Properties

Label 1336.2.a.e.1.9
Level $1336$
Weight $2$
Character 1336.1
Self dual yes
Analytic conductor $10.668$
Analytic rank $0$
Dimension $12$
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] = [1336,2,Mod(1,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, 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("1336.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.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: \(10.6680137100\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} - 2399 x^{3} + 148 x^{2} + 1224 x + 224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(1.94180\) of defining polynomial
Character \(\chi\) \(=\) 1336.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.94180 q^{3} -3.68233 q^{5} -2.51484 q^{7} +0.770577 q^{9} +O(q^{10})\) \(q+1.94180 q^{3} -3.68233 q^{5} -2.51484 q^{7} +0.770577 q^{9} +2.53131 q^{11} +1.40241 q^{13} -7.15035 q^{15} +6.24242 q^{17} +6.03220 q^{19} -4.88330 q^{21} -0.990062 q^{23} +8.55959 q^{25} -4.32909 q^{27} +2.81567 q^{29} +8.19863 q^{31} +4.91529 q^{33} +9.26047 q^{35} -5.90678 q^{37} +2.72320 q^{39} +11.9031 q^{41} -7.71591 q^{43} -2.83752 q^{45} -11.4013 q^{47} -0.675594 q^{49} +12.1215 q^{51} +13.3529 q^{53} -9.32112 q^{55} +11.7133 q^{57} +12.7570 q^{59} +10.0171 q^{61} -1.93788 q^{63} -5.16415 q^{65} +5.22542 q^{67} -1.92250 q^{69} +13.5563 q^{71} -15.1689 q^{73} +16.6210 q^{75} -6.36583 q^{77} +8.02509 q^{79} -10.7179 q^{81} -8.79046 q^{83} -22.9867 q^{85} +5.46746 q^{87} -8.44999 q^{89} -3.52684 q^{91} +15.9201 q^{93} -22.2126 q^{95} +3.29010 q^{97} +1.95057 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.94180 1.12110 0.560549 0.828122i \(-0.310590\pi\)
0.560549 + 0.828122i \(0.310590\pi\)
\(4\) 0 0
\(5\) −3.68233 −1.64679 −0.823395 0.567469i \(-0.807923\pi\)
−0.823395 + 0.567469i \(0.807923\pi\)
\(6\) 0 0
\(7\) −2.51484 −0.950519 −0.475260 0.879846i \(-0.657646\pi\)
−0.475260 + 0.879846i \(0.657646\pi\)
\(8\) 0 0
\(9\) 0.770577 0.256859
\(10\) 0 0
\(11\) 2.53131 0.763218 0.381609 0.924324i \(-0.375370\pi\)
0.381609 + 0.924324i \(0.375370\pi\)
\(12\) 0 0
\(13\) 1.40241 0.388959 0.194480 0.980907i \(-0.437698\pi\)
0.194480 + 0.980907i \(0.437698\pi\)
\(14\) 0 0
\(15\) −7.15035 −1.84621
\(16\) 0 0
\(17\) 6.24242 1.51401 0.757005 0.653409i \(-0.226662\pi\)
0.757005 + 0.653409i \(0.226662\pi\)
\(18\) 0 0
\(19\) 6.03220 1.38388 0.691940 0.721955i \(-0.256756\pi\)
0.691940 + 0.721955i \(0.256756\pi\)
\(20\) 0 0
\(21\) −4.88330 −1.06562
\(22\) 0 0
\(23\) −0.990062 −0.206442 −0.103221 0.994658i \(-0.532915\pi\)
−0.103221 + 0.994658i \(0.532915\pi\)
\(24\) 0 0
\(25\) 8.55959 1.71192
\(26\) 0 0
\(27\) −4.32909 −0.833133
\(28\) 0 0
\(29\) 2.81567 0.522857 0.261428 0.965223i \(-0.415806\pi\)
0.261428 + 0.965223i \(0.415806\pi\)
\(30\) 0 0
\(31\) 8.19863 1.47252 0.736259 0.676700i \(-0.236591\pi\)
0.736259 + 0.676700i \(0.236591\pi\)
\(32\) 0 0
\(33\) 4.91529 0.855642
\(34\) 0 0
\(35\) 9.26047 1.56531
\(36\) 0 0
\(37\) −5.90678 −0.971069 −0.485534 0.874218i \(-0.661375\pi\)
−0.485534 + 0.874218i \(0.661375\pi\)
\(38\) 0 0
\(39\) 2.72320 0.436061
\(40\) 0 0
\(41\) 11.9031 1.85896 0.929479 0.368876i \(-0.120257\pi\)
0.929479 + 0.368876i \(0.120257\pi\)
\(42\) 0 0
\(43\) −7.71591 −1.17666 −0.588332 0.808619i \(-0.700215\pi\)
−0.588332 + 0.808619i \(0.700215\pi\)
\(44\) 0 0
\(45\) −2.83752 −0.422993
\(46\) 0 0
\(47\) −11.4013 −1.66305 −0.831525 0.555487i \(-0.812532\pi\)
−0.831525 + 0.555487i \(0.812532\pi\)
\(48\) 0 0
\(49\) −0.675594 −0.0965134
\(50\) 0 0
\(51\) 12.1215 1.69735
\(52\) 0 0
\(53\) 13.3529 1.83416 0.917078 0.398707i \(-0.130541\pi\)
0.917078 + 0.398707i \(0.130541\pi\)
\(54\) 0 0
\(55\) −9.32112 −1.25686
\(56\) 0 0
\(57\) 11.7133 1.55146
\(58\) 0 0
\(59\) 12.7570 1.66082 0.830409 0.557155i \(-0.188107\pi\)
0.830409 + 0.557155i \(0.188107\pi\)
\(60\) 0 0
\(61\) 10.0171 1.28256 0.641281 0.767306i \(-0.278403\pi\)
0.641281 + 0.767306i \(0.278403\pi\)
\(62\) 0 0
\(63\) −1.93788 −0.244149
\(64\) 0 0
\(65\) −5.16415 −0.640534
\(66\) 0 0
\(67\) 5.22542 0.638387 0.319194 0.947690i \(-0.396588\pi\)
0.319194 + 0.947690i \(0.396588\pi\)
\(68\) 0 0
