Properties

Label 9405.2.a.bd.1.2
Level $9405$
Weight $2$
Character 9405.1
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
Dimension $7$
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] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, 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("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.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: \(75.0993031010\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.97792\) of defining polynomial
Character \(\chi\) \(=\) 9405.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.97792 q^{2} +1.91218 q^{4} +1.00000 q^{5} +1.06653 q^{7} +0.173707 q^{8} +O(q^{10})\) \(q-1.97792 q^{2} +1.91218 q^{4} +1.00000 q^{5} +1.06653 q^{7} +0.173707 q^{8} -1.97792 q^{10} -1.00000 q^{11} +0.939291 q^{13} -2.10952 q^{14} -4.16793 q^{16} -3.65839 q^{17} -1.00000 q^{19} +1.91218 q^{20} +1.97792 q^{22} -5.35035 q^{23} +1.00000 q^{25} -1.85784 q^{26} +2.03940 q^{28} +4.46966 q^{29} -2.71979 q^{31} +7.89643 q^{32} +7.23600 q^{34} +1.06653 q^{35} +0.932050 q^{37} +1.97792 q^{38} +0.173707 q^{40} +1.21816 q^{41} -3.58402 q^{43} -1.91218 q^{44} +10.5826 q^{46} +4.30223 q^{47} -5.86250 q^{49} -1.97792 q^{50} +1.79609 q^{52} +12.1575 q^{53} -1.00000 q^{55} +0.185265 q^{56} -8.84063 q^{58} +6.15610 q^{59} +2.13701 q^{61} +5.37953 q^{62} -7.28267 q^{64} +0.939291 q^{65} +4.93275 q^{67} -6.99548 q^{68} -2.10952 q^{70} -7.32299 q^{71} +1.14496 q^{73} -1.84352 q^{74} -1.91218 q^{76} -1.06653 q^{77} -6.71067 q^{79} -4.16793 q^{80} -2.40943 q^{82} -2.82737 q^{83} -3.65839 q^{85} +7.08891 q^{86} -0.173707 q^{88} +8.77091 q^{89} +1.00179 q^{91} -10.2308 q^{92} -8.50948 q^{94} -1.00000 q^{95} +13.3420 q^{97} +11.5956 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 7 q^{4} + 7 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} + 7 q^{4} + 7 q^{5} - q^{7} - 3 q^{8} - q^{10} - 7 q^{11} + q^{13} - 12 q^{14} + 3 q^{16} - q^{17} - 7 q^{19} + 7 q^{20} + q^{22} + 8 q^{23} + 7 q^{25} + 4 q^{28} - 11 q^{29} + 7 q^{31} - 12 q^{32} - 14 q^{34} - q^{35} - 17 q^{37} + q^{38} - 3 q^{40} - 17 q^{41} - 3 q^{43} - 7 q^{44} + 18 q^{46} - 14 q^{47} + 6 q^{49} - q^{50} - 17 q^{52} - 7 q^{53} - 7 q^{55} - 36 q^{56} - 15 q^{58} - 35 q^{59} + 17 q^{61} - 46 q^{62} + 5 q^{64} + q^{65} + 4 q^{67} + 35 q^{68} - 12 q^{70} - 10 q^{71} + 22 q^{73} + 11 q^{74} - 7 q^{76} + q^{77} + 11 q^{79} + 3 q^{80} - 14 q^{82} - 39 q^{83} - q^{85} + 24 q^{86} + 3 q^{88} - 18 q^{89} - 22 q^{91} + 51 q^{92} + 14 q^{94} - 7 q^{95} - 4 q^{97} + 26 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.97792 −1.39860 −0.699301 0.714827i \(-0.746505\pi\)
−0.699301 + 0.714827i \(0.746505\pi\)
\(3\) 0 0
\(4\) 1.91218 0.956088
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.06653 0.403112 0.201556 0.979477i \(-0.435400\pi\)
0.201556 + 0.979477i \(0.435400\pi\)
\(8\) 0.173707 0.0614148
\(9\) 0 0
\(10\) −1.97792 −0.625474
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.939291 0.260512 0.130256 0.991480i \(-0.458420\pi\)
0.130256 + 0.991480i \(0.458420\pi\)
\(14\) −2.10952 −0.563794
\(15\) 0 0
\(16\) −4.16793 −1.04198
\(17\) −3.65839 −0.887289 −0.443645 0.896203i \(-0.646315\pi\)
−0.443645 + 0.896203i \(0.646315\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 1.91218 0.427576
\(21\) 0 0
\(22\) 1.97792 0.421694
\(23\) −5.35035 −1.11563 −0.557813 0.829967i \(-0.688359\pi\)
−0.557813 + 0.829967i \(0.688359\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.85784 −0.364353
\(27\) 0 0
\(28\) 2.03940 0.385411
\(29\) 4.46966 0.829994 0.414997 0.909823i \(-0.363783\pi\)
0.414997 + 0.909823i \(0.363783\pi\)
\(30\) 0 0
\(31\) −2.71979 −0.488489 −0.244244 0.969714i \(-0.578540\pi\)
−0.244244 + 0.969714i \(0.578540\pi\)
\(32\) 7.89643 1.39591
\(33\) 0 0
\(34\) 7.23600 1.24096
\(35\) 1.06653 0.180277
\(36\) 0 0
\(37\) 0.932050 0.153228 0.0766141 0.997061i \(-0.475589\pi\)
0.0766141 + 0.997061i \(0.475589\pi\)
\(38\) 1.97792 0.320861
\(39\) 0 0
\(40\) 0.173707 0.0274656
\(41\) 1.21816 0.190245 0.0951227 0.995466i \(-0.469676\pi\)
0.0951227 + 0.995466i \(0.469676\pi\)
\(42\) 0 0
\(43\) −3.58402 −0.546558 −0.273279 0.961935i \(-0.588108\pi\)
−0.273279 + 0.961935i \(0.588108\pi\)
\(44\) −1.91218 −0.288272
\(45\) 0 0
\(46\) 10.5826 1.56032
\(47\) 4.30223 0.627545 0.313772 0.949498i \(-0.398407\pi\)
0.313772 + 0.949498i \(0.398407\pi\)
\(48\) 0 0
\(49\) −5.86250 −0.837501
\(50\) −1.97792 −0.279720
\(51\) 0 0
\(52\) 1.79609 0.249073
\(53\) 12.1575 1.66996 0.834980 0.550280i \(-0.185479\pi\)
