Properties

Label 3675.2.a.br.1.1
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $4$
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] = [3675,2,Mod(1,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.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: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.88404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 7x^{2} + 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.70285\) of defining polynomial
Character \(\chi\) \(=\) 3675.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.70285 q^{2} +1.00000 q^{3} +5.30542 q^{4} -2.70285 q^{6} -8.93406 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.70285 q^{2} +1.00000 q^{3} +5.30542 q^{4} -2.70285 q^{6} -8.93406 q^{8} +1.00000 q^{9} +2.82550 q^{11} +5.30542 q^{12} +4.47992 q^{13} +13.5366 q^{16} -4.23121 q^{17} -2.70285 q^{18} +2.10029 q^{19} -7.63692 q^{22} +6.61084 q^{23} -8.93406 q^{24} -12.1086 q^{26} +1.00000 q^{27} +4.00000 q^{29} +9.33150 q^{31} -18.7194 q^{32} +2.82550 q^{33} +11.4363 q^{34} +5.30542 q^{36} -4.20513 q^{37} -5.67677 q^{38} +4.47992 q^{39} +2.37963 q^{41} +3.13092 q^{43} +14.9905 q^{44} -17.8681 q^{46} -7.78534 q^{47} +13.5366 q^{48} -4.23121 q^{51} +23.7678 q^{52} -2.58021 q^{53} -2.70285 q^{54} +2.10029 q^{57} -10.8114 q^{58} -3.78534 q^{59} -5.00000 q^{61} -25.2217 q^{62} +23.5225 q^{64} -7.63692 q^{66} +3.30542 q^{67} -22.4483 q^{68} +6.61084 q^{69} -11.4223 q^{71} -8.93406 q^{72} -2.65100 q^{73} +11.3658 q^{74} +11.1429 q^{76} -12.1086 q^{78} +12.6369 q^{79} +1.00000 q^{81} -6.43179 q^{82} +3.02608 q^{83} -8.46242 q^{86} +4.00000 q^{87} -25.2432 q^{88} -8.23121 q^{89} +35.0733 q^{92} +9.33150 q^{93} +21.0426 q^{94} -18.7194 q^{96} +15.0165 q^{97} +2.82550 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 4 q^{3} + 7 q^{4} - q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 4 q^{3} + 7 q^{4} - q^{6} - 3 q^{8} + 4 q^{9} + 8 q^{11} + 7 q^{12} + 7 q^{13} + 17 q^{16} + 6 q^{17} - q^{18} + 3 q^{19} + 12 q^{22} - 2 q^{23} - 3 q^{24} - 19 q^{26} + 4 q^{27} + 16 q^{29} + 9 q^{31} - 17 q^{32} + 8 q^{33} + 14 q^{34} + 7 q^{36} - 8 q^{37} - 27 q^{38} + 7 q^{39} + 4 q^{41} - 5 q^{43} + 26 q^{44} - 6 q^{46} - 6 q^{47} + 17 q^{48} + 6 q^{51} + 35 q^{52} + 6 q^{53} - q^{54} + 3 q^{57} - 4 q^{58} + 10 q^{59} - 20 q^{61} - 44 q^{62} + 21 q^{64} + 12 q^{66} - q^{67} - 8 q^{68} - 2 q^{69} + 22 q^{71} - 3 q^{72} - 4 q^{73} - 21 q^{74} - 23 q^{76} - 19 q^{78} + 8 q^{79} + 4 q^{81} + 8 q^{82} - 2 q^{83} + 12 q^{86} + 16 q^{87} - 28 q^{88} - 10 q^{89} + 66 q^{92} + 9 q^{93} + 22 q^{94} - 17 q^{96} + 12 q^{97} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.70285 −1.91121 −0.955603 0.294657i \(-0.904795\pi\)
−0.955603 + 0.294657i \(0.904795\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.30542 2.65271
\(5\) 0 0
\(6\) −2.70285 −1.10344
\(7\) 0 0
\(8\) −8.93406 −3.15867
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.82550 0.851921 0.425960 0.904742i \(-0.359936\pi\)
0.425960 + 0.904742i \(0.359936\pi\)
\(12\) 5.30542 1.53154
\(13\) 4.47992 1.24251 0.621253 0.783610i \(-0.286624\pi\)
0.621253 + 0.783610i \(0.286624\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 13.5366 3.38416
\(17\) −4.23121 −1.02622 −0.513109 0.858323i \(-0.671506\pi\)
−0.513109 + 0.858323i \(0.671506\pi\)
\(18\) −2.70285 −0.637069
\(19\) 2.10029 0.481839 0.240920 0.970545i \(-0.422551\pi\)
0.240920 + 0.970545i \(0.422551\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −7.63692 −1.62820
\(23\) 6.61084 1.37845 0.689227 0.724545i \(-0.257950\pi\)
0.689227 + 0.724545i \(0.257950\pi\)
\(24\) −8.93406 −1.82366
\(25\) 0 0
\(26\) −12.1086 −2.37468
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 9.33150 1.67599 0.837993 0.545681i \(-0.183729\pi\)
0.837993 + 0.545681i \(0.183729\pi\)
\(32\) −18.7194 −3.30915
\(33\) 2.82550 0.491857
\(34\) 11.4363 1.96132
\(35\) 0 0
\(36\) 5.30542 0.884236
\(37\) −4.20513 −0.691319 −0.345659 0.938360i \(-0.612345\pi\)
−0.345659 + 0.938360i \(0.612345\pi\)
\(38\) −5.67677 −0.920894
\(39\) 4.47992 0.717361
\(40\) 0 0
\(41\) 2.37963 0.371635 0.185818 0.982584i \(-0.440507\pi\)
0.185818 + 0.982584i \(0.440507\pi\)
\(42\) 0 0
\(43\) 3.13092 0.477461 0.238730 0.971086i \(-0.423269\pi\)
0.238730 + 0.971086i \(0.423269\pi\)
\(44\) 14.9905 2.25990
\(45\) 0 0
\(46\) −17.8681 −2.63451
\(47\) −7.78534 −1.13561 −0.567804 0.823164i \(-0.692207\pi\)
−0.567804 + 0.823164i \(0.692207\pi\)
\(48\) 13.5366 1.95384
\(49\) 0 0
\(50\) 0 0
\(51\) −4.23121 −0.592488
\(52\) 23.7678 3.29601
\(53\) −2.58021 −0.354419 −0.177209 0.984173i \(-0.556707\pi\)
−0.177209 + 0.984173i \(0.556707\pi\)
\(54\) −2.70285 −0.367812
