Properties

Label 3584.2.m.e.2689.1
Level $3584$
Weight $2$
Character 3584.2689
Analytic conductor $28.618$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3584,2,Mod(897,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3584.897");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.m (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.6183840844\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 2689.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3584.2689
Dual form 3584.2.m.e.897.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.00000i) q^{3} +(-2.00000 + 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-1.00000 - 1.00000i) q^{3} +(-2.00000 + 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{9} +(2.00000 - 2.00000i) q^{11} +(-4.00000 - 4.00000i) q^{13} +4.00000 q^{15} -2.00000 q^{17} +(-5.00000 - 5.00000i) q^{19} +(-1.00000 + 1.00000i) q^{21} +4.00000i q^{23} -3.00000i q^{25} +(-4.00000 + 4.00000i) q^{27} +(1.00000 + 1.00000i) q^{29} +10.0000 q^{31} -4.00000 q^{33} +(2.00000 + 2.00000i) q^{35} +(5.00000 - 5.00000i) q^{37} +8.00000i q^{39} -6.00000i q^{41} +(6.00000 - 6.00000i) q^{43} +(2.00000 + 2.00000i) q^{45} -6.00000 q^{47} -1.00000 q^{49} +(2.00000 + 2.00000i) q^{51} +(-5.00000 + 5.00000i) q^{53} +8.00000i q^{55} +10.0000i q^{57} +(-7.00000 + 7.00000i) q^{59} +(-6.00000 - 6.00000i) q^{61} -1.00000 q^{63} +16.0000 q^{65} +(-4.00000 - 4.00000i) q^{67} +(4.00000 - 4.00000i) q^{69} +12.0000i q^{71} -6.00000i q^{73} +(-3.00000 + 3.00000i) q^{75} +(-2.00000 - 2.00000i) q^{77} +4.00000 q^{79} +5.00000 q^{81} +(5.00000 + 5.00000i) q^{83} +(4.00000 - 4.00000i) q^{85} -2.00000i q^{87} -14.0000i q^{89} +(-4.00000 + 4.00000i) q^{91} +(-10.0000 - 10.0000i) q^{93} +20.0000 q^{95} +10.0000 q^{97} +(-2.00000 - 2.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{5} + 4 q^{11} - 8 q^{13} + 8 q^{15} - 4 q^{17} - 10 q^{19} - 2 q^{21} - 8 q^{27} + 2 q^{29} + 20 q^{31} - 8 q^{33} + 4 q^{35} + 10 q^{37} + 12 q^{43} + 4 q^{45} - 12 q^{47} - 2 q^{49} + 4 q^{51} - 10 q^{53} - 14 q^{59} - 12 q^{61} - 2 q^{63} + 32 q^{65} - 8 q^{67} + 8 q^{69} - 6 q^{75} - 4 q^{77} + 8 q^{79} + 10 q^{81} + 10 q^{83} + 8 q^{85} - 8 q^{91} - 20 q^{93} + 40 q^{95} + 20 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −1.00000 1.00000i −0.577350 0.577350i 0.356822 0.934172i \(-0.383860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 0 0
\(5\) −2.00000 + 2.00000i −0.894427 + 0.894427i −0.994936 0.100509i \(-0.967953\pi\)
0.100509 + 0.994936i \(0.467953\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 2.00000 2.00000i 0.603023 0.603023i −0.338091 0.941113i \(-0.609781\pi\)
0.941113 + 0.338091i \(0.109781\pi\)
\(12\) 0 0
\(13\) −4.00000 4.00000i −1.10940 1.10940i −0.993229 0.116171i \(-0.962938\pi\)
−0.116171 0.993229i \(-0.537062\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −5.00000 5.00000i −1.14708 1.14708i −0.987124 0.159954i \(-0.948865\pi\)
−0.159954 0.987124i \(-0.551135\pi\)
\(20\) 0 0
\(21\) −1.00000 + 1.00000i −0.218218 + 0.218218i
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 3.00000i 0.600000i
\(26\) 0 0
\(27\) −4.00000 + 4.00000i −0.769800 + 0.769800i
\(28\) 0 0
\(29\) 1.00000 + 1.00000i 0.185695 + 0.185695i 0.793832 0.608137i \(-0.208083\pi\)
−0.608137 + 0.793832i \(0.708083\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 2.00000 + 2.00000i 0.338062 + 0.338062i
\(36\) 0 0
\(37\) 5.00000 5.00000i 0.821995 0.821995i −0.164399 0.986394i \(-0.552568\pi\)
0.986394 + 0.164399i \(0.0525685\pi\)
\(38\) 0 0
\(39\) 8.00000i 1.28103i
\(40\) 0 0
\(41\) 6.00000i 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) 6.00000 6.00000i 0.914991 0.914991i −0.0816682 0.996660i \(-0.526025\pi\)
0.996660 + 0.0816682i \(0.0260248\pi\)
\(44\) 0 0
\(45\) 2.00000 + 2.00000i 0.298142 + 0.298142i
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 2.00000 + 2.00000i 0.280056 + 0.280056i
\(52\) 0 0
\(53\) −5.00000 + 5.00000i −0.686803 + 0.686803i −0.961524 0.274721i \(-0.911414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 8.00000i 1.07872i
\(56\) 0 0
\(57\) 10.0000i 1.32453i
\(58\) 0 0
\(59\) −7.00000 + 7.00000i −0.911322 + 0.911322i −0.996376 0.0850540i \(-0.972894\pi\)
0.0850540 + 0.996376i \(0.472894\pi\)
\(60\) 0 0
\(61\) −6.00000 6.00000i −0.768221 0.768221i 0.209572 0.977793i \(-0.432793\pi\)
−0.977793 + 0.209572i \(0.932793\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 16.0000 1.98456
