Properties

Label 6003.2.a.p.1.13
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $0$
Dimension $14$
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] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, 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("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.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: \(47.9341963334\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 124 x^{10} - 216 x^{9} - 420 x^{8} + 647 x^{7} + 750 x^{6} - 939 x^{5} - 717 x^{4} + 604 x^{3} + 352 x^{2} - 128 x - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(2.31598\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.31598 q^{2} +3.36376 q^{4} +2.86683 q^{5} +3.60131 q^{7} +3.15843 q^{8} +O(q^{10})\) \(q+2.31598 q^{2} +3.36376 q^{4} +2.86683 q^{5} +3.60131 q^{7} +3.15843 q^{8} +6.63952 q^{10} -0.489865 q^{11} +2.88891 q^{13} +8.34055 q^{14} +0.587346 q^{16} +6.07526 q^{17} -0.208958 q^{19} +9.64332 q^{20} -1.13452 q^{22} +1.00000 q^{23} +3.21872 q^{25} +6.69064 q^{26} +12.1139 q^{28} +1.00000 q^{29} -6.93798 q^{31} -4.95658 q^{32} +14.0702 q^{34} +10.3243 q^{35} -5.06467 q^{37} -0.483943 q^{38} +9.05469 q^{40} -9.28334 q^{41} +7.52223 q^{43} -1.64779 q^{44} +2.31598 q^{46} +1.61063 q^{47} +5.96943 q^{49} +7.45448 q^{50} +9.71758 q^{52} -10.1414 q^{53} -1.40436 q^{55} +11.3745 q^{56} +2.31598 q^{58} -10.2020 q^{59} -9.55359 q^{61} -16.0682 q^{62} -12.6540 q^{64} +8.28200 q^{65} -12.1104 q^{67} +20.4357 q^{68} +23.9110 q^{70} -2.69908 q^{71} -4.65365 q^{73} -11.7297 q^{74} -0.702885 q^{76} -1.76416 q^{77} +7.85970 q^{79} +1.68382 q^{80} -21.5000 q^{82} +1.79638 q^{83} +17.4167 q^{85} +17.4213 q^{86} -1.54721 q^{88} +7.62479 q^{89} +10.4038 q^{91} +3.36376 q^{92} +3.73019 q^{94} -0.599048 q^{95} +5.18136 q^{97} +13.8251 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.31598 1.63764 0.818822 0.574047i \(-0.194627\pi\)
0.818822 + 0.574047i \(0.194627\pi\)
\(3\) 0 0
\(4\) 3.36376 1.68188
\(5\) 2.86683 1.28209 0.641043 0.767505i \(-0.278502\pi\)
0.641043 + 0.767505i \(0.278502\pi\)
\(6\) 0 0
\(7\) 3.60131 1.36117 0.680583 0.732671i \(-0.261726\pi\)
0.680583 + 0.732671i \(0.261726\pi\)
\(8\) 3.15843 1.11667
\(9\) 0 0
\(10\) 6.63952 2.09960
\(11\) −0.489865 −0.147700 −0.0738500 0.997269i \(-0.523529\pi\)
−0.0738500 + 0.997269i \(0.523529\pi\)
\(12\) 0 0
\(13\) 2.88891 0.801238 0.400619 0.916245i \(-0.368795\pi\)
0.400619 + 0.916245i \(0.368795\pi\)
\(14\) 8.34055 2.22911
\(15\) 0 0
\(16\) 0.587346 0.146837
\(17\) 6.07526 1.47347 0.736733 0.676184i \(-0.236368\pi\)
0.736733 + 0.676184i \(0.236368\pi\)
\(18\) 0 0
\(19\) −0.208958 −0.0479383 −0.0239692 0.999713i \(-0.507630\pi\)
−0.0239692 + 0.999713i \(0.507630\pi\)
\(20\) 9.64332 2.15631
\(21\) 0 0
\(22\) −1.13452 −0.241880
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 3.21872 0.643743
\(26\) 6.69064 1.31214
\(27\) 0 0
\(28\) 12.1139 2.28932
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −6.93798 −1.24610 −0.623049 0.782183i \(-0.714106\pi\)
−0.623049 + 0.782183i \(0.714106\pi\)
\(32\) −4.95658 −0.876208
\(33\) 0 0
\(34\) 14.0702 2.41301
\(35\) 10.3243 1.74513
\(36\) 0 0
\(37\) −5.06467 −0.832627 −0.416313 0.909221i \(-0.636678\pi\)
−0.416313 + 0.909221i \(0.636678\pi\)
\(38\) −0.483943 −0.0785060
\(39\) 0 0
\(40\) 9.05469 1.43167
\(41\) −9.28334 −1.44981 −0.724907 0.688847i \(-0.758117\pi\)
−0.724907 + 0.688847i \(0.758117\pi\)
\(42\) 0 0
\(43\) 7.52223 1.14713 0.573565 0.819160i \(-0.305560\pi\)
0.573565 + 0.819160i \(0.305560\pi\)
\(44\) −1.64779 −0.248413
\(45\) 0 0
\(46\) 2.31598 0.341472
\(47\) 1.61063 0.234935 0.117468 0.993077i \(-0.462522\pi\)
0.117468 + 0.993077i \(0.462522\pi\)
\(48\) 0 0
\(49\) 5.96943 0.852775
\(50\) 7.45448 1.05422
\(51\) 0 0
\(52\) 9.71758 1.34759
\(53\) −10.1414 −1.39303 −0.696516 0.717541i \(-0.745267\pi\)
−0.696516 + 0.717541i \(0.745267\pi\)
\(54\) 0 0
\(55\) −1.40436 −0.189364
\(56\) 11.3745 1.51998
\(57\) 0 0
\(58\) 2.31598 0.304103
\(59\) −10.2020 −1.32818 −0.664091 0.747652i \(-0.731181\pi\)
−0.664091 + 0.747652i \(0.731181\pi\)
\(60\) 0 0
\(61\) −9.55359 −1.22321 −0.611606 0.791163i \(-0.709476\pi\)
−0.611606 + 0.791163i \(0.709476\pi\)
\(62\) −16.0682 −2.04067
\(63\) 0 0
\(64\) −12.6540 −1.58175
\(65\) 8.28200 1.02726
\(66\) 0 0
\(67\) −12.1104 −1.47953 −0.739763 0.672867i \(-0.765062\pi\)
−0.739763 + 0.672867i \(0.765062\pi\)
\(68\) 20.4357 2.47819
\(69\) 0 0
\(70\) 23.9110 2.85791
