Properties

Label 165.4.a.h.1.3
Level $165$
Weight $4$
Character 165.1
Self dual yes
Analytic conductor $9.735$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1540841.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 27x^{2} - 18x + 92 \) 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 1.3
Root \(-2.63835\) 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)
 
\(f(q)\) \(=\) \(q+3.63835 q^{2} +3.00000 q^{3} +5.23763 q^{4} +5.00000 q^{5} +10.9151 q^{6} +20.8444 q^{7} -10.0505 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.63835 q^{2} +3.00000 q^{3} +5.23763 q^{4} +5.00000 q^{5} +10.9151 q^{6} +20.8444 q^{7} -10.0505 q^{8} +9.00000 q^{9} +18.1918 q^{10} -11.0000 q^{11} +15.7129 q^{12} +67.4988 q^{13} +75.8394 q^{14} +15.0000 q^{15} -78.4683 q^{16} -57.8120 q^{17} +32.7452 q^{18} -7.98646 q^{19} +26.1881 q^{20} +62.5333 q^{21} -40.0219 q^{22} +67.5185 q^{23} -30.1515 q^{24} +25.0000 q^{25} +245.585 q^{26} +27.0000 q^{27} +109.175 q^{28} -56.1558 q^{29} +54.5753 q^{30} -127.085 q^{31} -205.091 q^{32} -33.0000 q^{33} -210.341 q^{34} +104.222 q^{35} +47.1386 q^{36} -95.4222 q^{37} -29.0576 q^{38} +202.497 q^{39} -50.2525 q^{40} -485.903 q^{41} +227.518 q^{42} -146.216 q^{43} -57.6139 q^{44} +45.0000 q^{45} +245.656 q^{46} +164.296 q^{47} -235.405 q^{48} +91.4901 q^{49} +90.9589 q^{50} -173.436 q^{51} +353.534 q^{52} +431.492 q^{53} +98.2356 q^{54} -55.0000 q^{55} -209.497 q^{56} -23.9594 q^{57} -204.315 q^{58} -804.178 q^{59} +78.5644 q^{60} -120.847 q^{61} -462.381 q^{62} +187.600 q^{63} -118.449 q^{64} +337.494 q^{65} -120.066 q^{66} -371.469 q^{67} -302.798 q^{68} +202.555 q^{69} +379.197 q^{70} +529.835 q^{71} -90.4545 q^{72} +1059.19 q^{73} -347.180 q^{74} +75.0000 q^{75} -41.8301 q^{76} -229.289 q^{77} +736.754 q^{78} -168.663 q^{79} -392.341 q^{80} +81.0000 q^{81} -1767.89 q^{82} -144.130 q^{83} +327.526 q^{84} -289.060 q^{85} -531.986 q^{86} -168.468 q^{87} +110.555 q^{88} -1400.20 q^{89} +163.726 q^{90} +1406.97 q^{91} +353.637 q^{92} -381.256 q^{93} +597.767 q^{94} -39.9323 q^{95} -615.274 q^{96} +29.6912 q^{97} +332.873 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 34 q^{7} + 48 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 34 q^{7} + 48 q^{8} + 36 q^{9} + 20 q^{10} - 44 q^{11} + 78 q^{12} + 2 q^{13} - 52 q^{14} + 60 q^{15} + 66 q^{16} + 74 q^{17} + 36 q^{18} + 136 q^{19} + 130 q^{20} + 102 q^{21} - 44 q^{22} - 64 q^{23} + 144 q^{24} + 100 q^{25} - 320 q^{26} + 108 q^{27} - 20 q^{28} + 52 q^{29} + 60 q^{30} + 492 q^{31} + 208 q^{32} - 132 q^{33} + 244 q^{34} + 170 q^{35} + 234 q^{36} - 4 q^{37} - 404 q^{38} + 6 q^{39} + 240 q^{40} + 268 q^{41} - 156 q^{42} + 546 q^{43} - 286 q^{44} + 180 q^{45} + 368 q^{46} - 276 q^{47} + 198 q^{48} - 496 q^{49} + 100 q^{50} + 222 q^{51} - 1084 q^{52} - 184 q^{53} + 108 q^{54} - 220 q^{55} - 852 q^{56} + 408 q^{57} - 444 q^{58} - 1032 q^{59} + 390 q^{60} + 116 q^{61} - 1240 q^{62} + 306 q^{63} - 918 q^{64} + 10 q^{65} - 132 q^{66} - 552 q^{67} - 720 q^{68} - 192 q^{69} - 260 q^{70} - 920 q^{71} + 432 q^{72} + 926 q^{73} - 2856 q^{74} + 300 q^{75} + 1572 q^{76} - 374 q^{77} - 960 q^{78} + 1152 q^{79} + 330 q^{80} + 324 q^{81} - 1924 q^{82} - 134 q^{83} - 60 q^{84} + 370 q^{85} + 236 q^{86} + 156 q^{87} - 528 q^{88} - 1064 q^{89} + 180 q^{90} + 2780 q^{91} - 4896 q^{92} + 1476 q^{93} - 1432 q^{94} + 680 q^{95} + 624 q^{96} - 1648 q^{97} - 188 q^{98} - 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.63835 1.28635 0.643176 0.765718i \(-0.277616\pi\)
0.643176 + 0.765718i \(0.277616\pi\)
\(3\) 3.00000 0.577350
\(4\) 5.23763 0.654703
\(5\) 5.00000 0.447214
\(6\) 10.9151 0.742676
\(7\) 20.8444 1.12549 0.562747 0.826629i \(-0.309745\pi\)
0.562747 + 0.826629i \(0.309745\pi\)
\(8\) −10.0505 −0.444173
\(9\) 9.00000 0.333333
\(10\) 18.1918 0.575274
\(11\) −11.0000 −0.301511
\(12\) 15.7129 0.377993
\(13\) 67.4988 1.44006 0.720031 0.693942i \(-0.244128\pi\)
0.720031 + 0.693942i \(0.244128\pi\)
\(14\) 75.8394 1.44778
\(15\) 15.0000 0.258199
\(16\) −78.4683 −1.22607
\(17\) −57.8120 −0.824792 −0.412396 0.911005i \(-0.635308\pi\)
−0.412396 + 0.911005i \(0.635308\pi\)
\(18\) 32.7452 0.428784
\(19\) −7.98646 −0.0964326 −0.0482163 0.998837i \(-0.515354\pi\)
−0.0482163 + 0.998837i \(0.515354\pi\)
\(20\) 26.1881 0.292792
\(21\) 62.5333 0.649804
\(22\) −40.0219 −0.387850
\(23\) 67.5185 0.612112 0.306056 0.952013i \(-0.400990\pi\)
0.306056 + 0.952013i \(0.400990\pi\)
\(24\) −30.1515 −0.256444
\(25\) 25.0000 0.200000
\(26\) 245.585 1.85243
\(27\) 27.0000 0.192450
\(28\) 109.175 0.736864
\(29\) −56.1558 −0.359582 −0.179791 0.983705i \(-0.557542\pi\)
−0.179791 + 0.983705i \(0.557542\pi\)
\(30\) 54.5753 0.332135
\(31\) −127.085 −0.736296 −0.368148 0.929767i \(-0.620008\pi\)
−0.368148 + 0.929767i \(0.620008\pi\)
\(32\) −205.091 −1.13298
\(33\) −33.0000 −0.174078
\(34\) −210.341 −1.06097
\(35\) 104.222 0.503336
\(36\) 47.1386 0.218234
\(37\) −95.4222 −0.423981 −0.211991 0.977272i \(-0.567995\pi\)
−0.211991 + 0.977272i \(0.567995\pi\)
\(38\) −29.0576 −0.124046
\(39\) 202.497 0.831420
\(40\) −50.2525 −0.198640
\(41\) −485.903 −1.85086 −0.925431 0.378917i \(-0.876297\pi\)
−0.925431 + 0.378917i \(0.876297\pi\)
\(42\) 227.518 0.835877
\(43\) −146.216 −0.518552 −0.259276 0.965803i \(-0.583484\pi\)
