Properties

Label 2025.2.a.x.1.3
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 1.3
Root \(0.792287\) 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)
 
\(f(q)\) \(=\) \(q+0.792287 q^{2} -1.37228 q^{4} -3.46410 q^{7} -2.67181 q^{8} +O(q^{10})\) \(q+0.792287 q^{2} -1.37228 q^{4} -3.46410 q^{7} -2.67181 q^{8} +4.37228 q^{11} -5.84096 q^{13} -2.74456 q^{14} +0.627719 q^{16} +0.792287 q^{17} -0.372281 q^{19} +3.46410 q^{22} +1.58457 q^{23} -4.62772 q^{26} +4.75372 q^{28} +5.74456 q^{29} +6.37228 q^{31} +5.84096 q^{32} +0.627719 q^{34} +2.37686 q^{37} -0.294954 q^{38} -4.37228 q^{41} +3.46410 q^{43} -6.00000 q^{44} +1.25544 q^{46} +1.87953 q^{47} +5.00000 q^{49} +8.01544 q^{52} +11.9769 q^{53} +9.25544 q^{56} +4.55134 q^{58} -1.62772 q^{59} +9.37228 q^{61} +5.04868 q^{62} +3.37228 q^{64} -11.6819 q^{67} -1.08724 q^{68} -13.1168 q^{71} +2.37686 q^{73} +1.88316 q^{74} +0.510875 q^{76} -15.1460 q^{77} +6.74456 q^{79} -3.46410 q^{82} +11.9769 q^{83} +2.74456 q^{86} -11.6819 q^{88} +3.00000 q^{89} +20.2337 q^{91} -2.17448 q^{92} +1.48913 q^{94} +1.28962 q^{97} +3.96143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.792287 0.560232 0.280116 0.959966i \(-0.409627\pi\)
0.280116 + 0.959966i \(0.409627\pi\)
\(3\) 0 0
\(4\) −1.37228 −0.686141
\(5\) 0 0
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) −2.67181 −0.944629
\(9\) 0 0
\(10\) 0 0
\(11\) 4.37228 1.31829 0.659146 0.752015i \(-0.270918\pi\)
0.659146 + 0.752015i \(0.270918\pi\)
\(12\) 0 0
\(13\) −5.84096 −1.61999 −0.809996 0.586436i \(-0.800531\pi\)
−0.809996 + 0.586436i \(0.800531\pi\)
\(14\) −2.74456 −0.733515
\(15\) 0 0
\(16\) 0.627719 0.156930
\(17\) 0.792287 0.192158 0.0960789 0.995374i \(-0.469370\pi\)
0.0960789 + 0.995374i \(0.469370\pi\)
\(18\) 0 0
\(19\) −0.372281 −0.0854072 −0.0427036 0.999088i \(-0.513597\pi\)
−0.0427036 + 0.999088i \(0.513597\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.46410 0.738549
\(23\) 1.58457 0.330407 0.165203 0.986260i \(-0.447172\pi\)
0.165203 + 0.986260i \(0.447172\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.62772 −0.907570
\(27\) 0 0
\(28\) 4.75372 0.898369
\(29\) 5.74456 1.06674 0.533369 0.845883i \(-0.320926\pi\)
0.533369 + 0.845883i \(0.320926\pi\)
\(30\) 0 0
\(31\) 6.37228 1.14450 0.572248 0.820081i \(-0.306072\pi\)
0.572248 + 0.820081i \(0.306072\pi\)
\(32\) 5.84096 1.03255
\(33\) 0 0
\(34\) 0.627719 0.107653
\(35\) 0 0
\(36\) 0 0
\(37\) 2.37686 0.390754 0.195377 0.980728i \(-0.437407\pi\)
0.195377 + 0.980728i \(0.437407\pi\)
\(38\) −0.294954 −0.0478478
\(39\) 0 0
\(40\) 0 0
\(41\) −4.37228 −0.682836 −0.341418 0.939912i \(-0.610907\pi\)
−0.341418 + 0.939912i \(0.610907\pi\)
\(42\) 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
\(46\) 1.25544 0.185104
\(47\) 1.87953 0.274157 0.137079 0.990560i \(-0.456229\pi\)
0.137079 + 0.990560i \(0.456229\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 8.01544 1.11154
\(53\) 11.9769 1.64515 0.822575 0.568656i \(-0.192536\pi\)
0.822575 + 0.568656i \(0.192536\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 9.25544 1.23681
\(57\) 0 0
\(58\) 4.55134 0.597621
\(59\) −1.62772 −0.211911 −0.105955 0.994371i \(-0.533790\pi\)
−0.105955 + 0.994371i \(0.533790\pi\)
\(60\) 0 0
\(61\) 9.37228 1.20000 0.599999 0.800001i \(-0.295168\pi\)
0.599999 + 0.800001i \(0.295168\pi\)
\(62\) 5.04868 0.641182
\(63\) 0 0
\(64\) 3.37228 0.421535
\(65\) 0 0
\(66\) 0 0
\(67\) −11.6819 −1.42717 −0.713587 0.700566i \(-0.752931\pi\)
−0.713587 + 0.700566i \(0.752931\pi\)
\(68\) −1.08724 −0.131847
\(69\) 0 0
\(70\) 0 0
\(71\) −13.1168 −1.55668 −0.778341 0.627841i \(-0.783939\pi\)
−0.778341 + 0.627841i \(0.783939\pi\)
\(72\) 0 0
\(73\) 2.37686 0.278191 0.139095 0.990279i \(-0.455581\pi\)
0.139095 + 0.990279i \(0.455581\pi\)
\(74\) 1.88316 0.218912
\(75\) 0 0
\(76\) 0.510875 0.0586013
\(77\) −15.1460 −1.72605
\(78\) 0 0
\(79\) 6.74456 0.758823 0.379411 0.925228i \(-0.376127\pi\)
0.379411 + 0.925228i \(0.376127\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.46410 −0.382546
\(83\) 11.9769 1.31463 0.657317 0.753615i \(-0.271691\pi\)
0.657317 + 0.753615i \(0.271691\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.74456 0.295954
\(87\) 0 0
\(88\) −11.6819 −1.24530
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 20.2337 2.12107
