Properties

Label 2009.2.a.s.1.10
Level $2009$
Weight $2$
Character 2009.1
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $17$
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] = [2009,2,Mod(1,2009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, 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("2009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.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: \(16.0419457661\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 3 x^{16} - 25 x^{15} + 77 x^{14} + 247 x^{13} - 790 x^{12} - 1231 x^{11} + 4173 x^{10} + 3251 x^{9} - 12183 x^{8} - 4259 x^{7} + 19567 x^{6} + 2029 x^{5} - 16136 x^{4} + \cdots - 464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(1.11543\) of defining polynomial
Character \(\chi\) \(=\) 2009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.11543 q^{2} -2.75409 q^{3} -0.755825 q^{4} -1.34462 q^{5} -3.07199 q^{6} -3.07392 q^{8} +4.58503 q^{9} +O(q^{10})\) \(q+1.11543 q^{2} -2.75409 q^{3} -0.755825 q^{4} -1.34462 q^{5} -3.07199 q^{6} -3.07392 q^{8} +4.58503 q^{9} -1.49982 q^{10} -1.07728 q^{11} +2.08161 q^{12} -0.966204 q^{13} +3.70320 q^{15} -1.91708 q^{16} -6.52880 q^{17} +5.11426 q^{18} -7.07942 q^{19} +1.01630 q^{20} -1.20163 q^{22} +0.898247 q^{23} +8.46586 q^{24} -3.19200 q^{25} -1.07773 q^{26} -4.36532 q^{27} +2.82453 q^{29} +4.13065 q^{30} -1.38928 q^{31} +4.00948 q^{32} +2.96693 q^{33} -7.28240 q^{34} -3.46548 q^{36} -7.12505 q^{37} -7.89657 q^{38} +2.66102 q^{39} +4.13324 q^{40} +1.00000 q^{41} +1.23495 q^{43} +0.814235 q^{44} -6.16511 q^{45} +1.00193 q^{46} +7.85310 q^{47} +5.27981 q^{48} -3.56044 q^{50} +17.9809 q^{51} +0.730281 q^{52} +6.58939 q^{53} -4.86919 q^{54} +1.44853 q^{55} +19.4974 q^{57} +3.15055 q^{58} +2.72107 q^{59} -2.79897 q^{60} -11.0489 q^{61} -1.54964 q^{62} +8.30643 q^{64} +1.29917 q^{65} +3.30939 q^{66} +2.45673 q^{67} +4.93463 q^{68} -2.47385 q^{69} +2.40706 q^{71} -14.0940 q^{72} +8.12078 q^{73} -7.94747 q^{74} +8.79108 q^{75} +5.35081 q^{76} +2.96816 q^{78} -1.96133 q^{79} +2.57774 q^{80} -1.73260 q^{81} +1.11543 q^{82} -9.30225 q^{83} +8.77874 q^{85} +1.37750 q^{86} -7.77902 q^{87} +3.31147 q^{88} +12.7953 q^{89} -6.87672 q^{90} -0.678918 q^{92} +3.82622 q^{93} +8.75955 q^{94} +9.51912 q^{95} -11.0425 q^{96} +18.4064 q^{97} -4.93936 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 3 q^{2} + q^{3} + 25 q^{4} - q^{5} + 2 q^{6} + 9 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 3 q^{2} + q^{3} + 25 q^{4} - q^{5} + 2 q^{6} + 9 q^{8} + 26 q^{9} - 2 q^{10} + 15 q^{11} + 4 q^{12} - 5 q^{13} + 24 q^{15} + 33 q^{16} + 4 q^{17} + 10 q^{18} + 5 q^{19} - 26 q^{20} + 16 q^{22} + 12 q^{23} + 16 q^{24} + 24 q^{25} + 31 q^{26} - 11 q^{27} + 14 q^{29} - 33 q^{30} - 3 q^{31} + 16 q^{32} + 4 q^{33} - 24 q^{34} + 57 q^{36} + 24 q^{37} + 45 q^{39} + 36 q^{40} + 17 q^{41} + 14 q^{43} - 9 q^{44} - 21 q^{45} + 44 q^{46} + 19 q^{47} - 60 q^{48} - 4 q^{50} + 2 q^{51} - 25 q^{52} + 4 q^{53} + 68 q^{54} + 9 q^{55} - 12 q^{57} - q^{58} - 27 q^{59} + 66 q^{60} - q^{61} - 23 q^{62} + 75 q^{64} + 22 q^{65} - 16 q^{66} + 49 q^{67} + 45 q^{68} + 12 q^{69} + 40 q^{71} - 23 q^{72} - 14 q^{73} + 33 q^{74} + 27 q^{75} - 9 q^{76} - 12 q^{78} + 61 q^{79} - 82 q^{80} + 53 q^{81} + 3 q^{82} - 18 q^{83} - 13 q^{85} - 4 q^{86} - 17 q^{87} + 74 q^{88} + 18 q^{89} - 20 q^{90} + 28 q^{92} - 36 q^{93} - 5 q^{94} + 20 q^{95} + 148 q^{96} + 26 q^{97} + 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.11543 0.788725 0.394363 0.918955i \(-0.370965\pi\)
0.394363 + 0.918955i \(0.370965\pi\)
\(3\) −2.75409 −1.59008 −0.795038 0.606559i \(-0.792549\pi\)
−0.795038 + 0.606559i \(0.792549\pi\)
\(4\) −0.755825 −0.377913
\(5\) −1.34462 −0.601331 −0.300666 0.953730i \(-0.597209\pi\)
−0.300666 + 0.953730i \(0.597209\pi\)
\(6\) −3.07199 −1.25413
\(7\) 0 0
\(8\) −3.07392 −1.08679
\(9\) 4.58503 1.52834
\(10\) −1.49982 −0.474285
\(11\) −1.07728 −0.324812 −0.162406 0.986724i \(-0.551925\pi\)
−0.162406 + 0.986724i \(0.551925\pi\)
\(12\) 2.08161 0.600910
\(13\) −0.966204 −0.267977 −0.133988 0.990983i \(-0.542778\pi\)
−0.133988 + 0.990983i \(0.542778\pi\)
\(14\) 0 0
\(15\) 3.70320 0.956163
\(16\) −1.91708 −0.479269
\(17\) −6.52880 −1.58347 −0.791734 0.610866i \(-0.790821\pi\)
−0.791734 + 0.610866i \(0.790821\pi\)
\(18\) 5.11426 1.20544
\(19\) −7.07942 −1.62413 −0.812065 0.583566i \(-0.801657\pi\)
−0.812065 + 0.583566i \(0.801657\pi\)
\(20\) 1.01630 0.227251
\(21\) 0 0
\(22\) −1.20163 −0.256187
\(23\) 0.898247 0.187297 0.0936487 0.995605i \(-0.470147\pi\)
0.0936487 + 0.995605i \(0.470147\pi\)
\(24\) 8.46586 1.72809
\(25\) −3.19200 −0.638401
\(26\) −1.07773 −0.211360
\(27\) −4.36532 −0.840106
\(28\) 0 0
\(29\) 2.82453 0.524502 0.262251 0.965000i \(-0.415535\pi\)
0.262251 + 0.965000i \(0.415535\pi\)
\(30\) 4.13065 0.754149
\(31\) −1.38928 −0.249523 −0.124761 0.992187i \(-0.539817\pi\)
−0.124761 + 0.992187i \(0.539817\pi\)
\(32\) 4.00948 0.708783
\(33\) 2.96693 0.516476
\(34\) −7.28240 −1.24892
\(35\) 0 0
\(36\) −3.46548 −0.577580
