Properties

Label 8046.2.a.k.1.9
Level $8046$
Weight $2$
Character 8046.1
Self dual yes
Analytic conductor $64.248$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.a (trivial)

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: yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + 345 x^{3} - 1607 x^{2} - 594 x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(2.60206\) of defining polynomial
Character \(\chi\) \(=\) 8046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.60206 q^{5} -4.35416 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.60206 q^{5} -4.35416 q^{7} -1.00000 q^{8} -2.60206 q^{10} +0.265163 q^{11} +4.05522 q^{13} +4.35416 q^{14} +1.00000 q^{16} +4.33216 q^{17} +4.48387 q^{19} +2.60206 q^{20} -0.265163 q^{22} -7.28167 q^{23} +1.77072 q^{25} -4.05522 q^{26} -4.35416 q^{28} -4.41220 q^{29} -2.50928 q^{31} -1.00000 q^{32} -4.33216 q^{34} -11.3298 q^{35} -1.01155 q^{37} -4.48387 q^{38} -2.60206 q^{40} +7.56092 q^{41} +8.16581 q^{43} +0.265163 q^{44} +7.28167 q^{46} -4.04984 q^{47} +11.9587 q^{49} -1.77072 q^{50} +4.05522 q^{52} +12.0518 q^{53} +0.689971 q^{55} +4.35416 q^{56} +4.41220 q^{58} +2.90383 q^{59} -4.59376 q^{61} +2.50928 q^{62} +1.00000 q^{64} +10.5519 q^{65} -2.97293 q^{67} +4.33216 q^{68} +11.3298 q^{70} +6.06746 q^{71} -4.42989 q^{73} +1.01155 q^{74} +4.48387 q^{76} -1.15456 q^{77} +5.32897 q^{79} +2.60206 q^{80} -7.56092 q^{82} -14.1972 q^{83} +11.2725 q^{85} -8.16581 q^{86} -0.265163 q^{88} -2.33739 q^{89} -17.6571 q^{91} -7.28167 q^{92} +4.04984 q^{94} +11.6673 q^{95} +2.03002 q^{97} -11.9587 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} + 3 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} + 3 q^{5} - 6 q^{7} - 12 q^{8} - 3 q^{10} + 14 q^{11} - 3 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} - 4 q^{19} + 3 q^{20} - 14 q^{22} + 13 q^{23} + 11 q^{25} + 3 q^{26} - 6 q^{28} + 23 q^{29} - 14 q^{31} - 12 q^{32} - 8 q^{34} + 32 q^{35} - 19 q^{37} + 4 q^{38} - 3 q^{40} + 30 q^{41} - 15 q^{43} + 14 q^{44} - 13 q^{46} - q^{47} + 14 q^{49} - 11 q^{50} - 3 q^{52} + 16 q^{53} - 7 q^{55} + 6 q^{56} - 23 q^{58} + 26 q^{59} - 16 q^{61} + 14 q^{62} + 12 q^{64} + 8 q^{65} - 39 q^{67} + 8 q^{68} - 32 q^{70} + 15 q^{71} - 2 q^{73} + 19 q^{74} - 4 q^{76} + 34 q^{77} - 13 q^{79} + 3 q^{80} - 30 q^{82} + 6 q^{83} - 11 q^{85} + 15 q^{86} - 14 q^{88} + 18 q^{89} - 35 q^{91} + 13 q^{92} + q^{94} + 51 q^{95} + 19 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.60206 1.16368 0.581839 0.813304i \(-0.302334\pi\)
0.581839 + 0.813304i \(0.302334\pi\)
\(6\) 0 0
\(7\) −4.35416 −1.64572 −0.822859 0.568245i \(-0.807623\pi\)
−0.822859 + 0.568245i \(0.807623\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.60206 −0.822844
\(11\) 0.265163 0.0799497 0.0399749 0.999201i \(-0.487272\pi\)
0.0399749 + 0.999201i \(0.487272\pi\)
\(12\) 0 0
\(13\) 4.05522 1.12472 0.562358 0.826894i \(-0.309894\pi\)
0.562358 + 0.826894i \(0.309894\pi\)
\(14\) 4.35416 1.16370
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.33216 1.05070 0.525352 0.850885i \(-0.323934\pi\)
0.525352 + 0.850885i \(0.323934\pi\)
\(18\) 0 0
\(19\) 4.48387 1.02867 0.514336 0.857589i \(-0.328039\pi\)
0.514336 + 0.857589i \(0.328039\pi\)
\(20\) 2.60206 0.581839
\(21\) 0 0
\(22\) −0.265163 −0.0565330
\(23\) −7.28167 −1.51833 −0.759167 0.650896i \(-0.774393\pi\)
−0.759167 + 0.650896i \(0.774393\pi\)
\(24\) 0 0
\(25\) 1.77072 0.354144
\(26\) −4.05522 −0.795294
\(27\) 0 0
\(28\) −4.35416 −0.822859
\(29\) −4.41220 −0.819326 −0.409663 0.912237i \(-0.634354\pi\)
−0.409663 + 0.912237i \(0.634354\pi\)
\(30\) 0 0
\(31\) −2.50928 −0.450681 −0.225340 0.974280i \(-0.572349\pi\)
−0.225340 + 0.974280i \(0.572349\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.33216 −0.742959
\(35\) −11.3298 −1.91509
\(36\) 0 0
\(37\) −1.01155 −0.166297 −0.0831486 0.996537i \(-0.526498\pi\)
−0.0831486 + 0.996537i \(0.526498\pi\)
\(38\) −4.48387 −0.727380
\(39\) 0 0
\(40\) −2.60206 −0.411422
\(41\) 7.56092 1.18082 0.590409 0.807104i \(-0.298966\pi\)
0.590409 + 0.807104i \(0.298966\pi\)
\(42\) 0 0
\(43\) 8.16581 1.24527 0.622637 0.782511i \(-0.286061\pi\)
0.622637 + 0.782511i \(0.286061\pi\)
\(44\) 0.265163 0.0399749
\(45\) 0 0
\(46\) 7.28167 1.07362
\(47\) −4.04984 −0.590730 −0.295365 0.955385i \(-0.595441\pi\)
−0.295365 + 0.955385i \(0.595441\pi\)
\(48\) 0 0
\(49\) 11.9587 1.70839
\(50\) −1.77072 −0.250418
\(51\) 0 0
\(52\) 4.05522 0.562358
\(53\) 12.0518 1.65544 0.827721 0.561140i \(-0.189637\pi\)
0.827721 + 0.561140i \(0.189637\pi\)
\(54\) 0 0
