Properties

Label 2541.2.a.bq.1.4
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $0$
Dimension $10$
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] = [2541,2,Mod(1,2541)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2541, 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("2541.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.33330\) of defining polynomial
Character \(\chi\) \(=\) 2541.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.33330 q^{2} +1.00000 q^{3} -0.222305 q^{4} -0.873210 q^{5} -1.33330 q^{6} -1.00000 q^{7} +2.96300 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.33330 q^{2} +1.00000 q^{3} -0.222305 q^{4} -0.873210 q^{5} -1.33330 q^{6} -1.00000 q^{7} +2.96300 q^{8} +1.00000 q^{9} +1.16425 q^{10} -0.222305 q^{12} +0.0395440 q^{13} +1.33330 q^{14} -0.873210 q^{15} -3.50597 q^{16} +2.04633 q^{17} -1.33330 q^{18} -5.69421 q^{19} +0.194119 q^{20} -1.00000 q^{21} +5.97870 q^{23} +2.96300 q^{24} -4.23750 q^{25} -0.0527241 q^{26} +1.00000 q^{27} +0.222305 q^{28} -4.67254 q^{29} +1.16425 q^{30} +7.47891 q^{31} -1.25149 q^{32} -2.72838 q^{34} +0.873210 q^{35} -0.222305 q^{36} +11.5704 q^{37} +7.59210 q^{38} +0.0395440 q^{39} -2.58733 q^{40} -6.14882 q^{41} +1.33330 q^{42} -1.79689 q^{43} -0.873210 q^{45} -7.97142 q^{46} +6.04773 q^{47} -3.50597 q^{48} +1.00000 q^{49} +5.64987 q^{50} +2.04633 q^{51} -0.00879083 q^{52} -0.124401 q^{53} -1.33330 q^{54} -2.96300 q^{56} -5.69421 q^{57} +6.22991 q^{58} -14.3460 q^{59} +0.194119 q^{60} +7.55944 q^{61} -9.97165 q^{62} -1.00000 q^{63} +8.68056 q^{64} -0.0345302 q^{65} +4.85938 q^{67} -0.454910 q^{68} +5.97870 q^{69} -1.16425 q^{70} -11.2503 q^{71} +2.96300 q^{72} +0.612954 q^{73} -15.4269 q^{74} -4.23750 q^{75} +1.26585 q^{76} -0.0527241 q^{78} -0.155227 q^{79} +3.06145 q^{80} +1.00000 q^{81} +8.19824 q^{82} +8.91599 q^{83} +0.222305 q^{84} -1.78688 q^{85} +2.39580 q^{86} -4.67254 q^{87} +5.17689 q^{89} +1.16425 q^{90} -0.0395440 q^{91} -1.32910 q^{92} +7.47891 q^{93} -8.06345 q^{94} +4.97224 q^{95} -1.25149 q^{96} +7.94222 q^{97} -1.33330 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{3} + 18 q^{4} + 5 q^{5} - 10 q^{7} + 3 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{3} + 18 q^{4} + 5 q^{5} - 10 q^{7} + 3 q^{8} + 10 q^{9} + 6 q^{10} + 18 q^{12} - 6 q^{13} + 5 q^{15} + 38 q^{16} - 8 q^{17} + 7 q^{20} - 10 q^{21} + 3 q^{24} + 31 q^{25} + q^{26} + 10 q^{27} - 18 q^{28} + 14 q^{29} + 6 q^{30} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 18 q^{36} + 24 q^{37} + 8 q^{38} - 6 q^{39} + 5 q^{40} - 19 q^{41} + 6 q^{43} + 5 q^{45} + q^{46} + 15 q^{47} + 38 q^{48} + 10 q^{49} + q^{50} - 8 q^{51} + 25 q^{52} - q^{53} - 3 q^{56} + 11 q^{58} + 23 q^{59} + 7 q^{60} - 11 q^{62} - 10 q^{63} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} + 26 q^{71} + 3 q^{72} + q^{73} + 39 q^{74} + 31 q^{75} + 2 q^{76} + q^{78} - 5 q^{79} + 6 q^{80} + 10 q^{81} + 5 q^{82} - 6 q^{83} - 18 q^{84} + q^{85} - 41 q^{86} + 14 q^{87} - 9 q^{89} + 6 q^{90} + 6 q^{91} - 48 q^{92} + 26 q^{93} + 42 q^{95} + 41 q^{96} + 24 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.33330 −0.942787 −0.471394 0.881923i \(-0.656249\pi\)
−0.471394 + 0.881923i \(0.656249\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.222305 −0.111152
\(5\) −0.873210 −0.390512 −0.195256 0.980752i \(-0.562554\pi\)
−0.195256 + 0.980752i \(0.562554\pi\)
\(6\) −1.33330 −0.544318
\(7\) −1.00000 −0.377964
\(8\) 2.96300 1.04758
\(9\) 1.00000 0.333333
\(10\) 1.16425 0.368169
\(11\) 0 0
\(12\) −0.222305 −0.0641739
\(13\) 0.0395440 0.0109675 0.00548377 0.999985i \(-0.498254\pi\)
0.00548377 + 0.999985i \(0.498254\pi\)
\(14\) 1.33330 0.356340
\(15\) −0.873210 −0.225462
\(16\) −3.50597 −0.876493
\(17\) 2.04633 0.496308 0.248154 0.968721i \(-0.420176\pi\)
0.248154 + 0.968721i \(0.420176\pi\)
\(18\) −1.33330 −0.314262
\(19\) −5.69421 −1.30634 −0.653170 0.757211i \(-0.726561\pi\)
−0.653170 + 0.757211i \(0.726561\pi\)
\(20\) 0.194119 0.0434063
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 5.97870 1.24665 0.623323 0.781965i \(-0.285782\pi\)
0.623323 + 0.781965i \(0.285782\pi\)
\(24\) 2.96300 0.604821
\(25\) −4.23750 −0.847501
\(26\) −0.0527241 −0.0103401
\(27\) 1.00000 0.192450
\(28\) 0.222305 0.0420117
\(29\) −4.67254 −0.867669 −0.433834 0.900993i \(-0.642840\pi\)
−0.433834 + 0.900993i \(0.642840\pi\)
\(30\) 1.16425 0.212563
\(31\) 7.47891 1.34325 0.671626 0.740890i \(-0.265596\pi\)
0.671626 + 0.740890i \(0.265596\pi\)
\(32\) −1.25149 −0.221234
\(33\) 0 0
\(34\) −2.72838 −0.467913
\(35\) 0.873210 0.147600
\(36\) −0.222305 −0.0370508
\(37\) 11.5704 1.90217 0.951085 0.308930i \(-0.0999708\pi\)
0.951085 + 0.308930i \(0.0999708\pi\)
\(38\) 7.59210 1.23160
\(39\) 0.0395440 0.00633211
\(40\) −2.58733 −0.409092
\(41\) −6.14882 −0.960285 −0.480142 0.877191i \(-0.659415\pi\)
−0.480142 + 0.877191i \(0.659415\pi\)
\(42\) 1.33330 0.205733
\(43\) −1.79689 −0.274024 −0.137012 0.990569i \(-0.543750\pi\)
−0.137012 + 0.990569i \(0.543750\pi\)
\(44\) 0 0
\(45\) −0.873210 −0.130171
\(46\) −7.97142 −1.17532
\(47\) 6.04773 0.882152 0.441076 0.897470i \(-0.354597\pi\)
0.441076 + 0.897470i \(0.354597\pi\)
