Properties

Label 3024.2.r.b.2017.1
Level $3024$
Weight $2$
Character 3024.2017
Analytic conductor $24.147$
Analytic rank $0$
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] = [3024,2,Mod(1009,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.r (of order \(3\), degree \(2\), not 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: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2017.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3024.2017
Dual form 3024.2.r.b.1009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} +(3.00000 - 5.19615i) q^{11} +(-3.00000 - 5.19615i) q^{13} +2.00000 q^{17} -7.00000 q^{19} +(0.500000 + 0.866025i) q^{23} +(2.00000 - 3.46410i) q^{25} +(1.00000 - 1.73205i) q^{29} +(5.00000 + 8.66025i) q^{31} -1.00000 q^{35} -6.00000 q^{37} +(-4.00000 - 6.92820i) q^{41} +(-5.00000 + 8.66025i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-0.500000 - 0.866025i) q^{49} -2.00000 q^{53} -6.00000 q^{55} +(-3.50000 + 6.06218i) q^{61} +(-3.00000 + 5.19615i) q^{65} +(-6.00000 - 10.3923i) q^{67} +15.0000 q^{71} -2.00000 q^{73} +(-3.00000 - 5.19615i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-6.00000 + 10.3923i) q^{83} +(-1.00000 - 1.73205i) q^{85} -4.00000 q^{89} -6.00000 q^{91} +(3.50000 + 6.06218i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + q^{7} + 6 q^{11} - 6 q^{13} + 4 q^{17} - 14 q^{19} + q^{23} + 4 q^{25} + 2 q^{29} + 10 q^{31} - 2 q^{35} - 12 q^{37} - 8 q^{41} - 10 q^{43} - 8 q^{47} - q^{49} - 4 q^{53} - 12 q^{55} - 7 q^{61} - 6 q^{65} - 12 q^{67} + 30 q^{71} - 4 q^{73} - 6 q^{77} + q^{79} - 12 q^{83} - 2 q^{85} - 8 q^{89} - 12 q^{91} + 7 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i 0.732294 0.680989i \(-0.238450\pi\)
−0.955901 + 0.293691i \(0.905116\pi\)
\(6\) 0 0
\(7\) 0.500000 0.866025i 0.188982 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 5.19615i 0.904534 1.56670i 0.0829925 0.996550i \(-0.473552\pi\)
0.821541 0.570149i \(-0.193114\pi\)
\(12\) 0 0
\(13\) −3.00000 5.19615i −0.832050 1.44115i −0.896410 0.443227i \(-0.853834\pi\)
0.0643593 0.997927i \(-0.479500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.500000 + 0.866025i 0.104257 + 0.180579i 0.913434 0.406986i \(-0.133420\pi\)
−0.809177 + 0.587565i \(0.800087\pi\)
\(24\) 0 0
\(25\) 2.00000 3.46410i 0.400000 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 1.73205i 0.185695 0.321634i −0.758115 0.652121i \(-0.773880\pi\)
0.943811 + 0.330487i \(0.107213\pi\)
\(30\) 0 0
\(31\) 5.00000 + 8.66025i 0.898027 + 1.55543i 0.830014 + 0.557743i \(0.188333\pi\)
0.0680129 + 0.997684i \(0.478334\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 6.92820i −0.624695 1.08200i −0.988600 0.150567i \(-0.951890\pi\)
0.363905 0.931436i \(-0.381443\pi\)
\(42\) 0 0
\(43\) −5.00000 + 8.66025i −0.762493 + 1.32068i 0.179069 + 0.983836i \(0.442691\pi\)
−0.941562 + 0.336840i \(0.890642\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 + 6.92820i −0.583460 + 1.01058i 0.411606 + 0.911362i \(0.364968\pi\)
−0.995066 + 0.0992202i \(0.968365\pi\)
\(48\) 0 0
\(49\) −0.500000 0.866025i −0.0714286 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −3.50000 + 6.06218i −0.448129 + 0.776182i −0.998264 0.0588933i \(-0.981243\pi\)
0.550135 + 0.835076i \(0.314576\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) 0 0
\(67\) −6.00000 10.3923i −0.733017 1.26962i −0.955588 0.294706i \(-0.904778\pi\)
0.222571 0.974916i \(-0.428555\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 15.0000 1.78017 0.890086 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.00000 5.19615i −0.341882 0.592157i
\(78\) 0 0
\(79\) 0.500000 0.866025i 0.0562544 0.0974355i −0.836527 0.547926i \(-0.815418\pi\)
