Properties

Label 2025.2.a.y.1.1
Level $2025$
Weight $2$
Character 2025.1
Self dual yes
Analytic conductor $16.170$
Analytic rank $0$
Dimension $4$
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] = [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: 4.4.11661.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 225)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.63372\) 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)
 
\(f(q)\) \(=\) \(q-1.63372 q^{2} +0.669052 q^{4} -0.505348 q^{7} +2.17440 q^{8} +O(q^{10})\) \(q-1.63372 q^{2} +0.669052 q^{4} -0.505348 q^{7} +2.17440 q^{8} +3.10020 q^{11} +6.23927 q^{13} +0.825599 q^{14} -4.89047 q^{16} +6.10020 q^{17} -5.57022 q^{19} -5.06487 q^{22} +3.82560 q^{23} -10.1932 q^{26} -0.338104 q^{28} +2.45932 q^{29} +4.22858 q^{31} +3.64088 q^{32} -9.96604 q^{34} -6.72677 q^{37} +9.10020 q^{38} -5.44185 q^{41} -1.32741 q^{43} +2.07420 q^{44} -6.24997 q^{46} +3.70792 q^{47} -6.74462 q^{49} +4.17440 q^{52} -2.54205 q^{53} -1.09883 q^{56} -4.01785 q^{58} +2.88232 q^{59} -2.84345 q^{61} -6.90833 q^{62} +3.83276 q^{64} -2.40652 q^{67} +4.08135 q^{68} +5.54205 q^{71} +11.7988 q^{73} +10.9897 q^{74} -3.72677 q^{76} -1.56668 q^{77} +3.40298 q^{79} +8.89047 q^{82} +13.9012 q^{83} +2.16862 q^{86} +6.74108 q^{88} +3.38513 q^{89} -3.15301 q^{91} +2.55953 q^{92} -6.05772 q^{94} -11.0756 q^{97} +11.0188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{4} - q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 4 q^{4} - q^{7} + 9 q^{8} - q^{11} + 2 q^{13} + 3 q^{14} + 4 q^{16} + 11 q^{17} + 2 q^{19} + 3 q^{22} + 15 q^{23} - 10 q^{26} - 4 q^{28} + q^{29} - 4 q^{31} + 10 q^{32} + 9 q^{34} - q^{37} + 23 q^{38} - 5 q^{41} - 10 q^{43} + 22 q^{44} + 20 q^{47} - 3 q^{49} + 17 q^{52} + 20 q^{53} - 30 q^{56} - 18 q^{58} + 17 q^{59} - 13 q^{61} - 6 q^{62} + 19 q^{64} + 17 q^{67} + 34 q^{68} - 8 q^{71} + 2 q^{73} + 40 q^{74} + 11 q^{76} + 12 q^{77} - 7 q^{79} + 12 q^{82} + 30 q^{83} - 34 q^{86} + 9 q^{88} - 9 q^{89} - 17 q^{91} - 12 q^{92} + 3 q^{94} - 19 q^{97} + 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.63372 −1.15522 −0.577608 0.816314i \(-0.696014\pi\)
−0.577608 + 0.816314i \(0.696014\pi\)
\(3\) 0 0
\(4\) 0.669052 0.334526
\(5\) 0 0
\(6\) 0 0
\(7\) −0.505348 −0.191004 −0.0955019 0.995429i \(-0.530446\pi\)
−0.0955019 + 0.995429i \(0.530446\pi\)
\(8\) 2.17440 0.768767
\(9\) 0 0
\(10\) 0 0
\(11\) 3.10020 0.934746 0.467373 0.884060i \(-0.345200\pi\)
0.467373 + 0.884060i \(0.345200\pi\)
\(12\) 0 0
\(13\) 6.23927 1.73046 0.865232 0.501372i \(-0.167171\pi\)
0.865232 + 0.501372i \(0.167171\pi\)
\(14\) 0.825599 0.220651
\(15\) 0 0
\(16\) −4.89047 −1.22262
\(17\) 6.10020 1.47952 0.739758 0.672873i \(-0.234940\pi\)
0.739758 + 0.672873i \(0.234940\pi\)
\(18\) 0 0
\(19\) −5.57022 −1.27790 −0.638948 0.769250i \(-0.720630\pi\)
−0.638948 + 0.769250i \(0.720630\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.06487 −1.07983
\(23\) 3.82560 0.797693 0.398846 0.917018i \(-0.369411\pi\)
0.398846 + 0.917018i \(0.369411\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −10.1932 −1.99906
\(27\) 0 0
\(28\) −0.338104 −0.0638957
\(29\) 2.45932 0.456685 0.228342 0.973581i \(-0.426669\pi\)
0.228342 + 0.973581i \(0.426669\pi\)
\(30\) 0 0
\(31\) 4.22858 0.759475 0.379738 0.925094i \(-0.376014\pi\)
0.379738 + 0.925094i \(0.376014\pi\)
\(32\) 3.64088 0.643623
\(33\) 0 0
\(34\) −9.96604 −1.70916
\(35\) 0 0
\(36\) 0 0
\(37\) −6.72677 −1.10587 −0.552937 0.833223i \(-0.686493\pi\)
−0.552937 + 0.833223i \(0.686493\pi\)
\(38\) 9.10020 1.47625
\(39\) 0 0
\(40\) 0 0
\(41\) −5.44185 −0.849874 −0.424937 0.905223i \(-0.639704\pi\)
−0.424937 + 0.905223i \(0.639704\pi\)
\(42\) 0 0
\(43\) −1.32741 −0.202428 −0.101214 0.994865i \(-0.532273\pi\)
−0.101214 + 0.994865i \(0.532273\pi\)
\(44\) 2.07420 0.312697
\(45\) 0 0
\(46\) −6.24997 −0.921508
\(47\) 3.70792 0.540856 0.270428 0.962740i \(-0.412835\pi\)
0.270428 + 0.962740i \(0.412835\pi\)
\(48\) 0 0
\(49\) −6.74462 −0.963518
\(50\) 0 0
\(51\) 0 0
\(52\) 4.17440 0.578885
\(53\) −2.54205 −0.349177 −0.174589 0.984641i \(-0.555860\pi\)
−0.174589 + 0.984641i \(0.555860\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.09883 −0.146837
\(57\) 0 0
\(58\) −4.01785 −0.527570
\(59\) 2.88232 0.375246 0.187623 0.982241i \(-0.439922\pi\)
0.187623 + 0.982241i \(0.439922\pi\)
\(60\) 0 0
\(61\) −2.84345 −0.364067 −0.182033 0.983292i \(-0.558268\pi\)
−0.182033 + 0.983292i \(0.558268\pi\)
\(62\) −6.90833 −0.877358
\(63\) 0 0
\(64\) 3.83276 0.479095
