Properties

Label 845.2.a.n.1.4
Level $845$
Weight $2$
Character 845.1
Self dual yes
Analytic conductor $6.747$
Analytic rank $0$
Dimension $9$
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] = [845,2,Mod(1,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, 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("845.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 17x^{7} - 9x^{6} + 59x^{5} + 32x^{4} - 44x^{3} - 23x^{2} + x + 1 \) 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 1.4
Root \(-0.271062\) of defining polynomial
Character \(\chi\) \(=\) 845.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.53088 q^{2} +2.88726 q^{3} +0.343581 q^{4} -1.00000 q^{5} -4.42003 q^{6} -3.86493 q^{7} +2.53577 q^{8} +5.33625 q^{9} +O(q^{10})\) \(q-1.53088 q^{2} +2.88726 q^{3} +0.343581 q^{4} -1.00000 q^{5} -4.42003 q^{6} -3.86493 q^{7} +2.53577 q^{8} +5.33625 q^{9} +1.53088 q^{10} +4.75078 q^{11} +0.992006 q^{12} +5.91673 q^{14} -2.88726 q^{15} -4.56911 q^{16} -0.640094 q^{17} -8.16914 q^{18} +4.70598 q^{19} -0.343581 q^{20} -11.1591 q^{21} -7.27285 q^{22} -0.308077 q^{23} +7.32143 q^{24} +1.00000 q^{25} +6.74536 q^{27} -1.32792 q^{28} +2.48779 q^{29} +4.42003 q^{30} -0.635724 q^{31} +1.92320 q^{32} +13.7167 q^{33} +0.979904 q^{34} +3.86493 q^{35} +1.83343 q^{36} +5.15099 q^{37} -7.20428 q^{38} -2.53577 q^{40} +10.4049 q^{41} +17.0831 q^{42} +8.56187 q^{43} +1.63228 q^{44} -5.33625 q^{45} +0.471628 q^{46} +2.89673 q^{47} -13.1922 q^{48} +7.93772 q^{49} -1.53088 q^{50} -1.84812 q^{51} +1.28060 q^{53} -10.3263 q^{54} -4.75078 q^{55} -9.80059 q^{56} +13.5874 q^{57} -3.80850 q^{58} -4.06969 q^{59} -0.992006 q^{60} +0.335109 q^{61} +0.973215 q^{62} -20.6243 q^{63} +6.19404 q^{64} -20.9986 q^{66} +0.721556 q^{67} -0.219924 q^{68} -0.889497 q^{69} -5.91673 q^{70} -10.3222 q^{71} +13.5315 q^{72} -13.1133 q^{73} -7.88552 q^{74} +2.88726 q^{75} +1.61689 q^{76} -18.3614 q^{77} +10.7026 q^{79} +4.56911 q^{80} +3.46683 q^{81} -15.9287 q^{82} +8.94652 q^{83} -3.83404 q^{84} +0.640094 q^{85} -13.1072 q^{86} +7.18290 q^{87} +12.0469 q^{88} +0.0141672 q^{89} +8.16914 q^{90} -0.105849 q^{92} -1.83550 q^{93} -4.43454 q^{94} -4.70598 q^{95} +5.55278 q^{96} +10.1257 q^{97} -12.1517 q^{98} +25.3514 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9} + 3 q^{10} + 9 q^{11} + 12 q^{12} - 2 q^{14} - 7 q^{15} + 37 q^{16} - q^{17} + 10 q^{18} + 4 q^{19} - 17 q^{20} + q^{21} + 12 q^{22} + 14 q^{23} + 35 q^{24} + 9 q^{25} + 22 q^{27} - 18 q^{28} + 12 q^{29} - 2 q^{30} + 7 q^{31} - 22 q^{32} + 8 q^{33} + 30 q^{34} + 7 q^{35} + 3 q^{36} + 5 q^{37} - 47 q^{38} + 12 q^{40} + 10 q^{41} - 11 q^{42} + 39 q^{43} + 25 q^{44} - 16 q^{45} - 6 q^{46} - 36 q^{47} - 3 q^{48} + 16 q^{49} - 3 q^{50} + 43 q^{51} - 8 q^{53} + 2 q^{54} - 9 q^{55} - 29 q^{56} + 32 q^{57} - 21 q^{58} + 21 q^{59} - 12 q^{60} - 3 q^{61} - 10 q^{62} - 35 q^{63} + 34 q^{64} - 49 q^{66} - q^{67} - 20 q^{68} - 13 q^{69} + 2 q^{70} + q^{71} + 3 q^{72} - 15 q^{74} + 7 q^{75} + 5 q^{76} - 4 q^{77} + 39 q^{79} - 37 q^{80} + 29 q^{81} - 4 q^{82} - 7 q^{83} - 12 q^{84} + q^{85} + 24 q^{86} + 16 q^{87} + 42 q^{88} + 19 q^{89} - 10 q^{90} - 27 q^{92} - 31 q^{93} + 16 q^{94} - 4 q^{95} + 7 q^{96} + 34 q^{97} - 48 q^{98} + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53088 −1.08249 −0.541246 0.840864i \(-0.682047\pi\)
−0.541246 + 0.840864i \(0.682047\pi\)
\(3\) 2.88726 1.66696 0.833479 0.552551i \(-0.186345\pi\)
0.833479 + 0.552551i \(0.186345\pi\)
\(4\) 0.343581 0.171790
\(5\) −1.00000 −0.447214
\(6\) −4.42003 −1.80447
\(7\) −3.86493 −1.46081 −0.730404 0.683015i \(-0.760668\pi\)
−0.730404 + 0.683015i \(0.760668\pi\)
\(8\) 2.53577 0.896531
\(9\) 5.33625 1.77875
\(10\) 1.53088 0.484105
\(11\) 4.75078 1.43241 0.716207 0.697888i \(-0.245877\pi\)
0.716207 + 0.697888i \(0.245877\pi\)
\(12\) 0.992006 0.286368
\(13\) 0 0
\(14\) 5.91673 1.58131
\(15\) −2.88726 −0.745487
\(16\) −4.56911 −1.14228
\(17\) −0.640094 −0.155246 −0.0776228 0.996983i \(-0.524733\pi\)
−0.0776228 + 0.996983i \(0.524733\pi\)
\(18\) −8.16914 −1.92548
\(19\) 4.70598 1.07963 0.539813 0.841785i \(-0.318495\pi\)
0.539813 + 0.841785i \(0.318495\pi\)
\(20\) −0.343581 −0.0768270
\(21\) −11.1591 −2.43511
\(22\) −7.27285 −1.55058
\(23\) −0.308077 −0.0642385 −0.0321192 0.999484i \(-0.510226\pi\)
−0.0321192 + 0.999484i \(0.510226\pi\)
\(24\) 7.32143 1.49448
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 6.74536 1.29815
\(28\) −1.32792 −0.250953
\(29\) 2.48779 0.461972 0.230986 0.972957i \(-0.425805\pi\)
0.230986 + 0.972957i \(0.425805\pi\)
\(30\) 4.42003 0.806984
\(31\) −0.635724 −0.114179 −0.0570897 0.998369i \(-0.518182\pi\)
−0.0570897 + 0.998369i \(0.518182\pi\)
\(32\) 1.92320 0.339977
\(33\) 13.7167 2.38777
\(34\) 0.979904 0.168052
\(35\) 3.86493 0.653293
\(36\) 1.83343 0.305572
\(37\) 5.15099 0.846817 0.423409 0.905939i \(-0.360834\pi\)
0.423409 + 0.905939i \(0.360834\pi\)
\(38\) −7.20428 −1.16869
\(39\) 0 0
\(40\) −2.53577 −0.400941
\(41\) 10.4049 1.62498 0.812488 0.582977i \(-0.198112\pi\)
0.812488 + 0.582977i \(0.198112\pi\)
\(42\) 17.0831 2.63598
\(43\) 8.56187 1.30567 0.652836 0.757499i \(-0.273579\pi\)
0.652836 + 0.757499i \(0.273579\pi\)
\(44\) 1.63228 0.246075