\(71\) −2.69908 −0.320321 −0.160161 0.987091i \(-0.551201\pi\)
−0.160161 + 0.987091i \(0.551201\pi\)
\(72\) 0 0
\(73\) −4.65365 −0.544668 −0.272334 0.962203i \(-0.587796\pi\)
−0.272334 + 0.962203i \(0.587796\pi\)
\(74\) −11.7297 −1.36355
\(75\) 0 0
\(76\) −0.702885 −0.0806265
\(77\) −1.76416 −0.201044
\(78\) 0 0
\(79\) 7.85970 0.884285 0.442142 0.896945i \(-0.354219\pi\)
0.442142 + 0.896945i \(0.354219\pi\)
\(80\) 1.68382 0.188257
\(81\) 0 0
\(82\) −21.5000 −2.37428
\(83\) 1.79638 0.197178 0.0985889 0.995128i \(-0.468567\pi\)
0.0985889 + 0.995128i \(0.468567\pi\)
\(84\) 0 0
\(85\) 17.4167 1.88911
\(86\) 17.4213 1.87859
\(87\) 0 0
\(88\) −1.54721 −0.164933
\(89\) 7.62479 0.808227 0.404113 0.914709i \(-0.367580\pi\)
0.404113 + 0.914709i \(0.367580\pi\)
\(90\) 0 0
\(91\) 10.4038 1.09062
\(92\) 3.36376 0.350696
\(93\) 0 0
\(94\) 3.73019 0.384740
\(95\) −0.599048 −0.0614611
\(96\) 0 0
\(97\) 5.18136 0.526087 0.263044 0.964784i \(-0.415274\pi\)
0.263044 + 0.964784i \(0.415274\pi\)
\(98\) 13.8251 1.39654
\(99\) 0 0
\(100\) 10.8270 1.08270
\(101\) 13.7012 1.36332 0.681662 0.731667i \(-0.261258\pi\)
0.681662 + 0.731667i \(0.261258\pi\)
\(102\) 0 0
\(103\) 13.5310 1.33325 0.666626 0.745393i \(-0.267738\pi\)
0.666626 + 0.745393i \(0.267738\pi\)
\(104\) 9.12441 0.894722
\(105\) 0 0
\(106\) −23.4873 −2.28129
\(107\) −0.886283 −0.0856802 −0.0428401 0.999082i \(-0.513641\pi\)
−0.0428401 + 0.999082i \(0.513641\pi\)
\(108\) 0 0
\(109\) −2.81821 −0.269936 −0.134968 0.990850i \(-0.543093\pi\)
−0.134968 + 0.990850i \(0.543093\pi\)
\(110\) −3.25247 −0.310111
\(111\) 0 0
\(112\) 2.11521 0.199869
\(113\) 16.3262 1.53584 0.767920 0.640546i \(-0.221292\pi\)
0.767920 + 0.640546i \(0.221292\pi\)
\(114\) 0 0
\(115\) 2.86683 0.267333
\(116\) 3.36376 0.312317
\(117\) 0 0
\(118\) −23.6275 −2.17509
\(119\) 21.8789 2.00563
\(120\) 0 0
\(121\) −10.7600 −0.978185
\(122\) −22.1259 −2.00318
\(123\) 0 0
\(124\) −23.3377 −2.09579
\(125\) −5.10664 −0.456751
\(126\) 0 0
\(127\) −5.94903 −0.527891 −0.263945 0.964538i \(-0.585024\pi\)
−0.263945 + 0.964538i \(0.585024\pi\)
\(128\) −19.3933 −1.71414
\(129\) 0 0
\(130\) 19.1809 1.68228
\(131\) −8.74603 −0.764144 −0.382072 0.924133i \(-0.624789\pi\)
−0.382072 + 0.924133i \(0.624789\pi\)
\(132\) 0 0
\(133\) −0.752524 −0.0652521
\(134\) −28.0475 −2.42294
\(135\) 0 0
\(136\) 19.1883 1.64538
\(137\) 4.87871 0.416816 0.208408 0.978042i \(-0.433172\pi\)
0.208408 + 0.978042i \(0.433172\pi\)
\(138\) 0 0
\(139\) 8.31445 0.705223 0.352611 0.935770i \(-0.385294\pi\)
0.352611 + 0.935770i \(0.385294\pi\)
\(140\) 34.7286 2.93510
\(141\) 0 0
\(142\) −6.25100 −0.524573
\(143\) −1.41518 −0.118343
\(144\) 0 0
\(145\) 2.86683 0.238077
\(146\) −10.7777 −0.891973
\(147\) 0 0
\(148\) −17.0363 −1.40038
\(149\) 22.8015 1.86797 0.933987 0.357307i \(-0.116305\pi\)
0.933987 + 0.357307i \(0.116305\pi\)
\(150\) 0 0
\(151\) −17.4224 −1.41782 −0.708908 0.705301i \(-0.750812\pi\)
−0.708908 + 0.705301i \(0.750812\pi\)
\(152\) −0.659981 −0.0535315
\(153\) 0 0
\(154\) −4.08575 −0.329239
\(155\) −19.8900 −1.59760
\(156\) 0 0
\(157\) 9.18382 0.732949 0.366474 0.930428i \(-0.380565\pi\)
0.366474 + 0.930428i \(0.380565\pi\)
\(158\) 18.2029 1.44814
\(159\) 0 0
\(160\) −14.2097 −1.12337
\(161\) 3.60131 0.283823
\(162\) 0 0
\(163\) 15.9352 1.24814 0.624072 0.781367i \(-0.285477\pi\)
0.624072 + 0.781367i \(0.285477\pi\)
\(164\) −31.2269 −2.43841
\(165\) 0 0
\(166\) 4.16037 0.322907
\(167\) −21.1433 −1.63612 −0.818058 0.575136i \(-0.804949\pi\)
−0.818058 + 0.575136i \(0.804949\pi\)
\(168\) 0 0
\(169\) −4.65422 −0.358017
\(170\) 40.3368 3.09369
\(171\) 0 0
\(172\) 25.3030 1.92933
\(173\) −13.5153 −1.02755 −0.513776 0.857924i \(-0.671754\pi\)
−0.513776 + 0.857924i \(0.671754\pi\)
\(174\) 0 0
\(175\) 11.5916 0.876242
\(176\) −0.287721 −0.0216877
\(177\) 0 0
\(178\) 17.6589 1.32359
\(179\) 7.42060 0.554641 0.277321 0.960777i \(-0.410554\pi\)
0.277321 + 0.960777i \(0.410554\pi\)
\(180\) 0 0
\(181\) 4.32130 0.321200 0.160600 0.987020i \(-0.448657\pi\)
0.160600 + 0.987020i \(0.448657\pi\)
\(182\) 24.0951 1.78605
\(183\) 0 0
\(184\) 3.15843 0.232843
\(185\) −14.5196 −1.06750
\(186\) 0 0
\(187\) −2.97606 −0.217631
\(188\) 5.41778 0.395132
\(189\) 0 0
\(190\) −1.38738 −0.100651
\(191\) 1.47681 0.106858 0.0534291 0.998572i \(-0.482985\pi\)
0.0534291 + 0.998572i \(0.482985\pi\)
\(192\) 0 0
\(193\) −15.2102 −1.09485 −0.547427 0.836853i \(-0.684393\pi\)
−0.547427 + 0.836853i \(0.684393\pi\)
