Properties

Label 1815.2.a.q.1.3
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $1$
Dimension $4$
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] = [1815,2,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, 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("1815.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(14.4928479669\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.95630\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.209057 q^{2} -1.00000 q^{3} -1.95630 q^{4} +1.00000 q^{5} -0.209057 q^{6} -0.488830 q^{7} -0.827091 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.209057 q^{2} -1.00000 q^{3} -1.95630 q^{4} +1.00000 q^{5} -0.209057 q^{6} -0.488830 q^{7} -0.827091 q^{8} +1.00000 q^{9} +0.209057 q^{10} +1.95630 q^{12} -3.26755 q^{13} -0.102193 q^{14} -1.00000 q^{15} +3.73968 q^{16} +3.33070 q^{17} +0.209057 q^{18} +3.86889 q^{19} -1.95630 q^{20} +0.488830 q^{21} -0.267545 q^{23} +0.827091 q^{24} +1.00000 q^{25} -0.683103 q^{26} -1.00000 q^{27} +0.956295 q^{28} -6.31592 q^{29} -0.209057 q^{30} -5.83274 q^{31} +2.43599 q^{32} +0.696307 q^{34} -0.488830 q^{35} -1.95630 q^{36} +7.50361 q^{37} +0.808817 q^{38} +3.26755 q^{39} -0.827091 q^{40} -4.18769 q^{41} +0.102193 q^{42} +2.26657 q^{43} +1.00000 q^{45} -0.0559322 q^{46} -5.58347 q^{47} -3.73968 q^{48} -6.76105 q^{49} +0.209057 q^{50} -3.33070 q^{51} +6.39228 q^{52} -1.14301 q^{53} -0.209057 q^{54} +0.404307 q^{56} -3.86889 q^{57} -1.32039 q^{58} -4.51742 q^{59} +1.95630 q^{60} -12.5743 q^{61} -1.21937 q^{62} -0.488830 q^{63} -6.97010 q^{64} -3.26755 q^{65} +8.97575 q^{67} -6.51584 q^{68} +0.267545 q^{69} -0.102193 q^{70} +5.31592 q^{71} -0.827091 q^{72} -9.65885 q^{73} +1.56868 q^{74} -1.00000 q^{75} -7.56868 q^{76} +0.683103 q^{78} +1.50828 q^{79} +3.73968 q^{80} +1.00000 q^{81} -0.875466 q^{82} -17.6428 q^{83} -0.956295 q^{84} +3.33070 q^{85} +0.473842 q^{86} +6.31592 q^{87} +15.9226 q^{89} +0.209057 q^{90} +1.59727 q^{91} +0.523398 q^{92} +5.83274 q^{93} -1.16726 q^{94} +3.86889 q^{95} -2.43599 q^{96} -16.3364 q^{97} -1.41344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} + q^{4} + 4 q^{5} + q^{6} + 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 4 q^{3} + q^{4} + 4 q^{5} + q^{6} + 3 q^{8} + 4 q^{9} - q^{10} - q^{12} - 7 q^{13} - 5 q^{14} - 4 q^{15} - 9 q^{16} - 8 q^{17} - q^{18} - 11 q^{19} + q^{20} + 5 q^{23} - 3 q^{24} + 4 q^{25} - 12 q^{26} - 4 q^{27} - 5 q^{28} - 17 q^{29} + q^{30} - 5 q^{31} + 17 q^{34} + q^{36} + 15 q^{37} - q^{38} + 7 q^{39} + 3 q^{40} - 10 q^{41} + 5 q^{42} + 4 q^{43} + 4 q^{45} - 15 q^{46} - 8 q^{47} + 9 q^{48} - 8 q^{49} - q^{50} + 8 q^{51} + 7 q^{52} + 10 q^{53} + q^{54} + 10 q^{56} + 11 q^{57} - 7 q^{58} + 9 q^{59} - q^{60} - 37 q^{61} + 20 q^{62} - 7 q^{64} - 7 q^{65} + 3 q^{67} - 17 q^{68} - 5 q^{69} - 5 q^{70} + 13 q^{71} + 3 q^{72} - 15 q^{73} + 5 q^{74} - 4 q^{75} - 29 q^{76} + 12 q^{78} - 20 q^{79} - 9 q^{80} + 4 q^{81} + 5 q^{82} + 17 q^{83} + 5 q^{84} - 8 q^{85} + 34 q^{86} + 17 q^{87} - 24 q^{89} - q^{90} - 20 q^{91} + 10 q^{92} + 5 q^{93} - 23 q^{94} - 11 q^{95} - 32 q^{97} - 13 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.209057 0.147826 0.0739128 0.997265i \(-0.476451\pi\)
0.0739128 + 0.997265i \(0.476451\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.95630 −0.978148
\(5\) 1.00000 0.447214
\(6\) −0.209057 −0.0853471
\(7\) −0.488830 −0.184760 −0.0923801 0.995724i \(-0.529447\pi\)
−0.0923801 + 0.995724i \(0.529447\pi\)
\(8\) −0.827091 −0.292421
\(9\) 1.00000 0.333333
\(10\) 0.209057 0.0661096
\(11\) 0 0
\(12\) 1.95630 0.564734
\(13\) −3.26755 −0.906254 −0.453127 0.891446i \(-0.649692\pi\)
−0.453127 + 0.891446i \(0.649692\pi\)
\(14\) −0.102193 −0.0273123
\(15\) −1.00000 −0.258199
\(16\) 3.73968 0.934920
\(17\) 3.33070 0.807814 0.403907 0.914800i \(-0.367652\pi\)
0.403907 + 0.914800i \(0.367652\pi\)
\(18\) 0.209057 0.0492752
\(19\) 3.86889 0.887583 0.443792 0.896130i \(-0.353633\pi\)
0.443792 + 0.896130i \(0.353633\pi\)
\(20\) −1.95630 −0.437441
\(21\) 0.488830 0.106671
\(22\) 0 0
\(23\) −0.267545 −0.0557871 −0.0278935 0.999611i \(-0.508880\pi\)
−0.0278935 + 0.999611i \(0.508880\pi\)
\(24\) 0.827091 0.168829
\(25\) 1.00000 0.200000
\(26\) −0.683103 −0.133968
\(27\) −1.00000 −0.192450
\(28\) 0.956295 0.180723
\(29\) −6.31592 −1.17284 −0.586419 0.810008i \(-0.699463\pi\)
−0.586419 + 0.810008i \(0.699463\pi\)
\(30\) −0.209057 −0.0381684
\(31\) −5.83274 −1.04759 −0.523795 0.851844i \(-0.675484\pi\)
−0.523795 + 0.851844i \(0.675484\pi\)
\(32\) 2.43599 0.430626
\(33\) 0 0
\(34\) 0.696307 0.119416
\(35\) −0.488830 −0.0826273
\(36\) −1.95630 −0.326049
\(37\) 7.50361 1.23359 0.616793 0.787125i \(-0.288431\pi\)
0.616793 + 0.787125i \(0.288431\pi\)
\(38\) 0.808817 0.131207
\(39\) 3.26755 0.523226
\(40\) −0.827091 −0.130775
\(41\) −4.18769 −0.654008 −0.327004 0.945023i \(-0.606039\pi\)
−0.327004 + 0.945023i \(0.606039\pi\)
\(42\) 0.102193 0.0157688
\(43\) 2.26657 0.345649 0.172824 0.984953i \(-0.444711\pi\)
0.172824 + 0.984953i \(0.444711\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −0.0559322 −0.00824675
\(47\) −5.58347 −0.814432 −0.407216 0.913332i \(-0.633500\pi\)
−0.407216 + 0.913332i \(0.633500\pi\)
\(48\) −3.73968 −0.539777
