Properties

Label 630.2.d.d.629.4
Level $630$
Weight $2$
Character 630.629
Analytic conductor $5.031$
Analytic rank $0$
Dimension $4$
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] = [630,2,Mod(629,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.629");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.d (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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 629.4
Root \(-1.58114 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 630.629
Dual form 630.2.d.d.629.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.23607i q^{5} +(1.58114 - 2.12132i) q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.23607i q^{5} +(1.58114 - 2.12132i) q^{7} +1.00000 q^{8} +2.23607i q^{10} -1.41421i q^{11} +3.16228 q^{13} +(1.58114 - 2.12132i) q^{14} +1.00000 q^{16} +4.47214i q^{17} +2.23607i q^{20} -1.41421i q^{22} +6.00000 q^{23} -5.00000 q^{25} +3.16228 q^{26} +(1.58114 - 2.12132i) q^{28} +2.82843i q^{29} +1.00000 q^{32} +4.47214i q^{34} +(4.74342 + 3.53553i) q^{35} +4.24264i q^{37} +2.23607i q^{40} -9.48683 q^{41} -8.48528i q^{43} -1.41421i q^{44} +6.00000 q^{46} +4.47214i q^{47} +(-2.00000 - 6.70820i) q^{49} -5.00000 q^{50} +3.16228 q^{52} +6.00000 q^{53} +3.16228 q^{55} +(1.58114 - 2.12132i) q^{56} +2.82843i q^{58} -9.48683 q^{59} -13.4164i q^{61} +1.00000 q^{64} +7.07107i q^{65} +4.47214i q^{68} +(4.74342 + 3.53553i) q^{70} -5.65685i q^{71} -6.32456 q^{73} +4.24264i q^{74} +(-3.00000 - 2.23607i) q^{77} -4.00000 q^{79} +2.23607i q^{80} -9.48683 q^{82} -8.94427i q^{83} -10.0000 q^{85} -8.48528i q^{86} -1.41421i q^{88} +9.48683 q^{89} +(5.00000 - 6.70820i) q^{91} +6.00000 q^{92} +4.47214i q^{94} +12.6491 q^{97} +(-2.00000 - 6.70820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} + 24 q^{23} - 20 q^{25} + 4 q^{32} + 24 q^{46} - 8 q^{49} - 20 q^{50} + 24 q^{53} + 4 q^{64} - 12 q^{77} - 16 q^{79} - 40 q^{85} + 20 q^{91} + 24 q^{92} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) 1.58114 2.12132i 0.597614 0.801784i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.23607i 0.707107i
\(11\) 1.41421i 0.426401i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 3.16228 0.877058 0.438529 0.898717i \(-0.355500\pi\)
0.438529 + 0.898717i \(0.355500\pi\)
\(14\) 1.58114 2.12132i 0.422577 0.566947i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.47214i 1.08465i 0.840168 + 0.542326i \(0.182456\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 2.23607i 0.500000i
\(21\) 0 0
\(22\) 1.41421i 0.301511i
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 3.16228 0.620174
\(27\) 0 0
\(28\) 1.58114 2.12132i 0.298807 0.400892i
\(29\) 2.82843i 0.525226i 0.964901 + 0.262613i \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.47214i 0.766965i
\(35\) 4.74342 + 3.53553i 0.801784 + 0.597614i
\(36\) 0 0
\(37\) 4.24264i 0.697486i 0.937218 + 0.348743i \(0.113391\pi\)
−0.937218 + 0.348743i \(0.886609\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 2.23607i 0.353553i
\(41\) −9.48683 −1.48159 −0.740797 0.671729i \(-0.765552\pi\)
−0.740797 + 0.671729i \(0.765552\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i −0.762493 0.646997i \(-0.776025\pi\)
0.762493 0.646997i \(-0.223975\pi\)
\(44\) 1.41421i 0.213201i
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 4.47214i 0.652328i 0.945313 + 0.326164i \(0.105756\pi\)
−0.945313 + 0.326164i \(0.894244\pi\)
\(48\) 0 0
\(49\) −2.00000 6.70820i −0.285714 0.958315i
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) 3.16228 0.438529
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 3.16228 0.426401
\(56\) 1.58114 2.12132i 0.211289 0.283473i
\(57\) 0 0
\(58\) 2.82843i 0.371391i
\(59\) −9.48683 −1.23508 −0.617540 0.786539i \(-0.711871\pi\)
−0.617540 + 0.786539i \(0.711871\pi\)
\(60\) 0 0
\(61\) 13.4164i 1.71780i −0.512148 0.858898i \(-0.671150\pi\)
0.512148 0.858898i \(-0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.07107i 0.877058i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 4.47214i 0.542326i
\(69\) 0 0
\(70\) 4.74342 + 3.53553i 0.566947 + 0.422577i
\(71\) 5.65685i 0.671345i −0.941979 0.335673i \(-0.891036\pi\)
0.941979 0.335673i \(-0.108964\pi\)
\(72\) 0 0
\(73\) −6.32456 −0.740233 −0.370117 0.928985i \(-0.620682\pi\)
−0.370117 + 0.928985i \(0.620682\pi\)
\(74\) 4.24264i 0.493197i
