Properties

Label 2450.4.a.cx.1.2
Level $2450$
Weight $4$
Character 2450.1
Self dual yes
Analytic conductor $144.555$
Analytic rank $1$
Dimension $6$
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] = [2450,4,Mod(1,2450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2450.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2450.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: \(144.554679514\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 92x^{4} + 26x^{3} + 2116x^{2} + 80x - 7800 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.44232\) of defining polynomial
Character \(\chi\) \(=\) 2450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -6.44232 q^{3} +4.00000 q^{4} -12.8846 q^{6} +8.00000 q^{8} +14.5035 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -6.44232 q^{3} +4.00000 q^{4} -12.8846 q^{6} +8.00000 q^{8} +14.5035 q^{9} +48.3846 q^{11} -25.7693 q^{12} -93.4871 q^{13} +16.0000 q^{16} +20.2793 q^{17} +29.0071 q^{18} +31.0581 q^{19} +96.7692 q^{22} +21.0487 q^{23} -51.5386 q^{24} -186.974 q^{26} +80.5063 q^{27} +69.5116 q^{29} -161.116 q^{31} +32.0000 q^{32} -311.709 q^{33} +40.5585 q^{34} +58.0141 q^{36} -162.582 q^{37} +62.1162 q^{38} +602.274 q^{39} +365.073 q^{41} -254.518 q^{43} +193.538 q^{44} +42.0975 q^{46} -468.802 q^{47} -103.077 q^{48} -130.646 q^{51} -373.949 q^{52} +587.150 q^{53} +161.013 q^{54} -200.086 q^{57} +139.023 q^{58} -536.945 q^{59} +625.824 q^{61} -322.232 q^{62} +64.0000 q^{64} -623.419 q^{66} +123.074 q^{67} +81.1171 q^{68} -135.603 q^{69} +210.060 q^{71} +116.028 q^{72} +141.783 q^{73} -325.164 q^{74} +124.232 q^{76} +1204.55 q^{78} +513.424 q^{79} -910.243 q^{81} +730.146 q^{82} +117.376 q^{83} -509.037 q^{86} -447.816 q^{87} +387.077 q^{88} +61.2589 q^{89} +84.1950 q^{92} +1037.96 q^{93} -937.604 q^{94} -206.154 q^{96} +436.654 q^{97} +701.748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{2} - 7 q^{3} + 24 q^{4} - 14 q^{6} + 48 q^{8} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 12 q^{2} - 7 q^{3} + 24 q^{4} - 14 q^{6} + 48 q^{8} + 31 q^{9} + 31 q^{11} - 28 q^{12} - 59 q^{13} + 96 q^{16} - 68 q^{17} + 62 q^{18} - 93 q^{19} + 62 q^{22} - 94 q^{23} - 56 q^{24} - 118 q^{26} - 385 q^{27} + 169 q^{29} - 326 q^{31} + 192 q^{32} - 400 q^{33} - 136 q^{34} + 124 q^{36} - 253 q^{37} - 186 q^{38} - 434 q^{39} - 198 q^{41} - 99 q^{43} + 124 q^{44} - 188 q^{46} - 901 q^{47} - 112 q^{48} + 724 q^{51} - 236 q^{52} + 233 q^{53} - 770 q^{54} - 518 q^{57} + 338 q^{58} - 668 q^{59} - 157 q^{61} - 652 q^{62} + 384 q^{64} - 800 q^{66} + 1193 q^{67} - 272 q^{68} + 45 q^{69} + 1108 q^{71} + 248 q^{72} - 1458 q^{73} - 506 q^{74} - 372 q^{76} - 868 q^{78} - 886 q^{79} - 614 q^{81} - 396 q^{82} - 1799 q^{83} - 198 q^{86} - 789 q^{87} + 248 q^{88} - 3047 q^{89} - 376 q^{92} + 1902 q^{93} - 1802 q^{94} - 224 q^{96} - 3404 q^{97} + 4273 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −6.44232 −1.23983 −0.619913 0.784671i \(-0.712832\pi\)
−0.619913 + 0.784671i \(0.712832\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −12.8846 −0.876689
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 14.5035 0.537168
\(10\) 0 0
\(11\) 48.3846 1.32623 0.663114 0.748518i \(-0.269234\pi\)
0.663114 + 0.748518i \(0.269234\pi\)
\(12\) −25.7693 −0.619913
\(13\) −93.4871 −1.99451 −0.997256 0.0740250i \(-0.976416\pi\)
−0.997256 + 0.0740250i \(0.976416\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 20.2793 0.289320 0.144660 0.989481i \(-0.453791\pi\)
0.144660 + 0.989481i \(0.453791\pi\)
\(18\) 29.0071 0.379835
\(19\) 31.0581 0.375011 0.187506 0.982264i \(-0.439960\pi\)
0.187506 + 0.982264i \(0.439960\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 96.7692 0.937785
\(23\) 21.0487 0.190825 0.0954123 0.995438i \(-0.469583\pi\)
0.0954123 + 0.995438i \(0.469583\pi\)
\(24\) −51.5386 −0.438345
\(25\) 0 0
\(26\) −186.974 −1.41033
\(27\) 80.5063 0.573831
\(28\) 0 0
\(29\) 69.5116 0.445103 0.222551 0.974921i \(-0.428562\pi\)
0.222551 + 0.974921i \(0.428562\pi\)
\(30\) 0 0
\(31\) −161.116 −0.933462 −0.466731 0.884399i \(-0.654568\pi\)
−0.466731 + 0.884399i \(0.654568\pi\)
\(32\) 32.0000 0.176777
\(33\) −311.709 −1.64429
\(34\) 40.5585 0.204580
\(35\) 0 0
\(36\) 58.0141 0.268584
\(37\) −162.582 −0.722386 −0.361193 0.932491i \(-0.617630\pi\)
−0.361193 + 0.932491i \(0.617630\pi\)
\(38\) 62.1162 0.265173
\(39\) 602.274 2.47285
\(40\) 0 0
\(41\) 365.073 1.39060 0.695302 0.718717i \(-0.255270\pi\)
0.695302 + 0.718717i \(0.255270\pi\)
\(42\) 0 0
\(43\) −254.518 −0.902644 −0.451322 0.892361i \(-0.649047\pi\)
−0.451322 + 0.892361i \(0.649047\pi\)
\(44\) 193.538 0.663114
\(45\) 0 0
\(46\) 42.0975 0.134933
\(47\) −468.802 −1.45493 −0.727466 0.686144i \(-0.759302\pi\)
−0.727466 + 0.686144i \(0.759302\pi\)
