Properties

Label 2535.2.a.bb.1.2
Level $2535$
Weight $2$
Character 2535.1
Self dual yes
Analytic conductor $20.242$
Analytic rank $0$
Dimension $3$
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] = [2535,2,Mod(1,2535)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2535, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2535.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2535.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: \(20.2420769124\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.339877\) of defining polynomial
Character \(\chi\) \(=\) 2535.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.339877 q^{2} -1.00000 q^{3} -1.88448 q^{4} +1.00000 q^{5} +0.339877 q^{6} +0.660123 q^{7} +1.32025 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.339877 q^{2} -1.00000 q^{3} -1.88448 q^{4} +1.00000 q^{5} +0.339877 q^{6} +0.660123 q^{7} +1.32025 q^{8} +1.00000 q^{9} -0.339877 q^{10} +0.679754 q^{11} +1.88448 q^{12} -0.224361 q^{14} -1.00000 q^{15} +3.32025 q^{16} +7.42909 q^{17} -0.339877 q^{18} +0.115516 q^{19} -1.88448 q^{20} -0.660123 q^{21} -0.231033 q^{22} -7.76897 q^{23} -1.32025 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.24399 q^{28} +5.54461 q^{29} +0.339877 q^{30} -9.97370 q^{31} -3.76897 q^{32} -0.679754 q^{33} -2.52498 q^{34} +0.660123 q^{35} -1.88448 q^{36} +9.76897 q^{37} -0.0392613 q^{38} +1.32025 q^{40} +4.22436 q^{41} +0.224361 q^{42} -0.544607 q^{43} -1.28098 q^{44} +1.00000 q^{45} +2.64049 q^{46} -5.01963 q^{47} -3.32025 q^{48} -6.56424 q^{49} -0.339877 q^{50} -7.42909 q^{51} +0.679754 q^{53} +0.339877 q^{54} +0.679754 q^{55} +0.871525 q^{56} -0.115516 q^{57} -1.88448 q^{58} +2.22436 q^{59} +1.88448 q^{60} -4.20473 q^{61} +3.38983 q^{62} +0.660123 q^{63} -5.35951 q^{64} +0.231033 q^{66} -7.63382 q^{67} -14.0000 q^{68} +7.76897 q^{69} -0.224361 q^{70} +7.31357 q^{71} +1.32025 q^{72} +8.01963 q^{73} -3.32025 q^{74} -1.00000 q^{75} -0.217689 q^{76} +0.448721 q^{77} +9.97370 q^{79} +3.32025 q^{80} +1.00000 q^{81} -1.43576 q^{82} +1.76897 q^{83} +1.24399 q^{84} +7.42909 q^{85} +0.185099 q^{86} -5.54461 q^{87} +0.897442 q^{88} +13.5446 q^{89} -0.339877 q^{90} +14.6405 q^{92} +9.97370 q^{93} +1.70606 q^{94} +0.115516 q^{95} +3.76897 q^{96} +9.90411 q^{97} +2.23103 q^{98} +0.679754 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 6 q^{4} + 3 q^{5} + 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 6 q^{4} + 3 q^{5} + 3 q^{7} + 6 q^{8} + 3 q^{9} - 6 q^{12} + 12 q^{14} - 3 q^{15} + 12 q^{16} + 12 q^{19} + 6 q^{20} - 3 q^{21} - 24 q^{22} - 6 q^{24} + 3 q^{25} - 3 q^{27} + 12 q^{28} + 6 q^{29} + 3 q^{31} + 12 q^{32} + 3 q^{35} + 6 q^{36} + 6 q^{37} + 6 q^{38} + 6 q^{40} - 12 q^{42} + 9 q^{43} - 12 q^{44} + 3 q^{45} + 12 q^{46} - 12 q^{47} - 12 q^{48} - 6 q^{49} + 30 q^{56} - 12 q^{57} + 6 q^{58} - 6 q^{59} - 6 q^{60} - 3 q^{61} - 6 q^{62} + 3 q^{63} - 12 q^{64} + 24 q^{66} + 9 q^{67} - 42 q^{68} + 12 q^{70} - 12 q^{71} + 6 q^{72} + 21 q^{73} - 12 q^{74} - 3 q^{75} + 48 q^{76} - 24 q^{77} - 3 q^{79} + 12 q^{80} + 3 q^{81} - 18 q^{82} - 18 q^{83} - 12 q^{84} - 6 q^{86} - 6 q^{87} - 48 q^{88} + 30 q^{89} + 48 q^{92} - 3 q^{93} + 36 q^{94} + 12 q^{95} - 12 q^{96} + 15 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.339877 −0.240329 −0.120165 0.992754i \(-0.538342\pi\)
−0.120165 + 0.992754i \(0.538342\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.88448 −0.942242
\(5\) 1.00000 0.447214
\(6\) 0.339877 0.138754
\(7\) 0.660123 0.249503 0.124752 0.992188i \(-0.460187\pi\)
0.124752 + 0.992188i \(0.460187\pi\)
\(8\) 1.32025 0.466778
\(9\) 1.00000 0.333333
\(10\) −0.339877 −0.107479
\(11\) 0.679754 0.204953 0.102477 0.994735i \(-0.467323\pi\)
0.102477 + 0.994735i \(0.467323\pi\)
\(12\) 1.88448 0.544004
\(13\) 0 0
\(14\) −0.224361 −0.0599629
\(15\) −1.00000 −0.258199
\(16\) 3.32025 0.830062
\(17\) 7.42909 1.80182 0.900910 0.434007i \(-0.142901\pi\)
0.900910 + 0.434007i \(0.142901\pi\)
\(18\) −0.339877 −0.0801098
\(19\) 0.115516 0.0265013 0.0132506 0.999912i \(-0.495782\pi\)
0.0132506 + 0.999912i \(0.495782\pi\)
\(20\) −1.88448 −0.421383
\(21\) −0.660123 −0.144051
\(22\) −0.231033 −0.0492563
\(23\) −7.76897 −1.61994 −0.809971 0.586470i \(-0.800517\pi\)
−0.809971 + 0.586470i \(0.800517\pi\)
\(24\) −1.32025 −0.269494
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.24399 −0.235092
\(29\) 5.54461 1.02961 0.514804 0.857308i \(-0.327865\pi\)
0.514804 + 0.857308i \(0.327865\pi\)
\(30\) 0.339877 0.0620527
\(31\) −9.97370 −1.79133 −0.895664 0.444730i \(-0.853299\pi\)
−0.895664 + 0.444730i \(0.853299\pi\)
\(32\) −3.76897 −0.666266
\(33\) −0.679754 −0.118330
\(34\) −2.52498 −0.433030
\(35\) 0.660123 0.111581
\(36\) −1.88448 −0.314081
\(37\) 9.76897 1.60601 0.803004 0.595973i \(-0.203234\pi\)
0.803004 + 0.595973i \(0.203234\pi\)
\(38\) −0.0392613 −0.00636903
\(39\) 0 0
\(40\) 1.32025 0.208749
\(41\) 4.22436 0.659734 0.329867 0.944027i \(-0.392996\pi\)
0.329867 + 0.944027i \(0.392996\pi\)
\(42\) 0.224361 0.0346196
\(43\) −0.544607 −0.0830518 −0.0415259 0.999137i \(-0.513222\pi\)
−0.0415259 + 0.999137i \(0.513222\pi\)
\(44\) −1.28098 −0.193116
\(45\) 1.00000 0.149071
\(46\) 2.64049 0.389319
\(47\) −5.01963 −0.732188 −0.366094 0.930578i \(-0.619305\pi\)