\(75\) 0 0
\(76\) 0 0
\(77\) −3.00000 2.23607i −0.341882 0.254824i
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 2.23607i 0.250000i
\(81\) 0 0
\(82\) −9.48683 −1.04765
\(83\) 8.94427i 0.981761i −0.871227 0.490881i \(-0.836675\pi\)
0.871227 0.490881i \(-0.163325\pi\)
\(84\) 0 0
\(85\) −10.0000 −1.08465
\(86\) 8.48528i 0.914991i
\(87\) 0 0
\(88\) 1.41421i 0.150756i
\(89\) 9.48683 1.00560 0.502801 0.864402i \(-0.332303\pi\)
0.502801 + 0.864402i \(0.332303\pi\)
\(90\) 0 0
\(91\) 5.00000 6.70820i 0.524142 0.703211i
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 4.47214i 0.461266i
\(95\) 0 0
\(96\) 0 0
\(97\) 12.6491 1.28432 0.642161 0.766570i \(-0.278038\pi\)
0.642161 + 0.766570i \(0.278038\pi\)
\(98\) −2.00000 6.70820i −0.202031 0.677631i
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) −18.9737 −1.88795 −0.943975 0.330017i \(-0.892946\pi\)
−0.943975 + 0.330017i \(0.892946\pi\)
\(102\) 0 0
\(103\) −15.8114 −1.55794 −0.778971 0.627060i \(-0.784258\pi\)
−0.778971 + 0.627060i \(0.784258\pi\)
\(104\) 3.16228 0.310087
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 3.16228 0.301511
\(111\) 0 0
\(112\) 1.58114 2.12132i 0.149404 0.200446i
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 13.4164i 1.25109i
\(116\) 2.82843i 0.262613i
\(117\) 0 0
\(118\) −9.48683 −0.873334
\(119\) 9.48683 + 7.07107i 0.869657 + 0.648204i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 13.4164i 1.21466i
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 21.2132i 1.88237i 0.337895 + 0.941184i \(0.390285\pi\)
−0.337895 + 0.941184i \(0.609715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 7.07107i 0.620174i
\(131\) 9.48683 0.828868 0.414434 0.910079i \(-0.363979\pi\)
0.414434 + 0.910079i \(0.363979\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 4.47214i 0.383482i
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 13.4164i 1.13796i 0.822350 + 0.568982i \(0.192663\pi\)
−0.822350 + 0.568982i \(0.807337\pi\)
\(140\) 4.74342 + 3.53553i 0.400892 + 0.298807i
\(141\) 0 0
\(142\) 5.65685i 0.474713i
\(143\) 4.47214i 0.373979i
\(144\) 0 0
\(145\) −6.32456 −0.525226
\(146\) −6.32456 −0.523424
\(147\) 0 0
\(148\) 4.24264i 0.348743i
\(149\) 11.3137i 0.926855i 0.886135 + 0.463428i \(0.153381\pi\)
−0.886135 + 0.463428i \(0.846619\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −3.00000 2.23607i −0.241747 0.180187i
\(155\) 0 0
\(156\) 0 0
\(157\) −15.8114 −1.26189 −0.630943 0.775829i \(-0.717332\pi\)
−0.630943 + 0.775829i \(0.717332\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) 2.23607i 0.176777i
\(161\) 9.48683 12.7279i 0.747667 1.00310i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −9.48683 −0.740797
\(165\) 0 0
\(166\) 8.94427i 0.694210i
\(167\) 8.94427i 0.692129i −0.938211 0.346064i \(-0.887518\pi\)
0.938211 0.346064i \(-0.112482\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) −10.0000 −0.766965
\(171\) 0 0
\(172\) 8.48528i 0.646997i
\(173\) 8.94427i 0.680020i −0.940422 0.340010i \(-0.889569\pi\)
0.940422 0.340010i \(-0.110431\pi\)
\(174\) 0 0
\(175\) −7.90569 + 10.6066i −0.597614 + 0.801784i
\(176\) 1.41421i 0.106600i
\(177\) 0 0
\(178\) 9.48683 0.711068
\(179\) 18.3848i 1.37414i −0.726590 0.687071i \(-0.758896\pi\)
0.726590 0.687071i \(-0.241104\pi\)
\(180\) 0 0
\(181\) 13.4164i 0.997234i −0.866822 0.498617i \(-0.833841\pi\)
0.866822 0.498617i \(-0.166159\pi\)
\(182\) 5.00000 6.70820i 0.370625 0.497245i
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) −9.48683 −0.697486
\(186\) 0 0
\(187\) 6.32456 0.462497
\(188\) 4.47214i 0.326164i
\(189\) 0 0
\(190\) 0 0
\(191\) 22.6274i 1.63726i −0.574320 0.818631i \(-0.694733\pi\)
0.574320 0.818631i \(-0.305267\pi\)
\(192\) 0 0
\(193\) 8.48528i 0.610784i 0.952227 + 0.305392i \(0.0987875\pi\)
−0.952227 + 0.305392i \(0.901213\pi\)
\(194\) 12.6491 0.908153
\(195\) 0 0
\(196\) −2.00000 6.70820i −0.142857 0.479157i
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 26.8328i 1.90213i 0.308994 + 0.951064i \(0.400008\pi\)
−0.308994 + 0.951064i \(0.599992\pi\)
\(200\) −5.00000 −0.353553
\(201\) 0 0
\(202\) −18.9737 −1.33498
\(203\) 6.00000 + 4.47214i 0.421117 + 0.313882i
\(204\) 0 0
\(205\) 21.2132i 1.48159i
\(206\) −15.8114 −1.10163
\(207\) 0 0
\(208\) 3.16228 0.219265
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 18.9737 1.29399
