Properties

Label 2151.2.a.h.1.11
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $12$
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] = [2151,2,Mod(1,2151)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2151, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2151.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.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: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 256 x^{7} - 134 x^{6} + 571 x^{5} + 23 x^{4} - 479 x^{3} + 129 x^{2} + 88 x - 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 717)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-2.27963\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.27963 q^{2} +3.19672 q^{4} +2.95582 q^{5} +0.183248 q^{7} +2.72808 q^{8} +O(q^{10})\) \(q+2.27963 q^{2} +3.19672 q^{4} +2.95582 q^{5} +0.183248 q^{7} +2.72808 q^{8} +6.73817 q^{10} +2.01770 q^{11} +0.984021 q^{13} +0.417737 q^{14} -0.174418 q^{16} +6.12205 q^{17} -3.01138 q^{19} +9.44892 q^{20} +4.59962 q^{22} -3.53879 q^{23} +3.73684 q^{25} +2.24321 q^{26} +0.585791 q^{28} -1.30450 q^{29} -8.90832 q^{31} -5.85377 q^{32} +13.9560 q^{34} +0.541646 q^{35} +9.94350 q^{37} -6.86484 q^{38} +8.06371 q^{40} +8.60897 q^{41} +1.48207 q^{43} +6.45004 q^{44} -8.06713 q^{46} -7.82999 q^{47} -6.96642 q^{49} +8.51863 q^{50} +3.14564 q^{52} -1.11957 q^{53} +5.96396 q^{55} +0.499914 q^{56} -2.97377 q^{58} -5.43635 q^{59} +7.43259 q^{61} -20.3077 q^{62} -12.9956 q^{64} +2.90859 q^{65} +8.22250 q^{67} +19.5705 q^{68} +1.23475 q^{70} -12.2519 q^{71} +6.79732 q^{73} +22.6675 q^{74} -9.62655 q^{76} +0.369739 q^{77} +4.19772 q^{79} -0.515549 q^{80} +19.6253 q^{82} -3.07677 q^{83} +18.0957 q^{85} +3.37856 q^{86} +5.50446 q^{88} -11.9885 q^{89} +0.180320 q^{91} -11.3125 q^{92} -17.8495 q^{94} -8.90109 q^{95} +9.27936 q^{97} -15.8809 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.27963 1.61194 0.805972 0.591954i \(-0.201643\pi\)
0.805972 + 0.591954i \(0.201643\pi\)
\(3\) 0 0
\(4\) 3.19672 1.59836
\(5\) 2.95582 1.32188 0.660940 0.750438i \(-0.270158\pi\)
0.660940 + 0.750438i \(0.270158\pi\)
\(6\) 0 0
\(7\) 0.183248 0.0692611 0.0346305 0.999400i \(-0.488975\pi\)
0.0346305 + 0.999400i \(0.488975\pi\)
\(8\) 2.72808 0.964523
\(9\) 0 0
\(10\) 6.73817 2.13080
\(11\) 2.01770 0.608361 0.304180 0.952614i \(-0.401617\pi\)
0.304180 + 0.952614i \(0.401617\pi\)
\(12\) 0 0
\(13\) 0.984021 0.272918 0.136459 0.990646i \(-0.456428\pi\)
0.136459 + 0.990646i \(0.456428\pi\)
\(14\) 0.417737 0.111645
\(15\) 0 0
\(16\) −0.174418 −0.0436046
\(17\) 6.12205 1.48482 0.742408 0.669948i \(-0.233684\pi\)
0.742408 + 0.669948i \(0.233684\pi\)
\(18\) 0 0
\(19\) −3.01138 −0.690859 −0.345429 0.938445i \(-0.612267\pi\)
−0.345429 + 0.938445i \(0.612267\pi\)
\(20\) 9.44892 2.11284
\(21\) 0 0
\(22\) 4.59962 0.980643
\(23\) −3.53879 −0.737888 −0.368944 0.929452i \(-0.620281\pi\)
−0.368944 + 0.929452i \(0.620281\pi\)
\(24\) 0 0
\(25\) 3.73684 0.747369
\(26\) 2.24321 0.439929
\(27\) 0 0
\(28\) 0.585791 0.110704
\(29\) −1.30450 −0.242239 −0.121119 0.992638i \(-0.538648\pi\)
−0.121119 + 0.992638i \(0.538648\pi\)
\(30\) 0 0
\(31\) −8.90832 −1.59998 −0.799990 0.600013i \(-0.795162\pi\)
−0.799990 + 0.600013i \(0.795162\pi\)
\(32\) −5.85377 −1.03481
\(33\) 0 0
\(34\) 13.9560 2.39344
\(35\) 0.541646 0.0915549
\(36\) 0 0
\(37\) 9.94350 1.63470 0.817350 0.576141i \(-0.195442\pi\)
0.817350 + 0.576141i \(0.195442\pi\)
\(38\) −6.86484 −1.11362
\(39\) 0 0
\(40\) 8.06371 1.27498
\(41\) 8.60897 1.34450 0.672248 0.740326i \(-0.265329\pi\)
0.672248 + 0.740326i \(0.265329\pi\)
\(42\) 0 0
\(43\) 1.48207 0.226013 0.113006 0.993594i \(-0.463952\pi\)
0.113006 + 0.993594i \(0.463952\pi\)
\(44\) 6.45004 0.972380
\(45\) 0 0
\(46\) −8.06713 −1.18943
\(47\) −7.82999 −1.14212 −0.571061 0.820908i \(-0.693468\pi\)
−0.571061 + 0.820908i \(0.693468\pi\)
\(48\) 0 0
\(49\) −6.96642 −0.995203
\(50\) 8.51863 1.20472
\(51\) 0 0
\(52\) 3.14564 0.436222
\(53\) −1.11957 −0.153785 −0.0768925 0.997039i \(-0.524500\pi\)
−0.0768925 + 0.997039i \(0.524500\pi\)
\(54\) 0 0
\(55\) 5.96396 0.804181
\(56\) 0.499914 0.0668039
\(57\) 0 0
\(58\) −2.97377 −0.390475
\(59\) −5.43635 −0.707753 −0.353876 0.935292i \(-0.615137\pi\)
−0.353876 + 0.935292i \(0.615137\pi\)
\(60\) 0 0
\(61\) 7.43259 0.951646 0.475823 0.879541i \(-0.342150\pi\)
0.475823 + 0.879541i \(0.342150\pi\)
\(62\) −20.3077 −2.57908
\(63\) 0 0
\(64\) −12.9956 −1.62445
\(65\) 2.90859 0.360766
\(66\) 0 0
