Properties

Label 5225.2.a.x.1.13
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $15$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(2.59045\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.59045 q^{2} -2.99450 q^{3} +4.71045 q^{4} -7.75712 q^{6} +1.08607 q^{7} +7.02129 q^{8} +5.96706 q^{9} +O(q^{10})\) \(q+2.59045 q^{2} -2.99450 q^{3} +4.71045 q^{4} -7.75712 q^{6} +1.08607 q^{7} +7.02129 q^{8} +5.96706 q^{9} +1.00000 q^{11} -14.1055 q^{12} +3.56681 q^{13} +2.81342 q^{14} +8.76742 q^{16} -2.11143 q^{17} +15.4574 q^{18} -1.00000 q^{19} -3.25225 q^{21} +2.59045 q^{22} +4.81445 q^{23} -21.0253 q^{24} +9.23965 q^{26} -8.88487 q^{27} +5.11589 q^{28} -3.35232 q^{29} +4.72904 q^{31} +8.66902 q^{32} -2.99450 q^{33} -5.46957 q^{34} +28.1075 q^{36} +5.10670 q^{37} -2.59045 q^{38} -10.6808 q^{39} -8.37205 q^{41} -8.42480 q^{42} +7.95355 q^{43} +4.71045 q^{44} +12.4716 q^{46} +1.41079 q^{47} -26.2541 q^{48} -5.82045 q^{49} +6.32269 q^{51} +16.8013 q^{52} +9.64120 q^{53} -23.0158 q^{54} +7.62562 q^{56} +2.99450 q^{57} -8.68403 q^{58} -0.737428 q^{59} -5.70238 q^{61} +12.2504 q^{62} +6.48066 q^{63} +4.92184 q^{64} -7.75712 q^{66} -2.12759 q^{67} -9.94579 q^{68} -14.4169 q^{69} +10.6171 q^{71} +41.8964 q^{72} +11.0070 q^{73} +13.2287 q^{74} -4.71045 q^{76} +1.08607 q^{77} -27.6682 q^{78} -9.96615 q^{79} +8.70462 q^{81} -21.6874 q^{82} -14.0187 q^{83} -15.3195 q^{84} +20.6033 q^{86} +10.0385 q^{87} +7.02129 q^{88} +9.82782 q^{89} +3.87381 q^{91} +22.6782 q^{92} -14.1611 q^{93} +3.65458 q^{94} -25.9594 q^{96} +8.89758 q^{97} -15.0776 q^{98} +5.96706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 15 q^{7} + 15 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 15 q^{7} + 15 q^{8} + 19 q^{9} + 15 q^{11} + 9 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 11 q^{17} + 16 q^{18} - 15 q^{19} + 5 q^{22} + 10 q^{23} + 17 q^{24} - 17 q^{26} + 13 q^{27} + 30 q^{28} + 5 q^{29} + 6 q^{31} + 40 q^{32} + 4 q^{33} + 17 q^{34} + 28 q^{36} + 13 q^{37} - 5 q^{38} - 22 q^{39} + 30 q^{42} + 36 q^{43} + 17 q^{44} + 13 q^{46} - 6 q^{47} + 14 q^{48} + 16 q^{49} + 4 q^{51} + 50 q^{52} + 9 q^{53} + 9 q^{54} - 18 q^{56} - 4 q^{57} + 2 q^{58} - 7 q^{59} - 2 q^{61} + 11 q^{62} + 39 q^{63} + 17 q^{64} - q^{66} + 35 q^{67} - 18 q^{68} - 9 q^{69} + 13 q^{71} + 68 q^{72} + 2 q^{73} + 13 q^{74} - 17 q^{76} + 15 q^{77} - 10 q^{78} + 6 q^{79} + 11 q^{81} + 14 q^{82} + 30 q^{83} - 6 q^{84} - 25 q^{86} + 19 q^{87} + 15 q^{88} + 55 q^{89} + 26 q^{91} - 18 q^{92} + 14 q^{93} - 22 q^{94} - 17 q^{96} + 28 q^{97} - 22 q^{98} + 19 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.59045 1.83173 0.915863 0.401490i \(-0.131507\pi\)
0.915863 + 0.401490i \(0.131507\pi\)
\(3\) −2.99450 −1.72888 −0.864439 0.502738i \(-0.832326\pi\)
−0.864439 + 0.502738i \(0.832326\pi\)
\(4\) 4.71045 2.35522
\(5\) 0 0
\(6\) −7.75712 −3.16683
\(7\) 1.08607 0.410497 0.205248 0.978710i \(-0.434200\pi\)
0.205248 + 0.978710i \(0.434200\pi\)
\(8\) 7.02129 2.48240
\(9\) 5.96706 1.98902
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −14.1055 −4.07189
\(13\) 3.56681 0.989254 0.494627 0.869105i \(-0.335305\pi\)
0.494627 + 0.869105i \(0.335305\pi\)
\(14\) 2.81342 0.751918
\(15\) 0 0
\(16\) 8.76742 2.19185
\(17\) −2.11143 −0.512097 −0.256049 0.966664i \(-0.582421\pi\)
−0.256049 + 0.966664i \(0.582421\pi\)
\(18\) 15.4574 3.64334
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.25225 −0.709699
\(22\) 2.59045 0.552286
\(23\) 4.81445 1.00388 0.501941 0.864902i \(-0.332619\pi\)
0.501941 + 0.864902i \(0.332619\pi\)
\(24\) −21.0253 −4.29177
\(25\) 0 0
\(26\) 9.23965 1.81204
\(27\) −8.88487 −1.70989
\(28\) 5.11589 0.966811
\(29\) −3.35232 −0.622510 −0.311255 0.950326i \(-0.600749\pi\)
−0.311255 + 0.950326i \(0.600749\pi\)
\(30\) 0 0
\(31\) 4.72904 0.849361 0.424680 0.905343i \(-0.360386\pi\)
0.424680 + 0.905343i \(0.360386\pi\)
\(32\) 8.66902 1.53248
\(33\) −2.99450 −0.521276
\(34\) −5.46957 −0.938023
\(35\) 0 0
\(36\) 28.1075 4.68459
\(37\) 5.10670 0.839536 0.419768 0.907632i \(-0.362112\pi\)
0.419768 + 0.907632i \(0.362112\pi\)
\(38\) −2.59045 −0.420227
\(39\) −10.6808 −1.71030
\(40\) 0 0
\(41\) −8.37205 −1.30749 −0.653747 0.756713i \(-0.726804\pi\)
−0.653747 + 0.756713i \(0.726804\pi\)
\(42\) −8.42480 −1.29997
\(43\) 7.95355 1.21290 0.606452 0.795120i \(-0.292592\pi\)
0.606452 + 0.795120i \(0.292592\pi\)
\(44\) 4.71045 0.710127
\(45\) 0 0
\(46\) 12.4716 1.83884
\(47\) 1.41079 0.205785 0.102892 0.994693i \(-0.467190\pi\)
0.102892 + 0.994693i \(0.467190\pi\)
\(48\) −26.2541 −3.78945
\(49\) −5.82045 −0.831493
\(50\) 0 0
\(51\) 6.32269 0.885354
\(52\) 16.8013 2.32992
\(53\) 9.64120 1.32432 0.662160 0.749362i \(-0.269640\pi\)
0.662160 + 0.749362i \(0.269640\pi\)
\(54\) −23.0158 −3.13206
