Properties

Label 1183.2.a.r.1.8
Level $1183$
Weight $2$
Character 1183.1
Self dual yes
Analytic conductor $9.446$
Analytic rank $0$
Dimension $12$
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] = [1183,2,Mod(1,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, 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("1183.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.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: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 80 x^{8} - 246 x^{7} - 199 x^{6} + 562 x^{5} + 262 x^{4} - 542 x^{3} - 157 x^{2} + 183 x + 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.983820\) of defining polynomial
Character \(\chi\) \(=\) 1183.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.983820 q^{2} -1.57171 q^{3} -1.03210 q^{4} +0.398447 q^{5} -1.54628 q^{6} +1.00000 q^{7} -2.98304 q^{8} -0.529731 q^{9} +O(q^{10})\) \(q+0.983820 q^{2} -1.57171 q^{3} -1.03210 q^{4} +0.398447 q^{5} -1.54628 q^{6} +1.00000 q^{7} -2.98304 q^{8} -0.529731 q^{9} +0.392000 q^{10} +4.24206 q^{11} +1.62216 q^{12} +0.983820 q^{14} -0.626242 q^{15} -0.870575 q^{16} -5.10528 q^{17} -0.521160 q^{18} +2.12698 q^{19} -0.411236 q^{20} -1.57171 q^{21} +4.17342 q^{22} +2.19449 q^{23} +4.68847 q^{24} -4.84124 q^{25} +5.54771 q^{27} -1.03210 q^{28} +2.90902 q^{29} -0.616110 q^{30} +2.20466 q^{31} +5.10959 q^{32} -6.66728 q^{33} -5.02268 q^{34} +0.398447 q^{35} +0.546734 q^{36} +11.4227 q^{37} +2.09256 q^{38} -1.18858 q^{40} +5.07633 q^{41} -1.54628 q^{42} -0.328195 q^{43} -4.37822 q^{44} -0.211070 q^{45} +2.15899 q^{46} +6.62722 q^{47} +1.36829 q^{48} +1.00000 q^{49} -4.76291 q^{50} +8.02401 q^{51} +10.9423 q^{53} +5.45795 q^{54} +1.69024 q^{55} -2.98304 q^{56} -3.34299 q^{57} +2.86195 q^{58} -7.70620 q^{59} +0.646344 q^{60} +14.6039 q^{61} +2.16899 q^{62} -0.529731 q^{63} +6.76806 q^{64} -6.55941 q^{66} +1.22074 q^{67} +5.26915 q^{68} -3.44910 q^{69} +0.392000 q^{70} -1.75019 q^{71} +1.58021 q^{72} -6.11791 q^{73} +11.2379 q^{74} +7.60902 q^{75} -2.19525 q^{76} +4.24206 q^{77} +4.20871 q^{79} -0.346878 q^{80} -7.13019 q^{81} +4.99419 q^{82} +10.8154 q^{83} +1.62216 q^{84} -2.03418 q^{85} -0.322884 q^{86} -4.57213 q^{87} -12.6542 q^{88} -15.6973 q^{89} -0.207654 q^{90} -2.26493 q^{92} -3.46508 q^{93} +6.51999 q^{94} +0.847488 q^{95} -8.03079 q^{96} -14.4763 q^{97} +0.983820 q^{98} -2.24715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} + 8 q^{3} + 15 q^{4} - 4 q^{5} + 2 q^{6} + 12 q^{7} + 12 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} + 8 q^{3} + 15 q^{4} - 4 q^{5} + 2 q^{6} + 12 q^{7} + 12 q^{8} + 26 q^{9} - 6 q^{10} + 12 q^{11} + 13 q^{12} + 3 q^{14} + 11 q^{15} + 13 q^{16} + 31 q^{17} - 29 q^{18} - 3 q^{19} - 18 q^{20} + 8 q^{21} - 4 q^{22} + 18 q^{23} - 6 q^{24} + 32 q^{25} + 32 q^{27} + 15 q^{28} + 15 q^{29} - 10 q^{30} - 21 q^{31} - 3 q^{32} - 29 q^{33} + 3 q^{34} - 4 q^{35} + 49 q^{36} + 5 q^{37} + 45 q^{38} - 20 q^{40} - 16 q^{41} + 2 q^{42} - 22 q^{43} + 35 q^{44} + 5 q^{45} + 2 q^{46} - 4 q^{47} + 11 q^{48} + 12 q^{49} + 13 q^{50} + 18 q^{51} + 53 q^{53} - 5 q^{54} - 26 q^{55} + 12 q^{56} - 8 q^{57} + 32 q^{58} - 26 q^{59} + 38 q^{60} + 22 q^{61} + 19 q^{62} + 26 q^{63} + 2 q^{64} - 34 q^{66} + 12 q^{67} + 34 q^{68} + 3 q^{69} - 6 q^{70} + 21 q^{71} - 4 q^{72} - 15 q^{73} - 40 q^{74} + 15 q^{75} + 43 q^{76} + 12 q^{77} + 2 q^{79} + 13 q^{80} + 36 q^{81} - 32 q^{82} - 9 q^{83} + 13 q^{84} - 39 q^{85} + 44 q^{86} + 27 q^{87} - 48 q^{88} - 22 q^{89} - 26 q^{90} + 52 q^{92} - 53 q^{93} - 44 q^{94} + 29 q^{95} - 114 q^{96} + 9 q^{97} + 3 q^{98} + 37 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.983820 0.695666 0.347833 0.937557i \(-0.386918\pi\)
0.347833 + 0.937557i \(0.386918\pi\)
\(3\) −1.57171 −0.907427 −0.453713 0.891148i \(-0.649901\pi\)
−0.453713 + 0.891148i \(0.649901\pi\)
\(4\) −1.03210 −0.516049
\(5\) 0.398447 0.178191 0.0890954 0.996023i \(-0.471602\pi\)
0.0890954 + 0.996023i \(0.471602\pi\)
\(6\) −1.54628 −0.631265
\(7\) 1.00000 0.377964
\(8\) −2.98304 −1.05466
\(9\) −0.529731 −0.176577
\(10\) 0.392000 0.123961
\(11\) 4.24206 1.27903 0.639515 0.768779i \(-0.279135\pi\)
0.639515 + 0.768779i \(0.279135\pi\)
\(12\) 1.62216 0.468277
\(13\) 0 0
\(14\) 0.983820 0.262937
\(15\) −0.626242 −0.161695
\(16\) −0.870575 −0.217644
\(17\) −5.10528 −1.23821 −0.619106 0.785307i \(-0.712505\pi\)
−0.619106 + 0.785307i \(0.712505\pi\)
\(18\) −0.521160 −0.122838
\(19\) 2.12698 0.487962 0.243981 0.969780i \(-0.421546\pi\)
0.243981 + 0.969780i \(0.421546\pi\)
\(20\) −0.411236 −0.0919553
\(21\) −1.57171 −0.342975
\(22\) 4.17342 0.889777
\(23\) 2.19449 0.457583 0.228792 0.973475i \(-0.426523\pi\)
0.228792 + 0.973475i \(0.426523\pi\)
\(24\) 4.68847 0.957030
\(25\) −4.84124 −0.968248
\(26\) 0 0
\(27\) 5.54771 1.06766
\(28\) −1.03210 −0.195048
\(29\) 2.90902 0.540191 0.270095 0.962834i \(-0.412945\pi\)
0.270095 + 0.962834i \(0.412945\pi\)
\(30\) −0.616110 −0.112486
\(31\) 2.20466 0.395968 0.197984 0.980205i \(-0.436561\pi\)
0.197984 + 0.980205i \(0.436561\pi\)
\(32\) 5.10959 0.903256
\(33\) −6.66728 −1.16063
\(34\) −5.02268 −0.861382
\(35\) 0.398447 0.0673498
\(36\) 0.546734 0.0911224
\(37\) 11.4227 1.87788 0.938941 0.344077i \(-0.111808\pi\)
