Properties

Label 135.2.a.c.1.2
Level $135$
Weight $2$
Character 135.1
Self dual yes
Analytic conductor $1.078$
Analytic rank $0$
Dimension $2$
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] = [135,2,Mod(1,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("135.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.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: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 135.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.30278 q^{2} -0.302776 q^{4} +1.00000 q^{5} +4.60555 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.30278 q^{2} -0.302776 q^{4} +1.00000 q^{5} +4.60555 q^{7} -3.00000 q^{8} +1.30278 q^{10} -2.60555 q^{11} -0.605551 q^{13} +6.00000 q^{14} -3.30278 q^{16} -5.60555 q^{17} -3.60555 q^{19} -0.302776 q^{20} -3.39445 q^{22} -3.00000 q^{23} +1.00000 q^{25} -0.788897 q^{26} -1.39445 q^{28} +8.60555 q^{29} +1.60555 q^{31} +1.69722 q^{32} -7.30278 q^{34} +4.60555 q^{35} +2.00000 q^{37} -4.69722 q^{38} -3.00000 q^{40} +2.60555 q^{41} -6.60555 q^{43} +0.788897 q^{44} -3.90833 q^{46} +5.21110 q^{47} +14.2111 q^{49} +1.30278 q^{50} +0.183346 q^{52} -5.60555 q^{53} -2.60555 q^{55} -13.8167 q^{56} +11.2111 q^{58} -8.60555 q^{59} +10.2111 q^{61} +2.09167 q^{62} +8.81665 q^{64} -0.605551 q^{65} -15.2111 q^{67} +1.69722 q^{68} +6.00000 q^{70} +14.6056 q^{71} +5.39445 q^{73} +2.60555 q^{74} +1.09167 q^{76} -12.0000 q^{77} -4.39445 q^{79} -3.30278 q^{80} +3.39445 q^{82} +3.00000 q^{83} -5.60555 q^{85} -8.60555 q^{86} +7.81665 q^{88} +7.81665 q^{89} -2.78890 q^{91} +0.908327 q^{92} +6.78890 q^{94} -3.60555 q^{95} +8.00000 q^{97} +18.5139 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8} - q^{10} + 2 q^{11} + 6 q^{13} + 12 q^{14} - 3 q^{16} - 4 q^{17} + 3 q^{20} - 14 q^{22} - 6 q^{23} + 2 q^{25} - 16 q^{26} - 10 q^{28} + 10 q^{29} - 4 q^{31} + 7 q^{32} - 11 q^{34} + 2 q^{35} + 4 q^{37} - 13 q^{38} - 6 q^{40} - 2 q^{41} - 6 q^{43} + 16 q^{44} + 3 q^{46} - 4 q^{47} + 14 q^{49} - q^{50} + 22 q^{52} - 4 q^{53} + 2 q^{55} - 6 q^{56} + 8 q^{58} - 10 q^{59} + 6 q^{61} + 15 q^{62} - 4 q^{64} + 6 q^{65} - 16 q^{67} + 7 q^{68} + 12 q^{70} + 22 q^{71} + 18 q^{73} - 2 q^{74} + 13 q^{76} - 24 q^{77} - 16 q^{79} - 3 q^{80} + 14 q^{82} + 6 q^{83} - 4 q^{85} - 10 q^{86} - 6 q^{88} - 6 q^{89} - 20 q^{91} - 9 q^{92} + 28 q^{94} + 16 q^{97} + 19 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30278 0.921201 0.460601 0.887607i \(-0.347634\pi\)
0.460601 + 0.887607i \(0.347634\pi\)
\(3\) 0 0
\(4\) −0.302776 −0.151388
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.60555 1.74073 0.870367 0.492403i \(-0.163881\pi\)
0.870367 + 0.492403i \(0.163881\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 1.30278 0.411974
\(11\) −2.60555 −0.785603 −0.392802 0.919623i \(-0.628494\pi\)
−0.392802 + 0.919623i \(0.628494\pi\)
\(12\) 0 0
\(13\) −0.605551 −0.167950 −0.0839749 0.996468i \(-0.526762\pi\)
−0.0839749 + 0.996468i \(0.526762\pi\)
\(14\) 6.00000 1.60357
\(15\) 0 0
\(16\) −3.30278 −0.825694
\(17\) −5.60555 −1.35955 −0.679773 0.733423i \(-0.737922\pi\)
−0.679773 + 0.733423i \(0.737922\pi\)
\(18\) 0 0
\(19\) −3.60555 −0.827170 −0.413585 0.910465i \(-0.635724\pi\)
−0.413585 + 0.910465i \(0.635724\pi\)
\(20\) −0.302776 −0.0677027
\(21\) 0 0
\(22\) −3.39445 −0.723699
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −0.788897 −0.154716
\(27\) 0 0
\(28\) −1.39445 −0.263526
\(29\) 8.60555 1.59801 0.799005 0.601324i \(-0.205360\pi\)
0.799005 + 0.601324i \(0.205360\pi\)
\(30\) 0 0
\(31\) 1.60555 0.288366 0.144183 0.989551i \(-0.453945\pi\)
0.144183 + 0.989551i \(0.453945\pi\)
\(32\) 1.69722 0.300030
\(33\) 0 0
\(34\) −7.30278 −1.25242
\(35\) 4.60555 0.778480
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.69722 −0.761990
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 2.60555 0.406919 0.203459 0.979083i \(-0.434782\pi\)
0.203459 + 0.979083i \(0.434782\pi\)
\(42\) 0 0
\(43\) −6.60555 −1.00734 −0.503669 0.863897i \(-0.668017\pi\)
−0.503669 + 0.863897i \(0.668017\pi\)
\(44\) 0.788897 0.118931
\(45\) 0 0
\(46\) −3.90833 −0.576251
\(47\) 5.21110 0.760117 0.380059 0.924962i \(-0.375904\pi\)
0.380059 + 0.924962i \(0.375904\pi\)
\(48\) 0 0
\(49\) 14.2111 2.03016
\(50\) 1.30278 0.184240
\(51\) 0 0
\(52\) 0.183346 0.0254255
\(53\) −5.60555 −0.769982 −0.384991 0.922920i \(-0.625795\pi\)
−0.384991 + 0.922920i \(0.625795\pi\)
\(54\) 0 0
\(55\) −2.60555 −0.351332
\(56\) −13.8167 −1.84633
\(57\) 0 0
\(58\) 11.2111 1.47209
