Properties

Label 825.6.a.i.1.3
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,6,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 15x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.03932\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.07863 q^{2} -9.00000 q^{3} +33.2643 q^{4} -72.7077 q^{6} +39.3760 q^{7} +10.2141 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+8.07863 q^{2} -9.00000 q^{3} +33.2643 q^{4} -72.7077 q^{6} +39.3760 q^{7} +10.2141 q^{8} +81.0000 q^{9} +121.000 q^{11} -299.379 q^{12} +220.765 q^{13} +318.105 q^{14} -981.943 q^{16} +200.343 q^{17} +654.369 q^{18} +350.345 q^{19} -354.384 q^{21} +977.515 q^{22} -1385.09 q^{23} -91.9273 q^{24} +1783.48 q^{26} -729.000 q^{27} +1309.82 q^{28} +5506.64 q^{29} -2450.86 q^{31} -8259.61 q^{32} -1089.00 q^{33} +1618.50 q^{34} +2694.41 q^{36} +4060.46 q^{37} +2830.31 q^{38} -1986.88 q^{39} +527.283 q^{41} -2862.94 q^{42} +12078.9 q^{43} +4024.99 q^{44} -11189.6 q^{46} -563.023 q^{47} +8837.48 q^{48} -15256.5 q^{49} -1803.09 q^{51} +7343.59 q^{52} +37203.0 q^{53} -5889.32 q^{54} +402.192 q^{56} -3153.11 q^{57} +44486.2 q^{58} +2157.64 q^{59} -39938.0 q^{61} -19799.6 q^{62} +3189.46 q^{63} -35304.2 q^{64} -8797.63 q^{66} +38473.2 q^{67} +6664.29 q^{68} +12465.8 q^{69} -13725.3 q^{71} +827.345 q^{72} +39736.3 q^{73} +32803.0 q^{74} +11654.0 q^{76} +4764.50 q^{77} -16051.3 q^{78} +35672.9 q^{79} +6561.00 q^{81} +4259.73 q^{82} +79999.7 q^{83} -11788.4 q^{84} +97580.6 q^{86} -49559.8 q^{87} +1235.91 q^{88} -37783.0 q^{89} +8692.83 q^{91} -46074.1 q^{92} +22057.7 q^{93} -4548.46 q^{94} +74336.5 q^{96} +7616.35 q^{97} -123252. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9} + 363 q^{11} - 252 q^{12} - 450 q^{13} - 1504 q^{14} - 1360 q^{16} + 334 q^{17} + 162 q^{18} - 4036 q^{19} - 2088 q^{21} + 242 q^{22} + 7060 q^{23} + 216 q^{24} + 2932 q^{26} - 2187 q^{27} + 8320 q^{28} + 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3267 q^{33} - 3644 q^{34} + 2268 q^{36} - 2250 q^{37} + 12632 q^{38} + 4050 q^{39} + 10654 q^{41} + 13536 q^{42} + 35528 q^{43} + 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 12240 q^{48} + 7667 q^{49} - 3006 q^{51} + 14520 q^{52} + 12826 q^{53} - 1458 q^{54} + 17088 q^{56} + 36324 q^{57} + 17196 q^{58} - 81876 q^{59} - 62298 q^{61} - 109184 q^{62} + 18792 q^{63} - 72256 q^{64} - 2178 q^{66} + 46148 q^{67} + 35832 q^{68} - 63540 q^{69} - 64724 q^{71} - 1944 q^{72} - 810 q^{73} - 44796 q^{74} + 44656 q^{76} + 28072 q^{77} - 26388 q^{78} + 43876 q^{79} + 19683 q^{81} - 56060 q^{82} + 101024 q^{83} - 74880 q^{84} + 24128 q^{86} - 36378 q^{87} - 2904 q^{88} + 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 5472 q^{93} + 74552 q^{94} - 27936 q^{96} + 319746 q^{97} - 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.07863 1.42811 0.714057 0.700087i \(-0.246856\pi\)
0.714057 + 0.700087i \(0.246856\pi\)
\(3\) −9.00000 −0.577350
\(4\) 33.2643 1.03951
\(5\) 0 0
\(6\) −72.7077 −0.824522
\(7\) 39.3760 0.303730 0.151865 0.988401i \(-0.451472\pi\)
0.151865 + 0.988401i \(0.451472\pi\)
\(8\) 10.2141 0.0564257
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −299.379 −0.600162
\(13\) 220.765 0.362302 0.181151 0.983455i \(-0.442018\pi\)
