Properties

Label 2475.2.a.be.1.3
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,2,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.67513 q^{2} +5.15633 q^{4} -2.80606 q^{7} +8.44358 q^{8} +O(q^{10})\) \(q+2.67513 q^{2} +5.15633 q^{4} -2.80606 q^{7} +8.44358 q^{8} +1.00000 q^{11} +5.11871 q^{13} -7.50659 q^{14} +12.2750 q^{16} +4.54420 q^{17} -4.57452 q^{19} +2.67513 q^{22} +4.00000 q^{23} +13.6932 q^{26} -14.4690 q^{28} +2.38787 q^{29} -0.962389 q^{31} +15.9502 q^{32} +12.1563 q^{34} -1.61213 q^{37} -12.2374 q^{38} +2.38787 q^{41} -2.80606 q^{43} +5.15633 q^{44} +10.7005 q^{46} -4.31265 q^{47} +0.873992 q^{49} +26.3938 q^{52} -6.57452 q^{53} -23.6932 q^{56} +6.38787 q^{58} +13.2750 q^{59} +7.92478 q^{61} -2.57452 q^{62} +18.1187 q^{64} -10.7005 q^{67} +23.4314 q^{68} +7.35026 q^{71} -6.41819 q^{73} -4.31265 q^{74} -23.5877 q^{76} -2.80606 q^{77} +1.35026 q^{79} +6.38787 q^{82} -0.806063 q^{83} -7.50659 q^{86} +8.44358 q^{88} +2.96239 q^{89} -14.3634 q^{91} +20.6253 q^{92} -11.5369 q^{94} +9.92478 q^{97} +2.33804 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8} + 3 q^{11} - 6 q^{13} - 2 q^{14} + 5 q^{16} + 4 q^{17} - 2 q^{19} + 3 q^{22} + 12 q^{23} + 8 q^{26} - 12 q^{28} + 8 q^{29} + 8 q^{31} + 11 q^{32} + 26 q^{34} - 4 q^{37} + 6 q^{38} + 8 q^{41} - 8 q^{43} + 5 q^{44} + 12 q^{46} + 8 q^{47} + 11 q^{49} + 26 q^{52} - 8 q^{53} - 38 q^{56} + 20 q^{58} + 8 q^{59} + 2 q^{61} + 4 q^{62} + 33 q^{64} - 12 q^{67} + 28 q^{68} + 12 q^{71} - 18 q^{73} + 8 q^{74} - 18 q^{76} - 8 q^{77} - 6 q^{79} + 20 q^{82} - 2 q^{83} - 2 q^{86} + 9 q^{88} - 2 q^{89} + 8 q^{91} + 20 q^{92} - 12 q^{94} + 8 q^{97} - 29 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.67513 1.89160 0.945802 0.324745i \(-0.105279\pi\)
0.945802 + 0.324745i \(0.105279\pi\)
\(3\) 0 0
\(4\) 5.15633 2.57816
\(5\) 0 0
\(6\) 0 0
\(7\) −2.80606 −1.06059 −0.530296 0.847812i \(-0.677919\pi\)
−0.530296 + 0.847812i \(0.677919\pi\)
\(8\) 8.44358 2.98526
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.11871 1.41968 0.709838 0.704365i \(-0.248768\pi\)
0.709838 + 0.704365i \(0.248768\pi\)
\(14\) −7.50659 −2.00622
\(15\) 0 0
\(16\) 12.2750 3.06876
\(17\) 4.54420 1.10213 0.551065 0.834462i \(-0.314222\pi\)
0.551065 + 0.834462i \(0.314222\pi\)
\(18\) 0 0
\(19\) −4.57452 −1.04947 −0.524733 0.851267i \(-0.675835\pi\)
−0.524733 + 0.851267i \(0.675835\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.67513 0.570340
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 13.6932 2.68546
\(27\) 0 0
\(28\) −14.4690 −2.73438
\(29\) 2.38787 0.443417 0.221708 0.975113i \(-0.428837\pi\)
0.221708 + 0.975113i \(0.428837\pi\)
\(30\) 0 0
\(31\) −0.962389 −0.172850 −0.0864250 0.996258i \(-0.527544\pi\)
−0.0864250 + 0.996258i \(0.527544\pi\)
\(32\) 15.9502 2.81962
\(33\) 0 0
\(34\) 12.1563 2.08479
\(35\) 0 0
\(36\) 0 0
\(37\) −1.61213 −0.265032 −0.132516 0.991181i \(-0.542306\pi\)
−0.132516 + 0.991181i \(0.542306\pi\)
\(38\) −12.2374 −1.98517
\(39\) 0 0
\(40\) 0 0
\(41\) 2.38787 0.372923 0.186462 0.982462i \(-0.440298\pi\)
0.186462 + 0.982462i \(0.440298\pi\)
\(42\) 0 0
\(43\) −2.80606 −0.427921 −0.213960 0.976842i \(-0.568636\pi\)
−0.213960 + 0.976842i \(0.568636\pi\)
\(44\) 5.15633 0.777345
\(45\) 0 0
\(46\) 10.7005 1.57771
\(47\) −4.31265 −0.629065 −0.314532 0.949247i \(-0.601848\pi\)
−0.314532 + 0.949247i \(0.601848\pi\)
\(48\) 0 0
\(49\) 0.873992 0.124856
\(50\) 0 0
\(51\) 0 0
\(52\) 26.3938 3.66015
\(53\) −6.57452 −0.903079 −0.451540 0.892251i \(-0.649125\pi\)
−0.451540 + 0.892251i \(0.649125\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −23.6932 −3.16614
\(57\) 0 0
\(58\) 6.38787 0.838769
\(59\) 13.2750 1.72826 0.864131 0.503266i \(-0.167868\pi\)
0.864131 + 0.503266i \(0.167868\pi\)
\(60\) 0 0
\(61\) 7.92478 1.01466 0.507332 0.861751i \(-0.330632\pi\)
0.507332 + 0.861751i \(0.330632\pi\)
\(62\) −2.57452 −0.326964
\(63\) 0 0
\(64\) 18.1187 2.26484
\(65\) 0 0
\(66\) 0 0
\(67\) −10.7005 −1.30728 −0.653639 0.756807i \(-0.726758\pi\)
−0.653639 + 0.756807i \(0.726758\pi\)
\(68\) 23.4314 2.84147
\(69\) 0 0
\(70\) 0 0
\(71\) 7.35026 0.872316 0.436158 0.899870i \(-0.356339\pi\)
0.436158 + 0.899870i \(0.356339\pi\)
\(72\) 0 0
\(73\) −6.41819 −0.751192 −0.375596 0.926783i \(-0.622562\pi\)
−0.375596 + 0.926783i \(0.622562\pi\)
\(74\) −4.31265 −0.501335
\(75\) 0 0
\(76\) −23.5877 −2.70569