\(55\) 0 0
\(56\) 7.62562 1.01902
\(57\) 2.99450 0.396632
\(58\) −8.68403 −1.14027
\(59\) −0.737428 −0.0960050 −0.0480025 0.998847i \(-0.515286\pi\)
−0.0480025 + 0.998847i \(0.515286\pi\)
\(60\) 0 0
\(61\) −5.70238 −0.730114 −0.365057 0.930985i \(-0.618951\pi\)
−0.365057 + 0.930985i \(0.618951\pi\)
\(62\) 12.2504 1.55580
\(63\) 6.48066 0.816486
\(64\) 4.92184 0.615230
\(65\) 0 0
\(66\) −7.75712 −0.954836
\(67\) −2.12759 −0.259927 −0.129964 0.991519i \(-0.541486\pi\)
−0.129964 + 0.991519i \(0.541486\pi\)
\(68\) −9.94579 −1.20610
\(69\) −14.4169 −1.73559
\(70\) 0 0
\(71\) 10.6171 1.26002 0.630008 0.776588i \(-0.283051\pi\)
0.630008 + 0.776588i \(0.283051\pi\)
\(72\) 41.8964 4.93754
\(73\) 11.0070 1.28828 0.644138 0.764910i \(-0.277216\pi\)
0.644138 + 0.764910i \(0.277216\pi\)
\(74\) 13.2287 1.53780
\(75\) 0 0
\(76\) −4.71045 −0.540325
\(77\) 1.08607 0.123769
\(78\) −27.6682 −3.13280
\(79\) −9.96615 −1.12128 −0.560640 0.828060i \(-0.689445\pi\)
−0.560640 + 0.828060i \(0.689445\pi\)
\(80\) 0 0
\(81\) 8.70462 0.967179
\(82\) −21.6874 −2.39497
\(83\) −14.0187 −1.53875 −0.769377 0.638795i \(-0.779433\pi\)
−0.769377 + 0.638795i \(0.779433\pi\)
\(84\) −15.3195 −1.67150
\(85\) 0 0
\(86\) 20.6033 2.22171
\(87\) 10.0385 1.07624
\(88\) 7.02129 0.748472
\(89\) 9.82782 1.04175 0.520873 0.853634i \(-0.325606\pi\)
0.520873 + 0.853634i \(0.325606\pi\)
\(90\) 0 0
\(91\) 3.87381 0.406086
\(92\) 22.6782 2.36437
\(93\) −14.1611 −1.46844
\(94\) 3.65458 0.376941
\(95\) 0 0
\(96\) −25.9594 −2.64947
\(97\) 8.89758 0.903412 0.451706 0.892167i \(-0.350816\pi\)
0.451706 + 0.892167i \(0.350816\pi\)
\(98\) −15.0776 −1.52307
\(99\) 5.96706 0.599712
\(100\) 0 0
\(101\) −7.16344 −0.712788 −0.356394 0.934336i \(-0.615994\pi\)
−0.356394 + 0.934336i \(0.615994\pi\)
\(102\) 16.3786 1.62173
\(103\) −6.40667 −0.631268 −0.315634 0.948881i \(-0.602217\pi\)
−0.315634 + 0.948881i \(0.602217\pi\)
\(104\) 25.0436 2.45572
\(105\) 0 0
\(106\) 24.9751 2.42579
\(107\) −8.12343 −0.785321 −0.392661 0.919683i \(-0.628445\pi\)
−0.392661 + 0.919683i \(0.628445\pi\)
\(108\) −41.8517 −4.02718
\(109\) 5.30036 0.507683 0.253841 0.967246i \(-0.418306\pi\)
0.253841 + 0.967246i \(0.418306\pi\)
\(110\) 0 0
\(111\) −15.2920 −1.45145
\(112\) 9.52205 0.899749
\(113\) 18.0156 1.69477 0.847384 0.530980i \(-0.178176\pi\)
0.847384 + 0.530980i \(0.178176\pi\)
\(114\) 7.75712 0.726521
\(115\) 0 0
\(116\) −15.7909 −1.46615
\(117\) 21.2833 1.96765
\(118\) −1.91027 −0.175855
\(119\) −2.29317 −0.210214
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −14.7717 −1.33737
\(123\) 25.0701 2.26050
\(124\) 22.2759 2.00043
\(125\) 0 0
\(126\) 16.7878 1.49558
\(127\) 3.99191 0.354225 0.177112 0.984191i \(-0.443324\pi\)
0.177112 + 0.984191i \(0.443324\pi\)
\(128\) −4.58824 −0.405547
\(129\) −23.8169 −2.09696
\(130\) 0 0
\(131\) 11.9276 1.04212 0.521059 0.853521i \(-0.325537\pi\)
0.521059 + 0.853521i \(0.325537\pi\)
\(132\) −14.1055 −1.22772
\(133\) −1.08607 −0.0941744
\(134\) −5.51143 −0.476115
\(135\) 0 0
\(136\) −14.8250 −1.27123
\(137\) −21.8741 −1.86883 −0.934415 0.356187i \(-0.884077\pi\)
−0.934415 + 0.356187i \(0.884077\pi\)
\(138\) −37.3463 −3.17913
\(139\) 11.6552 0.988579 0.494290 0.869297i \(-0.335428\pi\)
0.494290 + 0.869297i \(0.335428\pi\)
\(140\) 0 0
\(141\) −4.22461 −0.355777
\(142\) 27.5031 2.30801
\(143\) 3.56681 0.298271
\(144\) 52.3157 4.35964
\(145\) 0 0
\(146\) 28.5132 2.35977
\(147\) 17.4294 1.43755
\(148\) 24.0548 1.97729
\(149\) 9.67476 0.792587 0.396294 0.918124i \(-0.370296\pi\)
0.396294 + 0.918124i \(0.370296\pi\)
\(150\) 0 0
\(151\) 14.6123 1.18913 0.594565 0.804047i \(-0.297324\pi\)
0.594565 + 0.804047i \(0.297324\pi\)
\(152\) −7.02129 −0.569502
\(153\) −12.5990 −1.01857
\(154\) 2.81342 0.226712
\(155\) 0 0
\(156\) −50.3114 −4.02814
\(157\) 7.58342 0.605223 0.302612 0.953114i \(-0.402142\pi\)
0.302612 + 0.953114i \(0.402142\pi\)
\(158\) −25.8168 −2.05388
\(159\) −28.8706 −2.28959
\(160\) 0 0
\(161\) 5.22884 0.412090
\(162\) 22.5489 1.77161
\(163\) 17.0674 1.33682 0.668411 0.743792i \(-0.266975\pi\)
0.668411 + 0.743792i \(0.266975\pi\)
\(164\) −39.4361 −3.07944
\(165\) 0 0
\(166\) −36.3148 −2.81858
\(167\) 6.96828 0.539222 0.269611 0.962969i \(-0.413105\pi\)
0.269611 + 0.962969i \(0.413105\pi\)
\(168\) −22.8350 −1.76176
\(169\) −0.277886 −0.0213758
\(170\) 0 0
\(171\) −5.96706 −0.456312
\(172\) 37.4648 2.85666
\(173\) 25.6458 1.94982 0.974908 0.222607i \(-0.0714567\pi\)
0.974908 + 0.222607i \(0.0714567\pi\)
\(174\) 26.0044 1.97139
\(175\) 0 0
\(176\) 8.76742 0.660869
\(177\) 2.20823 0.165981
\(178\) 25.4585 1.90820
\(179\) −15.6845 −1.17232 −0.586159 0.810196i \(-0.699361\pi\)
−0.586159 + 0.810196i \(0.699361\pi\)
\(180\) 0 0
\(181\) −10.2376 −0.760953 −0.380477 0.924791i \(-0.624240\pi\)
