Properties

Label 3549.2.a.l.1.1
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3549,2,Mod(1,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3549.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.473.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.12842\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.12842 q^{2} -1.00000 q^{3} +2.53017 q^{4} +2.65859 q^{5} +2.12842 q^{6} -1.00000 q^{7} -1.12842 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.12842 q^{2} -1.00000 q^{3} +2.53017 q^{4} +2.65859 q^{5} +2.12842 q^{6} -1.00000 q^{7} -1.12842 q^{8} +1.00000 q^{9} -5.65859 q^{10} +0.658587 q^{11} -2.53017 q^{12} +2.12842 q^{14} -2.65859 q^{15} -2.65859 q^{16} +6.12842 q^{17} -2.12842 q^{18} -5.25684 q^{19} +6.72667 q^{20} +1.00000 q^{21} -1.40175 q^{22} -3.53017 q^{23} +1.12842 q^{24} +2.06808 q^{25} -1.00000 q^{27} -2.53017 q^{28} +0.871581 q^{29} +5.65859 q^{30} +5.91542 q^{31} +7.91542 q^{32} -0.658587 q^{33} -13.0438 q^{34} -2.65859 q^{35} +2.53017 q^{36} -1.46208 q^{37} +11.1888 q^{38} -3.00000 q^{40} -10.5137 q^{41} -2.12842 q^{42} -3.53017 q^{43} +1.66633 q^{44} +2.65859 q^{45} +7.51368 q^{46} -8.71892 q^{47} +2.65859 q^{48} +1.00000 q^{49} -4.40175 q^{50} -6.12842 q^{51} -1.93966 q^{53} +2.12842 q^{54} +1.75091 q^{55} +1.12842 q^{56} +5.25684 q^{57} -1.85509 q^{58} -13.9077 q^{59} -6.72667 q^{60} +8.19650 q^{61} -12.5905 q^{62} -1.00000 q^{63} -11.5302 q^{64} +1.40175 q^{66} -13.3853 q^{67} +15.5059 q^{68} +3.53017 q^{69} +5.65859 q^{70} -4.27333 q^{71} -1.12842 q^{72} +3.46983 q^{73} +3.11193 q^{74} -2.06808 q^{75} -13.3007 q^{76} -0.658587 q^{77} +4.85509 q^{79} -7.06808 q^{80} +1.00000 q^{81} +22.3775 q^{82} -15.8551 q^{83} +2.53017 q^{84} +16.2929 q^{85} +7.51368 q^{86} -0.871581 q^{87} -0.743162 q^{88} +9.51368 q^{89} -5.65859 q^{90} -8.93192 q^{92} -5.91542 q^{93} +18.5575 q^{94} -13.9758 q^{95} -7.91542 q^{96} -2.97576 q^{97} -2.12842 q^{98} +0.658587 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 4 q^{4} - 2 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 4 q^{4} - 2 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 7 q^{10} - 8 q^{11} - 4 q^{12} + 2 q^{15} + 2 q^{16} + 12 q^{17} - 3 q^{19} + 11 q^{20} + 3 q^{21} - 7 q^{22} - 7 q^{23} - 3 q^{24} + 7 q^{25} - 3 q^{27} - 4 q^{28} + 9 q^{29} + 7 q^{30} - 5 q^{31} + q^{32} + 8 q^{33} - 10 q^{34} + 2 q^{35} + 4 q^{36} + 20 q^{38} - 9 q^{40} - 6 q^{41} - 7 q^{43} + 3 q^{44} - 2 q^{45} - 3 q^{46} - 9 q^{47} - 2 q^{48} + 3 q^{49} - 16 q^{50} - 12 q^{51} - 13 q^{53} + 26 q^{55} - 3 q^{56} + 3 q^{57} + 10 q^{58} - 11 q^{59} - 11 q^{60} + 19 q^{61} - 27 q^{62} - 3 q^{63} - 31 q^{64} + 7 q^{66} - 21 q^{67} + 13 q^{68} + 7 q^{69} + 7 q^{70} - 22 q^{71} + 3 q^{72} + 14 q^{73} - 19 q^{74} - 7 q^{75} + 2 q^{76} + 8 q^{77} - q^{79} - 22 q^{80} + 3 q^{81} + 40 q^{82} - 32 q^{83} + 4 q^{84} - q^{85} - 3 q^{86} - 9 q^{87} - 15 q^{88} + 3 q^{89} - 7 q^{90} - 26 q^{92} + 5 q^{93} + q^{94} - 12 q^{95} - q^{96} + 21 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.12842 −1.50502 −0.752510 0.658581i \(-0.771157\pi\)
−0.752510 + 0.658581i \(0.771157\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.53017 1.26508
\(5\) 2.65859 1.18896 0.594478 0.804112i \(-0.297359\pi\)
0.594478 + 0.804112i \(0.297359\pi\)
\(6\) 2.12842 0.868923
\(7\) −1.00000 −0.377964
\(8\) −1.12842 −0.398956
\(9\) 1.00000 0.333333
\(10\) −5.65859 −1.78940
\(11\) 0.658587 0.198571 0.0992857 0.995059i \(-0.468344\pi\)
0.0992857 + 0.995059i \(0.468344\pi\)
\(12\) −2.53017 −0.730397
\(13\) 0 0
\(14\) 2.12842 0.568844
\(15\) −2.65859 −0.686444
\(16\) −2.65859 −0.664647
\(17\) 6.12842 1.48636 0.743180 0.669092i \(-0.233317\pi\)
0.743180 + 0.669092i \(0.233317\pi\)
\(18\) −2.12842 −0.501673
\(19\) −5.25684 −1.20600 −0.603001 0.797741i \(-0.706028\pi\)
−0.603001 + 0.797741i \(0.706028\pi\)
\(20\) 6.72667 1.50413
\(21\) 1.00000 0.218218
\(22\) −1.40175 −0.298854
\(23\) −3.53017 −0.736091 −0.368045 0.929808i \(-0.619973\pi\)
−0.368045 + 0.929808i \(0.619973\pi\)
\(24\) 1.12842 0.230338
\(25\) 2.06808 0.413617
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −2.53017 −0.478157
\(29\) 0.871581 0.161849 0.0809243 0.996720i \(-0.474213\pi\)
0.0809243 + 0.996720i \(0.474213\pi\)
\(30\) 5.65859 1.03311
\(31\) 5.91542 1.06244 0.531221 0.847233i \(-0.321733\pi\)
0.531221 + 0.847233i \(0.321733\pi\)
\(32\) 7.91542 1.39926
\(33\) −0.658587 −0.114645
\(34\) −13.0438 −2.23700
\(35\) −2.65859 −0.449383
\(36\) 2.53017 0.421695
\(37\) −1.46208 −0.240365 −0.120183 0.992752i \(-0.538348\pi\)
−0.120183 + 0.992752i \(0.538348\pi\)
\(38\) 11.1888 1.81506
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −10.5137 −1.64196 −0.820980 0.570957i \(-0.806572\pi\)
−0.820980 + 0.570957i \(0.806572\pi\)
\(42\) −2.12842 −0.328422
\(43\) −3.53017 −0.538346 −0.269173 0.963092i \(-0.586750\pi\)
−0.269173 + 0.963092i \(0.586750\pi\)
\(44\) 1.66633 0.251209
\(45\) 2.65859 0.396319
\(46\) 7.51368 1.10783
\(47\) −8.71892 −1.27179 −0.635893 0.771777i \(-0.719368\pi\)
−0.635893 + 0.771777i \(0.719368\pi\)
\(48\) 2.65859 0.383734
\(49\) 1.00000 0.142857
