Properties

Label 1755.2.a.s.1.2
Level $1755$
Weight $2$
Character 1755.1
Self dual yes
Analytic conductor $14.014$
Analytic rank $0$
Dimension $4$
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] = [1755,2,Mod(1,1755)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1755, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1755.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1755.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: \(14.0137455547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1957.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.06150\) of defining polynomial
Character \(\chi\) \(=\) 1755.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.514916 q^{2} -1.73486 q^{4} -1.00000 q^{5} -4.44964 q^{7} -1.92314 q^{8} +O(q^{10})\) \(q+0.514916 q^{2} -1.73486 q^{4} -1.00000 q^{5} -4.44964 q^{7} -1.92314 q^{8} -0.514916 q^{10} -5.37278 q^{11} +1.00000 q^{13} -2.29119 q^{14} +2.47947 q^{16} +4.15845 q^{17} -2.50517 q^{19} +1.73486 q^{20} -2.76653 q^{22} +2.60625 q^{23} +1.00000 q^{25} +0.514916 q^{26} +7.71950 q^{28} -3.18450 q^{29} +4.19424 q^{31} +5.12300 q^{32} +2.14125 q^{34} +4.44964 q^{35} -5.57641 q^{37} -1.28995 q^{38} +1.92314 q^{40} -4.43244 q^{41} +8.18739 q^{43} +9.32102 q^{44} +1.34200 q^{46} +1.07975 q^{47} +12.7993 q^{49} +0.514916 q^{50} -1.73486 q^{52} +12.0816 q^{53} +5.37278 q^{55} +8.55727 q^{56} -1.63975 q^{58} +10.9790 q^{59} -6.66213 q^{61} +2.15968 q^{62} -2.32102 q^{64} -1.00000 q^{65} -3.83777 q^{67} -7.21433 q^{68} +2.29119 q^{70} +5.54842 q^{71} -13.2270 q^{73} -2.87139 q^{74} +4.34613 q^{76} +23.9069 q^{77} -6.82530 q^{79} -2.47947 q^{80} -2.28233 q^{82} -7.79453 q^{83} -4.15845 q^{85} +4.21582 q^{86} +10.3326 q^{88} +14.5602 q^{89} -4.44964 q^{91} -4.52148 q^{92} +0.555982 q^{94} +2.50517 q^{95} -1.49070 q^{97} +6.59054 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} + 3 q^{8} - 3 q^{10} + 4 q^{11} + 4 q^{13} + 3 q^{14} - 3 q^{16} + 14 q^{17} - 4 q^{19} - 3 q^{20} + 9 q^{22} + 5 q^{23} + 4 q^{25} + 3 q^{26} + 2 q^{28} + 12 q^{29} - q^{31} + 4 q^{32} - q^{34} + 3 q^{35} - 15 q^{37} + 14 q^{38} - 3 q^{40} + 4 q^{41} - 4 q^{43} + 27 q^{44} + 26 q^{46} + 3 q^{47} + 21 q^{49} + 3 q^{50} + 3 q^{52} + 35 q^{53} - 4 q^{55} - 16 q^{56} + 21 q^{58} + 13 q^{59} - q^{61} + 13 q^{62} + q^{64} - 4 q^{65} + 6 q^{67} - 6 q^{68} - 3 q^{70} + q^{71} - 25 q^{73} - 16 q^{74} + 33 q^{76} + 30 q^{77} + 25 q^{79} + 3 q^{80} - 26 q^{82} - 25 q^{83} - 14 q^{85} + 11 q^{86} + 26 q^{88} - 4 q^{89} - 3 q^{91} + 17 q^{92} + 29 q^{94} + 4 q^{95} - 17 q^{97} + 31 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.514916 0.364101 0.182050 0.983289i \(-0.441727\pi\)
0.182050 + 0.983289i \(0.441727\pi\)
\(3\) 0 0
\(4\) −1.73486 −0.867431
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.44964 −1.68180 −0.840902 0.541187i \(-0.817975\pi\)
−0.840902 + 0.541187i \(0.817975\pi\)
\(8\) −1.92314 −0.679933
\(9\) 0 0
\(10\) −0.514916 −0.162831
\(11\) −5.37278 −1.61995 −0.809976 0.586463i \(-0.800520\pi\)
−0.809976 + 0.586463i \(0.800520\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −2.29119 −0.612346
\(15\) 0 0
\(16\) 2.47947 0.619867
\(17\) 4.15845 1.00857 0.504286 0.863537i \(-0.331756\pi\)
0.504286 + 0.863537i \(0.331756\pi\)
\(18\) 0 0
\(19\) −2.50517 −0.574726 −0.287363 0.957822i \(-0.592779\pi\)
−0.287363 + 0.957822i \(0.592779\pi\)
\(20\) 1.73486 0.387927
\(21\) 0 0
\(22\) −2.76653 −0.589826
\(23\) 2.60625 0.543440 0.271720 0.962376i \(-0.412408\pi\)
0.271720 + 0.962376i \(0.412408\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.514916 0.100983
\(27\) 0 0
\(28\) 7.71950 1.45885
\(29\) −3.18450 −0.591346 −0.295673 0.955289i \(-0.595544\pi\)
−0.295673 + 0.955289i \(0.595544\pi\)
\(30\) 0 0
\(31\) 4.19424 0.753308 0.376654 0.926354i \(-0.377075\pi\)
0.376654 + 0.926354i \(0.377075\pi\)
\(32\) 5.12300 0.905627
\(33\) 0 0
\(34\) 2.14125 0.367221
\(35\) 4.44964 0.752126
\(36\) 0 0
\(37\) −5.57641 −0.916757 −0.458378 0.888757i \(-0.651570\pi\)
−0.458378 + 0.888757i \(0.651570\pi\)
\(38\) −1.28995 −0.209258
\(39\) 0 0
\(40\) 1.92314 0.304075
\(41\) −4.43244 −0.692231 −0.346115 0.938192i \(-0.612499\pi\)
−0.346115 + 0.938192i \(0.612499\pi\)
\(42\) 0 0
\(43\) 8.18739 1.24857 0.624283 0.781199i \(-0.285391\pi\)
0.624283 + 0.781199i \(0.285391\pi\)
\(44\) 9.32102 1.40520
\(45\) 0 0
\(46\) 1.34200 0.197867
\(47\) 1.07975 0.157498 0.0787490 0.996894i \(-0.474907\pi\)
0.0787490 + 0.996894i \(0.474907\pi\)
\(48\) 0 0
\(49\) 12.7993 1.82846
\(50\) 0.514916 0.0728201
\(51\) 0 0
\(52\) −1.73486 −0.240582