0.834980 + 0.550280i \(0.185479\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0.185265 0.0247571
\(57\) 0 0
\(58\) −8.84063 −1.16083
\(59\) 6.15610 0.801456 0.400728 0.916197i \(-0.368757\pi\)
0.400728 + 0.916197i \(0.368757\pi\)
\(60\) 0 0
\(61\) 2.13701 0.273616 0.136808 0.990598i \(-0.456316\pi\)
0.136808 + 0.990598i \(0.456316\pi\)
\(62\) 5.37953 0.683202
\(63\) 0 0
\(64\) −7.28267 −0.910333
\(65\) 0.939291 0.116505
\(66\) 0 0
\(67\) 4.93275 0.602631 0.301315 0.953525i \(-0.402574\pi\)
0.301315 + 0.953525i \(0.402574\pi\)
\(68\) −6.99548 −0.848327
\(69\) 0 0
\(70\) −2.10952 −0.252136
\(71\) −7.32299 −0.869079 −0.434539 0.900653i \(-0.643089\pi\)
−0.434539 + 0.900653i \(0.643089\pi\)
\(72\) 0 0
\(73\) 1.14496 0.134008 0.0670038 0.997753i \(-0.478656\pi\)
0.0670038 + 0.997753i \(0.478656\pi\)
\(74\) −1.84352 −0.214305
\(75\) 0 0
\(76\) −1.91218 −0.219342
\(77\) −1.06653 −0.121543
\(78\) 0 0
\(79\) −6.71067 −0.755009 −0.377505 0.926008i \(-0.623218\pi\)
−0.377505 + 0.926008i \(0.623218\pi\)
\(80\) −4.16793 −0.465989
\(81\) 0 0
\(82\) −2.40943 −0.266078
\(83\) −2.82737 −0.310344 −0.155172 0.987887i \(-0.549593\pi\)
−0.155172 + 0.987887i \(0.549593\pi\)
\(84\) 0 0
\(85\) −3.65839 −0.396808
\(86\) 7.08891 0.764417
\(87\) 0 0
\(88\) −0.173707 −0.0185173
\(89\) 8.77091 0.929714 0.464857 0.885386i \(-0.346106\pi\)
0.464857 + 0.885386i \(0.346106\pi\)
\(90\) 0 0
\(91\) 1.00179 0.105016
\(92\) −10.2308 −1.06664
\(93\) 0 0
\(94\) −8.50948 −0.877686
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 13.3420 1.35468 0.677339 0.735671i \(-0.263133\pi\)
0.677339 + 0.735671i \(0.263133\pi\)
\(98\) 11.5956 1.17133
\(99\) 0 0
\(100\) 1.91218 0.191218
\(101\) 0.926286 0.0921689 0.0460845 0.998938i \(-0.485326\pi\)
0.0460845 + 0.998938i \(0.485326\pi\)
\(102\) 0 0
\(103\) −10.4344 −1.02814 −0.514068 0.857749i \(-0.671862\pi\)
−0.514068 + 0.857749i \(0.671862\pi\)
\(104\) 0.163162 0.0159993
\(105\) 0 0
\(106\) −24.0466 −2.33561
\(107\) −13.5595 −1.31085 −0.655426 0.755260i \(-0.727511\pi\)
−0.655426 + 0.755260i \(0.727511\pi\)
\(108\) 0 0
\(109\) 5.85440 0.560750 0.280375 0.959891i \(-0.409541\pi\)
0.280375 + 0.959891i \(0.409541\pi\)
\(110\) 1.97792 0.188587
\(111\) 0 0
\(112\) −4.44525 −0.420036
\(113\) −8.68686 −0.817191 −0.408595 0.912716i \(-0.633981\pi\)
−0.408595 + 0.912716i \(0.633981\pi\)
\(114\) 0 0
\(115\) −5.35035 −0.498923
\(116\) 8.54677 0.793548
\(117\) 0 0
\(118\) −12.1763 −1.12092
\(119\) −3.90180 −0.357677
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −4.22684 −0.382680
\(123\) 0 0
\(124\) −5.20072 −0.467039
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −15.9545 −1.41573 −0.707866 0.706347i \(-0.750342\pi\)
−0.707866 + 0.706347i \(0.750342\pi\)
\(128\) −1.38832 −0.122711
\(129\) 0 0
\(130\) −1.85784 −0.162944
\(131\) −5.20014 −0.454338 −0.227169 0.973855i \(-0.572947\pi\)
−0.227169 + 0.973855i \(0.572947\pi\)
\(132\) 0 0
\(133\) −1.06653 −0.0924803
\(134\) −9.75659 −0.842841
\(135\) 0 0
\(136\) −0.635489 −0.0544927
\(137\) −7.54419 −0.644544 −0.322272 0.946647i \(-0.604447\pi\)
−0.322272 + 0.946647i \(0.604447\pi\)
\(138\) 0 0
\(139\) 0.357301 0.0303059 0.0151529 0.999885i \(-0.495176\pi\)
0.0151529 + 0.999885i \(0.495176\pi\)
\(140\) 2.03940 0.172361
\(141\) 0 0
\(142\) 14.4843 1.21550
\(143\) −0.939291 −0.0785475
\(144\) 0 0
\(145\) 4.46966 0.371185
\(146\) −2.26465 −0.187423
\(147\) 0 0
\(148\) 1.78224 0.146500
\(149\) 10.7634 0.881769 0.440884 0.897564i \(-0.354665\pi\)
0.440884 + 0.897564i \(0.354665\pi\)
\(150\) 0 0
\(151\) 7.13976 0.581025 0.290513 0.956871i \(-0.406174\pi\)
0.290513 + 0.956871i \(0.406174\pi\)
\(152\) −0.173707 −0.0140895
\(153\) 0 0
\(154\) 2.10952 0.169990
\(155\) −2.71979 −0.218459
\(156\) 0 0
\(157\) 12.7164 1.01488 0.507440 0.861687i \(-0.330592\pi\)
0.507440 + 0.861687i \(0.330592\pi\)
\(158\) 13.2732 1.05596
\(159\) 0 0
\(160\) 7.89643 0.624268
\(161\) −5.70633 −0.449722
\(162\) 0 0
\(163\) −4.45308 −0.348792 −0.174396 0.984676i \(-0.555797\pi\)
−0.174396 + 0.984676i \(0.555797\pi\)
\(164\) 2.32935 0.181891
\(165\) 0 0
\(166\) 5.59231 0.434047
\(167\) −10.0911 −0.780874 −0.390437 0.920630i \(-0.627676\pi\)
−0.390437 + 0.920630i \(0.627676\pi\)
\(168\) 0 0
\(169\) −12.1177 −0.932133
\(170\) 7.23600 0.554976
\(171\) 0 0
\(172\) −6.85328 −0.522558
\(173\) −23.8006 −1.80952 −0.904762 0.425917i \(-0.859952\pi\)
−0.904762 + 0.425917i \(0.859952\pi\)
\(174\) 0 0
\(175\) 1.06653 0.0806224
\(176\) 4.16793 0.314170
\(177\) 0 0
\(178\) −17.3482 −1.30030
\(179\) −20.0682 −1.49997 −0.749985 0.661455i \(-0.769939\pi\)
−0.749985 + 0.661455i \(0.769939\pi\)