\(69\) −1.92250 −0.231442
\(70\) 0 0
\(71\) 13.5563 1.60883 0.804417 0.594066i \(-0.202478\pi\)
0.804417 + 0.594066i \(0.202478\pi\)
\(72\) 0 0
\(73\) −15.1689 −1.77538 −0.887691 0.460439i \(-0.847692\pi\)
−0.887691 + 0.460439i \(0.847692\pi\)
\(74\) 0 0
\(75\) 16.6210 1.91923
\(76\) 0 0
\(77\) −6.36583 −0.725453
\(78\) 0 0
\(79\) 8.02509 0.902893 0.451447 0.892298i \(-0.350908\pi\)
0.451447 + 0.892298i \(0.350908\pi\)
\(80\) 0 0
\(81\) −10.7179 −1.19088
\(82\) 0 0
\(83\) −8.79046 −0.964879 −0.482439 0.875929i \(-0.660249\pi\)
−0.482439 + 0.875929i \(0.660249\pi\)
\(84\) 0 0
\(85\) −22.9867 −2.49326
\(86\) 0 0
\(87\) 5.46746 0.586173
\(88\) 0 0
\(89\) −8.44999 −0.895697 −0.447849 0.894109i \(-0.647810\pi\)
−0.447849 + 0.894109i \(0.647810\pi\)
\(90\) 0 0
\(91\) −3.52684 −0.369713
\(92\) 0 0
\(93\) 15.9201 1.65083
\(94\) 0 0
\(95\) −22.2126 −2.27896
\(96\) 0 0
\(97\) 3.29010 0.334059 0.167029 0.985952i \(-0.446583\pi\)
0.167029 + 0.985952i \(0.446583\pi\)
\(98\) 0 0
\(99\) 1.95057 0.196039
\(100\) 0 0
\(101\) −2.48311 −0.247078 −0.123539 0.992340i \(-0.539424\pi\)
−0.123539 + 0.992340i \(0.539424\pi\)
\(102\) 0 0
\(103\) 8.24951 0.812848 0.406424 0.913684i \(-0.366776\pi\)
0.406424 + 0.913684i \(0.366776\pi\)
\(104\) 0 0
\(105\) 17.9820 1.75486
\(106\) 0 0
\(107\) −1.81289 −0.175259 −0.0876295 0.996153i \(-0.527929\pi\)
−0.0876295 + 0.996153i \(0.527929\pi\)
\(108\) 0 0
\(109\) −11.5606 −1.10730 −0.553652 0.832748i \(-0.686766\pi\)
−0.553652 + 0.832748i \(0.686766\pi\)
\(110\) 0 0
\(111\) −11.4698 −1.08866
\(112\) 0 0
\(113\) −6.22979 −0.586049 −0.293025 0.956105i \(-0.594662\pi\)
−0.293025 + 0.956105i \(0.594662\pi\)
\(114\) 0 0
\(115\) 3.64574 0.339967
\(116\) 0 0
\(117\) 1.08067 0.0999077
\(118\) 0 0
\(119\) −15.6987 −1.43910
\(120\) 0 0
\(121\) −4.59248 −0.417498
\(122\) 0 0
\(123\) 23.1135 2.08407
\(124\) 0 0
\(125\) −13.1076 −1.17238
\(126\) 0 0
\(127\) 5.82117 0.516545 0.258273 0.966072i \(-0.416847\pi\)
0.258273 + 0.966072i \(0.416847\pi\)
\(128\) 0 0
\(129\) −14.9827 −1.31916
\(130\) 0 0
\(131\) 11.5722 1.01107 0.505534 0.862806i \(-0.331295\pi\)
0.505534 + 0.862806i \(0.331295\pi\)
\(132\) 0 0
\(133\) −15.1700 −1.31540
\(134\) 0 0
\(135\) 15.9411 1.37200
\(136\) 0 0
\(137\) 8.12029 0.693763 0.346882 0.937909i \(-0.387241\pi\)
0.346882 + 0.937909i \(0.387241\pi\)
\(138\) 0 0
\(139\) −19.0550 −1.61623 −0.808113 0.589027i \(-0.799511\pi\)
−0.808113 + 0.589027i \(0.799511\pi\)
\(140\) 0 0
\(141\) −22.1390 −1.86444
\(142\) 0 0
\(143\) 3.54994 0.296861
\(144\) 0 0
\(145\) −10.3682 −0.861035
\(146\) 0 0
\(147\) −1.31187 −0.108201
\(148\) 0 0
\(149\) −12.1870 −0.998396 −0.499198 0.866488i \(-0.666372\pi\)
−0.499198 + 0.866488i \(0.666372\pi\)
\(150\) 0 0
\(151\) 3.39930 0.276631 0.138315 0.990388i \(-0.455831\pi\)
0.138315 + 0.990388i \(0.455831\pi\)
\(152\) 0 0
\(153\) 4.81027 0.388887
\(154\) 0 0
\(155\) −30.1901 −2.42493
\(156\) 0 0
\(157\) −2.74134 −0.218783 −0.109391 0.993999i \(-0.534890\pi\)
−0.109391 + 0.993999i \(0.534890\pi\)
\(158\) 0 0
\(159\) 25.9286 2.05627
\(160\) 0 0
\(161\) 2.48985 0.196227
\(162\) 0 0
\(163\) 0.654254 0.0512451 0.0256225 0.999672i \(-0.491843\pi\)
0.0256225 + 0.999672i \(0.491843\pi\)
\(164\) 0 0
\(165\) −18.0997 −1.40906
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −11.0332 −0.848711
\(170\) 0 0
\(171\) 4.64827 0.355462
\(172\) 0 0
\(173\) 8.27460 0.629106 0.314553 0.949240i \(-0.398145\pi\)
0.314553 + 0.949240i \(0.398145\pi\)
\(174\) 0 0
\(175\) −21.5260 −1.62721
\(176\) 0 0
\(177\) 24.7715 1.86194
\(178\) 0 0
\(179\) 10.7422 0.802913 0.401457 0.915878i \(-0.368504\pi\)
0.401457 + 0.915878i \(0.368504\pi\)
\(180\) 0 0
\(181\) 6.32871 0.470409 0.235205 0.971946i \(-0.424424\pi\)
0.235205 + 0.971946i \(0.424424\pi\)
\(182\) 0 0
\(183\) 19.4512 1.43788
\(184\) 0 0
\(185\) 21.7507 1.59915
\(186\) 0 0
\(187\) 15.8015 1.15552
\(188\) 0 0
\(189\) 10.8870 0.791909
\(190\) 0 0
\(191\) 9.08743 0.657543 0.328772 0.944409i \(-0.393365\pi\)
0.328772 + 0.944409i \(0.393365\pi\)
\(192\) 0 0
\(193\) −11.7682 −0.847093 −0.423547 0.905874i \(-0.639215\pi\)
−0.423547 + 0.905874i \(0.639215\pi\)
\(194\) 0 0