−0.366094 + 0.930578i \(0.619305\pi\)
\(48\) −3.32025 −0.479236
\(49\) −6.56424 −0.937748
\(50\) −0.339877 −0.0480659
\(51\) −7.42909 −1.04028
\(52\) 0 0
\(53\) 0.679754 0.0933714 0.0466857 0.998910i \(-0.485134\pi\)
0.0466857 + 0.998910i \(0.485134\pi\)
\(54\) 0.339877 0.0462514
\(55\) 0.679754 0.0916580
\(56\) 0.871525 0.116462
\(57\) −0.115516 −0.0153005
\(58\) −1.88448 −0.247445
\(59\) 2.22436 0.289587 0.144794 0.989462i \(-0.453748\pi\)
0.144794 + 0.989462i \(0.453748\pi\)
\(60\) 1.88448 0.243286
\(61\) −4.20473 −0.538361 −0.269180 0.963090i \(-0.586753\pi\)
−0.269180 + 0.963090i \(0.586753\pi\)
\(62\) 3.38983 0.430509
\(63\) 0.660123 0.0831677
\(64\) −5.35951 −0.669938
\(65\) 0 0
\(66\) 0.231033 0.0284381
\(67\) −7.63382 −0.932620 −0.466310 0.884621i \(-0.654417\pi\)
−0.466310 + 0.884621i \(0.654417\pi\)
\(68\) −14.0000 −1.69775
\(69\) 7.76897 0.935274
\(70\) −0.224361 −0.0268162
\(71\) 7.31357 0.867962 0.433981 0.900922i \(-0.357109\pi\)
0.433981 + 0.900922i \(0.357109\pi\)
\(72\) 1.32025 0.155593
\(73\) 8.01963 0.938627 0.469313 0.883032i \(-0.344501\pi\)
0.469313 + 0.883032i \(0.344501\pi\)
\(74\) −3.32025 −0.385971
\(75\) −1.00000 −0.115470
\(76\) −0.217689 −0.0249706
\(77\) 0.448721 0.0511365
\(78\) 0 0
\(79\) 9.97370 1.12213 0.561064 0.827772i \(-0.310392\pi\)
0.561064 + 0.827772i \(0.310392\pi\)
\(80\) 3.32025 0.371215
\(81\) 1.00000 0.111111
\(82\) −1.43576 −0.158553
\(83\) 1.76897 0.194169 0.0970847 0.995276i \(-0.469048\pi\)
0.0970847 + 0.995276i \(0.469048\pi\)
\(84\) 1.24399 0.135731
\(85\) 7.42909 0.805798
\(86\) 0.185099 0.0199598
\(87\) −5.54461 −0.594444
\(88\) 0.897442 0.0956677
\(89\) 13.5446 1.43573 0.717863 0.696185i \(-0.245121\pi\)
0.717863 + 0.696185i \(0.245121\pi\)
\(90\) −0.339877 −0.0358262
\(91\) 0 0
\(92\) 14.6405 1.52638
\(93\) 9.97370 1.03422
\(94\) 1.70606 0.175966
\(95\) 0.115516 0.0118517
\(96\) 3.76897 0.384669
\(97\) 9.90411 1.00561 0.502805 0.864400i \(-0.332301\pi\)
0.502805 + 0.864400i \(0.332301\pi\)
\(98\) 2.23103 0.225368
\(99\) 0.679754 0.0683178
\(100\) −1.88448 −0.188448
\(101\) −9.35951 −0.931306 −0.465653 0.884967i \(-0.654180\pi\)
−0.465653 + 0.884967i \(0.654180\pi\)
\(102\) 2.52498 0.250010
\(103\) 4.50535 0.443925 0.221962 0.975055i \(-0.428754\pi\)
0.221962 + 0.975055i \(0.428754\pi\)
\(104\) 0 0
\(105\) −0.660123 −0.0644214
\(106\) −0.231033 −0.0224399
\(107\) 0.570909 0.0551919 0.0275960 0.999619i \(-0.491215\pi\)
0.0275960 + 0.999619i \(0.491215\pi\)
\(108\) 1.88448 0.181335
\(109\) 15.4095 1.47596 0.737979 0.674823i \(-0.235780\pi\)
0.737979 + 0.674823i \(0.235780\pi\)
\(110\) −0.231033 −0.0220281
\(111\) −9.76897 −0.927229
\(112\) 2.19177 0.207103
\(113\) −5.32025 −0.500487 −0.250243 0.968183i \(-0.580511\pi\)
−0.250243 + 0.968183i \(0.580511\pi\)
\(114\) 0.0392613 0.00367716
\(115\) −7.76897 −0.724460
\(116\) −10.4487 −0.970139
\(117\) 0 0
\(118\) −0.756009 −0.0695962
\(119\) 4.90411 0.449559
\(120\) −1.32025 −0.120521
\(121\) −10.5379 −0.957994
\(122\) 1.42909 0.129384
\(123\) −4.22436 −0.380898
\(124\) 18.7953 1.68787
\(125\) 1.00000 0.0894427
\(126\) −0.224361 −0.0199876
\(127\) −12.0196 −1.06657 −0.533285 0.845936i \(-0.679043\pi\)
−0.533285 + 0.845936i \(0.679043\pi\)
\(128\) 9.35951 0.827271
\(129\) 0.544607 0.0479500
\(130\) 0 0
\(131\) 10.9041 0.952697 0.476348 0.879257i \(-0.341960\pi\)
0.476348 + 0.879257i \(0.341960\pi\)
\(132\) 1.28098 0.111495
\(133\) 0.0762550 0.00661215
\(134\) 2.59456 0.224136
\(135\) −1.00000 −0.0860663
\(136\) 9.80823 0.841049
\(137\) 3.38983 0.289613 0.144806 0.989460i \(-0.453744\pi\)
0.144806 + 0.989460i \(0.453744\pi\)
\(138\) −2.64049 −0.224774
\(139\) 14.8082 1.25602 0.628009 0.778206i \(-0.283870\pi\)
0.628009 + 0.778206i \(0.283870\pi\)
\(140\) −1.24399 −0.105136
\(141\) 5.01963 0.422729
\(142\) −2.48571 −0.208597
\(143\) 0 0
\(144\) 3.32025 0.276687
\(145\) 5.54461 0.460455
\(146\) −2.72569 −0.225579
\(147\) 6.56424 0.541409
\(148\) −18.4095 −1.51325
\(149\) −17.0892 −1.40000 −0.700001 0.714141i \(-0.746817\pi\)
−0.700001 + 0.714141i \(0.746817\pi\)
\(150\) 0.339877 0.0277508
\(151\) 13.0130 1.05898 0.529490 0.848316i \(-0.322383\pi\)
0.529490 + 0.848316i \(0.322383\pi\)
\(152\) 0.152510 0.0123702
\(153\) 7.42909 0.600606
\(154\) −0.152510 −0.0122896
\(155\) −9.97370 −0.801107
\(156\) 0 0
\(157\) −0.775639 −0.0619028 −0.0309514 0.999521i \(-0.509854\pi\)
−0.0309514 + 0.999521i \(0.509854\pi\)
\(158\) −3.38983 −0.269680
\(159\) −0.679754 −0.0539080
\(160\) −3.76897 −0.297963
\(161\) −5.12847 −0.404180
\(162\) −0.339877 −0.0267033
\(163\) 12.1981 0.955426 0.477713 0.878516i \(-0.341466\pi\)
0.477713 + 0.878516i \(0.341466\pi\)
\(164\) −7.96074 −0.621629
\(165\) −0.679754 −0.0529188
\(166\) −0.601231 −0.0466646
\(167\) −1.59054 −0.123080 −0.0615398 0.998105i \(-0.519601\pi\)
−0.0615398 + 0.998105i \(0.519601\pi\)
\(168\) −0.871525 −0.0672396
\(169\) 0 0
\(170\) −2.52498 −0.193657
\(171\) 0.115516 0.00883375
\(172\) 1.02630 0.0782548
\(173\) 12.7493 0.969314 0.484657 0.874704i \(-0.338944\pi\)
0.484657 + 0.874704i \(0.338944\pi\)
\(174\) 1.88448 0.142862
\(175\) 0.660123 0.0499006
\(176\) 2.25695 0.170124
\(177\) −2.22436 −0.167193
\(178\) −4.60350 −0.345047
\(179\) 17.7623 1.32762 0.663808 0.747903i \(-0.268939\pi\)
