Properties

Label 2738.2.a.p.1.1
Level $2738$
Weight $2$
Character 2738.1
Self dual yes
Analytic conductor $21.863$
Analytic rank $0$
Dimension $3$
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] = [2738,2,Mod(1,2738)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2738, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2738.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.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: \(21.8630400734\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) 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 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 2738.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.87939 q^{3} +1.00000 q^{4} +3.53209 q^{5} -1.87939 q^{6} -0.879385 q^{7} +1.00000 q^{8} +0.532089 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.87939 q^{3} +1.00000 q^{4} +3.53209 q^{5} -1.87939 q^{6} -0.879385 q^{7} +1.00000 q^{8} +0.532089 q^{9} +3.53209 q^{10} +3.34730 q^{11} -1.87939 q^{12} +6.41147 q^{13} -0.879385 q^{14} -6.63816 q^{15} +1.00000 q^{16} +3.00000 q^{17} +0.532089 q^{18} -4.63816 q^{19} +3.53209 q^{20} +1.65270 q^{21} +3.34730 q^{22} -5.83750 q^{23} -1.87939 q^{24} +7.47565 q^{25} +6.41147 q^{26} +4.63816 q^{27} -0.879385 q^{28} +3.26857 q^{29} -6.63816 q^{30} +1.12061 q^{31} +1.00000 q^{32} -6.29086 q^{33} +3.00000 q^{34} -3.10607 q^{35} +0.532089 q^{36} -4.63816 q^{38} -12.0496 q^{39} +3.53209 q^{40} +8.58172 q^{41} +1.65270 q^{42} -5.61587 q^{43} +3.34730 q^{44} +1.87939 q^{45} -5.83750 q^{46} -5.12836 q^{47} -1.87939 q^{48} -6.22668 q^{49} +7.47565 q^{50} -5.63816 q^{51} +6.41147 q^{52} +0.268571 q^{53} +4.63816 q^{54} +11.8229 q^{55} -0.879385 q^{56} +8.71688 q^{57} +3.26857 q^{58} -7.61856 q^{59} -6.63816 q^{60} -2.12836 q^{61} +1.12061 q^{62} -0.467911 q^{63} +1.00000 q^{64} +22.6459 q^{65} -6.29086 q^{66} +8.53209 q^{67} +3.00000 q^{68} +10.9709 q^{69} -3.10607 q^{70} +13.1480 q^{71} +0.532089 q^{72} -8.57398 q^{73} -14.0496 q^{75} -4.63816 q^{76} -2.94356 q^{77} -12.0496 q^{78} +11.5817 q^{79} +3.53209 q^{80} -10.3131 q^{81} +8.58172 q^{82} +4.17024 q^{83} +1.65270 q^{84} +10.5963 q^{85} -5.61587 q^{86} -6.14290 q^{87} +3.34730 q^{88} -8.37464 q^{89} +1.87939 q^{90} -5.63816 q^{91} -5.83750 q^{92} -2.10607 q^{93} -5.12836 q^{94} -16.3824 q^{95} -1.87939 q^{96} +17.0642 q^{97} -6.22668 q^{98} +1.78106 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{9} + 6 q^{10} + 9 q^{11} + 9 q^{13} + 3 q^{14} - 3 q^{15} + 3 q^{16} + 9 q^{17} - 3 q^{18} + 3 q^{19} + 6 q^{20} + 6 q^{21} + 9 q^{22} - 15 q^{23} + 3 q^{25} + 9 q^{26} - 3 q^{27} + 3 q^{28} - 3 q^{30} + 9 q^{31} + 3 q^{32} - 3 q^{33} + 9 q^{34} + 3 q^{35} - 3 q^{36} + 3 q^{38} - 9 q^{39} + 6 q^{40} - 6 q^{41} + 6 q^{42} - 6 q^{43} + 9 q^{44} - 15 q^{46} + 3 q^{47} - 12 q^{49} + 3 q^{50} + 9 q^{52} - 9 q^{53} - 3 q^{54} + 15 q^{55} + 3 q^{56} + 18 q^{57} - 3 q^{59} - 3 q^{60} + 12 q^{61} + 9 q^{62} - 6 q^{63} + 3 q^{64} + 27 q^{65} - 3 q^{66} + 21 q^{67} + 9 q^{68} - 3 q^{69} + 3 q^{70} + 24 q^{71} - 3 q^{72} - 18 q^{73} - 15 q^{75} + 3 q^{76} + 6 q^{77} - 9 q^{78} + 3 q^{79} + 6 q^{80} - 9 q^{81} - 6 q^{82} - 9 q^{83} + 6 q^{84} + 18 q^{85} - 6 q^{86} - 18 q^{87} + 9 q^{88} - 3 q^{89} - 15 q^{92} + 6 q^{93} + 3 q^{94} - 3 q^{95} + 42 q^{97} - 12 q^{98} - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.87939 −1.08506 −0.542532 0.840035i \(-0.682534\pi\)
−0.542532 + 0.840035i \(0.682534\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.53209 1.57960 0.789799 0.613366i \(-0.210185\pi\)
0.789799 + 0.613366i \(0.210185\pi\)
\(6\) −1.87939 −0.767256
\(7\) −0.879385 −0.332376 −0.166188 0.986094i \(-0.553146\pi\)
−0.166188 + 0.986094i \(0.553146\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.532089 0.177363
\(10\) 3.53209 1.11694
\(11\) 3.34730 1.00925 0.504624 0.863339i \(-0.331631\pi\)
0.504624 + 0.863339i \(0.331631\pi\)
\(12\) −1.87939 −0.542532
\(13\) 6.41147 1.77822 0.889111 0.457691i \(-0.151323\pi\)
0.889111 + 0.457691i \(0.151323\pi\)
\(14\) −0.879385 −0.235026
\(15\) −6.63816 −1.71396
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0.532089 0.125415
\(19\) −4.63816 −1.06407 −0.532033 0.846724i \(-0.678572\pi\)
−0.532033 + 0.846724i \(0.678572\pi\)
\(20\) 3.53209 0.789799
\(21\) 1.65270 0.360650
\(22\) 3.34730 0.713646
\(23\) −5.83750 −1.21720 −0.608601 0.793476i \(-0.708269\pi\)
−0.608601 + 0.793476i \(0.708269\pi\)
\(24\) −1.87939 −0.383628
\(25\) 7.47565 1.49513
\(26\) 6.41147 1.25739
\(27\) 4.63816 0.892613
\(28\) −0.879385 −0.166188
\(29\) 3.26857 0.606958 0.303479 0.952838i \(-0.401852\pi\)
0.303479 + 0.952838i \(0.401852\pi\)
\(30\) −6.63816 −1.21196
\(31\) 1.12061 0.201268 0.100634 0.994923i \(-0.467913\pi\)
0.100634 + 0.994923i \(0.467913\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.29086 −1.09510
\(34\) 3.00000 0.514496
\(35\) −3.10607 −0.525021
\(36\) 0.532089 0.0886815
\(37\) 0 0
\(38\) −4.63816 −0.752408
\(39\) −12.0496 −1.92948
\(40\) 3.53209 0.558472
\(41\) 8.58172 1.34024 0.670120 0.742253i \(-0.266243\pi\)
0.670120 + 0.742253i \(0.266243\pi\)
\(42\) 1.65270 0.255018
\(43\) −5.61587 −0.856412 −0.428206 0.903681i \(-0.640854\pi\)
−0.428206 + 0.903681i \(0.640854\pi\)
\(44\) 3.34730 0.504624
