Properties

Label 1225.2.a.bc.1.3
Level $1225$
Weight $2$
Character 1225.1
Self dual yes
Analytic conductor $9.782$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1225,2,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, 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("1225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.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: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.28825\) of defining polynomial
Character \(\chi\) \(=\) 1225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.61803 q^{2} -2.28825 q^{3} +4.85410 q^{4} -5.99070 q^{6} +7.47214 q^{8} +2.23607 q^{9} +O(q^{10})\) \(q+2.61803 q^{2} -2.28825 q^{3} +4.85410 q^{4} -5.99070 q^{6} +7.47214 q^{8} +2.23607 q^{9} -2.23607 q^{11} -11.1074 q^{12} +5.45052 q^{13} +9.85410 q^{16} +1.08036 q^{17} +5.85410 q^{18} +4.24264 q^{19} -5.85410 q^{22} +8.23607 q^{23} -17.0981 q^{24} +14.2697 q^{26} +1.74806 q^{27} -2.23607 q^{29} -2.62210 q^{31} +10.8541 q^{32} +5.11667 q^{33} +2.82843 q^{34} +10.8541 q^{36} -2.70820 q^{37} +11.1074 q^{38} -12.4721 q^{39} -7.94510 q^{41} +5.00000 q^{43} -10.8541 q^{44} +21.5623 q^{46} +9.48683 q^{47} -22.5486 q^{48} -2.47214 q^{51} +26.4574 q^{52} -3.70820 q^{53} +4.57649 q^{54} -9.70820 q^{57} -5.85410 q^{58} +8.94665 q^{59} -12.5216 q^{61} -6.86474 q^{62} +8.70820 q^{64} +13.3956 q^{66} -4.70820 q^{67} +5.24419 q^{68} -18.8461 q^{69} +1.47214 q^{71} +16.7082 q^{72} -6.86474 q^{73} -7.09017 q^{74} +20.5942 q^{76} -32.6525 q^{78} +8.70820 q^{79} -10.7082 q^{81} -20.8005 q^{82} -2.62210 q^{83} +13.0902 q^{86} +5.11667 q^{87} -16.7082 q^{88} +4.78282 q^{89} +39.9787 q^{92} +6.00000 q^{93} +24.8369 q^{94} -24.8369 q^{96} -19.5927 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 6 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{2} + 6 q^{4} + 12 q^{8} + 26 q^{16} + 10 q^{18} - 10 q^{22} + 24 q^{23} + 30 q^{32} + 30 q^{36} + 16 q^{37} - 32 q^{39} + 20 q^{43} - 30 q^{44} + 46 q^{46} + 8 q^{51} + 12 q^{53} - 12 q^{57} - 10 q^{58} + 8 q^{64} + 8 q^{67} - 12 q^{71} + 40 q^{72} - 6 q^{74} - 68 q^{78} + 8 q^{79} - 16 q^{81} + 30 q^{86} - 40 q^{88} + 66 q^{92} + 24 q^{93} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.61803 1.85123 0.925615 0.378467i \(-0.123549\pi\)
0.925615 + 0.378467i \(0.123549\pi\)
\(3\) −2.28825 −1.32112 −0.660560 0.750774i \(-0.729681\pi\)
−0.660560 + 0.750774i \(0.729681\pi\)
\(4\) 4.85410 2.42705
\(5\) 0 0
\(6\) −5.99070 −2.44569
\(7\) 0 0
\(8\) 7.47214 2.64180
\(9\) 2.23607 0.745356
\(10\) 0 0
\(11\) −2.23607 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(12\) −11.1074 −3.20642
\(13\) 5.45052 1.51170 0.755852 0.654743i \(-0.227223\pi\)
0.755852 + 0.654743i \(0.227223\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 9.85410 2.46353
\(17\) 1.08036 0.262027 0.131013 0.991381i \(-0.458177\pi\)
0.131013 + 0.991381i \(0.458177\pi\)
\(18\) 5.85410 1.37983
\(19\) 4.24264 0.973329 0.486664 0.873589i \(-0.338214\pi\)
0.486664 + 0.873589i \(0.338214\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.85410 −1.24810
\(23\) 8.23607 1.71734 0.858669 0.512530i \(-0.171292\pi\)
0.858669 + 0.512530i \(0.171292\pi\)
\(24\) −17.0981 −3.49013
\(25\) 0 0
\(26\) 14.2697 2.79851
\(27\) 1.74806 0.336415
\(28\) 0 0
\(29\) −2.23607 −0.415227 −0.207614 0.978211i \(-0.566570\pi\)
−0.207614 + 0.978211i \(0.566570\pi\)
\(30\) 0 0
\(31\) −2.62210 −0.470942 −0.235471 0.971881i \(-0.575663\pi\)
−0.235471 + 0.971881i \(0.575663\pi\)
\(32\) 10.8541 1.91875
\(33\) 5.11667 0.890698
\(34\) 2.82843 0.485071
\(35\) 0 0
\(36\) 10.8541 1.80902
\(37\) −2.70820 −0.445226 −0.222613 0.974907i \(-0.571459\pi\)
−0.222613 + 0.974907i \(0.571459\pi\)
\(38\) 11.1074 1.80185
\(39\) −12.4721 −1.99714
\(40\) 0 0
\(41\) −7.94510 −1.24082 −0.620408 0.784279i \(-0.713033\pi\)
−0.620408 + 0.784279i \(0.713033\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −10.8541 −1.63632
\(45\) 0 0
\(46\) 21.5623 3.17919
\(47\) 9.48683 1.38380 0.691898 0.721995i \(-0.256775\pi\)
0.691898 + 0.721995i \(0.256775\pi\)
\(48\) −22.5486 −3.25461
\(49\) 0 0
\(50\) 0 0
\(51\) −2.47214 −0.346168
\(52\) 26.4574 3.66898
\(53\) −3.70820 −0.509361 −0.254680 0.967025i \(-0.581970\pi\)
−0.254680 + 0.967025i \(0.581970\pi\)
\(54\) 4.57649 0.622782
\(55\) 0 0
\(56\) 0 0
\(57\) −9.70820 −1.28588
\(58\) −5.85410 −0.768681
\(59\) 8.94665 1.16475 0.582377 0.812919i \(-0.302123\pi\)
0.582377 + 0.812919i \(0.302123\pi\)
\(60\) 0 0
\(61\) −12.5216 −1.60323 −0.801613 0.597844i \(-0.796024\pi\)
