Properties

Label 21.3.h.a.11.1
Level $21$
Weight $3$
Character 21.11
Analytic conductor $0.572$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,3,Mod(2,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), 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: \(0.572208555157\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 11.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.11
Dual form 21.3.h.a.2.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 - 2.59808i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-6.50000 + 2.59808i) q^{7} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\) \(q+(1.50000 - 2.59808i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-6.50000 + 2.59808i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(6.00000 + 10.3923i) q^{12} +23.0000 q^{13} +(-8.00000 - 13.8564i) q^{16} +(-5.50000 - 9.52628i) q^{19} +(-3.00000 + 20.7846i) q^{21} +(-12.5000 + 21.6506i) q^{25} -27.0000 q^{27} +(4.00000 - 27.7128i) q^{28} +(6.50000 - 11.2583i) q^{31} +36.0000 q^{36} +(36.5000 + 63.2199i) q^{37} +(34.5000 - 59.7558i) q^{39} -61.0000 q^{43} -48.0000 q^{48} +(35.5000 - 33.7750i) q^{49} +(-46.0000 + 79.6743i) q^{52} -33.0000 q^{57} +(-37.0000 - 64.0859i) q^{61} +(49.5000 + 38.9711i) q^{63} +64.0000 q^{64} +(6.50000 - 11.2583i) q^{67} +(48.5000 - 84.0045i) q^{73} +(37.5000 + 64.9519i) q^{75} +44.0000 q^{76} +(-5.50000 - 9.52628i) q^{79} +(-40.5000 + 70.1481i) q^{81} +(-66.0000 - 51.9615i) q^{84} +(-149.500 + 59.7558i) q^{91} +(-19.5000 - 33.7750i) q^{93} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 4 q^{4} - 13 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 4 q^{4} - 13 q^{7} - 9 q^{9} + 12 q^{12} + 46 q^{13} - 16 q^{16} - 11 q^{19} - 6 q^{21} - 25 q^{25} - 54 q^{27} + 8 q^{28} + 13 q^{31} + 72 q^{36} + 73 q^{37} + 69 q^{39} - 122 q^{43} - 96 q^{48} + 71 q^{49} - 92 q^{52} - 66 q^{57} - 74 q^{61} + 99 q^{63} + 128 q^{64} + 13 q^{67} + 97 q^{73} + 75 q^{75} + 88 q^{76} - 11 q^{79} - 81 q^{81} - 132 q^{84} - 299 q^{91} - 39 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) 1.50000 2.59808i 0.500000 0.866025i
\(4\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(8\) 0 0
\(9\) −4.50000 7.79423i −0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 6.00000 + 10.3923i 0.500000 + 0.866025i
\(13\) 23.0000 1.76923 0.884615 0.466321i \(-0.154421\pi\)
0.884615 + 0.466321i \(0.154421\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.00000 13.8564i −0.500000 0.866025i
\(17\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) −5.50000 9.52628i −0.289474 0.501383i 0.684211 0.729285i \(-0.260147\pi\)
−0.973684 + 0.227901i \(0.926814\pi\)
\(20\) 0 0
\(21\) −3.00000 + 20.7846i −0.142857 + 0.989743i
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) −27.0000 −1.00000
\(28\) 4.00000 27.7128i 0.142857 0.989743i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 6.50000 11.2583i 0.209677 0.363172i −0.741935 0.670471i \(-0.766092\pi\)
0.951613 + 0.307299i \(0.0994253\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 36.0000 1.00000
\(37\) 36.5000 + 63.2199i 0.986486 + 1.70864i 0.635135 + 0.772401i \(0.280944\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) 0 0
\(39\) 34.5000 59.7558i 0.884615 1.53220i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −61.0000 −1.41860 −0.709302 0.704904i \(-0.750990\pi\)
−0.709302 + 0.704904i \(0.750990\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) −48.0000 −1.00000
\(49\) 35.5000 33.7750i 0.724490 0.689286i
\(50\) 0 0
\(51\) 0 0