\(59\) −8.60555 −1.12035 −0.560174 0.828375i \(-0.689266\pi\)
−0.560174 + 0.828375i \(0.689266\pi\)
\(60\) 0 0
\(61\) 10.2111 1.30740 0.653699 0.756755i \(-0.273216\pi\)
0.653699 + 0.756755i \(0.273216\pi\)
\(62\) 2.09167 0.265643
\(63\) 0 0
\(64\) 8.81665 1.10208
\(65\) −0.605551 −0.0751094
\(66\) 0 0
\(67\) −15.2111 −1.85833 −0.929166 0.369663i \(-0.879473\pi\)
−0.929166 + 0.369663i \(0.879473\pi\)
\(68\) 1.69722 0.205819
\(69\) 0 0
\(70\) 6.00000 0.717137
\(71\) 14.6056 1.73336 0.866680 0.498864i \(-0.166249\pi\)
0.866680 + 0.498864i \(0.166249\pi\)
\(72\) 0 0
\(73\) 5.39445 0.631372 0.315686 0.948864i \(-0.397765\pi\)
0.315686 + 0.948864i \(0.397765\pi\)
\(74\) 2.60555 0.302889
\(75\) 0 0
\(76\) 1.09167 0.125223
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −4.39445 −0.494414 −0.247207 0.968963i \(-0.579513\pi\)
−0.247207 + 0.968963i \(0.579513\pi\)
\(80\) −3.30278 −0.369262
\(81\) 0 0
\(82\) 3.39445 0.374854
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) −5.60555 −0.608007
\(86\) −8.60555 −0.927960
\(87\) 0 0
\(88\) 7.81665 0.833258
\(89\) 7.81665 0.828564 0.414282 0.910149i \(-0.364033\pi\)
0.414282 + 0.910149i \(0.364033\pi\)
\(90\) 0 0
\(91\) −2.78890 −0.292356
\(92\) 0.908327 0.0946996
\(93\) 0 0
\(94\) 6.78890 0.700221
\(95\) −3.60555 −0.369922
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 18.5139 1.87018
\(99\) 0 0
\(100\) −0.302776 −0.0302776
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 1.81665 0.178138
\(105\) 0 0
\(106\) −7.30278 −0.709308
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) −3.39445 −0.323648
\(111\) 0 0
\(112\) −15.2111 −1.43731
\(113\) −0.788897 −0.0742132 −0.0371066 0.999311i \(-0.511814\pi\)
−0.0371066 + 0.999311i \(0.511814\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) −2.60555 −0.241919
\(117\) 0 0
\(118\) −11.2111 −1.03207
\(119\) −25.8167 −2.36661
\(120\) 0 0
\(121\) −4.21110 −0.382828
\(122\) 13.3028 1.20438
\(123\) 0 0
\(124\) −0.486122 −0.0436550
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.78890 −0.424946 −0.212473 0.977167i \(-0.568152\pi\)
−0.212473 + 0.977167i \(0.568152\pi\)
\(128\) 8.09167 0.715210
\(129\) 0 0
\(130\) −0.788897 −0.0691909
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −16.6056 −1.43988
\(134\) −19.8167 −1.71190
\(135\) 0 0
\(136\) 16.8167 1.44202
\(137\) −4.81665 −0.411515 −0.205757 0.978603i \(-0.565966\pi\)
−0.205757 + 0.978603i \(0.565966\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.39445 −0.117852
\(141\) 0 0
\(142\) 19.0278 1.59677
\(143\) 1.57779 0.131942
\(144\) 0 0
\(145\) 8.60555 0.714652
\(146\) 7.02776 0.581621
\(147\) 0 0
\(148\) −0.605551 −0.0497760
\(149\) −13.0278 −1.06728 −0.533638 0.845713i \(-0.679175\pi\)
−0.533638 + 0.845713i \(0.679175\pi\)
\(150\) 0 0
\(151\) −14.4222 −1.17366 −0.586831 0.809709i \(-0.699625\pi\)
−0.586831 + 0.809709i \(0.699625\pi\)
\(152\) 10.8167 0.877346
\(153\) 0 0
\(154\) −15.6333 −1.25977
\(155\) 1.60555 0.128961
\(156\) 0 0
\(157\) 3.81665 0.304602 0.152301 0.988334i \(-0.451332\pi\)
0.152301 + 0.988334i \(0.451332\pi\)
\(158\) −5.72498 −0.455455
\(159\) 0 0
\(160\) 1.69722 0.134177
\(161\) −13.8167 −1.08890
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −0.788897 −0.0616025
\(165\) 0 0
\(166\) 3.90833 0.303345
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) −12.6333 −0.971793
\(170\) −7.30278 −0.560097
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 10.8167 0.822375 0.411187 0.911551i \(-0.365114\pi\)
0.411187 + 0.911551i \(0.365114\pi\)
\(174\) 0 0
\(175\) 4.60555 0.348147
\(176\) 8.60555 0.648668
\(177\) 0 0
\(178\) 10.1833 0.763274
\(179\) −6.78890 −0.507426 −0.253713 0.967280i \(-0.581652\pi\)
−0.253713 + 0.967280i \(0.581652\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) −3.63331 −0.269319
\(183\) 0 0
\(184\) 9.00000 0.663489
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 14.6056 1.06806
\(188\) −1.57779 −0.115073
\(189\) 0 0
\(190\) −4.69722 −0.340772
\(191\) 16.4222 1.18827 0.594135 0.804366i \(-0.297495\pi\)
0.594135 + 0.804366i \(0.297495\pi\)
\(192\) 0 0
\(193\) 21.8167 1.57040 0.785199 0.619244i \(-0.212561\pi\)
0.785199 + 0.619244i \(0.212561\pi\)
\(194\) 10.4222 0.748271
\(195\) 0 0
\(196\) −4.30278 −0.307341
\(197\) −1.18335 −0.0843099 −0.0421550 0.999111i \(-0.513422\pi\)
−0.0421550 + 0.999111i \(0.513422\pi\)
\(198\) 0 0
\(199\) 13.2111 0.936510 0.468255 0.883593i \(-0.344883\pi\)
0.468255 + 0.883593i \(0.344883\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 15.6333 1.09996