−0.801613 + 0.597844i \(0.796024\pi\)
\(62\) −6.86474 −0.871822
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) 0 0
\(66\) 13.3956 1.64889
\(67\) −4.70820 −0.575199 −0.287599 0.957751i \(-0.592857\pi\)
−0.287599 + 0.957751i \(0.592857\pi\)
\(68\) 5.24419 0.635952
\(69\) −18.8461 −2.26881
\(70\) 0 0
\(71\) 1.47214 0.174710 0.0873552 0.996177i \(-0.472158\pi\)
0.0873552 + 0.996177i \(0.472158\pi\)
\(72\) 16.7082 1.96908
\(73\) −6.86474 −0.803457 −0.401728 0.915759i \(-0.631590\pi\)
−0.401728 + 0.915759i \(0.631590\pi\)
\(74\) −7.09017 −0.824216
\(75\) 0 0
\(76\) 20.5942 2.36232
\(77\) 0 0
\(78\) −32.6525 −3.69716
\(79\) 8.70820 0.979749 0.489875 0.871793i \(-0.337043\pi\)
0.489875 + 0.871793i \(0.337043\pi\)
\(80\) 0 0
\(81\) −10.7082 −1.18980
\(82\) −20.8005 −2.29704
\(83\) −2.62210 −0.287812 −0.143906 0.989591i \(-0.545966\pi\)
−0.143906 + 0.989591i \(0.545966\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 13.0902 1.41155
\(87\) 5.11667 0.548565
\(88\) −16.7082 −1.78110
\(89\) 4.78282 0.506978 0.253489 0.967338i \(-0.418422\pi\)
0.253489 + 0.967338i \(0.418422\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 39.9787 4.16807
\(93\) 6.00000 0.622171
\(94\) 24.8369 2.56173
\(95\) 0 0
\(96\) −24.8369 −2.53490
\(97\) −19.5927 −1.98933 −0.994667 0.103143i \(-0.967110\pi\)
−0.994667 + 0.103143i \(0.967110\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 1.62054 0.161250 0.0806251 0.996744i \(-0.474308\pi\)
0.0806251 + 0.996744i \(0.474308\pi\)
\(102\) −6.47214 −0.640837
\(103\) −9.48683 −0.934765 −0.467383 0.884055i \(-0.654803\pi\)
−0.467383 + 0.884055i \(0.654803\pi\)
\(104\) 40.7271 3.99362
\(105\) 0 0
\(106\) −9.70820 −0.942944
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 8.48528 0.816497
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 6.19704 0.588197
\(112\) 0 0
\(113\) 14.2361 1.33922 0.669608 0.742714i \(-0.266462\pi\)
0.669608 + 0.742714i \(0.266462\pi\)
\(114\) −25.4164 −2.38046
\(115\) 0 0
\(116\) −10.8541 −1.00778
\(117\) 12.1877 1.12676
\(118\) 23.4226 2.15623
\(119\) 0 0
\(120\) 0 0
\(121\) −6.00000 −0.545455
\(122\) −32.7820 −2.96794
\(123\) 18.1803 1.63927
\(124\) −12.7279 −1.14300
\(125\) 0 0
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 1.09017 0.0963583
\(129\) −11.4412 −1.00734
\(130\) 0 0
\(131\) −2.16073 −0.188784 −0.0943918 0.995535i \(-0.530091\pi\)
−0.0943918 + 0.995535i \(0.530091\pi\)
\(132\) 24.8369 2.16177
\(133\) 0 0
\(134\) −12.3262 −1.06482
\(135\) 0 0
\(136\) 8.07262 0.692221
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −49.3399 −4.20009
\(139\) −10.1058 −0.857165 −0.428582 0.903503i \(-0.640987\pi\)
−0.428582 + 0.903503i \(0.640987\pi\)
\(140\) 0 0
\(141\) −21.7082 −1.82816
\(142\) 3.85410 0.323429
\(143\) −12.1877 −1.01919
\(144\) 22.0344 1.83620
\(145\) 0 0
\(146\) −17.9721 −1.48738
\(147\) 0 0
\(148\) −13.1459 −1.08059
\(149\) −21.6525 −1.77384 −0.886920 0.461923i \(-0.847160\pi\)
−0.886920 + 0.461923i \(0.847160\pi\)
\(150\) 0 0
\(151\) −18.4164 −1.49871 −0.749353 0.662171i \(-0.769635\pi\)
−0.749353 + 0.662171i \(0.769635\pi\)
\(152\) 31.7016 2.57134
\(153\) 2.41577 0.195303
\(154\) 0 0
\(155\) 0 0
\(156\) −60.5410 −4.84716
\(157\) 4.03631 0.322133 0.161066 0.986944i \(-0.448507\pi\)
0.161066 + 0.986944i \(0.448507\pi\)
\(158\) 22.7984 1.81374
\(159\) 8.48528 0.672927
\(160\) 0 0
\(161\) 0 0
\(162\) −28.0344 −2.20259
\(163\) −15.7082 −1.23036 −0.615181 0.788386i \(-0.710917\pi\)
−0.615181 + 0.788386i \(0.710917\pi\)
\(164\) −38.5663 −3.01152
\(165\) 0 0
\(166\) −6.86474 −0.532807
\(167\) −1.54173 −0.119303 −0.0596514 0.998219i \(-0.518999\pi\)
−0.0596514 + 0.998219i \(0.518999\pi\)
\(168\) 0 0
\(169\) 16.7082 1.28525
\(170\) 0 0
\(171\) 9.48683 0.725476
\(172\) 24.2705 1.85061
\(173\) 4.24264 0.322562 0.161281 0.986909i \(-0.448437\pi\)
0.161281 + 0.986909i \(0.448437\pi\)
\(174\) 13.3956 1.01552
\(175\) 0 0
\(176\) −22.0344 −1.66091
\(177\) −20.4721 −1.53878
\(178\) 12.5216 0.938533
\(179\) −9.70820 −0.725625 −0.362813 0.931862i \(-0.618183\pi\)
−0.362813 + 0.931862i \(0.618183\pi\)
\(180\) 0 0
\(181\) 1.62054 0.120454 0.0602271 0.998185i \(-0.480818\pi\)
0.0602271 + 0.998185i \(0.480818\pi\)
\(182\) 0 0
\(183\) 28.6525 2.11805
\(184\) 61.5410 4.53686
\(185\) 0 0
\(186\) 15.7082 1.15178
\(187\) −2.41577 −0.176658
\(188\) 46.0501 3.35855
\(189\) 0 0
\(190\) 0 0
\(191\) 17.1246 1.23909 0.619547 0.784960i \(-0.287316\pi\)
0.619547 + 0.784960i \(0.287316\pi\)
\(192\) −19.9265 −1.43807
\(193\) 2.70820 0.194941 0.0974704 0.995238i \(-0.468925\pi\)
