Properties

Label 275.6.a.b.1.1
Level $275$
Weight $6$
Character 275.1
Self dual yes
Analytic conductor $44.106$
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] = [275,6,Mod(1,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("275.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(44.1055504486\)
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 \(\chi\) \(=\) 275.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.18772 q^{2} +3.48600 q^{3} +35.0388 q^{4} -28.5424 q^{6} -145.071 q^{7} -24.8808 q^{8} -230.848 q^{9} +O(q^{10})\) \(q-8.18772 q^{2} +3.48600 q^{3} +35.0388 q^{4} -28.5424 q^{6} -145.071 q^{7} -24.8808 q^{8} -230.848 q^{9} +121.000 q^{11} +122.145 q^{12} -615.772 q^{13} +1187.80 q^{14} -917.524 q^{16} -1840.68 q^{17} +1890.12 q^{18} +366.633 q^{19} -505.718 q^{21} -990.714 q^{22} +4516.38 q^{23} -86.7344 q^{24} +5041.77 q^{26} -1651.83 q^{27} -5083.12 q^{28} -1717.00 q^{29} -2650.54 q^{31} +8308.62 q^{32} +421.806 q^{33} +15070.9 q^{34} -8088.63 q^{36} -9660.61 q^{37} -3001.89 q^{38} -2146.58 q^{39} -11154.8 q^{41} +4140.68 q^{42} -8368.48 q^{43} +4239.69 q^{44} -36978.9 q^{46} +2221.22 q^{47} -3198.49 q^{48} +4238.64 q^{49} -6416.60 q^{51} -21575.9 q^{52} -23707.9 q^{53} +13524.8 q^{54} +3609.48 q^{56} +1278.08 q^{57} +14058.3 q^{58} +19517.8 q^{59} +20937.3 q^{61} +21701.9 q^{62} +33489.4 q^{63} -38667.9 q^{64} -3453.63 q^{66} +51707.7 q^{67} -64495.1 q^{68} +15744.1 q^{69} -1398.38 q^{71} +5743.67 q^{72} -72466.6 q^{73} +79098.4 q^{74} +12846.4 q^{76} -17553.6 q^{77} +17575.6 q^{78} +64632.2 q^{79} +50337.7 q^{81} +91332.4 q^{82} +96790.3 q^{83} -17719.8 q^{84} +68518.8 q^{86} -5985.47 q^{87} -3010.57 q^{88} -47614.1 q^{89} +89330.7 q^{91} +158249. q^{92} -9239.79 q^{93} -18186.7 q^{94} +28963.9 q^{96} +38399.6 q^{97} -34704.8 q^{98} -27932.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 34 q^{3} + 84 q^{4} - 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 34 q^{3} + 84 q^{4} - 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9} + 363 q^{11} - 992 q^{12} - 486 q^{13} - 1020 q^{14} + 1992 q^{16} - 1086 q^{17} + 3706 q^{18} + 1380 q^{19} - 908 q^{21} + 3066 q^{23} - 11748 q^{24} + 12132 q^{26} + 2990 q^{27} - 23712 q^{28} - 3426 q^{29} - 4098 q^{31} + 12408 q^{32} - 4114 q^{33} + 25320 q^{34} + 4756 q^{36} - 17724 q^{37} + 9240 q^{38} - 6560 q^{39} + 5994 q^{41} + 47828 q^{42} + 26208 q^{43} + 10164 q^{44} - 16806 q^{46} + 17232 q^{47} - 61064 q^{48} + 48531 q^{49} - 22724 q^{51} + 35304 q^{52} - 50586 q^{53} + 18814 q^{54} - 42312 q^{56} - 20160 q^{57} + 29172 q^{58} - 3738 q^{59} + 18486 q^{61} + 19974 q^{62} + 12496 q^{63} - 20352 q^{64} - 24926 q^{66} + 47754 q^{67} + 12600 q^{68} + 35042 q^{69} + 39282 q^{71} + 95040 q^{72} - 15426 q^{73} + 153294 q^{74} + 103920 q^{76} - 10164 q^{77} - 124984 q^{78} + 125148 q^{79} - 86917 q^{81} + 255372 q^{82} + 143928 q^{83} + 343616 q^{84} + 243060 q^{86} + 19368 q^{87} + 68244 q^{88} - 106824 q^{89} - 109632 q^{91} + 336528 q^{92} + 16622 q^{93} - 74928 q^{94} - 76456 q^{96} - 9684 q^{97} - 3480 q^{98} - 847 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −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 \(-0.464334\pi\)
0.111814 + 0.993729i \(0.464334\pi\)
\(4\) 35.0388 1.09496
\(5\) 0 0
\(6\) −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
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 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
\(15\) 0 0
\(16\) −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 \(-0.462833\pi\)
0.116498 + 0.993191i \(0.462833\pi\)
\(20\) 0 0
\(21\) −505.718 −0.250242
\(22\) −990.714 −0.436407
\(23\) 4516.38 1.78021 0.890104 0.455757i \(-0.150631\pi\)
0.890104 + 0.455757i \(0.150631\pi\)
\(24\) −86.7344 −0.0307371
\(25\) 0 0
\(26\) 5041.77 1.46268
\(27\) −1651.83 −0.436071
\(28\) −5083.12 −1.22528
\(29\) −1717.00 −0.379119 −0.189560 0.981869i \(-0.560706\pi\)
−0.189560 + 0.981869i \(0.560706\pi\)