0.663808 + 0.747903i \(0.268939\pi\)
\(180\) −1.88448 −0.140461
\(181\) 7.02630 0.522261 0.261130 0.965304i \(-0.415905\pi\)
0.261130 + 0.965304i \(0.415905\pi\)
\(182\) 0 0
\(183\) 4.20473 0.310823
\(184\) −10.2569 −0.756152
\(185\) 9.76897 0.718229
\(186\) −3.38983 −0.248554
\(187\) 5.04995 0.369289
\(188\) 9.45941 0.689899
\(189\) −0.660123 −0.0480169
\(190\) −0.0392613 −0.00284832
\(191\) −13.9541 −1.00968 −0.504840 0.863213i \(-0.668449\pi\)
−0.504840 + 0.863213i \(0.668449\pi\)
\(192\) 5.35951 0.386789
\(193\) 23.7493 1.70951 0.854757 0.519028i \(-0.173706\pi\)
0.854757 + 0.519028i \(0.173706\pi\)
\(194\) −3.36618 −0.241678
\(195\) 0 0
\(196\) 12.3702 0.883586
\(197\) 15.8082 1.12629 0.563145 0.826358i \(-0.309591\pi\)
0.563145 + 0.826358i \(0.309591\pi\)
\(198\) −0.231033 −0.0164188
\(199\) 8.76897 0.621616 0.310808 0.950473i \(-0.399400\pi\)
0.310808 + 0.950473i \(0.399400\pi\)
\(200\) 1.32025 0.0933555
\(201\) 7.63382 0.538448
\(202\) 3.18108 0.223820
\(203\) 3.66012 0.256890
\(204\) 14.0000 0.980196
\(205\) 4.22436 0.295042
\(206\) −1.53126 −0.106688
\(207\) −7.76897 −0.539981
\(208\) 0 0
\(209\) 0.0785226 0.00543152
\(210\) 0.224361 0.0154824
\(211\) 8.80823 0.606383 0.303192 0.952930i \(-0.401948\pi\)
0.303192 + 0.952930i \(0.401948\pi\)
\(212\) −1.28098 −0.0879784
\(213\) −7.31357 −0.501118
\(214\) −0.194039 −0.0132642
\(215\) −0.544607 −0.0371419
\(216\) −1.32025 −0.0898314
\(217\) −6.58387 −0.446942
\(218\) −5.23732 −0.354716
\(219\) −8.01963 −0.541916
\(220\) −1.28098 −0.0863640
\(221\) 0 0
\(222\) 3.32025 0.222840
\(223\) 10.0393 0.672279 0.336139 0.941812i \(-0.390879\pi\)
0.336139 + 0.941812i \(0.390879\pi\)
\(224\) −2.48798 −0.166235
\(225\) 1.00000 0.0666667
\(226\) 1.80823 0.120282
\(227\) 1.21140 0.0804036 0.0402018 0.999192i \(-0.487200\pi\)
0.0402018 + 0.999192i \(0.487200\pi\)
\(228\) 0.217689 0.0144168
\(229\) 19.2440 1.27168 0.635839 0.771821i \(-0.280654\pi\)
0.635839 + 0.771821i \(0.280654\pi\)
\(230\) 2.64049 0.174109
\(231\) −0.448721 −0.0295237
\(232\) 7.32025 0.480598
\(233\) −23.9081 −1.56627 −0.783137 0.621849i \(-0.786382\pi\)
−0.783137 + 0.621849i \(0.786382\pi\)
\(234\) 0 0
\(235\) −5.01963 −0.327445
\(236\) −4.19177 −0.272861
\(237\) −9.97370 −0.647861
\(238\) −1.66680 −0.108042
\(239\) 18.6798 1.20829 0.604146 0.796873i \(-0.293514\pi\)
0.604146 + 0.796873i \(0.293514\pi\)
\(240\) −3.32025 −0.214321
\(241\) 6.11552 0.393935 0.196968 0.980410i \(-0.436891\pi\)
0.196968 + 0.980410i \(0.436891\pi\)
\(242\) 3.58160 0.230234
\(243\) −1.00000 −0.0641500
\(244\) 7.92375 0.507266
\(245\) −6.56424 −0.419374
\(246\) 1.43576 0.0915409
\(247\) 0 0
\(248\) −13.1677 −0.836152
\(249\) −1.76897 −0.112104
\(250\) −0.339877 −0.0214957
\(251\) 3.35951 0.212050 0.106025 0.994363i \(-0.466188\pi\)
0.106025 + 0.994363i \(0.466188\pi\)
\(252\) −1.24399 −0.0783641
\(253\) −5.28098 −0.332013
\(254\) 4.08519 0.256328
\(255\) −7.42909 −0.465228
\(256\) 7.53793 0.471121
\(257\) −10.1088 −0.630572 −0.315286 0.948997i \(-0.602101\pi\)
−0.315286 + 0.948997i \(0.602101\pi\)
\(258\) −0.185099 −0.0115238
\(259\) 6.44872 0.400704
\(260\) 0 0
\(261\) 5.54461 0.343203
\(262\) −3.70606 −0.228961
\(263\) −22.5183 −1.38854 −0.694269 0.719716i \(-0.744272\pi\)
−0.694269 + 0.719716i \(0.744272\pi\)
\(264\) −0.897442 −0.0552338
\(265\) 0.679754 0.0417569
\(266\) −0.0259173 −0.00158909
\(267\) −13.5446 −0.828916
\(268\) 14.3858 0.878753
\(269\) 8.49465 0.517928 0.258964 0.965887i \(-0.416619\pi\)
0.258964 + 0.965887i \(0.416619\pi\)
\(270\) 0.339877 0.0206842
\(271\) 9.37020 0.569199 0.284600 0.958647i \(-0.408139\pi\)
0.284600 + 0.958647i \(0.408139\pi\)
\(272\) 24.6664 1.49562
\(273\) 0 0
\(274\) −1.15212 −0.0696024
\(275\) 0.679754 0.0409907
\(276\) −14.6405 −0.881254
\(277\) 0.719015 0.0432014 0.0216007 0.999767i \(-0.493124\pi\)
0.0216007 + 0.999767i \(0.493124\pi\)
\(278\) −5.03297 −0.301858
\(279\) −9.97370 −0.597110
\(280\) 0.871525 0.0520836
\(281\) 1.54461 0.0921435 0.0460718 0.998938i \(-0.485330\pi\)
0.0460718 + 0.998938i \(0.485330\pi\)
\(282\) −1.70606 −0.101594
\(283\) −18.1195 −1.07709 −0.538547 0.842595i \(-0.681027\pi\)
−0.538547 + 0.842595i \(0.681027\pi\)
\(284\) −13.7823 −0.817830
\(285\) −0.115516 −0.00684259
\(286\) 0 0
\(287\) 2.78860 0.164606
\(288\) −3.76897 −0.222089
\(289\) 38.1914 2.24655
\(290\) −1.88448 −0.110661
\(291\) −9.90411 −0.580589
\(292\) −15.1129 −0.884413
\(293\) −30.5050 −1.78212 −0.891059 0.453887i \(-0.850037\pi\)
−0.891059 + 0.453887i \(0.850037\pi\)
\(294\) −2.23103 −0.130116
\(295\) 2.22436 0.129507
\(296\) 12.8974 0.749649
\(297\) −0.679754 −0.0394433
\(298\) 5.80823 0.336462
\(299\) 0 0
\(300\) 1.88448 0.108801
\(301\) −0.359508 −0.0207217
\(302\) −4.42280 −0.254504
\(303\) 9.35951 0.537690
\(304\) 0.383543 0.0219977
\(305\) −4.20473 −0.240762
\(306\) −2.52498 −0.144343
\(307\) −4.77564 −0.272560 −0.136280 0.990670i \(-0.543515\pi\)
−0.136280 + 0.990670i \(0.543515\pi\)
\(308\) −0.845608 −0.0481830
\(309\) −4.50535 −0.256300
\(310\) 3.38983 0.192529
\(311\) 30.5812 1.73410 0.867051 0.498220i \(-0.166013\pi\)
0.867051 + 0.498220i \(0.166013\pi\)
\(312\) 0 0
\(313\) −26.7230 −1.51048 −0.755238 0.655451i \(-0.772479\pi\)
