Properties

Label 9405.2.a.bd.1.5
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 1.5
Root \(-1.08185\) 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)
 
\(f(q)\) \(=\) \(q+1.08185 q^{2} -0.829591 q^{4} +1.00000 q^{5} +4.02863 q^{7} -3.06121 q^{8} +O(q^{10})\) \(q+1.08185 q^{2} -0.829591 q^{4} +1.00000 q^{5} +4.02863 q^{7} -3.06121 q^{8} +1.08185 q^{10} -1.00000 q^{11} -3.57009 q^{13} +4.35839 q^{14} -1.65260 q^{16} -2.13712 q^{17} -1.00000 q^{19} -0.829591 q^{20} -1.08185 q^{22} +1.52642 q^{23} +1.00000 q^{25} -3.86231 q^{26} -3.34212 q^{28} -0.640237 q^{29} -2.79975 q^{31} +4.33454 q^{32} -2.31206 q^{34} +4.02863 q^{35} +6.88152 q^{37} -1.08185 q^{38} -3.06121 q^{40} -2.11443 q^{41} -11.2566 q^{43} +0.829591 q^{44} +1.65136 q^{46} +5.49461 q^{47} +9.22988 q^{49} +1.08185 q^{50} +2.96171 q^{52} -10.3553 q^{53} -1.00000 q^{55} -12.3325 q^{56} -0.692643 q^{58} -10.5403 q^{59} +7.73205 q^{61} -3.02892 q^{62} +7.99454 q^{64} -3.57009 q^{65} +7.97406 q^{67} +1.77294 q^{68} +4.35839 q^{70} -7.60336 q^{71} +3.48422 q^{73} +7.44481 q^{74} +0.829591 q^{76} -4.02863 q^{77} +5.50316 q^{79} -1.65260 q^{80} -2.28750 q^{82} -15.4236 q^{83} -2.13712 q^{85} -12.1781 q^{86} +3.06121 q^{88} -17.3913 q^{89} -14.3826 q^{91} -1.26630 q^{92} +5.94436 q^{94} -1.00000 q^{95} -6.28029 q^{97} +9.98538 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 7 q^{4} + 7 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.08185 0.764987 0.382493 0.923958i \(-0.375065\pi\)
0.382493 + 0.923958i \(0.375065\pi\)
\(3\) 0 0
\(4\) −0.829591 −0.414795
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.02863 1.52268 0.761340 0.648353i \(-0.224542\pi\)
0.761340 + 0.648353i \(0.224542\pi\)
\(8\) −3.06121 −1.08230
\(9\) 0 0
\(10\) 1.08185 0.342112
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.57009 −0.990164 −0.495082 0.868846i \(-0.664862\pi\)
−0.495082 + 0.868846i \(0.664862\pi\)
\(14\) 4.35839 1.16483
\(15\) 0 0
\(16\) −1.65260 −0.413149
\(17\) −2.13712 −0.518328 −0.259164 0.965833i \(-0.583447\pi\)
−0.259164 + 0.965833i \(0.583447\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −0.829591 −0.185502
\(21\) 0 0
\(22\) −1.08185 −0.230652
\(23\) 1.52642 0.318280 0.159140 0.987256i \(-0.449128\pi\)
0.159140 + 0.987256i \(0.449128\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.86231 −0.757462
\(27\) 0 0
\(28\) −3.34212 −0.631600
\(29\) −0.640237 −0.118889 −0.0594445 0.998232i \(-0.518933\pi\)
−0.0594445 + 0.998232i \(0.518933\pi\)
\(30\) 0 0
\(31\) −2.79975 −0.502850 −0.251425 0.967877i \(-0.580899\pi\)
−0.251425 + 0.967877i \(0.580899\pi\)
\(32\) 4.33454 0.766246
\(33\) 0 0
\(34\) −2.31206 −0.396514
\(35\) 4.02863 0.680963
\(36\) 0 0
\(37\) 6.88152 1.13132 0.565658 0.824640i \(-0.308622\pi\)
0.565658 + 0.824640i \(0.308622\pi\)
\(38\) −1.08185 −0.175500
\(39\) 0 0
\(40\) −3.06121 −0.484019
\(41\) −2.11443 −0.330218 −0.165109 0.986275i \(-0.552798\pi\)
−0.165109 + 0.986275i \(0.552798\pi\)
\(42\) 0 0
\(43\) −11.2566 −1.71662 −0.858311 0.513129i \(-0.828486\pi\)
−0.858311 + 0.513129i \(0.828486\pi\)
\(44\) 0.829591 0.125066
\(45\) 0 0
\(46\) 1.65136 0.243480
\(47\) 5.49461 0.801471 0.400735 0.916194i \(-0.368755\pi\)
0.400735 + 0.916194i \(0.368755\pi\)
\(48\) 0 0
\(49\) 9.22988 1.31855
\(50\) 1.08185 0.152997
\(51\) 0 0
\(52\) 2.96171 0.410715
\(53\) −10.3553 −1.42241 −0.711203 0.702987i \(-0.751849\pi\)
−0.711203 + 0.702987i \(0.751849\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −12.3325 −1.64800
\(57\) 0 0
\(58\) −0.692643 −0.0909485
\(59\) −10.5403 −1.37224 −0.686118 0.727490i \(-0.740687\pi\)
−0.686118 + 0.727490i \(0.740687\pi\)
\(60\) 0 0
\(61\) 7.73205 0.989987 0.494994 0.868897i \(-0.335170\pi\)
0.494994 + 0.868897i \(0.335170\pi\)
\(62\) −3.02892 −0.384674
\(63\) 0 0
\(64\) 7.99454 0.999317
\(65\) −3.57009 −0.442815
\(66\) 0 0
\(67\) 7.97406 0.974187 0.487094 0.873350i \(-0.338057\pi\)
0.487094 + 0.873350i \(0.338057\pi\)
\(68\) 1.77294 0.215000
\(69\) 0 0
\(70\) 4.35839 0.520928
\(71\) −7.60336 −0.902353 −0.451176 0.892435i \(-0.648996\pi\)
−0.451176 + 0.892435i \(0.648996\pi\)
\(72\) 0 0
\(73\) 3.48422 0.407797 0.203899 0.978992i \(-0.434639\pi\)
0.203899 + 0.978992i \(0.434639\pi\)
\(74\) 7.44481 0.865441
\(75\) 0 0
\(76\) 0.829591 0.0951606
\(77\) −4.02863 −0.459105
\(78\) 0 0
\(79\) 5.50316 0.619154 0.309577 0.950874i \(-0.399813\pi\)
0.309577 + 0.950874i \(0.399813\pi\)
\(80\) −1.65260 −0.184766
