Properties

Label 1305.2.a.o.1.1
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $1$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(1\)
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 145)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.67513 q^{2} +5.15633 q^{4} +1.00000 q^{5} -4.15633 q^{7} -8.44358 q^{8} +O(q^{10})\) \(q-2.67513 q^{2} +5.15633 q^{4} +1.00000 q^{5} -4.15633 q^{7} -8.44358 q^{8} -2.67513 q^{10} -2.80606 q^{11} +1.35026 q^{13} +11.1187 q^{14} +12.2750 q^{16} +7.11871 q^{17} +3.76845 q^{19} +5.15633 q^{20} +7.50659 q^{22} -4.80606 q^{23} +1.00000 q^{25} -3.61213 q^{26} -21.4314 q^{28} -1.00000 q^{29} +0.231548 q^{31} -15.9502 q^{32} -19.0435 q^{34} -4.15633 q^{35} -5.50659 q^{37} -10.0811 q^{38} -8.44358 q^{40} +6.96239 q^{41} -3.19394 q^{43} -14.4690 q^{44} +12.8568 q^{46} -6.41819 q^{47} +10.2750 q^{49} -2.67513 q^{50} +6.96239 q^{52} -6.96239 q^{53} -2.80606 q^{55} +35.0943 q^{56} +2.67513 q^{58} +2.57452 q^{59} +5.35026 q^{61} -0.619421 q^{62} +18.1187 q^{64} +1.35026 q^{65} -3.19394 q^{67} +36.7064 q^{68} +11.1187 q^{70} -11.3503 q^{71} -11.2447 q^{73} +14.7308 q^{74} +19.4314 q^{76} +11.6629 q^{77} -4.73084 q^{79} +12.2750 q^{80} -18.6253 q^{82} -2.54420 q^{83} +7.11871 q^{85} +8.54420 q^{86} +23.6932 q^{88} -14.3127 q^{89} -5.61213 q^{91} -24.7816 q^{92} +17.1695 q^{94} +3.76845 q^{95} -1.53102 q^{97} -27.4871 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 5 q^{4} + 3 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 5 q^{4} + 3 q^{5} - 2 q^{7} - 9 q^{8} - 3 q^{10} - 8 q^{11} - 6 q^{13} + 12 q^{14} + 5 q^{16} + 5 q^{20} + 2 q^{22} - 14 q^{23} + 3 q^{25} - 10 q^{26} - 22 q^{28} - 3 q^{29} + 12 q^{31} - 11 q^{32} - 14 q^{34} - 2 q^{35} + 4 q^{37} + 2 q^{38} - 9 q^{40} + 10 q^{41} - 10 q^{43} - 12 q^{44} + 8 q^{46} - 18 q^{47} - q^{49} - 3 q^{50} + 10 q^{52} - 10 q^{53} - 8 q^{55} + 32 q^{56} + 3 q^{58} - 4 q^{59} + 6 q^{61} - 14 q^{62} + 33 q^{64} - 6 q^{65} - 10 q^{67} + 36 q^{68} + 12 q^{70} - 24 q^{71} - 4 q^{73} + 22 q^{74} + 16 q^{76} + 4 q^{77} + 8 q^{79} + 5 q^{80} - 14 q^{82} + 2 q^{83} + 16 q^{86} + 38 q^{88} - 22 q^{89} - 16 q^{91} - 22 q^{92} - 36 q^{97} - 23 q^{98}+O(q^{100}) \) 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 \(-0.894721\pi\)
−0.945802 + 0.324745i \(0.894721\pi\)
\(3\) 0 0
\(4\) 5.15633 2.57816
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.15633 −1.57094 −0.785472 0.618898i \(-0.787580\pi\)
−0.785472 + 0.618898i \(0.787580\pi\)
\(8\) −8.44358 −2.98526
\(9\) 0 0
\(10\) −2.67513 −0.845951
\(11\) −2.80606 −0.846060 −0.423030 0.906116i \(-0.639034\pi\)
−0.423030 + 0.906116i \(0.639034\pi\)
\(12\) 0 0
\(13\) 1.35026 0.374495 0.187248 0.982313i \(-0.440043\pi\)
0.187248 + 0.982313i \(0.440043\pi\)
\(14\) 11.1187 2.97160
\(15\) 0 0
\(16\) 12.2750 3.06876
\(17\) 7.11871 1.72654 0.863271 0.504741i \(-0.168412\pi\)
0.863271 + 0.504741i \(0.168412\pi\)
\(18\) 0 0
\(19\) 3.76845 0.864542 0.432271 0.901744i \(-0.357712\pi\)
0.432271 + 0.901744i \(0.357712\pi\)
\(20\) 5.15633 1.15299
\(21\) 0 0
\(22\) 7.50659 1.60041
\(23\) −4.80606 −1.00213 −0.501067 0.865409i \(-0.667059\pi\)
−0.501067 + 0.865409i \(0.667059\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.61213 −0.708396
\(27\) 0 0
\(28\) −21.4314 −4.05015
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 0.231548 0.0415872 0.0207936 0.999784i \(-0.493381\pi\)
0.0207936 + 0.999784i \(0.493381\pi\)
\(32\) −15.9502 −2.81962
\(33\) 0 0
\(34\) −19.0435 −3.26593
\(35\) −4.15633 −0.702547
\(36\) 0 0
\(37\) −5.50659 −0.905277 −0.452639 0.891694i \(-0.649517\pi\)
−0.452639 + 0.891694i \(0.649517\pi\)
\(38\) −10.0811 −1.63537
\(39\) 0 0
\(40\) −8.44358 −1.33505
\(41\) 6.96239 1.08734 0.543671 0.839298i \(-0.317034\pi\)
0.543671 + 0.839298i \(0.317034\pi\)
\(42\) 0 0
\(43\) −3.19394 −0.487071 −0.243535 0.969892i \(-0.578307\pi\)
−0.243535 + 0.969892i \(0.578307\pi\)
\(44\) −14.4690 −2.18128
\(45\) 0 0
\(46\) 12.8568 1.89564
\(47\) −6.41819 −0.936189 −0.468095 0.883678i \(-0.655059\pi\)
−0.468095 + 0.883678i \(0.655059\pi\)
\(48\) 0 0
\(49\) 10.2750 1.46786
\(50\) −2.67513 −0.378321
\(51\) 0 0
\(52\) 6.96239 0.965510
\(53\) −6.96239 −0.956358 −0.478179 0.878263i \(-0.658703\pi\)
−0.478179 + 0.878263i \(0.658703\pi\)
\(54\) 0 0
\(55\) −2.80606 −0.378370
\(56\) 35.0943 4.68967
\(57\) 0 0
\(58\) 2.67513 0.351262
\(59\) 2.57452 0.335173 0.167587 0.985857i \(-0.446403\pi\)
0.167587 + 0.985857i \(0.446403\pi\)
\(60\) 0 0
\(61\) 5.35026 0.685031 0.342515 0.939512i \(-0.388721\pi\)
0.342515 + 0.939512i \(0.388721\pi\)