0.181151 + 0.983455i \(0.442018\pi\)
\(14\) 318.105 0.433761
\(15\) 0 0
\(16\) −981.943 −0.958928
\(17\) 200.343 0.168133 0.0840664 0.996460i \(-0.473209\pi\)
0.0840664 + 0.996460i \(0.473209\pi\)
\(18\) 654.369 0.476038
\(19\) 350.345 0.222645 0.111322 0.993784i \(-0.464491\pi\)
0.111322 + 0.993784i \(0.464491\pi\)
\(20\) 0 0
\(21\) −354.384 −0.175358
\(22\) 977.515 0.430593
\(23\) −1385.09 −0.545957 −0.272978 0.962020i \(-0.588009\pi\)
−0.272978 + 0.962020i \(0.588009\pi\)
\(24\) −91.9273 −0.0325774
\(25\) 0 0
\(26\) 1783.48 0.517409
\(27\) −729.000 −0.192450
\(28\) 1309.82 0.315730
\(29\) 5506.64 1.21588 0.607942 0.793982i \(-0.291995\pi\)
0.607942 + 0.793982i \(0.291995\pi\)
\(30\) 0 0
\(31\) −2450.86 −0.458051 −0.229025 0.973420i \(-0.573554\pi\)
−0.229025 + 0.973420i \(0.573554\pi\)
\(32\) −8259.61 −1.42588
\(33\) −1089.00 −0.174078
\(34\) 1618.50 0.240113
\(35\) 0 0
\(36\) 2694.41 0.346504
\(37\) 4060.46 0.487609 0.243804 0.969824i \(-0.421605\pi\)
0.243804 + 0.969824i \(0.421605\pi\)
\(38\) 2830.31 0.317962
\(39\) −1986.88 −0.209175
\(40\) 0 0
\(41\) 527.283 0.0489874 0.0244937 0.999700i \(-0.492203\pi\)
0.0244937 + 0.999700i \(0.492203\pi\)
\(42\) −2862.94 −0.250432
\(43\) 12078.9 0.996218 0.498109 0.867114i \(-0.334028\pi\)
0.498109 + 0.867114i \(0.334028\pi\)
\(44\) 4024.99 0.313424
\(45\) 0 0
\(46\) −11189.6 −0.779689
\(47\) −563.023 −0.0371776 −0.0185888 0.999827i \(-0.505917\pi\)
−0.0185888 + 0.999827i \(0.505917\pi\)
\(48\) 8837.48 0.553638
\(49\) −15256.5 −0.907748
\(50\) 0 0
\(51\) −1803.09 −0.0970715
\(52\) 7343.59 0.376617
\(53\) 37203.0 1.81923 0.909617 0.415449i \(-0.136375\pi\)
0.909617 + 0.415449i \(0.136375\pi\)
\(54\) −5889.32 −0.274841
\(55\) 0 0
\(56\) 402.192 0.0171381
\(57\) −3153.11 −0.128544
\(58\) 44486.2 1.73642
\(59\) 2157.64 0.0806955 0.0403477 0.999186i \(-0.487153\pi\)
0.0403477 + 0.999186i \(0.487153\pi\)
\(60\) 0 0
\(61\) −39938.0 −1.37424 −0.687119 0.726545i \(-0.741125\pi\)
−0.687119 + 0.726545i \(0.741125\pi\)
\(62\) −19799.6 −0.654149
\(63\) 3189.46 0.101243
\(64\) −35304.2 −1.07740
\(65\) 0 0
\(66\) −8797.63 −0.248603
\(67\) 38473.2 1.04706 0.523529 0.852008i \(-0.324615\pi\)
0.523529 + 0.852008i \(0.324615\pi\)
\(68\) 6664.29 0.174776
\(69\) 12465.8 0.315208
\(70\) 0 0
\(71\) −13725.3 −0.323129 −0.161564 0.986862i \(-0.551654\pi\)
−0.161564 + 0.986862i \(0.551654\pi\)
\(72\) 827.345 0.0188086
\(73\) 39736.3 0.872731 0.436366 0.899769i \(-0.356265\pi\)
0.436366 + 0.899769i \(0.356265\pi\)
\(74\) 32803.0 0.696361
\(75\) 0 0
\(76\) 11654.0 0.231441
\(77\) 4764.50 0.0915779
\(78\) −16051.3 −0.298726
\(79\) 35672.9 0.643088 0.321544 0.946895i \(-0.395798\pi\)
0.321544 + 0.946895i \(0.395798\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 4259.73 0.0699596
\(83\) 79999.7 1.27466 0.637328 0.770593i \(-0.280040\pi\)
0.637328 + 0.770593i \(0.280040\pi\)
\(84\) −11788.4 −0.182287
\(85\) 0 0
\(86\) 97580.6 1.42271
\(87\) −49559.8 −0.701991
\(88\) 1235.91 0.0170130
\(89\) −37783.0 −0.505616 −0.252808 0.967516i \(-0.581354\pi\)
−0.252808 + 0.967516i \(0.581354\pi\)
\(90\) 0 0
\(91\) 8692.83 0.110042
\(92\) −46074.1 −0.567528
\(93\) 22057.7 0.264456
\(94\) −4548.46 −0.0530939
\(95\) 0 0
