Properties

Label 3040.2.a.m.1.2
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.539189 q^{3} +1.00000 q^{5} -0.630898 q^{7} -2.70928 q^{9} +O(q^{10})\) \(q-0.539189 q^{3} +1.00000 q^{5} -0.630898 q^{7} -2.70928 q^{9} +1.70928 q^{11} -3.17009 q^{13} -0.539189 q^{15} +1.41855 q^{17} -1.00000 q^{19} +0.340173 q^{21} +4.04945 q^{23} +1.00000 q^{25} +3.07838 q^{27} +3.75872 q^{29} -1.41855 q^{31} -0.921622 q^{33} -0.630898 q^{35} -0.986669 q^{37} +1.70928 q^{39} -9.26180 q^{41} -5.70928 q^{43} -2.70928 q^{45} +4.04945 q^{47} -6.60197 q^{49} -0.764867 q^{51} -7.32684 q^{53} +1.70928 q^{55} +0.539189 q^{57} +13.0205 q^{59} -13.1278 q^{61} +1.70928 q^{63} -3.17009 q^{65} -6.14116 q^{67} -2.18342 q^{69} -7.94214 q^{71} -9.91548 q^{73} -0.539189 q^{75} -1.07838 q^{77} -5.02052 q^{79} +6.46800 q^{81} -3.86603 q^{83} +1.41855 q^{85} -2.02666 q^{87} +5.60197 q^{89} +2.00000 q^{91} +0.764867 q^{93} -1.00000 q^{95} -0.275126 q^{97} -4.63090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} + 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} + 2 q^{7} - q^{9} - 2 q^{11} - 4 q^{13} - 10 q^{17} - 3 q^{19} - 10 q^{21} - 6 q^{23} + 3 q^{25} + 6 q^{27} - 14 q^{29} + 10 q^{31} - 6 q^{33} + 2 q^{35} - 2 q^{37} - 2 q^{39} - 20 q^{41} - 10 q^{43} - q^{45} - 6 q^{47} - q^{49} - 12 q^{51} - 10 q^{53} - 2 q^{55} + 6 q^{59} - 18 q^{61} - 2 q^{63} - 4 q^{65} + 2 q^{67} - 2 q^{69} + 6 q^{71} + 2 q^{73} + 18 q^{79} - 13 q^{81} + 2 q^{83} - 10 q^{85} - 8 q^{87} - 2 q^{89} + 6 q^{91} + 12 q^{93} - 3 q^{95} + 6 q^{97} - 10 q^{99}+O(q^{100}) \) 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
\(3\) −0.539189 −0.311301 −0.155650 0.987812i \(-0.549747\pi\)
−0.155650 + 0.987812i \(0.549747\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.630898 −0.238457 −0.119228 0.992867i \(-0.538042\pi\)
−0.119228 + 0.992867i \(0.538042\pi\)
\(8\) 0 0
\(9\) −2.70928 −0.903092
\(10\) 0 0
\(11\) 1.70928 0.515366 0.257683 0.966230i \(-0.417041\pi\)
0.257683 + 0.966230i \(0.417041\pi\)
\(12\) 0 0
\(13\) −3.17009 −0.879224 −0.439612 0.898188i \(-0.644884\pi\)
−0.439612 + 0.898188i \(0.644884\pi\)
\(14\) 0 0
\(15\) −0.539189 −0.139218
\(16\) 0 0
\(17\) 1.41855 0.344049 0.172025 0.985093i \(-0.444969\pi\)
0.172025 + 0.985093i \(0.444969\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.340173 0.0742318
\(22\) 0 0
\(23\) 4.04945 0.844368 0.422184 0.906510i \(-0.361264\pi\)
0.422184 + 0.906510i \(0.361264\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.07838 0.592434
\(28\) 0 0
\(29\) 3.75872 0.697977 0.348989 0.937127i \(-0.386525\pi\)
0.348989 + 0.937127i \(0.386525\pi\)
\(30\) 0 0
\(31\) −1.41855 −0.254779 −0.127390 0.991853i \(-0.540660\pi\)
−0.127390 + 0.991853i \(0.540660\pi\)
\(32\) 0 0
\(33\) −0.921622 −0.160434
\(34\) 0 0
\(35\) −0.630898 −0.106641
\(36\) 0 0
\(37\) −0.986669 −0.162207 −0.0811037 0.996706i \(-0.525844\pi\)
−0.0811037 + 0.996706i \(0.525844\pi\)
\(38\) 0 0
\(39\) 1.70928 0.273703
\(40\) 0 0
\(41\) −9.26180 −1.44645 −0.723225 0.690613i \(-0.757341\pi\)
−0.723225 + 0.690613i \(0.757341\pi\)
\(42\) 0 0
\(43\) −5.70928 −0.870656 −0.435328 0.900272i \(-0.643368\pi\)
−0.435328 + 0.900272i \(0.643368\pi\)
\(44\) 0 0
\(45\) −2.70928 −0.403875
\(46\) 0 0
\(47\) 4.04945 0.590673 0.295336 0.955393i \(-0.404568\pi\)
0.295336 + 0.955393i \(0.404568\pi\)
\(48\) 0 0
\(49\) −6.60197 −0.943138
\(50\) 0 0
\(51\) −0.764867 −0.107103
\(52\) 0 0
\(53\) −7.32684 −1.00642 −0.503210 0.864164i \(-0.667848\pi\)
−0.503210 + 0.864164i \(0.667848\pi\)
\(54\) 0 0
\(55\) 1.70928 0.230479
\(56\) 0 0
\(57\) 0.539189 0.0714173
\(58\) 0 0
\(59\) 13.0205 1.69513 0.847564 0.530694i \(-0.178069\pi\)
0.847564 + 0.530694i \(0.178069\pi\)
\(60\) 0 0
\(61\) −13.1278 −1.68085 −0.840423 0.541931i \(-0.817693\pi\)
−0.840423 + 0.541931i \(0.817693\pi\)
\(62\) 0 0
\(63\) 1.70928 0.215348
\(64\) 0 0
\(65\) −3.17009 −0.393201
\(66\) 0 0
\(67\) −6.14116 −0.750262 −0.375131 0.926972i \(-0.622402\pi\)
−0.375131 + 0.926972i \(0.622402\pi\)
\(68\) 0 0
\(69\) −2.18342 −0.262853
\(70\) 0 0
\(71\) −7.94214 −0.942559 −0.471279 0.881984i \(-0.656208\pi\)
−0.471279 + 0.881984i \(0.656208\pi\)
\(72\) 0 0
