Properties

Label 2175.2.a.w.1.5
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
Analytic rank $1$
Dimension $5$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.71457\) of defining polynomial
Character \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.71457 q^{2} -1.00000 q^{3} +0.939748 q^{4} -1.71457 q^{6} +0.654317 q^{7} -1.81788 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.71457 q^{2} -1.00000 q^{3} +0.939748 q^{4} -1.71457 q^{6} +0.654317 q^{7} -1.81788 q^{8} +1.00000 q^{9} +0.163559 q^{11} -0.939748 q^{12} -2.65432 q^{13} +1.12187 q^{14} -4.99637 q^{16} +3.86328 q^{17} +1.71457 q^{18} -3.47954 q^{19} -0.654317 q^{21} +0.280433 q^{22} -7.69972 q^{23} +1.81788 q^{24} -4.55101 q^{26} -1.00000 q^{27} +0.614893 q^{28} +1.00000 q^{29} +5.05274 q^{31} -4.93087 q^{32} -0.163559 q^{33} +6.62385 q^{34} +0.939748 q^{36} -10.5904 q^{37} -5.96591 q^{38} +2.65432 q^{39} -6.17417 q^{41} -1.12187 q^{42} +10.5547 q^{43} +0.153704 q^{44} -13.2017 q^{46} -10.3036 q^{47} +4.99637 q^{48} -6.57187 q^{49} -3.86328 q^{51} -2.49439 q^{52} -4.04766 q^{53} -1.71457 q^{54} -1.18947 q^{56} +3.47954 q^{57} +1.71457 q^{58} -0.328734 q^{59} -5.72054 q^{61} +8.66328 q^{62} +0.654317 q^{63} +1.53842 q^{64} -0.280433 q^{66} -3.51985 q^{67} +3.63050 q^{68} +7.69972 q^{69} +11.7457 q^{71} -1.81788 q^{72} +1.12143 q^{73} -18.1580 q^{74} -3.26989 q^{76} +0.107019 q^{77} +4.55101 q^{78} -12.4074 q^{79} +1.00000 q^{81} -10.5860 q^{82} -7.89306 q^{83} -0.614893 q^{84} +18.0968 q^{86} -1.00000 q^{87} -0.297329 q^{88} +5.04702 q^{89} -1.73677 q^{91} -7.23579 q^{92} -5.05274 q^{93} -17.6663 q^{94} +4.93087 q^{96} -8.49076 q^{97} -11.2679 q^{98} +0.163559 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{2} - 5 q^{3} + 5 q^{4} + 3 q^{6} - 8 q^{7} - 9 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{2} - 5 q^{3} + 5 q^{4} + 3 q^{6} - 8 q^{7} - 9 q^{8} + 5 q^{9} + 12 q^{11} - 5 q^{12} - 2 q^{13} + 6 q^{14} + q^{16} - 3 q^{18} - 2 q^{19} + 8 q^{21} - 14 q^{22} - 8 q^{23} + 9 q^{24} - 5 q^{27} + 6 q^{28} + 5 q^{29} + 2 q^{31} - q^{32} - 12 q^{33} + 4 q^{34} + 5 q^{36} - 16 q^{37} + 14 q^{38} + 2 q^{39} - 14 q^{41} - 6 q^{42} + 20 q^{44} - 6 q^{46} - 2 q^{47} - q^{48} - 7 q^{49} - 16 q^{52} - 26 q^{53} + 3 q^{54} - 2 q^{56} + 2 q^{57} - 3 q^{58} + 4 q^{59} - 12 q^{61} - 8 q^{63} - 9 q^{64} + 14 q^{66} - 12 q^{67} + 20 q^{68} + 8 q^{69} + 30 q^{71} - 9 q^{72} + 12 q^{73} + 2 q^{74} - 44 q^{76} - 18 q^{77} - 18 q^{79} + 5 q^{81} - 10 q^{82} - 2 q^{83} - 6 q^{84} + 30 q^{86} - 5 q^{87} - 42 q^{88} - 22 q^{89} - 12 q^{91} - 20 q^{92} - 2 q^{93} - 50 q^{94} + q^{96} - 20 q^{97} + 9 q^{98} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.71457 1.21238 0.606192 0.795319i \(-0.292696\pi\)
0.606192 + 0.795319i \(0.292696\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.939748 0.469874
\(5\) 0 0
\(6\) −1.71457 −0.699970
\(7\) 0.654317 0.247309 0.123654 0.992325i \(-0.460539\pi\)
0.123654 + 0.992325i \(0.460539\pi\)
\(8\) −1.81788 −0.642716
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.163559 0.0493148 0.0246574 0.999696i \(-0.492151\pi\)
0.0246574 + 0.999696i \(0.492151\pi\)
\(12\) −0.939748 −0.271282
\(13\) −2.65432 −0.736175 −0.368088 0.929791i \(-0.619987\pi\)
−0.368088 + 0.929791i \(0.619987\pi\)
\(14\) 1.12187 0.299833
\(15\) 0 0
\(16\) −4.99637 −1.24909
\(17\) 3.86328 0.936982 0.468491 0.883468i \(-0.344798\pi\)
0.468491 + 0.883468i \(0.344798\pi\)
\(18\) 1.71457 0.404128
\(19\) −3.47954 −0.798260 −0.399130 0.916894i \(-0.630688\pi\)
−0.399130 + 0.916894i \(0.630688\pi\)
\(20\) 0 0
\(21\) −0.654317 −0.142784
\(22\) 0.280433 0.0597884
\(23\) −7.69972 −1.60550 −0.802751 0.596314i \(-0.796631\pi\)
−0.802751 + 0.596314i \(0.796631\pi\)
\(24\) 1.81788 0.371072
\(25\) 0 0
\(26\) −4.55101 −0.892527
\(27\) −1.00000 −0.192450
\(28\) 0.614893 0.116204
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 5.05274 0.907499 0.453750 0.891129i \(-0.350086\pi\)
0.453750 + 0.891129i \(0.350086\pi\)
\(32\) −4.93087 −0.871663
\(33\) −0.163559 −0.0284719
\(34\) 6.62385 1.13598
\(35\) 0 0
\(36\) 0.939748 0.156625
\(37\) −10.5904 −1.74106 −0.870528 0.492118i \(-0.836223\pi\)
−0.870528 + 0.492118i \(0.836223\pi\)
\(38\) −5.96591 −0.967798
\(39\) 2.65432 0.425031
\(40\) 0 0
\(41\) −6.17417 −0.964244 −0.482122 0.876104i \(-0.660134\pi\)
−0.482122 + 0.876104i \(0.660134\pi\)
\(42\) −1.12187 −0.173109
\(43\) 10.5547 1.60958 0.804790 0.593559i \(-0.202278\pi\)
0.804790 + 0.593559i \(0.202278\pi\)
\(44\) 0.153704 0.0231717
\(45\) 0 0
\(46\) −13.2017 −1.94648
\(47\) −10.3036 −1.50294 −0.751470 0.659767i \(-0.770655\pi\)
−0.751470 + 0.659767i \(0.770655\pi\)
\(48\) 4.99637 0.721164
\(49\) −6.57187 −0.938838
\(50\) 0 0
\(51\) −3.86328 −0.540967
\(52\) −2.49439 −0.345909
\(53\) −4.04766 −0.555989 −0.277995 0.960583i \(-0.589670\pi\)
−0.277995 + 0.960583i \(0.589670\pi\)