\(65\) 0 0
\(66\) 0 0
\(67\) −2.40652 −0.294003 −0.147002 0.989136i \(-0.546962\pi\)
−0.147002 + 0.989136i \(0.546962\pi\)
\(68\) 4.08135 0.494937
\(69\) 0 0
\(70\) 0 0
\(71\) 5.54205 0.657720 0.328860 0.944379i \(-0.393336\pi\)
0.328860 + 0.944379i \(0.393336\pi\)
\(72\) 0 0
\(73\) 11.7988 1.38095 0.690473 0.723359i \(-0.257403\pi\)
0.690473 + 0.723359i \(0.257403\pi\)
\(74\) 10.9897 1.27752
\(75\) 0 0
\(76\) −3.72677 −0.427490
\(77\) −1.56668 −0.178540
\(78\) 0 0
\(79\) 3.40298 0.382865 0.191433 0.981506i \(-0.438687\pi\)
0.191433 + 0.981506i \(0.438687\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 8.89047 0.981789
\(83\) 13.9012 1.52585 0.762926 0.646486i \(-0.223762\pi\)
0.762926 + 0.646486i \(0.223762\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.16862 0.233848
\(87\) 0 0
\(88\) 6.74108 0.718602
\(89\) 3.38513 0.358823 0.179411 0.983774i \(-0.442581\pi\)
0.179411 + 0.983774i \(0.442581\pi\)
\(90\) 0 0
\(91\) −3.15301 −0.330525
\(92\) 2.55953 0.266849
\(93\) 0 0
\(94\) −6.05772 −0.624806
\(95\) 0 0
\(96\) 0 0
\(97\) −11.0756 −1.12455 −0.562277 0.826949i \(-0.690074\pi\)
−0.562277 + 0.826949i \(0.690074\pi\)
\(98\) 11.0188 1.11307
\(99\) 0 0
\(100\) 0 0
\(101\) −17.3690 −1.72828 −0.864141 0.503249i \(-0.832138\pi\)
−0.864141 + 0.503249i \(0.832138\pi\)
\(102\) 0 0
\(103\) −0.832756 −0.0820539 −0.0410269 0.999158i \(-0.513063\pi\)
−0.0410269 + 0.999158i \(0.513063\pi\)
\(104\) 13.5667 1.33032
\(105\) 0 0
\(106\) 4.15301 0.403376
\(107\) 11.0684 1.07002 0.535012 0.844844i \(-0.320307\pi\)
0.535012 + 0.844844i \(0.320307\pi\)
\(108\) 0 0
\(109\) −4.65836 −0.446190 −0.223095 0.974797i \(-0.571616\pi\)
−0.223095 + 0.974797i \(0.571616\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.47139 0.233525
\(113\) 11.9942 1.12832 0.564160 0.825665i \(-0.309200\pi\)
0.564160 + 0.825665i \(0.309200\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.64542 0.152773
\(117\) 0 0
\(118\) −4.70892 −0.433491
\(119\) −3.08273 −0.282593
\(120\) 0 0
\(121\) −1.38874 −0.126249
\(122\) 4.64542 0.420576
\(123\) 0 0
\(124\) 2.82914 0.254064
\(125\) 0 0
\(126\) 0 0
\(127\) −3.22858 −0.286490 −0.143245 0.989687i \(-0.545754\pi\)
−0.143245 + 0.989687i \(0.545754\pi\)
\(128\) −13.5434 −1.19708
\(129\) 0 0
\(130\) 0 0
\(131\) −9.38513 −0.819982 −0.409991 0.912090i \(-0.634468\pi\)
−0.409991 + 0.912090i \(0.634468\pi\)
\(132\) 0 0
\(133\) 2.81490 0.244083
\(134\) 3.93159 0.339637
\(135\) 0 0
\(136\) 13.2643 1.13740
\(137\) 2.30955 0.197319 0.0986593 0.995121i \(-0.468545\pi\)
0.0986593 + 0.995121i \(0.468545\pi\)
\(138\) 0 0
\(139\) 10.8940 0.924018 0.462009 0.886875i \(-0.347129\pi\)
0.462009 + 0.886875i \(0.347129\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −9.05418 −0.759810
\(143\) 19.3430 1.61754
\(144\) 0 0
\(145\) 0 0
\(146\) −19.2760 −1.59529
\(147\) 0 0
\(148\) −4.50056 −0.369944
\(149\) −16.3430 −1.33887 −0.669436 0.742870i \(-0.733464\pi\)
−0.669436 + 0.742870i \(0.733464\pi\)
\(150\) 0 0
\(151\) 22.7827 1.85403 0.927015 0.375025i \(-0.122366\pi\)
0.927015 + 0.375025i \(0.122366\pi\)
\(152\) −12.1119 −0.982404
\(153\) 0 0
\(154\) 2.55953 0.206252
\(155\) 0 0
\(156\) 0 0
\(157\) 12.4607 0.994472 0.497236 0.867615i \(-0.334348\pi\)
0.497236 + 0.867615i \(0.334348\pi\)
\(158\) −5.55953 −0.442292
\(159\) 0 0
\(160\) 0 0
\(161\) −1.93326 −0.152362
\(162\) 0 0
\(163\) −7.57384 −0.593229 −0.296614 0.954997i \(-0.595858\pi\)
−0.296614 + 0.954997i \(0.595858\pi\)
\(164\) −3.64088 −0.284305
\(165\) 0 0
\(166\) −22.7107 −1.76269
\(167\) −2.97674 −0.230347 −0.115174 0.993345i \(-0.536742\pi\)
−0.115174 + 0.993345i \(0.536742\pi\)
\(168\) 0 0
\(169\) 25.9285 1.99450
\(170\) 0 0
\(171\) 0 0
\(172\) −0.888105 −0.0677174
\(173\) −15.8530 −1.20528 −0.602640 0.798013i \(-0.705884\pi\)
−0.602640 + 0.798013i \(0.705884\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −15.1615 −1.14284
\(177\) 0 0
\(178\) −5.53036 −0.414518
\(179\) 17.0841 1.27693 0.638463 0.769653i \(-0.279571\pi\)
0.638463 + 0.769653i \(0.279571\pi\)
\(180\) 0 0
\(181\) 13.3690 0.993712 0.496856 0.867833i \(-0.334488\pi\)
0.496856 + 0.867833i \(0.334488\pi\)
\(182\) 5.15114 0.381828
\(183\) 0 0
\(184\) 8.31839 0.613240
\(185\) 0 0
\(186\) 0 0
\(187\) 18.9119 1.38297
\(188\) 2.48079 0.180930
\(189\) 0 0
\(190\) 0 0
\(191\) 25.3372 1.83334 0.916669 0.399647i \(-0.130867\pi\)
0.916669 + 0.399647i \(0.130867\pi\)
\(192\) 0 0
\(193\) −9.55953 −0.688110 −0.344055 0.938950i \(-0.611801\pi\)
−0.344055 + 0.938950i \(0.611801\pi\)