\(30\) 0 0
\(31\) −2650.54 −0.495371 −0.247685 0.968841i \(-0.579670\pi\)
−0.247685 + 0.968841i \(0.579670\pi\)
\(32\) 8308.62 1.43435
\(33\) 421.806 0.0674261
\(34\) 15070.9 2.23585
\(35\) 0 0
\(36\) −8088.63 −1.04020
\(37\) −9660.61 −1.16011 −0.580057 0.814576i \(-0.696970\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(38\) −3001.89 −0.337238
\(39\) −2146.58 −0.225988
\(40\) 0 0
\(41\) −11154.8 −1.03634 −0.518170 0.855278i \(-0.673386\pi\)
−0.518170 + 0.855278i \(0.673386\pi\)
\(42\) 4140.68 0.362200
\(43\) −8368.48 −0.690201 −0.345100 0.938566i \(-0.612155\pi\)
−0.345100 + 0.938566i \(0.612155\pi\)
\(44\) 4239.69 0.330144
\(45\) 0 0
\(46\) −36978.9 −2.57667
\(47\) 2221.22 0.146672 0.0733360 0.997307i \(-0.476635\pi\)
0.0733360 + 0.997307i \(0.476635\pi\)
\(48\) −3198.49 −0.200374
\(49\) 4238.64 0.252195
\(50\) 0 0
\(51\) −6416.60 −0.345445
\(52\) −21575.9 −1.10652
\(53\) −23707.9 −1.15932 −0.579659 0.814859i \(-0.696814\pi\)
−0.579659 + 0.814859i \(0.696814\pi\)
\(54\) 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 \(0.381075\pi\)
\(60\) 0 0
\(61\) 20937.3 0.720436 0.360218 0.932868i \(-0.382702\pi\)
0.360218 + 0.932868i \(0.382702\pi\)
\(62\) 21701.9 0.716999
\(63\) 33489.4 1.06305
\(64\) −38667.9 −1.18005
\(65\) 0 0
\(66\) −3453.63 −0.0975924
\(67\) 51707.7 1.40724 0.703619 0.710577i \(-0.251566\pi\)
0.703619 + 0.710577i \(0.251566\pi\)
\(68\) −64495.1 −1.69143
\(69\) 15744.1 0.398103
\(70\) 0 0
\(71\) −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
\(75\) 0 0
\(76\) 12846.4 0.255122
\(77\) −17553.6 −0.337396
\(78\) 17575.6 0.327095
\(79\) 64632.2 1.16515 0.582574 0.812777i \(-0.302045\pi\)
0.582574 + 0.812777i \(0.302045\pi\)
\(80\) 0 0
\(81\) 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
\(85\) 0 0
\(86\) 68518.8 0.998995
\(87\) −5985.47 −0.0847814
\(88\) −3010.57 −0.0414422
\(89\) −47614.1 −0.637178 −0.318589 0.947893i \(-0.603209\pi\)
−0.318589 + 0.947893i \(0.603209\pi\)
\(90\) 0 0
\(91\) 89330.7 1.13083
\(92\) 158249. 1.94926
\(93\) −9239.79 −0.110778
\(94\) −18186.7 −0.212293
\(95\) 0 0
\(96\) 28963.9 0.320759
\(97\) 38399.6 0.414378 0.207189 0.978301i \(-0.433568\pi\)
0.207189 + 0.978301i \(0.433568\pi\)
\(98\) −34704.8 −0.365027
\(99\) −27932.6 −0.286433
\(100\) 0 0
\(101\) −41011.2 −0.400036 −0.200018 0.979792i \(-0.564100\pi\)
−0.200018 + 0.979792i \(0.564100\pi\)
\(102\) 52537.3 0.499997
\(103\) 49634.4 0.460988 0.230494 0.973074i \(-0.425966\pi\)
0.230494 + 0.973074i \(0.425966\pi\)
\(104\) 15320.9 0.138899
\(105\) 0 0
\(106\) 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 \(-0.372435\pi\)
0.390115 + 0.920766i \(0.372435\pi\)
\(110\) 0 0
\(111\) −33676.9 −0.259433
\(112\) 133106. 1.00266
\(113\) 212938. 1.56876 0.784379 0.620281i \(-0.212982\pi\)
0.784379 + 0.620281i \(0.212982\pi\)
\(114\) −10464.6 −0.0754155
\(115\) 0 0
\(116\) −60161.7 −0.415121
\(117\) 142149. 0.960021
\(118\) −159806. −1.05655
\(119\) 267029. 1.72859
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −171428. −1.04276
\(123\) −38885.6 −0.231754
\(124\) −92871.8 −0.542412
\(125\) 0 0
\(126\) −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
\(130\) 0 0
\(131\) 65299.5 0.332454 0.166227 0.986088i \(-0.446842\pi\)
0.166227 + 0.986088i \(0.446842\pi\)
\(132\) 14779.6 0.0738290
\(133\) −53187.9 −0.260726
\(134\) −423368. −2.03684
\(135\) 0 0
\(136\) 45797.4 0.212321
\(137\) −5322.74 −0.0242289 −0.0121145 0.999927i \(-0.503856\pi\)
−0.0121145 + 0.999927i \(0.503856\pi\)
\(138\) −128908. −0.576214
\(139\) 89967.1 0.394954 0.197477 0.980308i \(-0.436725\pi\)
0.197477 + 0.980308i \(0.436725\pi\)
\(140\) 0 0
\(141\) 7743.18 0.0327998
\(142\) 11449.6 0.0476506
\(143\) −74508.4 −0.304695
\(144\) 211808. 0.851211
\(145\) 0 0
\(146\) 593336. 2.30366
\(147\) 14775.9 0.0563977
\(148\) −338496. −1.27028
\(149\) 66489.8 0.245352 0.122676 0.992447i \(-0.460852\pi\)
0.122676 + 0.992447i \(0.460852\pi\)