0.892781 + 0.450490i \(0.148751\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.00000 + 10.3923i −0.658586 + 1.14070i 0.322396 + 0.946605i \(0.395512\pi\)
−0.980982 + 0.194099i \(0.937822\pi\)
\(84\) 0 0
\(85\) −1.00000 1.73205i −0.108465 0.187867i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.50000 + 6.06218i 0.359092 + 0.621966i
\(96\) 0 0
\(97\) 1.00000 1.73205i 0.101535 0.175863i −0.810782 0.585348i \(-0.800958\pi\)
0.912317 + 0.409484i \(0.134291\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.50000 2.59808i 0.149256 0.258518i −0.781697 0.623658i \(-0.785646\pi\)
0.930953 + 0.365140i \(0.118979\pi\)
\(102\) 0 0
\(103\) 3.00000 + 5.19615i 0.295599 + 0.511992i 0.975124 0.221660i \(-0.0711475\pi\)
−0.679525 + 0.733652i \(0.737814\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.50000 7.79423i −0.423324 0.733219i 0.572938 0.819599i \(-0.305804\pi\)
−0.996262 + 0.0863794i \(0.972470\pi\)
\(114\) 0 0
\(115\) 0.500000 0.866025i 0.0466252 0.0807573i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.00000 1.73205i 0.0916698 0.158777i
\(120\) 0 0
\(121\) −12.5000 21.6506i −1.13636 1.96824i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.50000 14.7224i −0.742648 1.28630i −0.951285 0.308312i \(-0.900236\pi\)
0.208637 0.977993i \(-0.433097\pi\)
\(132\) 0 0
\(133\) −3.50000 + 6.06218i −0.303488 + 0.525657i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.00000 12.1244i 0.598050 1.03585i −0.395058 0.918656i \(-0.629276\pi\)
0.993109 0.117198i \(-0.0373911\pi\)
\(138\) 0 0
\(139\) 1.50000 + 2.59808i 0.127228 + 0.220366i 0.922602 0.385754i \(-0.126059\pi\)
−0.795373 + 0.606120i \(0.792725\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −36.0000 −3.01047
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.00000 12.1244i −0.573462 0.993266i −0.996207 0.0870170i \(-0.972267\pi\)
0.422744 0.906249i \(-0.361067\pi\)
\(150\) 0 0
\(151\) −2.50000 + 4.33013i −0.203447 + 0.352381i −0.949637 0.313353i \(-0.898548\pi\)
0.746190 + 0.665733i \(0.231881\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.00000 8.66025i 0.401610 0.695608i
\(156\) 0 0
\(157\) 7.50000 + 12.9904i 0.598565 + 1.03675i 0.993033 + 0.117836i \(0.0375956\pi\)
−0.394468 + 0.918910i \(0.629071\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.00000 10.3923i −0.464294 0.804181i 0.534875 0.844931i \(-0.320359\pi\)
−0.999169 + 0.0407502i \(0.987025\pi\)
\(168\) 0 0
\(169\) −11.5000 + 19.9186i −0.884615 + 1.53220i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.00000 12.1244i 0.532200 0.921798i −0.467093 0.884208i \(-0.654699\pi\)
0.999293 0.0375896i \(-0.0119679\pi\)
\(174\) 0 0
\(175\) −2.00000 3.46410i −0.151186 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 + 5.19615i 0.220564 + 0.382029i
\(186\) 0 0
\(187\) 6.00000 10.3923i 0.438763 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.50000 4.33013i 0.180894 0.313317i −0.761291 0.648410i \(-0.775434\pi\)
0.942185 + 0.335093i \(0.108768\pi\)
\(192\) 0 0
\(193\) −4.50000 7.79423i −0.323917 0.561041i 0.657376 0.753563i \(-0.271667\pi\)
−0.981293 + 0.192522i \(0.938333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.00000 1.73205i −0.0701862 0.121566i
\(204\) 0 0
\(205\) −4.00000 + 6.92820i −0.279372 + 0.483887i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −21.0000 + 36.3731i −1.45260 + 2.51598i
\(210\) 0 0
\(211\) −2.00000 3.46410i −0.137686 0.238479i 0.788935 0.614477i \(-0.210633\pi\)
−0.926620 + 0.375999i \(0.877300\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 10.3923i −0.403604 0.699062i
\(222\) 0 0
\(223\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.500000 0.866025i 0.0331862 0.0574801i −0.848955 0.528465i \(-0.822768\pi\)
0.882141 + 0.470985i \(0.156101\pi\)
