Properties

Label 4598.2.a.by.1.8
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
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] = [4598,2,Mod(1,4598)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4598, 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("4598.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.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: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-2.75320\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.37860 q^{3} +1.00000 q^{4} -4.17017 q^{5} -3.37860 q^{6} +3.60700 q^{7} -1.00000 q^{8} +8.41492 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.37860 q^{3} +1.00000 q^{4} -4.17017 q^{5} -3.37860 q^{6} +3.60700 q^{7} -1.00000 q^{8} +8.41492 q^{9} +4.17017 q^{10} +3.37860 q^{12} +0.669426 q^{13} -3.60700 q^{14} -14.0893 q^{15} +1.00000 q^{16} -4.10493 q^{17} -8.41492 q^{18} +1.00000 q^{19} -4.17017 q^{20} +12.1866 q^{21} +6.35872 q^{23} -3.37860 q^{24} +12.3903 q^{25} -0.669426 q^{26} +18.2948 q^{27} +3.60700 q^{28} -3.27213 q^{29} +14.0893 q^{30} -1.55576 q^{31} -1.00000 q^{32} +4.10493 q^{34} -15.0418 q^{35} +8.41492 q^{36} +3.51430 q^{37} -1.00000 q^{38} +2.26172 q^{39} +4.17017 q^{40} -7.30933 q^{41} -12.1866 q^{42} +6.04669 q^{43} -35.0916 q^{45} -6.35872 q^{46} +0.862845 q^{47} +3.37860 q^{48} +6.01048 q^{49} -12.3903 q^{50} -13.8689 q^{51} +0.669426 q^{52} +6.79423 q^{53} -18.2948 q^{54} -3.60700 q^{56} +3.37860 q^{57} +3.27213 q^{58} -9.41317 q^{59} -14.0893 q^{60} +4.40660 q^{61} +1.55576 q^{62} +30.3526 q^{63} +1.00000 q^{64} -2.79162 q^{65} +1.80340 q^{67} -4.10493 q^{68} +21.4836 q^{69} +15.0418 q^{70} +11.9933 q^{71} -8.41492 q^{72} -9.91252 q^{73} -3.51430 q^{74} +41.8618 q^{75} +1.00000 q^{76} -2.26172 q^{78} -1.04552 q^{79} -4.17017 q^{80} +36.5661 q^{81} +7.30933 q^{82} -0.503831 q^{83} +12.1866 q^{84} +17.1183 q^{85} -6.04669 q^{86} -11.0552 q^{87} +8.23674 q^{89} +35.0916 q^{90} +2.41462 q^{91} +6.35872 q^{92} -5.25629 q^{93} -0.862845 q^{94} -4.17017 q^{95} -3.37860 q^{96} +0.809369 q^{97} -6.01048 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} - 4 q^{28} + 2 q^{29} - 4 q^{30} - 8 q^{32} - 4 q^{34} - 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} - 16 q^{39} - 8 q^{41} - 20 q^{42} - 8 q^{43} + 16 q^{45} - 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} - 36 q^{50} - 18 q^{51} + 12 q^{52} + 36 q^{53} - 32 q^{54} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} - 12 q^{61} - 24 q^{63} + 8 q^{64} - 16 q^{65} + 16 q^{67} + 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} - 22 q^{72} + 20 q^{73} - 24 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{78} + 12 q^{79} + 40 q^{81} + 8 q^{82} - 20 q^{83} + 20 q^{84} - 12 q^{85} + 8 q^{86} + 36 q^{87} + 8 q^{89} - 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} + 16 q^{94} - 8 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.37860 1.95063 0.975317 0.220810i \(-0.0708701\pi\)
0.975317 + 0.220810i \(0.0708701\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.17017 −1.86496 −0.932478 0.361228i \(-0.882358\pi\)
−0.932478 + 0.361228i \(0.882358\pi\)
\(6\) −3.37860 −1.37931
\(7\) 3.60700 1.36332 0.681660 0.731669i \(-0.261258\pi\)
0.681660 + 0.731669i \(0.261258\pi\)
\(8\) −1.00000 −0.353553
\(9\) 8.41492 2.80497
\(10\) 4.17017 1.31872
\(11\) 0 0
\(12\) 3.37860 0.975317
\(13\) 0.669426 0.185665 0.0928326 0.995682i \(-0.470408\pi\)
0.0928326 + 0.995682i \(0.470408\pi\)
\(14\) −3.60700 −0.964012
\(15\) −14.0893 −3.63784
\(16\) 1.00000 0.250000
\(17\) −4.10493 −0.995592 −0.497796 0.867294i \(-0.665857\pi\)
−0.497796 + 0.867294i \(0.665857\pi\)
\(18\) −8.41492 −1.98341
\(19\) 1.00000 0.229416
\(20\) −4.17017 −0.932478
\(21\) 12.1866 2.65934
\(22\) 0 0
\(23\) 6.35872 1.32589 0.662943 0.748670i \(-0.269307\pi\)
0.662943 + 0.748670i \(0.269307\pi\)
\(24\) −3.37860 −0.689653
\(25\) 12.3903 2.47806
\(26\) −0.669426 −0.131285
\(27\) 18.2948 3.52084
\(28\) 3.60700 0.681660
\(29\) −3.27213 −0.607619 −0.303809 0.952733i \(-0.598259\pi\)
−0.303809 + 0.952733i \(0.598259\pi\)
\(30\) 14.0893 2.57234
\(31\) −1.55576 −0.279423 −0.139712 0.990192i \(-0.544618\pi\)
−0.139712 + 0.990192i \(0.544618\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.10493 0.703990
\(35\) −15.0418 −2.54253
\(36\) 8.41492 1.40249
\(37\) 3.51430 0.577747 0.288873 0.957367i \(-0.406719\pi\)
0.288873 + 0.957367i \(0.406719\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.26172 0.362165
\(40\) 4.17017 0.659361
\(41\) −7.30933 −1.14153 −0.570763 0.821115i \(-0.693352\pi\)
−0.570763 + 0.821115i \(0.693352\pi\)
\(42\) −12.1866 −1.88044
\(43\) 6.04669 0.922112 0.461056 0.887371i \(-0.347471\pi\)
0.461056 + 0.887371i \(0.347471\pi\)
\(44\) 0 0
\(45\) −35.0916 −5.23115
\(46\) −6.35872 −0.937543
\(47\) 0.862845 0.125859 0.0629294 0.998018i \(-0.479956\pi\)
0.0629294 + 0.998018i \(0.479956\pi\)
\(48\) 3.37860 0.487658
\(49\) 6.01048 0.858640
\(50\) −12.3903 −1.75225
\(51\) −13.8689 −1.94204
\(52\) 0.669426 0.0928326
