Properties

Label 315.4.a.g.1.1
Level $315$
Weight $4$
Character 315.1
Self dual yes
Analytic conductor $18.586$
Analytic rank $1$
Dimension $2$
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] = [315,4,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(18.5856016518\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.70156 q^{2} +14.1047 q^{4} +5.00000 q^{5} +7.00000 q^{7} -28.7016 q^{8} +O(q^{10})\) \(q-4.70156 q^{2} +14.1047 q^{4} +5.00000 q^{5} +7.00000 q^{7} -28.7016 q^{8} -23.5078 q^{10} -24.5969 q^{11} -35.0156 q^{13} -32.9109 q^{14} +22.1047 q^{16} +18.4187 q^{17} -67.4031 q^{19} +70.5234 q^{20} +115.644 q^{22} +145.675 q^{23} +25.0000 q^{25} +164.628 q^{26} +98.7328 q^{28} -214.419 q^{29} -88.6594 q^{31} +125.686 q^{32} -86.5969 q^{34} +35.0000 q^{35} +162.125 q^{37} +316.900 q^{38} -143.508 q^{40} +337.769 q^{41} +122.156 q^{43} -346.931 q^{44} -684.900 q^{46} -354.219 q^{47} +49.0000 q^{49} -117.539 q^{50} -493.884 q^{52} -676.691 q^{53} -122.984 q^{55} -200.911 q^{56} +1008.10 q^{58} -501.319 q^{59} -708.931 q^{61} +416.837 q^{62} -767.758 q^{64} -175.078 q^{65} -907.956 q^{67} +259.791 q^{68} -164.555 q^{70} -430.334 q^{71} +41.3406 q^{73} -762.241 q^{74} -950.700 q^{76} -172.178 q^{77} +890.388 q^{79} +110.523 q^{80} -1588.04 q^{82} +1057.15 q^{83} +92.0937 q^{85} -574.325 q^{86} +705.969 q^{88} -1473.72 q^{89} -245.109 q^{91} +2054.70 q^{92} +1665.38 q^{94} -337.016 q^{95} +555.034 q^{97} -230.377 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 9 q^{4} + 10 q^{5} + 14 q^{7} - 51 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 9 q^{4} + 10 q^{5} + 14 q^{7} - 51 q^{8} - 15 q^{10} - 62 q^{11} - 6 q^{13} - 21 q^{14} + 25 q^{16} - 40 q^{17} - 122 q^{19} + 45 q^{20} + 52 q^{22} - 16 q^{23} + 50 q^{25} + 214 q^{26} + 63 q^{28} - 352 q^{29} + 66 q^{31} + 309 q^{32} - 186 q^{34} + 70 q^{35} - 188 q^{37} + 224 q^{38} - 255 q^{40} - 16 q^{41} - 396 q^{43} - 156 q^{44} - 960 q^{46} + 188 q^{47} + 98 q^{49} - 75 q^{50} - 642 q^{52} - 982 q^{53} - 310 q^{55} - 357 q^{56} + 774 q^{58} - 516 q^{59} - 880 q^{61} + 680 q^{62} - 479 q^{64} - 30 q^{65} - 356 q^{67} + 558 q^{68} - 105 q^{70} - 310 q^{71} + 326 q^{73} - 1358 q^{74} - 672 q^{76} - 434 q^{77} + 1832 q^{79} + 125 q^{80} - 2190 q^{82} + 680 q^{83} - 200 q^{85} - 1456 q^{86} + 1540 q^{88} - 796 q^{89} - 42 q^{91} + 2880 q^{92} + 2588 q^{94} - 610 q^{95} - 670 q^{97} - 147 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.70156 −1.66225 −0.831127 0.556083i \(-0.812304\pi\)
−0.831127 + 0.556083i \(0.812304\pi\)
\(3\) 0 0
\(4\) 14.1047 1.76309
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −28.7016 −1.26844
\(9\) 0 0
\(10\) −23.5078 −0.743382
\(11\) −24.5969 −0.674203 −0.337102 0.941468i \(-0.609447\pi\)
−0.337102 + 0.941468i \(0.609447\pi\)
\(12\) 0 0
\(13\) −35.0156 −0.747045 −0.373523 0.927621i \(-0.621850\pi\)
−0.373523 + 0.927621i \(0.621850\pi\)
\(14\) −32.9109 −0.628273
\(15\) 0 0
\(16\) 22.1047 0.345386
\(17\) 18.4187 0.262777 0.131388 0.991331i \(-0.458057\pi\)
0.131388 + 0.991331i \(0.458057\pi\)
\(18\) 0 0
\(19\) −67.4031 −0.813860 −0.406930 0.913459i \(-0.633401\pi\)
−0.406930 + 0.913459i \(0.633401\pi\)
\(20\) 70.5234 0.788476
\(21\) 0 0
\(22\) 115.644 1.12070
\(23\) 145.675 1.32067 0.660333 0.750973i \(-0.270415\pi\)
0.660333 + 0.750973i \(0.270415\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 164.628 1.24178
\(27\) 0 0
\(28\) 98.7328 0.666384
\(29\) −214.419 −1.37298 −0.686492 0.727137i \(-0.740851\pi\)
−0.686492 + 0.727137i \(0.740851\pi\)
\(30\) 0 0
\(31\) −88.6594 −0.513667 −0.256834 0.966456i \(-0.582679\pi\)
−0.256834 + 0.966456i \(0.582679\pi\)
\(32\) 125.686 0.694323
\(33\) 0 0
\(34\) −86.5969 −0.436801
\(35\) 35.0000 0.169031
\(36\) 0 0
\(37\) 162.125 0.720356 0.360178 0.932884i \(-0.382716\pi\)
0.360178 + 0.932884i \(0.382716\pi\)
\(38\) 316.900 1.35284
\(39\) 0 0
\(40\) −143.508 −0.567264
\(41\) 337.769 1.28660 0.643300 0.765614i \(-0.277565\pi\)
0.643300 + 0.765614i \(0.277565\pi\)
\(42\) 0 0
\(43\) 122.156 0.433224 0.216612 0.976258i \(-0.430499\pi\)
0.216612 + 0.976258i \(0.430499\pi\)
\(44\) −346.931 −1.18868
\(45\) 0 0
\(46\) −684.900 −2.19528
\(47\) −354.219 −1.09932 −0.549661 0.835388i \(-0.685243\pi\)
−0.549661 + 0.835388i \(0.685243\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) −117.539 −0.332451
\(51\) 0 0
\(52\) −493.884 −1.31710
\(53\) −676.691 −1.75378 −0.876892 0.480687i \(-0.840387\pi\)
−0.876892 + 0.480687i \(0.840387\pi\)
\(54\) 0 0
\(55\) −122.984 −0.301513
\(56\) −200.911 −0.479426
\(57\) 0 0
\(58\) 1008.10 2.28225
