Properties

Label 3174.2.a.ba.1.1
Level $3174$
Weight $2$
Character 3174.1
Self dual yes
Analytic conductor $25.345$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3174,2,Mod(1,3174)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3174, 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("3174.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3174.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: \(25.3445176016\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.30972\) of defining polynomial
Character \(\chi\) \(=\) 3174.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.88612 q^{5} -1.00000 q^{6} +4.16140 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.88612 q^{5} -1.00000 q^{6} +4.16140 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.88612 q^{10} +4.52825 q^{11} -1.00000 q^{12} -3.26315 q^{13} +4.16140 q^{14} +3.88612 q^{15} +1.00000 q^{16} -1.82862 q^{17} +1.00000 q^{18} +1.36564 q^{19} -3.88612 q^{20} -4.16140 q^{21} +4.52825 q^{22} -1.00000 q^{24} +10.1019 q^{25} -3.26315 q^{26} -1.00000 q^{27} +4.16140 q^{28} +0.561114 q^{29} +3.88612 q^{30} -4.63657 q^{31} +1.00000 q^{32} -4.52825 q^{33} -1.82862 q^{34} -16.1717 q^{35} +1.00000 q^{36} +2.51890 q^{37} +1.36564 q^{38} +3.26315 q^{39} -3.88612 q^{40} -6.11749 q^{41} -4.16140 q^{42} +9.85240 q^{43} +4.52825 q^{44} -3.88612 q^{45} +2.38128 q^{47} -1.00000 q^{48} +10.3172 q^{49} +10.1019 q^{50} +1.82862 q^{51} -3.26315 q^{52} +10.6102 q^{53} -1.00000 q^{54} -17.5973 q^{55} +4.16140 q^{56} -1.36564 q^{57} +0.561114 q^{58} -7.15262 q^{59} +3.88612 q^{60} -3.25684 q^{61} -4.63657 q^{62} +4.16140 q^{63} +1.00000 q^{64} +12.6810 q^{65} -4.52825 q^{66} -6.28762 q^{67} -1.82862 q^{68} -16.1717 q^{70} -6.45399 q^{71} +1.00000 q^{72} +12.3533 q^{73} +2.51890 q^{74} -10.1019 q^{75} +1.36564 q^{76} +18.8439 q^{77} +3.26315 q^{78} +8.43720 q^{79} -3.88612 q^{80} +1.00000 q^{81} -6.11749 q^{82} -0.377030 q^{83} -4.16140 q^{84} +7.10623 q^{85} +9.85240 q^{86} -0.561114 q^{87} +4.52825 q^{88} +15.2049 q^{89} -3.88612 q^{90} -13.5793 q^{91} +4.63657 q^{93} +2.38128 q^{94} -5.30706 q^{95} -1.00000 q^{96} +18.8759 q^{97} +10.3172 q^{98} +4.52825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} + 11 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} + 11 q^{7} + 5 q^{8} + 5 q^{9} - q^{10} + 11 q^{11} - 5 q^{12} + 12 q^{13} + 11 q^{14} + q^{15} + 5 q^{16} - q^{17} + 5 q^{18} + 15 q^{19} - q^{20} - 11 q^{21} + 11 q^{22} - 5 q^{24} + 6 q^{25} + 12 q^{26} - 5 q^{27} + 11 q^{28} + q^{29} + q^{30} - 18 q^{31} + 5 q^{32} - 11 q^{33} - q^{34} - 11 q^{35} + 5 q^{36} + 10 q^{37} + 15 q^{38} - 12 q^{39} - q^{40} - 16 q^{41} - 11 q^{42} + 18 q^{43} + 11 q^{44} - q^{45} - 4 q^{47} - 5 q^{48} + 20 q^{49} + 6 q^{50} + q^{51} + 12 q^{52} - q^{53} - 5 q^{54} - 22 q^{55} + 11 q^{56} - 15 q^{57} + q^{58} + 2 q^{59} + q^{60} - q^{61} - 18 q^{62} + 11 q^{63} + 5 q^{64} + 24 q^{65} - 11 q^{66} + 29 q^{67} - q^{68} - 11 q^{70} - 11 q^{71} + 5 q^{72} - 8 q^{73} + 10 q^{74} - 6 q^{75} + 15 q^{76} + 11 q^{77} - 12 q^{78} + 40 q^{79} - q^{80} + 5 q^{81} - 16 q^{82} + 8 q^{83} - 11 q^{84} - 13 q^{85} + 18 q^{86} - q^{87} + 11 q^{88} + 2 q^{89} - q^{90} + 33 q^{91} + 18 q^{93} - 4 q^{94} - 3 q^{95} - 5 q^{96} + 17 q^{97} + 20 q^{98} + 11 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.88612 −1.73793 −0.868963 0.494876i \(-0.835213\pi\)
−0.868963 + 0.494876i \(0.835213\pi\)
\(6\) −1.00000 −0.408248
\(7\) 4.16140 1.57286 0.786430 0.617679i \(-0.211927\pi\)
0.786430 + 0.617679i \(0.211927\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.88612 −1.22890
\(11\) 4.52825 1.36532 0.682659 0.730737i \(-0.260823\pi\)
0.682659 + 0.730737i \(0.260823\pi\)
\(12\) −1.00000 −0.288675
\(13\) −3.26315 −0.905036 −0.452518 0.891755i \(-0.649474\pi\)
−0.452518 + 0.891755i \(0.649474\pi\)
\(14\) 4.16140 1.11218
\(15\) 3.88612 1.00339
\(16\) 1.00000 0.250000
\(17\) −1.82862 −0.443505 −0.221753 0.975103i \(-0.571178\pi\)
−0.221753 + 0.975103i \(0.571178\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.36564 0.313300 0.156650 0.987654i \(-0.449931\pi\)
0.156650 + 0.987654i \(0.449931\pi\)
\(20\) −3.88612 −0.868963
\(21\) −4.16140 −0.908092
\(22\) 4.52825 0.965426
\(23\) 0 0
\(24\) −1.00000 −0.204124
\(25\) 10.1019 2.02039
\(26\) −3.26315 −0.639957
\(27\) −1.00000 −0.192450
\(28\) 4.16140 0.786430
\(29\) 0.561114 0.104196 0.0520981 0.998642i \(-0.483409\pi\)
0.0520981 + 0.998642i \(0.483409\pi\)
\(30\) 3.88612 0.709506
\(31\) −4.63657 −0.832752 −0.416376 0.909192i \(-0.636700\pi\)
−0.416376 + 0.909192i \(0.636700\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.52825 −0.788267
\(34\) −1.82862 −0.313605
\(35\) −16.1717 −2.73352
\(36\) 1.00000 0.166667
\(37\) 2.51890 0.414104 0.207052 0.978330i \(-0.433613\pi\)
0.207052 + 0.978330i \(0.433613\pi\)
\(38\) 1.36564 0.221537
\(39\) 3.26315 0.522523
\(40\) −3.88612 −0.614450
\(41\) −6.11749 −0.955392 −0.477696 0.878525i \(-0.658528\pi\)
−0.477696 + 0.878525i \(0.658528\pi\)
\(42\) −4.16140 −0.642118
\(43\) 9.85240 1.50248 0.751238 0.660031i \(-0.229457\pi\)
0.751238 + 0.660031i \(0.229457\pi\)
\(44\) 4.52825 0.682659
\(45\) −3.88612 −0.579309
\(46\) 0 0
\(47\) 2.38128 0.347345 0.173672 0.984803i \(-0.444437\pi\)
