Properties

Label 2240.2.h.f.671.20
Level $2240$
Weight $2$
Character 2240.671
Analytic conductor $17.886$
Analytic rank $0$
Dimension $24$
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] = [2240,2,Mod(671,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.671");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.h (of order \(2\), degree \(1\), 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 671.20
Character \(\chi\) \(=\) 2240.671
Dual form 2240.2.h.f.671.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.59614i q^{3} +1.00000 q^{5} +(2.28275 + 1.33755i) q^{7} -3.73996 q^{9} +O(q^{10})\) \(q+2.59614i q^{3} +1.00000 q^{5} +(2.28275 + 1.33755i) q^{7} -3.73996 q^{9} +3.05894 q^{11} -6.11716 q^{13} +2.59614i q^{15} +1.25746i q^{17} +3.37720i q^{19} +(-3.47247 + 5.92635i) q^{21} -4.24285i q^{23} +1.00000 q^{25} -1.92104i q^{27} +1.52586i q^{29} -9.46731 q^{31} +7.94145i q^{33} +(2.28275 + 1.33755i) q^{35} +3.12779i q^{37} -15.8810i q^{39} +3.19570i q^{41} -10.7755 q^{43} -3.73996 q^{45} +7.89284 q^{47} +(3.42191 + 6.10660i) q^{49} -3.26456 q^{51} +12.5951i q^{53} +3.05894 q^{55} -8.76770 q^{57} +14.5828i q^{59} +0.274335 q^{61} +(-8.53740 - 5.00239i) q^{63} -6.11716 q^{65} +10.7513 q^{67} +11.0150 q^{69} -7.23258i q^{71} -0.742408i q^{73} +2.59614i q^{75} +(6.98281 + 4.09149i) q^{77} -10.1284i q^{79} -6.23258 q^{81} +6.15502i q^{83} +1.25746i q^{85} -3.96135 q^{87} -5.78158i q^{89} +(-13.9640 - 8.18202i) q^{91} -24.5785i q^{93} +3.37720i q^{95} +14.3036i q^{97} -11.4403 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{5} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{5} - 36 q^{9} + 4 q^{13} + 8 q^{21} + 24 q^{25} - 36 q^{45} + 24 q^{57} + 56 q^{61} + 4 q^{65} + 96 q^{69} - 12 q^{77} + 24 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(3\) 2.59614i 1.49888i 0.662070 + 0.749442i \(0.269678\pi\)
−0.662070 + 0.749442i \(0.730322\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.28275 + 1.33755i 0.862799 + 0.505547i
\(8\) 0 0
\(9\) −3.73996 −1.24665
\(10\) 0 0
\(11\) 3.05894 0.922306 0.461153 0.887321i \(-0.347436\pi\)
0.461153 + 0.887321i \(0.347436\pi\)
\(12\) 0 0
\(13\) −6.11716 −1.69660 −0.848298 0.529519i \(-0.822372\pi\)
−0.848298 + 0.529519i \(0.822372\pi\)
\(14\) 0 0
\(15\) 2.59614i 0.670321i
\(16\) 0 0
\(17\) 1.25746i 0.304980i 0.988305 + 0.152490i \(0.0487292\pi\)
−0.988305 + 0.152490i \(0.951271\pi\)
\(18\) 0 0
\(19\) 3.37720i 0.774784i 0.921915 + 0.387392i \(0.126624\pi\)
−0.921915 + 0.387392i \(0.873376\pi\)
\(20\) 0 0
\(21\) −3.47247 + 5.92635i −0.757756 + 1.29324i
\(22\) 0 0
\(23\) 4.24285i 0.884695i −0.896844 0.442348i \(-0.854146\pi\)
0.896844 0.442348i \(-0.145854\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.92104i 0.369704i
\(28\) 0 0
\(29\) 1.52586i 0.283345i 0.989914 + 0.141673i \(0.0452480\pi\)
−0.989914 + 0.141673i \(0.954752\pi\)
\(30\) 0 0
\(31\) −9.46731 −1.70038 −0.850190 0.526476i \(-0.823513\pi\)
−0.850190 + 0.526476i \(0.823513\pi\)
\(32\) 0 0
\(33\) 7.94145i 1.38243i
\(34\) 0 0
\(35\) 2.28275 + 1.33755i 0.385856 + 0.226087i
\(36\) 0 0
\(37\) 3.12779i 0.514206i 0.966384 + 0.257103i \(0.0827680\pi\)
−0.966384 + 0.257103i \(0.917232\pi\)
\(38\) 0 0
\(39\) 15.8810i 2.54300i
\(40\) 0 0
\(41\) 3.19570i 0.499085i 0.968364 + 0.249543i \(0.0802803\pi\)
−0.968364 + 0.249543i \(0.919720\pi\)
\(42\) 0 0
\(43\) −10.7755 −1.64325 −0.821623 0.570031i \(-0.806931\pi\)
−0.821623 + 0.570031i \(0.806931\pi\)
\(44\) 0 0
\(45\) −3.73996 −0.557520
\(46\) 0 0
\(47\) 7.89284 1.15129 0.575645 0.817700i \(-0.304751\pi\)
0.575645 + 0.817700i \(0.304751\pi\)
\(48\) 0 0
\(49\) 3.42191 + 6.10660i 0.488845 + 0.872371i
\(50\) 0 0
\(51\) −3.26456 −0.457129
\(52\) 0 0
\(53\) 12.5951i 1.73007i 0.501710 + 0.865036i \(0.332704\pi\)
−0.501710 + 0.865036i \(0.667296\pi\)
\(54\) 0 0
\(55\) 3.05894 0.412468
\(56\) 0 0
\(57\) −8.76770 −1.16131
\(58\) 0 0
\(59\) 14.5828i 1.89852i 0.314497 + 0.949259i \(0.398164\pi\)
−0.314497 + 0.949259i \(0.601836\pi\)
\(60\) 0 0
\(61\) 0.274335 0.0351250 0.0175625 0.999846i \(-0.494409\pi\)
0.0175625 + 0.999846i \(0.494409\pi\)
\(62\) 0 0
\(63\) −8.53740 5.00239i −1.07561 0.630242i
\(64\) 0 0
\(65\) −6.11716 −0.758741
\(66\) 0 0
\(67\) 10.7513 1.31348 0.656740 0.754117i \(-0.271935\pi\)
0.656740 + 0.754117i \(0.271935\pi\)
\(68\) 0 0
\(69\) 11.0150 1.32606
\(70\) 0 0
\(71\) 7.23258i 0.858350i −0.903221 0.429175i \(-0.858804\pi\)
0.903221 0.429175i \(-0.141196\pi\)