\(48\) −103.077 −0.309956
\(49\) 0 0
\(50\) 0 0
\(51\) −130.646 −0.358707
\(52\) −373.949 −0.997256
\(53\) 587.150 1.52172 0.760861 0.648915i \(-0.224777\pi\)
0.760861 + 0.648915i \(0.224777\pi\)
\(54\) 161.013 0.405760
\(55\) 0 0
\(56\) 0 0
\(57\) −200.086 −0.464949
\(58\) 139.023 0.314735
\(59\) −536.945 −1.18482 −0.592409 0.805637i \(-0.701823\pi\)
−0.592409 + 0.805637i \(0.701823\pi\)
\(60\) 0 0
\(61\) 625.824 1.31358 0.656791 0.754072i \(-0.271913\pi\)
0.656791 + 0.754072i \(0.271913\pi\)
\(62\) −322.232 −0.660057
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −623.419 −1.16269
\(67\) 123.074 0.224416 0.112208 0.993685i \(-0.464208\pi\)
0.112208 + 0.993685i \(0.464208\pi\)
\(68\) 81.1171 0.144660
\(69\) −135.603 −0.236589
\(70\) 0 0
\(71\) 210.060 0.351121 0.175560 0.984469i \(-0.443826\pi\)
0.175560 + 0.984469i \(0.443826\pi\)
\(72\) 116.028 0.189917
\(73\) 141.783 0.227322 0.113661 0.993520i \(-0.463742\pi\)
0.113661 + 0.993520i \(0.463742\pi\)
\(74\) −325.164 −0.510804
\(75\) 0 0
\(76\) 124.232 0.187506
\(77\) 0 0
\(78\) 1204.55 1.74857
\(79\) 513.424 0.731198 0.365599 0.930772i \(-0.380864\pi\)
0.365599 + 0.930772i \(0.380864\pi\)
\(80\) 0 0
\(81\) −910.243 −1.24862
\(82\) 730.146 0.983306
\(83\) 117.376 0.155225 0.0776125 0.996984i \(-0.475270\pi\)
0.0776125 + 0.996984i \(0.475270\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −509.037 −0.638266
\(87\) −447.816 −0.551850
\(88\) 387.077 0.468892
\(89\) 61.2589 0.0729598 0.0364799 0.999334i \(-0.488386\pi\)
0.0364799 + 0.999334i \(0.488386\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 84.1950 0.0954123
\(93\) 1037.96 1.15733
\(94\) −937.604 −1.02879
\(95\) 0 0
\(96\) −206.154 −0.219172
\(97\) 436.654 0.457067 0.228533 0.973536i \(-0.426607\pi\)
0.228533 + 0.973536i \(0.426607\pi\)
\(98\) 0 0
\(99\) 701.748 0.712407
\(100\) 0 0
\(101\) −1768.21 −1.74201 −0.871007 0.491271i \(-0.836532\pi\)
−0.871007 + 0.491271i \(0.836532\pi\)
\(102\) −261.291 −0.253644
\(103\) 849.407 0.812568 0.406284 0.913747i \(-0.366824\pi\)
0.406284 + 0.913747i \(0.366824\pi\)
\(104\) −747.897 −0.705167
\(105\) 0 0
\(106\) 1174.30 1.07602
\(107\) −1282.58 −1.15880 −0.579399 0.815044i \(-0.696713\pi\)
−0.579399 + 0.815044i \(0.696713\pi\)
\(108\) 322.025 0.286916
\(109\) −598.398 −0.525836 −0.262918 0.964818i \(-0.584685\pi\)
−0.262918 + 0.964818i \(0.584685\pi\)
\(110\) 0 0
\(111\) 1047.40 0.895633
\(112\) 0 0
\(113\) −860.218 −0.716128 −0.358064 0.933697i \(-0.616563\pi\)
−0.358064 + 0.933697i \(0.616563\pi\)
\(114\) −400.172 −0.328768
\(115\) 0 0
\(116\) 278.046 0.222551
\(117\) −1355.89 −1.07139
\(118\) −1073.89 −0.837793
\(119\) 0 0
\(120\) 0 0
\(121\) 1010.07 0.758881
\(122\) 1251.65 0.928843
\(123\) −2351.92 −1.72411
\(124\) −644.465 −0.466731
\(125\) 0 0
\(126\) 0 0
\(127\) −971.435 −0.678748 −0.339374 0.940652i \(-0.610215\pi\)
−0.339374 + 0.940652i \(0.610215\pi\)
\(128\) 128.000 0.0883883
\(129\) 1639.69 1.11912
\(130\) 0 0
\(131\) −149.593 −0.0997713 −0.0498856 0.998755i \(-0.515886\pi\)
−0.0498856 + 0.998755i \(0.515886\pi\)
\(132\) −1246.84 −0.822146
\(133\) 0 0
\(134\) 246.147 0.158686
\(135\) 0 0
\(136\) 162.234 0.102290
\(137\) 495.245 0.308844 0.154422 0.988005i \(-0.450648\pi\)
0.154422 + 0.988005i \(0.450648\pi\)
\(138\) −271.206 −0.167294
\(139\) 466.520 0.284674 0.142337 0.989818i \(-0.454538\pi\)
0.142337 + 0.989818i \(0.454538\pi\)
\(140\) 0 0
\(141\) 3020.17 1.80386
\(142\) 420.121 0.248280
\(143\) −4523.34 −2.64518
\(144\) 232.056 0.134292
\(145\) 0 0
\(146\) 283.567 0.160741
\(147\) 0 0
\(148\) −650.327 −0.361193
\(149\) −178.554 −0.0981723 −0.0490862 0.998795i \(-0.515631\pi\)
−0.0490862 + 0.998795i \(0.515631\pi\)
\(150\) 0 0
\(151\) 831.668 0.448213 0.224107 0.974565i \(-0.428054\pi\)
0.224107 + 0.974565i \(0.428054\pi\)
\(152\) 248.465 0.132587
\(153\) 294.121 0.155413
\(154\) 0 0
\(155\) 0 0
\(156\) 2409.10 1.23642
\(157\) −167.697 −0.0852462 −0.0426231 0.999091i \(-0.513571\pi\)
−0.0426231 + 0.999091i \(0.513571\pi\)
\(158\) 1026.85 0.517035
\(159\) −3782.61 −1.88667
\(160\) 0 0
\(161\) 0 0
\(162\) −1820.49 −0.882907
\(163\) 1612.09 0.774655 0.387327 0.921942i \(-0.373398\pi\)
0.387327 + 0.921942i \(0.373398\pi\)
\(164\) 1460.29 0.695302
\(165\) 0 0
\(166\) 234.752 0.109761
\(167\) 1590.69 0.737073 0.368536 0.929613i \(-0.379859\pi\)
0.368536 + 0.929613i \(0.379859\pi\)
\(168\) 0 0
\(169\) 6542.84 2.97808
\(170\) 0 0
\(171\) 450.452 0.201444
\(172\) −1018.07 −0.451322
\(173\) −1742.41 −0.765739 −0.382870 0.923802i \(-0.625064\pi\)
−0.382870 + 0.923802i \(0.625064\pi\)
\(174\) −895.632 −0.390217
\(175\) 0 0
\(176\) 774.154 0.331557
\(177\) 3459.17 1.46897
\(178\) 122.518 0.0515904
\(179\) −2429.37 −1.01441 −0.507205 0.861825i \(-0.669321\pi\)
−0.507205 + 0.861825i \(0.669321\pi\)