0.173672 + 0.984803i \(0.444437\pi\)
\(48\) −1.00000 −0.144338
\(49\) 10.3172 1.47389
\(50\) 10.1019 1.42863
\(51\) 1.82862 0.256058
\(52\) −3.26315 −0.452518
\(53\) 10.6102 1.45743 0.728713 0.684820i \(-0.240119\pi\)
0.728713 + 0.684820i \(0.240119\pi\)
\(54\) −1.00000 −0.136083
\(55\) −17.5973 −2.37282
\(56\) 4.16140 0.556090
\(57\) −1.36564 −0.180884
\(58\) 0.561114 0.0736778
\(59\) −7.15262 −0.931192 −0.465596 0.884997i \(-0.654160\pi\)
−0.465596 + 0.884997i \(0.654160\pi\)
\(60\) 3.88612 0.501696
\(61\) −3.25684 −0.416995 −0.208498 0.978023i \(-0.566857\pi\)
−0.208498 + 0.978023i \(0.566857\pi\)
\(62\) −4.63657 −0.588845
\(63\) 4.16140 0.524287
\(64\) 1.00000 0.125000
\(65\) 12.6810 1.57289
\(66\) −4.52825 −0.557389
\(67\) −6.28762 −0.768154 −0.384077 0.923301i \(-0.625480\pi\)
−0.384077 + 0.923301i \(0.625480\pi\)
\(68\) −1.82862 −0.221753
\(69\) 0 0
\(70\) −16.1717 −1.93289
\(71\) −6.45399 −0.765948 −0.382974 0.923759i \(-0.625100\pi\)
−0.382974 + 0.923759i \(0.625100\pi\)
\(72\) 1.00000 0.117851
\(73\) 12.3533 1.44584 0.722920 0.690932i \(-0.242799\pi\)
0.722920 + 0.690932i \(0.242799\pi\)
\(74\) 2.51890 0.292816
\(75\) −10.1019 −1.16647
\(76\) 1.36564 0.156650
\(77\) 18.8439 2.14746
\(78\) 3.26315 0.369479
\(79\) 8.43720 0.949259 0.474629 0.880186i \(-0.342582\pi\)
0.474629 + 0.880186i \(0.342582\pi\)
\(80\) −3.88612 −0.434482
\(81\) 1.00000 0.111111
\(82\) −6.11749 −0.675564
\(83\) −0.377030 −0.0413845 −0.0206922 0.999786i \(-0.506587\pi\)
−0.0206922 + 0.999786i \(0.506587\pi\)
\(84\) −4.16140 −0.454046
\(85\) 7.10623 0.770779
\(86\) 9.85240 1.06241
\(87\) −0.561114 −0.0601577
\(88\) 4.52825 0.482713
\(89\) 15.2049 1.61172 0.805860 0.592106i \(-0.201703\pi\)
0.805860 + 0.592106i \(0.201703\pi\)
\(90\) −3.88612 −0.409633
\(91\) −13.5793 −1.42350
\(92\) 0 0
\(93\) 4.63657 0.480790
\(94\) 2.38128 0.245610
\(95\) −5.30706 −0.544493
\(96\) −1.00000 −0.102062
\(97\) 18.8759 1.91656 0.958278 0.285838i \(-0.0922721\pi\)
0.958278 + 0.285838i \(0.0922721\pi\)
\(98\) 10.3172 1.04220
\(99\) 4.52825 0.455106
\(100\) 10.1019 1.01019
\(101\) −1.10306 −0.109759 −0.0548793 0.998493i \(-0.517477\pi\)
−0.0548793 + 0.998493i \(0.517477\pi\)
\(102\) 1.82862 0.181060
\(103\) 3.36648 0.331709 0.165854 0.986150i \(-0.446962\pi\)
0.165854 + 0.986150i \(0.446962\pi\)
\(104\) −3.26315 −0.319978
\(105\) 16.1717 1.57820
\(106\) 10.6102 1.03056
\(107\) −10.4984 −1.01492 −0.507458 0.861676i \(-0.669415\pi\)
−0.507458 + 0.861676i \(0.669415\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −1.27668 −0.122284 −0.0611418 0.998129i \(-0.519474\pi\)
−0.0611418 + 0.998129i \(0.519474\pi\)
\(110\) −17.5973 −1.67784
\(111\) −2.51890 −0.239083
\(112\) 4.16140 0.393215
\(113\) 17.6913 1.66425 0.832127 0.554585i \(-0.187123\pi\)
0.832127 + 0.554585i \(0.187123\pi\)
\(114\) −1.36564 −0.127904
\(115\) 0 0
\(116\) 0.561114 0.0520981
\(117\) −3.26315 −0.301679
\(118\) −7.15262 −0.658452
\(119\) −7.60961 −0.697572
\(120\) 3.88612 0.354753
\(121\) 9.50505 0.864096
\(122\) −3.25684 −0.294860
\(123\) 6.11749 0.551596
\(124\) −4.63657 −0.416376
\(125\) −19.8268 −1.77336
\(126\) 4.16140 0.370727
\(127\) 10.0980 0.896050 0.448025 0.894021i \(-0.352128\pi\)
0.448025 + 0.894021i \(0.352128\pi\)
\(128\) 1.00000 0.0883883
\(129\) −9.85240 −0.867455
\(130\) 12.6810 1.11220
\(131\) 7.51893 0.656932 0.328466 0.944516i \(-0.393468\pi\)
0.328466 + 0.944516i \(0.393468\pi\)
\(132\) −4.52825 −0.394134
\(133\) 5.68299 0.492778
\(134\) −6.28762 −0.543167
\(135\) 3.88612 0.334464
\(136\) −1.82862 −0.156803
\(137\) −14.3005 −1.22178 −0.610888 0.791717i \(-0.709188\pi\)
−0.610888 + 0.791717i \(0.709188\pi\)
\(138\) 0 0
\(139\) 21.8003 1.84907 0.924537 0.381092i \(-0.124452\pi\)
0.924537 + 0.381092i \(0.124452\pi\)
\(140\) −16.1717 −1.36676
\(141\) −2.38128 −0.200540
\(142\) −6.45399 −0.541607
\(143\) −14.7764 −1.23566
\(144\) 1.00000 0.0833333
\(145\) −2.18056 −0.181085
\(146\) 12.3533 1.02236
\(147\) −10.3172 −0.850951
\(148\) 2.51890 0.207052
\(149\) −2.08094 −0.170477 −0.0852385 0.996361i \(-0.527165\pi\)
−0.0852385 + 0.996361i \(0.527165\pi\)
\(150\) −10.1019 −0.824821
\(151\) 18.7684 1.52735 0.763675 0.645600i \(-0.223393\pi\)
0.763675 + 0.645600i \(0.223393\pi\)
\(152\) 1.36564 0.110768
\(153\) −1.82862 −0.147835
\(154\) 18.8439 1.51848
\(155\) 18.0183 1.44726
\(156\) 3.26315 0.261261
\(157\) −1.99513 −0.159228 −0.0796142 0.996826i \(-0.525369\pi\)
−0.0796142 + 0.996826i \(0.525369\pi\)
\(158\) 8.43720 0.671227
\(159\) −10.6102 −0.841445
\(160\) −3.88612 −0.307225
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 0.698970 0.0547475 0.0273738 0.999625i \(-0.491286\pi\)
0.0273738 + 0.999625i \(0.491286\pi\)
\(164\) −6.11749 −0.477696
\(165\) 17.5973 1.36995
\(166\) −0.377030 −0.0292632
\(167\) 5.02836 0.389106 0.194553 0.980892i \(-0.437674\pi\)
0.194553 + 0.980892i \(0.437674\pi\)
\(168\) −4.16140 −0.321059
\(169\) −2.35183 −0.180910
\(170\) 7.10623 0.545023
\(171\) 1.36564 0.104433
\(172\) 9.85240 0.751238
\(173\) −8.68258 −0.660124 −0.330062 0.943959i \(-0.607070\pi\)
−0.330062 + 0.943959i \(0.607070\pi\)
\(174\) −0.561114 −0.0425379
\(175\) 42.0382 3.17779
\(176\) 4.52825 0.341330