\(66\) 0 0
\(67\) −4.00000 4.00000i −0.488678 0.488678i 0.419211 0.907889i \(-0.362307\pi\)
−0.907889 + 0.419211i \(0.862307\pi\)
\(68\) 0 0
\(69\) 4.00000 4.00000i 0.481543 0.481543i
\(70\) 0 0
\(71\) 12.0000i 1.42414i 0.702109 + 0.712069i \(0.252242\pi\)
−0.702109 + 0.712069i \(0.747758\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) −3.00000 + 3.00000i −0.346410 + 0.346410i
\(76\) 0 0
\(77\) −2.00000 2.00000i −0.227921 0.227921i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) 5.00000 + 5.00000i 0.548821 + 0.548821i 0.926100 0.377279i \(-0.123140\pi\)
−0.377279 + 0.926100i \(0.623140\pi\)
\(84\) 0 0
\(85\) 4.00000 4.00000i 0.433861 0.433861i
\(86\) 0 0
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) 14.0000i 1.48400i −0.670402 0.741999i \(-0.733878\pi\)
0.670402 0.741999i \(-0.266122\pi\)
\(90\) 0 0
\(91\) −4.00000 + 4.00000i −0.419314 + 0.419314i
\(92\) 0 0
\(93\) −10.0000 10.0000i −1.03695 1.03695i
\(94\) 0 0
\(95\) 20.0000 2.05196
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) −2.00000 2.00000i −0.201008 0.201008i
\(100\) 0 0
\(101\) −6.00000 + 6.00000i −0.597022 + 0.597022i −0.939519 0.342497i \(-0.888727\pi\)
0.342497 + 0.939519i \(0.388727\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i 0.955312 + 0.295599i \(0.0955191\pi\)
−0.955312 + 0.295599i \(0.904481\pi\)
\(104\) 0 0
\(105\) 4.00000i 0.390360i
\(106\) 0 0
\(107\) −8.00000 + 8.00000i −0.773389 + 0.773389i −0.978697 0.205308i \(-0.934180\pi\)
0.205308 + 0.978697i \(0.434180\pi\)
\(108\) 0 0
\(109\) 11.0000 + 11.0000i 1.05361 + 1.05361i 0.998479 + 0.0551297i \(0.0175572\pi\)
0.0551297 + 0.998479i \(0.482443\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) −8.00000 8.00000i −0.746004 0.746004i
\(116\) 0 0
\(117\) −4.00000 + 4.00000i −0.369800 + 0.369800i
\(118\) 0 0
\(119\) 2.00000i 0.183340i
\(120\) 0 0
\(121\) 3.00000i 0.272727i
\(122\) 0 0
\(123\) −6.00000 + 6.00000i −0.541002 + 0.541002i
\(124\) 0 0
\(125\) −4.00000 4.00000i −0.357771 0.357771i
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −1.00000 1.00000i −0.0873704 0.0873704i 0.662071 0.749441i \(-0.269678\pi\)
−0.749441 + 0.662071i \(0.769678\pi\)
\(132\) 0 0
\(133\) −5.00000 + 5.00000i −0.433555 + 0.433555i
\(134\) 0 0
\(135\) 16.0000i 1.37706i
\(136\) 0 0
\(137\) 4.00000i 0.341743i 0.985293 + 0.170872i \(0.0546583\pi\)
−0.985293 + 0.170872i \(0.945342\pi\)
\(138\) 0 0
\(139\) −13.0000 + 13.0000i −1.10265 + 1.10265i −0.108555 + 0.994090i \(0.534622\pi\)
−0.994090 + 0.108555i \(0.965378\pi\)
\(140\) 0 0
\(141\) 6.00000 + 6.00000i 0.505291 + 0.505291i
\(142\) 0 0
\(143\) −16.0000 −1.33799
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 1.00000 + 1.00000i 0.0824786 + 0.0824786i
\(148\) 0 0
\(149\) −9.00000 + 9.00000i −0.737309 + 0.737309i −0.972056 0.234748i \(-0.924574\pi\)
0.234748 + 0.972056i \(0.424574\pi\)
\(150\) 0 0
\(151\) 8.00000i 0.651031i 0.945537 + 0.325515i \(0.105538\pi\)
−0.945537 + 0.325515i \(0.894462\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) −20.0000 + 20.0000i −1.60644 + 1.60644i
\(156\) 0 0
\(157\) 12.0000 + 12.0000i 0.957704 + 0.957704i 0.999141 0.0414369i \(-0.0131935\pi\)
−0.0414369 + 0.999141i \(0.513194\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −10.0000 10.0000i −0.783260 0.783260i 0.197119 0.980380i \(-0.436841\pi\)
−0.980380 + 0.197119i \(0.936841\pi\)
\(164\) 0 0
\(165\) 8.00000 8.00000i 0.622799 0.622799i
\(166\) 0 0
\(167\) 2.00000i 0.154765i 0.997001 + 0.0773823i \(0.0246562\pi\)
−0.997001 + 0.0773823i \(0.975344\pi\)
\(168\) 0 0
\(169\) 19.0000i 1.46154i
\(170\) 0 0
\(171\) −5.00000 + 5.00000i −0.382360 + 0.382360i
\(172\) 0 0
\(173\) −12.0000 12.0000i −0.912343 0.912343i 0.0841131 0.996456i \(-0.473194\pi\)
−0.996456 + 0.0841131i \(0.973194\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) 14.0000 1.05230
\(178\) 0 0
\(179\) 12.0000 + 12.0000i 0.896922 + 0.896922i 0.995163 0.0982406i \(-0.0313215\pi\)
−0.0982406 + 0.995163i \(0.531321\pi\)
\(180\) 0 0
\(181\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(182\) 0 0
\(183\) 12.0000i 0.887066i
\(184\) 0 0
\(185\) 20.0000i 1.47043i
\(186\) 0 0
\(187\) −4.00000 + 4.00000i −0.292509 + 0.292509i
\(188\) 0 0
\(189\) 4.00000 + 4.00000i 0.290957 + 0.290957i
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) −16.0000 16.0000i −1.14578 1.14578i
\(196\) 0 0
\(197\) 9.00000 9.00000i 0.641223 0.641223i −0.309633 0.950856i \(-0.600206\pi\)