\(72\) 0 0
\(73\) 0.742408i 0.0868922i −0.999056 0.0434461i \(-0.986166\pi\)
0.999056 0.0434461i \(-0.0138337\pi\)
\(74\) 0 0
\(75\) 2.59614i 0.299777i
\(76\) 0 0
\(77\) 6.98281 + 4.09149i 0.795765 + 0.466269i
\(78\) 0 0
\(79\) 10.1284i 1.13953i −0.821807 0.569765i \(-0.807034\pi\)
0.821807 0.569765i \(-0.192966\pi\)
\(80\) 0 0
\(81\) −6.23258 −0.692509
\(82\) 0 0
\(83\) 6.15502i 0.675601i 0.941218 + 0.337801i \(0.109683\pi\)
−0.941218 + 0.337801i \(0.890317\pi\)
\(84\) 0 0
\(85\) 1.25746i 0.136391i
\(86\) 0 0
\(87\) −3.96135 −0.424702
\(88\) 0 0
\(89\) 5.78158i 0.612846i −0.951895 0.306423i \(-0.900868\pi\)
0.951895 0.306423i \(-0.0991322\pi\)
\(90\) 0 0
\(91\) −13.9640 8.18202i −1.46382 0.857709i
\(92\) 0 0
\(93\) 24.5785i 2.54867i
\(94\) 0 0
\(95\) 3.37720i 0.346494i
\(96\) 0 0
\(97\) 14.3036i 1.45231i 0.687533 + 0.726153i \(0.258694\pi\)
−0.687533 + 0.726153i \(0.741306\pi\)
\(98\) 0 0
\(99\) −11.4403 −1.14980
\(100\) 0 0
\(101\) −2.31463 −0.230314 −0.115157 0.993347i \(-0.536737\pi\)
−0.115157 + 0.993347i \(0.536737\pi\)
\(102\) 0 0
\(103\) 13.8344 1.36315 0.681573 0.731750i \(-0.261296\pi\)
0.681573 + 0.731750i \(0.261296\pi\)
\(104\) 0 0
\(105\) −3.47247 + 5.92635i −0.338879 + 0.578353i
\(106\) 0 0
\(107\) 13.6267 1.31734 0.658672 0.752430i \(-0.271118\pi\)
0.658672 + 0.752430i \(0.271118\pi\)
\(108\) 0 0
\(109\) 7.26068i 0.695447i −0.937597 0.347723i \(-0.886955\pi\)
0.937597 0.347723i \(-0.113045\pi\)
\(110\) 0 0
\(111\) −8.12020 −0.770736
\(112\) 0 0
\(113\) −9.04766 −0.851133 −0.425566 0.904927i \(-0.639925\pi\)
−0.425566 + 0.904927i \(0.639925\pi\)
\(114\) 0 0
\(115\) 4.24285i 0.395648i
\(116\) 0 0
\(117\) 22.8779 2.11507
\(118\) 0 0
\(119\) −1.68192 + 2.87048i −0.154182 + 0.263136i
\(120\) 0 0
\(121\) −1.64287 −0.149352
\(122\) 0 0
\(123\) −8.29651 −0.748071
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 9.27159i 0.822721i 0.911473 + 0.411360i \(0.134946\pi\)
−0.911473 + 0.411360i \(0.865054\pi\)
\(128\) 0 0
\(129\) 27.9747i 2.46303i
\(130\) 0 0
\(131\) 9.14476i 0.798981i −0.916737 0.399490i \(-0.869187\pi\)
0.916737 0.399490i \(-0.130813\pi\)
\(132\) 0 0
\(133\) −4.51718 + 7.70932i −0.391689 + 0.668483i
\(134\) 0 0
\(135\) 1.92104i 0.165337i
\(136\) 0 0
\(137\) −11.7481 −1.00371 −0.501853 0.864953i \(-0.667348\pi\)
−0.501853 + 0.864953i \(0.667348\pi\)
\(138\) 0 0
\(139\) 11.2894i 0.957552i −0.877937 0.478776i \(-0.841081\pi\)
0.877937 0.478776i \(-0.158919\pi\)
\(140\) 0 0
\(141\) 20.4910i 1.72565i
\(142\) 0 0
\(143\) −18.7121 −1.56478
\(144\) 0 0
\(145\) 1.52586i 0.126716i
\(146\) 0 0
\(147\) −15.8536 + 8.88378i −1.30758 + 0.732721i
\(148\) 0 0
\(149\) 18.6432i 1.52731i −0.645624 0.763655i \(-0.723403\pi\)
0.645624 0.763655i \(-0.276597\pi\)
\(150\) 0 0
\(151\) 3.29824i 0.268407i −0.990954 0.134204i \(-0.957152\pi\)
0.990954 0.134204i \(-0.0428476\pi\)
\(152\) 0 0
\(153\) 4.70286i 0.380204i
\(154\) 0 0
\(155\) −9.46731 −0.760433
\(156\) 0 0
\(157\) −0.308165 −0.0245943 −0.0122971 0.999924i \(-0.503914\pi\)
−0.0122971 + 0.999924i \(0.503914\pi\)
\(158\) 0 0
\(159\) −32.6987 −2.59318
\(160\) 0 0
\(161\) 5.67503 9.68537i 0.447255 0.763314i
\(162\) 0 0
\(163\) 12.8164 1.00386 0.501929 0.864909i \(-0.332624\pi\)
0.501929 + 0.864909i \(0.332624\pi\)
\(164\) 0 0
\(165\) 7.94145i 0.618241i
\(166\) 0 0
\(167\) 8.34267 0.645575 0.322788 0.946471i \(-0.395380\pi\)
0.322788 + 0.946471i \(0.395380\pi\)
\(168\) 0 0
\(169\) 24.4197 1.87844
\(170\) 0 0
\(171\) 12.6306i 0.965887i
\(172\) 0 0
\(173\) 0.924877 0.0703171 0.0351585 0.999382i \(-0.488806\pi\)
0.0351585 + 0.999382i \(0.488806\pi\)
\(174\) 0 0
\(175\) 2.28275 + 1.33755i 0.172560 + 0.101109i
\(176\) 0 0
\(177\) −37.8590 −2.84566
\(178\) 0 0
\(179\) −2.99825 −0.224100 −0.112050 0.993703i \(-0.535742\pi\)
−0.112050 + 0.993703i \(0.535742\pi\)
\(180\) 0 0
\(181\) −1.52021 −0.112997 −0.0564983 0.998403i \(-0.517994\pi\)
−0.0564983 + 0.998403i \(0.517994\pi\)
\(182\) 0 0
\(183\) 0.712214i 0.0526484i
\(184\) 0 0
\(185\) 3.12779i 0.229960i
\(186\) 0 0
\(187\) 3.84651i 0.281285i
\(188\) 0 0
\(189\) 2.56949 4.38526i 0.186903 0.318981i
\(190\) 0 0
\(191\) 4.77816i 0.345736i −0.984945 0.172868i \(-0.944697\pi\)
0.984945 0.172868i \(-0.0553034\pi\)
\(192\) 0 0
\(193\) 14.3979 1.03638 0.518191 0.855265i \(-0.326606\pi\)
0.518191 + 0.855265i \(0.326606\pi\)
\(194\) 0 0
\(195\) 15.8810i 1.13726i
\(196\) 0 0
\(197\) 20.8023i 1.48210i 0.671449 + 0.741050i \(0.265672\pi\)