\(45\) 1.87939 0.280162
\(46\) −5.83750 −0.860692
\(47\) −5.12836 −0.748048 −0.374024 0.927419i \(-0.622022\pi\)
−0.374024 + 0.927419i \(0.622022\pi\)
\(48\) −1.87939 −0.271266
\(49\) −6.22668 −0.889526
\(50\) 7.47565 1.05722
\(51\) −5.63816 −0.789500
\(52\) 6.41147 0.889111
\(53\) 0.268571 0.0368910 0.0184455 0.999830i \(-0.494128\pi\)
0.0184455 + 0.999830i \(0.494128\pi\)
\(54\) 4.63816 0.631173
\(55\) 11.8229 1.59421
\(56\) −0.879385 −0.117513
\(57\) 8.71688 1.15458
\(58\) 3.26857 0.429184
\(59\) −7.61856 −0.991851 −0.495926 0.868365i \(-0.665171\pi\)
−0.495926 + 0.868365i \(0.665171\pi\)
\(60\) −6.63816 −0.856982
\(61\) −2.12836 −0.272508 −0.136254 0.990674i \(-0.543506\pi\)
−0.136254 + 0.990674i \(0.543506\pi\)
\(62\) 1.12061 0.142318
\(63\) −0.467911 −0.0589513
\(64\) 1.00000 0.125000
\(65\) 22.6459 2.80888
\(66\) −6.29086 −0.774351
\(67\) 8.53209 1.04236 0.521180 0.853447i \(-0.325492\pi\)
0.521180 + 0.853447i \(0.325492\pi\)
\(68\) 3.00000 0.363803
\(69\) 10.9709 1.32074
\(70\) −3.10607 −0.371246
\(71\) 13.1480 1.56038 0.780188 0.625546i \(-0.215124\pi\)
0.780188 + 0.625546i \(0.215124\pi\)
\(72\) 0.532089 0.0627073
\(73\) −8.57398 −1.00351 −0.501754 0.865010i \(-0.667312\pi\)
−0.501754 + 0.865010i \(0.667312\pi\)
\(74\) 0 0
\(75\) −14.0496 −1.62231
\(76\) −4.63816 −0.532033
\(77\) −2.94356 −0.335450
\(78\) −12.0496 −1.36435
\(79\) 11.5817 1.30305 0.651523 0.758629i \(-0.274131\pi\)
0.651523 + 0.758629i \(0.274131\pi\)
\(80\) 3.53209 0.394900
\(81\) −10.3131 −1.14591
\(82\) 8.58172 0.947692
\(83\) 4.17024 0.457744 0.228872 0.973457i \(-0.426496\pi\)
0.228872 + 0.973457i \(0.426496\pi\)
\(84\) 1.65270 0.180325
\(85\) 10.5963 1.14933
\(86\) −5.61587 −0.605575
\(87\) −6.14290 −0.658588
\(88\) 3.34730 0.356823
\(89\) −8.37464 −0.887710 −0.443855 0.896099i \(-0.646389\pi\)
−0.443855 + 0.896099i \(0.646389\pi\)
\(90\) 1.87939 0.198105
\(91\) −5.63816 −0.591039
\(92\) −5.83750 −0.608601
\(93\) −2.10607 −0.218389
\(94\) −5.12836 −0.528949
\(95\) −16.3824 −1.68080
\(96\) −1.87939 −0.191814
\(97\) 17.0642 1.73260 0.866302 0.499520i \(-0.166490\pi\)
0.866302 + 0.499520i \(0.166490\pi\)
\(98\) −6.22668 −0.628990
\(99\) 1.78106 0.179003
\(100\) 7.47565 0.747565
\(101\) 1.08647 0.108107 0.0540537 0.998538i \(-0.482786\pi\)
0.0540537 + 0.998538i \(0.482786\pi\)
\(102\) −5.63816 −0.558261
\(103\) −11.2148 −1.10503 −0.552515 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(104\) 6.41147 0.628697
\(105\) 5.83750 0.569681
\(106\) 0.268571 0.0260859
\(107\) 9.59627 0.927706 0.463853 0.885912i \(-0.346467\pi\)
0.463853 + 0.885912i \(0.346467\pi\)
\(108\) 4.63816 0.446307
\(109\) 2.22163 0.212793 0.106397 0.994324i \(-0.466069\pi\)
0.106397 + 0.994324i \(0.466069\pi\)
\(110\) 11.8229 1.12727
\(111\) 0 0
\(112\) −0.879385 −0.0830941
\(113\) 4.63041 0.435593 0.217796 0.975994i \(-0.430113\pi\)
0.217796 + 0.975994i \(0.430113\pi\)
\(114\) 8.71688 0.816411
\(115\) −20.6186 −1.92269
\(116\) 3.26857 0.303479
\(117\) 3.41147 0.315391
\(118\) −7.61856 −0.701345
\(119\) −2.63816 −0.241839
\(120\) −6.63816 −0.605978
\(121\) 0.204393 0.0185812
\(122\) −2.12836 −0.192692
\(123\) −16.1284 −1.45424
\(124\) 1.12061 0.100634
\(125\) 8.74422 0.782107
\(126\) −0.467911 −0.0416848
\(127\) −5.08647 −0.451351 −0.225675 0.974203i \(-0.572459\pi\)
−0.225675 + 0.974203i \(0.572459\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.5544 0.929261
\(130\) 22.6459 1.98618
\(131\) −3.26083 −0.284900 −0.142450 0.989802i \(-0.545498\pi\)
−0.142450 + 0.989802i \(0.545498\pi\)
\(132\) −6.29086 −0.547549
\(133\) 4.07873 0.353670
\(134\) 8.53209 0.737060
\(135\) 16.3824 1.40997
\(136\) 3.00000 0.257248
\(137\) −1.59627 −0.136378 −0.0681891 0.997672i \(-0.521722\pi\)
−0.0681891 + 0.997672i \(0.521722\pi\)
\(138\) 10.9709 0.933905
\(139\) 11.8033 1.00115 0.500573 0.865694i \(-0.333123\pi\)
0.500573 + 0.865694i \(0.333123\pi\)
\(140\) −3.10607 −0.262511
\(141\) 9.63816 0.811679
\(142\) 13.1480 1.10335
\(143\) 21.4611 1.79467
\(144\) 0.532089 0.0443407
\(145\) 11.5449 0.958750
\(146\) −8.57398 −0.709587
\(147\) 11.7023 0.965192
\(148\) 0 0
\(149\) −12.6604 −1.03718 −0.518592 0.855022i \(-0.673544\pi\)
−0.518592 + 0.855022i \(0.673544\pi\)
\(150\) −14.0496 −1.14715
\(151\) −4.53983 −0.369446 −0.184723 0.982791i \(-0.559139\pi\)
−0.184723 + 0.982791i \(0.559139\pi\)
\(152\) −4.63816 −0.376204
\(153\) 1.59627 0.129051
\(154\) −2.94356 −0.237199
\(155\) 3.95811 0.317923
\(156\) −12.0496 −0.964742
\(157\) −17.8794 −1.42693 −0.713465 0.700691i \(-0.752875\pi\)
−0.713465 + 0.700691i \(0.752875\pi\)
\(158\) 11.5817 0.921392
\(159\) −0.504748 −0.0400291
\(160\) 3.53209 0.279236
\(161\) 5.13341 0.404569
\(162\) −10.3131 −0.810277
\(163\) 2.26857 0.177688 0.0888441 0.996046i \(-0.471683\pi\)
0.0888441 + 0.996046i \(0.471683\pi\)
\(164\) 8.58172 0.670120
\(165\) −22.2199 −1.72981
\(166\) 4.17024 0.323674
\(167\) 21.4466 1.65958 0.829792 0.558073i \(-0.188459\pi\)
0.829792 + 0.558073i \(0.188459\pi\)
\(168\) 1.65270 0.127509
\(169\) 28.1070 2.16208
\(170\) 10.5963 0.812697
\(171\) −2.46791 −0.188726
\(172\) −5.61587 −0.428206
\(173\) −11.2686 −0.856734 −0.428367 0.903605i \(-0.640911\pi\)
−0.428367 + 0.903605i \(0.640911\pi\)
\(174\) −6.14290 −0.465692