\(73\) −9.91548 −1.16052 −0.580260 0.814432i \(-0.697049\pi\)
−0.580260 + 0.814432i \(0.697049\pi\)
\(74\) 0 0
\(75\) −0.539189 −0.0622602
\(76\) 0 0
\(77\) −1.07838 −0.122893
\(78\) 0 0
\(79\) −5.02052 −0.564853 −0.282426 0.959289i \(-0.591139\pi\)
−0.282426 + 0.959289i \(0.591139\pi\)
\(80\) 0 0
\(81\) 6.46800 0.718667
\(82\) 0 0
\(83\) −3.86603 −0.424352 −0.212176 0.977231i \(-0.568055\pi\)
−0.212176 + 0.977231i \(0.568055\pi\)
\(84\) 0 0
\(85\) 1.41855 0.153863
\(86\) 0 0
\(87\) −2.02666 −0.217281
\(88\) 0 0
\(89\) 5.60197 0.593807 0.296904 0.954907i \(-0.404046\pi\)
0.296904 + 0.954907i \(0.404046\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0.764867 0.0793130
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −0.275126 −0.0279348 −0.0139674 0.999902i \(-0.504446\pi\)
−0.0139674 + 0.999902i \(0.504446\pi\)
\(98\) 0 0
\(99\) −4.63090 −0.465423
\(100\) 0 0
\(101\) −16.7298 −1.66468 −0.832338 0.554268i \(-0.812998\pi\)
−0.832338 + 0.554268i \(0.812998\pi\)
\(102\) 0 0
\(103\) 17.0628 1.68125 0.840623 0.541621i \(-0.182189\pi\)
0.840623 + 0.541621i \(0.182189\pi\)
\(104\) 0 0
\(105\) 0.340173 0.0331975
\(106\) 0 0
\(107\) 5.77432 0.558225 0.279112 0.960258i \(-0.409960\pi\)
0.279112 + 0.960258i \(0.409960\pi\)
\(108\) 0 0
\(109\) 1.44521 0.138426 0.0692131 0.997602i \(-0.477951\pi\)
0.0692131 + 0.997602i \(0.477951\pi\)
\(110\) 0 0
\(111\) 0.532001 0.0504953
\(112\) 0 0
\(113\) −3.93495 −0.370169 −0.185085 0.982723i \(-0.559256\pi\)
−0.185085 + 0.982723i \(0.559256\pi\)
\(114\) 0 0
\(115\) 4.04945 0.377613
\(116\) 0 0
\(117\) 8.58864 0.794020
\(118\) 0 0
\(119\) −0.894960 −0.0820409
\(120\) 0 0
\(121\) −8.07838 −0.734398
\(122\) 0 0
\(123\) 4.99386 0.450281
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 9.71646 0.862197 0.431098 0.902305i \(-0.358126\pi\)
0.431098 + 0.902305i \(0.358126\pi\)
\(128\) 0 0
\(129\) 3.07838 0.271036
\(130\) 0 0
\(131\) −14.6537 −1.28030 −0.640149 0.768251i \(-0.721127\pi\)
−0.640149 + 0.768251i \(0.721127\pi\)
\(132\) 0 0
\(133\) 0.630898 0.0547058
\(134\) 0 0
\(135\) 3.07838 0.264945
\(136\) 0 0
\(137\) 8.52359 0.728219 0.364110 0.931356i \(-0.381373\pi\)
0.364110 + 0.931356i \(0.381373\pi\)
\(138\) 0 0
\(139\) −22.3896 −1.89906 −0.949531 0.313672i \(-0.898441\pi\)
−0.949531 + 0.313672i \(0.898441\pi\)
\(140\) 0 0
\(141\) −2.18342 −0.183877
\(142\) 0 0
\(143\) −5.41855 −0.453122
\(144\) 0 0
\(145\) 3.75872 0.312145
\(146\) 0 0
\(147\) 3.55971 0.293600
\(148\) 0 0
\(149\) 6.38962 0.523458 0.261729 0.965141i \(-0.415707\pi\)
0.261729 + 0.965141i \(0.415707\pi\)
\(150\) 0 0
\(151\) 4.18342 0.340442 0.170221 0.985406i \(-0.445552\pi\)
0.170221 + 0.985406i \(0.445552\pi\)
\(152\) 0 0
\(153\) −3.84324 −0.310708
\(154\) 0 0
\(155\) −1.41855 −0.113941
\(156\) 0 0
\(157\) −15.3607 −1.22592 −0.612958 0.790115i \(-0.710021\pi\)
−0.612958 + 0.790115i \(0.710021\pi\)
\(158\) 0 0
\(159\) 3.95055 0.313299
\(160\) 0 0
\(161\) −2.55479 −0.201345
\(162\) 0 0
\(163\) −22.1217 −1.73270 −0.866352 0.499434i \(-0.833541\pi\)
−0.866352 + 0.499434i \(0.833541\pi\)
\(164\) 0 0
\(165\) −0.921622 −0.0717482
\(166\) 0 0
\(167\) −1.03612 −0.0801772 −0.0400886 0.999196i \(-0.512764\pi\)
−0.0400886 + 0.999196i \(0.512764\pi\)
\(168\) 0 0
\(169\) −2.95055 −0.226966
\(170\) 0 0
\(171\) 2.70928 0.207183
\(172\) 0 0
\(173\) −12.6153 −0.959123 −0.479562 0.877508i \(-0.659204\pi\)
−0.479562 + 0.877508i \(0.659204\pi\)
\(174\) 0 0
\(175\) −0.630898 −0.0476914
\(176\) 0 0
\(177\) −7.02052 −0.527695
\(178\) 0 0
\(179\) −23.4908 −1.75578 −0.877892 0.478859i \(-0.841051\pi\)
−0.877892 + 0.478859i \(0.841051\pi\)
\(180\) 0 0
\(181\) 3.36069 0.249798 0.124899 0.992169i \(-0.460139\pi\)
0.124899 + 0.992169i \(0.460139\pi\)
\(182\) 0 0
\(183\) 7.07838 0.523249
\(184\) 0 0
\(185\) −0.986669 −0.0725413
\(186\) 0 0
\(187\) 2.42469 0.177311
\(188\) 0 0
\(189\) −1.94214 −0.141270
\(190\) 0 0
\(191\) 21.4596 1.55276 0.776381 0.630264i \(-0.217053\pi\)
0.776381 + 0.630264i \(0.217053\pi\)
\(192\) 0 0
\(193\) 13.0856 0.941920 0.470960 0.882155i \(-0.343908\pi\)
0.470960 + 0.882155i \(0.343908\pi\)
\(194\) 0 0
\(195\) 1.70928 0.122404
\(196\) 0 0
