Properties

Label 6160.2.a.bn.1.3
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
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] = [6160,2,Mod(1,6160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("6160.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.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: \(49.1878476451\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.48119 q^{3} -1.00000 q^{5} +1.00000 q^{7} +3.15633 q^{9} +O(q^{10})\) \(q+2.48119 q^{3} -1.00000 q^{5} +1.00000 q^{7} +3.15633 q^{9} +1.00000 q^{11} +5.83146 q^{13} -2.48119 q^{15} +5.44358 q^{17} +1.35026 q^{19} +2.48119 q^{21} +3.19394 q^{23} +1.00000 q^{25} +0.387873 q^{27} -3.61213 q^{29} +5.28726 q^{31} +2.48119 q^{33} -1.00000 q^{35} -8.54420 q^{37} +14.4690 q^{39} -5.02539 q^{41} -5.89446 q^{43} -3.15633 q^{45} +11.8315 q^{47} +1.00000 q^{49} +13.5066 q^{51} -0.231548 q^{53} -1.00000 q^{55} +3.35026 q^{57} -13.5999 q^{59} -1.41327 q^{61} +3.15633 q^{63} -5.83146 q^{65} +10.8568 q^{67} +7.92478 q^{69} +15.5369 q^{71} -11.3684 q^{73} +2.48119 q^{75} +1.00000 q^{77} -1.96968 q^{79} -8.50659 q^{81} -10.6253 q^{83} -5.44358 q^{85} -8.96239 q^{87} +7.22425 q^{89} +5.83146 q^{91} +13.1187 q^{93} -1.35026 q^{95} -0.836381 q^{97} +3.15633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - q^{9} + 3 q^{11} + 2 q^{13} - 2 q^{15} - 6 q^{19} + 2 q^{21} + 10 q^{23} + 3 q^{25} + 2 q^{27} - 10 q^{29} + 10 q^{31} + 2 q^{33} - 3 q^{35} - 16 q^{37} + 12 q^{39} + 2 q^{43} + q^{45} + 20 q^{47} + 3 q^{49} + 20 q^{51} - 12 q^{53} - 3 q^{55} - 14 q^{59} + 10 q^{61} - q^{63} - 2 q^{65} + 2 q^{67} + 2 q^{69} + 24 q^{71} + 4 q^{73} + 2 q^{75} + 3 q^{77} - 8 q^{79} - 5 q^{81} + 10 q^{83} - 16 q^{87} + 20 q^{89} + 2 q^{91} + 18 q^{93} + 6 q^{95} - q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.48119 1.43252 0.716259 0.697834i \(-0.245853\pi\)
0.716259 + 0.697834i \(0.245853\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 3.15633 1.05211
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.83146 1.61735 0.808677 0.588252i \(-0.200184\pi\)
0.808677 + 0.588252i \(0.200184\pi\)
\(14\) 0 0
\(15\) −2.48119 −0.640642
\(16\) 0 0
\(17\) 5.44358 1.32026 0.660131 0.751150i \(-0.270501\pi\)
0.660131 + 0.751150i \(0.270501\pi\)
\(18\) 0 0
\(19\) 1.35026 0.309771 0.154886 0.987932i \(-0.450499\pi\)
0.154886 + 0.987932i \(0.450499\pi\)
\(20\) 0 0
\(21\) 2.48119 0.541441
\(22\) 0 0
\(23\) 3.19394 0.665982 0.332991 0.942930i \(-0.391942\pi\)
0.332991 + 0.942930i \(0.391942\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.387873 0.0746462
\(28\) 0 0
\(29\) −3.61213 −0.670755 −0.335378 0.942084i \(-0.608864\pi\)
−0.335378 + 0.942084i \(0.608864\pi\)
\(30\) 0 0
\(31\) 5.28726 0.949620 0.474810 0.880088i \(-0.342517\pi\)
0.474810 + 0.880088i \(0.342517\pi\)
\(32\) 0 0
\(33\) 2.48119 0.431920
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −8.54420 −1.40466 −0.702329 0.711853i \(-0.747856\pi\)
−0.702329 + 0.711853i \(0.747856\pi\)
\(38\) 0 0
\(39\) 14.4690 2.31689
\(40\) 0 0
\(41\) −5.02539 −0.784834 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(42\) 0 0
\(43\) −5.89446 −0.898897 −0.449448 0.893306i \(-0.648379\pi\)
−0.449448 + 0.893306i \(0.648379\pi\)
\(44\) 0 0
\(45\) −3.15633 −0.470517
\(46\) 0 0
\(47\) 11.8315 1.72580 0.862898 0.505379i \(-0.168647\pi\)
0.862898 + 0.505379i \(0.168647\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 13.5066 1.89130
\(52\) 0 0
\(53\) −0.231548 −0.0318056 −0.0159028 0.999874i \(-0.505062\pi\)
−0.0159028 + 0.999874i \(0.505062\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 3.35026 0.443753
\(58\) 0 0
\(59\) −13.5999 −1.77056 −0.885279 0.465061i \(-0.846032\pi\)
−0.885279 + 0.465061i \(0.846032\pi\)
\(60\) 0 0
\(61\) −1.41327 −0.180950 −0.0904751 0.995899i \(-0.528839\pi\)
−0.0904751 + 0.995899i \(0.528839\pi\)
\(62\) 0 0
\(63\) 3.15633 0.397660
\(64\) 0 0
\(65\) −5.83146 −0.723303
\(66\) 0 0
\(67\) 10.8568 1.32638 0.663188 0.748453i \(-0.269203\pi\)
0.663188 + 0.748453i \(0.269203\pi\)
\(68\) 0 0
\(69\) 7.92478 0.954031
\(70\) 0 0
\(71\) 15.5369 1.84389 0.921946 0.387319i \(-0.126599\pi\)
0.921946 + 0.387319i \(0.126599\pi\)
\(72\) 0 0
