Properties

Label 1600.2.o.h.543.4
Level $1600$
Weight $2$
Character 1600.543
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
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] = [1600,2,Mod(543,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.543");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.o (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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 543.4
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1600.543
Dual form 1600.2.o.h.607.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 + 0.707107i) q^{3} -2.00000i q^{9} +O(q^{10})\) \(q+(0.707107 + 0.707107i) q^{3} -2.00000i q^{9} +1.73205 q^{11} +(-2.82843 + 2.82843i) q^{13} +(1.22474 - 1.22474i) q^{17} -5.19615i q^{19} +(4.89898 - 4.89898i) q^{23} +(3.53553 - 3.53553i) q^{27} +6.92820 q^{29} +4.00000i q^{31} +(1.22474 + 1.22474i) q^{33} +(5.65685 + 5.65685i) q^{37} -4.00000 q^{39} +3.00000 q^{41} +(5.65685 + 5.65685i) q^{43} +(-4.89898 - 4.89898i) q^{47} -7.00000i q^{49} +1.73205 q^{51} +(-8.48528 + 8.48528i) q^{53} +(3.67423 - 3.67423i) q^{57} -10.3923i q^{59} +(4.94975 - 4.94975i) q^{67} +6.92820 q^{69} -12.0000i q^{71} +(6.12372 + 6.12372i) q^{73} -8.00000 q^{79} -1.00000 q^{81} +(10.6066 + 10.6066i) q^{83} +(4.89898 + 4.89898i) q^{87} -3.00000i q^{89} +(-2.82843 + 2.82843i) q^{93} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{39} + 24 q^{41} - 64 q^{79} - 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.707107 + 0.707107i 0.408248 + 0.408248i 0.881127 0.472879i \(-0.156785\pi\)
−0.472879 + 0.881127i \(0.656785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 2.00000i 0.666667i
\(10\) 0 0
\(11\) 1.73205 0.522233 0.261116 0.965307i \(-0.415909\pi\)
0.261116 + 0.965307i \(0.415909\pi\)
\(12\) 0 0
\(13\) −2.82843 + 2.82843i −0.784465 + 0.784465i −0.980581 0.196116i \(-0.937167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.22474 1.22474i 0.297044 0.297044i −0.542811 0.839855i \(-0.682640\pi\)
0.839855 + 0.542811i \(0.182640\pi\)
\(18\) 0 0
\(19\) 5.19615i 1.19208i −0.802955 0.596040i \(-0.796740\pi\)
0.802955 0.596040i \(-0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.89898 4.89898i 1.02151 1.02151i 0.0217443 0.999764i \(-0.493078\pi\)
0.999764 0.0217443i \(-0.00692196\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.53553 3.53553i 0.680414 0.680414i
\(28\) 0 0
\(29\) 6.92820 1.28654 0.643268 0.765641i \(-0.277578\pi\)
0.643268 + 0.765641i \(0.277578\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 1.22474 + 1.22474i 0.213201 + 0.213201i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.65685 + 5.65685i 0.929981 + 0.929981i 0.997704 0.0677230i \(-0.0215734\pi\)
−0.0677230 + 0.997704i \(0.521573\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 5.65685 + 5.65685i 0.862662 + 0.862662i 0.991647 0.128984i \(-0.0411717\pi\)
−0.128984 + 0.991647i \(0.541172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.89898 4.89898i −0.714590 0.714590i 0.252902 0.967492i \(-0.418615\pi\)
−0.967492 + 0.252902i \(0.918615\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 1.73205 0.242536
\(52\) 0 0
\(53\) −8.48528 + 8.48528i −1.16554 + 1.16554i −0.182300 + 0.983243i \(0.558354\pi\)
−0.983243 + 0.182300i \(0.941646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.67423 3.67423i 0.486664 0.486664i
\(58\) 0 0
\(59\) 10.3923i 1.35296i −0.736460 0.676481i \(-0.763504\pi\)
0.736460 0.676481i \(-0.236496\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.94975 4.94975i 0.604708 0.604708i −0.336850 0.941558i \(-0.609362\pi\)
0.941558 + 0.336850i \(0.109362\pi\)
\(68\) 0 0
\(69\) 6.92820 0.834058
\(70\) 0 0
\(71\) 12.0000i 1.42414i −0.702109 0.712069i \(-0.747758\pi\)
0.702109 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 6.12372 + 6.12372i 0.716728 + 0.716728i 0.967934 0.251206i \(-0.0808271\pi\)
−0.251206 + 0.967934i \(0.580827\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 10.6066 + 10.6066i 1.16423 + 1.16423i 0.983541 + 0.180685i \(0.0578314\pi\)
