Properties

Label 1225.2.a.x.1.1
Level $1225$
Weight $2$
Character 1225.1
Self dual yes
Analytic conductor $9.782$
Analytic rank $0$
Dimension $3$
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] = [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: \(3\)
Coefficient field: 3.3.257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.19869\) 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)
 
\(f(q)\) \(=\) \(q-2.19869 q^{2} +2.83424 q^{3} +2.83424 q^{4} -6.23163 q^{6} -1.83424 q^{8} +5.03293 q^{9} +O(q^{10})\) \(q-2.19869 q^{2} +2.83424 q^{3} +2.83424 q^{4} -6.23163 q^{6} -1.83424 q^{8} +5.03293 q^{9} -2.56314 q^{11} +8.03293 q^{12} -0.563139 q^{13} -1.63555 q^{16} +5.19869 q^{17} -11.0659 q^{18} +0.469795 q^{19} +5.63555 q^{22} +4.03293 q^{23} -5.19869 q^{24} +1.23817 q^{26} +5.76183 q^{27} +2.86718 q^{29} -3.86718 q^{31} +7.26456 q^{32} -7.26456 q^{33} -11.4303 q^{34} +14.2646 q^{36} +2.23163 q^{37} -1.03293 q^{38} -1.59607 q^{39} +3.30404 q^{41} +10.4303 q^{43} -7.26456 q^{44} -8.86718 q^{46} +9.36445 q^{47} -4.63555 q^{48} +14.7344 q^{51} -1.59607 q^{52} +10.7014 q^{53} -12.6685 q^{54} +1.33151 q^{57} -6.30404 q^{58} +3.96052 q^{59} -3.86718 q^{61} +8.50273 q^{62} -12.7014 q^{64} +15.9725 q^{66} -7.66849 q^{67} +14.7344 q^{68} +11.4303 q^{69} -8.73436 q^{71} -9.23163 q^{72} -11.3370 q^{73} -4.90666 q^{74} +1.33151 q^{76} +3.50927 q^{78} +5.15921 q^{79} +1.23163 q^{81} -7.26456 q^{82} +3.13282 q^{83} -22.9330 q^{86} +8.12628 q^{87} +4.70142 q^{88} -5.57514 q^{89} +11.4303 q^{92} -10.9605 q^{93} -20.5895 q^{94} +20.5895 q^{96} -15.2975 q^{97} -12.9001 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} + 3 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{3} + 3 q^{4} - 2 q^{6} + 4 q^{9} - 2 q^{11} + 13 q^{12} + 4 q^{13} - 5 q^{16} + 10 q^{17} - 11 q^{18} - 4 q^{19} + 17 q^{22} + q^{23} - 10 q^{24} + 15 q^{26} + 6 q^{27} - 8 q^{29} + 5 q^{31} - 6 q^{32} + 6 q^{33} - 12 q^{34} + 15 q^{36} - 10 q^{37} + 8 q^{38} + 12 q^{39} - q^{41} + 9 q^{43} + 6 q^{44} - 10 q^{46} + 28 q^{47} - 14 q^{48} + 11 q^{51} + 12 q^{52} + 10 q^{53} - 27 q^{54} + 15 q^{57} - 8 q^{58} - 5 q^{59} + 5 q^{61} + 9 q^{62} - 16 q^{64} + 26 q^{66} - 12 q^{67} + 11 q^{68} + 12 q^{69} + 7 q^{71} - 11 q^{72} - 12 q^{73} - 15 q^{74} + 15 q^{76} + 22 q^{78} - 7 q^{79} - 13 q^{81} + 6 q^{82} + 26 q^{83} - 30 q^{86} + 13 q^{87} - 8 q^{88} - 6 q^{89} + 12 q^{92} - 16 q^{93} - 17 q^{94} + 17 q^{96} - 7 q^{97} - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.19869 −1.55471 −0.777355 0.629062i \(-0.783439\pi\)
−0.777355 + 0.629062i \(0.783439\pi\)
\(3\) 2.83424 1.63635 0.818176 0.574969i \(-0.194986\pi\)
0.818176 + 0.574969i \(0.194986\pi\)
\(4\) 2.83424 1.41712
\(5\) 0 0
\(6\) −6.23163 −2.54405
\(7\) 0 0
\(8\) −1.83424 −0.648503
\(9\) 5.03293 1.67764
\(10\) 0 0
\(11\) −2.56314 −0.772816 −0.386408 0.922328i \(-0.626284\pi\)
−0.386408 + 0.922328i \(0.626284\pi\)
\(12\) 8.03293 2.31891
\(13\) −0.563139 −0.156187 −0.0780934 0.996946i \(-0.524883\pi\)
−0.0780934 + 0.996946i \(0.524883\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.63555 −0.408888
\(17\) 5.19869 1.26087 0.630434 0.776243i \(-0.282877\pi\)
0.630434 + 0.776243i \(0.282877\pi\)
\(18\) −11.0659 −2.60825
\(19\) 0.469795 0.107778 0.0538892 0.998547i \(-0.482838\pi\)
0.0538892 + 0.998547i \(0.482838\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.63555 1.20150
\(23\) 4.03293 0.840925 0.420462 0.907310i \(-0.361868\pi\)
0.420462 + 0.907310i \(0.361868\pi\)
\(24\) −5.19869 −1.06118
\(25\) 0 0
\(26\) 1.23817 0.242825
\(27\) 5.76183 1.10886
\(28\) 0 0
\(29\) 2.86718 0.532422 0.266211 0.963915i \(-0.414228\pi\)
0.266211 + 0.963915i \(0.414228\pi\)
\(30\) 0 0
\(31\) −3.86718 −0.694566 −0.347283 0.937760i \(-0.612896\pi\)
−0.347283 + 0.937760i \(0.612896\pi\)
\(32\) 7.26456 1.28420
\(33\) −7.26456 −1.26460
\(34\) −11.4303 −1.96028
\(35\) 0 0
\(36\) 14.2646 2.37743
\(37\) 2.23163 0.366877 0.183439 0.983031i \(-0.441277\pi\)
0.183439 + 0.983031i \(0.441277\pi\)
\(38\) −1.03293 −0.167564
\(39\) −1.59607 −0.255576
\(40\) 0 0
\(41\) 3.30404 0.516004 0.258002 0.966144i \(-0.416936\pi\)
0.258002 + 0.966144i \(0.416936\pi\)
\(42\) 0 0
\(43\) 10.4303 1.59061 0.795304 0.606211i \(-0.207311\pi\)
0.795304 + 0.606211i \(0.207311\pi\)
\(44\) −7.26456 −1.09517
\(45\) 0 0
\(46\) −8.86718 −1.30739
\(47\) 9.36445 1.36595 0.682973 0.730444i \(-0.260687\pi\)
0.682973 + 0.730444i \(0.260687\pi\)
\(48\) −4.63555 −0.669084
\(49\) 0 0
\(50\) 0 0
\(51\) 14.7344 2.06322
\(52\) −1.59607 −0.221336
\(53\) 10.7014 1.46995 0.734977 0.678092i \(-0.237193\pi\)
0.734977 + 0.678092i \(0.237193\pi\)
\(54\) −12.6685 −1.72396
\(55\) 0 0
\(56\) 0 0
\(57\) 1.33151 0.176363
