Properties

Label 2025.2.a.x.1.1
Level $2025$
Weight $2$
Character 2025.1
Self dual yes
Analytic conductor $16.170$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2025,2,Mod(1,2025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, 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("2025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.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: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 405)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.52434\) of defining polynomial
Character \(\chi\) \(=\) 2025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.52434 q^{2} +4.37228 q^{4} -3.46410 q^{7} -5.98844 q^{8} +O(q^{10})\) \(q-2.52434 q^{2} +4.37228 q^{4} -3.46410 q^{7} -5.98844 q^{8} -1.37228 q^{11} +4.10891 q^{13} +8.74456 q^{14} +6.37228 q^{16} -2.52434 q^{17} +5.37228 q^{19} +3.46410 q^{22} -5.04868 q^{23} -10.3723 q^{26} -15.1460 q^{28} -5.74456 q^{29} +0.627719 q^{31} -4.10891 q^{32} +6.37228 q^{34} -7.57301 q^{37} -13.5615 q^{38} +1.37228 q^{41} +3.46410 q^{43} -6.00000 q^{44} +12.7446 q^{46} +8.51278 q^{47} +5.00000 q^{49} +17.9653 q^{52} +5.34363 q^{53} +20.7446 q^{56} +14.5012 q^{58} -7.37228 q^{59} +3.62772 q^{61} -1.58457 q^{62} -2.37228 q^{64} +8.21782 q^{67} -11.0371 q^{68} +4.11684 q^{71} -7.57301 q^{73} +19.1168 q^{74} +23.4891 q^{76} +4.75372 q^{77} -4.74456 q^{79} -3.46410 q^{82} +5.34363 q^{83} -8.74456 q^{86} +8.21782 q^{88} +3.00000 q^{89} -14.2337 q^{91} -22.0742 q^{92} -21.4891 q^{94} -18.6101 q^{97} -12.6217 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} + 6 q^{11} + 12 q^{14} + 14 q^{16} + 10 q^{19} - 30 q^{26} + 14 q^{31} + 14 q^{34} - 6 q^{41} - 24 q^{44} + 28 q^{46} + 20 q^{49} + 60 q^{56} - 18 q^{59} + 26 q^{61} + 2 q^{64} - 18 q^{71} + 42 q^{74} + 48 q^{76} + 4 q^{79} - 12 q^{86} + 12 q^{89} + 12 q^{91} - 40 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.52434 −1.78498 −0.892488 0.451071i \(-0.851042\pi\)
−0.892488 + 0.451071i \(0.851042\pi\)
\(3\) 0 0
\(4\) 4.37228 2.18614
\(5\) 0 0
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) −5.98844 −2.11723
\(9\) 0 0
\(10\) 0 0
\(11\) −1.37228 −0.413758 −0.206879 0.978366i \(-0.566331\pi\)
−0.206879 + 0.978366i \(0.566331\pi\)
\(12\) 0 0
\(13\) 4.10891 1.13961 0.569804 0.821781i \(-0.307019\pi\)
0.569804 + 0.821781i \(0.307019\pi\)
\(14\) 8.74456 2.33708
\(15\) 0 0
\(16\) 6.37228 1.59307
\(17\) −2.52434 −0.612242 −0.306121 0.951993i \(-0.599031\pi\)
−0.306121 + 0.951993i \(0.599031\pi\)
\(18\) 0 0
\(19\) 5.37228 1.23249 0.616243 0.787556i \(-0.288654\pi\)
0.616243 + 0.787556i \(0.288654\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.46410 0.738549
\(23\) −5.04868 −1.05272 −0.526361 0.850261i \(-0.676444\pi\)
−0.526361 + 0.850261i \(0.676444\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −10.3723 −2.03417
\(27\) 0 0
\(28\) −15.1460 −2.86233
\(29\) −5.74456 −1.06674 −0.533369 0.845883i \(-0.679074\pi\)
−0.533369 + 0.845883i \(0.679074\pi\)
\(30\) 0 0
\(31\) 0.627719 0.112742 0.0563708 0.998410i \(-0.482047\pi\)
0.0563708 + 0.998410i \(0.482047\pi\)
\(32\) −4.10891 −0.726360
\(33\) 0 0
\(34\) 6.37228 1.09284
\(35\) 0 0
\(36\) 0 0
\(37\) −7.57301 −1.24500 −0.622498 0.782621i \(-0.713882\pi\)
−0.622498 + 0.782621i \(0.713882\pi\)
\(38\) −13.5615 −2.19996
\(39\) 0 0
\(40\) 0 0
\(41\) 1.37228 0.214314 0.107157 0.994242i \(-0.465825\pi\)
0.107157 + 0.994242i \(0.465825\pi\)
\(42\) 0 0
\(43\) 3.46410 0.528271 0.264135 0.964486i \(-0.414913\pi\)
0.264135 + 0.964486i \(0.414913\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 12.7446 1.87908
\(47\) 8.51278 1.24172 0.620858 0.783923i \(-0.286784\pi\)
0.620858 + 0.783923i \(0.286784\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 17.9653 2.49134
\(53\) 5.34363 0.734004 0.367002 0.930220i \(-0.380384\pi\)
0.367002 + 0.930220i \(0.380384\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 20.7446 2.77211
\(57\) 0 0
\(58\) 14.5012 1.90410
\(59\) −7.37228 −0.959789 −0.479895 0.877326i \(-0.659325\pi\)
−0.479895 + 0.877326i \(0.659325\pi\)
\(60\) 0 0
\(61\) 3.62772 0.464482 0.232241 0.972658i \(-0.425394\pi\)
0.232241 + 0.972658i \(0.425394\pi\)
\(62\) −1.58457 −0.201241
\(63\) 0 0
\(64\) −2.37228 −0.296535
\(65\) 0 0
\(66\) 0 0
\(67\) 8.21782 1.00397 0.501983 0.864877i \(-0.332604\pi\)
0.501983 + 0.864877i \(0.332604\pi\)