\(177\) 7.15262 0.537624
\(178\) 15.2049 1.13966
\(179\) −11.5753 −0.865176 −0.432588 0.901592i \(-0.642400\pi\)
−0.432588 + 0.901592i \(0.642400\pi\)
\(180\) −3.88612 −0.289654
\(181\) −2.61369 −0.194274 −0.0971370 0.995271i \(-0.530968\pi\)
−0.0971370 + 0.995271i \(0.530968\pi\)
\(182\) −13.5793 −1.00656
\(183\) 3.25684 0.240752
\(184\) 0 0
\(185\) −9.78874 −0.719683
\(186\) 4.63657 0.339970
\(187\) −8.28044 −0.605526
\(188\) 2.38128 0.173672
\(189\) −4.16140 −0.302697
\(190\) −5.30706 −0.385015
\(191\) 6.15128 0.445091 0.222546 0.974922i \(-0.428563\pi\)
0.222546 + 0.974922i \(0.428563\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.53015 0.614014 0.307007 0.951707i \(-0.400673\pi\)
0.307007 + 0.951707i \(0.400673\pi\)
\(194\) 18.8759 1.35521
\(195\) −12.6810 −0.908106
\(196\) 10.3172 0.736946
\(197\) 3.29008 0.234409 0.117204 0.993108i \(-0.462607\pi\)
0.117204 + 0.993108i \(0.462607\pi\)
\(198\) 4.52825 0.321809
\(199\) 20.0262 1.41962 0.709810 0.704393i \(-0.248781\pi\)
0.709810 + 0.704393i \(0.248781\pi\)
\(200\) 10.1019 0.714316
\(201\) 6.28762 0.443494
\(202\) −1.10306 −0.0776110
\(203\) 2.33502 0.163886
\(204\) 1.82862 0.128029
\(205\) 23.7733 1.66040
\(206\) 3.36648 0.234554
\(207\) 0 0
\(208\) −3.26315 −0.226259
\(209\) 6.18398 0.427755
\(210\) 16.1717 1.11595
\(211\) 2.62325 0.180592 0.0902959 0.995915i \(-0.471219\pi\)
0.0902959 + 0.995915i \(0.471219\pi\)
\(212\) 10.6102 0.728713
\(213\) 6.45399 0.442220
\(214\) −10.4984 −0.717654
\(215\) −38.2876 −2.61119
\(216\) −1.00000 −0.0680414
\(217\) −19.2946 −1.30980
\(218\) −1.27668 −0.0864676
\(219\) −12.3533 −0.834756
\(220\) −17.5973 −1.18641
\(221\) 5.96706 0.401388
\(222\) −2.51890 −0.169057
\(223\) 3.40900 0.228283 0.114142 0.993464i \(-0.463588\pi\)
0.114142 + 0.993464i \(0.463588\pi\)
\(224\) 4.16140 0.278045
\(225\) 10.1019 0.673463
\(226\) 17.6913 1.17681
\(227\) 14.8065 0.982745 0.491372 0.870950i \(-0.336495\pi\)
0.491372 + 0.870950i \(0.336495\pi\)
\(228\) −1.36564 −0.0904420
\(229\) 9.79565 0.647315 0.323657 0.946174i \(-0.395088\pi\)
0.323657 + 0.946174i \(0.395088\pi\)
\(230\) 0 0
\(231\) −18.8439 −1.23983
\(232\) 0.561114 0.0368389
\(233\) −3.95532 −0.259122 −0.129561 0.991571i \(-0.541357\pi\)
−0.129561 + 0.991571i \(0.541357\pi\)
\(234\) −3.26315 −0.213319
\(235\) −9.25393 −0.603660
\(236\) −7.15262 −0.465596
\(237\) −8.43720 −0.548055
\(238\) −7.60961 −0.493258
\(239\) −9.82751 −0.635689 −0.317844 0.948143i \(-0.602959\pi\)
−0.317844 + 0.948143i \(0.602959\pi\)
\(240\) 3.88612 0.250848
\(241\) 21.6867 1.39697 0.698483 0.715627i \(-0.253859\pi\)
0.698483 + 0.715627i \(0.253859\pi\)
\(242\) 9.50505 0.611008
\(243\) −1.00000 −0.0641500
\(244\) −3.25684 −0.208498
\(245\) −40.0941 −2.56151
\(246\) 6.11749 0.390037
\(247\) −4.45630 −0.283548
\(248\) −4.63657 −0.294422
\(249\) 0.377030 0.0238933
\(250\) −19.8268 −1.25396
\(251\) −7.45090 −0.470297 −0.235148 0.971959i \(-0.575558\pi\)
−0.235148 + 0.971959i \(0.575558\pi\)
\(252\) 4.16140 0.262143
\(253\) 0 0
\(254\) 10.0980 0.633603
\(255\) −7.10623 −0.445010
\(256\) 1.00000 0.0625000
\(257\) 4.10261 0.255914 0.127957 0.991780i \(-0.459158\pi\)
0.127957 + 0.991780i \(0.459158\pi\)
\(258\) −9.85240 −0.613383
\(259\) 10.4821 0.651328
\(260\) 12.6810 0.786443
\(261\) 0.561114 0.0347321
\(262\) 7.51893 0.464521
\(263\) −26.8046 −1.65284 −0.826420 0.563054i \(-0.809626\pi\)
−0.826420 + 0.563054i \(0.809626\pi\)
\(264\) −4.52825 −0.278695
\(265\) −41.2326 −2.53290
\(266\) 5.68299 0.348446
\(267\) −15.2049 −0.930527
\(268\) −6.28762 −0.384077
\(269\) −5.74299 −0.350156 −0.175078 0.984555i \(-0.556018\pi\)
−0.175078 + 0.984555i \(0.556018\pi\)
\(270\) 3.88612 0.236502
\(271\) 6.93596 0.421330 0.210665 0.977558i \(-0.432437\pi\)
0.210665 + 0.977558i \(0.432437\pi\)
\(272\) −1.82862 −0.110876
\(273\) 13.5793 0.821855
\(274\) −14.3005 −0.863927
\(275\) 45.7442 2.75848
\(276\) 0 0
\(277\) −3.81024 −0.228935 −0.114468 0.993427i \(-0.536516\pi\)
−0.114468 + 0.993427i \(0.536516\pi\)
\(278\) 21.8003 1.30749
\(279\) −4.63657 −0.277584
\(280\) −16.1717 −0.966444
\(281\) 18.1727 1.08409 0.542047 0.840348i \(-0.317649\pi\)
0.542047 + 0.840348i \(0.317649\pi\)
\(282\) −2.38128 −0.141803
\(283\) −17.0969 −1.01631 −0.508153 0.861267i \(-0.669672\pi\)
−0.508153 + 0.861267i \(0.669672\pi\)
\(284\) −6.45399 −0.382974
\(285\) 5.30706 0.314363
\(286\) −14.7764 −0.873745
\(287\) −25.4573 −1.50270
\(288\) 1.00000 0.0589256
\(289\) −13.6562 −0.803303
\(290\) −2.18056 −0.128047
\(291\) −18.8759 −1.10652
\(292\) 12.3533 0.722920
\(293\) −13.0545 −0.762650 −0.381325 0.924441i \(-0.624532\pi\)
−0.381325 + 0.924441i \(0.624532\pi\)
\(294\) −10.3172 −0.601714
\(295\) 27.7960 1.61834
\(296\) 2.51890 0.146408
\(297\) −4.52825 −0.262756
\(298\) −2.08094 −0.120545
\(299\) 0 0
\(300\) −10.1019 −0.583236
\(301\) 40.9997 2.36319
\(302\) 18.7684 1.08000
\(303\) 1.10306 0.0633691
\(304\) 1.36564 0.0783250
\(305\) 12.6565 0.724707
\(306\) −1.82862 −0.104535
\(307\) −24.0856 −1.37464 −0.687320 0.726355i \(-0.741213\pi\)
−0.687320 + 0.726355i \(0.741213\pi\)
\(308\) 18.8439 1.07373
\(309\) −3.36648 −0.191512
\(310\) 18.0183 1.02337
\(311\) 2.64709 0.150103 0.0750513 0.997180i \(-0.476088\pi\)