\(54\) −1.71457 −0.233323
\(55\) 0 0
\(56\) −1.18947 −0.158949
\(57\) 3.47954 0.460876
\(58\) 1.71457 0.225134
\(59\) −0.328734 −0.0427975 −0.0213988 0.999771i \(-0.506812\pi\)
−0.0213988 + 0.999771i \(0.506812\pi\)
\(60\) 0 0
\(61\) −5.72054 −0.732441 −0.366220 0.930528i \(-0.619348\pi\)
−0.366220 + 0.930528i \(0.619348\pi\)
\(62\) 8.66328 1.10024
\(63\) 0.654317 0.0824362
\(64\) 1.53842 0.192303
\(65\) 0 0
\(66\) −0.280433 −0.0345189
\(67\) −3.51985 −0.430019 −0.215009 0.976612i \(-0.568978\pi\)
−0.215009 + 0.976612i \(0.568978\pi\)
\(68\) 3.63050 0.440263
\(69\) 7.69972 0.926937
\(70\) 0 0
\(71\) 11.7457 1.39396 0.696981 0.717090i \(-0.254526\pi\)
0.696981 + 0.717090i \(0.254526\pi\)
\(72\) −1.81788 −0.214239
\(73\) 1.12143 0.131253 0.0656267 0.997844i \(-0.479095\pi\)
0.0656267 + 0.997844i \(0.479095\pi\)
\(74\) −18.1580 −2.11083
\(75\) 0 0
\(76\) −3.26989 −0.375082
\(77\) 0.107019 0.0121960
\(78\) 4.55101 0.515300
\(79\) −12.4074 −1.39594 −0.697970 0.716127i \(-0.745913\pi\)
−0.697970 + 0.716127i \(0.745913\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.5860 −1.16903
\(83\) −7.89306 −0.866376 −0.433188 0.901304i \(-0.642611\pi\)
−0.433188 + 0.901304i \(0.642611\pi\)
\(84\) −0.614893 −0.0670903
\(85\) 0 0
\(86\) 18.0968 1.95143
\(87\) −1.00000 −0.107211
\(88\) −0.297329 −0.0316954
\(89\) 5.04702 0.534983 0.267491 0.963560i \(-0.413805\pi\)
0.267491 + 0.963560i \(0.413805\pi\)
\(90\) 0 0
\(91\) −1.73677 −0.182062
\(92\) −7.23579 −0.754383
\(93\) −5.05274 −0.523945
\(94\) −17.6663 −1.82214
\(95\) 0 0
\(96\) 4.93087 0.503255
\(97\) −8.49076 −0.862106 −0.431053 0.902327i \(-0.641858\pi\)
−0.431053 + 0.902327i \(0.641858\pi\)
\(98\) −11.2679 −1.13823
\(99\) 0.163559 0.0164383
\(100\) 0 0
\(101\) 1.81126 0.180227 0.0901136 0.995931i \(-0.471277\pi\)
0.0901136 + 0.995931i \(0.471277\pi\)
\(102\) −6.62385 −0.655859
\(103\) −5.80807 −0.572286 −0.286143 0.958187i \(-0.592373\pi\)
−0.286143 + 0.958187i \(0.592373\pi\)
\(104\) 4.82522 0.473152
\(105\) 0 0
\(106\) −6.94000 −0.674072
\(107\) −3.48568 −0.336973 −0.168487 0.985704i \(-0.553888\pi\)
−0.168487 + 0.985704i \(0.553888\pi\)
\(108\) −0.939748 −0.0904273
\(109\) 8.11029 0.776825 0.388412 0.921486i \(-0.373024\pi\)
0.388412 + 0.921486i \(0.373024\pi\)
\(110\) 0 0
\(111\) 10.5904 1.00520
\(112\) −3.26921 −0.308911
\(113\) −2.06715 −0.194461 −0.0972307 0.995262i \(-0.530998\pi\)
−0.0972307 + 0.995262i \(0.530998\pi\)
\(114\) 5.96591 0.558758
\(115\) 0 0
\(116\) 0.939748 0.0872534
\(117\) −2.65432 −0.245392
\(118\) −0.563637 −0.0518870
\(119\) 2.52781 0.231724
\(120\) 0 0
\(121\) −10.9732 −0.997568
\(122\) −9.80827 −0.887999
\(123\) 6.17417 0.556706
\(124\) 4.74830 0.426410
\(125\) 0 0
\(126\) 1.12187 0.0999443
\(127\) 12.9962 1.15323 0.576613 0.817017i \(-0.304374\pi\)
0.576613 + 0.817017i \(0.304374\pi\)
\(128\) 12.4995 1.10481
\(129\) −10.5547 −0.929292
\(130\) 0 0
\(131\) 8.69003 0.759251 0.379626 0.925140i \(-0.376053\pi\)
0.379626 + 0.925140i \(0.376053\pi\)
\(132\) −0.153704 −0.0133782
\(133\) −2.27672 −0.197417
\(134\) −6.03504 −0.521348
\(135\) 0 0
\(136\) −7.02295 −0.602213
\(137\) −14.4550 −1.23498 −0.617489 0.786580i \(-0.711850\pi\)
−0.617489 + 0.786580i \(0.711850\pi\)
\(138\) 13.2017 1.12380
\(139\) −1.72208 −0.146065 −0.0730324 0.997330i \(-0.523268\pi\)
−0.0730324 + 0.997330i \(0.523268\pi\)
\(140\) 0 0
\(141\) 10.3036 0.867723
\(142\) 20.1389 1.69002
\(143\) −0.434137 −0.0363043
\(144\) −4.99637 −0.416364
\(145\) 0 0
\(146\) 1.92277 0.159129
\(147\) 6.57187 0.542039
\(148\) −9.95234 −0.818077
\(149\) 16.1461 1.32274 0.661369 0.750060i \(-0.269976\pi\)
0.661369 + 0.750060i \(0.269976\pi\)
\(150\) 0 0
\(151\) −0.733083 −0.0596574 −0.0298287 0.999555i \(-0.509496\pi\)
−0.0298287 + 0.999555i \(0.509496\pi\)
\(152\) 6.32536 0.513055
\(153\) 3.86328 0.312327
\(154\) 0.183492 0.0147862
\(155\) 0 0
\(156\) 2.49439 0.199711
\(157\) −10.8270 −0.864085 −0.432043 0.901853i \(-0.642207\pi\)
−0.432043 + 0.901853i \(0.642207\pi\)
\(158\) −21.2733 −1.69241
\(159\) 4.04766 0.321000
\(160\) 0 0
\(161\) −5.03806 −0.397054
\(162\) 1.71457 0.134709
\(163\) 13.5528 1.06154 0.530768 0.847517i \(-0.321904\pi\)
0.530768 + 0.847517i \(0.321904\pi\)
\(164\) −5.80216 −0.453073
\(165\) 0 0
\(166\) −13.5332 −1.05038
\(167\) −18.4671 −1.42902 −0.714512 0.699623i \(-0.753351\pi\)
−0.714512 + 0.699623i \(0.753351\pi\)
\(168\) 1.18947 0.0917694
\(169\) −5.95460 −0.458046
\(170\) 0 0
\(171\) −3.47954 −0.266087
\(172\) 9.91878 0.756300
\(173\) 8.92785 0.678772 0.339386 0.940647i \(-0.389781\pi\)
0.339386 + 0.940647i \(0.389781\pi\)
\(174\) −1.71457 −0.129981
\(175\) 0 0
\(176\) −0.817200 −0.0615987
\(177\) 0.328734 0.0247092
\(178\) 8.65346 0.648604
\(179\) −1.86896 −0.139693 −0.0698465 0.997558i \(-0.522251\pi\)
−0.0698465 + 0.997558i \(0.522251\pi\)
\(180\) 0 0
\(181\) 16.8852 1.25507 0.627533 0.778590i \(-0.284065\pi\)