\(45\) −5.33625 −0.795482
\(46\) 0.471628 0.0695377
\(47\) 2.89673 0.422532 0.211266 0.977429i \(-0.432241\pi\)
0.211266 + 0.977429i \(0.432241\pi\)
\(48\) −13.1922 −1.90413
\(49\) 7.93772 1.13396
\(50\) −1.53088 −0.216499
\(51\) −1.84812 −0.258788
\(52\) 0 0
\(53\) 1.28060 0.175903 0.0879517 0.996125i \(-0.471968\pi\)
0.0879517 + 0.996125i \(0.471968\pi\)
\(54\) −10.3263 −1.40523
\(55\) −4.75078 −0.640595
\(56\) −9.80059 −1.30966
\(57\) 13.5874 1.79969
\(58\) −3.80850 −0.500081
\(59\) −4.06969 −0.529828 −0.264914 0.964272i \(-0.585344\pi\)
−0.264914 + 0.964272i \(0.585344\pi\)
\(60\) −0.992006 −0.128067
\(61\) 0.335109 0.0429063 0.0214531 0.999770i \(-0.493171\pi\)
0.0214531 + 0.999770i \(0.493171\pi\)
\(62\) 0.973215 0.123598
\(63\) −20.6243 −2.59841
\(64\) 6.19404 0.774255
\(65\) 0 0
\(66\) −20.9986 −2.58475
\(67\) 0.721556 0.0881521 0.0440761 0.999028i \(-0.485966\pi\)
0.0440761 + 0.999028i \(0.485966\pi\)
\(68\) −0.219924 −0.0266697
\(69\) −0.889497 −0.107083
\(70\) −5.91673 −0.707185
\(71\) −10.3222 −1.22502 −0.612510 0.790463i \(-0.709840\pi\)
−0.612510 + 0.790463i \(0.709840\pi\)
\(72\) 13.5315 1.59470
\(73\) −13.1133 −1.53480 −0.767400 0.641168i \(-0.778450\pi\)
−0.767400 + 0.641168i \(0.778450\pi\)
\(74\) −7.88552 −0.916673
\(75\) 2.88726 0.333392
\(76\) 1.61689 0.185470
\(77\) −18.3614 −2.09248
\(78\) 0 0
\(79\) 10.7026 1.20414 0.602069 0.798444i \(-0.294343\pi\)
0.602069 + 0.798444i \(0.294343\pi\)
\(80\) 4.56911 0.510842
\(81\) 3.46683 0.385204
\(82\) −15.9287 −1.75903
\(83\) 8.94652 0.982008 0.491004 0.871157i \(-0.336630\pi\)
0.491004 + 0.871157i \(0.336630\pi\)
\(84\) −3.83404 −0.418328
\(85\) 0.640094 0.0694279
\(86\) −13.1072 −1.41338
\(87\) 7.18290 0.770088
\(88\) 12.0469 1.28420
\(89\) 0.0141672 0.00150172 0.000750858 1.00000i \(-0.499761\pi\)
0.000750858 1.00000i \(0.499761\pi\)
\(90\) 8.16914 0.861103
\(91\) 0 0
\(92\) −0.105849 −0.0110356
\(93\) −1.83550 −0.190332
\(94\) −4.43454 −0.457388
\(95\) −4.70598 −0.482824
\(96\) 5.55278 0.566728
\(97\) 10.1257 1.02811 0.514054 0.857758i \(-0.328143\pi\)
0.514054 + 0.857758i \(0.328143\pi\)
\(98\) −12.1517 −1.22750
\(99\) 25.3514 2.54791
\(100\) 0.343581 0.0343581
\(101\) 12.4363 1.23746 0.618730 0.785603i \(-0.287647\pi\)
0.618730 + 0.785603i \(0.287647\pi\)
\(102\) 2.82924 0.280136
\(103\) 10.2372 1.00871 0.504353 0.863498i \(-0.331731\pi\)
0.504353 + 0.863498i \(0.331731\pi\)
\(104\) 0 0
\(105\) 11.1591 1.08901
\(106\) −1.96043 −0.190414
\(107\) −15.3401 −1.48298 −0.741490 0.670964i \(-0.765881\pi\)
−0.741490 + 0.670964i \(0.765881\pi\)
\(108\) 2.31758 0.223009
\(109\) −0.515675 −0.0493927 −0.0246963 0.999695i \(-0.507862\pi\)
−0.0246963 + 0.999695i \(0.507862\pi\)
\(110\) 7.27285 0.693439
\(111\) 14.8722 1.41161
\(112\) 17.6593 1.66865
\(113\) −17.0097 −1.60013 −0.800067 0.599911i \(-0.795203\pi\)
−0.800067 + 0.599911i \(0.795203\pi\)
\(114\) −20.8006 −1.94815
\(115\) 0.308077 0.0287283
\(116\) 0.854759 0.0793624
\(117\) 0 0
\(118\) 6.23019 0.573535
\(119\) 2.47392 0.226784
\(120\) −7.32143 −0.668352
\(121\) 11.5699 1.05181
\(122\) −0.513010 −0.0464457
\(123\) 30.0417 2.70877
\(124\) −0.218423 −0.0196149
\(125\) −1.00000 −0.0894427
\(126\) 31.5732 2.81276
\(127\) −21.7877 −1.93335 −0.966675 0.256007i \(-0.917593\pi\)
−0.966675 + 0.256007i \(0.917593\pi\)
\(128\) −13.3287 −1.17810
\(129\) 24.7203 2.17650
\(130\) 0 0
\(131\) −10.8833 −0.950874 −0.475437 0.879750i \(-0.657710\pi\)
−0.475437 + 0.879750i \(0.657710\pi\)
\(132\) 4.71280 0.410197
\(133\) −18.1883 −1.57713
\(134\) −1.10461 −0.0954240
\(135\) −6.74536 −0.580548
\(136\) −1.62313 −0.139182
\(137\) −1.25583 −0.107293 −0.0536466 0.998560i \(-0.517084\pi\)
−0.0536466 + 0.998560i \(0.517084\pi\)
\(138\) 1.36171 0.115916
\(139\) 18.2380 1.54693 0.773463 0.633842i \(-0.218523\pi\)
0.773463 + 0.633842i \(0.218523\pi\)
\(140\) 1.32792 0.112230
\(141\) 8.36361 0.704343
\(142\) 15.8020 1.32608
\(143\) 0 0
\(144\) −24.3819 −2.03183
\(145\) −2.48779 −0.206600
\(146\) 20.0749 1.66141
\(147\) 22.9182 1.89026
\(148\) 1.76978 0.145475
\(149\) −21.0429 −1.72390 −0.861952 0.506991i \(-0.830758\pi\)
−0.861952 + 0.506991i \(0.830758\pi\)
\(150\) −4.42003 −0.360894
\(151\) 16.2381 1.32144 0.660720 0.750632i \(-0.270251\pi\)
0.660720 + 0.750632i \(0.270251\pi\)
\(152\) 11.9333 0.967918
\(153\) −3.41570 −0.276143
\(154\) 28.1091 2.26510
\(155\) 0.635724 0.0510626
\(156\) 0 0
\(157\) −2.30362 −0.183849 −0.0919246 0.995766i \(-0.529302\pi\)
−0.0919246 + 0.995766i \(0.529302\pi\)
\(158\) −16.3844 −1.30347
\(159\) 3.69741 0.293224
\(160\) −1.92320 −0.152042
\(161\) 1.19070 0.0938401
\(162\) −5.30729 −0.416980
\(163\) −15.3183 −1.19983 −0.599913 0.800065i \(-0.704798\pi\)
−0.599913 + 0.800065i \(0.704798\pi\)
\(164\) 3.57493 0.279156
\(165\) −13.7167 −1.06785
\(166\) −13.6960 −1.06302
\(167\) −11.9904 −0.927845 −0.463922 0.885876i \(-0.653558\pi\)
−0.463922 + 0.885876i \(0.653558\pi\)
\(168\) −28.2968 −2.18315
\(169\) 0 0
\(170\) −0.979904 −0.0751552
\(171\) 25.1123 1.92039
\(172\) 2.94169 0.224302
\(173\) −9.18608 −0.698404 −0.349202 0.937047i \(-0.613547\pi\)
−0.349202 + 0.937047i \(0.613547\pi\)
\(174\) −10.9961 −0.833615
\(175\) −3.86493 −0.292162
\(176\) −21.7068 −1.63622
\(177\) −11.7502 −0.883202