−0.755238 + 0.655451i \(0.772479\pi\)
\(314\) 0.263622 0.0148770
\(315\) 0.660123 0.0371937
\(316\) −18.7953 −1.05732
\(317\) −20.6271 −1.15854 −0.579268 0.815137i \(-0.696662\pi\)
−0.579268 + 0.815137i \(0.696662\pi\)
\(318\) 0.231033 0.0129557
\(319\) 3.76897 0.211022
\(320\) −5.35951 −0.299606
\(321\) −0.570909 −0.0318651
\(322\) 1.74305 0.0971364
\(323\) 0.858181 0.0477505
\(324\) −1.88448 −0.104694
\(325\) 0 0
\(326\) −4.14584 −0.229617
\(327\) −15.4095 −0.852145
\(328\) 5.57720 0.307949
\(329\) −3.31357 −0.182683
\(330\) 0.231033 0.0127179
\(331\) 5.37020 0.295173 0.147586 0.989049i \(-0.452850\pi\)
0.147586 + 0.989049i \(0.452850\pi\)
\(332\) −3.33359 −0.182955
\(333\) 9.76897 0.535336
\(334\) 0.540588 0.0295797
\(335\) −7.63382 −0.417080
\(336\) −2.19177 −0.119571
\(337\) 28.7623 1.56678 0.783391 0.621529i \(-0.213488\pi\)
0.783391 + 0.621529i \(0.213488\pi\)
\(338\) 0 0
\(339\) 5.32025 0.288956
\(340\) −14.0000 −0.759257
\(341\) −6.77966 −0.367139
\(342\) −0.0392613 −0.00212301
\(343\) −8.95407 −0.483474
\(344\) −0.719015 −0.0387667
\(345\) 7.76897 0.418267
\(346\) −4.33320 −0.232955
\(347\) 22.3961 1.20229 0.601143 0.799141i \(-0.294712\pi\)
0.601143 + 0.799141i \(0.294712\pi\)
\(348\) 10.4487 0.560110
\(349\) −11.9737 −0.640937 −0.320469 0.947259i \(-0.603840\pi\)
−0.320469 + 0.947259i \(0.603840\pi\)
\(350\) −0.224361 −0.0119926
\(351\) 0 0
\(352\) −2.56197 −0.136553
\(353\) −17.0366 −0.906767 −0.453384 0.891316i \(-0.649783\pi\)
−0.453384 + 0.891316i \(0.649783\pi\)
\(354\) 0.756009 0.0401814
\(355\) 7.31357 0.388164
\(356\) −25.5246 −1.35280
\(357\) −4.90411 −0.259553
\(358\) −6.03699 −0.319065
\(359\) −35.0825 −1.85159 −0.925793 0.378031i \(-0.876601\pi\)
−0.925793 + 0.378031i \(0.876601\pi\)
\(360\) 1.32025 0.0695831
\(361\) −18.9867 −0.999298
\(362\) −2.38808 −0.125515
\(363\) 10.5379 0.553098
\(364\) 0 0
\(365\) 8.01963 0.419767
\(366\) −1.42909 −0.0746998
\(367\) 20.0825 1.04830 0.524150 0.851626i \(-0.324383\pi\)
0.524150 + 0.851626i \(0.324383\pi\)
\(368\) −25.7949 −1.34465
\(369\) 4.22436 0.219911
\(370\) −3.32025 −0.172611
\(371\) 0.448721 0.0232964
\(372\) −18.7953 −0.974489
\(373\) −23.4790 −1.21570 −0.607849 0.794052i \(-0.707968\pi\)
−0.607849 + 0.794052i \(0.707968\pi\)
\(374\) −1.71636 −0.0887510
\(375\) −1.00000 −0.0516398
\(376\) −6.62715 −0.341769
\(377\) 0 0
\(378\) 0.224361 0.0115399
\(379\) 21.1414 1.08596 0.542981 0.839745i \(-0.317295\pi\)
0.542981 + 0.839745i \(0.317295\pi\)
\(380\) −0.217689 −0.0111672
\(381\) 12.0196 0.615784
\(382\) 4.74266 0.242656
\(383\) 1.73865 0.0888406 0.0444203 0.999013i \(-0.485856\pi\)
0.0444203 + 0.999013i \(0.485856\pi\)
\(384\) −9.35951 −0.477625
\(385\) 0.448721 0.0228689
\(386\) −8.07185 −0.410846
\(387\) −0.544607 −0.0276839
\(388\) −18.6641 −0.947528
\(389\) −26.6664 −1.35204 −0.676020 0.736883i \(-0.736297\pi\)
−0.676020 + 0.736883i \(0.736297\pi\)
\(390\) 0 0
\(391\) −57.7164 −2.91884
\(392\) −8.66641 −0.437720
\(393\) −10.9041 −0.550040
\(394\) −5.37285 −0.270680
\(395\) 9.97370 0.501831
\(396\) −1.28098 −0.0643719
\(397\) 14.5446 0.729973 0.364986 0.931013i \(-0.381074\pi\)
0.364986 + 0.931013i \(0.381074\pi\)
\(398\) −2.98037 −0.149392
\(399\) −0.0762550 −0.00381752
\(400\) 3.32025 0.166012
\(401\) −8.37020 −0.417988 −0.208994 0.977917i \(-0.567019\pi\)
−0.208994 + 0.977917i \(0.567019\pi\)
\(402\) −2.59456 −0.129405
\(403\) 0 0
\(404\) 17.6378 0.877515
\(405\) 1.00000 0.0496904
\(406\) −1.24399 −0.0617382
\(407\) 6.64049 0.329157
\(408\) −9.80823 −0.485580
\(409\) 17.2547 0.853189 0.426595 0.904443i \(-0.359713\pi\)
0.426595 + 0.904443i \(0.359713\pi\)
\(410\) −1.43576 −0.0709073
\(411\) −3.38983 −0.167208
\(412\) −8.49025 −0.418285
\(413\) 1.46835 0.0722529
\(414\) 2.64049 0.129773
\(415\) 1.76897 0.0868352
\(416\) 0 0
\(417\) −14.8082 −0.725162
\(418\) −0.0266880 −0.00130535
\(419\) 12.8649 0.628489 0.314245 0.949342i \(-0.398249\pi\)
0.314245 + 0.949342i \(0.398249\pi\)
\(420\) 1.24399 0.0607006
\(421\) 24.5616 1.19706 0.598529 0.801101i \(-0.295752\pi\)
0.598529 + 0.801101i \(0.295752\pi\)
\(422\) −2.99371 −0.145732
\(423\) −5.01963 −0.244063
\(424\) 0.897442 0.0435837
\(425\) 7.42909 0.360364
\(426\) 2.48571 0.120433
\(427\) −2.77564 −0.134323
\(428\) −1.07587 −0.0520041
\(429\) 0 0
\(430\) 0.185099 0.00892628
\(431\) 24.7797 1.19359 0.596797 0.802392i \(-0.296440\pi\)
0.596797 + 0.802392i \(0.296440\pi\)
\(432\) −3.32025 −0.159745
\(433\) −14.0589 −0.675627 −0.337814 0.941213i \(-0.609687\pi\)
−0.337814 + 0.941213i \(0.609687\pi\)
\(434\) 2.23770 0.107413
\(435\) −5.54461 −0.265844
\(436\) −29.0389 −1.39071
\(437\) −0.897442 −0.0429305
\(438\) 2.72569 0.130238
\(439\) 20.8452 0.994888 0.497444 0.867496i \(-0.334272\pi\)
0.497444 + 0.867496i \(0.334272\pi\)
\(440\) 0.897442 0.0427839
\(441\) −6.56424 −0.312583
\(442\) 0 0
\(443\) −11.4291 −0.543012 −0.271506 0.962437i \(-0.587522\pi\)
−0.271506 + 0.962437i \(0.587522\pi\)
\(444\) 18.4095 0.873674
\(445\) 13.5446 0.642076
\(446\) −3.41211 −0.161568
\(447\) 17.0892 0.808292
\(448\) −3.53793 −0.167152
\(449\) 25.9081 1.22268 0.611340 0.791368i \(-0.290631\pi\)
0.611340 + 0.791368i \(0.290631\pi\)
\(450\) −0.339877 −0.0160220
\(451\) 2.87153 0.135215
\(452\) 10.0259 0.471579