\(62\) −0.619421 −0.0786666
\(63\) 0 0
\(64\) 18.1187 2.26484
\(65\) 1.35026 0.167479
\(66\) 0 0
\(67\) −3.19394 −0.390201 −0.195101 0.980783i \(-0.562503\pi\)
−0.195101 + 0.980783i \(0.562503\pi\)
\(68\) 36.7064 4.45131
\(69\) 0 0
\(70\) 11.1187 1.32894
\(71\) −11.3503 −1.34703 −0.673514 0.739174i \(-0.735216\pi\)
−0.673514 + 0.739174i \(0.735216\pi\)
\(72\) 0 0
\(73\) −11.2447 −1.31610 −0.658048 0.752976i \(-0.728617\pi\)
−0.658048 + 0.752976i \(0.728617\pi\)
\(74\) 14.7308 1.71243
\(75\) 0 0
\(76\) 19.4314 2.22893
\(77\) 11.6629 1.32911
\(78\) 0 0
\(79\) −4.73084 −0.532261 −0.266131 0.963937i \(-0.585745\pi\)
−0.266131 + 0.963937i \(0.585745\pi\)
\(80\) 12.2750 1.37239
\(81\) 0 0
\(82\) −18.6253 −2.05682
\(83\) −2.54420 −0.279262 −0.139631 0.990204i \(-0.544592\pi\)
−0.139631 + 0.990204i \(0.544592\pi\)
\(84\) 0 0
\(85\) 7.11871 0.772133
\(86\) 8.54420 0.921345
\(87\) 0 0
\(88\) 23.6932 2.52571
\(89\) −14.3127 −1.51714 −0.758569 0.651593i \(-0.774101\pi\)
−0.758569 + 0.651593i \(0.774101\pi\)
\(90\) 0 0
\(91\) −5.61213 −0.588311
\(92\) −24.7816 −2.58366
\(93\) 0 0
\(94\) 17.1695 1.77090
\(95\) 3.76845 0.386635
\(96\) 0 0
\(97\) −1.53102 −0.155452 −0.0777260 0.996975i \(-0.524766\pi\)
−0.0777260 + 0.996975i \(0.524766\pi\)
\(98\) −27.4871 −2.77661
\(99\) 0 0
\(100\) 5.15633 0.515633
\(101\) −2.83638 −0.282230 −0.141115 0.989993i \(-0.545069\pi\)
−0.141115 + 0.989993i \(0.545069\pi\)
\(102\) 0 0
\(103\) −9.89446 −0.974930 −0.487465 0.873142i \(-0.662078\pi\)
−0.487465 + 0.873142i \(0.662078\pi\)
\(104\) −11.4010 −1.11796
\(105\) 0 0
\(106\) 18.6253 1.80905
\(107\) 11.6932 1.13043 0.565214 0.824945i \(-0.308794\pi\)
0.565214 + 0.824945i \(0.308794\pi\)
\(108\) 0 0
\(109\) −14.4993 −1.38878 −0.694390 0.719599i \(-0.744326\pi\)
−0.694390 + 0.719599i \(0.744326\pi\)
\(110\) 7.50659 0.715725
\(111\) 0 0
\(112\) −51.0191 −4.82085
\(113\) 16.3938 1.54219 0.771097 0.636717i \(-0.219708\pi\)
0.771097 + 0.636717i \(0.219708\pi\)
\(114\) 0 0
\(115\) −4.80606 −0.448168
\(116\) −5.15633 −0.478753
\(117\) 0 0
\(118\) −6.88717 −0.634015
\(119\) −29.5877 −2.71230
\(120\) 0 0
\(121\) −3.12601 −0.284183
\(122\) −14.3127 −1.29581
\(123\) 0 0
\(124\) 1.19394 0.107219
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 15.3561 1.36264 0.681319 0.731987i \(-0.261407\pi\)
0.681319 + 0.731987i \(0.261407\pi\)
\(128\) −16.5696 −1.46456
\(129\) 0 0
\(130\) −3.61213 −0.316804
\(131\) −1.38058 −0.120622 −0.0603109 0.998180i \(-0.519209\pi\)
−0.0603109 + 0.998180i \(0.519209\pi\)
\(132\) 0 0
\(133\) −15.6629 −1.35815
\(134\) 8.54420 0.738106
\(135\) 0 0
\(136\) −60.1075 −5.15417
\(137\) 6.49341 0.554770 0.277385 0.960759i \(-0.410532\pi\)
0.277385 + 0.960759i \(0.410532\pi\)
\(138\) 0 0
\(139\) −19.0132 −1.61268 −0.806338 0.591455i \(-0.798554\pi\)
−0.806338 + 0.591455i \(0.798554\pi\)
\(140\) −21.4314 −1.81128
\(141\) 0 0
\(142\) 30.3634 2.54804
\(143\) −3.78892 −0.316845
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 30.0811 2.48953
\(147\) 0 0
\(148\) −28.3938 −2.33395
\(149\) −6.62530 −0.542766 −0.271383 0.962471i \(-0.587481\pi\)
−0.271383 + 0.962471i \(0.587481\pi\)
\(150\) 0 0
\(151\) −9.27504 −0.754792 −0.377396 0.926052i \(-0.623180\pi\)
−0.377396 + 0.926052i \(0.623180\pi\)
\(152\) −31.8192 −2.58088
\(153\) 0 0
\(154\) −31.1998 −2.51415
\(155\) 0.231548 0.0185984
\(156\) 0 0
\(157\) −5.00729 −0.399626 −0.199813 0.979834i \(-0.564033\pi\)
−0.199813 + 0.979834i \(0.564033\pi\)
\(158\) 12.6556 1.00683
\(159\) 0 0
\(160\) −15.9502 −1.26097
\(161\) 19.9756 1.57429
\(162\) 0 0
\(163\) 7.50659 0.587961 0.293981 0.955811i \(-0.405020\pi\)
0.293981 + 0.955811i \(0.405020\pi\)
\(164\) 35.9003 2.80335
\(165\) 0 0
\(166\) 6.80606 0.528253
\(167\) −21.8945 −1.69424 −0.847122 0.531398i \(-0.821667\pi\)
−0.847122 + 0.531398i \(0.821667\pi\)
\(168\) 0 0
\(169\) −11.1768 −0.859753
\(170\) −19.0435 −1.46057
\(171\) 0 0
\(172\) −16.4690 −1.25575
\(173\) −7.02302 −0.533951 −0.266975 0.963703i \(-0.586024\pi\)
−0.266975 + 0.963703i \(0.586024\pi\)
\(174\) 0 0
\(175\) −4.15633 −0.314189
\(176\) −34.4445 −2.59635
\(177\) 0 0
\(178\) 38.2882 2.86982
\(179\) −4.77575 −0.356956 −0.178478 0.983944i \(-0.557117\pi\)
−0.178478 + 0.983944i \(0.557117\pi\)
\(180\) 0 0
\(181\) 1.87399 0.139293 0.0696464 0.997572i \(-0.477813\pi\)
0.0696464 + 0.997572i \(0.477813\pi\)
\(182\) 15.0132 1.11285
\(183\) 0 0
\(184\) 40.5804 2.99163
\(185\) −5.50659 −0.404852
\(186\) 0 0
\(187\) −19.9756 −1.46076
\(188\) −33.0943 −2.41365
\(189\) 0 0
\(190\) −10.0811 −0.731360