0.938941 + 0.344077i \(0.111808\pi\)
\(38\) 2.09256 0.339459
\(39\) 0 0
\(40\) −1.18858 −0.187931
\(41\) 5.07633 0.792790 0.396395 0.918080i \(-0.370261\pi\)
0.396395 + 0.918080i \(0.370261\pi\)
\(42\) −1.54628 −0.238596
\(43\) −0.328195 −0.0500492 −0.0250246 0.999687i \(-0.507966\pi\)
−0.0250246 + 0.999687i \(0.507966\pi\)
\(44\) −4.37822 −0.660042
\(45\) −0.211070 −0.0314644
\(46\) 2.15899 0.318325
\(47\) 6.62722 0.966679 0.483340 0.875433i \(-0.339424\pi\)
0.483340 + 0.875433i \(0.339424\pi\)
\(48\) 1.36829 0.197496
\(49\) 1.00000 0.142857
\(50\) −4.76291 −0.673577
\(51\) 8.02401 1.12359
\(52\) 0 0
\(53\) 10.9423 1.50304 0.751518 0.659712i \(-0.229322\pi\)
0.751518 + 0.659712i \(0.229322\pi\)
\(54\) 5.45795 0.742732
\(55\) 1.69024 0.227911
\(56\) −2.98304 −0.398625
\(57\) −3.34299 −0.442790
\(58\) 2.86195 0.375792
\(59\) −7.70620 −1.00326 −0.501631 0.865082i \(-0.667266\pi\)
−0.501631 + 0.865082i \(0.667266\pi\)
\(60\) 0.646344 0.0834427
\(61\) 14.6039 1.86984 0.934921 0.354857i \(-0.115470\pi\)
0.934921 + 0.354857i \(0.115470\pi\)
\(62\) 2.16899 0.275462
\(63\) −0.529731 −0.0667398
\(64\) 6.76806 0.846008
\(65\) 0 0
\(66\) −6.55941 −0.807407
\(67\) 1.22074 0.149137 0.0745685 0.997216i \(-0.476242\pi\)
0.0745685 + 0.997216i \(0.476242\pi\)
\(68\) 5.26915 0.638979
\(69\) −3.44910 −0.415223
\(70\) 0.392000 0.0468529
\(71\) −1.75019 −0.207709 −0.103854 0.994593i \(-0.533118\pi\)
−0.103854 + 0.994593i \(0.533118\pi\)
\(72\) 1.58021 0.186229
\(73\) −6.11791 −0.716047 −0.358023 0.933713i \(-0.616549\pi\)
−0.358023 + 0.933713i \(0.616549\pi\)
\(74\) 11.2379 1.30638
\(75\) 7.60902 0.878614
\(76\) −2.19525 −0.251813
\(77\) 4.24206 0.483428
\(78\) 0 0
\(79\) 4.20871 0.473517 0.236759 0.971569i \(-0.423915\pi\)
0.236759 + 0.971569i \(0.423915\pi\)
\(80\) −0.346878 −0.0387821
\(81\) −7.13019 −0.792244
\(82\) 4.99419 0.551517
\(83\) 10.8154 1.18714 0.593571 0.804782i \(-0.297718\pi\)
0.593571 + 0.804782i \(0.297718\pi\)
\(84\) 1.62216 0.176992
\(85\) −2.03418 −0.220638
\(86\) −0.322884 −0.0348175
\(87\) −4.57213 −0.490183
\(88\) −12.6542 −1.34895
\(89\) −15.6973 −1.66392 −0.831958 0.554839i \(-0.812780\pi\)
−0.831958 + 0.554839i \(0.812780\pi\)
\(90\) −0.207654 −0.0218887
\(91\) 0 0
\(92\) −2.26493 −0.236136
\(93\) −3.46508 −0.359312
\(94\) 6.51999 0.672485
\(95\) 0.847488 0.0869504
\(96\) −8.03079 −0.819639
\(97\) −14.4763 −1.46985 −0.734924 0.678149i \(-0.762782\pi\)
−0.734924 + 0.678149i \(0.762782\pi\)
\(98\) 0.983820 0.0993808
\(99\) −2.24715 −0.225847
\(100\) 4.99664 0.499664
\(101\) 7.31830 0.728198 0.364099 0.931360i \(-0.381377\pi\)
0.364099 + 0.931360i \(0.381377\pi\)
\(102\) 7.89418 0.781641
\(103\) −12.7643 −1.25770 −0.628851 0.777526i \(-0.716474\pi\)
−0.628851 + 0.777526i \(0.716474\pi\)
\(104\) 0 0
\(105\) −0.626242 −0.0611150
\(106\) 10.7652 1.04561
\(107\) 9.06986 0.876816 0.438408 0.898776i \(-0.355542\pi\)
0.438408 + 0.898776i \(0.355542\pi\)
\(108\) −5.72578 −0.550964
\(109\) −4.95091 −0.474211 −0.237106 0.971484i \(-0.576199\pi\)
−0.237106 + 0.971484i \(0.576199\pi\)
\(110\) 1.66289 0.158550
\(111\) −17.9532 −1.70404
\(112\) −0.870575 −0.0822616
\(113\) 2.39270 0.225086 0.112543 0.993647i \(-0.464100\pi\)
0.112543 + 0.993647i \(0.464100\pi\)
\(114\) −3.28890 −0.308034
\(115\) 0.874389 0.0815372
\(116\) −3.00239 −0.278765
\(117\) 0 0
\(118\) −7.58151 −0.697935
\(119\) −5.10528 −0.468000
\(120\) 1.86811 0.170534
\(121\) 6.99507 0.635915
\(122\) 14.3676 1.30078
\(123\) −7.97852 −0.719398
\(124\) −2.27543 −0.204339
\(125\) −3.92121 −0.350724
\(126\) −0.521160 −0.0464286
\(127\) 15.0587 1.33625 0.668123 0.744051i \(-0.267098\pi\)
0.668123 + 0.744051i \(0.267098\pi\)
\(128\) −3.56062 −0.314718
\(129\) 0.515826 0.0454160
\(130\) 0 0
\(131\) −11.6011 −1.01359 −0.506796 0.862066i \(-0.669170\pi\)
−0.506796 + 0.862066i \(0.669170\pi\)
\(132\) 6.88129 0.598940
\(133\) 2.12698 0.184432
\(134\) 1.20099 0.103750
\(135\) 2.21047 0.190247
\(136\) 15.2292 1.30590
\(137\) 0.410279 0.0350525 0.0175263 0.999846i \(-0.494421\pi\)
0.0175263 + 0.999846i \(0.494421\pi\)
\(138\) −3.39330 −0.288857
\(139\) −18.1715 −1.54129 −0.770644 0.637266i \(-0.780065\pi\)
−0.770644 + 0.637266i \(0.780065\pi\)
\(140\) −0.411236 −0.0347558
\(141\) −10.4161 −0.877190
\(142\) −1.72187 −0.144496
\(143\) 0 0
\(144\) 0.461170 0.0384309
\(145\) 1.15909 0.0962570
\(146\) −6.01892 −0.498129
\(147\) −1.57171 −0.129632
\(148\) −11.7894 −0.969080
\(149\) 10.9195 0.894561 0.447280 0.894394i \(-0.352393\pi\)
0.447280 + 0.894394i \(0.352393\pi\)
\(150\) 7.48590 0.611222
\(151\) 20.3174 1.65341 0.826703 0.562639i \(-0.190214\pi\)
0.826703 + 0.562639i \(0.190214\pi\)
\(152\) −6.34486 −0.514636
\(153\) 2.70442 0.218640
\(154\) 4.17342 0.336304
\(155\) 0.878439 0.0705579
\(156\) 0 0
\(157\) 23.4474 1.87131 0.935653 0.352922i \(-0.114812\pi\)
0.935653 + 0.352922i \(0.114812\pi\)
\(158\) 4.14062 0.329410
\(159\) −17.1981 −1.36390
\(160\) 2.03590 0.160952
\(161\) 2.19449 0.172950
\(162\) −7.01482 −0.551137
\(163\) 12.7006 0.994791 0.497396 0.867524i \(-0.334290\pi\)
0.497396 + 0.867524i \(0.334290\pi\)
\(164\) −5.23927 −0.409119
\(165\) −2.65656 −0.206813
\(166\) 10.6404 0.825853