\(195\) −10.0277 −0.718101
\(196\) 0 0
\(197\) −3.53903 −0.252145 −0.126073 0.992021i \(-0.540237\pi\)
−0.126073 + 0.992021i \(0.540237\pi\)
\(198\) 0 0
\(199\) 25.3303 1.79562 0.897808 0.440386i \(-0.145159\pi\)
0.897808 + 0.440386i \(0.145159\pi\)
\(200\) 0 0
\(201\) 10.1467 0.715694
\(202\) 0 0
\(203\) −7.08095 −0.496985
\(204\) 0 0
\(205\) −43.8313 −3.06131
\(206\) 0 0
\(207\) −0.762919 −0.0530266
\(208\) 0 0
\(209\) 15.2693 1.05620
\(210\) 0 0
\(211\) −9.18822 −0.632543 −0.316272 0.948669i \(-0.602431\pi\)
−0.316272 + 0.948669i \(0.602431\pi\)
\(212\) 0 0
\(213\) 26.3235 1.80366
\(214\) 0 0
\(215\) 28.4125 1.93772
\(216\) 0 0
\(217\) −20.6182 −1.39966
\(218\) 0 0
\(219\) −29.4549 −1.99038
\(220\) 0 0
\(221\) 8.75445 0.588888
\(222\) 0 0
\(223\) −0.936602 −0.0627195 −0.0313598 0.999508i \(-0.509984\pi\)
−0.0313598 + 0.999508i \(0.509984\pi\)
\(224\) 0 0
\(225\) 6.59582 0.439722
\(226\) 0 0
\(227\) −6.57719 −0.436544 −0.218272 0.975888i \(-0.570042\pi\)
−0.218272 + 0.975888i \(0.570042\pi\)
\(228\) 0 0
\(229\) −6.15337 −0.406626 −0.203313 0.979114i \(-0.565171\pi\)
−0.203313 + 0.979114i \(0.565171\pi\)
\(230\) 0 0
\(231\) −12.3611 −0.813304
\(232\) 0 0
\(233\) 2.88639 0.189094 0.0945469 0.995520i \(-0.469860\pi\)
0.0945469 + 0.995520i \(0.469860\pi\)
\(234\) 0 0
\(235\) 41.9834 2.73870
\(236\) 0 0
\(237\) 15.5831 1.01223
\(238\) 0 0
\(239\) 12.7704 0.826049 0.413025 0.910720i \(-0.364472\pi\)
0.413025 + 0.910720i \(0.364472\pi\)
\(240\) 0 0
\(241\) 10.8333 0.697832 0.348916 0.937154i \(-0.386550\pi\)
0.348916 + 0.937154i \(0.386550\pi\)
\(242\) 0 0
\(243\) −7.82481 −0.501962
\(244\) 0 0
\(245\) 2.48776 0.158937
\(246\) 0 0
\(247\) 8.45963 0.538273
\(248\) 0 0
\(249\) −17.0693 −1.08172
\(250\) 0 0
\(251\) 4.22678 0.266792 0.133396 0.991063i \(-0.457412\pi\)
0.133396 + 0.991063i \(0.457412\pi\)
\(252\) 0 0
\(253\) −2.50615 −0.157560
\(254\) 0 0
\(255\) −44.6355 −2.79518
\(256\) 0 0
\(257\) 29.7786 1.85754 0.928768 0.370662i \(-0.120869\pi\)
0.928768 + 0.370662i \(0.120869\pi\)
\(258\) 0 0
\(259\) 14.8546 0.923019
\(260\) 0 0
\(261\) 2.16969 0.134300
\(262\) 0 0
\(263\) 0.393566 0.0242683 0.0121342 0.999926i \(-0.496137\pi\)
0.0121342 + 0.999926i \(0.496137\pi\)
\(264\) 0 0
\(265\) −49.1697 −3.02047
\(266\) 0 0
\(267\) −16.4082 −1.00416
\(268\) 0 0
\(269\) −4.97029 −0.303044 −0.151522 0.988454i \(-0.548417\pi\)
−0.151522 + 0.988454i \(0.548417\pi\)
\(270\) 0 0
\(271\) −22.6033 −1.37306 −0.686528 0.727104i \(-0.740866\pi\)
−0.686528 + 0.727104i \(0.740866\pi\)
\(272\) 0 0
\(273\) −6.84841 −0.414485
\(274\) 0 0
\(275\) 21.6670 1.30657
\(276\) 0 0
\(277\) −19.3056 −1.15996 −0.579981 0.814630i \(-0.696940\pi\)
−0.579981 + 0.814630i \(0.696940\pi\)
\(278\) 0 0
\(279\) 6.31768 0.378229
\(280\) 0 0
\(281\) 29.7704 1.77595 0.887977 0.459888i \(-0.152111\pi\)
0.887977 + 0.459888i \(0.152111\pi\)
\(282\) 0 0
\(283\) 32.5284 1.93361 0.966807 0.255507i \(-0.0822425\pi\)
0.966807 + 0.255507i \(0.0822425\pi\)
\(284\) 0 0
\(285\) −43.1323 −2.55494
\(286\) 0 0
\(287\) −29.9344 −1.76697
\(288\) 0 0
\(289\) 21.9678 1.29223
\(290\) 0 0
\(291\) 6.38870 0.374512
\(292\) 0 0
\(293\) −14.8005 −0.864657 −0.432328 0.901716i \(-0.642308\pi\)
−0.432328 + 0.901716i \(0.642308\pi\)
\(294\) 0 0
\(295\) −46.9755 −2.73502
\(296\) 0 0
\(297\) −10.9583 −0.635862
\(298\) 0 0
\(299\) −1.38848 −0.0802976
\(300\) 0 0
\(301\) 19.4042 1.11844
\(302\) 0 0
\(303\) −4.82169 −0.276999
\(304\) 0 0
\(305\) −36.8864 −2.11211
\(306\) 0 0
\(307\) −16.7387 −0.955330 −0.477665 0.878542i \(-0.658517\pi\)
−0.477665 + 0.878542i \(0.658517\pi\)
\(308\) 0 0
\(309\) 16.0189 0.911282
\(310\) 0 0
\(311\) 16.8231 0.953951 0.476975 0.878917i \(-0.341733\pi\)
0.476975 + 0.878917i \(0.341733\pi\)
\(312\) 0 0
\(313\) −25.0431 −1.41552 −0.707759 0.706454i \(-0.750294\pi\)
−0.707759 + 0.706454i \(0.750294\pi\)
\(314\) 0 0
\(315\) 7.13591 0.402063
\(316\) 0 0
\(317\) −18.3304 −1.02954 −0.514768 0.857330i \(-0.672122\pi\)
−0.514768 + 0.857330i \(0.672122\pi\)
\(318\) 0 0
\(319\) 7.12733 0.399054
\(320\) 0 0
\(321\) −3.52027 −0.196482
\(322\) 0 0
\(323\) 37.6555 2.09521
\(324\) 0 0
\(325\) 12.0041 0.665866
