Properties

Label 1815.2.a.s.1.2
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $4$
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] = [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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.8112.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.07431\) 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)
 
\(f(q)\) \(=\) \(q-0.628052 q^{2} -1.00000 q^{3} -1.60555 q^{4} +1.00000 q^{5} +0.628052 q^{6} +4.14863 q^{7} +2.26447 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.628052 q^{2} -1.00000 q^{3} -1.60555 q^{4} +1.00000 q^{5} +0.628052 q^{6} +4.14863 q^{7} +2.26447 q^{8} +1.00000 q^{9} -0.628052 q^{10} +1.60555 q^{12} +5.40473 q^{13} -2.60555 q^{14} -1.00000 q^{15} +1.78890 q^{16} +4.14863 q^{17} -0.628052 q^{18} -1.25610 q^{19} -1.60555 q^{20} -4.14863 q^{21} +5.21110 q^{23} -2.26447 q^{24} +1.00000 q^{25} -3.39445 q^{26} -1.00000 q^{27} -6.66083 q^{28} -7.04115 q^{29} +0.628052 q^{30} +4.00000 q^{31} -5.65246 q^{32} -2.60555 q^{34} +4.14863 q^{35} -1.60555 q^{36} -7.21110 q^{37} +0.788897 q^{38} -5.40473 q^{39} +2.26447 q^{40} -9.55336 q^{41} +2.60555 q^{42} +6.66083 q^{43} +1.00000 q^{45} -3.27284 q^{46} -1.78890 q^{48} +10.2111 q^{49} -0.628052 q^{50} -4.14863 q^{51} -8.67757 q^{52} +11.2111 q^{53} +0.628052 q^{54} +9.39445 q^{56} +1.25610 q^{57} +4.42221 q^{58} -5.21110 q^{59} +1.60555 q^{60} +8.29725 q^{61} -2.51221 q^{62} +4.14863 q^{63} -0.0277564 q^{64} +5.40473 q^{65} -13.2111 q^{67} -6.66083 q^{68} -5.21110 q^{69} -2.60555 q^{70} +5.21110 q^{71} +2.26447 q^{72} -2.89252 q^{73} +4.52894 q^{74} -1.00000 q^{75} +2.01674 q^{76} +3.39445 q^{78} -1.25610 q^{79} +1.78890 q^{80} +1.00000 q^{81} +6.00000 q^{82} -13.7020 q^{83} +6.66083 q^{84} +4.14863 q^{85} -4.18335 q^{86} +7.04115 q^{87} +6.00000 q^{89} -0.628052 q^{90} +22.4222 q^{91} -8.36669 q^{92} -4.00000 q^{93} -1.25610 q^{95} +5.65246 q^{96} +19.2111 q^{97} -6.41310 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{9} - 8 q^{12} + 4 q^{14} - 4 q^{15} + 36 q^{16} + 8 q^{20} - 8 q^{23} + 4 q^{25} - 28 q^{26} - 4 q^{27} + 16 q^{31} + 4 q^{34} + 8 q^{36} + 32 q^{38} - 4 q^{42} + 4 q^{45} - 36 q^{48} + 12 q^{49} + 16 q^{53} + 52 q^{56} - 40 q^{58} + 8 q^{59} - 8 q^{60} + 72 q^{64} - 24 q^{67} + 8 q^{69} + 4 q^{70} - 8 q^{71} - 4 q^{75} + 28 q^{78} + 36 q^{80} + 4 q^{81} + 24 q^{82} - 60 q^{86} + 24 q^{89} + 32 q^{91} - 120 q^{92} - 16 q^{93} + 48 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.628052 −0.444099 −0.222050 0.975035i \(-0.571275\pi\)
−0.222050 + 0.975035i \(0.571275\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.60555 −0.802776
\(5\) 1.00000 0.447214
\(6\) 0.628052 0.256401
\(7\) 4.14863 1.56803 0.784017 0.620740i \(-0.213168\pi\)
0.784017 + 0.620740i \(0.213168\pi\)
\(8\) 2.26447 0.800612
\(9\) 1.00000 0.333333
\(10\) −0.628052 −0.198607
\(11\) 0 0
\(12\) 1.60555 0.463483
\(13\) 5.40473 1.49900 0.749501 0.662003i \(-0.230293\pi\)
0.749501 + 0.662003i \(0.230293\pi\)
\(14\) −2.60555 −0.696363
\(15\) −1.00000 −0.258199
\(16\) 1.78890 0.447224
\(17\) 4.14863 1.00619 0.503095 0.864231i \(-0.332195\pi\)
0.503095 + 0.864231i \(0.332195\pi\)
\(18\) −0.628052 −0.148033
\(19\) −1.25610 −0.288170 −0.144085 0.989565i \(-0.546024\pi\)
−0.144085 + 0.989565i \(0.546024\pi\)
\(20\) −1.60555 −0.359012
\(21\) −4.14863 −0.905305
\(22\) 0 0
\(23\) 5.21110 1.08659 0.543295 0.839542i \(-0.317177\pi\)
0.543295 + 0.839542i \(0.317177\pi\)
\(24\) −2.26447 −0.462233
\(25\) 1.00000 0.200000
\(26\) −3.39445 −0.665706
\(27\) −1.00000 −0.192450
\(28\) −6.66083 −1.25878
\(29\) −7.04115 −1.30751 −0.653754 0.756707i \(-0.726807\pi\)
−0.653754 + 0.756707i \(0.726807\pi\)
\(30\) 0.628052 0.114666
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −5.65246 −0.999224
\(33\) 0 0
\(34\) −2.60555 −0.446848
\(35\) 4.14863 0.701246
\(36\) −1.60555 −0.267592
\(37\) −7.21110 −1.18550 −0.592749 0.805387i \(-0.701957\pi\)
−0.592749 + 0.805387i \(0.701957\pi\)
\(38\) 0.788897 0.127976
\(39\) −5.40473 −0.865449
\(40\) 2.26447 0.358044
\(41\) −9.55336 −1.49198 −0.745992 0.665955i \(-0.768024\pi\)
−0.745992 + 0.665955i \(0.768024\pi\)
\(42\) 2.60555 0.402045
\(43\) 6.66083 1.01577 0.507884 0.861426i \(-0.330428\pi\)
0.507884 + 0.861426i \(0.330428\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −3.27284 −0.482554
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.78890 −0.258205
\(49\) 10.2111 1.45873
\(50\) −0.628052 −0.0888199
\(51\) −4.14863 −0.580924
\(52\) −8.67757 −1.20336
\(53\) 11.2111 1.53996 0.769982 0.638066i \(-0.220265\pi\)
0.769982 + 0.638066i \(0.220265\pi\)
\(54\) 0.628052 0.0854670
\(55\) 0 0
\(56\) 9.39445 1.25539
\(57\) 1.25610 0.166375
\(58\) 4.42221 0.580664
\(59\) −5.21110 −0.678428 −0.339214 0.940709i \(-0.610161\pi\)
−0.339214 + 0.940709i \(0.610161\pi\)