\(55\) 0.689971 0.0930357
\(56\) 4.35416 0.581849
\(57\) 0 0
\(58\) 4.41220 0.579351
\(59\) 2.90383 0.378046 0.189023 0.981973i \(-0.439468\pi\)
0.189023 + 0.981973i \(0.439468\pi\)
\(60\) 0 0
\(61\) −4.59376 −0.588171 −0.294086 0.955779i \(-0.595015\pi\)
−0.294086 + 0.955779i \(0.595015\pi\)
\(62\) 2.50928 0.318679
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 10.5519 1.30881
\(66\) 0 0
\(67\) −2.97293 −0.363201 −0.181601 0.983372i \(-0.558128\pi\)
−0.181601 + 0.983372i \(0.558128\pi\)
\(68\) 4.33216 0.525352
\(69\) 0 0
\(70\) 11.3298 1.35417
\(71\) 6.06746 0.720075 0.360037 0.932938i \(-0.382764\pi\)
0.360037 + 0.932938i \(0.382764\pi\)
\(72\) 0 0
\(73\) −4.42989 −0.518479 −0.259240 0.965813i \(-0.583472\pi\)
−0.259240 + 0.965813i \(0.583472\pi\)
\(74\) 1.01155 0.117590
\(75\) 0 0
\(76\) 4.48387 0.514336
\(77\) −1.15456 −0.131575
\(78\) 0 0
\(79\) 5.32897 0.599555 0.299778 0.954009i \(-0.403087\pi\)
0.299778 + 0.954009i \(0.403087\pi\)
\(80\) 2.60206 0.290919
\(81\) 0 0
\(82\) −7.56092 −0.834964
\(83\) −14.1972 −1.55835 −0.779173 0.626808i \(-0.784361\pi\)
−0.779173 + 0.626808i \(0.784361\pi\)
\(84\) 0 0
\(85\) 11.2725 1.22268
\(86\) −8.16581 −0.880542
\(87\) 0 0
\(88\) −0.265163 −0.0282665
\(89\) −2.33739 −0.247763 −0.123881 0.992297i \(-0.539534\pi\)
−0.123881 + 0.992297i \(0.539534\pi\)
\(90\) 0 0
\(91\) −17.6571 −1.85097
\(92\) −7.28167 −0.759167
\(93\) 0 0
\(94\) 4.04984 0.417709
\(95\) 11.6673 1.19704
\(96\) 0 0
\(97\) 2.03002 0.206117 0.103058 0.994675i \(-0.467137\pi\)
0.103058 + 0.994675i \(0.467137\pi\)
\(98\) −11.9587 −1.20801
\(99\) 0 0
\(100\) 1.77072 0.177072
\(101\) 4.87422 0.485003 0.242501 0.970151i \(-0.422032\pi\)
0.242501 + 0.970151i \(0.422032\pi\)
\(102\) 0 0
\(103\) 6.98778 0.688527 0.344263 0.938873i \(-0.388129\pi\)
0.344263 + 0.938873i \(0.388129\pi\)
\(104\) −4.05522 −0.397647
\(105\) 0 0
\(106\) −12.0518 −1.17057
\(107\) 0.355452 0.0343629 0.0171814 0.999852i \(-0.494531\pi\)
0.0171814 + 0.999852i \(0.494531\pi\)
\(108\) 0 0
\(109\) −5.84322 −0.559679 −0.279839 0.960047i \(-0.590281\pi\)
−0.279839 + 0.960047i \(0.590281\pi\)
\(110\) −0.689971 −0.0657862
\(111\) 0 0
\(112\) −4.35416 −0.411430
\(113\) 19.5256 1.83681 0.918407 0.395637i \(-0.129476\pi\)
0.918407 + 0.395637i \(0.129476\pi\)
\(114\) 0 0
\(115\) −18.9474 −1.76685
\(116\) −4.41220 −0.409663
\(117\) 0 0
\(118\) −2.90383 −0.267319
\(119\) −18.8629 −1.72916
\(120\) 0 0
\(121\) −10.9297 −0.993608
\(122\) 4.59376 0.415900
\(123\) 0 0
\(124\) −2.50928 −0.225340
\(125\) −8.40278 −0.751568
\(126\) 0 0
\(127\) −16.2907 −1.44556 −0.722782 0.691076i \(-0.757137\pi\)
−0.722782 + 0.691076i \(0.757137\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −10.5519 −0.925465
\(131\) 17.6406 1.54127 0.770635 0.637277i \(-0.219939\pi\)
0.770635 + 0.637277i \(0.219939\pi\)
\(132\) 0 0
\(133\) −19.5235 −1.69290
\(134\) 2.97293 0.256822
\(135\) 0 0
\(136\) −4.33216 −0.371480
\(137\) 17.1254 1.46312 0.731561 0.681776i \(-0.238792\pi\)
0.731561 + 0.681776i \(0.238792\pi\)
\(138\) 0 0
\(139\) 3.15986 0.268016 0.134008 0.990980i \(-0.457215\pi\)
0.134008 + 0.990980i \(0.457215\pi\)
\(140\) −11.3298 −0.957543
\(141\) 0 0
\(142\) −6.06746 −0.509170
\(143\) 1.07530 0.0899207
\(144\) 0 0
\(145\) −11.4808 −0.953431
\(146\) 4.42989 0.366620
\(147\) 0 0
\(148\) −1.01155 −0.0831486
\(149\) −1.00000 −0.0819232
\(150\) 0 0
\(151\) −11.4612 −0.932701 −0.466351 0.884600i \(-0.654431\pi\)
−0.466351 + 0.884600i \(0.654431\pi\)
\(152\) −4.48387 −0.363690
\(153\) 0 0
\(154\) 1.15456 0.0930374
\(155\) −6.52931 −0.524447
\(156\) 0 0
\(157\) 13.4362 1.07233 0.536164 0.844114i \(-0.319873\pi\)
0.536164 + 0.844114i \(0.319873\pi\)
\(158\) −5.32897 −0.423950
\(159\) 0 0
\(160\) −2.60206 −0.205711
\(161\) 31.7056 2.49875
\(162\) 0 0
\(163\) −2.50198 −0.195970 −0.0979852 0.995188i \(-0.531240\pi\)
−0.0979852 + 0.995188i \(0.531240\pi\)
\(164\) 7.56092 0.590409
\(165\) 0 0
\(166\) 14.1972 1.10192
\(167\) 12.6702 0.980451 0.490226 0.871596i \(-0.336914\pi\)
0.490226 + 0.871596i \(0.336914\pi\)
\(168\) 0 0
\(169\) 3.44481 0.264985
\(170\) −11.2725 −0.864565
\(171\) 0 0
\(172\) 8.16581 0.622637
\(173\) −8.32877 −0.633225 −0.316612 0.948555i \(-0.602545\pi\)
−0.316612 + 0.948555i \(0.602545\pi\)
\(174\) 0 0
\(175\) −7.71001 −0.582822
\(176\) 0.265163 0.0199874
\(177\) 0 0
\(178\) 2.33739 0.175195
\(179\) −7.90674 −0.590978 −0.295489 0.955346i \(-0.595482\pi\)
−0.295489 + 0.955346i \(0.595482\pi\)
\(180\) 0 0
\(181\) 23.3762 1.73754 0.868769 0.495217i \(-0.164912\pi\)
