Properties

Label 121.6.a.d.1.1
Level $121$
Weight $6$
Character 121.1
Self dual yes
Analytic conductor $19.406$
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] = [121,6,Mod(1,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 121.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: \(19.4064421974\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.29828\) of defining polynomial
Character \(\chi\) \(=\) 121.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.18772 q^{2} -3.48600 q^{3} +35.0388 q^{4} -59.8722 q^{5} +28.5424 q^{6} -145.071 q^{7} -24.8808 q^{8} -230.848 q^{9} +O(q^{10})\) \(q-8.18772 q^{2} -3.48600 q^{3} +35.0388 q^{4} -59.8722 q^{5} +28.5424 q^{6} -145.071 q^{7} -24.8808 q^{8} -230.848 q^{9} +490.217 q^{10} -122.145 q^{12} -615.772 q^{13} +1187.80 q^{14} +208.715 q^{15} -917.524 q^{16} -1840.68 q^{17} +1890.12 q^{18} -366.633 q^{19} -2097.85 q^{20} +505.718 q^{21} -4516.38 q^{23} +86.7344 q^{24} +459.685 q^{25} +5041.77 q^{26} +1651.83 q^{27} -5083.12 q^{28} +1717.00 q^{29} -1708.90 q^{30} -2650.54 q^{31} +8308.62 q^{32} +15070.9 q^{34} +8685.74 q^{35} -8088.63 q^{36} +9660.61 q^{37} +3001.89 q^{38} +2146.58 q^{39} +1489.67 q^{40} +11154.8 q^{41} -4140.68 q^{42} -8368.48 q^{43} +13821.4 q^{45} +36978.9 q^{46} -2221.22 q^{47} +3198.49 q^{48} +4238.64 q^{49} -3763.77 q^{50} +6416.60 q^{51} -21575.9 q^{52} +23707.9 q^{53} -13524.8 q^{54} +3609.48 q^{56} +1278.08 q^{57} -14058.3 q^{58} +19517.8 q^{59} +7313.11 q^{60} -20937.3 q^{61} +21701.9 q^{62} +33489.4 q^{63} -38667.9 q^{64} +36867.6 q^{65} -51707.7 q^{67} -64495.1 q^{68} +15744.1 q^{69} -71116.4 q^{70} -1398.38 q^{71} +5743.67 q^{72} -72466.6 q^{73} -79098.4 q^{74} -1602.46 q^{75} -12846.4 q^{76} -17575.6 q^{78} -64632.2 q^{79} +54934.2 q^{80} +50337.7 q^{81} -91332.4 q^{82} +96790.3 q^{83} +17719.8 q^{84} +110205. q^{85} +68518.8 q^{86} -5985.47 q^{87} -47614.1 q^{89} -113166. q^{90} +89330.7 q^{91} -158249. q^{92} +9239.79 q^{93} +18186.7 q^{94} +21951.2 q^{95} -28963.9 q^{96} -38399.6 q^{97} -34704.8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 34 q^{3} + 84 q^{4} + 24 q^{5} + 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 34 q^{3} + 84 q^{4} + 24 q^{5} + 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9} + 414 q^{10} + 992 q^{12} - 486 q^{13} - 1020 q^{14} + 1654 q^{15} + 1992 q^{16} - 1086 q^{17} + 3706 q^{18} - 1380 q^{19} - 3480 q^{20} + 908 q^{21} - 3066 q^{23} + 11748 q^{24} - 57 q^{25} + 12132 q^{26} - 2990 q^{27} - 23712 q^{28} + 3426 q^{29} - 2650 q^{30} - 4098 q^{31} + 12408 q^{32} + 25320 q^{34} + 24228 q^{35} + 4756 q^{36} + 17724 q^{37} - 9240 q^{38} + 6560 q^{39} + 15276 q^{40} - 5994 q^{41} - 47828 q^{42} + 26208 q^{43} + 18458 q^{45} + 16806 q^{46} - 17232 q^{47} + 61064 q^{48} + 48531 q^{49} - 41070 q^{50} + 22724 q^{51} + 35304 q^{52} + 50586 q^{53} - 18814 q^{54} - 42312 q^{56} - 20160 q^{57} - 29172 q^{58} - 3738 q^{59} - 13456 q^{60} - 18486 q^{61} + 19974 q^{62} + 12496 q^{63} - 20352 q^{64} + 7668 q^{65} - 47754 q^{67} + 12600 q^{68} + 35042 q^{69} - 123372 q^{70} + 39282 q^{71} + 95040 q^{72} - 15426 q^{73} - 153294 q^{74} - 21916 q^{75} - 103920 q^{76} + 124984 q^{78} - 125148 q^{79} + 118680 q^{80} - 86917 q^{81} - 255372 q^{82} + 143928 q^{83} - 343616 q^{84} + 104040 q^{85} + 243060 q^{86} + 19368 q^{87} - 106824 q^{89} - 103424 q^{90} - 109632 q^{91} - 336528 q^{92} - 16622 q^{93} + 74928 q^{94} + 22200 q^{95} + 76456 q^{96} + 9684 q^{97} - 3480 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.18772 −1.44740 −0.723699 0.690116i \(-0.757560\pi\)
−0.723699 + 0.690116i \(0.757560\pi\)
\(3\) −3.48600 −0.223627 −0.111814 0.993729i \(-0.535666\pi\)
−0.111814 + 0.993729i \(0.535666\pi\)
\(4\) 35.0388 1.09496
\(5\) −59.8722 −1.07103 −0.535514 0.844527i \(-0.679882\pi\)
−0.535514 + 0.844527i \(0.679882\pi\)
\(6\) 28.5424 0.323678
\(7\) −145.071 −1.11902 −0.559508 0.828825i \(-0.689010\pi\)
−0.559508 + 0.828825i \(0.689010\pi\)
\(8\) −24.8808 −0.137448
\(9\) −230.848 −0.949991
\(10\) 490.217 1.55020
\(11\) 0 0
\(12\) −122.145 −0.244863
\(13\) −615.772 −1.01056 −0.505279 0.862956i \(-0.668610\pi\)
−0.505279 + 0.862956i \(0.668610\pi\)
\(14\) 1187.80 1.61966
\(15\) 208.715 0.239511
\(16\) −917.524 −0.896020
\(17\) −1840.68 −1.54474 −0.772369 0.635174i \(-0.780929\pi\)
−0.772369 + 0.635174i \(0.780929\pi\)
\(18\) 1890.12 1.37502
\(19\) −366.633 −0.232996 −0.116498 0.993191i \(-0.537167\pi\)
−0.116498 + 0.993191i \(0.537167\pi\)
\(20\) −2097.85 −1.17273
\(21\) 505.718 0.250242
\(22\) 0 0
\(23\) −4516.38 −1.78021 −0.890104 0.455757i \(-0.849369\pi\)
−0.890104 + 0.455757i \(0.849369\pi\)
\(24\) 86.7344 0.0307371
\(25\) 459.685 0.147099
\(26\) 5041.77 1.46268
\(27\) 1651.83 0.436071
\(28\) −5083.12 −1.22528
\(29\) 1717.00 0.379119 0.189560 0.981869i \(-0.439294\pi\)
0.189560 + 0.981869i \(0.439294\pi\)
\(30\) −1708.90 −0.346667
\(31\) −2650.54 −0.495371 −0.247685 0.968841i \(-0.579670\pi\)
−0.247685 + 0.968841i \(0.579670\pi\)
\(32\) 8308.62 1.43435
\(33\) 0 0
\(34\) 15070.9 2.23585
\(35\) 8685.74 1.19850
\(36\) −8088.63 −1.04020
\(37\) 9660.61 1.16011 0.580057 0.814576i \(-0.303030\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(38\) 3001.89 0.337238
\(39\) 2146.58 0.225988
\(40\) 1489.67 0.147211
\(41\) 11154.8 1.03634 0.518170 0.855278i \(-0.326614\pi\)
