Properties

Label 9025.2.a.bg.1.3
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $4$
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] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, 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("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.7537.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 4x + 3 \) 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 1.3
Root \(-2.04717\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.19091 q^{2} -3.04717 q^{3} -0.581734 q^{4} -3.62891 q^{6} +0.609175 q^{7} -3.07461 q^{8} +6.28525 q^{9} +O(q^{10})\) \(q+1.19091 q^{2} -3.04717 q^{3} -0.581734 q^{4} -3.62891 q^{6} +0.609175 q^{7} -3.07461 q^{8} +6.28525 q^{9} +4.48517 q^{11} +1.77264 q^{12} -4.43800 q^{13} +0.725473 q^{14} -2.49812 q^{16} -2.90343 q^{17} +7.48517 q^{18} -1.85626 q^{21} +5.34143 q^{22} +2.84726 q^{23} +9.36887 q^{24} -5.28525 q^{26} -10.0107 q^{27} -0.354378 q^{28} +1.11630 q^{29} -6.22908 q^{31} +3.17419 q^{32} -13.6671 q^{33} -3.45773 q^{34} -3.65635 q^{36} +3.77264 q^{37} +13.5233 q^{39} -8.30369 q^{41} -2.21064 q^{42} +9.98877 q^{43} -2.60918 q^{44} +3.39082 q^{46} +5.88500 q^{47} +7.61219 q^{48} -6.62891 q^{49} +8.84726 q^{51} +2.58173 q^{52} -8.44872 q^{53} -11.9219 q^{54} -1.87298 q^{56} +1.32941 q^{58} +10.2359 q^{59} -4.98199 q^{61} -7.41827 q^{62} +3.82882 q^{63} +8.77641 q^{64} -16.2762 q^{66} -8.47616 q^{67} +1.68903 q^{68} -8.67608 q^{69} +11.6199 q^{71} -19.3247 q^{72} -3.72325 q^{73} +4.49288 q^{74} +2.73225 q^{77} +16.1051 q^{78} +9.03817 q^{79} +11.6486 q^{81} -9.88894 q^{82} +2.12178 q^{83} +1.07985 q^{84} +11.8957 q^{86} -3.40155 q^{87} -13.7901 q^{88} +7.93217 q^{89} -2.70352 q^{91} -1.65635 q^{92} +18.9811 q^{93} +7.00850 q^{94} -9.67231 q^{96} +9.67256 q^{97} -7.89443 q^{98} +28.1904 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} - 12 q^{8} + q^{9} - 2 q^{11} - 6 q^{12} - 7 q^{13} - q^{14} + 7 q^{16} + q^{17} + 10 q^{18} - 4 q^{21} - 2 q^{22} - 2 q^{23} + 23 q^{24} + 3 q^{26} - 12 q^{27} + 19 q^{28} - q^{29} - 30 q^{32} - 19 q^{33} + 15 q^{34} - 7 q^{36} + 2 q^{37} + 15 q^{39} - 8 q^{41} + 15 q^{42} - q^{43} - 12 q^{44} + 12 q^{46} + 12 q^{47} - 23 q^{48} - 10 q^{49} + 22 q^{51} + 3 q^{52} + 5 q^{53} - 34 q^{54} - 41 q^{56} + 27 q^{58} - 5 q^{59} - 37 q^{62} + 3 q^{63} + 56 q^{64} - 31 q^{66} - 4 q^{67} - 16 q^{68} - 9 q^{69} + 20 q^{71} - 17 q^{72} + 20 q^{73} + 25 q^{74} - 14 q^{77} + 18 q^{78} + 17 q^{79} + 12 q^{81} - 21 q^{82} - q^{83} - 20 q^{84} + 8 q^{86} + 16 q^{87} + 7 q^{88} + 11 q^{89} + 6 q^{91} + q^{92} + 8 q^{93} - 31 q^{94} + 21 q^{96} - q^{97} - 9 q^{98} + 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.19091 0.842100 0.421050 0.907037i \(-0.361662\pi\)
0.421050 + 0.907037i \(0.361662\pi\)
\(3\) −3.04717 −1.75929 −0.879643 0.475635i \(-0.842218\pi\)
−0.879643 + 0.475635i \(0.842218\pi\)
\(4\) −0.581734 −0.290867
\(5\) 0 0
\(6\) −3.62891 −1.48149
\(7\) 0.609175 0.230247 0.115123 0.993351i \(-0.463274\pi\)
0.115123 + 0.993351i \(0.463274\pi\)
\(8\) −3.07461 −1.08704
\(9\) 6.28525 2.09508
\(10\) 0 0
\(11\) 4.48517 1.35233 0.676164 0.736751i \(-0.263641\pi\)
0.676164 + 0.736751i \(0.263641\pi\)
\(12\) 1.77264 0.511718
\(13\) −4.43800 −1.23088 −0.615439 0.788184i \(-0.711021\pi\)
−0.615439 + 0.788184i \(0.711021\pi\)
\(14\) 0.725473 0.193891
\(15\) 0 0
\(16\) −2.49812 −0.624529
\(17\) −2.90343 −0.704186 −0.352093 0.935965i \(-0.614530\pi\)
−0.352093 + 0.935965i \(0.614530\pi\)
\(18\) 7.48517 1.76427
\(19\) 0 0
\(20\) 0 0
\(21\) −1.85626 −0.405069
\(22\) 5.34143 1.13880
\(23\) 2.84726 0.593694 0.296847 0.954925i \(-0.404065\pi\)
0.296847 + 0.954925i \(0.404065\pi\)
\(24\) 9.36887 1.91241
\(25\) 0 0
\(26\) −5.28525 −1.03652
\(27\) −10.0107 −1.92656
\(28\) −0.354378 −0.0669712
\(29\) 1.11630 0.207291 0.103646 0.994614i \(-0.466949\pi\)
0.103646 + 0.994614i \(0.466949\pi\)
\(30\) 0 0
\(31\) −6.22908 −1.11877 −0.559387 0.828906i \(-0.688964\pi\)
−0.559387 + 0.828906i \(0.688964\pi\)
\(32\) 3.17419 0.561123
\(33\) −13.6671 −2.37913
\(34\) −3.45773 −0.592995
\(35\) 0 0
\(36\) −3.65635 −0.609391
\(37\) 3.77264 0.620219 0.310109 0.950701i \(-0.399634\pi\)
0.310109 + 0.950701i \(0.399634\pi\)
\(38\) 0 0
\(39\) 13.5233 2.16547
\(40\) 0 0
\(41\) −8.30369 −1.29682 −0.648409 0.761292i \(-0.724565\pi\)
−0.648409 + 0.761292i \(0.724565\pi\)
\(42\) −2.21064 −0.341109
\(43\) 9.98877 1.52327 0.761637 0.648004i \(-0.224396\pi\)
0.761637 + 0.648004i \(0.224396\pi\)
\(44\) −2.60918 −0.393348
\(45\) 0 0
\(46\) 3.39082 0.499950
\(47\) 5.88500 0.858415 0.429208 0.903206i \(-0.358793\pi\)
0.429208 + 0.903206i \(0.358793\pi\)
\(48\) 7.61219 1.09872
\(49\) −6.62891 −0.946986
\(50\) 0 0
\(51\) 8.84726 1.23886
\(52\) 2.58173 0.358022
\(53\) −8.44872 −1.16052 −0.580261 0.814431i \(-0.697049\pi\)
−0.580261 + 0.814431i \(0.697049\pi\)
