Properties

Label 5225.2.a.p.1.3
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $9$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.92391\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.92391 q^{2} -0.621031 q^{3} +1.70143 q^{4} +1.19481 q^{6} -5.15278 q^{7} +0.574421 q^{8} -2.61432 q^{9} +O(q^{10})\) \(q-1.92391 q^{2} -0.621031 q^{3} +1.70143 q^{4} +1.19481 q^{6} -5.15278 q^{7} +0.574421 q^{8} -2.61432 q^{9} +1.00000 q^{11} -1.05664 q^{12} -0.138167 q^{13} +9.91350 q^{14} -4.50800 q^{16} -1.86183 q^{17} +5.02972 q^{18} +1.00000 q^{19} +3.20004 q^{21} -1.92391 q^{22} -0.167646 q^{23} -0.356733 q^{24} +0.265820 q^{26} +3.48667 q^{27} -8.76711 q^{28} +4.80913 q^{29} -1.73000 q^{31} +7.52414 q^{32} -0.621031 q^{33} +3.58200 q^{34} -4.44808 q^{36} -0.442598 q^{37} -1.92391 q^{38} +0.0858058 q^{39} -12.1984 q^{41} -6.15659 q^{42} +8.67909 q^{43} +1.70143 q^{44} +0.322537 q^{46} -2.21857 q^{47} +2.79960 q^{48} +19.5512 q^{49} +1.15626 q^{51} -0.235081 q^{52} -1.34355 q^{53} -6.70803 q^{54} -2.95987 q^{56} -0.621031 q^{57} -9.25233 q^{58} +3.15592 q^{59} +1.90706 q^{61} +3.32837 q^{62} +13.4710 q^{63} -5.45977 q^{64} +1.19481 q^{66} +10.7534 q^{67} -3.16778 q^{68} +0.104114 q^{69} +11.2883 q^{71} -1.50172 q^{72} +9.78050 q^{73} +0.851518 q^{74} +1.70143 q^{76} -5.15278 q^{77} -0.165083 q^{78} -6.14010 q^{79} +5.67763 q^{81} +23.4686 q^{82} +0.190459 q^{83} +5.44464 q^{84} -16.6978 q^{86} -2.98662 q^{87} +0.574421 q^{88} -8.94416 q^{89} +0.711944 q^{91} -0.285239 q^{92} +1.07439 q^{93} +4.26832 q^{94} -4.67272 q^{96} -6.74261 q^{97} -37.6147 q^{98} -2.61432 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} - 13 q^{7} - 9 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} - 13 q^{7} - 9 q^{8} + 12 q^{9} + 9 q^{11} + 5 q^{12} - 5 q^{13} - 2 q^{14} + q^{16} - 13 q^{17} - q^{18} + 9 q^{19} + q^{21} - 3 q^{22} - 8 q^{23} - 7 q^{24} + 8 q^{26} - 6 q^{27} - 10 q^{28} - 3 q^{29} - 9 q^{31} - 6 q^{32} - 3 q^{33} - 2 q^{34} - 15 q^{36} + 7 q^{37} - 3 q^{38} - 2 q^{39} + 9 q^{41} + 9 q^{42} - 23 q^{43} + 9 q^{44} - 32 q^{46} - 20 q^{47} + 18 q^{48} - 4 q^{49} + 8 q^{51} - 9 q^{52} + 5 q^{53} + 9 q^{54} + 4 q^{56} - 3 q^{57} - 3 q^{58} + 19 q^{59} + q^{61} - 18 q^{62} - 24 q^{63} + 23 q^{64} + 10 q^{67} - 9 q^{68} - 28 q^{69} - 24 q^{72} - 12 q^{73} + 5 q^{74} + 9 q^{76} - 13 q^{77} - 19 q^{78} - 21 q^{79} - 3 q^{81} + 14 q^{82} - 47 q^{83} - 11 q^{84} + 12 q^{86} + 2 q^{87} - 9 q^{88} - 2 q^{89} + 2 q^{91} + 19 q^{92} + 2 q^{93} + 26 q^{94} - 55 q^{96} + 32 q^{97} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.92391 −1.36041 −0.680205 0.733022i \(-0.738109\pi\)
−0.680205 + 0.733022i \(0.738109\pi\)
\(3\) −0.621031 −0.358552 −0.179276 0.983799i \(-0.557376\pi\)
−0.179276 + 0.983799i \(0.557376\pi\)
\(4\) 1.70143 0.850715
\(5\) 0 0
\(6\) 1.19481 0.487778
\(7\) −5.15278 −1.94757 −0.973785 0.227471i \(-0.926954\pi\)
−0.973785 + 0.227471i \(0.926954\pi\)
\(8\) 0.574421 0.203088
\(9\) −2.61432 −0.871440
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −1.05664 −0.305026
\(13\) −0.138167 −0.0383206 −0.0191603 0.999816i \(-0.506099\pi\)
−0.0191603 + 0.999816i \(0.506099\pi\)
\(14\) 9.91350 2.64949
\(15\) 0 0
\(16\) −4.50800 −1.12700
\(17\) −1.86183 −0.451561 −0.225780 0.974178i \(-0.572493\pi\)
−0.225780 + 0.974178i \(0.572493\pi\)
\(18\) 5.02972 1.18552
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.20004 0.698306
\(22\) −1.92391 −0.410179
\(23\) −0.167646 −0.0349567 −0.0174783 0.999847i \(-0.505564\pi\)
−0.0174783 + 0.999847i \(0.505564\pi\)
\(24\) −0.356733 −0.0728178
\(25\) 0 0
\(26\) 0.265820 0.0521317
\(27\) 3.48667 0.671009
\(28\) −8.76711 −1.65683
\(29\) 4.80913 0.893033 0.446516 0.894775i \(-0.352664\pi\)
0.446516 + 0.894775i \(0.352664\pi\)
\(30\) 0 0
\(31\) −1.73000 −0.310718 −0.155359 0.987858i \(-0.549653\pi\)
−0.155359 + 0.987858i \(0.549653\pi\)
\(32\) 7.52414 1.33009
\(33\) −0.621031 −0.108108
\(34\) 3.58200 0.614308
\(35\) 0 0
\(36\) −4.44808 −0.741347
\(37\) −0.442598 −0.0727626 −0.0363813 0.999338i \(-0.511583\pi\)
−0.0363813 + 0.999338i \(0.511583\pi\)
\(38\) −1.92391 −0.312099
\(39\) 0.0858058 0.0137399
\(40\) 0 0
\(41\) −12.1984 −1.90506 −0.952532 0.304437i \(-0.901532\pi\)
−0.952532 + 0.304437i \(0.901532\pi\)
\(42\) −6.15659 −0.949982
\(43\) 8.67909 1.32355 0.661774 0.749703i \(-0.269804\pi\)
0.661774 + 0.749703i \(0.269804\pi\)
\(44\) 1.70143 0.256500
\(45\) 0 0
\(46\) 0.322537 0.0475554
\(47\) −2.21857 −0.323611 −0.161806 0.986823i \(-0.551732\pi\)
−0.161806 + 0.986823i \(0.551732\pi\)
\(48\) 2.79960 0.404088
\(49\) 19.5512 2.79303
\(50\) 0 0
\(51\) 1.15626 0.161908
\(52\) −0.235081 −0.0325999
\(53\) −1.34355 −0.184550 −0.0922751 0.995734i \(-0.529414\pi\)
−0.0922751 + 0.995734i \(0.529414\pi\)
\(54\) −6.70803 −0.912848
\(55\) 0 0
\(56\) −2.95987 −0.395529
