Properties

Label 3465.2.a.bh.1.1
Level $3465$
Weight $2$
Character 3465.1
Self dual yes
Analytic conductor $27.668$
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] = [3465,2,Mod(1,3465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3465, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3465.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3465.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: \(27.6681643004\)
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 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 3465.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.21432 q^{2} -0.525428 q^{4} +1.00000 q^{5} -1.00000 q^{7} +3.06668 q^{8} +O(q^{10})\) \(q-1.21432 q^{2} -0.525428 q^{4} +1.00000 q^{5} -1.00000 q^{7} +3.06668 q^{8} -1.21432 q^{10} +1.00000 q^{11} -3.73975 q^{13} +1.21432 q^{14} -2.67307 q^{16} -0.0666765 q^{17} +6.42864 q^{19} -0.525428 q^{20} -1.21432 q^{22} +1.09679 q^{23} +1.00000 q^{25} +4.54125 q^{26} +0.525428 q^{28} +7.80642 q^{29} -5.59210 q^{31} -2.88739 q^{32} +0.0809666 q^{34} -1.00000 q^{35} +1.33185 q^{37} -7.80642 q^{38} +3.06668 q^{40} -6.64296 q^{41} -11.7605 q^{43} -0.525428 q^{44} -1.33185 q^{46} +2.26025 q^{47} +1.00000 q^{49} -1.21432 q^{50} +1.96497 q^{52} +1.71900 q^{53} +1.00000 q^{55} -3.06668 q^{56} -9.47949 q^{58} -2.54125 q^{59} +14.4494 q^{61} +6.79060 q^{62} +8.85236 q^{64} -3.73975 q^{65} +10.3827 q^{67} +0.0350337 q^{68} +1.21432 q^{70} +12.5620 q^{71} +1.17775 q^{73} -1.61729 q^{74} -3.37778 q^{76} -1.00000 q^{77} -8.51606 q^{79} -2.67307 q^{80} +8.06668 q^{82} +12.1017 q^{83} -0.0666765 q^{85} +14.2810 q^{86} +3.06668 q^{88} -15.6128 q^{89} +3.73975 q^{91} -0.576283 q^{92} -2.74467 q^{94} +6.42864 q^{95} -13.4193 q^{97} -1.21432 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8} + 3 q^{10} + 3 q^{11} + 2 q^{13} - 3 q^{14} + 5 q^{16} + 6 q^{19} + 5 q^{20} + 3 q^{22} + 10 q^{23} + 3 q^{25} + 20 q^{26} - 5 q^{28} + 10 q^{29} - 10 q^{31} + 11 q^{32} - 6 q^{34} - 3 q^{35} - 16 q^{37} - 10 q^{38} + 9 q^{40} - 2 q^{43} + 5 q^{44} + 16 q^{46} + 20 q^{47} + 3 q^{49} + 3 q^{50} + 32 q^{52} + 12 q^{53} + 3 q^{55} - 9 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} - 6 q^{62} + 33 q^{64} + 2 q^{65} - 2 q^{67} - 26 q^{68} - 3 q^{70} + 24 q^{71} + 4 q^{73} - 38 q^{74} - 10 q^{76} - 3 q^{77} + 8 q^{79} + 5 q^{80} + 24 q^{82} + 10 q^{83} + 36 q^{86} + 9 q^{88} - 20 q^{89} - 2 q^{91} + 18 q^{92} + 38 q^{94} + 6 q^{95} + 3 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21432 −0.858654 −0.429327 0.903149i \(-0.641249\pi\)
−0.429327 + 0.903149i \(0.641249\pi\)
\(3\) 0 0
\(4\) −0.525428 −0.262714
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 3.06668 1.08423
\(9\) 0 0
\(10\) −1.21432 −0.384002
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −3.73975 −1.03722 −0.518610 0.855011i \(-0.673550\pi\)
−0.518610 + 0.855011i \(0.673550\pi\)
\(14\) 1.21432 0.324541
\(15\) 0 0
\(16\) −2.67307 −0.668268
\(17\) −0.0666765 −0.0161714 −0.00808572 0.999967i \(-0.502574\pi\)
−0.00808572 + 0.999967i \(0.502574\pi\)
\(18\) 0 0
\(19\) 6.42864 1.47483 0.737416 0.675439i \(-0.236046\pi\)
0.737416 + 0.675439i \(0.236046\pi\)
\(20\) −0.525428 −0.117489
\(21\) 0 0
\(22\) −1.21432 −0.258894
\(23\) 1.09679 0.228696 0.114348 0.993441i \(-0.463522\pi\)
0.114348 + 0.993441i \(0.463522\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.54125 0.890612
\(27\) 0 0
\(28\) 0.525428 0.0992965
\(29\) 7.80642 1.44962 0.724808 0.688951i \(-0.241928\pi\)
0.724808 + 0.688951i \(0.241928\pi\)
\(30\) 0 0
\(31\) −5.59210 −1.00437 −0.502186 0.864760i \(-0.667471\pi\)
−0.502186 + 0.864760i \(0.667471\pi\)
\(32\) −2.88739 −0.510423
\(33\) 0 0
\(34\) 0.0809666 0.0138857
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 1.33185 0.218955 0.109478 0.993989i \(-0.465082\pi\)
0.109478 + 0.993989i \(0.465082\pi\)
\(38\) −7.80642 −1.26637
\(39\) 0 0
\(40\) 3.06668 0.484884
\(41\) −6.64296 −1.03746 −0.518728 0.854939i \(-0.673594\pi\)
−0.518728 + 0.854939i \(0.673594\pi\)
\(42\) 0 0
\(43\) −11.7605 −1.79346 −0.896729 0.442580i \(-0.854063\pi\)
−0.896729 + 0.442580i \(0.854063\pi\)
\(44\) −0.525428 −0.0792112
\(45\) 0 0
\(46\) −1.33185 −0.196371
\(47\) 2.26025 0.329692 0.164846 0.986319i \(-0.447287\pi\)
0.164846 + 0.986319i \(0.447287\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.21432 −0.171731
\(51\) 0 0
\(52\) 1.96497 0.272492
\(53\) 1.71900 0.236123 0.118062 0.993006i \(-0.462332\pi\)
0.118062 + 0.993006i \(0.462332\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −3.06668 −0.409802
\(57\) 0 0
\(58\) −9.47949 −1.24472