0.0974704 + 0.995238i \(0.468925\pi\)
\(194\) −51.2942 −3.68271
\(195\) 0 0
\(196\) 0 0
\(197\) 11.9443 0.850994 0.425497 0.904960i \(-0.360099\pi\)
0.425497 + 0.904960i \(0.360099\pi\)
\(198\) −13.0902 −0.930278
\(199\) 15.7627 1.11739 0.558693 0.829374i \(-0.311303\pi\)
0.558693 + 0.829374i \(0.311303\pi\)
\(200\) 0 0
\(201\) 10.7735 0.759906
\(202\) 4.24264 0.298511
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) −24.8369 −1.73047
\(207\) 18.4164 1.28003
\(208\) 53.7100 3.72412
\(209\) −9.48683 −0.656218
\(210\) 0 0
\(211\) −19.4164 −1.33668 −0.668340 0.743856i \(-0.732995\pi\)
−0.668340 + 0.743856i \(0.732995\pi\)
\(212\) −18.0000 −1.23625
\(213\) −3.36861 −0.230813
\(214\) 15.7082 1.07379
\(215\) 0 0
\(216\) 13.0618 0.888741
\(217\) 0 0
\(218\) −2.61803 −0.177316
\(219\) 15.7082 1.06146
\(220\) 0 0
\(221\) 5.88854 0.396106
\(222\) 16.2241 1.08889
\(223\) −15.3500 −1.02791 −0.513957 0.857816i \(-0.671821\pi\)
−0.513957 + 0.857816i \(0.671821\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 37.2705 2.47920
\(227\) −7.48373 −0.496713 −0.248356 0.968669i \(-0.579890\pi\)
−0.248356 + 0.968669i \(0.579890\pi\)
\(228\) −47.1246 −3.12090
\(229\) −7.07107 −0.467269 −0.233635 0.972324i \(-0.575062\pi\)
−0.233635 + 0.972324i \(0.575062\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −16.7082 −1.09695
\(233\) 13.4721 0.882589 0.441294 0.897362i \(-0.354519\pi\)
0.441294 + 0.897362i \(0.354519\pi\)
\(234\) 31.9079 2.08589
\(235\) 0 0
\(236\) 43.4280 2.82692
\(237\) −19.9265 −1.29437
\(238\) 0 0
\(239\) −11.1246 −0.719591 −0.359796 0.933031i \(-0.617154\pi\)
−0.359796 + 0.933031i \(0.617154\pi\)
\(240\) 0 0
\(241\) 23.2163 1.49549 0.747747 0.663984i \(-0.231136\pi\)
0.747747 + 0.663984i \(0.231136\pi\)
\(242\) −15.7082 −1.00976
\(243\) 19.2588 1.23545
\(244\) −60.7811 −3.89111
\(245\) 0 0
\(246\) 47.5967 3.03466
\(247\) 23.1246 1.47138
\(248\) −19.5927 −1.24414
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 20.0540 1.26580 0.632900 0.774234i \(-0.281865\pi\)
0.632900 + 0.774234i \(0.281865\pi\)
\(252\) 0 0
\(253\) −18.4164 −1.15783
\(254\) 18.3262 1.14989
\(255\) 0 0
\(256\) −14.5623 −0.910144
\(257\) −16.3516 −1.01998 −0.509991 0.860179i \(-0.670351\pi\)
−0.509991 + 0.860179i \(0.670351\pi\)
\(258\) −29.9535 −1.86482
\(259\) 0 0
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) −5.65685 −0.349482
\(263\) 12.0557 0.743388 0.371694 0.928355i \(-0.378777\pi\)
0.371694 + 0.928355i \(0.378777\pi\)
\(264\) 38.2325 2.35305
\(265\) 0 0
\(266\) 0 0
\(267\) −10.9443 −0.669779
\(268\) −22.8541 −1.39604
\(269\) −5.86319 −0.357485 −0.178742 0.983896i \(-0.557203\pi\)
−0.178742 + 0.983896i \(0.557203\pi\)
\(270\) 0 0
\(271\) −13.9358 −0.846540 −0.423270 0.906004i \(-0.639118\pi\)
−0.423270 + 0.906004i \(0.639118\pi\)
\(272\) 10.6460 0.645509
\(273\) 0 0
\(274\) −15.7082 −0.948967
\(275\) 0 0
\(276\) −91.4811 −5.50652
\(277\) −24.0000 −1.44202 −0.721010 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) −26.4574 −1.58681
\(279\) −5.86319 −0.351020
\(280\) 0 0
\(281\) 24.5967 1.46732 0.733659 0.679517i \(-0.237811\pi\)
0.733659 + 0.679517i \(0.237811\pi\)
\(282\) −56.8328 −3.38434
\(283\) −1.82688 −0.108596 −0.0542982 0.998525i \(-0.517292\pi\)
−0.0542982 + 0.998525i \(0.517292\pi\)
\(284\) 7.14590 0.424031
\(285\) 0 0
\(286\) −31.9079 −1.88675
\(287\) 0 0
\(288\) 24.2705 1.43015
\(289\) −15.8328 −0.931342
\(290\) 0 0
\(291\) 44.8328 2.62815
\(292\) −33.3221 −1.95003
\(293\) −22.6761 −1.32475 −0.662377 0.749171i \(-0.730452\pi\)
−0.662377 + 0.749171i \(0.730452\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −20.2361 −1.17620
\(297\) −3.90879 −0.226811
\(298\) −56.6869 −3.28378
\(299\) 44.8909 2.59611
\(300\) 0 0
\(301\) 0 0
\(302\) −48.2148 −2.77445
\(303\) −3.70820 −0.213031
\(304\) 41.8074 2.39782
\(305\) 0 0
\(306\) 6.32456 0.361551
\(307\) 33.5285 1.91357 0.956785 0.290795i \(-0.0939199\pi\)
0.956785 + 0.290795i \(0.0939199\pi\)
\(308\) 0 0
\(309\) 21.7082 1.23494
\(310\) 0 0
\(311\) −15.8902 −0.901051 −0.450525 0.892764i \(-0.648763\pi\)
−0.450525 + 0.892764i \(0.648763\pi\)
\(312\) −93.1935 −5.27604
\(313\) 21.2132 1.19904 0.599521 0.800359i \(-0.295358\pi\)
0.599521 + 0.800359i \(0.295358\pi\)
\(314\) 10.5672 0.596341
\(315\) 0 0
\(316\) 42.2705 2.37790
\(317\) 5.94427 0.333864 0.166932 0.985968i \(-0.446614\pi\)
0.166932 + 0.985968i \(0.446614\pi\)
\(318\) 22.2148 1.24574
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) −13.7295 −0.766304
\(322\) 0 0
\(323\) 4.58359 0.255038