\(326\) 0 0
\(327\) −22.4483 −1.24140
\(328\) 0 0
\(329\) 28.6724 1.58076
\(330\) 0 0
\(331\) −6.09542 −0.335035 −0.167517 0.985869i \(-0.553575\pi\)
−0.167517 + 0.985869i \(0.553575\pi\)
\(332\) 0 0
\(333\) −4.55163 −0.249428
\(334\) 0 0
\(335\) −19.2418 −1.05129
\(336\) 0 0
\(337\) 17.8720 0.973550 0.486775 0.873527i \(-0.338173\pi\)
0.486775 + 0.873527i \(0.338173\pi\)
\(338\) 0 0
\(339\) −12.0970 −0.657018
\(340\) 0 0
\(341\) 20.7533 1.12385
\(342\) 0 0
\(343\) 19.3029 1.04226
\(344\) 0 0
\(345\) 7.07929 0.381136
\(346\) 0 0
\(347\) −14.7608 −0.792401 −0.396201 0.918164i \(-0.629671\pi\)
−0.396201 + 0.918164i \(0.629671\pi\)
\(348\) 0 0
\(349\) −19.8320 −1.06158 −0.530792 0.847502i \(-0.678106\pi\)
−0.530792 + 0.847502i \(0.678106\pi\)
\(350\) 0 0
\(351\) −6.07117 −0.324055
\(352\) 0 0
\(353\) 15.8103 0.841498 0.420749 0.907177i \(-0.361767\pi\)
0.420749 + 0.907177i \(0.361767\pi\)
\(354\) 0 0
\(355\) −49.9187 −2.64941
\(356\) 0 0
\(357\) −30.4836 −1.61337
\(358\) 0 0
\(359\) −0.298461 −0.0157522 −0.00787608 0.999969i \(-0.502507\pi\)
−0.00787608 + 0.999969i \(0.502507\pi\)
\(360\) 0 0
\(361\) 17.3874 0.915125
\(362\) 0 0
\(363\) −8.91767 −0.468056
\(364\) 0 0
\(365\) 55.8569 2.92368
\(366\) 0 0
\(367\) −5.71709 −0.298430 −0.149215 0.988805i \(-0.547675\pi\)
−0.149215 + 0.988805i \(0.547675\pi\)
\(368\) 0 0
\(369\) 9.17228 0.477490
\(370\) 0 0
\(371\) −33.5803 −1.74340
\(372\) 0 0
\(373\) 28.6296 1.48238 0.741192 0.671293i \(-0.234261\pi\)
0.741192 + 0.671293i \(0.234261\pi\)
\(374\) 0 0
\(375\) −25.4523 −1.31435
\(376\) 0 0
\(377\) 3.94873 0.203370
\(378\) 0 0
\(379\) −21.5972 −1.10937 −0.554686 0.832060i \(-0.687162\pi\)
−0.554686 + 0.832060i \(0.687162\pi\)
\(380\) 0 0
\(381\) 11.3035 0.579098
\(382\) 0 0
\(383\) 8.00419 0.408995 0.204497 0.978867i \(-0.434444\pi\)
0.204497 + 0.978867i \(0.434444\pi\)
\(384\) 0 0
\(385\) 23.4411 1.19467
\(386\) 0 0
\(387\) −5.94570 −0.302237
\(388\) 0 0
\(389\) −15.5102 −0.786401 −0.393200 0.919453i \(-0.628632\pi\)
−0.393200 + 0.919453i \(0.628632\pi\)
\(390\) 0 0
\(391\) −6.18039 −0.312556
\(392\) 0 0
\(393\) 22.4709 1.13351
\(394\) 0 0
\(395\) −29.5511 −1.48688
\(396\) 0 0
\(397\) −8.55297 −0.429261 −0.214631 0.976695i \(-0.568855\pi\)
−0.214631 + 0.976695i \(0.568855\pi\)
\(398\) 0 0
\(399\) −29.4570 −1.47470
\(400\) 0 0
\(401\) −25.8505 −1.29091 −0.645457 0.763796i \(-0.723333\pi\)
−0.645457 + 0.763796i \(0.723333\pi\)
\(402\) 0 0
\(403\) 11.4979 0.572749
\(404\) 0 0
\(405\) 39.4671 1.96113
\(406\) 0 0
\(407\) −14.9519 −0.741137
\(408\) 0 0
\(409\) −11.7146 −0.579252 −0.289626 0.957140i \(-0.593531\pi\)
−0.289626 + 0.957140i \(0.593531\pi\)
\(410\) 0 0
\(411\) 15.7680 0.777776
\(412\) 0 0
\(413\) −32.0817 −1.57864
\(414\) 0 0
\(415\) 32.3694 1.58895
\(416\) 0 0
\(417\) −37.0010 −1.81195
\(418\) 0 0
\(419\) 2.43634 0.119023 0.0595116 0.998228i \(-0.481046\pi\)
0.0595116 + 0.998228i \(0.481046\pi\)
\(420\) 0 0
\(421\) −26.3748 −1.28543 −0.642714 0.766106i \(-0.722192\pi\)
−0.642714 + 0.766106i \(0.722192\pi\)
\(422\) 0 0
\(423\) −8.78558 −0.427170
\(424\) 0 0
\(425\) 53.4326 2.59186
\(426\) 0 0
\(427\) −25.1915 −1.21910
\(428\) 0 0
\(429\) 6.89326 0.332810
\(430\) 0 0
\(431\) −13.3288 −0.642024 −0.321012 0.947075i \(-0.604023\pi\)
−0.321012 + 0.947075i \(0.604023\pi\)
\(432\) 0 0
\(433\) 24.2184 1.16386 0.581930 0.813239i \(-0.302298\pi\)
0.581930 + 0.813239i \(0.302298\pi\)
\(434\) 0 0
\(435\) −20.1330 −0.965304
\(436\) 0 0
\(437\) −5.97225 −0.285691
\(438\) 0 0
\(439\) −34.2041 −1.63247 −0.816237 0.577717i \(-0.803944\pi\)
−0.816237 + 0.577717i \(0.803944\pi\)
\(440\) 0 0
\(441\) −0.520597 −0.0247903
\(442\) 0 0
\(443\) −6.02481 −0.286248 −0.143124 0.989705i \(-0.545715\pi\)
−0.143124 + 0.989705i \(0.545715\pi\)
\(444\) 0 0
\(445\) 31.1157 1.47503
\(446\) 0 0
\(447\) −23.6646 −1.11930
\(448\) 0 0
\(449\) −34.4523 −1.62590 −0.812952 0.582330i \(-0.802141\pi\)
−0.812952 + 0.582330i \(0.802141\pi\)
\(450\) 0 0
\(451\) 30.1305 1.41879
\(452\) 0 0
\(453\) 6.60075 0.310130
\(454\) 0 0
\(455\) 12.9870 0.608840
\(456\) 0 0
\(457\) −13.1048 −0.613017 −0.306509 0.951868i \(-0.599161\pi\)
