Properties

Label 2100.2.ce.c.1993.1
Level $2100$
Weight $2$
Character 2100.1993
Analytic conductor $16.769$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(157,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.ce (of order \(12\), degree \(4\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 1993.1
Character \(\chi\) \(=\) 2100.1993
Dual form 2100.2.ce.c.157.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.965926 - 0.258819i) q^{3} +(-0.588711 - 2.57942i) q^{7} +(0.866025 + 0.500000i) q^{9} +O(q^{10})\) \(q+(-0.965926 - 0.258819i) q^{3} +(-0.588711 - 2.57942i) q^{7} +(0.866025 + 0.500000i) q^{9} +(-0.401047 - 0.694634i) q^{11} +(2.60097 - 2.60097i) q^{13} +(0.207597 - 0.774764i) q^{17} +(-1.40765 + 2.43811i) q^{19} +(-0.0989528 + 2.64390i) q^{21} +(8.12259 - 2.17644i) q^{23} +(-0.707107 - 0.707107i) q^{27} +(-5.14126 + 2.96831i) q^{31} +(0.207597 + 0.774764i) q^{33} +(0.448288 + 1.67303i) q^{37} +(-3.18552 + 1.83916i) q^{39} -7.01985i q^{41} +(-2.23310 - 2.23310i) q^{43} +(-10.1949 + 2.73172i) q^{47} +(-6.30684 + 3.03707i) q^{49} +(-0.401047 + 0.694634i) q^{51} +(3.27417 - 12.2194i) q^{53} +(1.99071 - 1.99071i) q^{57} +(-3.50993 - 6.07937i) q^{59} +(8.78251 + 5.07059i) q^{61} +(0.779873 - 2.52820i) q^{63} +(-2.71936 - 0.728651i) q^{67} -8.40912 q^{69} -5.35664 q^{71} +(7.68957 + 2.06042i) q^{73} +(-1.55565 + 1.44341i) q^{77} +(-5.85687 - 3.38146i) q^{79} +(0.500000 + 0.866025i) q^{81} +(0.112245 - 0.112245i) q^{83} +(0.694634 - 1.20314i) q^{89} +(-8.24021 - 5.17777i) q^{91} +(5.73433 - 1.53651i) q^{93} +(-3.53553 - 3.53553i) q^{97} -0.802094i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 8 q^{11} - 4 q^{21} - 60 q^{31} - 8 q^{51} + 84 q^{61} + 112 q^{71} + 12 q^{81} - 136 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −0.965926 0.258819i −0.557678 0.149429i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.588711 2.57942i −0.222512 0.974930i
\(8\) 0 0
\(9\) 0.866025 + 0.500000i 0.288675 + 0.166667i
\(10\) 0 0
\(11\) −0.401047 0.694634i −0.120920 0.209440i 0.799211 0.601051i \(-0.205251\pi\)
−0.920131 + 0.391611i \(0.871918\pi\)
\(12\) 0 0
\(13\) 2.60097 2.60097i 0.721378 0.721378i −0.247508 0.968886i \(-0.579612\pi\)
0.968886 + 0.247508i \(0.0796115\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.207597 0.774764i 0.0503497 0.187908i −0.936171 0.351546i \(-0.885656\pi\)
0.986520 + 0.163638i \(0.0523229\pi\)
\(18\) 0 0
\(19\) −1.40765 + 2.43811i −0.322936 + 0.559342i −0.981092 0.193540i \(-0.938003\pi\)
0.658156 + 0.752881i \(0.271337\pi\)
\(20\) 0 0
\(21\) −0.0989528 + 2.64390i −0.0215933 + 0.576946i
\(22\) 0 0
\(23\) 8.12259 2.17644i 1.69368 0.453819i 0.722342 0.691536i \(-0.243066\pi\)
0.971334 + 0.237717i \(0.0763991\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 0.707107i −0.136083 0.136083i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −5.14126 + 2.96831i −0.923397 + 0.533123i −0.884717 0.466128i \(-0.845649\pi\)
−0.0386796 + 0.999252i \(0.512315\pi\)
\(32\) 0 0
\(33\) 0.207597 + 0.774764i 0.0361381 + 0.134869i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.448288 + 1.67303i 0.0736980 + 0.275045i 0.992935 0.118660i \(-0.0378599\pi\)
−0.919237 + 0.393705i \(0.871193\pi\)
\(38\) 0 0
\(39\) −3.18552 + 1.83916i −0.510092 + 0.294501i
\(40\) 0 0
\(41\) 7.01985i 1.09632i −0.836375 0.548158i \(-0.815329\pi\)
0.836375 0.548158i \(-0.184671\pi\)
\(42\) 0 0
\(43\) −2.23310 2.23310i −0.340544 0.340544i 0.516028 0.856572i \(-0.327410\pi\)
−0.856572 + 0.516028i \(0.827410\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.1949 + 2.73172i −1.48708 + 0.398462i −0.908750 0.417340i \(-0.862962\pi\)
−0.578331 + 0.815802i \(0.696296\pi\)
\(48\) 0 0
\(49\) −6.30684 + 3.03707i −0.900977 + 0.433867i
\(50\) 0 0
\(51\) −0.401047 + 0.694634i −0.0561578 + 0.0972682i
\(52\) 0 0
\(53\) 3.27417 12.2194i 0.449742 1.67846i −0.253360 0.967372i \(-0.581536\pi\)
0.703102 0.711089i \(-0.251798\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.99071 1.99071i 0.263676 0.263676i
\(58\) 0 0
\(59\) −3.50993 6.07937i −0.456953 0.791466i 0.541845 0.840479i \(-0.317726\pi\)
−0.998798 + 0.0490121i \(0.984393\pi\)
\(60\) 0 0
\(61\) 8.78251 + 5.07059i 1.12449 + 0.649222i 0.942542 0.334087i \(-0.108428\pi\)
0.181943 + 0.983309i \(0.441761\pi\)
\(62\) 0 0
\(63\) 0.779873 2.52820i 0.0982547 0.318523i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.71936 0.728651i −0.332223 0.0890189i 0.0888516 0.996045i \(-0.471680\pi\)
−0.421075 + 0.907026i \(0.638347\pi\)
\(68\) 0 0
\(69\) −8.40912 −1.01234
\(70\) 0 0
\(71\) −5.35664 −0.635717 −0.317858 0.948138i \(-0.602964\pi\)
−0.317858 + 0.948138i \(0.602964\pi\)
\(72\) 0 0
\(73\) 7.68957 + 2.06042i 0.899997 + 0.241153i 0.679015 0.734125i \(-0.262407\pi\)
0.220982 + 0.975278i \(0.429074\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.55565 + 1.44341i −0.177283 + 0.164492i
\(78\) 0 0
\(79\) −5.85687 3.38146i −0.658949 0.380444i 0.132927 0.991126i \(-0.457562\pi\)