\(197\) −2.49693 −0.177899 −0.0889494 0.996036i \(-0.528351\pi\)
−0.0889494 + 0.996036i \(0.528351\pi\)
\(198\) 0 0
\(199\) 12.6803 0.898886 0.449443 0.893309i \(-0.351623\pi\)
0.449443 + 0.893309i \(0.351623\pi\)
\(200\) 0 0
\(201\) 3.31124 0.233557
\(202\) 0 0
\(203\) −2.37137 −0.166438
\(204\) 0 0
\(205\) −9.26180 −0.646872
\(206\) 0 0
\(207\) −10.9711 −0.762542
\(208\) 0 0
\(209\) −1.70928 −0.118233
\(210\) 0 0
\(211\) −16.8638 −1.16095 −0.580475 0.814278i \(-0.697133\pi\)
−0.580475 + 0.814278i \(0.697133\pi\)
\(212\) 0 0
\(213\) 4.28231 0.293419
\(214\) 0 0
\(215\) −5.70928 −0.389369
\(216\) 0 0
\(217\) 0.894960 0.0607539
\(218\) 0 0
\(219\) 5.34632 0.361271
\(220\) 0 0
\(221\) −4.49693 −0.302496
\(222\) 0 0
\(223\) 8.22568 0.550832 0.275416 0.961325i \(-0.411184\pi\)
0.275416 + 0.961325i \(0.411184\pi\)
\(224\) 0 0
\(225\) −2.70928 −0.180618
\(226\) 0 0
\(227\) 9.19287 0.610152 0.305076 0.952328i \(-0.401318\pi\)
0.305076 + 0.952328i \(0.401318\pi\)
\(228\) 0 0
\(229\) 3.68261 0.243354 0.121677 0.992570i \(-0.461173\pi\)
0.121677 + 0.992570i \(0.461173\pi\)
\(230\) 0 0
\(231\) 0.581449 0.0382566
\(232\) 0 0
\(233\) 7.17727 0.470199 0.235099 0.971971i \(-0.424458\pi\)
0.235099 + 0.971971i \(0.424458\pi\)
\(234\) 0 0
\(235\) 4.04945 0.264157
\(236\) 0 0
\(237\) 2.70701 0.175839
\(238\) 0 0
\(239\) 10.1256 0.654968 0.327484 0.944857i \(-0.393799\pi\)
0.327484 + 0.944857i \(0.393799\pi\)
\(240\) 0 0
\(241\) −25.3607 −1.63363 −0.816813 0.576903i \(-0.804261\pi\)
−0.816813 + 0.576903i \(0.804261\pi\)
\(242\) 0 0
\(243\) −12.7226 −0.816156
\(244\) 0 0
\(245\) −6.60197 −0.421784
\(246\) 0 0
\(247\) 3.17009 0.201708
\(248\) 0 0
\(249\) 2.08452 0.132101
\(250\) 0 0
\(251\) 9.26180 0.584599 0.292300 0.956327i \(-0.405580\pi\)
0.292300 + 0.956327i \(0.405580\pi\)
\(252\) 0 0
\(253\) 6.92162 0.435159
\(254\) 0 0
\(255\) −0.764867 −0.0478978
\(256\) 0 0
\(257\) −6.34736 −0.395937 −0.197969 0.980208i \(-0.563434\pi\)
−0.197969 + 0.980208i \(0.563434\pi\)
\(258\) 0 0
\(259\) 0.622487 0.0386795
\(260\) 0 0
\(261\) −10.1834 −0.630338
\(262\) 0 0
\(263\) 1.79380 0.110610 0.0553051 0.998470i \(-0.482387\pi\)
0.0553051 + 0.998470i \(0.482387\pi\)
\(264\) 0 0
\(265\) −7.32684 −0.450084
\(266\) 0 0
\(267\) −3.02052 −0.184853
\(268\) 0 0
\(269\) −6.70701 −0.408933 −0.204467 0.978874i \(-0.565546\pi\)
−0.204467 + 0.978874i \(0.565546\pi\)
\(270\) 0 0
\(271\) −12.8143 −0.778414 −0.389207 0.921150i \(-0.627251\pi\)
−0.389207 + 0.921150i \(0.627251\pi\)
\(272\) 0 0
\(273\) −1.07838 −0.0652664
\(274\) 0 0
\(275\) 1.70928 0.103073
\(276\) 0 0
\(277\) 4.15676 0.249755 0.124878 0.992172i \(-0.460146\pi\)
0.124878 + 0.992172i \(0.460146\pi\)
\(278\) 0 0
\(279\) 3.84324 0.230089
\(280\) 0 0
\(281\) −29.0928 −1.73553 −0.867764 0.496976i \(-0.834444\pi\)
−0.867764 + 0.496976i \(0.834444\pi\)
\(282\) 0 0
\(283\) −12.1340 −0.721290 −0.360645 0.932703i \(-0.617443\pi\)
−0.360645 + 0.932703i \(0.617443\pi\)
\(284\) 0 0
\(285\) 0.539189 0.0319388
\(286\) 0 0
\(287\) 5.84324 0.344916
\(288\) 0 0
\(289\) −14.9877 −0.881630
\(290\) 0 0
\(291\) 0.148345 0.00869614
\(292\) 0 0
\(293\) 11.9227 0.696530 0.348265 0.937396i \(-0.386771\pi\)
0.348265 + 0.937396i \(0.386771\pi\)
\(294\) 0 0
\(295\) 13.0205 0.758084
\(296\) 0 0
\(297\) 5.26180 0.305320
\(298\) 0 0
\(299\) −12.8371 −0.742389
\(300\) 0 0
\(301\) 3.60197 0.207614
\(302\) 0 0
\(303\) 9.02052 0.518215
\(304\) 0 0
\(305\) −13.1278 −0.751697
\(306\) 0 0
\(307\) −3.02174 −0.172460 −0.0862299 0.996275i \(-0.527482\pi\)
−0.0862299 + 0.996275i \(0.527482\pi\)
\(308\) 0 0
\(309\) −9.20006 −0.523373
\(310\) 0 0
\(311\) 27.3835 1.55277 0.776387 0.630256i \(-0.217050\pi\)
0.776387 + 0.630256i \(0.217050\pi\)
\(312\) 0 0
\(313\) 15.1773 0.857870 0.428935 0.903335i \(-0.358889\pi\)
0.428935 + 0.903335i \(0.358889\pi\)
\(314\) 0 0
\(315\) 1.70928 0.0963068
\(316\) 0 0
\(317\) 13.6092 0.764366 0.382183 0.924087i \(-0.375172\pi\)
0.382183 + 0.924087i \(0.375172\pi\)
\(318\) 0 0
\(319\) 6.42469 0.359714
\(320\) 0 0
\(321\) −3.11345 −0.173776
\(322\) 0 0
\(323\) −1.41855 −0.0789303
\(324\) 0 0
\(325\) −3.17009 −0.175845
\(326\) 0 0
\(327\) −0.779243 −0.0430922
\(328\) 0 0
\(329\) −2.55479 −0.140850