\(324\) −51.9787 −2.88771
\(325\) 0 0
\(326\) −41.1246 −2.27768
\(327\) 2.28825 0.126540
\(328\) −59.3669 −3.27799
\(329\) 0 0
\(330\) 0 0
\(331\) −14.7082 −0.808436 −0.404218 0.914663i \(-0.632456\pi\)
−0.404218 + 0.914663i \(0.632456\pi\)
\(332\) −12.7279 −0.698535
\(333\) −6.05573 −0.331852
\(334\) −4.03631 −0.220857
\(335\) 0 0
\(336\) 0 0
\(337\) 17.1246 0.932837 0.466419 0.884564i \(-0.345544\pi\)
0.466419 + 0.884564i \(0.345544\pi\)
\(338\) 43.7426 2.37929
\(339\) −32.5756 −1.76926
\(340\) 0 0
\(341\) 5.86319 0.317509
\(342\) 24.8369 1.34302
\(343\) 0 0
\(344\) 37.3607 2.01435
\(345\) 0 0
\(346\) 11.1074 0.597136
\(347\) −3.65248 −0.196075 −0.0980376 0.995183i \(-0.531257\pi\)
−0.0980376 + 0.995183i \(0.531257\pi\)
\(348\) 24.8369 1.33139
\(349\) 17.9721 0.962025 0.481013 0.876714i \(-0.340269\pi\)
0.481013 + 0.876714i \(0.340269\pi\)
\(350\) 0 0
\(351\) 9.52786 0.508560
\(352\) −24.2705 −1.29362
\(353\) 16.3516 0.870306 0.435153 0.900356i \(-0.356694\pi\)
0.435153 + 0.900356i \(0.356694\pi\)
\(354\) −53.5967 −2.84864
\(355\) 0 0
\(356\) 23.2163 1.23046
\(357\) 0 0
\(358\) −25.4164 −1.34330
\(359\) 17.9443 0.947062 0.473531 0.880777i \(-0.342979\pi\)
0.473531 + 0.880777i \(0.342979\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 4.24264 0.222988
\(363\) 13.7295 0.720610
\(364\) 0 0
\(365\) 0 0
\(366\) 75.0132 3.92100
\(367\) −18.1784 −0.948907 −0.474454 0.880281i \(-0.657354\pi\)
−0.474454 + 0.880281i \(0.657354\pi\)
\(368\) 81.1591 4.23071
\(369\) −17.7658 −0.924850
\(370\) 0 0
\(371\) 0 0
\(372\) 29.1246 1.51004
\(373\) 24.1246 1.24913 0.624563 0.780975i \(-0.285277\pi\)
0.624563 + 0.780975i \(0.285277\pi\)
\(374\) −6.32456 −0.327035
\(375\) 0 0
\(376\) 70.8869 3.65571
\(377\) −12.1877 −0.627701
\(378\) 0 0
\(379\) −19.0000 −0.975964 −0.487982 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) 0 0
\(381\) −16.0177 −0.820613
\(382\) 44.8328 2.29385
\(383\) 8.94665 0.457153 0.228576 0.973526i \(-0.426593\pi\)
0.228576 + 0.973526i \(0.426593\pi\)
\(384\) −2.49458 −0.127301
\(385\) 0 0
\(386\) 7.09017 0.360880
\(387\) 11.1803 0.568329
\(388\) −95.1048 −4.82821
\(389\) 32.2361 1.63443 0.817217 0.576330i \(-0.195516\pi\)
0.817217 + 0.576330i \(0.195516\pi\)
\(390\) 0 0
\(391\) 8.89794 0.449988
\(392\) 0 0
\(393\) 4.94427 0.249406
\(394\) 31.2705 1.57539
\(395\) 0 0
\(396\) −24.2705 −1.21964
\(397\) 9.89949 0.496841 0.248421 0.968652i \(-0.420088\pi\)
0.248421 + 0.968652i \(0.420088\pi\)
\(398\) 41.2672 2.06854
\(399\) 0 0
\(400\) 0 0
\(401\) −35.0689 −1.75126 −0.875628 0.482986i \(-0.839552\pi\)
−0.875628 + 0.482986i \(0.839552\pi\)
\(402\) 28.2055 1.40676
\(403\) −14.2918 −0.711925
\(404\) 7.86629 0.391362
\(405\) 0 0
\(406\) 0 0
\(407\) 6.05573 0.300171
\(408\) −18.4721 −0.914507
\(409\) 23.2163 1.14797 0.573986 0.818865i \(-0.305396\pi\)
0.573986 + 0.818865i \(0.305396\pi\)
\(410\) 0 0
\(411\) 13.7295 0.677225
\(412\) −46.0501 −2.26872
\(413\) 0 0
\(414\) 48.2148 2.36963
\(415\) 0 0
\(416\) 59.1605 2.90058
\(417\) 23.1246 1.13242
\(418\) −24.8369 −1.21481
\(419\) 33.3221 1.62789 0.813946 0.580940i \(-0.197315\pi\)
0.813946 + 0.580940i \(0.197315\pi\)
\(420\) 0 0
\(421\) −22.7082 −1.10673 −0.553365 0.832939i \(-0.686657\pi\)
−0.553365 + 0.832939i \(0.686657\pi\)
\(422\) −50.8328 −2.47450
\(423\) 21.2132 1.03142
\(424\) −27.7082 −1.34563
\(425\) 0 0
\(426\) −8.81913 −0.427288
\(427\) 0 0
\(428\) 29.1246 1.40779
\(429\) 27.8885 1.34647
\(430\) 0 0
\(431\) 11.2361 0.541222 0.270611 0.962689i \(-0.412774\pi\)
0.270611 + 0.962689i \(0.412774\pi\)
\(432\) 17.2256 0.828767
\(433\) 24.2179 1.16384 0.581918 0.813247i \(-0.302302\pi\)
0.581918 + 0.813247i \(0.302302\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.85410 −0.232469
\(437\) 34.9427 1.67153
\(438\) 41.1246 1.96501
\(439\) −10.6947 −0.510431 −0.255215 0.966884i \(-0.582146\pi\)
−0.255215 + 0.966884i \(0.582146\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 15.4164 0.733284
\(443\) −2.18034 −0.103591 −0.0517955 0.998658i \(-0.516494\pi\)
−0.0517955 + 0.998658i \(0.516494\pi\)
\(444\) 30.0810 1.42758
\(445\) 0 0
\(446\) −40.1869 −1.90290
\(447\) 49.5462 2.34345
\(448\) 0 0
\(449\) 36.5967 1.72711 0.863554 0.504257i \(-0.168234\pi\)
0.863554 + 0.504257i \(0.168234\pi\)
\(450\) 0 0
\(451\) 17.7658 0.836558
\(452\) 69.1033 3.25035
\(453\) 42.1413 1.97997
\(454\) −19.5927 −0.919529
\(455\) 0 0
\(456\) −72.5410 −3.39704
\(457\) 20.7082 0.968689 0.484344 0.874877i \(-0.339058\pi\)
0.484344 + 0.874877i \(0.339058\pi\)
\(458\) −18.5123 −0.865023