\(37\) −7.12505 −1.17135 −0.585676 0.810545i \(-0.699171\pi\)
−0.585676 + 0.810545i \(0.699171\pi\)
\(38\) −7.89657 −1.28099
\(39\) 2.66102 0.426103
\(40\) 4.13324 0.653523
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) 1.23495 0.188328 0.0941642 0.995557i \(-0.469982\pi\)
0.0941642 + 0.995557i \(0.469982\pi\)
\(44\) 0.814235 0.122751
\(45\) −6.16511 −0.919040
\(46\) 1.00193 0.147726
\(47\) 7.85310 1.14549 0.572746 0.819733i \(-0.305878\pi\)
0.572746 + 0.819733i \(0.305878\pi\)
\(48\) 5.27981 0.762075
\(49\) 0 0
\(50\) −3.56044 −0.503523
\(51\) 17.9809 2.51783
\(52\) 0.730281 0.101272
\(53\) 6.58939 0.905122 0.452561 0.891733i \(-0.350510\pi\)
0.452561 + 0.891733i \(0.350510\pi\)
\(54\) −4.86919 −0.662613
\(55\) 1.44853 0.195320
\(56\) 0 0
\(57\) 19.4974 2.58249
\(58\) 3.15055 0.413688
\(59\) 2.72107 0.354253 0.177126 0.984188i \(-0.443320\pi\)
0.177126 + 0.984188i \(0.443320\pi\)
\(60\) −2.79897 −0.361346
\(61\) −11.0489 −1.41467 −0.707334 0.706880i \(-0.750102\pi\)
−0.707334 + 0.706880i \(0.750102\pi\)
\(62\) −1.54964 −0.196805
\(63\) 0 0
\(64\) 8.30643 1.03830
\(65\) 1.29917 0.161143
\(66\) 3.30939 0.407357
\(67\) 2.45673 0.300138 0.150069 0.988676i \(-0.452050\pi\)
0.150069 + 0.988676i \(0.452050\pi\)
\(68\) 4.93463 0.598412
\(69\) −2.47385 −0.297817
\(70\) 0 0
\(71\) 2.40706 0.285666 0.142833 0.989747i \(-0.454379\pi\)
0.142833 + 0.989747i \(0.454379\pi\)
\(72\) −14.0940 −1.66099
\(73\) 8.12078 0.950465 0.475232 0.879860i \(-0.342364\pi\)
0.475232 + 0.879860i \(0.342364\pi\)
\(74\) −7.94747 −0.923875
\(75\) 8.79108 1.01511
\(76\) 5.35081 0.613780
\(77\) 0 0
\(78\) 2.96816 0.336078
\(79\) −1.96133 −0.220666 −0.110333 0.993895i \(-0.535192\pi\)
−0.110333 + 0.993895i \(0.535192\pi\)
\(80\) 2.57774 0.288200
\(81\) −1.73260 −0.192511
\(82\) 1.11543 0.123178
\(83\) −9.30225 −1.02105 −0.510527 0.859862i \(-0.670550\pi\)
−0.510527 + 0.859862i \(0.670550\pi\)
\(84\) 0 0
\(85\) 8.77874 0.952188
\(86\) 1.37750 0.148539
\(87\) −7.77902 −0.833999
\(88\) 3.31147 0.353004
\(89\) 12.7953 1.35629 0.678147 0.734926i \(-0.262783\pi\)
0.678147 + 0.734926i \(0.262783\pi\)
\(90\) −6.87672 −0.724870
\(91\) 0 0
\(92\) −0.678918 −0.0707820
\(93\) 3.82622 0.396760
\(94\) 8.75955 0.903478
\(95\) 9.51912 0.976641
\(96\) −11.0425 −1.12702
\(97\) 18.4064 1.86888 0.934442 0.356117i \(-0.115899\pi\)
0.934442 + 0.356117i \(0.115899\pi\)
\(98\) 0 0
\(99\) −4.93936 −0.496424
\(100\) 2.41260 0.241260
\(101\) −7.26316 −0.722711 −0.361356 0.932428i \(-0.617686\pi\)
−0.361356 + 0.932428i \(0.617686\pi\)
\(102\) 20.0564 1.98588
\(103\) 19.0867 1.88066 0.940332 0.340257i \(-0.110514\pi\)
0.940332 + 0.340257i \(0.110514\pi\)
\(104\) 2.97003 0.291236
\(105\) 0 0
\(106\) 7.34997 0.713892
\(107\) −11.6695 −1.12813 −0.564065 0.825730i \(-0.690763\pi\)
−0.564065 + 0.825730i \(0.690763\pi\)
\(108\) 3.29942 0.317487
\(109\) 17.0444 1.63256 0.816281 0.577656i \(-0.196032\pi\)
0.816281 + 0.577656i \(0.196032\pi\)
\(110\) 1.61573 0.154053
\(111\) 19.6231 1.86254
\(112\) 0 0
\(113\) −4.74418 −0.446295 −0.223147 0.974785i \(-0.571633\pi\)
−0.223147 + 0.974785i \(0.571633\pi\)
\(114\) 21.7479 2.03688
\(115\) −1.20780 −0.112628
\(116\) −2.13485 −0.198216
\(117\) −4.43007 −0.409560
\(118\) 3.03515 0.279408
\(119\) 0 0
\(120\) −11.3833 −1.03915
\(121\) −9.83947 −0.894497
\(122\) −12.3242 −1.11578
\(123\) −2.75409 −0.248328
\(124\) 1.05006 0.0942978
\(125\) 11.0151 0.985222
\(126\) 0 0
\(127\) −9.95232 −0.883126 −0.441563 0.897230i \(-0.645576\pi\)
−0.441563 + 0.897230i \(0.645576\pi\)
\(128\) 1.24625 0.110154
\(129\) −3.40117 −0.299457
\(130\) 1.44913 0.127097
\(131\) 21.0937 1.84297 0.921484 0.388416i \(-0.126978\pi\)
0.921484 + 0.388416i \(0.126978\pi\)
\(132\) −2.24248 −0.195183
\(133\) 0 0
\(134\) 2.74030 0.236726
\(135\) 5.86968 0.505182
\(136\) 20.0690 1.72090
\(137\) −8.00151 −0.683616 −0.341808 0.939770i \(-0.611039\pi\)
−0.341808 + 0.939770i \(0.611039\pi\)
\(138\) −2.75940 −0.234896
\(139\) −7.13603 −0.605270 −0.302635 0.953107i \(-0.597866\pi\)
−0.302635 + 0.953107i \(0.597866\pi\)
\(140\) 0 0
\(141\) −21.6282 −1.82142
\(142\) 2.68490 0.225312
\(143\) 1.04087 0.0870420
\(144\) −8.78986 −0.732488
\(145\) −3.79791 −0.315400
\(146\) 9.05812 0.749656
\(147\) 0 0
\(148\) 5.38530 0.442669
\(149\) −10.9781 −0.899361 −0.449680 0.893190i \(-0.648462\pi\)
−0.449680 + 0.893190i \(0.648462\pi\)
\(150\) 9.80579 0.800640
\(151\) 19.8359 1.61422 0.807112 0.590399i \(-0.201029\pi\)
0.807112 + 0.590399i \(0.201029\pi\)
\(152\) 21.7616 1.76510
\(153\) −29.9348 −2.42008
\(154\) 0 0
\(155\) 1.86806 0.150046
\(156\) −2.01126 −0.161030
\(157\) −9.62144 −0.767875 −0.383937 0.923359i \(-0.625432\pi\)
−0.383937 + 0.923359i \(0.625432\pi\)
\(158\) −2.18771 −0.174045
\(159\) −18.1478 −1.43921
\(160\) −5.39122 −0.426213
\(161\) 0 0
\(162\) −1.93258 −0.151838
\(163\) 0.333705 0.0261378 0.0130689 0.999915i \(-0.495840\pi\)
0.0130689 + 0.999915i \(0.495840\pi\)
\(164\) −0.755825 −0.0590200
\(165\) −3.98938 −0.310573
\(166\) −10.3760 −0.805332