\(52\) −46.0000 + 79.6743i −0.884615 + 1.53220i
\(53\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −33.0000 −0.578947
\(58\) 0 0
\(59\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) −37.0000 64.0859i −0.606557 1.05059i −0.991803 0.127774i \(-0.959217\pi\)
0.385246 0.922814i \(-0.374117\pi\)
\(62\) 0 0
\(63\) 49.5000 + 38.9711i 0.785714 + 0.618590i
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.50000 11.2583i 0.0970149 0.168035i −0.813433 0.581659i \(-0.802404\pi\)
0.910448 + 0.413624i \(0.135737\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 48.5000 84.0045i 0.664384 1.15075i −0.315068 0.949069i \(-0.602027\pi\)
0.979452 0.201677i \(-0.0646392\pi\)
\(74\) 0 0
\(75\) 37.5000 + 64.9519i 0.500000 + 0.866025i
\(76\) 44.0000 0.578947
\(77\) 0 0
\(78\) 0 0
\(79\) −5.50000 9.52628i −0.0696203 0.120586i 0.829114 0.559080i \(-0.188845\pi\)
−0.898734 + 0.438494i \(0.855512\pi\)
\(80\) 0 0
\(81\) −40.5000 + 70.1481i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −66.0000 51.9615i −0.785714 0.618590i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) −149.500 + 59.7558i −1.64286 + 0.656657i
\(92\) 0 0
\(93\) −19.5000 33.7750i −0.209677 0.363172i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.0206186 0.0103093 0.999947i \(-0.496718\pi\)
0.0103093 + 0.999947i \(0.496718\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −50.0000 86.6025i −0.500000 0.866025i
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) 78.5000 + 135.966i 0.762136 + 1.32006i 0.941748 + 0.336321i \(0.109183\pi\)
−0.179612 + 0.983738i \(0.557484\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 54.0000 93.5307i 0.500000 0.866025i
\(109\) −35.5000 + 61.4878i −0.325688 + 0.564108i −0.981651 0.190684i \(-0.938929\pi\)
0.655963 + 0.754793i \(0.272263\pi\)
\(110\) 0 0
\(111\) 219.000 1.97297
\(112\) 88.0000 + 69.2820i 0.785714 + 0.618590i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −103.500 179.267i −0.884615 1.53220i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −60.5000 104.789i −0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 26.0000 + 45.0333i 0.209677 + 0.363172i
\(125\) 0 0
\(126\) 0 0
\(127\) 107.000 0.842520 0.421260 0.906940i \(-0.361588\pi\)
0.421260 + 0.906940i \(0.361588\pi\)
\(128\) 0 0
\(129\) −91.5000 + 158.483i −0.709302 + 1.22855i
\(130\) 0 0
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) 60.5000 + 47.6314i 0.454887 + 0.358131i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) −229.000 −1.64748 −0.823741 0.566966i \(-0.808117\pi\)
−0.823741 + 0.566966i \(0.808117\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −72.0000 + 124.708i −0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) −34.5000 142.894i −0.234694 0.972069i
\(148\) −292.000 −1.97297
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 143.000 247.683i 0.947020 1.64029i 0.195364 0.980731i \(-0.437411\pi\)
0.751656 0.659556i \(-0.229256\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 138.000 + 239.023i 0.884615 + 1.53220i
\(157\) 59.0000 102.191i 0.375796 0.650898i −0.614650 0.788800i \(-0.710703\pi\)
0.990446 + 0.137902i \(0.0440359\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 131.000 + 226.899i 0.803681 + 1.39202i 0.917178 + 0.398478i \(0.130461\pi\)
−0.113497 + 0.993538i \(0.536205\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 360.000 2.13018
\(170\) 0 0
\(171\) −49.5000 + 85.7365i −0.289474 + 0.501383i
\(172\) 122.000 211.310i 0.709302 1.22855i
