Properties

Label 165.6.a.f.1.2
Level $165$
Weight $6$
Character 165.1
Self dual yes
Analytic conductor $26.463$
Analytic rank $0$
Dimension $5$
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] = [165,6,Mod(1,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(26.4633302691\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 119x^{3} + 206x^{2} + 1428x - 1320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.28026\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.28026 q^{2} +9.00000 q^{3} -13.6794 q^{4} -25.0000 q^{5} -38.5223 q^{6} +202.127 q^{7} +195.520 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.28026 q^{2} +9.00000 q^{3} -13.6794 q^{4} -25.0000 q^{5} -38.5223 q^{6} +202.127 q^{7} +195.520 q^{8} +81.0000 q^{9} +107.006 q^{10} -121.000 q^{11} -123.115 q^{12} -636.662 q^{13} -865.154 q^{14} -225.000 q^{15} -399.133 q^{16} -1203.76 q^{17} -346.701 q^{18} +1153.75 q^{19} +341.985 q^{20} +1819.14 q^{21} +517.911 q^{22} +932.834 q^{23} +1759.68 q^{24} +625.000 q^{25} +2725.07 q^{26} +729.000 q^{27} -2764.97 q^{28} +1806.72 q^{29} +963.058 q^{30} +4326.37 q^{31} -4548.24 q^{32} -1089.00 q^{33} +5152.39 q^{34} -5053.16 q^{35} -1108.03 q^{36} +6690.43 q^{37} -4938.33 q^{38} -5729.95 q^{39} -4887.99 q^{40} +13597.5 q^{41} -7786.38 q^{42} +5496.81 q^{43} +1655.21 q^{44} -2025.00 q^{45} -3992.77 q^{46} -9481.06 q^{47} -3592.19 q^{48} +24048.2 q^{49} -2675.16 q^{50} -10833.8 q^{51} +8709.15 q^{52} +28001.9 q^{53} -3120.31 q^{54} +3025.00 q^{55} +39519.7 q^{56} +10383.7 q^{57} -7733.23 q^{58} +23054.7 q^{59} +3077.87 q^{60} -3225.94 q^{61} -18518.0 q^{62} +16372.3 q^{63} +32239.9 q^{64} +15916.5 q^{65} +4661.20 q^{66} +56889.2 q^{67} +16466.7 q^{68} +8395.51 q^{69} +21628.8 q^{70} -61686.0 q^{71} +15837.1 q^{72} +32580.0 q^{73} -28636.8 q^{74} +5625.00 q^{75} -15782.6 q^{76} -24457.3 q^{77} +24525.7 q^{78} +52398.7 q^{79} +9978.32 q^{80} +6561.00 q^{81} -58200.8 q^{82} -9259.82 q^{83} -24884.7 q^{84} +30093.9 q^{85} -23527.8 q^{86} +16260.5 q^{87} -23657.9 q^{88} -32806.6 q^{89} +8667.52 q^{90} -128686. q^{91} -12760.6 q^{92} +38937.3 q^{93} +40581.4 q^{94} -28843.7 q^{95} -40934.1 q^{96} +30708.3 q^{97} -102932. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 45 q^{3} + 82 q^{4} - 125 q^{5} - 18 q^{6} + 184 q^{7} + 24 q^{8} + 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + 45 q^{3} + 82 q^{4} - 125 q^{5} - 18 q^{6} + 184 q^{7} + 24 q^{8} + 405 q^{9} + 50 q^{10} - 605 q^{11} + 738 q^{12} + 1082 q^{13} + 432 q^{14} - 1125 q^{15} + 4770 q^{16} + 2174 q^{17} - 162 q^{18} + 1632 q^{19} - 2050 q^{20} + 1656 q^{21} + 242 q^{22} + 1212 q^{23} + 216 q^{24} + 3125 q^{25} + 5600 q^{26} + 3645 q^{27} + 16508 q^{28} + 82 q^{29} + 450 q^{30} + 12120 q^{31} - 4864 q^{32} - 5445 q^{33} - 4524 q^{34} - 4600 q^{35} + 6642 q^{36} - 6530 q^{37} - 15132 q^{38} + 9738 q^{39} - 600 q^{40} + 6782 q^{41} + 3888 q^{42} + 46184 q^{43} - 9922 q^{44} - 10125 q^{45} + 12048 q^{46} - 11692 q^{47} + 42930 q^{48} + 34445 q^{49} - 1250 q^{50} + 19566 q^{51} + 50020 q^{52} + 10314 q^{53} - 1458 q^{54} + 15125 q^{55} + 54928 q^{56} + 14688 q^{57} + 75048 q^{58} + 92892 q^{59} - 18450 q^{60} + 106 q^{61} + 97160 q^{62} + 14904 q^{63} + 44550 q^{64} - 27050 q^{65} + 2178 q^{66} + 100476 q^{67} + 119928 q^{68} + 10908 q^{69} - 10800 q^{70} - 13772 q^{71} + 1944 q^{72} + 94154 q^{73} - 47924 q^{74} + 28125 q^{75} - 51524 q^{76} - 22264 q^{77} + 50400 q^{78} + 178744 q^{79} - 119250 q^{80} + 32805 q^{81} - 299848 q^{82} - 100116 q^{83} + 148572 q^{84} - 54350 q^{85} - 167704 q^{86} + 738 q^{87} - 2904 q^{88} + 119410 q^{89} + 4050 q^{90} + 47536 q^{91} - 404560 q^{92} + 109080 q^{93} - 310288 q^{94} - 40800 q^{95} - 43776 q^{96} + 100682 q^{97} - 16434 q^{98} - 49005 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.28026 −0.756650 −0.378325 0.925673i \(-0.623500\pi\)
−0.378325 + 0.925673i \(0.623500\pi\)
\(3\) 9.00000 0.577350
\(4\) −13.6794 −0.427482
\(5\) −25.0000 −0.447214
\(6\) −38.5223 −0.436852
\(7\) 202.127 1.55912 0.779558 0.626330i \(-0.215444\pi\)
0.779558 + 0.626330i \(0.215444\pi\)
\(8\) 195.520 1.08010
\(9\) 81.0000 0.333333
\(10\) 107.006 0.338384
\(11\) −121.000 −0.301511
\(12\) −123.115 −0.246807
\(13\) −636.662 −1.04484 −0.522421 0.852688i \(-0.674971\pi\)
−0.522421 + 0.852688i \(0.674971\pi\)
\(14\) −865.154 −1.17970
\(15\) −225.000 −0.258199
\(16\) −399.133 −0.389778
\(17\) −1203.76 −1.01022 −0.505110 0.863055i \(-0.668548\pi\)
−0.505110 + 0.863055i \(0.668548\pi\)
\(18\) −346.701 −0.252217
\(19\) 1153.75 0.733207 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(20\) 341.985 0.191176
\(21\) 1819.14 0.900156
\(22\) 517.911 0.228138
\(23\) 932.834 0.367693 0.183846 0.982955i \(-0.441145\pi\)
0.183846 + 0.982955i \(0.441145\pi\)
\(24\) 1759.68 0.623598
\(25\) 625.000 0.200000
\(26\) 2725.07 0.790579
\(27\) 729.000 0.192450
\(28\) −2764.97 −0.666493
\(29\) 1806.72 0.398930 0.199465 0.979905i \(-0.436080\pi\)
0.199465 + 0.979905i \(0.436080\pi\)
\(30\) 963.058 0.195366
\(31\) 4326.37 0.808573 0.404286 0.914632i \(-0.367520\pi\)
0.404286 + 0.914632i \(0.367520\pi\)
\(32\) −4548.24 −0.785178
\(33\) −1089.00 −0.174078
\(34\) 5152.39 0.764383
\(35\) −5053.16 −0.697258
\(36\) −1108.03 −0.142494
\(37\) 6690.43 0.803434 0.401717 0.915764i \(-0.368414\pi\)
0.401717 + 0.915764i \(0.368414\pi\)
\(38\) −4938.33 −0.554780
\(39\) −5729.95 −0.603239
\(40\) −4887.99 −0.483037