\(150\) 0 0
\(151\) −130866. −0.467074 −0.233537 0.972348i \(-0.575030\pi\)
−0.233537 + 0.972348i \(0.575030\pi\)
\(152\) −9122.12 −0.0320248
\(153\) 424916. 1.46749
\(154\) 143724. 0.488346
\(155\) 0 0
\(156\) −75213.6 −0.247448
\(157\) 163297. 0.528723 0.264362 0.964424i \(-0.414839\pi\)
0.264362 + 0.964424i \(0.414839\pi\)
\(158\) −529191. −1.68643
\(159\) −82645.6 −0.259255
\(160\) 0 0
\(161\) −655197. −1.99208
\(162\) −412151. −1.23387
\(163\) 535758. 1.57943 0.789713 0.613477i \(-0.210229\pi\)
0.789713 + 0.613477i \(0.210229\pi\)
\(164\) −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
\(170\) 0 0
\(171\) −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
\(175\) 0 0
\(176\) −111020. −0.270160
\(177\) 68039.2 0.163240
\(178\) 389851. 0.922250
\(179\) 3030.33 0.00706900 0.00353450 0.999994i \(-0.498875\pi\)
0.00353450 + 0.999994i \(0.498875\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) 75652.8 0.160340
\(187\) −222722. −0.465756
\(188\) 77828.9 0.160600
\(189\) 239634. 0.487970
\(190\) 0 0
\(191\) −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
\(195\) 0 0
\(196\) 148517. 0.276144
\(197\) −574550. −1.05478 −0.527390 0.849623i \(-0.676829\pi\)
−0.527390 + 0.849623i \(0.676829\pi\)
\(198\) 228704. 0.414583
\(199\) 269926. 0.483183 0.241592 0.970378i \(-0.422331\pi\)
0.241592 + 0.970378i \(0.422331\pi\)
\(200\) 0 0
\(201\) 180253. 0.314697
\(202\) 335788. 0.579011
\(203\) 249088. 0.424240
\(204\) −224830. −0.378250
\(205\) 0 0
\(206\) −406393. −0.667234
\(207\) −1.04260e6 −1.69118
\(208\) 564985. 0.905480
\(209\) 44362.6 0.0702509
\(210\) 0 0
\(211\) −753372. −1.16494 −0.582470 0.812853i \(-0.697913\pi\)
−0.582470 + 0.812853i \(0.697913\pi\)
\(212\) −830695. −1.26941
\(213\) −4874.77 −0.00736216
\(214\) −55605.6 −0.0830011
\(215\) 0 0
\(216\) 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 \(-0.265545\pi\)
0.671745 + 0.740783i \(0.265545\pi\)
\(224\) −1.20534e6 −1.60506
\(225\) 0 0
\(226\) −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 \(0.544648\pi\)
\(230\) 0 0
\(231\) −61191.9 −0.0754508
\(232\) 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
\(235\) 0 0
\(236\) 683881. 0.799283
\(237\) 225308. 0.260559
\(238\) −2.18636e6 −2.50195
\(239\) −295471. −0.334595 −0.167298 0.985906i \(-0.553504\pi\)
−0.167298 + 0.985906i \(0.553504\pi\)
\(240\) 0 0
\(241\) 693153. 0.768753 0.384376 0.923176i \(-0.374416\pi\)
0.384376 + 0.923176i \(0.374416\pi\)
\(242\) −119876. −0.131582
\(243\) 576873. 0.626707
\(244\) 733616. 0.788850
\(245\) 0 0
\(246\) 318385. 0.335440
\(247\) −225762. −0.235456
\(248\) 65947.5 0.0680878
\(249\) 337411. 0.344875
\(250\) 0 0
\(251\) 533816. 0.534820 0.267410 0.963583i \(-0.413832\pi\)
0.267410 + 0.963583i \(0.413832\pi\)
\(252\) 1.17343e6 1.16400
\(253\) 546482. 0.536753
\(254\) −739874. −0.719571
\(255\) 0 0
\(256\) 822041. 0.783960
\(257\) 652296. 0.616044 0.308022 0.951379i \(-0.400333\pi\)
0.308022 + 0.951379i \(0.400333\pi\)
\(258\) 238857. 0.223402
\(259\) 1.40148e6 1.29818
\(260\) 0 0
\(261\) 396366. 0.360160
\(262\) −534654. −0.481193
\(263\) 622045. 0.554540 0.277270 0.960792i \(-0.410570\pi\)
0.277270 + 0.960792i \(0.410570\pi\)
\(264\) −10494.9 −0.00926759
\(265\) 0 0
\(266\) 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 \(0.565209\pi\)
\(270\) 0 0
\(271\) −1.10678e6 −0.915460 −0.457730 0.889091i \(-0.651337\pi\)
−0.457730 + 0.889091i \(0.651337\pi\)
\(272\) 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
\(280\) 0 0
\(281\) −257984. −0.194907 −0.0974534 0.995240i \(-0.531070\pi\)
−0.0974534 + 0.995240i \(0.531070\pi\)
\(282\) −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
\(285\) 0 0
\(286\) 610054. 0.441015
\(287\) 1.61824e6 1.15968
\(288\) −1.91803e6 −1.36262
\(289\) 1.96823e6 1.38622
\(290\) 0 0
\(291\) 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
\(295\) 0 0
\(296\) 240363. 0.159455
\(297\) −199872. −0.131480