−0.380477 + 0.924791i \(0.624240\pi\)
\(182\) 10.0349 0.743838
\(183\) 17.0758 1.26228
\(184\) 33.8036 2.49204
\(185\) 0 0
\(186\) −36.6838 −2.68978
\(187\) −2.11143 −0.154403
\(188\) 6.64544 0.484669
\(189\) −9.64961 −0.701906
\(190\) 0 0
\(191\) 11.0398 0.798812 0.399406 0.916774i \(-0.369216\pi\)
0.399406 + 0.916774i \(0.369216\pi\)
\(192\) −14.7385 −1.06366
\(193\) 15.4104 1.10926 0.554631 0.832097i \(-0.312860\pi\)
0.554631 + 0.832097i \(0.312860\pi\)
\(194\) 23.0488 1.65480
\(195\) 0 0
\(196\) −27.4169 −1.95835
\(197\) 15.7536 1.12239 0.561197 0.827682i \(-0.310341\pi\)
0.561197 + 0.827682i \(0.310341\pi\)
\(198\) 15.4574 1.09851
\(199\) 8.77418 0.621985 0.310992 0.950412i \(-0.399339\pi\)
0.310992 + 0.950412i \(0.399339\pi\)
\(200\) 0 0
\(201\) 6.37109 0.449382
\(202\) −18.5565 −1.30563
\(203\) −3.64086 −0.255538
\(204\) 29.7827 2.08521
\(205\) 0 0
\(206\) −16.5962 −1.15631
\(207\) 28.7281 1.99674
\(208\) 31.2717 2.16830
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 9.19454 0.632978 0.316489 0.948596i \(-0.397496\pi\)
0.316489 + 0.948596i \(0.397496\pi\)
\(212\) 45.4144 3.11907
\(213\) −31.7929 −2.17842
\(214\) −21.0434 −1.43849
\(215\) 0 0
\(216\) −62.3832 −4.24464
\(217\) 5.13608 0.348660
\(218\) 13.7303 0.929936
\(219\) −32.9606 −2.22727
\(220\) 0 0
\(221\) −7.53107 −0.506595
\(222\) −39.6133 −2.65867
\(223\) 16.3430 1.09441 0.547203 0.837000i \(-0.315692\pi\)
0.547203 + 0.837000i \(0.315692\pi\)
\(224\) 9.41517 0.629078
\(225\) 0 0
\(226\) 46.6687 3.10435
\(227\) 6.96435 0.462240 0.231120 0.972925i \(-0.425761\pi\)
0.231120 + 0.972925i \(0.425761\pi\)
\(228\) 14.1055 0.934157
\(229\) 6.67980 0.441413 0.220707 0.975340i \(-0.429164\pi\)
0.220707 + 0.975340i \(0.429164\pi\)
\(230\) 0 0
\(231\) −3.25225 −0.213982
\(232\) −23.5376 −1.54532
\(233\) −4.46110 −0.292256 −0.146128 0.989266i \(-0.546681\pi\)
−0.146128 + 0.989266i \(0.546681\pi\)
\(234\) 55.1335 3.60419
\(235\) 0 0
\(236\) −3.47362 −0.226113
\(237\) 29.8437 1.93856
\(238\) −5.94034 −0.385055
\(239\) 3.20600 0.207379 0.103690 0.994610i \(-0.466935\pi\)
0.103690 + 0.994610i \(0.466935\pi\)
\(240\) 0 0
\(241\) −18.5606 −1.19559 −0.597797 0.801647i \(-0.703957\pi\)
−0.597797 + 0.801647i \(0.703957\pi\)
\(242\) 2.59045 0.166521
\(243\) 0.588604 0.0377589
\(244\) −26.8607 −1.71958
\(245\) 0 0
\(246\) 64.9430 4.14062
\(247\) −3.56681 −0.226951
\(248\) 33.2039 2.10845
\(249\) 41.9791 2.66032
\(250\) 0 0
\(251\) 2.10322 0.132754 0.0663771 0.997795i \(-0.478856\pi\)
0.0663771 + 0.997795i \(0.478856\pi\)
\(252\) 30.5268 1.92301
\(253\) 4.81445 0.302682
\(254\) 10.3409 0.648843
\(255\) 0 0
\(256\) −21.7293 −1.35808
\(257\) −19.8087 −1.23563 −0.617817 0.786322i \(-0.711983\pi\)
−0.617817 + 0.786322i \(0.711983\pi\)
\(258\) −61.6967 −3.84107
\(259\) 5.54624 0.344627
\(260\) 0 0
\(261\) −20.0035 −1.23818
\(262\) 30.8978 1.90888
\(263\) −13.1192 −0.808967 −0.404483 0.914545i \(-0.632549\pi\)
−0.404483 + 0.914545i \(0.632549\pi\)
\(264\) −21.0253 −1.29402
\(265\) 0 0
\(266\) −2.81342 −0.172502
\(267\) −29.4295 −1.80105
\(268\) −10.0219 −0.612186
\(269\) −23.8316 −1.45304 −0.726520 0.687145i \(-0.758864\pi\)
−0.726520 + 0.687145i \(0.758864\pi\)
\(270\) 0 0
\(271\) 11.1137 0.675108 0.337554 0.941306i \(-0.390401\pi\)
0.337554 + 0.941306i \(0.390401\pi\)
\(272\) −18.5118 −1.12244
\(273\) −11.6001 −0.702072
\(274\) −56.6638 −3.42319
\(275\) 0 0
\(276\) −67.9100 −4.08770
\(277\) −16.3055 −0.979700 −0.489850 0.871807i \(-0.662948\pi\)
−0.489850 + 0.871807i \(0.662948\pi\)
\(278\) 30.1922 1.81081
\(279\) 28.2185 1.68940
\(280\) 0 0
\(281\) 5.40191 0.322251 0.161125 0.986934i \(-0.448488\pi\)
0.161125 + 0.986934i \(0.448488\pi\)
\(282\) −10.9437 −0.651685
\(283\) 3.52795 0.209715 0.104857 0.994487i \(-0.466561\pi\)
0.104857 + 0.994487i \(0.466561\pi\)
\(284\) 50.0112 2.96762
\(285\) 0 0
\(286\) 9.23965 0.546352
\(287\) −9.09265 −0.536722
\(288\) 51.7285 3.04813
\(289\) −12.5419 −0.737756
\(290\) 0 0
\(291\) −26.6438 −1.56189
\(292\) 51.8480 3.03418
\(293\) −30.9882 −1.81035 −0.905174 0.425040i \(-0.860260\pi\)
−0.905174 + 0.425040i \(0.860260\pi\)
\(294\) 45.1499 2.63320
\(295\) 0 0
\(296\) 35.8556 2.08406
\(297\) −8.88487 −0.515553
\(298\) 25.0620 1.45180
\(299\) 17.1722 0.993095
\(300\) 0 0
\(301\) 8.63813 0.497893
\(302\) 37.8524 2.17816
\(303\) 21.4509 1.23232
\(304\) −8.76742 −0.502846
\(305\) 0 0
\(306\) −32.6372 −1.86575
\(307\) −31.7550 −1.81236 −0.906178 0.422897i \(-0.861013\pi\)
−0.906178 + 0.422897i \(0.861013\pi\)
\(308\) 5.11589 0.291505
\(309\) 19.1848 1.09139
\(310\) 0 0
\(311\) −19.8017 −1.12285 −0.561426 0.827527i \(-0.689747\pi\)
−0.561426 + 0.827527i \(0.689747\pi\)
\(312\) −74.9931 −4.24565
\(313\) −20.7094 −1.17057 −0.585283 0.810829i \(-0.699017\pi\)