0.180685 + 0.983541i \(0.442169\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.89898 + 4.89898i 0.525226 + 0.525226i
\(88\) 0 0
\(89\) 3.00000i 0.317999i −0.987279 0.159000i \(-0.949173\pi\)
0.987279 0.159000i \(-0.0508269\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.82843 + 2.82843i −0.293294 + 0.293294i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) 13.8564i 1.37876i 0.724398 + 0.689382i \(0.242118\pi\)
−0.724398 + 0.689382i \(0.757882\pi\)
\(102\) 0 0
\(103\) 4.89898 4.89898i 0.482711 0.482711i −0.423286 0.905996i \(-0.639123\pi\)
0.905996 + 0.423286i \(0.139123\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.12132 + 2.12132i −0.205076 + 0.205076i −0.802171 0.597095i \(-0.796322\pi\)
0.597095 + 0.802171i \(0.296322\pi\)
\(108\) 0 0
\(109\) 13.8564 1.32720 0.663602 0.748086i \(-0.269027\pi\)
0.663602 + 0.748086i \(0.269027\pi\)
\(110\) 0 0
\(111\) 8.00000i 0.759326i
\(112\) 0 0
\(113\) 8.57321 + 8.57321i 0.806500 + 0.806500i 0.984102 0.177602i \(-0.0568340\pi\)
−0.177602 + 0.984102i \(0.556834\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.65685 + 5.65685i 0.522976 + 0.522976i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 0 0
\(123\) 2.12132 + 2.12132i 0.191273 + 0.191273i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −14.6969 14.6969i −1.30414 1.30414i −0.925573 0.378570i \(-0.876416\pi\)
−0.378570 0.925573i \(-0.623584\pi\)
\(128\) 0 0
\(129\) 8.00000i 0.704361i
\(130\) 0 0
\(131\) 3.46410 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.67423 + 3.67423i −0.313911 + 0.313911i −0.846423 0.532512i \(-0.821248\pi\)
0.532512 + 0.846423i \(0.321248\pi\)
\(138\) 0 0
\(139\) 1.73205i 0.146911i −0.997299 0.0734553i \(-0.976597\pi\)
0.997299 0.0734553i \(-0.0234026\pi\)
\(140\) 0 0
\(141\) 6.92820i 0.583460i
\(142\) 0 0
\(143\) −4.89898 + 4.89898i −0.409673 + 0.409673i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.94975 4.94975i 0.408248 0.408248i
\(148\) 0 0
\(149\) −6.92820 −0.567581 −0.283790 0.958886i \(-0.591592\pi\)
−0.283790 + 0.958886i \(0.591592\pi\)
\(150\) 0 0
\(151\) 16.0000i 1.30206i −0.759051 0.651031i \(-0.774337\pi\)
0.759051 0.651031i \(-0.225663\pi\)
\(152\) 0 0
\(153\) −2.44949 2.44949i −0.198030 0.198030i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.82843 + 2.82843i 0.225733 + 0.225733i 0.810907 0.585174i \(-0.198974\pi\)
−0.585174 + 0.810907i \(0.698974\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.94975 4.94975i −0.387694 0.387694i 0.486170 0.873864i \(-0.338394\pi\)
−0.873864 + 0.486170i \(0.838394\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.6969 + 14.6969i 1.13728 + 1.13728i 0.988935 + 0.148348i \(0.0473956\pi\)
0.148348 + 0.988935i \(0.452604\pi\)
\(168\) 0 0
\(169\) 3.00000i 0.230769i
\(170\) 0 0
\(171\) −10.3923 −0.794719
\(172\) 0 0
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.34847 7.34847i 0.552345 0.552345i
\(178\) 0 0
\(179\) 5.19615i 0.388379i 0.980964 + 0.194189i \(0.0622076\pi\)
−0.980964 + 0.194189i \(0.937792\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i −0.966282 0.257485i \(-0.917106\pi\)
0.966282 0.257485i \(-0.0828937\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.12132 2.12132i 0.155126 0.155126i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000i 0.868290i 0.900843 + 0.434145i \(0.142949\pi\)
−0.900843 + 0.434145i \(0.857051\pi\)
\(192\) 0 0
\(193\) −11.0227 11.0227i −0.793432 0.793432i 0.188619 0.982050i \(-0.439599\pi\)
−0.982050 + 0.188619i \(0.939599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.48528 8.48528i −0.604551 0.604551i 0.336966 0.941517i \(-0.390599\pi\)
−0.941517 + 0.336966i \(0.890599\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −9.79796 9.79796i −0.681005 0.681005i
\(208\) 0 0
\(209\) 9.00000i 0.622543i
\(210\) 0 0
\(211\) −8.66025 −0.596196 −0.298098 0.954535i \(-0.596352\pi\)
−0.298098 + 0.954535i \(0.596352\pi\)
\(212\) 0 0
\(213\) 8.48528 8.48528i 0.581402 0.581402i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 8.66025i 0.585206i