\(77\) −2.80606 −0.319781
\(78\) 0 0
\(79\) 1.35026 0.151916 0.0759582 0.997111i \(-0.475798\pi\)
0.0759582 + 0.997111i \(0.475798\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.38787 0.705423
\(83\) −0.806063 −0.0884770 −0.0442385 0.999021i \(-0.514086\pi\)
−0.0442385 + 0.999021i \(0.514086\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.50659 −0.809456
\(87\) 0 0
\(88\) 8.44358 0.900089
\(89\) 2.96239 0.314013 0.157006 0.987598i \(-0.449816\pi\)
0.157006 + 0.987598i \(0.449816\pi\)
\(90\) 0 0
\(91\) −14.3634 −1.50570
\(92\) 20.6253 2.15034
\(93\) 0 0
\(94\) −11.5369 −1.18994
\(95\) 0 0
\(96\) 0 0
\(97\) 9.92478 1.00771 0.503854 0.863789i \(-0.331915\pi\)
0.503854 + 0.863789i \(0.331915\pi\)
\(98\) 2.33804 0.236178
\(99\) 0 0
\(100\) 0 0
\(101\) 13.6121 1.35446 0.677229 0.735773i \(-0.263181\pi\)
0.677229 + 0.735773i \(0.263181\pi\)
\(102\) 0 0
\(103\) −16.3127 −1.60733 −0.803667 0.595080i \(-0.797120\pi\)
−0.803667 + 0.595080i \(0.797120\pi\)
\(104\) 43.2203 4.23810
\(105\) 0 0
\(106\) −17.5877 −1.70827
\(107\) −9.43136 −0.911764 −0.455882 0.890040i \(-0.650676\pi\)
−0.455882 + 0.890040i \(0.650676\pi\)
\(108\) 0 0
\(109\) −15.4010 −1.47515 −0.737576 0.675264i \(-0.764030\pi\)
−0.737576 + 0.675264i \(0.764030\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −34.4445 −3.25470
\(113\) −13.7381 −1.29238 −0.646188 0.763179i \(-0.723638\pi\)
−0.646188 + 0.763179i \(0.723638\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 12.3127 1.14320
\(117\) 0 0
\(118\) 35.5125 3.26919
\(119\) −12.7513 −1.16891
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 21.1998 1.91934
\(123\) 0 0
\(124\) −4.96239 −0.445636
\(125\) 0 0
\(126\) 0 0
\(127\) −12.4182 −1.10194 −0.550968 0.834526i \(-0.685741\pi\)
−0.550968 + 0.834526i \(0.685741\pi\)
\(128\) 16.5696 1.46456
\(129\) 0 0
\(130\) 0 0
\(131\) −5.92478 −0.517650 −0.258825 0.965924i \(-0.583335\pi\)
−0.258825 + 0.965924i \(0.583335\pi\)
\(132\) 0 0
\(133\) 12.8364 1.11306
\(134\) −28.6253 −2.47285
\(135\) 0 0
\(136\) 38.3693 3.29014
\(137\) 9.79877 0.837165 0.418583 0.908179i \(-0.362527\pi\)
0.418583 + 0.908179i \(0.362527\pi\)
\(138\) 0 0
\(139\) −2.12601 −0.180326 −0.0901628 0.995927i \(-0.528739\pi\)
−0.0901628 + 0.995927i \(0.528739\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 19.6629 1.65007
\(143\) 5.11871 0.428048
\(144\) 0 0
\(145\) 0 0
\(146\) −17.1695 −1.42096
\(147\) 0 0
\(148\) −8.31265 −0.683296
\(149\) −8.31265 −0.680999 −0.340499 0.940245i \(-0.610596\pi\)
−0.340499 + 0.940245i \(0.610596\pi\)
\(150\) 0 0
\(151\) −15.2750 −1.24307 −0.621533 0.783388i \(-0.713490\pi\)
−0.621533 + 0.783388i \(0.713490\pi\)
\(152\) −38.6253 −3.13293
\(153\) 0 0
\(154\) −7.50659 −0.604898
\(155\) 0 0
\(156\) 0 0
\(157\) −7.01317 −0.559712 −0.279856 0.960042i \(-0.590287\pi\)
−0.279856 + 0.960042i \(0.590287\pi\)
\(158\) 3.61213 0.287365
\(159\) 0 0
\(160\) 0 0
\(161\) −11.2243 −0.884595
\(162\) 0 0
\(163\) −6.76116 −0.529575 −0.264787 0.964307i \(-0.585302\pi\)
−0.264787 + 0.964307i \(0.585302\pi\)
\(164\) 12.3127 0.961456
\(165\) 0 0
\(166\) −2.15633 −0.167363
\(167\) −11.8192 −0.914600 −0.457300 0.889312i \(-0.651183\pi\)
−0.457300 + 0.889312i \(0.651183\pi\)
\(168\) 0 0
\(169\) 13.2012 1.01548
\(170\) 0 0
\(171\) 0 0
\(172\) −14.4690 −1.10325
\(173\) 6.99271 0.531646 0.265823 0.964022i \(-0.414356\pi\)
0.265823 + 0.964022i \(0.414356\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 12.2750 0.925266
\(177\) 0 0
\(178\) 7.92478 0.593987
\(179\) 2.70052 0.201847 0.100923 0.994894i \(-0.467820\pi\)
0.100923 + 0.994894i \(0.467820\pi\)
\(180\) 0 0
\(181\) −21.1998 −1.57577 −0.787885 0.615822i \(-0.788824\pi\)
−0.787885 + 0.615822i \(0.788824\pi\)
\(182\) −38.4241 −2.84818
\(183\) 0 0
\(184\) 33.7743 2.48988
\(185\) 0 0
\(186\) 0 0
\(187\) 4.54420 0.332305
\(188\) −22.2374 −1.62183
\(189\) 0 0
\(190\) 0 0
\(191\) −13.1490 −0.951430 −0.475715 0.879599i \(-0.657811\pi\)
−0.475715 + 0.879599i \(0.657811\pi\)
\(192\) 0 0
\(193\) −5.89446 −0.424293 −0.212146 0.977238i \(-0.568045\pi\)
−0.212146 + 0.977238i \(0.568045\pi\)
\(194\) 26.5501 1.90618
\(195\) 0 0
\(196\) 4.50659 0.321899
\(197\) 20.3938 1.45299 0.726497 0.687169i \(-0.241147\pi\)
0.726497 + 0.687169i \(0.241147\pi\)
\(198\) 0 0
\(199\) 17.4010 1.23353 0.616764 0.787148i \(-0.288443\pi\)