\(57\) −0.621031 −0.0822576
\(58\) −9.25233 −1.21489
\(59\) 3.15592 0.410866 0.205433 0.978671i \(-0.434140\pi\)
0.205433 + 0.978671i \(0.434140\pi\)
\(60\) 0 0
\(61\) 1.90706 0.244174 0.122087 0.992519i \(-0.461041\pi\)
0.122087 + 0.992519i \(0.461041\pi\)
\(62\) 3.32837 0.422703
\(63\) 13.4710 1.69719
\(64\) −5.45977 −0.682472
\(65\) 0 0
\(66\) 1.19481 0.147071
\(67\) 10.7534 1.31373 0.656867 0.754007i \(-0.271881\pi\)
0.656867 + 0.754007i \(0.271881\pi\)
\(68\) −3.16778 −0.384150
\(69\) 0.104114 0.0125338
\(70\) 0 0
\(71\) 11.2883 1.33968 0.669840 0.742505i \(-0.266363\pi\)
0.669840 + 0.742505i \(0.266363\pi\)
\(72\) −1.50172 −0.176979
\(73\) 9.78050 1.14472 0.572361 0.820002i \(-0.306028\pi\)
0.572361 + 0.820002i \(0.306028\pi\)
\(74\) 0.851518 0.0989870
\(75\) 0 0
\(76\) 1.70143 0.195167
\(77\) −5.15278 −0.587214
\(78\) −0.165083 −0.0186919
\(79\) −6.14010 −0.690816 −0.345408 0.938453i \(-0.612259\pi\)
−0.345408 + 0.938453i \(0.612259\pi\)
\(80\) 0 0
\(81\) 5.67763 0.630848
\(82\) 23.4686 2.59167
\(83\) 0.190459 0.0209056 0.0104528 0.999945i \(-0.496673\pi\)
0.0104528 + 0.999945i \(0.496673\pi\)
\(84\) 5.44464 0.594059
\(85\) 0 0
\(86\) −16.6978 −1.80057
\(87\) −2.98662 −0.320199
\(88\) 0.574421 0.0612335
\(89\) −8.94416 −0.948079 −0.474040 0.880504i \(-0.657205\pi\)
−0.474040 + 0.880504i \(0.657205\pi\)
\(90\) 0 0
\(91\) 0.711944 0.0746320
\(92\) −0.285239 −0.0297382
\(93\) 1.07439 0.111409
\(94\) 4.26832 0.440244
\(95\) 0 0
\(96\) −4.67272 −0.476908
\(97\) −6.74261 −0.684609 −0.342304 0.939589i \(-0.611207\pi\)
−0.342304 + 0.939589i \(0.611207\pi\)
\(98\) −37.6147 −3.79966
\(99\) −2.61432 −0.262749
\(100\) 0 0
\(101\) 2.79639 0.278251 0.139126 0.990275i \(-0.455571\pi\)
0.139126 + 0.990275i \(0.455571\pi\)
\(102\) −2.22453 −0.220262
\(103\) 7.49046 0.738057 0.369028 0.929418i \(-0.379691\pi\)
0.369028 + 0.929418i \(0.379691\pi\)
\(104\) −0.0793659 −0.00778246
\(105\) 0 0
\(106\) 2.58486 0.251064
\(107\) −3.26297 −0.315443 −0.157722 0.987484i \(-0.550415\pi\)
−0.157722 + 0.987484i \(0.550415\pi\)
\(108\) 5.93232 0.570838
\(109\) 1.46274 0.140105 0.0700526 0.997543i \(-0.477683\pi\)
0.0700526 + 0.997543i \(0.477683\pi\)
\(110\) 0 0
\(111\) 0.274867 0.0260892
\(112\) 23.2287 2.19491
\(113\) −8.58423 −0.807537 −0.403768 0.914861i \(-0.632300\pi\)
−0.403768 + 0.914861i \(0.632300\pi\)
\(114\) 1.19481 0.111904
\(115\) 0 0
\(116\) 8.18240 0.759717
\(117\) 0.361212 0.0333941
\(118\) −6.07171 −0.558946
\(119\) 9.59363 0.879446
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −3.66901 −0.332176
\(123\) 7.57556 0.683066
\(124\) −2.94348 −0.264332
\(125\) 0 0
\(126\) −25.9171 −2.30887
\(127\) 4.66887 0.414295 0.207148 0.978310i \(-0.433582\pi\)
0.207148 + 0.978310i \(0.433582\pi\)
\(128\) −4.54416 −0.401651
\(129\) −5.38998 −0.474561
\(130\) 0 0
\(131\) 0.940241 0.0821492 0.0410746 0.999156i \(-0.486922\pi\)
0.0410746 + 0.999156i \(0.486922\pi\)
\(132\) −1.05664 −0.0919688
\(133\) −5.15278 −0.446803
\(134\) −20.6885 −1.78722
\(135\) 0 0
\(136\) −1.06948 −0.0917068
\(137\) 12.8192 1.09522 0.547610 0.836734i \(-0.315538\pi\)
0.547610 + 0.836734i \(0.315538\pi\)
\(138\) −0.200305 −0.0170511
\(139\) −20.5940 −1.74676 −0.873379 0.487042i \(-0.838076\pi\)
−0.873379 + 0.487042i \(0.838076\pi\)
\(140\) 0 0
\(141\) 1.37780 0.116032
\(142\) −21.7178 −1.82251
\(143\) −0.138167 −0.0115541
\(144\) 11.7853 0.982112
\(145\) 0 0
\(146\) −18.8168 −1.55729
\(147\) −12.1419 −1.00145
\(148\) −0.753049 −0.0619003
\(149\) 19.5366 1.60050 0.800250 0.599666i \(-0.204700\pi\)
0.800250 + 0.599666i \(0.204700\pi\)
\(150\) 0 0
\(151\) −13.4871 −1.09756 −0.548781 0.835966i \(-0.684908\pi\)
−0.548781 + 0.835966i \(0.684908\pi\)
\(152\) 0.574421 0.0465917
\(153\) 4.86743 0.393508
\(154\) 9.91350 0.798852
\(155\) 0 0
\(156\) 0.145993 0.0116888
\(157\) 6.15820 0.491478 0.245739 0.969336i \(-0.420969\pi\)
0.245739 + 0.969336i \(0.420969\pi\)
\(158\) 11.8130 0.939793
\(159\) 0.834384 0.0661709
\(160\) 0 0
\(161\) 0.863846 0.0680806
\(162\) −10.9233 −0.858212
\(163\) 15.7457 1.23330 0.616649 0.787238i \(-0.288490\pi\)
0.616649 + 0.787238i \(0.288490\pi\)
\(164\) −20.7547 −1.62067
\(165\) 0 0
\(166\) −0.366426 −0.0284402
\(167\) 1.49742 0.115874 0.0579369 0.998320i \(-0.481548\pi\)
0.0579369 + 0.998320i \(0.481548\pi\)
\(168\) 1.83817 0.141818
\(169\) −12.9809 −0.998532
\(170\) 0 0
\(171\) −2.61432 −0.199922
\(172\) 14.7669 1.12596
\(173\) 25.3734 1.92910 0.964550 0.263898i \(-0.0850084\pi\)
0.964550 + 0.263898i \(0.0850084\pi\)
\(174\) 5.74598 0.435602
\(175\) 0 0
\(176\) −4.50800 −0.339803
\(177\) −1.95992 −0.147317
\(178\) 17.2078 1.28978
\(179\) −1.45676 −0.108883 −0.0544416 0.998517i \(-0.517338\pi\)
−0.0544416 + 0.998517i \(0.517338\pi\)
\(180\) 0 0
\(181\) −10.1387 −0.753605 −0.376803 0.926294i \(-0.622976\pi\)
