Properties

Label 2368.2.a.bd.1.3
Level $2368$
Weight $2$
Character 2368.1
Self dual yes
Analytic conductor $18.909$
Analytic rank $0$
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] = [2368,2,Mod(1,2368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2368, 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("2368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.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: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.66908\) of defining polynomial
Character \(\chi\) \(=\) 2368.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.66908 q^{3} -1.66908 q^{5} +0.545096 q^{7} +4.12398 q^{9} +O(q^{10})\) \(q+2.66908 q^{3} -1.66908 q^{5} +0.545096 q^{7} +4.12398 q^{9} +4.21417 q^{11} -0.123983 q^{13} -4.45490 q^{15} -0.909808 q^{17} +2.00000 q^{19} +1.45490 q^{21} +4.12398 q^{23} -2.21417 q^{25} +3.00000 q^{27} -1.21417 q^{29} +5.91705 q^{31} +11.2480 q^{33} -0.909808 q^{35} +1.00000 q^{37} -0.330921 q^{39} +1.33092 q^{41} +10.2480 q^{43} -6.88325 q^{45} -2.54510 q^{47} -6.70287 q^{49} -2.42835 q^{51} +6.79306 q^{53} -7.03379 q^{55} +5.33816 q^{57} +11.1578 q^{59} +2.75927 q^{61} +2.24797 q^{63} +0.206938 q^{65} +2.33092 q^{67} +11.0072 q^{69} -7.88325 q^{71} -13.5523 q^{73} -5.90981 q^{75} +2.29713 q^{77} -2.78583 q^{79} -4.36471 q^{81} -5.88325 q^{83} +1.51854 q^{85} -3.24073 q^{87} +9.09019 q^{89} -0.0675827 q^{91} +15.7931 q^{93} -3.33816 q^{95} +3.81962 q^{97} +17.3792 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} + 3 q^{7} + 3 q^{9} + 6 q^{11} + 9 q^{13} - 12 q^{15} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 9 q^{27} + 3 q^{29} - 9 q^{31} + 15 q^{33} + 3 q^{37} - 9 q^{39} + 12 q^{41} + 12 q^{43} - 6 q^{45} - 9 q^{47} + 6 q^{51} + 3 q^{53} - 9 q^{55} + 12 q^{59} + 3 q^{61} - 12 q^{63} + 18 q^{65} + 15 q^{67} + 9 q^{69} - 9 q^{71} - 18 q^{73} - 15 q^{75} + 27 q^{77} - 15 q^{79} - 9 q^{81} - 3 q^{83} - 6 q^{85} - 15 q^{87} + 30 q^{89} + 24 q^{91} + 30 q^{93} + 6 q^{95} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.66908 1.54099 0.770497 0.637444i \(-0.220008\pi\)
0.770497 + 0.637444i \(0.220008\pi\)
\(4\) 0 0
\(5\) −1.66908 −0.746435 −0.373217 0.927744i \(-0.621745\pi\)
−0.373217 + 0.927744i \(0.621745\pi\)
\(6\) 0 0
\(7\) 0.545096 0.206027 0.103013 0.994680i \(-0.467152\pi\)
0.103013 + 0.994680i \(0.467152\pi\)
\(8\) 0 0
\(9\) 4.12398 1.37466
\(10\) 0 0
\(11\) 4.21417 1.27062 0.635311 0.772257i \(-0.280872\pi\)
0.635311 + 0.772257i \(0.280872\pi\)
\(12\) 0 0
\(13\) −0.123983 −0.0343867 −0.0171934 0.999852i \(-0.505473\pi\)
−0.0171934 + 0.999852i \(0.505473\pi\)
\(14\) 0 0
\(15\) −4.45490 −1.15025
\(16\) 0 0
\(17\) −0.909808 −0.220661 −0.110330 0.993895i \(-0.535191\pi\)
−0.110330 + 0.993895i \(0.535191\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 1.45490 0.317486
\(22\) 0 0
\(23\) 4.12398 0.859910 0.429955 0.902850i \(-0.358529\pi\)
0.429955 + 0.902850i \(0.358529\pi\)
\(24\) 0 0
\(25\) −2.21417 −0.442835
\(26\) 0 0
\(27\) 3.00000 0.577350
\(28\) 0 0
\(29\) −1.21417 −0.225467 −0.112733 0.993625i \(-0.535961\pi\)
−0.112733 + 0.993625i \(0.535961\pi\)
\(30\) 0 0
\(31\) 5.91705 1.06273 0.531366 0.847142i \(-0.321679\pi\)
0.531366 + 0.847142i \(0.321679\pi\)
\(32\) 0 0
\(33\) 11.2480 1.95802
\(34\) 0 0
\(35\) −0.909808 −0.153786
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) −0.330921 −0.0529898
\(40\) 0 0
\(41\) 1.33092 0.207855 0.103927 0.994585i \(-0.466859\pi\)
0.103927 + 0.994585i \(0.466859\pi\)
\(42\) 0 0
\(43\) 10.2480 1.56280 0.781400 0.624030i \(-0.214506\pi\)
0.781400 + 0.624030i \(0.214506\pi\)
\(44\) 0 0
\(45\) −6.88325 −1.02609
\(46\) 0 0
\(47\) −2.54510 −0.371240 −0.185620 0.982622i \(-0.559429\pi\)
−0.185620 + 0.982622i \(0.559429\pi\)
\(48\) 0 0
\(49\) −6.70287 −0.957553
\(50\) 0 0
\(51\) −2.42835 −0.340037
\(52\) 0 0
\(53\) 6.79306 0.933099 0.466549 0.884495i \(-0.345497\pi\)
0.466549 + 0.884495i \(0.345497\pi\)
\(54\) 0 0
\(55\) −7.03379 −0.948436
\(56\) 0 0
\(57\) 5.33816 0.707056
\(58\) 0 0
\(59\) 11.1578 1.45262 0.726309 0.687368i \(-0.241234\pi\)
0.726309 + 0.687368i \(0.241234\pi\)
\(60\) 0 0
