Properties

Label 9405.2.a.bh.1.5
Level $9405$
Weight $2$
Character 9405.1
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
Dimension $9$
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] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, 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("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.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: \(75.0993031010\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.682920\) of defining polynomial
Character \(\chi\) \(=\) 9405.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.682920 q^{2} -1.53362 q^{4} -1.00000 q^{5} +0.856948 q^{7} +2.41318 q^{8} +O(q^{10})\) \(q-0.682920 q^{2} -1.53362 q^{4} -1.00000 q^{5} +0.856948 q^{7} +2.41318 q^{8} +0.682920 q^{10} -1.00000 q^{11} +5.50630 q^{13} -0.585228 q^{14} +1.41923 q^{16} +3.50630 q^{17} +1.00000 q^{19} +1.53362 q^{20} +0.682920 q^{22} -6.75237 q^{23} +1.00000 q^{25} -3.76036 q^{26} -1.31423 q^{28} +0.474680 q^{29} -5.19745 q^{31} -5.79558 q^{32} -2.39452 q^{34} -0.856948 q^{35} +0.420904 q^{37} -0.682920 q^{38} -2.41318 q^{40} -8.25141 q^{41} -1.53320 q^{43} +1.53362 q^{44} +4.61133 q^{46} -2.48849 q^{47} -6.26564 q^{49} -0.682920 q^{50} -8.44457 q^{52} +4.79753 q^{53} +1.00000 q^{55} +2.06797 q^{56} -0.324169 q^{58} -1.75917 q^{59} +3.42433 q^{61} +3.54944 q^{62} +1.11946 q^{64} -5.50630 q^{65} -5.12424 q^{67} -5.37733 q^{68} +0.585228 q^{70} +12.1937 q^{71} -12.7638 q^{73} -0.287444 q^{74} -1.53362 q^{76} -0.856948 q^{77} +10.8691 q^{79} -1.41923 q^{80} +5.63505 q^{82} -2.94669 q^{83} -3.50630 q^{85} +1.04705 q^{86} -2.41318 q^{88} -3.08736 q^{89} +4.71861 q^{91} +10.3556 q^{92} +1.69944 q^{94} -1.00000 q^{95} -8.88210 q^{97} +4.27893 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} + 9 q^{4} - 9 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{2} + 9 q^{4} - 9 q^{5} + 13 q^{7} - 9 q^{8} + 3 q^{10} - 9 q^{11} + 5 q^{13} + 2 q^{14} + q^{16} - 13 q^{17} + 9 q^{19} - 9 q^{20} + 3 q^{22} - 8 q^{23} + 9 q^{25} - 8 q^{26} + 10 q^{28} + 3 q^{29} - 9 q^{31} - 6 q^{32} - 2 q^{34} - 13 q^{35} - 7 q^{37} - 3 q^{38} + 9 q^{40} - 9 q^{41} + 23 q^{43} - 9 q^{44} - 32 q^{46} - 20 q^{47} - 4 q^{49} - 3 q^{50} + 9 q^{52} + 5 q^{53} + 9 q^{55} - 4 q^{56} + 3 q^{58} - 19 q^{59} + q^{61} - 18 q^{62} + 23 q^{64} - 5 q^{65} - 10 q^{67} - 9 q^{68} - 2 q^{70} + 12 q^{73} - 5 q^{74} + 9 q^{76} - 13 q^{77} - 21 q^{79} - q^{80} - 14 q^{82} - 47 q^{83} + 13 q^{85} - 12 q^{86} + 9 q^{88} + 2 q^{89} + 2 q^{91} + 19 q^{92} + 26 q^{94} - 9 q^{95} - 32 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.682920 −0.482898 −0.241449 0.970414i \(-0.577623\pi\)
−0.241449 + 0.970414i \(0.577623\pi\)
\(3\) 0 0
\(4\) −1.53362 −0.766810
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.856948 0.323896 0.161948 0.986799i \(-0.448222\pi\)
0.161948 + 0.986799i \(0.448222\pi\)
\(8\) 2.41318 0.853188
\(9\) 0 0
\(10\) 0.682920 0.215958
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 5.50630 1.52717 0.763586 0.645706i \(-0.223437\pi\)
0.763586 + 0.645706i \(0.223437\pi\)
\(14\) −0.585228 −0.156409
\(15\) 0 0
\(16\) 1.41923 0.354807
\(17\) 3.50630 0.850402 0.425201 0.905099i \(-0.360203\pi\)
0.425201 + 0.905099i \(0.360203\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 1.53362 0.342928
\(21\) 0 0
\(22\) 0.682920 0.145599
\(23\) −6.75237 −1.40797 −0.703984 0.710216i \(-0.748597\pi\)
−0.703984 + 0.710216i \(0.748597\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.76036 −0.737468
\(27\) 0 0
\(28\) −1.31423 −0.248367
\(29\) 0.474680 0.0881458 0.0440729 0.999028i \(-0.485967\pi\)
0.0440729 + 0.999028i \(0.485967\pi\)
\(30\) 0 0
\(31\) −5.19745 −0.933489 −0.466745 0.884392i \(-0.654573\pi\)
−0.466745 + 0.884392i \(0.654573\pi\)
\(32\) −5.79558 −1.02452
\(33\) 0 0
\(34\) −2.39452 −0.410657
\(35\) −0.856948 −0.144851
\(36\) 0 0
\(37\) 0.420904 0.0691961 0.0345981 0.999401i \(-0.488985\pi\)
0.0345981 + 0.999401i \(0.488985\pi\)
\(38\) −0.682920 −0.110784
\(39\) 0 0
\(40\) −2.41318 −0.381557
\(41\) −8.25141 −1.28865 −0.644327 0.764750i \(-0.722862\pi\)
−0.644327 + 0.764750i \(0.722862\pi\)
\(42\) 0 0
\(43\) −1.53320 −0.233811 −0.116905 0.993143i \(-0.537297\pi\)
−0.116905 + 0.993143i \(0.537297\pi\)
\(44\) 1.53362 0.231202
\(45\) 0 0
\(46\) 4.61133 0.679904
\(47\) −2.48849 −0.362984 −0.181492 0.983392i \(-0.558093\pi\)
−0.181492 + 0.983392i \(0.558093\pi\)
\(48\) 0 0
\(49\) −6.26564 −0.895091
\(50\) −0.682920 −0.0965795
\(51\) 0 0
\(52\) −8.44457 −1.17105
\(53\) 4.79753 0.658991 0.329495 0.944157i \(-0.393121\pi\)