\(203\) 39.6333 2.78171
\(204\) 0 0
\(205\) 2.60555 0.181980
\(206\) −5.21110 −0.363075
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 9.39445 0.649828
\(210\) 0 0
\(211\) 12.8167 0.882335 0.441167 0.897425i \(-0.354564\pi\)
0.441167 + 0.897425i \(0.354564\pi\)
\(212\) 1.69722 0.116566
\(213\) 0 0
\(214\) 0 0
\(215\) −6.60555 −0.450495
\(216\) 0 0
\(217\) 7.39445 0.501968
\(218\) −9.11943 −0.617646
\(219\) 0 0
\(220\) 0.788897 0.0531875
\(221\) 3.39445 0.228335
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 7.81665 0.522272
\(225\) 0 0
\(226\) −1.02776 −0.0683653
\(227\) −26.2111 −1.73969 −0.869846 0.493323i \(-0.835782\pi\)
−0.869846 + 0.493323i \(0.835782\pi\)
\(228\) 0 0
\(229\) −6.21110 −0.410441 −0.205221 0.978716i \(-0.565791\pi\)
−0.205221 + 0.978716i \(0.565791\pi\)
\(230\) −3.90833 −0.257707
\(231\) 0 0
\(232\) −25.8167 −1.69495
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 5.21110 0.339935
\(236\) 2.60555 0.169607
\(237\) 0 0
\(238\) −33.6333 −2.18012
\(239\) −0.788897 −0.0510295 −0.0255148 0.999674i \(-0.508122\pi\)
−0.0255148 + 0.999674i \(0.508122\pi\)
\(240\) 0 0
\(241\) 28.2111 1.81724 0.908618 0.417627i \(-0.137138\pi\)
0.908618 + 0.417627i \(0.137138\pi\)
\(242\) −5.48612 −0.352661
\(243\) 0 0
\(244\) −3.09167 −0.197924
\(245\) 14.2111 0.907914
\(246\) 0 0
\(247\) 2.18335 0.138923
\(248\) −4.81665 −0.305858
\(249\) 0 0
\(250\) 1.30278 0.0823948
\(251\) −15.6333 −0.986766 −0.493383 0.869812i \(-0.664240\pi\)
−0.493383 + 0.869812i \(0.664240\pi\)
\(252\) 0 0
\(253\) 7.81665 0.491429
\(254\) −6.23886 −0.391461
\(255\) 0 0
\(256\) −7.09167 −0.443230
\(257\) 22.8167 1.42326 0.711632 0.702553i \(-0.247956\pi\)
0.711632 + 0.702553i \(0.247956\pi\)
\(258\) 0 0
\(259\) 9.21110 0.572350
\(260\) 0.183346 0.0113706
\(261\) 0 0
\(262\) 7.81665 0.482914
\(263\) −17.2111 −1.06128 −0.530641 0.847597i \(-0.678049\pi\)
−0.530641 + 0.847597i \(0.678049\pi\)
\(264\) 0 0
\(265\) −5.60555 −0.344346
\(266\) −21.6333 −1.32642
\(267\) 0 0
\(268\) 4.60555 0.281329
\(269\) 11.2111 0.683553 0.341776 0.939781i \(-0.388971\pi\)
0.341776 + 0.939781i \(0.388971\pi\)
\(270\) 0 0
\(271\) −19.2389 −1.16868 −0.584339 0.811510i \(-0.698646\pi\)
−0.584339 + 0.811510i \(0.698646\pi\)
\(272\) 18.5139 1.12257
\(273\) 0 0
\(274\) −6.27502 −0.379088
\(275\) −2.60555 −0.157121
\(276\) 0 0
\(277\) −29.0278 −1.74411 −0.872054 0.489409i \(-0.837213\pi\)
−0.872054 + 0.489409i \(0.837213\pi\)
\(278\) −5.21110 −0.312541
\(279\) 0 0
\(280\) −13.8167 −0.825703
\(281\) 1.81665 0.108372 0.0541862 0.998531i \(-0.482744\pi\)
0.0541862 + 0.998531i \(0.482744\pi\)
\(282\) 0 0
\(283\) 10.6056 0.630435 0.315217 0.949020i \(-0.397923\pi\)
0.315217 + 0.949020i \(0.397923\pi\)
\(284\) −4.42221 −0.262410
\(285\) 0 0
\(286\) 2.05551 0.121545
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 14.4222 0.848365
\(290\) 11.2111 0.658339
\(291\) 0 0
\(292\) −1.63331 −0.0955821
\(293\) −28.8167 −1.68349 −0.841743 0.539878i \(-0.818470\pi\)
−0.841743 + 0.539878i \(0.818470\pi\)
\(294\) 0 0
\(295\) −8.60555 −0.501035
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −16.9722 −0.983176
\(299\) 1.81665 0.105060
\(300\) 0 0
\(301\) −30.4222 −1.75351
\(302\) −18.7889 −1.08118
\(303\) 0 0
\(304\) 11.9083 0.682989
\(305\) 10.2111 0.584686
\(306\) 0 0
\(307\) −20.4222 −1.16556 −0.582778 0.812631i \(-0.698034\pi\)
−0.582778 + 0.812631i \(0.698034\pi\)
\(308\) 3.63331 0.207027
\(309\) 0 0
\(310\) 2.09167 0.118799
\(311\) 13.8167 0.783471 0.391735 0.920078i \(-0.371875\pi\)
0.391735 + 0.920078i \(0.371875\pi\)
\(312\) 0 0
\(313\) 23.6333 1.33583 0.667917 0.744236i \(-0.267186\pi\)
0.667917 + 0.744236i \(0.267186\pi\)
\(314\) 4.97224 0.280600
\(315\) 0 0
\(316\) 1.33053 0.0748483
\(317\) −0.394449 −0.0221544 −0.0110772 0.999939i \(-0.503526\pi\)
−0.0110772 + 0.999939i \(0.503526\pi\)
\(318\) 0 0
\(319\) −22.4222 −1.25540
\(320\) 8.81665 0.492866
\(321\) 0 0
\(322\) −18.0000 −1.00310
\(323\) 20.2111 1.12458
\(324\) 0 0
\(325\) −0.605551 −0.0335899
\(326\) 2.60555 0.144308
\(327\) 0 0
\(328\) −7.81665 −0.431603
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 14.7889 0.812871 0.406436 0.913679i \(-0.366772\pi\)
0.406436 + 0.913679i \(0.366772\pi\)
\(332\) −0.908327 −0.0498509
\(333\) 0 0
\(334\) −3.90833 −0.213854
\(335\) −15.2111 −0.831071
\(336\) 0 0
\(337\) −0.605551 −0.0329865 −0.0164932 0.999864i \(-0.505250\pi\)
−0.0164932 + 0.999864i \(0.505250\pi\)
\(338\) −16.4584 −0.895217
