.303821i \(-0.0982625\pi\)
\(888\) 0 0
\(889\) −423.113 732.853i −0.475943 0.824357i
\(890\) 0 0
\(891\) 819.635 + 473.216i 0.919905 + 0.531107i
\(892\) 0 0
\(893\) 608.114 + 1053.28i 0.680979 + 1.17949i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −67.2985 + 116.564i −0.0750262 + 0.129949i
\(898\) 0 0
\(899\) 2305.76i 2.56481i
\(900\) 0 0
\(901\) 126.569 0.140476
\(902\) 0 0
\(903\) 471.810i 0.522492i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1499.08 865.496i 1.65279 0.954240i 0.676876 0.736097i \(-0.263333\pi\)
0.975917 0.218143i \(-0.0699999\pi\)
\(908\) 0 0
\(909\) 912.050i 1.00336i
\(910\) 0 0
\(911\) 210.720 121.659i 0.231306 0.133545i −0.379868 0.925041i \(-0.624031\pi\)
0.611174 + 0.791496i \(0.290697\pi\)
\(912\) 0 0
\(913\) −911.380 526.185i −0.998225 0.576326i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −353.450 −0.385442
\(918\) 0 0
\(919\) −449.145 −0.488733 −0.244366 0.969683i \(-0.578580\pi\)
−0.244366 + 0.969683i \(0.578580\pi\)
\(920\) 0 0
\(921\) 743.927 + 1288.52i 0.807739 + 1.39904i
\(922\) 0 0
\(923\) 115.635 200.286i 0.125282 0.216995i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 126.939 73.2883i 0.136935 0.0790597i
\(928\) 0 0
\(929\) 45.2420 26.1205i 0.0486997 0.0281168i −0.475452 0.879741i \(-0.657716\pi\)
0.524152 + 0.851625i \(0.324382\pi\)
\(930\) 0 0
\(931\) −467.623 + 809.947i −0.502280 + 0.869975i
\(932\) 0 0
\(933\) 678.140 0.726838
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 196.798i 0.210030i 0.994471 + 0.105015i \(0.0334891\pi\)
−0.994471 + 0.105015i \(0.966511\pi\)
\(938\) 0 0
\(939\) 490.736 849.979i 0.522615 0.905196i
\(940\) 0 0
\(941\) −822.375 474.798i −0.873937 0.504568i −0.00528258 0.999986i \(-0.501682\pi\)
−0.868655 + 0.495418i \(0.835015\pi\)
\(942\) 0 0
\(943\) 447.854 258.569i 0.474924 0.274198i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −814.378 1410.54i −0.859956 1.48949i −0.871970 0.489559i \(-0.837158\pi\)
0.0120142 0.999928i \(-0.496176\pi\)
\(948\) 0 0
\(949\) 15.5101 26.8643i 0.0163436 0.0283080i
\(950\) 0 0
\(951\) 21.2843 + 12.2885i 0.0223809 + 0.0129216i
\(952\) 0 0
\(953\) 789.836 0.828789 0.414395 0.910097i \(-0.363993\pi\)
0.414395 + 0.910097i \(0.363993\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1547.71i 1.61725i
\(958\) 0 0
\(959\) 672.756 + 388.416i 0.701518 + 0.405022i
\(960\) 0 0
\(961\) −883.056 1529.50i −0.918893 1.59157i
\(962\) 0 0
\(963\) 381.917 661.500i 0.396591 0.686916i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −456.689 263.669i −0.472274 0.272667i 0.244917 0.969544i \(-0.421239\pi\)
−0.717191 + 0.696877i \(0.754573\pi\)