−0.585283 + 0.810829i \(0.699017\pi\)
\(314\) 19.6445 1.10860
\(315\) 0 0
\(316\) −46.9450 −2.64086
\(317\) −32.8790 −1.84667 −0.923335 0.383996i \(-0.874548\pi\)
−0.923335 + 0.383996i \(0.874548\pi\)
\(318\) −74.7880 −4.19390
\(319\) −3.35232 −0.187694
\(320\) 0 0
\(321\) 24.3256 1.35772
\(322\) 13.5451 0.754837
\(323\) 2.11143 0.117483
\(324\) 41.0026 2.27792
\(325\) 0 0
\(326\) 44.2123 2.44869
\(327\) −15.8720 −0.877721
\(328\) −58.7826 −3.24572
\(329\) 1.53222 0.0844739
\(330\) 0 0
\(331\) −19.8918 −1.09335 −0.546677 0.837343i \(-0.684108\pi\)
−0.546677 + 0.837343i \(0.684108\pi\)
\(332\) −66.0344 −3.62411
\(333\) 30.4720 1.66985
\(334\) 18.0510 0.987707
\(335\) 0 0
\(336\) −28.5138 −1.55556
\(337\) 19.9888 1.08886 0.544429 0.838807i \(-0.316746\pi\)
0.544429 + 0.838807i \(0.316746\pi\)
\(338\) −0.719850 −0.0391547
\(339\) −53.9479 −2.93005
\(340\) 0 0
\(341\) 4.72904 0.256092
\(342\) −15.4574 −0.835840
\(343\) −13.9239 −0.751822
\(344\) 55.8442 3.01092
\(345\) 0 0
\(346\) 66.4343 3.57153
\(347\) 11.6347 0.624582 0.312291 0.949987i \(-0.398904\pi\)
0.312291 + 0.949987i \(0.398904\pi\)
\(348\) 47.2860 2.53480
\(349\) −33.9709 −1.81842 −0.909211 0.416335i \(-0.863314\pi\)
−0.909211 + 0.416335i \(0.863314\pi\)
\(350\) 0 0
\(351\) −31.6906 −1.69152
\(352\) 8.66902 0.462060
\(353\) −31.2710 −1.66439 −0.832194 0.554484i \(-0.812916\pi\)
−0.832194 + 0.554484i \(0.812916\pi\)
\(354\) 5.72032 0.304032
\(355\) 0 0
\(356\) 46.2934 2.45355
\(357\) 6.86690 0.363435
\(358\) −40.6301 −2.14737
\(359\) −9.02684 −0.476418 −0.238209 0.971214i \(-0.576560\pi\)
−0.238209 + 0.971214i \(0.576560\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −26.5200 −1.39386
\(363\) −2.99450 −0.157171
\(364\) 18.2474 0.956422
\(365\) 0 0
\(366\) 44.2340 2.31215
\(367\) 10.8245 0.565035 0.282517 0.959262i \(-0.408831\pi\)
0.282517 + 0.959262i \(0.408831\pi\)
\(368\) 42.2103 2.20036
\(369\) −49.9565 −2.60063
\(370\) 0 0
\(371\) 10.4710 0.543629
\(372\) −66.7053 −3.45851
\(373\) 21.7074 1.12397 0.561984 0.827148i \(-0.310038\pi\)
0.561984 + 0.827148i \(0.310038\pi\)
\(374\) −5.46957 −0.282824
\(375\) 0 0
\(376\) 9.90555 0.510840
\(377\) −11.9571 −0.615821
\(378\) −24.9969 −1.28570
\(379\) 20.2102 1.03813 0.519063 0.854736i \(-0.326281\pi\)
0.519063 + 0.854736i \(0.326281\pi\)
\(380\) 0 0
\(381\) −11.9538 −0.612412
\(382\) 28.5981 1.46321
\(383\) 34.9348 1.78509 0.892543 0.450962i \(-0.148919\pi\)
0.892543 + 0.450962i \(0.148919\pi\)
\(384\) 13.7395 0.701141
\(385\) 0 0
\(386\) 39.9198 2.03186
\(387\) 47.4593 2.41249
\(388\) 41.9116 2.12774
\(389\) −32.2359 −1.63443 −0.817213 0.576336i \(-0.804482\pi\)
−0.817213 + 0.576336i \(0.804482\pi\)
\(390\) 0 0
\(391\) −10.1654 −0.514086
\(392\) −40.8670 −2.06410
\(393\) −35.7172 −1.80169
\(394\) 40.8089 2.05592
\(395\) 0 0
\(396\) 28.1075 1.41246
\(397\) −21.6075 −1.08445 −0.542224 0.840234i \(-0.682418\pi\)
−0.542224 + 0.840234i \(0.682418\pi\)
\(398\) 22.7291 1.13931
\(399\) 3.25225 0.162816
\(400\) 0 0
\(401\) 32.9816 1.64702 0.823512 0.567299i \(-0.192012\pi\)
0.823512 + 0.567299i \(0.192012\pi\)
\(402\) 16.5040 0.823145
\(403\) 16.8676 0.840234
\(404\) −33.7430 −1.67878
\(405\) 0 0
\(406\) −9.43148 −0.468076
\(407\) 5.10670 0.253130
\(408\) 44.3934 2.19780
\(409\) 8.12644 0.401826 0.200913 0.979609i \(-0.435609\pi\)
0.200913 + 0.979609i \(0.435609\pi\)
\(410\) 0 0
\(411\) 65.5021 3.23098
\(412\) −30.1783 −1.48678
\(413\) −0.800900 −0.0394097
\(414\) 74.4188 3.65749
\(415\) 0 0
\(416\) 30.9207 1.51601
\(417\) −34.9015 −1.70913
\(418\) −2.59045 −0.126703
\(419\) 29.7694 1.45433 0.727166 0.686462i \(-0.240837\pi\)
0.727166 + 0.686462i \(0.240837\pi\)
\(420\) 0 0
\(421\) −2.95550 −0.144042 −0.0720211 0.997403i \(-0.522945\pi\)
−0.0720211 + 0.997403i \(0.522945\pi\)
\(422\) 23.8180 1.15944
\(423\) 8.41826 0.409310
\(424\) 67.6936 3.28749
\(425\) 0 0
\(426\) −82.3581 −3.99026
\(427\) −6.19319 −0.299709
\(428\) −38.2650 −1.84961
\(429\) −10.6808 −0.515675
\(430\) 0 0
\(431\) −15.7212 −0.757262 −0.378631 0.925548i \(-0.623605\pi\)
−0.378631 + 0.925548i \(0.623605\pi\)
\(432\) −77.8974 −3.74784
\(433\) −17.1336 −0.823386 −0.411693 0.911323i \(-0.635062\pi\)
−0.411693 + 0.911323i \(0.635062\pi\)
\(434\) 13.3048 0.638649
\(435\) 0 0
\(436\) 24.9671 1.19571
\(437\) −4.81445 −0.230306
\(438\) −85.3829 −4.07975
\(439\) 6.13113 0.292623 0.146311 0.989239i \(-0.453260\pi\)
0.146311 + 0.989239i \(0.453260\pi\)
\(440\) 0 0
\(441\) −34.7310 −1.65385
\(442\) −19.5089 −0.927943
\(443\) −16.9204 −0.803914 −0.401957 0.915659i \(-0.631670\pi\)
−0.401957 + 0.915659i \(0.631670\pi\)
\(444\) −72.0323 −3.41850
\(445\) 0 0
\(446\) 42.3357 2.00465
\(447\) −28.9711 −1.37029
\(448\) 5.34547 0.252550
\(449\) −10.5785 −0.499230 −0.249615 0.968345i \(-0.580304\pi\)