0.627533 + 0.778590i \(0.284065\pi\)
\(182\) −2.97780 −0.220730
\(183\) 5.72054 0.422875
\(184\) 13.9971 1.03188
\(185\) 0 0
\(186\) −8.66328 −0.635222
\(187\) 0.631872 0.0462071
\(188\) −9.68282 −0.706192
\(189\) −0.654317 −0.0475946
\(190\) 0 0
\(191\) 7.58677 0.548960 0.274480 0.961593i \(-0.411494\pi\)
0.274480 + 0.961593i \(0.411494\pi\)
\(192\) −1.53842 −0.111026
\(193\) −27.2794 −1.96362 −0.981808 0.189876i \(-0.939191\pi\)
−0.981808 + 0.189876i \(0.939191\pi\)
\(194\) −14.5580 −1.04520
\(195\) 0 0
\(196\) −6.17590 −0.441136
\(197\) 9.70626 0.691542 0.345771 0.938319i \(-0.387617\pi\)
0.345771 + 0.938319i \(0.387617\pi\)
\(198\) 0.280433 0.0199295
\(199\) 22.9520 1.62703 0.813513 0.581547i \(-0.197552\pi\)
0.813513 + 0.581547i \(0.197552\pi\)
\(200\) 0 0
\(201\) 3.51985 0.248271
\(202\) 3.10553 0.218504
\(203\) 0.654317 0.0459241
\(204\) −3.63050 −0.254186
\(205\) 0 0
\(206\) −9.95834 −0.693831
\(207\) −7.69972 −0.535167
\(208\) 13.2619 0.919551
\(209\) −0.569108 −0.0393660
\(210\) 0 0
\(211\) 18.0020 1.23931 0.619655 0.784874i \(-0.287273\pi\)
0.619655 + 0.784874i \(0.287273\pi\)
\(212\) −3.80378 −0.261245
\(213\) −11.7457 −0.804804
\(214\) −5.97644 −0.408541
\(215\) 0 0
\(216\) 1.81788 0.123691
\(217\) 3.30610 0.224432
\(218\) 13.9057 0.941810
\(219\) −1.12143 −0.0757792
\(220\) 0 0
\(221\) −10.2544 −0.689783
\(222\) 18.1580 1.21869
\(223\) 19.1383 1.28160 0.640799 0.767708i \(-0.278603\pi\)
0.640799 + 0.767708i \(0.278603\pi\)
\(224\) −3.22635 −0.215570
\(225\) 0 0
\(226\) −3.54428 −0.235762
\(227\) 13.3621 0.886872 0.443436 0.896306i \(-0.353759\pi\)
0.443436 + 0.896306i \(0.353759\pi\)
\(228\) 3.26989 0.216554
\(229\) 7.45145 0.492405 0.246203 0.969218i \(-0.420817\pi\)
0.246203 + 0.969218i \(0.420817\pi\)
\(230\) 0 0
\(231\) −0.107019 −0.00704135
\(232\) −1.81788 −0.119349
\(233\) −21.5080 −1.40904 −0.704518 0.709687i \(-0.748837\pi\)
−0.704518 + 0.709687i \(0.748837\pi\)
\(234\) −4.55101 −0.297509
\(235\) 0 0
\(236\) −0.308927 −0.0201094
\(237\) 12.4074 0.805946
\(238\) 4.33410 0.280938
\(239\) 24.1169 1.55999 0.779997 0.625784i \(-0.215221\pi\)
0.779997 + 0.625784i \(0.215221\pi\)
\(240\) 0 0
\(241\) −7.22805 −0.465600 −0.232800 0.972525i \(-0.574789\pi\)
−0.232800 + 0.972525i \(0.574789\pi\)
\(242\) −18.8144 −1.20944
\(243\) −1.00000 −0.0641500
\(244\) −5.37587 −0.344155
\(245\) 0 0
\(246\) 10.5860 0.674942
\(247\) 9.23579 0.587659
\(248\) −9.18526 −0.583264
\(249\) 7.89306 0.500203
\(250\) 0 0
\(251\) −8.74458 −0.551953 −0.275977 0.961164i \(-0.589001\pi\)
−0.275977 + 0.961164i \(0.589001\pi\)
\(252\) 0.614893 0.0387346
\(253\) −1.25936 −0.0791750
\(254\) 22.2829 1.39815
\(255\) 0 0
\(256\) 18.3544 1.14715
\(257\) 10.1877 0.635494 0.317747 0.948176i \(-0.397074\pi\)
0.317747 + 0.948176i \(0.397074\pi\)
\(258\) −18.0968 −1.12666
\(259\) −6.92950 −0.430578
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 14.8997 0.920504
\(263\) −11.6794 −0.720184 −0.360092 0.932917i \(-0.617255\pi\)
−0.360092 + 0.932917i \(0.617255\pi\)
\(264\) 0.297329 0.0182994
\(265\) 0 0
\(266\) −3.90359 −0.239345
\(267\) −5.04702 −0.308872
\(268\) −3.30778 −0.202055
\(269\) 23.0385 1.40468 0.702342 0.711839i \(-0.252138\pi\)
0.702342 + 0.711839i \(0.252138\pi\)
\(270\) 0 0
\(271\) −4.45505 −0.270625 −0.135312 0.990803i \(-0.543204\pi\)
−0.135312 + 0.990803i \(0.543204\pi\)
\(272\) −19.3024 −1.17038
\(273\) 1.73677 0.105114
\(274\) −24.7842 −1.49727
\(275\) 0 0
\(276\) 7.23579 0.435543
\(277\) −15.7000 −0.943321 −0.471661 0.881780i \(-0.656345\pi\)
−0.471661 + 0.881780i \(0.656345\pi\)
\(278\) −2.95262 −0.177087
\(279\) 5.05274 0.302500
\(280\) 0 0
\(281\) 13.4721 0.803677 0.401838 0.915711i \(-0.368371\pi\)
0.401838 + 0.915711i \(0.368371\pi\)
\(282\) 17.6663 1.05201
\(283\) −5.86481 −0.348627 −0.174313 0.984690i \(-0.555771\pi\)
−0.174313 + 0.984690i \(0.555771\pi\)
\(284\) 11.0380 0.654986
\(285\) 0 0
\(286\) −0.744357 −0.0440148
\(287\) −4.03987 −0.238466
\(288\) −4.93087 −0.290554
\(289\) −2.07510 −0.122065
\(290\) 0 0
\(291\) 8.49076 0.497737
\(292\) 1.05386 0.0616726
\(293\) 10.3642 0.605485 0.302742 0.953072i \(-0.402098\pi\)
0.302742 + 0.953072i \(0.402098\pi\)
\(294\) 11.2679 0.657159
\(295\) 0 0
\(296\) 19.2521 1.11901
\(297\) −0.163559 −0.00949064
\(298\) 27.6836 1.60367
\(299\) 20.4375 1.18193
\(300\) 0 0
\(301\) 6.90614 0.398063
\(302\) −1.25692 −0.0723277
\(303\) −1.81126 −0.104054
\(304\) 17.3850 0.997101
\(305\) 0 0
\(306\) 6.62385 0.378660
\(307\) 16.0760 0.917504 0.458752 0.888564i \(-0.348297\pi\)
0.458752 + 0.888564i \(0.348297\pi\)
\(308\) 0.100571 0.00573057
\(309\) 5.80807 0.330410
\(310\) 0 0
\(311\) 16.9976 0.963846 0.481923 0.876214i \(-0.339939\pi\)
0.481923 + 0.876214i \(0.339939\pi\)
\(312\) −4.82522 −0.273174
\(313\) −25.2391 −1.42660 −0.713300 0.700858i \(-0.752800\pi\)
−0.713300 + 0.700858i \(0.752800\pi\)