\(41\) 13597.5 1.26328 0.631640 0.775262i \(-0.282382\pi\)
0.631640 + 0.775262i \(0.282382\pi\)
\(42\) −7786.38 −0.681102
\(43\) 5496.81 0.453356 0.226678 0.973970i \(-0.427213\pi\)
0.226678 + 0.973970i \(0.427213\pi\)
\(44\) 1655.21 0.128891
\(45\) −2025.00 −0.149071
\(46\) −3992.77 −0.278214
\(47\) −9481.06 −0.626055 −0.313027 0.949744i \(-0.601343\pi\)
−0.313027 + 0.949744i \(0.601343\pi\)
\(48\) −3592.19 −0.225038
\(49\) 24048.2 1.43084
\(50\) −2675.16 −0.151330
\(51\) −10833.8 −0.583251
\(52\) 8709.15 0.446650
\(53\) 28001.9 1.36930 0.684648 0.728874i \(-0.259956\pi\)
0.684648 + 0.728874i \(0.259956\pi\)
\(54\) −3120.31 −0.145617
\(55\) 3025.00 0.134840
\(56\) 39519.7 1.68401
\(57\) 10383.7 0.423317
\(58\) −7733.23 −0.301850
\(59\) 23054.7 0.862242 0.431121 0.902294i \(-0.358118\pi\)
0.431121 + 0.902294i \(0.358118\pi\)
\(60\) 3077.87 0.110375
\(61\) −3225.94 −0.111002 −0.0555011 0.998459i \(-0.517676\pi\)
−0.0555011 + 0.998459i \(0.517676\pi\)
\(62\) −18518.0 −0.611806
\(63\) 16372.3 0.519705
\(64\) 32239.9 0.983882
\(65\) 15916.5 0.467267
\(66\) 4661.20 0.131716
\(67\) 56889.2 1.54826 0.774128 0.633029i \(-0.218189\pi\)
0.774128 + 0.633029i \(0.218189\pi\)
\(68\) 16466.7 0.431851
\(69\) 8395.51 0.212287
\(70\) 21628.8 0.527580
\(71\) −61686.0 −1.45225 −0.726124 0.687564i \(-0.758680\pi\)
−0.726124 + 0.687564i \(0.758680\pi\)
\(72\) 15837.1 0.360034
\(73\) 32580.0 0.715557 0.357779 0.933806i \(-0.383534\pi\)
0.357779 + 0.933806i \(0.383534\pi\)
\(74\) −28636.8 −0.607918
\(75\) 5625.00 0.115470
\(76\) −15782.6 −0.313432
\(77\) −24457.3 −0.470091
\(78\) 24525.7 0.456441
\(79\) 52398.7 0.944611 0.472305 0.881435i \(-0.343422\pi\)
0.472305 + 0.881435i \(0.343422\pi\)
\(80\) 9978.32 0.174314
\(81\) 6561.00 0.111111
\(82\) −58200.8 −0.955860
\(83\) −9259.82 −0.147539 −0.0737696 0.997275i \(-0.523503\pi\)
−0.0737696 + 0.997275i \(0.523503\pi\)
\(84\) −24884.7 −0.384800
\(85\) 30093.9 0.451785
\(86\) −23527.8 −0.343032
\(87\) 16260.5 0.230322
\(88\) −23657.9 −0.325663
\(89\) −32806.6 −0.439021 −0.219511 0.975610i \(-0.570446\pi\)
−0.219511 + 0.975610i \(0.570446\pi\)
\(90\) 8667.52 0.112795
\(91\) −128686. −1.62903
\(92\) −12760.6 −0.157182
\(93\) 38937.3 0.466830
\(94\) 40581.4 0.473704
\(95\) −28843.7 −0.327900
\(96\) −40934.1 −0.453323
\(97\) 30708.3 0.331380 0.165690 0.986178i \(-0.447015\pi\)
0.165690 + 0.986178i \(0.447015\pi\)
\(98\) −102932. −1.08265
\(99\) −9801.00 −0.100504
\(100\) −8549.63 −0.0854963
\(101\) −109736. −1.07040 −0.535201 0.844725i \(-0.679764\pi\)
−0.535201 + 0.844725i \(0.679764\pi\)
\(102\) 46371.5 0.441317
\(103\) 192972. 1.79226 0.896130 0.443792i \(-0.146367\pi\)
0.896130 + 0.443792i \(0.146367\pi\)
\(104\) −124480. −1.12854
\(105\) −45478.5 −0.402562
\(106\) −119855. −1.03608
\(107\) −145055. −1.22482 −0.612412 0.790539i \(-0.709801\pi\)
−0.612412 + 0.790539i \(0.709801\pi\)
\(108\) −9972.29 −0.0822689
\(109\) 221636. 1.78679 0.893396 0.449271i \(-0.148316\pi\)
0.893396 + 0.449271i \(0.148316\pi\)
\(110\) −12947.8 −0.102027
\(111\) 60213.9 0.463863
\(112\) −80675.3 −0.607709
\(113\) 105834. 0.779705 0.389853 0.920877i \(-0.372526\pi\)
0.389853 + 0.920877i \(0.372526\pi\)
\(114\) −44445.0 −0.320303
\(115\) −23320.9 −0.164437
\(116\) −24714.9 −0.170535
\(117\) −51569.6 −0.348280
\(118\) −98679.9 −0.652415
\(119\) −243311. −1.57505
\(120\) −43991.9 −0.278881
\(121\) 14641.0 0.0909091
\(122\) 13807.8 0.0839898
\(123\) 122377. 0.729355
\(124\) −59182.2 −0.345650
\(125\) −15625.0 −0.0894427
\(126\) −70077.4 −0.393235
\(127\) −240397. −1.32257 −0.661285 0.750134i \(-0.729989\pi\)
−0.661285 + 0.750134i \(0.729989\pi\)
\(128\) 7548.69 0.0407236
\(129\) 49471.3 0.261745
\(130\) −68126.9 −0.353557
\(131\) −175704. −0.894548 −0.447274 0.894397i \(-0.647605\pi\)
−0.447274 + 0.894397i \(0.647605\pi\)
\(132\) 14896.9 0.0744150
\(133\) 233203. 1.14315
\(134\) −243500. −1.17149
\(135\) −18225.0 −0.0860663
\(136\) −235358. −1.09114
\(137\) −223760. −1.01855 −0.509273 0.860605i \(-0.670086\pi\)
−0.509273 + 0.860605i \(0.670086\pi\)
\(138\) −35934.9 −0.160627
\(139\) 179526. 0.788116 0.394058 0.919085i \(-0.371071\pi\)
0.394058 + 0.919085i \(0.371071\pi\)
\(140\) 69124.3 0.298065
\(141\) −85329.6 −0.361453
\(142\) 264032. 1.09884
\(143\) 77036.0 0.315031
\(144\) −32329.8 −0.129926
\(145\) −45168.0 −0.178407
\(146\) −139451. −0.541426
\(147\) 216433. 0.826097
\(148\) −91521.2 −0.343453
\(149\) 475283. 1.75383 0.876914 0.480648i \(-0.159598\pi\)
0.876914 + 0.480648i \(0.159598\pi\)
\(150\) −24076.4 −0.0873704
\(151\) 263838. 0.941661 0.470830 0.882224i \(-0.343954\pi\)
0.470830 + 0.882224i \(0.343954\pi\)
\(152\) 225580. 0.791939
\(153\) −97504.3 −0.336740
\(154\) 104684. 0.355694
\(155\) −108159. −0.361605
\(156\) 78382.4 0.257874
\(157\) −100955. −0.326873 −0.163437 0.986554i \(-0.552258\pi\)
−0.163437 + 0.986554i \(0.552258\pi\)
\(158\) −224280. −0.714739
\(159\) 252017. 0.790563
\(160\) 113706. 0.351142
\(161\) 188551. 0.573275
\(162\) −28082.8 −0.0840722
\(163\) −560105. −1.65120 −0.825601 0.564254i \(-0.809164\pi\)
−0.825601 + 0.564254i \(0.809164\pi\)
\(164\) −186006. −0.540028
\(165\) 27225.0 0.0778499
\(166\) 39634.4 0.111635
\(167\) 442671. 1.22826 0.614129 0.789206i \(-0.289507\pi\)
0.614129 + 0.789206i \(0.289507\pi\)
\(168\) 355677. 0.972261
\(169\) 34044.9 0.0916929
\(170\) −128810. −0.341843
\(171\) 93453.5 0.244402
\(172\) −75193.1 −0.193801
\(173\) 345516. 0.877715 0.438857 0.898557i \(-0.355383\pi\)
