Properties

Label 1183.2.c.g.337.8
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1183,2,Mod(337,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.11667456256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 13x^{6} + 44x^{4} + 21x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.8
Root \(2.74108i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.g.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.74108i q^{2} +1.36482 q^{3} -5.51353 q^{4} -0.741082i q^{5} +3.74108i q^{6} -1.00000i q^{7} -9.63087i q^{8} -1.13727 q^{9} +O(q^{10})\) \(q+2.74108i q^{2} +1.36482 q^{3} -5.51353 q^{4} -0.741082i q^{5} +3.74108i q^{6} -1.00000i q^{7} -9.63087i q^{8} -1.13727 q^{9} +2.03137 q^{10} -1.36482i q^{11} -7.52497 q^{12} +2.74108 q^{14} -1.01144i q^{15} +15.3720 q^{16} +4.14871 q^{17} -3.11734i q^{18} -7.26606i q^{19} +4.08598i q^{20} -1.36482i q^{21} +3.74108 q^{22} +2.33345 q^{23} -13.1444i q^{24} +4.45080 q^{25} -5.64662 q^{27} +5.51353i q^{28} -0.407629 q^{29} +2.77245 q^{30} +2.77245i q^{31} +22.8740i q^{32} -1.86273i q^{33} +11.3720i q^{34} -0.741082 q^{35} +6.27036 q^{36} -6.10590i q^{37} +19.9169 q^{38} -7.13727 q^{40} -1.25461i q^{41} +3.74108 q^{42} +1.74108 q^{43} +7.52497i q^{44} +0.842809i q^{45} +6.39619i q^{46} -5.85843i q^{47} +20.9799 q^{48} -1.00000 q^{49} +12.2000i q^{50} +5.66224 q^{51} +4.56778 q^{53} -15.4779i q^{54} -1.01144 q^{55} -9.63087 q^{56} -9.91685i q^{57} -1.11734i q^{58} -10.9843i q^{59} +5.57662i q^{60} +6.52497 q^{61} -7.59951 q^{62} +1.13727i q^{63} -31.9557 q^{64} +5.10590 q^{66} +13.7597i q^{67} -22.8740 q^{68} +3.18474 q^{69} -2.03137i q^{70} +4.81526i q^{71} +10.9529i q^{72} +6.06987i q^{73} +16.7368 q^{74} +6.07453 q^{75} +40.0616i q^{76} -1.36482 q^{77} -9.12582 q^{79} -11.3919i q^{80} -4.29482 q^{81} +3.43900 q^{82} -11.7368i q^{83} +7.52497i q^{84} -3.07453i q^{85} +4.77245i q^{86} -0.556340 q^{87} -13.1444 q^{88} -1.76101i q^{89} -2.31021 q^{90} -12.8656 q^{92} +3.78389i q^{93} +16.0584 q^{94} -5.38474 q^{95} +31.2189i q^{96} -9.53381i q^{97} -2.74108i q^{98} +1.55217i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 10 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 10 q^{4} + 14 q^{9} + 22 q^{10} - 24 q^{12} + 2 q^{14} + 38 q^{16} + 8 q^{17} + 10 q^{22} + 4 q^{23} - 10 q^{25} - 52 q^{27} + 2 q^{29} + 8 q^{30} + 14 q^{35} + 68 q^{36} + 46 q^{38} - 34 q^{40} + 10 q^{42} - 6 q^{43} + 22 q^{48} - 8 q^{49} - 14 q^{51} + 4 q^{53} - 6 q^{55} - 12 q^{56} + 16 q^{61} + 10 q^{62} - 28 q^{64} + 12 q^{66} - 66 q^{68} + 36 q^{69} + 40 q^{74} + 14 q^{75} - 2 q^{77} - 52 q^{79} + 48 q^{81} + 28 q^{82} + 26 q^{87} - 6 q^{88} - 52 q^{90} + 24 q^{92} + 66 q^{94} - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.74108i 1.93824i 0.246594 + 0.969119i \(0.420689\pi\)
−0.246594 + 0.969119i \(0.579311\pi\)
\(3\) 1.36482 0.787979 0.393989 0.919115i \(-0.371095\pi\)
0.393989 + 0.919115i \(0.371095\pi\)
\(4\) −5.51353 −2.75677
\(5\) − 0.741082i − 0.331422i −0.986174 0.165711i \(-0.947008\pi\)
0.986174 0.165711i \(-0.0529919\pi\)
\(6\) 3.74108i 1.52729i
\(7\) − 1.00000i − 0.377964i
\(8\) − 9.63087i − 3.40503i
\(9\) −1.13727 −0.379089
\(10\) 2.03137 0.642374
\(11\) − 1.36482i − 0.411509i −0.978604 0.205754i \(-0.934035\pi\)
0.978604 0.205754i \(-0.0659648\pi\)
\(12\) −7.52497 −2.17227
\(13\) 0 0
\(14\) 2.74108 0.732585
\(15\) − 1.01144i − 0.261153i
\(16\) 15.3720 3.84299
\(17\) 4.14871 1.00621 0.503105 0.864225i \(-0.332191\pi\)
0.503105 + 0.864225i \(0.332191\pi\)
\(18\) − 3.11734i − 0.734765i
\(19\) − 7.26606i − 1.66695i −0.552559 0.833474i \(-0.686349\pi\)
0.552559 0.833474i \(-0.313651\pi\)
\(20\) 4.08598i 0.913652i
\(21\) − 1.36482i − 0.297828i
\(22\) 3.74108 0.797601
\(23\) 2.33345 0.486559 0.243279 0.969956i \(-0.421777\pi\)
0.243279 + 0.969956i \(0.421777\pi\)
\(24\) − 13.1444i − 2.68309i
\(25\) 4.45080 0.890159
\(26\) 0 0
\(27\) −5.64662 −1.08669
\(28\) 5.51353i 1.04196i
\(29\) −0.407629 −0.0756948 −0.0378474 0.999284i \(-0.512050\pi\)
−0.0378474 + 0.999284i \(0.512050\pi\)
\(30\) 2.77245 0.506178
\(31\) 2.77245i 0.497946i 0.968510 + 0.248973i \(0.0800931\pi\)
−0.968510 + 0.248973i \(0.919907\pi\)
\(32\) 22.8740i 4.04360i
\(33\) − 1.86273i − 0.324260i
\(34\) 11.3720i 1.95027i
\(35\) −0.741082 −0.125266
\(36\) 6.27036 1.04506
\(37\) − 6.10590i − 1.00380i −0.864924 0.501902i \(-0.832634\pi\)
0.864924 0.501902i \(-0.167366\pi\)
\(38\) 19.9169 3.23094
\(39\) 0 0
\(40\) −7.13727 −1.12850
\(41\) − 1.25461i − 0.195938i −0.995189 0.0979688i \(-0.968765\pi\)
0.995189 0.0979688i \(-0.0312345\pi\)
\(42\) 3.74108 0.577261
\(43\) 1.74108 0.265513 0.132756 0.991149i \(-0.457617\pi\)
0.132756 + 0.991149i \(0.457617\pi\)
\(44\) 7.52497i 1.13443i
\(45\) 0.842809i 0.125639i
\(46\) 6.39619i 0.943066i
\(47\) − 5.85843i − 0.854539i −0.904124 0.427270i \(-0.859475\pi\)
0.904124 0.427270i \(-0.140525\pi\)
\(48\) 20.9799 3.02819
\(49\) −1.00000 −0.142857
\(50\) 12.2000i 1.72534i
\(51\) 5.66224 0.792872
\(52\) 0 0
\(53\) 4.56778 0.627433 0.313717 0.949517i \(-0.398426\pi\)
0.313717 + 0.949517i \(0.398426\pi\)
\(54\) − 15.4779i − 2.10627i
\(55\) −1.01144 −0.136383
\(56\) −9.63087 −1.28698
\(57\) − 9.91685i − 1.31352i
\(58\) − 1.11734i − 0.146715i
\(59\) − 10.9843i − 1.43003i −0.699110 0.715014i \(-0.746420\pi\)
0.699110 0.715014i \(-0.253580\pi\)
\(60\) 5.57662i 0.719939i