0.868769 + 0.495217i \(0.164912\pi\)
\(182\) 17.6571 1.30883
\(183\) 0 0
\(184\) 7.28167 0.536812
\(185\) −2.63211 −0.193516
\(186\) 0 0
\(187\) 1.14873 0.0840034
\(188\) −4.04984 −0.295365
\(189\) 0 0
\(190\) −11.6673 −0.846436
\(191\) 14.9135 1.07910 0.539551 0.841953i \(-0.318594\pi\)
0.539551 + 0.841953i \(0.318594\pi\)
\(192\) 0 0
\(193\) −18.9804 −1.36624 −0.683120 0.730306i \(-0.739377\pi\)
−0.683120 + 0.730306i \(0.739377\pi\)
\(194\) −2.03002 −0.145747
\(195\) 0 0
\(196\) 11.9587 0.854195
\(197\) −7.79120 −0.555100 −0.277550 0.960711i \(-0.589522\pi\)
−0.277550 + 0.960711i \(0.589522\pi\)
\(198\) 0 0
\(199\) −8.84292 −0.626858 −0.313429 0.949612i \(-0.601478\pi\)
−0.313429 + 0.949612i \(0.601478\pi\)
\(200\) −1.77072 −0.125209
\(201\) 0 0
\(202\) −4.87422 −0.342949
\(203\) 19.2115 1.34838
\(204\) 0 0
\(205\) 19.6740 1.37409
\(206\) −6.98778 −0.486862
\(207\) 0 0
\(208\) 4.05522 0.281179
\(209\) 1.18896 0.0822420
\(210\) 0 0
\(211\) 17.6600 1.21576 0.607882 0.794028i \(-0.292019\pi\)
0.607882 + 0.794028i \(0.292019\pi\)
\(212\) 12.0518 0.827721
\(213\) 0 0
\(214\) −0.355452 −0.0242982
\(215\) 21.2479 1.44910
\(216\) 0 0
\(217\) 10.9258 0.741694
\(218\) 5.84322 0.395753
\(219\) 0 0
\(220\) 0.689971 0.0465178
\(221\) 17.5679 1.18174
\(222\) 0 0
\(223\) −11.3515 −0.760154 −0.380077 0.924955i \(-0.624102\pi\)
−0.380077 + 0.924955i \(0.624102\pi\)
\(224\) 4.35416 0.290925
\(225\) 0 0
\(226\) −19.5256 −1.29882
\(227\) −0.921850 −0.0611854 −0.0305927 0.999532i \(-0.509739\pi\)
−0.0305927 + 0.999532i \(0.509739\pi\)
\(228\) 0 0
\(229\) −4.88284 −0.322667 −0.161334 0.986900i \(-0.551579\pi\)
−0.161334 + 0.986900i \(0.551579\pi\)
\(230\) 18.9474 1.24935
\(231\) 0 0
\(232\) 4.41220 0.289675
\(233\) 15.2267 0.997532 0.498766 0.866737i \(-0.333787\pi\)
0.498766 + 0.866737i \(0.333787\pi\)
\(234\) 0 0
\(235\) −10.5379 −0.687418
\(236\) 2.90383 0.189023
\(237\) 0 0
\(238\) 18.8629 1.22270
\(239\) 17.2969 1.11884 0.559420 0.828884i \(-0.311024\pi\)
0.559420 + 0.828884i \(0.311024\pi\)
\(240\) 0 0
\(241\) 26.4918 1.70649 0.853243 0.521514i \(-0.174632\pi\)
0.853243 + 0.521514i \(0.174632\pi\)
\(242\) 10.9297 0.702587
\(243\) 0 0
\(244\) −4.59376 −0.294086
\(245\) 31.1174 1.98801
\(246\) 0 0
\(247\) 18.1831 1.15696
\(248\) 2.50928 0.159340
\(249\) 0 0
\(250\) 8.40278 0.531438
\(251\) 24.6534 1.55611 0.778055 0.628196i \(-0.216206\pi\)
0.778055 + 0.628196i \(0.216206\pi\)
\(252\) 0 0
\(253\) −1.93083 −0.121390
\(254\) 16.2907 1.02217
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.11123 −0.131695 −0.0658474 0.997830i \(-0.520975\pi\)
−0.0658474 + 0.997830i \(0.520975\pi\)
\(258\) 0 0
\(259\) 4.40444 0.273679
\(260\) 10.5519 0.654403
\(261\) 0 0
\(262\) −17.6406 −1.08984
\(263\) −3.80370 −0.234546 −0.117273 0.993100i \(-0.537415\pi\)
−0.117273 + 0.993100i \(0.537415\pi\)
\(264\) 0 0
\(265\) 31.3595 1.92640
\(266\) 19.5235 1.19706
\(267\) 0 0
\(268\) −2.97293 −0.181601
\(269\) 13.7256 0.836863 0.418432 0.908248i \(-0.362580\pi\)
0.418432 + 0.908248i \(0.362580\pi\)
\(270\) 0 0
\(271\) −17.0605 −1.03635 −0.518175 0.855275i \(-0.673388\pi\)
−0.518175 + 0.855275i \(0.673388\pi\)
\(272\) 4.33216 0.262676
\(273\) 0 0
\(274\) −17.1254 −1.03458
\(275\) 0.469530 0.0283137
\(276\) 0 0
\(277\) −10.2140 −0.613702 −0.306851 0.951758i \(-0.599275\pi\)
−0.306851 + 0.951758i \(0.599275\pi\)
\(278\) −3.15986 −0.189516
\(279\) 0 0
\(280\) 11.3298 0.677085
\(281\) −17.2972 −1.03186 −0.515931 0.856630i \(-0.672554\pi\)
−0.515931 + 0.856630i \(0.672554\pi\)
\(282\) 0 0
\(283\) −14.3288 −0.851758 −0.425879 0.904780i \(-0.640035\pi\)
−0.425879 + 0.904780i \(0.640035\pi\)
\(284\) 6.06746 0.360037
\(285\) 0 0
\(286\) −1.07530 −0.0635836
\(287\) −32.9215 −1.94329
\(288\) 0 0
\(289\) 1.76761 0.103977
\(290\) 11.4808 0.674177
\(291\) 0 0
\(292\) −4.42989 −0.259240
\(293\) 15.1302 0.883914 0.441957 0.897036i \(-0.354284\pi\)
0.441957 + 0.897036i \(0.354284\pi\)
\(294\) 0 0
\(295\) 7.55593 0.439923
\(296\) 1.01155 0.0587950
\(297\) 0 0
\(298\) 1.00000 0.0579284
\(299\) −29.5288 −1.70769
\(300\) 0 0
\(301\) −35.5553 −2.04937
\(302\) 11.4612 0.659519
\(303\) 0 0
\(304\) 4.48387 0.257168
\(305\) −11.9533 −0.684441
\(306\) 0 0
\(307\) −8.43123 −0.481196 −0.240598 0.970625i \(-0.577344\pi\)
−0.240598 + 0.970625i \(0.577344\pi\)
\(308\) −1.15456 −0.0657874
\(309\) 0 0
\(310\) 6.52931 0.370840
\(311\) 15.3131 0.868325 0.434162 0.900835i \(-0.357044\pi\)
0.434162 + 0.900835i \(0.357044\pi\)
\(312\) 0 0
\(313\) −9.41309 −0.532059 −0.266030 0.963965i \(-0.585712\pi\)
