Properties

Label 1305.2.a.s.1.2
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(0\)
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: \( x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.43890\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.43890 q^{2} +3.94822 q^{4} -1.00000 q^{5} +3.59688 q^{7} -4.75150 q^{8} +O(q^{10})\) \(q-2.43890 q^{2} +3.94822 q^{4} -1.00000 q^{5} +3.59688 q^{7} -4.75150 q^{8} +2.43890 q^{10} +6.58661 q^{11} +2.25623 q^{13} -8.77241 q^{14} +3.69199 q^{16} +0.256230 q^{17} +7.95099 q^{19} -3.94822 q^{20} -16.0641 q^{22} +0.961264 q^{23} +1.00000 q^{25} -5.50271 q^{26} +14.2013 q^{28} +1.00000 q^{29} -3.95099 q^{31} +0.498627 q^{32} -0.624919 q^{34} -3.59688 q^{35} -3.76964 q^{37} -19.3917 q^{38} +4.75150 q^{40} +5.00459 q^{41} -6.19835 q^{43} +26.0054 q^{44} -2.34442 q^{46} -7.02055 q^{47} +5.93752 q^{49} -2.43890 q^{50} +8.90809 q^{52} +6.38493 q^{53} -6.58661 q^{55} -17.0906 q^{56} -2.43890 q^{58} -3.19375 q^{59} +14.4072 q^{61} +9.63606 q^{62} -8.60008 q^{64} -2.25623 q^{65} -9.35246 q^{67} +1.01165 q^{68} +8.77241 q^{70} -11.9020 q^{71} -13.4965 q^{73} +9.19375 q^{74} +31.3923 q^{76} +23.6912 q^{77} -5.22222 q^{79} -3.69199 q^{80} -12.2057 q^{82} +0.195206 q^{83} -0.256230 q^{85} +15.1171 q^{86} -31.2963 q^{88} -3.72323 q^{89} +8.11538 q^{91} +3.79528 q^{92} +17.1224 q^{94} -7.95099 q^{95} -2.97486 q^{97} -14.4810 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7} + q^{10} + 3 q^{11} + 6 q^{13} + 9 q^{14} + 21 q^{16} - 8 q^{17} + 10 q^{19} - 13 q^{20} + 9 q^{22} - 11 q^{23} + 7 q^{25} - 3 q^{26} + 25 q^{28} + 7 q^{29} + 18 q^{31} - q^{32} - q^{34} - 10 q^{35} + 13 q^{37} - 12 q^{38} + 13 q^{41} + 9 q^{43} + 37 q^{44} - 8 q^{46} - 2 q^{47} + 21 q^{49} - q^{50} - q^{52} - 5 q^{53} - 3 q^{55} + 30 q^{56} - q^{58} + 8 q^{59} + 14 q^{61} + 8 q^{62} + 8 q^{64} - 6 q^{65} + 14 q^{67} - 27 q^{68} - 9 q^{70} + 8 q^{71} + 3 q^{73} + 34 q^{74} + 4 q^{76} + 28 q^{77} + 4 q^{79} - 21 q^{80} - 20 q^{82} - 17 q^{83} + 8 q^{85} - 4 q^{86} + 26 q^{88} + 20 q^{89} + 12 q^{91} - 60 q^{92} - 21 q^{94} - 10 q^{95} + 13 q^{97} + 20 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.43890 −1.72456 −0.862280 0.506431i \(-0.830964\pi\)
−0.862280 + 0.506431i \(0.830964\pi\)
\(3\) 0 0
\(4\) 3.94822 1.97411
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.59688 1.35949 0.679746 0.733448i \(-0.262090\pi\)
0.679746 + 0.733448i \(0.262090\pi\)
\(8\) −4.75150 −1.67991
\(9\) 0 0
\(10\) 2.43890 0.771247
\(11\) 6.58661 1.98594 0.992968 0.118382i \(-0.0377706\pi\)
0.992968 + 0.118382i \(0.0377706\pi\)
\(12\) 0 0
\(13\) 2.25623 0.625766 0.312883 0.949792i \(-0.398705\pi\)
0.312883 + 0.949792i \(0.398705\pi\)
\(14\) −8.77241 −2.34453
\(15\) 0 0
\(16\) 3.69199 0.922997
\(17\) 0.256230 0.0621449 0.0310725 0.999517i \(-0.490108\pi\)
0.0310725 + 0.999517i \(0.490108\pi\)
\(18\) 0 0
\(19\) 7.95099 1.82408 0.912041 0.410098i \(-0.134505\pi\)
0.912041 + 0.410098i \(0.134505\pi\)
\(20\) −3.94822 −0.882848
\(21\) 0 0
\(22\) −16.0641 −3.42487
\(23\) 0.961264 0.200437 0.100219 0.994965i \(-0.468046\pi\)
0.100219 + 0.994965i \(0.468046\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −5.50271 −1.07917
\(27\) 0 0
\(28\) 14.2013 2.68378
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −3.95099 −0.709619 −0.354810 0.934939i \(-0.615454\pi\)
−0.354810 + 0.934939i \(0.615454\pi\)
\(32\) 0.498627 0.0881456
\(33\) 0 0
\(34\) −0.624919 −0.107173
\(35\) −3.59688 −0.607983
\(36\) 0 0
\(37\) −3.76964 −0.619724 −0.309862 0.950781i \(-0.600283\pi\)
−0.309862 + 0.950781i \(0.600283\pi\)
\(38\) −19.3917 −3.14574
\(39\) 0 0
\(40\) 4.75150 0.751278
\(41\) 5.00459 0.781586 0.390793 0.920479i \(-0.372201\pi\)
0.390793 + 0.920479i \(0.372201\pi\)
\(42\) 0 0
\(43\) −6.19835 −0.945239 −0.472620 0.881267i \(-0.656691\pi\)
−0.472620 + 0.881267i \(0.656691\pi\)
\(44\) 26.0054 3.92045
\(45\) 0 0
\(46\) −2.34442 −0.345667
\(47\) −7.02055 −1.02405 −0.512026 0.858970i \(-0.671105\pi\)
−0.512026 + 0.858970i \(0.671105\pi\)
\(48\) 0 0
\(49\) 5.93752 0.848218
\(50\) −2.43890 −0.344912
\(51\) 0 0
\(52\) 8.90809 1.23533
\(53\) 6.38493 0.877038 0.438519 0.898722i \(-0.355503\pi\)
0.438519 + 0.898722i \(0.355503\pi\)
\(54\) 0 0
\(55\) −6.58661 −0.888138
\(56\) −17.0906 −2.28382
\(57\) 0 0
\(58\) −2.43890 −0.320243
\(59\) −3.19375 −0.415791 −0.207896 0.978151i \(-0.566661\pi\)
−0.207896 + 0.978151i \(0.566661\pi\)
\(60\) 0 0
\(61\) 14.4072 1.84465 0.922323 0.386419i \(-0.126288\pi\)
0.922323 + 0.386419i \(0.126288\pi\)