\(81\) 0 0
\(82\) −2.28750 −0.252613
\(83\) −15.4236 −1.69296 −0.846479 0.532422i \(-0.821282\pi\)
−0.846479 + 0.532422i \(0.821282\pi\)
\(84\) 0 0
\(85\) −2.13712 −0.231803
\(86\) −12.1781 −1.31319
\(87\) 0 0
\(88\) 3.06121 0.326326
\(89\) −17.3913 −1.84347 −0.921735 0.387820i \(-0.873228\pi\)
−0.921735 + 0.387820i \(0.873228\pi\)
\(90\) 0 0
\(91\) −14.3826 −1.50770
\(92\) −1.26630 −0.132021
\(93\) 0 0
\(94\) 5.94436 0.613114
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −6.28029 −0.637666 −0.318833 0.947811i \(-0.603291\pi\)
−0.318833 + 0.947811i \(0.603291\pi\)
\(98\) 9.98538 1.00868
\(99\) 0 0
\(100\) −0.829591 −0.0829591
\(101\) −9.34301 −0.929664 −0.464832 0.885399i \(-0.653885\pi\)
−0.464832 + 0.885399i \(0.653885\pi\)
\(102\) 0 0
\(103\) −14.5265 −1.43133 −0.715667 0.698442i \(-0.753877\pi\)
−0.715667 + 0.698442i \(0.753877\pi\)
\(104\) 10.9288 1.07165
\(105\) 0 0
\(106\) −11.2029 −1.08812
\(107\) −11.4273 −1.10472 −0.552361 0.833605i \(-0.686273\pi\)
−0.552361 + 0.833605i \(0.686273\pi\)
\(108\) 0 0
\(109\) 1.86373 0.178513 0.0892564 0.996009i \(-0.471551\pi\)
0.0892564 + 0.996009i \(0.471551\pi\)
\(110\) −1.08185 −0.103151
\(111\) 0 0
\(112\) −6.65771 −0.629094
\(113\) 11.8891 1.11843 0.559214 0.829023i \(-0.311103\pi\)
0.559214 + 0.829023i \(0.311103\pi\)
\(114\) 0 0
\(115\) 1.52642 0.142339
\(116\) 0.531134 0.0493146
\(117\) 0 0
\(118\) −11.4031 −1.04974
\(119\) −8.60968 −0.789248
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.36495 0.757327
\(123\) 0 0
\(124\) 2.32265 0.208580
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.5195 −0.933457 −0.466729 0.884401i \(-0.654567\pi\)
−0.466729 + 0.884401i \(0.654567\pi\)
\(128\) −0.0201534 −0.00178133
\(129\) 0 0
\(130\) −3.86231 −0.338747
\(131\) 4.29351 0.375126 0.187563 0.982253i \(-0.439941\pi\)
0.187563 + 0.982253i \(0.439941\pi\)
\(132\) 0 0
\(133\) −4.02863 −0.349327
\(134\) 8.62678 0.745240
\(135\) 0 0
\(136\) 6.54217 0.560987
\(137\) 14.8142 1.26566 0.632831 0.774290i \(-0.281893\pi\)
0.632831 + 0.774290i \(0.281893\pi\)
\(138\) 0 0
\(139\) 0.911248 0.0772910 0.0386455 0.999253i \(-0.487696\pi\)
0.0386455 + 0.999253i \(0.487696\pi\)
\(140\) −3.34212 −0.282460
\(141\) 0 0
\(142\) −8.22573 −0.690288
\(143\) 3.57009 0.298546
\(144\) 0 0
\(145\) −0.640237 −0.0531688
\(146\) 3.76942 0.311959
\(147\) 0 0
\(148\) −5.70885 −0.469264
\(149\) 2.83838 0.232529 0.116264 0.993218i \(-0.462908\pi\)
0.116264 + 0.993218i \(0.462908\pi\)
\(150\) 0 0
\(151\) −0.0486666 −0.00396043 −0.00198022 0.999998i \(-0.500630\pi\)
−0.00198022 + 0.999998i \(0.500630\pi\)
\(152\) 3.06121 0.248297
\(153\) 0 0
\(154\) −4.35839 −0.351209
\(155\) −2.79975 −0.224881
\(156\) 0 0
\(157\) 4.43937 0.354301 0.177150 0.984184i \(-0.443312\pi\)
0.177150 + 0.984184i \(0.443312\pi\)
\(158\) 5.95362 0.473644
\(159\) 0 0
\(160\) 4.33454 0.342676
\(161\) 6.14937 0.484638
\(162\) 0 0
\(163\) 19.4777 1.52561 0.762807 0.646626i \(-0.223820\pi\)
0.762807 + 0.646626i \(0.223820\pi\)
\(164\) 1.75411 0.136973
\(165\) 0 0
\(166\) −16.6861 −1.29509
\(167\) −10.8523 −0.839778 −0.419889 0.907575i \(-0.637931\pi\)
−0.419889 + 0.907575i \(0.637931\pi\)
\(168\) 0 0
\(169\) −0.254480 −0.0195753
\(170\) −2.31206 −0.177327
\(171\) 0 0
\(172\) 9.33841 0.712047
\(173\) 1.46585 0.111447 0.0557234 0.998446i \(-0.482254\pi\)
0.0557234 + 0.998446i \(0.482254\pi\)
\(174\) 0 0
\(175\) 4.02863 0.304536
\(176\) 1.65260 0.124569
\(177\) 0 0
\(178\) −18.8148 −1.41023
\(179\) −17.5848 −1.31435 −0.657175 0.753738i \(-0.728249\pi\)
−0.657175 + 0.753738i \(0.728249\pi\)
\(180\) 0 0
\(181\) 1.09957 0.0817307 0.0408654 0.999165i \(-0.486989\pi\)
0.0408654 + 0.999165i \(0.486989\pi\)
\(182\) −15.5598 −1.15337
\(183\) 0 0
\(184\) −4.67267 −0.344474
\(185\) 6.88152 0.505940
\(186\) 0 0
\(187\) 2.13712 0.156282
\(188\) −4.55827 −0.332446
\(189\) 0 0
\(190\) −1.08185 −0.0784860
\(191\) −20.0312 −1.44941 −0.724703 0.689061i \(-0.758023\pi\)
−0.724703 + 0.689061i \(0.758023\pi\)
\(192\) 0 0
\(193\) −22.7060 −1.63441 −0.817205 0.576347i \(-0.804478\pi\)
−0.817205 + 0.576347i \(0.804478\pi\)
\(194\) −6.79436 −0.487806
\(195\) 0 0
\(196\) −7.65702 −0.546930
\(197\) 0.862697 0.0614646 0.0307323 0.999528i \(-0.490216\pi\)
0.0307323 + 0.999528i \(0.490216\pi\)
\(198\) 0 0
\(199\) 18.2787 1.29574 0.647872 0.761749i \(-0.275659\pi\)
0.647872 + 0.761749i \(0.275659\pi\)
\(200\) −3.06121 −0.216460
\(201\) 0 0
\(202\) −10.1078 −0.711181