\(67\) 8.22250 1.00454 0.502269 0.864711i \(-0.332499\pi\)
0.502269 + 0.864711i \(0.332499\pi\)
\(68\) 19.5705 2.37327
\(69\) 0 0
\(70\) 1.23475 0.147581
\(71\) −12.2519 −1.45404 −0.727018 0.686619i \(-0.759094\pi\)
−0.727018 + 0.686619i \(0.759094\pi\)
\(72\) 0 0
\(73\) 6.79732 0.795567 0.397783 0.917479i \(-0.369780\pi\)
0.397783 + 0.917479i \(0.369780\pi\)
\(74\) 22.6675 2.63504
\(75\) 0 0
\(76\) −9.62655 −1.10424
\(77\) 0.369739 0.0421357
\(78\) 0 0
\(79\) 4.19772 0.472280 0.236140 0.971719i \(-0.424118\pi\)
0.236140 + 0.971719i \(0.424118\pi\)
\(80\) −0.515549 −0.0576401
\(81\) 0 0
\(82\) 19.6253 2.16725
\(83\) −3.07677 −0.337719 −0.168860 0.985640i \(-0.554008\pi\)
−0.168860 + 0.985640i \(0.554008\pi\)
\(84\) 0 0
\(85\) 18.0957 1.96275
\(86\) 3.37856 0.364320
\(87\) 0 0
\(88\) 5.50446 0.586778
\(89\) −11.9885 −1.27077 −0.635387 0.772194i \(-0.719159\pi\)
−0.635387 + 0.772194i \(0.719159\pi\)
\(90\) 0 0
\(91\) 0.180320 0.0189026
\(92\) −11.3125 −1.17941
\(93\) 0 0
\(94\) −17.8495 −1.84104
\(95\) −8.90109 −0.913233
\(96\) 0 0
\(97\) 9.27936 0.942176 0.471088 0.882086i \(-0.343861\pi\)
0.471088 + 0.882086i \(0.343861\pi\)
\(98\) −15.8809 −1.60421
\(99\) 0 0
\(100\) 11.9456 1.19456
\(101\) −1.73615 −0.172753 −0.0863765 0.996263i \(-0.527529\pi\)
−0.0863765 + 0.996263i \(0.527529\pi\)
\(102\) 0 0
\(103\) −12.2715 −1.20914 −0.604571 0.796551i \(-0.706656\pi\)
−0.604571 + 0.796551i \(0.706656\pi\)
\(104\) 2.68449 0.263236
\(105\) 0 0
\(106\) −2.55221 −0.247893
\(107\) 8.45372 0.817252 0.408626 0.912702i \(-0.366008\pi\)
0.408626 + 0.912702i \(0.366008\pi\)
\(108\) 0 0
\(109\) 3.16094 0.302763 0.151382 0.988475i \(-0.451628\pi\)
0.151382 + 0.988475i \(0.451628\pi\)
\(110\) 13.5956 1.29629
\(111\) 0 0
\(112\) −0.0319617 −0.00302010
\(113\) −13.0509 −1.22772 −0.613862 0.789413i \(-0.710385\pi\)
−0.613862 + 0.789413i \(0.710385\pi\)
\(114\) 0 0
\(115\) −10.4600 −0.975400
\(116\) −4.17011 −0.387185
\(117\) 0 0
\(118\) −12.3929 −1.14086
\(119\) 1.12185 0.102840
\(120\) 0 0
\(121\) −6.92887 −0.629897
\(122\) 16.9436 1.53400
\(123\) 0 0
\(124\) −28.4774 −2.55735
\(125\) −3.73365 −0.333948
\(126\) 0 0
\(127\) 8.13953 0.722267 0.361133 0.932514i \(-0.382390\pi\)
0.361133 + 0.932514i \(0.382390\pi\)
\(128\) −17.9177 −1.58371
\(129\) 0 0
\(130\) 6.63050 0.581534
\(131\) 12.9651 1.13276 0.566382 0.824143i \(-0.308343\pi\)
0.566382 + 0.824143i \(0.308343\pi\)
\(132\) 0 0
\(133\) −0.551829 −0.0478496
\(134\) 18.7443 1.61926
\(135\) 0 0
\(136\) 16.7015 1.43214
\(137\) 10.6813 0.912565 0.456282 0.889835i \(-0.349181\pi\)
0.456282 + 0.889835i \(0.349181\pi\)
\(138\) 0 0
\(139\) 13.6158 1.15488 0.577440 0.816433i \(-0.304052\pi\)
0.577440 + 0.816433i \(0.304052\pi\)
\(140\) 1.73149 0.146338
\(141\) 0 0
\(142\) −27.9299 −2.34382
\(143\) 1.98546 0.166033
\(144\) 0 0
\(145\) −3.85585 −0.320211
\(146\) 15.4954 1.28241
\(147\) 0 0
\(148\) 31.7866 2.61284
\(149\) −1.78257 −0.146034 −0.0730169 0.997331i \(-0.523263\pi\)
−0.0730169 + 0.997331i \(0.523263\pi\)
\(150\) 0 0
\(151\) 1.16896 0.0951282 0.0475641 0.998868i \(-0.484854\pi\)
0.0475641 + 0.998868i \(0.484854\pi\)
\(152\) −8.21530 −0.666349
\(153\) 0 0
\(154\) 0.842870 0.0679204
\(155\) −26.3313 −2.11498
\(156\) 0 0
\(157\) −21.6019 −1.72402 −0.862010 0.506891i \(-0.830795\pi\)
−0.862010 + 0.506891i \(0.830795\pi\)
\(158\) 9.56925 0.761289
\(159\) 0 0
\(160\) −17.3027 −1.36790
\(161\) −0.648474 −0.0511069
\(162\) 0 0
\(163\) 8.24197 0.645561 0.322780 0.946474i \(-0.395382\pi\)
0.322780 + 0.946474i \(0.395382\pi\)
\(164\) 27.5205 2.14899
\(165\) 0 0
\(166\) −7.01390 −0.544384
\(167\) −3.47700 −0.269059 −0.134529 0.990910i \(-0.542952\pi\)
−0.134529 + 0.990910i \(0.542952\pi\)
\(168\) 0 0
\(169\) −12.0317 −0.925516
\(170\) 41.2514 3.16384
\(171\) 0 0
\(172\) 4.73775 0.361250
\(173\) 4.07647 0.309929 0.154964 0.987920i \(-0.450474\pi\)
0.154964 + 0.987920i \(0.450474\pi\)
\(174\) 0 0
\(175\) 0.684768 0.0517636
\(176\) −0.351925 −0.0265273
\(177\) 0 0
\(178\) −27.3293 −2.04841
\(179\) −10.6107 −0.793084 −0.396542 0.918017i \(-0.629790\pi\)
−0.396542 + 0.918017i \(0.629790\pi\)
\(180\) 0 0
\(181\) −7.81115 −0.580598 −0.290299 0.956936i \(-0.593755\pi\)
−0.290299 + 0.956936i \(0.593755\pi\)
\(182\) 0.411062 0.0304700
\(183\) 0 0
\(184\) −9.65410 −0.711710
\(185\) 29.3911 2.16088
\(186\) 0 0
\(187\) 12.3525 0.903304
\(188\) −25.0303 −1.82552
\(189\) 0 0
\(190\) −20.2912 −1.47208
\(191\) −9.29194 −0.672341 −0.336171 0.941801i \(-0.609132\pi\)