−0.259276 + 0.965803i \(0.583484\pi\)
\(44\) −57.6139 −0.197400
\(45\) 45.0000 0.149071
\(46\) 245.656 0.787392
\(47\) 164.296 0.509894 0.254947 0.966955i \(-0.417942\pi\)
0.254947 + 0.966955i \(0.417942\pi\)
\(48\) −235.405 −0.707870
\(49\) 91.4901 0.266735
\(50\) 90.9589 0.257271
\(51\) −173.436 −0.476194
\(52\) 353.534 0.942813
\(53\) 431.492 1.11830 0.559151 0.829066i \(-0.311127\pi\)
0.559151 + 0.829066i \(0.311127\pi\)
\(54\) 98.2356 0.247559
\(55\) −55.0000 −0.134840
\(56\) −209.497 −0.499914
\(57\) −23.9594 −0.0556754
\(58\) −204.315 −0.462549
\(59\) −804.178 −1.77449 −0.887246 0.461296i \(-0.847385\pi\)
−0.887246 + 0.461296i \(0.847385\pi\)
\(60\) 78.5644 0.169044
\(61\) −120.847 −0.253653 −0.126826 0.991925i \(-0.540479\pi\)
−0.126826 + 0.991925i \(0.540479\pi\)
\(62\) −462.381 −0.947137
\(63\) 187.600 0.375164
\(64\) −118.449 −0.231346
\(65\) 337.494 0.644015
\(66\) −120.066 −0.223925
\(67\) −371.469 −0.677346 −0.338673 0.940904i \(-0.609978\pi\)
−0.338673 + 0.940904i \(0.609978\pi\)
\(68\) −302.798 −0.539994
\(69\) 202.555 0.353403
\(70\) 379.197 0.647467
\(71\) 529.835 0.885631 0.442816 0.896613i \(-0.353980\pi\)
0.442816 + 0.896613i \(0.353980\pi\)
\(72\) −90.4545 −0.148058
\(73\) 1059.19 1.69820 0.849100 0.528232i \(-0.177145\pi\)
0.849100 + 0.528232i \(0.177145\pi\)
\(74\) −347.180 −0.545389
\(75\) 75.0000 0.115470
\(76\) −41.8301 −0.0631348
\(77\) −229.289 −0.339349
\(78\) 736.754 1.06950
\(79\) −168.663 −0.240204 −0.120102 0.992762i \(-0.538322\pi\)
−0.120102 + 0.992762i \(0.538322\pi\)
\(80\) −392.341 −0.548314
\(81\) 81.0000 0.111111
\(82\) −1767.89 −2.38086
\(83\) −144.130 −0.190606 −0.0953032 0.995448i \(-0.530382\pi\)
−0.0953032 + 0.995448i \(0.530382\pi\)
\(84\) 327.526 0.425429
\(85\) −289.060 −0.368858
\(86\) −531.986 −0.667041
\(87\) −168.468 −0.207605
\(88\) 110.555 0.133923
\(89\) −1400.20 −1.66765 −0.833823 0.552032i \(-0.813853\pi\)
−0.833823 + 0.552032i \(0.813853\pi\)
\(90\) 163.726 0.191758
\(91\) 1406.97 1.62078
\(92\) 353.637 0.400752
\(93\) −381.256 −0.425101
\(94\) 597.767 0.655904
\(95\) −39.9323 −0.0431260
\(96\) −615.274 −0.654127
\(97\) 29.6912 0.0310792 0.0155396 0.999879i \(-0.495053\pi\)
0.0155396 + 0.999879i \(0.495053\pi\)
\(98\) 332.873 0.343115
\(99\) −99.0000 −0.100504
\(100\) 130.941 0.130941
\(101\) −17.5982 −0.0173375 −0.00866874 0.999962i \(-0.502759\pi\)
−0.00866874 + 0.999962i \(0.502759\pi\)
\(102\) −631.022 −0.612553
\(103\) 1423.03 1.36131 0.680655 0.732604i \(-0.261695\pi\)
0.680655 + 0.732604i \(0.261695\pi\)
\(104\) −678.397 −0.639637
\(105\) 312.666 0.290601
\(106\) 1569.92 1.43853
\(107\) −1335.15 −1.20629 −0.603147 0.797630i \(-0.706087\pi\)
−0.603147 + 0.797630i \(0.706087\pi\)
\(108\) 141.416 0.125998
\(109\) 1565.39 1.37557 0.687784 0.725916i \(-0.258584\pi\)
0.687784 + 0.725916i \(0.258584\pi\)
\(110\) −200.110 −0.173452
\(111\) −286.266 −0.244786
\(112\) −1635.63 −1.37993
\(113\) 1176.83 0.979709 0.489854 0.871804i \(-0.337050\pi\)
0.489854 + 0.871804i \(0.337050\pi\)
\(114\) −87.1728 −0.0716182
\(115\) 337.592 0.273745
\(116\) −294.123 −0.235420
\(117\) 607.490 0.480021
\(118\) −2925.89 −2.28262
\(119\) −1205.06 −0.928298
\(120\) −150.757 −0.114685
\(121\) 121.000 0.0909091
\(122\) −439.683 −0.326287
\(123\) −1457.71 −1.06860
\(124\) −665.625 −0.482056
\(125\) 125.000 0.0894427
\(126\) 682.555 0.482594
\(127\) 311.155 0.217406 0.108703 0.994074i \(-0.465330\pi\)
0.108703 + 0.994074i \(0.465330\pi\)
\(128\) 1209.77 0.835388
\(129\) −438.648 −0.299386
\(130\) 1227.92 0.828431
\(131\) 582.818 0.388711 0.194355 0.980931i \(-0.437739\pi\)
0.194355 + 0.980931i \(0.437739\pi\)
\(132\) −172.842 −0.113969
\(133\) −166.473 −0.108534
\(134\) −1351.54 −0.871306
\(135\) 135.000 0.0860663
\(136\) 581.039 0.366351
\(137\) −2367.98 −1.47672 −0.738358 0.674410i \(-0.764398\pi\)
−0.738358 + 0.674410i \(0.764398\pi\)
\(138\) 736.969 0.454601
\(139\) 2573.48 1.57036 0.785179 0.619269i \(-0.212571\pi\)
0.785179 + 0.619269i \(0.212571\pi\)
\(140\) 545.877 0.329536
\(141\) 492.888 0.294388
\(142\) 1927.73 1.13923
\(143\) −742.487 −0.434195
\(144\) −706.215 −0.408689
\(145\) −280.779 −0.160810
\(146\) 3853.70 2.18448
\(147\) 274.470 0.153999
\(148\) −499.786 −0.277582
\(149\) 1992.38 1.09545 0.547724 0.836659i \(-0.315494\pi\)
0.547724 + 0.836659i \(0.315494\pi\)
\(150\) 272.877 0.148535
\(151\) 2961.20 1.59589 0.797944 0.602731i \(-0.205921\pi\)
0.797944 + 0.602731i \(0.205921\pi\)
\(152\) 80.2679 0.0428328
\(153\) −520.308 −0.274931
\(154\) −834.234 −0.436522
\(155\) −635.426 −0.329282
\(156\) 1060.60 0.544334
\(157\) 1302.56 0.662139 0.331070 0.943606i \(-0.392591\pi\)
0.331070 + 0.943606i \(0.392591\pi\)
\(158\) −613.657 −0.308987
\(159\) 1294.48 0.645652
\(160\) −1025.46 −0.506685
\(161\) 1407.38 0.688928
\(162\) 294.707 0.142928
\(163\) −563.163 −0.270615 −0.135308 0.990804i \(-0.543202\pi\)
−0.135308 + 0.990804i \(0.543202\pi\)
\(164\) −2544.98 −1.21176
\(165\) −165.000 −0.0778499
\(166\) −524.396 −0.245187
\(167\) 1128.96 0.523124 0.261562 0.965187i \(-0.415762\pi\)
0.261562 + 0.965187i \(0.415762\pi\)
\(168\) −628.490 −0.288626
\(169\) 2359.09 1.07378
\(170\) −1051.70 −0.474482
\(171\) −71.8782 −0.0321442
\(172\) −765.825 −0.339498
\(173\) 3091.97 1.35883 0.679416 0.733754i \(-0.262233\pi\)