0.438857 + 0.898557i \(0.355383\pi\)
\(174\) −69599.1 −0.174273
\(175\) 126329. 0.311823
\(176\) 48295.1 0.117523
\(177\) 207492. 0.497815
\(178\) 140420. 0.332185
\(179\) −323189. −0.753917 −0.376959 0.926230i \(-0.623030\pi\)
−0.376959 + 0.926230i \(0.623030\pi\)
\(180\) 27700.8 0.0637252
\(181\) −73649.6 −0.167099 −0.0835495 0.996504i \(-0.526626\pi\)
−0.0835495 + 0.996504i \(0.526626\pi\)
\(182\) 550810. 1.23260
\(183\) −29033.5 −0.0640872
\(184\) 182387. 0.397146
\(185\) −167261. −0.359306
\(186\) −166662. −0.353227
\(187\) 145655. 0.304593
\(188\) 129695. 0.267627
\(189\) 147350. 0.300052
\(190\) 123458. 0.248105
\(191\) −993310. −1.97016 −0.985079 0.172100i \(-0.944945\pi\)
−0.985079 + 0.172100i \(0.944945\pi\)
\(192\) 290159. 0.568045
\(193\) −135815. −0.262455 −0.131227 0.991352i \(-0.541892\pi\)
−0.131227 + 0.991352i \(0.541892\pi\)
\(194\) −131440. −0.250739
\(195\) 143249. 0.269777
\(196\) −328965. −0.611658
\(197\) 236510. 0.434194 0.217097 0.976150i \(-0.430341\pi\)
0.217097 + 0.976150i \(0.430341\pi\)
\(198\) 41950.8 0.0760461
\(199\) −635807. −1.13813 −0.569066 0.822292i \(-0.692695\pi\)
−0.569066 + 0.822292i \(0.692695\pi\)
\(200\) 122200. 0.216021
\(201\) 512003. 0.893886
\(202\) 469700. 0.809919
\(203\) 365186. 0.621977
\(204\) 148200. 0.249329
\(205\) −339937. −0.564956
\(206\) −825969. −1.35611
\(207\) 75559.6 0.122564
\(208\) 254112. 0.407256
\(209\) −139603. −0.221070
\(210\) 194660. 0.304598
\(211\) −753940. −1.16582 −0.582909 0.812537i \(-0.698086\pi\)
−0.582909 + 0.812537i \(0.698086\pi\)
\(212\) −383049. −0.585348
\(213\) −555174. −0.838456
\(214\) 620873. 0.926763
\(215\) −137420. −0.202747
\(216\) 142534. 0.207866
\(217\) 874474. 1.26066
\(218\) −948658. −1.35197
\(219\) 293220. 0.413127
\(220\) −41380.2 −0.0576416
\(221\) 766386. 1.05552
\(222\) −257731. −0.350981
\(223\) 1.17395e6 1.58084 0.790420 0.612565i \(-0.209862\pi\)
0.790420 + 0.612565i \(0.209862\pi\)
\(224\) −919319. −1.22418
\(225\) 50625.0 0.0666667
\(226\) −452998. −0.589964
\(227\) 465701. 0.599850 0.299925 0.953963i \(-0.403038\pi\)
0.299925 + 0.953963i \(0.403038\pi\)
\(228\) −142043. −0.180960
\(229\) −1.01256e6 −1.27595 −0.637973 0.770059i \(-0.720227\pi\)
−0.637973 + 0.770059i \(0.720227\pi\)
\(230\) 99819.2 0.124421
\(231\) −220116. −0.271407
\(232\) 353249. 0.430885
\(233\) 1.02468e6 1.23651 0.618255 0.785977i \(-0.287840\pi\)
0.618255 + 0.785977i \(0.287840\pi\)
\(234\) 220731. 0.263526
\(235\) 237027. 0.279980
\(236\) −315374. −0.368592
\(237\) 471588. 0.545371
\(238\) 1.04143e6 1.19176
\(239\) 776375. 0.879178 0.439589 0.898199i \(-0.355124\pi\)
0.439589 + 0.898199i \(0.355124\pi\)
\(240\) 89804.9 0.100640
\(241\) 919762. 1.02008 0.510038 0.860152i \(-0.329631\pi\)
0.510038 + 0.860152i \(0.329631\pi\)
\(242\) −62667.2 −0.0687863
\(243\) 59049.0 0.0641500
\(244\) 44128.9 0.0474514
\(245\) −601204. −0.639892
\(246\) −523807. −0.551866
\(247\) −734546. −0.766084
\(248\) 845890. 0.873342
\(249\) −83338.4 −0.0851818
\(250\) 66879.0 0.0676768
\(251\) −293678. −0.294230 −0.147115 0.989119i \(-0.546999\pi\)
−0.147115 + 0.989119i \(0.546999\pi\)
\(252\) −223963. −0.222164
\(253\) −112873. −0.110864
\(254\) 1.02896e6 1.00072
\(255\) 270845. 0.260838
\(256\) −1.06399e6 −1.01470
\(257\) 469223. 0.443145 0.221573 0.975144i \(-0.428881\pi\)
0.221573 + 0.975144i \(0.428881\pi\)
\(258\) −211750. −0.198049
\(259\) 1.35231e6 1.25265
\(260\) −217729. −0.199748
\(261\) 146344. 0.132977
\(262\) 752059. 0.676859
\(263\) −1.22481e6 −1.09189 −0.545947 0.837820i \(-0.683830\pi\)
−0.545947 + 0.837820i \(0.683830\pi\)
\(264\) −212921. −0.188022
\(265\) −700046. −0.612367
\(266\) −998168. −0.864967
\(267\) −295259. −0.253469
\(268\) −778210. −0.661851
\(269\) 1.49623e6 1.26072 0.630358 0.776305i \(-0.282908\pi\)
0.630358 + 0.776305i \(0.282908\pi\)
\(270\) 78007.7 0.0651220
\(271\) −1.18393e6 −0.979269 −0.489634 0.871928i \(-0.662870\pi\)
−0.489634 + 0.871928i \(0.662870\pi\)
\(272\) 480459. 0.393762
\(273\) −1.15818e6 −0.940520
\(274\) 957749. 0.770682
\(275\) −75625.0 −0.0603023
\(276\) −114846. −0.0907490
\(277\) −234080. −0.183301 −0.0916505 0.995791i \(-0.529214\pi\)
−0.0916505 + 0.995791i \(0.529214\pi\)
\(278\) −768417. −0.596328
\(279\) 350436. 0.269524
\(280\) −987993. −0.753110
\(281\) −868631. −0.656250 −0.328125 0.944634i \(-0.606417\pi\)
−0.328125 + 0.944634i \(0.606417\pi\)
\(282\) 365232. 0.273493
\(283\) 2.15404e6 1.59877 0.799387 0.600816i \(-0.205158\pi\)
0.799387 + 0.600816i \(0.205158\pi\)
\(284\) 843828. 0.620809
\(285\) −259593. −0.189313
\(286\) −329734. −0.238368
\(287\) 2.74842e6 1.96960
\(288\) −368407. −0.261726
\(289\) 29173.0 0.0205464
\(290\) 193331. 0.134991
\(291\) 276375. 0.191323
\(292\) −445676. −0.305888
\(293\) −660182. −0.449257 −0.224628 0.974445i \(-0.572117\pi\)
−0.224628 + 0.974445i \(0.572117\pi\)
\(294\) −926390. −0.625066
\(295\) −576367. −0.385606
\(296\) 1.30811e6 0.867791
\(297\) −88209.0 −0.0580259
\(298\) −2.03433e6 −1.32703
\(299\) −593900. −0.384180
\(300\) −76946.7 −0.0493613
\(301\) 1.11105e6 0.706835
\(302\) −1.12929e6 −0.712507
\(303\) −987627. −0.617997
\(304\) −460498. −0.285788
\(305\) 80648.5 0.0496417
\(306\) 417343. 0.254794
\(307\) 2.74311e6 1.66111 0.830554 0.556938i \(-0.188024\pi\)
0.830554 + 0.556938i \(0.188024\pi\)
\(308\) 334562. 0.200955
\(309\) 1.73675e6 1.03476
\(310\) 462949. 0.273608
\(311\) −1.34303e6 −0.787381 −0.393690 0.919243i \(-0.628802\pi\)