\(314\) −18.5636 −1.04760
\(315\) 0 0
\(316\) −11.6598 −0.655916
\(317\) −3.06731 −0.172277 −0.0861386 0.996283i \(-0.527453\pi\)
−0.0861386 + 0.996283i \(0.527453\pi\)
\(318\) 6.94000 0.389176
\(319\) 0.163559 0.00915753
\(320\) 0 0
\(321\) 3.48568 0.194552
\(322\) −8.63810 −0.481382
\(323\) −13.4424 −0.747955
\(324\) 0.939748 0.0522082
\(325\) 0 0
\(326\) 23.2372 1.28699
\(327\) −8.11029 −0.448500
\(328\) 11.2239 0.619735
\(329\) −6.74185 −0.371690
\(330\) 0 0
\(331\) −19.9971 −1.09914 −0.549571 0.835447i \(-0.685209\pi\)
−0.549571 + 0.835447i \(0.685209\pi\)
\(332\) −7.41749 −0.407088
\(333\) −10.5904 −0.580352
\(334\) −31.6631 −1.73253
\(335\) 0 0
\(336\) 3.26921 0.178350
\(337\) 6.45613 0.351688 0.175844 0.984418i \(-0.443735\pi\)
0.175844 + 0.984418i \(0.443735\pi\)
\(338\) −10.2096 −0.555328
\(339\) 2.06715 0.112272
\(340\) 0 0
\(341\) 0.826420 0.0447531
\(342\) −5.96591 −0.322599
\(343\) −8.88031 −0.479491
\(344\) −19.1872 −1.03450
\(345\) 0 0
\(346\) 15.3074 0.822932
\(347\) 14.9415 0.802099 0.401050 0.916056i \(-0.368645\pi\)
0.401050 + 0.916056i \(0.368645\pi\)
\(348\) −0.939748 −0.0503758
\(349\) −12.1055 −0.647992 −0.323996 0.946058i \(-0.605026\pi\)
−0.323996 + 0.946058i \(0.605026\pi\)
\(350\) 0 0
\(351\) 2.65432 0.141677
\(352\) −0.806487 −0.0429859
\(353\) 18.1937 0.968355 0.484178 0.874970i \(-0.339119\pi\)
0.484178 + 0.874970i \(0.339119\pi\)
\(354\) 0.563637 0.0299570
\(355\) 0 0
\(356\) 4.74292 0.251374
\(357\) −2.52781 −0.133786
\(358\) −3.20447 −0.169361
\(359\) −35.1010 −1.85256 −0.926279 0.376838i \(-0.877011\pi\)
−0.926279 + 0.376838i \(0.877011\pi\)
\(360\) 0 0
\(361\) −6.89283 −0.362780
\(362\) 28.9508 1.52162
\(363\) 10.9732 0.575946
\(364\) −1.63212 −0.0855464
\(365\) 0 0
\(366\) 9.80827 0.512686
\(367\) −23.1943 −1.21073 −0.605367 0.795946i \(-0.706974\pi\)
−0.605367 + 0.795946i \(0.706974\pi\)
\(368\) 38.4706 2.00542
\(369\) −6.17417 −0.321415
\(370\) 0 0
\(371\) −2.64845 −0.137501
\(372\) −4.74830 −0.246188
\(373\) 4.99469 0.258615 0.129308 0.991605i \(-0.458725\pi\)
0.129308 + 0.991605i \(0.458725\pi\)
\(374\) 1.08339 0.0560207
\(375\) 0 0
\(376\) 18.7307 0.965964
\(377\) −2.65432 −0.136704
\(378\) −1.12187 −0.0577029
\(379\) −28.7738 −1.47801 −0.739006 0.673699i \(-0.764704\pi\)
−0.739006 + 0.673699i \(0.764704\pi\)
\(380\) 0 0
\(381\) −12.9962 −0.665816
\(382\) 13.0080 0.665550
\(383\) 24.9482 1.27479 0.637396 0.770537i \(-0.280012\pi\)
0.637396 + 0.770537i \(0.280012\pi\)
\(384\) −12.4995 −0.637861
\(385\) 0 0
\(386\) −46.7725 −2.38066
\(387\) 10.5547 0.536527
\(388\) −7.97917 −0.405081
\(389\) 31.0949 1.57657 0.788286 0.615308i \(-0.210969\pi\)
0.788286 + 0.615308i \(0.210969\pi\)
\(390\) 0 0
\(391\) −29.7461 −1.50433
\(392\) 11.9468 0.603407
\(393\) −8.69003 −0.438354
\(394\) 16.6421 0.838414
\(395\) 0 0
\(396\) 0.153704 0.00772391
\(397\) 12.1948 0.612038 0.306019 0.952025i \(-0.401003\pi\)
0.306019 + 0.952025i \(0.401003\pi\)
\(398\) 39.3528 1.97258
\(399\) 2.27672 0.113979
\(400\) 0 0
\(401\) 4.33039 0.216249 0.108125 0.994137i \(-0.465515\pi\)
0.108125 + 0.994137i \(0.465515\pi\)
\(402\) 6.03504 0.301000
\(403\) −13.4116 −0.668078
\(404\) 1.70213 0.0846841
\(405\) 0 0
\(406\) 1.12187 0.0556776
\(407\) −1.73216 −0.0858599
\(408\) 7.02295 0.347688
\(409\) 5.12178 0.253256 0.126628 0.991950i \(-0.459585\pi\)
0.126628 + 0.991950i \(0.459585\pi\)
\(410\) 0 0
\(411\) 14.4550 0.713015
\(412\) −5.45812 −0.268902
\(413\) −0.215096 −0.0105842
\(414\) −13.2017 −0.648828
\(415\) 0 0
\(416\) 13.0881 0.641697
\(417\) 1.72208 0.0843306
\(418\) −0.975776 −0.0477267
\(419\) 14.1945 0.693445 0.346723 0.937968i \(-0.387295\pi\)
0.346723 + 0.937968i \(0.387295\pi\)
\(420\) 0 0
\(421\) 4.83364 0.235577 0.117789 0.993039i \(-0.462419\pi\)
0.117789 + 0.993039i \(0.462419\pi\)
\(422\) 30.8657 1.50252
\(423\) −10.3036 −0.500980
\(424\) 7.35815 0.357343
\(425\) 0 0
\(426\) −20.1389 −0.975731
\(427\) −3.74305 −0.181139
\(428\) −3.27566 −0.158335
\(429\) 0.434137 0.0209603
\(430\) 0 0
\(431\) −18.0910 −0.871411 −0.435706 0.900089i \(-0.643501\pi\)
−0.435706 + 0.900089i \(0.643501\pi\)
\(432\) 4.99637 0.240388
\(433\) 1.71005 0.0821798 0.0410899 0.999155i \(-0.486917\pi\)
0.0410899 + 0.999155i \(0.486917\pi\)
\(434\) 5.66853 0.272098
\(435\) 0 0
\(436\) 7.62163 0.365010
\(437\) 26.7914 1.28161
\(438\) −1.92277 −0.0918734
\(439\) 24.3533 1.16232 0.581160 0.813789i \(-0.302599\pi\)
0.581160 + 0.813789i \(0.302599\pi\)
\(440\) 0 0
\(441\) −6.57187 −0.312946
\(442\) −17.5818 −0.836281
\(443\) −18.9066 −0.898280 −0.449140 0.893461i \(-0.648270\pi\)
−0.449140 + 0.893461i \(0.648270\pi\)
\(444\) 9.95234 0.472317
\(445\) 0 0
\(446\) 32.8140 1.55379
\(447\) −16.1461 −0.763684
\(448\) 1.00661 0.0475581
\(449\) 25.2808 1.19307 0.596537 0.802586i \(-0.296543\pi\)
0.596537 + 0.802586i \(0.296543\pi\)
\(450\) 0 0
\(451\) −1.00984 −0.0475515