\(60\) 1.60555 0.207276
\(61\) 8.29725 1.06235 0.531177 0.847261i \(-0.321750\pi\)
0.531177 + 0.847261i \(0.321750\pi\)
\(62\) −2.51221 −0.319050
\(63\) 4.14863 0.522678
\(64\) −0.0277564 −0.00346955
\(65\) 5.40473 0.670374
\(66\) 0 0
\(67\) −13.2111 −1.61399 −0.806997 0.590556i \(-0.798908\pi\)
−0.806997 + 0.590556i \(0.798908\pi\)
\(68\) −6.66083 −0.807745
\(69\) −5.21110 −0.627343
\(70\) −2.60555 −0.311423
\(71\) 5.21110 0.618444 0.309222 0.950990i \(-0.399931\pi\)
0.309222 + 0.950990i \(0.399931\pi\)
\(72\) 2.26447 0.266871
\(73\) −2.89252 −0.338544 −0.169272 0.985569i \(-0.554142\pi\)
−0.169272 + 0.985569i \(0.554142\pi\)
\(74\) 4.52894 0.526479
\(75\) −1.00000 −0.115470
\(76\) 2.01674 0.231336
\(77\) 0 0
\(78\) 3.39445 0.384346
\(79\) −1.25610 −0.141323 −0.0706613 0.997500i \(-0.522511\pi\)
−0.0706613 + 0.997500i \(0.522511\pi\)
\(80\) 1.78890 0.200005
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −13.7020 −1.50399 −0.751994 0.659170i \(-0.770908\pi\)
−0.751994 + 0.659170i \(0.770908\pi\)
\(84\) 6.66083 0.726756
\(85\) 4.14863 0.449982
\(86\) −4.18335 −0.451102
\(87\) 7.04115 0.754891
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −0.628052 −0.0662024
\(91\) 22.4222 2.35049
\(92\) −8.36669 −0.872288
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −1.25610 −0.128873
\(96\) 5.65246 0.576902
\(97\) 19.2111 1.95059 0.975296 0.220902i \(-0.0709002\pi\)
0.975296 + 0.220902i \(0.0709002\pi\)
\(98\) −6.41310 −0.647821
\(99\) 0 0
\(100\) −1.60555 −0.160555
\(101\) −1.25610 −0.124987 −0.0624935 0.998045i \(-0.519905\pi\)
−0.0624935 + 0.998045i \(0.519905\pi\)
\(102\) 2.60555 0.257988
\(103\) 1.21110 0.119333 0.0596667 0.998218i \(-0.480996\pi\)
0.0596667 + 0.998218i \(0.480996\pi\)
\(104\) 12.2389 1.20012
\(105\) −4.14863 −0.404864
\(106\) −7.04115 −0.683897
\(107\) 2.89252 0.279631 0.139815 0.990178i \(-0.455349\pi\)
0.139815 + 0.990178i \(0.455349\pi\)
\(108\) 1.60555 0.154494
\(109\) −16.5945 −1.58947 −0.794733 0.606960i \(-0.792389\pi\)
−0.794733 + 0.606960i \(0.792389\pi\)
\(110\) 0 0
\(111\) 7.21110 0.684448
\(112\) 7.42147 0.701263
\(113\) 11.2111 1.05465 0.527326 0.849663i \(-0.323195\pi\)
0.527326 + 0.849663i \(0.323195\pi\)
\(114\) −0.788897 −0.0738870
\(115\) 5.21110 0.485938
\(116\) 11.3049 1.04964
\(117\) 5.40473 0.499667
\(118\) 3.27284 0.301289
\(119\) 17.2111 1.57774
\(120\) −2.26447 −0.206717
\(121\) 0 0
\(122\) −5.21110 −0.471791
\(123\) 9.55336 0.861397
\(124\) −6.42221 −0.576731
\(125\) 1.00000 0.0894427
\(126\) −2.60555 −0.232121
\(127\) −1.63642 −0.145209 −0.0726044 0.997361i \(-0.523131\pi\)
−0.0726044 + 0.997361i \(0.523131\pi\)
\(128\) 11.3224 1.00076
\(129\) −6.66083 −0.586454
\(130\) −3.39445 −0.297713
\(131\) −5.78505 −0.505442 −0.252721 0.967539i \(-0.581325\pi\)
−0.252721 + 0.967539i \(0.581325\pi\)
\(132\) 0 0
\(133\) −5.21110 −0.451860
\(134\) 8.29725 0.716774
\(135\) −1.00000 −0.0860663
\(136\) 9.39445 0.805567
\(137\) −16.4222 −1.40304 −0.701522 0.712648i \(-0.747496\pi\)
−0.701522 + 0.712648i \(0.747496\pi\)
\(138\) 3.27284 0.278603
\(139\) 9.55336 0.810305 0.405153 0.914249i \(-0.367218\pi\)
0.405153 + 0.914249i \(0.367218\pi\)
\(140\) −6.66083 −0.562943
\(141\) 0 0
\(142\) −3.27284 −0.274651
\(143\) 0 0
\(144\) 1.78890 0.149075
\(145\) −7.04115 −0.584736
\(146\) 1.81665 0.150347
\(147\) −10.2111 −0.842198
\(148\) 11.5778 0.951689
\(149\) 22.8750 1.87399 0.936997 0.349336i \(-0.113593\pi\)
0.936997 + 0.349336i \(0.113593\pi\)
\(150\) 0.628052 0.0512802
\(151\) 15.3384 1.24822 0.624111 0.781336i \(-0.285461\pi\)
0.624111 + 0.781336i \(0.285461\pi\)
\(152\) −2.84441 −0.230712
\(153\) 4.14863 0.335397
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 8.67757 0.694762
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0.788897 0.0627613
\(159\) −11.2111 −0.889098
\(160\) −5.65246 −0.446866
\(161\) 21.6189 1.70381
\(162\) −0.628052 −0.0493444
\(163\) 13.2111 1.03477 0.517387 0.855752i \(-0.326905\pi\)
0.517387 + 0.855752i \(0.326905\pi\)
\(164\) 15.3384 1.19773
\(165\) 0 0
\(166\) 8.60555 0.667920
\(167\) −19.4870 −1.50795 −0.753976 0.656902i \(-0.771866\pi\)
−0.753976 + 0.656902i \(0.771866\pi\)
\(168\) −9.39445 −0.724797
\(169\) 16.2111 1.24701
\(170\) −2.60555 −0.199837
\(171\) −1.25610 −0.0960566
\(172\) −10.6943 −0.815433
\(173\) −12.4459 −0.946243 −0.473121 0.880997i \(-0.656873\pi\)
−0.473121 + 0.880997i \(0.656873\pi\)
\(174\) −4.42221 −0.335247
\(175\) 4.14863 0.313607
\(176\) 0 0
\(177\) 5.21110 0.391690
\(178\) −3.76831 −0.282447
\(179\) −22.4222 −1.67591 −0.837957 0.545736i \(-0.816250\pi\)
−0.837957 + 0.545736i \(0.816250\pi\)
\(180\) −1.60555 −0.119671
\(181\) 19.2111 1.42795 0.713975 0.700171i \(-0.246893\pi\)
0.713975 + 0.700171i \(0.246893\pi\)
\(182\) −14.0823 −1.04385
\(183\) −8.29725 −0.613351
\(184\) 11.8004 0.869937
\(185\) −7.21110 −0.530171
