Properties

Label 4012.2.a.i.1.8
Level $4012$
Weight $2$
Character 4012.1
Self dual yes
Analytic conductor $32.036$
Analytic rank $0$
Dimension $18$
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] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} - 17374 x^{10} - 893 x^{9} + 38112 x^{8} - 18700 x^{7} - 32137 x^{6} + 26381 x^{5} + \cdots - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.0880346\) of defining polynomial
Character \(\chi\) \(=\) 4012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.0880346 q^{3} -2.20461 q^{5} +0.606938 q^{7} -2.99225 q^{9} +O(q^{10})\) \(q+0.0880346 q^{3} -2.20461 q^{5} +0.606938 q^{7} -2.99225 q^{9} -5.81846 q^{11} -0.556004 q^{13} -0.194082 q^{15} -1.00000 q^{17} -7.49281 q^{19} +0.0534316 q^{21} +5.33989 q^{23} -0.139679 q^{25} -0.527525 q^{27} -5.38138 q^{29} +3.26105 q^{31} -0.512225 q^{33} -1.33806 q^{35} +1.04730 q^{37} -0.0489476 q^{39} +9.39425 q^{41} +0.399244 q^{43} +6.59675 q^{45} +5.48970 q^{47} -6.63163 q^{49} -0.0880346 q^{51} +5.91311 q^{53} +12.8274 q^{55} -0.659626 q^{57} -1.00000 q^{59} +9.48911 q^{61} -1.81611 q^{63} +1.22577 q^{65} -3.59463 q^{67} +0.470095 q^{69} +9.94804 q^{71} -5.60630 q^{73} -0.0122966 q^{75} -3.53144 q^{77} -2.91694 q^{79} +8.93031 q^{81} -9.42285 q^{83} +2.20461 q^{85} -0.473748 q^{87} -0.776620 q^{89} -0.337460 q^{91} +0.287085 q^{93} +16.5187 q^{95} -11.1488 q^{97} +17.4103 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) \) 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
\(3\) 0.0880346 0.0508268 0.0254134 0.999677i \(-0.491910\pi\)
0.0254134 + 0.999677i \(0.491910\pi\)
\(4\) 0 0
\(5\) −2.20461 −0.985933 −0.492967 0.870048i \(-0.664088\pi\)
−0.492967 + 0.870048i \(0.664088\pi\)
\(6\) 0 0
\(7\) 0.606938 0.229401 0.114701 0.993400i \(-0.463409\pi\)
0.114701 + 0.993400i \(0.463409\pi\)
\(8\) 0 0
\(9\) −2.99225 −0.997417
\(10\) 0 0
\(11\) −5.81846 −1.75433 −0.877165 0.480189i \(-0.840568\pi\)
−0.877165 + 0.480189i \(0.840568\pi\)
\(12\) 0 0
\(13\) −0.556004 −0.154208 −0.0771039 0.997023i \(-0.524567\pi\)
−0.0771039 + 0.997023i \(0.524567\pi\)
\(14\) 0 0
\(15\) −0.194082 −0.0501118
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −7.49281 −1.71897 −0.859484 0.511163i \(-0.829215\pi\)
−0.859484 + 0.511163i \(0.829215\pi\)
\(20\) 0 0
\(21\) 0.0534316 0.0116597
\(22\) 0 0
\(23\) 5.33989 1.11344 0.556722 0.830699i \(-0.312059\pi\)
0.556722 + 0.830699i \(0.312059\pi\)
\(24\) 0 0
\(25\) −0.139679 −0.0279358
\(26\) 0 0
\(27\) −0.527525 −0.101522
\(28\) 0 0
\(29\) −5.38138 −0.999298 −0.499649 0.866228i \(-0.666538\pi\)
−0.499649 + 0.866228i \(0.666538\pi\)
\(30\) 0 0
\(31\) 3.26105 0.585701 0.292851 0.956158i \(-0.405396\pi\)
0.292851 + 0.956158i \(0.405396\pi\)
\(32\) 0 0
\(33\) −0.512225 −0.0891670
\(34\) 0 0
\(35\) −1.33806 −0.226174
\(36\) 0 0
\(37\) 1.04730 0.172174 0.0860872 0.996288i \(-0.472564\pi\)
0.0860872 + 0.996288i \(0.472564\pi\)
\(38\) 0 0
\(39\) −0.0489476 −0.00783789
\(40\) 0 0
\(41\) 9.39425 1.46713 0.733567 0.679617i \(-0.237854\pi\)
0.733567 + 0.679617i \(0.237854\pi\)
\(42\) 0 0
\(43\) 0.399244 0.0608841 0.0304420 0.999537i \(-0.490309\pi\)
0.0304420 + 0.999537i \(0.490309\pi\)
\(44\) 0 0
\(45\) 6.59675 0.983386
\(46\) 0 0
\(47\) 5.48970 0.800755 0.400378 0.916350i \(-0.368879\pi\)
0.400378 + 0.916350i \(0.368879\pi\)
\(48\) 0 0
\(49\) −6.63163 −0.947375
\(50\) 0 0
\(51\) −0.0880346 −0.0123273
\(52\) 0 0
\(53\) 5.91311 0.812228 0.406114 0.913822i \(-0.366883\pi\)
0.406114 + 0.913822i \(0.366883\pi\)
\(54\) 0 0
\(55\) 12.8274 1.72965
\(56\) 0 0
\(57\) −0.659626 −0.0873696
\(58\) 0 0
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) 9.48911 1.21496 0.607478 0.794337i \(-0.292181\pi\)
0.607478 + 0.794337i \(0.292181\pi\)
\(62\) 0 0
\(63\) −1.81611 −0.228808
\(64\) 0 0
\(65\) 1.22577 0.152039
\(66\) 0 0
\(67\) −3.59463 −0.439154 −0.219577 0.975595i \(-0.570468\pi\)
−0.219577 + 0.975595i \(0.570468\pi\)
\(68\) 0 0
\(69\) 0.470095 0.0565928
\(70\) 0 0
\(71\) 9.94804 1.18062 0.590308 0.807178i \(-0.299006\pi\)
0.590308 + 0.807178i \(0.299006\pi\)
\(72\) 0 0
\(73\) −5.60630 −0.656168 −0.328084 0.944649i \(-0.606403\pi\)