\(49\) −6.76105 −0.965864
\(50\) 0.209057 0.0295651
\(51\) −3.33070 −0.466392
\(52\) 6.39228 0.886450
\(53\) −1.14301 −0.157005 −0.0785024 0.996914i \(-0.525014\pi\)
−0.0785024 + 0.996914i \(0.525014\pi\)
\(54\) −0.209057 −0.0284490
\(55\) 0 0
\(56\) 0.404307 0.0540277
\(57\) −3.86889 −0.512446
\(58\) −1.32039 −0.173375
\(59\) −4.51742 −0.588118 −0.294059 0.955787i \(-0.595006\pi\)
−0.294059 + 0.955787i \(0.595006\pi\)
\(60\) 1.95630 0.252557
\(61\) −12.5743 −1.60998 −0.804989 0.593290i \(-0.797829\pi\)
−0.804989 + 0.593290i \(0.797829\pi\)
\(62\) −1.21937 −0.154861
\(63\) −0.488830 −0.0615868
\(64\) −6.97010 −0.871263
\(65\) −3.26755 −0.405289
\(66\) 0 0
\(67\) 8.97575 1.09656 0.548281 0.836294i \(-0.315282\pi\)
0.548281 + 0.836294i \(0.315282\pi\)
\(68\) −6.51584 −0.790162
\(69\) 0.267545 0.0322087
\(70\) −0.102193 −0.0122144
\(71\) 5.31592 0.630884 0.315442 0.948945i \(-0.397847\pi\)
0.315442 + 0.948945i \(0.397847\pi\)
\(72\) −0.827091 −0.0974736
\(73\) −9.65885 −1.13048 −0.565242 0.824925i \(-0.691217\pi\)
−0.565242 + 0.824925i \(0.691217\pi\)
\(74\) 1.56868 0.182356
\(75\) −1.00000 −0.115470
\(76\) −7.56868 −0.868187
\(77\) 0 0
\(78\) 0.683103 0.0773462
\(79\) 1.50828 0.169695 0.0848476 0.996394i \(-0.472960\pi\)
0.0848476 + 0.996394i \(0.472960\pi\)
\(80\) 3.73968 0.418109
\(81\) 1.00000 0.111111
\(82\) −0.875466 −0.0966791
\(83\) −17.6428 −1.93655 −0.968275 0.249888i \(-0.919606\pi\)
−0.968275 + 0.249888i \(0.919606\pi\)
\(84\) −0.956295 −0.104340
\(85\) 3.33070 0.361266
\(86\) 0.473842 0.0510957
\(87\) 6.31592 0.677138
\(88\) 0 0
\(89\) 15.9226 1.68779 0.843895 0.536509i \(-0.180257\pi\)
0.843895 + 0.536509i \(0.180257\pi\)
\(90\) 0.209057 0.0220365
\(91\) 1.59727 0.167440
\(92\) 0.523398 0.0545680
\(93\) 5.83274 0.604827
\(94\) −1.16726 −0.120394
\(95\) 3.86889 0.396939
\(96\) −2.43599 −0.248622
\(97\) −16.3364 −1.65871 −0.829353 0.558726i \(-0.811290\pi\)
−0.829353 + 0.558726i \(0.811290\pi\)
\(98\) −1.41344 −0.142779
\(99\) 0 0
\(100\) −1.95630 −0.195630
\(101\) −17.8522 −1.77636 −0.888180 0.459496i \(-0.848030\pi\)
−0.888180 + 0.459496i \(0.848030\pi\)
\(102\) −0.696307 −0.0689446
\(103\) 15.3370 1.51120 0.755598 0.655036i \(-0.227347\pi\)
0.755598 + 0.655036i \(0.227347\pi\)
\(104\) 2.70256 0.265008
\(105\) 0.488830 0.0477049
\(106\) −0.238954 −0.0232093
\(107\) 5.83932 0.564508 0.282254 0.959340i \(-0.408918\pi\)
0.282254 + 0.959340i \(0.408918\pi\)
\(108\) 1.95630 0.188245
\(109\) −7.05284 −0.675540 −0.337770 0.941229i \(-0.609673\pi\)
−0.337770 + 0.941229i \(0.609673\pi\)
\(110\) 0 0
\(111\) −7.50361 −0.712211
\(112\) −1.82807 −0.172736
\(113\) −13.6019 −1.27956 −0.639782 0.768557i \(-0.720975\pi\)
−0.639782 + 0.768557i \(0.720975\pi\)
\(114\) −0.808817 −0.0757527
\(115\) −0.267545 −0.0249487
\(116\) 12.3558 1.14721
\(117\) −3.26755 −0.302085
\(118\) −0.944398 −0.0869389
\(119\) −1.62815 −0.149252
\(120\) 0.827091 0.0755027
\(121\) 0 0
\(122\) −2.62875 −0.237996
\(123\) 4.18769 0.377592
\(124\) 11.4106 1.02470
\(125\) 1.00000 0.0894427
\(126\) −0.102193 −0.00910410
\(127\) −22.0900 −1.96017 −0.980087 0.198568i \(-0.936371\pi\)
−0.980087 + 0.198568i \(0.936371\pi\)
\(128\) −6.32912 −0.559421
\(129\) −2.26657 −0.199560
\(130\) −0.683103 −0.0599121
\(131\) −3.87993 −0.338991 −0.169496 0.985531i \(-0.554214\pi\)
−0.169496 + 0.985531i \(0.554214\pi\)
\(132\) 0 0
\(133\) −1.89123 −0.163990
\(134\) 1.87644 0.162100
\(135\) −1.00000 −0.0860663
\(136\) −2.75480 −0.236222
\(137\) −5.36625 −0.458470 −0.229235 0.973371i \(-0.573622\pi\)
−0.229235 + 0.973371i \(0.573622\pi\)
\(138\) 0.0559322 0.00476127
\(139\) −12.6237 −1.07073 −0.535363 0.844622i \(-0.679825\pi\)
−0.535363 + 0.844622i \(0.679825\pi\)
\(140\) 0.956295 0.0808217
\(141\) 5.58347 0.470213
\(142\) 1.11133 0.0932607
\(143\) 0 0
\(144\) 3.73968 0.311640
\(145\) −6.31592 −0.524509
\(146\) −2.01925 −0.167114
\(147\) 6.76105 0.557642
\(148\) −14.6793 −1.20663
\(149\) 4.69947 0.384995 0.192498 0.981297i \(-0.438341\pi\)
0.192498 + 0.981297i \(0.438341\pi\)
\(150\) −0.209057 −0.0170694
\(151\) −14.5712 −1.18579 −0.592895 0.805280i \(-0.702015\pi\)
−0.592895 + 0.805280i \(0.702015\pi\)
\(152\) −3.19992 −0.259548
\(153\) 3.33070 0.269271
\(154\) 0 0
\(155\) −5.83274 −0.468497
\(156\) −6.39228 −0.511792
\(157\) 0.334194 0.0266716 0.0133358 0.999911i \(-0.495755\pi\)
0.0133358 + 0.999911i \(0.495755\pi\)
\(158\) 0.315317 0.0250853
\(159\) 1.14301 0.0906467
\(160\) 2.43599 0.192582
\(161\) 0.130784 0.0103072
\(162\) 0.209057 0.0164251
\(163\) 1.98935 0.155818 0.0779091 0.996960i \(-0.475176\pi\)
0.0779091 + 0.996960i \(0.475176\pi\)
\(164\) 8.19236 0.639716
\(165\) 0 0
\(166\) −3.68835 −0.286272
\(167\) 19.2322 1.48824 0.744118 0.668048i \(-0.232870\pi\)
0.744118 + 0.668048i \(0.232870\pi\)
\(168\) −0.404307 −0.0311929
\(169\) −2.32315 −0.178704
\(170\) 0.696307 0.0534043
\(171\) 3.86889 0.295861
\(172\) −4.43408 −0.338095
\(173\) −13.9713 −1.06222 −0.531108 0.847304i \(-0.678224\pi\)
−0.531108 + 0.847304i \(0.678224\pi\)
\(174\) 1.32039 0.100098
\(175\) −0.488830 −0.0369521
\(176\) 0 0
\(177\) 4.51742 0.339550
\(178\) 3.32873 0.249498
\(179\) −7.85995 −0.587480 −0.293740 0.955885i \(-0.594900\pi\)
−0.293740 + 0.955885i \(0.594900\pi\)