−0.306509 + 0.951868i \(0.599161\pi\)
\(458\) 0 0
\(459\) −27.0240 −1.26137
\(460\) 0 0
\(461\) −29.4582 −1.37201 −0.686003 0.727599i \(-0.740636\pi\)
−0.686003 + 0.727599i \(0.740636\pi\)
\(462\) 0 0
\(463\) 27.5619 1.28091 0.640456 0.767995i \(-0.278745\pi\)
0.640456 + 0.767995i \(0.278745\pi\)
\(464\) 0 0
\(465\) −58.6230 −2.71858
\(466\) 0 0
\(467\) −6.81919 −0.315555 −0.157777 0.987475i \(-0.550433\pi\)
−0.157777 + 0.987475i \(0.550433\pi\)
\(468\) 0 0
\(469\) −13.1411 −0.606799
\(470\) 0 0
\(471\) −5.32312 −0.245276
\(472\) 0 0
\(473\) −19.5313 −0.898052
\(474\) 0 0
\(475\) 51.6331 2.36909
\(476\) 0 0
\(477\) 10.2894 0.471120
\(478\) 0 0
\(479\) 14.3817 0.657118 0.328559 0.944483i \(-0.393437\pi\)
0.328559 + 0.944483i \(0.393437\pi\)
\(480\) 0 0
\(481\) −8.28375 −0.377706
\(482\) 0 0
\(483\) 4.83478 0.219990
\(484\) 0 0
\(485\) −12.1152 −0.550125
\(486\) 0 0
\(487\) 17.7508 0.804367 0.402184 0.915559i \(-0.368251\pi\)
0.402184 + 0.915559i \(0.368251\pi\)
\(488\) 0 0
\(489\) 1.27043 0.0574507
\(490\) 0 0
\(491\) 24.7017 1.11477 0.557387 0.830253i \(-0.311804\pi\)
0.557387 + 0.830253i \(0.311804\pi\)
\(492\) 0 0
\(493\) 17.5766 0.791610
\(494\) 0 0
\(495\) −7.18264 −0.322836
\(496\) 0 0
\(497\) −34.0918 −1.52923
\(498\) 0 0
\(499\) 12.4029 0.555230 0.277615 0.960692i \(-0.410456\pi\)
0.277615 + 0.960692i \(0.410456\pi\)
\(500\) 0 0
\(501\) 1.94180 0.0867531
\(502\) 0 0
\(503\) 3.58294 0.159755 0.0798777 0.996805i \(-0.474547\pi\)
0.0798777 + 0.996805i \(0.474547\pi\)
\(504\) 0 0
\(505\) 9.14363 0.406886
\(506\) 0 0
\(507\) −21.4243 −0.951487
\(508\) 0 0
\(509\) 15.8392 0.702062 0.351031 0.936364i \(-0.385831\pi\)
0.351031 + 0.936364i \(0.385831\pi\)
\(510\) 0 0
\(511\) 38.1472 1.68753
\(512\) 0 0
\(513\) −26.1139 −1.15296
\(514\) 0 0
\(515\) −30.3775 −1.33859
\(516\) 0 0
\(517\) −28.8602 −1.26927
\(518\) 0 0
\(519\) 16.0676 0.705289
\(520\) 0 0
\(521\) 35.4884 1.55478 0.777388 0.629021i \(-0.216544\pi\)
0.777388 + 0.629021i \(0.216544\pi\)
\(522\) 0 0
\(523\) −23.6911 −1.03594 −0.517970 0.855399i \(-0.673312\pi\)
−0.517970 + 0.855399i \(0.673312\pi\)
\(524\) 0 0
\(525\) −41.7991 −1.82426
\(526\) 0 0
\(527\) 51.1793 2.22941
\(528\) 0 0
\(529\) −22.0198 −0.957382
\(530\) 0 0
\(531\) 9.83024 0.426596
\(532\) 0 0
\(533\) 16.6931 0.723059
\(534\) 0 0
\(535\) 6.67568 0.288615
\(536\) 0 0
\(537\) 20.8593 0.900144
\(538\) 0 0
\(539\) −1.71014 −0.0736607
\(540\) 0 0
\(541\) 27.1705 1.16815 0.584076 0.811699i \(-0.301457\pi\)
0.584076 + 0.811699i \(0.301457\pi\)
\(542\) 0 0
\(543\) 12.2891 0.527375
\(544\) 0 0
\(545\) 42.5700 1.82350
\(546\) 0 0
\(547\) −16.1554 −0.690756 −0.345378 0.938464i \(-0.612249\pi\)
−0.345378 + 0.938464i \(0.612249\pi\)
\(548\) 0 0
\(549\) 7.71897 0.329438
\(550\) 0 0
\(551\) 16.9847 0.723571
\(552\) 0 0
\(553\) −20.1818 −0.858217
\(554\) 0 0
\(555\) 42.2355 1.79280
\(556\) 0 0
\(557\) 6.22505 0.263764 0.131882 0.991265i \(-0.457898\pi\)
0.131882 + 0.991265i \(0.457898\pi\)
\(558\) 0 0
\(559\) −10.8209 −0.457675
\(560\) 0 0
\(561\) 30.6833 1.29545
\(562\) 0 0
\(563\) 8.13126 0.342692 0.171346 0.985211i \(-0.445188\pi\)
0.171346 + 0.985211i \(0.445188\pi\)
\(564\) 0 0
\(565\) 22.9402 0.965100
\(566\) 0 0
\(567\) 26.9539 1.13196
\(568\) 0 0
\(569\) 45.2871 1.89854 0.949268 0.314469i \(-0.101826\pi\)
0.949268 + 0.314469i \(0.101826\pi\)
\(570\) 0 0
\(571\) 17.7671 0.743528 0.371764 0.928327i \(-0.378753\pi\)
0.371764 + 0.928327i \(0.378753\pi\)
\(572\) 0 0
\(573\) 17.6460 0.737170
\(574\) 0 0
\(575\) −8.47453 −0.353412
\(576\) 0 0
\(577\) 25.3505 1.05535 0.527677 0.849445i \(-0.323063\pi\)
0.527677 + 0.849445i \(0.323063\pi\)
\(578\) 0 0
\(579\) −22.8515 −0.949674
\(580\) 0 0
\(581\) 22.1066 0.917136
\(582\) 0 0
\(583\) 33.8002 1.39986
\(584\) 0 0
\(585\) −3.97938 −0.164527
\(586\) 0 0
\(587\) −6.23794 −0.257467 −0.128734 0.991679i \(-0.541091\pi\)
−0.128734 + 0.991679i \(0.541091\pi\)
\(588\) 0 0
\(589\) 49.4557 2.03779
\(590\) 0 0
\(591\) −6.87207 −0.282679
\(592\) 0 0
\(593\) 8.77480 0.360338 0.180169 0.983636i \(-0.442336\pi\)
0.180169 + 0.983636i \(0.442336\pi\)
\(594\) 0 0
\(595\) 57.8078 2.36989
\(596\) 0 0