\(59\) −2.54125 −0.330842 −0.165421 0.986223i \(-0.552898\pi\)
−0.165421 + 0.986223i \(0.552898\pi\)
\(60\) 0 0
\(61\) 14.4494 1.85005 0.925027 0.379901i \(-0.124042\pi\)
0.925027 + 0.379901i \(0.124042\pi\)
\(62\) 6.79060 0.862407
\(63\) 0 0
\(64\) 8.85236 1.10654
\(65\) −3.73975 −0.463859
\(66\) 0 0
\(67\) 10.3827 1.26845 0.634225 0.773149i \(-0.281319\pi\)
0.634225 + 0.773149i \(0.281319\pi\)
\(68\) 0.0350337 0.00424846
\(69\) 0 0
\(70\) 1.21432 0.145139
\(71\) 12.5620 1.49083 0.745417 0.666598i \(-0.232250\pi\)
0.745417 + 0.666598i \(0.232250\pi\)
\(72\) 0 0
\(73\) 1.17775 0.137846 0.0689229 0.997622i \(-0.478044\pi\)
0.0689229 + 0.997622i \(0.478044\pi\)
\(74\) −1.61729 −0.188007
\(75\) 0 0
\(76\) −3.37778 −0.387458
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −8.51606 −0.958132 −0.479066 0.877779i \(-0.659024\pi\)
−0.479066 + 0.877779i \(0.659024\pi\)
\(80\) −2.67307 −0.298858
\(81\) 0 0
\(82\) 8.06668 0.890815
\(83\) 12.1017 1.32834 0.664168 0.747584i \(-0.268786\pi\)
0.664168 + 0.747584i \(0.268786\pi\)
\(84\) 0 0
\(85\) −0.0666765 −0.00723209
\(86\) 14.2810 1.53996
\(87\) 0 0
\(88\) 3.06668 0.326909
\(89\) −15.6128 −1.65496 −0.827479 0.561496i \(-0.810226\pi\)
−0.827479 + 0.561496i \(0.810226\pi\)
\(90\) 0 0
\(91\) 3.73975 0.392032
\(92\) −0.576283 −0.0600816
\(93\) 0 0
\(94\) −2.74467 −0.283091
\(95\) 6.42864 0.659564
\(96\) 0 0
\(97\) −13.4193 −1.36252 −0.681260 0.732041i \(-0.738568\pi\)
−0.681260 + 0.732041i \(0.738568\pi\)
\(98\) −1.21432 −0.122665
\(99\) 0 0
\(100\) −0.525428 −0.0525428
\(101\) 8.44938 0.840745 0.420373 0.907352i \(-0.361899\pi\)
0.420373 + 0.907352i \(0.361899\pi\)
\(102\) 0 0
\(103\) −9.54617 −0.940612 −0.470306 0.882503i \(-0.655856\pi\)
−0.470306 + 0.882503i \(0.655856\pi\)
\(104\) −11.4686 −1.12459
\(105\) 0 0
\(106\) −2.08742 −0.202748
\(107\) 8.56199 0.827719 0.413860 0.910341i \(-0.364180\pi\)
0.413860 + 0.910341i \(0.364180\pi\)
\(108\) 0 0
\(109\) −6.99063 −0.669581 −0.334791 0.942293i \(-0.608666\pi\)
−0.334791 + 0.942293i \(0.608666\pi\)
\(110\) −1.21432 −0.115781
\(111\) 0 0
\(112\) 2.67307 0.252581
\(113\) 1.57136 0.147821 0.0739106 0.997265i \(-0.476452\pi\)
0.0739106 + 0.997265i \(0.476452\pi\)
\(114\) 0 0
\(115\) 1.09679 0.102276
\(116\) −4.10171 −0.380834
\(117\) 0 0
\(118\) 3.08589 0.284079
\(119\) 0.0666765 0.00611223
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −17.5462 −1.58856
\(123\) 0 0
\(124\) 2.93825 0.263862
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −13.3778 −1.18709 −0.593543 0.804802i \(-0.702271\pi\)
−0.593543 + 0.804802i \(0.702271\pi\)
\(128\) −4.97481 −0.439715
\(129\) 0 0
\(130\) 4.54125 0.398294
\(131\) 2.75557 0.240755 0.120378 0.992728i \(-0.461589\pi\)
0.120378 + 0.992728i \(0.461589\pi\)
\(132\) 0 0
\(133\) −6.42864 −0.557434
\(134\) −12.6079 −1.08916
\(135\) 0 0
\(136\) −0.204475 −0.0175336
\(137\) 17.8938 1.52877 0.764387 0.644758i \(-0.223042\pi\)
0.764387 + 0.644758i \(0.223042\pi\)
\(138\) 0 0
\(139\) −3.86665 −0.327965 −0.163982 0.986463i \(-0.552434\pi\)
−0.163982 + 0.986463i \(0.552434\pi\)
\(140\) 0.525428 0.0444067
\(141\) 0 0
\(142\) −15.2543 −1.28011
\(143\) −3.73975 −0.312733
\(144\) 0 0
\(145\) 7.80642 0.648288
\(146\) −1.43017 −0.118362
\(147\) 0 0
\(148\) −0.699791 −0.0575225
\(149\) −4.88892 −0.400516 −0.200258 0.979743i \(-0.564178\pi\)
−0.200258 + 0.979743i \(0.564178\pi\)
\(150\) 0 0
\(151\) 22.1891 1.80573 0.902863 0.429929i \(-0.141461\pi\)
0.902863 + 0.429929i \(0.141461\pi\)
\(152\) 19.7146 1.59906
\(153\) 0 0
\(154\) 1.21432 0.0978527
\(155\) −5.59210 −0.449169
\(156\) 0 0
\(157\) −4.81579 −0.384342 −0.192171 0.981361i \(-0.561553\pi\)
−0.192171 + 0.981361i \(0.561553\pi\)
\(158\) 10.3412 0.822703
\(159\) 0 0
\(160\) −2.88739 −0.228268
\(161\) −1.09679 −0.0864390
\(162\) 0 0
\(163\) 6.08742 0.476804 0.238402 0.971167i \(-0.423376\pi\)
0.238402 + 0.971167i \(0.423376\pi\)
\(164\) 3.49039 0.272554
\(165\) 0 0
\(166\) −14.6953 −1.14058
\(167\) 3.67307 0.284231 0.142115 0.989850i \(-0.454610\pi\)
0.142115 + 0.989850i \(0.454610\pi\)
\(168\) 0 0
\(169\) 0.985710 0.0758238
\(170\) 0.0809666 0.00620986
\(171\) 0 0
\(172\) 6.17929 0.471166
\(173\) 0.628669 0.0477968 0.0238984 0.999714i \(-0.492392\pi\)
0.0238984 + 0.999714i \(0.492392\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −2.67307 −0.201490
\(177\) 0 0
\(178\) 18.9590 1.42104
\(179\) 7.70471 0.575877 0.287939 0.957649i \(-0.407030\pi\)
0.287939 + 0.957649i \(0.407030\pi\)
\(180\) 0 0
\(181\) 10.9175 0.811492 0.405746 0.913986i \(-0.367012\pi\)
0.405746 + 0.913986i \(0.367012\pi\)
\(182\) −4.54125 −0.336620
\(183\) 0 0
\(184\) 3.36349 0.247960