\(92\) −2.17448 −0.226705
\(93\) 0 0
\(94\) 1.48913 0.153592
\(95\) 0 0
\(96\) 0 0
\(97\) 1.28962 0.130941 0.0654706 0.997855i \(-0.479145\pi\)
0.0654706 + 0.997855i \(0.479145\pi\)
\(98\) 3.96143 0.400165
\(99\) 0 0
\(100\) 0 0
\(101\) 16.3723 1.62910 0.814551 0.580091i \(-0.196983\pi\)
0.814551 + 0.580091i \(0.196983\pi\)
\(102\) 0 0
\(103\) 6.92820 0.682656 0.341328 0.939944i \(-0.389123\pi\)
0.341328 + 0.939944i \(0.389123\pi\)
\(104\) 15.6060 1.53029
\(105\) 0 0
\(106\) 9.48913 0.921665
\(107\) −13.2665 −1.28252 −0.641260 0.767323i \(-0.721588\pi\)
−0.641260 + 0.767323i \(0.721588\pi\)
\(108\) 0 0
\(109\) 9.74456 0.933360 0.466680 0.884426i \(-0.345450\pi\)
0.466680 + 0.884426i \(0.345450\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.17448 −0.205469
\(113\) 6.13592 0.577218 0.288609 0.957447i \(-0.406807\pi\)
0.288609 + 0.957447i \(0.406807\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −7.88316 −0.731933
\(117\) 0 0
\(118\) −1.28962 −0.118719
\(119\) −2.74456 −0.251594
\(120\) 0 0
\(121\) 8.11684 0.737895
\(122\) 7.42554 0.672276
\(123\) 0 0
\(124\) −8.74456 −0.785285
\(125\) 0 0
\(126\) 0 0
\(127\) 10.3923 0.922168 0.461084 0.887357i \(-0.347461\pi\)
0.461084 + 0.887357i \(0.347461\pi\)
\(128\) −9.01011 −0.796389
\(129\) 0 0
\(130\) 0 0
\(131\) −4.37228 −0.382008 −0.191004 0.981589i \(-0.561174\pi\)
−0.191004 + 0.981589i \(0.561174\pi\)
\(132\) 0 0
\(133\) 1.28962 0.111824
\(134\) −9.25544 −0.799548
\(135\) 0 0
\(136\) −2.11684 −0.181518
\(137\) −14.3537 −1.22632 −0.613161 0.789958i \(-0.710102\pi\)
−0.613161 + 0.789958i \(0.710102\pi\)
\(138\) 0 0
\(139\) 11.1168 0.942918 0.471459 0.881888i \(-0.343727\pi\)
0.471459 + 0.881888i \(0.343727\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.3923 −0.872103
\(143\) −25.5383 −2.13562
\(144\) 0 0
\(145\) 0 0
\(146\) 1.88316 0.155851
\(147\) 0 0
\(148\) −3.26172 −0.268112
\(149\) −10.6277 −0.870657 −0.435328 0.900272i \(-0.643368\pi\)
−0.435328 + 0.900272i \(0.643368\pi\)
\(150\) 0 0
\(151\) 0.883156 0.0718702 0.0359351 0.999354i \(-0.488559\pi\)
0.0359351 + 0.999354i \(0.488559\pi\)
\(152\) 0.994667 0.0806781
\(153\) 0 0
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) 11.4795 0.916167 0.458084 0.888909i \(-0.348536\pi\)
0.458084 + 0.888909i \(0.348536\pi\)
\(158\) 5.34363 0.425116
\(159\) 0 0
\(160\) 0 0
\(161\) −5.48913 −0.432604
\(162\) 0 0
\(163\) −15.1460 −1.18633 −0.593164 0.805082i \(-0.702122\pi\)
−0.593164 + 0.805082i \(0.702122\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 9.48913 0.736499
\(167\) 13.5615 1.04942 0.524708 0.851282i \(-0.324174\pi\)
0.524708 + 0.851282i \(0.324174\pi\)
\(168\) 0 0
\(169\) 21.1168 1.62437
\(170\) 0 0
\(171\) 0 0
\(172\) −4.75372 −0.362468
\(173\) −11.1846 −0.850349 −0.425174 0.905111i \(-0.639787\pi\)
−0.425174 + 0.905111i \(0.639787\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.74456 0.206879
\(177\) 0 0
\(178\) 2.37686 0.178153
\(179\) 4.88316 0.364984 0.182492 0.983207i \(-0.441584\pi\)
0.182492 + 0.983207i \(0.441584\pi\)
\(180\) 0 0
\(181\) −13.8614 −1.03031 −0.515155 0.857097i \(-0.672266\pi\)
−0.515155 + 0.857097i \(0.672266\pi\)
\(182\) 16.0309 1.18829
\(183\) 0 0
\(184\) −4.23369 −0.312112
\(185\) 0 0
\(186\) 0 0
\(187\) 3.46410 0.253320
\(188\) −2.57924 −0.188110
\(189\) 0 0
\(190\) 0 0
\(191\) −7.62772 −0.551922 −0.275961 0.961169i \(-0.588996\pi\)
−0.275961 + 0.961169i \(0.588996\pi\)
\(192\) 0 0
\(193\) −11.4795 −0.826316 −0.413158 0.910659i \(-0.635574\pi\)
−0.413158 + 0.910659i \(0.635574\pi\)
\(194\) 1.02175 0.0733573
\(195\) 0 0
\(196\) −6.86141 −0.490100
\(197\) 6.43087 0.458181 0.229090 0.973405i \(-0.426425\pi\)
0.229090 + 0.973405i \(0.426425\pi\)
\(198\) 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
\(202\) 12.9715 0.912675
\(203\) −19.8997 −1.39669
\(204\) 0 0
\(205\) 0 0
\(206\) 5.48913 0.382445
\(207\) 0 0
\(208\) −3.66648 −0.254225
\(209\) −1.62772 −0.112592
\(210\) 0 0
\(211\) −1.86141 −0.128145 −0.0640723 0.997945i \(-0.520409\pi\)
−0.0640723 + 0.997945i \(0.520409\pi\)
\(212\) −16.4356 −1.12880
\(213\) 0 0
\(214\) −10.5109 −0.718509
\(215\) 0 0
\(216\) 0 0
\(217\) −22.0742 −1.49850
\(218\) 7.72049 0.522898
\(219\) 0 0
\(220\) 0 0
\(221\) −4.62772 −0.311294