0.950856 + 0.309633i \(0.100206\pi\)
\(198\) 0 0
\(199\) 6.00000i 0.425329i −0.977125 0.212664i \(-0.931786\pi\)
0.977125 0.212664i \(-0.0682141\pi\)
\(200\) 0 0
\(201\) 8.00000i 0.564276i
\(202\) 0 0
\(203\) 1.00000 1.00000i 0.0701862 0.0701862i
\(204\) 0 0
\(205\) 12.0000 + 12.0000i 0.838116 + 0.838116i
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −20.0000 −1.38343
\(210\) 0 0
\(211\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(212\) 0 0
\(213\) 12.0000 12.0000i 0.822226 0.822226i
\(214\) 0 0
\(215\) 24.0000i 1.63679i
\(216\) 0 0
\(217\) 10.0000i 0.678844i
\(218\) 0 0
\(219\) −6.00000 + 6.00000i −0.405442 + 0.405442i
\(220\) 0 0
\(221\) 8.00000 + 8.00000i 0.538138 + 0.538138i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 7.00000 + 7.00000i 0.464606 + 0.464606i 0.900162 0.435556i \(-0.143448\pi\)
−0.435556 + 0.900162i \(0.643448\pi\)
\(228\) 0 0
\(229\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(230\) 0 0
\(231\) 4.00000i 0.263181i
\(232\) 0 0
\(233\) 22.0000i 1.44127i −0.693316 0.720634i \(-0.743851\pi\)
0.693316 0.720634i \(-0.256149\pi\)
\(234\) 0 0
\(235\) 12.0000 12.0000i 0.782794 0.782794i
\(236\) 0 0
\(237\) −4.00000 4.00000i −0.259828 0.259828i
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 7.00000 + 7.00000i 0.449050 + 0.449050i
\(244\) 0 0
\(245\) 2.00000 2.00000i 0.127775 0.127775i
\(246\) 0 0
\(247\) 40.0000i 2.54514i
\(248\) 0 0
\(249\) 10.0000i 0.633724i
\(250\) 0 0
\(251\) 5.00000 5.00000i 0.315597 0.315597i −0.531476 0.847073i \(-0.678362\pi\)
0.847073 + 0.531476i \(0.178362\pi\)
\(252\) 0 0
\(253\) 8.00000 + 8.00000i 0.502956 + 0.502956i
\(254\) 0 0
\(255\) −8.00000 −0.500979
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −5.00000 5.00000i −0.310685 0.310685i
\(260\) 0 0
\(261\) 1.00000 1.00000i 0.0618984 0.0618984i
\(262\) 0 0
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) 0 0
\(265\) 20.0000i 1.22859i
\(266\) 0 0
\(267\) −14.0000 + 14.0000i −0.856786 + 0.856786i
\(268\) 0 0
\(269\) 8.00000 + 8.00000i 0.487769 + 0.487769i 0.907601 0.419833i \(-0.137911\pi\)
−0.419833 + 0.907601i \(0.637911\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 0 0
\(275\) −6.00000 6.00000i −0.361814 0.361814i
\(276\) 0 0
\(277\) −3.00000 + 3.00000i −0.180253 + 0.180253i −0.791466 0.611213i \(-0.790682\pi\)
0.611213 + 0.791466i \(0.290682\pi\)
\(278\) 0 0
\(279\) 10.0000i 0.598684i
\(280\) 0 0
\(281\) 26.0000i 1.55103i 0.631329 + 0.775515i \(0.282510\pi\)
−0.631329 + 0.775515i \(0.717490\pi\)
\(282\) 0 0
\(283\) −15.0000 + 15.0000i −0.891657 + 0.891657i −0.994679 0.103022i \(-0.967149\pi\)
0.103022 + 0.994679i \(0.467149\pi\)
\(284\) 0 0
\(285\) −20.0000 20.0000i −1.18470 1.18470i
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −10.0000 10.0000i −0.586210 0.586210i
\(292\) 0 0
\(293\) −2.00000 + 2.00000i −0.116841 + 0.116841i −0.763110 0.646269i \(-0.776329\pi\)
0.646269 + 0.763110i \(0.276329\pi\)
\(294\) 0 0
\(295\) 28.0000i 1.63022i
\(296\) 0 0
\(297\) 16.0000i 0.928414i
\(298\) 0 0
\(299\) 16.0000 16.0000i 0.925304 0.925304i
\(300\) 0 0
\(301\) −6.00000 6.00000i −0.345834 0.345834i
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) 17.0000 + 17.0000i 0.970241 + 0.970241i 0.999570 0.0293286i \(-0.00933691\pi\)
−0.0293286 + 0.999570i \(0.509337\pi\)
\(308\) 0 0
\(309\) 6.00000 6.00000i 0.341328 0.341328i
\(310\) 0 0
\(311\) 24.0000i 1.36092i 0.732787 + 0.680458i \(0.238219\pi\)
−0.732787 + 0.680458i \(0.761781\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 0 0
\(315\) 2.00000 2.00000i 0.112687 0.112687i
\(316\) 0 0
\(317\) 5.00000 + 5.00000i 0.280828 + 0.280828i 0.833439 0.552611i \(-0.186369\pi\)
−0.552611 + 0.833439i \(0.686369\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 10.0000 + 10.0000i 0.556415 + 0.556415i
\(324\) 0 0
\(325\) −12.0000 + 12.0000i −0.665640 + 0.665640i
\(326\) 0 0
\(327\) 22.0000i 1.21660i
\(328\) 0 0
\(329\) 6.00000i 0.330791i
\(330\) 0 0
\(331\) 18.0000 18.0000i 0.989369 0.989369i −0.0105746 0.999944i \(-0.503366\pi\)
0.999944 + 0.0105746i \(0.00336607\pi\)
\(332\) 0 0
\(333\) −5.00000 5.00000i −0.273998 0.273998i
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) 12.0000 + 12.0000i 0.651751 + 0.651751i
\(340\) 0 0
\(341\) 20.0000 20.0000i 1.08306 1.08306i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 16.0000i 0.861411i
\(346\) 0 0
\(347\) 18.0000 18.0000i 0.966291 0.966291i −0.0331594 0.999450i \(-0.510557\pi\)