\(58\) −6.30404 −0.827761
\(59\) 3.96052 0.515616 0.257808 0.966196i \(-0.417000\pi\)
0.257808 + 0.966196i \(0.417000\pi\)
\(60\) 0 0
\(61\) −3.86718 −0.495141 −0.247571 0.968870i \(-0.579632\pi\)
−0.247571 + 0.968870i \(0.579632\pi\)
\(62\) 8.50273 1.07985
\(63\) 0 0
\(64\) −12.7014 −1.58768
\(65\) 0 0
\(66\) 15.9725 1.96608
\(67\) −7.66849 −0.936855 −0.468427 0.883502i \(-0.655179\pi\)
−0.468427 + 0.883502i \(0.655179\pi\)
\(68\) 14.7344 1.78680
\(69\) 11.4303 1.37605
\(70\) 0 0
\(71\) −8.73436 −1.03658 −0.518289 0.855206i \(-0.673431\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(72\) −9.23163 −1.08796
\(73\) −11.3370 −1.32689 −0.663446 0.748224i \(-0.730907\pi\)
−0.663446 + 0.748224i \(0.730907\pi\)
\(74\) −4.90666 −0.570387
\(75\) 0 0
\(76\) 1.33151 0.152735
\(77\) 0 0
\(78\) 3.50927 0.397347
\(79\) 5.15921 0.580457 0.290228 0.956957i \(-0.406269\pi\)
0.290228 + 0.956957i \(0.406269\pi\)
\(80\) 0 0
\(81\) 1.23163 0.136847
\(82\) −7.26456 −0.802236
\(83\) 3.13282 0.343872 0.171936 0.985108i \(-0.444998\pi\)
0.171936 + 0.985108i \(0.444998\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −22.9330 −2.47293
\(87\) 8.12628 0.871229
\(88\) 4.70142 0.501173
\(89\) −5.57514 −0.590964 −0.295482 0.955348i \(-0.595480\pi\)
−0.295482 + 0.955348i \(0.595480\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 11.4303 1.19169
\(93\) −10.9605 −1.13655
\(94\) −20.5895 −2.12365
\(95\) 0 0
\(96\) 20.5895 2.10141
\(97\) −15.2975 −1.55323 −0.776613 0.629978i \(-0.783064\pi\)
−0.776613 + 0.629978i \(0.783064\pi\)
\(98\) 0 0
\(99\) −12.9001 −1.29651
\(100\) 0 0
\(101\) 4.99346 0.496867 0.248434 0.968649i \(-0.420084\pi\)
0.248434 + 0.968649i \(0.420084\pi\)
\(102\) −32.3963 −3.20771
\(103\) 20.0264 1.97326 0.986629 0.162980i \(-0.0521104\pi\)
0.986629 + 0.162980i \(0.0521104\pi\)
\(104\) 1.03293 0.101288
\(105\) 0 0
\(106\) −23.5291 −2.28535
\(107\) −11.3699 −1.09917 −0.549585 0.835438i \(-0.685214\pi\)
−0.549585 + 0.835438i \(0.685214\pi\)
\(108\) 16.3304 1.57140
\(109\) 8.96052 0.858262 0.429131 0.903242i \(-0.358820\pi\)
0.429131 + 0.903242i \(0.358820\pi\)
\(110\) 0 0
\(111\) 6.32497 0.600340
\(112\) 0 0
\(113\) −11.8672 −1.11637 −0.558185 0.829716i \(-0.688502\pi\)
−0.558185 + 0.829716i \(0.688502\pi\)
\(114\) −2.92759 −0.274194
\(115\) 0 0
\(116\) 8.12628 0.754506
\(117\) −2.83424 −0.262026
\(118\) −8.70796 −0.801633
\(119\) 0 0
\(120\) 0 0
\(121\) −4.43032 −0.402756
\(122\) 8.50273 0.769801
\(123\) 9.36445 0.844364
\(124\) −10.9605 −0.984284
\(125\) 0 0
\(126\) 0 0
\(127\) −12.4303 −1.10301 −0.551506 0.834171i \(-0.685947\pi\)
−0.551506 + 0.834171i \(0.685947\pi\)
\(128\) 13.3974 1.18417
\(129\) 29.5621 2.60279
\(130\) 0 0
\(131\) 4.99346 0.436280 0.218140 0.975917i \(-0.430001\pi\)
0.218140 + 0.975917i \(0.430001\pi\)
\(132\) −20.5895 −1.79209
\(133\) 0 0
\(134\) 16.8606 1.45654
\(135\) 0 0
\(136\) −9.53566 −0.817676
\(137\) 0.509273 0.0435102 0.0217551 0.999763i \(-0.493075\pi\)
0.0217551 + 0.999763i \(0.493075\pi\)
\(138\) −25.1317 −2.13936
\(139\) 22.2975 1.89125 0.945624 0.325261i \(-0.105452\pi\)
0.945624 + 0.325261i \(0.105452\pi\)
\(140\) 0 0
\(141\) 26.5411 2.23517
\(142\) 19.2042 1.61158
\(143\) 1.44340 0.120704
\(144\) −8.23163 −0.685969
\(145\) 0 0
\(146\) 24.9265 2.06293
\(147\) 0 0
\(148\) 6.32497 0.519909
\(149\) 10.4698 0.857719 0.428860 0.903371i \(-0.358916\pi\)
0.428860 + 0.903371i \(0.358916\pi\)
\(150\) 0 0
\(151\) 9.99346 0.813256 0.406628 0.913594i \(-0.366705\pi\)
0.406628 + 0.913594i \(0.366705\pi\)
\(152\) −0.861719 −0.0698946
\(153\) 26.1647 2.11529
\(154\) 0 0
\(155\) 0 0
\(156\) −4.52366 −0.362183
\(157\) −6.62355 −0.528617 −0.264308 0.964438i \(-0.585144\pi\)
−0.264308 + 0.964438i \(0.585144\pi\)
\(158\) −11.3435 −0.902442
\(159\) 30.3304 2.40536
\(160\) 0 0
\(161\) 0 0
\(162\) −2.70796 −0.212758
\(163\) −7.21962 −0.565484 −0.282742 0.959196i \(-0.591244\pi\)
−0.282742 + 0.959196i \(0.591244\pi\)
\(164\) 9.36445 0.731241
\(165\) 0 0
\(166\) −6.88811 −0.534621
\(167\) 21.1712 1.63828 0.819139 0.573595i \(-0.194452\pi\)
0.819139 + 0.573595i \(0.194452\pi\)
\(168\) 0 0
\(169\) −12.6829 −0.975606
\(170\) 0 0
\(171\) 2.36445 0.180814
\(172\) 29.5621 2.25409
\(173\) 7.46980 0.567918 0.283959 0.958836i \(-0.408352\pi\)
0.283959 + 0.958836i \(0.408352\pi\)
\(174\) −17.8672 −1.35451
\(175\) 0 0
\(176\) 4.19215 0.315995
\(177\) 11.2251 0.843729
\(178\) 12.2580 0.918777
\(179\) −0.728896 −0.0544803 −0.0272401 0.999629i \(-0.508672\pi\)
−0.0272401 + 0.999629i \(0.508672\pi\)
\(180\) 0 0
\(181\) −2.00654 −0.149145 −0.0745726 0.997216i \(-0.523759\pi\)
−0.0745726 + 0.997216i \(0.523759\pi\)
\(182\) 0 0
\(183\) −10.9605 −0.810225
\(184\) −7.39738 −0.545342
\(185\) 0 0