−0.376803 + 0.926294i \(0.622976\pi\)
\(182\) −1.36972 −0.101530
\(183\) −1.18434 −0.0875490
\(184\) −0.0962996 −0.00709930
\(185\) 0 0
\(186\) −2.06702 −0.151561
\(187\) −1.86183 −0.136151
\(188\) −3.77474 −0.275301
\(189\) −17.9660 −1.30684
\(190\) 0 0
\(191\) −8.99086 −0.650556 −0.325278 0.945619i \(-0.605458\pi\)
−0.325278 + 0.945619i \(0.605458\pi\)
\(192\) 3.39069 0.244702
\(193\) 4.44729 0.320123 0.160061 0.987107i \(-0.448831\pi\)
0.160061 + 0.987107i \(0.448831\pi\)
\(194\) 12.9722 0.931348
\(195\) 0 0
\(196\) 33.2650 2.37607
\(197\) −21.9424 −1.56333 −0.781664 0.623699i \(-0.785629\pi\)
−0.781664 + 0.623699i \(0.785629\pi\)
\(198\) 5.02972 0.357446
\(199\) 13.7659 0.975840 0.487920 0.872888i \(-0.337756\pi\)
0.487920 + 0.872888i \(0.337756\pi\)
\(200\) 0 0
\(201\) −6.67818 −0.471042
\(202\) −5.38000 −0.378536
\(203\) −24.7804 −1.73924
\(204\) 1.96729 0.137738
\(205\) 0 0
\(206\) −14.4110 −1.00406
\(207\) 0.438282 0.0304627
\(208\) 0.622855 0.0431872
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −21.5479 −1.48342 −0.741710 0.670721i \(-0.765985\pi\)
−0.741710 + 0.670721i \(0.765985\pi\)
\(212\) −2.28595 −0.157000
\(213\) −7.01041 −0.480346
\(214\) 6.27766 0.429132
\(215\) 0 0
\(216\) 2.00281 0.136274
\(217\) 8.91433 0.605144
\(218\) −2.81418 −0.190600
\(219\) −6.07399 −0.410442
\(220\) 0 0
\(221\) 0.257243 0.0173041
\(222\) −0.528819 −0.0354920
\(223\) −25.5722 −1.71244 −0.856221 0.516610i \(-0.827194\pi\)
−0.856221 + 0.516610i \(0.827194\pi\)
\(224\) −38.7703 −2.59045
\(225\) 0 0
\(226\) 16.5153 1.09858
\(227\) −9.06540 −0.601692 −0.300846 0.953673i \(-0.597269\pi\)
−0.300846 + 0.953673i \(0.597269\pi\)
\(228\) −1.05664 −0.0699778
\(229\) 27.5774 1.82236 0.911182 0.412005i \(-0.135171\pi\)
0.911182 + 0.412005i \(0.135171\pi\)
\(230\) 0 0
\(231\) 3.20004 0.210547
\(232\) 2.76246 0.181365
\(233\) −13.5037 −0.884654 −0.442327 0.896854i \(-0.645847\pi\)
−0.442327 + 0.896854i \(0.645847\pi\)
\(234\) −0.694940 −0.0454296
\(235\) 0 0
\(236\) 5.36958 0.349530
\(237\) 3.81319 0.247694
\(238\) −18.4573 −1.19641
\(239\) 2.50739 0.162190 0.0810949 0.996706i \(-0.474158\pi\)
0.0810949 + 0.996706i \(0.474158\pi\)
\(240\) 0 0
\(241\) 24.4676 1.57610 0.788050 0.615611i \(-0.211091\pi\)
0.788050 + 0.615611i \(0.211091\pi\)
\(242\) −1.92391 −0.123674
\(243\) −13.9860 −0.897201
\(244\) 3.24472 0.207722
\(245\) 0 0
\(246\) −14.5747 −0.929249
\(247\) −0.138167 −0.00879134
\(248\) −0.993750 −0.0631032
\(249\) −0.118281 −0.00749576
\(250\) 0 0
\(251\) −9.98514 −0.630256 −0.315128 0.949049i \(-0.602048\pi\)
−0.315128 + 0.949049i \(0.602048\pi\)
\(252\) 22.9200 1.44383
\(253\) −0.167646 −0.0105398
\(254\) −8.98249 −0.563612
\(255\) 0 0
\(256\) 19.6621 1.22888
\(257\) 23.5112 1.46659 0.733294 0.679912i \(-0.237982\pi\)
0.733294 + 0.679912i \(0.237982\pi\)
\(258\) 10.3698 0.645598
\(259\) 2.28061 0.141710
\(260\) 0 0
\(261\) −12.5726 −0.778225
\(262\) −1.80894 −0.111757
\(263\) −22.8248 −1.40744 −0.703720 0.710478i \(-0.748479\pi\)
−0.703720 + 0.710478i \(0.748479\pi\)
\(264\) −0.356733 −0.0219554
\(265\) 0 0
\(266\) 9.91350 0.607835
\(267\) 5.55460 0.339936
\(268\) 18.2961 1.11761
\(269\) 18.7957 1.14599 0.572997 0.819557i \(-0.305781\pi\)
0.572997 + 0.819557i \(0.305781\pi\)
\(270\) 0 0
\(271\) 11.0978 0.674146 0.337073 0.941479i \(-0.390563\pi\)
0.337073 + 0.941479i \(0.390563\pi\)
\(272\) 8.39314 0.508909
\(273\) −0.442139 −0.0267595
\(274\) −24.6630 −1.48995
\(275\) 0 0
\(276\) 0.177142 0.0106627
\(277\) −27.1795 −1.63306 −0.816529 0.577305i \(-0.804104\pi\)
−0.816529 + 0.577305i \(0.804104\pi\)
\(278\) 39.6209 2.37631
\(279\) 4.52278 0.270772
\(280\) 0 0
\(281\) 11.4050 0.680364 0.340182 0.940360i \(-0.389511\pi\)
0.340182 + 0.940360i \(0.389511\pi\)
\(282\) −2.65076 −0.157850
\(283\) −21.8388 −1.29818 −0.649090 0.760711i \(-0.724850\pi\)
−0.649090 + 0.760711i \(0.724850\pi\)
\(284\) 19.2063 1.13969
\(285\) 0 0
\(286\) 0.265820 0.0157183
\(287\) 62.8556 3.71025
\(288\) −19.6705 −1.15910
\(289\) −13.5336 −0.796093
\(290\) 0 0
\(291\) 4.18737 0.245468
\(292\) 16.6408 0.973832
\(293\) 5.49779 0.321184 0.160592 0.987021i \(-0.448660\pi\)
0.160592 + 0.987021i \(0.448660\pi\)
\(294\) 23.3599 1.36238
\(295\) 0 0
\(296\) −0.254237 −0.0147772
\(297\) 3.48667 0.202317
\(298\) −37.5867 −2.17734
\(299\) 0.0231632 0.00133956
\(300\) 0 0
\(301\) −44.7215 −2.57770
\(302\) 25.9479 1.49314
\(303\) −1.73665 −0.0997677
\(304\) −4.50800 −0.258551
\(305\) 0 0
\(306\) −9.36450 −0.535333
\(307\) −9.90065 −0.565060 −0.282530 0.959258i \(-0.591174\pi\)
−0.282530 + 0.959258i \(0.591174\pi\)
\(308\) −8.76711 −0.499552
\(309\) −4.65181 −0.264632
\(310\) 0 0
\(311\) 27.6617 1.56855 0.784276 0.620412i \(-0.213035\pi\)
0.784276 + 0.620412i \(0.213035\pi\)
\(312\) 0.0492887 0.00279042
\(313\) 18.3016 1.03447 0.517234 0.855844i \(-0.326962\pi\)