\(167\) −16.6780 −1.29058 −0.645290 0.763938i \(-0.723263\pi\)
−0.645290 + 0.763938i \(0.723263\pi\)
\(168\) 0 0
\(169\) −12.0665 −0.928188
\(170\) 9.79204 0.751015
\(171\) −32.4594 −2.48223
\(172\) −0.933408 −0.0711717
\(173\) −12.9458 −0.984250 −0.492125 0.870524i \(-0.663780\pi\)
−0.492125 + 0.870524i \(0.663780\pi\)
\(174\) −8.67692 −0.657796
\(175\) 0 0
\(176\) 2.06523 0.155672
\(177\) −7.49408 −0.563289
\(178\) 14.2722 1.06974
\(179\) 17.0658 1.27556 0.637779 0.770219i \(-0.279853\pi\)
0.637779 + 0.770219i \(0.279853\pi\)
\(180\) 4.65975 0.347317
\(181\) −0.402992 −0.0299541 −0.0149771 0.999888i \(-0.504768\pi\)
−0.0149771 + 0.999888i \(0.504768\pi\)
\(182\) 0 0
\(183\) 30.4297 2.24943
\(184\) −2.76114 −0.203554
\(185\) 9.58047 0.704370
\(186\) 4.26786 0.312935
\(187\) 7.03334 0.514329
\(188\) −5.93557 −0.432896
\(189\) 0 0
\(190\) 10.6179 0.770301
\(191\) −13.6069 −0.984559 −0.492280 0.870437i \(-0.663836\pi\)
−0.492280 + 0.870437i \(0.663836\pi\)
\(192\) −22.8767 −1.65098
\(193\) −6.39685 −0.460456 −0.230228 0.973137i \(-0.573947\pi\)
−0.230228 + 0.973137i \(0.573947\pi\)
\(194\) 20.5309 1.47404
\(195\) −3.57805 −0.256229
\(196\) 0 0
\(197\) −21.8587 −1.55737 −0.778684 0.627416i \(-0.784113\pi\)
−0.778684 + 0.627416i \(0.784113\pi\)
\(198\) −5.50949 −0.391542
\(199\) 22.1759 1.57201 0.786005 0.618220i \(-0.212146\pi\)
0.786005 + 0.618220i \(0.212146\pi\)
\(200\) 9.81196 0.693810
\(201\) −6.76607 −0.477242
\(202\) −8.10151 −0.570020
\(203\) 0 0
\(204\) −13.5904 −0.951521
\(205\) −1.34462 −0.0939122
\(206\) 21.2898 1.48333
\(207\) 4.11849 0.286255
\(208\) 1.85229 0.128433
\(209\) 7.62652 0.527537
\(210\) 0 0
\(211\) 16.8283 1.15851 0.579254 0.815147i \(-0.303344\pi\)
0.579254 + 0.815147i \(0.303344\pi\)
\(212\) −4.98043 −0.342057
\(213\) −6.62927 −0.454230
\(214\) −13.0164 −0.889785
\(215\) −1.66054 −0.113248
\(216\) 13.4186 0.913022
\(217\) 0 0
\(218\) 19.0118 1.28764
\(219\) −22.3654 −1.51131
\(220\) −1.09483 −0.0738137
\(221\) 6.30815 0.424332
\(222\) 21.8881 1.46903
\(223\) 3.85768 0.258329 0.129165 0.991623i \(-0.458770\pi\)
0.129165 + 0.991623i \(0.458770\pi\)
\(224\) 0 0
\(225\) −14.6354 −0.975695
\(226\) −5.29178 −0.352004
\(227\) −28.1917 −1.87115 −0.935576 0.353126i \(-0.885119\pi\)
−0.935576 + 0.353126i \(0.885119\pi\)
\(228\) −14.7366 −0.975956
\(229\) −15.2236 −1.00601 −0.503003 0.864285i \(-0.667772\pi\)
−0.503003 + 0.864285i \(0.667772\pi\)
\(230\) −1.34721 −0.0888323
\(231\) 0 0
\(232\) −8.68238 −0.570026
\(233\) −29.0175 −1.90100 −0.950501 0.310722i \(-0.899429\pi\)
−0.950501 + 0.310722i \(0.899429\pi\)
\(234\) −4.94142 −0.323031
\(235\) −10.5594 −0.688820
\(236\) −2.05665 −0.133877
\(237\) 5.40168 0.350877
\(238\) 0 0
\(239\) 25.0381 1.61958 0.809792 0.586718i \(-0.199580\pi\)
0.809792 + 0.586718i \(0.199580\pi\)
\(240\) −7.09932 −0.458259
\(241\) −1.11382 −0.0717474 −0.0358737 0.999356i \(-0.511421\pi\)
−0.0358737 + 0.999356i \(0.511421\pi\)
\(242\) −10.9752 −0.705512
\(243\) 17.8677 1.14621
\(244\) 8.35104 0.534621
\(245\) 0 0
\(246\) −3.07199 −0.195863
\(247\) 6.84016 0.435229
\(248\) 4.27054 0.271180
\(249\) 25.6193 1.62356
\(250\) 12.2865 0.777069
\(251\) 7.27825 0.459399 0.229700 0.973262i \(-0.426226\pi\)
0.229700 + 0.973262i \(0.426226\pi\)
\(252\) 0 0
\(253\) −0.967663 −0.0608364
\(254\) −11.1011 −0.696544
\(255\) −24.1775 −1.51405
\(256\) −15.2228 −0.951423
\(257\) −8.89942 −0.555130 −0.277565 0.960707i \(-0.589527\pi\)
−0.277565 + 0.960707i \(0.589527\pi\)
\(258\) −3.79376 −0.236189
\(259\) 0 0
\(260\) −0.981949 −0.0608979
\(261\) 12.9506 0.801619
\(262\) 23.5285 1.45360
\(263\) −5.89870 −0.363729 −0.181865 0.983324i \(-0.558213\pi\)
−0.181865 + 0.983324i \(0.558213\pi\)
\(264\) −9.12009 −0.561303
\(265\) −8.86021 −0.544278
\(266\) 0 0
\(267\) −35.2393 −2.15661
\(268\) −1.85686 −0.113426
\(269\) 4.18733 0.255306 0.127653 0.991819i \(-0.459256\pi\)
0.127653 + 0.991819i \(0.459256\pi\)
\(270\) 6.54719 0.398450
\(271\) 10.0501 0.610499 0.305250 0.952272i \(-0.401260\pi\)
0.305250 + 0.952272i \(0.401260\pi\)
\(272\) 12.5162 0.758907
\(273\) 0 0
\(274\) −8.92510 −0.539185
\(275\) 3.43868 0.207360
\(276\) 1.86980 0.112549
\(277\) −6.16607 −0.370483 −0.185242 0.982693i \(-0.559307\pi\)
−0.185242 + 0.982693i \(0.559307\pi\)
\(278\) −7.95971 −0.477392
\(279\) −6.36991 −0.381356
\(280\) 0 0
\(281\) −6.64050 −0.396139 −0.198069 0.980188i \(-0.563467\pi\)
−0.198069 + 0.980188i \(0.563467\pi\)
\(282\) −24.1246 −1.43660
\(283\) 32.4418 1.92846 0.964232 0.265061i \(-0.0853921\pi\)
0.964232 + 0.265061i \(0.0853921\pi\)
\(284\) −1.81932 −0.107957
\(285\) −26.2165 −1.55293
\(286\) 1.16101 0.0686522
\(287\) 0 0
\(288\) 18.3836 1.08326
\(289\) 25.6253 1.50737
\(290\) −4.23629 −0.248764
\(291\) −50.6928 −2.97167
\(292\) −6.13789 −0.359193
\(293\) 3.63051 0.212097 0.106048 0.994361i \(-0.466180\pi\)
0.106048 + 0.994361i \(0.466180\pi\)
\(294\) 0 0
\(295\) −3.65880 −0.213023
\(296\) 21.9018 1.27302
\(297\) 4.70267 0.272876
\(298\) −12.2453 −0.709349
\(299\) −0.867889 −0.0501913
\(300\) −6.64452 −0.383621
