Properties

Label 1805.2.a.v.1.4
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
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] = [1805,2,Mod(1,1805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1805, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1805.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.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: \(14.4129975648\)
Analytic rank: \(0\)
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: \( x^{9} - 3x^{8} - 6x^{7} + 16x^{6} + 12x^{5} - 27x^{4} - 8x^{3} + 15x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.789016\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.210984 q^{2} +0.0798955 q^{3} -1.95549 q^{4} +1.00000 q^{5} +0.0168566 q^{6} -1.68723 q^{7} -0.834543 q^{8} -2.99362 q^{9} +O(q^{10})\) \(q+0.210984 q^{2} +0.0798955 q^{3} -1.95549 q^{4} +1.00000 q^{5} +0.0168566 q^{6} -1.68723 q^{7} -0.834543 q^{8} -2.99362 q^{9} +0.210984 q^{10} -2.88678 q^{11} -0.156235 q^{12} -6.46601 q^{13} -0.355977 q^{14} +0.0798955 q^{15} +3.73490 q^{16} +2.98649 q^{17} -0.631604 q^{18} -1.95549 q^{20} -0.134802 q^{21} -0.609064 q^{22} +8.25754 q^{23} -0.0666762 q^{24} +1.00000 q^{25} -1.36422 q^{26} -0.478863 q^{27} +3.29935 q^{28} +7.25727 q^{29} +0.0168566 q^{30} +4.05600 q^{31} +2.45709 q^{32} -0.230641 q^{33} +0.630100 q^{34} -1.68723 q^{35} +5.85398 q^{36} +7.96989 q^{37} -0.516605 q^{39} -0.834543 q^{40} +5.45595 q^{41} -0.0284410 q^{42} -5.33164 q^{43} +5.64506 q^{44} -2.99362 q^{45} +1.74221 q^{46} +1.64961 q^{47} +0.298402 q^{48} -4.15326 q^{49} +0.210984 q^{50} +0.238607 q^{51} +12.6442 q^{52} +1.91194 q^{53} -0.101032 q^{54} -2.88678 q^{55} +1.40806 q^{56} +1.53116 q^{58} +3.55276 q^{59} -0.156235 q^{60} -7.90014 q^{61} +0.855750 q^{62} +5.05091 q^{63} -6.95139 q^{64} -6.46601 q^{65} -0.0486615 q^{66} +2.26533 q^{67} -5.84003 q^{68} +0.659740 q^{69} -0.355977 q^{70} +10.4224 q^{71} +2.49830 q^{72} +1.41186 q^{73} +1.68152 q^{74} +0.0798955 q^{75} +4.87066 q^{77} -0.108995 q^{78} -2.41076 q^{79} +3.73490 q^{80} +8.94259 q^{81} +1.15112 q^{82} -11.4006 q^{83} +0.263603 q^{84} +2.98649 q^{85} -1.12489 q^{86} +0.579823 q^{87} +2.40914 q^{88} -10.1161 q^{89} -0.631604 q^{90} +10.9096 q^{91} -16.1475 q^{92} +0.324057 q^{93} +0.348041 q^{94} +0.196310 q^{96} -9.86820 q^{97} -0.876270 q^{98} +8.64192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 6 q^{2} + 9 q^{3} + 6 q^{4} + 9 q^{5} + 12 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 6 q^{2} + 9 q^{3} + 6 q^{4} + 9 q^{5} + 12 q^{6} + 6 q^{8} + 6 q^{9} + 6 q^{10} + 18 q^{12} + 9 q^{13} + 9 q^{15} + 12 q^{16} - 9 q^{17} + 24 q^{18} + 6 q^{20} + 12 q^{21} + 24 q^{22} - 12 q^{23} + 3 q^{24} + 9 q^{25} - 3 q^{26} + 24 q^{27} - 15 q^{28} + 9 q^{29} + 12 q^{30} + 18 q^{31} + 3 q^{32} - 9 q^{33} - 24 q^{34} + 18 q^{36} + 18 q^{37} + 18 q^{39} + 6 q^{40} + 6 q^{41} - 12 q^{43} + 48 q^{44} + 6 q^{45} - 9 q^{46} + 15 q^{47} - 21 q^{48} - 9 q^{49} + 6 q^{50} - 6 q^{51} + 33 q^{52} + 15 q^{53} + 63 q^{54} + 6 q^{58} + 21 q^{59} + 18 q^{60} - 12 q^{61} - 36 q^{62} + 21 q^{63} - 36 q^{64} + 9 q^{65} + 3 q^{66} + 60 q^{67} - 51 q^{68} - 15 q^{69} - 18 q^{71} + 27 q^{73} + 27 q^{74} + 9 q^{75} - 30 q^{77} - 6 q^{78} + 15 q^{79} + 12 q^{80} + 33 q^{81} + 24 q^{82} - 48 q^{84} - 9 q^{85} - 39 q^{86} + 15 q^{87} + 27 q^{88} - 39 q^{89} + 24 q^{90} + 21 q^{91} - 6 q^{92} + 15 q^{93} + 15 q^{94} - 33 q^{96} + 15 q^{97} - 15 q^{98} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.210984 0.149188 0.0745940 0.997214i \(-0.476234\pi\)
0.0745940 + 0.997214i \(0.476234\pi\)
\(3\) 0.0798955 0.0461277 0.0230639 0.999734i \(-0.492658\pi\)
0.0230639 + 0.999734i \(0.492658\pi\)
\(4\) −1.95549 −0.977743
\(5\) 1.00000 0.447214
\(6\) 0.0168566 0.00688170
\(7\) −1.68723 −0.637712 −0.318856 0.947803i \(-0.603299\pi\)
−0.318856 + 0.947803i \(0.603299\pi\)
\(8\) −0.834543 −0.295055
\(9\) −2.99362 −0.997872
\(10\) 0.210984 0.0667189
\(11\) −2.88678 −0.870398 −0.435199 0.900334i \(-0.643322\pi\)
−0.435199 + 0.900334i \(0.643322\pi\)
\(12\) −0.156235 −0.0451010
\(13\) −6.46601 −1.79335 −0.896674 0.442691i \(-0.854024\pi\)
−0.896674 + 0.442691i \(0.854024\pi\)
\(14\) −0.355977 −0.0951390
\(15\) 0.0798955 0.0206289
\(16\) 3.73490 0.933724
\(17\) 2.98649 0.724329 0.362165 0.932114i \(-0.382038\pi\)
0.362165 + 0.932114i \(0.382038\pi\)
\(18\) −0.631604 −0.148871
\(19\) 0 0
\(20\) −1.95549 −0.437260
\(21\) −0.134802 −0.0294162
\(22\) −0.609064 −0.129853
\(23\) 8.25754 1.72182 0.860908 0.508761i \(-0.169896\pi\)
0.860908 + 0.508761i \(0.169896\pi\)
\(24\) −0.0666762 −0.0136102
\(25\) 1.00000 0.200000
\(26\) −1.36422 −0.267546
\(27\) −0.478863 −0.0921573
\(28\) 3.29935 0.623519
\(29\) 7.25727 1.34764 0.673820 0.738895i \(-0.264652\pi\)
0.673820 + 0.738895i \(0.264652\pi\)
\(30\) 0.0168566 0.00307759
\(31\) 4.05600 0.728480 0.364240 0.931305i \(-0.381329\pi\)
0.364240 + 0.931305i \(0.381329\pi\)
\(32\) 2.45709 0.434356
\(33\) −0.230641 −0.0401495
\(34\) 0.630100 0.108061
\(35\) −1.68723 −0.285194
\(36\) 5.85398 0.975663
\(37\) 7.96989 1.31024 0.655121 0.755524i \(-0.272618\pi\)
0.655121 + 0.755524i \(0.272618\pi\)
\(38\) 0 0
\(39\) −0.516605 −0.0827230
\(40\) −0.834543 −0.131953
\(41\) 5.45595 0.852076 0.426038 0.904705i \(-0.359909\pi\)
0.426038 + 0.904705i \(0.359909\pi\)
\(42\) −0.0284410 −0.00438854