−0.249615 + 0.968345i \(0.580304\pi\)
\(450\) 0 0
\(451\) −8.37205 −0.394225
\(452\) 84.8617 3.99156
\(453\) −43.7565 −2.05586
\(454\) 18.0408 0.846698
\(455\) 0 0
\(456\) 21.0253 0.984599
\(457\) −0.207034 −0.00968465 −0.00484232 0.999988i \(-0.501541\pi\)
−0.00484232 + 0.999988i \(0.501541\pi\)
\(458\) 17.3037 0.808549
\(459\) 18.7598 0.875633
\(460\) 0 0
\(461\) −24.6854 −1.14971 −0.574856 0.818254i \(-0.694942\pi\)
−0.574856 + 0.818254i \(0.694942\pi\)
\(462\) −8.42480 −0.391957
\(463\) 32.3068 1.50142 0.750712 0.660630i \(-0.229711\pi\)
0.750712 + 0.660630i \(0.229711\pi\)
\(464\) −29.3912 −1.36445
\(465\) 0 0
\(466\) −11.5563 −0.535334
\(467\) −20.3410 −0.941269 −0.470634 0.882328i \(-0.655975\pi\)
−0.470634 + 0.882328i \(0.655975\pi\)
\(468\) 100.254 4.63425
\(469\) −2.31072 −0.106699
\(470\) 0 0
\(471\) −22.7086 −1.04636
\(472\) −5.17769 −0.238323
\(473\) 7.95355 0.365705
\(474\) 77.3086 3.55090
\(475\) 0 0
\(476\) −10.8018 −0.495102
\(477\) 57.5296 2.63410
\(478\) 8.30500 0.379862
\(479\) 6.75628 0.308702 0.154351 0.988016i \(-0.450671\pi\)
0.154351 + 0.988016i \(0.450671\pi\)
\(480\) 0 0
\(481\) 18.2146 0.830514
\(482\) −48.0804 −2.19000
\(483\) −15.6578 −0.712454
\(484\) 4.71045 0.214111
\(485\) 0 0
\(486\) 1.52475 0.0691641
\(487\) 22.2673 1.00903 0.504514 0.863403i \(-0.331672\pi\)
0.504514 + 0.863403i \(0.331672\pi\)
\(488\) −40.0380 −1.81244
\(489\) −51.1084 −2.31120
\(490\) 0 0
\(491\) 37.5046 1.69256 0.846278 0.532741i \(-0.178838\pi\)
0.846278 + 0.532741i \(0.178838\pi\)
\(492\) 118.092 5.32398
\(493\) 7.07819 0.318786
\(494\) −9.23965 −0.415711
\(495\) 0 0
\(496\) 41.4615 1.86168
\(497\) 11.5309 0.517233
\(498\) 108.745 4.87298
\(499\) −18.1313 −0.811670 −0.405835 0.913946i \(-0.633019\pi\)
−0.405835 + 0.913946i \(0.633019\pi\)
\(500\) 0 0
\(501\) −20.8665 −0.932248
\(502\) 5.44830 0.243169
\(503\) 6.96999 0.310776 0.155388 0.987853i \(-0.450337\pi\)
0.155388 + 0.987853i \(0.450337\pi\)
\(504\) 45.5025 2.02684
\(505\) 0 0
\(506\) 12.4716 0.554431
\(507\) 0.832131 0.0369562
\(508\) 18.8037 0.834279
\(509\) −27.9741 −1.23993 −0.619966 0.784629i \(-0.712854\pi\)
−0.619966 + 0.784629i \(0.712854\pi\)
\(510\) 0 0
\(511\) 11.9544 0.528833
\(512\) −47.1122 −2.08209
\(513\) 8.88487 0.392277
\(514\) −51.3135 −2.26334
\(515\) 0 0
\(516\) −112.188 −4.93882
\(517\) 1.41079 0.0620464
\(518\) 14.3673 0.631262
\(519\) −76.7966 −3.37100
\(520\) 0 0
\(521\) −1.35523 −0.0593736 −0.0296868 0.999559i \(-0.509451\pi\)
−0.0296868 + 0.999559i \(0.509451\pi\)
\(522\) −51.8181 −2.26802
\(523\) 18.1213 0.792391 0.396196 0.918166i \(-0.370330\pi\)
0.396196 + 0.918166i \(0.370330\pi\)
\(524\) 56.1842 2.45442
\(525\) 0 0
\(526\) −33.9848 −1.48181
\(527\) −9.98505 −0.434955
\(528\) −26.2541 −1.14256
\(529\) 0.178941 0.00778006
\(530\) 0 0
\(531\) −4.40028 −0.190956
\(532\) −5.11589 −0.221802
\(533\) −29.8615 −1.29344
\(534\) −76.2356 −3.29904
\(535\) 0 0
\(536\) −14.9384 −0.645243
\(537\) 46.9674 2.02679
\(538\) −61.7347 −2.66157
\(539\) −5.82045 −0.250704
\(540\) 0 0
\(541\) −8.41360 −0.361729 −0.180864 0.983508i \(-0.557890\pi\)
−0.180864 + 0.983508i \(0.557890\pi\)
\(542\) 28.7895 1.23661
\(543\) 30.6565 1.31560
\(544\) −18.3040 −0.784779
\(545\) 0 0
\(546\) −30.0496 −1.28601
\(547\) 20.0414 0.856907 0.428453 0.903564i \(-0.359059\pi\)
0.428453 + 0.903564i \(0.359059\pi\)
\(548\) −103.037 −4.40151
\(549\) −34.0264 −1.45221
\(550\) 0 0
\(551\) 3.35232 0.142814
\(552\) −101.225 −4.30843
\(553\) −10.8240 −0.460281
\(554\) −42.2385 −1.79454
\(555\) 0 0
\(556\) 54.9011 2.32833
\(557\) −16.5714 −0.702154 −0.351077 0.936347i \(-0.614184\pi\)
−0.351077 + 0.936347i \(0.614184\pi\)
\(558\) 73.0986 3.09451
\(559\) 28.3688 1.19987
\(560\) 0 0
\(561\) 6.32269 0.266944
\(562\) 13.9934 0.590275
\(563\) −22.8289 −0.962123 −0.481061 0.876687i \(-0.659749\pi\)
−0.481061 + 0.876687i \(0.659749\pi\)
\(564\) −19.8998 −0.837933
\(565\) 0 0
\(566\) 9.13898 0.384140
\(567\) 9.45384 0.397024
\(568\) 74.5456 3.12786
\(569\) −12.4875 −0.523502 −0.261751 0.965135i \(-0.584300\pi\)
−0.261751 + 0.965135i \(0.584300\pi\)
\(570\) 0 0
\(571\) −30.6254 −1.28163 −0.640816 0.767695i \(-0.721404\pi\)
−0.640816 + 0.767695i \(0.721404\pi\)
\(572\) 16.8013 0.702496
\(573\) −33.0587 −1.38105
\(574\) −23.5541 −0.983129
\(575\) 0 0
\(576\) 29.3689 1.22370
\(577\) 14.1736 0.590053 0.295027 0.955489i \(-0.404671\pi\)
0.295027 + 0.955489i \(0.404671\pi\)
\(578\) −32.4891 −1.35137
\(579\) −46.1464 −1.91778
\(580\) 0 0
\(581\) −15.2253 −0.631654
\(582\) −69.0196 −2.86096
\(583\) 9.64120 0.399298
\(584\) 77.2835 3.19801
\(585\) 0 0
\(586\) −80.2734 −3.31606
\(587\) −39.0045 −1.60989 −0.804943 0.593352i \(-0.797804\pi\)
−0.804943 + 0.593352i \(0.797804\pi\)