\(180\) −1.95630 −0.145814
\(181\) 7.39268 0.549494 0.274747 0.961517i \(-0.411406\pi\)
0.274747 + 0.961517i \(0.411406\pi\)
\(182\) 0.333921 0.0247519
\(183\) 12.5743 0.929521
\(184\) 0.221284 0.0163133
\(185\) 7.50361 0.551677
\(186\) 1.21937 0.0894089
\(187\) 0 0
\(188\) 10.9229 0.796635
\(189\) 0.488830 0.0355571
\(190\) 0.808817 0.0586778
\(191\) −0.429738 −0.0310947 −0.0155474 0.999879i \(-0.504949\pi\)
−0.0155474 + 0.999879i \(0.504949\pi\)
\(192\) 6.97010 0.503024
\(193\) −1.95004 −0.140367 −0.0701837 0.997534i \(-0.522359\pi\)
−0.0701837 + 0.997534i \(0.522359\pi\)
\(194\) −3.41523 −0.245199
\(195\) 3.26755 0.233994
\(196\) 13.2266 0.944757
\(197\) 22.2892 1.58804 0.794018 0.607894i \(-0.207985\pi\)
0.794018 + 0.607894i \(0.207985\pi\)
\(198\) 0 0
\(199\) 6.45077 0.457283 0.228642 0.973511i \(-0.426572\pi\)
0.228642 + 0.973511i \(0.426572\pi\)
\(200\) −0.827091 −0.0584842
\(201\) −8.97575 −0.633101
\(202\) −3.73212 −0.262591
\(203\) 3.08741 0.216694
\(204\) 6.51584 0.456200
\(205\) −4.18769 −0.292481
\(206\) 3.20630 0.223393
\(207\) −0.267545 −0.0185957
\(208\) −12.2196 −0.847275
\(209\) 0 0
\(210\) 0.102193 0.00705200
\(211\) −22.2379 −1.53092 −0.765460 0.643484i \(-0.777488\pi\)
−0.765460 + 0.643484i \(0.777488\pi\)
\(212\) 2.23607 0.153574
\(213\) −5.31592 −0.364241
\(214\) 1.22075 0.0834487
\(215\) 2.26657 0.154579
\(216\) 0.827091 0.0562764
\(217\) 2.85122 0.193553
\(218\) −1.47445 −0.0998620
\(219\) 9.65885 0.652685
\(220\) 0 0
\(221\) −10.8832 −0.732085
\(222\) −1.56868 −0.105283
\(223\) 13.7759 0.922501 0.461250 0.887270i \(-0.347401\pi\)
0.461250 + 0.887270i \(0.347401\pi\)
\(224\) −1.19078 −0.0795626
\(225\) 1.00000 0.0666667
\(226\) −2.84358 −0.189152
\(227\) 25.8554 1.71608 0.858040 0.513583i \(-0.171682\pi\)
0.858040 + 0.513583i \(0.171682\pi\)
\(228\) 7.56868 0.501248
\(229\) 14.3391 0.947555 0.473778 0.880645i \(-0.342890\pi\)
0.473778 + 0.880645i \(0.342890\pi\)
\(230\) −0.0559322 −0.00368806
\(231\) 0 0
\(232\) 5.22384 0.342962
\(233\) −3.66332 −0.239992 −0.119996 0.992774i \(-0.538288\pi\)
−0.119996 + 0.992774i \(0.538288\pi\)
\(234\) −0.683103 −0.0446558
\(235\) −5.58347 −0.364225
\(236\) 8.83741 0.575266
\(237\) −1.50828 −0.0979736
\(238\) −0.340375 −0.0220633
\(239\) −1.07794 −0.0697263 −0.0348632 0.999392i \(-0.511100\pi\)
−0.0348632 + 0.999392i \(0.511100\pi\)
\(240\) −3.73968 −0.241395
\(241\) −27.1803 −1.75084 −0.875420 0.483363i \(-0.839415\pi\)
−0.875420 + 0.483363i \(0.839415\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 24.5991 1.57480
\(245\) −6.76105 −0.431947
\(246\) 0.875466 0.0558177
\(247\) −12.6418 −0.804376
\(248\) 4.82420 0.306337
\(249\) 17.6428 1.11807
\(250\) 0.209057 0.0132219
\(251\) −2.60016 −0.164121 −0.0820603 0.996627i \(-0.526150\pi\)
−0.0820603 + 0.996627i \(0.526150\pi\)
\(252\) 0.956295 0.0602409
\(253\) 0 0
\(254\) −4.61808 −0.289764
\(255\) −3.33070 −0.208577
\(256\) 12.6171 0.788566
\(257\) 5.67343 0.353899 0.176949 0.984220i \(-0.443377\pi\)
0.176949 + 0.984220i \(0.443377\pi\)
\(258\) −0.473842 −0.0295001
\(259\) −3.66799 −0.227918
\(260\) 6.39228 0.396433
\(261\) −6.31592 −0.390946
\(262\) −0.811127 −0.0501116
\(263\) 23.4448 1.44567 0.722834 0.691022i \(-0.242839\pi\)
0.722834 + 0.691022i \(0.242839\pi\)
\(264\) 0 0
\(265\) −1.14301 −0.0702146
\(266\) −0.395374 −0.0242419
\(267\) −15.9226 −0.974446
\(268\) −17.5592 −1.07260
\(269\) −5.46398 −0.333144 −0.166572 0.986029i \(-0.553270\pi\)
−0.166572 + 0.986029i \(0.553270\pi\)
\(270\) −0.209057 −0.0127228
\(271\) −2.52319 −0.153273 −0.0766365 0.997059i \(-0.524418\pi\)
−0.0766365 + 0.997059i \(0.524418\pi\)
\(272\) 12.4558 0.755242
\(273\) −1.59727 −0.0966714
\(274\) −1.12185 −0.0677735
\(275\) 0 0
\(276\) −0.523398 −0.0315048
\(277\) 22.4067 1.34629 0.673145 0.739510i \(-0.264943\pi\)
0.673145 + 0.739510i \(0.264943\pi\)
\(278\) −2.63907 −0.158281
\(279\) −5.83274 −0.349197
\(280\) 0.404307 0.0241619
\(281\) 11.2790 0.672851 0.336426 0.941710i \(-0.390782\pi\)
0.336426 + 0.941710i \(0.390782\pi\)
\(282\) 1.16726 0.0695095
\(283\) 11.6925 0.695046 0.347523 0.937671i \(-0.387023\pi\)
0.347523 + 0.937671i \(0.387023\pi\)
\(284\) −10.3995 −0.617097
\(285\) −3.86889 −0.229173
\(286\) 0 0
\(287\) 2.04707 0.120835
\(288\) 2.43599 0.143542
\(289\) −5.90641 −0.347436
\(290\) −1.32039 −0.0775358
\(291\) 16.3364 0.957654
\(292\) 18.8956 1.10578
\(293\) 13.4918 0.788199 0.394100 0.919068i \(-0.371057\pi\)
0.394100 + 0.919068i \(0.371057\pi\)
\(294\) 1.41344 0.0824337
\(295\) −4.51742 −0.263014
\(296\) −6.20617 −0.360726
\(297\) 0 0
\(298\) 0.982456 0.0569121
\(299\) 0.874217 0.0505573
\(300\) 1.95630 0.112947
\(301\) −1.10797 −0.0638621
\(302\) −3.04622 −0.175290
\(303\) 17.8522 1.02558
\(304\) 14.4684 0.829820
\(305\) −12.5743 −0.720004
\(306\) 0.696307 0.0398052
\(307\) 0.364626 0.0208103 0.0104052 0.999946i \(-0.496688\pi\)
0.0104052 + 0.999946i \(0.496688\pi\)
\(308\) 0 0
\(309\) −15.3370 −0.872489
\(310\) −1.21937 −0.0692558
\(311\) 23.2542 1.31863 0.659313 0.751869i \(-0.270847\pi\)
0.659313 + 0.751869i \(0.270847\pi\)
\(312\) −2.70256 −0.153002
\(313\) −12.2326 −0.691430 −0.345715 0.938340i \(-0.612364\pi\)
−0.345715 + 0.938340i \(0.612364\pi\)
\(314\) 0.0698656 0.00394274