\(61\) 2.75927 0.353288 0.176644 0.984275i \(-0.443476\pi\)
0.176644 + 0.984275i \(0.443476\pi\)
\(62\) 0 0
\(63\) 2.24797 0.283217
\(64\) 0 0
\(65\) 0.206938 0.0256675
\(66\) 0 0
\(67\) 2.33092 0.284767 0.142384 0.989812i \(-0.454523\pi\)
0.142384 + 0.989812i \(0.454523\pi\)
\(68\) 0 0
\(69\) 11.0072 1.32512
\(70\) 0 0
\(71\) −7.88325 −0.935570 −0.467785 0.883842i \(-0.654948\pi\)
−0.467785 + 0.883842i \(0.654948\pi\)
\(72\) 0 0
\(73\) −13.5523 −1.58618 −0.793090 0.609104i \(-0.791529\pi\)
−0.793090 + 0.609104i \(0.791529\pi\)
\(74\) 0 0
\(75\) −5.90981 −0.682406
\(76\) 0 0
\(77\) 2.29713 0.261782
\(78\) 0 0
\(79\) −2.78583 −0.313430 −0.156715 0.987644i \(-0.550090\pi\)
−0.156715 + 0.987644i \(0.550090\pi\)
\(80\) 0 0
\(81\) −4.36471 −0.484968
\(82\) 0 0
\(83\) −5.88325 −0.645771 −0.322886 0.946438i \(-0.604653\pi\)
−0.322886 + 0.946438i \(0.604653\pi\)
\(84\) 0 0
\(85\) 1.51854 0.164709
\(86\) 0 0
\(87\) −3.24073 −0.347443
\(88\) 0 0
\(89\) 9.09019 0.963558 0.481779 0.876293i \(-0.339991\pi\)
0.481779 + 0.876293i \(0.339991\pi\)
\(90\) 0 0
\(91\) −0.0675827 −0.00708459
\(92\) 0 0
\(93\) 15.7931 1.63766
\(94\) 0 0
\(95\) −3.33816 −0.342488
\(96\) 0 0
\(97\) 3.81962 0.387823 0.193912 0.981019i \(-0.437882\pi\)
0.193912 + 0.981019i \(0.437882\pi\)
\(98\) 0 0
\(99\) 17.3792 1.74667
\(100\) 0 0
\(101\) 14.5596 1.44873 0.724366 0.689416i \(-0.242133\pi\)
0.724366 + 0.689416i \(0.242133\pi\)
\(102\) 0 0
\(103\) −5.84946 −0.576365 −0.288182 0.957576i \(-0.593051\pi\)
−0.288182 + 0.957576i \(0.593051\pi\)
\(104\) 0 0
\(105\) −2.42835 −0.236983
\(106\) 0 0
\(107\) −7.69563 −0.743965 −0.371982 0.928240i \(-0.621322\pi\)
−0.371982 + 0.928240i \(0.621322\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 2.66908 0.253338
\(112\) 0 0
\(113\) 6.72942 0.633051 0.316526 0.948584i \(-0.397484\pi\)
0.316526 + 0.948584i \(0.397484\pi\)
\(114\) 0 0
\(115\) −6.88325 −0.641867
\(116\) 0 0
\(117\) −0.511305 −0.0472701
\(118\) 0 0
\(119\) −0.495933 −0.0454621
\(120\) 0 0
\(121\) 6.75927 0.614479
\(122\) 0 0
\(123\) 3.55233 0.320303
\(124\) 0 0
\(125\) 12.0410 1.07698
\(126\) 0 0
\(127\) 16.6127 1.47414 0.737068 0.675818i \(-0.236210\pi\)
0.737068 + 0.675818i \(0.236210\pi\)
\(128\) 0 0
\(129\) 27.3526 2.40827
\(130\) 0 0
\(131\) −6.92428 −0.604977 −0.302489 0.953153i \(-0.597817\pi\)
−0.302489 + 0.953153i \(0.597817\pi\)
\(132\) 0 0
\(133\) 1.09019 0.0945316
\(134\) 0 0
\(135\) −5.00724 −0.430954
\(136\) 0 0
\(137\) −5.48870 −0.468931 −0.234465 0.972124i \(-0.575334\pi\)
−0.234465 + 0.972124i \(0.575334\pi\)
\(138\) 0 0
\(139\) 14.3719 1.21901 0.609506 0.792781i \(-0.291368\pi\)
0.609506 + 0.792781i \(0.291368\pi\)
\(140\) 0 0
\(141\) −6.79306 −0.572079
\(142\) 0 0
\(143\) −0.522487 −0.0436925
\(144\) 0 0
\(145\) 2.02655 0.168296
\(146\) 0 0
\(147\) −17.8905 −1.47558
\(148\) 0 0
\(149\) −12.3116 −1.00861 −0.504303 0.863527i \(-0.668251\pi\)
−0.504303 + 0.863527i \(0.668251\pi\)
\(150\) 0 0
\(151\) −18.0145 −1.46600 −0.732999 0.680230i \(-0.761880\pi\)
−0.732999 + 0.680230i \(0.761880\pi\)
\(152\) 0 0
\(153\) −3.75203 −0.303334
\(154\) 0 0
\(155\) −9.87602 −0.793261
\(156\) 0 0
\(157\) −3.15383 −0.251703 −0.125851 0.992049i \(-0.540166\pi\)
−0.125851 + 0.992049i \(0.540166\pi\)
\(158\) 0 0
\(159\) 18.1312 1.43790
\(160\) 0 0
\(161\) 2.24797 0.177165
\(162\) 0 0
\(163\) −9.76651 −0.764972 −0.382486 0.923961i \(-0.624932\pi\)
−0.382486 + 0.923961i \(0.624932\pi\)
\(164\) 0 0
\(165\) −18.7737 −1.46153
\(166\) 0 0
\(167\) −6.34540 −0.491021 −0.245511 0.969394i \(-0.578956\pi\)
−0.245511 + 0.969394i \(0.578956\pi\)
\(168\) 0 0
\(169\) −12.9846 −0.998818
\(170\) 0 0
\(171\) 8.24797 0.630738
\(172\) 0 0
\(173\) 22.7931 1.73292 0.866462 0.499243i \(-0.166388\pi\)
0.866462 + 0.499243i \(0.166388\pi\)
\(174\) 0 0
\(175\) −1.20694 −0.0912359
\(176\) 0 0
\(177\) 29.7810 2.23848
\(178\) 0 0
\(179\) −5.15777 −0.385510 −0.192755 0.981247i \(-0.561742\pi\)
−0.192755 + 0.981247i \(0.561742\pi\)
\(180\) 0 0
\(181\) −0.297130 −0.0220855 −0.0110427 0.999939i \(-0.503515\pi\)
−0.0110427 + 0.999939i \(0.503515\pi\)
\(182\) 0 0
\(183\) 7.36471 0.544415
\(184\) 0 0
\(185\) −1.66908 −0.122713
\(186\) 0 0
\(187\) −3.83409 −0.280376