\(180\) 0 0
\(181\) −2628.61 −1.07946 −0.539732 0.841837i \(-0.681474\pi\)
−0.539732 + 0.841837i \(0.681474\pi\)
\(182\) 0 0
\(183\) −4031.76 −1.62861
\(184\) 168.390 0.0674667
\(185\) 0 0
\(186\) 2075.92 0.818356
\(187\) 981.204 0.383705
\(188\) −1875.21 −0.727466
\(189\) 0 0
\(190\) 0 0
\(191\) 1124.82 0.426122 0.213061 0.977039i \(-0.431657\pi\)
0.213061 + 0.977039i \(0.431657\pi\)
\(192\) −412.309 −0.154978
\(193\) −2216.08 −0.826514 −0.413257 0.910615i \(-0.635609\pi\)
−0.413257 + 0.910615i \(0.635609\pi\)
\(194\) 873.308 0.323195
\(195\) 0 0
\(196\) 0 0
\(197\) 3125.56 1.13039 0.565194 0.824958i \(-0.308801\pi\)
0.565194 + 0.824958i \(0.308801\pi\)
\(198\) 1403.50 0.503748
\(199\) −4539.34 −1.61701 −0.808506 0.588488i \(-0.799724\pi\)
−0.808506 + 0.588488i \(0.799724\pi\)
\(200\) 0 0
\(201\) −792.881 −0.278236
\(202\) −3536.42 −1.23179
\(203\) 0 0
\(204\) −522.582 −0.179353
\(205\) 0 0
\(206\) 1698.81 0.574573
\(207\) 305.281 0.102505
\(208\) −1495.79 −0.498628
\(209\) 1502.73 0.497350
\(210\) 0 0
\(211\) 2520.40 0.822331 0.411165 0.911561i \(-0.365122\pi\)
0.411165 + 0.911561i \(0.365122\pi\)
\(212\) 2348.60 0.760861
\(213\) −1353.28 −0.435329
\(214\) −2565.15 −0.819394
\(215\) 0 0
\(216\) 644.051 0.202880
\(217\) 0 0
\(218\) −1196.80 −0.371822
\(219\) −913.415 −0.281839
\(220\) 0 0
\(221\) −1895.85 −0.577053
\(222\) 2094.81 0.633308
\(223\) −5800.58 −1.74186 −0.870931 0.491405i \(-0.836484\pi\)
−0.870931 + 0.491405i \(0.836484\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1720.44 −0.506379
\(227\) −6626.47 −1.93751 −0.968753 0.248026i \(-0.920218\pi\)
−0.968753 + 0.248026i \(0.920218\pi\)
\(228\) −800.345 −0.232474
\(229\) −507.845 −0.146547 −0.0732736 0.997312i \(-0.523345\pi\)
−0.0732736 + 0.997312i \(0.523345\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 556.093 0.157368
\(233\) −6307.82 −1.77356 −0.886779 0.462193i \(-0.847063\pi\)
−0.886779 + 0.462193i \(0.847063\pi\)
\(234\) −2711.79 −0.757586
\(235\) 0 0
\(236\) −2147.78 −0.592409
\(237\) −3307.64 −0.906559
\(238\) 0 0
\(239\) −4615.12 −1.24907 −0.624535 0.780997i \(-0.714711\pi\)
−0.624535 + 0.780997i \(0.714711\pi\)
\(240\) 0 0
\(241\) −1550.70 −0.414480 −0.207240 0.978290i \(-0.566448\pi\)
−0.207240 + 0.978290i \(0.566448\pi\)
\(242\) 2020.14 0.536610
\(243\) 3690.41 0.974238
\(244\) 2503.30 0.656791
\(245\) 0 0
\(246\) −4703.83 −1.21913
\(247\) −2903.53 −0.747965
\(248\) −1288.93 −0.330029
\(249\) −756.173 −0.192452
\(250\) 0 0
\(251\) −762.543 −0.191758 −0.0958790 0.995393i \(-0.530566\pi\)
−0.0958790 + 0.995393i \(0.530566\pi\)
\(252\) 0 0
\(253\) 1018.44 0.253077
\(254\) −1942.87 −0.479947
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −2302.36 −0.558822 −0.279411 0.960172i \(-0.590139\pi\)
−0.279411 + 0.960172i \(0.590139\pi\)
\(258\) 3279.38 0.791338
\(259\) 0 0
\(260\) 0 0
\(261\) 1008.16 0.239095
\(262\) −299.187 −0.0705490
\(263\) −4833.50 −1.13326 −0.566628 0.823974i \(-0.691752\pi\)
−0.566628 + 0.823974i \(0.691752\pi\)
\(264\) −2493.67 −0.581345
\(265\) 0 0
\(266\) 0 0
\(267\) −394.649 −0.0904575
\(268\) 492.295 0.112208
\(269\) −4830.35 −1.09484 −0.547420 0.836858i \(-0.684390\pi\)
−0.547420 + 0.836858i \(0.684390\pi\)
\(270\) 0 0
\(271\) 5950.37 1.33380 0.666899 0.745149i \(-0.267621\pi\)
0.666899 + 0.745149i \(0.267621\pi\)
\(272\) 324.468 0.0723301
\(273\) 0 0
\(274\) 990.491 0.218386
\(275\) 0 0
\(276\) −542.411 −0.118295
\(277\) −7668.90 −1.66346 −0.831732 0.555177i \(-0.812651\pi\)
−0.831732 + 0.555177i \(0.812651\pi\)
\(278\) 933.040 0.201295
\(279\) −2336.75 −0.501426
\(280\) 0 0
\(281\) −6314.18 −1.34047 −0.670235 0.742149i \(-0.733807\pi\)
−0.670235 + 0.742149i \(0.733807\pi\)
\(282\) 6040.35 1.27552
\(283\) −6122.72 −1.28607 −0.643035 0.765836i \(-0.722325\pi\)
−0.643035 + 0.765836i \(0.722325\pi\)
\(284\) 840.242 0.175560
\(285\) 0 0
\(286\) −9046.68 −1.87042
\(287\) 0 0
\(288\) 464.113 0.0949587
\(289\) −4501.75 −0.916294
\(290\) 0 0
\(291\) −2813.07 −0.566683
\(292\) 567.134 0.113661
\(293\) 1646.67 0.328326 0.164163 0.986433i \(-0.447508\pi\)
0.164163 + 0.986433i \(0.447508\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1300.65 −0.255402
\(297\) 3895.27 0.761031
\(298\) −357.107 −0.0694183
\(299\) −1967.79 −0.380602
\(300\) 0 0
\(301\) 0 0
\(302\) 1663.34 0.316934
\(303\) 11391.4 2.15979
\(304\) 496.929 0.0937528
\(305\) 0 0
\(306\) 588.242 0.109894
\(307\) 7890.42 1.46687 0.733436 0.679758i \(-0.237915\pi\)
0.733436 + 0.679758i \(0.237915\pi\)
\(308\) 0 0
\(309\) −5472.15 −1.00744
\(310\) 0 0
\(311\) 3436.98 0.626667 0.313333 0.949643i \(-0.398554\pi\)
0.313333 + 0.949643i \(0.398554\pi\)
\(312\) 4818.19 0.874284
\(313\) −7209.81 −1.30199 −0.650995 0.759082i \(-0.725648\pi\)
−0.650995 + 0.759082i \(0.725648\pi\)