\(53\) 12.0816 1.65953 0.829767 0.558110i \(-0.188473\pi\)
0.829767 + 0.558110i \(0.188473\pi\)
\(54\) 0 0
\(55\) 5.37278 0.724465
\(56\) 8.55727 1.14351
\(57\) 0 0
\(58\) −1.63975 −0.215310
\(59\) 10.9790 1.42935 0.714673 0.699458i \(-0.246575\pi\)
0.714673 + 0.699458i \(0.246575\pi\)
\(60\) 0 0
\(61\) −6.66213 −0.852998 −0.426499 0.904488i \(-0.640253\pi\)
−0.426499 + 0.904488i \(0.640253\pi\)
\(62\) 2.15968 0.274280
\(63\) 0 0
\(64\) −2.32102 −0.290128
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −3.83777 −0.468858 −0.234429 0.972133i \(-0.575322\pi\)
−0.234429 + 0.972133i \(0.575322\pi\)
\(68\) −7.21433 −0.874866
\(69\) 0 0
\(70\) 2.29119 0.273849
\(71\) 5.54842 0.658476 0.329238 0.944247i \(-0.393208\pi\)
0.329238 + 0.944247i \(0.393208\pi\)
\(72\) 0 0
\(73\) −13.2270 −1.54810 −0.774050 0.633125i \(-0.781772\pi\)
−0.774050 + 0.633125i \(0.781772\pi\)
\(74\) −2.87139 −0.333792
\(75\) 0 0
\(76\) 4.34613 0.498535
\(77\) 23.9069 2.72444
\(78\) 0 0
\(79\) −6.82530 −0.767907 −0.383953 0.923353i \(-0.625438\pi\)
−0.383953 + 0.923353i \(0.625438\pi\)
\(80\) −2.47947 −0.277213
\(81\) 0 0
\(82\) −2.28233 −0.252042
\(83\) −7.79453 −0.855560 −0.427780 0.903883i \(-0.640704\pi\)
−0.427780 + 0.903883i \(0.640704\pi\)
\(84\) 0 0
\(85\) −4.15845 −0.451047
\(86\) 4.21582 0.454603
\(87\) 0 0
\(88\) 10.3326 1.10146
\(89\) 14.5602 1.54337 0.771687 0.636002i \(-0.219413\pi\)
0.771687 + 0.636002i \(0.219413\pi\)
\(90\) 0 0
\(91\) −4.44964 −0.466448
\(92\) −4.52148 −0.471397
\(93\) 0 0
\(94\) 0.555982 0.0573451
\(95\) 2.50517 0.257025
\(96\) 0 0
\(97\) −1.49070 −0.151358 −0.0756789 0.997132i \(-0.524112\pi\)
−0.0756789 + 0.997132i \(0.524112\pi\)
\(98\) 6.59054 0.665745
\(99\) 0 0
\(100\) −1.73486 −0.173486
\(101\) 9.39972 0.935307 0.467653 0.883912i \(-0.345100\pi\)
0.467653 + 0.883912i \(0.345100\pi\)
\(102\) 0 0
\(103\) −7.66485 −0.755240 −0.377620 0.925961i \(-0.623258\pi\)
−0.377620 + 0.925961i \(0.623258\pi\)
\(104\) −1.92314 −0.188579
\(105\) 0 0
\(106\) 6.22100 0.604237
\(107\) −14.2422 −1.37685 −0.688423 0.725309i \(-0.741697\pi\)
−0.688423 + 0.725309i \(0.741697\pi\)
\(108\) 0 0
\(109\) 4.35452 0.417088 0.208544 0.978013i \(-0.433128\pi\)
0.208544 + 0.978013i \(0.433128\pi\)
\(110\) 2.76653 0.263778
\(111\) 0 0
\(112\) −11.0327 −1.04249
\(113\) 14.0078 1.31774 0.658871 0.752256i \(-0.271034\pi\)
0.658871 + 0.752256i \(0.271034\pi\)
\(114\) 0 0
\(115\) −2.60625 −0.243034
\(116\) 5.52466 0.512952
\(117\) 0 0
\(118\) 5.65327 0.520426
\(119\) −18.5036 −1.69622
\(120\) 0 0
\(121\) 17.8667 1.62425
\(122\) −3.43044 −0.310577
\(123\) 0 0
\(124\) −7.27643 −0.653442
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −18.7783 −1.66630 −0.833151 0.553045i \(-0.813466\pi\)
−0.833151 + 0.553045i \(0.813466\pi\)
\(128\) −11.4411 −1.01126
\(129\) 0 0
\(130\) −0.514916 −0.0451611
\(131\) −6.57369 −0.574346 −0.287173 0.957879i \(-0.592715\pi\)
−0.287173 + 0.957879i \(0.592715\pi\)
\(132\) 0 0
\(133\) 11.1471 0.966576
\(134\) −1.97613 −0.170712
\(135\) 0 0
\(136\) −7.99728 −0.685761
\(137\) −10.4744 −0.894893 −0.447446 0.894311i \(-0.647666\pi\)
−0.447446 + 0.894311i \(0.647666\pi\)
\(138\) 0 0
\(139\) 7.96639 0.675700 0.337850 0.941200i \(-0.390300\pi\)
0.337850 + 0.941200i \(0.390300\pi\)
\(140\) −7.71950 −0.652417
\(141\) 0 0
\(142\) 2.85697 0.239752
\(143\) −5.37278 −0.449294
\(144\) 0 0
\(145\) 3.18450 0.264458
\(146\) −6.81078 −0.563664
\(147\) 0 0
\(148\) 9.67431 0.795223
\(149\) 21.9611 1.79912 0.899562 0.436792i \(-0.143886\pi\)
0.899562 + 0.436792i \(0.143886\pi\)
\(150\) 0 0
\(151\) −16.0281 −1.30434 −0.652172 0.758071i \(-0.726142\pi\)
−0.652172 + 0.758071i \(0.726142\pi\)
\(152\) 4.81780 0.390775
\(153\) 0 0
\(154\) 12.3100 0.991971
\(155\) −4.19424 −0.336890
\(156\) 0 0
\(157\) 21.9483 1.75167 0.875833 0.482615i \(-0.160313\pi\)
0.875833 + 0.482615i \(0.160313\pi\)
\(158\) −3.51446 −0.279595
\(159\) 0 0
\(160\) −5.12300 −0.405009
\(161\) −11.5968 −0.913960
\(162\) 0 0
\(163\) 12.4245 0.973164 0.486582 0.873635i \(-0.338243\pi\)
0.486582 + 0.873635i \(0.338243\pi\)
\(164\) 7.68967 0.600462
\(165\) 0 0
\(166\) −4.01353 −0.311510
\(167\) 17.8284 1.37960 0.689800 0.724000i \(-0.257698\pi\)
0.689800 + 0.724000i \(0.257698\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −2.14125 −0.164226
\(171\) 0 0
\(172\) −14.2040 −1.08304
\(173\) 21.3699 1.62473 0.812363 0.583152i \(-0.198181\pi\)
0.812363 + 0.583152i \(0.198181\pi\)
\(174\) 0 0
\(175\) −4.44964 −0.336361
\(176\) −13.3216 −1.00415
\(177\) 0 0
\(178\) 7.49726 0.561944
\(179\) 9.24978 0.691361 0.345680 0.938352i \(-0.387648\pi\)
