Properties

Label 2738.2.a.r.1.2
Level $2738$
Weight $2$
Character 2738.1
Self dual yes
Analytic conductor $21.863$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2738,2,Mod(1,2738)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2738, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2738.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(21.8630400734\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.28558\) of defining polynomial
Character \(\chi\) \(=\) 2738.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.87939 q^{3} +1.00000 q^{4} +4.26414 q^{5} +1.87939 q^{6} +3.61144 q^{7} -1.00000 q^{8} +0.532089 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.87939 q^{3} +1.00000 q^{4} +4.26414 q^{5} +1.87939 q^{6} +3.61144 q^{7} -1.00000 q^{8} +0.532089 q^{9} -4.26414 q^{10} +2.11681 q^{11} -1.87939 q^{12} +2.88400 q^{13} -3.61144 q^{14} -8.01396 q^{15} +1.00000 q^{16} -0.0267197 q^{17} -0.532089 q^{18} +3.22847 q^{19} +4.26414 q^{20} -6.78728 q^{21} -2.11681 q^{22} -2.97244 q^{23} +1.87939 q^{24} +13.1829 q^{25} -2.88400 q^{26} +4.63816 q^{27} +3.61144 q^{28} -5.73300 q^{29} +8.01396 q^{30} +6.76932 q^{31} -1.00000 q^{32} -3.97829 q^{33} +0.0267197 q^{34} +15.3997 q^{35} +0.532089 q^{36} -3.22847 q^{38} -5.42015 q^{39} -4.26414 q^{40} -1.49152 q^{41} +6.78728 q^{42} +5.53737 q^{43} +2.11681 q^{44} +2.26890 q^{45} +2.97244 q^{46} +2.61309 q^{47} -1.87939 q^{48} +6.04247 q^{49} -13.1829 q^{50} +0.0502166 q^{51} +2.88400 q^{52} +1.90902 q^{53} -4.63816 q^{54} +9.02635 q^{55} -3.61144 q^{56} -6.06754 q^{57} +5.73300 q^{58} +5.62482 q^{59} -8.01396 q^{60} +5.74087 q^{61} -6.76932 q^{62} +1.92160 q^{63} +1.00000 q^{64} +12.2978 q^{65} +3.97829 q^{66} -6.92148 q^{67} -0.0267197 q^{68} +5.58637 q^{69} -15.3997 q^{70} -9.27923 q^{71} -0.532089 q^{72} -16.2707 q^{73} -24.7757 q^{75} +3.22847 q^{76} +7.64471 q^{77} +5.42015 q^{78} +1.41514 q^{79} +4.26414 q^{80} -10.3131 q^{81} +1.49152 q^{82} -1.69642 q^{83} -6.78728 q^{84} -0.113936 q^{85} -5.53737 q^{86} +10.7745 q^{87} -2.11681 q^{88} -8.27093 q^{89} -2.26890 q^{90} +10.4154 q^{91} -2.97244 q^{92} -12.7222 q^{93} -2.61309 q^{94} +13.7667 q^{95} +1.87939 q^{96} -6.52527 q^{97} -6.04247 q^{98} +1.12633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 6 q^{8} - 6 q^{9} - 6 q^{10} - 6 q^{11} + 12 q^{13} - 6 q^{15} + 6 q^{16} + 12 q^{17} + 6 q^{18} + 12 q^{19} + 6 q^{20} - 12 q^{21} + 6 q^{22} - 6 q^{23} + 6 q^{25} - 12 q^{26} - 6 q^{27} - 6 q^{29} + 6 q^{30} + 18 q^{31} - 6 q^{32} + 6 q^{33} - 12 q^{34} + 24 q^{35} - 6 q^{36} - 12 q^{38} + 6 q^{39} - 6 q^{40} - 12 q^{41} + 12 q^{42} - 6 q^{44} + 6 q^{45} + 6 q^{46} - 6 q^{47} - 12 q^{49} - 6 q^{50} + 24 q^{51} + 12 q^{52} - 24 q^{53} + 6 q^{54} + 36 q^{55} + 24 q^{57} + 6 q^{58} + 12 q^{59} - 6 q^{60} + 12 q^{61} - 18 q^{62} + 6 q^{63} + 6 q^{64} + 18 q^{65} - 6 q^{66} + 18 q^{67} + 12 q^{68} + 24 q^{69} - 24 q^{70} + 24 q^{71} + 6 q^{72} - 18 q^{75} + 12 q^{76} + 30 q^{77} - 6 q^{78} + 12 q^{79} + 6 q^{80} - 18 q^{81} + 12 q^{82} - 6 q^{83} - 12 q^{84} + 18 q^{85} + 6 q^{87} + 6 q^{88} - 6 q^{90} + 12 q^{91} - 6 q^{92} - 36 q^{93} + 6 q^{94} + 18 q^{95} - 12 q^{97} + 12 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.87939 −1.08506 −0.542532 0.840035i \(-0.682534\pi\)
−0.542532 + 0.840035i \(0.682534\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.26414 1.90698 0.953491 0.301423i \(-0.0974615\pi\)
0.953491 + 0.301423i \(0.0974615\pi\)
\(6\) 1.87939 0.767256
\(7\) 3.61144 1.36499 0.682497 0.730888i \(-0.260894\pi\)
0.682497 + 0.730888i \(0.260894\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0.532089 0.177363
\(10\) −4.26414 −1.34844
\(11\) 2.11681 0.638241 0.319120 0.947714i \(-0.396613\pi\)
0.319120 + 0.947714i \(0.396613\pi\)
\(12\) −1.87939 −0.542532
\(13\) 2.88400 0.799878 0.399939 0.916542i \(-0.369031\pi\)
0.399939 + 0.916542i \(0.369031\pi\)
\(14\) −3.61144 −0.965197
\(15\) −8.01396 −2.06920
\(16\) 1.00000 0.250000
\(17\) −0.0267197 −0.00648047 −0.00324024 0.999995i \(-0.501031\pi\)
−0.00324024 + 0.999995i \(0.501031\pi\)
\(18\) −0.532089 −0.125415
\(19\) 3.22847 0.740662 0.370331 0.928900i \(-0.379244\pi\)
0.370331 + 0.928900i \(0.379244\pi\)
\(20\) 4.26414 0.953491
\(21\) −6.78728 −1.48111
\(22\) −2.11681 −0.451304
\(23\) −2.97244 −0.619797 −0.309899 0.950770i \(-0.600295\pi\)
−0.309899 + 0.950770i \(0.600295\pi\)
\(24\) 1.87939 0.383628
\(25\) 13.1829 2.63658
\(26\) −2.88400 −0.565599
\(27\) 4.63816 0.892613
\(28\) 3.61144 0.682497
\(29\) −5.73300 −1.06459 −0.532296 0.846558i \(-0.678671\pi\)
−0.532296 + 0.846558i \(0.678671\pi\)
\(30\) 8.01396 1.46314
\(31\) 6.76932 1.21581 0.607903 0.794011i \(-0.292011\pi\)
0.607903 + 0.794011i \(0.292011\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.97829 −0.692532
\(34\) 0.0267197 0.00458239
\(35\) 15.3997 2.60302
\(36\) 0.532089 0.0886815
\(37\) 0 0
\(38\) −3.22847 −0.523727
\(39\) −5.42015 −0.867919
\(40\) −4.26414 −0.674220
\(41\) −1.49152 −0.232937 −0.116468 0.993194i \(-0.537157\pi\)
−0.116468 + 0.993194i \(0.537157\pi\)
\(42\) 6.78728 1.04730
\(43\) 5.53737 0.844441 0.422221 0.906493i \(-0.361251\pi\)
0.422221 + 0.906493i \(0.361251\pi\)
\(44\) 2.11681 0.319120
\(45\) 2.26890 0.338228
\(46\) 2.97244 0.438263