−0.336171 + 0.941801i \(0.609132\pi\)
\(192\) 0 0
\(193\) −6.49189 −0.467297 −0.233648 0.972321i \(-0.575066\pi\)
−0.233648 + 0.972321i \(0.575066\pi\)
\(194\) 21.1535 1.51873
\(195\) 0 0
\(196\) −22.2697 −1.59069
\(197\) −13.1871 −0.939540 −0.469770 0.882789i \(-0.655663\pi\)
−0.469770 + 0.882789i \(0.655663\pi\)
\(198\) 0 0
\(199\) −22.2646 −1.57830 −0.789148 0.614203i \(-0.789477\pi\)
−0.789148 + 0.614203i \(0.789477\pi\)
\(200\) 10.1944 0.720854
\(201\) 0 0
\(202\) −3.95777 −0.278468
\(203\) −0.239046 −0.0167777
\(204\) 0 0
\(205\) 25.4465 1.77726
\(206\) −27.9744 −1.94907
\(207\) 0 0
\(208\) −0.171631 −0.0119005
\(209\) −6.07608 −0.420291
\(210\) 0 0
\(211\) 15.5785 1.07247 0.536234 0.844069i \(-0.319846\pi\)
0.536234 + 0.844069i \(0.319846\pi\)
\(212\) −3.57896 −0.245804
\(213\) 0 0
\(214\) 19.2714 1.31736
\(215\) 4.38071 0.298762
\(216\) 0 0
\(217\) −1.63243 −0.110816
\(218\) 7.20579 0.488037
\(219\) 0 0
\(220\) 19.0651 1.28537
\(221\) 6.02423 0.405234
\(222\) 0 0
\(223\) −2.78655 −0.186601 −0.0933005 0.995638i \(-0.529742\pi\)
−0.0933005 + 0.995638i \(0.529742\pi\)
\(224\) −1.07269 −0.0716721
\(225\) 0 0
\(226\) −29.7512 −1.97902
\(227\) −8.74781 −0.580613 −0.290306 0.956934i \(-0.593757\pi\)
−0.290306 + 0.956934i \(0.593757\pi\)
\(228\) 0 0
\(229\) 5.72893 0.378578 0.189289 0.981921i \(-0.439382\pi\)
0.189289 + 0.981921i \(0.439382\pi\)
\(230\) −23.8449 −1.57229
\(231\) 0 0
\(232\) −3.55877 −0.233645
\(233\) 10.2904 0.674149 0.337074 0.941478i \(-0.390563\pi\)
0.337074 + 0.941478i \(0.390563\pi\)
\(234\) 0 0
\(235\) −23.1440 −1.50975
\(236\) −17.3785 −1.13124
\(237\) 0 0
\(238\) 2.55741 0.165772
\(239\) −1.00000 −0.0646846
\(240\) 0 0
\(241\) −8.38133 −0.539889 −0.269945 0.962876i \(-0.587005\pi\)
−0.269945 + 0.962876i \(0.587005\pi\)
\(242\) −15.7953 −1.01536
\(243\) 0 0
\(244\) 23.7599 1.52107
\(245\) −20.5915 −1.31554
\(246\) 0 0
\(247\) −2.96327 −0.188548
\(248\) −24.3026 −1.54322
\(249\) 0 0
\(250\) −8.51136 −0.538306
\(251\) −27.6155 −1.74308 −0.871538 0.490328i \(-0.836877\pi\)
−0.871538 + 0.490328i \(0.836877\pi\)
\(252\) 0 0
\(253\) −7.14022 −0.448902
\(254\) 18.5551 1.16425
\(255\) 0 0
\(256\) −14.8544 −0.928403
\(257\) 10.4899 0.654344 0.327172 0.944965i \(-0.393904\pi\)
0.327172 + 0.944965i \(0.393904\pi\)
\(258\) 0 0
\(259\) 1.82212 0.113221
\(260\) 9.29794 0.576634
\(261\) 0 0
\(262\) 29.5556 1.82595
\(263\) 8.63295 0.532331 0.266165 0.963927i \(-0.414243\pi\)
0.266165 + 0.963927i \(0.414243\pi\)
\(264\) 0 0
\(265\) −3.30925 −0.203286
\(266\) −1.25797 −0.0771308
\(267\) 0 0
\(268\) 26.2850 1.60561
\(269\) 15.2172 0.927812 0.463906 0.885885i \(-0.346448\pi\)
0.463906 + 0.885885i \(0.346448\pi\)
\(270\) 0 0
\(271\) 12.6574 0.768882 0.384441 0.923149i \(-0.374394\pi\)
0.384441 + 0.923149i \(0.374394\pi\)
\(272\) −1.06780 −0.0647448
\(273\) 0 0
\(274\) 24.3494 1.47100
\(275\) 7.53985 0.454670
\(276\) 0 0
\(277\) 1.31116 0.0787798 0.0393899 0.999224i \(-0.487459\pi\)
0.0393899 + 0.999224i \(0.487459\pi\)
\(278\) 31.0391 1.86160
\(279\) 0 0
\(280\) 1.47765 0.0883068
\(281\) −0.523608 −0.0312358 −0.0156179 0.999878i \(-0.504972\pi\)
−0.0156179 + 0.999878i \(0.504972\pi\)
\(282\) 0 0
\(283\) 27.6919 1.64611 0.823057 0.567959i \(-0.192267\pi\)
0.823057 + 0.567959i \(0.192267\pi\)
\(284\) −39.1660 −2.32407
\(285\) 0 0
\(286\) 4.52613 0.267636
\(287\) 1.57757 0.0931212
\(288\) 0 0
\(289\) 20.4795 1.20468
\(290\) −8.78992 −0.516162
\(291\) 0 0
\(292\) 21.7291 1.27160
\(293\) −1.42628 −0.0833244 −0.0416622 0.999132i \(-0.513265\pi\)
−0.0416622 + 0.999132i \(0.513265\pi\)
\(294\) 0 0
\(295\) −16.0689 −0.935565
\(296\) 27.1267 1.57671
\(297\) 0 0
\(298\) −4.06360 −0.235398
\(299\) −3.48224 −0.201383
\(300\) 0 0
\(301\) 0.271585 0.0156539
\(302\) 2.66479 0.153341
\(303\) 0 0
\(304\) 0.525241 0.0301246
\(305\) 21.9694 1.25796
\(306\) 0 0
\(307\) 30.6823 1.75113 0.875565 0.483100i \(-0.160489\pi\)
0.875565 + 0.483100i \(0.160489\pi\)
\(308\) 1.18195 0.0673481
\(309\) 0 0
\(310\) −60.0257 −3.40923
\(311\) 0.537158 0.0304594 0.0152297 0.999884i \(-0.495152\pi\)
0.0152297 + 0.999884i \(0.495152\pi\)
\(312\) 0 0
\(313\) −4.82980 −0.272996 −0.136498 0.990640i \(-0.543585\pi\)
−0.136498 + 0.990640i \(0.543585\pi\)
\(314\) −49.2444 −2.77902
\(315\) 0 0
\(316\) 13.4189 0.754874
\(317\) −32.0675 −1.80109 −0.900546 0.434761i \(-0.856833\pi\)
−0.900546 + 0.434761i \(0.856833\pi\)
\(318\) 0 0
\(319\) −2.63209 −0.147369
\(320\) −38.4126 −2.14733
\(321\) 0 0