\(61\) 6.52497 0.835437 0.417719 0.908576i \(-0.362830\pi\)
0.417719 + 0.908576i \(0.362830\pi\)
\(62\) −7.59951 −0.965139
\(63\) 1.13727i 0.143282i
\(64\) −31.9557 −3.99446
\(65\) 0 0
\(66\) 5.10590 0.628493
\(67\) 13.7597i 1.68101i 0.541804 + 0.840505i \(0.317742\pi\)
−0.541804 + 0.840505i \(0.682258\pi\)
\(68\) −22.8740 −2.77389
\(69\) 3.18474 0.383398
\(70\) − 2.03137i − 0.242795i
\(71\) 4.81526i 0.571466i 0.958309 + 0.285733i \(0.0922370\pi\)
−0.958309 + 0.285733i \(0.907763\pi\)
\(72\) 10.9529i 1.29081i
\(73\) 6.06987i 0.710425i 0.934786 + 0.355212i \(0.115591\pi\)
−0.934786 + 0.355212i \(0.884409\pi\)
\(74\) 16.7368 1.94561
\(75\) 6.07453 0.701427
\(76\) 40.0616i 4.59538i
\(77\) −1.36482 −0.155536
\(78\) 0 0
\(79\) −9.12582 −1.02674 −0.513368 0.858169i \(-0.671602\pi\)
−0.513368 + 0.858169i \(0.671602\pi\)
\(80\) − 11.3919i − 1.27365i
\(81\) −4.29482 −0.477202
\(82\) 3.43900 0.379774
\(83\) − 11.7368i − 1.28828i −0.764908 0.644139i \(-0.777216\pi\)
0.764908 0.644139i \(-0.222784\pi\)
\(84\) 7.52497i 0.821042i
\(85\) − 3.07453i − 0.333480i
\(86\) 4.77245i 0.514626i
\(87\) −0.556340 −0.0596459
\(88\) −13.1444 −1.40120
\(89\) − 1.76101i − 0.186666i −0.995635 0.0933331i \(-0.970248\pi\)
0.995635 0.0933331i \(-0.0297522\pi\)
\(90\) −2.31021 −0.243517
\(91\) 0 0
\(92\) −12.8656 −1.34133
\(93\) 3.78389i 0.392371i
\(94\) 16.0584 1.65630
\(95\) −5.38474 −0.552463
\(96\) 31.2189i 3.18627i
\(97\) − 9.53381i − 0.968012i −0.875065 0.484006i \(-0.839181\pi\)
0.875065 0.484006i \(-0.160819\pi\)
\(98\) − 2.74108i − 0.276891i
\(99\) 1.55217i 0.155998i
\(100\) −24.5396 −2.45396
\(101\) 7.49361 0.745642 0.372821 0.927903i \(-0.378391\pi\)
0.372821 + 0.927903i \(0.378391\pi\)
\(102\) 15.5207i 1.53678i
\(103\) −2.80848 −0.276728 −0.138364 0.990381i \(-0.544184\pi\)
−0.138364 + 0.990381i \(0.544184\pi\)
\(104\) 0 0
\(105\) −1.01144 −0.0987067
\(106\) 12.5207i 1.21611i
\(107\) 1.48647 0.143702 0.0718512 0.997415i \(-0.477109\pi\)
0.0718512 + 0.997415i \(0.477109\pi\)
\(108\) 31.1328 2.99576
\(109\) − 2.87121i − 0.275012i −0.990501 0.137506i \(-0.956091\pi\)
0.990501 0.137506i \(-0.0439087\pi\)
\(110\) − 2.77245i − 0.264343i
\(111\) − 8.33345i − 0.790976i
\(112\) − 15.3720i − 1.45251i
\(113\) −12.4194 −1.16832 −0.584161 0.811638i \(-0.698576\pi\)
−0.584161 + 0.811638i \(0.698576\pi\)
\(114\) 27.1829 2.54591
\(115\) − 1.72928i − 0.161256i
\(116\) 2.24747 0.208673
\(117\) 0 0
\(118\) 30.1087 2.77173
\(119\) − 4.14871i − 0.380312i
\(120\) −9.74108 −0.889235
\(121\) 9.13727 0.830661
\(122\) 17.8855i 1.61928i
\(123\) − 1.71232i − 0.154395i
\(124\) − 15.2860i − 1.37272i
\(125\) − 7.00382i − 0.626440i
\(126\) −3.11734 −0.277715
\(127\) −5.43052 −0.481880 −0.240940 0.970540i \(-0.577456\pi\)
−0.240940 + 0.970540i \(0.577456\pi\)
\(128\) − 41.8452i − 3.69862i
\(129\) 2.37626 0.209218
\(130\) 0 0
\(131\) −11.3220 −0.989209 −0.494604 0.869118i \(-0.664687\pi\)
−0.494604 + 0.869118i \(0.664687\pi\)
\(132\) 10.2702i 0.893909i
\(133\) −7.26606 −0.630047
\(134\) −37.7164 −3.25820
\(135\) 4.18461i 0.360154i
\(136\) − 39.9557i − 3.42617i
\(137\) − 13.9754i − 1.19400i −0.802241 0.597000i \(-0.796359\pi\)
0.802241 0.597000i \(-0.203641\pi\)
\(138\) 8.72964i 0.743116i
\(139\) 10.4309 0.884735 0.442368 0.896834i \(-0.354139\pi\)
0.442368 + 0.896834i \(0.354139\pi\)
\(140\) 4.08598 0.345328
\(141\) − 7.99569i − 0.673359i
\(142\) −13.1990 −1.10764
\(143\) 0 0
\(144\) −17.4820 −1.45684
\(145\) 0.302087i 0.0250869i
\(146\) −16.6380 −1.37697
\(147\) −1.36482 −0.112568
\(148\) 33.6651i 2.76725i
\(149\) − 8.16433i − 0.668848i −0.942423 0.334424i \(-0.891458\pi\)
0.942423 0.334424i \(-0.108542\pi\)
\(150\) 16.6508i 1.35953i
\(151\) − 2.46188i − 0.200345i −0.994970 0.100173i \(-0.968061\pi\)
0.994970 0.100173i \(-0.0319395\pi\)
\(152\) −69.9785 −5.67600
\(153\) −4.71820 −0.381444
\(154\) − 3.74108i − 0.301465i
\(155\) 2.05461 0.165030
\(156\) 0 0
\(157\) −12.9198 −1.03111 −0.515557 0.856855i \(-0.672415\pi\)
−0.515557 + 0.856855i \(0.672415\pi\)
\(158\) − 25.0146i − 1.99006i
\(159\) 6.23420 0.494404
\(160\) 16.9515 1.34014
\(161\) − 2.33345i − 0.183902i
\(162\) − 11.7724i − 0.924931i
\(163\) − 6.02742i − 0.472104i −0.971740 0.236052i \(-0.924146\pi\)
0.971740 0.236052i \(-0.0758536\pi\)
\(164\) 6.91734i 0.540154i
\(165\) −1.38044 −0.107467
\(166\) 32.1715 2.49699
\(167\) 7.65116i 0.592064i 0.955178 + 0.296032i \(0.0956636\pi\)
−0.955178 + 0.296032i \(0.904336\pi\)
\(168\) −13.1444 −1.01411
\(169\) 0 0
\(170\) 8.42755 0.646364
\(171\) 8.26345i 0.631922i
\(172\) −9.59951 −0.731956
\(173\) 0.164460 0.0125036 0.00625182 0.999980i \(-0.498010\pi\)
0.00625182 + 0.999980i \(0.498010\pi\)
\(174\) − 1.52497i − 0.115608i
\(175\) − 4.45080i − 0.336449i
\(176\) − 20.9799i − 1.58142i
\(177\) − 14.9915i − 1.12683i
\(178\) 4.82706 0.361803
\(179\) −0.768633 −0.0574503 −0.0287252 0.999587i \(-0.509145\pi\)
−0.0287252 + 0.999587i \(0.509145\pi\)
\(180\) − 4.64685i − 0.346356i
\(181\) 9.92152 0.737461 0.368730 0.929536i \(-0.379793\pi\)
0.368730 + 0.929536i \(0.379793\pi\)
\(182\) 0 0
\(183\) 8.90541 0.658307
\(184\) − 22.4732i − 1.65675i
\(185\) −4.52497 −0.332683
\(186\) −10.3720 −0.760509
\(187\) − 5.66224i − 0.414064i
\(188\) 32.3006i 2.35576i
\(189\) 5.64662i 0.410731i
\(190\) − 14.7600i − 1.07080i
\(191\) 9.89693 0.716117 0.358058 0.933699i \(-0.383439\pi\)
0.358058 + 0.933699i \(0.383439\pi\)