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) 25.0000 173.205i 0.142857 0.989743i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) −313.000 −1.72928 −0.864641 0.502390i \(-0.832454\pi\)
−0.864641 + 0.502390i \(0.832454\pi\)
\(182\) 0 0
\(183\) −222.000 −1.21311
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 175.500 70.1481i 0.928571 0.371154i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 96.0000 166.277i 0.500000 0.866025i
\(193\) −119.500 + 206.980i −0.619171 + 1.07244i 0.370466 + 0.928846i \(0.379198\pi\)
−0.989637 + 0.143590i \(0.954135\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 46.0000 + 190.526i 0.234694 + 0.972069i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −193.000 + 334.286i −0.969849 + 1.67983i −0.273869 + 0.961767i \(0.588304\pi\)
−0.695980 + 0.718061i \(0.745030\pi\)
\(200\) 0 0
\(201\) −19.5000 33.7750i −0.0970149 0.168035i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −184.000 318.697i −0.884615 1.53220i
\(209\) 0 0
\(210\) 0 0
\(211\) −166.000 −0.786730 −0.393365 0.919382i \(-0.628689\pi\)
−0.393365 + 0.919382i \(0.628689\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −13.0000 + 90.0666i −0.0599078 + 0.415054i
\(218\) 0 0
\(219\) −145.500 252.013i −0.664384 1.15075i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 338.000 1.51570 0.757848 0.652432i \(-0.226251\pi\)
0.757848 + 0.652432i \(0.226251\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(228\) 66.0000 114.315i 0.289474 0.501383i
\(229\) 204.500 + 354.204i 0.893013 + 1.54674i 0.836245 + 0.548357i \(0.184746\pi\)
0.0567686 + 0.998387i \(0.481920\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −33.0000 −0.139241
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 143.000 247.683i 0.593361 1.02773i −0.400415 0.916334i \(-0.631134\pi\)
0.993776 0.111397i \(-0.0355327\pi\)
\(242\) 0 0
\(243\) 121.500 + 210.444i 0.500000 + 0.866025i
\(244\) 296.000 1.21311
\(245\) 0 0
\(246\) 0 0
\(247\) −126.500 219.104i −0.512146 0.887062i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −234.000 + 93.5307i −0.928571 + 0.371154i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −128.000 + 221.703i −0.500000 + 0.866025i
\(257\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(258\) 0 0
\(259\) −401.500 316.099i −1.55019 1.22046i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 26.0000 + 45.0333i 0.0970149 + 0.168035i
\(269\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) −121.000 209.578i −0.446494 0.773351i 0.551661 0.834069i \(-0.313994\pi\)
−0.998155 + 0.0607176i \(0.980661\pi\)
\(272\) 0 0
\(273\) −69.0000 + 478.046i −0.252747 + 1.75108i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −203.500 + 352.472i −0.734657 + 1.27246i 0.220217 + 0.975451i \(0.429324\pi\)
−0.954874 + 0.297012i \(0.904010\pi\)
\(278\) 0 0
\(279\) −117.000 −0.419355
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 258.500 447.735i 0.913428 1.58210i 0.104240 0.994552i \(-0.466759\pi\)
0.809187 0.587551i \(-0.199908\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −144.500 250.281i −0.500000 0.866025i
\(290\) 0 0
\(291\) 3.00000 5.19615i 0.0103093 0.0178562i
\(292\) 194.000 + 336.018i 0.664384 + 1.15075i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −300.000 −1.00000
\(301\) 396.500 158.483i 1.31728 0.526520i
\(302\) 0 0
\(303\) 0 0
\(304\) −88.0000 + 152.420i −0.289474 + 0.501383i
\(305\) 0 0
\(306\) 0 0
\(307\) 611.000 1.99023 0.995114 0.0987325i \(-0.0314788\pi\)
0.995114 + 0.0987325i \(0.0314788\pi\)
\(308\) 0 0
\(309\) 471.000 1.52427