\(322\) −1.47828 −0.0823814
\(323\) −18.4358 −1.02580
\(324\) 0 0
\(325\) 3.67713 0.203971
\(326\) 18.7886 1.04061
\(327\) 0 0
\(328\) 23.4860 1.29680
\(329\) −1.43483 −0.0791046
\(330\) 0 0
\(331\) 29.6623 1.63038 0.815192 0.579190i \(-0.196631\pi\)
0.815192 + 0.579190i \(0.196631\pi\)
\(332\) −9.83556 −0.539797
\(333\) 0 0
\(334\) −7.92629 −0.433707
\(335\) 24.3042 1.32788
\(336\) 0 0
\(337\) 5.37623 0.292862 0.146431 0.989221i \(-0.453221\pi\)
0.146431 + 0.989221i \(0.453221\pi\)
\(338\) −27.4278 −1.49188
\(339\) 0 0
\(340\) 57.8468 3.13718
\(341\) −17.9743 −0.973366
\(342\) 0 0
\(343\) −2.55931 −0.138190
\(344\) 4.04320 0.217994
\(345\) 0 0
\(346\) 9.29286 0.499587
\(347\) 24.1031 1.29392 0.646961 0.762523i \(-0.276040\pi\)
0.646961 + 0.762523i \(0.276040\pi\)
\(348\) 0 0
\(349\) −21.6810 −1.16056 −0.580280 0.814417i \(-0.697057\pi\)
−0.580280 + 0.814417i \(0.697057\pi\)
\(350\) 1.56102 0.0834399
\(351\) 0 0
\(352\) −11.8112 −0.629538
\(353\) 6.15172 0.327423 0.163712 0.986508i \(-0.447653\pi\)
0.163712 + 0.986508i \(0.447653\pi\)
\(354\) 0 0
\(355\) −36.2144 −1.92206
\(356\) −38.3237 −2.03115
\(357\) 0 0
\(358\) −24.1886 −1.27841
\(359\) −30.5171 −1.61063 −0.805315 0.592847i \(-0.798004\pi\)
−0.805315 + 0.592847i \(0.798004\pi\)
\(360\) 0 0
\(361\) −9.93157 −0.522714
\(362\) −17.8065 −0.935891
\(363\) 0 0
\(364\) 0.576431 0.0302132
\(365\) 20.0916 1.05164
\(366\) 0 0
\(367\) 30.1076 1.57160 0.785801 0.618480i \(-0.212251\pi\)
0.785801 + 0.618480i \(0.212251\pi\)
\(368\) 0.617229 0.0321753
\(369\) 0 0
\(370\) 67.0010 3.48321
\(371\) −0.205159 −0.0106513
\(372\) 0 0
\(373\) −15.2921 −0.791796 −0.395898 0.918295i \(-0.629567\pi\)
−0.395898 + 0.918295i \(0.629567\pi\)
\(374\) 28.1591 1.45607
\(375\) 0 0
\(376\) −21.3609 −1.10160
\(377\) −1.28365 −0.0661115
\(378\) 0 0
\(379\) −11.3522 −0.583123 −0.291562 0.956552i \(-0.594175\pi\)
−0.291562 + 0.956552i \(0.594175\pi\)
\(380\) −28.4543 −1.45968
\(381\) 0 0
\(382\) −21.1822 −1.08378
\(383\) 3.50534 0.179114 0.0895572 0.995982i \(-0.471455\pi\)
0.0895572 + 0.995982i \(0.471455\pi\)
\(384\) 0 0
\(385\) 1.09288 0.0556984
\(386\) −14.7991 −0.753255
\(387\) 0 0
\(388\) 29.6635 1.50594
\(389\) 13.2532 0.671961 0.335981 0.941869i \(-0.390932\pi\)
0.335981 + 0.941869i \(0.390932\pi\)
\(390\) 0 0
\(391\) −21.6646 −1.09563
\(392\) −19.0050 −0.959896
\(393\) 0 0
\(394\) −30.0617 −1.51448
\(395\) 12.4077 0.624298
\(396\) 0 0
\(397\) 24.5340 1.23133 0.615663 0.788009i \(-0.288888\pi\)
0.615663 + 0.788009i \(0.288888\pi\)
\(398\) −50.7551 −2.54412
\(399\) 0 0
\(400\) −0.651774 −0.0325887
\(401\) 1.35306 0.0675688 0.0337844 0.999429i \(-0.489244\pi\)
0.0337844 + 0.999429i \(0.489244\pi\)
\(402\) 0 0
\(403\) −8.76597 −0.436664
\(404\) −5.54997 −0.276122
\(405\) 0 0
\(406\) −0.544936 −0.0270447
\(407\) 20.0630 0.994488
\(408\) 0 0
\(409\) −29.4139 −1.45443 −0.727213 0.686412i \(-0.759185\pi\)
−0.727213 + 0.686412i \(0.759185\pi\)
\(410\) 58.0087 2.86485
\(411\) 0 0
\(412\) −39.2284 −1.93265
\(413\) −0.996198 −0.0490197
\(414\) 0 0
\(415\) −9.09435 −0.446424
\(416\) −5.76024 −0.282419
\(417\) 0 0
\(418\) −13.8512 −0.677486
\(419\) −12.3351 −0.602608 −0.301304 0.953528i \(-0.597422\pi\)
−0.301304 + 0.953528i \(0.597422\pi\)
\(420\) 0 0
\(421\) −21.2394 −1.03515 −0.517573 0.855639i \(-0.673164\pi\)
−0.517573 + 0.855639i \(0.673164\pi\)
\(422\) 35.5132 1.72876
\(423\) 0 0
\(424\) −3.05429 −0.148329
\(425\) 22.8772 1.10971
\(426\) 0 0
\(427\) 1.36200 0.0659120
\(428\) 27.0242 1.30626
\(429\) 0 0
\(430\) 9.98641 0.481587
\(431\) 30.5427 1.47119 0.735595 0.677422i \(-0.236903\pi\)
0.735595 + 0.677422i \(0.236903\pi\)
\(432\) 0 0
\(433\) 29.6727 1.42598 0.712989 0.701175i \(-0.247341\pi\)
0.712989 + 0.701175i \(0.247341\pi\)
\(434\) −3.72133 −0.178630
\(435\) 0 0
\(436\) 10.1047 0.483925
\(437\) 10.6566 0.509776
\(438\) 0 0
\(439\) −14.4064 −0.687579 −0.343790 0.939047i \(-0.611711\pi\)
−0.343790 + 0.939047i \(0.611711\pi\)
\(440\) 16.2702 0.775650
\(441\) 0 0
\(442\) 13.7330 0.653213
\(443\) −24.9621 −1.18598 −0.592992 0.805209i \(-0.702053\pi\)
−0.592992 + 0.805209i \(0.702053\pi\)
\(444\) 0 0
\(445\) −35.4357 −1.67981
\(446\) −6.35230 −0.300790
\(447\) 0 0
\(448\) −2.38141 −0.112511
\(449\) −13.0255 −0.614712 −0.307356 0.951595i \(-0.599444\pi\)
−0.307356 + 0.951595i \(0.599444\pi\)
\(450\) 0 0
\(451\) 17.3704 0.817938
\(452\) −41.7201 −1.96235
\(453\) 0 0
\(454\) −19.9418 −0.935915
\(455\) 0.532991 0.0249870
\(456\) 0 0