\(228\) 0 0
\(229\) −2.50000 4.33013i −0.165205 0.286143i 0.771523 0.636201i \(-0.219495\pi\)
−0.936728 + 0.350058i \(0.886162\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.00000 0.458585 0.229293 0.973358i \(-0.426359\pi\)
0.229293 + 0.973358i \(0.426359\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.50000 4.33013i −0.161712 0.280093i 0.773771 0.633465i \(-0.218368\pi\)
−0.935483 + 0.353373i \(0.885035\pi\)
\(240\) 0 0
\(241\) −2.00000 + 3.46410i −0.128831 + 0.223142i −0.923224 0.384262i \(-0.874456\pi\)
0.794393 + 0.607404i \(0.207789\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.500000 + 0.866025i −0.0319438 + 0.0553283i
\(246\) 0 0
\(247\) 21.0000 + 36.3731i 1.33620 + 2.31436i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.0000 0.820553 0.410276 0.911961i \(-0.365432\pi\)
0.410276 + 0.911961i \(0.365432\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.00000 + 15.5885i 0.561405 + 0.972381i 0.997374 + 0.0724199i \(0.0230722\pi\)
−0.435970 + 0.899961i \(0.643595\pi\)
\(258\) 0 0
\(259\) −3.00000 + 5.19615i −0.186411 + 0.322873i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.50000 + 12.9904i −0.462470 + 0.801021i −0.999083 0.0428069i \(-0.986370\pi\)
0.536614 + 0.843828i \(0.319703\pi\)
\(264\) 0 0
\(265\) 1.00000 + 1.73205i 0.0614295 + 0.106399i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12.0000 20.7846i −0.723627 1.25336i
\(276\) 0 0
\(277\) −2.00000 + 3.46410i −0.120168 + 0.208138i −0.919834 0.392308i \(-0.871677\pi\)
0.799666 + 0.600446i \(0.205010\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.50000 + 2.59808i −0.0894825 + 0.154988i −0.907293 0.420500i \(-0.861855\pi\)
0.817810 + 0.575488i \(0.195188\pi\)
\(282\) 0 0
\(283\) −6.50000 11.2583i −0.386385 0.669238i 0.605575 0.795788i \(-0.292943\pi\)
−0.991960 + 0.126550i \(0.959610\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.50000 7.79423i −0.262893 0.455344i 0.704117 0.710084i \(-0.251343\pi\)
−0.967009 + 0.254741i \(0.918010\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00000 5.19615i 0.173494 0.300501i
\(300\) 0 0
\(301\) 5.00000 + 8.66025i 0.288195 + 0.499169i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.00000 0.400819
\(306\) 0 0
\(307\) 23.0000 1.31268 0.656340 0.754466i \(-0.272104\pi\)
0.656340 + 0.754466i \(0.272104\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 20.7846i −0.680458 1.17859i −0.974841 0.222900i \(-0.928448\pi\)
0.294384 0.955687i \(-0.404886\pi\)
\(312\) 0 0
\(313\) −1.00000 + 1.73205i −0.0565233 + 0.0979013i −0.892903 0.450250i \(-0.851335\pi\)
0.836379 + 0.548151i \(0.184668\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.00000 + 15.5885i −0.505490 + 0.875535i 0.494489 + 0.869184i \(0.335355\pi\)
−0.999980 + 0.00635137i \(0.997978\pi\)
\(318\) 0 0
\(319\) −6.00000 10.3923i −0.335936 0.581857i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −14.0000 −0.778981
\(324\) 0 0
\(325\) −24.0000 −1.33128
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.00000 + 6.92820i 0.220527 + 0.381964i
\(330\) 0 0
\(331\) 7.00000 12.1244i 0.384755 0.666415i −0.606980 0.794717i \(-0.707619\pi\)
0.991735 + 0.128302i \(0.0409527\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.00000 + 10.3923i −0.327815 + 0.567792i
\(336\) 0 0
\(337\) −1.00000 1.73205i −0.0544735 0.0943508i 0.837503 0.546433i \(-0.184015\pi\)
−0.891976 + 0.452082i \(0.850681\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 60.0000 3.24918
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 20.7846i −0.644194 1.11578i −0.984487 0.175457i \(-0.943860\pi\)
0.340293 0.940319i \(-0.389474\pi\)
\(348\) 0 0
\(349\) 5.00000 8.66025i 0.267644 0.463573i −0.700609 0.713545i \(-0.747088\pi\)