\(167\) −10.0996 −0.781531 −0.390765 0.920490i \(-0.627790\pi\)
−0.390765 + 0.920490i \(0.627790\pi\)
\(168\) 4.68847 0.361723
\(169\) 0 0
\(170\) −2.00127 −0.153490
\(171\) −1.12673 −0.0861629
\(172\) 0.338729 0.0258279
\(173\) 16.5810 1.26063 0.630316 0.776339i \(-0.282925\pi\)
0.630316 + 0.776339i \(0.282925\pi\)
\(174\) −4.49815 −0.341004
\(175\) −4.84124 −0.365963
\(176\) −3.69303 −0.278373
\(177\) 12.1119 0.910386
\(178\) −15.4434 −1.15753
\(179\) −23.5863 −1.76292 −0.881461 0.472257i \(-0.843439\pi\)
−0.881461 + 0.472257i \(0.843439\pi\)
\(180\) 0.217845 0.0162372
\(181\) −23.7219 −1.76324 −0.881619 0.471962i \(-0.843546\pi\)
−0.881619 + 0.471962i \(0.843546\pi\)
\(182\) 0 0
\(183\) −22.9531 −1.69674
\(184\) −6.54626 −0.482596
\(185\) 4.55134 0.334621
\(186\) −3.40902 −0.249961
\(187\) −21.6569 −1.58371
\(188\) −6.83994 −0.498854
\(189\) 5.54771 0.403537
\(190\) 0.833775 0.0604884
\(191\) 2.18467 0.158077 0.0790385 0.996872i \(-0.474815\pi\)
0.0790385 + 0.996872i \(0.474815\pi\)
\(192\) −10.6374 −0.767690
\(193\) 18.6395 1.34170 0.670851 0.741592i \(-0.265929\pi\)
0.670851 + 0.741592i \(0.265929\pi\)
\(194\) −14.2421 −1.02252
\(195\) 0 0
\(196\) −1.03210 −0.0737213
\(197\) −14.9820 −1.06743 −0.533713 0.845666i \(-0.679204\pi\)
−0.533713 + 0.845666i \(0.679204\pi\)
\(198\) −2.21079 −0.157114
\(199\) 8.58624 0.608662 0.304331 0.952566i \(-0.401567\pi\)
0.304331 + 0.952566i \(0.401567\pi\)
\(200\) 14.4416 1.02118
\(201\) −1.91865 −0.135331
\(202\) 7.19989 0.506583
\(203\) 2.90902 0.204173
\(204\) −8.28157 −0.579826
\(205\) 2.02265 0.141268
\(206\) −12.5578 −0.874940
\(207\) −1.16249 −0.0807987
\(208\) 0 0
\(209\) 9.02277 0.624118
\(210\) −0.616110 −0.0425156
\(211\) −28.3478 −1.95155 −0.975773 0.218785i \(-0.929791\pi\)
−0.975773 + 0.218785i \(0.929791\pi\)
\(212\) −11.2935 −0.775641
\(213\) 2.75078 0.188481
\(214\) 8.92310 0.609971
\(215\) −0.130768 −0.00891831
\(216\) −16.5490 −1.12602
\(217\) 2.20466 0.149662
\(218\) −4.87080 −0.329892
\(219\) 9.61557 0.649760
\(220\) −1.74449 −0.117613
\(221\) 0 0
\(222\) −17.6627 −1.18544
\(223\) −14.1295 −0.946181 −0.473091 0.881014i \(-0.656862\pi\)
−0.473091 + 0.881014i \(0.656862\pi\)
\(224\) 5.10959 0.341399
\(225\) 2.56455 0.170970
\(226\) 2.35399 0.156585
\(227\) −18.7325 −1.24332 −0.621661 0.783286i \(-0.713542\pi\)
−0.621661 + 0.783286i \(0.713542\pi\)
\(228\) 3.45030 0.228502
\(229\) −3.24595 −0.214499 −0.107249 0.994232i \(-0.534204\pi\)
−0.107249 + 0.994232i \(0.534204\pi\)
\(230\) 0.860241 0.0567226
\(231\) −6.66728 −0.438675
\(232\) −8.67771 −0.569719
\(233\) 17.4275 1.14171 0.570856 0.821050i \(-0.306612\pi\)
0.570856 + 0.821050i \(0.306612\pi\)
\(234\) 0 0
\(235\) 2.64059 0.172253
\(236\) 7.95356 0.517732
\(237\) −6.61487 −0.429682
\(238\) −5.02268 −0.325572
\(239\) −3.89803 −0.252142 −0.126071 0.992021i \(-0.540237\pi\)
−0.126071 + 0.992021i \(0.540237\pi\)
\(240\) 0.545191 0.0351919
\(241\) 17.6973 1.13998 0.569990 0.821652i \(-0.306947\pi\)
0.569990 + 0.821652i \(0.306947\pi\)
\(242\) 6.88189 0.442385
\(243\) −5.43654 −0.348754
\(244\) −15.0727 −0.964930
\(245\) 0.398447 0.0254558
\(246\) −7.84942 −0.500461
\(247\) 0 0
\(248\) −6.57658 −0.417613
\(249\) −16.9986 −1.07724
\(250\) −3.85776 −0.243986
\(251\) −1.70708 −0.107750 −0.0538750 0.998548i \(-0.517157\pi\)
−0.0538750 + 0.998548i \(0.517157\pi\)
\(252\) 0.546734 0.0344410
\(253\) 9.30917 0.585262
\(254\) 14.8151 0.929581
\(255\) 3.19714 0.200213
\(256\) −17.0391 −1.06495
\(257\) 23.2726 1.45171 0.725853 0.687849i \(-0.241445\pi\)
0.725853 + 0.687849i \(0.241445\pi\)
\(258\) 0.507480 0.0315943
\(259\) 11.4227 0.709773
\(260\) 0 0
\(261\) −1.54100 −0.0953852
\(262\) −11.4134 −0.705121
\(263\) 14.0596 0.866949 0.433475 0.901166i \(-0.357287\pi\)
0.433475 + 0.901166i \(0.357287\pi\)
\(264\) 19.8888 1.22407
\(265\) 4.35991 0.267827
\(266\) 2.09256 0.128303
\(267\) 24.6717 1.50988
\(268\) −1.25992 −0.0769621
\(269\) 3.44689 0.210161 0.105080 0.994464i \(-0.466490\pi\)
0.105080 + 0.994464i \(0.466490\pi\)
\(270\) 2.17470 0.132348
\(271\) −21.2277 −1.28949 −0.644744 0.764399i \(-0.723036\pi\)
−0.644744 + 0.764399i \(0.723036\pi\)
\(272\) 4.44453 0.269489
\(273\) 0 0
\(274\) 0.403641 0.0243848
\(275\) −20.5368 −1.23842
\(276\) 3.55982 0.214276
\(277\) 7.06655 0.424588 0.212294 0.977206i \(-0.431907\pi\)
0.212294 + 0.977206i \(0.431907\pi\)
\(278\) −17.8775 −1.07222
\(279\) −1.16788 −0.0699189
\(280\) −1.18858 −0.0710314
\(281\) 2.37834 0.141880 0.0709398 0.997481i \(-0.477400\pi\)
0.0709398 + 0.997481i \(0.477400\pi\)
\(282\) −10.2475 −0.610231
\(283\) −14.2529 −0.847248 −0.423624 0.905838i \(-0.639242\pi\)
−0.423624 + 0.905838i \(0.639242\pi\)
\(284\) 1.80636 0.107188
\(285\) −1.33200 −0.0789011
\(286\) 0 0
\(287\) 5.07633 0.299646
\(288\) −2.70671 −0.159494
\(289\) 9.06388 0.533169
\(290\) 1.14033 0.0669627
\(291\) 22.7526 1.33378
\(292\) 6.31428 0.369515
\(293\) −7.15874 −0.418218 −0.209109 0.977892i \(-0.567056\pi\)
−0.209109 + 0.977892i \(0.567056\pi\)
\(294\) −1.54628 −0.0901808
\(295\) −3.07051 −0.178772
\(296\) −34.0744 −1.98053
\(297\) 23.5337 1.36556
\(298\) 10.7428 0.622315
\(299\) 0 0
\(300\) −7.85326 −0.453408
\(301\) −0.328195 −0.0189168
\(302\) 19.9887 1.15022
