Properties

Label 3549.2.a.bd.1.7
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $8$
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] = [3549,2,Mod(1,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, 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("3549.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.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: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 14x^{5} + 25x^{4} - 24x^{3} - 16x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.45308\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.45308 q^{2} +1.00000 q^{3} +4.01758 q^{4} +0.0682999 q^{5} +2.45308 q^{6} +1.00000 q^{7} +4.94928 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.45308 q^{2} +1.00000 q^{3} +4.01758 q^{4} +0.0682999 q^{5} +2.45308 q^{6} +1.00000 q^{7} +4.94928 q^{8} +1.00000 q^{9} +0.167545 q^{10} -0.678230 q^{11} +4.01758 q^{12} +2.45308 q^{14} +0.0682999 q^{15} +4.10580 q^{16} -0.934958 q^{17} +2.45308 q^{18} +2.06767 q^{19} +0.274400 q^{20} +1.00000 q^{21} -1.66375 q^{22} +4.29948 q^{23} +4.94928 q^{24} -4.99534 q^{25} +1.00000 q^{27} +4.01758 q^{28} +8.01446 q^{29} +0.167545 q^{30} +10.0622 q^{31} +0.173279 q^{32} -0.678230 q^{33} -2.29352 q^{34} +0.0682999 q^{35} +4.01758 q^{36} -0.527492 q^{37} +5.07216 q^{38} +0.338035 q^{40} +5.13776 q^{41} +2.45308 q^{42} -1.78857 q^{43} -2.72485 q^{44} +0.0682999 q^{45} +10.5470 q^{46} +4.22889 q^{47} +4.10580 q^{48} +1.00000 q^{49} -12.2539 q^{50} -0.934958 q^{51} -3.21495 q^{53} +2.45308 q^{54} -0.0463231 q^{55} +4.94928 q^{56} +2.06767 q^{57} +19.6601 q^{58} -8.32746 q^{59} +0.274400 q^{60} +1.01053 q^{61} +24.6833 q^{62} +1.00000 q^{63} -7.78654 q^{64} -1.66375 q^{66} -7.30030 q^{67} -3.75627 q^{68} +4.29948 q^{69} +0.167545 q^{70} +9.91104 q^{71} +4.94928 q^{72} -3.42254 q^{73} -1.29398 q^{74} -4.99534 q^{75} +8.30704 q^{76} -0.678230 q^{77} -14.0484 q^{79} +0.280426 q^{80} +1.00000 q^{81} +12.6033 q^{82} -7.03411 q^{83} +4.01758 q^{84} -0.0638575 q^{85} -4.38750 q^{86} +8.01446 q^{87} -3.35675 q^{88} -15.7543 q^{89} +0.167545 q^{90} +17.2735 q^{92} +10.0622 q^{93} +10.3738 q^{94} +0.141222 q^{95} +0.173279 q^{96} +16.6705 q^{97} +2.45308 q^{98} -0.678230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 8 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 8 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9} - 4 q^{10} + 8 q^{11} + 6 q^{12} + 2 q^{14} + 2 q^{15} + 10 q^{16} + 10 q^{17} + 2 q^{18} + 18 q^{19} + 2 q^{20} + 8 q^{21} + 2 q^{22} - 2 q^{23} + 12 q^{24} - 6 q^{25} + 8 q^{27} + 6 q^{28} - 12 q^{29} - 4 q^{30} + 16 q^{31} + 26 q^{32} + 8 q^{33} + 24 q^{34} + 2 q^{35} + 6 q^{36} + 24 q^{37} + 16 q^{38} - 30 q^{40} - 4 q^{41} + 2 q^{42} - 10 q^{43} - 20 q^{44} + 2 q^{45} + 12 q^{46} + 10 q^{47} + 10 q^{48} + 8 q^{49} - 16 q^{50} + 10 q^{51} + 6 q^{53} + 2 q^{54} - 10 q^{55} + 12 q^{56} + 18 q^{57} + 16 q^{58} + 6 q^{59} + 2 q^{60} + 6 q^{61} + 16 q^{62} + 8 q^{63} - 8 q^{64} + 2 q^{66} + 24 q^{67} + 20 q^{68} - 2 q^{69} - 4 q^{70} + 42 q^{71} + 12 q^{72} + 32 q^{73} - 18 q^{74} - 6 q^{75} + 28 q^{76} + 8 q^{77} - 2 q^{79} - 40 q^{80} + 8 q^{81} - 18 q^{82} - 2 q^{83} + 6 q^{84} + 4 q^{85} + 26 q^{86} - 12 q^{87} - 2 q^{88} + 12 q^{89} - 4 q^{90} - 10 q^{92} + 16 q^{93} - 16 q^{94} - 4 q^{95} + 26 q^{96} + 64 q^{97} + 2 q^{98} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.45308 1.73459 0.867293 0.497797i \(-0.165858\pi\)
0.867293 + 0.497797i \(0.165858\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.01758 2.00879
\(5\) 0.0682999 0.0305446 0.0152723 0.999883i \(-0.495138\pi\)
0.0152723 + 0.999883i \(0.495138\pi\)
\(6\) 2.45308 1.00146
\(7\) 1.00000 0.377964
\(8\) 4.94928 1.74984
\(9\) 1.00000 0.333333
\(10\) 0.167545 0.0529823
\(11\) −0.678230 −0.204494 −0.102247 0.994759i \(-0.532603\pi\)
−0.102247 + 0.994759i \(0.532603\pi\)
\(12\) 4.01758 1.15978
\(13\) 0 0
\(14\) 2.45308 0.655612
\(15\) 0.0682999 0.0176350
\(16\) 4.10580 1.02645
\(17\) −0.934958 −0.226761 −0.113380 0.993552i \(-0.536168\pi\)
−0.113380 + 0.993552i \(0.536168\pi\)
\(18\) 2.45308 0.578196
\(19\) 2.06767 0.474356 0.237178 0.971466i \(-0.423777\pi\)
0.237178 + 0.971466i \(0.423777\pi\)
\(20\) 0.274400 0.0613578
\(21\) 1.00000 0.218218
\(22\) −1.66375 −0.354713
\(23\) 4.29948 0.896503 0.448252 0.893907i \(-0.352047\pi\)
0.448252 + 0.893907i \(0.352047\pi\)
\(24\) 4.94928 1.01027
\(25\) −4.99534 −0.999067
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 4.01758 0.759252
\(29\) 8.01446 1.48825 0.744124 0.668042i \(-0.232867\pi\)
0.744124 + 0.668042i \(0.232867\pi\)
\(30\) 0.167545 0.0305894
\(31\) 10.0622 1.80722 0.903609 0.428358i \(-0.140908\pi\)
0.903609 + 0.428358i \(0.140908\pi\)
\(32\) 0.173279 0.0306316
\(33\) −0.678230 −0.118065
\(34\) −2.29352 −0.393336
\(35\) 0.0682999 0.0115448
\(36\) 4.01758 0.669597
\(37\) −0.527492 −0.0867191 −0.0433596 0.999060i \(-0.513806\pi\)
−0.0433596 + 0.999060i \(0.513806\pi\)
\(38\) 5.07216 0.822812
\(39\) 0 0
\(40\) 0.338035 0.0534481
\(41\) 5.13776 0.802383 0.401191 0.915994i \(-0.368596\pi\)
0.401191 + 0.915994i \(0.368596\pi\)
\(42\) 2.45308 0.378518
\(43\) −1.78857 −0.272755 −0.136377 0.990657i \(-0.543546\pi\)