\(55\) 0 0
\(56\) 0 0
\(57\) 2.10029 0.278190
\(58\) −10.8114 −1.41961
\(59\) −3.78534 −0.492809 −0.246404 0.969167i \(-0.579249\pi\)
−0.246404 + 0.969167i \(0.579249\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) −25.2217 −3.20316
\(63\) 0 0
\(64\) 23.5225 2.94032
\(65\) 0 0
\(66\) −7.63692 −0.940039
\(67\) 3.30542 0.403821 0.201911 0.979404i \(-0.435285\pi\)
0.201911 + 0.979404i \(0.435285\pi\)
\(68\) −22.4483 −2.72226
\(69\) 6.61084 0.795851
\(70\) 0 0
\(71\) −11.4223 −1.35557 −0.677786 0.735259i \(-0.737060\pi\)
−0.677786 + 0.735259i \(0.737060\pi\)
\(72\) −8.93406 −1.05289
\(73\) −2.65100 −0.310276 −0.155138 0.987893i \(-0.549582\pi\)
−0.155138 + 0.987893i \(0.549582\pi\)
\(74\) 11.3658 1.32125
\(75\) 0 0
\(76\) 11.1429 1.27818
\(77\) 0 0
\(78\) −12.1086 −1.37102
\(79\) 12.6369 1.42176 0.710882 0.703311i \(-0.248296\pi\)
0.710882 + 0.703311i \(0.248296\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.43179 −0.710272
\(83\) 3.02608 0.332155 0.166078 0.986113i \(-0.446890\pi\)
0.166078 + 0.986113i \(0.446890\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.46242 −0.912526
\(87\) 4.00000 0.428845
\(88\) −25.2432 −2.69093
\(89\) −8.23121 −0.872506 −0.436253 0.899824i \(-0.643695\pi\)
−0.436253 + 0.899824i \(0.643695\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 35.0733 3.65664
\(93\) 9.33150 0.967631
\(94\) 21.0426 2.17038
\(95\) 0 0
\(96\) −18.7194 −1.91054
\(97\) 15.0165 1.52470 0.762350 0.647166i \(-0.224046\pi\)
0.762350 + 0.647166i \(0.224046\pi\)
\(98\) 0 0
\(99\) 2.82550 0.283974
\(100\) 0 0
\(101\) −2.23121 −0.222014 −0.111007 0.993820i \(-0.535408\pi\)
−0.111007 + 0.993820i \(0.535408\pi\)
\(102\) 11.4363 1.13237
\(103\) −7.84205 −0.772700 −0.386350 0.922352i \(-0.626264\pi\)
−0.386350 + 0.922352i \(0.626264\pi\)
\(104\) −40.0239 −3.92466
\(105\) 0 0
\(106\) 6.97392 0.677367
\(107\) 15.6369 1.51168 0.755839 0.654758i \(-0.227229\pi\)
0.755839 + 0.654758i \(0.227229\pi\)
\(108\) 5.30542 0.510514
\(109\) 3.07421 0.294456 0.147228 0.989103i \(-0.452965\pi\)
0.147228 + 0.989103i \(0.452965\pi\)
\(110\) 0 0
\(111\) −4.20513 −0.399133
\(112\) 0 0
\(113\) −0.348998 −0.0328310 −0.0164155 0.999865i \(-0.505225\pi\)
−0.0164155 + 0.999865i \(0.505225\pi\)
\(114\) −5.67677 −0.531679
\(115\) 0 0
\(116\) 21.2217 1.97038
\(117\) 4.47992 0.414169
\(118\) 10.2312 0.941859
\(119\) 0 0
\(120\) 0 0
\(121\) −3.01654 −0.274231
\(122\) 13.5143 1.22352
\(123\) 2.37963 0.214564
\(124\) 49.5075 4.44590
\(125\) 0 0
\(126\) 0 0
\(127\) 2.27137 0.201552 0.100776 0.994909i \(-0.467867\pi\)
0.100776 + 0.994909i \(0.467867\pi\)
\(128\) −26.1392 −2.31040
\(129\) 3.13092 0.275662
\(130\) 0 0
\(131\) 19.2217 1.67941 0.839703 0.543046i \(-0.182729\pi\)
0.839703 + 0.543046i \(0.182729\pi\)
\(132\) 14.9905 1.30475
\(133\) 0 0
\(134\) −8.93406 −0.771785
\(135\) 0 0
\(136\) 37.8019 3.24148
\(137\) −8.26184 −0.705857 −0.352928 0.935650i \(-0.614814\pi\)
−0.352928 + 0.935650i \(0.614814\pi\)
\(138\) −17.8681 −1.52104
\(139\) 0.985914 0.0836241 0.0418121 0.999125i \(-0.486687\pi\)
0.0418121 + 0.999125i \(0.486687\pi\)
\(140\) 0 0
\(141\) −7.78534 −0.655644
\(142\) 30.8727 2.59078
\(143\) 12.6580 1.05852
\(144\) 13.5366 1.12805
\(145\) 0 0
\(146\) 7.16527 0.593002
\(147\) 0 0
\(148\) −22.3100 −1.83387
\(149\) 7.82095 0.640717 0.320359 0.947296i \(-0.396197\pi\)
0.320359 + 0.947296i \(0.396197\pi\)
\(150\) 0 0
\(151\) −3.13433 −0.255068 −0.127534 0.991834i \(-0.540706\pi\)
−0.127534 + 0.991834i \(0.540706\pi\)
\(152\) −18.7641 −1.52197
\(153\) −4.23121 −0.342073
\(154\) 0 0
\(155\) 0 0
\(156\) 23.7678 1.90295
\(157\) −23.4790 −1.87383 −0.936913 0.349564i \(-0.886330\pi\)
−0.936913 + 0.349564i \(0.886330\pi\)
\(158\) −34.1557 −2.71728
\(159\) −2.58021 −0.204624
\(160\) 0 0
\(161\) 0 0
\(162\) −2.70285 −0.212356
\(163\) −12.4363 −0.974089 −0.487045 0.873377i \(-0.661925\pi\)
−0.487045 + 0.873377i \(0.661925\pi\)
\(164\) 12.6249 0.985841
\(165\) 0 0
\(166\) −8.17905 −0.634817
\(167\) 18.4483 1.42757 0.713787 0.700362i \(-0.246978\pi\)
0.713787 + 0.700362i \(0.246978\pi\)
\(168\) 0 0
\(169\) 7.06966 0.543820
\(170\) 0 0
\(171\) 2.10029 0.160613
\(172\) 16.6108 1.26656
\(173\) 7.65100 0.581695 0.290847 0.956769i \(-0.406063\pi\)
0.290847 + 0.956769i \(0.406063\pi\)
\(174\) −10.8114 −0.819611
\(175\) 0 0
\(176\) 38.2478 2.88303
\(177\) −3.78534 −0.284523
\(178\) 22.2478 1.66754
\(179\) −2.41026 −0.180151 −0.0900756 0.995935i \(-0.528711\pi\)
−0.0900756 + 0.995935i \(0.528711\pi\)
\(180\) 0 0
\(181\) −13.6970 −1.01809 −0.509046 0.860739i \(-0.670002\pi\)
−0.509046 + 0.860739i \(0.670002\pi\)
\(182\) 0 0
\(183\) −5.00000 −0.369611
\(184\) −59.0616 −4.35408
\(185\) 0 0
