Properties

Label 165.6.a.e.1.1
Level $165$
Weight $6$
Character 165.1
Self dual yes
Analytic conductor $26.463$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.307532.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 76x + 168 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-9.21967\) 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)
 
\(f(q)\) \(=\) \(q-7.21967 q^{2} -9.00000 q^{3} +20.1236 q^{4} -25.0000 q^{5} +64.9770 q^{6} +147.570 q^{7} +85.7438 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-7.21967 q^{2} -9.00000 q^{3} +20.1236 q^{4} -25.0000 q^{5} +64.9770 q^{6} +147.570 q^{7} +85.7438 q^{8} +81.0000 q^{9} +180.492 q^{10} +121.000 q^{11} -181.112 q^{12} -1124.39 q^{13} -1065.40 q^{14} +225.000 q^{15} -1263.00 q^{16} +1019.99 q^{17} -584.793 q^{18} -13.7131 q^{19} -503.090 q^{20} -1328.13 q^{21} -873.580 q^{22} -2116.94 q^{23} -771.694 q^{24} +625.000 q^{25} +8117.69 q^{26} -729.000 q^{27} +2969.63 q^{28} +3120.98 q^{29} -1624.42 q^{30} -9628.99 q^{31} +6374.61 q^{32} -1089.00 q^{33} -7363.96 q^{34} -3689.24 q^{35} +1630.01 q^{36} -3121.52 q^{37} +99.0039 q^{38} +10119.5 q^{39} -2143.59 q^{40} -5884.38 q^{41} +9588.63 q^{42} +20522.3 q^{43} +2434.95 q^{44} -2025.00 q^{45} +15283.6 q^{46} +24718.5 q^{47} +11367.0 q^{48} +4969.81 q^{49} -4512.29 q^{50} -9179.87 q^{51} -22626.7 q^{52} +22761.1 q^{53} +5263.14 q^{54} -3025.00 q^{55} +12653.2 q^{56} +123.418 q^{57} -22532.4 q^{58} -14533.5 q^{59} +4527.81 q^{60} +43984.9 q^{61} +69518.1 q^{62} +11953.1 q^{63} -5606.68 q^{64} +28109.6 q^{65} +7862.22 q^{66} -31488.3 q^{67} +20525.8 q^{68} +19052.5 q^{69} +26635.1 q^{70} -29173.2 q^{71} +6945.24 q^{72} +49583.0 q^{73} +22536.3 q^{74} -5625.00 q^{75} -275.956 q^{76} +17855.9 q^{77} -73059.2 q^{78} -85075.4 q^{79} +31574.9 q^{80} +6561.00 q^{81} +42483.3 q^{82} +77321.9 q^{83} -26726.7 q^{84} -25499.6 q^{85} -148164. q^{86} -28088.8 q^{87} +10375.0 q^{88} +70217.7 q^{89} +14619.8 q^{90} -165925. q^{91} -42600.5 q^{92} +86660.9 q^{93} -178459. q^{94} +342.827 q^{95} -57371.5 q^{96} -21080.0 q^{97} -35880.4 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} - 27 q^{3} + 73 q^{4} - 75 q^{5} - 63 q^{6} + 92 q^{7} + 231 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 7 q^{2} - 27 q^{3} + 73 q^{4} - 75 q^{5} - 63 q^{6} + 92 q^{7} + 231 q^{8} + 243 q^{9} - 175 q^{10} + 363 q^{11} - 657 q^{12} - 90 q^{13} - 784 q^{14} + 675 q^{15} - 415 q^{16} + 1934 q^{17} + 567 q^{18} + 2084 q^{19} - 1825 q^{20} - 828 q^{21} + 847 q^{22} + 1220 q^{23} - 2079 q^{24} + 1875 q^{25} + 17062 q^{26} - 2187 q^{27} + 11120 q^{28} + 4402 q^{29} + 1575 q^{30} - 10688 q^{31} + 12439 q^{32} - 3267 q^{33} - 4094 q^{34} - 2300 q^{35} + 5913 q^{36} - 8190 q^{37} + 13792 q^{38} + 810 q^{39} - 5775 q^{40} + 5974 q^{41} + 7056 q^{42} + 18868 q^{43} + 8833 q^{44} - 6075 q^{45} + 46220 q^{46} + 55500 q^{47} + 3735 q^{48} + 1907 q^{49} + 4375 q^{50} - 17406 q^{51} + 27330 q^{52} + 9206 q^{53} - 5103 q^{54} - 9075 q^{55} + 73248 q^{56} - 18756 q^{57} + 15366 q^{58} - 59196 q^{59} + 16425 q^{60} + 79902 q^{61} + 64616 q^{62} + 7452 q^{63} + 2129 q^{64} + 2250 q^{65} - 7623 q^{66} + 4468 q^{67} - 1218 q^{68} - 10980 q^{69} + 19600 q^{70} - 75164 q^{71} + 18711 q^{72} - 61290 q^{73} - 56766 q^{74} - 16875 q^{75} + 37816 q^{76} + 11132 q^{77} - 153558 q^{78} - 83564 q^{79} + 10375 q^{80} + 19683 q^{81} + 147410 q^{82} + 74764 q^{83} - 100080 q^{84} - 48350 q^{85} - 253432 q^{86} - 39618 q^{87} + 27951 q^{88} + 37342 q^{89} - 14175 q^{90} - 126488 q^{91} + 148164 q^{92} + 96192 q^{93} + 59252 q^{94} - 52100 q^{95} - 111951 q^{96} + 33486 q^{97} - 95249 q^{98} + 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.21967 −1.27627 −0.638134 0.769925i \(-0.720294\pi\)
−0.638134 + 0.769925i \(0.720294\pi\)
\(3\) −9.00000 −0.577350
\(4\) 20.1236 0.628862
\(5\) −25.0000 −0.447214
\(6\) 64.9770 0.736854
\(7\) 147.570 1.13829 0.569144 0.822238i \(-0.307275\pi\)
0.569144 + 0.822238i \(0.307275\pi\)
\(8\) 85.7438 0.473672
\(9\) 81.0000 0.333333
\(10\) 180.492 0.570765
\(11\) 121.000 0.301511
\(12\) −181.112 −0.363074
\(13\) −1124.39 −1.84526 −0.922628 0.385690i \(-0.873963\pi\)
−0.922628 + 0.385690i \(0.873963\pi\)
\(14\) −1065.40 −1.45276
\(15\) 225.000 0.258199
\(16\) −1263.00 −1.23339
\(17\) 1019.99 0.855996 0.427998 0.903780i \(-0.359219\pi\)
0.427998 + 0.903780i \(0.359219\pi\)
\(18\) −584.793 −0.425423
\(19\) −13.7131 −0.00871467 −0.00435734 0.999991i \(-0.501387\pi\)
−0.00435734 + 0.999991i \(0.501387\pi\)
\(20\) −503.090 −0.281236
\(21\) −1328.13 −0.657191
\(22\) −873.580 −0.384810
\(23\) −2116.94 −0.834430 −0.417215 0.908808i \(-0.636994\pi\)
−0.417215 + 0.908808i \(0.636994\pi\)
\(24\) −771.694 −0.273475
\(25\) 625.000 0.200000
\(26\) 8117.69 2.35504
\(27\) −729.000 −0.192450
\(28\) 2969.63 0.715826
\(29\) 3120.98 0.689122 0.344561 0.938764i \(-0.388028\pi\)
0.344561 + 0.938764i \(0.388028\pi\)
\(30\) −1624.42 −0.329531
\(31\) −9628.99 −1.79960 −0.899801 0.436301i \(-0.856288\pi\)
−0.899801 + 0.436301i \(0.856288\pi\)
\(32\) 6374.61 1.10047
\(33\) −1089.00 −0.174078
\(34\) −7363.96 −1.09248
\(35\) −3689.24 −0.509058
\(36\) 1630.01 0.209621
\(37\) −3121.52 −0.374854 −0.187427 0.982279i \(-0.560015\pi\)
−0.187427 + 0.982279i \(0.560015\pi\)
\(38\) 99.0039 0.0111223
\(39\) 10119.5 1.06536
\(40\) −2143.59 −0.211832
\(41\) −5884.38 −0.546690 −0.273345 0.961916i \(-0.588130\pi\)
−0.273345 + 0.961916i \(0.588130\pi\)