\(53\) 6.79423 0.933260 0.466630 0.884453i \(-0.345468\pi\)
0.466630 + 0.884453i \(0.345468\pi\)
\(54\) −18.2948 −2.48961
\(55\) 0 0
\(56\) −3.60700 −0.482006
\(57\) 3.37860 0.447506
\(58\) 3.27213 0.429651
\(59\) −9.41317 −1.22549 −0.612745 0.790280i \(-0.709935\pi\)
−0.612745 + 0.790280i \(0.709935\pi\)
\(60\) −14.0893 −1.81892
\(61\) 4.40660 0.564207 0.282103 0.959384i \(-0.408968\pi\)
0.282103 + 0.959384i \(0.408968\pi\)
\(62\) 1.55576 0.197582
\(63\) 30.3526 3.82407
\(64\) 1.00000 0.125000
\(65\) −2.79162 −0.346257
\(66\) 0 0
\(67\) 1.80340 0.220320 0.110160 0.993914i \(-0.464864\pi\)
0.110160 + 0.993914i \(0.464864\pi\)
\(68\) −4.10493 −0.497796
\(69\) 21.4836 2.58632
\(70\) 15.0418 1.79784
\(71\) 11.9933 1.42334 0.711669 0.702514i \(-0.247939\pi\)
0.711669 + 0.702514i \(0.247939\pi\)
\(72\) −8.41492 −0.991707
\(73\) −9.91252 −1.16017 −0.580087 0.814555i \(-0.696981\pi\)
−0.580087 + 0.814555i \(0.696981\pi\)
\(74\) −3.51430 −0.408529
\(75\) 41.8618 4.83378
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −2.26172 −0.256089
\(79\) −1.04552 −0.117630 −0.0588152 0.998269i \(-0.518732\pi\)
−0.0588152 + 0.998269i \(0.518732\pi\)
\(80\) −4.17017 −0.466239
\(81\) 36.5661 4.06290
\(82\) 7.30933 0.807180
\(83\) −0.503831 −0.0553026 −0.0276513 0.999618i \(-0.508803\pi\)
−0.0276513 + 0.999618i \(0.508803\pi\)
\(84\) 12.1866 1.32967
\(85\) 17.1183 1.85674
\(86\) −6.04669 −0.652032
\(87\) −11.0552 −1.18524
\(88\) 0 0
\(89\) 8.23674 0.873093 0.436546 0.899682i \(-0.356201\pi\)
0.436546 + 0.899682i \(0.356201\pi\)
\(90\) 35.0916 3.69898
\(91\) 2.41462 0.253121
\(92\) 6.35872 0.662943
\(93\) −5.25629 −0.545052
\(94\) −0.862845 −0.0889957
\(95\) −4.17017 −0.427850
\(96\) −3.37860 −0.344827
\(97\) 0.809369 0.0821790 0.0410895 0.999155i \(-0.486917\pi\)
0.0410895 + 0.999155i \(0.486917\pi\)
\(98\) −6.01048 −0.607150
\(99\) 0 0
\(100\) 12.3903 1.23903
\(101\) 12.0493 1.19895 0.599475 0.800393i \(-0.295376\pi\)
0.599475 + 0.800393i \(0.295376\pi\)
\(102\) 13.8689 1.37323
\(103\) 5.64096 0.555820 0.277910 0.960607i \(-0.410358\pi\)
0.277910 + 0.960607i \(0.410358\pi\)
\(104\) −0.669426 −0.0656426
\(105\) −50.8202 −4.95954
\(106\) −6.79423 −0.659914
\(107\) −15.1059 −1.46034 −0.730172 0.683264i \(-0.760560\pi\)
−0.730172 + 0.683264i \(0.760560\pi\)
\(108\) 18.2948 1.76042
\(109\) 11.6030 1.11137 0.555685 0.831393i \(-0.312456\pi\)
0.555685 + 0.831393i \(0.312456\pi\)
\(110\) 0 0
\(111\) 11.8734 1.12697
\(112\) 3.60700 0.340830
\(113\) −6.03660 −0.567876 −0.283938 0.958843i \(-0.591641\pi\)
−0.283938 + 0.958843i \(0.591641\pi\)
\(114\) −3.37860 −0.316435
\(115\) −26.5169 −2.47272
\(116\) −3.27213 −0.303809
\(117\) 5.63316 0.520786
\(118\) 9.41317 0.866553
\(119\) −14.8065 −1.35731
\(120\) 14.0893 1.28617
\(121\) 0 0
\(122\) −4.40660 −0.398954
\(123\) −24.6953 −2.22670
\(124\) −1.55576 −0.139712
\(125\) −30.8187 −2.75651
\(126\) −30.3526 −2.70403
\(127\) 7.06535 0.626949 0.313474 0.949597i \(-0.398507\pi\)
0.313474 + 0.949597i \(0.398507\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 20.4293 1.79870
\(130\) 2.79162 0.244841
\(131\) 3.26224 0.285023 0.142512 0.989793i \(-0.454482\pi\)
0.142512 + 0.989793i \(0.454482\pi\)
\(132\) 0 0
\(133\) 3.60700 0.312767
\(134\) −1.80340 −0.155790
\(135\) −76.2924 −6.56621
\(136\) 4.10493 0.351995
\(137\) 1.25379 0.107118 0.0535592 0.998565i \(-0.482943\pi\)
0.0535592 + 0.998565i \(0.482943\pi\)
\(138\) −21.4836 −1.82880
\(139\) −1.52427 −0.129287 −0.0646433 0.997908i \(-0.520591\pi\)
−0.0646433 + 0.997908i \(0.520591\pi\)
\(140\) −15.0418 −1.27127
\(141\) 2.91521 0.245505
\(142\) −11.9933 −1.00645
\(143\) 0 0
\(144\) 8.41492 0.701243
\(145\) 13.6453 1.13318
\(146\) 9.91252 0.820366
\(147\) 20.3070 1.67489
\(148\) 3.51430 0.288873
\(149\) 15.6872 1.28515 0.642575 0.766223i \(-0.277866\pi\)
0.642575 + 0.766223i \(0.277866\pi\)
\(150\) −41.8618 −3.41800
\(151\) −18.9865 −1.54510 −0.772548 0.634957i \(-0.781018\pi\)
−0.772548 + 0.634957i \(0.781018\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −34.5427 −2.79261
\(154\) 0 0
\(155\) 6.48779 0.521112
\(156\) 2.26172 0.181082
\(157\) −8.89570 −0.709954 −0.354977 0.934875i \(-0.615511\pi\)
−0.354977 + 0.934875i \(0.615511\pi\)
\(158\) 1.04552 0.0831772
\(159\) 22.9550 1.82045
\(160\) 4.17017 0.329681
\(161\) 22.9359 1.80761
\(162\) −36.5661 −2.87290
\(163\) −4.84499 −0.379489 −0.189744 0.981834i \(-0.560766\pi\)
−0.189744 + 0.981834i \(0.560766\pi\)
\(164\) −7.30933 −0.570763
\(165\) 0 0
\(166\) 0.503831 0.0391049
\(167\) 4.61536 0.357148 0.178574 0.983927i \(-0.442852\pi\)
0.178574 + 0.983927i \(0.442852\pi\)
\(168\) −12.1866 −0.940218
\(169\) −12.5519 −0.965528
\(170\) −17.1183 −1.31291
\(171\) 8.41492 0.643505
\(172\) 6.04669 0.461056
\(173\) 9.33836 0.709982 0.354991 0.934870i \(-0.384484\pi\)
0.354991 + 0.934870i \(0.384484\pi\)
\(174\) 11.0552 0.838092
\(175\) 44.6918 3.37839
\(176\) 0 0
\(177\) −31.8033 −2.39048
\(178\) −8.23674 −0.617370
\(179\) 16.7881 1.25480 0.627401 0.778696i \(-0.284119\pi\)