−0.791876 + 0.610681i \(0.790896\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 0.112245 0.112245i 0.0123205 0.0123205i −0.700920 0.713240i \(-0.747227\pi\)
0.713240 + 0.700920i \(0.247227\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.694634 1.20314i 0.0736311 0.127533i −0.826859 0.562409i \(-0.809875\pi\)
0.900490 + 0.434876i \(0.143208\pi\)
\(90\) 0 0
\(91\) −8.24021 5.17777i −0.863809 0.542778i
\(92\) 0 0
\(93\) 5.73433 1.53651i 0.594622 0.159328i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.53553 3.53553i −0.358979 0.358979i 0.504457 0.863437i \(-0.331693\pi\)
−0.863437 + 0.504457i \(0.831693\pi\)
\(98\) 0 0
\(99\) 0.802094i 0.0806135i
\(100\) 0 0
\(101\) 4.36188 2.51833i 0.434023 0.250583i −0.267036 0.963687i \(-0.586044\pi\)
0.701059 + 0.713103i \(0.252711\pi\)
\(102\) 0 0
\(103\) −3.30329 12.3280i −0.325483 1.21472i −0.913826 0.406107i \(-0.866886\pi\)
0.588343 0.808611i \(-0.299780\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.30359 4.86505i −0.126022 0.470322i 0.873852 0.486193i \(-0.161615\pi\)
−0.999874 + 0.0158708i \(0.994948\pi\)
\(108\) 0 0
\(109\) −7.48173 + 4.31958i −0.716620 + 0.413741i −0.813507 0.581555i \(-0.802445\pi\)
0.0968876 + 0.995295i \(0.469111\pi\)
\(110\) 0 0
\(111\) 1.73205i 0.164399i
\(112\) 0 0
\(113\) −7.34847 7.34847i −0.691286 0.691286i 0.271229 0.962515i \(-0.412570\pi\)
−0.962515 + 0.271229i \(0.912570\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.55299 0.952020i 0.328474 0.0880143i
\(118\) 0 0
\(119\) −2.12066 0.0793695i −0.194400 0.00727579i
\(120\) 0 0
\(121\) 5.17832 8.96912i 0.470757 0.815374i
\(122\) 0 0
\(123\) −1.81687 + 6.78066i −0.163822 + 0.611391i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.15325 8.15325i 0.723484 0.723484i −0.245829 0.969313i \(-0.579060\pi\)
0.969313 + 0.245829i \(0.0790602\pi\)
\(128\) 0 0
\(129\) 1.57904 + 2.73497i 0.139026 + 0.240801i
\(130\) 0 0
\(131\) −3.15874 1.82370i −0.275980 0.159337i 0.355622 0.934630i \(-0.384269\pi\)
−0.631602 + 0.775293i \(0.717602\pi\)
\(132\) 0 0
\(133\) 7.11762 + 2.19557i 0.617176 + 0.190380i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −21.6839 5.81018i −1.85258 0.496397i −0.852910 0.522059i \(-0.825164\pi\)
−0.999671 + 0.0256612i \(0.991831\pi\)
\(138\) 0 0
\(139\) −20.0131 −1.69749 −0.848743 0.528805i \(-0.822640\pi\)
−0.848743 + 0.528805i \(0.822640\pi\)
\(140\) 0 0
\(141\) 10.5546 0.888853
\(142\) 0 0
\(143\) −2.84983 0.763610i −0.238315 0.0638563i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.87799 1.30125i 0.567287 0.107325i
\(148\) 0 0
\(149\) 7.28005 + 4.20314i 0.596405 + 0.344335i 0.767626 0.640898i \(-0.221438\pi\)
−0.171221 + 0.985233i \(0.554771\pi\)
\(150\) 0 0
\(151\) 3.90105 + 6.75681i 0.317463 + 0.549862i 0.979958 0.199205i \(-0.0638358\pi\)
−0.662495 + 0.749066i \(0.730503\pi\)
\(152\) 0 0
\(153\) 0.567166 0.567166i 0.0458527 0.0458527i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.156376 + 0.583601i −0.0124801 + 0.0465765i −0.971885 0.235456i \(-0.924342\pi\)
0.959405 + 0.282032i \(0.0910085\pi\)
\(158\) 0 0
\(159\) −6.32522 + 10.9556i −0.501622 + 0.868835i
\(160\) 0 0
\(161\) −10.3958 19.6703i −0.819305 1.55024i
\(162\) 0 0
\(163\) 10.3693 2.77845i 0.812186 0.217625i 0.171259 0.985226i \(-0.445217\pi\)
0.640927 + 0.767601i \(0.278550\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.6539 14.6539i −1.13395 1.13395i −0.989514 0.144440i \(-0.953862\pi\)
−0.144440 0.989514i \(-0.546138\pi\)
\(168\) 0 0
\(169\) 0.530055i 0.0407735i
\(170\) 0 0
\(171\) −2.43811 + 1.40765i −0.186447 + 0.107645i
\(172\) 0 0
\(173\) 1.96811 + 7.34508i 0.149633 + 0.558436i 0.999505 + 0.0314484i \(0.0100120\pi\)
−0.849873 + 0.526988i \(0.823321\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.81687 + 6.78066i 0.136564 + 0.509665i
\(178\) 0 0
\(179\) 10.6672 6.15874i 0.797308 0.460326i −0.0452212 0.998977i \(-0.514399\pi\)
0.842529 + 0.538651i \(0.181066\pi\)
\(180\) 0 0
\(181\) 7.91980i 0.588674i −0.955702 0.294337i \(-0.904901\pi\)
0.955702 0.294337i \(-0.0950988\pi\)
\(182\) 0 0
\(183\) −7.17089 7.17089i −0.530087 0.530087i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.621434 + 0.166513i −0.0454437 + 0.0121766i
\(188\) 0 0
\(189\) −1.40765 + 2.24021i −0.102391 + 0.162951i
\(190\) 0 0
\(191\) −7.47518 + 12.9474i −0.540885 + 0.936840i 0.457968 + 0.888968i \(0.348577\pi\)
−0.998853 + 0.0478720i \(0.984756\pi\)
\(192\) 0 0
\(193\) 1.53651 5.73433i 0.110600 0.412766i −0.888320 0.459225i \(-0.848127\pi\)
0.998920 + 0.0464593i \(0.0147938\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.56052 + 6.56052i −0.467418 + 0.467418i −0.901077 0.433659i \(-0.857222\pi\)
0.433659 + 0.901077i \(0.357222\pi\)
\(198\) 0 0
\(199\) 6.45078 + 11.1731i 0.457284 + 0.792039i 0.998816 0.0486408i \(-0.0154890\pi\)
−0.541532 + 0.840680i \(0.682156\pi\)
\(200\) 0 0
\(201\) 2.43811 + 1.40765i 0.171971 + 0.0992877i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.12259 + 2.17644i 0.564559 + 0.151273i