0.517234 + 0.855844i \(0.326962\pi\)
\(314\) −11.8478 −0.668611
\(315\) 0 0
\(316\) −10.4470 −0.587687
\(317\) 3.72231 0.209066 0.104533 0.994521i \(-0.466665\pi\)
0.104533 + 0.994521i \(0.466665\pi\)
\(318\) −1.60528 −0.0900196
\(319\) 4.80913 0.269259
\(320\) 0 0
\(321\) 2.02641 0.113103
\(322\) −1.66196 −0.0926175
\(323\) −1.86183 −0.103595
\(324\) 9.66010 0.536672
\(325\) 0 0
\(326\) −30.2933 −1.67779
\(327\) −0.908408 −0.0502351
\(328\) −7.00700 −0.386897
\(329\) 11.4318 0.630255
\(330\) 0 0
\(331\) −10.3313 −0.567858 −0.283929 0.958845i \(-0.591638\pi\)
−0.283929 + 0.958845i \(0.591638\pi\)
\(332\) 0.324053 0.0177847
\(333\) 1.15709 0.0634083
\(334\) −2.88090 −0.157636
\(335\) 0 0
\(336\) −14.4258 −0.786990
\(337\) 9.86777 0.537532 0.268766 0.963206i \(-0.413384\pi\)
0.268766 + 0.963206i \(0.413384\pi\)
\(338\) 24.9741 1.35841
\(339\) 5.33107 0.289544
\(340\) 0 0
\(341\) −1.73000 −0.0936849
\(342\) 5.02972 0.271976
\(343\) −64.6736 −3.49205
\(344\) 4.98545 0.268797
\(345\) 0 0
\(346\) −48.8161 −2.62437
\(347\) −17.4270 −0.935529 −0.467764 0.883853i \(-0.654940\pi\)
−0.467764 + 0.883853i \(0.654940\pi\)
\(348\) −5.08152 −0.272398
\(349\) −0.187343 −0.0100282 −0.00501411 0.999987i \(-0.501596\pi\)
−0.00501411 + 0.999987i \(0.501596\pi\)
\(350\) 0 0
\(351\) −0.481741 −0.0257135
\(352\) 7.52414 0.401038
\(353\) 27.6872 1.47364 0.736821 0.676088i \(-0.236326\pi\)
0.736821 + 0.676088i \(0.236326\pi\)
\(354\) 3.77072 0.200411
\(355\) 0 0
\(356\) −15.2179 −0.806545
\(357\) −5.95794 −0.315328
\(358\) 2.80267 0.148126
\(359\) 11.8587 0.625878 0.312939 0.949773i \(-0.398686\pi\)
0.312939 + 0.949773i \(0.398686\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 19.5060 1.02521
\(363\) −0.621031 −0.0325957
\(364\) 1.21132 0.0634906
\(365\) 0 0
\(366\) 2.27857 0.119103
\(367\) −22.6382 −1.18170 −0.590852 0.806780i \(-0.701208\pi\)
−0.590852 + 0.806780i \(0.701208\pi\)
\(368\) 0.755749 0.0393962
\(369\) 31.8904 1.66015
\(370\) 0 0
\(371\) 6.92300 0.359425
\(372\) 1.82799 0.0947770
\(373\) −18.2379 −0.944322 −0.472161 0.881512i \(-0.656526\pi\)
−0.472161 + 0.881512i \(0.656526\pi\)
\(374\) 3.58200 0.185221
\(375\) 0 0
\(376\) −1.27439 −0.0657217
\(377\) −0.664462 −0.0342215
\(378\) 34.5651 1.77783
\(379\) 36.4789 1.87380 0.936898 0.349604i \(-0.113684\pi\)
0.936898 + 0.349604i \(0.113684\pi\)
\(380\) 0 0
\(381\) −2.89951 −0.148547
\(382\) 17.2976 0.885022
\(383\) −21.4254 −1.09479 −0.547393 0.836876i \(-0.684380\pi\)
−0.547393 + 0.836876i \(0.684380\pi\)
\(384\) 2.82206 0.144013
\(385\) 0 0
\(386\) −8.55619 −0.435498
\(387\) −22.6899 −1.15339
\(388\) −11.4721 −0.582407
\(389\) −7.30513 −0.370385 −0.185192 0.982702i \(-0.559291\pi\)
−0.185192 + 0.982702i \(0.559291\pi\)
\(390\) 0 0
\(391\) 0.312130 0.0157851
\(392\) 11.2306 0.567232
\(393\) −0.583919 −0.0294548
\(394\) 42.2151 2.12677
\(395\) 0 0
\(396\) −4.44808 −0.223525
\(397\) −27.3333 −1.37182 −0.685909 0.727688i \(-0.740595\pi\)
−0.685909 + 0.727688i \(0.740595\pi\)
\(398\) −26.4844 −1.32754
\(399\) 3.20004 0.160202
\(400\) 0 0
\(401\) 19.4498 0.971274 0.485637 0.874161i \(-0.338588\pi\)
0.485637 + 0.874161i \(0.338588\pi\)
\(402\) 12.8482 0.640810
\(403\) 0.239029 0.0119069
\(404\) 4.75786 0.236713
\(405\) 0 0
\(406\) 47.6753 2.36608
\(407\) −0.442598 −0.0219387
\(408\) 0.664178 0.0328817
\(409\) −4.01732 −0.198644 −0.0993219 0.995055i \(-0.531667\pi\)
−0.0993219 + 0.995055i \(0.531667\pi\)
\(410\) 0 0
\(411\) −7.96113 −0.392694
\(412\) 12.7445 0.627876
\(413\) −16.2618 −0.800190
\(414\) −0.843214 −0.0414417
\(415\) 0 0
\(416\) −1.03959 −0.0509699
\(417\) 12.7895 0.626304
\(418\) −1.92391 −0.0941015
\(419\) 23.9878 1.17188 0.585940 0.810354i \(-0.300725\pi\)
0.585940 + 0.810354i \(0.300725\pi\)
\(420\) 0 0
\(421\) −20.3725 −0.992894 −0.496447 0.868067i \(-0.665362\pi\)
−0.496447 + 0.868067i \(0.665362\pi\)
\(422\) 41.4562 2.01806
\(423\) 5.80004 0.282008
\(424\) −0.771761 −0.0374800
\(425\) 0 0
\(426\) 13.4874 0.653467
\(427\) −9.82665 −0.475545
\(428\) −5.55172 −0.268352
\(429\) 0.0858058 0.00414275
\(430\) 0 0
\(431\) −17.8520 −0.859901 −0.429950 0.902853i \(-0.641469\pi\)
−0.429950 + 0.902853i \(0.641469\pi\)
\(432\) −15.7179 −0.756227
\(433\) 37.5810 1.80603 0.903015 0.429609i \(-0.141349\pi\)
0.903015 + 0.429609i \(0.141349\pi\)
\(434\) −17.1504 −0.823244
\(435\) 0 0
\(436\) 2.48875 0.119190
\(437\) −0.167646 −0.00801962
\(438\) 11.6858 0.558370
\(439\) −17.9023 −0.854430 −0.427215 0.904150i \(-0.640505\pi\)
−0.427215 + 0.904150i \(0.640505\pi\)
\(440\) 0 0
\(441\) −51.1131 −2.43396
\(442\) −0.494913 −0.0235406
\(443\) 21.7096 1.03146 0.515728 0.856752i \(-0.327521\pi\)
0.515728 + 0.856752i \(0.327521\pi\)
\(444\) 0.467667 0.0221945
\(445\) 0 0
\(446\) 49.1986 2.32962
\(447\) −12.1328 −0.573863
\(448\) 28.1330 1.32916
\(449\) −35.7775 −1.68844 −0.844222 0.535994i \(-0.819937\pi\)