\(220\) 0 0
\(221\) 6.92820i 0.466041i
\(222\) 0 0
\(223\) −9.79796 + 9.79796i −0.656120 + 0.656120i −0.954460 0.298340i \(-0.903567\pi\)
0.298340 + 0.954460i \(0.403567\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −16.9706 + 16.9706i −1.12638 + 1.12638i −0.135614 + 0.990762i \(0.543301\pi\)
−0.990762 + 0.135614i \(0.956699\pi\)
\(228\) 0 0
\(229\) −13.8564 −0.915657 −0.457829 0.889041i \(-0.651373\pi\)
−0.457829 + 0.889041i \(0.651373\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −5.65685 5.65685i −0.367452 0.367452i
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 1.00000 0.0644157 0.0322078 0.999481i \(-0.489746\pi\)
0.0322078 + 0.999481i \(0.489746\pi\)
\(242\) 0 0
\(243\) −11.3137 11.3137i −0.725775 0.725775i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 14.6969 + 14.6969i 0.935144 + 0.935144i
\(248\) 0 0
\(249\) 15.0000i 0.950586i
\(250\) 0 0
\(251\) 29.4449 1.85854 0.929272 0.369397i \(-0.120436\pi\)
0.929272 + 0.369397i \(0.120436\pi\)
\(252\) 0 0
\(253\) 8.48528 8.48528i 0.533465 0.533465i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −19.5959 + 19.5959i −1.22236 + 1.22236i −0.255569 + 0.966791i \(0.582263\pi\)
−0.966791 + 0.255569i \(0.917737\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 13.8564i 0.857690i
\(262\) 0 0
\(263\) 14.6969 14.6969i 0.906252 0.906252i −0.0897154 0.995967i \(-0.528596\pi\)
0.995967 + 0.0897154i \(0.0285957\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.12132 2.12132i 0.129823 0.129823i
\(268\) 0 0
\(269\) −27.7128 −1.68968 −0.844840 0.535019i \(-0.820304\pi\)
−0.844840 + 0.535019i \(0.820304\pi\)
\(270\) 0 0
\(271\) 28.0000i 1.70088i 0.526073 + 0.850439i \(0.323664\pi\)
−0.526073 + 0.850439i \(0.676336\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.82843 2.82843i −0.169944 0.169944i 0.617011 0.786955i \(-0.288343\pi\)
−0.786955 + 0.617011i \(0.788343\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −20.5061 20.5061i −1.21896 1.21896i −0.967996 0.250965i \(-0.919252\pi\)
−0.250965 0.967996i \(-0.580748\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 14.0000i 0.823529i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 6.12372 6.12372i 0.355335 0.355335i
\(298\) 0 0
\(299\) 27.7128i 1.60267i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −9.79796 + 9.79796i −0.562878 + 0.562878i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.94975 + 4.94975i −0.282497 + 0.282497i −0.834104 0.551607i \(-0.814015\pi\)
0.551607 + 0.834104i \(0.314015\pi\)
\(308\) 0 0
\(309\) 6.92820 0.394132
\(310\) 0 0
\(311\) 24.0000i 1.36092i −0.732787 0.680458i \(-0.761781\pi\)
0.732787 0.680458i \(-0.238219\pi\)
\(312\) 0 0
\(313\) −19.5959 19.5959i −1.10763 1.10763i −0.993462 0.114165i \(-0.963581\pi\)
−0.114165 0.993462i \(-0.536419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −16.9706 16.9706i −0.953162 0.953162i 0.0457894 0.998951i \(-0.485420\pi\)
−0.998951 + 0.0457894i \(0.985420\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) −6.36396 6.36396i −0.354100 0.354100i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 9.79796 + 9.79796i 0.541828 + 0.541828i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −29.4449 −1.61844 −0.809218 0.587508i \(-0.800109\pi\)
−0.809218 + 0.587508i \(0.800109\pi\)
\(332\) 0 0
\(333\) 11.3137 11.3137i 0.619987 0.619987i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.57321 8.57321i 0.467013 0.467013i −0.433933 0.900945i \(-0.642874\pi\)
0.900945 + 0.433933i \(0.142874\pi\)
\(338\) 0 0
\(339\) 12.1244i 0.658505i
\(340\) 0 0
\(341\) 6.92820i 0.375183i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −19.0919 + 19.0919i −1.02491 + 1.02491i −0.0252242 + 0.999682i \(0.508030\pi\)
−0.999682 + 0.0252242i \(0.991970\pi\)
\(348\) 0 0
\(349\) −27.7128 −1.48343 −0.741716 0.670714i \(-0.765988\pi\)
−0.741716 + 0.670714i \(0.765988\pi\)
\(350\) 0 0
\(351\) 20.0000i 1.06752i
\(352\) 0 0
\(353\) −19.5959 19.5959i −1.04299 1.04299i −0.999034 0.0439518i \(-0.986005\pi\)