\(192\) −43.6138 −3.14755
\(193\) 8.70075i 0.626293i 0.949705 + 0.313147i \(0.101383\pi\)
−0.949705 + 0.313147i \(0.898617\pi\)
\(194\) 26.1330 1.87624
\(195\) 0 0
\(196\) 5.51353 0.393824
\(197\) 26.0186i 1.85375i 0.375374 + 0.926874i \(0.377514\pi\)
−0.375374 + 0.926874i \(0.622486\pi\)
\(198\) −4.25461 −0.302362
\(199\) −10.1330 −0.718307 −0.359153 0.933279i \(-0.616934\pi\)
−0.359153 + 0.933279i \(0.616934\pi\)
\(200\) − 42.8651i − 3.03102i
\(201\) 18.7795i 1.32460i
\(202\) 20.5406i 1.44523i
\(203\) 0.407629i 0.0286099i
\(204\) −31.2189 −2.18576
\(205\) −0.929771 −0.0649380
\(206\) − 7.69827i − 0.536364i
\(207\) −2.65376 −0.184449
\(208\) 0 0
\(209\) −9.91685 −0.685963
\(210\) − 2.77245i − 0.191317i
\(211\) 16.6782 1.14818 0.574088 0.818794i \(-0.305357\pi\)
0.574088 + 0.818794i \(0.305357\pi\)
\(212\) −25.1846 −1.72969
\(213\) 6.57196i 0.450303i
\(214\) 4.07453i 0.278529i
\(215\) − 1.29028i − 0.0879967i
\(216\) 54.3819i 3.70022i
\(217\) 2.77245 0.188206
\(218\) 7.87023 0.533039
\(219\) 8.28428i 0.559800i
\(220\) 5.57662 0.375976
\(221\) 0 0
\(222\) 22.8427 1.53310
\(223\) − 1.07036i − 0.0716766i −0.999358 0.0358383i \(-0.988590\pi\)
0.999358 0.0358383i \(-0.0114101\pi\)
\(224\) 22.8740 1.52834
\(225\) −5.06175 −0.337450
\(226\) − 34.0427i − 2.26449i
\(227\) 24.4664i 1.62389i 0.583732 + 0.811947i \(0.301592\pi\)
−0.583732 + 0.811947i \(0.698408\pi\)
\(228\) 54.6769i 3.62106i
\(229\) 4.72964i 0.312543i 0.987714 + 0.156272i \(0.0499475\pi\)
−0.987714 + 0.156272i \(0.950052\pi\)
\(230\) 4.74010 0.312553
\(231\) −1.86273 −0.122559
\(232\) 3.92582i 0.257743i
\(233\) −20.5507 −1.34632 −0.673160 0.739497i \(-0.735064\pi\)
−0.673160 + 0.739497i \(0.735064\pi\)
\(234\) 0 0
\(235\) −4.34157 −0.283213
\(236\) 60.5620i 3.94225i
\(237\) −12.4551 −0.809046
\(238\) 11.3720 0.737134
\(239\) − 6.25461i − 0.404577i −0.979326 0.202289i \(-0.935162\pi\)
0.979326 0.202289i \(-0.0648379\pi\)
\(240\) − 15.5479i − 1.00361i
\(241\) − 12.1444i − 0.782290i −0.920329 0.391145i \(-0.872079\pi\)
0.920329 0.391145i \(-0.127921\pi\)
\(242\) 25.0460i 1.61002i
\(243\) 11.0782 0.710668
\(244\) −35.9756 −2.30310
\(245\) 0.741082i 0.0473460i
\(246\) 4.69361 0.299254
\(247\) 0 0
\(248\) 26.7011 1.69552
\(249\) − 16.0186i − 1.01514i
\(250\) 19.1980 1.21419
\(251\) 6.31438 0.398560 0.199280 0.979943i \(-0.436140\pi\)
0.199280 + 0.979943i \(0.436140\pi\)
\(252\) − 6.27036i − 0.394996i
\(253\) − 3.18474i − 0.200223i
\(254\) − 14.8855i − 0.933999i
\(255\) − 4.19619i − 0.262775i
\(256\) 50.7896 3.17435
\(257\) −24.3562 −1.51930 −0.759649 0.650333i \(-0.774629\pi\)
−0.759649 + 0.650333i \(0.774629\pi\)
\(258\) 6.51353i 0.405515i
\(259\) −6.10590 −0.379402
\(260\) 0 0
\(261\) 0.463583 0.0286951
\(262\) − 31.0346i − 1.91732i
\(263\) −9.57910 −0.590672 −0.295336 0.955393i \(-0.595432\pi\)
−0.295336 + 0.955393i \(0.595432\pi\)
\(264\) −17.9397 −1.10411
\(265\) − 3.38510i − 0.207945i
\(266\) − 19.9169i − 1.22118i
\(267\) − 2.40345i − 0.147089i
\(268\) − 75.8643i − 4.63415i
\(269\) 29.3990 1.79249 0.896245 0.443560i \(-0.146285\pi\)
0.896245 + 0.443560i \(0.146285\pi\)
\(270\) −11.4704 −0.698064
\(271\) − 0.300385i − 0.0182471i −0.999958 0.00912354i \(-0.997096\pi\)
0.999958 0.00912354i \(-0.00290415\pi\)
\(272\) 63.7738 3.86686
\(273\) 0 0
\(274\) 38.3078 2.31426
\(275\) − 6.07453i − 0.366308i
\(276\) −17.5592 −1.05694
\(277\) 32.7710 1.96902 0.984509 0.175337i \(-0.0561014\pi\)
0.984509 + 0.175337i \(0.0561014\pi\)
\(278\) 28.5919i 1.71483i
\(279\) − 3.15302i − 0.188766i
\(280\) 7.13727i 0.426533i
\(281\) 4.29482i 0.256207i 0.991761 + 0.128104i \(0.0408890\pi\)
−0.991761 + 0.128104i \(0.959111\pi\)
\(282\) 21.9169 1.30513
\(283\) 21.1003 1.25428 0.627140 0.778907i \(-0.284225\pi\)
0.627140 + 0.778907i \(0.284225\pi\)
\(284\) − 26.5491i − 1.57540i
\(285\) −7.34920 −0.435329
\(286\) 0 0
\(287\) −1.25461 −0.0740574
\(288\) − 26.0139i − 1.53288i
\(289\) 0.211803 0.0124590
\(290\) −0.828044 −0.0486244
\(291\) − 13.0119i − 0.762773i
\(292\) − 33.4664i − 1.95847i
\(293\) 17.7638i 1.03777i 0.854843 + 0.518887i \(0.173654\pi\)
−0.854843 + 0.518887i \(0.826346\pi\)
\(294\) − 3.74108i − 0.218184i
\(295\) −8.14023 −0.473943
\(296\) −58.8052 −3.41798
\(297\) 7.70662i 0.447184i
\(298\) 22.3791 1.29639
\(299\) 0 0
\(300\) −33.4921 −1.93367
\(301\) − 1.74108i − 0.100354i
\(302\) 6.74822 0.388316
\(303\) 10.2274 0.587550
\(304\) − 111.693i − 6.40606i
\(305\) − 4.83554i − 0.276882i
\(306\) − 12.9330i − 0.739328i
\(307\) 18.0156i 1.02821i 0.857729 + 0.514103i \(0.171875\pi\)
−0.857729 + 0.514103i \(0.828125\pi\)
\(308\) 7.52497 0.428775
\(309\) −3.83307 −0.218056
\(310\) 5.63186i 0.319868i
\(311\) −17.2545 −0.978412 −0.489206 0.872168i \(-0.662713\pi\)
−0.489206 + 0.872168i \(0.662713\pi\)
\(312\) 0 0
\(313\) 6.81526 0.385221 0.192611 0.981275i \(-0.438305\pi\)
0.192611 + 0.981275i \(0.438305\pi\)
\(314\) − 35.4143i − 1.99854i
\(315\) 0.842809 0.0474869
\(316\) 50.3155 2.83047
\(317\) 25.0770i 1.40847i 0.709969 + 0.704233i \(0.248709\pi\)
−0.709969 + 0.704233i \(0.751291\pi\)
\(318\) 17.0885i 0.958273i
\(319\) 0.556340i 0.0311491i
\(320\) 23.6818i 1.32385i
\(321\) 2.02876 0.113234
\(322\) 6.39619 0.356446
\(323\) − 30.1448i − 1.67730i
\(324\) 23.6796 1.31553
\(325\) 0 0
\(326\) 16.5217 0.915050
\(327\) − 3.91869i − 0.216704i
\(328\) −12.0830 −0.667173
\(329\) −5.85843 −0.322986