\(208\) 0 0
\(209\) 2.25813 0.156198
\(210\) 0 0
\(211\) 19.1587 1.31894 0.659471 0.751730i \(-0.270780\pi\)
0.659471 + 0.751730i \(0.270780\pi\)
\(212\) 0 0
\(213\) 5.17412 + 1.38640i 0.354525 + 0.0949947i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 10.6832 + 11.5140i 0.725225 + 0.781621i
\(218\) 0 0
\(219\) −6.89428 3.98042i −0.465873 0.268972i
\(220\) 0 0
\(221\) −1.47518 2.55509i −0.0992314 0.171874i
\(222\) 0 0
\(223\) −17.7899 + 17.7899i −1.19130 + 1.19130i −0.214598 + 0.976702i \(0.568844\pi\)
−0.976702 + 0.214598i \(0.931156\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.32262 19.8643i 0.353275 1.31844i −0.529367 0.848393i \(-0.677571\pi\)
0.882642 0.470046i \(-0.155763\pi\)
\(228\) 0 0
\(229\) −3.49155 + 6.04754i −0.230728 + 0.399633i −0.958023 0.286693i \(-0.907444\pi\)
0.727295 + 0.686325i \(0.240777\pi\)
\(230\) 0 0
\(231\) 1.87623 0.991593i 0.123447 0.0652420i
\(232\) 0 0
\(233\) 8.69627 2.33016i 0.569711 0.152654i 0.0375471 0.999295i \(-0.488046\pi\)
0.532164 + 0.846641i \(0.321379\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.78211 + 4.78211i 0.310632 + 0.310632i
\(238\) 0 0
\(239\) 27.7525i 1.79516i −0.440855 0.897579i \(-0.645325\pi\)
0.440855 0.897579i \(-0.354675\pi\)
\(240\) 0 0
\(241\) −7.50000 + 4.33013i −0.483117 + 0.278928i −0.721715 0.692191i \(-0.756646\pi\)
0.238597 + 0.971119i \(0.423312\pi\)
\(242\) 0 0
\(243\) −0.258819 0.965926i −0.0166032 0.0619642i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.68021 + 10.0027i 0.170538 + 0.636456i
\(248\) 0 0
\(249\) −0.137472 + 0.0793695i −0.00871193 + 0.00502984i
\(250\) 0 0
\(251\) 6.15099i 0.388247i −0.980977 0.194124i \(-0.937814\pi\)
0.980977 0.194124i \(-0.0621863\pi\)
\(252\) 0 0
\(253\) −4.76937 4.76937i −0.299848 0.299848i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −13.2940 + 3.56211i −0.829255 + 0.222198i −0.648389 0.761310i \(-0.724557\pi\)
−0.180866 + 0.983508i \(0.557890\pi\)
\(258\) 0 0
\(259\) 4.05155 2.14126i 0.251751 0.133051i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.86617 18.1608i 0.300061 1.11984i −0.637054 0.770819i \(-0.719847\pi\)
0.937114 0.349022i \(-0.113486\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −0.982361 + 0.982361i −0.0601195 + 0.0601195i
\(268\) 0 0
\(269\) −10.1321 17.5493i −0.617765 1.07000i −0.989893 0.141819i \(-0.954705\pi\)
0.372127 0.928182i \(-0.378628\pi\)
\(270\) 0 0
\(271\) −0.688794 0.397675i −0.0418413 0.0241571i 0.478933 0.877851i \(-0.341024\pi\)
−0.520775 + 0.853694i \(0.674357\pi\)
\(272\) 0 0
\(273\) 6.61932 + 7.13407i 0.400620 + 0.431774i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 23.0613 + 6.17926i 1.38562 + 0.371276i 0.873160 0.487434i \(-0.162067\pi\)
0.512461 + 0.858710i \(0.328734\pi\)
\(278\) 0 0
\(279\) −5.93661 −0.355416
\(280\) 0 0
\(281\) 14.4063 0.859407 0.429703 0.902970i \(-0.358618\pi\)
0.429703 + 0.902970i \(0.358618\pi\)
\(282\) 0 0
\(283\) −19.9460 5.34452i −1.18567 0.317699i −0.388495 0.921451i \(-0.627005\pi\)
−0.797172 + 0.603752i \(0.793672\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −18.1072 + 4.13266i −1.06883 + 0.243943i
\(288\) 0 0
\(289\) 14.1653 + 8.17832i 0.833251 + 0.481078i
\(290\) 0 0
\(291\) 2.50000 + 4.33013i 0.146553 + 0.253837i
\(292\) 0 0
\(293\) −19.6239 + 19.6239i −1.14644 + 1.14644i −0.159196 + 0.987247i \(0.550890\pi\)
−0.987247 + 0.159196i \(0.949110\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.207597 + 0.774764i −0.0120460 + 0.0449563i
\(298\) 0 0
\(299\) 15.4657 26.7874i 0.894406 1.54916i
\(300\) 0 0
\(301\) −4.44545 + 7.07474i −0.256231 + 0.407781i
\(302\) 0 0
\(303\) −4.86505 + 1.30359i −0.279490 + 0.0748890i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.46843 + 2.46843i 0.140881 + 0.140881i 0.774030 0.633149i \(-0.218238\pi\)
−0.633149 + 0.774030i \(0.718238\pi\)
\(308\) 0 0
\(309\) 12.7629i 0.726058i
\(310\) 0 0
\(311\) −19.4413 + 11.2244i −1.10241 + 0.636478i −0.936854 0.349722i \(-0.886276\pi\)
−0.165559 + 0.986200i \(0.552943\pi\)
\(312\) 0 0
\(313\) 4.98054 + 18.5876i 0.281517 + 1.05064i 0.951347 + 0.308121i \(0.0997003\pi\)
−0.669830 + 0.742514i \(0.733633\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.60433 + 20.9156i 0.314771 + 1.17474i 0.924203 + 0.381902i \(0.124731\pi\)
−0.609432 + 0.792838i \(0.708603\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 5.03667i 0.281119i
\(322\) 0 0
\(323\) 1.59674 + 1.59674i 0.0888449 + 0.0888449i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 8.34478 2.23598i 0.461468 0.123650i
\(328\) 0 0
\(329\) 13.0481 + 24.6888i 0.719366 + 1.36114i
\(330\) 0 0
\(331\) −16.0598 + 27.8164i −0.882726 + 1.52893i −0.0344273 + 0.999407i \(0.510961\pi\)
−0.848298 + 0.529519i \(0.822373\pi\)
\(332\) 0 0
\(333\) −0.448288 + 1.67303i −0.0245660 + 0.0916816i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.76022 + 4.76022i −0.259305 + 0.259305i −0.824772 0.565466i \(-0.808696\pi\)
0.565466 + 0.824772i \(0.308696\pi\)
\(338\) 0 0