\(453\) −13.0130 −0.611402
\(454\) −0.411728 −0.0193233
\(455\) 0 0
\(456\) −0.152510 −0.00714193
\(457\) 5.90411 0.276183 0.138091 0.990419i \(-0.455903\pi\)
0.138091 + 0.990419i \(0.455903\pi\)
\(458\) −6.54059 −0.305622
\(459\) −7.42909 −0.346760
\(460\) 14.6405 0.682616
\(461\) −15.4920 −0.721534 −0.360767 0.932656i \(-0.617485\pi\)
−0.360767 + 0.932656i \(0.617485\pi\)
\(462\) 0.152510 0.00709541
\(463\) 17.4420 0.810601 0.405300 0.914184i \(-0.367167\pi\)
0.405300 + 0.914184i \(0.367167\pi\)
\(464\) 18.4095 0.854638
\(465\) 9.97370 0.462519
\(466\) 8.12582 0.376421
\(467\) −20.4790 −0.947657 −0.473829 0.880617i \(-0.657128\pi\)
−0.473829 + 0.880617i \(0.657128\pi\)
\(468\) 0 0
\(469\) −5.03926 −0.232691
\(470\) 1.70606 0.0786945
\(471\) 0.775639 0.0357396
\(472\) 2.93670 0.135173
\(473\) −0.370199 −0.0170217
\(474\) 3.38983 0.155700
\(475\) 0.115516 0.00530025
\(476\) −9.24172 −0.423594
\(477\) 0.679754 0.0311238
\(478\) −6.34882 −0.290388
\(479\) −40.9907 −1.87291 −0.936456 0.350785i \(-0.885915\pi\)
−0.936456 + 0.350785i \(0.885915\pi\)
\(480\) 3.76897 0.172029
\(481\) 0 0
\(482\) −2.07852 −0.0946741
\(483\) 5.12847 0.233354
\(484\) 19.8586 0.902662
\(485\) 9.90411 0.449723
\(486\) 0.339877 0.0154171
\(487\) 24.0629 1.09039 0.545197 0.838308i \(-0.316455\pi\)
0.545197 + 0.838308i \(0.316455\pi\)
\(488\) −5.55128 −0.251295
\(489\) −12.1981 −0.551615
\(490\) 2.23103 0.100788
\(491\) 36.7730 1.65954 0.829771 0.558104i \(-0.188471\pi\)
0.829771 + 0.558104i \(0.188471\pi\)
\(492\) 7.96074 0.358898
\(493\) 41.1914 1.85517
\(494\) 0 0
\(495\) 0.679754 0.0305527
\(496\) −33.1151 −1.48691
\(497\) 4.82786 0.216559
\(498\) 0.601231 0.0269418
\(499\) 34.8212 1.55881 0.779405 0.626520i \(-0.215521\pi\)
0.779405 + 0.626520i \(0.215521\pi\)
\(500\) −1.88448 −0.0842767
\(501\) 1.59054 0.0710601
\(502\) −1.14182 −0.0509619
\(503\) −32.0259 −1.42797 −0.713983 0.700164i \(-0.753110\pi\)
−0.713983 + 0.700164i \(0.753110\pi\)
\(504\) 0.871525 0.0388208
\(505\) −9.35951 −0.416493
\(506\) 1.79488 0.0797924
\(507\) 0 0
\(508\) 22.6508 1.00497
\(509\) −41.3462 −1.83264 −0.916318 0.400451i \(-0.868854\pi\)
−0.916318 + 0.400451i \(0.868854\pi\)
\(510\) 2.52498 0.111808
\(511\) 5.29394 0.234190
\(512\) −21.2810 −0.940496
\(513\) −0.115516 −0.00510017
\(514\) 3.43576 0.151545
\(515\) 4.50535 0.198529
\(516\) −1.02630 −0.0451805
\(517\) −3.41211 −0.150065
\(518\) −2.19177 −0.0963009
\(519\) −12.7493 −0.559634
\(520\) 0 0
\(521\) 4.40279 0.192890 0.0964448 0.995338i \(-0.469253\pi\)
0.0964448 + 0.995338i \(0.469253\pi\)
\(522\) −1.88448 −0.0824816
\(523\) −32.4487 −1.41888 −0.709442 0.704764i \(-0.751053\pi\)
−0.709442 + 0.704764i \(0.751053\pi\)
\(524\) −20.5486 −0.897671
\(525\) −0.660123 −0.0288101
\(526\) 7.65345 0.333706
\(527\) −74.0955 −3.22765
\(528\) −2.25695 −0.0982211
\(529\) 37.3569 1.62421
\(530\) −0.231033 −0.0100354
\(531\) 2.22436 0.0965290
\(532\) −0.143701 −0.00623024
\(533\) 0 0
\(534\) 4.60350 0.199213
\(535\) 0.570909 0.0246826
\(536\) −10.0785 −0.435326
\(537\) −17.7623 −0.766500
\(538\) −2.88714 −0.124473
\(539\) −4.46207 −0.192195
\(540\) 1.88448 0.0810953
\(541\) −8.57720 −0.368762 −0.184381 0.982855i \(-0.559028\pi\)
−0.184381 + 0.982855i \(0.559028\pi\)
\(542\) −3.18471 −0.136795
\(543\) −7.02630 −0.301528
\(544\) −28.0000 −1.20049
\(545\) 15.4095 0.660069
\(546\) 0 0
\(547\) −32.4920 −1.38926 −0.694629 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(548\) −6.38808 −0.272885
\(549\) −4.20473 −0.179454
\(550\) −0.231033 −0.00985126
\(551\) 0.640492 0.0272859
\(552\) 10.2569 0.436565
\(553\) 6.58387 0.279975
\(554\) −0.244377 −0.0103826
\(555\) −9.76897 −0.414670
\(556\) −27.9059 −1.18347
\(557\) −41.2150 −1.74634 −0.873169 0.487418i \(-0.837939\pi\)
−0.873169 + 0.487418i \(0.837939\pi\)
\(558\) 3.38983 0.143503
\(559\) 0 0
\(560\) 2.19177 0.0926192
\(561\) −5.04995 −0.213209
\(562\) −0.524976 −0.0221448
\(563\) −12.1784 −0.513260 −0.256630 0.966510i \(-0.582612\pi\)
−0.256630 + 0.966510i \(0.582612\pi\)
\(564\) −9.45941 −0.398313
\(565\) −5.32025 −0.223824
\(566\) 6.15841 0.258857
\(567\) 0.660123 0.0277226
\(568\) 9.65572 0.405145
\(569\) 6.94338 0.291081 0.145541 0.989352i \(-0.453508\pi\)
0.145541 + 0.989352i \(0.453508\pi\)
\(570\) 0.0392613 0.00164448
\(571\) 3.51429 0.147068 0.0735341 0.997293i \(-0.476572\pi\)
0.0735341 + 0.997293i \(0.476572\pi\)
\(572\) 0 0
\(573\) 13.9541 0.582939
\(574\) −0.947780 −0.0395596
\(575\) −7.76897 −0.323988
\(576\) −5.35951 −0.223313
\(577\) 3.14182 0.130796 0.0653978 0.997859i \(-0.479168\pi\)
0.0653978 + 0.997859i \(0.479168\pi\)
\(578\) −12.9804 −0.539912
\(579\) −23.7493 −0.986989
\(580\) −10.4487 −0.433860
\(581\) 1.16774 0.0484459
\(582\) 3.36618 0.139533
\(583\) 0.462065 0.0191368
\(584\) 10.5879 0.438130
\(585\) 0 0
\(586\) 10.3679 0.428295
\(587\) 37.9777 1.56751 0.783754 0.621071i \(-0.213302\pi\)
0.783754 + 0.621071i \(0.213302\pi\)
\(588\) −12.3702 −0.510138
\(589\) −1.15212 −0.0474725
\(590\) −0.756009 −0.0311244
\(591\) −15.8082 −0.650264
\(592\) 32.4354 1.33309
\(593\) 10.4487 0.429078 0.214539 0.976715i \(-0.431175\pi\)
0.214539 + 0.976715i \(0.431175\pi\)
\(594\) 0.231033 0.00947938
\(595\) 4.90411 0.201049
\(596\) 32.2043 1.31914