\(315\) −0.488830 −0.0275424
\(316\) −2.95065 −0.165987
\(317\) 32.0176 1.79829 0.899144 0.437653i \(-0.144190\pi\)
0.899144 + 0.437653i \(0.144190\pi\)
\(318\) 0.238954 0.0133999
\(319\) 0 0
\(320\) −6.97010 −0.389641
\(321\) −5.83932 −0.325919
\(322\) 0.0273413 0.00152367
\(323\) 12.8861 0.717003
\(324\) −1.95630 −0.108683
\(325\) −3.26755 −0.181251
\(326\) 0.415888 0.0230339
\(327\) 7.05284 0.390023
\(328\) 3.46360 0.191245
\(329\) 2.72936 0.150475
\(330\) 0 0
\(331\) −20.5705 −1.13066 −0.565328 0.824867i \(-0.691250\pi\)
−0.565328 + 0.824867i \(0.691250\pi\)
\(332\) 34.5145 1.89423
\(333\) 7.50361 0.411195
\(334\) 4.02063 0.219999
\(335\) 8.97575 0.490398
\(336\) 1.82807 0.0997292
\(337\) −22.3442 −1.21717 −0.608583 0.793491i \(-0.708262\pi\)
−0.608583 + 0.793491i \(0.708262\pi\)
\(338\) −0.485670 −0.0264170
\(339\) 13.6019 0.738756
\(340\) −6.51584 −0.353371
\(341\) 0 0
\(342\) 0.808817 0.0437358
\(343\) 6.72681 0.363213
\(344\) −1.87466 −0.101075
\(345\) 0.267545 0.0144042
\(346\) −2.92079 −0.157023
\(347\) 24.0053 1.28867 0.644336 0.764742i \(-0.277134\pi\)
0.644336 + 0.764742i \(0.277134\pi\)
\(348\) −12.3558 −0.662341
\(349\) −13.6284 −0.729509 −0.364754 0.931104i \(-0.618847\pi\)
−0.364754 + 0.931104i \(0.618847\pi\)
\(350\) −0.102193 −0.00546246
\(351\) 3.26755 0.174409
\(352\) 0 0
\(353\) −31.4058 −1.67156 −0.835780 0.549065i \(-0.814984\pi\)
−0.835780 + 0.549065i \(0.814984\pi\)
\(354\) 0.944398 0.0501942
\(355\) 5.31592 0.282140
\(356\) −31.1493 −1.65091
\(357\) 1.62815 0.0861707
\(358\) −1.64318 −0.0868446
\(359\) −2.87356 −0.151660 −0.0758302 0.997121i \(-0.524161\pi\)
−0.0758302 + 0.997121i \(0.524161\pi\)
\(360\) −0.827091 −0.0435915
\(361\) −4.03172 −0.212196
\(362\) 1.54549 0.0812292
\(363\) 0 0
\(364\) −3.12474 −0.163781
\(365\) −9.65885 −0.505567
\(366\) 2.62875 0.137407
\(367\) 24.0611 1.25598 0.627990 0.778221i \(-0.283878\pi\)
0.627990 + 0.778221i \(0.283878\pi\)
\(368\) −1.00053 −0.0521565
\(369\) −4.18769 −0.218003
\(370\) 1.56868 0.0815519
\(371\) 0.558738 0.0290082
\(372\) −11.4106 −0.591610
\(373\) −17.5348 −0.907915 −0.453958 0.891023i \(-0.649988\pi\)
−0.453958 + 0.891023i \(0.649988\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 4.61803 0.238157
\(377\) 20.6376 1.06289
\(378\) 0.102193 0.00525625
\(379\) −32.6393 −1.67657 −0.838285 0.545232i \(-0.816442\pi\)
−0.838285 + 0.545232i \(0.816442\pi\)
\(380\) −7.56868 −0.388265
\(381\) 22.0900 1.13171
\(382\) −0.0898397 −0.00459660
\(383\) 10.5167 0.537378 0.268689 0.963227i \(-0.413410\pi\)
0.268689 + 0.963227i \(0.413410\pi\)
\(384\) 6.32912 0.322982
\(385\) 0 0
\(386\) −0.407670 −0.0207499
\(387\) 2.26657 0.115216
\(388\) 31.9587 1.62246
\(389\) 23.3467 1.18373 0.591863 0.806038i \(-0.298393\pi\)
0.591863 + 0.806038i \(0.298393\pi\)
\(390\) 0.683103 0.0345903
\(391\) −0.891114 −0.0450656
\(392\) 5.59200 0.282439
\(393\) 3.87993 0.195717
\(394\) 4.65970 0.234752
\(395\) 1.50828 0.0758900
\(396\) 0 0
\(397\) 7.04740 0.353699 0.176849 0.984238i \(-0.443409\pi\)
0.176849 + 0.984238i \(0.443409\pi\)
\(398\) 1.34858 0.0675981
\(399\) 1.89123 0.0946797
\(400\) 3.73968 0.186984
\(401\) 22.4645 1.12182 0.560911 0.827876i \(-0.310451\pi\)
0.560911 + 0.827876i \(0.310451\pi\)
\(402\) −1.87644 −0.0935884
\(403\) 19.0587 0.949383
\(404\) 34.9242 1.73754
\(405\) 1.00000 0.0496904
\(406\) 0.645444 0.0320329
\(407\) 0 0
\(408\) 2.75480 0.136383
\(409\) −1.01918 −0.0503953 −0.0251976 0.999682i \(-0.508022\pi\)
−0.0251976 + 0.999682i \(0.508022\pi\)
\(410\) −0.875466 −0.0432362
\(411\) 5.36625 0.264698
\(412\) −30.0036 −1.47817
\(413\) 2.20825 0.108661
\(414\) −0.0559322 −0.00274892
\(415\) −17.6428 −0.866051
\(416\) −7.95970 −0.390256
\(417\) 12.6237 0.618184
\(418\) 0 0
\(419\) 7.54074 0.368389 0.184195 0.982890i \(-0.441032\pi\)
0.184195 + 0.982890i \(0.441032\pi\)
\(420\) −0.956295 −0.0466624
\(421\) 11.7585 0.573076 0.286538 0.958069i \(-0.407496\pi\)
0.286538 + 0.958069i \(0.407496\pi\)
\(422\) −4.64899 −0.226309
\(423\) −5.58347 −0.271477
\(424\) 0.945374 0.0459114
\(425\) 3.33070 0.161563
\(426\) −1.11133 −0.0538441
\(427\) 6.14671 0.297460
\(428\) −11.4234 −0.552172
\(429\) 0 0
\(430\) 0.473842 0.0228507
\(431\) 27.0506 1.30298 0.651491 0.758657i \(-0.274144\pi\)
0.651491 + 0.758657i \(0.274144\pi\)
\(432\) −3.73968 −0.179926
\(433\) −25.6704 −1.23364 −0.616821 0.787104i \(-0.711580\pi\)
−0.616821 + 0.787104i \(0.711580\pi\)
\(434\) 0.596066 0.0286121
\(435\) 6.31592 0.302825
\(436\) 13.7974 0.660778
\(437\) −1.03510 −0.0495157
\(438\) 2.01925 0.0964835
\(439\) −0.512954 −0.0244820 −0.0122410 0.999925i \(-0.503897\pi\)
−0.0122410 + 0.999925i \(0.503897\pi\)
\(440\) 0 0
\(441\) −6.76105 −0.321955
\(442\) −2.27521 −0.108221
\(443\) −33.2667 −1.58055 −0.790274 0.612754i \(-0.790062\pi\)
−0.790274 + 0.612754i \(0.790062\pi\)
\(444\) 14.6793 0.696648
\(445\) 15.9226 0.754803
\(446\) 2.87994 0.136369
\(447\) −4.69947 −0.222277
\(448\) 3.40719 0.160975
\(449\) 5.95841 0.281195 0.140597 0.990067i \(-0.455098\pi\)
0.140597 + 0.990067i \(0.455098\pi\)
\(450\) 0.209057 0.00985504
\(451\) 0 0
\(452\) 26.6094 1.25160
\(453\) 14.5712 0.684617
\(454\) 5.40524 0.253680
\(455\) 1.59727 0.0748813
\(456\) 3.19992 0.149850