−0.0439518 0.999034i \(-0.513995\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −8.00000 −0.421053
\(362\) 0 0
\(363\) −5.65685 5.65685i −0.296908 0.296908i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −9.79796 9.79796i −0.511449 0.511449i 0.403521 0.914970i \(-0.367786\pi\)
−0.914970 + 0.403521i \(0.867786\pi\)
\(368\) 0 0
\(369\) 6.00000i 0.312348i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 19.7990 19.7990i 1.02515 1.02515i 0.0254774 0.999675i \(-0.491889\pi\)
0.999675 0.0254774i \(-0.00811060\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −19.5959 + 19.5959i −1.00924 + 1.00924i
\(378\) 0 0
\(379\) 1.73205i 0.0889695i 0.999010 + 0.0444847i \(0.0141646\pi\)
−0.999010 + 0.0444847i \(0.985835\pi\)
\(380\) 0 0
\(381\) 20.7846i 1.06483i
\(382\) 0 0
\(383\) −19.5959 + 19.5959i −1.00130 + 1.00130i −0.00130548 + 0.999999i \(0.500416\pi\)
−0.999999 + 0.00130548i \(0.999584\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.3137 11.3137i 0.575108 0.575108i
\(388\) 0 0
\(389\) −6.92820 −0.351274 −0.175637 0.984455i \(-0.556198\pi\)
−0.175637 + 0.984455i \(0.556198\pi\)
\(390\) 0 0
\(391\) 12.0000i 0.606866i
\(392\) 0 0
\(393\) 2.44949 + 2.44949i 0.123560 + 0.123560i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.3137 + 11.3137i 0.567819 + 0.567819i 0.931517 0.363698i \(-0.118486\pi\)
−0.363698 + 0.931517i \(0.618486\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.0000 0.749064 0.374532 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) 0 0
\(403\) −11.3137 11.3137i −0.563576 0.563576i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.79796 + 9.79796i 0.485667 + 0.485667i
\(408\) 0 0
\(409\) 5.00000i 0.247234i 0.992330 + 0.123617i \(0.0394494\pi\)
−0.992330 + 0.123617i \(0.960551\pi\)
\(410\) 0 0
\(411\) −5.19615 −0.256307
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.22474 1.22474i 0.0599760 0.0599760i
\(418\) 0 0
\(419\) 19.0526i 0.930778i 0.885106 + 0.465389i \(0.154086\pi\)
−0.885106 + 0.465389i \(0.845914\pi\)
\(420\) 0 0
\(421\) 27.7128i 1.35064i −0.737525 0.675320i \(-0.764006\pi\)
0.737525 0.675320i \(-0.235994\pi\)
\(422\) 0 0
\(423\) −9.79796 + 9.79796i −0.476393 + 0.476393i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −6.92820 −0.334497
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 20.8207 + 20.8207i 1.00058 + 1.00058i 1.00000 0.000577367i \(0.000183782\pi\)
0.000577367 1.00000i \(0.499816\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −25.4558 25.4558i −1.21772 1.21772i
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) −14.0000 −0.666667
\(442\) 0 0
\(443\) −2.12132 2.12132i −0.100787 0.100787i 0.654915 0.755702i \(-0.272704\pi\)
−0.755702 + 0.654915i \(0.772704\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4.89898 4.89898i −0.231714 0.231714i
\(448\) 0 0
\(449\) 39.0000i 1.84052i 0.391303 + 0.920262i \(0.372024\pi\)
−0.391303 + 0.920262i \(0.627976\pi\)
\(450\) 0 0
\(451\) 5.19615 0.244677
\(452\) 0 0
\(453\) 11.3137 11.3137i 0.531564 0.531564i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.67423 3.67423i 0.171873 0.171873i −0.615929 0.787802i \(-0.711219\pi\)
0.787802 + 0.615929i \(0.211219\pi\)
\(458\) 0 0
\(459\) 8.66025i 0.404226i
\(460\) 0 0
\(461\) 41.5692i 1.93607i 0.250812 + 0.968036i \(0.419302\pi\)
−0.250812 + 0.968036i \(0.580698\pi\)
\(462\) 0 0
\(463\) −9.79796 + 9.79796i −0.455350 + 0.455350i −0.897126 0.441776i \(-0.854349\pi\)
0.441776 + 0.897126i \(0.354349\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16.9706 + 16.9706i −0.785304 + 0.785304i −0.980720 0.195416i \(-0.937394\pi\)
0.195416 + 0.980720i \(0.437394\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4.00000i 0.184310i
\(472\) 0 0
\(473\) 9.79796 + 9.79796i 0.450511 + 0.450511i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 16.9706 + 16.9706i 0.777029 + 0.777029i
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −32.0000 −1.45907
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 4.89898 + 4.89898i 0.221994 + 0.221994i 0.809338 0.587344i \(-0.199826\pi\)