0.0750513 + 0.997180i \(0.476088\pi\)
\(312\) 3.26315 0.184740
\(313\) −25.0150 −1.41393 −0.706965 0.707248i \(-0.749936\pi\)
−0.706965 + 0.707248i \(0.749936\pi\)
\(314\) −1.99513 −0.112591
\(315\) −16.1717 −0.911172
\(316\) 8.43720 0.474629
\(317\) −2.94022 −0.165139 −0.0825695 0.996585i \(-0.526313\pi\)
−0.0825695 + 0.996585i \(0.526313\pi\)
\(318\) −10.6102 −0.594991
\(319\) 2.54086 0.142261
\(320\) −3.88612 −0.217241
\(321\) 10.4984 0.585962
\(322\) 0 0
\(323\) −2.49724 −0.138950
\(324\) 1.00000 0.0555556
\(325\) −32.9642 −1.82853
\(326\) 0.698970 0.0387124
\(327\) 1.27668 0.0706005
\(328\) −6.11749 −0.337782
\(329\) 9.90943 0.546325
\(330\) 17.5973 0.968702
\(331\) −1.21838 −0.0669682 −0.0334841 0.999439i \(-0.510660\pi\)
−0.0334841 + 0.999439i \(0.510660\pi\)
\(332\) −0.377030 −0.0206922
\(333\) 2.51890 0.138035
\(334\) 5.02836 0.275140
\(335\) 24.4344 1.33500
\(336\) −4.16140 −0.227023
\(337\) 19.0636 1.03846 0.519230 0.854635i \(-0.326219\pi\)
0.519230 + 0.854635i \(0.326219\pi\)
\(338\) −2.35183 −0.127923
\(339\) −17.6913 −0.960858
\(340\) 7.10623 0.385390
\(341\) −20.9955 −1.13697
\(342\) 1.36564 0.0738456
\(343\) 13.8043 0.745365
\(344\) 9.85240 0.531206
\(345\) 0 0
\(346\) −8.68258 −0.466778
\(347\) −12.3566 −0.663340 −0.331670 0.943396i \(-0.607612\pi\)
−0.331670 + 0.943396i \(0.607612\pi\)
\(348\) −0.561114 −0.0300789
\(349\) −15.6813 −0.839400 −0.419700 0.907663i \(-0.637865\pi\)
−0.419700 + 0.907663i \(0.637865\pi\)
\(350\) 42.0382 2.24704
\(351\) 3.26315 0.174174
\(352\) 4.52825 0.241357
\(353\) 30.5143 1.62411 0.812057 0.583578i \(-0.198348\pi\)
0.812057 + 0.583578i \(0.198348\pi\)
\(354\) 7.15262 0.380158
\(355\) 25.0810 1.33116
\(356\) 15.2049 0.805860
\(357\) 7.60961 0.402743
\(358\) −11.5753 −0.611772
\(359\) 16.4977 0.870713 0.435357 0.900258i \(-0.356622\pi\)
0.435357 + 0.900258i \(0.356622\pi\)
\(360\) −3.88612 −0.204817
\(361\) −17.1350 −0.901843
\(362\) −2.61369 −0.137372
\(363\) −9.50505 −0.498886
\(364\) −13.5793 −0.711748
\(365\) −48.0063 −2.51276
\(366\) 3.25684 0.170238
\(367\) −27.6912 −1.44547 −0.722736 0.691125i \(-0.757116\pi\)
−0.722736 + 0.691125i \(0.757116\pi\)
\(368\) 0 0
\(369\) −6.11749 −0.318464
\(370\) −9.78874 −0.508892
\(371\) 44.1533 2.29233
\(372\) 4.63657 0.240395
\(373\) −20.3626 −1.05433 −0.527167 0.849762i \(-0.676746\pi\)
−0.527167 + 0.849762i \(0.676746\pi\)
\(374\) −8.28044 −0.428171
\(375\) 19.8268 1.02385
\(376\) 2.38128 0.122805
\(377\) −1.83100 −0.0943013
\(378\) −4.16140 −0.214039
\(379\) −15.3289 −0.787391 −0.393695 0.919241i \(-0.628803\pi\)
−0.393695 + 0.919241i \(0.628803\pi\)
\(380\) −5.30706 −0.272246
\(381\) −10.0980 −0.517335
\(382\) 6.15128 0.314727
\(383\) −20.7059 −1.05802 −0.529011 0.848615i \(-0.677437\pi\)
−0.529011 + 0.848615i \(0.677437\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −73.2295 −3.73212
\(386\) 8.53015 0.434173
\(387\) 9.85240 0.500825
\(388\) 18.8759 0.958278
\(389\) −34.4334 −1.74584 −0.872920 0.487863i \(-0.837777\pi\)
−0.872920 + 0.487863i \(0.837777\pi\)
\(390\) −12.6810 −0.642128
\(391\) 0 0
\(392\) 10.3172 0.521099
\(393\) −7.51893 −0.379280
\(394\) 3.29008 0.165752
\(395\) −32.7880 −1.64974
\(396\) 4.52825 0.227553
\(397\) −26.4490 −1.32744 −0.663718 0.747983i \(-0.731022\pi\)
−0.663718 + 0.747983i \(0.731022\pi\)
\(398\) 20.0262 1.00382
\(399\) −5.68299 −0.284505
\(400\) 10.1019 0.505097
\(401\) −17.8995 −0.893857 −0.446928 0.894570i \(-0.647482\pi\)
−0.446928 + 0.894570i \(0.647482\pi\)
\(402\) 6.28762 0.313598
\(403\) 15.1298 0.753671
\(404\) −1.10306 −0.0548793
\(405\) −3.88612 −0.193103
\(406\) 2.33502 0.115885
\(407\) 11.4062 0.565384
\(408\) 1.82862 0.0905301
\(409\) −9.47776 −0.468645 −0.234323 0.972159i \(-0.575287\pi\)
−0.234323 + 0.972159i \(0.575287\pi\)
\(410\) 23.7733 1.17408
\(411\) 14.3005 0.705393
\(412\) 3.36648 0.165854
\(413\) −29.7649 −1.46464
\(414\) 0 0
\(415\) 1.46519 0.0719232
\(416\) −3.26315 −0.159989
\(417\) −21.8003 −1.06756
\(418\) 6.18398 0.302468
\(419\) −22.4130 −1.09495 −0.547474 0.836822i \(-0.684411\pi\)
−0.547474 + 0.836822i \(0.684411\pi\)
\(420\) 16.1717 0.789098
\(421\) −1.53019 −0.0745770 −0.0372885 0.999305i \(-0.511872\pi\)
−0.0372885 + 0.999305i \(0.511872\pi\)
\(422\) 2.62325 0.127698
\(423\) 2.38128 0.115782
\(424\) 10.6102 0.515278
\(425\) −18.4726 −0.896053
\(426\) 6.45399 0.312697
\(427\) −13.5530 −0.655875
\(428\) −10.4984 −0.507458
\(429\) 14.7764 0.713410
\(430\) −38.2876 −1.84639
\(431\) 0.121649 0.00585962 0.00292981 0.999996i \(-0.499067\pi\)
0.00292981 + 0.999996i \(0.499067\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.9931 1.68166 0.840830 0.541300i \(-0.182068\pi\)
0.840830 + 0.541300i \(0.182068\pi\)
\(434\) −19.2946 −0.926171
\(435\) 2.18056 0.104550
\(436\) −1.27668 −0.0611418
\(437\) 0 0
\(438\) −12.3533 −0.590262
\(439\) −21.5673 −1.02935 −0.514676 0.857385i \(-0.672088\pi\)
−0.514676 + 0.857385i \(0.672088\pi\)
\(440\) −17.5973 −0.838920
\(441\) 10.3172 0.491297
\(442\) 5.96706 0.283824
\(443\) −3.30287 −0.156924 −0.0784620 0.996917i \(-0.525001\pi\)
−0.0784620 + 0.996917i \(0.525001\pi\)
\(444\) −2.51890 −0.119542
\(445\) −59.0883 −2.80105
\(446\) 3.40900 0.161421
\(447\) 2.08094 0.0984250
\(448\) 4.16140 0.196608