−0.266030 + 0.963965i \(0.585712\pi\)
\(314\) −13.4362 −0.758250
\(315\) 0 0
\(316\) 5.32897 0.299778
\(317\) 4.77010 0.267916 0.133958 0.990987i \(-0.457231\pi\)
0.133958 + 0.990987i \(0.457231\pi\)
\(318\) 0 0
\(319\) −1.16995 −0.0655049
\(320\) 2.60206 0.145460
\(321\) 0 0
\(322\) −31.7056 −1.76688
\(323\) 19.4249 1.08083
\(324\) 0 0
\(325\) 7.18066 0.398312
\(326\) 2.50198 0.138572
\(327\) 0 0
\(328\) −7.56092 −0.417482
\(329\) 17.6337 0.972175
\(330\) 0 0
\(331\) 22.1235 1.21601 0.608007 0.793931i \(-0.291969\pi\)
0.608007 + 0.793931i \(0.291969\pi\)
\(332\) −14.1972 −0.779173
\(333\) 0 0
\(334\) −12.6702 −0.693284
\(335\) −7.73574 −0.422649
\(336\) 0 0
\(337\) 33.4043 1.81965 0.909824 0.414995i \(-0.136217\pi\)
0.909824 + 0.414995i \(0.136217\pi\)
\(338\) −3.44481 −0.187373
\(339\) 0 0
\(340\) 11.2725 0.611340
\(341\) −0.665370 −0.0360318
\(342\) 0 0
\(343\) −21.5911 −1.16581
\(344\) −8.16581 −0.440271
\(345\) 0 0
\(346\) 8.32877 0.447758
\(347\) 35.4242 1.90167 0.950835 0.309697i \(-0.100228\pi\)
0.950835 + 0.309697i \(0.100228\pi\)
\(348\) 0 0
\(349\) 27.3976 1.46656 0.733280 0.679926i \(-0.237988\pi\)
0.733280 + 0.679926i \(0.237988\pi\)
\(350\) 7.71001 0.412117
\(351\) 0 0
\(352\) −0.265163 −0.0141333
\(353\) −14.5285 −0.773273 −0.386637 0.922232i \(-0.626363\pi\)
−0.386637 + 0.922232i \(0.626363\pi\)
\(354\) 0 0
\(355\) 15.7879 0.837934
\(356\) −2.33739 −0.123881
\(357\) 0 0
\(358\) 7.90674 0.417884
\(359\) −1.18115 −0.0623385 −0.0311693 0.999514i \(-0.509923\pi\)
−0.0311693 + 0.999514i \(0.509923\pi\)
\(360\) 0 0
\(361\) 1.10512 0.0581642
\(362\) −23.3762 −1.22863
\(363\) 0 0
\(364\) −17.6571 −0.925483
\(365\) −11.5268 −0.603342
\(366\) 0 0
\(367\) −11.0480 −0.576701 −0.288351 0.957525i \(-0.593107\pi\)
−0.288351 + 0.957525i \(0.593107\pi\)
\(368\) −7.28167 −0.379583
\(369\) 0 0
\(370\) 2.63211 0.136837
\(371\) −52.4755 −2.72439
\(372\) 0 0
\(373\) −38.1150 −1.97352 −0.986761 0.162181i \(-0.948147\pi\)
−0.986761 + 0.162181i \(0.948147\pi\)
\(374\) −1.14873 −0.0593994
\(375\) 0 0
\(376\) 4.04984 0.208854
\(377\) −17.8925 −0.921509
\(378\) 0 0
\(379\) −3.13216 −0.160888 −0.0804440 0.996759i \(-0.525634\pi\)
−0.0804440 + 0.996759i \(0.525634\pi\)
\(380\) 11.6673 0.598520
\(381\) 0 0
\(382\) −14.9135 −0.763040
\(383\) −12.2063 −0.623713 −0.311857 0.950129i \(-0.600951\pi\)
−0.311857 + 0.950129i \(0.600951\pi\)
\(384\) 0 0
\(385\) −3.00425 −0.153111
\(386\) 18.9804 0.966078
\(387\) 0 0
\(388\) 2.03002 0.103058
\(389\) −2.22633 −0.112880 −0.0564398 0.998406i \(-0.517975\pi\)
−0.0564398 + 0.998406i \(0.517975\pi\)
\(390\) 0 0
\(391\) −31.5454 −1.59532
\(392\) −11.9587 −0.604007
\(393\) 0 0
\(394\) 7.79120 0.392515
\(395\) 13.8663 0.697689
\(396\) 0 0
\(397\) 31.6782 1.58988 0.794941 0.606687i \(-0.207502\pi\)
0.794941 + 0.606687i \(0.207502\pi\)
\(398\) 8.84292 0.443255
\(399\) 0 0
\(400\) 1.77072 0.0885361
\(401\) 18.3003 0.913876 0.456938 0.889499i \(-0.348946\pi\)
0.456938 + 0.889499i \(0.348946\pi\)
\(402\) 0 0
\(403\) −10.1757 −0.506888
\(404\) 4.87422 0.242501
\(405\) 0 0
\(406\) −19.2115 −0.953449
\(407\) −0.268225 −0.0132954
\(408\) 0 0
\(409\) −5.20178 −0.257212 −0.128606 0.991696i \(-0.541050\pi\)
−0.128606 + 0.991696i \(0.541050\pi\)
\(410\) −19.6740 −0.971629
\(411\) 0 0
\(412\) 6.98778 0.344263
\(413\) −12.6437 −0.622157
\(414\) 0 0
\(415\) −36.9420 −1.81341
\(416\) −4.05522 −0.198824
\(417\) 0 0
\(418\) −1.18896 −0.0581539
\(419\) 6.23531 0.304615 0.152307 0.988333i \(-0.451330\pi\)
0.152307 + 0.988333i \(0.451330\pi\)
\(420\) 0 0
\(421\) −2.25873 −0.110084 −0.0550420 0.998484i \(-0.517529\pi\)
−0.0550420 + 0.998484i \(0.517529\pi\)
\(422\) −17.6600 −0.859675
\(423\) 0 0
\(424\) −12.0518 −0.585287
\(425\) 7.67105 0.372100
\(426\) 0 0
\(427\) 20.0020 0.967964
\(428\) 0.355452 0.0171814
\(429\) 0 0
\(430\) −21.2479 −1.02467
\(431\) 3.93223 0.189409 0.0947043 0.995505i \(-0.469809\pi\)
0.0947043 + 0.995505i \(0.469809\pi\)
\(432\) 0 0
\(433\) 38.5191 1.85111 0.925555 0.378613i \(-0.123599\pi\)
0.925555 + 0.378613i \(0.123599\pi\)
\(434\) −10.9258 −0.524457
\(435\) 0 0
\(436\) −5.84322 −0.279839
\(437\) −32.6501 −1.56187
\(438\) 0 0
\(439\) −9.87337 −0.471230 −0.235615 0.971846i \(-0.575710\pi\)
−0.235615 + 0.971846i \(0.575710\pi\)
\(440\) −0.689971 −0.0328931
\(441\) 0 0
\(442\) −17.5679 −0.835618
\(443\) 32.2395 1.53174 0.765872 0.642993i \(-0.222308\pi\)
0.765872 + 0.642993i \(0.222308\pi\)
\(444\) 0 0
\(445\) −6.08203 −0.288316
\(446\) 11.3515 0.537510
\(447\) 0 0
\(448\) −4.35416 −0.205715