\(203\) −2.57928 −0.181030
\(204\) 0 0
\(205\) −2.11443 −0.147678
\(206\) −15.7155 −1.09495
\(207\) 0 0
\(208\) 5.89992 0.409086
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 4.69164 0.322986 0.161493 0.986874i \(-0.448369\pi\)
0.161493 + 0.986874i \(0.448369\pi\)
\(212\) 8.59063 0.590007
\(213\) 0 0
\(214\) −12.3627 −0.845098
\(215\) −11.2566 −0.767697
\(216\) 0 0
\(217\) −11.2792 −0.765680
\(218\) 2.01628 0.136560
\(219\) 0 0
\(220\) 0.829591 0.0559310
\(221\) 7.62971 0.513230
\(222\) 0 0
\(223\) 0.856060 0.0573260 0.0286630 0.999589i \(-0.490875\pi\)
0.0286630 + 0.999589i \(0.490875\pi\)
\(224\) 17.4623 1.16675
\(225\) 0 0
\(226\) 12.8622 0.855583
\(227\) −4.53380 −0.300919 −0.150459 0.988616i \(-0.548075\pi\)
−0.150459 + 0.988616i \(0.548075\pi\)
\(228\) 0 0
\(229\) −28.5069 −1.88379 −0.941896 0.335905i \(-0.890958\pi\)
−0.941896 + 0.335905i \(0.890958\pi\)
\(230\) 1.65136 0.108887
\(231\) 0 0
\(232\) 1.95990 0.128674
\(233\) −5.61324 −0.367736 −0.183868 0.982951i \(-0.558862\pi\)
−0.183868 + 0.982951i \(0.558862\pi\)
\(234\) 0 0
\(235\) 5.49461 0.358429
\(236\) 8.74418 0.569197
\(237\) 0 0
\(238\) −9.31442 −0.603764
\(239\) 20.7190 1.34020 0.670101 0.742270i \(-0.266251\pi\)
0.670101 + 0.742270i \(0.266251\pi\)
\(240\) 0 0
\(241\) 11.4250 0.735952 0.367976 0.929835i \(-0.380051\pi\)
0.367976 + 0.929835i \(0.380051\pi\)
\(242\) 1.08185 0.0695442
\(243\) 0 0
\(244\) −6.41443 −0.410642
\(245\) 9.22988 0.589675
\(246\) 0 0
\(247\) 3.57009 0.227159
\(248\) 8.57061 0.544234
\(249\) 0 0
\(250\) 1.08185 0.0684225
\(251\) 10.3880 0.655685 0.327843 0.944732i \(-0.393678\pi\)
0.327843 + 0.944732i \(0.393678\pi\)
\(252\) 0 0
\(253\) −1.52642 −0.0959649
\(254\) −11.3806 −0.714082
\(255\) 0 0
\(256\) −16.0109 −1.00068
\(257\) −4.53951 −0.283167 −0.141583 0.989926i \(-0.545219\pi\)
−0.141583 + 0.989926i \(0.545219\pi\)
\(258\) 0 0
\(259\) 27.7231 1.72263
\(260\) 2.96171 0.183678
\(261\) 0 0
\(262\) 4.64495 0.286966
\(263\) 23.6937 1.46101 0.730507 0.682905i \(-0.239284\pi\)
0.730507 + 0.682905i \(0.239284\pi\)
\(264\) 0 0
\(265\) −10.3553 −0.636119
\(266\) −4.35839 −0.267230
\(267\) 0 0
\(268\) −6.61521 −0.404088
\(269\) −25.6963 −1.56673 −0.783365 0.621561i \(-0.786499\pi\)
−0.783365 + 0.621561i \(0.786499\pi\)
\(270\) 0 0
\(271\) 19.1460 1.16304 0.581519 0.813533i \(-0.302459\pi\)
0.581519 + 0.813533i \(0.302459\pi\)
\(272\) 3.53180 0.214147
\(273\) 0 0
\(274\) 16.0268 0.968214
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 12.6559 0.760421 0.380211 0.924900i \(-0.375851\pi\)
0.380211 + 0.924900i \(0.375851\pi\)
\(278\) 0.985838 0.0591266
\(279\) 0 0
\(280\) −12.3325 −0.737006
\(281\) −7.90646 −0.471660 −0.235830 0.971794i \(-0.575781\pi\)
−0.235830 + 0.971794i \(0.575781\pi\)
\(282\) 0 0
\(283\) 11.8035 0.701647 0.350823 0.936442i \(-0.385902\pi\)
0.350823 + 0.936442i \(0.385902\pi\)
\(284\) 6.30768 0.374292
\(285\) 0 0
\(286\) 3.86231 0.228383
\(287\) −8.51825 −0.502817
\(288\) 0 0
\(289\) −12.4327 −0.731336
\(290\) −0.692643 −0.0406734
\(291\) 0 0
\(292\) −2.89048 −0.169152
\(293\) −7.70878 −0.450352 −0.225176 0.974318i \(-0.572296\pi\)
−0.225176 + 0.974318i \(0.572296\pi\)
\(294\) 0 0
\(295\) −10.5403 −0.613683
\(296\) −21.0658 −1.22442
\(297\) 0 0
\(298\) 3.07071 0.177881
\(299\) −5.44944 −0.315149
\(300\) 0 0
\(301\) −45.3489 −2.61387
\(302\) −0.0526502 −0.00302968
\(303\) 0 0
\(304\) 1.65260 0.0947830
\(305\) 7.73205 0.442736
\(306\) 0 0
\(307\) −0.978502 −0.0558460 −0.0279230 0.999610i \(-0.508889\pi\)
−0.0279230 + 0.999610i \(0.508889\pi\)
\(308\) 3.34212 0.190435
\(309\) 0 0
\(310\) −3.02892 −0.172031
\(311\) 13.0195 0.738268 0.369134 0.929376i \(-0.379654\pi\)
0.369134 + 0.929376i \(0.379654\pi\)
\(312\) 0 0
\(313\) −35.0042 −1.97856 −0.989278 0.146047i \(-0.953345\pi\)
−0.989278 + 0.146047i \(0.953345\pi\)
\(314\) 4.80276 0.271035
\(315\) 0 0
\(316\) −4.56537 −0.256822
\(317\) −14.0969 −0.791763 −0.395882 0.918302i \(-0.629561\pi\)
−0.395882 + 0.918302i \(0.629561\pi\)
\(318\) 0 0
\(319\) 0.640237 0.0358464
\(320\) 7.99454 0.446908
\(321\) 0 0
\(322\) 6.65272 0.370742
\(323\) 2.13712 0.118913
\(324\) 0 0
\(325\) −3.57009 −0.198033
\(326\) 21.0721 1.16708
\(327\) 0 0
\(328\) 6.47270 0.357395
\(329\) 22.1357 1.22038
\(330\) 0 0
\(331\) 2.32171 0.127613 0.0638065 0.997962i \(-0.479676\pi\)
0.0638065 + 0.997962i \(0.479676\pi\)
\(332\) 12.7953 0.702231
\(333\) 0 0
\(334\) −11.7406 −0.642419