\(314\) −335.393 −0.0602782
\(315\) 0 0
\(316\) 2053.69 0.365599
\(317\) −885.804 −0.156945 −0.0784727 0.996916i \(-0.525004\pi\)
−0.0784727 + 0.996916i \(0.525004\pi\)
\(318\) −7565.22 −1.33408
\(319\) 3363.29 0.590308
\(320\) 0 0
\(321\) 8262.77 1.43671
\(322\) 0 0
\(323\) 629.835 0.108498
\(324\) −3640.97 −0.624309
\(325\) 0 0
\(326\) 3224.18 0.547764
\(327\) 3855.08 0.651945
\(328\) 2920.58 0.491653
\(329\) 0 0
\(330\) 0 0
\(331\) −1766.18 −0.293288 −0.146644 0.989189i \(-0.546847\pi\)
−0.146644 + 0.989189i \(0.546847\pi\)
\(332\) 469.504 0.0776125
\(333\) −2358.01 −0.388042
\(334\) 3181.38 0.521189
\(335\) 0 0
\(336\) 0 0
\(337\) −7521.34 −1.21577 −0.607883 0.794026i \(-0.707981\pi\)
−0.607883 + 0.794026i \(0.707981\pi\)
\(338\) 13085.7 2.10582
\(339\) 5541.80 0.887874
\(340\) 0 0
\(341\) −7795.54 −1.23798
\(342\) 900.904 0.142442
\(343\) 0 0
\(344\) −2036.15 −0.319133
\(345\) 0 0
\(346\) −3484.82 −0.541459
\(347\) −4658.53 −0.720700 −0.360350 0.932817i \(-0.617343\pi\)
−0.360350 + 0.932817i \(0.617343\pi\)
\(348\) −1791.26 −0.275925
\(349\) 7198.15 1.10403 0.552017 0.833833i \(-0.313858\pi\)
0.552017 + 0.833833i \(0.313858\pi\)
\(350\) 0 0
\(351\) −7526.30 −1.14451
\(352\) 1548.31 0.234446
\(353\) −3640.08 −0.548844 −0.274422 0.961609i \(-0.588486\pi\)
−0.274422 + 0.961609i \(0.588486\pi\)
\(354\) 6918.34 1.03872
\(355\) 0 0
\(356\) 245.035 0.0364799
\(357\) 0 0
\(358\) −4858.73 −0.717296
\(359\) −5469.89 −0.804150 −0.402075 0.915607i \(-0.631711\pi\)
−0.402075 + 0.915607i \(0.631711\pi\)
\(360\) 0 0
\(361\) −5894.40 −0.859367
\(362\) −5257.22 −0.763296
\(363\) −6507.20 −0.940880
\(364\) 0 0
\(365\) 0 0
\(366\) −8063.52 −1.15160
\(367\) −65.0871 −0.00925754 −0.00462877 0.999989i \(-0.501473\pi\)
−0.00462877 + 0.999989i \(0.501473\pi\)
\(368\) 336.780 0.0477062
\(369\) 5294.84 0.746988
\(370\) 0 0
\(371\) 0 0
\(372\) 4151.85 0.578665
\(373\) 10153.4 1.40944 0.704721 0.709484i \(-0.251072\pi\)
0.704721 + 0.709484i \(0.251072\pi\)
\(374\) 1962.41 0.271320
\(375\) 0 0
\(376\) −3750.41 −0.514396
\(377\) −6498.44 −0.887763
\(378\) 0 0
\(379\) 1453.96 0.197057 0.0985287 0.995134i \(-0.468586\pi\)
0.0985287 + 0.995134i \(0.468586\pi\)
\(380\) 0 0
\(381\) 6258.30 0.841529
\(382\) 2249.65 0.301314
\(383\) 577.301 0.0770201 0.0385101 0.999258i \(-0.487739\pi\)
0.0385101 + 0.999258i \(0.487739\pi\)
\(384\) −824.617 −0.109586
\(385\) 0 0
\(386\) −4432.16 −0.584433
\(387\) −3691.41 −0.484871
\(388\) 1746.62 0.228533
\(389\) −2071.00 −0.269933 −0.134966 0.990850i \(-0.543093\pi\)
−0.134966 + 0.990850i \(0.543093\pi\)
\(390\) 0 0
\(391\) 426.853 0.0552094
\(392\) 0 0
\(393\) 963.729 0.123699
\(394\) 6251.11 0.799306
\(395\) 0 0
\(396\) 2806.99 0.356203
\(397\) −5033.42 −0.636322 −0.318161 0.948037i \(-0.603065\pi\)
−0.318161 + 0.948037i \(0.603065\pi\)
\(398\) −9078.68 −1.14340
\(399\) 0 0
\(400\) 0 0
\(401\) −4946.88 −0.616048 −0.308024 0.951379i \(-0.599668\pi\)
−0.308024 + 0.951379i \(0.599668\pi\)
\(402\) −1585.76 −0.196743
\(403\) 15062.3 1.86180
\(404\) −7072.83 −0.871007
\(405\) 0 0
\(406\) 0 0
\(407\) −7866.46 −0.958049
\(408\) −1045.16 −0.126822
\(409\) 9389.84 1.13520 0.567601 0.823304i \(-0.307872\pi\)
0.567601 + 0.823304i \(0.307872\pi\)
\(410\) 0 0
\(411\) −3190.53 −0.382913
\(412\) 3397.63 0.406284
\(413\) 0 0
\(414\) 610.562 0.0724819
\(415\) 0 0
\(416\) −2991.59 −0.352583
\(417\) −3005.47 −0.352946
\(418\) 3005.47 0.351680
\(419\) −8423.77 −0.982167 −0.491084 0.871112i \(-0.663399\pi\)
−0.491084 + 0.871112i \(0.663399\pi\)
\(420\) 0 0
\(421\) 5680.15 0.657562 0.328781 0.944406i \(-0.393362\pi\)
0.328781 + 0.944406i \(0.393362\pi\)
\(422\) 5040.81 0.581476
\(423\) −6799.28 −0.781542
\(424\) 4697.20 0.538010
\(425\) 0 0
\(426\) −2706.55 −0.307824
\(427\) 0 0
\(428\) −5130.31 −0.579399
\(429\) 29140.8 3.27956
\(430\) 0 0
\(431\) 12348.4 1.38005 0.690023 0.723787i \(-0.257600\pi\)
0.690023 + 0.723787i \(0.257600\pi\)
\(432\) 1288.10 0.143458
\(433\) −9031.93 −1.00242 −0.501209 0.865326i \(-0.667111\pi\)
−0.501209 + 0.865326i \(0.667111\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2393.59 −0.262918
\(437\) 653.734 0.0715614
\(438\) −1826.83 −0.199291
\(439\) 10657.4 1.15866 0.579329 0.815094i \(-0.303315\pi\)
0.579329 + 0.815094i \(0.303315\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3791.70 −0.408038
\(443\) −901.320 −0.0966660 −0.0483330 0.998831i \(-0.515391\pi\)
−0.0483330 + 0.998831i \(0.515391\pi\)
\(444\) 4189.62 0.447816
\(445\) 0 0
\(446\) −11601.2 −1.23168
\(447\) 1150.30 0.121717
\(448\) 0 0
\(449\) −7771.88 −0.816877 −0.408438 0.912786i \(-0.633926\pi\)
−0.408438 + 0.912786i \(0.633926\pi\)
\(450\) 0 0
\(451\) 17663.9 1.84426
\(452\) −3440.87 −0.358064
\(453\) −5357.87 −0.555706
\(454\) −13252.9 −1.37002