\(43\) −5.33164 −0.813068 −0.406534 0.913636i \(-0.633263\pi\)
−0.406534 + 0.913636i \(0.633263\pi\)
\(44\) 5.64506 0.851025
\(45\) −2.99362 −0.446262
\(46\) 1.74221 0.256874
\(47\) 1.64961 0.240620 0.120310 0.992736i \(-0.461611\pi\)
0.120310 + 0.992736i \(0.461611\pi\)
\(48\) 0.298402 0.0430706
\(49\) −4.15326 −0.593323
\(50\) 0.210984 0.0298376
\(51\) 0.238607 0.0334117
\(52\) 12.6442 1.75343
\(53\) 1.91194 0.262625 0.131312 0.991341i \(-0.458081\pi\)
0.131312 + 0.991341i \(0.458081\pi\)
\(54\) −0.101032 −0.0137488
\(55\) −2.88678 −0.389254
\(56\) 1.40806 0.188160
\(57\) 0 0
\(58\) 1.53116 0.201052
\(59\) 3.55276 0.462530 0.231265 0.972891i \(-0.425714\pi\)
0.231265 + 0.972891i \(0.425714\pi\)
\(60\) −0.156235 −0.0201698
\(61\) −7.90014 −1.01151 −0.505755 0.862677i \(-0.668786\pi\)
−0.505755 + 0.862677i \(0.668786\pi\)
\(62\) 0.855750 0.108680
\(63\) 5.05091 0.636355
\(64\) −6.95139 −0.868924
\(65\) −6.46601 −0.802010
\(66\) −0.0486615 −0.00598982
\(67\) 2.26533 0.276754 0.138377 0.990380i \(-0.455811\pi\)
0.138377 + 0.990380i \(0.455811\pi\)
\(68\) −5.84003 −0.708208
\(69\) 0.659740 0.0794234
\(70\) −0.355977 −0.0425474
\(71\) 10.4224 1.23692 0.618458 0.785818i \(-0.287758\pi\)
0.618458 + 0.785818i \(0.287758\pi\)
\(72\) 2.49830 0.294428
\(73\) 1.41186 0.165246 0.0826231 0.996581i \(-0.473670\pi\)
0.0826231 + 0.996581i \(0.473670\pi\)
\(74\) 1.68152 0.195472
\(75\) 0.0798955 0.00922554
\(76\) 0 0
\(77\) 4.87066 0.555063
\(78\) −0.108995 −0.0123413
\(79\) −2.41076 −0.271232 −0.135616 0.990761i \(-0.543301\pi\)
−0.135616 + 0.990761i \(0.543301\pi\)
\(80\) 3.73490 0.417574
\(81\) 8.94259 0.993621
\(82\) 1.15112 0.127120
\(83\) −11.4006 −1.25138 −0.625688 0.780073i \(-0.715182\pi\)
−0.625688 + 0.780073i \(0.715182\pi\)
\(84\) 0.263603 0.0287615
\(85\) 2.98649 0.323930
\(86\) −1.12489 −0.121300
\(87\) 0.579823 0.0621636
\(88\) 2.40914 0.256816
\(89\) −10.1161 −1.07230 −0.536152 0.844121i \(-0.680123\pi\)
−0.536152 + 0.844121i \(0.680123\pi\)
\(90\) −0.631604 −0.0665769
\(91\) 10.9096 1.14364
\(92\) −16.1475 −1.68349
\(93\) 0.324057 0.0336031
\(94\) 0.348041 0.0358977
\(95\) 0 0
\(96\) 0.196310 0.0200358
\(97\) −9.86820 −1.00196 −0.500982 0.865458i \(-0.667028\pi\)
−0.500982 + 0.865458i \(0.667028\pi\)
\(98\) −0.876270 −0.0885167
\(99\) 8.64192 0.868546
\(100\) −1.95549 −0.195549
\(101\) 4.16975 0.414906 0.207453 0.978245i \(-0.433483\pi\)
0.207453 + 0.978245i \(0.433483\pi\)
\(102\) 0.0503422 0.00498462
\(103\) −12.5636 −1.23793 −0.618964 0.785419i \(-0.712448\pi\)
−0.618964 + 0.785419i \(0.712448\pi\)
\(104\) 5.39616 0.529137
\(105\) −0.134802 −0.0131553
\(106\) 0.403387 0.0391805
\(107\) 14.9423 1.44453 0.722264 0.691617i \(-0.243102\pi\)
0.722264 + 0.691617i \(0.243102\pi\)
\(108\) 0.936410 0.0901061
\(109\) 5.20496 0.498545 0.249272 0.968433i \(-0.419809\pi\)
0.249272 + 0.968433i \(0.419809\pi\)
\(110\) −0.609064 −0.0580720
\(111\) 0.636759 0.0604385
\(112\) −6.30162 −0.595447
\(113\) 8.57064 0.806258 0.403129 0.915143i \(-0.367923\pi\)
0.403129 + 0.915143i \(0.367923\pi\)
\(114\) 0 0
\(115\) 8.25754 0.770019
\(116\) −14.1915 −1.31765
\(117\) 19.3568 1.78953
\(118\) 0.749575 0.0690039
\(119\) −5.03888 −0.461914
\(120\) −0.0666762 −0.00608668
\(121\) −2.66648 −0.242407
\(122\) −1.66680 −0.150905
\(123\) 0.435906 0.0393043
\(124\) −7.93146 −0.712266
\(125\) 1.00000 0.0894427
\(126\) 1.06566 0.0949365
\(127\) 13.0369 1.15684 0.578421 0.815738i \(-0.303669\pi\)
0.578421 + 0.815738i \(0.303669\pi\)
\(128\) −6.38080 −0.563989
\(129\) −0.425974 −0.0375050
\(130\) −1.36422 −0.119650
\(131\) 0.187083 0.0163456 0.00817278 0.999967i \(-0.497398\pi\)
0.00817278 + 0.999967i \(0.497398\pi\)
\(132\) 0.451015 0.0392558
\(133\) 0 0
\(134\) 0.477948 0.0412884
\(135\) −0.478863 −0.0412140
\(136\) −2.49235 −0.213717
\(137\) 2.15559 0.184164 0.0920821 0.995751i \(-0.470648\pi\)
0.0920821 + 0.995751i \(0.470648\pi\)
\(138\) 0.139194 0.0118490
\(139\) −7.89712 −0.669825 −0.334913 0.942249i \(-0.608707\pi\)
−0.334913 + 0.942249i \(0.608707\pi\)
\(140\) 3.29935 0.278846
\(141\) 0.131796 0.0110993
\(142\) 2.19897 0.184533
\(143\) 18.6660 1.56093
\(144\) −11.1808 −0.931737
\(145\) 7.25727 0.602683
\(146\) 0.297880 0.0246527
\(147\) −0.331827 −0.0273686
\(148\) −15.5850 −1.28108
\(149\) 17.8966 1.46615 0.733074 0.680149i \(-0.238085\pi\)
0.733074 + 0.680149i \(0.238085\pi\)
\(150\) 0.0168566 0.00137634
\(151\) 2.93984 0.239241 0.119620 0.992820i \(-0.461832\pi\)
0.119620 + 0.992820i \(0.461832\pi\)
\(152\) 0 0
\(153\) −8.94040 −0.722788
\(154\) 1.02763 0.0828088
\(155\) 4.05600 0.325786
\(156\) 1.01021 0.0808819
\(157\) 6.76687 0.540055 0.270028 0.962853i \(-0.412967\pi\)
0.270028 + 0.962853i \(0.412967\pi\)
\(158\) −0.508631 −0.0404645
\(159\) 0.152755 0.0121143
\(160\) 2.45709 0.194250
\(161\) −13.9323 −1.09802
\(162\) 1.88674 0.148236
\(163\) −0.907834 −0.0711070 −0.0355535 0.999368i \(-0.511319\pi\)
−0.0355535 + 0.999368i \(0.511319\pi\)
\(164\) −10.6690 −0.833111
\(165\) −0.230641 −0.0179554
\(166\) −2.40534 −0.186690
\(167\) 14.2716 1.10437 0.552184 0.833722i \(-0.313795\pi\)
0.552184 + 0.833722i \(0.313795\pi\)
\(168\) 0.112498 0.00867941
\(169\) 28.8093 2.21610
\(170\) 0.630100 0.0483264
\(171\) 0 0
\(172\) 10.4260 0.794971
\(173\) 14.0774 1.07029 0.535144 0.844761i \(-0.320257\pi\)
0.535144 + 0.844761i \(0.320257\pi\)