\(452\) −1.94260 −0.0913723
\(453\) 0.733083 0.0344432
\(454\) 22.9102 1.07523
\(455\) 0 0
\(456\) −6.32536 −0.296212
\(457\) −4.25835 −0.199197 −0.0995986 0.995028i \(-0.531756\pi\)
−0.0995986 + 0.995028i \(0.531756\pi\)
\(458\) 12.7760 0.596984
\(459\) −3.86328 −0.180322
\(460\) 0 0
\(461\) 2.48353 0.115670 0.0578349 0.998326i \(-0.481580\pi\)
0.0578349 + 0.998326i \(0.481580\pi\)
\(462\) −0.183492 −0.00853682
\(463\) −27.4876 −1.27746 −0.638728 0.769433i \(-0.720539\pi\)
−0.638728 + 0.769433i \(0.720539\pi\)
\(464\) −4.99637 −0.231951
\(465\) 0 0
\(466\) −36.8769 −1.70829
\(467\) 6.24698 0.289076 0.144538 0.989499i \(-0.453830\pi\)
0.144538 + 0.989499i \(0.453830\pi\)
\(468\) −2.49439 −0.115303
\(469\) −2.30310 −0.106347
\(470\) 0 0
\(471\) 10.8270 0.498880
\(472\) 0.597598 0.0275067
\(473\) 1.72632 0.0793761
\(474\) 21.2733 0.977116
\(475\) 0 0
\(476\) 2.37550 0.108881
\(477\) −4.04766 −0.185330
\(478\) 41.3501 1.89131
\(479\) −11.5736 −0.528812 −0.264406 0.964412i \(-0.585176\pi\)
−0.264406 + 0.964412i \(0.585176\pi\)
\(480\) 0 0
\(481\) 28.1104 1.28172
\(482\) −12.3930 −0.564485
\(483\) 5.03806 0.229240
\(484\) −10.3121 −0.468731
\(485\) 0 0
\(486\) −1.71457 −0.0777744
\(487\) 31.1947 1.41357 0.706783 0.707431i \(-0.250146\pi\)
0.706783 + 0.707431i \(0.250146\pi\)
\(488\) 10.3992 0.470751
\(489\) −13.5528 −0.612878
\(490\) 0 0
\(491\) −7.93512 −0.358107 −0.179053 0.983839i \(-0.557303\pi\)
−0.179053 + 0.983839i \(0.557303\pi\)
\(492\) 5.80216 0.261582
\(493\) 3.86328 0.173993
\(494\) 15.8354 0.712469
\(495\) 0 0
\(496\) −25.2454 −1.13355
\(497\) 7.68543 0.344739
\(498\) 13.5332 0.606437
\(499\) −32.8984 −1.47274 −0.736368 0.676581i \(-0.763461\pi\)
−0.736368 + 0.676581i \(0.763461\pi\)
\(500\) 0 0
\(501\) 18.4671 0.825047
\(502\) −14.9932 −0.669179
\(503\) 40.7038 1.81489 0.907446 0.420168i \(-0.138029\pi\)
0.907446 + 0.420168i \(0.138029\pi\)
\(504\) −1.18947 −0.0529831
\(505\) 0 0
\(506\) −2.15925 −0.0959905
\(507\) 5.95460 0.264453
\(508\) 12.2132 0.541871
\(509\) −36.9407 −1.63737 −0.818684 0.574245i \(-0.805296\pi\)
−0.818684 + 0.574245i \(0.805296\pi\)
\(510\) 0 0
\(511\) 0.733771 0.0324601
\(512\) 6.47089 0.285976
\(513\) 3.47954 0.153625
\(514\) 17.4676 0.770462
\(515\) 0 0
\(516\) −9.91878 −0.436650
\(517\) −1.68525 −0.0741172
\(518\) −11.8811 −0.522026
\(519\) −8.92785 −0.391889
\(520\) 0 0
\(521\) −33.2555 −1.45695 −0.728476 0.685072i \(-0.759771\pi\)
−0.728476 + 0.685072i \(0.759771\pi\)
\(522\) 1.71457 0.0750447
\(523\) 0.341001 0.0149109 0.00745547 0.999972i \(-0.497627\pi\)
0.00745547 + 0.999972i \(0.497627\pi\)
\(524\) 8.16644 0.356752
\(525\) 0 0
\(526\) −20.0252 −0.873139
\(527\) 19.5201 0.850310
\(528\) 0.817200 0.0355640
\(529\) 36.2856 1.57764
\(530\) 0 0
\(531\) −0.328734 −0.0142658
\(532\) −2.13954 −0.0927609
\(533\) 16.3882 0.709852
\(534\) −8.65346 −0.374472
\(535\) 0 0
\(536\) 6.39866 0.276380
\(537\) 1.86896 0.0806518
\(538\) 39.5012 1.70302
\(539\) −1.07489 −0.0462986
\(540\) 0 0
\(541\) −18.4196 −0.791919 −0.395960 0.918268i \(-0.629588\pi\)
−0.395960 + 0.918268i \(0.629588\pi\)
\(542\) −7.63849 −0.328101
\(543\) −16.8852 −0.724613
\(544\) −19.0493 −0.816732
\(545\) 0 0
\(546\) 2.97780 0.127438
\(547\) −42.5936 −1.82117 −0.910586 0.413320i \(-0.864369\pi\)
−0.910586 + 0.413320i \(0.864369\pi\)
\(548\) −13.5841 −0.580284
\(549\) −5.72054 −0.244147
\(550\) 0 0
\(551\) −3.47954 −0.148233
\(552\) −13.9971 −0.595757
\(553\) −8.11836 −0.345228
\(554\) −26.9187 −1.14367
\(555\) 0 0
\(556\) −1.61832 −0.0686321
\(557\) −17.5805 −0.744908 −0.372454 0.928051i \(-0.621484\pi\)
−0.372454 + 0.928051i \(0.621484\pi\)
\(558\) 8.66328 0.366746
\(559\) −28.0156 −1.18493
\(560\) 0 0
\(561\) −0.631872 −0.0266777
\(562\) 23.0988 0.974365
\(563\) −6.29621 −0.265354 −0.132677 0.991159i \(-0.542357\pi\)
−0.132677 + 0.991159i \(0.542357\pi\)
\(564\) 9.68282 0.407720
\(565\) 0 0
\(566\) −10.0556 −0.422669
\(567\) 0.654317 0.0274787
\(568\) −21.3523 −0.895921
\(569\) −42.3981 −1.77742 −0.888710 0.458469i \(-0.848398\pi\)
−0.888710 + 0.458469i \(0.848398\pi\)
\(570\) 0 0
\(571\) −18.4882 −0.773708 −0.386854 0.922141i \(-0.626438\pi\)
−0.386854 + 0.922141i \(0.626438\pi\)
\(572\) −0.407979 −0.0170585
\(573\) −7.58677 −0.316942
\(574\) −6.92663 −0.289112
\(575\) 0 0
\(576\) 1.53842 0.0641009
\(577\) 22.0727 0.918897 0.459448 0.888204i \(-0.348047\pi\)
0.459448 + 0.888204i \(0.348047\pi\)
\(578\) −3.55791 −0.147990
\(579\) 27.2794 1.13369
\(580\) 0 0
\(581\) −5.16457 −0.214262
\(582\) 14.5580 0.603448
\(583\) −0.662030 −0.0274185
\(584\) −2.03862 −0.0843587
\(585\) 0 0
\(586\) 17.7702 0.734080
\(587\) 44.1932 1.82405 0.912024 0.410136i \(-0.134519\pi\)
0.912024 + 0.410136i \(0.134519\pi\)
\(588\) 6.17590 0.254690
\(589\) −17.5812 −0.724421
\(590\) 0 0
\(591\) −9.70626 −0.399262
\(592\) 52.9137 2.17474
\(593\) −17.4159 −0.715184 −0.357592 0.933878i \(-0.616402\pi\)