\(459\) 1.88854 0.0881497
\(460\) 0 0
\(461\) −30.2387 −1.40836 −0.704178 0.710024i \(-0.748684\pi\)
−0.704178 + 0.710024i \(0.748684\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) −22.0344 −1.02292
\(465\) 0 0
\(466\) 35.2705 1.63387
\(467\) −41.2672 −1.90962 −0.954810 0.297217i \(-0.903941\pi\)
−0.954810 + 0.297217i \(0.903941\pi\)
\(468\) 59.1605 2.73470
\(469\) 0 0
\(470\) 0 0
\(471\) −9.23607 −0.425576
\(472\) 66.8506 3.07705
\(473\) −11.1803 −0.514073
\(474\) −52.1683 −2.39617
\(475\) 0 0
\(476\) 0 0
\(477\) −8.29180 −0.379655
\(478\) −29.1246 −1.33213
\(479\) −12.1877 −0.556872 −0.278436 0.960455i \(-0.589816\pi\)
−0.278436 + 0.960455i \(0.589816\pi\)
\(480\) 0 0
\(481\) −14.7611 −0.673050
\(482\) 60.7811 2.76850
\(483\) 0 0
\(484\) −29.1246 −1.32385
\(485\) 0 0
\(486\) 50.4202 2.28711
\(487\) 15.2918 0.692937 0.346469 0.938062i \(-0.387381\pi\)
0.346469 + 0.938062i \(0.387381\pi\)
\(488\) −93.5630 −4.23540
\(489\) 35.9442 1.62545
\(490\) 0 0
\(491\) −1.47214 −0.0664366 −0.0332183 0.999448i \(-0.510576\pi\)
−0.0332183 + 0.999448i \(0.510576\pi\)
\(492\) 88.2492 3.97858
\(493\) −2.41577 −0.108801
\(494\) 60.5410 2.72387
\(495\) 0 0
\(496\) −25.8384 −1.16018
\(497\) 0 0
\(498\) 15.7082 0.703901
\(499\) 1.41641 0.0634071 0.0317036 0.999497i \(-0.489907\pi\)
0.0317036 + 0.999497i \(0.489907\pi\)
\(500\) 0 0
\(501\) 3.52786 0.157613
\(502\) 52.5021 2.34328
\(503\) −8.40647 −0.374826 −0.187413 0.982281i \(-0.560010\pi\)
−0.187413 + 0.982281i \(0.560010\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −48.2148 −2.14341
\(507\) −38.2325 −1.69796
\(508\) 33.9787 1.50756
\(509\) −10.0270 −0.444440 −0.222220 0.974997i \(-0.571330\pi\)
−0.222220 + 0.974997i \(0.571330\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −40.3050 −1.78124
\(513\) 7.41641 0.327442
\(514\) −42.8090 −1.88822
\(515\) 0 0
\(516\) −55.5369 −2.44488
\(517\) −21.2132 −0.932956
\(518\) 0 0
\(519\) −9.70820 −0.426143
\(520\) 0 0
\(521\) 23.3739 1.02403 0.512015 0.858976i \(-0.328899\pi\)
0.512015 + 0.858976i \(0.328899\pi\)
\(522\) −13.0902 −0.572941
\(523\) −22.2148 −0.971383 −0.485692 0.874130i \(-0.661432\pi\)
−0.485692 + 0.874130i \(0.661432\pi\)
\(524\) −10.4884 −0.458187
\(525\) 0 0
\(526\) 31.5623 1.37618
\(527\) −2.83282 −0.123399
\(528\) 50.4202 2.19426
\(529\) 44.8328 1.94925
\(530\) 0 0
\(531\) 20.0053 0.868157
\(532\) 0 0
\(533\) −43.3050 −1.87575
\(534\) −28.6525 −1.23991
\(535\) 0 0
\(536\) −35.1803 −1.51956
\(537\) 22.2148 0.958637
\(538\) −15.3500 −0.661786
\(539\) 0 0
\(540\) 0 0
\(541\) 4.70820 0.202421 0.101211 0.994865i \(-0.467728\pi\)
0.101211 + 0.994865i \(0.467728\pi\)
\(542\) −36.4844 −1.56714
\(543\) −3.70820 −0.159134
\(544\) 11.7264 0.502764
\(545\) 0 0
\(546\) 0 0
\(547\) 29.0000 1.23995 0.619975 0.784621i \(-0.287143\pi\)
0.619975 + 0.784621i \(0.287143\pi\)
\(548\) −29.1246 −1.24414
\(549\) −27.9991 −1.19497
\(550\) 0 0
\(551\) −9.48683 −0.404153
\(552\) −140.821 −5.99374
\(553\) 0 0
\(554\) −62.8328 −2.66951
\(555\) 0 0
\(556\) −49.0547 −2.08038
\(557\) −24.5967 −1.04220 −0.521099 0.853496i \(-0.674478\pi\)
−0.521099 + 0.853496i \(0.674478\pi\)
\(558\) −15.3500 −0.649818
\(559\) 27.2526 1.15266
\(560\) 0 0
\(561\) 5.52786 0.233387
\(562\) 64.3951 2.71634
\(563\) −20.1328 −0.848498 −0.424249 0.905546i \(-0.639462\pi\)
−0.424249 + 0.905546i \(0.639462\pi\)
\(564\) −105.374 −4.43704
\(565\) 0 0
\(566\) −4.78282 −0.201037
\(567\) 0 0
\(568\) 11.0000 0.461550
\(569\) −12.0557 −0.505402 −0.252701 0.967544i \(-0.581319\pi\)
−0.252701 + 0.967544i \(0.581319\pi\)
\(570\) 0 0
\(571\) −0.416408 −0.0174261 −0.00871306 0.999962i \(-0.502773\pi\)
−0.00871306 + 0.999962i \(0.502773\pi\)
\(572\) −59.1605 −2.47363
\(573\) −39.1853 −1.63699
\(574\) 0 0
\(575\) 0 0
\(576\) 19.4721 0.811339
\(577\) −33.7047 −1.40314 −0.701572 0.712598i \(-0.747518\pi\)
−0.701572 + 0.712598i \(0.747518\pi\)
\(578\) −41.4508 −1.72413
\(579\) −6.19704 −0.257540
\(580\) 0 0
\(581\) 0 0
\(582\) 117.374 4.86530
\(583\) 8.29180 0.343411
\(584\) −51.2942 −2.12257
\(585\) 0 0
\(586\) −59.3669 −2.45242
\(587\) 15.3500 0.633563 0.316782 0.948499i \(-0.397398\pi\)
0.316782 + 0.948499i \(0.397398\pi\)
\(588\) 0 0
\(589\) −11.1246 −0.458382
\(590\) 0 0
\(591\) −27.3314 −1.12426
\(592\) −26.6869 −1.09683
\(593\) −11.7264 −0.481544 −0.240772 0.970582i \(-0.577401\pi\)
−0.240772 + 0.970582i \(0.577401\pi\)
\(594\) −10.2333 −0.419879
\(595\) 0 0
\(596\) −105.103 −4.30520
\(597\) −36.0689 −1.47620
\(598\) 117.526 4.80599