\(298\) −544400. −0.355122
\(299\) −2.78106e6 −1.79900
\(300\) 0 0
\(301\) 1.21403e6 0.772345
\(302\) 1.07150e6 0.676042
\(303\) −142965. −0.0894589
\(304\) −336395. −0.208769
\(305\) 0 0
\(306\) −3.47909e6 −2.12404
\(307\) 1.30925e6 0.792826 0.396413 0.918072i \(-0.370255\pi\)
0.396413 + 0.918072i \(0.370255\pi\)
\(308\) −615057. −0.369436
\(309\) 173026. 0.103089
\(310\) 0 0
\(311\) 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 \(-0.165871\pi\)
0.867272 + 0.497835i \(0.165871\pi\)
\(314\) −1.33703e6 −0.765273
\(315\) 0 0
\(316\) 2.26464e6 1.27579
\(317\) −2.10147e6 −1.17456 −0.587279 0.809385i \(-0.699801\pi\)
−0.587279 + 0.809385i \(0.699801\pi\)
\(318\) 676679. 0.375245
\(319\) −207757. −0.114309
\(320\) 0 0
\(321\) 23674.6 0.0128239
\(322\) 5.36457e6 2.88333
\(323\) −674853. −0.359918
\(324\) 1.76377e6 0.933426
\(325\) 0 0
\(326\) −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
\(335\) 0 0
\(336\) 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
\(340\) 0 0
\(341\) −320715. −0.149360
\(342\) 692980. 0.320373
\(343\) 1.82331e6 0.836805
\(344\) 208214. 0.0948668
\(345\) 0 0
\(346\) 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 \(-0.674880\pi\)
−0.522177 + 0.852837i \(0.674880\pi\)
\(350\) 0 0
\(351\) 1.01715e6 0.440675
\(352\) 1.00534e6 0.432472
\(353\) −638696. −0.272808 −0.136404 0.990653i \(-0.543555\pi\)
−0.136404 + 0.990653i \(0.543555\pi\)
\(354\) −557086. −0.236273
\(355\) 0 0
\(356\) −1.66834e6 −0.697686
\(357\) 930864. 0.386559
\(358\) −24811.5 −0.0102317
\(359\) 1.50842e6 0.617712 0.308856 0.951109i \(-0.400054\pi\)
0.308856 + 0.951109i \(0.400054\pi\)
\(360\) 0 0
\(361\) −2.34168e6 −0.945713
\(362\) −6.23284e6 −2.49985
\(363\) 51038.5 0.0203297
\(364\) 3.13004e6 1.23822
\(365\) 0 0
\(366\) −597600. −0.233189
\(367\) −1.77368e6 −0.687403 −0.343701 0.939079i \(-0.611681\pi\)
−0.343701 + 0.939079i \(0.611681\pi\)
\(368\) −4.14389e6 −1.59510
\(369\) 2.57506e6 0.984513
\(370\) 0 0
\(371\) 3.43933e6 1.29729
\(372\) −323751. −0.121298
\(373\) −2.27176e6 −0.845456 −0.422728 0.906257i \(-0.638927\pi\)
−0.422728 + 0.906257i \(0.638927\pi\)
\(374\) 1.82358e6 0.674135
\(375\) 0 0
\(376\) −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 \(0.790457\pi\)
\(380\) 0 0
\(381\) 315009. 0.111176
\(382\) 3.52607e6 1.23632
\(383\) −2.37588e6 −0.827615 −0.413807 0.910364i \(-0.635801\pi\)
−0.413807 + 0.910364i \(0.635801\pi\)
\(384\) 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 \(0.646965\pi\)
\(390\) 0 0
\(391\) −8.31319e6 −2.74996
\(392\) −105461. −0.0346638
\(393\) 227634. 0.0743457
\(394\) 4.70425e6 1.52669
\(395\) 0 0
\(396\) −978724. −0.313633
\(397\) −2.15712e6 −0.686907 −0.343453 0.939170i \(-0.611597\pi\)
−0.343453 + 0.939170i \(0.611597\pi\)
\(398\) −2.21008e6 −0.699359
\(399\) −185413. −0.0583054
\(400\) 0 0
\(401\) −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
\(405\) 0 0
\(406\) −2.03946e6 −0.614045
\(407\) −1.16893e6 −0.349787
\(408\) 159650. 0.0474808
\(409\) 6.12831e6 1.81148 0.905738 0.423839i \(-0.139318\pi\)
0.905738 + 0.423839i \(0.139318\pi\)
\(410\) 0 0
\(411\) −18555.1 −0.00541824
\(412\) 1.73913e6 0.504765
\(413\) −2.83147e6 −0.816841
\(414\) 8.53649e6 2.44781
\(415\) 0 0
\(416\) −5.11621e6 −1.44949
\(417\) 313625. 0.0883225
\(418\) −363229. −0.101681
\(419\) −375626. −0.104525 −0.0522626 0.998633i \(-0.516643\pi\)
−0.0522626 + 0.998633i \(0.516643\pi\)
\(420\) 0 0
\(421\) 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
\(425\) 0 0
\(426\) 39913.3 0.0106560
\(427\) −3.03739e6 −0.806178
\(428\) 237960. 0.0627906
\(429\) −259736. −0.0681380
\(430\) 0 0
\(431\) 3.15287e6 0.817548 0.408774 0.912636i \(-0.365956\pi\)
0.408774 + 0.912636i \(0.365956\pi\)
\(432\) 1.51560e6 0.390728
\(433\) −1.62168e6 −0.415667 −0.207833 0.978164i \(-0.566641\pi\)
−0.207833 + 0.978164i \(0.566641\pi\)
\(434\) −3.14832e6 −0.802333
\(435\) 0 0
\(436\) 3.39108e6 0.854322
\(437\) 1.65586e6 0.414781