\(180\) 0 0
\(181\) 7.49807 0.557327 0.278663 0.960389i \(-0.410109\pi\)
0.278663 + 0.960389i \(0.410109\pi\)
\(182\) −1.98146 −0.146875
\(183\) 0 0
\(184\) −0.929396 −0.0685160
\(185\) 0.932050 0.0685257
\(186\) 0 0
\(187\) 3.65839 0.267528
\(188\) 8.22663 0.599988
\(189\) 0 0
\(190\) 1.97792 0.143494
\(191\) 4.21863 0.305250 0.152625 0.988284i \(-0.451227\pi\)
0.152625 + 0.988284i \(0.451227\pi\)
\(192\) 0 0
\(193\) 7.70714 0.554772 0.277386 0.960759i \(-0.410532\pi\)
0.277386 + 0.960759i \(0.410532\pi\)
\(194\) −26.3895 −1.89465
\(195\) 0 0
\(196\) −11.2101 −0.800725
\(197\) −18.4285 −1.31298 −0.656489 0.754336i \(-0.727959\pi\)
−0.656489 + 0.754336i \(0.727959\pi\)
\(198\) 0 0
\(199\) −21.6489 −1.53465 −0.767324 0.641260i \(-0.778412\pi\)
−0.767324 + 0.641260i \(0.778412\pi\)
\(200\) 0.173707 0.0122830
\(201\) 0 0
\(202\) −1.83212 −0.128908
\(203\) 4.76704 0.334581
\(204\) 0 0
\(205\) 1.21816 0.0850803
\(206\) 20.6385 1.43795
\(207\) 0 0
\(208\) −3.91490 −0.271450
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 23.9164 1.64647 0.823235 0.567700i \(-0.192167\pi\)
0.823235 + 0.567700i \(0.192167\pi\)
\(212\) 23.2473 1.59663
\(213\) 0 0
\(214\) 26.8197 1.83336
\(215\) −3.58402 −0.244428
\(216\) 0 0
\(217\) −2.90075 −0.196916
\(218\) −11.5795 −0.784266
\(219\) 0 0
\(220\) −1.91218 −0.128919
\(221\) −3.43629 −0.231150
\(222\) 0 0
\(223\) 4.18886 0.280506 0.140253 0.990116i \(-0.455208\pi\)
0.140253 + 0.990116i \(0.455208\pi\)
\(224\) 8.42182 0.562707
\(225\) 0 0
\(226\) 17.1819 1.14292
\(227\) 8.74816 0.580636 0.290318 0.956930i \(-0.406239\pi\)
0.290318 + 0.956930i \(0.406239\pi\)
\(228\) 0 0
\(229\) −16.8146 −1.11114 −0.555571 0.831469i \(-0.687500\pi\)
−0.555571 + 0.831469i \(0.687500\pi\)
\(230\) 10.5826 0.697795
\(231\) 0 0
\(232\) 0.776412 0.0509740
\(233\) 7.41167 0.485554 0.242777 0.970082i \(-0.421942\pi\)
0.242777 + 0.970082i \(0.421942\pi\)
\(234\) 0 0
\(235\) 4.30223 0.280647
\(236\) 11.7716 0.766263
\(237\) 0 0
\(238\) 7.71745 0.500248
\(239\) 19.7326 1.27640 0.638199 0.769871i \(-0.279680\pi\)
0.638199 + 0.769871i \(0.279680\pi\)
\(240\) 0 0
\(241\) 15.0994 0.972640 0.486320 0.873781i \(-0.338339\pi\)
0.486320 + 0.873781i \(0.338339\pi\)
\(242\) −1.97792 −0.127146
\(243\) 0 0
\(244\) 4.08634 0.261601
\(245\) −5.86250 −0.374542
\(246\) 0 0
\(247\) −0.939291 −0.0597657
\(248\) −0.472448 −0.0300005
\(249\) 0 0
\(250\) −1.97792 −0.125095
\(251\) −12.3858 −0.781782 −0.390891 0.920437i \(-0.627833\pi\)
−0.390891 + 0.920437i \(0.627833\pi\)
\(252\) 0 0
\(253\) 5.35035 0.336374
\(254\) 31.5567 1.98005
\(255\) 0 0
\(256\) 17.3113 1.08196
\(257\) −26.5496 −1.65612 −0.828060 0.560639i \(-0.810555\pi\)
−0.828060 + 0.560639i \(0.810555\pi\)
\(258\) 0 0
\(259\) 0.994064 0.0617681
\(260\) 1.79609 0.111389
\(261\) 0 0
\(262\) 10.2855 0.635438
\(263\) −15.0108 −0.925604 −0.462802 0.886462i \(-0.653156\pi\)
−0.462802 + 0.886462i \(0.653156\pi\)
\(264\) 0 0
\(265\) 12.1575 0.746829
\(266\) 2.10952 0.129343
\(267\) 0 0
\(268\) 9.43228 0.576168
\(269\) −11.8307 −0.721329 −0.360664 0.932696i \(-0.617450\pi\)
−0.360664 + 0.932696i \(0.617450\pi\)
\(270\) 0 0
\(271\) −13.4154 −0.814927 −0.407464 0.913221i \(-0.633587\pi\)
−0.407464 + 0.913221i \(0.633587\pi\)
\(272\) 15.2479 0.924540
\(273\) 0 0
\(274\) 14.9218 0.901461
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −0.515992 −0.0310030 −0.0155015 0.999880i \(-0.504934\pi\)
−0.0155015 + 0.999880i \(0.504934\pi\)
\(278\) −0.706713 −0.0423859
\(279\) 0 0
\(280\) 0.185265 0.0110717
\(281\) −21.9281 −1.30812 −0.654061 0.756441i \(-0.726936\pi\)
−0.654061 + 0.756441i \(0.726936\pi\)
\(282\) 0 0
\(283\) 18.6023 1.10579 0.552896 0.833250i \(-0.313522\pi\)
0.552896 + 0.833250i \(0.313522\pi\)
\(284\) −14.0028 −0.830916
\(285\) 0 0
\(286\) 1.85784 0.109857
\(287\) 1.29921 0.0766902
\(288\) 0 0
\(289\) −3.61621 −0.212718
\(290\) −8.84063 −0.519140
\(291\) 0 0
\(292\) 2.18937 0.128123
\(293\) 5.87287 0.343097 0.171548 0.985176i \(-0.445123\pi\)
0.171548 + 0.985176i \(0.445123\pi\)
\(294\) 0 0
\(295\) 6.15610 0.358422
\(296\) 0.161904 0.00941048
\(297\) 0 0
\(298\) −21.2891 −1.23324
\(299\) −5.02554 −0.290634
\(300\) 0 0
\(301\) −3.82248 −0.220324
\(302\) −14.1219 −0.812623
\(303\) 0 0
\(304\) 4.16793 0.239047
\(305\) 2.13701 0.122365
\(306\) 0 0
\(307\) 0.566566 0.0323356 0.0161678 0.999869i \(-0.494853\pi\)
0.0161678 + 0.999869i \(0.494853\pi\)
\(308\) −2.03940 −0.116206
\(309\) 0 0
\(310\) 5.37953 0.305537
\(311\) 22.1715 1.25723 0.628616 0.777716i \(-0.283622\pi\)
0.628616 + 0.777716i \(0.283622\pi\)
\(312\) 0 0