\(62\) 9.63606 1.22378
\(63\) 0 0
\(64\) −8.60008 −1.07501
\(65\) −2.25623 −0.279851
\(66\) 0 0
\(67\) −9.35246 −1.14259 −0.571293 0.820746i \(-0.693558\pi\)
−0.571293 + 0.820746i \(0.693558\pi\)
\(68\) 1.01165 0.122681
\(69\) 0 0
\(70\) 8.77241 1.04850
\(71\) −11.9020 −1.41251 −0.706253 0.707960i \(-0.749616\pi\)
−0.706253 + 0.707960i \(0.749616\pi\)
\(72\) 0 0
\(73\) −13.4965 −1.57965 −0.789824 0.613334i \(-0.789828\pi\)
−0.789824 + 0.613334i \(0.789828\pi\)
\(74\) 9.19375 1.06875
\(75\) 0 0
\(76\) 31.3923 3.60094
\(77\) 23.6912 2.69986
\(78\) 0 0
\(79\) −5.22222 −0.587545 −0.293773 0.955875i \(-0.594911\pi\)
−0.293773 + 0.955875i \(0.594911\pi\)
\(80\) −3.69199 −0.412777
\(81\) 0 0
\(82\) −12.2057 −1.34789
\(83\) 0.195206 0.0214266 0.0107133 0.999943i \(-0.496590\pi\)
0.0107133 + 0.999943i \(0.496590\pi\)
\(84\) 0 0
\(85\) −0.256230 −0.0277921
\(86\) 15.1171 1.63012
\(87\) 0 0
\(88\) −31.2963 −3.33619
\(89\) −3.72323 −0.394661 −0.197331 0.980337i \(-0.563227\pi\)
−0.197331 + 0.980337i \(0.563227\pi\)
\(90\) 0 0
\(91\) 8.11538 0.850723
\(92\) 3.79528 0.395685
\(93\) 0 0
\(94\) 17.1224 1.76604
\(95\) −7.95099 −0.815755
\(96\) 0 0
\(97\) −2.97486 −0.302052 −0.151026 0.988530i \(-0.548258\pi\)
−0.151026 + 0.988530i \(0.548258\pi\)
\(98\) −14.4810 −1.46280
\(99\) 0 0
\(100\) 3.94822 0.394822
\(101\) 14.4742 1.44024 0.720119 0.693850i \(-0.244087\pi\)
0.720119 + 0.693850i \(0.244087\pi\)
\(102\) 0 0
\(103\) 4.77210 0.470209 0.235105 0.971970i \(-0.424457\pi\)
0.235105 + 0.971970i \(0.424457\pi\)
\(104\) −10.7205 −1.05123
\(105\) 0 0
\(106\) −15.5722 −1.51251
\(107\) −11.9361 −1.15391 −0.576955 0.816776i \(-0.695759\pi\)
−0.576955 + 0.816776i \(0.695759\pi\)
\(108\) 0 0
\(109\) −4.30818 −0.412649 −0.206324 0.978484i \(-0.566150\pi\)
−0.206324 + 0.978484i \(0.566150\pi\)
\(110\) 16.0641 1.53165
\(111\) 0 0
\(112\) 13.2796 1.25481
\(113\) −9.98310 −0.939131 −0.469566 0.882898i \(-0.655590\pi\)
−0.469566 + 0.882898i \(0.655590\pi\)
\(114\) 0 0
\(115\) −0.961264 −0.0896384
\(116\) 3.94822 0.366583
\(117\) 0 0
\(118\) 7.78924 0.717057
\(119\) 0.921628 0.0844855
\(120\) 0 0
\(121\) 32.3834 2.94394
\(122\) −35.1376 −3.18121
\(123\) 0 0
\(124\) −15.5994 −1.40087
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.97486 0.618919 0.309460 0.950913i \(-0.399852\pi\)
0.309460 + 0.950913i \(0.399852\pi\)
\(128\) 19.9774 1.76577
\(129\) 0 0
\(130\) 5.50271 0.482620
\(131\) 21.0735 1.84120 0.920602 0.390502i \(-0.127699\pi\)
0.920602 + 0.390502i \(0.127699\pi\)
\(132\) 0 0
\(133\) 28.5987 2.47983
\(134\) 22.8097 1.97046
\(135\) 0 0
\(136\) −1.21748 −0.104398
\(137\) 12.9493 1.10634 0.553168 0.833069i \(-0.313419\pi\)
0.553168 + 0.833069i \(0.313419\pi\)
\(138\) 0 0
\(139\) −8.75391 −0.742497 −0.371249 0.928533i \(-0.621070\pi\)
−0.371249 + 0.928533i \(0.621070\pi\)
\(140\) −14.2013 −1.20022
\(141\) 0 0
\(142\) 29.0277 2.43595
\(143\) 14.8609 1.24273
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 32.9166 2.72420
\(147\) 0 0
\(148\) −14.8833 −1.22340
\(149\) −6.38751 −0.523285 −0.261643 0.965165i \(-0.584264\pi\)
−0.261643 + 0.965165i \(0.584264\pi\)
\(150\) 0 0
\(151\) 17.4892 1.42325 0.711626 0.702558i \(-0.247959\pi\)
0.711626 + 0.702558i \(0.247959\pi\)
\(152\) −37.7792 −3.06429
\(153\) 0 0
\(154\) −57.7804 −4.65608
\(155\) 3.95099 0.317351
\(156\) 0 0
\(157\) −6.42770 −0.512986 −0.256493 0.966546i \(-0.582567\pi\)
−0.256493 + 0.966546i \(0.582567\pi\)
\(158\) 12.7365 1.01326
\(159\) 0 0
\(160\) −0.498627 −0.0394199
\(161\) 3.45755 0.272493
\(162\) 0 0
\(163\) 17.2726 1.35290 0.676449 0.736490i \(-0.263518\pi\)
0.676449 + 0.736490i \(0.263518\pi\)
\(164\) 19.7592 1.54294
\(165\) 0 0
\(166\) −0.476087 −0.0369515
\(167\) 21.4846 1.66253 0.831264 0.555878i \(-0.187618\pi\)
0.831264 + 0.555878i \(0.187618\pi\)
\(168\) 0 0
\(169\) −7.90943 −0.608417
\(170\) 0.624919 0.0479291
\(171\) 0 0
\(172\) −24.4724 −1.86600
\(173\) 4.64188 0.352916 0.176458 0.984308i \(-0.443536\pi\)
0.176458 + 0.984308i \(0.443536\pi\)
\(174\) 0 0
\(175\) 3.59688 0.271898
\(176\) 24.3177 1.83301
\(177\) 0 0
\(178\) 9.08057 0.680617
\(179\) −12.3053 −0.919744 −0.459872 0.887985i \(-0.652105\pi\)
−0.459872 + 0.887985i \(0.652105\pi\)
\(180\) 0 0
\(181\) −15.1061 −1.12283 −0.561415 0.827534i \(-0.689743\pi\)
−0.561415 + 0.827534i \(0.689743\pi\)
\(182\) −19.7926 −1.46712
\(183\) 0 0
\(184\) −4.56745 −0.336717
\(185\) 3.76964 0.277149
\(186\) 0 0
\(187\) 1.68769 0.123416
\(188\) −27.7186 −2.02159
\(189\) 0 0