\(50\) −4.40175 −0.622501
\(51\) −6.12842 −0.858150
\(52\) 0 0
\(53\) −1.93966 −0.266433 −0.133217 0.991087i \(-0.542531\pi\)
−0.133217 + 0.991087i \(0.542531\pi\)
\(54\) 2.12842 0.289641
\(55\) 1.75091 0.236093
\(56\) 1.12842 0.150791
\(57\) 5.25684 0.696285
\(58\) −1.85509 −0.243585
\(59\) −13.9077 −1.81063 −0.905313 0.424746i \(-0.860363\pi\)
−0.905313 + 0.424746i \(0.860363\pi\)
\(60\) −6.72667 −0.868409
\(61\) 8.19650 1.04945 0.524727 0.851270i \(-0.324167\pi\)
0.524727 + 0.851270i \(0.324167\pi\)
\(62\) −12.5905 −1.59900
\(63\) −1.00000 −0.125988
\(64\) −11.5302 −1.44127
\(65\) 0 0
\(66\) 1.40175 0.172543
\(67\) −13.3853 −1.63527 −0.817635 0.575737i \(-0.804715\pi\)
−0.817635 + 0.575737i \(0.804715\pi\)
\(68\) 15.5059 1.88037
\(69\) 3.53017 0.424982
\(70\) 5.65859 0.676330
\(71\) −4.27333 −0.507151 −0.253575 0.967316i \(-0.581607\pi\)
−0.253575 + 0.967316i \(0.581607\pi\)
\(72\) −1.12842 −0.132985
\(73\) 3.46983 0.406113 0.203057 0.979167i \(-0.434912\pi\)
0.203057 + 0.979167i \(0.434912\pi\)
\(74\) 3.11193 0.361754
\(75\) −2.06808 −0.238802
\(76\) −13.3007 −1.52569
\(77\) −0.658587 −0.0750529
\(78\) 0 0
\(79\) 4.85509 0.546240 0.273120 0.961980i \(-0.411944\pi\)
0.273120 + 0.961980i \(0.411944\pi\)
\(80\) −7.06808 −0.790236
\(81\) 1.00000 0.111111
\(82\) 22.3775 2.47118
\(83\) −15.8551 −1.74032 −0.870161 0.492767i \(-0.835985\pi\)
−0.870161 + 0.492767i \(0.835985\pi\)
\(84\) 2.53017 0.276064
\(85\) 16.2929 1.76722
\(86\) 7.51368 0.810221
\(87\) −0.871581 −0.0934433
\(88\) −0.743162 −0.0792213
\(89\) 9.51368 1.00845 0.504224 0.863573i \(-0.331779\pi\)
0.504224 + 0.863573i \(0.331779\pi\)
\(90\) −5.65859 −0.596467
\(91\) 0 0
\(92\) −8.93192 −0.931217
\(93\) −5.91542 −0.613401
\(94\) 18.5575 1.91406
\(95\) −13.9758 −1.43388
\(96\) −7.91542 −0.807865
\(97\) −2.97576 −0.302143 −0.151071 0.988523i \(-0.548272\pi\)
−0.151071 + 0.988523i \(0.548272\pi\)
\(98\) −2.12842 −0.215003
\(99\) 0.658587 0.0661905
\(100\) 5.23260 0.523260
\(101\) −9.24035 −0.919449 −0.459724 0.888062i \(-0.652052\pi\)
−0.459724 + 0.888062i \(0.652052\pi\)
\(102\) 13.0438 1.29153
\(103\) 13.3775 1.31813 0.659063 0.752088i \(-0.270953\pi\)
0.659063 + 0.752088i \(0.270953\pi\)
\(104\) 0 0
\(105\) 2.65859 0.259452
\(106\) 4.12842 0.400988
\(107\) 9.84734 0.951978 0.475989 0.879451i \(-0.342090\pi\)
0.475989 + 0.879451i \(0.342090\pi\)
\(108\) −2.53017 −0.243466
\(109\) −6.32492 −0.605818 −0.302909 0.953020i \(-0.597958\pi\)
−0.302909 + 0.953020i \(0.597958\pi\)
\(110\) −3.72667 −0.355324
\(111\) 1.46208 0.138775
\(112\) 2.65859 0.251213
\(113\) 13.2326 1.24482 0.622409 0.782692i \(-0.286154\pi\)
0.622409 + 0.782692i \(0.286154\pi\)
\(114\) −11.1888 −1.04792
\(115\) −9.38526 −0.875180
\(116\) 2.20525 0.204752
\(117\) 0 0
\(118\) 29.6014 2.72503
\(119\) −6.12842 −0.561791
\(120\) 3.00000 0.273861
\(121\) −10.5663 −0.960569
\(122\) −17.4456 −1.57945
\(123\) 10.5137 0.947986
\(124\) 14.9670 1.34408
\(125\) −7.79475 −0.697184
\(126\) 2.12842 0.189615
\(127\) −12.1119 −1.07476 −0.537380 0.843340i \(-0.680586\pi\)
−0.537380 + 0.843340i \(0.680586\pi\)
\(128\) 8.71018 0.769878
\(129\) 3.53017 0.310814
\(130\) 0 0
\(131\) −3.34141 −0.291941 −0.145970 0.989289i \(-0.546630\pi\)
−0.145970 + 0.989289i \(0.546630\pi\)
\(132\) −1.66633 −0.145036
\(133\) 5.25684 0.455826
\(134\) 28.4894 2.46111
\(135\) −2.65859 −0.228815
\(136\) −6.91542 −0.592993
\(137\) −1.80350 −0.154083 −0.0770416 0.997028i \(-0.524547\pi\)
−0.0770416 + 0.997028i \(0.524547\pi\)
\(138\) −7.51368 −0.639607
\(139\) −1.12842 −0.0957113 −0.0478556 0.998854i \(-0.515239\pi\)
−0.0478556 + 0.998854i \(0.515239\pi\)
\(140\) −6.72667 −0.568507
\(141\) 8.71892 0.734266
\(142\) 9.09544 0.763272
\(143\) 0 0
\(144\) −2.65859 −0.221549
\(145\) 2.31717 0.192431
\(146\) −7.38526 −0.611208
\(147\) −1.00000 −0.0824786
\(148\) −3.69932 −0.304082
\(149\) −13.8396 −1.13378 −0.566892 0.823792i \(-0.691854\pi\)
−0.566892 + 0.823792i \(0.691854\pi\)
\(150\) 4.40175 0.359401
\(151\) 7.57401 0.616364 0.308182 0.951327i \(-0.400279\pi\)
0.308182 + 0.951327i \(0.400279\pi\)
\(152\) 5.93192 0.481142
\(153\) 6.12842 0.495453
\(154\) 1.40175 0.112956
\(155\) 15.7267 1.26320
\(156\) 0 0
\(157\) 7.36877 0.588092 0.294046 0.955791i \(-0.404998\pi\)
0.294046 + 0.955791i \(0.404998\pi\)
\(158\) −10.3337 −0.822102
\(159\) 1.93966 0.153825
\(160\) 21.0438 1.66366
\(161\) 3.53017 0.278216
\(162\) −2.12842 −0.167224
\(163\) 21.1207 1.65430 0.827149 0.561982i \(-0.189961\pi\)
0.827149 + 0.561982i \(0.189961\pi\)
\(164\) −26.6014 −2.07722
\(165\) −1.75091 −0.136308
\(166\) 33.7463 2.61922
\(167\) 7.12067 0.551014 0.275507 0.961299i \(-0.411154\pi\)
0.275507 + 0.961299i \(0.411154\pi\)
\(168\) −1.12842 −0.0870594
\(169\) 0 0
\(170\) −34.6782 −2.65970
\(171\) −5.25684 −0.402000
\(172\) −8.93192 −0.681052
\(173\) −4.59050 −0.349009 −0.174505 0.984656i \(-0.555832\pi\)
−0.174505 + 0.984656i \(0.555832\pi\)
\(174\) 1.85509 0.140634
\(175\) −2.06808 −0.156332
\(176\) −1.75091 −0.131980
\(177\) 13.9077 1.04536
\(178\) −20.2491 −1.51773
\(179\) 21.2161 1.58577 0.792883 0.609374i \(-0.208579\pi\)