−0.328084 + 0.944649i \(0.606403\pi\)
\(74\) 0 0
\(75\) −0.0122966 −0.00141989
\(76\) 0 0
\(77\) −3.53144 −0.402445
\(78\) 0 0
\(79\) −2.91694 −0.328182 −0.164091 0.986445i \(-0.552469\pi\)
−0.164091 + 0.986445i \(0.552469\pi\)
\(80\) 0 0
\(81\) 8.93031 0.992257
\(82\) 0 0
\(83\) −9.42285 −1.03429 −0.517146 0.855897i \(-0.673006\pi\)
−0.517146 + 0.855897i \(0.673006\pi\)
\(84\) 0 0
\(85\) 2.20461 0.239124
\(86\) 0 0
\(87\) −0.473748 −0.0507911
\(88\) 0 0
\(89\) −0.776620 −0.0823216 −0.0411608 0.999153i \(-0.513106\pi\)
−0.0411608 + 0.999153i \(0.513106\pi\)
\(90\) 0 0
\(91\) −0.337460 −0.0353754
\(92\) 0 0
\(93\) 0.287085 0.0297693
\(94\) 0 0
\(95\) 16.5187 1.69479
\(96\) 0 0
\(97\) −11.1488 −1.13199 −0.565996 0.824408i \(-0.691508\pi\)
−0.565996 + 0.824408i \(0.691508\pi\)
\(98\) 0 0
\(99\) 17.4103 1.74980
\(100\) 0 0
\(101\) 7.20456 0.716880 0.358440 0.933553i \(-0.383309\pi\)
0.358440 + 0.933553i \(0.383309\pi\)
\(102\) 0 0
\(103\) 11.8332 1.16595 0.582977 0.812488i \(-0.301888\pi\)
0.582977 + 0.812488i \(0.301888\pi\)
\(104\) 0 0
\(105\) −0.117796 −0.0114957
\(106\) 0 0
\(107\) −2.12416 −0.205350 −0.102675 0.994715i \(-0.532740\pi\)
−0.102675 + 0.994715i \(0.532740\pi\)
\(108\) 0 0
\(109\) 2.03598 0.195011 0.0975057 0.995235i \(-0.468914\pi\)
0.0975057 + 0.995235i \(0.468914\pi\)
\(110\) 0 0
\(111\) 0.0921982 0.00875107
\(112\) 0 0
\(113\) 9.35175 0.879739 0.439869 0.898062i \(-0.355025\pi\)
0.439869 + 0.898062i \(0.355025\pi\)
\(114\) 0 0
\(115\) −11.7724 −1.09778
\(116\) 0 0
\(117\) 1.66370 0.153809
\(118\) 0 0
\(119\) −0.606938 −0.0556379
\(120\) 0 0
\(121\) 22.8544 2.07767
\(122\) 0 0
\(123\) 0.827019 0.0745698
\(124\) 0 0
\(125\) 11.3310 1.01348
\(126\) 0 0
\(127\) 0.582629 0.0517000 0.0258500 0.999666i \(-0.491771\pi\)
0.0258500 + 0.999666i \(0.491771\pi\)
\(128\) 0 0
\(129\) 0.0351472 0.00309454
\(130\) 0 0
\(131\) 3.26541 0.285300 0.142650 0.989773i \(-0.454438\pi\)
0.142650 + 0.989773i \(0.454438\pi\)
\(132\) 0 0
\(133\) −4.54767 −0.394333
\(134\) 0 0
\(135\) 1.16299 0.100094
\(136\) 0 0
\(137\) −8.67211 −0.740908 −0.370454 0.928851i \(-0.620798\pi\)
−0.370454 + 0.928851i \(0.620798\pi\)
\(138\) 0 0
\(139\) −17.8261 −1.51199 −0.755997 0.654575i \(-0.772848\pi\)
−0.755997 + 0.654575i \(0.772848\pi\)
\(140\) 0 0
\(141\) 0.483284 0.0406998
\(142\) 0 0
\(143\) 3.23509 0.270531
\(144\) 0 0
\(145\) 11.8639 0.985241
\(146\) 0 0
\(147\) −0.583812 −0.0481520
\(148\) 0 0
\(149\) 2.54834 0.208768 0.104384 0.994537i \(-0.466713\pi\)
0.104384 + 0.994537i \(0.466713\pi\)
\(150\) 0 0
\(151\) −9.81464 −0.798704 −0.399352 0.916798i \(-0.630765\pi\)
−0.399352 + 0.916798i \(0.630765\pi\)
\(152\) 0 0
\(153\) 2.99225 0.241909
\(154\) 0 0
\(155\) −7.18935 −0.577462
\(156\) 0 0
\(157\) −6.46261 −0.515772 −0.257886 0.966175i \(-0.583026\pi\)
−0.257886 + 0.966175i \(0.583026\pi\)
\(158\) 0 0
\(159\) 0.520558 0.0412830
\(160\) 0 0
\(161\) 3.24099 0.255425
\(162\) 0 0
\(163\) 2.26184 0.177161 0.0885803 0.996069i \(-0.471767\pi\)
0.0885803 + 0.996069i \(0.471767\pi\)
\(164\) 0 0
\(165\) 1.12926 0.0879127
\(166\) 0 0
\(167\) 16.1845 1.25240 0.626198 0.779664i \(-0.284610\pi\)
0.626198 + 0.779664i \(0.284610\pi\)
\(168\) 0 0
\(169\) −12.6909 −0.976220
\(170\) 0 0
\(171\) 22.4203 1.71453
\(172\) 0 0
\(173\) −18.9302 −1.43923 −0.719617 0.694371i \(-0.755683\pi\)
−0.719617 + 0.694371i \(0.755683\pi\)
\(174\) 0 0
\(175\) −0.0847767 −0.00640851
\(176\) 0 0
\(177\) −0.0880346 −0.00661708
\(178\) 0 0
\(179\) 14.6054 1.09166 0.545829 0.837897i \(-0.316215\pi\)
0.545829 + 0.837897i \(0.316215\pi\)
\(180\) 0 0
\(181\) −10.8847 −0.809056 −0.404528 0.914526i \(-0.632564\pi\)
−0.404528 + 0.914526i \(0.632564\pi\)
\(182\) 0 0
\(183\) 0.835370 0.0617523
\(184\) 0 0
\(185\) −2.30888 −0.169752
\(186\) 0 0
\(187\) 5.81846 0.425488
\(188\) 0 0
\(189\) −0.320175 −0.0232893
\(190\) 0 0
\(191\) 6.36071 0.460245 0.230123 0.973162i \(-0.426087\pi\)
0.230123 + 0.973162i \(0.426087\pi\)
\(192\) 0 0
\(193\) 17.7850 1.28019 0.640097 0.768294i \(-0.278894\pi\)
0.640097 + 0.768294i \(0.278894\pi\)
\(194\) 0 0
\(195\) 0.107911 0.00772763
\(196\) 0 0
\(197\) −8.27138 −0.589312 −0.294656 0.955603i \(-0.595205\pi\)