0.345680 + 0.938352i \(0.387648\pi\)
\(180\) 0 0
\(181\) 3.01596 0.224175 0.112087 0.993698i \(-0.464246\pi\)
0.112087 + 0.993698i \(0.464246\pi\)
\(182\) −2.29119 −0.169834
\(183\) 0 0
\(184\) −5.01218 −0.369503
\(185\) 5.57641 0.409986
\(186\) 0 0
\(187\) −22.3424 −1.63384
\(188\) −1.87322 −0.136619
\(189\) 0 0
\(190\) 1.28995 0.0935830
\(191\) 13.4334 0.972006 0.486003 0.873957i \(-0.338454\pi\)
0.486003 + 0.873957i \(0.338454\pi\)
\(192\) 0 0
\(193\) −20.1970 −1.45381 −0.726907 0.686736i \(-0.759043\pi\)
−0.726907 + 0.686736i \(0.759043\pi\)
\(194\) −0.767586 −0.0551094
\(195\) 0 0
\(196\) −22.2049 −1.58607
\(197\) −3.86855 −0.275623 −0.137811 0.990458i \(-0.544007\pi\)
−0.137811 + 0.990458i \(0.544007\pi\)
\(198\) 0 0
\(199\) 15.8742 1.12529 0.562645 0.826699i \(-0.309784\pi\)
0.562645 + 0.826699i \(0.309784\pi\)
\(200\) −1.92314 −0.135987
\(201\) 0 0
\(202\) 4.84006 0.340546
\(203\) 14.1698 0.994528
\(204\) 0 0
\(205\) 4.43244 0.309575
\(206\) −3.94676 −0.274984
\(207\) 0 0
\(208\) 2.47947 0.171920
\(209\) 13.4597 0.931029
\(210\) 0 0
\(211\) −20.1776 −1.38909 −0.694543 0.719451i \(-0.744393\pi\)
−0.694543 + 0.719451i \(0.744393\pi\)
\(212\) −20.9599 −1.43953
\(213\) 0 0
\(214\) −7.33354 −0.501311
\(215\) −8.18739 −0.558375
\(216\) 0 0
\(217\) −18.6628 −1.26692
\(218\) 2.24221 0.151862
\(219\) 0 0
\(220\) −9.32102 −0.628423
\(221\) 4.15845 0.279727
\(222\) 0 0
\(223\) 16.6146 1.11260 0.556299 0.830982i \(-0.312221\pi\)
0.556299 + 0.830982i \(0.312221\pi\)
\(224\) −22.7955 −1.52309
\(225\) 0 0
\(226\) 7.21284 0.479791
\(227\) −21.1905 −1.40646 −0.703230 0.710962i \(-0.748260\pi\)
−0.703230 + 0.710962i \(0.748260\pi\)
\(228\) 0 0
\(229\) 12.1066 0.800025 0.400013 0.916510i \(-0.369006\pi\)
0.400013 + 0.916510i \(0.369006\pi\)
\(230\) −1.34200 −0.0884888
\(231\) 0 0
\(232\) 6.12423 0.402076
\(233\) 19.3182 1.26558 0.632788 0.774325i \(-0.281910\pi\)
0.632788 + 0.774325i \(0.281910\pi\)
\(234\) 0 0
\(235\) −1.07975 −0.0704353
\(236\) −19.0471 −1.23986
\(237\) 0 0
\(238\) −9.52779 −0.617595
\(239\) −23.8990 −1.54590 −0.772948 0.634469i \(-0.781219\pi\)
−0.772948 + 0.634469i \(0.781219\pi\)
\(240\) 0 0
\(241\) −1.92710 −0.124135 −0.0620677 0.998072i \(-0.519769\pi\)
−0.0620677 + 0.998072i \(0.519769\pi\)
\(242\) 9.19986 0.591389
\(243\) 0 0
\(244\) 11.5579 0.739917
\(245\) −12.7993 −0.817714
\(246\) 0 0
\(247\) −2.50517 −0.159400
\(248\) −8.06611 −0.512199
\(249\) 0 0
\(250\) −0.514916 −0.0325662
\(251\) 24.4095 1.54071 0.770357 0.637612i \(-0.220078\pi\)
0.770357 + 0.637612i \(0.220078\pi\)
\(252\) 0 0
\(253\) −14.0028 −0.880347
\(254\) −9.66924 −0.606702
\(255\) 0 0
\(256\) −1.24918 −0.0780736
\(257\) −3.50646 −0.218727 −0.109364 0.994002i \(-0.534881\pi\)
−0.109364 + 0.994002i \(0.534881\pi\)
\(258\) 0 0
\(259\) 24.8130 1.54181
\(260\) 1.73486 0.107592
\(261\) 0 0
\(262\) −3.38490 −0.209120
\(263\) −17.3816 −1.07180 −0.535899 0.844282i \(-0.680027\pi\)
−0.535899 + 0.844282i \(0.680027\pi\)
\(264\) 0 0
\(265\) −12.0816 −0.742166
\(266\) 5.73982 0.351931
\(267\) 0 0
\(268\) 6.65800 0.406702
\(269\) 0.127670 0.00778416 0.00389208 0.999992i \(-0.498761\pi\)
0.00389208 + 0.999992i \(0.498761\pi\)
\(270\) 0 0
\(271\) 8.61932 0.523586 0.261793 0.965124i \(-0.415686\pi\)
0.261793 + 0.965124i \(0.415686\pi\)
\(272\) 10.3107 0.625180
\(273\) 0 0
\(274\) −5.39346 −0.325831
\(275\) −5.37278 −0.323991
\(276\) 0 0
\(277\) 3.60519 0.216615 0.108307 0.994117i \(-0.465457\pi\)
0.108307 + 0.994117i \(0.465457\pi\)
\(278\) 4.10202 0.246023
\(279\) 0 0
\(280\) −8.55727 −0.511395
\(281\) −10.4581 −0.623881 −0.311940 0.950102i \(-0.600979\pi\)
−0.311940 + 0.950102i \(0.600979\pi\)
\(282\) 0 0
\(283\) −3.30153 −0.196256 −0.0981279 0.995174i \(-0.531285\pi\)
−0.0981279 + 0.995174i \(0.531285\pi\)
\(284\) −9.62574 −0.571182
\(285\) 0 0
\(286\) −2.76653 −0.163588
\(287\) 19.7227 1.16420
\(288\) 0 0
\(289\) 0.292678 0.0172163
\(290\) 1.63975 0.0962893
\(291\) 0 0
\(292\) 22.9470 1.34287
\(293\) 3.84339 0.224533 0.112267 0.993678i \(-0.464189\pi\)
0.112267 + 0.993678i \(0.464189\pi\)
\(294\) 0 0
\(295\) −10.9790 −0.639223
\(296\) 10.7242 0.623333
\(297\) 0 0
\(298\) 11.3081 0.655062
\(299\) 2.60625 0.150723
\(300\) 0 0
\(301\) −36.4309 −2.09984
\(302\) −8.25310 −0.474913
\(303\) 0 0
\(304\) −6.21149 −0.356253
\(305\) 6.66213 0.381472
\(306\) 0 0
\(307\) 17.8062 1.01625 0.508126 0.861283i \(-0.330339\pi\)
0.508126 + 0.861283i \(0.330339\pi\)
\(308\) −41.4751 −2.36327
\(309\) 0 0
\(310\) −2.15968 −0.122662
\(311\) −34.2196 −1.94041 −0.970207 0.242278i \(-0.922105\pi\)