0.616764 + 0.787148i \(0.288443\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 36.4142 2.56210
\(203\) −6.70052 −0.470285
\(204\) 0 0
\(205\) 0 0
\(206\) −43.6385 −3.04044
\(207\) 0 0
\(208\) 62.8324 4.35664
\(209\) −4.57452 −0.316426
\(210\) 0 0
\(211\) 18.1260 1.24785 0.623923 0.781486i \(-0.285538\pi\)
0.623923 + 0.781486i \(0.285538\pi\)
\(212\) −33.9003 −2.32828
\(213\) 0 0
\(214\) −25.2301 −1.72470
\(215\) 0 0
\(216\) 0 0
\(217\) 2.70052 0.183323
\(218\) −41.1998 −2.79040
\(219\) 0 0
\(220\) 0 0
\(221\) 23.2605 1.56467
\(222\) 0 0
\(223\) −23.6385 −1.58295 −0.791475 0.611202i \(-0.790686\pi\)
−0.791475 + 0.611202i \(0.790686\pi\)
\(224\) −44.7572 −2.99047
\(225\) 0 0
\(226\) −36.7513 −2.44466
\(227\) −1.26916 −0.0842371 −0.0421185 0.999113i \(-0.513411\pi\)
−0.0421185 + 0.999113i \(0.513411\pi\)
\(228\) 0 0
\(229\) 16.1768 1.06899 0.534496 0.845171i \(-0.320501\pi\)
0.534496 + 0.845171i \(0.320501\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 20.1622 1.32371
\(233\) −18.4690 −1.20994 −0.604971 0.796247i \(-0.706815\pi\)
−0.604971 + 0.796247i \(0.706815\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 68.4504 4.45574
\(237\) 0 0
\(238\) −34.1114 −2.21111
\(239\) −19.3258 −1.25008 −0.625042 0.780591i \(-0.714918\pi\)
−0.625042 + 0.780591i \(0.714918\pi\)
\(240\) 0 0
\(241\) 28.5501 1.83907 0.919536 0.393006i \(-0.128565\pi\)
0.919536 + 0.393006i \(0.128565\pi\)
\(242\) 2.67513 0.171964
\(243\) 0 0
\(244\) 40.8627 2.61597
\(245\) 0 0
\(246\) 0 0
\(247\) −23.4156 −1.48990
\(248\) −8.12601 −0.516002
\(249\) 0 0
\(250\) 0 0
\(251\) −23.1998 −1.46436 −0.732180 0.681112i \(-0.761497\pi\)
−0.732180 + 0.681112i \(0.761497\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −33.2203 −2.08443
\(255\) 0 0
\(256\) 8.08840 0.505525
\(257\) −10.8872 −0.679123 −0.339561 0.940584i \(-0.610279\pi\)
−0.339561 + 0.940584i \(0.610279\pi\)
\(258\) 0 0
\(259\) 4.52373 0.281091
\(260\) 0 0
\(261\) 0 0
\(262\) −15.8496 −0.979189
\(263\) 9.11871 0.562284 0.281142 0.959666i \(-0.409287\pi\)
0.281142 + 0.959666i \(0.409287\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 34.3390 2.10546
\(267\) 0 0
\(268\) −55.1754 −3.37037
\(269\) 11.2995 0.688941 0.344471 0.938797i \(-0.388058\pi\)
0.344471 + 0.938797i \(0.388058\pi\)
\(270\) 0 0
\(271\) 25.1998 1.53078 0.765390 0.643567i \(-0.222546\pi\)
0.765390 + 0.643567i \(0.222546\pi\)
\(272\) 55.7802 3.38217
\(273\) 0 0
\(274\) 26.2130 1.58358
\(275\) 0 0
\(276\) 0 0
\(277\) 2.41819 0.145295 0.0726475 0.997358i \(-0.476855\pi\)
0.0726475 + 0.997358i \(0.476855\pi\)
\(278\) −5.68735 −0.341105
\(279\) 0 0
\(280\) 0 0
\(281\) −30.4894 −1.81885 −0.909424 0.415870i \(-0.863477\pi\)
−0.909424 + 0.415870i \(0.863477\pi\)
\(282\) 0 0
\(283\) −8.35756 −0.496805 −0.248403 0.968657i \(-0.579906\pi\)
−0.248403 + 0.968657i \(0.579906\pi\)
\(284\) 37.9003 2.24897
\(285\) 0 0
\(286\) 13.6932 0.809698
\(287\) −6.70052 −0.395519
\(288\) 0 0
\(289\) 3.64974 0.214690
\(290\) 0 0
\(291\) 0 0
\(292\) −33.0943 −1.93670
\(293\) −23.0943 −1.34918 −0.674591 0.738192i \(-0.735680\pi\)
−0.674591 + 0.738192i \(0.735680\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −13.6121 −0.791189
\(297\) 0 0
\(298\) −22.2374 −1.28818
\(299\) 20.4749 1.18409
\(300\) 0 0
\(301\) 7.87399 0.453849
\(302\) −40.8627 −2.35139
\(303\) 0 0
\(304\) −56.1524 −3.22056
\(305\) 0 0
\(306\) 0 0
\(307\) −2.65562 −0.151564 −0.0757821 0.997124i \(-0.524145\pi\)
−0.0757821 + 0.997124i \(0.524145\pi\)
\(308\) −14.4690 −0.824446
\(309\) 0 0
\(310\) 0 0
\(311\) 15.9756 0.905891 0.452946 0.891538i \(-0.350373\pi\)
0.452946 + 0.891538i \(0.350373\pi\)
\(312\) 0 0
\(313\) 0.0606343 0.00342726 0.00171363 0.999999i \(-0.499455\pi\)
0.00171363 + 0.999999i \(0.499455\pi\)
\(314\) −18.7612 −1.05875
\(315\) 0 0
\(316\) 6.96239 0.391665
\(317\) 3.81336 0.214180 0.107090 0.994249i \(-0.465847\pi\)
0.107090 + 0.994249i \(0.465847\pi\)
\(318\) 0 0
\(319\) 2.38787 0.133695
\(320\) 0 0
\(321\) 0 0
\(322\) −30.0263 −1.67330
\(323\) −20.7875 −1.15665
\(324\) 0 0
\(325\) 0 0
\(326\) −18.0870 −1.00175
\(327\) 0 0
\(328\) 20.1622 1.11327
\(329\) 12.1016 0.667181
\(330\) 0 0
\(331\) 24.2882 1.33500 0.667500 0.744609i \(-0.267364\pi\)
0.667500 + 0.744609i \(0.267364\pi\)
\(332\) −4.15633 −0.228108
\(333\) 0 0
\(334\) −31.6180 −1.73006
\(335\) 0 0
\(336\) 0 0