\(186\) 2.51221 0.184204
\(187\) 0 0
\(188\) 0 0
\(189\) −4.14863 −0.301768
\(190\) 0.788897 0.0572326
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0.0277564 0.00200314
\(193\) 11.1898 0.805458 0.402729 0.915319i \(-0.368062\pi\)
0.402729 + 0.915319i \(0.368062\pi\)
\(194\) −12.0656 −0.866257
\(195\) −5.40473 −0.387041
\(196\) −16.3944 −1.17103
\(197\) 9.17304 0.653552 0.326776 0.945102i \(-0.394038\pi\)
0.326776 + 0.945102i \(0.394038\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 2.26447 0.160122
\(201\) 13.2111 0.931839
\(202\) 0.788897 0.0555066
\(203\) −29.2111 −2.05022
\(204\) 6.66083 0.466352
\(205\) −9.55336 −0.667235
\(206\) −0.760635 −0.0529959
\(207\) 5.21110 0.362197
\(208\) 9.66851 0.670390
\(209\) 0 0
\(210\) 2.60555 0.179800
\(211\) 3.76831 0.259421 0.129711 0.991552i \(-0.458595\pi\)
0.129711 + 0.991552i \(0.458595\pi\)
\(212\) −18.0000 −1.23625
\(213\) −5.21110 −0.357059
\(214\) −1.81665 −0.124184
\(215\) 6.66083 0.454265
\(216\) −2.26447 −0.154078
\(217\) 16.5945 1.12651
\(218\) 10.4222 0.705881
\(219\) 2.89252 0.195459
\(220\) 0 0
\(221\) 22.4222 1.50828
\(222\) −4.52894 −0.303963
\(223\) −2.42221 −0.162203 −0.0811014 0.996706i \(-0.525844\pi\)
−0.0811014 + 0.996706i \(0.525844\pi\)
\(224\) −23.4500 −1.56682
\(225\) 1.00000 0.0666667
\(226\) −7.04115 −0.468370
\(227\) −2.89252 −0.191984 −0.0959918 0.995382i \(-0.530602\pi\)
−0.0959918 + 0.995382i \(0.530602\pi\)
\(228\) −2.01674 −0.133562
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −3.27284 −0.215805
\(231\) 0 0
\(232\) −15.9445 −1.04681
\(233\) −25.7675 −1.68809 −0.844044 0.536274i \(-0.819831\pi\)
−0.844044 + 0.536274i \(0.819831\pi\)
\(234\) −3.39445 −0.221902
\(235\) 0 0
\(236\) 8.36669 0.544625
\(237\) 1.25610 0.0815927
\(238\) −10.8095 −0.700673
\(239\) −8.29725 −0.536705 −0.268352 0.963321i \(-0.586479\pi\)
−0.268352 + 0.963321i \(0.586479\pi\)
\(240\) −1.78890 −0.115473
\(241\) 3.27284 0.210822 0.105411 0.994429i \(-0.466384\pi\)
0.105411 + 0.994429i \(0.466384\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −13.3217 −0.852832
\(245\) 10.2111 0.652363
\(246\) −6.00000 −0.382546
\(247\) −6.78890 −0.431967
\(248\) 9.05789 0.575176
\(249\) 13.7020 0.868328
\(250\) −0.628052 −0.0397215
\(251\) −27.6333 −1.74420 −0.872099 0.489329i \(-0.837242\pi\)
−0.872099 + 0.489329i \(0.837242\pi\)
\(252\) −6.66083 −0.419593
\(253\) 0 0
\(254\) 1.02776 0.0644872
\(255\) −4.14863 −0.259797
\(256\) −7.05551 −0.440970
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 4.18335 0.260444
\(259\) −29.9162 −1.85890
\(260\) −8.67757 −0.538160
\(261\) −7.04115 −0.435836
\(262\) 3.63331 0.224466
\(263\) 7.91694 0.488179 0.244090 0.969753i \(-0.421511\pi\)
0.244090 + 0.969753i \(0.421511\pi\)
\(264\) 0 0
\(265\) 11.2111 0.688693
\(266\) 3.27284 0.200671
\(267\) −6.00000 −0.367194
\(268\) 21.2111 1.29567
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0.628052 0.0382220
\(271\) 22.8750 1.38956 0.694779 0.719223i \(-0.255502\pi\)
0.694779 + 0.719223i \(0.255502\pi\)
\(272\) 7.42147 0.449993
\(273\) −22.4222 −1.35705
\(274\) 10.3140 0.623091
\(275\) 0 0
\(276\) 8.36669 0.503616
\(277\) 10.4291 0.626626 0.313313 0.949650i \(-0.398561\pi\)
0.313313 + 0.949650i \(0.398561\pi\)
\(278\) −6.00000 −0.359856
\(279\) 4.00000 0.239474
\(280\) 9.39445 0.561426
\(281\) −26.1479 −1.55985 −0.779925 0.625873i \(-0.784743\pi\)
−0.779925 + 0.625873i \(0.784743\pi\)
\(282\) 0 0
\(283\) −31.5526 −1.87561 −0.937803 0.347167i \(-0.887144\pi\)
−0.937803 + 0.347167i \(0.887144\pi\)
\(284\) −8.36669 −0.496472
\(285\) 1.25610 0.0744051
\(286\) 0 0
\(287\) −39.6333 −2.33948
\(288\) −5.65246 −0.333075
\(289\) 0.211103 0.0124178
\(290\) 4.42221 0.259681
\(291\) −19.2111 −1.12617
\(292\) 4.64409 0.271775
\(293\) −20.7431 −1.21183 −0.605913 0.795531i \(-0.707192\pi\)
−0.605913 + 0.795531i \(0.707192\pi\)
\(294\) 6.41310 0.374020
\(295\) −5.21110 −0.303402
\(296\) −16.3293 −0.949124
\(297\) 0 0
\(298\) −14.3667 −0.832240
\(299\) 28.1646 1.62880
\(300\) 1.60555 0.0926965
\(301\) 27.6333 1.59276
\(302\) −9.63331 −0.554335
\(303\) 1.25610 0.0721612
\(304\) −2.24704 −0.128877
\(305\) 8.29725 0.475099
\(306\) −2.60555 −0.148949
\(307\) −7.42147 −0.423566 −0.211783 0.977317i \(-0.567927\pi\)
−0.211783 + 0.977317i \(0.567927\pi\)
\(308\) 0 0
\(309\) −1.21110 −0.0688972
\(310\) −2.51221 −0.142684
\(311\) 6.78890 0.384963 0.192482 0.981301i \(-0.438346\pi\)
0.192482 + 0.981301i \(0.438346\pi\)
\(312\) −12.2389 −0.692889
\(313\) 15.2111 0.859782 0.429891 0.902881i \(-0.358552\pi\)
0.429891 + 0.902881i \(0.358552\pi\)
\(314\) −6.28052 −0.354430
\(315\) 4.14863 0.233749
\(316\) 2.01674 0.113450
\(317\) −11.2111 −0.629678 −0.314839 0.949145i \(-0.601951\pi\)
−0.314839 + 0.949145i \(0.601951\pi\)
\(318\) 7.04115 0.394848
\(319\) 0 0
\(320\) −0.0277564 −0.00155163