\(185\) 1.33185 0.0979197
\(186\) 0 0
\(187\) −0.0666765 −0.00487587
\(188\) −1.18760 −0.0866146
\(189\) 0 0
\(190\) −7.80642 −0.566338
\(191\) 10.9175 0.789963 0.394981 0.918689i \(-0.370751\pi\)
0.394981 + 0.918689i \(0.370751\pi\)
\(192\) 0 0
\(193\) 11.7003 0.842204 0.421102 0.907013i \(-0.361643\pi\)
0.421102 + 0.907013i \(0.361643\pi\)
\(194\) 16.2953 1.16993
\(195\) 0 0
\(196\) −0.525428 −0.0375305
\(197\) −10.8430 −0.772531 −0.386265 0.922388i \(-0.626235\pi\)
−0.386265 + 0.922388i \(0.626235\pi\)
\(198\) 0 0
\(199\) 2.08097 0.147516 0.0737579 0.997276i \(-0.476501\pi\)
0.0737579 + 0.997276i \(0.476501\pi\)
\(200\) 3.06668 0.216847
\(201\) 0 0
\(202\) −10.2603 −0.721909
\(203\) −7.80642 −0.547904
\(204\) 0 0
\(205\) −6.64296 −0.463964
\(206\) 11.5921 0.807660
\(207\) 0 0
\(208\) 9.99661 0.693140
\(209\) 6.42864 0.444678
\(210\) 0 0
\(211\) 23.0923 1.58974 0.794871 0.606778i \(-0.207538\pi\)
0.794871 + 0.606778i \(0.207538\pi\)
\(212\) −0.903212 −0.0620328
\(213\) 0 0
\(214\) −10.3970 −0.710724
\(215\) −11.7605 −0.802059
\(216\) 0 0
\(217\) 5.59210 0.379617
\(218\) 8.48886 0.574938
\(219\) 0 0
\(220\) −0.525428 −0.0354243
\(221\) 0.249353 0.0167733
\(222\) 0 0
\(223\) −21.5462 −1.44284 −0.721419 0.692499i \(-0.756510\pi\)
−0.721419 + 0.692499i \(0.756510\pi\)
\(224\) 2.88739 0.192922
\(225\) 0 0
\(226\) −1.90813 −0.126927
\(227\) 27.2257 1.80703 0.903516 0.428553i \(-0.140977\pi\)
0.903516 + 0.428553i \(0.140977\pi\)
\(228\) 0 0
\(229\) 25.0005 1.65208 0.826039 0.563613i \(-0.190589\pi\)
0.826039 + 0.563613i \(0.190589\pi\)
\(230\) −1.33185 −0.0878197
\(231\) 0 0
\(232\) 23.9398 1.57172
\(233\) 3.65878 0.239695 0.119847 0.992792i \(-0.461759\pi\)
0.119847 + 0.992792i \(0.461759\pi\)
\(234\) 0 0
\(235\) 2.26025 0.147443
\(236\) 1.33524 0.0869169
\(237\) 0 0
\(238\) −0.0809666 −0.00524829
\(239\) −14.7052 −0.951200 −0.475600 0.879662i \(-0.657769\pi\)
−0.475600 + 0.879662i \(0.657769\pi\)
\(240\) 0 0
\(241\) 13.6938 0.882096 0.441048 0.897483i \(-0.354607\pi\)
0.441048 + 0.897483i \(0.354607\pi\)
\(242\) −1.21432 −0.0780594
\(243\) 0 0
\(244\) −7.59210 −0.486035
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −24.0415 −1.52972
\(248\) −17.1492 −1.08897
\(249\) 0 0
\(250\) −1.21432 −0.0768003
\(251\) 11.0114 0.695032 0.347516 0.937674i \(-0.387025\pi\)
0.347516 + 0.937674i \(0.387025\pi\)
\(252\) 0 0
\(253\) 1.09679 0.0689545
\(254\) 16.2449 1.01930
\(255\) 0 0
\(256\) −11.6637 −0.728981
\(257\) 15.9496 0.994910 0.497455 0.867490i \(-0.334268\pi\)
0.497455 + 0.867490i \(0.334268\pi\)
\(258\) 0 0
\(259\) −1.33185 −0.0827572
\(260\) 1.96497 0.121862
\(261\) 0 0
\(262\) −3.34614 −0.206725
\(263\) 5.11108 0.315163 0.157581 0.987506i \(-0.449630\pi\)
0.157581 + 0.987506i \(0.449630\pi\)
\(264\) 0 0
\(265\) 1.71900 0.105598
\(266\) 7.80642 0.478643
\(267\) 0 0
\(268\) −5.45536 −0.333239
\(269\) 18.2351 1.11181 0.555906 0.831245i \(-0.312372\pi\)
0.555906 + 0.831245i \(0.312372\pi\)
\(270\) 0 0
\(271\) 6.23506 0.378753 0.189377 0.981905i \(-0.439353\pi\)
0.189377 + 0.981905i \(0.439353\pi\)
\(272\) 0.178231 0.0108068
\(273\) 0 0
\(274\) −21.7288 −1.31269
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 6.32248 0.379881 0.189941 0.981796i \(-0.439170\pi\)
0.189941 + 0.981796i \(0.439170\pi\)
\(278\) 4.69535 0.281608
\(279\) 0 0
\(280\) −3.06668 −0.183269
\(281\) 25.2257 1.50484 0.752419 0.658684i \(-0.228887\pi\)
0.752419 + 0.658684i \(0.228887\pi\)
\(282\) 0 0
\(283\) 13.4193 0.797693 0.398846 0.917018i \(-0.369411\pi\)
0.398846 + 0.917018i \(0.369411\pi\)
\(284\) −6.60042 −0.391663
\(285\) 0 0
\(286\) 4.54125 0.268530
\(287\) 6.64296 0.392121
\(288\) 0 0
\(289\) −16.9956 −0.999738
\(290\) −9.47949 −0.556655
\(291\) 0 0
\(292\) −0.618825 −0.0362140
\(293\) −17.6064 −1.02858 −0.514288 0.857617i \(-0.671944\pi\)
−0.514288 + 0.857617i \(0.671944\pi\)
\(294\) 0 0
\(295\) −2.54125 −0.147957
\(296\) 4.08436 0.237398
\(297\) 0 0
\(298\) 5.93671 0.343905
\(299\) −4.10171 −0.237208
\(300\) 0 0
\(301\) 11.7605 0.677863
\(302\) −26.9447 −1.55049
\(303\) 0 0
\(304\) −17.1842 −0.985582
\(305\) 14.4494 0.827369
\(306\) 0 0
\(307\) 26.3368 1.50312 0.751560 0.659665i \(-0.229302\pi\)
0.751560 + 0.659665i \(0.229302\pi\)
\(308\) 0.525428 0.0299390
\(309\) 0 0
\(310\) 6.79060 0.385680
\(311\) −5.96052 −0.337990 −0.168995 0.985617i \(-0.554052\pi\)
−0.168995 + 0.985617i \(0.554052\pi\)
\(312\) 0 0
\(313\) −5.52098 −0.312064 −0.156032 0.987752i \(-0.549870\pi\)
−0.156032 + 0.987752i \(0.549870\pi\)
\(314\) 5.84791 0.330017
\(315\) 0 0
\(316\) 4.47457 0.251714
\(317\) 15.7146 0.882618 0.441309 0.897355i \(-0.354514\pi\)