\(178\) −0.0216882 −0.00162560
\(179\) −2.63815 −0.197185 −0.0985923 0.995128i \(-0.531434\pi\)
−0.0985923 + 0.995128i \(0.531434\pi\)
\(180\) −1.83343 −0.136656
\(181\) 9.36566 0.696144 0.348072 0.937468i \(-0.386837\pi\)
0.348072 + 0.937468i \(0.386837\pi\)
\(182\) 0 0
\(183\) 0.967545 0.0715230
\(184\) −0.781213 −0.0575918
\(185\) −5.15099 −0.378708
\(186\) 2.80992 0.206033
\(187\) −3.04094 −0.222376
\(188\) 0.995262 0.0725870
\(189\) −26.0704 −1.89634
\(190\) 7.20428 0.522653
\(191\) −6.30396 −0.456138 −0.228069 0.973645i \(-0.573241\pi\)
−0.228069 + 0.973645i \(0.573241\pi\)
\(192\) 17.8838 1.29065
\(193\) 8.22585 0.592110 0.296055 0.955171i \(-0.404329\pi\)
0.296055 + 0.955171i \(0.404329\pi\)
\(194\) −15.5012 −1.11292
\(195\) 0 0
\(196\) 2.72725 0.194803
\(197\) −0.462516 −0.0329529 −0.0164764 0.999864i \(-0.505245\pi\)
−0.0164764 + 0.999864i \(0.505245\pi\)
\(198\) −38.8098 −2.75809
\(199\) −19.3271 −1.37007 −0.685033 0.728512i \(-0.740212\pi\)
−0.685033 + 0.728512i \(0.740212\pi\)
\(200\) 2.53577 0.179306
\(201\) 2.08332 0.146946
\(202\) −19.0385 −1.33954
\(203\) −9.61516 −0.674852
\(204\) −0.634977 −0.0444573
\(205\) −10.4049 −0.726712
\(206\) −15.6719 −1.09192
\(207\) −1.64398 −0.114264
\(208\) 0 0
\(209\) 22.3571 1.54647
\(210\) −17.0831 −1.17885
\(211\) 18.8081 1.29480 0.647401 0.762149i \(-0.275856\pi\)
0.647401 + 0.762149i \(0.275856\pi\)
\(212\) 0.439988 0.0302185
\(213\) −29.8029 −2.04206
\(214\) 23.4837 1.60532
\(215\) −8.56187 −0.583914
\(216\) 17.1047 1.16383
\(217\) 2.45703 0.166794
\(218\) 0.789434 0.0534672
\(219\) −37.8616 −2.55845
\(220\) −1.63228 −0.110048
\(221\) 0 0
\(222\) −22.7675 −1.52806
\(223\) 3.50267 0.234556 0.117278 0.993099i \(-0.462583\pi\)
0.117278 + 0.993099i \(0.462583\pi\)
\(224\) −7.43305 −0.496642
\(225\) 5.33625 0.355750
\(226\) 26.0397 1.73213
\(227\) −24.5730 −1.63097 −0.815483 0.578781i \(-0.803529\pi\)
−0.815483 + 0.578781i \(0.803529\pi\)
\(228\) 4.66837 0.309170
\(229\) −17.8359 −1.17863 −0.589314 0.807904i \(-0.700602\pi\)
−0.589314 + 0.807904i \(0.700602\pi\)
\(230\) −0.471628 −0.0310982
\(231\) −53.0142 −3.48808
\(232\) 6.30848 0.414172
\(233\) 3.03322 0.198713 0.0993565 0.995052i \(-0.468322\pi\)
0.0993565 + 0.995052i \(0.468322\pi\)
\(234\) 0 0
\(235\) −2.89673 −0.188962
\(236\) −1.39827 −0.0910194
\(237\) 30.9012 2.00725
\(238\) −3.78727 −0.245492
\(239\) 6.93867 0.448825 0.224413 0.974494i \(-0.427954\pi\)
0.224413 + 0.974494i \(0.427954\pi\)
\(240\) 13.1922 0.851553
\(241\) 6.59429 0.424776 0.212388 0.977185i \(-0.431876\pi\)
0.212388 + 0.977185i \(0.431876\pi\)
\(242\) −17.7121 −1.13858
\(243\) −10.2264 −0.656027
\(244\) 0.115137 0.00737089
\(245\) −7.93772 −0.507122
\(246\) −45.9901 −2.93222
\(247\) 0 0
\(248\) −1.61205 −0.102365
\(249\) 25.8309 1.63697
\(250\) 1.53088 0.0968211
\(251\) −16.8636 −1.06442 −0.532211 0.846612i \(-0.678639\pi\)
−0.532211 + 0.846612i \(0.678639\pi\)
\(252\) −7.08610 −0.446383
\(253\) −1.46361 −0.0920161
\(254\) 33.3543 2.09284
\(255\) 1.84812 0.115733
\(256\) 8.01652 0.501033
\(257\) −10.6593 −0.664906 −0.332453 0.943120i \(-0.607876\pi\)
−0.332453 + 0.943120i \(0.607876\pi\)
\(258\) −37.8437 −2.35605
\(259\) −19.9082 −1.23704
\(260\) 0 0
\(261\) 13.2755 0.821733
\(262\) 16.6609 1.02931
\(263\) 14.2995 0.881745 0.440873 0.897570i \(-0.354669\pi\)
0.440873 + 0.897570i \(0.354669\pi\)
\(264\) 34.7825 2.14071
\(265\) −1.28060 −0.0786664
\(266\) 27.8441 1.70723
\(267\) 0.0409042 0.00250330
\(268\) 0.247913 0.0151437
\(269\) 10.7756 0.657001 0.328500 0.944504i \(-0.393457\pi\)
0.328500 + 0.944504i \(0.393457\pi\)
\(270\) 10.3263 0.628439
\(271\) 19.4273 1.18012 0.590062 0.807358i \(-0.299103\pi\)
0.590062 + 0.807358i \(0.299103\pi\)
\(272\) 2.92466 0.177334
\(273\) 0 0
\(274\) 1.92253 0.116144
\(275\) 4.75078 0.286483
\(276\) −0.305614 −0.0183958
\(277\) −1.04322 −0.0626810 −0.0313405 0.999509i \(-0.509978\pi\)
−0.0313405 + 0.999509i \(0.509978\pi\)
\(278\) −27.9201 −1.67454
\(279\) −3.39239 −0.203097
\(280\) 9.80059 0.585697
\(281\) 9.04835 0.539780 0.269890 0.962891i \(-0.413013\pi\)
0.269890 + 0.962891i \(0.413013\pi\)
\(282\) −12.8037 −0.762447
\(283\) −2.37857 −0.141392 −0.0706958 0.997498i \(-0.522522\pi\)
−0.0706958 + 0.997498i \(0.522522\pi\)
\(284\) −3.54651 −0.210447
\(285\) −13.5874 −0.804847
\(286\) 0 0
\(287\) −40.2144 −2.37378
\(288\) 10.2627 0.604735
\(289\) −16.5903 −0.975899
\(290\) 3.80850 0.223643
\(291\) 29.2355 1.71381
\(292\) −4.50549 −0.263664
\(293\) 14.1290 0.825423 0.412712 0.910862i \(-0.364582\pi\)
0.412712 + 0.910862i \(0.364582\pi\)
\(294\) −35.0850 −2.04620
\(295\) 4.06969 0.236946
\(296\) 13.0617 0.759198
\(297\) 32.0457 1.85948
\(298\) 32.2141 1.86611
\(299\) 0 0
\(300\) 0.992006 0.0572735
\(301\) −33.0910 −1.90734
\(302\) −24.8586 −1.43045
\(303\) 35.9069 2.06280
\(304\) −21.5022 −1.23323
\(305\) −0.335109 −0.0191883
\(306\) 5.22902 0.298923
\(307\) −24.3391 −1.38910 −0.694552 0.719442i \(-0.744397\pi\)
−0.694552 + 0.719442i \(0.744397\pi\)
\(308\) −6.30864 −0.359468
\(309\) 29.5576 1.68147
\(310\) −0.973215 −0.0552749
\(311\) 6.14513 0.348458 0.174229 0.984705i \(-0.444257\pi\)
0.174229 + 0.984705i \(0.444257\pi\)
\(312\) 0 0