0.627401 + 0.778696i \(0.284119\pi\)
\(180\) −35.0916 −2.61557
\(181\) 17.5877 1.30729 0.653643 0.756803i \(-0.273240\pi\)
0.653643 + 0.756803i \(0.273240\pi\)
\(182\) −2.41462 −0.178984
\(183\) 14.8881 1.10056
\(184\) −6.35872 −0.468771
\(185\) −14.6552 −1.07747
\(186\) 5.25629 0.385410
\(187\) 0 0
\(188\) 0.862845 0.0629294
\(189\) 65.9895 4.80003
\(190\) 4.17017 0.302536
\(191\) −2.76249 −0.199887 −0.0999435 0.994993i \(-0.531866\pi\)
−0.0999435 + 0.994993i \(0.531866\pi\)
\(192\) 3.37860 0.243829
\(193\) −21.4775 −1.54598 −0.772991 0.634417i \(-0.781240\pi\)
−0.772991 + 0.634417i \(0.781240\pi\)
\(194\) −0.809369 −0.0581093
\(195\) −9.43175 −0.675421
\(196\) 6.01048 0.429320
\(197\) 18.9032 1.34680 0.673399 0.739280i \(-0.264834\pi\)
0.673399 + 0.739280i \(0.264834\pi\)
\(198\) 0 0
\(199\) 7.76283 0.550292 0.275146 0.961402i \(-0.411274\pi\)
0.275146 + 0.961402i \(0.411274\pi\)
\(200\) −12.3903 −0.876126
\(201\) 6.09296 0.429764
\(202\) −12.0493 −0.847786
\(203\) −11.8026 −0.828378
\(204\) −13.8689 −0.971018
\(205\) 30.4811 2.12889
\(206\) −5.64096 −0.393024
\(207\) 53.5081 3.71907
\(208\) 0.669426 0.0464163
\(209\) 0 0
\(210\) 50.8202 3.50693
\(211\) 6.88377 0.473898 0.236949 0.971522i \(-0.423853\pi\)
0.236949 + 0.971522i \(0.423853\pi\)
\(212\) 6.79423 0.466630
\(213\) 40.5204 2.77641
\(214\) 15.1059 1.03262
\(215\) −25.2157 −1.71970
\(216\) −18.2948 −1.24480
\(217\) −5.61164 −0.380943
\(218\) −11.6030 −0.785857
\(219\) −33.4904 −2.26307
\(220\) 0 0
\(221\) −2.74795 −0.184847
\(222\) −11.8734 −0.796890
\(223\) 14.5212 0.972409 0.486204 0.873845i \(-0.338381\pi\)
0.486204 + 0.873845i \(0.338381\pi\)
\(224\) −3.60700 −0.241003
\(225\) 104.263 6.95088
\(226\) 6.03660 0.401549
\(227\) 18.1073 1.20182 0.600912 0.799315i \(-0.294804\pi\)
0.600912 + 0.799315i \(0.294804\pi\)
\(228\) 3.37860 0.223753
\(229\) −25.3580 −1.67570 −0.837850 0.545900i \(-0.816188\pi\)
−0.837850 + 0.545900i \(0.816188\pi\)
\(230\) 26.5169 1.74848
\(231\) 0 0
\(232\) 3.27213 0.214826
\(233\) 21.9886 1.44052 0.720261 0.693704i \(-0.244022\pi\)
0.720261 + 0.693704i \(0.244022\pi\)
\(234\) −5.63316 −0.368251
\(235\) −3.59821 −0.234721
\(236\) −9.41317 −0.612745
\(237\) −3.53240 −0.229454
\(238\) 14.8065 0.959764
\(239\) −15.2971 −0.989488 −0.494744 0.869039i \(-0.664738\pi\)
−0.494744 + 0.869039i \(0.664738\pi\)
\(240\) −14.0893 −0.909461
\(241\) −16.0409 −1.03328 −0.516642 0.856201i \(-0.672818\pi\)
−0.516642 + 0.856201i \(0.672818\pi\)
\(242\) 0 0
\(243\) 68.6575 4.40438
\(244\) 4.40660 0.282103
\(245\) −25.0647 −1.60133
\(246\) 24.6953 1.57451
\(247\) 0.669426 0.0425945
\(248\) 1.55576 0.0987910
\(249\) −1.70224 −0.107875
\(250\) 30.8187 1.94915
\(251\) −6.91197 −0.436279 −0.218140 0.975918i \(-0.569999\pi\)
−0.218140 + 0.975918i \(0.569999\pi\)
\(252\) 30.3526 1.91204
\(253\) 0 0
\(254\) −7.06535 −0.443320
\(255\) 57.8357 3.62181
\(256\) 1.00000 0.0625000
\(257\) −24.7512 −1.54394 −0.771968 0.635661i \(-0.780727\pi\)
−0.771968 + 0.635661i \(0.780727\pi\)
\(258\) −20.4293 −1.27187
\(259\) 12.6761 0.787654
\(260\) −2.79162 −0.173129
\(261\) −27.5347 −1.70435
\(262\) −3.26224 −0.201542
\(263\) −16.8155 −1.03689 −0.518444 0.855112i \(-0.673488\pi\)
−0.518444 + 0.855112i \(0.673488\pi\)
\(264\) 0 0
\(265\) −28.3331 −1.74049
\(266\) −3.60700 −0.221160
\(267\) 27.8286 1.70308
\(268\) 1.80340 0.110160
\(269\) 17.8288 1.08704 0.543521 0.839395i \(-0.317091\pi\)
0.543521 + 0.839395i \(0.317091\pi\)
\(270\) 76.2924 4.64301
\(271\) 30.2063 1.83490 0.917450 0.397850i \(-0.130244\pi\)
0.917450 + 0.397850i \(0.130244\pi\)
\(272\) −4.10493 −0.248898
\(273\) 8.15803 0.493747
\(274\) −1.25379 −0.0757441
\(275\) 0 0
\(276\) 21.4836 1.29316
\(277\) −32.7642 −1.96861 −0.984306 0.176473i \(-0.943531\pi\)
−0.984306 + 0.176473i \(0.943531\pi\)
\(278\) 1.52427 0.0914194
\(279\) −13.0916 −0.783774
\(280\) 15.0418 0.898920
\(281\) 9.70846 0.579158 0.289579 0.957154i \(-0.406485\pi\)
0.289579 + 0.957154i \(0.406485\pi\)
\(282\) −2.91521 −0.173598
\(283\) −14.5739 −0.866326 −0.433163 0.901316i \(-0.642603\pi\)
−0.433163 + 0.901316i \(0.642603\pi\)
\(284\) 11.9933 0.711669
\(285\) −14.0893 −0.834579
\(286\) 0 0
\(287\) −26.3648 −1.55626
\(288\) −8.41492 −0.495854
\(289\) −0.149526 −0.00879562
\(290\) −13.6453 −0.801281
\(291\) 2.73453 0.160301
\(292\) −9.91252 −0.580087
\(293\) 10.5241 0.614824 0.307412 0.951576i \(-0.400537\pi\)
0.307412 + 0.951576i \(0.400537\pi\)
\(294\) −20.3070 −1.18433
\(295\) 39.2545 2.28549
\(296\) −3.51430 −0.204264
\(297\) 0 0
\(298\) −15.6872 −0.908738
\(299\) 4.25669 0.246171
\(300\) 41.8618 2.41689
\(301\) 21.8104 1.25713
\(302\) 18.9865 1.09255
\(303\) 40.7097 2.33871
\(304\) 1.00000 0.0573539
\(305\) −18.3762 −1.05222
\(306\) 34.5427 1.97467
\(307\) 21.4131 1.22211 0.611054 0.791589i \(-0.290746\pi\)
0.611054 + 0.791589i \(0.290746\pi\)
\(308\) 0 0
\(309\) 19.0585 1.08420
\(310\) −6.48779 −0.368482
\(311\) 8.91440 0.505489 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(312\) −2.26172 −0.128045