\(310\) 0 0
\(311\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) −299.500 518.749i −0.956869 1.65735i −0.730032 0.683413i \(-0.760495\pi\)
−0.226837 0.973933i \(-0.572838\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 44.0000 0.139241
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −162.000 280.592i −0.500000 0.866025i
\(325\) −287.500 + 497.965i −0.884615 + 1.53220i
\(326\) 0 0
\(327\) 106.500 + 184.463i 0.325688 + 0.564108i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 330.500 + 572.443i 0.998489 + 1.72943i 0.546828 + 0.837245i \(0.315835\pi\)
0.451662 + 0.892189i \(0.350831\pi\)
\(332\) 0 0
\(333\) 328.500 568.979i 0.986486 1.70864i
\(334\) 0 0
\(335\) 0 0
\(336\) 312.000 124.708i 0.928571 0.371154i
\(337\) −649.000 −1.92582 −0.962908 0.269830i \(-0.913033\pi\)
−0.962908 + 0.269830i \(0.913033\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −143.000 + 311.769i −0.416910 + 0.908948i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) −502.000 −1.43840 −0.719198 0.694805i \(-0.755490\pi\)
−0.719198 + 0.694805i \(0.755490\pi\)
\(350\) 0 0
\(351\) −621.000 −1.76923
\(352\) 0 0
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) 120.000 207.846i 0.332410 0.575751i
\(362\) 0 0
\(363\) −363.000 −1.00000
\(364\) 92.0000 637.395i 0.252747 1.75108i
\(365\) 0 0
\(366\) 0 0
\(367\) −245.500 + 425.218i −0.668937 + 1.15863i 0.309264 + 0.950976i \(0.399917\pi\)
−0.978202 + 0.207657i \(0.933416\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 156.000 0.419355
\(373\) 288.500 + 499.697i 0.773458 + 1.33967i 0.935657 + 0.352911i \(0.114808\pi\)
−0.162198 + 0.986758i \(0.551858\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 611.000 1.61214 0.806069 0.591822i \(-0.201591\pi\)
0.806069 + 0.591822i \(0.201591\pi\)
\(380\) 0 0
\(381\) 160.500 277.994i 0.421260 0.729643i
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 274.500 + 475.448i 0.709302 + 1.22855i
\(388\) −4.00000 + 6.92820i −0.0103093 + 0.0178562i
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −215.500 373.257i −0.542821 0.940194i −0.998741 0.0501728i \(-0.984023\pi\)
0.455919 0.890021i \(-0.349311\pi\)
\(398\) 0 0
\(399\) 214.500 85.7365i 0.537594 0.214878i
\(400\) 400.000 1.00000
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) 149.500 258.942i 0.370968 0.642535i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 384.500 665.974i 0.940098 1.62830i 0.174817 0.984601i \(-0.444067\pi\)
0.765281 0.643696i \(-0.222600\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −628.000 −1.52427
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −343.500 + 594.959i −0.823741 + 1.42676i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −481.000 −1.14252 −0.571259 0.820770i \(-0.693545\pi\)
−0.571259 + 0.820770i \(0.693545\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 407.000 + 320.429i 0.953162 + 0.750420i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 216.000 + 374.123i 0.500000 + 0.866025i
\(433\) 359.000 0.829099 0.414550 0.910027i \(-0.363939\pi\)
0.414550 + 0.910027i \(0.363939\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −142.000 245.951i −0.325688 0.564108i
\(437\) 0 0
\(438\) 0 0
\(439\) 47.0000 + 81.4064i 0.107062 + 0.185436i 0.914579 0.404408i \(-0.132522\pi\)
−0.807517 + 0.589844i \(0.799189\pi\)
\(440\) 0 0
\(441\) −423.000 124.708i −0.959184 0.282784i
\(442\) 0 0