\(73\) −11.3684 −1.33057 −0.665283 0.746591i \(-0.731689\pi\)
−0.665283 + 0.746591i \(0.731689\pi\)
\(74\) 0 0
\(75\) 2.48119 0.286504
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −1.96968 −0.221607 −0.110803 0.993842i \(-0.535342\pi\)
−0.110803 + 0.993842i \(0.535342\pi\)
\(80\) 0 0
\(81\) −8.50659 −0.945176
\(82\) 0 0
\(83\) −10.6253 −1.16628 −0.583139 0.812372i \(-0.698176\pi\)
−0.583139 + 0.812372i \(0.698176\pi\)
\(84\) 0 0
\(85\) −5.44358 −0.590439
\(86\) 0 0
\(87\) −8.96239 −0.960869
\(88\) 0 0
\(89\) 7.22425 0.765769 0.382885 0.923796i \(-0.374931\pi\)
0.382885 + 0.923796i \(0.374931\pi\)
\(90\) 0 0
\(91\) 5.83146 0.611303
\(92\) 0 0
\(93\) 13.1187 1.36035
\(94\) 0 0
\(95\) −1.35026 −0.138534
\(96\) 0 0
\(97\) −0.836381 −0.0849216 −0.0424608 0.999098i \(-0.513520\pi\)
−0.0424608 + 0.999098i \(0.513520\pi\)
\(98\) 0 0
\(99\) 3.15633 0.317223
\(100\) 0 0
\(101\) 7.41327 0.737648 0.368824 0.929499i \(-0.379761\pi\)
0.368824 + 0.929499i \(0.379761\pi\)
\(102\) 0 0
\(103\) −4.21933 −0.415743 −0.207871 0.978156i \(-0.566654\pi\)
−0.207871 + 0.978156i \(0.566654\pi\)
\(104\) 0 0
\(105\) −2.48119 −0.242140
\(106\) 0 0
\(107\) 11.5369 1.11531 0.557657 0.830071i \(-0.311700\pi\)
0.557657 + 0.830071i \(0.311700\pi\)
\(108\) 0 0
\(109\) −2.18664 −0.209442 −0.104721 0.994502i \(-0.533395\pi\)
−0.104721 + 0.994502i \(0.533395\pi\)
\(110\) 0 0
\(111\) −21.1998 −2.01220
\(112\) 0 0
\(113\) −9.35026 −0.879599 −0.439799 0.898096i \(-0.644950\pi\)
−0.439799 + 0.898096i \(0.644950\pi\)
\(114\) 0 0
\(115\) −3.19394 −0.297836
\(116\) 0 0
\(117\) 18.4060 1.70163
\(118\) 0 0
\(119\) 5.44358 0.499012
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −12.4690 −1.12429
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.9624 1.50517 0.752584 0.658496i \(-0.228807\pi\)
0.752584 + 0.658496i \(0.228807\pi\)
\(128\) 0 0
\(129\) −14.6253 −1.28769
\(130\) 0 0
\(131\) 9.92478 0.867132 0.433566 0.901122i \(-0.357255\pi\)
0.433566 + 0.901122i \(0.357255\pi\)
\(132\) 0 0
\(133\) 1.35026 0.117083
\(134\) 0 0
\(135\) −0.387873 −0.0333828
\(136\) 0 0
\(137\) −10.9927 −0.939170 −0.469585 0.882887i \(-0.655596\pi\)
−0.469585 + 0.882887i \(0.655596\pi\)
\(138\) 0 0
\(139\) −6.88717 −0.584162 −0.292081 0.956394i \(-0.594348\pi\)
−0.292081 + 0.956394i \(0.594348\pi\)
\(140\) 0 0
\(141\) 29.3561 2.47223
\(142\) 0 0
\(143\) 5.83146 0.487651
\(144\) 0 0
\(145\) 3.61213 0.299971
\(146\) 0 0
\(147\) 2.48119 0.204645
\(148\) 0 0
\(149\) 22.8119 1.86883 0.934414 0.356190i \(-0.115924\pi\)
0.934414 + 0.356190i \(0.115924\pi\)
\(150\) 0 0
\(151\) 3.24472 0.264052 0.132026 0.991246i \(-0.457852\pi\)
0.132026 + 0.991246i \(0.457852\pi\)
\(152\) 0 0
\(153\) 17.1817 1.38906
\(154\) 0 0
\(155\) −5.28726 −0.424683
\(156\) 0 0
\(157\) −5.42548 −0.433001 −0.216500 0.976283i \(-0.569464\pi\)
−0.216500 + 0.976283i \(0.569464\pi\)
\(158\) 0 0
\(159\) −0.574515 −0.0455620
\(160\) 0 0
\(161\) 3.19394 0.251717
\(162\) 0 0
\(163\) −3.38058 −0.264787 −0.132394 0.991197i \(-0.542266\pi\)
−0.132394 + 0.991197i \(0.542266\pi\)
\(164\) 0 0
\(165\) −2.48119 −0.193161
\(166\) 0 0
\(167\) −11.2750 −0.872489 −0.436244 0.899828i \(-0.643692\pi\)
−0.436244 + 0.899828i \(0.643692\pi\)
\(168\) 0 0
\(169\) 21.0059 1.61584
\(170\) 0 0
\(171\) 4.26187 0.325913
\(172\) 0 0
\(173\) −8.98049 −0.682774 −0.341387 0.939923i \(-0.610897\pi\)
−0.341387 + 0.939923i \(0.610897\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −33.7440 −2.53636
\(178\) 0 0
\(179\) 26.2374 1.96108 0.980539 0.196326i \(-0.0629010\pi\)
0.980539 + 0.196326i \(0.0629010\pi\)
\(180\) 0 0
\(181\) −11.1998 −0.832476 −0.416238 0.909256i \(-0.636652\pi\)
−0.416238 + 0.909256i \(0.636652\pi\)
\(182\) 0 0
\(183\) −3.50659 −0.259214
\(184\) 0 0
\(185\) 8.54420 0.628182
\(186\) 0 0
\(187\) 5.44358 0.398074
\(188\) 0 0
\(189\) 0.387873 0.0282136
\(190\) 0 0
\(191\) −11.1998 −0.810390 −0.405195 0.914230i \(-0.632796\pi\)
−0.405195 + 0.914230i \(0.632796\pi\)
\(192\) 0 0
\(193\) 0.604833 0.0435368 0.0217684 0.999763i \(-0.493070\pi\)
0.0217684 + 0.999763i \(0.493070\pi\)
\(194\) 0 0
\(195\) −14.4690 −1.03614
\(196\) 0 0
\(197\) 15.3054 1.09046 0.545231 0.838286i \(-0.316442\pi\)