−0.393690 + 0.919243i \(0.628802\pi\)
\(312\) −1.12032e6 −0.651561
\(313\) −3.41274e6 −1.96898 −0.984492 0.175430i \(-0.943868\pi\)
−0.984492 + 0.175430i \(0.943868\pi\)
\(314\) 432114. 0.247329
\(315\) −409306. −0.232419
\(316\) −716783. −0.403804
\(317\) 1.74045e6 0.972775 0.486387 0.873743i \(-0.338314\pi\)
0.486387 + 0.873743i \(0.338314\pi\)
\(318\) −1.07870e6 −0.598179
\(319\) −218613. −0.120282
\(320\) −805997. −0.440006
\(321\) −1.30550e6 −0.707153
\(322\) −807045. −0.433769
\(323\) −1.38883e6 −0.740701
\(324\) −89750.6 −0.0474979
\(325\) −397913. −0.208968
\(326\) 2.39739e6 1.24938
\(327\) 1.99472e6 1.03160
\(328\) 2.65858e6 1.36447
\(329\) −1.91637e6 −0.976092
\(330\) −116530. −0.0589051
\(331\) 1.12360e6 0.563693 0.281847 0.959459i \(-0.409053\pi\)
0.281847 + 0.959459i \(0.409053\pi\)
\(332\) 126669. 0.0630703
\(333\) 541925. 0.267811
\(334\) −1.89474e6 −0.929361
\(335\) −1.42223e6 −0.692401
\(336\) −726078. −0.350861
\(337\) −2.21997e6 −1.06481 −0.532406 0.846489i \(-0.678712\pi\)
−0.532406 + 0.846489i \(0.678712\pi\)
\(338\) −145721. −0.0693794
\(339\) 952509. 0.450163
\(340\) −411667. −0.193130
\(341\) −523490. −0.243794
\(342\) −400005. −0.184927
\(343\) 1.46363e6 0.671732
\(344\) 1.07473e6 0.489671
\(345\) −209888. −0.0949378
\(346\) −1.47890e6 −0.664122
\(347\) −1.63788e6 −0.730228 −0.365114 0.930963i \(-0.618970\pi\)
−0.365114 + 0.930963i \(0.618970\pi\)
\(348\) −222434. −0.0984584
\(349\) 2.02682e6 0.890742 0.445371 0.895346i \(-0.353072\pi\)
0.445371 + 0.895346i \(0.353072\pi\)
\(350\) −540721. −0.235941
\(351\) −464126. −0.201080
\(352\) 550337. 0.236740
\(353\) −1.06927e6 −0.456719 −0.228359 0.973577i \(-0.573336\pi\)
−0.228359 + 0.973577i \(0.573336\pi\)
\(354\) −888119. −0.376672
\(355\) 1.54215e6 0.649465
\(356\) 448774. 0.187673
\(357\) −2.18980e6 −0.909356
\(358\) 1.38333e6 0.570451
\(359\) −2.61188e6 −1.06959 −0.534794 0.844982i \(-0.679611\pi\)
−0.534794 + 0.844982i \(0.679611\pi\)
\(360\) −395927. −0.161012
\(361\) −1.14497e6 −0.462408
\(362\) 315239. 0.126435
\(363\) 131769. 0.0524864
\(364\) 1.76035e6 0.696379
\(365\) −814501. −0.320007
\(366\) 124271. 0.0484915
\(367\) −2.28492e6 −0.885535 −0.442767 0.896636i \(-0.646003\pi\)
−0.442767 + 0.896636i \(0.646003\pi\)
\(368\) −372325. −0.143319
\(369\) 1.10140e6 0.421093
\(370\) 715919. 0.271869
\(371\) 5.65992e6 2.13489
\(372\) −532639. −0.199561
\(373\) −3.45093e6 −1.28429 −0.642146 0.766582i \(-0.721956\pi\)
−0.642146 + 0.766582i \(0.721956\pi\)
\(374\) −623439. −0.230470
\(375\) −140625. −0.0516398
\(376\) −1.85373e6 −0.676204
\(377\) −1.15027e6 −0.416818
\(378\) −630697. −0.227034
\(379\) −1.42060e6 −0.508011 −0.254005 0.967203i \(-0.581748\pi\)
−0.254005 + 0.967203i \(0.581748\pi\)
\(380\) 394564. 0.140171
\(381\) −2.16357e6 −0.763587
\(382\) 4.25162e6 1.49072
\(383\) 1.71667e6 0.597985 0.298993 0.954255i \(-0.403349\pi\)
0.298993 + 0.954255i \(0.403349\pi\)
\(384\) 67938.2 0.0235118
\(385\) 611433. 0.210231
\(386\) 581323. 0.198586
\(387\) 445242. 0.151119
\(388\) −420072. −0.141659
\(389\) −3.68417e6 −1.23443 −0.617214 0.786795i \(-0.711739\pi\)
−0.617214 + 0.786795i \(0.711739\pi\)
\(390\) −613142. −0.204126
\(391\) −1.12291e6 −0.371451
\(392\) 4.70188e6 1.54546
\(393\) −1.58134e6 −0.516468
\(394\) −1.01232e6 −0.328533
\(395\) −1.30997e6 −0.422443
\(396\) 134072. 0.0429635
\(397\) −1.65933e6 −0.528392 −0.264196 0.964469i \(-0.585107\pi\)
−0.264196 + 0.964469i \(0.585107\pi\)
\(398\) 2.72142e6 0.861167
\(399\) 2.09883e6 0.660000
\(400\) −249458. −0.0779556
\(401\) 3.70365e6 1.15019 0.575093 0.818088i \(-0.304966\pi\)
0.575093 + 0.818088i \(0.304966\pi\)
\(402\) −2.19150e6 −0.676358
\(403\) −2.75443e6 −0.844830
\(404\) 1.50113e6 0.457577
\(405\) −164025. −0.0496904
\(406\) −1.56309e6 −0.470619
\(407\) −809543. −0.242244
\(408\) −2.11822e6 −0.629972
\(409\) 312275. 0.0923057 0.0461529 0.998934i \(-0.485304\pi\)
0.0461529 + 0.998934i \(0.485304\pi\)
\(410\) 1.45502e6 0.427473
\(411\) −2.01384e6 −0.588058
\(412\) −2.63974e6 −0.766158
\(413\) 4.65996e6 1.34433
\(414\) −323414. −0.0927382
\(415\) 231496. 0.0659816
\(416\) 2.89569e6 0.820386
\(417\) 1.61573e6 0.455019
\(418\) 597538. 0.167273
\(419\) 4.46928e6 1.24366 0.621831 0.783152i \(-0.286389\pi\)
0.621831 + 0.783152i \(0.286389\pi\)
\(420\) 622119. 0.172088
\(421\) 1.60241e6 0.440624 0.220312 0.975429i \(-0.429293\pi\)
0.220312 + 0.975429i \(0.429293\pi\)
\(422\) 3.22706e6 0.882116
\(423\) −767966. −0.208685
\(424\) 5.47491e6 1.47898
\(425\) −752348. −0.202044
\(426\) 2.37629e6 0.634417
\(427\) −652048. −0.173065
\(428\) 1.98427e6 0.523590
\(429\) 693324. 0.181883
\(430\) 588194. 0.153408
\(431\) −4.62483e6 −1.19923 −0.599615 0.800289i \(-0.704680\pi\)
−0.599615 + 0.800289i \(0.704680\pi\)
\(432\) −290968. −0.0750128
\(433\) −10306.9 −0.00264184 −0.00132092 0.999999i \(-0.500420\pi\)
−0.00132092 + 0.999999i \(0.500420\pi\)
\(434\) −3.74297e6 −0.953877
\(435\) −406512. −0.103003
\(436\) −3.03185e6 −0.763820
\(437\) 1.07625e6 0.269595
\(438\) −1.25506e6 −0.312593
\(439\) −5.72201e6 −1.41706 −0.708529 0.705682i \(-0.750641\pi\)
−0.708529 + 0.705682i \(0.750641\pi\)
\(440\) 591447. 0.145641
\(441\) 1.94790e6 0.476947
\(442\) −3.28033e6 −0.798659
\(443\) 6.24922e6 1.51292 0.756460 0.654040i \(-0.226927\pi\)
0.756460 + 0.654040i \(0.226927\pi\)
\(444\) −823691. −0.198293
\(445\) 820164. 0.196336
\(446\) −5.02481e6 −1.19614
\(447\) 4.27755e6 1.01257
\(448\) 6.51653e6 1.53399