\(301\) 0 0
\(302\) 22.1255 1.27318
\(303\) 20.0034 1.14917
\(304\) 13.5718 0.778396
\(305\) 14.8566 0.850684
\(306\) −33.3900 −1.90878
\(307\) −0.641523 −0.0366136 −0.0183068 0.999832i \(-0.505828\pi\)
−0.0183068 + 0.999832i \(0.505828\pi\)
\(308\) 0 0
\(309\) −52.5664 −2.99040
\(310\) 2.08368 0.118345
\(311\) −11.8418 −0.671485 −0.335743 0.941954i \(-0.608987\pi\)
−0.335743 + 0.941954i \(0.608987\pi\)
\(312\) −8.17974 −0.463087
\(313\) 19.7303 1.11522 0.557612 0.830102i \(-0.311718\pi\)
0.557612 + 0.830102i \(0.311718\pi\)
\(314\) −10.7320 −0.605642
\(315\) 0 0
\(316\) 1.48242 0.0833926
\(317\) 24.5065 1.37642 0.688211 0.725510i \(-0.258396\pi\)
0.688211 + 0.725510i \(0.258396\pi\)
\(318\) −20.2425 −1.13514
\(319\) −3.04281 −0.170365
\(320\) −11.1690 −0.624365
\(321\) 32.1388 1.79381
\(322\) 0 0
\(323\) 46.2202 2.57176
\(324\) 1.30954 0.0727522
\(325\) 3.08413 0.171077
\(326\) 0.372223 0.0206155
\(327\) −46.9420 −2.59590
\(328\) −3.07392 −0.169729
\(329\) 0 0
\(330\) −4.44986 −0.244957
\(331\) 27.8301 1.52968 0.764841 0.644219i \(-0.222817\pi\)
0.764841 + 0.644219i \(0.222817\pi\)
\(332\) 7.03088 0.385869
\(333\) −32.6686 −1.79023
\(334\) −18.6030 −1.01791
\(335\) −3.30337 −0.180482
\(336\) 0 0
\(337\) 0.235392 0.0128226 0.00641130 0.999979i \(-0.497959\pi\)
0.00641130 + 0.999979i \(0.497959\pi\)
\(338\) −13.4592 −0.732086
\(339\) 13.0659 0.709643
\(340\) −6.63520 −0.359844
\(341\) 1.49665 0.0810480
\(342\) −36.2060 −1.95780
\(343\) 0 0
\(344\) −3.79614 −0.204674
\(345\) 3.32639 0.179087
\(346\) −14.4401 −0.776303
\(347\) −0.509008 −0.0273250 −0.0136625 0.999907i \(-0.504349\pi\)
−0.0136625 + 0.999907i \(0.504349\pi\)
\(348\) 5.87958 0.315179
\(349\) −23.8662 −1.27753 −0.638764 0.769403i \(-0.720554\pi\)
−0.638764 + 0.769403i \(0.720554\pi\)
\(350\) 0 0
\(351\) 4.21779 0.225129
\(352\) −4.31933 −0.230221
\(353\) 13.6030 0.724016 0.362008 0.932175i \(-0.382091\pi\)
0.362008 + 0.932175i \(0.382091\pi\)
\(354\) −8.35909 −0.444280
\(355\) −3.23658 −0.171780
\(356\) −9.67098 −0.512561
\(357\) 0 0
\(358\) 19.0356 1.00607
\(359\) −4.84897 −0.255919 −0.127959 0.991779i \(-0.540843\pi\)
−0.127959 + 0.991779i \(0.540843\pi\)
\(360\) 18.9510 0.998808
\(361\) 31.1182 1.63780
\(362\) −0.449507 −0.0236256
\(363\) 27.0988 1.42232
\(364\) 0 0
\(365\) −10.9193 −0.571544
\(366\) 33.9421 1.77418
\(367\) −20.2354 −1.05628 −0.528139 0.849158i \(-0.677110\pi\)
−0.528139 + 0.849158i \(0.677110\pi\)
\(368\) −1.72201 −0.0897659
\(369\) 4.58503 0.238687
\(370\) 10.6863 0.555555
\(371\) 0 0
\(372\) −2.89195 −0.149941
\(373\) −14.0870 −0.729399 −0.364700 0.931125i \(-0.618828\pi\)
−0.364700 + 0.931125i \(0.618828\pi\)
\(374\) 7.84517 0.405664
\(375\) −30.3366 −1.56658
\(376\) −24.1398 −1.24491
\(377\) −2.72907 −0.140554
\(378\) 0 0
\(379\) −2.35252 −0.120841 −0.0604205 0.998173i \(-0.519244\pi\)
−0.0604205 + 0.998173i \(0.519244\pi\)
\(380\) −7.19479 −0.369085
\(381\) 27.4096 1.40424
\(382\) −15.1775 −0.776547
\(383\) −11.8503 −0.605524 −0.302762 0.953066i \(-0.597909\pi\)
−0.302762 + 0.953066i \(0.597909\pi\)
\(384\) −3.43229 −0.175153
\(385\) 0 0
\(386\) −7.13522 −0.363173
\(387\) 5.66229 0.287830
\(388\) −13.9120 −0.706275
\(389\) −8.23104 −0.417330 −0.208665 0.977987i \(-0.566912\pi\)
−0.208665 + 0.977987i \(0.566912\pi\)
\(390\) −3.99105 −0.202094
\(391\) −5.86448 −0.296579
\(392\) 0 0
\(393\) −58.0941 −2.93046
\(394\) −24.3818 −1.22834
\(395\) 2.63723 0.132694
\(396\) 3.73329 0.187605
\(397\) −7.05555 −0.354108 −0.177054 0.984201i \(-0.556657\pi\)
−0.177054 + 0.984201i \(0.556657\pi\)
\(398\) 24.7356 1.23988
\(399\) 0 0
\(400\) 6.11932 0.305966
\(401\) 32.5397 1.62495 0.812477 0.582993i \(-0.198118\pi\)
0.812477 + 0.582993i \(0.198118\pi\)
\(402\) −7.54705 −0.376413
\(403\) 1.34233 0.0668663
\(404\) 5.48968 0.273122
\(405\) 2.32968 0.115763
\(406\) 0 0
\(407\) 7.67567 0.380469
\(408\) −55.2719 −2.73637
\(409\) −4.62355 −0.228620 −0.114310 0.993445i \(-0.536466\pi\)
−0.114310 + 0.993445i \(0.536466\pi\)
\(410\) −1.49982 −0.0740709
\(411\) 22.0369 1.08700
\(412\) −14.4262 −0.710727
\(413\) 0 0
\(414\) 4.59387 0.225776
\(415\) 12.5080 0.613992
\(416\) −3.87397 −0.189937
\(417\) 19.6533 0.962426
\(418\) 8.50681 0.416082
\(419\) −13.1741 −0.643598 −0.321799 0.946808i \(-0.604288\pi\)
−0.321799 + 0.946808i \(0.604288\pi\)
\(420\) 0 0
\(421\) 30.1903 1.47139 0.735693 0.677316i \(-0.236857\pi\)
0.735693 + 0.677316i \(0.236857\pi\)
\(422\) 18.7707 0.913745
\(423\) 36.0067 1.75070
\(424\) −20.2552 −0.983681
\(425\) 20.8400 1.01089
\(426\) −7.39446 −0.358263
\(427\) 0 0
\(428\) 8.82008 0.426335
\(429\) −2.86666 −0.138403
\(430\) −1.85221 −0.0893214
\(431\) −5.94948 −0.286576 −0.143288 0.989681i \(-0.545768\pi\)
−0.143288 + 0.989681i \(0.545768\pi\)
\(432\) 8.36865 0.402637
\(433\) 27.8876 1.34019 0.670096 0.742274i \(-0.266253\pi\)
0.670096 + 0.742274i \(0.266253\pi\)
\(434\) 0 0
\(435\) 10.4598 0.501509
\(436\) −12.8826 −0.616966
\(437\) −6.35907 −0.304195
\(438\) −24.9469 −1.19201
\(439\) 25.4479 1.21456 0.607281 0.794487i \(-0.292260\pi\)