\(47\) 2.61309 0.381158 0.190579 0.981672i \(-0.438964\pi\)
0.190579 + 0.981672i \(0.438964\pi\)
\(48\) −1.87939 −0.271266
\(49\) 6.04247 0.863210
\(50\) −13.1829 −1.86434
\(51\) 0.0502166 0.00703173
\(52\) 2.88400 0.399939
\(53\) 1.90902 0.262224 0.131112 0.991368i \(-0.458145\pi\)
0.131112 + 0.991368i \(0.458145\pi\)
\(54\) −4.63816 −0.631173
\(55\) 9.02635 1.21711
\(56\) −3.61144 −0.482598
\(57\) −6.06754 −0.803665
\(58\) 5.73300 0.752780
\(59\) 5.62482 0.732289 0.366144 0.930558i \(-0.380678\pi\)
0.366144 + 0.930558i \(0.380678\pi\)
\(60\) −8.01396 −1.03460
\(61\) 5.74087 0.735043 0.367522 0.930015i \(-0.380206\pi\)
0.367522 + 0.930015i \(0.380206\pi\)
\(62\) −6.76932 −0.859705
\(63\) 1.92160 0.242099
\(64\) 1.00000 0.125000
\(65\) 12.2978 1.52535
\(66\) 3.97829 0.489694
\(67\) −6.92148 −0.845593 −0.422796 0.906225i \(-0.638951\pi\)
−0.422796 + 0.906225i \(0.638951\pi\)
\(68\) −0.0267197 −0.00324024
\(69\) 5.58637 0.672519
\(70\) −15.3997 −1.84061
\(71\) −9.27923 −1.10124 −0.550621 0.834755i \(-0.685609\pi\)
−0.550621 + 0.834755i \(0.685609\pi\)
\(72\) −0.532089 −0.0627073
\(73\) −16.2707 −1.90434 −0.952169 0.305571i \(-0.901153\pi\)
−0.952169 + 0.305571i \(0.901153\pi\)
\(74\) 0 0
\(75\) −24.7757 −2.86085
\(76\) 3.22847 0.370331
\(77\) 7.64471 0.871195
\(78\) 5.42015 0.613711
\(79\) 1.41514 0.159216 0.0796078 0.996826i \(-0.474633\pi\)
0.0796078 + 0.996826i \(0.474633\pi\)
\(80\) 4.26414 0.476745
\(81\) −10.3131 −1.14591
\(82\) 1.49152 0.164711
\(83\) −1.69642 −0.186206 −0.0931030 0.995656i \(-0.529679\pi\)
−0.0931030 + 0.995656i \(0.529679\pi\)
\(84\) −6.78728 −0.740553
\(85\) −0.113936 −0.0123581
\(86\) −5.53737 −0.597110
\(87\) 10.7745 1.15515
\(88\) −2.11681 −0.225652
\(89\) −8.27093 −0.876717 −0.438358 0.898800i \(-0.644440\pi\)
−0.438358 + 0.898800i \(0.644440\pi\)
\(90\) −2.26890 −0.239163
\(91\) 10.4154 1.09183
\(92\) −2.97244 −0.309899
\(93\) −12.7222 −1.31923
\(94\) −2.61309 −0.269519
\(95\) 13.7667 1.41243
\(96\) 1.87939 0.191814
\(97\) −6.52527 −0.662541 −0.331270 0.943536i \(-0.607477\pi\)
−0.331270 + 0.943536i \(0.607477\pi\)
\(98\) −6.04247 −0.610382
\(99\) 1.12633 0.113200
\(100\) 13.1829 1.31829
\(101\) −16.5995 −1.65171 −0.825855 0.563882i \(-0.809307\pi\)
−0.825855 + 0.563882i \(0.809307\pi\)
\(102\) −0.0502166 −0.00497218
\(103\) −1.89481 −0.186701 −0.0933504 0.995633i \(-0.529758\pi\)
−0.0933504 + 0.995633i \(0.529758\pi\)
\(104\) −2.88400 −0.282800
\(105\) −28.9419 −2.82444
\(106\) −1.90902 −0.185421
\(107\) 0.382256 0.0369541 0.0184771 0.999829i \(-0.494118\pi\)
0.0184771 + 0.999829i \(0.494118\pi\)
\(108\) 4.63816 0.446307
\(109\) −0.965584 −0.0924862 −0.0462431 0.998930i \(-0.514725\pi\)
−0.0462431 + 0.998930i \(0.514725\pi\)
\(110\) −9.02635 −0.860629
\(111\) 0 0
\(112\) 3.61144 0.341249
\(113\) −1.89075 −0.177866 −0.0889332 0.996038i \(-0.528346\pi\)
−0.0889332 + 0.996038i \(0.528346\pi\)
\(114\) 6.06754 0.568277
\(115\) −12.6749 −1.18194
\(116\) −5.73300 −0.532296
\(117\) 1.53455 0.141869
\(118\) −5.62482 −0.517806
\(119\) −0.0964964 −0.00884581
\(120\) 8.01396 0.731571
\(121\) −6.51914 −0.592649
\(122\) −5.74087 −0.519754
\(123\) 2.80315 0.252751
\(124\) 6.76932 0.607903
\(125\) 34.8930 3.12092
\(126\) −1.92160 −0.171190
\(127\) −16.1144 −1.42992 −0.714961 0.699165i \(-0.753555\pi\)
−0.714961 + 0.699165i \(0.753555\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.4069 −0.916273
\(130\) −12.2978 −1.07859
\(131\) −0.822948 −0.0719013 −0.0359507 0.999354i \(-0.511446\pi\)
−0.0359507 + 0.999354i \(0.511446\pi\)
\(132\) −3.97829 −0.346266
\(133\) 11.6594 1.01100
\(134\) 6.92148 0.597924
\(135\) 19.7777 1.70220
\(136\) 0.0267197 0.00229119
\(137\) 17.3569 1.48290 0.741452 0.671006i \(-0.234138\pi\)
0.741452 + 0.671006i \(0.234138\pi\)
\(138\) −5.58637 −0.475543
\(139\) −9.22544 −0.782491 −0.391246 0.920286i \(-0.627956\pi\)
−0.391246 + 0.920286i \(0.627956\pi\)
\(140\) 15.3997 1.30151
\(141\) −4.91099 −0.413580
\(142\) 9.27923 0.778696
\(143\) 6.10487 0.510515
\(144\) 0.532089 0.0443407
\(145\) −24.4463 −2.03016
\(146\) 16.2707 1.34657
\(147\) −11.3561 −0.936638
\(148\) 0 0
\(149\) −7.73776 −0.633902 −0.316951 0.948442i \(-0.602659\pi\)
−0.316951 + 0.948442i \(0.602659\pi\)
\(150\) 24.7757 2.02293
\(151\) 12.6363 1.02833 0.514165 0.857691i \(-0.328102\pi\)
0.514165 + 0.857691i \(0.328102\pi\)
\(152\) −3.22847 −0.261864
\(153\) −0.0142172 −0.00114940
\(154\) −7.64471 −0.616028
\(155\) 28.8653 2.31852
\(156\) −5.42015 −0.433959
\(157\) 13.0467 1.04124 0.520621 0.853788i \(-0.325700\pi\)
0.520621 + 0.853788i \(0.325700\pi\)
\(158\) −1.41514 −0.112582
\(159\) −3.58779 −0.284530
\(160\) −4.26414 −0.337110
\(161\) −10.7348 −0.846020
\(162\) 10.3131 0.810277
\(163\) 25.2564 1.97823 0.989117 0.147133i \(-0.0470046\pi\)
0.989117 + 0.147133i \(0.0470046\pi\)
\(164\) −1.49152 −0.116468
\(165\) −16.9640 −1.32065
\(166\) 1.69642 0.131668
\(167\) 10.2810 0.795569 0.397785 0.917479i \(-0.369779\pi\)
0.397785 + 0.917479i \(0.369779\pi\)
\(168\) 6.78728 0.523650
\(169\) −4.68253 −0.360195
\(170\) 0.113936 0.00873853
\(171\) 1.71783 0.131366
\(172\) 5.53737 0.422221
\(173\) −23.6569 −1.79860 −0.899300 0.437332i \(-0.855923\pi\)
−0.899300 + 0.437332i \(0.855923\pi\)
\(174\) −10.7745 −0.816814
\(175\) 47.6092 3.59891