\(449\) −17.5915 −0.830193 −0.415097 0.909777i \(-0.636252\pi\)
−0.415097 + 0.909777i \(0.636252\pi\)
\(450\) 0 0
\(451\) 2.00488 0.0944061
\(452\) 19.5256 0.918407
\(453\) 0 0
\(454\) 0.921850 0.0432646
\(455\) −45.9448 −2.15393
\(456\) 0 0
\(457\) 32.5964 1.52480 0.762398 0.647109i \(-0.224022\pi\)
0.762398 + 0.647109i \(0.224022\pi\)
\(458\) 4.88284 0.228160
\(459\) 0 0
\(460\) −18.9474 −0.883425
\(461\) −2.92979 −0.136454 −0.0682269 0.997670i \(-0.521734\pi\)
−0.0682269 + 0.997670i \(0.521734\pi\)
\(462\) 0 0
\(463\) −33.4152 −1.55294 −0.776468 0.630157i \(-0.782991\pi\)
−0.776468 + 0.630157i \(0.782991\pi\)
\(464\) −4.41220 −0.204831
\(465\) 0 0
\(466\) −15.2267 −0.705362
\(467\) 5.39495 0.249649 0.124824 0.992179i \(-0.460163\pi\)
0.124824 + 0.992179i \(0.460163\pi\)
\(468\) 0 0
\(469\) 12.9446 0.597727
\(470\) 10.5379 0.486078
\(471\) 0 0
\(472\) −2.90383 −0.133659
\(473\) 2.16527 0.0995593
\(474\) 0 0
\(475\) 7.93969 0.364298
\(476\) −18.8629 −0.864581
\(477\) 0 0
\(478\) −17.2969 −0.791140
\(479\) 37.6688 1.72113 0.860565 0.509340i \(-0.170111\pi\)
0.860565 + 0.509340i \(0.170111\pi\)
\(480\) 0 0
\(481\) −4.10204 −0.187037
\(482\) −26.4918 −1.20667
\(483\) 0 0
\(484\) −10.9297 −0.496804
\(485\) 5.28223 0.239854
\(486\) 0 0
\(487\) −16.1912 −0.733693 −0.366846 0.930282i \(-0.619562\pi\)
−0.366846 + 0.930282i \(0.619562\pi\)
\(488\) 4.59376 0.207950
\(489\) 0 0
\(490\) −31.1174 −1.40574
\(491\) 27.8221 1.25559 0.627797 0.778377i \(-0.283957\pi\)
0.627797 + 0.778377i \(0.283957\pi\)
\(492\) 0 0
\(493\) −19.1144 −0.860868
\(494\) −18.1831 −0.818096
\(495\) 0 0
\(496\) −2.50928 −0.112670
\(497\) −26.4187 −1.18504
\(498\) 0 0
\(499\) 20.8407 0.932957 0.466479 0.884532i \(-0.345522\pi\)
0.466479 + 0.884532i \(0.345522\pi\)
\(500\) −8.40278 −0.375784
\(501\) 0 0
\(502\) −24.6534 −1.10034
\(503\) 27.9083 1.24437 0.622185 0.782870i \(-0.286245\pi\)
0.622185 + 0.782870i \(0.286245\pi\)
\(504\) 0 0
\(505\) 12.6830 0.564387
\(506\) 1.93083 0.0858360
\(507\) 0 0
\(508\) −16.2907 −0.722782
\(509\) 15.4793 0.686107 0.343054 0.939316i \(-0.388539\pi\)
0.343054 + 0.939316i \(0.388539\pi\)
\(510\) 0 0
\(511\) 19.2885 0.853271
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 2.11123 0.0931222
\(515\) 18.1826 0.801223
\(516\) 0 0
\(517\) −1.07387 −0.0472287
\(518\) −4.40444 −0.193520
\(519\) 0 0
\(520\) −10.5519 −0.462733
\(521\) 2.30576 0.101017 0.0505085 0.998724i \(-0.483916\pi\)
0.0505085 + 0.998724i \(0.483916\pi\)
\(522\) 0 0
\(523\) 12.0468 0.526768 0.263384 0.964691i \(-0.415161\pi\)
0.263384 + 0.964691i \(0.415161\pi\)
\(524\) 17.6406 0.770635
\(525\) 0 0
\(526\) 3.80370 0.165849
\(527\) −10.8706 −0.473532
\(528\) 0 0
\(529\) 30.0227 1.30534
\(530\) −31.3595 −1.36217
\(531\) 0 0
\(532\) −19.5235 −0.846452
\(533\) 30.6612 1.32808
\(534\) 0 0
\(535\) 0.924909 0.0399873
\(536\) 2.97293 0.128411
\(537\) 0 0
\(538\) −13.7256 −0.591752
\(539\) 3.17102 0.136585
\(540\) 0 0
\(541\) 22.8801 0.983691 0.491846 0.870682i \(-0.336322\pi\)
0.491846 + 0.870682i \(0.336322\pi\)
\(542\) 17.0605 0.732810
\(543\) 0 0
\(544\) −4.33216 −0.185740
\(545\) −15.2044 −0.651285
\(546\) 0 0
\(547\) 2.43374 0.104059 0.0520297 0.998646i \(-0.483431\pi\)
0.0520297 + 0.998646i \(0.483431\pi\)
\(548\) 17.1254 0.731561
\(549\) 0 0
\(550\) −0.469530 −0.0200208
\(551\) −19.7838 −0.842817
\(552\) 0 0
\(553\) −23.2032 −0.986700
\(554\) 10.2140 0.433953
\(555\) 0 0
\(556\) 3.15986 0.134008
\(557\) 5.06592 0.214650 0.107325 0.994224i \(-0.465771\pi\)
0.107325 + 0.994224i \(0.465771\pi\)
\(558\) 0 0
\(559\) 33.1142 1.40058
\(560\) −11.3298 −0.478771
\(561\) 0 0
\(562\) 17.2972 0.729637
\(563\) −45.2633 −1.90762 −0.953810 0.300411i \(-0.902876\pi\)
−0.953810 + 0.300411i \(0.902876\pi\)
\(564\) 0 0
\(565\) 50.8068 2.13746
\(566\) 14.3288 0.602284
\(567\) 0 0
\(568\) −6.06746 −0.254585
\(569\) 28.5576 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(570\) 0 0
\(571\) 8.04764 0.336783 0.168392 0.985720i \(-0.446143\pi\)
0.168392 + 0.985720i \(0.446143\pi\)
\(572\) 1.07530 0.0449604
\(573\) 0 0
\(574\) 32.9215 1.37412
\(575\) −12.8938 −0.537709
\(576\) 0 0
\(577\) 0.983491 0.0409432 0.0204716 0.999790i \(-0.493483\pi\)
0.0204716 + 0.999790i \(0.493483\pi\)
\(578\) −1.76761 −0.0735229
\(579\) 0 0
\(580\) −11.4808 −0.476715
\(581\) 61.8170 2.56460
\(582\) 0 0
\(583\) 3.19569 0.132352
\(584\) 4.42989 0.183310
\(585\) 0 0
\(586\) −15.1302 −0.625022
\(587\) −35.2018 −1.45294 −0.726468 0.687201i \(-0.758839\pi\)
−0.726468 + 0.687201i \(0.758839\pi\)
\(588\) 0 0