\(339\) 0 0
\(340\) 1.69722 0.0920449
\(341\) −4.18335 −0.226541
\(342\) 0 0
\(343\) 33.2111 1.79323
\(344\) 19.8167 1.06844
\(345\) 0 0
\(346\) 14.0917 0.757573
\(347\) 1.57779 0.0847005 0.0423502 0.999103i \(-0.486515\pi\)
0.0423502 + 0.999103i \(0.486515\pi\)
\(348\) 0 0
\(349\) 25.8444 1.38342 0.691710 0.722176i \(-0.256858\pi\)
0.691710 + 0.722176i \(0.256858\pi\)
\(350\) 6.00000 0.320713
\(351\) 0 0
\(352\) −4.42221 −0.235704
\(353\) −21.6333 −1.15142 −0.575712 0.817652i \(-0.695275\pi\)
−0.575712 + 0.817652i \(0.695275\pi\)
\(354\) 0 0
\(355\) 14.6056 0.775182
\(356\) −2.36669 −0.125434
\(357\) 0 0
\(358\) −8.84441 −0.467442
\(359\) 33.6333 1.77510 0.887549 0.460713i \(-0.152406\pi\)
0.887549 + 0.460713i \(0.152406\pi\)
\(360\) 0 0
\(361\) −6.00000 −0.315789
\(362\) −9.11943 −0.479307
\(363\) 0 0
\(364\) 0.844410 0.0442591
\(365\) 5.39445 0.282358
\(366\) 0 0
\(367\) 4.60555 0.240408 0.120204 0.992749i \(-0.461645\pi\)
0.120204 + 0.992749i \(0.461645\pi\)
\(368\) 9.90833 0.516507
\(369\) 0 0
\(370\) 2.60555 0.135456
\(371\) −25.8167 −1.34033
\(372\) 0 0
\(373\) −10.7889 −0.558628 −0.279314 0.960200i \(-0.590107\pi\)
−0.279314 + 0.960200i \(0.590107\pi\)
\(374\) 19.0278 0.983902
\(375\) 0 0
\(376\) −15.6333 −0.806226
\(377\) −5.21110 −0.268385
\(378\) 0 0
\(379\) 14.3944 0.739393 0.369697 0.929153i \(-0.379462\pi\)
0.369697 + 0.929153i \(0.379462\pi\)
\(380\) 1.09167 0.0560016
\(381\) 0 0
\(382\) 21.3944 1.09464
\(383\) −18.6333 −0.952118 −0.476059 0.879413i \(-0.657935\pi\)
−0.476059 + 0.879413i \(0.657935\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 28.4222 1.44665
\(387\) 0 0
\(388\) −2.42221 −0.122969
\(389\) 4.18335 0.212104 0.106052 0.994361i \(-0.466179\pi\)
0.106052 + 0.994361i \(0.466179\pi\)
\(390\) 0 0
\(391\) 16.8167 0.850455
\(392\) −42.6333 −2.15331
\(393\) 0 0
\(394\) −1.54163 −0.0776664
\(395\) −4.39445 −0.221109
\(396\) 0 0
\(397\) −12.6056 −0.632654 −0.316327 0.948650i \(-0.602450\pi\)
−0.316327 + 0.948650i \(0.602450\pi\)
\(398\) 17.2111 0.862715
\(399\) 0 0
\(400\) −3.30278 −0.165139
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −0.972244 −0.0484309
\(404\) −3.63331 −0.180764
\(405\) 0 0
\(406\) 51.6333 2.56252
\(407\) −5.21110 −0.258305
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 3.39445 0.167640
\(411\) 0 0
\(412\) 1.21110 0.0596667
\(413\) −39.6333 −1.95023
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) −1.02776 −0.0503899
\(417\) 0 0
\(418\) 12.2389 0.598622
\(419\) −13.0278 −0.636448 −0.318224 0.948016i \(-0.603086\pi\)
−0.318224 + 0.948016i \(0.603086\pi\)
\(420\) 0 0
\(421\) −23.4222 −1.14153 −0.570764 0.821114i \(-0.693353\pi\)
−0.570764 + 0.821114i \(0.693353\pi\)
\(422\) 16.6972 0.812808
\(423\) 0 0
\(424\) 16.8167 0.816689
\(425\) −5.60555 −0.271909
\(426\) 0 0
\(427\) 47.0278 2.27583
\(428\) 0 0
\(429\) 0 0
\(430\) −8.60555 −0.414997
\(431\) 25.8167 1.24354 0.621772 0.783198i \(-0.286413\pi\)
0.621772 + 0.783198i \(0.286413\pi\)
\(432\) 0 0
\(433\) −28.2389 −1.35707 −0.678536 0.734567i \(-0.737385\pi\)
−0.678536 + 0.734567i \(0.737385\pi\)
\(434\) 9.63331 0.462414
\(435\) 0 0
\(436\) 2.11943 0.101502
\(437\) 10.8167 0.517431
\(438\) 0 0
\(439\) 20.3944 0.973374 0.486687 0.873576i \(-0.338205\pi\)
0.486687 + 0.873576i \(0.338205\pi\)
\(440\) 7.81665 0.372644
\(441\) 0 0
\(442\) 4.42221 0.210343
\(443\) 18.6333 0.885295 0.442648 0.896696i \(-0.354039\pi\)
0.442648 + 0.896696i \(0.354039\pi\)
\(444\) 0 0
\(445\) 7.81665 0.370545
\(446\) −13.0278 −0.616882
\(447\) 0 0
\(448\) 40.6056 1.91843
\(449\) −12.2389 −0.577587 −0.288794 0.957391i \(-0.593254\pi\)
−0.288794 + 0.957391i \(0.593254\pi\)
\(450\) 0 0
\(451\) −6.78890 −0.319677
\(452\) 0.238859 0.0112350
\(453\) 0 0
\(454\) −34.1472 −1.60261
\(455\) −2.78890 −0.130746
\(456\) 0 0
\(457\) 1.21110 0.0566530 0.0283265 0.999599i \(-0.490982\pi\)
0.0283265 + 0.999599i \(0.490982\pi\)
\(458\) −8.09167 −0.378099
\(459\) 0 0
\(460\) 0.908327 0.0423510
\(461\) −21.6333 −1.00756 −0.503782 0.863831i \(-0.668058\pi\)
−0.503782 + 0.863831i \(0.668058\pi\)
\(462\) 0 0
\(463\) −15.2111 −0.706920 −0.353460 0.935450i \(-0.614995\pi\)
−0.353460 + 0.935450i \(0.614995\pi\)
\(464\) −28.4222 −1.31947
\(465\) 0 0
\(466\) −23.4500 −1.08630
\(467\) −2.21110 −0.102318 −0.0511588 0.998691i \(-0.516291\pi\)
−0.0511588 + 0.998691i \(0.516291\pi\)
\(468\) 0 0
\(469\) −70.0555 −3.23486
\(470\) 6.78890 0.313148
\(471\) 0 0
\(472\) 25.8167 1.18831
\(473\) 17.2111 0.791367