0.792883 + 0.609374i \(0.208579\pi\)
\(180\) 6.72667 0.501376
\(181\) −17.6179 −1.30952 −0.654762 0.755835i \(-0.727231\pi\)
−0.654762 + 0.755835i \(0.727231\pi\)
\(182\) 0 0
\(183\) −8.19650 −0.605903
\(184\) 3.98351 0.293668
\(185\) −3.88708 −0.285784
\(186\) 12.5905 0.923180
\(187\) 4.03610 0.295149
\(188\) −22.0603 −1.60892
\(189\) 1.00000 0.0727393
\(190\) 29.7463 2.15802
\(191\) −2.30068 −0.166472 −0.0832358 0.996530i \(-0.526525\pi\)
−0.0832358 + 0.996530i \(0.526525\pi\)
\(192\) 11.5302 0.832118
\(193\) 23.2249 1.67176 0.835881 0.548911i \(-0.184958\pi\)
0.835881 + 0.548911i \(0.184958\pi\)
\(194\) 6.33367 0.454731
\(195\) 0 0
\(196\) 2.53017 0.180726
\(197\) 11.3337 0.807490 0.403745 0.914872i \(-0.367708\pi\)
0.403745 + 0.914872i \(0.367708\pi\)
\(198\) −1.40175 −0.0996179
\(199\) −15.8144 −1.12105 −0.560525 0.828137i \(-0.689401\pi\)
−0.560525 + 0.828137i \(0.689401\pi\)
\(200\) −2.33367 −0.165015
\(201\) 13.3853 0.944123
\(202\) 19.6673 1.38379
\(203\) −0.871581 −0.0611730
\(204\) −15.5059 −1.08563
\(205\) −27.9515 −1.95222
\(206\) −28.4729 −1.98380
\(207\) −3.53017 −0.245364
\(208\) 0 0
\(209\) −3.46208 −0.239477
\(210\) −5.65859 −0.390480
\(211\) 6.60600 0.454776 0.227388 0.973804i \(-0.426981\pi\)
0.227388 + 0.973804i \(0.426981\pi\)
\(212\) −4.90768 −0.337061
\(213\) 4.27333 0.292804
\(214\) −20.9593 −1.43275
\(215\) −9.38526 −0.640069
\(216\) 1.12842 0.0767792
\(217\) −5.91542 −0.401565
\(218\) 13.4621 0.911767
\(219\) −3.46983 −0.234470
\(220\) 4.43010 0.298677
\(221\) 0 0
\(222\) −3.11193 −0.208859
\(223\) 9.80350 0.656491 0.328245 0.944592i \(-0.393543\pi\)
0.328245 + 0.944592i \(0.393543\pi\)
\(224\) −7.91542 −0.528872
\(225\) 2.06808 0.137872
\(226\) −28.1645 −1.87348
\(227\) 16.2929 1.08140 0.540700 0.841215i \(-0.318159\pi\)
0.540700 + 0.841215i \(0.318159\pi\)
\(228\) 13.3007 0.880859
\(229\) 23.1919 1.53256 0.766281 0.642506i \(-0.222105\pi\)
0.766281 + 0.642506i \(0.222105\pi\)
\(230\) 19.9758 1.31716
\(231\) 0.658587 0.0433318
\(232\) −0.983509 −0.0645705
\(233\) −2.60699 −0.170790 −0.0853949 0.996347i \(-0.527215\pi\)
−0.0853949 + 0.996347i \(0.527215\pi\)
\(234\) 0 0
\(235\) −23.1800 −1.51210
\(236\) −35.1888 −2.29059
\(237\) −4.85509 −0.315372
\(238\) 13.0438 0.845507
\(239\) −15.7432 −1.01834 −0.509170 0.860666i \(-0.670048\pi\)
−0.509170 + 0.860666i \(0.670048\pi\)
\(240\) 7.06808 0.456243
\(241\) −29.8669 −1.92390 −0.961950 0.273227i \(-0.911909\pi\)
−0.961950 + 0.273227i \(0.911909\pi\)
\(242\) 22.4894 1.44568
\(243\) −1.00000 −0.0641500
\(244\) 20.7385 1.32765
\(245\) 2.65859 0.169851
\(246\) −22.3775 −1.42674
\(247\) 0 0
\(248\) −6.67508 −0.423868
\(249\) 15.8551 1.00478
\(250\) 16.5905 1.04928
\(251\) −20.7189 −1.30777 −0.653883 0.756595i \(-0.726861\pi\)
−0.653883 + 0.756595i \(0.726861\pi\)
\(252\) −2.53017 −0.159386
\(253\) −2.32492 −0.146167
\(254\) 25.7793 1.61753
\(255\) −16.2929 −1.02030
\(256\) 4.52142 0.282589
\(257\) 10.6508 0.664381 0.332191 0.943212i \(-0.392212\pi\)
0.332191 + 0.943212i \(0.392212\pi\)
\(258\) −7.51368 −0.467781
\(259\) 1.46208 0.0908495
\(260\) 0 0
\(261\) 0.871581 0.0539495
\(262\) 7.11193 0.439376
\(263\) −23.6782 −1.46006 −0.730030 0.683415i \(-0.760494\pi\)
−0.730030 + 0.683415i \(0.760494\pi\)
\(264\) 0.743162 0.0457385
\(265\) −5.15677 −0.316778
\(266\) −11.1888 −0.686027
\(267\) −9.51368 −0.582228
\(268\) −33.8669 −2.06875
\(269\) −0.477581 −0.0291186 −0.0145593 0.999894i \(-0.504635\pi\)
−0.0145593 + 0.999894i \(0.504635\pi\)
\(270\) 5.65859 0.344371
\(271\) 20.6421 1.25392 0.626959 0.779052i \(-0.284299\pi\)
0.626959 + 0.779052i \(0.284299\pi\)
\(272\) −16.2929 −0.987904
\(273\) 0 0
\(274\) 3.83860 0.231898
\(275\) 1.36201 0.0821324
\(276\) 8.93192 0.537638
\(277\) −26.3600 −1.58382 −0.791910 0.610638i \(-0.790913\pi\)
−0.791910 + 0.610638i \(0.790913\pi\)
\(278\) 2.40175 0.144047
\(279\) 5.91542 0.354147
\(280\) 3.00000 0.179284
\(281\) −0.205246 −0.0122439 −0.00612197 0.999981i \(-0.501949\pi\)
−0.00612197 + 0.999981i \(0.501949\pi\)
\(282\) −18.5575 −1.10508
\(283\) 1.41824 0.0843056 0.0421528 0.999111i \(-0.486578\pi\)
0.0421528 + 0.999111i \(0.486578\pi\)
\(284\) −10.8122 −0.641588
\(285\) 13.9758 0.827853
\(286\) 0 0
\(287\) 10.5137 0.620603
\(288\) 7.91542 0.466421
\(289\) 20.5575 1.20927
\(290\) −4.93192 −0.289612
\(291\) 2.97576 0.174442
\(292\) 8.77926 0.513767
\(293\) −30.6256 −1.78917 −0.894583 0.446901i \(-0.852528\pi\)
−0.894583 + 0.446901i \(0.852528\pi\)
\(294\) 2.12842 0.124132
\(295\) −36.9748 −2.15275
\(296\) 1.64984 0.0958952
\(297\) −0.658587 −0.0382151
\(298\) 29.4565 1.70637
\(299\) 0 0
\(300\) −5.23260 −0.302104
\(301\) 3.53017 0.203475
\(302\) −16.1207 −0.927640
\(303\) 9.24035 0.530844
\(304\) 13.9758 0.801565
\(305\) 21.7911 1.24776
\(306\) −13.0438 −0.745667
\(307\) −4.47858 −0.255606 −0.127803 0.991800i \(-0.540792\pi\)
−0.127803 + 0.991800i \(0.540792\pi\)
\(308\) −1.66633 −0.0949482
\(309\) −13.3775 −0.761020
\(310\) −33.4729 −1.90114
\(311\) 0.675078 0.0382802 0.0191401 0.999817i \(-0.493907\pi\)
0.0191401 + 0.999817i \(0.493907\pi\)
\(312\) 0 0