\(176\) 2.11681 0.159560
\(177\) −10.5712 −0.794580
\(178\) 8.27093 0.619932
\(179\) 5.05746 0.378013 0.189006 0.981976i \(-0.439473\pi\)
0.189006 + 0.981976i \(0.439473\pi\)
\(180\) 2.26890 0.169114
\(181\) −1.04545 −0.0777075 −0.0388537 0.999245i \(-0.512371\pi\)
−0.0388537 + 0.999245i \(0.512371\pi\)
\(182\) −10.4154 −0.772040
\(183\) −10.7893 −0.797569
\(184\) 2.97244 0.219131
\(185\) 0 0
\(186\) 12.7222 0.932834
\(187\) −0.0565603 −0.00413610
\(188\) 2.61309 0.190579
\(189\) 16.7504 1.21841
\(190\) −13.7667 −0.998738
\(191\) 13.4633 0.974173 0.487087 0.873354i \(-0.338060\pi\)
0.487087 + 0.873354i \(0.338060\pi\)
\(192\) −1.87939 −0.135633
\(193\) 4.52228 0.325521 0.162760 0.986666i \(-0.447960\pi\)
0.162760 + 0.986666i \(0.447960\pi\)
\(194\) 6.52527 0.468487
\(195\) −23.1123 −1.65510
\(196\) 6.04247 0.431605
\(197\) −12.3720 −0.881469 −0.440735 0.897637i \(-0.645282\pi\)
−0.440735 + 0.897637i \(0.645282\pi\)
\(198\) −1.12633 −0.0800447
\(199\) −4.50965 −0.319681 −0.159840 0.987143i \(-0.551098\pi\)
−0.159840 + 0.987143i \(0.551098\pi\)
\(200\) −13.1829 −0.932171
\(201\) 13.0081 0.917522
\(202\) 16.5995 1.16794
\(203\) −20.7044 −1.45316
\(204\) 0.0502166 0.00351586
\(205\) −6.36006 −0.444206
\(206\) 1.89481 0.132017
\(207\) −1.58160 −0.109929
\(208\) 2.88400 0.199970
\(209\) 6.83404 0.472721
\(210\) 28.9419 1.99718
\(211\) −16.2460 −1.11842 −0.559209 0.829026i \(-0.688895\pi\)
−0.559209 + 0.829026i \(0.688895\pi\)
\(212\) 1.90902 0.131112
\(213\) 17.4392 1.19492
\(214\) −0.382256 −0.0261305
\(215\) 23.6121 1.61033
\(216\) −4.63816 −0.315587
\(217\) 24.4470 1.65957
\(218\) 0.965584 0.0653976
\(219\) 30.5789 2.06633
\(220\) 9.02635 0.608557
\(221\) −0.0770596 −0.00518359
\(222\) 0 0
\(223\) −0.847777 −0.0567713 −0.0283857 0.999597i \(-0.509037\pi\)
−0.0283857 + 0.999597i \(0.509037\pi\)
\(224\) −3.61144 −0.241299
\(225\) 7.01447 0.467631
\(226\) 1.89075 0.125770
\(227\) 12.1287 0.805013 0.402506 0.915417i \(-0.368139\pi\)
0.402506 + 0.915417i \(0.368139\pi\)
\(228\) −6.06754 −0.401833
\(229\) 13.2675 0.876741 0.438371 0.898794i \(-0.355556\pi\)
0.438371 + 0.898794i \(0.355556\pi\)
\(230\) 12.6749 0.835759
\(231\) −14.3673 −0.945302
\(232\) 5.73300 0.376390
\(233\) −4.30657 −0.282133 −0.141066 0.990000i \(-0.545053\pi\)
−0.141066 + 0.990000i \(0.545053\pi\)
\(234\) −1.53455 −0.100316
\(235\) 11.1426 0.726860
\(236\) 5.62482 0.366144
\(237\) −2.65959 −0.172759
\(238\) 0.0964964 0.00625493
\(239\) 14.2423 0.921258 0.460629 0.887593i \(-0.347624\pi\)
0.460629 + 0.887593i \(0.347624\pi\)
\(240\) −8.01396 −0.517299
\(241\) −14.2610 −0.918633 −0.459316 0.888273i \(-0.651906\pi\)
−0.459316 + 0.888273i \(0.651906\pi\)
\(242\) 6.51914 0.419066
\(243\) 5.46791 0.350767
\(244\) 5.74087 0.367522
\(245\) 25.7659 1.64613
\(246\) −2.80315 −0.178722
\(247\) 9.31092 0.592440
\(248\) −6.76932 −0.429852
\(249\) 3.18822 0.202045
\(250\) −34.8930 −2.20683
\(251\) −24.7076 −1.55953 −0.779764 0.626074i \(-0.784661\pi\)
−0.779764 + 0.626074i \(0.784661\pi\)
\(252\) 1.92160 0.121050
\(253\) −6.29208 −0.395580
\(254\) 16.1144 1.01111
\(255\) 0.214130 0.0134094
\(256\) 1.00000 0.0625000
\(257\) −16.4123 −1.02377 −0.511886 0.859053i \(-0.671053\pi\)
−0.511886 + 0.859053i \(0.671053\pi\)
\(258\) 10.4069 0.647903
\(259\) 0 0
\(260\) 12.2978 0.762676
\(261\) −3.05047 −0.188819
\(262\) 0.822948 0.0508419
\(263\) 28.1414 1.73527 0.867637 0.497198i \(-0.165638\pi\)
0.867637 + 0.497198i \(0.165638\pi\)
\(264\) 3.97829 0.244847
\(265\) 8.14034 0.500057
\(266\) −11.6594 −0.714885
\(267\) 15.5443 0.951293
\(268\) −6.92148 −0.422796
\(269\) 23.1703 1.41272 0.706359 0.707854i \(-0.250336\pi\)
0.706359 + 0.707854i \(0.250336\pi\)
\(270\) −19.7777 −1.20364
\(271\) 10.6088 0.644437 0.322219 0.946665i \(-0.395571\pi\)
0.322219 + 0.946665i \(0.395571\pi\)
\(272\) −0.0267197 −0.00162012
\(273\) −19.5745 −1.18470
\(274\) −17.3569 −1.04857
\(275\) 27.9056 1.68277
\(276\) 5.58637 0.336260
\(277\) 24.0416 1.44452 0.722261 0.691621i \(-0.243103\pi\)
0.722261 + 0.691621i \(0.243103\pi\)
\(278\) 9.22544 0.553305
\(279\) 3.60188 0.215639
\(280\) −15.3997 −0.920306
\(281\) 12.7248 0.759101 0.379550 0.925171i \(-0.376079\pi\)
0.379550 + 0.925171i \(0.376079\pi\)
\(282\) 4.91099 0.292445
\(283\) 0.257708 0.0153192 0.00765959 0.999971i \(-0.497562\pi\)
0.00765959 + 0.999971i \(0.497562\pi\)
\(284\) −9.27923 −0.550621
\(285\) −25.8728 −1.53257
\(286\) −6.10487 −0.360989
\(287\) −5.38654 −0.317957
\(288\) −0.532089 −0.0313536
\(289\) −16.9993 −0.999958
\(290\) 24.4463 1.43554
\(291\) 12.2635 0.718899
\(292\) −16.2707 −0.952169
\(293\) −8.98077 −0.524662 −0.262331 0.964978i \(-0.584491\pi\)
−0.262331 + 0.964978i \(0.584491\pi\)
\(294\) 11.3561 0.662303
\(295\) 23.9850 1.39646
\(296\) 0 0
\(297\) 9.81807 0.569702
\(298\) 7.73776 0.448236
\(299\) −8.57253 −0.495762
\(300\) −24.7757 −1.43043
\(301\) 19.9979 1.15266
\(302\) −12.6363 −0.727140
\(303\) 31.1968 1.79221
\(304\) 3.22847 0.185166
\(305\) 24.4799 1.40171
\(306\) 0.0142172 0.000812746 0
\(307\) 14.7982 0.844580 0.422290 0.906461i \(-0.361226\pi\)
0.422290 + 0.906461i \(0.361226\pi\)
\(308\) 7.64471 0.435598
\(309\) 3.56107 0.202582
\(310\) −28.8653 −1.63944
\(311\) −26.7159 −1.51492 −0.757460 0.652881i \(-0.773560\pi\)