\(174\) 0.122333 0.00927406
\(175\) −1.68723 −0.127542
\(176\) −10.7818 −0.812712
\(177\) 0.283850 0.0213355
\(178\) −2.13433 −0.159975
\(179\) 15.7498 1.17719 0.588596 0.808427i \(-0.299681\pi\)
0.588596 + 0.808427i \(0.299681\pi\)
\(180\) 5.85398 0.436330
\(181\) 4.55831 0.338816 0.169408 0.985546i \(-0.445814\pi\)
0.169408 + 0.985546i \(0.445814\pi\)
\(182\) 2.30175 0.170617
\(183\) −0.631186 −0.0466586
\(184\) −6.89127 −0.508031
\(185\) 7.96989 0.585958
\(186\) 0.0683706 0.00501318
\(187\) −8.62134 −0.630455
\(188\) −3.22579 −0.235265
\(189\) 0.807951 0.0587698
\(190\) 0 0
\(191\) −9.95887 −0.720599 −0.360299 0.932837i \(-0.617325\pi\)
−0.360299 + 0.932837i \(0.617325\pi\)
\(192\) −0.555385 −0.0400814
\(193\) −21.4012 −1.54049 −0.770247 0.637746i \(-0.779867\pi\)
−0.770247 + 0.637746i \(0.779867\pi\)
\(194\) −2.08203 −0.149481
\(195\) −0.516605 −0.0369949
\(196\) 8.12165 0.580118
\(197\) 1.36674 0.0973765 0.0486883 0.998814i \(-0.484496\pi\)
0.0486883 + 0.998814i \(0.484496\pi\)
\(198\) 1.82330 0.129577
\(199\) −17.9359 −1.27144 −0.635722 0.771918i \(-0.719298\pi\)
−0.635722 + 0.771918i \(0.719298\pi\)
\(200\) −0.834543 −0.0590111
\(201\) 0.180990 0.0127660
\(202\) 0.879749 0.0618989
\(203\) −12.2447 −0.859407
\(204\) −0.466592 −0.0326680
\(205\) 5.45595 0.381060
\(206\) −2.65072 −0.184684
\(207\) −24.7199 −1.71815
\(208\) −24.1499 −1.67449
\(209\) 0 0
\(210\) −0.0284410 −0.00196262
\(211\) 20.7678 1.42971 0.714857 0.699270i \(-0.246492\pi\)
0.714857 + 0.699270i \(0.246492\pi\)
\(212\) −3.73877 −0.256779
\(213\) 0.832707 0.0570561
\(214\) 3.15258 0.215506
\(215\) −5.33164 −0.363615
\(216\) 0.399632 0.0271915
\(217\) −6.84340 −0.464560
\(218\) 1.09816 0.0743769
\(219\) 0.112802 0.00762243
\(220\) 5.64506 0.380590
\(221\) −19.3107 −1.29898
\(222\) 0.134346 0.00901669
\(223\) 1.94996 0.130579 0.0652894 0.997866i \(-0.479203\pi\)
0.0652894 + 0.997866i \(0.479203\pi\)
\(224\) −4.14567 −0.276994
\(225\) −2.99362 −0.199574
\(226\) 1.80827 0.120284
\(227\) 12.6099 0.836950 0.418475 0.908228i \(-0.362565\pi\)
0.418475 + 0.908228i \(0.362565\pi\)
\(228\) 0 0
\(229\) −6.12765 −0.404926 −0.202463 0.979290i \(-0.564895\pi\)
−0.202463 + 0.979290i \(0.564895\pi\)
\(230\) 1.74221 0.114878
\(231\) 0.389144 0.0256038
\(232\) −6.05650 −0.397629
\(233\) 6.60713 0.432848 0.216424 0.976299i \(-0.430561\pi\)
0.216424 + 0.976299i \(0.430561\pi\)
\(234\) 4.08396 0.266977
\(235\) 1.64961 0.107609
\(236\) −6.94738 −0.452236
\(237\) −0.192609 −0.0125113
\(238\) −1.06312 −0.0689120
\(239\) −2.20382 −0.142553 −0.0712766 0.997457i \(-0.522707\pi\)
−0.0712766 + 0.997457i \(0.522707\pi\)
\(240\) 0.298402 0.0192617
\(241\) −0.124850 −0.00804232 −0.00402116 0.999992i \(-0.501280\pi\)
−0.00402116 + 0.999992i \(0.501280\pi\)
\(242\) −0.562584 −0.0361643
\(243\) 2.15106 0.137991
\(244\) 15.4486 0.988996
\(245\) −4.15326 −0.265342
\(246\) 0.0919690 0.00586373
\(247\) 0 0
\(248\) −3.38491 −0.214942
\(249\) −0.910855 −0.0577231
\(250\) 0.210984 0.0133438
\(251\) 3.71679 0.234602 0.117301 0.993096i \(-0.462576\pi\)
0.117301 + 0.993096i \(0.462576\pi\)
\(252\) −9.87699 −0.622192
\(253\) −23.8377 −1.49866
\(254\) 2.75058 0.172587
\(255\) 0.238607 0.0149421
\(256\) 12.5565 0.784783
\(257\) −26.9757 −1.68270 −0.841349 0.540492i \(-0.818238\pi\)
−0.841349 + 0.540492i \(0.818238\pi\)
\(258\) −0.0898736 −0.00559529
\(259\) −13.4470 −0.835557
\(260\) 12.6442 0.784159
\(261\) −21.7255 −1.34477
\(262\) 0.0394715 0.00243856
\(263\) 20.6759 1.27493 0.637464 0.770480i \(-0.279983\pi\)
0.637464 + 0.770480i \(0.279983\pi\)
\(264\) 0.192480 0.0118463
\(265\) 1.91194 0.117449
\(266\) 0 0
\(267\) −0.808231 −0.0494629
\(268\) −4.42982 −0.270594
\(269\) −13.7249 −0.836820 −0.418410 0.908258i \(-0.637413\pi\)
−0.418410 + 0.908258i \(0.637413\pi\)
\(270\) −0.101032 −0.00614863
\(271\) −5.38346 −0.327022 −0.163511 0.986542i \(-0.552282\pi\)
−0.163511 + 0.986542i \(0.552282\pi\)
\(272\) 11.1542 0.676324
\(273\) 0.871631 0.0527535
\(274\) 0.454794 0.0274751
\(275\) −2.88678 −0.174080
\(276\) −1.29011 −0.0776557
\(277\) 22.7985 1.36983 0.684916 0.728622i \(-0.259839\pi\)
0.684916 + 0.728622i \(0.259839\pi\)
\(278\) −1.66616 −0.0999299
\(279\) −12.1421 −0.726930
\(280\) 1.40806 0.0841479
\(281\) −17.2290 −1.02780 −0.513899 0.857851i \(-0.671799\pi\)
−0.513899 + 0.857851i \(0.671799\pi\)
\(282\) 0.0278069 0.00165588
\(283\) 16.2988 0.968866 0.484433 0.874828i \(-0.339026\pi\)
0.484433 + 0.874828i \(0.339026\pi\)
\(284\) −20.3809 −1.20939
\(285\) 0 0
\(286\) 3.93821 0.232871
\(287\) −9.20543 −0.543379
\(288\) −7.35558 −0.433432
\(289\) −8.08090 −0.475347
\(290\) 1.53116 0.0899131
\(291\) −0.788425 −0.0462183
\(292\) −2.76088 −0.161568
\(293\) 25.2882 1.47735 0.738675 0.674061i \(-0.235452\pi\)
0.738675 + 0.674061i \(0.235452\pi\)
\(294\) −0.0700101 −0.00408307
\(295\) 3.55276 0.206850
\(296\) −6.65122 −0.386594
\(297\) 1.38237 0.0802135
\(298\) 3.77589 0.218731
\(299\) −53.3933 −3.08781
\(300\) −0.156235 −0.00902021
\(301\) 8.99570 0.518503
\(302\) 0.620258 0.0356918
\(303\) 0.333144 0.0191386
\(304\) 0 0
\(305\) −7.90014 −0.452361
\(306\) −1.88628 −0.107831
\(307\) 17.2290 0.983309 0.491655 0.870790i \(-0.336392\pi\)
0.491655 + 0.870790i \(0.336392\pi\)
\(308\) −9.52451 −0.542709
\(309\) −1.00378 −0.0571028
\(310\) 0.855750 0.0486034
\(311\) 24.9724 1.41606 0.708028 0.706184i \(-0.249585\pi\)