0.329495 + 0.944157i \(0.393121\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 2.06797 0.276344
\(57\) 0 0
\(58\) −0.324169 −0.0425654
\(59\) −1.75917 −0.229025 −0.114513 0.993422i \(-0.536531\pi\)
−0.114513 + 0.993422i \(0.536531\pi\)
\(60\) 0 0
\(61\) 3.42433 0.438440 0.219220 0.975675i \(-0.429649\pi\)
0.219220 + 0.975675i \(0.429649\pi\)
\(62\) 3.54944 0.450780
\(63\) 0 0
\(64\) 1.11946 0.139933
\(65\) −5.50630 −0.682972
\(66\) 0 0
\(67\) −5.12424 −0.626026 −0.313013 0.949749i \(-0.601338\pi\)
−0.313013 + 0.949749i \(0.601338\pi\)
\(68\) −5.37733 −0.652097
\(69\) 0 0
\(70\) 0.585228 0.0699481
\(71\) 12.1937 1.44713 0.723563 0.690258i \(-0.242503\pi\)
0.723563 + 0.690258i \(0.242503\pi\)
\(72\) 0 0
\(73\) −12.7638 −1.49389 −0.746945 0.664886i \(-0.768480\pi\)
−0.746945 + 0.664886i \(0.768480\pi\)
\(74\) −0.287444 −0.0334146
\(75\) 0 0
\(76\) −1.53362 −0.175918
\(77\) −0.856948 −0.0976583
\(78\) 0 0
\(79\) 10.8691 1.22287 0.611437 0.791293i \(-0.290592\pi\)
0.611437 + 0.791293i \(0.290592\pi\)
\(80\) −1.41923 −0.158675
\(81\) 0 0
\(82\) 5.63505 0.622288
\(83\) −2.94669 −0.323441 −0.161721 0.986837i \(-0.551704\pi\)
−0.161721 + 0.986837i \(0.551704\pi\)
\(84\) 0 0
\(85\) −3.50630 −0.380311
\(86\) 1.04705 0.112907
\(87\) 0 0
\(88\) −2.41318 −0.257246
\(89\) −3.08736 −0.327259 −0.163629 0.986522i \(-0.552320\pi\)
−0.163629 + 0.986522i \(0.552320\pi\)
\(90\) 0 0
\(91\) 4.71861 0.494645
\(92\) 10.3556 1.07964
\(93\) 0 0
\(94\) 1.69944 0.175284
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −8.88210 −0.901840 −0.450920 0.892564i \(-0.648904\pi\)
−0.450920 + 0.892564i \(0.648904\pi\)
\(98\) 4.27893 0.432238
\(99\) 0 0
\(100\) −1.53362 −0.153362
\(101\) 2.09965 0.208923 0.104462 0.994529i \(-0.466688\pi\)
0.104462 + 0.994529i \(0.466688\pi\)
\(102\) 0 0
\(103\) −3.91556 −0.385812 −0.192906 0.981217i \(-0.561791\pi\)
−0.192906 + 0.981217i \(0.561791\pi\)
\(104\) 13.2877 1.30297
\(105\) 0 0
\(106\) −3.27633 −0.318225
\(107\) 18.0439 1.74437 0.872184 0.489178i \(-0.162703\pi\)
0.872184 + 0.489178i \(0.162703\pi\)
\(108\) 0 0
\(109\) 3.34558 0.320448 0.160224 0.987081i \(-0.448778\pi\)
0.160224 + 0.987081i \(0.448778\pi\)
\(110\) −0.682920 −0.0651139
\(111\) 0 0
\(112\) 1.21621 0.114921
\(113\) −7.26819 −0.683733 −0.341867 0.939748i \(-0.611059\pi\)
−0.341867 + 0.939748i \(0.611059\pi\)
\(114\) 0 0
\(115\) 6.75237 0.629662
\(116\) −0.727978 −0.0675911
\(117\) 0 0
\(118\) 1.20138 0.110596
\(119\) 3.00472 0.275442
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.33854 −0.211722
\(123\) 0 0
\(124\) 7.97091 0.715809
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.9303 −1.50232 −0.751162 0.660118i \(-0.770506\pi\)
−0.751162 + 0.660118i \(0.770506\pi\)
\(128\) 10.8267 0.956951
\(129\) 0 0
\(130\) 3.76036 0.329806
\(131\) −5.26619 −0.460109 −0.230055 0.973178i \(-0.573891\pi\)
−0.230055 + 0.973178i \(0.573891\pi\)
\(132\) 0 0
\(133\) 0.856948 0.0743069
\(134\) 3.49945 0.302307
\(135\) 0 0
\(136\) 8.46133 0.725553
\(137\) 20.6078 1.76064 0.880321 0.474378i \(-0.157327\pi\)
0.880321 + 0.474378i \(0.157327\pi\)
\(138\) 0 0
\(139\) 1.26192 0.107034 0.0535171 0.998567i \(-0.482957\pi\)
0.0535171 + 0.998567i \(0.482957\pi\)
\(140\) 1.31423 0.111073
\(141\) 0 0
\(142\) −8.32733 −0.698814
\(143\) −5.50630 −0.460460
\(144\) 0 0
\(145\) −0.474680 −0.0394200
\(146\) 8.71666 0.721396
\(147\) 0 0
\(148\) −0.645506 −0.0530603
\(149\) 11.3117 0.926692 0.463346 0.886177i \(-0.346649\pi\)
0.463346 + 0.886177i \(0.346649\pi\)
\(150\) 0 0
\(151\) −3.70074 −0.301162 −0.150581 0.988598i \(-0.548114\pi\)
−0.150581 + 0.988598i \(0.548114\pi\)
\(152\) 2.41318 0.195735
\(153\) 0 0
\(154\) 0.585228 0.0471590
\(155\) 5.19745 0.417469
\(156\) 0 0
\(157\) 2.91493 0.232637 0.116318 0.993212i \(-0.462891\pi\)
0.116318 + 0.993212i \(0.462891\pi\)
\(158\) −7.42276 −0.590523
\(159\) 0 0
\(160\) 5.79558 0.458181
\(161\) −5.78644 −0.456035
\(162\) 0 0
\(163\) 18.4278 1.44338 0.721690 0.692216i \(-0.243366\pi\)
0.721690 + 0.692216i \(0.243366\pi\)
\(164\) 12.6545 0.988152
\(165\) 0 0
\(166\) 2.01235 0.156189
\(167\) −15.1433 −1.17182 −0.585912 0.810374i \(-0.699264\pi\)
−0.585912 + 0.810374i \(0.699264\pi\)
\(168\) 0 0
\(169\) 17.3193 1.33226
\(170\) 2.39452 0.183652
\(171\) 0 0
\(172\) 2.35135 0.179288
\(173\) 0.110615 0.00840989 0.00420495 0.999991i \(-0.498662\pi\)
0.00420495 + 0.999991i \(0.498662\pi\)
\(174\) 0 0
\(175\) 0.856948 0.0647792
\(176\) −1.41923 −0.106978
\(177\) 0 0
\(178\) 2.10842 0.158033
\(179\) 9.42389 0.704375 0.352187 0.935930i \(-0.385438\pi\)
0.352187 + 0.935930i \(0.385438\pi\)