\(330\) 0 0
\(331\) 6.92162 0.380447 0.190223 0.981741i \(-0.439079\pi\)
0.190223 + 0.981741i \(0.439079\pi\)
\(332\) 0 0
\(333\) 2.67316 0.146488
\(334\) 0 0
\(335\) −6.14116 −0.335527
\(336\) 0 0
\(337\) 22.6875 1.23587 0.617934 0.786230i \(-0.287970\pi\)
0.617934 + 0.786230i \(0.287970\pi\)
\(338\) 0 0
\(339\) 2.12168 0.115234
\(340\) 0 0
\(341\) −2.42469 −0.131305
\(342\) 0 0
\(343\) 8.58145 0.463355
\(344\) 0 0
\(345\) −2.18342 −0.117551
\(346\) 0 0
\(347\) −28.8287 −1.54761 −0.773803 0.633427i \(-0.781648\pi\)
−0.773803 + 0.633427i \(0.781648\pi\)
\(348\) 0 0
\(349\) 10.2823 0.550400 0.275200 0.961387i \(-0.411256\pi\)
0.275200 + 0.961387i \(0.411256\pi\)
\(350\) 0 0
\(351\) −9.75872 −0.520882
\(352\) 0 0
\(353\) −21.6163 −1.15052 −0.575261 0.817970i \(-0.695099\pi\)
−0.575261 + 0.817970i \(0.695099\pi\)
\(354\) 0 0
\(355\) −7.94214 −0.421525
\(356\) 0 0
\(357\) 0.482553 0.0255394
\(358\) 0 0
\(359\) −15.0700 −0.795362 −0.397681 0.917524i \(-0.630185\pi\)
−0.397681 + 0.917524i \(0.630185\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 4.35577 0.228619
\(364\) 0 0
\(365\) −9.91548 −0.519000
\(366\) 0 0
\(367\) −29.3256 −1.53078 −0.765392 0.643564i \(-0.777455\pi\)
−0.765392 + 0.643564i \(0.777455\pi\)
\(368\) 0 0
\(369\) 25.0928 1.30628
\(370\) 0 0
\(371\) 4.62249 0.239988
\(372\) 0 0
\(373\) 0.0917087 0.00474850 0.00237425 0.999997i \(-0.499244\pi\)
0.00237425 + 0.999997i \(0.499244\pi\)
\(374\) 0 0
\(375\) −0.539189 −0.0278436
\(376\) 0 0
\(377\) −11.9155 −0.613678
\(378\) 0 0
\(379\) 11.1506 0.572768 0.286384 0.958115i \(-0.407547\pi\)
0.286384 + 0.958115i \(0.407547\pi\)
\(380\) 0 0
\(381\) −5.23901 −0.268403
\(382\) 0 0
\(383\) −14.9060 −0.761662 −0.380831 0.924645i \(-0.624362\pi\)
−0.380831 + 0.924645i \(0.624362\pi\)
\(384\) 0 0
\(385\) −1.07838 −0.0549592
\(386\) 0 0
\(387\) 15.4680 0.786283
\(388\) 0 0
\(389\) 2.71154 0.137481 0.0687403 0.997635i \(-0.478102\pi\)
0.0687403 + 0.997635i \(0.478102\pi\)
\(390\) 0 0
\(391\) 5.74435 0.290504
\(392\) 0 0
\(393\) 7.90110 0.398558
\(394\) 0 0
\(395\) −5.02052 −0.252610
\(396\) 0 0
\(397\) 33.3340 1.67299 0.836494 0.547977i \(-0.184602\pi\)
0.836494 + 0.547977i \(0.184602\pi\)
\(398\) 0 0
\(399\) −0.340173 −0.0170299
\(400\) 0 0
\(401\) 7.88882 0.393949 0.196974 0.980409i \(-0.436888\pi\)
0.196974 + 0.980409i \(0.436888\pi\)
\(402\) 0 0
\(403\) 4.49693 0.224008
\(404\) 0 0
\(405\) 6.46800 0.321397
\(406\) 0 0
\(407\) −1.68649 −0.0835962
\(408\) 0 0
\(409\) −1.97334 −0.0975753 −0.0487876 0.998809i \(-0.515536\pi\)
−0.0487876 + 0.998809i \(0.515536\pi\)
\(410\) 0 0
\(411\) −4.59583 −0.226695
\(412\) 0 0
\(413\) −8.21461 −0.404215
\(414\) 0 0
\(415\) −3.86603 −0.189776
\(416\) 0 0
\(417\) 12.0722 0.591180
\(418\) 0 0
\(419\) −16.9627 −0.828680 −0.414340 0.910122i \(-0.635988\pi\)
−0.414340 + 0.910122i \(0.635988\pi\)
\(420\) 0 0
\(421\) −0.366835 −0.0178784 −0.00893922 0.999960i \(-0.502845\pi\)
−0.00893922 + 0.999960i \(0.502845\pi\)
\(422\) 0 0
\(423\) −10.9711 −0.533432
\(424\) 0 0
\(425\) 1.41855 0.0688098
\(426\) 0 0
\(427\) 8.28231 0.400809
\(428\) 0 0
\(429\) 2.92162 0.141057
\(430\) 0 0
\(431\) −8.39803 −0.404519 −0.202259 0.979332i \(-0.564828\pi\)
−0.202259 + 0.979332i \(0.564828\pi\)
\(432\) 0 0
\(433\) 31.2690 1.50269 0.751346 0.659909i \(-0.229405\pi\)
0.751346 + 0.659909i \(0.229405\pi\)
\(434\) 0 0
\(435\) −2.02666 −0.0971710
\(436\) 0 0
\(437\) −4.04945 −0.193711
\(438\) 0 0
\(439\) 19.0784 0.910561 0.455281 0.890348i \(-0.349539\pi\)
0.455281 + 0.890348i \(0.349539\pi\)
\(440\) 0 0
\(441\) 17.8865 0.851740
\(442\) 0 0
\(443\) 8.04945 0.382441 0.191220 0.981547i \(-0.438755\pi\)
0.191220 + 0.981547i \(0.438755\pi\)
\(444\) 0 0
\(445\) 5.60197 0.265559
\(446\) 0 0
\(447\) −3.44521 −0.162953
\(448\) 0 0
\(449\) −6.92162 −0.326652 −0.163326 0.986572i \(-0.552222\pi\)
−0.163326 + 0.986572i \(0.552222\pi\)
\(450\) 0 0
\(451\) −15.8310 −0.745451
\(452\) 0 0
\(453\) −2.25565 −0.105980
\(454\) 0 0
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 14.6803 0.686718 0.343359 0.939204i \(-0.388435\pi\)
0.343359 + 0.939204i \(0.388435\pi\)
\(458\) 0 0
\(459\) 4.36683 0.203826