\(457\) −11.9643 −0.559668 −0.279834 0.960048i \(-0.590279\pi\)
−0.279834 + 0.960048i \(0.590279\pi\)
\(458\) 13.0599 0.610247
\(459\) 0 0
\(460\) −33.4377 −1.55904
\(461\) 16.4154 0.764542 0.382271 0.924050i \(-0.375142\pi\)
0.382271 + 0.924050i \(0.375142\pi\)
\(462\) 0 0
\(463\) −24.3043 −1.12952 −0.564758 0.825257i \(-0.691030\pi\)
−0.564758 + 0.825257i \(0.691030\pi\)
\(464\) 0.227528 0.0105627
\(465\) 0 0
\(466\) 23.4584 1.08669
\(467\) −25.6887 −1.18873 −0.594366 0.804195i \(-0.702597\pi\)
−0.594366 + 0.804195i \(0.702597\pi\)
\(468\) 0 0
\(469\) 1.50675 0.0695754
\(470\) −52.7598 −2.43363
\(471\) 0 0
\(472\) −14.8308 −0.682644
\(473\) 2.99037 0.137497
\(474\) 0 0
\(475\) −11.2531 −0.516326
\(476\) 3.58624 0.164375
\(477\) 0 0
\(478\) −2.27963 −0.104268
\(479\) 18.5787 0.848881 0.424441 0.905456i \(-0.360471\pi\)
0.424441 + 0.905456i \(0.360471\pi\)
\(480\) 0 0
\(481\) 9.78461 0.446140
\(482\) −19.1064 −0.870271
\(483\) 0 0
\(484\) −22.1497 −1.00680
\(485\) 27.4281 1.24544
\(486\) 0 0
\(487\) −8.99084 −0.407414 −0.203707 0.979032i \(-0.565299\pi\)
−0.203707 + 0.979032i \(0.565299\pi\)
\(488\) 20.2767 0.917884
\(489\) 0 0
\(490\) −46.9409 −2.12057
\(491\) 10.3500 0.467090 0.233545 0.972346i \(-0.424967\pi\)
0.233545 + 0.972346i \(0.424967\pi\)
\(492\) 0 0
\(493\) −7.98620 −0.359680
\(494\) −6.75515 −0.303929
\(495\) 0 0
\(496\) 1.55377 0.0697665
\(497\) −2.24513 −0.100708
\(498\) 0 0
\(499\) −18.9981 −0.850473 −0.425236 0.905082i \(-0.639809\pi\)
−0.425236 + 0.905082i \(0.639809\pi\)
\(500\) −11.9355 −0.533770
\(501\) 0 0
\(502\) −62.9532 −2.80974
\(503\) −35.2243 −1.57057 −0.785286 0.619133i \(-0.787484\pi\)
−0.785286 + 0.619133i \(0.787484\pi\)
\(504\) 0 0
\(505\) −5.13173 −0.228359
\(506\) −16.2771 −0.723605
\(507\) 0 0
\(508\) 26.0198 1.15444
\(509\) 31.4108 1.39226 0.696130 0.717915i \(-0.254904\pi\)
0.696130 + 0.717915i \(0.254904\pi\)
\(510\) 0 0
\(511\) 1.24559 0.0551018
\(512\) 1.97266 0.0871801
\(513\) 0 0
\(514\) 23.9132 1.05476
\(515\) −36.2722 −1.59834
\(516\) 0 0
\(517\) −15.7986 −0.694822
\(518\) 4.15377 0.182506
\(519\) 0 0
\(520\) 7.93486 0.347967
\(521\) −40.7369 −1.78471 −0.892357 0.451330i \(-0.850950\pi\)
−0.892357 + 0.451330i \(0.850950\pi\)
\(522\) 0 0
\(523\) −6.98147 −0.305278 −0.152639 0.988282i \(-0.548777\pi\)
−0.152639 + 0.988282i \(0.548777\pi\)
\(524\) 41.4458 1.81057
\(525\) 0 0
\(526\) 19.6800 0.858087
\(527\) −54.5372 −2.37568
\(528\) 0 0
\(529\) −10.4770 −0.455522
\(530\) −7.54387 −0.327685
\(531\) 0 0
\(532\) −1.76404 −0.0764809
\(533\) 8.47141 0.366938
\(534\) 0 0
\(535\) 24.9876 1.08031
\(536\) 22.4316 0.968900
\(537\) 0 0
\(538\) 34.6897 1.49558
\(539\) −14.0562 −0.605442
\(540\) 0 0
\(541\) 39.5536 1.70054 0.850272 0.526344i \(-0.176438\pi\)
0.850272 + 0.526344i \(0.176438\pi\)
\(542\) 28.8542 1.23939
\(543\) 0 0
\(544\) −35.8371 −1.53650
\(545\) 9.34316 0.400217
\(546\) 0 0
\(547\) 42.1343 1.80153 0.900766 0.434305i \(-0.143006\pi\)
0.900766 + 0.434305i \(0.143006\pi\)
\(548\) 34.1451 1.45861
\(549\) 0 0
\(550\) 17.1881 0.732902
\(551\) 3.92834 0.167353
\(552\) 0 0
\(553\) 0.769222 0.0327106
\(554\) 2.98896 0.126989
\(555\) 0 0
\(556\) 43.5260 1.84591
\(557\) −19.3767 −0.821016 −0.410508 0.911857i \(-0.634649\pi\)
−0.410508 + 0.911857i \(0.634649\pi\)
\(558\) 0 0
\(559\) 1.45838 0.0616831
\(560\) −0.0944730 −0.00399221
\(561\) 0 0
\(562\) −1.19363 −0.0503503
\(563\) −11.8935 −0.501252 −0.250626 0.968084i \(-0.580637\pi\)
−0.250626 + 0.968084i \(0.580637\pi\)
\(564\) 0 0
\(565\) −38.5760 −1.62291
\(566\) 63.1274 2.65344
\(567\) 0 0
\(568\) −33.4242 −1.40245
\(569\) 6.59449 0.276455 0.138228 0.990400i \(-0.455859\pi\)
0.138228 + 0.990400i \(0.455859\pi\)
\(570\) 0 0
\(571\) −4.55651 −0.190684 −0.0953420 0.995445i \(-0.530394\pi\)
−0.0953420 + 0.995445i \(0.530394\pi\)
\(572\) 6.34698 0.265380
\(573\) 0 0
\(574\) 3.59628 0.150106
\(575\) −13.2239 −0.551474
\(576\) 0 0
\(577\) 43.9337 1.82898 0.914491 0.404606i \(-0.132591\pi\)
0.914491 + 0.404606i \(0.132591\pi\)
\(578\) 46.6858 1.94187
\(579\) 0 0
\(580\) −12.3261 −0.511813
\(581\) −0.563810 −0.0233908
\(582\) 0 0
\(583\) −2.25897 −0.0935568
\(584\) 18.5437 0.767342
\(585\) 0 0
\(586\) −3.25140 −0.134314
\(587\) −20.5317 −0.847435 −0.423717 0.905794i \(-0.639275\pi\)
−0.423717 + 0.905794i \(0.639275\pi\)
\(588\) 0 0
\(589\) 26.8264 1.10536
\(590\) −36.6311 −1.50808
\(591\) 0 0
\(592\) −1.73433 −0.0712805
\(593\) 43.0656 1.76849 0.884246 0.467022i \(-0.154673\pi\)