\(449\) −2.01905e6 −0.472641 −0.236321 0.971675i \(-0.575942\pi\)
−0.236321 + 0.971675i \(0.575942\pi\)
\(450\) −216688. −0.0504433
\(451\) −1.64530e6 −0.380893
\(452\) −1.44775e6 −0.333310
\(453\) 2.37454e6 0.543668
\(454\) −1.99332e6 −0.453876
\(455\) 3.21716e6 0.728524
\(456\) 2.03022e6 0.457226
\(457\) −758581. −0.169907 −0.0849536 0.996385i \(-0.527074\pi\)
−0.0849536 + 0.996385i \(0.527074\pi\)
\(458\) 4.33402e6 0.965444
\(459\) −877539. −0.194417
\(460\) 319016. 0.0702938
\(461\) 5.39608e6 1.18257 0.591284 0.806463i \(-0.298621\pi\)
0.591284 + 0.806463i \(0.298621\pi\)
\(462\) 942152. 0.205360
\(463\) 227337. 0.0492852 0.0246426 0.999696i \(-0.492155\pi\)
0.0246426 + 0.999696i \(0.492155\pi\)
\(464\) −721122. −0.155494
\(465\) −973433. −0.208773
\(466\) −4.38589e6 −0.935605
\(467\) −1.48953e6 −0.316051 −0.158026 0.987435i \(-0.550513\pi\)
−0.158026 + 0.987435i \(0.550513\pi\)
\(468\) 705441. 0.148883
\(469\) 1.14988e7 2.41391
\(470\) −1.01453e6 −0.211847
\(471\) −908597. −0.188720
\(472\) 4.50764e6 0.931310
\(473\) −665114. −0.136692
\(474\) −2.01852e6 −0.412655
\(475\) 721092. 0.146641
\(476\) 3.32835e6 0.673305
\(477\) 2.26815e6 0.456432
\(478\) −3.32308e6 −0.665230
\(479\) −4.99978e6 −0.995662 −0.497831 0.867274i \(-0.665870\pi\)
−0.497831 + 0.867274i \(0.665870\pi\)
\(480\) 1.02335e6 0.202732
\(481\) −4.25954e6 −0.839460
\(482\) −3.93682e6 −0.771840
\(483\) 1.69696e6 0.330981
\(484\) −200280. −0.0388620
\(485\) −767708. −0.148198
\(486\) −252745. −0.0485391
\(487\) 1.41748e6 0.270828 0.135414 0.990789i \(-0.456763\pi\)
0.135414 + 0.990789i \(0.456763\pi\)
\(488\) −630734. −0.119894
\(489\) −5.04094e6 −0.953322
\(490\) 2.57331e6 0.484174
\(491\) 3.78615e6 0.708751 0.354376 0.935103i \(-0.384693\pi\)
0.354376 + 0.935103i \(0.384693\pi\)
\(492\) −1.67405e6 −0.311786
\(493\) −2.17485e6 −0.403007
\(494\) 3.14405e6 0.579657
\(495\) 245025. 0.0449467
\(496\) −1.72679e6 −0.315164
\(497\) −1.24684e7 −2.26422
\(498\) 356710. 0.0644528
\(499\) 2.47452e6 0.444877 0.222439 0.974947i \(-0.428598\pi\)
0.222439 + 0.974947i \(0.428598\pi\)
\(500\) 213741. 0.0382351
\(501\) 3.98404e6 0.709135
\(502\) 1.25702e6 0.222629
\(503\) 7.81639e6 1.37748 0.688742 0.725007i \(-0.258163\pi\)
0.688742 + 0.725007i \(0.258163\pi\)
\(504\) 3.20110e6 0.561335
\(505\) 2.74341e6 0.478698
\(506\) 483125. 0.0838848
\(507\) 306404. 0.0529389
\(508\) 3.28848e6 0.565375
\(509\) 1.95144e6 0.333856 0.166928 0.985969i \(-0.446615\pi\)
0.166928 + 0.985969i \(0.446615\pi\)
\(510\) −1.15929e6 −0.197363
\(511\) 6.58529e6 1.11564
\(512\) 4.31257e6 0.727046
\(513\) 841081. 0.141106
\(514\) −2.00839e6 −0.335306
\(515\) −4.82430e6 −0.801523
\(516\) −676738. −0.111891
\(517\) 1.14721e6 0.188763
\(518\) −5.78825e6 −0.947814
\(519\) 3.10965e6 0.506749
\(520\) 3.11199e6 0.504697
\(521\) 1.11720e6 0.180317 0.0901587 0.995927i \(-0.471263\pi\)
0.0901587 + 0.995927i \(0.471263\pi\)
\(522\) −626392. −0.100617
\(523\) 1.20699e7 1.92952 0.964760 0.263130i \(-0.0847548\pi\)
0.964760 + 0.263130i \(0.0847548\pi\)
\(524\) 2.40353e6 0.382403
\(525\) 1.13696e6 0.180031
\(526\) 5.24252e6 0.826181
\(527\) −5.20789e6 −0.816837
\(528\) 434656. 0.0678516
\(529\) −5.56616e6 −0.864802
\(530\) 2.99638e6 0.463348
\(531\) 1.86743e6 0.287414
\(532\) −3.19008e6 −0.488677
\(533\) −8.65700e6 −1.31993
\(534\) 1.26378e6 0.191787
\(535\) 3.62638e6 0.547758
\(536\) 1.11229e7 1.67228
\(537\) −2.90870e6 −0.435274
\(538\) −6.40424e6 −0.953920
\(539\) −2.90983e6 −0.431415
\(540\) 249307. 0.0367918
\(541\) −1.24911e7 −1.83488 −0.917439 0.397876i \(-0.869747\pi\)
−0.917439 + 0.397876i \(0.869747\pi\)
\(542\) 5.06751e6 0.740963
\(543\) −662846. −0.0964746
\(544\) 5.47497e6 0.793203
\(545\) −5.54090e6 −0.799077
\(546\) 4.95729e6 0.711644
\(547\) −6.59022e6 −0.941742 −0.470871 0.882202i \(-0.656060\pi\)
−0.470871 + 0.882202i \(0.656060\pi\)
\(548\) 3.06090e6 0.435409
\(549\) −261301. −0.0370007
\(550\) 323694. 0.0456277
\(551\) 2.08450e6 0.292498
\(552\) 1.64149e6 0.229292
\(553\) 1.05912e7 1.47276
\(554\) 1.00192e6 0.138695
\(555\) −1.50535e6 −0.207446
\(556\) −2.45581e6 −0.336905
\(557\) 1.15679e7 1.57986 0.789928 0.613200i \(-0.210118\pi\)
0.789928 + 0.613200i \(0.210118\pi\)
\(558\) −1.49995e6 −0.203935
\(559\) −3.49961e6 −0.473685
\(560\) 2.01688e6 0.271776
\(561\) 1.31089e6 0.175857
\(562\) 3.71796e6 0.496552
\(563\) −1.11114e7 −1.47740 −0.738702 0.674032i \(-0.764561\pi\)
−0.738702 + 0.674032i \(0.764561\pi\)
\(564\) 1.16726e6 0.154514
\(565\) −2.64586e6 −0.348695
\(566\) −9.21983e6 −1.20971
\(567\) 1.32615e6 0.173235
\(568\) −1.20608e7 −1.56858
\(569\) −4.82882e6 −0.625260 −0.312630 0.949875i \(-0.601210\pi\)
−0.312630 + 0.949875i \(0.601210\pi\)
\(570\) 1.11112e6 0.143244
\(571\) −9.35367e6 −1.20058 −0.600291 0.799782i \(-0.704948\pi\)
−0.600291 + 0.799782i \(0.704948\pi\)
\(572\) −1.05381e6 −0.134670
\(573\) −8.93979e6 −1.13747
\(574\) −1.17639e7 −1.49030
\(575\) 583021. 0.0735385
\(576\) 2.61143e6 0.327961
\(577\) −4.34302e6 −0.543065 −0.271533 0.962429i \(-0.587530\pi\)
−0.271533 + 0.962429i \(0.587530\pi\)
\(578\) −124868. −0.0155464
\(579\) −1.22234e6 −0.151528
\(580\) 617872. 0.0762656
\(581\) −1.87166e6 −0.230031
\(582\) −1.18296e6 −0.144764
\(583\) −3.38822e6 −0.412858
\(584\) 6.37003e6 0.772876
\(585\) 1.28924e6 0.155756
\(586\) 2.82575e6 0.339930
\(587\) −1.04623e7 −1.25324 −0.626618 0.779327i \(-0.715561\pi\)
−0.626618 + 0.779327i \(0.715561\pi\)
\(588\) −2.96068e6 −0.353141