\(186\) 24.0988 1.76701
\(187\) −13.3250 −0.974418
\(188\) 26.5411 1.93571
\(189\) 0 0
\(190\) 0 0
\(191\) 2.12628 0.153852 0.0769261 0.997037i \(-0.475489\pi\)
0.0769261 + 0.997037i \(0.475489\pi\)
\(192\) −35.9989 −2.59800
\(193\) −2.64101 −0.190104 −0.0950521 0.995472i \(-0.530302\pi\)
−0.0950521 + 0.995472i \(0.530302\pi\)
\(194\) 33.6345 2.41481
\(195\) 0 0
\(196\) 0 0
\(197\) 6.17122 0.439681 0.219840 0.975536i \(-0.429446\pi\)
0.219840 + 0.975536i \(0.429446\pi\)
\(198\) 28.3634 2.01570
\(199\) −22.7673 −1.61393 −0.806965 0.590599i \(-0.798892\pi\)
−0.806965 + 0.590599i \(0.798892\pi\)
\(200\) 0 0
\(201\) −21.7344 −1.53302
\(202\) −10.9791 −0.772485
\(203\) 0 0
\(204\) 41.7607 2.92384
\(205\) 0 0
\(206\) −44.0318 −3.06784
\(207\) 20.2975 1.41077
\(208\) 0.921044 0.0638629
\(209\) −1.20415 −0.0832928
\(210\) 0 0
\(211\) −15.8672 −1.09234 −0.546171 0.837674i \(-0.683915\pi\)
−0.546171 + 0.837674i \(0.683915\pi\)
\(212\) 30.3304 2.08310
\(213\) −24.7553 −1.69620
\(214\) 24.9989 1.70889
\(215\) 0 0
\(216\) −10.5686 −0.719102
\(217\) 0 0
\(218\) −19.7014 −1.33435
\(219\) −32.1317 −2.17126
\(220\) 0 0
\(221\) −2.92759 −0.196931
\(222\) −13.9067 −0.933354
\(223\) 7.56314 0.506465 0.253233 0.967405i \(-0.418506\pi\)
0.253233 + 0.967405i \(0.418506\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 26.0923 1.73563
\(227\) 13.3250 0.884409 0.442205 0.896914i \(-0.354196\pi\)
0.442205 + 0.896914i \(0.354196\pi\)
\(228\) 3.77383 0.249928
\(229\) −12.8212 −0.847246 −0.423623 0.905839i \(-0.639242\pi\)
−0.423623 + 0.905839i \(0.639242\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5.25910 −0.345277
\(233\) −20.6135 −1.35044 −0.675219 0.737617i \(-0.735951\pi\)
−0.675219 + 0.737617i \(0.735951\pi\)
\(234\) 6.23163 0.407374
\(235\) 0 0
\(236\) 11.2251 0.730691
\(237\) 14.6225 0.949831
\(238\) 0 0
\(239\) −15.4753 −1.00101 −0.500505 0.865733i \(-0.666852\pi\)
−0.500505 + 0.865733i \(0.666852\pi\)
\(240\) 0 0
\(241\) 11.0384 0.711045 0.355523 0.934668i \(-0.384303\pi\)
0.355523 + 0.934668i \(0.384303\pi\)
\(242\) 9.74090 0.626169
\(243\) −13.7948 −0.884935
\(244\) −10.9605 −0.701676
\(245\) 0 0
\(246\) −20.5895 −1.31274
\(247\) −0.264560 −0.0168336
\(248\) 7.09334 0.450428
\(249\) 8.87918 0.562695
\(250\) 0 0
\(251\) −27.8541 −1.75813 −0.879067 0.476698i \(-0.841834\pi\)
−0.879067 + 0.476698i \(0.841834\pi\)
\(252\) 0 0
\(253\) −10.3370 −0.649880
\(254\) 27.3304 1.71486
\(255\) 0 0
\(256\) −4.05387 −0.253367
\(257\) 14.5291 0.906302 0.453151 0.891434i \(-0.350300\pi\)
0.453151 + 0.891434i \(0.350300\pi\)
\(258\) −64.9978 −4.04659
\(259\) 0 0
\(260\) 0 0
\(261\) 14.4303 0.893214
\(262\) −10.9791 −0.678289
\(263\) −0.292035 −0.0180077 −0.00900384 0.999959i \(-0.502866\pi\)
−0.00900384 + 0.999959i \(0.502866\pi\)
\(264\) 13.3250 0.820095
\(265\) 0 0
\(266\) 0 0
\(267\) −15.8013 −0.967024
\(268\) −21.7344 −1.32764
\(269\) 8.53675 0.520495 0.260247 0.965542i \(-0.416196\pi\)
0.260247 + 0.965542i \(0.416196\pi\)
\(270\) 0 0
\(271\) 2.91211 0.176898 0.0884492 0.996081i \(-0.471809\pi\)
0.0884492 + 0.996081i \(0.471809\pi\)
\(272\) −8.50273 −0.515554
\(273\) 0 0
\(274\) −1.11973 −0.0676457
\(275\) 0 0
\(276\) 32.3963 1.95003
\(277\) −1.83970 −0.110537 −0.0552685 0.998472i \(-0.517601\pi\)
−0.0552685 + 0.998472i \(0.517601\pi\)
\(278\) −49.0253 −2.94034
\(279\) −19.4633 −1.16523
\(280\) 0 0
\(281\) −0.608077 −0.0362748 −0.0181374 0.999836i \(-0.505774\pi\)
−0.0181374 + 0.999836i \(0.505774\pi\)
\(282\) −58.3557 −3.47503
\(283\) 11.6949 0.695188 0.347594 0.937645i \(-0.386999\pi\)
0.347594 + 0.937645i \(0.386999\pi\)
\(284\) −24.7553 −1.46896
\(285\) 0 0
\(286\) −3.17360 −0.187659
\(287\) 0 0
\(288\) 36.5621 2.15444
\(289\) 10.0264 0.589788
\(290\) 0 0
\(291\) −43.3568 −2.54162
\(292\) −32.1317 −1.88037
\(293\) −28.0253 −1.63726 −0.818628 0.574324i \(-0.805265\pi\)
−0.818628 + 0.574324i \(0.805265\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −4.09334 −0.237921
\(297\) −14.7684 −0.856948
\(298\) −23.0198 −1.33350
\(299\) −2.27110 −0.131341
\(300\) 0 0
\(301\) 0 0
\(302\) −21.9725 −1.26438
\(303\) 14.1527 0.813050
\(304\) −0.768374 −0.0440693
\(305\) 0 0
\(306\) −57.5280 −3.28866
\(307\) −5.55660 −0.317132 −0.158566 0.987348i \(-0.550687\pi\)
−0.158566 + 0.987348i \(0.550687\pi\)
\(308\) 0 0
\(309\) 56.7597 3.22894
\(310\) 0 0
\(311\) 6.60808 0.374710 0.187355 0.982292i \(-0.440009\pi\)
0.187355 + 0.982292i \(0.440009\pi\)
\(312\) 2.92759 0.165742
\(313\) 11.9001 0.672634 0.336317 0.941749i \(-0.390819\pi\)
0.336317 + 0.941749i \(0.390819\pi\)
\(314\) 14.5631 0.821845
\(315\) 0 0
\(316\) 14.6225 0.822578
\(317\) −8.71035 −0.489222 −0.244611 0.969621i \(-0.578660\pi\)
−0.244611 + 0.969621i \(0.578660\pi\)