\(180\) 0 0
\(181\) −12.2876 −0.913334 −0.456667 0.889638i \(-0.650957\pi\)
−0.456667 + 0.889638i \(0.650957\pi\)
\(182\) −3.22244 −0.238863
\(183\) 0 0
\(184\) −16.2947 −1.20126
\(185\) −0.420904 −0.0309454
\(186\) 0 0
\(187\) −3.50630 −0.256406
\(188\) 3.81640 0.278340
\(189\) 0 0
\(190\) 0.682920 0.0495443
\(191\) −16.4558 −1.19070 −0.595351 0.803466i \(-0.702987\pi\)
−0.595351 + 0.803466i \(0.702987\pi\)
\(192\) 0 0
\(193\) 2.79958 0.201518 0.100759 0.994911i \(-0.467873\pi\)
0.100759 + 0.994911i \(0.467873\pi\)
\(194\) 6.06577 0.435497
\(195\) 0 0
\(196\) 9.60911 0.686365
\(197\) 7.19454 0.512589 0.256295 0.966599i \(-0.417498\pi\)
0.256295 + 0.966599i \(0.417498\pi\)
\(198\) 0 0
\(199\) −11.9287 −0.845607 −0.422803 0.906221i \(-0.638954\pi\)
−0.422803 + 0.906221i \(0.638954\pi\)
\(200\) 2.41318 0.170638
\(201\) 0 0
\(202\) −1.43390 −0.100889
\(203\) 0.406776 0.0285501
\(204\) 0 0
\(205\) 8.25141 0.576303
\(206\) 2.67402 0.186308
\(207\) 0 0
\(208\) 7.81470 0.541852
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 18.0430 1.24213 0.621065 0.783759i \(-0.286700\pi\)
0.621065 + 0.783759i \(0.286700\pi\)
\(212\) −7.35758 −0.505321
\(213\) 0 0
\(214\) −12.3225 −0.842351
\(215\) 1.53320 0.104563
\(216\) 0 0
\(217\) −4.45395 −0.302354
\(218\) −2.28476 −0.154744
\(219\) 0 0
\(220\) −1.53362 −0.103397
\(221\) 19.3067 1.29871
\(222\) 0 0
\(223\) −24.5001 −1.64065 −0.820323 0.571901i \(-0.806206\pi\)
−0.820323 + 0.571901i \(0.806206\pi\)
\(224\) −4.96652 −0.331839
\(225\) 0 0
\(226\) 4.96359 0.330173
\(227\) −16.6675 −1.10626 −0.553131 0.833094i \(-0.686567\pi\)
−0.553131 + 0.833094i \(0.686567\pi\)
\(228\) 0 0
\(229\) 14.8344 0.980282 0.490141 0.871643i \(-0.336945\pi\)
0.490141 + 0.871643i \(0.336945\pi\)
\(230\) −4.61133 −0.304062
\(231\) 0 0
\(232\) 1.14549 0.0752050
\(233\) −28.8509 −1.89009 −0.945043 0.326947i \(-0.893980\pi\)
−0.945043 + 0.326947i \(0.893980\pi\)
\(234\) 0 0
\(235\) 2.48849 0.162331
\(236\) 2.69790 0.175619
\(237\) 0 0
\(238\) −2.05198 −0.133010
\(239\) −8.29173 −0.536347 −0.268174 0.963371i \(-0.586420\pi\)
−0.268174 + 0.963371i \(0.586420\pi\)
\(240\) 0 0
\(241\) 3.48268 0.224339 0.112170 0.993689i \(-0.464220\pi\)
0.112170 + 0.993689i \(0.464220\pi\)
\(242\) −0.682920 −0.0438998
\(243\) 0 0
\(244\) −5.25162 −0.336200
\(245\) 6.26564 0.400297
\(246\) 0 0
\(247\) 5.50630 0.350357
\(248\) −12.5424 −0.796442
\(249\) 0 0
\(250\) 0.682920 0.0431917
\(251\) 28.8156 1.81883 0.909413 0.415895i \(-0.136532\pi\)
0.909413 + 0.415895i \(0.136532\pi\)
\(252\) 0 0
\(253\) 6.75237 0.424518
\(254\) 11.5621 0.725469
\(255\) 0 0
\(256\) −9.63267 −0.602042
\(257\) 8.44177 0.526583 0.263292 0.964716i \(-0.415192\pi\)
0.263292 + 0.964716i \(0.415192\pi\)
\(258\) 0 0
\(259\) 0.360693 0.0224124
\(260\) 8.44457 0.523710
\(261\) 0 0
\(262\) 3.59639 0.222186
\(263\) −21.4210 −1.32088 −0.660438 0.750881i \(-0.729629\pi\)
−0.660438 + 0.750881i \(0.729629\pi\)
\(264\) 0 0
\(265\) −4.79753 −0.294710
\(266\) −0.585228 −0.0358826
\(267\) 0 0
\(268\) 7.85864 0.480043
\(269\) −10.3642 −0.631916 −0.315958 0.948773i \(-0.602326\pi\)
−0.315958 + 0.948773i \(0.602326\pi\)
\(270\) 0 0
\(271\) −26.1079 −1.58594 −0.792972 0.609258i \(-0.791467\pi\)
−0.792972 + 0.609258i \(0.791467\pi\)
\(272\) 4.97624 0.301729
\(273\) 0 0
\(274\) −14.0735 −0.850210
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −18.7663 −1.12756 −0.563779 0.825925i \(-0.690653\pi\)
−0.563779 + 0.825925i \(0.690653\pi\)
\(278\) −0.861788 −0.0516866
\(279\) 0 0
\(280\) −2.06797 −0.123585
\(281\) −24.6322 −1.46944 −0.734718 0.678373i \(-0.762685\pi\)
−0.734718 + 0.678373i \(0.762685\pi\)
\(282\) 0 0
\(283\) −13.7870 −0.819550 −0.409775 0.912187i \(-0.634393\pi\)
−0.409775 + 0.912187i \(0.634393\pi\)
\(284\) −18.7005 −1.10967
\(285\) 0 0
\(286\) 3.76036 0.222355
\(287\) −7.07103 −0.417390
\(288\) 0 0
\(289\) −4.70587 −0.276816
\(290\) 0.324169 0.0190358
\(291\) 0 0
\(292\) 19.5748 1.14553
\(293\) 3.53033 0.206244 0.103122 0.994669i \(-0.467117\pi\)
0.103122 + 0.994669i \(0.467117\pi\)
\(294\) 0 0
\(295\) 1.75917 0.102423
\(296\) 1.01572 0.0590373
\(297\) 0 0
\(298\) −7.72501 −0.447498
\(299\) −37.1806 −2.15021
\(300\) 0 0
\(301\) −1.31387 −0.0757304
\(302\) 2.52731 0.145430
\(303\) 0 0
\(304\) 1.41923 0.0813984
\(305\) −3.42433 −0.196076
\(306\) 0 0
\(307\) 5.59192 0.319148 0.159574 0.987186i \(-0.448988\pi\)
0.159574 + 0.987186i \(0.448988\pi\)
\(308\) 1.31423 0.0748854
\(309\) 0 0
\(310\) −3.54944 −0.201595
\(311\) 3.38109 0.191724 0.0958621 0.995395i \(-0.469439\pi\)
0.0958621 + 0.995395i \(0.469439\pi\)