\(216\) 0 0
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) 3.16228 0.213201
\(221\) 14.1421i 0.951303i
\(222\) 0 0
\(223\) 22.1359 1.48233 0.741166 0.671322i \(-0.234273\pi\)
0.741166 + 0.671322i \(0.234273\pi\)
\(224\) 1.58114 2.12132i 0.105644 0.141737i
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 17.8885i 1.18730i 0.804722 + 0.593652i \(0.202314\pi\)
−0.804722 + 0.593652i \(0.797686\pi\)
\(228\) 0 0
\(229\) 13.4164i 0.886581i −0.896378 0.443291i \(-0.853811\pi\)
0.896378 0.443291i \(-0.146189\pi\)
\(230\) 13.4164i 0.884652i
\(231\) 0 0
\(232\) 2.82843i 0.185695i
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −10.0000 −0.652328
\(236\) −9.48683 −0.617540
\(237\) 0 0
\(238\) 9.48683 + 7.07107i 0.614940 + 0.458349i
\(239\) 11.3137i 0.731823i 0.930650 + 0.365911i \(0.119243\pi\)
−0.930650 + 0.365911i \(0.880757\pi\)
\(240\) 0 0
\(241\) 13.4164i 0.864227i 0.901819 + 0.432113i \(0.142232\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 9.00000 0.578542
\(243\) 0 0
\(244\) 13.4164i 0.858898i
\(245\) 15.0000 4.47214i 0.958315 0.285714i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 11.1803i 0.707107i
\(251\) 9.48683 0.598804 0.299402 0.954127i \(-0.403213\pi\)
0.299402 + 0.954127i \(0.403213\pi\)
\(252\) 0 0
\(253\) 8.48528i 0.533465i
\(254\) 21.2132i 1.33103i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.47214i 0.278964i 0.990225 + 0.139482i \(0.0445438\pi\)
−0.990225 + 0.139482i \(0.955456\pi\)
\(258\) 0 0
\(259\) 9.00000 + 6.70820i 0.559233 + 0.416828i
\(260\) 7.07107i 0.438529i
\(261\) 0 0
\(262\) 9.48683 0.586098
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 13.4164i 0.824163i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.9737 −1.15684 −0.578422 0.815737i \(-0.696331\pi\)
−0.578422 + 0.815737i \(0.696331\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 4.47214i 0.271163i
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 7.07107i 0.426401i
\(276\) 0 0
\(277\) 21.2132i 1.27458i −0.770625 0.637289i \(-0.780056\pi\)
0.770625 0.637289i \(-0.219944\pi\)
\(278\) 13.4164i 0.804663i
\(279\) 0 0
\(280\) 4.74342 + 3.53553i 0.283473 + 0.211289i
\(281\) 1.41421i 0.0843649i −0.999110 0.0421825i \(-0.986569\pi\)
0.999110 0.0421825i \(-0.0134311\pi\)
\(282\) 0 0
\(283\) −6.32456 −0.375956 −0.187978 0.982173i \(-0.560193\pi\)
−0.187978 + 0.982173i \(0.560193\pi\)
\(284\) 5.65685i 0.335673i
\(285\) 0 0
\(286\) 4.47214i 0.264443i
\(287\) −15.0000 + 20.1246i −0.885422 + 1.18792i
\(288\) 0 0
\(289\) −3.00000 −0.176471
\(290\) −6.32456 −0.371391
\(291\) 0 0
\(292\) −6.32456 −0.370117
\(293\) 4.47214i 0.261265i 0.991431 + 0.130632i \(0.0417008\pi\)
−0.991431 + 0.130632i \(0.958299\pi\)
\(294\) 0 0
\(295\) 21.2132i 1.23508i
\(296\) 4.24264i 0.246598i
\(297\) 0 0
\(298\) 11.3137i 0.655386i
\(299\) 18.9737 1.09728
\(300\) 0 0
\(301\) −18.0000 13.4164i −1.03750 0.773309i
\(302\) 20.0000 1.15087
\(303\) 0 0
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) 12.6491 0.721923 0.360961 0.932581i \(-0.382449\pi\)
0.360961 + 0.932581i \(0.382449\pi\)
\(308\) −3.00000 2.23607i −0.170941 0.127412i
\(309\) 0 0
\(310\) 0 0
\(311\) −18.9737 −1.07590 −0.537949 0.842977i \(-0.680801\pi\)
−0.537949 + 0.842977i \(0.680801\pi\)
\(312\) 0 0
\(313\) 31.6228 1.78743 0.893713 0.448640i \(-0.148091\pi\)
0.893713 + 0.448640i \(0.148091\pi\)
\(314\) −15.8114 −0.892288
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 2.23607i 0.125000i
\(321\) 0 0
\(322\) 9.48683 12.7279i 0.528681 0.709299i
\(323\) 0 0
\(324\) 0 0
\(325\) −15.8114 −0.877058
\(326\) 0 0
\(327\) 0 0
\(328\) −9.48683 −0.523823
\(329\) 9.48683 + 7.07107i 0.523026 + 0.389841i
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 8.94427i 0.490881i
\(333\) 0 0
\(334\) 8.94427i 0.489409i
\(335\) 0 0
\(336\) 0 0
\(337\) 25.4558i 1.38667i 0.720616 + 0.693334i \(0.243859\pi\)
−0.720616 + 0.693334i \(0.756141\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) −10.0000 −0.542326
\(341\) 0 0
\(342\) 0 0
\(343\) −17.3925 6.36396i −0.939108 0.343622i
\(344\) 8.48528i 0.457496i
\(345\) 0 0
\(346\) 8.94427i 0.480847i
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 13.4164i 0.718164i −0.933306 0.359082i \(-0.883090\pi\)
0.933306 0.359082i \(-0.116910\pi\)
\(350\) −7.90569 + 10.6066i −0.422577 + 0.566947i