\(191\) −19.1187 −1.38338 −0.691691 0.722194i \(-0.743134\pi\)
−0.691691 + 0.722194i \(0.743134\pi\)
\(192\) 0 0
\(193\) 19.8945 1.43203 0.716017 0.698083i \(-0.245963\pi\)
0.716017 + 0.698083i \(0.245963\pi\)
\(194\) 4.09569 0.294053
\(195\) 0 0
\(196\) 52.9814 3.78439
\(197\) 13.5369 0.964464 0.482232 0.876043i \(-0.339826\pi\)
0.482232 + 0.876043i \(0.339826\pi\)
\(198\) 0 0
\(199\) 2.57452 0.182503 0.0912513 0.995828i \(-0.470913\pi\)
0.0912513 + 0.995828i \(0.470913\pi\)
\(200\) −8.44358 −0.597051
\(201\) 0 0
\(202\) 7.58769 0.533868
\(203\) 4.15633 0.291717
\(204\) 0 0
\(205\) 6.96239 0.486274
\(206\) 26.4690 1.84418
\(207\) 0 0
\(208\) 16.5745 1.14924
\(209\) −10.5745 −0.731455
\(210\) 0 0
\(211\) 11.8945 0.818848 0.409424 0.912344i \(-0.365730\pi\)
0.409424 + 0.912344i \(0.365730\pi\)
\(212\) −35.9003 −2.46565
\(213\) 0 0
\(214\) −31.2809 −2.13832
\(215\) −3.19394 −0.217825
\(216\) 0 0
\(217\) −0.962389 −0.0653312
\(218\) 38.7875 2.62702
\(219\) 0 0
\(220\) −14.4690 −0.975498
\(221\) 9.61213 0.646582
\(222\) 0 0
\(223\) −1.11871 −0.0749146 −0.0374573 0.999298i \(-0.511926\pi\)
−0.0374573 + 0.999298i \(0.511926\pi\)
\(224\) 66.2941 4.42946
\(225\) 0 0
\(226\) −43.8554 −2.91722
\(227\) −0.0303172 −0.00201222 −0.00100611 0.999999i \(-0.500320\pi\)
−0.00100611 + 0.999999i \(0.500320\pi\)
\(228\) 0 0
\(229\) −5.84955 −0.386549 −0.193275 0.981145i \(-0.561911\pi\)
−0.193275 + 0.981145i \(0.561911\pi\)
\(230\) 12.8568 0.847755
\(231\) 0 0
\(232\) 8.44358 0.554348
\(233\) −26.1016 −1.70997 −0.854985 0.518652i \(-0.826434\pi\)
−0.854985 + 0.518652i \(0.826434\pi\)
\(234\) 0 0
\(235\) −6.41819 −0.418677
\(236\) 13.2750 0.864131
\(237\) 0 0
\(238\) 79.1509 5.13059
\(239\) −1.42548 −0.0922069 −0.0461035 0.998937i \(-0.514680\pi\)
−0.0461035 + 0.998937i \(0.514680\pi\)
\(240\) 0 0
\(241\) −0.0752228 −0.00484553 −0.00242276 0.999997i \(-0.500771\pi\)
−0.00242276 + 0.999997i \(0.500771\pi\)
\(242\) 8.36248 0.537561
\(243\) 0 0
\(244\) 27.5877 1.76612
\(245\) 10.2750 0.656448
\(246\) 0 0
\(247\) 5.08840 0.323767
\(248\) −1.95509 −0.124149
\(249\) 0 0
\(250\) −2.67513 −0.169190
\(251\) −16.9829 −1.07195 −0.535974 0.844234i \(-0.680056\pi\)
−0.535974 + 0.844234i \(0.680056\pi\)
\(252\) 0 0
\(253\) 13.4861 0.847865
\(254\) −41.0797 −2.57757
\(255\) 0 0
\(256\) 8.08840 0.505525
\(257\) −25.1998 −1.57192 −0.785961 0.618276i \(-0.787831\pi\)
−0.785961 + 0.618276i \(0.787831\pi\)
\(258\) 0 0
\(259\) 22.8872 1.42214
\(260\) 6.96239 0.431789
\(261\) 0 0
\(262\) 3.69323 0.228168
\(263\) 16.1319 0.994735 0.497367 0.867540i \(-0.334300\pi\)
0.497367 + 0.867540i \(0.334300\pi\)
\(264\) 0 0
\(265\) −6.96239 −0.427696
\(266\) 41.9003 2.56907
\(267\) 0 0
\(268\) −16.4690 −1.00600
\(269\) −5.28963 −0.322514 −0.161257 0.986912i \(-0.551555\pi\)
−0.161257 + 0.986912i \(0.551555\pi\)
\(270\) 0 0
\(271\) 1.13330 0.0688432 0.0344216 0.999407i \(-0.489041\pi\)
0.0344216 + 0.999407i \(0.489041\pi\)
\(272\) 87.3825 5.29834
\(273\) 0 0
\(274\) −17.3707 −1.04940
\(275\) −2.80606 −0.169212
\(276\) 0 0
\(277\) −16.3634 −0.983184 −0.491592 0.870826i \(-0.663585\pi\)
−0.491592 + 0.870826i \(0.663585\pi\)
\(278\) 50.8627 3.05054
\(279\) 0 0
\(280\) 35.0943 2.09728
\(281\) 24.8265 1.48103 0.740513 0.672042i \(-0.234582\pi\)
0.740513 + 0.672042i \(0.234582\pi\)
\(282\) 0 0
\(283\) −4.18076 −0.248521 −0.124260 0.992250i \(-0.539656\pi\)
−0.124260 + 0.992250i \(0.539656\pi\)
\(284\) −58.5256 −3.47286
\(285\) 0 0
\(286\) 10.1359 0.599346
\(287\) −28.9380 −1.70815
\(288\) 0 0
\(289\) 33.6761 1.98095
\(290\) 2.67513 0.157089
\(291\) 0 0
\(292\) −57.9814 −3.39311
\(293\) −23.6180 −1.37978 −0.689889 0.723915i \(-0.742341\pi\)
−0.689889 + 0.723915i \(0.742341\pi\)
\(294\) 0 0
\(295\) 2.57452 0.149894
\(296\) 46.4953 2.70249
\(297\) 0 0
\(298\) 17.7235 1.02670
\(299\) −6.48944 −0.375294
\(300\) 0 0
\(301\) 13.2750 0.765161
\(302\) 24.8119 1.42777
\(303\) 0 0
\(304\) 46.2579 2.65307
\(305\) 5.35026 0.306355
\(306\) 0 0
\(307\) 32.5052 1.85517 0.927584 0.373614i \(-0.121882\pi\)
0.927584 + 0.373614i \(0.121882\pi\)
\(308\) 60.1378 3.42667
\(309\) 0 0
\(310\) −0.619421 −0.0351808
\(311\) 9.31994 0.528486 0.264243 0.964456i \(-0.414878\pi\)
0.264243 + 0.964456i \(0.414878\pi\)
\(312\) 0 0
\(313\) 9.60228 0.542753 0.271376 0.962473i \(-0.412521\pi\)
0.271376 + 0.962473i \(0.412521\pi\)
\(314\) 13.3952 0.755933
\(315\) 0 0
\(316\) −24.3938 −1.37226
\(317\) −19.3707 −1.08797 −0.543984 0.839095i \(-0.683085\pi\)
−0.543984 + 0.839095i \(0.683085\pi\)
\(318\) 0 0
\(319\) 2.80606 0.157109
\(320\) 18.1187 1.01287