0.545231 + 0.838286i \(0.316442\pi\)
\(198\) 0 0
\(199\) 12.5623 0.890518 0.445259 0.895402i \(-0.353112\pi\)
0.445259 + 0.895402i \(0.353112\pi\)
\(200\) 0 0
\(201\) 26.9380 1.90006
\(202\) 0 0
\(203\) −3.61213 −0.253522
\(204\) 0 0
\(205\) 5.02539 0.350989
\(206\) 0 0
\(207\) 10.0811 0.700685
\(208\) 0 0
\(209\) 1.35026 0.0933996
\(210\) 0 0
\(211\) 4.43866 0.305570 0.152785 0.988259i \(-0.451176\pi\)
0.152785 + 0.988259i \(0.451176\pi\)
\(212\) 0 0
\(213\) 38.5501 2.64141
\(214\) 0 0
\(215\) 5.89446 0.401999
\(216\) 0 0
\(217\) 5.28726 0.358922
\(218\) 0 0
\(219\) −28.2071 −1.90606
\(220\) 0 0
\(221\) 31.7440 2.13533
\(222\) 0 0
\(223\) 7.78067 0.521032 0.260516 0.965469i \(-0.416107\pi\)
0.260516 + 0.965469i \(0.416107\pi\)
\(224\) 0 0
\(225\) 3.15633 0.210422
\(226\) 0 0
\(227\) 10.4485 0.693492 0.346746 0.937959i \(-0.387287\pi\)
0.346746 + 0.937959i \(0.387287\pi\)
\(228\) 0 0
\(229\) −29.4518 −1.94623 −0.973116 0.230316i \(-0.926024\pi\)
−0.973116 + 0.230316i \(0.926024\pi\)
\(230\) 0 0
\(231\) 2.48119 0.163251
\(232\) 0 0
\(233\) −8.73084 −0.571976 −0.285988 0.958233i \(-0.592322\pi\)
−0.285988 + 0.958233i \(0.592322\pi\)
\(234\) 0 0
\(235\) −11.8315 −0.771799
\(236\) 0 0
\(237\) −4.88717 −0.317456
\(238\) 0 0
\(239\) 21.2144 1.37225 0.686123 0.727486i \(-0.259311\pi\)
0.686123 + 0.727486i \(0.259311\pi\)
\(240\) 0 0
\(241\) −9.33804 −0.601516 −0.300758 0.953700i \(-0.597240\pi\)
−0.300758 + 0.953700i \(0.597240\pi\)
\(242\) 0 0
\(243\) −22.2701 −1.42863
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 7.87399 0.501010
\(248\) 0 0
\(249\) −26.3634 −1.67071
\(250\) 0 0
\(251\) −1.87636 −0.118435 −0.0592174 0.998245i \(-0.518861\pi\)
−0.0592174 + 0.998245i \(0.518861\pi\)
\(252\) 0 0
\(253\) 3.19394 0.200801
\(254\) 0 0
\(255\) −13.5066 −0.845815
\(256\) 0 0
\(257\) 27.1392 1.69290 0.846448 0.532472i \(-0.178737\pi\)
0.846448 + 0.532472i \(0.178737\pi\)
\(258\) 0 0
\(259\) −8.54420 −0.530911
\(260\) 0 0
\(261\) −11.4010 −0.705707
\(262\) 0 0
\(263\) −12.8119 −0.790018 −0.395009 0.918677i \(-0.629259\pi\)
−0.395009 + 0.918677i \(0.629259\pi\)
\(264\) 0 0
\(265\) 0.231548 0.0142239
\(266\) 0 0
\(267\) 17.9248 1.09698
\(268\) 0 0
\(269\) −6.26187 −0.381793 −0.190896 0.981610i \(-0.561139\pi\)
−0.190896 + 0.981610i \(0.561139\pi\)
\(270\) 0 0
\(271\) 5.73813 0.348567 0.174283 0.984696i \(-0.444239\pi\)
0.174283 + 0.984696i \(0.444239\pi\)
\(272\) 0 0
\(273\) 14.4690 0.875702
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −8.35756 −0.502157 −0.251078 0.967967i \(-0.580785\pi\)
−0.251078 + 0.967967i \(0.580785\pi\)
\(278\) 0 0
\(279\) 16.6883 0.999103
\(280\) 0 0
\(281\) −8.44851 −0.503996 −0.251998 0.967728i \(-0.581088\pi\)
−0.251998 + 0.967728i \(0.581088\pi\)
\(282\) 0 0
\(283\) −0.836381 −0.0497177 −0.0248588 0.999691i \(-0.507914\pi\)
−0.0248588 + 0.999691i \(0.507914\pi\)
\(284\) 0 0
\(285\) −3.35026 −0.198452
\(286\) 0 0
\(287\) −5.02539 −0.296640
\(288\) 0 0
\(289\) 12.6326 0.743094
\(290\) 0 0
\(291\) −2.07522 −0.121652
\(292\) 0 0
\(293\) −2.71862 −0.158824 −0.0794118 0.996842i \(-0.525304\pi\)
−0.0794118 + 0.996842i \(0.525304\pi\)
\(294\) 0 0
\(295\) 13.5999 0.791817
\(296\) 0 0
\(297\) 0.387873 0.0225067
\(298\) 0 0
\(299\) 18.6253 1.07713
\(300\) 0 0
\(301\) −5.89446 −0.339751
\(302\) 0 0
\(303\) 18.3938 1.05669
\(304\) 0 0
\(305\) 1.41327 0.0809234
\(306\) 0 0
\(307\) 8.36344 0.477326 0.238663 0.971102i \(-0.423291\pi\)
0.238663 + 0.971102i \(0.423291\pi\)
\(308\) 0 0
\(309\) −10.4690 −0.595559
\(310\) 0 0
\(311\) −4.43629 −0.251559 −0.125779 0.992058i \(-0.540143\pi\)
−0.125779 + 0.992058i \(0.540143\pi\)
\(312\) 0 0
\(313\) 29.7889 1.68377 0.841885 0.539658i \(-0.181446\pi\)
0.841885 + 0.539658i \(0.181446\pi\)
\(314\) 0 0
\(315\) −3.15633 −0.177839
\(316\) 0 0
\(317\) 15.4010 0.865009 0.432504 0.901632i \(-0.357630\pi\)
0.432504 + 0.901632i \(0.357630\pi\)
\(318\) 0 0
\(319\) −3.61213 −0.202240
\(320\) 0 0
\(321\) 28.6253 1.59771
\(322\) 0 0
\(323\) 7.35026 0.408980
\(324\) 0 0
\(325\) 5.83146 0.323471
\(326\) 0 0
\(327\) −5.42548 −0.300030
\(328\) 0 0
\(329\) 11.8315 0.652289
\(330\) 0 0
\(331\) −6.26187 −0.344183 −0.172092 0.985081i \(-0.555053\pi\)