\(335\) 7.97406 0.435670
\(336\) 0 0
\(337\) 25.4277 1.38514 0.692568 0.721353i \(-0.256479\pi\)
0.692568 + 0.721353i \(0.256479\pi\)
\(338\) −0.275310 −0.0149749
\(339\) 0 0
\(340\) 1.77294 0.0961510
\(341\) 2.79975 0.151615
\(342\) 0 0
\(343\) 8.98335 0.485055
\(344\) 34.4589 1.85790
\(345\) 0 0
\(346\) 1.58584 0.0852553
\(347\) −0.658323 −0.0353406 −0.0176703 0.999844i \(-0.505625\pi\)
−0.0176703 + 0.999844i \(0.505625\pi\)
\(348\) 0 0
\(349\) −25.4110 −1.36022 −0.680110 0.733110i \(-0.738068\pi\)
−0.680110 + 0.733110i \(0.738068\pi\)
\(350\) 4.35839 0.232966
\(351\) 0 0
\(352\) −4.33454 −0.231032
\(353\) −18.3240 −0.975287 −0.487643 0.873043i \(-0.662143\pi\)
−0.487643 + 0.873043i \(0.662143\pi\)
\(354\) 0 0
\(355\) −7.60336 −0.403545
\(356\) 14.4276 0.764663
\(357\) 0 0
\(358\) −19.0242 −1.00546
\(359\) 3.16020 0.166789 0.0833944 0.996517i \(-0.473424\pi\)
0.0833944 + 0.996517i \(0.473424\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 1.18958 0.0625229
\(363\) 0 0
\(364\) 11.9316 0.625388
\(365\) 3.48422 0.182372
\(366\) 0 0
\(367\) −31.3432 −1.63610 −0.818050 0.575147i \(-0.804945\pi\)
−0.818050 + 0.575147i \(0.804945\pi\)
\(368\) −2.52255 −0.131497
\(369\) 0 0
\(370\) 7.44481 0.387037
\(371\) −41.7176 −2.16587
\(372\) 0 0
\(373\) 16.6133 0.860206 0.430103 0.902780i \(-0.358477\pi\)
0.430103 + 0.902780i \(0.358477\pi\)
\(374\) 2.31206 0.119554
\(375\) 0 0
\(376\) −16.8201 −0.867431
\(377\) 2.28570 0.117720
\(378\) 0 0
\(379\) −24.6701 −1.26722 −0.633610 0.773653i \(-0.718428\pi\)
−0.633610 + 0.773653i \(0.718428\pi\)
\(380\) 0.829591 0.0425571
\(381\) 0 0
\(382\) −21.6708 −1.10878
\(383\) −34.0603 −1.74040 −0.870200 0.492699i \(-0.836010\pi\)
−0.870200 + 0.492699i \(0.836010\pi\)
\(384\) 0 0
\(385\) −4.02863 −0.205318
\(386\) −24.5645 −1.25030
\(387\) 0 0
\(388\) 5.21007 0.264501
\(389\) −14.4879 −0.734567 −0.367284 0.930109i \(-0.619712\pi\)
−0.367284 + 0.930109i \(0.619712\pi\)
\(390\) 0 0
\(391\) −3.26214 −0.164973
\(392\) −28.2545 −1.42707
\(393\) 0 0
\(394\) 0.933312 0.0470196
\(395\) 5.50316 0.276894
\(396\) 0 0
\(397\) 14.7742 0.741496 0.370748 0.928734i \(-0.379101\pi\)
0.370748 + 0.928734i \(0.379101\pi\)
\(398\) 19.7749 0.991227
\(399\) 0 0
\(400\) −1.65260 −0.0826299
\(401\) −0.975341 −0.0487062 −0.0243531 0.999703i \(-0.507753\pi\)
−0.0243531 + 0.999703i \(0.507753\pi\)
\(402\) 0 0
\(403\) 9.99535 0.497904
\(404\) 7.75088 0.385620
\(405\) 0 0
\(406\) −2.79040 −0.138485
\(407\) −6.88152 −0.341104
\(408\) 0 0
\(409\) 11.6767 0.577377 0.288689 0.957423i \(-0.406781\pi\)
0.288689 + 0.957423i \(0.406781\pi\)
\(410\) −2.28750 −0.112972
\(411\) 0 0
\(412\) 12.0510 0.593711
\(413\) −42.4632 −2.08948
\(414\) 0 0
\(415\) −15.4236 −0.757114
\(416\) −15.4747 −0.758709
\(417\) 0 0
\(418\) 1.08185 0.0529152
\(419\) 9.20529 0.449708 0.224854 0.974392i \(-0.427810\pi\)
0.224854 + 0.974392i \(0.427810\pi\)
\(420\) 0 0
\(421\) −26.1447 −1.27421 −0.637106 0.770776i \(-0.719869\pi\)
−0.637106 + 0.770776i \(0.719869\pi\)
\(422\) 5.07567 0.247080
\(423\) 0 0
\(424\) 31.6996 1.53947
\(425\) −2.13712 −0.103666
\(426\) 0 0
\(427\) 31.1496 1.50743
\(428\) 9.48002 0.458234
\(429\) 0 0
\(430\) −12.1781 −0.587278
\(431\) −17.5078 −0.843320 −0.421660 0.906754i \(-0.638552\pi\)
−0.421660 + 0.906754i \(0.638552\pi\)
\(432\) 0 0
\(433\) 23.2988 1.11967 0.559834 0.828605i \(-0.310865\pi\)
0.559834 + 0.828605i \(0.310865\pi\)
\(434\) −12.2024 −0.585735
\(435\) 0 0
\(436\) −1.54613 −0.0740463
\(437\) −1.52642 −0.0730183
\(438\) 0 0
\(439\) 22.5592 1.07669 0.538345 0.842725i \(-0.319050\pi\)
0.538345 + 0.842725i \(0.319050\pi\)
\(440\) 3.06121 0.145937
\(441\) 0 0
\(442\) 8.25424 0.392614
\(443\) 2.78821 0.132472 0.0662360 0.997804i \(-0.478901\pi\)
0.0662360 + 0.997804i \(0.478901\pi\)
\(444\) 0 0
\(445\) −17.3913 −0.824425
\(446\) 0.926132 0.0438536
\(447\) 0 0
\(448\) 32.2071 1.52164
\(449\) 28.2226 1.33191 0.665953 0.745994i \(-0.268025\pi\)
0.665953 + 0.745994i \(0.268025\pi\)
\(450\) 0 0
\(451\) 2.11443 0.0995645
\(452\) −9.86305 −0.463919
\(453\) 0 0
\(454\) −4.90491 −0.230199
\(455\) −14.3826 −0.674265
\(456\) 0 0
\(457\) −37.0632 −1.73374 −0.866872 0.498530i \(-0.833873\pi\)
−0.866872 + 0.498530i \(0.833873\pi\)
\(458\) −30.8404 −1.44108
\(459\) 0 0
\(460\) −1.26630 −0.0590415
\(461\) −19.8588 −0.924917 −0.462459 0.886641i \(-0.653033\pi\)
−0.462459 + 0.886641i \(0.653033\pi\)
\(462\) 0 0
\(463\) −14.9995 −0.697084 −0.348542 0.937293i \(-0.613323\pi\)