\(460\) 0 0
\(461\) −33.1506 −1.54398 −0.771989 0.635636i \(-0.780738\pi\)
−0.771989 + 0.635636i \(0.780738\pi\)
\(462\) 0 0
\(463\) 18.5197 0.860684 0.430342 0.902666i \(-0.358393\pi\)
0.430342 + 0.902666i \(0.358393\pi\)
\(464\) 0 0
\(465\) 0.764867 0.0354698
\(466\) 0 0
\(467\) 23.5669 1.09055 0.545273 0.838259i \(-0.316426\pi\)
0.545273 + 0.838259i \(0.316426\pi\)
\(468\) 0 0
\(469\) 3.87444 0.178905
\(470\) 0 0
\(471\) 8.28231 0.381629
\(472\) 0 0
\(473\) −9.75872 −0.448707
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 19.8504 0.908889
\(478\) 0 0
\(479\) 10.2907 0.470195 0.235098 0.971972i \(-0.424459\pi\)
0.235098 + 0.971972i \(0.424459\pi\)
\(480\) 0 0
\(481\) 3.12783 0.142617
\(482\) 0 0
\(483\) 1.37751 0.0626790
\(484\) 0 0
\(485\) −0.275126 −0.0124928
\(486\) 0 0
\(487\) 28.9926 1.31378 0.656891 0.753986i \(-0.271871\pi\)
0.656891 + 0.753986i \(0.271871\pi\)
\(488\) 0 0
\(489\) 11.9278 0.539392
\(490\) 0 0
\(491\) 18.5692 0.838015 0.419007 0.907983i \(-0.362378\pi\)
0.419007 + 0.907983i \(0.362378\pi\)
\(492\) 0 0
\(493\) 5.33194 0.240139
\(494\) 0 0
\(495\) −4.63090 −0.208143
\(496\) 0 0
\(497\) 5.01068 0.224760
\(498\) 0 0
\(499\) −19.7671 −0.884898 −0.442449 0.896794i \(-0.645890\pi\)
−0.442449 + 0.896794i \(0.645890\pi\)
\(500\) 0 0
\(501\) 0.558663 0.0249592
\(502\) 0 0
\(503\) −20.8287 −0.928705 −0.464353 0.885650i \(-0.653713\pi\)
−0.464353 + 0.885650i \(0.653713\pi\)
\(504\) 0 0
\(505\) −16.7298 −0.744466
\(506\) 0 0
\(507\) 1.59090 0.0706546
\(508\) 0 0
\(509\) 24.5380 1.08763 0.543813 0.839206i \(-0.316980\pi\)
0.543813 + 0.839206i \(0.316980\pi\)
\(510\) 0 0
\(511\) 6.25565 0.276734
\(512\) 0 0
\(513\) −3.07838 −0.135914
\(514\) 0 0
\(515\) 17.0628 0.751876
\(516\) 0 0
\(517\) 6.92162 0.304413
\(518\) 0 0
\(519\) 6.80203 0.298576
\(520\) 0 0
\(521\) −6.58145 −0.288339 −0.144169 0.989553i \(-0.546051\pi\)
−0.144169 + 0.989553i \(0.546051\pi\)
\(522\) 0 0
\(523\) −27.6586 −1.20943 −0.604713 0.796443i \(-0.706712\pi\)
−0.604713 + 0.796443i \(0.706712\pi\)
\(524\) 0 0
\(525\) 0.340173 0.0148464
\(526\) 0 0
\(527\) −2.01229 −0.0876566
\(528\) 0 0
\(529\) −6.60197 −0.287042
\(530\) 0 0
\(531\) −35.2762 −1.53086
\(532\) 0 0
\(533\) 29.3607 1.27175
\(534\) 0 0
\(535\) 5.77432 0.249646
\(536\) 0 0
\(537\) 12.6660 0.546577
\(538\) 0 0
\(539\) −11.2846 −0.486061
\(540\) 0 0
\(541\) −16.9444 −0.728497 −0.364249 0.931302i \(-0.618674\pi\)
−0.364249 + 0.931302i \(0.618674\pi\)
\(542\) 0 0
\(543\) −1.81205 −0.0777624
\(544\) 0 0
\(545\) 1.44521 0.0619061
\(546\) 0 0
\(547\) 15.4608 0.661057 0.330528 0.943796i \(-0.392773\pi\)
0.330528 + 0.943796i \(0.392773\pi\)
\(548\) 0 0
\(549\) 35.5669 1.51796
\(550\) 0 0
\(551\) −3.75872 −0.160127
\(552\) 0 0
\(553\) 3.16743 0.134693
\(554\) 0 0
\(555\) 0.532001 0.0225822
\(556\) 0 0
\(557\) 40.3012 1.70762 0.853809 0.520587i \(-0.174287\pi\)
0.853809 + 0.520587i \(0.174287\pi\)
\(558\) 0 0
\(559\) 18.0989 0.765502
\(560\) 0 0
\(561\) −1.30737 −0.0551971
\(562\) 0 0
\(563\) 18.2134 0.767603 0.383801 0.923416i \(-0.374615\pi\)
0.383801 + 0.923416i \(0.374615\pi\)
\(564\) 0 0
\(565\) −3.93495 −0.165545
\(566\) 0 0
\(567\) −4.08065 −0.171371
\(568\) 0 0
\(569\) −4.13009 −0.173143 −0.0865713 0.996246i \(-0.527591\pi\)
−0.0865713 + 0.996246i \(0.527591\pi\)
\(570\) 0 0
\(571\) 25.2267 1.05571 0.527853 0.849336i \(-0.322997\pi\)
0.527853 + 0.849336i \(0.322997\pi\)
\(572\) 0 0
\(573\) −11.5708 −0.483376
\(574\) 0 0
\(575\) 4.04945 0.168874
\(576\) 0 0
\(577\) 34.7480 1.44658 0.723290 0.690544i \(-0.242629\pi\)
0.723290 + 0.690544i \(0.242629\pi\)
\(578\) 0 0
\(579\) −7.05559 −0.293220
\(580\) 0 0
\(581\) 2.43907 0.101190
\(582\) 0 0
\(583\) −12.5236 −0.518674
\(584\) 0 0
\(585\) 8.58864 0.355096
\(586\) 0 0
\(587\) 7.91935 0.326867 0.163433 0.986554i \(-0.447743\pi\)
0.163433 + 0.986554i \(0.447743\pi\)
\(588\) 0 0
\(589\) 1.41855 0.0584504
\(590\) 0 0
\(591\) 1.34632 0.0553800
\(592\) 0 0
\(593\) 0.837101 0.0343756 0.0171878 0.999852i \(-0.494529\pi\)
0.0171878 + 0.999852i \(0.494529\pi\)
\(594\) 0 0
\(595\) −0.894960 −0.0366898
\(596\) 0 0
\(597\) −6.83710 −0.279824
\(598\) 0 0
\(599\) 24.9216 1.01827 0.509135 0.860687i \(-0.329965\pi\)