−0.970207 + 0.242278i \(0.922105\pi\)
\(312\) 0 0
\(313\) 16.8447 0.952118 0.476059 0.879413i \(-0.342065\pi\)
0.476059 + 0.879413i \(0.342065\pi\)
\(314\) 11.3015 0.637782
\(315\) 0 0
\(316\) 11.8410 0.666106
\(317\) 16.1049 0.904542 0.452271 0.891881i \(-0.350614\pi\)
0.452271 + 0.891881i \(0.350614\pi\)
\(318\) 0 0
\(319\) 17.1096 0.957953
\(320\) 2.32102 0.129749
\(321\) 0 0
\(322\) −5.97140 −0.332773
\(323\) −10.4176 −0.579652
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 6.39759 0.354330
\(327\) 0 0
\(328\) 8.52420 0.470670
\(329\) −4.80450 −0.264881
\(330\) 0 0
\(331\) 28.8658 1.58661 0.793305 0.608825i \(-0.208359\pi\)
0.793305 + 0.608825i \(0.208359\pi\)
\(332\) 13.5224 0.742139
\(333\) 0 0
\(334\) 9.18012 0.502314
\(335\) 3.83777 0.209680
\(336\) 0 0
\(337\) −30.1297 −1.64127 −0.820635 0.571452i \(-0.806380\pi\)
−0.820635 + 0.571452i \(0.806380\pi\)
\(338\) 0.514916 0.0280077
\(339\) 0 0
\(340\) 7.21433 0.391252
\(341\) −22.5347 −1.22032
\(342\) 0 0
\(343\) −25.8046 −1.39332
\(344\) −15.7455 −0.848940
\(345\) 0 0
\(346\) 11.0037 0.591564
\(347\) −6.65740 −0.357388 −0.178694 0.983905i \(-0.557187\pi\)
−0.178694 + 0.983905i \(0.557187\pi\)
\(348\) 0 0
\(349\) 33.5996 1.79855 0.899273 0.437387i \(-0.144096\pi\)
0.899273 + 0.437387i \(0.144096\pi\)
\(350\) −2.29119 −0.122469
\(351\) 0 0
\(352\) −27.5247 −1.46707
\(353\) 4.03385 0.214700 0.107350 0.994221i \(-0.465763\pi\)
0.107350 + 0.994221i \(0.465763\pi\)
\(354\) 0 0
\(355\) −5.54842 −0.294479
\(356\) −25.2599 −1.33877
\(357\) 0 0
\(358\) 4.76286 0.251725
\(359\) 15.3477 0.810019 0.405010 0.914312i \(-0.367268\pi\)
0.405010 + 0.914312i \(0.367268\pi\)
\(360\) 0 0
\(361\) −12.7241 −0.669690
\(362\) 1.55297 0.0816221
\(363\) 0 0
\(364\) 7.71950 0.404612
\(365\) 13.2270 0.692331
\(366\) 0 0
\(367\) −25.7870 −1.34607 −0.673034 0.739611i \(-0.735009\pi\)
−0.673034 + 0.739611i \(0.735009\pi\)
\(368\) 6.46210 0.336860
\(369\) 0 0
\(370\) 2.87139 0.149276
\(371\) −53.7587 −2.79101
\(372\) 0 0
\(373\) 14.0741 0.728731 0.364366 0.931256i \(-0.381286\pi\)
0.364366 + 0.931256i \(0.381286\pi\)
\(374\) −11.5045 −0.594881
\(375\) 0 0
\(376\) −2.07651 −0.107088
\(377\) −3.18450 −0.164010
\(378\) 0 0
\(379\) 11.7611 0.604126 0.302063 0.953288i \(-0.402325\pi\)
0.302063 + 0.953288i \(0.402325\pi\)
\(380\) −4.34613 −0.222952
\(381\) 0 0
\(382\) 6.91707 0.353908
\(383\) 2.54142 0.129861 0.0649304 0.997890i \(-0.479317\pi\)
0.0649304 + 0.997890i \(0.479317\pi\)
\(384\) 0 0
\(385\) −23.9069 −1.21841
\(386\) −10.3998 −0.529334
\(387\) 0 0
\(388\) 2.58616 0.131292
\(389\) −4.22869 −0.214403 −0.107201 0.994237i \(-0.534189\pi\)
−0.107201 + 0.994237i \(0.534189\pi\)
\(390\) 0 0
\(391\) 10.8379 0.548098
\(392\) −24.6148 −1.24323
\(393\) 0 0
\(394\) −1.99198 −0.100354
\(395\) 6.82530 0.343418
\(396\) 0 0
\(397\) 6.45665 0.324050 0.162025 0.986787i \(-0.448197\pi\)
0.162025 + 0.986787i \(0.448197\pi\)
\(398\) 8.17386 0.409719
\(399\) 0 0
\(400\) 2.47947 0.123973
\(401\) 29.6081 1.47856 0.739280 0.673398i \(-0.235166\pi\)
0.739280 + 0.673398i \(0.235166\pi\)
\(402\) 0 0
\(403\) 4.19424 0.208930
\(404\) −16.3072 −0.811314
\(405\) 0 0
\(406\) 7.29628 0.362108
\(407\) 29.9608 1.48510
\(408\) 0 0
\(409\) 31.1491 1.54022 0.770112 0.637909i \(-0.220200\pi\)
0.770112 + 0.637909i \(0.220200\pi\)
\(410\) 2.28233 0.112716
\(411\) 0 0
\(412\) 13.2975 0.655119
\(413\) −48.8526 −2.40388
\(414\) 0 0
\(415\) 7.79453 0.382618
\(416\) 5.12300 0.251176
\(417\) 0 0
\(418\) 6.93063 0.338988
\(419\) −30.6871 −1.49916 −0.749582 0.661911i \(-0.769745\pi\)
−0.749582 + 0.661911i \(0.769745\pi\)
\(420\) 0 0
\(421\) −7.27706 −0.354662 −0.177331 0.984151i \(-0.556746\pi\)
−0.177331 + 0.984151i \(0.556746\pi\)
\(422\) −10.3898 −0.505767
\(423\) 0 0
\(424\) −23.2346 −1.12837
\(425\) 4.15845 0.201714
\(426\) 0 0
\(427\) 29.6440 1.43458
\(428\) 24.7083 1.19432
\(429\) 0 0
\(430\) −4.21582 −0.203305
\(431\) 23.5582 1.13476 0.567380 0.823456i \(-0.307957\pi\)
0.567380 + 0.823456i \(0.307957\pi\)
\(432\) 0 0
\(433\) −36.4881 −1.75351 −0.876753 0.480941i \(-0.840295\pi\)
−0.876753 + 0.480941i \(0.840295\pi\)
\(434\) −9.60980 −0.461285
\(435\) 0 0
\(436\) −7.55449 −0.361795
\(437\) −6.52910 −0.312329
\(438\) 0 0
\(439\) −11.5625 −0.551846 −0.275923 0.961180i \(-0.588983\pi\)
−0.275923 + 0.961180i \(0.588983\pi\)
\(440\) −10.3326 −0.492587
\(441\) 0 0
\(442\) 2.14125 0.101849
\(443\) −1.09457 −0.0520045 −0.0260023 0.999662i \(-0.508278\pi\)
−0.0260023 + 0.999662i \(0.508278\pi\)
\(444\) 0 0
\(445\) −14.5602 −0.690218
\(446\) 8.55515 0.405098
\(447\) 0 0