0.679416 + 0.733754i \(0.262233\pi\)
\(174\) −612.945 −0.267053
\(175\) 521.111 0.225099
\(176\) 863.151 0.369673
\(177\) −2412.53 −1.02450
\(178\) −5094.41 −2.14518
\(179\) 2879.27 1.20227 0.601136 0.799147i \(-0.294715\pi\)
0.601136 + 0.799147i \(0.294715\pi\)
\(180\) 235.693 0.0975974
\(181\) −702.137 −0.288339 −0.144170 0.989553i \(-0.546051\pi\)
−0.144170 + 0.989553i \(0.546051\pi\)
\(182\) 5119.07 2.08489
\(183\) −362.540 −0.146447
\(184\) −678.594 −0.271884
\(185\) −477.111 −0.189610
\(186\) −1387.14 −0.546830
\(187\) 635.932 0.248684
\(188\) 860.521 0.333829
\(189\) 562.799 0.216601
\(190\) −145.288 −0.0554752
\(191\) 4294.28 1.62682 0.813412 0.581688i \(-0.197608\pi\)
0.813412 + 0.581688i \(0.197608\pi\)
\(192\) −355.348 −0.133568
\(193\) −3888.11 −1.45012 −0.725059 0.688687i \(-0.758187\pi\)
−0.725059 + 0.688687i \(0.758187\pi\)
\(194\) 108.027 0.0399788
\(195\) 1012.48 0.371822
\(196\) 479.191 0.174632
\(197\) −3062.26 −1.10750 −0.553748 0.832684i \(-0.686803\pi\)
−0.553748 + 0.832684i \(0.686803\pi\)
\(198\) −360.197 −0.129283
\(199\) 4874.64 1.73645 0.868226 0.496168i \(-0.165260\pi\)
0.868226 + 0.496168i \(0.165260\pi\)
\(200\) −251.262 −0.0888347
\(201\) −1114.41 −0.391066
\(202\) −64.0285 −0.0223021
\(203\) −1170.54 −0.404707
\(204\) −908.393 −0.311766
\(205\) −2429.52 −0.827730
\(206\) 5177.47 1.75112
\(207\) 607.666 0.204037
\(208\) −5296.52 −1.76561
\(209\) 87.8511 0.0290755
\(210\) 1137.59 0.373815
\(211\) −4324.02 −1.41080 −0.705398 0.708811i \(-0.749232\pi\)
−0.705398 + 0.708811i \(0.749232\pi\)
\(212\) 2259.99 0.732156
\(213\) 1589.50 0.511319
\(214\) −4857.74 −1.55172
\(215\) −731.081 −0.231904
\(216\) −271.363 −0.0854812
\(217\) −2649.02 −0.828697
\(218\) 5695.43 1.76947
\(219\) 3177.56 0.980456
\(220\) −288.069 −0.0882802
\(221\) −3902.24 −1.18775
\(222\) −1041.54 −0.314881
\(223\) −6205.69 −1.86351 −0.931757 0.363084i \(-0.881724\pi\)
−0.931757 + 0.363084i \(0.881724\pi\)
\(224\) −4275.01 −1.27516
\(225\) 225.000 0.0666667
\(226\) 4281.73 1.26025
\(227\) 1535.81 0.449055 0.224528 0.974468i \(-0.427916\pi\)
0.224528 + 0.974468i \(0.427916\pi\)
\(228\) −125.490 −0.0364509
\(229\) 94.7362 0.0273378 0.0136689 0.999907i \(-0.495649\pi\)
0.0136689 + 0.999907i \(0.495649\pi\)
\(230\) 1228.28 0.352132
\(231\) −687.866 −0.195923
\(232\) 564.394 0.159717
\(233\) 654.983 0.184160 0.0920801 0.995752i \(-0.470648\pi\)
0.0920801 + 0.995752i \(0.470648\pi\)
\(234\) 2210.26 0.617476
\(235\) 821.480 0.228032
\(236\) −4211.98 −1.16177
\(237\) −505.990 −0.138682
\(238\) −4384.43 −1.19412
\(239\) −5660.64 −1.53204 −0.766018 0.642819i \(-0.777765\pi\)
−0.766018 + 0.642819i \(0.777765\pi\)
\(240\) −1177.02 −0.316569
\(241\) −4156.88 −1.11107 −0.555536 0.831493i \(-0.687487\pi\)
−0.555536 + 0.831493i \(0.687487\pi\)
\(242\) 440.241 0.116941
\(243\) 243.000 0.0641500
\(244\) −632.950 −0.166067
\(245\) 457.450 0.119287
\(246\) −5303.66 −1.37459
\(247\) −539.077 −0.138869
\(248\) 1277.27 0.327043
\(249\) −432.390 −0.110047
\(250\) 454.794 0.115055
\(251\) −7156.50 −1.79966 −0.899829 0.436243i \(-0.856309\pi\)
−0.899829 + 0.436243i \(0.856309\pi\)
\(252\) 982.578 0.245621
\(253\) −742.703 −0.184559
\(254\) 1132.09 0.279660
\(255\) −867.180 −0.212960
\(256\) 5349.17 1.30595
\(257\) −2188.36 −0.531152 −0.265576 0.964090i \(-0.585562\pi\)
−0.265576 + 0.964090i \(0.585562\pi\)
\(258\) −1595.96 −0.385116
\(259\) −1989.02 −0.477188
\(260\) 1767.67 0.421639
\(261\) −505.403 −0.119861
\(262\) 2120.50 0.500019
\(263\) 757.388 0.177576 0.0887881 0.996051i \(-0.471701\pi\)
0.0887881 + 0.996051i \(0.471701\pi\)
\(264\) 331.666 0.0773207
\(265\) 2157.46 0.500120
\(266\) −605.689 −0.139613
\(267\) −4200.59 −0.962816
\(268\) −1945.62 −0.443461
\(269\) −6782.64 −1.53734 −0.768671 0.639645i \(-0.779081\pi\)
−0.768671 + 0.639645i \(0.779081\pi\)
\(270\) 491.178 0.110712
\(271\) 7040.87 1.57824 0.789119 0.614241i \(-0.210538\pi\)
0.789119 + 0.614241i \(0.210538\pi\)
\(272\) 4536.41 1.01125
\(273\) 4220.92 0.935758
\(274\) −8615.54 −1.89958
\(275\) −275.000 −0.0603023
\(276\) 1060.91 0.231374
\(277\) −3211.46 −0.696599 −0.348300 0.937383i \(-0.613241\pi\)
−0.348300 + 0.937383i \(0.613241\pi\)
\(278\) 9363.24 2.02003
\(279\) −1143.77 −0.245432
\(280\) −1047.48 −0.223568
\(281\) 3986.15 0.846241 0.423120 0.906073i \(-0.360935\pi\)
0.423120 + 0.906073i \(0.360935\pi\)
\(282\) 1793.30 0.378686
\(283\) 7838.75 1.64652 0.823260 0.567664i \(-0.192153\pi\)
0.823260 + 0.567664i \(0.192153\pi\)
\(284\) 2775.08 0.579826
\(285\) −119.797 −0.0248988
\(286\) −2701.43 −0.558528
\(287\) −10128.4 −2.08313
\(288\) −1845.82 −0.377660
\(289\) −1570.77 −0.319718
\(290\) −1021.57 −0.206858
\(291\) 89.0735 0.0179436
\(292\) 5547.63 1.11182
\(293\) −5430.65 −1.08280 −0.541402 0.840764i \(-0.682106\pi\)
−0.541402 + 0.840764i \(0.682106\pi\)
\(294\) 998.620 0.198098
\(295\) −4020.89 −0.793577
\(296\) 959.040 0.188321
\(297\) −297.000 −0.0580259
\(298\) 7248.97 1.40913
\(299\) 4557.42 0.881480
\(300\) 392.822 0.0755986
\(301\) −3047.79 −0.583627
\(302\) 10773.9 2.05288
\(303\) −52.7946 −0.0100098
\(304\) 626.684 0.118233
\(305\) −604.233 −0.113437
\(306\) −1893.06 −0.353658
\(307\) 8045.25 1.49566 0.747828 0.663892i \(-0.231097\pi\)
0.747828 + 0.663892i \(0.231097\pi\)
\(308\) −1200.93 −0.222173