0.999450 + 0.0331594i \(0.0105569\pi\)
\(348\) 0 0
\(349\) −8.00000 8.00000i −0.428230 0.428230i 0.459795 0.888025i \(-0.347923\pi\)
−0.888025 + 0.459795i \(0.847923\pi\)
\(350\) 0 0
\(351\) 32.0000 1.70803
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −24.0000 24.0000i −1.27379 1.27379i
\(356\) 0 0
\(357\) 2.00000 2.00000i 0.105851 0.105851i
\(358\) 0 0
\(359\) 8.00000i 0.422224i 0.977462 + 0.211112i \(0.0677085\pi\)
−0.977462 + 0.211112i \(0.932292\pi\)
\(360\) 0 0
\(361\) 31.0000i 1.63158i
\(362\) 0 0
\(363\) 3.00000 3.00000i 0.157459 0.157459i
\(364\) 0 0
\(365\) 12.0000 + 12.0000i 0.628109 + 0.628109i
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 5.00000 + 5.00000i 0.259587 + 0.259587i
\(372\) 0 0
\(373\) −15.0000 + 15.0000i −0.776671 + 0.776671i −0.979263 0.202593i \(-0.935063\pi\)
0.202593 + 0.979263i \(0.435063\pi\)
\(374\) 0 0
\(375\) 8.00000i 0.413118i
\(376\) 0 0
\(377\) 8.00000i 0.412021i
\(378\) 0 0
\(379\) −6.00000 + 6.00000i −0.308199 + 0.308199i −0.844211 0.536011i \(-0.819930\pi\)
0.536011 + 0.844211i \(0.319930\pi\)
\(380\) 0 0
\(381\) 8.00000 + 8.00000i 0.409852 + 0.409852i
\(382\) 0 0
\(383\) 14.0000 0.715367 0.357683 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 0 0
\(387\) −6.00000 6.00000i −0.304997 0.304997i
\(388\) 0 0
\(389\) −3.00000 + 3.00000i −0.152106 + 0.152106i −0.779058 0.626952i \(-0.784302\pi\)
0.626952 + 0.779058i \(0.284302\pi\)
\(390\) 0 0
\(391\) 8.00000i 0.404577i
\(392\) 0 0
\(393\) 2.00000i 0.100887i
\(394\) 0 0
\(395\) −8.00000 + 8.00000i −0.402524 + 0.402524i
\(396\) 0 0
\(397\) −24.0000 24.0000i −1.20453 1.20453i −0.972775 0.231750i \(-0.925555\pi\)
−0.231750 0.972775i \(-0.574445\pi\)
\(398\) 0 0
\(399\) 10.0000 0.500626
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −40.0000 40.0000i −1.99254 1.99254i
\(404\) 0 0
\(405\) −10.0000 + 10.0000i −0.496904 + 0.496904i
\(406\) 0 0
\(407\) 20.0000i 0.991363i
\(408\) 0 0
\(409\) 22.0000i 1.08783i −0.839140 0.543915i \(-0.816941\pi\)
0.839140 0.543915i \(-0.183059\pi\)
\(410\) 0 0
\(411\) 4.00000 4.00000i 0.197305 0.197305i
\(412\) 0 0
\(413\) 7.00000 + 7.00000i 0.344447 + 0.344447i
\(414\) 0 0
\(415\) −20.0000 −0.981761
\(416\) 0 0
\(417\) 26.0000 1.27323
\(418\) 0 0
\(419\) 3.00000 + 3.00000i 0.146560 + 0.146560i 0.776579 0.630020i \(-0.216953\pi\)
−0.630020 + 0.776579i \(0.716953\pi\)
\(420\) 0 0
\(421\) 3.00000 3.00000i 0.146211 0.146211i −0.630212 0.776423i \(-0.717032\pi\)
0.776423 + 0.630212i \(0.217032\pi\)
\(422\) 0 0
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) 6.00000i 0.291043i
\(426\) 0 0
\(427\) −6.00000 + 6.00000i −0.290360 + 0.290360i
\(428\) 0 0
\(429\) 16.0000 + 16.0000i 0.772487 + 0.772487i
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) 4.00000 + 4.00000i 0.191785 + 0.191785i
\(436\) 0 0
\(437\) 20.0000 20.0000i 0.956730 0.956730i
\(438\) 0 0
\(439\) 16.0000i 0.763638i 0.924237 + 0.381819i \(0.124702\pi\)
−0.924237 + 0.381819i \(0.875298\pi\)
\(440\) 0 0
\(441\) 1.00000i 0.0476190i
\(442\) 0 0
\(443\) 8.00000 8.00000i 0.380091 0.380091i −0.491044 0.871135i \(-0.663384\pi\)
0.871135 + 0.491044i \(0.163384\pi\)
\(444\) 0 0
\(445\) 28.0000 + 28.0000i 1.32733 + 1.32733i
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) −12.0000 12.0000i −0.565058 0.565058i
\(452\) 0 0
\(453\) 8.00000 8.00000i 0.375873 0.375873i
\(454\) 0 0
\(455\) 16.0000i 0.750092i
\(456\) 0 0
\(457\) 38.0000i 1.77757i −0.458329 0.888783i \(-0.651552\pi\)
0.458329 0.888783i \(-0.348448\pi\)
\(458\) 0 0
\(459\) 8.00000 8.00000i 0.373408 0.373408i
\(460\) 0 0
\(461\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 0 0
\(465\) 40.0000 1.85496
\(466\) 0 0
\(467\) 21.0000 + 21.0000i 0.971764 + 0.971764i 0.999612 0.0278481i \(-0.00886546\pi\)
−0.0278481 + 0.999612i \(0.508865\pi\)
\(468\) 0 0
\(469\) −4.00000 + 4.00000i −0.184703 + 0.184703i
\(470\) 0 0
\(471\) 24.0000i 1.10586i
\(472\) 0 0
\(473\) 24.0000i 1.10352i
\(474\) 0 0
\(475\) −15.0000 + 15.0000i −0.688247 + 0.688247i
\(476\) 0 0
\(477\) 5.00000 + 5.00000i 0.228934 + 0.228934i
\(478\) 0 0
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 0 0
\(483\) −4.00000 4.00000i −0.182006 0.182006i
\(484\) 0 0
\(485\) −20.0000 + 20.0000i −0.908153 + 0.908153i
\(486\) 0 0
\(487\) 28.0000i 1.26880i −0.773004 0.634401i \(-0.781247\pi\)
0.773004 0.634401i \(-0.218753\pi\)
\(488\) 0 0
\(489\) 20.0000i 0.904431i
\(490\) 0 0