−0.844222 + 0.535994i \(0.819937\pi\)
\(450\) 0 0
\(451\) −12.1984 −0.574399
\(452\) −14.6055 −0.686984
\(453\) 8.37589 0.393534
\(454\) 17.4410 0.818548
\(455\) 0 0
\(456\) −0.356733 −0.0167056
\(457\) −4.03494 −0.188747 −0.0943733 0.995537i \(-0.530085\pi\)
−0.0943733 + 0.995537i \(0.530085\pi\)
\(458\) −53.0564 −2.47916
\(459\) −6.49159 −0.303002
\(460\) 0 0
\(461\) −16.5876 −0.772563 −0.386281 0.922381i \(-0.626241\pi\)
−0.386281 + 0.922381i \(0.626241\pi\)
\(462\) −6.15659 −0.286430
\(463\) −20.1731 −0.937523 −0.468761 0.883325i \(-0.655300\pi\)
−0.468761 + 0.883325i \(0.655300\pi\)
\(464\) −21.6795 −1.00645
\(465\) 0 0
\(466\) 25.9798 1.20349
\(467\) −24.2739 −1.12326 −0.561632 0.827387i \(-0.689826\pi\)
−0.561632 + 0.827387i \(0.689826\pi\)
\(468\) 0.614578 0.0284089
\(469\) −55.4098 −2.55859
\(470\) 0 0
\(471\) −3.82443 −0.176221
\(472\) 1.81283 0.0834421
\(473\) 8.67909 0.399065
\(474\) −7.33624 −0.336965
\(475\) 0 0
\(476\) 16.3229 0.748158
\(477\) 3.51246 0.160825
\(478\) −4.82400 −0.220645
\(479\) −1.94923 −0.0890625 −0.0445312 0.999008i \(-0.514179\pi\)
−0.0445312 + 0.999008i \(0.514179\pi\)
\(480\) 0 0
\(481\) 0.0611523 0.00278830
\(482\) −47.0736 −2.14414
\(483\) −0.536475 −0.0244105
\(484\) 1.70143 0.0773378
\(485\) 0 0
\(486\) 26.9078 1.22056
\(487\) −18.1312 −0.821605 −0.410802 0.911724i \(-0.634751\pi\)
−0.410802 + 0.911724i \(0.634751\pi\)
\(488\) 1.09545 0.0495888
\(489\) −9.77857 −0.442202
\(490\) 0 0
\(491\) 20.7470 0.936298 0.468149 0.883649i \(-0.344921\pi\)
0.468149 + 0.883649i \(0.344921\pi\)
\(492\) 12.8893 0.581094
\(493\) −8.95379 −0.403259
\(494\) 0.265820 0.0119598
\(495\) 0 0
\(496\) 7.79884 0.350178
\(497\) −58.1664 −2.60912
\(498\) 0.227562 0.0101973
\(499\) −23.1410 −1.03593 −0.517967 0.855400i \(-0.673311\pi\)
−0.517967 + 0.855400i \(0.673311\pi\)
\(500\) 0 0
\(501\) −0.929944 −0.0415469
\(502\) 19.2105 0.857407
\(503\) −31.2204 −1.39205 −0.696024 0.718018i \(-0.745049\pi\)
−0.696024 + 0.718018i \(0.745049\pi\)
\(504\) 7.73804 0.344680
\(505\) 0 0
\(506\) 0.322537 0.0143385
\(507\) 8.06155 0.358026
\(508\) 7.94376 0.352447
\(509\) 11.1936 0.496147 0.248073 0.968741i \(-0.420203\pi\)
0.248073 + 0.968741i \(0.420203\pi\)
\(510\) 0 0
\(511\) −50.3968 −2.22942
\(512\) −28.7398 −1.27013
\(513\) 3.48667 0.153940
\(514\) −45.2334 −1.99516
\(515\) 0 0
\(516\) −9.17068 −0.403717
\(517\) −2.21857 −0.0975724
\(518\) −4.38769 −0.192784
\(519\) −15.7576 −0.691684
\(520\) 0 0
\(521\) 9.76912 0.427993 0.213996 0.976834i \(-0.431352\pi\)
0.213996 + 0.976834i \(0.431352\pi\)
\(522\) 24.1886 1.05870
\(523\) −10.2890 −0.449905 −0.224952 0.974370i \(-0.572223\pi\)
−0.224952 + 0.974370i \(0.572223\pi\)
\(524\) 1.59975 0.0698856
\(525\) 0 0
\(526\) 43.9129 1.91469
\(527\) 3.22098 0.140308
\(528\) 2.79960 0.121837
\(529\) −22.9719 −0.998778
\(530\) 0 0
\(531\) −8.25059 −0.358045
\(532\) −8.76711 −0.380102
\(533\) 1.68541 0.0730032
\(534\) −10.6866 −0.462452
\(535\) 0 0
\(536\) 6.17696 0.266804
\(537\) 0.904691 0.0390403
\(538\) −36.1613 −1.55902
\(539\) 19.5512 0.842129
\(540\) 0 0
\(541\) −21.0839 −0.906468 −0.453234 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(542\) −21.3512 −0.917115
\(543\) 6.29646 0.270207
\(544\) −14.0087 −0.600618
\(545\) 0 0
\(546\) 0.850636 0.0364039
\(547\) 14.5523 0.622212 0.311106 0.950375i \(-0.399301\pi\)
0.311106 + 0.950375i \(0.399301\pi\)
\(548\) 21.8110 0.931720
\(549\) −4.98566 −0.212783
\(550\) 0 0
\(551\) 4.80913 0.204876
\(552\) 0.0598050 0.00254547
\(553\) 31.6386 1.34541
\(554\) 52.2909 2.22163
\(555\) 0 0
\(556\) −35.0392 −1.48599
\(557\) −11.1213 −0.471226 −0.235613 0.971847i \(-0.575710\pi\)
−0.235613 + 0.971847i \(0.575710\pi\)
\(558\) −8.70143 −0.368361
\(559\) −1.19916 −0.0507191
\(560\) 0 0
\(561\) 1.15626 0.0488172
\(562\) −21.9422 −0.925574
\(563\) −22.9586 −0.967589 −0.483794 0.875182i \(-0.660742\pi\)
−0.483794 + 0.875182i \(0.660742\pi\)
\(564\) 2.34423 0.0987098
\(565\) 0 0
\(566\) 42.0159 1.76606
\(567\) −29.2556 −1.22862
\(568\) 6.48426 0.272074
\(569\) −35.9363 −1.50653 −0.753263 0.657719i \(-0.771521\pi\)
−0.753263 + 0.657719i \(0.771521\pi\)
\(570\) 0 0
\(571\) 13.4752 0.563919 0.281959 0.959426i \(-0.409016\pi\)
0.281959 + 0.959426i \(0.409016\pi\)
\(572\) −0.235081 −0.00982924
\(573\) 5.58360 0.233258
\(574\) −120.928 −5.04746
\(575\) 0 0
\(576\) 14.2736 0.594733
\(577\) 20.7453 0.863640 0.431820 0.901960i \(-0.357872\pi\)
0.431820 + 0.901960i \(0.357872\pi\)
\(578\) 26.0374 1.08301
\(579\) −2.76190 −0.114781
\(580\) 0 0
\(581\) −0.981395 −0.0407151
\(582\) −8.05613 −0.333937
\(583\) −1.34355 −0.0556440
\(584\) 5.61812 0.232480
\(585\) 0 0
\(586\) −10.5772 −0.436942
\(587\) −15.0468 −0.621048 −0.310524 0.950565i \(-0.600505\pi\)
−0.310524 + 0.950565i \(0.600505\pi\)
\(588\) −20.6586 −0.851946