\(186\) −25.2217 −1.84934
\(187\) −11.9553 −0.874257
\(188\) −41.3045 −3.01244
\(189\) 0 0
\(190\) 0 0
\(191\) −12.1343 −0.878010 −0.439005 0.898485i \(-0.644669\pi\)
−0.439005 + 0.898485i \(0.644669\pi\)
\(192\) 23.5225 1.69759
\(193\) 11.9258 0.858437 0.429219 0.903201i \(-0.358789\pi\)
0.429219 + 0.903201i \(0.358789\pi\)
\(194\) −40.5875 −2.91401
\(195\) 0 0
\(196\) 0 0
\(197\) −5.50258 −0.392043 −0.196021 0.980600i \(-0.562802\pi\)
−0.196021 + 0.980600i \(0.562802\pi\)
\(198\) −7.63692 −0.542732
\(199\) −11.3281 −0.803027 −0.401513 0.915853i \(-0.631516\pi\)
−0.401513 + 0.915853i \(0.631516\pi\)
\(200\) 0 0
\(201\) 3.30542 0.233146
\(202\) 6.03063 0.424314
\(203\) 0 0
\(204\) −22.4483 −1.57170
\(205\) 0 0
\(206\) 21.1959 1.47679
\(207\) 6.61084 0.459485
\(208\) 60.6430 4.20483
\(209\) 5.93437 0.410489
\(210\) 0 0
\(211\) 8.82095 0.607259 0.303630 0.952790i \(-0.401801\pi\)
0.303630 + 0.952790i \(0.401801\pi\)
\(212\) −13.6891 −0.940170
\(213\) −11.4223 −0.782640
\(214\) −42.2643 −2.88913
\(215\) 0 0
\(216\) −8.93406 −0.607886
\(217\) 0 0
\(218\) −8.30914 −0.562766
\(219\) −2.65100 −0.179138
\(220\) 0 0
\(221\) −18.9555 −1.27508
\(222\) 11.3658 0.762826
\(223\) 26.4885 1.77380 0.886900 0.461961i \(-0.152854\pi\)
0.886900 + 0.461961i \(0.152854\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.943291 0.0627468
\(227\) −17.7712 −1.17952 −0.589760 0.807579i \(-0.700778\pi\)
−0.589760 + 0.807579i \(0.700778\pi\)
\(228\) 11.1429 0.737957
\(229\) 18.6449 1.23209 0.616044 0.787712i \(-0.288734\pi\)
0.616044 + 0.787712i \(0.288734\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −35.7362 −2.34620
\(233\) −9.76879 −0.639975 −0.319987 0.947422i \(-0.603679\pi\)
−0.319987 + 0.947422i \(0.603679\pi\)
\(234\) −12.1086 −0.791561
\(235\) 0 0
\(236\) −20.0828 −1.30728
\(237\) 12.6369 0.820856
\(238\) 0 0
\(239\) −6.20058 −0.401082 −0.200541 0.979685i \(-0.564270\pi\)
−0.200541 + 0.979685i \(0.564270\pi\)
\(240\) 0 0
\(241\) −20.2382 −1.30366 −0.651829 0.758366i \(-0.725998\pi\)
−0.651829 + 0.758366i \(0.725998\pi\)
\(242\) 8.15328 0.524113
\(243\) 1.00000 0.0641500
\(244\) −26.5271 −1.69822
\(245\) 0 0
\(246\) −6.43179 −0.410076
\(247\) 9.40912 0.598688
\(248\) −83.3682 −5.29388
\(249\) 3.02608 0.191770
\(250\) 0 0
\(251\) −6.21466 −0.392266 −0.196133 0.980577i \(-0.562838\pi\)
−0.196133 + 0.980577i \(0.562838\pi\)
\(252\) 0 0
\(253\) 18.6789 1.17433
\(254\) −6.13919 −0.385207
\(255\) 0 0
\(256\) 23.6053 1.47533
\(257\) 5.65346 0.352653 0.176327 0.984332i \(-0.443579\pi\)
0.176327 + 0.984332i \(0.443579\pi\)
\(258\) −8.46242 −0.526847
\(259\) 0 0
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) −51.9534 −3.20969
\(263\) 25.0120 1.54231 0.771153 0.636650i \(-0.219680\pi\)
0.771153 + 0.636650i \(0.219680\pi\)
\(264\) −25.2432 −1.55361
\(265\) 0 0
\(266\) 0 0
\(267\) −8.23121 −0.503742
\(268\) 17.5366 1.07122
\(269\) −7.88221 −0.480587 −0.240293 0.970700i \(-0.577244\pi\)
−0.240293 + 0.970700i \(0.577244\pi\)
\(270\) 0 0
\(271\) 4.74271 0.288099 0.144050 0.989570i \(-0.453988\pi\)
0.144050 + 0.989570i \(0.453988\pi\)
\(272\) −57.2763 −3.47289
\(273\) 0 0
\(274\) 22.3305 1.34904
\(275\) 0 0
\(276\) 35.0733 2.11116
\(277\) 7.73721 0.464884 0.232442 0.972610i \(-0.425328\pi\)
0.232442 + 0.972610i \(0.425328\pi\)
\(278\) −2.66478 −0.159823
\(279\) 9.33150 0.558662
\(280\) 0 0
\(281\) 2.11779 0.126337 0.0631684 0.998003i \(-0.479879\pi\)
0.0631684 + 0.998003i \(0.479879\pi\)
\(282\) 21.0426 1.25307
\(283\) 4.39774 0.261419 0.130709 0.991421i \(-0.458275\pi\)
0.130709 + 0.991421i \(0.458275\pi\)
\(284\) −60.5998 −3.59594
\(285\) 0 0
\(286\) −34.2128 −2.02304
\(287\) 0 0
\(288\) −18.7194 −1.10305
\(289\) 0.903125 0.0531250
\(290\) 0 0
\(291\) 15.0165 0.880285
\(292\) −14.0647 −0.823073
\(293\) −14.2006 −0.829607 −0.414803 0.909911i \(-0.636150\pi\)
−0.414803 + 0.909911i \(0.636150\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 37.5689 2.18365
\(297\) 2.82550 0.163952
\(298\) −21.1389 −1.22454
\(299\) 29.6160 1.71274
\(300\) 0 0
\(301\) 0 0
\(302\) 8.47165 0.487488
\(303\) −2.23121 −0.128180
\(304\) 28.4308 1.63062
\(305\) 0 0
\(306\) 11.4363 0.653772
\(307\) −13.9423 −0.795731 −0.397866 0.917444i \(-0.630249\pi\)
−0.397866 + 0.917444i \(0.630249\pi\)
\(308\) 0 0
\(309\) −7.84205 −0.446118
\(310\) 0 0
\(311\) −3.79942 −0.215445 −0.107723 0.994181i \(-0.534356\pi\)
−0.107723 + 0.994181i \(0.534356\pi\)
\(312\) −40.0239 −2.26590
\(313\) 27.8976 1.57687 0.788433 0.615120i \(-0.210893\pi\)
0.788433 + 0.615120i \(0.210893\pi\)
\(314\) 63.4602 3.58127
\(315\) 0 0
\(316\) 67.0441 3.77153
\(317\) 34.4955 1.93746 0.968730 0.248116i \(-0.0798115\pi\)