\(968\) 0 0
\(969\) 811.923 468.764i 0.837898 0.483761i
\(970\) 0 0
\(971\) 179.700i 0.185067i 0.995710 + 0.0925337i \(0.0294966\pi\)
−0.995710 + 0.0925337i \(0.970503\pi\)
\(972\) 0 0
\(973\) 122.544i 0.125945i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 463.376 802.591i 0.474284 0.821485i −0.525282 0.850928i \(-0.676040\pi\)
0.999566 + 0.0294434i \(0.00937349\pi\)
\(978\) 0 0
\(979\) −556.379 963.677i −0.568314 0.984348i
\(980\) 0 0
\(981\) 728.855 1262.41i 0.742971 1.28686i
\(982\) 0 0
\(983\) −282.492 489.290i −0.287377 0.497752i 0.685806 0.727785i \(-0.259450\pi\)
−0.973183 + 0.230033i \(0.926117\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −511.025 −0.517755
\(988\) 0 0
\(989\) 835.673i 0.844967i
\(990\) 0 0
\(991\) −1815.42 −1.83191 −0.915953 0.401285i \(-0.868564\pi\)
−0.915953 + 0.401285i \(0.868564\pi\)
\(992\) 0 0
\(993\) 721.762 + 416.710i 0.726850 + 0.419647i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 590.961 341.192i 0.592739 0.342218i −0.173441 0.984844i \(-0.555488\pi\)
0.766180 + 0.642626i \(0.222155\pi\)
\(998\) 0 0
\(999\) −378.000 −0.378378
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 900.3.u.b.149.2 8
3.2 odd 2 2700.3.u.a.449.3 8
5.2 odd 4 900.3.p.b.401.2 4
5.3 odd 4 180.3.o.a.41.1 4
5.4 even 2 inner 900.3.u.b.149.3 8
9.2 odd 6 inner 900.3.u.b.749.3 8
9.7 even 3 2700.3.u.a.2249.2 8
15.2 even 4 2700.3.p.a.2501.2 4
15.8 even 4 540.3.o.a.341.2 4
15.14 odd 2 2700.3.u.a.449.2 8
20.3 even 4 720.3.bs.a.401.1 4
45.2 even 12 900.3.p.b.101.2 4
45.7 odd 12 2700.3.p.a.1601.2 4
45.13 odd 12 1620.3.g.a.161.4 4
45.23 even 12 1620.3.g.a.161.2 4
45.29 odd 6 inner 900.3.u.b.749.2 8
45.34 even 6 2700.3.u.a.2249.3 8
45.38 even 12 180.3.o.a.101.1 yes 4
45.43 odd 12 540.3.o.a.521.2 4
60.23 odd 4 2160.3.bs.a.881.2 4
180.43 even 12 2160.3.bs.a.1601.2 4
180.83 odd 12 720.3.bs.a.641.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.o.a.41.1 4 5.3 odd 4
180.3.o.a.101.1 yes 4 45.38 even 12
540.3.o.a.341.2 4 15.8 even 4
540.3.o.a.521.2 4 45.43 odd 12
720.3.bs.a.401.1 4 20.3 even 4
720.3.bs.a.641.1 4 180.83 odd 12
900.3.p.b.101.2 4 45.2 even 12
900.3.p.b.401.2 4 5.2 odd 4
900.3.u.b.149.2 8 1.1 even 1 trivial
900.3.u.b.149.3 8 5.4 even 2 inner
900.3.u.b.749.2 8 45.29 odd 6 inner
900.3.u.b.749.3 8 9.2 odd 6 inner
1620.3.g.a.161.2 4 45.23 even 12
1620.3.g.a.161.4 4 45.13 odd 12
2160.3.bs.a.881.2 4 60.23 odd 4
2160.3.bs.a.1601.2 4 180.43 even 12
2700.3.p.a.1601.2 4 45.7 odd 12
2700.3.p.a.2501.2 4 15.2 even 4
2700.3.u.a.449.2 8 15.14 odd 2
2700.3.u.a.449.3 8 3.2 odd 2
2700.3.u.a.2249.2 8 9.7 even 3
2700.3.u.a.2249.3 8 45.34 even 6