\(599\) −29.1803 −1.19228 −0.596138 0.802882i \(-0.703299\pi\)
−0.596138 + 0.802882i \(0.703299\pi\)
\(600\) 0 0
\(601\) 46.4326 1.89403 0.947013 0.321196i \(-0.104085\pi\)
0.947013 + 0.321196i \(0.104085\pi\)
\(602\) 0 0
\(603\) −10.5279 −0.428728
\(604\) −89.3951 −3.63744
\(605\) 0 0
\(606\) −9.70820 −0.394369
\(607\) 21.1831 0.859796 0.429898 0.902878i \(-0.358550\pi\)
0.429898 + 0.902878i \(0.358550\pi\)
\(608\) 46.0501 1.86758
\(609\) 0 0
\(610\) 0 0
\(611\) 51.7082 2.09189
\(612\) 11.7264 0.474010
\(613\) −12.4164 −0.501494 −0.250747 0.968053i \(-0.580676\pi\)
−0.250747 + 0.968053i \(0.580676\pi\)
\(614\) 87.7787 3.54246
\(615\) 0 0
\(616\) 0 0
\(617\) −25.3607 −1.02098 −0.510491 0.859883i \(-0.670536\pi\)
−0.510491 + 0.859883i \(0.670536\pi\)
\(618\) 56.8328 2.28615
\(619\) 19.1800 0.770909 0.385455 0.922727i \(-0.374045\pi\)
0.385455 + 0.922727i \(0.374045\pi\)
\(620\) 0 0
\(621\) 14.3972 0.577739
\(622\) −41.6011 −1.66805
\(623\) 0 0
\(624\) −122.902 −4.92001
\(625\) 0 0
\(626\) 55.5369 2.21970
\(627\) 21.7082 0.866942
\(628\) 19.5927 0.781832
\(629\) −2.92584 −0.116661
\(630\) 0 0
\(631\) 12.1246 0.482673 0.241337 0.970441i \(-0.422414\pi\)
0.241337 + 0.970441i \(0.422414\pi\)
\(632\) 65.0689 2.58830
\(633\) 44.4295 1.76591
\(634\) 15.5623 0.618058
\(635\) 0 0
\(636\) 41.1884 1.63323
\(637\) 0 0
\(638\) 13.0902 0.518245
\(639\) 3.29180 0.130221
\(640\) 0 0
\(641\) −25.4721 −1.00609 −0.503044 0.864261i \(-0.667787\pi\)
−0.503044 + 0.864261i \(0.667787\pi\)
\(642\) −35.9442 −1.41861
\(643\) 3.24109 0.127816 0.0639080 0.997956i \(-0.479644\pi\)
0.0639080 + 0.997956i \(0.479644\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 6.40337 0.251742 0.125871 0.992047i \(-0.459827\pi\)
0.125871 + 0.992047i \(0.459827\pi\)
\(648\) −80.0132 −3.14321
\(649\) −20.0053 −0.785278
\(650\) 0 0
\(651\) 0 0
\(652\) −76.2492 −2.98615
\(653\) −25.4164 −0.994621 −0.497310 0.867573i \(-0.665679\pi\)
−0.497310 + 0.867573i \(0.665679\pi\)
\(654\) 5.99070 0.234255
\(655\) 0 0
\(656\) −78.2918 −3.05678
\(657\) −15.3500 −0.598861
\(658\) 0 0
\(659\) 44.8328 1.74644 0.873219 0.487328i \(-0.162028\pi\)
0.873219 + 0.487328i \(0.162028\pi\)
\(660\) 0 0
\(661\) −36.9458 −1.43702 −0.718512 0.695514i \(-0.755177\pi\)
−0.718512 + 0.695514i \(0.755177\pi\)
\(662\) −38.5066 −1.49660
\(663\) −13.4744 −0.523304
\(664\) −19.5927 −0.760343
\(665\) 0 0
\(666\) −15.8541 −0.614334
\(667\) −18.4164 −0.713086
\(668\) −7.48373 −0.289554
\(669\) 35.1246 1.35800
\(670\) 0 0
\(671\) 27.9991 1.08089
\(672\) 0 0
\(673\) −0.875388 −0.0337437 −0.0168719 0.999858i \(-0.505371\pi\)
−0.0168719 + 0.999858i \(0.505371\pi\)
\(674\) 44.8328 1.72690
\(675\) 0 0
\(676\) 81.1033 3.11936
\(677\) −20.1328 −0.773768 −0.386884 0.922128i \(-0.626449\pi\)
−0.386884 + 0.922128i \(0.626449\pi\)
\(678\) −85.2841 −3.27532
\(679\) 0 0
\(680\) 0 0
\(681\) 17.1246 0.656217
\(682\) 15.3500 0.587783
\(683\) −17.0689 −0.653123 −0.326561 0.945176i \(-0.605890\pi\)
−0.326561 + 0.945176i \(0.605890\pi\)
\(684\) 46.0501 1.76077
\(685\) 0 0
\(686\) 0 0
\(687\) 16.1803 0.617318
\(688\) 49.2705 1.87842
\(689\) −20.2117 −0.770003
\(690\) 0 0
\(691\) −45.6374 −1.73613 −0.868064 0.496452i \(-0.834636\pi\)
−0.868064 + 0.496452i \(0.834636\pi\)
\(692\) 20.5942 0.782874
\(693\) 0 0
\(694\) −9.56231 −0.362980
\(695\) 0 0
\(696\) 38.2325 1.44920
\(697\) −8.58359 −0.325127
\(698\) 47.0516 1.78093
\(699\) −30.8276 −1.16601
\(700\) 0 0
\(701\) 4.58359 0.173120 0.0865599 0.996247i \(-0.472413\pi\)
0.0865599 + 0.996247i \(0.472413\pi\)
\(702\) 24.9443 0.941461
\(703\) −11.4899 −0.433351
\(704\) −19.4721 −0.733884
\(705\) 0 0
\(706\) 42.8090 1.61114
\(707\) 0 0
\(708\) −99.3738 −3.73470
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 19.4721 0.730262
\(712\) 35.7379 1.33933
\(713\) −21.5958 −0.808768
\(714\) 0 0
\(715\) 0 0
\(716\) −47.1246 −1.76113
\(717\) 25.4558 0.950666
\(718\) 46.9787 1.75323
\(719\) −14.2697 −0.532168 −0.266084 0.963950i \(-0.585730\pi\)
−0.266084 + 0.963950i \(0.585730\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2.61803 −0.0974331
\(723\) −53.1246 −1.97573
\(724\) 7.86629 0.292348
\(725\) 0 0
\(726\) 35.9442 1.33402
\(727\) 5.65685 0.209801 0.104901 0.994483i \(-0.466548\pi\)
0.104901 + 0.994483i \(0.466548\pi\)
\(728\) 0 0
\(729\) −11.9443 −0.442380
\(730\) 0 0
\(731\) 5.40182 0.199793
\(732\) 139.082 5.14062
\(733\) −44.4295 −1.64104 −0.820521 0.571617i \(-0.806316\pi\)
−0.820521 + 0.571617i \(0.806316\pi\)
\(734\) −47.5918 −1.75664
\(735\) 0 0
\(736\) 89.3951 3.29515
\(737\) 10.5279 0.387799