−0.172092 + 0.985081i \(0.555053\pi\)
\(332\) 0 0
\(333\) −26.9683 −1.47785
\(334\) 0 0
\(335\) −10.8568 −0.593173
\(336\) 0 0
\(337\) −15.8700 −0.864495 −0.432248 0.901755i \(-0.642279\pi\)
−0.432248 + 0.901755i \(0.642279\pi\)
\(338\) 0 0
\(339\) −23.1998 −1.26004
\(340\) 0 0
\(341\) 5.28726 0.286321
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −7.92478 −0.426656
\(346\) 0 0
\(347\) −6.79147 −0.364585 −0.182293 0.983244i \(-0.558352\pi\)
−0.182293 + 0.983244i \(0.558352\pi\)
\(348\) 0 0
\(349\) 26.7489 1.43184 0.715919 0.698183i \(-0.246008\pi\)
0.715919 + 0.698183i \(0.246008\pi\)
\(350\) 0 0
\(351\) 2.26187 0.120729
\(352\) 0 0
\(353\) −16.8627 −0.897512 −0.448756 0.893654i \(-0.648133\pi\)
−0.448756 + 0.893654i \(0.648133\pi\)
\(354\) 0 0
\(355\) −15.5369 −0.824613
\(356\) 0 0
\(357\) 13.5066 0.714844
\(358\) 0 0
\(359\) 3.79289 0.200181 0.100091 0.994978i \(-0.468087\pi\)
0.100091 + 0.994978i \(0.468087\pi\)
\(360\) 0 0
\(361\) −17.1768 −0.904042
\(362\) 0 0
\(363\) 2.48119 0.130229
\(364\) 0 0
\(365\) 11.3684 0.595047
\(366\) 0 0
\(367\) 6.36977 0.332500 0.166250 0.986084i \(-0.446834\pi\)
0.166250 + 0.986084i \(0.446834\pi\)
\(368\) 0 0
\(369\) −15.8618 −0.825731
\(370\) 0 0
\(371\) −0.231548 −0.0120214
\(372\) 0 0
\(373\) −21.3317 −1.10451 −0.552257 0.833674i \(-0.686233\pi\)
−0.552257 + 0.833674i \(0.686233\pi\)
\(374\) 0 0
\(375\) −2.48119 −0.128128
\(376\) 0 0
\(377\) −21.0640 −1.08485
\(378\) 0 0
\(379\) −24.7875 −1.27325 −0.636624 0.771174i \(-0.719670\pi\)
−0.636624 + 0.771174i \(0.719670\pi\)
\(380\) 0 0
\(381\) 42.0870 2.15618
\(382\) 0 0
\(383\) 5.45817 0.278900 0.139450 0.990229i \(-0.455467\pi\)
0.139450 + 0.990229i \(0.455467\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −18.6048 −0.945737
\(388\) 0 0
\(389\) −13.7235 −0.695811 −0.347906 0.937530i \(-0.613107\pi\)
−0.347906 + 0.937530i \(0.613107\pi\)
\(390\) 0 0
\(391\) 17.3865 0.879271
\(392\) 0 0
\(393\) 24.6253 1.24218
\(394\) 0 0
\(395\) 1.96968 0.0991055
\(396\) 0 0
\(397\) 2.11142 0.105969 0.0529846 0.998595i \(-0.483127\pi\)
0.0529846 + 0.998595i \(0.483127\pi\)
\(398\) 0 0
\(399\) 3.35026 0.167723
\(400\) 0 0
\(401\) 19.1490 0.956257 0.478128 0.878290i \(-0.341315\pi\)
0.478128 + 0.878290i \(0.341315\pi\)
\(402\) 0 0
\(403\) 30.8324 1.53587
\(404\) 0 0
\(405\) 8.50659 0.422696
\(406\) 0 0
\(407\) −8.54420 −0.423520
\(408\) 0 0
\(409\) −18.6883 −0.924077 −0.462039 0.886860i \(-0.652882\pi\)
−0.462039 + 0.886860i \(0.652882\pi\)
\(410\) 0 0
\(411\) −27.2750 −1.34538
\(412\) 0 0
\(413\) −13.5999 −0.669208
\(414\) 0 0
\(415\) 10.6253 0.521575
\(416\) 0 0
\(417\) −17.0884 −0.836822
\(418\) 0 0
\(419\) 0.773377 0.0377819 0.0188910 0.999822i \(-0.493986\pi\)
0.0188910 + 0.999822i \(0.493986\pi\)
\(420\) 0 0
\(421\) −10.5198 −0.512702 −0.256351 0.966584i \(-0.582520\pi\)
−0.256351 + 0.966584i \(0.582520\pi\)
\(422\) 0 0
\(423\) 37.3439 1.81572
\(424\) 0 0
\(425\) 5.44358 0.264053
\(426\) 0 0
\(427\) −1.41327 −0.0683927
\(428\) 0 0
\(429\) 14.4690 0.698569
\(430\) 0 0
\(431\) 24.7308 1.19124 0.595621 0.803265i \(-0.296906\pi\)
0.595621 + 0.803265i \(0.296906\pi\)
\(432\) 0 0
\(433\) −18.5599 −0.891933 −0.445967 0.895050i \(-0.647140\pi\)
−0.445967 + 0.895050i \(0.647140\pi\)
\(434\) 0 0
\(435\) 8.96239 0.429714
\(436\) 0 0
\(437\) 4.31265 0.206302
\(438\) 0 0
\(439\) 1.42548 0.0680347 0.0340173 0.999421i \(-0.489170\pi\)
0.0340173 + 0.999421i \(0.489170\pi\)
\(440\) 0 0
\(441\) 3.15633 0.150301
\(442\) 0 0
\(443\) −40.1925 −1.90960 −0.954802 0.297242i \(-0.903933\pi\)
−0.954802 + 0.297242i \(0.903933\pi\)
\(444\) 0 0
\(445\) −7.22425 −0.342462
\(446\) 0 0
\(447\) 56.6009 2.67713
\(448\) 0 0
\(449\) 12.6556 0.597256 0.298628 0.954370i \(-0.403471\pi\)
0.298628 + 0.954370i \(0.403471\pi\)
\(450\) 0 0
\(451\) −5.02539 −0.236636
\(452\) 0 0
\(453\) 8.05079 0.378259
\(454\) 0 0
\(455\) −5.83146 −0.273383
\(456\) 0 0
\(457\) −0.544198 −0.0254565 −0.0127283 0.999919i \(-0.504052\pi\)
−0.0127283 + 0.999919i \(0.504052\pi\)
\(458\) 0 0
\(459\) 2.11142 0.0985526
\(460\) 0 0
\(461\) −11.5755 −0.539123 −0.269562 0.962983i \(-0.586879\pi\)
−0.269562 + 0.962983i \(0.586879\pi\)
\(462\) 0 0