−0.757460 + 0.652881i \(0.773560\pi\)
\(312\) 5.42015 0.306856
\(313\) −9.52374 −0.538314 −0.269157 0.963096i \(-0.586745\pi\)
−0.269157 + 0.963096i \(0.586745\pi\)
\(314\) −13.0467 −0.736270
\(315\) 8.19399 0.461679
\(316\) 1.41514 0.0796078
\(317\) 15.1557 0.851226 0.425613 0.904905i \(-0.360058\pi\)
0.425613 + 0.904905i \(0.360058\pi\)
\(318\) 3.58779 0.201193
\(319\) −12.1357 −0.679466
\(320\) 4.26414 0.238373
\(321\) −0.718407 −0.0400976
\(322\) 10.7348 0.598226
\(323\) −0.0862637 −0.00479984
\(324\) −10.3131 −0.572953
\(325\) 38.0195 2.10894
\(326\) −25.2564 −1.39882
\(327\) 1.81470 0.100353
\(328\) 1.49152 0.0823556
\(329\) 9.43699 0.520278
\(330\) 16.9640 0.933837
\(331\) 2.06618 0.113568 0.0567838 0.998386i \(-0.481915\pi\)
0.0567838 + 0.998386i \(0.481915\pi\)
\(332\) −1.69642 −0.0931030
\(333\) 0 0
\(334\) −10.2810 −0.562552
\(335\) −29.5141 −1.61253
\(336\) −6.78728 −0.370276
\(337\) 13.9267 0.758637 0.379319 0.925266i \(-0.376158\pi\)
0.379319 + 0.925266i \(0.376158\pi\)
\(338\) 4.68253 0.254696
\(339\) 3.55344 0.192996
\(340\) −0.113936 −0.00617907
\(341\) 14.3293 0.775977
\(342\) −1.71783 −0.0928898
\(343\) −3.45806 −0.186718
\(344\) −5.53737 −0.298555
\(345\) 23.8210 1.28248
\(346\) 23.6569 1.27180
\(347\) −20.2115 −1.08501 −0.542505 0.840052i \(-0.682524\pi\)
−0.542505 + 0.840052i \(0.682524\pi\)
\(348\) 10.7745 0.577575
\(349\) 10.9899 0.588277 0.294138 0.955763i \(-0.404967\pi\)
0.294138 + 0.955763i \(0.404967\pi\)
\(350\) −47.6092 −2.54482
\(351\) 13.3765 0.713982
\(352\) −2.11681 −0.112826
\(353\) −4.35642 −0.231869 −0.115934 0.993257i \(-0.536986\pi\)
−0.115934 + 0.993257i \(0.536986\pi\)
\(354\) 10.5712 0.561853
\(355\) −39.5679 −2.10005
\(356\) −8.27093 −0.438358
\(357\) 0.181354 0.00959827
\(358\) −5.05746 −0.267295
\(359\) 7.68664 0.405686 0.202843 0.979211i \(-0.434982\pi\)
0.202843 + 0.979211i \(0.434982\pi\)
\(360\) −2.26890 −0.119582
\(361\) −8.57697 −0.451420
\(362\) 1.04545 0.0549475
\(363\) 12.2520 0.643061
\(364\) 10.4154 0.545915
\(365\) −69.3804 −3.63154
\(366\) 10.7893 0.563966
\(367\) 20.7176 1.08145 0.540726 0.841199i \(-0.318150\pi\)
0.540726 + 0.841199i \(0.318150\pi\)
\(368\) −2.97244 −0.154949
\(369\) −0.793623 −0.0413143
\(370\) 0 0
\(371\) 6.89431 0.357935
\(372\) −12.7222 −0.659614
\(373\) −15.6066 −0.808077 −0.404038 0.914742i \(-0.632394\pi\)
−0.404038 + 0.914742i \(0.632394\pi\)
\(374\) 0.0565603 0.00292467
\(375\) −65.5773 −3.38640
\(376\) −2.61309 −0.134760
\(377\) −16.5340 −0.851544
\(378\) −16.7504 −0.861548
\(379\) −25.1769 −1.29325 −0.646626 0.762807i \(-0.723820\pi\)
−0.646626 + 0.762807i \(0.723820\pi\)
\(380\) 13.7667 0.706214
\(381\) 30.2852 1.55156
\(382\) −13.4633 −0.688844
\(383\) −30.2135 −1.54384 −0.771919 0.635720i \(-0.780703\pi\)
−0.771919 + 0.635720i \(0.780703\pi\)
\(384\) 1.87939 0.0959070
\(385\) 32.5981 1.66135
\(386\) −4.52228 −0.230178
\(387\) 2.94637 0.149773
\(388\) −6.52527 −0.331270
\(389\) −25.0006 −1.26758 −0.633791 0.773505i \(-0.718502\pi\)
−0.633791 + 0.773505i \(0.718502\pi\)
\(390\) 23.1123 1.17034
\(391\) 0.0794227 0.00401658
\(392\) −6.04247 −0.305191
\(393\) 1.54664 0.0780175
\(394\) 12.3720 0.623293
\(395\) 6.03435 0.303621
\(396\) 1.12633 0.0566001
\(397\) −13.6736 −0.686260 −0.343130 0.939288i \(-0.611487\pi\)
−0.343130 + 0.939288i \(0.611487\pi\)
\(398\) 4.50965 0.226048
\(399\) −21.9125 −1.09700
\(400\) 13.1829 0.659144
\(401\) −1.62020 −0.0809089 −0.0404544 0.999181i \(-0.512881\pi\)
−0.0404544 + 0.999181i \(0.512881\pi\)
\(402\) −13.0081 −0.648786
\(403\) 19.5227 0.972497
\(404\) −16.5995 −0.825855
\(405\) −43.9767 −2.18522
\(406\) 20.7044 1.02754
\(407\) 0 0
\(408\) −0.0502166 −0.00248609
\(409\) 9.12867 0.451384 0.225692 0.974199i \(-0.427536\pi\)
0.225692 + 0.974199i \(0.427536\pi\)
\(410\) 6.36006 0.314101
\(411\) −32.6204 −1.60904
\(412\) −1.89481 −0.0933504
\(413\) 20.3137 0.999570
\(414\) 1.58160 0.0777316
\(415\) −7.23376 −0.355091
\(416\) −2.88400 −0.141400
\(417\) 17.3382 0.849053
\(418\) −6.83404 −0.334264
\(419\) 14.2480 0.696060 0.348030 0.937483i \(-0.386851\pi\)
0.348030 + 0.937483i \(0.386851\pi\)
\(420\) −28.9419 −1.41222
\(421\) 33.7943 1.64703 0.823516 0.567293i \(-0.192009\pi\)
0.823516 + 0.567293i \(0.192009\pi\)
\(422\) 16.2460 0.790842
\(423\) 1.39039 0.0676033
\(424\) −1.90902 −0.0927103
\(425\) −0.352242 −0.0170863
\(426\) −17.4392 −0.844934
\(427\) 20.7328 1.00333
\(428\) 0.382256 0.0184771
\(429\) −11.4734 −0.553941
\(430\) −23.6121 −1.13868
\(431\) −38.8388 −1.87080 −0.935399 0.353594i \(-0.884960\pi\)
−0.935399 + 0.353594i \(0.884960\pi\)
\(432\) 4.63816 0.223153
\(433\) 18.1122 0.870417 0.435208 0.900330i \(-0.356675\pi\)
0.435208 + 0.900330i \(0.356675\pi\)
\(434\) −24.4470 −1.17349
\(435\) 45.9441 2.20285
\(436\) −0.965584 −0.0462431
\(437\) −9.59645 −0.459060
\(438\) −30.5789 −1.46111
\(439\) −14.3517 −0.684967 −0.342484 0.939524i \(-0.611268\pi\)
−0.342484 + 0.939524i \(0.611268\pi\)
\(440\) −9.02635 −0.430315
\(441\) 3.21513 0.153101
\(442\) 0.0770596 0.00366535
\(443\) −5.43539 −0.258243 −0.129121 0.991629i \(-0.541216\pi\)
−0.129121 + 0.991629i \(0.541216\pi\)
\(444\) 0 0
\(445\) −35.2684 −1.67188
\(446\) 0.847777 0.0401434
\(447\) 14.5422 0.687824
\(448\) 3.61144 0.170624