\(337\) 22.5804 1.23003 0.615016 0.788514i \(-0.289149\pi\)
0.615016 + 0.788514i \(0.289149\pi\)
\(338\) 35.3150 1.92088
\(339\) 0 0
\(340\) 0 0
\(341\) −0.962389 −0.0521163
\(342\) 0 0
\(343\) 17.1900 0.928171
\(344\) −23.6932 −1.27745
\(345\) 0 0
\(346\) 18.7064 1.00566
\(347\) 20.1925 1.08399 0.541996 0.840381i \(-0.317669\pi\)
0.541996 + 0.840381i \(0.317669\pi\)
\(348\) 0 0
\(349\) −27.2506 −1.45869 −0.729346 0.684145i \(-0.760175\pi\)
−0.729346 + 0.684145i \(0.760175\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 15.9502 0.850147
\(353\) 5.02302 0.267349 0.133674 0.991025i \(-0.457322\pi\)
0.133674 + 0.991025i \(0.457322\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 15.2750 0.809575
\(357\) 0 0
\(358\) 7.22425 0.381814
\(359\) 18.1768 0.959334 0.479667 0.877450i \(-0.340757\pi\)
0.479667 + 0.877450i \(0.340757\pi\)
\(360\) 0 0
\(361\) 1.92619 0.101379
\(362\) −56.7123 −2.98073
\(363\) 0 0
\(364\) −74.0625 −3.88193
\(365\) 0 0
\(366\) 0 0
\(367\) 8.56467 0.447072 0.223536 0.974696i \(-0.428240\pi\)
0.223536 + 0.974696i \(0.428240\pi\)
\(368\) 49.1002 2.55952
\(369\) 0 0
\(370\) 0 0
\(371\) 18.4485 0.957799
\(372\) 0 0
\(373\) −7.81924 −0.404865 −0.202432 0.979296i \(-0.564885\pi\)
−0.202432 + 0.979296i \(0.564885\pi\)
\(374\) 12.1563 0.628589
\(375\) 0 0
\(376\) −36.4142 −1.87792
\(377\) 12.2228 0.629508
\(378\) 0 0
\(379\) 1.14903 0.0590218 0.0295109 0.999564i \(-0.490605\pi\)
0.0295109 + 0.999564i \(0.490605\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −35.1754 −1.79973
\(383\) 34.0870 1.74176 0.870882 0.491493i \(-0.163549\pi\)
0.870882 + 0.491493i \(0.163549\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −15.7685 −0.802593
\(387\) 0 0
\(388\) 51.1754 2.59804
\(389\) −23.7743 −1.20541 −0.602703 0.797965i \(-0.705910\pi\)
−0.602703 + 0.797965i \(0.705910\pi\)
\(390\) 0 0
\(391\) 18.1768 0.919240
\(392\) 7.37962 0.372727
\(393\) 0 0
\(394\) 54.5560 2.74849
\(395\) 0 0
\(396\) 0 0
\(397\) 4.15045 0.208305 0.104152 0.994561i \(-0.466787\pi\)
0.104152 + 0.994561i \(0.466787\pi\)
\(398\) 46.5501 2.33334
\(399\) 0 0
\(400\) 0 0
\(401\) 33.9149 1.69363 0.846815 0.531887i \(-0.178517\pi\)
0.846815 + 0.531887i \(0.178517\pi\)
\(402\) 0 0
\(403\) −4.92619 −0.245391
\(404\) 70.1886 3.49201
\(405\) 0 0
\(406\) −17.9248 −0.889592
\(407\) −1.61213 −0.0799102
\(408\) 0 0
\(409\) 8.55008 0.422774 0.211387 0.977402i \(-0.432202\pi\)
0.211387 + 0.977402i \(0.432202\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −84.1133 −4.14397
\(413\) −37.2506 −1.83298
\(414\) 0 0
\(415\) 0 0
\(416\) 81.6444 4.00294
\(417\) 0 0
\(418\) −12.2374 −0.598552
\(419\) −13.2995 −0.649722 −0.324861 0.945762i \(-0.605318\pi\)
−0.324861 + 0.945762i \(0.605318\pi\)
\(420\) 0 0
\(421\) 33.3014 1.62301 0.811505 0.584345i \(-0.198649\pi\)
0.811505 + 0.584345i \(0.198649\pi\)
\(422\) 48.4894 2.36043
\(423\) 0 0
\(424\) −55.5125 −2.69592
\(425\) 0 0
\(426\) 0 0
\(427\) −22.2374 −1.07614
\(428\) −48.6312 −2.35068
\(429\) 0 0
\(430\) 0 0
\(431\) −2.07522 −0.0999600 −0.0499800 0.998750i \(-0.515916\pi\)
−0.0499800 + 0.998750i \(0.515916\pi\)
\(432\) 0 0
\(433\) 37.4010 1.79738 0.898690 0.438585i \(-0.144520\pi\)
0.898690 + 0.438585i \(0.144520\pi\)
\(434\) 7.22425 0.346775
\(435\) 0 0
\(436\) −79.4128 −3.80318
\(437\) −18.2981 −0.875315
\(438\) 0 0
\(439\) −6.02444 −0.287531 −0.143765 0.989612i \(-0.545921\pi\)
−0.143765 + 0.989612i \(0.545921\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 62.2247 2.95973
\(443\) −10.1359 −0.481569 −0.240785 0.970579i \(-0.577405\pi\)
−0.240785 + 0.970579i \(0.577405\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −63.2360 −2.99431
\(447\) 0 0
\(448\) −50.8423 −2.40207
\(449\) 11.4861 0.542063 0.271032 0.962570i \(-0.412635\pi\)
0.271032 + 0.962570i \(0.412635\pi\)
\(450\) 0 0
\(451\) 2.38787 0.112441
\(452\) −70.8383 −3.33195
\(453\) 0 0
\(454\) −3.39517 −0.159343
\(455\) 0 0
\(456\) 0 0
\(457\) 15.0435 0.703705 0.351852 0.936055i \(-0.385552\pi\)
0.351852 + 0.936055i \(0.385552\pi\)
\(458\) 43.2750 2.02211
\(459\) 0 0
\(460\) 0 0
\(461\) 26.2374 1.22200 0.610999 0.791631i \(-0.290768\pi\)
0.610999 + 0.791631i \(0.290768\pi\)
\(462\) 0 0
\(463\) 5.46168 0.253826 0.126913 0.991914i \(-0.459493\pi\)
0.126913 + 0.991914i \(0.459493\pi\)
\(464\) 29.3112 1.36074
\(465\) 0 0
\(466\) −49.4069 −2.28873