\(448\) 10.3277 0.487938
\(449\) 36.8476 1.73895 0.869473 0.493980i \(-0.164458\pi\)
0.869473 + 0.493980i \(0.164458\pi\)
\(450\) 0 0
\(451\) 23.8145 1.12138
\(452\) −24.3016 −1.14305
\(453\) 0 0
\(454\) −10.9113 −0.512093
\(455\) 4.44964 0.208602
\(456\) 0 0
\(457\) −13.9478 −0.652450 −0.326225 0.945292i \(-0.605777\pi\)
−0.326225 + 0.945292i \(0.605777\pi\)
\(458\) 6.23387 0.291290
\(459\) 0 0
\(460\) 4.52148 0.210815
\(461\) 3.29468 0.153449 0.0767243 0.997052i \(-0.475554\pi\)
0.0767243 + 0.997052i \(0.475554\pi\)
\(462\) 0 0
\(463\) −29.6337 −1.37719 −0.688597 0.725144i \(-0.741773\pi\)
−0.688597 + 0.725144i \(0.741773\pi\)
\(464\) −7.89585 −0.366556
\(465\) 0 0
\(466\) 9.94724 0.460797
\(467\) 11.5765 0.535695 0.267848 0.963461i \(-0.413688\pi\)
0.267848 + 0.963461i \(0.413688\pi\)
\(468\) 0 0
\(469\) 17.0767 0.788528
\(470\) −0.555982 −0.0256455
\(471\) 0 0
\(472\) −21.1142 −0.971860
\(473\) −43.9890 −2.02262
\(474\) 0 0
\(475\) −2.50517 −0.114945
\(476\) 32.1011 1.47135
\(477\) 0 0
\(478\) −12.3060 −0.562862
\(479\) −2.77882 −0.126967 −0.0634837 0.997983i \(-0.520221\pi\)
−0.0634837 + 0.997983i \(0.520221\pi\)
\(480\) 0 0
\(481\) −5.57641 −0.254263
\(482\) −0.992295 −0.0451978
\(483\) 0 0
\(484\) −30.9963 −1.40892
\(485\) 1.49070 0.0676892
\(486\) 0 0
\(487\) −12.5009 −0.566471 −0.283236 0.959050i \(-0.591408\pi\)
−0.283236 + 0.959050i \(0.591408\pi\)
\(488\) 12.8122 0.579981
\(489\) 0 0
\(490\) −6.59054 −0.297730
\(491\) −9.51586 −0.429445 −0.214722 0.976675i \(-0.568885\pi\)
−0.214722 + 0.976675i \(0.568885\pi\)
\(492\) 0 0
\(493\) −13.2426 −0.596415
\(494\) −1.28995 −0.0580377
\(495\) 0 0
\(496\) 10.3995 0.466951
\(497\) −24.6884 −1.10743
\(498\) 0 0
\(499\) 18.1460 0.812326 0.406163 0.913801i \(-0.366867\pi\)
0.406163 + 0.913801i \(0.366867\pi\)
\(500\) 1.73486 0.0775854
\(501\) 0 0
\(502\) 12.5689 0.560975
\(503\) 6.11036 0.272448 0.136224 0.990678i \(-0.456503\pi\)
0.136224 + 0.990678i \(0.456503\pi\)
\(504\) 0 0
\(505\) −9.39972 −0.418282
\(506\) −7.21026 −0.320535
\(507\) 0 0
\(508\) 32.5777 1.44540
\(509\) 16.1158 0.714320 0.357160 0.934043i \(-0.383745\pi\)
0.357160 + 0.934043i \(0.383745\pi\)
\(510\) 0 0
\(511\) 58.8552 2.60360
\(512\) 22.2390 0.982836
\(513\) 0 0
\(514\) −1.80553 −0.0796387
\(515\) 7.66485 0.337754
\(516\) 0 0
\(517\) −5.80126 −0.255139
\(518\) 12.7766 0.561372
\(519\) 0 0
\(520\) 1.92314 0.0843353
\(521\) −12.1096 −0.530533 −0.265266 0.964175i \(-0.585460\pi\)
−0.265266 + 0.964175i \(0.585460\pi\)
\(522\) 0 0
\(523\) 27.1965 1.18922 0.594609 0.804015i \(-0.297307\pi\)
0.594609 + 0.804015i \(0.297307\pi\)
\(524\) 11.4044 0.498205
\(525\) 0 0
\(526\) −8.95008 −0.390242
\(527\) 17.4415 0.759765
\(528\) 0 0
\(529\) −16.2075 −0.704673
\(530\) −6.22100 −0.270223
\(531\) 0 0
\(532\) −19.3387 −0.838438
\(533\) −4.43244 −0.191990
\(534\) 0 0
\(535\) 14.2422 0.615745
\(536\) 7.38057 0.318792
\(537\) 0 0
\(538\) 0.0657392 0.00283422
\(539\) −68.7675 −2.96203
\(540\) 0 0
\(541\) 34.3549 1.47703 0.738517 0.674235i \(-0.235527\pi\)
0.738517 + 0.674235i \(0.235527\pi\)
\(542\) 4.43822 0.190638
\(543\) 0 0
\(544\) 21.3037 0.913389
\(545\) −4.35452 −0.186527
\(546\) 0 0
\(547\) −15.8457 −0.677512 −0.338756 0.940874i \(-0.610006\pi\)
−0.338756 + 0.940874i \(0.610006\pi\)
\(548\) 18.1717 0.776257
\(549\) 0 0
\(550\) −2.76653 −0.117965
\(551\) 7.97771 0.339862
\(552\) 0 0
\(553\) 30.3701 1.29147
\(554\) 1.85637 0.0788696
\(555\) 0 0
\(556\) −13.8206 −0.586123
\(557\) 3.13245 0.132726 0.0663631 0.997796i \(-0.478860\pi\)
0.0663631 + 0.997796i \(0.478860\pi\)
\(558\) 0 0
\(559\) 8.18739 0.346290
\(560\) 11.0327 0.466218
\(561\) 0 0
\(562\) −5.38507 −0.227155
\(563\) −0.650982 −0.0274356 −0.0137178 0.999906i \(-0.504367\pi\)
−0.0137178 + 0.999906i \(0.504367\pi\)
\(564\) 0 0
\(565\) −14.0078 −0.589312
\(566\) −1.70001 −0.0714568
\(567\) 0 0
\(568\) −10.6704 −0.447719
\(569\) 22.6287 0.948644 0.474322 0.880352i \(-0.342693\pi\)
0.474322 + 0.880352i \(0.342693\pi\)
\(570\) 0 0
\(571\) 20.5204 0.858753 0.429376 0.903126i \(-0.358733\pi\)
0.429376 + 0.903126i \(0.358733\pi\)
\(572\) 9.32102 0.389731
\(573\) 0 0
\(574\) 10.1556 0.423885
\(575\) 2.60625 0.108688
\(576\) 0 0
\(577\) −6.90761 −0.287568 −0.143784 0.989609i \(-0.545927\pi\)
−0.143784 + 0.989609i \(0.545927\pi\)
\(578\) 0.150704 0.00626848
\(579\) 0 0
\(580\) −5.52466 −0.229399
\(581\) 34.6828 1.43888
\(582\) 0 0
\(583\) −64.9116 −2.68837
\(584\) 25.4373 1.05260
\(585\) 0 0
\(586\) 1.97902 0.0817527
\(587\) 8.27601 0.341587 0.170794 0.985307i \(-0.445367\pi\)
0.170794 + 0.985307i \(0.445367\pi\)