\(589\) −11.2513 −0.463602
\(590\) −7.55593 −0.311073
\(591\) 0 0
\(592\) −1.01155 −0.0415743
\(593\) −30.4208 −1.24923 −0.624617 0.780931i \(-0.714745\pi\)
−0.624617 + 0.780931i \(0.714745\pi\)
\(594\) 0 0
\(595\) −49.0825 −2.01219
\(596\) −1.00000 −0.0409616
\(597\) 0 0
\(598\) 29.5288 1.20752
\(599\) −18.6711 −0.762880 −0.381440 0.924394i \(-0.624572\pi\)
−0.381440 + 0.924394i \(0.624572\pi\)
\(600\) 0 0
\(601\) 17.9550 0.732398 0.366199 0.930537i \(-0.380659\pi\)
0.366199 + 0.930537i \(0.380659\pi\)
\(602\) 35.5553 1.44912
\(603\) 0 0
\(604\) −11.4612 −0.466351
\(605\) −28.4397 −1.15624
\(606\) 0 0
\(607\) −32.4843 −1.31850 −0.659249 0.751924i \(-0.729126\pi\)
−0.659249 + 0.751924i \(0.729126\pi\)
\(608\) −4.48387 −0.181845
\(609\) 0 0
\(610\) 11.9533 0.483973
\(611\) −16.4230 −0.664403
\(612\) 0 0
\(613\) 32.1250 1.29752 0.648758 0.760995i \(-0.275289\pi\)
0.648758 + 0.760995i \(0.275289\pi\)
\(614\) 8.43123 0.340257
\(615\) 0 0
\(616\) 1.15456 0.0465187
\(617\) −24.3370 −0.979770 −0.489885 0.871787i \(-0.662961\pi\)
−0.489885 + 0.871787i \(0.662961\pi\)
\(618\) 0 0
\(619\) 45.5842 1.83218 0.916091 0.400969i \(-0.131327\pi\)
0.916091 + 0.400969i \(0.131327\pi\)
\(620\) −6.52931 −0.262223
\(621\) 0 0
\(622\) −15.3131 −0.613998
\(623\) 10.1774 0.407748
\(624\) 0 0
\(625\) −30.7182 −1.22873
\(626\) 9.41309 0.376223
\(627\) 0 0
\(628\) 13.4362 0.536164
\(629\) −4.38218 −0.174729
\(630\) 0 0
\(631\) 0.936949 0.0372993 0.0186497 0.999826i \(-0.494063\pi\)
0.0186497 + 0.999826i \(0.494063\pi\)
\(632\) −5.32897 −0.211975
\(633\) 0 0
\(634\) −4.77010 −0.189445
\(635\) −42.3893 −1.68217
\(636\) 0 0
\(637\) 48.4953 1.92145
\(638\) 1.16995 0.0463190
\(639\) 0 0
\(640\) −2.60206 −0.102855
\(641\) 11.5960 0.458013 0.229007 0.973425i \(-0.426452\pi\)
0.229007 + 0.973425i \(0.426452\pi\)
\(642\) 0 0
\(643\) 37.8312 1.49192 0.745958 0.665993i \(-0.231992\pi\)
0.745958 + 0.665993i \(0.231992\pi\)
\(644\) 31.7056 1.24938
\(645\) 0 0
\(646\) −19.4249 −0.764261
\(647\) 31.6105 1.24274 0.621368 0.783519i \(-0.286577\pi\)
0.621368 + 0.783519i \(0.286577\pi\)
\(648\) 0 0
\(649\) 0.769988 0.0302247
\(650\) −7.18066 −0.281649
\(651\) 0 0
\(652\) −2.50198 −0.0979852
\(653\) −15.1514 −0.592922 −0.296461 0.955045i \(-0.595806\pi\)
−0.296461 + 0.955045i \(0.595806\pi\)
\(654\) 0 0
\(655\) 45.9020 1.79354
\(656\) 7.56092 0.295205
\(657\) 0 0
\(658\) −17.6337 −0.687431
\(659\) −7.23718 −0.281920 −0.140960 0.990015i \(-0.545019\pi\)
−0.140960 + 0.990015i \(0.545019\pi\)
\(660\) 0 0
\(661\) −32.1606 −1.25090 −0.625452 0.780263i \(-0.715085\pi\)
−0.625452 + 0.780263i \(0.715085\pi\)
\(662\) −22.1235 −0.859852
\(663\) 0 0
\(664\) 14.1972 0.550959
\(665\) −50.8014 −1.96999
\(666\) 0 0
\(667\) 32.1282 1.24401
\(668\) 12.6702 0.490226
\(669\) 0 0
\(670\) 7.73574 0.298858
\(671\) −1.21810 −0.0470241
\(672\) 0 0
\(673\) −5.41204 −0.208619 −0.104309 0.994545i \(-0.533263\pi\)
−0.104309 + 0.994545i \(0.533263\pi\)
\(674\) −33.4043 −1.28669
\(675\) 0 0
\(676\) 3.44481 0.132493
\(677\) 17.8399 0.685641 0.342821 0.939401i \(-0.388618\pi\)
0.342821 + 0.939401i \(0.388618\pi\)
\(678\) 0 0
\(679\) −8.83902 −0.339211
\(680\) −11.2725 −0.432282
\(681\) 0 0
\(682\) 0.665370 0.0254783
\(683\) −1.65392 −0.0632856 −0.0316428 0.999499i \(-0.510074\pi\)
−0.0316428 + 0.999499i \(0.510074\pi\)
\(684\) 0 0
\(685\) 44.5613 1.70260
\(686\) 21.5911 0.824353
\(687\) 0 0
\(688\) 8.16581 0.311319
\(689\) 48.8727 1.86190
\(690\) 0 0
\(691\) 3.75941 0.143015 0.0715073 0.997440i \(-0.477219\pi\)
0.0715073 + 0.997440i \(0.477219\pi\)
\(692\) −8.32877 −0.316612
\(693\) 0 0
\(694\) −35.4242 −1.34468
\(695\) 8.22215 0.311884
\(696\) 0 0
\(697\) 32.7551 1.24069
\(698\) −27.3976 −1.03701
\(699\) 0 0
\(700\) −7.71001 −0.291411
\(701\) −21.4942 −0.811824 −0.405912 0.913912i \(-0.633046\pi\)
−0.405912 + 0.913912i \(0.633046\pi\)
\(702\) 0 0
\(703\) −4.53565 −0.171065
\(704\) 0.265163 0.00999372
\(705\) 0 0
\(706\) 14.5285 0.546787
\(707\) −21.2231 −0.798178
\(708\) 0 0
\(709\) −17.9742 −0.675037 −0.337519 0.941319i \(-0.609588\pi\)
−0.337519 + 0.941319i \(0.609588\pi\)
\(710\) −15.7879 −0.592509
\(711\) 0 0
\(712\) 2.33739 0.0875974
\(713\) 18.2718 0.684284
\(714\) 0 0
\(715\) 2.79798 0.104639
\(716\) −7.90674 −0.295489
\(717\) 0 0
\(718\) 1.18115 0.0440800
\(719\) −23.9192 −0.892035 −0.446017 0.895024i \(-0.647158\pi\)
−0.446017 + 0.895024i \(0.647158\pi\)
\(720\) 0 0
\(721\) −30.4259 −1.13312
\(722\) −1.10512 −0.0411283
\(723\) 0 0
\(724\) 23.3762 0.868769
\(725\) −7.81278 −0.290160