\(589\) 4.99153e6 0.592851
\(590\) 2.46700e6 0.291769
\(591\) 2.12859e6 0.250682
\(592\) −2.67037e6 −0.313161
\(593\) −1.34068e7 −1.56562 −0.782812 0.622259i \(-0.786215\pi\)
−0.782812 + 0.622259i \(0.786215\pi\)
\(594\) 377557. 0.0439053
\(595\) 6.08278e6 0.704384
\(596\) −6.50159e6 −0.749729
\(597\) −5.72226e6 −0.657101
\(598\) 2.54204e6 0.290690
\(599\) −6.14928e6 −0.700256 −0.350128 0.936702i \(-0.613862\pi\)
−0.350128 + 0.936702i \(0.613862\pi\)
\(600\) 1.09980e6 0.124720
\(601\) 1.14948e7 1.29812 0.649060 0.760737i \(-0.275162\pi\)
0.649060 + 0.760737i \(0.275162\pi\)
\(602\) −4.75558e6 −0.534826
\(603\) 4.60802e6 0.516085
\(604\) −3.60914e6 −0.402543
\(605\) −366025. −0.0406558
\(606\) 4.22730e6 0.467607
\(607\) −1.71751e7 −1.89202 −0.946012 0.324131i \(-0.894928\pi\)
−0.946012 + 0.324131i \(0.894928\pi\)
\(608\) −5.24751e6 −0.575698
\(609\) 3.28668e6 0.359099
\(610\) −345196. −0.0375614
\(611\) 6.03623e6 0.654128
\(612\) 1.33380e6 0.143950
\(613\) 4.69632e6 0.504785 0.252392 0.967625i \(-0.418783\pi\)
0.252392 + 0.967625i \(0.418783\pi\)
\(614\) −1.17412e7 −1.25688
\(615\) −3.05944e6 −0.326177
\(616\) −4.78188e6 −0.507747
\(617\) −3.73984e6 −0.395494 −0.197747 0.980253i \(-0.563362\pi\)
−0.197747 + 0.980253i \(0.563362\pi\)
\(618\) −7.43372e6 −0.782952
\(619\) −1.04630e7 −1.09756 −0.548780 0.835967i \(-0.684907\pi\)
−0.548780 + 0.835967i \(0.684907\pi\)
\(620\) 1.47955e6 0.154579
\(621\) 680036. 0.0707625
\(622\) 5.74851e6 0.595771
\(623\) −6.63108e6 −0.684485
\(624\) 2.28701e6 0.235129
\(625\) 390625. 0.0400000
\(626\) 1.46074e7 1.48983
\(627\) −1.25643e6 −0.127635
\(628\) 1.38101e6 0.139732
\(629\) −8.05365e6 −0.811646
\(630\) 1.75194e6 0.175860
\(631\) 5.70827e6 0.570731 0.285365 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(632\) 1.02450e7 1.02028
\(633\) −6.78546e6 −0.673085
\(634\) −7.44955e6 −0.736050
\(635\) 6.00991e6 0.591472
\(636\) −3.44744e6 −0.337951
\(637\) −1.53105e7 −1.49500
\(638\) 935721. 0.0910112
\(639\) −4.99657e6 −0.484083
\(640\) −188717. −0.0182122
\(641\) −1.24666e7 −1.19841 −0.599204 0.800597i \(-0.704516\pi\)
−0.599204 + 0.800597i \(0.704516\pi\)
\(642\) 5.58786e6 0.535067
\(643\) 1.70621e7 1.62744 0.813719 0.581258i \(-0.197439\pi\)
0.813719 + 0.581258i \(0.197439\pi\)
\(644\) −2.57926e6 −0.245065
\(645\) −1.23678e6 −0.117056
\(646\) 5.94455e6 0.560451
\(647\) 1.30654e7 1.22705 0.613524 0.789676i \(-0.289752\pi\)
0.613524 + 0.789676i \(0.289752\pi\)
\(648\) 1.28280e6 0.120011
\(649\) −2.78962e6 −0.259976
\(650\) 1.70317e6 0.158116
\(651\) 7.87027e6 0.727842
\(652\) 7.66190e6 0.705859
\(653\) −5.62419e6 −0.516151 −0.258076 0.966125i \(-0.583088\pi\)
−0.258076 + 0.966125i \(0.583088\pi\)
\(654\) −8.53792e6 −0.780563
\(655\) 4.39260e6 0.400054
\(656\) −5.42721e6 −0.492398
\(657\) 2.63898e6 0.238519
\(658\) 8.20257e6 0.738559
\(659\) 4.40853e6 0.395440 0.197720 0.980259i \(-0.436646\pi\)
0.197720 + 0.980259i \(0.436646\pi\)
\(660\) −372422. −0.0332794
\(661\) −8.03899e6 −0.715645 −0.357822 0.933790i \(-0.616481\pi\)
−0.357822 + 0.933790i \(0.616481\pi\)
\(662\) −4.80931e6 −0.426518
\(663\) 6.89747e6 0.609405
\(664\) −1.81048e6 −0.159358
\(665\) −5.83007e6 −0.511234
\(666\) −2.31958e6 −0.202639
\(667\) 1.68537e6 0.146683
\(668\) −6.05547e6 −0.525058
\(669\) 1.05656e7 0.912698
\(670\) 6.08751e6 0.523905
\(671\) 390339. 0.0334684
\(672\) −8.27387e6 −0.706782
\(673\) 1.79122e7 1.52445 0.762223 0.647314i \(-0.224108\pi\)
0.762223 + 0.647314i \(0.224108\pi\)
\(674\) 9.50205e6 0.805689
\(675\) 455625. 0.0384900
\(676\) −465714. −0.0391970
\(677\) −7.96540e6 −0.667937 −0.333969 0.942584i \(-0.608388\pi\)
−0.333969 + 0.942584i \(0.608388\pi\)
\(678\) −4.07698e6 −0.340616
\(679\) 6.20697e6 0.516660
\(680\) 5.88395e6 0.487974
\(681\) 4.19131e6 0.346323
\(682\) 2.24067e6 0.184467
\(683\) 1.72633e7 1.41602 0.708012 0.706200i \(-0.249592\pi\)
0.708012 + 0.706200i \(0.249592\pi\)
\(684\) −1.27839e6 −0.104477
\(685\) 5.59399e6 0.455507
\(686\) −6.26471e6 −0.508266
\(687\) −9.11305e6 −0.736668
\(688\) −2.19396e6 −0.176708
\(689\) −1.78277e7 −1.43070
\(690\) 898373. 0.0718347
\(691\) −6.08747e6 −0.485000 −0.242500 0.970151i \(-0.577967\pi\)
−0.242500 + 0.970151i \(0.577967\pi\)
\(692\) −4.72646e6 −0.375207
\(693\) −1.98104e6 −0.156697
\(694\) 7.01054e6 0.552526
\(695\) −4.48815e6 −0.352456
\(696\) 3.17924e6 0.248772
\(697\) −1.63681e7 −1.27619
\(698\) −8.67532e6 −0.673980
\(699\) 9.22211e6 0.713900
\(700\) −1.72811e6 −0.133299
\(701\) −6.99341e6 −0.537519 −0.268760 0.963207i \(-0.586614\pi\)
−0.268760 + 0.963207i \(0.586614\pi\)
\(702\) 1.98658e6 0.152147
\(703\) 7.71907e6 0.589083
\(704\) −3.90102e6 −0.296652
\(705\) 2.13324e6 0.161647
\(706\) 4.57673e6 0.345576
\(707\) −2.21806e7 −1.66888
\(708\) −2.83837e6 −0.212807
\(709\) −2.18416e7 −1.63181 −0.815905 0.578186i \(-0.803761\pi\)
−0.815905 + 0.578186i \(0.803761\pi\)
\(710\) −6.60080e6 −0.491417
\(711\) 4.24430e6 0.314870
\(712\) −6.41432e6 −0.474188
\(713\) 4.03578e6 0.297306
\(714\) 9.37291e6 0.688064
\(715\) −1.92590e6 −0.140886
\(716\) 4.42103e6 0.322286
\(717\) 6.98738e6 0.507594
\(718\) 1.11795e7 0.809304
\(719\) 1.36972e7 0.988118 0.494059 0.869428i \(-0.335513\pi\)
0.494059 + 0.869428i \(0.335513\pi\)
\(720\) 808244. 0.0581047
\(721\) 3.90047e7 2.79434
\(722\) 4.90076e6 0.349881
\(723\) 8.27786e6 0.588941
\(724\) 1.00748e6 0.0714317
\(725\) 1.12920e6 0.0797859
\(726\) −564005. −0.0397138