\(313\) −29.2280 −1.65206 −0.826031 0.563624i \(-0.809407\pi\)
−0.826031 + 0.563624i \(0.809407\pi\)
\(314\) 8.89570 0.502014
\(315\) −126.576 −7.13173
\(316\) −1.04552 −0.0588152
\(317\) −3.86251 −0.216940 −0.108470 0.994100i \(-0.534595\pi\)
−0.108470 + 0.994100i \(0.534595\pi\)
\(318\) −22.9550 −1.28725
\(319\) 0 0
\(320\) −4.17017 −0.233119
\(321\) −51.0368 −2.84860
\(322\) −22.9359 −1.27817
\(323\) −4.10493 −0.228405
\(324\) 36.5661 2.03145
\(325\) 8.29438 0.460089
\(326\) 4.84499 0.268339
\(327\) 39.2020 2.16788
\(328\) 7.30933 0.403590
\(329\) 3.11229 0.171586
\(330\) 0 0
\(331\) 30.7968 1.69275 0.846373 0.532590i \(-0.178781\pi\)
0.846373 + 0.532590i \(0.178781\pi\)
\(332\) −0.503831 −0.0276513
\(333\) 29.5725 1.62056
\(334\) −4.61536 −0.252542
\(335\) −7.52048 −0.410888
\(336\) 12.1866 0.664834
\(337\) 16.0941 0.876704 0.438352 0.898803i \(-0.355562\pi\)
0.438352 + 0.898803i \(0.355562\pi\)
\(338\) 12.5519 0.682732
\(339\) −20.3953 −1.10772
\(340\) 17.1183 0.928368
\(341\) 0 0
\(342\) −8.41492 −0.455027
\(343\) −3.56920 −0.192719
\(344\) −6.04669 −0.326016
\(345\) −89.5900 −4.82337
\(346\) −9.33836 −0.502033
\(347\) 31.7933 1.70676 0.853378 0.521292i \(-0.174550\pi\)
0.853378 + 0.521292i \(0.174550\pi\)
\(348\) −11.0552 −0.592621
\(349\) −11.5331 −0.617350 −0.308675 0.951168i \(-0.599886\pi\)
−0.308675 + 0.951168i \(0.599886\pi\)
\(350\) −44.6918 −2.38888
\(351\) 12.2470 0.653697
\(352\) 0 0
\(353\) −3.56825 −0.189919 −0.0949593 0.995481i \(-0.530272\pi\)
−0.0949593 + 0.995481i \(0.530272\pi\)
\(354\) 31.8033 1.69033
\(355\) −50.0139 −2.65446
\(356\) 8.23674 0.436546
\(357\) −50.0252 −2.64762
\(358\) −16.7881 −0.887279
\(359\) −21.8282 −1.15205 −0.576025 0.817432i \(-0.695397\pi\)
−0.576025 + 0.817432i \(0.695397\pi\)
\(360\) 35.0916 1.84949
\(361\) 1.00000 0.0526316
\(362\) −17.5877 −0.924391
\(363\) 0 0
\(364\) 2.41462 0.126561
\(365\) 41.3369 2.16367
\(366\) −14.8881 −0.778214
\(367\) 7.86931 0.410774 0.205387 0.978681i \(-0.434155\pi\)
0.205387 + 0.978681i \(0.434155\pi\)
\(368\) 6.35872 0.331471
\(369\) −61.5074 −3.20195
\(370\) 14.6552 0.761888
\(371\) 24.5068 1.27233
\(372\) −5.25629 −0.272526
\(373\) 2.90718 0.150528 0.0752639 0.997164i \(-0.476020\pi\)
0.0752639 + 0.997164i \(0.476020\pi\)
\(374\) 0 0
\(375\) −104.124 −5.37695
\(376\) −0.862845 −0.0444978
\(377\) −2.19045 −0.112814
\(378\) −65.9895 −3.39413
\(379\) 4.89402 0.251389 0.125694 0.992069i \(-0.459884\pi\)
0.125694 + 0.992069i \(0.459884\pi\)
\(380\) −4.17017 −0.213925
\(381\) 23.8710 1.22295
\(382\) 2.76249 0.141341
\(383\) −12.3879 −0.632994 −0.316497 0.948593i \(-0.602507\pi\)
−0.316497 + 0.948593i \(0.602507\pi\)
\(384\) −3.37860 −0.172413
\(385\) 0 0
\(386\) 21.4775 1.09317
\(387\) 50.8824 2.58650
\(388\) 0.809369 0.0410895
\(389\) −13.7527 −0.697291 −0.348645 0.937255i \(-0.613358\pi\)
−0.348645 + 0.937255i \(0.613358\pi\)
\(390\) 9.43175 0.477595
\(391\) −26.1021 −1.32004
\(392\) −6.01048 −0.303575
\(393\) 11.0218 0.555975
\(394\) −18.9032 −0.952330
\(395\) 4.36000 0.219375
\(396\) 0 0
\(397\) 10.0289 0.503338 0.251669 0.967813i \(-0.419021\pi\)
0.251669 + 0.967813i \(0.419021\pi\)
\(398\) −7.76283 −0.389115
\(399\) 12.1866 0.610094
\(400\) 12.3903 0.619515
\(401\) 22.2869 1.11296 0.556479 0.830862i \(-0.312152\pi\)
0.556479 + 0.830862i \(0.312152\pi\)
\(402\) −6.09296 −0.303889
\(403\) −1.04147 −0.0518792
\(404\) 12.0493 0.599475
\(405\) −152.487 −7.57712
\(406\) 11.8026 0.585752
\(407\) 0 0
\(408\) 13.8689 0.686613
\(409\) −5.88120 −0.290807 −0.145403 0.989372i \(-0.546448\pi\)
−0.145403 + 0.989372i \(0.546448\pi\)
\(410\) −30.4811 −1.50536
\(411\) 4.23604 0.208949
\(412\) 5.64096 0.277910
\(413\) −33.9534 −1.67074
\(414\) −53.5081 −2.62978
\(415\) 2.10106 0.103137
\(416\) −0.669426 −0.0328213
\(417\) −5.14988 −0.252191
\(418\) 0 0
\(419\) 3.24502 0.158530 0.0792648 0.996854i \(-0.474743\pi\)
0.0792648 + 0.996854i \(0.474743\pi\)
\(420\) −50.8202 −2.47977
\(421\) −19.2545 −0.938405 −0.469203 0.883091i \(-0.655459\pi\)
−0.469203 + 0.883091i \(0.655459\pi\)
\(422\) −6.88377 −0.335097
\(423\) 7.26077 0.353031
\(424\) −6.79423 −0.329957
\(425\) −50.8613 −2.46714
\(426\) −40.5204 −1.96322
\(427\) 15.8946 0.769194
\(428\) −15.1059 −0.730172
\(429\) 0 0
\(430\) 25.2157 1.21601
\(431\) 23.7828 1.14558 0.572789 0.819703i \(-0.305861\pi\)
0.572789 + 0.819703i \(0.305861\pi\)
\(432\) 18.2948 0.880210
\(433\) −17.1402 −0.823707 −0.411853 0.911250i \(-0.635118\pi\)
−0.411853 + 0.911250i \(0.635118\pi\)
\(434\) 5.61164 0.269367
\(435\) 46.1020 2.21042
\(436\) 11.6030 0.555685
\(437\) 6.35872 0.304179
\(438\) 33.4904 1.60023
\(439\) 0.144467 0.00689503 0.00344752 0.999994i \(-0.498903\pi\)
0.00344752 + 0.999994i \(0.498903\pi\)
\(440\) 0 0
\(441\) 50.5777 2.40846
\(442\) 2.74795 0.130707
\(443\) −19.6940 −0.935692 −0.467846 0.883810i \(-0.654970\pi\)
−0.467846 + 0.883810i \(0.654970\pi\)
\(444\) 11.8734 0.563486
\(445\) −34.3486 −1.62828
\(446\) −14.5212 −0.687597