\(222\) 0 0
\(223\) 18.6101 1.24623 0.623113 0.782132i \(-0.285868\pi\)
0.623113 + 0.782132i \(0.285868\pi\)
\(224\) −20.2337 −1.35192
\(225\) 0 0
\(226\) 4.86141 0.323376
\(227\) 13.5615 0.900105 0.450053 0.893002i \(-0.351405\pi\)
0.450053 + 0.893002i \(0.351405\pi\)
\(228\) 0 0
\(229\) −17.6060 −1.16344 −0.581718 0.813391i \(-0.697619\pi\)
−0.581718 + 0.813391i \(0.697619\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −15.3484 −1.00767
\(233\) 6.13592 0.401977 0.200989 0.979594i \(-0.435585\pi\)
0.200989 + 0.979594i \(0.435585\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2.23369 0.145401
\(237\) 0 0
\(238\) −2.17448 −0.140951
\(239\) −22.9783 −1.48634 −0.743170 0.669103i \(-0.766679\pi\)
−0.743170 + 0.669103i \(0.766679\pi\)
\(240\) 0 0
\(241\) −1.51087 −0.0973240 −0.0486620 0.998815i \(-0.515496\pi\)
−0.0486620 + 0.998815i \(0.515496\pi\)
\(242\) 6.43087 0.413392
\(243\) 0 0
\(244\) −12.8614 −0.823367
\(245\) 0 0
\(246\) 0 0
\(247\) 2.17448 0.138359
\(248\) −17.0256 −1.08112
\(249\) 0 0
\(250\) 0 0
\(251\) 20.2337 1.27714 0.638570 0.769564i \(-0.279526\pi\)
0.638570 + 0.769564i \(0.279526\pi\)
\(252\) 0 0
\(253\) 6.92820 0.435572
\(254\) 8.23369 0.516628
\(255\) 0 0
\(256\) −13.8832 −0.867697
\(257\) 6.43087 0.401147 0.200573 0.979679i \(-0.435720\pi\)
0.200573 + 0.979679i \(0.435720\pi\)
\(258\) 0 0
\(259\) −8.23369 −0.511616
\(260\) 0 0
\(261\) 0 0
\(262\) −3.46410 −0.214013
\(263\) −26.5330 −1.63609 −0.818047 0.575151i \(-0.804943\pi\)
−0.818047 + 0.575151i \(0.804943\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.02175 0.0626475
\(267\) 0 0
\(268\) 16.0309 0.979242
\(269\) 29.2337 1.78241 0.891205 0.453601i \(-0.149861\pi\)
0.891205 + 0.453601i \(0.149861\pi\)
\(270\) 0 0
\(271\) 5.25544 0.319245 0.159623 0.987178i \(-0.448972\pi\)
0.159623 + 0.987178i \(0.448972\pi\)
\(272\) 0.497333 0.0301553
\(273\) 0 0
\(274\) −11.3723 −0.687025
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0742 1.32631 0.663156 0.748481i \(-0.269217\pi\)
0.663156 + 0.748481i \(0.269217\pi\)
\(278\) 8.80773 0.528253
\(279\) 0 0
\(280\) 0 0
\(281\) 1.37228 0.0818634 0.0409317 0.999162i \(-0.486967\pi\)
0.0409317 + 0.999162i \(0.486967\pi\)
\(282\) 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
\(286\) −20.2337 −1.19644
\(287\) 15.1460 0.894042
\(288\) 0 0
\(289\) −16.3723 −0.963075
\(290\) 0 0
\(291\) 0 0
\(292\) −3.26172 −0.190878
\(293\) −0.792287 −0.0462859 −0.0231430 0.999732i \(-0.507367\pi\)
−0.0231430 + 0.999732i \(0.507367\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.35053 −0.369117
\(297\) 0 0
\(298\) −8.42020 −0.487769
\(299\) −9.25544 −0.535256
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0.699713 0.0402640
\(303\) 0 0
\(304\) −0.233688 −0.0134029
\(305\) 0 0
\(306\) 0 0
\(307\) 15.1460 0.864429 0.432215 0.901771i \(-0.357732\pi\)
0.432215 + 0.901771i \(0.357732\pi\)
\(308\) 20.7846 1.18431
\(309\) 0 0
\(310\) 0 0
\(311\) 15.8614 0.899418 0.449709 0.893175i \(-0.351528\pi\)
0.449709 + 0.893175i \(0.351528\pi\)
\(312\) 0 0
\(313\) −24.4511 −1.38206 −0.691029 0.722827i \(-0.742842\pi\)
−0.691029 + 0.722827i \(0.742842\pi\)
\(314\) 9.09509 0.513266
\(315\) 0 0
\(316\) −9.25544 −0.520659
\(317\) −29.4998 −1.65687 −0.828436 0.560084i \(-0.810769\pi\)
−0.828436 + 0.560084i \(0.810769\pi\)
\(318\) 0 0
\(319\) 25.1168 1.40627
\(320\) 0 0
\(321\) 0 0
\(322\) −4.34896 −0.242358
\(323\) −0.294954 −0.0164117
\(324\) 0 0
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) 11.6819 0.645026
\(329\) −6.51087 −0.358956
\(330\) 0 0
\(331\) 9.11684 0.501107 0.250554 0.968103i \(-0.419387\pi\)
0.250554 + 0.968103i \(0.419387\pi\)
\(332\) −16.4356 −0.902023
\(333\) 0 0
\(334\) 10.7446 0.587916
\(335\) 0 0
\(336\) 0 0
\(337\) −6.92820 −0.377403 −0.188702 0.982034i \(-0.560428\pi\)
−0.188702 + 0.982034i \(0.560428\pi\)
\(338\) 16.7306 0.910025
\(339\) 0 0
\(340\) 0 0
\(341\) 27.8614 1.50878
\(342\) 0 0
\(343\) 6.92820 0.374088
\(344\) −9.25544 −0.499020
\(345\) 0 0
\(346\) −8.86141 −0.476392
\(347\) −2.87419 −0.154295 −0.0771474 0.997020i \(-0.524581\pi\)
−0.0771474 + 0.997020i \(0.524581\pi\)
\(348\) 0 0
\(349\) −26.6060 −1.42418 −0.712092 0.702086i \(-0.752252\pi\)
−0.712092 + 0.702086i \(0.752252\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 25.5383 1.36120