\(68\) −11.0371 −1.33845
\(69\) 0 0
\(70\) 0 0
\(71\) 4.11684 0.488579 0.244290 0.969702i \(-0.421445\pi\)
0.244290 + 0.969702i \(0.421445\pi\)
\(72\) 0 0
\(73\) −7.57301 −0.886354 −0.443177 0.896434i \(-0.646149\pi\)
−0.443177 + 0.896434i \(0.646149\pi\)
\(74\) 19.1168 2.22229
\(75\) 0 0
\(76\) 23.4891 2.69439
\(77\) 4.75372 0.541737
\(78\) 0 0
\(79\) −4.74456 −0.533805 −0.266903 0.963724i \(-0.586000\pi\)
−0.266903 + 0.963724i \(0.586000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.46410 −0.382546
\(83\) 5.34363 0.586540 0.293270 0.956030i \(-0.405257\pi\)
0.293270 + 0.956030i \(0.405257\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.74456 −0.942950
\(87\) 0 0
\(88\) 8.21782 0.876023
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −14.2337 −1.49210
\(92\) −22.0742 −2.30140
\(93\) 0 0
\(94\) −21.4891 −2.21643
\(95\) 0 0
\(96\) 0 0
\(97\) −18.6101 −1.88957 −0.944786 0.327688i \(-0.893731\pi\)
−0.944786 + 0.327688i \(0.893731\pi\)
\(98\) −12.6217 −1.27498
\(99\) 0 0
\(100\) 0 0
\(101\) 10.6277 1.05750 0.528749 0.848778i \(-0.322661\pi\)
0.528749 + 0.848778i \(0.322661\pi\)
\(102\) 0 0
\(103\) 6.92820 0.682656 0.341328 0.939944i \(-0.389123\pi\)
0.341328 + 0.939944i \(0.389123\pi\)
\(104\) −24.6060 −2.41281
\(105\) 0 0
\(106\) −13.4891 −1.31018
\(107\) 13.2665 1.28252 0.641260 0.767323i \(-0.278412\pi\)
0.641260 + 0.767323i \(0.278412\pi\)
\(108\) 0 0
\(109\) −1.74456 −0.167099 −0.0835494 0.996504i \(-0.526626\pi\)
−0.0835494 + 0.996504i \(0.526626\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −22.0742 −2.08582
\(113\) 9.45254 0.889220 0.444610 0.895724i \(-0.353342\pi\)
0.444610 + 0.895724i \(0.353342\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −25.1168 −2.33204
\(117\) 0 0
\(118\) 18.6101 1.71320
\(119\) 8.74456 0.801613
\(120\) 0 0
\(121\) −9.11684 −0.828804
\(122\) −9.15759 −0.829089
\(123\) 0 0
\(124\) 2.74456 0.246469
\(125\) 0 0
\(126\) 0 0
\(127\) 10.3923 0.922168 0.461084 0.887357i \(-0.347461\pi\)
0.461084 + 0.887357i \(0.347461\pi\)
\(128\) 14.2063 1.25567
\(129\) 0 0
\(130\) 0 0
\(131\) 1.37228 0.119897 0.0599484 0.998201i \(-0.480906\pi\)
0.0599484 + 0.998201i \(0.480906\pi\)
\(132\) 0 0
\(133\) −18.6101 −1.61370
\(134\) −20.7446 −1.79206
\(135\) 0 0
\(136\) 15.1168 1.29626
\(137\) 2.22938 0.190469 0.0952346 0.995455i \(-0.469640\pi\)
0.0952346 + 0.995455i \(0.469640\pi\)
\(138\) 0 0
\(139\) −6.11684 −0.518824 −0.259412 0.965767i \(-0.583529\pi\)
−0.259412 + 0.965767i \(0.583529\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.3923 −0.872103
\(143\) −5.63858 −0.471522
\(144\) 0 0
\(145\) 0 0
\(146\) 19.1168 1.58212
\(147\) 0 0
\(148\) −33.1113 −2.72174
\(149\) −16.3723 −1.34127 −0.670635 0.741788i \(-0.733978\pi\)
−0.670635 + 0.741788i \(0.733978\pi\)
\(150\) 0 0
\(151\) 18.1168 1.47433 0.737164 0.675714i \(-0.236165\pi\)
0.737164 + 0.675714i \(0.236165\pi\)
\(152\) −32.1716 −2.60946
\(153\) 0 0
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) 21.4294 1.71025 0.855127 0.518419i \(-0.173479\pi\)
0.855127 + 0.518419i \(0.173479\pi\)
\(158\) 11.9769 0.952829
\(159\) 0 0
\(160\) 0 0
\(161\) 17.4891 1.37834
\(162\) 0 0
\(163\) 4.75372 0.372340 0.186170 0.982518i \(-0.440392\pi\)
0.186170 + 0.982518i \(0.440392\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −13.4891 −1.04696
\(167\) 0.294954 0.0228242 0.0114121 0.999935i \(-0.496367\pi\)
0.0114121 + 0.999935i \(0.496367\pi\)
\(168\) 0 0
\(169\) 3.88316 0.298704
\(170\) 0 0
\(171\) 0 0
\(172\) 15.1460 1.15487
\(173\) −7.86797 −0.598190 −0.299095 0.954223i \(-0.596685\pi\)
−0.299095 + 0.954223i \(0.596685\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −8.74456 −0.659146
\(177\) 0 0
\(178\) −7.57301 −0.567621
\(179\) 22.1168 1.65309 0.826545 0.562870i \(-0.190303\pi\)
0.826545 + 0.562870i \(0.190303\pi\)
\(180\) 0 0
\(181\) 14.8614 1.10464 0.552320 0.833632i \(-0.313743\pi\)
0.552320 + 0.833632i \(0.313743\pi\)
\(182\) 35.9306 2.66336
\(183\) 0 0
\(184\) 30.2337 2.22886
\(185\) 0 0
\(186\) 0 0
\(187\) 3.46410 0.253320
\(188\) 37.2203 2.71457
\(189\) 0 0
\(190\) 0 0
\(191\) −13.3723 −0.967584 −0.483792 0.875183i \(-0.660741\pi\)
−0.483792 + 0.875183i \(0.660741\pi\)
\(192\) 0 0
\(193\) −21.4294 −1.54252 −0.771262 0.636518i \(-0.780374\pi\)
−0.771262 + 0.636518i \(0.780374\pi\)
\(194\) 46.9783 3.37284
\(195\) 0 0
\(196\) 21.8614 1.56153