−0.587344 + 0.809338i \(0.699826\pi\)
\(488\) 0 0
\(489\) 7.00000i 0.316551i
\(490\) 0 0
\(491\) 10.3923 0.468998 0.234499 0.972116i \(-0.424655\pi\)
0.234499 + 0.972116i \(0.424655\pi\)
\(492\) 0 0
\(493\) 8.48528 8.48528i 0.382158 0.382158i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 24.2487i 1.08552i 0.839887 + 0.542761i \(0.182621\pi\)
−0.839887 + 0.542761i \(0.817379\pi\)
\(500\) 0 0
\(501\) 20.7846i 0.928588i
\(502\) 0 0
\(503\) 19.5959 19.5959i 0.873739 0.873739i −0.119139 0.992878i \(-0.538013\pi\)
0.992878 + 0.119139i \(0.0380134\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.12132 2.12132i 0.0942111 0.0942111i
\(508\) 0 0
\(509\) 6.92820 0.307087 0.153544 0.988142i \(-0.450931\pi\)
0.153544 + 0.988142i \(0.450931\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −18.3712 18.3712i −0.811107 0.811107i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −8.48528 8.48528i −0.373182 0.373182i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) 9.19239 + 9.19239i 0.401955 + 0.401955i 0.878922 0.476966i \(-0.158264\pi\)
−0.476966 + 0.878922i \(0.658264\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.89898 + 4.89898i 0.213403 + 0.213403i
\(528\) 0 0
\(529\) 25.0000i 1.08696i
\(530\) 0 0
\(531\) −20.7846 −0.901975
\(532\) 0 0
\(533\) −8.48528 + 8.48528i −0.367538 + 0.367538i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.67423 + 3.67423i −0.158555 + 0.158555i
\(538\) 0 0
\(539\) 12.1244i 0.522233i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 4.89898 4.89898i 0.210235 0.210235i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0208 12.0208i 0.513973 0.513973i −0.401768 0.915741i \(-0.631604\pi\)
0.915741 + 0.401768i \(0.131604\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 36.0000i 1.53365i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8.48528 8.48528i −0.359533 0.359533i 0.504108 0.863641i \(-0.331821\pi\)
−0.863641 + 0.504108i \(0.831821\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 3.00000 0.126660
\(562\) 0 0
\(563\) 16.9706 + 16.9706i 0.715224 + 0.715224i 0.967623 0.252399i \(-0.0812196\pi\)
−0.252399 + 0.967623i \(0.581220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.0000i 1.13190i 0.824440 + 0.565949i \(0.191490\pi\)
−0.824440 + 0.565949i \(0.808510\pi\)
\(570\) 0 0
\(571\) −10.3923 −0.434904 −0.217452 0.976071i \(-0.569775\pi\)
−0.217452 + 0.976071i \(0.569775\pi\)
\(572\) 0 0
\(573\) −8.48528 + 8.48528i −0.354478 + 0.354478i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −18.3712 + 18.3712i −0.764802 + 0.764802i −0.977186 0.212384i \(-0.931877\pi\)
0.212384 + 0.977186i \(0.431877\pi\)
\(578\) 0 0
\(579\) 15.5885i 0.647834i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −14.6969 + 14.6969i −0.608685 + 0.608685i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.0919 + 19.0919i −0.788006 + 0.788006i −0.981167 0.193161i \(-0.938126\pi\)
0.193161 + 0.981167i \(0.438126\pi\)
\(588\) 0 0
\(589\) 20.7846 0.856415
\(590\) 0 0
\(591\) 12.0000i 0.493614i
\(592\) 0 0
\(593\) 28.1691 + 28.1691i 1.15677 + 1.15677i 0.985167 + 0.171601i \(0.0548940\pi\)
0.171601 + 0.985167i \(0.445106\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.3137 + 11.3137i 0.463039 + 0.463039i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 19.0000 0.775026 0.387513 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(602\) 0 0
\(603\) −9.89949 9.89949i −0.403139 0.403139i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −19.5959 19.5959i −0.795374 0.795374i 0.186988 0.982362i \(-0.440127\pi\)
−0.982362 + 0.186988i \(0.940127\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 27.7128 1.12114
\(612\) 0 0
\(613\) −14.1421 + 14.1421i −0.571195 + 0.571195i −0.932462 0.361267i \(-0.882344\pi\)
0.361267 + 0.932462i \(0.382344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.5959 19.5959i 0.788902 0.788902i −0.192412 0.981314i \(-0.561631\pi\)
0.981314 + 0.192412i \(0.0616311\pi\)
\(618\) 0 0