\(353\) −1.87953 −0.100037 −0.0500186 0.998748i \(-0.515928\pi\)
−0.0500186 + 0.998748i \(0.515928\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.11684 −0.218192
\(357\) 0 0
\(358\) 3.86886 0.204476
\(359\) 10.8832 0.574391 0.287196 0.957872i \(-0.407277\pi\)
0.287196 + 0.957872i \(0.407277\pi\)
\(360\) 0 0
\(361\) −18.8614 −0.992706
\(362\) −10.9822 −0.577212
\(363\) 0 0
\(364\) −27.7663 −1.45535
\(365\) 0 0
\(366\) 0 0
\(367\) 16.4356 0.857934 0.428967 0.903320i \(-0.358878\pi\)
0.428967 + 0.903320i \(0.358878\pi\)
\(368\) 0.994667 0.0518506
\(369\) 0 0
\(370\) 0 0
\(371\) −41.4891 −2.15401
\(372\) 0 0
\(373\) −8.21782 −0.425503 −0.212751 0.977106i \(-0.568242\pi\)
−0.212751 + 0.977106i \(0.568242\pi\)
\(374\) 2.74456 0.141918
\(375\) 0 0
\(376\) −5.02175 −0.258977
\(377\) −33.5538 −1.72811
\(378\) 0 0
\(379\) −1.48913 −0.0764912 −0.0382456 0.999268i \(-0.512177\pi\)
−0.0382456 + 0.999268i \(0.512177\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −6.04334 −0.309204
\(383\) 11.0920 0.566776 0.283388 0.959005i \(-0.408542\pi\)
0.283388 + 0.959005i \(0.408542\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9.09509 −0.462928
\(387\) 0 0
\(388\) −1.76972 −0.0898440
\(389\) 0.510875 0.0259024 0.0129512 0.999916i \(-0.495877\pi\)
0.0129512 + 0.999916i \(0.495877\pi\)
\(390\) 0 0
\(391\) 1.25544 0.0634902
\(392\) −13.3591 −0.674735
\(393\) 0 0
\(394\) 5.09509 0.256687
\(395\) 0 0
\(396\) 0 0
\(397\) −17.5229 −0.879449 −0.439724 0.898133i \(-0.644924\pi\)
−0.439724 + 0.898133i \(0.644924\pi\)
\(398\) 12.6766 0.635420
\(399\) 0 0
\(400\) 0 0
\(401\) −25.3723 −1.26703 −0.633516 0.773730i \(-0.718389\pi\)
−0.633516 + 0.773730i \(0.718389\pi\)
\(402\) 0 0
\(403\) −37.2203 −1.85407
\(404\) −22.4674 −1.11779
\(405\) 0 0
\(406\) −15.7663 −0.782469
\(407\) 10.3923 0.515127
\(408\) 0 0
\(409\) 22.8614 1.13042 0.565212 0.824946i \(-0.308794\pi\)
0.565212 + 0.824946i \(0.308794\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −9.50744 −0.468398
\(413\) 5.63858 0.277457
\(414\) 0 0
\(415\) 0 0
\(416\) −34.1168 −1.67272
\(417\) 0 0
\(418\) −1.28962 −0.0630774
\(419\) 17.4891 0.854400 0.427200 0.904157i \(-0.359500\pi\)
0.427200 + 0.904157i \(0.359500\pi\)
\(420\) 0 0
\(421\) −21.7446 −1.05977 −0.529883 0.848071i \(-0.677764\pi\)
−0.529883 + 0.848071i \(0.677764\pi\)
\(422\) −1.47477 −0.0717906
\(423\) 0 0
\(424\) −32.0000 −1.55406
\(425\) 0 0
\(426\) 0 0
\(427\) −32.4665 −1.57117
\(428\) 18.2054 0.879990
\(429\) 0 0
\(430\) 0 0
\(431\) 33.3505 1.60644 0.803219 0.595683i \(-0.203119\pi\)
0.803219 + 0.595683i \(0.203119\pi\)
\(432\) 0 0
\(433\) 12.7692 0.613647 0.306823 0.951766i \(-0.400734\pi\)
0.306823 + 0.951766i \(0.400734\pi\)
\(434\) −17.4891 −0.839505
\(435\) 0 0
\(436\) −13.3723 −0.640416
\(437\) −0.589907 −0.0282191
\(438\) 0 0
\(439\) 31.8614 1.52066 0.760331 0.649536i \(-0.225037\pi\)
0.760331 + 0.649536i \(0.225037\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.66648 −0.174397
\(443\) 21.4843 1.02075 0.510375 0.859952i \(-0.329506\pi\)
0.510375 + 0.859952i \(0.329506\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 14.7446 0.698175
\(447\) 0 0
\(448\) −11.6819 −0.551919
\(449\) 10.8832 0.513608 0.256804 0.966464i \(-0.417331\pi\)
0.256804 + 0.966464i \(0.417331\pi\)
\(450\) 0 0
\(451\) −19.1168 −0.900177
\(452\) −8.42020 −0.396053
\(453\) 0 0
\(454\) 10.7446 0.504267
\(455\) 0 0
\(456\) 0 0
\(457\) −20.9870 −0.981730 −0.490865 0.871236i \(-0.663319\pi\)
−0.490865 + 0.871236i \(0.663319\pi\)
\(458\) −13.9490 −0.651793
\(459\) 0 0
\(460\) 0 0
\(461\) −24.0951 −1.12222 −0.561110 0.827741i \(-0.689626\pi\)
−0.561110 + 0.827741i \(0.689626\pi\)
\(462\) 0 0
\(463\) −13.8564 −0.643962 −0.321981 0.946746i \(-0.604349\pi\)
−0.321981 + 0.946746i \(0.604349\pi\)
\(464\) 3.60597 0.167403
\(465\) 0 0
\(466\) 4.86141 0.225200
\(467\) 18.3152 0.847525 0.423763 0.905773i \(-0.360709\pi\)
0.423763 + 0.905773i \(0.360709\pi\)
\(468\) 0 0
\(469\) 40.4674 1.86861
\(470\) 0 0
\(471\) 0 0
\(472\) 4.34896 0.200177
\(473\) 15.1460 0.696415
\(474\) 0 0
\(475\) 0 0
\(476\) 3.76631 0.172629
\(477\) 0 0
\(478\) −18.2054 −0.832694
\(479\) −18.6060 −0.850128 −0.425064 0.905163i \(-0.639748\pi\)
−0.425064 + 0.905163i \(0.639748\pi\)