\(318\) −66.6872 −3.73964
\(319\) −7.34898 −0.411464
\(320\) 0 0
\(321\) −32.2251 −1.79863
\(322\) 0 0
\(323\) 2.44232 0.135894
\(324\) 3.49073 0.193929
\(325\) 0 0
\(326\) 15.8737 0.879164
\(327\) 25.3963 1.40442
\(328\) −6.06041 −0.334630
\(329\) 0 0
\(330\) 0 0
\(331\) −12.4687 −0.685342 −0.342671 0.939455i \(-0.611332\pi\)
−0.342671 + 0.939455i \(0.611332\pi\)
\(332\) 8.87918 0.487308
\(333\) 11.2316 0.615489
\(334\) −46.5490 −2.54705
\(335\) 0 0
\(336\) 0 0
\(337\) 3.61462 0.196901 0.0984505 0.995142i \(-0.468611\pi\)
0.0984505 + 0.995142i \(0.468611\pi\)
\(338\) 27.8857 1.51678
\(339\) −33.6345 −1.82677
\(340\) 0 0
\(341\) 9.91211 0.536771
\(342\) −5.19869 −0.281113
\(343\) 0 0
\(344\) −19.1317 −1.03151
\(345\) 0 0
\(346\) −16.4238 −0.882948
\(347\) −21.8397 −1.17242 −0.586208 0.810160i \(-0.699380\pi\)
−0.586208 + 0.810160i \(0.699380\pi\)
\(348\) 23.0318 1.23464
\(349\) −3.69596 −0.197840 −0.0989201 0.995095i \(-0.531539\pi\)
−0.0989201 + 0.995095i \(0.531539\pi\)
\(350\) 0 0
\(351\) −3.24471 −0.173190
\(352\) −18.6201 −0.992454
\(353\) 31.6434 1.68421 0.842104 0.539315i \(-0.181317\pi\)
0.842104 + 0.539315i \(0.181317\pi\)
\(354\) −24.6805 −1.31175
\(355\) 0 0
\(356\) −15.8013 −0.837468
\(357\) 0 0
\(358\) 1.60262 0.0847010
\(359\) −17.1383 −0.904524 −0.452262 0.891885i \(-0.649383\pi\)
−0.452262 + 0.891885i \(0.649383\pi\)
\(360\) 0 0
\(361\) −18.7793 −0.988384
\(362\) 4.41177 0.231877
\(363\) −12.5566 −0.659050
\(364\) 0 0
\(365\) 0 0
\(366\) 24.0988 1.25966
\(367\) −11.7882 −0.615340 −0.307670 0.951493i \(-0.599549\pi\)
−0.307670 + 0.951493i \(0.599549\pi\)
\(368\) −6.59607 −0.343844
\(369\) 16.6290 0.865672
\(370\) 0 0
\(371\) 0 0
\(372\) −31.0648 −1.61063
\(373\) −9.40393 −0.486917 −0.243458 0.969911i \(-0.578282\pi\)
−0.243458 + 0.969911i \(0.578282\pi\)
\(374\) 29.2975 1.51494
\(375\) 0 0
\(376\) −17.1767 −0.885819
\(377\) −1.61462 −0.0831572
\(378\) 0 0
\(379\) −17.3490 −0.891157 −0.445579 0.895243i \(-0.647002\pi\)
−0.445579 + 0.895243i \(0.647002\pi\)
\(380\) 0 0
\(381\) −35.2305 −1.80492
\(382\) −4.67503 −0.239195
\(383\) −5.82116 −0.297447 −0.148724 0.988879i \(-0.547516\pi\)
−0.148724 + 0.988879i \(0.547516\pi\)
\(384\) 37.9714 1.93772
\(385\) 0 0
\(386\) 5.80677 0.295557
\(387\) 52.4951 2.66848
\(388\) −43.3568 −2.20111
\(389\) −26.6279 −1.35009 −0.675045 0.737777i \(-0.735876\pi\)
−0.675045 + 0.737777i \(0.735876\pi\)
\(390\) 0 0
\(391\) 20.9660 1.06030
\(392\) 0 0
\(393\) 14.1527 0.713908
\(394\) −13.5686 −0.683576
\(395\) 0 0
\(396\) −36.5621 −1.83731
\(397\) −35.2305 −1.76817 −0.884085 0.467326i \(-0.845217\pi\)
−0.884085 + 0.467326i \(0.845217\pi\)
\(398\) 50.0582 2.50919
\(399\) 0 0
\(400\) 0 0
\(401\) −24.4369 −1.22032 −0.610159 0.792279i \(-0.708895\pi\)
−0.610159 + 0.792279i \(0.708895\pi\)
\(402\) 47.7871 2.38341
\(403\) 2.17776 0.108482
\(404\) 14.1527 0.704122
\(405\) 0 0
\(406\) 0 0
\(407\) −5.71997 −0.283528
\(408\) −27.0264 −1.33801
\(409\) 29.4873 1.45805 0.729026 0.684487i \(-0.239974\pi\)
0.729026 + 0.684487i \(0.239974\pi\)
\(410\) 0 0
\(411\) 1.44340 0.0711979
\(412\) 56.7597 2.79635
\(413\) 0 0
\(414\) −44.6279 −2.19334
\(415\) 0 0
\(416\) −4.09096 −0.200576
\(417\) 63.1965 3.09475
\(418\) 2.64755 0.129496
\(419\) 9.74090 0.475874 0.237937 0.971281i \(-0.423529\pi\)
0.237937 + 0.971281i \(0.423529\pi\)
\(420\) 0 0
\(421\) −2.68942 −0.131074 −0.0655371 0.997850i \(-0.520876\pi\)
−0.0655371 + 0.997850i \(0.520876\pi\)
\(422\) 34.8870 1.69827
\(423\) 47.1307 2.29157
\(424\) −19.6290 −0.953269
\(425\) 0 0
\(426\) 54.4292 2.63710
\(427\) 0 0
\(428\) −32.2251 −1.55766
\(429\) 4.09096 0.197513
\(430\) 0 0
\(431\) −9.43686 −0.454558 −0.227279 0.973830i \(-0.572983\pi\)
−0.227279 + 0.973830i \(0.572983\pi\)
\(432\) −9.42377 −0.453401
\(433\) 20.8541 1.00218 0.501092 0.865394i \(-0.332932\pi\)
0.501092 + 0.865394i \(0.332932\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 25.3963 1.21626
\(437\) 1.89465 0.0906335
\(438\) 70.6478 3.37568
\(439\) −22.5621 −1.07683 −0.538414 0.842680i \(-0.680976\pi\)
−0.538414 + 0.842680i \(0.680976\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.43686 0.306170
\(443\) 30.5961 1.45366 0.726832 0.686816i \(-0.240992\pi\)
0.726832 + 0.686816i \(0.240992\pi\)
\(444\) 17.9265 0.850754
\(445\) 0 0
\(446\) −16.6290 −0.787406
\(447\) 29.6739 1.40353
\(448\) 0 0
\(449\) 1.25256 0.0591118 0.0295559 0.999563i \(-0.490591\pi\)
0.0295559 + 0.999563i \(0.490591\pi\)
\(450\) 0 0
\(451\) −8.46871 −0.398776
\(452\) −33.6345 −1.58203
\(453\) 28.3239 1.33077
\(454\) −29.2975 −1.37500
\(455\) 0 0
\(456\) −2.44232 −0.114372
\(457\) −23.7618 −1.11153 −0.555766 0.831339i \(-0.687575\pi\)
−0.555766 + 0.831339i \(0.687575\pi\)