\(194\) 18.0944 1.29910
\(195\) 0 0
\(196\) −4.51250 −0.322322
\(197\) −2.06841 −0.147368 −0.0736842 0.997282i \(-0.523476\pi\)
−0.0736842 + 0.997282i \(0.523476\pi\)
\(198\) 0 0
\(199\) 13.0970 0.928419 0.464210 0.885725i \(-0.346338\pi\)
0.464210 + 0.885725i \(0.346338\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 28.3762 1.99654
\(203\) −1.24281 −0.0872285
\(204\) 0 0
\(205\) 0 0
\(206\) 1.36049 0.0947900
\(207\) 0 0
\(208\) −30.5130 −2.11570
\(209\) −17.2688 −1.19451
\(210\) 0 0
\(211\) 11.1119 0.764974 0.382487 0.923961i \(-0.375068\pi\)
0.382487 + 0.923961i \(0.375068\pi\)
\(212\) −1.70076 −0.116809
\(213\) 0 0
\(214\) −18.0827 −1.23611
\(215\) 0 0
\(216\) 0 0
\(217\) −2.13690 −0.145063
\(218\) 7.61046 0.515446
\(219\) 0 0
\(220\) 0 0
\(221\) 38.0608 2.56025
\(222\) 0 0
\(223\) −3.89401 −0.260762 −0.130381 0.991464i \(-0.541620\pi\)
−0.130381 + 0.991464i \(0.541620\pi\)
\(224\) −1.83991 −0.122934
\(225\) 0 0
\(226\) −19.5952 −1.30346
\(227\) 12.8081 0.850105 0.425053 0.905169i \(-0.360256\pi\)
0.425053 + 0.905169i \(0.360256\pi\)
\(228\) 0 0
\(229\) 6.65295 0.439639 0.219820 0.975541i \(-0.429453\pi\)
0.219820 + 0.975541i \(0.429453\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.34755 0.351084
\(233\) −3.65836 −0.239667 −0.119833 0.992794i \(-0.538236\pi\)
−0.119833 + 0.992794i \(0.538236\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.92842 0.125530
\(237\) 0 0
\(238\) 5.03632 0.326456
\(239\) −15.6915 −1.01500 −0.507500 0.861652i \(-0.669430\pi\)
−0.507500 + 0.861652i \(0.669430\pi\)
\(240\) 0 0
\(241\) 11.2250 0.723063 0.361532 0.932360i \(-0.382254\pi\)
0.361532 + 0.932360i \(0.382254\pi\)
\(242\) 2.26882 0.145845
\(243\) 0 0
\(244\) −1.90242 −0.121790
\(245\) 0 0
\(246\) 0 0
\(247\) −34.7541 −2.21135
\(248\) 9.19462 0.583859
\(249\) 0 0
\(250\) 0 0
\(251\) 6.94042 0.438075 0.219038 0.975716i \(-0.429708\pi\)
0.219038 + 0.975716i \(0.429708\pi\)
\(252\) 0 0
\(253\) 11.8601 0.745640
\(254\) 5.27460 0.330958
\(255\) 0 0
\(256\) 14.4607 0.903794
\(257\) 18.3327 1.14356 0.571781 0.820406i \(-0.306253\pi\)
0.571781 + 0.820406i \(0.306253\pi\)
\(258\) 0 0
\(259\) 3.39936 0.211226
\(260\) 0 0
\(261\) 0 0
\(262\) 15.3327 0.947257
\(263\) 16.0766 0.991328 0.495664 0.868514i \(-0.334925\pi\)
0.495664 + 0.868514i \(0.334925\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.59877 −0.281969
\(267\) 0 0
\(268\) −1.61009 −0.0983517
\(269\) −18.2004 −1.10970 −0.554849 0.831951i \(-0.687224\pi\)
−0.554849 + 0.831951i \(0.687224\pi\)
\(270\) 0 0
\(271\) −2.48571 −0.150996 −0.0754979 0.997146i \(-0.524055\pi\)
−0.0754979 + 0.997146i \(0.524055\pi\)
\(272\) −29.8329 −1.80888
\(273\) 0 0
\(274\) −3.77317 −0.227946
\(275\) 0 0
\(276\) 0 0
\(277\) 7.66726 0.460681 0.230341 0.973110i \(-0.426016\pi\)
0.230341 + 0.973110i \(0.426016\pi\)
\(278\) −17.7978 −1.06744
\(279\) 0 0
\(280\) 0 0
\(281\) 0.273230 0.0162996 0.00814978 0.999967i \(-0.497406\pi\)
0.00814978 + 0.999967i \(0.497406\pi\)
\(282\) 0 0
\(283\) 3.37089 0.200379 0.100189 0.994968i \(-0.468055\pi\)
0.100189 + 0.994968i \(0.468055\pi\)
\(284\) 3.70792 0.220025
\(285\) 0 0
\(286\) −31.6011 −1.86861
\(287\) 2.75003 0.162329
\(288\) 0 0
\(289\) 20.2125 1.18897
\(290\) 0 0
\(291\) 0 0
\(292\) 7.89401 0.461962
\(293\) −5.64404 −0.329728 −0.164864 0.986316i \(-0.552719\pi\)
−0.164864 + 0.986316i \(0.552719\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −14.6267 −0.850159
\(297\) 0 0
\(298\) 26.7000 1.54669
\(299\) 23.8690 1.38038
\(300\) 0 0
\(301\) 0.670803 0.0386645
\(302\) −37.2206 −2.14181
\(303\) 0 0
\(304\) 27.2410 1.56238
\(305\) 0 0
\(306\) 0 0
\(307\) 5.44105 0.310537 0.155269 0.987872i \(-0.450376\pi\)
0.155269 + 0.987872i \(0.450376\pi\)
\(308\) −1.04819 −0.0597263
\(309\) 0 0
\(310\) 0 0
\(311\) −19.0797 −1.08191 −0.540955 0.841052i \(-0.681937\pi\)
−0.540955 + 0.841052i \(0.681937\pi\)
\(312\) 0 0
\(313\) −9.14231 −0.516754 −0.258377 0.966044i \(-0.583188\pi\)
−0.258377 + 0.966044i \(0.583188\pi\)
\(314\) −20.3573 −1.14883
\(315\) 0 0
\(316\) 2.27677 0.128078
\(317\) 14.2367 0.799614 0.399807 0.916599i \(-0.369077\pi\)
0.399807 + 0.916599i \(0.369077\pi\)
\(318\) 0 0
\(319\) 7.62440 0.426884
\(320\) 0 0
\(321\) 0 0
\(322\) 3.15841 0.176011
\(323\) −33.9795 −1.89067
\(324\) 0 0
\(325\) 0 0
\(326\) 12.3736 0.685308
\(327\) 0 0
\(328\) −11.8328 −0.653355
\(329\) −1.87379 −0.103305
\(330\) 0 0
\(331\) −12.2000 −0.670574 −0.335287 0.942116i \(-0.608833\pi\)
−0.335287 + 0.942116i \(0.608833\pi\)
\(332\) 9.30061 0.510437