\(588\) 82.1001 3.38575
\(589\) −4.72904 −0.194857
\(590\) 0 0
\(591\) −47.1741 −1.94048
\(592\) 44.7725 1.84014
\(593\) 43.5330 1.78769 0.893843 0.448381i \(-0.147999\pi\)
0.893843 + 0.448381i \(0.147999\pi\)
\(594\) −23.0158 −0.944352
\(595\) 0 0
\(596\) 45.5725 1.86672
\(597\) −26.2743 −1.07534
\(598\) 44.4838 1.81908
\(599\) 19.2721 0.787437 0.393719 0.919231i \(-0.371188\pi\)
0.393719 + 0.919231i \(0.371188\pi\)
\(600\) 0 0
\(601\) −21.9919 −0.897067 −0.448533 0.893766i \(-0.648053\pi\)
−0.448533 + 0.893766i \(0.648053\pi\)
\(602\) 22.3767 0.912005
\(603\) −12.6955 −0.517000
\(604\) 68.8304 2.80067
\(605\) 0 0
\(606\) 55.5677 2.25728
\(607\) −17.9582 −0.728902 −0.364451 0.931223i \(-0.618743\pi\)
−0.364451 + 0.931223i \(0.618743\pi\)
\(608\) −8.66902 −0.351575
\(609\) 10.9026 0.441795
\(610\) 0 0
\(611\) 5.03201 0.203573
\(612\) −59.3471 −2.39896
\(613\) −38.3600 −1.54935 −0.774673 0.632362i \(-0.782086\pi\)
−0.774673 + 0.632362i \(0.782086\pi\)
\(614\) −82.2599 −3.31974
\(615\) 0 0
\(616\) 7.62562 0.307245
\(617\) −37.0412 −1.49122 −0.745610 0.666382i \(-0.767842\pi\)
−0.745610 + 0.666382i \(0.767842\pi\)
\(618\) 49.6973 1.99912
\(619\) −42.9798 −1.72750 −0.863752 0.503917i \(-0.831892\pi\)
−0.863752 + 0.503917i \(0.831892\pi\)
\(620\) 0 0
\(621\) −42.7758 −1.71653
\(622\) −51.2954 −2.05676
\(623\) 10.6737 0.427634
\(624\) −93.6432 −3.74873
\(625\) 0 0
\(626\) −53.6468 −2.14416
\(627\) 2.99450 0.119589
\(628\) 35.7213 1.42544
\(629\) −10.7824 −0.429924
\(630\) 0 0
\(631\) −31.9210 −1.27076 −0.635378 0.772201i \(-0.719156\pi\)
−0.635378 + 0.772201i \(0.719156\pi\)
\(632\) −69.9752 −2.78346
\(633\) −27.5331 −1.09434
\(634\) −85.1716 −3.38259
\(635\) 0 0
\(636\) −135.994 −5.39250
\(637\) −20.7604 −0.822558
\(638\) −8.68403 −0.343804
\(639\) 63.3528 2.50620
\(640\) 0 0
\(641\) −18.8432 −0.744262 −0.372131 0.928180i \(-0.621373\pi\)
−0.372131 + 0.928180i \(0.621373\pi\)
\(642\) 63.0144 2.48698
\(643\) 15.4713 0.610129 0.305064 0.952332i \(-0.401322\pi\)
0.305064 + 0.952332i \(0.401322\pi\)
\(644\) 24.6302 0.970565
\(645\) 0 0
\(646\) 5.46957 0.215197
\(647\) −8.22385 −0.323313 −0.161656 0.986847i \(-0.551684\pi\)
−0.161656 + 0.986847i \(0.551684\pi\)
\(648\) 61.1176 2.40093
\(649\) −0.737428 −0.0289466
\(650\) 0 0
\(651\) −15.3800 −0.602790
\(652\) 80.3951 3.14852
\(653\) −9.29054 −0.363567 −0.181783 0.983339i \(-0.558187\pi\)
−0.181783 + 0.983339i \(0.558187\pi\)
\(654\) −41.1156 −1.60775
\(655\) 0 0
\(656\) −73.4013 −2.86584
\(657\) 65.6796 2.56240
\(658\) 3.96914 0.154733
\(659\) 31.5012 1.22711 0.613556 0.789651i \(-0.289738\pi\)
0.613556 + 0.789651i \(0.289738\pi\)
\(660\) 0 0
\(661\) 1.08128 0.0420569 0.0210285 0.999779i \(-0.493306\pi\)
0.0210285 + 0.999779i \(0.493306\pi\)
\(662\) −51.5289 −2.00273
\(663\) 22.5518 0.875840
\(664\) −98.4295 −3.81980
\(665\) 0 0
\(666\) 78.9362 3.05871
\(667\) −16.1396 −0.624927
\(668\) 32.8237 1.26999
\(669\) −48.9391 −1.89210
\(670\) 0 0
\(671\) −5.70238 −0.220138
\(672\) −28.1938 −1.08760
\(673\) −43.8982 −1.69215 −0.846075 0.533064i \(-0.821041\pi\)
−0.846075 + 0.533064i \(0.821041\pi\)
\(674\) 51.7800 1.99449
\(675\) 0 0
\(676\) −1.30897 −0.0503449
\(677\) 10.1422 0.389797 0.194898 0.980823i \(-0.437562\pi\)
0.194898 + 0.980823i \(0.437562\pi\)
\(678\) −139.750 −5.36705
\(679\) 9.66341 0.370848
\(680\) 0 0
\(681\) −20.8548 −0.799157
\(682\) 12.2504 0.469090
\(683\) −0.666978 −0.0255212 −0.0127606 0.999919i \(-0.504062\pi\)
−0.0127606 + 0.999919i \(0.504062\pi\)
\(684\) −28.1075 −1.07472
\(685\) 0 0
\(686\) −36.0693 −1.37713
\(687\) −20.0027 −0.763150
\(688\) 69.7321 2.65851
\(689\) 34.3883 1.31009
\(690\) 0 0
\(691\) 23.8707 0.908085 0.454042 0.890980i \(-0.349981\pi\)
0.454042 + 0.890980i \(0.349981\pi\)
\(692\) 120.803 4.59225
\(693\) 6.48066 0.246180
\(694\) 30.1391 1.14406
\(695\) 0 0
\(696\) 70.4834 2.67167
\(697\) 17.6770 0.669565
\(698\) −88.0001 −3.33085
\(699\) 13.3588 0.505276
\(700\) 0 0
\(701\) 42.6513 1.61092 0.805460 0.592651i \(-0.201919\pi\)
0.805460 + 0.592651i \(0.201919\pi\)
\(702\) −82.0931 −3.09840
\(703\) −5.10670 −0.192603
\(704\) 4.92184 0.185499
\(705\) 0 0
\(706\) −81.0061 −3.04871
\(707\) −7.78001 −0.292597
\(708\) 10.4018 0.390922
\(709\) −29.4487 −1.10597 −0.552985 0.833191i \(-0.686511\pi\)
−0.552985 + 0.833191i \(0.686511\pi\)
\(710\) 0 0
\(711\) −59.4686 −2.23025
\(712\) 69.0040 2.58603
\(713\) 22.7677 0.852658
\(714\) 17.7884 0.665713
\(715\) 0 0
\(716\) −73.8812 −2.76107
\(717\) −9.60039 −0.358533
\(718\) −23.3836 −0.872668
\(719\) −38.5175 −1.43646 −0.718230 0.695805i \(-0.755048\pi\)
−0.718230 + 0.695805i \(0.755048\pi\)
\(720\) 0 0
\(721\) −6.95810 −0.259133
\(722\) 2.59045 0.0964067
\(723\) 55.5799 2.06704
\(724\) −48.2236 −1.79222