\(438\) 2.06837e6 0.515161
\(439\) −2.48145e6 −0.614533 −0.307266 0.951624i \(-0.599414\pi\)
−0.307266 + 0.951624i \(0.599414\pi\)
\(440\) 0 0
\(441\) −978482. −0.239583
\(442\) −9.28026e6 −2.25946
\(443\) −3.75466e6 −0.908994 −0.454497 0.890748i \(-0.650181\pi\)
−0.454497 + 0.890748i \(0.650181\pi\)
\(444\) −1.18000e6 −0.284069
\(445\) 0 0
\(446\) −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 \(0.690311\pi\)
\(450\) 0 0
\(451\) −1.34973e6 −0.312468
\(452\) 7.46108e6 1.71773
\(453\) −456200. −0.104450
\(454\) −4.05467e6 −0.923244
\(455\) 0 0
\(456\) −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
\(460\) 0 0
\(461\) 8.12745e6 1.78116 0.890578 0.454830i \(-0.150300\pi\)
0.890578 + 0.454830i \(0.150300\pi\)
\(462\) 501022. 0.109207
\(463\) 2.67361e6 0.579623 0.289812 0.957084i \(-0.406407\pi\)
0.289812 + 0.957084i \(0.406407\pi\)
\(464\) 1.57539e6 0.339699
\(465\) 0 0
\(466\) 5.07168e6 1.08190
\(467\) 4.32733e6 0.918180 0.459090 0.888390i \(-0.348175\pi\)
0.459090 + 0.888390i \(0.348175\pi\)
\(468\) 4.98075e6 1.05119
\(469\) −7.50129e6 −1.57472
\(470\) 0 0
\(471\) 569253. 0.118237
\(472\) −485618. −0.100332
\(473\) −1.01259e6 −0.208103
\(474\) −1.84476e6 −0.377133
\(475\) 0 0
\(476\) 9.35637e6 1.89274
\(477\) 5.47291e6 1.10134
\(478\) 2.41923e6 0.484293
\(479\) 1.55878e6 0.310417 0.155208 0.987882i \(-0.450395\pi\)
0.155208 + 0.987882i \(0.450395\pi\)
\(480\) 0 0
\(481\) 5.94873e6 1.17236
\(482\) −5.67535e6 −1.11269
\(483\) −2.28402e6 −0.445483
\(484\) 513003. 0.0995420
\(485\) 0 0
\(486\) −4.72328e6 −0.907095
\(487\) 7.63818e6 1.45938 0.729689 0.683779i \(-0.239665\pi\)
0.729689 + 0.683779i \(0.239665\pi\)
\(488\) −520935. −0.0990225
\(489\) 1.86765e6 0.353202
\(490\) 0 0
\(491\) 3.60872e6 0.675537 0.337768 0.941229i \(-0.390328\pi\)
0.337768 + 0.941229i \(0.390328\pi\)
\(492\) −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 \(0.775095\pi\)
\(500\) 0 0
\(501\) −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
\(505\) 0 0
\(506\) −4.47444e6 −0.776896
\(507\) 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 \(0.725328\pi\)
\(510\) 0 0
\(511\) 1.05128e7 1.78101
\(512\) −8.35388e6 −1.40836
\(513\) −605618. −0.101603
\(514\) −5.34082e6 −0.891661
\(515\) 0 0
\(516\) −1.02217e6 −0.169005
\(517\) 268768. 0.0442233
\(518\) −1.14749e7 −1.87899
\(519\) −930667. −0.151662
\(520\) 0 0
\(521\) 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
\(525\) 0 0
\(526\) −5.09313e6 −0.802640
\(527\) 4.87879e6 0.765218
\(528\) −387018. −0.0604151
\(529\) 1.39613e7 2.16914
\(530\) 0 0
\(531\) −4.50565e6 −0.693459
\(532\) −1.86364e6 −0.285485
\(533\) 6.86881e6 1.04728
\(534\) 1.35902e6 0.206240
\(535\) 0 0
\(536\) −1.28653e6 −0.193422
\(537\) 10563.8 0.00158082
\(538\) 3.95354e6 0.588885
\(539\) 512876. 0.0760397
\(540\) 0 0
\(541\) −4.34177e6 −0.637784 −0.318892 0.947791i \(-0.603311\pi\)
−0.318892 + 0.947791i \(0.603311\pi\)
\(542\) 9.06203e6 1.32503
\(543\) 2.65369e6 0.386234
\(544\) −1.52935e7 −2.21569
\(545\) 0 0
\(546\) −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
\(555\) 0 0
\(556\) 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
\(560\) 0 0
\(561\) −776409. −0.104156
\(562\) 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
\(565\) 0 0
\(566\) 8.43265e6 1.10643
\(567\) −7.30255e6 −0.953931
\(568\) 34792.9 0.00452501
\(569\) −1.19642e7 −1.54919 −0.774595 0.632458i \(-0.782046\pi\)
−0.774595 + 0.632458i \(0.782046\pi\)
\(570\) 0 0
\(571\) −7.97842e6 −1.02406 −0.512032 0.858967i \(-0.671107\pi\)
−0.512032 + 0.858967i \(0.671107\pi\)
\(572\) −2.61068e6 −0.333629
\(573\) −1.50126e6 −0.191016
\(574\) −1.32497e7 −1.67852
\(575\) 0 0
\(576\) 8.92640e6 1.12104
\(577\) −5.90743e6 −0.738685 −0.369342 0.929293i \(-0.620417\pi\)
−0.369342 + 0.929293i \(0.620417\pi\)
\(578\) −1.61153e7 −2.00641
\(579\) 949186. 0.117667
\(580\) 0 0
\(581\) −1.40415e7 −1.72573
\(582\) −1.09602e6 −0.134125