\(48\) −3.50597 −0.506043
\(49\) 1.00000 0.142857
\(50\) 5.64987 0.799013
\(51\) 2.04633 0.286544
\(52\) −0.00879083 −0.00121907
\(53\) −0.124401 −0.0170878 −0.00854390 0.999964i \(-0.502720\pi\)
−0.00854390 + 0.999964i \(0.502720\pi\)
\(54\) −1.33330 −0.181439
\(55\) 0 0
\(56\) −2.96300 −0.395948
\(57\) −5.69421 −0.754216
\(58\) 6.22991 0.818027
\(59\) −14.3460 −1.86769 −0.933845 0.357679i \(-0.883568\pi\)
−0.933845 + 0.357679i \(0.883568\pi\)
\(60\) 0.194119 0.0250607
\(61\) 7.55944 0.967887 0.483944 0.875099i \(-0.339204\pi\)
0.483944 + 0.875099i \(0.339204\pi\)
\(62\) −9.97165 −1.26640
\(63\) −1.00000 −0.125988
\(64\) 8.68056 1.08507
\(65\) −0.0345302 −0.00428295
\(66\) 0 0
\(67\) 4.85938 0.593668 0.296834 0.954929i \(-0.404069\pi\)
0.296834 + 0.954929i \(0.404069\pi\)
\(68\) −0.454910 −0.0551659
\(69\) 5.97870 0.719751
\(70\) −1.16425 −0.139155
\(71\) −11.2503 −1.33516 −0.667582 0.744536i \(-0.732671\pi\)
−0.667582 + 0.744536i \(0.732671\pi\)
\(72\) 2.96300 0.349193
\(73\) 0.612954 0.0717409 0.0358704 0.999356i \(-0.488580\pi\)
0.0358704 + 0.999356i \(0.488580\pi\)
\(74\) −15.4269 −1.79334
\(75\) −4.23750 −0.489305
\(76\) 1.26585 0.145203
\(77\) 0 0
\(78\) −0.0527241 −0.00596983
\(79\) −0.155227 −0.0174644 −0.00873220 0.999962i \(-0.502780\pi\)
−0.00873220 + 0.999962i \(0.502780\pi\)
\(80\) 3.06145 0.342281
\(81\) 1.00000 0.111111
\(82\) 8.19824 0.905344
\(83\) 8.91599 0.978657 0.489328 0.872100i \(-0.337242\pi\)
0.489328 + 0.872100i \(0.337242\pi\)
\(84\) 0.222305 0.0242555
\(85\) −1.78688 −0.193814
\(86\) 2.39580 0.258346
\(87\) −4.67254 −0.500949
\(88\) 0 0
\(89\) 5.17689 0.548749 0.274374 0.961623i \(-0.411529\pi\)
0.274374 + 0.961623i \(0.411529\pi\)
\(90\) 1.16425 0.122723
\(91\) −0.0395440 −0.00414534
\(92\) −1.32910 −0.138568
\(93\) 7.47891 0.775527
\(94\) −8.06345 −0.831682
\(95\) 4.97224 0.510141
\(96\) −1.25149 −0.127730
\(97\) 7.94222 0.806410 0.403205 0.915110i \(-0.367896\pi\)
0.403205 + 0.915110i \(0.367896\pi\)
\(98\) −1.33330 −0.134684
\(99\) 0 0
\(100\) 0.942018 0.0942018
\(101\) −9.09560 −0.905046 −0.452523 0.891753i \(-0.649476\pi\)
−0.452523 + 0.891753i \(0.649476\pi\)
\(102\) −2.72838 −0.270150
\(103\) 3.43750 0.338707 0.169354 0.985555i \(-0.445832\pi\)
0.169354 + 0.985555i \(0.445832\pi\)
\(104\) 0.117169 0.0114894
\(105\) 0.873210 0.0852166
\(106\) 0.165864 0.0161102
\(107\) 3.29161 0.318212 0.159106 0.987262i \(-0.449139\pi\)
0.159106 + 0.987262i \(0.449139\pi\)
\(108\) −0.222305 −0.0213913
\(109\) 9.99510 0.957357 0.478678 0.877990i \(-0.341116\pi\)
0.478678 + 0.877990i \(0.341116\pi\)
\(110\) 0 0
\(111\) 11.5704 1.09822
\(112\) 3.50597 0.331283
\(113\) 4.40261 0.414163 0.207081 0.978324i \(-0.433603\pi\)
0.207081 + 0.978324i \(0.433603\pi\)
\(114\) 7.59210 0.711065
\(115\) −5.22067 −0.486830
\(116\) 1.03873 0.0964435
\(117\) 0.0395440 0.00365585
\(118\) 19.1275 1.76083
\(119\) −2.04633 −0.187587
\(120\) −2.58733 −0.236190
\(121\) 0 0
\(122\) −10.0790 −0.912512
\(123\) −6.14882 −0.554421
\(124\) −1.66260 −0.149306
\(125\) 8.06628 0.721470
\(126\) 1.33330 0.118780
\(127\) 12.9823 1.15200 0.575998 0.817451i \(-0.304614\pi\)
0.575998 + 0.817451i \(0.304614\pi\)
\(128\) −9.07082 −0.801755
\(129\) −1.79689 −0.158208
\(130\) 0.0460393 0.00403791
\(131\) 18.1255 1.58363 0.791817 0.610758i \(-0.209135\pi\)
0.791817 + 0.610758i \(0.209135\pi\)
\(132\) 0 0
\(133\) 5.69421 0.493750
\(134\) −6.47902 −0.559702
\(135\) −0.873210 −0.0751540
\(136\) 6.06329 0.519923
\(137\) 5.20699 0.444863 0.222431 0.974948i \(-0.428601\pi\)
0.222431 + 0.974948i \(0.428601\pi\)
\(138\) −7.97142 −0.678572
\(139\) −18.1482 −1.53931 −0.769657 0.638458i \(-0.779573\pi\)
−0.769657 + 0.638458i \(0.779573\pi\)
\(140\) −0.194119 −0.0164061
\(141\) 6.04773 0.509311
\(142\) 15.0000 1.25878
\(143\) 0 0
\(144\) −3.50597 −0.292164
\(145\) 4.08011 0.338835
\(146\) −0.817253 −0.0676364
\(147\) 1.00000 0.0824786
\(148\) −2.57217 −0.211431
\(149\) 11.0495 0.905212 0.452606 0.891711i \(-0.350494\pi\)
0.452606 + 0.891711i \(0.350494\pi\)
\(150\) 5.64987 0.461310
\(151\) 15.9314 1.29648 0.648240 0.761436i \(-0.275505\pi\)
0.648240 + 0.761436i \(0.275505\pi\)
\(152\) −16.8720 −1.36850
\(153\) 2.04633 0.165436
\(154\) 0 0
\(155\) −6.53066 −0.524555
\(156\) −0.00879083 −0.000703830 0
\(157\) 16.0035 1.27722 0.638610 0.769531i \(-0.279510\pi\)
0.638610 + 0.769531i \(0.279510\pi\)
\(158\) 0.206964 0.0164652
\(159\) −0.124401 −0.00986564
\(160\) 1.09281 0.0863945
\(161\) −5.97870 −0.471188
\(162\) −1.33330 −0.104754
\(163\) 22.0034 1.72344 0.861721 0.507383i \(-0.169387\pi\)
0.861721 + 0.507383i \(0.169387\pi\)
\(164\) 1.36691 0.106738
\(165\) 0 0
\(166\) −11.8877 −0.922665
\(167\) −4.72101 −0.365323 −0.182661 0.983176i \(-0.558471\pi\)
−0.182661 + 0.983176i \(0.558471\pi\)
\(168\) −2.96300 −0.228601
\(169\) −12.9984 −0.999880
\(170\) 2.38245 0.182725
\(171\) −5.69421 −0.435447
\(172\) 0.399458 0.0304584
\(173\) −9.47572 −0.720426 −0.360213 0.932870i \(-0.617296\pi\)
−0.360213 + 0.932870i \(0.617296\pi\)
\(174\) 6.22991 0.472288
\(175\) 4.23750 0.320325
\(176\) 0 0
\(177\) −14.3460 −1.07831
\(178\) −6.90235 −0.517353