0.968730 + 0.248116i \(0.0798115\pi\)
\(318\) 6.97392 0.391078
\(319\) 11.3020 0.632791
\(320\) 0 0
\(321\) 15.6369 0.872768
\(322\) 0 0
\(323\) −8.88676 −0.494473
\(324\) 5.30542 0.294745
\(325\) 0 0
\(326\) 33.6136 1.86169
\(327\) 3.07421 0.170004
\(328\) −21.2597 −1.17387
\(329\) 0 0
\(330\) 0 0
\(331\) −29.8942 −1.64313 −0.821567 0.570112i \(-0.806900\pi\)
−0.821567 + 0.570112i \(0.806900\pi\)
\(332\) 16.0546 0.881112
\(333\) −4.20513 −0.230440
\(334\) −49.8631 −2.72839
\(335\) 0 0
\(336\) 0 0
\(337\) 19.3480 1.05395 0.526977 0.849879i \(-0.323325\pi\)
0.526977 + 0.849879i \(0.323325\pi\)
\(338\) −19.1083 −1.03935
\(339\) −0.348998 −0.0189550
\(340\) 0 0
\(341\) 26.3662 1.42781
\(342\) −5.67677 −0.306965
\(343\) 0 0
\(344\) −27.9718 −1.50814
\(345\) 0 0
\(346\) −20.6795 −1.11174
\(347\) 26.2999 1.41185 0.705927 0.708285i \(-0.250531\pi\)
0.705927 + 0.708285i \(0.250531\pi\)
\(348\) 21.2217 1.13760
\(349\) −14.9083 −0.798022 −0.399011 0.916946i \(-0.630647\pi\)
−0.399011 + 0.916946i \(0.630647\pi\)
\(350\) 0 0
\(351\) 4.47992 0.239120
\(352\) −52.8917 −2.81914
\(353\) −15.8210 −0.842064 −0.421032 0.907046i \(-0.638332\pi\)
−0.421032 + 0.907046i \(0.638332\pi\)
\(354\) 10.2312 0.543783
\(355\) 0 0
\(356\) −43.6700 −2.31451
\(357\) 0 0
\(358\) 6.51458 0.344306
\(359\) 16.6771 0.880183 0.440091 0.897953i \(-0.354946\pi\)
0.440091 + 0.897953i \(0.354946\pi\)
\(360\) 0 0
\(361\) −14.5888 −0.767831
\(362\) 37.0211 1.94579
\(363\) −3.01654 −0.158328
\(364\) 0 0
\(365\) 0 0
\(366\) 13.5143 0.706402
\(367\) 28.1876 1.47138 0.735691 0.677317i \(-0.236858\pi\)
0.735691 + 0.677317i \(0.236858\pi\)
\(368\) 89.4884 4.66491
\(369\) 2.37963 0.123878
\(370\) 0 0
\(371\) 0 0
\(372\) 49.5075 2.56684
\(373\) 23.9990 1.24262 0.621312 0.783564i \(-0.286600\pi\)
0.621312 + 0.783564i \(0.286600\pi\)
\(374\) 32.3134 1.67089
\(375\) 0 0
\(376\) 69.5547 3.58701
\(377\) 17.9197 0.922910
\(378\) 0 0
\(379\) 13.8550 0.711683 0.355842 0.934546i \(-0.384194\pi\)
0.355842 + 0.934546i \(0.384194\pi\)
\(380\) 0 0
\(381\) 2.27137 0.116366
\(382\) 32.7973 1.67806
\(383\) −30.7764 −1.57260 −0.786301 0.617844i \(-0.788006\pi\)
−0.786301 + 0.617844i \(0.788006\pi\)
\(384\) −26.1392 −1.33391
\(385\) 0 0
\(386\) −32.2337 −1.64065
\(387\) 3.13092 0.159154
\(388\) 79.6690 4.04458
\(389\) 38.4127 1.94760 0.973801 0.227402i \(-0.0730231\pi\)
0.973801 + 0.227402i \(0.0730231\pi\)
\(390\) 0 0
\(391\) −27.9718 −1.41460
\(392\) 0 0
\(393\) 19.2217 0.969605
\(394\) 14.8727 0.749275
\(395\) 0 0
\(396\) 14.9905 0.753299
\(397\) −33.4615 −1.67938 −0.839691 0.543064i \(-0.817264\pi\)
−0.839691 + 0.543064i \(0.817264\pi\)
\(398\) 30.6182 1.53475
\(399\) 0 0
\(400\) 0 0
\(401\) 14.8420 0.741176 0.370588 0.928797i \(-0.379156\pi\)
0.370588 + 0.928797i \(0.379156\pi\)
\(402\) −8.93406 −0.445591
\(403\) 41.8043 2.08242
\(404\) −11.8375 −0.588937
\(405\) 0 0
\(406\) 0 0
\(407\) −11.8816 −0.588949
\(408\) 37.8019 1.87147
\(409\) 8.44492 0.417574 0.208787 0.977961i \(-0.433048\pi\)
0.208787 + 0.977961i \(0.433048\pi\)
\(410\) 0 0
\(411\) −8.26184 −0.407526
\(412\) −41.6053 −2.04975
\(413\) 0 0
\(414\) −17.8681 −0.878170
\(415\) 0 0
\(416\) −83.8614 −4.11164
\(417\) 0.985914 0.0482804
\(418\) −16.0397 −0.784529
\(419\) −18.8427 −0.920524 −0.460262 0.887783i \(-0.652245\pi\)
−0.460262 + 0.887783i \(0.652245\pi\)
\(420\) 0 0
\(421\) −9.09687 −0.443355 −0.221677 0.975120i \(-0.571153\pi\)
−0.221677 + 0.975120i \(0.571153\pi\)
\(422\) −23.8417 −1.16060
\(423\) −7.78534 −0.378536
\(424\) 23.0517 1.11949
\(425\) 0 0
\(426\) 30.8727 1.49579
\(427\) 0 0
\(428\) 82.9604 4.01004
\(429\) 12.6580 0.611135
\(430\) 0 0
\(431\) −4.95983 −0.238907 −0.119453 0.992840i \(-0.538114\pi\)
−0.119453 + 0.992840i \(0.538114\pi\)
\(432\) 13.5366 0.651281
\(433\) −2.61696 −0.125763 −0.0628815 0.998021i \(-0.520029\pi\)
−0.0628815 + 0.998021i \(0.520029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 16.3100 0.781106
\(437\) 13.8847 0.664194
\(438\) 7.16527 0.342370
\(439\) −26.5050 −1.26502 −0.632508 0.774554i \(-0.717975\pi\)
−0.632508 + 0.774554i \(0.717975\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 51.2338 2.43695
\(443\) 39.4903 1.87624 0.938121 0.346307i \(-0.112564\pi\)
0.938121 + 0.346307i \(0.112564\pi\)
\(444\) −22.3100 −1.05878
\(445\) 0 0
\(446\) −71.5945 −3.39010
\(447\) 7.82095 0.369918
\(448\) 0 0
\(449\) 21.3045 1.00542 0.502710 0.864455i \(-0.332336\pi\)
0.502710 + 0.864455i \(0.332336\pi\)
\(450\) 0 0
\(451\) 6.72364 0.316604
\(452\) −1.85158 −0.0870910
\(453\) −3.13433 −0.147264
\(454\) 48.0331 2.25430
\(455\) 0 0
\(456\) −18.7641 −0.878710
\(457\) 27.8801 1.30418 0.652088 0.758143i \(-0.273893\pi\)