\(303\) −11.5022 −0.660787
\(304\) −1.85169 −0.106202
\(305\) 5.81889 0.333189
\(306\) 2.66067 0.152100
\(307\) 24.4217 1.39382 0.696910 0.717159i \(-0.254558\pi\)
0.696910 + 0.717159i \(0.254558\pi\)
\(308\) −4.37822 −0.249472
\(309\) 20.0617 1.14127
\(310\) 0.864226 0.0490847
\(311\) 23.7212 1.34510 0.672552 0.740050i \(-0.265198\pi\)
0.672552 + 0.740050i \(0.265198\pi\)
\(312\) 0 0
\(313\) −15.0876 −0.852803 −0.426401 0.904534i \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(314\) 23.0680 1.30180
\(315\) −0.211070 −0.0118924
\(316\) −4.34381 −0.244358
\(317\) 31.5913 1.77434 0.887172 0.461440i \(-0.152667\pi\)
0.887172 + 0.461440i \(0.152667\pi\)
\(318\) −16.9198 −0.948815
\(319\) 12.3402 0.690920
\(320\) 2.69671 0.150751
\(321\) −14.2552 −0.795646
\(322\) 2.15899 0.120316
\(323\) −10.8588 −0.604201
\(324\) 7.35906 0.408837
\(325\) 0 0
\(326\) 12.4951 0.692042
\(327\) 7.78139 0.430312
\(328\) −15.1429 −0.836126
\(329\) 6.62722 0.365370
\(330\) −2.61357 −0.143873
\(331\) 19.8976 1.09367 0.546835 0.837240i \(-0.315832\pi\)
0.546835 + 0.837240i \(0.315832\pi\)
\(332\) −11.1625 −0.612623
\(333\) −6.05096 −0.331591
\(334\) −9.93619 −0.543684
\(335\) 0.486400 0.0265749
\(336\) 1.36829 0.0746464
\(337\) −2.71943 −0.148137 −0.0740685 0.997253i \(-0.523598\pi\)
−0.0740685 + 0.997253i \(0.523598\pi\)
\(338\) 0 0
\(339\) −3.76063 −0.204249
\(340\) 2.09948 0.113860
\(341\) 9.35229 0.506455
\(342\) −1.10850 −0.0599406
\(343\) 1.00000 0.0539949
\(344\) 0.979017 0.0527851
\(345\) −1.37428 −0.0739890
\(346\) 16.3127 0.876978
\(347\) −12.8135 −0.687865 −0.343932 0.938994i \(-0.611759\pi\)
−0.343932 + 0.938994i \(0.611759\pi\)
\(348\) 4.71889 0.252959
\(349\) −9.05354 −0.484625 −0.242312 0.970198i \(-0.577906\pi\)
−0.242312 + 0.970198i \(0.577906\pi\)
\(350\) −4.76291 −0.254588
\(351\) 0 0
\(352\) 21.6752 1.15529
\(353\) −9.43537 −0.502194 −0.251097 0.967962i \(-0.580791\pi\)
−0.251097 + 0.967962i \(0.580791\pi\)
\(354\) 11.9159 0.633324
\(355\) −0.697356 −0.0370118
\(356\) 16.2012 0.858662
\(357\) 8.02401 0.424676
\(358\) −23.2047 −1.22640
\(359\) 9.68118 0.510953 0.255477 0.966815i \(-0.417768\pi\)
0.255477 + 0.966815i \(0.417768\pi\)
\(360\) 0.629629 0.0331843
\(361\) −14.4760 −0.761893
\(362\) −23.3381 −1.22662
\(363\) −10.9942 −0.577047
\(364\) 0 0
\(365\) −2.43766 −0.127593
\(366\) −22.5817 −1.18037
\(367\) −15.3650 −0.802047 −0.401023 0.916068i \(-0.631345\pi\)
−0.401023 + 0.916068i \(0.631345\pi\)
\(368\) −1.91047 −0.0995901
\(369\) −2.68909 −0.139988
\(370\) 4.47770 0.232785
\(371\) 10.9423 0.568094
\(372\) 3.57631 0.185423
\(373\) 12.6878 0.656948 0.328474 0.944513i \(-0.393466\pi\)
0.328474 + 0.944513i \(0.393466\pi\)
\(374\) −21.3065 −1.10173
\(375\) 6.16300 0.318256
\(376\) −19.7692 −1.01952
\(377\) 0 0
\(378\) 5.45795 0.280726
\(379\) 6.59026 0.338519 0.169259 0.985572i \(-0.445862\pi\)
0.169259 + 0.985572i \(0.445862\pi\)
\(380\) −0.874691 −0.0448707
\(381\) −23.6679 −1.21255
\(382\) 2.14932 0.109969
\(383\) −31.3776 −1.60332 −0.801660 0.597781i \(-0.796049\pi\)
−0.801660 + 0.597781i \(0.796049\pi\)
\(384\) 5.59626 0.285583
\(385\) 1.69024 0.0861424
\(386\) 18.3379 0.933376
\(387\) 0.173855 0.00883754
\(388\) 14.9410 0.758514
\(389\) 16.4265 0.832857 0.416428 0.909169i \(-0.363282\pi\)
0.416428 + 0.909169i \(0.363282\pi\)
\(390\) 0 0
\(391\) −11.2035 −0.566585
\(392\) −2.98304 −0.150666
\(393\) 18.2335 0.919760
\(394\) −14.7396 −0.742572
\(395\) 1.67695 0.0843764
\(396\) 2.31928 0.116548
\(397\) −6.94037 −0.348327 −0.174163 0.984717i \(-0.555722\pi\)
−0.174163 + 0.984717i \(0.555722\pi\)
\(398\) 8.44731 0.423425
\(399\) −3.34299 −0.167359
\(400\) 4.21466 0.210733
\(401\) −0.890125 −0.0444507 −0.0222254 0.999753i \(-0.507075\pi\)
−0.0222254 + 0.999753i \(0.507075\pi\)
\(402\) −1.88760 −0.0941451
\(403\) 0 0
\(404\) −7.55321 −0.375786
\(405\) −2.84100 −0.141171
\(406\) 2.86195 0.142036
\(407\) 48.4558 2.40187
\(408\) −23.9359 −1.18501
\(409\) 2.00173 0.0989793 0.0494897 0.998775i \(-0.484241\pi\)
0.0494897 + 0.998775i \(0.484241\pi\)
\(410\) 1.98992 0.0982752
\(411\) −0.644840 −0.0318076
\(412\) 13.1740 0.649036
\(413\) −7.70620 −0.379197
\(414\) −1.14368 −0.0562088
\(415\) 4.30935 0.211538
\(416\) 0 0
\(417\) 28.5603 1.39861
\(418\) 8.87678 0.434178
\(419\) 32.3340 1.57962 0.789809 0.613353i \(-0.210180\pi\)
0.789809 + 0.613353i \(0.210180\pi\)
\(420\) 0.646344 0.0315384
\(421\) 9.10353 0.443679 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(422\) −27.8892 −1.35762
\(423\) −3.51064 −0.170693
\(424\) −32.6412 −1.58520
\(425\) 24.7159 1.19890
\(426\) 2.70627 0.131119
\(427\) 14.6039 0.706734
\(428\) −9.36099 −0.452480
\(429\) 0 0
\(430\) −0.128652 −0.00620416
\(431\) −15.8655 −0.764216 −0.382108 0.924118i \(-0.624802\pi\)
−0.382108 + 0.924118i \(0.624802\pi\)
\(432\) −4.82970 −0.232369
\(433\) 15.8736 0.762835 0.381418 0.924403i \(-0.375436\pi\)
0.381418 + 0.924403i \(0.375436\pi\)
\(434\) 2.16899 0.104115
\(435\) −1.82175 −0.0873462
\(436\) 5.10983 0.244716
\(437\) 4.66764 0.223283
\(438\) 9.45999 0.452016
\(439\) 5.37687 0.256624 0.128312 0.991734i \(-0.459044\pi\)
0.128312 + 0.991734i \(0.459044\pi\)
\(440\) −5.04204 −0.240370
\(441\) −0.529731 −0.0252253