\(179\) 17.7185 1.32434 0.662170 0.749353i \(-0.269636\pi\)
0.662170 + 0.749353i \(0.269636\pi\)
\(180\) 0.194119 0.0144688
\(181\) 16.2742 1.20965 0.604826 0.796358i \(-0.293243\pi\)
0.604826 + 0.796358i \(0.293243\pi\)
\(182\) 0.0527241 0.00390817
\(183\) 7.55944 0.558810
\(184\) 17.7149 1.30596
\(185\) −10.1034 −0.742819
\(186\) −9.97165 −0.731157
\(187\) 0 0
\(188\) −1.34444 −0.0980534
\(189\) −1.00000 −0.0727393
\(190\) −6.62950 −0.480954
\(191\) 6.44604 0.466419 0.233210 0.972426i \(-0.425077\pi\)
0.233210 + 0.972426i \(0.425077\pi\)
\(192\) 8.68056 0.626465
\(193\) −1.81074 −0.130340 −0.0651701 0.997874i \(-0.520759\pi\)
−0.0651701 + 0.997874i \(0.520759\pi\)
\(194\) −10.5894 −0.760273
\(195\) −0.0345302 −0.00247276
\(196\) −0.222305 −0.0158789
\(197\) −6.85772 −0.488592 −0.244296 0.969701i \(-0.578557\pi\)
−0.244296 + 0.969701i \(0.578557\pi\)
\(198\) 0 0
\(199\) −5.07400 −0.359687 −0.179843 0.983695i \(-0.557559\pi\)
−0.179843 + 0.983695i \(0.557559\pi\)
\(200\) −12.5557 −0.887825
\(201\) 4.85938 0.342754
\(202\) 12.1272 0.853266
\(203\) 4.67254 0.327948
\(204\) −0.454910 −0.0318500
\(205\) 5.36922 0.375002
\(206\) −4.58323 −0.319329
\(207\) 5.97870 0.415549
\(208\) −0.138640 −0.00961296
\(209\) 0 0
\(210\) −1.16425 −0.0803411
\(211\) 8.10688 0.558101 0.279050 0.960276i \(-0.409980\pi\)
0.279050 + 0.960276i \(0.409980\pi\)
\(212\) 0.0276550 0.00189935
\(213\) −11.2503 −0.770857
\(214\) −4.38871 −0.300006
\(215\) 1.56907 0.107009
\(216\) 2.96300 0.201607
\(217\) −7.47891 −0.507702
\(218\) −13.3265 −0.902583
\(219\) 0.612954 0.0414196
\(220\) 0 0
\(221\) 0.0809201 0.00544328
\(222\) −15.4269 −1.03539
\(223\) −16.2050 −1.08517 −0.542583 0.840002i \(-0.682554\pi\)
−0.542583 + 0.840002i \(0.682554\pi\)
\(224\) 1.25149 0.0836187
\(225\) −4.23750 −0.282500
\(226\) −5.87001 −0.390467
\(227\) 23.6742 1.57131 0.785655 0.618665i \(-0.212326\pi\)
0.785655 + 0.618665i \(0.212326\pi\)
\(228\) 1.26585 0.0838330
\(229\) 10.1567 0.671176 0.335588 0.942009i \(-0.391065\pi\)
0.335588 + 0.942009i \(0.391065\pi\)
\(230\) 6.96073 0.458977
\(231\) 0 0
\(232\) −13.8448 −0.908952
\(233\) 23.5158 1.54057 0.770287 0.637697i \(-0.220113\pi\)
0.770287 + 0.637697i \(0.220113\pi\)
\(234\) −0.0527241 −0.00344668
\(235\) −5.28094 −0.344491
\(236\) 3.18919 0.207598
\(237\) −0.155227 −0.0100831
\(238\) 2.72838 0.176854
\(239\) 7.82554 0.506192 0.253096 0.967441i \(-0.418551\pi\)
0.253096 + 0.967441i \(0.418551\pi\)
\(240\) 3.06145 0.197616
\(241\) −6.86139 −0.441981 −0.220991 0.975276i \(-0.570929\pi\)
−0.220991 + 0.975276i \(0.570929\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −1.68050 −0.107583
\(245\) −0.873210 −0.0557874
\(246\) 8.19824 0.522701
\(247\) −0.225172 −0.0143273
\(248\) 22.1600 1.40716
\(249\) 8.91599 0.565028
\(250\) −10.7548 −0.680193
\(251\) −2.56709 −0.162033 −0.0810166 0.996713i \(-0.525817\pi\)
−0.0810166 + 0.996713i \(0.525817\pi\)
\(252\) 0.222305 0.0140039
\(253\) 0 0
\(254\) −17.3094 −1.08609
\(255\) −1.78688 −0.111899
\(256\) −5.26696 −0.329185
\(257\) −19.5915 −1.22208 −0.611042 0.791598i \(-0.709250\pi\)
−0.611042 + 0.791598i \(0.709250\pi\)
\(258\) 2.39580 0.149156
\(259\) −11.5704 −0.718953
\(260\) 0.00767624 0.000476060 0
\(261\) −4.67254 −0.289223
\(262\) −24.1668 −1.49303
\(263\) −26.5306 −1.63595 −0.817974 0.575255i \(-0.804903\pi\)
−0.817974 + 0.575255i \(0.804903\pi\)
\(264\) 0 0
\(265\) 0.108628 0.00667298
\(266\) −7.59210 −0.465501
\(267\) 5.17689 0.316820
\(268\) −1.08026 −0.0659876
\(269\) −17.1248 −1.04412 −0.522058 0.852910i \(-0.674836\pi\)
−0.522058 + 0.852910i \(0.674836\pi\)
\(270\) 1.16425 0.0708542
\(271\) 5.29554 0.321681 0.160840 0.986980i \(-0.448580\pi\)
0.160840 + 0.986980i \(0.448580\pi\)
\(272\) −7.17438 −0.435010
\(273\) −0.0395440 −0.00239331
\(274\) −6.94249 −0.419411
\(275\) 0 0
\(276\) −1.32910 −0.0800021
\(277\) 2.45004 0.147209 0.0736044 0.997288i \(-0.476550\pi\)
0.0736044 + 0.997288i \(0.476550\pi\)
\(278\) 24.1971 1.45125
\(279\) 7.47891 0.447751
\(280\) 2.58733 0.154622
\(281\) 0.301746 0.0180006 0.00900031 0.999959i \(-0.497135\pi\)
0.00900031 + 0.999959i \(0.497135\pi\)
\(282\) −8.06345 −0.480172
\(283\) 24.0094 1.42721 0.713605 0.700548i \(-0.247061\pi\)
0.713605 + 0.700548i \(0.247061\pi\)
\(284\) 2.50100 0.148407
\(285\) 4.97224 0.294530
\(286\) 0 0
\(287\) 6.14882 0.362954
\(288\) −1.25149 −0.0737448
\(289\) −12.8125 −0.753678
\(290\) −5.44002 −0.319449
\(291\) 7.94222 0.465581
\(292\) −0.136263 −0.00797418
\(293\) −1.96403 −0.114740 −0.0573700 0.998353i \(-0.518271\pi\)
−0.0573700 + 0.998353i \(0.518271\pi\)
\(294\) −1.33330 −0.0777598
\(295\) 12.5271 0.729354
\(296\) 34.2833 1.99268
\(297\) 0 0
\(298\) −14.7324 −0.853422
\(299\) 0.236422 0.0136726
\(300\) 0.942018 0.0543874
\(301\) 1.79689 0.103571
\(302\) −21.2414 −1.22230
\(303\) −9.09560 −0.522529
\(304\) 19.9637 1.14500
\(305\) −6.60098 −0.377971
\(306\) −2.72838 −0.155971
\(307\) 20.0180 1.14249 0.571243 0.820781i \(-0.306462\pi\)
0.571243 + 0.820781i \(0.306462\pi\)
\(308\) 0 0
\(309\) 3.43750 0.195553
\(310\) 8.70735 0.494544
\(311\) 17.7041 1.00391 0.501954 0.864894i \(-0.332615\pi\)
0.501954 + 0.864894i \(0.332615\pi\)
\(312\) 0.117169 0.00663339