0.708028 + 0.706184i \(0.249585\pi\)
\(312\) 0.431129 0.0244079
\(313\) −16.8633 −0.953167 −0.476584 0.879129i \(-0.658125\pi\)
−0.476584 + 0.879129i \(0.658125\pi\)
\(314\) 1.42770 0.0805697
\(315\) 5.05091 0.284587
\(316\) 4.71421 0.265195
\(317\) 24.1211 1.35478 0.677389 0.735625i \(-0.263111\pi\)
0.677389 + 0.735625i \(0.263111\pi\)
\(318\) 0.0322288 0.00180730
\(319\) −20.9502 −1.17298
\(320\) −6.95139 −0.388594
\(321\) 1.19382 0.0666328
\(322\) −2.93950 −0.163812
\(323\) 0 0
\(324\) −17.4871 −0.971506
\(325\) −6.46601 −0.358670
\(326\) −0.191538 −0.0106083
\(327\) 0.415853 0.0229967
\(328\) −4.55322 −0.251410
\(329\) −2.78327 −0.153447
\(330\) −0.0486615 −0.00267873
\(331\) 15.5403 0.854174 0.427087 0.904211i \(-0.359540\pi\)
0.427087 + 0.904211i \(0.359540\pi\)
\(332\) 22.2937 1.22352
\(333\) −23.8588 −1.30745
\(334\) 3.01107 0.164758
\(335\) 2.26533 0.123768
\(336\) −0.503471 −0.0274666
\(337\) 11.0057 0.599516 0.299758 0.954015i \(-0.403094\pi\)
0.299758 + 0.954015i \(0.403094\pi\)
\(338\) 6.07829 0.330615
\(339\) 0.684756 0.0371908
\(340\) −5.84003 −0.316720
\(341\) −11.7088 −0.634067
\(342\) 0 0
\(343\) 18.8181 1.01608
\(344\) 4.44948 0.239900
\(345\) 0.659740 0.0355192
\(346\) 2.97011 0.159674
\(347\) 30.4697 1.63570 0.817849 0.575433i \(-0.195167\pi\)
0.817849 + 0.575433i \(0.195167\pi\)
\(348\) −1.13384 −0.0607800
\(349\) 5.61744 0.300695 0.150347 0.988633i \(-0.451961\pi\)
0.150347 + 0.988633i \(0.451961\pi\)
\(350\) −0.355977 −0.0190278
\(351\) 3.09633 0.165270
\(352\) −7.09308 −0.378062
\(353\) −7.48617 −0.398449 −0.199224 0.979954i \(-0.563842\pi\)
−0.199224 + 0.979954i \(0.563842\pi\)
\(354\) 0.0598877 0.00318299
\(355\) 10.4224 0.553166
\(356\) 19.7819 1.04844
\(357\) −0.402584 −0.0213070
\(358\) 3.32294 0.175623
\(359\) −25.4854 −1.34507 −0.672534 0.740066i \(-0.734794\pi\)
−0.672534 + 0.740066i \(0.734794\pi\)
\(360\) 2.49830 0.131672
\(361\) 0 0
\(362\) 0.961728 0.0505473
\(363\) −0.213040 −0.0111817
\(364\) −21.3336 −1.11819
\(365\) 1.41186 0.0739003
\(366\) −0.133170 −0.00696090
\(367\) −33.4662 −1.74692 −0.873461 0.486894i \(-0.838130\pi\)
−0.873461 + 0.486894i \(0.838130\pi\)
\(368\) 30.8410 1.60770
\(369\) −16.3330 −0.850263
\(370\) 1.68152 0.0874179
\(371\) −3.22587 −0.167479
\(372\) −0.633688 −0.0328552
\(373\) −10.4787 −0.542566 −0.271283 0.962500i \(-0.587448\pi\)
−0.271283 + 0.962500i \(0.587448\pi\)
\(374\) −1.81896 −0.0940563
\(375\) 0.0798955 0.00412579
\(376\) −1.37667 −0.0709964
\(377\) −46.9256 −2.41679
\(378\) 0.170464 0.00876775
\(379\) −14.2962 −0.734344 −0.367172 0.930153i \(-0.619674\pi\)
−0.367172 + 0.930153i \(0.619674\pi\)
\(380\) 0 0
\(381\) 1.04159 0.0533625
\(382\) −2.10116 −0.107505
\(383\) 36.9815 1.88967 0.944833 0.327552i \(-0.106224\pi\)
0.944833 + 0.327552i \(0.106224\pi\)
\(384\) −0.509798 −0.0260155
\(385\) 4.87066 0.248232
\(386\) −4.51531 −0.229823
\(387\) 15.9609 0.811338
\(388\) 19.2971 0.979663
\(389\) −16.6074 −0.842029 −0.421015 0.907054i \(-0.638326\pi\)
−0.421015 + 0.907054i \(0.638326\pi\)
\(390\) −0.108995 −0.00551919
\(391\) 24.6610 1.24716
\(392\) 3.46608 0.175063
\(393\) 0.0149471 0.000753983 0
\(394\) 0.288361 0.0145274
\(395\) −2.41076 −0.121299
\(396\) −16.8992 −0.849215
\(397\) −26.6383 −1.33694 −0.668470 0.743739i \(-0.733051\pi\)
−0.668470 + 0.743739i \(0.733051\pi\)
\(398\) −3.78419 −0.189684
\(399\) 0 0
\(400\) 3.73490 0.186745
\(401\) 1.29475 0.0646570 0.0323285 0.999477i \(-0.489708\pi\)
0.0323285 + 0.999477i \(0.489708\pi\)
\(402\) 0.0381859 0.00190454
\(403\) −26.2262 −1.30642
\(404\) −8.15389 −0.405671
\(405\) 8.94259 0.444361
\(406\) −2.58342 −0.128213
\(407\) −23.0074 −1.14043
\(408\) −0.199128 −0.00985829
\(409\) 27.0318 1.33664 0.668318 0.743876i \(-0.267015\pi\)
0.668318 + 0.743876i \(0.267015\pi\)
\(410\) 1.15112 0.0568496
\(411\) 0.172222 0.00849507
\(412\) 24.5680 1.21038
\(413\) −5.99432 −0.294961
\(414\) −5.21549 −0.256328
\(415\) −11.4006 −0.559632
\(416\) −15.8876 −0.778951
\(417\) −0.630945 −0.0308975
\(418\) 0 0
\(419\) −4.15498 −0.202984 −0.101492 0.994836i \(-0.532362\pi\)
−0.101492 + 0.994836i \(0.532362\pi\)
\(420\) 0.263603 0.0128625
\(421\) −26.4050 −1.28690 −0.643449 0.765489i \(-0.722497\pi\)
−0.643449 + 0.765489i \(0.722497\pi\)
\(422\) 4.38167 0.213296
\(423\) −4.93830 −0.240108
\(424\) −1.59559 −0.0774889
\(425\) 2.98649 0.144866
\(426\) 0.175687 0.00851209
\(427\) 13.3293 0.645052
\(428\) −29.2195 −1.41238
\(429\) 1.49133 0.0720020
\(430\) −1.12489 −0.0542470
\(431\) −9.72137 −0.468262 −0.234131 0.972205i \(-0.575224\pi\)
−0.234131 + 0.972205i \(0.575224\pi\)
\(432\) −1.78850 −0.0860495
\(433\) −0.659842 −0.0317100 −0.0158550 0.999874i \(-0.505047\pi\)
−0.0158550 + 0.999874i \(0.505047\pi\)
\(434\) −1.44385 −0.0693068
\(435\) 0.579823 0.0278004
\(436\) −10.1782 −0.487449
\(437\) 0 0
\(438\) 0.0237993 0.00113717
\(439\) 27.1898 1.29770 0.648849 0.760917i \(-0.275251\pi\)
0.648849 + 0.760917i \(0.275251\pi\)
\(440\) 2.40914 0.114851
\(441\) 12.4333 0.592061
\(442\) −4.07423 −0.193791
\(443\) 37.7880 1.79536 0.897682 0.440644i \(-0.145250\pi\)
0.897682 + 0.440644i \(0.145250\pi\)
\(444\) −1.24517 −0.0590933
\(445\) −10.1161 −0.479549
\(446\) 0.411409 0.0194808
\(447\) 1.42986 0.0676300
\(448\) 11.7286 0.554123
\(449\) −16.5499 −0.781039 −0.390520 0.920595i \(-0.627705\pi\)
−0.390520 + 0.920595i \(0.627705\pi\)