\(442\) 0 0
\(443\) −25.4188 −1.20768 −0.603841 0.797105i \(-0.706364\pi\)
−0.603841 + 0.797105i \(0.706364\pi\)
\(444\) 18.5295 0.879369
\(445\) −6.25456 −0.296494
\(446\) −13.9009 −0.658226
\(447\) −17.1623 −0.811748
\(448\) 6.76806 0.319761
\(449\) −15.4369 −0.728512 −0.364256 0.931299i \(-0.618677\pi\)
−0.364256 + 0.931299i \(0.618677\pi\)
\(450\) 2.52306 0.118938
\(451\) 21.5341 1.01400
\(452\) −2.46950 −0.116156
\(453\) −31.9330 −1.50034
\(454\) −18.4294 −0.864936
\(455\) 0 0
\(456\) 9.97227 0.466995
\(457\) 30.2784 1.41636 0.708181 0.706031i \(-0.249516\pi\)
0.708181 + 0.706031i \(0.249516\pi\)
\(458\) −3.19343 −0.149219
\(459\) −28.3226 −1.32199
\(460\) −0.902455 −0.0420772
\(461\) 25.3431 1.18035 0.590173 0.807276i \(-0.299059\pi\)
0.590173 + 0.807276i \(0.299059\pi\)
\(462\) −6.55941 −0.305171
\(463\) −2.93975 −0.136622 −0.0683109 0.997664i \(-0.521761\pi\)
−0.0683109 + 0.997664i \(0.521761\pi\)
\(464\) −2.53252 −0.117569
\(465\) −1.38065 −0.0640262
\(466\) 17.1455 0.794250
\(467\) 24.0967 1.11506 0.557531 0.830156i \(-0.311749\pi\)
0.557531 + 0.830156i \(0.311749\pi\)
\(468\) 0 0
\(469\) 1.22074 0.0563685
\(470\) 2.59787 0.119831
\(471\) −36.8525 −1.69807
\(472\) 22.9879 1.05810
\(473\) −1.39222 −0.0640144
\(474\) −6.50784 −0.298915
\(475\) −10.2972 −0.472469
\(476\) 5.26915 0.241511
\(477\) −5.79646 −0.265402
\(478\) −3.83495 −0.175407
\(479\) 11.1069 0.507488 0.253744 0.967271i \(-0.418338\pi\)
0.253744 + 0.967271i \(0.418338\pi\)
\(480\) −3.19984 −0.146052
\(481\) 0 0
\(482\) 17.4109 0.793045
\(483\) −3.44910 −0.156940
\(484\) −7.21960 −0.328164
\(485\) −5.76805 −0.261913
\(486\) −5.34858 −0.242616
\(487\) −29.1520 −1.32100 −0.660501 0.750825i \(-0.729656\pi\)
−0.660501 + 0.750825i \(0.729656\pi\)
\(488\) −43.5641 −1.97205
\(489\) −19.9617 −0.902700
\(490\) 0.392000 0.0177087
\(491\) 0.374507 0.0169013 0.00845063 0.999964i \(-0.497310\pi\)
0.00845063 + 0.999964i \(0.497310\pi\)
\(492\) 8.23462 0.371245
\(493\) −14.8513 −0.668871
\(494\) 0 0
\(495\) −0.895370 −0.0402439
\(496\) −1.91932 −0.0861800
\(497\) −1.75019 −0.0785066
\(498\) −16.7236 −0.749401
\(499\) −0.402490 −0.0180179 −0.00900896 0.999959i \(-0.502868\pi\)
−0.00900896 + 0.999959i \(0.502868\pi\)
\(500\) 4.04708 0.180991
\(501\) 15.8736 0.709182
\(502\) −1.67946 −0.0749580
\(503\) 14.1098 0.629124 0.314562 0.949237i \(-0.398142\pi\)
0.314562 + 0.949237i \(0.398142\pi\)
\(504\) 1.58021 0.0703880
\(505\) 2.91596 0.129758
\(506\) 9.15854 0.407147
\(507\) 0 0
\(508\) −15.5421 −0.689569
\(509\) 37.0510 1.64225 0.821127 0.570745i \(-0.193345\pi\)
0.821127 + 0.570745i \(0.193345\pi\)
\(510\) 3.14541 0.139281
\(511\) −6.11791 −0.270640
\(512\) −9.64220 −0.426129
\(513\) 11.7999 0.520977
\(514\) 22.8961 1.00990
\(515\) −5.08589 −0.224111
\(516\) −0.532384 −0.0234369
\(517\) 28.1131 1.23641
\(518\) 11.2379 0.493765
\(519\) −26.0605 −1.14393
\(520\) 0 0
\(521\) 28.5515 1.25086 0.625432 0.780279i \(-0.284923\pi\)
0.625432 + 0.780279i \(0.284923\pi\)
\(522\) −1.51606 −0.0663562
\(523\) −7.85498 −0.343474 −0.171737 0.985143i \(-0.554938\pi\)
−0.171737 + 0.985143i \(0.554938\pi\)
\(524\) 11.9735 0.523063
\(525\) 7.60902 0.332085
\(526\) 13.8321 0.603107
\(527\) −11.2554 −0.490293
\(528\) 5.80437 0.252603
\(529\) −18.1842 −0.790617
\(530\) 4.28937 0.186318
\(531\) 4.08221 0.177153
\(532\) −2.19525 −0.0951763
\(533\) 0 0
\(534\) 24.2725 1.05037
\(535\) 3.61386 0.156241
\(536\) −3.64151 −0.157289
\(537\) 37.0708 1.59972
\(538\) 3.39112 0.146202
\(539\) 4.24206 0.182718
\(540\) −2.28142 −0.0981767
\(541\) −36.0835 −1.55135 −0.775676 0.631131i \(-0.782591\pi\)
−0.775676 + 0.631131i \(0.782591\pi\)
\(542\) −20.8842 −0.897052
\(543\) 37.2840 1.60001
\(544\) −26.0859 −1.11842
\(545\) −1.97267 −0.0845001
\(546\) 0 0
\(547\) −30.0255 −1.28380 −0.641899 0.766789i \(-0.721853\pi\)
−0.641899 + 0.766789i \(0.721853\pi\)
\(548\) −0.423449 −0.0180888
\(549\) −7.73615 −0.330171
\(550\) −20.2045 −0.861524
\(551\) 6.18742 0.263593
\(552\) 10.2888 0.437921
\(553\) 4.20871 0.178973
\(554\) 6.95221 0.295371
\(555\) −7.15339 −0.303644
\(556\) 18.7548 0.795380
\(557\) −11.6919 −0.495400 −0.247700 0.968837i \(-0.579675\pi\)
−0.247700 + 0.968837i \(0.579675\pi\)
\(558\) −1.14898 −0.0486402
\(559\) 0 0
\(560\) −0.346878 −0.0146583
\(561\) 34.0383 1.43710
\(562\) 2.33985 0.0987008
\(563\) −24.4062 −1.02860 −0.514300 0.857610i \(-0.671948\pi\)
−0.514300 + 0.857610i \(0.671948\pi\)
\(564\) 10.7504 0.452674
\(565\) 0.953364 0.0401083
\(566\) −14.0223 −0.589401
\(567\) −7.13019 −0.299440
\(568\) 5.22087 0.219063
\(569\) 21.2322 0.890099 0.445050 0.895506i \(-0.353186\pi\)
0.445050 + 0.895506i \(0.353186\pi\)
\(570\) −1.31045 −0.0548888
\(571\) 29.0906 1.21741 0.608703 0.793398i \(-0.291690\pi\)
0.608703 + 0.793398i \(0.291690\pi\)
\(572\) 0 0
\(573\) −3.43366 −0.143443
\(574\) 4.99419 0.208454
\(575\) −10.6241 −0.443054
\(576\) −3.58525 −0.149385
\(577\) −25.1627 −1.04754 −0.523768 0.851861i \(-0.675474\pi\)
−0.523768 + 0.851861i \(0.675474\pi\)
\(578\) 8.91723 0.370908
\(579\) −29.2959 −1.21750
\(580\) −1.19629 −0.0496734
\(581\) 10.8154 0.448697
\(582\) 22.3844 0.927864
\(583\) 46.4178 1.92243
\(584\) 18.2499 0.755188
\(585\) 0 0