\(42\) 9588.63 0.838752
\(43\) 20522.3 1.69260 0.846302 0.532704i \(-0.178824\pi\)
0.846302 + 0.532704i \(0.178824\pi\)
\(44\) 2434.95 0.189609
\(45\) −2025.00 −0.149071
\(46\) 15283.6 1.06496
\(47\) 24718.5 1.63222 0.816108 0.577899i \(-0.196127\pi\)
0.816108 + 0.577899i \(0.196127\pi\)
\(48\) 11367.0 0.712101
\(49\) 4969.81 0.295699
\(50\) −4512.29 −0.255254
\(51\) −9179.87 −0.494210
\(52\) −22626.7 −1.16041
\(53\) 22761.1 1.11302 0.556511 0.830840i \(-0.312140\pi\)
0.556511 + 0.830840i \(0.312140\pi\)
\(54\) 5263.14 0.245618
\(55\) −3025.00 −0.134840
\(56\) 12653.2 0.539175
\(57\) 123.418 0.00503142
\(58\) −22532.4 −0.879505
\(59\) −14533.5 −0.543553 −0.271776 0.962360i \(-0.587611\pi\)
−0.271776 + 0.962360i \(0.587611\pi\)
\(60\) 4527.81 0.162371
\(61\) 43984.9 1.51349 0.756743 0.653712i \(-0.226789\pi\)
0.756743 + 0.653712i \(0.226789\pi\)
\(62\) 69518.1 2.29677
\(63\) 11953.1 0.379429
\(64\) −5606.68 −0.171102
\(65\) 28109.6 0.825224
\(66\) 7862.22 0.222170
\(67\) −31488.3 −0.856964 −0.428482 0.903550i \(-0.640951\pi\)
−0.428482 + 0.903550i \(0.640951\pi\)
\(68\) 20525.8 0.538303
\(69\) 19052.5 0.481758
\(70\) 26635.1 0.649694
\(71\) −29173.2 −0.686812 −0.343406 0.939187i \(-0.611581\pi\)
−0.343406 + 0.939187i \(0.611581\pi\)
\(72\) 6945.24 0.157891
\(73\) 49583.0 1.08899 0.544497 0.838763i \(-0.316721\pi\)
0.544497 + 0.838763i \(0.316721\pi\)
\(74\) 22536.3 0.478414
\(75\) −5625.00 −0.115470
\(76\) −275.956 −0.00548033
\(77\) 17855.9 0.343207
\(78\) −73059.2 −1.35968
\(79\) −85075.4 −1.53369 −0.766843 0.641835i \(-0.778174\pi\)
−0.766843 + 0.641835i \(0.778174\pi\)
\(80\) 31574.9 0.551591
\(81\) 6561.00 0.111111
\(82\) 42483.3 0.697724
\(83\) 77321.9 1.23199 0.615995 0.787750i \(-0.288754\pi\)
0.615995 + 0.787750i \(0.288754\pi\)
\(84\) −26726.7 −0.413282
\(85\) −25499.6 −0.382813
\(86\) −148164. −2.16022
\(87\) −28088.8 −0.397865
\(88\) 10375.0 0.142817
\(89\) 70217.7 0.939661 0.469831 0.882757i \(-0.344315\pi\)
0.469831 + 0.882757i \(0.344315\pi\)
\(90\) 14619.8 0.190255
\(91\) −165925. −2.10043
\(92\) −42600.5 −0.524741
\(93\) 86660.9 1.03900
\(94\) −178459. −2.08315
\(95\) 342.827 0.00389732
\(96\) −57371.5 −0.635357
\(97\) −21080.0 −0.227479 −0.113739 0.993511i \(-0.536283\pi\)
−0.113739 + 0.993511i \(0.536283\pi\)
\(98\) −35880.4 −0.377391
\(99\) 9801.00 0.100504
\(100\) 12577.2 0.125772
\(101\) −60279.4 −0.587984 −0.293992 0.955808i \(-0.594984\pi\)
−0.293992 + 0.955808i \(0.594984\pi\)
\(102\) 66275.6 0.630744
\(103\) 83535.5 0.775851 0.387925 0.921691i \(-0.373192\pi\)
0.387925 + 0.921691i \(0.373192\pi\)
\(104\) −96409.0 −0.874046
\(105\) 33203.2 0.293905
\(106\) −164328. −1.42052
\(107\) 159288. 1.34500 0.672502 0.740095i \(-0.265219\pi\)
0.672502 + 0.740095i \(0.265219\pi\)
\(108\) −14670.1 −0.121025
\(109\) −31166.3 −0.251258 −0.125629 0.992077i \(-0.540095\pi\)
−0.125629 + 0.992077i \(0.540095\pi\)
\(110\) 21839.5 0.172092
\(111\) 28093.7 0.216422
\(112\) −186380. −1.40396
\(113\) 141850. 1.04504 0.522521 0.852626i \(-0.324992\pi\)
0.522521 + 0.852626i \(0.324992\pi\)
\(114\) −891.035 −0.00642144
\(115\) 52923.6 0.373168
\(116\) 62805.3 0.433363
\(117\) −91075.2 −0.615086
\(118\) 104927. 0.693719
\(119\) 150519. 0.974370
\(120\) 19292.3 0.122302
\(121\) 14641.0 0.0909091
\(122\) −317556. −1.93162
\(123\) 52959.5 0.315632
\(124\) −193770. −1.13170
\(125\) −15625.0 −0.0894427
\(126\) −86297.7 −0.484254
\(127\) −92350.0 −0.508075 −0.254037 0.967194i \(-0.581759\pi\)
−0.254037 + 0.967194i \(0.581759\pi\)
\(128\) −163509. −0.882098
\(129\) −184701. −0.977225
\(130\) −202942. −1.05321
\(131\) 210539. 1.07190 0.535949 0.844250i \(-0.319954\pi\)
0.535949 + 0.844250i \(0.319954\pi\)
\(132\) −21914.6 −0.109471
\(133\) −2023.64 −0.00991981
\(134\) 227335. 1.09372
\(135\) 18225.0 0.0860663
\(136\) 87457.4 0.405461
\(137\) 23859.0 0.108605 0.0543027 0.998525i \(-0.482706\pi\)
0.0543027 + 0.998525i \(0.482706\pi\)
\(138\) −137553. −0.614853
\(139\) 292430. 1.28377 0.641883 0.766803i \(-0.278154\pi\)
0.641883 + 0.766803i \(0.278154\pi\)
\(140\) −74240.8 −0.320127
\(141\) −222467. −0.942361
\(142\) 210621. 0.876557
\(143\) −136051. −0.556366
\(144\) −102303. −0.411132
\(145\) −78024.5 −0.308185
\(146\) −357973. −1.38985
\(147\) −44728.3 −0.170722
\(148\) −62816.2 −0.235731
\(149\) 350530. 1.29348 0.646740 0.762710i \(-0.276132\pi\)
0.646740 + 0.762710i \(0.276132\pi\)
\(150\) 40610.6 0.147371
\(151\) −513284. −1.83196 −0.915979 0.401225i \(-0.868584\pi\)
−0.915979 + 0.401225i \(0.868584\pi\)
\(152\) −1175.81 −0.00412790
\(153\) 82618.8 0.285332
\(154\) −128914. −0.438024
\(155\) 240725. 0.804806
\(156\) 203640. 0.669964
\(157\) 154810. 0.501244 0.250622 0.968085i \(-0.419365\pi\)
0.250622 + 0.968085i \(0.419365\pi\)
\(158\) 614216. 1.95740
\(159\) −204850. −0.642604
\(160\) −159365. −0.492146
\(161\) −312397. −0.949821
\(162\) −47368.2 −0.141808
\(163\) 184152. 0.542883 0.271442 0.962455i \(-0.412500\pi\)
0.271442 + 0.962455i \(0.412500\pi\)
\(164\) −118415. −0.343793
\(165\) 27225.0 0.0778499
\(166\) −558238. −1.57235
\(167\) −72082.3 −0.200003 −0.100002 0.994987i \(-0.531885\pi\)
−0.100002 + 0.994987i \(0.531885\pi\)
\(168\) −113879. −0.311293
\(169\) 892949. 2.40497
\(170\) 184099. 0.488572
\(171\) −1110.76 −0.00290489
\(172\) 412983. 1.06441
\(173\) −374895. −0.952345 −0.476173 0.879352i \(-0.657976\pi\)
−0.476173 + 0.879352i \(0.657976\pi\)
\(174\) 202792. 0.507782