\(449\) 0.130238 0.00614633 0.00307317 0.999995i \(-0.499022\pi\)
0.00307317 + 0.999995i \(0.499022\pi\)
\(450\) −7.01447 −0.330665
\(451\) −3.15726 −0.148670
\(452\) −1.89075 −0.0889332
\(453\) −23.7485 −1.11580
\(454\) −12.1287 −0.569230
\(455\) 44.4127 2.08210
\(456\) 6.06754 0.284139
\(457\) 28.2675 1.32230 0.661150 0.750254i \(-0.270069\pi\)
0.661150 + 0.750254i \(0.270069\pi\)
\(458\) −13.2675 −0.619950
\(459\) −0.123930 −0.00578456
\(460\) −12.6749 −0.590971
\(461\) 8.01301 0.373203 0.186601 0.982436i \(-0.440253\pi\)
0.186601 + 0.982436i \(0.440253\pi\)
\(462\) 14.3673 0.668430
\(463\) 34.2291 1.59076 0.795381 0.606110i \(-0.207271\pi\)
0.795381 + 0.606110i \(0.207271\pi\)
\(464\) −5.73300 −0.266148
\(465\) −54.2491 −2.51574
\(466\) 4.30657 0.199498
\(467\) 20.5376 0.950365 0.475182 0.879887i \(-0.342382\pi\)
0.475182 + 0.879887i \(0.342382\pi\)
\(468\) 1.53455 0.0709344
\(469\) −24.9965 −1.15423
\(470\) −11.1426 −0.513968
\(471\) −24.5198 −1.12981
\(472\) −5.62482 −0.258903
\(473\) 11.7215 0.538957
\(474\) 2.65959 0.122159
\(475\) 42.5606 1.95281
\(476\) −0.0964964 −0.00442291
\(477\) 1.01577 0.0465089
\(478\) −14.2423 −0.651428
\(479\) −6.29987 −0.287848 −0.143924 0.989589i \(-0.545972\pi\)
−0.143924 + 0.989589i \(0.545972\pi\)
\(480\) 8.01396 0.365786
\(481\) 0 0
\(482\) 14.2610 0.649572
\(483\) 20.1748 0.917985
\(484\) −6.51914 −0.296324
\(485\) −27.8247 −1.26345
\(486\) −5.46791 −0.248029
\(487\) −27.5903 −1.25024 −0.625118 0.780530i \(-0.714949\pi\)
−0.625118 + 0.780530i \(0.714949\pi\)
\(488\) −5.74087 −0.259877
\(489\) −47.4665 −2.14651
\(490\) −25.7659 −1.16399
\(491\) 33.0201 1.49018 0.745088 0.666967i \(-0.232408\pi\)
0.745088 + 0.666967i \(0.232408\pi\)
\(492\) 2.80315 0.126376
\(493\) 0.153184 0.00689906
\(494\) −9.31092 −0.418918
\(495\) 4.80282 0.215871
\(496\) 6.76932 0.303952
\(497\) −33.5113 −1.50319
\(498\) −3.18822 −0.142868
\(499\) 21.0861 0.943944 0.471972 0.881613i \(-0.343542\pi\)
0.471972 + 0.881613i \(0.343542\pi\)
\(500\) 34.8930 1.56046
\(501\) −19.3220 −0.863243
\(502\) 24.7076 1.10275
\(503\) 17.6544 0.787172 0.393586 0.919288i \(-0.371234\pi\)
0.393586 + 0.919288i \(0.371234\pi\)
\(504\) −1.92160 −0.0855951
\(505\) −70.7825 −3.14978
\(506\) 6.29208 0.279717
\(507\) 8.80028 0.390834
\(508\) −16.1144 −0.714961
\(509\) −41.8141 −1.85338 −0.926688 0.375831i \(-0.877357\pi\)
−0.926688 + 0.375831i \(0.877357\pi\)
\(510\) −0.214130 −0.00948186
\(511\) −58.7605 −2.59941
\(512\) −1.00000 −0.0441942
\(513\) 14.9742 0.661125
\(514\) 16.4123 0.723916
\(515\) −8.07971 −0.356035
\(516\) −10.4069 −0.458136
\(517\) 5.53139 0.243270
\(518\) 0 0
\(519\) 44.4604 1.95160
\(520\) −12.2978 −0.539294
\(521\) −35.9808 −1.57635 −0.788173 0.615453i \(-0.788973\pi\)
−0.788173 + 0.615453i \(0.788973\pi\)
\(522\) 3.05047 0.133515
\(523\) 32.8099 1.43468 0.717338 0.696725i \(-0.245360\pi\)
0.717338 + 0.696725i \(0.245360\pi\)
\(524\) −0.822948 −0.0359507
\(525\) −89.4759 −3.90505
\(526\) −28.1414 −1.22702
\(527\) −0.180874 −0.00787900
\(528\) −3.97829 −0.173133
\(529\) −14.1646 −0.615851
\(530\) −8.14034 −0.353594
\(531\) 2.99290 0.129881
\(532\) 11.6594 0.505500
\(533\) −4.30156 −0.186321
\(534\) −15.5443 −0.672666
\(535\) 1.63000 0.0704708
\(536\) 6.92148 0.298962
\(537\) −9.50492 −0.410168
\(538\) −23.1703 −0.998943
\(539\) 12.7907 0.550936
\(540\) 19.7777 0.851099
\(541\) −37.5209 −1.61315 −0.806576 0.591131i \(-0.798682\pi\)
−0.806576 + 0.591131i \(0.798682\pi\)
\(542\) −10.6088 −0.455686
\(543\) 1.96480 0.0843175
\(544\) 0.0267197 0.00114560
\(545\) −4.11739 −0.176369
\(546\) 19.5745 0.837713
\(547\) 26.0779 1.11501 0.557505 0.830174i \(-0.311759\pi\)
0.557505 + 0.830174i \(0.311759\pi\)
\(548\) 17.3569 0.741452
\(549\) 3.05465 0.130369
\(550\) −27.9056 −1.18990
\(551\) −18.5088 −0.788503
\(552\) −5.58637 −0.237772
\(553\) 5.11068 0.217328
\(554\) −24.0416 −1.02143
\(555\) 0 0
\(556\) −9.22544 −0.391246
\(557\) 6.50687 0.275705 0.137852 0.990453i \(-0.455980\pi\)
0.137852 + 0.990453i \(0.455980\pi\)
\(558\) −3.60188 −0.152480
\(559\) 15.9698 0.675450
\(560\) 15.3997 0.650755
\(561\) 0.106299 0.00448793
\(562\) −12.7248 −0.536765
\(563\) −38.7491 −1.63308 −0.816540 0.577289i \(-0.804111\pi\)
−0.816540 + 0.577289i \(0.804111\pi\)
\(564\) −4.91099 −0.206790
\(565\) −8.06240 −0.339188
\(566\) −0.257708 −0.0108323
\(567\) −37.2453 −1.56415
\(568\) 9.27923 0.389348
\(569\) 12.9149 0.541420 0.270710 0.962661i \(-0.412742\pi\)
0.270710 + 0.962661i \(0.412742\pi\)
\(570\) 25.8728 1.08369
\(571\) 2.99965 0.125531 0.0627657 0.998028i \(-0.480008\pi\)
0.0627657 + 0.998028i \(0.480008\pi\)
\(572\) 6.10487 0.255257
\(573\) −25.3028 −1.05704
\(574\) 5.38654 0.224830
\(575\) −39.1854 −1.63414
\(576\) 0.532089 0.0221704
\(577\) 28.8518 1.20112 0.600559 0.799581i \(-0.294945\pi\)
0.600559 + 0.799581i \(0.294945\pi\)
\(578\) 16.9993 0.707077
\(579\) −8.49911 −0.353211
\(580\) −24.4463 −1.01508
\(581\) −6.12650 −0.254170
\(582\) −12.2635 −0.508338
\(583\) 4.04103 0.167362
\(584\) 16.2707 0.673285
\(585\) 6.54352 0.270541
\(586\) 8.98077 0.370992
\(587\) 24.2762 1.00199 0.500993 0.865451i \(-0.332968\pi\)
0.500993 + 0.865451i \(0.332968\pi\)
\(588\) −11.3561 −0.468319
\(589\) 21.8546 0.900502
\(590\) −23.9850 −0.987447
\(591\) 23.2518 0.956450