\(586\) −7.04291 −0.290940
\(587\) 21.7327 0.897006 0.448503 0.893781i \(-0.351957\pi\)
0.448503 + 0.893781i \(0.351957\pi\)
\(588\) 1.62216 0.0668967
\(589\) 4.68926 0.193218
\(590\) −3.02083 −0.124366
\(591\) 23.5474 0.968611
\(592\) −9.94433 −0.408709
\(593\) −30.7678 −1.26348 −0.631741 0.775180i \(-0.717659\pi\)
−0.631741 + 0.775180i \(0.717659\pi\)
\(594\) 23.1529 0.949976
\(595\) −2.03418 −0.0833934
\(596\) −11.2700 −0.461637
\(597\) −13.4951 −0.552316
\(598\) 0 0
\(599\) −17.7791 −0.726436 −0.363218 0.931704i \(-0.618322\pi\)
−0.363218 + 0.931704i \(0.618322\pi\)
\(600\) −22.6980 −0.926642
\(601\) −7.17744 −0.292774 −0.146387 0.989227i \(-0.546764\pi\)
−0.146387 + 0.989227i \(0.546764\pi\)
\(602\) −0.322884 −0.0131598
\(603\) −0.646663 −0.0263342
\(604\) −20.9696 −0.853239
\(605\) 2.78716 0.113314
\(606\) −11.3161 −0.459687
\(607\) 27.2937 1.10782 0.553908 0.832578i \(-0.313136\pi\)
0.553908 + 0.832578i \(0.313136\pi\)
\(608\) 10.8680 0.440755
\(609\) −4.57213 −0.185272
\(610\) 5.72474 0.231788
\(611\) 0 0
\(612\) −2.79123 −0.112829
\(613\) −19.4824 −0.786888 −0.393444 0.919349i \(-0.628716\pi\)
−0.393444 + 0.919349i \(0.628716\pi\)
\(614\) 24.0265 0.969632
\(615\) −3.17901 −0.128190
\(616\) −12.6542 −0.509853
\(617\) 7.14474 0.287636 0.143818 0.989604i \(-0.454062\pi\)
0.143818 + 0.989604i \(0.454062\pi\)
\(618\) 19.7371 0.793944
\(619\) −22.9703 −0.923255 −0.461628 0.887074i \(-0.652734\pi\)
−0.461628 + 0.887074i \(0.652734\pi\)
\(620\) −0.906636 −0.0364114
\(621\) 12.1744 0.488542
\(622\) 23.3374 0.935743
\(623\) −15.6973 −0.628901
\(624\) 0 0
\(625\) 22.6438 0.905752
\(626\) −14.8435 −0.593266
\(627\) −14.1812 −0.566341
\(628\) −24.2000 −0.965686
\(629\) −58.3161 −2.32522
\(630\) −0.207654 −0.00827315
\(631\) −30.1524 −1.20035 −0.600173 0.799870i \(-0.704902\pi\)
−0.600173 + 0.799870i \(0.704902\pi\)
\(632\) −12.5548 −0.499401
\(633\) 44.5546 1.77089
\(634\) 31.0801 1.23435
\(635\) 6.00010 0.238107
\(636\) 17.7501 0.703837
\(637\) 0 0
\(638\) 12.1406 0.480649
\(639\) 0.927127 0.0366766
\(640\) −1.41872 −0.0560798
\(641\) −40.2705 −1.59059 −0.795295 0.606223i \(-0.792684\pi\)
−0.795295 + 0.606223i \(0.792684\pi\)
\(642\) −14.0245 −0.553504
\(643\) −13.6321 −0.537598 −0.268799 0.963196i \(-0.586627\pi\)
−0.268799 + 0.963196i \(0.586627\pi\)
\(644\) −2.26493 −0.0892509
\(645\) 0.205529 0.00809271
\(646\) −10.6831 −0.420322
\(647\) −2.57274 −0.101145 −0.0505724 0.998720i \(-0.516105\pi\)
−0.0505724 + 0.998720i \(0.516105\pi\)
\(648\) 21.2696 0.835550
\(649\) −32.6902 −1.28320
\(650\) 0 0
\(651\) −3.46508 −0.135807
\(652\) −13.1083 −0.513361
\(653\) 38.5454 1.50840 0.754198 0.656647i \(-0.228026\pi\)
0.754198 + 0.656647i \(0.228026\pi\)
\(654\) 7.65548 0.299353
\(655\) −4.62241 −0.180613
\(656\) −4.41933 −0.172546
\(657\) 3.24084 0.126437
\(658\) 6.51999 0.254176
\(659\) 15.1716 0.591000 0.295500 0.955343i \(-0.404514\pi\)
0.295500 + 0.955343i \(0.404514\pi\)
\(660\) 2.74183 0.106726
\(661\) −3.88736 −0.151201 −0.0756003 0.997138i \(-0.524087\pi\)
−0.0756003 + 0.997138i \(0.524087\pi\)
\(662\) 19.5756 0.760829
\(663\) 0 0
\(664\) −32.2627 −1.25203
\(665\) 0.847488 0.0328642
\(666\) −5.95306 −0.230676
\(667\) 6.38381 0.247182
\(668\) 10.4238 0.403308
\(669\) 22.2075 0.858590
\(670\) 0.478530 0.0184872
\(671\) 61.9507 2.39158
\(672\) −8.03079 −0.309794
\(673\) −31.7515 −1.22393 −0.611965 0.790885i \(-0.709621\pi\)
−0.611965 + 0.790885i \(0.709621\pi\)
\(674\) −2.67543 −0.103054
\(675\) −26.8578 −1.03376
\(676\) 0 0
\(677\) −11.8303 −0.454677 −0.227338 0.973816i \(-0.573002\pi\)
−0.227338 + 0.973816i \(0.573002\pi\)
\(678\) −3.69978 −0.142089
\(679\) −14.4763 −0.555550
\(680\) 6.06804 0.232699
\(681\) 29.4421 1.12822
\(682\) 9.20097 0.352323
\(683\) −12.6451 −0.483852 −0.241926 0.970295i \(-0.577779\pi\)
−0.241926 + 0.970295i \(0.577779\pi\)
\(684\) 1.16289 0.0444643
\(685\) 0.163474 0.00624604
\(686\) 0.983820 0.0375624
\(687\) 5.10169 0.194642
\(688\) 0.285718 0.0108929
\(689\) 0 0
\(690\) −1.35205 −0.0514716
\(691\) 14.6468 0.557190 0.278595 0.960409i \(-0.410131\pi\)
0.278595 + 0.960409i \(0.410131\pi\)
\(692\) −17.1132 −0.650548
\(693\) −2.24715 −0.0853622
\(694\) −12.6062 −0.478524
\(695\) −7.24038 −0.274643
\(696\) 13.6388 0.516979
\(697\) −25.9161 −0.981642
\(698\) −8.90705 −0.337137
\(699\) −27.3909 −1.03602
\(700\) 4.99664 0.188855
\(701\) 7.16794 0.270729 0.135365 0.990796i \(-0.456779\pi\)
0.135365 + 0.990796i \(0.456779\pi\)
\(702\) 0 0
\(703\) 24.2959 0.916336
\(704\) 28.7105 1.08207
\(705\) −4.15025 −0.156307
\(706\) −9.28271 −0.349359
\(707\) 7.31830 0.275233
\(708\) −12.5007 −0.469804
\(709\) 2.14832 0.0806817 0.0403409 0.999186i \(-0.487156\pi\)
0.0403409 + 0.999186i \(0.487156\pi\)
\(710\) −0.686073 −0.0257478
\(711\) −2.22948 −0.0836122
\(712\) 46.8258 1.75487
\(713\) 4.83811 0.181189
\(714\) 7.89418 0.295432
\(715\) 0 0
\(716\) 24.3434 0.909755
\(717\) 6.12656 0.228801
\(718\) 9.52454 0.355453
\(719\) 9.34269 0.348424 0.174212 0.984708i \(-0.444262\pi\)
0.174212 + 0.984708i \(0.444262\pi\)
\(720\) 0.183752 0.00684803
\(721\) −12.7643 −0.475367
\(722\) −14.2417 −0.530023
\(723\) −27.8149 −1.03445
\(724\) 24.4834 0.909918
\(725\) −14.0832 −0.523039