\(725\) 0 0
\(726\) −7.75712 −0.287894
\(727\) 13.9379 0.516929 0.258464 0.966021i \(-0.416784\pi\)
0.258464 + 0.966021i \(0.416784\pi\)
\(728\) 27.1991 1.00807
\(729\) −27.8764 −1.03246
\(730\) 0 0
\(731\) −16.7934 −0.621126
\(732\) 80.4346 2.97295
\(733\) −23.2536 −0.858891 −0.429445 0.903093i \(-0.641291\pi\)
−0.429445 + 0.903093i \(0.641291\pi\)
\(734\) 28.0404 1.03499
\(735\) 0 0
\(736\) 41.7366 1.53843
\(737\) −2.12759 −0.0783709
\(738\) −129.410 −4.76365
\(739\) 28.2385 1.03877 0.519385 0.854540i \(-0.326161\pi\)
0.519385 + 0.854540i \(0.326161\pi\)
\(740\) 0 0
\(741\) 10.6808 0.392370
\(742\) 27.1247 0.995780
\(743\) 24.5547 0.900823 0.450412 0.892821i \(-0.351277\pi\)
0.450412 + 0.892821i \(0.351277\pi\)
\(744\) −99.4294 −3.64526
\(745\) 0 0
\(746\) 56.2321 2.05880
\(747\) −83.6505 −3.06061
\(748\) −9.94579 −0.363654
\(749\) −8.82263 −0.322372
\(750\) 0 0
\(751\) −39.8070 −1.45258 −0.726290 0.687389i \(-0.758757\pi\)
−0.726290 + 0.687389i \(0.758757\pi\)
\(752\) 12.3690 0.451050
\(753\) −6.29811 −0.229516
\(754\) −30.9742 −1.12802
\(755\) 0 0
\(756\) −45.4540 −1.65315
\(757\) −13.9769 −0.508000 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(758\) 52.3535 1.90156
\(759\) −14.4169 −0.523300
\(760\) 0 0
\(761\) 30.8110 1.11690 0.558449 0.829539i \(-0.311397\pi\)
0.558449 + 0.829539i \(0.311397\pi\)
\(762\) −30.9658 −1.12177
\(763\) 5.75657 0.208402
\(764\) 52.0024 1.88138
\(765\) 0 0
\(766\) 90.4970 3.26979
\(767\) −2.63026 −0.0949733
\(768\) 65.0685 2.34796
\(769\) 41.5445 1.49813 0.749066 0.662496i \(-0.230503\pi\)
0.749066 + 0.662496i \(0.230503\pi\)
\(770\) 0 0
\(771\) 59.3173 2.13626
\(772\) 72.5896 2.61256
\(773\) 16.7752 0.603363 0.301681 0.953409i \(-0.402452\pi\)
0.301681 + 0.953409i \(0.402452\pi\)
\(774\) 122.941 4.41903
\(775\) 0 0
\(776\) 62.4724 2.24263
\(777\) −16.6082 −0.595817
\(778\) −83.5057 −2.99382
\(779\) 8.37205 0.299960
\(780\) 0 0
\(781\) 10.6171 0.379909
\(782\) −26.3330 −0.941665
\(783\) 29.7849 1.06443
\(784\) −51.0303 −1.82251
\(785\) 0 0
\(786\) −92.5237 −3.30021
\(787\) 45.4889 1.62150 0.810752 0.585390i \(-0.199059\pi\)
0.810752 + 0.585390i \(0.199059\pi\)
\(788\) 74.2063 2.64349
\(789\) 39.2856 1.39860
\(790\) 0 0
\(791\) 19.5663 0.695697
\(792\) 41.8964 1.48872
\(793\) −20.3393 −0.722269
\(794\) −55.9731 −1.98641
\(795\) 0 0
\(796\) 41.3303 1.46491
\(797\) −1.55392 −0.0550428 −0.0275214 0.999621i \(-0.508761\pi\)
−0.0275214 + 0.999621i \(0.508761\pi\)
\(798\) 8.42480 0.298235
\(799\) −2.97878 −0.105382
\(800\) 0 0
\(801\) 58.6432 2.07206
\(802\) 85.4373 3.01690
\(803\) 11.0070 0.388430
\(804\) 30.0107 1.05840
\(805\) 0 0
\(806\) 43.6947 1.53908
\(807\) 71.3639 2.51213
\(808\) −50.2965 −1.76943
\(809\) 10.4343 0.366849 0.183425 0.983034i \(-0.441282\pi\)
0.183425 + 0.983034i \(0.441282\pi\)
\(810\) 0 0
\(811\) 47.3513 1.66273 0.831364 0.555729i \(-0.187561\pi\)
0.831364 + 0.555729i \(0.187561\pi\)
\(812\) −17.1501 −0.601850
\(813\) −33.2800 −1.16718
\(814\) 13.2287 0.463664
\(815\) 0 0
\(816\) 55.4337 1.94057
\(817\) −7.95355 −0.278259
\(818\) 21.0511 0.736036
\(819\) 23.1152 0.807712
\(820\) 0 0
\(821\) −15.8887 −0.554521 −0.277260 0.960795i \(-0.589426\pi\)
−0.277260 + 0.960795i \(0.589426\pi\)
\(822\) 169.680 5.91827
\(823\) −15.6533 −0.545640 −0.272820 0.962065i \(-0.587956\pi\)
−0.272820 + 0.962065i \(0.587956\pi\)
\(824\) −44.9831 −1.56706
\(825\) 0 0
\(826\) −2.07469 −0.0721878
\(827\) −42.8097 −1.48864 −0.744319 0.667824i \(-0.767226\pi\)
−0.744319 + 0.667824i \(0.767226\pi\)
\(828\) 135.322 4.70277
\(829\) −27.2954 −0.948007 −0.474004 0.880523i \(-0.657192\pi\)
−0.474004 + 0.880523i \(0.657192\pi\)
\(830\) 0 0
\(831\) 48.8268 1.69378
\(832\) 17.5553 0.608619
\(833\) 12.2895 0.425805
\(834\) −90.4107 −3.13066
\(835\) 0 0
\(836\) −4.71045 −0.162914
\(837\) −42.0169 −1.45232
\(838\) 77.1163 2.66394
\(839\) −4.39518 −0.151739 −0.0758693 0.997118i \(-0.524173\pi\)
−0.0758693 + 0.997118i \(0.524173\pi\)
\(840\) 0 0
\(841\) −17.7620 −0.612481
\(842\) −7.65608 −0.263846
\(843\) −16.1760 −0.557132
\(844\) 43.3104 1.49080
\(845\) 0 0
\(846\) 21.8071 0.749743
\(847\) 1.08607 0.0373179
\(848\) 84.5285 2.90272
\(849\) −10.5645 −0.362571
\(850\) 0 0
\(851\) 24.5859 0.842795
\(852\) −149.759 −5.13065
\(853\) 0.792470 0.0271337 0.0135668 0.999908i \(-0.495681\pi\)
0.0135668 + 0.999908i \(0.495681\pi\)
\(854\) −16.0432 −0.548986
\(855\) 0 0
\(856\) −57.0369 −1.94948
\(857\) 35.8171 1.22349 0.611744 0.791056i \(-0.290468\pi\)
0.611744 + 0.791056i \(0.290468\pi\)
\(858\) −27.6682 −0.944576
\(859\) −45.8663 −1.56494 −0.782470 0.622689i \(-0.786040\pi\)
−0.782470 + 0.622689i \(0.786040\pi\)
\(860\) 0 0
\(861\) 27.2280 0.927927
\(862\) −40.7250 −1.38710
\(863\) 1.85336 0.0630892 0.0315446 0.999502i \(-0.489957\pi\)