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) −438.000 + 758.638i −0.986486 + 1.70864i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −416.000 + 166.277i −0.928571 + 0.371154i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −429.000 743.050i −0.947020 1.64029i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −383.500 664.241i −0.839168 1.45348i −0.890591 0.454805i \(-0.849709\pi\)
0.0514223 0.998677i \(-0.483625\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −397.000 −0.857451 −0.428726 0.903435i \(-0.641037\pi\)
−0.428726 + 0.903435i \(0.641037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(468\) 828.000 1.76923
\(469\) −13.0000 + 90.0666i −0.0277186 + 0.192040i
\(470\) 0 0
\(471\) −177.000 306.573i −0.375796 0.650898i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 275.000 0.578947
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 839.500 + 1454.06i 1.74532 + 3.02299i
\(482\) 0 0
\(483\) 0 0
\(484\) 484.000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 174.500 302.243i 0.358316 0.620622i −0.629363 0.777111i \(-0.716684\pi\)
0.987680 + 0.156489i \(0.0500176\pi\)
\(488\) 0 0
\(489\) 786.000 1.60736
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −208.000 −0.419355
\(497\) 0 0
\(498\) 0 0
\(499\) −425.500 736.988i −0.852705 1.47693i −0.878758 0.477269i \(-0.841627\pi\)
0.0260521 0.999661i \(-0.491706\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 540.000 935.307i 1.06509 1.84479i
\(508\) −214.000 + 370.659i −0.421260 + 0.729643i
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) −97.0000 + 672.036i −0.189824 + 1.31514i
\(512\) 0 0
\(513\) 148.500 + 257.210i 0.289474 + 0.501383i
\(514\) 0 0
\(515\) 0 0
\(516\) −366.000 633.931i −0.709302 1.22855i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) −89.5000 155.019i −0.171128 0.296403i 0.767686 0.640826i \(-0.221408\pi\)
−0.938815 + 0.344423i \(0.888075\pi\)
\(524\) 0 0
\(525\) −412.500 324.760i −0.785714 0.618590i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −264.500 + 458.127i −0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) −286.000 + 114.315i −0.537594 + 0.214878i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 120.500 + 208.712i 0.222736 + 0.385790i 0.955638 0.294545i \(-0.0951680\pi\)
−0.732902 + 0.680334i \(0.761835\pi\)
\(542\) 0 0
\(543\) −469.500 + 813.198i −0.864641 + 1.49760i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 506.000 0.925046 0.462523 0.886607i \(-0.346944\pi\)
0.462523 + 0.886607i \(0.346944\pi\)
\(548\) 0 0
\(549\) −333.000 + 576.773i −0.606557 + 1.05059i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 60.5000 + 47.6314i 0.109403 + 0.0861327i
\(554\) 0 0
\(555\) 0 0
\(556\) 458.000 793.279i 0.823741 1.42676i
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) −1403.00 −2.50984
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 81.0000 561.184i 0.142857 0.989743i
\(568\) 0 0
\(569\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) 90.5000 156.751i 0.158494 0.274519i −0.775832 0.630940i \(-0.782670\pi\)
0.934326 + 0.356420i \(0.116003\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −288.000 498.831i −0.500000 0.866025i
\(577\) −35.5000 + 61.4878i −0.0615251 + 0.106565i −0.895147 0.445770i \(-0.852930\pi\)
0.833622 + 0.552335i \(0.186263\pi\)
\(578\) 0 0
\(579\) 358.500 + 620.940i 0.619171 + 1.07244i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 564.000 + 166.277i 0.959184 + 0.282784i
\(589\) −143.000 −0.242784
\(590\) 0 0
\(591\) 0 0
\(592\) 584.000 1011.52i 0.986486 1.70864i
\(593\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 579.000 + 1002.86i 0.969849 + 1.67983i