\(54\) −11.9219 −1.62236
\(55\) 0 0
\(56\) −1.87298 −0.250287
\(57\) 0 0
\(58\) 1.32941 0.174560
\(59\) 10.2359 1.33259 0.666297 0.745686i \(-0.267878\pi\)
0.666297 + 0.745686i \(0.267878\pi\)
\(60\) 0 0
\(61\) −4.98199 −0.637878 −0.318939 0.947775i \(-0.603327\pi\)
−0.318939 + 0.947775i \(0.603327\pi\)
\(62\) −7.41827 −0.942121
\(63\) 3.82882 0.482386
\(64\) 8.77641 1.09705
\(65\) 0 0
\(66\) −16.2762 −2.00347
\(67\) −8.47616 −1.03553 −0.517764 0.855524i \(-0.673235\pi\)
−0.517764 + 0.855524i \(0.673235\pi\)
\(68\) 1.68903 0.204825
\(69\) −8.67608 −1.04448
\(70\) 0 0
\(71\) 11.6199 1.37903 0.689514 0.724272i \(-0.257824\pi\)
0.689514 + 0.724272i \(0.257824\pi\)
\(72\) −19.3247 −2.27744
\(73\) −3.72325 −0.435773 −0.217887 0.975974i \(-0.569916\pi\)
−0.217887 + 0.975974i \(0.569916\pi\)
\(74\) 4.49288 0.522286
\(75\) 0 0
\(76\) 0 0
\(77\) 2.73225 0.311369
\(78\) 16.1051 1.82354
\(79\) 9.03817 1.01687 0.508437 0.861099i \(-0.330224\pi\)
0.508437 + 0.861099i \(0.330224\pi\)
\(80\) 0 0
\(81\) 11.6486 1.29429
\(82\) −9.88894 −1.09205
\(83\) 2.12178 0.232896 0.116448 0.993197i \(-0.462849\pi\)
0.116448 + 0.993197i \(0.462849\pi\)
\(84\) 1.07985 0.117821
\(85\) 0 0
\(86\) 11.8957 1.28275
\(87\) −3.40155 −0.364684
\(88\) −13.7901 −1.47003
\(89\) 7.93217 0.840808 0.420404 0.907337i \(-0.361888\pi\)
0.420404 + 0.907337i \(0.361888\pi\)
\(90\) 0 0
\(91\) −2.70352 −0.283406
\(92\) −1.65635 −0.172686
\(93\) 18.9811 1.96824
\(94\) 7.00850 0.722872
\(95\) 0 0
\(96\) −9.67231 −0.987176
\(97\) 9.67256 0.982099 0.491050 0.871132i \(-0.336613\pi\)
0.491050 + 0.871132i \(0.336613\pi\)
\(98\) −7.89443 −0.797458
\(99\) 28.1904 2.83324
\(100\) 0 0
\(101\) −0.971265 −0.0966444 −0.0483222 0.998832i \(-0.515387\pi\)
−0.0483222 + 0.998832i \(0.515387\pi\)
\(102\) 10.5363 1.04325
\(103\) 3.34143 0.329241 0.164620 0.986357i \(-0.447360\pi\)
0.164620 + 0.986357i \(0.447360\pi\)
\(104\) 13.6451 1.33801
\(105\) 0 0
\(106\) −10.0617 −0.977275
\(107\) −9.51655 −0.920000 −0.460000 0.887919i \(-0.652151\pi\)
−0.460000 + 0.887919i \(0.652151\pi\)
\(108\) 5.82358 0.560374
\(109\) 5.54357 0.530978 0.265489 0.964114i \(-0.414467\pi\)
0.265489 + 0.964114i \(0.414467\pi\)
\(110\) 0 0
\(111\) −11.4959 −1.09114
\(112\) −1.52179 −0.143796
\(113\) −1.54134 −0.144997 −0.0724987 0.997369i \(-0.523097\pi\)
−0.0724987 + 0.997369i \(0.523097\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.649388 −0.0602942
\(117\) −27.8939 −2.57879
\(118\) 12.1900 1.12218
\(119\) −1.76870 −0.162136
\(120\) 0 0
\(121\) 9.11672 0.828793
\(122\) −5.93310 −0.537158
\(123\) 25.3028 2.28147
\(124\) 3.62367 0.325415
\(125\) 0 0
\(126\) 4.55978 0.406217
\(127\) −2.30549 −0.204579 −0.102289 0.994755i \(-0.532617\pi\)
−0.102289 + 0.994755i \(0.532617\pi\)
\(128\) 4.10353 0.362704
\(129\) −30.4375 −2.67987
\(130\) 0 0
\(131\) −12.9181 −1.12866 −0.564330 0.825549i \(-0.690865\pi\)
−0.564330 + 0.825549i \(0.690865\pi\)
\(132\) 7.95060 0.692011
\(133\) 0 0
\(134\) −10.0943 −0.872018
\(135\) 0 0
\(136\) 8.92693 0.765478
\(137\) 12.7335 1.08790 0.543950 0.839118i \(-0.316928\pi\)
0.543950 + 0.839118i \(0.316928\pi\)
\(138\) −10.3324 −0.879554
\(139\) 10.6087 0.899816 0.449908 0.893075i \(-0.351457\pi\)
0.449908 + 0.893075i \(0.351457\pi\)
\(140\) 0 0
\(141\) −17.9326 −1.51020
\(142\) 13.8383 1.16128
\(143\) −19.9052 −1.66455
\(144\) −15.7013 −1.30844
\(145\) 0 0
\(146\) −4.43405 −0.366965
\(147\) 20.1994 1.66602
\(148\) −2.19468 −0.180401
\(149\) 3.77307 0.309102 0.154551 0.987985i \(-0.450607\pi\)
0.154551 + 0.987985i \(0.450607\pi\)
\(150\) 0 0
\(151\) −9.51562 −0.774370 −0.387185 0.922002i \(-0.626553\pi\)
−0.387185 + 0.922002i \(0.626553\pi\)
\(152\) 0 0
\(153\) −18.2488 −1.47533
\(154\) 3.25387 0.262204
\(155\) 0 0
\(156\) −7.86699 −0.629863
\(157\) 3.45643 0.275853 0.137927 0.990442i \(-0.455956\pi\)
0.137927 + 0.990442i \(0.455956\pi\)
\(158\) 10.7636 0.856309
\(159\) 25.7447 2.04169
\(160\) 0 0
\(161\) 1.73448 0.136696
\(162\) 13.8725 1.08992
\(163\) −6.65283 −0.521090 −0.260545 0.965462i \(-0.583902\pi\)
−0.260545 + 0.965462i \(0.583902\pi\)
\(164\) 4.83054 0.377202
\(165\) 0 0
\(166\) 2.52685 0.196122
\(167\) 16.4555 1.27336 0.636682 0.771126i \(-0.280306\pi\)
0.636682 + 0.771126i \(0.280306\pi\)
\(168\) 5.70728 0.440327
\(169\) 6.69581 0.515062
\(170\) 0 0
\(171\) 0 0
\(172\) −5.81081 −0.443070
\(173\) 22.7824 1.73212 0.866058 0.499943i \(-0.166646\pi\)
0.866058 + 0.499943i \(0.166646\pi\)
\(174\) −4.05094 −0.307101
\(175\) 0 0
\(176\) −11.2045 −0.844569
\(177\) −31.1904 −2.34441
\(178\) 9.44650 0.708045
\(179\) −2.32916 −0.174090 −0.0870449 0.996204i \(-0.527742\pi\)
−0.0870449 + 0.996204i \(0.527742\pi\)
\(180\) 0 0
\(181\) −22.3392 −1.66046 −0.830230 0.557421i \(-0.811791\pi\)
−0.830230 + 0.557421i \(0.811791\pi\)
\(182\) −3.21965 −0.238656
\(183\) 15.1810 1.12221