\(321\) −2.89252 −0.161445
\(322\) −13.5778 −0.756661
\(323\) −5.21110 −0.289954
\(324\) −1.60555 −0.0891973
\(325\) 5.40473 0.299800
\(326\) −8.29725 −0.459542
\(327\) 16.5945 0.917678
\(328\) −21.6333 −1.19450
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 21.9992 1.20736
\(333\) −7.21110 −0.395166
\(334\) 12.2389 0.669681
\(335\) −13.2111 −0.721800
\(336\) −7.42147 −0.404874
\(337\) 8.67757 0.472697 0.236349 0.971668i \(-0.424049\pi\)
0.236349 + 0.971668i \(0.424049\pi\)
\(338\) −10.1814 −0.553796
\(339\) −11.2111 −0.608904
\(340\) −6.66083 −0.361234
\(341\) 0 0
\(342\) 0.788897 0.0426587
\(343\) 13.3217 0.719302
\(344\) 15.0833 0.813235
\(345\) −5.21110 −0.280556
\(346\) 7.81665 0.420226
\(347\) 29.5359 1.58557 0.792784 0.609503i \(-0.208631\pi\)
0.792784 + 0.609503i \(0.208631\pi\)
\(348\) −11.3049 −0.606008
\(349\) −13.3217 −0.713092 −0.356546 0.934278i \(-0.616046\pi\)
−0.356546 + 0.934278i \(0.616046\pi\)
\(350\) −2.60555 −0.139273
\(351\) −5.40473 −0.288483
\(352\) 0 0
\(353\) 12.7889 0.680684 0.340342 0.940302i \(-0.389457\pi\)
0.340342 + 0.940302i \(0.389457\pi\)
\(354\) −3.27284 −0.173950
\(355\) 5.21110 0.276577
\(356\) −9.63331 −0.510564
\(357\) −17.2111 −0.910908
\(358\) 14.0823 0.744273
\(359\) 15.8339 0.835680 0.417840 0.908521i \(-0.362787\pi\)
0.417840 + 0.908521i \(0.362787\pi\)
\(360\) 2.26447 0.119348
\(361\) −17.4222 −0.916958
\(362\) −12.0656 −0.634152
\(363\) 0 0
\(364\) −36.0000 −1.88691
\(365\) −2.89252 −0.151402
\(366\) 5.21110 0.272389
\(367\) 18.4222 0.961631 0.480816 0.876822i \(-0.340341\pi\)
0.480816 + 0.876822i \(0.340341\pi\)
\(368\) 9.32213 0.485950
\(369\) −9.55336 −0.497328
\(370\) 4.52894 0.235449
\(371\) 46.5107 2.41471
\(372\) 6.42221 0.332976
\(373\) −7.91694 −0.409923 −0.204962 0.978770i \(-0.565707\pi\)
−0.204962 + 0.978770i \(0.565707\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −38.0555 −1.95996
\(378\) 2.60555 0.134015
\(379\) −14.4222 −0.740819 −0.370409 0.928869i \(-0.620783\pi\)
−0.370409 + 0.928869i \(0.620783\pi\)
\(380\) 2.01674 0.103456
\(381\) 1.63642 0.0838364
\(382\) 0 0
\(383\) −5.21110 −0.266275 −0.133137 0.991098i \(-0.542505\pi\)
−0.133137 + 0.991098i \(0.542505\pi\)
\(384\) −11.3224 −0.577792
\(385\) 0 0
\(386\) −7.02776 −0.357703
\(387\) 6.66083 0.338589
\(388\) −30.8444 −1.56589
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 3.39445 0.171885
\(391\) 21.6189 1.09332
\(392\) 23.1228 1.16788
\(393\) 5.78505 0.291817
\(394\) −5.76114 −0.290242
\(395\) −1.25610 −0.0632014
\(396\) 0 0
\(397\) −13.6333 −0.684236 −0.342118 0.939657i \(-0.611144\pi\)
−0.342118 + 0.939657i \(0.611144\pi\)
\(398\) 5.02441 0.251851
\(399\) 5.21110 0.260881
\(400\) 1.78890 0.0894449
\(401\) 28.4222 1.41934 0.709669 0.704536i \(-0.248845\pi\)
0.709669 + 0.704536i \(0.248845\pi\)
\(402\) −8.29725 −0.413829
\(403\) 21.6189 1.07692
\(404\) 2.01674 0.100336
\(405\) 1.00000 0.0496904
\(406\) 18.3461 0.910501
\(407\) 0 0
\(408\) −9.39445 −0.465095
\(409\) −7.53662 −0.372662 −0.186331 0.982487i \(-0.559660\pi\)
−0.186331 + 0.982487i \(0.559660\pi\)
\(410\) 6.00000 0.296319
\(411\) 16.4222 0.810048
\(412\) −1.94449 −0.0957980
\(413\) −21.6189 −1.06380
\(414\) −3.27284 −0.160851
\(415\) −13.7020 −0.672604
\(416\) −30.5500 −1.49784
\(417\) −9.55336 −0.467830
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 6.66083 0.325015
\(421\) 39.2111 1.91103 0.955516 0.294939i \(-0.0952993\pi\)
0.955516 + 0.294939i \(0.0952993\pi\)
\(422\) −2.36669 −0.115209
\(423\) 0 0
\(424\) 25.3872 1.23291
\(425\) 4.14863 0.201238
\(426\) 3.27284 0.158570
\(427\) 34.4222 1.66581
\(428\) −4.64409 −0.224481
\(429\) 0 0
\(430\) −4.18335 −0.201739
\(431\) −2.51221 −0.121009 −0.0605044 0.998168i \(-0.519271\pi\)
−0.0605044 + 0.998168i \(0.519271\pi\)
\(432\) −1.78890 −0.0860684
\(433\) −4.78890 −0.230140 −0.115070 0.993357i \(-0.536709\pi\)
−0.115070 + 0.993357i \(0.536709\pi\)
\(434\) −10.4222 −0.500282
\(435\) 7.04115 0.337597
\(436\) 26.6433 1.27598
\(437\) −6.54568 −0.313122
\(438\) −1.81665 −0.0868031
\(439\) 2.01674 0.0962536 0.0481268 0.998841i \(-0.484675\pi\)
0.0481268 + 0.998841i \(0.484675\pi\)
\(440\) 0 0
\(441\) 10.2111 0.486243
\(442\) −14.0823 −0.669827
\(443\) 27.6333 1.31290 0.656449 0.754370i \(-0.272058\pi\)
0.656449 + 0.754370i \(0.272058\pi\)
\(444\) −11.5778 −0.549458
\(445\) 6.00000 0.284427
\(446\) 1.52127 0.0720342
\(447\) −22.8750 −1.08195
\(448\) −0.115151 −0.00544037
\(449\) 16.4222 0.775012 0.387506 0.921867i \(-0.373337\pi\)
0.387506 + 0.921867i \(0.373337\pi\)
\(450\) −0.628052 −0.0296066
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) −15.3384 −0.720661
\(454\) 1.81665 0.0852598
\(455\) 22.4222 1.05117
\(456\) 2.84441 0.133202
\(457\) −2.13189 −0.0997255 −0.0498628 0.998756i \(-0.515878\pi\)
−0.0498628 + 0.998756i \(0.515878\pi\)
\(458\) −6.28052 −0.293469