−0.136377 + 0.990657i \(0.543546\pi\)
\(44\) −2.72485 −0.410786
\(45\) 0.0682999 0.0101815
\(46\) 10.5470 1.55506
\(47\) 4.22889 0.616846 0.308423 0.951249i \(-0.400199\pi\)
0.308423 + 0.951249i \(0.400199\pi\)
\(48\) 4.10580 0.592621
\(49\) 1.00000 0.142857
\(50\) −12.2539 −1.73297
\(51\) −0.934958 −0.130920
\(52\) 0 0
\(53\) −3.21495 −0.441608 −0.220804 0.975318i \(-0.570868\pi\)
−0.220804 + 0.975318i \(0.570868\pi\)
\(54\) 2.45308 0.333821
\(55\) −0.0463231 −0.00624620
\(56\) 4.94928 0.661376
\(57\) 2.06767 0.273870
\(58\) 19.6601 2.58149
\(59\) −8.32746 −1.08414 −0.542071 0.840332i \(-0.682360\pi\)
−0.542071 + 0.840332i \(0.682360\pi\)
\(60\) 0.274400 0.0354249
\(61\) 1.01053 0.129386 0.0646928 0.997905i \(-0.479393\pi\)
0.0646928 + 0.997905i \(0.479393\pi\)
\(62\) 24.6833 3.13478
\(63\) 1.00000 0.125988
\(64\) −7.78654 −0.973317
\(65\) 0 0
\(66\) −1.66375 −0.204794
\(67\) −7.30030 −0.891874 −0.445937 0.895064i \(-0.647129\pi\)
−0.445937 + 0.895064i \(0.647129\pi\)
\(68\) −3.75627 −0.455515
\(69\) 4.29948 0.517597
\(70\) 0.167545 0.0200254
\(71\) 9.91104 1.17622 0.588112 0.808779i \(-0.299871\pi\)
0.588112 + 0.808779i \(0.299871\pi\)
\(72\) 4.94928 0.583278
\(73\) −3.42254 −0.400578 −0.200289 0.979737i \(-0.564188\pi\)
−0.200289 + 0.979737i \(0.564188\pi\)
\(74\) −1.29398 −0.150422
\(75\) −4.99534 −0.576812
\(76\) 8.30704 0.952883
\(77\) −0.678230 −0.0772915
\(78\) 0 0
\(79\) −14.0484 −1.58057 −0.790283 0.612742i \(-0.790066\pi\)
−0.790283 + 0.612742i \(0.790066\pi\)
\(80\) 0.280426 0.0313526
\(81\) 1.00000 0.111111
\(82\) 12.6033 1.39180
\(83\) −7.03411 −0.772094 −0.386047 0.922479i \(-0.626160\pi\)
−0.386047 + 0.922479i \(0.626160\pi\)
\(84\) 4.01758 0.438354
\(85\) −0.0638575 −0.00692632
\(86\) −4.38750 −0.473117
\(87\) 8.01446 0.859240
\(88\) −3.35675 −0.357831
\(89\) −15.7543 −1.66995 −0.834975 0.550288i \(-0.814518\pi\)
−0.834975 + 0.550288i \(0.814518\pi\)
\(90\) 0.167545 0.0176608
\(91\) 0 0
\(92\) 17.2735 1.80089
\(93\) 10.0622 1.04340
\(94\) 10.3738 1.06997
\(95\) 0.141222 0.0144890
\(96\) 0.173279 0.0176852
\(97\) 16.6705 1.69263 0.846314 0.532684i \(-0.178817\pi\)
0.846314 + 0.532684i \(0.178817\pi\)
\(98\) 2.45308 0.247798
\(99\) −0.678230 −0.0681647
\(100\) −20.0692 −2.00692
\(101\) −1.50429 −0.149682 −0.0748412 0.997195i \(-0.523845\pi\)
−0.0748412 + 0.997195i \(0.523845\pi\)
\(102\) −2.29352 −0.227093
\(103\) −2.21721 −0.218468 −0.109234 0.994016i \(-0.534840\pi\)
−0.109234 + 0.994016i \(0.534840\pi\)
\(104\) 0 0
\(105\) 0.0682999 0.00666539
\(106\) −7.88652 −0.766007
\(107\) −19.7860 −1.91278 −0.956390 0.292092i \(-0.905649\pi\)
−0.956390 + 0.292092i \(0.905649\pi\)
\(108\) 4.01758 0.386592
\(109\) 1.71086 0.163870 0.0819352 0.996638i \(-0.473890\pi\)
0.0819352 + 0.996638i \(0.473890\pi\)
\(110\) −0.113634 −0.0108346
\(111\) −0.527492 −0.0500673
\(112\) 4.10580 0.387962
\(113\) −13.6392 −1.28307 −0.641536 0.767093i \(-0.721702\pi\)
−0.641536 + 0.767093i \(0.721702\pi\)
\(114\) 5.07216 0.475051
\(115\) 0.293654 0.0273834
\(116\) 32.1987 2.98958
\(117\) 0 0
\(118\) −20.4279 −1.88054
\(119\) −0.934958 −0.0857075
\(120\) 0.338035 0.0308583
\(121\) −10.5400 −0.958182
\(122\) 2.47892 0.224430
\(123\) 5.13776 0.463256
\(124\) 40.4256 3.63032
\(125\) −0.682680 −0.0610608
\(126\) 2.45308 0.218537
\(127\) −8.00439 −0.710275 −0.355138 0.934814i \(-0.615566\pi\)
−0.355138 + 0.934814i \(0.615566\pi\)
\(128\) −19.4475 −1.71893
\(129\) −1.78857 −0.157475
\(130\) 0 0
\(131\) −10.2912 −0.899146 −0.449573 0.893244i \(-0.648424\pi\)
−0.449573 + 0.893244i \(0.648424\pi\)
\(132\) −2.72485 −0.237167
\(133\) 2.06767 0.179290
\(134\) −17.9082 −1.54703
\(135\) 0.0682999 0.00587832
\(136\) −4.62737 −0.396794
\(137\) −3.61956 −0.309239 −0.154620 0.987974i \(-0.549415\pi\)
−0.154620 + 0.987974i \(0.549415\pi\)
\(138\) 10.5470 0.897816
\(139\) 1.21815 0.103322 0.0516610 0.998665i \(-0.483548\pi\)
0.0516610 + 0.998665i \(0.483548\pi\)
\(140\) 0.274400 0.0231911
\(141\) 4.22889 0.356136
\(142\) 24.3125 2.04026
\(143\) 0 0
\(144\) 4.10580 0.342150
\(145\) 0.547387 0.0454580
\(146\) −8.39574 −0.694836
\(147\) 1.00000 0.0824786
\(148\) −2.11924 −0.174201
\(149\) 16.5273 1.35397 0.676983 0.735999i \(-0.263287\pi\)
0.676983 + 0.735999i \(0.263287\pi\)
\(150\) −12.2539 −1.00053
\(151\) −20.3543 −1.65641 −0.828206 0.560424i \(-0.810638\pi\)
−0.828206 + 0.560424i \(0.810638\pi\)
\(152\) 10.2335 0.830046
\(153\) −0.934958 −0.0755869
\(154\) −1.66375 −0.134069
\(155\) 0.687245 0.0552008
\(156\) 0 0
\(157\) 15.1583 1.20976 0.604881 0.796316i \(-0.293221\pi\)
0.604881 + 0.796316i \(0.293221\pi\)
\(158\) −34.4617 −2.74163
\(159\) −3.21495 −0.254962
\(160\) 0.0118349 0.000935632 0
\(161\) 4.29948 0.338846
\(162\) 2.45308 0.192732
\(163\) 16.8412 1.31910 0.659551 0.751660i \(-0.270747\pi\)
0.659551 + 0.751660i \(0.270747\pi\)
\(164\) 20.6414 1.61182
\(165\) −0.0463231 −0.00360625
\(166\) −17.2552 −1.33926
\(167\) −1.65129 −0.127780 −0.0638902 0.997957i \(-0.520351\pi\)
−0.0638902 + 0.997957i \(0.520351\pi\)
\(168\) 4.94928 0.381845
\(169\) 0 0
\(170\) −0.156647 −0.0120143
\(171\) 2.06767 0.158119
\(172\) −7.18574 −0.547907
\(173\) 24.7787 1.88389 0.941943 0.335772i \(-0.108997\pi\)