\(351\) 0 0
\(352\) 1.41421i 0.0753778i
\(353\) 31.3050i 1.66619i 0.553127 + 0.833097i \(0.313435\pi\)
−0.553127 + 0.833097i \(0.686565\pi\)
\(354\) 0 0
\(355\) 12.6491 0.671345
\(356\) 9.48683 0.502801
\(357\) 0 0
\(358\) 18.3848i 0.971666i
\(359\) 31.1127i 1.64207i −0.570881 0.821033i \(-0.693398\pi\)
0.570881 0.821033i \(-0.306602\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 13.4164i 0.705151i
\(363\) 0 0
\(364\) 5.00000 6.70820i 0.262071 0.351605i
\(365\) 14.1421i 0.740233i
\(366\) 0 0
\(367\) 22.1359 1.15549 0.577743 0.816218i \(-0.303933\pi\)
0.577743 + 0.816218i \(0.303933\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) −9.48683 −0.493197
\(371\) 9.48683 12.7279i 0.492532 0.660801i
\(372\) 0 0
\(373\) 21.2132i 1.09838i 0.835698 + 0.549189i \(0.185063\pi\)
−0.835698 + 0.549189i \(0.814937\pi\)
\(374\) 6.32456 0.327035
\(375\) 0 0
\(376\) 4.47214i 0.230633i
\(377\) 8.94427i 0.460653i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 22.6274i 1.15772i
\(383\) 4.47214i 0.228515i 0.993451 + 0.114258i \(0.0364490\pi\)
−0.993451 + 0.114258i \(0.963551\pi\)
\(384\) 0 0
\(385\) 5.00000 6.70820i 0.254824 0.341882i
\(386\) 8.48528i 0.431889i
\(387\) 0 0
\(388\) 12.6491 0.642161
\(389\) 11.3137i 0.573628i 0.957986 + 0.286814i \(0.0925961\pi\)
−0.957986 + 0.286814i \(0.907404\pi\)
\(390\) 0 0
\(391\) 26.8328i 1.35699i
\(392\) −2.00000 6.70820i −0.101015 0.338815i
\(393\) 0 0
\(394\) 12.0000 0.604551
\(395\) 8.94427i 0.450035i
\(396\) 0 0
\(397\) 22.1359 1.11097 0.555486 0.831526i \(-0.312532\pi\)
0.555486 + 0.831526i \(0.312532\pi\)
\(398\) 26.8328i 1.34501i
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 1.41421i 0.0706225i −0.999376 0.0353112i \(-0.988758\pi\)
0.999376 0.0353112i \(-0.0112422\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −18.9737 −0.943975
\(405\) 0 0
\(406\) 6.00000 + 4.47214i 0.297775 + 0.221948i
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 21.2132i 1.04765i
\(411\) 0 0
\(412\) −15.8114 −0.778971
\(413\) −15.0000 + 20.1246i −0.738102 + 0.990267i
\(414\) 0 0
\(415\) 20.0000 0.981761
\(416\) 3.16228 0.155043
\(417\) 0 0
\(418\) 0 0
\(419\) −9.48683 −0.463462 −0.231731 0.972780i \(-0.574439\pi\)
−0.231731 + 0.972780i \(0.574439\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 22.3607i 1.08465i
\(426\) 0 0
\(427\) −28.4605 21.2132i −1.37730 1.02658i
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 18.9737 0.914991
\(431\) 36.7696i 1.77113i 0.464518 + 0.885564i \(0.346227\pi\)
−0.464518 + 0.885564i \(0.653773\pi\)
\(432\) 0 0
\(433\) −6.32456 −0.303939 −0.151969 0.988385i \(-0.548562\pi\)
−0.151969 + 0.988385i \(0.548562\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 3.16228 0.150756
\(441\) 0 0
\(442\) 14.1421i 0.672673i
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 21.2132i 1.00560i
\(446\) 22.1359 1.04817
\(447\) 0 0
\(448\) 1.58114 2.12132i 0.0747018 0.100223i
\(449\) 18.3848i 0.867631i −0.901002 0.433816i \(-0.857167\pi\)
0.901002 0.433816i \(-0.142833\pi\)
\(450\) 0 0
\(451\) 13.4164i 0.631754i
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 17.8885i 0.839551i
\(455\) 15.0000 + 11.1803i 0.703211 + 0.524142i
\(456\) 0 0
\(457\) 16.9706i 0.793849i 0.917851 + 0.396925i \(0.129923\pi\)
−0.917851 + 0.396925i \(0.870077\pi\)
\(458\) 13.4164i 0.626908i
\(459\) 0 0
\(460\) 13.4164i 0.625543i
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 21.2132i 0.985861i −0.870069 0.492931i \(-0.835926\pi\)
0.870069 0.492931i \(-0.164074\pi\)
\(464\) 2.82843i 0.131306i
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 8.94427i 0.413892i −0.978352 0.206946i \(-0.933648\pi\)
0.978352 0.206946i \(-0.0663524\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −10.0000 −0.461266
\(471\) 0 0
\(472\) −9.48683 −0.436667
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 9.48683 + 7.07107i 0.434828 + 0.324102i
\(477\) 0 0
\(478\) 11.3137i 0.517477i
\(479\) 18.9737 0.866929 0.433464 0.901171i \(-0.357291\pi\)
0.433464 + 0.901171i \(0.357291\pi\)
\(480\) 0 0
\(481\) 13.4164i 0.611736i
\(482\) 13.4164i 0.611101i
\(483\) 0 0
\(484\) 9.00000 0.409091
\(485\) 28.2843i 1.28432i
\(486\) 0 0
\(487\) 4.24264i 0.192252i −0.995369 0.0961262i \(-0.969355\pi\)
0.995369 0.0961262i \(-0.0306452\pi\)
\(488\) 13.4164i 0.607332i
\(489\) 0 0