\(459\) −4.14863 −0.193641
\(460\) −8.36669 −0.390099
\(461\) 6.28052 0.292513 0.146256 0.989247i \(-0.453278\pi\)
0.146256 + 0.989247i \(0.453278\pi\)
\(462\) 0 0
\(463\) −26.4222 −1.22794 −0.613972 0.789328i \(-0.710429\pi\)
−0.613972 + 0.789328i \(0.710429\pi\)
\(464\) −12.5959 −0.584750
\(465\) −4.00000 −0.185496
\(466\) 16.1833 0.749679
\(467\) −17.2111 −0.796435 −0.398217 0.917291i \(-0.630371\pi\)
−0.398217 + 0.917291i \(0.630371\pi\)
\(468\) −8.67757 −0.401121
\(469\) −54.8079 −2.53080
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) −11.8004 −0.543157
\(473\) 0 0
\(474\) −0.788897 −0.0362353
\(475\) −1.25610 −0.0576340
\(476\) −27.6333 −1.26657
\(477\) 11.2111 0.513321
\(478\) 5.21110 0.238350
\(479\) 3.27284 0.149540 0.0747700 0.997201i \(-0.476178\pi\)
0.0747700 + 0.997201i \(0.476178\pi\)
\(480\) 5.65246 0.257998
\(481\) −38.9741 −1.77706
\(482\) −2.05551 −0.0936260
\(483\) −21.6189 −0.983695
\(484\) 0 0
\(485\) 19.2111 0.872331
\(486\) 0.628052 0.0284890
\(487\) −1.21110 −0.0548803 −0.0274401 0.999623i \(-0.508736\pi\)
−0.0274401 + 0.999623i \(0.508736\pi\)
\(488\) 18.7889 0.850533
\(489\) −13.2111 −0.597427
\(490\) −6.41310 −0.289714
\(491\) −40.7256 −1.83792 −0.918961 0.394348i \(-0.870970\pi\)
−0.918961 + 0.394348i \(0.870970\pi\)
\(492\) −15.3384 −0.691509
\(493\) −29.2111 −1.31560
\(494\) 4.26378 0.191836
\(495\) 0 0
\(496\) 7.15559 0.321295
\(497\) 21.6189 0.969741
\(498\) −8.60555 −0.385624
\(499\) 14.4222 0.645627 0.322813 0.946463i \(-0.395371\pi\)
0.322813 + 0.946463i \(0.395371\pi\)
\(500\) −1.60555 −0.0718024
\(501\) 19.4870 0.870616
\(502\) 17.3551 0.774598
\(503\) 18.7264 0.834969 0.417484 0.908684i \(-0.362912\pi\)
0.417484 + 0.908684i \(0.362912\pi\)
\(504\) 9.39445 0.418462
\(505\) −1.25610 −0.0558959
\(506\) 0 0
\(507\) −16.2111 −0.719960
\(508\) 2.62736 0.116570
\(509\) 16.4222 0.727901 0.363951 0.931418i \(-0.381428\pi\)
0.363951 + 0.931418i \(0.381428\pi\)
\(510\) 2.60555 0.115376
\(511\) −12.0000 −0.530849
\(512\) −18.2135 −0.804930
\(513\) 1.25610 0.0554583
\(514\) −3.76831 −0.166213
\(515\) 1.21110 0.0533676
\(516\) 10.6943 0.470791
\(517\) 0 0
\(518\) 18.7889 0.825537
\(519\) 12.4459 0.546313
\(520\) 12.2389 0.536709
\(521\) 7.57779 0.331989 0.165995 0.986127i \(-0.446917\pi\)
0.165995 + 0.986127i \(0.446917\pi\)
\(522\) 4.42221 0.193555
\(523\) −25.7675 −1.12674 −0.563368 0.826206i \(-0.690495\pi\)
−0.563368 + 0.826206i \(0.690495\pi\)
\(524\) 9.28819 0.405756
\(525\) −4.14863 −0.181061
\(526\) −4.97224 −0.216800
\(527\) 16.5945 0.722868
\(528\) 0 0
\(529\) 4.15559 0.180678
\(530\) −7.04115 −0.305848
\(531\) −5.21110 −0.226143
\(532\) 8.36669 0.362742
\(533\) −51.6333 −2.23649
\(534\) 3.76831 0.163071
\(535\) 2.89252 0.125055
\(536\) −29.9162 −1.29218
\(537\) 22.4222 0.967590
\(538\) 11.3049 0.487390
\(539\) 0 0
\(540\) 1.60555 0.0690919
\(541\) 33.1890 1.42691 0.713454 0.700703i \(-0.247130\pi\)
0.713454 + 0.700703i \(0.247130\pi\)
\(542\) −14.3667 −0.617102
\(543\) −19.2111 −0.824427
\(544\) −23.4500 −1.00541
\(545\) −16.5945 −0.710831
\(546\) 14.0823 0.602667
\(547\) −25.7675 −1.10174 −0.550870 0.834591i \(-0.685704\pi\)
−0.550870 + 0.834591i \(0.685704\pi\)
\(548\) 26.3667 1.12633
\(549\) 8.29725 0.354118
\(550\) 0 0
\(551\) 8.84441 0.376785
\(552\) −11.8004 −0.502258
\(553\) −5.21110 −0.221599
\(554\) −6.55004 −0.278284
\(555\) 7.21110 0.306094
\(556\) −15.3384 −0.650493
\(557\) −3.38799 −0.143554 −0.0717769 0.997421i \(-0.522867\pi\)
−0.0717769 + 0.997421i \(0.522867\pi\)
\(558\) −2.51221 −0.106350
\(559\) 36.0000 1.52264
\(560\) 7.42147 0.313614
\(561\) 0 0
\(562\) 16.4222 0.692729
\(563\) 22.7599 0.959214 0.479607 0.877483i \(-0.340779\pi\)
0.479607 + 0.877483i \(0.340779\pi\)
\(564\) 0 0
\(565\) 11.2111 0.471655
\(566\) 19.8167 0.832956
\(567\) 4.14863 0.174226
\(568\) 11.8004 0.495134
\(569\) −1.25610 −0.0526586 −0.0263293 0.999653i \(-0.508382\pi\)
−0.0263293 + 0.999653i \(0.508382\pi\)
\(570\) −0.788897 −0.0330433
\(571\) −31.1723 −1.30452 −0.652260 0.757996i \(-0.726179\pi\)
−0.652260 + 0.757996i \(0.726179\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 24.8918 1.03896
\(575\) 5.21110 0.217318
\(576\) −0.0277564 −0.00115652
\(577\) 11.5778 0.481990 0.240995 0.970526i \(-0.422526\pi\)
0.240995 + 0.970526i \(0.422526\pi\)
\(578\) −0.132583 −0.00551474
\(579\) −11.1898 −0.465031
\(580\) 11.3049 0.469412
\(581\) −56.8444 −2.35830
\(582\) 12.0656 0.500134
\(583\) 0 0
\(584\) −6.55004 −0.271043
\(585\) 5.40473 0.223458
\(586\) 13.0278 0.538172
\(587\) 17.2111 0.710378 0.355189 0.934794i \(-0.384416\pi\)
0.355189 + 0.934794i \(0.384416\pi\)
\(588\) 16.3944 0.676096
\(589\) −5.02441 −0.207027
\(590\) 3.27284 0.134741
\(591\) −9.17304 −0.377328
\(592\) −12.8999 −0.530184
\(593\) −9.93367 −0.407927 −0.203964 0.978978i \(-0.565382\pi\)