−0.671449 + 0.741050i \(0.734328\pi\)
\(198\) 0 0
\(199\) 4.94052 0.350224 0.175112 0.984549i \(-0.443971\pi\)
0.175112 + 0.984549i \(0.443971\pi\)
\(200\) 0 0
\(201\) 27.9119i 1.96875i
\(202\) 0 0
\(203\) −2.04092 + 3.48316i −0.143244 + 0.244470i
\(204\) 0 0
\(205\) 3.19570i 0.223198i
\(206\) 0 0
\(207\) 15.8681i 1.10291i
\(208\) 0 0
\(209\) 10.3307i 0.714588i
\(210\) 0 0
\(211\) −4.89937 −0.337287 −0.168643 0.985677i \(-0.553939\pi\)
−0.168643 + 0.985677i \(0.553939\pi\)
\(212\) 0 0
\(213\) 18.7768 1.28657
\(214\) 0 0
\(215\) −10.7755 −0.734882
\(216\) 0 0
\(217\) −21.6115 12.6630i −1.46709 0.859622i
\(218\) 0 0
\(219\) 1.92740 0.130241
\(220\) 0 0
\(221\) 7.69211i 0.517428i
\(222\) 0 0
\(223\) 3.08814 0.206797 0.103399 0.994640i \(-0.467028\pi\)
0.103399 + 0.994640i \(0.467028\pi\)
\(224\) 0 0
\(225\) −3.73996 −0.249331
\(226\) 0 0
\(227\) 2.04747i 0.135896i 0.997689 + 0.0679478i \(0.0216451\pi\)
−0.997689 + 0.0679478i \(0.978355\pi\)
\(228\) 0 0
\(229\) −9.06904 −0.599299 −0.299649 0.954049i \(-0.596870\pi\)
−0.299649 + 0.954049i \(0.596870\pi\)
\(230\) 0 0
\(231\) −10.6221 + 18.1284i −0.698883 + 1.19276i
\(232\) 0 0
\(233\) −12.8301 −0.840529 −0.420265 0.907402i \(-0.638063\pi\)
−0.420265 + 0.907402i \(0.638063\pi\)
\(234\) 0 0
\(235\) 7.89284 0.514872
\(236\) 0 0
\(237\) 26.2947 1.70802
\(238\) 0 0
\(239\) 14.4982i 0.937811i −0.883248 0.468905i \(-0.844649\pi\)
0.883248 0.468905i \(-0.155351\pi\)
\(240\) 0 0
\(241\) 0.632410i 0.0407371i −0.999793 0.0203686i \(-0.993516\pi\)
0.999793 0.0203686i \(-0.00648396\pi\)
\(242\) 0 0
\(243\) 21.9438i 1.40770i
\(244\) 0 0
\(245\) 3.42191 + 6.10660i 0.218618 + 0.390136i
\(246\) 0 0
\(247\) 20.6589i 1.31449i
\(248\) 0 0
\(249\) −15.9793 −1.01265
\(250\) 0 0
\(251\) 9.77158i 0.616777i −0.951261 0.308388i \(-0.900210\pi\)
0.951261 0.308388i \(-0.0997897\pi\)
\(252\) 0 0
\(253\) 12.9786i 0.815960i
\(254\) 0 0
\(255\) −3.26456 −0.204434
\(256\) 0 0
\(257\) 3.31038i 0.206496i 0.994656 + 0.103248i \(0.0329235\pi\)
−0.994656 + 0.103248i \(0.967077\pi\)
\(258\) 0 0
\(259\) −4.18359 + 7.13998i −0.259955 + 0.443657i
\(260\) 0 0
\(261\) 5.70666i 0.353233i
\(262\) 0 0
\(263\) 28.2314i 1.74082i −0.492324 0.870412i \(-0.663852\pi\)
0.492324 0.870412i \(-0.336148\pi\)
\(264\) 0 0
\(265\) 12.5951i 0.773711i
\(266\) 0 0
\(267\) 15.0098 0.918585
\(268\) 0 0
\(269\) −28.0436 −1.70985 −0.854925 0.518751i \(-0.826397\pi\)
−0.854925 + 0.518751i \(0.826397\pi\)
\(270\) 0 0
\(271\) 8.79282 0.534126 0.267063 0.963679i \(-0.413947\pi\)
0.267063 + 0.963679i \(0.413947\pi\)
\(272\) 0 0
\(273\) 21.2417 36.2525i 1.28561 2.19410i
\(274\) 0 0
\(275\) 3.05894 0.184461
\(276\) 0 0
\(277\) 18.3990i 1.10549i −0.833351 0.552744i \(-0.813581\pi\)
0.833351 0.552744i \(-0.186419\pi\)
\(278\) 0 0
\(279\) 35.4074 2.11978
\(280\) 0 0
\(281\) 31.1469 1.85807 0.929035 0.369991i \(-0.120639\pi\)
0.929035 + 0.369991i \(0.120639\pi\)
\(282\) 0 0
\(283\) 29.8362i 1.77358i 0.462172 + 0.886790i \(0.347070\pi\)
−0.462172 + 0.886790i \(0.652930\pi\)
\(284\) 0 0
\(285\) −8.76770 −0.519354
\(286\) 0 0
\(287\) −4.27442 + 7.29500i −0.252311 + 0.430610i
\(288\) 0 0
\(289\) 15.4188 0.906987
\(290\) 0 0
\(291\) −37.1341 −2.17684
\(292\) 0 0
\(293\) 16.5550 0.967156 0.483578 0.875301i \(-0.339337\pi\)
0.483578 + 0.875301i \(0.339337\pi\)
\(294\) 0 0
\(295\) 14.5828i 0.849043i
\(296\) 0 0
\(297\) 5.87635i 0.340981i
\(298\) 0 0
\(299\) 25.9542i 1.50097i
\(300\) 0 0
\(301\) −24.5978 14.4128i −1.41779 0.830738i
\(302\) 0 0
\(303\) 6.00911i 0.345215i
\(304\) 0 0
\(305\) 0.274335 0.0157084
\(306\) 0 0
\(307\) 22.4879i 1.28345i 0.766933 + 0.641727i \(0.221782\pi\)
−0.766933 + 0.641727i \(0.778218\pi\)
\(308\) 0 0
\(309\) 35.9161i 2.04320i
\(310\) 0 0
\(311\) −3.02754 −0.171676 −0.0858380 0.996309i \(-0.527357\pi\)
−0.0858380 + 0.996309i \(0.527357\pi\)
\(312\) 0 0
\(313\) 23.0675i 1.30385i −0.758282 0.651926i \(-0.773961\pi\)
0.758282 0.651926i \(-0.226039\pi\)
\(314\) 0 0
\(315\) −8.53740 5.00239i −0.481028 0.281853i
\(316\) 0 0
\(317\) 10.8809i 0.611135i −0.952170 0.305567i \(-0.901154\pi\)
0.952170 0.305567i \(-0.0988462\pi\)
\(318\) 0 0
\(319\) 4.66752i 0.261331i
\(320\) 0 0
\(321\) 35.3769i 1.97455i
\(322\) 0 0
\(323\) −4.24671 −0.236293
\(324\) 0 0
\(325\) −6.11716 −0.339319
\(326\) 0 0
\(327\) 18.8498 1.04239
\(328\) 0 0
\(329\) 18.0174 + 10.5571i 0.993332 + 0.582031i
\(330\) 0 0
\(331\) 33.3053 1.83062 0.915312 0.402745i \(-0.131944\pi\)