\(490\) 15.0000 4.47214i 0.677631 0.202031i
\(491\) 15.5563i 0.702048i 0.936366 + 0.351024i \(0.114166\pi\)
−0.936366 + 0.351024i \(0.885834\pi\)
\(492\) 0 0
\(493\) −12.6491 −0.569687
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 8.94427i −0.538274 0.401205i
\(498\) 0 0
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 11.1803i 0.500000i
\(501\) 0 0
\(502\) 9.48683 0.423418
\(503\) 22.3607i 0.997013i −0.866886 0.498507i \(-0.833882\pi\)
0.866886 0.498507i \(-0.166118\pi\)
\(504\) 0 0
\(505\) 42.4264i 1.88795i
\(506\) 8.48528i 0.377217i
\(507\) 0 0
\(508\) 21.2132i 0.941184i
\(509\) −18.9737 −0.840993 −0.420496 0.907294i \(-0.638144\pi\)
−0.420496 + 0.907294i \(0.638144\pi\)
\(510\) 0 0
\(511\) −10.0000 + 13.4164i −0.442374 + 0.593507i
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 4.47214i 0.197257i
\(515\) 35.3553i 1.55794i
\(516\) 0 0
\(517\) 6.32456 0.278154
\(518\) 9.00000 + 6.70820i 0.395437 + 0.294742i
\(519\) 0 0
\(520\) 7.07107i 0.310087i
\(521\) 28.4605 1.24688 0.623439 0.781872i \(-0.285735\pi\)
0.623439 + 0.781872i \(0.285735\pi\)
\(522\) 0 0
\(523\) −6.32456 −0.276553 −0.138277 0.990394i \(-0.544156\pi\)
−0.138277 + 0.990394i \(0.544156\pi\)
\(524\) 9.48683 0.414434
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 13.4164i 0.582772i
\(531\) 0 0
\(532\) 0 0
\(533\) −30.0000 −1.29944
\(534\) 0 0
\(535\) 26.8328i 1.16008i
\(536\) 0 0
\(537\) 0 0
\(538\) −18.9737 −0.818013
\(539\) −9.48683 + 2.82843i −0.408627 + 0.121829i
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 4.47214i 0.191741i
\(545\) 22.3607i 0.957826i
\(546\) 0 0
\(547\) 42.4264i 1.81402i 0.421107 + 0.907011i \(0.361642\pi\)
−0.421107 + 0.907011i \(0.638358\pi\)
\(548\) −18.0000 −0.768922
\(549\) 0 0
\(550\) 7.07107i 0.301511i
\(551\) 0 0
\(552\) 0 0
\(553\) −6.32456 + 8.48528i −0.268947 + 0.360831i
\(554\) 21.2132i 0.901263i
\(555\) 0 0
\(556\) 13.4164i 0.568982i
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 26.8328i 1.13491i
\(560\) 4.74342 + 3.53553i 0.200446 + 0.149404i
\(561\) 0 0
\(562\) 1.41421i 0.0596550i
\(563\) 35.7771i 1.50782i −0.656975 0.753912i \(-0.728164\pi\)
0.656975 0.753912i \(-0.271836\pi\)
\(564\) 0 0
\(565\) 13.4164i 0.564433i
\(566\) −6.32456 −0.265841
\(567\) 0 0
\(568\) 5.65685i 0.237356i
\(569\) 9.89949i 0.415008i −0.978234 0.207504i \(-0.933466\pi\)
0.978234 0.207504i \(-0.0665341\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 4.47214i 0.186989i
\(573\) 0 0
\(574\) −15.0000 + 20.1246i −0.626088 + 0.839985i
\(575\) −30.0000 −1.25109
\(576\) 0 0
\(577\) −25.2982 −1.05318 −0.526589 0.850120i \(-0.676529\pi\)
−0.526589 + 0.850120i \(0.676529\pi\)
\(578\) −3.00000 −0.124784
\(579\) 0 0
\(580\) −6.32456 −0.262613
\(581\) −18.9737 14.1421i −0.787160 0.586715i
\(582\) 0 0
\(583\) 8.48528i 0.351424i
\(584\) −6.32456 −0.261712
\(585\) 0 0
\(586\) 4.47214i 0.184742i
\(587\) 8.94427i 0.369170i −0.982817 0.184585i \(-0.940906\pi\)
0.982817 0.184585i \(-0.0590940\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 21.2132i 0.873334i
\(591\) 0 0
\(592\) 4.24264i 0.174371i
\(593\) 22.3607i 0.918243i −0.888373 0.459122i \(-0.848164\pi\)
0.888373 0.459122i \(-0.151836\pi\)
\(594\) 0 0
\(595\) −15.8114 + 21.2132i −0.648204 + 0.869657i
\(596\) 11.3137i 0.463428i
\(597\) 0 0
\(598\) 18.9737 0.775891
\(599\) 2.82843i 0.115566i 0.998329 + 0.0577832i \(0.0184032\pi\)
−0.998329 + 0.0577832i \(0.981597\pi\)
\(600\) 0 0
\(601\) 13.4164i 0.547267i 0.961834 + 0.273633i \(0.0882255\pi\)
−0.961834 + 0.273633i \(0.911775\pi\)
\(602\) −18.0000 13.4164i −0.733625 0.546812i
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) 20.1246i 0.818182i
\(606\) 0 0
\(607\) −15.8114 −0.641764 −0.320882 0.947119i \(-0.603979\pi\)
−0.320882 + 0.947119i \(0.603979\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 30.0000 1.21466
\(611\) 14.1421i 0.572130i
\(612\) 0 0
\(613\) 21.2132i 0.856793i 0.903591 + 0.428397i \(0.140921\pi\)
−0.903591 + 0.428397i \(0.859079\pi\)
\(614\) 12.6491 0.510477
\(615\) 0 0
\(616\) −3.00000 2.23607i −0.120873 0.0900937i
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 40.2492i 1.61775i −0.587979 0.808876i \(-0.700076\pi\)
0.587979 0.808876i \(-0.299924\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −18.9737 −0.760775