\(450\) −0.631604 −0.0297741
\(451\) −15.7501 −0.741645
\(452\) −16.7598 −0.788313
\(453\) 0.234880 0.0110356
\(454\) 2.66049 0.124863
\(455\) 10.9096 0.511451
\(456\) 0 0
\(457\) 11.4492 0.535571 0.267785 0.963479i \(-0.413708\pi\)
0.267785 + 0.963479i \(0.413708\pi\)
\(458\) −1.29283 −0.0604101
\(459\) −1.43012 −0.0667522
\(460\) −16.1475 −0.752881
\(461\) −10.7597 −0.501127 −0.250563 0.968100i \(-0.580616\pi\)
−0.250563 + 0.968100i \(0.580616\pi\)
\(462\) 0.0821030 0.00381978
\(463\) 17.0882 0.794155 0.397078 0.917785i \(-0.370024\pi\)
0.397078 + 0.917785i \(0.370024\pi\)
\(464\) 27.1051 1.25832
\(465\) 0.324057 0.0150278
\(466\) 1.39400 0.0645757
\(467\) 31.8559 1.47412 0.737058 0.675829i \(-0.236214\pi\)
0.737058 + 0.675829i \(0.236214\pi\)
\(468\) −37.8519 −1.74970
\(469\) −3.82213 −0.176489
\(470\) 0.348041 0.0160539
\(471\) 0.540643 0.0249115
\(472\) −2.96493 −0.136472
\(473\) 15.3913 0.707693
\(474\) −0.0406374 −0.00186654
\(475\) 0 0
\(476\) 9.85346 0.451633
\(477\) −5.72361 −0.262066
\(478\) −0.464970 −0.0212672
\(479\) −3.85027 −0.175923 −0.0879616 0.996124i \(-0.528035\pi\)
−0.0879616 + 0.996124i \(0.528035\pi\)
\(480\) 0.196310 0.00896030
\(481\) −51.5334 −2.34972
\(482\) −0.0263414 −0.00119982
\(483\) −1.11313 −0.0506493
\(484\) 5.21427 0.237012
\(485\) −9.86820 −0.448092
\(486\) 0.453839 0.0205866
\(487\) 35.4075 1.60447 0.802233 0.597011i \(-0.203645\pi\)
0.802233 + 0.597011i \(0.203645\pi\)
\(488\) 6.59300 0.298451
\(489\) −0.0725319 −0.00328000
\(490\) −0.876270 −0.0395859
\(491\) 20.2571 0.914192 0.457096 0.889417i \(-0.348890\pi\)
0.457096 + 0.889417i \(0.348890\pi\)
\(492\) −0.852408 −0.0384295
\(493\) 21.6737 0.976136
\(494\) 0 0
\(495\) 8.64192 0.388426
\(496\) 15.1488 0.680199
\(497\) −17.5850 −0.788797
\(498\) −0.192176 −0.00861159
\(499\) 0.589427 0.0263864 0.0131932 0.999913i \(-0.495800\pi\)
0.0131932 + 0.999913i \(0.495800\pi\)
\(500\) −1.95549 −0.0874520
\(501\) 1.14024 0.0509420
\(502\) 0.784181 0.0349997
\(503\) −11.3500 −0.506073 −0.253037 0.967457i \(-0.581429\pi\)
−0.253037 + 0.967457i \(0.581429\pi\)
\(504\) −4.21520 −0.187760
\(505\) 4.16975 0.185551
\(506\) −5.02937 −0.223583
\(507\) 2.30173 0.102224
\(508\) −25.4936 −1.13109
\(509\) 11.8654 0.525926 0.262963 0.964806i \(-0.415300\pi\)
0.262963 + 0.964806i \(0.415300\pi\)
\(510\) 0.0503422 0.00222919
\(511\) −2.38214 −0.105379
\(512\) 15.4108 0.681069
\(513\) 0 0
\(514\) −5.69143 −0.251038
\(515\) −12.5636 −0.553619
\(516\) 0.832987 0.0366702
\(517\) −4.76207 −0.209436
\(518\) −2.83710 −0.124655
\(519\) 1.12472 0.0493699
\(520\) 5.39616 0.236637
\(521\) −24.5221 −1.07433 −0.537166 0.843476i \(-0.680505\pi\)
−0.537166 + 0.843476i \(0.680505\pi\)
\(522\) −4.58372 −0.200624
\(523\) 21.2338 0.928488 0.464244 0.885707i \(-0.346326\pi\)
0.464244 + 0.885707i \(0.346326\pi\)
\(524\) −0.365839 −0.0159817
\(525\) −0.134802 −0.00588324
\(526\) 4.36227 0.190204
\(527\) 12.1132 0.527659
\(528\) −0.861421 −0.0374885
\(529\) 45.1869 1.96465
\(530\) 0.403387 0.0175220
\(531\) −10.6356 −0.461546
\(532\) 0 0
\(533\) −35.2782 −1.52807
\(534\) −0.170524 −0.00737928
\(535\) 14.9423 0.646013
\(536\) −1.89051 −0.0816578
\(537\) 1.25833 0.0543012
\(538\) −2.89572 −0.124844
\(539\) 11.9896 0.516427
\(540\) 0.936410 0.0402967
\(541\) 14.8259 0.637414 0.318707 0.947853i \(-0.396751\pi\)
0.318707 + 0.947853i \(0.396751\pi\)
\(542\) −1.13582 −0.0487877
\(543\) 0.364188 0.0156288
\(544\) 7.33806 0.314617
\(545\) 5.20496 0.222956
\(546\) 0.183900 0.00787019
\(547\) 7.96145 0.340407 0.170203 0.985409i \(-0.445558\pi\)
0.170203 + 0.985409i \(0.445558\pi\)
\(548\) −4.21522 −0.180065
\(549\) 23.6500 1.00936
\(550\) −0.609064 −0.0259706
\(551\) 0 0
\(552\) −0.550581 −0.0234343
\(553\) 4.06750 0.172968
\(554\) 4.81012 0.204362
\(555\) 0.636759 0.0270289
\(556\) 15.4427 0.654917
\(557\) −8.10522 −0.343429 −0.171715 0.985147i \(-0.554931\pi\)
−0.171715 + 0.985147i \(0.554931\pi\)
\(558\) −2.56179 −0.108449
\(559\) 34.4745 1.45811
\(560\) −6.30162 −0.266292
\(561\) −0.688806 −0.0290814
\(562\) −3.63504 −0.153335
\(563\) 8.29823 0.349729 0.174864 0.984593i \(-0.444051\pi\)
0.174864 + 0.984593i \(0.444051\pi\)
\(564\) −0.257726 −0.0108522
\(565\) 8.57064 0.360570
\(566\) 3.43879 0.144543
\(567\) −15.0882 −0.633644
\(568\) −8.69798 −0.364959
\(569\) −7.28643 −0.305463 −0.152732 0.988268i \(-0.548807\pi\)
−0.152732 + 0.988268i \(0.548807\pi\)
\(570\) 0 0
\(571\) −20.6974 −0.866159 −0.433080 0.901356i \(-0.642573\pi\)
−0.433080 + 0.901356i \(0.642573\pi\)
\(572\) −36.5010 −1.52619
\(573\) −0.795669 −0.0332396
\(574\) −1.94219 −0.0810657
\(575\) 8.25754 0.344363
\(576\) 20.8098 0.867075
\(577\) 14.5909 0.607429 0.303715 0.952763i \(-0.401773\pi\)
0.303715 + 0.952763i \(0.401773\pi\)
\(578\) −1.70494 −0.0709160
\(579\) −1.70986 −0.0710594
\(580\) −14.1915 −0.589269
\(581\) 19.2354 0.798018
\(582\) −0.166345 −0.00689521
\(583\) −5.51935 −0.228588
\(584\) −1.17826 −0.0487568
\(585\) 19.3568 0.800303
\(586\) 5.33539 0.220403
\(587\) −10.5077 −0.433699 −0.216849 0.976205i \(-0.569578\pi\)
−0.216849 + 0.976205i \(0.569578\pi\)
\(588\) 0.648883 0.0267595
\(589\) 0 0
\(590\) 0.749575 0.0308595
\(591\) 0.109197 0.00449175
\(592\) 29.7667 1.22340
\(593\) 24.3159 0.998535 0.499267 0.866448i \(-0.333603\pi\)
0.499267 + 0.866448i \(0.333603\pi\)
\(594\) 0.291658 0.0119669