\(491\) 20.0000 20.0000i 0.902587 0.902587i −0.0930720 0.995659i \(-0.529669\pi\)
0.995659 + 0.0930720i \(0.0296687\pi\)
\(492\) 0 0
\(493\) −2.00000 2.00000i −0.0900755 0.0900755i
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 20.0000 + 20.0000i 0.895323 + 0.895323i 0.995018 0.0996951i \(-0.0317867\pi\)
−0.0996951 + 0.995018i \(0.531787\pi\)
\(500\) 0 0
\(501\) 2.00000 2.00000i 0.0893534 0.0893534i
\(502\) 0 0
\(503\) 8.00000i 0.356702i 0.983967 + 0.178351i \(0.0570763\pi\)
−0.983967 + 0.178351i \(0.942924\pi\)
\(504\) 0 0
\(505\) 24.0000i 1.06799i
\(506\) 0 0
\(507\) 19.0000 19.0000i 0.843820 0.843820i
\(508\) 0 0
\(509\) −16.0000 16.0000i −0.709188 0.709188i 0.257177 0.966364i \(-0.417208\pi\)
−0.966364 + 0.257177i \(0.917208\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 0 0
\(513\) 40.0000 1.76604
\(514\) 0 0
\(515\) −12.0000 12.0000i −0.528783 0.528783i
\(516\) 0 0
\(517\) −12.0000 + 12.0000i −0.527759 + 0.527759i
\(518\) 0 0
\(519\) 24.0000i 1.05348i
\(520\) 0 0
\(521\) 6.00000i 0.262865i −0.991325 0.131432i \(-0.958042\pi\)
0.991325 0.131432i \(-0.0419576\pi\)
\(522\) 0 0
\(523\) −5.00000 + 5.00000i −0.218635 + 0.218635i −0.807923 0.589288i \(-0.799408\pi\)
0.589288 + 0.807923i \(0.299408\pi\)
\(524\) 0 0
\(525\) 3.00000 + 3.00000i 0.130931 + 0.130931i
\(526\) 0 0
\(527\) −20.0000 −0.871214
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 7.00000 + 7.00000i 0.303774 + 0.303774i
\(532\) 0 0
\(533\) −24.0000 + 24.0000i −1.03956 + 1.03956i
\(534\) 0 0
\(535\) 32.0000i 1.38348i
\(536\) 0 0
\(537\) 24.0000i 1.03568i
\(538\) 0 0
\(539\) −2.00000 + 2.00000i −0.0861461 + 0.0861461i
\(540\) 0 0
\(541\) −3.00000 3.00000i −0.128980 0.128980i 0.639670 0.768650i \(-0.279071\pi\)
−0.768650 + 0.639670i \(0.779071\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −44.0000 −1.88475
\(546\) 0 0
\(547\) 2.00000 + 2.00000i 0.0855138 + 0.0855138i 0.748570 0.663056i \(-0.230741\pi\)
−0.663056 + 0.748570i \(0.730741\pi\)
\(548\) 0 0
\(549\) −6.00000 + 6.00000i −0.256074 + 0.256074i
\(550\) 0 0
\(551\) 10.0000i 0.426014i
\(552\) 0 0
\(553\) 4.00000i 0.170097i
\(554\) 0 0
\(555\) 20.0000 20.0000i 0.848953 0.848953i
\(556\) 0 0
\(557\) −13.0000 13.0000i −0.550828 0.550828i 0.375852 0.926680i \(-0.377350\pi\)
−0.926680 + 0.375852i \(0.877350\pi\)
\(558\) 0 0
\(559\) −48.0000 −2.03018
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 25.0000 + 25.0000i 1.05362 + 1.05362i 0.998478 + 0.0551461i \(0.0175625\pi\)
0.0551461 + 0.998478i \(0.482438\pi\)
\(564\) 0 0
\(565\) 24.0000 24.0000i 1.00969 1.00969i
\(566\) 0 0
\(567\) 5.00000i 0.209980i
\(568\) 0 0
\(569\) 22.0000i 0.922288i −0.887325 0.461144i \(-0.847439\pi\)
0.887325 0.461144i \(-0.152561\pi\)
\(570\) 0 0
\(571\) 6.00000 6.00000i 0.251092 0.251092i −0.570326 0.821418i \(-0.693183\pi\)
0.821418 + 0.570326i \(0.193183\pi\)
\(572\) 0 0
\(573\) −12.0000 12.0000i −0.501307 0.501307i
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) 14.0000 + 14.0000i 0.581820 + 0.581820i
\(580\) 0 0
\(581\) 5.00000 5.00000i 0.207435 0.207435i
\(582\) 0 0
\(583\) 20.0000i 0.828315i
\(584\) 0 0
\(585\) 16.0000i 0.661519i
\(586\) 0 0
\(587\) −11.0000 + 11.0000i −0.454019 + 0.454019i −0.896686 0.442667i \(-0.854032\pi\)
0.442667 + 0.896686i \(0.354032\pi\)
\(588\) 0 0
\(589\) −50.0000 50.0000i −2.06021 2.06021i
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) −4.00000 4.00000i −0.163984 0.163984i
\(596\) 0 0
\(597\) −6.00000 + 6.00000i −0.245564 + 0.245564i
\(598\) 0 0
\(599\) 16.0000i 0.653742i 0.945069 + 0.326871i \(0.105994\pi\)
−0.945069 + 0.326871i \(0.894006\pi\)
\(600\) 0 0
\(601\) 2.00000i 0.0815817i 0.999168 + 0.0407909i \(0.0129877\pi\)
−0.999168 + 0.0407909i \(0.987012\pi\)
\(602\) 0 0
\(603\) −4.00000 + 4.00000i −0.162893 + 0.162893i
\(604\) 0 0
\(605\) −6.00000 6.00000i −0.243935 0.243935i
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 24.0000 + 24.0000i 0.970936 + 0.970936i
\(612\) 0 0
\(613\) −11.0000 + 11.0000i −0.444286 + 0.444286i −0.893449 0.449164i \(-0.851722\pi\)
0.449164 + 0.893449i \(0.351722\pi\)
\(614\) 0 0
\(615\) 24.0000i 0.967773i
\(616\) 0 0
\(617\) 24.0000i 0.966204i 0.875564 + 0.483102i \(0.160490\pi\)
−0.875564 + 0.483102i \(0.839510\pi\)
\(618\) 0 0
\(619\) 21.0000 21.0000i 0.844061 0.844061i −0.145323 0.989384i \(-0.546422\pi\)
0.989384 + 0.145323i \(0.0464221\pi\)
\(620\) 0 0
\(621\) −16.0000 16.0000i −0.642058 0.642058i