\(175\) 92231.0 0.227658
\(176\) −152823. −0.371882
\(177\) 130802. 0.313820
\(178\) −506948. −1.19926
\(179\) 354545. 0.827064 0.413532 0.910490i \(-0.364295\pi\)
0.413532 + 0.910490i \(0.364295\pi\)
\(180\) −40750.3 −0.0937452
\(181\) 403999. 0.916609 0.458305 0.888795i \(-0.348457\pi\)
0.458305 + 0.888795i \(0.348457\pi\)
\(182\) 1.19792e6 2.68072
\(183\) −395864. −0.873812
\(184\) −181515. −0.395246
\(185\) 78038.0 0.167640
\(186\) −625663. −1.32604
\(187\) 123418. 0.258093
\(188\) 497425. 1.02644
\(189\) −107578. −0.219064
\(190\) −2475.10 −0.00497403
\(191\) 181282. 0.359560 0.179780 0.983707i \(-0.442461\pi\)
0.179780 + 0.983707i \(0.442461\pi\)
\(192\) 50460.2 0.0987860
\(193\) 683762. 1.32133 0.660665 0.750681i \(-0.270274\pi\)
0.660665 + 0.750681i \(0.270274\pi\)
\(194\) 152190. 0.290324
\(195\) −252987. −0.476443
\(196\) 100010. 0.185954
\(197\) −442098. −0.811620 −0.405810 0.913958i \(-0.633010\pi\)
−0.405810 + 0.913958i \(0.633010\pi\)
\(198\) −70760.0 −0.128270
\(199\) 925615. 1.65691 0.828453 0.560059i \(-0.189222\pi\)
0.828453 + 0.560059i \(0.189222\pi\)
\(200\) 53589.8 0.0947344
\(201\) 283395. 0.494768
\(202\) 435197. 0.750425
\(203\) 460562. 0.784419
\(204\) −184732. −0.310790
\(205\) 147110. 0.244487
\(206\) −603099. −0.990194
\(207\) −171472. −0.278143
\(208\) 1.42009e6 2.27593
\(209\) −1659.28 −0.00262757
\(210\) −239716. −0.375101
\(211\) 605880. 0.936873 0.468436 0.883497i \(-0.344818\pi\)
0.468436 + 0.883497i \(0.344818\pi\)
\(212\) 458035. 0.699938
\(213\) 262559. 0.396531
\(214\) −1.15001e6 −1.71659
\(215\) −513058. −0.756955
\(216\) −62507.2 −0.0911582
\(217\) −1.42095e6 −2.04846
\(218\) 225011. 0.320673
\(219\) −446247. −0.628731
\(220\) −60873.8 −0.0847957
\(221\) −1.14686e6 −1.57953
\(222\) −202827. −0.276213
\(223\) 581147. 0.782571 0.391286 0.920269i \(-0.372030\pi\)
0.391286 + 0.920269i \(0.372030\pi\)
\(224\) 940699. 1.25265
\(225\) 50625.0 0.0666667
\(226\) −1.02411e6 −1.33376
\(227\) 1.51651e6 1.95335 0.976674 0.214726i \(-0.0688859\pi\)
0.976674 + 0.214726i \(0.0688859\pi\)
\(228\) 2483.61 0.00316407
\(229\) 916540. 1.15495 0.577474 0.816409i \(-0.304038\pi\)
0.577474 + 0.816409i \(0.304038\pi\)
\(230\) −382091. −0.476263
\(231\) −160703. −0.198150
\(232\) 267605. 0.326418
\(233\) −650940. −0.785509 −0.392755 0.919643i \(-0.628478\pi\)
−0.392755 + 0.919643i \(0.628478\pi\)
\(234\) 657533. 0.785014
\(235\) −617963. −0.729950
\(236\) −292467. −0.341820
\(237\) 765679. 0.885474
\(238\) −1.08670e6 −1.24356
\(239\) −1.06473e6 −1.20572 −0.602860 0.797847i \(-0.705972\pi\)
−0.602860 + 0.797847i \(0.705972\pi\)
\(240\) −284174. −0.318461
\(241\) −1.21958e6 −1.35259 −0.676297 0.736629i \(-0.736417\pi\)
−0.676297 + 0.736629i \(0.736417\pi\)
\(242\) −105703. −0.116024
\(243\) −59049.0 −0.0641500
\(244\) 885133. 0.951774
\(245\) −124245. −0.132240
\(246\) −382350. −0.402831
\(247\) 15418.8 0.0160808
\(248\) −825625. −0.852420
\(249\) −695897. −0.711290
\(250\) 112807. 0.114153
\(251\) 1.82454e6 1.82797 0.913987 0.405743i \(-0.132987\pi\)
0.913987 + 0.405743i \(0.132987\pi\)
\(252\) 240540. 0.238609
\(253\) −256150. −0.251590
\(254\) 666736. 0.648440
\(255\) 229497. 0.221017
\(256\) 1.35990e6 1.29690
\(257\) −475038. −0.448637 −0.224319 0.974516i \(-0.572016\pi\)
−0.224319 + 0.974516i \(0.572016\pi\)
\(258\) 1.33348e6 1.24720
\(259\) −460642. −0.426691
\(260\) 565667. 0.518952
\(261\) 252799. 0.229707
\(262\) −1.52002e6 −1.36803
\(263\) −1.07504e6 −0.958376 −0.479188 0.877712i \(-0.659069\pi\)
−0.479188 + 0.877712i \(0.659069\pi\)
\(264\) −93375.0 −0.0824557
\(265\) −569028. −0.497759
\(266\) 14610.0 0.0126603
\(267\) −631959. −0.542514
\(268\) −633658. −0.538912
\(269\) 180662. 0.152225 0.0761126 0.997099i \(-0.475749\pi\)
0.0761126 + 0.997099i \(0.475749\pi\)
\(270\) −131578. −0.109844
\(271\) 302884. 0.250526 0.125263 0.992124i \(-0.460023\pi\)
0.125263 + 0.992124i \(0.460023\pi\)
\(272\) −1.28824e6 −1.05578
\(273\) 1.49333e6 1.21269
\(274\) −172254. −0.138610
\(275\) 75625.0 0.0603023
\(276\) 383404. 0.302959
\(277\) 1.08898e6 0.852745 0.426372 0.904548i \(-0.359791\pi\)
0.426372 + 0.904548i \(0.359791\pi\)
\(278\) −2.11125e6 −1.63843
\(279\) −779948. −0.599867
\(280\) −316329. −0.241126
\(281\) −942590. −0.712127 −0.356063 0.934462i \(-0.615881\pi\)
−0.356063 + 0.934462i \(0.615881\pi\)
\(282\) 1.60614e6 1.20271
\(283\) 848952. 0.630111 0.315055 0.949073i \(-0.397977\pi\)
0.315055 + 0.949073i \(0.397977\pi\)
\(284\) −587069. −0.431910
\(285\) −3085.44 −0.00225012
\(286\) 982240. 0.710072
\(287\) −868357. −0.622291
\(288\) 516343. 0.366824
\(289\) −379486. −0.267271
\(290\) 563311. 0.393327
\(291\) 189720. 0.131335
\(292\) 997788. 0.684827
\(293\) 628367. 0.427606 0.213803 0.976877i \(-0.431415\pi\)
0.213803 + 0.976877i \(0.431415\pi\)
\(294\) 322923. 0.217887
\(295\) 363339. 0.243084
\(296\) −267651. −0.177558
\(297\) −88209.0 −0.0580259
\(298\) −2.53071e6 −1.65083
\(299\) 2.38026e6 1.53974
\(300\) −113195. −0.0726147
\(301\) 3.02847e6 1.92667
\(302\) 3.70574e6 2.33807
\(303\) 542514. 0.339473
\(304\) 17319.6 0.0107486
\(305\) −1.09962e6 −0.676852
\(306\) −596480. −0.364160
\(307\) 1.88248e6 1.13994 0.569972 0.821664i \(-0.306954\pi\)
0.569972 + 0.821664i \(0.306954\pi\)
\(308\) 359325. 0.215830
\(309\) −751820. −0.447938
\(310\) −1.73795e6 −1.02715
\(311\) −1.53497e6 −0.899908 −0.449954 0.893052i \(-0.648560\pi\)
−0.449954 + 0.893052i \(0.648560\pi\)