0.968253 + 0.249973i \(0.0804216\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.00000 5.19615i 0.159674 0.276563i −0.775077 0.631867i \(-0.782289\pi\)
0.934751 + 0.355303i \(0.115622\pi\)
\(354\) 0 0
\(355\) −7.50000 12.9904i −0.398059 0.689458i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 33.0000 1.74167 0.870837 0.491572i \(-0.163578\pi\)
0.870837 + 0.491572i \(0.163578\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.00000 + 1.73205i 0.0523424 + 0.0906597i
\(366\) 0 0
\(367\) 16.0000 27.7128i 0.835193 1.44660i −0.0586798 0.998277i \(-0.518689\pi\)
0.893873 0.448320i \(-0.147978\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.00000 + 1.73205i −0.0519174 + 0.0899236i
\(372\) 0 0
\(373\) −1.00000 1.73205i −0.0517780 0.0896822i 0.838975 0.544170i \(-0.183156\pi\)
−0.890753 + 0.454488i \(0.849822\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.00000 + 5.19615i 0.153293 + 0.265511i 0.932436 0.361335i \(-0.117679\pi\)
−0.779143 + 0.626846i \(0.784346\pi\)
\(384\) 0 0
\(385\) −3.00000 + 5.19615i −0.152894 + 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.0000 + 17.3205i −0.507020 + 0.878185i 0.492947 + 0.870059i \(0.335920\pi\)
−0.999967 + 0.00812520i \(0.997414\pi\)
\(390\) 0 0
\(391\) 1.00000 + 1.73205i 0.0505722 + 0.0875936i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.00000 −0.0503155
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.5000 18.1865i −0.524345 0.908192i −0.999598 0.0283431i \(-0.990977\pi\)
0.475253 0.879849i \(-0.342356\pi\)
\(402\) 0 0
\(403\) 30.0000 51.9615i 1.49441 2.58839i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −18.0000 + 31.1769i −0.892227 + 1.54538i
\(408\) 0 0
\(409\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.50000 + 6.06218i 0.170986 + 0.296157i 0.938765 0.344558i \(-0.111971\pi\)
−0.767779 + 0.640715i \(0.778638\pi\)
\(420\) 0 0
\(421\) −5.00000 + 8.66025i −0.243685 + 0.422075i −0.961761 0.273890i \(-0.911690\pi\)
0.718076 + 0.695965i \(0.245023\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.00000 6.92820i 0.194029 0.336067i
\(426\) 0 0
\(427\) 3.50000 + 6.06218i 0.169377 + 0.293369i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.50000 6.06218i −0.167428 0.289993i
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.0000 29.4449i 0.807694 1.39897i −0.106763 0.994285i \(-0.534048\pi\)
0.914457 0.404683i \(-0.132618\pi\)
\(444\) 0 0
\(445\) 2.00000 + 3.46410i 0.0948091 + 0.164214i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.00000 −0.0471929 −0.0235965 0.999722i \(-0.507512\pi\)
−0.0235965 + 0.999722i \(0.507512\pi\)
\(450\) 0 0
\(451\) −48.0000 −2.26023
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.00000 + 5.19615i 0.140642 + 0.243599i
\(456\) 0 0
\(457\) 5.50000 9.52628i 0.257279 0.445621i −0.708233 0.705979i \(-0.750507\pi\)
0.965512 + 0.260358i \(0.0838407\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.50000 + 2.59808i −0.0698620 + 0.121004i −0.898840 0.438276i \(-0.855589\pi\)
0.828978 + 0.559281i \(0.188923\pi\)
\(462\) 0 0
\(463\) 11.5000 + 19.9186i 0.534450 + 0.925695i 0.999190 + 0.0402476i \(0.0128147\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 30.0000 + 51.9615i 1.37940 + 2.38919i
\(474\) 0 0
\(475\) −14.0000 + 24.2487i −0.642364 + 1.11261i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −17.0000 + 29.4449i −0.776750 + 1.34537i 0.157056 + 0.987590i \(0.449800\pi\)
−0.933806 + 0.357780i \(0.883534\pi\)
\(480\) 0 0
\(481\) 18.0000 + 31.1769i 0.820729 + 1.42154i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −1.00000 −0.0453143 −0.0226572 0.999743i \(-0.507213\pi\)
−0.0226572 + 0.999743i \(0.507213\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.00000 + 5.19615i 0.135388 + 0.234499i 0.925746 0.378147i \(-0.123439\pi\)