\(188\) 0 0
\(189\) 1.63529 0.118950
\(190\) 0 0
\(191\) −12.1650 −0.880229 −0.440115 0.897942i \(-0.645062\pi\)
−0.440115 + 0.897942i \(0.645062\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 0 0
\(195\) 0.552333 0.0395534
\(196\) 0 0
\(197\) −1.93636 −0.137960 −0.0689801 0.997618i \(-0.521974\pi\)
−0.0689801 + 0.997618i \(0.521974\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 6.22141 0.438825
\(202\) 0 0
\(203\) −0.661842 −0.0464522
\(204\) 0 0
\(205\) −2.22141 −0.155150
\(206\) 0 0
\(207\) 17.0072 1.18208
\(208\) 0 0
\(209\) 8.42835 0.583001
\(210\) 0 0
\(211\) −15.9807 −1.10016 −0.550078 0.835113i \(-0.685402\pi\)
−0.550078 + 0.835113i \(0.685402\pi\)
\(212\) 0 0
\(213\) −21.0410 −1.44171
\(214\) 0 0
\(215\) −17.1047 −1.16653
\(216\) 0 0
\(217\) 3.22536 0.218952
\(218\) 0 0
\(219\) −36.1722 −2.44429
\(220\) 0 0
\(221\) 0.112801 0.00758781
\(222\) 0 0
\(223\) −6.18433 −0.414133 −0.207067 0.978327i \(-0.566392\pi\)
−0.207067 + 0.978327i \(0.566392\pi\)
\(224\) 0 0
\(225\) −9.13122 −0.608748
\(226\) 0 0
\(227\) −25.1578 −1.66978 −0.834890 0.550417i \(-0.814469\pi\)
−0.834890 + 0.550417i \(0.814469\pi\)
\(228\) 0 0
\(229\) −29.7173 −1.96378 −0.981889 0.189459i \(-0.939327\pi\)
−0.981889 + 0.189459i \(0.939327\pi\)
\(230\) 0 0
\(231\) 6.13122 0.403405
\(232\) 0 0
\(233\) −9.89049 −0.647948 −0.323974 0.946066i \(-0.605019\pi\)
−0.323974 + 0.946066i \(0.605019\pi\)
\(234\) 0 0
\(235\) 4.24797 0.277107
\(236\) 0 0
\(237\) −7.43559 −0.482993
\(238\) 0 0
\(239\) 20.5934 1.33207 0.666037 0.745919i \(-0.267989\pi\)
0.666037 + 0.745919i \(0.267989\pi\)
\(240\) 0 0
\(241\) −23.1722 −1.49266 −0.746328 0.665578i \(-0.768185\pi\)
−0.746328 + 0.665578i \(0.768185\pi\)
\(242\) 0 0
\(243\) −20.6498 −1.32468
\(244\) 0 0
\(245\) 11.1876 0.714751
\(246\) 0 0
\(247\) −0.247966 −0.0157777
\(248\) 0 0
\(249\) −15.7029 −0.995129
\(250\) 0 0
\(251\) 7.63923 0.482184 0.241092 0.970502i \(-0.422494\pi\)
0.241092 + 0.970502i \(0.422494\pi\)
\(252\) 0 0
\(253\) 17.3792 1.09262
\(254\) 0 0
\(255\) 4.05311 0.253815
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0.545096 0.0338706
\(260\) 0 0
\(261\) −5.00724 −0.309940
\(262\) 0 0
\(263\) 25.9508 1.60020 0.800099 0.599869i \(-0.204780\pi\)
0.800099 + 0.599869i \(0.204780\pi\)
\(264\) 0 0
\(265\) −11.3382 −0.696498
\(266\) 0 0
\(267\) 24.2624 1.48484
\(268\) 0 0
\(269\) −16.2624 −0.991539 −0.495769 0.868454i \(-0.665114\pi\)
−0.495769 + 0.868454i \(0.665114\pi\)
\(270\) 0 0
\(271\) −14.6127 −0.887657 −0.443829 0.896112i \(-0.646380\pi\)
−0.443829 + 0.896112i \(0.646380\pi\)
\(272\) 0 0
\(273\) −0.180384 −0.0109173
\(274\) 0 0
\(275\) −9.33092 −0.562676
\(276\) 0 0
\(277\) −23.0217 −1.38324 −0.691620 0.722261i \(-0.743103\pi\)
−0.691620 + 0.722261i \(0.743103\pi\)
\(278\) 0 0
\(279\) 24.4018 1.46090
\(280\) 0 0
\(281\) −12.8567 −0.766966 −0.383483 0.923548i \(-0.625276\pi\)
−0.383483 + 0.923548i \(0.625276\pi\)
\(282\) 0 0
\(283\) −7.57165 −0.450088 −0.225044 0.974349i \(-0.572253\pi\)
−0.225044 + 0.974349i \(0.572253\pi\)
\(284\) 0 0
\(285\) −8.90981 −0.527771
\(286\) 0 0
\(287\) 0.725480 0.0428237
\(288\) 0 0
\(289\) −16.1722 −0.951309
\(290\) 0 0
\(291\) 10.1949 0.597633
\(292\) 0 0
\(293\) 22.7439 1.32871 0.664356 0.747416i \(-0.268706\pi\)
0.664356 + 0.747416i \(0.268706\pi\)
\(294\) 0 0
\(295\) −18.6232 −1.08429
\(296\) 0 0
\(297\) 12.6425 0.733594
\(298\) 0 0
\(299\) −0.511305 −0.0295695
\(300\) 0 0
\(301\) 5.58612 0.321979
\(302\) 0 0
\(303\) 38.8606 2.23249
\(304\) 0 0
\(305\) −4.60544 −0.263707
\(306\) 0 0
\(307\) −3.16501 −0.180637 −0.0903184 0.995913i \(-0.528788\pi\)
−0.0903184 + 0.995913i \(0.528788\pi\)
\(308\) 0 0
\(309\) −15.6127 −0.888174
\(310\) 0 0
\(311\) 19.2962 1.09419 0.547094 0.837071i \(-0.315734\pi\)
0.547094 + 0.837071i \(0.315734\pi\)
\(312\) 0 0
\(313\) 22.6908 1.28256 0.641280 0.767307i \(-0.278404\pi\)
0.641280 + 0.767307i \(0.278404\pi\)
\(314\) 0 0
\(315\) −3.75203 −0.211403
\(316\) 0 0
\(317\) 7.76651 0.436211 0.218105 0.975925i \(-0.430012\pi\)
0.218105 + 0.975925i \(0.430012\pi\)
\(318\) 0 0
\(319\) −5.11675 −0.286483
\(320\) 0 0
\(321\) −20.5403 −1.14645
\(322\) 0 0