\(312\) 867681. 0.504631
\(313\) −2.22901e6 −1.28603 −0.643015 0.765854i \(-0.722317\pi\)
−0.643015 + 0.765854i \(0.722317\pi\)
\(314\) −1.11767e6 −0.639722
\(315\) −298829. −0.169686
\(316\) −1.71202e6 −0.964477
\(317\) 1.00659e6 0.562605 0.281303 0.959619i \(-0.409234\pi\)
0.281303 + 0.959619i \(0.409234\pi\)
\(318\) 1.47895e6 0.820135
\(319\) 377639. 0.207778
\(320\) 140167. 0.0765193
\(321\) −1.43359e6 −0.776539
\(322\) 2.25540e6 1.21223
\(323\) −13987.2 −0.00745973
\(324\) 132031. 0.0698736
\(325\) −702741. −0.369051
\(326\) −1.32951e6 −0.692865
\(327\) 280497. 0.145064
\(328\) −504549. −0.258952
\(329\) 3.64770e6 1.85793
\(330\) −196555. −0.0993574
\(331\) −2.96963e6 −1.48982 −0.744908 0.667167i \(-0.767507\pi\)
−0.744908 + 0.667167i \(0.767507\pi\)
\(332\) 1.55599e6 0.774752
\(333\) −252843. −0.124951
\(334\) 520410. 0.255258
\(335\) 787208. 0.383246
\(336\) 1.67742e6 0.810575
\(337\) 69782.5 0.0334712 0.0167356 0.999860i \(-0.494673\pi\)
0.0167356 + 0.999860i \(0.494673\pi\)
\(338\) −6.44680e6 −3.06939
\(339\) −1.27665e6 −0.603356
\(340\) −513144. −0.240737
\(341\) −1.16511e6 −0.542600
\(342\) 8019.32 0.00370742
\(343\) −1.74681e6 −0.801697
\(344\) 1.75966e6 0.801739
\(345\) −476312. −0.215449
\(346\) 2.70662e6 1.21545
\(347\) −1.04233e6 −0.464708 −0.232354 0.972631i \(-0.574643\pi\)
−0.232354 + 0.972631i \(0.574643\pi\)
\(348\) −565248. −0.250202
\(349\) −3.75012e6 −1.64809 −0.824047 0.566522i \(-0.808289\pi\)
−0.824047 + 0.566522i \(0.808289\pi\)
\(350\) −665877. −0.290552
\(351\) 819677. 0.355120
\(352\) 771328. 0.331805
\(353\) 800792. 0.342045 0.171022 0.985267i \(-0.445293\pi\)
0.171022 + 0.985267i \(0.445293\pi\)
\(354\) −944346. −0.400519
\(355\) 729330. 0.307152
\(356\) 1.41303e6 0.590917
\(357\) −1.35467e6 −0.562553
\(358\) −2.55970e6 −1.05556
\(359\) 267422. 0.109512 0.0547559 0.998500i \(-0.482562\pi\)
0.0547559 + 0.998500i \(0.482562\pi\)
\(360\) −173631. −0.0706108
\(361\) −2.47591e6 −0.999924
\(362\) −2.91674e6 −1.16984
\(363\) −131769. −0.0524864
\(364\) −3.33901e6 −1.32088
\(365\) −1.23957e6 −0.487013
\(366\) 2.85800e6 1.11522
\(367\) 2.63334e6 1.02057 0.510283 0.860007i \(-0.329541\pi\)
0.510283 + 0.860007i \(0.329541\pi\)
\(368\) 2.67369e6 1.02918
\(369\) −476635. −0.182230
\(370\) −563408. −0.213953
\(371\) 3.35885e6 1.26694
\(372\) 1.74393e6 0.653388
\(373\) −3.93389e6 −1.46403 −0.732015 0.681289i \(-0.761420\pi\)
−0.732015 + 0.681289i \(0.761420\pi\)
\(374\) −891039. −0.329395
\(375\) 140625. 0.0516398
\(376\) 2.11946e6 0.773135
\(377\) −3.50918e6 −1.27161
\(378\) 776679. 0.279584
\(379\) −1.80355e6 −0.644956 −0.322478 0.946577i \(-0.604516\pi\)
−0.322478 + 0.946577i \(0.604516\pi\)
\(380\) 6898.91 0.00245088
\(381\) 831150. 0.293337
\(382\) −1.30880e6 −0.458895
\(383\) 2.19398e6 0.764252 0.382126 0.924110i \(-0.375192\pi\)
0.382126 + 0.924110i \(0.375192\pi\)
\(384\) 1.47158e6 0.509280
\(385\) −446398. −0.153487
\(386\) −4.93653e6 −1.68637
\(387\) 1.66231e6 0.564201
\(388\) −424204. −0.143053
\(389\) 354221. 0.118686 0.0593430 0.998238i \(-0.481099\pi\)
0.0593430 + 0.998238i \(0.481099\pi\)
\(390\) 1.82648e6 0.608070
\(391\) −2.15925e6 −0.714268
\(392\) 426130. 0.140064
\(393\) −1.89485e6 −0.618861
\(394\) 3.19180e6 1.03584
\(395\) 2.12689e6 0.685885
\(396\) 197231. 0.0632030
\(397\) 168373. 0.0536162 0.0268081 0.999641i \(-0.491466\pi\)
0.0268081 + 0.999641i \(0.491466\pi\)
\(398\) −6.68263e6 −2.11466
\(399\) 18212.7 0.00572720
\(400\) −789373. −0.246679
\(401\) −4.59458e6 −1.42687 −0.713436 0.700721i \(-0.752862\pi\)
−0.713436 + 0.700721i \(0.752862\pi\)
\(402\) −2.04602e6 −0.631457
\(403\) 1.08267e7 3.32073
\(404\) −1.21304e6 −0.369761
\(405\) −164025. −0.0496904
\(406\) −3.32510e6 −1.00113
\(407\) −377704. −0.113023
\(408\) −787117. −0.234093
\(409\) 3.59299e6 1.06206 0.531028 0.847354i \(-0.321806\pi\)
0.531028 + 0.847354i \(0.321806\pi\)
\(410\) −1.06208e6 −0.312032
\(411\) −214731. −0.0627033
\(412\) 1.68103e6 0.487903
\(413\) −2.14471e6 −0.618719
\(414\) 1.23797e6 0.354986
\(415\) −1.93305e6 −0.550963
\(416\) −7.16752e6 −2.03065
\(417\) −2.63187e6 −0.741182
\(418\) 11979.5 0.00335349
\(419\) −4.10701e6 −1.14285 −0.571427 0.820653i \(-0.693610\pi\)
−0.571427 + 0.820653i \(0.693610\pi\)
\(420\) 668167. 0.184825
\(421\) 5.36278e6 1.47464 0.737318 0.675546i \(-0.236092\pi\)
0.737318 + 0.675546i \(0.236092\pi\)
\(422\) −4.37425e6 −1.19570
\(423\) 2.00220e6 0.544072
\(424\) 1.95162e6 0.527208
\(425\) 637491. 0.171199
\(426\) −1.89559e6 −0.506081
\(427\) 6.49083e6 1.72278
\(428\) 3.20545e6 0.845822
\(429\) 1.22446e6 0.321218
\(430\) 3.70411e6 0.966078
\(431\) 2.52160e6 0.653857 0.326929 0.945049i \(-0.393986\pi\)
0.326929 + 0.945049i \(0.393986\pi\)
\(432\) 920724. 0.237367
\(433\) 2.99694e6 0.768171 0.384085 0.923298i \(-0.374517\pi\)
0.384085 + 0.923298i \(0.374517\pi\)
\(434\) 1.02588e7 2.61439
\(435\) 702221. 0.177931
\(436\) −627179. −0.158007
\(437\) 29029.8 0.00727178
\(438\) 3.22175e6 0.802430
\(439\) 1.93618e6 0.479495 0.239748 0.970835i \(-0.422935\pi\)
0.239748 + 0.970835i \(0.422935\pi\)
\(440\) −259375. −0.0638699
\(441\) 402554. 0.0985662
\(442\) 8.27992e6 2.01591
\(443\) −7.28132e6 −1.76279 −0.881395 0.472380i \(-0.843395\pi\)
−0.881395 + 0.472380i \(0.843395\pi\)
\(444\) 565346. 0.136100
\(445\) −1.75544e6 −0.420229
\(446\) −4.19569e6 −0.998771
\(447\) −3.15477e6 −0.746791
\(448\) −827377. −0.194764
\(449\) 6.15007e6 1.43967 0.719837 0.694143i \(-0.244216\pi\)