\(59\) −501.319 −1.10621 −0.553103 0.833113i \(-0.686556\pi\)
−0.553103 + 0.833113i \(0.686556\pi\)
\(60\) 0 0
\(61\) −708.931 −1.48802 −0.744011 0.668167i \(-0.767079\pi\)
−0.744011 + 0.668167i \(0.767079\pi\)
\(62\) 416.837 0.853845
\(63\) 0 0
\(64\) −767.758 −1.49953
\(65\) −175.078 −0.334089
\(66\) 0 0
\(67\) −907.956 −1.65559 −0.827795 0.561031i \(-0.810405\pi\)
−0.827795 + 0.561031i \(0.810405\pi\)
\(68\) 259.791 0.463298
\(69\) 0 0
\(70\) −164.555 −0.280972
\(71\) −430.334 −0.719314 −0.359657 0.933085i \(-0.617106\pi\)
−0.359657 + 0.933085i \(0.617106\pi\)
\(72\) 0 0
\(73\) 41.3406 0.0662816 0.0331408 0.999451i \(-0.489449\pi\)
0.0331408 + 0.999451i \(0.489449\pi\)
\(74\) −762.241 −1.19741
\(75\) 0 0
\(76\) −950.700 −1.43490
\(77\) −172.178 −0.254825
\(78\) 0 0
\(79\) 890.388 1.26806 0.634028 0.773310i \(-0.281400\pi\)
0.634028 + 0.773310i \(0.281400\pi\)
\(80\) 110.523 0.154461
\(81\) 0 0
\(82\) −1588.04 −2.13866
\(83\) 1057.15 1.39804 0.699020 0.715102i \(-0.253620\pi\)
0.699020 + 0.715102i \(0.253620\pi\)
\(84\) 0 0
\(85\) 92.0937 0.117517
\(86\) −574.325 −0.720129
\(87\) 0 0
\(88\) 705.969 0.855188
\(89\) −1473.72 −1.75522 −0.877610 0.479376i \(-0.840863\pi\)
−0.877610 + 0.479376i \(0.840863\pi\)
\(90\) 0 0
\(91\) −245.109 −0.282356
\(92\) 2054.70 2.32845
\(93\) 0 0
\(94\) 1665.38 1.82735
\(95\) −337.016 −0.363969
\(96\) 0 0
\(97\) 555.034 0.580981 0.290491 0.956878i \(-0.406181\pi\)
0.290491 + 0.956878i \(0.406181\pi\)
\(98\) −230.377 −0.237465
\(99\) 0 0
\(100\) 352.617 0.352617
\(101\) −1890.14 −1.86214 −0.931071 0.364838i \(-0.881124\pi\)
−0.931071 + 0.364838i \(0.881124\pi\)
\(102\) 0 0
\(103\) 662.700 0.633959 0.316979 0.948432i \(-0.397331\pi\)
0.316979 + 0.948432i \(0.397331\pi\)
\(104\) 1005.00 0.947583
\(105\) 0 0
\(106\) 3181.50 2.91523
\(107\) −1614.53 −1.45872 −0.729358 0.684132i \(-0.760181\pi\)
−0.729358 + 0.684132i \(0.760181\pi\)
\(108\) 0 0
\(109\) 217.206 0.190868 0.0954339 0.995436i \(-0.469576\pi\)
0.0954339 + 0.995436i \(0.469576\pi\)
\(110\) 578.219 0.501191
\(111\) 0 0
\(112\) 154.733 0.130544
\(113\) 1658.20 1.38044 0.690221 0.723598i \(-0.257513\pi\)
0.690221 + 0.723598i \(0.257513\pi\)
\(114\) 0 0
\(115\) 728.375 0.590620
\(116\) −3024.31 −2.42069
\(117\) 0 0
\(118\) 2356.98 1.83879
\(119\) 128.931 0.0993202
\(120\) 0 0
\(121\) −725.994 −0.545450
\(122\) 3333.08 2.47347
\(123\) 0 0
\(124\) −1250.51 −0.905640
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1108.81 −0.774734 −0.387367 0.921926i \(-0.626615\pi\)
−0.387367 + 0.921926i \(0.626615\pi\)
\(128\) 2604.17 1.79827
\(129\) 0 0
\(130\) 823.141 0.555340
\(131\) −185.488 −0.123711 −0.0618554 0.998085i \(-0.519702\pi\)
−0.0618554 + 0.998085i \(0.519702\pi\)
\(132\) 0 0
\(133\) −471.822 −0.307610
\(134\) 4268.81 2.75201
\(135\) 0 0
\(136\) −528.647 −0.333317
\(137\) 37.9907 0.0236917 0.0118458 0.999930i \(-0.496229\pi\)
0.0118458 + 0.999930i \(0.496229\pi\)
\(138\) 0 0
\(139\) 183.609 0.112040 0.0560199 0.998430i \(-0.482159\pi\)
0.0560199 + 0.998430i \(0.482159\pi\)
\(140\) 493.664 0.298016
\(141\) 0 0
\(142\) 2023.24 1.19568
\(143\) 861.275 0.503660
\(144\) 0 0
\(145\) −1072.09 −0.614018
\(146\) −194.366 −0.110177
\(147\) 0 0
\(148\) 2286.72 1.27005
\(149\) 1383.34 0.760587 0.380293 0.924866i \(-0.375823\pi\)
0.380293 + 0.924866i \(0.375823\pi\)
\(150\) 0 0
\(151\) 765.256 0.412422 0.206211 0.978508i \(-0.433887\pi\)
0.206211 + 0.978508i \(0.433887\pi\)
\(152\) 1934.57 1.03233
\(153\) 0 0
\(154\) 809.506 0.423584
\(155\) −443.297 −0.229719
\(156\) 0 0
\(157\) −2366.76 −1.20311 −0.601554 0.798832i \(-0.705452\pi\)
−0.601554 + 0.798832i \(0.705452\pi\)
\(158\) −4186.21 −2.10783
\(159\) 0 0
\(160\) 628.430 0.310511
\(161\) 1019.72 0.499165
\(162\) 0 0
\(163\) −3137.69 −1.50775 −0.753875 0.657018i \(-0.771817\pi\)
−0.753875 + 0.657018i \(0.771817\pi\)
\(164\) 4764.12 2.26839
\(165\) 0 0
\(166\) −4970.26 −2.32390
\(167\) −146.469 −0.0678688 −0.0339344 0.999424i \(-0.510804\pi\)
−0.0339344 + 0.999424i \(0.510804\pi\)
\(168\) 0 0
\(169\) −970.906 −0.441924
\(170\) −432.984 −0.195343
\(171\) 0 0
\(172\) 1722.98 0.763812
\(173\) 1424.12 0.625860 0.312930 0.949776i \(-0.398689\pi\)
0.312930 + 0.949776i \(0.398689\pi\)
\(174\) 0 0
\(175\) 175.000 0.0755929
\(176\) −543.706 −0.232860
\(177\) 0 0
\(178\) 6928.81 2.91762
\(179\) −1244.70 −0.519737 −0.259869 0.965644i \(-0.583679\pi\)
−0.259869 + 0.965644i \(0.583679\pi\)
\(180\) 0 0
\(181\) −3879.09 −1.59299 −0.796493 0.604648i \(-0.793314\pi\)
−0.796493 + 0.604648i \(0.793314\pi\)
\(182\) 1152.40 0.469348
\(183\) 0 0