\(474\) 0 0
\(475\) −3.60555 −0.165434
\(476\) 7.81665 0.358276
\(477\) 0 0
\(478\) −1.02776 −0.0470085
\(479\) −16.1833 −0.739436 −0.369718 0.929144i \(-0.620546\pi\)
−0.369718 + 0.929144i \(0.620546\pi\)
\(480\) 0 0
\(481\) −1.21110 −0.0552215
\(482\) 36.7527 1.67404
\(483\) 0 0
\(484\) 1.27502 0.0579554
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −8.18335 −0.370823 −0.185411 0.982661i \(-0.559362\pi\)
−0.185411 + 0.982661i \(0.559362\pi\)
\(488\) −30.6333 −1.38670
\(489\) 0 0
\(490\) 18.5139 0.836372
\(491\) 12.7889 0.577155 0.288577 0.957457i \(-0.406818\pi\)
0.288577 + 0.957457i \(0.406818\pi\)
\(492\) 0 0
\(493\) −48.2389 −2.17257
\(494\) 2.84441 0.127976
\(495\) 0 0
\(496\) −5.30278 −0.238102
\(497\) 67.2666 3.01732
\(498\) 0 0
\(499\) −27.6056 −1.23579 −0.617897 0.786259i \(-0.712015\pi\)
−0.617897 + 0.786259i \(0.712015\pi\)
\(500\) −0.302776 −0.0135405
\(501\) 0 0
\(502\) −20.3667 −0.909010
\(503\) 31.4222 1.40105 0.700523 0.713629i \(-0.252950\pi\)
0.700523 + 0.713629i \(0.252950\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 10.1833 0.452705
\(507\) 0 0
\(508\) 1.44996 0.0643316
\(509\) 32.8444 1.45580 0.727901 0.685682i \(-0.240496\pi\)
0.727901 + 0.685682i \(0.240496\pi\)
\(510\) 0 0
\(511\) 24.8444 1.09905
\(512\) −25.4222 −1.12351
\(513\) 0 0
\(514\) 29.7250 1.31111
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −13.5778 −0.597151
\(518\) 12.0000 0.527250
\(519\) 0 0
\(520\) 1.81665 0.0796655
\(521\) −21.3944 −0.937308 −0.468654 0.883382i \(-0.655261\pi\)
−0.468654 + 0.883382i \(0.655261\pi\)
\(522\) 0 0
\(523\) 23.3944 1.02297 0.511484 0.859293i \(-0.329096\pi\)
0.511484 + 0.859293i \(0.329096\pi\)
\(524\) −1.81665 −0.0793609
\(525\) 0 0
\(526\) −22.4222 −0.977655
\(527\) −9.00000 −0.392046
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −7.30278 −0.317212
\(531\) 0 0
\(532\) 5.02776 0.217981
\(533\) −1.57779 −0.0683419
\(534\) 0 0
\(535\) 0 0
\(536\) 45.6333 1.97106
\(537\) 0 0
\(538\) 14.6056 0.629690
\(539\) −37.0278 −1.59490
\(540\) 0 0
\(541\) −11.5778 −0.497768 −0.248884 0.968533i \(-0.580064\pi\)
−0.248884 + 0.968533i \(0.580064\pi\)
\(542\) −25.0639 −1.07659
\(543\) 0 0
\(544\) −9.51388 −0.407904
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) −6.60555 −0.282433 −0.141216 0.989979i \(-0.545101\pi\)
−0.141216 + 0.989979i \(0.545101\pi\)
\(548\) 1.45837 0.0622983
\(549\) 0 0
\(550\) −3.39445 −0.144740
\(551\) −31.0278 −1.32183
\(552\) 0 0
\(553\) −20.2389 −0.860644
\(554\) −37.8167 −1.60668
\(555\) 0 0
\(556\) 1.21110 0.0513622
\(557\) −33.6333 −1.42509 −0.712544 0.701627i \(-0.752457\pi\)
−0.712544 + 0.701627i \(0.752457\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) −15.2111 −0.642786
\(561\) 0 0
\(562\) 2.36669 0.0998329
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −0.788897 −0.0331892
\(566\) 13.8167 0.580757
\(567\) 0 0
\(568\) −43.8167 −1.83851
\(569\) 37.8167 1.58536 0.792678 0.609640i \(-0.208686\pi\)
0.792678 + 0.609640i \(0.208686\pi\)
\(570\) 0 0
\(571\) −36.4500 −1.52538 −0.762692 0.646762i \(-0.776123\pi\)
−0.762692 + 0.646762i \(0.776123\pi\)
\(572\) −0.477718 −0.0199744
\(573\) 0 0
\(574\) 15.6333 0.652522
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) 27.8167 1.15802 0.579011 0.815320i \(-0.303439\pi\)
0.579011 + 0.815320i \(0.303439\pi\)
\(578\) 18.7889 0.781515
\(579\) 0 0
\(580\) −2.60555 −0.108190
\(581\) 13.8167 0.573211
\(582\) 0 0
\(583\) 14.6056 0.604900
\(584\) −16.1833 −0.669672
\(585\) 0 0
\(586\) −37.5416 −1.55083
\(587\) 21.0000 0.866763 0.433381 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 0 0
\(589\) −5.78890 −0.238527
\(590\) −11.2111 −0.461554
\(591\) 0 0
\(592\) −6.60555 −0.271486
\(593\) 21.2389 0.872175 0.436088 0.899904i \(-0.356364\pi\)
0.436088 + 0.899904i \(0.356364\pi\)
\(594\) 0 0
\(595\) −25.8167 −1.05838
\(596\) 3.94449 0.161572
\(597\) 0 0
\(598\) 2.36669 0.0967812
\(599\) −15.3944 −0.629000 −0.314500 0.949257i \(-0.601837\pi\)
−0.314500 + 0.949257i \(0.601837\pi\)
\(600\) 0 0
\(601\) 32.6333 1.33114 0.665570 0.746335i \(-0.268188\pi\)
0.665570 + 0.746335i \(0.268188\pi\)
\(602\) −39.6333 −1.61533
\(603\) 0 0
\(604\) 4.36669 0.177678
\(605\) −4.21110 −0.171206
\(606\) 0 0
\(607\) 17.3944 0.706019 0.353009 0.935620i \(-0.385158\pi\)
0.353009 + 0.935620i \(0.385158\pi\)
\(608\) −6.11943 −0.248176
\(609\) 0 0
\(610\) 13.3028 0.538614
\(611\) −3.15559 −0.127661
\(612\) 0 0
\(613\) 28.8444 1.16501 0.582507 0.812825i \(-0.302072\pi\)