\(463\) −23.7948 −1.10584 −0.552919 0.833235i \(-0.686486\pi\)
−0.552919 + 0.833235i \(0.686486\pi\)
\(464\) 0 0
\(465\) −13.1187 −0.608366
\(466\) 0 0
\(467\) 2.66784 0.123453 0.0617264 0.998093i \(-0.480339\pi\)
0.0617264 + 0.998093i \(0.480339\pi\)
\(468\) 0 0
\(469\) 10.8568 0.501323
\(470\) 0 0
\(471\) −13.4617 −0.620282
\(472\) 0 0
\(473\) −5.89446 −0.271028
\(474\) 0 0
\(475\) 1.35026 0.0619543
\(476\) 0 0
\(477\) −0.730841 −0.0334629
\(478\) 0 0
\(479\) 10.7104 0.489369 0.244685 0.969603i \(-0.421316\pi\)
0.244685 + 0.969603i \(0.421316\pi\)
\(480\) 0 0
\(481\) −49.8251 −2.27183
\(482\) 0 0
\(483\) 7.92478 0.360590
\(484\) 0 0
\(485\) 0.836381 0.0379781
\(486\) 0 0
\(487\) −17.4314 −0.789891 −0.394945 0.918705i \(-0.629236\pi\)
−0.394945 + 0.918705i \(0.629236\pi\)
\(488\) 0 0
\(489\) −8.38787 −0.379313
\(490\) 0 0
\(491\) 28.3693 1.28029 0.640145 0.768254i \(-0.278874\pi\)
0.640145 + 0.768254i \(0.278874\pi\)
\(492\) 0 0
\(493\) −19.6629 −0.885573
\(494\) 0 0
\(495\) −3.15633 −0.141866
\(496\) 0 0
\(497\) 15.5369 0.696925
\(498\) 0 0
\(499\) −27.4763 −1.23001 −0.615003 0.788524i \(-0.710845\pi\)
−0.615003 + 0.788524i \(0.710845\pi\)
\(500\) 0 0
\(501\) −27.9756 −1.24986
\(502\) 0 0
\(503\) −20.2981 −0.905046 −0.452523 0.891753i \(-0.649476\pi\)
−0.452523 + 0.891753i \(0.649476\pi\)
\(504\) 0 0
\(505\) −7.41327 −0.329886
\(506\) 0 0
\(507\) 52.1197 2.31472
\(508\) 0 0
\(509\) −24.2619 −1.07539 −0.537694 0.843140i \(-0.680704\pi\)
−0.537694 + 0.843140i \(0.680704\pi\)
\(510\) 0 0
\(511\) −11.3684 −0.502907
\(512\) 0 0
\(513\) 0.523730 0.0231233
\(514\) 0 0
\(515\) 4.21933 0.185926
\(516\) 0 0
\(517\) 11.8315 0.520347
\(518\) 0 0
\(519\) −22.2823 −0.978086
\(520\) 0 0
\(521\) 2.20123 0.0964377 0.0482188 0.998837i \(-0.484646\pi\)
0.0482188 + 0.998837i \(0.484646\pi\)
\(522\) 0 0
\(523\) 22.1378 0.968017 0.484008 0.875063i \(-0.339180\pi\)
0.484008 + 0.875063i \(0.339180\pi\)
\(524\) 0 0
\(525\) 2.48119 0.108288
\(526\) 0 0
\(527\) 28.7816 1.25375
\(528\) 0 0
\(529\) −12.7988 −0.556468
\(530\) 0 0
\(531\) −42.9257 −1.86282
\(532\) 0 0
\(533\) −29.3054 −1.26936
\(534\) 0 0
\(535\) −11.5369 −0.498784
\(536\) 0 0
\(537\) 65.1002 2.80928
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 23.0640 0.991597 0.495799 0.868438i \(-0.334875\pi\)
0.495799 + 0.868438i \(0.334875\pi\)
\(542\) 0 0
\(543\) −27.7889 −1.19254
\(544\) 0 0
\(545\) 2.18664 0.0936655
\(546\) 0 0
\(547\) −21.3766 −0.913998 −0.456999 0.889467i \(-0.651076\pi\)
−0.456999 + 0.889467i \(0.651076\pi\)
\(548\) 0 0
\(549\) −4.46073 −0.190379
\(550\) 0 0
\(551\) −4.87732 −0.207781
\(552\) 0 0
\(553\) −1.96968 −0.0837594
\(554\) 0 0
\(555\) 21.1998 0.899882
\(556\) 0 0
\(557\) −9.19394 −0.389560 −0.194780 0.980847i \(-0.562399\pi\)
−0.194780 + 0.980847i \(0.562399\pi\)
\(558\) 0 0
\(559\) −34.3733 −1.45384
\(560\) 0 0
\(561\) 13.5066 0.570249
\(562\) 0 0
\(563\) 9.79877 0.412969 0.206484 0.978450i \(-0.433798\pi\)
0.206484 + 0.978450i \(0.433798\pi\)
\(564\) 0 0
\(565\) 9.35026 0.393368
\(566\) 0 0
\(567\) −8.50659 −0.357243
\(568\) 0 0
\(569\) −33.5125 −1.40492 −0.702458 0.711725i \(-0.747914\pi\)
−0.702458 + 0.711725i \(0.747914\pi\)
\(570\) 0 0
\(571\) −43.1392 −1.80532 −0.902659 0.430356i \(-0.858388\pi\)
−0.902659 + 0.430356i \(0.858388\pi\)
\(572\) 0 0
\(573\) −27.7889 −1.16090
\(574\) 0 0
\(575\) 3.19394 0.133196
\(576\) 0 0
\(577\) −14.8510 −0.618254 −0.309127 0.951021i \(-0.600037\pi\)
−0.309127 + 0.951021i \(0.600037\pi\)
\(578\) 0 0
\(579\) 1.50071 0.0623673
\(580\) 0 0
\(581\) −10.6253 −0.440812
\(582\) 0 0
\(583\) −0.231548 −0.00958974
\(584\) 0 0
\(585\) −18.4060 −0.760993
\(586\) 0 0
\(587\) −14.7938 −0.610607 −0.305304 0.952255i \(-0.598758\pi\)
−0.305304 + 0.952255i \(0.598758\pi\)
\(588\) 0 0
\(589\) 7.13918 0.294165
\(590\) 0 0
\(591\) 37.9756 1.56211
\(592\) 0 0
\(593\) −27.4191 −1.12597 −0.562985 0.826467i \(-0.690347\pi\)
−0.562985 + 0.826467i \(0.690347\pi\)
\(594\) 0 0
\(595\) −5.44358 −0.223165
\(596\) 0 0
\(597\) 31.1695 1.27568
\(598\) 0 0
\(599\) −11.3258 −0.462761 −0.231380 0.972863i \(-0.574324\pi\)
−0.231380 + 0.972863i \(0.574324\pi\)
\(600\) 0 0
\(601\) 15.5393 0.633860 0.316930 0.948449i \(-0.397348\pi\)