\(339\) 5.19615 + 9.00000i 0.282216 + 0.488813i
\(340\) 0 0
\(341\) 4.12377 + 2.38086i 0.223315 + 0.128931i
\(342\) 0 0
\(343\) 11.5468 + 14.4800i 0.623468 + 0.781849i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.46452 + 2.53601i 0.508082 + 0.136140i 0.503748 0.863850i \(-0.331954\pi\)
0.00433355 + 0.999991i \(0.498621\pi\)
\(348\) 0 0
\(349\) −1.11432 −0.0596484 −0.0298242 0.999555i \(-0.509495\pi\)
−0.0298242 + 0.999555i \(0.509495\pi\)
\(350\) 0 0
\(351\) −3.67832 −0.196334
\(352\) 0 0
\(353\) 15.0624 + 4.03595i 0.801689 + 0.214812i 0.636326 0.771421i \(-0.280454\pi\)
0.165364 + 0.986233i \(0.447120\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 2.02786 + 0.625532i 0.107325 + 0.0331066i
\(358\) 0 0
\(359\) 31.5893 + 18.2381i 1.66722 + 0.962571i 0.969126 + 0.246565i \(0.0793020\pi\)
0.698095 + 0.716005i \(0.254031\pi\)
\(360\) 0 0
\(361\) 5.53707 + 9.59048i 0.291425 + 0.504762i
\(362\) 0 0
\(363\) −7.32325 + 7.32325i −0.384371 + 0.384371i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.28139 12.2463i 0.171287 0.639253i −0.825867 0.563865i \(-0.809314\pi\)
0.997154 0.0753878i \(-0.0240195\pi\)
\(368\) 0 0
\(369\) 3.50993 6.07937i 0.182719 0.316479i
\(370\) 0 0
\(371\) −33.4465 1.25180i −1.73646 0.0649900i
\(372\) 0 0
\(373\) 24.1377 6.46767i 1.24980 0.334883i 0.427542 0.903996i \(-0.359380\pi\)
0.822260 + 0.569112i \(0.192713\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 13.9154i 0.714786i −0.933954 0.357393i \(-0.883666\pi\)
0.933954 0.357393i \(-0.116334\pi\)
\(380\) 0 0
\(381\) −9.98565 + 5.76522i −0.511580 + 0.295361i
\(382\) 0 0
\(383\) −2.93932 10.9697i −0.150192 0.560524i −0.999469 0.0325771i \(-0.989629\pi\)
0.849277 0.527947i \(-0.177038\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.817370 3.05047i −0.0415492 0.155064i
\(388\) 0 0
\(389\) 23.9756 13.8423i 1.21561 0.701832i 0.251634 0.967823i \(-0.419032\pi\)
0.963976 + 0.265990i \(0.0856989\pi\)
\(390\) 0 0
\(391\) 6.74491i 0.341105i
\(392\) 0 0
\(393\) 2.57910 + 2.57910i 0.130098 + 0.130098i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.35171 0.898088i 0.168217 0.0450737i −0.173727 0.984794i \(-0.555581\pi\)
0.341945 + 0.939720i \(0.388914\pi\)
\(398\) 0 0
\(399\) −6.30684 3.96293i −0.315737 0.198395i
\(400\) 0 0
\(401\) 13.0402 22.5863i 0.651197 1.12791i −0.331636 0.943407i \(-0.607601\pi\)
0.982833 0.184498i \(-0.0590660\pi\)
\(402\) 0 0
\(403\) −5.65177 + 21.0927i −0.281535 + 1.05070i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.982361 0.982361i 0.0486938 0.0486938i
\(408\) 0 0
\(409\) 3.49155 + 6.04754i 0.172646 + 0.299032i 0.939344 0.342976i \(-0.111435\pi\)
−0.766698 + 0.642008i \(0.778102\pi\)
\(410\) 0 0
\(411\) 19.4413 + 11.2244i 0.958966 + 0.553659i
\(412\) 0 0
\(413\) −13.6149 + 12.6326i −0.669947 + 0.621608i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 19.3311 + 5.17976i 0.946650 + 0.253654i
\(418\) 0 0
\(419\) 30.2640 1.47849 0.739247 0.673434i \(-0.235182\pi\)
0.739247 + 0.673434i \(0.235182\pi\)
\(420\) 0 0
\(421\) 30.3566 1.47949 0.739746 0.672886i \(-0.234946\pi\)
0.739746 + 0.672886i \(0.234946\pi\)
\(422\) 0 0
\(423\) −10.1949 2.73172i −0.495694 0.132821i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 7.90882 25.6389i 0.382735 1.24075i
\(428\) 0 0
\(429\) 2.55509 + 1.47518i 0.123361 + 0.0712224i
\(430\) 0 0
\(431\) 8.46994 + 14.6704i 0.407983 + 0.706647i 0.994664 0.103171i \(-0.0328990\pi\)
−0.586681 + 0.809818i \(0.699566\pi\)
\(432\) 0 0
\(433\) −9.25962 + 9.25962i −0.444989 + 0.444989i −0.893685 0.448696i \(-0.851889\pi\)
0.448696 + 0.893685i \(0.351889\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.12731 + 22.8674i −0.293109 + 1.09390i
\(438\) 0 0
\(439\) −5.45919 + 9.45560i −0.260553 + 0.451291i −0.966389 0.257084i \(-0.917238\pi\)
0.705836 + 0.708375i \(0.250572\pi\)
\(440\) 0 0
\(441\) −6.98042 0.523243i −0.332401 0.0249163i
\(442\) 0 0
\(443\) −6.20698 + 1.66315i −0.294902 + 0.0790188i −0.403237 0.915096i \(-0.632115\pi\)
0.108335 + 0.994114i \(0.465448\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −5.94414 5.94414i −0.281148 0.281148i
\(448\) 0 0
\(449\) 14.9504i 0.705551i −0.935708 0.352776i \(-0.885238\pi\)
0.935708 0.352776i \(-0.114762\pi\)
\(450\) 0 0
\(451\) −4.87623 + 2.81529i −0.229613 + 0.132567i
\(452\) 0 0
\(453\) −2.01933 7.53624i −0.0948764 0.354084i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.41033 + 12.7275i 0.159529 + 0.595369i 0.998675 + 0.0514631i \(0.0163885\pi\)
−0.839146 + 0.543906i \(0.816945\pi\)
\(458\) 0 0
\(459\) −0.694634 + 0.401047i −0.0324227 + 0.0187193i
\(460\) 0 0
\(461\) 35.6932i 1.66240i −0.555976 0.831198i \(-0.687655\pi\)
0.555976 0.831198i \(-0.312345\pi\)
\(462\) 0 0
\(463\) −28.9259 28.9259i −1.34430 1.34430i −0.891725 0.452578i \(-0.850505\pi\)
−0.452578 0.891725i \(-0.649495\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.89461 2.38330i 0.411593 0.110286i −0.0470798 0.998891i \(-0.514992\pi\)
0.458673 + 0.888605i \(0.348325\pi\)