\(313\) −30.3230 −1.71396 −0.856979 0.515351i \(-0.827662\pi\)
−0.856979 + 0.515351i \(0.827662\pi\)
\(314\) −25.1521 −1.41941
\(315\) 0 0
\(316\) −12.8320 −0.721856
\(317\) 13.6662 0.767569 0.383785 0.923423i \(-0.374621\pi\)
0.383785 + 0.923423i \(0.374621\pi\)
\(318\) 0 0
\(319\) −4.46966 −0.250253
\(320\) −7.28267 −0.407113
\(321\) 0 0
\(322\) 11.2867 0.628982
\(323\) 3.65839 0.203558
\(324\) 0 0
\(325\) 0.939291 0.0521025
\(326\) 8.80785 0.487822
\(327\) 0 0
\(328\) 0.211604 0.0116839
\(329\) 4.58848 0.252971
\(330\) 0 0
\(331\) 1.04337 0.0573489 0.0286745 0.999589i \(-0.490871\pi\)
0.0286745 + 0.999589i \(0.490871\pi\)
\(332\) −5.40642 −0.296716
\(333\) 0 0
\(334\) 19.9595 1.09213
\(335\) 4.93275 0.269505
\(336\) 0 0
\(337\) −32.3936 −1.76459 −0.882296 0.470696i \(-0.844003\pi\)
−0.882296 + 0.470696i \(0.844003\pi\)
\(338\) 23.9679 1.30368
\(339\) 0 0
\(340\) −6.99548 −0.379383
\(341\) 2.71979 0.147285
\(342\) 0 0
\(343\) −13.7183 −0.740719
\(344\) −0.622571 −0.0335668
\(345\) 0 0
\(346\) 47.0757 2.53081
\(347\) −0.837115 −0.0449387 −0.0224693 0.999748i \(-0.507153\pi\)
−0.0224693 + 0.999748i \(0.507153\pi\)
\(348\) 0 0
\(349\) −1.98491 −0.106250 −0.0531250 0.998588i \(-0.516918\pi\)
−0.0531250 + 0.998588i \(0.516918\pi\)
\(350\) −2.10952 −0.112759
\(351\) 0 0
\(352\) −7.89643 −0.420881
\(353\) −0.0707574 −0.00376604 −0.00188302 0.999998i \(-0.500599\pi\)
−0.00188302 + 0.999998i \(0.500599\pi\)
\(354\) 0 0
\(355\) −7.32299 −0.388664
\(356\) 16.7715 0.888889
\(357\) 0 0
\(358\) 39.6934 2.09786
\(359\) 23.4992 1.24024 0.620120 0.784507i \(-0.287084\pi\)
0.620120 + 0.784507i \(0.287084\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.8306 −0.779479
\(363\) 0 0
\(364\) 1.91559 0.100404
\(365\) 1.14496 0.0599301
\(366\) 0 0
\(367\) −16.4763 −0.860057 −0.430029 0.902815i \(-0.641497\pi\)
−0.430029 + 0.902815i \(0.641497\pi\)
\(368\) 22.2999 1.16246
\(369\) 0 0
\(370\) −1.84352 −0.0958402
\(371\) 12.9664 0.673181
\(372\) 0 0
\(373\) 1.92401 0.0996217 0.0498108 0.998759i \(-0.484138\pi\)
0.0498108 + 0.998759i \(0.484138\pi\)
\(374\) −7.23600 −0.374165
\(375\) 0 0
\(376\) 0.747330 0.0385406
\(377\) 4.19831 0.216224
\(378\) 0 0
\(379\) −29.8162 −1.53156 −0.765779 0.643104i \(-0.777646\pi\)
−0.765779 + 0.643104i \(0.777646\pi\)
\(380\) −1.91218 −0.0980926
\(381\) 0 0
\(382\) −8.34413 −0.426923
\(383\) 19.4719 0.994969 0.497484 0.867473i \(-0.334257\pi\)
0.497484 + 0.867473i \(0.334257\pi\)
\(384\) 0 0
\(385\) −1.06653 −0.0543556
\(386\) −15.2441 −0.775906
\(387\) 0 0
\(388\) 25.5123 1.29519
\(389\) 11.7829 0.597415 0.298708 0.954345i \(-0.403444\pi\)
0.298708 + 0.954345i \(0.403444\pi\)
\(390\) 0 0
\(391\) 19.5737 0.989882
\(392\) −1.01836 −0.0514350
\(393\) 0 0
\(394\) 36.4502 1.83633
\(395\) −6.71067 −0.337650
\(396\) 0 0
\(397\) 35.1177 1.76251 0.881254 0.472643i \(-0.156700\pi\)
0.881254 + 0.472643i \(0.156700\pi\)
\(398\) 42.8198 2.14636
\(399\) 0 0
\(400\) −4.16793 −0.208397
\(401\) −27.0103 −1.34883 −0.674414 0.738353i \(-0.735604\pi\)
−0.674414 + 0.738353i \(0.735604\pi\)
\(402\) 0 0
\(403\) −2.55468 −0.127257
\(404\) 1.77122 0.0881216
\(405\) 0 0
\(406\) −9.42884 −0.467946
\(407\) −0.932050 −0.0462000
\(408\) 0 0
\(409\) 33.0664 1.63503 0.817515 0.575907i \(-0.195351\pi\)
0.817515 + 0.575907i \(0.195351\pi\)
\(410\) −2.40943 −0.118994
\(411\) 0 0
\(412\) −19.9525 −0.982989
\(413\) 6.56570 0.323077
\(414\) 0 0
\(415\) −2.82737 −0.138790
\(416\) 7.41705 0.363651
\(417\) 0 0
\(418\) −1.97792 −0.0967433
\(419\) −7.98503 −0.390094 −0.195047 0.980794i \(-0.562486\pi\)
−0.195047 + 0.980794i \(0.562486\pi\)
\(420\) 0 0
\(421\) −28.8735 −1.40721 −0.703604 0.710592i \(-0.748427\pi\)
−0.703604 + 0.710592i \(0.748427\pi\)
\(422\) −47.3047 −2.30276
\(423\) 0 0
\(424\) 2.11185 0.102560
\(425\) −3.65839 −0.177458
\(426\) 0 0
\(427\) 2.27919 0.110298
\(428\) −25.9283 −1.25329
\(429\) 0 0
\(430\) 7.08891 0.341858
\(431\) −27.5088 −1.32505 −0.662527 0.749038i \(-0.730516\pi\)
−0.662527 + 0.749038i \(0.730516\pi\)
\(432\) 0 0
\(433\) −33.5624 −1.61291 −0.806453 0.591299i \(-0.798615\pi\)
−0.806453 + 0.591299i \(0.798615\pi\)
\(434\) 5.73746 0.275407
\(435\) 0 0
\(436\) 11.1946 0.536126
\(437\) 5.35035 0.255942
\(438\) 0 0
\(439\) 25.8608 1.23427 0.617134 0.786858i \(-0.288294\pi\)
0.617134 + 0.786858i \(0.288294\pi\)
\(440\) −0.173707 −0.00828118
\(441\) 0 0
\(442\) 6.79672 0.323287
\(443\) −2.34708 −0.111513 −0.0557566 0.998444i \(-0.517757\pi\)
−0.0557566 + 0.998444i \(0.517757\pi\)
\(444\) 0 0
\(445\) 8.77091 0.415781
\(446\) −8.28523 −0.392317
\(447\) 0 0