\(333\) 0 0
\(334\) 4.86317 0.266101
\(335\) 0 0
\(336\) 0 0
\(337\) 4.58986 0.250026 0.125013 0.992155i \(-0.460103\pi\)
0.125013 + 0.992155i \(0.460103\pi\)
\(338\) −42.3601 −2.30408
\(339\) 0 0
\(340\) 0 0
\(341\) 13.1094 0.709917
\(342\) 0 0
\(343\) 6.94582 0.375039
\(344\) −2.88632 −0.155620
\(345\) 0 0
\(346\) 25.8994 1.39236
\(347\) 33.4603 1.79624 0.898121 0.439748i \(-0.144932\pi\)
0.898121 + 0.439748i \(0.144932\pi\)
\(348\) 0 0
\(349\) −28.0816 −1.50317 −0.751586 0.659636i \(-0.770711\pi\)
−0.751586 + 0.659636i \(0.770711\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 11.2875 0.601624
\(353\) −1.84170 −0.0980239 −0.0490119 0.998798i \(-0.515607\pi\)
−0.0490119 + 0.998798i \(0.515607\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.26483 0.120036
\(357\) 0 0
\(358\) −27.9107 −1.47513
\(359\) −12.1119 −0.639241 −0.319621 0.947546i \(-0.603555\pi\)
−0.319621 + 0.947546i \(0.603555\pi\)
\(360\) 0 0
\(361\) 12.0274 0.633020
\(362\) −21.8413 −1.14795
\(363\) 0 0
\(364\) −2.10953 −0.110569
\(365\) 0 0
\(366\) 0 0
\(367\) −14.5738 −0.760744 −0.380372 0.924834i \(-0.624204\pi\)
−0.380372 + 0.924834i \(0.624204\pi\)
\(368\) −18.7090 −0.975274
\(369\) 0 0
\(370\) 0 0
\(371\) 1.28462 0.0666942
\(372\) 0 0
\(373\) −9.44646 −0.489119 −0.244560 0.969634i \(-0.578643\pi\)
−0.244560 + 0.969634i \(0.578643\pi\)
\(374\) −30.8968 −1.59763
\(375\) 0 0
\(376\) 8.06251 0.415792
\(377\) 15.3444 0.790276
\(378\) 0 0
\(379\) −28.5541 −1.46673 −0.733363 0.679837i \(-0.762051\pi\)
−0.733363 + 0.679837i \(0.762051\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −41.3940 −2.11790
\(383\) 1.46541 0.0748788 0.0374394 0.999299i \(-0.488080\pi\)
0.0374394 + 0.999299i \(0.488080\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 15.6176 0.794916
\(387\) 0 0
\(388\) −7.41014 −0.376193
\(389\) −12.9101 −0.654569 −0.327284 0.944926i \(-0.606134\pi\)
−0.327284 + 0.944926i \(0.606134\pi\)
\(390\) 0 0
\(391\) 23.3369 1.18020
\(392\) −14.6655 −0.740720
\(393\) 0 0
\(394\) 3.37922 0.170242
\(395\) 0 0
\(396\) 0 0
\(397\) −0.868386 −0.0435831 −0.0217915 0.999763i \(-0.506937\pi\)
−0.0217915 + 0.999763i \(0.506937\pi\)
\(398\) −21.3968 −1.07253
\(399\) 0 0
\(400\) 0 0
\(401\) 33.4125 1.66854 0.834270 0.551356i \(-0.185889\pi\)
0.834270 + 0.551356i \(0.185889\pi\)
\(402\) 0 0
\(403\) 26.3833 1.31424
\(404\) −11.6208 −0.578156
\(405\) 0 0
\(406\) 2.03042 0.100768
\(407\) −20.8544 −1.03371
\(408\) 0 0
\(409\) 5.05535 0.249971 0.124985 0.992159i \(-0.460112\pi\)
0.124985 + 0.992159i \(0.460112\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.557157 −0.0274492
\(413\) −1.45658 −0.0716734
\(414\) 0 0
\(415\) 0 0
\(416\) 22.7165 1.11377
\(417\) 0 0
\(418\) 28.2125 1.37992
\(419\) 10.9576 0.535313 0.267657 0.963514i \(-0.413751\pi\)
0.267657 + 0.963514i \(0.413751\pi\)
\(420\) 0 0
\(421\) −10.6386 −0.518495 −0.259248 0.965811i \(-0.583475\pi\)
−0.259248 + 0.965811i \(0.583475\pi\)
\(422\) −18.1538 −0.883711
\(423\) 0 0
\(424\) −5.52744 −0.268436
\(425\) 0 0
\(426\) 0 0
\(427\) 1.43693 0.0695381
\(428\) 7.40535 0.357951
\(429\) 0 0
\(430\) 0 0
\(431\) −37.3529 −1.79923 −0.899613 0.436687i \(-0.856152\pi\)
−0.899613 + 0.436687i \(0.856152\pi\)
\(432\) 0 0
\(433\) −17.2125 −0.827179 −0.413589 0.910464i \(-0.635725\pi\)
−0.413589 + 0.910464i \(0.635725\pi\)
\(434\) 3.49111 0.167579
\(435\) 0 0
\(436\) −3.11668 −0.149262
\(437\) −21.3094 −1.01937
\(438\) 0 0
\(439\) −31.7487 −1.51528 −0.757642 0.652670i \(-0.773649\pi\)
−0.757642 + 0.652670i \(0.773649\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −62.1809 −2.95764
\(443\) −0.355958 −0.0169121 −0.00845603 0.999964i \(-0.502692\pi\)
−0.00845603 + 0.999964i \(0.502692\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 6.36174 0.301237
\(447\) 0 0
\(448\) −1.93688 −0.0915088
\(449\) −7.85632 −0.370762 −0.185381 0.982667i \(-0.559352\pi\)
−0.185381 + 0.982667i \(0.559352\pi\)
\(450\) 0 0
\(451\) −16.8708 −0.794416
\(452\) 8.02476 0.377453
\(453\) 0 0
\(454\) −20.9249 −0.982056
\(455\) 0 0
\(456\) 0 0
\(457\) 21.4910 1.00531 0.502654 0.864488i \(-0.332357\pi\)
0.502654 + 0.864488i \(0.332357\pi\)
\(458\) −10.8691 −0.507879
\(459\) 0 0
\(460\) 0 0
\(461\) 40.9927 1.90922 0.954611 0.297856i \(-0.0962715\pi\)
0.954611 + 0.297856i \(0.0962715\pi\)
\(462\) 0 0
\(463\) 42.1339 1.95813 0.979063 0.203555i \(-0.0652497\pi\)
0.979063 + 0.203555i \(0.0652497\pi\)
\(464\) −12.0273 −0.558351
\(465\) 0 0
\(466\) 5.97674 0.276867
\(467\) −22.5376 −1.04292 −0.521459 0.853276i \(-0.674612\pi\)