0.915312 + 0.402745i \(0.131944\pi\)
\(332\) 0 0
\(333\) 11.6978i 0.641037i
\(334\) 0 0
\(335\) 10.7513 0.587406
\(336\) 0 0
\(337\) 14.3149 0.779783 0.389892 0.920861i \(-0.372512\pi\)
0.389892 + 0.920861i \(0.372512\pi\)
\(338\) 0 0
\(339\) 23.4890i 1.27575i
\(340\) 0 0
\(341\) −28.9600 −1.56827
\(342\) 0 0
\(343\) −0.356507 + 18.5168i −0.0192496 + 0.999815i
\(344\) 0 0
\(345\) 11.0150 0.593030
\(346\) 0 0
\(347\) 19.7204 1.05865 0.529324 0.848420i \(-0.322446\pi\)
0.529324 + 0.848420i \(0.322446\pi\)
\(348\) 0 0
\(349\) 9.99879 0.535223 0.267612 0.963527i \(-0.413766\pi\)
0.267612 + 0.963527i \(0.413766\pi\)
\(350\) 0 0
\(351\) 11.7513i 0.627239i
\(352\) 0 0
\(353\) 37.0563i 1.97231i 0.165839 + 0.986153i \(0.446967\pi\)
−0.165839 + 0.986153i \(0.553033\pi\)
\(354\) 0 0
\(355\) 7.23258i 0.383866i
\(356\) 0 0
\(357\) −7.45217 4.36651i −0.394411 0.231100i
\(358\) 0 0
\(359\) 14.8233i 0.782343i 0.920318 + 0.391172i \(0.127930\pi\)
−0.920318 + 0.391172i \(0.872070\pi\)
\(360\) 0 0
\(361\) 7.59449 0.399710
\(362\) 0 0
\(363\) 4.26512i 0.223861i
\(364\) 0 0
\(365\) 0.742408i 0.0388594i
\(366\) 0 0
\(367\) 28.3172 1.47814 0.739072 0.673626i \(-0.235264\pi\)
0.739072 + 0.673626i \(0.235264\pi\)
\(368\) 0 0
\(369\) 11.9518i 0.622186i
\(370\) 0 0
\(371\) −16.8466 + 28.7515i −0.874632 + 1.49270i
\(372\) 0 0
\(373\) 8.43536i 0.436767i 0.975863 + 0.218383i \(0.0700783\pi\)
−0.975863 + 0.218383i \(0.929922\pi\)
\(374\) 0 0
\(375\) 2.59614i 0.134064i
\(376\) 0 0
\(377\) 9.33394i 0.480722i
\(378\) 0 0
\(379\) 10.1855 0.523196 0.261598 0.965177i \(-0.415751\pi\)
0.261598 + 0.965177i \(0.415751\pi\)
\(380\) 0 0
\(381\) −24.0704 −1.23316
\(382\) 0 0
\(383\) 37.5763 1.92006 0.960031 0.279895i \(-0.0902998\pi\)
0.960031 + 0.279895i \(0.0902998\pi\)
\(384\) 0 0
\(385\) 6.98281 + 4.09149i 0.355877 + 0.208522i
\(386\) 0 0
\(387\) 40.2999 2.04856
\(388\) 0 0
\(389\) 4.88157i 0.247505i −0.992313 0.123753i \(-0.960507\pi\)
0.992313 0.123753i \(-0.0394929\pi\)
\(390\) 0 0
\(391\) 5.33523 0.269814
\(392\) 0 0
\(393\) 23.7411 1.19758
\(394\) 0 0
\(395\) 10.1284i 0.509614i
\(396\) 0 0
\(397\) −22.4288 −1.12567 −0.562835 0.826569i \(-0.690289\pi\)
−0.562835 + 0.826569i \(0.690289\pi\)
\(398\) 0 0
\(399\) −20.0145 11.7273i −1.00198 0.587097i
\(400\) 0 0
\(401\) 4.25212 0.212341 0.106170 0.994348i \(-0.466141\pi\)
0.106170 + 0.994348i \(0.466141\pi\)
\(402\) 0 0
\(403\) 57.9131 2.88486
\(404\) 0 0
\(405\) −6.23258 −0.309699
\(406\) 0 0
\(407\) 9.56775i 0.474256i
\(408\) 0 0
\(409\) 27.7723i 1.37325i −0.727010 0.686627i \(-0.759091\pi\)
0.727010 0.686627i \(-0.240909\pi\)
\(410\) 0 0
\(411\) 30.4997i 1.50444i
\(412\) 0 0
\(413\) −19.5052 + 33.2889i −0.959789 + 1.63804i
\(414\) 0 0
\(415\) 6.15502i 0.302138i
\(416\) 0 0
\(417\) 29.3088 1.43526
\(418\) 0 0
\(419\) 9.91246i 0.484255i 0.970244 + 0.242128i \(0.0778453\pi\)
−0.970244 + 0.242128i \(0.922155\pi\)
\(420\) 0 0
\(421\) 33.1235i 1.61434i 0.590318 + 0.807171i \(0.299002\pi\)
−0.590318 + 0.807171i \(0.700998\pi\)
\(422\) 0 0
\(423\) −29.5189 −1.43526
\(424\) 0 0
\(425\) 1.25746i 0.0609960i
\(426\) 0 0
\(427\) 0.626240 + 0.366938i 0.0303059 + 0.0177574i
\(428\) 0 0
\(429\) 48.5792i 2.34542i
\(430\) 0 0
\(431\) 24.9585i 1.20221i 0.799171 + 0.601104i \(0.205272\pi\)
−0.799171 + 0.601104i \(0.794728\pi\)
\(432\) 0 0
\(433\) 4.83379i 0.232297i 0.993232 + 0.116149i \(0.0370549\pi\)
−0.993232 + 0.116149i \(0.962945\pi\)
\(434\) 0 0
\(435\) −3.96135 −0.189932
\(436\) 0 0
\(437\) 14.3290 0.685447
\(438\) 0 0
\(439\) −18.2643 −0.871708 −0.435854 0.900017i \(-0.643554\pi\)
−0.435854 + 0.900017i \(0.643554\pi\)
\(440\) 0 0
\(441\) −12.7978 22.8384i −0.609420 1.08754i
\(442\) 0 0
\(443\) −26.2979 −1.24945 −0.624725 0.780844i \(-0.714789\pi\)
−0.624725 + 0.780844i \(0.714789\pi\)
\(444\) 0 0
\(445\) 5.78158i 0.274073i
\(446\) 0 0
\(447\) 48.4004 2.28926
\(448\) 0 0
\(449\) −38.9639 −1.83882 −0.919411 0.393299i \(-0.871334\pi\)
−0.919411 + 0.393299i \(0.871334\pi\)
\(450\) 0 0
\(451\) 9.77548i 0.460309i
\(452\) 0 0
\(453\) 8.56271 0.402311
\(454\) 0 0
\(455\) −13.9640 8.18202i −0.654641 0.383579i
\(456\) 0 0
\(457\) 35.8688 1.67787 0.838936 0.544230i \(-0.183178\pi\)
0.838936 + 0.544230i \(0.183178\pi\)
\(458\) 0 0
\(459\) 2.41564 0.112752
\(460\) 0 0
\(461\) −33.9750 −1.58237 −0.791187 0.611574i \(-0.790537\pi\)
−0.791187 + 0.611574i \(0.790537\pi\)
\(462\) 0 0
\(463\) 15.8826i 0.738125i 0.929405 + 0.369063i \(0.120321\pi\)