\(184\) −8.75421 −0.645369
\(185\) 0 0
\(186\) 22.6047 1.65746
\(187\) −13.0224 −0.952291
\(188\) −3.42350 −0.249685
\(189\) −6.09829 −0.443585
\(190\) 0 0
\(191\) 2.23766 0.161911 0.0809556 0.996718i \(-0.474203\pi\)
0.0809556 + 0.996718i \(0.474203\pi\)
\(192\) −26.7432 −1.93003
\(193\) 4.54306 0.327017 0.163508 0.986542i \(-0.447719\pi\)
0.163508 + 0.986542i \(0.447719\pi\)
\(194\) 11.5191 0.827026
\(195\) 0 0
\(196\) 3.85626 0.275447
\(197\) 19.2236 1.36962 0.684812 0.728720i \(-0.259884\pi\)
0.684812 + 0.728720i \(0.259884\pi\)
\(198\) 33.5722 2.38587
\(199\) −6.15094 −0.436029 −0.218014 0.975946i \(-0.569958\pi\)
−0.218014 + 0.975946i \(0.569958\pi\)
\(200\) 0 0
\(201\) 25.8283 1.82179
\(202\) −1.15669 −0.0813843
\(203\) 0.680021 0.0477281
\(204\) −5.14675 −0.360345
\(205\) 0 0
\(206\) 3.97934 0.277254
\(207\) 17.8957 1.24384
\(208\) 11.0866 0.768720
\(209\) 0 0
\(210\) 0 0
\(211\) 12.6932 0.873837 0.436919 0.899501i \(-0.356070\pi\)
0.436919 + 0.899501i \(0.356070\pi\)
\(212\) 4.91491 0.337557
\(213\) −35.4078 −2.42610
\(214\) −11.3334 −0.774732
\(215\) 0 0
\(216\) 30.7791 2.09425
\(217\) −3.79460 −0.257594
\(218\) 6.60189 0.447136
\(219\) 11.3454 0.766649
\(220\) 0 0
\(221\) 12.8854 0.866767
\(222\) −13.6906 −0.918851
\(223\) −22.5376 −1.50923 −0.754614 0.656169i \(-0.772176\pi\)
−0.754614 + 0.656169i \(0.772176\pi\)
\(224\) 1.93364 0.129197
\(225\) 0 0
\(226\) −1.83560 −0.122102
\(227\) −18.1124 −1.20216 −0.601080 0.799189i \(-0.705263\pi\)
−0.601080 + 0.799189i \(0.705263\pi\)
\(228\) 0 0
\(229\) −9.41604 −0.622229 −0.311115 0.950372i \(-0.600702\pi\)
−0.311115 + 0.950372i \(0.600702\pi\)
\(230\) 0 0
\(231\) −8.32564 −0.547787
\(232\) −3.43218 −0.225334
\(233\) −15.7000 −1.02854 −0.514271 0.857628i \(-0.671937\pi\)
−0.514271 + 0.857628i \(0.671937\pi\)
\(234\) −33.2191 −2.17160
\(235\) 0 0
\(236\) −5.95455 −0.387608
\(237\) −27.5408 −1.78897
\(238\) −2.10636 −0.136535
\(239\) −23.4610 −1.51757 −0.758783 0.651344i \(-0.774205\pi\)
−0.758783 + 0.651344i \(0.774205\pi\)
\(240\) 0 0
\(241\) 13.1694 0.848314 0.424157 0.905589i \(-0.360570\pi\)
0.424157 + 0.905589i \(0.360570\pi\)
\(242\) 10.8572 0.697927
\(243\) −5.46321 −0.350465
\(244\) 2.89819 0.185538
\(245\) 0 0
\(246\) 30.1333 1.92123
\(247\) 0 0
\(248\) 19.1520 1.21615
\(249\) −6.46544 −0.409730
\(250\) 0 0
\(251\) −17.3251 −1.09355 −0.546776 0.837279i \(-0.684145\pi\)
−0.546776 + 0.837279i \(0.684145\pi\)
\(252\) −2.22736 −0.140310
\(253\) 12.7704 0.802869
\(254\) −2.74563 −0.172276
\(255\) 0 0
\(256\) −12.6659 −0.791618
\(257\) 5.67960 0.354283 0.177142 0.984185i \(-0.443315\pi\)
0.177142 + 0.984185i \(0.443315\pi\)
\(258\) −36.2483 −2.25672
\(259\) 2.29820 0.142803
\(260\) 0 0
\(261\) 7.01621 0.434293
\(262\) −15.3843 −0.950445
\(263\) −5.65764 −0.348865 −0.174433 0.984669i \(-0.555809\pi\)
−0.174433 + 0.984669i \(0.555809\pi\)
\(264\) 42.0209 2.58621
\(265\) 0 0
\(266\) 0 0
\(267\) −24.1707 −1.47922
\(268\) 4.93087 0.301201
\(269\) −23.9918 −1.46280 −0.731402 0.681946i \(-0.761134\pi\)
−0.731402 + 0.681946i \(0.761134\pi\)
\(270\) 0 0
\(271\) −21.2995 −1.29385 −0.646926 0.762553i \(-0.723946\pi\)
−0.646926 + 0.762553i \(0.723946\pi\)
\(272\) 7.25311 0.439785
\(273\) 8.23808 0.498591
\(274\) 15.1645 0.916121
\(275\) 0 0
\(276\) 5.04717 0.303804
\(277\) 0.821109 0.0493357 0.0246678 0.999696i \(-0.492147\pi\)
0.0246678 + 0.999696i \(0.492147\pi\)
\(278\) 12.6340 0.757735
\(279\) −39.1513 −2.34393
\(280\) 0 0
\(281\) −0.587479 −0.0350461 −0.0175230 0.999846i \(-0.505578\pi\)
−0.0175230 + 0.999846i \(0.505578\pi\)
\(282\) −21.3561 −1.27174
\(283\) −30.9424 −1.83933 −0.919667 0.392699i \(-0.871541\pi\)
−0.919667 + 0.392699i \(0.871541\pi\)
\(284\) −6.75969 −0.401114
\(285\) 0 0
\(286\) −23.7052 −1.40172
\(287\) −5.05840 −0.298588
\(288\) 19.9506 1.17560
\(289\) −8.57008 −0.504122
\(290\) 0 0
\(291\) −29.4739 −1.72779
\(292\) 2.16594 0.126752
\(293\) −3.76271 −0.219820 −0.109910 0.993942i \(-0.535056\pi\)
−0.109910 + 0.993942i \(0.535056\pi\)
\(294\) 24.0557 1.40296
\(295\) 0 0
\(296\) −11.5994 −0.674202
\(297\) −44.8998 −2.60535
\(298\) 4.49338 0.260295
\(299\) −12.6361 −0.730765
\(300\) 0 0
\(301\) 6.08491 0.350728
\(302\) −11.3322 −0.652097
\(303\) 2.95961 0.170025
\(304\) 0 0
\(305\) 0 0
\(306\) −21.7327 −1.24237
\(307\) −20.3419 −1.16097 −0.580485 0.814271i \(-0.697137\pi\)
−0.580485 + 0.814271i \(0.697137\pi\)
\(308\) −1.58945 −0.0905670
\(309\) −10.1819 −0.579228
\(310\) 0 0
\(311\) −7.67830 −0.435397 −0.217698 0.976016i \(-0.569855\pi\)
−0.217698 + 0.976016i \(0.569855\pi\)
\(312\) −41.5790 −2.35395
\(313\) −23.9927 −1.35615 −0.678074 0.734993i \(-0.737185\pi\)
−0.678074 + 0.734993i \(0.737185\pi\)
\(314\) 4.11630 0.232296
\(315\) 0 0
\(316\) −5.25781 −0.295775
\(317\) −1.03904 −0.0583580 −0.0291790 0.999574i \(-0.509289\pi\)
−0.0291790 + 0.999574i \(0.509289\pi\)