\(457\) −20.4665 −0.957384 −0.478692 0.877983i \(-0.658889\pi\)
−0.478692 + 0.877983i \(0.658889\pi\)
\(458\) 2.99769 0.140073
\(459\) −3.33070 −0.155464
\(460\) 0.523398 0.0244035
\(461\) −31.1778 −1.45210 −0.726048 0.687644i \(-0.758645\pi\)
−0.726048 + 0.687644i \(0.758645\pi\)
\(462\) 0 0
\(463\) 10.9989 0.511164 0.255582 0.966787i \(-0.417733\pi\)
0.255582 + 0.966787i \(0.417733\pi\)
\(464\) −23.6195 −1.09651
\(465\) 5.83274 0.270487
\(466\) −0.765842 −0.0354770
\(467\) −35.4060 −1.63839 −0.819196 0.573513i \(-0.805580\pi\)
−0.819196 + 0.573513i \(0.805580\pi\)
\(468\) 6.39228 0.295483
\(469\) −4.38761 −0.202601
\(470\) −1.16726 −0.0538418
\(471\) −0.334194 −0.0153989
\(472\) 3.73632 0.171978
\(473\) 0 0
\(474\) −0.315317 −0.0144830
\(475\) 3.86889 0.177517
\(476\) 3.18514 0.145990
\(477\) −1.14301 −0.0523349
\(478\) −0.225351 −0.0103073
\(479\) −19.3816 −0.885570 −0.442785 0.896628i \(-0.646009\pi\)
−0.442785 + 0.896628i \(0.646009\pi\)
\(480\) −2.43599 −0.111187
\(481\) −24.5184 −1.11794
\(482\) −5.68224 −0.258819
\(483\) −0.130784 −0.00595088
\(484\) 0 0
\(485\) −16.3364 −0.741795
\(486\) −0.209057 −0.00948301
\(487\) −16.3500 −0.740891 −0.370445 0.928854i \(-0.620795\pi\)
−0.370445 + 0.928854i \(0.620795\pi\)
\(488\) 10.4001 0.470791
\(489\) −1.98935 −0.0899616
\(490\) −1.41344 −0.0638529
\(491\) 13.7855 0.622131 0.311065 0.950388i \(-0.399314\pi\)
0.311065 + 0.950388i \(0.399314\pi\)
\(492\) −8.19236 −0.369340
\(493\) −21.0365 −0.947435
\(494\) −2.64285 −0.118907
\(495\) 0 0
\(496\) −21.8126 −0.979414
\(497\) −2.59858 −0.116562
\(498\) 3.68835 0.165279
\(499\) 19.7828 0.885599 0.442799 0.896621i \(-0.353985\pi\)
0.442799 + 0.896621i \(0.353985\pi\)
\(500\) −1.95630 −0.0874882
\(501\) −19.2322 −0.859233
\(502\) −0.543581 −0.0242612
\(503\) 12.0127 0.535622 0.267811 0.963471i \(-0.413700\pi\)
0.267811 + 0.963471i \(0.413700\pi\)
\(504\) 0.404307 0.0180092
\(505\) −17.8522 −0.794412
\(506\) 0 0
\(507\) 2.32315 0.103175
\(508\) 43.2146 1.91734
\(509\) −7.98102 −0.353753 −0.176876 0.984233i \(-0.556599\pi\)
−0.176876 + 0.984233i \(0.556599\pi\)
\(510\) −0.696307 −0.0308330
\(511\) 4.72153 0.208868
\(512\) 15.2959 0.675991
\(513\) −3.86889 −0.170815
\(514\) 1.18607 0.0523153
\(515\) 15.3370 0.675827
\(516\) 4.43408 0.195199
\(517\) 0 0
\(518\) −0.766819 −0.0336921
\(519\) 13.9713 0.613271
\(520\) 2.70256 0.118515
\(521\) −44.4424 −1.94705 −0.973527 0.228571i \(-0.926595\pi\)
−0.973527 + 0.228571i \(0.926595\pi\)
\(522\) −1.32039 −0.0577918
\(523\) 12.0607 0.527376 0.263688 0.964608i \(-0.415061\pi\)
0.263688 + 0.964608i \(0.415061\pi\)
\(524\) 7.59029 0.331584
\(525\) 0.488830 0.0213343
\(526\) 4.90130 0.213707
\(527\) −19.4271 −0.846259
\(528\) 0 0
\(529\) −22.9284 −0.996888
\(530\) −0.238954 −0.0103795
\(531\) −4.51742 −0.196039
\(532\) 3.69980 0.160407
\(533\) 13.6835 0.592697
\(534\) −3.32873 −0.144048
\(535\) 5.83932 0.252456
\(536\) −7.42376 −0.320658
\(537\) 7.85995 0.339182
\(538\) −1.14228 −0.0492473
\(539\) 0 0
\(540\) 1.95630 0.0841855
\(541\) −37.7990 −1.62510 −0.812552 0.582888i \(-0.801923\pi\)
−0.812552 + 0.582888i \(0.801923\pi\)
\(542\) −0.527491 −0.0226577
\(543\) −7.39268 −0.317250
\(544\) 8.11356 0.347866
\(545\) −7.05284 −0.302111
\(546\) −0.333921 −0.0142905
\(547\) 14.9409 0.638825 0.319412 0.947616i \(-0.396515\pi\)
0.319412 + 0.947616i \(0.396515\pi\)
\(548\) 10.4980 0.448451
\(549\) −12.5743 −0.536659
\(550\) 0 0
\(551\) −24.4356 −1.04099
\(552\) −0.221284 −0.00941849
\(553\) −0.737294 −0.0313529
\(554\) 4.68428 0.199016
\(555\) −7.50361 −0.318511
\(556\) 24.6956 1.04733
\(557\) −7.30060 −0.309336 −0.154668 0.987966i \(-0.549431\pi\)
−0.154668 + 0.987966i \(0.549431\pi\)
\(558\) −1.21937 −0.0516202
\(559\) −7.40612 −0.313245
\(560\) −1.82807 −0.0772499
\(561\) 0 0
\(562\) 2.35796 0.0994646
\(563\) −23.3440 −0.983832 −0.491916 0.870643i \(-0.663703\pi\)
−0.491916 + 0.870643i \(0.663703\pi\)
\(564\) −10.9229 −0.459937
\(565\) −13.6019 −0.572238
\(566\) 2.44440 0.102746
\(567\) −0.488830 −0.0205289
\(568\) −4.39675 −0.184484
\(569\) −44.4220 −1.86227 −0.931134 0.364678i \(-0.881179\pi\)
−0.931134 + 0.364678i \(0.881179\pi\)
\(570\) −0.808817 −0.0338776
\(571\) 0.347203 0.0145300 0.00726499 0.999974i \(-0.497687\pi\)
0.00726499 + 0.999974i \(0.497687\pi\)
\(572\) 0 0
\(573\) 0.429738 0.0179526
\(574\) 0.427954 0.0178624
\(575\) −0.267545 −0.0111574
\(576\) −6.97010 −0.290421
\(577\) 9.94354 0.413955 0.206978 0.978346i \(-0.433637\pi\)
0.206978 + 0.978346i \(0.433637\pi\)
\(578\) −1.23478 −0.0513599
\(579\) 1.95004 0.0810411
\(580\) 12.3558 0.513047
\(581\) 8.62433 0.357797
\(582\) 3.41523 0.141566
\(583\) 0 0
\(584\) 7.98875 0.330577
\(585\) −3.26755 −0.135096
\(586\) 2.82055 0.116516
\(587\) −8.07388 −0.333245 −0.166622 0.986021i \(-0.553286\pi\)
−0.166622 + 0.986021i \(0.553286\pi\)
\(588\) −13.2266 −0.545456
\(589\) −22.5662 −0.929824
\(590\) −0.944398 −0.0388803
\(591\) −22.2892 −0.916853
\(592\) 28.0611 1.15331
\(593\) 25.5393 1.04877 0.524387 0.851480i \(-0.324295\pi\)
0.524387 + 0.851480i \(0.324295\pi\)
\(594\) 0 0
\(595\) −1.62815 −0.0667475
\(596\) −9.19354 −0.376582
\(597\) −6.45077 −0.264013
\(598\) 0.182761 0.00747365
\(599\) 3.77156 0.154102 0.0770508 0.997027i \(-0.475450\pi\)