\(622\) 0 0
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) 31.0000 1.24000
\(626\) 0 0
\(627\) 20.0000 + 20.0000i 0.798723 + 0.798723i
\(628\) 0 0
\(629\) −10.0000 + 10.0000i −0.398726 + 0.398726i
\(630\) 0 0
\(631\) 20.0000i 0.796187i −0.917345 0.398094i \(-0.869672\pi\)
0.917345 0.398094i \(-0.130328\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 16.0000 16.0000i 0.634941 0.634941i
\(636\) 0 0
\(637\) 4.00000 + 4.00000i 0.158486 + 0.158486i
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) −1.00000 1.00000i −0.0394362 0.0394362i 0.687114 0.726550i \(-0.258877\pi\)
−0.726550 + 0.687114i \(0.758877\pi\)
\(644\) 0 0
\(645\) 24.0000 24.0000i 0.944999 0.944999i
\(646\) 0 0
\(647\) 6.00000i 0.235884i −0.993020 0.117942i \(-0.962370\pi\)
0.993020 0.117942i \(-0.0376297\pi\)
\(648\) 0 0
\(649\) 28.0000i 1.09910i
\(650\) 0 0
\(651\) −10.0000 + 10.0000i −0.391931 + 0.391931i
\(652\) 0 0
\(653\) 29.0000 + 29.0000i 1.13486 + 1.13486i 0.989358 + 0.145499i \(0.0464789\pi\)
0.145499 + 0.989358i \(0.453521\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −30.0000 30.0000i −1.16863 1.16863i −0.982530 0.186104i \(-0.940414\pi\)
−0.186104 0.982530i \(-0.559586\pi\)
\(660\) 0 0
\(661\) 6.00000 6.00000i 0.233373 0.233373i −0.580726 0.814099i \(-0.697231\pi\)
0.814099 + 0.580726i \(0.197231\pi\)
\(662\) 0 0
\(663\) 16.0000i 0.621389i
\(664\) 0 0
\(665\) 20.0000i 0.775567i
\(666\) 0 0
\(667\) −4.00000 + 4.00000i −0.154881 + 0.154881i
\(668\) 0 0
\(669\) 8.00000 + 8.00000i 0.309298 + 0.309298i
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 20.0000 0.770943 0.385472 0.922720i \(-0.374039\pi\)
0.385472 + 0.922720i \(0.374039\pi\)
\(674\) 0 0
\(675\) 12.0000 + 12.0000i 0.461880 + 0.461880i
\(676\) 0 0
\(677\) −20.0000 + 20.0000i −0.768662 + 0.768662i −0.977871 0.209209i \(-0.932911\pi\)
0.209209 + 0.977871i \(0.432911\pi\)
\(678\) 0 0
\(679\) 10.0000i 0.383765i
\(680\) 0 0
\(681\) 14.0000i 0.536481i
\(682\) 0 0
\(683\) 8.00000 8.00000i 0.306111 0.306111i −0.537288 0.843399i \(-0.680551\pi\)
0.843399 + 0.537288i \(0.180551\pi\)
\(684\) 0 0
\(685\) −8.00000 8.00000i −0.305664 0.305664i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 40.0000 1.52388
\(690\) 0 0
\(691\) 9.00000 + 9.00000i 0.342376 + 0.342376i 0.857260 0.514884i \(-0.172165\pi\)
−0.514884 + 0.857260i \(0.672165\pi\)
\(692\) 0 0
\(693\) −2.00000 + 2.00000i −0.0759737 + 0.0759737i
\(694\) 0 0
\(695\) 52.0000i 1.97247i
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) 0 0
\(699\) −22.0000 + 22.0000i −0.832116 + 0.832116i
\(700\) 0 0
\(701\) −29.0000 29.0000i −1.09531 1.09531i −0.994951 0.100364i \(-0.967999\pi\)
−0.100364 0.994951i \(-0.532001\pi\)
\(702\) 0 0
\(703\) −50.0000 −1.88579
\(704\) 0 0
\(705\) −24.0000 −0.903892
\(706\) 0 0
\(707\) 6.00000 + 6.00000i 0.225653 + 0.225653i
\(708\) 0 0
\(709\) 13.0000 13.0000i 0.488225 0.488225i −0.419521 0.907746i \(-0.637802\pi\)
0.907746 + 0.419521i \(0.137802\pi\)
\(710\) 0 0
\(711\) 4.00000i 0.150012i
\(712\) 0 0
\(713\) 40.0000i 1.49801i
\(714\) 0 0
\(715\) 32.0000 32.0000i 1.19673 1.19673i
\(716\) 0 0
\(717\) 20.0000 + 20.0000i 0.746914 + 0.746914i
\(718\) 0 0
\(719\) −50.0000 −1.86469 −0.932343 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) 0 0
\(721\) 6.00000 0.223452
\(722\) 0 0
\(723\) −14.0000 14.0000i −0.520666 0.520666i
\(724\) 0 0
\(725\) 3.00000 3.00000i 0.111417 0.111417i
\(726\) 0 0
\(727\) 10.0000i 0.370879i 0.982656 + 0.185440i \(0.0593710\pi\)
−0.982656 + 0.185440i \(0.940629\pi\)
\(728\) 0 0
\(729\) 29.0000i 1.07407i
\(730\) 0 0
\(731\) −12.0000 + 12.0000i −0.443836 + 0.443836i
\(732\) 0 0
\(733\) 6.00000 + 6.00000i 0.221615 + 0.221615i 0.809178 0.587563i \(-0.199913\pi\)
−0.587563 + 0.809178i \(0.699913\pi\)
\(734\) 0 0
\(735\) −4.00000 −0.147542
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 34.0000 + 34.0000i 1.25071 + 1.25071i 0.955402 + 0.295308i \(0.0954223\pi\)
0.295308 + 0.955402i \(0.404578\pi\)
\(740\) 0 0
\(741\) 40.0000 40.0000i 1.46944 1.46944i
\(742\) 0 0
\(743\) 16.0000i 0.586983i −0.955962 0.293492i \(-0.905183\pi\)
0.955962 0.293492i \(-0.0948173\pi\)
\(744\) 0 0
\(745\) 36.0000i 1.31894i
\(746\) 0 0
\(747\) 5.00000 5.00000i 0.182940 0.182940i
\(748\) 0 0
\(749\) 8.00000 + 8.00000i 0.292314 + 0.292314i
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) −10.0000 −0.364420
\(754\) 0 0
\(755\) −16.0000 16.0000i −0.582300 0.582300i
\(756\) 0 0