\(318\) 30.6596 1.71931
\(319\) 5.00678 0.280326
\(320\) 0 0
\(321\) 28.9986 1.61854
\(322\) 2.06561 0.115112
\(323\) 0 0
\(324\) −6.77641 −0.376467
\(325\) 0 0
\(326\) −7.92292 −0.438810
\(327\) −16.8922 −0.934141
\(328\) 25.5306 1.40969
\(329\) 3.58500 0.197647
\(330\) 0 0
\(331\) 30.8316 1.69466 0.847328 0.531069i \(-0.178210\pi\)
0.847328 + 0.531069i \(0.178210\pi\)
\(332\) −1.23431 −0.0677418
\(333\) 23.7120 1.29941
\(334\) 19.5970 1.07230
\(335\) 0 0
\(336\) 4.63716 0.252978
\(337\) −20.7153 −1.12843 −0.564217 0.825627i \(-0.690822\pi\)
−0.564217 + 0.825627i \(0.690822\pi\)
\(338\) 7.97410 0.433734
\(339\) 4.69674 0.255092
\(340\) 0 0
\(341\) −27.9384 −1.51295
\(342\) 0 0
\(343\) −8.30239 −0.448287
\(344\) −30.7116 −1.65586
\(345\) 0 0
\(346\) 27.1318 1.45862
\(347\) −8.22136 −0.441346 −0.220673 0.975348i \(-0.570825\pi\)
−0.220673 + 0.975348i \(0.570825\pi\)
\(348\) 1.97880 0.106075
\(349\) 11.9216 0.638150 0.319075 0.947730i \(-0.396628\pi\)
0.319075 + 0.947730i \(0.396628\pi\)
\(350\) 0 0
\(351\) 44.4276 2.37137
\(352\) 14.2368 0.758823
\(353\) 11.7983 0.627959 0.313980 0.949430i \(-0.398338\pi\)
0.313980 + 0.949430i \(0.398338\pi\)
\(354\) −37.1450 −1.97423
\(355\) 0 0
\(356\) −4.61441 −0.244563
\(357\) 5.38953 0.285244
\(358\) −2.77382 −0.146601
\(359\) 0.110812 0.00584841 0.00292420 0.999996i \(-0.499069\pi\)
0.00292420 + 0.999996i \(0.499069\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −26.6040 −1.39827
\(363\) −27.7802 −1.45808
\(364\) 1.57273 0.0824334
\(365\) 0 0
\(366\) 18.0792 0.945013
\(367\) −11.7397 −0.612808 −0.306404 0.951902i \(-0.599126\pi\)
−0.306404 + 0.951902i \(0.599126\pi\)
\(368\) −7.11278 −0.370779
\(369\) −52.1908 −2.71694
\(370\) 0 0
\(371\) −5.14675 −0.267206
\(372\) −11.0419 −0.572498
\(373\) −14.5190 −0.751763 −0.375882 0.926668i \(-0.622660\pi\)
−0.375882 + 0.926668i \(0.622660\pi\)
\(374\) −15.5085 −0.801924
\(375\) 0 0
\(376\) −18.0941 −0.933131
\(377\) −4.95412 −0.255150
\(378\) −7.26251 −0.373543
\(379\) −6.59023 −0.338518 −0.169259 0.985572i \(-0.554137\pi\)
−0.169259 + 0.985572i \(0.554137\pi\)
\(380\) 0 0
\(381\) 7.02522 0.359913
\(382\) 2.66485 0.136345
\(383\) 2.86921 0.146610 0.0733049 0.997310i \(-0.476645\pi\)
0.0733049 + 0.997310i \(0.476645\pi\)
\(384\) −12.5041 −0.638099
\(385\) 0 0
\(386\) 5.41038 0.275381
\(387\) 62.7819 3.19138
\(388\) −5.62686 −0.285660
\(389\) −6.33149 −0.321019 −0.160510 0.987034i \(-0.551314\pi\)
−0.160510 + 0.987034i \(0.551314\pi\)
\(390\) 0 0
\(391\) −8.26682 −0.418071
\(392\) 20.3813 1.02941
\(393\) 39.3637 1.98563
\(394\) 22.8936 1.15336
\(395\) 0 0
\(396\) −16.3993 −0.824097
\(397\) 30.5498 1.53325 0.766626 0.642094i \(-0.221934\pi\)
0.766626 + 0.642094i \(0.221934\pi\)
\(398\) −7.32522 −0.367180
\(399\) 0 0
\(400\) 0 0
\(401\) 30.3422 1.51522 0.757609 0.652709i \(-0.226367\pi\)
0.757609 + 0.652709i \(0.226367\pi\)
\(402\) 30.7592 1.53413
\(403\) 27.6446 1.37708
\(404\) 0.565018 0.0281107
\(405\) 0 0
\(406\) 0.809843 0.0401919
\(407\) 16.9209 0.838740
\(408\) −27.2019 −1.34669
\(409\) −14.9726 −0.740345 −0.370173 0.928963i \(-0.620701\pi\)
−0.370173 + 0.928963i \(0.620701\pi\)
\(410\) 0 0
\(411\) −38.8013 −1.91393
\(412\) −1.94382 −0.0957653
\(413\) 6.23543 0.306825
\(414\) 21.3122 1.04744
\(415\) 0 0
\(416\) −14.0871 −0.690675
\(417\) −32.3264 −1.58303
\(418\) 0 0
\(419\) −6.17419 −0.301629 −0.150815 0.988562i \(-0.548190\pi\)
−0.150815 + 0.988562i \(0.548190\pi\)
\(420\) 0 0
\(421\) −27.5428 −1.34235 −0.671177 0.741297i \(-0.734211\pi\)
−0.671177 + 0.741297i \(0.734211\pi\)
\(422\) 15.1165 0.735858
\(423\) 36.9887 1.79845
\(424\) 25.9765 1.26153
\(425\) 0 0
\(426\) −42.1675 −2.04302
\(427\) −3.03490 −0.146869
\(428\) 5.53610 0.267598
\(429\) 60.6544 2.92842
\(430\) 0 0
\(431\) −15.0588 −0.725358 −0.362679 0.931914i \(-0.618138\pi\)
−0.362679 + 0.931914i \(0.618138\pi\)
\(432\) 25.0080 1.20320
\(433\) 0.970840 0.0466556 0.0233278 0.999728i \(-0.492574\pi\)
0.0233278 + 0.999728i \(0.492574\pi\)
\(434\) −4.51902 −0.216920
\(435\) 0 0
\(436\) −3.22488 −0.154444
\(437\) 0 0
\(438\) 13.5113 0.645596
\(439\) −27.4375 −1.30952 −0.654760 0.755837i \(-0.727230\pi\)
−0.654760 + 0.755837i \(0.727230\pi\)
\(440\) 0 0
\(441\) −41.6643 −1.98402
\(442\) 15.3454 0.729905
\(443\) 8.76544 0.416459 0.208229 0.978080i \(-0.433230\pi\)
0.208229 + 0.978080i \(0.433230\pi\)
\(444\) 6.68755 0.317377
\(445\) 0 0
\(446\) −26.8402 −1.27092
\(447\) −11.4972 −0.543798
\(448\) 5.34637 0.252592
\(449\) −9.63397 −0.454655 −0.227327 0.973818i \(-0.572999\pi\)
−0.227327 + 0.973818i \(0.572999\pi\)
\(450\) 0 0
\(451\) −37.2434 −1.75372
\(452\) 0.896652 0.0421750
\(453\) 28.9957 1.36234
\(454\) −21.5702 −1.01234
\(455\) 0 0
\(456\) 0 0
\(457\) 10.6708 0.499161 0.249580 0.968354i \(-0.419707\pi\)
0.249580 + 0.968354i \(0.419707\pi\)
\(458\) −11.2137 −0.523980