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 1199.00 1.99501 0.997504 0.0706077i \(-0.0224939\pi\)
0.997504 + 0.0706077i \(0.0224939\pi\)
\(602\) 0 0
\(603\) −117.000 −0.194030
\(604\) 572.000 + 990.733i 0.947020 + 1.64029i
\(605\) 0 0
\(606\) 0 0
\(607\) −593.500 1027.97i −0.977759 1.69353i −0.670511 0.741900i \(-0.733925\pi\)
−0.307249 0.951629i \(-0.599408\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 563.000 975.145i 0.918434 1.59077i 0.116639 0.993174i \(-0.462788\pi\)
0.801794 0.597600i \(-0.203879\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −581.500 + 1007.19i −0.939418 + 1.62712i −0.172859 + 0.984947i \(0.555301\pi\)
−0.766559 + 0.642174i \(0.778033\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −1104.00 −1.76923
\(625\) −312.500 541.266i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 236.000 + 408.764i 0.375796 + 0.650898i
\(629\) 0 0
\(630\) 0 0
\(631\) 674.000 1.06815 0.534073 0.845438i \(-0.320661\pi\)
0.534073 + 0.845438i \(0.320661\pi\)
\(632\) 0 0
\(633\) −249.000 + 431.281i −0.393365 + 0.681328i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 816.500 776.825i 1.28179 1.21951i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) −1237.00 −1.92379 −0.961897 0.273411i \(-0.911848\pi\)
−0.961897 + 0.273411i \(0.911848\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 214.500 + 168.875i 0.329493 + 0.259409i
\(652\) −1048.00 −1.60736
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −873.000 −1.32877
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −539.500 + 934.441i −0.816188 + 1.41368i 0.0922844 + 0.995733i \(0.470583\pi\)
−0.908472 + 0.417946i \(0.862750\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 507.000 878.150i 0.757848 1.31263i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 23.0000 0.0341753 0.0170877 0.999854i \(-0.494561\pi\)
0.0170877 + 0.999854i \(0.494561\pi\)
\(674\) 0 0
\(675\) 337.500 584.567i 0.500000 0.866025i
\(676\) −720.000 + 1247.08i −1.06509 + 1.84479i
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) −13.0000 + 5.19615i −0.0191458 + 0.00765265i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) −198.000 342.946i −0.289474 0.501383i
\(685\) 0 0
\(686\) 0 0
\(687\) 1227.00 1.78603
\(688\) 488.000 + 845.241i 0.709302 + 1.22855i
\(689\) 0 0
\(690\) 0 0
\(691\) −509.500 882.480i −0.737337 1.27711i −0.953690 0.300790i \(-0.902750\pi\)
0.216353 0.976315i \(-0.430584\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 550.000 + 433.013i 0.785714 + 0.618590i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 401.500 695.418i 0.571124 0.989215i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 467.000 + 808.868i 0.658674 + 1.14086i 0.980959 + 0.194214i \(0.0622158\pi\)
−0.322285 + 0.946643i \(0.604451\pi\)
\(710\) 0 0
\(711\) −49.5000 + 85.7365i −0.0696203 + 0.120586i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) −863.500 679.830i −1.19764 0.942899i
\(722\) 0 0
\(723\) −429.000 743.050i −0.593361 1.02773i
\(724\) 626.000 1084.26i 0.864641 1.49760i
\(725\) 0 0
\(726\) 0 0
\(727\) 947.000 1.30261 0.651307 0.758815i \(-0.274221\pi\)
0.651307 + 0.758815i \(0.274221\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 444.000 769.031i 0.606557 1.05059i
\(733\) 708.500 + 1227.16i 0.966576 + 1.67416i 0.705321 + 0.708888i \(0.250803\pi\)
0.261255 + 0.965270i \(0.415864\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −665.500 + 1152.68i −0.900541 + 1.55978i −0.0737483 + 0.997277i \(0.523496\pi\)