0.316930 + 0.948449i \(0.397348\pi\)
\(602\) 0 0
\(603\) 34.2677 1.39549
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −17.7235 −0.719377 −0.359688 0.933073i \(-0.617117\pi\)
−0.359688 + 0.933073i \(0.617117\pi\)
\(608\) 0 0
\(609\) −8.96239 −0.363174
\(610\) 0 0
\(611\) 68.9946 2.79122
\(612\) 0 0
\(613\) 22.2941 0.900450 0.450225 0.892915i \(-0.351344\pi\)
0.450225 + 0.892915i \(0.351344\pi\)
\(614\) 0 0
\(615\) 12.4690 0.502798
\(616\) 0 0
\(617\) −30.9438 −1.24575 −0.622876 0.782321i \(-0.714036\pi\)
−0.622876 + 0.782321i \(0.714036\pi\)
\(618\) 0 0
\(619\) −32.4119 −1.30274 −0.651371 0.758759i \(-0.725806\pi\)
−0.651371 + 0.758759i \(0.725806\pi\)
\(620\) 0 0
\(621\) 1.23884 0.0497130
\(622\) 0 0
\(623\) 7.22425 0.289434
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.35026 0.133797
\(628\) 0 0
\(629\) −46.5111 −1.85452
\(630\) 0 0
\(631\) 27.3258 1.08782 0.543912 0.839142i \(-0.316943\pi\)
0.543912 + 0.839142i \(0.316943\pi\)
\(632\) 0 0
\(633\) 11.0132 0.437734
\(634\) 0 0
\(635\) −16.9624 −0.673132
\(636\) 0 0
\(637\) 5.83146 0.231051
\(638\) 0 0
\(639\) 49.0395 1.93997
\(640\) 0 0
\(641\) 19.4460 0.768069 0.384034 0.923319i \(-0.374534\pi\)
0.384034 + 0.923319i \(0.374534\pi\)
\(642\) 0 0
\(643\) −5.29314 −0.208741 −0.104370 0.994538i \(-0.533283\pi\)
−0.104370 + 0.994538i \(0.533283\pi\)
\(644\) 0 0
\(645\) 14.6253 0.575871
\(646\) 0 0
\(647\) −35.0966 −1.37979 −0.689896 0.723909i \(-0.742344\pi\)
−0.689896 + 0.723909i \(0.742344\pi\)
\(648\) 0 0
\(649\) −13.5999 −0.533843
\(650\) 0 0
\(651\) 13.1187 0.514163
\(652\) 0 0
\(653\) 27.7988 1.08785 0.543925 0.839134i \(-0.316938\pi\)
0.543925 + 0.839134i \(0.316938\pi\)
\(654\) 0 0
\(655\) −9.92478 −0.387793
\(656\) 0 0
\(657\) −35.8822 −1.39990
\(658\) 0 0
\(659\) −19.6180 −0.764209 −0.382105 0.924119i \(-0.624801\pi\)
−0.382105 + 0.924119i \(0.624801\pi\)
\(660\) 0 0
\(661\) 21.5633 0.838713 0.419357 0.907822i \(-0.362256\pi\)
0.419357 + 0.907822i \(0.362256\pi\)
\(662\) 0 0
\(663\) 78.7631 3.05890
\(664\) 0 0
\(665\) −1.35026 −0.0523609
\(666\) 0 0
\(667\) −11.5369 −0.446711
\(668\) 0 0
\(669\) 19.3054 0.746388
\(670\) 0 0
\(671\) −1.41327 −0.0545585
\(672\) 0 0
\(673\) −21.0679 −0.812109 −0.406054 0.913849i \(-0.633096\pi\)
−0.406054 + 0.913849i \(0.633096\pi\)
\(674\) 0 0
\(675\) 0.387873 0.0149292
\(676\) 0 0
\(677\) 34.5174 1.32661 0.663306 0.748349i \(-0.269153\pi\)
0.663306 + 0.748349i \(0.269153\pi\)
\(678\) 0 0
\(679\) −0.836381 −0.0320973
\(680\) 0 0
\(681\) 25.9248 0.993440
\(682\) 0 0
\(683\) −33.7802 −1.29256 −0.646282 0.763099i \(-0.723677\pi\)
−0.646282 + 0.763099i \(0.723677\pi\)
\(684\) 0 0
\(685\) 10.9927 0.420010
\(686\) 0 0
\(687\) −73.0757 −2.78801
\(688\) 0 0
\(689\) −1.35026 −0.0514409
\(690\) 0 0
\(691\) 13.8618 0.527327 0.263663 0.964615i \(-0.415069\pi\)
0.263663 + 0.964615i \(0.415069\pi\)
\(692\) 0 0
\(693\) 3.15633 0.119899
\(694\) 0 0
\(695\) 6.88717 0.261245
\(696\) 0 0
\(697\) −27.3561 −1.03619
\(698\) 0 0
\(699\) −21.6629 −0.819367
\(700\) 0 0
\(701\) 40.5256 1.53063 0.765316 0.643655i \(-0.222583\pi\)
0.765316 + 0.643655i \(0.222583\pi\)
\(702\) 0 0
\(703\) −11.5369 −0.435123
\(704\) 0 0
\(705\) −29.3561 −1.10562
\(706\) 0 0
\(707\) 7.41327 0.278805
\(708\) 0 0
\(709\) −0.850969 −0.0319588 −0.0159794 0.999872i \(-0.505087\pi\)
−0.0159794 + 0.999872i \(0.505087\pi\)
\(710\) 0 0
\(711\) −6.21696 −0.233154
\(712\) 0 0
\(713\) 16.8872 0.632429
\(714\) 0 0
\(715\) −5.83146 −0.218084
\(716\) 0 0
\(717\) 52.6371 1.96577
\(718\) 0 0
\(719\) −22.5769 −0.841976 −0.420988 0.907066i \(-0.638317\pi\)
−0.420988 + 0.907066i \(0.638317\pi\)
\(720\) 0 0
\(721\) −4.21933 −0.157136
\(722\) 0 0
\(723\) −23.1695 −0.861683
\(724\) 0 0
\(725\) −3.61213 −0.134151
\(726\) 0 0
\(727\) −12.5174 −0.464244 −0.232122 0.972687i \(-0.574567\pi\)
−0.232122 + 0.972687i \(0.574567\pi\)
\(728\) 0 0
\(729\) −29.7367 −1.10136
\(730\) 0 0
\(731\) −32.0870 −1.18678
\(732\) 0 0
\(733\) 16.6678 0.615641 0.307820 0.951445i \(-0.400400\pi\)
0.307820 + 0.951445i \(0.400400\pi\)
\(734\) 0 0
\(735\) −2.48119 −0.0915202
\(736\) 0 0
\(737\) 10.8568 0.399917
\(738\) 0 0
\(739\) −42.7005 −1.57076 −0.785382 0.619011i \(-0.787533\pi\)