0.582507 + 0.812825i \(0.302072\pi\)
\(614\) −26.6056 −1.07371
\(615\) 0 0
\(616\) 36.0000 1.45048
\(617\) −26.4500 −1.06484 −0.532418 0.846482i \(-0.678716\pi\)
−0.532418 + 0.846482i \(0.678716\pi\)
\(618\) 0 0
\(619\) −7.63331 −0.306809 −0.153404 0.988164i \(-0.549024\pi\)
−0.153404 + 0.988164i \(0.549024\pi\)
\(620\) −0.486122 −0.0195231
\(621\) 0 0
\(622\) 18.0000 0.721734
\(623\) 36.0000 1.44231
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 30.7889 1.23057
\(627\) 0 0
\(628\) −1.15559 −0.0461131
\(629\) −11.2111 −0.447016
\(630\) 0 0
\(631\) 30.0278 1.19539 0.597693 0.801725i \(-0.296084\pi\)
0.597693 + 0.801725i \(0.296084\pi\)
\(632\) 13.1833 0.524405
\(633\) 0 0
\(634\) −0.513878 −0.0204087
\(635\) −4.78890 −0.190042
\(636\) 0 0
\(637\) −8.60555 −0.340964
\(638\) −29.2111 −1.15648
\(639\) 0 0
\(640\) 8.09167 0.319851
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) −23.0278 −0.908126 −0.454063 0.890970i \(-0.650026\pi\)
−0.454063 + 0.890970i \(0.650026\pi\)
\(644\) 4.18335 0.164847
\(645\) 0 0
\(646\) 26.3305 1.03596
\(647\) 38.2111 1.50223 0.751117 0.660169i \(-0.229516\pi\)
0.751117 + 0.660169i \(0.229516\pi\)
\(648\) 0 0
\(649\) 22.4222 0.880149
\(650\) −0.788897 −0.0309431
\(651\) 0 0
\(652\) −0.605551 −0.0237152
\(653\) 27.2389 1.06594 0.532969 0.846134i \(-0.321076\pi\)
0.532969 + 0.846134i \(0.321076\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) −8.60555 −0.335990
\(657\) 0 0
\(658\) 31.2666 1.21890
\(659\) −20.6056 −0.802678 −0.401339 0.915930i \(-0.631455\pi\)
−0.401339 + 0.915930i \(0.631455\pi\)
\(660\) 0 0
\(661\) 22.8444 0.888545 0.444272 0.895892i \(-0.353462\pi\)
0.444272 + 0.895892i \(0.353462\pi\)
\(662\) 19.2666 0.748818
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) −16.6056 −0.643936
\(666\) 0 0
\(667\) −25.8167 −0.999625
\(668\) 0.908327 0.0351442
\(669\) 0 0
\(670\) −19.8167 −0.765584
\(671\) −26.6056 −1.02710
\(672\) 0 0
\(673\) −11.0278 −0.425089 −0.212544 0.977151i \(-0.568175\pi\)
−0.212544 + 0.977151i \(0.568175\pi\)
\(674\) −0.788897 −0.0303872
\(675\) 0 0
\(676\) 3.82506 0.147118
\(677\) −21.6333 −0.831436 −0.415718 0.909494i \(-0.636470\pi\)
−0.415718 + 0.909494i \(0.636470\pi\)
\(678\) 0 0
\(679\) 36.8444 1.41396
\(680\) 16.8167 0.644889
\(681\) 0 0
\(682\) −5.44996 −0.208690
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 0 0
\(685\) −4.81665 −0.184035
\(686\) 43.2666 1.65193
\(687\) 0 0
\(688\) 21.8167 0.831752
\(689\) 3.39445 0.129318
\(690\) 0 0
\(691\) 2.39445 0.0910891 0.0455446 0.998962i \(-0.485498\pi\)
0.0455446 + 0.998962i \(0.485498\pi\)
\(692\) −3.27502 −0.124498
\(693\) 0 0
\(694\) 2.05551 0.0780262
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −14.6056 −0.553225
\(698\) 33.6695 1.27441
\(699\) 0 0
\(700\) −1.39445 −0.0527052
\(701\) 40.4222 1.52673 0.763363 0.645970i \(-0.223547\pi\)
0.763363 + 0.645970i \(0.223547\pi\)
\(702\) 0 0
\(703\) −7.21110 −0.271972
\(704\) −22.9722 −0.865799
\(705\) 0 0
\(706\) −28.1833 −1.06069
\(707\) 55.2666 2.07851
\(708\) 0 0
\(709\) 34.8444 1.30861 0.654305 0.756231i \(-0.272961\pi\)
0.654305 + 0.756231i \(0.272961\pi\)
\(710\) 19.0278 0.714099
\(711\) 0 0
\(712\) −23.4500 −0.878824
\(713\) −4.81665 −0.180385
\(714\) 0 0
\(715\) 1.57779 0.0590062
\(716\) 2.05551 0.0768181
\(717\) 0 0
\(718\) 43.8167 1.63522
\(719\) −49.2666 −1.83733 −0.918667 0.395032i \(-0.870733\pi\)
−0.918667 + 0.395032i \(0.870733\pi\)
\(720\) 0 0
\(721\) −18.4222 −0.686079
\(722\) −7.81665 −0.290906
\(723\) 0 0
\(724\) 2.11943 0.0787680
\(725\) 8.60555 0.319602
\(726\) 0 0
\(727\) −7.63331 −0.283104 −0.141552 0.989931i \(-0.545209\pi\)
−0.141552 + 0.989931i \(0.545209\pi\)
\(728\) 8.36669 0.310090
\(729\) 0 0
\(730\) 7.02776 0.260109
\(731\) 37.0278 1.36952
\(732\) 0 0
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) 6.00000 0.221464
\(735\) 0 0
\(736\) −5.09167 −0.187682
\(737\) 39.6333 1.45991
\(738\) 0 0
\(739\) 30.0278 1.10459 0.552294 0.833649i \(-0.313752\pi\)
0.552294 + 0.833649i \(0.313752\pi\)
\(740\) −0.605551 −0.0222605
\(741\) 0 0
\(742\) −33.6333 −1.23472
\(743\) −34.4222 −1.26283 −0.631414 0.775446i \(-0.717525\pi\)
−0.631414 + 0.775446i \(0.717525\pi\)
\(744\) 0 0
\(745\) −13.0278 −0.477300
\(746\) −14.0555 −0.514609
\(747\) 0 0
\(748\) −4.42221 −0.161692
\(749\) 0 0
\(750\) 0 0
\(751\) 6.02776 0.219956 0.109978 0.993934i \(-0.464922\pi\)
0.109978 + 0.993934i \(0.464922\pi\)
\(752\) −17.2111 −0.627624
\(753\) 0 0
\(754\) −6.78890 −0.247237
\(755\) −14.4222 −0.524878