\(321\) 0 0
\(322\) −53.4372 −2.97794
\(323\) 26.8265 1.49267
\(324\) 0 0
\(325\) 1.35026 0.0748990
\(326\) −20.0811 −1.11219
\(327\) 0 0
\(328\) −58.7875 −3.24600
\(329\) 26.6761 1.47070
\(330\) 0 0
\(331\) −29.5428 −1.62382 −0.811909 0.583784i \(-0.801572\pi\)
−0.811909 + 0.583784i \(0.801572\pi\)
\(332\) −13.1187 −0.719983
\(333\) 0 0
\(334\) 58.5705 3.20484
\(335\) −3.19394 −0.174503
\(336\) 0 0
\(337\) 12.5442 0.683326 0.341663 0.939823i \(-0.389010\pi\)
0.341663 + 0.939823i \(0.389010\pi\)
\(338\) 29.8994 1.62631
\(339\) 0 0
\(340\) 36.7064 1.99068
\(341\) −0.649738 −0.0351853
\(342\) 0 0
\(343\) −13.6121 −0.734986
\(344\) 26.9683 1.45403
\(345\) 0 0
\(346\) 18.7875 1.01002
\(347\) 25.0943 1.34713 0.673566 0.739127i \(-0.264762\pi\)
0.673566 + 0.739127i \(0.264762\pi\)
\(348\) 0 0
\(349\) −17.0738 −0.913940 −0.456970 0.889482i \(-0.651065\pi\)
−0.456970 + 0.889482i \(0.651065\pi\)
\(350\) 11.1187 0.594320
\(351\) 0 0
\(352\) 44.7572 2.38557
\(353\) −5.66291 −0.301406 −0.150703 0.988579i \(-0.548154\pi\)
−0.150703 + 0.988579i \(0.548154\pi\)
\(354\) 0 0
\(355\) −11.3503 −0.602409
\(356\) −73.8007 −3.91143
\(357\) 0 0
\(358\) 12.7757 0.675219
\(359\) −0.755278 −0.0398621 −0.0199310 0.999801i \(-0.506345\pi\)
−0.0199310 + 0.999801i \(0.506345\pi\)
\(360\) 0 0
\(361\) −4.79877 −0.252567
\(362\) −5.01317 −0.263487
\(363\) 0 0
\(364\) −28.9380 −1.51676
\(365\) −11.2447 −0.588576
\(366\) 0 0
\(367\) 11.4460 0.597474 0.298737 0.954335i \(-0.403435\pi\)
0.298737 + 0.954335i \(0.403435\pi\)
\(368\) −58.9946 −3.07531
\(369\) 0 0
\(370\) 14.7308 0.765820
\(371\) 28.9380 1.50238
\(372\) 0 0
\(373\) 3.86414 0.200078 0.100039 0.994984i \(-0.468103\pi\)
0.100039 + 0.994984i \(0.468103\pi\)
\(374\) 53.4372 2.76317
\(375\) 0 0
\(376\) 54.1925 2.79477
\(377\) −1.35026 −0.0695420
\(378\) 0 0
\(379\) 12.1055 0.621820 0.310910 0.950439i \(-0.399366\pi\)
0.310910 + 0.950439i \(0.399366\pi\)
\(380\) 19.4314 0.996808
\(381\) 0 0
\(382\) 51.1451 2.61681
\(383\) 10.0205 0.512022 0.256011 0.966674i \(-0.417592\pi\)
0.256011 + 0.966674i \(0.417592\pi\)
\(384\) 0 0
\(385\) 11.6629 0.594397
\(386\) −53.2203 −2.70884
\(387\) 0 0
\(388\) −7.89446 −0.400780
\(389\) 25.6629 1.30116 0.650581 0.759437i \(-0.274526\pi\)
0.650581 + 0.759437i \(0.274526\pi\)
\(390\) 0 0
\(391\) −34.2130 −1.73023
\(392\) −86.7581 −4.38195
\(393\) 0 0
\(394\) −36.2130 −1.82438
\(395\) −4.73084 −0.238034
\(396\) 0 0
\(397\) 27.7137 1.39091 0.695455 0.718569i \(-0.255203\pi\)
0.695455 + 0.718569i \(0.255203\pi\)
\(398\) −6.88717 −0.345222
\(399\) 0 0
\(400\) 12.2750 0.613752
\(401\) −7.42548 −0.370811 −0.185406 0.982662i \(-0.559360\pi\)
−0.185406 + 0.982662i \(0.559360\pi\)
\(402\) 0 0
\(403\) 0.312650 0.0155742
\(404\) −14.6253 −0.727636
\(405\) 0 0
\(406\) −11.1187 −0.551812
\(407\) 15.4518 0.765919
\(408\) 0 0
\(409\) −33.1998 −1.64163 −0.820813 0.571198i \(-0.806479\pi\)
−0.820813 + 0.571198i \(0.806479\pi\)
\(410\) −18.6253 −0.919838
\(411\) 0 0
\(412\) −51.0191 −2.51353
\(413\) −10.7005 −0.526538
\(414\) 0 0
\(415\) −2.54420 −0.124890
\(416\) −21.5369 −1.05593
\(417\) 0 0
\(418\) 28.2882 1.38362
\(419\) −16.5599 −0.809005 −0.404503 0.914537i \(-0.632555\pi\)
−0.404503 + 0.914537i \(0.632555\pi\)
\(420\) 0 0
\(421\) −8.82653 −0.430179 −0.215089 0.976594i \(-0.569004\pi\)
−0.215089 + 0.976594i \(0.569004\pi\)
\(422\) −31.8192 −1.54894
\(423\) 0 0
\(424\) 58.7875 2.85497
\(425\) 7.11871 0.345308
\(426\) 0 0
\(427\) −22.2374 −1.07614
\(428\) 60.2941 2.91442
\(429\) 0 0
\(430\) 8.54420 0.412038
\(431\) −4.25202 −0.204812 −0.102406 0.994743i \(-0.532654\pi\)
−0.102406 + 0.994743i \(0.532654\pi\)
\(432\) 0 0
\(433\) −1.81924 −0.0874270 −0.0437135 0.999044i \(-0.513919\pi\)
−0.0437135 + 0.999044i \(0.513919\pi\)
\(434\) 2.57452 0.123581
\(435\) 0 0
\(436\) −74.7631 −3.58050
\(437\) −18.1114 −0.866387
\(438\) 0 0
\(439\) 14.1114 0.673501 0.336751 0.941594i \(-0.390672\pi\)
0.336751 + 0.941594i \(0.390672\pi\)
\(440\) 23.6932 1.12953
\(441\) 0 0
\(442\) −25.7137 −1.22308
\(443\) 17.2809 0.821041 0.410521 0.911851i \(-0.365347\pi\)
0.410521 + 0.911851i \(0.365347\pi\)
\(444\) 0 0
\(445\) −14.3127 −0.678485
\(446\) 2.99271 0.141709
\(447\) 0 0
\(448\) −75.3073 −3.55793
\(449\) −9.35026 −0.441266 −0.220633 0.975357i \(-0.570812\pi\)
−0.220633 + 0.975357i \(0.570812\pi\)
\(450\) 0 0
\(451\) −19.5369 −0.919957
\(452\) 84.5315 3.97603
\(453\) 0 0
\(454\) 0.0811024 0.00380632
\(455\) −5.61213 −0.263101
\(456\) 0 0
\(457\) −17.6629 −0.826236 −0.413118 0.910677i \(-0.635560\pi\)
−0.413118 + 0.910677i \(0.635560\pi\)