0.518170 + 0.855278i \(0.326614\pi\)
\(42\) −4140.68 −0.362200
\(43\) −8368.48 −0.690201 −0.345100 0.938566i \(-0.612155\pi\)
−0.345100 + 0.938566i \(0.612155\pi\)
\(44\) 0 0
\(45\) 13821.4 1.01747
\(46\) 36978.9 2.57667
\(47\) −2221.22 −0.146672 −0.0733360 0.997307i \(-0.523365\pi\)
−0.0733360 + 0.997307i \(0.523365\pi\)
\(48\) 3198.49 0.200374
\(49\) 4238.64 0.252195
\(50\) −3763.77 −0.212911
\(51\) 6416.60 0.345445
\(52\) −21575.9 −1.10652
\(53\) 23707.9 1.15932 0.579659 0.814859i \(-0.303186\pi\)
0.579659 + 0.814859i \(0.303186\pi\)
\(54\) −13524.8 −0.631168
\(55\) 0 0
\(56\) 3609.48 0.153807
\(57\) 1278.08 0.0521042
\(58\) −14058.3 −0.548737
\(59\) 19517.8 0.729964 0.364982 0.931015i \(-0.381075\pi\)
0.364982 + 0.931015i \(0.381075\pi\)
\(60\) 7313.11 0.262255
\(61\) −20937.3 −0.720436 −0.360218 0.932868i \(-0.617298\pi\)
−0.360218 + 0.932868i \(0.617298\pi\)
\(62\) 21701.9 0.716999
\(63\) 33489.4 1.06305
\(64\) −38667.9 −1.18005
\(65\) 36867.6 1.08233
\(66\) 0 0
\(67\) −51707.7 −1.40724 −0.703619 0.710577i \(-0.748434\pi\)
−0.703619 + 0.710577i \(0.748434\pi\)
\(68\) −64495.1 −1.69143
\(69\) 15744.1 0.398103
\(70\) −71116.4 −1.73470
\(71\) −1398.38 −0.0329216 −0.0164608 0.999865i \(-0.505240\pi\)
−0.0164608 + 0.999865i \(0.505240\pi\)
\(72\) 5743.67 0.130574
\(73\) −72466.6 −1.59159 −0.795794 0.605567i \(-0.792946\pi\)
−0.795794 + 0.605567i \(0.792946\pi\)
\(74\) −79098.4 −1.67915
\(75\) −1602.46 −0.0328954
\(76\) −12846.4 −0.255122
\(77\) 0 0
\(78\) −17575.6 −0.327095
\(79\) −64632.2 −1.16515 −0.582574 0.812777i \(-0.697955\pi\)
−0.582574 + 0.812777i \(0.697955\pi\)
\(80\) 54934.2 0.959662
\(81\) 50337.7 0.852474
\(82\) −91332.4 −1.50000
\(83\) 96790.3 1.54219 0.771093 0.636723i \(-0.219710\pi\)
0.771093 + 0.636723i \(0.219710\pi\)
\(84\) 17719.8 0.274006
\(85\) 110205. 1.65446
\(86\) 68518.8 0.998995
\(87\) −5985.47 −0.0847814
\(88\) 0 0
\(89\) −47614.1 −0.637178 −0.318589 0.947893i \(-0.603209\pi\)
−0.318589 + 0.947893i \(0.603209\pi\)
\(90\) −113166. −1.47268
\(91\) 89330.7 1.13083
\(92\) −158249. −1.94926
\(93\) 9239.79 0.110778
\(94\) 18186.7 0.212293
\(95\) 21951.2 0.249545
\(96\) −28963.9 −0.320759
\(97\) −38399.6 −0.414378 −0.207189 0.978301i \(-0.566432\pi\)
−0.207189 + 0.978301i \(0.566432\pi\)
\(98\) −34704.8 −0.365027
\(99\) 0 0
\(100\) 16106.8 0.161068
\(101\) 41011.2 0.400036 0.200018 0.979792i \(-0.435900\pi\)
0.200018 + 0.979792i \(0.435900\pi\)
\(102\) −52537.3 −0.499997
\(103\) −49634.4 −0.460988 −0.230494 0.973074i \(-0.574034\pi\)
−0.230494 + 0.973074i \(0.574034\pi\)
\(104\) 15320.9 0.138899
\(105\) −30278.5 −0.268016
\(106\) −194113. −1.67800
\(107\) 6791.34 0.0573450 0.0286725 0.999589i \(-0.490872\pi\)
0.0286725 + 0.999589i \(0.490872\pi\)
\(108\) 57878.3 0.477481
\(109\) −96780.7 −0.780230 −0.390115 0.920766i \(-0.627565\pi\)
−0.390115 + 0.920766i \(0.627565\pi\)
\(110\) 0 0
\(111\) −33676.9 −0.259433
\(112\) 133106. 1.00266
\(113\) −212938. −1.56876 −0.784379 0.620281i \(-0.787018\pi\)
−0.784379 + 0.620281i \(0.787018\pi\)
\(114\) −10464.6 −0.0754155
\(115\) 270406. 1.90665
\(116\) 60161.7 0.415121
\(117\) 142149. 0.960021
\(118\) −159806. −1.05655
\(119\) 267029. 1.72859
\(120\) −5192.98 −0.0329203
\(121\) 0 0
\(122\) 171428. 1.04276
\(123\) −38885.6 −0.231754
\(124\) −92871.8 −0.542412
\(125\) 159578. 0.913480
\(126\) −274202. −1.53866
\(127\) 90363.9 0.497148 0.248574 0.968613i \(-0.420038\pi\)
0.248574 + 0.968613i \(0.420038\pi\)
\(128\) 50726.1 0.273657
\(129\) 29172.5 0.154348
\(130\) −301862. −1.56657
\(131\) −65299.5 −0.332454 −0.166227 0.986088i \(-0.553158\pi\)
−0.166227 + 0.986088i \(0.553158\pi\)
\(132\) 0 0
\(133\) 53187.9 0.260726
\(134\) 423368. 2.03684
\(135\) −98899.0 −0.467044
\(136\) 45797.4 0.212321
\(137\) 5322.74 0.0242289 0.0121145 0.999927i \(-0.496144\pi\)
0.0121145 + 0.999927i \(0.496144\pi\)
\(138\) −128908. −0.576214
\(139\) −89967.1 −0.394954 −0.197477 0.980308i \(-0.563275\pi\)
−0.197477 + 0.980308i \(0.563275\pi\)
\(140\) 304338. 1.31231
\(141\) 7743.18 0.0327998
\(142\) 11449.6 0.0476506
\(143\) 0 0
\(144\) 211808. 0.851211
\(145\) −102801. −0.406047
\(146\) 593336. 2.30366
\(147\) −14775.9 −0.0563977
\(148\) 338496. 1.27028
\(149\) −66489.8 −0.245352 −0.122676 0.992447i \(-0.539148\pi\)
−0.122676 + 0.992447i \(0.539148\pi\)
\(150\) 13120.5 0.0476127
\(151\) 130866. 0.467074 0.233537 0.972348i \(-0.424970\pi\)
0.233537 + 0.972348i \(0.424970\pi\)
\(152\) 9122.12 0.0320248
\(153\) 424916. 1.46749
\(154\) 0 0
\(155\) 158694. 0.530555
\(156\) 75213.6 0.247448
\(157\) −163297. −0.528723 −0.264362 0.964424i \(-0.585161\pi\)
−0.264362 + 0.964424i \(0.585161\pi\)
\(158\) 529191. 1.68643
\(159\) −82645.6 −0.259255
\(160\) −497456. −1.53622
\(161\) 655197. 1.99208
\(162\) −412151. −1.23387
\(163\) −535758. −1.57943 −0.789713 0.613477i \(-0.789771\pi\)
−0.789713 + 0.613477i \(0.789771\pi\)
\(164\) 390851. 1.13475
\(165\) 0 0
\(166\) −792492. −2.23216
\(167\) −553587. −1.53601 −0.768005 0.640443i \(-0.778751\pi\)
−0.768005 + 0.640443i \(0.778751\pi\)
\(168\) −12582.7 −0.0343953
\(169\) 7881.54 0.0212273
\(170\) −902331. −2.39466
\(171\) 84636.5 0.221344
\(172\) −293221. −0.755744
\(173\) −266973. −0.678190 −0.339095 0.940752i \(-0.610121\pi\)
−0.339095 + 0.940752i \(0.610121\pi\)
\(174\) 49007.4 0.122712