\(726\) −10.8163 −0.401432
\(727\) 41.2539 1.53002 0.765011 0.644017i \(-0.222733\pi\)
0.765011 + 0.644017i \(0.222733\pi\)
\(728\) 0 0
\(729\) 29.9352 1.10871
\(730\) −2.39822 −0.0887620
\(731\) 1.67553 0.0619715
\(732\) 23.6899 0.875604
\(733\) 11.6062 0.428686 0.214343 0.976758i \(-0.431239\pi\)
0.214343 + 0.976758i \(0.431239\pi\)
\(734\) −15.1164 −0.557956
\(735\) −0.626242 −0.0230993
\(736\) 11.2130 0.413315
\(737\) 5.17845 0.190751
\(738\) −2.64558 −0.0973851
\(739\) −7.80551 −0.287130 −0.143565 0.989641i \(-0.545857\pi\)
−0.143565 + 0.989641i \(0.545857\pi\)
\(740\) −4.69744 −0.172681
\(741\) 0 0
\(742\) 10.7652 0.395204
\(743\) 41.5158 1.52307 0.761533 0.648126i \(-0.224447\pi\)
0.761533 + 0.648126i \(0.224447\pi\)
\(744\) 10.3365 0.378954
\(745\) 4.35084 0.159402
\(746\) 12.4825 0.457016
\(747\) −5.72924 −0.209622
\(748\) 22.3521 0.817272
\(749\) 9.06986 0.331405
\(750\) 6.06328 0.221400
\(751\) −22.7184 −0.829005 −0.414503 0.910048i \(-0.636044\pi\)
−0.414503 + 0.910048i \(0.636044\pi\)
\(752\) −5.76949 −0.210392
\(753\) 2.68304 0.0977753
\(754\) 0 0
\(755\) 8.09540 0.294622
\(756\) −5.72578 −0.208245
\(757\) −4.60782 −0.167474 −0.0837370 0.996488i \(-0.526686\pi\)
−0.0837370 + 0.996488i \(0.526686\pi\)
\(758\) 6.48362 0.235496
\(759\) −14.6313 −0.531083
\(760\) −2.52809 −0.0917034
\(761\) 21.2169 0.769112 0.384556 0.923102i \(-0.374355\pi\)
0.384556 + 0.923102i \(0.374355\pi\)
\(762\) −23.2850 −0.843526
\(763\) −4.95091 −0.179235
\(764\) −2.25479 −0.0815755
\(765\) 1.07757 0.0389596
\(766\) −30.8699 −1.11537
\(767\) 0 0
\(768\) 26.7806 0.966360
\(769\) −15.0781 −0.543731 −0.271865 0.962335i \(-0.587641\pi\)
−0.271865 + 0.962335i \(0.587641\pi\)
\(770\) 1.66289 0.0599263
\(771\) −36.5778 −1.31732
\(772\) −19.2378 −0.692385
\(773\) 4.39415 0.158047 0.0790233 0.996873i \(-0.474820\pi\)
0.0790233 + 0.996873i \(0.474820\pi\)
\(774\) 0.171042 0.00614797
\(775\) −10.6733 −0.383396
\(776\) 43.1834 1.55020
\(777\) −17.9532 −0.644067
\(778\) 16.1607 0.579390
\(779\) 10.7972 0.386852
\(780\) 0 0
\(781\) −7.42439 −0.265666
\(782\) −11.0222 −0.394154
\(783\) 16.1384 0.576739
\(784\) −0.870575 −0.0310920
\(785\) 9.34254 0.333449
\(786\) 17.9385 0.639845
\(787\) −28.1360 −1.00294 −0.501470 0.865175i \(-0.667207\pi\)
−0.501470 + 0.865175i \(0.667207\pi\)
\(788\) 15.4629 0.550845
\(789\) −22.0975 −0.786693
\(790\) 1.64982 0.0586978
\(791\) 2.39270 0.0850746
\(792\) 6.70333 0.238193
\(793\) 0 0
\(794\) −6.82807 −0.242319
\(795\) −6.85251 −0.243034
\(796\) −8.86185 −0.314100
\(797\) 14.7844 0.523692 0.261846 0.965110i \(-0.415669\pi\)
0.261846 + 0.965110i \(0.415669\pi\)
\(798\) −3.28890 −0.116426
\(799\) −33.8338 −1.19695
\(800\) −24.7367 −0.874576
\(801\) 8.31537 0.293809
\(802\) −0.875723 −0.0309229
\(803\) −25.9525 −0.915845
\(804\) 1.98023 0.0698375
\(805\) 0.874389 0.0308181
\(806\) 0 0
\(807\) −5.41751 −0.190705
\(808\) −21.8308 −0.768004
\(809\) 23.3097 0.819525 0.409763 0.912192i \(-0.365612\pi\)
0.409763 + 0.912192i \(0.365612\pi\)
\(810\) −2.79503 −0.0982075
\(811\) −13.6662 −0.479886 −0.239943 0.970787i \(-0.577129\pi\)
−0.239943 + 0.970787i \(0.577129\pi\)
\(812\) −3.00239 −0.105363
\(813\) 33.3637 1.17012
\(814\) 47.6718 1.67090
\(815\) 5.06053 0.177263
\(816\) −6.98550 −0.244542
\(817\) −0.698063 −0.0244221
\(818\) 1.96934 0.0688565
\(819\) 0 0
\(820\) −2.08757 −0.0729012
\(821\) 11.8254 0.412708 0.206354 0.978477i \(-0.433840\pi\)
0.206354 + 0.978477i \(0.433840\pi\)
\(822\) −0.634406 −0.0221274
\(823\) −12.6805 −0.442015 −0.221007 0.975272i \(-0.570935\pi\)
−0.221007 + 0.975272i \(0.570935\pi\)
\(824\) 38.0763 1.32645
\(825\) 32.2779 1.12377
\(826\) −7.58151 −0.263794
\(827\) 19.7897 0.688156 0.344078 0.938941i \(-0.388191\pi\)
0.344078 + 0.938941i \(0.388191\pi\)
\(828\) 1.19980 0.0416961
\(829\) −15.7027 −0.545378 −0.272689 0.962102i \(-0.587913\pi\)
−0.272689 + 0.962102i \(0.587913\pi\)
\(830\) 4.23962 0.147160
\(831\) −11.1066 −0.385282
\(832\) 0 0
\(833\) −5.10528 −0.176887
\(834\) 28.0982 0.972962
\(835\) −4.02416 −0.139262
\(836\) −9.31239 −0.322076
\(837\) 12.2308 0.422759
\(838\) 31.8108 1.09889
\(839\) 18.8485 0.650722 0.325361 0.945590i \(-0.394514\pi\)
0.325361 + 0.945590i \(0.394514\pi\)
\(840\) 1.86811 0.0644558
\(841\) −20.5376 −0.708194
\(842\) 8.95623 0.308652
\(843\) −3.73805 −0.128745
\(844\) 29.2578 1.00709
\(845\) 0 0
\(846\) −3.45384 −0.118745
\(847\) 6.99507 0.240353
\(848\) −9.52606 −0.327126
\(849\) 22.4014 0.768815
\(850\) 24.3160 0.834031
\(851\) 25.0671 0.859288
\(852\) −2.83908 −0.0972653
\(853\) −2.42144 −0.0829084 −0.0414542 0.999140i \(-0.513199\pi\)
−0.0414542 + 0.999140i \(0.513199\pi\)
\(854\) 14.3676 0.491650
\(855\) −0.448940 −0.0153534
\(856\) −27.0557 −0.924746
\(857\) 39.9800 1.36569 0.682845 0.730564i \(-0.260743\pi\)
0.682845 + 0.730564i \(0.260743\pi\)
\(858\) 0 0
\(859\) −2.31038 −0.0788292 −0.0394146 0.999223i \(-0.512549\pi\)
−0.0394146 + 0.999223i \(0.512549\pi\)
\(860\) 0.134966 0.00460229
\(861\) −7.97852 −0.271907
\(862\) −15.6088 −0.531639
\(863\) 46.1232 1.57005 0.785027 0.619462i \(-0.212649\pi\)
0.785027 + 0.619462i \(0.212649\pi\)
\(864\) 28.3465 0.964368
\(865\) 6.60665 0.224633
\(866\) 15.6167 0.530678