−0.521459 + 0.853276i \(0.674612\pi\)
\(468\) 0 0
\(469\) 1.21613 0.0561557
\(470\) 0 0
\(471\) 0 0
\(472\) 6.26732 0.288477
\(473\) −4.11523 −0.189219
\(474\) 0 0
\(475\) 0 0
\(476\) −2.06251 −0.0945348
\(477\) 0 0
\(478\) 25.6356 1.17255
\(479\) −33.2880 −1.52097 −0.760483 0.649358i \(-0.775038\pi\)
−0.760483 + 0.649358i \(0.775038\pi\)
\(480\) 0 0
\(481\) −41.9702 −1.91367
\(482\) −18.3385 −0.835295
\(483\) 0 0
\(484\) −0.929141 −0.0422337
\(485\) 0 0
\(486\) 0 0
\(487\) 23.7703 1.07713 0.538566 0.842583i \(-0.318966\pi\)
0.538566 + 0.842583i \(0.318966\pi\)
\(488\) −6.18281 −0.279882
\(489\) 0 0
\(490\) 0 0
\(491\) −4.60563 −0.207849 −0.103925 0.994585i \(-0.533140\pi\)
−0.103925 + 0.994585i \(0.533140\pi\)
\(492\) 0 0
\(493\) 15.0024 0.675673
\(494\) 56.7787 2.55459
\(495\) 0 0
\(496\) −20.6797 −0.928548
\(497\) −2.80067 −0.125627
\(498\) 0 0
\(499\) −18.8976 −0.845971 −0.422985 0.906137i \(-0.639018\pi\)
−0.422985 + 0.906137i \(0.639018\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −11.3387 −0.506072
\(503\) 35.7581 1.59438 0.797188 0.603731i \(-0.206320\pi\)
0.797188 + 0.603731i \(0.206320\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −19.3762 −0.861376
\(507\) 0 0
\(508\) −2.16009 −0.0958384
\(509\) 24.4067 1.08181 0.540904 0.841084i \(-0.318082\pi\)
0.540904 + 0.841084i \(0.318082\pi\)
\(510\) 0 0
\(511\) −5.96250 −0.263766
\(512\) 3.46207 0.153003
\(513\) 0 0
\(514\) −29.9506 −1.32106
\(515\) 0 0
\(516\) 0 0
\(517\) 11.4953 0.505563
\(518\) −5.55362 −0.244012
\(519\) 0 0
\(520\) 0 0
\(521\) 33.3968 1.46314 0.731571 0.681766i \(-0.238788\pi\)
0.731571 + 0.681766i \(0.238788\pi\)
\(522\) 0 0
\(523\) −37.3654 −1.63388 −0.816938 0.576726i \(-0.804330\pi\)
−0.816938 + 0.576726i \(0.804330\pi\)
\(524\) −6.27914 −0.274305
\(525\) 0 0
\(526\) −26.2648 −1.14520
\(527\) 25.7952 1.12366
\(528\) 0 0
\(529\) −8.36479 −0.363686
\(530\) 0 0
\(531\) 0 0
\(532\) 1.88332 0.0816521
\(533\) −33.9532 −1.47068
\(534\) 0 0
\(535\) 0 0
\(536\) −5.23274 −0.226020
\(537\) 0 0
\(538\) 29.7344 1.28194
\(539\) −20.9097 −0.900645
\(540\) 0 0
\(541\) 28.2560 1.21482 0.607409 0.794389i \(-0.292209\pi\)
0.607409 + 0.794389i \(0.292209\pi\)
\(542\) 4.06096 0.174433
\(543\) 0 0
\(544\) 22.2101 0.952250
\(545\) 0 0
\(546\) 0 0
\(547\) −38.5452 −1.64807 −0.824036 0.566537i \(-0.808283\pi\)
−0.824036 + 0.566537i \(0.808283\pi\)
\(548\) 1.54521 0.0660082
\(549\) 0 0
\(550\) 0 0
\(551\) −13.6990 −0.583596
\(552\) 0 0
\(553\) −1.71969 −0.0731286
\(554\) −12.5262 −0.532187
\(555\) 0 0
\(556\) 7.28866 0.309108
\(557\) −27.4125 −1.16151 −0.580753 0.814080i \(-0.697242\pi\)
−0.580753 + 0.814080i \(0.697242\pi\)
\(558\) 0 0
\(559\) −8.28206 −0.350294
\(560\) 0 0
\(561\) 0 0
\(562\) −0.446383 −0.0188295
\(563\) −27.6392 −1.16485 −0.582427 0.812883i \(-0.697897\pi\)
−0.582427 + 0.812883i \(0.697897\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −5.50710 −0.231481
\(567\) 0 0
\(568\) 12.0506 0.505634
\(569\) −14.7161 −0.616933 −0.308467 0.951235i \(-0.599816\pi\)
−0.308467 + 0.951235i \(0.599816\pi\)
\(570\) 0 0
\(571\) −28.3006 −1.18434 −0.592172 0.805812i \(-0.701729\pi\)
−0.592172 + 0.805812i \(0.701729\pi\)
\(572\) 12.9415 0.541111
\(573\) 0 0
\(574\) −4.49279 −0.187525
\(575\) 0 0
\(576\) 0 0
\(577\) −40.7976 −1.69843 −0.849214 0.528049i \(-0.822924\pi\)
−0.849214 + 0.528049i \(0.822924\pi\)
\(578\) −33.0216 −1.37352
\(579\) 0 0
\(580\) 0 0
\(581\) −7.02493 −0.291443
\(582\) 0 0
\(583\) −7.88087 −0.326392
\(584\) 25.6553 1.06162
\(585\) 0 0
\(586\) 9.22080 0.380908
\(587\) 2.78033 0.114756 0.0573782 0.998353i \(-0.481726\pi\)
0.0573782 + 0.998353i \(0.481726\pi\)
\(588\) 0 0
\(589\) −23.5541 −0.970531
\(590\) 0 0
\(591\) 0 0
\(592\) 32.8971 1.35206
\(593\) −14.8084 −0.608109 −0.304055 0.952655i \(-0.598341\pi\)
−0.304055 + 0.952655i \(0.598341\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10.9343 −0.447888
\(597\) 0 0
\(598\) −38.9953 −1.59464
\(599\) −16.3430 −0.667758 −0.333879 0.942616i \(-0.608358\pi\)
−0.333879 + 0.942616i \(0.608358\pi\)
\(600\) 0 0
\(601\) −6.62371 −0.270187 −0.135093 0.990833i \(-0.543133\pi\)
−0.135093 + 0.990833i \(0.543133\pi\)
\(602\) −1.09591 −0.0446658
\(603\) 0 0
\(604\) 15.2428 0.620221
\(605\) 0 0
\(606\) 0 0
\(607\) 30.3094 1.23022 0.615110 0.788442i \(-0.289112\pi\)
0.615110 + 0.788442i \(0.289112\pi\)
\(608\) −20.2805 −0.822483
\(609\) 0 0
\(610\) 0 0
\(611\) 23.1347 0.935931
\(612\) 0 0
\(613\) 14.7803 0.596969 0.298484 0.954415i \(-0.403519\pi\)