\(447\) 53.0009 2.50686
\(448\) 3.60700 0.170415
\(449\) −30.0122 −1.41636 −0.708182 0.706030i \(-0.750484\pi\)
−0.708182 + 0.706030i \(0.750484\pi\)
\(450\) −104.263 −4.91502
\(451\) 0 0
\(452\) −6.03660 −0.283938
\(453\) −64.1476 −3.01392
\(454\) −18.1073 −0.849817
\(455\) −10.0694 −0.472060
\(456\) −3.37860 −0.158217
\(457\) −0.810437 −0.0379106 −0.0189553 0.999820i \(-0.506034\pi\)
−0.0189553 + 0.999820i \(0.506034\pi\)
\(458\) 25.3580 1.18490
\(459\) −75.0990 −3.50532
\(460\) −26.5169 −1.23636
\(461\) −29.4451 −1.37139 −0.685697 0.727888i \(-0.740502\pi\)
−0.685697 + 0.727888i \(0.740502\pi\)
\(462\) 0 0
\(463\) 24.9830 1.16106 0.580529 0.814239i \(-0.302846\pi\)
0.580529 + 0.814239i \(0.302846\pi\)
\(464\) −3.27213 −0.151905
\(465\) 21.9196 1.01650
\(466\) −21.9886 −1.01860
\(467\) −21.6826 −1.00335 −0.501676 0.865056i \(-0.667283\pi\)
−0.501676 + 0.865056i \(0.667283\pi\)
\(468\) 5.63316 0.260393
\(469\) 6.50487 0.300367
\(470\) 3.59821 0.165973
\(471\) −30.0550 −1.38486
\(472\) 9.41317 0.433276
\(473\) 0 0
\(474\) 3.53240 0.162248
\(475\) 12.3903 0.568506
\(476\) −14.8065 −0.678655
\(477\) 57.1729 2.61777
\(478\) 15.2971 0.699674
\(479\) −30.8590 −1.40998 −0.704992 0.709215i \(-0.749049\pi\)
−0.704992 + 0.709215i \(0.749049\pi\)
\(480\) 14.0893 0.643086
\(481\) 2.35256 0.107268
\(482\) 16.0409 0.730643
\(483\) 77.4913 3.52598
\(484\) 0 0
\(485\) −3.37520 −0.153260
\(486\) −68.6575 −3.11437
\(487\) −5.68001 −0.257386 −0.128693 0.991685i \(-0.541078\pi\)
−0.128693 + 0.991685i \(0.541078\pi\)
\(488\) −4.40660 −0.199477
\(489\) −16.3693 −0.740244
\(490\) 25.0647 1.13231
\(491\) −29.9204 −1.35029 −0.675145 0.737685i \(-0.735919\pi\)
−0.675145 + 0.737685i \(0.735919\pi\)
\(492\) −24.6953 −1.11335
\(493\) 13.4319 0.604941
\(494\) −0.669426 −0.0301189
\(495\) 0 0
\(496\) −1.55576 −0.0698558
\(497\) 43.2598 1.94047
\(498\) 1.70224 0.0762793
\(499\) −2.89019 −0.129383 −0.0646913 0.997905i \(-0.520606\pi\)
−0.0646913 + 0.997905i \(0.520606\pi\)
\(500\) −30.8187 −1.37826
\(501\) 15.5935 0.696664
\(502\) 6.91197 0.308496
\(503\) 27.9653 1.24691 0.623456 0.781858i \(-0.285728\pi\)
0.623456 + 0.781858i \(0.285728\pi\)
\(504\) −30.3526 −1.35201
\(505\) −50.2476 −2.23599
\(506\) 0 0
\(507\) −42.4077 −1.88339
\(508\) 7.06535 0.313474
\(509\) −34.7436 −1.53998 −0.769992 0.638054i \(-0.779740\pi\)
−0.769992 + 0.638054i \(0.779740\pi\)
\(510\) −57.8357 −2.56101
\(511\) −35.7545 −1.58169
\(512\) −1.00000 −0.0441942
\(513\) 18.2948 0.807736
\(514\) 24.7512 1.09173
\(515\) −23.5237 −1.03658
\(516\) 20.4293 0.899351
\(517\) 0 0
\(518\) −12.6761 −0.556955
\(519\) 31.5506 1.38492
\(520\) 2.79162 0.122420
\(521\) 26.9219 1.17947 0.589734 0.807597i \(-0.299232\pi\)
0.589734 + 0.807597i \(0.299232\pi\)
\(522\) 27.5347 1.20516
\(523\) 2.23979 0.0979394 0.0489697 0.998800i \(-0.484406\pi\)
0.0489697 + 0.998800i \(0.484406\pi\)
\(524\) 3.26224 0.142512
\(525\) 150.996 6.58999
\(526\) 16.8155 0.733190
\(527\) 6.38630 0.278192
\(528\) 0 0
\(529\) 17.4334 0.757973
\(530\) 28.3331 1.23071
\(531\) −79.2110 −3.43747
\(532\) 3.60700 0.156383
\(533\) −4.89305 −0.211942
\(534\) −27.8286 −1.20426
\(535\) 62.9942 2.72348
\(536\) −1.80340 −0.0778950
\(537\) 56.7203 2.44766
\(538\) −17.8288 −0.768655
\(539\) 0 0
\(540\) −76.2924 −3.28310
\(541\) 9.52299 0.409425 0.204713 0.978822i \(-0.434374\pi\)
0.204713 + 0.978822i \(0.434374\pi\)
\(542\) −30.2063 −1.29747
\(543\) 59.4219 2.55004
\(544\) 4.10493 0.175998
\(545\) −48.3866 −2.07265
\(546\) −8.15803 −0.349132
\(547\) 17.0265 0.727999 0.364000 0.931399i \(-0.381411\pi\)
0.364000 + 0.931399i \(0.381411\pi\)
\(548\) 1.25379 0.0535592
\(549\) 37.0811 1.58258
\(550\) 0 0
\(551\) −3.27213 −0.139397
\(552\) −21.4836 −0.914401
\(553\) −3.77120 −0.160368
\(554\) 32.7642 1.39202
\(555\) −49.5140 −2.10175
\(556\) −1.52427 −0.0646433
\(557\) 14.4105 0.610593 0.305296 0.952257i \(-0.401244\pi\)
0.305296 + 0.952257i \(0.401244\pi\)
\(558\) 13.0916 0.554212
\(559\) 4.04781 0.171204
\(560\) −15.0418 −0.635633
\(561\) 0 0
\(562\) −9.70846 −0.409527
\(563\) 14.3589 0.605156 0.302578 0.953125i \(-0.402153\pi\)
0.302578 + 0.953125i \(0.402153\pi\)
\(564\) 2.91521 0.122752
\(565\) 25.1736 1.05906
\(566\) 14.5739 0.612585
\(567\) 131.894 5.53902
\(568\) −11.9933 −0.503226
\(569\) −27.8880 −1.16912 −0.584562 0.811349i \(-0.698734\pi\)
−0.584562 + 0.811349i \(0.698734\pi\)
\(570\) 14.0893 0.590136
\(571\) 40.1622 1.68073 0.840367 0.542018i \(-0.182340\pi\)
0.840367 + 0.542018i \(0.182340\pi\)
\(572\) 0 0
\(573\) −9.33335 −0.389906
\(574\) 26.3648 1.10044
\(575\) 78.7864 3.28562
\(576\) 8.41492 0.350621
\(577\) 3.68171 0.153271 0.0766357 0.997059i \(-0.475582\pi\)
0.0766357 + 0.997059i \(0.475582\pi\)
\(578\) 0.149526 0.00621944
\(579\) −72.5637 −3.01564
\(580\) 13.6453 0.566591
\(581\) −1.81732 −0.0753952
\(582\) −2.73453 −0.113350
\(583\) 0 0
\(584\) 9.91252 0.410183
\(585\) −23.4912 −0.971242
\(586\) −10.5241 −0.434746
\(587\) −44.1246 −1.82122 −0.910608 0.413272i \(-0.864386\pi\)