\(184\) −4181.10 −1.67519
\(185\) 810.625 0.322153
\(186\) 0 0
\(187\) −453.044 −0.177165
\(188\) −4996.14 −1.93820
\(189\) 0 0
\(190\) 1584.50 0.605009
\(191\) −1574.90 −0.596628 −0.298314 0.954468i \(-0.596424\pi\)
−0.298314 + 0.954468i \(0.596424\pi\)
\(192\) 0 0
\(193\) −4775.67 −1.78114 −0.890572 0.454843i \(-0.849695\pi\)
−0.890572 + 0.454843i \(0.849695\pi\)
\(194\) −2609.53 −0.965738
\(195\) 0 0
\(196\) 691.130 0.251869
\(197\) 2803.58 1.01394 0.506971 0.861963i \(-0.330765\pi\)
0.506971 + 0.861963i \(0.330765\pi\)
\(198\) 0 0
\(199\) 4102.92 1.46155 0.730774 0.682620i \(-0.239159\pi\)
0.730774 + 0.682620i \(0.239159\pi\)
\(200\) −717.539 −0.253688
\(201\) 0 0
\(202\) 8886.63 3.09535
\(203\) −1500.93 −0.518940
\(204\) 0 0
\(205\) 1688.84 0.575385
\(206\) −3115.72 −1.05380
\(207\) 0 0
\(208\) −774.009 −0.258019
\(209\) 1657.91 0.548707
\(210\) 0 0
\(211\) −823.512 −0.268687 −0.134343 0.990935i \(-0.542893\pi\)
−0.134343 + 0.990935i \(0.542893\pi\)
\(212\) −9544.51 −3.09207
\(213\) 0 0
\(214\) 7590.82 2.42476
\(215\) 610.781 0.193744
\(216\) 0 0
\(217\) −620.616 −0.194148
\(218\) −1021.21 −0.317271
\(219\) 0 0
\(220\) −1734.66 −0.531593
\(221\) −644.944 −0.196306
\(222\) 0 0
\(223\) 817.194 0.245396 0.122698 0.992444i \(-0.460845\pi\)
0.122698 + 0.992444i \(0.460845\pi\)
\(224\) 879.802 0.262430
\(225\) 0 0
\(226\) −7796.12 −2.29465
\(227\) −3655.85 −1.06893 −0.534465 0.845190i \(-0.679487\pi\)
−0.534465 + 0.845190i \(0.679487\pi\)
\(228\) 0 0
\(229\) 939.393 0.271078 0.135539 0.990772i \(-0.456723\pi\)
0.135539 + 0.990772i \(0.456723\pi\)
\(230\) −3424.50 −0.981760
\(231\) 0 0
\(232\) 6154.15 1.74155
\(233\) 7.64701 0.00215010 0.00107505 0.999999i \(-0.499658\pi\)
0.00107505 + 0.999999i \(0.499658\pi\)
\(234\) 0 0
\(235\) −1771.09 −0.491631
\(236\) −7070.94 −1.95034
\(237\) 0 0
\(238\) −606.178 −0.165095
\(239\) 889.115 0.240636 0.120318 0.992735i \(-0.461609\pi\)
0.120318 + 0.992735i \(0.461609\pi\)
\(240\) 0 0
\(241\) 2140.23 0.572051 0.286026 0.958222i \(-0.407666\pi\)
0.286026 + 0.958222i \(0.407666\pi\)
\(242\) 3413.30 0.906676
\(243\) 0 0
\(244\) −9999.25 −2.62351
\(245\) 245.000 0.0638877
\(246\) 0 0
\(247\) 2360.16 0.607990
\(248\) 2544.66 0.651557
\(249\) 0 0
\(250\) −587.695 −0.148676
\(251\) 6749.81 1.69739 0.848693 0.528886i \(-0.177390\pi\)
0.848693 + 0.528886i \(0.177390\pi\)
\(252\) 0 0
\(253\) −3583.15 −0.890398
\(254\) 5213.15 1.28780
\(255\) 0 0
\(256\) −6101.62 −1.48965
\(257\) −3068.64 −0.744811 −0.372405 0.928070i \(-0.621467\pi\)
−0.372405 + 0.928070i \(0.621467\pi\)
\(258\) 0 0
\(259\) 1134.87 0.272269
\(260\) −2469.42 −0.589027
\(261\) 0 0
\(262\) 872.081 0.205639
\(263\) 4674.12 1.09589 0.547944 0.836515i \(-0.315411\pi\)
0.547944 + 0.836515i \(0.315411\pi\)
\(264\) 0 0
\(265\) −3383.45 −0.784316
\(266\) 2218.30 0.511326
\(267\) 0 0
\(268\) −12806.4 −2.91895
\(269\) −2417.38 −0.547919 −0.273960 0.961741i \(-0.588333\pi\)
−0.273960 + 0.961741i \(0.588333\pi\)
\(270\) 0 0
\(271\) 7724.30 1.73143 0.865715 0.500537i \(-0.166864\pi\)
0.865715 + 0.500537i \(0.166864\pi\)
\(272\) 407.141 0.0907593
\(273\) 0 0
\(274\) −178.616 −0.0393816
\(275\) −614.922 −0.134841
\(276\) 0 0
\(277\) −4576.17 −0.992620 −0.496310 0.868145i \(-0.665312\pi\)
−0.496310 + 0.868145i \(0.665312\pi\)
\(278\) −863.250 −0.186239
\(279\) 0 0
\(280\) −1004.55 −0.214406
\(281\) 1358.56 0.288415 0.144208 0.989547i \(-0.453937\pi\)
0.144208 + 0.989547i \(0.453937\pi\)
\(282\) 0 0
\(283\) 3885.04 0.816048 0.408024 0.912971i \(-0.366218\pi\)
0.408024 + 0.912971i \(0.366218\pi\)
\(284\) −6069.73 −1.26821
\(285\) 0 0
\(286\) −4049.34 −0.837211
\(287\) 2364.38 0.486289
\(288\) 0 0
\(289\) −4573.75 −0.930948
\(290\) 5040.52 1.02065
\(291\) 0 0
\(292\) 583.097 0.116860
\(293\) 4033.91 0.804312 0.402156 0.915571i \(-0.368261\pi\)
0.402156 + 0.915571i \(0.368261\pi\)
\(294\) 0 0
\(295\) −2506.59 −0.494710
\(296\) −4653.24 −0.913730
\(297\) 0 0
\(298\) −6503.85 −1.26429
\(299\) −5100.90 −0.986598
\(300\) 0 0
\(301\) 855.093 0.163743
\(302\) −3597.90 −0.685549
\(303\) 0 0
\(304\) −1489.92 −0.281096
\(305\) −3544.66 −0.665464
\(306\) 0 0
\(307\) −4620.36 −0.858950 −0.429475 0.903079i \(-0.641301\pi\)
−0.429475 + 0.903079i \(0.641301\pi\)
\(308\) −2428.52 −0.449278
\(309\) 0 0
\(310\) 2084.19 0.381851
\(311\) −6675.89 −1.21722 −0.608609 0.793470i \(-0.708272\pi\)
−0.608609 + 0.793470i \(0.708272\pi\)
\(312\) 0 0
\(313\) 2836.78 0.512283 0.256141 0.966639i \(-0.417549\pi\)
0.256141 + 0.966639i \(0.417549\pi\)
\(314\) 11127.5 1.99987
\(315\) 0 0
\(316\) 12558.6 2.23569