−0.203964 + 0.978978i \(0.565382\pi\)
\(594\) 0 0
\(595\) 17.2111 0.705586
\(596\) −36.7270 −1.50440
\(597\) 8.00000 0.327418
\(598\) −17.6888 −0.723350
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) −2.26447 −0.0924467
\(601\) −43.9985 −1.79474 −0.897368 0.441284i \(-0.854523\pi\)
−0.897368 + 0.441284i \(0.854523\pi\)
\(602\) −17.3551 −0.707343
\(603\) −13.2111 −0.537998
\(604\) −24.6266 −1.00204
\(605\) 0 0
\(606\) −0.788897 −0.0320468
\(607\) −18.2309 −0.739970 −0.369985 0.929038i \(-0.620637\pi\)
−0.369985 + 0.929038i \(0.620637\pi\)
\(608\) 7.10008 0.287946
\(609\) 29.2111 1.18369
\(610\) −5.21110 −0.210991
\(611\) 0 0
\(612\) −6.66083 −0.269248
\(613\) 41.8666 1.69098 0.845488 0.533995i \(-0.179310\pi\)
0.845488 + 0.533995i \(0.179310\pi\)
\(614\) 4.66106 0.188105
\(615\) 9.55336 0.385229
\(616\) 0 0
\(617\) 24.7889 0.997963 0.498982 0.866613i \(-0.333707\pi\)
0.498982 + 0.866613i \(0.333707\pi\)
\(618\) 0.760635 0.0305972
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) −6.42221 −0.257922
\(621\) −5.21110 −0.209114
\(622\) −4.26378 −0.170962
\(623\) 24.8918 0.997267
\(624\) −9.66851 −0.387050
\(625\) 1.00000 0.0400000
\(626\) −9.55336 −0.381829
\(627\) 0 0
\(628\) −16.0555 −0.640685
\(629\) −29.9162 −1.19284
\(630\) −2.60555 −0.103808
\(631\) −28.8444 −1.14828 −0.574139 0.818758i \(-0.694663\pi\)
−0.574139 + 0.818758i \(0.694663\pi\)
\(632\) −2.84441 −0.113145
\(633\) −3.76831 −0.149777
\(634\) 7.04115 0.279640
\(635\) −1.63642 −0.0649394
\(636\) 18.0000 0.713746
\(637\) 55.1882 2.18664
\(638\) 0 0
\(639\) 5.21110 0.206148
\(640\) 11.3224 0.447556
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 1.81665 0.0716976
\(643\) −0.366692 −0.0144609 −0.00723047 0.999974i \(-0.502302\pi\)
−0.00723047 + 0.999974i \(0.502302\pi\)
\(644\) −34.7103 −1.36778
\(645\) −6.66083 −0.262270
\(646\) 3.27284 0.128768
\(647\) 5.21110 0.204870 0.102435 0.994740i \(-0.467337\pi\)
0.102435 + 0.994740i \(0.467337\pi\)
\(648\) 2.26447 0.0889569
\(649\) 0 0
\(650\) −3.39445 −0.133141
\(651\) −16.5945 −0.650390
\(652\) −21.2111 −0.830691
\(653\) 4.42221 0.173054 0.0865271 0.996249i \(-0.472423\pi\)
0.0865271 + 0.996249i \(0.472423\pi\)
\(654\) −10.4222 −0.407540
\(655\) −5.78505 −0.226040
\(656\) −17.0900 −0.667251
\(657\) −2.89252 −0.112848
\(658\) 0 0
\(659\) −33.1890 −1.29286 −0.646430 0.762973i \(-0.723739\pi\)
−0.646430 + 0.762973i \(0.723739\pi\)
\(660\) 0 0
\(661\) 30.8444 1.19971 0.599854 0.800109i \(-0.295225\pi\)
0.599854 + 0.800109i \(0.295225\pi\)
\(662\) −5.02441 −0.195279
\(663\) −22.4222 −0.870806
\(664\) −31.0278 −1.20411
\(665\) −5.21110 −0.202078
\(666\) 4.52894 0.175493
\(667\) −36.6922 −1.42073
\(668\) 31.2874 1.21055
\(669\) 2.42221 0.0936479
\(670\) 8.29725 0.320551
\(671\) 0 0
\(672\) 23.4500 0.904602
\(673\) −15.4536 −0.595691 −0.297845 0.954614i \(-0.596268\pi\)
−0.297845 + 0.954614i \(0.596268\pi\)
\(674\) −5.44996 −0.209925
\(675\) −1.00000 −0.0384900
\(676\) −26.0278 −1.00107
\(677\) 3.38799 0.130211 0.0651056 0.997878i \(-0.479262\pi\)
0.0651056 + 0.997878i \(0.479262\pi\)
\(678\) 7.04115 0.270414
\(679\) 79.6997 3.05859
\(680\) 9.39445 0.360261
\(681\) 2.89252 0.110842
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 2.01674 0.0771119
\(685\) −16.4222 −0.627460
\(686\) −8.36669 −0.319442
\(687\) −10.0000 −0.381524
\(688\) 11.9155 0.454276
\(689\) 60.5930 2.30841
\(690\) 3.27284 0.124595
\(691\) 9.57779 0.364356 0.182178 0.983266i \(-0.441685\pi\)
0.182178 + 0.983266i \(0.441685\pi\)
\(692\) 19.9825 0.759621
\(693\) 0 0
\(694\) −18.5500 −0.704150
\(695\) 9.55336 0.362379
\(696\) 15.9445 0.604374
\(697\) −39.6333 −1.50122
\(698\) 8.36669 0.316684
\(699\) 25.7675 0.974618
\(700\) −6.66083 −0.251756
\(701\) 7.04115 0.265941 0.132970 0.991120i \(-0.457549\pi\)
0.132970 + 0.991120i \(0.457549\pi\)
\(702\) 3.39445 0.128115
\(703\) 9.05789 0.341625
\(704\) 0 0
\(705\) 0 0
\(706\) −8.03209 −0.302292
\(707\) −5.21110 −0.195984
\(708\) −8.36669 −0.314440
\(709\) −5.63331 −0.211563 −0.105782 0.994389i \(-0.533734\pi\)
−0.105782 + 0.994389i \(0.533734\pi\)
\(710\) −3.27284 −0.122828
\(711\) −1.25610 −0.0471075
\(712\) 13.5868 0.509188
\(713\) 20.8444 0.780629
\(714\) 10.8095 0.404534
\(715\) 0 0
\(716\) 36.0000 1.34538
\(717\) 8.29725 0.309867
\(718\) −9.94449 −0.371125
\(719\) −10.4222 −0.388683 −0.194341 0.980934i \(-0.562257\pi\)
−0.194341 + 0.980934i \(0.562257\pi\)
\(720\) 1.78890 0.0666683
\(721\) 5.02441 0.187119
\(722\) 10.9420 0.407221
\(723\) −3.27284 −0.121718
\(724\) −30.8444 −1.14632
\(725\) −7.04115 −0.261502
\(726\) 0 0
\(727\) −21.5778 −0.800276 −0.400138 0.916455i \(-0.631038\pi\)
−0.400138 + 0.916455i \(0.631038\pi\)
\(728\) 50.7745 1.88183
\(729\) 1.00000 0.0370370
\(730\) 1.81665 0.0672374
\(731\) 27.6333 1.02205
\(732\) 13.3217 0.492383
\(733\) 16.2142 0.598885 0.299442 0.954114i \(-0.403199\pi\)