\(468\) 0 0
\(469\) −0.278581 + 7.44335i −0.0128637 + 0.343702i
\(470\) 0 0
\(471\) 0.302094 0.523243i 0.0139198 0.0241098i
\(472\) 0 0
\(473\) −0.655608 + 2.44676i −0.0301449 + 0.112502i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 8.94521 8.94521i 0.409573 0.409573i
\(478\) 0 0
\(479\) 17.9473 + 31.0856i 0.820033 + 1.42034i 0.905657 + 0.424012i \(0.139378\pi\)
−0.0856235 + 0.996328i \(0.527288\pi\)
\(480\) 0 0
\(481\) 5.51748 + 3.18552i 0.251576 + 0.145247i
\(482\) 0 0
\(483\) 4.95054 + 21.6907i 0.225257 + 0.986960i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 9.74260 + 2.61052i 0.441479 + 0.118294i 0.472710 0.881218i \(-0.343276\pi\)
−0.0312305 + 0.999512i \(0.509943\pi\)
\(488\) 0 0
\(489\) −10.7351 −0.485457
\(490\) 0 0
\(491\) 7.83079 0.353399 0.176699 0.984265i \(-0.443458\pi\)
0.176699 + 0.984265i \(0.443458\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.15351 + 13.8170i 0.141454 + 0.619779i
\(498\) 0 0
\(499\) 11.5121 + 6.64649i 0.515351 + 0.297538i 0.735030 0.678034i \(-0.237168\pi\)
−0.219680 + 0.975572i \(0.570501\pi\)
\(500\) 0 0
\(501\) 10.3619 + 17.9473i 0.462935 + 0.801826i
\(502\) 0 0
\(503\) −25.4558 + 25.4558i −1.13502 + 1.13502i −0.145690 + 0.989330i \(0.546540\pi\)
−0.989330 + 0.145690i \(0.953460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.137188 + 0.511994i −0.00609275 + 0.0227384i
\(508\) 0 0
\(509\) 16.4206 28.4413i 0.727829 1.26064i −0.229970 0.973198i \(-0.573863\pi\)
0.957799 0.287439i \(-0.0928037\pi\)
\(510\) 0 0
\(511\) 0.787747 21.0476i 0.0348479 0.931093i
\(512\) 0 0
\(513\) 2.71936 0.728651i 0.120063 0.0321707i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.98619 + 5.98619i 0.263272 + 0.263272i
\(518\) 0 0
\(519\) 7.60419i 0.333787i
\(520\) 0 0
\(521\) 5.56502 3.21297i 0.243808 0.140763i −0.373118 0.927784i \(-0.621711\pi\)
0.616926 + 0.787021i \(0.288378\pi\)
\(522\) 0 0
\(523\) 6.30288 + 23.5227i 0.275606 + 1.02857i 0.955434 + 0.295203i \(0.0953873\pi\)
−0.679829 + 0.733371i \(0.737946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.23242 + 4.59947i 0.0536852 + 0.200356i
\(528\) 0 0
\(529\) 41.3209 23.8566i 1.79656 1.03725i
\(530\) 0 0
\(531\) 7.01985i 0.304636i
\(532\) 0 0
\(533\) −18.2584 18.2584i −0.790859 0.790859i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −11.8978 + 3.18800i −0.513427 + 0.137572i
\(538\) 0 0
\(539\) 4.63899 + 3.16294i 0.199815 + 0.136237i
\(540\) 0 0
\(541\) −21.9979 + 38.1015i −0.945764 + 1.63811i −0.191549 + 0.981483i \(0.561351\pi\)
−0.754215 + 0.656627i \(0.771982\pi\)
\(542\) 0 0
\(543\) −2.04979 + 7.64994i −0.0879651 + 0.328290i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 16.4881 16.4881i 0.704980 0.704980i −0.260495 0.965475i \(-0.583886\pi\)
0.965475 + 0.260495i \(0.0838858\pi\)
\(548\) 0 0
\(549\) 5.07059 + 8.78251i 0.216407 + 0.374828i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −5.27422 + 17.0980i −0.224283 + 0.727083i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −31.7221 8.49991i −1.34411 0.360153i −0.486151 0.873875i \(-0.661600\pi\)
−0.857957 + 0.513722i \(0.828266\pi\)
\(558\) 0 0
\(559\) −11.6164 −0.491322
\(560\) 0 0
\(561\) 0.643355 0.0271625
\(562\) 0 0
\(563\) 42.7973 + 11.4675i 1.80369 + 0.483297i 0.994545 0.104311i \(-0.0332637\pi\)
0.809146 + 0.587608i \(0.199930\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.93949 1.79955i 0.0814509 0.0755740i
\(568\) 0 0
\(569\) −4.77646 2.75769i −0.200240 0.115608i 0.396528 0.918023i \(-0.370215\pi\)
−0.596767 + 0.802414i \(0.703548\pi\)
\(570\) 0 0
\(571\) 10.6465 + 18.4403i 0.445542 + 0.771701i 0.998090 0.0617801i \(-0.0196777\pi\)
−0.552548 + 0.833481i \(0.686344\pi\)
\(572\) 0 0
\(573\) 10.5715 10.5715i 0.441631 0.441631i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −8.60401 + 32.1106i −0.358190 + 1.33678i 0.518233 + 0.855239i \(0.326590\pi\)
−0.876423 + 0.481543i \(0.840077\pi\)
\(578\) 0 0
\(579\) −2.96831 + 5.14126i −0.123359 + 0.213663i
\(580\) 0 0
\(581\) −0.355608 0.223448i −0.0147531 0.00927020i
\(582\) 0 0
\(583\) −9.80110 + 2.62620i −0.405920 + 0.108766i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.5720 + 22.5720i 0.931647 + 0.931647i 0.997809 0.0661617i \(-0.0210753\pi\)
−0.0661617 + 0.997809i \(0.521075\pi\)
\(588\) 0 0
\(589\) 16.7133i 0.688659i
\(590\) 0 0
\(591\) 8.03497 4.63899i 0.330514 0.190823i
\(592\) 0 0
\(593\) 3.96917 + 14.8131i 0.162994 + 0.608303i 0.998288 + 0.0584983i \(0.0186312\pi\)
−0.835293 + 0.549805i \(0.814702\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.33917 12.4620i −0.136663 0.510034i
\(598\) 0 0
\(599\) −41.5104 + 23.9661i −1.69607 + 0.979227i −0.746652 + 0.665215i \(0.768340\pi\)
−0.949419 + 0.314012i \(0.898327\pi\)
\(600\) 0 0
\(601\) 36.2503i 1.47868i −0.673331 0.739341i \(-0.735137\pi\)
0.673331 0.739341i \(-0.264863\pi\)
\(602\) 0 0
\(603\) −1.99071 1.99071i −0.0810680 0.0810680i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −0.955811 + 0.256109i −0.0387952 + 0.0103951i −0.278164 0.960533i \(-0.589726\pi\)