\(317\) −4010.63 −0.710597 −0.355299 0.934753i \(-0.615621\pi\)
−0.355299 + 0.934753i \(0.615621\pi\)
\(318\) 0 0
\(319\) 5274.03 0.925671
\(320\) −3838.79 −0.670609
\(321\) 0 0
\(322\) −4794.30 −0.829739
\(323\) −1241.48 −0.213863
\(324\) 0 0
\(325\) −875.391 −0.149409
\(326\) 14752.1 2.50626
\(327\) 0 0
\(328\) −9694.49 −1.63198
\(329\) −2479.53 −0.415504
\(330\) 0 0
\(331\) 11087.5 1.84117 0.920583 0.390546i \(-0.127714\pi\)
0.920583 + 0.390546i \(0.127714\pi\)
\(332\) 14910.8 2.46486
\(333\) 0 0
\(334\) 688.631 0.112815
\(335\) −4539.78 −0.740402
\(336\) 0 0
\(337\) 12118.7 1.95890 0.979450 0.201689i \(-0.0646431\pi\)
0.979450 + 0.201689i \(0.0646431\pi\)
\(338\) 4564.78 0.734589
\(339\) 0 0
\(340\) 1298.95 0.207193
\(341\) 2180.74 0.346316
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) −3506.07 −0.549520
\(345\) 0 0
\(346\) −6695.58 −1.04034
\(347\) 6361.22 0.984116 0.492058 0.870562i \(-0.336245\pi\)
0.492058 + 0.870562i \(0.336245\pi\)
\(348\) 0 0
\(349\) −3115.18 −0.477799 −0.238899 0.971044i \(-0.576787\pi\)
−0.238899 + 0.971044i \(0.576787\pi\)
\(350\) −822.773 −0.125655
\(351\) 0 0
\(352\) −3091.48 −0.468115
\(353\) 11927.4 1.79839 0.899194 0.437550i \(-0.144154\pi\)
0.899194 + 0.437550i \(0.144154\pi\)
\(354\) 0 0
\(355\) −2151.67 −0.321687
\(356\) −20786.4 −3.09460
\(357\) 0 0
\(358\) 5852.02 0.863935
\(359\) 6143.95 0.903245 0.451623 0.892209i \(-0.350845\pi\)
0.451623 + 0.892209i \(0.350845\pi\)
\(360\) 0 0
\(361\) −2315.82 −0.337632
\(362\) 18237.8 2.64794
\(363\) 0 0
\(364\) −3457.19 −0.497819
\(365\) 206.703 0.0296420
\(366\) 0 0
\(367\) −1927.67 −0.274178 −0.137089 0.990559i \(-0.543775\pi\)
−0.137089 + 0.990559i \(0.543775\pi\)
\(368\) 3220.10 0.456139
\(369\) 0 0
\(370\) −3811.20 −0.535500
\(371\) −4736.83 −0.662868
\(372\) 0 0
\(373\) 10452.0 1.45090 0.725449 0.688276i \(-0.241632\pi\)
0.725449 + 0.688276i \(0.241632\pi\)
\(374\) 2130.01 0.294493
\(375\) 0 0
\(376\) 10166.6 1.39443
\(377\) 7508.01 1.02568
\(378\) 0 0
\(379\) 7066.43 0.957726 0.478863 0.877890i \(-0.341049\pi\)
0.478863 + 0.877890i \(0.341049\pi\)
\(380\) −4753.50 −0.641709
\(381\) 0 0
\(382\) 7404.51 0.991747
\(383\) −7168.04 −0.956318 −0.478159 0.878273i \(-0.658696\pi\)
−0.478159 + 0.878273i \(0.658696\pi\)
\(384\) 0 0
\(385\) −860.891 −0.113961
\(386\) 22453.1 2.96071
\(387\) 0 0
\(388\) 7828.58 1.02432
\(389\) 7414.06 0.966344 0.483172 0.875525i \(-0.339485\pi\)
0.483172 + 0.875525i \(0.339485\pi\)
\(390\) 0 0
\(391\) 2683.15 0.347040
\(392\) −1406.38 −0.181206
\(393\) 0 0
\(394\) −13181.2 −1.68543
\(395\) 4451.94 0.567092
\(396\) 0 0
\(397\) −8936.01 −1.12969 −0.564843 0.825198i \(-0.691063\pi\)
−0.564843 + 0.825198i \(0.691063\pi\)
\(398\) −19290.1 −2.42946
\(399\) 0 0
\(400\) 552.617 0.0690771
\(401\) −1782.91 −0.222031 −0.111015 0.993819i \(-0.535410\pi\)
−0.111015 + 0.993819i \(0.535410\pi\)
\(402\) 0 0
\(403\) 3104.46 0.383733
\(404\) −26659.9 −3.28312
\(405\) 0 0
\(406\) 7056.72 0.862609
\(407\) −3987.77 −0.485667
\(408\) 0 0
\(409\) −8759.92 −1.05905 −0.529524 0.848295i \(-0.677629\pi\)
−0.529524 + 0.848295i \(0.677629\pi\)
\(410\) −7940.20 −0.956436
\(411\) 0 0
\(412\) 9347.17 1.11772
\(413\) −3509.23 −0.418106
\(414\) 0 0
\(415\) 5285.75 0.625222
\(416\) −4400.97 −0.518691
\(417\) 0 0
\(418\) −7794.75 −0.912090
\(419\) 3212.74 0.374588 0.187294 0.982304i \(-0.440028\pi\)
0.187294 + 0.982304i \(0.440028\pi\)
\(420\) 0 0
\(421\) 15757.8 1.82420 0.912101 0.409965i \(-0.134459\pi\)
0.912101 + 0.409965i \(0.134459\pi\)
\(422\) 3871.79 0.446626
\(423\) 0 0
\(424\) 19422.1 2.22457
\(425\) 460.469 0.0525553
\(426\) 0 0
\(427\) −4962.52 −0.562419
\(428\) −22772.5 −2.57184
\(429\) 0 0
\(430\) −2871.63 −0.322051
\(431\) 405.917 0.0453650 0.0226825 0.999743i \(-0.492779\pi\)
0.0226825 + 0.999743i \(0.492779\pi\)
\(432\) 0 0
\(433\) −7845.25 −0.870713 −0.435357 0.900258i \(-0.643378\pi\)
−0.435357 + 0.900258i \(0.643378\pi\)
\(434\) 2917.86 0.322723
\(435\) 0 0
\(436\) 3063.63 0.336516
\(437\) −9818.95 −1.07484
\(438\) 0 0
\(439\) 423.029 0.0459911 0.0229955 0.999736i \(-0.492680\pi\)
0.0229955 + 0.999736i \(0.492680\pi\)
\(440\) 3529.84 0.382452
\(441\) 0 0
\(442\) 3032.24 0.326310
\(443\) 16058.7 1.72229 0.861143 0.508362i \(-0.169749\pi\)
0.861143 + 0.508362i \(0.169749\pi\)
\(444\) 0 0
\(445\) −7368.62 −0.784958
\(446\) −3842.09 −0.407911
\(447\) 0 0
\(448\) −5374.30 −0.566768
\(449\) −2186.75 −0.229842 −0.114921 0.993375i \(-0.536662\pi\)
−0.114921 + 0.993375i \(0.536662\pi\)
\(450\) 0 0
\(451\) −8308.05 −0.867430
\(452\) 23388.3 2.43384
\(453\) 0 0