\(309\) 4269.08 0.785952
\(310\) −2311.91 −0.423573
\(311\) −4712.06 −0.859152 −0.429576 0.903031i \(-0.641337\pi\)
−0.429576 + 0.903031i \(0.641337\pi\)
\(312\) −2035.19 −0.369295
\(313\) −1425.19 −0.257369 −0.128684 0.991686i \(-0.541075\pi\)
−0.128684 + 0.991686i \(0.541075\pi\)
\(314\) 4739.19 0.851744
\(315\) 937.999 0.167779
\(316\) −883.395 −0.157262
\(317\) −2031.54 −0.359944 −0.179972 0.983672i \(-0.557601\pi\)
−0.179972 + 0.983672i \(0.557601\pi\)
\(318\) 4709.76 0.830536
\(319\) 617.714 0.108418
\(320\) −592.247 −0.103461
\(321\) −4005.44 −0.696454
\(322\) 5120.56 0.886204
\(323\) 461.713 0.0795369
\(324\) 424.248 0.0727448
\(325\) 1687.47 0.288012
\(326\) −2048.99 −0.348107
\(327\) 4696.16 0.794184
\(328\) 4883.57 0.822103
\(329\) 3424.65 0.573882
\(330\) −600.329 −0.100142
\(331\) 316.790 0.0526054 0.0263027 0.999654i \(-0.491627\pi\)
0.0263027 + 0.999654i \(0.491627\pi\)
\(332\) −754.899 −0.124791
\(333\) −858.799 −0.141327
\(334\) 4107.57 0.672922
\(335\) −1857.35 −0.302918
\(336\) −4906.88 −0.796703
\(337\) 4441.65 0.717958 0.358979 0.933346i \(-0.383125\pi\)
0.358979 + 0.933346i \(0.383125\pi\)
\(338\) 8583.22 1.38126
\(339\) 3530.50 0.565635
\(340\) −1513.99 −0.241493
\(341\) 1397.94 0.222002
\(342\) −261.518 −0.0413488
\(343\) −5242.58 −0.825285
\(344\) 1469.54 0.230327
\(345\) 1012.78 0.158047
\(346\) 11249.7 1.74794
\(347\) 258.695 0.0400215 0.0200108 0.999800i \(-0.493630\pi\)
0.0200108 + 0.999800i \(0.493630\pi\)
\(348\) −882.370 −0.135920
\(349\) 9929.62 1.52298 0.761491 0.648176i \(-0.224468\pi\)
0.761491 + 0.648176i \(0.224468\pi\)
\(350\) 1895.99 0.289556
\(351\) 1822.47 0.277140
\(352\) 2256.01 0.341607
\(353\) −1008.73 −0.152095 −0.0760474 0.997104i \(-0.524230\pi\)
−0.0760474 + 0.997104i \(0.524230\pi\)
\(354\) −8777.66 −1.31787
\(355\) 2649.17 0.396066
\(356\) −7333.70 −1.09181
\(357\) −3615.17 −0.535953
\(358\) 10475.8 1.54655
\(359\) 5813.26 0.854630 0.427315 0.904103i \(-0.359460\pi\)
0.427315 + 0.904103i \(0.359460\pi\)
\(360\) −452.272 −0.0662135
\(361\) −6795.22 −0.990701
\(362\) −2554.62 −0.370906
\(363\) 363.000 0.0524864
\(364\) 7369.21 1.06113
\(365\) 5295.94 0.759458
\(366\) −1319.05 −0.188382
\(367\) −10247.0 −1.45747 −0.728733 0.684798i \(-0.759890\pi\)
−0.728733 + 0.684798i \(0.759890\pi\)
\(368\) −5298.06 −0.750490
\(369\) −4373.13 −0.616954
\(370\) −1735.90 −0.243906
\(371\) 8994.20 1.25864
\(372\) −1996.88 −0.278315
\(373\) −2202.63 −0.305759 −0.152879 0.988245i \(-0.548855\pi\)
−0.152879 + 0.988245i \(0.548855\pi\)
\(374\) 2313.75 0.319896
\(375\) 375.000 0.0516398
\(376\) −1651.26 −0.226481
\(377\) −3790.45 −0.517821
\(378\) 2047.66 0.278626
\(379\) −1851.13 −0.250887 −0.125444 0.992101i \(-0.540035\pi\)
−0.125444 + 0.992101i \(0.540035\pi\)
\(380\) −209.151 −0.0282347
\(381\) 933.464 0.125519
\(382\) 15624.1 2.09267
\(383\) 5880.65 0.784562 0.392281 0.919846i \(-0.371686\pi\)
0.392281 + 0.919846i \(0.371686\pi\)
\(384\) 3629.31 0.482311
\(385\) −1146.44 −0.151761
\(386\) −14146.3 −1.86536
\(387\) −1315.95 −0.172851
\(388\) 155.511 0.0203476
\(389\) −6963.66 −0.907639 −0.453820 0.891094i \(-0.649939\pi\)
−0.453820 + 0.891094i \(0.649939\pi\)
\(390\) 3683.77 0.478295
\(391\) −3903.38 −0.504865
\(392\) −919.520 −0.118477
\(393\) 1748.46 0.224422
\(394\) −11141.6 −1.42463
\(395\) −843.316 −0.107422
\(396\) −518.525 −0.0658002
\(397\) 3024.31 0.382332 0.191166 0.981558i \(-0.438773\pi\)
0.191166 + 0.981558i \(0.438773\pi\)
\(398\) 17735.7 2.23369
\(399\) −499.420 −0.0626623
\(400\) −1961.71 −0.245213
\(401\) −13392.7 −1.66783 −0.833914 0.551895i \(-0.813905\pi\)
−0.833914 + 0.551895i \(0.813905\pi\)
\(402\) −4054.61 −0.503049
\(403\) −8578.11 −1.06031
\(404\) −92.1728 −0.0113509
\(405\) 405.000 0.0496904
\(406\) −4258.83 −0.520596
\(407\) 1049.64 0.127835
\(408\) 1743.12 0.211513
\(409\) 7637.16 0.923308 0.461654 0.887060i \(-0.347256\pi\)
0.461654 + 0.887060i \(0.347256\pi\)
\(410\) −8839.44 −1.06475
\(411\) −7103.93 −0.852582
\(412\) 7453.28 0.891254
\(413\) −16762.6 −1.99718
\(414\) 2210.91 0.262464
\(415\) −720.650 −0.0852417
\(416\) −13843.4 −1.63156
\(417\) 7720.44 0.906647
\(418\) 319.633 0.0374014
\(419\) 12523.9 1.46022 0.730112 0.683328i \(-0.239468\pi\)
0.730112 + 0.683328i \(0.239468\pi\)
\(420\) 1637.63 0.190257
\(421\) −11150.0 −1.29078 −0.645391 0.763852i \(-0.723306\pi\)
−0.645391 + 0.763852i \(0.723306\pi\)
\(422\) −15732.3 −1.81478
\(423\) 1478.66 0.169965
\(424\) −4336.71 −0.496720
\(425\) −1445.30 −0.164958
\(426\) 5783.18 0.657737
\(427\) −2518.98 −0.285485
\(428\) −6993.00 −0.789765
\(429\) −2227.46 −0.250683
\(430\) −2659.93 −0.298310
\(431\) −2093.77 −0.233998 −0.116999 0.993132i \(-0.537327\pi\)
−0.116999 + 0.993132i \(0.537327\pi\)
\(432\) −2118.64 −0.235957
\(433\) −4391.47 −0.487392 −0.243696 0.969852i \(-0.578360\pi\)
−0.243696 + 0.969852i \(0.578360\pi\)
\(434\) −9638.07 −1.06600
\(435\) −842.338 −0.0928437
\(436\) 8198.91 0.900589
\(437\) −539.234 −0.0590276
\(438\) 11561.1 1.26121
\(439\) 15614.6 1.69759 0.848797 0.528719i \(-0.177328\pi\)
0.848797 + 0.528719i \(0.177328\pi\)
\(440\) 552.777 0.0598923
\(441\) 823.411 0.0889116
\(442\) −14197.7 −1.52787
\(443\) 7909.57 0.848296 0.424148 0.905593i \(-0.360574\pi\)
0.424148 + 0.905593i \(0.360574\pi\)
\(444\) −1499.36 −0.160262