0.509135 + 0.860687i \(0.329965\pi\)
\(600\) 0 0
\(601\) −17.7275 −0.723121 −0.361560 0.932349i \(-0.617756\pi\)
−0.361560 + 0.932349i \(0.617756\pi\)
\(602\) 0 0
\(603\) 16.6381 0.677555
\(604\) 0 0
\(605\) −8.07838 −0.328433
\(606\) 0 0
\(607\) −9.40749 −0.381838 −0.190919 0.981606i \(-0.561147\pi\)
−0.190919 + 0.981606i \(0.561147\pi\)
\(608\) 0 0
\(609\) 1.27862 0.0518121
\(610\) 0 0
\(611\) −12.8371 −0.519334
\(612\) 0 0
\(613\) 15.3919 0.621673 0.310836 0.950463i \(-0.399391\pi\)
0.310836 + 0.950463i \(0.399391\pi\)
\(614\) 0 0
\(615\) 4.99386 0.201372
\(616\) 0 0
\(617\) 5.28846 0.212905 0.106453 0.994318i \(-0.466051\pi\)
0.106453 + 0.994318i \(0.466051\pi\)
\(618\) 0 0
\(619\) −28.9132 −1.16212 −0.581060 0.813861i \(-0.697362\pi\)
−0.581060 + 0.813861i \(0.697362\pi\)
\(620\) 0 0
\(621\) 12.4657 0.500233
\(622\) 0 0
\(623\) −3.53427 −0.141597
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.921622 0.0368060
\(628\) 0 0
\(629\) −1.39964 −0.0558073
\(630\) 0 0
\(631\) 6.38962 0.254367 0.127183 0.991879i \(-0.459406\pi\)
0.127183 + 0.991879i \(0.459406\pi\)
\(632\) 0 0
\(633\) 9.09275 0.361405
\(634\) 0 0
\(635\) 9.71646 0.385586
\(636\) 0 0
\(637\) 20.9288 0.829230
\(638\) 0 0
\(639\) 21.5174 0.851217
\(640\) 0 0
\(641\) 1.04718 0.0413612 0.0206806 0.999786i \(-0.493417\pi\)
0.0206806 + 0.999786i \(0.493417\pi\)
\(642\) 0 0
\(643\) 7.46800 0.294509 0.147255 0.989099i \(-0.452956\pi\)
0.147255 + 0.989099i \(0.452956\pi\)
\(644\) 0 0
\(645\) 3.07838 0.121211
\(646\) 0 0
\(647\) −2.70313 −0.106271 −0.0531355 0.998587i \(-0.516922\pi\)
−0.0531355 + 0.998587i \(0.516922\pi\)
\(648\) 0 0
\(649\) 22.2557 0.873611
\(650\) 0 0
\(651\) −0.482553 −0.0189127
\(652\) 0 0
\(653\) −21.4329 −0.838735 −0.419368 0.907817i \(-0.637748\pi\)
−0.419368 + 0.907817i \(0.637748\pi\)
\(654\) 0 0
\(655\) −14.6537 −0.572567
\(656\) 0 0
\(657\) 26.8638 1.04806
\(658\) 0 0
\(659\) −19.8576 −0.773543 −0.386772 0.922176i \(-0.626410\pi\)
−0.386772 + 0.922176i \(0.626410\pi\)
\(660\) 0 0
\(661\) −2.83710 −0.110350 −0.0551752 0.998477i \(-0.517572\pi\)
−0.0551752 + 0.998477i \(0.517572\pi\)
\(662\) 0 0
\(663\) 2.42469 0.0941673
\(664\) 0 0
\(665\) 0.630898 0.0244652
\(666\) 0 0
\(667\) 15.2208 0.589350
\(668\) 0 0
\(669\) −4.43519 −0.171475
\(670\) 0 0
\(671\) −22.4391 −0.866251
\(672\) 0 0
\(673\) 24.0338 0.926437 0.463218 0.886244i \(-0.346695\pi\)
0.463218 + 0.886244i \(0.346695\pi\)
\(674\) 0 0
\(675\) 3.07838 0.118487
\(676\) 0 0
\(677\) 19.4114 0.746039 0.373020 0.927823i \(-0.378322\pi\)
0.373020 + 0.927823i \(0.378322\pi\)
\(678\) 0 0
\(679\) 0.173576 0.00666125
\(680\) 0 0
\(681\) −4.95669 −0.189941
\(682\) 0 0
\(683\) −0.821503 −0.0314339 −0.0157170 0.999876i \(-0.505003\pi\)
−0.0157170 + 0.999876i \(0.505003\pi\)
\(684\) 0 0
\(685\) 8.52359 0.325670
\(686\) 0 0
\(687\) −1.98562 −0.0757563
\(688\) 0 0
\(689\) 23.2267 0.884868
\(690\) 0 0
\(691\) 4.54638 0.172952 0.0864762 0.996254i \(-0.472439\pi\)
0.0864762 + 0.996254i \(0.472439\pi\)
\(692\) 0 0
\(693\) 2.92162 0.110983
\(694\) 0 0
\(695\) −22.3896 −0.849287
\(696\) 0 0
\(697\) −13.1383 −0.497650
\(698\) 0 0
\(699\) −3.86991 −0.146373
\(700\) 0 0
\(701\) 12.6453 0.477605 0.238803 0.971068i \(-0.423245\pi\)
0.238803 + 0.971068i \(0.423245\pi\)
\(702\) 0 0
\(703\) 0.986669 0.0372129
\(704\) 0 0
\(705\) −2.18342 −0.0822323
\(706\) 0 0
\(707\) 10.5548 0.396954
\(708\) 0 0
\(709\) −12.8059 −0.480936 −0.240468 0.970657i \(-0.577301\pi\)
−0.240468 + 0.970657i \(0.577301\pi\)
\(710\) 0 0
\(711\) 13.6020 0.510114
\(712\) 0 0
\(713\) −5.74435 −0.215128
\(714\) 0 0
\(715\) −5.41855 −0.202642
\(716\) 0 0
\(717\) −5.45959 −0.203892
\(718\) 0 0
\(719\) 1.90707 0.0711217 0.0355608 0.999368i \(-0.488678\pi\)
0.0355608 + 0.999368i \(0.488678\pi\)
\(720\) 0 0
\(721\) −10.7649 −0.400905
\(722\) 0 0
\(723\) 13.6742 0.508549
\(724\) 0 0
\(725\) 3.75872 0.139595
\(726\) 0 0
\(727\) −25.9239 −0.961464 −0.480732 0.876868i \(-0.659629\pi\)
−0.480732 + 0.876868i \(0.659629\pi\)
\(728\) 0 0
\(729\) −12.5441 −0.464597
\(730\) 0 0
\(731\) −8.09890 −0.299549
\(732\) 0 0
\(733\) 39.1605 1.44642 0.723212 0.690626i \(-0.242665\pi\)