\(313\) 1.39881 0.0790652 0.0395326 0.999218i \(-0.487413\pi\)
0.0395326 + 0.999218i \(0.487413\pi\)
\(314\) 3.52656 0.199015
\(315\) 20.6243 1.16205
\(316\) 3.67721 0.206859
\(317\) 8.92070 0.501036 0.250518 0.968112i \(-0.419399\pi\)
0.250518 + 0.968112i \(0.419399\pi\)
\(318\) −5.66027 −0.317412
\(319\) 11.8190 0.661735
\(320\) −6.19404 −0.346258
\(321\) −44.2907 −2.47207
\(322\) −1.82281 −0.101581
\(323\) −3.01227 −0.167607
\(324\) 1.19114 0.0661743
\(325\) 0 0
\(326\) 23.4505 1.29880
\(327\) −1.48889 −0.0823356
\(328\) 26.3845 1.45684
\(329\) −11.1957 −0.617238
\(330\) 20.9986 1.15593
\(331\) −19.7654 −1.08640 −0.543202 0.839602i \(-0.682788\pi\)
−0.543202 + 0.839602i \(0.682788\pi\)
\(332\) 3.07385 0.168700
\(333\) 27.4870 1.50628
\(334\) 18.3558 1.00439
\(335\) −0.721556 −0.0394228
\(336\) 50.9870 2.78157
\(337\) 20.2361 1.10233 0.551166 0.834396i \(-0.314183\pi\)
0.551166 + 0.834396i \(0.314183\pi\)
\(338\) 0 0
\(339\) −49.1112 −2.66736
\(340\) 0.219924 0.0119271
\(341\) −3.02019 −0.163552
\(342\) −38.4438 −2.07880
\(343\) −3.62422 −0.195689
\(344\) 21.7109 1.17058
\(345\) 0.889497 0.0478889
\(346\) 14.0627 0.756018
\(347\) 29.2678 1.57118 0.785589 0.618749i \(-0.212360\pi\)
0.785589 + 0.618749i \(0.212360\pi\)
\(348\) 2.46791 0.132294
\(349\) 0.195008 0.0104386 0.00521928 0.999986i \(-0.498339\pi\)
0.00521928 + 0.999986i \(0.498339\pi\)
\(350\) 5.91673 0.316263
\(351\) 0 0
\(352\) 9.13671 0.486988
\(353\) −12.2379 −0.651358 −0.325679 0.945480i \(-0.605593\pi\)
−0.325679 + 0.945480i \(0.605593\pi\)
\(354\) 17.9881 0.956059
\(355\) 10.3222 0.547846
\(356\) 0.00486757 0.000257981 0
\(357\) 7.14285 0.378040
\(358\) 4.03868 0.213451
\(359\) 2.43702 0.128621 0.0643104 0.997930i \(-0.479515\pi\)
0.0643104 + 0.997930i \(0.479515\pi\)
\(360\) −13.5315 −0.713174
\(361\) 3.14628 0.165593
\(362\) −14.3377 −0.753571
\(363\) 33.4053 1.75332
\(364\) 0 0
\(365\) 13.1133 0.686384
\(366\) −1.48119 −0.0774231
\(367\) −28.8213 −1.50446 −0.752229 0.658901i \(-0.771022\pi\)
−0.752229 + 0.658901i \(0.771022\pi\)
\(368\) 1.40764 0.0733783
\(369\) 55.5233 2.89043
\(370\) 7.88552 0.409949
\(371\) −4.94942 −0.256961
\(372\) −0.630643 −0.0326973
\(373\) 3.72479 0.192862 0.0964311 0.995340i \(-0.469257\pi\)
0.0964311 + 0.995340i \(0.469257\pi\)
\(374\) 4.65531 0.240720
\(375\) −2.88726 −0.149097
\(376\) 7.34546 0.378813
\(377\) 0 0
\(378\) 39.9105 2.05278
\(379\) −27.1916 −1.39674 −0.698370 0.715737i \(-0.746091\pi\)
−0.698370 + 0.715737i \(0.746091\pi\)
\(380\) −1.61689 −0.0829445
\(381\) −62.9068 −3.22281
\(382\) 9.65057 0.493766
\(383\) −33.2900 −1.70104 −0.850519 0.525945i \(-0.823712\pi\)
−0.850519 + 0.525945i \(0.823712\pi\)
\(384\) −38.4834 −1.96385
\(385\) 18.3614 0.935786
\(386\) −12.5928 −0.640955
\(387\) 45.6883 2.32247
\(388\) 3.47899 0.176619
\(389\) −23.5358 −1.19331 −0.596655 0.802498i \(-0.703504\pi\)
−0.596655 + 0.802498i \(0.703504\pi\)
\(390\) 0 0
\(391\) 0.197198 0.00997274
\(392\) 20.1282 1.01663
\(393\) −31.4227 −1.58507
\(394\) 0.708054 0.0356713
\(395\) −10.7026 −0.538507
\(396\) 8.71024 0.437706
\(397\) −22.4582 −1.12714 −0.563572 0.826067i \(-0.690573\pi\)
−0.563572 + 0.826067i \(0.690573\pi\)
\(398\) 29.5875 1.48309
\(399\) −52.5143 −2.62901
\(400\) −4.56911 −0.228456
\(401\) −0.0227333 −0.00113525 −0.000567624 1.00000i \(-0.500181\pi\)
−0.000567624 1.00000i \(0.500181\pi\)
\(402\) −3.18930 −0.159068
\(403\) 0 0
\(404\) 4.27288 0.212584
\(405\) −3.46683 −0.172268
\(406\) 14.7196 0.730523
\(407\) 24.4712 1.21299
\(408\) −4.68640 −0.232011
\(409\) −28.7890 −1.42352 −0.711761 0.702421i \(-0.752102\pi\)
−0.711761 + 0.702421i \(0.752102\pi\)
\(410\) 15.9287 0.786660
\(411\) −3.62592 −0.178853
\(412\) 3.51732 0.173286
\(413\) 15.7291 0.773977
\(414\) 2.51672 0.123690
\(415\) −8.94652 −0.439167
\(416\) 0 0
\(417\) 52.6578 2.57866
\(418\) −34.2259 −1.67404
\(419\) 17.4064 0.850358 0.425179 0.905109i \(-0.360211\pi\)
0.425179 + 0.905109i \(0.360211\pi\)
\(420\) 3.83404 0.187082
\(421\) −30.1642 −1.47011 −0.735056 0.678007i \(-0.762844\pi\)
−0.735056 + 0.678007i \(0.762844\pi\)
\(422\) −28.7929 −1.40161
\(423\) 15.4577 0.751579
\(424\) 3.24730 0.157703
\(425\) −0.640094 −0.0310491
\(426\) 45.6245 2.21051
\(427\) −1.29517 −0.0626778
\(428\) −5.27055 −0.254762
\(429\) 0 0
\(430\) 13.1072 0.632083
\(431\) 10.1531 0.489056 0.244528 0.969642i \(-0.421367\pi\)
0.244528 + 0.969642i \(0.421367\pi\)
\(432\) −30.8203 −1.48284
\(433\) 22.2687 1.07017 0.535083 0.844799i \(-0.320280\pi\)
0.535083 + 0.844799i \(0.320280\pi\)
\(434\) −3.76141 −0.180554
\(435\) −7.18290 −0.344394
\(436\) −0.177176 −0.00848519
\(437\) −1.44981 −0.0693536
\(438\) 57.9614 2.76950
\(439\) 37.8809 1.80796 0.903980 0.427576i \(-0.140632\pi\)
0.903980 + 0.427576i \(0.140632\pi\)
\(440\) −12.0469 −0.574313
\(441\) 42.3577 2.01703
\(442\) 0 0
\(443\) −0.918139 −0.0436221 −0.0218110 0.999762i \(-0.506943\pi\)
−0.0218110 + 0.999762i \(0.506943\pi\)
\(444\) 5.10981 0.242501
\(445\) −0.0141672 −0.000671588 0
\(446\) −5.36216 −0.253906
\(447\) −60.7563 −2.87368
\(448\) −23.9396 −1.13104
\(449\) 34.7819 1.64146 0.820730 0.571317i \(-0.193567\pi\)
0.820730 + 0.571317i \(0.193567\pi\)
\(450\) −8.16914 −0.385097
\(451\) 49.4315 2.32764
\(452\) −5.84419 −0.274888