\(445\) −7000.98 −0.745794
\(446\) −22578.5 −2.39714
\(447\) 5977.13 0.632458
\(448\) −2469.01 −0.260379
\(449\) −14958.7 −1.57226 −0.786128 0.618064i \(-0.787917\pi\)
−0.786128 + 0.618064i \(0.787917\pi\)
\(450\) 818.630 0.0857568
\(451\) 5344.93 0.558056
\(452\) 6163.81 0.641419
\(453\) 8883.61 0.921387
\(454\) 5587.84 0.577644
\(455\) 7034.87 0.724835
\(456\) 240.804 0.0247295
\(457\) 14584.6 1.49286 0.746430 0.665464i \(-0.231766\pi\)
0.746430 + 0.665464i \(0.231766\pi\)
\(458\) 344.684 0.0351660
\(459\) −1560.92 −0.158731
\(460\) 1768.18 0.179222
\(461\) −1802.35 −0.182091 −0.0910454 0.995847i \(-0.529021\pi\)
−0.0910454 + 0.995847i \(0.529021\pi\)
\(462\) −2502.70 −0.252026
\(463\) 6753.79 0.677916 0.338958 0.940801i \(-0.389926\pi\)
0.338958 + 0.940801i \(0.389926\pi\)
\(464\) 4406.45 0.440872
\(465\) −1906.28 −0.190111
\(466\) 2383.06 0.236895
\(467\) −1744.55 −0.172866 −0.0864329 0.996258i \(-0.527547\pi\)
−0.0864329 + 0.996258i \(0.527547\pi\)
\(468\) 3181.80 0.314271
\(469\) −7743.06 −0.762349
\(470\) 2988.83 0.293329
\(471\) 3907.69 0.382286
\(472\) 8082.39 0.788182
\(473\) 1608.38 0.156349
\(474\) −1840.97 −0.178394
\(475\) −199.662 −0.0192865
\(476\) −6311.64 −0.607760
\(477\) 3883.43 0.372767
\(478\) −20595.4 −1.97074
\(479\) 16930.6 1.61499 0.807495 0.589874i \(-0.200823\pi\)
0.807495 + 0.589874i \(0.200823\pi\)
\(480\) −3076.37 −0.292534
\(481\) −6440.88 −0.610559
\(482\) −15124.2 −1.42923
\(483\) 4222.15 0.397753
\(484\) 633.753 0.0595185
\(485\) 148.456 0.0138990
\(486\) 884.120 0.0825196
\(487\) −3932.10 −0.365874 −0.182937 0.983125i \(-0.558560\pi\)
−0.182937 + 0.983125i \(0.558560\pi\)
\(488\) 1214.57 0.112666
\(489\) −1689.49 −0.156240
\(490\) 1664.37 0.153446
\(491\) 11477.9 1.05497 0.527484 0.849565i \(-0.323136\pi\)
0.527484 + 0.849565i \(0.323136\pi\)
\(492\) −7634.94 −0.699613
\(493\) 3246.48 0.296580
\(494\) −1961.35 −0.178635
\(495\) −495.000 −0.0449467
\(496\) 9972.16 0.902749
\(497\) 11044.1 0.996772
\(498\) −1573.19 −0.141559
\(499\) 12806.6 1.14890 0.574451 0.818539i \(-0.305215\pi\)
0.574451 + 0.818539i \(0.305215\pi\)
\(500\) 654.703 0.0585584
\(501\) 3386.89 0.302026
\(502\) −26037.9 −2.31500
\(503\) 1319.41 0.116958 0.0584789 0.998289i \(-0.481375\pi\)
0.0584789 + 0.998289i \(0.481375\pi\)
\(504\) −1885.47 −0.166638
\(505\) −87.9910 −0.00775356
\(506\) −2702.22 −0.237408
\(507\) 7077.28 0.619947
\(508\) 1629.71 0.142336
\(509\) 1658.45 0.144419 0.0722097 0.997389i \(-0.476995\pi\)
0.0722097 + 0.997389i \(0.476995\pi\)
\(510\) −3155.11 −0.273942
\(511\) 22078.2 1.91131
\(512\) 9784.02 0.844524
\(513\) −215.635 −0.0185585
\(514\) −7962.02 −0.683248
\(515\) 7115.13 0.608796
\(516\) −2297.48 −0.196009
\(517\) −1807.26 −0.153739
\(518\) −7236.76 −0.613832
\(519\) 9275.90 0.784522
\(520\) −3391.98 −0.286054
\(521\) 2790.40 0.234644 0.117322 0.993094i \(-0.462569\pi\)
0.117322 + 0.993094i \(0.462569\pi\)
\(522\) −1838.83 −0.154183
\(523\) −2440.70 −0.204062 −0.102031 0.994781i \(-0.532534\pi\)
−0.102031 + 0.994781i \(0.532534\pi\)
\(524\) 3052.59 0.254490
\(525\) 1563.33 0.129961
\(526\) 2755.64 0.228426
\(527\) 7347.05 0.607292
\(528\) 2589.45 0.213431
\(529\) −7608.25 −0.625319
\(530\) 7849.61 0.643330
\(531\) −7237.60 −0.591498
\(532\) −871.925 −0.0710578
\(533\) −32797.9 −2.66536
\(534\) −15283.2 −1.23852
\(535\) −6675.73 −0.539471
\(536\) 3733.45 0.300859
\(537\) 8637.81 0.694132
\(538\) −24677.6 −1.97756
\(539\) −1006.39 −0.0804236
\(540\) 707.080 0.0563479
\(541\) −7756.41 −0.616403 −0.308202 0.951321i \(-0.599727\pi\)
−0.308202 + 0.951321i \(0.599727\pi\)
\(542\) 25617.2 2.03017
\(543\) −2106.41 −0.166473
\(544\) 11856.7 0.934474
\(545\) 7826.93 0.615173
\(546\) 15357.2 1.20371
\(547\) −13056.6 −1.02058 −0.510290 0.860002i \(-0.670462\pi\)
−0.510290 + 0.860002i \(0.670462\pi\)
\(548\) −12402.6 −0.966810
\(549\) −1087.62 −0.0845510
\(550\) −1000.55 −0.0775700
\(551\) 448.487 0.0346754
\(552\) −2035.78 −0.156972
\(553\) −3515.69 −0.270348
\(554\) −11684.4 −0.896072
\(555\) −1431.33 −0.109471
\(556\) 13478.9 1.02812
\(557\) −1837.48 −0.139778 −0.0698892 0.997555i \(-0.522265\pi\)
−0.0698892 + 0.997555i \(0.522265\pi\)
\(558\) −4161.43 −0.315712
\(559\) −9869.42 −0.746748
\(560\) −8178.13 −0.617123
\(561\) 1907.80 0.143578
\(562\) 14503.0 1.08856
\(563\) 11473.6 0.858890 0.429445 0.903093i \(-0.358709\pi\)
0.429445 + 0.903093i \(0.358709\pi\)
\(564\) 2581.56 0.192736
\(565\) 5884.16 0.438139
\(566\) 28520.1 2.11801
\(567\) 1688.40 0.125055
\(568\) −5325.10 −0.393374
\(569\) 9698.61 0.714564 0.357282 0.933997i \(-0.383703\pi\)
0.357282 + 0.933997i \(0.383703\pi\)
\(570\) −435.864 −0.0320286
\(571\) 14019.8 1.02752 0.513758 0.857935i \(-0.328253\pi\)
0.513758 + 0.857935i \(0.328253\pi\)
\(572\) −3888.87 −0.284269
\(573\) 12882.8 0.939247
\(574\) −36850.6 −2.67964
\(575\) 1687.96 0.122422
\(576\) −1066.04 −0.0771155
\(577\) 3912.25 0.282269 0.141134 0.989990i \(-0.454925\pi\)
0.141134 + 0.989990i \(0.454925\pi\)
\(578\) −5715.03 −0.411270
\(579\) −11664.3 −0.837226
\(580\) −1470.62 −0.105283
\(581\) −3004.31 −0.214526
\(582\) 324.081 0.0230818
\(583\) −4746.41 −0.337181
\(584\) −10645.4 −0.754295
\(585\) 3037.45 0.214672
\(586\) −19758.6 −1.39287
\(587\) 11148.5 0.783899 0.391949 0.919987i \(-0.371801\pi\)