0.298484 + 0.954415i \(0.403519\pi\)
\(614\) −8.88918 −0.358738
\(615\) 0 0
\(616\) −3.40660 −0.137256
\(617\) −33.8512 −1.36280 −0.681399 0.731912i \(-0.738628\pi\)
−0.681399 + 0.731912i \(0.738628\pi\)
\(618\) 0 0
\(619\) 11.6887 0.469807 0.234903 0.972019i \(-0.424523\pi\)
0.234903 + 0.972019i \(0.424523\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 31.1709 1.24984
\(623\) −1.71067 −0.0685364
\(624\) 0 0
\(625\) 0 0
\(626\) 14.9360 0.596963
\(627\) 0 0
\(628\) 8.33686 0.332677
\(629\) −41.0347 −1.63616
\(630\) 0 0
\(631\) −38.1357 −1.51816 −0.759078 0.650999i \(-0.774350\pi\)
−0.759078 + 0.650999i \(0.774350\pi\)
\(632\) 7.39944 0.294334
\(633\) 0 0
\(634\) −23.2589 −0.923728
\(635\) 0 0
\(636\) 0 0
\(637\) −42.0816 −1.66733
\(638\) −12.4562 −0.493144
\(639\) 0 0
\(640\) 0 0
\(641\) −34.7654 −1.37315 −0.686576 0.727058i \(-0.740887\pi\)
−0.686576 + 0.727058i \(0.740887\pi\)
\(642\) 0 0
\(643\) 2.68515 0.105892 0.0529461 0.998597i \(-0.483139\pi\)
0.0529461 + 0.998597i \(0.483139\pi\)
\(644\) −1.29345 −0.0509692
\(645\) 0 0
\(646\) 55.5131 2.18413
\(647\) 40.5103 1.59262 0.796311 0.604887i \(-0.206782\pi\)
0.796311 + 0.604887i \(0.206782\pi\)
\(648\) 0 0
\(649\) 8.93578 0.350760
\(650\) 0 0
\(651\) 0 0
\(652\) −5.06729 −0.198451
\(653\) 13.3354 0.521856 0.260928 0.965358i \(-0.415971\pi\)
0.260928 + 0.965358i \(0.415971\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 26.6132 1.03907
\(657\) 0 0
\(658\) 3.06126 0.119340
\(659\) 31.1543 1.21360 0.606800 0.794855i \(-0.292453\pi\)
0.606800 + 0.794855i \(0.292453\pi\)
\(660\) 0 0
\(661\) 6.31788 0.245737 0.122869 0.992423i \(-0.460791\pi\)
0.122869 + 0.992423i \(0.460791\pi\)
\(662\) 19.9315 0.774659
\(663\) 0 0
\(664\) 30.2267 1.17302
\(665\) 0 0
\(666\) 0 0
\(667\) 9.40838 0.364294
\(668\) −1.99160 −0.0770571
\(669\) 0 0
\(670\) 0 0
\(671\) −8.81528 −0.340310
\(672\) 0 0
\(673\) −6.59220 −0.254110 −0.127055 0.991896i \(-0.540553\pi\)
−0.127055 + 0.991896i \(0.540553\pi\)
\(674\) −7.49857 −0.288834
\(675\) 0 0
\(676\) 17.3476 0.667214
\(677\) 34.8947 1.34111 0.670556 0.741859i \(-0.266056\pi\)
0.670556 + 0.741859i \(0.266056\pi\)
\(678\) 0 0
\(679\) 5.59702 0.214794
\(680\) 0 0
\(681\) 0 0
\(682\) −21.4172 −0.820108
\(683\) 26.0958 0.998528 0.499264 0.866450i \(-0.333604\pi\)
0.499264 + 0.866450i \(0.333604\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −11.3476 −0.433252
\(687\) 0 0
\(688\) 6.49165 0.247492
\(689\) −15.8606 −0.604239
\(690\) 0 0
\(691\) −29.3058 −1.11485 −0.557423 0.830229i \(-0.688210\pi\)
−0.557423 + 0.830229i \(0.688210\pi\)
\(692\) −10.6065 −0.403197
\(693\) 0 0
\(694\) −54.6648 −2.07505
\(695\) 0 0
\(696\) 0 0
\(697\) −33.1964 −1.25740
\(698\) 45.8775 1.73649
\(699\) 0 0
\(700\) 0 0
\(701\) 15.3891 0.581239 0.290620 0.956839i \(-0.406139\pi\)
0.290620 + 0.956839i \(0.406139\pi\)
\(702\) 0 0
\(703\) 37.4696 1.41319
\(704\) 11.8823 0.447832
\(705\) 0 0
\(706\) 3.00883 0.113239
\(707\) 8.77741 0.330108
\(708\) 0 0
\(709\) −7.73991 −0.290678 −0.145339 0.989382i \(-0.546427\pi\)
−0.145339 + 0.989382i \(0.546427\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 7.36062 0.275851
\(713\) 16.1768 0.605828
\(714\) 0 0
\(715\) 0 0
\(716\) 11.4302 0.427165
\(717\) 0 0
\(718\) 19.7875 0.738462
\(719\) −15.1316 −0.564313 −0.282156 0.959368i \(-0.591050\pi\)
−0.282156 + 0.959368i \(0.591050\pi\)
\(720\) 0 0
\(721\) 0.420832 0.0156726
\(722\) −19.6494 −0.731275
\(723\) 0 0
\(724\) 8.94457 0.332422
\(725\) 0 0
\(726\) 0 0
\(727\) 0.161876 0.00600366 0.00300183 0.999995i \(-0.499044\pi\)
0.00300183 + 0.999995i \(0.499044\pi\)
\(728\) −6.85590 −0.254097
\(729\) 0 0
\(730\) 0 0
\(731\) −8.09746 −0.299495
\(732\) 0 0
\(733\) −50.0332 −1.84802 −0.924009 0.382371i \(-0.875107\pi\)
−0.924009 + 0.382371i \(0.875107\pi\)
\(734\) 23.8095 0.878825
\(735\) 0 0
\(736\) 13.9285 0.513413
\(737\) −7.46070 −0.274818
\(738\) 0 0
\(739\) 30.5505 1.12382 0.561909 0.827199i \(-0.310067\pi\)
0.561909 + 0.827199i \(0.310067\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.09872 −0.0770463
\(743\) 5.96684 0.218902 0.109451 0.993992i \(-0.465091\pi\)
0.109451 + 0.993992i \(0.465091\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 15.4329 0.565039
\(747\) 0 0
\(748\) 12.6530 0.462640
\(749\) −5.59340 −0.204379
\(750\) 0 0
\(751\) 34.3976 1.25519 0.627593 0.778542i \(-0.284040\pi\)
0.627593 + 0.778542i \(0.284040\pi\)
\(752\) −18.1335 −0.661260
\(753\) 0 0
\(754\) −25.0685 −0.912941
\(755\) 0 0
\(756\) 0 0