0.884246 + 0.467022i \(0.154673\pi\)
\(594\) 0 0
\(595\) 3.31598 0.135942
\(596\) −5.69838 −0.233415
\(597\) 0 0
\(598\) −7.93823 −0.324618
\(599\) −8.12826 −0.332112 −0.166056 0.986116i \(-0.553103\pi\)
−0.166056 + 0.986116i \(0.553103\pi\)
\(600\) 0 0
\(601\) 28.2747 1.15335 0.576674 0.816975i \(-0.304350\pi\)
0.576674 + 0.816975i \(0.304350\pi\)
\(602\) 0.619113 0.0252332
\(603\) 0 0
\(604\) 3.73682 0.152049
\(605\) −20.4805 −0.832649
\(606\) 0 0
\(607\) −19.1530 −0.777395 −0.388697 0.921365i \(-0.627075\pi\)
−0.388697 + 0.921365i \(0.627075\pi\)
\(608\) 17.6280 0.714908
\(609\) 0 0
\(610\) 50.0821 2.02776
\(611\) −7.70488 −0.311706
\(612\) 0 0
\(613\) −30.8608 −1.24646 −0.623229 0.782040i \(-0.714179\pi\)
−0.623229 + 0.782040i \(0.714179\pi\)
\(614\) 69.9443 2.82272
\(615\) 0 0
\(616\) 1.00868 0.0406409
\(617\) 41.4736 1.66966 0.834832 0.550504i \(-0.185565\pi\)
0.834832 + 0.550504i \(0.185565\pi\)
\(618\) 0 0
\(619\) 48.0533 1.93143 0.965713 0.259612i \(-0.0835946\pi\)
0.965713 + 0.259612i \(0.0835946\pi\)
\(620\) −84.1739 −3.38051
\(621\) 0 0
\(622\) 1.22452 0.0490989
\(623\) −2.19685 −0.0880151
\(624\) 0 0
\(625\) −29.7202 −1.18881
\(626\) −11.0102 −0.440055
\(627\) 0 0
\(628\) −69.0553 −2.75561
\(629\) 60.8746 2.42723
\(630\) 0 0
\(631\) 2.36423 0.0941183 0.0470592 0.998892i \(-0.485015\pi\)
0.0470592 + 0.998892i \(0.485015\pi\)
\(632\) 11.4517 0.455525
\(633\) 0 0
\(634\) −73.1021 −2.90326
\(635\) 24.0590 0.954750
\(636\) 0 0
\(637\) −6.85511 −0.271609
\(638\) −6.00019 −0.237550
\(639\) 0 0
\(640\) −52.9613 −2.09348
\(641\) 18.2702 0.721628 0.360814 0.932638i \(-0.382499\pi\)
0.360814 + 0.932638i \(0.382499\pi\)
\(642\) 0 0
\(643\) −46.0250 −1.81505 −0.907524 0.420001i \(-0.862030\pi\)
−0.907524 + 0.420001i \(0.862030\pi\)
\(644\) −2.07299 −0.0816872
\(645\) 0 0
\(646\) −42.0269 −1.65353
\(647\) 5.66198 0.222595 0.111298 0.993787i \(-0.464499\pi\)
0.111298 + 0.993787i \(0.464499\pi\)
\(648\) 0 0
\(649\) −10.9690 −0.430569
\(650\) 8.38251 0.328789
\(651\) 0 0
\(652\) 26.3473 1.03184
\(653\) 20.6692 0.808849 0.404424 0.914571i \(-0.367472\pi\)
0.404424 + 0.914571i \(0.367472\pi\)
\(654\) 0 0
\(655\) 38.3224 1.49738
\(656\) −1.50156 −0.0586262
\(657\) 0 0
\(658\) −3.27088 −0.127512
\(659\) 33.1249 1.29036 0.645182 0.764029i \(-0.276781\pi\)
0.645182 + 0.764029i \(0.276781\pi\)
\(660\) 0 0
\(661\) 40.5473 1.57711 0.788553 0.614967i \(-0.210831\pi\)
0.788553 + 0.614967i \(0.210831\pi\)
\(662\) 67.6190 2.62809
\(663\) 0 0
\(664\) −8.39367 −0.325738
\(665\) −1.63110 −0.0632515
\(666\) 0 0
\(667\) 4.61633 0.178745
\(668\) −11.1150 −0.430053
\(669\) 0 0
\(670\) 55.4046 2.14047
\(671\) 14.9968 0.578944
\(672\) 0 0
\(673\) 25.1681 0.970158 0.485079 0.874470i \(-0.338791\pi\)
0.485079 + 0.874470i \(0.338791\pi\)
\(674\) 12.2558 0.472076
\(675\) 0 0
\(676\) −38.4620 −1.47931
\(677\) 33.7809 1.29831 0.649153 0.760658i \(-0.275123\pi\)
0.649153 + 0.760658i \(0.275123\pi\)
\(678\) 0 0
\(679\) 1.70042 0.0652561
\(680\) 49.3664 1.89312
\(681\) 0 0
\(682\) −40.9749 −1.56901
\(683\) −9.09130 −0.347869 −0.173934 0.984757i \(-0.555648\pi\)
−0.173934 + 0.984757i \(0.555648\pi\)
\(684\) 0 0
\(685\) 31.5719 1.20630
\(686\) −5.83429 −0.222754
\(687\) 0 0
\(688\) −0.258499 −0.00985520
\(689\) −1.10168 −0.0419708
\(690\) 0 0
\(691\) 19.0041 0.722951 0.361476 0.932382i \(-0.382273\pi\)
0.361476 + 0.932382i \(0.382273\pi\)
\(692\) 13.0313 0.495378
\(693\) 0 0
\(694\) 54.9462 2.08573
\(695\) 40.2459 1.52661
\(696\) 0 0
\(697\) 52.7046 1.99633
\(698\) −49.4248 −1.87076
\(699\) 0 0
\(700\) 2.18901 0.0827368
\(701\) −5.72366 −0.216179 −0.108090 0.994141i \(-0.534473\pi\)
−0.108090 + 0.994141i \(0.534473\pi\)
\(702\) 0 0
\(703\) −29.9437 −1.12935
\(704\) −26.2213 −0.988253
\(705\) 0 0
\(706\) 14.0237 0.527788
\(707\) −0.318145 −0.0119651
\(708\) 0 0
\(709\) 26.9353 1.01158 0.505788 0.862658i \(-0.331202\pi\)
0.505788 + 0.862658i \(0.331202\pi\)
\(710\) −82.5555 −3.09825
\(711\) 0 0
\(712\) −32.7055 −1.22569
\(713\) 31.5246 1.18061
\(714\) 0 0
\(715\) 5.86867 0.219476
\(716\) −33.9196 −1.26763
\(717\) 0 0
\(718\) −69.5677 −2.59624
\(719\) 43.7251 1.63067 0.815335 0.578989i \(-0.196553\pi\)
0.815335 + 0.578989i \(0.196553\pi\)
\(720\) 0 0
\(721\) −2.24871 −0.0837465
\(722\) −22.6403 −0.842586
\(723\) 0 0
\(724\) −24.9701 −0.928005
\(725\) −4.87470 −0.181042
\(726\) 0 0
\(727\) 48.0927 1.78366 0.891829 0.452373i \(-0.149422\pi\)
0.891829 + 0.452373i \(0.149422\pi\)
\(728\) 0.491927 0.0182320