0.391949 + 0.919987i \(0.371801\pi\)
\(588\) 1437.57 0.100824
\(589\) 1014.96 0.0710030
\(590\) −14629.4 −1.02082
\(591\) −9186.77 −0.639413
\(592\) 7487.61 0.519829
\(593\) −17155.0 −1.18798 −0.593989 0.804473i \(-0.702448\pi\)
−0.593989 + 0.804473i \(0.702448\pi\)
\(594\) −1080.59 −0.0746418
\(595\) −6025.29 −0.415147
\(596\) 10435.3 0.717194
\(597\) 14623.9 1.00254
\(598\) 16581.5 1.13389
\(599\) −2891.03 −0.197203 −0.0986014 0.995127i \(-0.531437\pi\)
−0.0986014 + 0.995127i \(0.531437\pi\)
\(600\) −753.787 −0.0512887
\(601\) 11439.4 0.776412 0.388206 0.921573i \(-0.373095\pi\)
0.388206 + 0.921573i \(0.373095\pi\)
\(602\) −11088.9 −0.750750
\(603\) −3343.22 −0.225782
\(604\) 15509.7 1.04483
\(605\) 605.000 0.0406558
\(606\) −192.085 −0.0128761
\(607\) −7768.78 −0.519481 −0.259741 0.965678i \(-0.583637\pi\)
−0.259741 + 0.965678i \(0.583637\pi\)
\(608\) 1637.96 0.109256
\(609\) −3511.61 −0.233658
\(610\) −2198.41 −0.145920
\(611\) 11089.8 0.734279
\(612\) −2725.18 −0.179998
\(613\) −6469.74 −0.426281 −0.213140 0.977022i \(-0.568369\pi\)
−0.213140 + 0.977022i \(0.568369\pi\)
\(614\) 29271.5 1.92394
\(615\) −7288.55 −0.477890
\(616\) 2304.46 0.150730
\(617\) −18323.7 −1.19560 −0.597799 0.801646i \(-0.703958\pi\)
−0.597799 + 0.801646i \(0.703958\pi\)
\(618\) 15532.4 1.01101
\(619\) −16697.1 −1.08419 −0.542096 0.840317i \(-0.682369\pi\)
−0.542096 + 0.840317i \(0.682369\pi\)
\(620\) −3328.13 −0.215582
\(621\) 1823.00 0.117801
\(622\) −17144.1 −1.10517
\(623\) −29186.3 −1.87692
\(624\) −15889.6 −1.01938
\(625\) 625.000 0.0400000
\(626\) −5185.35 −0.331067
\(627\) 263.553 0.0167868
\(628\) 6822.34 0.433505
\(629\) 5516.54 0.349696
\(630\) 3412.77 0.215822
\(631\) 20458.8 1.29073 0.645367 0.763873i \(-0.276705\pi\)
0.645367 + 0.763873i \(0.276705\pi\)
\(632\) 1695.15 0.106692
\(633\) −12972.1 −0.814524
\(634\) −7391.45 −0.463016
\(635\) 1555.77 0.0972267
\(636\) 6779.98 0.422710
\(637\) 6175.47 0.384115
\(638\) 2247.46 0.139464
\(639\) 4768.51 0.295210
\(640\) 6048.86 0.373597
\(641\) 4676.71 0.288173 0.144087 0.989565i \(-0.453976\pi\)
0.144087 + 0.989565i \(0.453976\pi\)
\(642\) −14573.2 −0.895886
\(643\) −2321.97 −0.142410 −0.0712049 0.997462i \(-0.522684\pi\)
−0.0712049 + 0.997462i \(0.522684\pi\)
\(644\) 7371.35 0.451043
\(645\) −2193.24 −0.133890
\(646\) 1679.88 0.102312
\(647\) 18149.3 1.10282 0.551408 0.834236i \(-0.314091\pi\)
0.551408 + 0.834236i \(0.314091\pi\)
\(648\) −814.090 −0.0493526
\(649\) 8845.96 0.535030
\(650\) 6139.62 0.370486
\(651\) −7947.06 −0.478448
\(652\) −2949.64 −0.177173
\(653\) 23089.9 1.38374 0.691868 0.722024i \(-0.256788\pi\)
0.691868 + 0.722024i \(0.256788\pi\)
\(654\) 17086.3 1.02160
\(655\) 2914.09 0.173837
\(656\) 38128.0 2.26928
\(657\) 9532.69 0.566067
\(658\) 12460.1 0.738215
\(659\) 415.639 0.0245690 0.0122845 0.999925i \(-0.496090\pi\)
0.0122845 + 0.999925i \(0.496090\pi\)
\(660\) −864.208 −0.0509686
\(661\) 12044.0 0.708710 0.354355 0.935111i \(-0.384700\pi\)
0.354355 + 0.935111i \(0.384700\pi\)
\(662\) 1152.60 0.0676691
\(663\) −11706.7 −0.685749
\(664\) 1448.58 0.0846622
\(665\) −832.366 −0.0485380
\(666\) −3124.62 −0.181796
\(667\) −3791.56 −0.220105
\(668\) 5913.09 0.342491
\(669\) −18617.1 −1.07590
\(670\) −6757.69 −0.389660
\(671\) 1329.31 0.0764792
\(672\) −12825.0 −0.736215
\(673\) 8699.88 0.498300 0.249150 0.968465i \(-0.419849\pi\)
0.249150 + 0.968465i \(0.419849\pi\)
\(674\) 16160.3 0.923547
\(675\) 675.000 0.0384900
\(676\) 12356.0 0.703007
\(677\) −29389.8 −1.66845 −0.834225 0.551424i \(-0.814085\pi\)
−0.834225 + 0.551424i \(0.814085\pi\)
\(678\) 12845.2 0.727606
\(679\) 618.895 0.0349794
\(680\) 2905.20 0.163837
\(681\) 4607.44 0.259262
\(682\) 5086.20 0.285573
\(683\) 12261.0 0.686905 0.343452 0.939170i \(-0.388404\pi\)
0.343452 + 0.939170i \(0.388404\pi\)
\(684\) −376.471 −0.0210449
\(685\) −11839.9 −0.660407
\(686\) −19074.4 −1.06161
\(687\) 284.209 0.0157835
\(688\) 11473.3 0.635780
\(689\) 29125.2 1.61042
\(690\) 3684.84 0.203304
\(691\) −22711.4 −1.25034 −0.625168 0.780490i \(-0.714970\pi\)
−0.625168 + 0.780490i \(0.714970\pi\)
\(692\) 16194.6 0.889631
\(693\) −2063.60 −0.113116
\(694\) 941.224 0.0514818
\(695\) 12867.4 0.702286
\(696\) 1693.18 0.0922125
\(697\) 28091.0 1.52658
\(698\) 36127.5 1.95909
\(699\) 1964.95 0.106325
\(700\) 2729.38 0.147373
\(701\) −21070.9 −1.13529 −0.567643 0.823275i \(-0.692145\pi\)
−0.567643 + 0.823275i \(0.692145\pi\)
\(702\) 6630.79 0.356500
\(703\) 762.086 0.0408856
\(704\) 1302.94 0.0697536
\(705\) 2464.44 0.131654
\(706\) −3670.13 −0.195648
\(707\) −366.824 −0.0195132
\(708\) −12636.0 −0.670746
\(709\) 8521.30 0.451374 0.225687 0.974200i \(-0.427537\pi\)
0.225687 + 0.974200i \(0.427537\pi\)
\(710\) 9638.63 0.509481
\(711\) −1517.97 −0.0800679
\(712\) 14072.7 0.740724
\(713\) −8580.61 −0.450696
\(714\) −13153.3 −0.689425
\(715\) −3712.44 −0.194178
\(716\) 15080.5 0.787132
\(717\) −16981.9 −0.884521
\(718\) 21150.7 1.09936
\(719\) 8217.19 0.426216 0.213108 0.977029i \(-0.431641\pi\)
0.213108 + 0.977029i \(0.431641\pi\)
\(720\) −3531.07 −0.182771
\(721\) 29662.1 1.53214
\(722\) −24723.4 −1.27439
\(723\) −12470.6 −0.641478
\(724\) −3677.53 −0.188777
\(725\) −1403.90 −0.0719164
\(726\) 1320.72 0.0675160
\(727\) −20870.8 −1.06473 −0.532363 0.846516i \(-0.678696\pi\)