0.441309 + 0.897355i \(0.354514\pi\)
\(318\) 0 0
\(319\) 7.80642 0.437076
\(320\) 8.85236 0.494862
\(321\) 0 0
\(322\) 1.33185 0.0742212
\(323\) −0.428639 −0.0238501
\(324\) 0 0
\(325\) −3.73975 −0.207444
\(326\) −7.39207 −0.409409
\(327\) 0 0
\(328\) −20.3718 −1.12484
\(329\) −2.26025 −0.124612
\(330\) 0 0
\(331\) 18.2351 1.00229 0.501145 0.865363i \(-0.332912\pi\)
0.501145 + 0.865363i \(0.332912\pi\)
\(332\) −6.35857 −0.348972
\(333\) 0 0
\(334\) −4.46028 −0.244056
\(335\) 10.3827 0.567268
\(336\) 0 0
\(337\) 32.2908 1.75899 0.879497 0.475904i \(-0.157879\pi\)
0.879497 + 0.475904i \(0.157879\pi\)
\(338\) −1.19697 −0.0651064
\(339\) 0 0
\(340\) 0.0350337 0.00189997
\(341\) −5.59210 −0.302829
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −36.0656 −1.94453
\(345\) 0 0
\(346\) −0.763405 −0.0410409
\(347\) −22.6909 −1.21811 −0.609056 0.793127i \(-0.708451\pi\)
−0.609056 + 0.793127i \(0.708451\pi\)
\(348\) 0 0
\(349\) 16.9097 0.905154 0.452577 0.891725i \(-0.350505\pi\)
0.452577 + 0.891725i \(0.350505\pi\)
\(350\) 1.21432 0.0649081
\(351\) 0 0
\(352\) −2.88739 −0.153898
\(353\) −24.3970 −1.29852 −0.649261 0.760566i \(-0.724922\pi\)
−0.649261 + 0.760566i \(0.724922\pi\)
\(354\) 0 0
\(355\) 12.5620 0.666721
\(356\) 8.20342 0.434780
\(357\) 0 0
\(358\) −9.35599 −0.494479
\(359\) 32.8113 1.73172 0.865858 0.500289i \(-0.166773\pi\)
0.865858 + 0.500289i \(0.166773\pi\)
\(360\) 0 0
\(361\) 22.3274 1.17513
\(362\) −13.2573 −0.696790
\(363\) 0 0
\(364\) −1.96497 −0.102992
\(365\) 1.17775 0.0616465
\(366\) 0 0
\(367\) −6.94269 −0.362406 −0.181203 0.983446i \(-0.557999\pi\)
−0.181203 + 0.983446i \(0.557999\pi\)
\(368\) −2.93179 −0.152830
\(369\) 0 0
\(370\) −1.61729 −0.0840791
\(371\) −1.71900 −0.0892462
\(372\) 0 0
\(373\) 36.9733 1.91440 0.957202 0.289421i \(-0.0934627\pi\)
0.957202 + 0.289421i \(0.0934627\pi\)
\(374\) 0.0809666 0.00418669
\(375\) 0 0
\(376\) 6.93146 0.357463
\(377\) −29.1941 −1.50357
\(378\) 0 0
\(379\) −23.6414 −1.21438 −0.607189 0.794557i \(-0.707703\pi\)
−0.607189 + 0.794557i \(0.707703\pi\)
\(380\) −3.37778 −0.173277
\(381\) 0 0
\(382\) −13.2573 −0.678304
\(383\) −13.7210 −0.701111 −0.350555 0.936542i \(-0.614007\pi\)
−0.350555 + 0.936542i \(0.614007\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −14.2079 −0.723161
\(387\) 0 0
\(388\) 7.05086 0.357953
\(389\) 15.5526 0.788549 0.394275 0.918993i \(-0.370996\pi\)
0.394275 + 0.918993i \(0.370996\pi\)
\(390\) 0 0
\(391\) −0.0731300 −0.00369835
\(392\) 3.06668 0.154891
\(393\) 0 0
\(394\) 13.1669 0.663337
\(395\) −8.51606 −0.428489
\(396\) 0 0
\(397\) −0.253799 −0.0127378 −0.00636891 0.999980i \(-0.502027\pi\)
−0.00636891 + 0.999980i \(0.502027\pi\)
\(398\) −2.52696 −0.126665
\(399\) 0 0
\(400\) −2.67307 −0.133654
\(401\) −20.3684 −1.01715 −0.508575 0.861018i \(-0.669828\pi\)
−0.508575 + 0.861018i \(0.669828\pi\)
\(402\) 0 0
\(403\) 20.9131 1.04175
\(404\) −4.43954 −0.220875
\(405\) 0 0
\(406\) 9.47949 0.470459
\(407\) 1.33185 0.0660174
\(408\) 0 0
\(409\) 12.1225 0.599417 0.299708 0.954031i \(-0.403111\pi\)
0.299708 + 0.954031i \(0.403111\pi\)
\(410\) 8.06668 0.398385
\(411\) 0 0
\(412\) 5.01582 0.247112
\(413\) 2.54125 0.125047
\(414\) 0 0
\(415\) 12.1017 0.594050
\(416\) 10.7981 0.529421
\(417\) 0 0
\(418\) −7.80642 −0.381825
\(419\) 21.4400 1.04741 0.523707 0.851899i \(-0.324549\pi\)
0.523707 + 0.851899i \(0.324549\pi\)
\(420\) 0 0
\(421\) 29.8622 1.45539 0.727697 0.685898i \(-0.240591\pi\)
0.727697 + 0.685898i \(0.240591\pi\)
\(422\) −28.0415 −1.36504
\(423\) 0 0
\(424\) 5.27163 0.256013
\(425\) −0.0666765 −0.00323429
\(426\) 0 0
\(427\) −14.4494 −0.699255
\(428\) −4.49871 −0.217453
\(429\) 0 0
\(430\) 14.2810 0.688691
\(431\) 19.6588 0.946930 0.473465 0.880813i \(-0.343003\pi\)
0.473465 + 0.880813i \(0.343003\pi\)
\(432\) 0 0
\(433\) −32.9719 −1.58453 −0.792264 0.610178i \(-0.791098\pi\)
−0.792264 + 0.610178i \(0.791098\pi\)
\(434\) −6.79060 −0.325959
\(435\) 0 0
\(436\) 3.67307 0.175908
\(437\) 7.05086 0.337288
\(438\) 0 0
\(439\) −0.815792 −0.0389356 −0.0194678 0.999810i \(-0.506197\pi\)
−0.0194678 + 0.999810i \(0.506197\pi\)
\(440\) 3.06668 0.146198
\(441\) 0 0
\(442\) −0.302795 −0.0144025
\(443\) −24.9763 −1.18666 −0.593331 0.804959i \(-0.702187\pi\)
−0.593331 + 0.804959i \(0.702187\pi\)
\(444\) 0 0
\(445\) −15.6128 −0.740120
\(446\) 26.1639 1.23890
\(447\) 0 0
\(448\) −8.85236 −0.418235
\(449\) −0.414349 −0.0195544 −0.00977718 0.999952i \(-0.503112\pi\)
−0.00977718 + 0.999952i \(0.503112\pi\)
\(450\) 0 0
\(451\) −6.64296 −0.312805
\(452\) −0.825636 −0.0388347
\(453\) 0 0
\(454\) −33.0607 −1.55162