\(459\) 29.0655 1.35666
\(460\) 0 0
\(461\) 5.68680 0.264861 0.132430 0.991192i \(-0.457722\pi\)
0.132430 + 0.991192i \(0.457722\pi\)
\(462\) −9.91509 −0.461292
\(463\) −35.3550 −1.64309 −0.821543 0.570147i \(-0.806886\pi\)
−0.821543 + 0.570147i \(0.806886\pi\)
\(464\) −2.78864 −0.129459
\(465\) 0 0
\(466\) −18.6973 −0.866135
\(467\) 32.9071 1.52276 0.761380 0.648306i \(-0.224522\pi\)
0.761380 + 0.648306i \(0.224522\pi\)
\(468\) 16.2269 0.750086
\(469\) −5.16347 −0.238427
\(470\) 0 0
\(471\) −10.5323 −0.485305
\(472\) −31.4713 −1.44858
\(473\) 44.8013 2.05997
\(474\) −32.7986 −1.50649
\(475\) 0 0
\(476\) 1.02891 0.0471602
\(477\) −53.1023 −2.43139
\(478\) −27.9399 −1.27794
\(479\) −9.05721 −0.413835 −0.206917 0.978358i \(-0.566343\pi\)
−0.206917 + 0.978358i \(0.566343\pi\)
\(480\) 0 0
\(481\) −16.7430 −0.763414
\(482\) 15.6835 0.714366
\(483\) −5.28525 −0.240487
\(484\) −5.30351 −0.241069
\(485\) 0 0
\(486\) −6.50619 −0.295127
\(487\) 16.5206 0.748620 0.374310 0.927304i \(-0.377880\pi\)
0.374310 + 0.927304i \(0.377880\pi\)
\(488\) 15.3177 0.693399
\(489\) 20.2723 0.916745
\(490\) 0 0
\(491\) −1.39125 −0.0627862 −0.0313931 0.999507i \(-0.509994\pi\)
−0.0313931 + 0.999507i \(0.509994\pi\)
\(492\) −14.7195 −0.663605
\(493\) −3.24109 −0.145972
\(494\) 0 0
\(495\) 0 0
\(496\) 15.5610 0.698708
\(497\) 7.07856 0.317517
\(498\) −7.69975 −0.345034
\(499\) −16.6651 −0.746033 −0.373016 0.927825i \(-0.621676\pi\)
−0.373016 + 0.927825i \(0.621676\pi\)
\(500\) 0 0
\(501\) −50.1427 −2.24021
\(502\) −20.6327 −0.920881
\(503\) −15.6391 −0.697314 −0.348657 0.937250i \(-0.613362\pi\)
−0.348657 + 0.937250i \(0.613362\pi\)
\(504\) −11.7721 −0.524373
\(505\) 0 0
\(506\) 15.2084 0.676096
\(507\) −20.4033 −0.906141
\(508\) 1.34118 0.0595053
\(509\) −19.1540 −0.848988 −0.424494 0.905431i \(-0.639548\pi\)
−0.424494 + 0.905431i \(0.639548\pi\)
\(510\) 0 0
\(511\) −2.26811 −0.100335
\(512\) −23.2910 −1.02933
\(513\) 0 0
\(514\) 6.76389 0.298342
\(515\) 0 0
\(516\) 17.7065 0.779487
\(517\) 26.3952 1.16086
\(518\) 2.73695 0.120255
\(519\) −69.4220 −3.04729
\(520\) 0 0
\(521\) −19.2394 −0.842892 −0.421446 0.906853i \(-0.638477\pi\)
−0.421446 + 0.906853i \(0.638477\pi\)
\(522\) 8.35567 0.365718
\(523\) −6.63945 −0.290323 −0.145161 0.989408i \(-0.546370\pi\)
−0.145161 + 0.989408i \(0.546370\pi\)
\(524\) 7.51490 0.328290
\(525\) 0 0
\(526\) −6.73774 −0.293779
\(527\) 18.0857 0.787825
\(528\) 34.1419 1.48584
\(529\) −14.8931 −0.647528
\(530\) 0 0
\(531\) 64.3349 2.79190
\(532\) 0 0
\(533\) 36.8517 1.59623
\(534\) −28.7851 −1.24565
\(535\) 0 0
\(536\) 26.0609 1.12566
\(537\) 7.09736 0.306274
\(538\) −28.5720 −1.23183
\(539\) −29.7317 −1.28064
\(540\) 0 0
\(541\) 41.7511 1.79502 0.897510 0.440994i \(-0.145374\pi\)
0.897510 + 0.440994i \(0.145374\pi\)
\(542\) −25.3658 −1.08955
\(543\) 68.0714 2.92122
\(544\) −9.21606 −0.395135
\(545\) 0 0
\(546\) 9.81081 0.419864
\(547\) −12.2052 −0.521855 −0.260927 0.965358i \(-0.584028\pi\)
−0.260927 + 0.965358i \(0.584028\pi\)
\(548\) −7.40754 −0.316434
\(549\) −31.3131 −1.33641
\(550\) 0 0
\(551\) 0 0
\(552\) 26.6756 1.13539
\(553\) 5.50583 0.234132
\(554\) 0.977867 0.0415456
\(555\) 0 0
\(556\) −6.17143 −0.261727
\(557\) −35.1548 −1.48956 −0.744779 0.667311i \(-0.767445\pi\)
−0.744779 + 0.667311i \(0.767445\pi\)
\(558\) −46.6257 −1.97382
\(559\) −44.3301 −1.87496
\(560\) 0 0
\(561\) 39.6814 1.67535
\(562\) −0.699634 −0.0295123
\(563\) −17.8406 −0.751891 −0.375945 0.926642i \(-0.622682\pi\)
−0.375945 + 0.926642i \(0.622682\pi\)
\(564\) 10.4320 0.439267
\(565\) 0 0
\(566\) −36.8496 −1.54890
\(567\) 7.09606 0.298007
\(568\) −35.7267 −1.49906
\(569\) 31.6042 1.32492 0.662459 0.749098i \(-0.269513\pi\)
0.662459 + 0.749098i \(0.269513\pi\)
\(570\) 0 0
\(571\) −4.73053 −0.197967 −0.0989833 0.995089i \(-0.531559\pi\)
−0.0989833 + 0.995089i \(0.531559\pi\)
\(572\) 11.5795 0.484164
\(573\) −6.81852 −0.284848
\(574\) −6.02410 −0.251441
\(575\) 0 0
\(576\) 55.1620 2.29841
\(577\) −24.4074 −1.01609 −0.508047 0.861330i \(-0.669632\pi\)
−0.508047 + 0.861330i \(0.669632\pi\)
\(578\) −10.2062 −0.424522
\(579\) −13.8435 −0.575316
\(580\) 0 0
\(581\) 1.29254 0.0536235
\(582\) −35.1008 −1.45497
\(583\) −37.8939 −1.56941
\(584\) 11.4475 0.473703
\(585\) 0 0
\(586\) −4.48104 −0.185110
\(587\) 28.6154 1.18109 0.590543 0.807006i \(-0.298914\pi\)
0.590543 + 0.807006i \(0.298914\pi\)
\(588\) −11.7507 −0.484590
\(589\) 0 0
\(590\) 0 0
\(591\) −58.5776 −2.40956
\(592\) −9.42450 −0.387345
\(593\) −3.71511 −0.152561 −0.0762807 0.997086i \(-0.524305\pi\)
−0.0762807 + 0.997086i \(0.524305\pi\)
\(594\) −53.4716 −2.19397
\(595\) 0 0
\(596\) −2.19492 −0.0899076
\(597\) 18.7430 0.767099
\(598\) −15.0485 −0.615378
\(599\) 7.09940 0.290074 0.145037 0.989426i \(-0.453670\pi\)
0.145037 + 0.989426i \(0.453670\pi\)
\(600\) 0 0