\(175\) −66687.0 −0.164606
\(176\) 0 0
\(177\) −68039.2 −0.163240
\(178\) 389851. 0.922250
\(179\) 3030.33 0.00706900 0.00353450 0.999994i \(-0.498875\pi\)
0.00353450 + 0.999994i \(0.498875\pi\)
\(180\) 484284. 1.11409
\(181\) 761242. 1.72714 0.863568 0.504233i \(-0.168225\pi\)
0.863568 + 0.504233i \(0.168225\pi\)
\(182\) −731415. −1.63676
\(183\) 72987.3 0.161109
\(184\) 112371. 0.244686
\(185\) −578402. −1.24251
\(186\) −75652.8 −0.160340
\(187\) 0 0
\(188\) −77828.9 −0.160600
\(189\) −239634. −0.487970
\(190\) −179730. −0.361191
\(191\) −430653. −0.854170 −0.427085 0.904212i \(-0.640459\pi\)
−0.427085 + 0.904212i \(0.640459\pi\)
\(192\) 134796. 0.263891
\(193\) 272285. 0.526175 0.263088 0.964772i \(-0.415259\pi\)
0.263088 + 0.964772i \(0.415259\pi\)
\(194\) 314405. 0.599770
\(195\) −128521. −0.242039
\(196\) 148517. 0.276144
\(197\) −574550. −1.05478 −0.527390 0.849623i \(-0.676829\pi\)
−0.527390 + 0.849623i \(0.676829\pi\)
\(198\) 0 0
\(199\) 269926. 0.483183 0.241592 0.970378i \(-0.422331\pi\)
0.241592 + 0.970378i \(0.422331\pi\)
\(200\) −11437.3 −0.0202185
\(201\) 180253. 0.314697
\(202\) −335788. −0.579011
\(203\) −249088. −0.424240
\(204\) 224830. 0.378250
\(205\) −667863. −1.10995
\(206\) 406393. 0.667234
\(207\) 1.04260e6 1.69118
\(208\) 564985. 0.905480
\(209\) 0 0
\(210\) 247912. 0.387926
\(211\) 753372. 1.16494 0.582470 0.812853i \(-0.302087\pi\)
0.582470 + 0.812853i \(0.302087\pi\)
\(212\) 830695. 1.26941
\(213\) 4874.77 0.00736216
\(214\) −55605.6 −0.0830011
\(215\) 501040. 0.739224
\(216\) −41098.9 −0.0599371
\(217\) 384517. 0.554327
\(218\) 792414. 1.12930
\(219\) 252619. 0.355922
\(220\) 0 0
\(221\) 1.13344e6 1.56105
\(222\) 275737. 0.375503
\(223\) −997692. −1.34349 −0.671745 0.740783i \(-0.734455\pi\)
−0.671745 + 0.740783i \(0.734455\pi\)
\(224\) −1.20534e6 −1.60506
\(225\) −106117. −0.139743
\(226\) 1.74347e6 2.27062
\(227\) 495214. 0.637864 0.318932 0.947778i \(-0.396676\pi\)
0.318932 + 0.947778i \(0.396676\pi\)
\(228\) 44782.5 0.0570521
\(229\) −221893. −0.279611 −0.139806 0.990179i \(-0.544648\pi\)
−0.139806 + 0.990179i \(0.544648\pi\)
\(230\) −2.21401e6 −2.75968
\(231\) 0 0
\(232\) −42720.4 −0.0521093
\(233\) −619425. −0.747479 −0.373739 0.927534i \(-0.621925\pi\)
−0.373739 + 0.927534i \(0.621925\pi\)
\(234\) −1.16388e6 −1.38953
\(235\) 132989. 0.157090
\(236\) 683881. 0.799283
\(237\) 225308. 0.260559
\(238\) −2.18636e6 −2.50195
\(239\) 295471. 0.334595 0.167298 0.985906i \(-0.446496\pi\)
0.167298 + 0.985906i \(0.446496\pi\)
\(240\) −191501. −0.214606
\(241\) −693153. −0.768753 −0.384376 0.923176i \(-0.625584\pi\)
−0.384376 + 0.923176i \(0.625584\pi\)
\(242\) 0 0
\(243\) −576873. −0.626707
\(244\) −733616. −0.788850
\(245\) −253777. −0.270108
\(246\) 318385. 0.335440
\(247\) 225762. 0.235456
\(248\) 65947.5 0.0680878
\(249\) −337411. −0.344875
\(250\) −1.30658e6 −1.32217
\(251\) 533816. 0.534820 0.267410 0.963583i \(-0.413832\pi\)
0.267410 + 0.963583i \(0.413832\pi\)
\(252\) 1.17343e6 1.16400
\(253\) 0 0
\(254\) −739874. −0.719571
\(255\) −384176. −0.369981
\(256\) 822041. 0.783960
\(257\) −652296. −0.616044 −0.308022 0.951379i \(-0.599667\pi\)
−0.308022 + 0.951379i \(0.599667\pi\)
\(258\) −238857. −0.223402
\(259\) −1.40148e6 −1.29818
\(260\) 1.29180e6 1.18512
\(261\) −396366. −0.360160
\(262\) 534654. 0.481193
\(263\) 622045. 0.554540 0.277270 0.960792i \(-0.410570\pi\)
0.277270 + 0.960792i \(0.410570\pi\)
\(264\) 0 0
\(265\) −1.41944e6 −1.24166
\(266\) −435488. −0.377374
\(267\) 165983. 0.142490
\(268\) −1.81177e6 −1.54087
\(269\) −482862. −0.406858 −0.203429 0.979090i \(-0.565209\pi\)
−0.203429 + 0.979090i \(0.565209\pi\)
\(270\) 809758. 0.675998
\(271\) 1.10678e6 0.915460 0.457730 0.889091i \(-0.348663\pi\)
0.457730 + 0.889091i \(0.348663\pi\)
\(272\) 1.68887e6 1.38412
\(273\) −311407. −0.252884
\(274\) −43581.1 −0.0350689
\(275\) 0 0
\(276\) 551655. 0.435908
\(277\) −639062. −0.500430 −0.250215 0.968190i \(-0.580501\pi\)
−0.250215 + 0.968190i \(0.580501\pi\)
\(278\) 736626. 0.571656
\(279\) 611872. 0.470598
\(280\) −216108. −0.164731
\(281\) 257984. 0.194907 0.0974534 0.995240i \(-0.468930\pi\)
0.0974534 + 0.995240i \(0.468930\pi\)
\(282\) −63399.0 −0.0474744
\(283\) −1.02991e6 −0.764425 −0.382213 0.924074i \(-0.624838\pi\)
−0.382213 + 0.924074i \(0.624838\pi\)
\(284\) −48997.7 −0.0360479
\(285\) −76521.8 −0.0558050
\(286\) 0 0
\(287\) −1.61824e6 −1.15968
\(288\) −1.91803e6 −1.36262
\(289\) 1.96823e6 1.38622
\(290\) 841704. 0.587712
\(291\) 133861. 0.0926662
\(292\) −2.53914e6 −1.74273
\(293\) −877712. −0.597287 −0.298644 0.954365i \(-0.596534\pi\)
−0.298644 + 0.954365i \(0.596534\pi\)
\(294\) 120981. 0.0816299
\(295\) −1.16858e6 −0.781811
\(296\) −240363. −0.159455
\(297\) 0 0
\(298\) 544400. 0.355122
\(299\) 2.78106e6 1.79900
\(300\) −56148.4 −0.0360192
\(301\) 1.21403e6 0.772345
\(302\) −1.07150e6 −0.676042
\(303\) −142965. −0.0894589
\(304\) 336395. 0.208769
\(305\) 1.25356e6 0.771606
\(306\) −3.47909e6 −2.12404
\(307\) 1.30925e6 0.792826 0.396413 0.918072i \(-0.370255\pi\)
0.396413 + 0.918072i \(0.370255\pi\)
\(308\) 0 0
\(309\) 173026. 0.103089
\(310\) −1.29934e6 −0.767925
\(311\) 3.35930e6 1.96946 0.984731 0.174083i \(-0.0556962\pi\)
0.984731 + 0.174083i \(0.0556962\pi\)
\(312\) −53408.6 −0.0310616
\(313\) −3.00640e6 −1.73454 −0.867272 0.497835i \(-0.834129\pi\)