−0.790358 + 0.612646i \(0.790105\pi\)
\(492\) 0 0
\(493\) 2.00000 3.46410i 0.0900755 0.156015i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.50000 12.9904i 0.336421 0.582698i
\(498\) 0 0
\(499\) 8.00000 + 13.8564i 0.358129 + 0.620298i 0.987648 0.156687i \(-0.0500814\pi\)
−0.629519 + 0.776985i \(0.716748\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −32.0000 −1.42681 −0.713405 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) 0 0
\(505\) −3.00000 −0.133498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −15.0000 25.9808i −0.664863 1.15158i −0.979322 0.202306i \(-0.935156\pi\)
0.314459 0.949271i \(-0.398177\pi\)
\(510\) 0 0
\(511\) −1.00000 + 1.73205i −0.0442374 + 0.0766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.00000 5.19615i 0.132196 0.228970i
\(516\) 0 0
\(517\) 24.0000 + 41.5692i 1.05552 + 1.82821i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) −29.0000 −1.26808 −0.634041 0.773300i \(-0.718605\pi\)
−0.634041 + 0.773300i \(0.718605\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.0000 + 17.3205i 0.435607 + 0.754493i
\(528\) 0 0
\(529\) 11.0000 19.0526i 0.478261 0.828372i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −24.0000 + 41.5692i −1.03956 + 1.80056i
\(534\) 0 0
\(535\) −6.00000 10.3923i −0.259403 0.449299i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.00000 8.66025i −0.214176 0.370965i
\(546\) 0 0
\(547\) −4.00000 + 6.92820i −0.171028 + 0.296229i −0.938779 0.344519i \(-0.888042\pi\)
0.767752 + 0.640747i \(0.221375\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7.00000 + 12.1244i −0.298210 + 0.516515i
\(552\) 0 0
\(553\) −0.500000 0.866025i −0.0212622 0.0368271i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 44.0000 1.86434 0.932170 0.362021i \(-0.117913\pi\)
0.932170 + 0.362021i \(0.117913\pi\)
\(558\) 0 0
\(559\) 60.0000 2.53773
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11.5000 + 19.9186i 0.484667 + 0.839468i 0.999845 0.0176152i \(-0.00560738\pi\)
−0.515178 + 0.857083i \(0.672274\pi\)
\(564\) 0 0
\(565\) −4.50000 + 7.79423i −0.189316 + 0.327906i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.0000 25.9808i 0.628833 1.08917i −0.358954 0.933355i \(-0.616866\pi\)
0.987786 0.155815i \(-0.0498003\pi\)
\(570\) 0 0
\(571\) −20.0000 34.6410i −0.836974 1.44968i −0.892413 0.451219i \(-0.850989\pi\)
0.0554391 0.998462i \(-0.482344\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 36.0000 1.49870 0.749350 0.662174i \(-0.230366\pi\)
0.749350 + 0.662174i \(0.230366\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6.00000 + 10.3923i 0.248922 + 0.431145i
\(582\) 0 0
\(583\) −6.00000 + 10.3923i −0.248495 + 0.430405i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.50000 4.33013i 0.103186 0.178723i −0.809810 0.586693i \(-0.800430\pi\)
0.912996 + 0.407969i \(0.133763\pi\)
\(588\) 0 0
\(589\) −35.0000 60.6218i −1.44215 2.49788i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16.0000 + 27.7128i 0.653742 + 1.13231i 0.982208 + 0.187799i \(0.0601353\pi\)
−0.328465 + 0.944516i \(0.606531\pi\)
\(600\) 0 0
\(601\) 7.00000 12.1244i 0.285536 0.494563i −0.687203 0.726465i \(-0.741162\pi\)
0.972739 + 0.231903i \(0.0744951\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −12.5000 + 21.6506i −0.508197 + 0.880223i
\(606\) 0 0
\(607\) −1.00000 1.73205i −0.0405887 0.0703018i 0.845017 0.534739i \(-0.179590\pi\)
−0.885606 + 0.464437i \(0.846257\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.0000 25.9808i −0.603877 1.04595i −0.992228 0.124434i \(-0.960288\pi\)
0.388351 0.921512i \(-0.373045\pi\)
\(618\) 0 0
\(619\) −16.5000 + 28.5788i −0.663191 + 1.14868i 0.316581 + 0.948565i \(0.397465\pi\)