\(312\) 0 0
\(313\) 13.5676 0.766886 0.383443 0.923565i \(-0.374738\pi\)
0.383443 + 0.923565i \(0.374738\pi\)
\(314\) −1.99066 −0.112340
\(315\) 0 0
\(316\) −16.6691 −0.937712
\(317\) 9.02947 0.507145 0.253573 0.967316i \(-0.418394\pi\)
0.253573 + 0.967316i \(0.418394\pi\)
\(318\) 0 0
\(319\) −0.474680 −0.0265770
\(320\) −1.11946 −0.0625799
\(321\) 0 0
\(322\) 3.95167 0.220218
\(323\) 3.50630 0.195096
\(324\) 0 0
\(325\) 5.50630 0.305434
\(326\) −12.5847 −0.697005
\(327\) 0 0
\(328\) −19.9121 −1.09946
\(329\) −2.13251 −0.117569
\(330\) 0 0
\(331\) −18.6966 −1.02766 −0.513829 0.857892i \(-0.671774\pi\)
−0.513829 + 0.857892i \(0.671774\pi\)
\(332\) 4.51910 0.248018
\(333\) 0 0
\(334\) 10.3417 0.565871
\(335\) 5.12424 0.279967
\(336\) 0 0
\(337\) 29.3264 1.59751 0.798755 0.601657i \(-0.205492\pi\)
0.798755 + 0.601657i \(0.205492\pi\)
\(338\) −11.8277 −0.643343
\(339\) 0 0
\(340\) 5.37733 0.291627
\(341\) 5.19745 0.281458
\(342\) 0 0
\(343\) −11.3680 −0.613813
\(344\) −3.69989 −0.199485
\(345\) 0 0
\(346\) −0.0755411 −0.00406112
\(347\) −11.0048 −0.590768 −0.295384 0.955379i \(-0.595448\pi\)
−0.295384 + 0.955379i \(0.595448\pi\)
\(348\) 0 0
\(349\) −24.1430 −1.29235 −0.646174 0.763190i \(-0.723632\pi\)
−0.646174 + 0.763190i \(0.723632\pi\)
\(350\) −0.585228 −0.0312817
\(351\) 0 0
\(352\) 5.79558 0.308906
\(353\) −13.1068 −0.697602 −0.348801 0.937197i \(-0.613411\pi\)
−0.348801 + 0.937197i \(0.613411\pi\)
\(354\) 0 0
\(355\) −12.1937 −0.647175
\(356\) 4.73483 0.250945
\(357\) 0 0
\(358\) −6.43577 −0.340141
\(359\) 11.2347 0.592942 0.296471 0.955042i \(-0.404190\pi\)
0.296471 + 0.955042i \(0.404190\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 8.39149 0.441047
\(363\) 0 0
\(364\) −7.23656 −0.379299
\(365\) 12.7638 0.668088
\(366\) 0 0
\(367\) 0.796321 0.0415676 0.0207838 0.999784i \(-0.493384\pi\)
0.0207838 + 0.999784i \(0.493384\pi\)
\(368\) −9.58316 −0.499557
\(369\) 0 0
\(370\) 0.287444 0.0149435
\(371\) 4.11123 0.213445
\(372\) 0 0
\(373\) 4.22370 0.218695 0.109347 0.994004i \(-0.465124\pi\)
0.109347 + 0.994004i \(0.465124\pi\)
\(374\) 2.39452 0.123818
\(375\) 0 0
\(376\) −6.00518 −0.309694
\(377\) 2.61373 0.134614
\(378\) 0 0
\(379\) −22.5751 −1.15961 −0.579803 0.814757i \(-0.696870\pi\)
−0.579803 + 0.814757i \(0.696870\pi\)
\(380\) 1.53362 0.0786730
\(381\) 0 0
\(382\) 11.2380 0.574987
\(383\) −32.5159 −1.66148 −0.830742 0.556658i \(-0.812083\pi\)
−0.830742 + 0.556658i \(0.812083\pi\)
\(384\) 0 0
\(385\) 0.856948 0.0436741
\(386\) −1.91189 −0.0973128
\(387\) 0 0
\(388\) 13.6218 0.691540
\(389\) 35.4867 1.79925 0.899624 0.436665i \(-0.143840\pi\)
0.899624 + 0.436665i \(0.143840\pi\)
\(390\) 0 0
\(391\) −23.6758 −1.19734
\(392\) −15.1201 −0.763681
\(393\) 0 0
\(394\) −4.91329 −0.247528
\(395\) −10.8691 −0.546886
\(396\) 0 0
\(397\) 11.7223 0.588325 0.294162 0.955755i \(-0.404959\pi\)
0.294162 + 0.955755i \(0.404959\pi\)
\(398\) 8.14639 0.408341
\(399\) 0 0
\(400\) 1.41923 0.0709614
\(401\) 33.5093 1.67337 0.836687 0.547681i \(-0.184489\pi\)
0.836687 + 0.547681i \(0.184489\pi\)
\(402\) 0 0
\(403\) −28.6187 −1.42560
\(404\) −3.22007 −0.160205
\(405\) 0 0
\(406\) −0.277796 −0.0137868
\(407\) −0.420904 −0.0208634
\(408\) 0 0
\(409\) −6.50358 −0.321581 −0.160791 0.986989i \(-0.551404\pi\)
−0.160791 + 0.986989i \(0.551404\pi\)
\(410\) −5.63505 −0.278295
\(411\) 0 0
\(412\) 6.00499 0.295844
\(413\) −1.50752 −0.0741803
\(414\) 0 0
\(415\) 2.94669 0.144647
\(416\) −31.9122 −1.56462
\(417\) 0 0
\(418\) 0.682920 0.0334027
\(419\) 7.49790 0.366296 0.183148 0.983085i \(-0.441371\pi\)
0.183148 + 0.983085i \(0.441371\pi\)
\(420\) 0 0
\(421\) 11.5280 0.561840 0.280920 0.959731i \(-0.409360\pi\)
0.280920 + 0.959731i \(0.409360\pi\)
\(422\) −12.3219 −0.599822
\(423\) 0 0
\(424\) 11.5773 0.562243
\(425\) 3.50630 0.170080
\(426\) 0 0
\(427\) 2.93447 0.142009
\(428\) −27.6724 −1.33760
\(429\) 0 0
\(430\) −1.04705 −0.0504934
\(431\) −22.4459 −1.08118 −0.540590 0.841286i \(-0.681799\pi\)
−0.540590 + 0.841286i \(0.681799\pi\)
\(432\) 0 0
\(433\) −27.4999 −1.32156 −0.660781 0.750579i \(-0.729775\pi\)
−0.660781 + 0.750579i \(0.729775\pi\)
\(434\) 3.04169 0.146006
\(435\) 0 0
\(436\) −5.13084 −0.245723
\(437\) −6.75237 −0.323010
\(438\) 0 0
\(439\) 8.18489 0.390644 0.195322 0.980739i \(-0.437425\pi\)
0.195322 + 0.980739i \(0.437425\pi\)
\(440\) 2.41318 0.115044
\(441\) 0 0
\(442\) −13.1850 −0.627144
\(443\) 11.1175 0.528209 0.264104 0.964494i \(-0.414924\pi\)
0.264104 + 0.964494i \(0.414924\pi\)
\(444\) 0 0
\(445\) 3.08736 0.146355
\(446\) 16.7316 0.792264
\(447\) 0 0
\(448\) 0.959323 0.0453237