\(738\) −46.5114 −1.71211
\(739\) −20.7082 −0.761764 −0.380882 0.924624i \(-0.624380\pi\)
−0.380882 + 0.924624i \(0.624380\pi\)
\(740\) 0 0
\(741\) −52.9148 −1.94387
\(742\) 0 0
\(743\) 20.2918 0.744434 0.372217 0.928146i \(-0.378598\pi\)
0.372217 + 0.928146i \(0.378598\pi\)
\(744\) 44.8328 1.64365
\(745\) 0 0
\(746\) 63.1591 2.31242
\(747\) −5.86319 −0.214523
\(748\) −11.7264 −0.428759
\(749\) 0 0
\(750\) 0 0
\(751\) −26.8328 −0.979143 −0.489572 0.871963i \(-0.662847\pi\)
−0.489572 + 0.871963i \(0.662847\pi\)
\(752\) 93.4842 3.40902
\(753\) −45.8885 −1.67227
\(754\) −31.9079 −1.16202
\(755\) 0 0
\(756\) 0 0
\(757\) 14.4164 0.523973 0.261987 0.965072i \(-0.415622\pi\)
0.261987 + 0.965072i \(0.415622\pi\)
\(758\) −49.7426 −1.80673
\(759\) 42.1413 1.52963
\(760\) 0 0
\(761\) 14.9675 0.542570 0.271285 0.962499i \(-0.412551\pi\)
0.271285 + 0.962499i \(0.412551\pi\)
\(762\) −41.9349 −1.51914
\(763\) 0 0
\(764\) 83.1246 3.00734
\(765\) 0 0
\(766\) 23.4226 0.846294
\(767\) 48.7639 1.76076
\(768\) 33.3221 1.20241
\(769\) 18.1784 0.655532 0.327766 0.944759i \(-0.393704\pi\)
0.327766 + 0.944759i \(0.393704\pi\)
\(770\) 0 0
\(771\) 37.4164 1.34752
\(772\) 13.1459 0.473131
\(773\) 5.86319 0.210884 0.105442 0.994425i \(-0.466374\pi\)
0.105442 + 0.994425i \(0.466374\pi\)
\(774\) 29.2705 1.05211
\(775\) 0 0
\(776\) −146.399 −5.25542
\(777\) 0 0
\(778\) 84.3951 3.02571
\(779\) −33.7082 −1.20772
\(780\) 0 0
\(781\) −3.29180 −0.117790
\(782\) 23.2951 0.833032
\(783\) −3.90879 −0.139689
\(784\) 0 0
\(785\) 0 0
\(786\) 12.9443 0.461707
\(787\) 12.1089 0.431637 0.215818 0.976434i \(-0.430758\pi\)
0.215818 + 0.976434i \(0.430758\pi\)
\(788\) 57.9787 2.06541
\(789\) −27.5865 −0.982104
\(790\) 0 0
\(791\) 0 0
\(792\) −37.3607 −1.32755
\(793\) −68.2492 −2.42360
\(794\) 25.9172 0.919768
\(795\) 0 0
\(796\) 76.5137 2.71195
\(797\) 30.5424 1.08187 0.540934 0.841065i \(-0.318071\pi\)
0.540934 + 0.841065i \(0.318071\pi\)
\(798\) 0 0
\(799\) 10.2492 0.362591
\(800\) 0 0
\(801\) 10.6947 0.377879
\(802\) −91.8115 −3.24198
\(803\) 15.3500 0.541690
\(804\) 52.2958 1.84433
\(805\) 0 0
\(806\) −37.4164 −1.31794
\(807\) 13.4164 0.472280
\(808\) 12.1089 0.425991
\(809\) −29.0689 −1.02201 −0.511004 0.859578i \(-0.670726\pi\)
−0.511004 + 0.859578i \(0.670726\pi\)
\(810\) 0 0
\(811\) 7.07107 0.248299 0.124149 0.992264i \(-0.460380\pi\)
0.124149 + 0.992264i \(0.460380\pi\)
\(812\) 0 0
\(813\) 31.8885 1.11838
\(814\) 15.8541 0.555686
\(815\) 0 0
\(816\) −24.3607 −0.852794
\(817\) 21.2132 0.742156
\(818\) 60.7811 2.12516
\(819\) 0 0
\(820\) 0 0
\(821\) 37.4164 1.30584 0.652921 0.757426i \(-0.273543\pi\)
0.652921 + 0.757426i \(0.273543\pi\)
\(822\) 35.9442 1.25370
\(823\) 37.5410 1.30860 0.654299 0.756236i \(-0.272964\pi\)
0.654299 + 0.756236i \(0.272964\pi\)
\(824\) −70.8869 −2.46946
\(825\) 0 0
\(826\) 0 0
\(827\) 7.47214 0.259832 0.129916 0.991525i \(-0.458529\pi\)
0.129916 + 0.991525i \(0.458529\pi\)
\(828\) 89.3951 3.10670
\(829\) −46.8453 −1.62700 −0.813502 0.581562i \(-0.802442\pi\)
−0.813502 + 0.581562i \(0.802442\pi\)
\(830\) 0 0
\(831\) 54.9179 1.90508
\(832\) 47.4643 1.64553
\(833\) 0 0
\(834\) 60.5410 2.09636
\(835\) 0 0
\(836\) −46.0501 −1.59267
\(837\) −4.58359 −0.158432
\(838\) 87.2385 3.01360
\(839\) −40.8059 −1.40877 −0.704387 0.709816i \(-0.748778\pi\)
−0.704387 + 0.709816i \(0.748778\pi\)
\(840\) 0 0
\(841\) −24.0000 −0.827586
\(842\) −59.4508 −2.04881
\(843\) −56.2834 −1.93850
\(844\) −94.2492 −3.24419
\(845\) 0 0
\(846\) 55.5369 1.90940
\(847\) 0 0
\(848\) −36.5410 −1.25482
\(849\) 4.18034 0.143469
\(850\) 0 0
\(851\) −22.3050 −0.764604
\(852\) −16.3516 −0.560196
\(853\) −3.24109 −0.110973 −0.0554864 0.998459i \(-0.517671\pi\)
−0.0554864 + 0.998459i \(0.517671\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 44.8328 1.53235
\(857\) 17.5107 0.598156 0.299078 0.954229i \(-0.403321\pi\)
0.299078 + 0.954229i \(0.403321\pi\)
\(858\) 73.0132 2.49263
\(859\) −17.1468 −0.585041 −0.292520 0.956259i \(-0.594494\pi\)
−0.292520 + 0.956259i \(0.594494\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 29.4164 1.00193
\(863\) 43.3607 1.47601 0.738007 0.674793i \(-0.235767\pi\)
0.738007 + 0.674793i \(0.235767\pi\)
\(864\) 18.9737 0.645497
\(865\) 0 0
\(866\) 63.4032 2.15453
\(867\) 36.2294 1.23041
\(868\) 0 0
\(869\) −19.4721 −0.660547
\(870\) 0 0
\(871\) −25.6622 −0.869530
\(872\) −7.47214 −0.253038
\(873\) −43.8105 −1.48276
\(874\) 91.4811 3.09439
\(875\) 0 0
\(876\) 76.2492 2.57622
\(877\) −51.7082 −1.74606 −0.873031 0.487665i \(-0.837849\pi\)