−0.294656 + 0.955603i \(0.595205\pi\)
\(198\) 0 0
\(199\) −4.16843 −0.295493 −0.147746 0.989025i \(-0.547202\pi\)
−0.147746 + 0.989025i \(0.547202\pi\)
\(200\) 0 0
\(201\) −0.316451 −0.0223208
\(202\) 0 0
\(203\) −3.26617 −0.229240
\(204\) 0 0
\(205\) −20.7107 −1.44650
\(206\) 0 0
\(207\) −15.9783 −1.11057
\(208\) 0 0
\(209\) 43.5966 3.01564
\(210\) 0 0
\(211\) −6.60899 −0.454982 −0.227491 0.973780i \(-0.573052\pi\)
−0.227491 + 0.973780i \(0.573052\pi\)
\(212\) 0 0
\(213\) 0.875772 0.0600069
\(214\) 0 0
\(215\) −0.880178 −0.0600276
\(216\) 0 0
\(217\) 1.97925 0.134361
\(218\) 0 0
\(219\) −0.493549 −0.0333509
\(220\) 0 0
\(221\) 0.556004 0.0374009
\(222\) 0 0
\(223\) 14.9658 1.00219 0.501093 0.865394i \(-0.332932\pi\)
0.501093 + 0.865394i \(0.332932\pi\)
\(224\) 0 0
\(225\) 0.417955 0.0278637
\(226\) 0 0
\(227\) −15.0729 −1.00042 −0.500212 0.865903i \(-0.666744\pi\)
−0.500212 + 0.865903i \(0.666744\pi\)
\(228\) 0 0
\(229\) 12.3127 0.813645 0.406823 0.913507i \(-0.366637\pi\)
0.406823 + 0.913507i \(0.366637\pi\)
\(230\) 0 0
\(231\) −0.310889 −0.0204550
\(232\) 0 0
\(233\) −9.63811 −0.631413 −0.315707 0.948857i \(-0.602242\pi\)
−0.315707 + 0.948857i \(0.602242\pi\)
\(234\) 0 0
\(235\) −12.1027 −0.789491
\(236\) 0 0
\(237\) −0.256792 −0.0166804
\(238\) 0 0
\(239\) 20.4945 1.32568 0.662839 0.748762i \(-0.269351\pi\)
0.662839 + 0.748762i \(0.269351\pi\)
\(240\) 0 0
\(241\) 11.7543 0.757160 0.378580 0.925569i \(-0.376412\pi\)
0.378580 + 0.925569i \(0.376412\pi\)
\(242\) 0 0
\(243\) 2.36875 0.151955
\(244\) 0 0
\(245\) 14.6202 0.934049
\(246\) 0 0
\(247\) 4.16603 0.265078
\(248\) 0 0
\(249\) −0.829537 −0.0525698
\(250\) 0 0
\(251\) 6.08421 0.384032 0.192016 0.981392i \(-0.438497\pi\)
0.192016 + 0.981392i \(0.438497\pi\)
\(252\) 0 0
\(253\) −31.0699 −1.95335
\(254\) 0 0
\(255\) 0.194082 0.0121539
\(256\) 0 0
\(257\) 8.27216 0.516003 0.258001 0.966145i \(-0.416936\pi\)
0.258001 + 0.966145i \(0.416936\pi\)
\(258\) 0 0
\(259\) 0.635644 0.0394970
\(260\) 0 0
\(261\) 16.1024 0.996716
\(262\) 0 0
\(263\) −21.3149 −1.31433 −0.657167 0.753745i \(-0.728245\pi\)
−0.657167 + 0.753745i \(0.728245\pi\)
\(264\) 0 0
\(265\) −13.0361 −0.800803
\(266\) 0 0
\(267\) −0.0683695 −0.00418414
\(268\) 0 0
\(269\) 17.9867 1.09667 0.548333 0.836260i \(-0.315263\pi\)
0.548333 + 0.836260i \(0.315263\pi\)
\(270\) 0 0
\(271\) 8.56872 0.520513 0.260256 0.965540i \(-0.416193\pi\)
0.260256 + 0.965540i \(0.416193\pi\)
\(272\) 0 0
\(273\) −0.0297082 −0.00179802
\(274\) 0 0
\(275\) 0.812717 0.0490087
\(276\) 0 0
\(277\) 3.39045 0.203713 0.101856 0.994799i \(-0.467522\pi\)
0.101856 + 0.994799i \(0.467522\pi\)
\(278\) 0 0
\(279\) −9.75787 −0.584188
\(280\) 0 0
\(281\) 13.6604 0.814913 0.407456 0.913225i \(-0.366416\pi\)
0.407456 + 0.913225i \(0.366416\pi\)
\(282\) 0 0
\(283\) −12.6437 −0.751589 −0.375794 0.926703i \(-0.622630\pi\)
−0.375794 + 0.926703i \(0.622630\pi\)
\(284\) 0 0
\(285\) 1.45422 0.0861406
\(286\) 0 0
\(287\) 5.70173 0.336562
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −0.981483 −0.0575356
\(292\) 0 0
\(293\) 31.1243 1.81830 0.909152 0.416465i \(-0.136731\pi\)
0.909152 + 0.416465i \(0.136731\pi\)
\(294\) 0 0
\(295\) 2.20461 0.128358
\(296\) 0 0
\(297\) 3.06938 0.178104
\(298\) 0 0
\(299\) −2.96900 −0.171702
\(300\) 0 0
\(301\) 0.242316 0.0139669
\(302\) 0 0
\(303\) 0.634250 0.0364367
\(304\) 0 0
\(305\) −20.9198 −1.19787
\(306\) 0 0
\(307\) −7.40991 −0.422906 −0.211453 0.977388i \(-0.567820\pi\)
−0.211453 + 0.977388i \(0.567820\pi\)
\(308\) 0 0
\(309\) 1.04173 0.0592617
\(310\) 0 0
\(311\) −14.9674 −0.848722 −0.424361 0.905493i \(-0.639501\pi\)
−0.424361 + 0.905493i \(0.639501\pi\)
\(312\) 0 0
\(313\) 3.82878 0.216415 0.108208 0.994128i \(-0.465489\pi\)
0.108208 + 0.994128i \(0.465489\pi\)
\(314\) 0 0
\(315\) 4.00382 0.225590
\(316\) 0 0
\(317\) 14.7445 0.828132 0.414066 0.910247i \(-0.364108\pi\)
0.414066 + 0.910247i \(0.364108\pi\)
\(318\) 0 0
\(319\) 31.3113 1.75310
\(320\) 0 0
\(321\) −0.187000 −0.0104373
\(322\) 0 0
\(323\) 7.49281 0.416911
\(324\) 0 0
\(325\) 0.0776622 0.00430793
\(326\) 0 0
\(327\) 0.179237 0.00991181
\(328\) 0 0
\(329\) 3.33191 0.183694