\(330\) − 3.78389i − 0.208296i
\(331\) 2.99534i 0.164639i 0.996606 + 0.0823193i \(0.0262327\pi\)
−0.996606 + 0.0823193i \(0.973767\pi\)
\(332\) 64.7111i 3.55148i
\(333\) 6.94405i 0.380531i
\(334\) −20.9724 −1.14756
\(335\) 10.1970 0.557124
\(336\) − 20.9799i − 1.14455i
\(337\) 29.4888 1.60636 0.803179 0.595738i \(-0.203140\pi\)
0.803179 + 0.595738i \(0.203140\pi\)
\(338\) 0 0
\(339\) −16.9503 −0.920613
\(340\) 16.9515i 0.919326i
\(341\) 3.78389 0.204909
\(342\) −22.6508 −1.22481
\(343\) 1.00000i 0.0539949i
\(344\) − 16.7681i − 0.904078i
\(345\) − 2.36015i − 0.127066i
\(346\) 0.450797i 0.0242350i
\(347\) −5.98686 −0.321391 −0.160696 0.987004i \(-0.551374\pi\)
−0.160696 + 0.987004i \(0.551374\pi\)
\(348\) 3.06740 0.164430
\(349\) − 30.3362i − 1.62386i −0.583757 0.811929i \(-0.698418\pi\)
0.583757 0.811929i \(-0.301582\pi\)
\(350\) 12.2000 0.652117
\(351\) 0 0
\(352\) 31.2189 1.66398
\(353\) 28.6063i 1.52256i 0.648424 + 0.761280i \(0.275428\pi\)
−0.648424 + 0.761280i \(0.724572\pi\)
\(354\) 41.0930 2.18407
\(355\) 3.56850 0.189396
\(356\) 9.70936i 0.514595i
\(357\) − 5.66224i − 0.299678i
\(358\) − 2.10689i − 0.111352i
\(359\) 23.4618i 1.23826i 0.785287 + 0.619132i \(0.212515\pi\)
−0.785287 + 0.619132i \(0.787485\pi\)
\(360\) 8.11699 0.427803
\(361\) −33.7956 −1.77871
\(362\) 27.1957i 1.42937i
\(363\) 12.4707 0.654543
\(364\) 0 0
\(365\) 4.49827 0.235450
\(366\) 24.4105i 1.27596i
\(367\) 36.5197 1.90631 0.953156 0.302479i \(-0.0978143\pi\)
0.953156 + 0.302479i \(0.0978143\pi\)
\(368\) 35.8697 1.86984
\(369\) 1.42683i 0.0742778i
\(370\) − 12.4033i − 0.644818i
\(371\) − 4.56778i − 0.237147i
\(372\) − 20.8626i − 1.08168i
\(373\) −13.0498 −0.675694 −0.337847 0.941201i \(-0.609699\pi\)
−0.337847 + 0.941201i \(0.609699\pi\)
\(374\) 15.5207 0.802555
\(375\) − 9.55894i − 0.493622i
\(376\) −56.4218 −2.90973
\(377\) 0 0
\(378\) −15.4779 −0.796095
\(379\) − 30.6037i − 1.57201i −0.618223 0.786003i \(-0.712147\pi\)
0.618223 0.786003i \(-0.287853\pi\)
\(380\) 29.6889 1.52301
\(381\) −7.41167 −0.379711
\(382\) 27.1283i 1.38800i
\(383\) 4.88598i 0.249662i 0.992178 + 0.124831i \(0.0398388\pi\)
−0.992178 + 0.124831i \(0.960161\pi\)
\(384\) − 57.1111i − 2.91444i
\(385\) 1.01144i 0.0515479i
\(386\) −23.8495 −1.21391
\(387\) −1.98008 −0.100653
\(388\) 52.5650i 2.66858i
\(389\) 1.85425 0.0940143 0.0470072 0.998895i \(-0.485032\pi\)
0.0470072 + 0.998895i \(0.485032\pi\)
\(390\) 0 0
\(391\) 9.68082 0.489580
\(392\) 9.63087i 0.486433i
\(393\) −15.4525 −0.779475
\(394\) −71.3191 −3.59300
\(395\) 6.76298i 0.340283i
\(396\) − 8.55791i − 0.430051i
\(397\) 21.5134i 1.07973i 0.841753 + 0.539863i \(0.181524\pi\)
−0.841753 + 0.539863i \(0.818476\pi\)
\(398\) − 27.7753i − 1.39225i
\(399\) −9.91685 −0.496464
\(400\) 68.4175 3.42087
\(401\) − 14.5653i − 0.727357i −0.931525 0.363678i \(-0.881521\pi\)
0.931525 0.363678i \(-0.118479\pi\)
\(402\) −51.4760 −2.56739
\(403\) 0 0
\(404\) −41.3162 −2.05556
\(405\) 3.18281i 0.158155i
\(406\) −1.11734 −0.0554529
\(407\) −8.33345 −0.413074
\(408\) − 54.5323i − 2.69975i
\(409\) 22.1290i 1.09421i 0.837064 + 0.547105i \(0.184270\pi\)
−0.837064 + 0.547105i \(0.815730\pi\)
\(410\) − 2.54858i − 0.125865i
\(411\) − 19.0739i − 0.940847i
\(412\) 15.4846 0.762873
\(413\) −10.9843 −0.540500
\(414\) − 7.27418i − 0.357506i
\(415\) −8.69791 −0.426964
\(416\) 0 0
\(417\) 14.2363 0.697153
\(418\) − 27.1829i − 1.32956i
\(419\) 3.37590 0.164924 0.0824618 0.996594i \(-0.473722\pi\)
0.0824618 + 0.996594i \(0.473722\pi\)
\(420\) 5.57662 0.272111
\(421\) 25.1101i 1.22379i 0.790939 + 0.611895i \(0.209593\pi\)
−0.790939 + 0.611895i \(0.790407\pi\)
\(422\) 45.7164i 2.22544i
\(423\) 6.66260i 0.323947i
\(424\) − 43.9917i − 2.13643i
\(425\) 18.4651 0.895688
\(426\) −18.0143 −0.872794
\(427\) − 6.52497i − 0.315766i
\(428\) −8.19570 −0.396154
\(429\) 0 0
\(430\) 3.53678 0.170558
\(431\) − 10.7948i − 0.519969i −0.965613 0.259985i \(-0.916282\pi\)
0.965613 0.259985i \(-0.0837175\pi\)
\(432\) −86.7997 −4.17615
\(433\) 14.5182 0.697700 0.348850 0.937179i \(-0.386572\pi\)
0.348850 + 0.937179i \(0.386572\pi\)
\(434\) 7.59951i 0.364788i
\(435\) 0.412294i 0.0197680i
\(436\) 15.8305i 0.758144i
\(437\) − 16.9550i − 0.811068i
\(438\) −22.7079 −1.08502
\(439\) 14.4309 0.688748 0.344374 0.938833i \(-0.388091\pi\)
0.344374 + 0.938833i \(0.388091\pi\)
\(440\) 9.74108i 0.464388i
\(441\) 1.13727 0.0541556
\(442\) 0 0
\(443\) −30.2430 −1.43689 −0.718445 0.695584i \(-0.755146\pi\)
−0.718445 + 0.695584i \(0.755146\pi\)
\(444\) 45.9467i 2.18054i
\(445\) −1.30505 −0.0618653
\(446\) 2.93395 0.138926
\(447\) − 11.1428i − 0.527038i
\(448\) 31.9557i 1.50977i
\(449\) 31.2760i 1.47601i 0.674797 + 0.738003i \(0.264231\pi\)
−0.674797 + 0.738003i \(0.735769\pi\)
\(450\) − 13.8747i − 0.654058i
\(451\) −1.71232 −0.0806300
\(452\) 68.4749 3.22079
\(453\) − 3.36002i − 0.157868i
\(454\) −67.0645 −3.14749
\(455\) 0 0
\(456\) −95.5080 −4.47257
\(457\) − 20.6466i − 0.965808i −0.875673 0.482904i \(-0.839582\pi\)
0.875673 0.482904i \(-0.160418\pi\)
\(458\) −12.9643 −0.605783
\(459\) −23.4262 −1.09344
\(460\) 9.53444i 0.444545i
\(461\) − 27.2961i − 1.27131i −0.771975 0.635653i \(-0.780731\pi\)
0.771975 0.635653i \(-0.219269\pi\)
\(462\) − 5.10590i − 0.237548i
\(463\) 5.65977i 0.263032i 0.991314 + 0.131516i \(0.0419844\pi\)
−0.991314 + 0.131516i \(0.958016\pi\)
\(464\) −6.26606 −0.290894