\(480\) 0 0
\(481\) −13.8832 −0.633017
\(482\) −1.19705 −0.0545240
\(483\) 0 0
\(484\) −11.1386 −0.506300
\(485\) 0 0
\(486\) 0 0
\(487\) 9.50744 0.430823 0.215412 0.976523i \(-0.430891\pi\)
0.215412 + 0.976523i \(0.430891\pi\)
\(488\) −25.0410 −1.13355
\(489\) 0 0
\(490\) 0 0
\(491\) 31.6277 1.42734 0.713669 0.700483i \(-0.247032\pi\)
0.713669 + 0.700483i \(0.247032\pi\)
\(492\) 0 0
\(493\) 4.55134 0.204982
\(494\) 1.72281 0.0775130
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 45.4381 2.03818
\(498\) 0 0
\(499\) −24.3723 −1.09105 −0.545527 0.838094i \(-0.683670\pi\)
−0.545527 + 0.838094i \(0.683670\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 16.0309 0.715494
\(503\) −3.16915 −0.141305 −0.0706527 0.997501i \(-0.522508\pi\)
−0.0706527 + 0.997501i \(0.522508\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 5.48913 0.244021
\(507\) 0 0
\(508\) −14.2612 −0.632737
\(509\) −0.510875 −0.0226441 −0.0113221 0.999936i \(-0.503604\pi\)
−0.0113221 + 0.999936i \(0.503604\pi\)
\(510\) 0 0
\(511\) −8.23369 −0.364237
\(512\) 7.02078 0.310277
\(513\) 0 0
\(514\) 5.09509 0.224735
\(515\) 0 0
\(516\) 0 0
\(517\) 8.21782 0.361419
\(518\) −6.52344 −0.286624
\(519\) 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
\(523\) 4.75372 0.207866 0.103933 0.994584i \(-0.466857\pi\)
0.103933 + 0.994584i \(0.466857\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −21.0217 −0.916592
\(527\) 5.04868 0.219924
\(528\) 0 0
\(529\) −20.4891 −0.890832
\(530\) 0 0
\(531\) 0 0
\(532\) −1.76972 −0.0767272
\(533\) 25.5383 1.10619
\(534\) 0 0
\(535\) 0 0
\(536\) 31.2119 1.34815
\(537\) 0 0
\(538\) 23.1615 0.998562
\(539\) 21.8614 0.941637
\(540\) 0 0
\(541\) 19.2337 0.826921 0.413460 0.910522i \(-0.364320\pi\)
0.413460 + 0.910522i \(0.364320\pi\)
\(542\) 4.16381 0.178851
\(543\) 0 0
\(544\) 4.62772 0.198412
\(545\) 0 0
\(546\) 0 0
\(547\) 6.92820 0.296229 0.148114 0.988970i \(-0.452680\pi\)
0.148114 + 0.988970i \(0.452680\pi\)
\(548\) 19.6974 0.841430
\(549\) 0 0
\(550\) 0 0
\(551\) −2.13859 −0.0911071
\(552\) 0 0
\(553\) −23.3639 −0.993532
\(554\) 17.4891 0.743042
\(555\) 0 0
\(556\) −15.2554 −0.646975
\(557\) 25.0410 1.06102 0.530511 0.847678i \(-0.322000\pi\)
0.530511 + 0.847678i \(0.322000\pi\)
\(558\) 0 0
\(559\) −20.2337 −0.855794
\(560\) 0 0
\(561\) 0 0
\(562\) 1.08724 0.0458625
\(563\) −32.1716 −1.35587 −0.677935 0.735122i \(-0.737125\pi\)
−0.677935 + 0.735122i \(0.737125\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 21.9565 0.922901
\(567\) 0 0
\(568\) 35.0458 1.47049
\(569\) −20.4891 −0.858949 −0.429474 0.903079i \(-0.641301\pi\)
−0.429474 + 0.903079i \(0.641301\pi\)
\(570\) 0 0
\(571\) 4.13859 0.173195 0.0865974 0.996243i \(-0.472401\pi\)
0.0865974 + 0.996243i \(0.472401\pi\)
\(572\) 35.0458 1.46534
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) −18.4077 −0.766325 −0.383162 0.923681i \(-0.625165\pi\)
−0.383162 + 0.923681i \(0.625165\pi\)
\(578\) −12.9715 −0.539545
\(579\) 0 0
\(580\) 0 0
\(581\) −41.4891 −1.72126
\(582\) 0 0
\(583\) 52.3663 2.16879
\(584\) −6.35053 −0.262787
\(585\) 0 0
\(586\) −0.627719 −0.0259308
\(587\) −16.7306 −0.690546 −0.345273 0.938502i \(-0.612214\pi\)
−0.345273 + 0.938502i \(0.612214\pi\)
\(588\) 0 0
\(589\) −2.37228 −0.0977481
\(590\) 0 0
\(591\) 0 0
\(592\) 1.49200 0.0613208
\(593\) −1.67715 −0.0688722 −0.0344361 0.999407i \(-0.510964\pi\)
−0.0344361 + 0.999407i \(0.510964\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 14.5842 0.597393
\(597\) 0 0
\(598\) −7.33296 −0.299867
\(599\) −1.62772 −0.0665068 −0.0332534 0.999447i \(-0.510587\pi\)
−0.0332534 + 0.999447i \(0.510587\pi\)
\(600\) 0 0
\(601\) −17.9783 −0.733348 −0.366674 0.930349i \(-0.619504\pi\)
−0.366674 + 0.930349i \(0.619504\pi\)
\(602\) −9.50744 −0.387494
\(603\) 0 0
\(604\) −1.21194 −0.0493131
\(605\) 0 0
\(606\) 0 0
\(607\) 43.2636 1.75602 0.878008 0.478647i \(-0.158872\pi\)
0.878008 + 0.478647i \(0.158872\pi\)
\(608\) −2.17448 −0.0881869
\(609\) 0 0
\(610\) 0 0
\(611\) −10.9783 −0.444132
\(612\) 0 0
\(613\) 30.2921 1.22348 0.611742 0.791057i \(-0.290469\pi\)
0.611742 + 0.791057i \(0.290469\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 40.4674 1.63048
\(617\) 20.6920 0.833030 0.416515 0.909129i \(-0.363251\pi\)
0.416515 + 0.909129i \(0.363251\pi\)