\(330\) 0 0
\(331\) 17.1023 0.940026 0.470013 0.882659i \(-0.344249\pi\)
0.470013 + 0.882659i \(0.344249\pi\)
\(332\) 0 0
\(333\) −3.13377 −0.171730
\(334\) 0 0
\(335\) 7.92476 0.432976
\(336\) 0 0
\(337\) −26.0556 −1.41934 −0.709669 0.704535i \(-0.751156\pi\)
−0.709669 + 0.704535i \(0.751156\pi\)
\(338\) 0 0
\(339\) 0.823277 0.0447143
\(340\) 0 0
\(341\) −18.9743 −1.02751
\(342\) 0 0
\(343\) −8.27356 −0.446730
\(344\) 0 0
\(345\) −1.03638 −0.0557967
\(346\) 0 0
\(347\) 22.7388 1.22068 0.610340 0.792140i \(-0.291033\pi\)
0.610340 + 0.792140i \(0.291033\pi\)
\(348\) 0 0
\(349\) −11.5643 −0.619025 −0.309512 0.950895i \(-0.600166\pi\)
−0.309512 + 0.950895i \(0.600166\pi\)
\(350\) 0 0
\(351\) 0.293306 0.0156555
\(352\) 0 0
\(353\) −34.5310 −1.83790 −0.918951 0.394373i \(-0.870962\pi\)
−0.918951 + 0.394373i \(0.870962\pi\)
\(354\) 0 0
\(355\) −21.9316 −1.16401
\(356\) 0 0
\(357\) −0.0534316 −0.00282790
\(358\) 0 0
\(359\) 30.1188 1.58961 0.794805 0.606865i \(-0.207573\pi\)
0.794805 + 0.606865i \(0.207573\pi\)
\(360\) 0 0
\(361\) 37.1421 1.95485
\(362\) 0 0
\(363\) 2.01198 0.105602
\(364\) 0 0
\(365\) 12.3597 0.646938
\(366\) 0 0
\(367\) −16.7791 −0.875863 −0.437932 0.899008i \(-0.644289\pi\)
−0.437932 + 0.899008i \(0.644289\pi\)
\(368\) 0 0
\(369\) −28.1099 −1.46334
\(370\) 0 0
\(371\) 3.58889 0.186326
\(372\) 0 0
\(373\) −29.5242 −1.52871 −0.764353 0.644798i \(-0.776942\pi\)
−0.764353 + 0.644798i \(0.776942\pi\)
\(374\) 0 0
\(375\) 0.997520 0.0515117
\(376\) 0 0
\(377\) 2.99207 0.154100
\(378\) 0 0
\(379\) −9.08720 −0.466778 −0.233389 0.972383i \(-0.574982\pi\)
−0.233389 + 0.972383i \(0.574982\pi\)
\(380\) 0 0
\(381\) 0.0512915 0.00262774
\(382\) 0 0
\(383\) 22.9004 1.17015 0.585077 0.810978i \(-0.301064\pi\)
0.585077 + 0.810978i \(0.301064\pi\)
\(384\) 0 0
\(385\) 7.78547 0.396784
\(386\) 0 0
\(387\) −1.19464 −0.0607268
\(388\) 0 0
\(389\) −3.57905 −0.181465 −0.0907325 0.995875i \(-0.528921\pi\)
−0.0907325 + 0.995875i \(0.528921\pi\)
\(390\) 0 0
\(391\) −5.33989 −0.270050
\(392\) 0 0
\(393\) 0.287469 0.0145009
\(394\) 0 0
\(395\) 6.43073 0.323565
\(396\) 0 0
\(397\) −13.3379 −0.669411 −0.334705 0.942323i \(-0.608637\pi\)
−0.334705 + 0.942323i \(0.608637\pi\)
\(398\) 0 0
\(399\) −0.400352 −0.0200427
\(400\) 0 0
\(401\) 2.35489 0.117598 0.0587988 0.998270i \(-0.481273\pi\)
0.0587988 + 0.998270i \(0.481273\pi\)
\(402\) 0 0
\(403\) −1.81316 −0.0903197
\(404\) 0 0
\(405\) −19.6879 −0.978299
\(406\) 0 0
\(407\) −6.09364 −0.302051
\(408\) 0 0
\(409\) −25.9520 −1.28324 −0.641622 0.767021i \(-0.721738\pi\)
−0.641622 + 0.767021i \(0.721738\pi\)
\(410\) 0 0
\(411\) −0.763446 −0.0376580
\(412\) 0 0
\(413\) −0.606938 −0.0298655
\(414\) 0 0
\(415\) 20.7738 1.01974
\(416\) 0 0
\(417\) −1.56932 −0.0768498
\(418\) 0 0
\(419\) 16.7079 0.816236 0.408118 0.912929i \(-0.366185\pi\)
0.408118 + 0.912929i \(0.366185\pi\)
\(420\) 0 0
\(421\) 30.7099 1.49671 0.748354 0.663300i \(-0.230845\pi\)
0.748354 + 0.663300i \(0.230845\pi\)
\(422\) 0 0
\(423\) −16.4266 −0.798687
\(424\) 0 0
\(425\) 0.139679 0.00677544
\(426\) 0 0
\(427\) 5.75930 0.278712
\(428\) 0 0
\(429\) 0.284799 0.0137502
\(430\) 0 0
\(431\) 14.0193 0.675288 0.337644 0.941274i \(-0.390370\pi\)
0.337644 + 0.941274i \(0.390370\pi\)
\(432\) 0 0
\(433\) 19.6414 0.943905 0.471953 0.881624i \(-0.343549\pi\)
0.471953 + 0.881624i \(0.343549\pi\)
\(434\) 0 0
\(435\) 1.04443 0.0500766
\(436\) 0 0
\(437\) −40.0108 −1.91398
\(438\) 0 0
\(439\) 16.2666 0.776361 0.388181 0.921583i \(-0.373104\pi\)
0.388181 + 0.921583i \(0.373104\pi\)
\(440\) 0 0
\(441\) 19.8435 0.944928
\(442\) 0 0
\(443\) 11.5740 0.549898 0.274949 0.961459i \(-0.411339\pi\)
0.274949 + 0.961459i \(0.411339\pi\)
\(444\) 0 0
\(445\) 1.71215 0.0811636
\(446\) 0 0
\(447\) 0.224342 0.0106110
\(448\) 0 0
\(449\) −13.6693 −0.645096 −0.322548 0.946553i \(-0.604539\pi\)
−0.322548 + 0.946553i \(0.604539\pi\)
\(450\) 0 0
\(451\) −54.6600 −2.57384
\(452\) 0 0
\(453\) −0.864028 −0.0405956
\(454\) 0 0
\(455\) 0.743969 0.0348778
\(456\) 0 0
\(457\) 21.1850 0.990993 0.495496 0.868610i \(-0.334986\pi\)
0.495496 + 0.868610i \(0.334986\pi\)
\(458\) 0 0
\(459\) 0.527525 0.0246228