\(729\) 0 0
\(730\) 45.8015 1.69519
\(731\) 9.07328 0.335587
\(732\) 0 0
\(733\) −45.9464 −1.69707 −0.848535 0.529139i \(-0.822515\pi\)
−0.848535 + 0.529139i \(0.822515\pi\)
\(734\) 68.6341 2.53333
\(735\) 0 0
\(736\) 20.7153 0.763574
\(737\) 16.5906 0.611121
\(738\) 0 0
\(739\) −35.2656 −1.29727 −0.648633 0.761101i \(-0.724659\pi\)
−0.648633 + 0.761101i \(0.724659\pi\)
\(740\) 93.9553 3.45386
\(741\) 0 0
\(742\) −0.467687 −0.0171693
\(743\) −27.5772 −1.01171 −0.505855 0.862619i \(-0.668823\pi\)
−0.505855 + 0.862619i \(0.668823\pi\)
\(744\) 0 0
\(745\) −5.26895 −0.193039
\(746\) −34.8604 −1.27633
\(747\) 0 0
\(748\) 39.4875 1.44380
\(749\) 1.54912 0.0566038
\(750\) 0 0
\(751\) −1.38797 −0.0506479 −0.0253239 0.999679i \(-0.508062\pi\)
−0.0253239 + 0.999679i \(0.508062\pi\)
\(752\) 1.36569 0.0498018
\(753\) 0 0
\(754\) −2.92626 −0.106568
\(755\) 3.45522 0.125748
\(756\) 0 0
\(757\) −9.69778 −0.352472 −0.176236 0.984348i \(-0.556392\pi\)
−0.176236 + 0.984348i \(0.556392\pi\)
\(758\) −25.8788 −0.939961
\(759\) 0 0
\(760\) −24.2829 −0.880834
\(761\) −47.9029 −1.73648 −0.868240 0.496144i \(-0.834749\pi\)
−0.868240 + 0.496144i \(0.834749\pi\)
\(762\) 0 0
\(763\) 0.579235 0.0209697
\(764\) −29.7037 −1.07464
\(765\) 0 0
\(766\) 7.99088 0.288722
\(767\) −5.34949 −0.193159
\(768\) 0 0
\(769\) 35.1787 1.26858 0.634288 0.773097i \(-0.281293\pi\)
0.634288 + 0.773097i \(0.281293\pi\)
\(770\) 2.49137 0.0897826
\(771\) 0 0
\(772\) −20.7528 −0.746908
\(773\) 49.0826 1.76538 0.882689 0.469958i \(-0.155731\pi\)
0.882689 + 0.469958i \(0.155731\pi\)
\(774\) 0 0
\(775\) −33.2890 −1.19578
\(776\) 25.3149 0.908750
\(777\) 0 0
\(778\) 30.2123 1.08316
\(779\) −25.9249 −0.928856
\(780\) 0 0
\(781\) −24.7208 −0.884578
\(782\) −49.3874 −1.76609
\(783\) 0 0
\(784\) 1.21507 0.0433954
\(785\) −63.8513 −2.27895
\(786\) 0 0
\(787\) 23.4029 0.834222 0.417111 0.908856i \(-0.363043\pi\)
0.417111 + 0.908856i \(0.363043\pi\)
\(788\) −42.1554 −1.50172
\(789\) 0 0
\(790\) 28.2849 1.00633
\(791\) −2.39154 −0.0850335
\(792\) 0 0
\(793\) 7.31383 0.259722
\(794\) 55.9285 1.98483
\(795\) 0 0
\(796\) −71.1737 −2.52268
\(797\) 40.7729 1.44425 0.722126 0.691762i \(-0.243165\pi\)
0.722126 + 0.691762i \(0.243165\pi\)
\(798\) 0 0
\(799\) −47.9356 −1.69584
\(800\) −21.8746 −0.773385
\(801\) 0 0
\(802\) 3.08449 0.108917
\(803\) 13.7150 0.483992
\(804\) 0 0
\(805\) −1.91677 −0.0675572
\(806\) −19.9832 −0.703878
\(807\) 0 0
\(808\) −4.73635 −0.166624
\(809\) −5.76489 −0.202683 −0.101341 0.994852i \(-0.532313\pi\)
−0.101341 + 0.994852i \(0.532313\pi\)
\(810\) 0 0
\(811\) 26.4033 0.927146 0.463573 0.886059i \(-0.346567\pi\)
0.463573 + 0.886059i \(0.346567\pi\)
\(812\) −0.764163 −0.0268169
\(813\) 0 0
\(814\) 45.7363 1.60306
\(815\) 24.3617 0.853354
\(816\) 0 0
\(817\) −4.46307 −0.156143
\(818\) −67.0529 −2.34445
\(819\) 0 0
\(820\) 81.3454 2.84071
\(821\) 17.6703 0.616697 0.308348 0.951273i \(-0.400224\pi\)
0.308348 + 0.951273i \(0.400224\pi\)
\(822\) 0 0
\(823\) −0.687817 −0.0239758 −0.0119879 0.999928i \(-0.503816\pi\)
−0.0119879 + 0.999928i \(0.503816\pi\)
\(824\) −33.4776 −1.16625
\(825\) 0 0
\(826\) −2.27096 −0.0790170
\(827\) 30.2380 1.05148 0.525739 0.850646i \(-0.323789\pi\)
0.525739 + 0.850646i \(0.323789\pi\)
\(828\) 0 0
\(829\) 55.7717 1.93703 0.968516 0.248953i \(-0.0800864\pi\)
0.968516 + 0.248953i \(0.0800864\pi\)
\(830\) −20.7318 −0.719611
\(831\) 0 0
\(832\) −12.7880 −0.443343
\(833\) −42.6488 −1.47769
\(834\) 0 0
\(835\) −10.2774 −0.355663
\(836\) −19.4235 −0.671777
\(837\) 0 0
\(838\) −28.1195 −0.971370
\(839\) 43.3319 1.49598 0.747991 0.663709i \(-0.231019\pi\)
0.747991 + 0.663709i \(0.231019\pi\)
\(840\) 0 0
\(841\) −27.2983 −0.941320
\(842\) −48.4180 −1.66860
\(843\) 0 0
\(844\) 49.8001 1.71419
\(845\) −35.5635 −1.22342
\(846\) 0 0
\(847\) −1.26970 −0.0436273
\(848\) 0.195274 0.00670574
\(849\) 0 0
\(850\) 52.1515 1.78878
\(851\) −35.1879 −1.20623
\(852\) 0 0
\(853\) 51.6565 1.76869 0.884343 0.466837i \(-0.154607\pi\)
0.884343 + 0.466837i \(0.154607\pi\)
\(854\) 3.10487 0.106246
\(855\) 0 0
\(856\) 23.0625 0.788258
\(857\) 3.12227 0.106655 0.0533274 0.998577i \(-0.483017\pi\)
0.0533274 + 0.998577i \(0.483017\pi\)
\(858\) 0 0
\(859\) 24.7795 0.845466 0.422733 0.906254i \(-0.361071\pi\)
0.422733 + 0.906254i \(0.361071\pi\)
\(860\) 14.0039 0.477529
\(861\) 0 0
\(862\) 69.6261 2.37147
\(863\) 34.0270 1.15829 0.579146 0.815224i \(-0.303386\pi\)
0.579146 + 0.815224i \(0.303386\pi\)