\(448\) −7.76722 −0.366966
\(449\) −18.4815 −0.872195 −0.436097 0.899899i \(-0.643640\pi\)
−0.436097 + 0.899899i \(0.643640\pi\)
\(450\) 0 0
\(451\) −1.21816 −0.0573611
\(452\) −16.6108 −0.781307
\(453\) 0 0
\(454\) −17.3032 −0.812079
\(455\) 1.00179 0.0469645
\(456\) 0 0
\(457\) 17.1365 0.801612 0.400806 0.916163i \(-0.368730\pi\)
0.400806 + 0.916163i \(0.368730\pi\)
\(458\) 33.2581 1.55405
\(459\) 0 0
\(460\) −10.2308 −0.477014
\(461\) 16.3205 0.760122 0.380061 0.924961i \(-0.375903\pi\)
0.380061 + 0.924961i \(0.375903\pi\)
\(462\) 0 0
\(463\) −18.2993 −0.850441 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(464\) −18.6292 −0.864840
\(465\) 0 0
\(466\) −14.6597 −0.679098
\(467\) −3.89977 −0.180460 −0.0902299 0.995921i \(-0.528760\pi\)
−0.0902299 + 0.995921i \(0.528760\pi\)
\(468\) 0 0
\(469\) 5.26094 0.242928
\(470\) −8.50948 −0.392513
\(471\) 0 0
\(472\) 1.06936 0.0492213
\(473\) 3.58402 0.164793
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) −7.46092 −0.341971
\(477\) 0 0
\(478\) −39.0296 −1.78517
\(479\) −16.4447 −0.751377 −0.375688 0.926746i \(-0.622594\pi\)
−0.375688 + 0.926746i \(0.622594\pi\)
\(480\) 0 0
\(481\) 0.875467 0.0399178
\(482\) −29.8655 −1.36034
\(483\) 0 0
\(484\) 1.91218 0.0869171
\(485\) 13.3420 0.605830
\(486\) 0 0
\(487\) −28.4329 −1.28842 −0.644209 0.764849i \(-0.722813\pi\)
−0.644209 + 0.764849i \(0.722813\pi\)
\(488\) 0.371214 0.0168041
\(489\) 0 0
\(490\) 11.5956 0.523835
\(491\) 6.25955 0.282490 0.141245 0.989975i \(-0.454890\pi\)
0.141245 + 0.989975i \(0.454890\pi\)
\(492\) 0 0
\(493\) −16.3517 −0.736445
\(494\) 1.85784 0.0835884
\(495\) 0 0
\(496\) 11.3359 0.508997
\(497\) −7.81022 −0.350336
\(498\) 0 0
\(499\) −10.4851 −0.469378 −0.234689 0.972070i \(-0.575407\pi\)
−0.234689 + 0.972070i \(0.575407\pi\)
\(500\) 1.91218 0.0855151
\(501\) 0 0
\(502\) 24.4981 1.09340
\(503\) −18.4277 −0.821651 −0.410825 0.911714i \(-0.634759\pi\)
−0.410825 + 0.911714i \(0.634759\pi\)
\(504\) 0 0
\(505\) 0.926286 0.0412192
\(506\) −10.5826 −0.470453
\(507\) 0 0
\(508\) −30.5078 −1.35357
\(509\) −14.0013 −0.620596 −0.310298 0.950639i \(-0.600429\pi\)
−0.310298 + 0.950639i \(0.600429\pi\)
\(510\) 0 0
\(511\) 1.22114 0.0540201
\(512\) −31.4638 −1.39052
\(513\) 0 0
\(514\) 52.5131 2.31625
\(515\) −10.4344 −0.459796
\(516\) 0 0
\(517\) −4.30223 −0.189212
\(518\) −1.96618 −0.0863891
\(519\) 0 0
\(520\) 0.163162 0.00715512
\(521\) 36.3013 1.59039 0.795195 0.606353i \(-0.207368\pi\)
0.795195 + 0.606353i \(0.207368\pi\)
\(522\) 0 0
\(523\) 23.2657 1.01734 0.508669 0.860962i \(-0.330138\pi\)
0.508669 + 0.860962i \(0.330138\pi\)
\(524\) −9.94358 −0.434387
\(525\) 0 0
\(526\) 29.6901 1.29455
\(527\) 9.95005 0.433431
\(528\) 0 0
\(529\) 5.62625 0.244620
\(530\) −24.0466 −1.04452
\(531\) 0 0
\(532\) −2.03940 −0.0884193
\(533\) 1.14421 0.0495613
\(534\) 0 0
\(535\) −13.5595 −0.586230
\(536\) 0.856854 0.0370105
\(537\) 0 0
\(538\) 23.4001 1.00885
\(539\) 5.86250 0.252516
\(540\) 0 0
\(541\) 30.6302 1.31689 0.658447 0.752627i \(-0.271214\pi\)
0.658447 + 0.752627i \(0.271214\pi\)
\(542\) 26.5346 1.13976
\(543\) 0 0
\(544\) −28.8882 −1.23857
\(545\) 5.85440 0.250775
\(546\) 0 0
\(547\) −14.9225 −0.638040 −0.319020 0.947748i \(-0.603354\pi\)
−0.319020 + 0.947748i \(0.603354\pi\)
\(548\) −14.4258 −0.616241
\(549\) 0 0
\(550\) 1.97792 0.0843389
\(551\) −4.46966 −0.190414
\(552\) 0 0
\(553\) −7.15716 −0.304354
\(554\) 1.02059 0.0433608
\(555\) 0 0
\(556\) 0.683222 0.0289751
\(557\) −8.54485 −0.362057 −0.181028 0.983478i \(-0.557943\pi\)
−0.181028 + 0.983478i \(0.557943\pi\)
\(558\) 0 0
\(559\) −3.36644 −0.142385
\(560\) −4.44525 −0.187846
\(561\) 0 0
\(562\) 43.3721 1.82954
\(563\) −17.7855 −0.749571 −0.374786 0.927111i \(-0.622284\pi\)
−0.374786 + 0.927111i \(0.622284\pi\)
\(564\) 0 0
\(565\) −8.68686 −0.365459
\(566\) −36.7939 −1.54656
\(567\) 0 0
\(568\) −1.27206 −0.0533743
\(569\) 29.7657 1.24785 0.623923 0.781486i \(-0.285538\pi\)
0.623923 + 0.781486i \(0.285538\pi\)
\(570\) 0 0
\(571\) 22.2184 0.929810 0.464905 0.885360i \(-0.346088\pi\)
0.464905 + 0.885360i \(0.346088\pi\)
\(572\) −1.79609 −0.0750983
\(573\) 0 0
\(574\) −2.56975 −0.107259
\(575\) −5.35035 −0.223125
\(576\) 0 0
\(577\) −12.0936 −0.503464 −0.251732 0.967797i \(-0.581000\pi\)
−0.251732 + 0.967797i \(0.581000\pi\)
\(578\) 7.15258 0.297508
\(579\) 0 0
\(580\) 8.54677 0.354885
\(581\) −3.01548 −0.125103
\(582\) 0 0
\(583\) −12.1575 −0.503512
\(584\) 0.198888 0.00823006
\(585\) 0 0
\(586\) −11.6161 −0.479856
\(587\) 14.3554 0.592510 0.296255 0.955109i \(-0.404262\pi\)
0.296255 + 0.955109i \(0.404262\pi\)