\(601\) −11.0596 −0.451131 −0.225566 0.974228i \(-0.572423\pi\)
−0.225566 + 0.974228i \(0.572423\pi\)
\(602\) 7.24658 0.295349
\(603\) −53.2748 −2.16952
\(604\) 5.53556 0.225239
\(605\) 0 0
\(606\) 3.52463 0.143178
\(607\) −27.1193 −1.10074 −0.550369 0.834921i \(-0.685513\pi\)
−0.550369 + 0.834921i \(0.685513\pi\)
\(608\) 0 0
\(609\) −2.07214 −0.0839673
\(610\) 0 0
\(611\) −26.1176 −1.05660
\(612\) 10.6160 0.429125
\(613\) −41.8312 −1.68954 −0.844772 0.535126i \(-0.820264\pi\)
−0.844772 + 0.535126i \(0.820264\pi\)
\(614\) −24.2253 −0.977654
\(615\) 0 0
\(616\) −8.40062 −0.338471
\(617\) 17.7052 0.712786 0.356393 0.934336i \(-0.384006\pi\)
0.356393 + 0.934336i \(0.384006\pi\)
\(618\) −12.1257 −0.487768
\(619\) −39.8064 −1.59995 −0.799976 0.600032i \(-0.795155\pi\)
−0.799976 + 0.600032i \(0.795155\pi\)
\(620\) 0 0
\(621\) −28.5031 −1.14379
\(622\) −9.14416 −0.366648
\(623\) 4.83208 0.193593
\(624\) −33.7829 −1.35240
\(625\) 0 0
\(626\) −28.5732 −1.14201
\(627\) 0 0
\(628\) −2.01072 −0.0802366
\(629\) −10.9536 −0.436749
\(630\) 0 0
\(631\) 46.1980 1.83911 0.919557 0.392956i \(-0.128547\pi\)
0.919557 + 0.392956i \(0.128547\pi\)
\(632\) −27.7889 −1.10538
\(633\) −38.6784 −1.53733
\(634\) −1.23740 −0.0491433
\(635\) 0 0
\(636\) −14.9766 −0.593860
\(637\) 29.4191 1.16563
\(638\) 5.96262 0.236062
\(639\) 73.0340 2.88918
\(640\) 0 0
\(641\) 6.61348 0.261217 0.130608 0.991434i \(-0.458307\pi\)
0.130608 + 0.991434i \(0.458307\pi\)
\(642\) 34.5347 1.36297
\(643\) 30.6152 1.20735 0.603673 0.797232i \(-0.293703\pi\)
0.603673 + 0.797232i \(0.293703\pi\)
\(644\) −1.00901 −0.0397604
\(645\) 0 0
\(646\) 0 0
\(647\) 11.8979 0.467753 0.233877 0.972266i \(-0.424859\pi\)
0.233877 + 0.972266i \(0.424859\pi\)
\(648\) −35.8150 −1.40695
\(649\) 45.9095 1.80211
\(650\) 0 0
\(651\) 11.5628 0.453182
\(652\) 3.87018 0.151568
\(653\) 1.42899 0.0559207 0.0279604 0.999609i \(-0.491099\pi\)
0.0279604 + 0.999609i \(0.491099\pi\)
\(654\) −20.1171 −0.786640
\(655\) 0 0
\(656\) 20.7436 0.809901
\(657\) −23.4015 −0.912981
\(658\) 4.26941 0.166439
\(659\) −24.4970 −0.954268 −0.477134 0.878831i \(-0.658324\pi\)
−0.477134 + 0.878831i \(0.658324\pi\)
\(660\) 0 0
\(661\) −3.22606 −0.125479 −0.0627396 0.998030i \(-0.519984\pi\)
−0.0627396 + 0.998030i \(0.519984\pi\)
\(662\) 36.7176 1.42707
\(663\) −39.2641 −1.52489
\(664\) −6.52366 −0.253167
\(665\) 0 0
\(666\) 28.2389 1.09423
\(667\) 3.17838 0.123068
\(668\) −9.57273 −0.370380
\(669\) 68.6759 2.65516
\(670\) 0 0
\(671\) −22.3451 −0.862621
\(672\) −5.89213 −0.227294
\(673\) −37.1424 −1.43173 −0.715866 0.698237i \(-0.753968\pi\)
−0.715866 + 0.698237i \(0.753968\pi\)
\(674\) −24.6700 −0.950254
\(675\) 0 0
\(676\) −3.89518 −0.149815
\(677\) 24.7550 0.951412 0.475706 0.879604i \(-0.342193\pi\)
0.475706 + 0.879604i \(0.342193\pi\)
\(678\) 5.59339 0.214813
\(679\) 5.89228 0.226125
\(680\) 0 0
\(681\) 55.1914 2.11494
\(682\) −33.2722 −1.27406
\(683\) −40.1153 −1.53497 −0.767484 0.641068i \(-0.778492\pi\)
−0.767484 + 0.641068i \(0.778492\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −9.88740 −0.377503
\(687\) 28.6923 1.09468
\(688\) −24.9531 −0.951328
\(689\) 37.4954 1.42846
\(690\) 0 0
\(691\) 39.4963 1.50251 0.751254 0.660013i \(-0.229449\pi\)
0.751254 + 0.660013i \(0.229449\pi\)
\(692\) −13.2533 −0.503816
\(693\) 17.1729 0.652344
\(694\) −9.79090 −0.371658
\(695\) 0 0
\(696\) 10.4584 0.396426
\(697\) 24.1092 0.913201
\(698\) 14.1976 0.537386
\(699\) 47.8406 1.80950
\(700\) 0 0
\(701\) 0.0219552 0.000829236 0 0.000414618 1.00000i \(-0.499868\pi\)
0.000414618 1.00000i \(0.499868\pi\)
\(702\) 52.9092 1.99693
\(703\) 0 0
\(704\) 39.3637 1.48357
\(705\) 0 0
\(706\) 14.0507 0.528805
\(707\) −0.591670 −0.0222521
\(708\) 18.1445 0.681913
\(709\) −17.8017 −0.668558 −0.334279 0.942474i \(-0.608493\pi\)
−0.334279 + 0.942474i \(0.608493\pi\)
\(710\) 0 0
\(711\) 56.8071 2.13043
\(712\) −24.3883 −0.913992
\(713\) −17.7358 −0.664210
\(714\) 6.41844 0.240204
\(715\) 0 0
\(716\) 1.35495 0.0506370
\(717\) 71.4896 2.66983
\(718\) 0.131967 0.00492495
\(719\) 18.8103 0.701506 0.350753 0.936468i \(-0.385926\pi\)
0.350753 + 0.936468i \(0.385926\pi\)
\(720\) 0 0
\(721\) 2.03552 0.0758066
\(722\) 0 0
\(723\) −40.1294 −1.49243
\(724\) 12.9955 0.482973
\(725\) 0 0
\(726\) −33.0837 −1.22785
\(727\) −5.00301 −0.185552 −0.0927758 0.995687i \(-0.529574\pi\)
−0.0927758 + 0.995687i \(0.529574\pi\)
\(728\) 8.31227 0.308073
\(729\) −18.2986 −0.677725
\(730\) 0 0
\(731\) −29.0017 −1.07267
\(732\) −8.83129 −0.326414
\(733\) 23.5259 0.868950 0.434475 0.900684i \(-0.356934\pi\)
0.434475 + 0.900684i \(0.356934\pi\)
\(734\) −13.9809 −0.516046
\(735\) 0 0
\(736\) 9.03774 0.333136
\(737\) −38.0170 −1.40037
\(738\) −62.1545 −2.28794
\(739\) −37.3836 −1.37518 −0.687590 0.726099i \(-0.741331\pi\)
−0.687590 + 0.726099i \(0.741331\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −6.12932 −0.225014