−0.785382 + 0.619011i \(0.787533\pi\)
\(740\) 0 0
\(741\) 19.5369 0.717706
\(742\) 0 0
\(743\) 19.6873 0.722259 0.361129 0.932516i \(-0.382391\pi\)
0.361129 + 0.932516i \(0.382391\pi\)
\(744\) 0 0
\(745\) −22.8119 −0.835765
\(746\) 0 0
\(747\) −33.5369 −1.22705
\(748\) 0 0
\(749\) 11.5369 0.421549
\(750\) 0 0
\(751\) 5.85940 0.213813 0.106906 0.994269i \(-0.465906\pi\)
0.106906 + 0.994269i \(0.465906\pi\)
\(752\) 0 0
\(753\) −4.65562 −0.169660
\(754\) 0 0
\(755\) −3.24472 −0.118088
\(756\) 0 0
\(757\) −40.5863 −1.47513 −0.737567 0.675274i \(-0.764025\pi\)
−0.737567 + 0.675274i \(0.764025\pi\)
\(758\) 0 0
\(759\) 7.92478 0.287651
\(760\) 0 0
\(761\) 21.8472 0.791960 0.395980 0.918259i \(-0.370405\pi\)
0.395980 + 0.918259i \(0.370405\pi\)
\(762\) 0 0
\(763\) −2.18664 −0.0791618
\(764\) 0 0
\(765\) −17.1817 −0.621206
\(766\) 0 0
\(767\) −79.3073 −2.86362
\(768\) 0 0
\(769\) 45.2892 1.63317 0.816585 0.577226i \(-0.195865\pi\)
0.816585 + 0.577226i \(0.195865\pi\)
\(770\) 0 0
\(771\) 67.3376 2.42510
\(772\) 0 0
\(773\) −33.8153 −1.21625 −0.608125 0.793841i \(-0.708078\pi\)
−0.608125 + 0.793841i \(0.708078\pi\)
\(774\) 0 0
\(775\) 5.28726 0.189924
\(776\) 0 0
\(777\) −21.1998 −0.760539
\(778\) 0 0
\(779\) −6.78560 −0.243119
\(780\) 0 0
\(781\) 15.5369 0.555954
\(782\) 0 0
\(783\) −1.40105 −0.0500693
\(784\) 0 0
\(785\) 5.42548 0.193644
\(786\) 0 0
\(787\) −1.27504 −0.0454502 −0.0227251 0.999742i \(-0.507234\pi\)
−0.0227251 + 0.999742i \(0.507234\pi\)
\(788\) 0 0
\(789\) −31.7889 −1.13172
\(790\) 0 0
\(791\) −9.35026 −0.332457
\(792\) 0 0
\(793\) −8.24140 −0.292661
\(794\) 0 0
\(795\) 0.574515 0.0203760
\(796\) 0 0
\(797\) −42.5256 −1.50634 −0.753168 0.657829i \(-0.771475\pi\)
−0.753168 + 0.657829i \(0.771475\pi\)
\(798\) 0 0
\(799\) 64.4055 2.27850
\(800\) 0 0
\(801\) 22.8021 0.805672
\(802\) 0 0
\(803\) −11.3684 −0.401181
\(804\) 0 0
\(805\) −3.19394 −0.112571
\(806\) 0 0
\(807\) −15.5369 −0.546925
\(808\) 0 0
\(809\) −14.7151 −0.517356 −0.258678 0.965964i \(-0.583287\pi\)
−0.258678 + 0.965964i \(0.583287\pi\)
\(810\) 0 0
\(811\) −51.7743 −1.81804 −0.909021 0.416750i \(-0.863169\pi\)
−0.909021 + 0.416750i \(0.863169\pi\)
\(812\) 0 0
\(813\) 14.2374 0.499328
\(814\) 0 0
\(815\) 3.38058 0.118417
\(816\) 0 0
\(817\) −7.95906 −0.278452
\(818\) 0 0
\(819\) 18.4060 0.643157
\(820\) 0 0
\(821\) 2.64974 0.0924765 0.0462383 0.998930i \(-0.485277\pi\)
0.0462383 + 0.998930i \(0.485277\pi\)
\(822\) 0 0
\(823\) −5.76845 −0.201076 −0.100538 0.994933i \(-0.532056\pi\)
−0.100538 + 0.994933i \(0.532056\pi\)
\(824\) 0 0
\(825\) 2.48119 0.0863841
\(826\) 0 0
\(827\) 13.4920 0.469163 0.234581 0.972096i \(-0.424628\pi\)
0.234581 + 0.972096i \(0.424628\pi\)
\(828\) 0 0
\(829\) −4.70052 −0.163256 −0.0816280 0.996663i \(-0.526012\pi\)
−0.0816280 + 0.996663i \(0.526012\pi\)
\(830\) 0 0
\(831\) −20.7367 −0.719349
\(832\) 0 0
\(833\) 5.44358 0.188609
\(834\) 0 0
\(835\) 11.2750 0.390189
\(836\) 0 0
\(837\) 2.05079 0.0708855
\(838\) 0 0
\(839\) 38.8045 1.33968 0.669839 0.742506i \(-0.266363\pi\)
0.669839 + 0.742506i \(0.266363\pi\)
\(840\) 0 0
\(841\) −15.9525 −0.550088
\(842\) 0 0
\(843\) −20.9624 −0.721983
\(844\) 0 0
\(845\) −21.0059 −0.722624
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −2.07522 −0.0712215
\(850\) 0 0
\(851\) −27.2896 −0.935476
\(852\) 0 0
\(853\) 20.6824 0.708153 0.354076 0.935217i \(-0.384795\pi\)
0.354076 + 0.935217i \(0.384795\pi\)
\(854\) 0 0
\(855\) −4.26187 −0.145753
\(856\) 0 0
\(857\) 26.3453 0.899940 0.449970 0.893044i \(-0.351435\pi\)
0.449970 + 0.893044i \(0.351435\pi\)
\(858\) 0 0
\(859\) 8.51151 0.290409 0.145205 0.989402i \(-0.453616\pi\)
0.145205 + 0.989402i \(0.453616\pi\)
\(860\) 0 0
\(861\) −12.4690 −0.424942
\(862\) 0 0
\(863\) −7.56722 −0.257591 −0.128796 0.991671i \(-0.541111\pi\)
−0.128796 + 0.991671i \(0.541111\pi\)
\(864\) 0 0
\(865\) 8.98049 0.305346
\(866\) 0 0
\(867\) 31.3439 1.06450
\(868\) 0 0
\(869\) −1.96968 −0.0668169
\(870\) 0 0
\(871\) 63.3112 2.14522
\(872\) 0 0
\(873\) −2.63989 −0.0893467
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 17.2955 0.584028 0.292014 0.956414i \(-0.405675\pi\)
0.292014 + 0.956414i \(0.405675\pi\)