\(323\) −1.81962 −0.101246
\(324\) 0 0
\(325\) 0.274520 0.0152277
\(326\) 0 0
\(327\) 26.6908 1.47600
\(328\) 0 0
\(329\) −1.38732 −0.0764855
\(330\) 0 0
\(331\) 5.58612 0.307041 0.153521 0.988145i \(-0.450939\pi\)
0.153521 + 0.988145i \(0.450939\pi\)
\(332\) 0 0
\(333\) 4.12398 0.225993
\(334\) 0 0
\(335\) −3.89049 −0.212560
\(336\) 0 0
\(337\) 15.8003 0.860697 0.430349 0.902663i \(-0.358391\pi\)
0.430349 + 0.902663i \(0.358391\pi\)
\(338\) 0 0
\(339\) 17.9614 0.975528
\(340\) 0 0
\(341\) 24.9355 1.35033
\(342\) 0 0
\(343\) −7.46938 −0.403309
\(344\) 0 0
\(345\) −18.3719 −0.989113
\(346\) 0 0
\(347\) −6.01447 −0.322874 −0.161437 0.986883i \(-0.551613\pi\)
−0.161437 + 0.986883i \(0.551613\pi\)
\(348\) 0 0
\(349\) 6.74390 0.360993 0.180496 0.983576i \(-0.442230\pi\)
0.180496 + 0.983576i \(0.442230\pi\)
\(350\) 0 0
\(351\) −0.371950 −0.0198532
\(352\) 0 0
\(353\) 20.1949 1.07486 0.537432 0.843307i \(-0.319395\pi\)
0.537432 + 0.843307i \(0.319395\pi\)
\(354\) 0 0
\(355\) 13.1578 0.698342
\(356\) 0 0
\(357\) −1.32368 −0.0700568
\(358\) 0 0
\(359\) −3.88325 −0.204950 −0.102475 0.994736i \(-0.532676\pi\)
−0.102475 + 0.994736i \(0.532676\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 18.0410 0.946908
\(364\) 0 0
\(365\) 22.6199 1.18398
\(366\) 0 0
\(367\) −32.7439 −1.70922 −0.854609 0.519272i \(-0.826203\pi\)
−0.854609 + 0.519272i \(0.826203\pi\)
\(368\) 0 0
\(369\) 5.48870 0.285730
\(370\) 0 0
\(371\) 3.70287 0.192243
\(372\) 0 0
\(373\) −31.3035 −1.62083 −0.810416 0.585855i \(-0.800759\pi\)
−0.810416 + 0.585855i \(0.800759\pi\)
\(374\) 0 0
\(375\) 32.1385 1.65962
\(376\) 0 0
\(377\) 0.150537 0.00775306
\(378\) 0 0
\(379\) 20.0072 1.02770 0.513851 0.857879i \(-0.328218\pi\)
0.513851 + 0.857879i \(0.328218\pi\)
\(380\) 0 0
\(381\) 44.3406 2.27164
\(382\) 0 0
\(383\) 3.17225 0.162094 0.0810472 0.996710i \(-0.474174\pi\)
0.0810472 + 0.996710i \(0.474174\pi\)
\(384\) 0 0
\(385\) −3.83409 −0.195403
\(386\) 0 0
\(387\) 42.2624 2.14832
\(388\) 0 0
\(389\) −35.5297 −1.80143 −0.900714 0.434412i \(-0.856956\pi\)
−0.900714 + 0.434412i \(0.856956\pi\)
\(390\) 0 0
\(391\) −3.75203 −0.189748
\(392\) 0 0
\(393\) −18.4815 −0.932266
\(394\) 0 0
\(395\) 4.64976 0.233955
\(396\) 0 0
\(397\) 2.31160 0.116016 0.0580080 0.998316i \(-0.481525\pi\)
0.0580080 + 0.998316i \(0.481525\pi\)
\(398\) 0 0
\(399\) 2.90981 0.145673
\(400\) 0 0
\(401\) −32.8712 −1.64151 −0.820754 0.571282i \(-0.806446\pi\)
−0.820754 + 0.571282i \(0.806446\pi\)
\(402\) 0 0
\(403\) −0.733614 −0.0365439
\(404\) 0 0
\(405\) 7.28505 0.361997
\(406\) 0 0
\(407\) 4.21417 0.208889
\(408\) 0 0
\(409\) −32.9243 −1.62800 −0.814001 0.580864i \(-0.802715\pi\)
−0.814001 + 0.580864i \(0.802715\pi\)
\(410\) 0 0
\(411\) −14.6498 −0.722619
\(412\) 0 0
\(413\) 6.08206 0.299278
\(414\) 0 0
\(415\) 9.81962 0.482026
\(416\) 0 0
\(417\) 38.3599 1.87849
\(418\) 0 0
\(419\) 18.4356 0.900637 0.450319 0.892868i \(-0.351310\pi\)
0.450319 + 0.892868i \(0.351310\pi\)
\(420\) 0 0
\(421\) 40.6489 1.98110 0.990552 0.137136i \(-0.0437897\pi\)
0.990552 + 0.137136i \(0.0437897\pi\)
\(422\) 0 0
\(423\) −10.4959 −0.510330
\(424\) 0 0
\(425\) 2.01447 0.0977164
\(426\) 0 0
\(427\) 1.50407 0.0727869
\(428\) 0 0
\(429\) −1.39456 −0.0673299
\(430\) 0 0
\(431\) 26.4428 1.27371 0.636853 0.770985i \(-0.280236\pi\)
0.636853 + 0.770985i \(0.280236\pi\)
\(432\) 0 0
\(433\) −24.8639 −1.19488 −0.597442 0.801912i \(-0.703816\pi\)
−0.597442 + 0.801912i \(0.703816\pi\)
\(434\) 0 0
\(435\) 5.40903 0.259343
\(436\) 0 0
\(437\) 8.24797 0.394554
\(438\) 0 0
\(439\) −6.05640 −0.289056 −0.144528 0.989501i \(-0.546166\pi\)
−0.144528 + 0.989501i \(0.546166\pi\)
\(440\) 0 0
\(441\) −27.6425 −1.31631
\(442\) 0 0
\(443\) −39.7472 −1.88845 −0.944223 0.329307i \(-0.893185\pi\)
−0.944223 + 0.329307i \(0.893185\pi\)
\(444\) 0 0
\(445\) −15.1722 −0.719234
\(446\) 0 0
\(447\) −32.8606 −1.55426
\(448\) 0 0
\(449\) 27.3526 1.29085 0.645425 0.763823i \(-0.276680\pi\)
0.645425 + 0.763823i \(0.276680\pi\)
\(450\) 0 0
\(451\) 5.60873 0.264105
\(452\) 0 0
\(453\) −48.0821 −2.25909
\(454\) 0 0
\(455\) 0.112801 0.00528819
\(456\) 0 0
\(457\) 17.6006 0.823321 0.411661 0.911337i \(-0.364949\pi\)