\(455\) 3.73975 0.175322
\(456\) 0 0
\(457\) 9.33185 0.436526 0.218263 0.975890i \(-0.429961\pi\)
0.218263 + 0.975890i \(0.429961\pi\)
\(458\) −30.3586 −1.41856
\(459\) 0 0
\(460\) −0.576283 −0.0268693
\(461\) −29.9891 −1.39673 −0.698366 0.715741i \(-0.746089\pi\)
−0.698366 + 0.715741i \(0.746089\pi\)
\(462\) 0 0
\(463\) −31.5353 −1.46557 −0.732784 0.680461i \(-0.761780\pi\)
−0.732784 + 0.680461i \(0.761780\pi\)
\(464\) −20.8671 −0.968732
\(465\) 0 0
\(466\) −4.44293 −0.205815
\(467\) 5.67952 0.262817 0.131409 0.991328i \(-0.458050\pi\)
0.131409 + 0.991328i \(0.458050\pi\)
\(468\) 0 0
\(469\) −10.3827 −0.479429
\(470\) −2.74467 −0.126602
\(471\) 0 0
\(472\) −7.79319 −0.358711
\(473\) −11.7605 −0.540748
\(474\) 0 0
\(475\) 6.42864 0.294966
\(476\) −0.0350337 −0.00160577
\(477\) 0 0
\(478\) 17.8568 0.816751
\(479\) 39.4608 1.80301 0.901504 0.432771i \(-0.142464\pi\)
0.901504 + 0.432771i \(0.142464\pi\)
\(480\) 0 0
\(481\) −4.98079 −0.227104
\(482\) −16.6287 −0.757415
\(483\) 0 0
\(484\) −0.525428 −0.0238831
\(485\) −13.4193 −0.609338
\(486\) 0 0
\(487\) −3.19850 −0.144938 −0.0724689 0.997371i \(-0.523088\pi\)
−0.0724689 + 0.997371i \(0.523088\pi\)
\(488\) 44.3116 2.00589
\(489\) 0 0
\(490\) −1.21432 −0.0548574
\(491\) −26.3511 −1.18921 −0.594603 0.804019i \(-0.702691\pi\)
−0.594603 + 0.804019i \(0.702691\pi\)
\(492\) 0 0
\(493\) −0.520505 −0.0234424
\(494\) 29.1941 1.31350
\(495\) 0 0
\(496\) 14.9481 0.671189
\(497\) −12.5620 −0.563482
\(498\) 0 0
\(499\) 3.52987 0.158019 0.0790094 0.996874i \(-0.474824\pi\)
0.0790094 + 0.996874i \(0.474824\pi\)
\(500\) −0.525428 −0.0234978
\(501\) 0 0
\(502\) −13.3713 −0.596792
\(503\) −22.7368 −1.01379 −0.506893 0.862009i \(-0.669206\pi\)
−0.506893 + 0.862009i \(0.669206\pi\)
\(504\) 0 0
\(505\) 8.44938 0.375993
\(506\) −1.33185 −0.0592080
\(507\) 0 0
\(508\) 7.02906 0.311864
\(509\) 36.2351 1.60609 0.803045 0.595918i \(-0.203212\pi\)
0.803045 + 0.595918i \(0.203212\pi\)
\(510\) 0 0
\(511\) −1.17775 −0.0521008
\(512\) 24.1131 1.06566
\(513\) 0 0
\(514\) −19.3679 −0.854283
\(515\) −9.54617 −0.420655
\(516\) 0 0
\(517\) 2.26025 0.0994058
\(518\) 1.61729 0.0710598
\(519\) 0 0
\(520\) −11.4686 −0.502931
\(521\) 6.79706 0.297784 0.148892 0.988853i \(-0.452429\pi\)
0.148892 + 0.988853i \(0.452429\pi\)
\(522\) 0 0
\(523\) 34.0701 1.48978 0.744890 0.667187i \(-0.232502\pi\)
0.744890 + 0.667187i \(0.232502\pi\)
\(524\) −1.44785 −0.0632497
\(525\) 0 0
\(526\) −6.20648 −0.270616
\(527\) 0.372862 0.0162421
\(528\) 0 0
\(529\) −21.7971 −0.947698
\(530\) −2.08742 −0.0906717
\(531\) 0 0
\(532\) 3.37778 0.146446
\(533\) 24.8430 1.07607
\(534\) 0 0
\(535\) 8.56199 0.370167
\(536\) 31.8404 1.37530
\(537\) 0 0
\(538\) −22.1432 −0.954661
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −27.1941 −1.16916 −0.584582 0.811335i \(-0.698741\pi\)
−0.584582 + 0.811335i \(0.698741\pi\)
\(542\) −7.57136 −0.325218
\(543\) 0 0
\(544\) 0.192521 0.00825428
\(545\) −6.99063 −0.299446
\(546\) 0 0
\(547\) −40.2449 −1.72075 −0.860374 0.509663i \(-0.829770\pi\)
−0.860374 + 0.509663i \(0.829770\pi\)
\(548\) −9.40192 −0.401630
\(549\) 0 0
\(550\) −1.21432 −0.0517788
\(551\) 50.1847 2.13794
\(552\) 0 0
\(553\) 8.51606 0.362140
\(554\) −7.67752 −0.326186
\(555\) 0 0
\(556\) 2.03164 0.0861608
\(557\) 7.09679 0.300701 0.150350 0.988633i \(-0.451960\pi\)
0.150350 + 0.988633i \(0.451960\pi\)
\(558\) 0 0
\(559\) 43.9813 1.86021
\(560\) 2.67307 0.112958
\(561\) 0 0
\(562\) −30.6321 −1.29214
\(563\) 18.7971 0.792201 0.396101 0.918207i \(-0.370363\pi\)
0.396101 + 0.918207i \(0.370363\pi\)
\(564\) 0 0
\(565\) 1.57136 0.0661076
\(566\) −16.2953 −0.684942
\(567\) 0 0
\(568\) 38.5236 1.61641
\(569\) 0.0316429 0.00132654 0.000663269 1.00000i \(-0.499789\pi\)
0.000663269 1.00000i \(0.499789\pi\)
\(570\) 0 0
\(571\) 0.0503787 0.00210828 0.00105414 0.999999i \(-0.499664\pi\)
0.00105414 + 0.999999i \(0.499664\pi\)
\(572\) 1.96497 0.0821594
\(573\) 0 0
\(574\) −8.06668 −0.336697
\(575\) 1.09679 0.0457392
\(576\) 0 0
\(577\) −13.6316 −0.567490 −0.283745 0.958900i \(-0.591577\pi\)
−0.283745 + 0.958900i \(0.591577\pi\)
\(578\) 20.6380 0.858429
\(579\) 0 0
\(580\) −4.10171 −0.170314
\(581\) −12.1017 −0.502064
\(582\) 0 0
\(583\) 1.71900 0.0711939
\(584\) 3.61179 0.149457
\(585\) 0 0
\(586\) 21.3798 0.883191
\(587\) −1.63804 −0.0676090 −0.0338045 0.999428i \(-0.510762\pi\)
−0.0338045 + 0.999428i \(0.510762\pi\)
\(588\) 0 0
\(589\) −35.9496 −1.48128
\(590\) 3.08589 0.127044
\(591\) 0 0
\(592\) −3.56013 −0.146321
\(593\) −8.46367 −0.347561 −0.173781 0.984784i \(-0.555598\pi\)
−0.173781 + 0.984784i \(0.555598\pi\)
\(594\) 0 0