\(878\) 0 0
\(879\) −6.74543 −0.227518
\(880\) 0 0
\(881\) 20.4504 0.688992 0.344496 0.938788i \(-0.388050\pi\)
0.344496 + 0.938788i \(0.388050\pi\)
\(882\) 0 0
\(883\) 49.6589 1.67116 0.835578 0.549371i \(-0.185133\pi\)
0.835578 + 0.549371i \(0.185133\pi\)
\(884\) 0 0
\(885\) 33.7440 1.13429
\(886\) 0 0
\(887\) 47.1100 1.58180 0.790900 0.611946i \(-0.209613\pi\)
0.790900 + 0.611946i \(0.209613\pi\)
\(888\) 0 0
\(889\) 16.9624 0.568900
\(890\) 0 0
\(891\) −8.50659 −0.284981
\(892\) 0 0
\(893\) 15.9756 0.534602
\(894\) 0 0
\(895\) −26.2374 −0.877020
\(896\) 0 0
\(897\) 46.2130 1.54301
\(898\) 0 0
\(899\) −19.0982 −0.636962
\(900\) 0 0
\(901\) −1.26045 −0.0419917
\(902\) 0 0
\(903\) −14.6253 −0.486700
\(904\) 0 0
\(905\) 11.1998 0.372294
\(906\) 0 0
\(907\) −14.4591 −0.480107 −0.240054 0.970760i \(-0.577165\pi\)
−0.240054 + 0.970760i \(0.577165\pi\)
\(908\) 0 0
\(909\) 23.3987 0.776085
\(910\) 0 0
\(911\) 31.5369 1.04486 0.522432 0.852681i \(-0.325025\pi\)
0.522432 + 0.852681i \(0.325025\pi\)
\(912\) 0 0
\(913\) −10.6253 −0.351646
\(914\) 0 0
\(915\) 3.50659 0.115924
\(916\) 0 0
\(917\) 9.92478 0.327745
\(918\) 0 0
\(919\) −5.26328 −0.173620 −0.0868098 0.996225i \(-0.527667\pi\)
−0.0868098 + 0.996225i \(0.527667\pi\)
\(920\) 0 0
\(921\) 20.7513 0.683779
\(922\) 0 0
\(923\) 90.6028 2.98223
\(924\) 0 0
\(925\) −8.54420 −0.280932
\(926\) 0 0
\(927\) −13.3176 −0.437407
\(928\) 0 0
\(929\) 26.0508 0.854699 0.427349 0.904087i \(-0.359447\pi\)
0.427349 + 0.904087i \(0.359447\pi\)
\(930\) 0 0
\(931\) 1.35026 0.0442530
\(932\) 0 0
\(933\) −11.0073 −0.360363
\(934\) 0 0
\(935\) −5.44358 −0.178024
\(936\) 0 0
\(937\) −29.3439 −0.958624 −0.479312 0.877645i \(-0.659114\pi\)
−0.479312 + 0.877645i \(0.659114\pi\)
\(938\) 0 0
\(939\) 73.9121 2.41203
\(940\) 0 0
\(941\) −28.6375 −0.933556 −0.466778 0.884374i \(-0.654585\pi\)
−0.466778 + 0.884374i \(0.654585\pi\)
\(942\) 0 0
\(943\) −16.0508 −0.522685
\(944\) 0 0
\(945\) −0.387873 −0.0126175
\(946\) 0 0
\(947\) 52.8178 1.71635 0.858174 0.513358i \(-0.171599\pi\)
0.858174 + 0.513358i \(0.171599\pi\)
\(948\) 0 0
\(949\) −66.2941 −2.15200
\(950\) 0 0
\(951\) 38.2130 1.23914
\(952\) 0 0
\(953\) 37.1939 1.20483 0.602415 0.798183i \(-0.294205\pi\)
0.602415 + 0.798183i \(0.294205\pi\)
\(954\) 0 0
\(955\) 11.1998 0.362418
\(956\) 0 0
\(957\) −8.96239 −0.289713
\(958\) 0 0
\(959\) −10.9927 −0.354973
\(960\) 0 0
\(961\) −3.04491 −0.0982228
\(962\) 0 0
\(963\) 36.4142 1.17343
\(964\) 0 0
\(965\) −0.604833 −0.0194703
\(966\) 0 0
\(967\) −4.07125 −0.130923 −0.0654613 0.997855i \(-0.520852\pi\)
−0.0654613 + 0.997855i \(0.520852\pi\)
\(968\) 0 0
\(969\) 18.2374 0.585871
\(970\) 0 0
\(971\) −0.773377 −0.0248188 −0.0124094 0.999923i \(-0.503950\pi\)
−0.0124094 + 0.999923i \(0.503950\pi\)
\(972\) 0 0
\(973\) −6.88717 −0.220792
\(974\) 0 0
\(975\) 14.4690 0.463378
\(976\) 0 0
\(977\) −37.8740 −1.21170 −0.605848 0.795580i \(-0.707166\pi\)
−0.605848 + 0.795580i \(0.707166\pi\)
\(978\) 0 0
\(979\) 7.22425 0.230888
\(980\) 0 0
\(981\) −6.90175 −0.220356
\(982\) 0 0
\(983\) 15.5794 0.496907 0.248453 0.968644i \(-0.420078\pi\)
0.248453 + 0.968644i \(0.420078\pi\)
\(984\) 0 0
\(985\) −15.3054 −0.487669
\(986\) 0 0
\(987\) 29.3561 0.934416
\(988\) 0 0
\(989\) −18.8265 −0.598649
\(990\) 0 0
\(991\) −27.0982 −0.860804 −0.430402 0.902637i \(-0.641628\pi\)
−0.430402 + 0.902637i \(0.641628\pi\)
\(992\) 0 0
\(993\) −15.5369 −0.493049
\(994\) 0 0
\(995\) −12.5623 −0.398252
\(996\) 0 0
\(997\) 50.4060 1.59637 0.798187 0.602410i \(-0.205793\pi\)
0.798187 + 0.602410i \(0.205793\pi\)
\(998\) 0 0
\(999\) −3.31406 −0.104852
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 6160.2.a.bn.1.3 3
4.3 odd 2 385.2.a.f.1.1 3
12.11 even 2 3465.2.a.bh.1.3 3
20.3 even 4 1925.2.b.n.1849.6 6
20.7 even 4 1925.2.b.n.1849.1 6
20.19 odd 2 1925.2.a.v.1.3 3
28.27 even 2 2695.2.a.g.1.1 3
44.43 even 2 4235.2.a.q.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.f.1.1 3 4.3 odd 2
1925.2.a.v.1.3 3 20.19 odd 2
1925.2.b.n.1849.1 6 20.7 even 4
1925.2.b.n.1849.6 6 20.3 even 4
2695.2.a.g.1.1 3 28.27 even 2
3465.2.a.bh.1.3 3 12.11 even 2
4235.2.a.q.1.3 3 44.43 even 2
6160.2.a.bn.1.3 3 1.1 even 1 trivial