\(623\) 15.0000 20.1246i 0.600962 0.806276i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 31.6228 1.26390
\(627\) 0 0
\(628\) −15.8114 −0.630943
\(629\) −18.9737 −0.756530
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) 12.0000 0.476581
\(635\) −47.4342 −1.88237
\(636\) 0 0
\(637\) −6.32456 21.2132i −0.250588 0.840498i
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) 2.23607i 0.0883883i
\(641\) 15.5563i 0.614439i 0.951639 + 0.307219i \(0.0993986\pi\)
−0.951639 + 0.307219i \(0.900601\pi\)
\(642\) 0 0
\(643\) −6.32456 −0.249416 −0.124708 0.992193i \(-0.539799\pi\)
−0.124708 + 0.992193i \(0.539799\pi\)
\(644\) 9.48683 12.7279i 0.373834 0.501550i
\(645\) 0 0
\(646\) 0 0
\(647\) 22.3607i 0.879089i −0.898221 0.439545i \(-0.855140\pi\)
0.898221 0.439545i \(-0.144860\pi\)
\(648\) 0 0
\(649\) 13.4164i 0.526640i
\(650\) −15.8114 −0.620174
\(651\) 0 0
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) 21.2132i 0.828868i
\(656\) −9.48683 −0.370399
\(657\) 0 0
\(658\) 9.48683 + 7.07107i 0.369835 + 0.275659i
\(659\) 32.5269i 1.26707i 0.773715 + 0.633534i \(0.218396\pi\)
−0.773715 + 0.633534i \(0.781604\pi\)
\(660\) 0 0
\(661\) 40.2492i 1.56551i −0.622328 0.782757i \(-0.713813\pi\)
0.622328 0.782757i \(-0.286187\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) 8.94427i 0.347105i
\(665\) 0 0
\(666\) 0 0
\(667\) 16.9706i 0.657103i
\(668\) 8.94427i 0.346064i
\(669\) 0 0
\(670\) 0 0
\(671\) −18.9737 −0.732470
\(672\) 0 0
\(673\) 8.48528i 0.327084i 0.986536 + 0.163542i \(0.0522919\pi\)
−0.986536 + 0.163542i \(0.947708\pi\)
\(674\) 25.4558i 0.980522i
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) 22.3607i 0.859391i −0.902974 0.429695i \(-0.858621\pi\)
0.902974 0.429695i \(-0.141379\pi\)
\(678\) 0 0
\(679\) 20.0000 26.8328i 0.767530 1.02975i
\(680\) −10.0000 −0.383482
\(681\) 0 0
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 40.2492i 1.53784i
\(686\) −17.3925 6.36396i −0.664050 0.242977i
\(687\) 0 0
\(688\) 8.48528i 0.323498i
\(689\) 18.9737 0.722839
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 8.94427i 0.340010i
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −30.0000 −1.13796
\(696\) 0 0
\(697\) 42.4264i 1.60701i
\(698\) 13.4164i 0.507819i
\(699\) 0 0
\(700\) −7.90569 + 10.6066i −0.298807 + 0.400892i
\(701\) 36.7696i 1.38877i 0.719605 + 0.694383i \(0.244323\pi\)
−0.719605 + 0.694383i \(0.755677\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.41421i 0.0533002i
\(705\) 0 0
\(706\) 31.3050i 1.17818i
\(707\) −30.0000 + 40.2492i −1.12827 + 1.51373i
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 12.6491 0.474713
\(711\) 0 0
\(712\) 9.48683 0.355534
\(713\) 0 0
\(714\) 0 0
\(715\) 10.0000 0.373979
\(716\) 18.3848i 0.687071i
\(717\) 0 0
\(718\) 31.1127i 1.16112i
\(719\) −37.9473 −1.41520 −0.707598 0.706615i \(-0.750221\pi\)
−0.707598 + 0.706615i \(0.750221\pi\)
\(720\) 0 0
\(721\) −25.0000 + 33.5410i −0.931049 + 1.24913i
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) 13.4164i 0.498617i
\(725\) 14.1421i 0.525226i
\(726\) 0 0
\(727\) 3.16228 0.117282 0.0586412 0.998279i \(-0.481323\pi\)
0.0586412 + 0.998279i \(0.481323\pi\)
\(728\) 5.00000 6.70820i 0.185312 0.248623i
\(729\) 0 0
\(730\) 14.1421i 0.523424i
\(731\) 37.9473 1.40353
\(732\) 0 0
\(733\) 41.1096 1.51842 0.759209 0.650847i \(-0.225586\pi\)
0.759209 + 0.650847i \(0.225586\pi\)
\(734\) 22.1359 0.817053
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) −9.48683 −0.348743
\(741\) 0 0
\(742\) 9.48683 12.7279i 0.348273 0.467257i
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −25.2982 −0.926855
\(746\) 21.2132i 0.776671i
\(747\) 0 0
\(748\) 6.32456 0.231249
\(749\) −18.9737 + 25.4558i −0.693283 + 0.930136i
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 4.47214i 0.163082i
\(753\) 0 0
\(754\) 8.94427i 0.325731i
\(755\) 44.7214i 1.62758i
\(756\) 0 0
\(757\) 46.6690i 1.69622i 0.529824 + 0.848108i \(0.322258\pi\)
−0.529824 + 0.848108i \(0.677742\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) 9.48683 0.343897 0.171949 0.985106i \(-0.444994\pi\)
0.171949 + 0.985106i \(0.444994\pi\)
\(762\) 0 0
\(763\) −15.8114 + 21.2132i −0.572411 + 0.767970i
\(764\) 22.6274i 0.818631i
\(765\) 0 0
\(766\) 4.47214i 0.161585i
\(767\) −30.0000 −1.08324
\(768\) 0 0
\(769\) 40.2492i 1.45142i 0.687999 + 0.725712i \(0.258490\pi\)