\(588\) 0 0
\(589\) 2.71979 0.112067
\(590\) −12.1763 −0.501290
\(591\) 0 0
\(592\) −3.88472 −0.159661
\(593\) −7.64570 −0.313971 −0.156986 0.987601i \(-0.550178\pi\)
−0.156986 + 0.987601i \(0.550178\pi\)
\(594\) 0 0
\(595\) −3.90180 −0.159958
\(596\) 20.5814 0.843049
\(597\) 0 0
\(598\) 9.94012 0.406482
\(599\) −22.5450 −0.921165 −0.460583 0.887617i \(-0.652359\pi\)
−0.460583 + 0.887617i \(0.652359\pi\)
\(600\) 0 0
\(601\) 8.27574 0.337574 0.168787 0.985653i \(-0.446015\pi\)
0.168787 + 0.985653i \(0.446015\pi\)
\(602\) 7.56057 0.308146
\(603\) 0 0
\(604\) 13.6525 0.555511
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −12.8552 −0.521778 −0.260889 0.965369i \(-0.584016\pi\)
−0.260889 + 0.965369i \(0.584016\pi\)
\(608\) −7.89643 −0.320243
\(609\) 0 0
\(610\) −4.22684 −0.171140
\(611\) 4.04105 0.163483
\(612\) 0 0
\(613\) −17.0191 −0.687394 −0.343697 0.939081i \(-0.611679\pi\)
−0.343697 + 0.939081i \(0.611679\pi\)
\(614\) −1.12062 −0.0452247
\(615\) 0 0
\(616\) −0.185265 −0.00746454
\(617\) −21.5265 −0.866625 −0.433313 0.901244i \(-0.642655\pi\)
−0.433313 + 0.901244i \(0.642655\pi\)
\(618\) 0 0
\(619\) 34.9987 1.40672 0.703359 0.710835i \(-0.251683\pi\)
0.703359 + 0.710835i \(0.251683\pi\)
\(620\) −5.20072 −0.208866
\(621\) 0 0
\(622\) −43.8536 −1.75837
\(623\) 9.35448 0.374779
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 59.9766 2.39715
\(627\) 0 0
\(628\) 24.3160 0.970315
\(629\) −3.40980 −0.135958
\(630\) 0 0
\(631\) 26.6365 1.06038 0.530191 0.847878i \(-0.322120\pi\)
0.530191 + 0.847878i \(0.322120\pi\)
\(632\) −1.16569 −0.0463688
\(633\) 0 0
\(634\) −27.0306 −1.07352
\(635\) −15.9545 −0.633135
\(636\) 0 0
\(637\) −5.50660 −0.218179
\(638\) 8.84063 0.350004
\(639\) 0 0
\(640\) −1.38832 −0.0548781
\(641\) 18.3904 0.726376 0.363188 0.931716i \(-0.381688\pi\)
0.363188 + 0.931716i \(0.381688\pi\)
\(642\) 0 0
\(643\) 10.1583 0.400603 0.200302 0.979734i \(-0.435808\pi\)
0.200302 + 0.979734i \(0.435808\pi\)
\(644\) −10.9115 −0.429974
\(645\) 0 0
\(646\) −7.23600 −0.284697
\(647\) −24.9283 −0.980034 −0.490017 0.871713i \(-0.663009\pi\)
−0.490017 + 0.871713i \(0.663009\pi\)
\(648\) 0 0
\(649\) −6.15610 −0.241648
\(650\) −1.85784 −0.0728707
\(651\) 0 0
\(652\) −8.51508 −0.333476
\(653\) −3.18346 −0.124579 −0.0622893 0.998058i \(-0.519840\pi\)
−0.0622893 + 0.998058i \(0.519840\pi\)
\(654\) 0 0
\(655\) −5.20014 −0.203186
\(656\) −5.07723 −0.198232
\(657\) 0 0
\(658\) −9.07566 −0.353806
\(659\) −44.5078 −1.73378 −0.866888 0.498503i \(-0.833883\pi\)
−0.866888 + 0.498503i \(0.833883\pi\)
\(660\) 0 0
\(661\) 5.31054 0.206556 0.103278 0.994653i \(-0.467067\pi\)
0.103278 + 0.994653i \(0.467067\pi\)
\(662\) −2.06371 −0.0802083
\(663\) 0 0
\(664\) −0.491134 −0.0190597
\(665\) −1.06653 −0.0413584
\(666\) 0 0
\(667\) −23.9142 −0.925963
\(668\) −19.2960 −0.746585
\(669\) 0 0
\(670\) −9.75659 −0.376930
\(671\) −2.13701 −0.0824983
\(672\) 0 0
\(673\) −27.9535 −1.07753 −0.538765 0.842456i \(-0.681109\pi\)
−0.538765 + 0.842456i \(0.681109\pi\)
\(674\) 64.0720 2.46796
\(675\) 0 0
\(676\) −23.1712 −0.891202
\(677\) 41.9276 1.61141 0.805704 0.592319i \(-0.201787\pi\)
0.805704 + 0.592319i \(0.201787\pi\)
\(678\) 0 0
\(679\) 14.2297 0.546087
\(680\) −0.635489 −0.0243699
\(681\) 0 0
\(682\) −5.37953 −0.205993
\(683\) −21.6527 −0.828516 −0.414258 0.910159i \(-0.635959\pi\)
−0.414258 + 0.910159i \(0.635959\pi\)
\(684\) 0 0
\(685\) −7.54419 −0.288249
\(686\) 27.1337 1.03597
\(687\) 0 0
\(688\) 14.9380 0.569504
\(689\) 11.4194 0.435045
\(690\) 0 0
\(691\) 45.5121 1.73136 0.865681 0.500597i \(-0.166886\pi\)
0.865681 + 0.500597i \(0.166886\pi\)
\(692\) −45.5109 −1.73007
\(693\) 0 0
\(694\) 1.65575 0.0628514
\(695\) 0.357301 0.0135532
\(696\) 0 0
\(697\) −4.45652 −0.168803
\(698\) 3.92601 0.148602
\(699\) 0 0
\(700\) 2.03940 0.0770822
\(701\) 27.3430 1.03273 0.516365 0.856369i \(-0.327285\pi\)
0.516365 + 0.856369i \(0.327285\pi\)
\(702\) 0 0
\(703\) −0.932050 −0.0351529
\(704\) 7.28267 0.274476
\(705\) 0 0
\(706\) 0.139953 0.00526719
\(707\) 0.987916 0.0371544
\(708\) 0 0
\(709\) 3.38760 0.127224 0.0636120 0.997975i \(-0.479738\pi\)
0.0636120 + 0.997975i \(0.479738\pi\)
\(710\) 14.4843 0.543586
\(711\) 0 0
\(712\) 1.52357 0.0570983
\(713\) 14.5518 0.544970
\(714\) 0 0
\(715\) −0.939291 −0.0351275
\(716\) −38.3740 −1.43410
\(717\) 0 0
\(718\) −46.4796 −1.73460
\(719\) −33.5021 −1.24942 −0.624709 0.780858i \(-0.714782\pi\)
−0.624709 + 0.780858i \(0.714782\pi\)
\(720\) 0 0
\(721\) −11.1287 −0.414454
\(722\) −1.97792 −0.0736106
\(723\) 0 0
\(724\) 14.3376 0.532854
\(725\) 4.46966 0.165999