\(465\) 2.80417 0.130040
\(466\) − 56.3311i − 2.60949i
\(467\) −42.2145 −1.95345 −0.976727 0.214486i \(-0.931192\pi\)
−0.976727 + 0.214486i \(0.931192\pi\)
\(468\) 0 0
\(469\) 13.7597 0.635362
\(470\) − 11.9006i − 0.548934i
\(471\) −17.6332 −0.812496
\(472\) −105.788 −4.86929
\(473\) − 2.37626i − 0.109261i
\(474\) − 34.1405i − 1.56812i
\(475\) − 32.3397i − 1.48385i
\(476\) 22.8740i 1.04843i
\(477\) −5.19479 −0.237853
\(478\) 17.1444 0.784167
\(479\) 32.5316i 1.48641i 0.669065 + 0.743204i \(0.266695\pi\)
−0.669065 + 0.743204i \(0.733305\pi\)
\(480\) 23.1358 1.05600
\(481\) 0 0
\(482\) 33.2888 1.51626
\(483\) − 3.18474i − 0.144911i
\(484\) −50.3786 −2.28994
\(485\) −7.06534 −0.320820
\(486\) 30.3663i 1.37744i
\(487\) − 26.8583i − 1.21707i −0.793529 0.608533i \(-0.791758\pi\)
0.793529 0.608533i \(-0.208242\pi\)
\(488\) − 62.8412i − 2.84469i
\(489\) − 8.22634i − 0.372008i
\(490\) −2.03137 −0.0917678
\(491\) −43.6878 −1.97160 −0.985802 0.167912i \(-0.946298\pi\)
−0.985802 + 0.167912i \(0.946298\pi\)
\(492\) 9.44093i 0.425630i
\(493\) −1.69113 −0.0761649
\(494\) 0 0
\(495\) 1.15028 0.0517013
\(496\) 42.6180i 1.91360i
\(497\) 4.81526 0.215994
\(498\) 43.9082 1.96758
\(499\) 18.1020i 0.810355i 0.914238 + 0.405177i \(0.132790\pi\)
−0.914238 + 0.405177i \(0.867210\pi\)
\(500\) 38.6158i 1.72695i
\(501\) 10.4424i 0.466534i
\(502\) 17.3082i 0.772505i
\(503\) −28.4155 −1.26698 −0.633492 0.773749i \(-0.718379\pi\)
−0.633492 + 0.773749i \(0.718379\pi\)
\(504\) 10.9529 0.487880
\(505\) − 5.55338i − 0.247122i
\(506\) 8.72964 0.388080
\(507\) 0 0
\(508\) 29.9413 1.32843
\(509\) − 17.4791i − 0.774748i −0.921923 0.387374i \(-0.873382\pi\)
0.921923 0.387374i \(-0.126618\pi\)
\(510\) 11.5021 0.509321
\(511\) 6.06987 0.268515
\(512\) 55.5280i 2.45402i
\(513\) 41.0287i 1.81146i
\(514\) − 66.7624i − 2.94476i
\(515\) 2.08131i 0.0917136i
\(516\) −13.1016 −0.576766
\(517\) −7.99569 −0.351650
\(518\) − 16.7368i − 0.735372i
\(519\) 0.224458 0.00985260
\(520\) 0 0
\(521\) 19.3087 0.845931 0.422966 0.906146i \(-0.360989\pi\)
0.422966 + 0.906146i \(0.360989\pi\)
\(522\) 1.27072i 0.0556179i
\(523\) −10.0229 −0.438270 −0.219135 0.975695i \(-0.570324\pi\)
−0.219135 + 0.975695i \(0.570324\pi\)
\(524\) 62.4242 2.72702
\(525\) − 6.07453i − 0.265114i
\(526\) − 26.2571i − 1.14486i
\(527\) 11.5021i 0.501039i
\(528\) − 28.6338i − 1.24613i
\(529\) −17.5550 −0.763261
\(530\) 9.27884 0.403047
\(531\) 12.4920i 0.542108i
\(532\) 40.0616 1.73689
\(533\) 0 0
\(534\) 6.58807 0.285093
\(535\) − 1.10160i − 0.0476261i
\(536\) 132.518 5.72389
\(537\) −1.04904 −0.0452696
\(538\) 80.5851i 3.47427i
\(539\) 1.36482i 0.0587869i
\(540\) − 23.0720i − 0.992860i
\(541\) 17.6153i 0.757339i 0.925532 + 0.378670i \(0.123618\pi\)
−0.925532 + 0.378670i \(0.876382\pi\)
\(542\) 0.823379 0.0353672
\(543\) 13.5411 0.581103
\(544\) 94.8978i 4.06871i
\(545\) −2.12780 −0.0911451
\(546\) 0 0
\(547\) 2.98425 0.127597 0.0637987 0.997963i \(-0.479678\pi\)
0.0637987 + 0.997963i \(0.479678\pi\)
\(548\) 77.0539i 3.29158i
\(549\) −7.42064 −0.316705
\(550\) 16.6508 0.709992
\(551\) 2.96185i 0.126179i
\(552\) − 30.6719i − 1.30548i
\(553\) 9.12582i 0.388070i
\(554\) 89.8279i 3.81642i
\(555\) −6.17577 −0.262147
\(556\) −57.5109 −2.43901
\(557\) 7.25596i 0.307445i 0.988114 + 0.153722i \(0.0491261\pi\)
−0.988114 + 0.153722i \(0.950874\pi\)
\(558\) 8.64268 0.365874
\(559\) 0 0
\(560\) −11.3919 −0.481395
\(561\) − 7.72794i − 0.326274i
\(562\) −11.7724 −0.496591
\(563\) −4.27933 −0.180352 −0.0901762 0.995926i \(-0.528743\pi\)
−0.0901762 + 0.995926i \(0.528743\pi\)
\(564\) 44.0845i 1.85629i
\(565\) 9.20382i 0.387207i
\(566\) 57.8375i 2.43109i
\(567\) 4.29482i 0.180365i
\(568\) 46.3751 1.94586
\(569\) 19.7626 0.828492 0.414246 0.910165i \(-0.364045\pi\)
0.414246 + 0.910165i \(0.364045\pi\)
\(570\) − 20.1448i − 0.843771i
\(571\) 19.9236 0.833778 0.416889 0.908957i \(-0.363120\pi\)
0.416889 + 0.908957i \(0.363120\pi\)
\(572\) 0 0
\(573\) 13.5075 0.564285
\(574\) − 3.43900i − 0.143541i
\(575\) 10.3857 0.433115
\(576\) 36.3422 1.51426
\(577\) 28.7300i 1.19605i 0.801479 + 0.598023i \(0.204047\pi\)
−0.801479 + 0.598023i \(0.795953\pi\)
\(578\) 0.580569i 0.0241485i
\(579\) 11.8749i 0.493506i
\(580\) − 1.66556i − 0.0691587i
\(581\) −11.7368 −0.486924
\(582\) 35.6668 1.47844
\(583\) − 6.23420i − 0.258194i
\(584\) 58.4582 2.41902
\(585\) 0 0
\(586\) −48.6921 −2.01145
\(587\) − 31.5388i − 1.30174i −0.759188 0.650872i \(-0.774403\pi\)
0.759188 0.650872i \(-0.225597\pi\)
\(588\) 7.52497 0.310325
\(589\) 20.1448 0.830051
\(590\) − 22.3130i − 0.918613i
\(591\) 35.5107i 1.46071i
\(592\) − 93.8597i − 3.85761i
\(593\) − 11.1181i − 0.456564i −0.973595 0.228282i \(-0.926689\pi\)
0.973595 0.228282i \(-0.0733108\pi\)
\(594\) −21.1245 −0.866748
\(595\) −3.07453 −0.126044
\(596\) 45.0143i 1.84386i
\(597\) −13.8297 −0.566010
\(598\) 0 0
\(599\) −7.29572 −0.298095 −0.149048 0.988830i \(-0.547621\pi\)
−0.149048 + 0.988830i \(0.547621\pi\)
\(600\) − 58.5031i − 2.38838i
\(601\) −1.17258 −0.0478306 −0.0239153 0.999714i \(-0.507613\pi\)
−0.0239153 + 0.999714i \(0.507613\pi\)
\(602\) 4.77245 0.194510
\(603\) − 15.6484i − 0.637253i
\(604\) 13.5737i 0.552304i
\(605\) − 6.77146i − 0.275299i
\(606\) 28.0342i 1.13881i
\(607\) 0.633838 0.0257267 0.0128633 0.999917i \(-0.495905\pi\)
0.0128633 + 0.999917i \(0.495905\pi\)
\(608\) 166.204 6.74047