\(589\) −1.73000 −0.0712835
\(590\) 0 0
\(591\) 13.6269 0.560535
\(592\) 1.99523 0.0820034
\(593\) 12.7120 0.522019 0.261009 0.965336i \(-0.415945\pi\)
0.261009 + 0.965336i \(0.415945\pi\)
\(594\) −6.70803 −0.275234
\(595\) 0 0
\(596\) 33.2402 1.36157
\(597\) −8.54906 −0.349890
\(598\) −0.0445639 −0.00182235
\(599\) −17.3589 −0.709267 −0.354634 0.935005i \(-0.615394\pi\)
−0.354634 + 0.935005i \(0.615394\pi\)
\(600\) 0 0
\(601\) −9.95232 −0.405964 −0.202982 0.979183i \(-0.565063\pi\)
−0.202982 + 0.979183i \(0.565063\pi\)
\(602\) 86.0401 3.50673
\(603\) −28.1128 −1.14484
\(604\) −22.9473 −0.933713
\(605\) 0 0
\(606\) 3.34115 0.135725
\(607\) −31.0381 −1.25980 −0.629900 0.776677i \(-0.716904\pi\)
−0.629900 + 0.776677i \(0.716904\pi\)
\(608\) 7.52414 0.305144
\(609\) 15.3894 0.623610
\(610\) 0 0
\(611\) 0.306532 0.0124010
\(612\) 8.28159 0.334764
\(613\) 18.2820 0.738401 0.369201 0.929350i \(-0.379631\pi\)
0.369201 + 0.929350i \(0.379631\pi\)
\(614\) 19.0480 0.768713
\(615\) 0 0
\(616\) −2.95987 −0.119256
\(617\) −35.8243 −1.44223 −0.721115 0.692815i \(-0.756370\pi\)
−0.721115 + 0.692815i \(0.756370\pi\)
\(618\) 8.94966 0.360008
\(619\) −33.3475 −1.34035 −0.670175 0.742203i \(-0.733781\pi\)
−0.670175 + 0.742203i \(0.733781\pi\)
\(620\) 0 0
\(621\) −0.584527 −0.0234563
\(622\) −53.2187 −2.13387
\(623\) 46.0873 1.84645
\(624\) −0.386812 −0.0154849
\(625\) 0 0
\(626\) −35.2106 −1.40730
\(627\) −0.621031 −0.0248016
\(628\) 10.4778 0.418108
\(629\) 0.824043 0.0328567
\(630\) 0 0
\(631\) −41.2198 −1.64094 −0.820468 0.571692i \(-0.806287\pi\)
−0.820468 + 0.571692i \(0.806287\pi\)
\(632\) −3.52700 −0.140297
\(633\) 13.3819 0.531883
\(634\) −7.16140 −0.284415
\(635\) 0 0
\(636\) 1.41965 0.0562926
\(637\) −2.70132 −0.107030
\(638\) −9.25233 −0.366303
\(639\) −29.5114 −1.16745
\(640\) 0 0
\(641\) 12.8023 0.505662 0.252831 0.967510i \(-0.418638\pi\)
0.252831 + 0.967510i \(0.418638\pi\)
\(642\) −3.89862 −0.153866
\(643\) −24.6826 −0.973386 −0.486693 0.873573i \(-0.661797\pi\)
−0.486693 + 0.873573i \(0.661797\pi\)
\(644\) 1.46977 0.0579172
\(645\) 0 0
\(646\) 3.58200 0.140932
\(647\) 27.4253 1.07820 0.539101 0.842241i \(-0.318764\pi\)
0.539101 + 0.842241i \(0.318764\pi\)
\(648\) 3.26135 0.128118
\(649\) 3.15592 0.123881
\(650\) 0 0
\(651\) −5.53608 −0.216976
\(652\) 26.7902 1.04919
\(653\) −44.1678 −1.72842 −0.864210 0.503131i \(-0.832181\pi\)
−0.864210 + 0.503131i \(0.832181\pi\)
\(654\) 1.74769 0.0683403
\(655\) 0 0
\(656\) 54.9902 2.14701
\(657\) −25.5694 −0.997556
\(658\) −21.9937 −0.857406
\(659\) −19.2583 −0.750198 −0.375099 0.926985i \(-0.622391\pi\)
−0.375099 + 0.926985i \(0.622391\pi\)
\(660\) 0 0
\(661\) 30.1994 1.17462 0.587311 0.809361i \(-0.300186\pi\)
0.587311 + 0.809361i \(0.300186\pi\)
\(662\) 19.8764 0.772519
\(663\) −0.159756 −0.00620442
\(664\) 0.109404 0.00424569
\(665\) 0 0
\(666\) −2.22614 −0.0862612
\(667\) −0.806233 −0.0312175
\(668\) 2.54776 0.0985757
\(669\) 15.8811 0.614000
\(670\) 0 0
\(671\) 1.90706 0.0736211
\(672\) 24.0775 0.928811
\(673\) −30.5163 −1.17632 −0.588159 0.808746i \(-0.700147\pi\)
−0.588159 + 0.808746i \(0.700147\pi\)
\(674\) −18.9847 −0.731264
\(675\) 0 0
\(676\) −22.0861 −0.849466
\(677\) −26.1491 −1.00499 −0.502497 0.864579i \(-0.667585\pi\)
−0.502497 + 0.864579i \(0.667585\pi\)
\(678\) −10.2565 −0.393899
\(679\) 34.7432 1.33332
\(680\) 0 0
\(681\) 5.62989 0.215738
\(682\) 3.32837 0.127450
\(683\) −6.55898 −0.250972 −0.125486 0.992095i \(-0.540049\pi\)
−0.125486 + 0.992095i \(0.540049\pi\)
\(684\) −4.44808 −0.170077
\(685\) 0 0
\(686\) 124.426 4.75061
\(687\) −17.1264 −0.653413
\(688\) −39.1253 −1.49164
\(689\) 0.185633 0.00707207
\(690\) 0 0
\(691\) 26.3501 1.00240 0.501202 0.865330i \(-0.332891\pi\)
0.501202 + 0.865330i \(0.332891\pi\)
\(692\) 43.1710 1.64112
\(693\) 13.4710 0.511722
\(694\) 33.5279 1.27270
\(695\) 0 0
\(696\) −1.71558 −0.0650287
\(697\) 22.7113 0.860253
\(698\) 0.360431 0.0136425
\(699\) 8.38619 0.317195
\(700\) 0 0
\(701\) 13.5154 0.510470 0.255235 0.966879i \(-0.417847\pi\)
0.255235 + 0.966879i \(0.417847\pi\)
\(702\) 0.926827 0.0349808
\(703\) −0.442598 −0.0166929
\(704\) −5.45977 −0.205773
\(705\) 0 0
\(706\) −53.2677 −2.00476
\(707\) −14.4092 −0.541914
\(708\) −3.33468 −0.125325
\(709\) −17.7736 −0.667503 −0.333751 0.942661i \(-0.608315\pi\)
−0.333751 + 0.942661i \(0.608315\pi\)
\(710\) 0 0
\(711\) 16.0522 0.602005
\(712\) −5.13771 −0.192544
\(713\) 0.290029 0.0108617
\(714\) 11.4625 0.428975
\(715\) 0 0
\(716\) −2.47857 −0.0926285
\(717\) −1.55717 −0.0581535
\(718\) −22.8151 −0.851450
\(719\) 27.2414 1.01593 0.507966 0.861377i \(-0.330397\pi\)
0.507966 + 0.861377i \(0.330397\pi\)
\(720\) 0 0
\(721\) −38.5967 −1.43742
\(722\) −1.92391 −0.0716005
\(723\) −15.1952 −0.565114
\(724\) −17.2503 −0.641103
\(725\) 0 0
\(726\) 1.19481 0.0443435