\(458\) 15.6483 0.731198
\(459\) 0 0
\(460\) −24.7816 −1.15545
\(461\) −15.5633 −0.724853 −0.362426 0.932012i \(-0.618052\pi\)
−0.362426 + 0.932012i \(0.618052\pi\)
\(462\) 0 0
\(463\) 2.98286 0.138625 0.0693126 0.997595i \(-0.477919\pi\)
0.0693126 + 0.997595i \(0.477919\pi\)
\(464\) −12.2750 −0.569854
\(465\) 0 0
\(466\) 69.8251 3.23459
\(467\) −34.5804 −1.60019 −0.800095 0.599873i \(-0.795218\pi\)
−0.800095 + 0.599873i \(0.795218\pi\)
\(468\) 0 0
\(469\) 13.2750 0.612984
\(470\) 17.1695 0.791970
\(471\) 0 0
\(472\) −21.7381 −1.00058
\(473\) 8.96239 0.412091
\(474\) 0 0
\(475\) 3.76845 0.172908
\(476\) −152.564 −6.99275
\(477\) 0 0
\(478\) 3.81336 0.174419
\(479\) 34.1925 1.56230 0.781148 0.624346i \(-0.214634\pi\)
0.781148 + 0.624346i \(0.214634\pi\)
\(480\) 0 0
\(481\) −7.43533 −0.339022
\(482\) 0.201231 0.00916581
\(483\) 0 0
\(484\) −16.1187 −0.732669
\(485\) −1.53102 −0.0695202
\(486\) 0 0
\(487\) 38.4953 1.74439 0.872195 0.489159i \(-0.162696\pi\)
0.872195 + 0.489159i \(0.162696\pi\)
\(488\) −45.1754 −2.04499
\(489\) 0 0
\(490\) −27.4871 −1.24174
\(491\) 27.4676 1.23959 0.619797 0.784762i \(-0.287215\pi\)
0.619797 + 0.784762i \(0.287215\pi\)
\(492\) 0 0
\(493\) −7.11871 −0.320611
\(494\) −13.6121 −0.612439
\(495\) 0 0
\(496\) 2.84226 0.127621
\(497\) 47.1754 2.11610
\(498\) 0 0
\(499\) −32.1016 −1.43706 −0.718532 0.695494i \(-0.755186\pi\)
−0.718532 + 0.695494i \(0.755186\pi\)
\(500\) 5.15633 0.230598
\(501\) 0 0
\(502\) 45.4314 2.02770
\(503\) −9.74401 −0.434464 −0.217232 0.976120i \(-0.569703\pi\)
−0.217232 + 0.976120i \(0.569703\pi\)
\(504\) 0 0
\(505\) −2.83638 −0.126217
\(506\) −36.0771 −1.60382
\(507\) 0 0
\(508\) 79.1813 3.51310
\(509\) 8.57452 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(510\) 0 0
\(511\) 46.7367 2.06751
\(512\) 11.5017 0.508306
\(513\) 0 0
\(514\) 67.4128 2.97345
\(515\) −9.89446 −0.436002
\(516\) 0 0
\(517\) 18.0098 0.792072
\(518\) −61.2262 −2.69012
\(519\) 0 0
\(520\) −11.4010 −0.499969
\(521\) 37.8251 1.65715 0.828574 0.559879i \(-0.189152\pi\)
0.828574 + 0.559879i \(0.189152\pi\)
\(522\) 0 0
\(523\) −13.5818 −0.593891 −0.296946 0.954894i \(-0.595968\pi\)
−0.296946 + 0.954894i \(0.595968\pi\)
\(524\) −7.11871 −0.310982
\(525\) 0 0
\(526\) −43.1549 −1.88164
\(527\) 1.64832 0.0718021
\(528\) 0 0
\(529\) 0.0982457 0.00427155
\(530\) 18.6253 0.809031
\(531\) 0 0
\(532\) −80.7631 −3.50152
\(533\) 9.40105 0.407205
\(534\) 0 0
\(535\) 11.6932 0.505542
\(536\) 26.9683 1.16485
\(537\) 0 0
\(538\) 14.1504 0.610069
\(539\) −28.8324 −1.24190
\(540\) 0 0
\(541\) 42.3127 1.81916 0.909581 0.415526i \(-0.136402\pi\)
0.909581 + 0.415526i \(0.136402\pi\)
\(542\) −3.03173 −0.130224
\(543\) 0 0
\(544\) −113.545 −4.86819
\(545\) −14.4993 −0.621081
\(546\) 0 0
\(547\) −36.4690 −1.55930 −0.779650 0.626215i \(-0.784603\pi\)
−0.779650 + 0.626215i \(0.784603\pi\)
\(548\) 33.4821 1.43029
\(549\) 0 0
\(550\) 7.50659 0.320082
\(551\) −3.76845 −0.160541
\(552\) 0 0
\(553\) 19.6629 0.836152
\(554\) 43.7743 1.85979
\(555\) 0 0
\(556\) −98.0381 −4.15774
\(557\) −19.5223 −0.827187 −0.413594 0.910462i \(-0.635727\pi\)
−0.413594 + 0.910462i \(0.635727\pi\)
\(558\) 0 0
\(559\) −4.31265 −0.182406
\(560\) −51.0191 −2.15595
\(561\) 0 0
\(562\) −66.4142 −2.80151
\(563\) 2.94192 0.123987 0.0619936 0.998077i \(-0.480254\pi\)
0.0619936 + 0.998077i \(0.480254\pi\)
\(564\) 0 0
\(565\) 16.3938 0.689690
\(566\) 11.1841 0.470102
\(567\) 0 0
\(568\) 95.8369 4.02123
\(569\) −2.49929 −0.104776 −0.0523879 0.998627i \(-0.516683\pi\)
−0.0523879 + 0.998627i \(0.516683\pi\)
\(570\) 0 0
\(571\) 43.8007 1.83300 0.916501 0.400033i \(-0.131001\pi\)
0.916501 + 0.400033i \(0.131001\pi\)
\(572\) −19.5369 −0.816879
\(573\) 0 0
\(574\) 77.4128 3.23115
\(575\) −4.80606 −0.200427
\(576\) 0 0
\(577\) −23.9062 −0.995229 −0.497614 0.867398i \(-0.665791\pi\)
−0.497614 + 0.867398i \(0.665791\pi\)
\(578\) −90.0879 −3.74716
\(579\) 0 0
\(580\) −5.15633 −0.214105
\(581\) 10.5745 0.438705
\(582\) 0 0
\(583\) 19.5369 0.809136
\(584\) 94.9457 3.92888
\(585\) 0 0
\(586\) 63.1813 2.60999
\(587\) 3.71767 0.153445 0.0767223 0.997053i \(-0.475555\pi\)
0.0767223 + 0.997053i \(0.475555\pi\)
\(588\) 0 0
\(589\) 0.872577 0.0359539
\(590\) −6.88717 −0.283540
\(591\) 0 0
\(592\) −67.5936 −2.77808
\(593\) 25.5125 1.04767 0.523836 0.851819i \(-0.324501\pi\)
0.523836 + 0.851819i \(0.324501\pi\)
\(594\) 0 0
\(595\) −29.5877 −1.21298
\(596\) −34.1622 −1.39934
\(597\) 0 0
\(598\) 17.3601 0.709908
\(599\) 15.6834 0.640806 0.320403 0.947281i \(-0.396182\pi\)
0.320403 + 0.947281i \(0.396182\pi\)