−0.357592 + 0.933878i \(0.616402\pi\)
\(594\) −0.280433 −0.0115063
\(595\) 0 0
\(596\) 15.1732 0.621520
\(597\) −22.9520 −0.939364
\(598\) 35.0415 1.43295
\(599\) 7.70544 0.314836 0.157418 0.987532i \(-0.449683\pi\)
0.157418 + 0.987532i \(0.449683\pi\)
\(600\) 0 0
\(601\) −31.5500 −1.28695 −0.643476 0.765466i \(-0.722508\pi\)
−0.643476 + 0.765466i \(0.722508\pi\)
\(602\) 11.8410 0.482605
\(603\) −3.51985 −0.143340
\(604\) −0.688913 −0.0280315
\(605\) 0 0
\(606\) −3.10553 −0.126154
\(607\) 13.2516 0.537864 0.268932 0.963159i \(-0.413329\pi\)
0.268932 + 0.963159i \(0.413329\pi\)
\(608\) 17.1571 0.695814
\(609\) −0.654317 −0.0265143
\(610\) 0 0
\(611\) 27.3491 1.10643
\(612\) 3.63050 0.146754
\(613\) −29.4609 −1.18991 −0.594957 0.803758i \(-0.702831\pi\)
−0.594957 + 0.803758i \(0.702831\pi\)
\(614\) 27.5634 1.11237
\(615\) 0 0
\(616\) −0.194548 −0.00783855
\(617\) 5.61068 0.225877 0.112939 0.993602i \(-0.463974\pi\)
0.112939 + 0.993602i \(0.463974\pi\)
\(618\) 9.95834 0.400583
\(619\) 40.5248 1.62883 0.814415 0.580282i \(-0.197058\pi\)
0.814415 + 0.580282i \(0.197058\pi\)
\(620\) 0 0
\(621\) 7.69972 0.308979
\(622\) 29.1436 1.16855
\(623\) 3.30235 0.132306
\(624\) −13.2619 −0.530903
\(625\) 0 0
\(626\) −43.2743 −1.72959
\(627\) 0.569108 0.0227280
\(628\) −10.1746 −0.406011
\(629\) −40.9138 −1.63134
\(630\) 0 0
\(631\) −20.2197 −0.804935 −0.402467 0.915434i \(-0.631847\pi\)
−0.402467 + 0.915434i \(0.631847\pi\)
\(632\) 22.5551 0.897193
\(633\) −18.0020 −0.715516
\(634\) −5.25911 −0.208866
\(635\) 0 0
\(636\) 3.80378 0.150830
\(637\) 17.4438 0.691149
\(638\) 0.280433 0.0111024
\(639\) 11.7457 0.464654
\(640\) 0 0
\(641\) −44.0804 −1.74107 −0.870535 0.492107i \(-0.836227\pi\)
−0.870535 + 0.492107i \(0.836227\pi\)
\(642\) 5.97644 0.235871
\(643\) −3.56303 −0.140512 −0.0702561 0.997529i \(-0.522382\pi\)
−0.0702561 + 0.997529i \(0.522382\pi\)
\(644\) −4.73450 −0.186566
\(645\) 0 0
\(646\) −23.0479 −0.906809
\(647\) 40.7523 1.60214 0.801069 0.598573i \(-0.204265\pi\)
0.801069 + 0.598573i \(0.204265\pi\)
\(648\) −1.81788 −0.0714129
\(649\) −0.0537673 −0.00211055
\(650\) 0 0
\(651\) −3.30610 −0.129576
\(652\) 12.7362 0.498788
\(653\) 11.6902 0.457472 0.228736 0.973489i \(-0.426541\pi\)
0.228736 + 0.973489i \(0.426541\pi\)
\(654\) −13.9057 −0.543754
\(655\) 0 0
\(656\) 30.8484 1.20443
\(657\) 1.12143 0.0437511
\(658\) −11.5594 −0.450631
\(659\) −28.1795 −1.09772 −0.548858 0.835915i \(-0.684937\pi\)
−0.548858 + 0.835915i \(0.684937\pi\)
\(660\) 0 0
\(661\) −20.6736 −0.804108 −0.402054 0.915616i \(-0.631704\pi\)
−0.402054 + 0.915616i \(0.631704\pi\)
\(662\) −34.2865 −1.33258
\(663\) 10.2544 0.398246
\(664\) 14.3486 0.556834
\(665\) 0 0
\(666\) −18.1580 −0.703610
\(667\) −7.69972 −0.298134
\(668\) −17.3544 −0.671461
\(669\) −19.1383 −0.739931
\(670\) 0 0
\(671\) −0.935645 −0.0361202
\(672\) 3.22635 0.124459
\(673\) −44.7915 −1.72659 −0.863293 0.504703i \(-0.831602\pi\)
−0.863293 + 0.504703i \(0.831602\pi\)
\(674\) 11.0695 0.426380
\(675\) 0 0
\(676\) −5.59582 −0.215224
\(677\) −16.8746 −0.648542 −0.324271 0.945964i \(-0.605119\pi\)
−0.324271 + 0.945964i \(0.605119\pi\)
\(678\) 3.54428 0.136117
\(679\) −5.55565 −0.213206
\(680\) 0 0
\(681\) −13.3621 −0.512036
\(682\) 1.41695 0.0542580
\(683\) 28.6207 1.09514 0.547571 0.836759i \(-0.315553\pi\)
0.547571 + 0.836759i \(0.315553\pi\)
\(684\) −3.26989 −0.125027
\(685\) 0 0
\(686\) −15.2259 −0.581328
\(687\) −7.45145 −0.284290
\(688\) −52.7353 −2.01051
\(689\) 10.7438 0.409305
\(690\) 0 0
\(691\) −28.5928 −1.08772 −0.543861 0.839175i \(-0.683038\pi\)
−0.543861 + 0.839175i \(0.683038\pi\)
\(692\) 8.38993 0.318937
\(693\) 0.107019 0.00406532
\(694\) 25.6182 0.972452
\(695\) 0 0
\(696\) 1.81788 0.0689064
\(697\) −23.8525 −0.903479
\(698\) −20.7557 −0.785615
\(699\) 21.5080 0.813507
\(700\) 0 0
\(701\) 49.6636 1.87577 0.937885 0.346946i \(-0.112781\pi\)
0.937885 + 0.346946i \(0.112781\pi\)
\(702\) 4.55101 0.171767
\(703\) 36.8498 1.38982
\(704\) 0.251622 0.00948336
\(705\) 0 0
\(706\) 31.1944 1.17402
\(707\) 1.18514 0.0445717
\(708\) 0.308927 0.0116102
\(709\) −19.4755 −0.731419 −0.365710 0.930729i \(-0.619174\pi\)
−0.365710 + 0.930729i \(0.619174\pi\)
\(710\) 0 0
\(711\) −12.4074 −0.465313
\(712\) −9.17485 −0.343842
\(713\) −38.9047 −1.45699
\(714\) −4.33410 −0.162200
\(715\) 0 0
\(716\) −1.75636 −0.0656381
\(717\) −24.1169 −0.900663
\(718\) −60.1830 −2.24601
\(719\) −34.3195 −1.27990 −0.639951 0.768415i \(-0.721046\pi\)
−0.639951 + 0.768415i \(0.721046\pi\)
\(720\) 0 0
\(721\) −3.80032 −0.141531
\(722\) −11.8182 −0.439829
\(723\) 7.22805 0.268814
\(724\) 15.8678 0.589723
\(725\) 0 0
\(726\) 18.8144 0.698268
\(727\) −19.3206 −0.716560 −0.358280 0.933614i \(-0.616637\pi\)
−0.358280 + 0.933614i \(0.616637\pi\)
\(728\) 3.15722 0.117014
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 40.7758 1.50815