0.0770508 + 0.997027i \(0.475450\pi\)
\(600\) 0.827091 0.0337658
\(601\) −2.48063 −0.101187 −0.0505934 0.998719i \(-0.516111\pi\)
−0.0505934 + 0.998719i \(0.516111\pi\)
\(602\) −0.231628 −0.00944045
\(603\) 8.97575 0.365521
\(604\) 28.5056 1.15988
\(605\) 0 0
\(606\) 3.73212 0.151607
\(607\) 7.44537 0.302198 0.151099 0.988519i \(-0.451719\pi\)
0.151099 + 0.988519i \(0.451719\pi\)
\(608\) 9.42456 0.382216
\(609\) −3.08741 −0.125108
\(610\) −2.62875 −0.106435
\(611\) 18.2442 0.738082
\(612\) −6.51584 −0.263387
\(613\) 28.9464 1.16914 0.584568 0.811345i \(-0.301264\pi\)
0.584568 + 0.811345i \(0.301264\pi\)
\(614\) 0.0762276 0.00307630
\(615\) 4.18769 0.168864
\(616\) 0 0
\(617\) −18.7581 −0.755172 −0.377586 0.925975i \(-0.623246\pi\)
−0.377586 + 0.925975i \(0.623246\pi\)
\(618\) −3.20630 −0.128976
\(619\) −29.4451 −1.18350 −0.591749 0.806123i \(-0.701562\pi\)
−0.591749 + 0.806123i \(0.701562\pi\)
\(620\) 11.4106 0.458259
\(621\) 0.267545 0.0107362
\(622\) 4.86145 0.194927
\(623\) −7.78343 −0.311836
\(624\) 12.2196 0.489175
\(625\) 1.00000 0.0400000
\(626\) −2.55732 −0.102211
\(627\) 0 0
\(628\) −0.653783 −0.0260888
\(629\) 24.9923 0.996509
\(630\) −0.102193 −0.00407148
\(631\) 40.6457 1.61808 0.809040 0.587754i \(-0.199988\pi\)
0.809040 + 0.587754i \(0.199988\pi\)
\(632\) −1.24749 −0.0496224
\(633\) 22.2379 0.883877
\(634\) 6.69350 0.265833
\(635\) −22.0900 −0.876617
\(636\) −2.23607 −0.0886659
\(637\) 22.0920 0.875318
\(638\) 0 0
\(639\) 5.31592 0.210295
\(640\) −6.32912 −0.250181
\(641\) 22.8967 0.904365 0.452183 0.891925i \(-0.350645\pi\)
0.452183 + 0.891925i \(0.350645\pi\)
\(642\) −1.22075 −0.0481792
\(643\) 30.4536 1.20097 0.600486 0.799635i \(-0.294974\pi\)
0.600486 + 0.799635i \(0.294974\pi\)
\(644\) −0.255852 −0.0100820
\(645\) −2.26657 −0.0892461
\(646\) 2.69393 0.105991
\(647\) −14.4006 −0.566145 −0.283072 0.959099i \(-0.591354\pi\)
−0.283072 + 0.959099i \(0.591354\pi\)
\(648\) −0.827091 −0.0324912
\(649\) 0 0
\(650\) −0.683103 −0.0267935
\(651\) −2.85122 −0.111748
\(652\) −3.89176 −0.152413
\(653\) 30.3188 1.18646 0.593232 0.805031i \(-0.297852\pi\)
0.593232 + 0.805031i \(0.297852\pi\)
\(654\) 1.47445 0.0576554
\(655\) −3.87993 −0.151602
\(656\) −15.6606 −0.611445
\(657\) −9.65885 −0.376828
\(658\) 0.570592 0.0222440
\(659\) 25.8802 1.00815 0.504076 0.863659i \(-0.331833\pi\)
0.504076 + 0.863659i \(0.331833\pi\)
\(660\) 0 0
\(661\) −9.41852 −0.366338 −0.183169 0.983081i \(-0.558636\pi\)
−0.183169 + 0.983081i \(0.558636\pi\)
\(662\) −4.30040 −0.167140
\(663\) 10.8832 0.422670
\(664\) 14.5922 0.566287
\(665\) −1.89123 −0.0733386
\(666\) 1.56868 0.0607852
\(667\) 1.68980 0.0654291
\(668\) −37.6240 −1.45571
\(669\) −13.7759 −0.532606
\(670\) 1.87644 0.0724933
\(671\) 0 0
\(672\) 1.19078 0.0459355
\(673\) −13.3259 −0.513676 −0.256838 0.966454i \(-0.582681\pi\)
−0.256838 + 0.966454i \(0.582681\pi\)
\(674\) −4.67121 −0.179928
\(675\) −1.00000 −0.0384900
\(676\) 4.54476 0.174799
\(677\) −5.08029 −0.195252 −0.0976258 0.995223i \(-0.531125\pi\)
−0.0976258 + 0.995223i \(0.531125\pi\)
\(678\) 2.84358 0.109207
\(679\) 7.98569 0.306463
\(680\) −2.75480 −0.105642
\(681\) −25.8554 −0.990779
\(682\) 0 0
\(683\) 19.0765 0.729942 0.364971 0.931019i \(-0.381079\pi\)
0.364971 + 0.931019i \(0.381079\pi\)
\(684\) −7.56868 −0.289396
\(685\) −5.36625 −0.205034
\(686\) 1.40629 0.0536922
\(687\) −14.3391 −0.547071
\(688\) 8.47625 0.323154
\(689\) 3.73484 0.142286
\(690\) 0.0559322 0.00212930
\(691\) 8.69891 0.330922 0.165461 0.986216i \(-0.447089\pi\)
0.165461 + 0.986216i \(0.447089\pi\)
\(692\) 27.3320 1.03900
\(693\) 0 0
\(694\) 5.01848 0.190499
\(695\) −12.6237 −0.478844
\(696\) −5.22384 −0.198009
\(697\) −13.9480 −0.528317
\(698\) −2.84910 −0.107840
\(699\) 3.66332 0.138559
\(700\) 0.956295 0.0361446
\(701\) 40.2941 1.52189 0.760944 0.648818i \(-0.224736\pi\)
0.760944 + 0.648818i \(0.224736\pi\)
\(702\) 0.683103 0.0257821
\(703\) 29.0306 1.09491
\(704\) 0 0
\(705\) 5.58347 0.210285
\(706\) −6.56559 −0.247099
\(707\) 8.72668 0.328201
\(708\) −8.83741 −0.332130
\(709\) 23.6968 0.889950 0.444975 0.895543i \(-0.353212\pi\)
0.444975 + 0.895543i \(0.353212\pi\)
\(710\) 1.11133 0.0417075
\(711\) 1.50828 0.0565651
\(712\) −13.1694 −0.493545
\(713\) 1.56052 0.0584420
\(714\) 0.340375 0.0127382
\(715\) 0 0
\(716\) 15.3764 0.574643
\(717\) 1.07794 0.0402565
\(718\) −0.600737 −0.0224193
\(719\) −17.3576 −0.647330 −0.323665 0.946172i \(-0.604915\pi\)
−0.323665 + 0.946172i \(0.604915\pi\)
\(720\) 3.73968 0.139370
\(721\) −7.49716 −0.279209
\(722\) −0.842860 −0.0313680
\(723\) 27.1803 1.01085
\(724\) −14.4623 −0.537486
\(725\) −6.31592 −0.234567
\(726\) 0 0
\(727\) 28.5895 1.06033 0.530163 0.847896i \(-0.322131\pi\)
0.530163 + 0.847896i \(0.322131\pi\)
\(728\) −1.32109 −0.0489629
\(729\) 1.00000 0.0370370
\(730\) −2.01925 −0.0747358
\(731\) 7.54927 0.279220
\(732\) −24.5991 −0.909209
\(733\) 32.4058 1.19694 0.598468 0.801146i \(-0.295776\pi\)
0.598468 + 0.801146i \(0.295776\pi\)
\(734\) 5.03014 0.185666
\(735\) 6.76105 0.249385
\(736\) −0.651737 −0.0240234
\(737\) 0 0
\(738\) −0.875466 −0.0322264
\(739\) −4.16917 −0.153365 −0.0766827 0.997056i \(-0.524433\pi\)
−0.0766827 + 0.997056i \(0.524433\pi\)