−0.979772 + 0.200115i \(0.935868\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.00000 + 3.46410i −0.0801283 + 0.138786i
\(624\) 0 0
\(625\) −5.50000 9.52628i −0.220000 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −5.00000 −0.199047 −0.0995234 0.995035i \(-0.531732\pi\)
−0.0995234 + 0.995035i \(0.531732\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.50000 + 4.33013i 0.0992095 + 0.171836i
\(636\) 0 0
\(637\) −3.00000 + 5.19615i −0.118864 + 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 21.5000 37.2391i 0.849199 1.47086i −0.0327252 0.999464i \(-0.510419\pi\)
0.881924 0.471391i \(-0.156248\pi\)
\(642\) 0 0
\(643\) 2.00000 + 3.46410i 0.0788723 + 0.136611i 0.902764 0.430137i \(-0.141535\pi\)
−0.823891 + 0.566748i \(0.808201\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.0000 + 38.1051i 0.860927 + 1.49117i 0.871036 + 0.491220i \(0.163449\pi\)
−0.0101092 + 0.999949i \(0.503218\pi\)
\(654\) 0 0
\(655\) −8.50000 + 14.7224i −0.332122 + 0.575253i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 17.0000 29.4449i 0.662226 1.14701i −0.317803 0.948157i \(-0.602945\pi\)
0.980029 0.198852i \(-0.0637214\pi\)
\(660\) 0 0
\(661\) 12.5000 + 21.6506i 0.486194 + 0.842112i 0.999874 0.0158695i \(-0.00505163\pi\)
−0.513680 + 0.857982i \(0.671718\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.00000 0.271448
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 21.0000 + 36.3731i 0.810696 + 1.40417i
\(672\) 0 0
\(673\) −9.50000 + 16.4545i −0.366198 + 0.634274i −0.988968 0.148132i \(-0.952674\pi\)
0.622770 + 0.782405i \(0.286007\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.0000 + 36.3731i −0.807096 + 1.39793i 0.107772 + 0.994176i \(0.465628\pi\)
−0.914867 + 0.403755i \(0.867705\pi\)
\(678\) 0 0
\(679\) −1.00000 1.73205i −0.0383765 0.0664700i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.0000 0.688751 0.344375 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) 0 0
\(685\) −14.0000 −0.534913
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.00000 + 10.3923i 0.228582 + 0.395915i
\(690\) 0 0
\(691\) 20.5000 35.5070i 0.779857 1.35075i −0.152167 0.988355i \(-0.548625\pi\)
0.932024 0.362397i \(-0.118041\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.50000 2.59808i 0.0568982 0.0985506i
\(696\) 0 0
\(697\) −8.00000 13.8564i −0.303022 0.524849i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 42.0000 1.58406
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.50000 2.59808i −0.0564133 0.0977107i
\(708\) 0 0
\(709\) 22.0000 38.1051i 0.826227 1.43107i −0.0747503 0.997202i \(-0.523816\pi\)
0.900978 0.433865i \(-0.142851\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.00000 + 8.66025i −0.187251 + 0.324329i
\(714\) 0 0
\(715\) 18.0000 + 31.1769i 0.673162 + 1.16595i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) 6.00000 0.223452
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.00000 6.92820i −0.148556 0.257307i
\(726\) 0 0
\(727\) −14.0000 + 24.2487i −0.519231 + 0.899335i 0.480519 + 0.876984i \(0.340448\pi\)
−0.999750 + 0.0223506i \(0.992885\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10.0000 + 17.3205i −0.369863 + 0.640622i
\(732\) 0 0
\(733\) −15.5000 26.8468i −0.572506 0.991609i −0.996308 0.0858539i \(-0.972638\pi\)
0.423802 0.905755i \(-0.360695\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −72.0000 −2.65215
\(738\) 0 0
\(739\) 38.0000 1.39785 0.698926 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −22.0000 38.1051i −0.807102 1.39794i −0.914863 0.403764i \(-0.867702\pi\)
0.107761 0.994177i \(-0.465632\pi\)
\(744\) 0 0
\(745\) −7.00000 + 12.1244i −0.256460 + 0.444202i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6.00000 10.3923i 0.219235 0.379727i