\(197\) 23.0140 1.63968 0.819840 0.572593i \(-0.194063\pi\)
0.819840 + 0.572593i \(0.194063\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −26.8280 −1.88761
\(203\) 19.8997 1.39669
\(204\) 0 0
\(205\) 0 0
\(206\) −17.4891 −1.21853
\(207\) 0 0
\(208\) 26.1831 1.81547
\(209\) −7.37228 −0.509951
\(210\) 0 0
\(211\) 26.8614 1.84922 0.924608 0.380921i \(-0.124393\pi\)
0.924608 + 0.380921i \(0.124393\pi\)
\(212\) 23.3639 1.60464
\(213\) 0 0
\(214\) −33.4891 −2.28927
\(215\) 0 0
\(216\) 0 0
\(217\) −2.17448 −0.147613
\(218\) 4.40387 0.298267
\(219\) 0 0
\(220\) 0 0
\(221\) −10.3723 −0.697715
\(222\) 0 0
\(223\) −1.28962 −0.0863594 −0.0431797 0.999067i \(-0.513749\pi\)
−0.0431797 + 0.999067i \(0.513749\pi\)
\(224\) 14.2337 0.951028
\(225\) 0 0
\(226\) −23.8614 −1.58724
\(227\) 0.294954 0.0195768 0.00978838 0.999952i \(-0.496884\pi\)
0.00978838 + 0.999952i \(0.496884\pi\)
\(228\) 0 0
\(229\) 22.6060 1.49384 0.746922 0.664911i \(-0.231531\pi\)
0.746922 + 0.664911i \(0.231531\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 34.4010 2.25853
\(233\) 9.45254 0.619257 0.309628 0.950858i \(-0.399795\pi\)
0.309628 + 0.950858i \(0.399795\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −32.2337 −2.09823
\(237\) 0 0
\(238\) −22.0742 −1.43086
\(239\) 22.9783 1.48634 0.743170 0.669103i \(-0.233321\pi\)
0.743170 + 0.669103i \(0.233321\pi\)
\(240\) 0 0
\(241\) −24.4891 −1.57748 −0.788742 0.614725i \(-0.789267\pi\)
−0.788742 + 0.614725i \(0.789267\pi\)
\(242\) 23.0140 1.47940
\(243\) 0 0
\(244\) 15.8614 1.01542
\(245\) 0 0
\(246\) 0 0
\(247\) 22.0742 1.40455
\(248\) −3.75906 −0.238700
\(249\) 0 0
\(250\) 0 0
\(251\) −14.2337 −0.898422 −0.449211 0.893426i \(-0.648295\pi\)
−0.449211 + 0.893426i \(0.648295\pi\)
\(252\) 0 0
\(253\) 6.92820 0.435572
\(254\) −26.2337 −1.64605
\(255\) 0 0
\(256\) −31.1168 −1.94480
\(257\) 23.0140 1.43557 0.717787 0.696263i \(-0.245155\pi\)
0.717787 + 0.696263i \(0.245155\pi\)
\(258\) 0 0
\(259\) 26.2337 1.63008
\(260\) 0 0
\(261\) 0 0
\(262\) −3.46410 −0.214013
\(263\) 26.5330 1.63609 0.818047 0.575151i \(-0.195057\pi\)
0.818047 + 0.575151i \(0.195057\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 46.9783 2.88042
\(267\) 0 0
\(268\) 35.9306 2.19481
\(269\) −5.23369 −0.319104 −0.159552 0.987190i \(-0.551005\pi\)
−0.159552 + 0.987190i \(0.551005\pi\)
\(270\) 0 0
\(271\) 16.7446 1.01716 0.508580 0.861015i \(-0.330171\pi\)
0.508580 + 0.861015i \(0.330171\pi\)
\(272\) −16.0858 −0.975344
\(273\) 0 0
\(274\) −5.62772 −0.339983
\(275\) 0 0
\(276\) 0 0
\(277\) 2.17448 0.130652 0.0653260 0.997864i \(-0.479191\pi\)
0.0653260 + 0.997864i \(0.479191\pi\)
\(278\) 15.4410 0.926088
\(279\) 0 0
\(280\) 0 0
\(281\) −4.37228 −0.260828 −0.130414 0.991460i \(-0.541631\pi\)
−0.130414 + 0.991460i \(0.541631\pi\)
\(282\) 0 0
\(283\) 27.7128 1.64736 0.823678 0.567058i \(-0.191918\pi\)
0.823678 + 0.567058i \(0.191918\pi\)
\(284\) 18.0000 1.06810
\(285\) 0 0
\(286\) 14.2337 0.841656
\(287\) −4.75372 −0.280603
\(288\) 0 0
\(289\) −10.6277 −0.625160
\(290\) 0 0
\(291\) 0 0
\(292\) −33.1113 −1.93769
\(293\) 2.52434 0.147473 0.0737367 0.997278i \(-0.476508\pi\)
0.0737367 + 0.997278i \(0.476508\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 45.3505 2.63595
\(297\) 0 0
\(298\) 41.3292 2.39413
\(299\) −20.7446 −1.19969
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) −45.7330 −2.63164
\(303\) 0 0
\(304\) 34.2337 1.96344
\(305\) 0 0
\(306\) 0 0
\(307\) −4.75372 −0.271309 −0.135655 0.990756i \(-0.543314\pi\)
−0.135655 + 0.990756i \(0.543314\pi\)
\(308\) 20.7846 1.18431
\(309\) 0 0
\(310\) 0 0
\(311\) −12.8614 −0.729303 −0.364652 0.931144i \(-0.618812\pi\)
−0.364652 + 0.931144i \(0.618812\pi\)
\(312\) 0 0
\(313\) 5.39853 0.305143 0.152572 0.988292i \(-0.451245\pi\)
0.152572 + 0.988292i \(0.451245\pi\)
\(314\) −54.0951 −3.05276
\(315\) 0 0
\(316\) −20.7446 −1.16697
\(317\) 6.98311 0.392210 0.196105 0.980583i \(-0.437171\pi\)
0.196105 + 0.980583i \(0.437171\pi\)
\(318\) 0 0
\(319\) 7.88316 0.441372
\(320\) 0 0
\(321\) 0 0
\(322\) −44.1485 −2.46030
\(323\) −13.5615 −0.754579
\(324\) 0 0
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) −8.21782 −0.453753
\(329\) −29.4891 −1.62579
\(330\) 0 0
\(331\) −8.11684 −0.446142 −0.223071 0.974802i \(-0.571608\pi\)
−0.223071 + 0.974802i \(0.571608\pi\)
\(332\) 23.3639 1.28226
\(333\) 0 0