\(595\) 0.0666765 0.00273347
\(596\) 2.56877 0.105221
\(597\) 0 0
\(598\) 4.98079 0.203680
\(599\) 26.9590 1.10151 0.550757 0.834665i \(-0.314339\pi\)
0.550757 + 0.834665i \(0.314339\pi\)
\(600\) 0 0
\(601\) −16.4909 −0.672677 −0.336338 0.941741i \(-0.609189\pi\)
−0.336338 + 0.941741i \(0.609189\pi\)
\(602\) −14.2810 −0.582050
\(603\) 0 0
\(604\) −11.6588 −0.474389
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 19.5526 0.793617 0.396808 0.917902i \(-0.370118\pi\)
0.396808 + 0.917902i \(0.370118\pi\)
\(608\) −18.5620 −0.752788
\(609\) 0 0
\(610\) −17.5462 −0.710424
\(611\) −8.45277 −0.341963
\(612\) 0 0
\(613\) −39.5955 −1.59925 −0.799623 0.600502i \(-0.794968\pi\)
−0.799623 + 0.600502i \(0.794968\pi\)
\(614\) −31.9813 −1.29066
\(615\) 0 0
\(616\) −3.06668 −0.123560
\(617\) −23.1669 −0.932662 −0.466331 0.884610i \(-0.654425\pi\)
−0.466331 + 0.884610i \(0.654425\pi\)
\(618\) 0 0
\(619\) 3.43017 0.137870 0.0689351 0.997621i \(-0.478040\pi\)
0.0689351 + 0.997621i \(0.478040\pi\)
\(620\) 2.93825 0.118003
\(621\) 0 0
\(622\) 7.23798 0.290216
\(623\) 15.6128 0.625516
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.70424 0.267955
\(627\) 0 0
\(628\) 2.53035 0.100972
\(629\) −0.0888033 −0.00354082
\(630\) 0 0
\(631\) 10.9590 0.436270 0.218135 0.975919i \(-0.430003\pi\)
0.218135 + 0.975919i \(0.430003\pi\)
\(632\) −26.1160 −1.03884
\(633\) 0 0
\(634\) −19.0825 −0.757863
\(635\) −13.3778 −0.530881
\(636\) 0 0
\(637\) −3.73975 −0.148174
\(638\) −9.47949 −0.375297
\(639\) 0 0
\(640\) −4.97481 −0.196647
\(641\) 14.9862 0.591919 0.295959 0.955201i \(-0.404361\pi\)
0.295959 + 0.955201i \(0.404361\pi\)
\(642\) 0 0
\(643\) −14.4222 −0.568755 −0.284378 0.958712i \(-0.591787\pi\)
−0.284378 + 0.958712i \(0.591787\pi\)
\(644\) 0.576283 0.0227087
\(645\) 0 0
\(646\) 0.520505 0.0204790
\(647\) 33.7309 1.32610 0.663048 0.748577i \(-0.269262\pi\)
0.663048 + 0.748577i \(0.269262\pi\)
\(648\) 0 0
\(649\) −2.54125 −0.0997527
\(650\) 4.54125 0.178122
\(651\) 0 0
\(652\) −3.19850 −0.125263
\(653\) −36.7971 −1.43998 −0.719990 0.693984i \(-0.755854\pi\)
−0.719990 + 0.693984i \(0.755854\pi\)
\(654\) 0 0
\(655\) 2.75557 0.107669
\(656\) 17.7571 0.693298
\(657\) 0 0
\(658\) 2.74467 0.106998
\(659\) −3.79213 −0.147721 −0.0738603 0.997269i \(-0.523532\pi\)
−0.0738603 + 0.997269i \(0.523532\pi\)
\(660\) 0 0
\(661\) −35.2543 −1.37123 −0.685616 0.727963i \(-0.740467\pi\)
−0.685616 + 0.727963i \(0.740467\pi\)
\(662\) −22.1432 −0.860620
\(663\) 0 0
\(664\) 37.1120 1.44023
\(665\) −6.42864 −0.249292
\(666\) 0 0
\(667\) 8.56199 0.331522
\(668\) −1.92993 −0.0746713
\(669\) 0 0
\(670\) −12.6079 −0.487087
\(671\) 14.4494 0.557812
\(672\) 0 0
\(673\) −35.1383 −1.35448 −0.677240 0.735762i \(-0.736824\pi\)
−0.677240 + 0.735762i \(0.736824\pi\)
\(674\) −39.2114 −1.51037
\(675\) 0 0
\(676\) −0.517919 −0.0199200
\(677\) −23.1907 −0.891290 −0.445645 0.895210i \(-0.647026\pi\)
−0.445645 + 0.895210i \(0.647026\pi\)
\(678\) 0 0
\(679\) 13.4193 0.514984
\(680\) −0.204475 −0.00784127
\(681\) 0 0
\(682\) 6.79060 0.260026
\(683\) 7.74758 0.296453 0.148227 0.988953i \(-0.452644\pi\)
0.148227 + 0.988953i \(0.452644\pi\)
\(684\) 0 0
\(685\) 17.8938 0.683689
\(686\) 1.21432 0.0463629
\(687\) 0 0
\(688\) 31.4366 1.19851
\(689\) −6.42864 −0.244912
\(690\) 0 0
\(691\) −14.7763 −0.562117 −0.281059 0.959691i \(-0.590686\pi\)
−0.281059 + 0.959691i \(0.590686\pi\)
\(692\) −0.330320 −0.0125569
\(693\) 0 0
\(694\) 27.5540 1.04594
\(695\) −3.86665 −0.146670
\(696\) 0 0
\(697\) 0.442930 0.0167772
\(698\) −20.5337 −0.777214
\(699\) 0 0
\(700\) 0.525428 0.0198593
\(701\) 19.8765 0.750725 0.375362 0.926878i \(-0.377518\pi\)
0.375362 + 0.926878i \(0.377518\pi\)
\(702\) 0 0
\(703\) 8.56199 0.322922
\(704\) 8.85236 0.333636
\(705\) 0 0
\(706\) 29.6258 1.11498
\(707\) −8.44938 −0.317772
\(708\) 0 0
\(709\) 0.368416 0.0138362 0.00691808 0.999976i \(-0.497798\pi\)
0.00691808 + 0.999976i \(0.497798\pi\)
\(710\) −15.2543 −0.572483
\(711\) 0 0
\(712\) −47.8796 −1.79436
\(713\) −6.13335 −0.229696
\(714\) 0 0
\(715\) −3.73975 −0.139859
\(716\) −4.04827 −0.151291
\(717\) 0 0
\(718\) −39.8435 −1.48694
\(719\) 5.86865 0.218864 0.109432 0.993994i \(-0.465097\pi\)
0.109432 + 0.993994i \(0.465097\pi\)
\(720\) 0 0
\(721\) 9.54617 0.355518
\(722\) −27.1126 −1.00903
\(723\) 0 0
\(724\) −5.73636 −0.213190
\(725\) 7.80642 0.289923
\(726\) 0 0
\(727\) 1.19066 0.0441592 0.0220796 0.999756i \(-0.492971\pi\)
0.0220796 + 0.999756i \(0.492971\pi\)
\(728\) 11.4686 0.425054
\(729\) 0 0
\(730\) −1.43017 −0.0529330
\(731\) 0.784149 0.0290028
\(732\) 0 0
\(733\) 19.6795 0.726880 0.363440 0.931618i \(-0.381602\pi\)