−0.929405 + 0.369063i \(0.879679\pi\)
\(464\) 0 0
\(465\) 24.5785i 1.13980i
\(466\) 0 0
\(467\) 13.1435i 0.608209i 0.952639 + 0.304105i \(0.0983572\pi\)
−0.952639 + 0.304105i \(0.901643\pi\)
\(468\) 0 0
\(469\) 24.5425 + 14.3804i 1.13327 + 0.664026i
\(470\) 0 0
\(471\) 0.800041i 0.0368639i
\(472\) 0 0
\(473\) −32.9616 −1.51558
\(474\) 0 0
\(475\) 3.37720i 0.154957i
\(476\) 0 0
\(477\) 47.1052i 2.15680i
\(478\) 0 0
\(479\) 22.5908 1.03220 0.516100 0.856528i \(-0.327383\pi\)
0.516100 + 0.856528i \(0.327383\pi\)
\(480\) 0 0
\(481\) 19.1332i 0.872400i
\(482\) 0 0
\(483\) 25.1446 + 14.7332i 1.14412 + 0.670383i
\(484\) 0 0
\(485\) 14.3036i 0.649491i
\(486\) 0 0
\(487\) 25.3997i 1.15097i 0.817812 + 0.575485i \(0.195187\pi\)
−0.817812 + 0.575485i \(0.804813\pi\)
\(488\) 0 0
\(489\) 33.2732i 1.50467i
\(490\) 0 0
\(491\) 12.4146 0.560264 0.280132 0.959961i \(-0.409622\pi\)
0.280132 + 0.959961i \(0.409622\pi\)
\(492\) 0 0
\(493\) −1.91872 −0.0864146
\(494\) 0 0
\(495\) −11.4403 −0.514204
\(496\) 0 0
\(497\) 9.67395 16.5102i 0.433936 0.740583i
\(498\) 0 0
\(499\) 20.3343 0.910290 0.455145 0.890417i \(-0.349587\pi\)
0.455145 + 0.890417i \(0.349587\pi\)
\(500\) 0 0
\(501\) 21.6588i 0.967642i
\(502\) 0 0
\(503\) 3.07370 0.137050 0.0685248 0.997649i \(-0.478171\pi\)
0.0685248 + 0.997649i \(0.478171\pi\)
\(504\) 0 0
\(505\) −2.31463 −0.103000
\(506\) 0 0
\(507\) 63.3970i 2.81556i
\(508\) 0 0
\(509\) 32.3148 1.43233 0.716163 0.697933i \(-0.245897\pi\)
0.716163 + 0.697933i \(0.245897\pi\)
\(510\) 0 0
\(511\) 0.993008 1.69473i 0.0439281 0.0749706i
\(512\) 0 0
\(513\) 6.48775 0.286441
\(514\) 0 0
\(515\) 13.8344 0.609618
\(516\) 0 0
\(517\) 24.1438 1.06184
\(518\) 0 0
\(519\) 2.40111i 0.105397i
\(520\) 0 0
\(521\) 25.7947i 1.13009i 0.825062 + 0.565043i \(0.191140\pi\)
−0.825062 + 0.565043i \(0.808860\pi\)
\(522\) 0 0
\(523\) 11.7945i 0.515736i −0.966180 0.257868i \(-0.916980\pi\)
0.966180 0.257868i \(-0.0830200\pi\)
\(524\) 0 0
\(525\) −3.47247 + 5.92635i −0.151551 + 0.258647i
\(526\) 0 0
\(527\) 11.9048i 0.518582i
\(528\) 0 0
\(529\) 4.99824 0.217315
\(530\) 0 0
\(531\) 54.5390i 2.36679i
\(532\) 0 0
\(533\) 19.5486i 0.846746i
\(534\) 0 0
\(535\) 13.6267 0.589134
\(536\) 0 0
\(537\) 7.78390i 0.335900i
\(538\) 0 0
\(539\) 10.4674 + 18.6797i 0.450864 + 0.804593i
\(540\) 0 0
\(541\) 2.07521i 0.0892204i 0.999004 + 0.0446102i \(0.0142046\pi\)
−0.999004 + 0.0446102i \(0.985795\pi\)
\(542\) 0 0
\(543\) 3.94669i 0.169369i
\(544\) 0 0
\(545\) 7.26068i 0.311013i
\(546\) 0 0
\(547\) −17.5339 −0.749697 −0.374849 0.927086i \(-0.622305\pi\)
−0.374849 + 0.927086i \(0.622305\pi\)
\(548\) 0 0
\(549\) −1.02600 −0.0437887
\(550\) 0 0
\(551\) −5.15314 −0.219531
\(552\) 0 0
\(553\) 13.5472 23.1206i 0.576086 0.983186i
\(554\) 0 0
\(555\) −8.12020 −0.344683
\(556\) 0 0
\(557\) 17.9855i 0.762068i −0.924561 0.381034i \(-0.875568\pi\)
0.924561 0.381034i \(-0.124432\pi\)
\(558\) 0 0
\(559\) 65.9154 2.78792
\(560\) 0 0
\(561\) −9.98609 −0.421613
\(562\) 0 0
\(563\) 26.5159i 1.11751i −0.829332 0.558756i \(-0.811279\pi\)
0.829332 0.558756i \(-0.188721\pi\)
\(564\) 0 0
\(565\) −9.04766 −0.380638
\(566\) 0 0
\(567\) −14.2274 8.33640i −0.597496 0.350096i
\(568\) 0 0
\(569\) −9.60242 −0.402554 −0.201277 0.979534i \(-0.564509\pi\)
−0.201277 + 0.979534i \(0.564509\pi\)
\(570\) 0 0
\(571\) −14.4225 −0.603563 −0.301782 0.953377i \(-0.597581\pi\)
−0.301782 + 0.953377i \(0.597581\pi\)
\(572\) 0 0
\(573\) 12.4048 0.518218
\(574\) 0 0
\(575\) 4.24285i 0.176939i
\(576\) 0 0
\(577\) 33.6880i 1.40245i −0.712940 0.701225i \(-0.752637\pi\)
0.712940 0.701225i \(-0.247363\pi\)
\(578\) 0 0
\(579\) 37.3789i 1.55342i
\(580\) 0 0
\(581\) −8.23266 + 14.0504i −0.341548 + 0.582908i
\(582\) 0 0
\(583\) 38.5277i 1.59566i
\(584\) 0 0
\(585\) 22.8779 0.945887
\(586\) 0 0
\(587\) 33.5627i 1.38528i −0.721283 0.692641i \(-0.756447\pi\)
0.721283 0.692641i \(-0.243553\pi\)
\(588\) 0 0
\(589\) 31.9731i 1.31743i
\(590\) 0 0
\(591\) −54.0057 −2.22150
\(592\) 0 0
\(593\) 24.5203i 1.00693i 0.864016 + 0.503464i \(0.167941\pi\)
−0.864016 + 0.503464i \(0.832059\pi\)
\(594\) 0 0
\(595\) −1.68192 + 2.87048i −0.0689521 + 0.117678i
\(596\) 0 0
\(597\) 12.8263i 0.524945i
\(598\) 0 0
\(599\) 3.17590i 0.129764i −0.997893 0.0648818i \(-0.979333\pi\)
0.997893 0.0648818i \(-0.0206670\pi\)
\(600\) 0 0
\(601\) 0.334920i 0.0136617i −0.999977 0.00683083i \(-0.997826\pi\)
0.999977 0.00683083i \(-0.00217434\pi\)