\(600\) 0 0
\(601\) 13.1392 0.535958 0.267979 0.963425i \(-0.413644\pi\)
0.267979 + 0.963425i \(0.413644\pi\)
\(602\) −35.5125 −1.44738
\(603\) 0 0
\(604\) −47.8251 −1.94598
\(605\) −3.12601 −0.127090
\(606\) 0 0
\(607\) 18.1465 0.736543 0.368271 0.929718i \(-0.379950\pi\)
0.368271 + 0.929718i \(0.379950\pi\)
\(608\) −60.1075 −2.43768
\(609\) 0 0
\(610\) −14.3127 −0.579502
\(611\) −8.66624 −0.350598
\(612\) 0 0
\(613\) −32.5501 −1.31469 −0.657343 0.753592i \(-0.728320\pi\)
−0.657343 + 0.753592i \(0.728320\pi\)
\(614\) −86.9556 −3.50924
\(615\) 0 0
\(616\) −98.4768 −3.96774
\(617\) −20.2433 −0.814965 −0.407482 0.913213i \(-0.633593\pi\)
−0.407482 + 0.913213i \(0.633593\pi\)
\(618\) 0 0
\(619\) −16.2071 −0.651419 −0.325709 0.945470i \(-0.605603\pi\)
−0.325709 + 0.945470i \(0.605603\pi\)
\(620\) 1.19394 0.0479496
\(621\) 0 0
\(622\) −24.9321 −0.999685
\(623\) 59.4880 2.38334
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −25.6873 −1.02667
\(627\) 0 0
\(628\) −25.8192 −1.03030
\(629\) −39.1998 −1.56300
\(630\) 0 0
\(631\) 2.13586 0.0850271 0.0425136 0.999096i \(-0.486463\pi\)
0.0425136 + 0.999096i \(0.486463\pi\)
\(632\) 39.9452 1.58894
\(633\) 0 0
\(634\) 51.8192 2.05800
\(635\) 15.3561 0.609390
\(636\) 0 0
\(637\) 13.8740 0.549708
\(638\) −7.50659 −0.297189
\(639\) 0 0
\(640\) −16.5696 −0.654971
\(641\) 14.0362 0.554396 0.277198 0.960813i \(-0.410594\pi\)
0.277198 + 0.960813i \(0.410594\pi\)
\(642\) 0 0
\(643\) −43.8799 −1.73045 −0.865227 0.501381i \(-0.832826\pi\)
−0.865227 + 0.501381i \(0.832826\pi\)
\(644\) 103.000 4.05879
\(645\) 0 0
\(646\) −71.7645 −2.82354
\(647\) 22.5560 0.886766 0.443383 0.896332i \(-0.353778\pi\)
0.443383 + 0.896332i \(0.353778\pi\)
\(648\) 0 0
\(649\) −7.22425 −0.283577
\(650\) −3.61213 −0.141679
\(651\) 0 0
\(652\) 38.7064 1.51586
\(653\) −27.3054 −1.06854 −0.534271 0.845314i \(-0.679414\pi\)
−0.534271 + 0.845314i \(0.679414\pi\)
\(654\) 0 0
\(655\) −1.38058 −0.0539437
\(656\) 85.4636 3.33679
\(657\) 0 0
\(658\) −71.3620 −2.78198
\(659\) −14.0665 −0.547954 −0.273977 0.961736i \(-0.588339\pi\)
−0.273977 + 0.961736i \(0.588339\pi\)
\(660\) 0 0
\(661\) 38.9741 1.51592 0.757959 0.652302i \(-0.226197\pi\)
0.757959 + 0.652302i \(0.226197\pi\)
\(662\) 79.0308 3.07162
\(663\) 0 0
\(664\) 21.4821 0.833669
\(665\) −15.6629 −0.607382
\(666\) 0 0
\(667\) 4.80606 0.186092
\(668\) −112.895 −4.36804
\(669\) 0 0
\(670\) 8.54420 0.330091
\(671\) −15.0132 −0.579577
\(672\) 0 0
\(673\) 20.3390 0.784011 0.392005 0.919963i \(-0.371781\pi\)
0.392005 + 0.919963i \(0.371781\pi\)
\(674\) −33.5574 −1.29258
\(675\) 0 0
\(676\) −57.6312 −2.21658
\(677\) 19.1841 0.737304 0.368652 0.929567i \(-0.379819\pi\)
0.368652 + 0.929567i \(0.379819\pi\)
\(678\) 0 0
\(679\) 6.36344 0.244206
\(680\) −60.1075 −2.30502
\(681\) 0 0
\(682\) 1.73813 0.0665566
\(683\) 24.1319 0.923381 0.461691 0.887041i \(-0.347243\pi\)
0.461691 + 0.887041i \(0.347243\pi\)
\(684\) 0 0
\(685\) 6.49341 0.248101
\(686\) 36.4142 1.39030
\(687\) 0 0
\(688\) −39.2057 −1.49470
\(689\) −9.40105 −0.358151
\(690\) 0 0
\(691\) −4.28821 −0.163131 −0.0815657 0.996668i \(-0.525992\pi\)
−0.0815657 + 0.996668i \(0.525992\pi\)
\(692\) −36.2130 −1.37661
\(693\) 0 0
\(694\) −67.1305 −2.54824
\(695\) −19.0132 −0.721211
\(696\) 0 0
\(697\) 49.5633 1.87734
\(698\) 45.6747 1.72881
\(699\) 0 0
\(700\) −21.4314 −0.810029
\(701\) 14.1260 0.533532 0.266766 0.963761i \(-0.414045\pi\)
0.266766 + 0.963761i \(0.414045\pi\)
\(702\) 0 0
\(703\) −20.7513 −0.782650
\(704\) −50.8423 −1.91619
\(705\) 0 0
\(706\) 15.1490 0.570141
\(707\) 11.7889 0.443368
\(708\) 0 0
\(709\) 6.75131 0.253551 0.126775 0.991931i \(-0.459537\pi\)
0.126775 + 0.991931i \(0.459537\pi\)
\(710\) 30.3634 1.13952
\(711\) 0 0
\(712\) 120.850 4.52905
\(713\) −1.11283 −0.0416760
\(714\) 0 0
\(715\) −3.78892 −0.141698
\(716\) −24.6253 −0.920291
\(717\) 0 0
\(718\) 2.02047 0.0754032
\(719\) 43.8251 1.63440 0.817201 0.576353i \(-0.195525\pi\)
0.817201 + 0.576353i \(0.195525\pi\)
\(720\) 0 0
\(721\) 41.1246 1.53156
\(722\) 12.8373 0.477756
\(723\) 0 0
\(724\) 9.66291 0.359119
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −14.8813 −0.551916 −0.275958 0.961170i \(-0.588995\pi\)
−0.275958 + 0.961170i \(0.588995\pi\)
\(728\) 47.3865 1.75626
\(729\) 0 0
\(730\) 30.0811 1.11335
\(731\) −22.7367 −0.840948
\(732\) 0 0
\(733\) 7.17935 0.265175 0.132588 0.991171i \(-0.457671\pi\)
0.132588 + 0.991171i \(0.457671\pi\)
\(734\) −30.6194 −1.13018
\(735\) 0 0
\(736\) 76.6575 2.82563
\(737\) 8.96239 0.330134
\(738\) 0 0