\(175\) −6.57398 −0.496946
\(176\) 3.34730 0.252312
\(177\) 14.3182 1.07622
\(178\) −8.37464 −0.627706
\(179\) 6.43107 0.480681 0.240341 0.970689i \(-0.422741\pi\)
0.240341 + 0.970689i \(0.422741\pi\)
\(180\) 1.87939 0.140081
\(181\) 3.50299 0.260375 0.130188 0.991489i \(-0.458442\pi\)
0.130188 + 0.991489i \(0.458442\pi\)
\(182\) −5.63816 −0.417928
\(183\) 4.00000 0.295689
\(184\) −5.83750 −0.430346
\(185\) 0 0
\(186\) −2.10607 −0.154424
\(187\) 10.0419 0.734336
\(188\) −5.12836 −0.374024
\(189\) −4.07873 −0.296684
\(190\) −16.3824 −1.18850
\(191\) 13.9240 1.00750 0.503751 0.863849i \(-0.331953\pi\)
0.503751 + 0.863849i \(0.331953\pi\)
\(192\) −1.87939 −0.135633
\(193\) 11.5767 0.833307 0.416653 0.909065i \(-0.363203\pi\)
0.416653 + 0.909065i \(0.363203\pi\)
\(194\) 17.0642 1.22514
\(195\) −42.5604 −3.04781
\(196\) −6.22668 −0.444763
\(197\) 14.4561 1.02995 0.514976 0.857205i \(-0.327801\pi\)
0.514976 + 0.857205i \(0.327801\pi\)
\(198\) 1.78106 0.126574
\(199\) 5.71688 0.405259 0.202629 0.979255i \(-0.435051\pi\)
0.202629 + 0.979255i \(0.435051\pi\)
\(200\) 7.47565 0.528608
\(201\) −16.0351 −1.13103
\(202\) 1.08647 0.0764435
\(203\) −2.87433 −0.201739
\(204\) −5.63816 −0.394750
\(205\) 30.3114 2.11704
\(206\) −11.2148 −0.781374
\(207\) −3.10607 −0.215887
\(208\) 6.41147 0.444556
\(209\) −15.5253 −1.07391
\(210\) 5.83750 0.402826
\(211\) −7.66550 −0.527715 −0.263857 0.964562i \(-0.584995\pi\)
−0.263857 + 0.964562i \(0.584995\pi\)
\(212\) 0.268571 0.0184455
\(213\) −24.7101 −1.69311
\(214\) 9.59627 0.655987
\(215\) −19.8357 −1.35279
\(216\) 4.63816 0.315587
\(217\) −0.985452 −0.0668968
\(218\) 2.22163 0.150468
\(219\) 16.1138 1.08887
\(220\) 11.8229 0.797103
\(221\) 19.2344 1.29385
\(222\) 0 0
\(223\) −1.99226 −0.133412 −0.0667058 0.997773i \(-0.521249\pi\)
−0.0667058 + 0.997773i \(0.521249\pi\)
\(224\) −0.879385 −0.0587564
\(225\) 3.97771 0.265181
\(226\) 4.63041 0.308011
\(227\) 19.4834 1.29316 0.646579 0.762847i \(-0.276199\pi\)
0.646579 + 0.762847i \(0.276199\pi\)
\(228\) 8.71688 0.577290
\(229\) −12.8152 −0.846853 −0.423426 0.905931i \(-0.639173\pi\)
−0.423426 + 0.905931i \(0.639173\pi\)
\(230\) −20.6186 −1.35955
\(231\) 5.53209 0.363985
\(232\) 3.26857 0.214592
\(233\) 2.91622 0.191048 0.0955240 0.995427i \(-0.469547\pi\)
0.0955240 + 0.995427i \(0.469547\pi\)
\(234\) 3.41147 0.223015
\(235\) −18.1138 −1.18161
\(236\) −7.61856 −0.495926
\(237\) −21.7665 −1.41389
\(238\) −2.63816 −0.171006
\(239\) −16.0128 −1.03578 −0.517891 0.855447i \(-0.673283\pi\)
−0.517891 + 0.855447i \(0.673283\pi\)
\(240\) −6.63816 −0.428491
\(241\) −9.78880 −0.630552 −0.315276 0.949000i \(-0.602097\pi\)
−0.315276 + 0.949000i \(0.602097\pi\)
\(242\) 0.204393 0.0131389
\(243\) 5.46791 0.350767
\(244\) −2.12836 −0.136254
\(245\) −21.9932 −1.40509
\(246\) −16.1284 −1.02831
\(247\) −29.7374 −1.89215
\(248\) 1.12061 0.0711591
\(249\) −7.83750 −0.496681
\(250\) 8.74422 0.553033
\(251\) −16.7442 −1.05689 −0.528443 0.848969i \(-0.677224\pi\)
−0.528443 + 0.848969i \(0.677224\pi\)
\(252\) −0.467911 −0.0294756
\(253\) −19.5398 −1.22846
\(254\) −5.08647 −0.319153
\(255\) −19.9145 −1.24709
\(256\) 1.00000 0.0625000
\(257\) 21.7716 1.35807 0.679036 0.734105i \(-0.262398\pi\)
0.679036 + 0.734105i \(0.262398\pi\)
\(258\) 10.5544 0.657087
\(259\) 0 0
\(260\) 22.6459 1.40444
\(261\) 1.73917 0.107652
\(262\) −3.26083 −0.201455
\(263\) 21.1857 1.30637 0.653184 0.757199i \(-0.273433\pi\)
0.653184 + 0.757199i \(0.273433\pi\)
\(264\) −6.29086 −0.387176
\(265\) 0.948615 0.0582730
\(266\) 4.07873 0.250083
\(267\) 15.7392 0.963222
\(268\) 8.53209 0.521180
\(269\) 20.2317 1.23355 0.616775 0.787139i \(-0.288439\pi\)
0.616775 + 0.787139i \(0.288439\pi\)
\(270\) 16.3824 0.997000
\(271\) −13.3405 −0.810377 −0.405189 0.914233i \(-0.632794\pi\)
−0.405189 + 0.914233i \(0.632794\pi\)
\(272\) 3.00000 0.181902
\(273\) 10.5963 0.641315
\(274\) −1.59627 −0.0964340
\(275\) 25.0232 1.50896
\(276\) 10.9709 0.660371
\(277\) −32.2627 −1.93848 −0.969239 0.246122i \(-0.920844\pi\)
−0.969239 + 0.246122i \(0.920844\pi\)
\(278\) 11.8033 0.707918
\(279\) 0.596267 0.0356976
\(280\) −3.10607 −0.185623
\(281\) −4.31727 −0.257547 −0.128773 0.991674i \(-0.541104\pi\)
−0.128773 + 0.991674i \(0.541104\pi\)
\(282\) 9.63816 0.573944
\(283\) −1.34224 −0.0797881 −0.0398941 0.999204i \(-0.512702\pi\)
−0.0398941 + 0.999204i \(0.512702\pi\)
\(284\) 13.1480 0.780188
\(285\) 30.7888 1.82377
\(286\) 21.4611 1.26902
\(287\) −7.54664 −0.445464
\(288\) 0.532089 0.0313536
\(289\) −8.00000 −0.470588
\(290\) 11.5449 0.677939
\(291\) −32.0702 −1.87999
\(292\) −8.57398 −0.501754
\(293\) 2.75877 0.161169 0.0805845 0.996748i \(-0.474321\pi\)
0.0805845 + 0.996748i \(0.474321\pi\)
\(294\) 11.7023 0.682494
\(295\) −26.9094 −1.56673
\(296\) 0 0
\(297\) 15.5253 0.900868
\(298\) −12.6604 −0.733400
\(299\) −37.4270 −2.16446
\(300\) −14.0496 −0.811156
\(301\) 4.93851 0.284651
\(302\) −4.53983 −0.261238
\(303\) −2.04189 −0.117303
\(304\) −4.63816 −0.266016
\(305\) −7.51754 −0.430453
\(306\) 1.59627 0.0912525
\(307\) −6.58853 −0.376027 −0.188014 0.982166i \(-0.560205\pi\)
−0.188014 + 0.982166i \(0.560205\pi\)
\(308\) −2.94356 −0.167725
\(309\) 21.0770 1.19903
\(310\) 3.95811 0.224806
\(311\) 19.0351 1.07938 0.539690 0.841864i \(-0.318541\pi\)