\(597\) 49.1863 2.01306
\(598\) 0 0
\(599\) 33.6324 1.37418 0.687092 0.726570i \(-0.258887\pi\)
0.687092 + 0.726570i \(0.258887\pi\)
\(600\) 0 0
\(601\) −12.0193 −0.490278 −0.245139 0.969488i \(-0.578834\pi\)
−0.245139 + 0.969488i \(0.578834\pi\)
\(602\) 0 0
\(603\) 4.02659 0.163975
\(604\) 0 0
\(605\) 16.9111 0.687532
\(606\) 0 0
\(607\) −11.2311 −0.455855 −0.227928 0.973678i \(-0.573195\pi\)
−0.227928 + 0.973678i \(0.573195\pi\)
\(608\) 0 0
\(609\) −13.7498 −0.557169
\(610\) 0 0
\(611\) −15.9893 −0.646859
\(612\) 0 0
\(613\) −41.5509 −1.67823 −0.839113 0.543957i \(-0.816925\pi\)
−0.839113 + 0.543957i \(0.816925\pi\)
\(614\) 0 0
\(615\) −85.1116 −3.43203
\(616\) 0 0
\(617\) −6.87722 −0.276866 −0.138433 0.990372i \(-0.544207\pi\)
−0.138433 + 0.990372i \(0.544207\pi\)
\(618\) 0 0
\(619\) −6.08552 −0.244598 −0.122299 0.992493i \(-0.539027\pi\)
−0.122299 + 0.992493i \(0.539027\pi\)
\(620\) 0 0
\(621\) 4.28607 0.171994
\(622\) 0 0
\(623\) 21.2503 0.851377
\(624\) 0 0
\(625\) 5.46862 0.218745
\(626\) 0 0
\(627\) 29.6500 1.18411
\(628\) 0 0
\(629\) −36.8726 −1.47021
\(630\) 0 0
\(631\) −6.91686 −0.275356 −0.137678 0.990477i \(-0.543964\pi\)
−0.137678 + 0.990477i \(0.543964\pi\)
\(632\) 0 0
\(633\) −17.8417 −0.709143
\(634\) 0 0
\(635\) −21.4355 −0.850642
\(636\) 0 0
\(637\) −0.947461 −0.0375398
\(638\) 0 0
\(639\) 10.4462 0.413243
\(640\) 0 0
\(641\) −35.1119 −1.38684 −0.693419 0.720534i \(-0.743897\pi\)
−0.693419 + 0.720534i \(0.743897\pi\)
\(642\) 0 0
\(643\) −5.58940 −0.220425 −0.110212 0.993908i \(-0.535153\pi\)
−0.110212 + 0.993908i \(0.535153\pi\)
\(644\) 0 0
\(645\) 55.1714 2.17237
\(646\) 0 0
\(647\) −25.7389 −1.01190 −0.505949 0.862563i \(-0.668858\pi\)
−0.505949 + 0.862563i \(0.668858\pi\)
\(648\) 0 0
\(649\) 32.2918 1.26757
\(650\) 0 0
\(651\) −40.0364 −1.56915
\(652\) 0 0
\(653\) −4.21923 −0.165111 −0.0825557 0.996586i \(-0.526308\pi\)
−0.0825557 + 0.996586i \(0.526308\pi\)
\(654\) 0 0
\(655\) −42.6128 −1.66502
\(656\) 0 0
\(657\) −11.6888 −0.456023
\(658\) 0 0
\(659\) 0.00339238 0.000132148 0 6.60741e−5 1.00000i \(-0.499979\pi\)
6.60741e−5 1.00000i \(0.499979\pi\)
\(660\) 0 0
\(661\) 0.251775 0.00979290 0.00489645 0.999988i \(-0.498441\pi\)
0.00489645 + 0.999988i \(0.498441\pi\)
\(662\) 0 0
\(663\) 16.9994 0.660201
\(664\) 0 0
\(665\) 55.8610 2.16620
\(666\) 0 0
\(667\) −2.78769 −0.107940
\(668\) 0 0
\(669\) −1.81869 −0.0703147
\(670\) 0 0
\(671\) 25.3564 0.978874
\(672\) 0 0
\(673\) −3.29797 −0.127127 −0.0635637 0.997978i \(-0.520247\pi\)
−0.0635637 + 0.997978i \(0.520247\pi\)
\(674\) 0 0
\(675\) −37.0552 −1.42626
\(676\) 0 0
\(677\) −30.5673 −1.17480 −0.587398 0.809298i \(-0.699848\pi\)
−0.587398 + 0.809298i \(0.699848\pi\)
\(678\) 0 0
\(679\) −8.27406 −0.317529
\(680\) 0 0
\(681\) −12.7716 −0.489408
\(682\) 0 0
\(683\) −4.05889 −0.155309 −0.0776546 0.996980i \(-0.524743\pi\)
−0.0776546 + 0.996980i \(0.524743\pi\)
\(684\) 0 0
\(685\) −29.9016 −1.14248
\(686\) 0 0
\(687\) −11.9486 −0.455867
\(688\) 0 0
\(689\) 18.7262 0.713412
\(690\) 0 0
\(691\) −32.1912 −1.22461 −0.612305 0.790622i \(-0.709758\pi\)
−0.612305 + 0.790622i \(0.709758\pi\)
\(692\) 0 0
\(693\) −4.90536 −0.186339
\(694\) 0 0
\(695\) 70.1670 2.66159
\(696\) 0 0
\(697\) 74.3044 2.81448
\(698\) 0 0
\(699\) 5.60479 0.211993
\(700\) 0 0
\(701\) 20.4257 0.771469 0.385735 0.922610i \(-0.373948\pi\)
0.385735 + 0.922610i \(0.373948\pi\)
\(702\) 0 0
\(703\) −35.6309 −1.34384
\(704\) 0 0
\(705\) 81.5233 3.07034
\(706\) 0 0
\(707\) 6.24461 0.234853
\(708\) 0 0
\(709\) −30.7428 −1.15457 −0.577285 0.816542i \(-0.695888\pi\)
−0.577285 + 0.816542i \(0.695888\pi\)
\(710\) 0 0
\(711\) 6.18395 0.231916
\(712\) 0 0
\(713\) −8.11715 −0.303990
\(714\) 0 0
\(715\) −13.0721 −0.488867
\(716\) 0 0
\(717\) 24.7976 0.926082
\(718\) 0 0
\(719\) 6.99493 0.260867 0.130433 0.991457i \(-0.458363\pi\)
0.130433 + 0.991457i \(0.458363\pi\)
\(720\) 0 0
\(721\) −20.7462 −0.772628
\(722\) 0 0
\(723\) 21.0360 0.782338
\(724\) 0 0
\(725\) 24.1010 0.895088
\(726\) 0 0
\(727\) 21.8854 0.811685 0.405842 0.913943i \(-0.366978\pi\)
0.405842 + 0.913943i \(0.366978\pi\)
\(728\) 0 0
\(729\) 16.9596 0.628134