−0.867272 + 0.497835i \(0.834129\pi\)
\(314\) 1.33703e6 0.765273
\(315\) −2.00508e6 −1.13856
\(316\) −2.26464e6 −1.27579
\(317\) 2.10147e6 1.17456 0.587279 0.809385i \(-0.300199\pi\)
0.587279 + 0.809385i \(0.300199\pi\)
\(318\) 676679. 0.375245
\(319\) 0 0
\(320\) 2.31513e6 1.26387
\(321\) −23674.6 −0.0128239
\(322\) −5.36457e6 −2.88333
\(323\) 674853. 0.359918
\(324\) 1.76377e6 0.933426
\(325\) −283061. −0.148652
\(326\) 4.38663e6 2.28606
\(327\) 337378. 0.174481
\(328\) −277540. −0.142443
\(329\) 322235. 0.164128
\(330\) 0 0
\(331\) 1.23338e6 0.618766 0.309383 0.950938i \(-0.399878\pi\)
0.309383 + 0.950938i \(0.399878\pi\)
\(332\) 3.39142e6 1.68864
\(333\) −2.23013e6 −1.10210
\(334\) 4.53261e6 2.22322
\(335\) 3.09585e6 1.50719
\(336\) −464009. −0.224222
\(337\) −679319. −0.325836 −0.162918 0.986640i \(-0.552091\pi\)
−0.162918 + 0.986640i \(0.552091\pi\)
\(338\) −64531.9 −0.0307243
\(339\) 742301. 0.350817
\(340\) 3.86146e6 1.81157
\(341\) 0 0
\(342\) −692980. −0.320373
\(343\) 1.82331e6 0.836805
\(344\) 208214. 0.0948668
\(345\) −942635. −0.426379
\(346\) 2.18590e6 0.981611
\(347\) −2.67540e6 −1.19279 −0.596397 0.802690i \(-0.703402\pi\)
−0.596397 + 0.802690i \(0.703402\pi\)
\(348\) −209724. −0.0928324
\(349\) 2.37636e6 1.04435 0.522177 0.852837i \(-0.325120\pi\)
0.522177 + 0.852837i \(0.325120\pi\)
\(350\) 546015. 0.238251
\(351\) −1.01715e6 −0.440675
\(352\) 0 0
\(353\) 638696. 0.272808 0.136404 0.990653i \(-0.456445\pi\)
0.136404 + 0.990653i \(0.456445\pi\)
\(354\) 557086. 0.236273
\(355\) 83724.4 0.0352599
\(356\) −1.66834e6 −0.697686
\(357\) −930864. −0.386559
\(358\) −24811.5 −0.0102317
\(359\) −1.50842e6 −0.617712 −0.308856 0.951109i \(-0.599946\pi\)
−0.308856 + 0.951109i \(0.599946\pi\)
\(360\) −343886. −0.139849
\(361\) −2.34168e6 −0.945713
\(362\) −6.23284e6 −2.49985
\(363\) 0 0
\(364\) 3.13004e6 1.23822
\(365\) 4.33874e6 1.70463
\(366\) −597600. −0.233189
\(367\) 1.77368e6 0.687403 0.343701 0.939079i \(-0.388319\pi\)
0.343701 + 0.939079i \(0.388319\pi\)
\(368\) 4.14389e6 1.59510
\(369\) −2.57506e6 −0.984513
\(370\) 4.73580e6 1.79841
\(371\) −3.43933e6 −1.29729
\(372\) 323751. 0.121298
\(373\) −2.27176e6 −0.845456 −0.422728 0.906257i \(-0.638927\pi\)
−0.422728 + 0.906257i \(0.638927\pi\)
\(374\) 0 0
\(375\) −556290. −0.204279
\(376\) 55265.7 0.0201598
\(377\) −1.05728e6 −0.383122
\(378\) 1.96205e6 0.706287
\(379\) −4.42409e6 −1.58207 −0.791035 0.611771i \(-0.790457\pi\)
−0.791035 + 0.611771i \(0.790457\pi\)
\(380\) 769142. 0.273242
\(381\) −315009. −0.111176
\(382\) 3.52607e6 1.23632
\(383\) 2.37588e6 0.827615 0.413807 0.910364i \(-0.364199\pi\)
0.413807 + 0.910364i \(0.364199\pi\)
\(384\) −176831. −0.0611971
\(385\) 0 0
\(386\) −2.22939e6 −0.761586
\(387\) 1.93185e6 0.655684
\(388\) −1.34547e6 −0.453728
\(389\) −2.65905e6 −0.890949 −0.445474 0.895295i \(-0.646965\pi\)
−0.445474 + 0.895295i \(0.646965\pi\)
\(390\) 1.05229e6 0.350328
\(391\) 8.31319e6 2.74996
\(392\) −105461. −0.0346638
\(393\) 227634. 0.0743457
\(394\) 4.70425e6 1.52669
\(395\) 3.86968e6 1.24791
\(396\) 0 0
\(397\) 2.15712e6 0.686907 0.343453 0.939170i \(-0.388403\pi\)
0.343453 + 0.939170i \(0.388403\pi\)
\(398\) −2.21008e6 −0.699359
\(399\) −185413. −0.0583054
\(400\) −421772. −0.131804
\(401\) −2.43031e6 −0.754744 −0.377372 0.926062i \(-0.623172\pi\)
−0.377372 + 0.926062i \(0.623172\pi\)
\(402\) −1.47586e6 −0.455492
\(403\) 1.63213e6 0.500601
\(404\) 1.43698e6 0.438024
\(405\) −3.01383e6 −0.913022
\(406\) 2.03946e6 0.614045
\(407\) 0 0
\(408\) −159650. −0.0474808
\(409\) −6.12831e6 −1.81148 −0.905738 0.423839i \(-0.860682\pi\)
−0.905738 + 0.423839i \(0.860682\pi\)
\(410\) 5.46827e6 1.60654
\(411\) −18555.1 −0.00541824
\(412\) −1.73913e6 −0.504765
\(413\) −2.83147e6 −0.816841
\(414\) −8.53649e6 −2.44781
\(415\) −5.79505e6 −1.65172
\(416\) −5.11621e6 −1.44949
\(417\) 313625. 0.0883225
\(418\) 0 0
\(419\) −375626. −0.104525 −0.0522626 0.998633i \(-0.516643\pi\)
−0.0522626 + 0.998633i \(0.516643\pi\)
\(420\) −1.06092e6 −0.293468
\(421\) 3.52333e6 0.968831 0.484416 0.874838i \(-0.339032\pi\)
0.484416 + 0.874838i \(0.339032\pi\)
\(422\) −6.16840e6 −1.68613
\(423\) 512764. 0.139337
\(424\) −589870. −0.159346
\(425\) −846131. −0.227230
\(426\) −39913.3 −0.0106560
\(427\) 3.03739e6 0.806178
\(428\) 237960. 0.0627906
\(429\) 0 0
\(430\) −4.10237e6 −1.06995
\(431\) −3.15287e6 −0.817548 −0.408774 0.912636i \(-0.634044\pi\)
−0.408774 + 0.912636i \(0.634044\pi\)
\(432\) −1.51560e6 −0.390728
\(433\) 1.62168e6 0.415667 0.207833 0.978164i \(-0.433359\pi\)
0.207833 + 0.978164i \(0.433359\pi\)
\(434\) −3.14832e6 −0.802333
\(435\) 358364. 0.0908032
\(436\) −3.39108e6 −0.854322
\(437\) 1.65586e6 0.414781
\(438\) −2.06837e6 −0.515161
\(439\) 2.48145e6 0.614533 0.307266 0.951624i \(-0.400586\pi\)
0.307266 + 0.951624i \(0.400586\pi\)
\(440\) 0 0
\(441\) −978482. −0.239583
\(442\) −9.28026e6 −2.25946
\(443\) 3.75466e6 0.908994 0.454497 0.890748i \(-0.349819\pi\)
0.454497 + 0.890748i \(0.349819\pi\)
\(444\) −1.18000e6 −0.284069
\(445\) 2.85076e6 0.682435
\(446\) 8.16882e6 1.94456
\(447\) 231783. 0.0548673
\(448\) 5.60960e6 1.32049
\(449\) −4.80916e6 −1.12578 −0.562890 0.826532i \(-0.690311\pi\)
−0.562890 + 0.826532i \(0.690311\pi\)
\(450\) 868859. 0.202264
\(451\) 0 0
\(452\) −7.46108e6 −1.71773