\(190\) 19.3917 1.40682
\(191\) 13.3833 0.968380 0.484190 0.874963i \(-0.339114\pi\)
0.484190 + 0.874963i \(0.339114\pi\)
\(192\) 0 0
\(193\) −5.12776 −0.369104 −0.184552 0.982823i \(-0.559083\pi\)
−0.184552 + 0.982823i \(0.559083\pi\)
\(194\) 7.25539 0.520907
\(195\) 0 0
\(196\) 23.4426 1.67447
\(197\) 9.95230 0.709072 0.354536 0.935042i \(-0.384639\pi\)
0.354536 + 0.935042i \(0.384639\pi\)
\(198\) 0 0
\(199\) −21.3405 −1.51279 −0.756395 0.654115i \(-0.773041\pi\)
−0.756395 + 0.654115i \(0.773041\pi\)
\(200\) −4.75150 −0.335982
\(201\) 0 0
\(202\) −35.3011 −2.48378
\(203\) 3.59688 0.252451
\(204\) 0 0
\(205\) −5.00459 −0.349536
\(206\) −11.6387 −0.810904
\(207\) 0 0
\(208\) 8.32997 0.577580
\(209\) 52.3701 3.62251
\(210\) 0 0
\(211\) 17.9602 1.23643 0.618215 0.786009i \(-0.287856\pi\)
0.618215 + 0.786009i \(0.287856\pi\)
\(212\) 25.2091 1.73137
\(213\) 0 0
\(214\) 29.1110 1.98999
\(215\) 6.19835 0.422724
\(216\) 0 0
\(217\) −14.2112 −0.964722
\(218\) 10.5072 0.711637
\(219\) 0 0
\(220\) −26.0054 −1.75328
\(221\) 0.578114 0.0388882
\(222\) 0 0
\(223\) 10.9175 0.731088 0.365544 0.930794i \(-0.380883\pi\)
0.365544 + 0.930794i \(0.380883\pi\)
\(224\) 1.79350 0.119833
\(225\) 0 0
\(226\) 24.3478 1.61959
\(227\) −27.1891 −1.80461 −0.902303 0.431103i \(-0.858125\pi\)
−0.902303 + 0.431103i \(0.858125\pi\)
\(228\) 0 0
\(229\) 3.93198 0.259832 0.129916 0.991525i \(-0.458529\pi\)
0.129916 + 0.991525i \(0.458529\pi\)
\(230\) 2.34442 0.154587
\(231\) 0 0
\(232\) −4.75150 −0.311951
\(233\) −19.3343 −1.26663 −0.633315 0.773894i \(-0.718306\pi\)
−0.633315 + 0.773894i \(0.718306\pi\)
\(234\) 0 0
\(235\) 7.02055 0.457970
\(236\) −12.6096 −0.820817
\(237\) 0 0
\(238\) −2.24776 −0.145700
\(239\) 10.8473 0.701656 0.350828 0.936440i \(-0.385900\pi\)
0.350828 + 0.936440i \(0.385900\pi\)
\(240\) 0 0
\(241\) −22.4834 −1.44828 −0.724142 0.689651i \(-0.757764\pi\)
−0.724142 + 0.689651i \(0.757764\pi\)
\(242\) −78.9797 −5.07701
\(243\) 0 0
\(244\) 56.8826 3.64153
\(245\) −5.93752 −0.379334
\(246\) 0 0
\(247\) 17.9393 1.14145
\(248\) 18.7732 1.19210
\(249\) 0 0
\(250\) 2.43890 0.154249
\(251\) −16.7271 −1.05581 −0.527903 0.849304i \(-0.677022\pi\)
−0.527903 + 0.849304i \(0.677022\pi\)
\(252\) 0 0
\(253\) 6.33147 0.398056
\(254\) −17.0110 −1.06736
\(255\) 0 0
\(256\) −31.5228 −1.97017
\(257\) −22.3892 −1.39660 −0.698299 0.715806i \(-0.746060\pi\)
−0.698299 + 0.715806i \(0.746060\pi\)
\(258\) 0 0
\(259\) −13.5589 −0.842510
\(260\) −8.90809 −0.552456
\(261\) 0 0
\(262\) −51.3962 −3.17527
\(263\) 1.48565 0.0916090 0.0458045 0.998950i \(-0.485415\pi\)
0.0458045 + 0.998950i \(0.485415\pi\)
\(264\) 0 0
\(265\) −6.38493 −0.392223
\(266\) −69.7494 −4.27661
\(267\) 0 0
\(268\) −36.9256 −2.25559
\(269\) −2.08929 −0.127386 −0.0636931 0.997970i \(-0.520288\pi\)
−0.0636931 + 0.997970i \(0.520288\pi\)
\(270\) 0 0
\(271\) 14.2890 0.867992 0.433996 0.900915i \(-0.357103\pi\)
0.433996 + 0.900915i \(0.357103\pi\)
\(272\) 0.945998 0.0573596
\(273\) 0 0
\(274\) −31.5821 −1.90794
\(275\) 6.58661 0.397187
\(276\) 0 0
\(277\) −13.5685 −0.815255 −0.407627 0.913148i \(-0.633644\pi\)
−0.407627 + 0.913148i \(0.633644\pi\)
\(278\) 21.3499 1.28048
\(279\) 0 0
\(280\) 17.0906 1.02136
\(281\) −16.5216 −0.985599 −0.492799 0.870143i \(-0.664026\pi\)
−0.492799 + 0.870143i \(0.664026\pi\)
\(282\) 0 0
\(283\) −20.7310 −1.23233 −0.616166 0.787617i \(-0.711315\pi\)
−0.616166 + 0.787617i \(0.711315\pi\)
\(284\) −46.9916 −2.78844
\(285\) 0 0
\(286\) −36.2442 −2.14316
\(287\) 18.0009 1.06256
\(288\) 0 0
\(289\) −16.9343 −0.996138
\(290\) 2.43890 0.143217
\(291\) 0 0
\(292\) −53.2872 −3.11840
\(293\) −4.86682 −0.284323 −0.142161 0.989843i \(-0.545405\pi\)
−0.142161 + 0.989843i \(0.545405\pi\)
\(294\) 0 0
\(295\) 3.19375 0.185948
\(296\) 17.9114 1.04108
\(297\) 0 0
\(298\) 15.5785 0.902437
\(299\) 2.16883 0.125427
\(300\) 0 0
\(301\) −22.2947 −1.28504
\(302\) −42.6544 −2.45448
\(303\) 0 0
\(304\) 29.3550 1.68362
\(305\) −14.4072 −0.824951
\(306\) 0 0
\(307\) 13.3624 0.762631 0.381315 0.924445i \(-0.375471\pi\)
0.381315 + 0.924445i \(0.375471\pi\)
\(308\) 93.5381 5.32983
\(309\) 0 0
\(310\) −9.63606 −0.547292
\(311\) 1.27926 0.0725400 0.0362700 0.999342i \(-0.488452\pi\)
0.0362700 + 0.999342i \(0.488452\pi\)
\(312\) 0 0
\(313\) 29.2467 1.65312 0.826560 0.562849i \(-0.190295\pi\)
0.826560 + 0.562849i \(0.190295\pi\)
\(314\) 15.6765 0.884675
\(315\) 0 0
\(316\) −20.6185 −1.15988
\(317\) 25.3909 1.42609 0.713047 0.701116i \(-0.247315\pi\)
0.713047 + 0.701116i \(0.247315\pi\)