\(592\) 0 0
\(593\) 12.0052 0.492995 0.246497 0.969143i \(-0.420720\pi\)
0.246497 + 0.969143i \(0.420720\pi\)
\(594\) −9.81807 −0.402840
\(595\) −0.411474 −0.0168688
\(596\) −7.73776 −0.316951
\(597\) 8.47538 0.346874
\(598\) 8.57253 0.350557
\(599\) −19.0516 −0.778426 −0.389213 0.921148i \(-0.627253\pi\)
−0.389213 + 0.921148i \(0.627253\pi\)
\(600\) 24.7757 1.01146
\(601\) −22.7644 −0.928579 −0.464290 0.885683i \(-0.653690\pi\)
−0.464290 + 0.885683i \(0.653690\pi\)
\(602\) −19.9979 −0.815052
\(603\) −3.68284 −0.149977
\(604\) 12.6363 0.514165
\(605\) −27.7985 −1.13017
\(606\) −31.1968 −1.26728
\(607\) 19.6049 0.795739 0.397869 0.917442i \(-0.369750\pi\)
0.397869 + 0.917442i \(0.369750\pi\)
\(608\) −3.22847 −0.130932
\(609\) 38.9115 1.57677
\(610\) −24.4799 −0.991161
\(611\) 7.53614 0.304880
\(612\) −0.0142172 −0.000574698 0
\(613\) 2.53879 0.102541 0.0512704 0.998685i \(-0.483673\pi\)
0.0512704 + 0.998685i \(0.483673\pi\)
\(614\) −14.7982 −0.597208
\(615\) 11.9530 0.481992
\(616\) −7.64471 −0.308014
\(617\) 18.9730 0.763825 0.381912 0.924199i \(-0.375266\pi\)
0.381912 + 0.924199i \(0.375266\pi\)
\(618\) −3.56107 −0.143247
\(619\) −2.51358 −0.101029 −0.0505147 0.998723i \(-0.516086\pi\)
−0.0505147 + 0.998723i \(0.516086\pi\)
\(620\) 28.8653 1.15926
\(621\) −13.7867 −0.553239
\(622\) 26.7159 1.07121
\(623\) −29.8699 −1.19671
\(624\) −5.42015 −0.216980
\(625\) 82.8741 3.31496
\(626\) 9.52374 0.380645
\(627\) −12.8438 −0.512932
\(628\) 13.0467 0.520621
\(629\) 0 0
\(630\) −8.19399 −0.326456
\(631\) 6.64305 0.264456 0.132228 0.991219i \(-0.457787\pi\)
0.132228 + 0.991219i \(0.457787\pi\)
\(632\) −1.41514 −0.0562912
\(633\) 30.5324 1.21356
\(634\) −15.1557 −0.601908
\(635\) −68.7140 −2.72683
\(636\) −3.58779 −0.142265
\(637\) 17.4265 0.690463
\(638\) 12.1357 0.480455
\(639\) −4.93737 −0.195320
\(640\) −4.26414 −0.168555
\(641\) 8.23442 0.325240 0.162620 0.986689i \(-0.448006\pi\)
0.162620 + 0.986689i \(0.448006\pi\)
\(642\) 0.718407 0.0283533
\(643\) −14.2192 −0.560750 −0.280375 0.959891i \(-0.590459\pi\)
−0.280375 + 0.959891i \(0.590459\pi\)
\(644\) −10.7348 −0.423010
\(645\) −44.3763 −1.74731
\(646\) 0.0862637 0.00339400
\(647\) 39.8752 1.56766 0.783828 0.620977i \(-0.213264\pi\)
0.783828 + 0.620977i \(0.213264\pi\)
\(648\) 10.3131 0.405139
\(649\) 11.9066 0.467377
\(650\) −38.0195 −1.49125
\(651\) −45.9453 −1.80074
\(652\) 25.2564 0.989117
\(653\) −8.11503 −0.317566 −0.158783 0.987314i \(-0.550757\pi\)
−0.158783 + 0.987314i \(0.550757\pi\)
\(654\) −1.81470 −0.0709606
\(655\) −3.50917 −0.137114
\(656\) −1.49152 −0.0582342
\(657\) −8.65745 −0.337759
\(658\) −9.43699 −0.367892
\(659\) 28.0448 1.09247 0.546235 0.837632i \(-0.316061\pi\)
0.546235 + 0.837632i \(0.316061\pi\)
\(660\) −16.9640 −0.660323
\(661\) −4.52498 −0.176001 −0.0880007 0.996120i \(-0.528048\pi\)
−0.0880007 + 0.996120i \(0.528048\pi\)
\(662\) −2.06618 −0.0803044
\(663\) 0.144825 0.00562452
\(664\) 1.69642 0.0658338
\(665\) 49.7174 1.92796
\(666\) 0 0
\(667\) 17.0410 0.659831
\(668\) 10.2810 0.397785
\(669\) 1.59330 0.0616005
\(670\) 29.5141 1.14023
\(671\) 12.1523 0.469135
\(672\) 6.78728 0.261825
\(673\) 10.4858 0.404199 0.202100 0.979365i \(-0.435224\pi\)
0.202100 + 0.979365i \(0.435224\pi\)
\(674\) −13.9267 −0.536438
\(675\) 61.1443 2.35344
\(676\) −4.68253 −0.180097
\(677\) −26.8553 −1.03213 −0.516067 0.856549i \(-0.672604\pi\)
−0.516067 + 0.856549i \(0.672604\pi\)
\(678\) −3.55344 −0.136469
\(679\) −23.5656 −0.904365
\(680\) 0.113936 0.00436926
\(681\) −22.7946 −0.873490
\(682\) −14.3293 −0.548699
\(683\) −17.8721 −0.683858 −0.341929 0.939726i \(-0.611080\pi\)
−0.341929 + 0.939726i \(0.611080\pi\)
\(684\) 1.71783 0.0656830
\(685\) 74.0124 2.82787
\(686\) 3.45806 0.132029
\(687\) −24.9347 −0.951320
\(688\) 5.53737 0.211110
\(689\) 5.50563 0.209748
\(690\) −23.8210 −0.906852
\(691\) −21.1083 −0.802999 −0.401499 0.915859i \(-0.631511\pi\)
−0.401499 + 0.915859i \(0.631511\pi\)
\(692\) −23.6569 −0.899300
\(693\) 4.06766 0.154518
\(694\) 20.2115 0.767218
\(695\) −39.3386 −1.49220
\(696\) −10.7745 −0.408407
\(697\) 0.0398530 0.00150954
\(698\) −10.9899 −0.415974
\(699\) 8.09370 0.306132
\(700\) 47.6092 1.79946
\(701\) 4.57469 0.172784 0.0863919 0.996261i \(-0.472466\pi\)
0.0863919 + 0.996261i \(0.472466\pi\)
\(702\) −13.3765 −0.504862
\(703\) 0 0
\(704\) 2.11681 0.0797801
\(705\) −20.9412 −0.788690
\(706\) 4.35642 0.163956
\(707\) −59.9480 −2.25458
\(708\) −10.5712 −0.397290
\(709\) −4.67695 −0.175647 −0.0878233 0.996136i \(-0.527991\pi\)
−0.0878233 + 0.996136i \(0.527991\pi\)
\(710\) 39.5679 1.48496
\(711\) 0.752980 0.0282389
\(712\) 8.27093 0.309966
\(713\) −20.1214 −0.753553
\(714\) −0.181354 −0.00678700
\(715\) 26.0320 0.973542
\(716\) 5.05746 0.189006
\(717\) −26.7668 −0.999624
\(718\) −7.68664 −0.286863
\(719\) 46.0624 1.71784 0.858919 0.512112i \(-0.171137\pi\)
0.858919 + 0.512112i \(0.171137\pi\)
\(720\) 2.26890 0.0845570
\(721\) −6.84297 −0.254845
\(722\) 8.57697 0.319202
\(723\) 26.8019 0.996775
\(724\) −1.04545 −0.0388537
\(725\) −75.5775 −2.80688
\(726\) −12.2520 −0.454713
\(727\) 43.5632 1.61567 0.807835 0.589408i \(-0.200639\pi\)
0.807835 + 0.589408i \(0.200639\pi\)
\(728\) −10.4154 −0.386020
\(729\) 20.6631 0.765301
\(730\) 69.3804 2.56789