\(334\) −0.744563 −0.0407407
\(335\) 0 0
\(336\) 0 0
\(337\) −6.92820 −0.377403 −0.188702 0.982034i \(-0.560428\pi\)
−0.188702 + 0.982034i \(0.560428\pi\)
\(338\) −9.80240 −0.533180
\(339\) 0 0
\(340\) 0 0
\(341\) −0.861407 −0.0466478
\(342\) 0 0
\(343\) 6.92820 0.374088
\(344\) −20.7446 −1.11847
\(345\) 0 0
\(346\) 19.8614 1.06776
\(347\) 23.6588 1.27007 0.635036 0.772483i \(-0.280985\pi\)
0.635036 + 0.772483i \(0.280985\pi\)
\(348\) 0 0
\(349\) 13.6060 0.728311 0.364155 0.931338i \(-0.381358\pi\)
0.364155 + 0.931338i \(0.381358\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.63858 0.300537
\(353\) −8.51278 −0.453089 −0.226545 0.974001i \(-0.572743\pi\)
−0.226545 + 0.974001i \(0.572743\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 13.1168 0.695191
\(357\) 0 0
\(358\) −55.8304 −2.95073
\(359\) 28.1168 1.48395 0.741975 0.670427i \(-0.233889\pi\)
0.741975 + 0.670427i \(0.233889\pi\)
\(360\) 0 0
\(361\) 9.86141 0.519021
\(362\) −37.5152 −1.97176
\(363\) 0 0
\(364\) −62.2337 −3.26193
\(365\) 0 0
\(366\) 0 0
\(367\) −23.3639 −1.21958 −0.609792 0.792562i \(-0.708747\pi\)
−0.609792 + 0.792562i \(0.708747\pi\)
\(368\) −32.1716 −1.67706
\(369\) 0 0
\(370\) 0 0
\(371\) −18.5109 −0.961037
\(372\) 0 0
\(373\) 11.6819 0.604867 0.302434 0.953170i \(-0.402201\pi\)
0.302434 + 0.953170i \(0.402201\pi\)
\(374\) −8.74456 −0.452171
\(375\) 0 0
\(376\) −50.9783 −2.62900
\(377\) −23.6039 −1.21566
\(378\) 0 0
\(379\) 21.4891 1.10382 0.551911 0.833903i \(-0.313899\pi\)
0.551911 + 0.833903i \(0.313899\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 33.7562 1.72712
\(383\) −35.3407 −1.80583 −0.902913 0.429822i \(-0.858576\pi\)
−0.902913 + 0.429822i \(0.858576\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 54.0951 2.75337
\(387\) 0 0
\(388\) −81.3687 −4.13087
\(389\) 23.4891 1.19095 0.595473 0.803375i \(-0.296965\pi\)
0.595473 + 0.803375i \(0.296965\pi\)
\(390\) 0 0
\(391\) 12.7446 0.644520
\(392\) −29.9422 −1.51231
\(393\) 0 0
\(394\) −58.0951 −2.92679
\(395\) 0 0
\(396\) 0 0
\(397\) 12.3267 0.618661 0.309331 0.950955i \(-0.399895\pi\)
0.309331 + 0.950955i \(0.399895\pi\)
\(398\) −40.3894 −2.02454
\(399\) 0 0
\(400\) 0 0
\(401\) −19.6277 −0.980161 −0.490081 0.871677i \(-0.663033\pi\)
−0.490081 + 0.871677i \(0.663033\pi\)
\(402\) 0 0
\(403\) 2.57924 0.128481
\(404\) 46.4674 2.31184
\(405\) 0 0
\(406\) −50.2337 −2.49306
\(407\) 10.3923 0.515127
\(408\) 0 0
\(409\) −5.86141 −0.289828 −0.144914 0.989444i \(-0.546291\pi\)
−0.144914 + 0.989444i \(0.546291\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 30.2921 1.49238
\(413\) 25.5383 1.25666
\(414\) 0 0
\(415\) 0 0
\(416\) −16.8832 −0.827765
\(417\) 0 0
\(418\) 18.6101 0.910251
\(419\) −5.48913 −0.268161 −0.134081 0.990970i \(-0.542808\pi\)
−0.134081 + 0.990970i \(0.542808\pi\)
\(420\) 0 0
\(421\) −10.2554 −0.499819 −0.249910 0.968269i \(-0.580401\pi\)
−0.249910 + 0.968269i \(0.580401\pi\)
\(422\) −67.8073 −3.30081
\(423\) 0 0
\(424\) −32.0000 −1.55406
\(425\) 0 0
\(426\) 0 0
\(427\) −12.5668 −0.608149
\(428\) 58.0049 2.80377
\(429\) 0 0
\(430\) 0 0
\(431\) −18.3505 −0.883914 −0.441957 0.897036i \(-0.645716\pi\)
−0.441957 + 0.897036i \(0.645716\pi\)
\(432\) 0 0
\(433\) 2.81929 0.135487 0.0677433 0.997703i \(-0.478420\pi\)
0.0677433 + 0.997703i \(0.478420\pi\)
\(434\) 5.48913 0.263486
\(435\) 0 0
\(436\) −7.62772 −0.365301
\(437\) −27.1229 −1.29746
\(438\) 0 0
\(439\) 3.13859 0.149797 0.0748984 0.997191i \(-0.476137\pi\)
0.0748984 + 0.997191i \(0.476137\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 26.1831 1.24541
\(443\) −24.9484 −1.18534 −0.592668 0.805447i \(-0.701925\pi\)
−0.592668 + 0.805447i \(0.701925\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.25544 0.154149
\(447\) 0 0
\(448\) 8.21782 0.388256
\(449\) 28.1168 1.32692 0.663458 0.748214i \(-0.269088\pi\)
0.663458 + 0.748214i \(0.269088\pi\)
\(450\) 0 0
\(451\) −1.88316 −0.0886744
\(452\) 41.3292 1.94396
\(453\) 0 0
\(454\) −0.744563 −0.0349441
\(455\) 0 0
\(456\) 0 0
\(457\) 8.86263 0.414577 0.207288 0.978280i \(-0.433536\pi\)
0.207288 + 0.978280i \(0.433536\pi\)
\(458\) −57.0651 −2.66648
\(459\) 0 0
\(460\) 0 0
\(461\) 39.0951 1.82084 0.910420 0.413685i \(-0.135759\pi\)
0.910420 + 0.413685i \(0.135759\pi\)
\(462\) 0 0
\(463\) −13.8564 −0.643962 −0.321981 0.946746i \(-0.604349\pi\)
−0.321981 + 0.946746i \(0.604349\pi\)
\(464\) −36.6060 −1.69939