\(618\) 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
\(622\) 12.5668 0.503882
\(623\) −10.3923 −0.416359
\(624\) 0 0
\(625\) 0 0
\(626\) −19.3723 −0.774272
\(627\) 0 0
\(628\) −15.7532 −0.628620
\(629\) 1.88316 0.0750863
\(630\) 0 0
\(631\) 6.37228 0.253677 0.126838 0.991923i \(-0.459517\pi\)
0.126838 + 0.991923i \(0.459517\pi\)
\(632\) −18.0202 −0.716806
\(633\) 0 0
\(634\) −23.3723 −0.928232
\(635\) 0 0
\(636\) 0 0
\(637\) −29.2048 −1.15714
\(638\) 19.8997 0.787839
\(639\) 0 0
\(640\) 0 0
\(641\) −7.97825 −0.315122 −0.157561 0.987509i \(-0.550363\pi\)
−0.157561 + 0.987509i \(0.550363\pi\)
\(642\) 0 0
\(643\) 1.28962 0.0508577 0.0254288 0.999677i \(-0.491905\pi\)
0.0254288 + 0.999677i \(0.491905\pi\)
\(644\) 7.53262 0.296827
\(645\) 0 0
\(646\) −0.233688 −0.00919433
\(647\) 28.7075 1.12861 0.564304 0.825567i \(-0.309145\pi\)
0.564304 + 0.825567i \(0.309145\pi\)
\(648\) 0 0
\(649\) −7.11684 −0.279361
\(650\) 0 0
\(651\) 0 0
\(652\) 20.7846 0.813988
\(653\) 6.33830 0.248037 0.124018 0.992280i \(-0.460422\pi\)
0.124018 + 0.992280i \(0.460422\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.74456 −0.107157
\(657\) 0 0
\(658\) −5.15848 −0.201099
\(659\) −21.2554 −0.827994 −0.413997 0.910278i \(-0.635868\pi\)
−0.413997 + 0.910278i \(0.635868\pi\)
\(660\) 0 0
\(661\) −15.2337 −0.592522 −0.296261 0.955107i \(-0.595740\pi\)
−0.296261 + 0.955107i \(0.595740\pi\)
\(662\) 7.22316 0.280736
\(663\) 0 0
\(664\) −32.0000 −1.24184
\(665\) 0 0
\(666\) 0 0
\(667\) 9.10268 0.352457
\(668\) −18.6101 −0.720047
\(669\) 0 0
\(670\) 0 0
\(671\) 40.9783 1.58195
\(672\) 0 0
\(673\) 10.5947 0.408395 0.204198 0.978930i \(-0.434542\pi\)
0.204198 + 0.978930i \(0.434542\pi\)
\(674\) −5.48913 −0.211433
\(675\) 0 0
\(676\) −28.9783 −1.11455
\(677\) 26.5330 1.01975 0.509873 0.860250i \(-0.329692\pi\)
0.509873 + 0.860250i \(0.329692\pi\)
\(678\) 0 0
\(679\) −4.46738 −0.171442
\(680\) 0 0
\(681\) 0 0
\(682\) 22.0742 0.845266
\(683\) 27.1229 1.03783 0.518915 0.854826i \(-0.326336\pi\)
0.518915 + 0.854826i \(0.326336\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 5.48913 0.209576
\(687\) 0 0
\(688\) 2.17448 0.0829013
\(689\) −69.9565 −2.66513
\(690\) 0 0
\(691\) 45.7228 1.73938 0.869689 0.493600i \(-0.164319\pi\)
0.869689 + 0.493600i \(0.164319\pi\)
\(692\) 15.3484 0.583459
\(693\) 0 0
\(694\) −2.27719 −0.0864408
\(695\) 0 0
\(696\) 0 0
\(697\) −3.46410 −0.131212
\(698\) −21.0796 −0.797873
\(699\) 0 0
\(700\) 0 0
\(701\) −17.2337 −0.650907 −0.325454 0.945558i \(-0.605517\pi\)
−0.325454 + 0.945558i \(0.605517\pi\)
\(702\) 0 0
\(703\) −0.884861 −0.0333732
\(704\) 14.7446 0.555707
\(705\) 0 0
\(706\) −1.48913 −0.0560440
\(707\) −56.7152 −2.13300
\(708\) 0 0
\(709\) 52.3505 1.96607 0.983033 0.183430i \(-0.0587201\pi\)
0.983033 + 0.183430i \(0.0587201\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −8.01544 −0.300391
\(713\) 10.0974 0.378149
\(714\) 0 0
\(715\) 0 0
\(716\) −6.70106 −0.250431
\(717\) 0 0
\(718\) 8.62258 0.321792
\(719\) −10.8832 −0.405873 −0.202937 0.979192i \(-0.565049\pi\)
−0.202937 + 0.979192i \(0.565049\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) −14.9436 −0.556145
\(723\) 0 0
\(724\) 19.0217 0.706938
\(725\) 0 0
\(726\) 0 0
\(727\) −48.9022 −1.81368 −0.906841 0.421473i \(-0.861513\pi\)
−0.906841 + 0.421473i \(0.861513\pi\)
\(728\) −54.0607 −2.00362
\(729\) 0 0
\(730\) 0 0
\(731\) 2.74456 0.101511
\(732\) 0 0
\(733\) 16.4356 0.607064 0.303532 0.952821i \(-0.401834\pi\)
0.303532 + 0.952821i \(0.401834\pi\)
\(734\) 13.0217 0.480642
\(735\) 0 0
\(736\) 9.25544 0.341160
\(737\) −51.0767 −1.88143
\(738\) 0 0
\(739\) 5.62772 0.207019 0.103509 0.994628i \(-0.466993\pi\)
0.103509 + 0.994628i \(0.466993\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −32.8713 −1.20674
\(743\) 9.39764 0.344766 0.172383 0.985030i \(-0.444853\pi\)
0.172383 + 0.985030i \(0.444853\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −6.51087 −0.238380
\(747\) 0 0
\(748\) −4.75372 −0.173813
\(749\) 45.9565 1.67921
\(750\) 0 0
\(751\) 21.7228 0.792677 0.396338 0.918105i \(-0.370281\pi\)
0.396338 + 0.918105i \(0.370281\pi\)
\(752\) 1.17981 0.0430234
\(753\) 0 0
\(754\) −26.5842 −0.968140
\(755\) 0 0
\(756\) 0 0
\(757\) 41.5692 1.51086 0.755429 0.655230i \(-0.227428\pi\)