\(449\) −11.1426 −0.525854 −0.262927 0.964816i \(-0.584688\pi\)
−0.262927 + 0.964816i \(0.584688\pi\)
\(450\) 0 0
\(451\) 8.25141 0.388544
\(452\) 11.1466 0.524294
\(453\) 0 0
\(454\) 11.3826 0.534212
\(455\) −4.71861 −0.221212
\(456\) 0 0
\(457\) 18.3241 0.857164 0.428582 0.903503i \(-0.359013\pi\)
0.428582 + 0.903503i \(0.359013\pi\)
\(458\) −10.1307 −0.473376
\(459\) 0 0
\(460\) −10.3556 −0.482831
\(461\) −31.6216 −1.47276 −0.736382 0.676566i \(-0.763467\pi\)
−0.736382 + 0.676566i \(0.763467\pi\)
\(462\) 0 0
\(463\) −7.76077 −0.360674 −0.180337 0.983605i \(-0.557719\pi\)
−0.180337 + 0.983605i \(0.557719\pi\)
\(464\) 0.673679 0.0312748
\(465\) 0 0
\(466\) 19.7029 0.912718
\(467\) −31.2308 −1.44519 −0.722594 0.691273i \(-0.757050\pi\)
−0.722594 + 0.691273i \(0.757050\pi\)
\(468\) 0 0
\(469\) −4.39121 −0.202767
\(470\) −1.69944 −0.0783894
\(471\) 0 0
\(472\) −4.24521 −0.195401
\(473\) 1.53320 0.0704966
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) −4.60809 −0.211212
\(477\) 0 0
\(478\) 5.66259 0.259001
\(479\) 18.2169 0.832353 0.416177 0.909284i \(-0.363370\pi\)
0.416177 + 0.909284i \(0.363370\pi\)
\(480\) 0 0
\(481\) 2.31762 0.105674
\(482\) −2.37840 −0.108333
\(483\) 0 0
\(484\) −1.53362 −0.0697100
\(485\) 8.88210 0.403315
\(486\) 0 0
\(487\) 39.2194 1.77720 0.888600 0.458683i \(-0.151679\pi\)
0.888600 + 0.458683i \(0.151679\pi\)
\(488\) 8.26352 0.374072
\(489\) 0 0
\(490\) −4.27893 −0.193302
\(491\) −36.5478 −1.64938 −0.824688 0.565587i \(-0.808650\pi\)
−0.824688 + 0.565587i \(0.808650\pi\)
\(492\) 0 0
\(493\) 1.66437 0.0749594
\(494\) −3.76036 −0.169187
\(495\) 0 0
\(496\) −7.37637 −0.331209
\(497\) 10.4494 0.468719
\(498\) 0 0
\(499\) 31.4475 1.40778 0.703892 0.710307i \(-0.251444\pi\)
0.703892 + 0.710307i \(0.251444\pi\)
\(500\) 1.53362 0.0685856
\(501\) 0 0
\(502\) −19.6788 −0.878307
\(503\) −20.7020 −0.923057 −0.461528 0.887125i \(-0.652699\pi\)
−0.461528 + 0.887125i \(0.652699\pi\)
\(504\) 0 0
\(505\) −2.09965 −0.0934334
\(506\) −4.61133 −0.204999
\(507\) 0 0
\(508\) 25.9647 1.15200
\(509\) −23.5196 −1.04249 −0.521245 0.853407i \(-0.674532\pi\)
−0.521245 + 0.853407i \(0.674532\pi\)
\(510\) 0 0
\(511\) −10.9379 −0.483865
\(512\) −15.0750 −0.666226
\(513\) 0 0
\(514\) −5.76506 −0.254286
\(515\) 3.91556 0.172540
\(516\) 0 0
\(517\) 2.48849 0.109444
\(518\) −0.246324 −0.0108229
\(519\) 0 0
\(520\) −13.2877 −0.582704
\(521\) −28.6887 −1.25687 −0.628437 0.777860i \(-0.716305\pi\)
−0.628437 + 0.777860i \(0.716305\pi\)
\(522\) 0 0
\(523\) 5.70672 0.249537 0.124769 0.992186i \(-0.460181\pi\)
0.124769 + 0.992186i \(0.460181\pi\)
\(524\) 8.07634 0.352816
\(525\) 0 0
\(526\) 14.6288 0.637848
\(527\) −18.2238 −0.793842
\(528\) 0 0
\(529\) 22.5945 0.982371
\(530\) 3.27633 0.142315
\(531\) 0 0
\(532\) −1.31423 −0.0569792
\(533\) −45.4347 −1.96800
\(534\) 0 0
\(535\) −18.0439 −0.780105
\(536\) −12.3657 −0.534118
\(537\) 0 0
\(538\) 7.07792 0.305151
\(539\) 6.26564 0.269880
\(540\) 0 0
\(541\) 3.50382 0.150641 0.0753205 0.997159i \(-0.476002\pi\)
0.0753205 + 0.997159i \(0.476002\pi\)
\(542\) 17.8296 0.765849
\(543\) 0 0
\(544\) −20.3210 −0.871257
\(545\) −3.34558 −0.143309
\(546\) 0 0
\(547\) 28.3788 1.21339 0.606696 0.794934i \(-0.292495\pi\)
0.606696 + 0.794934i \(0.292495\pi\)
\(548\) −31.6045 −1.35008
\(549\) 0 0
\(550\) 0.682920 0.0291198
\(551\) 0.474680 0.0202220
\(552\) 0 0
\(553\) 9.31430 0.396084
\(554\) 12.8159 0.544495
\(555\) 0 0
\(556\) −1.93530 −0.0820749
\(557\) 0.440370 0.0186591 0.00932953 0.999956i \(-0.497030\pi\)
0.00932953 + 0.999956i \(0.497030\pi\)
\(558\) 0 0
\(559\) −8.44226 −0.357069
\(560\) −1.21621 −0.0513941
\(561\) 0 0
\(562\) 16.8219 0.709587
\(563\) −44.5147 −1.87607 −0.938036 0.346537i \(-0.887358\pi\)
−0.938036 + 0.346537i \(0.887358\pi\)
\(564\) 0 0
\(565\) 7.26819 0.305775
\(566\) 9.41540 0.395759
\(567\) 0 0
\(568\) 29.4256 1.23467
\(569\) −5.32611 −0.223282 −0.111641 0.993749i \(-0.535611\pi\)
−0.111641 + 0.993749i \(0.535611\pi\)
\(570\) 0 0
\(571\) 11.2850 0.472264 0.236132 0.971721i \(-0.424120\pi\)
0.236132 + 0.971721i \(0.424120\pi\)
\(572\) 8.44457 0.353085
\(573\) 0 0
\(574\) 4.82895 0.201557
\(575\) −6.75237 −0.281593
\(576\) 0 0
\(577\) 44.9805 1.87256 0.936281 0.351251i \(-0.114243\pi\)
0.936281 + 0.351251i \(0.114243\pi\)
\(578\) 3.21374 0.133674
\(579\) 0 0
\(580\) 0.727978 0.0302277
\(581\) −2.52516 −0.104761
\(582\) 0 0
\(583\) −4.79753 −0.198693
\(584\) −30.8014 −1.27457
\(585\) 0 0
\(586\) −2.41094 −0.0995948
\(587\) 12.8765 0.531470 0.265735 0.964046i \(-0.414385\pi\)
0.265735 + 0.964046i \(0.414385\pi\)