0.607281 + 0.794487i \(0.292260\pi\)
\(440\) −4.45266 −0.212272
\(441\) 0 0
\(442\) 7.03628 0.334682
\(443\) 9.42035 0.447575 0.223787 0.974638i \(-0.428158\pi\)
0.223787 + 0.974638i \(0.428158\pi\)
\(444\) −14.8316 −0.703877
\(445\) −17.2047 −0.815582
\(446\) 4.30295 0.203751
\(447\) 30.2347 1.43005
\(448\) 0 0
\(449\) −23.2678 −1.09808 −0.549038 0.835798i \(-0.685006\pi\)
−0.549038 + 0.835798i \(0.685006\pi\)
\(450\) −16.3247 −0.769555
\(451\) −1.07728 −0.0507271
\(452\) 3.58577 0.168660
\(453\) −54.6300 −2.56674
\(454\) −31.4458 −1.47582
\(455\) 0 0
\(456\) −59.9334 −2.80664
\(457\) 4.86796 0.227713 0.113857 0.993497i \(-0.463680\pi\)
0.113857 + 0.993497i \(0.463680\pi\)
\(458\) −16.9808 −0.793462
\(459\) 28.5003 1.33028
\(460\) 0.912884 0.0425635
\(461\) 16.1888 0.753987 0.376993 0.926216i \(-0.376958\pi\)
0.376993 + 0.926216i \(0.376958\pi\)
\(462\) 0 0
\(463\) −12.0971 −0.562200 −0.281100 0.959679i \(-0.590699\pi\)
−0.281100 + 0.959679i \(0.590699\pi\)
\(464\) −5.41484 −0.251378
\(465\) −5.14480 −0.238584
\(466\) −32.3669 −1.49937
\(467\) 3.23514 0.149704 0.0748521 0.997195i \(-0.476152\pi\)
0.0748521 + 0.997195i \(0.476152\pi\)
\(468\) 3.34836 0.154778
\(469\) 0 0
\(470\) −11.7782 −0.543290
\(471\) 26.4983 1.22098
\(472\) −8.36434 −0.385000
\(473\) −1.33039 −0.0611713
\(474\) 6.02517 0.276745
\(475\) 22.5975 1.03685
\(476\) 0 0
\(477\) 30.2125 1.38334
\(478\) 27.9282 1.27741
\(479\) −12.2257 −0.558605 −0.279303 0.960203i \(-0.590103\pi\)
−0.279303 + 0.960203i \(0.590103\pi\)
\(480\) 14.8479 0.677711
\(481\) 6.88425 0.313895
\(482\) −1.24238 −0.0565890
\(483\) 0 0
\(484\) 7.43692 0.338042
\(485\) −24.7495 −1.12382
\(486\) 19.9301 0.904047
\(487\) 3.36737 0.152590 0.0762951 0.997085i \(-0.475691\pi\)
0.0762951 + 0.997085i \(0.475691\pi\)
\(488\) 33.9634 1.53745
\(489\) −0.919055 −0.0415611
\(490\) 0 0
\(491\) −1.58497 −0.0715286 −0.0357643 0.999360i \(-0.511387\pi\)
−0.0357643 + 0.999360i \(0.511387\pi\)
\(492\) 2.08161 0.0938464
\(493\) −18.4408 −0.830532
\(494\) 7.62970 0.343276
\(495\) 6.64155 0.298515
\(496\) 2.66336 0.119589
\(497\) 0 0
\(498\) 28.5764 1.28054
\(499\) −27.5970 −1.23541 −0.617706 0.786409i \(-0.711938\pi\)
−0.617706 + 0.786409i \(0.711938\pi\)
\(500\) −8.32550 −0.372328
\(501\) 45.9326 2.05212
\(502\) 8.11835 0.362340
\(503\) −0.387957 −0.0172982 −0.00864908 0.999963i \(-0.502753\pi\)
−0.00864908 + 0.999963i \(0.502753\pi\)
\(504\) 0 0
\(505\) 9.76617 0.434589
\(506\) −1.07936 −0.0479832
\(507\) 33.2321 1.47589
\(508\) 7.52222 0.333744
\(509\) −38.7667 −1.71830 −0.859151 0.511722i \(-0.829008\pi\)
−0.859151 + 0.511722i \(0.829008\pi\)
\(510\) −26.9682 −1.19417
\(511\) 0 0
\(512\) −19.4724 −0.860565
\(513\) 30.9039 1.36444
\(514\) −9.92664 −0.437845
\(515\) −25.6643 −1.13090
\(516\) 2.57069 0.113168
\(517\) −8.45998 −0.372069
\(518\) 0 0
\(519\) 35.6539 1.56503
\(520\) −3.99356 −0.175129
\(521\) −0.619049 −0.0271210 −0.0135605 0.999908i \(-0.504317\pi\)
−0.0135605 + 0.999908i \(0.504317\pi\)
\(522\) 14.4454 0.632257
\(523\) −1.03218 −0.0451341 −0.0225670 0.999745i \(-0.507184\pi\)
−0.0225670 + 0.999745i \(0.507184\pi\)
\(524\) −15.9432 −0.696481
\(525\) 0 0
\(526\) −6.57956 −0.286883
\(527\) 9.07036 0.395111
\(528\) −5.68783 −0.247531
\(529\) −22.1932 −0.964920
\(530\) −9.88290 −0.429286
\(531\) 12.4762 0.541420
\(532\) 0 0
\(533\) −0.966204 −0.0418509
\(534\) −39.3069 −1.70097
\(535\) 15.6910 0.678380
\(536\) −7.55180 −0.326188
\(537\) −47.0008 −2.02824
\(538\) 4.67065 0.201366
\(539\) 0 0
\(540\) −4.43645 −0.190915
\(541\) 38.3318 1.64801 0.824007 0.566580i \(-0.191734\pi\)
0.824007 + 0.566580i \(0.191734\pi\)
\(542\) 11.2101 0.481516
\(543\) 1.10988 0.0476294
\(544\) −26.1771 −1.12233
\(545\) −22.9183 −0.981710
\(546\) 0 0
\(547\) 1.37593 0.0588306 0.0294153 0.999567i \(-0.490635\pi\)
0.0294153 + 0.999567i \(0.490635\pi\)
\(548\) 6.04775 0.258347
\(549\) −50.6596 −2.16210
\(550\) 3.83559 0.163550
\(551\) −19.9960 −0.851860
\(552\) 7.60443 0.323666
\(553\) 0 0
\(554\) −6.87780 −0.292210
\(555\) −26.3855 −1.12000
\(556\) 5.39359 0.228739
\(557\) −14.8018 −0.627174 −0.313587 0.949560i \(-0.601531\pi\)
−0.313587 + 0.949560i \(0.601531\pi\)
\(558\) −7.10516 −0.300785
\(559\) −1.19322 −0.0504676
\(560\) 0 0
\(561\) −19.3705 −0.817823
\(562\) −7.40698 −0.312445
\(563\) 32.7991 1.38232 0.691158 0.722704i \(-0.257101\pi\)
0.691158 + 0.722704i \(0.257101\pi\)
\(564\) 16.3471 0.688337
\(565\) 6.37910 0.268371
\(566\) 36.1864 1.52103
\(567\) 0 0
\(568\) −7.39911 −0.310460
\(569\) 12.7274 0.533562 0.266781 0.963757i \(-0.414040\pi\)
0.266781 + 0.963757i \(0.414040\pi\)
\(570\) −29.2426 −1.22484
\(571\) 25.8904 1.08348 0.541740 0.840546i \(-0.317766\pi\)
0.541740 + 0.840546i \(0.317766\pi\)
\(572\) −0.786717 −0.0328943
\(573\) 37.4746 1.56552
\(574\) 0 0
\(575\) −2.86721 −0.119571
\(576\) 38.0852 1.58688
\(577\) −15.7044 −0.653784 −0.326892 0.945062i \(-0.606001\pi\)
−0.326892 + 0.945062i \(0.606001\pi\)
\(578\) 28.5831 1.18890
\(579\) 17.6175 0.732160
\(580\) 2.87056 0.119193