0.539690 + 0.841864i \(0.318541\pi\)
\(312\) −12.0496 −0.682176
\(313\) 12.8152 0.724358 0.362179 0.932108i \(-0.382033\pi\)
0.362179 + 0.932108i \(0.382033\pi\)
\(314\) −17.8794 −1.00899
\(315\) −1.65270 −0.0931193
\(316\) 11.5817 0.651523
\(317\) −19.5449 −1.09775 −0.548875 0.835904i \(-0.684944\pi\)
−0.548875 + 0.835904i \(0.684944\pi\)
\(318\) −0.504748 −0.0283048
\(319\) 10.9409 0.612571
\(320\) 3.53209 0.197450
\(321\) −18.0351 −1.00662
\(322\) 5.13341 0.286074
\(323\) −13.9145 −0.774222
\(324\) −10.3131 −0.572953
\(325\) 47.9299 2.65868
\(326\) 2.26857 0.125645
\(327\) −4.17530 −0.230894
\(328\) 8.58172 0.473846
\(329\) 4.50980 0.248633
\(330\) −22.2199 −1.22316
\(331\) −13.6973 −0.752871 −0.376435 0.926443i \(-0.622850\pi\)
−0.376435 + 0.926443i \(0.622850\pi\)
\(332\) 4.17024 0.228872
\(333\) 0 0
\(334\) 21.4466 1.17350
\(335\) 30.1361 1.64651
\(336\) 1.65270 0.0901624
\(337\) −5.11381 −0.278567 −0.139283 0.990253i \(-0.544480\pi\)
−0.139283 + 0.990253i \(0.544480\pi\)
\(338\) 28.1070 1.52882
\(339\) −8.70233 −0.472646
\(340\) 10.5963 0.574663
\(341\) 3.75103 0.203130
\(342\) −2.46791 −0.133449
\(343\) 11.6313 0.628034
\(344\) −5.61587 −0.302787
\(345\) 38.7502 2.08624
\(346\) −11.2686 −0.605802
\(347\) −14.9632 −0.803265 −0.401632 0.915801i \(-0.631557\pi\)
−0.401632 + 0.915801i \(0.631557\pi\)
\(348\) −6.14290 −0.329294
\(349\) −13.3628 −0.715293 −0.357647 0.933857i \(-0.616421\pi\)
−0.357647 + 0.933857i \(0.616421\pi\)
\(350\) −6.57398 −0.351394
\(351\) 29.7374 1.58727
\(352\) 3.34730 0.178411
\(353\) −8.15301 −0.433941 −0.216970 0.976178i \(-0.569618\pi\)
−0.216970 + 0.976178i \(0.569618\pi\)
\(354\) 14.3182 0.761004
\(355\) 46.4397 2.46477
\(356\) −8.37464 −0.443855
\(357\) 4.95811 0.262411
\(358\) 6.43107 0.339893
\(359\) 1.08647 0.0573415 0.0286708 0.999589i \(-0.490873\pi\)
0.0286708 + 0.999589i \(0.490873\pi\)
\(360\) 1.87939 0.0990523
\(361\) 2.51249 0.132236
\(362\) 3.50299 0.184113
\(363\) −0.384133 −0.0201618
\(364\) −5.63816 −0.295520
\(365\) −30.2841 −1.58514
\(366\) 4.00000 0.209083
\(367\) −31.7124 −1.65538 −0.827688 0.561189i \(-0.810344\pi\)
−0.827688 + 0.561189i \(0.810344\pi\)
\(368\) −5.83750 −0.304301
\(369\) 4.56624 0.237709
\(370\) 0 0
\(371\) −0.236177 −0.0122617
\(372\) −2.10607 −0.109194
\(373\) 3.52023 0.182271 0.0911353 0.995839i \(-0.470950\pi\)
0.0911353 + 0.995839i \(0.470950\pi\)
\(374\) 10.0419 0.519254
\(375\) −16.4338 −0.848636
\(376\) −5.12836 −0.264475
\(377\) 20.9564 1.07931
\(378\) −4.07873 −0.209787
\(379\) −12.3277 −0.633231 −0.316616 0.948554i \(-0.602547\pi\)
−0.316616 + 0.948554i \(0.602547\pi\)
\(380\) −16.3824 −0.840398
\(381\) 9.55943 0.489744
\(382\) 13.9240 0.712412
\(383\) −24.4593 −1.24981 −0.624907 0.780699i \(-0.714863\pi\)
−0.624907 + 0.780699i \(0.714863\pi\)
\(384\) −1.87939 −0.0959070
\(385\) −10.3969 −0.529876
\(386\) 11.5767 0.589237
\(387\) −2.98814 −0.151896
\(388\) 17.0642 0.866302
\(389\) −14.0865 −0.714212 −0.357106 0.934064i \(-0.616237\pi\)
−0.357106 + 0.934064i \(0.616237\pi\)
\(390\) −42.5604 −2.15513
\(391\) −17.5125 −0.885645
\(392\) −6.22668 −0.314495
\(393\) 6.12836 0.309135
\(394\) 14.4561 0.728285
\(395\) 40.9077 2.05829
\(396\) 1.78106 0.0895016
\(397\) −16.3482 −0.820494 −0.410247 0.911974i \(-0.634558\pi\)
−0.410247 + 0.911974i \(0.634558\pi\)
\(398\) 5.71688 0.286561
\(399\) −7.66550 −0.383755
\(400\) 7.47565 0.373783
\(401\) −32.2763 −1.61180 −0.805901 0.592050i \(-0.798319\pi\)
−0.805901 + 0.592050i \(0.798319\pi\)
\(402\) −16.0351 −0.799757
\(403\) 7.18479 0.357900
\(404\) 1.08647 0.0540537
\(405\) −36.4270 −1.81007
\(406\) −2.87433 −0.142651
\(407\) 0 0
\(408\) −5.63816 −0.279130
\(409\) 14.9932 0.741366 0.370683 0.928760i \(-0.379124\pi\)
0.370683 + 0.928760i \(0.379124\pi\)
\(410\) 30.3114 1.49697
\(411\) 3.00000 0.147979
\(412\) −11.2148 −0.552515
\(413\) 6.69965 0.329668
\(414\) −3.10607 −0.152655
\(415\) 14.7297 0.723051
\(416\) 6.41147 0.314348
\(417\) −22.1830 −1.08631
\(418\) −15.5253 −0.759366
\(419\) −34.7401 −1.69717 −0.848583 0.529063i \(-0.822544\pi\)
−0.848583 + 0.529063i \(0.822544\pi\)
\(420\) 5.83750 0.284841
\(421\) −15.8452 −0.772250 −0.386125 0.922447i \(-0.626187\pi\)
−0.386125 + 0.922447i \(0.626187\pi\)
\(422\) −7.66550 −0.373151
\(423\) −2.72874 −0.132676
\(424\) 0.268571 0.0130429
\(425\) 22.4270 1.08787
\(426\) −24.7101 −1.19721
\(427\) 1.87164 0.0905752
\(428\) 9.59627 0.463853
\(429\) −40.3337 −1.94733
\(430\) −19.8357 −0.956564
\(431\) 22.4688 1.08229 0.541143 0.840931i \(-0.317992\pi\)
0.541143 + 0.840931i \(0.317992\pi\)
\(432\) 4.63816 0.223153
\(433\) −4.86247 −0.233676 −0.116838 0.993151i \(-0.537276\pi\)
−0.116838 + 0.993151i \(0.537276\pi\)
\(434\) −0.985452 −0.0473032
\(435\) −21.6973 −1.04031
\(436\) 2.22163 0.106397
\(437\) 27.0752 1.29518
\(438\) 16.1138 0.769948
\(439\) 32.3655 1.54472 0.772360 0.635185i \(-0.219076\pi\)
0.772360 + 0.635185i \(0.219076\pi\)
\(440\) 11.8229 0.563637
\(441\) −3.31315 −0.157769
\(442\) 19.2344 0.914888
\(443\) −19.8844 −0.944738 −0.472369 0.881401i \(-0.656601\pi\)
−0.472369 + 0.881401i \(0.656601\pi\)
\(444\) 0 0
\(445\) −29.5800 −1.40222
\(446\) −1.99226 −0.0943362
\(447\) 23.7939 1.12541
\(448\) −0.879385 −0.0415470