0.941943 + 0.335772i \(0.108997\pi\)
\(174\) 19.6601 1.49043
\(175\) −4.99534 −0.377612
\(176\) −2.78468 −0.209903
\(177\) −8.32746 −0.625930
\(178\) −38.6465 −2.89667
\(179\) −6.30284 −0.471097 −0.235548 0.971863i \(-0.575689\pi\)
−0.235548 + 0.971863i \(0.575689\pi\)
\(180\) 0.274400 0.0204526
\(181\) −18.9468 −1.40830 −0.704152 0.710049i \(-0.748673\pi\)
−0.704152 + 0.710049i \(0.748673\pi\)
\(182\) 0 0
\(183\) 1.01053 0.0747008
\(184\) 21.2793 1.56873
\(185\) −0.0360276 −0.00264880
\(186\) 24.6833 1.80986
\(187\) 0.634117 0.0463712
\(188\) 16.9899 1.23912
\(189\) 1.00000 0.0727393
\(190\) 0.346428 0.0251325
\(191\) −8.79410 −0.636319 −0.318159 0.948037i \(-0.603065\pi\)
−0.318159 + 0.948037i \(0.603065\pi\)
\(192\) −7.78654 −0.561945
\(193\) −6.19901 −0.446215 −0.223107 0.974794i \(-0.571620\pi\)
−0.223107 + 0.974794i \(0.571620\pi\)
\(194\) 40.8939 2.93601
\(195\) 0 0
\(196\) 4.01758 0.286970
\(197\) −15.1434 −1.07892 −0.539460 0.842011i \(-0.681371\pi\)
−0.539460 + 0.842011i \(0.681371\pi\)
\(198\) −1.66375 −0.118238
\(199\) 15.0372 1.06596 0.532979 0.846128i \(-0.321072\pi\)
0.532979 + 0.846128i \(0.321072\pi\)
\(200\) −24.7233 −1.74820
\(201\) −7.30030 −0.514924
\(202\) −3.69014 −0.259637
\(203\) 8.01446 0.562505
\(204\) −3.75627 −0.262992
\(205\) 0.350908 0.0245085
\(206\) −5.43898 −0.378952
\(207\) 4.29948 0.298834
\(208\) 0 0
\(209\) −1.40236 −0.0970031
\(210\) 0.167545 0.0115617
\(211\) 25.4741 1.75371 0.876856 0.480752i \(-0.159636\pi\)
0.876856 + 0.480752i \(0.159636\pi\)
\(212\) −12.9163 −0.887097
\(213\) 9.91104 0.679093
\(214\) −48.5365 −3.31788
\(215\) −0.122159 −0.00833119
\(216\) 4.94928 0.336756
\(217\) 10.0622 0.683064
\(218\) 4.19686 0.284248
\(219\) −3.42254 −0.231274
\(220\) −0.186107 −0.0125473
\(221\) 0 0
\(222\) −1.29398 −0.0868461
\(223\) −7.00654 −0.469192 −0.234596 0.972093i \(-0.575377\pi\)
−0.234596 + 0.972093i \(0.575377\pi\)
\(224\) 0.173279 0.0115777
\(225\) −4.99534 −0.333022
\(226\) −33.4581 −2.22560
\(227\) −27.7571 −1.84230 −0.921151 0.389206i \(-0.872749\pi\)
−0.921151 + 0.389206i \(0.872749\pi\)
\(228\) 8.30704 0.550147
\(229\) 3.61011 0.238563 0.119281 0.992860i \(-0.461941\pi\)
0.119281 + 0.992860i \(0.461941\pi\)
\(230\) 0.720355 0.0474988
\(231\) −0.678230 −0.0446243
\(232\) 39.6658 2.60419
\(233\) −15.6176 −1.02315 −0.511573 0.859240i \(-0.670937\pi\)
−0.511573 + 0.859240i \(0.670937\pi\)
\(234\) 0 0
\(235\) 0.288832 0.0188413
\(236\) −33.4562 −2.17782
\(237\) −14.0484 −0.912540
\(238\) −2.29352 −0.148667
\(239\) 27.9831 1.81008 0.905038 0.425332i \(-0.139843\pi\)
0.905038 + 0.425332i \(0.139843\pi\)
\(240\) 0.280426 0.0181014
\(241\) 25.9732 1.67308 0.836541 0.547904i \(-0.184574\pi\)
0.836541 + 0.547904i \(0.184574\pi\)
\(242\) −25.8554 −1.66205
\(243\) 1.00000 0.0641500
\(244\) 4.05990 0.259909
\(245\) 0.0682999 0.00436352
\(246\) 12.6033 0.803558
\(247\) 0 0
\(248\) 49.8005 3.16233
\(249\) −7.03411 −0.445769
\(250\) −1.67467 −0.105915
\(251\) −6.38525 −0.403033 −0.201517 0.979485i \(-0.564587\pi\)
−0.201517 + 0.979485i \(0.564587\pi\)
\(252\) 4.01758 0.253084
\(253\) −2.91604 −0.183330
\(254\) −19.6354 −1.23203
\(255\) −0.0638575 −0.00399891
\(256\) −32.1332 −2.00832
\(257\) −13.8179 −0.861934 −0.430967 0.902368i \(-0.641827\pi\)
−0.430967 + 0.902368i \(0.641827\pi\)
\(258\) −4.38750 −0.273154
\(259\) −0.527492 −0.0327767
\(260\) 0 0
\(261\) 8.01446 0.496083
\(262\) −25.2451 −1.55965
\(263\) −11.3364 −0.699033 −0.349517 0.936930i \(-0.613654\pi\)
−0.349517 + 0.936930i \(0.613654\pi\)
\(264\) −3.35675 −0.206594
\(265\) −0.219581 −0.0134887
\(266\) 5.07216 0.310994
\(267\) −15.7543 −0.964146
\(268\) −29.3296 −1.79159
\(269\) 21.0002 1.28040 0.640202 0.768206i \(-0.278851\pi\)
0.640202 + 0.768206i \(0.278851\pi\)
\(270\) 0.167545 0.0101965
\(271\) 29.2450 1.77651 0.888253 0.459354i \(-0.151919\pi\)
0.888253 + 0.459354i \(0.151919\pi\)
\(272\) −3.83875 −0.232759
\(273\) 0 0
\(274\) −8.87904 −0.536403
\(275\) 3.38799 0.204303
\(276\) 17.2735 1.03974
\(277\) 16.5036 0.991603 0.495801 0.868436i \(-0.334874\pi\)
0.495801 + 0.868436i \(0.334874\pi\)
\(278\) 2.98821 0.179221
\(279\) 10.0622 0.602406
\(280\) 0.338035 0.0202015
\(281\) 1.12161 0.0669094 0.0334547 0.999440i \(-0.489349\pi\)
0.0334547 + 0.999440i \(0.489349\pi\)
\(282\) 10.3738 0.617749
\(283\) 10.9412 0.650387 0.325194 0.945647i \(-0.394571\pi\)
0.325194 + 0.945647i \(0.394571\pi\)
\(284\) 39.8184 2.36279
\(285\) 0.141222 0.00836525
\(286\) 0 0
\(287\) 5.13776 0.303272
\(288\) 0.173279 0.0102105
\(289\) −16.1259 −0.948580
\(290\) 1.34278 0.0788508
\(291\) 16.6705 0.977239
\(292\) −13.7503 −0.804677
\(293\) −15.5503 −0.908457 −0.454229 0.890885i \(-0.650085\pi\)
−0.454229 + 0.890885i \(0.650085\pi\)
\(294\) 2.45308 0.143066
\(295\) −0.568764 −0.0331147
\(296\) −2.61071 −0.151744
\(297\) −0.678230 −0.0393549
\(298\) 40.5426 2.34857
\(299\) 0 0
\(300\) −20.0692 −1.15869
\(301\) −1.78857 −0.103092
\(302\) −49.9307 −2.87319
\(303\) −1.50429 −0.0864192
\(304\) 8.48945 0.486903
\(305\) 0.0690193 0.00395203
\(306\) −2.29352 −0.131112
\(307\) 0.119003 0.00679184 0.00339592 0.999994i \(-0.498919\pi\)
0.00339592 + 0.999994i \(0.498919\pi\)