\(313\) 28.3315 1.60139 0.800697 0.599070i \(-0.204463\pi\)
0.800697 + 0.599070i \(0.204463\pi\)
\(314\) −21.3375 −1.20415
\(315\) 0.873210 0.0491998
\(316\) 0.0345077 0.00194121
\(317\) −21.9554 −1.23314 −0.616570 0.787300i \(-0.711478\pi\)
−0.616570 + 0.787300i \(0.711478\pi\)
\(318\) 0.165864 0.00930120
\(319\) 0 0
\(320\) −7.57995 −0.423732
\(321\) 3.29161 0.183720
\(322\) 7.97142 0.444230
\(323\) −11.6522 −0.648347
\(324\) −0.222305 −0.0123503
\(325\) −0.167568 −0.00929499
\(326\) −29.3372 −1.62484
\(327\) 9.99510 0.552730
\(328\) −18.2190 −1.00598
\(329\) −6.04773 −0.333422
\(330\) 0 0
\(331\) 23.1465 1.27225 0.636123 0.771587i \(-0.280537\pi\)
0.636123 + 0.771587i \(0.280537\pi\)
\(332\) −1.98207 −0.108780
\(333\) 11.5704 0.634057
\(334\) 6.29454 0.344422
\(335\) −4.24326 −0.231834
\(336\) 3.50597 0.191266
\(337\) 35.0111 1.90717 0.953587 0.301117i \(-0.0973595\pi\)
0.953587 + 0.301117i \(0.0973595\pi\)
\(338\) 17.3308 0.942674
\(339\) 4.40261 0.239117
\(340\) 0.397232 0.0215429
\(341\) 0 0
\(342\) 7.59210 0.410534
\(343\) −1.00000 −0.0539949
\(344\) −5.32420 −0.287062
\(345\) −5.22067 −0.281071
\(346\) 12.6340 0.679208
\(347\) −3.75958 −0.201825 −0.100912 0.994895i \(-0.532176\pi\)
−0.100912 + 0.994895i \(0.532176\pi\)
\(348\) 1.03873 0.0556817
\(349\) −12.1740 −0.651658 −0.325829 0.945429i \(-0.605643\pi\)
−0.325829 + 0.945429i \(0.605643\pi\)
\(350\) −5.64987 −0.301998
\(351\) 0.0395440 0.00211070
\(352\) 0 0
\(353\) −5.79059 −0.308202 −0.154101 0.988055i \(-0.549248\pi\)
−0.154101 + 0.988055i \(0.549248\pi\)
\(354\) 19.1275 1.01662
\(355\) 9.82387 0.521397
\(356\) −1.15085 −0.0609948
\(357\) −2.04633 −0.108303
\(358\) −23.6241 −1.24857
\(359\) 5.76086 0.304046 0.152023 0.988377i \(-0.451421\pi\)
0.152023 + 0.988377i \(0.451421\pi\)
\(360\) −2.58733 −0.136364
\(361\) 13.4240 0.706525
\(362\) −21.6984 −1.14044
\(363\) 0 0
\(364\) 0.00879083 0.000460765 0
\(365\) −0.535238 −0.0280156
\(366\) −10.0790 −0.526839
\(367\) 14.2580 0.744263 0.372132 0.928180i \(-0.378627\pi\)
0.372132 + 0.928180i \(0.378627\pi\)
\(368\) −20.9612 −1.09268
\(369\) −6.14882 −0.320095
\(370\) 13.4709 0.700320
\(371\) 0.124401 0.00645858
\(372\) −1.66260 −0.0862017
\(373\) 32.2223 1.66841 0.834203 0.551458i \(-0.185928\pi\)
0.834203 + 0.551458i \(0.185928\pi\)
\(374\) 0 0
\(375\) 8.06628 0.416541
\(376\) 17.9195 0.924125
\(377\) −0.184771 −0.00951619
\(378\) 1.33330 0.0685777
\(379\) −28.5774 −1.46792 −0.733960 0.679193i \(-0.762330\pi\)
−0.733960 + 0.679193i \(0.762330\pi\)
\(380\) −1.10535 −0.0567035
\(381\) 12.9823 0.665105
\(382\) −8.59452 −0.439734
\(383\) −20.1613 −1.03019 −0.515097 0.857132i \(-0.672244\pi\)
−0.515097 + 0.857132i \(0.672244\pi\)
\(384\) −9.07082 −0.462894
\(385\) 0 0
\(386\) 2.41427 0.122883
\(387\) −1.79689 −0.0913412
\(388\) −1.76559 −0.0896345
\(389\) −16.5608 −0.839663 −0.419832 0.907602i \(-0.637911\pi\)
−0.419832 + 0.907602i \(0.637911\pi\)
\(390\) 0.0460393 0.00233129
\(391\) 12.2344 0.618721
\(392\) 2.96300 0.149654
\(393\) 18.1255 0.914312
\(394\) 9.14342 0.460639
\(395\) 0.135546 0.00682005
\(396\) 0 0
\(397\) 30.9920 1.55545 0.777723 0.628607i \(-0.216375\pi\)
0.777723 + 0.628607i \(0.216375\pi\)
\(398\) 6.76518 0.339108
\(399\) 5.69421 0.285067
\(400\) 14.8566 0.742828
\(401\) 3.48264 0.173915 0.0869574 0.996212i \(-0.472286\pi\)
0.0869574 + 0.996212i \(0.472286\pi\)
\(402\) −6.47902 −0.323144
\(403\) 0.295746 0.0147322
\(404\) 2.02200 0.100598
\(405\) −0.873210 −0.0433902
\(406\) −6.22991 −0.309185
\(407\) 0 0
\(408\) 6.06329 0.300177
\(409\) −26.5883 −1.31471 −0.657353 0.753583i \(-0.728324\pi\)
−0.657353 + 0.753583i \(0.728324\pi\)
\(410\) −7.15879 −0.353547
\(411\) 5.20699 0.256842
\(412\) −0.764174 −0.0376481
\(413\) 14.3460 0.705920
\(414\) −7.97142 −0.391774
\(415\) −7.78553 −0.382177
\(416\) −0.0494889 −0.00242639
\(417\) −18.1482 −0.888723
\(418\) 0 0
\(419\) 29.1491 1.42402 0.712012 0.702167i \(-0.247784\pi\)
0.712012 + 0.702167i \(0.247784\pi\)
\(420\) −0.194119 −0.00947204
\(421\) −19.9909 −0.974297 −0.487148 0.873319i \(-0.661963\pi\)
−0.487148 + 0.873319i \(0.661963\pi\)
\(422\) −10.8089 −0.526170
\(423\) 6.04773 0.294051
\(424\) −0.368601 −0.0179008
\(425\) −8.67134 −0.420622
\(426\) 15.0000 0.726754
\(427\) −7.55944 −0.365827
\(428\) −0.731740 −0.0353700
\(429\) 0 0
\(430\) −2.09204 −0.100887
\(431\) 16.9285 0.815417 0.407709 0.913112i \(-0.366328\pi\)
0.407709 + 0.913112i \(0.366328\pi\)
\(432\) −3.50597 −0.168681
\(433\) −8.55148 −0.410958 −0.205479 0.978662i \(-0.565875\pi\)
−0.205479 + 0.978662i \(0.565875\pi\)
\(434\) 9.97165 0.478654
\(435\) 4.08011 0.195626
\(436\) −2.22196 −0.106413
\(437\) −34.0440 −1.62854
\(438\) −0.817253 −0.0390499
\(439\) −33.9408 −1.61990 −0.809952 0.586496i \(-0.800507\pi\)
−0.809952 + 0.586496i \(0.800507\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −0.107891 −0.00513185
\(443\) −35.1601 −1.67051 −0.835254 0.549864i \(-0.814680\pi\)
−0.835254 + 0.549864i \(0.814680\pi\)
\(444\) −2.57217 −0.122070
\(445\) −4.52051 −0.214293
\(446\) 21.6061 1.02308
\(447\) 11.0495 0.522624
\(448\) −8.68056 −0.410118
\(449\) −1.92800 −0.0909879 −0.0454939 0.998965i \(-0.514486\pi\)