\(460\) 0 0
\(461\) 15.9707 0.743832 0.371916 0.928266i \(-0.378701\pi\)
0.371916 + 0.928266i \(0.378701\pi\)
\(462\) 0 0
\(463\) −21.2268 −0.986493 −0.493247 0.869889i \(-0.664190\pi\)
−0.493247 + 0.869889i \(0.664190\pi\)
\(464\) 0 0
\(465\) −0.632911 −0.0293505
\(466\) 0 0
\(467\) 28.2175 1.30575 0.652875 0.757466i \(-0.273563\pi\)
0.652875 + 0.757466i \(0.273563\pi\)
\(468\) 0 0
\(469\) −2.18172 −0.100742
\(470\) 0 0
\(471\) −0.568933 −0.0262151
\(472\) 0 0
\(473\) −2.32298 −0.106811
\(474\) 0 0
\(475\) 1.04659 0.0480208
\(476\) 0 0
\(477\) −17.6935 −0.810130
\(478\) 0 0
\(479\) −3.77349 −0.172415 −0.0862076 0.996277i \(-0.527475\pi\)
−0.0862076 + 0.996277i \(0.527475\pi\)
\(480\) 0 0
\(481\) −0.582301 −0.0265506
\(482\) 0 0
\(483\) 0.285319 0.0129825
\(484\) 0 0
\(485\) 24.5789 1.11607
\(486\) 0 0
\(487\) 28.5598 1.29417 0.647085 0.762418i \(-0.275988\pi\)
0.647085 + 0.762418i \(0.275988\pi\)
\(488\) 0 0
\(489\) 0.199120 0.00900451
\(490\) 0 0
\(491\) −15.7356 −0.710139 −0.355069 0.934840i \(-0.615543\pi\)
−0.355069 + 0.934840i \(0.615543\pi\)
\(492\) 0 0
\(493\) 5.38138 0.242365
\(494\) 0 0
\(495\) −38.3829 −1.72518
\(496\) 0 0
\(497\) 6.03785 0.270835
\(498\) 0 0
\(499\) 28.8155 1.28996 0.644979 0.764200i \(-0.276866\pi\)
0.644979 + 0.764200i \(0.276866\pi\)
\(500\) 0 0
\(501\) 1.42480 0.0636553
\(502\) 0 0
\(503\) −10.7016 −0.477162 −0.238581 0.971123i \(-0.576682\pi\)
−0.238581 + 0.971123i \(0.576682\pi\)
\(504\) 0 0
\(505\) −15.8833 −0.706796
\(506\) 0 0
\(507\) −1.11723 −0.0496181
\(508\) 0 0
\(509\) 0.197801 0.00876736 0.00438368 0.999990i \(-0.498605\pi\)
0.00438368 + 0.999990i \(0.498605\pi\)
\(510\) 0 0
\(511\) −3.40268 −0.150526
\(512\) 0 0
\(513\) 3.95264 0.174513
\(514\) 0 0
\(515\) −26.0875 −1.14955
\(516\) 0 0
\(517\) −31.9416 −1.40479
\(518\) 0 0
\(519\) −1.66651 −0.0731516
\(520\) 0 0
\(521\) −34.7114 −1.52073 −0.760366 0.649495i \(-0.774981\pi\)
−0.760366 + 0.649495i \(0.774981\pi\)
\(522\) 0 0
\(523\) 0.587145 0.0256741 0.0128370 0.999918i \(-0.495914\pi\)
0.0128370 + 0.999918i \(0.495914\pi\)
\(524\) 0 0
\(525\) −0.00746328 −0.000325724 0
\(526\) 0 0
\(527\) −3.26105 −0.142053
\(528\) 0 0
\(529\) 5.51446 0.239759
\(530\) 0 0
\(531\) 2.99225 0.129853
\(532\) 0 0
\(533\) −5.22324 −0.226244
\(534\) 0 0
\(535\) 4.68296 0.202462
\(536\) 0 0
\(537\) 1.28578 0.0554855
\(538\) 0 0
\(539\) 38.5858 1.66201
\(540\) 0 0
\(541\) −38.2713 −1.64541 −0.822705 0.568468i \(-0.807536\pi\)
−0.822705 + 0.568468i \(0.807536\pi\)
\(542\) 0 0
\(543\) −0.958233 −0.0411217
\(544\) 0 0
\(545\) −4.48855 −0.192268
\(546\) 0 0
\(547\) 27.6049 1.18030 0.590151 0.807293i \(-0.299068\pi\)
0.590151 + 0.807293i \(0.299068\pi\)
\(548\) 0 0
\(549\) −28.3938 −1.21182
\(550\) 0 0
\(551\) 40.3217 1.71776
\(552\) 0 0
\(553\) −1.77040 −0.0752852
\(554\) 0 0
\(555\) −0.203261 −0.00862797
\(556\) 0 0
\(557\) −7.16580 −0.303625 −0.151812 0.988409i \(-0.548511\pi\)
−0.151812 + 0.988409i \(0.548511\pi\)
\(558\) 0 0
\(559\) −0.221981 −0.00938880
\(560\) 0 0
\(561\) 0.512225 0.0216262
\(562\) 0 0
\(563\) 43.9228 1.85113 0.925563 0.378594i \(-0.123592\pi\)
0.925563 + 0.378594i \(0.123592\pi\)
\(564\) 0 0
\(565\) −20.6170 −0.867363
\(566\) 0 0
\(567\) 5.42015 0.227625
\(568\) 0 0
\(569\) −26.8409 −1.12523 −0.562615 0.826719i \(-0.690205\pi\)
−0.562615 + 0.826719i \(0.690205\pi\)
\(570\) 0 0
\(571\) −21.7315 −0.909433 −0.454717 0.890636i \(-0.650259\pi\)
−0.454717 + 0.890636i \(0.650259\pi\)
\(572\) 0 0
\(573\) 0.559963 0.0233928
\(574\) 0 0
\(575\) −0.745872 −0.0311050
\(576\) 0 0
\(577\) 18.3570 0.764211 0.382106 0.924119i \(-0.375199\pi\)
0.382106 + 0.924119i \(0.375199\pi\)
\(578\) 0 0
\(579\) 1.56570 0.0650681
\(580\) 0 0
\(581\) −5.71909 −0.237268
\(582\) 0 0
\(583\) −34.4052 −1.42492
\(584\) 0 0
\(585\) −3.66782 −0.151646
\(586\) 0 0
\(587\) 18.5526 0.765747 0.382873 0.923801i \(-0.374935\pi\)
0.382873 + 0.923801i \(0.374935\pi\)
\(588\) 0 0
\(589\) −24.4344 −1.00680
\(590\) 0 0
\(591\) −0.728168 −0.0299528
\(592\) 0 0
\(593\) −43.4689 −1.78505 −0.892527 0.450993i \(-0.851070\pi\)
−0.892527 + 0.450993i \(0.851070\pi\)
\(594\) 0 0
\(595\) 1.33806 0.0548553