\(602\) 0 0
\(603\) −40.2094 −1.63745
\(604\) 0 0
\(605\) −1.64287 −0.0667921
\(606\) 0 0
\(607\) −5.55422 −0.225439 −0.112719 0.993627i \(-0.535956\pi\)
−0.112719 + 0.993627i \(0.535956\pi\)
\(608\) 0 0
\(609\) −9.04279 5.29851i −0.366432 0.214707i
\(610\) 0 0
\(611\) −48.2818 −1.95327
\(612\) 0 0
\(613\) 0.100256i 0.00404931i 0.999998 + 0.00202465i \(0.000644468\pi\)
−0.999998 + 0.00202465i \(0.999356\pi\)
\(614\) 0 0
\(615\) −8.29651 −0.334547
\(616\) 0 0
\(617\) 13.6533 0.549660 0.274830 0.961493i \(-0.411378\pi\)
0.274830 + 0.961493i \(0.411378\pi\)
\(618\) 0 0
\(619\) 21.7703i 0.875024i −0.899213 0.437512i \(-0.855860\pi\)
0.899213 0.437512i \(-0.144140\pi\)
\(620\) 0 0
\(621\) −8.15068 −0.327076
\(622\) 0 0
\(623\) 7.73316 13.1979i 0.309822 0.528763i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −26.8199 −1.07108
\(628\) 0 0
\(629\) −3.93309 −0.156823
\(630\) 0 0
\(631\) 22.1194i 0.880559i 0.897861 + 0.440280i \(0.145121\pi\)
−0.897861 + 0.440280i \(0.854879\pi\)
\(632\) 0 0
\(633\) 12.7195i 0.505554i
\(634\) 0 0
\(635\) 9.27159i 0.367932i
\(636\) 0 0
\(637\) −20.9324 37.3550i −0.829372 1.48006i
\(638\) 0 0
\(639\) 27.0496i 1.07006i
\(640\) 0 0
\(641\) 26.4852 1.04610 0.523050 0.852302i \(-0.324794\pi\)
0.523050 + 0.852302i \(0.324794\pi\)
\(642\) 0 0
\(643\) 39.9374i 1.57498i −0.616330 0.787488i \(-0.711381\pi\)
0.616330 0.787488i \(-0.288619\pi\)
\(644\) 0 0
\(645\) 27.9747i 1.10150i
\(646\) 0 0
\(647\) −45.2321 −1.77826 −0.889128 0.457658i \(-0.848688\pi\)
−0.889128 + 0.457658i \(0.848688\pi\)
\(648\) 0 0
\(649\) 44.6079i 1.75101i
\(650\) 0 0
\(651\) 32.8750 56.1066i 1.28847 2.19899i
\(652\) 0 0
\(653\) 9.97322i 0.390282i −0.980775 0.195141i \(-0.937484\pi\)
0.980775 0.195141i \(-0.0625165\pi\)
\(654\) 0 0
\(655\) 9.14476i 0.357315i
\(656\) 0 0
\(657\) 2.77657i 0.108324i
\(658\) 0 0
\(659\) 33.7753 1.31570 0.657850 0.753149i \(-0.271466\pi\)
0.657850 + 0.753149i \(0.271466\pi\)
\(660\) 0 0
\(661\) −18.0667 −0.702715 −0.351357 0.936241i \(-0.614280\pi\)
−0.351357 + 0.936241i \(0.614280\pi\)
\(662\) 0 0
\(663\) 19.9698 0.775564
\(664\) 0 0
\(665\) −4.51718 + 7.70932i −0.175169 + 0.298955i
\(666\) 0 0
\(667\) 6.47400 0.250674
\(668\) 0 0
\(669\) 8.01727i 0.309965i
\(670\) 0 0
\(671\) 0.839176 0.0323960
\(672\) 0 0
\(673\) −37.7732 −1.45605 −0.728024 0.685551i \(-0.759561\pi\)
−0.728024 + 0.685551i \(0.759561\pi\)
\(674\) 0 0
\(675\) 1.92104i 0.0739409i
\(676\) 0 0
\(677\) 36.0391 1.38510 0.692548 0.721371i \(-0.256488\pi\)
0.692548 + 0.721371i \(0.256488\pi\)
\(678\) 0 0
\(679\) −19.1317 + 32.6515i −0.734209 + 1.25305i
\(680\) 0 0
\(681\) −5.31553 −0.203692
\(682\) 0 0
\(683\) −11.2784 −0.431554 −0.215777 0.976443i \(-0.569228\pi\)
−0.215777 + 0.976443i \(0.569228\pi\)
\(684\) 0 0
\(685\) −11.7481 −0.448871
\(686\) 0 0
\(687\) 23.5445i 0.898280i
\(688\) 0 0
\(689\) 77.0463i 2.93523i
\(690\) 0 0
\(691\) 21.4353i 0.815438i −0.913107 0.407719i \(-0.866324\pi\)
0.913107 0.407719i \(-0.133676\pi\)
\(692\) 0 0
\(693\) −26.1154 15.3020i −0.992043 0.581276i
\(694\) 0 0
\(695\) 11.2894i 0.428230i
\(696\) 0 0
\(697\) −4.01848 −0.152211
\(698\) 0 0
\(699\) 33.3088i 1.25986i
\(700\) 0 0
\(701\) 1.09861i 0.0414940i −0.999785 0.0207470i \(-0.993396\pi\)
0.999785 0.0207470i \(-0.00660446\pi\)
\(702\) 0 0
\(703\) −10.5632 −0.398399
\(704\) 0 0
\(705\) 20.4910i 0.771734i
\(706\) 0 0
\(707\) −5.28373 3.09594i −0.198715 0.116435i
\(708\) 0 0
\(709\) 20.7287i 0.778481i 0.921136 + 0.389240i \(0.127262\pi\)
−0.921136 + 0.389240i \(0.872738\pi\)
\(710\) 0 0
\(711\) 37.8797i 1.42060i
\(712\) 0 0
\(713\) 40.1684i 1.50432i
\(714\) 0 0
\(715\) −18.7121 −0.699791
\(716\) 0 0
\(717\) 37.6394 1.40567
\(718\) 0 0
\(719\) 26.5825 0.991361 0.495680 0.868505i \(-0.334919\pi\)
0.495680 + 0.868505i \(0.334919\pi\)
\(720\) 0 0
\(721\) 31.5806 + 18.5043i 1.17612 + 0.689134i
\(722\) 0 0
\(723\) 1.64183 0.0610602
\(724\) 0 0
\(725\) 1.52586i 0.0566691i
\(726\) 0 0
\(727\) −2.08151 −0.0771989 −0.0385995 0.999255i \(-0.512290\pi\)
−0.0385995 + 0.999255i \(0.512290\pi\)
\(728\) 0 0
\(729\) 38.2715 1.41746
\(730\) 0 0
\(731\) 13.5498i 0.501157i
\(732\) 0 0
\(733\) −2.11229 −0.0780191 −0.0390096 0.999239i \(-0.512420\pi\)
−0.0390096 + 0.999239i \(0.512420\pi\)
\(734\) 0 0
\(735\) −15.8536 + 8.88378i −0.584769 + 0.327683i
\(736\) 0 0
\(737\) 32.8876 1.21143
\(738\) 0 0
\(739\) −13.6264 −0.501255 −0.250628 0.968084i \(-0.580637\pi\)
−0.250628 + 0.968084i \(0.580637\pi\)