−0.532363 + 0.846516i \(0.678696\pi\)
\(728\) −14140.8 −0.719907
\(729\) 729.000 0.0370370
\(730\) 19268.5 0.976931
\(731\) 8453.04 0.427698
\(732\) −1898.85 −0.0958790
\(733\) −33776.4 −1.70199 −0.850997 0.525171i \(-0.824002\pi\)
−0.850997 + 0.525171i \(0.824002\pi\)
\(734\) −37282.3 −1.87482
\(735\) 1372.35 0.0688706
\(736\) −13847.5 −0.693512
\(737\) 4086.16 0.204228
\(738\) −15911.0 −0.793620
\(739\) −14342.3 −0.713922 −0.356961 0.934119i \(-0.616187\pi\)
−0.356961 + 0.934119i \(0.616187\pi\)
\(740\) −2498.93 −0.124138
\(741\) −1617.23 −0.0801761
\(742\) 32724.1 1.61906
\(743\) −19225.5 −0.949279 −0.474640 0.880180i \(-0.657422\pi\)
−0.474640 + 0.880180i \(0.657422\pi\)
\(744\) 3831.81 0.188819
\(745\) 9961.88 0.489900
\(746\) −8013.96 −0.393313
\(747\) −1297.17 −0.0635354
\(748\) 3330.77 0.162814
\(749\) −27830.4 −1.35768
\(750\) 1364.38 0.0664270
\(751\) −19629.4 −0.953776 −0.476888 0.878964i \(-0.658235\pi\)
−0.476888 + 0.878964i \(0.658235\pi\)
\(752\) −12892.0 −0.625164
\(753\) −21469.5 −1.03903
\(754\) −13791.0 −0.666100
\(755\) 14806.0 0.713703
\(756\) 2947.73 0.141810
\(757\) −26801.5 −1.28681 −0.643406 0.765525i \(-0.722479\pi\)
−0.643406 + 0.765525i \(0.722479\pi\)
\(758\) −6735.07 −0.322729
\(759\) −2228.11 −0.106555
\(760\) 401.340 0.0191554
\(761\) 6490.31 0.309164 0.154582 0.987980i \(-0.450597\pi\)
0.154582 + 0.987980i \(0.450597\pi\)
\(762\) 3396.27 0.161462
\(763\) 32629.6 1.54819
\(764\) 22491.8 1.06509
\(765\) −2601.54 −0.122953
\(766\) 21395.9 1.00922
\(767\) −54281.1 −2.55538
\(768\) 16047.5 0.753991
\(769\) 14356.1 0.673204 0.336602 0.941647i \(-0.390722\pi\)
0.336602 + 0.941647i \(0.390722\pi\)
\(770\) −4171.17 −0.195219
\(771\) −6565.07 −0.306661
\(772\) −20364.5 −0.949397
\(773\) 19127.6 0.890004 0.445002 0.895529i \(-0.353203\pi\)
0.445002 + 0.895529i \(0.353203\pi\)
\(774\) −4787.88 −0.222347
\(775\) −3177.13 −0.147259
\(776\) −298.411 −0.0138045
\(777\) −5967.06 −0.275505
\(778\) −25336.3 −1.16754
\(779\) 3880.65 0.178483
\(780\) 5303.01 0.243433
\(781\) −5828.18 −0.267028
\(782\) −14201.9 −0.649435
\(783\) −1516.21 −0.0692016
\(784\) −7179.07 −0.327035
\(785\) 6512.81 0.296118
\(786\) 6361.50 0.288686
\(787\) 23509.7 1.06484 0.532420 0.846480i \(-0.321283\pi\)
0.532420 + 0.846480i \(0.321283\pi\)
\(788\) −16039.0 −0.725082
\(789\) 2272.16 0.102524
\(790\) −3068.28 −0.138183
\(791\) 24530.4 1.10266
\(792\) 994.999 0.0446411
\(793\) −8157.01 −0.365276
\(794\) 11003.5 0.491814
\(795\) 6472.38 0.288744
\(796\) 25531.5 1.13686
\(797\) 43145.1 1.91754 0.958769 0.284187i \(-0.0917236\pi\)
0.958769 + 0.284187i \(0.0917236\pi\)
\(798\) −1817.07 −0.0806058
\(799\) −9498.27 −0.420557
\(800\) −5127.29 −0.226596
\(801\) −12601.8 −0.555882
\(802\) −48727.3 −2.14541
\(803\) −11651.1 −0.512027
\(804\) −5836.85 −0.256032
\(805\) 7036.92 0.308098
\(806\) −31210.2 −1.36394
\(807\) −20347.9 −0.887584
\(808\) 176.871 0.00770085
\(809\) −23470.4 −1.02000 −0.509998 0.860176i \(-0.670354\pi\)
−0.509998 + 0.860176i \(0.670354\pi\)
\(810\) 1473.53 0.0639194
\(811\) −4906.52 −0.212443 −0.106221 0.994343i \(-0.533875\pi\)
−0.106221 + 0.994343i \(0.533875\pi\)
\(812\) −6130.83 −0.264963
\(813\) 21122.6 0.911196
\(814\) 3818.98 0.164441
\(815\) −2815.81 −0.121023
\(816\) 13609.2 0.583846
\(817\) 1167.75 0.0500054
\(818\) 27786.7 1.18770
\(819\) 12662.8 0.540260
\(820\) −12724.9 −0.541918
\(821\) 45229.6 1.92268 0.961342 0.275356i \(-0.0887958\pi\)
0.961342 + 0.275356i \(0.0887958\pi\)
\(822\) −25846.6 −1.09672
\(823\) 24192.1 1.02465 0.512323 0.858793i \(-0.328785\pi\)
0.512323 + 0.858793i \(0.328785\pi\)
\(824\) −14302.1 −0.604657
\(825\) −825.000 −0.0348155
\(826\) −60988.4 −2.56908
\(827\) 2031.70 0.0854282 0.0427141 0.999087i \(-0.486400\pi\)
0.0427141 + 0.999087i \(0.486400\pi\)
\(828\) 3182.73 0.133584
\(829\) 17010.6 0.712671 0.356335 0.934358i \(-0.384026\pi\)
0.356335 + 0.934358i \(0.384026\pi\)
\(830\) −2621.98 −0.109651
\(831\) −9634.38 −0.402182
\(832\) −7995.20 −0.333153
\(833\) −5289.22 −0.220001
\(834\) 28089.7 1.16627
\(835\) 5644.81 0.233948
\(836\) 460.131 0.0190358
\(837\) −3431.30 −0.141700
\(838\) 45566.5 1.87836
\(839\) 3184.10 0.131022 0.0655109 0.997852i \(-0.479132\pi\)
0.0655109 + 0.997852i \(0.479132\pi\)
\(840\) −3142.45 −0.129077
\(841\) −21235.5 −0.870701
\(842\) −40567.8 −1.66040
\(843\) 11958.4 0.488577
\(844\) −22647.6 −0.923653
\(845\) 11795.5 0.480209
\(846\) 5379.90 0.218635
\(847\) 2522.18 0.102318
\(848\) −33858.4 −1.37111
\(849\) 23516.2 0.950619
\(850\) −5258.51 −0.212195
\(851\) −6442.76 −0.259524
\(852\) 8325.23 0.334762
\(853\) 20566.8 0.825548 0.412774 0.910833i \(-0.364560\pi\)
0.412774 + 0.910833i \(0.364560\pi\)
\(854\) −9164.94 −0.367234
\(855\) −359.391 −0.0143753
\(856\) 13418.9 0.535804
\(857\) 48125.8 1.91826 0.959129 0.282971i \(-0.0913199\pi\)
0.959129 + 0.282971i \(0.0913199\pi\)
\(858\) −8104.30 −0.322466
\(859\) −22013.7 −0.874387 −0.437194 0.899367i \(-0.644028\pi\)
−0.437194 + 0.899367i \(0.644028\pi\)
\(860\) −3829.13 −0.151828
\(861\) −30385.1 −1.20270
\(862\) −7617.86 −0.301004
\(863\) −23417.0 −0.923667 −0.461834 0.886967i \(-0.652808\pi\)
−0.461834 + 0.886967i \(0.652808\pi\)
\(864\) −5537.47 −0.218042
\(865\) 15459.8 0.607688
\(866\) −15977.7 −0.626958
\(867\) −4712.32 −0.184589