\(727\) 2.14780e7 1.50715 0.753577 0.657359i \(-0.228327\pi\)
0.753577 + 0.657359i \(0.228327\pi\)
\(728\) −2.51607e7 −1.75952
\(729\) 531441. 0.0370370
\(730\) 3.48627e6 0.242133
\(731\) −6.61682e6 −0.457990
\(732\) 397161. 0.0273961
\(733\) 1.75804e7 1.20856 0.604282 0.796771i \(-0.293460\pi\)
0.604282 + 0.796771i \(0.293460\pi\)
\(734\) 9.78004e6 0.670040
\(735\) −5.41083e6 −0.369442
\(736\) −4.24275e6 −0.288704
\(737\) −6.88359e6 −0.466817
\(738\) −4.71426e6 −0.318620
\(739\) 1.04421e7 0.703360 0.351680 0.936120i \(-0.385611\pi\)
0.351680 + 0.936120i \(0.385611\pi\)
\(740\) 2.28803e6 0.153597
\(741\) −6.61091e6 −0.442299
\(742\) −2.42259e7 −1.61536
\(743\) 3.35925e6 0.223239 0.111620 0.993751i \(-0.464396\pi\)
0.111620 + 0.993751i \(0.464396\pi\)
\(744\) 7.61301e6 0.504224
\(745\) −1.18821e7 −0.784335
\(746\) 1.47709e7 0.971760
\(747\) −750046. −0.0491797
\(748\) −1.99247e6 −0.130208
\(749\) −2.93195e7 −1.90964
\(750\) 601911. 0.0390732
\(751\) 2.94949e7 1.90830 0.954152 0.299322i \(-0.0967606\pi\)
0.954152 + 0.299322i \(0.0967606\pi\)
\(752\) 3.78420e6 0.244022
\(753\) −2.64310e6 −0.169874
\(754\) 4.92345e6 0.315385
\(755\) −6.59594e6 −0.421123
\(756\) −2.01566e6 −0.128267
\(757\) −1.30403e7 −0.827083 −0.413542 0.910485i \(-0.635708\pi\)
−0.413542 + 0.910485i \(0.635708\pi\)
\(758\) 6.08052e6 0.384386
\(759\) −1.01586e6 −0.0640071
\(760\) −5.63950e6 −0.354166
\(761\) 5.20428e6 0.325761 0.162880 0.986646i \(-0.447922\pi\)
0.162880 + 0.986646i \(0.447922\pi\)
\(762\) 9.26063e6 0.577767
\(763\) 4.47985e7 2.78581
\(764\) 1.35879e7 0.842207
\(765\) 2.43761e6 0.150595
\(766\) −7.34780e6 −0.452465
\(767\) −1.46780e7 −0.900906
\(768\) −9.57587e6 −0.585835
\(769\) 2.36072e7 1.43956 0.719778 0.694204i \(-0.244244\pi\)
0.719778 + 0.694204i \(0.244244\pi\)
\(770\) −2.61709e6 −0.159071
\(771\) 4.22300e6 0.255850
\(772\) 1.85787e6 0.112195
\(773\) −1.51766e7 −0.913536 −0.456768 0.889586i \(-0.650993\pi\)
−0.456768 + 0.889586i \(0.650993\pi\)
\(774\) −1.90575e6 −0.114344
\(775\) 2.70398e6 0.161715
\(776\) 6.00408e6 0.357925
\(777\) 1.21708e7 0.723215
\(778\) 1.57692e7 0.934029
\(779\) 1.56881e7 0.926245
\(780\) −1.95956e6 −0.115325
\(781\) 7.46401e6 0.437869
\(782\) 4.80632e6 0.281058
\(783\) 1.31710e6 0.0767740
\(784\) −9.59841e6 −0.557711
\(785\) 2.52388e6 0.146182
\(786\) 6.76853e6 0.390785
\(787\) 1.45626e7 0.838110 0.419055 0.907961i \(-0.362361\pi\)
0.419055 + 0.907961i \(0.362361\pi\)
\(788\) −3.23532e6 −0.185610
\(789\) −1.10233e7 −0.630405
\(790\) 5.60700e6 0.319641
\(791\) 2.13919e7 1.21565
\(792\) −1.91629e6 −0.108554
\(793\) 2.05383e6 0.115980
\(794\) 7.10235e6 0.399807
\(795\) −6.30042e6 −0.353550
\(796\) 8.69747e6 0.486530
\(797\) −924949. −0.0515789 −0.0257894 0.999667i \(-0.508210\pi\)
−0.0257894 + 0.999667i \(0.508210\pi\)
\(798\) −8.98351e6 −0.499389
\(799\) 1.14129e7 0.632454
\(800\) −2.84265e6 −0.157036
\(801\) −2.65733e6 −0.146340
\(802\) −1.58525e7 −0.870288
\(803\) −3.94218e6 −0.215749
\(804\) −7.00389e6 −0.382120
\(805\) −4.71377e6 −0.256377
\(806\) 1.17897e7 0.639240
\(807\) 1.34661e7 0.727875
\(808\) −2.14556e7 −1.15614
\(809\) 1.15962e7 0.622938 0.311469 0.950256i \(-0.399179\pi\)
0.311469 + 0.950256i \(0.399179\pi\)
\(810\) 702069. 0.0375982
\(811\) −1.69320e7 −0.903974 −0.451987 0.892025i \(-0.649285\pi\)
−0.451987 + 0.892025i \(0.649285\pi\)
\(812\) −4.99553e6 −0.265884
\(813\) −1.06553e7 −0.565381
\(814\) 3.46505e6 0.183294
\(815\) 1.40026e7 0.738440
\(816\) 4.32413e6 0.227339
\(817\) 6.34192e6 0.332404
\(818\) −1.33662e6 −0.0698431
\(819\) −1.04236e7 −0.543009
\(820\) 4.65014e6 0.241508
\(821\) 2.60647e7 1.34957 0.674783 0.738016i \(-0.264237\pi\)
0.674783 + 0.738016i \(0.264237\pi\)
\(822\) 8.61974e6 0.444954
\(823\) 1.20408e6 0.0619663 0.0309831 0.999520i \(-0.490136\pi\)
0.0309831 + 0.999520i \(0.490136\pi\)
\(824\) 3.77298e7 1.93583
\(825\) −680625. −0.0348155
\(826\) −1.99458e7 −1.01719
\(827\) 3.53147e7 1.79553 0.897764 0.440478i \(-0.145191\pi\)
0.897764 + 0.440478i \(0.145191\pi\)
\(828\) −1.03361e6 −0.0523939
\(829\) −1.35902e7 −0.686813 −0.343406 0.939187i \(-0.611581\pi\)
−0.343406 + 0.939187i \(0.611581\pi\)
\(830\) −990861. −0.0499249
\(831\) −2.10672e6 −0.105829
\(832\) −2.05259e7 −1.02800
\(833\) −2.89481e7 −1.44547
\(834\) −6.91576e6 −0.344290
\(835\) −1.10668e7 −0.549294
\(836\) 1.90969e6 0.0945034
\(837\) 3.15392e6 0.155610
\(838\) −1.91296e7 −0.941016
\(839\) 1.01361e7 0.497125 0.248562 0.968616i \(-0.420042\pi\)
0.248562 + 0.968616i \(0.420042\pi\)
\(840\) −8.89193e6 −0.434808
\(841\) −1.72469e7 −0.840855
\(842\) −6.85871e6 −0.333398
\(843\) −7.81768e6 −0.378886
\(844\) 1.03135e7 0.498366
\(845\) −851123. −0.0410063
\(846\) 3.28709e6 0.157901
\(847\) 2.95934e6 0.141738
\(848\) −1.11765e7 −0.533721
\(849\) 1.93863e7 0.923053
\(850\) 3.22024e6 0.152877
\(851\) 6.24107e6 0.295417
\(852\) 7.59445e6 0.358424
\(853\) −1.64131e7 −0.772358 −0.386179 0.922424i \(-0.626205\pi\)
−0.386179 + 0.922424i \(0.626205\pi\)
\(854\) 2.79093e6 0.130950
\(855\) −2.33634e6 −0.109300
\(856\) −2.83611e7 −1.32294
\(857\) −1.90891e7 −0.887839 −0.443919 0.896067i \(-0.646412\pi\)
−0.443919 + 0.896067i \(0.646412\pi\)
\(858\) −2.96761e6 −0.137622
\(859\) −3.79193e7 −1.75339 −0.876694 0.481049i \(-0.840256\pi\)
−0.876694 + 0.481049i \(0.840256\pi\)
\(860\) 1.87983e6 0.0866706
\(861\) 2.47357e7 1.13715
\(862\) 1.97954e7 0.907396
\(863\) −2.22911e7 −1.01884 −0.509419 0.860519i \(-0.670140\pi\)