\(609\) 0.556340i 0.0225440i
\(610\) 13.2546 0.536664
\(611\) 0 0
\(612\) 26.0139 1.05155
\(613\) 30.8550i 1.24622i 0.782134 + 0.623110i \(0.214131\pi\)
−0.782134 + 0.623110i \(0.785869\pi\)
\(614\) −49.3823 −1.99291
\(615\) −1.26897 −0.0511698
\(616\) 13.1444i 0.529603i
\(617\) − 33.8209i − 1.36158i −0.732479 0.680790i \(-0.761637\pi\)
0.732479 0.680790i \(-0.238363\pi\)
\(618\) − 10.5068i − 0.422644i
\(619\) − 0.404797i − 0.0162702i −0.999967 0.00813509i \(-0.997410\pi\)
0.999967 0.00813509i \(-0.00258951\pi\)
\(620\) −11.3282 −0.454950
\(621\) −13.1761 −0.528740
\(622\) − 47.2959i − 1.89639i
\(623\) −1.76101 −0.0705532
\(624\) 0 0
\(625\) 17.0636 0.682543
\(626\) 18.6812i 0.746650i
\(627\) −13.5347 −0.540524
\(628\) 71.2338 2.84254
\(629\) − 25.3316i − 1.01004i
\(630\) 2.31021i 0.0920409i
\(631\) 30.4508i 1.21223i 0.795378 + 0.606114i \(0.207272\pi\)
−0.795378 + 0.606114i \(0.792728\pi\)
\(632\) 87.8897i 3.49606i
\(633\) 22.7628 0.904738
\(634\) −68.7381 −2.72994
\(635\) 4.02446i 0.159706i
\(636\) −34.3724 −1.36296
\(637\) 0 0
\(638\) −1.52497 −0.0603743
\(639\) − 5.47624i − 0.216637i
\(640\) −31.0107 −1.22581
\(641\) −9.64564 −0.380980 −0.190490 0.981689i \(-0.561008\pi\)
−0.190490 + 0.981689i \(0.561008\pi\)
\(642\) 5.56100i 0.219475i
\(643\) − 8.13296i − 0.320733i −0.987058 0.160366i \(-0.948732\pi\)
0.987058 0.160366i \(-0.0512676\pi\)
\(644\) 12.8656i 0.506974i
\(645\) − 1.76101i − 0.0693395i
\(646\) 82.6293 3.25101
\(647\) 11.5227 0.453005 0.226503 0.974011i \(-0.427271\pi\)
0.226503 + 0.974011i \(0.427271\pi\)
\(648\) 41.3629i 1.62489i
\(649\) −14.9915 −0.588469
\(650\) 0 0
\(651\) 3.78389 0.148302
\(652\) 33.2324i 1.30148i
\(653\) 42.4039 1.65939 0.829697 0.558215i \(-0.188513\pi\)
0.829697 + 0.558215i \(0.188513\pi\)
\(654\) 10.7414 0.420024
\(655\) 8.39054i 0.327845i
\(656\) − 19.2858i − 0.752986i
\(657\) − 6.90307i − 0.269314i
\(658\) − 16.0584i − 0.626023i
\(659\) −2.50088 −0.0974203 −0.0487101 0.998813i \(-0.515511\pi\)
−0.0487101 + 0.998813i \(0.515511\pi\)
\(660\) 7.61108 0.296261
\(661\) 14.8394i 0.577184i 0.957452 + 0.288592i \(0.0931871\pi\)
−0.957452 + 0.288592i \(0.906813\pi\)
\(662\) −8.21046 −0.319109
\(663\) 0 0
\(664\) −113.035 −4.38663
\(665\) 5.38474i 0.208811i
\(666\) −19.0342 −0.737560
\(667\) −0.951183 −0.0368300
\(668\) − 42.1849i − 1.63218i
\(669\) − 1.46085i − 0.0564797i
\(670\) 27.9509i 1.07984i
\(671\) − 8.90541i − 0.343790i
\(672\) 31.2189 1.20430
\(673\) −39.6091 −1.52682 −0.763410 0.645915i \(-0.776476\pi\)
−0.763410 + 0.645915i \(0.776476\pi\)
\(674\) 80.8313i 3.11350i
\(675\) −25.1320 −0.967330
\(676\) 0 0
\(677\) 17.0321 0.654596 0.327298 0.944921i \(-0.393862\pi\)
0.327298 + 0.944921i \(0.393862\pi\)
\(678\) − 46.4621i − 1.78437i
\(679\) −9.53381 −0.365874
\(680\) −29.6105 −1.13551
\(681\) 33.3922i 1.27959i
\(682\) 10.3720i 0.397163i
\(683\) 32.8912i 1.25854i 0.777185 + 0.629272i \(0.216647\pi\)
−0.777185 + 0.629272i \(0.783353\pi\)
\(684\) − 45.5608i − 1.74206i
\(685\) −10.3569 −0.395718
\(686\) −2.74108 −0.104655
\(687\) 6.45510i 0.246278i
\(688\) 26.7638 1.02036
\(689\) 0 0
\(690\) 6.46938 0.246285
\(691\) 23.7922i 0.905099i 0.891739 + 0.452550i \(0.149486\pi\)
−0.891739 + 0.452550i \(0.850514\pi\)
\(692\) −0.906754 −0.0344696
\(693\) 1.55217 0.0589619
\(694\) − 16.4105i − 0.622933i
\(695\) − 7.73013i − 0.293221i
\(696\) 5.35804i 0.203096i
\(697\) − 5.20502i − 0.197154i
\(698\) 83.1539 3.14742
\(699\) −28.0480 −1.06087
\(700\) 24.5396i 0.927510i
\(701\) −29.7796 −1.12476 −0.562380 0.826879i \(-0.690114\pi\)
−0.562380 + 0.826879i \(0.690114\pi\)
\(702\) 0 0
\(703\) −44.3658 −1.67329
\(704\) 43.6138i 1.64376i
\(705\) −5.92547 −0.223166
\(706\) −78.4122 −2.95108
\(707\) − 7.49361i − 0.281826i
\(708\) 82.6562i 3.10641i
\(709\) 11.9304i 0.448054i 0.974583 + 0.224027i \(0.0719204\pi\)
−0.974583 + 0.224027i \(0.928080\pi\)
\(710\) 9.78155i 0.367095i
\(711\) 10.3785 0.389224
\(712\) −16.9600 −0.635604
\(713\) 6.46938i 0.242280i
\(714\) 15.5207 0.580846
\(715\) 0 0
\(716\) 4.23788 0.158377
\(717\) − 8.53642i − 0.318798i
\(718\) −64.3106 −2.40005
\(719\) −32.3638 −1.20697 −0.603484 0.797375i \(-0.706221\pi\)
−0.603484 + 0.797375i \(0.706221\pi\)
\(720\) 12.9556i 0.482827i
\(721\) 2.80848i 0.104593i
\(722\) − 92.6364i − 3.44757i
\(723\) − 16.5749i − 0.616428i
\(724\) −54.7026 −2.03301
\(725\) −1.81427 −0.0673805
\(726\) 34.1833i 1.26866i
\(727\) 31.4897 1.16789 0.583943 0.811794i \(-0.301509\pi\)
0.583943 + 0.811794i \(0.301509\pi\)
\(728\) 0 0
\(729\) 28.0042 1.03719
\(730\) 12.3301i 0.456359i
\(731\) 7.22325 0.267161
\(732\) −49.1003 −1.81480
\(733\) − 2.66224i − 0.0983321i −0.998791 0.0491661i \(-0.984344\pi\)
0.998791 0.0491661i \(-0.0156564\pi\)
\(734\) 100.103i 3.69489i
\(735\) 1.01144i 0.0373076i
\(736\) 53.3755i 1.96745i
\(737\) 18.7795 0.691750
\(738\) −3.91106 −0.143968
\(739\) − 35.5828i − 1.30893i −0.756091 0.654467i \(-0.772893\pi\)
0.756091 0.654467i \(-0.227107\pi\)
\(740\) 24.9486 0.917128
\(741\) 0 0
\(742\) 12.5207 0.459648
\(743\) − 24.2406i − 0.889300i −0.895704 0.444650i \(-0.853328\pi\)
0.895704 0.444650i \(-0.146672\pi\)
\(744\) 36.4422 1.33604
\(745\) −6.05044 −0.221671
\(746\) − 35.7706i − 1.30966i
\(747\) 13.3479i 0.488373i
\(748\) 31.2189i 1.14148i
\(749\) − 1.48647i − 0.0543144i
\(750\) 26.2018 0.956756
\(751\) 29.1410 1.06337 0.531684 0.846943i \(-0.321559\pi\)