\(739\) 16.1709 0.594857 0.297428 0.954744i \(-0.403871\pi\)
0.297428 + 0.954744i \(0.403871\pi\)
\(740\) −28.3938 −1.04378
\(741\) 0 0
\(742\) −77.4128 −2.84191
\(743\) −27.8192 −1.02059 −0.510294 0.860000i \(-0.670464\pi\)
−0.510294 + 0.860000i \(0.670464\pi\)
\(744\) 0 0
\(745\) −6.62530 −0.242732
\(746\) −10.3371 −0.378468
\(747\) 0 0
\(748\) −103.000 −3.76607
\(749\) −48.6009 −1.77584
\(750\) 0 0
\(751\) −17.6326 −0.643423 −0.321711 0.946838i \(-0.604258\pi\)
−0.321711 + 0.946838i \(0.604258\pi\)
\(752\) −78.7835 −2.87294
\(753\) 0 0
\(754\) 3.61213 0.131546
\(755\) −9.27504 −0.337553
\(756\) 0 0
\(757\) 1.53102 0.0556460 0.0278230 0.999613i \(-0.491143\pi\)
0.0278230 + 0.999613i \(0.491143\pi\)
\(758\) −32.3839 −1.17624
\(759\) 0 0
\(760\) −31.8192 −1.15421
\(761\) 34.4749 1.24971 0.624856 0.780740i \(-0.285158\pi\)
0.624856 + 0.780740i \(0.285158\pi\)
\(762\) 0 0
\(763\) 60.2638 2.18170
\(764\) −98.5823 −3.56658
\(765\) 0 0
\(766\) −26.8061 −0.968542
\(767\) 3.47627 0.125521
\(768\) 0 0
\(769\) −19.3404 −0.697433 −0.348717 0.937228i \(-0.613382\pi\)
−0.348717 + 0.937228i \(0.613382\pi\)
\(770\) −31.1998 −1.12436
\(771\) 0 0
\(772\) 102.582 3.69202
\(773\) −36.2677 −1.30446 −0.652230 0.758021i \(-0.726166\pi\)
−0.652230 + 0.758021i \(0.726166\pi\)
\(774\) 0 0
\(775\) 0.231548 0.00831745
\(776\) 12.9273 0.464064
\(777\) 0 0
\(778\) −68.6516 −2.46128
\(779\) 26.2374 0.940053
\(780\) 0 0
\(781\) 31.8496 1.13967
\(782\) 91.5242 3.27290
\(783\) 0 0
\(784\) 126.127 4.50452
\(785\) −5.00729 −0.178718
\(786\) 0 0
\(787\) 32.0059 1.14089 0.570443 0.821337i \(-0.306771\pi\)
0.570443 + 0.821337i \(0.306771\pi\)
\(788\) 69.8007 2.48655
\(789\) 0 0
\(790\) 12.6556 0.450267
\(791\) −68.1378 −2.42270
\(792\) 0 0
\(793\) 7.22425 0.256541
\(794\) −74.1378 −2.63105
\(795\) 0 0
\(796\) 13.2750 0.470521
\(797\) 41.6932 1.47685 0.738425 0.674336i \(-0.235570\pi\)
0.738425 + 0.674336i \(0.235570\pi\)
\(798\) 0 0
\(799\) −45.6893 −1.61637
\(800\) −15.9502 −0.563924
\(801\) 0 0
\(802\) 19.8641 0.701427
\(803\) 31.5534 1.11350
\(804\) 0 0
\(805\) 19.9756 0.704046
\(806\) −0.836381 −0.0294603
\(807\) 0 0
\(808\) 23.9492 0.842530
\(809\) −30.6371 −1.07714 −0.538571 0.842580i \(-0.681036\pi\)
−0.538571 + 0.842580i \(0.681036\pi\)
\(810\) 0 0
\(811\) 25.4617 0.894081 0.447040 0.894514i \(-0.352478\pi\)
0.447040 + 0.894514i \(0.352478\pi\)
\(812\) 21.4314 0.752093
\(813\) 0 0
\(814\) −41.3357 −1.44881
\(815\) 7.50659 0.262944
\(816\) 0 0
\(817\) −12.0362 −0.421093
\(818\) 88.8139 3.10530
\(819\) 0 0
\(820\) 35.9003 1.25369
\(821\) 32.7005 1.14126 0.570628 0.821209i \(-0.306700\pi\)
0.570628 + 0.821209i \(0.306700\pi\)
\(822\) 0 0
\(823\) −31.1041 −1.08422 −0.542111 0.840307i \(-0.682375\pi\)
−0.542111 + 0.840307i \(0.682375\pi\)
\(824\) 83.5447 2.91042
\(825\) 0 0
\(826\) 28.6253 0.996002
\(827\) 1.58181 0.0550049 0.0275025 0.999622i \(-0.491245\pi\)
0.0275025 + 0.999622i \(0.491245\pi\)
\(828\) 0 0
\(829\) −0.111420 −0.00386976 −0.00193488 0.999998i \(-0.500616\pi\)
−0.00193488 + 0.999998i \(0.500616\pi\)
\(830\) 6.80606 0.236242
\(831\) 0 0
\(832\) 24.4650 0.848171
\(833\) 73.1451 2.53433
\(834\) 0 0
\(835\) −21.8945 −0.757689
\(836\) −54.5256 −1.88581
\(837\) 0 0
\(838\) 44.3000 1.53032
\(839\) −28.9829 −1.00060 −0.500300 0.865852i \(-0.666777\pi\)
−0.500300 + 0.865852i \(0.666777\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 23.6121 0.813728
\(843\) 0 0
\(844\) 61.3317 2.11112
\(845\) −11.1768 −0.384493
\(846\) 0 0
\(847\) 12.9927 0.446435
\(848\) −85.4636 −2.93483
\(849\) 0 0
\(850\) −19.0435 −0.653186
\(851\) 26.4650 0.907209
\(852\) 0 0
\(853\) −7.77319 −0.266149 −0.133075 0.991106i \(-0.542485\pi\)
−0.133075 + 0.991106i \(0.542485\pi\)
\(854\) 59.4880 2.03564
\(855\) 0 0
\(856\) −98.7328 −3.37462
\(857\) 13.8740 0.473927 0.236963 0.971519i \(-0.423848\pi\)
0.236963 + 0.971519i \(0.423848\pi\)
\(858\) 0 0
\(859\) −15.2809 −0.521378 −0.260689 0.965423i \(-0.583950\pi\)
−0.260689 + 0.965423i \(0.583950\pi\)
\(860\) −16.4690 −0.561587
\(861\) 0 0
\(862\) 11.3747 0.387424
\(863\) −31.2301 −1.06309 −0.531543 0.847031i \(-0.678388\pi\)
−0.531543 + 0.847031i \(0.678388\pi\)
\(864\) 0 0
\(865\) −7.02302 −0.238790
\(866\) 4.86670 0.165377
\(867\) 0 0
\(868\) −4.96239 −0.168434
\(869\) 13.2750 0.450325
\(870\) 0 0
\(871\) −4.31265 −0.146129
\(872\) 122.426 4.14587
\(873\) 0 0
\(874\) 48.4504 1.63886
\(875\) −4.15633 −0.140509
\(876\) 0 0
\(877\) −4.26187 −0.143913 −0.0719565 0.997408i \(-0.522924\pi\)
−0.0719565 + 0.997408i \(0.522924\pi\)
\(878\) −37.7499 −1.27400
\(879\) 0 0