\(453\) 46.8837 2.20279
\(454\) 37.6182 1.76551
\(455\) 0 0
\(456\) 34.4545 1.61348
\(457\) −13.7197 −0.641781 −0.320890 0.947116i \(-0.603982\pi\)
−0.320890 + 0.947116i \(0.603982\pi\)
\(458\) 27.3045 1.27586
\(459\) −4.31767 −0.201531
\(460\) 0.105849 0.00493525
\(461\) 32.5057 1.51394 0.756971 0.653449i \(-0.226679\pi\)
0.756971 + 0.653449i \(0.226679\pi\)
\(462\) 81.1582 3.77582
\(463\) 34.0054 1.58037 0.790183 0.612871i \(-0.209985\pi\)
0.790183 + 0.612871i \(0.209985\pi\)
\(464\) −11.3670 −0.527701
\(465\) 1.83550 0.0851192
\(466\) −4.64349 −0.215105
\(467\) −0.0693333 −0.00320836 −0.00160418 0.999999i \(-0.500511\pi\)
−0.00160418 + 0.999999i \(0.500511\pi\)
\(468\) 0 0
\(469\) −2.78877 −0.128773
\(470\) 4.43454 0.204550
\(471\) −6.65115 −0.306469
\(472\) −10.3198 −0.475007
\(473\) 40.6755 1.87026
\(474\) −47.3059 −2.17283
\(475\) 4.70598 0.215925
\(476\) 0.849992 0.0389593
\(477\) 6.83358 0.312888
\(478\) −10.6222 −0.485850
\(479\) 9.16068 0.418562 0.209281 0.977856i \(-0.432888\pi\)
0.209281 + 0.977856i \(0.432888\pi\)
\(480\) −5.55278 −0.253449
\(481\) 0 0
\(482\) −10.0950 −0.459816
\(483\) 3.43785 0.156428
\(484\) 3.97520 0.180691
\(485\) −10.1257 −0.459784
\(486\) 15.6554 0.710144
\(487\) 27.2691 1.23568 0.617841 0.786303i \(-0.288008\pi\)
0.617841 + 0.786303i \(0.288008\pi\)
\(488\) 0.849759 0.0384668
\(489\) −44.2280 −2.00006
\(490\) 12.1517 0.548956
\(491\) −17.9855 −0.811676 −0.405838 0.913945i \(-0.633020\pi\)
−0.405838 + 0.913945i \(0.633020\pi\)
\(492\) 10.3218 0.465341
\(493\) −1.59242 −0.0717191
\(494\) 0 0
\(495\) −25.3514 −1.13946
\(496\) 2.90470 0.130425
\(497\) 39.8946 1.78952
\(498\) −39.5439 −1.77200
\(499\) −12.9963 −0.581792 −0.290896 0.956755i \(-0.593953\pi\)
−0.290896 + 0.956755i \(0.593953\pi\)
\(500\) −0.343581 −0.0153654
\(501\) −34.6194 −1.54668
\(502\) 25.8161 1.15223
\(503\) −6.92255 −0.308661 −0.154331 0.988019i \(-0.549322\pi\)
−0.154331 + 0.988019i \(0.549322\pi\)
\(504\) −52.2984 −2.32956
\(505\) −12.4363 −0.553409
\(506\) 2.24060 0.0996068
\(507\) 0 0
\(508\) −7.48585 −0.332131
\(509\) −1.39866 −0.0619945 −0.0309972 0.999519i \(-0.509868\pi\)
−0.0309972 + 0.999519i \(0.509868\pi\)
\(510\) −2.82924 −0.125281
\(511\) 50.6822 2.24205
\(512\) 14.3851 0.635739
\(513\) 31.7436 1.40151
\(514\) 16.3180 0.719756
\(515\) −10.2372 −0.451107
\(516\) 8.49343 0.373902
\(517\) 13.7617 0.605241
\(518\) 30.4770 1.33908
\(519\) −26.5226 −1.16421
\(520\) 0 0
\(521\) −19.4736 −0.853154 −0.426577 0.904451i \(-0.640281\pi\)
−0.426577 + 0.904451i \(0.640281\pi\)
\(522\) −20.3231 −0.889520
\(523\) −8.05342 −0.352152 −0.176076 0.984377i \(-0.556340\pi\)
−0.176076 + 0.984377i \(0.556340\pi\)
\(524\) −3.73928 −0.163351
\(525\) −11.1591 −0.487021
\(526\) −21.8908 −0.954483
\(527\) 0.406923 0.0177259
\(528\) −62.6732 −2.72750
\(529\) −22.9051 −0.995873
\(530\) 1.96043 0.0851558
\(531\) −21.7169 −0.942432
\(532\) −6.24916 −0.270935
\(533\) 0 0
\(534\) −0.0626193 −0.00270980
\(535\) 15.3401 0.663209
\(536\) 1.82970 0.0790311
\(537\) −7.61702 −0.328699
\(538\) −16.4961 −0.711199
\(539\) 37.7103 1.62430
\(540\) −2.31758 −0.0997327
\(541\) 7.04652 0.302954 0.151477 0.988461i \(-0.451597\pi\)
0.151477 + 0.988461i \(0.451597\pi\)
\(542\) −29.7408 −1.27748
\(543\) 27.0411 1.16044
\(544\) −1.23103 −0.0527800
\(545\) 0.515675 0.0220891
\(546\) 0 0
\(547\) 39.5338 1.69034 0.845172 0.534494i \(-0.179498\pi\)
0.845172 + 0.534494i \(0.179498\pi\)
\(548\) −0.431481 −0.0184319
\(549\) 1.78822 0.0763196
\(550\) −7.27285 −0.310115
\(551\) 11.7075 0.498757
\(552\) −2.25556 −0.0960031
\(553\) −41.3649 −1.75901
\(554\) 1.59704 0.0678518
\(555\) −14.8722 −0.631291
\(556\) 6.26622 0.265747
\(557\) 8.81508 0.373507 0.186753 0.982407i \(-0.440203\pi\)
0.186753 + 0.982407i \(0.440203\pi\)
\(558\) 5.19332 0.219851
\(559\) 0 0
\(560\) −17.6593 −0.746243
\(561\) −8.77999 −0.370691
\(562\) −13.8519 −0.584307
\(563\) 10.7679 0.453813 0.226907 0.973917i \(-0.427139\pi\)
0.226907 + 0.973917i \(0.427139\pi\)
\(564\) 2.87358 0.120999
\(565\) 17.0097 0.715602
\(566\) 3.64130 0.153055
\(567\) −13.3991 −0.562709
\(568\) −26.1748 −1.09827
\(569\) −22.5894 −0.946995 −0.473498 0.880795i \(-0.657009\pi\)
−0.473498 + 0.880795i \(0.657009\pi\)
\(570\) 20.8006 0.871241
\(571\) −5.95207 −0.249087 −0.124543 0.992214i \(-0.539747\pi\)
−0.124543 + 0.992214i \(0.539747\pi\)
\(572\) 0 0
\(573\) −18.2011 −0.760364
\(574\) 61.5632 2.56960
\(575\) −0.308077 −0.0128477
\(576\) 33.0530 1.37721
\(577\) 21.0425 0.876010 0.438005 0.898972i \(-0.355685\pi\)
0.438005 + 0.898972i \(0.355685\pi\)
\(578\) 25.3977 1.05640
\(579\) 23.7501 0.987022
\(580\) −0.854759 −0.0354919
\(581\) −34.5777 −1.43452
\(582\) −44.7559 −1.85519
\(583\) 6.08383 0.251966
\(584\) −33.2524 −1.37600
\(585\) 0 0
\(586\) −21.6297 −0.893515
\(587\) −19.8969 −0.821232 −0.410616 0.911808i \(-0.634686\pi\)
−0.410616 + 0.911808i \(0.634686\pi\)
\(588\) 7.87427 0.324729
\(589\) −2.99171 −0.123271
\(590\) −6.23019 −0.256493
\(591\) −1.33540 −0.0549311
\(592\) −23.5354 −0.967301
\(593\) −35.4858 −1.45723 −0.728614 0.684925i \(-0.759835\pi\)
−0.728614 + 0.684925i \(0.759835\pi\)
\(594\) −49.0580 −2.01287
\(595\) −2.47392 −0.101421
\(596\) −7.22995 −0.296150