\(757\) 40.6873 1.47881 0.739403 0.673263i \(-0.235108\pi\)
0.739403 + 0.673263i \(0.235108\pi\)
\(758\) 46.6495 1.69439
\(759\) 0 0
\(760\) 0 0
\(761\) 28.2596 1.02441 0.512204 0.858864i \(-0.328829\pi\)
0.512204 + 0.858864i \(0.328829\pi\)
\(762\) 0 0
\(763\) 2.35409 0.0852239
\(764\) 16.9519 0.613299
\(765\) 0 0
\(766\) −2.39407 −0.0865013
\(767\) 17.9836 0.649350
\(768\) 0 0
\(769\) −46.9036 −1.69139 −0.845694 0.533668i \(-0.820813\pi\)
−0.845694 + 0.533668i \(0.820813\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.39582 −0.230191
\(773\) 9.19641 0.330772 0.165386 0.986229i \(-0.447113\pi\)
0.165386 + 0.986229i \(0.447113\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −24.0827 −0.864520
\(777\) 0 0
\(778\) 21.0916 0.756169
\(779\) 30.3123 1.08605
\(780\) 0 0
\(781\) 17.1815 0.614802
\(782\) −38.1261 −1.36339
\(783\) 0 0
\(784\) 32.9844 1.17801
\(785\) 0 0
\(786\) 0 0
\(787\) 5.74637 0.204836 0.102418 0.994741i \(-0.467342\pi\)
0.102418 + 0.994741i \(0.467342\pi\)
\(788\) −1.38388 −0.0492986
\(789\) 0 0
\(790\) 0 0
\(791\) −6.06126 −0.215514
\(792\) 0 0
\(793\) −17.7411 −0.630004
\(794\) 1.41870 0.0503479
\(795\) 0 0
\(796\) 8.76255 0.310580
\(797\) −7.07450 −0.250592 −0.125296 0.992119i \(-0.539988\pi\)
−0.125296 + 0.992119i \(0.539988\pi\)
\(798\) 0 0
\(799\) 22.6191 0.800205
\(800\) 0 0
\(801\) 0 0
\(802\) −54.5868 −1.92753
\(803\) 36.5787 1.29083
\(804\) 0 0
\(805\) 0 0
\(806\) −43.1029 −1.51824
\(807\) 0 0
\(808\) −37.7672 −1.32865
\(809\) 38.1075 1.33979 0.669894 0.742457i \(-0.266340\pi\)
0.669894 + 0.742457i \(0.266340\pi\)
\(810\) 0 0
\(811\) −1.44105 −0.0506022 −0.0253011 0.999680i \(-0.508054\pi\)
−0.0253011 + 0.999680i \(0.508054\pi\)
\(812\) −0.831508 −0.0291802
\(813\) 0 0
\(814\) 34.0702 1.19416
\(815\) 0 0
\(816\) 0 0
\(817\) 7.39396 0.258682
\(818\) −8.25904 −0.288771
\(819\) 0 0
\(820\) 0 0
\(821\) 22.5143 0.785753 0.392876 0.919591i \(-0.371480\pi\)
0.392876 + 0.919591i \(0.371480\pi\)
\(822\) 0 0
\(823\) −41.4589 −1.44517 −0.722583 0.691284i \(-0.757045\pi\)
−0.722583 + 0.691284i \(0.757045\pi\)
\(824\) −1.81075 −0.0630803
\(825\) 0 0
\(826\) 2.37964 0.0827984
\(827\) 27.8133 0.967164 0.483582 0.875299i \(-0.339336\pi\)
0.483582 + 0.875299i \(0.339336\pi\)
\(828\) 0 0
\(829\) 20.7232 0.719745 0.359872 0.933002i \(-0.382820\pi\)
0.359872 + 0.933002i \(0.382820\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 23.9136 0.829056
\(833\) −41.1436 −1.42554
\(834\) 0 0
\(835\) 0 0
\(836\) −11.5537 −0.399595
\(837\) 0 0
\(838\) −17.9017 −0.618403
\(839\) 18.1451 0.626437 0.313218 0.949681i \(-0.398593\pi\)
0.313218 + 0.949681i \(0.398593\pi\)
\(840\) 0 0
\(841\) −22.9517 −0.791439
\(842\) 17.3806 0.598975
\(843\) 0 0
\(844\) 7.43444 0.255904
\(845\) 0 0
\(846\) 0 0
\(847\) 0.701798 0.0241141
\(848\) 12.4318 0.426911
\(849\) 0 0
\(850\) 0 0
\(851\) −25.7339 −0.882148
\(852\) 0 0
\(853\) −4.00708 −0.137200 −0.0685999 0.997644i \(-0.521853\pi\)
−0.0685999 + 0.997644i \(0.521853\pi\)
\(854\) −2.34755 −0.0803316
\(855\) 0 0
\(856\) 24.0672 0.822599
\(857\) −9.08971 −0.310499 −0.155249 0.987875i \(-0.549618\pi\)
−0.155249 + 0.987875i \(0.549618\pi\)
\(858\) 0 0
\(859\) 16.3870 0.559116 0.279558 0.960129i \(-0.409812\pi\)
0.279558 + 0.960129i \(0.409812\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 61.0243 2.07850
\(863\) 23.7967 0.810050 0.405025 0.914306i \(-0.367263\pi\)
0.405025 + 0.914306i \(0.367263\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 28.1204 0.955571
\(867\) 0 0
\(868\) −1.42970 −0.0485272
\(869\) 10.5499 0.357882
\(870\) 0 0
\(871\) −15.0149 −0.508762
\(872\) −10.1291 −0.343016
\(873\) 0 0
\(874\) 34.8137 1.17759
\(875\) 0 0
\(876\) 0 0
\(877\) 36.0744 1.21815 0.609073 0.793114i \(-0.291542\pi\)
0.609073 + 0.793114i \(0.291542\pi\)
\(878\) 51.8687 1.75048
\(879\) 0 0
\(880\) 0 0
\(881\) −35.4575 −1.19459 −0.597297 0.802020i \(-0.703759\pi\)
−0.597297 + 0.802020i \(0.703759\pi\)
\(882\) 0 0
\(883\) −39.1320 −1.31690 −0.658448 0.752626i \(-0.728787\pi\)
−0.658448 + 0.752626i \(0.728787\pi\)
\(884\) 25.4647 0.856470
\(885\) 0 0
\(886\) 0.581537 0.0195371
\(887\) 51.2833 1.72192 0.860962 0.508670i \(-0.169863\pi\)
0.860962 + 0.508670i \(0.169863\pi\)
\(888\) 0 0
\(889\) 1.63156 0.0547207
\(890\) 0 0
\(891\) 0 0
\(892\) −2.60530 −0.0872318
\(893\) −20.6539 −0.691158
\(894\) 0 0
\(895\) 0 0
\(896\) 6.84415 0.228647
\(897\) 0 0
\(898\) 12.8350 0.428311
\(899\) 10.3994 0.346841
\(900\) 0 0
\(901\) −15.5070 −0.516614
\(902\) 27.5623 0.917723
\(903\) 0 0