\(449\) −15.2713 −0.720695 −0.360348 0.932818i \(-0.617342\pi\)
−0.360348 + 0.932818i \(0.617342\pi\)
\(450\) 3.97771 0.187511
\(451\) 28.7256 1.35263
\(452\) 4.63041 0.217796
\(453\) 8.53209 0.400873
\(454\) 19.4834 0.914401
\(455\) −19.9145 −0.933605
\(456\) 8.71688 0.408205
\(457\) 13.4602 0.629640 0.314820 0.949151i \(-0.398056\pi\)
0.314820 + 0.949151i \(0.398056\pi\)
\(458\) −12.8152 −0.598815
\(459\) 13.9145 0.649472
\(460\) −20.6186 −0.961345
\(461\) 13.2422 0.616749 0.308375 0.951265i \(-0.400215\pi\)
0.308375 + 0.951265i \(0.400215\pi\)
\(462\) 5.53209 0.257376
\(463\) −10.7065 −0.497571 −0.248786 0.968559i \(-0.580031\pi\)
−0.248786 + 0.968559i \(0.580031\pi\)
\(464\) 3.26857 0.151740
\(465\) −7.43882 −0.344967
\(466\) 2.91622 0.135091
\(467\) 15.7229 0.727568 0.363784 0.931483i \(-0.381485\pi\)
0.363784 + 0.931483i \(0.381485\pi\)
\(468\) 3.41147 0.157695
\(469\) −7.50299 −0.346456
\(470\) −18.1138 −0.835528
\(471\) 33.6023 1.54831
\(472\) −7.61856 −0.350672
\(473\) −18.7980 −0.864332
\(474\) −21.7665 −0.999769
\(475\) −34.6732 −1.59092
\(476\) −2.63816 −0.120920
\(477\) 0.142903 0.00654310
\(478\) −16.0128 −0.732408
\(479\) −29.8093 −1.36202 −0.681012 0.732273i \(-0.738460\pi\)
−0.681012 + 0.732273i \(0.738460\pi\)
\(480\) −6.63816 −0.302989
\(481\) 0 0
\(482\) −9.78880 −0.445868
\(483\) −9.64765 −0.438983
\(484\) 0.204393 0.00929059
\(485\) 60.2722 2.73682
\(486\) 5.46791 0.248029
\(487\) 11.2550 0.510011 0.255005 0.966940i \(-0.417923\pi\)
0.255005 + 0.966940i \(0.417923\pi\)
\(488\) −2.12836 −0.0963461
\(489\) −4.26352 −0.192803
\(490\) −21.9932 −0.993551
\(491\) 1.03952 0.0469131 0.0234566 0.999725i \(-0.492533\pi\)
0.0234566 + 0.999725i \(0.492533\pi\)
\(492\) −16.1284 −0.727122
\(493\) 9.80571 0.441627
\(494\) −29.7374 −1.33795
\(495\) 6.29086 0.282753
\(496\) 1.12061 0.0503171
\(497\) −11.5621 −0.518632
\(498\) −7.83750 −0.351207
\(499\) −35.6810 −1.59730 −0.798650 0.601796i \(-0.794452\pi\)
−0.798650 + 0.601796i \(0.794452\pi\)
\(500\) 8.74422 0.391054
\(501\) −40.3063 −1.80075
\(502\) −16.7442 −0.747331
\(503\) −25.4260 −1.13369 −0.566845 0.823824i \(-0.691836\pi\)
−0.566845 + 0.823824i \(0.691836\pi\)
\(504\) −0.467911 −0.0208424
\(505\) 3.83750 0.170766
\(506\) −19.5398 −0.868651
\(507\) −52.8239 −2.34599
\(508\) −5.08647 −0.225675
\(509\) 0.00680713 0.000301721 0 0.000150860 1.00000i \(-0.499952\pi\)
0.000150860 1.00000i \(0.499952\pi\)
\(510\) −19.9145 −0.881827
\(511\) 7.53983 0.333542
\(512\) 1.00000 0.0441942
\(513\) −21.5125 −0.949800
\(514\) 21.7716 0.960303
\(515\) −39.6117 −1.74550
\(516\) 10.5544 0.464631
\(517\) −17.1661 −0.754965
\(518\) 0 0
\(519\) 21.1780 0.929610
\(520\) 22.6459 0.993088
\(521\) 6.97864 0.305740 0.152870 0.988246i \(-0.451148\pi\)
0.152870 + 0.988246i \(0.451148\pi\)
\(522\) 1.73917 0.0761214
\(523\) −39.5408 −1.72900 −0.864498 0.502636i \(-0.832364\pi\)
−0.864498 + 0.502636i \(0.832364\pi\)
\(524\) −3.26083 −0.142450
\(525\) 12.3550 0.539218
\(526\) 21.1857 0.923742
\(527\) 3.36184 0.146444
\(528\) −6.29086 −0.273775
\(529\) 11.0764 0.481581
\(530\) 0.948615 0.0412052
\(531\) −4.05375 −0.175918
\(532\) 4.07873 0.176835
\(533\) 55.0215 2.38324
\(534\) 15.7392 0.681101
\(535\) 33.8949 1.46540
\(536\) 8.53209 0.368530
\(537\) −12.0865 −0.521570
\(538\) 20.2317 0.872252
\(539\) −20.8425 −0.897752
\(540\) 16.3824 0.704985
\(541\) 11.8111 0.507798 0.253899 0.967231i \(-0.418287\pi\)
0.253899 + 0.967231i \(0.418287\pi\)
\(542\) −13.3405 −0.573023
\(543\) −6.58347 −0.282524
\(544\) 3.00000 0.128624
\(545\) 7.84699 0.336128
\(546\) 10.5963 0.453478
\(547\) −42.1165 −1.80077 −0.900386 0.435093i \(-0.856716\pi\)
−0.900386 + 0.435093i \(0.856716\pi\)
\(548\) −1.59627 −0.0681891
\(549\) −1.13247 −0.0483328
\(550\) 25.0232 1.06699
\(551\) −15.1601 −0.645844
\(552\) 10.9709 0.466953
\(553\) −10.1848 −0.433101
\(554\) −32.2627 −1.37071
\(555\) 0 0
\(556\) 11.8033 0.500573
\(557\) −12.3301 −0.522441 −0.261221 0.965279i \(-0.584125\pi\)
−0.261221 + 0.965279i \(0.584125\pi\)
\(558\) 0.596267 0.0252420
\(559\) −36.0060 −1.52289
\(560\) −3.10607 −0.131255
\(561\) −18.8726 −0.796801
\(562\) −4.31727 −0.182113
\(563\) −8.37733 −0.353062 −0.176531 0.984295i \(-0.556488\pi\)
−0.176531 + 0.984295i \(0.556488\pi\)
\(564\) 9.63816 0.405840
\(565\) 16.3550 0.688062
\(566\) −1.34224 −0.0564187
\(567\) 9.06923 0.380872
\(568\) 13.1480 0.551676
\(569\) 8.29086 0.347571 0.173785 0.984784i \(-0.444400\pi\)
0.173785 + 0.984784i \(0.444400\pi\)
\(570\) 30.7888 1.28960
\(571\) 0.0196004 0.000820250 0 0.000410125 1.00000i \(-0.499869\pi\)
0.000410125 1.00000i \(0.499869\pi\)
\(572\) 21.4611 0.897334
\(573\) −26.1685 −1.09320
\(574\) −7.54664 −0.314991
\(575\) −43.6391 −1.81988
\(576\) 0.532089 0.0221704
\(577\) −5.80840 −0.241807 −0.120903 0.992664i \(-0.538579\pi\)
−0.120903 + 0.992664i \(0.538579\pi\)
\(578\) −8.00000 −0.332756
\(579\) −21.7570 −0.904191
\(580\) 11.5449 0.479375
\(581\) −3.66725 −0.152143
\(582\) −32.0702 −1.32935
\(583\) 0.898986 0.0372322
\(584\) −8.57398 −0.354794
\(585\) 12.0496 0.498191
\(586\) 2.75877 0.113964
\(587\) 1.60307 0.0661659 0.0330830 0.999453i \(-0.489467\pi\)
0.0330830 + 0.999453i \(0.489467\pi\)
\(588\) 11.7023 0.482596
\(589\) −5.19759 −0.214163