\(449\) 22.2893 1.05190 0.525949 0.850516i \(-0.323710\pi\)
0.525949 + 0.850516i \(0.323710\pi\)
\(450\) 10.1019 0.476210
\(451\) −27.7015 −1.30441
\(452\) 17.6913 0.832127
\(453\) −18.7684 −0.881816
\(454\) 14.8065 0.694906
\(455\) 52.7707 2.47393
\(456\) −1.36564 −0.0639521
\(457\) −36.1129 −1.68929 −0.844645 0.535327i \(-0.820188\pi\)
−0.844645 + 0.535327i \(0.820188\pi\)
\(458\) 9.79565 0.457721
\(459\) 1.82862 0.0853526
\(460\) 0 0
\(461\) 31.5662 1.47018 0.735091 0.677968i \(-0.237139\pi\)
0.735091 + 0.677968i \(0.237139\pi\)
\(462\) −18.8439 −0.876695
\(463\) −27.8475 −1.29418 −0.647092 0.762412i \(-0.724015\pi\)
−0.647092 + 0.762412i \(0.724015\pi\)
\(464\) 0.561114 0.0260490
\(465\) −18.0183 −0.835577
\(466\) −3.95532 −0.183227
\(467\) 0.589854 0.0272952 0.0136476 0.999907i \(-0.495656\pi\)
0.0136476 + 0.999907i \(0.495656\pi\)
\(468\) −3.26315 −0.150839
\(469\) −26.1653 −1.20820
\(470\) −9.25393 −0.426852
\(471\) 1.99513 0.0919305
\(472\) −7.15262 −0.329226
\(473\) 44.6141 2.05136
\(474\) −8.43720 −0.387533
\(475\) 13.7957 0.632988
\(476\) −7.60961 −0.348786
\(477\) 10.6102 0.485808
\(478\) −9.82751 −0.449500
\(479\) −10.2856 −0.469960 −0.234980 0.972000i \(-0.575503\pi\)
−0.234980 + 0.972000i \(0.575503\pi\)
\(480\) 3.88612 0.177376
\(481\) −8.21955 −0.374779
\(482\) 21.6867 0.987804
\(483\) 0 0
\(484\) 9.50505 0.432048
\(485\) −73.3540 −3.33083
\(486\) −1.00000 −0.0453609
\(487\) −27.5042 −1.24633 −0.623167 0.782089i \(-0.714154\pi\)
−0.623167 + 0.782089i \(0.714154\pi\)
\(488\) −3.25684 −0.147430
\(489\) −0.698970 −0.0316085
\(490\) −40.0941 −1.81126
\(491\) 8.79100 0.396732 0.198366 0.980128i \(-0.436436\pi\)
0.198366 + 0.980128i \(0.436436\pi\)
\(492\) 6.11749 0.275798
\(493\) −1.02606 −0.0462115
\(494\) −4.45630 −0.200499
\(495\) −17.5973 −0.790941
\(496\) −4.63657 −0.208188
\(497\) −26.8576 −1.20473
\(498\) 0.377030 0.0168951
\(499\) 23.5520 1.05433 0.527167 0.849762i \(-0.323254\pi\)
0.527167 + 0.849762i \(0.323254\pi\)
\(500\) −19.8268 −0.886682
\(501\) −5.02836 −0.224651
\(502\) −7.45090 −0.332550
\(503\) 6.82809 0.304450 0.152225 0.988346i \(-0.451356\pi\)
0.152225 + 0.988346i \(0.451356\pi\)
\(504\) 4.16140 0.185363
\(505\) 4.28663 0.190752
\(506\) 0 0
\(507\) 2.35183 0.104449
\(508\) 10.0980 0.448025
\(509\) 18.7374 0.830522 0.415261 0.909702i \(-0.363690\pi\)
0.415261 + 0.909702i \(0.363690\pi\)
\(510\) −7.10623 −0.314669
\(511\) 51.4069 2.27411
\(512\) 1.00000 0.0441942
\(513\) −1.36564 −0.0602946
\(514\) 4.10261 0.180958
\(515\) −13.0825 −0.576486
\(516\) −9.85240 −0.433728
\(517\) 10.7830 0.474236
\(518\) 10.4821 0.460558
\(519\) 8.68258 0.381123
\(520\) 12.6810 0.556099
\(521\) 31.2416 1.36872 0.684360 0.729144i \(-0.260082\pi\)
0.684360 + 0.729144i \(0.260082\pi\)
\(522\) 0.561114 0.0245593
\(523\) 13.6962 0.598892 0.299446 0.954113i \(-0.403198\pi\)
0.299446 + 0.954113i \(0.403198\pi\)
\(524\) 7.51893 0.328466
\(525\) −42.0382 −1.83470
\(526\) −26.8046 −1.16873
\(527\) 8.47851 0.369330
\(528\) −4.52825 −0.197067
\(529\) 0 0
\(530\) −41.2326 −1.79103
\(531\) −7.15262 −0.310397
\(532\) 5.68299 0.246389
\(533\) 19.9623 0.864664
\(534\) −15.2049 −0.657982
\(535\) 40.7980 1.76385
\(536\) −6.28762 −0.271584
\(537\) 11.5753 0.499510
\(538\) −5.74299 −0.247598
\(539\) 46.7190 2.01233
\(540\) 3.88612 0.167232
\(541\) 23.8228 1.02422 0.512111 0.858919i \(-0.328864\pi\)
0.512111 + 0.858919i \(0.328864\pi\)
\(542\) 6.93596 0.297925
\(543\) 2.61369 0.112164
\(544\) −1.82862 −0.0784014
\(545\) 4.96133 0.212520
\(546\) 13.5793 0.581140
\(547\) −22.0310 −0.941977 −0.470988 0.882139i \(-0.656103\pi\)
−0.470988 + 0.882139i \(0.656103\pi\)
\(548\) −14.3005 −0.610888
\(549\) −3.25684 −0.138998
\(550\) 45.7442 1.95054
\(551\) 0.766281 0.0326447
\(552\) 0 0
\(553\) 35.1105 1.49305
\(554\) −3.81024 −0.161882
\(555\) 9.78874 0.415509
\(556\) 21.8003 0.924537
\(557\) 10.0072 0.424019 0.212010 0.977268i \(-0.431999\pi\)
0.212010 + 0.977268i \(0.431999\pi\)
\(558\) −4.63657 −0.196282
\(559\) −32.1499 −1.35979
\(560\) −16.1717 −0.683379
\(561\) 8.28044 0.349600
\(562\) 18.1727 0.766571
\(563\) 13.6494 0.575252 0.287626 0.957743i \(-0.407134\pi\)
0.287626 + 0.957743i \(0.407134\pi\)
\(564\) −2.38128 −0.100270
\(565\) −68.7504 −2.89235
\(566\) −17.0969 −0.718638
\(567\) 4.16140 0.174762
\(568\) −6.45399 −0.270804
\(569\) 20.5236 0.860394 0.430197 0.902735i \(-0.358444\pi\)
0.430197 + 0.902735i \(0.358444\pi\)
\(570\) 5.30706 0.222288
\(571\) −10.3256 −0.432114 −0.216057 0.976381i \(-0.569320\pi\)
−0.216057 + 0.976381i \(0.569320\pi\)
\(572\) −14.7764 −0.617831
\(573\) −6.15128 −0.256974
\(574\) −25.4573 −1.06257
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 40.8152 1.69916 0.849579 0.527461i \(-0.176856\pi\)
0.849579 + 0.527461i \(0.176856\pi\)
\(578\) −13.6562 −0.568021
\(579\) −8.53015 −0.354501
\(580\) −2.18056 −0.0905427
\(581\) −1.56897 −0.0650920
\(582\) −18.8759 −0.782431
\(583\) 48.0457 1.98985
\(584\) 12.3533 0.511182
\(585\) 12.6810 0.524295
\(586\) −13.0545 −0.539275
\(587\) 42.9852 1.77419 0.887095 0.461587i \(-0.152720\pi\)
0.887095 + 0.461587i \(0.152720\pi\)
\(588\) −10.3172 −0.425476
\(589\) −6.33190 −0.260901
\(590\) 27.7960 1.14434
\(591\) −3.29008 −0.135336