\(318\) 0 0
\(319\) 6.58661 0.368779
\(320\) 8.60008 0.480759
\(321\) 0 0
\(322\) −8.43261 −0.469931
\(323\) 2.03728 0.113357
\(324\) 0 0
\(325\) 2.25623 0.125153
\(326\) −42.1262 −2.33315
\(327\) 0 0
\(328\) −23.7793 −1.31299
\(329\) −25.2520 −1.39219
\(330\) 0 0
\(331\) −19.8405 −1.09053 −0.545266 0.838263i \(-0.683571\pi\)
−0.545266 + 0.838263i \(0.683571\pi\)
\(332\) 0.770716 0.0422985
\(333\) 0 0
\(334\) −52.3987 −2.86713
\(335\) 9.35246 0.510980
\(336\) 0 0
\(337\) 21.7192 1.18312 0.591560 0.806261i \(-0.298512\pi\)
0.591560 + 0.806261i \(0.298512\pi\)
\(338\) 19.2903 1.04925
\(339\) 0 0
\(340\) −1.01165 −0.0548645
\(341\) −26.0236 −1.40926
\(342\) 0 0
\(343\) −3.82160 −0.206347
\(344\) 29.4515 1.58792
\(345\) 0 0
\(346\) −11.3211 −0.608624
\(347\) −9.47114 −0.508437 −0.254219 0.967147i \(-0.581818\pi\)
−0.254219 + 0.967147i \(0.581818\pi\)
\(348\) 0 0
\(349\) 25.8065 1.38139 0.690696 0.723145i \(-0.257304\pi\)
0.690696 + 0.723145i \(0.257304\pi\)
\(350\) −8.77241 −0.468905
\(351\) 0 0
\(352\) 3.28426 0.175052
\(353\) −19.9743 −1.06312 −0.531561 0.847020i \(-0.678394\pi\)
−0.531561 + 0.847020i \(0.678394\pi\)
\(354\) 0 0
\(355\) 11.9020 0.631692
\(356\) −14.7001 −0.779104
\(357\) 0 0
\(358\) 30.0114 1.58615
\(359\) −28.6956 −1.51450 −0.757248 0.653128i \(-0.773457\pi\)
−0.757248 + 0.653128i \(0.773457\pi\)
\(360\) 0 0
\(361\) 44.2183 2.32728
\(362\) 36.8423 1.93639
\(363\) 0 0
\(364\) 32.0413 1.67942
\(365\) 13.4965 0.706440
\(366\) 0 0
\(367\) 19.0084 0.992230 0.496115 0.868257i \(-0.334759\pi\)
0.496115 + 0.868257i \(0.334759\pi\)
\(368\) 3.54898 0.185003
\(369\) 0 0
\(370\) −9.19375 −0.477961
\(371\) 22.9658 1.19233
\(372\) 0 0
\(373\) 2.88462 0.149360 0.0746800 0.997208i \(-0.476206\pi\)
0.0746800 + 0.997208i \(0.476206\pi\)
\(374\) −4.11609 −0.212838
\(375\) 0 0
\(376\) 33.3581 1.72031
\(377\) 2.25623 0.116202
\(378\) 0 0
\(379\) −21.1117 −1.08444 −0.542218 0.840238i \(-0.682415\pi\)
−0.542218 + 0.840238i \(0.682415\pi\)
\(380\) −31.3923 −1.61039
\(381\) 0 0
\(382\) −32.6404 −1.67003
\(383\) 2.99855 0.153219 0.0766093 0.997061i \(-0.475591\pi\)
0.0766093 + 0.997061i \(0.475591\pi\)
\(384\) 0 0
\(385\) −23.6912 −1.20742
\(386\) 12.5061 0.636542
\(387\) 0 0
\(388\) −11.7454 −0.596283
\(389\) −10.9069 −0.552999 −0.276500 0.961014i \(-0.589174\pi\)
−0.276500 + 0.961014i \(0.589174\pi\)
\(390\) 0 0
\(391\) 0.246305 0.0124562
\(392\) −28.2122 −1.42493
\(393\) 0 0
\(394\) −24.2726 −1.22284
\(395\) 5.22222 0.262758
\(396\) 0 0
\(397\) −3.44546 −0.172923 −0.0864613 0.996255i \(-0.527556\pi\)
−0.0864613 + 0.996255i \(0.527556\pi\)
\(398\) 52.0474 2.60890
\(399\) 0 0
\(400\) 3.69199 0.184599
\(401\) −3.05491 −0.152555 −0.0762775 0.997087i \(-0.524303\pi\)
−0.0762775 + 0.997087i \(0.524303\pi\)
\(402\) 0 0
\(403\) −8.91435 −0.444055
\(404\) 57.1474 2.84319
\(405\) 0 0
\(406\) −8.77241 −0.435367
\(407\) −24.8291 −1.23073
\(408\) 0 0
\(409\) 24.0609 1.18974 0.594868 0.803823i \(-0.297204\pi\)
0.594868 + 0.803823i \(0.297204\pi\)
\(410\) 12.2057 0.602796
\(411\) 0 0
\(412\) 18.8413 0.928244
\(413\) −11.4875 −0.565265
\(414\) 0 0
\(415\) −0.195206 −0.00958228
\(416\) 1.12502 0.0551585
\(417\) 0 0
\(418\) −127.725 −6.24724
\(419\) −31.7868 −1.55289 −0.776443 0.630188i \(-0.782978\pi\)
−0.776443 + 0.630188i \(0.782978\pi\)
\(420\) 0 0
\(421\) −21.4413 −1.04498 −0.522492 0.852644i \(-0.674997\pi\)
−0.522492 + 0.852644i \(0.674997\pi\)
\(422\) −43.8030 −2.13230
\(423\) 0 0
\(424\) −30.3380 −1.47334
\(425\) 0.256230 0.0124290
\(426\) 0 0
\(427\) 51.8208 2.50778
\(428\) −47.1264 −2.27794
\(429\) 0 0
\(430\) −15.1171 −0.729013
\(431\) −19.8462 −0.955959 −0.477980 0.878371i \(-0.658631\pi\)
−0.477980 + 0.878371i \(0.658631\pi\)
\(432\) 0 0
\(433\) 2.40179 0.115422 0.0577112 0.998333i \(-0.481620\pi\)
0.0577112 + 0.998333i \(0.481620\pi\)
\(434\) 34.6597 1.66372
\(435\) 0 0
\(436\) −17.0096 −0.814613
\(437\) 7.64301 0.365615
\(438\) 0 0
\(439\) 3.51627 0.167822 0.0839111 0.996473i \(-0.473259\pi\)
0.0839111 + 0.996473i \(0.473259\pi\)
\(440\) 31.2963 1.49199
\(441\) 0 0
\(442\) −1.40996 −0.0670650
\(443\) 40.2174 1.91079 0.955393 0.295338i \(-0.0954321\pi\)
0.955393 + 0.295338i \(0.0954321\pi\)
\(444\) 0 0
\(445\) 3.72323 0.176498
\(446\) −26.6266 −1.26081
\(447\) 0 0
\(448\) −30.9334 −1.46147
\(449\) −12.8868 −0.608163 −0.304082 0.952646i \(-0.598350\pi\)
−0.304082 + 0.952646i \(0.598350\pi\)
\(450\) 0 0
\(451\) 32.9633 1.55218
\(452\) −39.4155 −1.85395
\(453\) 0 0
\(454\) 66.3115 3.11215