\(595\) −5.03888 −0.206574
\(596\) −34.9966 −1.43351
\(597\) −1.43300 −0.0586488
\(598\) −11.2651 −0.460665
\(599\) −37.9095 −1.54894 −0.774471 0.632609i \(-0.781984\pi\)
−0.774471 + 0.632609i \(0.781984\pi\)
\(600\) −0.0666762 −0.00272205
\(601\) 25.9476 1.05843 0.529213 0.848489i \(-0.322487\pi\)
0.529213 + 0.848489i \(0.322487\pi\)
\(602\) 1.89794 0.0773544
\(603\) −6.78153 −0.276165
\(604\) −5.74881 −0.233916
\(605\) −2.66648 −0.108408
\(606\) 0.0702880 0.00285525
\(607\) −11.5300 −0.467988 −0.233994 0.972238i \(-0.575180\pi\)
−0.233994 + 0.972238i \(0.575180\pi\)
\(608\) 0 0
\(609\) −0.978294 −0.0396425
\(610\) −1.66680 −0.0674868
\(611\) −10.6664 −0.431516
\(612\) 17.4828 0.706701
\(613\) 22.6581 0.915154 0.457577 0.889170i \(-0.348717\pi\)
0.457577 + 0.889170i \(0.348717\pi\)
\(614\) 3.63503 0.146698
\(615\) 0.435906 0.0175774
\(616\) −4.06477 −0.163774
\(617\) 5.97675 0.240615 0.120307 0.992737i \(-0.461612\pi\)
0.120307 + 0.992737i \(0.461612\pi\)
\(618\) −0.211780 −0.00851905
\(619\) −34.2062 −1.37486 −0.687431 0.726249i \(-0.741262\pi\)
−0.687431 + 0.726249i \(0.741262\pi\)
\(620\) −7.93146 −0.318535
\(621\) −3.95423 −0.158678
\(622\) 5.26877 0.211259
\(623\) 17.0682 0.683822
\(624\) −1.92947 −0.0772405
\(625\) 1.00000 0.0400000
\(626\) −3.55787 −0.142201
\(627\) 0 0
\(628\) −13.2325 −0.528035
\(629\) 23.8020 0.949047
\(630\) 1.06566 0.0424569
\(631\) −14.2510 −0.567325 −0.283662 0.958924i \(-0.591549\pi\)
−0.283662 + 0.958924i \(0.591549\pi\)
\(632\) 2.01188 0.0800284
\(633\) 1.65925 0.0659494
\(634\) 5.08916 0.202117
\(635\) 13.0369 0.517355
\(636\) −0.298711 −0.0118446
\(637\) 26.8550 1.06404
\(638\) −4.42014 −0.174995
\(639\) −31.2008 −1.23428
\(640\) −6.38080 −0.252223
\(641\) −39.7444 −1.56981 −0.784904 0.619617i \(-0.787288\pi\)
−0.784904 + 0.619617i \(0.787288\pi\)
\(642\) 0.251877 0.00994081
\(643\) 15.9065 0.627290 0.313645 0.949540i \(-0.398450\pi\)
0.313645 + 0.949540i \(0.398450\pi\)
\(644\) 27.2445 1.07358
\(645\) −0.425974 −0.0167727
\(646\) 0 0
\(647\) 18.8549 0.741262 0.370631 0.928780i \(-0.379142\pi\)
0.370631 + 0.928780i \(0.379142\pi\)
\(648\) −7.46297 −0.293173
\(649\) −10.2561 −0.402585
\(650\) −1.36422 −0.0535092
\(651\) −0.546757 −0.0214291
\(652\) 1.77526 0.0695244
\(653\) −42.5252 −1.66414 −0.832070 0.554670i \(-0.812844\pi\)
−0.832070 + 0.554670i \(0.812844\pi\)
\(654\) 0.0877382 0.00343083
\(655\) 0.187083 0.00730995
\(656\) 20.3774 0.795604
\(657\) −4.22658 −0.164895
\(658\) −0.587224 −0.0228924
\(659\) −45.8272 −1.78517 −0.892587 0.450874i \(-0.851112\pi\)
−0.892587 + 0.450874i \(0.851112\pi\)
\(660\) 0.451015 0.0175557
\(661\) −46.1676 −1.79571 −0.897856 0.440289i \(-0.854876\pi\)
−0.897856 + 0.440289i \(0.854876\pi\)
\(662\) 3.27876 0.127432
\(663\) −1.54283 −0.0599187
\(664\) 9.51427 0.369225
\(665\) 0 0
\(666\) −5.03382 −0.195056
\(667\) 59.9271 2.32039
\(668\) −27.9079 −1.07979
\(669\) 0.155793 0.00602330
\(670\) 0.477948 0.0184647
\(671\) 22.8060 0.880415
\(672\) −0.331220 −0.0127771
\(673\) 5.48820 0.211555 0.105777 0.994390i \(-0.466267\pi\)
0.105777 + 0.994390i \(0.466267\pi\)
\(674\) 2.32201 0.0894406
\(675\) −0.478863 −0.0184315
\(676\) −56.3362 −2.16678
\(677\) −29.4307 −1.13111 −0.565557 0.824709i \(-0.691339\pi\)
−0.565557 + 0.824709i \(0.691339\pi\)
\(678\) 0.144472 0.00554843
\(679\) 16.6499 0.638965
\(680\) −2.49235 −0.0955773
\(681\) 1.00748 0.0386066
\(682\) −2.47037 −0.0945952
\(683\) −27.4543 −1.05051 −0.525254 0.850945i \(-0.676030\pi\)
−0.525254 + 0.850945i \(0.676030\pi\)
\(684\) 0 0
\(685\) 2.15559 0.0823608
\(686\) 3.97031 0.151587
\(687\) −0.489571 −0.0186783
\(688\) −19.9131 −0.759181
\(689\) −12.3626 −0.470978
\(690\) 0.139194 0.00529904
\(691\) 3.42779 0.130399 0.0651997 0.997872i \(-0.479232\pi\)
0.0651997 + 0.997872i \(0.479232\pi\)
\(692\) −27.5282 −1.04647
\(693\) −14.5809 −0.553882
\(694\) 6.42860 0.244026
\(695\) −7.89712 −0.299555
\(696\) −0.483887 −0.0183417
\(697\) 16.2941 0.617184
\(698\) 1.18519 0.0448600
\(699\) 0.527880 0.0199663
\(700\) 3.29935 0.124704
\(701\) −3.26100 −0.123166 −0.0615832 0.998102i \(-0.519615\pi\)
−0.0615832 + 0.998102i \(0.519615\pi\)
\(702\) 0.653276 0.0246563
\(703\) 0 0
\(704\) 20.0672 0.756309
\(705\) 0.131796 0.00496374
\(706\) −1.57946 −0.0594437
\(707\) −7.03532 −0.264590
\(708\) −0.555064 −0.0208606
\(709\) −17.1381 −0.643633 −0.321817 0.946802i \(-0.604293\pi\)
−0.321817 + 0.946802i \(0.604293\pi\)
\(710\) 2.19897 0.0825257
\(711\) 7.21690 0.270655
\(712\) 8.44232 0.316389
\(713\) 33.4926 1.25431
\(714\) −0.0849387 −0.00317875
\(715\) 18.6660 0.698068
\(716\) −30.7984 −1.15099
\(717\) −0.176075 −0.00657565
\(718\) −5.37701 −0.200668
\(719\) 6.22549 0.232172 0.116086 0.993239i \(-0.462965\pi\)
0.116086 + 0.993239i \(0.462965\pi\)
\(720\) −11.1808 −0.416686
\(721\) 21.1977 0.789442
\(722\) 0 0
\(723\) −0.00997498 −0.000370974 0
\(724\) −8.91371 −0.331275
\(725\) 7.25727 0.269528
\(726\) −0.0449480 −0.00166817
\(727\) −42.3189 −1.56952 −0.784760 0.619800i \(-0.787214\pi\)
−0.784760 + 0.619800i \(0.787214\pi\)
\(728\) −9.10455 −0.337437
\(729\) −26.6559 −0.987256
\(730\) 0.297880 0.0110250
\(731\) −15.9229 −0.588929
\(732\) 1.23427 0.0456201
\(733\) −22.7630 −0.840771 −0.420386 0.907346i \(-0.638105\pi\)
−0.420386 + 0.907346i \(0.638105\pi\)
\(734\) −7.06082 −0.260620
\(735\) −0.331827 −0.0122396