\(454\) 17188.2 1.77683
\(455\) −1225.55 −0.126274
\(456\) 0 0
\(457\) −5799.22 −0.593602 −0.296801 0.954939i \(-0.595920\pi\)
−0.296801 + 0.954939i \(0.595920\pi\)
\(458\) −4416.62 −0.450600
\(459\) 0 0
\(460\) 10273.5 1.04131
\(461\) −9873.35 −0.997500 −0.498750 0.866746i \(-0.666207\pi\)
−0.498750 + 0.866746i \(0.666207\pi\)
\(462\) 0 0
\(463\) −6181.84 −0.620506 −0.310253 0.950654i \(-0.600414\pi\)
−0.310253 + 0.950654i \(0.600414\pi\)
\(464\) −4739.66 −0.474209
\(465\) 0 0
\(466\) −35.9529 −0.00357400
\(467\) −6145.50 −0.608950 −0.304475 0.952520i \(-0.598481\pi\)
−0.304475 + 0.952520i \(0.598481\pi\)
\(468\) 0 0
\(469\) −6355.69 −0.625754
\(470\) 8326.91 0.817216
\(471\) 0 0
\(472\) 14388.6 1.40316
\(473\) −3004.66 −0.292081
\(474\) 0 0
\(475\) −1685.08 −0.162772
\(476\) 1818.53 0.175110
\(477\) 0 0
\(478\) −4180.23 −0.399999
\(479\) −10879.4 −1.03777 −0.518887 0.854843i \(-0.673653\pi\)
−0.518887 + 0.854843i \(0.673653\pi\)
\(480\) 0 0
\(481\) −5676.91 −0.538139
\(482\) −10062.4 −0.950894
\(483\) 0 0
\(484\) −10239.9 −0.961675
\(485\) 2775.17 0.259823
\(486\) 0 0
\(487\) 8087.51 0.752526 0.376263 0.926513i \(-0.377209\pi\)
0.376263 + 0.926513i \(0.377209\pi\)
\(488\) 20347.4 1.88747
\(489\) 0 0
\(490\) −1151.88 −0.106197
\(491\) 6959.90 0.639707 0.319853 0.947467i \(-0.396366\pi\)
0.319853 + 0.947467i \(0.396366\pi\)
\(492\) 0 0
\(493\) −3949.32 −0.360788
\(494\) −11096.4 −1.01063
\(495\) 0 0
\(496\) −1959.79 −0.177413
\(497\) −3012.34 −0.271875
\(498\) 0 0
\(499\) 18632.0 1.67151 0.835756 0.549101i \(-0.185030\pi\)
0.835756 + 0.549101i \(0.185030\pi\)
\(500\) 1763.09 0.157695
\(501\) 0 0
\(502\) −31734.6 −2.82149
\(503\) −4627.62 −0.410209 −0.205105 0.978740i \(-0.565753\pi\)
−0.205105 + 0.978740i \(0.565753\pi\)
\(504\) 0 0
\(505\) −9450.72 −0.832775
\(506\) 16846.4 1.48007
\(507\) 0 0
\(508\) −15639.4 −1.36592
\(509\) 11351.8 0.988528 0.494264 0.869312i \(-0.335438\pi\)
0.494264 + 0.869312i \(0.335438\pi\)
\(510\) 0 0
\(511\) 289.384 0.0250521
\(512\) 7853.76 0.677911
\(513\) 0 0
\(514\) 14427.4 1.23806
\(515\) 3313.50 0.283515
\(516\) 0 0
\(517\) 8712.67 0.741166
\(518\) −5335.68 −0.452580
\(519\) 0 0
\(520\) 5025.02 0.423772
\(521\) −19096.1 −1.60579 −0.802893 0.596123i \(-0.796707\pi\)
−0.802893 + 0.596123i \(0.796707\pi\)
\(522\) 0 0
\(523\) −3145.11 −0.262956 −0.131478 0.991319i \(-0.541972\pi\)
−0.131478 + 0.991319i \(0.541972\pi\)
\(524\) −2616.24 −0.218113
\(525\) 0 0
\(526\) −21975.7 −1.82164
\(527\) −1632.99 −0.134980
\(528\) 0 0
\(529\) 9054.20 0.744160
\(530\) 15907.5 1.30373
\(531\) 0 0
\(532\) −6654.90 −0.542343
\(533\) −11827.2 −0.961148
\(534\) 0 0
\(535\) −8072.66 −0.652358
\(536\) 26059.8 2.10002
\(537\) 0 0
\(538\) 11365.5 0.910781
\(539\) −1205.25 −0.0963148
\(540\) 0 0
\(541\) 8776.12 0.697440 0.348720 0.937227i \(-0.386616\pi\)
0.348720 + 0.937227i \(0.386616\pi\)
\(542\) −36316.3 −2.87808
\(543\) 0 0
\(544\) 2314.98 0.182452
\(545\) 1086.03 0.0853587
\(546\) 0 0
\(547\) −13695.1 −1.07049 −0.535247 0.844696i \(-0.679781\pi\)
−0.535247 + 0.844696i \(0.679781\pi\)
\(548\) 535.847 0.0417705
\(549\) 0 0
\(550\) 2891.09 0.224139
\(551\) 14452.5 1.11742
\(552\) 0 0
\(553\) 6232.71 0.479280
\(554\) 21515.2 1.64999
\(555\) 0 0
\(556\) 2589.75 0.197536
\(557\) −7850.44 −0.597188 −0.298594 0.954380i \(-0.596518\pi\)
−0.298594 + 0.954380i \(0.596518\pi\)
\(558\) 0 0
\(559\) −4277.38 −0.323638
\(560\) 773.664 0.0583808
\(561\) 0 0
\(562\) −6387.33 −0.479419
\(563\) 4948.81 0.370457 0.185229 0.982695i \(-0.440697\pi\)
0.185229 + 0.982695i \(0.440697\pi\)
\(564\) 0 0
\(565\) 8290.98 0.617353
\(566\) −18265.7 −1.35648
\(567\) 0 0
\(568\) 12351.3 0.912408
\(569\) 8115.76 0.597945 0.298972 0.954262i \(-0.403356\pi\)
0.298972 + 0.954262i \(0.403356\pi\)
\(570\) 0 0
\(571\) 5656.42 0.414560 0.207280 0.978282i \(-0.433539\pi\)
0.207280 + 0.978282i \(0.433539\pi\)
\(572\) 12148.0 0.887996
\(573\) 0 0
\(574\) −11116.3 −0.808336
\(575\) 3641.87 0.264133
\(576\) 0 0
\(577\) −9536.77 −0.688078 −0.344039 0.938955i \(-0.611795\pi\)
−0.344039 + 0.938955i \(0.611795\pi\)
\(578\) 21503.8 1.54747
\(579\) 0 0
\(580\) −15121.5 −1.08257
\(581\) 7400.05 0.528409
\(582\) 0 0
\(583\) 16644.5 1.18241
\(584\) −1186.54 −0.0840743
\(585\) 0 0
\(586\) −18965.7 −1.33697
\(587\) 13089.6 0.920383 0.460191 0.887820i \(-0.347781\pi\)
0.460191 + 0.887820i \(0.347781\pi\)
\(588\) 0 0
\(589\) 5975.92 0.418053
\(590\) 11784.9 0.822334
\(591\) 0 0
\(592\) 3583.72 0.248801
\(593\) −4281.96 −0.296524 −0.148262 0.988948i \(-0.547368\pi\)
−0.148262 + 0.988948i \(0.547368\pi\)