\(743\) −10.3886 −0.381121 −0.190560 0.981675i \(-0.561030\pi\)
−0.190560 + 0.981675i \(0.561030\pi\)
\(744\) −58.3594 −2.13956
\(745\) 0 0
\(746\) −17.2908 −0.633060
\(747\) 13.3359 0.487937
\(748\) 7.57556 0.276990
\(749\) −5.79725 −0.211827
\(750\) 0 0
\(751\) −31.2825 −1.14152 −0.570758 0.821119i \(-0.693350\pi\)
−0.570758 + 0.821119i \(0.693350\pi\)
\(752\) −14.7014 −0.536105
\(753\) 52.7927 1.92387
\(754\) −5.89991 −0.214862
\(755\) 0 0
\(756\) 3.54758 0.129024
\(757\) 12.6241 0.458830 0.229415 0.973329i \(-0.426319\pi\)
0.229415 + 0.973329i \(0.426319\pi\)
\(758\) −7.84837 −0.285066
\(759\) −38.9137 −1.41248
\(760\) 0 0
\(761\) 11.5495 0.418668 0.209334 0.977844i \(-0.432870\pi\)
0.209334 + 0.977844i \(0.432870\pi\)
\(762\) 8.36640 0.303083
\(763\) 3.37701 0.122256
\(764\) −1.30172 −0.0470946
\(765\) 0 0
\(766\) 3.41697 0.123460
\(767\) −45.4267 −1.64026
\(768\) 38.5951 1.39268
\(769\) −26.9207 −0.970784 −0.485392 0.874297i \(-0.661323\pi\)
−0.485392 + 0.874297i \(0.661323\pi\)
\(770\) 0 0
\(771\) −17.3067 −0.623286
\(772\) −2.64286 −0.0951184
\(773\) −21.3331 −0.767299 −0.383649 0.923479i \(-0.625333\pi\)
−0.383649 + 0.923479i \(0.625333\pi\)
\(774\) 74.7676 2.68747
\(775\) 0 0
\(776\) −29.7394 −1.06758
\(777\) −7.00301 −0.251232
\(778\) −7.54024 −0.270331
\(779\) 0 0
\(780\) 0 0
\(781\) 52.1172 1.86490
\(782\) −9.84503 −0.352058
\(783\) −11.1749 −0.399360
\(784\) 16.5598 0.591421
\(785\) 0 0
\(786\) 46.8786 1.67210
\(787\) −3.52489 −0.125649 −0.0628243 0.998025i \(-0.520011\pi\)
−0.0628243 + 0.998025i \(0.520011\pi\)
\(788\) −11.1830 −0.398379
\(789\) 17.2398 0.613753
\(790\) 0 0
\(791\) −0.938948 −0.0333852
\(792\) −86.6746 −3.07985
\(793\) 22.1100 0.785151
\(794\) 36.3821 1.29115
\(795\) 0 0
\(796\) 3.57821 0.126826
\(797\) 39.0084 1.38175 0.690875 0.722974i \(-0.257226\pi\)
0.690875 + 0.722974i \(0.257226\pi\)
\(798\) 0 0
\(799\) −17.0867 −0.604484
\(800\) 0 0
\(801\) 49.8557 1.76156
\(802\) 36.1348 1.27597
\(803\) −16.6994 −0.589309
\(804\) −15.0252 −0.529899
\(805\) 0 0
\(806\) 32.9222 1.15964
\(807\) 73.1071 2.57349
\(808\) 2.98626 0.105056
\(809\) 50.7196 1.78321 0.891604 0.452816i \(-0.149581\pi\)
0.891604 + 0.452816i \(0.149581\pi\)
\(810\) 0 0
\(811\) 29.0376 1.01965 0.509824 0.860279i \(-0.329711\pi\)
0.509824 + 0.860279i \(0.329711\pi\)
\(812\) −0.395591 −0.0138825
\(813\) 64.9032 2.27625
\(814\) 20.1513 0.706303
\(815\) 0 0
\(816\) −22.1015 −0.773706
\(817\) 0 0
\(818\) −17.8310 −0.623445
\(819\) −16.9923 −0.593759
\(820\) 0 0
\(821\) 32.9878 1.15128 0.575640 0.817703i \(-0.304753\pi\)
0.575640 + 0.817703i \(0.304753\pi\)
\(822\) −46.2088 −1.61172
\(823\) 26.5533 0.925591 0.462796 0.886465i \(-0.346846\pi\)
0.462796 + 0.886465i \(0.346846\pi\)
\(824\) −10.2736 −0.357898
\(825\) 0 0
\(826\) 7.42583 0.258378
\(827\) 32.7666 1.13941 0.569703 0.821851i \(-0.307058\pi\)
0.569703 + 0.821851i \(0.307058\pi\)
\(828\) −10.4106 −0.361792
\(829\) 32.4548 1.12720 0.563601 0.826047i \(-0.309416\pi\)
0.563601 + 0.826047i \(0.309416\pi\)
\(830\) 0 0
\(831\) −2.50206 −0.0867955
\(832\) −38.9497 −1.35034
\(833\) 19.2466 0.666854
\(834\) −38.4979 −1.33307
\(835\) 0 0
\(836\) 0 0
\(837\) 62.3576 2.15539
\(838\) −7.35291 −0.254002
\(839\) −35.2098 −1.21558 −0.607788 0.794099i \(-0.707943\pi\)
−0.607788 + 0.794099i \(0.707943\pi\)
\(840\) 0 0
\(841\) −27.7539 −0.957030
\(842\) −32.8010 −1.13040
\(843\) 1.79015 0.0616560
\(844\) −7.38408 −0.254171
\(845\) 0 0
\(846\) 44.0502 1.51448
\(847\) 5.55368 0.190827
\(848\) 21.1059 0.724779
\(849\) 94.2867 3.23591
\(850\) 0 0
\(851\) 10.7417 0.368220
\(852\) 20.5979 0.705674
\(853\) 1.78825 0.0612286 0.0306143 0.999531i \(-0.490254\pi\)
0.0306143 + 0.999531i \(0.490254\pi\)
\(854\) −3.61430 −0.123679
\(855\) 0 0
\(856\) 29.2597 1.00008
\(857\) −3.61379 −0.123445 −0.0617224 0.998093i \(-0.519659\pi\)
−0.0617224 + 0.998093i \(0.519659\pi\)
\(858\) 72.2339 2.46603
\(859\) −25.3648 −0.865437 −0.432719 0.901529i \(-0.642446\pi\)
−0.432719 + 0.901529i \(0.642446\pi\)
\(860\) 0 0
\(861\) 15.4138 0.525301
\(862\) −17.9337 −0.610824
\(863\) 29.9878 1.02080 0.510398 0.859939i \(-0.329498\pi\)
0.510398 + 0.859939i \(0.329498\pi\)
\(864\) −31.7760 −1.08104
\(865\) 0 0
\(866\) 1.15618 0.0392887
\(867\) 26.1145 0.886895
\(868\) 2.20745 0.0749257
\(869\) 40.5377 1.37515
\(870\) 0 0
\(871\) 37.6172 1.27461
\(872\) −17.0443 −0.577194
\(873\) 60.7945 2.05758
\(874\) 0 0
\(875\) 0 0
\(876\) −6.59999 −0.222993
\(877\) −10.6478 −0.359550 −0.179775 0.983708i \(-0.557537\pi\)
−0.179775 + 0.983708i \(0.557537\pi\)
\(878\) −32.6756 −1.10275
\(879\) 11.4656 0.386726
\(880\) 0 0
\(881\) −31.5797 −1.06395 −0.531973 0.846761i \(-0.678549\pi\)
−0.531973 + 0.846761i \(0.678549\pi\)
\(882\) −49.6185 −1.67074
\(883\) −17.0174 −0.572682 −0.286341 0.958128i \(-0.592439\pi\)
−0.286341 + 0.958128i \(0.592439\pi\)