0.299442 + 0.954114i \(0.403199\pi\)
\(734\) −11.5701 −0.427060
\(735\) −10.2111 −0.376642
\(736\) −29.4556 −1.08575
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) −42.7424 −1.57230 −0.786152 0.618034i \(-0.787930\pi\)
−0.786152 + 0.618034i \(0.787930\pi\)
\(740\) 11.5778 0.425608
\(741\) 6.78890 0.249396
\(742\) −29.2111 −1.07237
\(743\) 33.5693 1.23154 0.615770 0.787926i \(-0.288845\pi\)
0.615770 + 0.787926i \(0.288845\pi\)
\(744\) −9.05789 −0.332078
\(745\) 22.8750 0.838076
\(746\) 4.97224 0.182047
\(747\) −13.7020 −0.501329
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0.628052 0.0229332
\(751\) 36.8444 1.34447 0.672236 0.740337i \(-0.265334\pi\)
0.672236 + 0.740337i \(0.265334\pi\)
\(752\) 0 0
\(753\) 27.6333 1.00701
\(754\) 23.9008 0.870417
\(755\) 15.3384 0.558222
\(756\) 6.66083 0.242252
\(757\) 20.4222 0.742258 0.371129 0.928581i \(-0.378971\pi\)
0.371129 + 0.928581i \(0.378971\pi\)
\(758\) 9.05789 0.328997
\(759\) 0 0
\(760\) −2.84441 −0.103178
\(761\) 26.9085 0.975432 0.487716 0.873002i \(-0.337830\pi\)
0.487716 + 0.873002i \(0.337830\pi\)
\(762\) −1.02776 −0.0372317
\(763\) −68.8444 −2.49233
\(764\) 0 0
\(765\) 4.14863 0.149994
\(766\) 3.27284 0.118253
\(767\) −28.1646 −1.01696
\(768\) 7.05551 0.254594
\(769\) −10.8095 −0.389799 −0.194900 0.980823i \(-0.562438\pi\)
−0.194900 + 0.980823i \(0.562438\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) −17.9658 −0.646602
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) −4.18335 −0.150367
\(775\) 4.00000 0.143684
\(776\) 43.5030 1.56167
\(777\) 29.9162 1.07324
\(778\) −11.3049 −0.405301
\(779\) 12.0000 0.429945
\(780\) 8.67757 0.310707
\(781\) 0 0
\(782\) −13.5778 −0.485541
\(783\) 7.04115 0.251630
\(784\) 18.2666 0.652379
\(785\) 10.0000 0.356915
\(786\) −3.63331 −0.129596
\(787\) 6.66083 0.237433 0.118717 0.992928i \(-0.462122\pi\)
0.118717 + 0.992928i \(0.462122\pi\)
\(788\) −14.7278 −0.524656
\(789\) −7.91694 −0.281850
\(790\) 0.788897 0.0280677
\(791\) 46.5107 1.65373
\(792\) 0 0
\(793\) 44.8444 1.59247
\(794\) 8.56242 0.303869
\(795\) −11.2111 −0.397617
\(796\) 12.8444 0.455258
\(797\) 54.4777 1.92970 0.964850 0.262802i \(-0.0846465\pi\)
0.964850 + 0.262802i \(0.0846465\pi\)
\(798\) −3.27284 −0.115857
\(799\) 0 0
\(800\) −5.65246 −0.199845
\(801\) 6.00000 0.212000
\(802\) −17.8506 −0.630327
\(803\) 0 0
\(804\) −21.2111 −0.748058
\(805\) 21.6189 0.761967
\(806\) −13.5778 −0.478257
\(807\) 18.0000 0.633630
\(808\) −2.84441 −0.100066
\(809\) −12.0656 −0.424203 −0.212101 0.977248i \(-0.568031\pi\)
−0.212101 + 0.977248i \(0.568031\pi\)
\(810\) −0.628052 −0.0220675
\(811\) −51.8003 −1.81895 −0.909477 0.415755i \(-0.863517\pi\)
−0.909477 + 0.415755i \(0.863517\pi\)
\(812\) 46.8999 1.64586
\(813\) −22.8750 −0.802262
\(814\) 0 0
\(815\) 13.2111 0.462765
\(816\) −7.42147 −0.259803
\(817\) −8.36669 −0.292714
\(818\) 4.73338 0.165499
\(819\) 22.4222 0.783495
\(820\) 15.3384 0.535640
\(821\) 37.7180 1.31637 0.658183 0.752858i \(-0.271325\pi\)
0.658183 + 0.752858i \(0.271325\pi\)
\(822\) −10.3140 −0.359742
\(823\) 4.36669 0.152213 0.0761067 0.997100i \(-0.475751\pi\)
0.0761067 + 0.997100i \(0.475751\pi\)
\(824\) 2.74251 0.0955398
\(825\) 0 0
\(826\) 13.5778 0.472432
\(827\) −13.7020 −0.476465 −0.238232 0.971208i \(-0.576568\pi\)
−0.238232 + 0.971208i \(0.576568\pi\)
\(828\) −8.36669 −0.290763
\(829\) −36.0555 −1.25226 −0.626130 0.779719i \(-0.715362\pi\)
−0.626130 + 0.779719i \(0.715362\pi\)
\(830\) 8.60555 0.298703
\(831\) −10.4291 −0.361783
\(832\) −0.150016 −0.00520086
\(833\) 42.3621 1.46776
\(834\) 6.00000 0.207763
\(835\) −19.4870 −0.674376
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 7.53662 0.260348
\(839\) −30.7889 −1.06295 −0.531475 0.847074i \(-0.678362\pi\)
−0.531475 + 0.847074i \(0.678362\pi\)
\(840\) −9.39445 −0.324139
\(841\) 20.5778 0.709579
\(842\) −24.6266 −0.848688
\(843\) 26.1479 0.900580
\(844\) −6.05021 −0.208257
\(845\) 16.2111 0.557679
\(846\) 0 0
\(847\) 0 0
\(848\) 20.0555 0.688709
\(849\) 31.5526 1.08288
\(850\) −2.60555 −0.0893697
\(851\) −37.5778 −1.28815
\(852\) 8.36669 0.286638
\(853\) −19.4870 −0.667223 −0.333612 0.942711i \(-0.608267\pi\)
−0.333612 + 0.942711i \(0.608267\pi\)
\(854\) −21.6189 −0.739784
\(855\) −1.25610 −0.0429578
\(856\) 6.55004 0.223876
\(857\) −40.6105 −1.38723 −0.693614 0.720347i \(-0.743983\pi\)
−0.693614 + 0.720347i \(0.743983\pi\)
\(858\) 0 0
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) −10.6943 −0.364673
\(861\) 39.6333 1.35070
\(862\) 1.57779 0.0537399
\(863\) −15.6333 −0.532164 −0.266082 0.963950i \(-0.585729\pi\)
−0.266082 + 0.963950i \(0.585729\pi\)
\(864\) 5.65246 0.192301
\(865\) −12.4459 −0.423173
\(866\) 3.00767 0.102205
\(867\) −0.211103 −0.00716942
\(868\) −26.6433 −0.904334
\(869\) 0 0
\(870\) −4.42221 −0.149927
\(871\) −71.4024 −2.41938