\(313\) 10.0242 0.566604 0.283302 0.959031i \(-0.408570\pi\)
0.283302 + 0.959031i \(0.408570\pi\)
\(314\) −15.6838 −0.885089
\(315\) −2.65859 −0.149794
\(316\) 12.2842 0.691040
\(317\) −27.1558 −1.52522 −0.762610 0.646859i \(-0.776082\pi\)
−0.762610 + 0.646859i \(0.776082\pi\)
\(318\) −4.12842 −0.231510
\(319\) 0.574012 0.0321385
\(320\) −30.6540 −1.71361
\(321\) −9.84734 −0.549625
\(322\) −7.51368 −0.418721
\(323\) −32.2161 −1.79255
\(324\) 2.53017 0.140565
\(325\) 0 0
\(326\) −44.9536 −2.48975
\(327\) 6.32492 0.349769
\(328\) 11.8638 0.655071
\(329\) 8.71892 0.480690
\(330\) 3.72667 0.205146
\(331\) −23.8112 −1.30878 −0.654392 0.756155i \(-0.727075\pi\)
−0.654392 + 0.756155i \(0.727075\pi\)
\(332\) −40.1160 −2.20165
\(333\) −1.46208 −0.0801217
\(334\) −15.1558 −0.829287
\(335\) −35.5859 −1.94426
\(336\) −2.65859 −0.145038
\(337\) −21.4894 −1.17060 −0.585302 0.810815i \(-0.699024\pi\)
−0.585302 + 0.810815i \(0.699024\pi\)
\(338\) 0 0
\(339\) −13.2326 −0.718696
\(340\) 41.2239 2.23568
\(341\) 3.89582 0.210971
\(342\) 11.1888 0.605019
\(343\) −1.00000 −0.0539949
\(344\) 3.98351 0.214776
\(345\) 9.38526 0.505285
\(346\) 9.77051 0.525266
\(347\) −19.6266 −1.05361 −0.526806 0.849986i \(-0.676610\pi\)
−0.526806 + 0.849986i \(0.676610\pi\)
\(348\) −2.20525 −0.118214
\(349\) −27.0799 −1.44956 −0.724778 0.688983i \(-0.758058\pi\)
−0.724778 + 0.688983i \(0.758058\pi\)
\(350\) 4.40175 0.235283
\(351\) 0 0
\(352\) 5.21299 0.277854
\(353\) 1.67408 0.0891025 0.0445512 0.999007i \(-0.485814\pi\)
0.0445512 + 0.999007i \(0.485814\pi\)
\(354\) −29.6014 −1.57329
\(355\) −11.3610 −0.602980
\(356\) 24.0712 1.27577
\(357\) 6.12842 0.324350
\(358\) −45.1568 −2.38661
\(359\) −20.1207 −1.06193 −0.530964 0.847394i \(-0.678170\pi\)
−0.530964 + 0.847394i \(0.678170\pi\)
\(360\) −3.00000 −0.158114
\(361\) 8.63435 0.454439
\(362\) 37.4982 1.97086
\(363\) 10.5663 0.554585
\(364\) 0 0
\(365\) 9.22485 0.482851
\(366\) 17.4456 0.911896
\(367\) 16.7112 0.872316 0.436158 0.899870i \(-0.356339\pi\)
0.436158 + 0.899870i \(0.356339\pi\)
\(368\) 9.38526 0.489240
\(369\) −10.5137 −0.547320
\(370\) 8.27333 0.430110
\(371\) 1.93966 0.100702
\(372\) −14.9670 −0.776004
\(373\) 16.2852 0.843215 0.421607 0.906778i \(-0.361466\pi\)
0.421607 + 0.906778i \(0.361466\pi\)
\(374\) −8.59050 −0.444204
\(375\) 7.79475 0.402519
\(376\) 9.83860 0.507387
\(377\) 0 0
\(378\) −2.12842 −0.109474
\(379\) −27.5255 −1.41389 −0.706946 0.707268i \(-0.749928\pi\)
−0.706946 + 0.707268i \(0.749928\pi\)
\(380\) −35.3610 −1.81398
\(381\) 12.1119 0.620513
\(382\) 4.89682 0.250543
\(383\) −12.2413 −0.625503 −0.312752 0.949835i \(-0.601251\pi\)
−0.312752 + 0.949835i \(0.601251\pi\)
\(384\) −8.71018 −0.444489
\(385\) −1.75091 −0.0892346
\(386\) −49.4322 −2.51603
\(387\) −3.53017 −0.179449
\(388\) −7.52917 −0.382236
\(389\) −2.77051 −0.140471 −0.0702353 0.997530i \(-0.522375\pi\)
−0.0702353 + 0.997530i \(0.522375\pi\)
\(390\) 0 0
\(391\) −21.6343 −1.09410
\(392\) −1.12842 −0.0569938
\(393\) 3.34141 0.168552
\(394\) −24.1228 −1.21529
\(395\) 12.9077 0.649456
\(396\) 1.66633 0.0837365
\(397\) −0.812241 −0.0407652 −0.0203826 0.999792i \(-0.506488\pi\)
−0.0203826 + 0.999792i \(0.506488\pi\)
\(398\) 33.6596 1.68720
\(399\) −5.25684 −0.263171
\(400\) −5.49818 −0.274909
\(401\) 9.76277 0.487529 0.243765 0.969834i \(-0.421618\pi\)
0.243765 + 0.969834i \(0.421618\pi\)
\(402\) −28.4894 −1.42092
\(403\) 0 0
\(404\) −23.3796 −1.16318
\(405\) 2.65859 0.132106
\(406\) 1.85509 0.0920665
\(407\) −0.962909 −0.0477296
\(408\) 6.91542 0.342365
\(409\) −18.8231 −0.930742 −0.465371 0.885116i \(-0.654079\pi\)
−0.465371 + 0.885116i \(0.654079\pi\)
\(410\) 59.4925 2.93813
\(411\) 1.80350 0.0889600
\(412\) 33.8473 1.66754
\(413\) 13.9077 0.684352
\(414\) 7.51368 0.369277
\(415\) −42.1521 −2.06917
\(416\) 0 0
\(417\) 1.12842 0.0552589
\(418\) 7.36877 0.360418
\(419\) −12.5224 −0.611760 −0.305880 0.952070i \(-0.598951\pi\)
−0.305880 + 0.952070i \(0.598951\pi\)
\(420\) 6.72667 0.328228
\(421\) −27.5771 −1.34403 −0.672013 0.740539i \(-0.734570\pi\)
−0.672013 + 0.740539i \(0.734570\pi\)
\(422\) −14.0603 −0.684446
\(423\) −8.71892 −0.423929
\(424\) 2.18875 0.106295
\(425\) 12.6741 0.614783
\(426\) −9.09544 −0.440675
\(427\) −8.19650 −0.396657
\(428\) 24.9154 1.20433
\(429\) 0 0
\(430\) 19.9758 0.963317
\(431\) −0.888072 −0.0427769 −0.0213885 0.999771i \(-0.506809\pi\)
−0.0213885 + 0.999771i \(0.506809\pi\)
\(432\) 2.65859 0.127911
\(433\) 26.8308 1.28941 0.644704 0.764432i \(-0.276981\pi\)
0.644704 + 0.764432i \(0.276981\pi\)
\(434\) 12.5905 0.604363
\(435\) −2.31717 −0.111100
\(436\) −16.0031 −0.766410
\(437\) 18.5575 0.887727
\(438\) 7.38526 0.352881
\(439\) −34.0041 −1.62293 −0.811464 0.584403i \(-0.801329\pi\)
−0.811464 + 0.584403i \(0.801329\pi\)
\(440\) −1.97576 −0.0941907
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −20.9670 −0.996173 −0.498087 0.867127i \(-0.665964\pi\)
−0.498087 + 0.867127i \(0.665964\pi\)
\(444\) 3.69932 0.175562
\(445\) 25.2929 1.19900
\(446\) −20.8660 −0.988031
\(447\) 13.8396 0.654590
\(448\) 11.5302 0.544749