\(597\) −8.76897 −0.358890
\(598\) 0 0
\(599\) −45.5705 −1.86196 −0.930981 0.365069i \(-0.881045\pi\)
−0.930981 + 0.365069i \(0.881045\pi\)
\(600\) −1.32025 −0.0538988
\(601\) −14.7819 −0.602967 −0.301484 0.953471i \(-0.597482\pi\)
−0.301484 + 0.953471i \(0.597482\pi\)
\(602\) 0.122188 0.00498002
\(603\) −7.63382 −0.310873
\(604\) −24.5227 −0.997815
\(605\) −10.5379 −0.428428
\(606\) −3.18108 −0.129223
\(607\) −18.9344 −0.768525 −0.384263 0.923224i \(-0.625544\pi\)
−0.384263 + 0.923224i \(0.625544\pi\)
\(608\) −0.435377 −0.0176569
\(609\) −3.66012 −0.148316
\(610\) 1.42909 0.0578622
\(611\) 0 0
\(612\) −14.0000 −0.565916
\(613\) −2.58387 −0.104361 −0.0521807 0.998638i \(-0.516617\pi\)
−0.0521807 + 0.998638i \(0.516617\pi\)
\(614\) 1.62313 0.0655042
\(615\) −4.22436 −0.170343
\(616\) 0.592422 0.0238694
\(617\) 14.0393 0.565199 0.282600 0.959238i \(-0.408803\pi\)
0.282600 + 0.959238i \(0.408803\pi\)
\(618\) 1.53126 0.0615964
\(619\) −17.8582 −0.717781 −0.358890 0.933380i \(-0.616845\pi\)
−0.358890 + 0.933380i \(0.616845\pi\)
\(620\) 18.7953 0.754836
\(621\) 7.76897 0.311758
\(622\) −10.3938 −0.416755
\(623\) 8.94111 0.358218
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 9.08254 0.363011
\(627\) −0.0785226 −0.00313589
\(628\) 1.46168 0.0583274
\(629\) 72.5745 2.89374
\(630\) −0.224361 −0.00893874
\(631\) −24.9630 −0.993762 −0.496881 0.867819i \(-0.665521\pi\)
−0.496881 + 0.867819i \(0.665521\pi\)
\(632\) 13.1677 0.523784
\(633\) −8.80823 −0.350096
\(634\) 7.01069 0.278430
\(635\) −12.0196 −0.476984
\(636\) 1.28098 0.0507944
\(637\) 0 0
\(638\) −1.28098 −0.0507147
\(639\) 7.31357 0.289321
\(640\) 9.35951 0.369967
\(641\) 23.3069 0.920567 0.460284 0.887772i \(-0.347748\pi\)
0.460284 + 0.887772i \(0.347748\pi\)
\(642\) 0.194039 0.00765811
\(643\) 45.4790 1.79352 0.896759 0.442519i \(-0.145915\pi\)
0.896759 + 0.442519i \(0.145915\pi\)
\(644\) 9.66453 0.380836
\(645\) 0.544607 0.0214439
\(646\) −0.291676 −0.0114758
\(647\) 6.40946 0.251982 0.125991 0.992031i \(-0.459789\pi\)
0.125991 + 0.992031i \(0.459789\pi\)
\(648\) 1.32025 0.0518642
\(649\) 1.51202 0.0593519
\(650\) 0 0
\(651\) 6.58387 0.258042
\(652\) −22.9870 −0.900242
\(653\) 1.48170 0.0579832 0.0289916 0.999580i \(-0.490770\pi\)
0.0289916 + 0.999580i \(0.490770\pi\)
\(654\) 5.23732 0.204795
\(655\) 10.9041 0.426059
\(656\) 14.0259 0.547620
\(657\) 8.01963 0.312876
\(658\) 1.12621 0.0439041
\(659\) −7.08921 −0.276157 −0.138078 0.990421i \(-0.544093\pi\)
−0.138078 + 0.990421i \(0.544093\pi\)
\(660\) 1.28098 0.0498623
\(661\) 1.17843 0.0458355 0.0229178 0.999737i \(-0.492704\pi\)
0.0229178 + 0.999737i \(0.492704\pi\)
\(662\) −1.82521 −0.0709387
\(663\) 0 0
\(664\) 2.33547 0.0906339
\(665\) 0.0762550 0.00295704
\(666\) −3.32025 −0.128657
\(667\) −43.0759 −1.66790
\(668\) 2.99735 0.115971
\(669\) −10.0393 −0.388140
\(670\) 2.59456 0.100237
\(671\) −2.85818 −0.110339
\(672\) 2.48798 0.0959760
\(673\) 13.5576 0.522606 0.261303 0.965257i \(-0.415848\pi\)
0.261303 + 0.965257i \(0.415848\pi\)
\(674\) −9.77564 −0.376544
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 9.53793 0.366573 0.183286 0.983060i \(-0.441326\pi\)
0.183286 + 0.983060i \(0.441326\pi\)
\(678\) −1.80823 −0.0694446
\(679\) 6.53793 0.250903
\(680\) 9.80823 0.376128
\(681\) −1.21140 −0.0464210
\(682\) 2.30425 0.0882343
\(683\) −44.5183 −1.70345 −0.851723 0.523993i \(-0.824442\pi\)
−0.851723 + 0.523993i \(0.824442\pi\)
\(684\) −0.217689 −0.00832353
\(685\) 3.38983 0.129519
\(686\) 3.04328 0.116193
\(687\) −19.2440 −0.734204
\(688\) −1.80823 −0.0689381
\(689\) 0 0
\(690\) −2.64049 −0.100522
\(691\) −5.28060 −0.200883 −0.100442 0.994943i \(-0.532026\pi\)
−0.100442 + 0.994943i \(0.532026\pi\)
\(692\) −24.0259 −0.913328
\(693\) 0.448721 0.0170455
\(694\) −7.61192 −0.288945
\(695\) 14.8082 0.561708
\(696\) −7.32025 −0.277473
\(697\) 31.3832 1.18872
\(698\) 4.06958 0.154036
\(699\) 23.9081 0.904289
\(700\) −1.24399 −0.0470184
\(701\) 20.1392 0.760646 0.380323 0.924854i \(-0.375813\pi\)
0.380323 + 0.924854i \(0.375813\pi\)
\(702\) 0 0
\(703\) 1.12847 0.0425612
\(704\) −3.64315 −0.137306
\(705\) 5.01963 0.189050
\(706\) 5.79035 0.217923
\(707\) −6.17843 −0.232364
\(708\) 4.19177 0.157536
\(709\) 1.54059 0.0578580 0.0289290 0.999581i \(-0.490790\pi\)
0.0289290 + 0.999581i \(0.490790\pi\)
\(710\) −2.48571 −0.0932872
\(711\) 9.97370 0.374043
\(712\) 17.8822 0.670164
\(713\) 77.4853 2.90185
\(714\) 1.66680 0.0623782
\(715\) 0 0
\(716\) −33.4728 −1.25094
\(717\) −18.6798 −0.697608
\(718\) 11.9237 0.444990
\(719\) 37.0040 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(720\) 3.32025 0.123738
\(721\) 2.97408 0.110761
\(722\) 6.45313 0.240160
\(723\) −6.11552 −0.227438
\(724\) −13.2410 −0.492096
\(725\) 5.54461 0.205922
\(726\) −3.58160 −0.132926
\(727\) −44.9015 −1.66530 −0.832652 0.553797i \(-0.813178\pi\)
−0.832652 + 0.553797i \(0.813178\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.72569 −0.100882
\(731\) −4.04593 −0.149644
\(732\) −7.92375 −0.292870
\(733\) −4.94072 −0.182490 −0.0912449 0.995828i \(-0.529085\pi\)
−0.0912449 + 0.995828i \(0.529085\pi\)
\(734\) −6.82559 −0.251937
\(735\) 6.56424 0.242126
\(736\) 29.2810 1.07931
\(737\) −5.18912 −0.191144
\(738\) −1.43576 −0.0528511