\(465\) 0 0
\(466\) −23.8614 −1.10536
\(467\) −14.8511 −0.687226 −0.343613 0.939111i \(-0.611651\pi\)
−0.343613 + 0.939111i \(0.611651\pi\)
\(468\) 0 0
\(469\) −28.4674 −1.31450
\(470\) 0 0
\(471\) 0 0
\(472\) 44.1485 2.03210
\(473\) −4.75372 −0.218576
\(474\) 0 0
\(475\) 0 0
\(476\) 38.2337 1.75244
\(477\) 0 0
\(478\) −58.0049 −2.65308
\(479\) 21.6060 0.987202 0.493601 0.869688i \(-0.335680\pi\)
0.493601 + 0.869688i \(0.335680\pi\)
\(480\) 0 0
\(481\) −31.1168 −1.41881
\(482\) 61.8188 2.81577
\(483\) 0 0
\(484\) −39.8614 −1.81188
\(485\) 0 0
\(486\) 0 0
\(487\) −30.2921 −1.37266 −0.686332 0.727288i \(-0.740780\pi\)
−0.686332 + 0.727288i \(0.740780\pi\)
\(488\) −21.7244 −0.983416
\(489\) 0 0
\(490\) 0 0
\(491\) 37.3723 1.68659 0.843294 0.537453i \(-0.180613\pi\)
0.843294 + 0.537453i \(0.180613\pi\)
\(492\) 0 0
\(493\) 14.5012 0.653102
\(494\) −55.7228 −2.50709
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −14.2612 −0.639701
\(498\) 0 0
\(499\) −18.6277 −0.833891 −0.416946 0.908931i \(-0.636899\pi\)
−0.416946 + 0.908931i \(0.636899\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 35.9306 1.60366
\(503\) 10.0974 0.450219 0.225109 0.974334i \(-0.427726\pi\)
0.225109 + 0.974334i \(0.427726\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −17.4891 −0.777486
\(507\) 0 0
\(508\) 45.4381 2.01599
\(509\) −23.4891 −1.04114 −0.520569 0.853820i \(-0.674280\pi\)
−0.520569 + 0.853820i \(0.674280\pi\)
\(510\) 0 0
\(511\) 26.2337 1.16051
\(512\) 50.1369 2.21576
\(513\) 0 0
\(514\) −58.0951 −2.56246
\(515\) 0 0
\(516\) 0 0
\(517\) −11.6819 −0.513770
\(518\) −66.2227 −2.90966
\(519\) 0 0
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −15.1460 −0.662290 −0.331145 0.943580i \(-0.607435\pi\)
−0.331145 + 0.943580i \(0.607435\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −66.9783 −2.92039
\(527\) −1.58457 −0.0690251
\(528\) 0 0
\(529\) 2.48913 0.108223
\(530\) 0 0
\(531\) 0 0
\(532\) −81.3687 −3.52778
\(533\) 5.63858 0.244234
\(534\) 0 0
\(535\) 0 0
\(536\) −49.2119 −2.12563
\(537\) 0 0
\(538\) 13.2116 0.569592
\(539\) −6.86141 −0.295542
\(540\) 0 0
\(541\) −15.2337 −0.654947 −0.327474 0.944860i \(-0.606197\pi\)
−0.327474 + 0.944860i \(0.606197\pi\)
\(542\) −42.2689 −1.81561
\(543\) 0 0
\(544\) 10.3723 0.444708
\(545\) 0 0
\(546\) 0 0
\(547\) 6.92820 0.296229 0.148114 0.988970i \(-0.452680\pi\)
0.148114 + 0.988970i \(0.452680\pi\)
\(548\) 9.74749 0.416392
\(549\) 0 0
\(550\) 0 0
\(551\) −30.8614 −1.31474
\(552\) 0 0
\(553\) 16.4356 0.698915
\(554\) −5.48913 −0.233211
\(555\) 0 0
\(556\) −26.7446 −1.13422
\(557\) 21.7244 0.920491 0.460246 0.887792i \(-0.347761\pi\)
0.460246 + 0.887792i \(0.347761\pi\)
\(558\) 0 0
\(559\) 14.2337 0.602021
\(560\) 0 0
\(561\) 0 0
\(562\) 11.0371 0.465573
\(563\) 0.994667 0.0419202 0.0209601 0.999780i \(-0.493328\pi\)
0.0209601 + 0.999780i \(0.493328\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −69.9565 −2.94049
\(567\) 0 0
\(568\) −24.6535 −1.03444
\(569\) 2.48913 0.104350 0.0521748 0.998638i \(-0.483385\pi\)
0.0521748 + 0.998638i \(0.483385\pi\)
\(570\) 0 0
\(571\) 32.8614 1.37521 0.687604 0.726086i \(-0.258663\pi\)
0.687604 + 0.726086i \(0.258663\pi\)
\(572\) −24.6535 −1.03081
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) −28.3576 −1.18054 −0.590272 0.807205i \(-0.700979\pi\)
−0.590272 + 0.807205i \(0.700979\pi\)
\(578\) 26.8280 1.11590
\(579\) 0 0
\(580\) 0 0
\(581\) −18.5109 −0.767960
\(582\) 0 0
\(583\) −7.33296 −0.303700
\(584\) 45.3505 1.87662
\(585\) 0 0
\(586\) −6.37228 −0.263237
\(587\) 9.80240 0.404588 0.202294 0.979325i \(-0.435160\pi\)
0.202294 + 0.979325i \(0.435160\pi\)
\(588\) 0 0
\(589\) 3.37228 0.138952
\(590\) 0 0
\(591\) 0 0
\(592\) −48.2574 −1.98337
\(593\) −38.1600 −1.56704 −0.783522 0.621364i \(-0.786579\pi\)
−0.783522 + 0.621364i \(0.786579\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −71.5842 −2.93220
\(597\) 0 0
\(598\) 52.3663 2.14142
\(599\) −7.37228 −0.301223 −0.150612 0.988593i \(-0.548124\pi\)
−0.150612 + 0.988593i \(0.548124\pi\)
\(600\) 0 0
\(601\) 27.9783 1.14126 0.570628 0.821208i \(-0.306700\pi\)
0.570628 + 0.821208i \(0.306700\pi\)
\(602\) 30.2921 1.23461
\(603\) 0 0
\(604\) 79.2119 3.22309
\(605\) 0 0
\(606\) 0 0
\(607\) −36.3354 −1.47481 −0.737404 0.675452i \(-0.763949\pi\)
−0.737404 + 0.675452i \(0.763949\pi\)
\(608\) −22.0742 −0.895228