\(590\) −26.9094 −1.10784
\(591\) −27.1685 −1.11756
\(592\) 0 0
\(593\) 26.7543 1.09867 0.549334 0.835603i \(-0.314881\pi\)
0.549334 + 0.835603i \(0.314881\pi\)
\(594\) 15.5253 0.637010
\(595\) −9.31820 −0.382009
\(596\) −12.6604 −0.518592
\(597\) −10.7442 −0.439732
\(598\) −37.4270 −1.53050
\(599\) −14.3746 −0.587332 −0.293666 0.955908i \(-0.594875\pi\)
−0.293666 + 0.955908i \(0.594875\pi\)
\(600\) −14.0496 −0.573574
\(601\) 1.92303 0.0784420 0.0392210 0.999231i \(-0.487512\pi\)
0.0392210 + 0.999231i \(0.487512\pi\)
\(602\) 4.93851 0.201279
\(603\) 4.53983 0.184876
\(604\) −4.53983 −0.184723
\(605\) 0.721934 0.0293508
\(606\) −2.04189 −0.0829461
\(607\) −27.0678 −1.09865 −0.549324 0.835609i \(-0.685115\pi\)
−0.549324 + 0.835609i \(0.685115\pi\)
\(608\) −4.63816 −0.188102
\(609\) 5.40198 0.218899
\(610\) −7.51754 −0.304376
\(611\) −32.8803 −1.33020
\(612\) 1.59627 0.0645253
\(613\) 30.1516 1.21781 0.608905 0.793243i \(-0.291609\pi\)
0.608905 + 0.793243i \(0.291609\pi\)
\(614\) −6.58853 −0.265891
\(615\) −56.9668 −2.29712
\(616\) −2.94356 −0.118600
\(617\) −18.7050 −0.753036 −0.376518 0.926409i \(-0.622879\pi\)
−0.376518 + 0.926409i \(0.622879\pi\)
\(618\) 21.0770 0.847840
\(619\) −23.3783 −0.939652 −0.469826 0.882759i \(-0.655683\pi\)
−0.469826 + 0.882759i \(0.655683\pi\)
\(620\) 3.95811 0.158962
\(621\) −27.0752 −1.08649
\(622\) 19.0351 0.763237
\(623\) 7.36453 0.295054
\(624\) −12.0496 −0.482371
\(625\) −6.49289 −0.259716
\(626\) 12.8152 0.512199
\(627\) 29.1780 1.16526
\(628\) −17.8794 −0.713465
\(629\) 0 0
\(630\) −1.65270 −0.0658453
\(631\) −21.5493 −0.857865 −0.428933 0.903337i \(-0.641110\pi\)
−0.428933 + 0.903337i \(0.641110\pi\)
\(632\) 11.5817 0.460696
\(633\) 14.4064 0.572604
\(634\) −19.5449 −0.776226
\(635\) −17.9659 −0.712953
\(636\) −0.504748 −0.0200145
\(637\) −39.9222 −1.58178
\(638\) 10.9409 0.433153
\(639\) 6.99588 0.276753
\(640\) 3.53209 0.139618
\(641\) −5.67324 −0.224079 −0.112040 0.993704i \(-0.535738\pi\)
−0.112040 + 0.993704i \(0.535738\pi\)
\(642\) −18.0351 −0.711788
\(643\) 25.8958 1.02123 0.510615 0.859809i \(-0.329418\pi\)
0.510615 + 0.859809i \(0.329418\pi\)
\(644\) 5.13341 0.202285
\(645\) 37.2790 1.46786
\(646\) −13.9145 −0.547457
\(647\) −10.2558 −0.403196 −0.201598 0.979468i \(-0.564613\pi\)
−0.201598 + 0.979468i \(0.564613\pi\)
\(648\) −10.3131 −0.405139
\(649\) −25.5016 −1.00102
\(650\) 47.9299 1.87997
\(651\) 1.85204 0.0725873
\(652\) 2.26857 0.0888441
\(653\) 33.1557 1.29748 0.648741 0.761009i \(-0.275296\pi\)
0.648741 + 0.761009i \(0.275296\pi\)
\(654\) −4.17530 −0.163267
\(655\) −11.5175 −0.450028
\(656\) 8.58172 0.335060
\(657\) −4.56212 −0.177985
\(658\) 4.50980 0.175810
\(659\) 12.2335 0.476549 0.238275 0.971198i \(-0.423418\pi\)
0.238275 + 0.971198i \(0.423418\pi\)
\(660\) −22.2199 −0.864907
\(661\) −16.5098 −0.642157 −0.321078 0.947053i \(-0.604045\pi\)
−0.321078 + 0.947053i \(0.604045\pi\)
\(662\) −13.6973 −0.532360
\(663\) −36.1489 −1.40391
\(664\) 4.17024 0.161837
\(665\) 14.4064 0.558657
\(666\) 0 0
\(667\) −19.0803 −0.738791
\(668\) 21.4466 0.829792
\(669\) 3.74422 0.144760
\(670\) 30.1361 1.16426
\(671\) −7.12424 −0.275028
\(672\) 1.65270 0.0637544
\(673\) −47.4475 −1.82897 −0.914483 0.404624i \(-0.867402\pi\)
−0.914483 + 0.404624i \(0.867402\pi\)
\(674\) −5.11381 −0.196977
\(675\) 34.6732 1.33457
\(676\) 28.1070 1.08104
\(677\) −8.83244 −0.339458 −0.169729 0.985491i \(-0.554289\pi\)
−0.169729 + 0.985491i \(0.554289\pi\)
\(678\) −8.70233 −0.334211
\(679\) −15.0060 −0.575877
\(680\) 10.5963 0.406348
\(681\) −36.6168 −1.40316
\(682\) 3.75103 0.143634
\(683\) −41.0077 −1.56912 −0.784559 0.620054i \(-0.787111\pi\)
−0.784559 + 0.620054i \(0.787111\pi\)
\(684\) −2.46791 −0.0943629
\(685\) −5.63816 −0.215423
\(686\) 11.6313 0.444087
\(687\) 24.0847 0.918889
\(688\) −5.61587 −0.214103
\(689\) 1.72193 0.0656005
\(690\) 38.7502 1.47520
\(691\) 46.3779 1.76430 0.882150 0.470969i \(-0.156096\pi\)
0.882150 + 0.470969i \(0.156096\pi\)
\(692\) −11.2686 −0.428367
\(693\) −1.56624 −0.0594964
\(694\) −14.9632 −0.567994
\(695\) 41.6905 1.58141
\(696\) −6.14290 −0.232846
\(697\) 25.7452 0.975167
\(698\) −13.3628 −0.505789
\(699\) −5.48070 −0.207299
\(700\) −6.57398 −0.248473
\(701\) −6.57129 −0.248194 −0.124097 0.992270i \(-0.539603\pi\)
−0.124097 + 0.992270i \(0.539603\pi\)
\(702\) 29.7374 1.12237
\(703\) 0 0
\(704\) 3.34730 0.126156
\(705\) 34.0428 1.28213
\(706\) −8.15301 −0.306843
\(707\) −0.955423 −0.0359324
\(708\) 14.3182 0.538111
\(709\) 13.3105 0.499885 0.249942 0.968261i \(-0.419588\pi\)
0.249942 + 0.968261i \(0.419588\pi\)
\(710\) 46.4397 1.74285
\(711\) 6.16250 0.231112
\(712\) −8.37464 −0.313853
\(713\) −6.54158 −0.244984
\(714\) 4.95811 0.185553
\(715\) 75.8025 2.83485
\(716\) 6.43107 0.240341
\(717\) 30.0942 1.12389
\(718\) 1.08647 0.0405466
\(719\) 18.5280 0.690977 0.345488 0.938423i \(-0.387713\pi\)
0.345488 + 0.938423i \(0.387713\pi\)
\(720\) 1.87939 0.0700406
\(721\) 9.86215 0.367286
\(722\) 2.51249 0.0935051
\(723\) 18.3969 0.684189
\(724\) 3.50299 0.130188
\(725\) 24.4347 0.907482
\(726\) −0.384133 −0.0142565
\(727\) 22.4279 0.831804 0.415902 0.909409i \(-0.363466\pi\)
0.415902 + 0.909409i \(0.363466\pi\)
\(728\) −5.63816 −0.208964