0.411661 + 0.911337i \(0.364949\pi\)
\(458\) 0 0
\(459\) −2.72942 −0.127399
\(460\) 0 0
\(461\) −25.1722 −1.17239 −0.586194 0.810171i \(-0.699374\pi\)
−0.586194 + 0.810171i \(0.699374\pi\)
\(462\) 0 0
\(463\) 9.21417 0.428219 0.214110 0.976810i \(-0.431315\pi\)
0.214110 + 0.976810i \(0.431315\pi\)
\(464\) 0 0
\(465\) −26.3599 −1.22241
\(466\) 0 0
\(467\) −0.262441 −0.0121443 −0.00607216 0.999982i \(-0.501933\pi\)
−0.00607216 + 0.999982i \(0.501933\pi\)
\(468\) 0 0
\(469\) 1.27058 0.0586697
\(470\) 0 0
\(471\) −8.41782 −0.387873
\(472\) 0 0
\(473\) 43.1867 1.98573
\(474\) 0 0
\(475\) −4.42835 −0.203187
\(476\) 0 0
\(477\) 28.0145 1.28269
\(478\) 0 0
\(479\) −9.21417 −0.421006 −0.210503 0.977593i \(-0.567510\pi\)
−0.210503 + 0.977593i \(0.567510\pi\)
\(480\) 0 0
\(481\) −0.123983 −0.00565315
\(482\) 0 0
\(483\) 6.00000 0.273009
\(484\) 0 0
\(485\) −6.37524 −0.289485
\(486\) 0 0
\(487\) −32.0289 −1.45137 −0.725685 0.688027i \(-0.758477\pi\)
−0.725685 + 0.688027i \(0.758477\pi\)
\(488\) 0 0
\(489\) −26.0676 −1.17882
\(490\) 0 0
\(491\) −11.9436 −0.539007 −0.269504 0.962999i \(-0.586860\pi\)
−0.269504 + 0.962999i \(0.586860\pi\)
\(492\) 0 0
\(493\) 1.10467 0.0497517
\(494\) 0 0
\(495\) −29.0072 −1.30378
\(496\) 0 0
\(497\) −4.29713 −0.192753
\(498\) 0 0
\(499\) −11.4202 −0.511239 −0.255620 0.966777i \(-0.582279\pi\)
−0.255620 + 0.966777i \(0.582279\pi\)
\(500\) 0 0
\(501\) −16.9364 −0.756661
\(502\) 0 0
\(503\) 21.5176 0.959424 0.479712 0.877426i \(-0.340741\pi\)
0.479712 + 0.877426i \(0.340741\pi\)
\(504\) 0 0
\(505\) −24.3011 −1.08138
\(506\) 0 0
\(507\) −34.6570 −1.53917
\(508\) 0 0
\(509\) 0.778588 0.0345103 0.0172551 0.999851i \(-0.494507\pi\)
0.0172551 + 0.999851i \(0.494507\pi\)
\(510\) 0 0
\(511\) −7.38732 −0.326796
\(512\) 0 0
\(513\) 6.00000 0.264906
\(514\) 0 0
\(515\) 9.76322 0.430219
\(516\) 0 0
\(517\) −10.7255 −0.471706
\(518\) 0 0
\(519\) 60.8365 2.67043
\(520\) 0 0
\(521\) 35.6498 1.56184 0.780922 0.624628i \(-0.214749\pi\)
0.780922 + 0.624628i \(0.214749\pi\)
\(522\) 0 0
\(523\) 31.7810 1.38969 0.694843 0.719162i \(-0.255474\pi\)
0.694843 + 0.719162i \(0.255474\pi\)
\(524\) 0 0
\(525\) −3.22141 −0.140594
\(526\) 0 0
\(527\) −5.38338 −0.234504
\(528\) 0 0
\(529\) −5.99276 −0.260555
\(530\) 0 0
\(531\) 46.0145 1.99686
\(532\) 0 0
\(533\) −0.165012 −0.00714745
\(534\) 0 0
\(535\) 12.8446 0.555321
\(536\) 0 0
\(537\) −13.7665 −0.594069
\(538\) 0 0
\(539\) −28.2471 −1.21669
\(540\) 0 0
\(541\) −30.8269 −1.32535 −0.662675 0.748907i \(-0.730579\pi\)
−0.662675 + 0.748907i \(0.730579\pi\)
\(542\) 0 0
\(543\) −0.793062 −0.0340336
\(544\) 0 0
\(545\) −16.6908 −0.714955
\(546\) 0 0
\(547\) 2.97739 0.127304 0.0636520 0.997972i \(-0.479725\pi\)
0.0636520 + 0.997972i \(0.479725\pi\)
\(548\) 0 0
\(549\) 11.3792 0.485652
\(550\) 0 0
\(551\) −2.42835 −0.103451
\(552\) 0 0
\(553\) −1.51854 −0.0645750
\(554\) 0 0
\(555\) −4.45490 −0.189100
\(556\) 0 0
\(557\) −41.7656 −1.76967 −0.884833 0.465909i \(-0.845728\pi\)
−0.884833 + 0.465909i \(0.845728\pi\)
\(558\) 0 0
\(559\) −1.27058 −0.0537396
\(560\) 0 0
\(561\) −10.2335 −0.432058
\(562\) 0 0
\(563\) −7.45885 −0.314353 −0.157177 0.987571i \(-0.550239\pi\)
−0.157177 + 0.987571i \(0.550239\pi\)
\(564\) 0 0
\(565\) −11.2319 −0.472531
\(566\) 0 0
\(567\) −2.37919 −0.0999165
\(568\) 0 0
\(569\) 14.9098 0.625052 0.312526 0.949909i \(-0.398825\pi\)
0.312526 + 0.949909i \(0.398825\pi\)
\(570\) 0 0
\(571\) 21.0830 0.882294 0.441147 0.897435i \(-0.354572\pi\)
0.441147 + 0.897435i \(0.354572\pi\)
\(572\) 0 0
\(573\) −32.4694 −1.35643
\(574\) 0 0
\(575\) −9.13122 −0.380798
\(576\) 0 0
\(577\) 20.0145 0.833213 0.416607 0.909087i \(-0.363219\pi\)
0.416607 + 0.909087i \(0.363219\pi\)
\(578\) 0 0
\(579\) 42.7053 1.77477
\(580\) 0 0
\(581\) −3.20694 −0.133046
\(582\) 0 0
\(583\) 28.6272 1.18562
\(584\) 0 0
\(585\) 0.853408 0.0352841
\(586\) 0 0
\(587\) −9.17225 −0.378579 −0.189290 0.981921i \(-0.560618\pi\)
−0.189290 + 0.981921i \(0.560618\pi\)
\(588\) 0 0
\(589\) 11.8341 0.487615
\(590\) 0 0
\(591\) −5.16830 −0.212596
\(592\) 0 0
\(593\) 26.8374 1.10208 0.551040 0.834479i \(-0.314231\pi\)