\(756\) 0 0
\(757\) 31.2111 1.13439 0.567193 0.823585i \(-0.308029\pi\)
0.567193 + 0.823585i \(0.308029\pi\)
\(758\) 18.7527 0.681130
\(759\) 0 0
\(760\) 10.8167 0.392361
\(761\) −53.4500 −1.93756 −0.968780 0.247923i \(-0.920252\pi\)
−0.968780 + 0.247923i \(0.920252\pi\)
\(762\) 0 0
\(763\) −32.2389 −1.16713
\(764\) −4.97224 −0.179889
\(765\) 0 0
\(766\) −24.2750 −0.877092
\(767\) 5.21110 0.188162
\(768\) 0 0
\(769\) 41.0000 1.47850 0.739249 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) −15.6333 −0.563385
\(771\) 0 0
\(772\) −6.60555 −0.237739
\(773\) 16.8167 0.604853 0.302426 0.953173i \(-0.402203\pi\)
0.302426 + 0.953173i \(0.402203\pi\)
\(774\) 0 0
\(775\) 1.60555 0.0576731
\(776\) −24.0000 −0.861550
\(777\) 0 0
\(778\) 5.44996 0.195391
\(779\) −9.39445 −0.336591
\(780\) 0 0
\(781\) −38.0555 −1.36173
\(782\) 21.9083 0.783440
\(783\) 0 0
\(784\) −46.9361 −1.67629
\(785\) 3.81665 0.136222
\(786\) 0 0
\(787\) −2.97224 −0.105949 −0.0529745 0.998596i \(-0.516870\pi\)
−0.0529745 + 0.998596i \(0.516870\pi\)
\(788\) 0.358288 0.0127635
\(789\) 0 0
\(790\) −5.72498 −0.203686
\(791\) −3.63331 −0.129186
\(792\) 0 0
\(793\) −6.18335 −0.219577
\(794\) −16.4222 −0.582802
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −37.6611 −1.33402 −0.667012 0.745047i \(-0.732427\pi\)
−0.667012 + 0.745047i \(0.732427\pi\)
\(798\) 0 0
\(799\) −29.2111 −1.03341
\(800\) 1.69722 0.0600059
\(801\) 0 0
\(802\) −39.0833 −1.38008
\(803\) −14.0555 −0.496008
\(804\) 0 0
\(805\) −13.8167 −0.486973
\(806\) −1.26662 −0.0446146
\(807\) 0 0
\(808\) −36.0000 −1.26648
\(809\) 50.6056 1.77920 0.889598 0.456744i \(-0.150984\pi\)
0.889598 + 0.456744i \(0.150984\pi\)
\(810\) 0 0
\(811\) 42.4222 1.48965 0.744823 0.667263i \(-0.232534\pi\)
0.744823 + 0.667263i \(0.232534\pi\)
\(812\) −12.0000 −0.421117
\(813\) 0 0
\(814\) −6.78890 −0.237951
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) 23.8167 0.833239
\(818\) 6.51388 0.227752
\(819\) 0 0
\(820\) −0.788897 −0.0275495
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 3.81665 0.133040 0.0665201 0.997785i \(-0.478810\pi\)
0.0665201 + 0.997785i \(0.478810\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) −51.6333 −1.79655
\(827\) −33.7889 −1.17496 −0.587478 0.809240i \(-0.699879\pi\)
−0.587478 + 0.809240i \(0.699879\pi\)
\(828\) 0 0
\(829\) −27.2111 −0.945081 −0.472540 0.881309i \(-0.656663\pi\)
−0.472540 + 0.881309i \(0.656663\pi\)
\(830\) 3.90833 0.135660
\(831\) 0 0
\(832\) −5.33894 −0.185094
\(833\) −79.6611 −2.76009
\(834\) 0 0
\(835\) −3.00000 −0.103819
\(836\) −2.84441 −0.0983760
\(837\) 0 0
\(838\) −16.9722 −0.586296
\(839\) 18.7889 0.648665 0.324332 0.945943i \(-0.394860\pi\)
0.324332 + 0.945943i \(0.394860\pi\)
\(840\) 0 0
\(841\) 45.0555 1.55364
\(842\) −30.5139 −1.05158
\(843\) 0 0
\(844\) −3.88057 −0.133575
\(845\) −12.6333 −0.434599
\(846\) 0 0
\(847\) −19.3944 −0.666401
\(848\) 18.5139 0.635769
\(849\) 0 0
\(850\) −7.30278 −0.250483
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) 14.7889 0.506362 0.253181 0.967419i \(-0.418523\pi\)
0.253181 + 0.967419i \(0.418523\pi\)
\(854\) 61.2666 2.09650
\(855\) 0 0
\(856\) 0 0
\(857\) −45.2389 −1.54533 −0.772665 0.634814i \(-0.781077\pi\)
−0.772665 + 0.634814i \(0.781077\pi\)
\(858\) 0 0
\(859\) 6.02776 0.205664 0.102832 0.994699i \(-0.467210\pi\)
0.102832 + 0.994699i \(0.467210\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) 33.6333 1.14556
\(863\) 15.7889 0.537460 0.268730 0.963216i \(-0.413396\pi\)
0.268730 + 0.963216i \(0.413396\pi\)
\(864\) 0 0
\(865\) 10.8167 0.367777
\(866\) −36.7889 −1.25014
\(867\) 0 0
\(868\) −2.23886 −0.0759918
\(869\) 11.4500 0.388413
\(870\) 0 0
\(871\) 9.21110 0.312106
\(872\) 21.0000 0.711150
\(873\) 0 0
\(874\) 14.0917 0.476658
\(875\) 4.60555 0.155696
\(876\) 0 0
\(877\) −56.6611 −1.91331 −0.956654 0.291227i \(-0.905937\pi\)
−0.956654 + 0.291227i \(0.905937\pi\)
\(878\) 26.5694 0.896674
\(879\) 0 0
\(880\) 8.60555 0.290093
\(881\) −32.6056 −1.09851 −0.549254 0.835655i \(-0.685088\pi\)
−0.549254 + 0.835655i \(0.685088\pi\)
\(882\) 0 0
\(883\) −5.81665 −0.195746 −0.0978730 0.995199i \(-0.531204\pi\)
−0.0978730 + 0.995199i \(0.531204\pi\)
\(884\) −1.02776 −0.0345672
\(885\) 0 0
\(886\) 24.2750 0.815535
\(887\) 12.6333 0.424185 0.212092 0.977250i \(-0.431972\pi\)
0.212092 + 0.977250i \(0.431972\pi\)
\(888\) 0 0
\(889\) −22.0555 −0.739718
\(890\) 10.1833 0.341347
\(891\) 0 0
\(892\) 3.02776 0.101377
\(893\) −18.7889 −0.628746
\(894\) 0 0