\(726\) 0 0
\(727\) −45.5324 −1.68871 −0.844353 0.535788i \(-0.820015\pi\)
−0.844353 + 0.535788i \(0.820015\pi\)
\(728\) 0.174018 0.00644953
\(729\) 0 0
\(730\) −2.26465 −0.0838183
\(731\) 13.1117 0.484955
\(732\) 0 0
\(733\) −4.71433 −0.174128 −0.0870640 0.996203i \(-0.527748\pi\)
−0.0870640 + 0.996203i \(0.527748\pi\)
\(734\) 32.5889 1.20288
\(735\) 0 0
\(736\) −42.2487 −1.55731
\(737\) −4.93275 −0.181700
\(738\) 0 0
\(739\) 35.7265 1.31422 0.657110 0.753795i \(-0.271779\pi\)
0.657110 + 0.753795i \(0.271779\pi\)
\(740\) 1.78224 0.0655166
\(741\) 0 0
\(742\) −25.6465 −0.941513
\(743\) 26.9341 0.988118 0.494059 0.869428i \(-0.335513\pi\)
0.494059 + 0.869428i \(0.335513\pi\)
\(744\) 0 0
\(745\) 10.7634 0.394339
\(746\) −3.80555 −0.139331
\(747\) 0 0
\(748\) 6.99548 0.255780
\(749\) −14.4617 −0.528420
\(750\) 0 0
\(751\) −27.7767 −1.01358 −0.506792 0.862068i \(-0.669169\pi\)
−0.506792 + 0.862068i \(0.669169\pi\)
\(752\) −17.9314 −0.653891
\(753\) 0 0
\(754\) −8.30393 −0.302411
\(755\) 7.13976 0.259842
\(756\) 0 0
\(757\) −23.4521 −0.852380 −0.426190 0.904634i \(-0.640145\pi\)
−0.426190 + 0.904634i \(0.640145\pi\)
\(758\) 58.9742 2.14204
\(759\) 0 0
\(760\) −0.173707 −0.00630103
\(761\) 15.9940 0.579783 0.289892 0.957059i \(-0.406381\pi\)
0.289892 + 0.957059i \(0.406381\pi\)
\(762\) 0 0
\(763\) 6.24392 0.226045
\(764\) 8.06678 0.291846
\(765\) 0 0
\(766\) −38.5139 −1.39157
\(767\) 5.78237 0.208789
\(768\) 0 0
\(769\) 19.9815 0.720552 0.360276 0.932846i \(-0.382683\pi\)
0.360276 + 0.932846i \(0.382683\pi\)
\(770\) 2.10952 0.0760219
\(771\) 0 0
\(772\) 14.7374 0.530411
\(773\) −53.4366 −1.92198 −0.960991 0.276581i \(-0.910799\pi\)
−0.960991 + 0.276581i \(0.910799\pi\)
\(774\) 0 0
\(775\) −2.71979 −0.0976978
\(776\) 2.31761 0.0831973
\(777\) 0 0
\(778\) −23.3056 −0.835546
\(779\) −1.21816 −0.0436453
\(780\) 0 0
\(781\) 7.32299 0.262037
\(782\) −38.7152 −1.38445
\(783\) 0 0
\(784\) 24.4345 0.872662
\(785\) 12.7164 0.453868
\(786\) 0 0
\(787\) 6.90402 0.246102 0.123051 0.992400i \(-0.460732\pi\)
0.123051 + 0.992400i \(0.460732\pi\)
\(788\) −35.2386 −1.25532
\(789\) 0 0
\(790\) 13.2732 0.472239
\(791\) −9.26484 −0.329420
\(792\) 0 0
\(793\) 2.00727 0.0712804
\(794\) −69.4601 −2.46505
\(795\) 0 0
\(796\) −41.3965 −1.46726
\(797\) −28.5820 −1.01243 −0.506213 0.862409i \(-0.668955\pi\)
−0.506213 + 0.862409i \(0.668955\pi\)
\(798\) 0 0
\(799\) −15.7392 −0.556814
\(800\) 7.89643 0.279181
\(801\) 0 0
\(802\) 53.4242 1.88647
\(803\) −1.14496 −0.0404048
\(804\) 0 0
\(805\) −5.70633 −0.201122
\(806\) 5.05295 0.177983
\(807\) 0 0
\(808\) 0.160903 0.00566054
\(809\) −45.3303 −1.59373 −0.796864 0.604158i \(-0.793509\pi\)
−0.796864 + 0.604158i \(0.793509\pi\)
\(810\) 0 0
\(811\) 11.1668 0.392119 0.196060 0.980592i \(-0.437185\pi\)
0.196060 + 0.980592i \(0.437185\pi\)
\(812\) 9.11543 0.319889
\(813\) 0 0
\(814\) 1.84352 0.0646155
\(815\) −4.45308 −0.155985
\(816\) 0 0
\(817\) 3.58402 0.125389
\(818\) −65.4029 −2.28676
\(819\) 0 0
\(820\) 2.32935 0.0813443
\(821\) 23.2715 0.812180 0.406090 0.913833i \(-0.366892\pi\)
0.406090 + 0.913833i \(0.366892\pi\)
\(822\) 0 0
\(823\) −36.1529 −1.26021 −0.630105 0.776510i \(-0.716988\pi\)
−0.630105 + 0.776510i \(0.716988\pi\)
\(824\) −1.81254 −0.0631428
\(825\) 0 0
\(826\) −12.9864 −0.451856
\(827\) −10.7797 −0.374848 −0.187424 0.982279i \(-0.560014\pi\)
−0.187424 + 0.982279i \(0.560014\pi\)
\(828\) 0 0
\(829\) −42.2859 −1.46865 −0.734326 0.678797i \(-0.762501\pi\)
−0.734326 + 0.678797i \(0.762501\pi\)
\(830\) 5.59231 0.194112
\(831\) 0 0
\(832\) −6.84054 −0.237153
\(833\) 21.4473 0.743105
\(834\) 0 0
\(835\) −10.0911 −0.349218
\(836\) 1.91218 0.0661340
\(837\) 0 0
\(838\) 15.7938 0.545587
\(839\) −27.2267 −0.939971 −0.469986 0.882674i \(-0.655741\pi\)
−0.469986 + 0.882674i \(0.655741\pi\)
\(840\) 0 0
\(841\) −9.02217 −0.311109
\(842\) 57.1095 1.96812
\(843\) 0 0
\(844\) 45.7323 1.57417
\(845\) −12.1177 −0.416863
\(846\) 0 0
\(847\) 1.06653 0.0366466
\(848\) −50.6716 −1.74007
\(849\) 0 0
\(850\) 7.23600 0.248193
\(851\) −4.98680 −0.170945
\(852\) 0 0
\(853\) −11.3022 −0.386981 −0.193490 0.981102i \(-0.561981\pi\)
−0.193490 + 0.981102i \(0.561981\pi\)
\(854\) −4.50807 −0.154263
\(855\) 0 0
\(856\) −2.35539 −0.0805057
\(857\) −1.20558 −0.0411817 −0.0205908 0.999788i \(-0.506555\pi\)
−0.0205908 + 0.999788i \(0.506555\pi\)
\(858\) 0 0
\(859\) 28.1617 0.960865 0.480432 0.877032i \(-0.340480\pi\)
0.480432 + 0.877032i \(0.340480\pi\)
\(860\) −6.85328 −0.233695
\(861\) 0 0
\(862\) 54.4103 1.85322
\(863\) −8.48764 −0.288923 −0.144461 0.989510i \(-0.546145\pi\)