\(609\) 0 0
\(610\) 0 0
\(611\) 34.9783 1.41507
\(612\) 0 0
\(613\) −9.50744 −0.384002 −0.192001 0.981395i \(-0.561498\pi\)
−0.192001 + 0.981395i \(0.561498\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) −28.4674 −1.14698
\(617\) −22.4241 −0.902760 −0.451380 0.892332i \(-0.649068\pi\)
−0.451380 + 0.892332i \(0.649068\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 32.4665 1.30179
\(623\) −10.3923 −0.416359
\(624\) 0 0
\(625\) 0 0
\(626\) −13.6277 −0.544673
\(627\) 0 0
\(628\) 93.6955 3.73886
\(629\) 19.1168 0.762238
\(630\) 0 0
\(631\) 0.627719 0.0249891 0.0124945 0.999922i \(-0.496023\pi\)
0.0124945 + 0.999922i \(0.496023\pi\)
\(632\) 28.4125 1.13019
\(633\) 0 0
\(634\) −17.6277 −0.700086
\(635\) 0 0
\(636\) 0 0
\(637\) 20.5446 0.814005
\(638\) −19.8997 −0.787839
\(639\) 0 0
\(640\) 0 0
\(641\) 37.9783 1.50005 0.750025 0.661409i \(-0.230041\pi\)
0.750025 + 0.661409i \(0.230041\pi\)
\(642\) 0 0
\(643\) −18.6101 −0.733912 −0.366956 0.930238i \(-0.619600\pi\)
−0.366956 + 0.930238i \(0.619600\pi\)
\(644\) 76.4674 3.01324
\(645\) 0 0
\(646\) 34.2337 1.34691
\(647\) −4.45877 −0.175292 −0.0876461 0.996152i \(-0.527934\pi\)
−0.0876461 + 0.996152i \(0.527934\pi\)
\(648\) 0 0
\(649\) 10.1168 0.397121
\(650\) 0 0
\(651\) 0 0
\(652\) 20.7846 0.813988
\(653\) −20.1947 −0.790280 −0.395140 0.918621i \(-0.629304\pi\)
−0.395140 + 0.918621i \(0.629304\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 8.74456 0.341418
\(657\) 0 0
\(658\) 74.4405 2.90199
\(659\) −32.7446 −1.27555 −0.637774 0.770224i \(-0.720144\pi\)
−0.637774 + 0.770224i \(0.720144\pi\)
\(660\) 0 0
\(661\) 19.2337 0.748104 0.374052 0.927408i \(-0.377968\pi\)
0.374052 + 0.927408i \(0.377968\pi\)
\(662\) 20.4897 0.796353
\(663\) 0 0
\(664\) −32.0000 −1.24184
\(665\) 0 0
\(666\) 0 0
\(667\) 29.0024 1.12298
\(668\) 1.28962 0.0498969
\(669\) 0 0
\(670\) 0 0
\(671\) −4.97825 −0.192183
\(672\) 0 0
\(673\) −19.2549 −0.742223 −0.371112 0.928588i \(-0.621023\pi\)
−0.371112 + 0.928588i \(0.621023\pi\)
\(674\) 17.4891 0.673656
\(675\) 0 0
\(676\) 16.9783 0.653010
\(677\) −26.5330 −1.01975 −0.509873 0.860250i \(-0.670308\pi\)
−0.509873 + 0.860250i \(0.670308\pi\)
\(678\) 0 0
\(679\) 64.4674 2.47403
\(680\) 0 0
\(681\) 0 0
\(682\) 2.17448 0.0832652
\(683\) 0.589907 0.0225722 0.0112861 0.999936i \(-0.496407\pi\)
0.0112861 + 0.999936i \(0.496407\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −17.4891 −0.667738
\(687\) 0 0
\(688\) 22.0742 0.841572
\(689\) 21.9565 0.836476
\(690\) 0 0
\(691\) −11.7228 −0.445957 −0.222978 0.974823i \(-0.571578\pi\)
−0.222978 + 0.974823i \(0.571578\pi\)
\(692\) −34.4010 −1.30773
\(693\) 0 0
\(694\) −59.7228 −2.26705
\(695\) 0 0
\(696\) 0 0
\(697\) −3.46410 −0.131212
\(698\) −34.3461 −1.30002
\(699\) 0 0
\(700\) 0 0
\(701\) 17.2337 0.650907 0.325454 0.945558i \(-0.394483\pi\)
0.325454 + 0.945558i \(0.394483\pi\)
\(702\) 0 0
\(703\) −40.6844 −1.53444
\(704\) 3.25544 0.122694
\(705\) 0 0
\(706\) 21.4891 0.808754
\(707\) −36.8155 −1.38459
\(708\) 0 0
\(709\) 0.649468 0.0243913 0.0121956 0.999926i \(-0.496118\pi\)
0.0121956 + 0.999926i \(0.496118\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −17.9653 −0.673279
\(713\) −3.16915 −0.118686
\(714\) 0 0
\(715\) 0 0
\(716\) 96.7011 3.61389
\(717\) 0 0
\(718\) −70.9764 −2.64882
\(719\) −28.1168 −1.04858 −0.524291 0.851539i \(-0.675669\pi\)
−0.524291 + 0.851539i \(0.675669\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) −24.8935 −0.926441
\(723\) 0 0
\(724\) 64.9783 2.41490
\(725\) 0 0
\(726\) 0 0
\(727\) 10.7971 0.400441 0.200220 0.979751i \(-0.435834\pi\)
0.200220 + 0.979751i \(0.435834\pi\)
\(728\) 85.2376 3.15911
\(729\) 0 0
\(730\) 0 0
\(731\) −8.74456 −0.323429
\(732\) 0 0
\(733\) −23.3639 −0.862964 −0.431482 0.902122i \(-0.642009\pi\)
−0.431482 + 0.902122i \(0.642009\pi\)
\(734\) 58.9783 2.17693
\(735\) 0 0
\(736\) 20.7446 0.764655
\(737\) −11.2772 −0.415400
\(738\) 0 0
\(739\) 11.3723 0.418336 0.209168 0.977880i \(-0.432924\pi\)
0.209168 + 0.977880i \(0.432924\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 46.7277 1.71543
\(743\) 42.5639 1.56152 0.780759 0.624833i \(-0.214833\pi\)
0.780759 + 0.624833i \(0.214833\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −29.4891 −1.07967
\(747\) 0 0
\(748\) 15.1460 0.553794
\(749\) −45.9565 −1.67921
\(750\) 0 0
\(751\) −35.7228 −1.30354 −0.651772 0.758415i \(-0.725974\pi\)