\(588\) 0 0
\(589\) −5.19745 −0.214157
\(590\) −1.20138 −0.0494599
\(591\) 0 0
\(592\) 0.597359 0.0245513
\(593\) 25.1716 1.03367 0.516837 0.856084i \(-0.327109\pi\)
0.516837 + 0.856084i \(0.327109\pi\)
\(594\) 0 0
\(595\) −3.00472 −0.123181
\(596\) −17.3479 −0.710597
\(597\) 0 0
\(598\) 25.3914 1.03833
\(599\) −30.2614 −1.23645 −0.618224 0.786002i \(-0.712147\pi\)
−0.618224 + 0.786002i \(0.712147\pi\)
\(600\) 0 0
\(601\) 33.1672 1.35292 0.676458 0.736481i \(-0.263514\pi\)
0.676458 + 0.736481i \(0.263514\pi\)
\(602\) 0.897271 0.0365700
\(603\) 0 0
\(604\) 5.67552 0.230934
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −10.4625 −0.424660 −0.212330 0.977198i \(-0.568105\pi\)
−0.212330 + 0.977198i \(0.568105\pi\)
\(608\) −5.79558 −0.235042
\(609\) 0 0
\(610\) 2.33854 0.0946848
\(611\) −13.7024 −0.554339
\(612\) 0 0
\(613\) −9.24317 −0.373328 −0.186664 0.982424i \(-0.559768\pi\)
−0.186664 + 0.982424i \(0.559768\pi\)
\(614\) −3.81884 −0.154116
\(615\) 0 0
\(616\) −2.06797 −0.0833210
\(617\) 1.95689 0.0787814 0.0393907 0.999224i \(-0.487458\pi\)
0.0393907 + 0.999224i \(0.487458\pi\)
\(618\) 0 0
\(619\) −48.3987 −1.94531 −0.972653 0.232263i \(-0.925387\pi\)
−0.972653 + 0.232263i \(0.925387\pi\)
\(620\) −7.97091 −0.320119
\(621\) 0 0
\(622\) −2.30902 −0.0925832
\(623\) −2.64570 −0.105998
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −9.26559 −0.370327
\(627\) 0 0
\(628\) −4.47039 −0.178388
\(629\) 1.47581 0.0588445
\(630\) 0 0
\(631\) 30.8039 1.22628 0.613141 0.789973i \(-0.289906\pi\)
0.613141 + 0.789973i \(0.289906\pi\)
\(632\) 26.2292 1.04334
\(633\) 0 0
\(634\) −6.16641 −0.244899
\(635\) 16.9303 0.671860
\(636\) 0 0
\(637\) −34.5005 −1.36696
\(638\) 0.324169 0.0128340
\(639\) 0 0
\(640\) −10.8267 −0.427961
\(641\) 7.12059 0.281246 0.140623 0.990063i \(-0.455089\pi\)
0.140623 + 0.990063i \(0.455089\pi\)
\(642\) 0 0
\(643\) −43.8135 −1.72783 −0.863917 0.503634i \(-0.831996\pi\)
−0.863917 + 0.503634i \(0.831996\pi\)
\(644\) 8.87419 0.349692
\(645\) 0 0
\(646\) −2.39452 −0.0942112
\(647\) −19.4584 −0.764988 −0.382494 0.923958i \(-0.624935\pi\)
−0.382494 + 0.923958i \(0.624935\pi\)
\(648\) 0 0
\(649\) 1.75917 0.0690536
\(650\) −3.76036 −0.147494
\(651\) 0 0
\(652\) −28.2613 −1.10680
\(653\) 35.4429 1.38699 0.693493 0.720463i \(-0.256071\pi\)
0.693493 + 0.720463i \(0.256071\pi\)
\(654\) 0 0
\(655\) 5.26619 0.205767
\(656\) −11.7106 −0.457224
\(657\) 0 0
\(658\) 1.45633 0.0567738
\(659\) 7.91185 0.308202 0.154101 0.988055i \(-0.450752\pi\)
0.154101 + 0.988055i \(0.450752\pi\)
\(660\) 0 0
\(661\) −37.9285 −1.47525 −0.737625 0.675211i \(-0.764053\pi\)
−0.737625 + 0.675211i \(0.764053\pi\)
\(662\) 12.7683 0.496254
\(663\) 0 0
\(664\) −7.11090 −0.275956
\(665\) −0.856948 −0.0332310
\(666\) 0 0
\(667\) −3.20522 −0.124106
\(668\) 23.2241 0.898567
\(669\) 0 0
\(670\) −3.49945 −0.135196
\(671\) −3.42433 −0.132195
\(672\) 0 0
\(673\) −5.19834 −0.200381 −0.100191 0.994968i \(-0.531945\pi\)
−0.100191 + 0.994968i \(0.531945\pi\)
\(674\) −20.0276 −0.771434
\(675\) 0 0
\(676\) −26.5613 −1.02159
\(677\) 12.7010 0.488137 0.244069 0.969758i \(-0.421518\pi\)
0.244069 + 0.969758i \(0.421518\pi\)
\(678\) 0 0
\(679\) −7.61150 −0.292103
\(680\) −8.46133 −0.324477
\(681\) 0 0
\(682\) −3.54944 −0.135915
\(683\) −45.3282 −1.73444 −0.867219 0.497928i \(-0.834095\pi\)
−0.867219 + 0.497928i \(0.834095\pi\)
\(684\) 0 0
\(685\) −20.6078 −0.787383
\(686\) 7.76342 0.296409
\(687\) 0 0
\(688\) −2.17596 −0.0829578
\(689\) 26.4166 1.00639
\(690\) 0 0
\(691\) 30.5149 1.16084 0.580421 0.814317i \(-0.302888\pi\)
0.580421 + 0.814317i \(0.302888\pi\)
\(692\) −0.169641 −0.00644879
\(693\) 0 0
\(694\) 7.51539 0.285280
\(695\) −1.26192 −0.0478672
\(696\) 0 0
\(697\) −28.9319 −1.09587
\(698\) 16.4878 0.624072
\(699\) 0 0
\(700\) −1.31423 −0.0496733
\(701\) −0.621590 −0.0234771 −0.0117386 0.999931i \(-0.503737\pi\)
−0.0117386 + 0.999931i \(0.503737\pi\)
\(702\) 0 0
\(703\) 0.420904 0.0158747
\(704\) −1.11946 −0.0421914
\(705\) 0 0
\(706\) 8.95087 0.336870
\(707\) 1.79930 0.0676695
\(708\) 0 0
\(709\) −21.9142 −0.823004 −0.411502 0.911409i \(-0.634996\pi\)
−0.411502 + 0.911409i \(0.634996\pi\)
\(710\) 8.32733 0.312519
\(711\) 0 0
\(712\) −7.45035 −0.279214
\(713\) 35.0951 1.31432
\(714\) 0 0
\(715\) 5.50630 0.205924
\(716\) −14.4527 −0.540121
\(717\) 0 0
\(718\) −7.67238 −0.286331
\(719\) 43.0880 1.60691 0.803455 0.595365i \(-0.202993\pi\)
0.803455 + 0.595365i \(0.202993\pi\)
\(720\) 0 0
\(721\) −3.35544 −0.124963
\(722\) −0.682920 −0.0254157
\(723\) 0 0
\(724\) 18.8446 0.700353
\(725\) 0.474680 0.0176292
\(726\) 0 0