\(592\) 2.51890 0.103526
\(593\) −24.9472 −1.02446 −0.512229 0.858849i \(-0.671180\pi\)
−0.512229 + 0.858849i \(0.671180\pi\)
\(594\) −4.52825 −0.185796
\(595\) 29.5719 1.21233
\(596\) −2.08094 −0.0852385
\(597\) −20.0262 −0.819618
\(598\) 0 0
\(599\) 33.7284 1.37811 0.689053 0.724711i \(-0.258027\pi\)
0.689053 + 0.724711i \(0.258027\pi\)
\(600\) −10.1019 −0.412410
\(601\) 39.8604 1.62594 0.812970 0.582306i \(-0.197849\pi\)
0.812970 + 0.582306i \(0.197849\pi\)
\(602\) 40.9997 1.67102
\(603\) −6.28762 −0.256051
\(604\) 18.7684 0.763675
\(605\) −36.9378 −1.50174
\(606\) 1.10306 0.0448087
\(607\) 28.8082 1.16929 0.584645 0.811289i \(-0.301234\pi\)
0.584645 + 0.811289i \(0.301234\pi\)
\(608\) 1.36564 0.0553842
\(609\) −2.33502 −0.0946197
\(610\) 12.6565 0.512445
\(611\) −7.77047 −0.314359
\(612\) −1.82862 −0.0739175
\(613\) −28.9620 −1.16977 −0.584883 0.811118i \(-0.698860\pi\)
−0.584883 + 0.811118i \(0.698860\pi\)
\(614\) −24.0856 −0.972017
\(615\) −23.7733 −0.958633
\(616\) 18.8439 0.759241
\(617\) −26.0167 −1.04739 −0.523696 0.851905i \(-0.675447\pi\)
−0.523696 + 0.851905i \(0.675447\pi\)
\(618\) −3.36648 −0.135420
\(619\) 21.2470 0.853989 0.426995 0.904254i \(-0.359572\pi\)
0.426995 + 0.904254i \(0.359572\pi\)
\(620\) 18.0183 0.723631
\(621\) 0 0
\(622\) 2.64709 0.106139
\(623\) 63.2738 2.53501
\(624\) 3.26315 0.130631
\(625\) 26.5396 1.06159
\(626\) −25.0150 −0.999800
\(627\) −6.18398 −0.246964
\(628\) −1.99513 −0.0796142
\(629\) −4.60610 −0.183657
\(630\) −16.1717 −0.644296
\(631\) 17.1772 0.683813 0.341906 0.939734i \(-0.388927\pi\)
0.341906 + 0.939734i \(0.388927\pi\)
\(632\) 8.43720 0.335614
\(633\) −2.62325 −0.104265
\(634\) −2.94022 −0.116771
\(635\) −39.2420 −1.55727
\(636\) −10.6102 −0.420722
\(637\) −33.6667 −1.33392
\(638\) 2.54086 0.100594
\(639\) −6.45399 −0.255316
\(640\) −3.88612 −0.153612
\(641\) −2.37406 −0.0937697 −0.0468848 0.998900i \(-0.514929\pi\)
−0.0468848 + 0.998900i \(0.514929\pi\)
\(642\) 10.4984 0.414338
\(643\) 46.5640 1.83630 0.918152 0.396229i \(-0.129681\pi\)
0.918152 + 0.396229i \(0.129681\pi\)
\(644\) 0 0
\(645\) 38.2876 1.50757
\(646\) −2.49724 −0.0982526
\(647\) −9.57063 −0.376260 −0.188130 0.982144i \(-0.560243\pi\)
−0.188130 + 0.982144i \(0.560243\pi\)
\(648\) 1.00000 0.0392837
\(649\) −32.3889 −1.27137
\(650\) −32.9642 −1.29296
\(651\) 19.2946 0.756215
\(652\) 0.698970 0.0273738
\(653\) −32.1688 −1.25886 −0.629432 0.777056i \(-0.716712\pi\)
−0.629432 + 0.777056i \(0.716712\pi\)
\(654\) 1.27668 0.0499221
\(655\) −29.2195 −1.14170
\(656\) −6.11749 −0.238848
\(657\) 12.3533 0.481947
\(658\) 9.90943 0.386310
\(659\) 1.81861 0.0708431 0.0354216 0.999372i \(-0.488723\pi\)
0.0354216 + 0.999372i \(0.488723\pi\)
\(660\) 17.5973 0.684975
\(661\) 40.7121 1.58352 0.791760 0.610833i \(-0.209165\pi\)
0.791760 + 0.610833i \(0.209165\pi\)
\(662\) −1.21838 −0.0473537
\(663\) −5.96706 −0.231741
\(664\) −0.377030 −0.0146316
\(665\) −22.0848 −0.856411
\(666\) 2.51890 0.0976053
\(667\) 0 0
\(668\) 5.02836 0.194553
\(669\) −3.40900 −0.131799
\(670\) 24.4344 0.943985
\(671\) −14.7478 −0.569331
\(672\) −4.16140 −0.160529
\(673\) −12.9279 −0.498335 −0.249167 0.968460i \(-0.580157\pi\)
−0.249167 + 0.968460i \(0.580157\pi\)
\(674\) 19.0636 0.734302
\(675\) −10.1019 −0.388824
\(676\) −2.35183 −0.0904551
\(677\) 6.82478 0.262298 0.131149 0.991363i \(-0.458133\pi\)
0.131149 + 0.991363i \(0.458133\pi\)
\(678\) −17.6913 −0.679429
\(679\) 78.5501 3.01448
\(680\) 7.10623 0.272512
\(681\) −14.8065 −0.567388
\(682\) −20.9955 −0.803961
\(683\) 1.89490 0.0725064 0.0362532 0.999343i \(-0.488458\pi\)
0.0362532 + 0.999343i \(0.488458\pi\)
\(684\) 1.36564 0.0522167
\(685\) 55.5736 2.12336
\(686\) 13.8043 0.527052
\(687\) −9.79565 −0.373727
\(688\) 9.85240 0.375619
\(689\) −34.6228 −1.31902
\(690\) 0 0
\(691\) −14.4219 −0.548635 −0.274317 0.961639i \(-0.588452\pi\)
−0.274317 + 0.961639i \(0.588452\pi\)
\(692\) −8.68258 −0.330062
\(693\) 18.8439 0.715819
\(694\) −12.3566 −0.469052
\(695\) −84.7185 −3.21356
\(696\) −0.561114 −0.0212690
\(697\) 11.1866 0.423721
\(698\) −15.6813 −0.593545
\(699\) 3.95532 0.149604
\(700\) 42.0382 1.58890
\(701\) 36.1516 1.36543 0.682713 0.730687i \(-0.260800\pi\)
0.682713 + 0.730687i \(0.260800\pi\)
\(702\) 3.26315 0.123160
\(703\) 3.43992 0.129739
\(704\) 4.52825 0.170665
\(705\) 9.25393 0.348523
\(706\) 30.5143 1.14842
\(707\) −4.59027 −0.172635
\(708\) 7.15262 0.268812
\(709\) 19.0207 0.714337 0.357168 0.934040i \(-0.383742\pi\)
0.357168 + 0.934040i \(0.383742\pi\)
\(710\) 25.0810 0.941274
\(711\) 8.43720 0.316420
\(712\) 15.2049 0.569829
\(713\) 0 0
\(714\) 7.60961 0.284782
\(715\) 57.4228 2.14749
\(716\) −11.5753 −0.432588
\(717\) 9.82751 0.367015
\(718\) 16.4977 0.615687
\(719\) −11.0635 −0.412600 −0.206300 0.978489i \(-0.566142\pi\)
−0.206300 + 0.978489i \(0.566142\pi\)
\(720\) −3.88612 −0.144827
\(721\) 14.0093 0.521732
\(722\) −17.1350 −0.637699
\(723\) −21.6867 −0.806538
\(724\) −2.61369 −0.0971370
\(725\) 5.66834 0.210517
\(726\) −9.50505 −0.352766
\(727\) −5.79278 −0.214843 −0.107421 0.994214i \(-0.534259\pi\)
−0.107421 + 0.994214i \(0.534259\pi\)
\(728\) −13.5793 −0.503282
\(729\) 1.00000 0.0370370
\(730\) −48.0063 −1.77679