−0.348542 + 0.937293i \(0.613323\pi\)
\(464\) 1.05805 0.0491189
\(465\) 0 0
\(466\) −6.07271 −0.281313
\(467\) −34.4593 −1.59459 −0.797293 0.603592i \(-0.793736\pi\)
−0.797293 + 0.603592i \(0.793736\pi\)
\(468\) 0 0
\(469\) 32.1246 1.48337
\(470\) 5.94436 0.274193
\(471\) 0 0
\(472\) 32.2662 1.48517
\(473\) 11.2566 0.517581
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 7.14251 0.327376
\(477\) 0 0
\(478\) 22.4150 1.02524
\(479\) −16.8146 −0.768280 −0.384140 0.923275i \(-0.625502\pi\)
−0.384140 + 0.923275i \(0.625502\pi\)
\(480\) 0 0
\(481\) −24.5676 −1.12019
\(482\) 12.3602 0.562993
\(483\) 0 0
\(484\) −0.829591 −0.0377087
\(485\) −6.28029 −0.285173
\(486\) 0 0
\(487\) 31.1763 1.41273 0.706366 0.707847i \(-0.250333\pi\)
0.706366 + 0.707847i \(0.250333\pi\)
\(488\) −23.6694 −1.07146
\(489\) 0 0
\(490\) 9.98538 0.451094
\(491\) −9.96801 −0.449850 −0.224925 0.974376i \(-0.572214\pi\)
−0.224925 + 0.974376i \(0.572214\pi\)
\(492\) 0 0
\(493\) 1.36826 0.0616235
\(494\) 3.86231 0.173774
\(495\) 0 0
\(496\) 4.62686 0.207752
\(497\) −30.6311 −1.37399
\(498\) 0 0
\(499\) 14.1464 0.633279 0.316640 0.948546i \(-0.397445\pi\)
0.316640 + 0.948546i \(0.397445\pi\)
\(500\) −0.829591 −0.0371004
\(501\) 0 0
\(502\) 11.2383 0.501590
\(503\) 22.9128 1.02163 0.510816 0.859690i \(-0.329343\pi\)
0.510816 + 0.859690i \(0.329343\pi\)
\(504\) 0 0
\(505\) −9.34301 −0.415759
\(506\) −1.65136 −0.0734119
\(507\) 0 0
\(508\) 8.72690 0.387194
\(509\) 32.1228 1.42382 0.711908 0.702273i \(-0.247831\pi\)
0.711908 + 0.702273i \(0.247831\pi\)
\(510\) 0 0
\(511\) 14.0366 0.620945
\(512\) −17.2811 −0.763726
\(513\) 0 0
\(514\) −4.91109 −0.216619
\(515\) −14.5265 −0.640112
\(516\) 0 0
\(517\) −5.49461 −0.241653
\(518\) 29.9924 1.31779
\(519\) 0 0
\(520\) 10.9288 0.479258
\(521\) −22.6591 −0.992713 −0.496356 0.868119i \(-0.665329\pi\)
−0.496356 + 0.868119i \(0.665329\pi\)
\(522\) 0 0
\(523\) −1.68836 −0.0738269 −0.0369134 0.999318i \(-0.511753\pi\)
−0.0369134 + 0.999318i \(0.511753\pi\)
\(524\) −3.56186 −0.155600
\(525\) 0 0
\(526\) 25.6331 1.11766
\(527\) 5.98341 0.260641
\(528\) 0 0
\(529\) −20.6701 −0.898698
\(530\) −11.2029 −0.486623
\(531\) 0 0
\(532\) 3.34212 0.144899
\(533\) 7.54869 0.326970
\(534\) 0 0
\(535\) −11.4273 −0.494047
\(536\) −24.4103 −1.05436
\(537\) 0 0
\(538\) −27.7997 −1.19853
\(539\) −9.22988 −0.397559
\(540\) 0 0
\(541\) 0.0300570 0.00129225 0.000646125 1.00000i \(-0.499794\pi\)
0.000646125 1.00000i \(0.499794\pi\)
\(542\) 20.7132 0.889708
\(543\) 0 0
\(544\) −9.26344 −0.397167
\(545\) 1.86373 0.0798334
\(546\) 0 0
\(547\) −18.4920 −0.790662 −0.395331 0.918539i \(-0.629370\pi\)
−0.395331 + 0.918539i \(0.629370\pi\)
\(548\) −12.2897 −0.524990
\(549\) 0 0
\(550\) −1.08185 −0.0461304
\(551\) 0.640237 0.0272750
\(552\) 0 0
\(553\) 22.1702 0.942773
\(554\) 13.6919 0.581712
\(555\) 0 0
\(556\) −0.755963 −0.0320600
\(557\) 40.9492 1.73507 0.867537 0.497372i \(-0.165702\pi\)
0.867537 + 0.497372i \(0.165702\pi\)
\(558\) 0 0
\(559\) 40.1872 1.69974
\(560\) −6.65771 −0.281340
\(561\) 0 0
\(562\) −8.55364 −0.360814
\(563\) 8.92339 0.376076 0.188038 0.982162i \(-0.439787\pi\)
0.188038 + 0.982162i \(0.439787\pi\)
\(564\) 0 0
\(565\) 11.8891 0.500176
\(566\) 12.7697 0.536750
\(567\) 0 0
\(568\) 23.2755 0.976616
\(569\) 47.1486 1.97657 0.988286 0.152613i \(-0.0487688\pi\)
0.988286 + 0.152613i \(0.0487688\pi\)
\(570\) 0 0
\(571\) 29.7980 1.24701 0.623504 0.781820i \(-0.285709\pi\)
0.623504 + 0.781820i \(0.285709\pi\)
\(572\) −2.96171 −0.123835
\(573\) 0 0
\(574\) −9.21551 −0.384648
\(575\) 1.52642 0.0636559
\(576\) 0 0
\(577\) −1.81737 −0.0756582 −0.0378291 0.999284i \(-0.512044\pi\)
−0.0378291 + 0.999284i \(0.512044\pi\)
\(578\) −13.4504 −0.559462
\(579\) 0 0
\(580\) 0.531134 0.0220542
\(581\) −62.1359 −2.57783
\(582\) 0 0
\(583\) 10.3553 0.428871
\(584\) −10.6659 −0.441359
\(585\) 0 0
\(586\) −8.33978 −0.344513
\(587\) 17.9910 0.742567 0.371284 0.928519i \(-0.378918\pi\)
0.371284 + 0.928519i \(0.378918\pi\)
\(588\) 0 0
\(589\) 2.79975 0.115362
\(590\) −11.4031 −0.469459
\(591\) 0 0
\(592\) −11.3724 −0.467402
\(593\) 28.7694 1.18142 0.590710 0.806884i \(-0.298848\pi\)
0.590710 + 0.806884i \(0.298848\pi\)
\(594\) 0 0
\(595\) −8.60968 −0.352962
\(596\) −2.35469 −0.0964518
\(597\) 0 0
\(598\) −5.89550 −0.241085
\(599\) 9.51516 0.388779 0.194389 0.980924i \(-0.437727\pi\)
0.194389 + 0.980924i \(0.437727\pi\)
\(600\) 0 0