\(588\) 0 0
\(589\) −10.5073 −0.432945
\(590\) −5.65327 −0.232742
\(591\) 0 0
\(592\) −13.8265 −0.568267
\(593\) 1.17530 0.0482637 0.0241318 0.999709i \(-0.492318\pi\)
0.0241318 + 0.999709i \(0.492318\pi\)
\(594\) 0 0
\(595\) 18.5036 0.758572
\(596\) −38.0995 −1.56062
\(597\) 0 0
\(598\) 1.34200 0.0548784
\(599\) 33.1386 1.35401 0.677003 0.735980i \(-0.263278\pi\)
0.677003 + 0.735980i \(0.263278\pi\)
\(600\) 0 0
\(601\) 19.6751 0.802563 0.401282 0.915955i \(-0.368565\pi\)
0.401282 + 0.915955i \(0.368565\pi\)
\(602\) −18.7589 −0.764554
\(603\) 0 0
\(604\) 27.8065 1.13143
\(605\) −17.8667 −0.726385
\(606\) 0 0
\(607\) −1.81610 −0.0737133 −0.0368567 0.999321i \(-0.511734\pi\)
−0.0368567 + 0.999321i \(0.511734\pi\)
\(608\) −12.8340 −0.520487
\(609\) 0 0
\(610\) 3.43044 0.138894
\(611\) 1.07975 0.0436821
\(612\) 0 0
\(613\) 14.1676 0.572226 0.286113 0.958196i \(-0.407637\pi\)
0.286113 + 0.958196i \(0.407637\pi\)
\(614\) 9.16868 0.370018
\(615\) 0 0
\(616\) −45.9763 −1.85244
\(617\) −46.9789 −1.89130 −0.945649 0.325189i \(-0.894572\pi\)
−0.945649 + 0.325189i \(0.894572\pi\)
\(618\) 0 0
\(619\) 2.62966 0.105695 0.0528475 0.998603i \(-0.483170\pi\)
0.0528475 + 0.998603i \(0.483170\pi\)
\(620\) 7.27643 0.292228
\(621\) 0 0
\(622\) −17.6202 −0.706506
\(623\) −64.7874 −2.59565
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.67360 0.346667
\(627\) 0 0
\(628\) −38.0773 −1.51945
\(629\) −23.1892 −0.924615
\(630\) 0 0
\(631\) 36.4482 1.45098 0.725490 0.688233i \(-0.241613\pi\)
0.725490 + 0.688233i \(0.241613\pi\)
\(632\) 13.1260 0.522125
\(633\) 0 0
\(634\) 8.29268 0.329344
\(635\) 18.7783 0.745193
\(636\) 0 0
\(637\) 12.7993 0.507125
\(638\) 8.81000 0.348791
\(639\) 0 0
\(640\) 11.4411 0.452250
\(641\) 40.8350 1.61289 0.806444 0.591311i \(-0.201389\pi\)
0.806444 + 0.591311i \(0.201389\pi\)
\(642\) 0 0
\(643\) −25.6445 −1.01132 −0.505660 0.862733i \(-0.668751\pi\)
−0.505660 + 0.862733i \(0.668751\pi\)
\(644\) 20.1189 0.792797
\(645\) 0 0
\(646\) −5.36420 −0.211052
\(647\) −17.8129 −0.700298 −0.350149 0.936694i \(-0.613869\pi\)
−0.350149 + 0.936694i \(0.613869\pi\)
\(648\) 0 0
\(649\) −58.9878 −2.31547
\(650\) 0.514916 0.0201967
\(651\) 0 0
\(652\) −21.5548 −0.844153
\(653\) 21.7875 0.852611 0.426305 0.904579i \(-0.359815\pi\)
0.426305 + 0.904579i \(0.359815\pi\)
\(654\) 0 0
\(655\) 6.57369 0.256855
\(656\) −10.9901 −0.429091
\(657\) 0 0
\(658\) −2.47392 −0.0964433
\(659\) −15.8255 −0.616475 −0.308238 0.951309i \(-0.599739\pi\)
−0.308238 + 0.951309i \(0.599739\pi\)
\(660\) 0 0
\(661\) −10.1484 −0.394725 −0.197363 0.980331i \(-0.563238\pi\)
−0.197363 + 0.980331i \(0.563238\pi\)
\(662\) 14.8635 0.577685
\(663\) 0 0
\(664\) 14.9900 0.581723
\(665\) −11.1471 −0.432266
\(666\) 0 0
\(667\) −8.29958 −0.321361
\(668\) −30.9298 −1.19671
\(669\) 0 0
\(670\) 1.97613 0.0763446
\(671\) 35.7941 1.38182
\(672\) 0 0
\(673\) 26.8157 1.03367 0.516835 0.856085i \(-0.327110\pi\)
0.516835 + 0.856085i \(0.327110\pi\)
\(674\) −15.5143 −0.597588
\(675\) 0 0
\(676\) −1.73486 −0.0667254
\(677\) 20.8477 0.801241 0.400620 0.916244i \(-0.368795\pi\)
0.400620 + 0.916244i \(0.368795\pi\)
\(678\) 0 0
\(679\) 6.63307 0.254554
\(680\) 7.99728 0.306682
\(681\) 0 0
\(682\) −11.6035 −0.444320
\(683\) −30.5434 −1.16871 −0.584356 0.811498i \(-0.698653\pi\)
−0.584356 + 0.811498i \(0.698653\pi\)
\(684\) 0 0
\(685\) 10.4744 0.400208
\(686\) −13.2872 −0.507307
\(687\) 0 0
\(688\) 20.3004 0.773944
\(689\) 12.0816 0.460272
\(690\) 0 0
\(691\) 22.3125 0.848808 0.424404 0.905473i \(-0.360484\pi\)
0.424404 + 0.905473i \(0.360484\pi\)
\(692\) −37.0739 −1.40934
\(693\) 0 0
\(694\) −3.42800 −0.130125
\(695\) −7.96639 −0.302182
\(696\) 0 0
\(697\) −18.4321 −0.698164
\(698\) 17.3010 0.654852
\(699\) 0 0
\(700\) 7.71950 0.291770
\(701\) −38.2280 −1.44385 −0.721925 0.691971i \(-0.756743\pi\)
−0.721925 + 0.691971i \(0.756743\pi\)
\(702\) 0 0
\(703\) 13.9699 0.526884
\(704\) 12.4703 0.469993
\(705\) 0 0
\(706\) 2.07709 0.0781724
\(707\) −41.8253 −1.57300
\(708\) 0 0
\(709\) 16.8893 0.634290 0.317145 0.948377i \(-0.397276\pi\)
0.317145 + 0.948377i \(0.397276\pi\)
\(710\) −2.85697 −0.107220
\(711\) 0 0
\(712\) −28.0012 −1.04939
\(713\) 10.9312 0.409378
\(714\) 0 0
\(715\) 5.37278 0.200930
\(716\) −16.0471 −0.599708
\(717\) 0 0
\(718\) 7.90276 0.294928
\(719\) 12.0803 0.450518 0.225259 0.974299i \(-0.427677\pi\)
0.225259 + 0.974299i \(0.427677\pi\)
\(720\) 0 0
\(721\) 34.1058 1.27017
\(722\) −6.55185 −0.243835
\(723\) 0 0
\(724\) −5.23227 −0.194456
\(725\) −3.18450 −0.118269
\(726\) 0 0