\(453\) −456200. −0.104450
\(454\) −4.05467e6 −0.923244
\(455\) −5.34843e6 −1.21115
\(456\) −31799.7 −0.00716162
\(457\) −7.09951e6 −1.59015 −0.795075 0.606512i \(-0.792568\pi\)
−0.795075 + 0.606512i \(0.792568\pi\)
\(458\) 1.81680e6 0.404709
\(459\) −3.04049e6 −0.673616
\(460\) 9.47469e6 2.08771
\(461\) −8.12745e6 −1.78116 −0.890578 0.454830i \(-0.849700\pi\)
−0.890578 + 0.454830i \(0.849700\pi\)
\(462\) 0 0
\(463\) −2.67361e6 −0.579623 −0.289812 0.957084i \(-0.593593\pi\)
−0.289812 + 0.957084i \(0.593593\pi\)
\(464\) −1.57539e6 −0.339699
\(465\) −553207. −0.118647
\(466\) 5.07168e6 1.08190
\(467\) −4.32733e6 −0.918180 −0.459090 0.888390i \(-0.651825\pi\)
−0.459090 + 0.888390i \(0.651825\pi\)
\(468\) 4.98075e6 1.05119
\(469\) 7.50129e6 1.57472
\(470\) −1.08888e6 −0.227371
\(471\) 569253. 0.118237
\(472\) −485618. −0.100332
\(473\) 0 0
\(474\) −1.84476e6 −0.377133
\(475\) −168536. −0.0342735
\(476\) 9.35637e6 1.89274
\(477\) −5.47291e6 −1.10134
\(478\) −2.41923e6 −0.484293
\(479\) −1.55878e6 −0.310417 −0.155208 0.987882i \(-0.549605\pi\)
−0.155208 + 0.987882i \(0.549605\pi\)
\(480\) 1.73413e6 0.343541
\(481\) −5.94873e6 −1.17236
\(482\) 5.67535e6 1.11269
\(483\) −2.28402e6 −0.445483
\(484\) 0 0
\(485\) 2.29907e6 0.443810
\(486\) 4.72328e6 0.907095
\(487\) −7.63818e6 −1.45938 −0.729689 0.683779i \(-0.760335\pi\)
−0.729689 + 0.683779i \(0.760335\pi\)
\(488\) 520935. 0.0990225
\(489\) 1.86765e6 0.353202
\(490\) 2.07786e6 0.390954
\(491\) −3.60872e6 −0.675537 −0.337768 0.941229i \(-0.609672\pi\)
−0.337768 + 0.941229i \(0.609672\pi\)
\(492\) −1.36251e6 −0.253761
\(493\) −3.16045e6 −0.585640
\(494\) −1.84848e6 −0.340798
\(495\) 0 0
\(496\) 2.43194e6 0.443862
\(497\) 202865. 0.0368397
\(498\) 2.76263e6 0.499171
\(499\) −8.46131e6 −1.52120 −0.760599 0.649221i \(-0.775095\pi\)
−0.760599 + 0.649221i \(0.775095\pi\)
\(500\) 5.59143e6 1.00023
\(501\) 1.92980e6 0.343494
\(502\) −4.37074e6 −0.774098
\(503\) 8.28353e6 1.45981 0.729904 0.683550i \(-0.239565\pi\)
0.729904 + 0.683550i \(0.239565\pi\)
\(504\) −833241. −0.146115
\(505\) −2.45543e6 −0.428449
\(506\) 0 0
\(507\) −27475.1 −0.00474700
\(508\) 3.16624e6 0.544358
\(509\) −7.60138e6 −1.30046 −0.650232 0.759736i \(-0.725328\pi\)
−0.650232 + 0.759736i \(0.725328\pi\)
\(510\) 3.14553e6 0.535511
\(511\) 1.05128e7 1.78101
\(512\) −8.35388e6 −1.40836
\(513\) −605618. −0.101603
\(514\) 5.34082e6 0.891661
\(515\) 2.97172e6 0.493731
\(516\) 1.02217e6 0.169005
\(517\) 0 0
\(518\) 1.14749e7 1.87899
\(519\) 930667. 0.151662
\(520\) −917295. −0.148765
\(521\) 9.60432e6 1.55015 0.775073 0.631872i \(-0.217713\pi\)
0.775073 + 0.631872i \(0.217713\pi\)
\(522\) 3.24534e6 0.521295
\(523\) 9.97831e6 1.59515 0.797577 0.603217i \(-0.206115\pi\)
0.797577 + 0.603217i \(0.206115\pi\)
\(524\) −2.28802e6 −0.364025
\(525\) 232471. 0.0368104
\(526\) −5.09313e6 −0.802640
\(527\) 4.87879e6 0.765218
\(528\) 0 0
\(529\) 1.39613e7 2.16914
\(530\) 1.16220e7 1.79718
\(531\) −4.50565e6 −0.693459
\(532\) 1.86364e6 0.285485
\(533\) −6.86881e6 −1.04728
\(534\) −1.35902e6 −0.206240
\(535\) −406612. −0.0614181
\(536\) 1.28653e6 0.193422
\(537\) −10563.8 −0.00158082
\(538\) 3.95354e6 0.588885
\(539\) 0 0
\(540\) −3.46530e6 −0.511395
\(541\) 4.34177e6 0.637784 0.318892 0.947791i \(-0.396689\pi\)
0.318892 + 0.947791i \(0.396689\pi\)
\(542\) −9.06203e6 −1.32503
\(543\) −2.65369e6 −0.386234
\(544\) −1.52935e7 −2.21569
\(545\) 5.79448e6 0.835647
\(546\) 2.54971e6 0.366024
\(547\) −1.14668e7 −1.63860 −0.819302 0.573363i \(-0.805639\pi\)
−0.819302 + 0.573363i \(0.805639\pi\)
\(548\) 186503. 0.0265298
\(549\) 4.83332e6 0.684407
\(550\) 0 0
\(551\) −629511. −0.0883332
\(552\) −391726. −0.0547185
\(553\) 9.37627e6 1.30382
\(554\) 5.23246e6 0.724322
\(555\) 2.01631e6 0.277860
\(556\) −3.15234e6 −0.432460
\(557\) 1.57100e6 0.214555 0.107278 0.994229i \(-0.465787\pi\)
0.107278 + 0.994229i \(0.465787\pi\)
\(558\) −5.00983e6 −0.681142
\(559\) 5.15307e6 0.697488
\(560\) −7.96938e6 −1.07388
\(561\) 0 0
\(562\) −2.11230e6 −0.282108
\(563\) 850908. 0.113139 0.0565694 0.998399i \(-0.481984\pi\)
0.0565694 + 0.998399i \(0.481984\pi\)
\(564\) 271312. 0.0359146
\(565\) 1.27490e7 1.68018
\(566\) 8.43265e6 1.10643
\(567\) −7.30255e6 −0.953931
\(568\) 34792.9 0.00452501
\(569\) 1.19642e7 1.54919 0.774595 0.632458i \(-0.217954\pi\)
0.774595 + 0.632458i \(0.217954\pi\)
\(570\) 626539. 0.0807721
\(571\) 7.97842e6 1.02406 0.512032 0.858967i \(-0.328893\pi\)
0.512032 + 0.858967i \(0.328893\pi\)
\(572\) 0 0
\(573\) 1.50126e6 0.191016
\(574\) 1.32497e7 1.67852
\(575\) −2.07611e6 −0.261867
\(576\) 8.92640e6 1.12104
\(577\) 5.90743e6 0.738685 0.369342 0.929293i \(-0.379583\pi\)
0.369342 + 0.929293i \(0.379583\pi\)
\(578\) −1.61153e7 −2.00641
\(579\) −949186. −0.117667
\(580\) −3.60202e6 −0.444606
\(581\) −1.40415e7 −1.72573
\(582\) −1.09602e6 −0.134125
\(583\) 0 0
\(584\) 1.80303e6 0.218761
\(585\) −8.51081e6 −1.02821
\(586\) 7.18646e6 0.864512
\(587\) 1.34766e6 0.161430 0.0807151 0.996737i \(-0.474280\pi\)
0.0807151 + 0.996737i \(0.474280\pi\)
\(588\) −517730. −0.0617533
\(589\) 971777. 0.115419
\(590\) 9.56797e6 1.13159
\(591\) 2.00288e6 0.235877
\(592\) −8.86385e6 −1.03948
\(593\) −1.05883e7 −1.23649 −0.618243 0.785987i \(-0.712155\pi\)
−0.618243 + 0.785987i \(0.712155\pi\)
\(594\) 0 0