\(596\) 0 0
\(597\) −0.366966 −0.0150189
\(598\) 0 0
\(599\) 31.2451 1.27664 0.638320 0.769771i \(-0.279630\pi\)
0.638320 + 0.769771i \(0.279630\pi\)
\(600\) 0 0
\(601\) −37.5430 −1.53141 −0.765706 0.643191i \(-0.777610\pi\)
−0.765706 + 0.643191i \(0.777610\pi\)
\(602\) 0 0
\(603\) 10.7560 0.438019
\(604\) 0 0
\(605\) −50.3852 −2.04845
\(606\) 0 0
\(607\) −4.78102 −0.194055 −0.0970277 0.995282i \(-0.530934\pi\)
−0.0970277 + 0.995282i \(0.530934\pi\)
\(608\) 0 0
\(609\) −0.287536 −0.0116515
\(610\) 0 0
\(611\) −3.05230 −0.123483
\(612\) 0 0
\(613\) 21.3232 0.861234 0.430617 0.902535i \(-0.358296\pi\)
0.430617 + 0.902535i \(0.358296\pi\)
\(614\) 0 0
\(615\) −1.82326 −0.0735208
\(616\) 0 0
\(617\) −38.0873 −1.53334 −0.766668 0.642043i \(-0.778087\pi\)
−0.766668 + 0.642043i \(0.778087\pi\)
\(618\) 0 0
\(619\) −2.53037 −0.101704 −0.0508522 0.998706i \(-0.516194\pi\)
−0.0508522 + 0.998706i \(0.516194\pi\)
\(620\) 0 0
\(621\) −2.81693 −0.113039
\(622\) 0 0
\(623\) −0.471361 −0.0188847
\(624\) 0 0
\(625\) −24.2821 −0.971284
\(626\) 0 0
\(627\) 3.83800 0.153275
\(628\) 0 0
\(629\) −1.04730 −0.0417584
\(630\) 0 0
\(631\) −4.80239 −0.191180 −0.0955900 0.995421i \(-0.530474\pi\)
−0.0955900 + 0.995421i \(0.530474\pi\)
\(632\) 0 0
\(633\) −0.581820 −0.0231253
\(634\) 0 0
\(635\) −1.28447 −0.0509727
\(636\) 0 0
\(637\) 3.68721 0.146093
\(638\) 0 0
\(639\) −29.7670 −1.17757
\(640\) 0 0
\(641\) −33.6244 −1.32808 −0.664042 0.747695i \(-0.731160\pi\)
−0.664042 + 0.747695i \(0.731160\pi\)
\(642\) 0 0
\(643\) 21.5015 0.847937 0.423969 0.905677i \(-0.360637\pi\)
0.423969 + 0.905677i \(0.360637\pi\)
\(644\) 0 0
\(645\) −0.0774861 −0.00305101
\(646\) 0 0
\(647\) 17.8583 0.702082 0.351041 0.936360i \(-0.385828\pi\)
0.351041 + 0.936360i \(0.385828\pi\)
\(648\) 0 0
\(649\) 5.81846 0.228394
\(650\) 0 0
\(651\) 0.174243 0.00682911
\(652\) 0 0
\(653\) 2.48258 0.0971510 0.0485755 0.998820i \(-0.484532\pi\)
0.0485755 + 0.998820i \(0.484532\pi\)
\(654\) 0 0
\(655\) −7.19896 −0.281287
\(656\) 0 0
\(657\) 16.7755 0.654473
\(658\) 0 0
\(659\) −14.5036 −0.564979 −0.282489 0.959270i \(-0.591160\pi\)
−0.282489 + 0.959270i \(0.591160\pi\)
\(660\) 0 0
\(661\) −21.8368 −0.849352 −0.424676 0.905345i \(-0.639612\pi\)
−0.424676 + 0.905345i \(0.639612\pi\)
\(662\) 0 0
\(663\) 0.0489476 0.00190097
\(664\) 0 0
\(665\) 10.0259 0.388786
\(666\) 0 0
\(667\) −28.7360 −1.11266
\(668\) 0 0
\(669\) 1.31751 0.0509378
\(670\) 0 0
\(671\) −55.2120 −2.13143
\(672\) 0 0
\(673\) −3.01503 −0.116221 −0.0581104 0.998310i \(-0.518508\pi\)
−0.0581104 + 0.998310i \(0.518508\pi\)
\(674\) 0 0
\(675\) 0.0736843 0.00283611
\(676\) 0 0
\(677\) −37.9748 −1.45949 −0.729744 0.683720i \(-0.760361\pi\)
−0.729744 + 0.683720i \(0.760361\pi\)
\(678\) 0 0
\(679\) −6.76665 −0.259680
\(680\) 0 0
\(681\) −1.32694 −0.0508483
\(682\) 0 0
\(683\) −6.15378 −0.235468 −0.117734 0.993045i \(-0.537563\pi\)
−0.117734 + 0.993045i \(0.537563\pi\)
\(684\) 0 0
\(685\) 19.1187 0.730486
\(686\) 0 0
\(687\) 1.08394 0.0413550
\(688\) 0 0
\(689\) −3.28772 −0.125252
\(690\) 0 0
\(691\) 19.1340 0.727893 0.363946 0.931420i \(-0.381429\pi\)
0.363946 + 0.931420i \(0.381429\pi\)
\(692\) 0 0
\(693\) 10.5670 0.401406
\(694\) 0 0
\(695\) 39.2998 1.49072
\(696\) 0 0
\(697\) −9.39425 −0.355832
\(698\) 0 0
\(699\) −0.848487 −0.0320927
\(700\) 0 0
\(701\) 19.1965 0.725042 0.362521 0.931976i \(-0.381916\pi\)
0.362521 + 0.931976i \(0.381916\pi\)
\(702\) 0 0
\(703\) −7.84718 −0.295962
\(704\) 0 0
\(705\) −1.06545 −0.0401273
\(706\) 0 0
\(707\) 4.37272 0.164453
\(708\) 0 0
\(709\) 44.5762 1.67409 0.837046 0.547132i \(-0.184280\pi\)
0.837046 + 0.547132i \(0.184280\pi\)
\(710\) 0 0
\(711\) 8.72822 0.327334
\(712\) 0 0
\(713\) 17.4136 0.652146
\(714\) 0 0
\(715\) −7.13211 −0.266726
\(716\) 0 0
\(717\) 1.80422 0.0673800
\(718\) 0 0
\(719\) −24.8206 −0.925652 −0.462826 0.886449i \(-0.653164\pi\)
−0.462826 + 0.886449i \(0.653164\pi\)
\(720\) 0 0
\(721\) 7.18199 0.267471
\(722\) 0 0
\(723\) 1.03478 0.0384840
\(724\) 0 0
\(725\) 0.751668 0.0279162
\(726\) 0 0
\(727\) 36.5971 1.35731 0.678655 0.734457i \(-0.262563\pi\)
0.678655 + 0.734457i \(0.262563\pi\)
\(728\) 0 0