\(597\) −55.8024 −2.28384
\(598\) 0 0
\(599\) 12.5055 0.510962 0.255481 0.966814i \(-0.417766\pi\)
0.255481 + 0.966814i \(0.417766\pi\)
\(600\) 7.32143 0.298896
\(601\) −10.9255 −0.445661 −0.222831 0.974857i \(-0.571530\pi\)
−0.222831 + 0.974857i \(0.571530\pi\)
\(602\) 50.6583 2.06468
\(603\) 3.85041 0.156801
\(604\) 5.57911 0.227011
\(605\) −11.5699 −0.470383
\(606\) −54.9690 −2.23296
\(607\) −2.90532 −0.117923 −0.0589616 0.998260i \(-0.518779\pi\)
−0.0589616 + 0.998260i \(0.518779\pi\)
\(608\) 9.05056 0.367049
\(609\) −27.7614 −1.12495
\(610\) 0.513010 0.0207712
\(611\) 0 0
\(612\) −1.17357 −0.0474388
\(613\) −19.2621 −0.777990 −0.388995 0.921240i \(-0.627178\pi\)
−0.388995 + 0.921240i \(0.627178\pi\)
\(614\) 37.2601 1.50370
\(615\) −30.0417 −1.21140
\(616\) −46.5604 −1.87597
\(617\) −0.902301 −0.0363253 −0.0181626 0.999835i \(-0.505782\pi\)
−0.0181626 + 0.999835i \(0.505782\pi\)
\(618\) −45.2489 −1.82018
\(619\) 23.8784 0.959752 0.479876 0.877336i \(-0.340682\pi\)
0.479876 + 0.877336i \(0.340682\pi\)
\(620\) 0.218423 0.00877207
\(621\) −2.07809 −0.0833909
\(622\) −9.40743 −0.377204
\(623\) −0.0547552 −0.00219372
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −2.14140 −0.0855875
\(627\) 64.5506 2.57790
\(628\) −0.791481 −0.0315835
\(629\) −3.29712 −0.131465
\(630\) −31.5732 −1.25791
\(631\) 45.4774 1.81043 0.905213 0.424958i \(-0.139711\pi\)
0.905213 + 0.424958i \(0.139711\pi\)
\(632\) 27.1394 1.07955
\(633\) 54.3038 2.15838
\(634\) −13.6565 −0.542368
\(635\) 21.7877 0.864620
\(636\) 1.27036 0.0503730
\(637\) 0 0
\(638\) −18.0934 −0.716323
\(639\) −55.0819 −2.17901
\(640\) 13.3287 0.526864
\(641\) 38.7591 1.53089 0.765447 0.643499i \(-0.222518\pi\)
0.765447 + 0.643499i \(0.222518\pi\)
\(642\) 67.8036 2.67599
\(643\) 14.5186 0.572559 0.286279 0.958146i \(-0.407581\pi\)
0.286279 + 0.958146i \(0.407581\pi\)
\(644\) 0.409101 0.0161208
\(645\) −24.7203 −0.973361
\(646\) 4.61141 0.181434
\(647\) −5.86833 −0.230708 −0.115354 0.993324i \(-0.536800\pi\)
−0.115354 + 0.993324i \(0.536800\pi\)
\(648\) 8.79110 0.345347
\(649\) −19.3342 −0.758933
\(650\) 0 0
\(651\) 7.09408 0.278039
\(652\) −5.26309 −0.206119
\(653\) −13.9308 −0.545154 −0.272577 0.962134i \(-0.587876\pi\)
−0.272577 + 0.962134i \(0.587876\pi\)
\(654\) 2.27930 0.0891276
\(655\) 10.8833 0.425244
\(656\) −47.5413 −1.85618
\(657\) −69.9761 −2.73003
\(658\) 17.1392 0.668156
\(659\) 32.4177 1.26281 0.631406 0.775452i \(-0.282478\pi\)
0.631406 + 0.775452i \(0.282478\pi\)
\(660\) −4.71280 −0.183446
\(661\) −42.0999 −1.63750 −0.818748 0.574154i \(-0.805331\pi\)
−0.818748 + 0.574154i \(0.805331\pi\)
\(662\) 30.2584 1.17603
\(663\) 0 0
\(664\) 22.6863 0.880400
\(665\) 18.1883 0.705313
\(666\) −42.0791 −1.63053
\(667\) −0.766432 −0.0296764
\(668\) −4.11967 −0.159395
\(669\) 10.1131 0.390996
\(670\) 1.10461 0.0426749
\(671\) 1.59203 0.0614595
\(672\) −21.4611 −0.827881
\(673\) 9.29450 0.358277 0.179138 0.983824i \(-0.442669\pi\)
0.179138 + 0.983824i \(0.442669\pi\)
\(674\) −30.9790 −1.19327
\(675\) 6.74536 0.259629
\(676\) 0 0
\(677\) −45.6554 −1.75468 −0.877340 0.479870i \(-0.840684\pi\)
−0.877340 + 0.479870i \(0.840684\pi\)
\(678\) 75.1832 2.88739
\(679\) −39.1351 −1.50187
\(680\) 1.62313 0.0622443
\(681\) −70.9485 −2.71875
\(682\) 4.62353 0.177044
\(683\) 12.5289 0.479403 0.239702 0.970847i \(-0.422950\pi\)
0.239702 + 0.970847i \(0.422950\pi\)
\(684\) 8.62811 0.329904
\(685\) 1.25583 0.0479830
\(686\) 5.54822 0.211832
\(687\) −51.4968 −1.96472
\(688\) −39.1201 −1.49144
\(689\) 0 0
\(690\) −1.36171 −0.0518394
\(691\) 5.09243 0.193725 0.0968625 0.995298i \(-0.469119\pi\)
0.0968625 + 0.995298i \(0.469119\pi\)
\(692\) −3.15616 −0.119979
\(693\) −97.9813 −3.72200
\(694\) −44.8054 −1.70079
\(695\) −18.2380 −0.691806
\(696\) 18.2142 0.690408
\(697\) −6.66013 −0.252270
\(698\) −0.298534 −0.0112997
\(699\) 8.75769 0.331246
\(700\) −1.32792 −0.0501906
\(701\) −33.0018 −1.24646 −0.623230 0.782039i \(-0.714180\pi\)
−0.623230 + 0.782039i \(0.714180\pi\)
\(702\) 0 0
\(703\) 24.2405 0.914246
\(704\) 29.4265 1.10905
\(705\) −8.36361 −0.314992
\(706\) 18.7347 0.705090
\(707\) −48.0656 −1.80769
\(708\) −4.03716 −0.151726
\(709\) 9.50378 0.356922 0.178461 0.983947i \(-0.442888\pi\)
0.178461 + 0.983947i \(0.442888\pi\)
\(710\) −15.8020 −0.593039
\(711\) 57.1119 2.14186
\(712\) 0.0359247 0.00134633
\(713\) 0.195852 0.00733472
\(714\) −10.9348 −0.409225
\(715\) 0 0
\(716\) −0.906418 −0.0338744
\(717\) 20.0337 0.748174
\(718\) −3.73077 −0.139231
\(719\) 32.8065 1.22348 0.611739 0.791060i \(-0.290470\pi\)
0.611739 + 0.791060i \(0.290470\pi\)
\(720\) 24.3819 0.908662
\(721\) −39.5663 −1.47353
\(722\) −4.81656 −0.179254
\(723\) 19.0394 0.708083
\(724\) 3.21786 0.119591
\(725\) 2.48779 0.0923944
\(726\) −51.1393 −1.89796
\(727\) −23.3248 −0.865070 −0.432535 0.901617i \(-0.642381\pi\)
−0.432535 + 0.901617i \(0.642381\pi\)
\(728\) 0 0
\(729\) −39.9269 −1.47877
\(730\) −20.0749 −0.743005
\(731\) −5.48040 −0.202700
\(732\) 0.332430 0.0122870
\(733\) 41.5378 1.53423 0.767117 0.641507i \(-0.221690\pi\)
0.767117 + 0.641507i \(0.221690\pi\)
\(734\) 44.1218 1.62857
\(735\) −22.9182 −0.845352
\(736\) −0.592494 −0.0218396
\(737\) 3.42795 0.126270
\(738\) −84.9993 −3.12887