\(736\) 20.2895 0.747881
\(737\) −6.53952 −0.240886
\(738\) −3.44600 −0.126849
\(739\) 17.9424 0.660023 0.330012 0.943977i \(-0.392947\pi\)
0.330012 + 0.943977i \(0.392947\pi\)
\(740\) −15.5850 −0.572916
\(741\) 0 0
\(742\) −0.680606 −0.0249858
\(743\) −17.3309 −0.635810 −0.317905 0.948123i \(-0.602979\pi\)
−0.317905 + 0.948123i \(0.602979\pi\)
\(744\) −0.270439 −0.00991478
\(745\) 17.8966 0.655681
\(746\) −2.21083 −0.0809443
\(747\) 34.1290 1.24871
\(748\) 16.8589 0.616423
\(749\) −25.2111 −0.921193
\(750\) 0.0168566 0.000615518 0
\(751\) −2.18627 −0.0797781 −0.0398890 0.999204i \(-0.512700\pi\)
−0.0398890 + 0.999204i \(0.512700\pi\)
\(752\) 6.16113 0.224673
\(753\) 0.296955 0.0108216
\(754\) −9.90053 −0.360556
\(755\) 2.93984 0.106992
\(756\) −1.57994 −0.0574618
\(757\) −15.6387 −0.568397 −0.284198 0.958766i \(-0.591727\pi\)
−0.284198 + 0.958766i \(0.591727\pi\)
\(758\) −3.01625 −0.109555
\(759\) −1.90453 −0.0691299
\(760\) 0 0
\(761\) 26.2993 0.953349 0.476675 0.879080i \(-0.341842\pi\)
0.476675 + 0.879080i \(0.341842\pi\)
\(762\) 0.219759 0.00796104
\(763\) −8.78195 −0.317928
\(764\) 19.4744 0.704560
\(765\) −8.94040 −0.323241
\(766\) 7.80249 0.281915
\(767\) −22.9722 −0.829478
\(768\) 1.00321 0.0362002
\(769\) 0.0847497 0.00305615 0.00152808 0.999999i \(-0.499514\pi\)
0.00152808 + 0.999999i \(0.499514\pi\)
\(770\) 1.02763 0.0370332
\(771\) −2.15524 −0.0776190
\(772\) 41.8498 1.50621
\(773\) −0.0114752 −0.000412734 0 −0.000206367 1.00000i \(-0.500066\pi\)
−0.000206367 1.00000i \(0.500066\pi\)
\(774\) 3.36749 0.121042
\(775\) 4.05600 0.145696
\(776\) 8.23544 0.295635
\(777\) −1.07436 −0.0385423
\(778\) −3.50389 −0.125621
\(779\) 0 0
\(780\) 1.01021 0.0361715
\(781\) −30.0873 −1.07661
\(782\) 5.20307 0.186061
\(783\) −3.47524 −0.124195
\(784\) −15.5120 −0.554000
\(785\) 6.76687 0.241520
\(786\) 0.00315360 0.000112485 0
\(787\) 30.1068 1.07319 0.536596 0.843839i \(-0.319710\pi\)
0.536596 + 0.843839i \(0.319710\pi\)
\(788\) −2.67265 −0.0952092
\(789\) 1.65191 0.0588095
\(790\) −0.508631 −0.0180963
\(791\) −14.4606 −0.514161
\(792\) −7.21205 −0.256269
\(793\) 51.0824 1.81399
\(794\) −5.62026 −0.199455
\(795\) 0.152755 0.00541767
\(796\) 35.0735 1.24315
\(797\) 38.2339 1.35432 0.677158 0.735838i \(-0.263211\pi\)
0.677158 + 0.735838i \(0.263211\pi\)
\(798\) 0 0
\(799\) 4.92654 0.174288
\(800\) 2.45709 0.0868712
\(801\) 30.2837 1.07002
\(802\) 0.273172 0.00964604
\(803\) −4.07574 −0.143830
\(804\) −0.353923 −0.0124819
\(805\) −13.9323 −0.491051
\(806\) −5.53329 −0.194902
\(807\) −1.09656 −0.0386006
\(808\) −3.47983 −0.122420
\(809\) 3.70003 0.130086 0.0650431 0.997882i \(-0.479282\pi\)
0.0650431 + 0.997882i \(0.479282\pi\)
\(810\) 1.88674 0.0662933
\(811\) 40.9256 1.43709 0.718545 0.695480i \(-0.244808\pi\)
0.718545 + 0.695480i \(0.244808\pi\)
\(812\) 23.9443 0.840279
\(813\) −0.430114 −0.0150848
\(814\) −4.85417 −0.170139
\(815\) −0.907834 −0.0318000
\(816\) 0.891172 0.0311973
\(817\) 0 0
\(818\) 5.70327 0.199410
\(819\) −32.6593 −1.14121
\(820\) −10.6690 −0.372579
\(821\) 17.4402 0.608666 0.304333 0.952566i \(-0.401566\pi\)
0.304333 + 0.952566i \(0.401566\pi\)
\(822\) 0.0363360 0.00126736
\(823\) −6.50473 −0.226741 −0.113370 0.993553i \(-0.536165\pi\)
−0.113370 + 0.993553i \(0.536165\pi\)
\(824\) 10.4849 0.365258
\(825\) −0.230641 −0.00802989
\(826\) −1.26470 −0.0440046
\(827\) 1.62225 0.0564112 0.0282056 0.999602i \(-0.491021\pi\)
0.0282056 + 0.999602i \(0.491021\pi\)
\(828\) 48.3394 1.67991
\(829\) 32.6503 1.13399 0.566996 0.823721i \(-0.308105\pi\)
0.566996 + 0.823721i \(0.308105\pi\)
\(830\) −2.40534 −0.0834904
\(831\) 1.82150 0.0631872
\(832\) 44.9477 1.55828
\(833\) −12.4037 −0.429761
\(834\) −0.133119 −0.00460953
\(835\) 14.2716 0.493888
\(836\) 0 0
\(837\) −1.94227 −0.0671347
\(838\) −0.876632 −0.0302827
\(839\) −11.5599 −0.399093 −0.199547 0.979888i \(-0.563947\pi\)
−0.199547 + 0.979888i \(0.563947\pi\)
\(840\) 0.112498 0.00388155
\(841\) 23.6679 0.816135
\(842\) −5.57101 −0.191990
\(843\) −1.37652 −0.0474099
\(844\) −40.6111 −1.39789
\(845\) 28.8093 0.991070
\(846\) −1.04190 −0.0358213
\(847\) 4.49896 0.154586
\(848\) 7.14089 0.245219
\(849\) 1.30220 0.0446915
\(850\) 0.630100 0.0216122
\(851\) 65.8117 2.25600
\(852\) −1.62835 −0.0557862
\(853\) 47.7141 1.63370 0.816850 0.576850i \(-0.195718\pi\)
0.816850 + 0.576850i \(0.195718\pi\)
\(854\) 2.81227 0.0962339
\(855\) 0 0
\(856\) −12.4700 −0.426216
\(857\) −38.5358 −1.31636 −0.658179 0.752862i \(-0.728673\pi\)
−0.658179 + 0.752862i \(0.728673\pi\)
\(858\) 0.314646 0.0107418
\(859\) 19.1921 0.654826 0.327413 0.944881i \(-0.393823\pi\)
0.327413 + 0.944881i \(0.393823\pi\)
\(860\) 10.4260 0.355522
\(861\) −0.735473 −0.0250648
\(862\) −2.05105 −0.0698591
\(863\) −9.16794 −0.312080 −0.156040 0.987751i \(-0.549873\pi\)
−0.156040 + 0.987751i \(0.549873\pi\)
\(864\) −1.17661 −0.0400290
\(865\) 14.0774 0.478647
\(866\) −0.139216 −0.00473075
\(867\) −0.645628 −0.0219267
\(868\) 13.3822 0.454221
\(869\) 6.95935 0.236080
\(870\) 0.122333 0.00414748
\(871\) −14.6476 −0.496317
\(872\) −4.34376 −0.147098
\(873\) 29.5416 0.999832
\(874\) 0 0
\(875\) −1.68723 −0.0570387
\(876\) −0.220582 −0.00745277
\(877\) −3.62989 −0.122573 −0.0612864 0.998120i \(-0.519520\pi\)
−0.0612864 + 0.998120i \(0.519520\pi\)
\(878\) 5.73660 0.193601
\(879\) 2.02041 0.0681468
\(880\) −10.7818 −0.363456