\(895\) −6.78890 −0.226928
\(896\) 37.2666 1.24499
\(897\) 0 0
\(898\) −15.9445 −0.532074
\(899\) 13.8167 0.460811
\(900\) 0 0
\(901\) 31.4222 1.04683
\(902\) −8.84441 −0.294487
\(903\) 0 0
\(904\) 2.36669 0.0787150
\(905\) −7.00000 −0.232688
\(906\) 0 0
\(907\) 30.4222 1.01015 0.505076 0.863075i \(-0.331464\pi\)
0.505076 + 0.863075i \(0.331464\pi\)
\(908\) 7.93608 0.263368
\(909\) 0 0
\(910\) −3.63331 −0.120443
\(911\) 31.0278 1.02800 0.513998 0.857792i \(-0.328164\pi\)
0.513998 + 0.857792i \(0.328164\pi\)
\(912\) 0 0
\(913\) −7.81665 −0.258693
\(914\) 1.57779 0.0521888
\(915\) 0 0
\(916\) 1.88057 0.0621358
\(917\) 27.6333 0.912532
\(918\) 0 0
\(919\) −2.42221 −0.0799012 −0.0399506 0.999202i \(-0.512720\pi\)
−0.0399506 + 0.999202i \(0.512720\pi\)
\(920\) 9.00000 0.296721
\(921\) 0 0
\(922\) −28.1833 −0.928169
\(923\) −8.84441 −0.291117
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −19.8167 −0.651216
\(927\) 0 0
\(928\) 14.6056 0.479451
\(929\) 26.6056 0.872900 0.436450 0.899729i \(-0.356236\pi\)
0.436450 + 0.899729i \(0.356236\pi\)
\(930\) 0 0
\(931\) −51.2389 −1.67929
\(932\) 5.44996 0.178519
\(933\) 0 0
\(934\) −2.88057 −0.0942551
\(935\) 14.6056 0.477653
\(936\) 0 0
\(937\) 26.7889 0.875155 0.437578 0.899181i \(-0.355837\pi\)
0.437578 + 0.899181i \(0.355837\pi\)
\(938\) −91.2666 −2.97996
\(939\) 0 0
\(940\) −1.57779 −0.0514620
\(941\) 28.4222 0.926537 0.463269 0.886218i \(-0.346676\pi\)
0.463269 + 0.886218i \(0.346676\pi\)
\(942\) 0 0
\(943\) −7.81665 −0.254545
\(944\) 28.4222 0.925064
\(945\) 0 0
\(946\) 22.4222 0.729009
\(947\) 39.0000 1.26733 0.633665 0.773608i \(-0.281550\pi\)
0.633665 + 0.773608i \(0.281550\pi\)
\(948\) 0 0
\(949\) −3.26662 −0.106039
\(950\) −4.69722 −0.152398
\(951\) 0 0
\(952\) 77.4500 2.51017
\(953\) 26.8444 0.869576 0.434788 0.900533i \(-0.356823\pi\)
0.434788 + 0.900533i \(0.356823\pi\)
\(954\) 0 0
\(955\) 16.4222 0.531410
\(956\) 0.238859 0.00772525
\(957\) 0 0
\(958\) −21.0833 −0.681170
\(959\) −22.1833 −0.716338
\(960\) 0 0
\(961\) −28.4222 −0.916845
\(962\) −1.57779 −0.0508701
\(963\) 0 0
\(964\) −8.54163 −0.275108
\(965\) 21.8167 0.702303
\(966\) 0 0
\(967\) 50.0000 1.60789 0.803946 0.594703i \(-0.202730\pi\)
0.803946 + 0.594703i \(0.202730\pi\)
\(968\) 12.6333 0.406050
\(969\) 0 0
\(970\) 10.4222 0.334637
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 0 0
\(973\) −18.4222 −0.590589
\(974\) −10.6611 −0.341603
\(975\) 0 0
\(976\) −33.7250 −1.07951
\(977\) −0.788897 −0.0252391 −0.0126195 0.999920i \(-0.504017\pi\)
−0.0126195 + 0.999920i \(0.504017\pi\)
\(978\) 0 0
\(979\) −20.3667 −0.650922
\(980\) −4.30278 −0.137447
\(981\) 0 0
\(982\) 16.6611 0.531676
\(983\) 0.633308 0.0201994 0.0100997 0.999949i \(-0.496785\pi\)
0.0100997 + 0.999949i \(0.496785\pi\)
\(984\) 0 0
\(985\) −1.18335 −0.0377045
\(986\) −62.8444 −2.00137
\(987\) 0 0
\(988\) −0.661064 −0.0210312
\(989\) 19.8167 0.630133
\(990\) 0 0
\(991\) −50.8167 −1.61424 −0.807122 0.590385i \(-0.798976\pi\)
−0.807122 + 0.590385i \(0.798976\pi\)
\(992\) 2.72498 0.0865182
\(993\) 0 0
\(994\) 87.6333 2.77956
\(995\) 13.2111 0.418820
\(996\) 0 0
\(997\) 53.8722 1.70615 0.853074 0.521789i \(-0.174735\pi\)
0.853074 + 0.521789i \(0.174735\pi\)
\(998\) −35.9638 −1.13842
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 135.2.a.c.1.2 2
3.2 odd 2 135.2.a.d.1.1 yes 2
4.3 odd 2 2160.2.a.ba.1.1 2
5.2 odd 4 675.2.b.i.649.3 4
5.3 odd 4 675.2.b.i.649.2 4
5.4 even 2 675.2.a.p.1.1 2
7.6 odd 2 6615.2.a.p.1.2 2
8.3 odd 2 8640.2.a.ck.1.1 2
8.5 even 2 8640.2.a.cr.1.2 2
9.2 odd 6 405.2.e.j.271.2 4
9.4 even 3 405.2.e.k.136.1 4
9.5 odd 6 405.2.e.j.136.2 4
9.7 even 3 405.2.e.k.271.1 4
12.11 even 2 2160.2.a.y.1.1 2
15.2 even 4 675.2.b.h.649.2 4
15.8 even 4 675.2.b.h.649.3 4
15.14 odd 2 675.2.a.k.1.2 2
21.20 even 2 6615.2.a.v.1.1 2
24.5 odd 2 8640.2.a.df.1.2 2
24.11 even 2 8640.2.a.cy.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.a.c.1.2 2 1.1 even 1 trivial
135.2.a.d.1.1 yes 2 3.2 odd 2
405.2.e.j.136.2 4 9.5 odd 6
405.2.e.j.271.2 4 9.2 odd 6
405.2.e.k.136.1 4 9.4 even 3
405.2.e.k.271.1 4 9.7 even 3
675.2.a.k.1.2 2 15.14 odd 2
675.2.a.p.1.1 2 5.4 even 2
675.2.b.h.649.2 4 15.2 even 4
675.2.b.h.649.3 4 15.8 even 4
675.2.b.i.649.2 4 5.3 odd 4
675.2.b.i.649.3 4 5.2 odd 4
2160.2.a.y.1.1 2 12.11 even 2
2160.2.a.ba.1.1 2 4.3 odd 2
6615.2.a.p.1.2 2 7.6 odd 2
6615.2.a.v.1.1 2 21.20 even 2
8640.2.a.ck.1.1 2 8.3 odd 2
8640.2.a.cr.1.2 2 8.5 even 2
8640.2.a.cy.1.1 2 24.11 even 2
8640.2.a.df.1.2 2 24.5 odd 2