\(194\) 11.9999 0.861543
\(195\) 0 0
\(196\) 20.0797 1.43426
\(197\) 2.57123 0.183192 0.0915961 0.995796i \(-0.470803\pi\)
0.0915961 + 0.995796i \(0.470803\pi\)
\(198\) 0 0
\(199\) −15.7035 −1.11319 −0.556596 0.830783i \(-0.687893\pi\)
−0.556596 + 0.830783i \(0.687893\pi\)
\(200\) 10.1661 0.718852
\(201\) 0 0
\(202\) 31.7318 2.23264
\(203\) 3.60131 0.252762
\(204\) 0 0
\(205\) −26.6138 −1.85879
\(206\) 31.3376 2.18339
\(207\) 0 0
\(208\) 1.69679 0.117651
\(209\) 0.102361 0.00708049
\(210\) 0 0
\(211\) −23.4076 −1.61145 −0.805723 0.592292i \(-0.798223\pi\)
−0.805723 + 0.592292i \(0.798223\pi\)
\(212\) −34.1133 −2.34291
\(213\) 0 0
\(214\) −2.05261 −0.140314
\(215\) 21.5650 1.47072
\(216\) 0 0
\(217\) −24.9858 −1.69615
\(218\) −6.52692 −0.442059
\(219\) 0 0
\(220\) −4.72393 −0.318487
\(221\) 17.5508 1.18060
\(222\) 0 0
\(223\) −15.7410 −1.05410 −0.527048 0.849836i \(-0.676701\pi\)
−0.527048 + 0.849836i \(0.676701\pi\)
\(224\) −17.8502 −1.19267
\(225\) 0 0
\(226\) 37.8111 2.51516
\(227\) 22.0224 1.46168 0.730840 0.682549i \(-0.239129\pi\)
0.730840 + 0.682549i \(0.239129\pi\)
\(228\) 0 0
\(229\) −16.1546 −1.06752 −0.533762 0.845635i \(-0.679222\pi\)
−0.533762 + 0.845635i \(0.679222\pi\)
\(230\) 6.63952 0.437797
\(231\) 0 0
\(232\) 3.15843 0.207361
\(233\) 18.8801 1.23688 0.618440 0.785832i \(-0.287765\pi\)
0.618440 + 0.785832i \(0.287765\pi\)
\(234\) 0 0
\(235\) 4.61741 0.301207
\(236\) −34.3169 −2.23384
\(237\) 0 0
\(238\) 50.6710 3.28451
\(239\) 13.8197 0.893921 0.446960 0.894554i \(-0.352507\pi\)
0.446960 + 0.894554i \(0.352507\pi\)
\(240\) 0 0
\(241\) −4.07448 −0.262460 −0.131230 0.991352i \(-0.541893\pi\)
−0.131230 + 0.991352i \(0.541893\pi\)
\(242\) −24.9200 −1.60192
\(243\) 0 0
\(244\) −32.1359 −2.05729
\(245\) 17.1133 1.09333
\(246\) 0 0
\(247\) −0.603661 −0.0384100
\(248\) −21.9131 −1.39149
\(249\) 0 0
\(250\) −11.8269 −0.747996
\(251\) −6.49386 −0.409889 −0.204945 0.978774i \(-0.565701\pi\)
−0.204945 + 0.978774i \(0.565701\pi\)
\(252\) 0 0
\(253\) −0.489865 −0.0307976
\(254\) −13.7778 −0.864498
\(255\) 0 0
\(256\) −19.6064 −1.22540
\(257\) −15.0388 −0.938097 −0.469049 0.883172i \(-0.655403\pi\)
−0.469049 + 0.883172i \(0.655403\pi\)
\(258\) 0 0
\(259\) −18.2394 −1.13334
\(260\) 27.8586 1.72772
\(261\) 0 0
\(262\) −20.2556 −1.25140
\(263\) 18.5844 1.14597 0.572983 0.819567i \(-0.305786\pi\)
0.572983 + 0.819567i \(0.305786\pi\)
\(264\) 0 0
\(265\) −29.0737 −1.78599
\(266\) −1.74283 −0.106860
\(267\) 0 0
\(268\) −40.7366 −2.48838
\(269\) 10.7377 0.654686 0.327343 0.944905i \(-0.393847\pi\)
0.327343 + 0.944905i \(0.393847\pi\)
\(270\) 0 0
\(271\) 2.66418 0.161837 0.0809187 0.996721i \(-0.474215\pi\)
0.0809187 + 0.996721i \(0.474215\pi\)
\(272\) 3.56828 0.216359
\(273\) 0 0
\(274\) 11.2990 0.682597
\(275\) −1.57674 −0.0950809
\(276\) 0 0
\(277\) 17.2762 1.03803 0.519013 0.854767i \(-0.326300\pi\)
0.519013 + 0.854767i \(0.326300\pi\)
\(278\) 19.2561 1.15490
\(279\) 0 0
\(280\) 32.6087 1.94874
\(281\) 26.7094 1.59335 0.796674 0.604409i \(-0.206591\pi\)
0.796674 + 0.604409i \(0.206591\pi\)
\(282\) 0 0
\(283\) 25.3486 1.50682 0.753410 0.657551i \(-0.228408\pi\)
0.753410 + 0.657551i \(0.228408\pi\)
\(284\) −9.07904 −0.538742
\(285\) 0 0
\(286\) −3.27752 −0.193804
\(287\) −33.4322 −1.97344
\(288\) 0 0
\(289\) 19.9087 1.17110
\(290\) 6.63952 0.389886
\(291\) 0 0
\(292\) −15.6537 −0.916066
\(293\) 5.13931 0.300242 0.150121 0.988668i \(-0.452034\pi\)
0.150121 + 0.988668i \(0.452034\pi\)
\(294\) 0 0
\(295\) −29.2473 −1.70284
\(296\) −15.9964 −0.929773
\(297\) 0 0
\(298\) 52.8079 3.05908
\(299\) 2.88891 0.167070
\(300\) 0 0
\(301\) 27.0899 1.56144
\(302\) −40.3500 −2.32188
\(303\) 0 0
\(304\) −0.122731 −0.00703910
\(305\) −27.3885 −1.56826
\(306\) 0 0
\(307\) 19.5783 1.11739 0.558695 0.829373i \(-0.311302\pi\)
0.558695 + 0.829373i \(0.311302\pi\)
\(308\) −5.93419 −0.338132
\(309\) 0 0
\(310\) −46.0648 −2.61631
\(311\) 34.2429 1.94174 0.970868 0.239617i \(-0.0770218\pi\)
0.970868 + 0.239617i \(0.0770218\pi\)
\(312\) 0 0
\(313\) −6.49992 −0.367397 −0.183699 0.982983i \(-0.558807\pi\)
−0.183699 + 0.982983i \(0.558807\pi\)
\(314\) 21.2695 1.20031
\(315\) 0 0
\(316\) 26.4381 1.48726
\(317\) 15.0420 0.844842 0.422421 0.906400i \(-0.361180\pi\)
0.422421 + 0.906400i \(0.361180\pi\)
\(318\) 0 0
\(319\) −0.489865 −0.0274272
\(320\) −36.2770 −2.02794
\(321\) 0 0
\(322\) 8.34055 0.464801
\(323\) −1.26948 −0.0706355
\(324\) 0 0
\(325\) 9.29857 0.515792