\(730\) 0 0
\(731\) −48.1659 −1.78148
\(732\) 0 0
\(733\) 27.1139 1.00147 0.500737 0.865599i \(-0.333062\pi\)
0.500737 + 0.865599i \(0.333062\pi\)
\(734\) 0 0
\(735\) 4.83073 0.178184
\(736\) 0 0
\(737\) 13.2272 0.487228
\(738\) 0 0
\(739\) 7.59944 0.279550 0.139775 0.990183i \(-0.455362\pi\)
0.139775 + 0.990183i \(0.455362\pi\)
\(740\) 0 0
\(741\) 16.4269 0.603457
\(742\) 0 0
\(743\) 52.3660 1.92112 0.960562 0.278066i \(-0.0896935\pi\)
0.960562 + 0.278066i \(0.0896935\pi\)
\(744\) 0 0
\(745\) 44.8765 1.64415
\(746\) 0 0
\(747\) −6.77373 −0.247838
\(748\) 0 0
\(749\) 4.55913 0.166587
\(750\) 0 0
\(751\) −13.8593 −0.505734 −0.252867 0.967501i \(-0.581374\pi\)
−0.252867 + 0.967501i \(0.581374\pi\)
\(752\) 0 0
\(753\) 8.20754 0.299100
\(754\) 0 0
\(755\) −12.5174 −0.455553
\(756\) 0 0
\(757\) −0.489528 −0.0177922 −0.00889610 0.999960i \(-0.502832\pi\)
−0.00889610 + 0.999960i \(0.502832\pi\)
\(758\) 0 0
\(759\) −4.86644 −0.176641
\(760\) 0 0
\(761\) −16.0553 −0.582005 −0.291002 0.956722i \(-0.593989\pi\)
−0.291002 + 0.956722i \(0.593989\pi\)
\(762\) 0 0
\(763\) 29.0730 1.05251
\(764\) 0 0
\(765\) −17.7130 −0.640415
\(766\) 0 0
\(767\) 17.8906 0.645990
\(768\) 0 0
\(769\) −11.9309 −0.430239 −0.215120 0.976588i \(-0.569014\pi\)
−0.215120 + 0.976588i \(0.569014\pi\)
\(770\) 0 0
\(771\) 57.8239 2.08248
\(772\) 0 0
\(773\) −44.5255 −1.60147 −0.800736 0.599017i \(-0.795558\pi\)
−0.800736 + 0.599017i \(0.795558\pi\)
\(774\) 0 0
\(775\) 70.1769 2.52083
\(776\) 0 0
\(777\) 28.8446 1.03479
\(778\) 0 0
\(779\) 71.8020 2.57257
\(780\) 0 0
\(781\) 34.3151 1.22789
\(782\) 0 0
\(783\) −12.1893 −0.435609
\(784\) 0 0
\(785\) 10.0945 0.360289
\(786\) 0 0
\(787\) −24.8006 −0.884045 −0.442022 0.897004i \(-0.645739\pi\)
−0.442022 + 0.897004i \(0.645739\pi\)
\(788\) 0 0
\(789\) 0.764225 0.0272071
\(790\) 0 0
\(791\) 15.6669 0.557051
\(792\) 0 0
\(793\) 14.0482 0.498865
\(794\) 0 0
\(795\) −95.4776 −3.38624
\(796\) 0 0
\(797\) −52.5955 −1.86303 −0.931515 0.363704i \(-0.881512\pi\)
−0.931515 + 0.363704i \(0.881512\pi\)
\(798\) 0 0
\(799\) −71.1717 −2.51787
\(800\) 0 0
\(801\) −6.51137 −0.230068
\(802\) 0 0
\(803\) −38.3971 −1.35500
\(804\) 0 0
\(805\) −9.16844 −0.323145
\(806\) 0 0
\(807\) −9.65130 −0.339742
\(808\) 0 0
\(809\) 20.8948 0.734621 0.367310 0.930098i \(-0.380279\pi\)
0.367310 + 0.930098i \(0.380279\pi\)
\(810\) 0 0
\(811\) −23.0344 −0.808846 −0.404423 0.914572i \(-0.632528\pi\)
−0.404423 + 0.914572i \(0.632528\pi\)
\(812\) 0 0
\(813\) −43.8911 −1.53933
\(814\) 0 0
\(815\) −2.40918 −0.0843899
\(816\) 0 0
\(817\) −46.5438 −1.62836
\(818\) 0 0
\(819\) −2.71770 −0.0949642
\(820\) 0 0
\(821\) −14.0016 −0.488658 −0.244329 0.969692i \(-0.578568\pi\)
−0.244329 + 0.969692i \(0.578568\pi\)
\(822\) 0 0
\(823\) 21.5218 0.750202 0.375101 0.926984i \(-0.377608\pi\)
0.375101 + 0.926984i \(0.377608\pi\)
\(824\) 0 0
\(825\) 42.0728 1.46479
\(826\) 0 0
\(827\) 46.0650 1.60184 0.800918 0.598774i \(-0.204345\pi\)
0.800918 + 0.598774i \(0.204345\pi\)
\(828\) 0 0
\(829\) −28.4817 −0.989210 −0.494605 0.869118i \(-0.664687\pi\)
−0.494605 + 0.869118i \(0.664687\pi\)
\(830\) 0 0
\(831\) −37.4876 −1.30043
\(832\) 0 0
\(833\) −4.21734 −0.146122
\(834\) 0 0
\(835\) −3.68233 −0.127432
\(836\) 0 0
\(837\) −35.4926 −1.22680
\(838\) 0 0
\(839\) −29.5227 −1.01924 −0.509619 0.860400i \(-0.670213\pi\)
−0.509619 + 0.860400i \(0.670213\pi\)
\(840\) 0 0
\(841\) −21.0720 −0.726621
\(842\) 0 0
\(843\) 57.8081 1.99102
\(844\) 0 0
\(845\) 40.6281 1.39765
\(846\) 0 0
\(847\) 11.5493 0.396840
\(848\) 0 0
\(849\) 63.1636 2.16777
\(850\) 0 0
\(851\) 5.84808 0.200470
\(852\) 0 0
\(853\) 5.35509 0.183355 0.0916773 0.995789i \(-0.470777\pi\)
0.0916773 + 0.995789i \(0.470777\pi\)
\(854\) 0 0
\(855\) −17.1165 −0.585372
\(856\) 0 0
\(857\) −13.2566 −0.452835 −0.226418 0.974030i \(-0.572701\pi\)
−0.226418 + 0.974030i \(0.572701\pi\)
\(858\) 0 0
\(859\) 2.80254 0.0956215 0.0478107 0.998856i \(-0.484776\pi\)
0.0478107 + 0.998856i \(0.484776\pi\)
\(860\) 0 0
\(861\) −58.1266 −1.98095
\(862\) 0 0
\(863\) −33.3178 −1.13415 −0.567075 0.823666i \(-0.691925\pi\)
−0.567075 + 0.823666i \(0.691925\pi\)
\(864\) 0 0