\(740\) −14.6793 −0.539621
\(741\) 12.6418 0.464407
\(742\) 0.116808 0.00428816
\(743\) −48.8869 −1.79349 −0.896743 0.442551i \(-0.854074\pi\)
−0.896743 + 0.442551i \(0.854074\pi\)
\(744\) −4.82420 −0.176864
\(745\) 4.69947 0.172175
\(746\) −3.66576 −0.134213
\(747\) −17.6428 −0.645516
\(748\) 0 0
\(749\) −2.85443 −0.104299
\(750\) −0.209057 −0.00763368
\(751\) −40.5342 −1.47912 −0.739558 0.673093i \(-0.764965\pi\)
−0.739558 + 0.673093i \(0.764965\pi\)
\(752\) −20.8804 −0.761429
\(753\) 2.60016 0.0947551
\(754\) 4.31442 0.157122
\(755\) −14.5712 −0.530302
\(756\) −0.956295 −0.0347801
\(757\) −21.8143 −0.792854 −0.396427 0.918066i \(-0.629750\pi\)
−0.396427 + 0.918066i \(0.629750\pi\)
\(758\) −6.82348 −0.247840
\(759\) 0 0
\(760\) −3.19992 −0.116073
\(761\) −28.6123 −1.03720 −0.518598 0.855018i \(-0.673546\pi\)
−0.518598 + 0.855018i \(0.673546\pi\)
\(762\) 4.61808 0.167295
\(763\) 3.44764 0.124813
\(764\) 0.840694 0.0304152
\(765\) 3.33070 0.120422
\(766\) 2.19859 0.0794382
\(767\) 14.7609 0.532984
\(768\) −12.6171 −0.455279
\(769\) 11.2690 0.406372 0.203186 0.979140i \(-0.434870\pi\)
0.203186 + 0.979140i \(0.434870\pi\)
\(770\) 0 0
\(771\) −5.67343 −0.204324
\(772\) 3.81486 0.137300
\(773\) −15.7132 −0.565166 −0.282583 0.959243i \(-0.591191\pi\)
−0.282583 + 0.959243i \(0.591191\pi\)
\(774\) 0.473842 0.0170319
\(775\) −5.83274 −0.209518
\(776\) 13.5116 0.485040
\(777\) 3.66799 0.131588
\(778\) 4.88080 0.174985
\(779\) −16.2017 −0.580486
\(780\) −6.39228 −0.228880
\(781\) 0 0
\(782\) −0.186294 −0.00666185
\(783\) 6.31592 0.225713
\(784\) −25.2842 −0.903006
\(785\) 0.334194 0.0119279
\(786\) 0.811127 0.0289319
\(787\) −18.4148 −0.656416 −0.328208 0.944606i \(-0.606445\pi\)
−0.328208 + 0.944606i \(0.606445\pi\)
\(788\) −43.6042 −1.55333
\(789\) −23.4448 −0.834657
\(790\) 0.315317 0.0112185
\(791\) 6.64903 0.236412
\(792\) 0 0
\(793\) 41.0872 1.45905
\(794\) 1.47331 0.0522857
\(795\) 1.14301 0.0405384
\(796\) −12.6196 −0.447290
\(797\) 25.2719 0.895176 0.447588 0.894240i \(-0.352283\pi\)
0.447588 + 0.894240i \(0.352283\pi\)
\(798\) 0.395374 0.0139961
\(799\) −18.5969 −0.657910
\(800\) 2.43599 0.0861252
\(801\) 15.9226 0.562597
\(802\) 4.69635 0.165834
\(803\) 0 0
\(804\) 17.5592 0.619266
\(805\) 0.130784 0.00460953
\(806\) 3.98436 0.140343
\(807\) 5.46398 0.192341
\(808\) 14.7654 0.519444
\(809\) 11.2635 0.396002 0.198001 0.980202i \(-0.436555\pi\)
0.198001 + 0.980202i \(0.436555\pi\)
\(810\) 0.209057 0.00734551
\(811\) 24.3405 0.854710 0.427355 0.904084i \(-0.359445\pi\)
0.427355 + 0.904084i \(0.359445\pi\)
\(812\) −6.03988 −0.211958
\(813\) 2.52319 0.0884923
\(814\) 0 0
\(815\) 1.98935 0.0696840
\(816\) −12.4558 −0.436039
\(817\) 8.76910 0.306792
\(818\) −0.213067 −0.00744971
\(819\) 1.59727 0.0558132
\(820\) 8.19236 0.286090
\(821\) −3.53529 −0.123383 −0.0616913 0.998095i \(-0.519649\pi\)
−0.0616913 + 0.998095i \(0.519649\pi\)
\(822\) 1.12185 0.0391291
\(823\) −46.9230 −1.63563 −0.817817 0.575478i \(-0.804816\pi\)
−0.817817 + 0.575478i \(0.804816\pi\)
\(824\) −12.6851 −0.441905
\(825\) 0 0
\(826\) 0.461650 0.0160629
\(827\) 43.7647 1.52185 0.760923 0.648842i \(-0.224746\pi\)
0.760923 + 0.648842i \(0.224746\pi\)
\(828\) 0.523398 0.0181893
\(829\) −20.2447 −0.703126 −0.351563 0.936164i \(-0.614350\pi\)
−0.351563 + 0.936164i \(0.614350\pi\)
\(830\) −3.68835 −0.128025
\(831\) −22.4067 −0.777281
\(832\) 22.7751 0.789585
\(833\) −22.5190 −0.780239
\(834\) 2.63907 0.0913834
\(835\) 19.2322 0.665559
\(836\) 0 0
\(837\) 5.83274 0.201609
\(838\) 1.57644 0.0544573
\(839\) 39.0495 1.34814 0.674070 0.738668i \(-0.264545\pi\)
0.674070 + 0.738668i \(0.264545\pi\)
\(840\) −0.404307 −0.0139499
\(841\) 10.8909 0.375547
\(842\) 2.45820 0.0847153
\(843\) −11.2790 −0.388471
\(844\) 43.5039 1.49747
\(845\) −2.32315 −0.0799187
\(846\) −1.16726 −0.0401313
\(847\) 0 0
\(848\) −4.27450 −0.146787
\(849\) −11.6925 −0.401285
\(850\) 0.696307 0.0238831
\(851\) −2.00756 −0.0688182
\(852\) 10.3995 0.356281
\(853\) 41.1973 1.41057 0.705285 0.708924i \(-0.250819\pi\)
0.705285 + 0.708924i \(0.250819\pi\)
\(854\) 1.28501 0.0439722
\(855\) 3.86889 0.132313
\(856\) −4.82965 −0.165074
\(857\) −39.6853 −1.35563 −0.677813 0.735235i \(-0.737072\pi\)
−0.677813 + 0.735235i \(0.737072\pi\)
\(858\) 0 0
\(859\) 22.6446 0.772625 0.386313 0.922368i \(-0.373749\pi\)
0.386313 + 0.922368i \(0.373749\pi\)
\(860\) −4.43408 −0.151201
\(861\) −2.04707 −0.0697639
\(862\) 5.65512 0.192614
\(863\) −34.0135 −1.15783 −0.578916 0.815387i \(-0.696524\pi\)
−0.578916 + 0.815387i \(0.696524\pi\)
\(864\) −2.43599 −0.0828740
\(865\) −13.9713 −0.475038
\(866\) −5.36658 −0.182364
\(867\) 5.90641 0.200592
\(868\) −5.57782 −0.189324
\(869\) 0 0
\(870\) 1.32039 0.0447653
\(871\) −29.3287 −0.993764
\(872\) 5.83334 0.197542
\(873\) −16.3364 −0.552902
\(874\) −0.216395 −0.00731968
\(875\) −0.488830 −0.0165255
\(876\) −18.8956 −0.638422
\(877\) −27.4935 −0.928389 −0.464195 0.885733i \(-0.653656\pi\)
−0.464195 + 0.885733i \(0.653656\pi\)
\(878\) −0.107237 −0.00361906
\(879\) −13.4918 −0.455067
\(880\) 0 0
\(881\) 41.2585 1.39003 0.695017 0.718993i \(-0.255397\pi\)
0.695017 + 0.718993i \(0.255397\pi\)
\(882\) −1.41344 −0.0475931