−0.0454939 + 0.998965i \(0.514486\pi\)
\(450\) 5.64987 0.266338
\(451\) 0 0
\(452\) −0.978723 −0.0460352
\(453\) 15.9314 0.748523
\(454\) −31.5648 −1.48141
\(455\) 0.0345302 0.00161880
\(456\) −16.8720 −0.790102
\(457\) −7.65166 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(458\) −13.5420 −0.632776
\(459\) 2.04633 0.0955146
\(460\) 1.16058 0.0541123
\(461\) 40.7525 1.89803 0.949017 0.315224i \(-0.102080\pi\)
0.949017 + 0.315224i \(0.102080\pi\)
\(462\) 0 0
\(463\) −8.37407 −0.389176 −0.194588 0.980885i \(-0.562337\pi\)
−0.194588 + 0.980885i \(0.562337\pi\)
\(464\) 16.3818 0.760505
\(465\) −6.53066 −0.302852
\(466\) −31.3537 −1.45243
\(467\) 9.49614 0.439429 0.219714 0.975564i \(-0.429487\pi\)
0.219714 + 0.975564i \(0.429487\pi\)
\(468\) −0.00879083 −0.000406356 0
\(469\) −4.85938 −0.224385
\(470\) 7.04109 0.324781
\(471\) 16.0035 0.737403
\(472\) −42.5072 −1.95655
\(473\) 0 0
\(474\) 0.206964 0.00950619
\(475\) 24.1292 1.10712
\(476\) 0.454910 0.0208507
\(477\) −0.124401 −0.00569593
\(478\) −10.4338 −0.477232
\(479\) −36.2983 −1.65851 −0.829256 0.558869i \(-0.811235\pi\)
−0.829256 + 0.558869i \(0.811235\pi\)
\(480\) 1.09281 0.0498799
\(481\) 0.457542 0.0208621
\(482\) 9.14831 0.416694
\(483\) −5.97870 −0.272040
\(484\) 0 0
\(485\) −6.93523 −0.314912
\(486\) −1.33330 −0.0604798
\(487\) 5.36736 0.243218 0.121609 0.992578i \(-0.461195\pi\)
0.121609 + 0.992578i \(0.461195\pi\)
\(488\) 22.3987 1.01394
\(489\) 22.0034 0.995030
\(490\) 1.16425 0.0525956
\(491\) 23.0614 1.04075 0.520373 0.853939i \(-0.325793\pi\)
0.520373 + 0.853939i \(0.325793\pi\)
\(492\) 1.36691 0.0616252
\(493\) −9.56156 −0.430631
\(494\) 0.300222 0.0135076
\(495\) 0 0
\(496\) −26.2208 −1.17735
\(497\) 11.2503 0.504644
\(498\) −11.8877 −0.532701
\(499\) 33.1392 1.48351 0.741756 0.670670i \(-0.233993\pi\)
0.741756 + 0.670670i \(0.233993\pi\)
\(500\) −1.79318 −0.0801932
\(501\) −4.72101 −0.210919
\(502\) 3.42270 0.152763
\(503\) −18.1118 −0.807563 −0.403782 0.914855i \(-0.632304\pi\)
−0.403782 + 0.914855i \(0.632304\pi\)
\(504\) −2.96300 −0.131983
\(505\) 7.94237 0.353431
\(506\) 0 0
\(507\) −12.9984 −0.577281
\(508\) −2.88604 −0.128047
\(509\) −15.5919 −0.691100 −0.345550 0.938400i \(-0.612308\pi\)
−0.345550 + 0.938400i \(0.612308\pi\)
\(510\) 2.38245 0.105497
\(511\) −0.612954 −0.0271155
\(512\) 25.1641 1.11211
\(513\) −5.69421 −0.251405
\(514\) 26.1214 1.15217
\(515\) −3.00166 −0.132269
\(516\) 0.399458 0.0175852
\(517\) 0 0
\(518\) 15.4269 0.677819
\(519\) −9.47572 −0.415938
\(520\) −0.102313 −0.00448673
\(521\) −15.4692 −0.677716 −0.338858 0.940838i \(-0.610041\pi\)
−0.338858 + 0.940838i \(0.610041\pi\)
\(522\) 6.22991 0.272676
\(523\) −17.3464 −0.758504 −0.379252 0.925293i \(-0.623819\pi\)
−0.379252 + 0.925293i \(0.623819\pi\)
\(524\) −4.02940 −0.176025
\(525\) 4.23750 0.184940
\(526\) 35.3733 1.54235
\(527\) 15.3043 0.666667
\(528\) 0 0
\(529\) 12.7449 0.554126
\(530\) −0.144834 −0.00629120
\(531\) −14.3460 −0.622563
\(532\) −1.26585 −0.0548816
\(533\) −0.243149 −0.0105320
\(534\) −6.90235 −0.298694
\(535\) −2.87427 −0.124265
\(536\) 14.3984 0.621914
\(537\) 17.7185 0.764608
\(538\) 22.8325 0.984380
\(539\) 0 0
\(540\) 0.194119 0.00835355
\(541\) 30.2945 1.30246 0.651232 0.758879i \(-0.274252\pi\)
0.651232 + 0.758879i \(0.274252\pi\)
\(542\) −7.06055 −0.303277
\(543\) 16.2742 0.698393
\(544\) −2.56096 −0.109800
\(545\) −8.72782 −0.373859
\(546\) 0.0527241 0.00225638
\(547\) −30.4490 −1.30191 −0.650954 0.759118i \(-0.725631\pi\)
−0.650954 + 0.759118i \(0.725631\pi\)
\(548\) −1.15754 −0.0494476
\(549\) 7.55944 0.322629
\(550\) 0 0
\(551\) 26.6064 1.13347
\(552\) 17.7149 0.753997
\(553\) 0.155227 0.00660092
\(554\) −3.26665 −0.138787
\(555\) −10.1034 −0.428867
\(556\) 4.03445 0.171099
\(557\) 7.17456 0.303996 0.151998 0.988381i \(-0.451429\pi\)
0.151998 + 0.988381i \(0.451429\pi\)
\(558\) −9.97165 −0.422134
\(559\) −0.0710563 −0.00300536
\(560\) −3.06145 −0.129370
\(561\) 0 0
\(562\) −0.402318 −0.0169708
\(563\) 9.01364 0.379880 0.189940 0.981796i \(-0.439171\pi\)
0.189940 + 0.981796i \(0.439171\pi\)
\(564\) −1.34444 −0.0566112
\(565\) −3.84441 −0.161735
\(566\) −32.0118 −1.34556
\(567\) −1.00000 −0.0419961
\(568\) −33.3347 −1.39869
\(569\) 7.45999 0.312739 0.156370 0.987699i \(-0.450021\pi\)
0.156370 + 0.987699i \(0.450021\pi\)
\(570\) −6.62950 −0.277679
\(571\) −37.5921 −1.57318 −0.786590 0.617475i \(-0.788156\pi\)
−0.786590 + 0.617475i \(0.788156\pi\)
\(572\) 0 0
\(573\) 6.44604 0.269287
\(574\) −8.19824 −0.342188
\(575\) −25.3348 −1.05653
\(576\) 8.68056 0.361690
\(577\) −12.1230 −0.504688 −0.252344 0.967638i \(-0.581201\pi\)
−0.252344 + 0.967638i \(0.581201\pi\)
\(578\) 17.0830 0.710558
\(579\) −1.81074 −0.0752519
\(580\) −0.907029 −0.0376623
\(581\) −8.91599 −0.369897
\(582\) −10.5894 −0.438944
\(583\) 0 0
\(584\) 1.81619 0.0751543
\(585\) −0.0345302 −0.00142765
\(586\) 2.61865 0.108175
\(587\) −1.28073 −0.0528615 −0.0264307 0.999651i \(-0.508414\pi\)
−0.0264307 + 0.999651i \(0.508414\pi\)
\(588\) −0.222305 −0.00916770
\(589\) −42.5865 −1.75474
\(590\) −16.7024 −0.687626
\(591\) −6.85772 −0.282089
\(592\) −40.5656 −1.66724