\(757\) −3.00000 + 3.00000i −0.109037 + 0.109037i −0.759520 0.650484i \(-0.774566\pi\)
0.650484 + 0.759520i \(0.274566\pi\)
\(758\) 0 0
\(759\) 16.0000i 0.580763i
\(760\) 0 0
\(761\) 50.0000i 1.81250i 0.422744 + 0.906249i \(0.361067\pi\)
−0.422744 + 0.906249i \(0.638933\pi\)
\(762\) 0 0
\(763\) 11.0000 11.0000i 0.398227 0.398227i
\(764\) 0 0
\(765\) −4.00000 4.00000i −0.144620 0.144620i
\(766\) 0 0
\(767\) 56.0000 2.02204
\(768\) 0 0
\(769\) 38.0000 1.37032 0.685158 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) 0 0
\(771\) 14.0000 + 14.0000i 0.504198 + 0.504198i
\(772\) 0 0
\(773\) −18.0000 + 18.0000i −0.647415 + 0.647415i −0.952368 0.304953i \(-0.901359\pi\)
0.304953 + 0.952368i \(0.401359\pi\)
\(774\) 0 0
\(775\) 30.0000i 1.07763i
\(776\) 0 0
\(777\) 10.0000i 0.358748i
\(778\) 0 0
\(779\) −30.0000 + 30.0000i −1.07486 + 1.07486i
\(780\) 0 0
\(781\) 24.0000 + 24.0000i 0.858788 + 0.858788i
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) −48.0000 −1.71319
\(786\) 0 0
\(787\) −27.0000 27.0000i −0.962446 0.962446i 0.0368739 0.999320i \(-0.488260\pi\)
−0.999320 + 0.0368739i \(0.988260\pi\)
\(788\) 0 0
\(789\) 12.0000 12.0000i 0.427211 0.427211i
\(790\) 0 0
\(791\) 12.0000i 0.426671i
\(792\) 0 0
\(793\) 48.0000i 1.70453i
\(794\) 0 0
\(795\) −20.0000 + 20.0000i −0.709327 + 0.709327i
\(796\) 0 0
\(797\) 16.0000 + 16.0000i 0.566749 + 0.566749i 0.931216 0.364467i \(-0.118749\pi\)
−0.364467 + 0.931216i \(0.618749\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) −12.0000 12.0000i −0.423471 0.423471i
\(804\) 0 0
\(805\) −8.00000 + 8.00000i −0.281963 + 0.281963i
\(806\) 0 0
\(807\) 16.0000i 0.563227i
\(808\) 0 0
\(809\) 42.0000i 1.47664i −0.674450 0.738321i \(-0.735619\pi\)
0.674450 0.738321i \(-0.264381\pi\)
\(810\) 0 0
\(811\) 25.0000 25.0000i 0.877869 0.877869i −0.115445 0.993314i \(-0.536829\pi\)
0.993314 + 0.115445i \(0.0368294\pi\)
\(812\) 0 0
\(813\) −32.0000 32.0000i −1.12229 1.12229i
\(814\) 0 0
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) −60.0000 −2.09913
\(818\) 0 0
\(819\) 4.00000 + 4.00000i 0.139771 + 0.139771i
\(820\) 0 0
\(821\) 5.00000 5.00000i 0.174501 0.174501i −0.614453 0.788954i \(-0.710623\pi\)
0.788954 + 0.614453i \(0.210623\pi\)
\(822\) 0 0
\(823\) 48.0000i 1.67317i −0.547833 0.836587i \(-0.684547\pi\)
0.547833 0.836587i \(-0.315453\pi\)
\(824\) 0 0
\(825\) 12.0000i 0.417786i
\(826\) 0 0
\(827\) 4.00000 4.00000i 0.139094 0.139094i −0.634132 0.773225i \(-0.718642\pi\)
0.773225 + 0.634132i \(0.218642\pi\)
\(828\) 0 0
\(829\) −22.0000 22.0000i −0.764092 0.764092i 0.212968 0.977059i \(-0.431687\pi\)
−0.977059 + 0.212968i \(0.931687\pi\)
\(830\) 0 0
\(831\) 6.00000 0.208138
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −4.00000 4.00000i −0.138426 0.138426i
\(836\) 0 0
\(837\) −40.0000 + 40.0000i −1.38260 + 1.38260i
\(838\) 0 0
\(839\) 50.0000i 1.72619i −0.505040 0.863096i \(-0.668522\pi\)
0.505040 0.863096i \(-0.331478\pi\)
\(840\) 0 0
\(841\) 27.0000i 0.931034i
\(842\) 0 0
\(843\) 26.0000 26.0000i 0.895488 0.895488i
\(844\) 0 0
\(845\) −38.0000 38.0000i −1.30724 1.30724i
\(846\) 0 0
\(847\) 3.00000 0.103081
\(848\) 0 0
\(849\) 30.0000 1.02960
\(850\) 0 0
\(851\) 20.0000 + 20.0000i 0.685591 + 0.685591i
\(852\) 0 0
\(853\) −4.00000 + 4.00000i −0.136957 + 0.136957i −0.772262 0.635304i \(-0.780875\pi\)
0.635304 + 0.772262i \(0.280875\pi\)
\(854\) 0 0
\(855\) 20.0000i 0.683986i
\(856\) 0 0
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) 15.0000 15.0000i 0.511793 0.511793i −0.403282 0.915076i \(-0.632131\pi\)
0.915076 + 0.403282i \(0.132131\pi\)
\(860\) 0 0
\(861\) 6.00000 + 6.00000i 0.204479 + 0.204479i
\(862\) 0 0
\(863\) 20.0000 0.680808 0.340404 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 0 0
\(865\) 48.0000 1.63205
\(866\) 0 0
\(867\) 13.0000 + 13.0000i 0.441503 + 0.441503i
\(868\) 0 0
\(869\) 8.00000 8.00000i 0.271381 0.271381i
\(870\) 0 0
\(871\) 32.0000i 1.08428i
\(872\) 0 0
\(873\) 10.0000i 0.338449i
\(874\) 0 0
\(875\) −4.00000 + 4.00000i −0.135225 + 0.135225i
\(876\) 0 0
\(877\) 15.0000 + 15.0000i 0.506514 + 0.506514i 0.913455 0.406941i \(-0.133404\pi\)
−0.406941 + 0.913455i \(0.633404\pi\)
\(878\) 0 0
\(879\) 4.00000 0.134917
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) −12.0000 12.0000i −0.403832 0.403832i 0.475749 0.879581i \(-0.342177\pi\)
−0.879581 + 0.475749i \(0.842177\pi\)
\(884\) 0 0
\(885\) −28.0000 + 28.0000i −0.941210 + 0.941210i