0.0315446 + 0.999502i \(0.489957\pi\)
\(864\) −77.0231 −2.62038
\(865\) 0 0
\(866\) −44.3837 −1.50822
\(867\) 37.5566 1.27549
\(868\) 24.1932 0.821172
\(869\) −9.96615 −0.338078
\(870\) 0 0
\(871\) −7.58872 −0.257134
\(872\) 37.2154 1.26027
\(873\) 53.0924 1.79690
\(874\) −12.4716 −0.421859
\(875\) 0 0
\(876\) −155.259 −5.24572
\(877\) 28.7139 0.969598 0.484799 0.874626i \(-0.338893\pi\)
0.484799 + 0.874626i \(0.338893\pi\)
\(878\) 15.8824 0.536005
\(879\) 92.7943 3.12987
\(880\) 0 0
\(881\) 14.9317 0.503061 0.251531 0.967849i \(-0.419066\pi\)
0.251531 + 0.967849i \(0.419066\pi\)
\(882\) −89.9689 −3.02941
\(883\) −28.3711 −0.954765 −0.477382 0.878696i \(-0.658414\pi\)
−0.477382 + 0.878696i \(0.658414\pi\)
\(884\) −35.4747 −1.19314
\(885\) 0 0
\(886\) −43.8316 −1.47255
\(887\) 2.80865 0.0943053 0.0471527 0.998888i \(-0.484985\pi\)
0.0471527 + 0.998888i \(0.484985\pi\)
\(888\) −107.370 −3.60309
\(889\) 4.33550 0.145408
\(890\) 0 0
\(891\) 8.70462 0.291616
\(892\) 76.9828 2.57757
\(893\) −1.41079 −0.0472102
\(894\) −75.0483 −2.50999
\(895\) 0 0
\(896\) −4.98316 −0.166476
\(897\) −51.4223 −1.71694
\(898\) −27.4031 −0.914453
\(899\) −15.8533 −0.528736
\(900\) 0 0
\(901\) −20.3567 −0.678181
\(902\) −21.6874 −0.722112
\(903\) −25.8669 −0.860797
\(904\) 126.493 4.20709
\(905\) 0 0
\(906\) −113.349 −3.76578
\(907\) 47.3996 1.57388 0.786939 0.617031i \(-0.211665\pi\)
0.786939 + 0.617031i \(0.211665\pi\)
\(908\) 32.8052 1.08868
\(909\) −42.7446 −1.41775
\(910\) 0 0
\(911\) −21.0979 −0.699005 −0.349502 0.936935i \(-0.613649\pi\)
−0.349502 + 0.936935i \(0.613649\pi\)
\(912\) 26.2541 0.869359
\(913\) −14.0187 −0.463952
\(914\) −0.536312 −0.0177396
\(915\) 0 0
\(916\) 31.4648 1.03963
\(917\) 12.9542 0.427786
\(918\) 48.5964 1.60392
\(919\) 19.3565 0.638512 0.319256 0.947668i \(-0.396567\pi\)
0.319256 + 0.947668i \(0.396567\pi\)
\(920\) 0 0
\(921\) 95.0906 3.13334
\(922\) −63.9463 −2.10596
\(923\) 37.8691 1.24648
\(924\) −15.3195 −0.503976
\(925\) 0 0
\(926\) 83.6892 2.75020
\(927\) −38.2290 −1.25560
\(928\) −29.0613 −0.953984
\(929\) −9.90582 −0.324999 −0.162500 0.986709i \(-0.551956\pi\)
−0.162500 + 0.986709i \(0.551956\pi\)
\(930\) 0 0
\(931\) 5.82045 0.190757
\(932\) −21.0138 −0.688329
\(933\) 59.2963 1.94127
\(934\) −52.6924 −1.72415
\(935\) 0 0
\(936\) 149.436 4.88448
\(937\) 41.1901 1.34562 0.672811 0.739814i \(-0.265087\pi\)
0.672811 + 0.739814i \(0.265087\pi\)
\(938\) −5.98581 −0.195444
\(939\) 62.0145 2.02377
\(940\) 0 0
\(941\) −46.3419 −1.51070 −0.755351 0.655320i \(-0.772534\pi\)
−0.755351 + 0.655320i \(0.772534\pi\)
\(942\) −58.8256 −1.91664
\(943\) −40.3068 −1.31257
\(944\) −6.46534 −0.210429
\(945\) 0 0
\(946\) 20.6033 0.669871
\(947\) −24.4233 −0.793650 −0.396825 0.917894i \(-0.629888\pi\)
−0.396825 + 0.917894i \(0.629888\pi\)
\(948\) 140.577 4.56573
\(949\) 39.2599 1.27443
\(950\) 0 0
\(951\) 98.4564 3.19267
\(952\) −16.1010 −0.521836
\(953\) 10.5765 0.342607 0.171303 0.985218i \(-0.445202\pi\)
0.171303 + 0.985218i \(0.445202\pi\)
\(954\) 149.028 4.82495
\(955\) 0 0
\(956\) 15.1017 0.488424
\(957\) 10.0385 0.324500
\(958\) 17.5018 0.565458
\(959\) −23.7568 −0.767148
\(960\) 0 0
\(961\) −8.63618 −0.278586
\(962\) 47.1841 1.52128
\(963\) −48.4730 −1.56202
\(964\) −87.4288 −2.81589
\(965\) 0 0
\(966\) −40.5608 −1.30502
\(967\) −16.1131 −0.518161 −0.259081 0.965856i \(-0.583420\pi\)
−0.259081 + 0.965856i \(0.583420\pi\)
\(968\) 7.02129 0.225673
\(969\) −6.32269 −0.203114
\(970\) 0 0
\(971\) −57.4053 −1.84222 −0.921111 0.389299i \(-0.872717\pi\)
−0.921111 + 0.389299i \(0.872717\pi\)
\(972\) 2.77259 0.0889308
\(973\) 12.6584 0.405808
\(974\) 57.6825 1.84827
\(975\) 0 0
\(976\) −49.9951 −1.60030
\(977\) 12.2856 0.393050 0.196525 0.980499i \(-0.437034\pi\)
0.196525 + 0.980499i \(0.437034\pi\)
\(978\) −132.394 −4.23349
\(979\) 9.82782 0.314099
\(980\) 0 0
\(981\) 31.6276 1.00979
\(982\) 97.1538 3.10030
\(983\) −2.24299 −0.0715404 −0.0357702 0.999360i \(-0.511388\pi\)
−0.0357702 + 0.999360i \(0.511388\pi\)
\(984\) 176.025 5.61146
\(985\) 0 0
\(986\) 18.3357 0.583929
\(987\) −4.58823 −0.146045
\(988\) −16.8013 −0.534519
\(989\) 38.2920 1.21761
\(990\) 0 0
\(991\) 31.9640 1.01537 0.507685 0.861543i \(-0.330502\pi\)
0.507685 + 0.861543i \(0.330502\pi\)
\(992\) 40.9961 1.30163
\(993\) 59.5662 1.89028
\(994\) 29.8703 0.947429
\(995\) 0 0
\(996\) 197.740 6.26565
\(997\) 10.1521 0.321519 0.160759 0.986994i \(-0.448606\pi\)
0.160759 + 0.986994i \(0.448606\pi\)
\(998\) −46.9683 −1.48676
\(999\) −45.3723 −1.43552
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 5225.2.a.x.1.13 yes 15
5.4 even 2 5225.2.a.s.1.3 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.s.1.3 15 5.4 even 2
5225.2.a.x.1.13 yes 15 1.1 even 1 trivial