\(731\) −0.147957 −0.00547238
\(732\) −10.7893 −0.398784
\(733\) 27.8003 1.02683 0.513414 0.858141i \(-0.328380\pi\)
0.513414 + 0.858141i \(0.328380\pi\)
\(734\) −20.7176 −0.764702
\(735\) −48.4241 −1.78615
\(736\) 2.97244 0.109566
\(737\) −14.6514 −0.539692
\(738\) 0.793623 0.0292137
\(739\) −26.4463 −0.972843 −0.486421 0.873724i \(-0.661698\pi\)
−0.486421 + 0.873724i \(0.661698\pi\)
\(740\) 0 0
\(741\) −17.4988 −0.642835
\(742\) −6.89431 −0.253098
\(743\) −40.5404 −1.48728 −0.743641 0.668579i \(-0.766903\pi\)
−0.743641 + 0.668579i \(0.766903\pi\)
\(744\) 12.7222 0.466417
\(745\) −32.9949 −1.20884
\(746\) 15.6066 0.571397
\(747\) −0.902645 −0.0330261
\(748\) −0.0565603 −0.00206805
\(749\) 1.38049 0.0504422
\(750\) 65.5773 2.39455
\(751\) −10.2833 −0.375242 −0.187621 0.982241i \(-0.560078\pi\)
−0.187621 + 0.982241i \(0.560078\pi\)
\(752\) 2.61309 0.0952894
\(753\) 46.4350 1.69219
\(754\) 16.5340 0.602133
\(755\) 53.8831 1.96101
\(756\) 16.7504 0.609206
\(757\) 35.4525 1.28854 0.644271 0.764797i \(-0.277161\pi\)
0.644271 + 0.764797i \(0.277161\pi\)
\(758\) 25.1769 0.914467
\(759\) 11.8252 0.429229
\(760\) −13.7667 −0.499369
\(761\) −4.99043 −0.180903 −0.0904515 0.995901i \(-0.528831\pi\)
−0.0904515 + 0.995901i \(0.528831\pi\)
\(762\) −30.2852 −1.09712
\(763\) −3.48715 −0.126243
\(764\) 13.4633 0.487087
\(765\) −0.0606243 −0.00219188
\(766\) 30.2135 1.09166
\(767\) 16.2220 0.585742
\(768\) −1.87939 −0.0678165
\(769\) 35.2184 1.27001 0.635004 0.772509i \(-0.280999\pi\)
0.635004 + 0.772509i \(0.280999\pi\)
\(770\) −32.5981 −1.17475
\(771\) 30.8451 1.11086
\(772\) 4.52228 0.162760
\(773\) −10.7155 −0.385409 −0.192704 0.981257i \(-0.561726\pi\)
−0.192704 + 0.981257i \(0.561726\pi\)
\(774\) −2.94637 −0.105905
\(775\) 89.2392 3.20557
\(776\) 6.52527 0.234244
\(777\) 0 0
\(778\) 25.0006 0.896316
\(779\) −4.81534 −0.172527
\(780\) −23.1123 −0.827552
\(781\) −19.6423 −0.702857
\(782\) −0.0794227 −0.00284015
\(783\) −26.5906 −0.950269
\(784\) 6.04247 0.215803
\(785\) 55.6331 1.98563
\(786\) −1.54664 −0.0551667
\(787\) 27.2203 0.970297 0.485149 0.874432i \(-0.338766\pi\)
0.485149 + 0.874432i \(0.338766\pi\)
\(788\) −12.3720 −0.440735
\(789\) −52.8886 −1.88288
\(790\) −6.03435 −0.214693
\(791\) −6.82831 −0.242787
\(792\) −1.12633 −0.0400223
\(793\) 16.5567 0.587945
\(794\) 13.6736 0.485259
\(795\) −15.2988 −0.542594
\(796\) −4.50965 −0.159840
\(797\) 3.10673 0.110046 0.0550231 0.998485i \(-0.482477\pi\)
0.0550231 + 0.998485i \(0.482477\pi\)
\(798\) 21.9125 0.775695
\(799\) −0.0698208 −0.00247008
\(800\) −13.1829 −0.466085
\(801\) −4.40087 −0.155497
\(802\) 1.62020 0.0572112
\(803\) −34.4419 −1.21543
\(804\) 13.0081 0.458761
\(805\) −45.7746 −1.61334
\(806\) −19.5227 −0.687659
\(807\) −43.5459 −1.53289
\(808\) 16.5995 0.583968
\(809\) −5.19497 −0.182645 −0.0913227 0.995821i \(-0.529109\pi\)
−0.0913227 + 0.995821i \(0.529109\pi\)
\(810\) 43.9767 1.54518
\(811\) 31.9817 1.12303 0.561514 0.827467i \(-0.310219\pi\)
0.561514 + 0.827467i \(0.310219\pi\)
\(812\) −20.7044 −0.726581
\(813\) −19.9380 −0.699255
\(814\) 0 0
\(815\) 107.697 3.77245
\(816\) 0.0502166 0.00175793
\(817\) 17.8772 0.625446
\(818\) −9.12867 −0.319176
\(819\) 5.54191 0.193650
\(820\) −6.36006 −0.222103
\(821\) 7.49467 0.261566 0.130783 0.991411i \(-0.458251\pi\)
0.130783 + 0.991411i \(0.458251\pi\)
\(822\) 32.6204 1.13777
\(823\) −40.9456 −1.42727 −0.713637 0.700516i \(-0.752953\pi\)
−0.713637 + 0.700516i \(0.752953\pi\)
\(824\) 1.89481 0.0660087
\(825\) −52.4454 −1.82591
\(826\) −20.3137 −0.706803
\(827\) 43.8214 1.52382 0.761909 0.647684i \(-0.224262\pi\)
0.761909 + 0.647684i \(0.224262\pi\)
\(828\) −1.58160 −0.0549645
\(829\) −36.3976 −1.26414 −0.632071 0.774911i \(-0.717795\pi\)
−0.632071 + 0.774911i \(0.717795\pi\)
\(830\) 7.23376 0.251088
\(831\) −45.1835 −1.56740
\(832\) 2.88400 0.0999848
\(833\) −0.161453 −0.00559401
\(834\) −17.3382 −0.600371
\(835\) 43.8397 1.51714
\(836\) 6.83404 0.236360
\(837\) 31.3972 1.08525
\(838\) −14.2480 −0.492189
\(839\) −22.1035 −0.763099 −0.381549 0.924348i \(-0.624609\pi\)
−0.381549 + 0.924348i \(0.624609\pi\)
\(840\) 28.9419 0.998591
\(841\) 3.86732 0.133356
\(842\) −33.7943 −1.16463
\(843\) −23.9149 −0.823673
\(844\) −16.2460 −0.559209
\(845\) −19.9670 −0.686885
\(846\) −1.39039 −0.0478027
\(847\) −23.5434 −0.808962
\(848\) 1.90902 0.0655561
\(849\) −0.484333 −0.0166223
\(850\) 0.352242 0.0120818
\(851\) 0 0
\(852\) 17.4392 0.597459
\(853\) −41.7706 −1.43020 −0.715100 0.699023i \(-0.753619\pi\)
−0.715100 + 0.699023i \(0.753619\pi\)
\(854\) −20.7328 −0.709462
\(855\) 7.32508 0.250513
\(856\) −0.382256 −0.0130653
\(857\) 2.80039 0.0956595 0.0478297 0.998856i \(-0.484770\pi\)
0.0478297 + 0.998856i \(0.484770\pi\)
\(858\) 11.4734 0.391696
\(859\) −33.5977 −1.14634 −0.573169 0.819437i \(-0.694286\pi\)
−0.573169 + 0.819437i \(0.694286\pi\)
\(860\) 23.6121 0.805167
\(861\) 10.1234 0.345004
\(862\) 38.8388 1.32285
\(863\) −25.4084 −0.864913 −0.432456 0.901655i \(-0.642353\pi\)
−0.432456 + 0.901655i \(0.642353\pi\)
\(864\) −4.63816 −0.157793
\(865\) −100.876 −3.42990
\(866\) −18.1122 −0.615478
\(867\) 31.9482 1.08502
\(868\) 24.4470 0.829784
\(869\) 2.99557 0.101618
\(870\) −45.9441 −1.55765
\(871\) −19.9616 −0.676371