−0.910608 + 0.413272i \(0.864386\pi\)
\(588\) 20.3070 0.837446
\(589\) −1.55576 −0.0641041
\(590\) −39.2545 −1.61608
\(591\) 63.8663 2.62711
\(592\) 3.51430 0.144437
\(593\) 2.42622 0.0996330 0.0498165 0.998758i \(-0.484136\pi\)
0.0498165 + 0.998758i \(0.484136\pi\)
\(594\) 0 0
\(595\) 61.7456 2.53132
\(596\) 15.6872 0.642575
\(597\) 26.2275 1.07342
\(598\) −4.25669 −0.174069
\(599\) 30.0748 1.22882 0.614412 0.788986i \(-0.289393\pi\)
0.614412 + 0.788986i \(0.289393\pi\)
\(600\) −41.8618 −1.70900
\(601\) −43.6023 −1.77858 −0.889288 0.457348i \(-0.848799\pi\)
−0.889288 + 0.457348i \(0.848799\pi\)
\(602\) −21.8104 −0.888927
\(603\) 15.1755 0.617992
\(604\) −18.9865 −0.772548
\(605\) 0 0
\(606\) −40.7097 −1.65372
\(607\) 3.95907 0.160694 0.0803468 0.996767i \(-0.474397\pi\)
0.0803468 + 0.996767i \(0.474397\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −39.8761 −1.61586
\(610\) 18.3762 0.744032
\(611\) 0.577611 0.0233676
\(612\) −34.5427 −1.39630
\(613\) −33.3417 −1.34666 −0.673329 0.739343i \(-0.735136\pi\)
−0.673329 + 0.739343i \(0.735136\pi\)
\(614\) −21.4131 −0.864161
\(615\) 102.983 4.15269
\(616\) 0 0
\(617\) −46.8084 −1.88443 −0.942217 0.335004i \(-0.891262\pi\)
−0.942217 + 0.335004i \(0.891262\pi\)
\(618\) −19.0585 −0.766646
\(619\) −3.44952 −0.138648 −0.0693239 0.997594i \(-0.522084\pi\)
−0.0693239 + 0.997594i \(0.522084\pi\)
\(620\) 6.48779 0.260556
\(621\) 116.332 4.66823
\(622\) −8.91440 −0.357435
\(623\) 29.7100 1.19030
\(624\) 2.26172 0.0905412
\(625\) 66.5679 2.66271
\(626\) 29.2280 1.16818
\(627\) 0 0
\(628\) −8.89570 −0.354977
\(629\) −14.4260 −0.575200
\(630\) 126.576 5.04289
\(631\) −3.45479 −0.137533 −0.0687665 0.997633i \(-0.521906\pi\)
−0.0687665 + 0.997633i \(0.521906\pi\)
\(632\) 1.04552 0.0415886
\(633\) 23.2575 0.924402
\(634\) 3.86251 0.153400
\(635\) −29.4637 −1.16923
\(636\) 22.9550 0.910224
\(637\) 4.02357 0.159420
\(638\) 0 0
\(639\) 100.922 3.99243
\(640\) 4.17017 0.164840
\(641\) 4.16142 0.164366 0.0821831 0.996617i \(-0.473811\pi\)
0.0821831 + 0.996617i \(0.473811\pi\)
\(642\) 51.0368 2.01426
\(643\) −27.0695 −1.06752 −0.533759 0.845637i \(-0.679221\pi\)
−0.533759 + 0.845637i \(0.679221\pi\)
\(644\) 22.9359 0.903803
\(645\) −85.1937 −3.35450
\(646\) 4.10493 0.161506
\(647\) −4.31680 −0.169711 −0.0848554 0.996393i \(-0.527043\pi\)
−0.0848554 + 0.996393i \(0.527043\pi\)
\(648\) −36.5661 −1.43645
\(649\) 0 0
\(650\) −8.29438 −0.325332
\(651\) −18.9595 −0.743080
\(652\) −4.84499 −0.189744
\(653\) −5.85277 −0.229037 −0.114518 0.993421i \(-0.536532\pi\)
−0.114518 + 0.993421i \(0.536532\pi\)
\(654\) −39.2020 −1.53292
\(655\) −13.6041 −0.531555
\(656\) −7.30933 −0.285381
\(657\) −83.4131 −3.25425
\(658\) −3.11229 −0.121330
\(659\) −16.4116 −0.639307 −0.319653 0.947535i \(-0.603566\pi\)
−0.319653 + 0.947535i \(0.603566\pi\)
\(660\) 0 0
\(661\) −25.8425 −1.00516 −0.502578 0.864532i \(-0.667615\pi\)
−0.502578 + 0.864532i \(0.667615\pi\)
\(662\) −30.7968 −1.19695
\(663\) −9.28421 −0.360569
\(664\) 0.503831 0.0195524
\(665\) −15.0418 −0.583296
\(666\) −29.5725 −1.14591
\(667\) −20.8066 −0.805633
\(668\) 4.61536 0.178574
\(669\) 49.0611 1.89681
\(670\) 7.52048 0.290541
\(671\) 0 0
\(672\) −12.1866 −0.470109
\(673\) −4.23989 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(674\) −16.0941 −0.619923
\(675\) 226.678 8.72484
\(676\) −12.5519 −0.482764
\(677\) −49.7388 −1.91162 −0.955809 0.293988i \(-0.905017\pi\)
−0.955809 + 0.293988i \(0.905017\pi\)
\(678\) 20.3953 0.783275
\(679\) 2.91940 0.112036
\(680\) −17.1183 −0.656455
\(681\) 61.1772 2.34432
\(682\) 0 0
\(683\) 38.4329 1.47059 0.735297 0.677745i \(-0.237043\pi\)
0.735297 + 0.677745i \(0.237043\pi\)
\(684\) 8.41492 0.321752
\(685\) −5.22851 −0.199771
\(686\) 3.56920 0.136273
\(687\) −85.6743 −3.26868
\(688\) 6.04669 0.230528
\(689\) 4.54823 0.173274
\(690\) 89.5900 3.41063
\(691\) 26.5880 1.01145 0.505727 0.862693i \(-0.331224\pi\)
0.505727 + 0.862693i \(0.331224\pi\)
\(692\) 9.33836 0.354991
\(693\) 0 0
\(694\) −31.7933 −1.20686
\(695\) 6.35644 0.241114
\(696\) 11.0552 0.419046
\(697\) 30.0043 1.13649
\(698\) 11.5331 0.436532
\(699\) 74.2906 2.80993
\(700\) 44.6918 1.68919
\(701\) 14.6368 0.552826 0.276413 0.961039i \(-0.410854\pi\)
0.276413 + 0.961039i \(0.410854\pi\)
\(702\) −12.2470 −0.462234
\(703\) 3.51430 0.132544
\(704\) 0 0
\(705\) −12.1569 −0.457855
\(706\) 3.56825 0.134293
\(707\) 43.4619 1.63455
\(708\) −31.8033 −1.19524
\(709\) 14.7490 0.553911 0.276956 0.960883i \(-0.410674\pi\)
0.276956 + 0.960883i \(0.410674\pi\)
\(710\) 50.0139 1.87699
\(711\) −8.79798 −0.329950
\(712\) −8.23674 −0.308685
\(713\) −9.89266 −0.370483
\(714\) 50.0252 1.87215
\(715\) 0 0
\(716\) 16.7881 0.627401
\(717\) −51.6828 −1.93013
\(718\) 21.8282 0.814622
\(719\) 8.82348 0.329060 0.164530 0.986372i \(-0.447389\pi\)
0.164530 + 0.986372i \(0.447389\pi\)
\(720\) −35.0916 −1.30779
\(721\) 20.3470 0.757760
\(722\) −1.00000 −0.0372161
\(723\) −54.1957 −2.01556
\(724\) 17.5877 0.653643
\(725\) −40.5426 −1.50571
\(726\) 0 0