\(458\) 28.1898 1.31722
\(459\) 29.9540 1.39813
\(460\) 0 0
\(461\) −31.0833 −1.44770 −0.723848 0.689960i \(-0.757628\pi\)
−0.723848 + 0.689960i \(0.757628\pi\)
\(462\) 0 0
\(463\) 24.8606 1.15537 0.577686 0.816259i \(-0.303956\pi\)
0.577686 + 0.816259i \(0.303956\pi\)
\(464\) −4.68942 −0.217701
\(465\) 0 0
\(466\) 45.3228 2.09954
\(467\) −13.6674 −0.632452 −0.316226 0.948684i \(-0.602416\pi\)
−0.316226 + 0.948684i \(0.602416\pi\)
\(468\) −8.03293 −0.371323
\(469\) 0 0
\(470\) 0 0
\(471\) −18.7727 −0.865003
\(472\) −7.26456 −0.334378
\(473\) −26.7344 −1.22925
\(474\) −32.1503 −1.47671
\(475\) 0 0
\(476\) 0 0
\(477\) 53.8595 2.46606
\(478\) 34.0253 1.55628
\(479\) −4.16576 −0.190338 −0.0951691 0.995461i \(-0.530339\pi\)
−0.0951691 + 0.995461i \(0.530339\pi\)
\(480\) 0 0
\(481\) −1.25672 −0.0573013
\(482\) −24.2700 −1.10547
\(483\) 0 0
\(484\) −12.5566 −0.570754
\(485\) 0 0
\(486\) 30.3304 1.37582
\(487\) −14.1801 −0.642564 −0.321282 0.946984i \(-0.604114\pi\)
−0.321282 + 0.946984i \(0.604114\pi\)
\(488\) 7.09334 0.321101
\(489\) −20.4622 −0.925331
\(490\) 0 0
\(491\) 18.8737 0.851759 0.425880 0.904780i \(-0.359965\pi\)
0.425880 + 0.904780i \(0.359965\pi\)
\(492\) 26.5411 1.19657
\(493\) 14.9056 0.671313
\(494\) 0.581686 0.0261713
\(495\) 0 0
\(496\) 6.32497 0.284000
\(497\) 0 0
\(498\) −19.5226 −0.874828
\(499\) −10.5237 −0.471104 −0.235552 0.971862i \(-0.575690\pi\)
−0.235552 + 0.971862i \(0.575690\pi\)
\(500\) 0 0
\(501\) 60.0044 2.68080
\(502\) 61.2425 2.73339
\(503\) 19.3854 0.864351 0.432176 0.901789i \(-0.357746\pi\)
0.432176 + 0.901789i \(0.357746\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 22.7278 1.01037
\(507\) −35.9463 −1.59643
\(508\) −35.2305 −1.56310
\(509\) 19.0713 0.845322 0.422661 0.906288i \(-0.361096\pi\)
0.422661 + 0.906288i \(0.361096\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −17.8816 −0.790261
\(513\) 2.70688 0.119512
\(514\) −31.9450 −1.40904
\(515\) 0 0
\(516\) 83.7861 3.68848
\(517\) −24.0024 −1.05562
\(518\) 0 0
\(519\) 21.1712 0.929313
\(520\) 0 0
\(521\) −9.64448 −0.422532 −0.211266 0.977429i \(-0.567759\pi\)
−0.211266 + 0.977429i \(0.567759\pi\)
\(522\) −31.7278 −1.38869
\(523\) −30.6135 −1.33864 −0.669318 0.742976i \(-0.733414\pi\)
−0.669318 + 0.742976i \(0.733414\pi\)
\(524\) 14.1527 0.618262
\(525\) 0 0
\(526\) 0.642096 0.0279967
\(527\) −20.1043 −0.875755
\(528\) 11.8816 0.517079
\(529\) −6.73544 −0.292845
\(530\) 0 0
\(531\) 19.9330 0.865021
\(532\) 0 0
\(533\) −1.86063 −0.0805930
\(534\) 34.7422 1.50344
\(535\) 0 0
\(536\) 14.0659 0.607553
\(537\) −2.06587 −0.0891488
\(538\) −18.7697 −0.809218
\(539\) 0 0
\(540\) 0 0
\(541\) −31.6829 −1.36215 −0.681077 0.732212i \(-0.738488\pi\)
−0.681077 + 0.732212i \(0.738488\pi\)
\(542\) −6.40284 −0.275026
\(543\) −5.68703 −0.244054
\(544\) 37.7662 1.61921
\(545\) 0 0
\(546\) 0 0
\(547\) −5.91866 −0.253064 −0.126532 0.991963i \(-0.540385\pi\)
−0.126532 + 0.991963i \(0.540385\pi\)
\(548\) 1.44340 0.0616592
\(549\) −19.4633 −0.830671
\(550\) 0 0
\(551\) 1.34699 0.0573835
\(552\) −20.9660 −0.892371
\(553\) 0 0
\(554\) 4.04494 0.171853
\(555\) 0 0
\(556\) 63.1965 2.68013
\(557\) −31.5806 −1.33811 −0.669057 0.743212i \(-0.733302\pi\)
−0.669057 + 0.743212i \(0.733302\pi\)
\(558\) 42.7937 1.81160
\(559\) −5.87372 −0.248432
\(560\) 0 0
\(561\) −37.7662 −1.59449
\(562\) 1.33697 0.0563968
\(563\) −20.9320 −0.882177 −0.441089 0.897464i \(-0.645408\pi\)
−0.441089 + 0.897464i \(0.645408\pi\)
\(564\) 75.2240 3.16750
\(565\) 0 0
\(566\) −25.7134 −1.08082
\(567\) 0 0
\(568\) 16.0209 0.672223
\(569\) 26.7224 1.12026 0.560130 0.828405i \(-0.310751\pi\)
0.560130 + 0.828405i \(0.310751\pi\)
\(570\) 0 0
\(571\) 23.2844 0.974422 0.487211 0.873284i \(-0.338014\pi\)
0.487211 + 0.873284i \(0.338014\pi\)
\(572\) 4.09096 0.171052
\(573\) 6.02639 0.251756
\(574\) 0 0
\(575\) 0 0
\(576\) −63.9254 −2.66356
\(577\) 14.7937 0.615869 0.307934 0.951408i \(-0.400362\pi\)
0.307934 + 0.951408i \(0.400362\pi\)
\(578\) −22.0449 −0.916949
\(579\) −7.48527 −0.311077
\(580\) 0 0
\(581\) 0 0
\(582\) 95.3283 3.95148
\(583\) −27.4292 −1.13600
\(584\) 20.7948 0.860493
\(585\) 0 0
\(586\) 61.6190 2.54546
\(587\) −34.6334 −1.42947 −0.714736 0.699394i \(-0.753453\pi\)
−0.714736 + 0.699394i \(0.753453\pi\)
\(588\) 0 0
\(589\) −1.81678 −0.0748592
\(590\) 0 0
\(591\) 17.4907 0.719472
\(592\) −3.64994 −0.150012
\(593\) 5.85517 0.240443 0.120222 0.992747i \(-0.461639\pi\)
0.120222 + 0.992747i \(0.461639\pi\)
\(594\) 32.4711 1.33231
\(595\) 0 0
\(596\) 29.6739 1.21549
\(597\) −64.5280 −2.64096
\(598\) 4.99346 0.204198
\(599\) 32.7267 1.33718 0.668589 0.743632i \(-0.266899\pi\)
0.668589 + 0.743632i \(0.266899\pi\)
\(600\) 0 0