0.652088 + 0.758143i \(0.273893\pi\)
\(458\) −50.3944 −2.35478
\(459\) −4.23121 −0.197496
\(460\) 0 0
\(461\) 22.5237 1.04903 0.524516 0.851401i \(-0.324246\pi\)
0.524516 + 0.851401i \(0.324246\pi\)
\(462\) 0 0
\(463\) −1.69301 −0.0786809 −0.0393405 0.999226i \(-0.512526\pi\)
−0.0393405 + 0.999226i \(0.512526\pi\)
\(464\) 54.1465 2.51369
\(465\) 0 0
\(466\) 26.4036 1.22312
\(467\) −0.0803308 −0.00371727 −0.00185863 0.999998i \(-0.500592\pi\)
−0.00185863 + 0.999998i \(0.500592\pi\)
\(468\) 23.7678 1.09867
\(469\) 0 0
\(470\) 0 0
\(471\) −23.4790 −1.08185
\(472\) 33.8184 1.55662
\(473\) 8.84642 0.406759
\(474\) −34.1557 −1.56882
\(475\) 0 0
\(476\) 0 0
\(477\) −2.58021 −0.118140
\(478\) 16.7593 0.766551
\(479\) −38.7783 −1.77182 −0.885912 0.463853i \(-0.846466\pi\)
−0.885912 + 0.463853i \(0.846466\pi\)
\(480\) 0 0
\(481\) −18.8386 −0.858968
\(482\) 54.7009 2.49156
\(483\) 0 0
\(484\) −16.0040 −0.727456
\(485\) 0 0
\(486\) −2.70285 −0.122604
\(487\) −27.1309 −1.22942 −0.614710 0.788753i \(-0.710727\pi\)
−0.614710 + 0.788753i \(0.710727\pi\)
\(488\) 44.6703 2.02213
\(489\) −12.4363 −0.562391
\(490\) 0 0
\(491\) 16.7243 0.754755 0.377378 0.926059i \(-0.376826\pi\)
0.377378 + 0.926059i \(0.376826\pi\)
\(492\) 12.6249 0.569175
\(493\) −16.9248 −0.762256
\(494\) −25.4315 −1.14422
\(495\) 0 0
\(496\) 126.317 5.67180
\(497\) 0 0
\(498\) −8.17905 −0.366512
\(499\) 19.7844 0.885670 0.442835 0.896603i \(-0.353973\pi\)
0.442835 + 0.896603i \(0.353973\pi\)
\(500\) 0 0
\(501\) 18.4483 0.824211
\(502\) 16.7973 0.749701
\(503\) −6.75015 −0.300975 −0.150487 0.988612i \(-0.548084\pi\)
−0.150487 + 0.988612i \(0.548084\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −50.4864 −2.24439
\(507\) 7.06966 0.313975
\(508\) 12.0506 0.534658
\(509\) −40.3471 −1.78835 −0.894177 0.447715i \(-0.852238\pi\)
−0.894177 + 0.447715i \(0.852238\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11.5234 −0.509266
\(513\) 2.10029 0.0927300
\(514\) −15.2805 −0.673993
\(515\) 0 0
\(516\) 16.6108 0.731251
\(517\) −21.9975 −0.967448
\(518\) 0 0
\(519\) 7.65100 0.335842
\(520\) 0 0
\(521\) 37.6038 1.64745 0.823725 0.566989i \(-0.191892\pi\)
0.823725 + 0.566989i \(0.191892\pi\)
\(522\) −10.8114 −0.473203
\(523\) 39.3124 1.71901 0.859506 0.511125i \(-0.170771\pi\)
0.859506 + 0.511125i \(0.170771\pi\)
\(524\) 101.979 4.45497
\(525\) 0 0
\(526\) −67.6038 −2.94766
\(527\) −39.4835 −1.71993
\(528\) 38.2478 1.66452
\(529\) 20.7032 0.900137
\(530\) 0 0
\(531\) −3.78534 −0.164270
\(532\) 0 0
\(533\) 10.6605 0.461759
\(534\) 22.2478 0.962754
\(535\) 0 0
\(536\) −29.5308 −1.27554
\(537\) −2.41026 −0.104010
\(538\) 21.3045 0.918501
\(539\) 0 0
\(540\) 0 0
\(541\) 21.5722 0.927463 0.463732 0.885976i \(-0.346510\pi\)
0.463732 + 0.885976i \(0.346510\pi\)
\(542\) −12.8189 −0.550617
\(543\) −13.6970 −0.587796
\(544\) 79.2057 3.39592
\(545\) 0 0
\(546\) 0 0
\(547\) −21.3707 −0.913745 −0.456873 0.889532i \(-0.651031\pi\)
−0.456873 + 0.889532i \(0.651031\pi\)
\(548\) −43.8325 −1.87243
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) 8.40116 0.357901
\(552\) −59.0616 −2.51383
\(553\) 0 0
\(554\) −20.9125 −0.888488
\(555\) 0 0
\(556\) 5.23069 0.221830
\(557\) 14.2618 0.604293 0.302147 0.953261i \(-0.402297\pi\)
0.302147 + 0.953261i \(0.402297\pi\)
\(558\) −25.2217 −1.06772
\(559\) 14.0263 0.593248
\(560\) 0 0
\(561\) −11.9553 −0.504752
\(562\) −5.72408 −0.241456
\(563\) −2.36308 −0.0995921 −0.0497961 0.998759i \(-0.515857\pi\)
−0.0497961 + 0.998759i \(0.515857\pi\)
\(564\) −41.3045 −1.73923
\(565\) 0 0
\(566\) −11.8865 −0.499625
\(567\) 0 0
\(568\) 102.047 4.28180
\(569\) −12.7458 −0.534331 −0.267166 0.963651i \(-0.586087\pi\)
−0.267166 + 0.963651i \(0.586087\pi\)
\(570\) 0 0
\(571\) 0.361514 0.0151289 0.00756444 0.999971i \(-0.497592\pi\)
0.00756444 + 0.999971i \(0.497592\pi\)
\(572\) 67.1560 2.80794
\(573\) −12.1343 −0.506919
\(574\) 0 0
\(575\) 0 0
\(576\) 23.5225 0.980106
\(577\) 9.59334 0.399376 0.199688 0.979860i \(-0.436007\pi\)
0.199688 + 0.979860i \(0.436007\pi\)
\(578\) −2.44102 −0.101533
\(579\) 11.9258 0.495619
\(580\) 0 0
\(581\) 0 0
\(582\) −40.5875 −1.68241
\(583\) −7.29038 −0.301937
\(584\) 23.6842 0.980060
\(585\) 0 0
\(586\) 38.3821 1.58555
\(587\) 19.2217 0.793363 0.396682 0.917956i \(-0.370162\pi\)
0.396682 + 0.917956i \(0.370162\pi\)
\(588\) 0 0
\(589\) 19.5988 0.807556
\(590\) 0 0
\(591\) −5.50258 −0.226346
\(592\) −56.9233 −2.33953
\(593\) 11.7994 0.484544 0.242272 0.970208i \(-0.422107\pi\)
0.242272 + 0.970208i \(0.422107\pi\)
\(594\) −7.63692 −0.313346
\(595\) 0 0
\(596\) 41.4934 1.69964
\(597\) −11.3281 −0.463628
\(598\) −80.0477 −3.27339
\(599\) 40.3962 1.65054 0.825271 0.564736i \(-0.191022\pi\)