\(731\) −18.0163 −0.666356
\(732\) 3.25684 0.120376
\(733\) −14.8050 −0.546836 −0.273418 0.961895i \(-0.588154\pi\)
−0.273418 + 0.961895i \(0.588154\pi\)
\(734\) −27.6912 −1.02210
\(735\) 40.0941 1.47889
\(736\) 0 0
\(737\) −28.4719 −1.04878
\(738\) −6.11749 −0.225188
\(739\) 3.68448 0.135536 0.0677679 0.997701i \(-0.478412\pi\)
0.0677679 + 0.997701i \(0.478412\pi\)
\(740\) −9.78874 −0.359841
\(741\) 4.45630 0.163706
\(742\) 44.1533 1.62092
\(743\) −40.5510 −1.48767 −0.743835 0.668363i \(-0.766995\pi\)
−0.743835 + 0.668363i \(0.766995\pi\)
\(744\) 4.63657 0.169985
\(745\) 8.08678 0.296277
\(746\) −20.3626 −0.745526
\(747\) −0.377030 −0.0137948
\(748\) −8.28044 −0.302763
\(749\) −43.6879 −1.59632
\(750\) 19.8268 0.723972
\(751\) 16.0575 0.585947 0.292974 0.956120i \(-0.405355\pi\)
0.292974 + 0.956120i \(0.405355\pi\)
\(752\) 2.38128 0.0868362
\(753\) 7.45090 0.271526
\(754\) −1.83100 −0.0666811
\(755\) −72.9363 −2.65442
\(756\) −4.16140 −0.151349
\(757\) −46.6087 −1.69402 −0.847011 0.531575i \(-0.821600\pi\)
−0.847011 + 0.531575i \(0.821600\pi\)
\(758\) −15.3289 −0.556769
\(759\) 0 0
\(760\) −5.30706 −0.192507
\(761\) −24.2888 −0.880467 −0.440234 0.897883i \(-0.645104\pi\)
−0.440234 + 0.897883i \(0.645104\pi\)
\(762\) −10.0980 −0.365811
\(763\) −5.31277 −0.192335
\(764\) 6.15128 0.222546
\(765\) 7.10623 0.256926
\(766\) −20.7059 −0.748135
\(767\) 23.3401 0.842762
\(768\) −1.00000 −0.0360844
\(769\) 18.4202 0.664250 0.332125 0.943235i \(-0.392234\pi\)
0.332125 + 0.943235i \(0.392234\pi\)
\(770\) −73.2295 −2.63901
\(771\) −4.10261 −0.147752
\(772\) 8.53015 0.307007
\(773\) 0.914757 0.0329015 0.0164508 0.999865i \(-0.494763\pi\)
0.0164508 + 0.999865i \(0.494763\pi\)
\(774\) 9.85240 0.354137
\(775\) −46.8384 −1.68248
\(776\) 18.8759 0.677605
\(777\) −10.4821 −0.376044
\(778\) −34.4334 −1.23450
\(779\) −8.35432 −0.299324
\(780\) −12.6810 −0.454053
\(781\) −29.2253 −1.04576
\(782\) 0 0
\(783\) −0.561114 −0.0200526
\(784\) 10.3172 0.368473
\(785\) 7.75330 0.276727
\(786\) −7.51893 −0.268191
\(787\) −23.8906 −0.851608 −0.425804 0.904815i \(-0.640009\pi\)
−0.425804 + 0.904815i \(0.640009\pi\)
\(788\) 3.29008 0.117204
\(789\) 26.8046 0.954268
\(790\) −32.7880 −1.16654
\(791\) 73.6204 2.61764
\(792\) 4.52825 0.160904
\(793\) 10.6276 0.377395
\(794\) −26.4490 −0.938638
\(795\) 41.2326 1.46237
\(796\) 20.0262 0.709810
\(797\) −32.1264 −1.13798 −0.568988 0.822346i \(-0.692665\pi\)
−0.568988 + 0.822346i \(0.692665\pi\)
\(798\) −5.68299 −0.201176
\(799\) −4.35444 −0.154049
\(800\) 10.1019 0.357158
\(801\) 15.2049 0.537240
\(802\) −17.8995 −0.632052
\(803\) 55.9387 1.97403
\(804\) 6.28762 0.221747
\(805\) 0 0
\(806\) 15.1298 0.532926
\(807\) 5.74299 0.202163
\(808\) −1.10306 −0.0388055
\(809\) −52.3359 −1.84003 −0.920017 0.391879i \(-0.871825\pi\)
−0.920017 + 0.391879i \(0.871825\pi\)
\(810\) −3.88612 −0.136544
\(811\) −46.4728 −1.63188 −0.815941 0.578135i \(-0.803781\pi\)
−0.815941 + 0.578135i \(0.803781\pi\)
\(812\) 2.33502 0.0819431
\(813\) −6.93596 −0.243255
\(814\) 11.4062 0.399787
\(815\) −2.71628 −0.0951472
\(816\) 1.82862 0.0640144
\(817\) 13.4549 0.470726
\(818\) −9.47776 −0.331382
\(819\) −13.5793 −0.474498
\(820\) 23.7733 0.830201
\(821\) −16.0165 −0.558981 −0.279491 0.960148i \(-0.590166\pi\)
−0.279491 + 0.960148i \(0.590166\pi\)
\(822\) 14.3005 0.498788
\(823\) 48.9112 1.70494 0.852469 0.522778i \(-0.175104\pi\)
0.852469 + 0.522778i \(0.175104\pi\)
\(824\) 3.36648 0.117277
\(825\) −45.7442 −1.59261
\(826\) −29.7649 −1.03565
\(827\) 10.4217 0.362400 0.181200 0.983446i \(-0.442002\pi\)
0.181200 + 0.983446i \(0.442002\pi\)
\(828\) 0 0
\(829\) −0.753780 −0.0261799 −0.0130899 0.999914i \(-0.504167\pi\)
−0.0130899 + 0.999914i \(0.504167\pi\)
\(830\) 1.46519 0.0508574
\(831\) 3.81024 0.132176
\(832\) −3.26315 −0.113129
\(833\) −18.8663 −0.653678
\(834\) −21.8003 −0.754882
\(835\) −19.5408 −0.676239
\(836\) 6.18398 0.213877
\(837\) 4.63657 0.160263
\(838\) −22.4130 −0.774246
\(839\) 9.19175 0.317335 0.158667 0.987332i \(-0.449280\pi\)
0.158667 + 0.987332i \(0.449280\pi\)
\(840\) 16.1717 0.557977
\(841\) −28.6852 −0.989143
\(842\) −1.53019 −0.0527339
\(843\) −18.1727 −0.625902
\(844\) 2.62325 0.0902959
\(845\) 9.13951 0.314409
\(846\) 2.38128 0.0818699
\(847\) 39.5543 1.35910
\(848\) 10.6102 0.364356
\(849\) 17.0969 0.586765
\(850\) −18.4726 −0.633605
\(851\) 0 0
\(852\) 6.45399 0.221110
\(853\) −37.5833 −1.28683 −0.643414 0.765518i \(-0.722483\pi\)
−0.643414 + 0.765518i \(0.722483\pi\)
\(854\) −13.5530 −0.463774
\(855\) −5.30706 −0.181498
\(856\) −10.4984 −0.358827
\(857\) 39.2383 1.34036 0.670178 0.742200i \(-0.266218\pi\)
0.670178 + 0.742200i \(0.266218\pi\)
\(858\) 14.7764 0.504457
\(859\) 25.8456 0.881842 0.440921 0.897546i \(-0.354652\pi\)
0.440921 + 0.897546i \(0.354652\pi\)
\(860\) −38.2876 −1.30560
\(861\) 25.4573 0.867583
\(862\) 0.121649 0.00414337
\(863\) −54.5197 −1.85587 −0.927936 0.372740i \(-0.878419\pi\)
−0.927936 + 0.372740i \(0.878419\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 33.7416 1.14725
\(866\) 34.9931 1.18911
\(867\) 13.6562 0.463787
\(868\) −19.2946 −0.654902
\(869\) 38.2057 1.29604
\(870\) 2.18056 0.0739278
\(871\) 20.5175 0.695207
\(872\) −1.27668 −0.0432338