\(865\) −30.4699 −1.03601
\(866\) 0 0
\(867\) 42.6571 1.44871
\(868\) 0 0
\(869\) 20.3140 0.689104
\(870\) 0 0
\(871\) 7.32820 0.248307
\(872\) 0 0
\(873\) 2.53527 0.0858060
\(874\) 0 0
\(875\) 32.9635 1.11437
\(876\) 0 0
\(877\) 52.6528 1.77796 0.888979 0.457947i \(-0.151415\pi\)
0.888979 + 0.457947i \(0.151415\pi\)
\(878\) 0 0
\(879\) −28.7396 −0.969364
\(880\) 0 0
\(881\) −5.06476 −0.170636 −0.0853181 0.996354i \(-0.527191\pi\)
−0.0853181 + 0.996354i \(0.527191\pi\)
\(882\) 0 0
\(883\) 2.48451 0.0836105 0.0418052 0.999126i \(-0.486689\pi\)
0.0418052 + 0.999126i \(0.486689\pi\)
\(884\) 0 0
\(885\) −91.2168 −3.06622
\(886\) 0 0
\(887\) 21.7059 0.728811 0.364406 0.931240i \(-0.381272\pi\)
0.364406 + 0.931240i \(0.381272\pi\)
\(888\) 0 0
\(889\) −14.6393 −0.490986
\(890\) 0 0
\(891\) −27.1304 −0.908903
\(892\) 0 0
\(893\) −68.7749 −2.30146
\(894\) 0 0
\(895\) −39.5566 −1.32223
\(896\) 0 0
\(897\) −2.69614 −0.0900215
\(898\) 0 0
\(899\) 23.0846 0.769915
\(900\) 0 0
\(901\) 83.3542 2.77693
\(902\) 0 0
\(903\) 37.6791 1.25388
\(904\) 0 0
\(905\) −23.3044 −0.774666
\(906\) 0 0
\(907\) 6.87267 0.228203 0.114102 0.993469i \(-0.463601\pi\)
0.114102 + 0.993469i \(0.463601\pi\)
\(908\) 0 0
\(909\) −1.91342 −0.0634643
\(910\) 0 0
\(911\) 37.4501 1.24078 0.620389 0.784294i \(-0.286975\pi\)
0.620389 + 0.784294i \(0.286975\pi\)
\(912\) 0 0
\(913\) −22.2514 −0.736413
\(914\) 0 0
\(915\) −71.6260 −2.36788
\(916\) 0 0
\(917\) −29.1022 −0.961040
\(918\) 0 0
\(919\) 28.1748 0.929402 0.464701 0.885468i \(-0.346162\pi\)
0.464701 + 0.885468i \(0.346162\pi\)
\(920\) 0 0
\(921\) −32.5032 −1.07102
\(922\) 0 0
\(923\) 19.0115 0.625771
\(924\) 0 0
\(925\) −50.5596 −1.66239
\(926\) 0 0
\(927\) 6.35688 0.208787
\(928\) 0 0
\(929\) −25.4167 −0.833896 −0.416948 0.908930i \(-0.636900\pi\)
−0.416948 + 0.908930i \(0.636900\pi\)
\(930\) 0 0
\(931\) −4.07531 −0.133563
\(932\) 0 0
\(933\) 32.6671 1.06947
\(934\) 0 0
\(935\) −58.1864 −1.90290
\(936\) 0 0
\(937\) −15.3436 −0.501254 −0.250627 0.968084i \(-0.580637\pi\)
−0.250627 + 0.968084i \(0.580637\pi\)
\(938\) 0 0
\(939\) −48.6286 −1.58693
\(940\) 0 0
\(941\) −29.7781 −0.970739 −0.485369 0.874309i \(-0.661315\pi\)
−0.485369 + 0.874309i \(0.661315\pi\)
\(942\) 0 0
\(943\) −11.7848 −0.383767
\(944\) 0 0
\(945\) −40.0894 −1.30411
\(946\) 0 0
\(947\) 25.4473 0.826926 0.413463 0.910521i \(-0.364319\pi\)
0.413463 + 0.910521i \(0.364319\pi\)
\(948\) 0 0
\(949\) −21.2730 −0.690552
\(950\) 0 0
\(951\) −35.5938 −1.15421
\(952\) 0 0
\(953\) 38.0702 1.23322 0.616608 0.787271i \(-0.288506\pi\)
0.616608 + 0.787271i \(0.288506\pi\)
\(954\) 0 0
\(955\) −33.4630 −1.08284
\(956\) 0 0
\(957\) 13.8398 0.447378
\(958\) 0 0
\(959\) −20.4212 −0.659435
\(960\) 0 0
\(961\) 36.2175 1.16831
\(962\) 0 0
\(963\) −1.39697 −0.0450168
\(964\) 0 0
\(965\) 43.3344 1.39499
\(966\) 0 0
\(967\) −29.0135 −0.933010 −0.466505 0.884519i \(-0.654487\pi\)
−0.466505 + 0.884519i \(0.654487\pi\)
\(968\) 0 0
\(969\) 73.1194 2.34893
\(970\) 0 0
\(971\) −8.12130 −0.260625 −0.130312 0.991473i \(-0.541598\pi\)
−0.130312 + 0.991473i \(0.541598\pi\)
\(972\) 0 0
\(973\) 47.9203 1.53625
\(974\) 0 0
\(975\) 23.3095 0.746501
\(976\) 0 0
\(977\) −19.9592 −0.638551 −0.319275 0.947662i \(-0.603440\pi\)
−0.319275 + 0.947662i \(0.603440\pi\)
\(978\) 0 0
\(979\) −21.3895 −0.683612
\(980\) 0 0
\(981\) −8.90834 −0.284421
\(982\) 0 0
\(983\) −51.7024 −1.64905 −0.824525 0.565826i \(-0.808557\pi\)
−0.824525 + 0.565826i \(0.808557\pi\)
\(984\) 0 0
\(985\) 13.0319 0.415230
\(986\) 0 0
\(987\) 55.6760 1.77219
\(988\) 0 0
\(989\) 7.63923 0.242913
\(990\) 0 0
\(991\) 0.944892 0.0300155 0.0150077 0.999887i \(-0.495223\pi\)
0.0150077 + 0.999887i \(0.495223\pi\)
\(992\) 0 0
\(993\) −11.8361 −0.375607
\(994\) 0 0
\(995\) −93.2746 −2.95700
\(996\) 0 0
\(997\) 1.18542 0.0375428 0.0187714 0.999824i \(-0.494025\pi\)
0.0187714 + 0.999824i \(0.494025\pi\)
\(998\) 0 0
\(999\) 25.5710 0.809030
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 1336.2.a.e.1.9 12
4.3 odd 2 2672.2.a.n.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1336.2.a.e.1.9 12 1.1 even 1 trivial
2672.2.a.n.1.4 12 4.3 odd 2