\(619\) 10.3923i 0.417702i 0.977947 + 0.208851i \(0.0669724\pi\)
−0.977947 + 0.208851i \(0.933028\pi\)
\(620\) 0 0
\(621\) 34.6410i 1.39010i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 6.36396 6.36396i 0.254152 0.254152i
\(628\) 0 0
\(629\) 13.8564 0.552491
\(630\) 0 0
\(631\) 28.0000i 1.11466i 0.830290 + 0.557331i \(0.188175\pi\)
−0.830290 + 0.557331i \(0.811825\pi\)
\(632\) 0 0
\(633\) −6.12372 6.12372i −0.243396 0.243396i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 19.7990 + 19.7990i 0.784465 + 0.784465i
\(638\) 0 0
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 5.65685 + 5.65685i 0.223085 + 0.223085i 0.809796 0.586711i \(-0.199578\pi\)
−0.586711 + 0.809796i \(0.699578\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) 18.0000i 0.706562i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −16.9706 + 16.9706i −0.664109 + 0.664109i −0.956346 0.292237i \(-0.905601\pi\)
0.292237 + 0.956346i \(0.405601\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 12.2474 12.2474i 0.477818 0.477818i
\(658\) 0 0
\(659\) 36.3731i 1.41689i −0.705764 0.708447i \(-0.749396\pi\)
0.705764 0.708447i \(-0.250604\pi\)
\(660\) 0 0
\(661\) 6.92820i 0.269476i −0.990881 0.134738i \(-0.956981\pi\)
0.990881 0.134738i \(-0.0430193\pi\)
\(662\) 0 0
\(663\) −4.89898 + 4.89898i −0.190261 + 0.190261i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 33.9411 33.9411i 1.31421 1.31421i
\(668\) 0 0
\(669\) −13.8564 −0.535720
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −19.5959 19.5959i −0.755367 0.755367i 0.220108 0.975475i \(-0.429359\pi\)
−0.975475 + 0.220108i \(0.929359\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) 31.8198 + 31.8198i 1.21755 + 1.21755i 0.968487 + 0.249064i \(0.0801229\pi\)
0.249064 + 0.968487i \(0.419877\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −9.79796 9.79796i −0.373815 0.373815i
\(688\) 0 0
\(689\) 48.0000i 1.82865i
\(690\) 0 0
\(691\) −5.19615 −0.197671 −0.0988355 0.995104i \(-0.531512\pi\)
−0.0988355 + 0.995104i \(0.531512\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.67423 3.67423i 0.139172 0.139172i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 34.6410i 1.30837i 0.756333 + 0.654187i \(0.226989\pi\)
−0.756333 + 0.654187i \(0.773011\pi\)
\(702\) 0 0
\(703\) 29.3939 29.3939i 1.10861 1.10861i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 41.5692 1.56116 0.780582 0.625053i \(-0.214923\pi\)
0.780582 + 0.625053i \(0.214923\pi\)
\(710\) 0 0
\(711\) 16.0000i 0.600047i
\(712\) 0 0
\(713\) 19.5959 + 19.5959i 0.733873 + 0.733873i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8.48528 + 8.48528i 0.316889 + 0.316889i
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0.707107 + 0.707107i 0.0262976 + 0.0262976i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −4.89898 4.89898i −0.181693 0.181693i 0.610400 0.792093i \(-0.291009\pi\)
−0.792093 + 0.610400i \(0.791009\pi\)
\(728\) 0 0
\(729\) 13.0000i 0.481481i
\(730\) 0 0
\(731\) 13.8564 0.512498
\(732\) 0 0
\(733\) 2.82843 2.82843i 0.104470 0.104470i −0.652940 0.757410i \(-0.726464\pi\)
0.757410 + 0.652940i \(0.226464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.57321 8.57321i 0.315798 0.315798i
\(738\) 0 0
\(739\) 51.9615i 1.91144i −0.294285 0.955718i \(-0.595081\pi\)
0.294285 0.955718i \(-0.404919\pi\)
\(740\) 0 0
\(741\) 20.7846i 0.763542i
\(742\) 0 0
\(743\) 24.4949 24.4949i 0.898631 0.898631i −0.0966845 0.995315i \(-0.530824\pi\)
0.995315 + 0.0966845i \(0.0308238\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 21.2132 21.2132i 0.776151 0.776151i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.00000i 0.145962i 0.997333 + 0.0729810i \(0.0232513\pi\)
−0.997333 + 0.0729810i \(0.976749\pi\)
\(752\) 0 0
\(753\) 20.8207 + 20.8207i 0.758747 + 0.758747i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 28.2843 + 28.2843i 1.02801 + 1.02801i 0.999596 + 0.0284131i \(0.00904537\pi\)
0.0284131 + 0.999596i \(0.490955\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 29.3939 + 29.3939i 1.06135 + 1.06135i