\(732\) 5.37587 0.198698
\(733\) −3.10588 −0.114718 −0.0573591 0.998354i \(-0.518268\pi\)
−0.0573591 + 0.998354i \(0.518268\pi\)
\(734\) −39.7683 −1.46787
\(735\) 0 0
\(736\) 37.9663 1.39946
\(737\) −0.575703 −0.0212063
\(738\) −10.5860 −0.389678
\(739\) 38.7106 1.42399 0.711997 0.702183i \(-0.247791\pi\)
0.711997 + 0.702183i \(0.247791\pi\)
\(740\) 0 0
\(741\) −9.23579 −0.339285
\(742\) −4.54096 −0.166704
\(743\) 8.41130 0.308581 0.154290 0.988026i \(-0.450691\pi\)
0.154290 + 0.988026i \(0.450691\pi\)
\(744\) 9.18526 0.336748
\(745\) 0 0
\(746\) 8.56373 0.313541
\(747\) −7.89306 −0.288792
\(748\) 0.593801 0.0217115
\(749\) −2.28074 −0.0833364
\(750\) 0 0
\(751\) −30.5930 −1.11635 −0.558177 0.829722i \(-0.688499\pi\)
−0.558177 + 0.829722i \(0.688499\pi\)
\(752\) 51.4808 1.87731
\(753\) 8.74458 0.318670
\(754\) −4.55101 −0.165738
\(755\) 0 0
\(756\) −0.614893 −0.0223634
\(757\) 41.6526 1.51389 0.756945 0.653479i \(-0.226691\pi\)
0.756945 + 0.653479i \(0.226691\pi\)
\(758\) −49.3347 −1.79192
\(759\) 1.25936 0.0457117
\(760\) 0 0
\(761\) −28.3957 −1.02934 −0.514671 0.857388i \(-0.672086\pi\)
−0.514671 + 0.857388i \(0.672086\pi\)
\(762\) −22.2829 −0.807224
\(763\) 5.30670 0.192115
\(764\) 7.12965 0.257942
\(765\) 0 0
\(766\) 42.7753 1.54554
\(767\) 0.872565 0.0315065
\(768\) −18.3544 −0.662306
\(769\) −48.6782 −1.75538 −0.877690 0.479229i \(-0.840916\pi\)
−0.877690 + 0.479229i \(0.840916\pi\)
\(770\) 0 0
\(771\) −10.1877 −0.366902
\(772\) −25.6358 −0.922652
\(773\) 54.9974 1.97812 0.989060 0.147513i \(-0.0471267\pi\)
0.989060 + 0.147513i \(0.0471267\pi\)
\(774\) 18.0968 0.650476
\(775\) 0 0
\(776\) 15.4351 0.554089
\(777\) 6.92950 0.248595
\(778\) 53.3143 1.91141
\(779\) 21.4833 0.769717
\(780\) 0 0
\(781\) 1.92112 0.0687429
\(782\) −51.0018 −1.82382
\(783\) −1.00000 −0.0357371
\(784\) 32.8355 1.17270
\(785\) 0 0
\(786\) −14.8997 −0.531453
\(787\) −32.7472 −1.16731 −0.583655 0.812002i \(-0.698378\pi\)
−0.583655 + 0.812002i \(0.698378\pi\)
\(788\) 9.12143 0.324938
\(789\) 11.6794 0.415798
\(790\) 0 0
\(791\) −1.35257 −0.0480920
\(792\) −0.297329 −0.0105651
\(793\) 15.1841 0.539205
\(794\) 20.9088 0.742025
\(795\) 0 0
\(796\) 21.5691 0.764497
\(797\) −6.54714 −0.231912 −0.115956 0.993254i \(-0.536993\pi\)
−0.115956 + 0.993254i \(0.536993\pi\)
\(798\) 3.90359 0.138186
\(799\) −39.8058 −1.40823
\(800\) 0 0
\(801\) 5.04702 0.178328
\(802\) 7.42475 0.262177
\(803\) 0.183420 0.00647274
\(804\) 3.30778 0.116656
\(805\) 0 0
\(806\) −22.9951 −0.809967
\(807\) −23.0385 −0.810995
\(808\) −3.29265 −0.115835
\(809\) 40.5849 1.42689 0.713444 0.700712i \(-0.247134\pi\)
0.713444 + 0.700712i \(0.247134\pi\)
\(810\) 0 0
\(811\) 15.4588 0.542832 0.271416 0.962462i \(-0.412508\pi\)
0.271416 + 0.962462i \(0.412508\pi\)
\(812\) 0.614893 0.0215785
\(813\) 4.45505 0.156245
\(814\) −2.96990 −0.104095
\(815\) 0 0
\(816\) 19.3024 0.675717
\(817\) −36.7255 −1.28486
\(818\) 8.78165 0.307043
\(819\) −1.73677 −0.0606875
\(820\) 0 0
\(821\) −22.4828 −0.784656 −0.392328 0.919825i \(-0.628330\pi\)
−0.392328 + 0.919825i \(0.628330\pi\)
\(822\) 24.7842 0.864448
\(823\) −45.1129 −1.57254 −0.786268 0.617886i \(-0.787989\pi\)
−0.786268 + 0.617886i \(0.787989\pi\)
\(824\) 10.5584 0.367818
\(825\) 0 0
\(826\) −0.368798 −0.0128321
\(827\) −9.44115 −0.328301 −0.164150 0.986435i \(-0.552488\pi\)
−0.164150 + 0.986435i \(0.552488\pi\)
\(828\) −7.23579 −0.251461
\(829\) −55.4844 −1.92705 −0.963527 0.267612i \(-0.913765\pi\)
−0.963527 + 0.267612i \(0.913765\pi\)
\(830\) 0 0
\(831\) 15.7000 0.544627
\(832\) −4.08346 −0.141568
\(833\) −25.3889 −0.879675
\(834\) 2.95262 0.102241
\(835\) 0 0
\(836\) −0.534818 −0.0184971
\(837\) −5.05274 −0.174648
\(838\) 24.3374 0.840721
\(839\) 36.6797 1.26632 0.633162 0.774019i \(-0.281757\pi\)
0.633162 + 0.774019i \(0.281757\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 8.28762 0.285610
\(843\) −13.4721 −0.464003
\(844\) 16.9174 0.582319
\(845\) 0 0
\(846\) −17.6663 −0.607380
\(847\) −7.17998 −0.246707
\(848\) 20.2236 0.694482
\(849\) 5.86481 0.201280
\(850\) 0 0
\(851\) 81.5433 2.79527
\(852\) −11.0380 −0.378156
\(853\) 16.2128 0.555114 0.277557 0.960709i \(-0.410475\pi\)
0.277557 + 0.960709i \(0.410475\pi\)
\(854\) −6.41772 −0.219610
\(855\) 0 0
\(856\) 6.33653 0.216578
\(857\) −52.9150 −1.80754 −0.903772 0.428015i \(-0.859213\pi\)
−0.903772 + 0.428015i \(0.859213\pi\)
\(858\) 0.744357 0.0254119
\(859\) −13.7986 −0.470803 −0.235401 0.971898i \(-0.575641\pi\)
−0.235401 + 0.971898i \(0.575641\pi\)
\(860\) 0 0
\(861\) 4.03987 0.137678
\(862\) −31.0182 −1.05648
\(863\) 4.25394 0.144806 0.0724029 0.997375i \(-0.476933\pi\)
0.0724029 + 0.997375i \(0.476933\pi\)
\(864\) 4.93087 0.167752
\(865\) 0 0
\(866\) 2.93200 0.0996334
\(867\) 2.07510 0.0704743
\(868\) 3.10690 0.105455
\(869\) −2.02934 −0.0688405
\(870\) 0 0
\(871\) 9.34281 0.316569
\(872\) −14.7435 −0.499278