\(467\) −2.70052 −0.124965 −0.0624827 0.998046i \(-0.519902\pi\)
−0.0624827 + 0.998046i \(0.519902\pi\)
\(468\) 0 0
\(469\) 30.0263 1.38649
\(470\) 0 0
\(471\) 0 0
\(472\) 112.089 5.15931
\(473\) −2.80606 −0.129023
\(474\) 0 0
\(475\) 0 0
\(476\) −65.7499 −3.01364
\(477\) 0 0
\(478\) −51.6991 −2.36466
\(479\) −16.7757 −0.766503 −0.383252 0.923644i \(-0.625196\pi\)
−0.383252 + 0.923644i \(0.625196\pi\)
\(480\) 0 0
\(481\) −8.25202 −0.376260
\(482\) 76.3752 3.47879
\(483\) 0 0
\(484\) 5.15633 0.234378
\(485\) 0 0
\(486\) 0 0
\(487\) 34.3996 1.55880 0.779398 0.626529i \(-0.215525\pi\)
0.779398 + 0.626529i \(0.215525\pi\)
\(488\) 66.9135 3.02903
\(489\) 0 0
\(490\) 0 0
\(491\) −14.9525 −0.674799 −0.337399 0.941362i \(-0.609547\pi\)
−0.337399 + 0.941362i \(0.609547\pi\)
\(492\) 0 0
\(493\) 10.8510 0.488703
\(494\) −62.6399 −2.81830
\(495\) 0 0
\(496\) −11.8134 −0.530435
\(497\) −20.6253 −0.925171
\(498\) 0 0
\(499\) −2.85097 −0.127627 −0.0638135 0.997962i \(-0.520326\pi\)
−0.0638135 + 0.997962i \(0.520326\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −62.0625 −2.76999
\(503\) 1.48024 0.0660006 0.0330003 0.999455i \(-0.489494\pi\)
0.0330003 + 0.999455i \(0.489494\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 10.7005 0.475696
\(507\) 0 0
\(508\) −64.0322 −2.84097
\(509\) 23.2995 1.03273 0.516366 0.856368i \(-0.327285\pi\)
0.516366 + 0.856368i \(0.327285\pi\)
\(510\) 0 0
\(511\) 18.0098 0.796709
\(512\) −11.5017 −0.508306
\(513\) 0 0
\(514\) −29.1246 −1.28463
\(515\) 0 0
\(516\) 0 0
\(517\) −4.31265 −0.189670
\(518\) 12.1016 0.531712
\(519\) 0 0
\(520\) 0 0
\(521\) 6.81194 0.298437 0.149218 0.988804i \(-0.452324\pi\)
0.149218 + 0.988804i \(0.452324\pi\)
\(522\) 0 0
\(523\) 16.0567 0.702109 0.351054 0.936355i \(-0.385823\pi\)
0.351054 + 0.936355i \(0.385823\pi\)
\(524\) −30.5501 −1.33459
\(525\) 0 0
\(526\) 24.3938 1.06362
\(527\) −4.37328 −0.190503
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 66.1886 2.86964
\(533\) 12.2228 0.529430
\(534\) 0 0
\(535\) 0 0
\(536\) −90.3508 −3.90256
\(537\) 0 0
\(538\) 30.2276 1.30320
\(539\) 0.873992 0.0376455
\(540\) 0 0
\(541\) 6.62530 0.284844 0.142422 0.989806i \(-0.454511\pi\)
0.142422 + 0.989806i \(0.454511\pi\)
\(542\) 67.4128 2.89563
\(543\) 0 0
\(544\) 72.4807 3.10759
\(545\) 0 0
\(546\) 0 0
\(547\) 5.75860 0.246220 0.123110 0.992393i \(-0.460713\pi\)
0.123110 + 0.992393i \(0.460713\pi\)
\(548\) 50.5256 2.15835
\(549\) 0 0
\(550\) 0 0
\(551\) −10.9234 −0.465351
\(552\) 0 0
\(553\) −3.78892 −0.161121
\(554\) 6.46898 0.274840
\(555\) 0 0
\(556\) −10.9624 −0.464909
\(557\) 19.8700 0.841920 0.420960 0.907079i \(-0.361693\pi\)
0.420960 + 0.907079i \(0.361693\pi\)
\(558\) 0 0
\(559\) −14.3634 −0.607509
\(560\) 0 0
\(561\) 0 0
\(562\) −81.5633 −3.44054
\(563\) 5.83383 0.245866 0.122933 0.992415i \(-0.460770\pi\)
0.122933 + 0.992415i \(0.460770\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −22.3576 −0.939758
\(567\) 0 0
\(568\) 62.0625 2.60409
\(569\) −46.7123 −1.95828 −0.979140 0.203185i \(-0.934871\pi\)
−0.979140 + 0.203185i \(0.934871\pi\)
\(570\) 0 0
\(571\) −24.4241 −1.02212 −0.511058 0.859546i \(-0.670746\pi\)
−0.511058 + 0.859546i \(0.670746\pi\)
\(572\) 26.3938 1.10358
\(573\) 0 0
\(574\) −17.9248 −0.748166
\(575\) 0 0
\(576\) 0 0
\(577\) −16.5647 −0.689596 −0.344798 0.938677i \(-0.612053\pi\)
−0.344798 + 0.938677i \(0.612053\pi\)
\(578\) 9.76353 0.406109
\(579\) 0 0
\(580\) 0 0
\(581\) 2.26187 0.0938380
\(582\) 0 0
\(583\) −6.57452 −0.272289
\(584\) −54.1925 −2.24250
\(585\) 0 0
\(586\) −61.7802 −2.55212
\(587\) 27.3258 1.12786 0.563929 0.825823i \(-0.309289\pi\)
0.563929 + 0.825823i \(0.309289\pi\)
\(588\) 0 0
\(589\) 4.40246 0.181400
\(590\) 0 0
\(591\) 0 0
\(592\) −19.7889 −0.813320
\(593\) −12.6048 −0.517618 −0.258809 0.965928i \(-0.583330\pi\)
−0.258809 + 0.965928i \(0.583330\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −42.8627 −1.75573
\(597\) 0 0
\(598\) 54.7729 2.23983
\(599\) −10.5990 −0.433061 −0.216531 0.976276i \(-0.569474\pi\)
−0.216531 + 0.976276i \(0.569474\pi\)
\(600\) 0 0
\(601\) −22.4749 −0.916768 −0.458384 0.888754i \(-0.651572\pi\)
−0.458384 + 0.888754i \(0.651572\pi\)
\(602\) 21.0640 0.858503
\(603\) 0 0
\(604\) −78.7631 −3.20482
\(605\) 0 0
\(606\) 0 0
\(607\) −33.5183 −1.36047 −0.680234 0.732995i \(-0.738122\pi\)
−0.680234 + 0.732995i \(0.738122\pi\)