0.755429 + 0.655230i \(0.227428\pi\)
\(758\) −1.17981 −0.0428528
\(759\) 0 0
\(760\) 0 0
\(761\) −32.4891 −1.17773 −0.588865 0.808231i \(-0.700425\pi\)
−0.588865 + 0.808231i \(0.700425\pi\)
\(762\) 0 0
\(763\) −33.7562 −1.22205
\(764\) 10.4674 0.378696
\(765\) 0 0
\(766\) 8.78806 0.317526
\(767\) 9.50744 0.343294
\(768\) 0 0
\(769\) −16.4891 −0.594613 −0.297307 0.954782i \(-0.596088\pi\)
−0.297307 + 0.954782i \(0.596088\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 15.7532 0.566969
\(773\) −21.5769 −0.776067 −0.388034 0.921645i \(-0.626846\pi\)
−0.388034 + 0.921645i \(0.626846\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.44563 −0.123691
\(777\) 0 0
\(778\) 0.404759 0.0145113
\(779\) 1.62772 0.0583191
\(780\) 0 0
\(781\) −57.3505 −2.05216
\(782\) 0.994667 0.0355692
\(783\) 0 0
\(784\) 3.13859 0.112093
\(785\) 0 0
\(786\) 0 0
\(787\) −17.3205 −0.617409 −0.308705 0.951158i \(-0.599895\pi\)
−0.308705 + 0.951158i \(0.599895\pi\)
\(788\) −8.82496 −0.314376
\(789\) 0 0
\(790\) 0 0
\(791\) −21.2554 −0.755756
\(792\) 0 0
\(793\) −54.7431 −1.94399
\(794\) −13.8832 −0.492695
\(795\) 0 0
\(796\) −21.9565 −0.778228
\(797\) −25.6309 −0.907893 −0.453947 0.891029i \(-0.649984\pi\)
−0.453947 + 0.891029i \(0.649984\pi\)
\(798\) 0 0
\(799\) 1.48913 0.0526815
\(800\) 0 0
\(801\) 0 0
\(802\) −20.1021 −0.709831
\(803\) 10.3923 0.366736
\(804\) 0 0
\(805\) 0 0
\(806\) −29.4891 −1.03871
\(807\) 0 0
\(808\) −43.7437 −1.53890
\(809\) −21.0000 −0.738321 −0.369160 0.929366i \(-0.620355\pi\)
−0.369160 + 0.929366i \(0.620355\pi\)
\(810\) 0 0
\(811\) 24.8832 0.873766 0.436883 0.899518i \(-0.356082\pi\)
0.436883 + 0.899518i \(0.356082\pi\)
\(812\) 27.3081 0.958325
\(813\) 0 0
\(814\) 8.23369 0.288591
\(815\) 0 0
\(816\) 0 0
\(817\) −1.28962 −0.0451181
\(818\) 18.1128 0.633299
\(819\) 0 0
\(820\) 0 0
\(821\) 40.7228 1.42124 0.710618 0.703578i \(-0.248415\pi\)
0.710618 + 0.703578i \(0.248415\pi\)
\(822\) 0 0
\(823\) −4.34896 −0.151595 −0.0757977 0.997123i \(-0.524150\pi\)
−0.0757977 + 0.997123i \(0.524150\pi\)
\(824\) −18.5109 −0.644857
\(825\) 0 0
\(826\) 4.46738 0.155440
\(827\) −22.3692 −0.777853 −0.388926 0.921269i \(-0.627154\pi\)
−0.388926 + 0.921269i \(0.627154\pi\)
\(828\) 0 0
\(829\) −23.3505 −0.810997 −0.405499 0.914096i \(-0.632902\pi\)
−0.405499 + 0.914096i \(0.632902\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −19.6974 −0.682883
\(833\) 3.96143 0.137256
\(834\) 0 0
\(835\) 0 0
\(836\) 2.23369 0.0772537
\(837\) 0 0
\(838\) 13.8564 0.478662
\(839\) 45.8614 1.58331 0.791656 0.610967i \(-0.209219\pi\)
0.791656 + 0.610967i \(0.209219\pi\)
\(840\) 0 0
\(841\) 4.00000 0.137931
\(842\) −17.2279 −0.593714
\(843\) 0 0
\(844\) 2.55437 0.0879252
\(845\) 0 0
\(846\) 0 0
\(847\) −28.1176 −0.966131
\(848\) 7.51811 0.258173
\(849\) 0 0
\(850\) 0 0
\(851\) 3.76631 0.129108
\(852\) 0 0
\(853\) −10.7971 −0.369684 −0.184842 0.982768i \(-0.559177\pi\)
−0.184842 + 0.982768i \(0.559177\pi\)
\(854\) −25.7228 −0.880217
\(855\) 0 0
\(856\) 35.4456 1.21151
\(857\) −49.3995 −1.68746 −0.843728 0.536772i \(-0.819644\pi\)
−0.843728 + 0.536772i \(0.819644\pi\)
\(858\) 0 0
\(859\) 48.3288 1.64896 0.824478 0.565893i \(-0.191469\pi\)
0.824478 + 0.565893i \(0.191469\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 26.4232 0.899978
\(863\) 16.7306 0.569516 0.284758 0.958599i \(-0.408087\pi\)
0.284758 + 0.958599i \(0.408087\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 10.1168 0.343784
\(867\) 0 0
\(868\) 30.2921 1.02818
\(869\) 29.4891 1.00035
\(870\) 0 0
\(871\) 68.2337 2.31201
\(872\) −26.0357 −0.881679
\(873\) 0 0
\(874\) −0.467376 −0.0158092
\(875\) 0 0
\(876\) 0 0
\(877\) 1.08724 0.0367135 0.0183568 0.999832i \(-0.494157\pi\)
0.0183568 + 0.999832i \(0.494157\pi\)
\(878\) 25.2434 0.851923
\(879\) 0 0
\(880\) 0 0
\(881\) −10.8832 −0.366663 −0.183331 0.983051i \(-0.558688\pi\)
−0.183331 + 0.983051i \(0.558688\pi\)
\(882\) 0 0
\(883\) −54.9455 −1.84906 −0.924532 0.381104i \(-0.875544\pi\)
−0.924532 + 0.381104i \(0.875544\pi\)
\(884\) 6.35053 0.213592
\(885\) 0 0
\(886\) 17.0217 0.571857
\(887\) 43.8535 1.47246 0.736228 0.676733i \(-0.236605\pi\)
0.736228 + 0.676733i \(0.236605\pi\)
\(888\) 0 0
\(889\) −36.0000 −1.20740
\(890\) 0 0
\(891\) 0 0
\(892\) −25.5383 −0.855087