\(593\) −24.7324 −1.01564 −0.507819 0.861464i \(-0.669548\pi\)
−0.507819 + 0.861464i \(0.669548\pi\)
\(594\) 0 0
\(595\) 1.78688 0.0732548
\(596\) −2.45636 −0.100617
\(597\) −5.07400 −0.207665
\(598\) −0.315222 −0.0128904
\(599\) −27.9003 −1.13998 −0.569988 0.821653i \(-0.693052\pi\)
−0.569988 + 0.821653i \(0.693052\pi\)
\(600\) −12.5557 −0.512586
\(601\) −34.4256 −1.40425 −0.702124 0.712055i \(-0.747765\pi\)
−0.702124 + 0.712055i \(0.747765\pi\)
\(602\) −2.39580 −0.0976456
\(603\) 4.85938 0.197889
\(604\) −3.54163 −0.144107
\(605\) 0 0
\(606\) 12.1272 0.492633
\(607\) −26.4243 −1.07253 −0.536264 0.844050i \(-0.680165\pi\)
−0.536264 + 0.844050i \(0.680165\pi\)
\(608\) 7.12624 0.289007
\(609\) 4.67254 0.189341
\(610\) 8.80111 0.356346
\(611\) 0.239152 0.00967503
\(612\) −0.454910 −0.0183886
\(613\) −39.6007 −1.59946 −0.799729 0.600361i \(-0.795024\pi\)
−0.799729 + 0.600361i \(0.795024\pi\)
\(614\) −26.6900 −1.07712
\(615\) 5.36922 0.216508
\(616\) 0 0
\(617\) 17.0640 0.686971 0.343486 0.939158i \(-0.388392\pi\)
0.343486 + 0.939158i \(0.388392\pi\)
\(618\) −4.58323 −0.184364
\(619\) 29.2807 1.17689 0.588445 0.808538i \(-0.299741\pi\)
0.588445 + 0.808538i \(0.299741\pi\)
\(620\) 1.45180 0.0583056
\(621\) 5.97870 0.239917
\(622\) −23.6049 −0.946471
\(623\) −5.17689 −0.207408
\(624\) −0.138640 −0.00555005
\(625\) 14.1440 0.565758
\(626\) −37.7745 −1.50977
\(627\) 0 0
\(628\) −3.55766 −0.141966
\(629\) 23.6770 0.944062
\(630\) −1.16425 −0.0463850
\(631\) −40.2944 −1.60410 −0.802048 0.597260i \(-0.796256\pi\)
−0.802048 + 0.597260i \(0.796256\pi\)
\(632\) −0.459938 −0.0182954
\(633\) 8.10688 0.322220
\(634\) 29.2732 1.16259
\(635\) −11.3363 −0.449867
\(636\) 0.0276550 0.00109659
\(637\) 0.0395440 0.00156679
\(638\) 0 0
\(639\) −11.2503 −0.445055
\(640\) 7.92074 0.313095
\(641\) −12.5572 −0.495980 −0.247990 0.968763i \(-0.579770\pi\)
−0.247990 + 0.968763i \(0.579770\pi\)
\(642\) −4.38871 −0.173208
\(643\) −4.13705 −0.163149 −0.0815747 0.996667i \(-0.525995\pi\)
−0.0815747 + 0.996667i \(0.525995\pi\)
\(644\) 1.32910 0.0523737
\(645\) 1.56907 0.0617819
\(646\) 15.5359 0.611254
\(647\) −37.6088 −1.47856 −0.739278 0.673400i \(-0.764833\pi\)
−0.739278 + 0.673400i \(0.764833\pi\)
\(648\) 2.96300 0.116398
\(649\) 0 0
\(650\) 0.223419 0.00876320
\(651\) −7.47891 −0.293122
\(652\) −4.89147 −0.191565
\(653\) 6.02137 0.235634 0.117817 0.993035i \(-0.462410\pi\)
0.117817 + 0.993035i \(0.462410\pi\)
\(654\) −13.3265 −0.521107
\(655\) −15.8274 −0.618428
\(656\) 21.5576 0.841682
\(657\) 0.612954 0.0239136
\(658\) 8.06345 0.314346
\(659\) −38.6192 −1.50439 −0.752195 0.658940i \(-0.771005\pi\)
−0.752195 + 0.658940i \(0.771005\pi\)
\(660\) 0 0
\(661\) 43.0833 1.67575 0.837874 0.545864i \(-0.183798\pi\)
0.837874 + 0.545864i \(0.183798\pi\)
\(662\) −30.8613 −1.19946
\(663\) 0.0809201 0.00314268
\(664\) 26.4181 1.02522
\(665\) −4.97224 −0.192815
\(666\) −15.4269 −0.597780
\(667\) −27.9357 −1.08168
\(668\) 1.04950 0.0406066
\(669\) −16.2050 −0.626521
\(670\) 5.65755 0.218570
\(671\) 0 0
\(672\) 1.25149 0.0482773
\(673\) −21.0171 −0.810149 −0.405075 0.914284i \(-0.632754\pi\)
−0.405075 + 0.914284i \(0.632754\pi\)
\(674\) −46.6803 −1.79806
\(675\) −4.23750 −0.163102
\(676\) 2.88962 0.111139
\(677\) −23.8886 −0.918115 −0.459057 0.888407i \(-0.651813\pi\)
−0.459057 + 0.888407i \(0.651813\pi\)
\(678\) −5.87001 −0.225437
\(679\) −7.94222 −0.304794
\(680\) −5.29453 −0.203036
\(681\) 23.6742 0.907196
\(682\) 0 0
\(683\) −13.5672 −0.519134 −0.259567 0.965725i \(-0.583580\pi\)
−0.259567 + 0.965725i \(0.583580\pi\)
\(684\) 1.26585 0.0484010
\(685\) −4.54679 −0.173724
\(686\) 1.33330 0.0509057
\(687\) 10.1567 0.387504
\(688\) 6.29985 0.240180
\(689\) −0.00491932 −0.000187411 0
\(690\) 6.96073 0.264990
\(691\) −4.91171 −0.186850 −0.0934251 0.995626i \(-0.529782\pi\)
−0.0934251 + 0.995626i \(0.529782\pi\)
\(692\) 2.10650 0.0800771
\(693\) 0 0
\(694\) 5.01266 0.190278
\(695\) 15.8472 0.601120
\(696\) −13.8448 −0.524784
\(697\) −12.5825 −0.476597
\(698\) 16.2316 0.614375
\(699\) 23.5158 0.889451
\(700\) −0.942018 −0.0356049
\(701\) 36.9242 1.39461 0.697304 0.716775i \(-0.254383\pi\)
0.697304 + 0.716775i \(0.254383\pi\)
\(702\) −0.0527241 −0.00198994
\(703\) −65.8845 −2.48488
\(704\) 0 0
\(705\) −5.28094 −0.198892
\(706\) 7.72061 0.290569
\(707\) 9.09560 0.342075
\(708\) 3.18919 0.119857
\(709\) 0.489376 0.0183789 0.00918945 0.999958i \(-0.497075\pi\)
0.00918945 + 0.999958i \(0.497075\pi\)
\(710\) −13.0982 −0.491566
\(711\) −0.155227 −0.00582146
\(712\) 15.3391 0.574858
\(713\) 44.7142 1.67456
\(714\) 2.72838 0.102107
\(715\) 0 0
\(716\) −3.93890 −0.147204
\(717\) 7.82554 0.292250
\(718\) −7.68096 −0.286651
\(719\) 10.1392 0.378127 0.189064 0.981965i \(-0.439455\pi\)
0.189064 + 0.981965i \(0.439455\pi\)
\(720\) 3.06145 0.114094
\(721\) −3.43750 −0.128019
\(722\) −17.8982 −0.666103
\(723\) −6.86139 −0.255178
\(724\) −3.61784 −0.134456
\(725\) 19.7999 0.735350
\(726\) 0 0
\(727\) −10.5834 −0.392516 −0.196258 0.980552i \(-0.562879\pi\)
−0.196258 + 0.980552i \(0.562879\pi\)
\(728\) −0.117169 −0.00434257
\(729\) 1.00000 0.0370370
\(730\) 0.713634 0.0264128