\(583\) −2.86865e6 −0.349548
\(584\) 1.80303e6 0.218761
\(585\) 0 0
\(586\) 7.18646e6 0.864512
\(587\) −1.34766e6 −0.161430 −0.0807151 0.996737i \(-0.525720\pi\)
−0.0807151 + 0.996737i \(0.525720\pi\)
\(588\) 517730. 0.0617533
\(589\) −971777. −0.115419
\(590\) 0 0
\(591\) −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 \(0.712155\pi\)
\(594\) 1.63650e6 0.190304
\(595\) 0 0
\(596\) 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 \(0.436439\pi\)
\(600\) 0 0
\(601\) −6.41433e6 −0.724378 −0.362189 0.932105i \(-0.617970\pi\)
−0.362189 + 0.932105i \(0.617970\pi\)
\(602\) −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
\(610\) 0 0
\(611\) −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
\(615\) 0 0
\(616\) 436747. 0.0463744
\(617\) 1.00683e6 0.106474 0.0532371 0.998582i \(-0.483046\pi\)
0.0532371 + 0.998582i \(0.483046\pi\)
\(618\) −1.41669e6 −0.149212
\(619\) 1.27458e7 1.33703 0.668513 0.743700i \(-0.266931\pi\)
0.668513 + 0.743700i \(0.266931\pi\)
\(620\) 0 0
\(621\) −7.46031e6 −0.776297
\(622\) −2.75050e7 −2.85060
\(623\) 6.90744e6 0.713012
\(624\) 1.96954e6 0.202490
\(625\) 0 0
\(626\) −2.46155e7 −2.51058
\(627\) 154648. 0.0157100
\(628\) 5.72172e6 0.578932
\(629\) 1.77821e7 1.79207
\(630\) 0 0
\(631\) 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
\(635\) 0 0
\(636\) −2.89580e6 −0.283874
\(637\) −2.61004e6 −0.254858
\(638\) 1.70106e6 0.165450
\(639\) 322814. 0.0312752
\(640\) 0 0
\(641\) −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 \(-0.621587\pi\)
−0.372757 + 0.927929i \(0.621587\pi\)
\(644\) −2.29573e7 −2.18125
\(645\) 0 0
\(646\) 5.52551e6 0.520944
\(647\) −2.01624e7 −1.89357 −0.946786 0.321863i \(-0.895691\pi\)
−0.946786 + 0.321863i \(0.895691\pi\)
\(648\) −1.25244e6 −0.117171
\(649\) 2.36166e6 0.220092
\(650\) 0 0
\(651\) 1.34043e6 0.123963
\(652\) 1.87723e7 1.72941
\(653\) −324619. −0.0297914 −0.0148957 0.999889i \(-0.504742\pi\)
−0.0148957 + 0.999889i \(0.504742\pi\)
\(654\) −2.76235e6 −0.252543
\(655\) 0 0
\(656\) 1.02348e7 0.928581
\(657\) 1.67288e7 1.51199
\(658\) 2.63837e6 0.237559
\(659\) −1.07107e7 −0.960740 −0.480370 0.877066i \(-0.659498\pi\)
−0.480370 + 0.877066i \(0.659498\pi\)
\(660\) 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
\(665\) 0 0
\(666\) −1.82597e7 −1.59517
\(667\) −7.75464e6 −0.674912
\(668\) −1.93970e7 −1.68187
\(669\) 3.47795e6 0.300441
\(670\) 0 0
\(671\) 2.53341e6 0.217219
\(672\) −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
\(675\) 0 0
\(676\) 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
\(680\) 0 0
\(681\) 1.72632e6 0.142644
\(682\) 2.62593e6 0.216183
\(683\) −4.40512e6 −0.361332 −0.180666 0.983545i \(-0.557825\pi\)
−0.180666 + 0.983545i \(0.557825\pi\)
\(684\) −2.96556e6 −0.242363
\(685\) 0 0
\(686\) −1.49287e7 −1.21119
\(687\) −773519. −0.0625287
\(688\) 7.67828e6 0.618434
\(689\) 1.45986e7 1.17156
\(690\) 0 0
\(691\) −5.86199e6 −0.467035 −0.233518 0.972353i \(-0.575024\pi\)
−0.233518 + 0.972353i \(0.575024\pi\)
\(692\) −9.35440e6 −0.742592
\(693\) 4.05221e6 0.320523
\(694\) 2.19054e7 1.72645
\(695\) 0 0
\(696\) 148923. 0.0116530
\(697\) 2.05324e7 1.60087
\(698\) 1.94569e7 1.51160
\(699\) −2.15932e6 −0.167157
\(700\) 0 0
\(701\) −8.02106e6 −0.616505 −0.308253 0.951305i \(-0.599744\pi\)
−0.308253 + 0.951305i \(0.599744\pi\)
\(702\) −8.32816e6 −0.637832
\(703\) −3.54190e6 −0.270301
\(704\) −4.67881e6 −0.355799
\(705\) 0 0
\(706\) 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 \(0.197318\pi\)
\(710\) 0 0
\(711\) −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 \(0.378238\pi\)
\(720\) 0 0
\(721\) −7.20052e6 −0.515853
\(722\) 1.91730e7 1.36882
\(723\) 2.41633e6 0.171914
\(724\) 2.66730e7 1.89115
\(725\) 0 0
\(726\) −417889. −0.0294252
\(727\) 2.03348e7 1.42693 0.713466 0.700690i \(-0.247124\pi\)
0.713466 + 0.700690i \(0.247124\pi\)
\(728\) −2.22262e6 −0.155430
\(729\) −1.02211e7 −0.712325
\(730\) 0 0