\(581\) 0 0
\(582\) −56.5441 −2.34383
\(583\) −7.09861 −0.293994
\(584\) −24.9626 −1.03296
\(585\) 5.95675 0.246281
\(586\) 4.04957 0.167286
\(587\) −11.2577 −0.464653 −0.232327 0.972638i \(-0.574634\pi\)
−0.232327 + 0.972638i \(0.574634\pi\)
\(588\) 0 0
\(589\) 9.83533 0.405258
\(590\) −4.08112 −0.168017
\(591\) 60.2009 2.47634
\(592\) 13.6593 0.561393
\(593\) −3.77210 −0.154902 −0.0774508 0.996996i \(-0.524678\pi\)
−0.0774508 + 0.996996i \(0.524678\pi\)
\(594\) 5.24548 0.215224
\(595\) 0 0
\(596\) 8.29752 0.339880
\(597\) −61.0746 −2.49962
\(598\) −0.968066 −0.0395872
\(599\) 22.1317 0.904278 0.452139 0.891947i \(-0.350661\pi\)
0.452139 + 0.891947i \(0.350661\pi\)
\(600\) −27.0231 −1.10321
\(601\) 16.4817 0.672301 0.336151 0.941808i \(-0.390875\pi\)
0.336151 + 0.941808i \(0.390875\pi\)
\(602\) 0 0
\(603\) 11.2642 0.458713
\(604\) −14.9925 −0.610036
\(605\) 13.2303 0.537889
\(606\) 22.3123 0.906376
\(607\) 14.0566 0.570539 0.285270 0.958447i \(-0.407917\pi\)
0.285270 + 0.958447i \(0.407917\pi\)
\(608\) −28.3848 −1.15116
\(609\) 0 0
\(610\) 16.5714 0.670956
\(611\) −7.58769 −0.306965
\(612\) 22.6254 0.914579
\(613\) −36.8757 −1.48939 −0.744697 0.667403i \(-0.767406\pi\)
−0.744697 + 0.667403i \(0.767406\pi\)
\(614\) −0.715571 −0.0288781
\(615\) 3.70320 0.149328
\(616\) 0 0
\(617\) 7.53517 0.303354 0.151677 0.988430i \(-0.451533\pi\)
0.151677 + 0.988430i \(0.451533\pi\)
\(618\) −58.6340 −2.35860
\(619\) 22.5002 0.904358 0.452179 0.891927i \(-0.350647\pi\)
0.452179 + 0.891927i \(0.350647\pi\)
\(620\) −1.41192 −0.0567042
\(621\) −3.92113 −0.157350
\(622\) −13.2086 −0.529617
\(623\) 0 0
\(624\) −5.10137 −0.204218
\(625\) 1.14891 0.0459564
\(626\) 22.0077 0.879606
\(627\) −21.0041 −0.838824
\(628\) 7.27213 0.290190
\(629\) 46.5181 1.85480
\(630\) 0 0
\(631\) 25.4663 1.01380 0.506899 0.862005i \(-0.330792\pi\)
0.506899 + 0.862005i \(0.330792\pi\)
\(632\) 6.02896 0.239819
\(633\) −46.3467 −1.84212
\(634\) 27.3352 1.08562
\(635\) 13.3821 0.531051
\(636\) 13.7166 0.543897
\(637\) 0 0
\(638\) −3.39403 −0.134371
\(639\) 11.0365 0.436595
\(640\) −1.67573 −0.0662390
\(641\) 45.2070 1.78557 0.892785 0.450483i \(-0.148748\pi\)
0.892785 + 0.450483i \(0.148748\pi\)
\(642\) 35.8485 1.41483
\(643\) 7.05984 0.278413 0.139206 0.990263i \(-0.455545\pi\)
0.139206 + 0.990263i \(0.455545\pi\)
\(644\) 0 0
\(645\) 4.57328 0.180073
\(646\) 51.5552 2.02841
\(647\) −0.683172 −0.0268583 −0.0134291 0.999910i \(-0.504275\pi\)
−0.0134291 + 0.999910i \(0.504275\pi\)
\(648\) 5.32586 0.209220
\(649\) −2.93135 −0.115066
\(650\) 3.44011 0.134932
\(651\) 0 0
\(652\) −0.252223 −0.00987780
\(653\) −17.1829 −0.672419 −0.336210 0.941787i \(-0.609145\pi\)
−0.336210 + 0.941787i \(0.609145\pi\)
\(654\) −52.3603 −2.04745
\(655\) −28.3630 −1.10823
\(656\) −1.91708 −0.0748493
\(657\) 37.2340 1.45264
\(658\) 0 0
\(659\) −7.11888 −0.277312 −0.138656 0.990341i \(-0.544278\pi\)
−0.138656 + 0.990341i \(0.544278\pi\)
\(660\) 3.01528 0.117369
\(661\) −24.3519 −0.947180 −0.473590 0.880745i \(-0.657042\pi\)
−0.473590 + 0.880745i \(0.657042\pi\)
\(662\) 31.0424 1.20650
\(663\) −17.3732 −0.674721
\(664\) 28.5944 1.10968
\(665\) 0 0
\(666\) −36.4394 −1.41200
\(667\) 2.53713 0.0982379
\(668\) 12.6056 0.487726
\(669\) −10.6244 −0.410763
\(670\) −3.68466 −0.142351
\(671\) 11.9028 0.459501
\(672\) 0 0
\(673\) 45.7116 1.76205 0.881027 0.473066i \(-0.156853\pi\)
0.881027 + 0.473066i \(0.156853\pi\)
\(674\) 0.262562 0.0101135
\(675\) 13.9341 0.536324
\(676\) 9.12013 0.350774
\(677\) 22.1983 0.853151 0.426576 0.904452i \(-0.359720\pi\)
0.426576 + 0.904452i \(0.359720\pi\)
\(678\) 14.5740 0.559713
\(679\) 0 0
\(680\) −26.9851 −1.03483
\(681\) 77.6427 2.97527
\(682\) 1.66940 0.0639246
\(683\) 11.3820 0.435522 0.217761 0.976002i \(-0.430125\pi\)
0.217761 + 0.976002i \(0.430125\pi\)
\(684\) 24.5336 0.938066
\(685\) 10.7590 0.411079
\(686\) 0 0
\(687\) 41.9273 1.59963
\(688\) −2.36750 −0.0902601
\(689\) −6.36669 −0.242552
\(690\) 3.71034 0.141250
\(691\) −38.3097 −1.45737 −0.728685 0.684849i \(-0.759868\pi\)
−0.728685 + 0.684849i \(0.759868\pi\)
\(692\) 9.78476 0.371961
\(693\) 0 0
\(694\) −0.567761 −0.0215519
\(695\) 9.59523 0.363968
\(696\) 23.9121 0.906385
\(697\) −6.52880 −0.247296
\(698\) −26.6210 −1.00762
\(699\) 79.9170 3.02274
\(700\) 0 0
\(701\) −31.1871 −1.17792 −0.588961 0.808162i \(-0.700463\pi\)
−0.588961 + 0.808162i \(0.700463\pi\)
\(702\) 4.70463 0.177565
\(703\) 50.4413 1.90243
\(704\) −8.94835 −0.337254
\(705\) 29.0816 1.09528
\(706\) 15.1732 0.571050
\(707\) 0 0
\(708\) 5.66421 0.212874
\(709\) 24.9158 0.935733 0.467867 0.883799i \(-0.345023\pi\)
0.467867 + 0.883799i \(0.345023\pi\)
\(710\) −3.61016 −0.135487
\(711\) −8.99274 −0.337254
\(712\) −39.3316 −1.47401
\(713\) −1.24792 −0.0467349
\(714\) 0 0
\(715\) −1.39957 −0.0523411
\(716\) −12.8988 −0.482050
\(717\) −68.9574 −2.57526
\(718\) −5.40867 −0.201850
\(719\) 41.1707 1.53541 0.767704 0.640804i \(-0.221399\pi\)
0.767704 + 0.640804i \(0.221399\pi\)
\(720\) 11.8190 0.440468
\(721\) 0 0