\(873\) −8.49076 −0.287369
\(874\) 45.9358 1.55380
\(875\) 0 0
\(876\) −1.05386 −0.0356067
\(877\) −49.9185 −1.68563 −0.842814 0.538205i \(-0.819103\pi\)
−0.842814 + 0.538205i \(0.819103\pi\)
\(878\) 41.7554 1.40918
\(879\) −10.3642 −0.349577
\(880\) 0 0
\(881\) −11.5561 −0.389334 −0.194667 0.980869i \(-0.562363\pi\)
−0.194667 + 0.980869i \(0.562363\pi\)
\(882\) −11.2679 −0.379411
\(883\) −10.5007 −0.353375 −0.176688 0.984267i \(-0.556538\pi\)
−0.176688 + 0.984267i \(0.556538\pi\)
\(884\) −9.63651 −0.324111
\(885\) 0 0
\(886\) −32.4167 −1.08906
\(887\) 13.1447 0.441356 0.220678 0.975347i \(-0.429173\pi\)
0.220678 + 0.975347i \(0.429173\pi\)
\(888\) −19.2521 −0.646058
\(889\) 8.50364 0.285203
\(890\) 0 0
\(891\) 0.163559 0.00547942
\(892\) 17.9852 0.602190
\(893\) 35.8519 1.19974
\(894\) −27.6836 −0.925877
\(895\) 0 0
\(896\) 8.17862 0.273228
\(897\) −20.4375 −0.682388
\(898\) 43.3456 1.44646
\(899\) 5.05274 0.168518
\(900\) 0 0
\(901\) −15.6372 −0.520952
\(902\) −1.73144 −0.0576506
\(903\) −6.90614 −0.229822
\(904\) 3.75783 0.124983
\(905\) 0 0
\(906\) 1.25692 0.0417584
\(907\) 8.45271 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(908\) 12.5570 0.416718
\(909\) 1.81126 0.0600757
\(910\) 0 0
\(911\) −23.3000 −0.771965 −0.385982 0.922506i \(-0.626137\pi\)
−0.385982 + 0.922506i \(0.626137\pi\)
\(912\) −17.3850 −0.575676
\(913\) −1.29098 −0.0427252
\(914\) −7.30123 −0.241503
\(915\) 0 0
\(916\) 7.00248 0.231368
\(917\) 5.68603 0.187769
\(918\) −6.62385 −0.218620
\(919\) −37.7491 −1.24523 −0.622614 0.782529i \(-0.713929\pi\)
−0.622614 + 0.782529i \(0.713929\pi\)
\(920\) 0 0
\(921\) −16.0760 −0.529721
\(922\) 4.25819 0.140236
\(923\) −31.1769 −1.02620
\(924\) −0.100571 −0.00330855
\(925\) 0 0
\(926\) −47.1294 −1.54877
\(927\) −5.80807 −0.190762
\(928\) −4.93087 −0.161864
\(929\) 34.3843 1.12811 0.564055 0.825737i \(-0.309240\pi\)
0.564055 + 0.825737i \(0.309240\pi\)
\(930\) 0 0
\(931\) 22.8671 0.749437
\(932\) −20.2121 −0.662069
\(933\) −16.9976 −0.556476
\(934\) 10.7109 0.350471
\(935\) 0 0
\(936\) 4.82522 0.157717
\(937\) 8.85110 0.289153 0.144576 0.989494i \(-0.453818\pi\)
0.144576 + 0.989494i \(0.453818\pi\)
\(938\) −3.94883 −0.128934
\(939\) 25.2391 0.823648
\(940\) 0 0
\(941\) −20.4890 −0.667921 −0.333960 0.942587i \(-0.608385\pi\)
−0.333960 + 0.942587i \(0.608385\pi\)
\(942\) 18.5636 0.604834
\(943\) 47.5394 1.54810
\(944\) 1.64248 0.0534581
\(945\) 0 0
\(946\) 2.95989 0.0962343
\(947\) 38.8206 1.26150 0.630750 0.775986i \(-0.282747\pi\)
0.630750 + 0.775986i \(0.282747\pi\)
\(948\) 11.6598 0.378693
\(949\) −2.97663 −0.0966255
\(950\) 0 0
\(951\) 3.06731 0.0994643
\(952\) −4.59524 −0.148933
\(953\) −33.3755 −1.08114 −0.540570 0.841299i \(-0.681791\pi\)
−0.540570 + 0.841299i \(0.681791\pi\)
\(954\) −6.94000 −0.224691
\(955\) 0 0
\(956\) 22.6638 0.733000
\(957\) −0.163559 −0.00528710
\(958\) −19.8437 −0.641122
\(959\) −9.45818 −0.305421
\(960\) 0 0
\(961\) −5.46979 −0.176445
\(962\) 48.1972 1.55394
\(963\) −3.48568 −0.112324
\(964\) −6.79254 −0.218773
\(965\) 0 0
\(966\) 8.63810 0.277926
\(967\) −14.2745 −0.459036 −0.229518 0.973304i \(-0.573715\pi\)
−0.229518 + 0.973304i \(0.573715\pi\)
\(968\) 19.9480 0.641153
\(969\) 13.4424 0.431832
\(970\) 0 0
\(971\) 58.2228 1.86846 0.934229 0.356675i \(-0.116090\pi\)
0.934229 + 0.356675i \(0.116090\pi\)
\(972\) −0.939748 −0.0301424
\(973\) −1.12679 −0.0361231
\(974\) 53.4854 1.71378
\(975\) 0 0
\(976\) 28.5820 0.914886
\(977\) −57.0456 −1.82505 −0.912526 0.409019i \(-0.865871\pi\)
−0.912526 + 0.409019i \(0.865871\pi\)
\(978\) −23.2372 −0.743043
\(979\) 0.825483 0.0263826
\(980\) 0 0
\(981\) 8.11029 0.258942
\(982\) −13.6053 −0.434163
\(983\) −2.76691 −0.0882506 −0.0441253 0.999026i \(-0.514050\pi\)
−0.0441253 + 0.999026i \(0.514050\pi\)
\(984\) −11.2239 −0.357804
\(985\) 0 0
\(986\) 6.62385 0.210946
\(987\) 6.74185 0.214595
\(988\) 8.67932 0.276126
\(989\) −81.2684 −2.58418
\(990\) 0 0
\(991\) 26.2090 0.832557 0.416278 0.909237i \(-0.363334\pi\)
0.416278 + 0.909237i \(0.363334\pi\)
\(992\) −24.9144 −0.791034
\(993\) 19.9971 0.634590
\(994\) 13.1772 0.417955
\(995\) 0 0
\(996\) 7.41749 0.235032
\(997\) −53.0937 −1.68149 −0.840747 0.541428i \(-0.817884\pi\)
−0.840747 + 0.541428i \(0.817884\pi\)
\(998\) −56.4066 −1.78552
\(999\) 10.5904 0.335067
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 2175.2.a.w.1.5 5
3.2 odd 2 6525.2.a.bs.1.1 5
5.2 odd 4 435.2.c.e.349.8 yes 10
5.3 odd 4 435.2.c.e.349.3 10
5.4 even 2 2175.2.a.z.1.1 5
15.2 even 4 1305.2.c.j.784.3 10
15.8 even 4 1305.2.c.j.784.8 10
15.14 odd 2 6525.2.a.bl.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.e.349.3 10 5.3 odd 4
435.2.c.e.349.8 yes 10 5.2 odd 4
1305.2.c.j.784.3 10 15.2 even 4
1305.2.c.j.784.8 10 15.8 even 4
2175.2.a.w.1.5 5 1.1 even 1 trivial
2175.2.a.z.1.1 5 5.4 even 2
6525.2.a.bl.1.5 5 15.14 odd 2
6525.2.a.bs.1.1 5 3.2 odd 2