0.551040 + 0.834479i \(0.314231\pi\)
\(594\) 0 0
\(595\) 0.827751 0.0339345
\(596\) 0 0
\(597\) −42.7053 −1.74781
\(598\) 0 0
\(599\) −26.1988 −1.07045 −0.535227 0.844708i \(-0.679774\pi\)
−0.535227 + 0.844708i \(0.679774\pi\)
\(600\) 0 0
\(601\) 18.1795 0.741557 0.370778 0.928721i \(-0.379091\pi\)
0.370778 + 0.928721i \(0.379091\pi\)
\(602\) 0 0
\(603\) 9.61268 0.391459
\(604\) 0 0
\(605\) −11.2818 −0.458669
\(606\) 0 0
\(607\) −40.1240 −1.62858 −0.814291 0.580457i \(-0.802874\pi\)
−0.814291 + 0.580457i \(0.802874\pi\)
\(608\) 0 0
\(609\) −1.76651 −0.0715825
\(610\) 0 0
\(611\) 0.315549 0.0127657
\(612\) 0 0
\(613\) −5.68840 −0.229752 −0.114876 0.993380i \(-0.536647\pi\)
−0.114876 + 0.993380i \(0.536647\pi\)
\(614\) 0 0
\(615\) −5.92913 −0.239085
\(616\) 0 0
\(617\) 5.06848 0.204049 0.102025 0.994782i \(-0.467468\pi\)
0.102025 + 0.994782i \(0.467468\pi\)
\(618\) 0 0
\(619\) 47.0257 1.89012 0.945060 0.326896i \(-0.106003\pi\)
0.945060 + 0.326896i \(0.106003\pi\)
\(620\) 0 0
\(621\) 12.3719 0.496469
\(622\) 0 0
\(623\) 4.95503 0.198519
\(624\) 0 0
\(625\) −9.02655 −0.361062
\(626\) 0 0
\(627\) 22.4959 0.898401
\(628\) 0 0
\(629\) −0.909808 −0.0362764
\(630\) 0 0
\(631\) −34.1916 −1.36114 −0.680572 0.732681i \(-0.738269\pi\)
−0.680572 + 0.732681i \(0.738269\pi\)
\(632\) 0 0
\(633\) −42.6537 −1.69533
\(634\) 0 0
\(635\) −27.7279 −1.10035
\(636\) 0 0
\(637\) 0.831043 0.0329271
\(638\) 0 0
\(639\) −32.5104 −1.28609
\(640\) 0 0
\(641\) −8.48870 −0.335283 −0.167642 0.985848i \(-0.553615\pi\)
−0.167642 + 0.985848i \(0.553615\pi\)
\(642\) 0 0
\(643\) −21.6006 −0.851844 −0.425922 0.904760i \(-0.640050\pi\)
−0.425922 + 0.904760i \(0.640050\pi\)
\(644\) 0 0
\(645\) −45.6537 −1.79761
\(646\) 0 0
\(647\) −1.71011 −0.0672313 −0.0336156 0.999435i \(-0.510702\pi\)
−0.0336156 + 0.999435i \(0.510702\pi\)
\(648\) 0 0
\(649\) 47.0208 1.84573
\(650\) 0 0
\(651\) 8.60873 0.337403
\(652\) 0 0
\(653\) −22.5668 −0.883107 −0.441554 0.897235i \(-0.645573\pi\)
−0.441554 + 0.897235i \(0.645573\pi\)
\(654\) 0 0
\(655\) 11.5572 0.451576
\(656\) 0 0
\(657\) −55.8896 −2.18046
\(658\) 0 0
\(659\) 14.9170 0.581086 0.290543 0.956862i \(-0.406164\pi\)
0.290543 + 0.956862i \(0.406164\pi\)
\(660\) 0 0
\(661\) −13.1997 −0.513409 −0.256704 0.966490i \(-0.582637\pi\)
−0.256704 + 0.966490i \(0.582637\pi\)
\(662\) 0 0
\(663\) 0.301075 0.0116928
\(664\) 0 0
\(665\) −1.81962 −0.0705617
\(666\) 0 0
\(667\) −5.00724 −0.193881
\(668\) 0 0
\(669\) −16.5065 −0.638177
\(670\) 0 0
\(671\) 11.6281 0.448896
\(672\) 0 0
\(673\) −7.16501 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(674\) 0 0
\(675\) −6.64252 −0.255671
\(676\) 0 0
\(677\) 46.2664 1.77816 0.889081 0.457750i \(-0.151345\pi\)
0.889081 + 0.457750i \(0.151345\pi\)
\(678\) 0 0
\(679\) 2.08206 0.0799020
\(680\) 0 0
\(681\) −67.1481 −2.57312
\(682\) 0 0
\(683\) 49.2543 1.88466 0.942332 0.334680i \(-0.108628\pi\)
0.942332 + 0.334680i \(0.108628\pi\)
\(684\) 0 0
\(685\) 9.16107 0.350026
\(686\) 0 0
\(687\) −79.3179 −3.02617
\(688\) 0 0
\(689\) −0.842225 −0.0320862
\(690\) 0 0
\(691\) 23.5330 0.895238 0.447619 0.894224i \(-0.352272\pi\)
0.447619 + 0.894224i \(0.352272\pi\)
\(692\) 0 0
\(693\) 9.47332 0.359862
\(694\) 0 0
\(695\) −23.9879 −0.909914
\(696\) 0 0
\(697\) −1.21088 −0.0458655
\(698\) 0 0
\(699\) −26.3985 −0.998483
\(700\) 0 0
\(701\) 13.7367 0.518827 0.259413 0.965766i \(-0.416471\pi\)
0.259413 + 0.965766i \(0.416471\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 11.3382 0.427020
\(706\) 0 0
\(707\) 7.93636 0.298478
\(708\) 0 0
\(709\) −20.2857 −0.761846 −0.380923 0.924607i \(-0.624394\pi\)
−0.380923 + 0.924607i \(0.624394\pi\)
\(710\) 0 0
\(711\) −11.4887 −0.430860
\(712\) 0 0
\(713\) 24.4018 0.913854
\(714\) 0 0
\(715\) 0.872072 0.0326136
\(716\) 0 0
\(717\) 54.9653 2.05272
\(718\) 0 0
\(719\) −11.3421 −0.422989 −0.211495 0.977379i \(-0.567833\pi\)
−0.211495 + 0.977379i \(0.567833\pi\)
\(720\) 0 0
\(721\) −3.18852 −0.118747
\(722\) 0 0
\(723\) −61.8486 −2.30017
\(724\) 0 0
\(725\) 2.68840 0.0998445
\(726\) 0 0
\(727\) −15.1345 −0.561308 −0.280654 0.959809i \(-0.590551\pi\)
−0.280654 + 0.959809i \(0.590551\pi\)
\(728\) 0 0
\(729\) −42.0217 −1.55636