\(726\) 0 0
\(727\) −36.8898 −1.36817 −0.684083 0.729404i \(-0.739798\pi\)
−0.684083 + 0.729404i \(0.739798\pi\)
\(728\) 17.6571 0.654415
\(729\) 0 0
\(730\) 11.5268 0.426627
\(731\) 35.3756 1.30841
\(732\) 0 0
\(733\) −12.0334 −0.444463 −0.222231 0.974994i \(-0.571334\pi\)
−0.222231 + 0.974994i \(0.571334\pi\)
\(734\) 11.0480 0.407789
\(735\) 0 0
\(736\) 7.28167 0.268406
\(737\) −0.788312 −0.0290378
\(738\) 0 0
\(739\) 0.356220 0.0131038 0.00655189 0.999979i \(-0.497914\pi\)
0.00655189 + 0.999979i \(0.497914\pi\)
\(740\) −2.63211 −0.0967582
\(741\) 0 0
\(742\) 52.4755 1.92644
\(743\) 10.8681 0.398710 0.199355 0.979927i \(-0.436115\pi\)
0.199355 + 0.979927i \(0.436115\pi\)
\(744\) 0 0
\(745\) −2.60206 −0.0953321
\(746\) 38.1150 1.39549
\(747\) 0 0
\(748\) 1.14873 0.0420017
\(749\) −1.54770 −0.0565516
\(750\) 0 0
\(751\) −34.1809 −1.24728 −0.623640 0.781712i \(-0.714347\pi\)
−0.623640 + 0.781712i \(0.714347\pi\)
\(752\) −4.04984 −0.147682
\(753\) 0 0
\(754\) 17.8925 0.651605
\(755\) −29.8228 −1.08536
\(756\) 0 0
\(757\) 46.4945 1.68987 0.844936 0.534868i \(-0.179639\pi\)
0.844936 + 0.534868i \(0.179639\pi\)
\(758\) 3.13216 0.113765
\(759\) 0 0
\(760\) −11.6673 −0.423218
\(761\) 4.48038 0.162413 0.0812067 0.996697i \(-0.474123\pi\)
0.0812067 + 0.996697i \(0.474123\pi\)
\(762\) 0 0
\(763\) 25.4423 0.921074
\(764\) 14.9135 0.539551
\(765\) 0 0
\(766\) 12.2063 0.441032
\(767\) 11.7757 0.425194
\(768\) 0 0
\(769\) 30.6664 1.10586 0.552929 0.833229i \(-0.313510\pi\)
0.552929 + 0.833229i \(0.313510\pi\)
\(770\) 3.00425 0.108266
\(771\) 0 0
\(772\) −18.9804 −0.683120
\(773\) −9.53310 −0.342882 −0.171441 0.985194i \(-0.554842\pi\)
−0.171441 + 0.985194i \(0.554842\pi\)
\(774\) 0 0
\(775\) −4.44324 −0.159606
\(776\) −2.03002 −0.0728734
\(777\) 0 0
\(778\) 2.22633 0.0798179
\(779\) 33.9022 1.21467
\(780\) 0 0
\(781\) 1.60887 0.0575698
\(782\) 31.5454 1.12806
\(783\) 0 0
\(784\) 11.9587 0.427098
\(785\) 34.9619 1.24784
\(786\) 0 0
\(787\) 18.0080 0.641916 0.320958 0.947093i \(-0.395995\pi\)
0.320958 + 0.947093i \(0.395995\pi\)
\(788\) −7.79120 −0.277550
\(789\) 0 0
\(790\) −13.8663 −0.493341
\(791\) −85.0177 −3.02288
\(792\) 0 0
\(793\) −18.6287 −0.661525
\(794\) −31.6782 −1.12422
\(795\) 0 0
\(796\) −8.84292 −0.313429
\(797\) 1.60934 0.0570057 0.0285029 0.999594i \(-0.490926\pi\)
0.0285029 + 0.999594i \(0.490926\pi\)
\(798\) 0 0
\(799\) −17.5445 −0.620681
\(800\) −1.77072 −0.0626044
\(801\) 0 0
\(802\) −18.3003 −0.646208
\(803\) −1.17464 −0.0414523
\(804\) 0 0
\(805\) 82.4999 2.90774
\(806\) 10.1757 0.358424
\(807\) 0 0
\(808\) −4.87422 −0.171474
\(809\) 29.1199 1.02380 0.511900 0.859045i \(-0.328942\pi\)
0.511900 + 0.859045i \(0.328942\pi\)
\(810\) 0 0
\(811\) 55.9566 1.96490 0.982451 0.186520i \(-0.0597208\pi\)
0.982451 + 0.186520i \(0.0597208\pi\)
\(812\) 19.2115 0.674190
\(813\) 0 0
\(814\) 0.268225 0.00940128
\(815\) −6.51031 −0.228046
\(816\) 0 0
\(817\) 36.6145 1.28098
\(818\) 5.20178 0.181876
\(819\) 0 0
\(820\) 19.6740 0.687045
\(821\) 41.8838 1.46175 0.730877 0.682509i \(-0.239111\pi\)
0.730877 + 0.682509i \(0.239111\pi\)
\(822\) 0 0
\(823\) −9.09408 −0.317000 −0.158500 0.987359i \(-0.550666\pi\)
−0.158500 + 0.987359i \(0.550666\pi\)
\(824\) −6.98778 −0.243431
\(825\) 0 0
\(826\) 12.6437 0.439932
\(827\) 32.6643 1.13585 0.567924 0.823081i \(-0.307747\pi\)
0.567924 + 0.823081i \(0.307747\pi\)
\(828\) 0 0
\(829\) −30.7027 −1.06635 −0.533174 0.846006i \(-0.679001\pi\)
−0.533174 + 0.846006i \(0.679001\pi\)
\(830\) 36.9420 1.28228
\(831\) 0 0
\(832\) 4.05522 0.140589
\(833\) 51.8072 1.79501
\(834\) 0 0
\(835\) 32.9687 1.14093
\(836\) 1.18896 0.0411210
\(837\) 0 0
\(838\) −6.23531 −0.215395
\(839\) −35.8451 −1.23751 −0.618755 0.785584i \(-0.712363\pi\)
−0.618755 + 0.785584i \(0.712363\pi\)
\(840\) 0 0
\(841\) −9.53245 −0.328705
\(842\) 2.25873 0.0778411
\(843\) 0 0
\(844\) 17.6600 0.607882
\(845\) 8.96360 0.308357
\(846\) 0 0
\(847\) 47.5896 1.63520
\(848\) 12.0518 0.413860
\(849\) 0 0
\(850\) −7.67105 −0.263115
\(851\) 7.36575 0.252495
\(852\) 0 0
\(853\) 52.0443 1.78196 0.890982 0.454039i \(-0.150017\pi\)
0.890982 + 0.454039i \(0.150017\pi\)
\(854\) −20.0020 −0.684454
\(855\) 0 0
\(856\) −0.355452 −0.0121491
\(857\) −15.0154 −0.512918 −0.256459 0.966555i \(-0.582556\pi\)
−0.256459 + 0.966555i \(0.582556\pi\)
\(858\) 0 0
\(859\) −19.4136 −0.662383 −0.331192 0.943564i \(-0.607451\pi\)
−0.331192 + 0.943564i \(0.607451\pi\)
\(860\) 21.2479 0.724548
\(861\) 0 0
\(862\) −3.93223 −0.133932
\(863\) −24.8730 −0.846687 −0.423343 0.905969i \(-0.639144\pi\)