0.825271 + 0.564736i \(0.191022\pi\)
\(600\) 0 0
\(601\) −20.9073 −0.852828 −0.426414 0.904528i \(-0.640223\pi\)
−0.426414 + 0.904528i \(0.640223\pi\)
\(602\) 0 0
\(603\) 3.30542 0.134607
\(604\) −16.6290 −0.676622
\(605\) 0 0
\(606\) 6.03063 0.244978
\(607\) −4.20759 −0.170781 −0.0853903 0.996348i \(-0.527214\pi\)
−0.0853903 + 0.996348i \(0.527214\pi\)
\(608\) −39.3161 −1.59448
\(609\) 0 0
\(610\) 0 0
\(611\) −34.8777 −1.41100
\(612\) −22.4483 −0.907420
\(613\) −16.4268 −0.663472 −0.331736 0.943372i \(-0.607634\pi\)
−0.331736 + 0.943372i \(0.607634\pi\)
\(614\) 37.6841 1.52081
\(615\) 0 0
\(616\) 0 0
\(617\) −21.9037 −0.881811 −0.440906 0.897553i \(-0.645343\pi\)
−0.440906 + 0.897553i \(0.645343\pi\)
\(618\) 21.1959 0.852624
\(619\) 0.319504 0.0128420 0.00642098 0.999979i \(-0.497956\pi\)
0.00642098 + 0.999979i \(0.497956\pi\)
\(620\) 0 0
\(621\) 6.61084 0.265284
\(622\) 10.2693 0.411761
\(623\) 0 0
\(624\) 60.6430 2.42766
\(625\) 0 0
\(626\) −75.4032 −3.01372
\(627\) 5.93437 0.236996
\(628\) −124.566 −4.97071
\(629\) 17.7928 0.709445
\(630\) 0 0
\(631\) 17.3090 0.689061 0.344530 0.938775i \(-0.388038\pi\)
0.344530 + 0.938775i \(0.388038\pi\)
\(632\) −112.899 −4.49088
\(633\) 8.82095 0.350601
\(634\) −93.2363 −3.70289
\(635\) 0 0
\(636\) −13.6891 −0.542807
\(637\) 0 0
\(638\) −30.5477 −1.20939
\(639\) −11.4223 −0.451857
\(640\) 0 0
\(641\) 48.2858 1.90718 0.953588 0.301115i \(-0.0973589\pi\)
0.953588 + 0.301115i \(0.0973589\pi\)
\(642\) −42.2643 −1.66804
\(643\) −12.6460 −0.498710 −0.249355 0.968412i \(-0.580219\pi\)
−0.249355 + 0.968412i \(0.580219\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0196 0.945039
\(647\) 15.8534 0.623262 0.311631 0.950203i \(-0.399125\pi\)
0.311631 + 0.950203i \(0.399125\pi\)
\(648\) −8.93406 −0.350963
\(649\) −10.6955 −0.419834
\(650\) 0 0
\(651\) 0 0
\(652\) −65.9800 −2.58398
\(653\) −1.15795 −0.0453143 −0.0226571 0.999743i \(-0.507213\pi\)
−0.0226571 + 0.999743i \(0.507213\pi\)
\(654\) −8.30914 −0.324913
\(655\) 0 0
\(656\) 32.2121 1.25767
\(657\) −2.65100 −0.103425
\(658\) 0 0
\(659\) −34.5617 −1.34633 −0.673167 0.739490i \(-0.735067\pi\)
−0.673167 + 0.739490i \(0.735067\pi\)
\(660\) 0 0
\(661\) −1.28030 −0.0497977 −0.0248989 0.999690i \(-0.507926\pi\)
−0.0248989 + 0.999690i \(0.507926\pi\)
\(662\) 80.7997 3.14037
\(663\) −18.9555 −0.736169
\(664\) −27.0352 −1.04917
\(665\) 0 0
\(666\) 11.3658 0.440418
\(667\) 26.4433 1.02389
\(668\) 97.8761 3.78694
\(669\) 26.4885 1.02410
\(670\) 0 0
\(671\) −14.1275 −0.545386
\(672\) 0 0
\(673\) −45.9058 −1.76954 −0.884769 0.466031i \(-0.845684\pi\)
−0.884769 + 0.466031i \(0.845684\pi\)
\(674\) −52.2949 −2.01433
\(675\) 0 0
\(676\) 37.5075 1.44260
\(677\) −19.7100 −0.757516 −0.378758 0.925496i \(-0.623649\pi\)
−0.378758 + 0.925496i \(0.623649\pi\)
\(678\) 0.943291 0.0362269
\(679\) 0 0
\(680\) 0 0
\(681\) −17.7712 −0.680996
\(682\) −71.2639 −2.72883
\(683\) −1.44317 −0.0552212 −0.0276106 0.999619i \(-0.508790\pi\)
−0.0276106 + 0.999619i \(0.508790\pi\)
\(684\) 11.1429 0.426060
\(685\) 0 0
\(686\) 0 0
\(687\) 18.6449 0.711347
\(688\) 42.3821 1.61580
\(689\) −11.5591 −0.440367
\(690\) 0 0
\(691\) −30.9221 −1.17633 −0.588167 0.808740i \(-0.700150\pi\)
−0.588167 + 0.808740i \(0.700150\pi\)
\(692\) 40.5918 1.54307
\(693\) 0 0
\(694\) −71.0848 −2.69834
\(695\) 0 0
\(696\) −35.7362 −1.35458
\(697\) −10.0687 −0.381379
\(698\) 40.2949 1.52519
\(699\) −9.76879 −0.369490
\(700\) 0 0
\(701\) 50.6746 1.91395 0.956976 0.290168i \(-0.0937111\pi\)
0.956976 + 0.290168i \(0.0937111\pi\)
\(702\) −12.1086 −0.457008
\(703\) −8.83199 −0.333105
\(704\) 66.4630 2.50492
\(705\) 0 0
\(706\) 42.7617 1.60936
\(707\) 0 0
\(708\) −20.0828 −0.754757
\(709\) −20.1770 −0.757762 −0.378881 0.925445i \(-0.623691\pi\)
−0.378881 + 0.925445i \(0.623691\pi\)
\(710\) 0 0
\(711\) 12.6369 0.473921
\(712\) 73.5381 2.75596
\(713\) 61.6890 2.31027
\(714\) 0 0
\(715\) 0 0
\(716\) −12.7874 −0.477889
\(717\) −6.20058 −0.231565
\(718\) −45.0757 −1.68221
\(719\) 45.3278 1.69044 0.845221 0.534416i \(-0.179468\pi\)
0.845221 + 0.534416i \(0.179468\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 39.4314 1.46748
\(723\) −20.2382 −0.752667
\(724\) −72.6685 −2.70070
\(725\) 0 0
\(726\) 8.15328 0.302597
\(727\) 16.9117 0.627220 0.313610 0.949552i \(-0.398461\pi\)
0.313610 + 0.949552i \(0.398461\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −13.2476 −0.489979
\(732\) −26.5271 −0.980470
\(733\) 7.16558 0.264667 0.132333 0.991205i \(-0.457753\pi\)
0.132333 + 0.991205i \(0.457753\pi\)
\(734\) −76.1870 −2.81211
\(735\) 0 0
\(736\) −123.751 −4.56152
\(737\) 9.33946 0.344024
\(738\) −6.43179 −0.236757