\(449\) −17.8221 −0.841077 −0.420539 0.907275i \(-0.638159\pi\)
−0.420539 + 0.907275i \(0.638159\pi\)
\(450\) −4.40175 −0.207500
\(451\) −6.92417 −0.326046
\(452\) 33.4807 1.57480
\(453\) −7.57401 −0.355858
\(454\) −34.6782 −1.62753
\(455\) 0 0
\(456\) −5.93192 −0.277787
\(457\) 41.0557 1.92050 0.960252 0.279133i \(-0.0900472\pi\)
0.960252 + 0.279133i \(0.0900472\pi\)
\(458\) −49.3620 −2.30654
\(459\) −6.12842 −0.286050
\(460\) −23.7463 −1.10718
\(461\) 35.9876 1.67611 0.838055 0.545585i \(-0.183693\pi\)
0.838055 + 0.545585i \(0.183693\pi\)
\(462\) −1.40175 −0.0652152
\(463\) 13.1527 0.611256 0.305628 0.952151i \(-0.401134\pi\)
0.305628 + 0.952151i \(0.401134\pi\)
\(464\) −2.31717 −0.107572
\(465\) −15.7267 −0.729307
\(466\) 5.54878 0.257042
\(467\) 19.4729 0.901100 0.450550 0.892751i \(-0.351228\pi\)
0.450550 + 0.892751i \(0.351228\pi\)
\(468\) 0 0
\(469\) 13.3853 0.618074
\(470\) 49.3368 2.27574
\(471\) −7.36877 −0.339535
\(472\) 15.6937 0.722361
\(473\) −2.32492 −0.106900
\(474\) 10.3337 0.474641
\(475\) −10.8716 −0.498822
\(476\) −15.5059 −0.710713
\(477\) −1.93966 −0.0888111
\(478\) 33.5080 1.53262
\(479\) −5.04384 −0.230459 −0.115230 0.993339i \(-0.536760\pi\)
−0.115230 + 0.993339i \(0.536760\pi\)
\(480\) −21.0438 −0.960516
\(481\) 0 0
\(482\) 63.5694 2.89551
\(483\) −3.53017 −0.160628
\(484\) −26.7344 −1.21520
\(485\) −7.91132 −0.359234
\(486\) 2.12842 0.0965470
\(487\) 12.4941 0.566160 0.283080 0.959096i \(-0.408644\pi\)
0.283080 + 0.959096i \(0.408644\pi\)
\(488\) −9.24909 −0.418687
\(489\) −21.1207 −0.955110
\(490\) −5.65859 −0.255629
\(491\) −0.0680836 −0.00307257 −0.00153629 0.999999i \(-0.500489\pi\)
−0.00153629 + 0.999999i \(0.500489\pi\)
\(492\) 26.6014 1.19928
\(493\) 5.34141 0.240565
\(494\) 0 0
\(495\) 1.75091 0.0786976
\(496\) −15.7267 −0.706148
\(497\) 4.27333 0.191685
\(498\) −33.7463 −1.51221
\(499\) 2.14392 0.0959748 0.0479874 0.998848i \(-0.484719\pi\)
0.0479874 + 0.998848i \(0.484719\pi\)
\(500\) −19.7220 −0.881996
\(501\) −7.12067 −0.318128
\(502\) 44.0985 1.96821
\(503\) −22.7824 −1.01582 −0.507908 0.861411i \(-0.669581\pi\)
−0.507908 + 0.861411i \(0.669581\pi\)
\(504\) 1.12842 0.0502638
\(505\) −24.5663 −1.09318
\(506\) 4.94841 0.219984
\(507\) 0 0
\(508\) −30.6452 −1.35966
\(509\) −25.9515 −1.15028 −0.575140 0.818055i \(-0.695053\pi\)
−0.575140 + 0.818055i \(0.695053\pi\)
\(510\) 34.6782 1.53558
\(511\) −3.46983 −0.153496
\(512\) −27.0438 −1.19518
\(513\) 5.25684 0.232095
\(514\) −22.6694 −0.999906
\(515\) 35.5653 1.56719
\(516\) 8.93192 0.393206
\(517\) −5.74217 −0.252540
\(518\) −3.11193 −0.136730
\(519\) 4.59050 0.201501
\(520\) 0 0
\(521\) −11.0077 −0.482258 −0.241129 0.970493i \(-0.577518\pi\)
−0.241129 + 0.970493i \(0.577518\pi\)
\(522\) −1.85509 −0.0811951
\(523\) −24.0031 −1.04958 −0.524791 0.851231i \(-0.675857\pi\)
−0.524791 + 0.851231i \(0.675857\pi\)
\(524\) −8.45434 −0.369329
\(525\) 2.06808 0.0902586
\(526\) 50.3971 2.19742
\(527\) 36.2522 1.57917
\(528\) 1.75091 0.0761986
\(529\) −10.5379 −0.458170
\(530\) 10.9758 0.476757
\(531\) −13.9077 −0.603542
\(532\) 13.3007 0.576658
\(533\) 0 0
\(534\) 20.2491 0.876264
\(535\) 26.1800 1.13186
\(536\) 15.1042 0.652401
\(537\) −21.2161 −0.915543
\(538\) 1.01649 0.0438241
\(539\) 0.658587 0.0283673
\(540\) −6.72667 −0.289470
\(541\) 7.12842 0.306475 0.153237 0.988189i \(-0.451030\pi\)
0.153237 + 0.988189i \(0.451030\pi\)
\(542\) −43.9350 −1.88717
\(543\) 17.6179 0.756055
\(544\) 48.5090 2.07981
\(545\) −16.8154 −0.720291
\(546\) 0 0
\(547\) 29.6540 1.26791 0.633956 0.773369i \(-0.281430\pi\)
0.633956 + 0.773369i \(0.281430\pi\)
\(548\) −4.56315 −0.194928
\(549\) 8.19650 0.349818
\(550\) −2.89893 −0.123611
\(551\) −4.58176 −0.195190
\(552\) −3.98351 −0.169549
\(553\) −4.85509 −0.206459
\(554\) 56.1052 2.38368
\(555\) 3.88708 0.164997
\(556\) −2.85509 −0.121083
\(557\) −8.61474 −0.365018 −0.182509 0.983204i \(-0.558422\pi\)
−0.182509 + 0.983204i \(0.558422\pi\)
\(558\) −12.5905 −0.532998
\(559\) 0 0
\(560\) 7.06808 0.298681
\(561\) −4.03610 −0.170404
\(562\) 0.436849 0.0184274
\(563\) 35.6694 1.50329 0.751644 0.659569i \(-0.229261\pi\)
0.751644 + 0.659569i \(0.229261\pi\)
\(564\) 22.0603 0.928908
\(565\) 35.1800 1.48003
\(566\) −3.01861 −0.126882
\(567\) −1.00000 −0.0419961
\(568\) 4.82211 0.202331
\(569\) −11.3765 −0.476928 −0.238464 0.971151i \(-0.576644\pi\)
−0.238464 + 0.971151i \(0.576644\pi\)
\(570\) −29.7463 −1.24593
\(571\) 24.6266 1.03059 0.515296 0.857013i \(-0.327682\pi\)
0.515296 + 0.857013i \(0.327682\pi\)
\(572\) 0 0
\(573\) 2.30068 0.0961124
\(574\) −22.3775 −0.934019
\(575\) −7.30068 −0.304459
\(576\) −11.5302 −0.480424
\(577\) 18.0438 0.751175 0.375588 0.926787i \(-0.377441\pi\)
0.375588 + 0.926787i \(0.377441\pi\)
\(578\) −43.7550 −1.81997
\(579\) −23.2249 −0.965192
\(580\) 5.86284 0.243441
\(581\) 15.8551 0.657780
\(582\) −6.33367 −0.262539
\(583\) −1.27744 −0.0529061
\(584\) −3.91542 −0.162021
\(585\) 0 0
\(586\) 65.1841 2.69273
\(587\) −46.1723 −1.90573 −0.952867 0.303390i \(-0.901882\pi\)
−0.952867 + 0.303390i \(0.901882\pi\)
\(588\) −2.53017 −0.104342
\(589\) −31.0964 −1.28131