−0.687999 + 0.725712i \(0.741510\pi\)
\(770\) 5.00000 6.70820i 0.180187 0.241747i
\(771\) 0 0
\(772\) 8.48528i 0.305392i
\(773\) 31.3050i 1.12596i 0.826470 + 0.562980i \(0.190345\pi\)
−0.826470 + 0.562980i \(0.809655\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12.6491 0.454077
\(777\) 0 0
\(778\) 11.3137i 0.405616i
\(779\) 0 0
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 26.8328i 0.959540i
\(783\) 0 0
\(784\) −2.00000 6.70820i −0.0714286 0.239579i
\(785\) 35.3553i 1.26189i
\(786\) 0 0
\(787\) 31.6228 1.12723 0.563615 0.826038i \(-0.309410\pi\)
0.563615 + 0.826038i \(0.309410\pi\)
\(788\) 12.0000 0.427482
\(789\) 0 0
\(790\) 8.94427i 0.318223i
\(791\) −9.48683 + 12.7279i −0.337313 + 0.452553i
\(792\) 0 0
\(793\) 42.4264i 1.50661i
\(794\) 22.1359 0.785575
\(795\) 0 0
\(796\) 26.8328i 0.951064i
\(797\) 8.94427i 0.316822i −0.987373 0.158411i \(-0.949363\pi\)
0.987373 0.158411i \(-0.0506372\pi\)
\(798\) 0 0
\(799\) −20.0000 −0.707549
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) 1.41421i 0.0499376i
\(803\) 8.94427i 0.315637i
\(804\) 0 0
\(805\) 28.4605 + 21.2132i 1.00310 + 0.747667i
\(806\) 0 0
\(807\) 0 0
\(808\) −18.9737 −0.667491
\(809\) 52.3259i 1.83968i −0.392293 0.919840i \(-0.628318\pi\)
0.392293 0.919840i \(-0.371682\pi\)
\(810\) 0 0
\(811\) 40.2492i 1.41334i −0.707543 0.706671i \(-0.750196\pi\)
0.707543 0.706671i \(-0.249804\pi\)
\(812\) 6.00000 + 4.47214i 0.210559 + 0.156941i
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 21.2132i 0.740797i
\(821\) 19.7990i 0.690990i 0.938421 + 0.345495i \(0.112289\pi\)
−0.938421 + 0.345495i \(0.887711\pi\)
\(822\) 0 0
\(823\) 21.2132i 0.739446i −0.929142 0.369723i \(-0.879453\pi\)
0.929142 0.369723i \(-0.120547\pi\)
\(824\) −15.8114 −0.550816
\(825\) 0 0
\(826\) −15.0000 + 20.1246i −0.521917 + 0.700225i
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 0 0
\(829\) 13.4164i 0.465971i −0.972480 0.232986i \(-0.925151\pi\)
0.972480 0.232986i \(-0.0748495\pi\)
\(830\) 20.0000 0.694210
\(831\) 0 0
\(832\) 3.16228 0.109632
\(833\) 30.0000 8.94427i 1.03944 0.309901i
\(834\) 0 0
\(835\) 20.0000 0.692129
\(836\) 0 0
\(837\) 0 0
\(838\) −9.48683 −0.327717
\(839\) 37.9473 1.31009 0.655044 0.755591i \(-0.272650\pi\)
0.655044 + 0.755591i \(0.272650\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) 6.70820i 0.230769i
\(846\) 0 0
\(847\) 14.2302 19.0919i 0.488957 0.656005i
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 22.3607i 0.766965i
\(851\) 25.4558i 0.872615i
\(852\) 0 0
\(853\) 3.16228 0.108274 0.0541372 0.998534i \(-0.482759\pi\)
0.0541372 + 0.998534i \(0.482759\pi\)
\(854\) −28.4605 21.2132i −0.973898 0.725901i
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 31.3050i 1.06936i 0.845056 + 0.534678i \(0.179567\pi\)
−0.845056 + 0.534678i \(0.820433\pi\)
\(858\) 0 0
\(859\) 13.4164i 0.457762i 0.973454 + 0.228881i \(0.0735067\pi\)
−0.973454 + 0.228881i \(0.926493\pi\)
\(860\) 18.9737 0.646997
\(861\) 0 0
\(862\) 36.7696i 1.25238i
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) 20.0000 0.680020
\(866\) −6.32456 −0.214917
\(867\) 0 0
\(868\) 0 0
\(869\) 5.65685i 0.191896i
\(870\) 0 0
\(871\) 0 0
\(872\) −10.0000 −0.338643
\(873\) 0 0
\(874\) 0 0
\(875\) −23.7171 17.6777i −0.801784 0.597614i
\(876\) 0 0
\(877\) 21.2132i 0.716319i −0.933660 0.358159i \(-0.883404\pi\)
0.933660 0.358159i \(-0.116596\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 3.16228 0.106600
\(881\) 47.4342 1.59810 0.799049 0.601266i \(-0.205337\pi\)
0.799049 + 0.601266i \(0.205337\pi\)
\(882\) 0 0
\(883\) 33.9411i 1.14221i −0.820877 0.571105i \(-0.806515\pi\)
0.820877 0.571105i \(-0.193485\pi\)
\(884\) 14.1421i 0.475651i
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 44.7214i 1.50160i 0.660532 + 0.750798i \(0.270331\pi\)
−0.660532 + 0.750798i \(0.729669\pi\)
\(888\) 0 0
\(889\) 45.0000 + 33.5410i 1.50925 + 1.12493i
\(890\) 21.2132i 0.711068i
\(891\) 0 0
\(892\) 22.1359 0.741166
\(893\) 0 0
\(894\) 0 0
\(895\) 41.1096 1.37414
\(896\) 1.58114 2.12132i 0.0528221 0.0708683i
\(897\) 0 0
\(898\) 18.3848i 0.613508i
\(899\) 0 0
\(900\) 0 0
\(901\) 26.8328i 0.893931i
\(902\) 13.4164i 0.446718i
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 30.0000 0.997234
\(906\) 0 0
\(907\) 25.4558i 0.845247i −0.906305 0.422624i \(-0.861109\pi\)