\(308\) −2.72485 −0.155263
\(309\) −2.21721 −0.126133
\(310\) 1.68586 0.0957506
\(311\) −31.4738 −1.78472 −0.892358 0.451327i \(-0.850951\pi\)
−0.892358 + 0.451327i \(0.850951\pi\)
\(312\) 0 0
\(313\) 14.1243 0.798355 0.399178 0.916874i \(-0.369296\pi\)
0.399178 + 0.916874i \(0.369296\pi\)
\(314\) 37.1844 2.09844
\(315\) 0.0682999 0.00384826
\(316\) −56.4405 −3.17503
\(317\) 18.6470 1.04732 0.523661 0.851927i \(-0.324566\pi\)
0.523661 + 0.851927i \(0.324566\pi\)
\(318\) −7.88652 −0.442254
\(319\) −5.43565 −0.304338
\(320\) −0.531820 −0.0297296
\(321\) −19.7860 −1.10434
\(322\) 10.5470 0.587759
\(323\) −1.93319 −0.107565
\(324\) 4.01758 0.223199
\(325\) 0 0
\(326\) 41.3126 2.28810
\(327\) 1.71086 0.0946107
\(328\) 25.4282 1.40404
\(329\) 4.22889 0.233146
\(330\) −0.113634 −0.00625535
\(331\) 22.0843 1.21386 0.606930 0.794755i \(-0.292401\pi\)
0.606930 + 0.794755i \(0.292401\pi\)
\(332\) −28.2601 −1.55098
\(333\) −0.527492 −0.0289064
\(334\) −4.05073 −0.221646
\(335\) −0.498610 −0.0272420
\(336\) 4.10580 0.223990
\(337\) −32.5355 −1.77232 −0.886161 0.463377i \(-0.846638\pi\)
−0.886161 + 0.463377i \(0.846638\pi\)
\(338\) 0 0
\(339\) −13.6392 −0.740782
\(340\) −0.256553 −0.0139135
\(341\) −6.82447 −0.369566
\(342\) 5.07216 0.274271
\(343\) 1.00000 0.0539949
\(344\) −8.85215 −0.477276
\(345\) 0.293654 0.0158098
\(346\) 60.7839 3.26776
\(347\) 4.41223 0.236861 0.118431 0.992962i \(-0.462214\pi\)
0.118431 + 0.992962i \(0.462214\pi\)
\(348\) 32.1987 1.72603
\(349\) 7.12421 0.381350 0.190675 0.981653i \(-0.438932\pi\)
0.190675 + 0.981653i \(0.438932\pi\)
\(350\) −12.2539 −0.655000
\(351\) 0 0
\(352\) −0.117523 −0.00626399
\(353\) 3.33664 0.177591 0.0887957 0.996050i \(-0.471698\pi\)
0.0887957 + 0.996050i \(0.471698\pi\)
\(354\) −20.4279 −1.08573
\(355\) 0.676923 0.0359273
\(356\) −63.2941 −3.35458
\(357\) −0.934958 −0.0494832
\(358\) −15.4614 −0.817158
\(359\) −24.5869 −1.29765 −0.648823 0.760940i \(-0.724738\pi\)
−0.648823 + 0.760940i \(0.724738\pi\)
\(360\) 0.338035 0.0178160
\(361\) −14.7247 −0.774986
\(362\) −46.4779 −2.44282
\(363\) −10.5400 −0.553207
\(364\) 0 0
\(365\) −0.233759 −0.0122355
\(366\) 2.47892 0.129575
\(367\) −15.0766 −0.786991 −0.393495 0.919327i \(-0.628734\pi\)
−0.393495 + 0.919327i \(0.628734\pi\)
\(368\) 17.6528 0.920216
\(369\) 5.13776 0.267461
\(370\) −0.0883785 −0.00459458
\(371\) −3.21495 −0.166912
\(372\) 40.4256 2.09597
\(373\) −25.2471 −1.30725 −0.653623 0.756820i \(-0.726752\pi\)
−0.653623 + 0.756820i \(0.726752\pi\)
\(374\) 1.55554 0.0804349
\(375\) −0.682680 −0.0352535
\(376\) 20.9299 1.07938
\(377\) 0 0
\(378\) 2.45308 0.126173
\(379\) 24.0432 1.23502 0.617508 0.786565i \(-0.288142\pi\)
0.617508 + 0.786565i \(0.288142\pi\)
\(380\) 0.567370 0.0291055
\(381\) −8.00439 −0.410077
\(382\) −21.5726 −1.10375
\(383\) −4.53923 −0.231944 −0.115972 0.993252i \(-0.536998\pi\)
−0.115972 + 0.993252i \(0.536998\pi\)
\(384\) −19.4475 −0.992427
\(385\) −0.0463231 −0.00236084
\(386\) −15.2067 −0.773998
\(387\) −1.78857 −0.0909182
\(388\) 66.9749 3.40014
\(389\) 19.7847 1.00313 0.501563 0.865121i \(-0.332758\pi\)
0.501563 + 0.865121i \(0.332758\pi\)
\(390\) 0 0
\(391\) −4.01983 −0.203292
\(392\) 4.94928 0.249976
\(393\) −10.2912 −0.519122
\(394\) −37.1478 −1.87148
\(395\) −0.959502 −0.0482778
\(396\) −2.72485 −0.136929
\(397\) 10.4584 0.524892 0.262446 0.964947i \(-0.415471\pi\)
0.262446 + 0.964947i \(0.415471\pi\)
\(398\) 36.8874 1.84900
\(399\) 2.06767 0.103513
\(400\) −20.5099 −1.02549
\(401\) 22.4236 1.11978 0.559891 0.828566i \(-0.310843\pi\)
0.559891 + 0.828566i \(0.310843\pi\)
\(402\) −17.9082 −0.893180
\(403\) 0 0
\(404\) −6.04361 −0.300681
\(405\) 0.0682999 0.00339385
\(406\) 19.6601 0.975713
\(407\) 0.357761 0.0177336
\(408\) −4.62737 −0.229089
\(409\) 24.4405 1.20851 0.604253 0.796793i \(-0.293472\pi\)
0.604253 + 0.796793i \(0.293472\pi\)
\(410\) 0.860805 0.0425121
\(411\) −3.61956 −0.178539
\(412\) −8.90782 −0.438857
\(413\) −8.32746 −0.409767
\(414\) 10.5470 0.518354
\(415\) −0.480429 −0.0235833
\(416\) 0 0
\(417\) 1.21815 0.0596529
\(418\) −3.44009 −0.168260
\(419\) 27.7961 1.35793 0.678965 0.734170i \(-0.262429\pi\)
0.678965 + 0.734170i \(0.262429\pi\)
\(420\) 0.274400 0.0133894
\(421\) −15.8324 −0.771626 −0.385813 0.922577i \(-0.626079\pi\)
−0.385813 + 0.922577i \(0.626079\pi\)
\(422\) 62.4900 3.04197
\(423\) 4.22889 0.205615
\(424\) −15.9117 −0.772741
\(425\) 4.67043 0.226549
\(426\) 24.3125 1.17795
\(427\) 1.01053 0.0489031
\(428\) −79.4917 −3.84238
\(429\) 0 0
\(430\) −0.299666 −0.0144512
\(431\) 18.2098 0.877135 0.438567 0.898698i \(-0.355486\pi\)
0.438567 + 0.898698i \(0.355486\pi\)
\(432\) 4.10580 0.197540
\(433\) −9.40449 −0.451951 −0.225975 0.974133i \(-0.572557\pi\)
−0.225975 + 0.974133i \(0.572557\pi\)
\(434\) 24.6833 1.18483
\(435\) 0.547387 0.0262452
\(436\) 6.87351 0.329182
\(437\) 8.88991 0.425262
\(438\) −8.39574 −0.401164
\(439\) −19.2339 −0.917984 −0.458992 0.888440i \(-0.651789\pi\)
−0.458992 + 0.888440i \(0.651789\pi\)
\(440\) −0.229266 −0.0109298
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −5.40445 −0.256773 −0.128387 0.991724i \(-0.540980\pi\)
−0.128387 + 0.991724i \(0.540980\pi\)
\(444\) −2.11924 −0.100575