\(727\) −6.66912 −0.247344 −0.123672 0.992323i \(-0.539467\pi\)
−0.123672 + 0.992323i \(0.539467\pi\)
\(728\) 8.55727 0.317154
\(729\) 0 0
\(730\) 6.81078 0.252078
\(731\) 34.0468 1.25927
\(732\) 0 0
\(733\) 8.95393 0.330721 0.165360 0.986233i \(-0.447121\pi\)
0.165360 + 0.986233i \(0.447121\pi\)
\(734\) −13.2781 −0.490104
\(735\) 0 0
\(736\) 13.3518 0.492154
\(737\) 20.6195 0.759528
\(738\) 0 0
\(739\) −23.9855 −0.882322 −0.441161 0.897428i \(-0.645433\pi\)
−0.441161 + 0.897428i \(0.645433\pi\)
\(740\) −9.67431 −0.355635
\(741\) 0 0
\(742\) −27.6812 −1.01621
\(743\) −4.48597 −0.164574 −0.0822872 0.996609i \(-0.526222\pi\)
−0.0822872 + 0.996609i \(0.526222\pi\)
\(744\) 0 0
\(745\) −21.9611 −0.804593
\(746\) 7.24700 0.265331
\(747\) 0 0
\(748\) 38.7610 1.41724
\(749\) 63.3727 2.31559
\(750\) 0 0
\(751\) −14.5617 −0.531365 −0.265682 0.964061i \(-0.585597\pi\)
−0.265682 + 0.964061i \(0.585597\pi\)
\(752\) 2.67721 0.0976278
\(753\) 0 0
\(754\) −1.63975 −0.0597161
\(755\) 16.0281 0.583321
\(756\) 0 0
\(757\) −5.01931 −0.182430 −0.0912150 0.995831i \(-0.529075\pi\)
−0.0912150 + 0.995831i \(0.529075\pi\)
\(758\) 6.05597 0.219963
\(759\) 0 0
\(760\) −4.81780 −0.174760
\(761\) 8.77091 0.317945 0.158973 0.987283i \(-0.449182\pi\)
0.158973 + 0.987283i \(0.449182\pi\)
\(762\) 0 0
\(763\) −19.3760 −0.701460
\(764\) −23.3051 −0.843147
\(765\) 0 0
\(766\) 1.30862 0.0472824
\(767\) 10.9790 0.396430
\(768\) 0 0
\(769\) −3.35285 −0.120907 −0.0604535 0.998171i \(-0.519255\pi\)
−0.0604535 + 0.998171i \(0.519255\pi\)
\(770\) −12.3100 −0.443623
\(771\) 0 0
\(772\) 35.0390 1.26108
\(773\) −32.3940 −1.16513 −0.582566 0.812783i \(-0.697951\pi\)
−0.582566 + 0.812783i \(0.697951\pi\)
\(774\) 0 0
\(775\) 4.19424 0.150662
\(776\) 2.86683 0.102913
\(777\) 0 0
\(778\) −2.17742 −0.0780642
\(779\) 11.1040 0.397843
\(780\) 0 0
\(781\) −29.8104 −1.06670
\(782\) 5.58063 0.199563
\(783\) 0 0
\(784\) 31.7353 1.13340
\(785\) −21.9483 −0.783368
\(786\) 0 0
\(787\) 15.7890 0.562819 0.281409 0.959588i \(-0.409198\pi\)
0.281409 + 0.959588i \(0.409198\pi\)
\(788\) 6.71140 0.239084
\(789\) 0 0
\(790\) 3.51446 0.125039
\(791\) −62.3296 −2.21618
\(792\) 0 0
\(793\) −6.66213 −0.236579
\(794\) 3.32464 0.117987
\(795\) 0 0
\(796\) −27.5395 −0.976111
\(797\) −5.98197 −0.211892 −0.105946 0.994372i \(-0.533787\pi\)
−0.105946 + 0.994372i \(0.533787\pi\)
\(798\) 0 0
\(799\) 4.49009 0.158848
\(800\) 5.12300 0.181125
\(801\) 0 0
\(802\) 15.2457 0.538345
\(803\) 71.0655 2.50785
\(804\) 0 0
\(805\) 11.5968 0.408735
\(806\) 2.15968 0.0760716
\(807\) 0 0
\(808\) −18.0770 −0.635946
\(809\) 17.5389 0.616636 0.308318 0.951283i \(-0.400234\pi\)
0.308318 + 0.951283i \(0.400234\pi\)
\(810\) 0 0
\(811\) 27.2599 0.957223 0.478612 0.878027i \(-0.341140\pi\)
0.478612 + 0.878027i \(0.341140\pi\)
\(812\) −24.5827 −0.862684
\(813\) 0 0
\(814\) 15.4273 0.540727
\(815\) −12.4245 −0.435212
\(816\) 0 0
\(817\) −20.5108 −0.717583
\(818\) 16.0392 0.560797
\(819\) 0 0
\(820\) −7.68967 −0.268535
\(821\) 27.6454 0.964832 0.482416 0.875942i \(-0.339759\pi\)
0.482416 + 0.875942i \(0.339759\pi\)
\(822\) 0 0
\(823\) −28.5070 −0.993692 −0.496846 0.867839i \(-0.665509\pi\)
−0.496846 + 0.867839i \(0.665509\pi\)
\(824\) 14.7406 0.513513
\(825\) 0 0
\(826\) −25.1550 −0.875255
\(827\) −12.6993 −0.441599 −0.220799 0.975319i \(-0.570867\pi\)
−0.220799 + 0.975319i \(0.570867\pi\)
\(828\) 0 0
\(829\) 0.935064 0.0324761 0.0162381 0.999868i \(-0.494831\pi\)
0.0162381 + 0.999868i \(0.494831\pi\)
\(830\) 4.01353 0.139312
\(831\) 0 0
\(832\) −2.32102 −0.0804669
\(833\) 53.2250 1.84414
\(834\) 0 0
\(835\) −17.8284 −0.616976
\(836\) −23.3508 −0.807603
\(837\) 0 0
\(838\) −15.8013 −0.545847
\(839\) −32.1537 −1.11007 −0.555034 0.831828i \(-0.687295\pi\)
−0.555034 + 0.831828i \(0.687295\pi\)
\(840\) 0 0
\(841\) −18.8590 −0.650310
\(842\) −3.74708 −0.129133
\(843\) 0 0
\(844\) 35.0054 1.20494
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −79.5004 −2.73166
\(848\) 29.9559 1.02869
\(849\) 0 0
\(850\) 2.14125 0.0734443
\(851\) −14.5335 −0.498202
\(852\) 0 0
\(853\) −24.7177 −0.846319 −0.423159 0.906055i \(-0.639079\pi\)
−0.423159 + 0.906055i \(0.639079\pi\)
\(854\) 15.2642 0.522330
\(855\) 0 0
\(856\) 27.3898 0.936163
\(857\) 27.6460 0.944368 0.472184 0.881500i \(-0.343466\pi\)
0.472184 + 0.881500i \(0.343466\pi\)
\(858\) 0 0
\(859\) −30.0622 −1.02571 −0.512855 0.858475i \(-0.671412\pi\)
−0.512855 + 0.858475i \(0.671412\pi\)
\(860\) 14.2040 0.484352
\(861\) 0 0
\(862\) 12.1305 0.413167
\(863\) 28.9538 0.985599 0.492799 0.870143i \(-0.335974\pi\)
0.492799 + 0.870143i \(0.335974\pi\)