\(868\) −13874.6 −0.542550
\(869\) 1855.30 0.0724241
\(870\) −3064.72 −0.119430
\(871\) −25073.7 −0.975421
\(872\) −15732.9 −0.610991
\(873\) 267.221 0.0103597
\(874\) −1961.92 −0.0759303
\(875\) 2605.55 0.100667
\(876\) 16642.9 0.641908
\(877\) −29017.9 −1.11729 −0.558645 0.829407i \(-0.688679\pi\)
−0.558645 + 0.829407i \(0.688679\pi\)
\(878\) 56811.4 2.18370
\(879\) −16291.9 −0.625158
\(880\) 4315.76 0.165323
\(881\) −33241.1 −1.27119 −0.635597 0.772021i \(-0.719246\pi\)
−0.635597 + 0.772021i \(0.719246\pi\)
\(882\) 2995.86 0.114372
\(883\) −5618.39 −0.214127 −0.107063 0.994252i \(-0.534145\pi\)
−0.107063 + 0.994252i \(0.534145\pi\)
\(884\) −20438.5 −0.777625
\(885\) −12062.7 −0.458172
\(886\) 28777.8 1.09121
\(887\) −14334.2 −0.542612 −0.271306 0.962493i \(-0.587455\pi\)
−0.271306 + 0.962493i \(0.587455\pi\)
\(888\) 2877.12 0.108727
\(889\) 6485.84 0.244688
\(890\) −25472.1 −0.959354
\(891\) −891.000 −0.0335013
\(892\) −32503.1 −1.22005
\(893\) −1312.14 −0.0491704
\(894\) 21746.9 0.813564
\(895\) 14396.3 0.537672
\(896\) 25217.0 0.940223
\(897\) 13672.3 0.508922
\(898\) −54424.9 −2.02247
\(899\) 7136.58 0.264759
\(900\) 1178.47 0.0436469
\(901\) −24945.4 −0.922367
\(902\) 19446.8 0.717856
\(903\) −9143.37 −0.336957
\(904\) −11827.7 −0.435161
\(905\) −3510.68 −0.128949
\(906\) 32321.7 1.18523
\(907\) −36659.3 −1.34207 −0.671033 0.741428i \(-0.734149\pi\)
−0.671033 + 0.741428i \(0.734149\pi\)
\(908\) 8044.02 0.293998
\(909\) −158.384 −0.00577916
\(910\) 25595.4 0.932393
\(911\) −9050.97 −0.329168 −0.164584 0.986363i \(-0.552628\pi\)
−0.164584 + 0.986363i \(0.552628\pi\)
\(912\) 1880.05 0.0682618
\(913\) 1585.43 0.0574700
\(914\) 53063.8 1.92035
\(915\) −1812.70 −0.0654929
\(916\) 496.193 0.0178981
\(917\) 12148.5 0.437491
\(918\) −5679.19 −0.204184
\(919\) −46444.5 −1.66710 −0.833549 0.552445i \(-0.813695\pi\)
−0.833549 + 0.552445i \(0.813695\pi\)
\(920\) −3392.97 −0.121590
\(921\) 24135.7 0.863518
\(922\) −6557.59 −0.234233
\(923\) 35763.2 1.27536
\(924\) −3602.79 −0.128272
\(925\) −2385.55 −0.0847962
\(926\) 24572.7 0.872039
\(927\) 12807.2 0.453770
\(928\) 11517.1 0.407400
\(929\) −18695.8 −0.660267 −0.330133 0.943934i \(-0.607094\pi\)
−0.330133 + 0.943934i \(0.607094\pi\)
\(930\) −6935.72 −0.244550
\(931\) −730.682 −0.0257219
\(932\) 3430.55 0.120570
\(933\) −14136.2 −0.496032
\(934\) −6347.30 −0.222366
\(935\) 3179.66 0.111215
\(936\) −6105.57 −0.213212
\(937\) −13323.6 −0.464530 −0.232265 0.972653i \(-0.574614\pi\)
−0.232265 + 0.972653i \(0.574614\pi\)
\(938\) −28172.0 −0.980649
\(939\) −4275.57 −0.148592
\(940\) 4302.60 0.149293
\(941\) −9500.50 −0.329126 −0.164563 0.986367i \(-0.552621\pi\)
−0.164563 + 0.986367i \(0.552621\pi\)
\(942\) 14217.6 0.491755
\(943\) −32807.4 −1.13293
\(944\) 63102.5 2.17565
\(945\) 2814.00 0.0968670
\(946\) 5851.85 0.201120
\(947\) 15179.4 0.520869 0.260435 0.965491i \(-0.416134\pi\)
0.260435 + 0.965491i \(0.416134\pi\)
\(948\) −2650.18 −0.0907953
\(949\) 71494.0 2.44551
\(950\) −726.440 −0.0248093
\(951\) −6094.61 −0.207814
\(952\) 12111.4 0.412325
\(953\) 5051.55 0.171706 0.0858530 0.996308i \(-0.472638\pi\)
0.0858530 + 0.996308i \(0.472638\pi\)
\(954\) 14129.3 0.479510
\(955\) 21471.4 0.727538
\(956\) −29648.3 −1.00303
\(957\) 1853.14 0.0625952
\(958\) 61599.6 2.07745
\(959\) −49359.1 −1.66203
\(960\) −1776.74 −0.0597334
\(961\) −13640.3 −0.457867
\(962\) −23434.2 −0.785395
\(963\) −12016.3 −0.402098
\(964\) −21772.2 −0.727422
\(965\) −19440.6 −0.648512
\(966\) 15361.7 0.511650
\(967\) 15341.1 0.510172 0.255086 0.966918i \(-0.417896\pi\)
0.255086 + 0.966918i \(0.417896\pi\)
\(968\) −1216.11 −0.0403794
\(969\) 1385.14 0.0459206
\(970\) 540.135 0.0178791
\(971\) −5397.65 −0.178392 −0.0891961 0.996014i \(-0.528430\pi\)
−0.0891961 + 0.996014i \(0.528430\pi\)
\(972\) 1272.74 0.0419992
\(973\) 53642.7 1.76743
\(974\) −14306.4 −0.470643
\(975\) 5062.41 0.166284
\(976\) 9482.63 0.310995
\(977\) 24180.4 0.791812 0.395906 0.918291i \(-0.370431\pi\)
0.395906 + 0.918291i \(0.370431\pi\)
\(978\) −6146.96 −0.200980
\(979\) 15402.2 0.502814
\(980\) 2395.95 0.0780979
\(981\) 14088.5 0.458523
\(982\) 41760.5 1.35706
\(983\) 15159.6 0.491877 0.245938 0.969285i \(-0.420904\pi\)
0.245938 + 0.969285i \(0.420904\pi\)
\(984\) 14650.7 0.474642
\(985\) −15311.3 −0.495288
\(986\) 11811.9 0.381507
\(987\) 10274.0 0.331331
\(988\) −2823.48 −0.0909180
\(989\) −9872.29 −0.317412
\(990\) −1800.99 −0.0578173
\(991\) −3654.97 −0.117158 −0.0585792 0.998283i \(-0.518657\pi\)
−0.0585792 + 0.998283i \(0.518657\pi\)
\(992\) 26064.1 0.834210
\(993\) 950.371 0.0303717
\(994\) 40182.4 1.28220
\(995\) 24373.2 0.776565
\(996\) −2264.70 −0.0720479
\(997\) 46219.9 1.46820 0.734102 0.679040i \(-0.237604\pi\)
0.734102 + 0.679040i \(0.237604\pi\)
\(998\) 46595.0 1.47789
\(999\) −2576.40 −0.0815952
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 165.4.a.h.1.3 4
3.2 odd 2 495.4.a.m.1.2 4
5.2 odd 4 825.4.c.p.199.6 8
5.3 odd 4 825.4.c.p.199.3 8
5.4 even 2 825.4.a.t.1.2 4
11.10 odd 2 1815.4.a.t.1.2 4
15.14 odd 2 2475.4.a.be.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.h.1.3 4 1.1 even 1 trivial
495.4.a.m.1.2 4 3.2 odd 2
825.4.a.t.1.2 4 5.4 even 2
825.4.c.p.199.3 8 5.3 odd 4
825.4.c.p.199.6 8 5.2 odd 4
1815.4.a.t.1.2 4 11.10 odd 2
2475.4.a.be.1.3 4 15.14 odd 2