\(455\) −8.11538 −0.380455
\(456\) 0 0
\(457\) 11.1142 0.519901 0.259950 0.965622i \(-0.416294\pi\)
0.259950 + 0.965622i \(0.416294\pi\)
\(458\) −9.58969 −0.448097
\(459\) 0 0
\(460\) −3.79528 −0.176956
\(461\) −15.3222 −0.713625 −0.356813 0.934176i \(-0.616137\pi\)
−0.356813 + 0.934176i \(0.616137\pi\)
\(462\) 0 0
\(463\) 41.2367 1.91643 0.958216 0.286046i \(-0.0923410\pi\)
0.958216 + 0.286046i \(0.0923410\pi\)
\(464\) 3.69199 0.171396
\(465\) 0 0
\(466\) 47.1543 2.18438
\(467\) −28.0043 −1.29588 −0.647941 0.761690i \(-0.724370\pi\)
−0.647941 + 0.761690i \(0.724370\pi\)
\(468\) 0 0
\(469\) −33.6397 −1.55334
\(470\) −17.1224 −0.789797
\(471\) 0 0
\(472\) 15.1751 0.698492
\(473\) −40.8261 −1.87718
\(474\) 0 0
\(475\) 7.95099 0.364817
\(476\) 3.63879 0.166784
\(477\) 0 0
\(478\) −26.4555 −1.21005
\(479\) −6.36308 −0.290737 −0.145368 0.989378i \(-0.546437\pi\)
−0.145368 + 0.989378i \(0.546437\pi\)
\(480\) 0 0
\(481\) −8.50517 −0.387802
\(482\) 54.8347 2.49765
\(483\) 0 0
\(484\) 127.857 5.81166
\(485\) 2.97486 0.135082
\(486\) 0 0
\(487\) 3.88748 0.176159 0.0880793 0.996113i \(-0.471927\pi\)
0.0880793 + 0.996113i \(0.471927\pi\)
\(488\) −68.4556 −3.09884
\(489\) 0 0
\(490\) 14.4810 0.654185
\(491\) 9.96901 0.449895 0.224948 0.974371i \(-0.427779\pi\)
0.224948 + 0.974371i \(0.427779\pi\)
\(492\) 0 0
\(493\) 0.256230 0.0115400
\(494\) −43.7520 −1.96850
\(495\) 0 0
\(496\) −14.5870 −0.654976
\(497\) −42.8100 −1.92029
\(498\) 0 0
\(499\) 6.60090 0.295497 0.147749 0.989025i \(-0.452797\pi\)
0.147749 + 0.989025i \(0.452797\pi\)
\(500\) −3.94822 −0.176570
\(501\) 0 0
\(502\) 40.7957 1.82080
\(503\) −32.5121 −1.44964 −0.724821 0.688937i \(-0.758078\pi\)
−0.724821 + 0.688937i \(0.758078\pi\)
\(504\) 0 0
\(505\) −14.4742 −0.644094
\(506\) −15.4418 −0.686472
\(507\) 0 0
\(508\) 27.5383 1.22181
\(509\) 4.46788 0.198035 0.0990177 0.995086i \(-0.468430\pi\)
0.0990177 + 0.995086i \(0.468430\pi\)
\(510\) 0 0
\(511\) −48.5453 −2.14752
\(512\) 36.9259 1.63191
\(513\) 0 0
\(514\) 54.6049 2.40852
\(515\) −4.77210 −0.210284
\(516\) 0 0
\(517\) −46.2416 −2.03370
\(518\) 33.0688 1.45296
\(519\) 0 0
\(520\) 10.7205 0.470124
\(521\) 11.1982 0.490604 0.245302 0.969447i \(-0.421113\pi\)
0.245302 + 0.969447i \(0.421113\pi\)
\(522\) 0 0
\(523\) 21.3797 0.934867 0.467434 0.884028i \(-0.345179\pi\)
0.467434 + 0.884028i \(0.345179\pi\)
\(524\) 83.2029 3.63474
\(525\) 0 0
\(526\) −3.62334 −0.157985
\(527\) −1.01236 −0.0440992
\(528\) 0 0
\(529\) −22.0760 −0.959825
\(530\) 15.5722 0.676413
\(531\) 0 0
\(532\) 112.914 4.89545
\(533\) 11.2915 0.489090
\(534\) 0 0
\(535\) 11.9361 0.516044
\(536\) 44.4383 1.91944
\(537\) 0 0
\(538\) 5.09556 0.219685
\(539\) 39.1081 1.68451
\(540\) 0 0
\(541\) 6.28388 0.270165 0.135083 0.990834i \(-0.456870\pi\)
0.135083 + 0.990834i \(0.456870\pi\)
\(542\) −34.8493 −1.49691
\(543\) 0 0
\(544\) 0.127763 0.00547780
\(545\) 4.30818 0.184542
\(546\) 0 0
\(547\) 27.3897 1.17110 0.585551 0.810636i \(-0.300878\pi\)
0.585551 + 0.810636i \(0.300878\pi\)
\(548\) 51.1268 2.18403
\(549\) 0 0
\(550\) −16.0641 −0.684973
\(551\) 7.95099 0.338724
\(552\) 0 0
\(553\) −18.7837 −0.798763
\(554\) 33.0923 1.40596
\(555\) 0 0
\(556\) −34.5624 −1.46577
\(557\) 26.3046 1.11456 0.557280 0.830325i \(-0.311845\pi\)
0.557280 + 0.830325i \(0.311845\pi\)
\(558\) 0 0
\(559\) −13.9849 −0.591498
\(560\) −13.2796 −0.561167
\(561\) 0 0
\(562\) 40.2946 1.69972
\(563\) 21.8053 0.918984 0.459492 0.888182i \(-0.348031\pi\)
0.459492 + 0.888182i \(0.348031\pi\)
\(564\) 0 0
\(565\) 9.98310 0.419992
\(566\) 50.5608 2.12523
\(567\) 0 0
\(568\) 56.5523 2.37288
\(569\) 7.77161 0.325803 0.162901 0.986642i \(-0.447915\pi\)
0.162901 + 0.986642i \(0.447915\pi\)
\(570\) 0 0
\(571\) −16.9439 −0.709079 −0.354539 0.935041i \(-0.615362\pi\)
−0.354539 + 0.935041i \(0.615362\pi\)
\(572\) 58.6741 2.45329
\(573\) 0 0
\(574\) −43.9023 −1.83245
\(575\) 0.961264 0.0400875
\(576\) 0 0
\(577\) −7.67750 −0.319619 −0.159809 0.987148i \(-0.551088\pi\)
−0.159809 + 0.987148i \(0.551088\pi\)
\(578\) 41.3011 1.71790
\(579\) 0 0
\(580\) −3.94822 −0.163941
\(581\) 0.702132 0.0291293
\(582\) 0 0
\(583\) 42.0550 1.74174
\(584\) 64.1287 2.65366
\(585\) 0 0
\(586\) 11.8697 0.490332
\(587\) −7.58590 −0.313104 −0.156552 0.987670i \(-0.550038\pi\)
−0.156552 + 0.987670i \(0.550038\pi\)
\(588\) 0 0
\(589\) −31.4143 −1.29440
\(590\) −7.78924 −0.320678
\(591\) 0 0
\(592\) −13.9175 −0.572004
\(593\) −3.02067 −0.124044 −0.0620220 0.998075i \(-0.519755\pi\)
−0.0620220 + 0.998075i \(0.519755\pi\)