\(727\) −20.2128 −0.749651 −0.374826 0.927095i \(-0.622297\pi\)
−0.374826 + 0.927095i \(0.622297\pi\)
\(728\) 0.408955 0.0151569
\(729\) −8.34717 −0.309154
\(730\) 0 0
\(731\) −16.1590 −0.597663
\(732\) −2.01507 −0.0744793
\(733\) 27.9563 1.03259 0.516295 0.856411i \(-0.327311\pi\)
0.516295 + 0.856411i \(0.327311\pi\)
\(734\) 43.5538 1.60760
\(735\) 0 0
\(736\) −1.26139 −0.0464956
\(737\) 10.7534 0.396105
\(738\) −61.3543 −2.25848
\(739\) −38.5940 −1.41971 −0.709853 0.704350i \(-0.751238\pi\)
−0.709853 + 0.704350i \(0.751238\pi\)
\(740\) 0 0
\(741\) 0.0858058 0.00315216
\(742\) −13.3192 −0.488965
\(743\) 32.7873 1.20285 0.601425 0.798929i \(-0.294600\pi\)
0.601425 + 0.798929i \(0.294600\pi\)
\(744\) 0.617149 0.0226258
\(745\) 0 0
\(746\) 35.0881 1.28467
\(747\) −0.497921 −0.0182180
\(748\) −3.16778 −0.115826
\(749\) 16.8134 0.614348
\(750\) 0 0
\(751\) 3.21182 0.117201 0.0586005 0.998282i \(-0.481336\pi\)
0.0586005 + 0.998282i \(0.481336\pi\)
\(752\) 10.0013 0.364709
\(753\) 6.20108 0.225980
\(754\) 1.27836 0.0465553
\(755\) 0 0
\(756\) −30.5680 −1.11175
\(757\) 19.6023 0.712459 0.356230 0.934398i \(-0.384062\pi\)
0.356230 + 0.934398i \(0.384062\pi\)
\(758\) −70.1821 −2.54913
\(759\) 0.104114 0.00377909
\(760\) 0 0
\(761\) 14.2807 0.517674 0.258837 0.965921i \(-0.416661\pi\)
0.258837 + 0.965921i \(0.416661\pi\)
\(762\) 5.57841 0.202084
\(763\) −7.53719 −0.272865
\(764\) −15.2973 −0.553437
\(765\) 0 0
\(766\) 41.2205 1.48936
\(767\) −0.436043 −0.0157446
\(768\) −12.2108 −0.440618
\(769\) 21.9005 0.789751 0.394875 0.918735i \(-0.370788\pi\)
0.394875 + 0.918735i \(0.370788\pi\)
\(770\) 0 0
\(771\) −14.6012 −0.525849
\(772\) 7.56676 0.272333
\(773\) 49.0337 1.76362 0.881810 0.471605i \(-0.156325\pi\)
0.881810 + 0.471605i \(0.156325\pi\)
\(774\) 43.6534 1.56909
\(775\) 0 0
\(776\) −3.87310 −0.139036
\(777\) −1.41633 −0.0508105
\(778\) 14.0544 0.503875
\(779\) −12.1984 −0.437052
\(780\) 0 0
\(781\) 11.2883 0.403929
\(782\) −0.600510 −0.0214742
\(783\) 16.7678 0.599233
\(784\) −88.1367 −3.14774
\(785\) 0 0
\(786\) 1.12341 0.0400706
\(787\) −46.8687 −1.67069 −0.835345 0.549726i \(-0.814732\pi\)
−0.835345 + 0.549726i \(0.814732\pi\)
\(788\) −37.3334 −1.32995
\(789\) 14.1749 0.504641
\(790\) 0 0
\(791\) 44.2327 1.57273
\(792\) −1.50172 −0.0533613
\(793\) −0.263492 −0.00935687
\(794\) 52.5868 1.86623
\(795\) 0 0
\(796\) 23.4217 0.830162
\(797\) −38.5922 −1.36701 −0.683503 0.729947i \(-0.739545\pi\)
−0.683503 + 0.729947i \(0.739545\pi\)
\(798\) −6.15659 −0.217941
\(799\) 4.13060 0.146130
\(800\) 0 0
\(801\) 23.3829 0.826194
\(802\) −37.4196 −1.32133
\(803\) 9.78050 0.345146
\(804\) −11.3625 −0.400723
\(805\) 0 0
\(806\) −0.459870 −0.0161982
\(807\) −11.6727 −0.410899
\(808\) 1.60630 0.0565096
\(809\) −24.9581 −0.877478 −0.438739 0.898614i \(-0.644575\pi\)
−0.438739 + 0.898614i \(0.644575\pi\)
\(810\) 0 0
\(811\) 2.91522 0.102367 0.0511836 0.998689i \(-0.483701\pi\)
0.0511836 + 0.998689i \(0.483701\pi\)
\(812\) −42.1621 −1.47960
\(813\) −6.89210 −0.241717
\(814\) 0.851518 0.0298457
\(815\) 0 0
\(816\) −5.21240 −0.182470
\(817\) 8.67909 0.303643
\(818\) 7.72897 0.270237
\(819\) −1.86125 −0.0650373
\(820\) 0 0
\(821\) −44.1959 −1.54245 −0.771225 0.636563i \(-0.780355\pi\)
−0.771225 + 0.636563i \(0.780355\pi\)
\(822\) 15.3165 0.534224
\(823\) 7.80327 0.272005 0.136002 0.990709i \(-0.456575\pi\)
0.136002 + 0.990709i \(0.456575\pi\)
\(824\) 4.30267 0.149891
\(825\) 0 0
\(826\) 31.2862 1.08859
\(827\) −30.7160 −1.06810 −0.534051 0.845452i \(-0.679331\pi\)
−0.534051 + 0.845452i \(0.679331\pi\)
\(828\) 0.745706 0.0259151
\(829\) 30.3940 1.05563 0.527813 0.849361i \(-0.323012\pi\)
0.527813 + 0.849361i \(0.323012\pi\)
\(830\) 0 0
\(831\) 16.8793 0.585537
\(832\) 0.754359 0.0261527
\(833\) −36.4011 −1.26122
\(834\) −24.6058 −0.852030
\(835\) 0 0
\(836\) 1.70143 0.0588452
\(837\) −6.03194 −0.208494
\(838\) −46.1504 −1.59424
\(839\) −37.9883 −1.31150 −0.655750 0.754978i \(-0.727648\pi\)
−0.655750 + 0.754978i \(0.727648\pi\)
\(840\) 0 0
\(841\) −5.87228 −0.202493
\(842\) 39.1948 1.35074
\(843\) −7.08285 −0.243946
\(844\) −36.6623 −1.26197
\(845\) 0 0
\(846\) −11.1588 −0.383646
\(847\) −5.15278 −0.177052
\(848\) 6.05670 0.207988
\(849\) 13.5626 0.465466
\(850\) 0 0
\(851\) 0.0741999 0.00254354
\(852\) −11.9277 −0.408637
\(853\) 25.5855 0.876032 0.438016 0.898967i \(-0.355681\pi\)
0.438016 + 0.898967i \(0.355681\pi\)
\(854\) 18.9056 0.646936
\(855\) 0 0
\(856\) −1.87432 −0.0640629
\(857\) 47.7536 1.63123 0.815616 0.578593i \(-0.196398\pi\)
0.815616 + 0.578593i \(0.196398\pi\)
\(858\) −0.165083 −0.00563583
\(859\) 26.0637 0.889282 0.444641 0.895709i \(-0.353331\pi\)
0.444641 + 0.895709i \(0.353331\pi\)
\(860\) 0 0
\(861\) −39.0352 −1.33032
\(862\) 34.3457 1.16982
\(863\) −16.9616 −0.577380 −0.288690 0.957423i \(-0.593220\pi\)