\(739\) −1.05857 −0.0389401 −0.0194701 0.999810i \(-0.506198\pi\)
−0.0194701 + 0.999810i \(0.506198\pi\)
\(740\) −1.76978 −0.0650584
\(741\) 0 0
\(742\) 7.57695 0.278158
\(743\) −19.0752 −0.699800 −0.349900 0.936787i \(-0.613784\pi\)
−0.349900 + 0.936787i \(0.613784\pi\)
\(744\) −4.65441 −0.170639
\(745\) 21.0429 0.770953
\(746\) −5.70219 −0.208772
\(747\) 47.7409 1.74675
\(748\) −1.04481 −0.0382021
\(749\) 59.2884 2.16635
\(750\) 4.42003 0.161397
\(751\) 46.7588 1.70625 0.853127 0.521703i \(-0.174703\pi\)
0.853127 + 0.521703i \(0.174703\pi\)
\(752\) −13.2355 −0.482649
\(753\) −48.6896 −1.77435
\(754\) 0 0
\(755\) −16.2381 −0.590966
\(756\) −8.95729 −0.325773
\(757\) 15.0982 0.548752 0.274376 0.961623i \(-0.411529\pi\)
0.274376 + 0.961623i \(0.411529\pi\)
\(758\) 41.6270 1.51196
\(759\) −4.22581 −0.153387
\(760\) −11.9333 −0.432866
\(761\) −20.5311 −0.744253 −0.372127 0.928182i \(-0.621371\pi\)
−0.372127 + 0.928182i \(0.621371\pi\)
\(762\) 96.3025 3.48867
\(763\) 1.99305 0.0721532
\(764\) −2.16592 −0.0783602
\(765\) 3.41570 0.123495
\(766\) 50.9628 1.84136
\(767\) 0 0
\(768\) 23.1458 0.835201
\(769\) 33.9725 1.22508 0.612540 0.790440i \(-0.290148\pi\)
0.612540 + 0.790440i \(0.290148\pi\)
\(770\) −28.1091 −1.01298
\(771\) −30.7760 −1.10837
\(772\) 2.82625 0.101719
\(773\) 10.8481 0.390177 0.195089 0.980786i \(-0.437501\pi\)
0.195089 + 0.980786i \(0.437501\pi\)
\(774\) −69.9431 −2.51405
\(775\) −0.635724 −0.0228359
\(776\) 25.6764 0.921731
\(777\) −57.4802 −2.06209
\(778\) 36.0303 1.29175
\(779\) 48.9654 1.75437
\(780\) 0 0
\(781\) −49.0385 −1.75474
\(782\) −0.301886 −0.0107954
\(783\) 16.7811 0.599707
\(784\) −36.2683 −1.29530
\(785\) 2.30362 0.0822198
\(786\) 48.1043 1.71582
\(787\) 9.91801 0.353539 0.176769 0.984252i \(-0.443435\pi\)
0.176769 + 0.984252i \(0.443435\pi\)
\(788\) −0.158912 −0.00566099
\(789\) 41.2863 1.46983
\(790\) 16.3844 0.582930
\(791\) 65.7412 2.33749
\(792\) 64.2853 2.28428
\(793\) 0 0
\(794\) 34.3807 1.22013
\(795\) −3.69741 −0.131134
\(796\) −6.64044 −0.235364
\(797\) −33.0159 −1.16948 −0.584742 0.811219i \(-0.698804\pi\)
−0.584742 + 0.811219i \(0.698804\pi\)
\(798\) 80.3929 2.84588
\(799\) −1.85418 −0.0655962
\(800\) 1.92320 0.0679955
\(801\) 0.0755996 0.00267118
\(802\) 0.0348019 0.00122890
\(803\) −62.2986 −2.19847
\(804\) 0.715788 0.0252439
\(805\) −1.19070 −0.0419666
\(806\) 0 0
\(807\) 31.1120 1.09519
\(808\) 31.5357 1.10942
\(809\) −11.0248 −0.387610 −0.193805 0.981040i \(-0.562083\pi\)
−0.193805 + 0.981040i \(0.562083\pi\)
\(810\) 5.30729 0.186479
\(811\) −31.2682 −1.09797 −0.548987 0.835831i \(-0.684986\pi\)
−0.548987 + 0.835831i \(0.684986\pi\)
\(812\) −3.30359 −0.115933
\(813\) 56.0916 1.96722
\(814\) −37.4624 −1.31306
\(815\) 15.3183 0.536578
\(816\) 8.44425 0.295608
\(817\) 40.2920 1.40964
\(818\) 44.0723 1.54095
\(819\) 0 0
\(820\) −3.57493 −0.124842
\(821\) −12.3719 −0.431782 −0.215891 0.976417i \(-0.569266\pi\)
−0.215891 + 0.976417i \(0.569266\pi\)
\(822\) 5.55083 0.193607
\(823\) −47.6643 −1.66147 −0.830737 0.556665i \(-0.812081\pi\)
−0.830737 + 0.556665i \(0.812081\pi\)
\(824\) 25.9593 0.904336
\(825\) 13.7167 0.477555
\(826\) −24.0793 −0.837825
\(827\) −18.9966 −0.660575 −0.330287 0.943880i \(-0.607146\pi\)
−0.330287 + 0.943880i \(0.607146\pi\)
\(828\) −0.564839 −0.0196295
\(829\) −44.5434 −1.54706 −0.773528 0.633762i \(-0.781510\pi\)
−0.773528 + 0.633762i \(0.781510\pi\)
\(830\) 13.6960 0.475395
\(831\) −3.01205 −0.104487
\(832\) 0 0
\(833\) −5.08088 −0.176042
\(834\) −80.6125 −2.79138
\(835\) 11.9904 0.414945
\(836\) 7.68147 0.265669
\(837\) −4.28819 −0.148222
\(838\) −26.6470 −0.920506
\(839\) 23.6568 0.816724 0.408362 0.912820i \(-0.366100\pi\)
0.408362 + 0.912820i \(0.366100\pi\)
\(840\) 28.2968 0.976333
\(841\) −22.8109 −0.786582
\(842\) 46.1776 1.59138
\(843\) 26.1249 0.899790
\(844\) 6.46210 0.222435
\(845\) 0 0
\(846\) −23.6638 −0.813579
\(847\) −44.7169 −1.53649
\(848\) −5.85119 −0.200931
\(849\) −6.86756 −0.235694
\(850\) 0.979904 0.0336104
\(851\) −1.58690 −0.0543983
\(852\) −10.2397 −0.350806
\(853\) 1.85306 0.0634474 0.0317237 0.999497i \(-0.489900\pi\)
0.0317237 + 0.999497i \(0.489900\pi\)
\(854\) 1.98275 0.0678483
\(855\) −25.1123 −0.858823
\(856\) −38.8989 −1.32954
\(857\) 32.9142 1.12433 0.562163 0.827026i \(-0.309969\pi\)
0.562163 + 0.827026i \(0.309969\pi\)
\(858\) 0 0
\(859\) 31.9147 1.08892 0.544458 0.838788i \(-0.316735\pi\)
0.544458 + 0.838788i \(0.316735\pi\)
\(860\) −2.94169 −0.100311
\(861\) −116.109 −3.95699
\(862\) −15.5431 −0.529400
\(863\) 11.5883 0.394469 0.197235 0.980356i \(-0.436804\pi\)
0.197235 + 0.980356i \(0.436804\pi\)
\(864\) 12.9727 0.441340
\(865\) 9.18608 0.312336
\(866\) −34.0906 −1.15845
\(867\) −47.9004 −1.62678
\(868\) 0.844189 0.0286537
\(869\) 50.8457 1.72482
\(870\) 10.9961 0.372804
\(871\) 0 0
\(872\) −1.30763 −0.0442821
\(873\) 54.0333 1.82875
\(874\) 2.21947 0.0750748
\(875\) 3.86493 0.130659
\(876\) −13.0085 −0.439517
\(877\) 48.0221 1.62159 0.810796 0.585328i \(-0.199034\pi\)
0.810796 + 0.585328i \(0.199034\pi\)
\(878\) −57.9910 −1.95710
\(879\) 40.7940 1.37595
\(880\) 21.7068 0.731738
\(881\) −32.4131 −1.09202 −0.546012 0.837777i \(-0.683855\pi\)
−0.546012 + 0.837777i \(0.683855\pi\)