\(739\) −20.3848 −0.749867 −0.374933 0.927052i \(-0.622334\pi\)
−0.374933 + 0.927052i \(0.622334\pi\)
\(740\) 0 0
\(741\) 9.40912 0.345653
\(742\) 0 0
\(743\) 16.9267 0.620982 0.310491 0.950576i \(-0.399507\pi\)
0.310491 + 0.950576i \(0.399507\pi\)
\(744\) −83.3682 −3.05643
\(745\) 0 0
\(746\) −64.8659 −2.37491
\(747\) 3.02608 0.110718
\(748\) −63.4278 −2.31915
\(749\) 0 0
\(750\) 0 0
\(751\) 39.0877 1.42633 0.713165 0.700996i \(-0.247261\pi\)
0.713165 + 0.700996i \(0.247261\pi\)
\(752\) −105.387 −3.84308
\(753\) −6.21466 −0.226475
\(754\) −48.4342 −1.76387
\(755\) 0 0
\(756\) 0 0
\(757\) −22.6675 −0.823866 −0.411933 0.911214i \(-0.635146\pi\)
−0.411933 + 0.911214i \(0.635146\pi\)
\(758\) −37.4480 −1.36017
\(759\) 18.6789 0.678002
\(760\) 0 0
\(761\) 10.3275 0.374370 0.187185 0.982325i \(-0.440064\pi\)
0.187185 + 0.982325i \(0.440064\pi\)
\(762\) −6.13919 −0.222399
\(763\) 0 0
\(764\) −64.3777 −2.32910
\(765\) 0 0
\(766\) 83.1841 3.00557
\(767\) −16.9580 −0.612318
\(768\) 23.6053 0.851784
\(769\) −25.8785 −0.933204 −0.466602 0.884467i \(-0.654522\pi\)
−0.466602 + 0.884467i \(0.654522\pi\)
\(770\) 0 0
\(771\) 5.65346 0.203604
\(772\) 63.2713 2.27718
\(773\) 9.35416 0.336446 0.168223 0.985749i \(-0.446197\pi\)
0.168223 + 0.985749i \(0.446197\pi\)
\(774\) −8.46242 −0.304175
\(775\) 0 0
\(776\) −134.159 −4.81602
\(777\) 0 0
\(778\) −103.824 −3.72227
\(779\) 4.99791 0.179069
\(780\) 0 0
\(781\) −32.2736 −1.15484
\(782\) 75.6038 2.70358
\(783\) 4.00000 0.142948
\(784\) 0 0
\(785\) 0 0
\(786\) −51.9534 −1.85312
\(787\) −2.80188 −0.0998762 −0.0499381 0.998752i \(-0.515902\pi\)
−0.0499381 + 0.998752i \(0.515902\pi\)
\(788\) −29.1935 −1.03998
\(789\) 25.0120 0.890451
\(790\) 0 0
\(791\) 0 0
\(792\) −25.2432 −0.896978
\(793\) −22.3996 −0.795433
\(794\) 90.4414 3.20965
\(795\) 0 0
\(796\) −60.1002 −2.13020
\(797\) 13.0211 0.461231 0.230615 0.973045i \(-0.425926\pi\)
0.230615 + 0.973045i \(0.425926\pi\)
\(798\) 0 0
\(799\) 32.9414 1.16538
\(800\) 0 0
\(801\) −8.23121 −0.290835
\(802\) −40.1159 −1.41654
\(803\) −7.49041 −0.264331
\(804\) 17.5366 0.618469
\(805\) 0 0
\(806\) −112.991 −3.97994
\(807\) −7.88221 −0.277467
\(808\) 19.9338 0.701267
\(809\) 21.0211 0.739062 0.369531 0.929218i \(-0.379518\pi\)
0.369531 + 0.929218i \(0.379518\pi\)
\(810\) 0 0
\(811\) −46.2591 −1.62438 −0.812189 0.583395i \(-0.801724\pi\)
−0.812189 + 0.583395i \(0.801724\pi\)
\(812\) 0 0
\(813\) 4.74271 0.166334
\(814\) 32.1142 1.12560
\(815\) 0 0
\(816\) −57.2763 −2.00507
\(817\) 6.57584 0.230059
\(818\) −22.8254 −0.798070
\(819\) 0 0
\(820\) 0 0
\(821\) 16.9383 0.591151 0.295575 0.955319i \(-0.404489\pi\)
0.295575 + 0.955319i \(0.404489\pi\)
\(822\) 22.3305 0.778867
\(823\) 18.5332 0.646027 0.323014 0.946394i \(-0.395304\pi\)
0.323014 + 0.946394i \(0.395304\pi\)
\(824\) 70.0613 2.44070
\(825\) 0 0
\(826\) 0 0
\(827\) −32.5496 −1.13186 −0.565930 0.824453i \(-0.691483\pi\)
−0.565930 + 0.824453i \(0.691483\pi\)
\(828\) 35.0733 1.21888
\(829\) −10.9553 −0.380493 −0.190246 0.981736i \(-0.560929\pi\)
−0.190246 + 0.981736i \(0.560929\pi\)
\(830\) 0 0
\(831\) 7.73721 0.268401
\(832\) 105.379 3.65336
\(833\) 0 0
\(834\) −2.66478 −0.0922738
\(835\) 0 0
\(836\) 31.4843 1.08891
\(837\) 9.33150 0.322544
\(838\) 50.9290 1.75931
\(839\) 1.24074 0.0428352 0.0214176 0.999771i \(-0.493182\pi\)
0.0214176 + 0.999771i \(0.493182\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 24.5875 0.847342
\(843\) 2.11779 0.0729405
\(844\) 46.7988 1.61088
\(845\) 0 0
\(846\) 21.0426 0.723460
\(847\) 0 0
\(848\) −34.9273 −1.19941
\(849\) 4.39774 0.150930
\(850\) 0 0
\(851\) −27.7994 −0.952952
\(852\) −60.5998 −2.07612
\(853\) −44.8500 −1.53564 −0.767818 0.640669i \(-0.778657\pi\)
−0.767818 + 0.640669i \(0.778657\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −139.701 −4.77489
\(857\) −38.6108 −1.31892 −0.659461 0.751739i \(-0.729215\pi\)
−0.659461 + 0.751739i \(0.729215\pi\)
\(858\) −34.2128 −1.16800
\(859\) 35.1083 1.19788 0.598939 0.800795i \(-0.295589\pi\)
0.598939 + 0.800795i \(0.295589\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 13.4057 0.456600
\(863\) −4.94575 −0.168355 −0.0841776 0.996451i \(-0.526826\pi\)
−0.0841776 + 0.996451i \(0.526826\pi\)
\(864\) −18.7194 −0.636847
\(865\) 0 0
\(866\) 7.07325 0.240359
\(867\) 0.903125 0.0306717
\(868\) 0 0
\(869\) 35.7056 1.21123
\(870\) 0 0
\(871\) 14.8080 0.501750
\(872\) −27.4652 −0.930088
\(873\) 15.0165 0.508233
\(874\) −37.5282 −1.26941
\(875\) 0 0
\(876\) −14.0647 −0.475201
\(877\) 10.7123 0.361727 0.180864 0.983508i \(-0.442111\pi\)
0.180864 + 0.983508i \(0.442111\pi\)
\(878\) 71.6392 2.41771
\(879\) −14.2006 −0.478974
\(880\) 0 0