−0.423343 + 0.905969i \(0.639144\pi\)
\(864\) 0 0
\(865\) −21.6720 −0.736869
\(866\) −38.5191 −1.30893
\(867\) 0 0
\(868\) 10.9258 0.370847
\(869\) 1.41305 0.0479343
\(870\) 0 0
\(871\) −12.0559 −0.408498
\(872\) 5.84322 0.197876
\(873\) 0 0
\(874\) 32.6501 1.10441
\(875\) 36.5871 1.23687
\(876\) 0 0
\(877\) −30.5738 −1.03240 −0.516202 0.856467i \(-0.672655\pi\)
−0.516202 + 0.856467i \(0.672655\pi\)
\(878\) 9.87337 0.333210
\(879\) 0 0
\(880\) 0.689971 0.0232589
\(881\) 35.1220 1.18329 0.591644 0.806199i \(-0.298479\pi\)
0.591644 + 0.806199i \(0.298479\pi\)
\(882\) 0 0
\(883\) −26.3339 −0.886205 −0.443103 0.896471i \(-0.646122\pi\)
−0.443103 + 0.896471i \(0.646122\pi\)
\(884\) 17.5679 0.590871
\(885\) 0 0
\(886\) −32.2395 −1.08311
\(887\) 49.0364 1.64648 0.823241 0.567692i \(-0.192163\pi\)
0.823241 + 0.567692i \(0.192163\pi\)
\(888\) 0 0
\(889\) 70.9323 2.37899
\(890\) 6.08203 0.203870
\(891\) 0 0
\(892\) −11.3515 −0.380077
\(893\) −18.1590 −0.607666
\(894\) 0 0
\(895\) −20.5738 −0.687707
\(896\) 4.35416 0.145462
\(897\) 0 0
\(898\) 17.5915 0.587035
\(899\) 11.0715 0.369254
\(900\) 0 0
\(901\) 52.2103 1.73938
\(902\) −2.00488 −0.0667552
\(903\) 0 0
\(904\) −19.5256 −0.649412
\(905\) 60.8263 2.02193
\(906\) 0 0
\(907\) −16.0934 −0.534372 −0.267186 0.963645i \(-0.586094\pi\)
−0.267186 + 0.963645i \(0.586094\pi\)
\(908\) −0.921850 −0.0305927
\(909\) 0 0
\(910\) 45.9448 1.52306
\(911\) 0.00281244 9.31803e−5 0 4.65902e−5 1.00000i \(-0.499985\pi\)
4.65902e−5 1.00000i \(0.499985\pi\)
\(912\) 0 0
\(913\) −3.76458 −0.124589
\(914\) −32.5964 −1.07819
\(915\) 0 0
\(916\) −4.88284 −0.161334
\(917\) −76.8102 −2.53650
\(918\) 0 0
\(919\) 24.5632 0.810266 0.405133 0.914258i \(-0.367225\pi\)
0.405133 + 0.914258i \(0.367225\pi\)
\(920\) 18.9474 0.624676
\(921\) 0 0
\(922\) 2.92979 0.0964874
\(923\) 24.6049 0.809879
\(924\) 0 0
\(925\) −1.79117 −0.0588932
\(926\) 33.4152 1.09809
\(927\) 0 0
\(928\) 4.41220 0.144838
\(929\) −35.7462 −1.17279 −0.586397 0.810024i \(-0.699454\pi\)
−0.586397 + 0.810024i \(0.699454\pi\)
\(930\) 0 0
\(931\) 53.6215 1.75737
\(932\) 15.2267 0.498766
\(933\) 0 0
\(934\) −5.39495 −0.176528
\(935\) 2.98907 0.0977529
\(936\) 0 0
\(937\) −32.5134 −1.06217 −0.531083 0.847320i \(-0.678215\pi\)
−0.531083 + 0.847320i \(0.678215\pi\)
\(938\) −12.9446 −0.422657
\(939\) 0 0
\(940\) −10.5379 −0.343709
\(941\) −8.80935 −0.287176 −0.143588 0.989638i \(-0.545864\pi\)
−0.143588 + 0.989638i \(0.545864\pi\)
\(942\) 0 0
\(943\) −55.0562 −1.79288
\(944\) 2.90383 0.0945115
\(945\) 0 0
\(946\) −2.16527 −0.0703991
\(947\) 11.0349 0.358585 0.179292 0.983796i \(-0.442619\pi\)
0.179292 + 0.983796i \(0.442619\pi\)
\(948\) 0 0
\(949\) −17.9642 −0.583142
\(950\) −7.93969 −0.257598
\(951\) 0 0
\(952\) 18.8629 0.611351
\(953\) −25.3792 −0.822112 −0.411056 0.911610i \(-0.634840\pi\)
−0.411056 + 0.911610i \(0.634840\pi\)
\(954\) 0 0
\(955\) 38.8058 1.25573
\(956\) 17.2969 0.559420
\(957\) 0 0
\(958\) −37.6688 −1.21702
\(959\) −74.5668 −2.40789
\(960\) 0 0
\(961\) −24.7035 −0.796887
\(962\) 4.10204 0.132255
\(963\) 0 0
\(964\) 26.4918 0.853243
\(965\) −49.3882 −1.58986
\(966\) 0 0
\(967\) −9.02425 −0.290200 −0.145100 0.989417i \(-0.546350\pi\)
−0.145100 + 0.989417i \(0.546350\pi\)
\(968\) 10.9297 0.351293
\(969\) 0 0
\(970\) −5.28223 −0.169602
\(971\) 43.0296 1.38089 0.690443 0.723387i \(-0.257415\pi\)
0.690443 + 0.723387i \(0.257415\pi\)
\(972\) 0 0
\(973\) −13.7585 −0.441079
\(974\) 16.1912 0.518799
\(975\) 0 0
\(976\) −4.59376 −0.147043
\(977\) −32.8446 −1.05079 −0.525396 0.850858i \(-0.676083\pi\)
−0.525396 + 0.850858i \(0.676083\pi\)
\(978\) 0 0
\(979\) −0.619790 −0.0198086
\(980\) 31.1174 0.994007
\(981\) 0 0
\(982\) −27.8221 −0.887839
\(983\) −36.3307 −1.15877 −0.579385 0.815054i \(-0.696707\pi\)
−0.579385 + 0.815054i \(0.696707\pi\)
\(984\) 0 0
\(985\) −20.2732 −0.645957
\(986\) 19.1144 0.608726
\(987\) 0 0
\(988\) 18.1831 0.578481
\(989\) −59.4607 −1.89074
\(990\) 0 0
\(991\) −37.1719 −1.18080 −0.590402 0.807109i \(-0.701031\pi\)
−0.590402 + 0.807109i \(0.701031\pi\)
\(992\) 2.50928 0.0796699
\(993\) 0 0
\(994\) 26.4187 0.837950
\(995\) −23.0098 −0.729460
\(996\) 0 0
\(997\) 1.67330 0.0529938 0.0264969 0.999649i \(-0.491565\pi\)
0.0264969 + 0.999649i \(0.491565\pi\)
\(998\) −20.8407 −0.659700
\(999\) 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 8046.2.a.k.1.9 12
3.2 odd 2 8046.2.a.n.1.4 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8046.2.a.k.1.9 12 1.1 even 1 trivial
8046.2.a.n.1.4 yes 12 3.2 odd 2