\(608\) −72.9643 −2.95909
\(609\) 0 0
\(610\) 0 0
\(611\) −22.0752 −0.893068
\(612\) 0 0
\(613\) −11.5672 −0.467196 −0.233598 0.972333i \(-0.575050\pi\)
−0.233598 + 0.972333i \(0.575050\pi\)
\(614\) −7.10413 −0.286699
\(615\) 0 0
\(616\) −23.6932 −0.954627
\(617\) −33.3357 −1.34204 −0.671022 0.741438i \(-0.734144\pi\)
−0.671022 + 0.741438i \(0.734144\pi\)
\(618\) 0 0
\(619\) 4.43866 0.178405 0.0892024 0.996014i \(-0.471568\pi\)
0.0892024 + 0.996014i \(0.471568\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 42.7367 1.71359
\(623\) −8.31265 −0.333039
\(624\) 0 0
\(625\) 0 0
\(626\) 0.162205 0.00648301
\(627\) 0 0
\(628\) −36.1622 −1.44303
\(629\) −7.32582 −0.292100
\(630\) 0 0
\(631\) −39.8496 −1.58639 −0.793193 0.608971i \(-0.791583\pi\)
−0.793193 + 0.608971i \(0.791583\pi\)
\(632\) 11.4010 0.453509
\(633\) 0 0
\(634\) 10.2012 0.405143
\(635\) 0 0
\(636\) 0 0
\(637\) 4.47371 0.177255
\(638\) 6.38787 0.252898
\(639\) 0 0
\(640\) 0 0
\(641\) 3.88858 0.153590 0.0767948 0.997047i \(-0.475531\pi\)
0.0767948 + 0.997047i \(0.475531\pi\)
\(642\) 0 0
\(643\) −40.9380 −1.61444 −0.807218 0.590254i \(-0.799028\pi\)
−0.807218 + 0.590254i \(0.799028\pi\)
\(644\) −57.8759 −2.28063
\(645\) 0 0
\(646\) −55.6093 −2.18792
\(647\) 1.76257 0.0692939 0.0346469 0.999400i \(-0.488969\pi\)
0.0346469 + 0.999400i \(0.488969\pi\)
\(648\) 0 0
\(649\) 13.2750 0.521091
\(650\) 0 0
\(651\) 0 0
\(652\) −34.8627 −1.36533
\(653\) −7.03761 −0.275403 −0.137702 0.990474i \(-0.543971\pi\)
−0.137702 + 0.990474i \(0.543971\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 29.3112 1.14441
\(657\) 0 0
\(658\) 32.3733 1.26204
\(659\) 12.6253 0.491812 0.245906 0.969294i \(-0.420915\pi\)
0.245906 + 0.969294i \(0.420915\pi\)
\(660\) 0 0
\(661\) 13.2243 0.514364 0.257182 0.966363i \(-0.417206\pi\)
0.257182 + 0.966363i \(0.417206\pi\)
\(662\) 64.9741 2.52529
\(663\) 0 0
\(664\) −6.80606 −0.264126
\(665\) 0 0
\(666\) 0 0
\(667\) 9.55149 0.369835
\(668\) −60.9438 −2.35799
\(669\) 0 0
\(670\) 0 0
\(671\) 7.92478 0.305933
\(672\) 0 0
\(673\) −18.2677 −0.704170 −0.352085 0.935968i \(-0.614527\pi\)
−0.352085 + 0.935968i \(0.614527\pi\)
\(674\) 60.4055 2.32673
\(675\) 0 0
\(676\) 68.0698 2.61807
\(677\) −33.0191 −1.26903 −0.634513 0.772912i \(-0.718799\pi\)
−0.634513 + 0.772912i \(0.718799\pi\)
\(678\) 0 0
\(679\) −27.8496 −1.06877
\(680\) 0 0
\(681\) 0 0
\(682\) −2.57452 −0.0985833
\(683\) 30.8627 1.18093 0.590465 0.807063i \(-0.298944\pi\)
0.590465 + 0.807063i \(0.298944\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 45.9854 1.75573
\(687\) 0 0
\(688\) −34.4445 −1.31319
\(689\) −33.6531 −1.28208
\(690\) 0 0
\(691\) 27.6991 1.05372 0.526862 0.849951i \(-0.323369\pi\)
0.526862 + 0.849951i \(0.323369\pi\)
\(692\) 36.0567 1.37067
\(693\) 0 0
\(694\) 54.0176 2.05048
\(695\) 0 0
\(696\) 0 0
\(697\) 10.8510 0.411010
\(698\) −72.8989 −2.75927
\(699\) 0 0
\(700\) 0 0
\(701\) 38.9643 1.47166 0.735831 0.677166i \(-0.236792\pi\)
0.735831 + 0.677166i \(0.236792\pi\)
\(702\) 0 0
\(703\) 7.37470 0.278142
\(704\) 18.1187 0.682875
\(705\) 0 0
\(706\) 13.4372 0.505717
\(707\) −38.1965 −1.43653
\(708\) 0 0
\(709\) −4.32250 −0.162335 −0.0811674 0.996700i \(-0.525865\pi\)
−0.0811674 + 0.996700i \(0.525865\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 25.0132 0.937408
\(713\) −3.84955 −0.144167
\(714\) 0 0
\(715\) 0 0
\(716\) 13.9248 0.520393
\(717\) 0 0
\(718\) 48.6253 1.81468
\(719\) 38.3996 1.43206 0.716032 0.698067i \(-0.245956\pi\)
0.716032 + 0.698067i \(0.245956\pi\)
\(720\) 0 0
\(721\) 45.7743 1.70473
\(722\) 5.15282 0.191768
\(723\) 0 0
\(724\) −109.313 −4.06259
\(725\) 0 0
\(726\) 0 0
\(727\) 21.6728 0.803798 0.401899 0.915684i \(-0.368350\pi\)
0.401899 + 0.915684i \(0.368350\pi\)
\(728\) −121.279 −4.49489
\(729\) 0 0
\(730\) 0 0
\(731\) −12.7513 −0.471624
\(732\) 0 0
\(733\) −25.0698 −0.925976 −0.462988 0.886365i \(-0.653223\pi\)
−0.462988 + 0.886365i \(0.653223\pi\)
\(734\) 22.9116 0.845683
\(735\) 0 0
\(736\) 63.8007 2.35172
\(737\) −10.7005 −0.394159
\(738\) 0 0
\(739\) −18.9018 −0.695312 −0.347656 0.937622i \(-0.613022\pi\)
−0.347656 + 0.937622i \(0.613022\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 49.3522 1.81178
\(743\) 43.6688 1.60205 0.801026 0.598629i \(-0.204288\pi\)
0.801026 + 0.598629i \(0.204288\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −20.9175 −0.765843