\(904\) 26.0802 0.867416
\(905\) 0 0
\(906\) 0 0
\(907\) 47.8588 1.58913 0.794563 0.607181i \(-0.207700\pi\)
0.794563 + 0.607181i \(0.207700\pi\)
\(908\) 8.56930 0.284382
\(909\) 0 0
\(910\) 0 0
\(911\) 18.0431 0.597793 0.298897 0.954285i \(-0.403381\pi\)
0.298897 + 0.954285i \(0.403381\pi\)
\(912\) 0 0
\(913\) 43.0964 1.42628
\(914\) −35.1104 −1.16135
\(915\) 0 0
\(916\) 4.45117 0.147071
\(917\) 4.74276 0.156620
\(918\) 0 0
\(919\) 10.3976 0.342984 0.171492 0.985185i \(-0.445141\pi\)
0.171492 + 0.985185i \(0.445141\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −66.9708 −2.20557
\(923\) 34.5784 1.13816
\(924\) 0 0
\(925\) 0 0
\(926\) −68.8351 −2.26206
\(927\) 0 0
\(928\) 8.95410 0.293933
\(929\) −36.0216 −1.18183 −0.590915 0.806734i \(-0.701233\pi\)
−0.590915 + 0.806734i \(0.701233\pi\)
\(930\) 0 0
\(931\) 37.5691 1.23128
\(932\) −2.44763 −0.0801748
\(933\) 0 0
\(934\) 36.8203 1.20480
\(935\) 0 0
\(936\) 0 0
\(937\) 24.0326 0.785111 0.392555 0.919728i \(-0.371591\pi\)
0.392555 + 0.919728i \(0.371591\pi\)
\(938\) −1.98682 −0.0648720
\(939\) 0 0
\(940\) 0 0
\(941\) 16.6676 0.543348 0.271674 0.962389i \(-0.412423\pi\)
0.271674 + 0.962389i \(0.412423\pi\)
\(942\) 0 0
\(943\) −20.8183 −0.677938
\(944\) −14.0959 −0.458783
\(945\) 0 0
\(946\) 6.72315 0.218589
\(947\) −27.5400 −0.894928 −0.447464 0.894302i \(-0.647673\pi\)
−0.447464 + 0.894302i \(0.647673\pi\)
\(948\) 0 0
\(949\) 73.6160 2.38968
\(950\) 0 0
\(951\) 0 0
\(952\) −6.70308 −0.217248
\(953\) −18.1344 −0.587432 −0.293716 0.955893i \(-0.594892\pi\)
−0.293716 + 0.955893i \(0.594892\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −10.4984 −0.339544
\(957\) 0 0
\(958\) 54.3833 1.75705
\(959\) −1.16713 −0.0376886
\(960\) 0 0
\(961\) −13.1191 −0.423198
\(962\) 68.5676 2.21071
\(963\) 0 0
\(964\) 7.51009 0.241884
\(965\) 0 0
\(966\) 0 0
\(967\) 36.1875 1.16371 0.581855 0.813292i \(-0.302327\pi\)
0.581855 + 0.813292i \(0.302327\pi\)
\(968\) −3.01968 −0.0970562
\(969\) 0 0
\(970\) 0 0
\(971\) −34.6173 −1.11092 −0.555461 0.831542i \(-0.687458\pi\)
−0.555461 + 0.831542i \(0.687458\pi\)
\(972\) 0 0
\(973\) −5.50527 −0.176491
\(974\) −38.8340 −1.24432
\(975\) 0 0
\(976\) 13.9058 0.445115
\(977\) −29.4331 −0.941650 −0.470825 0.882227i \(-0.656044\pi\)
−0.470825 + 0.882227i \(0.656044\pi\)
\(978\) 0 0
\(979\) 10.4946 0.335408
\(980\) 0 0
\(981\) 0 0
\(982\) 7.52432 0.240111
\(983\) 24.4911 0.781145 0.390573 0.920572i \(-0.372277\pi\)
0.390573 + 0.920572i \(0.372277\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −24.5097 −0.780549
\(987\) 0 0
\(988\) −23.2523 −0.739755
\(989\) −5.07813 −0.161475
\(990\) 0 0
\(991\) −13.2821 −0.421919 −0.210959 0.977495i \(-0.567659\pi\)
−0.210959 + 0.977495i \(0.567659\pi\)
\(992\) 15.3957 0.488815
\(993\) 0 0
\(994\) 4.57551 0.145126
\(995\) 0 0
\(996\) 0 0
\(997\) 39.1530 1.23999 0.619994 0.784606i \(-0.287135\pi\)
0.619994 + 0.784606i \(0.287135\pi\)
\(998\) 30.8734 0.977280
\(999\) 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.y.1.1 4
3.2 odd 2 2025.2.a.p.1.4 4
5.2 odd 4 2025.2.b.n.649.2 8
5.3 odd 4 2025.2.b.n.649.7 8
5.4 even 2 2025.2.a.q.1.4 4
9.2 odd 6 675.2.e.e.226.1 8
9.4 even 3 225.2.e.c.151.4 yes 8
9.5 odd 6 675.2.e.e.451.1 8
9.7 even 3 225.2.e.c.76.4 8
15.2 even 4 2025.2.b.o.649.7 8
15.8 even 4 2025.2.b.o.649.2 8
15.14 odd 2 2025.2.a.z.1.1 4
45.2 even 12 675.2.k.c.199.7 16
45.4 even 6 225.2.e.e.151.1 yes 8
45.7 odd 12 225.2.k.c.49.2 16
45.13 odd 12 225.2.k.c.124.2 16
45.14 odd 6 675.2.e.c.451.4 8
45.22 odd 12 225.2.k.c.124.7 16
45.23 even 12 675.2.k.c.424.7 16
45.29 odd 6 675.2.e.c.226.4 8
45.32 even 12 675.2.k.c.424.2 16
45.34 even 6 225.2.e.e.76.1 yes 8
45.38 even 12 675.2.k.c.199.2 16
45.43 odd 12 225.2.k.c.49.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.e.c.76.4 8 9.7 even 3
225.2.e.c.151.4 yes 8 9.4 even 3
225.2.e.e.76.1 yes 8 45.34 even 6
225.2.e.e.151.1 yes 8 45.4 even 6
225.2.k.c.49.2 16 45.7 odd 12
225.2.k.c.49.7 16 45.43 odd 12
225.2.k.c.124.2 16 45.13 odd 12
225.2.k.c.124.7 16 45.22 odd 12
675.2.e.c.226.4 8 45.29 odd 6
675.2.e.c.451.4 8 45.14 odd 6
675.2.e.e.226.1 8 9.2 odd 6
675.2.e.e.451.1 8 9.5 odd 6
675.2.k.c.199.2 16 45.38 even 12
675.2.k.c.199.7 16 45.2 even 12
675.2.k.c.424.2 16 45.32 even 12
675.2.k.c.424.7 16 45.23 even 12
2025.2.a.p.1.4 4 3.2 odd 2
2025.2.a.q.1.4 4 5.4 even 2
2025.2.a.y.1.1 4 1.1 even 1 trivial
2025.2.a.z.1.1 4 15.14 odd 2
2025.2.b.n.649.2 8 5.2 odd 4
2025.2.b.n.649.7 8 5.3 odd 4
2025.2.b.o.649.2 8 15.8 even 4
2025.2.b.o.649.7 8 15.2 even 4