\(750\) 0 0
\(751\) 2.50000 + 4.33013i 0.0912263 + 0.158009i 0.908027 0.418911i \(-0.137588\pi\)
−0.816801 + 0.576919i \(0.804255\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.00000 + 5.19615i 0.108750 + 0.188360i 0.915264 0.402854i \(-0.131982\pi\)
−0.806514 + 0.591215i \(0.798649\pi\)
\(762\) 0 0
\(763\) 5.00000 8.66025i 0.181012 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −5.00000 8.66025i −0.180305 0.312297i 0.761680 0.647954i \(-0.224375\pi\)
−0.941984 + 0.335657i \(0.891042\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −31.0000 −1.11499 −0.557496 0.830179i \(-0.688238\pi\)
−0.557496 + 0.830179i \(0.688238\pi\)
\(774\) 0 0
\(775\) 40.0000 1.43684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 28.0000 + 48.4974i 1.00320 + 1.73760i
\(780\) 0 0
\(781\) 45.0000 77.9423i 1.61023 2.78899i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.50000 12.9904i 0.267686 0.463647i
\(786\) 0 0
\(787\) −10.0000 17.3205i −0.356462 0.617409i 0.630905 0.775860i \(-0.282684\pi\)
−0.987367 + 0.158450i \(0.949350\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 0 0
\(793\) 42.0000 1.49146
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.50000 + 7.79423i 0.159398 + 0.276086i 0.934652 0.355564i \(-0.115711\pi\)
−0.775254 + 0.631650i \(0.782378\pi\)
\(798\) 0 0
\(799\) −8.00000 + 13.8564i −0.283020 + 0.490204i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.00000 + 10.3923i −0.211735 + 0.366736i
\(804\) 0 0
\(805\) −0.500000 0.866025i −0.0176227 0.0305234i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.00000 + 8.66025i 0.175142 + 0.303355i
\(816\) 0 0
\(817\) 35.0000 60.6218i 1.22449 2.12089i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(822\) 0 0
\(823\) −8.00000 13.8564i −0.278862 0.483004i 0.692240 0.721668i \(-0.256624\pi\)
−0.971102 + 0.238664i \(0.923291\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.00000 1.73205i −0.0346479 0.0600120i
\(834\) 0 0
\(835\) −6.00000 + 10.3923i −0.207639 + 0.359641i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 6.00000 10.3923i 0.207143 0.358782i −0.743670 0.668546i \(-0.766917\pi\)
0.950813 + 0.309764i \(0.100250\pi\)
\(840\) 0 0
\(841\) 12.5000 + 21.6506i 0.431034 + 0.746574i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) −25.0000 −0.859010
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.00000 5.19615i −0.102839 0.178122i
\(852\) 0 0
\(853\) 11.5000 19.9186i 0.393753 0.681999i −0.599189 0.800608i \(-0.704510\pi\)
0.992941 + 0.118609i \(0.0378434\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.00000 + 5.19615i −0.102478 + 0.177497i −0.912705 0.408619i \(-0.866010\pi\)
0.810227 + 0.586116i \(0.199344\pi\)
\(858\) 0 0
\(859\) −10.0000 17.3205i −0.341196 0.590968i 0.643459 0.765480i \(-0.277499\pi\)
−0.984655 + 0.174512i \(0.944165\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −43.0000 −1.46374 −0.731869 0.681446i \(-0.761351\pi\)
−0.731869 + 0.681446i \(0.761351\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −3.00000 5.19615i −0.101768 0.176267i
\(870\) 0 0
\(871\) −36.0000 + 62.3538i −1.21981 + 2.11278i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.50000 + 7.79423i −0.152128 + 0.263493i
\(876\) 0 0
\(877\) −7.00000 12.1244i −0.236373 0.409410i 0.723298 0.690536i \(-0.242625\pi\)
−0.959671 + 0.281126i \(0.909292\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) 26.0000 0.874970 0.437485 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(888\) 0 0
\(889\) −2.50000 + 4.33013i −0.0838473 + 0.145228i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 28.0000 48.4974i 0.936984 1.62290i
\(894\) 0 0