\(739\) 6.03926 0.222158 0.111079 0.993812i \(-0.464569\pi\)
0.111079 + 0.993812i \(0.464569\pi\)
\(740\) −18.4095 −0.676745
\(741\) 0 0
\(742\) −0.152510 −0.00559882
\(743\) −13.0589 −0.479084 −0.239542 0.970886i \(-0.576997\pi\)
−0.239542 + 0.970886i \(0.576997\pi\)
\(744\) 13.1677 0.482753
\(745\) −17.0892 −0.626100
\(746\) 7.97998 0.292168
\(747\) 1.76897 0.0647231
\(748\) −9.51655 −0.347960
\(749\) 0.376871 0.0137706
\(750\) 0.339877 0.0124105
\(751\) 48.0236 1.75241 0.876204 0.481941i \(-0.160068\pi\)
0.876204 + 0.481941i \(0.160068\pi\)
\(752\) −16.6664 −0.607761
\(753\) −3.35951 −0.122427
\(754\) 0 0
\(755\) 13.0130 0.473590
\(756\) 1.24399 0.0452435
\(757\) −4.56424 −0.165890 −0.0829450 0.996554i \(-0.526433\pi\)
−0.0829450 + 0.996554i \(0.526433\pi\)
\(758\) −7.18548 −0.260989
\(759\) 5.28098 0.191688
\(760\) 0.152510 0.00553212
\(761\) 21.0892 0.764483 0.382242 0.924062i \(-0.375152\pi\)
0.382242 + 0.924062i \(0.375152\pi\)
\(762\) −4.08519 −0.147991
\(763\) 10.1721 0.368256
\(764\) 26.2962 0.951364
\(765\) 7.42909 0.268599
\(766\) −0.590926 −0.0213510
\(767\) 0 0
\(768\) −7.53793 −0.272002
\(769\) 42.3332 1.52657 0.763287 0.646059i \(-0.223584\pi\)
0.763287 + 0.646059i \(0.223584\pi\)
\(770\) −0.152510 −0.00549608
\(771\) 10.1088 0.364061
\(772\) −44.7552 −1.61078
\(773\) 50.2436 1.80714 0.903568 0.428444i \(-0.140938\pi\)
0.903568 + 0.428444i \(0.140938\pi\)
\(774\) 0.185099 0.00665326
\(775\) −9.97370 −0.358266
\(776\) 13.0759 0.469396
\(777\) −6.44872 −0.231347
\(778\) 9.06330 0.324935
\(779\) 0.487982 0.0174838
\(780\) 0 0
\(781\) 4.97143 0.177892
\(782\) 19.6165 0.701483
\(783\) −5.54461 −0.198148
\(784\) −21.7949 −0.778389
\(785\) −0.775639 −0.0276838
\(786\) 3.70606 0.132191
\(787\) −10.5816 −0.377193 −0.188597 0.982055i \(-0.560394\pi\)
−0.188597 + 0.982055i \(0.560394\pi\)
\(788\) −29.7903 −1.06124
\(789\) 22.5183 0.801673
\(790\) −3.38983 −0.120605
\(791\) −3.51202 −0.124873
\(792\) 0.897442 0.0318892
\(793\) 0 0
\(794\) −4.94338 −0.175434
\(795\) −0.679754 −0.0241084
\(796\) −16.5250 −0.585712
\(797\) −20.1481 −0.713683 −0.356841 0.934165i \(-0.616146\pi\)
−0.356841 + 0.934165i \(0.616146\pi\)
\(798\) 0.0259173 0.000917463 0
\(799\) −37.2913 −1.31927
\(800\) −3.76897 −0.133253
\(801\) 13.5446 0.478575
\(802\) 2.84484 0.100455
\(803\) 5.45137 0.192375
\(804\) −14.3858 −0.507348
\(805\) −5.12847 −0.180755
\(806\) 0 0
\(807\) −8.49465 −0.299026
\(808\) −12.3569 −0.434713
\(809\) 12.9108 0.453919 0.226960 0.973904i \(-0.427121\pi\)
0.226960 + 0.973904i \(0.427121\pi\)
\(810\) −0.339877 −0.0119421
\(811\) −15.8974 −0.558235 −0.279117 0.960257i \(-0.590042\pi\)
−0.279117 + 0.960257i \(0.590042\pi\)
\(812\) −6.89744 −0.242053
\(813\) −9.37020 −0.328627
\(814\) −2.25695 −0.0791061
\(815\) 12.1981 0.427279
\(816\) −24.6664 −0.863497
\(817\) −0.0629110 −0.00220098
\(818\) −5.86447 −0.205046
\(819\) 0 0
\(820\) −7.96074 −0.278001
\(821\) −37.2284 −1.29928 −0.649640 0.760242i \(-0.725080\pi\)
−0.649640 + 0.760242i \(0.725080\pi\)
\(822\) 1.15212 0.0401850
\(823\) −4.89744 −0.170714 −0.0853571 0.996350i \(-0.527203\pi\)
−0.0853571 + 0.996350i \(0.527203\pi\)
\(824\) 5.94817 0.207214
\(825\) −0.679754 −0.0236660
\(826\) −0.499059 −0.0173645
\(827\) 37.6334 1.30864 0.654321 0.756217i \(-0.272954\pi\)
0.654321 + 0.756217i \(0.272954\pi\)
\(828\) 14.6405 0.508792
\(829\) −48.9104 −1.69873 −0.849364 0.527807i \(-0.823014\pi\)
−0.849364 + 0.527807i \(0.823014\pi\)
\(830\) −0.601231 −0.0208690
\(831\) −0.719015 −0.0249424
\(832\) 0 0
\(833\) −48.7663 −1.68965
\(834\) 5.03297 0.174278
\(835\) −1.59054 −0.0550429
\(836\) −0.147975 −0.00511781
\(837\) 9.97370 0.344741
\(838\) −4.37247 −0.151044
\(839\) 48.2783 1.66675 0.833377 0.552706i \(-0.186405\pi\)
0.833377 + 0.552706i \(0.186405\pi\)
\(840\) −0.871525 −0.0300705
\(841\) 1.74266 0.0600919
\(842\) −8.34791 −0.287688
\(843\) −1.54461 −0.0531991
\(844\) −16.5990 −0.571360
\(845\) 0 0
\(846\) 1.70606 0.0586554
\(847\) −6.95633 −0.239022
\(848\) 2.25695 0.0775040
\(849\) 18.1195 0.621861
\(850\) −2.52498 −0.0866060
\(851\) −75.8948 −2.60164
\(852\) 13.7823 0.472174
\(853\) −14.4291 −0.494043 −0.247021 0.969010i \(-0.579452\pi\)
−0.247021 + 0.969010i \(0.579452\pi\)
\(854\) 0.943376 0.0322817
\(855\) 0.115516 0.00395057
\(856\) 0.753741 0.0257623
\(857\) 3.64678 0.124572 0.0622858 0.998058i \(-0.480161\pi\)
0.0622858 + 0.998058i \(0.480161\pi\)
\(858\) 0 0
\(859\) 21.7427 0.741850 0.370925 0.928663i \(-0.379041\pi\)
0.370925 + 0.928663i \(0.379041\pi\)
\(860\) 1.02630 0.0349966
\(861\) −2.78860 −0.0950352
\(862\) −8.42203 −0.286856
\(863\) −17.0892 −0.581724 −0.290862 0.956765i \(-0.593942\pi\)
−0.290862 + 0.956765i \(0.593942\pi\)
\(864\) 3.76897 0.128223
\(865\) 12.7493 0.433490
\(866\) 4.77829 0.162373
\(867\) −38.1914 −1.29705
\(868\) 12.4072 0.421128
\(869\) 6.77966 0.229984
\(870\) 1.88448 0.0638900
\(871\) 0 0
\(872\) 20.3443 0.688944
\(873\) 9.90411 0.335203
\(874\) 0.305020 0.0103175
\(875\) 0.660123 0.0223162
\(876\) 15.1129 0.510616
\(877\) −4.85818 −0.164049 −0.0820246 0.996630i \(-0.526139\pi\)
−0.0820246 + 0.996630i \(0.526139\pi\)
\(878\) −7.08481 −0.239101
\(879\) 30.5050 1.02891
\(880\) 2.25695 0.0760818
\(881\) −23.3202 −0.785679 −0.392840 0.919607i \(-0.628507\pi\)