0.531684 + 0.846943i \(0.321559\pi\)
\(752\) − 90.0555i − 3.28399i
\(753\) 8.61799 0.314057
\(754\) 0 0
\(755\) −1.82446 −0.0663988
\(756\) − 31.1328i − 1.13229i
\(757\) −4.98990 −0.181361 −0.0906805 0.995880i \(-0.528904\pi\)
−0.0906805 + 0.995880i \(0.528904\pi\)
\(758\) 83.8872 3.04692
\(759\) − 4.34660i − 0.157772i
\(760\) 51.8598i 1.88115i
\(761\) − 11.8372i − 0.429097i −0.976713 0.214548i \(-0.931172\pi\)
0.976713 0.214548i \(-0.0688280\pi\)
\(762\) − 20.3160i − 0.735971i
\(763\) −2.87121 −0.103945
\(764\) −54.5670 −1.97417
\(765\) 3.49657i 0.126419i
\(766\) −13.3929 −0.483904
\(767\) 0 0
\(768\) 69.3186 2.50132
\(769\) 35.8183i 1.29164i 0.763489 + 0.645821i \(0.223485\pi\)
−0.763489 + 0.645821i \(0.776515\pi\)
\(770\) −2.77245 −0.0999121
\(771\) −33.2418 −1.19718
\(772\) − 47.9718i − 1.72654i
\(773\) 19.9534i 0.717673i 0.933400 + 0.358837i \(0.116826\pi\)
−0.933400 + 0.358837i \(0.883174\pi\)
\(774\) − 5.42755i − 0.195089i
\(775\) 12.3396i 0.443252i
\(776\) −91.8190 −3.29611
\(777\) −8.33345 −0.298961
\(778\) 5.08266i 0.182222i
\(779\) −9.11608 −0.326618
\(780\) 0 0
\(781\) 6.57196 0.235163
\(782\) 26.5359i 0.948923i
\(783\) 2.30173 0.0822570
\(784\) −15.3720 −0.548998
\(785\) 9.57464i 0.341734i
\(786\) − 42.3566i − 1.51081i
\(787\) 2.79619i 0.0996733i 0.998757 + 0.0498367i \(0.0158701\pi\)
−0.998757 + 0.0498367i \(0.984130\pi\)
\(788\) − 143.454i − 5.11035i
\(789\) −13.0737 −0.465437
\(790\) −18.5379 −0.659549
\(791\) 12.4194i 0.441584i
\(792\) 14.9487 0.531179
\(793\) 0 0
\(794\) −58.9700 −2.09277
\(795\) − 4.62005i − 0.163856i
\(796\) 55.8684 1.98020
\(797\) −1.68562 −0.0597076 −0.0298538 0.999554i \(-0.509504\pi\)
−0.0298538 + 0.999554i \(0.509504\pi\)
\(798\) − 27.1829i − 0.962265i
\(799\) − 24.3049i − 0.859846i
\(800\) 101.808i 3.59945i
\(801\) 2.00273i 0.0707632i
\(802\) 39.9247 1.40979
\(803\) 8.28428 0.292346
\(804\) − 103.541i − 3.65161i
\(805\) −1.72928 −0.0609491
\(806\) 0 0
\(807\) 40.1244 1.41244
\(808\) − 72.1700i − 2.53893i
\(809\) −43.4372 −1.52717 −0.763585 0.645708i \(-0.776562\pi\)
−0.763585 + 0.645708i \(0.776562\pi\)
\(810\) −8.72435 −0.306542
\(811\) − 5.60812i − 0.196928i −0.995141 0.0984639i \(-0.968607\pi\)
0.995141 0.0984639i \(-0.0313929\pi\)
\(812\) − 2.24747i − 0.0788709i
\(813\) − 0.409971i − 0.0143783i
\(814\) − 22.8427i − 0.800635i
\(815\) −4.46681 −0.156466
\(816\) 87.0397 3.04700
\(817\) − 12.6508i − 0.442595i
\(818\) −60.6574 −2.12084
\(819\) 0 0
\(820\) 5.12632 0.179019
\(821\) 23.9448i 0.835678i 0.908521 + 0.417839i \(0.137212\pi\)
−0.908521 + 0.417839i \(0.862788\pi\)
\(822\) 52.2832 1.82358
\(823\) −35.4117 −1.23437 −0.617187 0.786817i \(-0.711728\pi\)
−0.617187 + 0.786817i \(0.711728\pi\)
\(824\) 27.0481i 0.942266i
\(825\) − 8.29064i − 0.288643i
\(826\) − 30.1087i − 1.04762i
\(827\) − 16.1563i − 0.561811i −0.959735 0.280905i \(-0.909365\pi\)
0.959735 0.280905i \(-0.0906347\pi\)
\(828\) 14.6316 0.508483
\(829\) 52.7010 1.83038 0.915190 0.403022i \(-0.132040\pi\)
0.915190 + 0.403022i \(0.132040\pi\)
\(830\) − 23.8417i − 0.827557i
\(831\) 44.7265 1.55154
\(832\) 0 0
\(833\) −4.14871 −0.143744
\(834\) 39.0228i 1.35125i
\(835\) 5.67013 0.196223
\(836\) 54.6769 1.89104
\(837\) − 15.6550i − 0.541115i
\(838\) 9.25363i 0.319661i
\(839\) − 22.4338i − 0.774502i −0.921974 0.387251i \(-0.873425\pi\)
0.921974 0.387251i \(-0.126575\pi\)
\(840\) 9.74108i 0.336099i
\(841\) −28.8338 −0.994270
\(842\) −68.8288 −2.37200
\(843\) 5.86165i 0.201886i
\(844\) −91.9559 −3.16525
\(845\) 0 0
\(846\) −18.2627 −0.627886
\(847\) − 9.13727i − 0.313960i
\(848\) 70.2158 2.41122
\(849\) 28.7980 0.988346
\(850\) 50.6143i 1.73606i
\(851\) − 14.2478i − 0.488409i
\(852\) − 36.2347i − 1.24138i
\(853\) 30.8521i 1.05635i 0.849134 + 0.528177i \(0.177124\pi\)
−0.849134 + 0.528177i \(0.822876\pi\)
\(854\) 17.8855 0.612029
\(855\) 6.12390 0.209433
\(856\) − 14.3160i − 0.489311i
\(857\) −26.7400 −0.913420 −0.456710 0.889616i \(-0.650972\pi\)
−0.456710 + 0.889616i \(0.650972\pi\)
\(858\) 0 0
\(859\) −5.15804 −0.175990 −0.0879950 0.996121i \(-0.528046\pi\)
−0.0879950 + 0.996121i \(0.528046\pi\)
\(860\) 7.11402i 0.242586i
\(861\) −1.71232 −0.0583557
\(862\) 29.5896 1.00782
\(863\) − 8.16814i − 0.278047i −0.990289 0.139023i \(-0.955604\pi\)
0.990289 0.139023i \(-0.0443963\pi\)
\(864\) − 129.161i − 4.39415i
\(865\) − 0.121878i − 0.00414398i
\(866\) 39.7956i 1.35231i
\(867\) 0.289073 0.00981742
\(868\) −15.2860 −0.518840
\(869\) 12.4551i 0.422510i
\(870\) −1.13013 −0.0383150
\(871\) 0 0
\(872\) −27.6523 −0.936425
\(873\) 10.8425i 0.366963i
\(874\) 46.4750 1.57204
\(875\) −7.00382 −0.236772
\(876\) − 45.6756i − 1.54324i
\(877\) − 15.6184i − 0.527398i −0.964605 0.263699i \(-0.915058\pi\)
0.964605 0.263699i \(-0.0849425\pi\)
\(878\) 39.5562i 1.33496i
\(879\) 24.2444i 0.817744i
\(880\) −15.5479 −0.524118
\(881\) 46.4375 1.56452 0.782260 0.622952i \(-0.214067\pi\)
0.782260 + 0.622952i \(0.214067\pi\)
\(882\) 3.11734i 0.104966i
\(883\) −15.6588 −0.526960 −0.263480 0.964665i \(-0.584870\pi\)
−0.263480 + 0.964665i \(0.584870\pi\)
\(884\) 0 0
\(885\) −11.1099 −0.373457
\(886\) − 82.8986i − 2.78503i
\(887\) 31.5107 1.05803 0.529013 0.848614i \(-0.322562\pi\)
0.529013 + 0.848614i \(0.322562\pi\)
\(888\) −80.2584 −2.69330
\(889\) 5.43052i 0.182134i
\(890\) − 3.57725i − 0.119910i
\(891\) 5.86165i 0.196373i
\(892\) 5.90146i 0.197596i