\(445\) −1.07602 −0.0510080
\(446\) −17.1876 −0.813855
\(447\) 16.5273 0.781713
\(448\) −7.78654 −0.367879
\(449\) −13.5534 −0.639623 −0.319812 0.947481i \(-0.603620\pi\)
−0.319812 + 0.947481i \(0.603620\pi\)
\(450\) −12.2539 −0.577656
\(451\) −3.48458 −0.164083
\(452\) −54.7968 −2.57742
\(453\) −20.3543 −0.956329
\(454\) −68.0902 −3.19563
\(455\) 0 0
\(456\) 10.2335 0.479227
\(457\) −29.8229 −1.39506 −0.697528 0.716558i \(-0.745717\pi\)
−0.697528 + 0.716558i \(0.745717\pi\)
\(458\) 8.85588 0.413808
\(459\) −0.934958 −0.0436401
\(460\) 1.17978 0.0550075
\(461\) 2.22803 0.103770 0.0518849 0.998653i \(-0.483477\pi\)
0.0518849 + 0.998653i \(0.483477\pi\)
\(462\) −1.66375 −0.0774047
\(463\) 21.6086 1.00424 0.502118 0.864799i \(-0.332554\pi\)
0.502118 + 0.864799i \(0.332554\pi\)
\(464\) 32.9058 1.52761
\(465\) 0.687245 0.0318702
\(466\) −38.3112 −1.77473
\(467\) −24.0068 −1.11090 −0.555452 0.831549i \(-0.687455\pi\)
−0.555452 + 0.831549i \(0.687455\pi\)
\(468\) 0 0
\(469\) −7.30030 −0.337097
\(470\) 0.708528 0.0326819
\(471\) 15.1583 0.698456
\(472\) −41.2149 −1.89707
\(473\) 1.21306 0.0557767
\(474\) −34.4617 −1.58288
\(475\) −10.3287 −0.473914
\(476\) −3.75627 −0.172168
\(477\) −3.21495 −0.147203
\(478\) 68.6446 3.13973
\(479\) 30.6921 1.40236 0.701179 0.712985i \(-0.252657\pi\)
0.701179 + 0.712985i \(0.252657\pi\)
\(480\) 0.0118349 0.000540187 0
\(481\) 0 0
\(482\) 63.7143 2.90211
\(483\) 4.29948 0.195633
\(484\) −42.3453 −1.92479
\(485\) 1.13859 0.0517007
\(486\) 2.45308 0.111274
\(487\) −17.8707 −0.809798 −0.404899 0.914361i \(-0.632693\pi\)
−0.404899 + 0.914361i \(0.632693\pi\)
\(488\) 5.00142 0.226403
\(489\) 16.8412 0.761583
\(490\) 0.167545 0.00756890
\(491\) −13.5696 −0.612388 −0.306194 0.951969i \(-0.599056\pi\)
−0.306194 + 0.951969i \(0.599056\pi\)
\(492\) 20.6414 0.930585
\(493\) −7.49318 −0.337476
\(494\) 0 0
\(495\) −0.0463231 −0.00208207
\(496\) 41.3132 1.85502
\(497\) 9.91104 0.444571
\(498\) −17.2552 −0.773225
\(499\) 10.4240 0.466644 0.233322 0.972399i \(-0.425040\pi\)
0.233322 + 0.972399i \(0.425040\pi\)
\(500\) −2.74272 −0.122658
\(501\) −1.65129 −0.0737740
\(502\) −15.6635 −0.699096
\(503\) 5.04813 0.225085 0.112543 0.993647i \(-0.464101\pi\)
0.112543 + 0.993647i \(0.464101\pi\)
\(504\) 4.94928 0.220459
\(505\) −0.102743 −0.00457199
\(506\) −7.15326 −0.318001
\(507\) 0 0
\(508\) −32.1583 −1.42679
\(509\) −8.99245 −0.398583 −0.199292 0.979940i \(-0.563864\pi\)
−0.199292 + 0.979940i \(0.563864\pi\)
\(510\) −0.156647 −0.00693646
\(511\) −3.42254 −0.151404
\(512\) −39.9301 −1.76468
\(513\) 2.06767 0.0912899
\(514\) −33.8962 −1.49510
\(515\) −0.151435 −0.00667303
\(516\) −7.18574 −0.316334
\(517\) −2.86816 −0.126141
\(518\) −1.29398 −0.0568541
\(519\) 24.7787 1.08766
\(520\) 0 0
\(521\) −9.21666 −0.403789 −0.201895 0.979407i \(-0.564710\pi\)
−0.201895 + 0.979407i \(0.564710\pi\)
\(522\) 19.6601 0.860498
\(523\) 18.6656 0.816191 0.408096 0.912939i \(-0.366193\pi\)
0.408096 + 0.912939i \(0.366193\pi\)
\(524\) −41.3457 −1.80620
\(525\) −4.99534 −0.218014
\(526\) −27.8091 −1.21253
\(527\) −9.40770 −0.409806
\(528\) −2.78468 −0.121188
\(529\) −4.51447 −0.196281
\(530\) −0.538648 −0.0233974
\(531\) −8.32746 −0.361381
\(532\) 8.30704 0.360156
\(533\) 0 0
\(534\) −38.6465 −1.67240
\(535\) −1.35138 −0.0584252
\(536\) −36.1313 −1.56063
\(537\) −6.30284 −0.271988
\(538\) 51.5151 2.22097
\(539\) −0.678230 −0.0292135
\(540\) 0.274400 0.0118083
\(541\) 16.3383 0.702437 0.351219 0.936293i \(-0.385767\pi\)
0.351219 + 0.936293i \(0.385767\pi\)
\(542\) 71.7402 3.08150
\(543\) −18.9468 −0.813085
\(544\) −0.162008 −0.00694605
\(545\) 0.116851 0.00500536
\(546\) 0 0
\(547\) 5.07719 0.217085 0.108542 0.994092i \(-0.465382\pi\)
0.108542 + 0.994092i \(0.465382\pi\)
\(548\) −14.5419 −0.621197
\(549\) 1.01053 0.0431285
\(550\) 8.31099 0.354382
\(551\) 16.5713 0.705960
\(552\) 21.2793 0.905709
\(553\) −14.0484 −0.597398
\(554\) 40.4845 1.72002
\(555\) −0.0360276 −0.00152929
\(556\) 4.89401 0.207552
\(557\) 1.28369 0.0543917 0.0271959 0.999630i \(-0.491342\pi\)
0.0271959 + 0.999630i \(0.491342\pi\)
\(558\) 24.6833 1.04493
\(559\) 0 0
\(560\) 0.280426 0.0118502
\(561\) 0.634117 0.0267725
\(562\) 2.75138 0.116060
\(563\) 16.9798 0.715613 0.357807 0.933796i \(-0.383525\pi\)
0.357807 + 0.933796i \(0.383525\pi\)
\(564\) 16.9899 0.715404
\(565\) −0.931558 −0.0391909
\(566\) 26.8396 1.12815
\(567\) 1.00000 0.0419961
\(568\) 49.0525 2.05820
\(569\) −35.9628 −1.50764 −0.753820 0.657080i \(-0.771791\pi\)
−0.753820 + 0.657080i \(0.771791\pi\)
\(570\) 0.346428 0.0145103
\(571\) −20.6811 −0.865475 −0.432738 0.901520i \(-0.642452\pi\)
−0.432738 + 0.901520i \(0.642452\pi\)
\(572\) 0 0
\(573\) −8.79410 −0.367379
\(574\) 12.6033 0.526052
\(575\) −21.4773 −0.895667
\(576\) −7.78654 −0.324439
\(577\) −3.67781 −0.153109 −0.0765546 0.997065i \(-0.524392\pi\)
−0.0765546 + 0.997065i \(0.524392\pi\)
\(578\) −39.5579 −1.64539
\(579\) −6.19901 −0.257622
\(580\) 2.19917 0.0913156
\(581\) −7.03411 −0.291824
\(582\) 40.8939 1.69511
\(583\) 2.18048 0.0903062
\(584\) −16.9391 −0.700945
\(585\) 0 0
\(586\) −38.1460 −1.57580
\(587\) 12.0433 0.497078 0.248539 0.968622i \(-0.420050\pi\)