\(740\) 0 0
\(741\) 53.6335 1.97028
\(742\) 0 0
\(743\) 42.8070i 1.57044i 0.619219 + 0.785218i \(0.287449\pi\)
−0.619219 + 0.785218i \(0.712551\pi\)
\(744\) 0 0
\(745\) 18.6432i 0.683034i
\(746\) 0 0
\(747\) 23.0195i 0.842241i
\(748\) 0 0
\(749\) 31.1064 + 18.2264i 1.13660 + 0.665979i
\(750\) 0 0
\(751\) 25.3283i 0.924244i 0.886816 + 0.462122i \(0.152912\pi\)
−0.886816 + 0.462122i \(0.847088\pi\)
\(752\) 0 0
\(753\) 25.3684 0.924477
\(754\) 0 0
\(755\) 3.29824i 0.120035i
\(756\) 0 0
\(757\) 37.2852i 1.35516i −0.735451 0.677578i \(-0.763030\pi\)
0.735451 0.677578i \(-0.236970\pi\)
\(758\) 0 0
\(759\) 33.6944 1.22303
\(760\) 0 0
\(761\) 28.5390i 1.03454i −0.855823 0.517268i \(-0.826949\pi\)
0.855823 0.517268i \(-0.173051\pi\)
\(762\) 0 0
\(763\) 9.71153 16.5743i 0.351581 0.600031i
\(764\) 0 0
\(765\) 4.70286i 0.170032i
\(766\) 0 0
\(767\) 89.2053i 3.22102i
\(768\) 0 0
\(769\) 25.7803i 0.929660i −0.885400 0.464830i \(-0.846115\pi\)
0.885400 0.464830i \(-0.153885\pi\)
\(770\) 0 0
\(771\) −8.59421 −0.309513
\(772\) 0 0
\(773\) −29.8642 −1.07414 −0.537070 0.843538i \(-0.680469\pi\)
−0.537070 + 0.843538i \(0.680469\pi\)
\(774\) 0 0
\(775\) −9.46731 −0.340076
\(776\) 0 0
\(777\) −18.5364 10.8612i −0.664990 0.389643i
\(778\) 0 0
\(779\) −10.7925 −0.386683
\(780\) 0 0
\(781\) 22.1241i 0.791661i
\(782\) 0 0
\(783\) 2.93124 0.104754
\(784\) 0 0
\(785\) −0.308165 −0.0109989
\(786\) 0 0
\(787\) 0.839063i 0.0299094i −0.999888 0.0149547i \(-0.995240\pi\)
0.999888 0.0149547i \(-0.00476040\pi\)
\(788\) 0 0
\(789\) 73.2928 2.60929
\(790\) 0 0
\(791\) −20.6536 12.1017i −0.734357 0.430287i
\(792\) 0 0
\(793\) −1.67815 −0.0595930
\(794\) 0 0
\(795\) −32.6987 −1.15970
\(796\) 0 0
\(797\) 2.39992 0.0850096 0.0425048 0.999096i \(-0.486466\pi\)
0.0425048 + 0.999096i \(0.486466\pi\)
\(798\) 0 0
\(799\) 9.92497i 0.351120i
\(800\) 0 0
\(801\) 21.6229i 0.764007i
\(802\) 0 0
\(803\) 2.27098i 0.0801412i
\(804\) 0 0
\(805\) 5.67503 9.68537i 0.200018 0.341364i
\(806\) 0 0
\(807\) 72.8053i 2.56287i
\(808\) 0 0
\(809\) −31.4961 −1.10734 −0.553672 0.832735i \(-0.686774\pi\)
−0.553672 + 0.832735i \(0.686774\pi\)
\(810\) 0 0
\(811\) 8.89901i 0.312486i 0.987719 + 0.156243i \(0.0499384\pi\)
−0.987719 + 0.156243i \(0.950062\pi\)
\(812\) 0 0
\(813\) 22.8274i 0.800592i
\(814\) 0 0
\(815\) 12.8164 0.448939
\(816\) 0 0
\(817\) 36.3910i 1.27316i
\(818\) 0 0
\(819\) 52.2247 + 30.6004i 1.82488 + 1.06927i
\(820\) 0 0
\(821\) 24.8802i 0.868326i 0.900834 + 0.434163i \(0.142956\pi\)
−0.900834 + 0.434163i \(0.857044\pi\)
\(822\) 0 0
\(823\) 38.0624i 1.32677i 0.748277 + 0.663386i \(0.230881\pi\)
−0.748277 + 0.663386i \(0.769119\pi\)
\(824\) 0 0
\(825\) 7.94145i 0.276486i
\(826\) 0 0
\(827\) −28.3847 −0.987031 −0.493516 0.869737i \(-0.664288\pi\)
−0.493516 + 0.869737i \(0.664288\pi\)
\(828\) 0 0
\(829\) −31.3879 −1.09015 −0.545074 0.838388i \(-0.683498\pi\)
−0.545074 + 0.838388i \(0.683498\pi\)
\(830\) 0 0
\(831\) 47.7664 1.65700
\(832\) 0 0
\(833\) −7.67882 + 4.30293i −0.266056 + 0.149088i
\(834\) 0 0
\(835\) 8.34267 0.288710
\(836\) 0 0
\(837\) 18.1871i 0.628638i
\(838\) 0 0
\(839\) −10.4588 −0.361079 −0.180539 0.983568i \(-0.557784\pi\)
−0.180539 + 0.983568i \(0.557784\pi\)
\(840\) 0 0
\(841\) 26.6717 0.919715
\(842\) 0 0
\(843\) 80.8619i 2.78503i
\(844\) 0 0
\(845\) 24.4197 0.840063
\(846\) 0 0
\(847\) −3.75026 2.19742i −0.128861 0.0755043i
\(848\) 0 0
\(849\) −77.4592 −2.65839
\(850\) 0 0
\(851\) 13.2708 0.454916
\(852\) 0 0
\(853\) 17.0347 0.583256 0.291628 0.956532i \(-0.405803\pi\)
0.291628 + 0.956532i \(0.405803\pi\)
\(854\) 0 0
\(855\) 12.6306i 0.431958i
\(856\) 0 0
\(857\) 43.2852i 1.47859i −0.673379 0.739297i \(-0.735158\pi\)
0.673379 0.739297i \(-0.264842\pi\)
\(858\) 0 0
\(859\) 23.0541i 0.786597i −0.919411 0.393298i \(-0.871334\pi\)
0.919411 0.393298i \(-0.128666\pi\)
\(860\) 0 0
\(861\) −18.9389 11.0970i −0.645435 0.378185i
\(862\) 0 0
\(863\) 24.3924i 0.830327i 0.909747 + 0.415163i \(0.136276\pi\)
−0.909747 + 0.415163i \(0.863724\pi\)
\(864\) 0 0
\(865\) 0.924877 0.0314468
\(866\) 0 0
\(867\) 40.0294i 1.35947i
\(868\) 0 0
\(869\) 30.9821i 1.05100i
\(870\) 0 0
\(871\) −65.7675 −2.22845
\(872\) 0 0
\(873\) 53.4947i 1.81052i
\(874\) 0 0
\(875\) 2.28275 + 1.33755i 0.0771711 + 0.0452175i
\(876\) 0 0
\(877\) 57.3223i 1.93564i 0.251649 + 0.967819i \(0.419027\pi\)
−0.251649 + 0.967819i \(0.580973\pi\)
\(878\) 0 0
\(879\) 42.9793i 1.44965i
\(880\) 0 0