\(455\) 0 0
\(456\) −1600.69 −0.164384
\(457\) 2760.35 0.282547 0.141273 0.989971i \(-0.454880\pi\)
0.141273 + 0.989971i \(0.454880\pi\)
\(458\) −1015.69 −0.103625
\(459\) 1632.61 0.166021
\(460\) 0 0
\(461\) 16482.3 1.66520 0.832602 0.553872i \(-0.186850\pi\)
0.832602 + 0.553872i \(0.186850\pi\)
\(462\) 0 0
\(463\) 9949.16 0.998653 0.499327 0.866414i \(-0.333581\pi\)
0.499327 + 0.866414i \(0.333581\pi\)
\(464\) 1112.19 0.111276
\(465\) 0 0
\(466\) −12615.6 −1.25410
\(467\) −2101.55 −0.208240 −0.104120 0.994565i \(-0.533203\pi\)
−0.104120 + 0.994565i \(0.533203\pi\)
\(468\) −5423.57 −0.535694
\(469\) 0 0
\(470\) 0 0
\(471\) 1080.36 0.105690
\(472\) −4295.56 −0.418896
\(473\) −12314.8 −1.19711
\(474\) −6615.28 −0.641034
\(475\) 0 0
\(476\) 0 0
\(477\) 8515.74 0.817420
\(478\) −9230.25 −0.883225
\(479\) 8009.97 0.764060 0.382030 0.924150i \(-0.375225\pi\)
0.382030 + 0.924150i \(0.375225\pi\)
\(480\) 0 0
\(481\) 15199.3 1.44081
\(482\) −3101.41 −0.293082
\(483\) 0 0
\(484\) 4040.28 0.379441
\(485\) 0 0
\(486\) 7380.82 0.688890
\(487\) 5382.59 0.500839 0.250420 0.968137i \(-0.419431\pi\)
0.250420 + 0.968137i \(0.419431\pi\)
\(488\) 5006.59 0.464422
\(489\) −10385.6 −0.960437
\(490\) 0 0
\(491\) −3456.93 −0.317737 −0.158869 0.987300i \(-0.550785\pi\)
−0.158869 + 0.987300i \(0.550785\pi\)
\(492\) −9407.67 −0.862054
\(493\) 1409.64 0.128777
\(494\) −5807.06 −0.528891
\(495\) 0 0
\(496\) −2577.86 −0.233365
\(497\) 0 0
\(498\) −1512.35 −0.136084
\(499\) −12098.2 −1.08535 −0.542674 0.839943i \(-0.682588\pi\)
−0.542674 + 0.839943i \(0.682588\pi\)
\(500\) 0 0
\(501\) −10247.7 −0.913842
\(502\) −1525.09 −0.135593
\(503\) −1508.10 −0.133684 −0.0668418 0.997764i \(-0.521292\pi\)
−0.0668418 + 0.997764i \(0.521292\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2036.87 0.178952
\(507\) −42151.1 −3.69230
\(508\) −3885.74 −0.339374
\(509\) 10363.5 0.902460 0.451230 0.892408i \(-0.350985\pi\)
0.451230 + 0.892408i \(0.350985\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) 2500.37 0.215193
\(514\) −4604.72 −0.395147
\(515\) 0 0
\(516\) 6558.76 0.559561
\(517\) −22682.8 −1.92957
\(518\) 0 0
\(519\) 11225.2 0.949383
\(520\) 0 0
\(521\) −4650.70 −0.391077 −0.195538 0.980696i \(-0.562645\pi\)
−0.195538 + 0.980696i \(0.562645\pi\)
\(522\) 2016.33 0.169066
\(523\) −6457.71 −0.539916 −0.269958 0.962872i \(-0.587010\pi\)
−0.269958 + 0.962872i \(0.587010\pi\)
\(524\) −598.374 −0.0498856
\(525\) 0 0
\(526\) −9667.00 −0.801333
\(527\) −3267.32 −0.270069
\(528\) −4987.35 −0.411073
\(529\) −11724.0 −0.963586
\(530\) 0 0
\(531\) −7787.59 −0.636446
\(532\) 0 0
\(533\) −34129.6 −2.77358
\(534\) −789.299 −0.0639631
\(535\) 0 0
\(536\) 984.590 0.0793429
\(537\) 15650.8 1.25769
\(538\) −9660.70 −0.774168
\(539\) 0 0
\(540\) 0 0
\(541\) −23030.2 −1.83021 −0.915106 0.403214i \(-0.867893\pi\)
−0.915106 + 0.403214i \(0.867893\pi\)
\(542\) 11900.7 0.943137
\(543\) 16934.4 1.33835
\(544\) 648.936 0.0511451
\(545\) 0 0
\(546\) 0 0
\(547\) −6553.72 −0.512280 −0.256140 0.966640i \(-0.582451\pi\)
−0.256140 + 0.966640i \(0.582451\pi\)
\(548\) 1980.98 0.154422
\(549\) 9076.66 0.705614
\(550\) 0 0
\(551\) 2158.90 0.166919
\(552\) −1084.82 −0.0836469
\(553\) 0 0
\(554\) −15337.8 −1.17625
\(555\) 0 0
\(556\) 1866.08 0.142337
\(557\) 5254.73 0.399731 0.199865 0.979823i \(-0.435949\pi\)
0.199865 + 0.979823i \(0.435949\pi\)
\(558\) −4673.51 −0.354561
\(559\) 23794.2 1.80034
\(560\) 0 0
\(561\) −6321.24 −0.475727
\(562\) −12628.4 −0.947856
\(563\) −9759.11 −0.730546 −0.365273 0.930900i \(-0.619024\pi\)
−0.365273 + 0.930900i \(0.619024\pi\)
\(564\) 12080.7 0.901931
\(565\) 0 0
\(566\) −12245.4 −0.909389
\(567\) 0 0
\(568\) 1680.48 0.124140
\(569\) 4063.44 0.299382 0.149691 0.988733i \(-0.452172\pi\)
0.149691 + 0.988733i \(0.452172\pi\)
\(570\) 0 0
\(571\) −23907.0 −1.75215 −0.876075 0.482175i \(-0.839847\pi\)
−0.876075 + 0.482175i \(0.839847\pi\)
\(572\) −18093.4 −1.32259
\(573\) −7246.48 −0.528317
\(574\) 0 0
\(575\) 0 0
\(576\) 928.226 0.0671460
\(577\) 15145.4 1.09274 0.546369 0.837545i \(-0.316010\pi\)
0.546369 + 0.837545i \(0.316010\pi\)
\(578\) −9003.50 −0.647918
\(579\) 14276.7 1.02473
\(580\) 0 0
\(581\) 0 0
\(582\) −5626.13 −0.400706
\(583\) 28409.0 2.01815
\(584\) 1134.27 0.0803704
\(585\) 0 0
\(586\) 3293.34 0.232162
\(587\) 10865.9 0.764025 0.382013 0.924157i \(-0.375231\pi\)
0.382013 + 0.924157i \(0.375231\pi\)
\(588\) 0 0
\(589\) −5003.96 −0.350059
\(590\) 0 0
\(591\) −20135.8 −1.40149
\(592\) −2601.31 −0.180596
\(593\) 4578.29 0.317045 0.158523 0.987355i \(-0.449327\pi\)
0.158523 + 0.987355i \(0.449327\pi\)
\(594\) 7790.53 0.538130
\(595\) 0 0
\(596\) −714.214 −0.0490862
\(597\) 29243.9 2.00481
\(598\) −3935.57 −0.269126
\(599\) −20561.5 −1.40254 −0.701270 0.712896i \(-0.747383\pi\)