\(601\) 26.9485 1.09925 0.549627 0.835410i \(-0.314770\pi\)
0.549627 + 0.835410i \(0.314770\pi\)
\(602\) 0 0
\(603\) −38.5950 −1.57171
\(604\) 28.3239 1.15248
\(605\) 0 0
\(606\) −31.1173 −1.26406
\(607\) 15.4094 0.625448 0.312724 0.949844i \(-0.398759\pi\)
0.312724 + 0.949844i \(0.398759\pi\)
\(608\) 3.41285 0.138410
\(609\) 0 0
\(610\) 0 0
\(611\) −5.27349 −0.213343
\(612\) 74.1570 2.99762
\(613\) 39.8726 1.61044 0.805220 0.592976i \(-0.202047\pi\)
0.805220 + 0.592976i \(0.202047\pi\)
\(614\) 12.2172 0.493048
\(615\) 0 0
\(616\) 0 0
\(617\) 36.7409 1.47913 0.739566 0.673084i \(-0.235031\pi\)
0.739566 + 0.673084i \(0.235031\pi\)
\(618\) −124.797 −5.02007
\(619\) −25.7289 −1.03413 −0.517066 0.855946i \(-0.672976\pi\)
−0.517066 + 0.855946i \(0.672976\pi\)
\(620\) 0 0
\(621\) 23.2371 0.932472
\(622\) −14.5291 −0.582565
\(623\) 0 0
\(624\) 2.61046 0.104502
\(625\) 0 0
\(626\) −26.1647 −1.04575
\(627\) −3.41285 −0.136296
\(628\) −18.7727 −0.749114
\(629\) 11.6015 0.462583
\(630\) 0 0
\(631\) 2.47525 0.0985383 0.0492692 0.998786i \(-0.484311\pi\)
0.0492692 + 0.998786i \(0.484311\pi\)
\(632\) −9.46325 −0.376428
\(633\) −44.9714 −1.78745
\(634\) 19.1514 0.760598
\(635\) 0 0
\(636\) 85.9638 3.40869
\(637\) 0 0
\(638\) 16.1581 0.639706
\(639\) −43.9594 −1.73901
\(640\) 0 0
\(641\) −35.3359 −1.39568 −0.697842 0.716252i \(-0.745856\pi\)
−0.697842 + 0.716252i \(0.745856\pi\)
\(642\) 70.8530 2.79635
\(643\) 2.98691 0.117792 0.0588962 0.998264i \(-0.481242\pi\)
0.0588962 + 0.998264i \(0.481242\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −5.36991 −0.211276
\(647\) 26.0275 1.02325 0.511623 0.859210i \(-0.329045\pi\)
0.511623 + 0.859210i \(0.329045\pi\)
\(648\) −2.25910 −0.0887459
\(649\) −10.1514 −0.398476
\(650\) 0 0
\(651\) 0 0
\(652\) −20.4622 −0.801360
\(653\) 17.8936 0.700229 0.350115 0.936707i \(-0.386143\pi\)
0.350115 + 0.936707i \(0.386143\pi\)
\(654\) −55.8386 −2.18346
\(655\) 0 0
\(656\) −5.40393 −0.210988
\(657\) −57.0582 −2.22605
\(658\) 0 0
\(659\) −15.2076 −0.592405 −0.296202 0.955125i \(-0.595720\pi\)
−0.296202 + 0.955125i \(0.595720\pi\)
\(660\) 0 0
\(661\) −5.48180 −0.213217 −0.106609 0.994301i \(-0.533999\pi\)
−0.106609 + 0.994301i \(0.533999\pi\)
\(662\) 27.4148 1.06551
\(663\) −8.29749 −0.322248
\(664\) −5.74636 −0.223002
\(665\) 0 0
\(666\) −24.6949 −0.956907
\(667\) 11.5631 0.447727
\(668\) 60.0044 2.32164
\(669\) 21.4358 0.828755
\(670\) 0 0
\(671\) 9.91211 0.382653
\(672\) 0 0
\(673\) −30.6763 −1.18249 −0.591243 0.806494i \(-0.701363\pi\)
−0.591243 + 0.806494i \(0.701363\pi\)
\(674\) −7.94743 −0.306124
\(675\) 0 0
\(676\) −35.9463 −1.38255
\(677\) −5.54112 −0.212963 −0.106481 0.994315i \(-0.533958\pi\)
−0.106481 + 0.994315i \(0.533958\pi\)
\(678\) 73.9518 2.84010
\(679\) 0 0
\(680\) 0 0
\(681\) 37.7662 1.44720
\(682\) −21.7937 −0.834523
\(683\) −1.90666 −0.0729562 −0.0364781 0.999334i \(-0.511614\pi\)
−0.0364781 + 0.999334i \(0.511614\pi\)
\(684\) 6.70142 0.256235
\(685\) 0 0
\(686\) 0 0
\(687\) −36.3383 −1.38639
\(688\) −17.0593 −0.650381
\(689\) −6.02639 −0.229587
\(690\) 0 0
\(691\) −0.905571 −0.0344495 −0.0172248 0.999852i \(-0.505483\pi\)
−0.0172248 + 0.999852i \(0.505483\pi\)
\(692\) 21.1712 0.804809
\(693\) 0 0
\(694\) 48.0188 1.82277
\(695\) 0 0
\(696\) −14.9056 −0.564994
\(697\) 17.1767 0.650613
\(698\) 8.12628 0.307584
\(699\) −58.4238 −2.20979
\(700\) 0 0
\(701\) 36.8541 1.39196 0.695980 0.718061i \(-0.254970\pi\)
0.695980 + 0.718061i \(0.254970\pi\)
\(702\) 7.13412 0.269260
\(703\) 1.04841 0.0395414
\(704\) 32.5555 1.22698
\(705\) 0 0
\(706\) −69.5741 −2.61845
\(707\) 0 0
\(708\) 31.8146 1.19567
\(709\) −41.6818 −1.56539 −0.782696 0.622404i \(-0.786156\pi\)
−0.782696 + 0.622404i \(0.786156\pi\)
\(710\) 0 0
\(711\) 25.9660 0.973800
\(712\) 10.2262 0.383242
\(713\) −15.5961 −0.584078
\(714\) 0 0
\(715\) 0 0
\(716\) −2.06587 −0.0772051
\(717\) −43.8606 −1.63801
\(718\) 37.6818 1.40627
\(719\) −44.8595 −1.67298 −0.836489 0.547983i \(-0.815396\pi\)
−0.836489 + 0.547983i \(0.815396\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 41.2899 1.53665
\(723\) 31.2855 1.16352
\(724\) −5.68703 −0.211357
\(725\) 0 0
\(726\) 27.6081 1.02463
\(727\) 9.08134 0.336808 0.168404 0.985718i \(-0.446139\pi\)
0.168404 + 0.985718i \(0.446139\pi\)
\(728\) 0 0
\(729\) −42.7926 −1.58491
\(730\) 0 0
\(731\) 54.2240 2.00555
\(732\) −31.0648 −1.14819
\(733\) 28.4203 1.04973 0.524864 0.851186i \(-0.324116\pi\)
0.524864 + 0.851186i \(0.324116\pi\)
\(734\) 25.9187 0.956675
\(735\) 0 0
\(736\) 29.2975 1.07992
\(737\) 19.6554 0.724016
\(738\) −36.5621 −1.34587
\(739\) 5.41047 0.199027 0.0995137 0.995036i \(-0.468271\pi\)
0.0995137 + 0.995036i \(0.468271\pi\)
\(740\) 0 0
\(741\) −0.749828 −0.0275456
\(742\) 0 0