−0.288690 + 0.957423i \(0.593220\pi\)
\(864\) 26.2342 0.892504
\(865\) 0 0
\(866\) −72.3026 −2.45694
\(867\) 8.40477 0.285441
\(868\) 15.1671 0.514805
\(869\) −6.14010 −0.208289
\(870\) 0 0
\(871\) −1.48576 −0.0503430
\(872\) 0.840229 0.0284537
\(873\) 17.6274 0.596595
\(874\) 0.322537 0.0109100
\(875\) 0 0
\(876\) −10.3345 −0.349170
\(877\) 4.50622 0.152164 0.0760822 0.997102i \(-0.475759\pi\)
0.0760822 + 0.997102i \(0.475759\pi\)
\(878\) 34.4424 1.16238
\(879\) −3.41429 −0.115161
\(880\) 0 0
\(881\) 6.05500 0.203998 0.101999 0.994784i \(-0.467476\pi\)
0.101999 + 0.994784i \(0.467476\pi\)
\(882\) 98.3370 3.31118
\(883\) 40.0852 1.34897 0.674487 0.738287i \(-0.264365\pi\)
0.674487 + 0.738287i \(0.264365\pi\)
\(884\) 0.437682 0.0147208
\(885\) 0 0
\(886\) −41.7674 −1.40320
\(887\) 9.84825 0.330672 0.165336 0.986237i \(-0.447129\pi\)
0.165336 + 0.986237i \(0.447129\pi\)
\(888\) 0.157889 0.00529842
\(889\) −24.0577 −0.806869
\(890\) 0 0
\(891\) 5.67763 0.190208
\(892\) −43.5093 −1.45680
\(893\) −2.21857 −0.0742415
\(894\) 23.3425 0.780689
\(895\) 0 0
\(896\) 23.4151 0.782243
\(897\) −0.0143850 −0.000480303 0
\(898\) 68.8327 2.29698
\(899\) −8.31980 −0.277481
\(900\) 0 0
\(901\) 2.50146 0.0833357
\(902\) 23.4686 0.781418
\(903\) 27.7734 0.924241
\(904\) −4.93096 −0.164001
\(905\) 0 0
\(906\) −16.1145 −0.535367
\(907\) −7.41910 −0.246347 −0.123174 0.992385i \(-0.539307\pi\)
−0.123174 + 0.992385i \(0.539307\pi\)
\(908\) −15.4242 −0.511868
\(909\) −7.31066 −0.242479
\(910\) 0 0
\(911\) 41.2334 1.36612 0.683062 0.730360i \(-0.260648\pi\)
0.683062 + 0.730360i \(0.260648\pi\)
\(912\) 2.79960 0.0927042
\(913\) 0.190459 0.00630328
\(914\) 7.76287 0.256773
\(915\) 0 0
\(916\) 46.9210 1.55031
\(917\) −4.84486 −0.159991
\(918\) 12.4892 0.412206
\(919\) −2.28384 −0.0753370 −0.0376685 0.999290i \(-0.511993\pi\)
−0.0376685 + 0.999290i \(0.511993\pi\)
\(920\) 0 0
\(921\) 6.14861 0.202604
\(922\) 31.9131 1.05100
\(923\) −1.55967 −0.0513373
\(924\) 5.44464 0.179116
\(925\) 0 0
\(926\) 38.8112 1.27542
\(927\) −19.5825 −0.643172
\(928\) 36.1845 1.18782
\(929\) 0.727324 0.0238627 0.0119314 0.999929i \(-0.496202\pi\)
0.0119314 + 0.999929i \(0.496202\pi\)
\(930\) 0 0
\(931\) 19.5512 0.640764
\(932\) −22.9755 −0.752589
\(933\) −17.1788 −0.562408
\(934\) 46.7009 1.52810
\(935\) 0 0
\(936\) 0.207488 0.00678195
\(937\) −7.67144 −0.250615 −0.125308 0.992118i \(-0.539992\pi\)
−0.125308 + 0.992118i \(0.539992\pi\)
\(938\) 106.603 3.48073
\(939\) −11.3659 −0.370911
\(940\) 0 0
\(941\) 33.4104 1.08915 0.544574 0.838713i \(-0.316691\pi\)
0.544574 + 0.838713i \(0.316691\pi\)
\(942\) 7.35787 0.239732
\(943\) 2.04501 0.0665948
\(944\) −14.2269 −0.463045
\(945\) 0 0
\(946\) −16.6978 −0.542892
\(947\) 53.7974 1.74818 0.874091 0.485762i \(-0.161458\pi\)
0.874091 + 0.485762i \(0.161458\pi\)
\(948\) 6.48789 0.210717
\(949\) −1.35134 −0.0438664
\(950\) 0 0
\(951\) −2.31167 −0.0749611
\(952\) 5.51078 0.178605
\(953\) 14.6240 0.473719 0.236859 0.971544i \(-0.423882\pi\)
0.236859 + 0.971544i \(0.423882\pi\)
\(954\) −6.75766 −0.218787
\(955\) 0 0
\(956\) 4.26615 0.137977
\(957\) −2.98662 −0.0965436
\(958\) 3.75014 0.121161
\(959\) −66.0547 −2.13302
\(960\) 0 0
\(961\) −28.0071 −0.903455
\(962\) −0.117651 −0.00379324
\(963\) 8.53045 0.274890
\(964\) 41.6300 1.34081
\(965\) 0 0
\(966\) 1.03213 0.0332082
\(967\) 35.2412 1.13328 0.566640 0.823965i \(-0.308243\pi\)
0.566640 + 0.823965i \(0.308243\pi\)
\(968\) 0.574421 0.0184626
\(969\) 1.15626 0.0371443
\(970\) 0 0
\(971\) 30.8760 0.990858 0.495429 0.868649i \(-0.335011\pi\)
0.495429 + 0.868649i \(0.335011\pi\)
\(972\) −23.7962 −0.763263
\(973\) 106.116 3.40193
\(974\) 34.8829 1.11772
\(975\) 0 0
\(976\) −8.59700 −0.275183
\(977\) 43.8813 1.40389 0.701943 0.712233i \(-0.252316\pi\)
0.701943 + 0.712233i \(0.252316\pi\)
\(978\) 18.8131 0.601576
\(979\) −8.94416 −0.285857
\(980\) 0 0
\(981\) −3.82407 −0.122093
\(982\) −39.9153 −1.27375
\(983\) 59.9692 1.91272 0.956360 0.292191i \(-0.0943842\pi\)
0.956360 + 0.292191i \(0.0943842\pi\)
\(984\) 4.35156 0.138723
\(985\) 0 0
\(986\) 17.2263 0.548597
\(987\) −7.09950 −0.225980
\(988\) −0.235081 −0.00747893
\(989\) −1.45502 −0.0462669
\(990\) 0 0
\(991\) −44.2019 −1.40412 −0.702060 0.712118i \(-0.747736\pi\)
−0.702060 + 0.712118i \(0.747736\pi\)
\(992\) −13.0168 −0.413283
\(993\) 6.41604 0.203607
\(994\) 111.907 3.54947
\(995\) 0 0
\(996\) −0.201247 −0.00637675
\(997\) −55.6039 −1.76099 −0.880496 0.474054i \(-0.842790\pi\)
−0.880496 + 0.474054i \(0.842790\pi\)
\(998\) 44.5213 1.40930
\(999\) −1.54319 −0.0488244
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 5225.2.a.p.1.3 9
5.4 even 2 1045.2.a.k.1.7 9
15.14 odd 2 9405.2.a.bh.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.k.1.7 9 5.4 even 2
5225.2.a.p.1.3 9 1.1 even 1 trivial
9405.2.a.bh.1.3 9 15.14 odd 2