−0.701270 + 0.712896i \(0.747383\pi\)
\(600\) 0 0
\(601\) 1254.44 0.0851406 0.0425703 0.999093i \(-0.486445\pi\)
0.0425703 + 0.999093i \(0.486445\pi\)
\(602\) 0 0
\(603\) 1785.00 0.120549
\(604\) 3326.67 0.224107
\(605\) 0 0
\(606\) 22782.7 1.52720
\(607\) −10861.4 −0.726277 −0.363139 0.931735i \(-0.618295\pi\)
−0.363139 + 0.931735i \(0.618295\pi\)
\(608\) 993.859 0.0662933
\(609\) 0 0
\(610\) 0 0
\(611\) 43826.9 2.90188
\(612\) 1176.48 0.0777067
\(613\) −21385.7 −1.40907 −0.704536 0.709668i \(-0.748845\pi\)
−0.704536 + 0.709668i \(0.748845\pi\)
\(614\) 15780.8 1.03724
\(615\) 0 0
\(616\) 0 0
\(617\) 11863.3 0.774066 0.387033 0.922066i \(-0.373500\pi\)
0.387033 + 0.922066i \(0.373500\pi\)
\(618\) −10944.3 −0.712370
\(619\) 8098.60 0.525864 0.262932 0.964814i \(-0.415310\pi\)
0.262932 + 0.964814i \(0.415310\pi\)
\(620\) 0 0
\(621\) 1694.56 0.109501
\(622\) 6873.96 0.443120
\(623\) 0 0
\(624\) 9636.39 0.618212
\(625\) 0 0
\(626\) −14419.6 −0.920645
\(627\) −9681.10 −0.616628
\(628\) −670.787 −0.0426231
\(629\) −3297.04 −0.209001
\(630\) 0 0
\(631\) 28405.4 1.79208 0.896038 0.443977i \(-0.146433\pi\)
0.896038 + 0.443977i \(0.146433\pi\)
\(632\) 4107.39 0.258518
\(633\) −16237.3 −1.01955
\(634\) −1771.61 −0.110977
\(635\) 0 0
\(636\) −15130.4 −0.943335
\(637\) 0 0
\(638\) 6726.58 0.417411
\(639\) 3046.62 0.188611
\(640\) 0 0
\(641\) 4854.85 0.299150 0.149575 0.988750i \(-0.452209\pi\)
0.149575 + 0.988750i \(0.452209\pi\)
\(642\) 16525.5 1.01591
\(643\) 23838.8 1.46207 0.731033 0.682342i \(-0.239038\pi\)
0.731033 + 0.682342i \(0.239038\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1259.67 0.0767199
\(647\) 16912.8 1.02768 0.513841 0.857886i \(-0.328222\pi\)
0.513841 + 0.857886i \(0.328222\pi\)
\(648\) −7281.94 −0.441453
\(649\) −25979.9 −1.57134
\(650\) 0 0
\(651\) 0 0
\(652\) 6448.37 0.387327
\(653\) −14301.5 −0.857059 −0.428530 0.903528i \(-0.640968\pi\)
−0.428530 + 0.903528i \(0.640968\pi\)
\(654\) 7710.15 0.460995
\(655\) 0 0
\(656\) 5841.17 0.347651
\(657\) 2056.36 0.122110
\(658\) 0 0
\(659\) −8601.03 −0.508420 −0.254210 0.967149i \(-0.581815\pi\)
−0.254210 + 0.967149i \(0.581815\pi\)
\(660\) 0 0
\(661\) 794.288 0.0467386 0.0233693 0.999727i \(-0.492561\pi\)
0.0233693 + 0.999727i \(0.492561\pi\)
\(662\) −3532.37 −0.207386
\(663\) 12213.7 0.715445
\(664\) 939.007 0.0548803
\(665\) 0 0
\(666\) −4716.02 −0.274387
\(667\) 1463.13 0.0849366
\(668\) 6362.75 0.368536
\(669\) 37369.2 2.15961
\(670\) 0 0
\(671\) 30280.3 1.74211
\(672\) 0 0
\(673\) −3100.85 −0.177606 −0.0888031 0.996049i \(-0.528304\pi\)
−0.0888031 + 0.996049i \(0.528304\pi\)
\(674\) −15042.7 −0.859677
\(675\) 0 0
\(676\) 26171.4 1.48904
\(677\) 3236.73 0.183748 0.0918742 0.995771i \(-0.470714\pi\)
0.0918742 + 0.995771i \(0.470714\pi\)
\(678\) 11083.6 0.627822
\(679\) 0 0
\(680\) 0 0
\(681\) 42689.8 2.40217
\(682\) −15591.1 −0.875386
\(683\) −9415.00 −0.527460 −0.263730 0.964597i \(-0.584953\pi\)
−0.263730 + 0.964597i \(0.584953\pi\)
\(684\) 1801.81 0.100722
\(685\) 0 0
\(686\) 0 0
\(687\) 3271.70 0.181693
\(688\) −4072.29 −0.225661
\(689\) −54891.0 −3.03509
\(690\) 0 0
\(691\) 22798.5 1.25513 0.627565 0.778564i \(-0.284052\pi\)
0.627565 + 0.778564i \(0.284052\pi\)
\(692\) −6969.63 −0.382870
\(693\) 0 0
\(694\) −9317.06 −0.509612
\(695\) 0 0
\(696\) −3582.53 −0.195108
\(697\) 7403.41 0.402330
\(698\) 14396.3 0.780670
\(699\) 40637.0 2.19890
\(700\) 0 0
\(701\) 33797.9 1.82101 0.910506 0.413496i \(-0.135692\pi\)
0.910506 + 0.413496i \(0.135692\pi\)
\(702\) −15052.6 −0.809294
\(703\) −5049.48 −0.270903
\(704\) 3096.62 0.165779
\(705\) 0 0
\(706\) −7280.16 −0.388091
\(707\) 0 0
\(708\) 13836.7 0.734484
\(709\) −26226.1 −1.38920 −0.694599 0.719397i \(-0.744418\pi\)
−0.694599 + 0.719397i \(0.744418\pi\)
\(710\) 0 0
\(711\) 7446.45 0.392776
\(712\) 490.071 0.0257952
\(713\) −3391.29 −0.178128
\(714\) 0 0
\(715\) 0 0
\(716\) −9717.47 −0.507205
\(717\) 29732.1 1.54863
\(718\) −10939.8 −0.568620
\(719\) 8593.89 0.445755 0.222878 0.974846i \(-0.428455\pi\)
0.222878 + 0.974846i \(0.428455\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −11788.8 −0.607664
\(723\) 9990.14 0.513883
\(724\) −10514.4 −0.539732
\(725\) 0 0
\(726\) −13014.4 −0.665303
\(727\) −26368.9 −1.34521 −0.672606 0.740001i \(-0.734825\pi\)
−0.672606 + 0.740001i \(0.734825\pi\)
\(728\) 0 0
\(729\) 801.754 0.0407333
\(730\) 0 0
\(731\) −5161.45 −0.261153
\(732\) −16127.0 −0.814307
\(733\) −16495.5 −0.831207 −0.415604 0.909546i \(-0.636430\pi\)
−0.415604 + 0.909546i \(0.636430\pi\)
\(734\) −130.174 −0.00654607
\(735\) 0 0
\(736\) 673.560 0.0337333
\(737\) 5954.87 0.297626
\(738\) 10589.7 0.528200
\(739\) 8670.04 0.431573 0.215786 0.976441i \(-0.430768\pi\)
0.215786 + 0.976441i \(0.430768\pi\)
\(740\) 0 0