\(883\) −15.1814 −0.510895 −0.255447 0.966823i \(-0.582223\pi\)
−0.255447 + 0.966823i \(0.582223\pi\)
\(884\) 21.2908 0.716087
\(885\) 4.51742 0.151851
\(886\) −6.95463 −0.233645
\(887\) −34.1080 −1.14523 −0.572617 0.819823i \(-0.694072\pi\)
−0.572617 + 0.819823i \(0.694072\pi\)
\(888\) 6.20617 0.208265
\(889\) 10.7983 0.362162
\(890\) 3.32873 0.111579
\(891\) 0 0
\(892\) −26.9497 −0.902342
\(893\) −21.6018 −0.722876
\(894\) −0.982456 −0.0328582
\(895\) −7.85995 −0.262729
\(896\) 3.09386 0.103359
\(897\) −0.874217 −0.0291892
\(898\) 1.24565 0.0415678
\(899\) 36.8391 1.22865
\(900\) −1.95630 −0.0652098
\(901\) −3.80703 −0.126831
\(902\) 0 0
\(903\) 1.10797 0.0368708
\(904\) 11.2500 0.374171
\(905\) 7.39268 0.245741
\(906\) 3.04622 0.101204
\(907\) −17.9230 −0.595123 −0.297561 0.954703i \(-0.596173\pi\)
−0.297561 + 0.954703i \(0.596173\pi\)
\(908\) −50.5807 −1.67858
\(909\) −17.8522 −0.592120
\(910\) 0.333921 0.0110694
\(911\) −21.9045 −0.725727 −0.362864 0.931842i \(-0.618201\pi\)
−0.362864 + 0.931842i \(0.618201\pi\)
\(912\) −14.4684 −0.479097
\(913\) 0 0
\(914\) −4.27867 −0.141526
\(915\) 12.5743 0.415695
\(916\) −28.0515 −0.926849
\(917\) 1.89663 0.0626321
\(918\) −0.696307 −0.0229815
\(919\) −4.04370 −0.133390 −0.0666948 0.997773i \(-0.521245\pi\)
−0.0666948 + 0.997773i \(0.521245\pi\)
\(920\) 0.221284 0.00729553
\(921\) −0.364626 −0.0120148
\(922\) −6.51794 −0.214657
\(923\) −17.3700 −0.571741
\(924\) 0 0
\(925\) 7.50361 0.246717
\(926\) 2.29940 0.0755631
\(927\) 15.3370 0.503732
\(928\) −15.3855 −0.505054
\(929\) −10.3334 −0.339027 −0.169514 0.985528i \(-0.554220\pi\)
−0.169514 + 0.985528i \(0.554220\pi\)
\(930\) 1.21937 0.0399849
\(931\) −26.1577 −0.857284
\(932\) 7.16653 0.234748
\(933\) −23.2542 −0.761309
\(934\) −7.40186 −0.242196
\(935\) 0 0
\(936\) 2.70256 0.0883358
\(937\) −32.5450 −1.06320 −0.531600 0.846996i \(-0.678409\pi\)
−0.531600 + 0.846996i \(0.678409\pi\)
\(938\) −0.917261 −0.0299496
\(939\) 12.2326 0.399197
\(940\) 10.9229 0.356266
\(941\) 12.3962 0.404104 0.202052 0.979375i \(-0.435239\pi\)
0.202052 + 0.979375i \(0.435239\pi\)
\(942\) −0.0698656 −0.00227634
\(943\) 1.12040 0.0364852
\(944\) −16.8937 −0.549844
\(945\) 0.488830 0.0159016
\(946\) 0 0
\(947\) 44.7602 1.45451 0.727256 0.686366i \(-0.240795\pi\)
0.727256 + 0.686366i \(0.240795\pi\)
\(948\) 2.95065 0.0958326
\(949\) 31.5607 1.02450
\(950\) 0.808817 0.0262415
\(951\) −32.0176 −1.03824
\(952\) 1.34663 0.0436444
\(953\) 21.3562 0.691794 0.345897 0.938273i \(-0.387575\pi\)
0.345897 + 0.938273i \(0.387575\pi\)
\(954\) −0.238954 −0.00773644
\(955\) −0.429738 −0.0139060
\(956\) 2.10877 0.0682026
\(957\) 0 0
\(958\) −4.05187 −0.130910
\(959\) 2.62318 0.0847070
\(960\) 6.97010 0.224959
\(961\) 3.02083 0.0974461
\(962\) −5.12574 −0.165261
\(963\) 5.83932 0.188169
\(964\) 53.1728 1.71258
\(965\) −1.95004 −0.0627742
\(966\) −0.0273413 −0.000879693 0
\(967\) 31.3251 1.00735 0.503674 0.863894i \(-0.331981\pi\)
0.503674 + 0.863894i \(0.331981\pi\)
\(968\) 0 0
\(969\) −12.8861 −0.413962
\(970\) −3.41523 −0.109656
\(971\) 15.7125 0.504239 0.252119 0.967696i \(-0.418872\pi\)
0.252119 + 0.967696i \(0.418872\pi\)
\(972\) 1.95630 0.0627482
\(973\) 6.17083 0.197828
\(974\) −3.41809 −0.109523
\(975\) 3.26755 0.104645
\(976\) −47.0240 −1.50520
\(977\) −22.5622 −0.721828 −0.360914 0.932599i \(-0.617535\pi\)
−0.360914 + 0.932599i \(0.617535\pi\)
\(978\) −0.415888 −0.0132986
\(979\) 0 0
\(980\) 13.2266 0.422508
\(981\) −7.05284 −0.225180
\(982\) 2.88195 0.0919669
\(983\) 45.9795 1.46652 0.733259 0.679950i \(-0.237998\pi\)
0.733259 + 0.679950i \(0.237998\pi\)
\(984\) −3.46360 −0.110416
\(985\) 22.2892 0.710192
\(986\) −4.39782 −0.140055
\(987\) −2.72936 −0.0868766
\(988\) 24.7310 0.786798
\(989\) −0.606410 −0.0192827
\(990\) 0 0
\(991\) 5.48050 0.174094 0.0870469 0.996204i \(-0.472257\pi\)
0.0870469 + 0.996204i \(0.472257\pi\)
\(992\) −14.2085 −0.451120
\(993\) 20.5705 0.652784
\(994\) −0.543251 −0.0172309
\(995\) 6.45077 0.204503
\(996\) −34.5145 −1.09363
\(997\) −18.2558 −0.578167 −0.289083 0.957304i \(-0.593350\pi\)
−0.289083 + 0.957304i \(0.593350\pi\)
\(998\) 4.13573 0.130914
\(999\) −7.50361 −0.237404
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 1815.2.a.q.1.3 4
3.2 odd 2 5445.2.a.bq.1.2 4
5.4 even 2 9075.2.a.df.1.2 4
11.7 odd 10 165.2.m.c.16.1 8
11.8 odd 10 165.2.m.c.31.1 yes 8
11.10 odd 2 1815.2.a.u.1.2 4
33.8 even 10 495.2.n.c.361.2 8
33.29 even 10 495.2.n.c.181.2 8
33.32 even 2 5445.2.a.bj.1.3 4
55.7 even 20 825.2.bx.e.49.3 16
55.8 even 20 825.2.bx.e.724.3 16
55.18 even 20 825.2.bx.e.49.2 16
55.19 odd 10 825.2.n.j.526.2 8
55.29 odd 10 825.2.n.j.676.2 8
55.52 even 20 825.2.bx.e.724.2 16
55.54 odd 2 9075.2.a.co.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.c.16.1 8 11.7 odd 10
165.2.m.c.31.1 yes 8 11.8 odd 10
495.2.n.c.181.2 8 33.29 even 10
495.2.n.c.361.2 8 33.8 even 10
825.2.n.j.526.2 8 55.19 odd 10
825.2.n.j.676.2 8 55.29 odd 10
825.2.bx.e.49.2 16 55.18 even 20
825.2.bx.e.49.3 16 55.7 even 20
825.2.bx.e.724.2 16 55.52 even 20
825.2.bx.e.724.3 16 55.8 even 20
1815.2.a.q.1.3 4 1.1 even 1 trivial
1815.2.a.u.1.2 4 11.10 odd 2
5445.2.a.bj.1.3 4 33.32 even 2
5445.2.a.bq.1.2 4 3.2 odd 2
9075.2.a.co.1.3 4 55.54 odd 2
9075.2.a.df.1.2 4 5.4 even 2