\(722\) 34.7101 1.29178
\(723\) 3.06756 0.114084
\(724\) 0.304591 0.0113200
\(725\) −9.01591 −0.334843
\(726\) 30.2267 1.12182
\(727\) −2.84985 −0.105695 −0.0528475 0.998603i \(-0.516830\pi\)
−0.0528475 + 0.998603i \(0.516830\pi\)
\(728\) 0 0
\(729\) −44.0115 −1.63005
\(730\) −12.1797 −0.450791
\(731\) −8.06276 −0.298212
\(732\) −22.9996 −0.850088
\(733\) 36.0067 1.32994 0.664968 0.746872i \(-0.268445\pi\)
0.664968 + 0.746872i \(0.268445\pi\)
\(734\) −22.5711 −0.833113
\(735\) 0 0
\(736\) 3.60150 0.132753
\(737\) −2.64659 −0.0974883
\(738\) 5.11426 0.188258
\(739\) −5.90631 −0.217267 −0.108634 0.994082i \(-0.534648\pi\)
−0.108634 + 0.994082i \(0.534648\pi\)
\(740\) −7.24116 −0.266190
\(741\) −18.8385 −0.692048
\(742\) 0 0
\(743\) −47.1256 −1.72887 −0.864436 0.502744i \(-0.832324\pi\)
−0.864436 + 0.502744i \(0.832324\pi\)
\(744\) −11.7615 −0.431197
\(745\) 14.7613 0.540814
\(746\) −15.7130 −0.575295
\(747\) −42.6511 −1.56052
\(748\) −5.31598 −0.194371
\(749\) 0 0
\(750\) −33.8383 −1.23560
\(751\) −3.99429 −0.145754 −0.0728768 0.997341i \(-0.523218\pi\)
−0.0728768 + 0.997341i \(0.523218\pi\)
\(752\) −15.0550 −0.548999
\(753\) −20.0450 −0.730480
\(754\) −3.04408 −0.110859
\(755\) −26.6717 −0.970683
\(756\) 0 0
\(757\) 0.0523804 0.00190380 0.000951899 1.00000i \(-0.499697\pi\)
0.000951899 1.00000i \(0.499697\pi\)
\(758\) −2.62407 −0.0953104
\(759\) 2.66503 0.0967346
\(760\) −29.2610 −1.06141
\(761\) 47.4215 1.71903 0.859515 0.511111i \(-0.170766\pi\)
0.859515 + 0.511111i \(0.170766\pi\)
\(762\) 30.5734 1.10756
\(763\) 0 0
\(764\) 10.2844 0.372077
\(765\) 40.2508 1.45527
\(766\) −13.2182 −0.477592
\(767\) −2.62911 −0.0949315
\(768\) 41.9249 1.51283
\(769\) −24.0845 −0.868508 −0.434254 0.900790i \(-0.642988\pi\)
−0.434254 + 0.900790i \(0.642988\pi\)
\(770\) 0 0
\(771\) 24.5098 0.882700
\(772\) 4.83490 0.174012
\(773\) 25.7537 0.926297 0.463148 0.886281i \(-0.346720\pi\)
0.463148 + 0.886281i \(0.346720\pi\)
\(774\) 6.31587 0.227019
\(775\) 4.43460 0.159295
\(776\) −56.5797 −2.03109
\(777\) 0 0
\(778\) −9.18111 −0.329159
\(779\) −7.07942 −0.253647
\(780\) 2.70438 0.0968323
\(781\) −2.59308 −0.0927877
\(782\) −6.54139 −0.233920
\(783\) −12.3300 −0.440637
\(784\) 0 0
\(785\) 12.9372 0.461747
\(786\) −64.7997 −2.31133
\(787\) −30.9026 −1.10156 −0.550780 0.834651i \(-0.685670\pi\)
−0.550780 + 0.834651i \(0.685670\pi\)
\(788\) 16.5214 0.588549
\(789\) 16.2456 0.578358
\(790\) 2.94164 0.104659
\(791\) 0 0
\(792\) 15.1832 0.539511
\(793\) 10.6755 0.379098
\(794\) −7.86994 −0.279294
\(795\) 24.4018 0.865444
\(796\) −16.7611 −0.594083
\(797\) −39.1277 −1.38598 −0.692988 0.720949i \(-0.743706\pi\)
−0.692988 + 0.720949i \(0.743706\pi\)
\(798\) 0 0
\(799\) −51.2713 −1.81385
\(800\) −12.7983 −0.452487
\(801\) 58.6666 2.07288
\(802\) 36.2956 1.28164
\(803\) −8.74834 −0.308722
\(804\) 5.11397 0.180356
\(805\) 0 0
\(806\) 1.49727 0.0527391
\(807\) −11.5323 −0.405956
\(808\) 22.3264 0.785438
\(809\) 15.7632 0.554205 0.277102 0.960840i \(-0.410626\pi\)
0.277102 + 0.960840i \(0.410626\pi\)
\(810\) 2.59858 0.0913049
\(811\) −32.9193 −1.15595 −0.577976 0.816054i \(-0.696157\pi\)
−0.577976 + 0.816054i \(0.696157\pi\)
\(812\) 0 0
\(813\) −27.6789 −0.970740
\(814\) 8.56164 0.300085
\(815\) −0.448706 −0.0157175
\(816\) −34.4708 −1.20672
\(817\) −8.74275 −0.305870
\(818\) −5.15722 −0.180318
\(819\) 0 0
\(820\) 1.01630 0.0354906
\(821\) −41.7837 −1.45826 −0.729131 0.684375i \(-0.760075\pi\)
−0.729131 + 0.684375i \(0.760075\pi\)
\(822\) 24.5805 0.857345
\(823\) 35.0046 1.22018 0.610091 0.792331i \(-0.291133\pi\)
0.610091 + 0.792331i \(0.291133\pi\)
\(824\) −58.6708 −2.04390
\(825\) −9.47044 −0.329719
\(826\) 0 0
\(827\) 16.3017 0.566866 0.283433 0.958992i \(-0.408527\pi\)
0.283433 + 0.958992i \(0.408527\pi\)
\(828\) −3.11286 −0.108179
\(829\) 21.1961 0.736171 0.368085 0.929792i \(-0.380013\pi\)
0.368085 + 0.929792i \(0.380013\pi\)
\(830\) 13.9517 0.484271
\(831\) 16.9819 0.589097
\(832\) −8.02571 −0.278241
\(833\) 0 0
\(834\) 21.9218 0.759089
\(835\) 22.4255 0.776065
\(836\) −5.76431 −0.199363
\(837\) 6.06466 0.209625
\(838\) −14.6948 −0.507622
\(839\) 33.6120 1.16042 0.580208 0.814468i \(-0.302971\pi\)
0.580208 + 0.814468i \(0.302971\pi\)
\(840\) 0 0
\(841\) −21.0220 −0.724897
\(842\) 33.6750 1.16052
\(843\) 18.2885 0.629891
\(844\) −12.7193 −0.437815
\(845\) 16.2248 0.558149
\(846\) 40.1628 1.38082
\(847\) 0 0
\(848\) −12.6324 −0.433797
\(849\) −89.3477 −3.06640
\(850\) 23.2454 0.797312
\(851\) −6.40006 −0.219391
\(852\) 5.01057 0.171659
\(853\) −49.5839 −1.69772 −0.848860 0.528618i \(-0.822711\pi\)
−0.848860 + 0.528618i \(0.822711\pi\)
\(854\) 0 0
\(855\) 43.6454 1.49264
\(856\) 35.8710 1.22605
\(857\) −40.8291 −1.39470 −0.697348 0.716732i \(-0.745637\pi\)
−0.697348 + 0.716732i \(0.745637\pi\)
\(858\) −3.19754 −0.109162
\(859\) −5.75495 −0.196356 −0.0981782 0.995169i \(-0.531302\pi\)
−0.0981782 + 0.995169i \(0.531302\pi\)
\(860\) 1.25508 0.0427978
\(861\) 0 0
\(862\) −6.63621 −0.226030
\(863\) −5.44313 −0.185286 −0.0926432 0.995699i \(-0.529532\pi\)