\(741\) 18705.5 0.927346
\(742\) 0 0
\(743\) 30923.0 1.52686 0.763430 0.645891i \(-0.223514\pi\)
0.763430 + 0.645891i \(0.223514\pi\)
\(744\) 8303.70 0.409178
\(745\) 0 0
\(746\) 20306.8 0.996627
\(747\) 1702.36 0.0833819
\(748\) 3924.82 0.191852
\(749\) 0 0
\(750\) 0 0
\(751\) 13168.0 0.639822 0.319911 0.947448i \(-0.396347\pi\)
0.319911 + 0.947448i \(0.396347\pi\)
\(752\) −7500.83 −0.363733
\(753\) 4912.55 0.237747
\(754\) −12996.9 −0.627743
\(755\) 0 0
\(756\) 0 0
\(757\) −31713.3 −1.52264 −0.761321 0.648375i \(-0.775449\pi\)
−0.761321 + 0.648375i \(0.775449\pi\)
\(758\) 2907.91 0.139341
\(759\) −6561.09 −0.313771
\(760\) 0 0
\(761\) 818.651 0.0389962 0.0194981 0.999810i \(-0.493793\pi\)
0.0194981 + 0.999810i \(0.493793\pi\)
\(762\) 12516.6 0.595051
\(763\) 0 0
\(764\) 4499.29 0.213061
\(765\) 0 0
\(766\) 1154.60 0.0544615
\(767\) 50197.4 2.36313
\(768\) −1649.23 −0.0774891
\(769\) −28514.9 −1.33716 −0.668578 0.743642i \(-0.733097\pi\)
−0.668578 + 0.743642i \(0.733097\pi\)
\(770\) 0 0
\(771\) 14832.6 0.692842
\(772\) −8864.33 −0.413257
\(773\) 28791.0 1.33964 0.669819 0.742525i \(-0.266372\pi\)
0.669819 + 0.742525i \(0.266372\pi\)
\(774\) −7382.83 −0.342856
\(775\) 0 0
\(776\) 3493.23 0.161598
\(777\) 0 0
\(778\) −4142.00 −0.190871
\(779\) 11338.5 0.521493
\(780\) 0 0
\(781\) 10163.7 0.465666
\(782\) 853.706 0.0390390
\(783\) 5596.12 0.255414
\(784\) 0 0
\(785\) 0 0
\(786\) 1927.46 0.0874684
\(787\) 7485.80 0.339059 0.169530 0.985525i \(-0.445775\pi\)
0.169530 + 0.985525i \(0.445775\pi\)
\(788\) 12502.2 0.565194
\(789\) 31139.0 1.40504
\(790\) 0 0
\(791\) 0 0
\(792\) 5613.98 0.251874
\(793\) −58506.5 −2.61996
\(794\) −10066.8 −0.449948
\(795\) 0 0
\(796\) −18157.4 −0.808506
\(797\) 2323.41 0.103261 0.0516307 0.998666i \(-0.483558\pi\)
0.0516307 + 0.998666i \(0.483558\pi\)
\(798\) 0 0
\(799\) −9506.96 −0.420941
\(800\) 0 0
\(801\) 888.469 0.0391917
\(802\) −9893.76 −0.435612
\(803\) 6860.14 0.301481
\(804\) −3171.52 −0.139118
\(805\) 0 0
\(806\) 30124.6 1.31649
\(807\) 31118.7 1.35741
\(808\) −14145.7 −0.615895
\(809\) −9103.69 −0.395635 −0.197818 0.980239i \(-0.563385\pi\)
−0.197818 + 0.980239i \(0.563385\pi\)
\(810\) 0 0
\(811\) 4516.19 0.195542 0.0977711 0.995209i \(-0.468829\pi\)
0.0977711 + 0.995209i \(0.468829\pi\)
\(812\) 0 0
\(813\) −38334.2 −1.65368
\(814\) −15732.9 −0.677443
\(815\) 0 0
\(816\) −2090.33 −0.0896767
\(817\) −7904.86 −0.338502
\(818\) 18779.7 0.802709
\(819\) 0 0
\(820\) 0 0
\(821\) −6046.44 −0.257031 −0.128515 0.991708i \(-0.541021\pi\)
−0.128515 + 0.991708i \(0.541021\pi\)
\(822\) −6381.06 −0.270760
\(823\) 44591.3 1.88864 0.944322 0.329022i \(-0.106719\pi\)
0.944322 + 0.329022i \(0.106719\pi\)
\(824\) 6795.25 0.287286
\(825\) 0 0
\(826\) 0 0
\(827\) 17170.6 0.721982 0.360991 0.932569i \(-0.382439\pi\)
0.360991 + 0.932569i \(0.382439\pi\)
\(828\) 1221.12 0.0512524
\(829\) 30509.9 1.27823 0.639115 0.769111i \(-0.279301\pi\)
0.639115 + 0.769111i \(0.279301\pi\)
\(830\) 0 0
\(831\) 49405.5 2.06241
\(832\) −5983.18 −0.249314
\(833\) 0 0
\(834\) −6010.95 −0.249571
\(835\) 0 0
\(836\) 6010.93 0.248675
\(837\) −12970.9 −0.535650
\(838\) −16847.5 −0.694497
\(839\) 34089.6 1.40275 0.701373 0.712794i \(-0.252571\pi\)
0.701373 + 0.712794i \(0.252571\pi\)
\(840\) 0 0
\(841\) −19557.1 −0.801884
\(842\) 11360.3 0.464966
\(843\) 40678.0 1.66195
\(844\) 10081.6 0.411165
\(845\) 0 0
\(846\) −13598.6 −0.552634
\(847\) 0 0
\(848\) 9394.40 0.380430
\(849\) 39444.6 1.59450
\(850\) 0 0
\(851\) −3422.14 −0.137849
\(852\) −5413.11 −0.217664
\(853\) 5078.39 0.203846 0.101923 0.994792i \(-0.467500\pi\)
0.101923 + 0.994792i \(0.467500\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −10260.6 −0.409697
\(857\) 85.5262 0.00340901 0.00170450 0.999999i \(-0.499457\pi\)
0.00170450 + 0.999999i \(0.499457\pi\)
\(858\) 58281.6 2.31900
\(859\) −32320.1 −1.28376 −0.641878 0.766807i \(-0.721844\pi\)
−0.641878 + 0.766807i \(0.721844\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 24696.7 0.975840
\(863\) 20936.2 0.825813 0.412907 0.910773i \(-0.364514\pi\)
0.412907 + 0.910773i \(0.364514\pi\)
\(864\) 2576.20 0.101440
\(865\) 0 0
\(866\) −18063.9 −0.708816
\(867\) 29001.7 1.13604
\(868\) 0 0
\(869\) 24841.8 0.969736
\(870\) 0 0
\(871\) −11505.8 −0.447600
\(872\) −4787.19 −0.185911
\(873\) 6333.02 0.245522
\(874\) 1307.47 0.0506015
\(875\) 0 0
\(876\) −3653.66 −0.140920
\(877\) −24078.1 −0.927093 −0.463546 0.886073i \(-0.653423\pi\)
−0.463546 + 0.886073i \(0.653423\pi\)
\(878\) 21314.8 0.819295
\(879\) −10608.4 −0.407067
\(880\) 0 0
\(881\) −23196.8 −0.887081 −0.443541 0.896254i \(-0.646278\pi\)
−0.443541 + 0.896254i \(0.646278\pi\)
\(882\) 0 0
\(883\) 31800.2 1.21196 0.605981 0.795479i \(-0.292781\pi\)
0.605981 + 0.795479i \(0.292781\pi\)