0.719837 + 0.694143i \(0.244216\pi\)
\(450\) −365496. −0.0850846
\(451\) −712011. −0.164833
\(452\) 2.85454e6 0.657188
\(453\) 4.61956e6 1.05768
\(454\) −1.09487e7 −2.49300
\(455\) 4.14813e6 0.939342
\(456\) 10582.3 0.00238324
\(457\) −5.60231e6 −1.25481 −0.627403 0.778695i \(-0.715882\pi\)
−0.627403 + 0.778695i \(0.715882\pi\)
\(458\) −6.61711e6 −1.47402
\(459\) −743570. −0.164737
\(460\) 1.06501e6 0.234671
\(461\) 2.85697e6 0.626113 0.313057 0.949734i \(-0.398647\pi\)
0.313057 + 0.949734i \(0.398647\pi\)
\(462\) 1.16022e6 0.252893
\(463\) 2.42615e6 0.525976 0.262988 0.964799i \(-0.415292\pi\)
0.262988 + 0.964799i \(0.415292\pi\)
\(464\) −3.94179e6 −0.849959
\(465\) −2.16652e6 −0.464655
\(466\) 4.69957e6 1.00252
\(467\) 167522. 0.0355451 0.0177726 0.999842i \(-0.494343\pi\)
0.0177726 + 0.999842i \(0.494343\pi\)
\(468\) −1.83276e6 −0.386804
\(469\) −4.64672e6 −0.975471
\(470\) 4.46149e6 0.931612
\(471\) −1.39329e6 −0.289393
\(472\) −1.24616e6 −0.257466
\(473\) 2.48320e6 0.510339
\(474\) −5.52795e6 −1.13010
\(475\) −8570.68 −0.00174293
\(476\) 3.02898e6 0.612744
\(477\) 1.84365e6 0.371008
\(478\) 7.68702e6 1.53882
\(479\) 5.39034e6 1.07344 0.536719 0.843761i \(-0.319663\pi\)
0.536719 + 0.843761i \(0.319663\pi\)
\(480\) 1.43429e6 0.284140
\(481\) 3.50979e6 0.691701
\(482\) 8.80496e6 1.72627
\(483\) 2.81157e6 0.548379
\(484\) 294629. 0.0571693
\(485\) 526999. 0.101732
\(486\) 426314. 0.0818727
\(487\) 1.38574e6 0.264764 0.132382 0.991199i \(-0.457737\pi\)
0.132382 + 0.991199i \(0.457737\pi\)
\(488\) 3.77143e6 0.716896
\(489\) −1.65736e6 −0.313434
\(490\) 897009. 0.168774
\(491\) −3.39592e6 −0.635703 −0.317851 0.948141i \(-0.602961\pi\)
−0.317851 + 0.948141i \(0.602961\pi\)
\(492\) 1.06573e6 0.198489
\(493\) 3.18336e6 0.589886
\(494\) −111319. −0.0205234
\(495\) −245025. −0.0449467
\(496\) 1.21614e7 2.21962
\(497\) −4.30508e6 −0.781790
\(498\) 5.02414e6 0.907797
\(499\) 2.83293e6 0.509313 0.254657 0.967032i \(-0.418038\pi\)
0.254657 + 0.967032i \(0.418038\pi\)
\(500\) −314431. −0.0562471
\(501\) 648741. 0.115472
\(502\) −1.31726e7 −2.33299
\(503\) 9.23328e6 1.62718 0.813591 0.581437i \(-0.197509\pi\)
0.813591 + 0.581437i \(0.197509\pi\)
\(504\) 1.02491e6 0.179725
\(505\) 1.50698e6 0.262954
\(506\) 1.84932e6 0.321096
\(507\) −8.03654e6 −1.38851
\(508\) −1.85841e6 −0.319509
\(509\) −5.70039e6 −0.975238 −0.487619 0.873057i \(-0.662134\pi\)
−0.487619 + 0.873057i \(0.662134\pi\)
\(510\) −1.65689e6 −0.282077
\(511\) 7.31695e6 1.23959
\(512\) −4.58570e6 −0.773091
\(513\) 9996.84 0.00167714
\(514\) 3.42961e6 0.572582
\(515\) −2.08839e6 −0.346971
\(516\) −3.71684e6 −0.614540
\(517\) 2.99094e6 0.492132
\(518\) 3.32568e6 0.544573
\(519\) 3.37406e6 0.549837
\(520\) 2.41023e6 0.390885
\(521\) −1.35063e6 −0.217994 −0.108997 0.994042i \(-0.534764\pi\)
−0.108997 + 0.994042i \(0.534764\pi\)
\(522\) −1.82513e6 −0.293168
\(523\) 440837. 0.0704731 0.0352366 0.999379i \(-0.488782\pi\)
0.0352366 + 0.999379i \(0.488782\pi\)
\(524\) 4.23679e6 0.674076
\(525\) −830079. −0.131438
\(526\) 7.76144e6 1.22314
\(527\) −9.82143e6 −1.54045
\(528\) 1.37540e6 0.214706
\(529\) −1.95489e6 −0.303727
\(530\) 4.10819e6 0.635274
\(531\) −1.17722e6 −0.181184
\(532\) −40722.8 −0.00623819
\(533\) 6.61632e6 1.00878
\(534\) 4.56253e6 0.692393
\(535\) −3.98220e6 −0.601504
\(536\) −2.69993e6 −0.405919
\(537\) −3.19091e6 −0.477506
\(538\) −1.30432e6 −0.194280
\(539\) 601347. 0.0891565
\(540\) 366752. 0.0541238
\(541\) 7.01375e6 1.03028 0.515142 0.857105i \(-0.327739\pi\)
0.515142 + 0.857105i \(0.327739\pi\)
\(542\) −2.18672e6 −0.319738
\(543\) −3.63599e6 −0.529205
\(544\) 6.50201e6 0.941999
\(545\) 779159. 0.112366
\(546\) −1.07813e7 −1.54771
\(547\) −1.01822e7 −1.45503 −0.727517 0.686090i \(-0.759326\pi\)
−0.727517 + 0.686090i \(0.759326\pi\)
\(548\) 480129. 0.0682978
\(549\) 3.56277e6 0.504496
\(550\) −545987. −0.0769619
\(551\) −42798.3 −0.00600547
\(552\) 1.63363e6 0.228195
\(553\) −1.25546e7 −1.74578
\(554\) −7.86205e6 −1.08833
\(555\) −702342. −0.0967868
\(556\) 5.88475e6 0.807311
\(557\) −1.09154e7 −1.49074 −0.745371 0.666650i \(-0.767728\pi\)
−0.745371 + 0.666650i \(0.767728\pi\)
\(558\) 5.63096e6 0.765592
\(559\) −2.30750e7 −3.12329
\(560\) 4.65950e6 0.627869
\(561\) −1.11076e6 −0.149010
\(562\) 6.80519e6 0.908865
\(563\) 1.25187e6 0.166452 0.0832261 0.996531i \(-0.473478\pi\)
0.0832261 + 0.996531i \(0.473478\pi\)
\(564\) −4.47683e6 −0.592615
\(565\) −3.54626e6 −0.467357
\(566\) −6.12915e6 −0.804191
\(567\) 968205. 0.126476
\(568\) −2.50142e6 −0.325324
\(569\) 3.75381e6 0.486061 0.243031 0.970019i \(-0.421858\pi\)
0.243031 + 0.970019i \(0.421858\pi\)
\(570\) 22275.9 0.00287176
\(571\) −7.39573e6 −0.949273 −0.474636 0.880182i \(-0.657420\pi\)
−0.474636 + 0.880182i \(0.657420\pi\)
\(572\) −2.73783e6 −0.349877
\(573\) −1.63154e6 −0.207592
\(574\) 6.26925e6 0.794210
\(575\) −1.32309e6 −0.166886
\(576\) −454141. −0.0570341
\(577\) −6.45181e6 −0.806755 −0.403378 0.915034i \(-0.632164\pi\)
−0.403378 + 0.915034i \(0.632164\pi\)
\(578\) 2.73976e6 0.341109
\(579\) −6.15386e6 −0.762871
\(580\) −1.57013e6 −0.193806
\(581\) 1.14104e7 1.40236
\(582\) −1.36971e6 −0.167619
\(583\) 2.75410e6 0.335589
\(584\) 4.25143e6 0.515826
\(585\) 2.27688e6 0.275075
\(586\) −4.53660e6 −0.545741
\(587\) 5.91956e6 0.709078 0.354539 0.935041i \(-0.384638\pi\)
0.354539 + 0.935041i \(0.384638\pi\)
\(588\) −900093. −0.107360