\(873\) 18.8759 0.638852
\(874\) 0 0
\(875\) −82.5072 −2.78925
\(876\) −12.3533 −0.417378
\(877\) −24.0009 −0.810451 −0.405226 0.914217i \(-0.632807\pi\)
−0.405226 + 0.914217i \(0.632807\pi\)
\(878\) −21.5673 −0.727862
\(879\) 13.0545 0.440316
\(880\) −17.5973 −0.593206
\(881\) 19.3576 0.652175 0.326088 0.945340i \(-0.394270\pi\)
0.326088 + 0.945340i \(0.394270\pi\)
\(882\) 10.3172 0.347399
\(883\) −37.7107 −1.26907 −0.634534 0.772895i \(-0.718808\pi\)
−0.634534 + 0.772895i \(0.718808\pi\)
\(884\) 5.96706 0.200694
\(885\) −27.7960 −0.934351
\(886\) −3.30287 −0.110962
\(887\) 5.07878 0.170529 0.0852643 0.996358i \(-0.472827\pi\)
0.0852643 + 0.996358i \(0.472827\pi\)
\(888\) −2.51890 −0.0845286
\(889\) 42.0217 1.40936
\(890\) −59.0883 −1.98064
\(891\) 4.52825 0.151702
\(892\) 3.40900 0.114142
\(893\) 3.25197 0.108823
\(894\) 2.08094 0.0695970
\(895\) 44.9829 1.50361
\(896\) 4.16140 0.139023
\(897\) 0 0
\(898\) 22.2893 0.743804
\(899\) −2.60164 −0.0867696
\(900\) 10.1019 0.336732
\(901\) −19.4020 −0.646375
\(902\) −27.7015 −0.922360
\(903\) −40.9997 −1.36439
\(904\) 17.6913 0.588403
\(905\) 10.1571 0.337634
\(906\) −18.7684 −0.623538
\(907\) −36.5683 −1.21423 −0.607115 0.794614i \(-0.707673\pi\)
−0.607115 + 0.794614i \(0.707673\pi\)
\(908\) 14.8065 0.491372
\(909\) −1.10306 −0.0365862
\(910\) 52.7707 1.74933
\(911\) 28.6820 0.950277 0.475139 0.879911i \(-0.342398\pi\)
0.475139 + 0.879911i \(0.342398\pi\)
\(912\) −1.36564 −0.0452210
\(913\) −1.70729 −0.0565030
\(914\) −36.1129 −1.19451
\(915\) −12.6565 −0.418410
\(916\) 9.79565 0.323657
\(917\) 31.2893 1.03326
\(918\) 1.82862 0.0603534
\(919\) −16.5221 −0.545013 −0.272507 0.962154i \(-0.587853\pi\)
−0.272507 + 0.962154i \(0.587853\pi\)
\(920\) 0 0
\(921\) 24.0856 0.793649
\(922\) 31.5662 1.03958
\(923\) 21.0604 0.693211
\(924\) −18.8439 −0.619917
\(925\) 25.4458 0.836652
\(926\) −27.8475 −0.915126
\(927\) 3.36648 0.110570
\(928\) 0.561114 0.0184195
\(929\) −3.55998 −0.116799 −0.0583996 0.998293i \(-0.518600\pi\)
−0.0583996 + 0.998293i \(0.518600\pi\)
\(930\) −18.0183 −0.590842
\(931\) 14.0897 0.461770
\(932\) −3.95532 −0.129561
\(933\) −2.64709 −0.0866618
\(934\) 0.589854 0.0193006
\(935\) 32.1788 1.05236
\(936\) −3.26315 −0.106659
\(937\) −50.5231 −1.65052 −0.825258 0.564756i \(-0.808970\pi\)
−0.825258 + 0.564756i \(0.808970\pi\)
\(938\) −26.1653 −0.854326
\(939\) 25.0150 0.816333
\(940\) −9.25393 −0.301830
\(941\) 60.0089 1.95623 0.978116 0.208059i \(-0.0667146\pi\)
0.978116 + 0.208059i \(0.0667146\pi\)
\(942\) 1.99513 0.0650047
\(943\) 0 0
\(944\) −7.15262 −0.232798
\(945\) 16.1717 0.526066
\(946\) 44.6141 1.45053
\(947\) 10.3111 0.335067 0.167533 0.985866i \(-0.446420\pi\)
0.167533 + 0.985866i \(0.446420\pi\)
\(948\) −8.43720 −0.274027
\(949\) −40.3106 −1.30854
\(950\) 13.7957 0.447590
\(951\) 2.94022 0.0953431
\(952\) −7.60961 −0.246629
\(953\) 42.2831 1.36968 0.684842 0.728692i \(-0.259871\pi\)
0.684842 + 0.728692i \(0.259871\pi\)
\(954\) 10.6102 0.343518
\(955\) −23.9046 −0.773536
\(956\) −9.82751 −0.317844
\(957\) −2.54086 −0.0821344
\(958\) −10.2856 −0.332312
\(959\) −59.5102 −1.92168
\(960\) 3.88612 0.125424
\(961\) −9.50223 −0.306524
\(962\) −8.21955 −0.265009
\(963\) −10.4984 −0.338305
\(964\) 21.6867 0.698483
\(965\) −33.1492 −1.06711
\(966\) 0 0
\(967\) −18.6393 −0.599399 −0.299700 0.954034i \(-0.596886\pi\)
−0.299700 + 0.954034i \(0.596886\pi\)
\(968\) 9.50505 0.305504
\(969\) 2.49724 0.0802229
\(970\) −73.3540 −2.35526
\(971\) 7.59445 0.243718 0.121859 0.992547i \(-0.461115\pi\)
0.121859 + 0.992547i \(0.461115\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 90.7196 2.90834
\(974\) −27.5042 −0.881291
\(975\) 32.9642 1.05570
\(976\) −3.25684 −0.104249
\(977\) 12.9508 0.414332 0.207166 0.978306i \(-0.433576\pi\)
0.207166 + 0.978306i \(0.433576\pi\)
\(978\) −0.698970 −0.0223506
\(979\) 68.8518 2.20051
\(980\) −40.0941 −1.28076
\(981\) −1.27668 −0.0407612
\(982\) 8.79100 0.280532
\(983\) −36.3780 −1.16028 −0.580139 0.814518i \(-0.697002\pi\)
−0.580139 + 0.814518i \(0.697002\pi\)
\(984\) 6.11749 0.195019
\(985\) −12.7857 −0.407385
\(986\) −1.02606 −0.0326765
\(987\) −9.90943 −0.315421
\(988\) −4.45630 −0.141774
\(989\) 0 0
\(990\) −17.5973 −0.559280
\(991\) −31.3165 −0.994801 −0.497400 0.867521i \(-0.665712\pi\)
−0.497400 + 0.867521i \(0.665712\pi\)
\(992\) −4.63657 −0.147211
\(993\) 1.21838 0.0386641
\(994\) −26.8576 −0.851873
\(995\) −77.8243 −2.46720
\(996\) 0.377030 0.0119467
\(997\) −19.4478 −0.615920 −0.307960 0.951399i \(-0.599646\pi\)
−0.307960 + 0.951399i \(0.599646\pi\)
\(998\) 23.5520 0.745526
\(999\) −2.51890 −0.0796944
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 3174.2.a.ba.1.1 5
3.2 odd 2 9522.2.a.bs.1.5 5
23.4 even 11 138.2.e.b.85.1 yes 10
23.6 even 11 138.2.e.b.13.1 10
23.22 odd 2 3174.2.a.bb.1.5 5
69.29 odd 22 414.2.i.e.289.1 10
69.50 odd 22 414.2.i.e.361.1 10
69.68 even 2 9522.2.a.br.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.b.13.1 10 23.6 even 11
138.2.e.b.85.1 yes 10 23.4 even 11
414.2.i.e.289.1 10 69.29 odd 22
414.2.i.e.361.1 10 69.50 odd 22
3174.2.a.ba.1.1 5 1.1 even 1 trivial
3174.2.a.bb.1.5 5 23.22 odd 2
9522.2.a.br.1.1 5 69.68 even 2
9522.2.a.bs.1.5 5 3.2 odd 2