\(594\) 0 0
\(595\) 644.656 0.0444173
\(596\) 19511.5 1.34098
\(597\) 0 0
\(598\) 23982.2 1.63997
\(599\) −3699.92 −0.252378 −0.126189 0.992006i \(-0.540275\pi\)
−0.126189 + 0.992006i \(0.540275\pi\)
\(600\) 0 0
\(601\) −17286.1 −1.17323 −0.586616 0.809865i \(-0.699540\pi\)
−0.586616 + 0.809865i \(0.699540\pi\)
\(602\) −4020.28 −0.272183
\(603\) 0 0
\(604\) 10793.7 0.727135
\(605\) −3629.97 −0.243933
\(606\) 0 0
\(607\) 14456.7 0.966689 0.483344 0.875430i \(-0.339422\pi\)
0.483344 + 0.875430i \(0.339422\pi\)
\(608\) −8471.63 −0.565082
\(609\) 0 0
\(610\) 16665.4 1.10617
\(611\) 12403.2 0.821243
\(612\) 0 0
\(613\) 17981.9 1.18480 0.592400 0.805644i \(-0.298181\pi\)
0.592400 + 0.805644i \(0.298181\pi\)
\(614\) 21722.9 1.42779
\(615\) 0 0
\(616\) 4941.78 0.323231
\(617\) −19614.7 −1.27983 −0.639916 0.768445i \(-0.721031\pi\)
−0.639916 + 0.768445i \(0.721031\pi\)
\(618\) 0 0
\(619\) −10462.9 −0.679385 −0.339692 0.940537i \(-0.610323\pi\)
−0.339692 + 0.940537i \(0.610323\pi\)
\(620\) −6252.56 −0.405014
\(621\) 0 0
\(622\) 31387.1 2.02332
\(623\) −10316.1 −0.663411
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −13337.3 −0.851544
\(627\) 0 0
\(628\) −33382.4 −2.12118
\(629\) 2986.14 0.189293
\(630\) 0 0
\(631\) 24481.9 1.54454 0.772272 0.635292i \(-0.219120\pi\)
0.772272 + 0.635292i \(0.219120\pi\)
\(632\) −25555.5 −1.60846
\(633\) 0 0
\(634\) 18856.2 1.18119
\(635\) −5544.06 −0.346471
\(636\) 0 0
\(637\) −1715.77 −0.106721
\(638\) −24796.2 −1.53870
\(639\) 0 0
\(640\) 13020.9 0.804211
\(641\) 1109.39 0.0683595 0.0341797 0.999416i \(-0.489118\pi\)
0.0341797 + 0.999416i \(0.489118\pi\)
\(642\) 0 0
\(643\) 30112.5 1.84684 0.923422 0.383787i \(-0.125380\pi\)
0.923422 + 0.383787i \(0.125380\pi\)
\(644\) 14382.9 0.880071
\(645\) 0 0
\(646\) 5836.90 0.355495
\(647\) 4260.27 0.258869 0.129435 0.991588i \(-0.458684\pi\)
0.129435 + 0.991588i \(0.458684\pi\)
\(648\) 0 0
\(649\) 12330.9 0.745808
\(650\) 4115.70 0.248356
\(651\) 0 0
\(652\) −44256.2 −2.65829
\(653\) 10576.8 0.633844 0.316922 0.948452i \(-0.397351\pi\)
0.316922 + 0.948452i \(0.397351\pi\)
\(654\) 0 0
\(655\) −927.438 −0.0553252
\(656\) 7466.27 0.444373
\(657\) 0 0
\(658\) 11657.7 0.690674
\(659\) −3394.70 −0.200666 −0.100333 0.994954i \(-0.531991\pi\)
−0.100333 + 0.994954i \(0.531991\pi\)
\(660\) 0 0
\(661\) −33174.4 −1.95210 −0.976048 0.217554i \(-0.930192\pi\)
−0.976048 + 0.217554i \(0.930192\pi\)
\(662\) −52128.7 −3.06048
\(663\) 0 0
\(664\) −30341.9 −1.77333
\(665\) −2359.11 −0.137567
\(666\) 0 0
\(667\) −31235.4 −1.81326
\(668\) −2065.89 −0.119658
\(669\) 0 0
\(670\) 21344.1 1.23074
\(671\) 17437.5 1.00323
\(672\) 0 0
\(673\) 753.881 0.0431797 0.0215899 0.999767i \(-0.493127\pi\)
0.0215899 + 0.999767i \(0.493127\pi\)
\(674\) −56976.9 −3.25619
\(675\) 0 0
\(676\) −13694.3 −0.779149
\(677\) −15668.8 −0.889511 −0.444756 0.895652i \(-0.646709\pi\)
−0.444756 + 0.895652i \(0.646709\pi\)
\(678\) 0 0
\(679\) 3885.24 0.219590
\(680\) −2643.23 −0.149064
\(681\) 0 0
\(682\) −10252.9 −0.575665
\(683\) 11557.4 0.647485 0.323742 0.946145i \(-0.395059\pi\)
0.323742 + 0.946145i \(0.395059\pi\)
\(684\) 0 0
\(685\) 189.953 0.0105952
\(686\) −1612.64 −0.0897532
\(687\) 0 0
\(688\) 2700.22 0.149630
\(689\) 23694.7 1.31016
\(690\) 0 0
\(691\) −18503.1 −1.01866 −0.509328 0.860572i \(-0.670106\pi\)
−0.509328 + 0.860572i \(0.670106\pi\)
\(692\) 20086.8 1.10344
\(693\) 0 0
\(694\) −29907.7 −1.63585
\(695\) 918.046 0.0501057
\(696\) 0 0
\(697\) 6221.28 0.338088
\(698\) 14646.2 0.794223
\(699\) 0 0
\(700\) 2468.32 0.133277
\(701\) −22580.4 −1.21662 −0.608311 0.793699i \(-0.708153\pi\)
−0.608311 + 0.793699i \(0.708153\pi\)
\(702\) 0 0
\(703\) −10927.7 −0.586269
\(704\) 18884.4 1.01099
\(705\) 0 0
\(706\) −56077.4 −2.98938
\(707\) −13231.0 −0.703823
\(708\) 0 0
\(709\) −27426.6 −1.45279 −0.726394 0.687278i \(-0.758805\pi\)
−0.726394 + 0.687278i \(0.758805\pi\)
\(710\) 10116.2 0.534725
\(711\) 0 0
\(712\) 42298.2 2.22639
\(713\) −12915.5 −0.678383
\(714\) 0 0
\(715\) 4306.37 0.225244
\(716\) −17556.1 −0.916342
\(717\) 0 0
\(718\) −28886.1 −1.50142
\(719\) −19383.0 −1.00538 −0.502688 0.864468i \(-0.667656\pi\)
−0.502688 + 0.864468i \(0.667656\pi\)
\(720\) 0 0
\(721\) 4638.90 0.239614
\(722\) 10888.0 0.561230
\(723\) 0 0
\(724\) −54713.3 −2.80857
\(725\) −5360.47 −0.274597
\(726\) 0 0
\(727\) −12317.3 −0.628368 −0.314184 0.949362i \(-0.601731\pi\)
−0.314184 + 0.949362i \(0.601731\pi\)
\(728\) 7035.02 0.358153
\(729\) 0 0
\(730\) −971.828 −0.0492726
\(731\) 2249.96 0.113841
\(732\) 0 0
\(733\) 1234.02 0.0621822 0.0310911 0.999517i \(-0.490102\pi\)