\(881\) 44.0539i 1.48421i 0.670282 + 0.742107i \(0.266173\pi\)
−0.670282 + 0.742107i \(0.733827\pi\)
\(882\) 0 0
\(883\) 44.5580 1.49950 0.749749 0.661723i \(-0.230174\pi\)
0.749749 + 0.661723i \(0.230174\pi\)
\(884\) 0 0
\(885\) −37.8590 −1.27262
\(886\) 0 0
\(887\) −2.89650 −0.0972550 −0.0486275 0.998817i \(-0.515485\pi\)
−0.0486275 + 0.998817i \(0.515485\pi\)
\(888\) 0 0
\(889\) −12.4012 + 21.1647i −0.415924 + 0.709843i
\(890\) 0 0
\(891\) −19.0651 −0.638705
\(892\) 0 0
\(893\) 26.6557i 0.892000i
\(894\) 0 0
\(895\) −2.99825 −0.100221
\(896\) 0 0
\(897\) −67.3808 −2.24978
\(898\) 0 0
\(899\) 14.4458i 0.481795i
\(900\) 0 0
\(901\) −15.8379 −0.527637
\(902\) 0 0
\(903\) 37.4176 63.8593i 1.24518 2.12510i
\(904\) 0 0
\(905\) −1.52021 −0.0505336
\(906\) 0 0
\(907\) 26.2835 0.872728 0.436364 0.899770i \(-0.356266\pi\)
0.436364 + 0.899770i \(0.356266\pi\)
\(908\) 0 0
\(909\) 8.65663 0.287122
\(910\) 0 0
\(911\) 33.5911i 1.11292i −0.830873 0.556462i \(-0.812158\pi\)
0.830873 0.556462i \(-0.187842\pi\)
\(912\) 0 0
\(913\) 18.8279i 0.623111i
\(914\) 0 0
\(915\) 0.712214i 0.0235451i
\(916\) 0 0
\(917\) 12.2316 20.8752i 0.403922 0.689360i
\(918\) 0 0
\(919\) 30.6535i 1.01117i 0.862778 + 0.505583i \(0.168723\pi\)
−0.862778 + 0.505583i \(0.831277\pi\)
\(920\) 0 0
\(921\) −58.3819 −1.92375
\(922\) 0 0
\(923\) 44.2429i 1.45627i
\(924\) 0 0
\(925\) 3.12779i 0.102841i
\(926\) 0 0
\(927\) −51.7402 −1.69937
\(928\) 0 0
\(929\) 59.1269i 1.93989i 0.243322 + 0.969946i \(0.421763\pi\)
−0.243322 + 0.969946i \(0.578237\pi\)
\(930\) 0 0
\(931\) −20.6232 + 11.5565i −0.675899 + 0.378749i
\(932\) 0 0
\(933\) 7.85992i 0.257322i
\(934\) 0 0
\(935\) 3.84651i 0.125794i
\(936\) 0 0
\(937\) 16.3377i 0.533730i 0.963734 + 0.266865i \(0.0859878\pi\)
−0.963734 + 0.266865i \(0.914012\pi\)
\(938\) 0 0
\(939\) 59.8866 1.95432
\(940\) 0 0
\(941\) 24.6773 0.804457 0.402228 0.915539i \(-0.368236\pi\)
0.402228 + 0.915539i \(0.368236\pi\)
\(942\) 0 0
\(943\) 13.5589 0.441538
\(944\) 0 0
\(945\) 2.56949 4.38526i 0.0835855 0.142652i
\(946\) 0 0
\(947\) −28.5781 −0.928664 −0.464332 0.885661i \(-0.653706\pi\)
−0.464332 + 0.885661i \(0.653706\pi\)
\(948\) 0 0
\(949\) 4.54143i 0.147421i
\(950\) 0 0
\(951\) 28.2485 0.916020
\(952\) 0 0
\(953\) −20.9275 −0.677908 −0.338954 0.940803i \(-0.610073\pi\)
−0.338954 + 0.940803i \(0.610073\pi\)
\(954\) 0 0
\(955\) 4.77816i 0.154618i
\(956\) 0 0
\(957\) −12.1176 −0.391705
\(958\) 0 0
\(959\) −26.8179 15.7137i −0.865996 0.507420i
\(960\) 0 0
\(961\) 58.6300 1.89129
\(962\) 0 0
\(963\) −50.9634 −1.64227
\(964\) 0 0
\(965\) 14.3979 0.463484
\(966\) 0 0
\(967\) 21.8079i 0.701295i −0.936508 0.350647i \(-0.885962\pi\)
0.936508 0.350647i \(-0.114038\pi\)
\(968\) 0 0
\(969\) 11.0251i 0.354176i
\(970\) 0 0
\(971\) 32.1900i 1.03303i −0.856279 0.516513i \(-0.827230\pi\)
0.856279 0.516513i \(-0.172770\pi\)
\(972\) 0 0
\(973\) 15.1001 25.7708i 0.484088 0.826175i
\(974\) 0 0
\(975\) 15.8810i 0.508600i
\(976\) 0 0
\(977\) 19.4242 0.621435 0.310717 0.950502i \(-0.399431\pi\)
0.310717 + 0.950502i \(0.399431\pi\)
\(978\) 0 0
\(979\) 17.6855i 0.565232i
\(980\) 0 0
\(981\) 27.1546i 0.866981i
\(982\) 0 0
\(983\) −57.5903 −1.83685 −0.918423 0.395599i \(-0.870537\pi\)
−0.918423 + 0.395599i \(0.870537\pi\)
\(984\) 0 0
\(985\) 20.8023i 0.662816i
\(986\) 0 0
\(987\) −27.4077 + 46.7758i −0.872397 + 1.48889i
\(988\) 0 0
\(989\) 45.7187i 1.45377i
\(990\) 0 0
\(991\) 61.5330i 1.95466i −0.211725 0.977329i \(-0.567908\pi\)
0.211725 0.977329i \(-0.432092\pi\)
\(992\) 0 0
\(993\) 86.4653i 2.74389i
\(994\) 0 0
\(995\) 4.94052 0.156625
\(996\) 0 0
\(997\) −29.2889 −0.927589 −0.463795 0.885943i \(-0.653512\pi\)
−0.463795 + 0.885943i \(0.653512\pi\)
\(998\) 0 0
\(999\) 6.00862 0.190104
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 2240.2.h.f.671.20 yes 24
4.3 odd 2 inner 2240.2.h.f.671.5 yes 24
7.6 odd 2 2240.2.h.e.671.5 24
8.3 odd 2 2240.2.h.e.671.6 yes 24
8.5 even 2 2240.2.h.e.671.19 yes 24
28.27 even 2 2240.2.h.e.671.20 yes 24
56.13 odd 2 inner 2240.2.h.f.671.6 yes 24
56.27 even 2 inner 2240.2.h.f.671.19 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2240.2.h.e.671.5 24 7.6 odd 2
2240.2.h.e.671.6 yes 24 8.3 odd 2
2240.2.h.e.671.19 yes 24 8.5 even 2
2240.2.h.e.671.20 yes 24 28.27 even 2
2240.2.h.f.671.5 yes 24 4.3 odd 2 inner
2240.2.h.f.671.6 yes 24 56.13 odd 2 inner
2240.2.h.f.671.19 yes 24 56.27 even 2 inner
2240.2.h.f.671.20 yes 24 1.1 even 1 trivial