\(872\) −37.5778 −1.27254
\(873\) 19.2111 0.650197
\(874\) 4.11103 0.139058
\(875\) 4.14863 0.140249
\(876\) −4.64409 −0.156909
\(877\) 13.7020 0.462683 0.231342 0.972873i \(-0.425689\pi\)
0.231342 + 0.972873i \(0.425689\pi\)
\(878\) −1.26662 −0.0427462
\(879\) 20.7431 0.699649
\(880\) 0 0
\(881\) −49.2666 −1.65983 −0.829917 0.557887i \(-0.811612\pi\)
−0.829917 + 0.557887i \(0.811612\pi\)
\(882\) −6.41310 −0.215940
\(883\) −6.42221 −0.216124 −0.108062 0.994144i \(-0.534465\pi\)
−0.108062 + 0.994144i \(0.534465\pi\)
\(884\) −36.0000 −1.21081
\(885\) 5.21110 0.175169
\(886\) −17.3551 −0.583057
\(887\) −38.5937 −1.29585 −0.647926 0.761704i \(-0.724363\pi\)
−0.647926 + 0.761704i \(0.724363\pi\)
\(888\) 16.3293 0.547977
\(889\) −6.78890 −0.227692
\(890\) −3.76831 −0.126314
\(891\) 0 0
\(892\) 3.88897 0.130212
\(893\) 0 0
\(894\) 14.3667 0.480494
\(895\) −22.4222 −0.749492
\(896\) 46.9722 1.56923
\(897\) −28.1646 −0.940389
\(898\) −10.3140 −0.344182
\(899\) −28.1646 −0.939342
\(900\) −1.60555 −0.0535184
\(901\) 46.5107 1.54950
\(902\) 0 0
\(903\) −27.6333 −0.919579
\(904\) 25.3872 0.844367
\(905\) 19.2111 0.638599
\(906\) 9.63331 0.320045
\(907\) −14.4222 −0.478881 −0.239441 0.970911i \(-0.576964\pi\)
−0.239441 + 0.970911i \(0.576964\pi\)
\(908\) 4.64409 0.154120
\(909\) −1.25610 −0.0416623
\(910\) −14.0823 −0.466824
\(911\) 10.4222 0.345303 0.172652 0.984983i \(-0.444767\pi\)
0.172652 + 0.984983i \(0.444767\pi\)
\(912\) 2.24704 0.0744069
\(913\) 0 0
\(914\) 1.33894 0.0442881
\(915\) −8.29725 −0.274299
\(916\) −16.0555 −0.530489
\(917\) −24.0000 −0.792550
\(918\) 2.60555 0.0859960
\(919\) 18.6112 0.613928 0.306964 0.951721i \(-0.400687\pi\)
0.306964 + 0.951721i \(0.400687\pi\)
\(920\) 11.8004 0.389048
\(921\) 7.42147 0.244546
\(922\) −3.94449 −0.129905
\(923\) 28.1646 0.927049
\(924\) 0 0
\(925\) −7.21110 −0.237100
\(926\) 16.5945 0.545329
\(927\) 1.21110 0.0397778
\(928\) 39.7998 1.30649
\(929\) 16.4222 0.538795 0.269398 0.963029i \(-0.413175\pi\)
0.269398 + 0.963029i \(0.413175\pi\)
\(930\) 2.51221 0.0823785
\(931\) −12.8262 −0.420362
\(932\) 41.3711 1.35516
\(933\) −6.78890 −0.222259
\(934\) 10.8095 0.353696
\(935\) 0 0
\(936\) 12.2389 0.400040
\(937\) 10.4291 0.340705 0.170353 0.985383i \(-0.445509\pi\)
0.170353 + 0.985383i \(0.445509\pi\)
\(938\) 34.4222 1.12392
\(939\) −15.2111 −0.496396
\(940\) 0 0
\(941\) −6.28052 −0.204739 −0.102369 0.994746i \(-0.532642\pi\)
−0.102369 + 0.994746i \(0.532642\pi\)
\(942\) 6.28052 0.204630
\(943\) −49.7835 −1.62117
\(944\) −9.32213 −0.303409
\(945\) −4.14863 −0.134955
\(946\) 0 0
\(947\) −1.57779 −0.0512714 −0.0256357 0.999671i \(-0.508161\pi\)
−0.0256357 + 0.999671i \(0.508161\pi\)
\(948\) −2.01674 −0.0655006
\(949\) −15.6333 −0.507479
\(950\) 0.788897 0.0255952
\(951\) 11.2111 0.363545
\(952\) 38.9741 1.26316
\(953\) −34.8254 −1.12811 −0.564053 0.825738i \(-0.690759\pi\)
−0.564053 + 0.825738i \(0.690759\pi\)
\(954\) −7.04115 −0.227966
\(955\) 0 0
\(956\) 13.3217 0.430853
\(957\) 0 0
\(958\) −2.05551 −0.0664106
\(959\) −68.1296 −2.20002
\(960\) 0.0277564 0.000895833 0
\(961\) −15.0000 −0.483871
\(962\) 24.4777 0.789193
\(963\) 2.89252 0.0932103
\(964\) −5.25471 −0.169243
\(965\) 11.1898 0.360212
\(966\) 13.5778 0.436858
\(967\) −13.2065 −0.424693 −0.212346 0.977194i \(-0.568111\pi\)
−0.212346 + 0.977194i \(0.568111\pi\)
\(968\) 0 0
\(969\) 5.21110 0.167405
\(970\) −12.0656 −0.387402
\(971\) −58.4222 −1.87486 −0.937429 0.348177i \(-0.886801\pi\)
−0.937429 + 0.348177i \(0.886801\pi\)
\(972\) 1.60555 0.0514981
\(973\) 39.6333 1.27059
\(974\) 0.760635 0.0243723
\(975\) −5.40473 −0.173090
\(976\) 14.8429 0.475111
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 8.29725 0.265317
\(979\) 0 0
\(980\) −16.3944 −0.523701
\(981\) −16.5945 −0.529822
\(982\) 25.5778 0.816220
\(983\) 26.0555 0.831042 0.415521 0.909584i \(-0.363599\pi\)
0.415521 + 0.909584i \(0.363599\pi\)
\(984\) 21.6333 0.689645
\(985\) 9.17304 0.292277
\(986\) 18.3461 0.584258
\(987\) 0 0
\(988\) 10.8999 0.346773
\(989\) 34.7103 1.10372
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −22.6099 −0.717864
\(993\) −8.00000 −0.253872
\(994\) −13.5778 −0.430662
\(995\) −8.00000 −0.253617
\(996\) −21.9992 −0.697072
\(997\) 30.2965 0.959499 0.479750 0.877405i \(-0.340727\pi\)
0.479750 + 0.877405i \(0.340727\pi\)
\(998\) −9.05789 −0.286722
\(999\) 7.21110 0.228149
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.s.1.2 4
3.2 odd 2 5445.2.a.bo.1.3 4
5.4 even 2 9075.2.a.dc.1.3 4
11.10 odd 2 inner 1815.2.a.s.1.3 yes 4
33.32 even 2 5445.2.a.bo.1.2 4
55.54 odd 2 9075.2.a.dc.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.s.1.2 4 1.1 even 1 trivial
1815.2.a.s.1.3 yes 4 11.10 odd 2 inner
5445.2.a.bo.1.2 4 33.32 even 2
5445.2.a.bo.1.3 4 3.2 odd 2
9075.2.a.dc.1.2 4 55.54 odd 2
9075.2.a.dc.1.3 4 5.4 even 2