\(729\) 20.6631 0.765301
\(730\) −30.2841 −1.12086
\(731\) −16.8476 −0.623131
\(732\) 4.00000 0.147844
\(733\) −25.0300 −0.924505 −0.462253 0.886748i \(-0.652959\pi\)
−0.462253 + 0.886748i \(0.652959\pi\)
\(734\) −31.7124 −1.17053
\(735\) 41.3337 1.52462
\(736\) −5.83750 −0.215173
\(737\) 28.5594 1.05200
\(738\) 4.56624 0.168086
\(739\) 27.2327 1.00177 0.500885 0.865514i \(-0.333008\pi\)
0.500885 + 0.865514i \(0.333008\pi\)
\(740\) 0 0
\(741\) 55.8881 2.05310
\(742\) −0.236177 −0.00867033
\(743\) −2.23349 −0.0819388 −0.0409694 0.999160i \(-0.513045\pi\)
−0.0409694 + 0.999160i \(0.513045\pi\)
\(744\) −2.10607 −0.0772122
\(745\) −44.7178 −1.63833
\(746\) 3.52023 0.128885
\(747\) 2.21894 0.0811868
\(748\) 10.0419 0.367168
\(749\) −8.43882 −0.308348
\(750\) −16.4338 −0.600076
\(751\) −0.167881 −0.00612605 −0.00306302 0.999995i \(-0.500975\pi\)
−0.00306302 + 0.999995i \(0.500975\pi\)
\(752\) −5.12836 −0.187012
\(753\) 31.4688 1.14679
\(754\) 20.9564 0.763185
\(755\) −16.0351 −0.583576
\(756\) −4.07873 −0.148342
\(757\) 34.9195 1.26917 0.634586 0.772852i \(-0.281171\pi\)
0.634586 + 0.772852i \(0.281171\pi\)
\(758\) −12.3277 −0.447762
\(759\) 36.7229 1.33296
\(760\) −16.3824 −0.594251
\(761\) −2.21719 −0.0803729 −0.0401865 0.999192i \(-0.512795\pi\)
−0.0401865 + 0.999192i \(0.512795\pi\)
\(762\) 9.55943 0.346302
\(763\) −1.95367 −0.0707275
\(764\) 13.9240 0.503751
\(765\) 5.63816 0.203848
\(766\) −24.4593 −0.883752
\(767\) −48.8462 −1.76373
\(768\) −1.87939 −0.0678165
\(769\) 52.1421 1.88029 0.940146 0.340772i \(-0.110689\pi\)
0.940146 + 0.340772i \(0.110689\pi\)
\(770\) −10.3969 −0.374679
\(771\) −40.9172 −1.47360
\(772\) 11.5767 0.416653
\(773\) 44.9786 1.61777 0.808885 0.587967i \(-0.200072\pi\)
0.808885 + 0.587967i \(0.200072\pi\)
\(774\) −2.98814 −0.107406
\(775\) 8.37733 0.300922
\(776\) 17.0642 0.612568
\(777\) 0 0
\(778\) −14.0865 −0.505024
\(779\) −39.8033 −1.42610
\(780\) −42.5604 −1.52391
\(781\) 44.0101 1.57481
\(782\) −17.5125 −0.626245
\(783\) 15.1601 0.541779
\(784\) −6.22668 −0.222381
\(785\) −63.1516 −2.25398
\(786\) 6.12836 0.218591
\(787\) 54.9968 1.96042 0.980212 0.197949i \(-0.0634280\pi\)
0.980212 + 0.197949i \(0.0634280\pi\)
\(788\) 14.4561 0.514976
\(789\) −39.8161 −1.41749
\(790\) 40.9077 1.45543
\(791\) −4.07192 −0.144781
\(792\) 1.78106 0.0632872
\(793\) −13.6459 −0.484580
\(794\) −16.3482 −0.580177
\(795\) −1.78281 −0.0632299
\(796\) 5.71688 0.202629
\(797\) 21.1138 0.747889 0.373945 0.927451i \(-0.378005\pi\)
0.373945 + 0.927451i \(0.378005\pi\)
\(798\) −7.66550 −0.271356
\(799\) −15.3851 −0.544285
\(800\) 7.47565 0.264304
\(801\) −4.45605 −0.157447
\(802\) −32.2763 −1.13972
\(803\) −28.6996 −1.01279
\(804\) −16.0351 −0.565514
\(805\) 18.1317 0.639057
\(806\) 7.18479 0.253074
\(807\) −38.0232 −1.33848
\(808\) 1.08647 0.0382218
\(809\) −47.1634 −1.65818 −0.829089 0.559117i \(-0.811140\pi\)
−0.829089 + 0.559117i \(0.811140\pi\)
\(810\) −36.4270 −1.27991
\(811\) −17.7466 −0.623167 −0.311583 0.950219i \(-0.600859\pi\)
−0.311583 + 0.950219i \(0.600859\pi\)
\(812\) −2.87433 −0.100869
\(813\) 25.0719 0.879311
\(814\) 0 0
\(815\) 8.01279 0.280676
\(816\) −5.63816 −0.197375
\(817\) 26.0473 0.911278
\(818\) 14.9932 0.524225
\(819\) −3.00000 −0.104828
\(820\) 30.3114 1.05852
\(821\) −0.352349 −0.0122971 −0.00614853 0.999981i \(-0.501957\pi\)
−0.00614853 + 0.999981i \(0.501957\pi\)
\(822\) 3.00000 0.104637
\(823\) 45.1661 1.57439 0.787196 0.616703i \(-0.211532\pi\)
0.787196 + 0.616703i \(0.211532\pi\)
\(824\) −11.2148 −0.390687
\(825\) −47.0283 −1.63731
\(826\) 6.69965 0.233110
\(827\) 1.03003 0.0358176 0.0179088 0.999840i \(-0.494299\pi\)
0.0179088 + 0.999840i \(0.494299\pi\)
\(828\) −3.10607 −0.107943
\(829\) 45.5553 1.58220 0.791101 0.611686i \(-0.209508\pi\)
0.791101 + 0.611686i \(0.209508\pi\)
\(830\) 14.7297 0.511274
\(831\) 60.6340 2.10337
\(832\) 6.41147 0.222278
\(833\) −18.6800 −0.647225
\(834\) −22.1830 −0.768136
\(835\) 75.7511 2.62148
\(836\) −15.5253 −0.536953
\(837\) 5.19759 0.179655
\(838\) −34.7401 −1.20008
\(839\) −38.8786 −1.34224 −0.671119 0.741350i \(-0.734186\pi\)
−0.671119 + 0.741350i \(0.734186\pi\)
\(840\) 5.83750 0.201413
\(841\) −18.3164 −0.631602
\(842\) −15.8452 −0.546063
\(843\) 8.11381 0.279454
\(844\) −7.66550 −0.263857
\(845\) 99.2764 3.41521
\(846\) −2.72874 −0.0938160
\(847\) −0.179740 −0.00617594
\(848\) 0.268571 0.00922275
\(849\) 2.52259 0.0865752
\(850\) 22.4270 0.769238
\(851\) 0 0
\(852\) −24.7101 −0.846553
\(853\) 38.6783 1.32432 0.662160 0.749363i \(-0.269640\pi\)
0.662160 + 0.749363i \(0.269640\pi\)
\(854\) 1.87164 0.0640464
\(855\) −8.71688 −0.298111
\(856\) 9.59627 0.327994
\(857\) −43.6965 −1.49264 −0.746321 0.665586i \(-0.768182\pi\)
−0.746321 + 0.665586i \(0.768182\pi\)
\(858\) −40.3337 −1.37697
\(859\) 43.3364 1.47862 0.739309 0.673366i \(-0.235152\pi\)
0.739309 + 0.673366i \(0.235152\pi\)
\(860\) −19.8357 −0.676393
\(861\) 14.1830 0.483357
\(862\) 22.4688 0.765292
\(863\) −40.0847 −1.36450 −0.682250 0.731119i \(-0.738998\pi\)
−0.682250 + 0.731119i \(0.738998\pi\)
\(864\) 4.63816 0.157793
\(865\) −39.8016 −1.35329
\(866\) −4.86247 −0.165234
\(867\) 15.0351 0.510618
\(868\) −0.985452 −0.0334484
\(869\) 38.7674 1.31510
\(870\) −21.6973 −0.735607