\(326\) 36.9057 2.04402
\(327\) 0 0
\(328\) −29.3208 −1.61897
\(329\) 5.80039 0.319786
\(330\) 0 0
\(331\) −2.36598 −0.130046 −0.0650229 0.997884i \(-0.520712\pi\)
−0.0650229 + 0.997884i \(0.520712\pi\)
\(332\) 6.04257 0.331629
\(333\) 0 0
\(334\) −48.9674 −2.67938
\(335\) −34.7186 −1.89688
\(336\) 0 0
\(337\) 9.98588 0.543965 0.271983 0.962302i \(-0.412321\pi\)
0.271983 + 0.962302i \(0.412321\pi\)
\(338\) −10.7791 −0.586304
\(339\) 0 0
\(340\) 58.5856 3.17725
\(341\) 3.39868 0.184049
\(342\) 0 0
\(343\) −3.71141 −0.200397
\(344\) 23.7585 1.28097
\(345\) 0 0
\(346\) −31.3012 −1.68277
\(347\) −6.98651 −0.375055 −0.187528 0.982259i \(-0.560047\pi\)
−0.187528 + 0.982259i \(0.560047\pi\)
\(348\) 0 0
\(349\) 26.5857 1.42310 0.711550 0.702636i \(-0.247994\pi\)
0.711550 + 0.702636i \(0.247994\pi\)
\(350\) 26.8459 1.43497
\(351\) 0 0
\(352\) 2.42806 0.129416
\(353\) −7.02757 −0.374040 −0.187020 0.982356i \(-0.559883\pi\)
−0.187020 + 0.982356i \(0.559883\pi\)
\(354\) 0 0
\(355\) −7.73780 −0.410680
\(356\) 25.6480 1.35934
\(357\) 0 0
\(358\) 17.1859 0.908305
\(359\) 26.5303 1.40022 0.700108 0.714037i \(-0.253135\pi\)
0.700108 + 0.714037i \(0.253135\pi\)
\(360\) 0 0
\(361\) −18.9563 −0.997702
\(362\) 10.0080 0.526011
\(363\) 0 0
\(364\) 34.9960 1.83429
\(365\) −13.3412 −0.698311
\(366\) 0 0
\(367\) −34.2926 −1.79006 −0.895029 0.446007i \(-0.852845\pi\)
−0.895029 + 0.446007i \(0.852845\pi\)
\(368\) 0.587346 0.0306175
\(369\) 0 0
\(370\) −33.6270 −1.74818
\(371\) −36.5224 −1.89615
\(372\) 0 0
\(373\) −14.1764 −0.734028 −0.367014 0.930215i \(-0.619620\pi\)
−0.367014 + 0.930215i \(0.619620\pi\)
\(374\) −6.89249 −0.356402
\(375\) 0 0
\(376\) 5.08707 0.262346
\(377\) 2.88891 0.148786
\(378\) 0 0
\(379\) −25.6733 −1.31875 −0.659374 0.751815i \(-0.729178\pi\)
−0.659374 + 0.751815i \(0.729178\pi\)
\(380\) −2.01505 −0.103370
\(381\) 0 0
\(382\) 3.42026 0.174996
\(383\) −17.9881 −0.919147 −0.459573 0.888140i \(-0.651998\pi\)
−0.459573 + 0.888140i \(0.651998\pi\)
\(384\) 0 0
\(385\) −5.05754 −0.257756
\(386\) −35.2265 −1.79298
\(387\) 0 0
\(388\) 17.4288 0.884814
\(389\) 23.3777 1.18530 0.592648 0.805462i \(-0.298083\pi\)
0.592648 + 0.805462i \(0.298083\pi\)
\(390\) 0 0
\(391\) 6.07526 0.307239
\(392\) 18.8540 0.952272
\(393\) 0 0
\(394\) 5.95491 0.300004
\(395\) 22.5324 1.13373
\(396\) 0 0
\(397\) 34.1121 1.71204 0.856018 0.516945i \(-0.172931\pi\)
0.856018 + 0.516945i \(0.172931\pi\)
\(398\) −36.3690 −1.82301
\(399\) 0 0
\(400\) 1.89050 0.0945250
\(401\) 4.90447 0.244918 0.122459 0.992474i \(-0.460922\pi\)
0.122459 + 0.992474i \(0.460922\pi\)
\(402\) 0 0
\(403\) −20.0432 −0.998422
\(404\) 46.0876 2.29294
\(405\) 0 0
\(406\) 8.34055 0.413935
\(407\) 2.48101 0.122979
\(408\) 0 0
\(409\) −6.13942 −0.303575 −0.151787 0.988413i \(-0.548503\pi\)
−0.151787 + 0.988413i \(0.548503\pi\)
\(410\) −61.6369 −3.04403
\(411\) 0 0
\(412\) 45.5151 2.24237
\(413\) −36.7404 −1.80788
\(414\) 0 0
\(415\) 5.14990 0.252799
\(416\) −14.3191 −0.702052
\(417\) 0 0
\(418\) 0.237067 0.0115953
\(419\) −1.46153 −0.0714006 −0.0357003 0.999363i \(-0.511366\pi\)
−0.0357003 + 0.999363i \(0.511366\pi\)
\(420\) 0 0
\(421\) 32.7873 1.59796 0.798978 0.601360i \(-0.205374\pi\)
0.798978 + 0.601360i \(0.205374\pi\)
\(422\) −54.2115 −2.63898
\(423\) 0 0
\(424\) −32.0310 −1.55556
\(425\) 19.5545 0.948534
\(426\) 0 0
\(427\) −34.4054 −1.66499
\(428\) −2.98124 −0.144104
\(429\) 0 0
\(430\) 49.9440 2.40851
\(431\) 27.1771 1.30908 0.654539 0.756029i \(-0.272863\pi\)
0.654539 + 0.756029i \(0.272863\pi\)
\(432\) 0 0
\(433\) 30.6675 1.47379 0.736893 0.676009i \(-0.236292\pi\)
0.736893 + 0.676009i \(0.236292\pi\)
\(434\) −57.8666 −2.77769
\(435\) 0 0
\(436\) −9.47978 −0.453999
\(437\) −0.208958 −0.00999584
\(438\) 0 0
\(439\) 3.50454 0.167263 0.0836313 0.996497i \(-0.473348\pi\)
0.0836313 + 0.996497i \(0.473348\pi\)
\(440\) −4.43558 −0.211458
\(441\) 0 0
\(442\) 40.6474 1.93340
\(443\) 5.91295 0.280933 0.140466 0.990085i \(-0.455140\pi\)
0.140466 + 0.990085i \(0.455140\pi\)
\(444\) 0 0
\(445\) 21.8590 1.03622
\(446\) −36.4558 −1.72623
\(447\) 0 0
\(448\) −45.5711 −2.15303
\(449\) −22.8726 −1.07942 −0.539711 0.841850i \(-0.681467\pi\)
−0.539711 + 0.841850i \(0.681467\pi\)
\(450\) 0 0
\(451\) 4.54759 0.214137
\(452\) 54.9174 2.58310
\(453\) 0 0
\(454\) 51.0035 2.39371
\(455\) 29.8261 1.39827
\(456\) 0 0
\(457\) 16.5249 0.773001 0.386500 0.922289i \(-0.373684\pi\)
0.386500 + 0.922289i \(0.373684\pi\)