\(867\) −14.2458 −0.483812
\(868\) −2.27543 −0.0772330
\(869\) 17.8536 0.605642
\(870\) −1.79227 −0.0607637
\(871\) 0 0
\(872\) 14.7688 0.500133
\(873\) 7.66856 0.259541
\(874\) 4.59212 0.155331
\(875\) −3.92121 −0.132561
\(876\) −9.92421 −0.335308
\(877\) −22.7695 −0.768870 −0.384435 0.923152i \(-0.625604\pi\)
−0.384435 + 0.923152i \(0.625604\pi\)
\(878\) 5.28987 0.178524
\(879\) 11.2515 0.379502
\(880\) −1.47148 −0.0496035
\(881\) 20.0984 0.677131 0.338566 0.940943i \(-0.390058\pi\)
0.338566 + 0.940943i \(0.390058\pi\)
\(882\) −0.521160 −0.0175484
\(883\) −22.1650 −0.745911 −0.372956 0.927849i \(-0.621656\pi\)
−0.372956 + 0.927849i \(0.621656\pi\)
\(884\) 0 0
\(885\) 4.82595 0.162222
\(886\) −25.0075 −0.840143
\(887\) 6.35544 0.213395 0.106697 0.994292i \(-0.465972\pi\)
0.106697 + 0.994292i \(0.465972\pi\)
\(888\) 53.5550 1.79719
\(889\) 15.0587 0.505054
\(890\) −6.15336 −0.206261
\(891\) −30.2467 −1.01330
\(892\) 14.5830 0.488276
\(893\) 14.0960 0.471703
\(894\) −16.8846 −0.564705
\(895\) −9.39788 −0.314136
\(896\) −3.56062 −0.118952
\(897\) 0 0
\(898\) −15.1871 −0.506801
\(899\) 6.41339 0.213898
\(900\) −2.64687 −0.0882291
\(901\) −55.8633 −1.86108
\(902\) 21.1857 0.705406
\(903\) 0.515826 0.0171656
\(904\) −7.13752 −0.237390
\(905\) −9.45193 −0.314193
\(906\) −31.4163 −1.04374
\(907\) −29.7539 −0.987964 −0.493982 0.869472i \(-0.664459\pi\)
−0.493982 + 0.869472i \(0.664459\pi\)
\(908\) 19.3338 0.641615
\(909\) −3.87673 −0.128583
\(910\) 0 0
\(911\) −34.3649 −1.13856 −0.569279 0.822144i \(-0.692778\pi\)
−0.569279 + 0.822144i \(0.692778\pi\)
\(912\) 2.91032 0.0963705
\(913\) 45.8795 1.51839
\(914\) 29.7884 0.985314
\(915\) −9.14560 −0.302344
\(916\) 3.35014 0.110692
\(917\) −11.6011 −0.383101
\(918\) −27.8643 −0.919660
\(919\) −44.0300 −1.45241 −0.726207 0.687476i \(-0.758719\pi\)
−0.726207 + 0.687476i \(0.758719\pi\)
\(920\) −2.60833 −0.0859943
\(921\) −38.3838 −1.26479
\(922\) 24.9331 0.821127
\(923\) 0 0
\(924\) 6.88129 0.226378
\(925\) −55.3001 −1.81826
\(926\) −2.89218 −0.0950430
\(927\) 6.76163 0.222081
\(928\) 14.8639 0.487931
\(929\) −33.8451 −1.11042 −0.555210 0.831710i \(-0.687362\pi\)
−0.555210 + 0.831710i \(0.687362\pi\)
\(930\) −1.35831 −0.0445408
\(931\) 2.12698 0.0697089
\(932\) −17.9869 −0.589180
\(933\) −37.2828 −1.22058
\(934\) 23.7068 0.775710
\(935\) −8.62912 −0.282203
\(936\) 0 0
\(937\) 24.0910 0.787017 0.393509 0.919321i \(-0.371261\pi\)
0.393509 + 0.919321i \(0.371261\pi\)
\(938\) 1.20099 0.0392136
\(939\) 23.7134 0.773856
\(940\) −2.72535 −0.0888912
\(941\) 29.5369 0.962874 0.481437 0.876481i \(-0.340115\pi\)
0.481437 + 0.876481i \(0.340115\pi\)
\(942\) −36.2562 −1.18129
\(943\) 11.1400 0.362767
\(944\) 6.70882 0.218354
\(945\) 2.21047 0.0719065
\(946\) −1.36969 −0.0445326
\(947\) 22.6950 0.737490 0.368745 0.929531i \(-0.379788\pi\)
0.368745 + 0.929531i \(0.379788\pi\)
\(948\) 6.82720 0.221737
\(949\) 0 0
\(950\) −10.1306 −0.328680
\(951\) −49.6523 −1.61009
\(952\) 15.2292 0.493583
\(953\) 49.0720 1.58960 0.794799 0.606872i \(-0.207576\pi\)
0.794799 + 0.606872i \(0.207576\pi\)
\(954\) −5.70267 −0.184631
\(955\) 0.870474 0.0281679
\(956\) 4.02315 0.130118
\(957\) −19.3952 −0.626959
\(958\) 10.9272 0.353042
\(959\) 0.410279 0.0132486
\(960\) −4.23845 −0.136795
\(961\) −26.1395 −0.843209
\(962\) 0 0
\(963\) −4.80458 −0.154825
\(964\) −18.2653 −0.588286
\(965\) 7.42686 0.239079
\(966\) −3.39330 −0.109178
\(967\) 23.1018 0.742905 0.371452 0.928452i \(-0.378860\pi\)
0.371452 + 0.928452i \(0.378860\pi\)
\(968\) −20.8666 −0.670677
\(969\) 17.0669 0.548268
\(970\) −5.67472 −0.182204
\(971\) −30.3111 −0.972729 −0.486365 0.873756i \(-0.661677\pi\)
−0.486365 + 0.873756i \(0.661677\pi\)
\(972\) 5.61105 0.179974
\(973\) −18.1715 −0.582552
\(974\) −28.6803 −0.918976
\(975\) 0 0
\(976\) −12.7138 −0.406959
\(977\) −2.18246 −0.0698230 −0.0349115 0.999390i \(-0.511115\pi\)
−0.0349115 + 0.999390i \(0.511115\pi\)
\(978\) −19.6387 −0.627977
\(979\) −66.5891 −2.12820
\(980\) −0.411236 −0.0131365
\(981\) 2.62265 0.0837347
\(982\) 0.368447 0.0117576
\(983\) 2.60962 0.0832339 0.0416170 0.999134i \(-0.486749\pi\)
0.0416170 + 0.999134i \(0.486749\pi\)
\(984\) 23.8002 0.758723
\(985\) −5.96955 −0.190206
\(986\) −14.6110 −0.465310
\(987\) −10.4161 −0.331547
\(988\) 0 0
\(989\) −0.720221 −0.0229017
\(990\) −0.880882 −0.0279963
\(991\) 12.5621 0.399048 0.199524 0.979893i \(-0.436060\pi\)
0.199524 + 0.979893i \(0.436060\pi\)
\(992\) 11.2649 0.357661
\(993\) −31.2732 −0.992426
\(994\) −1.72187 −0.0546143
\(995\) 3.42116 0.108458
\(996\) 17.5443 0.555911
\(997\) −32.9839 −1.04461 −0.522305 0.852759i \(-0.674928\pi\)
−0.522305 + 0.852759i \(0.674928\pi\)
\(998\) −0.395978 −0.0125345
\(999\) 63.3699 2.00493
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 1183.2.a.r.1.8 yes 12
7.6 odd 2 8281.2.a.cq.1.8 12
13.5 odd 4 1183.2.c.j.337.9 24
13.8 odd 4 1183.2.c.j.337.16 24
13.12 even 2 1183.2.a.q.1.5 12
91.90 odd 2 8281.2.a.cn.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1183.2.a.q.1.5 12 13.12 even 2
1183.2.a.r.1.8 yes 12 1.1 even 1 trivial
1183.2.c.j.337.9 24 13.5 odd 4
1183.2.c.j.337.16 24 13.8 odd 4
8281.2.a.cn.1.5 12 91.90 odd 2
8281.2.a.cq.1.8 12 7.6 odd 2