0.723212 + 0.690626i \(0.242665\pi\)
\(734\) 0 0
\(735\) 3.55971 0.131302
\(736\) 0 0
\(737\) −10.4969 −0.386659
\(738\) 0 0
\(739\) −8.31351 −0.305817 −0.152909 0.988240i \(-0.548864\pi\)
−0.152909 + 0.988240i \(0.548864\pi\)
\(740\) 0 0
\(741\) −1.70928 −0.0627918
\(742\) 0 0
\(743\) 47.5597 1.74480 0.872398 0.488796i \(-0.162564\pi\)
0.872398 + 0.488796i \(0.162564\pi\)
\(744\) 0 0
\(745\) 6.38962 0.234098
\(746\) 0 0
\(747\) 10.4741 0.383229
\(748\) 0 0
\(749\) −3.64301 −0.133113
\(750\) 0 0
\(751\) 23.8622 0.870742 0.435371 0.900251i \(-0.356617\pi\)
0.435371 + 0.900251i \(0.356617\pi\)
\(752\) 0 0
\(753\) −4.99386 −0.181986
\(754\) 0 0
\(755\) 4.18342 0.152250
\(756\) 0 0
\(757\) 44.3402 1.61157 0.805785 0.592208i \(-0.201743\pi\)
0.805785 + 0.592208i \(0.201743\pi\)
\(758\) 0 0
\(759\) −3.73206 −0.135465
\(760\) 0 0
\(761\) 0.577574 0.0209370 0.0104685 0.999945i \(-0.496668\pi\)
0.0104685 + 0.999945i \(0.496668\pi\)
\(762\) 0 0
\(763\) −0.911781 −0.0330087
\(764\) 0 0
\(765\) −3.84324 −0.138953
\(766\) 0 0
\(767\) −41.2762 −1.49040
\(768\) 0 0
\(769\) 21.7938 0.785904 0.392952 0.919559i \(-0.371454\pi\)
0.392952 + 0.919559i \(0.371454\pi\)
\(770\) 0 0
\(771\) 3.42243 0.123256
\(772\) 0 0
\(773\) −43.4135 −1.56147 −0.780737 0.624860i \(-0.785156\pi\)
−0.780737 + 0.624860i \(0.785156\pi\)
\(774\) 0 0
\(775\) −1.41855 −0.0509558
\(776\) 0 0
\(777\) −0.335638 −0.0120410
\(778\) 0 0
\(779\) 9.26180 0.331838
\(780\) 0 0
\(781\) −13.5753 −0.485763
\(782\) 0 0
\(783\) 11.5708 0.413506
\(784\) 0 0
\(785\) −15.3607 −0.548247
\(786\) 0 0
\(787\) −14.2134 −0.506653 −0.253326 0.967381i \(-0.581525\pi\)
−0.253326 + 0.967381i \(0.581525\pi\)
\(788\) 0 0
\(789\) −0.967195 −0.0344331
\(790\) 0 0
\(791\) 2.48255 0.0882694
\(792\) 0 0
\(793\) 41.6163 1.47784
\(794\) 0 0
\(795\) 3.95055 0.140112
\(796\) 0 0
\(797\) −22.0482 −0.780988 −0.390494 0.920605i \(-0.627696\pi\)
−0.390494 + 0.920605i \(0.627696\pi\)
\(798\) 0 0
\(799\) 5.74435 0.203220
\(800\) 0 0
\(801\) −15.1773 −0.536263
\(802\) 0 0
\(803\) −16.9483 −0.598092
\(804\) 0 0
\(805\) −2.55479 −0.0900444
\(806\) 0 0
\(807\) 3.61634 0.127301
\(808\) 0 0
\(809\) 7.49854 0.263635 0.131817 0.991274i \(-0.457919\pi\)
0.131817 + 0.991274i \(0.457919\pi\)
\(810\) 0 0
\(811\) −43.4063 −1.52420 −0.762100 0.647459i \(-0.775832\pi\)
−0.762100 + 0.647459i \(0.775832\pi\)
\(812\) 0 0
\(813\) 6.90934 0.242321
\(814\) 0 0
\(815\) −22.1217 −0.774889
\(816\) 0 0
\(817\) 5.70928 0.199742
\(818\) 0 0
\(819\) −5.41855 −0.189339
\(820\) 0 0
\(821\) −26.7115 −0.932239 −0.466120 0.884722i \(-0.654348\pi\)
−0.466120 + 0.884722i \(0.654348\pi\)
\(822\) 0 0
\(823\) −22.2206 −0.774561 −0.387280 0.921962i \(-0.626585\pi\)
−0.387280 + 0.921962i \(0.626585\pi\)
\(824\) 0 0
\(825\) −0.921622 −0.0320868
\(826\) 0 0
\(827\) 31.7431 1.10382 0.551908 0.833905i \(-0.313900\pi\)
0.551908 + 0.833905i \(0.313900\pi\)
\(828\) 0 0
\(829\) 40.9093 1.42084 0.710420 0.703778i \(-0.248505\pi\)
0.710420 + 0.703778i \(0.248505\pi\)
\(830\) 0 0
\(831\) −2.24128 −0.0777490
\(832\) 0 0
\(833\) −9.36523 −0.324486
\(834\) 0 0
\(835\) −1.03612 −0.0358563
\(836\) 0 0
\(837\) −4.36683 −0.150940
\(838\) 0 0
\(839\) 14.3980 0.497075 0.248538 0.968622i \(-0.420050\pi\)
0.248538 + 0.968622i \(0.420050\pi\)
\(840\) 0 0
\(841\) −14.8720 −0.512827
\(842\) 0 0
\(843\) 15.6865 0.540271
\(844\) 0 0
\(845\) −2.95055 −0.101502
\(846\) 0 0
\(847\) 5.09663 0.175122
\(848\) 0 0
\(849\) 6.54250 0.224538
\(850\) 0 0
\(851\) −3.99547 −0.136963
\(852\) 0 0
\(853\) −24.1399 −0.826536 −0.413268 0.910610i \(-0.635613\pi\)
−0.413268 + 0.910610i \(0.635613\pi\)
\(854\) 0 0
\(855\) 2.70928 0.0926553
\(856\) 0 0
\(857\) 47.4089 1.61946 0.809729 0.586804i \(-0.199614\pi\)
0.809729 + 0.586804i \(0.199614\pi\)
\(858\) 0 0
\(859\) 38.9504 1.32897 0.664485 0.747302i \(-0.268651\pi\)
0.664485 + 0.747302i \(0.268651\pi\)
\(860\) 0 0
\(861\) −3.15061 −0.107373
\(862\) 0 0
\(863\) −20.5392 −0.699162 −0.349581 0.936906i \(-0.613676\pi\)
−0.349581 + 0.936906i \(0.613676\pi\)
\(864\) 0 0
\(865\) −12.6153 −0.428933
\(866\) 0 0
\(867\) 8.08121 0.274452
\(868\) 0 0
\(869\) −8.58145 −0.291106