\(884\) −7583.40 −0.288526
\(885\) 0 0
\(886\) −1802.64 −0.0683532
\(887\) 34200.2 1.29462 0.647311 0.762226i \(-0.275893\pi\)
0.647311 + 0.762226i \(0.275893\pi\)
\(888\) 8379.24 0.316654
\(889\) 0 0
\(890\) 0 0
\(891\) −44041.8 −1.65595
\(892\) −23202.3 −0.870931
\(893\) −14560.1 −0.545616
\(894\) 2300.60 0.0860666
\(895\) 0 0
\(896\) 0 0
\(897\) 12677.1 0.471880
\(898\) −15543.8 −0.577619
\(899\) −11199.4 −0.415486
\(900\) 0 0
\(901\) 11907.0 0.440265
\(902\) 35327.8 1.30409
\(903\) 0 0
\(904\) −6881.74 −0.253190
\(905\) 0 0
\(906\) −10715.7 −0.392944
\(907\) −4793.39 −0.175482 −0.0877409 0.996143i \(-0.527965\pi\)
−0.0877409 + 0.996143i \(0.527965\pi\)
\(908\) −26505.9 −0.968753
\(909\) −25645.3 −0.935753
\(910\) 0 0
\(911\) −25153.4 −0.914787 −0.457393 0.889264i \(-0.651217\pi\)
−0.457393 + 0.889264i \(0.651217\pi\)
\(912\) −3201.38 −0.116237
\(913\) 5679.19 0.205864
\(914\) 5520.70 0.199791
\(915\) 0 0
\(916\) −2031.38 −0.0732736
\(917\) 0 0
\(918\) 3265.22 0.117395
\(919\) −20437.0 −0.733573 −0.366786 0.930305i \(-0.619542\pi\)
−0.366786 + 0.930305i \(0.619542\pi\)
\(920\) 0 0
\(921\) −50832.6 −1.81867
\(922\) 32964.7 1.17748
\(923\) −19637.9 −0.700315
\(924\) 0 0
\(925\) 0 0
\(926\) 19898.3 0.706155
\(927\) 12319.4 0.436485
\(928\) 2224.37 0.0786838
\(929\) −1282.13 −0.0452802 −0.0226401 0.999744i \(-0.507207\pi\)
−0.0226401 + 0.999744i \(0.507207\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −25231.3 −0.886779
\(933\) −22142.1 −0.776957
\(934\) −4203.11 −0.147248
\(935\) 0 0
\(936\) −10847.1 −0.378793
\(937\) −16878.5 −0.588472 −0.294236 0.955733i \(-0.595065\pi\)
−0.294236 + 0.955733i \(0.595065\pi\)
\(938\) 0 0
\(939\) 46447.9 1.61424
\(940\) 0 0
\(941\) −14931.6 −0.517275 −0.258637 0.965974i \(-0.583274\pi\)
−0.258637 + 0.965974i \(0.583274\pi\)
\(942\) 2160.71 0.0747344
\(943\) 7684.33 0.265362
\(944\) −8591.11 −0.296204
\(945\) 0 0
\(946\) −24629.5 −0.846486
\(947\) −37545.7 −1.28835 −0.644177 0.764877i \(-0.722800\pi\)
−0.644177 + 0.764877i \(0.722800\pi\)
\(948\) −13230.6 −0.453279
\(949\) −13254.9 −0.453396
\(950\) 0 0
\(951\) 5706.63 0.194585
\(952\) 0 0
\(953\) −437.260 −0.0148628 −0.00743140 0.999972i \(-0.502366\pi\)
−0.00743140 + 0.999972i \(0.502366\pi\)
\(954\) 17031.5 0.578003
\(955\) 0 0
\(956\) −18460.5 −0.624535
\(957\) −21667.4 −0.731879
\(958\) 16019.9 0.540272
\(959\) 0 0
\(960\) 0 0
\(961\) −3832.58 −0.128649
\(962\) 30398.6 1.01881
\(963\) −18601.9 −0.622469
\(964\) −6202.82 −0.207240
\(965\) 0 0
\(966\) 0 0
\(967\) 52372.7 1.74167 0.870834 0.491577i \(-0.163579\pi\)
0.870834 + 0.491577i \(0.163579\pi\)
\(968\) 8080.57 0.268305
\(969\) −4057.60 −0.134519
\(970\) 0 0
\(971\) −40759.4 −1.34710 −0.673548 0.739144i \(-0.735231\pi\)
−0.673548 + 0.739144i \(0.735231\pi\)
\(972\) 14761.6 0.487119
\(973\) 0 0
\(974\) 10765.2 0.354147
\(975\) 0 0
\(976\) 10013.2 0.328396
\(977\) 37637.9 1.23249 0.616246 0.787554i \(-0.288653\pi\)
0.616246 + 0.787554i \(0.288653\pi\)
\(978\) −20771.2 −0.679132
\(979\) 2963.99 0.0967614
\(980\) 0 0
\(981\) −8678.89 −0.282462
\(982\) −6913.86 −0.224674
\(983\) 8799.74 0.285522 0.142761 0.989757i \(-0.454402\pi\)
0.142761 + 0.989757i \(0.454402\pi\)
\(984\) −18815.3 −0.609564
\(985\) 0 0
\(986\) 2819.29 0.0910592
\(987\) 0 0
\(988\) −11614.1 −0.373982
\(989\) −5357.29 −0.172247
\(990\) 0 0
\(991\) −43972.1 −1.40950 −0.704752 0.709454i \(-0.748942\pi\)
−0.704752 + 0.709454i \(0.748942\pi\)
\(992\) −5155.72 −0.165014
\(993\) 11378.3 0.363626
\(994\) 0 0
\(995\) 0 0
\(996\) −3024.69 −0.0962260
\(997\) −10052.7 −0.319329 −0.159664 0.987171i \(-0.551041\pi\)
−0.159664 + 0.987171i \(0.551041\pi\)
\(998\) −24196.3 −0.767457
\(999\) −13088.9 −0.414528
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 2450.4.a.cx.1.2 6
5.2 odd 4 490.4.c.e.99.11 12
5.3 odd 4 490.4.c.e.99.2 12
5.4 even 2 2450.4.a.cw.1.5 6
7.3 odd 6 350.4.e.n.51.2 12
7.5 odd 6 350.4.e.n.151.2 12
7.6 odd 2 2450.4.a.cy.1.5 6
35.3 even 12 70.4.i.a.9.11 yes 24
35.12 even 12 70.4.i.a.39.11 yes 24
35.13 even 4 490.4.c.f.99.5 12
35.17 even 12 70.4.i.a.9.2 24
35.19 odd 6 350.4.e.o.151.5 12
35.24 odd 6 350.4.e.o.51.5 12
35.27 even 4 490.4.c.f.99.8 12
35.33 even 12 70.4.i.a.39.2 yes 24
35.34 odd 2 2450.4.a.cv.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.4.i.a.9.2 24 35.17 even 12
70.4.i.a.9.11 yes 24 35.3 even 12
70.4.i.a.39.2 yes 24 35.33 even 12
70.4.i.a.39.11 yes 24 35.12 even 12
350.4.e.n.51.2 12 7.3 odd 6
350.4.e.n.151.2 12 7.5 odd 6
350.4.e.o.51.5 12 35.24 odd 6
350.4.e.o.151.5 12 35.19 odd 6
490.4.c.e.99.2 12 5.3 odd 4
490.4.c.e.99.11 12 5.2 odd 4
490.4.c.f.99.5 12 35.13 even 4
490.4.c.f.99.8 12 35.27 even 4
2450.4.a.cv.1.2 6 35.34 odd 2
2450.4.a.cw.1.5 6 5.4 even 2
2450.4.a.cx.1.2 6 1.1 even 1 trivial
2450.4.a.cy.1.5 6 7.6 odd 2