\(589\) 132043. 0.0156829
\(590\) −2.62318e6 −0.310241
\(591\) 3.97888e6 0.468589
\(592\) 3.94247e6 0.462343
\(593\) −2.54171e6 −0.296817 −0.148409 0.988926i \(-0.547415\pi\)
−0.148409 + 0.988926i \(0.547415\pi\)
\(594\) 636840. 0.0740566
\(595\) −3.76297e6 −0.435751
\(596\) 7.05393e6 0.813421
\(597\) −8.33053e6 −0.956615
\(598\) −1.71847e7 −1.96512
\(599\) 1.17309e7 1.33587 0.667935 0.744220i \(-0.267179\pi\)
0.667935 + 0.744220i \(0.267179\pi\)
\(600\) −482309. −0.0546949
\(601\) −2.88018e6 −0.325262 −0.162631 0.986687i \(-0.551998\pi\)
−0.162631 + 0.986687i \(0.551998\pi\)
\(602\) −2.18646e7 −2.45895
\(603\) −2.55055e6 −0.285655
\(604\) −1.03291e7 −1.15205
\(605\) −366025. −0.0406558
\(606\) −3.91677e6 −0.433258
\(607\) −1.33207e7 −1.46742 −0.733712 0.679460i \(-0.762214\pi\)
−0.733712 + 0.679460i \(0.762214\pi\)
\(608\) −87415.6 −0.00959025
\(609\) −4.14506e6 −0.452885
\(610\) 7.93890e6 0.863845
\(611\) −2.77931e7 −3.01186
\(612\) 1.66259e6 0.179434
\(613\) −8.85926e6 −0.952239 −0.476120 0.879381i \(-0.657957\pi\)
−0.476120 + 0.879381i \(0.657957\pi\)
\(614\) −1.35908e7 −1.45487
\(615\) −1.32399e6 −0.141155
\(616\) 1.53103e6 0.162567
\(617\) 1.75148e7 1.85222 0.926110 0.377253i \(-0.123131\pi\)
0.926110 + 0.377253i \(0.123131\pi\)
\(618\) 5.42789e6 0.571689
\(619\) 3.94768e6 0.414109 0.207055 0.978329i \(-0.433612\pi\)
0.207055 + 0.978329i \(0.433612\pi\)
\(620\) 4.84424e6 0.506112
\(621\) 1.54325e6 0.160586
\(622\) 1.10820e7 1.14852
\(623\) 1.03620e7 1.06960
\(624\) −1.27808e7 −1.31401
\(625\) 390625. 0.0400000
\(626\) 1.60927e7 1.64132
\(627\) 14933.6 0.00151703
\(628\) 3.11533e6 0.315213
\(629\) −3.18391e6 −0.320873
\(630\) 2.15744e6 0.216565
\(631\) −5.78181e6 −0.578083 −0.289042 0.957317i \(-0.593337\pi\)
−0.289042 + 0.957317i \(0.593337\pi\)
\(632\) −7.29469e6 −0.726464
\(633\) −5.45292e6 −0.540904
\(634\) −7.26723e6 −0.718035
\(635\) 2.30875e6 0.227218
\(636\) −4.12232e6 −0.404109
\(637\) −5.58798e6 −0.545640
\(638\) −2.72643e6 −0.265181
\(639\) −2.36303e6 −0.228937
\(640\) 4.08773e6 0.394486
\(641\) −4.31966e6 −0.415245 −0.207623 0.978209i \(-0.566573\pi\)
−0.207623 + 0.978209i \(0.566573\pi\)
\(642\) 1.03501e7 0.991072
\(643\) 1.95083e7 1.86076 0.930381 0.366593i \(-0.119476\pi\)
0.930381 + 0.366593i \(0.119476\pi\)
\(644\) −6.28654e6 −0.597306
\(645\) 4.61752e6 0.437028
\(646\) 100983. 0.00952062
\(647\) −1.08322e7 −1.01732 −0.508660 0.860967i \(-0.669859\pi\)
−0.508660 + 0.860967i \(0.669859\pi\)
\(648\) 562565. 0.0526302
\(649\) −1.75856e6 −0.163887
\(650\) 5.07355e6 0.471009
\(651\) 1.27885e7 1.18268
\(652\) 3.70579e6 0.341399
\(653\) 1.83320e6 0.168239 0.0841197 0.996456i \(-0.473192\pi\)
0.0841197 + 0.996456i \(0.473192\pi\)
\(654\) −2.02510e6 −0.185140
\(655\) −5.26346e6 −0.479367
\(656\) 7.43195e6 0.674285
\(657\) 4.01622e6 0.362998
\(658\) −2.63352e7 −2.37122
\(659\) 1.77259e7 1.58999 0.794994 0.606618i \(-0.207474\pi\)
0.794994 + 0.606618i \(0.207474\pi\)
\(660\) 547865. 0.0489568
\(661\) −5.89810e6 −0.525060 −0.262530 0.964924i \(-0.584557\pi\)
−0.262530 + 0.964924i \(0.584557\pi\)
\(662\) 2.14398e7 1.90141
\(663\) 1.03217e7 0.911943
\(664\) 6.62987e6 0.583559
\(665\) 50590.9 0.00443627
\(666\) 1.82544e6 0.159471
\(667\) −6.60694e6 −0.575024
\(668\) −1.45055e6 −0.125775
\(669\) −5.23032e6 −0.451818
\(670\) −5.68338e6 −0.489125
\(671\) 5.32217e6 0.456333
\(672\) −8.46629e6 −0.723219
\(673\) −6.98177e6 −0.594193 −0.297097 0.954847i \(-0.596018\pi\)
−0.297097 + 0.954847i \(0.596018\pi\)
\(674\) −503806. −0.0427183
\(675\) −455625. −0.0384900
\(676\) 1.79693e7 1.51240
\(677\) 5.68896e6 0.477047 0.238524 0.971137i \(-0.423337\pi\)
0.238524 + 0.971137i \(0.423337\pi\)
\(678\) 9.21700e6 0.770044
\(679\) −3.11076e6 −0.258936
\(680\) −2.18644e6 −0.181328
\(681\) −1.36486e7 −1.12777
\(682\) 8.41169e6 0.692504
\(683\) 1.65539e6 0.135784 0.0678920 0.997693i \(-0.478373\pi\)
0.0678920 + 0.997693i \(0.478373\pi\)
\(684\) −22352.5 −0.00182678
\(685\) −596475. −0.0485698
\(686\) 1.26114e7 1.02318
\(687\) −8.24886e6 −0.666810
\(688\) −2.59196e7 −2.08765
\(689\) −2.55923e7 −2.05381
\(690\) 3.43882e6 0.274971
\(691\) −8.36041e6 −0.666090 −0.333045 0.942911i \(-0.608076\pi\)
−0.333045 + 0.942911i \(0.608076\pi\)
\(692\) −7.54423e6 −0.598894
\(693\) 1.44633e6 0.114402
\(694\) 7.52524e6 0.593092
\(695\) −7.31076e6 −0.574117
\(696\) −2.40844e6 −0.188457
\(697\) −6.00199e6 −0.467965
\(698\) 2.70746e7 2.10341
\(699\) 5.85846e6 0.453514
\(700\) 1.85602e6 0.143165
\(701\) 1.07851e7 0.828948 0.414474 0.910061i \(-0.363966\pi\)
0.414474 + 0.910061i \(0.363966\pi\)
\(702\) −5.91779e6 −0.453228
\(703\) 42805.7 0.00326673
\(704\) −678409. −0.0515893
\(705\) 5.56167e6 0.421437
\(706\) −5.78145e6 −0.436541
\(707\) −8.89541e6 −0.669295
\(708\) 2.63220e6 0.197350
\(709\) −1.37457e7 −1.02695 −0.513476 0.858104i \(-0.671642\pi\)
−0.513476 + 0.858104i \(0.671642\pi\)
\(710\) −5.26552e6 −0.392008
\(711\) −6.89111e6 −0.511229
\(712\) 6.02073e6 0.445091
\(713\) 2.03840e7 1.50164
\(714\) 9.78027e6 0.717968
\(715\) 3.40127e6 0.248814
\(716\) 7.13472e6 0.520109
\(717\) 9.58260e6 0.696122
\(718\) −1.93070e6 −0.139767
\(719\) 2.23290e6 0.161082 0.0805409 0.996751i \(-0.474335\pi\)
0.0805409 + 0.996751i \(0.474335\pi\)
\(720\) 2.55757e6 0.183864
\(721\) 1.23273e7 0.883141
\(722\) 1.78753e7 1.27617
\(723\) 1.09762e7 0.780921
\(724\) 8.12991e6 0.576421
\(725\) 1.95061e6 0.137824
\(726\) 951328. 0.0669867