\(601\) 1.39250 0.0568011 0.0284006 0.999597i \(-0.490959\pi\)
0.0284006 + 0.999597i \(0.490959\pi\)
\(602\) −49.0609 −1.99957
\(603\) 0 0
\(604\) 0.0403734 0.00164277
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −47.4420 −1.92561 −0.962805 0.270198i \(-0.912911\pi\)
−0.962805 + 0.270198i \(0.912911\pi\)
\(608\) −4.33454 −0.175789
\(609\) 0 0
\(610\) 8.36495 0.338687
\(611\) −19.6162 −0.793587
\(612\) 0 0
\(613\) −7.35538 −0.297081 −0.148540 0.988906i \(-0.547458\pi\)
−0.148540 + 0.988906i \(0.547458\pi\)
\(614\) −1.05860 −0.0427215
\(615\) 0 0
\(616\) 12.3325 0.496889
\(617\) 25.6134 1.03116 0.515578 0.856843i \(-0.327577\pi\)
0.515578 + 0.856843i \(0.327577\pi\)
\(618\) 0 0
\(619\) −15.6734 −0.629969 −0.314984 0.949097i \(-0.601999\pi\)
−0.314984 + 0.949097i \(0.601999\pi\)
\(620\) 2.32265 0.0932797
\(621\) 0 0
\(622\) 14.0852 0.564765
\(623\) −70.0630 −2.80702
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −37.8695 −1.51357
\(627\) 0 0
\(628\) −3.68286 −0.146962
\(629\) −14.7067 −0.586393
\(630\) 0 0
\(631\) −11.7074 −0.466065 −0.233032 0.972469i \(-0.574865\pi\)
−0.233032 + 0.972469i \(0.574865\pi\)
\(632\) −16.8463 −0.670110
\(633\) 0 0
\(634\) −15.2508 −0.605689
\(635\) −10.5195 −0.417455
\(636\) 0 0
\(637\) −32.9515 −1.30558
\(638\) 0.692643 0.0274220
\(639\) 0 0
\(640\) −0.0201534 −0.000796634 0
\(641\) −22.6311 −0.893874 −0.446937 0.894566i \(-0.647485\pi\)
−0.446937 + 0.894566i \(0.647485\pi\)
\(642\) 0 0
\(643\) −50.5556 −1.99372 −0.996860 0.0791848i \(-0.974768\pi\)
−0.996860 + 0.0791848i \(0.974768\pi\)
\(644\) −5.10146 −0.201026
\(645\) 0 0
\(646\) 2.31206 0.0909666
\(647\) 29.2983 1.15183 0.575917 0.817508i \(-0.304645\pi\)
0.575917 + 0.817508i \(0.304645\pi\)
\(648\) 0 0
\(649\) 10.5403 0.413745
\(650\) −3.86231 −0.151492
\(651\) 0 0
\(652\) −16.1586 −0.632818
\(653\) 4.84850 0.189736 0.0948682 0.995490i \(-0.469757\pi\)
0.0948682 + 0.995490i \(0.469757\pi\)
\(654\) 0 0
\(655\) 4.29351 0.167761
\(656\) 3.49430 0.136429
\(657\) 0 0
\(658\) 23.9477 0.933577
\(659\) −15.7032 −0.611708 −0.305854 0.952078i \(-0.598942\pi\)
−0.305854 + 0.952078i \(0.598942\pi\)
\(660\) 0 0
\(661\) −13.3369 −0.518744 −0.259372 0.965777i \(-0.583516\pi\)
−0.259372 + 0.965777i \(0.583516\pi\)
\(662\) 2.51176 0.0976222
\(663\) 0 0
\(664\) 47.2148 1.83229
\(665\) −4.02863 −0.156224
\(666\) 0 0
\(667\) −0.977267 −0.0378399
\(668\) 9.00299 0.348336
\(669\) 0 0
\(670\) 8.62678 0.333282
\(671\) −7.73205 −0.298492
\(672\) 0 0
\(673\) 44.6837 1.72243 0.861214 0.508242i \(-0.169704\pi\)
0.861214 + 0.508242i \(0.169704\pi\)
\(674\) 27.5091 1.05961
\(675\) 0 0
\(676\) 0.211114 0.00811976
\(677\) −11.2362 −0.431843 −0.215922 0.976411i \(-0.569276\pi\)
−0.215922 + 0.976411i \(0.569276\pi\)
\(678\) 0 0
\(679\) −25.3010 −0.970962
\(680\) 6.54217 0.250881
\(681\) 0 0
\(682\) 3.02892 0.115983
\(683\) 40.5671 1.55226 0.776128 0.630575i \(-0.217181\pi\)
0.776128 + 0.630575i \(0.217181\pi\)
\(684\) 0 0
\(685\) 14.8142 0.566021
\(686\) 9.71867 0.371061
\(687\) 0 0
\(688\) 18.6027 0.709222
\(689\) 36.9692 1.40841
\(690\) 0 0
\(691\) 9.10082 0.346212 0.173106 0.984903i \(-0.444620\pi\)
0.173106 + 0.984903i \(0.444620\pi\)
\(692\) −1.21606 −0.0462276
\(693\) 0 0
\(694\) −0.712209 −0.0270351
\(695\) 0.911248 0.0345656
\(696\) 0 0
\(697\) 4.51879 0.171161
\(698\) −27.4910 −1.04055
\(699\) 0 0
\(700\) −3.34212 −0.126320
\(701\) −24.5747 −0.928175 −0.464088 0.885789i \(-0.653618\pi\)
−0.464088 + 0.885789i \(0.653618\pi\)
\(702\) 0 0
\(703\) −6.88152 −0.259542
\(704\) −7.99454 −0.301306
\(705\) 0 0
\(706\) −19.8239 −0.746081
\(707\) −37.6396 −1.41558
\(708\) 0 0
\(709\) −6.77169 −0.254316 −0.127158 0.991882i \(-0.540586\pi\)
−0.127158 + 0.991882i \(0.540586\pi\)
\(710\) −8.22573 −0.308706
\(711\) 0 0
\(712\) 53.2382 1.99519
\(713\) −4.27358 −0.160047
\(714\) 0 0
\(715\) 3.57009 0.133514
\(716\) 14.5882 0.545186
\(717\) 0 0
\(718\) 3.41887 0.127591
\(719\) −36.1548 −1.34835 −0.674173 0.738574i \(-0.735500\pi\)
−0.674173 + 0.738574i \(0.735500\pi\)
\(720\) 0 0
\(721\) −58.5217 −2.17946
\(722\) 1.08185 0.0402625
\(723\) 0 0
\(724\) −0.912197 −0.0339015
\(725\) −0.640237 −0.0237778
\(726\) 0 0
\(727\) 10.4462 0.387427 0.193714 0.981058i \(-0.437947\pi\)
0.193714 + 0.981058i \(0.437947\pi\)
\(728\) 44.0280 1.63179
\(729\) 0 0
\(730\) 3.76942 0.139513
\(731\) 24.0568 0.889774
\(732\) 0 0
\(733\) −0.656143 −0.0242352 −0.0121176 0.999927i \(-0.503857\pi\)