\(768\) 0 0
\(769\) 41.0000i 1.47850i 0.673432 + 0.739249i \(0.264819\pi\)
−0.673432 + 0.739249i \(0.735181\pi\)
\(770\) 0 0
\(771\) −27.7128 −0.998053
\(772\) 0 0
\(773\) −25.4558 + 25.4558i −0.915583 + 0.915583i −0.996704 0.0811212i \(-0.974150\pi\)
0.0811212 + 0.996704i \(0.474150\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 15.5885i 0.558514i
\(780\) 0 0
\(781\) 20.7846i 0.743732i
\(782\) 0 0
\(783\) 24.4949 24.4949i 0.875376 0.875376i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5.65685 5.65685i 0.201645 0.201645i −0.599059 0.800705i \(-0.704459\pi\)
0.800705 + 0.599059i \(0.204459\pi\)
\(788\) 0 0
\(789\) 20.7846 0.739952
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −25.4558 25.4558i −0.901692 0.901692i 0.0938903 0.995583i \(-0.470070\pi\)
−0.995583 + 0.0938903i \(0.970070\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 10.6066 + 10.6066i 0.374299 + 0.374299i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −19.5959 19.5959i −0.689809 0.689809i
\(808\) 0 0
\(809\) 42.0000i 1.47664i −0.674450 0.738321i \(-0.735619\pi\)
0.674450 0.738321i \(-0.264381\pi\)
\(810\) 0 0
\(811\) 38.1051 1.33805 0.669026 0.743239i \(-0.266712\pi\)
0.669026 + 0.743239i \(0.266712\pi\)
\(812\) 0 0
\(813\) −19.7990 + 19.7990i −0.694381 + 0.694381i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 29.3939 29.3939i 1.02836 1.02836i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 20.7846i 0.725388i −0.931908 0.362694i \(-0.881857\pi\)
0.931908 0.362694i \(-0.118143\pi\)
\(822\) 0 0
\(823\) −9.79796 + 9.79796i −0.341535 + 0.341535i −0.856944 0.515409i \(-0.827640\pi\)
0.515409 + 0.856944i \(0.327640\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.12132 2.12132i 0.0737655 0.0737655i −0.669261 0.743027i \(-0.733389\pi\)
0.743027 + 0.669261i \(0.233389\pi\)
\(828\) 0 0
\(829\) 34.6410 1.20313 0.601566 0.798823i \(-0.294544\pi\)
0.601566 + 0.798823i \(0.294544\pi\)
\(830\) 0 0
\(831\) 4.00000i 0.138758i
\(832\) 0 0
\(833\) −8.57321 8.57321i −0.297044 0.297044i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 14.1421 + 14.1421i 0.488824 + 0.488824i
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) 0 0
\(843\) 4.24264 + 4.24264i 0.146124 + 0.146124i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 29.0000i 0.995277i
\(850\) 0 0
\(851\) 55.4256 1.89997
\(852\) 0 0
\(853\) −31.1127 + 31.1127i −1.06528 + 1.06528i −0.0675635 + 0.997715i \(0.521523\pi\)
−0.997715 + 0.0675635i \(0.978477\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −25.7196 + 25.7196i −0.878566 + 0.878566i −0.993386 0.114820i \(-0.963371\pi\)
0.114820 + 0.993386i \(0.463371\pi\)
\(858\) 0 0
\(859\) 43.3013i 1.47742i −0.674023 0.738710i \(-0.735435\pi\)
0.674023 0.738710i \(-0.264565\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.89898 4.89898i 0.166763 0.166763i −0.618792 0.785555i \(-0.712378\pi\)
0.785555 + 0.618792i \(0.212378\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −9.89949 + 9.89949i −0.336204 + 0.336204i
\(868\) 0 0
\(869\) −13.8564 −0.470046
\(870\) 0 0
\(871\) 28.0000i 0.948744i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 5.65685 + 5.65685i 0.191018 + 0.191018i 0.796136 0.605118i \(-0.206874\pi\)
−0.605118 + 0.796136i \(0.706874\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 17.6777 + 17.6777i 0.594901 + 0.594901i 0.938951 0.344050i \(-0.111799\pi\)
−0.344050 + 0.938951i \(0.611799\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −19.5959 19.5959i −0.657967 0.657967i 0.296932 0.954899i \(-0.404037\pi\)
−0.954899 + 0.296932i \(0.904037\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.73205 −0.0580259
\(892\) 0 0
\(893\) −25.4558 + 25.4558i −0.851847 + 0.851847i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −19.5959 + 19.5959i −0.654289 + 0.654289i
\(898\) 0 0
\(899\) 27.7128i 0.924274i
\(900\) 0 0
\(901\) 20.7846i 0.692436i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −5.65685 + 5.65685i −0.187833 + 0.187833i −0.794759 0.606926i \(-0.792403\pi\)