−0.392840 + 0.919607i \(0.628507\pi\)
\(882\) 2.23103 0.0751228
\(883\) −12.8934 −0.433898 −0.216949 0.976183i \(-0.569611\pi\)
−0.216949 + 0.976183i \(0.569611\pi\)
\(884\) 0 0
\(885\) −2.22436 −0.0747711
\(886\) 3.88448 0.130502
\(887\) 30.9278 1.03845 0.519226 0.854637i \(-0.326220\pi\)
0.519226 + 0.854637i \(0.326220\pi\)
\(888\) −12.8974 −0.432810
\(889\) −7.93444 −0.266112
\(890\) −4.60350 −0.154310
\(891\) 0.679754 0.0227726
\(892\) −18.9188 −0.633449
\(893\) −0.579849 −0.0194039
\(894\) −5.80823 −0.194256
\(895\) 17.7623 0.593728
\(896\) 6.17843 0.206407
\(897\) 0 0
\(898\) −8.80558 −0.293846
\(899\) −55.3002 −1.84437
\(900\) −1.88448 −0.0628161
\(901\) 5.04995 0.168238
\(902\) −0.975965 −0.0324961
\(903\) 0.359508 0.0119637
\(904\) −7.02404 −0.233616
\(905\) 7.02630 0.233562
\(906\) 4.42280 0.146938
\(907\) −16.9604 −0.563159 −0.281580 0.959538i \(-0.590858\pi\)
−0.281580 + 0.959538i \(0.590858\pi\)
\(908\) −2.28287 −0.0757596
\(909\) −9.35951 −0.310435
\(910\) 0 0
\(911\) 37.7297 1.25004 0.625020 0.780608i \(-0.285091\pi\)
0.625020 + 0.780608i \(0.285091\pi\)
\(912\) −0.383543 −0.0127004
\(913\) 1.20246 0.0397957
\(914\) −2.00667 −0.0663748
\(915\) 4.20473 0.139004
\(916\) −36.2650 −1.19823
\(917\) 7.19806 0.237701
\(918\) 2.52498 0.0833366
\(919\) 40.1628 1.32485 0.662425 0.749129i \(-0.269528\pi\)
0.662425 + 0.749129i \(0.269528\pi\)
\(920\) −10.2569 −0.338162
\(921\) 4.77564 0.157363
\(922\) 5.26537 0.173406
\(923\) 0 0
\(924\) 0.845608 0.0278185
\(925\) 9.76897 0.321202
\(926\) −5.92815 −0.194811
\(927\) 4.50535 0.147975
\(928\) −20.8974 −0.685992
\(929\) 23.7690 0.779835 0.389917 0.920850i \(-0.372504\pi\)
0.389917 + 0.920850i \(0.372504\pi\)
\(930\) −3.38983 −0.111157
\(931\) −0.758276 −0.0248515
\(932\) 45.0545 1.47581
\(933\) −30.5812 −1.00118
\(934\) 6.96035 0.227750
\(935\) 5.04995 0.165151
\(936\) 0 0
\(937\) −7.43803 −0.242990 −0.121495 0.992592i \(-0.538769\pi\)
−0.121495 + 0.992592i \(0.538769\pi\)
\(938\) 1.71273 0.0559226
\(939\) 26.7230 0.872073
\(940\) 9.45941 0.308532
\(941\) −19.3528 −0.630884 −0.315442 0.948945i \(-0.602153\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(942\) −0.263622 −0.00858927
\(943\) −32.8189 −1.06873
\(944\) 7.38542 0.240375
\(945\) −0.660123 −0.0214738
\(946\) 0.125822 0.00409082
\(947\) 13.9171 0.452244 0.226122 0.974099i \(-0.427395\pi\)
0.226122 + 0.974099i \(0.427395\pi\)
\(948\) 18.7953 0.610442
\(949\) 0 0
\(950\) −0.0392613 −0.00127381
\(951\) 20.6271 0.668881
\(952\) 6.47464 0.209844
\(953\) −10.6271 −0.344247 −0.172124 0.985075i \(-0.555063\pi\)
−0.172124 + 0.985075i \(0.555063\pi\)
\(954\) −0.231033 −0.00747996
\(955\) −13.9541 −0.451543
\(956\) −35.2017 −1.13850
\(957\) −3.76897 −0.121833
\(958\) 13.9318 0.450115
\(959\) 2.23770 0.0722593
\(960\) 5.35951 0.172977
\(961\) 68.4746 2.20886
\(962\) 0 0
\(963\) 0.570909 0.0183973
\(964\) −11.5246 −0.371182
\(965\) 23.7493 0.764518
\(966\) −1.74305 −0.0560817
\(967\) 51.6771 1.66182 0.830912 0.556404i \(-0.187819\pi\)
0.830912 + 0.556404i \(0.187819\pi\)
\(968\) −13.9127 −0.447170
\(969\) −0.858181 −0.0275687
\(970\) −3.36618 −0.108082
\(971\) −42.8974 −1.37664 −0.688322 0.725405i \(-0.741652\pi\)
−0.688322 + 0.725405i \(0.741652\pi\)
\(972\) 1.88448 0.0604448
\(973\) 9.77525 0.313380
\(974\) −8.17843 −0.262054
\(975\) 0 0
\(976\) −13.9607 −0.446872
\(977\) 18.1651 0.581153 0.290576 0.956852i \(-0.406153\pi\)
0.290576 + 0.956852i \(0.406153\pi\)
\(978\) 4.14584 0.132569
\(979\) 9.20700 0.294257
\(980\) 12.3702 0.395151
\(981\) 15.4095 0.491986
\(982\) −12.4983 −0.398836
\(983\) −10.8582 −0.346322 −0.173161 0.984894i \(-0.555398\pi\)
−0.173161 + 0.984894i \(0.555398\pi\)
\(984\) −5.57720 −0.177795
\(985\) 15.8082 0.503692
\(986\) −14.0000 −0.445851
\(987\) 3.31357 0.105472
\(988\) 0 0
\(989\) 4.23103 0.134539
\(990\) −0.231033 −0.00734270
\(991\) −22.3725 −0.710685 −0.355342 0.934736i \(-0.615636\pi\)
−0.355342 + 0.934736i \(0.615636\pi\)
\(992\) 37.5905 1.19350
\(993\) −5.37020 −0.170418
\(994\) −1.64088 −0.0520455
\(995\) 8.76897 0.277995
\(996\) 3.33359 0.105629
\(997\) −24.2373 −0.767604 −0.383802 0.923415i \(-0.625385\pi\)
−0.383802 + 0.923415i \(0.625385\pi\)
\(998\) −11.8349 −0.374628
\(999\) −9.76897 −0.309076
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 2535.2.a.bb.1.2 3
3.2 odd 2 7605.2.a.bv.1.2 3
13.3 even 3 195.2.i.d.61.2 yes 6
13.9 even 3 195.2.i.d.16.2 6
13.12 even 2 2535.2.a.ba.1.2 3
39.29 odd 6 585.2.j.f.451.2 6
39.35 odd 6 585.2.j.f.406.2 6
39.38 odd 2 7605.2.a.bw.1.2 3
65.3 odd 12 975.2.bb.k.724.3 12
65.9 even 6 975.2.i.l.601.2 6
65.22 odd 12 975.2.bb.k.874.3 12
65.29 even 6 975.2.i.l.451.2 6
65.42 odd 12 975.2.bb.k.724.4 12
65.48 odd 12 975.2.bb.k.874.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.i.d.16.2 6 13.9 even 3
195.2.i.d.61.2 yes 6 13.3 even 3
585.2.j.f.406.2 6 39.35 odd 6
585.2.j.f.451.2 6 39.29 odd 6
975.2.i.l.451.2 6 65.29 even 6
975.2.i.l.601.2 6 65.9 even 6
975.2.bb.k.724.3 12 65.3 odd 12
975.2.bb.k.724.4 12 65.42 odd 12
975.2.bb.k.874.3 12 65.22 odd 12
975.2.bb.k.874.4 12 65.48 odd 12
2535.2.a.ba.1.2 3 13.12 even 2
2535.2.a.bb.1.2 3 1.1 even 1 trivial
7605.2.a.bv.1.2 3 3.2 odd 2
7605.2.a.bw.1.2 3 39.38 odd 2