\(727\) 7.70500e6 0.540675 0.270338 0.962766i \(-0.412865\pi\)
0.270338 + 0.962766i \(0.412865\pi\)
\(728\) −1.42270e7 −0.994916
\(729\) 531441. 0.0370370
\(730\) 8.94932e6 0.621560
\(731\) 2.09325e7 1.44886
\(732\) −7.96620e6 −0.549507
\(733\) −5.23205e6 −0.359676 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(734\) −1.90118e7 −1.30252
\(735\) 1.11821e6 0.0763491
\(736\) −1.34947e7 −0.918266
\(737\) −3.81009e6 −0.258384
\(738\) 3.44115e6 0.232575
\(739\) 1.90698e7 1.28450 0.642252 0.766494i \(-0.278000\pi\)
0.642252 + 0.766494i \(0.278000\pi\)
\(740\) 1.57040e6 0.105422
\(741\) −138769. −0.00928426
\(742\) −2.42498e7 −1.61696
\(743\) 2.31407e7 1.53781 0.768907 0.639361i \(-0.220801\pi\)
0.768907 + 0.639361i \(0.220801\pi\)
\(744\) 7.43063e6 0.492145
\(745\) −8.76326e6 −0.578462
\(746\) 2.84013e7 1.86849
\(747\) 6.26307e6 0.410663
\(748\) 2.48362e6 0.162305
\(749\) 2.35061e7 1.53100
\(750\) −1.01527e6 −0.0659062
\(751\) 1.96879e7 1.27380 0.636898 0.770948i \(-0.280217\pi\)
0.636898 + 0.770948i \(0.280217\pi\)
\(752\) −3.12194e7 −2.01317
\(753\) −1.64209e7 −1.05538
\(754\) 2.53351e7 1.62291
\(755\) 1.28321e7 0.819277
\(756\) −2.16486e6 −0.137761
\(757\) −8.22162e6 −0.521456 −0.260728 0.965412i \(-0.583963\pi\)
−0.260728 + 0.965412i \(0.583963\pi\)
\(758\) 1.30210e7 0.823137
\(759\) 2.30535e6 0.145256
\(760\) 29395.3 0.00184605
\(761\) 2.18719e6 0.136907 0.0684534 0.997654i \(-0.478194\pi\)
0.0684534 + 0.997654i \(0.478194\pi\)
\(762\) −6.00063e6 −0.374377
\(763\) −4.59921e6 −0.286004
\(764\) 3.64805e6 0.226114
\(765\) −2.06547e6 −0.127604
\(766\) −1.58398e7 −0.975391
\(767\) 1.63413e7 1.00299
\(768\) −1.22391e7 −0.748764
\(769\) 3.97950e6 0.242668 0.121334 0.992612i \(-0.461283\pi\)
0.121334 + 0.992612i \(0.461283\pi\)
\(770\) 3.22285e6 0.195890
\(771\) 4.27534e6 0.259021
\(772\) 1.37597e7 0.830935
\(773\) −1.92979e7 −1.16161 −0.580807 0.814042i \(-0.697263\pi\)
−0.580807 + 0.814042i \(0.697263\pi\)
\(774\) −1.20013e7 −0.720072
\(775\) −6.01812e6 −0.359920
\(776\) −1.80748e6 −0.107750
\(777\) 4.14578e6 0.246350
\(778\) −2.55735e6 −0.151475
\(779\) 80693.1 0.00476423
\(780\) −5.09100e6 −0.299617
\(781\) −3.52996e6 −0.207082
\(782\) 1.55891e7 0.911599
\(783\) −2.27519e6 −0.132622
\(784\) −6.27685e6 −0.364713
\(785\) −3.87024e6 −0.224163
\(786\) 1.36802e7 0.789832
\(787\) −1.54391e7 −0.888556 −0.444278 0.895889i \(-0.646540\pi\)
−0.444278 + 0.895889i \(0.646540\pi\)
\(788\) −8.89659e6 −0.510397
\(789\) 9.67537e6 0.553318
\(790\) −1.53554e7 −0.875374
\(791\) 2.09328e7 1.18956
\(792\) 840375. 0.0476058
\(793\) −4.94559e7 −2.79277
\(794\) −1.21560e6 −0.0684287
\(795\) 5.12125e6 0.287381
\(796\) 1.86267e7 1.04196
\(797\) 2.66071e7 1.48372 0.741860 0.670555i \(-0.233944\pi\)
0.741860 + 0.670555i \(0.233944\pi\)
\(798\) −131490. −0.00730945
\(799\) 2.52125e7 1.39717
\(800\) 3.98413e6 0.220094
\(801\) 5.68763e6 0.313220
\(802\) 3.31713e7 1.82107
\(803\) 5.99954e6 0.328344
\(804\) 5.70292e6 0.311141
\(805\) 7.80992e6 0.424773
\(806\) −7.81651e7 −4.23814
\(807\) −1.62596e6 −0.0878872
\(808\) −5.16858e6 −0.278511
\(809\) −3.15651e7 −1.69565 −0.847825 0.530276i \(-0.822088\pi\)
−0.847825 + 0.530276i \(0.822088\pi\)
\(810\) 1.18421e6 0.0634183
\(811\) 3.50230e7 1.86982 0.934912 0.354880i \(-0.115478\pi\)
0.934912 + 0.354880i \(0.115478\pi\)
\(812\) 9.26816e6 0.493291
\(813\) −2.72595e6 −0.144641
\(814\) 2.72690e6 0.144247
\(815\) −4.60379e6 −0.242785
\(816\) 1.15941e7 0.609555
\(817\) −281424. −0.0147505
\(818\) −2.59402e7 −1.35547
\(819\) −1.34399e7 −0.700144
\(820\) 2.96037e6 0.153749
\(821\) −6.00202e6 −0.310770 −0.155385 0.987854i \(-0.549662\pi\)
−0.155385 + 0.987854i \(0.549662\pi\)
\(822\) 1.55029e6 0.0800263
\(823\) −2.98462e7 −1.53599 −0.767996 0.640455i \(-0.778746\pi\)
−0.767996 + 0.640455i \(0.778746\pi\)
\(824\) 7.16265e6 0.367499
\(825\) −680625. −0.0348155
\(826\) 1.54841e7 0.789652
\(827\) −3.81477e6 −0.193956 −0.0969782 0.995287i \(-0.530918\pi\)
−0.0969782 + 0.995287i \(0.530918\pi\)
\(828\) −3.45064e6 −0.174914
\(829\) 7.23404e6 0.365590 0.182795 0.983151i \(-0.441486\pi\)
0.182795 + 0.983151i \(0.441486\pi\)
\(830\) 1.39560e7 0.703177
\(831\) −9.80079e6 −0.492332
\(832\) 6.30407e6 0.315728
\(833\) 5.06913e6 0.253117
\(834\) 1.90013e7 0.945948
\(835\) 1.80206e6 0.0894442
\(836\) −33390.7 −0.00165238
\(837\) 7.01953e6 0.346333
\(838\) 2.96512e7 1.45859
\(839\) −2.18515e6 −0.107171 −0.0535853 0.998563i \(-0.517065\pi\)
−0.0535853 + 0.998563i \(0.517065\pi\)
\(840\) 2.84697e6 0.139214
\(841\) −1.07706e7 −0.525111
\(842\) −3.87175e7 −1.88203
\(843\) 8.48331e6 0.411146
\(844\) 1.21925e7 0.589164
\(845\) −2.23237e7 −1.07554
\(846\) −1.44552e7 −0.694383
\(847\) 2.16057e6 0.103481
\(848\) −2.87472e7 −1.37280
\(849\) −7.64056e6 −0.363795
\(850\) −4.60247e6 −0.218496
\(851\) 6.60808e6 0.312789
\(852\) 5.28362e6 0.249363
\(853\) −1.41107e7 −0.664014 −0.332007 0.943277i \(-0.607726\pi\)
−0.332007 + 0.943277i \(0.607726\pi\)
\(854\) −4.68616e7 −2.19873
\(855\) 27769.0 0.00129911
\(856\) 1.36580e7 0.637091
\(857\) 3.50068e7 1.62817 0.814087 0.580743i \(-0.197238\pi\)
0.814087 + 0.580743i \(0.197238\pi\)
\(858\) −8.84016e6 −0.409960
\(859\) 2.72242e6 0.125885 0.0629423 0.998017i \(-0.479952\pi\)
0.0629423 + 0.998017i \(0.479952\pi\)
\(860\) −1.03246e7 −0.476020
\(861\) 7.81521e6 0.359280
\(862\) −1.82051e7 −0.834498
\(863\) 976336. 0.0446244 0.0223122 0.999751i \(-0.492897\pi\)
0.0223122 + 0.999751i \(0.492897\pi\)