0.248539 + 0.968622i \(0.420050\pi\)
\(588\) 4.01758 0.165682
\(589\) 20.8052 0.857265
\(590\) −1.39522 −0.0574404
\(591\) −15.1434 −0.622914
\(592\) −2.16578 −0.0890129
\(593\) −24.4737 −1.00501 −0.502507 0.864573i \(-0.667589\pi\)
−0.502507 + 0.864573i \(0.667589\pi\)
\(594\) −1.66375 −0.0682645
\(595\) −0.0638575 −0.00261790
\(596\) 66.3996 2.71984
\(597\) 15.0372 0.615431
\(598\) 0 0
\(599\) −21.8012 −0.890773 −0.445387 0.895338i \(-0.646934\pi\)
−0.445387 + 0.895338i \(0.646934\pi\)
\(600\) −24.7233 −1.00933
\(601\) 6.58213 0.268491 0.134245 0.990948i \(-0.457139\pi\)
0.134245 + 0.990948i \(0.457139\pi\)
\(602\) −4.38750 −0.178821
\(603\) −7.30030 −0.297291
\(604\) −81.7752 −3.32738
\(605\) −0.719881 −0.0292673
\(606\) −3.69014 −0.149902
\(607\) 45.6933 1.85463 0.927317 0.374278i \(-0.122109\pi\)
0.927317 + 0.374278i \(0.122109\pi\)
\(608\) 0.358283 0.0145303
\(609\) 8.01446 0.324762
\(610\) 0.169310 0.00685515
\(611\) 0 0
\(612\) −3.75627 −0.151838
\(613\) −41.9953 −1.69618 −0.848088 0.529855i \(-0.822246\pi\)
−0.848088 + 0.529855i \(0.822246\pi\)
\(614\) 0.291923 0.0117810
\(615\) 0.350908 0.0141500
\(616\) −3.35675 −0.135247
\(617\) 24.4374 0.983811 0.491906 0.870648i \(-0.336300\pi\)
0.491906 + 0.870648i \(0.336300\pi\)
\(618\) −5.43898 −0.218788
\(619\) 24.1245 0.969644 0.484822 0.874613i \(-0.338884\pi\)
0.484822 + 0.874613i \(0.338884\pi\)
\(620\) 2.76106 0.110887
\(621\) 4.29948 0.172532
\(622\) −77.2077 −3.09575
\(623\) −15.7543 −0.631182
\(624\) 0 0
\(625\) 24.9300 0.997202
\(626\) 34.6481 1.38482
\(627\) −1.40236 −0.0560048
\(628\) 60.8996 2.43016
\(629\) 0.493183 0.0196645
\(630\) 0.167545 0.00667514
\(631\) 23.5737 0.938456 0.469228 0.883077i \(-0.344532\pi\)
0.469228 + 0.883077i \(0.344532\pi\)
\(632\) −69.5294 −2.76573
\(633\) 25.4741 1.01251
\(634\) 45.7426 1.81667
\(635\) −0.546699 −0.0216951
\(636\) −12.9163 −0.512166
\(637\) 0 0
\(638\) −13.3341 −0.527901
\(639\) 9.91104 0.392075
\(640\) −1.32826 −0.0525042
\(641\) 36.8958 1.45730 0.728648 0.684889i \(-0.240149\pi\)
0.728648 + 0.684889i \(0.240149\pi\)
\(642\) −48.5365 −1.91558
\(643\) 25.1777 0.992911 0.496455 0.868062i \(-0.334635\pi\)
0.496455 + 0.868062i \(0.334635\pi\)
\(644\) 17.2735 0.680672
\(645\) −0.122159 −0.00481002
\(646\) −4.74225 −0.186581
\(647\) 5.00172 0.196638 0.0983189 0.995155i \(-0.468653\pi\)
0.0983189 + 0.995155i \(0.468653\pi\)
\(648\) 4.94928 0.194426
\(649\) 5.64794 0.221701
\(650\) 0 0
\(651\) 10.0622 0.394367
\(652\) 67.6607 2.64980
\(653\) −27.8545 −1.09003 −0.545015 0.838426i \(-0.683476\pi\)
−0.545015 + 0.838426i \(0.683476\pi\)
\(654\) 4.19686 0.164110
\(655\) −0.702887 −0.0274641
\(656\) 21.0946 0.823606
\(657\) −3.42254 −0.133526
\(658\) 10.3738 0.404412
\(659\) 4.79936 0.186957 0.0934783 0.995621i \(-0.470201\pi\)
0.0934783 + 0.995621i \(0.470201\pi\)
\(660\) −0.186107 −0.00724419
\(661\) −3.05390 −0.118783 −0.0593914 0.998235i \(-0.518916\pi\)
−0.0593914 + 0.998235i \(0.518916\pi\)
\(662\) 54.1744 2.10555
\(663\) 0 0
\(664\) −34.8138 −1.35104
\(665\) 0.141222 0.00547634
\(666\) −1.29398 −0.0501406
\(667\) 34.4580 1.33422
\(668\) −6.63418 −0.256684
\(669\) −7.00654 −0.270888
\(670\) −1.22313 −0.0472535
\(671\) −0.685375 −0.0264586
\(672\) 0.173279 0.00668437
\(673\) 12.7900 0.493016 0.246508 0.969141i \(-0.420717\pi\)
0.246508 + 0.969141i \(0.420717\pi\)
\(674\) −79.8121 −3.07425
\(675\) −4.99534 −0.192271
\(676\) 0 0
\(677\) 21.6803 0.833240 0.416620 0.909081i \(-0.363215\pi\)
0.416620 + 0.909081i \(0.363215\pi\)
\(678\) −33.4581 −1.28495
\(679\) 16.6705 0.639753
\(680\) −0.316049 −0.0121199
\(681\) −27.7571 −1.06365
\(682\) −16.7409 −0.641043
\(683\) 33.9669 1.29971 0.649854 0.760059i \(-0.274830\pi\)
0.649854 + 0.760059i \(0.274830\pi\)
\(684\) 8.30704 0.317628
\(685\) −0.247215 −0.00944561
\(686\) 2.45308 0.0936589
\(687\) 3.61011 0.137734
\(688\) −7.34352 −0.279969
\(689\) 0 0
\(690\) 0.720355 0.0274235
\(691\) −43.2170 −1.64405 −0.822026 0.569450i \(-0.807156\pi\)
−0.822026 + 0.569450i \(0.807156\pi\)
\(692\) 99.5503 3.78433
\(693\) −0.678230 −0.0257638
\(694\) 10.8235 0.410856
\(695\) 0.0831993 0.00315593
\(696\) 39.6658 1.50353
\(697\) −4.80359 −0.181949
\(698\) 17.4762 0.661485
\(699\) −15.6176 −0.590713
\(700\) −20.0692 −0.758543
\(701\) −0.456718 −0.0172500 −0.00862500 0.999963i \(-0.502745\pi\)
−0.00862500 + 0.999963i \(0.502745\pi\)
\(702\) 0 0
\(703\) −1.09068 −0.0411358
\(704\) 5.28107 0.199038
\(705\) 0.288832 0.0108781
\(706\) 8.18503 0.308048
\(707\) −1.50429 −0.0565746
\(708\) −33.4562 −1.25736
\(709\) 14.8844 0.558994 0.279497 0.960147i \(-0.409832\pi\)
0.279497 + 0.960147i \(0.409832\pi\)
\(710\) 1.66054 0.0623191
\(711\) −14.0484 −0.526855
\(712\) −77.9724 −2.92214
\(713\) 43.2621 1.62018
\(714\) −2.29352 −0.0858330
\(715\) 0 0
\(716\) −25.3222 −0.946335
\(717\) 27.9831 1.04505
\(718\) −60.3135 −2.25088
\(719\) 9.97486 0.372000 0.186000 0.982550i \(-0.440448\pi\)
0.186000 + 0.982550i \(0.440448\pi\)
\(720\) 0.280426 0.0104509
\(721\) −2.21721 −0.0825732
\(722\) −36.1209 −1.34428
\(723\) 25.9732 0.965955
\(724\) −76.1203 −2.82899
\(725\) −40.0349 −1.48686
\(726\) −25.8554 −0.959585
\(727\) 10.4392 0.387169 0.193585 0.981084i \(-0.437989\pi\)