\(882\) −64.8443 −2.18342
\(883\) 53.6030 1.80389 0.901943 0.431856i \(-0.142141\pi\)
0.901943 + 0.431856i \(0.142141\pi\)
\(884\) 0 0
\(885\) 11.7502 0.394980
\(886\) 1.40556 0.0472206
\(887\) −39.8878 −1.33930 −0.669650 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(888\) 37.7126 1.26555
\(889\) 84.2082 2.82425
\(890\) 0.0216882 0.000726989 0
\(891\) 16.4702 0.551771
\(892\) 1.20345 0.0402946
\(893\) 13.6320 0.456177
\(894\) 93.0104 3.11073
\(895\) 2.63815 0.0881837
\(896\) 51.5146 1.72098
\(897\) 0 0
\(898\) −53.2468 −1.77687
\(899\) −1.58155 −0.0527477
\(900\) 1.83343 0.0611145
\(901\) −0.819702 −0.0273082
\(902\) −75.6735 −2.51965
\(903\) −95.5424 −3.17945
\(904\) −43.1326 −1.43457
\(905\) −9.36566 −0.311325
\(906\) −71.7731 −2.38450
\(907\) 33.5672 1.11458 0.557290 0.830318i \(-0.311841\pi\)
0.557290 + 0.830318i \(0.311841\pi\)
\(908\) −8.44281 −0.280185
\(909\) 66.3634 2.20113
\(910\) 0 0
\(911\) −25.2448 −0.836399 −0.418199 0.908355i \(-0.637339\pi\)
−0.418199 + 0.908355i \(0.637339\pi\)
\(912\) −62.0823 −2.05575
\(913\) 42.5029 1.40664
\(914\) 21.0032 0.694723
\(915\) −0.967545 −0.0319861
\(916\) −6.12807 −0.202477
\(917\) 42.0631 1.38904
\(918\) 6.60981 0.218156
\(919\) 30.1551 0.994726 0.497363 0.867543i \(-0.334302\pi\)
0.497363 + 0.867543i \(0.334302\pi\)
\(920\) 0.781213 0.0257558
\(921\) −70.2732 −2.31558
\(922\) −49.7622 −1.63883
\(923\) 0 0
\(924\) −18.2147 −0.599219
\(925\) 5.15099 0.169363
\(926\) −52.0581 −1.71074
\(927\) 54.6285 1.79424
\(928\) 4.78453 0.157060
\(929\) 34.6023 1.13526 0.567632 0.823283i \(-0.307860\pi\)
0.567632 + 0.823283i \(0.307860\pi\)
\(930\) −2.80992 −0.0921410
\(931\) 37.3548 1.22425
\(932\) 1.04216 0.0341370
\(933\) 17.7426 0.580866
\(934\) 0.106141 0.00347303
\(935\) 3.04094 0.0994495
\(936\) 0 0
\(937\) −15.7809 −0.515541 −0.257771 0.966206i \(-0.582988\pi\)
−0.257771 + 0.966206i \(0.582988\pi\)
\(938\) 4.26926 0.139396
\(939\) 4.03871 0.131798
\(940\) −0.995262 −0.0324619
\(941\) 22.5246 0.734280 0.367140 0.930166i \(-0.380337\pi\)
0.367140 + 0.930166i \(0.380337\pi\)
\(942\) 10.1821 0.331750
\(943\) −3.20552 −0.104386
\(944\) 18.5949 0.605211
\(945\) 26.0704 0.848070
\(946\) −62.2692 −2.02455
\(947\) 5.97656 0.194212 0.0971060 0.995274i \(-0.469041\pi\)
0.0971060 + 0.995274i \(0.469041\pi\)
\(948\) 10.6171 0.344826
\(949\) 0 0
\(950\) −7.20428 −0.233738
\(951\) 25.7563 0.835207
\(952\) 6.27330 0.203319
\(953\) −0.933220 −0.0302300 −0.0151150 0.999886i \(-0.504811\pi\)
−0.0151150 + 0.999886i \(0.504811\pi\)
\(954\) −10.4614 −0.338699
\(955\) 6.30396 0.203991
\(956\) 2.38400 0.0771039
\(957\) 34.1244 1.10308
\(958\) −14.0239 −0.453091
\(959\) 4.85372 0.156735
\(960\) −17.8838 −0.577197
\(961\) −30.5959 −0.986963
\(962\) 0 0
\(963\) −81.8585 −2.63785
\(964\) 2.26567 0.0729724
\(965\) −8.22585 −0.264800
\(966\) −5.26292 −0.169332
\(967\) 39.4196 1.26765 0.633825 0.773477i \(-0.281484\pi\)
0.633825 + 0.773477i \(0.281484\pi\)
\(968\) 29.3386 0.942979
\(969\) −8.69720 −0.279394
\(970\) 15.5012 0.497713
\(971\) −37.6269 −1.20750 −0.603752 0.797172i \(-0.706328\pi\)
−0.603752 + 0.797172i \(0.706328\pi\)
\(972\) −3.51361 −0.112699
\(973\) −70.4886 −2.25976
\(974\) −41.7456 −1.33762
\(975\) 0 0
\(976\) −1.53115 −0.0490109
\(977\) 62.0293 1.98449 0.992247 0.124282i \(-0.0396627\pi\)
0.992247 + 0.124282i \(0.0396627\pi\)
\(978\) 67.7076 2.16505
\(979\) 0.0673051 0.00215108
\(980\) −2.72725 −0.0871187
\(981\) −2.75177 −0.0878573
\(982\) 27.5336 0.878633
\(983\) 27.9591 0.891757 0.445879 0.895093i \(-0.352891\pi\)
0.445879 + 0.895093i \(0.352891\pi\)
\(984\) 76.1789 2.42849
\(985\) 0.462516 0.0147370
\(986\) 2.43780 0.0776354
\(987\) −32.3248 −1.02891
\(988\) 0 0
\(989\) −2.63771 −0.0838744
\(990\) 38.8098 1.23346
\(991\) 56.5077 1.79503 0.897513 0.440988i \(-0.145372\pi\)
0.897513 + 0.440988i \(0.145372\pi\)
\(992\) −1.22263 −0.0388184
\(993\) −57.0678 −1.81099
\(994\) −61.0738 −1.93714
\(995\) 19.3271 0.612712
\(996\) 8.87500 0.281215
\(997\) −6.01115 −0.190375 −0.0951875 0.995459i \(-0.530345\pi\)
−0.0951875 + 0.995459i \(0.530345\pi\)
\(998\) 19.8957 0.629786
\(999\) 34.7453 1.09929
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 845.2.a.n.1.4 9
3.2 odd 2 7605.2.a.cs.1.6 9
5.4 even 2 4225.2.a.bt.1.6 9
13.2 odd 12 845.2.m.j.316.6 36
13.3 even 3 845.2.e.p.191.6 18
13.4 even 6 845.2.e.o.146.4 18
13.5 odd 4 845.2.c.h.506.13 18
13.6 odd 12 845.2.m.j.361.13 36
13.7 odd 12 845.2.m.j.361.6 36
13.8 odd 4 845.2.c.h.506.6 18
13.9 even 3 845.2.e.p.146.6 18
13.10 even 6 845.2.e.o.191.4 18
13.11 odd 12 845.2.m.j.316.13 36
13.12 even 2 845.2.a.o.1.6 yes 9
39.38 odd 2 7605.2.a.cp.1.4 9
65.64 even 2 4225.2.a.bs.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
845.2.a.n.1.4 9 1.1 even 1 trivial
845.2.a.o.1.6 yes 9 13.12 even 2
845.2.c.h.506.6 18 13.8 odd 4
845.2.c.h.506.13 18 13.5 odd 4
845.2.e.o.146.4 18 13.4 even 6
845.2.e.o.191.4 18 13.10 even 6
845.2.e.p.146.6 18 13.9 even 3
845.2.e.p.191.6 18 13.3 even 3
845.2.m.j.316.6 36 13.2 odd 12
845.2.m.j.316.13 36 13.11 odd 12
845.2.m.j.361.6 36 13.7 odd 12
845.2.m.j.361.13 36 13.6 odd 12
4225.2.a.bs.1.4 9 65.64 even 2
4225.2.a.bt.1.6 9 5.4 even 2
7605.2.a.cp.1.4 9 39.38 odd 2
7605.2.a.cs.1.6 9 3.2 odd 2