\(590\) 78.6978 3.23994
\(591\) −11.3337 −0.466205
\(592\) 3.88708 0.159758
\(593\) 37.8505 1.55433 0.777166 0.629296i \(-0.216657\pi\)
0.777166 + 0.629296i \(0.216657\pi\)
\(594\) 1.40175 0.0575144
\(595\) −16.2929 −0.667945
\(596\) −35.0165 −1.43433
\(597\) 15.8144 0.647239
\(598\) 0 0
\(599\) −3.68283 −0.150476 −0.0752381 0.997166i \(-0.523972\pi\)
−0.0752381 + 0.997166i \(0.523972\pi\)
\(600\) 2.33367 0.0952715
\(601\) 2.65859 0.108446 0.0542230 0.998529i \(-0.482732\pi\)
0.0542230 + 0.998529i \(0.482732\pi\)
\(602\) −7.51368 −0.306235
\(603\) −13.3853 −0.545090
\(604\) 19.1635 0.779753
\(605\) −28.0913 −1.14207
\(606\) −19.6673 −0.798931
\(607\) 44.7298 1.81553 0.907763 0.419484i \(-0.137789\pi\)
0.907763 + 0.419484i \(0.137789\pi\)
\(608\) −41.6101 −1.68751
\(609\) 0.871581 0.0353182
\(610\) −46.3806 −1.87790
\(611\) 0 0
\(612\) 15.5059 0.626790
\(613\) 43.0031 1.73688 0.868440 0.495795i \(-0.165123\pi\)
0.868440 + 0.495795i \(0.165123\pi\)
\(614\) 9.53229 0.384692
\(615\) 27.9515 1.12711
\(616\) 0.743162 0.0299428
\(617\) −13.1723 −0.530295 −0.265148 0.964208i \(-0.585421\pi\)
−0.265148 + 0.964208i \(0.585421\pi\)
\(618\) 28.4729 1.14535
\(619\) −24.7705 −0.995611 −0.497806 0.867289i \(-0.665861\pi\)
−0.497806 + 0.867289i \(0.665861\pi\)
\(620\) 39.7911 1.59805
\(621\) 3.53017 0.141661
\(622\) −1.43685 −0.0576124
\(623\) −9.51368 −0.381157
\(624\) 0 0
\(625\) −31.0634 −1.24254
\(626\) −21.3358 −0.852749
\(627\) 3.46208 0.138262
\(628\) 18.6442 0.743985
\(629\) −8.96026 −0.357269
\(630\) 5.65859 0.225443
\(631\) −4.43374 −0.176504 −0.0882521 0.996098i \(-0.528128\pi\)
−0.0882521 + 0.996098i \(0.528128\pi\)
\(632\) −5.47858 −0.217926
\(633\) −6.60600 −0.262565
\(634\) 57.7989 2.29549
\(635\) −32.2006 −1.27784
\(636\) 4.90768 0.194602
\(637\) 0 0
\(638\) −1.22174 −0.0483690
\(639\) −4.27333 −0.169050
\(640\) 23.1568 0.915352
\(641\) −12.4698 −0.492529 −0.246264 0.969203i \(-0.579203\pi\)
−0.246264 + 0.969203i \(0.579203\pi\)
\(642\) 20.9593 0.827196
\(643\) −38.0274 −1.49965 −0.749826 0.661635i \(-0.769863\pi\)
−0.749826 + 0.661635i \(0.769863\pi\)
\(644\) 8.93192 0.351967
\(645\) 9.38526 0.369544
\(646\) 68.5694 2.69783
\(647\) 12.1888 0.479189 0.239595 0.970873i \(-0.422985\pi\)
0.239595 + 0.970873i \(0.422985\pi\)
\(648\) −1.12842 −0.0443285
\(649\) −9.15941 −0.359538
\(650\) 0 0
\(651\) 5.91542 0.231844
\(652\) 53.4388 2.09283
\(653\) 11.3290 0.443339 0.221670 0.975122i \(-0.428849\pi\)
0.221670 + 0.975122i \(0.428849\pi\)
\(654\) −13.4621 −0.526409
\(655\) −8.88344 −0.347105
\(656\) 27.9515 1.09132
\(657\) 3.46983 0.135371
\(658\) −18.5575 −0.723447
\(659\) 42.2929 1.64750 0.823749 0.566954i \(-0.191878\pi\)
0.823749 + 0.566954i \(0.191878\pi\)
\(660\) −4.43010 −0.172441
\(661\) 15.3533 0.597173 0.298586 0.954383i \(-0.403485\pi\)
0.298586 + 0.954383i \(0.403485\pi\)
\(662\) 50.6803 1.96975
\(663\) 0 0
\(664\) 17.8912 0.694313
\(665\) 13.9758 0.541957
\(666\) 3.11193 0.120585
\(667\) −3.07683 −0.119135
\(668\) 18.0165 0.697079
\(669\) −9.80350 −0.379025
\(670\) 75.7416 2.92615
\(671\) 5.39811 0.208392
\(672\) 7.91542 0.305344
\(673\) 38.7814 1.49491 0.747456 0.664311i \(-0.231275\pi\)
0.747456 + 0.664311i \(0.231275\pi\)
\(674\) 45.7385 1.76178
\(675\) −2.06808 −0.0796006
\(676\) 0 0
\(677\) −21.4729 −0.825272 −0.412636 0.910896i \(-0.635392\pi\)
−0.412636 + 0.910896i \(0.635392\pi\)
\(678\) 28.1645 1.08165
\(679\) 2.97576 0.114199
\(680\) −18.3853 −0.705042
\(681\) −16.2929 −0.624347
\(682\) −8.29194 −0.317515
\(683\) −5.15266 −0.197161 −0.0985805 0.995129i \(-0.531430\pi\)
−0.0985805 + 0.995129i \(0.531430\pi\)
\(684\) −13.3007 −0.508564
\(685\) −4.79475 −0.183198
\(686\) 2.12842 0.0812634
\(687\) −23.1919 −0.884825
\(688\) 9.38526 0.357810
\(689\) 0 0
\(690\) −19.9758 −0.760464
\(691\) −24.2852 −0.923852 −0.461926 0.886918i \(-0.652841\pi\)
−0.461926 + 0.886918i \(0.652841\pi\)
\(692\) −11.6147 −0.441526
\(693\) −0.658587 −0.0250176
\(694\) 41.7736 1.58571
\(695\) −3.00000 −0.113796
\(696\) 0.983509 0.0372798
\(697\) −64.4322 −2.44054
\(698\) 57.6375 2.18161
\(699\) 2.60699 0.0986056
\(700\) −5.23260 −0.197774
\(701\) −1.31307 −0.0495938 −0.0247969 0.999693i \(-0.507894\pi\)
−0.0247969 + 0.999693i \(0.507894\pi\)
\(702\) 0 0
\(703\) 7.68594 0.289881
\(704\) −7.59362 −0.286195
\(705\) 23.1800 0.873010
\(706\) −3.56315 −0.134101
\(707\) 9.24035 0.347519
\(708\) 35.1888 1.32247
\(709\) −32.7715 −1.23076 −0.615380 0.788231i \(-0.710997\pi\)
−0.615380 + 0.788231i \(0.710997\pi\)
\(710\) 24.1810 0.907497
\(711\) 4.85509 0.182080
\(712\) −10.7354 −0.402327
\(713\) −20.8824 −0.782054
\(714\) −13.0438 −0.488154
\(715\) 0 0
\(716\) 53.6803 2.00613
\(717\) 15.7432 0.587939
\(718\) 42.8252 1.59822
\(719\) −46.4642 −1.73282 −0.866411 0.499331i \(-0.833579\pi\)
−0.866411 + 0.499331i \(0.833579\pi\)
\(720\) −7.06808 −0.263412
\(721\) −13.3775 −0.498204
\(722\) −18.3775 −0.683940
\(723\) 29.8669 1.11076
\(724\) −44.5761 −1.65666
\(725\) 1.80250 0.0669433
\(726\) −22.4894 −0.834661
\(727\) 19.5059 0.723435 0.361717 0.932288i \(-0.382191\pi\)