\(595\) −1.59876e7 −1.85136
\(596\) −2.32972e6 −0.268651
\(597\) −940962. −0.108053
\(598\) −2.27705e7 −2.60388
\(599\) 3.48377e6 0.396718 0.198359 0.980129i \(-0.436439\pi\)
0.198359 + 0.980129i \(0.436439\pi\)
\(600\) 39870.5 0.00452141
\(601\) 6.41433e6 0.724378 0.362189 0.932105i \(-0.382030\pi\)
0.362189 + 0.932105i \(0.382030\pi\)
\(602\) −9.94010e6 −1.11789
\(603\) 1.19366e7 1.33686
\(604\) 4.58540e6 0.511428
\(605\) 0 0
\(606\) 1.17056e6 0.129483
\(607\) 700912. 0.0772132 0.0386066 0.999254i \(-0.487708\pi\)
0.0386066 + 0.999254i \(0.487708\pi\)
\(608\) −3.04622e6 −0.334197
\(609\) 868320. 0.0948717
\(610\) −1.02638e7 −1.11682
\(611\) 1.36776e6 0.148221
\(612\) 1.48885e7 1.60684
\(613\) 1.17591e7 1.26393 0.631966 0.774996i \(-0.282248\pi\)
0.631966 + 0.774996i \(0.282248\pi\)
\(614\) −1.07198e7 −1.14753
\(615\) 2.32817e6 0.248214
\(616\) 0 0
\(617\) −1.00683e6 −0.106474 −0.0532371 0.998582i \(-0.516954\pi\)
−0.0532371 + 0.998582i \(0.516954\pi\)
\(618\) −1.41669e6 −0.149212
\(619\) 1.27458e7 1.33703 0.668513 0.743700i \(-0.266931\pi\)
0.668513 + 0.743700i \(0.266931\pi\)
\(620\) 5.56044e6 0.580938
\(621\) −7.46031e6 −0.776297
\(622\) −2.75050e7 −2.85060
\(623\) 6.90744e6 0.713012
\(624\) −1.96954e6 −0.202490
\(625\) −1.09908e7 −1.12546
\(626\) 2.46155e7 2.51058
\(627\) 0 0
\(628\) −5.72172e6 −0.578932
\(629\) −1.77821e7 −1.79207
\(630\) 1.64171e7 1.64795
\(631\) 1.41284e7 1.41260 0.706299 0.707913i \(-0.250363\pi\)
0.706299 + 0.707913i \(0.250363\pi\)
\(632\) 1.60810e6 0.160148
\(633\) −2.62626e6 −0.260512
\(634\) −1.72062e7 −1.70005
\(635\) −5.41029e6 −0.532459
\(636\) −2.89580e6 −0.283874
\(637\) −2.61004e6 −0.254858
\(638\) 0 0
\(639\) 322814. 0.0312752
\(640\) −3.03708e6 −0.293094
\(641\) −4.36680e6 −0.419777 −0.209888 0.977725i \(-0.567310\pi\)
−0.209888 + 0.977725i \(0.567310\pi\)
\(642\) 193841. 0.0185613
\(643\) 7.81597e6 0.745513 0.372757 0.927929i \(-0.378413\pi\)
0.372757 + 0.927929i \(0.378413\pi\)
\(644\) 2.29573e7 2.18125
\(645\) −1.74662e6 −0.165310
\(646\) −5.52551e6 −0.520944
\(647\) 2.01624e7 1.89357 0.946786 0.321863i \(-0.104309\pi\)
0.946786 + 0.321863i \(0.104309\pi\)
\(648\) −1.25244e6 −0.117171
\(649\) 0 0
\(650\) 2.31762e6 0.215159
\(651\) −1.34043e6 −0.123963
\(652\) −1.87723e7 −1.72941
\(653\) 324619. 0.0297914 0.0148957 0.999889i \(-0.495258\pi\)
0.0148957 + 0.999889i \(0.495258\pi\)
\(654\) −2.76235e6 −0.252543
\(655\) 3.90963e6 0.356067
\(656\) −1.02348e7 −0.928581
\(657\) 1.67288e7 1.51199
\(658\) −2.63837e6 −0.237559
\(659\) 1.07107e7 0.960740 0.480370 0.877066i \(-0.340502\pi\)
0.480370 + 0.877066i \(0.340502\pi\)
\(660\) 0 0
\(661\) −1.11064e7 −0.988712 −0.494356 0.869260i \(-0.664596\pi\)
−0.494356 + 0.869260i \(0.664596\pi\)
\(662\) −1.00986e7 −0.895601
\(663\) −3.95116e6 −0.349093
\(664\) −2.40822e6 −0.211971
\(665\) −3.18448e6 −0.279244
\(666\) 1.82597e7 1.59517
\(667\) −7.75464e6 −0.674912
\(668\) −1.93970e7 −1.68187
\(669\) 3.47795e6 0.300441
\(670\) −2.53480e7 −2.18151
\(671\) 0 0
\(672\) 4.20182e6 0.358934
\(673\) 1.38137e7 1.17564 0.587818 0.808993i \(-0.299987\pi\)
0.587818 + 0.808993i \(0.299987\pi\)
\(674\) 5.56208e6 0.471615
\(675\) 759323. 0.0641457
\(676\) 276160. 0.0232431
\(677\) −2.29090e6 −0.192103 −0.0960514 0.995376i \(-0.530621\pi\)
−0.0960514 + 0.995376i \(0.530621\pi\)
\(678\) −6.07775e6 −0.507772
\(679\) 5.57067e6 0.463695
\(680\) −2.74200e6 −0.227402
\(681\) −1.72632e6 −0.142644
\(682\) 0 0
\(683\) 4.40512e6 0.361332 0.180666 0.983545i \(-0.442175\pi\)
0.180666 + 0.983545i \(0.442175\pi\)
\(684\) 2.96556e6 0.242363
\(685\) −318685. −0.0259498
\(686\) −1.49287e7 −1.21119
\(687\) 773519. 0.0625287
\(688\) 7.67828e6 0.618434
\(689\) −1.45986e7 −1.17156
\(690\) 7.71803e6 0.617140
\(691\) −5.86199e6 −0.467035 −0.233518 0.972353i \(-0.575024\pi\)
−0.233518 + 0.972353i \(0.575024\pi\)
\(692\) −9.35440e6 −0.742592
\(693\) 0 0
\(694\) 2.19054e7 1.72645
\(695\) 5.38653e6 0.423007
\(696\) 148923. 0.0116530
\(697\) −2.05324e7 −1.60087
\(698\) −1.94569e7 −1.51160
\(699\) 2.15932e6 0.167157
\(700\) −2.33663e6 −0.180238
\(701\) 8.02106e6 0.616505 0.308253 0.951305i \(-0.400256\pi\)
0.308253 + 0.951305i \(0.400256\pi\)
\(702\) 8.32816e6 0.637832
\(703\) −3.54190e6 −0.270301
\(704\) 0 0
\(705\) −463602. −0.0351295
\(706\) −5.22946e6 −0.394862
\(707\) −5.94954e6 −0.447646
\(708\) −2.38401e6 −0.178741
\(709\) 2.17891e7 1.62788 0.813941 0.580948i \(-0.197318\pi\)
0.813941 + 0.580948i \(0.197318\pi\)
\(710\) −685512. −0.0510351
\(711\) 1.49202e7 1.10688
\(712\) 1.18468e6 0.0875789
\(713\) 1.19709e7 0.881863
\(714\) 7.62165e6 0.559505
\(715\) 0 0
\(716\) 106179. 0.00774029
\(717\) −1.03001e6 −0.0748246
\(718\) 1.23505e7 0.894076
\(719\) 1.03483e7 0.746531 0.373266 0.927724i \(-0.378238\pi\)
0.373266 + 0.927724i \(0.378238\pi\)
\(720\) −1.26814e7 −0.911670
\(721\) 7.20052e6 0.515853
\(722\) 1.91730e7 1.36882
\(723\) 2.41633e6 0.171914
\(724\) 2.66730e7 1.89115
\(725\) 789280. 0.0557682
\(726\) 0 0
\(727\) −2.03348e7 −1.42693 −0.713466 0.700690i \(-0.752876\pi\)
−0.713466 + 0.700690i \(0.752876\pi\)
\(728\) −2.22262e6 −0.155430
\(729\) −1.02211e7 −0.712325
\(730\) −3.55244e7 −2.46729
\(731\) 1.54037e7 1.06618
\(732\) 2.55739e6 0.176408
\(733\) −4.78280e6 −0.328793 −0.164396 0.986394i \(-0.552568\pi\)
−0.164396 + 0.986394i \(0.552568\pi\)