\(881\) −8.34382 −0.281111 −0.140555 0.990073i \(-0.544889\pi\)
−0.140555 + 0.990073i \(0.544889\pi\)
\(882\) 2.62322 0.0883283
\(883\) 28.2122 0.949415 0.474708 0.880144i \(-0.342554\pi\)
0.474708 + 0.880144i \(0.342554\pi\)
\(884\) 37.7617 1.27006
\(885\) 0.283850 0.00954151
\(886\) 7.97266 0.267847
\(887\) −8.84194 −0.296883 −0.148442 0.988921i \(-0.547426\pi\)
−0.148442 + 0.988921i \(0.547426\pi\)
\(888\) −0.531402 −0.0178327
\(889\) −21.9963 −0.737732
\(890\) −2.13433 −0.0715430
\(891\) −25.8153 −0.864846
\(892\) −3.81312 −0.127673
\(893\) 0 0
\(894\) 0.301677 0.0100896
\(895\) 15.7498 0.526456
\(896\) 10.7659 0.359662
\(897\) −4.26589 −0.142434
\(898\) −3.49176 −0.116522
\(899\) 29.4355 0.981729
\(900\) 5.85398 0.195133
\(901\) 5.70997 0.190227
\(902\) −3.32302 −0.110645
\(903\) 0.718716 0.0239174
\(904\) −7.15257 −0.237891
\(905\) 4.55831 0.151523
\(906\) 0.0495558 0.00164638
\(907\) 4.22931 0.140432 0.0702159 0.997532i \(-0.477631\pi\)
0.0702159 + 0.997532i \(0.477631\pi\)
\(908\) −24.6585 −0.818322
\(909\) −12.4826 −0.414023
\(910\) 2.30175 0.0763024
\(911\) 10.1182 0.335231 0.167616 0.985852i \(-0.446393\pi\)
0.167616 + 0.985852i \(0.446393\pi\)
\(912\) 0 0
\(913\) 32.9110 1.08920
\(914\) 2.41559 0.0799007
\(915\) −0.631186 −0.0208664
\(916\) 11.9825 0.395914
\(917\) −0.315652 −0.0104238
\(918\) −0.301732 −0.00995863
\(919\) −47.8757 −1.57927 −0.789637 0.613575i \(-0.789731\pi\)
−0.789637 + 0.613575i \(0.789731\pi\)
\(920\) −6.89127 −0.227198
\(921\) 1.37652 0.0453578
\(922\) −2.27011 −0.0747621
\(923\) −67.3916 −2.21822
\(924\) −0.760966 −0.0250339
\(925\) 7.96989 0.262048
\(926\) 3.60533 0.118478
\(927\) 37.6106 1.23529
\(928\) 17.8317 0.585356
\(929\) −21.6302 −0.709662 −0.354831 0.934930i \(-0.615462\pi\)
−0.354831 + 0.934930i \(0.615462\pi\)
\(930\) 0.0683706 0.00224196
\(931\) 0 0
\(932\) −12.9202 −0.423214
\(933\) 1.99518 0.0653194
\(934\) 6.72108 0.219920
\(935\) −8.62134 −0.281948
\(936\) −16.1540 −0.528011
\(937\) 36.0833 1.17879 0.589395 0.807845i \(-0.299366\pi\)
0.589395 + 0.807845i \(0.299366\pi\)
\(938\) −0.806406 −0.0263301
\(939\) −1.34730 −0.0439674
\(940\) −3.22579 −0.105214
\(941\) −48.4806 −1.58042 −0.790212 0.612834i \(-0.790030\pi\)
−0.790212 + 0.612834i \(0.790030\pi\)
\(942\) 0.114067 0.00371650
\(943\) 45.0527 1.46712
\(944\) 13.2692 0.431876
\(945\) 0.807951 0.0262827
\(946\) 3.24731 0.105579
\(947\) 35.2103 1.14418 0.572091 0.820190i \(-0.306132\pi\)
0.572091 + 0.820190i \(0.306132\pi\)
\(948\) 0.376644 0.0122328
\(949\) −9.12913 −0.296344
\(950\) 0 0
\(951\) 1.92717 0.0624928
\(952\) 4.20516 0.136290
\(953\) 44.5937 1.44453 0.722265 0.691616i \(-0.243101\pi\)
0.722265 + 0.691616i \(0.243101\pi\)
\(954\) −1.20759 −0.0390971
\(955\) −9.95887 −0.322262
\(956\) 4.30954 0.139380
\(957\) −1.67382 −0.0541070
\(958\) −0.812343 −0.0262456
\(959\) −3.63697 −0.117444
\(960\) −0.555385 −0.0179250
\(961\) −14.5488 −0.469317
\(962\) −10.8727 −0.350550
\(963\) −44.7316 −1.44145
\(964\) 0.244143 0.00786332
\(965\) −21.4012 −0.688930
\(966\) −0.234853 −0.00755626
\(967\) 2.70529 0.0869963 0.0434981 0.999054i \(-0.486150\pi\)
0.0434981 + 0.999054i \(0.486150\pi\)
\(968\) 2.22529 0.0715236
\(969\) 0 0
\(970\) −2.08203 −0.0668499
\(971\) −7.14238 −0.229210 −0.114605 0.993411i \(-0.536560\pi\)
−0.114605 + 0.993411i \(0.536560\pi\)
\(972\) −4.20637 −0.134919
\(973\) 13.3242 0.427156
\(974\) 7.47040 0.239367
\(975\) −0.516605 −0.0165446
\(976\) −29.5062 −0.944471
\(977\) −28.8990 −0.924560 −0.462280 0.886734i \(-0.652968\pi\)
−0.462280 + 0.886734i \(0.652968\pi\)
\(978\) −0.0153030 −0.000489337 0
\(979\) 29.2030 0.933332
\(980\) 8.12165 0.259436
\(981\) −15.5817 −0.497484
\(982\) 4.27392 0.136386
\(983\) 10.3410 0.329827 0.164914 0.986308i \(-0.447265\pi\)
0.164914 + 0.986308i \(0.447265\pi\)
\(984\) −0.363782 −0.0115970
\(985\) 1.36674 0.0435481
\(986\) 4.57280 0.145628
\(987\) −0.222371 −0.00707814
\(988\) 0 0
\(989\) −44.0262 −1.39995
\(990\) 1.82330 0.0579484
\(991\) 31.6203 1.00445 0.502226 0.864736i \(-0.332514\pi\)
0.502226 + 0.864736i \(0.332514\pi\)
\(992\) 9.96596 0.316419
\(993\) 1.24160 0.0394011
\(994\) −3.71015 −0.117679
\(995\) −17.9359 −0.568607
\(996\) 1.78116 0.0564384
\(997\) 7.87725 0.249475 0.124737 0.992190i \(-0.460191\pi\)
0.124737 + 0.992190i \(0.460191\pi\)
\(998\) 0.124359 0.00393653
\(999\) −3.81649 −0.120748
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 1805.2.a.v.1.4 9
5.4 even 2 9025.2.a.cc.1.6 9
19.6 even 9 95.2.k.a.36.2 18
19.16 even 9 95.2.k.a.66.2 yes 18
19.18 odd 2 1805.2.a.s.1.6 9
57.35 odd 18 855.2.bs.c.541.2 18
57.44 odd 18 855.2.bs.c.226.2 18
95.44 even 18 475.2.l.c.226.2 18
95.54 even 18 475.2.l.c.351.2 18
95.63 odd 36 475.2.u.b.74.3 36
95.73 odd 36 475.2.u.b.199.4 36
95.82 odd 36 475.2.u.b.74.4 36
95.92 odd 36 475.2.u.b.199.3 36
95.94 odd 2 9025.2.a.cf.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.k.a.36.2 18 19.6 even 9
95.2.k.a.66.2 yes 18 19.16 even 9
475.2.l.c.226.2 18 95.44 even 18
475.2.l.c.351.2 18 95.54 even 18
475.2.u.b.74.3 36 95.63 odd 36
475.2.u.b.74.4 36 95.82 odd 36
475.2.u.b.199.3 36 95.92 odd 36
475.2.u.b.199.4 36 95.73 odd 36
855.2.bs.c.226.2 18 57.44 odd 18
855.2.bs.c.541.2 18 57.35 odd 18
1805.2.a.s.1.6 9 19.18 odd 2
1805.2.a.v.1.4 9 1.1 even 1 trivial
9025.2.a.cc.1.6 9 5.4 even 2
9025.2.a.cf.1.4 9 95.94 odd 2