\(730\) 0 0
\(731\) −9.32368 −0.344849
\(732\) 0 0
\(733\) 7.02655 0.259532 0.129766 0.991545i \(-0.458577\pi\)
0.129766 + 0.991545i \(0.458577\pi\)
\(734\) 0 0
\(735\) 29.8606 1.10143
\(736\) 0 0
\(737\) 9.82291 0.361831
\(738\) 0 0
\(739\) −32.6425 −1.20077 −0.600387 0.799709i \(-0.704987\pi\)
−0.600387 + 0.799709i \(0.704987\pi\)
\(740\) 0 0
\(741\) −0.661842 −0.0243134
\(742\) 0 0
\(743\) −17.3487 −0.636462 −0.318231 0.948013i \(-0.603089\pi\)
−0.318231 + 0.948013i \(0.603089\pi\)
\(744\) 0 0
\(745\) 20.5490 0.752859
\(746\) 0 0
\(747\) −24.2624 −0.887716
\(748\) 0 0
\(749\) −4.19486 −0.153277
\(750\) 0 0
\(751\) −31.8833 −1.16344 −0.581718 0.813390i \(-0.697619\pi\)
−0.581718 + 0.813390i \(0.697619\pi\)
\(752\) 0 0
\(753\) 20.3897 0.743043
\(754\) 0 0
\(755\) 30.0676 1.09427
\(756\) 0 0
\(757\) −41.2021 −1.49752 −0.748758 0.662844i \(-0.769349\pi\)
−0.748758 + 0.662844i \(0.769349\pi\)
\(758\) 0 0
\(759\) 46.3864 1.68372
\(760\) 0 0
\(761\) −31.9010 −1.15641 −0.578206 0.815891i \(-0.696247\pi\)
−0.578206 + 0.815891i \(0.696247\pi\)
\(762\) 0 0
\(763\) 5.45096 0.197338
\(764\) 0 0
\(765\) 6.26244 0.226419
\(766\) 0 0
\(767\) −1.38338 −0.0499508
\(768\) 0 0
\(769\) 32.0821 1.15691 0.578454 0.815715i \(-0.303656\pi\)
0.578454 + 0.815715i \(0.303656\pi\)
\(770\) 0 0
\(771\) 16.0145 0.576747
\(772\) 0 0
\(773\) 39.0024 1.40282 0.701409 0.712759i \(-0.252555\pi\)
0.701409 + 0.712759i \(0.252555\pi\)
\(774\) 0 0
\(775\) −13.1014 −0.470615
\(776\) 0 0
\(777\) 1.45490 0.0521944
\(778\) 0 0
\(779\) 2.66184 0.0953704
\(780\) 0 0
\(781\) −33.2214 −1.18876
\(782\) 0 0
\(783\) −3.64252 −0.130173
\(784\) 0 0
\(785\) 5.26399 0.187880
\(786\) 0 0
\(787\) 7.34210 0.261718 0.130859 0.991401i \(-0.458227\pi\)
0.130859 + 0.991401i \(0.458227\pi\)
\(788\) 0 0
\(789\) 69.2648 2.46589
\(790\) 0 0
\(791\) 3.66818 0.130426
\(792\) 0 0
\(793\) −0.342103 −0.0121484
\(794\) 0 0
\(795\) −30.2624 −1.07330
\(796\) 0 0
\(797\) 5.44767 0.192966 0.0964831 0.995335i \(-0.469241\pi\)
0.0964831 + 0.995335i \(0.469241\pi\)
\(798\) 0 0
\(799\) 2.31555 0.0819182
\(800\) 0 0
\(801\) 37.4878 1.32457
\(802\) 0 0
\(803\) −57.1119 −2.01544
\(804\) 0 0
\(805\) −3.75203 −0.132242
\(806\) 0 0
\(807\) −43.4057 −1.52795
\(808\) 0 0
\(809\) 37.3382 1.31274 0.656370 0.754439i \(-0.272091\pi\)
0.656370 + 0.754439i \(0.272091\pi\)
\(810\) 0 0
\(811\) −20.3107 −0.713205 −0.356603 0.934256i \(-0.616065\pi\)
−0.356603 + 0.934256i \(0.616065\pi\)
\(812\) 0 0
\(813\) −39.0024 −1.36787
\(814\) 0 0
\(815\) 16.3011 0.571002
\(816\) 0 0
\(817\) 20.4959 0.717062
\(818\) 0 0
\(819\) −0.278710 −0.00973892
\(820\) 0 0
\(821\) 4.47751 0.156266 0.0781331 0.996943i \(-0.475104\pi\)
0.0781331 + 0.996943i \(0.475104\pi\)
\(822\) 0 0
\(823\) −20.1352 −0.701868 −0.350934 0.936400i \(-0.614136\pi\)
−0.350934 + 0.936400i \(0.614136\pi\)
\(824\) 0 0
\(825\) −24.9050 −0.867080
\(826\) 0 0
\(827\) 6.66184 0.231655 0.115827 0.993269i \(-0.463048\pi\)
0.115827 + 0.993269i \(0.463048\pi\)
\(828\) 0 0
\(829\) 26.3044 0.913588 0.456794 0.889572i \(-0.348998\pi\)
0.456794 + 0.889572i \(0.348998\pi\)
\(830\) 0 0
\(831\) −61.4468 −2.13156
\(832\) 0 0
\(833\) 6.09833 0.211294
\(834\) 0 0
\(835\) 10.5910 0.366516
\(836\) 0 0
\(837\) 17.7511 0.613569
\(838\) 0 0
\(839\) 52.3300 1.80663 0.903317 0.428975i \(-0.141125\pi\)
0.903317 + 0.428975i \(0.141125\pi\)
\(840\) 0 0
\(841\) −27.5258 −0.949165
\(842\) 0 0
\(843\) −34.3155 −1.18189
\(844\) 0 0
\(845\) 21.6724 0.745552
\(846\) 0 0
\(847\) 3.68445 0.126599
\(848\) 0 0
\(849\) −20.2093 −0.693583
\(850\) 0 0
\(851\) 4.12398 0.141368
\(852\) 0 0
\(853\) 31.9170 1.09282 0.546409 0.837518i \(-0.315994\pi\)
0.546409 + 0.837518i \(0.315994\pi\)
\(854\) 0 0
\(855\) −13.7665 −0.470805
\(856\) 0 0
\(857\) −34.7584 −1.18732 −0.593662 0.804715i \(-0.702318\pi\)
−0.593662 + 0.804715i \(0.702318\pi\)
\(858\) 0 0
\(859\) −37.7955 −1.28956 −0.644782 0.764366i \(-0.723052\pi\)
−0.644782 + 0.764366i \(0.723052\pi\)
\(860\) 0 0
\(861\) 1.93636 0.0659910
\(862\) 0 0
\(863\) 39.6682 1.35032 0.675160 0.737671i \(-0.264074\pi\)
0.675160 + 0.737671i \(0.264074\pi\)
\(864\) 0 0
\(865\) −38.0434 −1.29352