\(872\) 0.965584 0.0326988
\(873\) −3.47202 −0.117510
\(874\) 9.59645 0.324605
\(875\) 126.014 4.26004
\(876\) 30.5789 1.03316
\(877\) 21.1796 0.715183 0.357591 0.933878i \(-0.383598\pi\)
0.357591 + 0.933878i \(0.383598\pi\)
\(878\) 14.3517 0.484345
\(879\) 16.8783 0.569292
\(880\) 9.02635 0.304278
\(881\) 14.5051 0.488690 0.244345 0.969688i \(-0.421427\pi\)
0.244345 + 0.969688i \(0.421427\pi\)
\(882\) −3.21513 −0.108259
\(883\) 37.1477 1.25012 0.625060 0.780577i \(-0.285075\pi\)
0.625060 + 0.780577i \(0.285075\pi\)
\(884\) −0.0770596 −0.00259179
\(885\) −45.0771 −1.51525
\(886\) 5.43539 0.182605
\(887\) −2.08616 −0.0700464 −0.0350232 0.999386i \(-0.511151\pi\)
−0.0350232 + 0.999386i \(0.511151\pi\)
\(888\) 0 0
\(889\) −58.1961 −1.95183
\(890\) 35.2684 1.18220
\(891\) −21.8309 −0.731364
\(892\) −0.847777 −0.0283857
\(893\) 8.43627 0.282309
\(894\) −14.5422 −0.486365
\(895\) 21.5657 0.720863
\(896\) −3.61144 −0.120650
\(897\) 16.1111 0.537934
\(898\) −0.130238 −0.00434611
\(899\) −38.8085 −1.29434
\(900\) 7.01447 0.233816
\(901\) −0.0510085 −0.00169934
\(902\) 3.15726 0.105125
\(903\) −37.5837 −1.25071
\(904\) 1.89075 0.0628852
\(905\) −4.45793 −0.148187
\(906\) 23.7485 0.788993
\(907\) −10.3856 −0.344849 −0.172425 0.985023i \(-0.555160\pi\)
−0.172425 + 0.985023i \(0.555160\pi\)
\(908\) 12.1287 0.402506
\(909\) −8.83240 −0.292952
\(910\) −44.4127 −1.47227
\(911\) 6.68612 0.221521 0.110761 0.993847i \(-0.464671\pi\)
0.110761 + 0.993847i \(0.464671\pi\)
\(912\) −6.06754 −0.200916
\(913\) −3.59099 −0.118844
\(914\) −28.2675 −0.935007
\(915\) −46.0071 −1.52095
\(916\) 13.2675 0.438371
\(917\) −2.97202 −0.0981449
\(918\) 0.123930 0.00409030
\(919\) −48.7881 −1.60937 −0.804685 0.593702i \(-0.797666\pi\)
−0.804685 + 0.593702i \(0.797666\pi\)
\(920\) 12.6749 0.417879
\(921\) −27.8116 −0.916423
\(922\) −8.01301 −0.263894
\(923\) −26.7613 −0.880859
\(924\) −14.3673 −0.472651
\(925\) 0 0
\(926\) −34.2291 −1.12484
\(927\) −1.00820 −0.0331138
\(928\) 5.73300 0.188195
\(929\) 24.1954 0.793825 0.396913 0.917856i \(-0.370082\pi\)
0.396913 + 0.917856i \(0.370082\pi\)
\(930\) 54.2491 1.77890
\(931\) 19.5079 0.639347
\(932\) −4.30657 −0.141066
\(933\) 50.2095 1.64378
\(934\) −20.5376 −0.672009
\(935\) −0.241181 −0.00788747
\(936\) −1.53455 −0.0501582
\(937\) −58.5924 −1.91413 −0.957065 0.289873i \(-0.906387\pi\)
−0.957065 + 0.289873i \(0.906387\pi\)
\(938\) 24.9965 0.816164
\(939\) 17.8988 0.584105
\(940\) 11.1426 0.363430
\(941\) −47.0814 −1.53481 −0.767405 0.641163i \(-0.778452\pi\)
−0.767405 + 0.641163i \(0.778452\pi\)
\(942\) 24.5198 0.798900
\(943\) 4.43347 0.144374
\(944\) 5.62482 0.183072
\(945\) 71.4261 2.32349
\(946\) −11.7215 −0.381100
\(947\) 19.1273 0.621553 0.310776 0.950483i \(-0.399411\pi\)
0.310776 + 0.950483i \(0.399411\pi\)
\(948\) −2.65959 −0.0863795
\(949\) −46.9247 −1.52324
\(950\) −42.5606 −1.38085
\(951\) −28.4833 −0.923635
\(952\) 0.0964964 0.00312747
\(953\) −0.858570 −0.0278118 −0.0139059 0.999903i \(-0.504427\pi\)
−0.0139059 + 0.999903i \(0.504427\pi\)
\(954\) −1.01577 −0.0328868
\(955\) 57.4096 1.85773
\(956\) 14.2423 0.460629
\(957\) 22.8076 0.737264
\(958\) 6.29987 0.203540
\(959\) 62.6835 2.02416
\(960\) −8.01396 −0.258649
\(961\) 14.8237 0.478185
\(962\) 0 0
\(963\) 0.203394 0.00655429
\(964\) −14.2610 −0.459316
\(965\) 19.2836 0.620762
\(966\) −20.1748 −0.649114
\(967\) −60.3645 −1.94119 −0.970596 0.240714i \(-0.922618\pi\)
−0.970596 + 0.240714i \(0.922618\pi\)
\(968\) 6.51914 0.209533
\(969\) 0.162123 0.00520813
\(970\) 27.8247 0.893396
\(971\) −35.3779 −1.13533 −0.567665 0.823260i \(-0.692153\pi\)
−0.567665 + 0.823260i \(0.692153\pi\)
\(972\) 5.46791 0.175383
\(973\) −33.3171 −1.06810
\(974\) 27.5903 0.884050
\(975\) −71.4532 −2.28834
\(976\) 5.74087 0.183761
\(977\) 34.2781 1.09665 0.548327 0.836264i \(-0.315265\pi\)
0.548327 + 0.836264i \(0.315265\pi\)
\(978\) 47.4665 1.51781
\(979\) −17.5079 −0.559556
\(980\) 25.7659 0.823063
\(981\) −0.513777 −0.0164036
\(982\) −33.0201 −1.05371
\(983\) −16.2324 −0.517734 −0.258867 0.965913i \(-0.583349\pi\)
−0.258867 + 0.965913i \(0.583349\pi\)
\(984\) −2.80315 −0.0893610
\(985\) −52.7560 −1.68095
\(986\) −0.153184 −0.00487837
\(987\) −17.7357 −0.564535
\(988\) 9.31092 0.296220
\(989\) −16.4595 −0.523382
\(990\) −4.80282 −0.152644
\(991\) 3.38215 0.107437 0.0537187 0.998556i \(-0.482893\pi\)
0.0537187 + 0.998556i \(0.482893\pi\)
\(992\) −6.76932 −0.214926
\(993\) −3.88315 −0.123228
\(994\) 33.5113 1.06292
\(995\) −19.2298 −0.609625
\(996\) 3.18822 0.101023
\(997\) −40.0933 −1.26977 −0.634883 0.772608i \(-0.718952\pi\)
−0.634883 + 0.772608i \(0.718952\pi\)
\(998\) −21.0861 −0.667469
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2738.2.a.r.1.2 6
37.5 odd 36 74.2.h.a.25.2 yes 12
37.15 odd 36 74.2.h.a.3.2 12
37.36 even 2 2738.2.a.s.1.1 6
111.5 even 36 666.2.bj.c.469.1 12
111.89 even 36 666.2.bj.c.595.1 12
148.15 even 36 592.2.bq.b.225.1 12
148.79 even 36 592.2.bq.b.321.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.3.2 12 37.15 odd 36
74.2.h.a.25.2 yes 12 37.5 odd 36
592.2.bq.b.225.1 12 148.15 even 36
592.2.bq.b.321.1 12 148.79 even 36
666.2.bj.c.469.1 12 111.5 even 36
666.2.bj.c.595.1 12 111.89 even 36
2738.2.a.r.1.2 6 1.1 even 1 trivial
2738.2.a.s.1.1 6 37.36 even 2