\(731\) 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
\(735\) 0 0
\(736\) 3.75249e7 2.55344
\(737\) 6.25663e6 0.424298
\(738\) −2.10839e7 −1.42498
\(739\) −1.08737e7 −0.732429 −0.366215 0.930530i \(-0.619346\pi\)
−0.366215 + 0.930530i \(0.619346\pi\)
\(740\) 0 0
\(741\) −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
\(745\) 0 0
\(746\) 1.86006e7 1.22371
\(747\) −2.23438e7 −1.46506
\(748\) −7.80390e6 −0.509986
\(749\) −985227. −0.0641700
\(750\) 0 0
\(751\) 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
\(755\) 0 0
\(756\) 8.39647e6 0.534309
\(757\) 1.33506e7 0.846759 0.423380 0.905952i \(-0.360844\pi\)
0.423380 + 0.905952i \(0.360844\pi\)
\(758\) 3.62232e7 2.28988
\(759\) 1.90504e6 0.120033
\(760\) 0 0
\(761\) 1.28109e7 0.801898 0.400949 0.916100i \(-0.368680\pi\)
0.400949 + 0.916100i \(0.368680\pi\)
\(762\) −2.57920e6 −0.160916
\(763\) −1.40401e7 −0.873089
\(764\) −1.50896e7 −0.935284
\(765\) 0 0
\(766\) 1.94531e7 1.19789
\(767\) −1.20185e7 −0.737671
\(768\) 2.86564e6 0.175315
\(769\) 1.90629e7 1.16245 0.581224 0.813743i \(-0.302574\pi\)
0.581224 + 0.813743i \(0.302574\pi\)
\(770\) 0 0
\(771\) 2.27390e6 0.137764
\(772\) 9.54054e6 0.576142
\(773\) 2.34434e6 0.141115 0.0705574 0.997508i \(-0.477522\pi\)
0.0705574 + 0.997508i \(0.477522\pi\)
\(774\) −1.58174e7 −0.949037
\(775\) 0 0
\(776\) −955410. −0.0569555
\(777\) 4.88555e6 0.290309
\(778\) 2.17716e7 1.28956
\(779\) −4.08972e6 −0.241463
\(780\) 0 0
\(781\) −169204. −0.00992623
\(782\) 6.80661e7 3.98028
\(783\) 2.83620e6 0.165323
\(784\) −3.88906e6 −0.225972
\(785\) 0 0
\(786\) −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
\(790\) 0 0
\(791\) −3.08911e7 −1.75547
\(792\) 694984. 0.0393697
\(793\) −1.28926e7 −0.728042
\(794\) 1.76619e7 0.994228
\(795\) 0 0
\(796\) 9.45788e6 0.529068
\(797\) 2.70618e7 1.50907 0.754537 0.656257i \(-0.227861\pi\)
0.754537 + 0.656257i \(0.227861\pi\)
\(798\) 1.51811e6 0.0843911
\(799\) −4.08855e6 −0.226570
\(800\) 0 0
\(801\) 1.09916e7 0.605313
\(802\) 1.98987e7 1.09242
\(803\) −8.76846e6 −0.479882
\(804\) 6.31585e6 0.344581
\(805\) 0 0
\(806\) −1.33634e7 −0.724569
\(807\) −1.68326e6 −0.0909844
\(808\) 1.02039e6 0.0549842
\(809\) −2.25663e6 −0.121224 −0.0606121 0.998161i \(-0.519305\pi\)
−0.0606121 + 0.998161i \(0.519305\pi\)
\(810\) 0 0
\(811\) 4.49121e6 0.239779 0.119890 0.992787i \(-0.461746\pi\)
0.119890 + 0.992787i \(0.461746\pi\)
\(812\) 8.72773e6 0.464527
\(813\) −3.85825e6 −0.204722
\(814\) 9.57091e6 0.506282
\(815\) 0 0
\(816\) 5.88739e6 0.309526
\(817\) −3.06816e6 −0.160814
\(818\) −5.01769e7 −2.62193
\(819\) −2.06218e7 −1.07428
\(820\) 0 0
\(821\) 592581. 0.0306825 0.0153412 0.999882i \(-0.495117\pi\)
0.0153412 + 0.999882i \(0.495117\pi\)
\(822\) 151924. 0.00784236
\(823\) 1.14748e7 0.590533 0.295266 0.955415i \(-0.404592\pi\)
0.295266 + 0.955415i \(0.404592\pi\)
\(824\) −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 \(0.631497\pi\)
\(830\) 0 0
\(831\) −2.22777e6 −0.111910
\(832\) 2.38106e7 1.19251
\(833\) −7.80197e6 −0.389576
\(834\) −2.56788e6 −0.127838
\(835\) 0 0
\(836\) 1.55441e6 0.0769221
\(837\) 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 \(0.272812\pi\)
\(840\) 0 0
\(841\) −1.75631e7 −0.856268
\(842\) −2.88480e7 −1.40228
\(843\) −899333. −0.0435864
\(844\) −2.63972e7 −1.27556
\(845\) 0 0
\(846\) 4.19837e6 0.201676
\(847\) −2.12399e6 −0.101729
\(848\) 2.17525e7 1.03877
\(849\) −3.59028e6 −0.170946
\(850\) 0 0
\(851\) −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
\(855\) 0 0
\(856\) −168974. −0.00788197
\(857\) 6.00941e6 0.279499 0.139749 0.990187i \(-0.455370\pi\)
0.139749 + 0.990187i \(0.455370\pi\)
\(858\) 2.12665e6 0.0986228
\(859\) −1.74629e7 −0.807484 −0.403742 0.914873i \(-0.632291\pi\)
−0.403742 + 0.914873i \(0.632291\pi\)
\(860\) 0 0
\(861\) 5.64118e6 0.259336
\(862\) −2.58149e7 −1.18332
\(863\) −2.34431e6 −0.107149 −0.0535746 0.998564i \(-0.517061\pi\)
−0.0535746 + 0.998564i \(0.517061\pi\)