0.363440 + 0.931618i \(0.381602\pi\)
\(734\) 8.43065 0.311181
\(735\) 0 0
\(736\) −3.16686 −0.116732
\(737\) 10.3827 0.382452
\(738\) 0 0
\(739\) 27.1427 0.998461 0.499231 0.866469i \(-0.333616\pi\)
0.499231 + 0.866469i \(0.333616\pi\)
\(740\) −0.699791 −0.0257248
\(741\) 0 0
\(742\) 2.08742 0.0766316
\(743\) 31.0509 1.13915 0.569573 0.821941i \(-0.307109\pi\)
0.569573 + 0.821941i \(0.307109\pi\)
\(744\) 0 0
\(745\) −4.88892 −0.179116
\(746\) −44.8974 −1.64381
\(747\) 0 0
\(748\) 0.0350337 0.00128096
\(749\) −8.56199 −0.312848
\(750\) 0 0
\(751\) −35.8292 −1.30743 −0.653713 0.756743i \(-0.726789\pi\)
−0.653713 + 0.756743i \(0.726789\pi\)
\(752\) −6.04182 −0.220322
\(753\) 0 0
\(754\) 35.4509 1.29105
\(755\) 22.1891 0.807545
\(756\) 0 0
\(757\) −1.15563 −0.0420020 −0.0210010 0.999779i \(-0.506685\pi\)
−0.0210010 + 0.999779i \(0.506685\pi\)
\(758\) 28.7083 1.04273
\(759\) 0 0
\(760\) 19.7146 0.715122
\(761\) −36.5640 −1.32544 −0.662722 0.748866i \(-0.730599\pi\)
−0.662722 + 0.748866i \(0.730599\pi\)
\(762\) 0 0
\(763\) 6.99063 0.253078
\(764\) −5.73636 −0.207534
\(765\) 0 0
\(766\) 16.6617 0.602012
\(767\) 9.50363 0.343156
\(768\) 0 0
\(769\) −38.7545 −1.39752 −0.698762 0.715354i \(-0.746265\pi\)
−0.698762 + 0.715354i \(0.746265\pi\)
\(770\) 1.21432 0.0437610
\(771\) 0 0
\(772\) −6.14764 −0.221259
\(773\) −55.3372 −1.99034 −0.995171 0.0981537i \(-0.968706\pi\)
−0.995171 + 0.0981537i \(0.968706\pi\)
\(774\) 0 0
\(775\) −5.59210 −0.200874
\(776\) −41.1526 −1.47729
\(777\) 0 0
\(778\) −18.8859 −0.677091
\(779\) −42.7052 −1.53007
\(780\) 0 0
\(781\) 12.5620 0.449503
\(782\) 0.0888033 0.00317560
\(783\) 0 0
\(784\) −2.67307 −0.0954668
\(785\) −4.81579 −0.171883
\(786\) 0 0
\(787\) −13.6731 −0.487392 −0.243696 0.969852i \(-0.578360\pi\)
−0.243696 + 0.969852i \(0.578360\pi\)
\(788\) 5.69721 0.202955
\(789\) 0 0
\(790\) 10.3412 0.367924
\(791\) −1.57136 −0.0558711
\(792\) 0 0
\(793\) −54.0370 −1.91891
\(794\) 0.308193 0.0109374
\(795\) 0 0
\(796\) −1.09340 −0.0387544
\(797\) −17.8765 −0.633218 −0.316609 0.948556i \(-0.602544\pi\)
−0.316609 + 0.948556i \(0.602544\pi\)
\(798\) 0 0
\(799\) −0.150706 −0.00533159
\(800\) −2.88739 −0.102085
\(801\) 0 0
\(802\) 24.7338 0.873380
\(803\) 1.17775 0.0415621
\(804\) 0 0
\(805\) −1.09679 −0.0386567
\(806\) −25.3951 −0.894506
\(807\) 0 0
\(808\) 25.9115 0.911564
\(809\) −14.6450 −0.514890 −0.257445 0.966293i \(-0.582881\pi\)
−0.257445 + 0.966293i \(0.582881\pi\)
\(810\) 0 0
\(811\) 30.2667 1.06281 0.531404 0.847119i \(-0.321665\pi\)
0.531404 + 0.847119i \(0.321665\pi\)
\(812\) 4.10171 0.143942
\(813\) 0 0
\(814\) −1.61729 −0.0566861
\(815\) 6.08742 0.213233
\(816\) 0 0
\(817\) −75.6040 −2.64505
\(818\) −14.7205 −0.514691
\(819\) 0 0
\(820\) 3.49039 0.121890
\(821\) −10.4286 −0.363962 −0.181981 0.983302i \(-0.558251\pi\)
−0.181981 + 0.983302i \(0.558251\pi\)
\(822\) 0 0
\(823\) 4.28100 0.149226 0.0746131 0.997213i \(-0.476228\pi\)
0.0746131 + 0.997213i \(0.476228\pi\)
\(824\) −29.2750 −1.01984
\(825\) 0 0
\(826\) −3.08589 −0.107372
\(827\) 13.8336 0.481042 0.240521 0.970644i \(-0.422682\pi\)
0.240521 + 0.970644i \(0.422682\pi\)
\(828\) 0 0
\(829\) 10.8573 0.377089 0.188544 0.982065i \(-0.439623\pi\)
0.188544 + 0.982065i \(0.439623\pi\)
\(830\) −14.6953 −0.510083
\(831\) 0 0
\(832\) −33.1056 −1.14773
\(833\) −0.0666765 −0.00231021
\(834\) 0 0
\(835\) 3.67307 0.127112
\(836\) −3.37778 −0.116823
\(837\) 0 0
\(838\) −26.0350 −0.899365
\(839\) −52.4820 −1.81188 −0.905940 0.423407i \(-0.860834\pi\)
−0.905940 + 0.423407i \(0.860834\pi\)
\(840\) 0 0
\(841\) 31.9403 1.10139
\(842\) −36.2623 −1.24968
\(843\) 0 0
\(844\) −12.1334 −0.417647
\(845\) 0.985710 0.0339095
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −4.59502 −0.157794
\(849\) 0 0
\(850\) 0.0809666 0.00277713
\(851\) 1.46076 0.0500742
\(852\) 0 0
\(853\) 9.89184 0.338690 0.169345 0.985557i \(-0.445835\pi\)
0.169345 + 0.985557i \(0.445835\pi\)
\(854\) 17.5462 0.600418
\(855\) 0 0
\(856\) 26.2569 0.897441
\(857\) 3.58766 0.122552 0.0612760 0.998121i \(-0.480483\pi\)
0.0612760 + 0.998121i \(0.480483\pi\)
\(858\) 0 0
\(859\) −17.2050 −0.587025 −0.293513 0.955955i \(-0.594824\pi\)
−0.293513 + 0.955955i \(0.594824\pi\)
\(860\) 6.17929 0.210712
\(861\) 0 0
\(862\) −23.8720 −0.813085
\(863\) −15.0781 −0.513263 −0.256631 0.966509i \(-0.582613\pi\)
−0.256631 + 0.966509i \(0.582613\pi\)
\(864\) 0 0
\(865\) 0.628669 0.0213754
\(866\) 40.0384 1.36056
\(867\) 0 0
\(868\) −2.93825 −0.0997306
\(869\) −8.51606 −0.288888
\(870\) 0 0
\(871\) −38.8287 −1.31566
\(872\) −21.4380 −0.725983
\(873\) 0 0