−0.144461 + 0.989510i \(0.546145\pi\)
\(864\) 0 0
\(865\) −23.8006 −0.809244
\(866\) 66.3838 2.25581
\(867\) 0 0
\(868\) −5.54675 −0.188269
\(869\) 6.71067 0.227644
\(870\) 0 0
\(871\) 4.63328 0.156993
\(872\) 1.01695 0.0344383
\(873\) 0 0
\(874\) −10.5826 −0.357961
\(875\) 1.06653 0.0360555
\(876\) 0 0
\(877\) 34.1439 1.15296 0.576478 0.817112i \(-0.304427\pi\)
0.576478 + 0.817112i \(0.304427\pi\)
\(878\) −51.1506 −1.72625
\(879\) 0 0
\(880\) 4.16793 0.140501
\(881\) 42.0557 1.41689 0.708446 0.705765i \(-0.249397\pi\)
0.708446 + 0.705765i \(0.249397\pi\)
\(882\) 0 0
\(883\) −12.9984 −0.437430 −0.218715 0.975789i \(-0.570187\pi\)
−0.218715 + 0.975789i \(0.570187\pi\)
\(884\) −6.57079 −0.221000
\(885\) 0 0
\(886\) 4.64235 0.155963
\(887\) −58.3513 −1.95925 −0.979623 0.200847i \(-0.935631\pi\)
−0.979623 + 0.200847i \(0.935631\pi\)
\(888\) 0 0
\(889\) −17.0160 −0.570699
\(890\) −17.3482 −0.581512
\(891\) 0 0
\(892\) 8.00983 0.268189
\(893\) −4.30223 −0.143969
\(894\) 0 0
\(895\) −20.0682 −0.670807
\(896\) −1.48069 −0.0494664
\(897\) 0 0
\(898\) 36.5549 1.21985
\(899\) −12.1565 −0.405443
\(900\) 0 0
\(901\) −44.4768 −1.48174
\(902\) 2.40943 0.0802254
\(903\) 0 0
\(904\) −1.50897 −0.0501876
\(905\) 7.49807 0.249244
\(906\) 0 0
\(907\) −58.8388 −1.95371 −0.976855 0.213904i \(-0.931382\pi\)
−0.976855 + 0.213904i \(0.931382\pi\)
\(908\) 16.7280 0.555139
\(909\) 0 0
\(910\) −1.98146 −0.0656846
\(911\) 23.1241 0.766137 0.383068 0.923720i \(-0.374867\pi\)
0.383068 + 0.923720i \(0.374867\pi\)
\(912\) 0 0
\(913\) 2.82737 0.0935721
\(914\) −33.8947 −1.12114
\(915\) 0 0
\(916\) −32.1526 −1.06235
\(917\) −5.54613 −0.183149
\(918\) 0 0
\(919\) −32.2210 −1.06287 −0.531437 0.847098i \(-0.678348\pi\)
−0.531437 + 0.847098i \(0.678348\pi\)
\(920\) −0.929396 −0.0306413
\(921\) 0 0
\(922\) −32.2807 −1.06311
\(923\) −6.87842 −0.226406
\(924\) 0 0
\(925\) 0.932050 0.0306456
\(926\) 36.1946 1.18943
\(927\) 0 0
\(928\) 35.2943 1.15859
\(929\) 0.527392 0.0173032 0.00865158 0.999963i \(-0.497246\pi\)
0.00865158 + 0.999963i \(0.497246\pi\)
\(930\) 0 0
\(931\) 5.86250 0.192136
\(932\) 14.1724 0.464233
\(933\) 0 0
\(934\) 7.71344 0.252392
\(935\) 3.65839 0.119642
\(936\) 0 0
\(937\) −41.3338 −1.35032 −0.675158 0.737673i \(-0.735925\pi\)
−0.675158 + 0.737673i \(0.735925\pi\)
\(938\) −10.4057 −0.339759
\(939\) 0 0
\(940\) 8.22663 0.268323
\(941\) −25.4253 −0.828840 −0.414420 0.910086i \(-0.636016\pi\)
−0.414420 + 0.910086i \(0.636016\pi\)
\(942\) 0 0
\(943\) −6.51761 −0.212242
\(944\) −25.6582 −0.835104
\(945\) 0 0
\(946\) −7.08891 −0.230480
\(947\) 13.9372 0.452897 0.226448 0.974023i \(-0.427289\pi\)
0.226448 + 0.974023i \(0.427289\pi\)
\(948\) 0 0
\(949\) 1.07545 0.0349107
\(950\) 1.97792 0.0641723
\(951\) 0 0
\(952\) −0.677771 −0.0219667
\(953\) −40.7458 −1.31989 −0.659943 0.751316i \(-0.729420\pi\)
−0.659943 + 0.751316i \(0.729420\pi\)
\(954\) 0 0
\(955\) 4.21863 0.136512
\(956\) 37.7323 1.22035
\(957\) 0 0
\(958\) 32.5263 1.05088
\(959\) −8.04614 −0.259824
\(960\) 0 0
\(961\) −23.6027 −0.761379
\(962\) −1.73160 −0.0558292
\(963\) 0 0
\(964\) 28.8728 0.929930
\(965\) 7.70714 0.248102
\(966\) 0 0
\(967\) −21.5806 −0.693985 −0.346993 0.937868i \(-0.612797\pi\)
−0.346993 + 0.937868i \(0.612797\pi\)
\(968\) 0.173707 0.00558317
\(969\) 0 0
\(970\) −26.3895 −0.847315
\(971\) 40.5213 1.30039 0.650194 0.759768i \(-0.274687\pi\)
0.650194 + 0.759768i \(0.274687\pi\)
\(972\) 0 0
\(973\) 0.381074 0.0122167
\(974\) 56.2381 1.80199
\(975\) 0 0
\(976\) −8.90691 −0.285103
\(977\) 51.7112 1.65439 0.827194 0.561917i \(-0.189936\pi\)
0.827194 + 0.561917i \(0.189936\pi\)
\(978\) 0 0
\(979\) −8.77091 −0.280319
\(980\) −11.2101 −0.358095
\(981\) 0 0
\(982\) −12.3809 −0.395090
\(983\) −39.5850 −1.26256 −0.631282 0.775553i \(-0.717471\pi\)
−0.631282 + 0.775553i \(0.717471\pi\)
\(984\) 0 0
\(985\) −18.4285 −0.587181
\(986\) 32.3425 1.02999
\(987\) 0 0
\(988\) −1.79609 −0.0571413
\(989\) 19.1758 0.609754
\(990\) 0 0
\(991\) 42.2901 1.34339 0.671694 0.740828i \(-0.265567\pi\)
0.671694 + 0.740828i \(0.265567\pi\)
\(992\) −21.4766 −0.681884
\(993\) 0 0
\(994\) 15.4480 0.489981
\(995\) −21.6489 −0.686315
\(996\) 0 0
\(997\) −40.6774 −1.28827 −0.644133 0.764914i \(-0.722782\pi\)
−0.644133 + 0.764914i \(0.722782\pi\)
\(998\) 20.7387 0.656473
\(999\) 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 9405.2.a.bd.1.2 7
3.2 odd 2 1045.2.a.h.1.6 7
15.14 odd 2 5225.2.a.m.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.h.1.6 7 3.2 odd 2
5225.2.a.m.1.2 7 15.14 odd 2
9405.2.a.bd.1.2 7 1.1 even 1 trivial