0.0310911 + 0.999517i \(0.490102\pi\)
\(734\) 9063.05 0.455754
\(735\) 0 0
\(736\) 18309.3 0.916970
\(737\) 22332.9 1.11620
\(738\) 0 0
\(739\) −15257.3 −0.759473 −0.379736 0.925095i \(-0.623985\pi\)
−0.379736 + 0.925095i \(0.623985\pi\)
\(740\) 11433.6 0.567984
\(741\) 0 0
\(742\) 22270.5 1.10186
\(743\) 35565.1 1.75606 0.878032 0.478602i \(-0.158856\pi\)
0.878032 + 0.478602i \(0.158856\pi\)
\(744\) 0 0
\(745\) 6916.69 0.340145
\(746\) −49140.8 −2.41176
\(747\) 0 0
\(748\) −6390.04 −0.312357
\(749\) −11301.7 −0.551343
\(750\) 0 0
\(751\) 14266.7 0.693209 0.346605 0.938011i \(-0.387335\pi\)
0.346605 + 0.938011i \(0.387335\pi\)
\(752\) −7829.89 −0.379690
\(753\) 0 0
\(754\) −35299.4 −1.70494
\(755\) 3826.28 0.184441
\(756\) 0 0
\(757\) −15927.9 −0.764744 −0.382372 0.924009i \(-0.624893\pi\)
−0.382372 + 0.924009i \(0.624893\pi\)
\(758\) −33223.3 −1.59198
\(759\) 0 0
\(760\) 9672.87 0.461674
\(761\) 2566.48 0.122253 0.0611266 0.998130i \(-0.480531\pi\)
0.0611266 + 0.998130i \(0.480531\pi\)
\(762\) 0 0
\(763\) 1520.44 0.0721413
\(764\) −22213.5 −1.05191
\(765\) 0 0
\(766\) 33701.0 1.58964
\(767\) 17554.0 0.826386
\(768\) 0 0
\(769\) 14433.1 0.676816 0.338408 0.940999i \(-0.390112\pi\)
0.338408 + 0.940999i \(0.390112\pi\)
\(770\) 4047.53 0.189432
\(771\) 0 0
\(772\) −67359.4 −3.14031
\(773\) 29443.2 1.36999 0.684993 0.728550i \(-0.259805\pi\)
0.684993 + 0.728550i \(0.259805\pi\)
\(774\) 0 0
\(775\) −2216.48 −0.102733
\(776\) −15930.4 −0.736941
\(777\) 0 0
\(778\) −34857.7 −1.60631
\(779\) −22766.7 −1.04711
\(780\) 0 0
\(781\) 10584.9 0.484964
\(782\) −12615.0 −0.576869
\(783\) 0 0
\(784\) 1083.13 0.0493408
\(785\) −11833.8 −0.538046
\(786\) 0 0
\(787\) 26390.6 1.19533 0.597664 0.801747i \(-0.296096\pi\)
0.597664 + 0.801747i \(0.296096\pi\)
\(788\) 39543.6 1.78767
\(789\) 0 0
\(790\) −20931.1 −0.942650
\(791\) 11607.4 0.521758
\(792\) 0 0
\(793\) 24823.7 1.11162
\(794\) 42013.2 1.87783
\(795\) 0 0
\(796\) 57870.3 2.57683
\(797\) 3738.33 0.166146 0.0830730 0.996543i \(-0.473527\pi\)
0.0830730 + 0.996543i \(0.473527\pi\)
\(798\) 0 0
\(799\) −6524.26 −0.288876
\(800\) 3142.15 0.138865
\(801\) 0 0
\(802\) 8382.48 0.369072
\(803\) −1016.85 −0.0446873
\(804\) 0 0
\(805\) 5098.62 0.223233
\(806\) −14595.8 −0.637861
\(807\) 0 0
\(808\) 54250.1 2.36202
\(809\) −43204.1 −1.87760 −0.938798 0.344468i \(-0.888059\pi\)
−0.938798 + 0.344468i \(0.888059\pi\)
\(810\) 0 0
\(811\) −30192.4 −1.30727 −0.653637 0.756809i \(-0.726758\pi\)
−0.653637 + 0.756809i \(0.726758\pi\)
\(812\) −21170.2 −0.914935
\(813\) 0 0
\(814\) 18748.7 0.807301
\(815\) −15688.5 −0.674286
\(816\) 0 0
\(817\) −8233.71 −0.352584
\(818\) 41185.3 1.76041
\(819\) 0 0
\(820\) 23820.6 1.01445
\(821\) 40274.7 1.71206 0.856028 0.516929i \(-0.172925\pi\)
0.856028 + 0.516929i \(0.172925\pi\)
\(822\) 0 0
\(823\) 25184.2 1.06667 0.533334 0.845905i \(-0.320939\pi\)
0.533334 + 0.845905i \(0.320939\pi\)
\(824\) −19020.5 −0.804140
\(825\) 0 0
\(826\) 16498.9 0.694999
\(827\) 38941.7 1.63741 0.818703 0.574218i \(-0.194694\pi\)
0.818703 + 0.574218i \(0.194694\pi\)
\(828\) 0 0
\(829\) −8327.05 −0.348867 −0.174433 0.984669i \(-0.555809\pi\)
−0.174433 + 0.984669i \(0.555809\pi\)
\(830\) −24851.3 −1.03928
\(831\) 0 0
\(832\) 26883.5 1.12021
\(833\) 902.519 0.0375395
\(834\) 0 0
\(835\) −732.343 −0.0303518
\(836\) 23384.2 0.967418
\(837\) 0 0
\(838\) −15104.9 −0.622660
\(839\) −8784.41 −0.361468 −0.180734 0.983532i \(-0.557847\pi\)
−0.180734 + 0.983532i \(0.557847\pi\)
\(840\) 0 0
\(841\) 21586.4 0.885087
\(842\) −74086.4 −3.03229
\(843\) 0 0
\(844\) −11615.4 −0.473718
\(845\) −4854.53 −0.197634
\(846\) 0 0
\(847\) −5081.96 −0.206161
\(848\) −14958.0 −0.605732
\(849\) 0 0
\(850\) −2164.92 −0.0873602
\(851\) 23617.6 0.951350
\(852\) 0 0
\(853\) −9076.15 −0.364316 −0.182158 0.983269i \(-0.558308\pi\)
−0.182158 + 0.983269i \(0.558308\pi\)
\(854\) 23331.6 0.934884
\(855\) 0 0
\(856\) 46339.6 1.85030
\(857\) −36396.7 −1.45074 −0.725372 0.688357i \(-0.758332\pi\)
−0.725372 + 0.688357i \(0.758332\pi\)
\(858\) 0 0
\(859\) 8915.27 0.354115 0.177058 0.984200i \(-0.443342\pi\)
0.177058 + 0.984200i \(0.443342\pi\)
\(860\) 8614.88 0.341587
\(861\) 0 0
\(862\) −1908.44 −0.0754081
\(863\) 6148.26 0.242514 0.121257 0.992621i \(-0.461308\pi\)
0.121257 + 0.992621i \(0.461308\pi\)
\(864\) 0 0
\(865\) 7120.59 0.279893
\(866\) 36884.9 1.44735
\(867\) 0 0
\(868\) −8753.59 −0.342300
\(869\) −21900.8 −0.854928
\(870\) 0 0
\(871\) 31792.6 1.23680
\(872\) −6234.16 −0.242105
\(873\) 0 0
\(874\) 46164.4 1.78665
\(875\) 875.000 0.0338062