0.361717 + 0.932288i \(0.382191\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −19.6343 −0.726700
\(731\) −21.6343 −0.800175
\(732\) −20.7385 −0.766518
\(733\) 21.6849 0.800952 0.400476 0.916307i \(-0.368845\pi\)
0.400476 + 0.916307i \(0.368845\pi\)
\(734\) −35.5684 −1.31285
\(735\) −2.65859 −0.0980635
\(736\) −27.9428 −1.02998
\(737\) −8.81535 −0.324718
\(738\) 22.3775 0.823727
\(739\) 44.3554 1.63164 0.815820 0.578306i \(-0.196286\pi\)
0.815820 + 0.578306i \(0.196286\pi\)
\(740\) −9.83496 −0.361540
\(741\) 0 0
\(742\) −4.12842 −0.151559
\(743\) 15.8112 0.580058 0.290029 0.957018i \(-0.406335\pi\)
0.290029 + 0.957018i \(0.406335\pi\)
\(744\) 6.67508 0.244720
\(745\) −36.7938 −1.34802
\(746\) −34.6617 −1.26905
\(747\) −15.8551 −0.580107
\(748\) 10.2120 0.373388
\(749\) −9.84734 −0.359814
\(750\) −16.5905 −0.605800
\(751\) −11.4008 −0.416019 −0.208010 0.978127i \(-0.566699\pi\)
−0.208010 + 0.978127i \(0.566699\pi\)
\(752\) 23.1800 0.845288
\(753\) 20.7189 0.755039
\(754\) 0 0
\(755\) 20.1362 0.732830
\(756\) 2.53017 0.0920213
\(757\) 0.623486 0.0226610 0.0113305 0.999936i \(-0.496393\pi\)
0.0113305 + 0.999936i \(0.496393\pi\)
\(758\) 58.5859 2.12793
\(759\) 2.32492 0.0843893
\(760\) 15.7705 0.572057
\(761\) 36.2929 1.31562 0.657809 0.753185i \(-0.271483\pi\)
0.657809 + 0.753185i \(0.271483\pi\)
\(762\) −25.7793 −0.933884
\(763\) 6.32492 0.228978
\(764\) −5.82111 −0.210600
\(765\) 16.2929 0.589072
\(766\) 26.0547 0.941395
\(767\) 0 0
\(768\) −4.52142 −0.163153
\(769\) −35.1305 −1.26684 −0.633420 0.773808i \(-0.718349\pi\)
−0.633420 + 0.773808i \(0.718349\pi\)
\(770\) 3.72667 0.134300
\(771\) −10.6508 −0.383581
\(772\) 58.7628 2.11492
\(773\) 51.7901 1.86276 0.931381 0.364047i \(-0.118605\pi\)
0.931381 + 0.364047i \(0.118605\pi\)
\(774\) 7.51368 0.270074
\(775\) 12.2336 0.439444
\(776\) 3.35790 0.120542
\(777\) −1.46208 −0.0524520
\(778\) 5.89682 0.211411
\(779\) 55.2687 1.98021
\(780\) 0 0
\(781\) −2.81436 −0.100706
\(782\) 46.0470 1.64664
\(783\) −0.871581 −0.0311478
\(784\) −2.65859 −0.0949495
\(785\) 19.5905 0.699215
\(786\) −7.11193 −0.253674
\(787\) 32.0041 1.14082 0.570412 0.821359i \(-0.306784\pi\)
0.570412 + 0.821359i \(0.306784\pi\)
\(788\) 28.6761 1.02154
\(789\) 23.6782 0.842966
\(790\) −27.4729 −0.977444
\(791\) −13.2326 −0.470497
\(792\) −0.743162 −0.0264071
\(793\) 0 0
\(794\) 1.72879 0.0613524
\(795\) 5.15677 0.182892
\(796\) −40.0130 −1.41822
\(797\) 15.0264 0.532261 0.266130 0.963937i \(-0.414255\pi\)
0.266130 + 0.963937i \(0.414255\pi\)
\(798\) 11.1888 0.396078
\(799\) −53.4332 −1.89033
\(800\) 16.3698 0.578758
\(801\) 9.51368 0.336149
\(802\) −20.7793 −0.733741
\(803\) 2.28519 0.0806425
\(804\) 33.8669 1.19440
\(805\) 9.38526 0.330787
\(806\) 0 0
\(807\) 0.477581 0.0168116
\(808\) 10.4270 0.366820
\(809\) −52.4729 −1.84485 −0.922425 0.386176i \(-0.873796\pi\)
−0.922425 + 0.386176i \(0.873796\pi\)
\(810\) −5.65859 −0.198822
\(811\) 29.6627 1.04160 0.520799 0.853679i \(-0.325634\pi\)
0.520799 + 0.853679i \(0.325634\pi\)
\(812\) −2.20525 −0.0773890
\(813\) −20.6421 −0.723950
\(814\) 2.04947 0.0718340
\(815\) 56.1511 1.96689
\(816\) 16.2929 0.570367
\(817\) 18.5575 0.649245
\(818\) 40.0634 1.40079
\(819\) 0 0
\(820\) −70.7220 −2.46972
\(821\) 5.77614 0.201589 0.100794 0.994907i \(-0.467862\pi\)
0.100794 + 0.994907i \(0.467862\pi\)
\(822\) −3.83860 −0.133886
\(823\) −11.2568 −0.392389 −0.196194 0.980565i \(-0.562858\pi\)
−0.196194 + 0.980565i \(0.562858\pi\)
\(824\) −15.0954 −0.525874
\(825\) −1.36201 −0.0474192
\(826\) −29.6014 −1.02996
\(827\) −20.0644 −0.697709 −0.348855 0.937177i \(-0.613429\pi\)
−0.348855 + 0.937177i \(0.613429\pi\)
\(828\) −8.93192 −0.310406
\(829\) −31.6617 −1.09966 −0.549828 0.835278i \(-0.685307\pi\)
−0.549828 + 0.835278i \(0.685307\pi\)
\(830\) 89.7174 3.11414
\(831\) 26.3600 0.914419
\(832\) 0 0
\(833\) 6.12842 0.212337
\(834\) −2.40175 −0.0831658
\(835\) 18.9309 0.655132
\(836\) −8.75965 −0.302959
\(837\) −5.91542 −0.204467
\(838\) 26.6530 0.920711
\(839\) 12.0506 0.416033 0.208016 0.978125i \(-0.433299\pi\)
0.208016 + 0.978125i \(0.433299\pi\)
\(840\) −3.00000 −0.103510
\(841\) −28.2403 −0.973805
\(842\) 58.6957 2.02279
\(843\) 0.205246 0.00706905
\(844\) 16.7143 0.575329
\(845\) 0 0
\(846\) 18.5575 0.638021
\(847\) 10.5663 0.363061
\(848\) 5.15677 0.177084
\(849\) −1.41824 −0.0486739
\(850\) −26.9758 −0.925261
\(851\) 5.16140 0.176931
\(852\) 10.8122 0.370421
\(853\) −36.1764 −1.23866 −0.619328 0.785133i \(-0.712595\pi\)
−0.619328 + 0.785133i \(0.712595\pi\)
\(854\) 17.4456 0.596976
\(855\) −13.9758 −0.477961
\(856\) −11.1119 −0.379798
\(857\) −30.6869 −1.04825 −0.524123 0.851643i \(-0.675607\pi\)
−0.524123 + 0.851643i \(0.675607\pi\)
\(858\) 0 0
\(859\) −5.93291 −0.202428 −0.101214 0.994865i \(-0.532273\pi\)
−0.101214 + 0.994865i \(0.532273\pi\)
\(860\) −23.7463 −0.809741
\(861\) −10.5137 −0.358305
\(862\) 1.89019 0.0643801
\(863\) −16.8299 −0.572895 −0.286447 0.958096i \(-0.592474\pi\)
−0.286447 + 0.958096i \(0.592474\pi\)
\(864\) −7.91542 −0.269288
\(865\) −12.2043 −0.414957
\(866\) −57.1073 −1.94058
\(867\) −20.5575 −0.698170
\(868\) −14.9670 −0.508014