\(727\) 0.938013 0.0347890 0.0173945 0.999849i \(-0.494463\pi\)
0.0173945 + 0.999849i \(0.494463\pi\)
\(728\) −2.41462 −0.0894918
\(729\) 122.268 4.52844
\(730\) −41.3369 −1.52995
\(731\) −24.8213 −0.918048
\(732\) 14.8881 0.550280
\(733\) −2.58558 −0.0955007 −0.0477504 0.998859i \(-0.515205\pi\)
−0.0477504 + 0.998859i \(0.515205\pi\)
\(734\) −7.86931 −0.290461
\(735\) −84.6835 −3.12360
\(736\) −6.35872 −0.234386
\(737\) 0 0
\(738\) 61.5074 2.26412
\(739\) 1.56553 0.0575888 0.0287944 0.999585i \(-0.490833\pi\)
0.0287944 + 0.999585i \(0.490833\pi\)
\(740\) −14.6552 −0.538736
\(741\) 2.26172 0.0830863
\(742\) −24.5068 −0.899674
\(743\) −35.3325 −1.29623 −0.648113 0.761545i \(-0.724441\pi\)
−0.648113 + 0.761545i \(0.724441\pi\)
\(744\) 5.25629 0.192705
\(745\) −65.4184 −2.39675
\(746\) −2.90718 −0.106439
\(747\) −4.23970 −0.155122
\(748\) 0 0
\(749\) −54.4871 −1.99092
\(750\) 104.124 3.80208
\(751\) −11.3947 −0.415799 −0.207900 0.978150i \(-0.566663\pi\)
−0.207900 + 0.978150i \(0.566663\pi\)
\(752\) 0.862845 0.0314647
\(753\) −23.3527 −0.851021
\(754\) 2.19045 0.0797713
\(755\) 79.1767 2.88153
\(756\) 65.9895 2.40001
\(757\) −6.68211 −0.242865 −0.121433 0.992600i \(-0.538749\pi\)
−0.121433 + 0.992600i \(0.538749\pi\)
\(758\) −4.89402 −0.177759
\(759\) 0 0
\(760\) 4.17017 0.151268
\(761\) −2.33169 −0.0845239 −0.0422619 0.999107i \(-0.513456\pi\)
−0.0422619 + 0.999107i \(0.513456\pi\)
\(762\) −23.8710 −0.864754
\(763\) 41.8522 1.51515
\(764\) −2.76249 −0.0999435
\(765\) 144.049 5.20809
\(766\) 12.3879 0.447594
\(767\) −6.30142 −0.227531
\(768\) 3.37860 0.121915
\(769\) −28.8994 −1.04214 −0.521070 0.853514i \(-0.674467\pi\)
−0.521070 + 0.853514i \(0.674467\pi\)
\(770\) 0 0
\(771\) −83.6242 −3.01165
\(772\) −21.4775 −0.772991
\(773\) −25.5506 −0.918991 −0.459495 0.888180i \(-0.651970\pi\)
−0.459495 + 0.888180i \(0.651970\pi\)
\(774\) −50.8824 −1.82893
\(775\) −19.2763 −0.692427
\(776\) −0.809369 −0.0290546
\(777\) 42.8274 1.53642
\(778\) 13.7527 0.493059
\(779\) −7.30933 −0.261884
\(780\) −9.43175 −0.337711
\(781\) 0 0
\(782\) 26.1021 0.933411
\(783\) −59.8630 −2.13933
\(784\) 6.01048 0.214660
\(785\) 37.0966 1.32403
\(786\) −11.0218 −0.393134
\(787\) −10.5042 −0.374436 −0.187218 0.982318i \(-0.559947\pi\)
−0.187218 + 0.982318i \(0.559947\pi\)
\(788\) 18.9032 0.673399
\(789\) −56.8128 −2.02259
\(790\) −4.36000 −0.155122
\(791\) −21.7741 −0.774196
\(792\) 0 0
\(793\) 2.94989 0.104754
\(794\) −10.0289 −0.355914
\(795\) −95.7261 −3.39505
\(796\) 7.76283 0.275146
\(797\) −6.50041 −0.230256 −0.115128 0.993351i \(-0.536728\pi\)
−0.115128 + 0.993351i \(0.536728\pi\)
\(798\) −12.1866 −0.431401
\(799\) −3.54192 −0.125304
\(800\) −12.3903 −0.438063
\(801\) 69.3115 2.44900
\(802\) −22.2869 −0.786979
\(803\) 0 0
\(804\) 6.09296 0.214882
\(805\) −95.6467 −3.37110
\(806\) 1.04147 0.0366841
\(807\) 60.2364 2.12042
\(808\) −12.0493 −0.423893
\(809\) 30.5543 1.07423 0.537116 0.843509i \(-0.319514\pi\)
0.537116 + 0.843509i \(0.319514\pi\)
\(810\) 152.487 5.35783
\(811\) −5.81000 −0.204017 −0.102008 0.994784i \(-0.532527\pi\)
−0.102008 + 0.994784i \(0.532527\pi\)
\(812\) −11.8026 −0.414189
\(813\) 102.055 3.57922
\(814\) 0 0
\(815\) 20.2044 0.707730
\(816\) −13.8689 −0.485509
\(817\) 6.04669 0.211547
\(818\) 5.88120 0.205631
\(819\) 20.3188 0.709997
\(820\) 30.4811 1.06445
\(821\) 43.0690 1.50312 0.751559 0.659666i \(-0.229302\pi\)
0.751559 + 0.659666i \(0.229302\pi\)
\(822\) −4.23604 −0.147749
\(823\) −20.6760 −0.720722 −0.360361 0.932813i \(-0.617346\pi\)
−0.360361 + 0.932813i \(0.617346\pi\)
\(824\) −5.64096 −0.196512
\(825\) 0 0
\(826\) 33.9534 1.18139
\(827\) −35.8535 −1.24675 −0.623374 0.781924i \(-0.714239\pi\)
−0.623374 + 0.781924i \(0.714239\pi\)
\(828\) 53.5081 1.85954
\(829\) 10.6519 0.369956 0.184978 0.982743i \(-0.440779\pi\)
0.184978 + 0.982743i \(0.440779\pi\)
\(830\) −2.10106 −0.0729288
\(831\) −110.697 −3.84004
\(832\) 0.669426 0.0232082
\(833\) −24.6726 −0.854856
\(834\) 5.14988 0.178326
\(835\) −19.2468 −0.666064
\(836\) 0 0
\(837\) −28.4624 −0.983804
\(838\) −3.24502 −0.112097
\(839\) −46.8329 −1.61685 −0.808425 0.588599i \(-0.799680\pi\)
−0.808425 + 0.588599i \(0.799680\pi\)
\(840\) 50.8202 1.75346
\(841\) −18.2932 −0.630799
\(842\) 19.2545 0.663553
\(843\) 32.8010 1.12973
\(844\) 6.88377 0.236949
\(845\) 52.3434 1.80067
\(846\) −7.26077 −0.249630
\(847\) 0 0
\(848\) 6.79423 0.233315
\(849\) −49.2392 −1.68988
\(850\) 50.8613 1.74453
\(851\) 22.3464 0.766026
\(852\) 40.5204 1.38821
\(853\) −35.3351 −1.20985 −0.604926 0.796282i \(-0.706797\pi\)
−0.604926 + 0.796282i \(0.706797\pi\)
\(854\) −15.8946 −0.543902
\(855\) −35.0916 −1.20011
\(856\) 15.1059 0.516309
\(857\) 4.94229 0.168826 0.0844128 0.996431i \(-0.473099\pi\)
0.0844128 + 0.996431i \(0.473099\pi\)
\(858\) 0 0
\(859\) 10.1131 0.345054 0.172527 0.985005i \(-0.444807\pi\)
0.172527 + 0.985005i \(0.444807\pi\)
\(860\) −25.2157 −0.859849
\(861\) −89.0760 −3.03570
\(862\) −23.7828 −0.810046