\(870\) 0 0
\(871\) 19.4680 0.659648
\(872\) 0 0
\(873\) 0.745393 0.0252277
\(874\) 0 0
\(875\) −0.630898 −0.0213282
\(876\) 0 0
\(877\) 2.56198 0.0865118 0.0432559 0.999064i \(-0.486227\pi\)
0.0432559 + 0.999064i \(0.486227\pi\)
\(878\) 0 0
\(879\) −6.42857 −0.216830
\(880\) 0 0
\(881\) −25.0289 −0.843246 −0.421623 0.906771i \(-0.638539\pi\)
−0.421623 + 0.906771i \(0.638539\pi\)
\(882\) 0 0
\(883\) −57.2411 −1.92632 −0.963158 0.268936i \(-0.913328\pi\)
−0.963158 + 0.268936i \(0.913328\pi\)
\(884\) 0 0
\(885\) −7.02052 −0.235992
\(886\) 0 0
\(887\) −31.0049 −1.04104 −0.520522 0.853848i \(-0.674263\pi\)
−0.520522 + 0.853848i \(0.674263\pi\)
\(888\) 0 0
\(889\) −6.13009 −0.205597
\(890\) 0 0
\(891\) 11.0556 0.370376
\(892\) 0 0
\(893\) −4.04945 −0.135510
\(894\) 0 0
\(895\) −23.4908 −0.785210
\(896\) 0 0
\(897\) 6.92162 0.231106
\(898\) 0 0
\(899\) −5.33194 −0.177830
\(900\) 0 0
\(901\) −10.3935 −0.346258
\(902\) 0 0
\(903\) −1.94214 −0.0646304
\(904\) 0 0
\(905\) 3.36069 0.111713
\(906\) 0 0
\(907\) 24.9783 0.829389 0.414695 0.909961i \(-0.363888\pi\)
0.414695 + 0.909961i \(0.363888\pi\)
\(908\) 0 0
\(909\) 45.3256 1.50336
\(910\) 0 0
\(911\) −54.4678 −1.80460 −0.902300 0.431109i \(-0.858122\pi\)
−0.902300 + 0.431109i \(0.858122\pi\)
\(912\) 0 0
\(913\) −6.60811 −0.218697
\(914\) 0 0
\(915\) 7.07838 0.234004
\(916\) 0 0
\(917\) 9.24497 0.305296
\(918\) 0 0
\(919\) −19.7998 −0.653134 −0.326567 0.945174i \(-0.605892\pi\)
−0.326567 + 0.945174i \(0.605892\pi\)
\(920\) 0 0
\(921\) 1.62929 0.0536869
\(922\) 0 0
\(923\) 25.1773 0.828720
\(924\) 0 0
\(925\) −0.986669 −0.0324415
\(926\) 0 0
\(927\) −46.2278 −1.51832
\(928\) 0 0
\(929\) −7.78992 −0.255579 −0.127790 0.991801i \(-0.540788\pi\)
−0.127790 + 0.991801i \(0.540788\pi\)
\(930\) 0 0
\(931\) 6.60197 0.216371
\(932\) 0 0
\(933\) −14.7649 −0.483380
\(934\) 0 0
\(935\) 2.42469 0.0792960
\(936\) 0 0
\(937\) 2.88058 0.0941046 0.0470523 0.998892i \(-0.485017\pi\)
0.0470523 + 0.998892i \(0.485017\pi\)
\(938\) 0 0
\(939\) −8.18342 −0.267056
\(940\) 0 0
\(941\) −53.6163 −1.74784 −0.873921 0.486067i \(-0.838431\pi\)
−0.873921 + 0.486067i \(0.838431\pi\)
\(942\) 0 0
\(943\) −37.5052 −1.22134
\(944\) 0 0
\(945\) −1.94214 −0.0631779
\(946\) 0 0
\(947\) −26.7708 −0.869935 −0.434968 0.900446i \(-0.643240\pi\)
−0.434968 + 0.900446i \(0.643240\pi\)
\(948\) 0 0
\(949\) 31.4329 1.02036
\(950\) 0 0
\(951\) −7.33791 −0.237948
\(952\) 0 0
\(953\) 16.1061 0.521727 0.260864 0.965376i \(-0.415993\pi\)
0.260864 + 0.965376i \(0.415993\pi\)
\(954\) 0 0
\(955\) 21.4596 0.694416
\(956\) 0 0
\(957\) −3.46412 −0.111979
\(958\) 0 0
\(959\) −5.37751 −0.173649
\(960\) 0 0
\(961\) −28.9877 −0.935088
\(962\) 0 0
\(963\) −15.6442 −0.504128
\(964\) 0 0
\(965\) 13.0856 0.421239
\(966\) 0 0
\(967\) 17.8537 0.574138 0.287069 0.957910i \(-0.407319\pi\)
0.287069 + 0.957910i \(0.407319\pi\)
\(968\) 0 0
\(969\) 0.764867 0.0245711
\(970\) 0 0
\(971\) 58.1978 1.86766 0.933828 0.357722i \(-0.116447\pi\)
0.933828 + 0.357722i \(0.116447\pi\)
\(972\) 0 0
\(973\) 14.1256 0.452845
\(974\) 0 0
\(975\) 1.70928 0.0547406
\(976\) 0 0
\(977\) 12.7766 0.408759 0.204380 0.978892i \(-0.434482\pi\)
0.204380 + 0.978892i \(0.434482\pi\)
\(978\) 0 0
\(979\) 9.57531 0.306028
\(980\) 0 0
\(981\) −3.91548 −0.125012
\(982\) 0 0
\(983\) 40.0833 1.27846 0.639229 0.769016i \(-0.279253\pi\)
0.639229 + 0.769016i \(0.279253\pi\)
\(984\) 0 0
\(985\) −2.49693 −0.0795588
\(986\) 0 0
\(987\) 1.37751 0.0438467
\(988\) 0 0
\(989\) −23.1194 −0.735155
\(990\) 0 0
\(991\) 27.5318 0.874577 0.437289 0.899321i \(-0.355939\pi\)
0.437289 + 0.899321i \(0.355939\pi\)
\(992\) 0 0
\(993\) −3.73206 −0.118433
\(994\) 0 0
\(995\) 12.6803 0.401994
\(996\) 0 0
\(997\) 20.1399 0.637838 0.318919 0.947782i \(-0.396680\pi\)
0.318919 + 0.947782i \(0.396680\pi\)
\(998\) 0 0
\(999\) −3.03734 −0.0960972
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 3040.2.a.m.1.2 yes 3
4.3 odd 2 3040.2.a.l.1.2 3
8.3 odd 2 6080.2.a.bt.1.2 3
8.5 even 2 6080.2.a.bu.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.l.1.2 3 4.3 odd 2
3040.2.a.m.1.2 yes 3 1.1 even 1 trivial
6080.2.a.bt.1.2 3 8.3 odd 2
6080.2.a.bu.1.2 3 8.5 even 2