\(876\) 0 0
\(877\) −14287.0 −0.550101 −0.275050 0.961430i \(-0.588694\pi\)
−0.275050 + 0.961430i \(0.588694\pi\)
\(878\) −1988.90 −0.0764488
\(879\) 0 0
\(880\) −2718.53 −0.104138
\(881\) 13315.9 0.509221 0.254610 0.967044i \(-0.418053\pi\)
0.254610 + 0.967044i \(0.418053\pi\)
\(882\) 0 0
\(883\) −5271.78 −0.200917 −0.100458 0.994941i \(-0.532031\pi\)
−0.100458 + 0.994941i \(0.532031\pi\)
\(884\) −9096.73 −0.346104
\(885\) 0 0
\(886\) −75501.1 −2.86288
\(887\) −2606.07 −0.0986507 −0.0493253 0.998783i \(-0.515707\pi\)
−0.0493253 + 0.998783i \(0.515707\pi\)
\(888\) 0 0
\(889\) −7761.69 −0.292822
\(890\) 34644.0 1.30480
\(891\) 0 0
\(892\) 11526.3 0.432654
\(893\) 23875.4 0.894694
\(894\) 0 0
\(895\) −6223.48 −0.232434
\(896\) 18229.2 0.679682
\(897\) 0 0
\(898\) 10281.1 0.382056
\(899\) 19010.2 0.705258
\(900\) 0 0
\(901\) −12463.8 −0.460854
\(902\) 39060.8 1.44189
\(903\) 0 0
\(904\) −47592.8 −1.75101
\(905\) −19395.4 −0.712405
\(906\) 0 0
\(907\) 18610.6 0.681317 0.340659 0.940187i \(-0.389350\pi\)
0.340659 + 0.940187i \(0.389350\pi\)
\(908\) −51564.6 −1.88462
\(909\) 0 0
\(910\) 5761.98 0.209899
\(911\) −41091.7 −1.49443 −0.747216 0.664581i \(-0.768610\pi\)
−0.747216 + 0.664581i \(0.768610\pi\)
\(912\) 0 0
\(913\) −26002.6 −0.942563
\(914\) 27265.4 0.986716
\(915\) 0 0
\(916\) 13249.8 0.477934
\(917\) −1298.41 −0.0467583
\(918\) 0 0
\(919\) 38891.3 1.39598 0.697990 0.716107i \(-0.254078\pi\)
0.697990 + 0.716107i \(0.254078\pi\)
\(920\) −20905.5 −0.749167
\(921\) 0 0
\(922\) 46420.2 1.65810
\(923\) 15068.4 0.537360
\(924\) 0 0
\(925\) 4053.12 0.144071
\(926\) 29064.3 1.03144
\(927\) 0 0
\(928\) −26949.4 −0.953295
\(929\) −18699.4 −0.660396 −0.330198 0.943912i \(-0.607115\pi\)
−0.330198 + 0.943912i \(0.607115\pi\)
\(930\) 0 0
\(931\) −3302.75 −0.116266
\(932\) 107.859 0.00379080
\(933\) 0 0
\(934\) 28893.4 1.01223
\(935\) −2265.22 −0.0792305
\(936\) 0 0
\(937\) 21509.6 0.749933 0.374967 0.927038i \(-0.377654\pi\)
0.374967 + 0.927038i \(0.377654\pi\)
\(938\) 29881.7 1.04016
\(939\) 0 0
\(940\) −24980.7 −0.866788
\(941\) 11241.7 0.389448 0.194724 0.980858i \(-0.437619\pi\)
0.194724 + 0.980858i \(0.437619\pi\)
\(942\) 0 0
\(943\) 49204.5 1.69917
\(944\) −11081.5 −0.382068
\(945\) 0 0
\(946\) 14126.6 0.485513
\(947\) 36556.3 1.25441 0.627203 0.778856i \(-0.284200\pi\)
0.627203 + 0.778856i \(0.284200\pi\)
\(948\) 0 0
\(949\) −1447.57 −0.0495153
\(950\) 7922.50 0.270568
\(951\) 0 0
\(952\) −3700.53 −0.125982
\(953\) 36633.4 1.24520 0.622598 0.782542i \(-0.286077\pi\)
0.622598 + 0.782542i \(0.286077\pi\)
\(954\) 0 0
\(955\) −7874.52 −0.266820
\(956\) 12540.7 0.424263
\(957\) 0 0
\(958\) 51150.3 1.72504
\(959\) 265.935 0.00895462
\(960\) 0 0
\(961\) −21930.5 −0.736146
\(962\) 26690.3 0.894523
\(963\) 0 0
\(964\) 30187.3 1.00858
\(965\) −23878.4 −0.796551
\(966\) 0 0
\(967\) −35515.8 −1.18109 −0.590544 0.807006i \(-0.701087\pi\)
−0.590544 + 0.807006i \(0.701087\pi\)
\(968\) 20837.2 0.691871
\(969\) 0 0
\(970\) −13047.6 −0.431891
\(971\) 39661.0 1.31080 0.655398 0.755283i \(-0.272501\pi\)
0.655398 + 0.755283i \(0.272501\pi\)
\(972\) 0 0
\(973\) 1285.26 0.0423471
\(974\) −38023.9 −1.25089
\(975\) 0 0
\(976\) −15670.7 −0.513942
\(977\) −50325.3 −1.64795 −0.823977 0.566624i \(-0.808249\pi\)
−0.823977 + 0.566624i \(0.808249\pi\)
\(978\) 0 0
\(979\) 36249.0 1.18337
\(980\) 3455.65 0.112639
\(981\) 0 0
\(982\) −32722.4 −1.06335
\(983\) −51189.0 −1.66091 −0.830456 0.557084i \(-0.811920\pi\)
−0.830456 + 0.557084i \(0.811920\pi\)
\(984\) 0 0
\(985\) 14017.9 0.453449
\(986\) 18568.0 0.599721
\(987\) 0 0
\(988\) 33289.3 1.07194
\(989\) 17795.1 0.572145
\(990\) 0 0
\(991\) −55137.3 −1.76740 −0.883700 0.468054i \(-0.844955\pi\)
−0.883700 + 0.468054i \(0.844955\pi\)
\(992\) −11143.2 −0.356651
\(993\) 0 0
\(994\) 14162.7 0.451925
\(995\) 20514.6 0.653624
\(996\) 0 0
\(997\) 41606.5 1.32166 0.660828 0.750537i \(-0.270205\pi\)
0.660828 + 0.750537i \(0.270205\pi\)
\(998\) −87599.6 −2.77848
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 315.4.a.g.1.1 2
3.2 odd 2 105.4.a.g.1.2 2
5.4 even 2 1575.4.a.y.1.2 2
7.6 odd 2 2205.4.a.v.1.1 2
12.11 even 2 1680.4.a.y.1.2 2
15.2 even 4 525.4.d.j.274.4 4
15.8 even 4 525.4.d.j.274.1 4
15.14 odd 2 525.4.a.i.1.1 2
21.20 even 2 735.4.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.g.1.2 2 3.2 odd 2
315.4.a.g.1.1 2 1.1 even 1 trivial
525.4.a.i.1.1 2 15.14 odd 2
525.4.d.j.274.1 4 15.8 even 4
525.4.d.j.274.4 4 15.2 even 4
735.4.a.q.1.2 2 21.20 even 2
1575.4.a.y.1.2 2 5.4 even 2
1680.4.a.y.1.2 2 12.11 even 2
2205.4.a.v.1.1 2 7.6 odd 2