\(727\) 30.8306 1.14344 0.571721 0.820448i \(-0.306276\pi\)
0.571721 + 0.820448i \(0.306276\pi\)
\(728\) 11.3869 0.422025
\(729\) 0 0
\(730\) −8.71666 −0.322618
\(731\) −5.37586 −0.198833
\(732\) 0 0
\(733\) −17.6688 −0.652610 −0.326305 0.945264i \(-0.605804\pi\)
−0.326305 + 0.945264i \(0.605804\pi\)
\(734\) −0.543824 −0.0200729
\(735\) 0 0
\(736\) 39.1339 1.44250
\(737\) 5.12424 0.188754
\(738\) 0 0
\(739\) −6.72966 −0.247555 −0.123777 0.992310i \(-0.539501\pi\)
−0.123777 + 0.992310i \(0.539501\pi\)
\(740\) 0.645506 0.0237293
\(741\) 0 0
\(742\) −2.80764 −0.103072
\(743\) −48.4780 −1.77848 −0.889242 0.457437i \(-0.848767\pi\)
−0.889242 + 0.457437i \(0.848767\pi\)
\(744\) 0 0
\(745\) −11.3117 −0.414429
\(746\) −2.88445 −0.105607
\(747\) 0 0
\(748\) 5.37733 0.196615
\(749\) 15.4627 0.564994
\(750\) 0 0
\(751\) −36.2684 −1.32345 −0.661726 0.749746i \(-0.730176\pi\)
−0.661726 + 0.749746i \(0.730176\pi\)
\(752\) −3.53174 −0.128789
\(753\) 0 0
\(754\) −1.78497 −0.0650047
\(755\) 3.70074 0.134684
\(756\) 0 0
\(757\) −5.13939 −0.186794 −0.0933972 0.995629i \(-0.529773\pi\)
−0.0933972 + 0.995629i \(0.529773\pi\)
\(758\) 15.4170 0.559971
\(759\) 0 0
\(760\) −2.41318 −0.0875353
\(761\) −10.2119 −0.370179 −0.185090 0.982722i \(-0.559258\pi\)
−0.185090 + 0.982722i \(0.559258\pi\)
\(762\) 0 0
\(763\) 2.86699 0.103792
\(764\) 25.2370 0.913042
\(765\) 0 0
\(766\) 22.2057 0.802326
\(767\) −9.68654 −0.349761
\(768\) 0 0
\(769\) −49.2015 −1.77425 −0.887126 0.461527i \(-0.847302\pi\)
−0.887126 + 0.461527i \(0.847302\pi\)
\(770\) −0.585228 −0.0210901
\(771\) 0 0
\(772\) −4.29350 −0.154526
\(773\) 40.7240 1.46474 0.732371 0.680906i \(-0.238414\pi\)
0.732371 + 0.680906i \(0.238414\pi\)
\(774\) 0 0
\(775\) −5.19745 −0.186698
\(776\) −21.4341 −0.769440
\(777\) 0 0
\(778\) −24.2346 −0.868853
\(779\) −8.25141 −0.295637
\(780\) 0 0
\(781\) −12.1937 −0.436325
\(782\) 16.1687 0.578192
\(783\) 0 0
\(784\) −8.89238 −0.317585
\(785\) −2.91493 −0.104038
\(786\) 0 0
\(787\) 7.23417 0.257870 0.128935 0.991653i \(-0.458844\pi\)
0.128935 + 0.991653i \(0.458844\pi\)
\(788\) −11.0337 −0.393059
\(789\) 0 0
\(790\) 7.42276 0.264090
\(791\) −6.22846 −0.221459
\(792\) 0 0
\(793\) 18.8554 0.669574
\(794\) −8.00539 −0.284101
\(795\) 0 0
\(796\) 18.2942 0.648420
\(797\) −43.9784 −1.55779 −0.778897 0.627152i \(-0.784220\pi\)
−0.778897 + 0.627152i \(0.784220\pi\)
\(798\) 0 0
\(799\) −8.72540 −0.308682
\(800\) −5.79558 −0.204905
\(801\) 0 0
\(802\) −22.8842 −0.808068
\(803\) 12.7638 0.450425
\(804\) 0 0
\(805\) 5.78644 0.203945
\(806\) 19.5443 0.688419
\(807\) 0 0
\(808\) 5.06685 0.178251
\(809\) −11.9798 −0.421187 −0.210593 0.977574i \(-0.567540\pi\)
−0.210593 + 0.977574i \(0.567540\pi\)
\(810\) 0 0
\(811\) 30.0059 1.05365 0.526825 0.849974i \(-0.323382\pi\)
0.526825 + 0.849974i \(0.323382\pi\)
\(812\) −0.623840 −0.0218925
\(813\) 0 0
\(814\) 0.287444 0.0100749
\(815\) −18.4278 −0.645499
\(816\) 0 0
\(817\) −1.53320 −0.0536399
\(818\) 4.44143 0.155291
\(819\) 0 0
\(820\) −12.6545 −0.441915
\(821\) −26.5123 −0.925284 −0.462642 0.886545i \(-0.653099\pi\)
−0.462642 + 0.886545i \(0.653099\pi\)
\(822\) 0 0
\(823\) 13.2244 0.460972 0.230486 0.973076i \(-0.425968\pi\)
0.230486 + 0.973076i \(0.425968\pi\)
\(824\) −9.44896 −0.329170
\(825\) 0 0
\(826\) 1.02952 0.0358215
\(827\) −25.4465 −0.884861 −0.442431 0.896803i \(-0.645884\pi\)
−0.442431 + 0.896803i \(0.645884\pi\)
\(828\) 0 0
\(829\) −28.6999 −0.996789 −0.498395 0.866950i \(-0.666077\pi\)
−0.498395 + 0.866950i \(0.666077\pi\)
\(830\) −2.01235 −0.0698498
\(831\) 0 0
\(832\) 6.16410 0.213702
\(833\) −21.9692 −0.761188
\(834\) 0 0
\(835\) 15.1433 0.524056
\(836\) 1.53362 0.0530413
\(837\) 0 0
\(838\) −5.12047 −0.176884
\(839\) 4.10582 0.141749 0.0708743 0.997485i \(-0.477421\pi\)
0.0708743 + 0.997485i \(0.477421\pi\)
\(840\) 0 0
\(841\) −28.7747 −0.992230
\(842\) −7.87271 −0.271311
\(843\) 0 0
\(844\) −27.6711 −0.952478
\(845\) −17.3193 −0.595803
\(846\) 0 0
\(847\) 0.856948 0.0294451
\(848\) 6.80879 0.233815
\(849\) 0 0
\(850\) −2.39452 −0.0821315
\(851\) −2.84210 −0.0974259
\(852\) 0 0
\(853\) −51.2689 −1.75541 −0.877707 0.479198i \(-0.840928\pi\)
−0.877707 + 0.479198i \(0.840928\pi\)
\(854\) −2.00401 −0.0685758
\(855\) 0 0
\(856\) 43.5431 1.48827
\(857\) 34.2836 1.17110 0.585552 0.810635i \(-0.300878\pi\)
0.585552 + 0.810635i \(0.300878\pi\)
\(858\) 0 0
\(859\) 39.5431 1.34919 0.674596 0.738187i \(-0.264318\pi\)
0.674596 + 0.738187i \(0.264318\pi\)
\(860\) −2.35135 −0.0801802
\(861\) 0 0
\(862\) 15.3288 0.522100
\(863\) 13.1157 0.446462 0.223231 0.974766i \(-0.428340\pi\)