\(864\) −4.64709e6 −0.211786
\(865\) 9.37238e6 0.425902
\(866\) −2.16369e7 −0.980392
\(867\) 3.41538e6 0.154309
\(868\) −2.85945e7 −1.28820
\(869\) −1.02941e7 −0.462424
\(870\) −5.06980e6 −0.227087
\(871\) 3.54050e7 1.58132
\(872\) −2.67232e6 −0.119014
\(873\) −1.70748e6 −0.0758262
\(874\) −209586. −0.00928075
\(875\) −2.30578e6 −0.101812
\(876\) −8.98009e6 −0.395385
\(877\) 3.04885e7 1.33856 0.669279 0.743011i \(-0.266603\pi\)
0.669279 + 0.743011i \(0.266603\pi\)
\(878\) −1.39786e7 −0.611965
\(879\) −5.65530e6 −0.246879
\(880\) 3.82056e6 0.166311
\(881\) 3.54112e7 1.53710 0.768548 0.639792i \(-0.220980\pi\)
0.768548 + 0.639792i \(0.220980\pi\)
\(882\) −2.90631e6 −0.125797
\(883\) 7.34069e6 0.316836 0.158418 0.987372i \(-0.449361\pi\)
0.158418 + 0.987372i \(0.449361\pi\)
\(884\) −2.30789e7 −0.993308
\(885\) −3.27005e6 −0.140345
\(886\) 5.25687e7 2.24979
\(887\) 8.51565e6 0.363420 0.181710 0.983352i \(-0.441837\pi\)
0.181710 + 0.983352i \(0.441837\pi\)
\(888\) 2.40886e6 0.102513
\(889\) −1.36281e7 −0.578335
\(890\) 1.26737e7 0.536326
\(891\) 793881. 0.0335013
\(892\) 1.16948e7 0.492129
\(893\) −338967. −0.0142242
\(894\) 2.27764e7 0.953107
\(895\) −8.86363e6 −0.369874
\(896\) −2.41290e7 −1.00408
\(897\) −2.14223e7 −0.888967
\(898\) −4.44015e7 −1.83741
\(899\) −3.00519e7 −1.24014
\(900\) 1.01876e6 0.0419241
\(901\) 2.32160e7 0.952743
\(902\) 5.14048e6 0.210372
\(903\) −2.72562e7 −1.11236
\(904\) 1.21628e7 0.495007
\(905\) −1.01000e7 −0.409920
\(906\) −3.33517e7 −1.34989
\(907\) 1.84383e7 0.744224 0.372112 0.928188i \(-0.378634\pi\)
0.372112 + 0.928188i \(0.378634\pi\)
\(908\) 3.05176e7 1.22839
\(909\) −4.88263e6 −0.195995
\(910\) −2.99481e7 −1.19885
\(911\) 3.20540e7 1.27964 0.639819 0.768526i \(-0.279009\pi\)
0.639819 + 0.768526i \(0.279009\pi\)
\(912\) −155876. −0.00620573
\(913\) 9.35595e6 0.371459
\(914\) 4.04468e7 1.60147
\(915\) 9.89659e6 0.390781
\(916\) 1.84441e7 0.726303
\(917\) 3.10691e7 1.22013
\(918\) 5.36832e6 0.210248
\(919\) 2.41323e7 0.942563 0.471282 0.881983i \(-0.343792\pi\)
0.471282 + 0.881983i \(0.343792\pi\)
\(920\) 4.53787e6 0.176759
\(921\) −1.69423e7 −0.658147
\(922\) −2.06264e7 −0.799089
\(923\) 3.28019e7 1.26735
\(924\) −3.23393e6 −0.124609
\(925\) −1.95095e6 −0.0749708
\(926\) −1.75160e7 −0.671287
\(927\) 6.76638e6 0.258617
\(928\) 1.98950e7 0.758359
\(929\) −2.12500e7 −0.807828 −0.403914 0.914797i \(-0.632350\pi\)
−0.403914 + 0.914797i \(0.632350\pi\)
\(930\) 1.56416e7 0.593025
\(931\) −68151.4 −0.00257692
\(932\) −1.30993e7 −0.493977
\(933\) 1.38147e7 0.519562
\(934\) −1.20945e6 −0.0453652
\(935\) −3.08546e6 −0.115422
\(936\) −7.80913e6 −0.291349
\(937\) −2.96484e7 −1.10320 −0.551598 0.834110i \(-0.685982\pi\)
−0.551598 + 0.834110i \(0.685982\pi\)
\(938\) 3.35478e7 1.24496
\(939\) 2.00611e7 0.742490
\(940\) −1.24356e7 −0.459038
\(941\) −4.32235e7 −1.59128 −0.795638 0.605772i \(-0.792864\pi\)
−0.795638 + 0.605772i \(0.792864\pi\)
\(942\) 1.00591e7 0.369344
\(943\) 1.24569e7 0.456175
\(944\) 1.83558e7 0.670415
\(945\) 2.68946e6 0.0979682
\(946\) −1.79279e7 −0.651330
\(947\) 1.03051e7 0.373402 0.186701 0.982417i \(-0.440220\pi\)
0.186701 + 0.982417i \(0.440220\pi\)
\(948\) 1.54082e7 0.556841
\(949\) −5.57504e7 −2.00947
\(950\) 61877.4 0.00222445
\(951\) −9.05929e6 −0.324820
\(952\) 1.29061e7 0.461531
\(953\) 2.76302e7 0.985488 0.492744 0.870174i \(-0.335994\pi\)
0.492744 + 0.870174i \(0.335994\pi\)
\(954\) −1.33105e7 −0.473505
\(955\) −4.53205e6 −0.160800
\(956\) −2.14263e7 −0.758231
\(957\) −3.39875e6 −0.119961
\(958\) −3.89164e7 −1.37000
\(959\) 3.52087e6 0.123624
\(960\) −1.26150e6 −0.0441785
\(961\) 6.40882e7 2.23856
\(962\) −2.53395e7 −0.882797
\(963\) 1.29023e7 0.448335
\(964\) −2.45423e7 −0.850596
\(965\) −1.70940e7 −0.590917
\(966\) −2.02986e7 −0.699879
\(967\) −1.92779e7 −0.662968 −0.331484 0.943461i \(-0.607549\pi\)
−0.331484 + 0.943461i \(0.607549\pi\)
\(968\) 1.25537e6 0.0430611
\(969\) 125884. 0.00430688
\(970\) −3.80476e6 −0.129837
\(971\) −2.79869e7 −0.952592 −0.476296 0.879285i \(-0.658021\pi\)
−0.476296 + 0.879285i \(0.658021\pi\)
\(972\) −1.18828e6 −0.0403415
\(973\) 4.31539e7 1.46129
\(974\) −1.00046e7 −0.337910
\(975\) 6.32467e6 0.213072
\(976\) −5.55527e7 −1.86673
\(977\) 3.40783e7 1.14220 0.571099 0.820881i \(-0.306517\pi\)
0.571099 + 0.820881i \(0.306517\pi\)
\(978\) 1.19656e7 0.400026
\(979\) 8.49634e6 0.283319
\(980\) −2.50026e6 −0.0831610
\(981\) −2.52447e6 −0.0837526
\(982\) 2.45174e7 0.811328
\(983\) −5.50760e7 −1.81794 −0.908968 0.416866i \(-0.863128\pi\)
−0.908968 + 0.416866i \(0.863128\pi\)
\(984\) 4.54094e6 0.149506
\(985\) 1.10524e7 0.362967
\(986\) −2.29828e7 −0.752853
\(987\) −3.28293e7 −1.07268
\(988\) 310281. 0.0101126
\(989\) −4.34446e7 −1.41236
\(990\) 1.76900e6 0.0573640
\(991\) −6.15387e6 −0.199051 −0.0995255 0.995035i \(-0.531732\pi\)
−0.0995255 + 0.995035i \(0.531732\pi\)
\(992\) −6.13810e7 −1.98041
\(993\) 2.67267e7 0.860146
\(994\) 3.10812e7 0.997774
\(995\) −2.31404e7 −0.740990
\(996\) −1.40039e7 −0.447303
\(997\) 1.41497e7 0.450828 0.225414 0.974263i \(-0.427627\pi\)
0.225414 + 0.974263i \(0.427627\pi\)
\(998\) −2.04528e7 −0.650021
\(999\) 2.27559e6 0.0721406
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.e.1.1 3
3.2 odd 2 495.6.a.a.1.3 3
5.4 even 2 825.6.a.f.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.e.1.1 3 1.1 even 1 trivial
495.6.a.a.1.3 3 3.2 odd 2
825.6.a.f.1.3 3 5.4 even 2