\(869\) 3.19750 0.108468
\(870\) 4.93192 0.167208
\(871\) 0 0
\(872\) 7.13716 0.241695
\(873\) −2.97576 −0.100714
\(874\) −39.4982 −1.33605
\(875\) 7.79475 0.263511
\(876\) −8.77926 −0.296624
\(877\) 51.8679 1.75146 0.875728 0.482805i \(-0.160382\pi\)
0.875728 + 0.482805i \(0.160382\pi\)
\(878\) 72.3750 2.44254
\(879\) 30.6256 1.03298
\(880\) −4.65495 −0.156918
\(881\) −16.2609 −0.547845 −0.273923 0.961752i \(-0.588321\pi\)
−0.273923 + 0.961752i \(0.588321\pi\)
\(882\) −2.12842 −0.0716676
\(883\) −21.5049 −0.723699 −0.361849 0.932237i \(-0.617855\pi\)
−0.361849 + 0.932237i \(0.617855\pi\)
\(884\) 0 0
\(885\) 36.9748 1.24289
\(886\) 44.6266 1.49926
\(887\) 0.634347 0.0212993 0.0106496 0.999943i \(-0.496610\pi\)
0.0106496 + 0.999943i \(0.496610\pi\)
\(888\) −1.64984 −0.0553651
\(889\) 12.1119 0.406221
\(890\) −53.8340 −1.80452
\(891\) 0.658587 0.0220635
\(892\) 24.8045 0.830516
\(893\) 45.8340 1.53378
\(894\) −29.4565 −0.985171
\(895\) 56.4049 1.88541
\(896\) −8.71018 −0.290987
\(897\) 0 0
\(898\) 37.9329 1.26584
\(899\) 5.15577 0.171955
\(900\) 5.23260 0.174420
\(901\) −11.8871 −0.396016
\(902\) 14.7375 0.490706
\(903\) −3.53017 −0.117477
\(904\) −14.9319 −0.496628
\(905\) −46.8386 −1.55697
\(906\) 16.1207 0.535573
\(907\) −24.3533 −0.808637 −0.404319 0.914618i \(-0.632491\pi\)
−0.404319 + 0.914618i \(0.632491\pi\)
\(908\) 41.2239 1.36806
\(909\) −9.24035 −0.306483
\(910\) 0 0
\(911\) −2.71217 −0.0898582 −0.0449291 0.998990i \(-0.514306\pi\)
−0.0449291 + 0.998990i \(0.514306\pi\)
\(912\) −13.9758 −0.462784
\(913\) −10.4420 −0.345578
\(914\) −87.3837 −2.89040
\(915\) −21.7911 −0.720392
\(916\) 58.6793 1.93882
\(917\) 3.34141 0.110343
\(918\) 13.0438 0.430511
\(919\) 4.97576 0.164135 0.0820676 0.996627i \(-0.473848\pi\)
0.0820676 + 0.996627i \(0.473848\pi\)
\(920\) 10.5905 0.349159
\(921\) 4.47858 0.147574
\(922\) −76.5967 −2.52258
\(923\) 0 0
\(924\) 1.66633 0.0548184
\(925\) −3.02371 −0.0994190
\(926\) −27.9944 −0.919952
\(927\) 13.3775 0.439375
\(928\) 6.89893 0.226469
\(929\) −50.6334 −1.66123 −0.830613 0.556850i \(-0.812010\pi\)
−0.830613 + 0.556850i \(0.812010\pi\)
\(930\) 33.4729 1.09762
\(931\) −5.25684 −0.172286
\(932\) −6.59613 −0.216064
\(933\) −0.675078 −0.0221011
\(934\) −41.4466 −1.35617
\(935\) 10.7303 0.350919
\(936\) 0 0
\(937\) 19.5018 0.637097 0.318548 0.947907i \(-0.396805\pi\)
0.318548 + 0.947907i \(0.396805\pi\)
\(938\) −28.4894 −0.930213
\(939\) −10.0242 −0.327129
\(940\) −58.6493 −1.91293
\(941\) 9.95827 0.324630 0.162315 0.986739i \(-0.448104\pi\)
0.162315 + 0.986739i \(0.448104\pi\)
\(942\) 15.6838 0.511007
\(943\) 37.1150 1.20863
\(944\) 36.9748 1.20343
\(945\) 2.65859 0.0864838
\(946\) 4.94841 0.160887
\(947\) −60.0186 −1.95034 −0.975171 0.221452i \(-0.928921\pi\)
−0.975171 + 0.221452i \(0.928921\pi\)
\(948\) −12.2842 −0.398972
\(949\) 0 0
\(950\) 23.1393 0.750737
\(951\) 27.1558 0.880586
\(952\) 6.91542 0.224130
\(953\) −1.59925 −0.0518047 −0.0259023 0.999664i \(-0.508246\pi\)
−0.0259023 + 0.999664i \(0.508246\pi\)
\(954\) 4.12842 0.133663
\(955\) −6.11656 −0.197927
\(956\) −39.8328 −1.28829
\(957\) −0.574012 −0.0185552
\(958\) 10.7354 0.346845
\(959\) 1.80350 0.0582380
\(960\) 30.6540 0.989352
\(961\) 3.99225 0.128782
\(962\) 0 0
\(963\) 9.84734 0.317326
\(964\) −75.5684 −2.43389
\(965\) 61.7453 1.98765
\(966\) 7.51368 0.241749
\(967\) −29.7705 −0.957355 −0.478678 0.877991i \(-0.658884\pi\)
−0.478678 + 0.877991i \(0.658884\pi\)
\(968\) 11.9232 0.383225
\(969\) 32.2161 1.03493
\(970\) 16.8386 0.540655
\(971\) 36.1810 1.16110 0.580552 0.814223i \(-0.302837\pi\)
0.580552 + 0.814223i \(0.302837\pi\)
\(972\) −2.53017 −0.0811552
\(973\) 1.12842 0.0361755
\(974\) −26.5926 −0.852083
\(975\) 0 0
\(976\) −21.7911 −0.697517
\(977\) 4.30068 0.137591 0.0687955 0.997631i \(-0.478084\pi\)
0.0687955 + 0.997631i \(0.478084\pi\)
\(978\) 44.9536 1.43746
\(979\) 6.26558 0.200249
\(980\) 6.72667 0.214876
\(981\) −6.32492 −0.201939
\(982\) 0.144911 0.00462428
\(983\) −7.42599 −0.236852 −0.118426 0.992963i \(-0.537785\pi\)
−0.118426 + 0.992963i \(0.537785\pi\)
\(984\) −11.8638 −0.378205
\(985\) 30.1315 0.960070
\(986\) −11.3688 −0.362055
\(987\) −8.71892 −0.277526
\(988\) 0 0
\(989\) 12.4621 0.396271
\(990\) −3.72667 −0.118441
\(991\) 10.2764 0.326442 0.163221 0.986590i \(-0.447812\pi\)
0.163221 + 0.986590i \(0.447812\pi\)
\(992\) 46.8231 1.48663
\(993\) 23.8112 0.755627
\(994\) −9.09544 −0.288490
\(995\) −42.0438 −1.33288
\(996\) 40.1160 1.27113
\(997\) −23.3894 −0.740749 −0.370374 0.928883i \(-0.620771\pi\)
−0.370374 + 0.928883i \(0.620771\pi\)
\(998\) −4.56315 −0.144444
\(999\) 1.46208 0.0462583
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3549.2.a.l.1.1 3
13.4 even 6 273.2.k.b.211.1 yes 6
13.10 even 6 273.2.k.b.22.1 6
13.12 even 2 3549.2.a.m.1.3 3
39.17 odd 6 819.2.o.f.757.3 6
39.23 odd 6 819.2.o.f.568.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.k.b.22.1 6 13.10 even 6
273.2.k.b.211.1 yes 6 13.4 even 6
819.2.o.f.568.3 6 39.23 odd 6
819.2.o.f.757.3 6 39.17 odd 6
3549.2.a.l.1.1 3 1.1 even 1 trivial
3549.2.a.m.1.3 3 13.12 even 2