\(734\) −1.45224e7 −0.994946
\(735\) 884668. 0.0604034
\(736\) −3.75249e7 −2.55344
\(737\) 0 0
\(738\) 2.10839e7 1.42498
\(739\) 1.08737e7 0.732429 0.366215 0.930530i \(-0.380654\pi\)
0.366215 + 0.930530i \(0.380654\pi\)
\(740\) −2.02665e7 −1.36050
\(741\) −787008. −0.0526543
\(742\) 2.81603e7 1.87770
\(743\) 1.01036e7 0.671434 0.335717 0.941963i \(-0.391021\pi\)
0.335717 + 0.941963i \(0.391021\pi\)
\(744\) −229893. −0.0152263
\(745\) 3.98089e6 0.262778
\(746\) 1.86006e7 1.22371
\(747\) −2.23438e7 −1.46506
\(748\) 0 0
\(749\) −985227. −0.0641700
\(750\) 4.55475e6 0.295673
\(751\) 9.91947e6 0.641784 0.320892 0.947116i \(-0.396017\pi\)
0.320892 + 0.947116i \(0.396017\pi\)
\(752\) 2.03802e6 0.131421
\(753\) −1.86088e6 −0.119600
\(754\) 8.65673e6 0.554530
\(755\) −7.83526e6 −0.500249
\(756\) −8.39647e6 −0.534309
\(757\) −1.33506e7 −0.846759 −0.423380 0.905952i \(-0.639156\pi\)
−0.423380 + 0.905952i \(0.639156\pi\)
\(758\) 3.62232e7 2.28988
\(759\) 0 0
\(760\) −546162. −0.0342995
\(761\) −1.28109e7 −0.801898 −0.400949 0.916100i \(-0.631320\pi\)
−0.400949 + 0.916100i \(0.631320\pi\)
\(762\) 2.57920e6 0.160916
\(763\) 1.40401e7 0.873089
\(764\) −1.50896e7 −0.935284
\(765\) −2.54407e7 −1.57172
\(766\) −1.94531e7 −1.19789
\(767\) −1.20185e7 −0.737671
\(768\) −2.86564e6 −0.175315
\(769\) −1.90629e7 −1.16245 −0.581224 0.813743i \(-0.697426\pi\)
−0.581224 + 0.813743i \(0.697426\pi\)
\(770\) 0 0
\(771\) 2.27390e6 0.137764
\(772\) 9.54054e6 0.576142
\(773\) −2.34434e6 −0.141115 −0.0705574 0.997508i \(-0.522478\pi\)
−0.0705574 + 0.997508i \(0.522478\pi\)
\(774\) −1.58174e7 −0.949037
\(775\) −1.21841e6 −0.0728686
\(776\) 955410. 0.0569555
\(777\) 4.88555e6 0.290309
\(778\) 2.17716e7 1.28956
\(779\) −4.08972e6 −0.241463
\(780\) −4.50321e6 −0.265024
\(781\) 0 0
\(782\) −6.80661e7 −3.98028
\(783\) 2.83620e6 0.165323
\(784\) −3.88906e6 −0.225972
\(785\) 9.77694e6 0.566277
\(786\) −1.86381e6 −0.107608
\(787\) 1.52283e7 0.876425 0.438212 0.898871i \(-0.355612\pi\)
0.438212 + 0.898871i \(0.355612\pi\)
\(788\) −2.01315e7 −1.15494
\(789\) −2.16845e6 −0.124010
\(790\) −3.16838e7 −1.80622
\(791\) 3.08911e7 1.75547
\(792\) 0 0
\(793\) 1.28926e7 0.728042
\(794\) −1.76619e7 −0.994228
\(795\) 4.94818e6 0.277669
\(796\) 9.45788e6 0.529068
\(797\) −2.70618e7 −1.50907 −0.754537 0.656257i \(-0.772139\pi\)
−0.754537 + 0.656257i \(0.772139\pi\)
\(798\) 1.51811e6 0.0843911
\(799\) 4.08855e6 0.226570
\(800\) 3.81935e6 0.210991
\(801\) 1.09916e7 0.605313
\(802\) 1.98987e7 1.09242
\(803\) 0 0
\(804\) 6.31585e6 0.344581
\(805\) −3.92281e7 −2.13357
\(806\) −1.33634e7 −0.724569
\(807\) 1.68326e6 0.0909844
\(808\) −1.02039e6 −0.0549842
\(809\) 2.25663e6 0.121224 0.0606121 0.998161i \(-0.480695\pi\)
0.0606121 + 0.998161i \(0.480695\pi\)
\(810\) 2.46764e7 1.32151
\(811\) −4.49121e6 −0.239779 −0.119890 0.992787i \(-0.538254\pi\)
−0.119890 + 0.992787i \(0.538254\pi\)
\(812\) −8.72773e6 −0.464527
\(813\) −3.85825e6 −0.204722
\(814\) 0 0
\(815\) 3.20770e7 1.69161
\(816\) −5.88739e6 −0.309526
\(817\) 3.06816e6 0.160814
\(818\) 5.01769e7 2.62193
\(819\) −2.06218e7 −1.07428
\(820\) −2.34011e7 −1.21535
\(821\) −592581. −0.0306825 −0.0153412 0.999882i \(-0.504883\pi\)
−0.0153412 + 0.999882i \(0.504883\pi\)
\(822\) 151924. 0.00784236
\(823\) −1.14748e7 −0.590533 −0.295266 0.955415i \(-0.595408\pi\)
−0.295266 + 0.955415i \(0.595408\pi\)
\(824\) 1.23494e6 0.0633620
\(825\) 0 0
\(826\) 2.31833e7 1.18229
\(827\) 8.47060e6 0.430676 0.215338 0.976540i \(-0.430915\pi\)
0.215338 + 0.976540i \(0.430915\pi\)
\(828\) 3.65313e7 1.85178
\(829\) −1.58876e7 −0.802919 −0.401460 0.915877i \(-0.631497\pi\)
−0.401460 + 0.915877i \(0.631497\pi\)
\(830\) 4.74483e7 2.39070
\(831\) 2.22777e6 0.111910
\(832\) 2.38106e7 1.19251
\(833\) −7.80197e6 −0.389576
\(834\) −2.56788e6 −0.127838
\(835\) 3.31445e7 1.64511
\(836\) 0 0
\(837\) −4.37825e6 −0.216017
\(838\) 3.07552e6 0.151290
\(839\) 2.66963e7 1.30932 0.654659 0.755924i \(-0.272812\pi\)
0.654659 + 0.755924i \(0.272812\pi\)
\(840\) 753352. 0.0368383
\(841\) −1.75631e7 −0.856268
\(842\) −2.88480e7 −1.40228
\(843\) −899333. −0.0435864
\(844\) 2.63972e7 1.27556
\(845\) −471886. −0.0227350
\(846\) −4.19837e6 −0.201676
\(847\) 0 0
\(848\) −2.17525e7 −1.03877
\(849\) 3.59028e6 0.170946
\(850\) 6.92789e6 0.328892
\(851\) −4.36310e7 −2.06524
\(852\) 170806. 0.00806128
\(853\) −3.58773e6 −0.168829 −0.0844146 0.996431i \(-0.526902\pi\)
−0.0844146 + 0.996431i \(0.526902\pi\)
\(854\) −2.48693e7 −1.16686
\(855\) −5.06738e6 −0.237065
\(856\) −168974. −0.00788197
\(857\) 6.00941e6 0.279499 0.139749 0.990187i \(-0.455370\pi\)
0.139749 + 0.990187i \(0.455370\pi\)
\(858\) 0 0
\(859\) −1.74629e7 −0.807484 −0.403742 0.914873i \(-0.632291\pi\)
−0.403742 + 0.914873i \(0.632291\pi\)
\(860\) 1.75558e7 0.809422
\(861\) 5.64118e6 0.259336
\(862\) 2.58149e7 1.18332
\(863\) 2.34431e6 0.107149 0.0535746 0.998564i \(-0.482939\pi\)
0.0535746 + 0.998564i \(0.482939\pi\)
\(864\) 1.37245e7 0.625476
\(865\) 1.59842e7 0.726360
\(866\) −1.32779e7 −0.601635
\(867\) −6.86126e6 −0.309996
\(868\) 1.34730e7 0.606968
\(869\) 0 0
\(870\) −2.93418e6 −0.131428
\(871\) 3.18401e7 1.42210
\(872\) 2.40798e6 0.107241
\(873\) 8.86445e6 0.393655
\(874\) −1.35577e7 −0.600354
\(875\) −2.31502e7 −1.02220
\(876\) 8.85145e6 0.389721