\(731\) −3.67704 −0.136000
\(732\) −1.68050 −0.0621131
\(733\) 12.5752 0.464475 0.232238 0.972659i \(-0.425395\pi\)
0.232238 + 0.972659i \(0.425395\pi\)
\(734\) −19.0103 −0.701682
\(735\) −0.873210 −0.0322089
\(736\) −7.48229 −0.275801
\(737\) 0 0
\(738\) 8.19824 0.301781
\(739\) −1.39740 −0.0514043 −0.0257022 0.999670i \(-0.508182\pi\)
−0.0257022 + 0.999670i \(0.508182\pi\)
\(740\) 2.24604 0.0825662
\(741\) −0.225172 −0.00827189
\(742\) −0.165864 −0.00608907
\(743\) 24.1170 0.884767 0.442383 0.896826i \(-0.354133\pi\)
0.442383 + 0.896826i \(0.354133\pi\)
\(744\) 22.1600 0.812427
\(745\) −9.64856 −0.353496
\(746\) −42.9620 −1.57295
\(747\) 8.91599 0.326219
\(748\) 0 0
\(749\) −3.29161 −0.120273
\(750\) −10.7548 −0.392710
\(751\) −19.2248 −0.701523 −0.350762 0.936465i \(-0.614077\pi\)
−0.350762 + 0.936465i \(0.614077\pi\)
\(752\) −21.2032 −0.773200
\(753\) −2.56709 −0.0935499
\(754\) 0.246355 0.00897174
\(755\) −13.9115 −0.506290
\(756\) 0.222305 0.00808515
\(757\) −42.4814 −1.54401 −0.772006 0.635615i \(-0.780747\pi\)
−0.772006 + 0.635615i \(0.780747\pi\)
\(758\) 38.1023 1.38394
\(759\) 0 0
\(760\) 14.7328 0.534414
\(761\) 14.7018 0.532939 0.266470 0.963843i \(-0.414143\pi\)
0.266470 + 0.963843i \(0.414143\pi\)
\(762\) −17.3094 −0.627052
\(763\) −9.99510 −0.361847
\(764\) −1.43299 −0.0518437
\(765\) −1.78688 −0.0646047
\(766\) 26.8811 0.971254
\(767\) −0.567298 −0.0204839
\(768\) −5.26696 −0.190055
\(769\) −45.4931 −1.64052 −0.820262 0.571988i \(-0.806172\pi\)
−0.820262 + 0.571988i \(0.806172\pi\)
\(770\) 0 0
\(771\) −19.5915 −0.705571
\(772\) 0.402537 0.0144876
\(773\) 6.05381 0.217741 0.108870 0.994056i \(-0.465277\pi\)
0.108870 + 0.994056i \(0.465277\pi\)
\(774\) 2.39580 0.0861153
\(775\) −31.6919 −1.13841
\(776\) 23.5328 0.844779
\(777\) −11.5704 −0.415087
\(778\) 22.0805 0.791624
\(779\) 35.0127 1.25446
\(780\) 0.00767624 0.000274854 0
\(781\) 0 0
\(782\) −16.3122 −0.583322
\(783\) −4.67254 −0.166983
\(784\) −3.50597 −0.125213
\(785\) −13.9744 −0.498769
\(786\) −24.1668 −0.862002
\(787\) 54.7630 1.95209 0.976046 0.217566i \(-0.0698117\pi\)
0.976046 + 0.217566i \(0.0698117\pi\)
\(788\) 1.52451 0.0543083
\(789\) −26.5306 −0.944515
\(790\) −0.180723 −0.00642985
\(791\) −4.40261 −0.156539
\(792\) 0 0
\(793\) 0.298931 0.0106153
\(794\) −41.3218 −1.46645
\(795\) 0.108628 0.00385265
\(796\) 1.12798 0.0399801
\(797\) −43.2922 −1.53349 −0.766744 0.641953i \(-0.778124\pi\)
−0.766744 + 0.641953i \(0.778124\pi\)
\(798\) −7.59210 −0.268757
\(799\) 12.3757 0.437819
\(800\) 5.30319 0.187496
\(801\) 5.17689 0.182916
\(802\) −4.64342 −0.163965
\(803\) 0 0
\(804\) −1.08026 −0.0380980
\(805\) 5.22067 0.184004
\(806\) −0.394319 −0.0138893
\(807\) −17.1248 −0.602821
\(808\) −26.9503 −0.948108
\(809\) −15.5173 −0.545559 −0.272779 0.962077i \(-0.587943\pi\)
−0.272779 + 0.962077i \(0.587943\pi\)
\(810\) 1.16425 0.0409077
\(811\) 41.4421 1.45523 0.727615 0.685986i \(-0.240629\pi\)
0.727615 + 0.685986i \(0.240629\pi\)
\(812\) −1.03873 −0.0364522
\(813\) 5.29554 0.185723
\(814\) 0 0
\(815\) −19.2136 −0.673024
\(816\) −7.17438 −0.251153
\(817\) 10.2319 0.357968
\(818\) 35.4502 1.23949
\(819\) −0.0395440 −0.00138178
\(820\) −1.19360 −0.0416824
\(821\) −40.0922 −1.39923 −0.699613 0.714522i \(-0.746644\pi\)
−0.699613 + 0.714522i \(0.746644\pi\)
\(822\) −6.94249 −0.242147
\(823\) 4.23033 0.147460 0.0737301 0.997278i \(-0.476510\pi\)
0.0737301 + 0.997278i \(0.476510\pi\)
\(824\) 10.1853 0.354823
\(825\) 0 0
\(826\) −19.1275 −0.665532
\(827\) 47.6889 1.65830 0.829152 0.559023i \(-0.188824\pi\)
0.829152 + 0.559023i \(0.188824\pi\)
\(828\) −1.32910 −0.0461893
\(829\) 4.68523 0.162725 0.0813624 0.996685i \(-0.474073\pi\)
0.0813624 + 0.996685i \(0.474073\pi\)
\(830\) 10.3805 0.360311
\(831\) 2.45004 0.0849911
\(832\) 0.343264 0.0119005
\(833\) 2.04633 0.0709012
\(834\) 24.1971 0.837877
\(835\) 4.12244 0.142663
\(836\) 0 0
\(837\) 7.47891 0.258509
\(838\) −38.8645 −1.34255
\(839\) −6.93707 −0.239494 −0.119747 0.992804i \(-0.538208\pi\)
−0.119747 + 0.992804i \(0.538208\pi\)
\(840\) 2.58733 0.0892712
\(841\) −7.16738 −0.247151
\(842\) 26.6539 0.918555
\(843\) 0.301746 0.0103927
\(844\) −1.80220 −0.0620343
\(845\) 11.3504 0.390465
\(846\) −8.06345 −0.277227
\(847\) 0 0
\(848\) 0.436146 0.0149773
\(849\) 24.0094 0.824000
\(850\) 11.5615 0.396557
\(851\) 69.1763 2.37133
\(852\) 2.50100 0.0856827
\(853\) −6.23883 −0.213613 −0.106807 0.994280i \(-0.534063\pi\)
−0.106807 + 0.994280i \(0.534063\pi\)
\(854\) 10.0790 0.344897
\(855\) 4.97224 0.170047
\(856\) 9.75304 0.333352
\(857\) 26.5684 0.907559 0.453779 0.891114i \(-0.350075\pi\)
0.453779 + 0.891114i \(0.350075\pi\)
\(858\) 0 0
\(859\) 44.6475 1.52335 0.761677 0.647957i \(-0.224377\pi\)
0.761677 + 0.647957i \(0.224377\pi\)
\(860\) −0.348811 −0.0118944
\(861\) 6.14882 0.209551
\(862\) −22.5708 −0.768765
\(863\) −21.2309 −0.722707 −0.361353 0.932429i \(-0.617685\pi\)
−0.361353 + 0.932429i \(0.617685\pi\)
\(864\) −1.25149 −0.0425766
\(865\) 8.27430 0.281335
\(866\) 11.4017 0.387446
\(867\) −12.8125 −0.435136
\(868\) 1.66260 0.0564323
\(869\) 0 0
\(870\) −5.44002 −0.184434
\(871\) 0.192159 0.00651107
\(872\) 29.6155 1.00291