0.239369 + 0.970929i \(0.423059\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −19.4115 + 33.6217i −0.785306 + 1.36019i
\(612\) 0 0
\(613\) −3.61472 + 13.4903i −0.145997 + 0.544868i 0.853712 + 0.520745i \(0.174346\pi\)
−0.999709 + 0.0241228i \(0.992321\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.2946 13.2946i 0.535221 0.535221i −0.386900 0.922121i \(-0.626454\pi\)
0.922121 + 0.386900i \(0.126454\pi\)
\(618\) 0 0
\(619\) −4.23484 7.33496i −0.170213 0.294817i 0.768281 0.640112i \(-0.221112\pi\)
−0.938494 + 0.345295i \(0.887779\pi\)
\(620\) 0 0
\(621\) −7.28251 4.20456i −0.292237 0.168723i
\(622\) 0 0
\(623\) −3.51235 1.08345i −0.140719 0.0434076i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −2.18119 0.584447i −0.0871082 0.0233406i
\(628\) 0 0
\(629\) 1.38927 0.0553937
\(630\) 0 0
\(631\) −26.4371 −1.05244 −0.526221 0.850348i \(-0.676392\pi\)
−0.526221 + 0.850348i \(0.676392\pi\)
\(632\) 0 0
\(633\) −18.5059 4.95865i −0.735544 0.197089i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −8.50457 + 24.3032i −0.336963 + 0.962927i
\(638\) 0 0
\(639\) −4.63899 2.67832i −0.183516 0.105953i
\(640\) 0 0
\(641\) −13.2329 22.9200i −0.522667 0.905286i −0.999652 0.0263745i \(-0.991604\pi\)
0.476985 0.878911i \(-0.341730\pi\)
\(642\) 0 0
\(643\) −22.4173 + 22.4173i −0.884051 + 0.884051i −0.993943 0.109892i \(-0.964949\pi\)
0.109892 + 0.993943i \(0.464949\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.06784 7.71729i 0.0812953 0.303398i −0.913292 0.407306i \(-0.866468\pi\)
0.994587 + 0.103908i \(0.0331348\pi\)
\(648\) 0 0
\(649\) −2.81529 + 4.87623i −0.110510 + 0.191409i
\(650\) 0 0
\(651\) −7.33916 13.8867i −0.287644 0.544262i
\(652\) 0 0
\(653\) 2.41828 0.647977i 0.0946348 0.0253573i −0.211191 0.977445i \(-0.567734\pi\)
0.305826 + 0.952088i \(0.401068\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.62916 + 5.62916i 0.219614 + 0.219614i
\(658\) 0 0
\(659\) 21.5937i 0.841172i −0.907253 0.420586i \(-0.861825\pi\)
0.907253 0.420586i \(-0.138175\pi\)
\(660\) 0 0
\(661\) 24.8524 14.3486i 0.966648 0.558095i 0.0684353 0.997656i \(-0.478199\pi\)
0.898213 + 0.439561i \(0.144866\pi\)
\(662\) 0 0
\(663\) 0.763610 + 2.84983i 0.0296561 + 0.110678i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 21.7881 12.5794i 0.842377 0.486347i
\(670\) 0 0
\(671\) 8.13418i 0.314016i
\(672\) 0 0
\(673\) −11.9403 11.9403i −0.460263 0.460263i 0.438478 0.898742i \(-0.355518\pi\)
−0.898742 + 0.438478i \(0.855518\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.0113 5.62995i 0.807528 0.216377i 0.168642 0.985677i \(-0.446062\pi\)
0.638887 + 0.769301i \(0.279395\pi\)
\(678\) 0 0
\(679\) −7.03823 + 11.2010i −0.270102 + 0.429857i
\(680\) 0 0
\(681\) −10.2825 + 17.8098i −0.394027 + 0.682474i
\(682\) 0 0
\(683\) 6.76559 25.2495i 0.258878 0.966147i −0.707013 0.707200i \(-0.749958\pi\)
0.965892 0.258947i \(-0.0833754\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.93779 4.93779i 0.188389 0.188389i
\(688\) 0 0
\(689\) −23.2662 40.2982i −0.886371 1.53524i
\(690\) 0 0
\(691\) 5.86188 + 3.38436i 0.222996 + 0.128747i 0.607337 0.794444i \(-0.292238\pi\)
−0.384341 + 0.923191i \(0.625571\pi\)
\(692\) 0 0
\(693\) −2.06894 + 0.472201i −0.0785925 + 0.0179375i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −5.43873 1.45730i −0.206006 0.0551993i
\(698\) 0 0
\(699\) −9.00304 −0.340526
\(700\) 0 0
\(701\) 38.7238 1.46258 0.731288 0.682069i \(-0.238919\pi\)
0.731288 + 0.682069i \(0.238919\pi\)
\(702\) 0 0
\(703\) −4.71007 1.26206i −0.177644 0.0475995i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −9.06373 9.76856i −0.340877 0.367385i
\(708\) 0 0
\(709\) 17.2866 + 9.98042i 0.649211 + 0.374822i 0.788154 0.615478i \(-0.211037\pi\)
−0.138943 + 0.990300i \(0.544370\pi\)
\(710\) 0 0
\(711\) −3.38146 5.85687i −0.126815 0.219650i
\(712\) 0 0
\(713\) −35.2999 + 35.2999i −1.32199 + 1.32199i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −7.18286 + 26.8068i −0.268249 + 1.00112i
\(718\) 0 0
\(719\) 1.68623 2.92063i 0.0628856 0.108921i −0.832868 0.553471i \(-0.813303\pi\)
0.895754 + 0.444550i \(0.146636\pi\)
\(720\) 0 0
\(721\) −29.8545 + 15.7782i −1.11184 + 0.587612i
\(722\) 0 0
\(723\) 8.36516 2.24144i 0.311104 0.0833600i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −10.0642 10.0642i −0.373259 0.373259i 0.495404 0.868663i \(-0.335020\pi\)
−0.868663 + 0.495404i \(0.835020\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) −2.19371 + 1.26654i −0.0811371 + 0.0468446i
\(732\) 0 0
\(733\) −11.8753 44.3191i −0.438623 1.63696i −0.732245 0.681041i \(-0.761527\pi\)
0.293622 0.955922i \(-0.405139\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.584447 + 2.18119i 0.0215284 + 0.0803450i
\(738\) 0 0
\(739\) 2.94993 1.70314i 0.108515 0.0626511i −0.444760 0.895650i \(-0.646711\pi\)
0.553275 + 0.832999i \(0.313378\pi\)
\(740\) 0 0
\(741\) 10.3555i 0.380421i
\(742\) 0 0
\(743\) −21.8718 21.8718i −0.802400 0.802400i 0.181070 0.983470i \(-0.442044\pi\)