\(871\) 54.7033 1.85355
\(872\) 2.22163 0.0752339
\(873\) 9.07966 0.307300
\(874\) 27.0752 0.915833
\(875\) −7.68954 −0.259954
\(876\) 16.1138 0.544435
\(877\) −52.2731 −1.76514 −0.882569 0.470183i \(-0.844188\pi\)
−0.882569 + 0.470183i \(0.844188\pi\)
\(878\) 32.3655 1.09228
\(879\) −5.18479 −0.174879
\(880\) 11.8229 0.398552
\(881\) −23.9195 −0.805869 −0.402935 0.915229i \(-0.632010\pi\)
−0.402935 + 0.915229i \(0.632010\pi\)
\(882\) −3.31315 −0.111559
\(883\) −13.4311 −0.451992 −0.225996 0.974128i \(-0.572564\pi\)
−0.225996 + 0.974128i \(0.572564\pi\)
\(884\) 19.2344 0.646924
\(885\) 50.5732 1.70000
\(886\) −19.8844 −0.668031
\(887\) 49.3242 1.65614 0.828072 0.560622i \(-0.189438\pi\)
0.828072 + 0.560622i \(0.189438\pi\)
\(888\) 0 0
\(889\) 4.47296 0.150018
\(890\) −29.5800 −0.991523
\(891\) −34.5212 −1.15650
\(892\) −1.99226 −0.0667058
\(893\) 23.7861 0.795972
\(894\) 23.7939 0.795785
\(895\) 22.7151 0.759283
\(896\) −0.879385 −0.0293782
\(897\) 70.3397 2.34857
\(898\) −15.2713 −0.509609
\(899\) 3.66281 0.122162
\(900\) 3.97771 0.132590
\(901\) 0.805712 0.0268422
\(902\) 28.7256 0.956456
\(903\) −9.28136 −0.308864
\(904\) 4.63041 0.154005
\(905\) 12.3729 0.411289
\(906\) 8.53209 0.283460
\(907\) 43.9522 1.45941 0.729705 0.683762i \(-0.239657\pi\)
0.729705 + 0.683762i \(0.239657\pi\)
\(908\) 19.4834 0.646579
\(909\) 0.578097 0.0191743
\(910\) −19.9145 −0.660158
\(911\) −16.0597 −0.532083 −0.266041 0.963962i \(-0.585716\pi\)
−0.266041 + 0.963962i \(0.585716\pi\)
\(912\) 8.71688 0.288645
\(913\) 13.9590 0.461977
\(914\) 13.4602 0.445223
\(915\) 14.1284 0.467069
\(916\) −12.8152 −0.423426
\(917\) 2.86753 0.0946940
\(918\) 13.9145 0.459246
\(919\) −34.6468 −1.14289 −0.571447 0.820639i \(-0.693618\pi\)
−0.571447 + 0.820639i \(0.693618\pi\)
\(920\) −20.6186 −0.679774
\(921\) 12.3824 0.408013
\(922\) 13.2422 0.436107
\(923\) 84.2978 2.77470
\(924\) 5.53209 0.181992
\(925\) 0 0
\(926\) −10.7065 −0.351836
\(927\) −5.96728 −0.195991
\(928\) 3.26857 0.107296
\(929\) −48.5577 −1.59313 −0.796563 0.604556i \(-0.793351\pi\)
−0.796563 + 0.604556i \(0.793351\pi\)
\(930\) −7.43882 −0.243928
\(931\) 28.8803 0.946514
\(932\) 2.91622 0.0955240
\(933\) −35.7743 −1.17120
\(934\) 15.7229 0.514468
\(935\) 35.4688 1.15996
\(936\) 3.41147 0.111508
\(937\) 48.9617 1.59951 0.799755 0.600326i \(-0.204963\pi\)
0.799755 + 0.600326i \(0.204963\pi\)
\(938\) −7.50299 −0.244981
\(939\) −24.0847 −0.785975
\(940\) −18.1138 −0.590807
\(941\) −17.5012 −0.570524 −0.285262 0.958450i \(-0.592081\pi\)
−0.285262 + 0.958450i \(0.592081\pi\)
\(942\) 33.6023 1.09482
\(943\) −50.0958 −1.63134
\(944\) −7.61856 −0.247963
\(945\) −14.4064 −0.468641
\(946\) −18.7980 −0.611175
\(947\) 17.3223 0.562900 0.281450 0.959576i \(-0.409185\pi\)
0.281450 + 0.959576i \(0.409185\pi\)
\(948\) −21.7665 −0.706943
\(949\) −54.9718 −1.78446
\(950\) −34.6732 −1.12495
\(951\) 36.7324 1.19113
\(952\) −2.63816 −0.0855031
\(953\) −38.8522 −1.25854 −0.629272 0.777185i \(-0.716647\pi\)
−0.629272 + 0.777185i \(0.716647\pi\)
\(954\) 0.142903 0.00462667
\(955\) 49.1807 1.59145
\(956\) −16.0128 −0.517891
\(957\) −20.5621 −0.664679
\(958\) −29.8093 −0.963096
\(959\) 1.40373 0.0453289
\(960\) −6.63816 −0.214246
\(961\) −29.7442 −0.959491
\(962\) 0 0
\(963\) 5.10607 0.164541
\(964\) −9.78880 −0.315276
\(965\) 40.8898 1.31629
\(966\) −9.64765 −0.310408
\(967\) 44.5827 1.43368 0.716841 0.697237i \(-0.245587\pi\)
0.716841 + 0.697237i \(0.245587\pi\)
\(968\) 0.204393 0.00656944
\(969\) 26.1506 0.840080
\(970\) 60.2722 1.93522
\(971\) −57.1525 −1.83411 −0.917056 0.398759i \(-0.869441\pi\)
−0.917056 + 0.398759i \(0.869441\pi\)
\(972\) 5.46791 0.175383
\(973\) −10.3797 −0.332758
\(974\) 11.2550 0.360632
\(975\) −90.0788 −2.88483
\(976\) −2.12836 −0.0681270
\(977\) 34.8489 1.11491 0.557457 0.830206i \(-0.311777\pi\)
0.557457 + 0.830206i \(0.311777\pi\)
\(978\) −4.26352 −0.136332
\(979\) −28.0324 −0.895919
\(980\) −21.9932 −0.702547
\(981\) 1.18210 0.0377417
\(982\) 1.03952 0.0331726
\(983\) −1.06955 −0.0341135 −0.0170567 0.999855i \(-0.505430\pi\)
−0.0170567 + 0.999855i \(0.505430\pi\)
\(984\) −16.1284 −0.514153
\(985\) 51.0601 1.62691
\(986\) 9.80571 0.312277
\(987\) −8.47565 −0.269783
\(988\) −29.7374 −0.946073
\(989\) 32.7826 1.04243
\(990\) 6.29086 0.199937
\(991\) 20.3250 0.645645 0.322823 0.946460i \(-0.395368\pi\)
0.322823 + 0.946460i \(0.395368\pi\)
\(992\) 1.12061 0.0355796
\(993\) 25.7425 0.816913
\(994\) −11.5621 −0.366728
\(995\) 20.1925 0.640146
\(996\) −7.83750 −0.248341
\(997\) 46.2404 1.46445 0.732224 0.681064i \(-0.238482\pi\)
0.732224 + 0.681064i \(0.238482\pi\)
\(998\) −35.6810 −1.12946
\(999\) 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 2738.2.a.p.1.1 3
37.21 even 18 74.2.f.a.71.1 yes 6
37.30 even 18 74.2.f.a.49.1 6
37.36 even 2 2738.2.a.m.1.1 3
111.95 odd 18 666.2.x.c.145.1 6
111.104 odd 18 666.2.x.c.271.1 6
148.67 odd 18 592.2.bc.b.49.1 6
148.95 odd 18 592.2.bc.b.145.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.a.49.1 6 37.30 even 18
74.2.f.a.71.1 yes 6 37.21 even 18
592.2.bc.b.49.1 6 148.67 odd 18
592.2.bc.b.145.1 6 148.95 odd 18
666.2.x.c.145.1 6 111.95 odd 18
666.2.x.c.271.1 6 111.104 odd 18
2738.2.a.m.1.1 3 37.36 even 2
2738.2.a.p.1.1 3 1.1 even 1 trivial