\(729\) −26.5824 −0.984533
\(730\) 0 0
\(731\) −0.399244 −0.0147666
\(732\) 0 0
\(733\) −8.47046 −0.312863 −0.156432 0.987689i \(-0.549999\pi\)
−0.156432 + 0.987689i \(0.549999\pi\)
\(734\) 0 0
\(735\) 1.28708 0.0474747
\(736\) 0 0
\(737\) 20.9152 0.770420
\(738\) 0 0
\(739\) 24.3741 0.896617 0.448309 0.893879i \(-0.352027\pi\)
0.448309 + 0.893879i \(0.352027\pi\)
\(740\) 0 0
\(741\) 0.366755 0.0134731
\(742\) 0 0
\(743\) 30.5396 1.12039 0.560194 0.828361i \(-0.310727\pi\)
0.560194 + 0.828361i \(0.310727\pi\)
\(744\) 0 0
\(745\) −5.61811 −0.205832
\(746\) 0 0
\(747\) 28.1955 1.03162
\(748\) 0 0
\(749\) −1.28924 −0.0471076
\(750\) 0 0
\(751\) 15.2444 0.556276 0.278138 0.960541i \(-0.410283\pi\)
0.278138 + 0.960541i \(0.410283\pi\)
\(752\) 0 0
\(753\) 0.535621 0.0195191
\(754\) 0 0
\(755\) 21.6375 0.787469
\(756\) 0 0
\(757\) −32.4647 −1.17995 −0.589974 0.807422i \(-0.700862\pi\)
−0.589974 + 0.807422i \(0.700862\pi\)
\(758\) 0 0
\(759\) −2.73523 −0.0992825
\(760\) 0 0
\(761\) 47.5122 1.72231 0.861157 0.508339i \(-0.169740\pi\)
0.861157 + 0.508339i \(0.169740\pi\)
\(762\) 0 0
\(763\) 1.23571 0.0447359
\(764\) 0 0
\(765\) −6.59675 −0.238506
\(766\) 0 0
\(767\) 0.556004 0.0200761
\(768\) 0 0
\(769\) 3.98091 0.143555 0.0717776 0.997421i \(-0.477133\pi\)
0.0717776 + 0.997421i \(0.477133\pi\)
\(770\) 0 0
\(771\) 0.728236 0.0262268
\(772\) 0 0
\(773\) −12.1856 −0.438286 −0.219143 0.975693i \(-0.570326\pi\)
−0.219143 + 0.975693i \(0.570326\pi\)
\(774\) 0 0
\(775\) −0.455500 −0.0163621
\(776\) 0 0
\(777\) 0.0559586 0.00200750
\(778\) 0 0
\(779\) −70.3893 −2.52196
\(780\) 0 0
\(781\) −57.8823 −2.07119
\(782\) 0 0
\(783\) 2.83882 0.101451
\(784\) 0 0
\(785\) 14.2476 0.508517
\(786\) 0 0
\(787\) 14.3667 0.512118 0.256059 0.966661i \(-0.417576\pi\)
0.256059 + 0.966661i \(0.417576\pi\)
\(788\) 0 0
\(789\) −1.87645 −0.0668033
\(790\) 0 0
\(791\) 5.67594 0.201813
\(792\) 0 0
\(793\) −5.27598 −0.187356
\(794\) 0 0
\(795\) −1.14763 −0.0407022
\(796\) 0 0
\(797\) 41.2362 1.46066 0.730331 0.683093i \(-0.239366\pi\)
0.730331 + 0.683093i \(0.239366\pi\)
\(798\) 0 0
\(799\) −5.48970 −0.194212
\(800\) 0 0
\(801\) 2.32384 0.0821089
\(802\) 0 0
\(803\) 32.6200 1.15114
\(804\) 0 0
\(805\) −7.14512 −0.251832
\(806\) 0 0
\(807\) 1.58345 0.0557400
\(808\) 0 0
\(809\) 3.35950 0.118114 0.0590569 0.998255i \(-0.481191\pi\)
0.0590569 + 0.998255i \(0.481191\pi\)
\(810\) 0 0
\(811\) 35.1318 1.23364 0.616822 0.787103i \(-0.288420\pi\)
0.616822 + 0.787103i \(0.288420\pi\)
\(812\) 0 0
\(813\) 0.754344 0.0264560
\(814\) 0 0
\(815\) −4.98647 −0.174669
\(816\) 0 0
\(817\) −2.99145 −0.104658
\(818\) 0 0
\(819\) 1.00977 0.0352841
\(820\) 0 0
\(821\) −38.9929 −1.36086 −0.680432 0.732812i \(-0.738208\pi\)
−0.680432 + 0.732812i \(0.738208\pi\)
\(822\) 0 0
\(823\) −33.0567 −1.15229 −0.576143 0.817349i \(-0.695443\pi\)
−0.576143 + 0.817349i \(0.695443\pi\)
\(824\) 0 0
\(825\) 0.0715472 0.00249095
\(826\) 0 0
\(827\) −24.7228 −0.859697 −0.429849 0.902901i \(-0.641433\pi\)
−0.429849 + 0.902901i \(0.641433\pi\)
\(828\) 0 0
\(829\) 45.0698 1.56534 0.782669 0.622439i \(-0.213858\pi\)
0.782669 + 0.622439i \(0.213858\pi\)
\(830\) 0 0
\(831\) 0.298477 0.0103541
\(832\) 0 0
\(833\) 6.63163 0.229772
\(834\) 0 0
\(835\) −35.6806 −1.23478
\(836\) 0 0
\(837\) −1.72028 −0.0594617
\(838\) 0 0
\(839\) 2.25823 0.0779628 0.0389814 0.999240i \(-0.487589\pi\)
0.0389814 + 0.999240i \(0.487589\pi\)
\(840\) 0 0
\(841\) −0.0407033 −0.00140356
\(842\) 0 0
\(843\) 1.20259 0.0414194
\(844\) 0 0
\(845\) 27.9784 0.962488
\(846\) 0 0
\(847\) 13.8712 0.476621
\(848\) 0 0
\(849\) −1.11308 −0.0382008
\(850\) 0 0
\(851\) 5.59245 0.191707
\(852\) 0 0
\(853\) 10.1564 0.347748 0.173874 0.984768i \(-0.444371\pi\)
0.173874 + 0.984768i \(0.444371\pi\)
\(854\) 0 0
\(855\) −49.4282 −1.69041
\(856\) 0 0
\(857\) 14.6624 0.500860 0.250430 0.968135i \(-0.419428\pi\)
0.250430 + 0.968135i \(0.419428\pi\)
\(858\) 0 0
\(859\) 51.8702 1.76979 0.884895 0.465791i \(-0.154230\pi\)
0.884895 + 0.465791i \(0.154230\pi\)
\(860\) 0 0
\(861\) 0.501949 0.0171064
\(862\) 0 0
\(863\) 7.25436 0.246941 0.123471 0.992348i \(-0.460597\pi\)
0.123471 + 0.992348i \(0.460597\pi\)