\(877\) −1.98979e7 −0.873591 −0.436796 0.899561i \(-0.643887\pi\)
−0.436796 + 0.899561i \(0.643887\pi\)
\(878\) −2.03175e7 −0.889474
\(879\) 3.05971e6 0.133570
\(880\) 0 0
\(881\) 2.32718e7 1.01016 0.505081 0.863072i \(-0.331463\pi\)
0.505081 + 0.863072i \(0.331463\pi\)
\(882\) 8.01154e6 0.346772
\(883\) −2.71777e7 −1.17304 −0.586518 0.809936i \(-0.699502\pi\)
−0.586518 + 0.809936i \(0.699502\pi\)
\(884\) 3.97142e7 1.70929
\(885\) 4.07366e6 0.174834
\(886\) −3.07421e7 −1.31568
\(887\) 1.39671e7 0.596069 0.298034 0.954555i \(-0.403669\pi\)
0.298034 + 0.954555i \(0.403669\pi\)
\(888\) 837907. 0.0356585
\(889\) −1.31092e7 −0.556316
\(890\) −2.33413e7 −0.987755
\(891\) 0 0
\(892\) −3.49579e7 −1.47107
\(893\) 814374. 0.0341740
\(894\) −1.89778e6 −0.0794149
\(895\) −181433. −0.00757109
\(896\) −7.35889e6 −0.306226
\(897\) −9.69477e6 −0.402306
\(898\) 3.93761e7 1.62945
\(899\) −4.55099e6 −0.187805
\(900\) −3.71822e6 −0.153013
\(901\) −4.36385e7 −1.79084
\(902\) 0 0
\(903\) −4.23209e6 −0.172717
\(904\) 5.29805e6 0.215623
\(905\) −4.55773e7 −1.84981
\(906\) 3.73524e6 0.151181
\(907\) 1.77875e7 0.717954 0.358977 0.933346i \(-0.383126\pi\)
0.358977 + 0.933346i \(0.383126\pi\)
\(908\) 1.73517e7 0.698437
\(909\) −9.46734e6 −0.380031
\(910\) 4.37914e7 1.75302
\(911\) 30398.8 0.00121356 0.000606780 1.00000i \(-0.499807\pi\)
0.000606780 1.00000i \(0.499807\pi\)
\(912\) −1.17267e6 −0.0466864
\(913\) 0 0
\(914\) 5.81288e7 2.30158
\(915\) −4.36991e6 −0.172552
\(916\) −7.77486e6 −0.306164
\(917\) 9.47308e6 0.372021
\(918\) 2.48947e7 0.974990
\(919\) 4.10055e6 0.160160 0.0800798 0.996788i \(-0.474482\pi\)
0.0800798 + 0.996788i \(0.474482\pi\)
\(920\) −6.72790e6 −0.262066
\(921\) −4.56406e6 −0.177297
\(922\) 6.65453e7 2.57804
\(923\) 861085. 0.0332692
\(924\) 0 0
\(925\) 4.44084e6 0.170652
\(926\) 2.18908e7 0.838946
\(927\) 1.14580e7 0.437935
\(928\) 1.42659e7 0.543788
\(929\) −1.46532e7 −0.557048 −0.278524 0.960429i \(-0.589845\pi\)
−0.278524 + 0.960429i \(0.589845\pi\)
\(930\) 4.52950e6 0.171729
\(931\) −1.55403e6 −0.0587604
\(932\) −2.17039e7 −0.818461
\(933\) −1.17105e7 −0.440425
\(934\) 3.54310e7 1.32897
\(935\) 0 0
\(936\) −3.53679e6 −0.131953
\(937\) 3.97538e7 1.47921 0.739604 0.673042i \(-0.235013\pi\)
0.739604 + 0.673042i \(0.235013\pi\)
\(938\) −6.14185e7 −2.27925
\(939\) 1.04803e7 0.387891
\(940\) 4.65979e6 0.172007
\(941\) −5.32850e6 −0.196169 −0.0980847 0.995178i \(-0.531272\pi\)
−0.0980847 + 0.995178i \(0.531272\pi\)
\(942\) −4.66088e6 −0.171136
\(943\) −5.03793e7 −1.84490
\(944\) −1.79081e7 −0.654062
\(945\) 1.43474e7 0.522629
\(946\) 0 0
\(947\) 3.11430e7 1.12846 0.564230 0.825618i \(-0.309173\pi\)
0.564230 + 0.825618i \(0.309173\pi\)
\(948\) 7.89452e6 0.285302
\(949\) 4.46229e7 1.60839
\(950\) 1.37993e6 0.0496074
\(951\) −7.32571e6 −0.262663
\(952\) −6.64389e6 −0.237591
\(953\) −4.87227e7 −1.73780 −0.868899 0.494990i \(-0.835172\pi\)
−0.868899 + 0.494990i \(0.835172\pi\)
\(954\) 4.48106e7 1.59408
\(955\) 2.57842e7 0.914839
\(956\) 1.03529e7 0.366369
\(957\) 0 0
\(958\) 1.27628e7 0.449297
\(959\) −772177. −0.0271125
\(960\) −8.07056e6 −0.282635
\(961\) −2.16038e7 −0.754608
\(962\) 4.87065e7 1.69687
\(963\) −1.56776e6 −0.0544773
\(964\) −2.42873e7 −0.841755
\(965\) −1.63023e7 −0.563548
\(966\) 1.87009e7 0.644792
\(967\) −4.85436e7 −1.66942 −0.834711 0.550688i \(-0.814365\pi\)
−0.834711 + 0.550688i \(0.814365\pi\)
\(968\) 0 0
\(969\) −2.35254e6 −0.0804873
\(970\) −1.88241e7 −0.642370
\(971\) −3.15035e7 −1.07229 −0.536143 0.844127i \(-0.680119\pi\)
−0.536143 + 0.844127i \(0.680119\pi\)
\(972\) −2.02129e7 −0.686221
\(973\) 1.30516e7 0.441960
\(974\) 6.25393e7 2.11230
\(975\) 986751. 0.0332427
\(976\) 1.92104e7 0.645525
\(977\) −5.35354e7 −1.79434 −0.897170 0.441684i \(-0.854381\pi\)
−0.897170 + 0.441684i \(0.854381\pi\)
\(978\) −1.52918e7 −0.511225
\(979\) 0 0
\(980\) −8.89204e6 −0.295758
\(981\) 2.23416e7 0.741211
\(982\) 2.95472e7 0.977771
\(983\) −5.40925e7 −1.78547 −0.892736 0.450580i \(-0.851217\pi\)
−0.892736 + 0.450580i \(0.851217\pi\)
\(984\) 967505. 0.0318541
\(985\) 3.43996e7 1.12970
\(986\) 2.58769e7 0.847655
\(987\) −1.12331e6 −0.0367035
\(988\) 7.91044e6 0.257815
\(989\) 3.77952e7 1.22870
\(990\) 0 0
\(991\) 2.31007e7 0.747208 0.373604 0.927588i \(-0.378122\pi\)
0.373604 + 0.927588i \(0.378122\pi\)
\(992\) −2.20223e7 −0.710533
\(993\) −4.29956e6 −0.138373
\(994\) −1.66100e6 −0.0533218
\(995\) −1.61611e7 −0.517502
\(996\) −1.18225e7 −0.377625
\(997\) 4.54061e7 1.44669 0.723346 0.690486i \(-0.242603\pi\)
0.723346 + 0.690486i \(0.242603\pi\)
\(998\) 6.92788e7 2.20178
\(999\) 1.59577e7 0.505891
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 121.6.a.d.1.1 3
3.2 odd 2 1089.6.a.r.1.3 3
11.10 odd 2 11.6.a.b.1.3 3
33.32 even 2 99.6.a.g.1.1 3
44.43 even 2 176.6.a.i.1.3 3
55.32 even 4 275.6.b.b.199.5 6
55.43 even 4 275.6.b.b.199.2 6
55.54 odd 2 275.6.a.b.1.1 3
77.76 even 2 539.6.a.e.1.3 3
88.21 odd 2 704.6.a.q.1.3 3
88.43 even 2 704.6.a.t.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.3 3 11.10 odd 2
99.6.a.g.1.1 3 33.32 even 2
121.6.a.d.1.1 3 1.1 even 1 trivial
176.6.a.i.1.3 3 44.43 even 2
275.6.a.b.1.1 3 55.54 odd 2
275.6.b.b.199.2 6 55.43 even 4
275.6.b.b.199.5 6 55.32 even 4
539.6.a.e.1.3 3 77.76 even 2
704.6.a.q.1.3 3 88.21 odd 2
704.6.a.t.1.1 3 88.43 even 2
1089.6.a.r.1.3 3 3.2 odd 2