\(743\) 30.1712 1.10687 0.553437 0.832891i \(-0.313316\pi\)
0.553437 + 0.832891i \(0.313316\pi\)
\(744\) 20.1043 0.737058
\(745\) 0 0
\(746\) 20.6763 0.757014
\(747\) 15.7673 0.576895
\(748\) −37.7662 −1.38087
\(749\) 0 0
\(750\) 0 0
\(751\) 43.9056 1.60214 0.801069 0.598573i \(-0.204265\pi\)
0.801069 + 0.598573i \(0.204265\pi\)
\(752\) −15.3160 −0.558519
\(753\) −78.9453 −2.87693
\(754\) 3.55005 0.129285
\(755\) 0 0
\(756\) 0 0
\(757\) −38.6914 −1.40626 −0.703132 0.711060i \(-0.748216\pi\)
−0.703132 + 0.711060i \(0.748216\pi\)
\(758\) 38.1450 1.38549
\(759\) −29.2975 −1.06343
\(760\) 0 0
\(761\) −31.3293 −1.13569 −0.567844 0.823136i \(-0.692222\pi\)
−0.567844 + 0.823136i \(0.692222\pi\)
\(762\) 77.4611 2.80612
\(763\) 0 0
\(764\) 6.02639 0.218027
\(765\) 0 0
\(766\) 12.7989 0.462444
\(767\) −2.23033 −0.0805324
\(768\) −11.4896 −0.414597
\(769\) 9.00654 0.324784 0.162392 0.986726i \(-0.448079\pi\)
0.162392 + 0.986726i \(0.448079\pi\)
\(770\) 0 0
\(771\) 41.1791 1.48303
\(772\) −7.48527 −0.269401
\(773\) −15.7673 −0.567110 −0.283555 0.958956i \(-0.591514\pi\)
−0.283555 + 0.958956i \(0.591514\pi\)
\(774\) −115.421 −4.14870
\(775\) 0 0
\(776\) 28.0593 1.00727
\(777\) 0 0
\(778\) 58.5466 2.09900
\(779\) 1.55222 0.0556141
\(780\) 0 0
\(781\) 22.3874 0.801083
\(782\) −46.0977 −1.64845
\(783\) 16.5202 0.590383
\(784\) 0 0
\(785\) 0 0
\(786\) −31.1173 −1.10992
\(787\) 35.6225 1.26980 0.634902 0.772593i \(-0.281041\pi\)
0.634902 + 0.772593i \(0.281041\pi\)
\(788\) 17.4907 0.623081
\(789\) −0.827699 −0.0294669
\(790\) 0 0
\(791\) 0 0
\(792\) 23.6619 0.840791
\(793\) 2.17776 0.0773345
\(794\) 77.4611 2.74899
\(795\) 0 0
\(796\) −64.5280 −2.28714
\(797\) 39.6268 1.40365 0.701827 0.712347i \(-0.252368\pi\)
0.701827 + 0.712347i \(0.252368\pi\)
\(798\) 0 0
\(799\) 48.6829 1.72228
\(800\) 0 0
\(801\) −28.0593 −0.991428
\(802\) 53.7291 1.89724
\(803\) 29.0582 1.02544
\(804\) −61.6004 −2.17248
\(805\) 0 0
\(806\) −4.78822 −0.168658
\(807\) 24.1952 0.851712
\(808\) −9.15921 −0.322220
\(809\) −6.82531 −0.239965 −0.119983 0.992776i \(-0.538284\pi\)
−0.119983 + 0.992776i \(0.538284\pi\)
\(810\) 0 0
\(811\) 46.4303 1.63039 0.815194 0.579187i \(-0.196630\pi\)
0.815194 + 0.579187i \(0.196630\pi\)
\(812\) 0 0
\(813\) 8.25364 0.289468
\(814\) 12.5764 0.440804
\(815\) 0 0
\(816\) −24.0988 −0.843627
\(817\) 4.90011 0.171433
\(818\) −64.8334 −2.26685
\(819\) 0 0
\(820\) 0 0
\(821\) 38.4491 1.34188 0.670941 0.741511i \(-0.265890\pi\)
0.670941 + 0.741511i \(0.265890\pi\)
\(822\) −3.17360 −0.110692
\(823\) 22.5939 0.787574 0.393787 0.919202i \(-0.371165\pi\)
0.393787 + 0.919202i \(0.371165\pi\)
\(824\) −36.7333 −1.27966
\(825\) 0 0
\(826\) 0 0
\(827\) −15.2460 −0.530156 −0.265078 0.964227i \(-0.585398\pi\)
−0.265078 + 0.964227i \(0.585398\pi\)
\(828\) 57.5280 1.99924
\(829\) 12.9671 0.450365 0.225182 0.974317i \(-0.427702\pi\)
0.225182 + 0.974317i \(0.427702\pi\)
\(830\) 0 0
\(831\) −5.21416 −0.180877
\(832\) 7.15267 0.247974
\(833\) 0 0
\(834\) −138.950 −4.81143
\(835\) 0 0
\(836\) −3.41285 −0.118036
\(837\) −22.2820 −0.770179
\(838\) −21.4172 −0.739846
\(839\) 19.4818 0.672586 0.336293 0.941757i \(-0.390827\pi\)
0.336293 + 0.941757i \(0.390827\pi\)
\(840\) 0 0
\(841\) −20.7793 −0.716527
\(842\) 5.91320 0.203782
\(843\) −1.72344 −0.0593583
\(844\) −44.9714 −1.54798
\(845\) 0 0
\(846\) −103.626 −3.56273
\(847\) 0 0
\(848\) −17.5027 −0.601046
\(849\) 33.1461 1.13757
\(850\) 0 0
\(851\) 9.00000 0.308516
\(852\) −70.1625 −2.40373
\(853\) −20.0449 −0.686326 −0.343163 0.939276i \(-0.611498\pi\)
−0.343163 + 0.939276i \(0.611498\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 20.8552 0.712815
\(857\) −50.3383 −1.71952 −0.859761 0.510696i \(-0.829388\pi\)
−0.859761 + 0.510696i \(0.829388\pi\)
\(858\) −8.99476 −0.307076
\(859\) 38.1396 1.30131 0.650653 0.759375i \(-0.274495\pi\)
0.650653 + 0.759375i \(0.274495\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 20.7487 0.706705
\(863\) −0.864103 −0.0294144 −0.0147072 0.999892i \(-0.504682\pi\)
−0.0147072 + 0.999892i \(0.504682\pi\)
\(864\) 41.8572 1.42401
\(865\) 0 0
\(866\) −45.8517 −1.55810
\(867\) 28.4172 0.965100
\(868\) 0 0
\(869\) −13.2238 −0.448586
\(870\) 0 0
\(871\) 4.31843 0.146324
\(872\) −16.4358 −0.556586
\(873\) −76.9913 −2.60576
\(874\) −4.16576 −0.140909
\(875\) 0 0
\(876\) −91.0692 −3.07694
\(877\) 9.31297 0.314477 0.157238 0.987561i \(-0.449741\pi\)
0.157238 + 0.987561i \(0.449741\pi\)
\(878\) 49.6070 1.67415
\(879\) −79.4305 −2.67913
\(880\) 0 0
\(881\) −19.4602 −0.655630 −0.327815 0.944742i \(-0.606312\pi\)
−0.327815 + 0.944742i \(0.606312\pi\)
\(882\) 0 0
\(883\) −43.1712 −1.45283 −0.726414 0.687258i \(-0.758814\pi\)
−0.726414 + 0.687258i \(0.758814\pi\)