−0.651772 + 0.758415i \(0.725974\pi\)
\(752\) 54.2458 1.97814
\(753\) 0 0
\(754\) 59.5842 2.16993
\(755\) 0 0
\(756\) 0 0
\(757\) 41.5692 1.51086 0.755429 0.655230i \(-0.227428\pi\)
0.755429 + 0.655230i \(0.227428\pi\)
\(758\) −54.2458 −1.97030
\(759\) 0 0
\(760\) 0 0
\(761\) −9.51087 −0.344769 −0.172384 0.985030i \(-0.555147\pi\)
−0.172384 + 0.985030i \(0.555147\pi\)
\(762\) 0 0
\(763\) 6.04334 0.218784
\(764\) −58.4674 −2.11528
\(765\) 0 0
\(766\) 89.2119 3.22336
\(767\) −30.2921 −1.09378
\(768\) 0 0
\(769\) 6.48913 0.234004 0.117002 0.993132i \(-0.462672\pi\)
0.117002 + 0.993132i \(0.462672\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −93.6955 −3.37217
\(773\) −18.2603 −0.656776 −0.328388 0.944543i \(-0.606505\pi\)
−0.328388 + 0.944543i \(0.606505\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 111.446 4.00066
\(777\) 0 0
\(778\) −59.2945 −2.12581
\(779\) 7.37228 0.264139
\(780\) 0 0
\(781\) −5.64947 −0.202154
\(782\) −32.1716 −1.15045
\(783\) 0 0
\(784\) 31.8614 1.13791
\(785\) 0 0
\(786\) 0 0
\(787\) −17.3205 −0.617409 −0.308705 0.951158i \(-0.599895\pi\)
−0.308705 + 0.951158i \(0.599895\pi\)
\(788\) 100.624 3.58457
\(789\) 0 0
\(790\) 0 0
\(791\) −32.7446 −1.16426
\(792\) 0 0
\(793\) 14.9060 0.529327
\(794\) −31.1168 −1.10430
\(795\) 0 0
\(796\) 69.9565 2.47954
\(797\) −48.8473 −1.73026 −0.865130 0.501548i \(-0.832764\pi\)
−0.865130 + 0.501548i \(0.832764\pi\)
\(798\) 0 0
\(799\) −21.4891 −0.760231
\(800\) 0 0
\(801\) 0 0
\(802\) 49.5470 1.74957
\(803\) 10.3923 0.366736
\(804\) 0 0
\(805\) 0 0
\(806\) −6.51087 −0.229336
\(807\) 0 0
\(808\) −63.6434 −2.23897
\(809\) −21.0000 −0.738321 −0.369160 0.929366i \(-0.620355\pi\)
−0.369160 + 0.929366i \(0.620355\pi\)
\(810\) 0 0
\(811\) 42.1168 1.47892 0.739461 0.673199i \(-0.235080\pi\)
0.739461 + 0.673199i \(0.235080\pi\)
\(812\) 87.0073 3.05336
\(813\) 0 0
\(814\) −26.2337 −0.919490
\(815\) 0 0
\(816\) 0 0
\(817\) 18.6101 0.651086
\(818\) 14.7962 0.517336
\(819\) 0 0
\(820\) 0 0
\(821\) −16.7228 −0.583630 −0.291815 0.956475i \(-0.594259\pi\)
−0.291815 + 0.956475i \(0.594259\pi\)
\(822\) 0 0
\(823\) −44.1485 −1.53892 −0.769459 0.638696i \(-0.779474\pi\)
−0.769459 + 0.638696i \(0.779474\pi\)
\(824\) −41.4891 −1.44534
\(825\) 0 0
\(826\) −64.4674 −2.24311
\(827\) −15.7359 −0.547192 −0.273596 0.961845i \(-0.588213\pi\)
−0.273596 + 0.961845i \(0.588213\pi\)
\(828\) 0 0
\(829\) 28.3505 0.984655 0.492327 0.870410i \(-0.336146\pi\)
0.492327 + 0.870410i \(0.336146\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −9.74749 −0.337934
\(833\) −12.6217 −0.437316
\(834\) 0 0
\(835\) 0 0
\(836\) −32.2337 −1.11483
\(837\) 0 0
\(838\) 13.8564 0.478662
\(839\) 17.1386 0.591690 0.295845 0.955236i \(-0.404399\pi\)
0.295845 + 0.955236i \(0.404399\pi\)
\(840\) 0 0
\(841\) 4.00000 0.137931
\(842\) 25.8882 0.892166
\(843\) 0 0
\(844\) 117.446 4.04265
\(845\) 0 0
\(846\) 0 0
\(847\) 31.5817 1.08516
\(848\) 34.0511 1.16932
\(849\) 0 0
\(850\) 0 0
\(851\) 38.2337 1.31063
\(852\) 0 0
\(853\) 48.9022 1.67438 0.837189 0.546913i \(-0.184197\pi\)
0.837189 + 0.546913i \(0.184197\pi\)
\(854\) 31.7228 1.08553
\(855\) 0 0
\(856\) −79.4456 −2.71540
\(857\) 26.8829 0.918301 0.459150 0.888359i \(-0.348154\pi\)
0.459150 + 0.888359i \(0.348154\pi\)
\(858\) 0 0
\(859\) −49.3288 −1.68308 −0.841538 0.540198i \(-0.818349\pi\)
−0.841538 + 0.540198i \(0.818349\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 46.3229 1.57777
\(863\) −9.80240 −0.333677 −0.166839 0.985984i \(-0.553356\pi\)
−0.166839 + 0.985984i \(0.553356\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −7.11684 −0.241840
\(867\) 0 0
\(868\) −9.50744 −0.322704
\(869\) 6.51087 0.220866
\(870\) 0 0
\(871\) 33.7663 1.14413
\(872\) 10.4472 0.353787
\(873\) 0 0
\(874\) 68.4674 2.31594
\(875\) 0 0
\(876\) 0 0
\(877\) 11.0371 0.372697 0.186348 0.982484i \(-0.440335\pi\)
0.186348 + 0.982484i \(0.440335\pi\)
\(878\) −7.92287 −0.267384
\(879\) 0 0
\(880\) 0 0
\(881\) −28.1168 −0.947281 −0.473640 0.880718i \(-0.657060\pi\)
−0.473640 + 0.880718i \(0.657060\pi\)
\(882\) 0 0
\(883\) 44.5532 1.49934 0.749668 0.661815i \(-0.230213\pi\)
0.749668 + 0.661815i \(0.230213\pi\)
\(884\) −45.3505 −1.52530
\(885\) 0 0
\(886\) 62.9783 2.11580
\(887\) −9.21249 −0.309325 −0.154663 0.987967i \(-0.549429\pi\)
−0.154663 + 0.987967i \(0.549429\pi\)
\(888\) 0 0