\(458\) −37.4136 −1.74822
\(459\) 0 0
\(460\) 9.64332 0.449622
\(461\) 6.64056 0.309282 0.154641 0.987971i \(-0.450578\pi\)
0.154641 + 0.987971i \(0.450578\pi\)
\(462\) 0 0
\(463\) −18.7173 −0.869868 −0.434934 0.900462i \(-0.643228\pi\)
−0.434934 + 0.900462i \(0.643228\pi\)
\(464\) 0.587346 0.0272669
\(465\) 0 0
\(466\) 43.7260 2.02557
\(467\) −7.74364 −0.358333 −0.179167 0.983819i \(-0.557340\pi\)
−0.179167 + 0.983819i \(0.557340\pi\)
\(468\) 0 0
\(469\) −43.6135 −2.01388
\(470\) 10.6938 0.493270
\(471\) 0 0
\(472\) −32.2222 −1.48315
\(473\) −3.68488 −0.169431
\(474\) 0 0
\(475\) −0.672578 −0.0308600
\(476\) 73.5952 3.37323
\(477\) 0 0
\(478\) 32.0061 1.46392
\(479\) −19.4249 −0.887547 −0.443773 0.896139i \(-0.646360\pi\)
−0.443773 + 0.896139i \(0.646360\pi\)
\(480\) 0 0
\(481\) −14.6314 −0.667133
\(482\) −9.43640 −0.429816
\(483\) 0 0
\(484\) −36.1941 −1.64519
\(485\) 14.8541 0.674489
\(486\) 0 0
\(487\) 10.6818 0.484040 0.242020 0.970271i \(-0.422190\pi\)
0.242020 + 0.970271i \(0.422190\pi\)
\(488\) −30.1743 −1.36593
\(489\) 0 0
\(490\) 39.6341 1.79049
\(491\) −1.90753 −0.0860855 −0.0430427 0.999073i \(-0.513705\pi\)
−0.0430427 + 0.999073i \(0.513705\pi\)
\(492\) 0 0
\(493\) 6.07526 0.273616
\(494\) −1.39807 −0.0629020
\(495\) 0 0
\(496\) −4.07499 −0.182973
\(497\) −9.72021 −0.436011
\(498\) 0 0
\(499\) −13.2192 −0.591774 −0.295887 0.955223i \(-0.595615\pi\)
−0.295887 + 0.955223i \(0.595615\pi\)
\(500\) −17.1775 −0.768200
\(501\) 0 0
\(502\) −15.0397 −0.671253
\(503\) −27.6893 −1.23460 −0.617302 0.786726i \(-0.711774\pi\)
−0.617302 + 0.786726i \(0.711774\pi\)
\(504\) 0 0
\(505\) 39.2791 1.74790
\(506\) −1.13452 −0.0504355
\(507\) 0 0
\(508\) −20.0111 −0.887848
\(509\) −9.69520 −0.429732 −0.214866 0.976644i \(-0.568931\pi\)
−0.214866 + 0.976644i \(0.568931\pi\)
\(510\) 0 0
\(511\) −16.7592 −0.741384
\(512\) −6.62141 −0.292628
\(513\) 0 0
\(514\) −34.8296 −1.53627
\(515\) 38.7911 1.70934
\(516\) 0 0
\(517\) −0.788994 −0.0346999
\(518\) −42.2422 −1.85601
\(519\) 0 0
\(520\) 26.1581 1.14711
\(521\) −24.2177 −1.06100 −0.530498 0.847686i \(-0.677995\pi\)
−0.530498 + 0.847686i \(0.677995\pi\)
\(522\) 0 0
\(523\) −32.5030 −1.42126 −0.710629 0.703567i \(-0.751590\pi\)
−0.710629 + 0.703567i \(0.751590\pi\)
\(524\) −29.4195 −1.28520
\(525\) 0 0
\(526\) 43.0412 1.87668
\(527\) −42.1500 −1.83608
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −67.3342 −2.92481
\(531\) 0 0
\(532\) −2.53131 −0.109746
\(533\) −26.8187 −1.16165
\(534\) 0 0
\(535\) −2.54082 −0.109849
\(536\) −38.2500 −1.65215
\(537\) 0 0
\(538\) 24.8682 1.07214
\(539\) −2.92422 −0.125955
\(540\) 0 0
\(541\) −2.49329 −0.107195 −0.0535974 0.998563i \(-0.517069\pi\)
−0.0535974 + 0.998563i \(0.517069\pi\)
\(542\) 6.17019 0.265032
\(543\) 0 0
\(544\) −30.1125 −1.29106
\(545\) −8.07934 −0.346081
\(546\) 0 0
\(547\) −44.0449 −1.88322 −0.941612 0.336699i \(-0.890690\pi\)
−0.941612 + 0.336699i \(0.890690\pi\)
\(548\) 16.4108 0.701035
\(549\) 0 0
\(550\) −3.65169 −0.155709
\(551\) −0.208958 −0.00890193
\(552\) 0 0
\(553\) 28.3052 1.20366
\(554\) 40.0113 1.69992
\(555\) 0 0
\(556\) 27.9678 1.18610
\(557\) −1.56699 −0.0663954 −0.0331977 0.999449i \(-0.510569\pi\)
−0.0331977 + 0.999449i \(0.510569\pi\)
\(558\) 0 0
\(559\) 21.7310 0.919124
\(560\) 6.06396 0.256249
\(561\) 0 0
\(562\) 61.8584 2.60934
\(563\) 24.2279 1.02109 0.510543 0.859852i \(-0.329444\pi\)
0.510543 + 0.859852i \(0.329444\pi\)
\(564\) 0 0
\(565\) 46.8045 1.96908
\(566\) 58.7069 2.46763
\(567\) 0 0
\(568\) −8.52485 −0.357695
\(569\) 14.7118 0.616753 0.308376 0.951264i \(-0.400214\pi\)
0.308376 + 0.951264i \(0.400214\pi\)
\(570\) 0 0
\(571\) 17.6613 0.739103 0.369551 0.929210i \(-0.379511\pi\)
0.369551 + 0.929210i \(0.379511\pi\)
\(572\) −4.76031 −0.199038
\(573\) 0 0
\(574\) −77.4282 −3.23179
\(575\) 3.21872 0.134230
\(576\) 0 0
\(577\) 12.7470 0.530665 0.265333 0.964157i \(-0.414518\pi\)
0.265333 + 0.964157i \(0.414518\pi\)
\(578\) 46.1082 1.91785
\(579\) 0 0
\(580\) 9.64332 0.400417
\(581\) 6.46930 0.268392
\(582\) 0 0
\(583\) 4.96793 0.205751
\(584\) −14.6982 −0.608217
\(585\) 0 0
\(586\) 11.9025 0.491689
\(587\) 0.658885 0.0271951 0.0135976 0.999908i \(-0.495672\pi\)
0.0135976 + 0.999908i \(0.495672\pi\)
\(588\) 0 0
\(589\) 1.44975 0.0597359
\(590\) −67.7361 −2.78865
\(591\) 0 0
\(592\) −2.97471 −0.122260
\(593\) 34.1532 1.40250 0.701251 0.712914i \(-0.252625\pi\)
0.701251 + 0.712914i \(0.252625\pi\)
\(594\) 0 0
\(595\) 62.7230 2.57139