−0.0926432 + 0.995699i \(0.529532\pi\)
\(864\) −17.5027 −0.595452
\(865\) 17.4071 0.591860
\(866\) 31.1066 1.05704
\(867\) −70.5744 −2.39683
\(868\) 0 0
\(869\) 2.11290 0.0716751
\(870\) 11.6671 0.395553
\(871\) −2.37370 −0.0804299
\(872\) −52.3932 −1.77426
\(873\) 84.3937 2.85629
\(874\) −7.09307 −0.239927
\(875\) 0 0
\(876\) 16.9043 0.571144
\(877\) −0.0377361 −0.00127426 −0.000637128 1.00000i \(-0.500203\pi\)
−0.000637128 1.00000i \(0.500203\pi\)
\(878\) 28.3852 0.957956
\(879\) −9.99877 −0.337250
\(880\) −2.77694 −0.0936107
\(881\) 43.4095 1.46250 0.731252 0.682108i \(-0.238936\pi\)
0.731252 + 0.682108i \(0.238936\pi\)
\(882\) 0 0
\(883\) 29.9703 1.00858 0.504291 0.863534i \(-0.331754\pi\)
0.504291 + 0.863534i \(0.331754\pi\)
\(884\) −4.76786 −0.160361
\(885\) 10.0767 0.338723
\(886\) 10.5077 0.353013
\(887\) −19.5093 −0.655059 −0.327529 0.944841i \(-0.606216\pi\)
−0.327529 + 0.944841i \(0.606216\pi\)
\(888\) −60.3197 −2.02420
\(889\) 0 0
\(890\) −19.1906 −0.643270
\(891\) 1.86649 0.0625298
\(892\) −2.91573 −0.0976259
\(893\) −55.5954 −1.86043
\(894\) 33.7246 1.12792
\(895\) −22.9470 −0.767033
\(896\) 0 0
\(897\) 2.39025 0.0798081
\(898\) −25.9535 −0.866080
\(899\) −3.92407 −0.130875
\(900\) 11.0618 0.368728
\(901\) −43.0208 −1.43323
\(902\) −1.20163 −0.0400097
\(903\) 0 0
\(904\) 14.5832 0.485031
\(905\) 0.541870 0.0180124
\(906\) −60.9357 −2.02445
\(907\) 35.2788 1.17141 0.585707 0.810523i \(-0.300817\pi\)
0.585707 + 0.810523i \(0.300817\pi\)
\(908\) 21.3080 0.707132
\(909\) −33.3018 −1.10455
\(910\) 0 0
\(911\) 15.2689 0.505882 0.252941 0.967482i \(-0.418602\pi\)
0.252941 + 0.967482i \(0.418602\pi\)
\(912\) −37.3780 −1.23771
\(913\) 10.0211 0.331651
\(914\) 5.42984 0.179603
\(915\) −40.9163 −1.35265
\(916\) 11.5064 0.380182
\(917\) 0 0
\(918\) 31.7900 1.04923
\(919\) 52.3612 1.72724 0.863619 0.504145i \(-0.168192\pi\)
0.863619 + 0.504145i \(0.168192\pi\)
\(920\) 3.71267 0.122403
\(921\) 1.76681 0.0582185
\(922\) 18.0574 0.594688
\(923\) −2.32571 −0.0765518
\(924\) 0 0
\(925\) 22.7432 0.747792
\(926\) −13.4934 −0.443421
\(927\) 87.5129 2.87430
\(928\) 11.3249 0.371758
\(929\) 46.0886 1.51212 0.756060 0.654503i \(-0.227122\pi\)
0.756060 + 0.654503i \(0.227122\pi\)
\(930\) −5.73864 −0.188177
\(931\) 0 0
\(932\) 21.9322 0.718413
\(933\) 32.6133 1.06771
\(934\) 3.60856 0.118076
\(935\) −9.45716 −0.309282
\(936\) 13.6177 0.445108
\(937\) −33.6725 −1.10003 −0.550016 0.835154i \(-0.685378\pi\)
−0.550016 + 0.835154i \(0.685378\pi\)
\(938\) 0 0
\(939\) −54.3392 −1.77329
\(940\) 7.98107 0.260314
\(941\) −9.44269 −0.307823 −0.153911 0.988085i \(-0.549187\pi\)
−0.153911 + 0.988085i \(0.549187\pi\)
\(942\) 29.5569 0.963017
\(943\) 0.898247 0.0292509
\(944\) −5.21650 −0.169783
\(945\) 0 0
\(946\) −1.48395 −0.0482474
\(947\) −30.4919 −0.990853 −0.495427 0.868650i \(-0.664988\pi\)
−0.495427 + 0.868650i \(0.664988\pi\)
\(948\) −4.08272 −0.132601
\(949\) −7.84632 −0.254702
\(950\) 25.2059 0.817787
\(951\) −67.4932 −2.18862
\(952\) 0 0
\(953\) −60.9536 −1.97448 −0.987241 0.159235i \(-0.949097\pi\)
−0.987241 + 0.159235i \(0.949097\pi\)
\(954\) 33.6998 1.09107
\(955\) 18.2961 0.592046
\(956\) −18.9245 −0.612061
\(957\) 8.38018 0.270893
\(958\) −13.6368 −0.440586
\(959\) 0 0
\(960\) 30.7604 0.992787
\(961\) −29.0699 −0.937738
\(962\) 7.67887 0.247577
\(963\) −53.5049 −1.72417
\(964\) 0.841853 0.0271143
\(965\) 8.60132 0.276886
\(966\) 0 0
\(967\) 13.3405 0.429001 0.214501 0.976724i \(-0.431188\pi\)
0.214501 + 0.976724i \(0.431188\pi\)
\(968\) 30.2457 0.972135
\(969\) −127.295 −4.08929
\(970\) −27.6063 −0.886383
\(971\) 14.3391 0.460163 0.230081 0.973171i \(-0.426101\pi\)
0.230081 + 0.973171i \(0.426101\pi\)
\(972\) −13.5048 −0.433168
\(973\) 0 0
\(974\) 3.75605 0.120352
\(975\) −8.49397 −0.272025
\(976\) 21.1816 0.678007
\(977\) 46.5148 1.48814 0.744069 0.668102i \(-0.232893\pi\)
0.744069 + 0.668102i \(0.232893\pi\)
\(978\) −1.02514 −0.0327803
\(979\) −13.7841 −0.440541
\(980\) 0 0
\(981\) 78.1493 2.49511
\(982\) −1.76791 −0.0564164
\(983\) −43.7876 −1.39661 −0.698303 0.715802i \(-0.746061\pi\)
−0.698303 + 0.715802i \(0.746061\pi\)
\(984\) 8.46586 0.269882
\(985\) 29.3916 0.936494
\(986\) −20.5693 −0.655062
\(987\) 0 0
\(988\) −5.16997 −0.164479
\(989\) 1.10929 0.0352734
\(990\) 7.40815 0.235447
\(991\) −8.16795 −0.259463 −0.129732 0.991549i \(-0.541412\pi\)
−0.129732 + 0.991549i \(0.541412\pi\)
\(992\) −5.57030 −0.176857
\(993\) −76.6468 −2.43231
\(994\) 0 0
\(995\) −29.8182 −0.945299
\(996\) −19.3637 −0.613562
\(997\) −44.3813 −1.40557 −0.702786 0.711402i \(-0.748061\pi\)
−0.702786 + 0.711402i \(0.748061\pi\)
\(998\) −30.7824 −0.974401
\(999\) 31.1031 0.984059
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 2009.2.a.s.1.10 17
7.2 even 3 287.2.e.d.165.8 34
7.4 even 3 287.2.e.d.247.8 yes 34
7.6 odd 2 2009.2.a.r.1.10 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
287.2.e.d.165.8 34 7.2 even 3
287.2.e.d.247.8 yes 34 7.4 even 3
2009.2.a.r.1.10 17 7.6 odd 2
2009.2.a.s.1.10 17 1.1 even 1 trivial