\(884\) −8.29749 −0.279075
\(885\) 0 0
\(886\) −67.2713 −2.26002
\(887\) 17.4907 0.587281 0.293641 0.955916i \(-0.405133\pi\)
0.293641 + 0.955916i \(0.405133\pi\)
\(888\) −11.6015 −0.389322
\(889\) 0 0
\(890\) 0 0
\(891\) −3.15683 −0.105758
\(892\) 21.4358 0.717723
\(893\) 4.39937 0.147219
\(894\) −65.2438 −2.18208
\(895\) 0 0
\(896\) 0 0
\(897\) −6.43686 −0.214921
\(898\) −2.75399 −0.0919017
\(899\) −11.0879 −0.369802
\(900\) 0 0
\(901\) 55.6334 1.85342
\(902\) 18.6201 0.619981
\(903\) 0 0
\(904\) 21.7673 0.723969
\(905\) 0 0
\(906\) −62.2755 −2.06896
\(907\) −16.7453 −0.556018 −0.278009 0.960579i \(-0.589674\pi\)
−0.278009 + 0.960579i \(0.589674\pi\)
\(908\) 37.7662 1.25332
\(909\) 25.1317 0.833567
\(910\) 0 0
\(911\) −27.9485 −0.925976 −0.462988 0.886365i \(-0.653223\pi\)
−0.462988 + 0.886365i \(0.653223\pi\)
\(912\) −2.17776 −0.0721128
\(913\) −8.02986 −0.265750
\(914\) 52.2449 1.72811
\(915\) 0 0
\(916\) −36.3383 −1.20065
\(917\) 0 0
\(918\) −65.8595 −2.17369
\(919\) 23.2940 0.768399 0.384199 0.923250i \(-0.374477\pi\)
0.384199 + 0.923250i \(0.374477\pi\)
\(920\) 0 0
\(921\) −15.7487 −0.518939
\(922\) 68.3426 2.25075
\(923\) 4.91866 0.161900
\(924\) 0 0
\(925\) 0 0
\(926\) −54.6609 −1.79627
\(927\) 100.792 3.31043
\(928\) 20.8288 0.683738
\(929\) −39.6949 −1.30235 −0.651173 0.758929i \(-0.725723\pi\)
−0.651173 + 0.758929i \(0.725723\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −58.4238 −1.91373
\(933\) 18.7289 0.613157
\(934\) 30.0504 0.983279
\(935\) 0 0
\(936\) 5.19869 0.169925
\(937\) −2.00654 −0.0655509 −0.0327755 0.999463i \(-0.510435\pi\)
−0.0327755 + 0.999463i \(0.510435\pi\)
\(938\) 0 0
\(939\) 33.7278 1.10067
\(940\) 0 0
\(941\) −44.0318 −1.43540 −0.717699 0.696354i \(-0.754804\pi\)
−0.717699 + 0.696354i \(0.754804\pi\)
\(942\) 41.2755 1.34483
\(943\) 13.3250 0.433921
\(944\) −6.47764 −0.210829
\(945\) 0 0
\(946\) 58.7806 1.91112
\(947\) −24.6609 −0.801370 −0.400685 0.916216i \(-0.631228\pi\)
−0.400685 + 0.916216i \(0.631228\pi\)
\(948\) 41.4436 1.34603
\(949\) 6.38429 0.207243
\(950\) 0 0
\(951\) −24.6872 −0.800539
\(952\) 0 0
\(953\) 35.5435 1.15137 0.575684 0.817673i \(-0.304736\pi\)
0.575684 + 0.817673i \(0.304736\pi\)
\(954\) −118.421 −3.83401
\(955\) 0 0
\(956\) −43.8606 −1.41855
\(957\) −20.8288 −0.673299
\(958\) 9.15921 0.295921
\(959\) 0 0
\(960\) 0 0
\(961\) −16.0449 −0.517579
\(962\) 2.76313 0.0890869
\(963\) −57.2240 −1.84402
\(964\) 31.2855 1.00764
\(965\) 0 0
\(966\) 0 0
\(967\) 38.3228 1.23238 0.616189 0.787598i \(-0.288676\pi\)
0.616189 + 0.787598i \(0.288676\pi\)
\(968\) 8.12628 0.261188
\(969\) 6.92213 0.222371
\(970\) 0 0
\(971\) −47.5819 −1.52698 −0.763488 0.645822i \(-0.776515\pi\)
−0.763488 + 0.645822i \(0.776515\pi\)
\(972\) −39.0977 −1.25406
\(973\) 0 0
\(974\) 31.1778 0.999000
\(975\) 0 0
\(976\) 6.32497 0.202457
\(977\) −10.4842 −0.335419 −0.167709 0.985836i \(-0.553637\pi\)
−0.167709 + 0.985836i \(0.553637\pi\)
\(978\) 44.9900 1.43862
\(979\) 14.2899 0.456706
\(980\) 0 0
\(981\) 45.0977 1.43986
\(982\) −41.4975 −1.32424
\(983\) 10.2262 0.326164 0.163082 0.986613i \(-0.447856\pi\)
0.163082 + 0.986613i \(0.447856\pi\)
\(984\) −17.1767 −0.547572
\(985\) 0 0
\(986\) −32.7727 −1.04370
\(987\) 0 0
\(988\) −0.749828 −0.0238552
\(989\) 42.0648 1.33758
\(990\) 0 0
\(991\) 1.27873 0.0406203 0.0203101 0.999794i \(-0.493535\pi\)
0.0203101 + 0.999794i \(0.493535\pi\)
\(992\) −28.0933 −0.891965
\(993\) −35.3394 −1.12146
\(994\) 0 0
\(995\) 0 0
\(996\) 25.1658 0.797408
\(997\) −27.7200 −0.877900 −0.438950 0.898511i \(-0.644650\pi\)
−0.438950 + 0.898511i \(0.644650\pi\)
\(998\) 23.1383 0.732430
\(999\) 12.8582 0.406817
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.x.1.1 3
5.2 odd 4 1225.2.b.l.99.1 6
5.3 odd 4 1225.2.b.l.99.6 6
5.4 even 2 1225.2.a.y.1.3 3
7.2 even 3 175.2.e.e.151.3 yes 6
7.4 even 3 175.2.e.e.51.3 yes 6
7.6 odd 2 1225.2.a.w.1.1 3
35.2 odd 12 175.2.k.b.74.1 12
35.4 even 6 175.2.e.d.51.1 6
35.9 even 6 175.2.e.d.151.1 yes 6
35.13 even 4 1225.2.b.m.99.6 6
35.18 odd 12 175.2.k.b.149.1 12
35.23 odd 12 175.2.k.b.74.6 12
35.27 even 4 1225.2.b.m.99.1 6
35.32 odd 12 175.2.k.b.149.6 12
35.34 odd 2 1225.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.e.d.51.1 6 35.4 even 6
175.2.e.d.151.1 yes 6 35.9 even 6
175.2.e.e.51.3 yes 6 7.4 even 3
175.2.e.e.151.3 yes 6 7.2 even 3
175.2.k.b.74.1 12 35.2 odd 12
175.2.k.b.74.6 12 35.23 odd 12
175.2.k.b.149.1 12 35.18 odd 12
175.2.k.b.149.6 12 35.32 odd 12
1225.2.a.w.1.1 3 7.6 odd 2
1225.2.a.x.1.1 3 1.1 even 1 trivial
1225.2.a.y.1.3 3 5.4 even 2
1225.2.a.z.1.3 3 35.34 odd 2
1225.2.b.l.99.1 6 5.2 odd 4
1225.2.b.l.99.6 6 5.3 odd 4
1225.2.b.m.99.1 6 35.27 even 4
1225.2.b.m.99.6 6 35.13 even 4