\(886\) 0 0
\(887\) 14.0000i 0.470074i −0.971986 0.235037i \(-0.924479\pi\)
0.971986 0.235037i \(-0.0755211\pi\)
\(888\) 0 0
\(889\) 8.00000i 0.268311i
\(890\) 0 0
\(891\) 10.0000 10.0000i 0.335013 0.335013i
\(892\) 0 0
\(893\) 30.0000 + 30.0000i 1.00391 + 1.00391i
\(894\) 0 0
\(895\) −48.0000 −1.60446
\(896\) 0 0
\(897\) −32.0000 −1.06845
\(898\) 0 0
\(899\) 10.0000 + 10.0000i 0.333519 + 0.333519i
\(900\) 0 0
\(901\) 10.0000 10.0000i 0.333148 0.333148i
\(902\) 0 0
\(903\) 12.0000i 0.399335i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20.0000 20.0000i 0.664089 0.664089i −0.292252 0.956341i \(-0.594405\pi\)
0.956341 + 0.292252i \(0.0944047\pi\)
\(908\) 0 0
\(909\) 6.00000 + 6.00000i 0.199007 + 0.199007i
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 20.0000 0.661903
\(914\) 0 0
\(915\) −24.0000 24.0000i −0.793416 0.793416i
\(916\) 0 0
\(917\) −1.00000 + 1.00000i −0.0330229 + 0.0330229i
\(918\) 0 0
\(919\) 16.0000i 0.527791i 0.964551 + 0.263896i \(0.0850075\pi\)
−0.964551 + 0.263896i \(0.914993\pi\)
\(920\) 0 0
\(921\) 34.0000i 1.12034i
\(922\) 0 0
\(923\) 48.0000 48.0000i 1.57994 1.57994i
\(924\) 0 0
\(925\) −15.0000 15.0000i −0.493197 0.493197i
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) 5.00000 + 5.00000i 0.163868 + 0.163868i
\(932\) 0 0
\(933\) 24.0000 24.0000i 0.785725 0.785725i
\(934\) 0 0
\(935\) 16.0000i 0.523256i
\(936\) 0 0
\(937\) 14.0000i 0.457360i −0.973502 0.228680i \(-0.926559\pi\)
0.973502 0.228680i \(-0.0734410\pi\)
\(938\) 0 0
\(939\) 14.0000 14.0000i 0.456873 0.456873i
\(940\) 0 0
\(941\) 14.0000 + 14.0000i 0.456387 + 0.456387i 0.897468 0.441081i \(-0.145405\pi\)
−0.441081 + 0.897468i \(0.645405\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) −16.0000 −0.520480
\(946\) 0 0
\(947\) 18.0000 + 18.0000i 0.584921 + 0.584921i 0.936252 0.351330i \(-0.114271\pi\)
−0.351330 + 0.936252i \(0.614271\pi\)
\(948\) 0 0
\(949\) −24.0000 + 24.0000i −0.779073 + 0.779073i
\(950\) 0 0
\(951\) 10.0000i 0.324272i
\(952\) 0 0
\(953\) 48.0000i 1.55487i −0.628962 0.777436i \(-0.716520\pi\)
0.628962 0.777436i \(-0.283480\pi\)
\(954\) 0 0
\(955\) −24.0000 + 24.0000i −0.776622 + 0.776622i
\(956\) 0 0
\(957\) −4.00000 4.00000i −0.129302 0.129302i
\(958\) 0 0
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 8.00000 + 8.00000i 0.257796 + 0.257796i
\(964\) 0 0
\(965\) 28.0000 28.0000i 0.901352 0.901352i
\(966\) 0 0
\(967\) 44.0000i 1.41494i 0.706741 + 0.707472i \(0.250165\pi\)
−0.706741 + 0.707472i \(0.749835\pi\)
\(968\) 0 0
\(969\) 20.0000i 0.642493i
\(970\) 0 0
\(971\) −41.0000 + 41.0000i −1.31575 + 1.31575i −0.398649 + 0.917104i \(0.630521\pi\)
−0.917104 + 0.398649i \(0.869479\pi\)
\(972\) 0 0
\(973\) 13.0000 + 13.0000i 0.416761 + 0.416761i
\(974\) 0 0
\(975\) 24.0000 0.768615
\(976\) 0 0
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 0 0
\(979\) −28.0000 28.0000i −0.894884 0.894884i
\(980\) 0 0
\(981\) 11.0000 11.0000i 0.351203 0.351203i
\(982\) 0 0
\(983\) 6.00000i 0.191370i −0.995412 0.0956851i \(-0.969496\pi\)
0.995412 0.0956851i \(-0.0305042\pi\)
\(984\) 0 0
\(985\) 36.0000i 1.14706i
\(986\) 0 0
\(987\) 6.00000 6.00000i 0.190982 0.190982i
\(988\) 0 0
\(989\) 24.0000 + 24.0000i 0.763156 + 0.763156i
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) −36.0000 −1.14243
\(994\) 0 0
\(995\) 12.0000 + 12.0000i 0.380426 + 0.380426i
\(996\) 0 0
\(997\) −34.0000 + 34.0000i −1.07679 + 1.07679i −0.0799956 + 0.996795i \(0.525491\pi\)
−0.996795 + 0.0799956i \(0.974509\pi\)
\(998\) 0 0
\(999\) 40.0000i 1.26554i
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 3584.2.m.e.2689.1 yes 2
4.3 odd 2 3584.2.m.q.2689.1 yes 2
8.3 odd 2 3584.2.m.l.2689.1 yes 2
8.5 even 2 3584.2.m.x.2689.1 yes 2
16.3 odd 4 3584.2.m.l.897.1 yes 2
16.5 even 4 inner 3584.2.m.e.897.1 2
16.11 odd 4 3584.2.m.q.897.1 yes 2
16.13 even 4 3584.2.m.x.897.1 yes 2
32.5 even 8 7168.2.a.m.1.2 2
32.11 odd 8 7168.2.a.g.1.2 2
32.21 even 8 7168.2.a.m.1.1 2
32.27 odd 8 7168.2.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3584.2.m.e.897.1 2 16.5 even 4 inner
3584.2.m.e.2689.1 yes 2 1.1 even 1 trivial
3584.2.m.l.897.1 yes 2 16.3 odd 4
3584.2.m.l.2689.1 yes 2 8.3 odd 2
3584.2.m.q.897.1 yes 2 16.11 odd 4
3584.2.m.q.2689.1 yes 2 4.3 odd 2
3584.2.m.x.897.1 yes 2 16.13 even 4
3584.2.m.x.2689.1 yes 2 8.5 even 2
7168.2.a.g.1.1 2 32.27 odd 8
7168.2.a.g.1.2 2 32.11 odd 8
7168.2.a.m.1.1 2 32.21 even 8
7168.2.a.m.1.2 2 32.5 even 8