\(880\) −34.4445 −1.16113
\(881\) −15.2144 −0.512586 −0.256293 0.966599i \(-0.582501\pi\)
−0.256293 + 0.966599i \(0.582501\pi\)
\(882\) 0 0
\(883\) −13.7078 −0.461305 −0.230652 0.973036i \(-0.574086\pi\)
−0.230652 + 0.973036i \(0.574086\pi\)
\(884\) 49.5633 1.66699
\(885\) 0 0
\(886\) −46.2287 −1.55308
\(887\) −12.6556 −0.424934 −0.212467 0.977168i \(-0.568150\pi\)
−0.212467 + 0.977168i \(0.568150\pi\)
\(888\) 0 0
\(889\) −63.8251 −2.14063
\(890\) 38.2882 1.28342
\(891\) 0 0
\(892\) −5.76845 −0.193142
\(893\) −24.1866 −0.809375
\(894\) 0 0
\(895\) −4.77575 −0.159636
\(896\) 68.8686 2.30074
\(897\) 0 0
\(898\) 25.0132 0.834700
\(899\) −0.231548 −0.00772256
\(900\) 0 0
\(901\) −49.5633 −1.65119
\(902\) 52.2638 1.74019
\(903\) 0 0
\(904\) −138.422 −4.60385
\(905\) 1.87399 0.0622936
\(906\) 0 0
\(907\) −12.5540 −0.416850 −0.208425 0.978038i \(-0.566834\pi\)
−0.208425 + 0.978038i \(0.566834\pi\)
\(908\) −0.156325 −0.00518783
\(909\) 0 0
\(910\) 15.0132 0.497682
\(911\) −22.8714 −0.757765 −0.378882 0.925445i \(-0.623691\pi\)
−0.378882 + 0.925445i \(0.623691\pi\)
\(912\) 0 0
\(913\) 7.13918 0.236272
\(914\) 47.2506 1.56291
\(915\) 0 0
\(916\) −30.1622 −0.996587
\(917\) 5.73813 0.189490
\(918\) 0 0
\(919\) 9.67750 0.319231 0.159616 0.987179i \(-0.448975\pi\)
0.159616 + 0.987179i \(0.448975\pi\)
\(920\) 40.5804 1.33790
\(921\) 0 0
\(922\) 41.6337 1.37113
\(923\) −15.3258 −0.504456
\(924\) 0 0
\(925\) −5.50659 −0.181055
\(926\) −7.97953 −0.262224
\(927\) 0 0
\(928\) 15.9502 0.523590
\(929\) −51.9248 −1.70360 −0.851798 0.523870i \(-0.824488\pi\)
−0.851798 + 0.523870i \(0.824488\pi\)
\(930\) 0 0
\(931\) 38.7210 1.26903
\(932\) −134.588 −4.40858
\(933\) 0 0
\(934\) 92.5071 3.02692
\(935\) −19.9756 −0.653271
\(936\) 0 0
\(937\) 3.58769 0.117205 0.0586024 0.998281i \(-0.481336\pi\)
0.0586024 + 0.998281i \(0.481336\pi\)
\(938\) −35.5125 −1.15952
\(939\) 0 0
\(940\) −33.0943 −1.07942
\(941\) 18.6253 0.607167 0.303584 0.952805i \(-0.401817\pi\)
0.303584 + 0.952805i \(0.401817\pi\)
\(942\) 0 0
\(943\) −33.4617 −1.08966
\(944\) 31.6023 1.02857
\(945\) 0 0
\(946\) −23.9756 −0.779513
\(947\) 16.5950 0.539265 0.269632 0.962963i \(-0.413098\pi\)
0.269632 + 0.962963i \(0.413098\pi\)
\(948\) 0 0
\(949\) −15.1833 −0.492871
\(950\) −10.0811 −0.327074
\(951\) 0 0
\(952\) 249.826 8.09691
\(953\) 12.7005 0.411410 0.205705 0.978614i \(-0.434051\pi\)
0.205705 + 0.978614i \(0.434051\pi\)
\(954\) 0 0
\(955\) −19.1187 −0.618667
\(956\) −7.35026 −0.237724
\(957\) 0 0
\(958\) −91.4695 −2.95524
\(959\) −26.9887 −0.871512
\(960\) 0 0
\(961\) −30.9464 −0.998271
\(962\) 19.8905 0.641295
\(963\) 0 0
\(964\) −0.387873 −0.0124926
\(965\) 19.8945 0.640425
\(966\) 0 0
\(967\) 37.4314 1.20371 0.601856 0.798605i \(-0.294428\pi\)
0.601856 + 0.798605i \(0.294428\pi\)
\(968\) 26.3947 0.848358
\(969\) 0 0
\(970\) 4.09569 0.131505
\(971\) −7.51644 −0.241214 −0.120607 0.992700i \(-0.538484\pi\)
−0.120607 + 0.992700i \(0.538484\pi\)
\(972\) 0 0
\(973\) 79.0249 2.53342
\(974\) −102.980 −3.29969
\(975\) 0 0
\(976\) 65.6747 2.10220
\(977\) 2.52847 0.0808929 0.0404465 0.999182i \(-0.487122\pi\)
0.0404465 + 0.999182i \(0.487122\pi\)
\(978\) 0 0
\(979\) 40.1622 1.28359
\(980\) 52.9814 1.69243
\(981\) 0 0
\(982\) −73.4793 −2.34482
\(983\) 9.32979 0.297574 0.148787 0.988869i \(-0.452463\pi\)
0.148787 + 0.988869i \(0.452463\pi\)
\(984\) 0 0
\(985\) 13.5369 0.431322
\(986\) 19.0435 0.606468
\(987\) 0 0
\(988\) 26.2374 0.834724
\(989\) 15.3503 0.488110
\(990\) 0 0
\(991\) 38.4241 1.22058 0.610290 0.792178i \(-0.291053\pi\)
0.610290 + 0.792178i \(0.291053\pi\)
\(992\) −3.69323 −0.117260
\(993\) 0 0
\(994\) −126.200 −4.00283
\(995\) 2.57452 0.0816176
\(996\) 0 0
\(997\) −18.8423 −0.596740 −0.298370 0.954450i \(-0.596443\pi\)
−0.298370 + 0.954450i \(0.596443\pi\)
\(998\) 85.8759 2.71835
\(999\) 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 1305.2.a.o.1.1 3
3.2 odd 2 145.2.a.d.1.3 3
5.4 even 2 6525.2.a.bh.1.3 3
12.11 even 2 2320.2.a.s.1.2 3
15.2 even 4 725.2.b.d.349.6 6
15.8 even 4 725.2.b.d.349.1 6
15.14 odd 2 725.2.a.d.1.1 3
21.20 even 2 7105.2.a.p.1.3 3
24.5 odd 2 9280.2.a.bu.1.2 3
24.11 even 2 9280.2.a.bm.1.2 3
87.86 odd 2 4205.2.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.3 3 3.2 odd 2
725.2.a.d.1.1 3 15.14 odd 2
725.2.b.d.349.1 6 15.8 even 4
725.2.b.d.349.6 6 15.2 even 4
1305.2.a.o.1.1 3 1.1 even 1 trivial
2320.2.a.s.1.2 3 12.11 even 2
4205.2.a.e.1.1 3 87.86 odd 2
6525.2.a.bh.1.3 3 5.4 even 2
7105.2.a.p.1.3 3 21.20 even 2
9280.2.a.bm.1.2 3 24.11 even 2
9280.2.a.bu.1.2 3 24.5 odd 2