\(884\) −7.49589 −0.252114
\(885\) 0 0
\(886\) 10.4388 0.350700
\(887\) 13.2290 0.444187 0.222093 0.975025i \(-0.428711\pi\)
0.222093 + 0.975025i \(0.428711\pi\)
\(888\) 35.3454 1.18611
\(889\) −1.40445 −0.0471036
\(890\) 0 0
\(891\) 52.2461 1.75031
\(892\) 13.1109 0.438985
\(893\) 0 0
\(894\) −13.6921 −0.457933
\(895\) 0 0
\(896\) 2.49977 0.0835113
\(897\) 38.5044 1.28562
\(898\) −11.4732 −0.382865
\(899\) −6.95350 −0.231912
\(900\) 0 0
\(901\) 24.5303 0.817222
\(902\) −44.3536 −1.47681
\(903\) −18.5418 −0.617031
\(904\) 4.73903 0.157618
\(905\) 0 0
\(906\) 34.5313 1.14723
\(907\) 14.8904 0.494428 0.247214 0.968961i \(-0.420485\pi\)
0.247214 + 0.968961i \(0.420485\pi\)
\(908\) 10.5366 0.349669
\(909\) −6.10464 −0.202478
\(910\) 0 0
\(911\) −49.5480 −1.64160 −0.820800 0.571216i \(-0.806472\pi\)
−0.820800 + 0.571216i \(0.806472\pi\)
\(912\) 0 0
\(913\) 9.51655 0.314952
\(914\) 12.7080 0.420343
\(915\) 0 0
\(916\) 5.47763 0.180986
\(917\) −7.86939 −0.259870
\(918\) 34.6143 1.14244
\(919\) 27.3835 0.903300 0.451650 0.892195i \(-0.350836\pi\)
0.451650 + 0.892195i \(0.350836\pi\)
\(920\) 0 0
\(921\) 61.9851 2.04248
\(922\) 6.77247 0.223039
\(923\) −51.5691 −1.69742
\(924\) 4.84331 0.159333
\(925\) 0 0
\(926\) −42.1046 −1.38364
\(927\) 21.0017 0.689787
\(928\) 3.54334 0.116316
\(929\) 52.1894 1.71228 0.856139 0.516745i \(-0.172857\pi\)
0.856139 + 0.516745i \(0.172857\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 9.13323 0.299169
\(933\) 23.3971 0.765987
\(934\) 39.1894 1.28232
\(935\) 0 0
\(936\) 85.7630 2.80325
\(937\) −15.5798 −0.508968 −0.254484 0.967077i \(-0.581906\pi\)
−0.254484 + 0.967077i \(0.581906\pi\)
\(938\) −6.14922 −0.200779
\(939\) 73.1099 2.38585
\(940\) 0 0
\(941\) −58.6553 −1.91211 −0.956055 0.293189i \(-0.905284\pi\)
−0.956055 + 0.293189i \(0.905284\pi\)
\(942\) −12.5431 −0.408675
\(943\) −23.6427 −0.769913
\(944\) −25.5704 −0.832244
\(945\) 0 0
\(946\) 53.3543 1.73470
\(947\) −12.3065 −0.399908 −0.199954 0.979805i \(-0.564079\pi\)
−0.199954 + 0.979805i \(0.564079\pi\)
\(948\) 16.0214 0.520352
\(949\) 16.5238 0.536384
\(950\) 0 0
\(951\) 3.16612 0.102668
\(952\) 5.43807 0.176249
\(953\) −8.26997 −0.267891 −0.133945 0.990989i \(-0.542765\pi\)
−0.133945 + 0.990989i \(0.542765\pi\)
\(954\) −63.2401 −2.04747
\(955\) 0 0
\(956\) 13.6481 0.441410
\(957\) −15.2565 −0.493173
\(958\) −10.7863 −0.348490
\(959\) 7.75696 0.250485
\(960\) 0 0
\(961\) 7.80138 0.251657
\(962\) −19.9394 −0.642871
\(963\) −59.8139 −1.92748
\(964\) −7.66108 −0.246747
\(965\) 0 0
\(966\) −6.29426 −0.202514
\(967\) −14.2247 −0.457436 −0.228718 0.973493i \(-0.573453\pi\)
−0.228718 + 0.973493i \(0.573453\pi\)
\(968\) −28.0304 −0.900931
\(969\) 0 0
\(970\) 0 0
\(971\) 9.73861 0.312527 0.156263 0.987715i \(-0.450055\pi\)
0.156263 + 0.987715i \(0.450055\pi\)
\(972\) 3.17814 0.101939
\(973\) 6.46254 0.207180
\(974\) 19.6745 0.630413
\(975\) 0 0
\(976\) 12.4456 0.398374
\(977\) 6.35308 0.203253 0.101627 0.994823i \(-0.467595\pi\)
0.101627 + 0.994823i \(0.467595\pi\)
\(978\) 24.1425 0.771991
\(979\) 35.5771 1.13705
\(980\) 0 0
\(981\) 34.8427 1.11244
\(982\) −1.65685 −0.0528723
\(983\) −9.88087 −0.315151 −0.157575 0.987507i \(-0.550368\pi\)
−0.157575 + 0.987507i \(0.550368\pi\)
\(984\) −77.7962 −2.48005
\(985\) 0 0
\(986\) −3.85985 −0.122923
\(987\) −10.9241 −0.347718
\(988\) 0 0
\(989\) 28.4406 0.904358
\(990\) 0 0
\(991\) 37.3350 1.18599 0.592993 0.805208i \(-0.297946\pi\)
0.592993 + 0.805208i \(0.297946\pi\)
\(992\) −19.7723 −0.627771
\(993\) −93.9491 −2.98138
\(994\) 8.42992 0.267381
\(995\) 0 0
\(996\) 3.76117 0.119177
\(997\) 43.6948 1.38383 0.691914 0.721980i \(-0.256768\pi\)
0.691914 + 0.721980i \(0.256768\pi\)
\(998\) −19.8466 −0.628234
\(999\) −37.7669 −1.19489
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 9025.2.a.bg.1.3 4
5.4 even 2 1805.2.a.o.1.2 4
19.7 even 3 475.2.e.e.201.2 8
19.11 even 3 475.2.e.e.26.2 8
19.18 odd 2 9025.2.a.bp.1.2 4
95.7 odd 12 475.2.j.c.49.3 16
95.49 even 6 95.2.e.c.26.3 yes 8
95.64 even 6 95.2.e.c.11.3 8
95.68 odd 12 475.2.j.c.349.3 16
95.83 odd 12 475.2.j.c.49.6 16
95.87 odd 12 475.2.j.c.349.6 16
95.94 odd 2 1805.2.a.i.1.3 4
285.239 odd 6 855.2.k.h.406.2 8
285.254 odd 6 855.2.k.h.676.2 8
380.159 odd 6 1520.2.q.o.961.4 8
380.239 odd 6 1520.2.q.o.881.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.c.11.3 8 95.64 even 6
95.2.e.c.26.3 yes 8 95.49 even 6
475.2.e.e.26.2 8 19.11 even 3
475.2.e.e.201.2 8 19.7 even 3
475.2.j.c.49.3 16 95.7 odd 12
475.2.j.c.49.6 16 95.83 odd 12
475.2.j.c.349.3 16 95.68 odd 12
475.2.j.c.349.6 16 95.87 odd 12
855.2.k.h.406.2 8 285.239 odd 6
855.2.k.h.676.2 8 285.254 odd 6
1520.2.q.o.881.4 8 380.239 odd 6
1520.2.q.o.961.4 8 380.159 odd 6
1805.2.a.i.1.3 4 95.94 odd 2
1805.2.a.o.1.2 4 5.4 even 2
9025.2.a.bg.1.3 4 1.1 even 1 trivial
9025.2.a.bp.1.2 4 19.18 odd 2