−0.0121176 + 0.999927i \(0.503857\pi\)
\(734\) −33.9087 −1.25159
\(735\) 0 0
\(736\) 6.61631 0.243880
\(737\) −7.97406 −0.293728
\(738\) 0 0
\(739\) −42.0114 −1.54541 −0.772707 0.634762i \(-0.781098\pi\)
−0.772707 + 0.634762i \(0.781098\pi\)
\(740\) −5.70885 −0.209861
\(741\) 0 0
\(742\) −45.1323 −1.65686
\(743\) 6.16676 0.226236 0.113118 0.993582i \(-0.463916\pi\)
0.113118 + 0.993582i \(0.463916\pi\)
\(744\) 0 0
\(745\) 2.83838 0.103990
\(746\) 17.9732 0.658046
\(747\) 0 0
\(748\) −1.77294 −0.0648250
\(749\) −46.0366 −1.68214
\(750\) 0 0
\(751\) 37.9420 1.38452 0.692262 0.721646i \(-0.256614\pi\)
0.692262 + 0.721646i \(0.256614\pi\)
\(752\) −9.08037 −0.331127
\(753\) 0 0
\(754\) 2.47280 0.0900539
\(755\) −0.0486666 −0.00177116
\(756\) 0 0
\(757\) −18.3569 −0.667192 −0.333596 0.942716i \(-0.608262\pi\)
−0.333596 + 0.942716i \(0.608262\pi\)
\(758\) −26.6895 −0.969407
\(759\) 0 0
\(760\) 3.06121 0.111042
\(761\) −18.0726 −0.655130 −0.327565 0.944829i \(-0.606228\pi\)
−0.327565 + 0.944829i \(0.606228\pi\)
\(762\) 0 0
\(763\) 7.50828 0.271818
\(764\) 16.6177 0.601207
\(765\) 0 0
\(766\) −36.8483 −1.33138
\(767\) 37.6300 1.35874
\(768\) 0 0
\(769\) 35.1639 1.26804 0.634021 0.773316i \(-0.281403\pi\)
0.634021 + 0.773316i \(0.281403\pi\)
\(770\) −4.35839 −0.157066
\(771\) 0 0
\(772\) 18.8366 0.677946
\(773\) −15.9468 −0.573568 −0.286784 0.957995i \(-0.592586\pi\)
−0.286784 + 0.957995i \(0.592586\pi\)
\(774\) 0 0
\(775\) −2.79975 −0.100570
\(776\) 19.2252 0.690146
\(777\) 0 0
\(778\) −15.6738 −0.561934
\(779\) 2.11443 0.0757573
\(780\) 0 0
\(781\) 7.60336 0.272070
\(782\) −3.52916 −0.126202
\(783\) 0 0
\(784\) −15.2533 −0.544760
\(785\) 4.43937 0.158448
\(786\) 0 0
\(787\) 52.0777 1.85637 0.928185 0.372119i \(-0.121369\pi\)
0.928185 + 0.372119i \(0.121369\pi\)
\(788\) −0.715685 −0.0254952
\(789\) 0 0
\(790\) 5.95362 0.211820
\(791\) 47.8966 1.70301
\(792\) 0 0
\(793\) −27.6041 −0.980250
\(794\) 15.9835 0.567234
\(795\) 0 0
\(796\) −15.1639 −0.537469
\(797\) 41.6490 1.47528 0.737641 0.675193i \(-0.235940\pi\)
0.737641 + 0.675193i \(0.235940\pi\)
\(798\) 0 0
\(799\) −11.7426 −0.415425
\(800\) 4.33454 0.153249
\(801\) 0 0
\(802\) −1.05518 −0.0372596
\(803\) −3.48422 −0.122955
\(804\) 0 0
\(805\) 6.14937 0.216737
\(806\) 10.8135 0.380890
\(807\) 0 0
\(808\) 28.6009 1.00618
\(809\) −32.6440 −1.14770 −0.573851 0.818960i \(-0.694551\pi\)
−0.573851 + 0.818960i \(0.694551\pi\)
\(810\) 0 0
\(811\) −7.82261 −0.274689 −0.137345 0.990523i \(-0.543857\pi\)
−0.137345 + 0.990523i \(0.543857\pi\)
\(812\) 2.13975 0.0750903
\(813\) 0 0
\(814\) −7.44481 −0.260940
\(815\) 19.4777 0.682276
\(816\) 0 0
\(817\) 11.2566 0.393820
\(818\) 12.6325 0.441686
\(819\) 0 0
\(820\) 1.75411 0.0612562
\(821\) −38.0387 −1.32756 −0.663780 0.747927i \(-0.731049\pi\)
−0.663780 + 0.747927i \(0.731049\pi\)
\(822\) 0 0
\(823\) 39.7091 1.38417 0.692087 0.721815i \(-0.256692\pi\)
0.692087 + 0.721815i \(0.256692\pi\)
\(824\) 44.4685 1.54913
\(825\) 0 0
\(826\) −45.9390 −1.59842
\(827\) 38.3592 1.33388 0.666940 0.745111i \(-0.267604\pi\)
0.666940 + 0.745111i \(0.267604\pi\)
\(828\) 0 0
\(829\) 3.48941 0.121192 0.0605960 0.998162i \(-0.480700\pi\)
0.0605960 + 0.998162i \(0.480700\pi\)
\(830\) −16.6861 −0.579182
\(831\) 0 0
\(832\) −28.5412 −0.989488
\(833\) −19.7254 −0.683444
\(834\) 0 0
\(835\) −10.8523 −0.375560
\(836\) −0.829591 −0.0286920
\(837\) 0 0
\(838\) 9.95878 0.344020
\(839\) −39.6707 −1.36959 −0.684793 0.728738i \(-0.740107\pi\)
−0.684793 + 0.728738i \(0.740107\pi\)
\(840\) 0 0
\(841\) −28.5901 −0.985865
\(842\) −28.2847 −0.974756
\(843\) 0 0
\(844\) −3.89214 −0.133973
\(845\) −0.254480 −0.00875436
\(846\) 0 0
\(847\) 4.02863 0.138425
\(848\) 17.1131 0.587666
\(849\) 0 0
\(850\) −2.31206 −0.0793029
\(851\) 10.5041 0.360075
\(852\) 0 0
\(853\) 9.19215 0.314733 0.157367 0.987540i \(-0.449700\pi\)
0.157367 + 0.987540i \(0.449700\pi\)
\(854\) 33.6993 1.15317
\(855\) 0 0
\(856\) 34.9814 1.19564
\(857\) 40.0597 1.36841 0.684207 0.729288i \(-0.260149\pi\)
0.684207 + 0.729288i \(0.260149\pi\)
\(858\) 0 0
\(859\) 13.2980 0.453722 0.226861 0.973927i \(-0.427154\pi\)
0.226861 + 0.973927i \(0.427154\pi\)
\(860\) 9.33841 0.318437
\(861\) 0 0
\(862\) −18.9409 −0.645129
\(863\) 35.6269 1.21275 0.606377 0.795177i \(-0.292622\pi\)
0.606377 + 0.795177i \(0.292622\pi\)
\(864\) 0 0
\(865\) 1.46585 0.0498405
\(866\) 25.2059 0.856530
\(867\) 0 0
\(868\) 9.35709 0.317600