−0.983470 + 0.181070i \(0.942044\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.153330 0.0410847i 0.00561005 0.00150321i
\(748\) 0 0
\(749\) −11.7816 + 6.22660i −0.430489 + 0.227515i
\(750\) 0 0
\(751\) 12.4164 21.5059i 0.453082 0.784761i −0.545494 0.838115i \(-0.683658\pi\)
0.998576 + 0.0533539i \(0.0169911\pi\)
\(752\) 0 0
\(753\) −1.59199 + 5.94140i −0.0580155 + 0.216517i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.662347 0.662347i 0.0240734 0.0240734i −0.694968 0.719041i \(-0.744581\pi\)
0.719041 + 0.694968i \(0.244581\pi\)
\(758\) 0 0
\(759\) 3.37245 + 5.84126i 0.122412 + 0.212024i
\(760\) 0 0
\(761\) 14.0507 + 8.11216i 0.509337 + 0.294066i 0.732561 0.680701i \(-0.238325\pi\)
−0.223224 + 0.974767i \(0.571658\pi\)
\(762\) 0 0
\(763\) 15.5466 + 16.7556i 0.562824 + 0.606592i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.9414 6.68304i −0.900583 0.241311i
\(768\) 0 0
\(769\) −54.1886 −1.95409 −0.977045 0.213033i \(-0.931666\pi\)
−0.977045 + 0.213033i \(0.931666\pi\)
\(770\) 0 0
\(771\) 13.7629 0.495660
\(772\) 0 0
\(773\) 23.1822 + 6.21166i 0.833806 + 0.223418i 0.650374 0.759614i \(-0.274612\pi\)
0.183433 + 0.983032i \(0.441279\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −4.46769 + 1.01968i −0.160278 + 0.0365807i
\(778\) 0 0
\(779\) 17.1152 + 9.88146i 0.613216 + 0.354040i
\(780\) 0 0
\(781\) 2.14827 + 3.72091i 0.0768710 + 0.133145i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.41425 9.01011i 0.0860588 0.321176i −0.909454 0.415805i \(-0.863500\pi\)
0.995513 + 0.0946293i \(0.0301666\pi\)
\(788\) 0 0
\(789\) −9.40071 + 16.2825i −0.334674 + 0.579673i
\(790\) 0 0
\(791\) −14.6287 + 23.2809i −0.520136 + 0.827775i
\(792\) 0 0
\(793\) 36.0314 9.65460i 1.27951 0.342845i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.4334 11.4334i −0.404990 0.404990i 0.474997 0.879987i \(-0.342449\pi\)
−0.879987 + 0.474997i \(0.842449\pi\)
\(798\) 0 0
\(799\) 8.46575i 0.299497i
\(800\) 0 0
\(801\) 1.20314 0.694634i 0.0425109 0.0245437i
\(802\) 0 0
\(803\) −1.65265 6.16776i −0.0583207 0.217656i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5.24476 + 19.5737i 0.184624 + 0.689027i
\(808\) 0 0
\(809\) −23.2809 + 13.4413i −0.818514 + 0.472569i −0.849904 0.526938i \(-0.823340\pi\)
0.0313899 + 0.999507i \(0.490007\pi\)
\(810\) 0 0
\(811\) 30.3319i 1.06510i −0.846400 0.532548i \(-0.821235\pi\)
0.846400 0.532548i \(-0.178765\pi\)
\(812\) 0 0
\(813\) 0.562398 + 0.562398i 0.0197242 + 0.0197242i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.58795 2.30113i 0.300454 0.0805065i
\(818\) 0 0
\(819\) −4.54734 8.60419i −0.158897 0.300655i
\(820\) 0 0
\(821\) −8.59895 + 14.8938i −0.300106 + 0.519798i −0.976160 0.217054i \(-0.930355\pi\)
0.676054 + 0.736852i \(0.263689\pi\)
\(822\) 0 0
\(823\) 6.89036 25.7152i 0.240183 0.896374i −0.735561 0.677459i \(-0.763081\pi\)
0.975744 0.218916i \(-0.0702520\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.59580 4.59580i 0.159812 0.159812i −0.622672 0.782483i \(-0.713953\pi\)
0.782483 + 0.622672i \(0.213953\pi\)
\(828\) 0 0
\(829\) 18.1555 + 31.4462i 0.630565 + 1.09217i 0.987436 + 0.158017i \(0.0505100\pi\)
−0.356872 + 0.934153i \(0.616157\pi\)
\(830\) 0 0
\(831\) −20.6762 11.9374i −0.717250 0.414105i
\(832\) 0 0
\(833\) 1.04373 + 5.51680i 0.0361630 + 0.191146i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5.73433 + 1.53651i 0.198207 + 0.0531095i
\(838\) 0 0
\(839\) −13.7207 −0.473692 −0.236846 0.971547i \(-0.576114\pi\)
−0.236846 + 0.971547i \(0.576114\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) −13.9154 3.72862i −0.479272 0.128421i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −26.1837 8.07687i −0.899682 0.277524i
\(848\) 0 0
\(849\) 17.8831 + 10.3248i 0.613746 + 0.354347i
\(850\) 0 0
\(851\) 7.28251 + 12.6137i 0.249641 + 0.432391i
\(852\) 0 0
\(853\) 27.1341 27.1341i 0.929053 0.929053i −0.0685915 0.997645i \(-0.521851\pi\)
0.997645 + 0.0685915i \(0.0218505\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.4787 39.1071i 0.357946 1.33587i −0.518789 0.854902i \(-0.673617\pi\)
0.876736 0.480972i \(-0.159716\pi\)
\(858\) 0 0
\(859\) 18.6419 32.2888i 0.636055 1.10168i −0.350236 0.936662i \(-0.613899\pi\)
0.986291 0.165018i \(-0.0527681\pi\)
\(860\) 0 0
\(861\) 18.5598 + 0.694634i 0.632516 + 0.0236731i
\(862\) 0 0
\(863\) 16.5107 4.42404i 0.562032 0.150596i 0.0333925 0.999442i \(-0.489369\pi\)
0.528640 + 0.848846i \(0.322702\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −11.5659 11.5659i −0.392798 0.392798i
\(868\) 0 0
\(869\) 5.42451i 0.184014i
\(870\) 0 0
\(871\) −8.96817 + 5.17777i −0.303875 + 0.175442i
\(872\) 0 0
\(873\) −1.29410 4.82963i −0.0437985 0.163458i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −10.6304 39.6731i −0.358962 1.33967i −0.875424 0.483355i \(-0.839418\pi\)
0.516462 0.856310i \(-0.327249\pi\)
\(878\) 0 0
\(879\) 24.0343 13.8762i 0.810658 0.468034i
\(880\) 0 0
\(881\) 18.8014i 0.633436i −0.948520 0.316718i \(-0.897419\pi\)