\(866\) 0 0
\(867\) −43.1650 −1.46596
\(868\) 0 0
\(869\) −11.7400 −0.398251
\(870\) 0 0
\(871\) −0.288995 −0.00979222
\(872\) 0 0
\(873\) 15.7520 0.533126
\(874\) 0 0
\(875\) 6.56352 0.221887
\(876\) 0 0
\(877\) 13.5330 0.456977 0.228489 0.973547i \(-0.426622\pi\)
0.228489 + 0.973547i \(0.426622\pi\)
\(878\) 0 0
\(879\) 60.7053 2.04754
\(880\) 0 0
\(881\) −7.47422 −0.251813 −0.125906 0.992042i \(-0.540184\pi\)
−0.125906 + 0.992042i \(0.540184\pi\)
\(882\) 0 0
\(883\) 35.2995 1.18792 0.593962 0.804493i \(-0.297563\pi\)
0.593962 + 0.804493i \(0.297563\pi\)
\(884\) 0 0
\(885\) −49.7068 −1.67088
\(886\) 0 0
\(887\) −24.3792 −0.818573 −0.409286 0.912406i \(-0.634222\pi\)
−0.409286 + 0.912406i \(0.634222\pi\)
\(888\) 0 0
\(889\) 9.05550 0.303712
\(890\) 0 0
\(891\) −18.3937 −0.616211
\(892\) 0 0
\(893\) −5.09019 −0.170337
\(894\) 0 0
\(895\) 8.60873 0.287758
\(896\) 0 0
\(897\) −1.36471 −0.0455664
\(898\) 0 0
\(899\) −7.18433 −0.239611
\(900\) 0 0
\(901\) −6.18038 −0.205898
\(902\) 0 0
\(903\) 14.9098 0.496167
\(904\) 0 0
\(905\) 0.495933 0.0164854
\(906\) 0 0
\(907\) −5.14330 −0.170780 −0.0853902 0.996348i \(-0.527214\pi\)
−0.0853902 + 0.996348i \(0.527214\pi\)
\(908\) 0 0
\(909\) 60.0434 1.99151
\(910\) 0 0
\(911\) −31.7665 −1.05247 −0.526236 0.850339i \(-0.676397\pi\)
−0.526236 + 0.850339i \(0.676397\pi\)
\(912\) 0 0
\(913\) −24.7931 −0.820531
\(914\) 0 0
\(915\) −12.2923 −0.406370
\(916\) 0 0
\(917\) −3.77440 −0.124642
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) −8.44767 −0.278360
\(922\) 0 0
\(923\) 0.977391 0.0321712
\(924\) 0 0
\(925\) −2.21417 −0.0728016
\(926\) 0 0
\(927\) −24.1231 −0.792306
\(928\) 0 0
\(929\) 38.8413 1.27434 0.637171 0.770722i \(-0.280104\pi\)
0.637171 + 0.770722i \(0.280104\pi\)
\(930\) 0 0
\(931\) −13.4057 −0.439355
\(932\) 0 0
\(933\) 51.5032 1.68614
\(934\) 0 0
\(935\) 6.39940 0.209283
\(936\) 0 0
\(937\) −18.7964 −0.614050 −0.307025 0.951701i \(-0.599334\pi\)
−0.307025 + 0.951701i \(0.599334\pi\)
\(938\) 0 0
\(939\) 60.5635 1.97642
\(940\) 0 0
\(941\) 53.7810 1.75321 0.876605 0.481211i \(-0.159803\pi\)
0.876605 + 0.481211i \(0.159803\pi\)
\(942\) 0 0
\(943\) 5.48870 0.178737
\(944\) 0 0
\(945\) −2.72942 −0.0887882
\(946\) 0 0
\(947\) −32.7294 −1.06356 −0.531782 0.846881i \(-0.678477\pi\)
−0.531782 + 0.846881i \(0.678477\pi\)
\(948\) 0 0
\(949\) 1.68026 0.0545436
\(950\) 0 0
\(951\) 20.7294 0.672198
\(952\) 0 0
\(953\) −47.9952 −1.55472 −0.777358 0.629059i \(-0.783441\pi\)
−0.777358 + 0.629059i \(0.783441\pi\)
\(954\) 0 0
\(955\) 20.3044 0.657034
\(956\) 0 0
\(957\) −13.6570 −0.441468
\(958\) 0 0
\(959\) −2.99187 −0.0966124
\(960\) 0 0
\(961\) 4.01143 0.129401
\(962\) 0 0
\(963\) −31.7367 −1.02270
\(964\) 0 0
\(965\) −26.7053 −0.859673
\(966\) 0 0
\(967\) 59.1635 1.90257 0.951284 0.308315i \(-0.0997651\pi\)
0.951284 + 0.308315i \(0.0997651\pi\)
\(968\) 0 0
\(969\) −4.85670 −0.156020
\(970\) 0 0
\(971\) 42.9928 1.37970 0.689852 0.723951i \(-0.257676\pi\)
0.689852 + 0.723951i \(0.257676\pi\)
\(972\) 0 0
\(973\) 7.83409 0.251149
\(974\) 0 0
\(975\) 0.732717 0.0234657
\(976\) 0 0
\(977\) 38.7584 1.23999 0.619995 0.784606i \(-0.287135\pi\)
0.619995 + 0.784606i \(0.287135\pi\)
\(978\) 0 0
\(979\) 38.3077 1.22432
\(980\) 0 0
\(981\) 41.2398 1.31669
\(982\) 0 0
\(983\) −6.50646 −0.207524 −0.103762 0.994602i \(-0.533088\pi\)
−0.103762 + 0.994602i \(0.533088\pi\)
\(984\) 0 0
\(985\) 3.23194 0.102978
\(986\) 0 0
\(987\) −3.70287 −0.117864
\(988\) 0 0
\(989\) 42.2624 1.34387
\(990\) 0 0
\(991\) 34.9276 1.10951 0.554755 0.832013i \(-0.312812\pi\)
0.554755 + 0.832013i \(0.312812\pi\)
\(992\) 0 0
\(993\) 14.9098 0.473148
\(994\) 0 0
\(995\) 26.7053 0.846614
\(996\) 0 0
\(997\) 4.41388 0.139789 0.0698944 0.997554i \(-0.477734\pi\)
0.0698944 + 0.997554i \(0.477734\pi\)
\(998\) 0 0
\(999\) 3.00000 0.0949158
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 2368.2.a.bd.1.3 3
4.3 odd 2 2368.2.a.bc.1.1 3
8.3 odd 2 1184.2.a.l.1.3 3
8.5 even 2 1184.2.a.m.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1184.2.a.l.1.3 3 8.3 odd 2
1184.2.a.m.1.1 yes 3 8.5 even 2
2368.2.a.bc.1.1 3 4.3 odd 2
2368.2.a.bd.1.3 3 1.1 even 1 trivial