\(864\) −1.37245e7 −0.625476
\(865\) 0 0
\(866\) 1.32779e7 0.601635
\(867\) 6.86126e6 0.309996
\(868\) 1.34730e7 0.606968
\(869\) 7.82050e6 0.351306
\(870\) 0 0
\(871\) −3.18401e7 −1.42210
\(872\) −2.40798e6 −0.107241
\(873\) −8.86445e6 −0.393655
\(874\) −1.35577e7 −0.600354
\(875\) 0 0
\(876\) −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 \(-0.300498\pi\)
0.586518 + 0.809936i \(0.300498\pi\)
\(884\) 3.97142e7 1.70929
\(885\) 0 0
\(886\) 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
\(890\) 0 0
\(891\) 6.09086e6 0.257030
\(892\) 3.49579e7 1.47107
\(893\) 814374. 0.0341740
\(894\) −1.89778e6 −0.0794149
\(895\) 0 0
\(896\) −7.35889e6 −0.306226
\(897\) −9.69477e6 −0.402306
\(898\) 3.93761e7 1.62945
\(899\) 4.55099e6 0.187805
\(900\) 0 0
\(901\) 4.36385e7 1.79084
\(902\) 1.10512e7 0.452266
\(903\) 4.23209e6 0.172717
\(904\) −5.29805e6 −0.215623
\(905\) 0 0
\(906\) 3.73524e6 0.151181
\(907\) −1.77875e7 −0.717954 −0.358977 0.933346i \(-0.616874\pi\)
−0.358977 + 0.933346i \(0.616874\pi\)
\(908\) 1.73517e7 0.698437
\(909\) 9.46734e6 0.380031
\(910\) 0 0
\(911\) 30398.8 0.00121356 0.000606780 1.00000i \(-0.499807\pi\)
0.000606780 1.00000i \(0.499807\pi\)
\(912\) −1.17267e6 −0.0466864
\(913\) 1.17116e7 0.464987
\(914\) 5.81288e7 2.30158
\(915\) 0 0
\(916\) −7.77486e6 −0.306164
\(917\) −9.47308e6 −0.372021
\(918\) −2.48947e7 −0.974990
\(919\) −4.10055e6 −0.160160 −0.0800798 0.996788i \(-0.525518\pi\)
−0.0800798 + 0.996788i \(0.525518\pi\)
\(920\) 0 0
\(921\) 4.56406e6 0.177297
\(922\) −6.65453e7 −2.57804
\(923\) 861085. 0.0332692
\(924\) −2.14409e6 −0.0826158
\(925\) 0 0
\(926\) −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 \(0.589845\pi\)
\(930\) 0 0
\(931\) 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
\(940\) 0 0
\(941\) 5.32850e6 0.196169 0.0980847 0.995178i \(-0.468728\pi\)
0.0980847 + 0.995178i \(0.468728\pi\)
\(942\) −4.66088e6 −0.171136
\(943\) −5.03793e7 −1.84490
\(944\) −1.79081e7 −0.654062
\(945\) 0 0
\(946\) 8.29077e6 0.301208
\(947\) −3.11430e7 −1.12846 −0.564230 0.825618i \(-0.690827\pi\)
−0.564230 + 0.825618i \(0.690827\pi\)
\(948\) 7.89452e6 0.285302
\(949\) 4.46229e7 1.60839
\(950\) 0 0
\(951\) −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
\(955\) 0 0
\(956\) −1.03529e7 −0.366369
\(957\) −724242. −0.0255625
\(958\) −1.27628e7 −0.449297
\(959\) 772177. 0.0271125
\(960\) 0 0
\(961\) −2.16038e7 −0.754608
\(962\) −4.87065e7 −1.69687
\(963\) −1.56776e6 −0.0544773
\(964\) 2.42873e7 0.841755
\(965\) 0 0
\(966\) 1.87009e7 0.644792
\(967\) −4.85436e7 −1.66942 −0.834711 0.550688i \(-0.814365\pi\)
−0.834711 + 0.550688i \(0.814365\pi\)
\(968\) −364279. −0.0124953
\(969\) −2.35254e6 −0.0804873
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) −1.92104e7 −0.645525
\(977\) 5.35354e7 1.79434 0.897170 0.441684i \(-0.145619\pi\)
0.897170 + 0.441684i \(0.145619\pi\)
\(978\) −1.52918e7 −0.511225
\(979\) −5.76131e6 −0.192116
\(980\) 0 0
\(981\) −2.23416e7 −0.741211
\(982\) −2.95472e7 −0.977771
\(983\) 5.40925e7 1.78547 0.892736 0.450580i \(-0.148783\pi\)
0.892736 + 0.450580i \(0.148783\pi\)
\(984\) 967505. 0.0318541
\(985\) 0 0
\(986\) −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
\(995\) 0 0
\(996\) 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 275.6.a.b.1.1 3
5.2 odd 4 275.6.b.b.199.2 6
5.3 odd 4 275.6.b.b.199.5 6
5.4 even 2 11.6.a.b.1.3 3
15.14 odd 2 99.6.a.g.1.1 3
20.19 odd 2 176.6.a.i.1.3 3
35.34 odd 2 539.6.a.e.1.3 3
40.19 odd 2 704.6.a.t.1.1 3
40.29 even 2 704.6.a.q.1.3 3
55.54 odd 2 121.6.a.d.1.1 3
165.164 even 2 1089.6.a.r.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.3 3 5.4 even 2
99.6.a.g.1.1 3 15.14 odd 2
121.6.a.d.1.1 3 55.54 odd 2
176.6.a.i.1.3 3 20.19 odd 2
275.6.a.b.1.1 3 1.1 even 1 trivial
275.6.b.b.199.2 6 5.2 odd 4
275.6.b.b.199.5 6 5.3 odd 4
539.6.a.e.1.3 3 35.34 odd 2
704.6.a.q.1.3 3 40.29 even 2
704.6.a.t.1.1 3 40.19 odd 2
1089.6.a.r.1.3 3 165.164 even 2