\(747\) 0 0
\(748\) 23.4314 0.856736
\(749\) 26.4650 0.967010
\(750\) 0 0
\(751\) −47.0541 −1.71703 −0.858514 0.512789i \(-0.828612\pi\)
−0.858514 + 0.512789i \(0.828612\pi\)
\(752\) −52.9380 −1.93045
\(753\) 0 0
\(754\) 32.6977 1.19078
\(755\) 0 0
\(756\) 0 0
\(757\) 26.9116 0.978119 0.489059 0.872250i \(-0.337340\pi\)
0.489059 + 0.872250i \(0.337340\pi\)
\(758\) 3.07381 0.111646
\(759\) 0 0
\(760\) 0 0
\(761\) −16.6859 −0.604865 −0.302432 0.953171i \(-0.597799\pi\)
−0.302432 + 0.953171i \(0.597799\pi\)
\(762\) 0 0
\(763\) 43.2163 1.56454
\(764\) −67.8007 −2.45294
\(765\) 0 0
\(766\) 91.1871 3.29473
\(767\) 67.9511 2.45357
\(768\) 0 0
\(769\) 23.2995 0.840201 0.420100 0.907478i \(-0.361995\pi\)
0.420100 + 0.907478i \(0.361995\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −30.3938 −1.09390
\(773\) −1.63656 −0.0588631 −0.0294316 0.999567i \(-0.509370\pi\)
−0.0294316 + 0.999567i \(0.509370\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 83.8007 3.00827
\(777\) 0 0
\(778\) −63.5994 −2.28015
\(779\) −10.9234 −0.391370
\(780\) 0 0
\(781\) 7.35026 0.263013
\(782\) 48.6253 1.73884
\(783\) 0 0
\(784\) 10.7283 0.383153
\(785\) 0 0
\(786\) 0 0
\(787\) 2.09095 0.0745344 0.0372672 0.999305i \(-0.488135\pi\)
0.0372672 + 0.999305i \(0.488135\pi\)
\(788\) 105.157 3.74606
\(789\) 0 0
\(790\) 0 0
\(791\) 38.5501 1.37068
\(792\) 0 0
\(793\) 40.5647 1.44049
\(794\) 11.1030 0.394030
\(795\) 0 0
\(796\) 89.7255 3.18023
\(797\) −26.8872 −0.952392 −0.476196 0.879339i \(-0.657985\pi\)
−0.476196 + 0.879339i \(0.657985\pi\)
\(798\) 0 0
\(799\) −19.5975 −0.693311
\(800\) 0 0
\(801\) 0 0
\(802\) 90.7269 3.20368
\(803\) −6.41819 −0.226493
\(804\) 0 0
\(805\) 0 0
\(806\) −13.1782 −0.464183
\(807\) 0 0
\(808\) 114.935 4.04340
\(809\) 45.4128 1.59663 0.798315 0.602241i \(-0.205725\pi\)
0.798315 + 0.602241i \(0.205725\pi\)
\(810\) 0 0
\(811\) 34.3488 1.20615 0.603076 0.797684i \(-0.293942\pi\)
0.603076 + 0.797684i \(0.293942\pi\)
\(812\) −34.5501 −1.21247
\(813\) 0 0
\(814\) −4.31265 −0.151158
\(815\) 0 0
\(816\) 0 0
\(817\) 12.8364 0.449088
\(818\) 22.8726 0.799721
\(819\) 0 0
\(820\) 0 0
\(821\) −18.7612 −0.654769 −0.327384 0.944891i \(-0.606167\pi\)
−0.327384 + 0.944891i \(0.606167\pi\)
\(822\) 0 0
\(823\) 33.0592 1.15237 0.576186 0.817319i \(-0.304540\pi\)
0.576186 + 0.817319i \(0.304540\pi\)
\(824\) −137.737 −4.79830
\(825\) 0 0
\(826\) −99.6502 −3.46728
\(827\) −10.8813 −0.378379 −0.189190 0.981941i \(-0.560586\pi\)
−0.189190 + 0.981941i \(0.560586\pi\)
\(828\) 0 0
\(829\) 42.7221 1.48380 0.741900 0.670510i \(-0.233925\pi\)
0.741900 + 0.670510i \(0.233925\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 92.7445 3.21534
\(833\) 3.97159 0.137608
\(834\) 0 0
\(835\) 0 0
\(836\) −23.5877 −0.815797
\(837\) 0 0
\(838\) −35.5778 −1.22902
\(839\) 21.1735 0.730989 0.365495 0.930813i \(-0.380900\pi\)
0.365495 + 0.930813i \(0.380900\pi\)
\(840\) 0 0
\(841\) −23.2981 −0.803381
\(842\) 89.0856 3.07009
\(843\) 0 0
\(844\) 93.4636 3.21715
\(845\) 0 0
\(846\) 0 0
\(847\) −2.80606 −0.0964175
\(848\) −80.7024 −2.77133
\(849\) 0 0
\(850\) 0 0
\(851\) −6.44851 −0.221052
\(852\) 0 0
\(853\) −54.4709 −1.86505 −0.932524 0.361109i \(-0.882398\pi\)
−0.932524 + 0.361109i \(0.882398\pi\)
\(854\) −59.4880 −2.03564
\(855\) 0 0
\(856\) −79.6345 −2.72185
\(857\) 38.5705 1.31754 0.658772 0.752342i \(-0.271076\pi\)
0.658772 + 0.752342i \(0.271076\pi\)
\(858\) 0 0
\(859\) −18.5139 −0.631685 −0.315843 0.948812i \(-0.602287\pi\)
−0.315843 + 0.948812i \(0.602287\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −5.55149 −0.189085
\(863\) 22.5383 0.767213 0.383607 0.923497i \(-0.374682\pi\)
0.383607 + 0.923497i \(0.374682\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 100.053 3.39993
\(867\) 0 0
\(868\) 13.9248 0.472638
\(869\) 1.35026 0.0458045
\(870\) 0 0
\(871\) −54.7729 −1.85591
\(872\) −130.040 −4.40371
\(873\) 0 0
\(874\) −48.9497 −1.65575
\(875\) 0 0
\(876\) 0 0
\(877\) −18.9419 −0.639623 −0.319812 0.947481i \(-0.603620\pi\)
−0.319812 + 0.947481i \(0.603620\pi\)
\(878\) −16.1162 −0.543894
\(879\) 0 0
\(880\) 0 0
\(881\) 20.0263 0.674705 0.337352 0.941378i \(-0.390469\pi\)
0.337352 + 0.941378i \(0.390469\pi\)
\(882\) 0 0
\(883\) −22.2981 −0.750390 −0.375195 0.926946i \(-0.622424\pi\)
−0.375195 + 0.926946i \(0.622424\pi\)
\(884\) 119.938 4.03397