\(596\) 76.6988 3.14171
\(597\) 0 0
\(598\) 6.69064 0.273601
\(599\) 11.7324 0.479373 0.239687 0.970850i \(-0.422955\pi\)
0.239687 + 0.970850i \(0.422955\pi\)
\(600\) 0 0
\(601\) −22.1962 −0.905400 −0.452700 0.891663i \(-0.649539\pi\)
−0.452700 + 0.891663i \(0.649539\pi\)
\(602\) 62.7396 2.55707
\(603\) 0 0
\(604\) −58.6048 −2.38460
\(605\) −30.8472 −1.25412
\(606\) 0 0
\(607\) −48.0676 −1.95100 −0.975501 0.219995i \(-0.929396\pi\)
−0.975501 + 0.219995i \(0.929396\pi\)
\(608\) 1.03572 0.0420040
\(609\) 0 0
\(610\) −63.4312 −2.56825
\(611\) 4.65297 0.188239
\(612\) 0 0
\(613\) 7.40933 0.299260 0.149630 0.988742i \(-0.452192\pi\)
0.149630 + 0.988742i \(0.452192\pi\)
\(614\) 45.3428 1.82989
\(615\) 0 0
\(616\) −5.57197 −0.224501
\(617\) 13.8020 0.555649 0.277824 0.960632i \(-0.410387\pi\)
0.277824 + 0.960632i \(0.410387\pi\)
\(618\) 0 0
\(619\) −15.4327 −0.620294 −0.310147 0.950689i \(-0.600378\pi\)
−0.310147 + 0.950689i \(0.600378\pi\)
\(620\) −66.9052 −2.68698
\(621\) 0 0
\(622\) 79.3058 3.17987
\(623\) 27.4592 1.10013
\(624\) 0 0
\(625\) −30.7334 −1.22934
\(626\) −15.0537 −0.601666
\(627\) 0 0
\(628\) 30.8921 1.23273
\(629\) −30.7692 −1.22685
\(630\) 0 0
\(631\) 29.3091 1.16678 0.583388 0.812193i \(-0.301727\pi\)
0.583388 + 0.812193i \(0.301727\pi\)
\(632\) 24.8243 0.987458
\(633\) 0 0
\(634\) 34.8369 1.38355
\(635\) −17.0549 −0.676801
\(636\) 0 0
\(637\) 17.2451 0.683276
\(638\) −1.13452 −0.0449160
\(639\) 0 0
\(640\) −55.5973 −2.19768
\(641\) −1.75725 −0.0694071 −0.0347035 0.999398i \(-0.511049\pi\)
−0.0347035 + 0.999398i \(0.511049\pi\)
\(642\) 0 0
\(643\) −33.6057 −1.32528 −0.662639 0.748939i \(-0.730564\pi\)
−0.662639 + 0.748939i \(0.730564\pi\)
\(644\) 12.1139 0.477356
\(645\) 0 0
\(646\) −2.94008 −0.115676
\(647\) −20.4559 −0.804204 −0.402102 0.915595i \(-0.631720\pi\)
−0.402102 + 0.915595i \(0.631720\pi\)
\(648\) 0 0
\(649\) 4.99759 0.196172
\(650\) 21.5353 0.844684
\(651\) 0 0
\(652\) 53.6022 2.09923
\(653\) −11.8457 −0.463559 −0.231780 0.972768i \(-0.574455\pi\)
−0.231780 + 0.972768i \(0.574455\pi\)
\(654\) 0 0
\(655\) −25.0734 −0.979698
\(656\) −5.45253 −0.212886
\(657\) 0 0
\(658\) 13.4336 0.523695
\(659\) −1.45749 −0.0567757 −0.0283878 0.999597i \(-0.509037\pi\)
−0.0283878 + 0.999597i \(0.509037\pi\)
\(660\) 0 0
\(661\) −3.17925 −0.123659 −0.0618293 0.998087i \(-0.519693\pi\)
−0.0618293 + 0.998087i \(0.519693\pi\)
\(662\) −5.47955 −0.212969
\(663\) 0 0
\(664\) 5.67373 0.220183
\(665\) −2.15736 −0.0836588
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) −71.1208 −2.75175
\(669\) 0 0
\(670\) −80.4075 −3.10641
\(671\) 4.67997 0.180668
\(672\) 0 0
\(673\) −35.6189 −1.37301 −0.686503 0.727127i \(-0.740855\pi\)
−0.686503 + 0.727127i \(0.740855\pi\)
\(674\) 23.1271 0.890822
\(675\) 0 0
\(676\) −15.6557 −0.602141
\(677\) 34.7094 1.33399 0.666994 0.745063i \(-0.267580\pi\)
0.666994 + 0.745063i \(0.267580\pi\)
\(678\) 0 0
\(679\) 18.6597 0.716092
\(680\) 55.0095 2.10952
\(681\) 0 0
\(682\) 7.87126 0.301406
\(683\) −38.2453 −1.46342 −0.731708 0.681618i \(-0.761277\pi\)
−0.731708 + 0.681618i \(0.761277\pi\)
\(684\) 0 0
\(685\) 13.9864 0.534394
\(686\) −8.59555 −0.328179
\(687\) 0 0
\(688\) 4.41815 0.168441
\(689\) −29.2976 −1.11615
\(690\) 0 0
\(691\) −18.9352 −0.720331 −0.360165 0.932888i \(-0.617280\pi\)
−0.360165 + 0.932888i \(0.617280\pi\)
\(692\) −45.4623 −1.72822
\(693\) 0 0
\(694\) −16.1806 −0.614207
\(695\) 23.8361 0.904156
\(696\) 0 0
\(697\) −56.3987 −2.13625
\(698\) 61.5719 2.33053
\(699\) 0 0
\(700\) 38.9913 1.47373
\(701\) −23.3256 −0.880995 −0.440498 0.897754i \(-0.645198\pi\)
−0.440498 + 0.897754i \(0.645198\pi\)
\(702\) 0 0
\(703\) 1.05831 0.0399148
\(704\) 6.19877 0.233625
\(705\) 0 0
\(706\) −16.2757 −0.612544
\(707\) 49.3424 1.85571
\(708\) 0 0
\(709\) −24.5991 −0.923839 −0.461919 0.886922i \(-0.652839\pi\)
−0.461919 + 0.886922i \(0.652839\pi\)
\(710\) −17.9206 −0.672547
\(711\) 0 0
\(712\) 24.0824 0.902526
\(713\) −6.93798 −0.259829
\(714\) 0 0
\(715\) −4.05707 −0.151726
\(716\) 24.9611 0.932839
\(717\) 0 0
\(718\) 61.4437 2.29306
\(719\) −23.4718 −0.875352 −0.437676 0.899133i \(-0.644198\pi\)
−0.437676 + 0.899133i \(0.644198\pi\)
\(720\) 0 0
\(721\) 48.7294 1.81478
\(722\) −43.9025 −1.63388
\(723\) 0 0
\(724\) 14.5358 0.540219
\(725\) 3.21872 0.119540
\(726\) 0 0
\(727\) −22.5482 −0.836266 −0.418133 0.908386i \(-0.637315\pi\)
−0.418133 + 0.908386i \(0.637315\pi\)
\(728\) 32.8598 1.21787
\(729\) 0 0