\(864\) 0 0
\(865\) −21.3699 −0.726600
\(866\) −18.7883 −0.638453
\(867\) 0 0
\(868\) 32.3774 1.09896
\(869\) 36.6708 1.24397
\(870\) 0 0
\(871\) −3.83777 −0.130038
\(872\) −8.37436 −0.283591
\(873\) 0 0
\(874\) −3.36194 −0.113719
\(875\) 4.44964 0.150425
\(876\) 0 0
\(877\) 22.0412 0.744280 0.372140 0.928177i \(-0.378624\pi\)
0.372140 + 0.928177i \(0.378624\pi\)
\(878\) −5.95369 −0.200927
\(879\) 0 0
\(880\) 13.3216 0.449072
\(881\) 21.2728 0.716699 0.358350 0.933587i \(-0.383340\pi\)
0.358350 + 0.933587i \(0.383340\pi\)
\(882\) 0 0
\(883\) 35.5673 1.19694 0.598468 0.801147i \(-0.295777\pi\)
0.598468 + 0.801147i \(0.295777\pi\)
\(884\) −7.21433 −0.242644
\(885\) 0 0
\(886\) −0.563611 −0.0189349
\(887\) 26.8853 0.902720 0.451360 0.892342i \(-0.350939\pi\)
0.451360 + 0.892342i \(0.350939\pi\)
\(888\) 0 0
\(889\) 83.5565 2.80239
\(890\) −7.49726 −0.251309
\(891\) 0 0
\(892\) −28.8241 −0.965102
\(893\) −2.70496 −0.0905182
\(894\) 0 0
\(895\) −9.24978 −0.309186
\(896\) 50.9088 1.70074
\(897\) 0 0
\(898\) 18.9734 0.633152
\(899\) −13.3565 −0.445466
\(900\) 0 0
\(901\) 50.2406 1.67376
\(902\) 12.2625 0.408296
\(903\) 0 0
\(904\) −26.9390 −0.895976
\(905\) −3.01596 −0.100254
\(906\) 0 0
\(907\) 19.0379 0.632143 0.316071 0.948735i \(-0.397636\pi\)
0.316071 + 0.948735i \(0.397636\pi\)
\(908\) 36.7625 1.22001
\(909\) 0 0
\(910\) 2.29119 0.0759522
\(911\) 52.1035 1.72626 0.863132 0.504978i \(-0.168499\pi\)
0.863132 + 0.504978i \(0.168499\pi\)
\(912\) 0 0
\(913\) 41.8782 1.38597
\(914\) −7.18194 −0.237557
\(915\) 0 0
\(916\) −21.0032 −0.693967
\(917\) 29.2505 0.965937
\(918\) 0 0
\(919\) −45.5433 −1.50233 −0.751167 0.660113i \(-0.770509\pi\)
−0.751167 + 0.660113i \(0.770509\pi\)
\(920\) 5.01218 0.165247
\(921\) 0 0
\(922\) 1.69648 0.0558707
\(923\) 5.54842 0.182628
\(924\) 0 0
\(925\) −5.57641 −0.183351
\(926\) −15.2589 −0.501438
\(927\) 0 0
\(928\) −16.3142 −0.535539
\(929\) −10.1464 −0.332894 −0.166447 0.986050i \(-0.553229\pi\)
−0.166447 + 0.986050i \(0.553229\pi\)
\(930\) 0 0
\(931\) −32.0643 −1.05087
\(932\) −33.5144 −1.09780
\(933\) 0 0
\(934\) 5.96091 0.195047
\(935\) 22.3424 0.730675
\(936\) 0 0
\(937\) 27.9000 0.911452 0.455726 0.890120i \(-0.349380\pi\)
0.455726 + 0.890120i \(0.349380\pi\)
\(938\) 8.79306 0.287104
\(939\) 0 0
\(940\) 1.87322 0.0610977
\(941\) −51.4135 −1.67603 −0.838017 0.545645i \(-0.816285\pi\)
−0.838017 + 0.545645i \(0.816285\pi\)
\(942\) 0 0
\(943\) −11.5520 −0.376186
\(944\) 27.2221 0.886005
\(945\) 0 0
\(946\) −22.6506 −0.736436
\(947\) −2.63195 −0.0855269 −0.0427635 0.999085i \(-0.513616\pi\)
−0.0427635 + 0.999085i \(0.513616\pi\)
\(948\) 0 0
\(949\) −13.2270 −0.429365
\(950\) −1.28995 −0.0418516
\(951\) 0 0
\(952\) 35.5850 1.15332
\(953\) −19.4156 −0.628933 −0.314466 0.949269i \(-0.601826\pi\)
−0.314466 + 0.949269i \(0.601826\pi\)
\(954\) 0 0
\(955\) −13.4334 −0.434694
\(956\) 41.4614 1.34096
\(957\) 0 0
\(958\) −1.43086 −0.0462289
\(959\) 46.6075 1.50503
\(960\) 0 0
\(961\) −13.4083 −0.432527
\(962\) −2.87139 −0.0925772
\(963\) 0 0
\(964\) 3.34325 0.107679
\(965\) 20.1970 0.650165
\(966\) 0 0
\(967\) 40.8150 1.31252 0.656260 0.754535i \(-0.272137\pi\)
0.656260 + 0.754535i \(0.272137\pi\)
\(968\) −34.3602 −1.10438
\(969\) 0 0
\(970\) 0.767586 0.0246457
\(971\) 34.5822 1.10980 0.554898 0.831919i \(-0.312757\pi\)
0.554898 + 0.831919i \(0.312757\pi\)
\(972\) 0 0
\(973\) −35.4475 −1.13639
\(974\) −6.43693 −0.206253
\(975\) 0 0
\(976\) −16.5185 −0.528745
\(977\) 55.3685 1.77140 0.885698 0.464262i \(-0.153681\pi\)
0.885698 + 0.464262i \(0.153681\pi\)
\(978\) 0 0
\(979\) −78.2285 −2.50019
\(980\) 22.2049 0.709310
\(981\) 0 0
\(982\) −4.89987 −0.156361
\(983\) 38.6556 1.23292 0.616461 0.787386i \(-0.288566\pi\)
0.616461 + 0.787386i \(0.288566\pi\)
\(984\) 0 0
\(985\) 3.86855 0.123262
\(986\) −6.81881 −0.217155
\(987\) 0 0
\(988\) 4.34613 0.138269
\(989\) 21.3384 0.678520
\(990\) 0 0
\(991\) 8.73406 0.277446 0.138723 0.990331i \(-0.455700\pi\)
0.138723 + 0.990331i \(0.455700\pi\)
\(992\) 21.4871 0.682216
\(993\) 0 0
\(994\) −12.7125 −0.403215
\(995\) −15.8742 −0.503245
\(996\) 0 0
\(997\) −16.3184 −0.516807 −0.258404 0.966037i \(-0.583196\pi\)
−0.258404 + 0.966037i \(0.583196\pi\)
\(998\) 9.34366 0.295768
\(999\) 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 1755.2.a.s.1.2 yes 4
3.2 odd 2 1755.2.a.m.1.3 4
5.4 even 2 8775.2.a.bh.1.3 4
15.14 odd 2 8775.2.a.bt.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1755.2.a.m.1.3 4 3.2 odd 2
1755.2.a.s.1.2 yes 4 1.1 even 1 trivial
8775.2.a.bh.1.3 4 5.4 even 2
8775.2.a.bt.1.2 4 15.14 odd 2