−0.826793 + 0.562506i \(0.809837\pi\)
\(740\) 0 0
\(741\) −759.000 −1.02429
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −89.5000 155.019i −0.119174 0.206416i 0.800266 0.599645i \(-0.204691\pi\)
−0.919441 + 0.393229i \(0.871358\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −108.000 + 748.246i −0.142857 + 0.989743i
\(757\) −838.000 −1.10700 −0.553501 0.832849i \(-0.686708\pi\)
−0.553501 + 0.832849i \(0.686708\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) 71.0000 491.902i 0.0930537 0.644695i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 384.000 + 665.108i 0.500000 + 0.866025i
\(769\) 863.000 1.12224 0.561118 0.827736i \(-0.310371\pi\)
0.561118 + 0.827736i \(0.310371\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −478.000 827.920i −0.619171 1.07244i
\(773\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 0 0
\(775\) 162.500 + 281.458i 0.209677 + 0.363172i
\(776\) 0 0
\(777\) −1423.50 + 568.979i −1.83205 + 0.732276i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −752.000 221.703i −0.959184 0.282784i
\(785\) 0 0
\(786\) 0 0
\(787\) −781.000 + 1352.73i −0.992376 + 1.71885i −0.389454 + 0.921046i \(0.627336\pi\)
−0.602922 + 0.797800i \(0.705997\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −851.000 1473.98i −1.07314 1.85873i
\(794\) 0 0
\(795\) 0 0
\(796\) −772.000 1337.14i −0.969849 1.67983i
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 156.000 0.194030
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) 1514.00 1.86683 0.933416 0.358797i \(-0.116813\pi\)
0.933416 + 0.358797i \(0.116813\pi\)
\(812\) 0 0
\(813\) −726.000 −0.892989
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 335.500 + 581.103i 0.410649 + 0.711264i
\(818\) 0 0
\(819\) 1138.50 + 896.336i 1.39011 + 1.09443i
\(820\) 0 0
\(821\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(822\) 0 0
\(823\) −529.000 + 916.255i −0.642770 + 1.11331i 0.342041 + 0.939685i \(0.388882\pi\)
−0.984812 + 0.173626i \(0.944452\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 804.500 1393.43i 0.970446 1.68086i 0.276236 0.961090i \(-0.410913\pi\)
0.694210 0.719773i \(-0.255754\pi\)
\(830\) 0 0
\(831\) 610.500 + 1057.42i 0.734657 + 1.27246i
\(832\) 1472.00 1.76923
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −175.500 + 303.975i −0.209677 + 0.363172i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 332.000 575.041i 0.393365 0.681328i
\(845\) 0 0
\(846\) 0 0
\(847\) 665.500 + 523.945i 0.785714 + 0.618590i
\(848\) 0 0
\(849\) −775.500 1343.21i −0.913428 1.58210i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −481.000 −0.563892 −0.281946 0.959430i \(-0.590980\pi\)
−0.281946 + 0.959430i \(0.590980\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(858\) 0 0
\(859\) −709.000 1228.02i −0.825378 1.42960i −0.901630 0.432509i \(-0.857629\pi\)
0.0762515 0.997089i \(-0.475705\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −867.000 −1.00000
\(868\) −286.000 225.167i −0.329493 0.259409i
\(869\) 0 0
\(870\) 0 0
\(871\) 149.500 258.942i 0.171642 0.297292i
\(872\) 0 0
\(873\) −9.00000 15.5885i −0.0103093 0.0178562i
\(874\) 0 0
\(875\) 0 0
\(876\) 1164.00 1.32877
\(877\) 299.000 + 517.883i 0.340935 + 0.590517i 0.984607 0.174785i \(-0.0559231\pi\)
−0.643672 + 0.765302i \(0.722590\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 443.000 0.501699 0.250849 0.968026i \(-0.419290\pi\)
0.250849 + 0.968026i \(0.419290\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) 0 0
\(889\) −695.500 + 277.994i −0.782340 + 0.312704i
\(890\) 0 0