\(730\) −30.8980 −1.14359
\(731\) 45.6995 1.69026
\(732\) 0 0
\(733\) −18.3331 −0.677149 −0.338574 0.940940i \(-0.609945\pi\)
−0.338574 + 0.940940i \(0.609945\pi\)
\(734\) −79.4209 −2.93148
\(735\) 0 0
\(736\) −4.95658 −0.182702
\(737\) 5.93249 0.218526
\(738\) 0 0
\(739\) 33.6701 1.23858 0.619288 0.785164i \(-0.287421\pi\)
0.619288 + 0.785164i \(0.287421\pi\)
\(740\) −48.8402 −1.79540
\(741\) 0 0
\(742\) −84.5851 −3.10522
\(743\) 18.9673 0.695843 0.347921 0.937524i \(-0.386888\pi\)
0.347921 + 0.937524i \(0.386888\pi\)
\(744\) 0 0
\(745\) 65.3681 2.39490
\(746\) −32.8323 −1.20208
\(747\) 0 0
\(748\) −10.0107 −0.366029
\(749\) −3.19178 −0.116625
\(750\) 0 0
\(751\) −1.06184 −0.0387470 −0.0193735 0.999812i \(-0.506167\pi\)
−0.0193735 + 0.999812i \(0.506167\pi\)
\(752\) 0.945999 0.0344970
\(753\) 0 0
\(754\) 6.69064 0.243659
\(755\) −49.9471 −1.81776
\(756\) 0 0
\(757\) −14.5206 −0.527761 −0.263880 0.964555i \(-0.585002\pi\)
−0.263880 + 0.964555i \(0.585002\pi\)
\(758\) −59.4587 −2.15964
\(759\) 0 0
\(760\) −1.89205 −0.0686320
\(761\) 40.3345 1.46213 0.731063 0.682310i \(-0.239025\pi\)
0.731063 + 0.682310i \(0.239025\pi\)
\(762\) 0 0
\(763\) −10.1493 −0.367428
\(764\) 4.96763 0.179723
\(765\) 0 0
\(766\) −41.6600 −1.50524
\(767\) −29.4725 −1.06419
\(768\) 0 0
\(769\) −10.0707 −0.363158 −0.181579 0.983376i \(-0.558121\pi\)
−0.181579 + 0.983376i \(0.558121\pi\)
\(770\) −11.7132 −0.422113
\(771\) 0 0
\(772\) −51.1634 −1.84141
\(773\) −40.1438 −1.44387 −0.721937 0.691959i \(-0.756748\pi\)
−0.721937 + 0.691959i \(0.756748\pi\)
\(774\) 0 0
\(775\) −22.3314 −0.802167
\(776\) 16.3650 0.587468
\(777\) 0 0
\(778\) 54.1422 1.94109
\(779\) 1.93983 0.0695017
\(780\) 0 0
\(781\) 1.32218 0.0473115
\(782\) 14.0702 0.503148
\(783\) 0 0
\(784\) 3.50612 0.125219
\(785\) 26.3285 0.939703
\(786\) 0 0
\(787\) −1.18087 −0.0420933 −0.0210467 0.999778i \(-0.506700\pi\)
−0.0210467 + 0.999778i \(0.506700\pi\)
\(788\) 8.64898 0.308107
\(789\) 0 0
\(790\) 52.1846 1.85664
\(791\) 58.7957 2.09053
\(792\) 0 0
\(793\) −27.5994 −0.980084
\(794\) 79.0029 2.80371
\(795\) 0 0
\(796\) −52.8228 −1.87225
\(797\) 27.8099 0.985077 0.492538 0.870291i \(-0.336069\pi\)
0.492538 + 0.870291i \(0.336069\pi\)
\(798\) 0 0
\(799\) 9.78501 0.346169
\(800\) −15.9538 −0.564053
\(801\) 0 0
\(802\) 11.3586 0.401088
\(803\) 2.27966 0.0804475
\(804\) 0 0
\(805\) 10.3243 0.363885
\(806\) −46.4196 −1.63506
\(807\) 0 0
\(808\) 43.2744 1.52239
\(809\) −1.32667 −0.0466432 −0.0233216 0.999728i \(-0.507424\pi\)
−0.0233216 + 0.999728i \(0.507424\pi\)
\(810\) 0 0
\(811\) −17.3003 −0.607497 −0.303748 0.952752i \(-0.598238\pi\)
−0.303748 + 0.952752i \(0.598238\pi\)
\(812\) 12.1139 0.425116
\(813\) 0 0
\(814\) 5.74596 0.201396
\(815\) 45.6836 1.60023
\(816\) 0 0
\(817\) −1.57183 −0.0549915
\(818\) −14.2188 −0.497147
\(819\) 0 0
\(820\) −89.5222 −3.12625
\(821\) −6.86557 −0.239610 −0.119805 0.992797i \(-0.538227\pi\)
−0.119805 + 0.992797i \(0.538227\pi\)
\(822\) 0 0
\(823\) 12.8510 0.447959 0.223980 0.974594i \(-0.428095\pi\)
0.223980 + 0.974594i \(0.428095\pi\)
\(824\) 42.7368 1.48881
\(825\) 0 0
\(826\) −85.0900 −2.96066
\(827\) −26.4853 −0.920983 −0.460491 0.887664i \(-0.652327\pi\)
−0.460491 + 0.887664i \(0.652327\pi\)
\(828\) 0 0
\(829\) 29.5767 1.02724 0.513621 0.858017i \(-0.328304\pi\)
0.513621 + 0.858017i \(0.328304\pi\)
\(830\) 11.9271 0.413994
\(831\) 0 0
\(832\) −36.5563 −1.26736
\(833\) 36.2658 1.25654
\(834\) 0 0
\(835\) −60.6142 −2.09764
\(836\) 0.344319 0.0119085
\(837\) 0 0
\(838\) −3.38488 −0.116929
\(839\) −18.3216 −0.632531 −0.316265 0.948671i \(-0.602429\pi\)
−0.316265 + 0.948671i \(0.602429\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 75.9348 2.61688
\(843\) 0 0
\(844\) −78.7375 −2.71026
\(845\) −13.3429 −0.459008
\(846\) 0 0
\(847\) −38.7502 −1.33147
\(848\) −5.95653 −0.204548
\(849\) 0 0
\(850\) 45.2879 1.55336
\(851\) −5.06467 −0.173615
\(852\) 0 0
\(853\) 36.3737 1.24541 0.622706 0.782456i \(-0.286033\pi\)
0.622706 + 0.782456i \(0.286033\pi\)
\(854\) −79.6822 −2.72667
\(855\) 0 0
\(856\) −2.79927 −0.0956769
\(857\) −12.8909 −0.440344 −0.220172 0.975461i \(-0.570662\pi\)
−0.220172 + 0.975461i \(0.570662\pi\)
\(858\) 0 0
\(859\) 13.1919 0.450102 0.225051 0.974347i \(-0.427745\pi\)
0.225051 + 0.974347i \(0.427745\pi\)
\(860\) 72.5393 2.47357
\(861\) 0 0
\(862\) 62.9417 2.14380
\(863\) 22.4029 0.762603 0.381302 0.924451i \(-0.375476\pi\)
0.381302 + 0.924451i \(0.375476\pi\)