\(863\) 12.9844 0.441995 0.220998 0.975274i \(-0.429069\pi\)
0.220998 + 0.975274i \(0.429069\pi\)
\(864\) −18.2948 −0.622402
\(865\) −38.9425 −1.32409
\(866\) 17.1402 0.582449
\(867\) −0.505186 −0.0171570
\(868\) −5.61164 −0.190471
\(869\) 0 0
\(870\) −46.1020 −1.56300
\(871\) 1.20724 0.0409058
\(872\) −11.6030 −0.392929
\(873\) 6.81077 0.230510
\(874\) −6.35872 −0.215087
\(875\) −111.163 −3.75801
\(876\) −33.4904 −1.13154
\(877\) −44.8363 −1.51401 −0.757007 0.653407i \(-0.773339\pi\)
−0.757007 + 0.653407i \(0.773339\pi\)
\(878\) −0.144467 −0.00487552
\(879\) 35.5567 1.19930
\(880\) 0 0
\(881\) −21.0954 −0.710722 −0.355361 0.934729i \(-0.615642\pi\)
−0.355361 + 0.934729i \(0.615642\pi\)
\(882\) −50.5777 −1.70304
\(883\) 53.8142 1.81099 0.905496 0.424355i \(-0.139499\pi\)
0.905496 + 0.424355i \(0.139499\pi\)
\(884\) −2.74795 −0.0924235
\(885\) 132.625 4.45814
\(886\) 19.6940 0.661634
\(887\) 28.4930 0.956701 0.478351 0.878169i \(-0.341235\pi\)
0.478351 + 0.878169i \(0.341235\pi\)
\(888\) −11.8734 −0.398445
\(889\) 25.4848 0.854731
\(890\) 34.3486 1.15137
\(891\) 0 0
\(892\) 14.5212 0.486204
\(893\) 0.862845 0.0288740
\(894\) −53.0009 −1.77261
\(895\) −70.0092 −2.34015
\(896\) −3.60700 −0.120502
\(897\) 14.3816 0.480189
\(898\) 30.0122 1.00152
\(899\) 5.09065 0.169783
\(900\) 104.263 3.47544
\(901\) −27.8899 −0.929146
\(902\) 0 0
\(903\) 73.6887 2.45221
\(904\) 6.03660 0.200774
\(905\) −73.3438 −2.43803
\(906\) 64.1476 2.13116
\(907\) −19.0581 −0.632815 −0.316408 0.948623i \(-0.602477\pi\)
−0.316408 + 0.948623i \(0.602477\pi\)
\(908\) 18.1073 0.600912
\(909\) 101.394 3.36302
\(910\) 10.0694 0.333796
\(911\) −8.41671 −0.278858 −0.139429 0.990232i \(-0.544527\pi\)
−0.139429 + 0.990232i \(0.544527\pi\)
\(912\) 3.37860 0.111877
\(913\) 0 0
\(914\) 0.810437 0.0268069
\(915\) −62.0859 −2.05250
\(916\) −25.3580 −0.837850
\(917\) 11.7669 0.388577
\(918\) 75.0990 2.47864
\(919\) 10.5289 0.347315 0.173658 0.984806i \(-0.444441\pi\)
0.173658 + 0.984806i \(0.444441\pi\)
\(920\) 26.5169 0.874238
\(921\) 72.3461 2.38388
\(922\) 29.4451 0.969721
\(923\) 8.02860 0.264265
\(924\) 0 0
\(925\) 43.5432 1.43169
\(926\) −24.9830 −0.820992
\(927\) 47.4682 1.55906
\(928\) 3.27213 0.107413
\(929\) −19.5004 −0.639787 −0.319893 0.947454i \(-0.603647\pi\)
−0.319893 + 0.947454i \(0.603647\pi\)
\(930\) −21.9196 −0.718773
\(931\) 6.01048 0.196986
\(932\) 21.9886 0.720261
\(933\) 30.1182 0.986024
\(934\) 21.6826 0.709477
\(935\) 0 0
\(936\) −5.63316 −0.184126
\(937\) −46.4018 −1.51588 −0.757941 0.652323i \(-0.773794\pi\)
−0.757941 + 0.652323i \(0.773794\pi\)
\(938\) −6.50487 −0.212392
\(939\) −98.7495 −3.22257
\(940\) −3.59821 −0.117361
\(941\) 17.4235 0.567991 0.283996 0.958826i \(-0.408340\pi\)
0.283996 + 0.958826i \(0.408340\pi\)
\(942\) 30.0550 0.979245
\(943\) −46.4780 −1.51353
\(944\) −9.41317 −0.306373
\(945\) −275.187 −8.95184
\(946\) 0 0
\(947\) −18.8103 −0.611254 −0.305627 0.952151i \(-0.598866\pi\)
−0.305627 + 0.952151i \(0.598866\pi\)
\(948\) −3.53240 −0.114727
\(949\) −6.63570 −0.215404
\(950\) −12.3903 −0.401994
\(951\) −13.0499 −0.423171
\(952\) 14.8065 0.479882
\(953\) −54.4674 −1.76437 −0.882187 0.470900i \(-0.843929\pi\)
−0.882187 + 0.470900i \(0.843929\pi\)
\(954\) −57.1729 −1.85104
\(955\) 11.5201 0.372780
\(956\) −15.2971 −0.494744
\(957\) 0 0
\(958\) 30.8590 0.997009
\(959\) 4.52242 0.146037
\(960\) −14.0893 −0.454731
\(961\) −28.5796 −0.921923
\(962\) −2.35256 −0.0758496
\(963\) −127.115 −4.09622
\(964\) −16.0409 −0.516642
\(965\) 89.5646 2.88319
\(966\) −77.4913 −2.49324
\(967\) −51.7050 −1.66272 −0.831361 0.555733i \(-0.812438\pi\)
−0.831361 + 0.555733i \(0.812438\pi\)
\(968\) 0 0
\(969\) −13.8689 −0.445534
\(970\) 3.37520 0.108371
\(971\) 41.0651 1.31784 0.658920 0.752213i \(-0.271013\pi\)
0.658920 + 0.752213i \(0.271013\pi\)
\(972\) 68.6575 2.20219
\(973\) −5.49803 −0.176259
\(974\) 5.68001 0.181999
\(975\) 28.0234 0.897466
\(976\) 4.40660 0.141052
\(977\) −15.0675 −0.482052 −0.241026 0.970519i \(-0.577484\pi\)
−0.241026 + 0.970519i \(0.577484\pi\)
\(978\) 16.3693 0.523431
\(979\) 0 0
\(980\) −25.0647 −0.800663
\(981\) 97.6386 3.11736
\(982\) 29.9204 0.954799
\(983\) −24.2754 −0.774263 −0.387132 0.922024i \(-0.626534\pi\)
−0.387132 + 0.922024i \(0.626534\pi\)
\(984\) 24.6953 0.787257
\(985\) −78.8295 −2.51172
\(986\) −13.4319 −0.427758
\(987\) 10.5152 0.334701
\(988\) 0.669426 0.0212973
\(989\) 38.4492 1.22261
\(990\) 0 0
\(991\) −18.0803 −0.574339 −0.287169 0.957880i \(-0.592714\pi\)
−0.287169 + 0.957880i \(0.592714\pi\)
\(992\) 1.55576 0.0493955
\(993\) 104.050 3.30193
\(994\) −43.2598 −1.37212
\(995\) −32.3723 −1.02627
\(996\) −1.70224 −0.0539376
\(997\) −17.0357 −0.539527 −0.269764 0.962927i \(-0.586946\pi\)
−0.269764 + 0.962927i \(0.586946\pi\)
\(998\) 2.89019 0.0914873
\(999\) 64.2934 2.03415
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 4598.2.a.by.1.8 8
11.10 odd 2 4598.2.a.cb.1.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.by.1.8 8 1.1 even 1 trivial
4598.2.a.cb.1.8 yes 8 11.10 odd 2