−0.873031 + 0.487665i \(0.837849\pi\)
\(878\) −27.9991 −0.944925
\(879\) 51.8885 1.75016
\(880\) 0 0
\(881\) −11.1074 −0.374217 −0.187109 0.982339i \(-0.559912\pi\)
−0.187109 + 0.982339i \(0.559912\pi\)
\(882\) 0 0
\(883\) −19.2918 −0.649221 −0.324610 0.945848i \(-0.605233\pi\)
−0.324610 + 0.945848i \(0.605233\pi\)
\(884\) 28.5836 0.961370
\(885\) 0 0
\(886\) −5.70820 −0.191771
\(887\) −47.1304 −1.58248 −0.791242 0.611503i \(-0.790565\pi\)
−0.791242 + 0.611503i \(0.790565\pi\)
\(888\) 46.3051 1.55390
\(889\) 0 0
\(890\) 0 0
\(891\) 23.9443 0.802163
\(892\) −74.5106 −2.49480
\(893\) 40.2492 1.34689
\(894\) 129.714 4.33827
\(895\) 0 0
\(896\) 0 0
\(897\) −102.721 −3.42977
\(898\) 95.8115 3.19727
\(899\) 5.86319 0.195548
\(900\) 0 0
\(901\) −4.00621 −0.133466
\(902\) 46.5114 1.54866
\(903\) 0 0
\(904\) 106.374 3.53794
\(905\) 0 0
\(906\) 110.327 3.66538
\(907\) 38.8328 1.28942 0.644711 0.764426i \(-0.276978\pi\)
0.644711 + 0.764426i \(0.276978\pi\)
\(908\) −36.3268 −1.20555
\(909\) 3.62365 0.120189
\(910\) 0 0
\(911\) 26.2361 0.869240 0.434620 0.900614i \(-0.356883\pi\)
0.434620 + 0.900614i \(0.356883\pi\)
\(912\) −95.6656 −3.16781
\(913\) 5.86319 0.194043
\(914\) 54.2148 1.79327
\(915\) 0 0
\(916\) −34.3237 −1.13409
\(917\) 0 0
\(918\) 4.94427 0.163185
\(919\) −21.8328 −0.720198 −0.360099 0.932914i \(-0.617257\pi\)
−0.360099 + 0.932914i \(0.617257\pi\)
\(920\) 0 0
\(921\) −76.7214 −2.52805
\(922\) −79.1659 −2.60719
\(923\) 8.02391 0.264110
\(924\) 0 0
\(925\) 0 0
\(926\) −36.6525 −1.20448
\(927\) −21.2132 −0.696733
\(928\) −24.2705 −0.796719
\(929\) 0.382559 0.0125513 0.00627567 0.999980i \(-0.498002\pi\)
0.00627567 + 0.999980i \(0.498002\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 65.3951 2.14209
\(933\) 36.3607 1.19040
\(934\) −108.039 −3.53514
\(935\) 0 0
\(936\) 91.0685 2.97667
\(937\) −37.9774 −1.24067 −0.620334 0.784337i \(-0.713003\pi\)
−0.620334 + 0.784337i \(0.713003\pi\)
\(938\) 0 0
\(939\) −48.5410 −1.58408
\(940\) 0 0
\(941\) −24.8369 −0.809658 −0.404829 0.914392i \(-0.632669\pi\)
−0.404829 + 0.914392i \(0.632669\pi\)
\(942\) −24.1803 −0.787838
\(943\) −65.4364 −2.13090
\(944\) 88.1612 2.86940
\(945\) 0 0
\(946\) −29.2705 −0.951666
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −96.7253 −3.14149
\(949\) −37.4164 −1.21459
\(950\) 0 0
\(951\) −13.6020 −0.441074
\(952\) 0 0
\(953\) 12.5967 0.408049 0.204024 0.978966i \(-0.434598\pi\)
0.204024 + 0.978966i \(0.434598\pi\)
\(954\) −21.7082 −0.702829
\(955\) 0 0
\(956\) −54.0000 −1.74648
\(957\) −11.4412 −0.369842
\(958\) −31.9079 −1.03090
\(959\) 0 0
\(960\) 0 0
\(961\) −24.1246 −0.778213
\(962\) −38.6451 −1.24597
\(963\) 13.4164 0.432338
\(964\) 112.694 3.62964
\(965\) 0 0
\(966\) 0 0
\(967\) 34.2492 1.10138 0.550690 0.834710i \(-0.314365\pi\)
0.550690 + 0.834710i \(0.314365\pi\)
\(968\) −44.8328 −1.44098
\(969\) −10.4884 −0.336935
\(970\) 0 0
\(971\) −12.7279 −0.408458 −0.204229 0.978923i \(-0.565469\pi\)
−0.204229 + 0.978923i \(0.565469\pi\)
\(972\) 93.4842 2.99851
\(973\) 0 0
\(974\) 40.0344 1.28279
\(975\) 0 0
\(976\) −123.389 −3.94959
\(977\) −32.2361 −1.03132 −0.515662 0.856792i \(-0.672454\pi\)
−0.515662 + 0.856792i \(0.672454\pi\)
\(978\) 94.1032 3.00909
\(979\) −10.6947 −0.341805
\(980\) 0 0
\(981\) −2.23607 −0.0713922
\(982\) −3.85410 −0.122989
\(983\) 55.5369 1.77135 0.885676 0.464304i \(-0.153696\pi\)
0.885676 + 0.464304i \(0.153696\pi\)
\(984\) 135.846 4.33061
\(985\) 0 0
\(986\) −6.32456 −0.201415
\(987\) 0 0
\(988\) 112.249 3.57112
\(989\) 41.1803 1.30946
\(990\) 0 0
\(991\) −5.00000 −0.158830 −0.0794151 0.996842i \(-0.525305\pi\)
−0.0794151 + 0.996842i \(0.525305\pi\)
\(992\) −28.4605 −0.903622
\(993\) 33.6560 1.06804
\(994\) 0 0
\(995\) 0 0
\(996\) 29.1246 0.922849
\(997\) −10.7248 −0.339658 −0.169829 0.985474i \(-0.554322\pi\)
−0.169829 + 0.985474i \(0.554322\pi\)
\(998\) 3.70820 0.117381
\(999\) −4.73411 −0.149781
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 1225.2.a.bc.1.3 yes 4
5.2 odd 4 1225.2.b.n.99.8 8
5.3 odd 4 1225.2.b.n.99.1 8
5.4 even 2 1225.2.a.ba.1.2 yes 4
7.6 odd 2 inner 1225.2.a.bc.1.4 yes 4
35.13 even 4 1225.2.b.n.99.2 8
35.27 even 4 1225.2.b.n.99.7 8
35.34 odd 2 1225.2.a.ba.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1225.2.a.ba.1.1 4 35.34 odd 2
1225.2.a.ba.1.2 yes 4 5.4 even 2
1225.2.a.bc.1.3 yes 4 1.1 even 1 trivial
1225.2.a.bc.1.4 yes 4 7.6 odd 2 inner
1225.2.b.n.99.1 8 5.3 odd 4
1225.2.b.n.99.2 8 35.13 even 4
1225.2.b.n.99.7 8 35.27 even 4
1225.2.b.n.99.8 8 5.2 odd 4