0.193585 + 0.981084i \(0.437989\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −0.573428 −0.0212235
\(731\) 1.67224 0.0618500
\(732\) 4.05990 0.150058
\(733\) 22.7887 0.841718 0.420859 0.907126i \(-0.361729\pi\)
0.420859 + 0.907126i \(0.361729\pi\)
\(734\) −36.9840 −1.36510
\(735\) 0.0682999 0.00251928
\(736\) 0.745008 0.0274614
\(737\) 4.95129 0.182383
\(738\) 12.6033 0.463934
\(739\) −42.2796 −1.55528 −0.777640 0.628710i \(-0.783583\pi\)
−0.777640 + 0.628710i \(0.783583\pi\)
\(740\) −0.144744 −0.00532089
\(741\) 0 0
\(742\) −7.88652 −0.289523
\(743\) −6.98202 −0.256145 −0.128073 0.991765i \(-0.540879\pi\)
−0.128073 + 0.991765i \(0.540879\pi\)
\(744\) 49.8005 1.82577
\(745\) 1.12881 0.0413564
\(746\) −61.9331 −2.26753
\(747\) −7.03411 −0.257365
\(748\) 2.54762 0.0931501
\(749\) −19.7860 −0.722963
\(750\) −1.67467 −0.0611502
\(751\) 35.5530 1.29735 0.648673 0.761067i \(-0.275324\pi\)
0.648673 + 0.761067i \(0.275324\pi\)
\(752\) 17.3630 0.633162
\(753\) −6.38525 −0.232691
\(754\) 0 0
\(755\) −1.39020 −0.0505945
\(756\) 4.01758 0.146118
\(757\) −13.1171 −0.476749 −0.238374 0.971173i \(-0.576615\pi\)
−0.238374 + 0.971173i \(0.576615\pi\)
\(758\) 58.9798 2.14224
\(759\) −2.91604 −0.105845
\(760\) 0.698946 0.0253534
\(761\) −3.28595 −0.119116 −0.0595578 0.998225i \(-0.518969\pi\)
−0.0595578 + 0.998225i \(0.518969\pi\)
\(762\) −19.6354 −0.711315
\(763\) 1.71086 0.0619372
\(764\) −35.3310 −1.27823
\(765\) −0.0638575 −0.00230877
\(766\) −11.1351 −0.402327
\(767\) 0 0
\(768\) −32.1332 −1.15951
\(769\) 1.28809 0.0464498 0.0232249 0.999730i \(-0.492607\pi\)
0.0232249 + 0.999730i \(0.492607\pi\)
\(770\) −0.113634 −0.00409508
\(771\) −13.8179 −0.497638
\(772\) −24.9051 −0.896352
\(773\) 22.0147 0.791813 0.395907 0.918291i \(-0.370430\pi\)
0.395907 + 0.918291i \(0.370430\pi\)
\(774\) −4.38750 −0.157706
\(775\) −50.2639 −1.80553
\(776\) 82.5068 2.96182
\(777\) −0.527492 −0.0189237
\(778\) 48.5335 1.74001
\(779\) 10.6232 0.380615
\(780\) 0 0
\(781\) −6.72197 −0.240531
\(782\) −9.86096 −0.352627
\(783\) 8.01446 0.286413
\(784\) 4.10580 0.146636
\(785\) 1.03531 0.0369517
\(786\) −25.2451 −0.900462
\(787\) −5.09721 −0.181696 −0.0908479 0.995865i \(-0.528958\pi\)
−0.0908479 + 0.995865i \(0.528958\pi\)
\(788\) −60.8397 −2.16732
\(789\) −11.3364 −0.403587
\(790\) −2.35373 −0.0837420
\(791\) −13.6392 −0.484955
\(792\) −3.35675 −0.119277
\(793\) 0 0
\(794\) 25.6553 0.910471
\(795\) −0.219581 −0.00778773
\(796\) 60.4132 2.14129
\(797\) −41.5555 −1.47197 −0.735986 0.676997i \(-0.763281\pi\)
−0.735986 + 0.676997i \(0.763281\pi\)
\(798\) 5.07216 0.179552
\(799\) −3.95383 −0.139877
\(800\) −0.865585 −0.0306031
\(801\) −15.7543 −0.556650
\(802\) 55.0068 1.94236
\(803\) 2.32127 0.0819158
\(804\) −29.3296 −1.03437
\(805\) 0.293654 0.0103499
\(806\) 0 0
\(807\) 21.0002 0.739242
\(808\) −7.44515 −0.261920
\(809\) 22.0300 0.774532 0.387266 0.921968i \(-0.373419\pi\)
0.387266 + 0.921968i \(0.373419\pi\)
\(810\) 0.167545 0.00588692
\(811\) 20.0797 0.705094 0.352547 0.935794i \(-0.385316\pi\)
0.352547 + 0.935794i \(0.385316\pi\)
\(812\) 32.1987 1.12995
\(813\) 29.2450 1.02567
\(814\) 0.877615 0.0307604
\(815\) 1.15025 0.0402915
\(816\) −3.83875 −0.134383
\(817\) −3.69818 −0.129383
\(818\) 59.9544 2.09626
\(819\) 0 0
\(820\) 1.40980 0.0492324
\(821\) −13.8647 −0.483882 −0.241941 0.970291i \(-0.577784\pi\)
−0.241941 + 0.970291i \(0.577784\pi\)
\(822\) −8.87904 −0.309692
\(823\) −14.7804 −0.515213 −0.257606 0.966250i \(-0.582934\pi\)
−0.257606 + 0.966250i \(0.582934\pi\)
\(824\) −10.9736 −0.382283
\(825\) 3.38799 0.117955
\(826\) −20.4279 −0.710777
\(827\) −26.2554 −0.912988 −0.456494 0.889726i \(-0.650895\pi\)
−0.456494 + 0.889726i \(0.650895\pi\)
\(828\) 17.2735 0.600296
\(829\) 14.3216 0.497408 0.248704 0.968579i \(-0.419995\pi\)
0.248704 + 0.968579i \(0.419995\pi\)
\(830\) −1.17853 −0.0409073
\(831\) 16.5036 0.572502
\(832\) 0 0
\(833\) −0.934958 −0.0323944
\(834\) 2.98821 0.103473
\(835\) −0.112783 −0.00390301
\(836\) −5.63409 −0.194859
\(837\) 10.0622 0.347799
\(838\) 68.1861 2.35545
\(839\) 34.5482 1.19274 0.596368 0.802711i \(-0.296610\pi\)
0.596368 + 0.802711i \(0.296610\pi\)
\(840\) 0.338035 0.0116633
\(841\) 35.2315 1.21488
\(842\) −38.8382 −1.33845
\(843\) 1.12161 0.0386302
\(844\) 102.344 3.52284
\(845\) 0 0
\(846\) 10.3738 0.356658
\(847\) −10.5400 −0.362159
\(848\) −13.2000 −0.453288
\(849\) 10.9412 0.375501
\(850\) 11.4569 0.392969
\(851\) −2.26794 −0.0777440
\(852\) 39.8184 1.36416
\(853\) −40.2048 −1.37659 −0.688294 0.725432i \(-0.741640\pi\)
−0.688294 + 0.725432i \(0.741640\pi\)
\(854\) 2.47892 0.0848267
\(855\) 0.141222 0.00482968
\(856\) −97.9263 −3.34705
\(857\) 24.2988 0.830031 0.415015 0.909814i \(-0.363776\pi\)
0.415015 + 0.909814i \(0.363776\pi\)
\(858\) 0 0
\(859\) −38.8589 −1.32585 −0.662924 0.748687i \(-0.730685\pi\)
−0.662924 + 0.748687i \(0.730685\pi\)
\(860\) −0.490785 −0.0167356
\(861\) 5.13776 0.175094
\(862\) 44.6700 1.52147
\(863\) 30.0838 1.02406 0.512032 0.858966i \(-0.328893\pi\)
0.512032 + 0.858966i \(0.328893\pi\)
\(864\) 0.173279 0.00589506
\(865\) 1.69238 0.0575426
\(866\) −23.0699 −0.783948
\(867\) −16.1259 −0.547663