\(873\) 7.94222 0.268803
\(874\) 45.3909 1.53537
\(875\) −8.06628 −0.272690
\(876\) −0.136263 −0.00460389
\(877\) 9.63512 0.325355 0.162677 0.986679i \(-0.447987\pi\)
0.162677 + 0.986679i \(0.447987\pi\)
\(878\) 45.2533 1.52722
\(879\) −1.96403 −0.0662452
\(880\) 0 0
\(881\) −38.9952 −1.31378 −0.656891 0.753985i \(-0.728129\pi\)
−0.656891 + 0.753985i \(0.728129\pi\)
\(882\) −1.33330 −0.0448946
\(883\) 14.8304 0.499081 0.249541 0.968364i \(-0.419720\pi\)
0.249541 + 0.968364i \(0.419720\pi\)
\(884\) −0.0179889 −0.000605034 0
\(885\) 12.5271 0.421093
\(886\) 46.8791 1.57493
\(887\) −17.6091 −0.591254 −0.295627 0.955303i \(-0.595529\pi\)
−0.295627 + 0.955303i \(0.595529\pi\)
\(888\) 34.2833 1.15047
\(889\) −12.9823 −0.435413
\(890\) 6.02721 0.202032
\(891\) 0 0
\(892\) 3.60245 0.120619
\(893\) −34.4370 −1.15239
\(894\) −14.7324 −0.492724
\(895\) −15.4720 −0.517170
\(896\) 9.07082 0.303035
\(897\) 0.236422 0.00789390
\(898\) 2.57060 0.0857822
\(899\) −34.9455 −1.16550
\(900\) 0.942018 0.0314006
\(901\) −0.254566 −0.00848081
\(902\) 0 0
\(903\) 1.79689 0.0597968
\(904\) 13.0450 0.433869
\(905\) −14.2108 −0.472383
\(906\) −21.2414 −0.705698
\(907\) 46.0126 1.52782 0.763911 0.645322i \(-0.223277\pi\)
0.763911 + 0.645322i \(0.223277\pi\)
\(908\) −5.26289 −0.174655
\(909\) −9.09560 −0.301682
\(910\) −0.0460393 −0.00152619
\(911\) 19.5000 0.646064 0.323032 0.946388i \(-0.395298\pi\)
0.323032 + 0.946388i \(0.395298\pi\)
\(912\) 19.9637 0.661065
\(913\) 0 0
\(914\) 10.2020 0.337451
\(915\) −6.60098 −0.218222
\(916\) −2.25789 −0.0746029
\(917\) −18.1255 −0.598558
\(918\) −2.72838 −0.0900499
\(919\) 11.7377 0.387191 0.193596 0.981081i \(-0.437985\pi\)
0.193596 + 0.981081i \(0.437985\pi\)
\(920\) −15.4689 −0.509993
\(921\) 20.0180 0.659614
\(922\) −54.3354 −1.78944
\(923\) −0.444882 −0.0146435
\(924\) 0 0
\(925\) −49.0298 −1.61209
\(926\) 11.1652 0.366910
\(927\) 3.43750 0.112902
\(928\) 5.84764 0.191958
\(929\) 35.6903 1.17096 0.585480 0.810687i \(-0.300906\pi\)
0.585480 + 0.810687i \(0.300906\pi\)
\(930\) 8.70735 0.285525
\(931\) −5.69421 −0.186620
\(932\) −5.22769 −0.171239
\(933\) 17.7041 0.579606
\(934\) −12.6612 −0.414288
\(935\) 0 0
\(936\) 0.117169 0.00382979
\(937\) 35.3541 1.15497 0.577485 0.816402i \(-0.304034\pi\)
0.577485 + 0.816402i \(0.304034\pi\)
\(938\) 6.47902 0.211548
\(939\) 28.3315 0.924565
\(940\) 1.17398 0.0382910
\(941\) −30.5431 −0.995678 −0.497839 0.867270i \(-0.665873\pi\)
−0.497839 + 0.867270i \(0.665873\pi\)
\(942\) −21.3375 −0.695214
\(943\) −36.7620 −1.19713
\(944\) 50.2966 1.63702
\(945\) 0.873210 0.0284055
\(946\) 0 0
\(947\) 38.9527 1.26579 0.632897 0.774236i \(-0.281866\pi\)
0.632897 + 0.774236i \(0.281866\pi\)
\(948\) 0.0345077 0.00112076
\(949\) 0.0242387 0.000786821 0
\(950\) −32.1715 −1.04378
\(951\) −21.9554 −0.711953
\(952\) −6.06329 −0.196512
\(953\) 35.2846 1.14298 0.571490 0.820609i \(-0.306366\pi\)
0.571490 + 0.820609i \(0.306366\pi\)
\(954\) 0.165864 0.00537005
\(955\) −5.62875 −0.182142
\(956\) −1.73966 −0.0562645
\(957\) 0 0
\(958\) 48.3966 1.56362
\(959\) −5.20699 −0.168142
\(960\) −7.57995 −0.244642
\(961\) 24.9341 0.804326
\(962\) −0.610042 −0.0196685
\(963\) 3.29161 0.106071
\(964\) 1.52532 0.0491273
\(965\) 1.58116 0.0508993
\(966\) 7.97142 0.256476
\(967\) 31.4791 1.01230 0.506150 0.862445i \(-0.331068\pi\)
0.506150 + 0.862445i \(0.331068\pi\)
\(968\) 0 0
\(969\) −11.6522 −0.374324
\(970\) 9.24675 0.296895
\(971\) −34.4771 −1.10642 −0.553211 0.833041i \(-0.686598\pi\)
−0.553211 + 0.833041i \(0.686598\pi\)
\(972\) −0.222305 −0.00713044
\(973\) 18.1482 0.581806
\(974\) −7.15631 −0.229303
\(975\) −0.167568 −0.00536647
\(976\) −26.5032 −0.848346
\(977\) 5.51932 0.176579 0.0882894 0.996095i \(-0.471860\pi\)
0.0882894 + 0.996095i \(0.471860\pi\)
\(978\) −29.3372 −0.938101
\(979\) 0 0
\(980\) 0.194119 0.00620090
\(981\) 9.99510 0.319119
\(982\) −30.7478 −0.981202
\(983\) −33.0076 −1.05278 −0.526390 0.850243i \(-0.676455\pi\)
−0.526390 + 0.850243i \(0.676455\pi\)
\(984\) −18.2190 −0.580800
\(985\) 5.98823 0.190801
\(986\) 12.7485 0.405993
\(987\) −6.04773 −0.192501
\(988\) 0.0500568 0.00159252
\(989\) −10.7431 −0.341610
\(990\) 0 0
\(991\) −40.9399 −1.30050 −0.650249 0.759721i \(-0.725335\pi\)
−0.650249 + 0.759721i \(0.725335\pi\)
\(992\) −9.35978 −0.297173
\(993\) 23.1465 0.734532
\(994\) −15.0000 −0.475772
\(995\) 4.43067 0.140462
\(996\) −1.98207 −0.0628042
\(997\) −37.8795 −1.19966 −0.599828 0.800129i \(-0.704764\pi\)
−0.599828 + 0.800129i \(0.704764\pi\)
\(998\) −44.1845 −1.39864
\(999\) 11.5704 0.366073
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 2541.2.a.bq.1.4 10
3.2 odd 2 7623.2.a.cx.1.7 10
11.5 even 5 231.2.j.g.190.4 yes 20
11.9 even 5 231.2.j.g.169.4 20
11.10 odd 2 2541.2.a.br.1.7 10
33.5 odd 10 693.2.m.j.190.2 20
33.20 odd 10 693.2.m.j.631.2 20
33.32 even 2 7623.2.a.cy.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.g.169.4 20 11.9 even 5
231.2.j.g.190.4 yes 20 11.5 even 5
693.2.m.j.190.2 20 33.5 odd 10
693.2.m.j.631.2 20 33.20 odd 10
2541.2.a.bq.1.4 10 1.1 even 1 trivial
2541.2.a.br.1.7 10 11.10 odd 2
7623.2.a.cx.1.7 10 3.2 odd 2
7623.2.a.cy.1.4 10 33.32 even 2