\(889\) −36.0000 −1.20740
\(890\) 0 0
\(891\) 0 0
\(892\) −5.63858 −0.188794
\(893\) 45.7330 1.53040
\(894\) 0 0
\(895\) 0 0
\(896\) −49.2119 −1.64406
\(897\) 0 0
\(898\) −70.9764 −2.36851
\(899\) −3.60597 −0.120266
\(900\) 0 0
\(901\) −13.4891 −0.449388
\(902\) 4.75372 0.158282
\(903\) 0 0
\(904\) −56.6060 −1.88269
\(905\) 0 0
\(906\) 0 0
\(907\) 58.0049 1.92602 0.963010 0.269466i \(-0.0868471\pi\)
0.963010 + 0.269466i \(0.0868471\pi\)
\(908\) 1.28962 0.0427976
\(909\) 0 0
\(910\) 0 0
\(911\) −21.6060 −0.715838 −0.357919 0.933753i \(-0.616514\pi\)
−0.357919 + 0.933753i \(0.616514\pi\)
\(912\) 0 0
\(913\) −7.33296 −0.242686
\(914\) −22.3723 −0.740009
\(915\) 0 0
\(916\) 98.8397 3.26575
\(917\) −4.75372 −0.156982
\(918\) 0 0
\(919\) 3.64947 0.120385 0.0601924 0.998187i \(-0.480829\pi\)
0.0601924 + 0.998187i \(0.480829\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −98.6892 −3.25016
\(923\) 16.9157 0.556789
\(924\) 0 0
\(925\) 0 0
\(926\) 34.9783 1.14946
\(927\) 0 0
\(928\) 23.6039 0.774836
\(929\) 26.4891 0.869080 0.434540 0.900653i \(-0.356911\pi\)
0.434540 + 0.900653i \(0.356911\pi\)
\(930\) 0 0
\(931\) 26.8614 0.880347
\(932\) 41.3292 1.35378
\(933\) 0 0
\(934\) 37.4891 1.22668
\(935\) 0 0
\(936\) 0 0
\(937\) −27.4728 −0.897496 −0.448748 0.893658i \(-0.648130\pi\)
−0.448748 + 0.893658i \(0.648130\pi\)
\(938\) 71.8613 2.34635
\(939\) 0 0
\(940\) 0 0
\(941\) −21.8614 −0.712661 −0.356331 0.934360i \(-0.615972\pi\)
−0.356331 + 0.934360i \(0.615972\pi\)
\(942\) 0 0
\(943\) −6.92820 −0.225613
\(944\) −46.9783 −1.52901
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) −36.5205 −1.18676 −0.593379 0.804923i \(-0.702206\pi\)
−0.593379 + 0.804923i \(0.702206\pi\)
\(948\) 0 0
\(949\) −31.1168 −1.01010
\(950\) 0 0
\(951\) 0 0
\(952\) −52.3663 −1.69720
\(953\) −21.7244 −0.703721 −0.351861 0.936052i \(-0.614451\pi\)
−0.351861 + 0.936052i \(0.614451\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 100.467 3.24935
\(957\) 0 0
\(958\) −54.5408 −1.76213
\(959\) −7.72281 −0.249383
\(960\) 0 0
\(961\) −30.6060 −0.987289
\(962\) 78.5494 2.53254
\(963\) 0 0
\(964\) −107.073 −3.44860
\(965\) 0 0
\(966\) 0 0
\(967\) −36.3354 −1.16847 −0.584234 0.811585i \(-0.698605\pi\)
−0.584234 + 0.811585i \(0.698605\pi\)
\(968\) 54.5957 1.75477
\(969\) 0 0
\(970\) 0 0
\(971\) 44.5842 1.43078 0.715388 0.698728i \(-0.246250\pi\)
0.715388 + 0.698728i \(0.246250\pi\)
\(972\) 0 0
\(973\) 21.1894 0.679300
\(974\) 76.4674 2.45017
\(975\) 0 0
\(976\) 23.1168 0.739952
\(977\) 54.8357 1.75435 0.877175 0.480171i \(-0.159425\pi\)
0.877175 + 0.480171i \(0.159425\pi\)
\(978\) 0 0
\(979\) −4.11684 −0.131575
\(980\) 0 0
\(981\) 0 0
\(982\) −94.3403 −3.01052
\(983\) −57.0102 −1.81834 −0.909171 0.416422i \(-0.863284\pi\)
−0.909171 + 0.416422i \(0.863284\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −36.6060 −1.16577
\(987\) 0 0
\(988\) 96.5147 3.07054
\(989\) −17.4891 −0.556122
\(990\) 0 0
\(991\) −1.60597 −0.0510153 −0.0255076 0.999675i \(-0.508120\pi\)
−0.0255076 + 0.999675i \(0.508120\pi\)
\(992\) −2.57924 −0.0818910
\(993\) 0 0
\(994\) 36.0000 1.14185
\(995\) 0 0
\(996\) 0 0
\(997\) 10.6324 0.336730 0.168365 0.985725i \(-0.446151\pi\)
0.168365 + 0.985725i \(0.446151\pi\)
\(998\) 47.0227 1.48848
\(999\) 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 2025.2.a.x.1.1 4
3.2 odd 2 2025.2.a.w.1.4 4
5.2 odd 4 405.2.b.b.244.1 yes 4
5.3 odd 4 405.2.b.b.244.4 yes 4
5.4 even 2 inner 2025.2.a.x.1.4 4
15.2 even 4 405.2.b.a.244.4 yes 4
15.8 even 4 405.2.b.a.244.1 4
15.14 odd 2 2025.2.a.w.1.1 4
45.2 even 12 405.2.j.e.109.2 4
45.7 odd 12 405.2.j.a.109.1 4
45.13 odd 12 405.2.j.a.379.1 4
45.22 odd 12 405.2.j.d.379.2 4
45.23 even 12 405.2.j.e.379.2 4
45.32 even 12 405.2.j.b.379.1 4
45.38 even 12 405.2.j.b.109.1 4
45.43 odd 12 405.2.j.d.109.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.b.a.244.1 4 15.8 even 4
405.2.b.a.244.4 yes 4 15.2 even 4
405.2.b.b.244.1 yes 4 5.2 odd 4
405.2.b.b.244.4 yes 4 5.3 odd 4
405.2.j.a.109.1 4 45.7 odd 12
405.2.j.a.379.1 4 45.13 odd 12
405.2.j.b.109.1 4 45.38 even 12
405.2.j.b.379.1 4 45.32 even 12
405.2.j.d.109.2 4 45.43 odd 12
405.2.j.d.379.2 4 45.22 odd 12
405.2.j.e.109.2 4 45.2 even 12
405.2.j.e.379.2 4 45.23 even 12
2025.2.a.w.1.1 4 15.14 odd 2
2025.2.a.w.1.4 4 3.2 odd 2
2025.2.a.x.1.1 4 1.1 even 1 trivial
2025.2.a.x.1.4 4 5.4 even 2 inner