0.948520 0.316718i \(-0.102581\pi\)
\(882\) 0 0
\(883\) 30.9855 + 30.9855i 1.04274 + 1.04274i 0.999045 + 0.0436994i \(0.0139144\pi\)
0.0436994 + 0.999045i \(0.486086\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.08950 2.16758i 0.271619 0.0727801i −0.120439 0.992721i \(-0.538430\pi\)
0.392057 + 0.919941i \(0.371763\pi\)
\(888\) 0 0
\(889\) −25.8306 16.2308i −0.866330 0.544363i
\(890\) 0 0
\(891\) 0.401047 0.694634i 0.0134356 0.0232711i
\(892\) 0 0
\(893\) 7.69058 28.7017i 0.257356 0.960464i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −21.8718 + 21.8718i −0.730279 + 0.730279i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −8.78742 5.07342i −0.292752 0.169020i
\(902\) 0 0
\(903\) 6.12505 5.68311i 0.203829 0.189122i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 35.2222 + 9.43776i 1.16953 + 0.313376i 0.790768 0.612116i \(-0.209682\pi\)
0.378767 + 0.925492i \(0.376348\pi\)
\(908\) 0 0
\(909\) 5.03667 0.167056
\(910\) 0 0
\(911\) 46.0699 1.52637 0.763183 0.646183i \(-0.223636\pi\)
0.763183 + 0.646183i \(0.223636\pi\)
\(912\) 0 0
\(913\) −0.122985 0.0329538i −0.00407022 0.00109061i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.84451 + 9.22135i −0.0939339 + 0.304516i
\(918\) 0 0
\(919\) −24.0828 13.9042i −0.794417 0.458657i 0.0470984 0.998890i \(-0.485003\pi\)
−0.841515 + 0.540234i \(0.818336\pi\)
\(920\) 0 0
\(921\) −1.74544 3.02320i −0.0575143 0.0996178i
\(922\) 0 0
\(923\) −13.9325 + 13.9325i −0.458592 + 0.458592i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 3.30329 12.3280i 0.108494 0.404906i
\(928\) 0 0
\(929\) −29.6061 + 51.2793i −0.971346 + 1.68242i −0.279845 + 0.960045i \(0.590283\pi\)
−0.691501 + 0.722376i \(0.743050\pi\)
\(930\) 0 0
\(931\) 1.47308 19.6519i 0.0482783 0.644065i
\(932\) 0 0
\(933\) 21.6839 5.81018i 0.709899 0.190217i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −24.5855 24.5855i −0.803174 0.803174i 0.180416 0.983590i \(-0.442256\pi\)
−0.983590 + 0.180416i \(0.942256\pi\)
\(938\) 0 0
\(939\) 19.2433i 0.627983i
\(940\) 0 0
\(941\) −25.9713 + 14.9945i −0.846640 + 0.488808i −0.859516 0.511109i \(-0.829235\pi\)
0.0128755 + 0.999917i \(0.495901\pi\)
\(942\) 0 0
\(943\) −15.2783 57.0193i −0.497529 1.85681i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.88519 18.2318i −0.158747 0.592453i −0.998755 0.0498792i \(-0.984116\pi\)
0.840008 0.542574i \(-0.182550\pi\)
\(948\) 0 0
\(949\) 25.3594 14.6413i 0.823201 0.475275i
\(950\) 0 0
\(951\) 21.6535i 0.702162i
\(952\) 0 0
\(953\) −12.3642 12.3642i −0.400517 0.400517i 0.477898 0.878415i \(-0.341399\pi\)
−0.878415 + 0.477898i \(0.841399\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.22137 + 59.3524i −0.0717319 + 1.91659i
\(960\) 0 0
\(961\) 2.12167 3.67484i 0.0684410 0.118543i
\(962\) 0 0
\(963\) 1.30359 4.86505i 0.0420074 0.156774i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −30.6395 + 30.6395i −0.985299 + 0.985299i −0.999893 0.0145946i \(-0.995354\pi\)
0.0145946 + 0.999893i \(0.495354\pi\)
\(968\) 0 0
\(969\) −1.12906 1.95560i −0.0362708 0.0628228i
\(970\) 0 0
\(971\) −10.1395 5.85403i −0.325391 0.187865i 0.328402 0.944538i \(-0.393490\pi\)
−0.653793 + 0.756673i \(0.726823\pi\)
\(972\) 0 0
\(973\) 11.7819 + 51.6222i 0.377711 + 1.65493i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 14.5951 + 3.91076i 0.466940 + 0.125116i 0.484615 0.874728i \(-0.338960\pi\)
−0.0176751 + 0.999844i \(0.505626\pi\)
\(978\) 0 0
\(979\) −1.11432 −0.0356140
\(980\) 0 0
\(981\) −8.63916 −0.275827
\(982\) 0 0
\(983\) 53.0225 + 14.2073i 1.69116 + 0.453144i 0.970688 0.240344i \(-0.0772601\pi\)
0.720469 + 0.693488i \(0.243927\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −6.21358 27.2246i −0.197780 0.866570i
\(988\) 0 0
\(989\) −22.9987 13.2783i −0.731316 0.422226i
\(990\) 0 0
\(991\) −5.58251 9.66918i −0.177334 0.307152i 0.763632 0.645651i \(-0.223414\pi\)
−0.940967 + 0.338499i \(0.890081\pi\)
\(992\) 0 0
\(993\) 22.7120 22.7120i 0.720743 0.720743i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −8.64347 + 32.2579i −0.273742 + 1.02162i 0.682939 + 0.730476i \(0.260702\pi\)
−0.956680 + 0.291141i \(0.905965\pi\)
\(998\) 0 0
\(999\) 0.866025 1.50000i 0.0273998 0.0474579i
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 2100.2.ce.c.1993.1 yes 24
5.2 odd 4 inner 2100.2.ce.c.1657.3 yes 24
5.3 odd 4 inner 2100.2.ce.c.1657.5 yes 24
5.4 even 2 inner 2100.2.ce.c.1993.6 yes 24
7.3 odd 6 inner 2100.2.ce.c.493.3 yes 24
35.3 even 12 inner 2100.2.ce.c.157.6 yes 24
35.17 even 12 inner 2100.2.ce.c.157.1 24
35.24 odd 6 inner 2100.2.ce.c.493.5 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.ce.c.157.1 24 35.17 even 12 inner
2100.2.ce.c.157.6 yes 24 35.3 even 12 inner
2100.2.ce.c.493.3 yes 24 7.3 odd 6 inner
2100.2.ce.c.493.5 yes 24 35.24 odd 6 inner
2100.2.ce.c.1657.3 yes 24 5.2 odd 4 inner
2100.2.ce.c.1657.5 yes 24 5.3 odd 4 inner
2100.2.ce.c.1993.1 yes 24 1.1 even 1 trivial
2100.2.ce.c.1993.6 yes 24 5.4 even 2 inner