\(864\) 0 0
\(865\) −38.7462 −1.31741
\(866\) 71.0253 2.41354
\(867\) 0 0
\(868\) −84.0462 −2.85271
\(869\) −3.85019 −0.130609
\(870\) 0 0
\(871\) −34.9859 −1.18545
\(872\) −8.90113 −0.301430
\(873\) 0 0
\(874\) −0.483943 −0.0163696
\(875\) −18.3906 −0.621715
\(876\) 0 0
\(877\) 24.5509 0.829024 0.414512 0.910044i \(-0.363952\pi\)
0.414512 + 0.910044i \(0.363952\pi\)
\(878\) 8.11644 0.273917
\(879\) 0 0
\(880\) −0.824846 −0.0278056
\(881\) 31.0084 1.04470 0.522350 0.852731i \(-0.325056\pi\)
0.522350 + 0.852731i \(0.325056\pi\)
\(882\) 0 0
\(883\) 38.3023 1.28898 0.644488 0.764615i \(-0.277071\pi\)
0.644488 + 0.764615i \(0.277071\pi\)
\(884\) 59.0368 1.98562
\(885\) 0 0
\(886\) 13.6943 0.460068
\(887\) 1.22792 0.0412294 0.0206147 0.999787i \(-0.493438\pi\)
0.0206147 + 0.999787i \(0.493438\pi\)
\(888\) 0 0
\(889\) −21.4243 −0.718548
\(890\) 50.6250 1.69695
\(891\) 0 0
\(892\) −52.9489 −1.77286
\(893\) −0.336555 −0.0112624
\(894\) 0 0
\(895\) 21.2736 0.711098
\(896\) −69.8413 −2.33323
\(897\) 0 0
\(898\) −52.9723 −1.76771
\(899\) −6.93798 −0.231395
\(900\) 0 0
\(901\) −61.6117 −2.05258
\(902\) 10.5321 0.350681
\(903\) 0 0
\(904\) 51.5652 1.71503
\(905\) 12.3884 0.411806
\(906\) 0 0
\(907\) −46.6379 −1.54859 −0.774293 0.632828i \(-0.781894\pi\)
−0.774293 + 0.632828i \(0.781894\pi\)
\(908\) 74.0781 2.45837
\(909\) 0 0
\(910\) 69.0765 2.28986
\(911\) −14.2047 −0.470623 −0.235311 0.971920i \(-0.575611\pi\)
−0.235311 + 0.971920i \(0.575611\pi\)
\(912\) 0 0
\(913\) −0.879982 −0.0291232
\(914\) 38.2712 1.26590
\(915\) 0 0
\(916\) −54.3400 −1.79544
\(917\) −31.4971 −1.04013
\(918\) 0 0
\(919\) 27.6942 0.913546 0.456773 0.889583i \(-0.349005\pi\)
0.456773 + 0.889583i \(0.349005\pi\)
\(920\) 9.05469 0.298524
\(921\) 0 0
\(922\) 15.3794 0.506493
\(923\) −7.79738 −0.256654
\(924\) 0 0
\(925\) −16.3017 −0.535998
\(926\) −43.3489 −1.42453
\(927\) 0 0
\(928\) −4.95658 −0.162708
\(929\) 31.4646 1.03232 0.516160 0.856492i \(-0.327361\pi\)
0.516160 + 0.856492i \(0.327361\pi\)
\(930\) 0 0
\(931\) −1.24736 −0.0408806
\(932\) 63.5082 2.08028
\(933\) 0 0
\(934\) −17.9341 −0.586822
\(935\) −8.53185 −0.279021
\(936\) 0 0
\(937\) −42.5528 −1.39014 −0.695070 0.718942i \(-0.744627\pi\)
−0.695070 + 0.718942i \(0.744627\pi\)
\(938\) −101.008 −3.29802
\(939\) 0 0
\(940\) 15.5319 0.506593
\(941\) 53.6015 1.74736 0.873680 0.486501i \(-0.161727\pi\)
0.873680 + 0.486501i \(0.161727\pi\)
\(942\) 0 0
\(943\) −9.28334 −0.302307
\(944\) −5.99208 −0.195026
\(945\) 0 0
\(946\) −8.53411 −0.277468
\(947\) −6.75896 −0.219637 −0.109818 0.993952i \(-0.535027\pi\)
−0.109818 + 0.993952i \(0.535027\pi\)
\(948\) 0 0
\(949\) −13.4440 −0.436409
\(950\) −1.55768 −0.0505377
\(951\) 0 0
\(952\) 69.1029 2.23964
\(953\) 2.12857 0.0689513 0.0344756 0.999406i \(-0.489024\pi\)
0.0344756 + 0.999406i \(0.489024\pi\)
\(954\) 0 0
\(955\) 4.23377 0.137001
\(956\) 46.4860 1.50347
\(957\) 0 0
\(958\) −44.9877 −1.45349
\(959\) 17.5697 0.567357
\(960\) 0 0
\(961\) 17.1356 0.552760
\(962\) −33.8859 −1.09253
\(963\) 0 0
\(964\) −13.7056 −0.441426
\(965\) −43.6051 −1.40370
\(966\) 0 0
\(967\) 49.9771 1.60715 0.803577 0.595201i \(-0.202928\pi\)
0.803577 + 0.595201i \(0.202928\pi\)
\(968\) −33.9848 −1.09231
\(969\) 0 0
\(970\) 34.4017 1.10457
\(971\) −3.70618 −0.118937 −0.0594684 0.998230i \(-0.518941\pi\)
−0.0594684 + 0.998230i \(0.518941\pi\)
\(972\) 0 0
\(973\) 29.9429 0.959926
\(974\) 24.7389 0.792685
\(975\) 0 0
\(976\) −5.61126 −0.179612
\(977\) −36.6110 −1.17129 −0.585645 0.810568i \(-0.699159\pi\)
−0.585645 + 0.810568i \(0.699159\pi\)
\(978\) 0 0
\(979\) −3.73512 −0.119375
\(980\) 57.5651 1.83885
\(981\) 0 0
\(982\) −4.41779 −0.140977
\(983\) 2.20529 0.0703378 0.0351689 0.999381i \(-0.488803\pi\)
0.0351689 + 0.999381i \(0.488803\pi\)
\(984\) 0 0
\(985\) 7.37127 0.234868
\(986\) 14.0702 0.448085
\(987\) 0 0
\(988\) −2.03057 −0.0646010
\(989\) 7.52223 0.239193
\(990\) 0 0
\(991\) −29.6667 −0.942394 −0.471197 0.882028i \(-0.656178\pi\)
−0.471197 + 0.882028i \(0.656178\pi\)
\(992\) 34.3887 1.09184
\(993\) 0 0
\(994\) −22.5118 −0.714031
\(995\) −45.0193 −1.42721
\(996\) 0 0
\(997\) 26.9367 0.853094 0.426547 0.904465i \(-0.359730\pi\)
0.426547 + 0.904465i \(0.359730\pi\)
\(998\) −30.6155 −0.969116
\(999\) 0 0
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 6003.2.a.p.1.13 14
3.2 odd 2 2001.2.a.m.1.2 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.m.1.2 14 3.2 odd 2
6003.2.a.p.1.13 14 1.1 even 1 trivial