\(874\) −8.56199 −0.289614
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −31.4750 −1.06284 −0.531418 0.847109i \(-0.678341\pi\)
−0.531418 + 0.847109i \(0.678341\pi\)
\(878\) 0.990632 0.0334322
\(879\) 0 0
\(880\) −2.67307 −0.0901092
\(881\) 47.1209 1.58754 0.793772 0.608215i \(-0.208114\pi\)
0.793772 + 0.608215i \(0.208114\pi\)
\(882\) 0 0
\(883\) 33.8118 1.13786 0.568929 0.822386i \(-0.307358\pi\)
0.568929 + 0.822386i \(0.307358\pi\)
\(884\) −0.131017 −0.00440658
\(885\) 0 0
\(886\) 30.3293 1.01893
\(887\) 31.6258 1.06189 0.530944 0.847407i \(-0.321837\pi\)
0.530944 + 0.847407i \(0.321837\pi\)
\(888\) 0 0
\(889\) 13.3778 0.448676
\(890\) 18.9590 0.635507
\(891\) 0 0
\(892\) 11.3210 0.379054
\(893\) 14.5303 0.486240
\(894\) 0 0
\(895\) 7.70471 0.257540
\(896\) 4.97481 0.166197
\(897\) 0 0
\(898\) 0.503153 0.0167904
\(899\) −43.6543 −1.45595
\(900\) 0 0
\(901\) −0.114617 −0.00381845
\(902\) 8.06668 0.268591
\(903\) 0 0
\(904\) 4.81885 0.160273
\(905\) 10.9175 0.362910
\(906\) 0 0
\(907\) −46.8943 −1.55710 −0.778550 0.627582i \(-0.784045\pi\)
−0.778550 + 0.627582i \(0.784045\pi\)
\(908\) −14.3051 −0.474732
\(909\) 0 0
\(910\) −4.54125 −0.150541
\(911\) 28.5620 0.946301 0.473151 0.880982i \(-0.343117\pi\)
0.473151 + 0.880982i \(0.343117\pi\)
\(912\) 0 0
\(913\) 12.1017 0.400508
\(914\) −11.3319 −0.374824
\(915\) 0 0
\(916\) −13.1359 −0.434024
\(917\) −2.75557 −0.0909969
\(918\) 0 0
\(919\) 30.3555 1.00134 0.500668 0.865639i \(-0.333088\pi\)
0.500668 + 0.865639i \(0.333088\pi\)
\(920\) 3.36349 0.110891
\(921\) 0 0
\(922\) 36.4164 1.19931
\(923\) −46.9787 −1.54632
\(924\) 0 0
\(925\) 1.33185 0.0437910
\(926\) 38.2939 1.25842
\(927\) 0 0
\(928\) −22.5402 −0.739918
\(929\) −2.71408 −0.0890461 −0.0445231 0.999008i \(-0.514177\pi\)
−0.0445231 + 0.999008i \(0.514177\pi\)
\(930\) 0 0
\(931\) 6.42864 0.210690
\(932\) −1.92242 −0.0629711
\(933\) 0 0
\(934\) −6.89676 −0.225669
\(935\) −0.0666765 −0.00218056
\(936\) 0 0
\(937\) 13.7081 0.447824 0.223912 0.974609i \(-0.428117\pi\)
0.223912 + 0.974609i \(0.428117\pi\)
\(938\) 12.6079 0.411663
\(939\) 0 0
\(940\) −1.18760 −0.0387352
\(941\) 21.1635 0.689909 0.344955 0.938619i \(-0.387894\pi\)
0.344955 + 0.938619i \(0.387894\pi\)
\(942\) 0 0
\(943\) −7.28592 −0.237262
\(944\) 6.79294 0.221091
\(945\) 0 0
\(946\) 14.2810 0.464315
\(947\) 14.8746 0.483361 0.241680 0.970356i \(-0.422301\pi\)
0.241680 + 0.970356i \(0.422301\pi\)
\(948\) 0 0
\(949\) −4.40451 −0.142976
\(950\) −7.80642 −0.253274
\(951\) 0 0
\(952\) 0.204475 0.00662709
\(953\) −35.0968 −1.13690 −0.568448 0.822719i \(-0.692456\pi\)
−0.568448 + 0.822719i \(0.692456\pi\)
\(954\) 0 0
\(955\) 10.9175 0.353282
\(956\) 7.72651 0.249893
\(957\) 0 0
\(958\) −47.9180 −1.54816
\(959\) −17.8938 −0.577822
\(960\) 0 0
\(961\) 0.271628 0.00876221
\(962\) 6.04827 0.195004
\(963\) 0 0
\(964\) −7.19511 −0.231739
\(965\) 11.7003 0.376645
\(966\) 0 0
\(967\) −53.0879 −1.70719 −0.853596 0.520936i \(-0.825583\pi\)
−0.853596 + 0.520936i \(0.825583\pi\)
\(968\) 3.06668 0.0985667
\(969\) 0 0
\(970\) 16.2953 0.523210
\(971\) −21.4400 −0.688043 −0.344021 0.938962i \(-0.611789\pi\)
−0.344021 + 0.938962i \(0.611789\pi\)
\(972\) 0 0
\(973\) 3.86665 0.123959
\(974\) 3.88400 0.124451
\(975\) 0 0
\(976\) −38.6242 −1.23633
\(977\) 54.0415 1.72894 0.864470 0.502684i \(-0.167654\pi\)
0.864470 + 0.502684i \(0.167654\pi\)
\(978\) 0 0
\(979\) −15.6128 −0.498989
\(980\) −0.525428 −0.0167842
\(981\) 0 0
\(982\) 31.9986 1.02112
\(983\) 38.3432 1.22296 0.611480 0.791260i \(-0.290575\pi\)
0.611480 + 0.791260i \(0.290575\pi\)
\(984\) 0 0
\(985\) −10.8430 −0.345486
\(986\) 0.632060 0.0201289
\(987\) 0 0
\(988\) 12.6321 0.401879
\(989\) −12.8988 −0.410157
\(990\) 0 0
\(991\) 51.6543 1.64085 0.820427 0.571751i \(-0.193736\pi\)
0.820427 + 0.571751i \(0.193736\pi\)
\(992\) 16.1466 0.512655
\(993\) 0 0
\(994\) 15.2543 0.483836
\(995\) 2.08097 0.0659711
\(996\) 0 0
\(997\) 41.4445 1.31256 0.656280 0.754518i \(-0.272129\pi\)
0.656280 + 0.754518i \(0.272129\pi\)
\(998\) −4.28639 −0.135683
\(999\) 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 3465.2.a.bh.1.1 3
3.2 odd 2 385.2.a.f.1.3 3
12.11 even 2 6160.2.a.bn.1.2 3
15.2 even 4 1925.2.b.n.1849.4 6
15.8 even 4 1925.2.b.n.1849.3 6
15.14 odd 2 1925.2.a.v.1.1 3
21.20 even 2 2695.2.a.g.1.3 3
33.32 even 2 4235.2.a.q.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.f.1.3 3 3.2 odd 2
1925.2.a.v.1.1 3 15.14 odd 2
1925.2.b.n.1849.3 6 15.8 even 4
1925.2.b.n.1849.4 6 15.2 even 4
2695.2.a.g.1.3 3 21.20 even 2
3465.2.a.bh.1.1 3 1.1 even 1 trivial
4235.2.a.q.1.1 3 33.32 even 2
6160.2.a.bn.1.2 3 12.11 even 2