\(594\) 0 0
\(595\) −0.921628 −0.0377831
\(596\) −25.2193 −1.03302
\(597\) 0 0
\(598\) −5.28956 −0.216306
\(599\) 28.2147 1.15282 0.576410 0.817161i \(-0.304453\pi\)
0.576410 + 0.817161i \(0.304453\pi\)
\(600\) 0 0
\(601\) 7.40917 0.302226 0.151113 0.988516i \(-0.451714\pi\)
0.151113 + 0.988516i \(0.451714\pi\)
\(602\) 54.3744 2.21614
\(603\) 0 0
\(604\) 69.0512 2.80965
\(605\) −32.3834 −1.31657
\(606\) 0 0
\(607\) −31.4998 −1.27854 −0.639269 0.768983i \(-0.720763\pi\)
−0.639269 + 0.768983i \(0.720763\pi\)
\(608\) 3.96458 0.160785
\(609\) 0 0
\(610\) 35.1376 1.42268
\(611\) −15.8400 −0.640816
\(612\) 0 0
\(613\) 26.6641 1.07695 0.538477 0.842640i \(-0.319000\pi\)
0.538477 + 0.842640i \(0.319000\pi\)
\(614\) −32.5894 −1.31520
\(615\) 0 0
\(616\) −112.569 −4.53553
\(617\) 42.6460 1.71686 0.858431 0.512929i \(-0.171440\pi\)
0.858431 + 0.512929i \(0.171440\pi\)
\(618\) 0 0
\(619\) −23.6243 −0.949541 −0.474771 0.880110i \(-0.657469\pi\)
−0.474771 + 0.880110i \(0.657469\pi\)
\(620\) 15.5994 0.626486
\(621\) 0 0
\(622\) −3.11997 −0.125100
\(623\) −13.3920 −0.536539
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −71.3296 −2.85090
\(627\) 0 0
\(628\) −25.3779 −1.01269
\(629\) −0.965894 −0.0385127
\(630\) 0 0
\(631\) 22.5629 0.898213 0.449107 0.893478i \(-0.351742\pi\)
0.449107 + 0.893478i \(0.351742\pi\)
\(632\) 24.8134 0.987023
\(633\) 0 0
\(634\) −61.9257 −2.45939
\(635\) −6.97486 −0.276789
\(636\) 0 0
\(637\) 13.3964 0.530785
\(638\) −16.0641 −0.635982
\(639\) 0 0
\(640\) −19.9774 −0.789678
\(641\) 6.65061 0.262683 0.131342 0.991337i \(-0.458072\pi\)
0.131342 + 0.991337i \(0.458072\pi\)
\(642\) 0 0
\(643\) 13.5539 0.534514 0.267257 0.963625i \(-0.413883\pi\)
0.267257 + 0.963625i \(0.413883\pi\)
\(644\) 13.6512 0.537931
\(645\) 0 0
\(646\) −4.96872 −0.195492
\(647\) −6.91189 −0.271734 −0.135867 0.990727i \(-0.543382\pi\)
−0.135867 + 0.990727i \(0.543382\pi\)
\(648\) 0 0
\(649\) −21.0360 −0.825735
\(650\) −5.50271 −0.215834
\(651\) 0 0
\(652\) 68.1961 2.67077
\(653\) 22.0261 0.861947 0.430974 0.902364i \(-0.358170\pi\)
0.430974 + 0.902364i \(0.358170\pi\)
\(654\) 0 0
\(655\) −21.0735 −0.823411
\(656\) 18.4769 0.721402
\(657\) 0 0
\(658\) 61.5871 2.40092
\(659\) −24.9205 −0.970766 −0.485383 0.874302i \(-0.661320\pi\)
−0.485383 + 0.874302i \(0.661320\pi\)
\(660\) 0 0
\(661\) −24.5688 −0.955617 −0.477809 0.878464i \(-0.658569\pi\)
−0.477809 + 0.878464i \(0.658569\pi\)
\(662\) 48.3889 1.88069
\(663\) 0 0
\(664\) −0.927522 −0.0359948
\(665\) −28.5987 −1.10901
\(666\) 0 0
\(667\) 0.961264 0.0372203
\(668\) 84.8258 3.28201
\(669\) 0 0
\(670\) −22.8097 −0.881215
\(671\) 94.8942 3.66335
\(672\) 0 0
\(673\) 20.8663 0.804337 0.402169 0.915566i \(-0.368257\pi\)
0.402169 + 0.915566i \(0.368257\pi\)
\(674\) −52.9709 −2.04036
\(675\) 0 0
\(676\) −31.2281 −1.20108
\(677\) 12.9901 0.499250 0.249625 0.968343i \(-0.419693\pi\)
0.249625 + 0.968343i \(0.419693\pi\)
\(678\) 0 0
\(679\) −10.7002 −0.410637
\(680\) 1.21748 0.0466881
\(681\) 0 0
\(682\) 63.4690 2.43035
\(683\) −25.4388 −0.973388 −0.486694 0.873573i \(-0.661797\pi\)
−0.486694 + 0.873573i \(0.661797\pi\)
\(684\) 0 0
\(685\) −12.9493 −0.494769
\(686\) 9.32048 0.355858
\(687\) 0 0
\(688\) −22.8842 −0.872453
\(689\) 14.4059 0.548820
\(690\) 0 0
\(691\) 7.63166 0.290322 0.145161 0.989408i \(-0.453630\pi\)
0.145161 + 0.989408i \(0.453630\pi\)
\(692\) 18.3271 0.696694
\(693\) 0 0
\(694\) 23.0991 0.876831
\(695\) 8.75391 0.332055
\(696\) 0 0
\(697\) 1.28233 0.0485716
\(698\) −62.9395 −2.38230
\(699\) 0 0
\(700\) 14.2013 0.536757
\(701\) 20.1233 0.760045 0.380023 0.924977i \(-0.375916\pi\)
0.380023 + 0.924977i \(0.375916\pi\)
\(702\) 0 0
\(703\) −29.9724 −1.13043
\(704\) −56.6453 −2.13490
\(705\) 0 0
\(706\) 48.7152 1.83342
\(707\) 52.0620 1.95799
\(708\) 0 0
\(709\) −0.0528446 −0.00198462 −0.000992311 1.00000i \(-0.500316\pi\)
−0.000992311 1.00000i \(0.500316\pi\)
\(710\) −29.0277 −1.08939
\(711\) 0 0
\(712\) 17.6909 0.662995
\(713\) −3.79795 −0.142234
\(714\) 0 0
\(715\) −14.8609 −0.555766
\(716\) −48.5841 −1.81567
\(717\) 0 0
\(718\) 69.9856 2.61184
\(719\) 3.64149 0.135805 0.0679024 0.997692i \(-0.478369\pi\)
0.0679024 + 0.997692i \(0.478369\pi\)
\(720\) 0 0
\(721\) 17.1647 0.639246
\(722\) −107.844 −4.01353
\(723\) 0 0
\(724\) −59.6423 −2.21659
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −7.68378 −0.284976 −0.142488 0.989797i \(-0.545510\pi\)
−0.142488 + 0.989797i \(0.545510\pi\)
\(728\) −38.5603 −1.42914
\(729\) 0 0