\(864\) 0 0
\(865\) 12.0493 0.409689
\(866\) 67.6428 2.29860
\(867\) 0 0
\(868\) −5.21841 −0.177124
\(869\) 8.46976 0.287317
\(870\) 0 0
\(871\) 8.09111 0.274157
\(872\) 8.62331 0.292022
\(873\) 0 0
\(874\) 24.2932 0.821730
\(875\) −0.684183 −0.0231296
\(876\) 0 0
\(877\) 1.41045 0.0476275 0.0238138 0.999716i \(-0.492419\pi\)
0.0238138 + 0.999716i \(0.492419\pi\)
\(878\) −32.8412 −1.10834
\(879\) 0 0
\(880\) −1.04022 −0.0350660
\(881\) −3.11367 −0.104902 −0.0524511 0.998623i \(-0.516703\pi\)
−0.0524511 + 0.998623i \(0.516703\pi\)
\(882\) 0 0
\(883\) −0.280218 −0.00943009 −0.00471505 0.999989i \(-0.501501\pi\)
−0.00471505 + 0.999989i \(0.501501\pi\)
\(884\) 19.2578 0.647709
\(885\) 0 0
\(886\) −56.9043 −1.91174
\(887\) −51.2584 −1.72109 −0.860545 0.509375i \(-0.829877\pi\)
−0.860545 + 0.509375i \(0.829877\pi\)
\(888\) 0 0
\(889\) 1.49155 0.0500250
\(890\) −80.7802 −2.70776
\(891\) 0 0
\(892\) −8.90781 −0.298255
\(893\) 23.5791 0.789045
\(894\) 0 0
\(895\) −31.3634 −1.04836
\(896\) −3.28337 −0.109690
\(897\) 0 0
\(898\) −29.6934 −0.990881
\(899\) 11.6209 0.387578
\(900\) 0 0
\(901\) −6.85408 −0.228343
\(902\) 39.5980 1.31847
\(903\) 0 0
\(904\) −35.6039 −1.18417
\(905\) −23.0883 −0.767482
\(906\) 0 0
\(907\) 30.2941 1.00590 0.502949 0.864316i \(-0.332248\pi\)
0.502949 + 0.864316i \(0.332248\pi\)
\(908\) −27.9643 −0.928029
\(909\) 0 0
\(910\) 1.21502 0.0402776
\(911\) −16.8868 −0.559485 −0.279742 0.960075i \(-0.590249\pi\)
−0.279742 + 0.960075i \(0.590249\pi\)
\(912\) 0 0
\(913\) −6.20801 −0.205455
\(914\) −27.2743 −0.902153
\(915\) 0 0
\(916\) 18.3138 0.605105
\(917\) 2.37582 0.0784565
\(918\) 0 0
\(919\) −4.94560 −0.163140 −0.0815701 0.996668i \(-0.525993\pi\)
−0.0815701 + 0.996668i \(0.525993\pi\)
\(920\) −28.5357 −0.940795
\(921\) 0 0
\(922\) 37.4211 1.23240
\(923\) −12.0562 −0.396833
\(924\) 0 0
\(925\) 37.1573 1.22172
\(926\) −55.4048 −1.82071
\(927\) 0 0
\(928\) 7.63623 0.250672
\(929\) 30.1457 0.989048 0.494524 0.869164i \(-0.335342\pi\)
0.494524 + 0.869164i \(0.335342\pi\)
\(930\) 0 0
\(931\) 20.9786 0.687545
\(932\) 32.8956 1.07753
\(933\) 0 0
\(934\) −58.5608 −1.91617
\(935\) 36.5117 1.19406
\(936\) 0 0
\(937\) −47.7140 −1.55875 −0.779374 0.626559i \(-0.784463\pi\)
−0.779374 + 0.626559i \(0.784463\pi\)
\(938\) 3.43484 0.112152
\(939\) 0 0
\(940\) −73.9849 −2.41312
\(941\) −29.1888 −0.951527 −0.475763 0.879573i \(-0.657828\pi\)
−0.475763 + 0.879573i \(0.657828\pi\)
\(942\) 0 0
\(943\) −30.4653 −0.992087
\(944\) 0.948200 0.0308613
\(945\) 0 0
\(946\) 6.81694 0.221638
\(947\) −52.9404 −1.72033 −0.860166 0.510013i \(-0.829640\pi\)
−0.860166 + 0.510013i \(0.829640\pi\)
\(948\) 0 0
\(949\) 6.68871 0.217125
\(950\) −25.6529 −0.832289
\(951\) 0 0
\(952\) 3.06050 0.0991914
\(953\) 53.2278 1.72422 0.862109 0.506722i \(-0.169143\pi\)
0.862109 + 0.506722i \(0.169143\pi\)
\(954\) 0 0
\(955\) −27.4653 −0.888755
\(956\) −3.19672 −0.103389
\(957\) 0 0
\(958\) 42.3525 1.36835
\(959\) 1.95732 0.0632052
\(960\) 0 0
\(961\) 48.3581 1.55994
\(962\) 22.3053 0.719152
\(963\) 0 0
\(964\) −26.7928 −0.862937
\(965\) −19.1888 −0.617710
\(966\) 0 0
\(967\) 43.6823 1.40473 0.702365 0.711817i \(-0.252128\pi\)
0.702365 + 0.711817i \(0.252128\pi\)
\(968\) −18.9025 −0.607550
\(969\) 0 0
\(970\) 62.5259 2.00759
\(971\) 19.1283 0.613857 0.306928 0.951733i \(-0.400699\pi\)
0.306928 + 0.951733i \(0.400699\pi\)
\(972\) 0 0
\(973\) 2.49507 0.0799882
\(974\) −20.4958 −0.656728
\(975\) 0 0
\(976\) −1.29638 −0.0414961
\(977\) −35.6262 −1.13978 −0.569892 0.821720i \(-0.693015\pi\)
−0.569892 + 0.821720i \(0.693015\pi\)
\(978\) 0 0
\(979\) −24.1892 −0.773089
\(980\) −65.8251 −2.10271
\(981\) 0 0
\(982\) 23.5942 0.752922
\(983\) 39.5475 1.26137 0.630684 0.776039i \(-0.282774\pi\)
0.630684 + 0.776039i \(0.282774\pi\)
\(984\) 0 0
\(985\) −38.9785 −1.24196
\(986\) −18.2056 −0.579784
\(987\) 0 0
\(988\) −9.47273 −0.301368
\(989\) −5.24471 −0.166772
\(990\) 0 0
\(991\) 62.2695 1.97806 0.989028 0.147728i \(-0.0471960\pi\)
0.989028 + 0.147728i \(0.0471960\pi\)
\(992\) 52.1473 1.65568
\(993\) 0 0
\(994\) −5.11808 −0.162336
\(995\) −65.8100 −2.08632
\(996\) 0 0
\(997\) 6.16335 0.195195 0.0975976 0.995226i \(-0.468884\pi\)
0.0975976 + 0.995226i \(0.468884\pi\)
\(998\) −43.3087 −1.37091
\(999\) 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 2151.2.a.h.1.11 12
3.2 odd 2 717.2.a.g.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
717.2.a.g.1.2 12 3.2 odd 2
2151.2.a.h.1.11 12 1.1 even 1 trivial