\(893\) −0.699713 −0.0234150
\(894\) 0 0
\(895\) 0 0
\(896\) 31.2119 1.04272
\(897\) 0 0
\(898\) 8.62258 0.287739
\(899\) 36.6060 1.22088
\(900\) 0 0
\(901\) 9.48913 0.316129
\(902\) −15.1460 −0.504308
\(903\) 0 0
\(904\) −16.3940 −0.545257
\(905\) 0 0
\(906\) 0 0
\(907\) 18.2054 0.604499 0.302250 0.953229i \(-0.402262\pi\)
0.302250 + 0.953229i \(0.402262\pi\)
\(908\) −18.6101 −0.617599
\(909\) 0 0
\(910\) 0 0
\(911\) 18.6060 0.616443 0.308222 0.951315i \(-0.400266\pi\)
0.308222 + 0.951315i \(0.400266\pi\)
\(912\) 0 0
\(913\) 52.3663 1.73307
\(914\) −16.6277 −0.549996
\(915\) 0 0
\(916\) 24.1603 0.798280
\(917\) 15.1460 0.500166
\(918\) 0 0
\(919\) 55.3505 1.82585 0.912923 0.408132i \(-0.133820\pi\)
0.912923 + 0.408132i \(0.133820\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −19.0902 −0.628703
\(923\) 76.6150 2.52181
\(924\) 0 0
\(925\) 0 0
\(926\) −10.9783 −0.360768
\(927\) 0 0
\(928\) 33.5538 1.10146
\(929\) 3.51087 0.115188 0.0575940 0.998340i \(-0.481657\pi\)
0.0575940 + 0.998340i \(0.481657\pi\)
\(930\) 0 0
\(931\) −1.86141 −0.0610051
\(932\) −8.42020 −0.275813
\(933\) 0 0
\(934\) 14.5109 0.474810
\(935\) 0 0
\(936\) 0 0
\(937\) 22.2766 0.727745 0.363873 0.931449i \(-0.381454\pi\)
0.363873 + 0.931449i \(0.381454\pi\)
\(938\) 32.0618 1.04685
\(939\) 0 0
\(940\) 0 0
\(941\) 6.86141 0.223675 0.111838 0.993726i \(-0.464326\pi\)
0.111838 + 0.993726i \(0.464326\pi\)
\(942\) 0 0
\(943\) −6.92820 −0.225613
\(944\) −1.02175 −0.0332551
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) −43.1538 −1.40231 −0.701155 0.713009i \(-0.747332\pi\)
−0.701155 + 0.713009i \(0.747332\pi\)
\(948\) 0 0
\(949\) −13.8832 −0.450666
\(950\) 0 0
\(951\) 0 0
\(952\) 7.33296 0.237663
\(953\) −25.0410 −0.811158 −0.405579 0.914060i \(-0.632930\pi\)
−0.405579 + 0.914060i \(0.632930\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 31.5326 1.01984
\(957\) 0 0
\(958\) −14.7413 −0.476269
\(959\) 49.7228 1.60563
\(960\) 0 0
\(961\) 9.60597 0.309870
\(962\) −10.9994 −0.354636
\(963\) 0 0
\(964\) 2.07335 0.0667780
\(965\) 0 0
\(966\) 0 0
\(967\) 43.2636 1.39126 0.695632 0.718399i \(-0.255125\pi\)
0.695632 + 0.718399i \(0.255125\pi\)
\(968\) −21.6867 −0.697037
\(969\) 0 0
\(970\) 0 0
\(971\) −41.5842 −1.33450 −0.667251 0.744833i \(-0.732529\pi\)
−0.667251 + 0.744833i \(0.732529\pi\)
\(972\) 0 0
\(973\) −38.5099 −1.23457
\(974\) 7.53262 0.241361
\(975\) 0 0
\(976\) 5.88316 0.188315
\(977\) 28.3027 0.905484 0.452742 0.891642i \(-0.350446\pi\)
0.452742 + 0.891642i \(0.350446\pi\)
\(978\) 0 0
\(979\) 13.1168 0.419216
\(980\) 0 0
\(981\) 0 0
\(982\) 25.0582 0.799640
\(983\) −50.3770 −1.60678 −0.803388 0.595456i \(-0.796971\pi\)
−0.803388 + 0.595456i \(0.796971\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 3.60597 0.114837
\(987\) 0 0
\(988\) −2.98400 −0.0949337
\(989\) 5.48913 0.174544
\(990\) 0 0
\(991\) 38.6060 1.22636 0.613180 0.789944i \(-0.289890\pi\)
0.613180 + 0.789944i \(0.289890\pi\)
\(992\) 37.2203 1.18174
\(993\) 0 0
\(994\) 36.0000 1.14185
\(995\) 0 0
\(996\) 0 0
\(997\) 60.3817 1.91231 0.956154 0.292864i \(-0.0946082\pi\)
0.956154 + 0.292864i \(0.0946082\pi\)
\(998\) −19.3098 −0.611242
\(999\) 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.1.3 4
3.2 odd 2 2025.2.a.w.1.2 4
5.2 odd 4 405.2.b.b.244.3 yes 4
5.3 odd 4 405.2.b.b.244.2 yes 4
5.4 even 2 inner 2025.2.a.x.1.2 4
15.2 even 4 405.2.b.a.244.2 4
15.8 even 4 405.2.b.a.244.3 yes 4
15.14 odd 2 2025.2.a.w.1.3 4
45.2 even 12 405.2.j.e.109.1 4
45.7 odd 12 405.2.j.a.109.2 4
45.13 odd 12 405.2.j.a.379.2 4
45.22 odd 12 405.2.j.d.379.1 4
45.23 even 12 405.2.j.e.379.1 4
45.32 even 12 405.2.j.b.379.2 4
45.38 even 12 405.2.j.b.109.2 4
45.43 odd 12 405.2.j.d.109.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.b.a.244.2 4 15.2 even 4
405.2.b.a.244.3 yes 4 15.8 even 4
405.2.b.b.244.2 yes 4 5.3 odd 4
405.2.b.b.244.3 yes 4 5.2 odd 4
405.2.j.a.109.2 4 45.7 odd 12
405.2.j.a.379.2 4 45.13 odd 12
405.2.j.b.109.2 4 45.38 even 12
405.2.j.b.379.2 4 45.32 even 12
405.2.j.d.109.1 4 45.43 odd 12
405.2.j.d.379.1 4 45.22 odd 12
405.2.j.e.109.1 4 45.2 even 12
405.2.j.e.379.1 4 45.23 even 12
2025.2.a.w.1.2 4 3.2 odd 2
2025.2.a.w.1.3 4 15.14 odd 2
2025.2.a.x.1.2 4 5.4 even 2 inner
2025.2.a.x.1.3 4 1.1 even 1 trivial