\(869\) −5.50316 −0.186682
\(870\) 0 0
\(871\) −28.4681 −0.964605
\(872\) −5.70526 −0.193204
\(873\) 0 0
\(874\) −1.65136 −0.0558581
\(875\) 4.02863 0.136193
\(876\) 0 0
\(877\) −36.2257 −1.22325 −0.611627 0.791146i \(-0.709485\pi\)
−0.611627 + 0.791146i \(0.709485\pi\)
\(878\) 24.4057 0.823654
\(879\) 0 0
\(880\) 1.65260 0.0557091
\(881\) 5.12526 0.172674 0.0863372 0.996266i \(-0.472484\pi\)
0.0863372 + 0.996266i \(0.472484\pi\)
\(882\) 0 0
\(883\) 38.1184 1.28279 0.641394 0.767212i \(-0.278357\pi\)
0.641394 + 0.767212i \(0.278357\pi\)
\(884\) −6.32954 −0.212885
\(885\) 0 0
\(886\) 3.01644 0.101339
\(887\) −31.1569 −1.04615 −0.523074 0.852287i \(-0.675215\pi\)
−0.523074 + 0.852287i \(0.675215\pi\)
\(888\) 0 0
\(889\) −42.3793 −1.42136
\(890\) −18.8148 −0.630674
\(891\) 0 0
\(892\) −0.710179 −0.0237786
\(893\) −5.49461 −0.183870
\(894\) 0 0
\(895\) −17.5848 −0.587795
\(896\) −0.0811907 −0.00271239
\(897\) 0 0
\(898\) 30.5327 1.01889
\(899\) 1.79250 0.0597833
\(900\) 0 0
\(901\) 22.1305 0.737273
\(902\) 2.28750 0.0761655
\(903\) 0 0
\(904\) −36.3948 −1.21047
\(905\) 1.09957 0.0365511
\(906\) 0 0
\(907\) −23.4382 −0.778253 −0.389126 0.921184i \(-0.627223\pi\)
−0.389126 + 0.921184i \(0.627223\pi\)
\(908\) 3.76120 0.124820
\(909\) 0 0
\(910\) −15.5598 −0.515804
\(911\) 32.7985 1.08666 0.543331 0.839519i \(-0.317163\pi\)
0.543331 + 0.839519i \(0.317163\pi\)
\(912\) 0 0
\(913\) 15.4236 0.510446
\(914\) −40.0970 −1.32629
\(915\) 0 0
\(916\) 23.6491 0.781388
\(917\) 17.2970 0.571196
\(918\) 0 0
\(919\) 11.6985 0.385898 0.192949 0.981209i \(-0.438195\pi\)
0.192949 + 0.981209i \(0.438195\pi\)
\(920\) −4.67267 −0.154053
\(921\) 0 0
\(922\) −21.4843 −0.707549
\(923\) 27.1447 0.893477
\(924\) 0 0
\(925\) 6.88152 0.226263
\(926\) −16.2272 −0.533260
\(927\) 0 0
\(928\) −2.77513 −0.0910982
\(929\) 4.68472 0.153701 0.0768503 0.997043i \(-0.475514\pi\)
0.0768503 + 0.997043i \(0.475514\pi\)
\(930\) 0 0
\(931\) −9.22988 −0.302497
\(932\) 4.65669 0.152535
\(933\) 0 0
\(934\) −37.2800 −1.21984
\(935\) 2.13712 0.0698914
\(936\) 0 0
\(937\) 2.22853 0.0728029 0.0364015 0.999337i \(-0.488410\pi\)
0.0364015 + 0.999337i \(0.488410\pi\)
\(938\) 34.7541 1.13476
\(939\) 0 0
\(940\) −4.55827 −0.148675
\(941\) −22.5997 −0.736730 −0.368365 0.929681i \(-0.620082\pi\)
−0.368365 + 0.929681i \(0.620082\pi\)
\(942\) 0 0
\(943\) −3.22750 −0.105102
\(944\) 17.4190 0.566939
\(945\) 0 0
\(946\) 12.1781 0.395943
\(947\) −49.0872 −1.59512 −0.797560 0.603239i \(-0.793876\pi\)
−0.797560 + 0.603239i \(0.793876\pi\)
\(948\) 0 0
\(949\) −12.4390 −0.403786
\(950\) −1.08185 −0.0351000
\(951\) 0 0
\(952\) 26.3560 0.854203
\(953\) −1.08440 −0.0351272 −0.0175636 0.999846i \(-0.505591\pi\)
−0.0175636 + 0.999846i \(0.505591\pi\)
\(954\) 0 0
\(955\) −20.0312 −0.648194
\(956\) −17.1883 −0.555910
\(957\) 0 0
\(958\) −18.1910 −0.587724
\(959\) 59.6809 1.92720
\(960\) 0 0
\(961\) −23.1614 −0.747142
\(962\) −26.5786 −0.856929
\(963\) 0 0
\(964\) −9.47811 −0.305269
\(965\) −22.7060 −0.730931
\(966\) 0 0
\(967\) −51.3313 −1.65070 −0.825351 0.564620i \(-0.809023\pi\)
−0.825351 + 0.564620i \(0.809023\pi\)
\(968\) −3.06121 −0.0983909
\(969\) 0 0
\(970\) −6.79436 −0.218154
\(971\) 22.9688 0.737104 0.368552 0.929607i \(-0.379854\pi\)
0.368552 + 0.929607i \(0.379854\pi\)
\(972\) 0 0
\(973\) 3.67108 0.117690
\(974\) 33.7282 1.08072
\(975\) 0 0
\(976\) −12.7780 −0.409013
\(977\) 21.2040 0.678376 0.339188 0.940719i \(-0.389848\pi\)
0.339188 + 0.940719i \(0.389848\pi\)
\(978\) 0 0
\(979\) 17.3913 0.555827
\(980\) −7.65702 −0.244594
\(981\) 0 0
\(982\) −10.7839 −0.344129
\(983\) −5.95713 −0.190003 −0.0950014 0.995477i \(-0.530286\pi\)
−0.0950014 + 0.995477i \(0.530286\pi\)
\(984\) 0 0
\(985\) 0.862697 0.0274878
\(986\) 1.48026 0.0471412
\(987\) 0 0
\(988\) −2.96171 −0.0942246
\(989\) −17.1823 −0.546366
\(990\) 0 0
\(991\) −4.63897 −0.147362 −0.0736809 0.997282i \(-0.523475\pi\)
−0.0736809 + 0.997282i \(0.523475\pi\)
\(992\) −12.1356 −0.385307
\(993\) 0 0
\(994\) −33.1384 −1.05109
\(995\) 18.2787 0.579475
\(996\) 0 0
\(997\) −30.8647 −0.977495 −0.488747 0.872425i \(-0.662546\pi\)
−0.488747 + 0.872425i \(0.662546\pi\)
\(998\) 15.3043 0.484450
\(999\) 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.1.5 7
3.2 odd 2 1045.2.a.h.1.3 7
15.14 odd 2 5225.2.a.m.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.h.1.3 7 3.2 odd 2
5225.2.a.m.1.5 7 15.14 odd 2
9405.2.a.bd.1.5 7 1.1 even 1 trivial