\(730\) −32.9166 −1.21830
\(731\) −1.58820 −0.0587418
\(732\) 0 0
\(733\) −43.4669 −1.60549 −0.802743 0.596325i \(-0.796627\pi\)
−0.802743 + 0.596325i \(0.796627\pi\)
\(734\) −46.3595 −1.71116
\(735\) 0 0
\(736\) 0.479312 0.0176677
\(737\) −61.6010 −2.26910
\(738\) 0 0
\(739\) 38.2405 1.40670 0.703350 0.710844i \(-0.251687\pi\)
0.703350 + 0.710844i \(0.251687\pi\)
\(740\) 14.8833 0.547123
\(741\) 0 0
\(742\) −56.0113 −2.05624
\(743\) −42.7317 −1.56767 −0.783836 0.620968i \(-0.786740\pi\)
−0.783836 + 0.620968i \(0.786740\pi\)
\(744\) 0 0
\(745\) 6.38751 0.234020
\(746\) −7.03529 −0.257580
\(747\) 0 0
\(748\) 6.66335 0.243636
\(749\) −42.9328 −1.56873
\(750\) 0 0
\(751\) 30.7352 1.12154 0.560772 0.827971i \(-0.310505\pi\)
0.560772 + 0.827971i \(0.310505\pi\)
\(752\) −25.9198 −0.945197
\(753\) 0 0
\(754\) −5.50271 −0.200397
\(755\) −17.4892 −0.636498
\(756\) 0 0
\(757\) −47.7626 −1.73596 −0.867980 0.496599i \(-0.834582\pi\)
−0.867980 + 0.496599i \(0.834582\pi\)
\(758\) 51.4893 1.87018
\(759\) 0 0
\(760\) 37.7792 1.37039
\(761\) 7.02492 0.254653 0.127327 0.991861i \(-0.459360\pi\)
0.127327 + 0.991861i \(0.459360\pi\)
\(762\) 0 0
\(763\) −15.4960 −0.560992
\(764\) 52.8401 1.91169
\(765\) 0 0
\(766\) −7.31315 −0.264235
\(767\) −7.20584 −0.260188
\(768\) 0 0
\(769\) −25.1269 −0.906100 −0.453050 0.891485i \(-0.649664\pi\)
−0.453050 + 0.891485i \(0.649664\pi\)
\(770\) 57.7804 2.08226
\(771\) 0 0
\(772\) −20.2455 −0.728652
\(773\) 10.8598 0.390599 0.195300 0.980744i \(-0.437432\pi\)
0.195300 + 0.980744i \(0.437432\pi\)
\(774\) 0 0
\(775\) −3.95099 −0.141924
\(776\) 14.1351 0.507420
\(777\) 0 0
\(778\) 26.6007 0.953681
\(779\) 39.7915 1.42568
\(780\) 0 0
\(781\) −78.3937 −2.80515
\(782\) −0.600712 −0.0214814
\(783\) 0 0
\(784\) 21.9213 0.782902
\(785\) 6.42770 0.229414
\(786\) 0 0
\(787\) 16.3114 0.581438 0.290719 0.956809i \(-0.406106\pi\)
0.290719 + 0.956809i \(0.406106\pi\)
\(788\) 39.2939 1.39979
\(789\) 0 0
\(790\) −12.7365 −0.453143
\(791\) −35.9080 −1.27674
\(792\) 0 0
\(793\) 32.5059 1.15432
\(794\) 8.40312 0.298216
\(795\) 0 0
\(796\) −84.2571 −2.98641
\(797\) −4.55126 −0.161214 −0.0806070 0.996746i \(-0.525686\pi\)
−0.0806070 + 0.996746i \(0.525686\pi\)
\(798\) 0 0
\(799\) −1.79887 −0.0636396
\(800\) 0.498627 0.0176291
\(801\) 0 0
\(802\) 7.45061 0.263090
\(803\) −88.8962 −3.13708
\(804\) 0 0
\(805\) −3.45755 −0.121863
\(806\) 21.7412 0.765800
\(807\) 0 0
\(808\) −68.7743 −2.41947
\(809\) 2.72709 0.0958793 0.0479396 0.998850i \(-0.484734\pi\)
0.0479396 + 0.998850i \(0.484734\pi\)
\(810\) 0 0
\(811\) −39.1620 −1.37516 −0.687581 0.726107i \(-0.741328\pi\)
−0.687581 + 0.726107i \(0.741328\pi\)
\(812\) 14.2013 0.498366
\(813\) 0 0
\(814\) 60.5556 2.12247
\(815\) −17.2726 −0.605034
\(816\) 0 0
\(817\) −49.2830 −1.72419
\(818\) −58.6821 −2.05177
\(819\) 0 0
\(820\) −19.7592 −0.690022
\(821\) 6.22315 0.217190 0.108595 0.994086i \(-0.465365\pi\)
0.108595 + 0.994086i \(0.465365\pi\)
\(822\) 0 0
\(823\) 43.5896 1.51944 0.759718 0.650253i \(-0.225337\pi\)
0.759718 + 0.650253i \(0.225337\pi\)
\(824\) −22.6747 −0.789909
\(825\) 0 0
\(826\) 28.0169 0.974833
\(827\) 0.982635 0.0341696 0.0170848 0.999854i \(-0.494561\pi\)
0.0170848 + 0.999854i \(0.494561\pi\)
\(828\) 0 0
\(829\) −11.5045 −0.399568 −0.199784 0.979840i \(-0.564024\pi\)
−0.199784 + 0.979840i \(0.564024\pi\)
\(830\) 0.476087 0.0165252
\(831\) 0 0
\(832\) −19.4037 −0.672704
\(833\) 1.52137 0.0527124
\(834\) 0 0
\(835\) −21.4846 −0.743505
\(836\) 206.768 7.15123
\(837\) 0 0
\(838\) 77.5247 2.67805
\(839\) 23.6588 0.816791 0.408396 0.912805i \(-0.366088\pi\)
0.408396 + 0.912805i \(0.366088\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 52.2930 1.80214
\(843\) 0 0
\(844\) 70.9107 2.44085
\(845\) 7.90943 0.272093
\(846\) 0 0
\(847\) 116.479 4.00227
\(848\) 23.5731 0.809503
\(849\) 0 0
\(850\) −0.624919 −0.0214345
\(851\) −3.62362 −0.124216
\(852\) 0 0
\(853\) −7.87029 −0.269474 −0.134737 0.990881i \(-0.543019\pi\)
−0.134737 + 0.990881i \(0.543019\pi\)
\(854\) −126.385 −4.32482
\(855\) 0 0
\(856\) 56.7145 1.93846
\(857\) −18.9403 −0.646988 −0.323494 0.946230i \(-0.604858\pi\)
−0.323494 + 0.946230i \(0.604858\pi\)
\(858\) 0 0
\(859\) −16.6320 −0.567477 −0.283739 0.958902i \(-0.591575\pi\)
−0.283739 + 0.958902i \(0.591575\pi\)
\(860\) 24.4724 0.834503
\(861\) 0 0
\(862\) 48.4029 1.64861
\(863\) −8.81066 −0.299918 −0.149959 0.988692i \(-0.547914\pi\)
−0.149959 + 0.988692i \(0.547914\pi\)
\(864\) 0 0
\(865\) −4.64188 −0.157829
\(866\) −5.85771 −0.199053