\(895\) 5.00000 + 8.66025i 0.167132 + 0.289480i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.50000 4.33013i −0.0831028 0.143938i
\(906\) 0 0
\(907\) −14.0000 + 24.2487i −0.464862 + 0.805165i −0.999195 0.0401089i \(-0.987230\pi\)
0.534333 + 0.845274i \(0.320563\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −9.50000 + 16.4545i −0.314749 + 0.545161i −0.979384 0.202007i \(-0.935254\pi\)
0.664635 + 0.747168i \(0.268587\pi\)
\(912\) 0 0
\(913\) 36.0000 + 62.3538i 1.19143 + 2.06361i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −17.0000 −0.561389
\(918\) 0 0
\(919\) 17.0000 0.560778 0.280389 0.959886i \(-0.409536\pi\)
0.280389 + 0.959886i \(0.409536\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −45.0000 77.9423i −1.48119 2.56550i
\(924\) 0 0
\(925\) −12.0000 + 20.7846i −0.394558 + 0.683394i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.00000 3.46410i 0.0656179 0.113653i −0.831350 0.555749i \(-0.812431\pi\)
0.896968 + 0.442096i \(0.145765\pi\)
\(930\) 0 0
\(931\) 3.50000 + 6.06218i 0.114708 + 0.198680i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −16.5000 28.5788i −0.537885 0.931644i −0.999018 0.0443125i \(-0.985890\pi\)
0.461133 0.887331i \(-0.347443\pi\)
\(942\) 0 0
\(943\) 4.00000 6.92820i 0.130258 0.225613i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14.0000 + 24.2487i −0.454939 + 0.787977i −0.998685 0.0512727i \(-0.983672\pi\)
0.543746 + 0.839250i \(0.317006\pi\)
\(948\) 0 0
\(949\) 6.00000 + 10.3923i 0.194768 + 0.337348i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 0 0
\(955\) −5.00000 −0.161796
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −7.00000 12.1244i −0.226042 0.391516i
\(960\) 0 0
\(961\) −34.5000 + 59.7558i −1.11290 + 1.92760i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −4.50000 + 7.79423i −0.144860 + 0.250905i
\(966\) 0 0
\(967\) −1.50000 2.59808i −0.0482367 0.0835485i 0.840899 0.541192i \(-0.182027\pi\)
−0.889136 + 0.457644i \(0.848694\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 55.0000 1.76503 0.882517 0.470281i \(-0.155847\pi\)
0.882517 + 0.470281i \(0.155847\pi\)
\(972\) 0 0
\(973\) 3.00000 0.0961756
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.00000 + 5.19615i 0.0959785 + 0.166240i 0.910017 0.414572i \(-0.136069\pi\)
−0.814038 + 0.580812i \(0.802735\pi\)
\(978\) 0 0
\(979\) −12.0000 + 20.7846i −0.383522 + 0.664279i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −9.00000 + 15.5885i −0.287055 + 0.497195i −0.973106 0.230360i \(-0.926010\pi\)
0.686050 + 0.727554i \(0.259343\pi\)
\(984\) 0 0
\(985\) 3.00000 + 5.19615i 0.0955879 + 0.165563i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −10.0000 −0.317982
\(990\) 0 0
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.00000 + 1.73205i 0.0317021 + 0.0549097i
\(996\) 0 0
\(997\) −9.50000 + 16.4545i −0.300868 + 0.521119i −0.976333 0.216274i \(-0.930610\pi\)
0.675465 + 0.737392i \(0.263943\pi\)
\(998\) 0 0
\(999\) 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 3024.2.r.b.2017.1 2
3.2 odd 2 1008.2.r.c.673.1 2
4.3 odd 2 1512.2.r.a.505.1 2
9.2 odd 6 9072.2.a.i.1.1 1
9.4 even 3 inner 3024.2.r.b.1009.1 2
9.5 odd 6 1008.2.r.c.337.1 2
9.7 even 3 9072.2.a.n.1.1 1
12.11 even 2 504.2.r.a.169.1 2
36.7 odd 6 4536.2.a.g.1.1 1
36.11 even 6 4536.2.a.d.1.1 1
36.23 even 6 504.2.r.a.337.1 yes 2
36.31 odd 6 1512.2.r.a.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.a.169.1 2 12.11 even 2
504.2.r.a.337.1 yes 2 36.23 even 6
1008.2.r.c.337.1 2 9.5 odd 6
1008.2.r.c.673.1 2 3.2 odd 2
1512.2.r.a.505.1 2 4.3 odd 2
1512.2.r.a.1009.1 2 36.31 odd 6
3024.2.r.b.1009.1 2 9.4 even 3 inner
3024.2.r.b.2017.1 2 1.1 even 1 trivial
4536.2.a.d.1.1 1 36.11 even 6
4536.2.a.g.1.1 1 36.7 odd 6
9072.2.a.i.1.1 1 9.2 odd 6
9072.2.a.n.1.1 1 9.7 even 3