\(864\) 0 0
\(865\) 41.7337 1.41899
\(866\) 0 0
\(867\) 0.0880346 0.00298981
\(868\) 0 0
\(869\) 16.9721 0.575739
\(870\) 0 0
\(871\) 1.99863 0.0677209
\(872\) 0 0
\(873\) 33.3601 1.12907
\(874\) 0 0
\(875\) 6.87722 0.232493
\(876\) 0 0
\(877\) −45.8317 −1.54763 −0.773813 0.633414i \(-0.781653\pi\)
−0.773813 + 0.633414i \(0.781653\pi\)
\(878\) 0 0
\(879\) 2.74002 0.0924185
\(880\) 0 0
\(881\) 13.8391 0.466250 0.233125 0.972447i \(-0.425105\pi\)
0.233125 + 0.972447i \(0.425105\pi\)
\(882\) 0 0
\(883\) 30.4604 1.02507 0.512536 0.858666i \(-0.328706\pi\)
0.512536 + 0.858666i \(0.328706\pi\)
\(884\) 0 0
\(885\) 0.194082 0.00652400
\(886\) 0 0
\(887\) 14.0747 0.472581 0.236291 0.971682i \(-0.424068\pi\)
0.236291 + 0.971682i \(0.424068\pi\)
\(888\) 0 0
\(889\) 0.353620 0.0118600
\(890\) 0 0
\(891\) −51.9606 −1.74075
\(892\) 0 0
\(893\) −41.1333 −1.37647
\(894\) 0 0
\(895\) −32.1992 −1.07630
\(896\) 0 0
\(897\) −0.261375 −0.00872705
\(898\) 0 0
\(899\) −17.5489 −0.585290
\(900\) 0 0
\(901\) −5.91311 −0.196994
\(902\) 0 0
\(903\) 0.0213322 0.000709891 0
\(904\) 0 0
\(905\) 23.9966 0.797675
\(906\) 0 0
\(907\) −30.9486 −1.02763 −0.513816 0.857901i \(-0.671769\pi\)
−0.513816 + 0.857901i \(0.671769\pi\)
\(908\) 0 0
\(909\) −21.5578 −0.715028
\(910\) 0 0
\(911\) 22.9041 0.758845 0.379423 0.925223i \(-0.376123\pi\)
0.379423 + 0.925223i \(0.376123\pi\)
\(912\) 0 0
\(913\) 54.8265 1.81449
\(914\) 0 0
\(915\) −1.84167 −0.0608836
\(916\) 0 0
\(917\) 1.98190 0.0654481
\(918\) 0 0
\(919\) −0.0673641 −0.00222214 −0.00111107 0.999999i \(-0.500354\pi\)
−0.00111107 + 0.999999i \(0.500354\pi\)
\(920\) 0 0
\(921\) −0.652329 −0.0214950
\(922\) 0 0
\(923\) −5.53115 −0.182060
\(924\) 0 0
\(925\) −0.146285 −0.00480984
\(926\) 0 0
\(927\) −35.4077 −1.16294
\(928\) 0 0
\(929\) 51.0077 1.67351 0.836754 0.547579i \(-0.184451\pi\)
0.836754 + 0.547579i \(0.184451\pi\)
\(930\) 0 0
\(931\) 49.6895 1.62851
\(932\) 0 0
\(933\) −1.31765 −0.0431378
\(934\) 0 0
\(935\) −12.8274 −0.419502
\(936\) 0 0
\(937\) −11.2385 −0.367146 −0.183573 0.983006i \(-0.558766\pi\)
−0.183573 + 0.983006i \(0.558766\pi\)
\(938\) 0 0
\(939\) 0.337065 0.0109997
\(940\) 0 0
\(941\) −4.51945 −0.147330 −0.0736649 0.997283i \(-0.523470\pi\)
−0.0736649 + 0.997283i \(0.523470\pi\)
\(942\) 0 0
\(943\) 50.1643 1.63357
\(944\) 0 0
\(945\) 0.705863 0.0229617
\(946\) 0 0
\(947\) 24.1090 0.783438 0.391719 0.920085i \(-0.371881\pi\)
0.391719 + 0.920085i \(0.371881\pi\)
\(948\) 0 0
\(949\) 3.11713 0.101186
\(950\) 0 0
\(951\) 1.29802 0.0420913
\(952\) 0 0
\(953\) −26.9162 −0.871902 −0.435951 0.899970i \(-0.643588\pi\)
−0.435951 + 0.899970i \(0.643588\pi\)
\(954\) 0 0
\(955\) −14.0229 −0.453771
\(956\) 0 0
\(957\) 2.75648 0.0891044
\(958\) 0 0
\(959\) −5.26344 −0.169965
\(960\) 0 0
\(961\) −20.3656 −0.656954
\(962\) 0 0
\(963\) 6.35602 0.204820
\(964\) 0 0
\(965\) −39.2091 −1.26218
\(966\) 0 0
\(967\) −56.9043 −1.82992 −0.914959 0.403547i \(-0.867777\pi\)
−0.914959 + 0.403547i \(0.867777\pi\)
\(968\) 0 0
\(969\) 0.659626 0.0211902
\(970\) 0 0
\(971\) 0.957032 0.0307126 0.0153563 0.999882i \(-0.495112\pi\)
0.0153563 + 0.999882i \(0.495112\pi\)
\(972\) 0 0
\(973\) −10.8194 −0.346853
\(974\) 0 0
\(975\) 0.00683696 0.000218958 0
\(976\) 0 0
\(977\) −17.0859 −0.546628 −0.273314 0.961925i \(-0.588120\pi\)
−0.273314 + 0.961925i \(0.588120\pi\)
\(978\) 0 0
\(979\) 4.51873 0.144419
\(980\) 0 0
\(981\) −6.09216 −0.194508
\(982\) 0 0
\(983\) −49.6978 −1.58511 −0.792557 0.609798i \(-0.791251\pi\)
−0.792557 + 0.609798i \(0.791251\pi\)
\(984\) 0 0
\(985\) 18.2352 0.581022
\(986\) 0 0
\(987\) 0.293323 0.00933658
\(988\) 0 0
\(989\) 2.13192 0.0677911
\(990\) 0 0
\(991\) 57.7871 1.83567 0.917834 0.396964i \(-0.129936\pi\)
0.917834 + 0.396964i \(0.129936\pi\)
\(992\) 0 0
\(993\) 1.50559 0.0477785
\(994\) 0 0
\(995\) 9.18979 0.291336
\(996\) 0 0
\(997\) −33.8749 −1.07283 −0.536415 0.843954i \(-0.680222\pi\)
−0.536415 + 0.843954i \(0.680222\pi\)
\(998\) 0 0
\(999\) −0.552475 −0.0174795
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 4012.2.a.i.1.8 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.a.i.1.8 18 1.1 even 1 trivial