\(891\) 0 0
\(892\) −676.000 + 1170.87i −0.757848 + 1.31263i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −450.000 + 779.423i −0.500000 + 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) 183.000 1267.86i 0.202658 1.40405i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −833.500 + 1443.66i −0.918964 + 1.59169i −0.117971 + 0.993017i \(0.537639\pi\)
−0.800992 + 0.598675i \(0.795694\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 264.000 + 457.261i 0.289474 + 0.501383i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −1636.00 −1.78603
\(917\) 0 0
\(918\) 0 0
\(919\) 918.500 + 1590.89i 0.999456 + 1.73111i 0.528292 + 0.849063i \(0.322833\pi\)
0.471164 + 0.882045i \(0.343834\pi\)
\(920\) 0 0
\(921\) 916.500 1587.42i 0.995114 1.72359i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1825.00 −1.97297
\(926\) 0 0
\(927\) 706.500 1223.69i 0.762136 1.32006i
\(928\) 0 0
\(929\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(930\) 0 0
\(931\) −517.000 152.420i −0.555317 0.163717i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −649.000 −0.692636 −0.346318 0.938117i \(-0.612568\pi\)
−0.346318 + 0.938117i \(0.612568\pi\)
\(938\) 0 0
\(939\) −1797.00 −1.91374
\(940\) 0 0
\(941\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(948\) 66.0000 114.315i 0.0696203 0.120586i
\(949\) 1115.50 1932.10i 1.17545 2.03594i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 396.000 + 685.892i 0.412071 + 0.713727i
\(962\) 0 0
\(963\) 0 0
\(964\) 572.000 + 990.733i 0.593361 + 1.02773i
\(965\) 0 0
\(966\) 0 0
\(967\) 1787.00 1.84798 0.923992 0.382412i \(-0.124907\pi\)
0.923992 + 0.382412i \(0.124907\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(972\) −972.000 −1.00000
\(973\) 1488.50 594.959i 1.52980 0.611469i
\(974\) 0 0
\(975\) 862.500 + 1493.89i 0.884615 + 1.53220i
\(976\) −592.000 + 1025.37i −0.606557 + 1.05059i
\(977\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 639.000 0.651376
\(982\) 0 0
\(983\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1012.00 1.02429
\(989\) 0 0
\(990\) 0 0
\(991\) 846.500 1466.18i 0.854188 1.47950i −0.0232089 0.999731i \(-0.507388\pi\)
0.877397 0.479766i \(-0.159278\pi\)
\(992\) 0 0
\(993\) 1983.00 1.99698
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −203.500 + 352.472i −0.204112 + 0.353533i −0.949850 0.312707i \(-0.898764\pi\)
0.745737 + 0.666240i \(0.232097\pi\)
\(998\) 0 0
\(999\) −985.500 1706.94i −0.986486 1.70864i
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 21.3.h.a.11.1 yes 2
3.2 odd 2 CM 21.3.h.a.11.1 yes 2
4.3 odd 2 336.3.bn.b.305.1 2
7.2 even 3 inner 21.3.h.a.2.1 2
7.3 odd 6 147.3.b.b.50.1 1
7.4 even 3 147.3.b.a.50.1 1
7.5 odd 6 147.3.h.a.128.1 2
7.6 odd 2 147.3.h.a.116.1 2
12.11 even 2 336.3.bn.b.305.1 2
21.2 odd 6 inner 21.3.h.a.2.1 2
21.5 even 6 147.3.h.a.128.1 2
21.11 odd 6 147.3.b.a.50.1 1
21.17 even 6 147.3.b.b.50.1 1
21.20 even 2 147.3.h.a.116.1 2
28.23 odd 6 336.3.bn.b.65.1 2
84.23 even 6 336.3.bn.b.65.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.3.h.a.2.1 2 7.2 even 3 inner
21.3.h.a.2.1 2 21.2 odd 6 inner
21.3.h.a.11.1 yes 2 1.1 even 1 trivial
21.3.h.a.11.1 yes 2 3.2 odd 2 CM
147.3.b.a.50.1 1 7.4 even 3
147.3.b.a.50.1 1 21.11 odd 6
147.3.b.b.50.1 1 7.3 odd 6
147.3.b.b.50.1 1 21.17 even 6
147.3.h.a.116.1 2 7.6 odd 2
147.3.h.a.116.1 2 21.20 even 2
147.3.h.a.128.1 2 7.5 odd 6
147.3.h.a.128.1 2 21.5 even 6
336.3.bn.b.65.1 2 28.23 odd 6
336.3.bn.b.65.1 2 84.23 even 6
336.3.bn.b.305.1 2 4.3 odd 2
336.3.bn.b.305.1 2 12.11 even 2