\(881\) 40.2527 1.35615 0.678075 0.734993i \(-0.262815\pi\)
0.678075 + 0.734993i \(0.262815\pi\)
\(882\) 0 0
\(883\) 26.2228 0.882468 0.441234 0.897392i \(-0.354541\pi\)
0.441234 + 0.897392i \(0.354541\pi\)
\(884\) −100.567 −3.38242
\(885\) 0 0
\(886\) −106.737 −3.58589
\(887\) −49.3964 −1.65857 −0.829284 0.558828i \(-0.811251\pi\)
−0.829284 + 0.558828i \(0.811251\pi\)
\(888\) 37.5689 1.26073
\(889\) 0 0
\(890\) 0 0
\(891\) 2.82550 0.0946578
\(892\) 140.533 4.70538
\(893\) −16.3515 −0.547181
\(894\) −21.1389 −0.706990
\(895\) 0 0
\(896\) 0 0
\(897\) 29.6160 0.988850
\(898\) −57.5828 −1.92156
\(899\) 37.3260 1.24489
\(900\) 0 0
\(901\) 10.9174 0.363711
\(902\) −18.1730 −0.605095
\(903\) 0 0
\(904\) 3.11797 0.103702
\(905\) 0 0
\(906\) 8.47165 0.281452
\(907\) −56.1629 −1.86486 −0.932429 0.361354i \(-0.882315\pi\)
−0.932429 + 0.361354i \(0.882315\pi\)
\(908\) −94.2839 −3.12892
\(909\) −2.23121 −0.0740045
\(910\) 0 0
\(911\) −35.9981 −1.19267 −0.596335 0.802736i \(-0.703377\pi\)
−0.596335 + 0.802736i \(0.703377\pi\)
\(912\) 28.4308 0.941439
\(913\) 8.55019 0.282970
\(914\) −75.3559 −2.49255
\(915\) 0 0
\(916\) 98.9189 3.26837
\(917\) 0 0
\(918\) 11.4363 0.377455
\(919\) −7.54801 −0.248986 −0.124493 0.992221i \(-0.539730\pi\)
−0.124493 + 0.992221i \(0.539730\pi\)
\(920\) 0 0
\(921\) −13.9423 −0.459416
\(922\) −60.8782 −2.00492
\(923\) −51.1707 −1.68431
\(924\) 0 0
\(925\) 0 0
\(926\) 4.57596 0.150375
\(927\) −7.84205 −0.257567
\(928\) −74.8776 −2.45798
\(929\) 5.19104 0.170313 0.0851563 0.996368i \(-0.472861\pi\)
0.0851563 + 0.996368i \(0.472861\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −51.8275 −1.69767
\(933\) −3.79942 −0.124387
\(934\) 0.217122 0.00710446
\(935\) 0 0
\(936\) −40.0239 −1.30822
\(937\) 5.23690 0.171082 0.0855410 0.996335i \(-0.472738\pi\)
0.0855410 + 0.996335i \(0.472738\pi\)
\(938\) 0 0
\(939\) 27.8976 0.910404
\(940\) 0 0
\(941\) −12.9439 −0.421959 −0.210980 0.977490i \(-0.567665\pi\)
−0.210980 + 0.977490i \(0.567665\pi\)
\(942\) 63.4602 2.06764
\(943\) 15.7313 0.512283
\(944\) −51.2407 −1.66774
\(945\) 0 0
\(946\) −23.9106 −0.777400
\(947\) 23.0143 0.747863 0.373932 0.927456i \(-0.378009\pi\)
0.373932 + 0.927456i \(0.378009\pi\)
\(948\) 67.0441 2.17749
\(949\) −11.8763 −0.385520
\(950\) 0 0
\(951\) 34.4955 1.11859
\(952\) 0 0
\(953\) 15.1560 0.490952 0.245476 0.969403i \(-0.421056\pi\)
0.245476 + 0.969403i \(0.421056\pi\)
\(954\) 6.97392 0.225789
\(955\) 0 0
\(956\) −32.8967 −1.06395
\(957\) 11.3020 0.365342
\(958\) 104.812 3.38632
\(959\) 0 0
\(960\) 0 0
\(961\) 56.0768 1.80893
\(962\) 50.9181 1.64166
\(963\) 15.6369 0.503893
\(964\) −107.372 −3.45823
\(965\) 0 0
\(966\) 0 0
\(967\) 45.5145 1.46365 0.731824 0.681494i \(-0.238669\pi\)
0.731824 + 0.681494i \(0.238669\pi\)
\(968\) 26.9500 0.866206
\(969\) −8.88676 −0.285484
\(970\) 0 0
\(971\) −7.60192 −0.243957 −0.121979 0.992533i \(-0.538924\pi\)
−0.121979 + 0.992533i \(0.538924\pi\)
\(972\) 5.30542 0.170171
\(973\) 0 0
\(974\) 73.3309 2.34967
\(975\) 0 0
\(976\) −67.6831 −2.16648
\(977\) 34.6936 1.10995 0.554974 0.831868i \(-0.312728\pi\)
0.554974 + 0.831868i \(0.312728\pi\)
\(978\) 33.6136 1.07484
\(979\) −23.2573 −0.743306
\(980\) 0 0
\(981\) 3.07421 0.0981520
\(982\) −45.2032 −1.44249
\(983\) −15.5779 −0.496859 −0.248429 0.968650i \(-0.579914\pi\)
−0.248429 + 0.968650i \(0.579914\pi\)
\(984\) −21.2597 −0.677736
\(985\) 0 0
\(986\) 45.7454 1.45683
\(987\) 0 0
\(988\) 49.9193 1.58815
\(989\) 20.6980 0.658158
\(990\) 0 0
\(991\) −45.9589 −1.45993 −0.729966 0.683484i \(-0.760464\pi\)
−0.729966 + 0.683484i \(0.760464\pi\)
\(992\) −174.680 −5.54610
\(993\) −29.8942 −0.948664
\(994\) 0 0
\(995\) 0 0
\(996\) 16.0546 0.508710
\(997\) −41.0687 −1.30066 −0.650329 0.759652i \(-0.725369\pi\)
−0.650329 + 0.759652i \(0.725369\pi\)
\(998\) −53.4743 −1.69270
\(999\) −4.20513 −0.133044
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 3675.2.a.br.1.1 4
5.4 even 2 3675.2.a.bw.1.4 4
7.2 even 3 525.2.i.j.151.4 yes 8
7.4 even 3 525.2.i.j.226.4 yes 8
7.6 odd 2 3675.2.a.bq.1.1 4
35.2 odd 12 525.2.r.h.424.1 16
35.4 even 6 525.2.i.i.226.1 yes 8
35.9 even 6 525.2.i.i.151.1 8
35.18 odd 12 525.2.r.h.499.1 16
35.23 odd 12 525.2.r.h.424.8 16
35.32 odd 12 525.2.r.h.499.8 16
35.34 odd 2 3675.2.a.bx.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.2.i.i.151.1 8 35.9 even 6
525.2.i.i.226.1 yes 8 35.4 even 6
525.2.i.j.151.4 yes 8 7.2 even 3
525.2.i.j.226.4 yes 8 7.4 even 3
525.2.r.h.424.1 16 35.2 odd 12
525.2.r.h.424.8 16 35.23 odd 12
525.2.r.h.499.1 16 35.18 odd 12
525.2.r.h.499.8 16 35.32 odd 12
3675.2.a.bq.1.1 4 7.6 odd 2
3675.2.a.br.1.1 4 1.1 even 1 trivial
3675.2.a.bw.1.4 4 5.4 even 2
3675.2.a.bx.1.4 4 35.34 odd 2