0.223231 + 0.974766i \(0.428340\pi\)
\(864\) 0 0
\(865\) −0.110615 −0.00376102
\(866\) 18.7802 0.638179
\(867\) 0 0
\(868\) 6.83066 0.231848
\(869\) −10.8691 −0.368711
\(870\) 0 0
\(871\) −28.2156 −0.956050
\(872\) 8.07349 0.273403
\(873\) 0 0
\(874\) 4.61133 0.155981
\(875\) −0.856948 −0.0289701
\(876\) 0 0
\(877\) −34.1036 −1.15160 −0.575798 0.817592i \(-0.695309\pi\)
−0.575798 + 0.817592i \(0.695309\pi\)
\(878\) −5.58963 −0.188641
\(879\) 0 0
\(880\) 1.41923 0.0478422
\(881\) −39.1343 −1.31847 −0.659234 0.751938i \(-0.729119\pi\)
−0.659234 + 0.751938i \(0.729119\pi\)
\(882\) 0 0
\(883\) 44.1281 1.48503 0.742515 0.669829i \(-0.233633\pi\)
0.742515 + 0.669829i \(0.233633\pi\)
\(884\) −29.6092 −0.995864
\(885\) 0 0
\(886\) −7.59238 −0.255071
\(887\) 48.8255 1.63940 0.819700 0.572793i \(-0.194140\pi\)
0.819700 + 0.572793i \(0.194140\pi\)
\(888\) 0 0
\(889\) −14.5084 −0.486597
\(890\) −2.10842 −0.0706743
\(891\) 0 0
\(892\) 37.5738 1.25806
\(893\) −2.48849 −0.0832742
\(894\) 0 0
\(895\) −9.42389 −0.315006
\(896\) 9.27789 0.309953
\(897\) 0 0
\(898\) 7.60954 0.253934
\(899\) −2.46712 −0.0822832
\(900\) 0 0
\(901\) 16.8216 0.560407
\(902\) −5.63505 −0.187627
\(903\) 0 0
\(904\) −17.5394 −0.583353
\(905\) 12.2876 0.408455
\(906\) 0 0
\(907\) −25.4676 −0.845638 −0.422819 0.906214i \(-0.638959\pi\)
−0.422819 + 0.906214i \(0.638959\pi\)
\(908\) 25.5617 0.848293
\(909\) 0 0
\(910\) 3.22244 0.106823
\(911\) 23.4140 0.775740 0.387870 0.921714i \(-0.373211\pi\)
0.387870 + 0.921714i \(0.373211\pi\)
\(912\) 0 0
\(913\) 2.94669 0.0975212
\(914\) −12.5139 −0.413923
\(915\) 0 0
\(916\) −22.7503 −0.751690
\(917\) −4.51286 −0.149028
\(918\) 0 0
\(919\) −18.7511 −0.618542 −0.309271 0.950974i \(-0.600085\pi\)
−0.309271 + 0.950974i \(0.600085\pi\)
\(920\) 16.2947 0.537220
\(921\) 0 0
\(922\) 21.5950 0.711194
\(923\) 67.1422 2.21001
\(924\) 0 0
\(925\) 0.420904 0.0138392
\(926\) 5.29999 0.174169
\(927\) 0 0
\(928\) −2.75105 −0.0903075
\(929\) −40.7701 −1.33762 −0.668812 0.743432i \(-0.733197\pi\)
−0.668812 + 0.743432i \(0.733197\pi\)
\(930\) 0 0
\(931\) −6.26564 −0.205348
\(932\) 44.2463 1.44934
\(933\) 0 0
\(934\) 21.3281 0.697878
\(935\) 3.50630 0.114668
\(936\) 0 0
\(937\) −6.54322 −0.213758 −0.106879 0.994272i \(-0.534086\pi\)
−0.106879 + 0.994272i \(0.534086\pi\)
\(938\) 2.99885 0.0979159
\(939\) 0 0
\(940\) −3.81640 −0.124477
\(941\) 34.5714 1.12700 0.563498 0.826117i \(-0.309455\pi\)
0.563498 + 0.826117i \(0.309455\pi\)
\(942\) 0 0
\(943\) 55.7166 1.81438
\(944\) −2.49667 −0.0812597
\(945\) 0 0
\(946\) −1.04705 −0.0340427
\(947\) −17.6584 −0.573822 −0.286911 0.957957i \(-0.592628\pi\)
−0.286911 + 0.957957i \(0.592628\pi\)
\(948\) 0 0
\(949\) −70.2813 −2.28143
\(950\) −0.682920 −0.0221569
\(951\) 0 0
\(952\) 7.25093 0.235004
\(953\) 42.7340 1.38429 0.692145 0.721758i \(-0.256666\pi\)
0.692145 + 0.721758i \(0.256666\pi\)
\(954\) 0 0
\(955\) 16.4558 0.532498
\(956\) 12.7164 0.411276
\(957\) 0 0
\(958\) −12.4407 −0.401941
\(959\) 17.6598 0.570265
\(960\) 0 0
\(961\) −3.98652 −0.128597
\(962\) −1.58275 −0.0510299
\(963\) 0 0
\(964\) −5.34111 −0.172026
\(965\) −2.79958 −0.0901218
\(966\) 0 0
\(967\) −3.62541 −0.116585 −0.0582927 0.998300i \(-0.518566\pi\)
−0.0582927 + 0.998300i \(0.518566\pi\)
\(968\) 2.41318 0.0775626
\(969\) 0 0
\(970\) −6.06577 −0.194760
\(971\) −2.86101 −0.0918142 −0.0459071 0.998946i \(-0.514618\pi\)
−0.0459071 + 0.998946i \(0.514618\pi\)
\(972\) 0 0
\(973\) 1.08140 0.0346680
\(974\) −26.7837 −0.858206
\(975\) 0 0
\(976\) 4.85990 0.155562
\(977\) −29.4990 −0.943756 −0.471878 0.881664i \(-0.656424\pi\)
−0.471878 + 0.881664i \(0.656424\pi\)
\(978\) 0 0
\(979\) 3.08736 0.0986723
\(980\) −9.60911 −0.306952
\(981\) 0 0
\(982\) 24.9592 0.796480
\(983\) −31.1737 −0.994287 −0.497144 0.867668i \(-0.665618\pi\)
−0.497144 + 0.867668i \(0.665618\pi\)
\(984\) 0 0
\(985\) −7.19454 −0.229237
\(986\) −1.13663 −0.0361977
\(987\) 0 0
\(988\) −8.44457 −0.268657
\(989\) 10.3527 0.329198
\(990\) 0 0
\(991\) −55.9145 −1.77618 −0.888092 0.459666i \(-0.847969\pi\)
−0.888092 + 0.459666i \(0.847969\pi\)
\(992\) 30.1222 0.956382
\(993\) 0 0
\(994\) −7.13609 −0.226343
\(995\) 11.9287 0.378167
\(996\) 0 0
\(997\) 42.1768 1.33575 0.667877 0.744272i \(-0.267203\pi\)
0.667877 + 0.744272i \(0.267203\pi\)
\(998\) −21.4762 −0.679816
\(999\) 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 9405.2.a.bh.1.5 9
3.2 odd 2 1045.2.a.k.1.5 9
15.14 odd 2 5225.2.a.p.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.k.1.5 9 3.2 odd 2
5225.2.a.p.1.5 9 15.14 odd 2
9405.2.a.bh.1.5 9 1.1 even 1 trivial