0.906305 0.422624i \(-0.138891\pi\)
\(908\) 17.8885i 0.593652i
\(909\) 0 0
\(910\) 15.0000 + 11.1803i 0.497245 + 0.370625i
\(911\) 48.0833i 1.59307i −0.604593 0.796535i \(-0.706664\pi\)
0.604593 0.796535i \(-0.293336\pi\)
\(912\) 0 0
\(913\) −12.6491 −0.418624
\(914\) 16.9706i 0.561336i
\(915\) 0 0
\(916\) 13.4164i 0.443291i
\(917\) 15.0000 20.1246i 0.495344 0.664573i
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 13.4164i 0.442326i
\(921\) 0 0
\(922\) 0 0
\(923\) 17.8885i 0.588809i
\(924\) 0 0
\(925\) 21.2132i 0.697486i
\(926\) 21.2132i 0.697109i
\(927\) 0 0
\(928\) 2.82843i 0.0928477i
\(929\) −9.48683 −0.311253 −0.155626 0.987816i \(-0.549740\pi\)
−0.155626 + 0.987816i \(0.549740\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) 8.94427i 0.292666i
\(935\) 14.1421i 0.462497i
\(936\) 0 0
\(937\) 12.6491 0.413228 0.206614 0.978422i \(-0.433755\pi\)
0.206614 + 0.978422i \(0.433755\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −10.0000 −0.326164
\(941\) 37.9473 1.23705 0.618524 0.785766i \(-0.287731\pi\)
0.618524 + 0.785766i \(0.287731\pi\)
\(942\) 0 0
\(943\) −56.9210 −1.85360
\(944\) −9.48683 −0.308770
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) 0 0
\(952\) 9.48683 + 7.07107i 0.307470 + 0.229175i
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 50.5964 1.63726
\(956\) 11.3137i 0.365911i
\(957\) 0 0
\(958\) 18.9737 0.613011
\(959\) −28.4605 + 38.1838i −0.919037 + 1.23302i
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 13.4164i 0.432562i
\(963\) 0 0
\(964\) 13.4164i 0.432113i
\(965\) −18.9737 −0.610784
\(966\) 0 0
\(967\) 21.2132i 0.682171i 0.940032 + 0.341085i \(0.110795\pi\)
−0.940032 + 0.341085i \(0.889205\pi\)
\(968\) 9.00000 0.289271
\(969\) 0 0
\(970\) 28.2843i 0.908153i
\(971\) −28.4605 −0.913341 −0.456670 0.889636i \(-0.650958\pi\)
−0.456670 + 0.889636i \(0.650958\pi\)
\(972\) 0 0
\(973\) 28.4605 + 21.2132i 0.912402 + 0.680064i
\(974\) 4.24264i 0.135943i
\(975\) 0 0
\(976\) 13.4164i 0.429449i
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 13.4164i 0.428790i
\(980\) 15.0000 4.47214i 0.479157 0.142857i
\(981\) 0 0
\(982\) 15.5563i 0.496423i
\(983\) 4.47214i 0.142639i 0.997454 + 0.0713195i \(0.0227210\pi\)
−0.997454 + 0.0713195i \(0.977279\pi\)
\(984\) 0 0
\(985\) 26.8328i 0.854965i
\(986\) −12.6491 −0.402830
\(987\) 0 0
\(988\) 0 0
\(989\) 50.9117i 1.61890i
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −12.0000 8.94427i −0.380617 0.283695i
\(995\) −60.0000 −1.90213
\(996\) 0 0
\(997\) −53.7587 −1.70256 −0.851278 0.524715i \(-0.824172\pi\)
−0.851278 + 0.524715i \(0.824172\pi\)
\(998\) −10.0000 −0.316544
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 630.2.d.d.629.4 yes 4
3.2 odd 2 630.2.d.a.629.2 yes 4
4.3 odd 2 5040.2.k.a.1889.3 4
5.2 odd 4 3150.2.b.c.251.8 8
5.3 odd 4 3150.2.b.c.251.1 8
5.4 even 2 630.2.d.a.629.1 4
7.6 odd 2 inner 630.2.d.d.629.1 yes 4
12.11 even 2 5040.2.k.d.1889.1 4
15.2 even 4 3150.2.b.c.251.4 8
15.8 even 4 3150.2.b.c.251.5 8
15.14 odd 2 inner 630.2.d.d.629.3 yes 4
20.19 odd 2 5040.2.k.d.1889.2 4
21.20 even 2 630.2.d.a.629.3 yes 4
28.27 even 2 5040.2.k.a.1889.2 4
35.13 even 4 3150.2.b.c.251.2 8
35.27 even 4 3150.2.b.c.251.7 8
35.34 odd 2 630.2.d.a.629.4 yes 4
60.59 even 2 5040.2.k.a.1889.4 4
84.83 odd 2 5040.2.k.d.1889.4 4
105.62 odd 4 3150.2.b.c.251.3 8
105.83 odd 4 3150.2.b.c.251.6 8
105.104 even 2 inner 630.2.d.d.629.2 yes 4
140.139 even 2 5040.2.k.d.1889.3 4
420.419 odd 2 5040.2.k.a.1889.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.d.a.629.1 4 5.4 even 2
630.2.d.a.629.2 yes 4 3.2 odd 2
630.2.d.a.629.3 yes 4 21.20 even 2
630.2.d.a.629.4 yes 4 35.34 odd 2
630.2.d.d.629.1 yes 4 7.6 odd 2 inner
630.2.d.d.629.2 yes 4 105.104 even 2 inner
630.2.d.d.629.3 yes 4 15.14 odd 2 inner
630.2.d.d.629.4 yes 4 1.1 even 1 trivial
3150.2.b.c.251.1 8 5.3 odd 4
3150.2.b.c.251.2 8 35.13 even 4
3150.2.b.c.251.3 8 105.62 odd 4
3150.2.b.c.251.4 8 15.2 even 4
3150.2.b.c.251.5 8 15.8 even 4
3150.2.b.c.251.6 8 105.83 odd 4
3150.2.b.c.251.7 8 35.27 even 4
3150.2.b.c.251.8 8 5.2 odd 4
5040.2.k.a.1889.1 4 420.419 odd 2
5040.2.k.a.1889.2 4 28.27 even 2
5040.2.k.a.1889.3 4 4.3 odd 2
5040.2.k.a.1889.4 4 60.59 even 2
5040.2.k.d.1889.1 4 12.11 even 2
5040.2.k.d.1889.2 4 20.19 odd 2
5040.2.k.d.1889.3 4 140.139 even 2
5040.2.k.d.1889.4 4 84.83 odd 2