\(868\) 40.4256 1.37213
\(869\) 9.52804 0.323216
\(870\) 1.34278 0.0455245
\(871\) 0 0
\(872\) 8.46752 0.286746
\(873\) 16.6705 0.564209
\(874\) 21.8076 0.737654
\(875\) −0.682680 −0.0230788
\(876\) −13.7503 −0.464580
\(877\) 14.7532 0.498179 0.249090 0.968480i \(-0.419869\pi\)
0.249090 + 0.968480i \(0.419869\pi\)
\(878\) −47.1822 −1.59232
\(879\) −15.5503 −0.524498
\(880\) −0.190193 −0.00641141
\(881\) 34.5537 1.16414 0.582072 0.813137i \(-0.302242\pi\)
0.582072 + 0.813137i \(0.302242\pi\)
\(882\) 2.45308 0.0825994
\(883\) 7.47582 0.251581 0.125791 0.992057i \(-0.459853\pi\)
0.125791 + 0.992057i \(0.459853\pi\)
\(884\) 0 0
\(885\) −0.568764 −0.0191188
\(886\) −13.2575 −0.445396
\(887\) −20.3565 −0.683503 −0.341752 0.939790i \(-0.611020\pi\)
−0.341752 + 0.939790i \(0.611020\pi\)
\(888\) −2.61071 −0.0876095
\(889\) −8.00439 −0.268459
\(890\) −2.63955 −0.0884778
\(891\) −0.678230 −0.0227216
\(892\) −28.1493 −0.942509
\(893\) 8.74395 0.292605
\(894\) 40.5426 1.35595
\(895\) −0.430484 −0.0143895
\(896\) −19.4475 −0.649696
\(897\) 0 0
\(898\) −33.2475 −1.10948
\(899\) 80.6428 2.68959
\(900\) −20.0692 −0.668972
\(901\) 3.00585 0.100139
\(902\) −8.54795 −0.284616
\(903\) −1.78857 −0.0595200
\(904\) −67.5044 −2.24516
\(905\) −1.29406 −0.0430161
\(906\) −49.9307 −1.65884
\(907\) −9.91111 −0.329093 −0.164546 0.986369i \(-0.552616\pi\)
−0.164546 + 0.986369i \(0.552616\pi\)
\(908\) −111.516 −3.70080
\(909\) −1.50429 −0.0498941
\(910\) 0 0
\(911\) −11.7109 −0.387998 −0.193999 0.981002i \(-0.562146\pi\)
−0.193999 + 0.981002i \(0.562146\pi\)
\(912\) 8.48945 0.281114
\(913\) 4.77075 0.157889
\(914\) −73.1578 −2.41984
\(915\) 0.0690193 0.00228171
\(916\) 14.5039 0.479223
\(917\) −10.2912 −0.339845
\(918\) −2.29352 −0.0756976
\(919\) −20.7949 −0.685962 −0.342981 0.939342i \(-0.611437\pi\)
−0.342981 + 0.939342i \(0.611437\pi\)
\(920\) 1.45338 0.0479164
\(921\) 0.119003 0.00392127
\(922\) 5.46553 0.179998
\(923\) 0 0
\(924\) −2.72485 −0.0896409
\(925\) 2.63500 0.0866382
\(926\) 53.0075 1.74194
\(927\) −2.21721 −0.0728227
\(928\) 1.38873 0.0455875
\(929\) −51.1824 −1.67924 −0.839620 0.543174i \(-0.817222\pi\)
−0.839620 + 0.543174i \(0.817222\pi\)
\(930\) 1.68586 0.0552816
\(931\) 2.06767 0.0677652
\(932\) −62.7451 −2.05528
\(933\) −31.4738 −1.03041
\(934\) −58.8906 −1.92696
\(935\) 0.0433101 0.00141639
\(936\) 0 0
\(937\) 46.1728 1.50840 0.754200 0.656645i \(-0.228025\pi\)
0.754200 + 0.656645i \(0.228025\pi\)
\(938\) −17.9082 −0.584723
\(939\) 14.1243 0.460931
\(940\) 1.16041 0.0378483
\(941\) −31.6254 −1.03096 −0.515480 0.856902i \(-0.672386\pi\)
−0.515480 + 0.856902i \(0.672386\pi\)
\(942\) 37.1844 1.21153
\(943\) 22.0897 0.719339
\(944\) −34.1909 −1.11282
\(945\) 0.0682999 0.00222180
\(946\) 2.97574 0.0967496
\(947\) −19.2256 −0.624747 −0.312373 0.949959i \(-0.601124\pi\)
−0.312373 + 0.949959i \(0.601124\pi\)
\(948\) −56.4405 −1.83310
\(949\) 0 0
\(950\) −25.3371 −0.822045
\(951\) 18.6470 0.604671
\(952\) −4.62737 −0.149974
\(953\) 14.4415 0.467806 0.233903 0.972260i \(-0.424850\pi\)
0.233903 + 0.972260i \(0.424850\pi\)
\(954\) −7.88652 −0.255336
\(955\) −0.600636 −0.0194361
\(956\) 112.424 3.63606
\(957\) −5.43565 −0.175710
\(958\) 75.2901 2.43251
\(959\) −3.61956 −0.116882
\(960\) −0.531820 −0.0171644
\(961\) 70.2471 2.26604
\(962\) 0 0
\(963\) −19.7860 −0.637593
\(964\) 104.350 3.36087
\(965\) −0.423392 −0.0136295
\(966\) 10.5470 0.339343
\(967\) 3.28457 0.105625 0.0528123 0.998604i \(-0.483181\pi\)
0.0528123 + 0.998604i \(0.483181\pi\)
\(968\) −52.1655 −1.67666
\(969\) −1.93319 −0.0621029
\(970\) 2.79305 0.0896794
\(971\) −15.9721 −0.512568 −0.256284 0.966601i \(-0.582498\pi\)
−0.256284 + 0.966601i \(0.582498\pi\)
\(972\) 4.01758 0.128864
\(973\) 1.21815 0.0390520
\(974\) −43.8382 −1.40467
\(975\) 0 0
\(976\) 4.14905 0.132808
\(977\) −21.9335 −0.701714 −0.350857 0.936429i \(-0.614110\pi\)
−0.350857 + 0.936429i \(0.614110\pi\)
\(978\) 41.3126 1.32103
\(979\) 10.6850 0.341495
\(980\) 0.274400 0.00876540
\(981\) 1.71086 0.0546235
\(982\) −33.2873 −1.06224
\(983\) −22.0867 −0.704456 −0.352228 0.935914i \(-0.614576\pi\)
−0.352228 + 0.935914i \(0.614576\pi\)
\(984\) 25.4282 0.810622
\(985\) −1.03429 −0.0329552
\(986\) −18.3814 −0.585382
\(987\) 4.22889 0.134607
\(988\) 0 0
\(989\) −7.68993 −0.244526
\(990\) −0.113634 −0.00361153
\(991\) 4.36458 0.138645 0.0693227 0.997594i \(-0.477916\pi\)
0.0693227 + 0.997594i \(0.477916\pi\)
\(992\) 1.74356 0.0553580
\(993\) 22.0843 0.700823
\(994\) 24.3125 0.771147
\(995\) 1.02704 0.0325593
\(996\) −28.2601 −0.895456
\(997\) 55.3319 1.75238 0.876189 0.481967i \(-0.160078\pi\)
0.876189 + 0.481967i \(0.160078\pi\)
\(998\) 25.5710 0.809435
\(999\) −0.527492 −0.0166891
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 3549.2.a.bd.1.7 8
13.2 odd 12 273.2.bd.a.43.8 16
13.7 odd 12 273.2.bd.a.127.8 yes 16
13.12 even 2 3549.2.a.bb.1.2 8
39.2 even 12 819.2.ct.b.316.1 16
39.20 even 12 819.2.ct.b.127.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.bd.a.43.8 16 13.2 odd 12
273.2.bd.a.127.8 yes 16 13.7 odd 12
819.2.ct.b.127.1 16 39.20 even 12
819.2.ct.b.316.1 16 39.2 even 12
3549.2.a.bb.1.2 8 13.12 even 2
3549.2.a.bd.1.7 8 1.1 even 1 trivial