\(885\) 0 0
\(886\) −27.1147 −0.910938
\(887\) 3.10413 0.104226 0.0521132 0.998641i \(-0.483404\pi\)
0.0521132 + 0.998641i \(0.483404\pi\)
\(888\) 0 0
\(889\) 34.8462 1.16871
\(890\) 0 0
\(891\) 0 0
\(892\) −121.888 −4.08110
\(893\) 19.7283 0.660182
\(894\) 0 0
\(895\) 0 0
\(896\) −46.4953 −1.55330
\(897\) 0 0
\(898\) 30.7269 1.02537
\(899\) −2.29806 −0.0766447
\(900\) 0 0
\(901\) −29.8759 −0.995311
\(902\) 6.38787 0.212693
\(903\) 0 0
\(904\) −115.999 −3.85807
\(905\) 0 0
\(906\) 0 0
\(907\) −10.8627 −0.360691 −0.180345 0.983603i \(-0.557722\pi\)
−0.180345 + 0.983603i \(0.557722\pi\)
\(908\) −6.54420 −0.217177
\(909\) 0 0
\(910\) 0 0
\(911\) 5.82321 0.192931 0.0964657 0.995336i \(-0.469246\pi\)
0.0964657 + 0.995336i \(0.469246\pi\)
\(912\) 0 0
\(913\) −0.806063 −0.0266768
\(914\) 40.2433 1.33113
\(915\) 0 0
\(916\) 83.4128 2.75604
\(917\) 16.6253 0.549016
\(918\) 0 0
\(919\) −31.2750 −1.03167 −0.515834 0.856688i \(-0.672518\pi\)
−0.515834 + 0.856688i \(0.672518\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 70.1886 2.31154
\(923\) 37.6239 1.23841
\(924\) 0 0
\(925\) 0 0
\(926\) 14.6107 0.480138
\(927\) 0 0
\(928\) 38.0870 1.25027
\(929\) 21.2243 0.696345 0.348173 0.937430i \(-0.386802\pi\)
0.348173 + 0.937430i \(0.386802\pi\)
\(930\) 0 0
\(931\) −3.99809 −0.131032
\(932\) −95.2320 −3.11943
\(933\) 0 0
\(934\) −7.22425 −0.236385
\(935\) 0 0
\(936\) 0 0
\(937\) −16.4328 −0.536835 −0.268418 0.963303i \(-0.586501\pi\)
−0.268418 + 0.963303i \(0.586501\pi\)
\(938\) 80.3244 2.62268
\(939\) 0 0
\(940\) 0 0
\(941\) −5.86414 −0.191166 −0.0955828 0.995421i \(-0.530471\pi\)
−0.0955828 + 0.995421i \(0.530471\pi\)
\(942\) 0 0
\(943\) 9.55149 0.311039
\(944\) 162.952 5.30362
\(945\) 0 0
\(946\) −7.50659 −0.244060
\(947\) −38.7415 −1.25893 −0.629464 0.777030i \(-0.716726\pi\)
−0.629464 + 0.777030i \(0.716726\pi\)
\(948\) 0 0
\(949\) −32.8529 −1.06645
\(950\) 0 0
\(951\) 0 0
\(952\) −107.667 −3.48950
\(953\) −6.09569 −0.197459 −0.0987294 0.995114i \(-0.531478\pi\)
−0.0987294 + 0.995114i \(0.531478\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −99.6502 −3.22292
\(957\) 0 0
\(958\) −44.8773 −1.44992
\(959\) −27.4960 −0.887891
\(960\) 0 0
\(961\) −30.0738 −0.970123
\(962\) −22.0752 −0.711734
\(963\) 0 0
\(964\) 147.213 4.74143
\(965\) 0 0
\(966\) 0 0
\(967\) −37.9697 −1.22102 −0.610511 0.792008i \(-0.709036\pi\)
−0.610511 + 0.792008i \(0.709036\pi\)
\(968\) 8.44358 0.271387
\(969\) 0 0
\(970\) 0 0
\(971\) 60.8773 1.95365 0.976823 0.214049i \(-0.0686651\pi\)
0.976823 + 0.214049i \(0.0686651\pi\)
\(972\) 0 0
\(973\) 5.96571 0.191252
\(974\) 92.0235 2.94862
\(975\) 0 0
\(976\) 97.2769 3.11376
\(977\) 21.3963 0.684529 0.342264 0.939604i \(-0.388806\pi\)
0.342264 + 0.939604i \(0.388806\pi\)
\(978\) 0 0
\(979\) 2.96239 0.0946784
\(980\) 0 0
\(981\) 0 0
\(982\) −40.0000 −1.27645
\(983\) 18.8919 0.602558 0.301279 0.953536i \(-0.402586\pi\)
0.301279 + 0.953536i \(0.402586\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 29.0278 0.924432
\(987\) 0 0
\(988\) −120.739 −3.84121
\(989\) −11.2243 −0.356911
\(990\) 0 0
\(991\) −1.33567 −0.0424291 −0.0212145 0.999775i \(-0.506753\pi\)
−0.0212145 + 0.999775i \(0.506753\pi\)
\(992\) −15.3503 −0.487371
\(993\) 0 0
\(994\) −55.1754 −1.75006
\(995\) 0 0
\(996\) 0 0
\(997\) 48.9076 1.54892 0.774460 0.632623i \(-0.218022\pi\)
0.774460 + 0.632623i \(0.218022\pi\)
\(998\) −7.62672 −0.241419
\(999\) 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 2475.2.a.be.1.3 3
3.2 odd 2 825.2.a.h.1.1 3
5.2 odd 4 495.2.c.d.199.6 6
5.3 odd 4 495.2.c.d.199.1 6
5.4 even 2 2475.2.a.y.1.1 3
15.2 even 4 165.2.c.a.34.1 6
15.8 even 4 165.2.c.a.34.6 yes 6
15.14 odd 2 825.2.a.n.1.3 3
33.32 even 2 9075.2.a.ck.1.3 3
60.23 odd 4 2640.2.d.i.529.1 6
60.47 odd 4 2640.2.d.i.529.4 6
165.32 odd 4 1815.2.c.d.364.6 6
165.98 odd 4 1815.2.c.d.364.1 6
165.164 even 2 9075.2.a.cc.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.a.34.1 6 15.2 even 4
165.2.c.a.34.6 yes 6 15.8 even 4
495.2.c.d.199.1 6 5.3 odd 4
495.2.c.d.199.6 6 5.2 odd 4
825.2.a.h.1.1 3 3.2 odd 2
825.2.a.n.1.3 3 15.14 odd 2
1815.2.c.d.364.1 6 165.98 odd 4
1815.2.c.d.364.6 6 165.32 odd 4
2475.2.a.y.1.1 3 5.4 even 2
2475.2.a.be.1.3 3 1.1 even 1 trivial
2640.2.d.i.529.1 6 60.23 odd 4
2640.2.d.i.529.4 6 60.47 odd 4
9075.2.a.cc.1.1 3 165.164 even 2
9075.2.a.ck.1.3 3 33.32 even 2