−0.509419 + 0.860519i \(0.670140\pi\)
\(864\) −3.31566e6 −0.151108
\(865\) −8.63791e6 −0.392526
\(866\) 44116.0 0.00199895
\(867\) 262557. 0.0118625
\(868\) −1.19623e7 −0.538908
\(869\) −6.34024e6 −0.284811
\(870\) 1.73998e6 0.0779373
\(871\) −3.62192e7 −1.61768
\(872\) 4.33341e7 1.92992
\(873\) 2.48737e6 0.110460
\(874\) −4.60664e6 −0.203989
\(875\) −3.15823e6 −0.139452
\(876\) −4.01108e6 −0.176604
\(877\) 3.18988e7 1.40047 0.700237 0.713910i \(-0.253078\pi\)
0.700237 + 0.713910i \(0.253078\pi\)
\(878\) 2.44917e7 1.07222
\(879\) −5.94164e6 −0.259378
\(880\) −1.20738e6 −0.0525577
\(881\) −4.66914e6 −0.202673 −0.101337 0.994852i \(-0.532312\pi\)
−0.101337 + 0.994852i \(0.532312\pi\)
\(882\) −8.33751e6 −0.360882
\(883\) −7.83553e6 −0.338195 −0.169097 0.985599i \(-0.554085\pi\)
−0.169097 + 0.985599i \(0.554085\pi\)
\(884\) −1.04837e7 −0.451215
\(885\) −5.18730e6 −0.222630
\(886\) −2.67482e7 −1.14475
\(887\) 2.31623e7 0.988491 0.494245 0.869322i \(-0.335444\pi\)
0.494245 + 0.869322i \(0.335444\pi\)
\(888\) 1.17730e7 0.501019
\(889\) −4.85905e7 −2.06204
\(890\) −3.51051e6 −0.148558
\(891\) −793881. −0.0335013
\(892\) −1.60590e7 −0.675780
\(893\) −1.09387e7 −0.459028
\(894\) −1.83090e7 −0.766163
\(895\) 8.07972e6 0.337162
\(896\) 1.52579e6 0.0634929
\(897\) −5.34510e6 −0.221807
\(898\) 8.64206e6 0.357624
\(899\) 7.81654e6 0.322564
\(900\) −692520. −0.0284988
\(901\) −3.37074e7 −1.38329
\(902\) 7.04229e6 0.288203
\(903\) 9.99946e6 0.408091
\(904\) 2.06927e7 0.842162
\(905\) 1.84124e6 0.0747289
\(906\) −1.01636e7 −0.411366
\(907\) 1.45366e7 0.586738 0.293369 0.955999i \(-0.405223\pi\)
0.293369 + 0.955999i \(0.405223\pi\)
\(908\) −6.37051e6 −0.256425
\(909\) −8.88864e6 −0.356801
\(910\) −1.37702e7 −0.551237
\(911\) 1.98780e7 0.793554 0.396777 0.917915i \(-0.370129\pi\)
0.396777 + 0.917915i \(0.370129\pi\)
\(912\) −4.14448e6 −0.165000
\(913\) 1.12044e6 0.0444848
\(914\) 3.24692e6 0.128560
\(915\) 725836. 0.0286607
\(916\) 1.38512e7 0.545443
\(917\) −3.55145e7 −1.39470
\(918\) 3.75609e6 0.147106
\(919\) 2.87364e7 1.12239 0.561195 0.827684i \(-0.310342\pi\)
0.561195 + 0.827684i \(0.310342\pi\)
\(920\) −4.55968e6 −0.177609
\(921\) 2.46880e7 0.959041
\(922\) −2.30966e7 −0.894790
\(923\) 3.92731e7 1.51737
\(924\) 3.01105e6 0.116022
\(925\) 4.18152e6 0.160687
\(926\) −973059. −0.0372917
\(927\) 1.56307e7 0.597420
\(928\) −8.21740e6 −0.313231
\(929\) 3.50219e7 1.33138 0.665688 0.746231i \(-0.268138\pi\)
0.665688 + 0.746231i \(0.268138\pi\)
\(930\) 4.16654e6 0.157968
\(931\) 2.77455e7 1.04910
\(932\) −1.40170e7 −0.528585
\(933\) −1.20873e7 −0.454594
\(934\) 6.37558e6 0.239140
\(935\) −3.64136e6 −0.136218
\(936\) −1.00829e7 −0.376179
\(937\) 4.31664e7 1.60619 0.803095 0.595851i \(-0.203185\pi\)
0.803095 + 0.595851i \(0.203185\pi\)
\(938\) −4.92179e7 −1.82648
\(939\) −3.07146e7 −1.13679
\(940\) −3.24238e6 −0.119686
\(941\) 1.82030e7 0.670147 0.335073 0.942192i \(-0.391239\pi\)
0.335073 + 0.942192i \(0.391239\pi\)
\(942\) 3.88903e6 0.142795
\(943\) 1.26842e7 0.464498
\(944\) −9.20188e6 −0.336083
\(945\) −3.68376e6 −0.134187
\(946\) 2.84686e6 0.103428
\(947\) 4.35637e7 1.57852 0.789259 0.614060i \(-0.210465\pi\)
0.789259 + 0.614060i \(0.210465\pi\)
\(948\) −6.45105e6 −0.233136
\(949\) −2.07425e7 −0.747644
\(950\) −3.08646e6 −0.110956
\(951\) 1.56640e7 0.561632
\(952\) −4.75721e7 −1.70122
\(953\) 2.03171e6 0.0724653 0.0362326 0.999343i \(-0.488464\pi\)
0.0362326 + 0.999343i \(0.488464\pi\)
\(954\) −9.70826e6 −0.345359
\(955\) 2.48327e7 0.881082
\(956\) −1.06204e7 −0.375832
\(957\) −1.96752e6 −0.0694447
\(958\) 2.14003e7 0.753367
\(959\) −4.52278e7 −1.58803
\(960\) −7.25397e6 −0.254037
\(961\) −9.91169e6 −0.346210
\(962\) 1.82319e7 0.635177
\(963\) −1.17495e7 −0.408275
\(964\) −1.25818e7 −0.436064
\(965\) 3.39538e6 0.117373
\(966\) −7.26340e6 −0.250436
\(967\) 2.09486e7 0.720427 0.360213 0.932870i \(-0.382704\pi\)
0.360213 + 0.932870i \(0.382704\pi\)
\(968\) 2.86260e6 0.0981912
\(969\) −1.24995e7 −0.427644
\(970\) 3.28599e6 0.112134
\(971\) −2.57967e7 −0.878043 −0.439022 0.898477i \(-0.644675\pi\)
−0.439022 + 0.898477i \(0.644675\pi\)
\(972\) −807755. −0.0274230
\(973\) 3.62870e7 1.22876
\(974\) −6.06717e6 −0.204922
\(975\) −3.58122e6 −0.120648
\(976\) 1.28758e6 0.0432662
\(977\) −5.53752e6 −0.185600 −0.0928002 0.995685i \(-0.529582\pi\)
−0.0928002 + 0.995685i \(0.529582\pi\)
\(978\) 2.15765e7 0.721331
\(979\) 3.96959e6 0.132370
\(980\) 8.22411e6 0.273542
\(981\) 1.79525e7 0.595597
\(982\) −1.62057e7 −0.536276
\(983\) −2.67527e7 −0.883048 −0.441524 0.897249i \(-0.645562\pi\)
−0.441524 + 0.897249i \(0.645562\pi\)
\(984\) 2.39272e7 0.787778
\(985\) −5.91275e6 −0.194178
\(986\) 9.30893e6 0.304935
\(987\) −1.72474e7 −0.563547
\(988\) 1.00482e7 0.327487
\(989\) 5.12761e6 0.166696
\(990\) −1.04877e6 −0.0340089
\(991\) −4.26868e7 −1.38073 −0.690366 0.723460i \(-0.742551\pi\)
−0.690366 + 0.723460i \(0.742551\pi\)
\(992\) −1.96773e7 −0.634874
\(993\) 1.01124e7 0.325448
\(994\) 5.33679e7 1.71322
\(995\) 1.58952e7 0.508988
\(996\) 1.14002e6 0.0364137
\(997\) 1.07810e7 0.343495 0.171748 0.985141i \(-0.445059\pi\)
0.171748 + 0.985141i \(0.445059\pi\)
\(998\) −1.05916e7 −0.336616
\(999\) 4.87733e6 0.154621
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 165.6.a.f.1.2 5
3.2 odd 2 495.6.a.j.1.4 5
5.4 even 2 825.6.a.l.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.f.1.2 5 1.1 even 1 trivial
495.6.a.j.1.4 5 3.2 odd 2
825.6.a.l.1.4 5 5.4 even 2