0.606926 + 0.794759i \(0.292403\pi\)
\(908\) 0 0
\(909\) 27.7128 0.919176
\(910\) 0 0
\(911\) 24.0000i 0.795155i −0.917568 0.397578i \(-0.869851\pi\)
0.917568 0.397578i \(-0.130149\pi\)
\(912\) 0 0
\(913\) 18.3712 + 18.3712i 0.607997 + 0.607997i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 0 0
\(923\) 33.9411 + 33.9411i 1.11719 + 1.11719i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −9.79796 9.79796i −0.321807 0.321807i
\(928\) 0 0
\(929\) 18.0000i 0.590561i 0.955411 + 0.295280i \(0.0954131\pi\)
−0.955411 + 0.295280i \(0.904587\pi\)
\(930\) 0 0
\(931\) −36.3731 −1.19208
\(932\) 0 0
\(933\) 16.9706 16.9706i 0.555591 0.555591i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 25.7196 25.7196i 0.840224 0.840224i −0.148664 0.988888i \(-0.547497\pi\)
0.988888 + 0.148664i \(0.0474972\pi\)
\(938\) 0 0
\(939\) 27.7128i 0.904373i
\(940\) 0 0
\(941\) 34.6410i 1.12926i 0.825342 + 0.564632i \(0.190982\pi\)
−0.825342 + 0.564632i \(0.809018\pi\)
\(942\) 0 0
\(943\) 14.6969 14.6969i 0.478598 0.478598i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16.9706 16.9706i 0.551469 0.551469i −0.375396 0.926865i \(-0.622493\pi\)
0.926865 + 0.375396i \(0.122493\pi\)
\(948\) 0 0
\(949\) −34.6410 −1.12449
\(950\) 0 0
\(951\) 24.0000i 0.778253i
\(952\) 0 0
\(953\) −13.4722 13.4722i −0.436407 0.436407i 0.454394 0.890801i \(-0.349856\pi\)
−0.890801 + 0.454394i \(0.849856\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8.48528 + 8.48528i 0.274290 + 0.274290i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) 4.24264 + 4.24264i 0.136717 + 0.136717i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −9.79796 9.79796i −0.315081 0.315081i 0.531793 0.846874i \(-0.321518\pi\)
−0.846874 + 0.531793i \(0.821518\pi\)
\(968\) 0 0
\(969\) 9.00000i 0.289122i
\(970\) 0 0
\(971\) −29.4449 −0.944931 −0.472465 0.881349i \(-0.656636\pi\)
−0.472465 + 0.881349i \(0.656636\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.22474 1.22474i 0.0391831 0.0391831i −0.687244 0.726427i \(-0.741180\pi\)
0.726427 + 0.687244i \(0.241180\pi\)
\(978\) 0 0
\(979\) 5.19615i 0.166070i
\(980\) 0 0
\(981\) 27.7128i 0.884802i
\(982\) 0 0
\(983\) 9.79796 9.79796i 0.312506 0.312506i −0.533373 0.845880i \(-0.679076\pi\)
0.845880 + 0.533373i \(0.179076\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 55.4256 1.76243
\(990\) 0 0
\(991\) 40.0000i 1.27064i −0.772248 0.635321i \(-0.780868\pi\)
0.772248 0.635321i \(-0.219132\pi\)
\(992\) 0 0
\(993\) −20.8207 20.8207i −0.660724 0.660724i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 11.3137 + 11.3137i 0.358309 + 0.358309i 0.863189 0.504881i \(-0.168463\pi\)
−0.504881 + 0.863189i \(0.668463\pi\)
\(998\) 0 0
\(999\) 40.0000 1.26554
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 1600.2.o.h.543.4 yes 8
4.3 odd 2 1600.2.o.i.543.1 yes 8
5.2 odd 4 1600.2.o.i.607.4 yes 8
5.3 odd 4 1600.2.o.i.607.2 yes 8
5.4 even 2 inner 1600.2.o.h.543.2 yes 8
8.3 odd 2 1600.2.o.i.543.4 yes 8
8.5 even 2 inner 1600.2.o.h.543.1 8
20.3 even 4 inner 1600.2.o.h.607.3 yes 8
20.7 even 4 inner 1600.2.o.h.607.1 yes 8
20.19 odd 2 1600.2.o.i.543.3 yes 8
40.3 even 4 inner 1600.2.o.h.607.2 yes 8
40.13 odd 4 1600.2.o.i.607.3 yes 8
40.19 odd 2 1600.2.o.i.543.2 yes 8
40.27 even 4 inner 1600.2.o.h.607.4 yes 8
40.29 even 2 inner 1600.2.o.h.543.3 yes 8
40.37 odd 4 1600.2.o.i.607.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.o.h.543.1 8 8.5 even 2 inner
1600.2.o.h.543.2 yes 8 5.4 even 2 inner
1600.2.o.h.543.3 yes 8 40.29 even 2 inner
1600.2.o.h.543.4 yes 8 1.1 even 1 trivial
1600.2.o.h.607.1 yes 8 20.7 even 4 inner
1600.2.o.h.607.2 yes 8 40.3 even 4 inner
1600.2.o.h.607.3 yes 8 20.3 even 4 inner
1600.2.o.h.607.4 yes 8 40.27 even 4 inner
1600.2.o.i.543.1 yes 8 4.3 odd 2
1600.2.o.i.543.2 yes 8 40.19 odd 2
1600.2.o.i.543.3 yes 8 20.19 odd 2
1600.2.o.i.543.4 yes 8 8.3 odd 2
1600.2.o.i.607.1 yes 8 40.37 odd 4
1600.2.o.i.607.2 yes 8 5.3 odd 4
1600.2.o.i.607.3 yes 8 40.13 odd 4
1600.2.o.i.607.4 yes 8 5.2 odd 4