\(867\) 0 0
\(868\) −56.1091 −1.90447
\(869\) −34.3967 −1.16683
\(870\) 0 0
\(871\) −21.1013 −0.714991
\(872\) 20.4703 0.693212
\(873\) 0 0
\(874\) −18.6405 −0.630524
\(875\) −3.59688 −0.121597
\(876\) 0 0
\(877\) −16.2007 −0.547060 −0.273530 0.961863i \(-0.588191\pi\)
−0.273530 + 0.961863i \(0.588191\pi\)
\(878\) −8.57581 −0.289419
\(879\) 0 0
\(880\) −24.3177 −0.819748
\(881\) −54.7427 −1.84433 −0.922164 0.386799i \(-0.873581\pi\)
−0.922164 + 0.386799i \(0.873581\pi\)
\(882\) 0 0
\(883\) 42.5560 1.43212 0.716062 0.698036i \(-0.245943\pi\)
0.716062 + 0.698036i \(0.245943\pi\)
\(884\) 2.28252 0.0767695
\(885\) 0 0
\(886\) −98.0861 −3.29527
\(887\) −52.9851 −1.77907 −0.889533 0.456871i \(-0.848970\pi\)
−0.889533 + 0.456871i \(0.848970\pi\)
\(888\) 0 0
\(889\) 25.0877 0.841415
\(890\) −9.08057 −0.304381
\(891\) 0 0
\(892\) 43.1046 1.44325
\(893\) −55.8203 −1.86796
\(894\) 0 0
\(895\) 12.3053 0.411322
\(896\) 71.8564 2.40055
\(897\) 0 0
\(898\) 31.4295 1.04881
\(899\) −3.95099 −0.131773
\(900\) 0 0
\(901\) 1.63601 0.0545035
\(902\) −80.3940 −2.67683
\(903\) 0 0
\(904\) 47.4347 1.57766
\(905\) 15.1061 0.502145
\(906\) 0 0
\(907\) −54.3483 −1.80461 −0.902303 0.431103i \(-0.858124\pi\)
−0.902303 + 0.431103i \(0.858124\pi\)
\(908\) −107.349 −3.56249
\(909\) 0 0
\(910\) 19.7926 0.656118
\(911\) 46.7376 1.54849 0.774243 0.632888i \(-0.218131\pi\)
0.774243 + 0.632888i \(0.218131\pi\)
\(912\) 0 0
\(913\) 1.28574 0.0425519
\(914\) −27.1064 −0.896600
\(915\) 0 0
\(916\) 15.5243 0.512937
\(917\) 75.7989 2.50310
\(918\) 0 0
\(919\) 10.9218 0.360277 0.180139 0.983641i \(-0.442345\pi\)
0.180139 + 0.983641i \(0.442345\pi\)
\(920\) 4.56745 0.150584
\(921\) 0 0
\(922\) 37.3692 1.23069
\(923\) −26.8536 −0.883898
\(924\) 0 0
\(925\) −3.76964 −0.123945
\(926\) −100.572 −3.30500
\(927\) 0 0
\(928\) 0.498627 0.0163682
\(929\) 52.9110 1.73595 0.867976 0.496606i \(-0.165421\pi\)
0.867976 + 0.496606i \(0.165421\pi\)
\(930\) 0 0
\(931\) 47.2092 1.54722
\(932\) −76.3359 −2.50047
\(933\) 0 0
\(934\) 68.2995 2.23483
\(935\) −1.68769 −0.0551932
\(936\) 0 0
\(937\) 27.0005 0.882067 0.441033 0.897491i \(-0.354612\pi\)
0.441033 + 0.897491i \(0.354612\pi\)
\(938\) 82.0437 2.67882
\(939\) 0 0
\(940\) 27.7186 0.904082
\(941\) 9.49673 0.309584 0.154792 0.987947i \(-0.450529\pi\)
0.154792 + 0.987947i \(0.450529\pi\)
\(942\) 0 0
\(943\) 4.81074 0.156659
\(944\) −11.7913 −0.383774
\(945\) 0 0
\(946\) 99.5706 3.23732
\(947\) −25.3434 −0.823549 −0.411775 0.911286i \(-0.635091\pi\)
−0.411775 + 0.911286i \(0.635091\pi\)
\(948\) 0 0
\(949\) −30.4512 −0.988489
\(950\) −19.3917 −0.629148
\(951\) 0 0
\(952\) −4.37912 −0.141928
\(953\) −0.515249 −0.0166906 −0.00834528 0.999965i \(-0.502656\pi\)
−0.00834528 + 0.999965i \(0.502656\pi\)
\(954\) 0 0
\(955\) −13.3833 −0.433073
\(956\) 42.8277 1.38515
\(957\) 0 0
\(958\) 15.5189 0.501393
\(959\) 46.5772 1.50406
\(960\) 0 0
\(961\) −15.3897 −0.496440
\(962\) 20.7432 0.668788
\(963\) 0 0
\(964\) −88.7694 −2.85907
\(965\) 5.12776 0.165068
\(966\) 0 0
\(967\) −41.0700 −1.32072 −0.660360 0.750949i \(-0.729597\pi\)
−0.660360 + 0.750949i \(0.729597\pi\)
\(968\) −153.870 −4.94556
\(969\) 0 0
\(970\) −7.25539 −0.232956
\(971\) 35.9638 1.15413 0.577066 0.816697i \(-0.304198\pi\)
0.577066 + 0.816697i \(0.304198\pi\)
\(972\) 0 0
\(973\) −31.4868 −1.00942
\(974\) −9.48117 −0.303796
\(975\) 0 0
\(976\) 53.1910 1.70260
\(977\) 56.4987 1.80755 0.903777 0.428004i \(-0.140783\pi\)
0.903777 + 0.428004i \(0.140783\pi\)
\(978\) 0 0
\(979\) −24.5234 −0.783772
\(980\) −23.4426 −0.748848
\(981\) 0 0
\(982\) −24.3134 −0.775871
\(983\) −3.49908 −0.111603 −0.0558017 0.998442i \(-0.517771\pi\)
−0.0558017 + 0.998442i \(0.517771\pi\)
\(984\) 0 0
\(985\) −9.95230 −0.317107
\(986\) −0.624919 −0.0199015
\(987\) 0 0
\(988\) 70.8281 2.25334
\(989\) −5.95825 −0.189461
\(990\) 0 0
\(991\) 2.42787 0.0771237 0.0385618 0.999256i \(-0.487722\pi\)
0.0385618 + 0.999256i \(0.487722\pi\)
\(992\) −1.97007 −0.0625498
\(993\) 0 0
\(994\) 104.409 3.31166
\(995\) 21.3405 0.676541
\(996\) 0 0
\(997\) −28.5463 −0.904070 −0.452035 0.892000i \(-0.649302\pi\)
−0.452035 + 0.892000i \(0.649302\pi\)
\(998\) −16.0989 −0.509602
\(999\) 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 1305.2.a.s.1.2 7
3.2 odd 2 1305.2.a.t.1.6 yes 7
5.4 even 2 6525.2.a.bw.1.6 7
15.14 odd 2 6525.2.a.bv.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.s.1.2 7 1.1 even 1 trivial
1305.2.a.t.1.6 yes 7 3.2 odd 2
6525.2.a.bv.1.2 7 15.14 odd 2
6525.2.a.bw.1.6 7 5.4 even 2