\(893\) −42.5677 −1.42447
\(894\) 30.5434 1.02152
\(895\) 0.569620i 0.0190403i
\(896\) −41.8452 −1.39795
\(897\) 0 0
\(898\) −85.7301 −2.86085
\(899\) − 1.13013i − 0.0376920i
\(900\) 27.9081 0.930270
\(901\) 18.9504 0.631330
\(902\) − 4.69361i − 0.156280i
\(903\) − 2.37626i − 0.0790771i
\(904\) 119.610i 3.97817i
\(905\) − 7.35266i − 0.244411i
\(906\) 9.21010 0.305985
\(907\) 0.747991 0.0248366 0.0124183 0.999923i \(-0.496047\pi\)
0.0124183 + 0.999923i \(0.496047\pi\)
\(908\) − 134.896i − 4.47669i
\(909\) −8.52224 −0.282665
\(910\) 0 0
\(911\) 24.9000 0.824973 0.412486 0.910964i \(-0.364660\pi\)
0.412486 + 0.910964i \(0.364660\pi\)
\(912\) − 152.441i − 5.04784i
\(913\) −16.0186 −0.530138
\(914\) 56.5941 1.87197
\(915\) − 6.59964i − 0.218177i
\(916\) − 26.0770i − 0.861609i
\(917\) 11.3220i 0.373886i
\(918\) − 64.2132i − 2.11935i
\(919\) 0.586495 0.0193467 0.00967334 0.999953i \(-0.496921\pi\)
0.00967334 + 0.999953i \(0.496921\pi\)
\(920\) −16.6545 −0.549082
\(921\) 24.5881i 0.810204i
\(922\) 74.8208 2.46409
\(923\) 0 0
\(924\) 10.2702 0.337866
\(925\) − 27.1761i − 0.893546i
\(926\) −15.5139 −0.509818
\(927\) 3.19399 0.104905
\(928\) − 9.32412i − 0.306079i
\(929\) 50.1949i 1.64684i 0.567431 + 0.823421i \(0.307938\pi\)
−0.567431 + 0.823421i \(0.692062\pi\)
\(930\) 7.68647i 0.252049i
\(931\) 7.26606i 0.238135i
\(932\) 113.307 3.71149
\(933\) −23.5493 −0.770968
\(934\) − 115.713i − 3.78626i
\(935\) −4.19619 −0.137230
\(936\) 0 0
\(937\) 22.7130 0.742003 0.371001 0.928632i \(-0.379015\pi\)
0.371001 + 0.928632i \(0.379015\pi\)
\(938\) 37.7164i 1.23148i
\(939\) 9.30160 0.303546
\(940\) 23.9374 0.780752
\(941\) 49.8734i 1.62583i 0.582384 + 0.812914i \(0.302120\pi\)
−0.582384 + 0.812914i \(0.697880\pi\)
\(942\) − 48.3341i − 1.57481i
\(943\) − 2.92758i − 0.0953351i
\(944\) − 168.849i − 5.49558i
\(945\) 4.18461 0.136125
\(946\) 6.51353 0.211773
\(947\) 0.266414i 0.00865731i 0.999991 + 0.00432865i \(0.00137786\pi\)
−0.999991 + 0.00432865i \(0.998622\pi\)
\(948\) 68.6716 2.23035
\(949\) 0 0
\(950\) 88.6459 2.87605
\(951\) 34.2256i 1.10984i
\(952\) −39.9557 −1.29497
\(953\) −7.82029 −0.253324 −0.126662 0.991946i \(-0.540426\pi\)
−0.126662 + 0.991946i \(0.540426\pi\)
\(954\) − 14.2394i − 0.461016i
\(955\) − 7.33444i − 0.237337i
\(956\) 34.4850i 1.11532i
\(957\) 0.759304i 0.0245448i
\(958\) −89.1718 −2.88101
\(959\) −13.9754 −0.451290
\(960\) 32.3214i 1.04317i
\(961\) 23.3135 0.752049
\(962\) 0 0
\(963\) −1.69051 −0.0544761
\(964\) 66.9585i 2.15659i
\(965\) 6.44797 0.207567
\(966\) 8.72964 0.280872
\(967\) − 39.8224i − 1.28060i −0.768124 0.640301i \(-0.778810\pi\)
0.768124 0.640301i \(-0.221190\pi\)
\(968\) − 87.9999i − 2.82842i
\(969\) − 41.1422i − 1.32168i
\(970\) − 19.3667i − 0.621826i
\(971\) −45.9295 −1.47395 −0.736974 0.675921i \(-0.763746\pi\)
−0.736974 + 0.675921i \(0.763746\pi\)
\(972\) −61.0801 −1.95915
\(973\) − 10.4309i − 0.334398i
\(974\) 73.6208 2.35896
\(975\) 0 0
\(976\) 100.302 3.21058
\(977\) 28.6627i 0.917002i 0.888694 + 0.458501i \(0.151613\pi\)
−0.888694 + 0.458501i \(0.848387\pi\)
\(978\) 22.5491 0.721040
\(979\) −2.40345 −0.0768147
\(980\) − 4.08598i − 0.130522i
\(981\) 3.26534i 0.104254i
\(982\) − 119.752i − 3.82144i
\(983\) − 46.1200i − 1.47100i −0.677524 0.735500i \(-0.736947\pi\)
0.677524 0.735500i \(-0.263053\pi\)
\(984\) −16.4911 −0.525718
\(985\) 19.2819 0.614372
\(986\) − 4.63554i − 0.147626i
\(987\) −7.99569 −0.254506
\(988\) 0 0
\(989\) 4.06273 0.129187
\(990\) 3.15302i 0.100209i
\(991\) 37.8249 1.20155 0.600773 0.799419i \(-0.294859\pi\)
0.600773 + 0.799419i \(0.294859\pi\)
\(992\) −63.4171 −2.01350
\(993\) 4.08809i 0.129732i
\(994\) 13.1990i 0.418647i
\(995\) 7.50936i 0.238063i
\(996\) 88.3189i 2.79849i
\(997\) 38.3748 1.21534 0.607671 0.794189i \(-0.292104\pi\)
0.607671 + 0.794189i \(0.292104\pi\)
\(998\) −49.6189 −1.57066
\(999\) 34.4777i 1.09083i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1183.2.c.g.337.8 8
13.5 odd 4 1183.2.a.l.1.4 4
13.7 odd 12 91.2.f.c.29.4 yes 8
13.8 odd 4 1183.2.a.k.1.1 4
13.11 odd 12 91.2.f.c.22.4 8
13.12 even 2 inner 1183.2.c.g.337.1 8
39.11 even 12 819.2.o.h.568.1 8
39.20 even 12 819.2.o.h.757.1 8
52.7 even 12 1456.2.s.q.1121.3 8
52.11 even 12 1456.2.s.q.113.3 8
91.11 odd 12 637.2.g.k.373.4 8
91.20 even 12 637.2.f.i.393.4 8
91.24 even 12 637.2.g.j.373.4 8
91.33 even 12 637.2.g.j.263.4 8
91.34 even 4 8281.2.a.bp.1.1 4
91.37 odd 12 637.2.h.h.165.1 8
91.46 odd 12 637.2.h.h.471.1 8
91.59 even 12 637.2.h.i.471.1 8
91.72 odd 12 637.2.g.k.263.4 8
91.76 even 12 637.2.f.i.295.4 8
91.83 even 4 8281.2.a.bt.1.4 4
91.89 even 12 637.2.h.i.165.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.4 8 13.11 odd 12
91.2.f.c.29.4 yes 8 13.7 odd 12
637.2.f.i.295.4 8 91.76 even 12
637.2.f.i.393.4 8 91.20 even 12
637.2.g.j.263.4 8 91.33 even 12
637.2.g.j.373.4 8 91.24 even 12
637.2.g.k.263.4 8 91.72 odd 12
637.2.g.k.373.4 8 91.11 odd 12
637.2.h.h.165.1 8 91.37 odd 12
637.2.h.h.471.1 8 91.46 odd 12
637.2.h.i.165.1 8 91.89 even 12
637.2.h.i.471.1 8 91.59 even 12
819.2.o.h.568.1 8 39.11 even 12
819.2.o.h.757.1 8 39.20 even 12
1183.2.a.k.1.1 4 13.8 odd 4
1183.2.a.l.1.4 4 13.5 odd 4
1183.2.c.g.337.1 8 13.12 even 2 inner
1183.2.c.g.337.8 8 1.1 even 1 trivial
1456.2.s.q.113.3 8 52.11 even 12
1456.2.s.q.1121.3 8 52.7 even 12
8281.2.a.bp.1.1 4 91.34 even 4
8281.2.a.bt.1.4 4 91.83 even 4