\(96\) 74336.5 0.823235
\(97\) 7616.35 0.0821897 0.0410948 0.999155i \(-0.486915\pi\)
0.0410948 + 0.999155i \(0.486915\pi\)
\(98\) −123252. −1.29637
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 125963. 1.22868 0.614341 0.789041i \(-0.289422\pi\)
0.614341 + 0.789041i \(0.289422\pi\)
\(102\) −14566.5 −0.138629
\(103\) 165220. 1.53451 0.767257 0.641340i \(-0.221621\pi\)
0.767257 + 0.641340i \(0.221621\pi\)
\(104\) 2254.92 0.0204431
\(105\) 0 0
\(106\) 300550. 2.59807
\(107\) 13152.3 0.111056 0.0555279 0.998457i \(-0.482316\pi\)
0.0555279 + 0.998457i \(0.482316\pi\)
\(108\) −24249.7 −0.200054
\(109\) 37970.9 0.306115 0.153057 0.988217i \(-0.451088\pi\)
0.153057 + 0.988217i \(0.451088\pi\)
\(110\) 0 0
\(111\) −36544.2 −0.281521
\(112\) −38665.0 −0.291255
\(113\) 85189.7 0.627612 0.313806 0.949487i \(-0.398396\pi\)
0.313806 + 0.949487i \(0.398396\pi\)
\(114\) −25472.8 −0.183575
\(115\) 0 0
\(116\) 183175. 1.26392
\(117\) 17881.9 0.120767
\(118\) 17430.8 0.115242
\(119\) 7888.73 0.0510669
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −322645. −1.96257
\(123\) −4745.55 −0.0282829
\(124\) −81526.1 −0.476148
\(125\) 0 0
\(126\) 25766.5 0.144587
\(127\) 141237. 0.777033 0.388517 0.921442i \(-0.372988\pi\)
0.388517 + 0.921442i \(0.372988\pi\)
\(128\) −20902.2 −0.112763
\(129\) −108710. −0.575167
\(130\) 0 0
\(131\) 210469. 1.07155 0.535773 0.844362i \(-0.320020\pi\)
0.535773 + 0.844362i \(0.320020\pi\)
\(132\) −36224.9 −0.180956
\(133\) 13795.2 0.0676237
\(134\) 310811. 1.49532
\(135\) 0 0
\(136\) 2046.33 0.00948701
\(137\) −19565.1 −0.0890594 −0.0445297 0.999008i \(-0.514179\pi\)
−0.0445297 + 0.999008i \(0.514179\pi\)
\(138\) 100707. 0.450154
\(139\) 132498. 0.581663 0.290831 0.956774i \(-0.406068\pi\)
0.290831 + 0.956774i \(0.406068\pi\)
\(140\) 0 0
\(141\) 5067.21 0.0214645
\(142\) −110882. −0.461465
\(143\) 26712.5 0.109238
\(144\) −79537.3 −0.319643
\(145\) 0 0
\(146\) 321015. 1.24636
\(147\) 137309. 0.524089
\(148\) 135069. 0.506874
\(149\) −153316. −0.565748 −0.282874 0.959157i \(-0.591288\pi\)
−0.282874 + 0.959157i \(0.591288\pi\)
\(150\) 0 0
\(151\) 248151. 0.885672 0.442836 0.896603i \(-0.353972\pi\)
0.442836 + 0.896603i \(0.353972\pi\)
\(152\) 3578.47 0.0125629
\(153\) 16227.8 0.0560443
\(154\) 38490.7 0.130784
\(155\) 0 0
\(156\) −66092.3 −0.217440
\(157\) 180692. 0.585047 0.292523 0.956258i \(-0.405505\pi\)
0.292523 + 0.956258i \(0.405505\pi\)
\(158\) 288188. 0.918403
\(159\) −334827. −1.05033
\(160\) 0 0
\(161\) −54539.4 −0.165823
\(162\) 53003.9 0.158679
\(163\) 176045. 0.518985 0.259493 0.965745i \(-0.416445\pi\)
0.259493 + 0.965745i \(0.416445\pi\)
\(164\) 17539.7 0.0509229
\(165\) 0 0
\(166\) 646288. 1.82035
\(167\) −293818. −0.815244 −0.407622 0.913151i \(-0.633642\pi\)
−0.407622 + 0.913151i \(0.633642\pi\)
\(168\) −3619.73 −0.00989471
\(169\) −322556. −0.868737
\(170\) 0 0
\(171\) 28378.0 0.0742148
\(172\) 401795. 1.03558
\(173\) 20784.9 0.0527999 0.0263999 0.999651i \(-0.491596\pi\)
0.0263999 + 0.999651i \(0.491596\pi\)
\(174\) −400375. −1.00252
\(175\) 0 0
\(176\) −118815. −0.289128
\(177\) −19418.8 −0.0465896
\(178\) −305235. −0.722078
\(179\) −229326. −0.534958 −0.267479 0.963564i \(-0.586191\pi\)
−0.267479 + 0.963564i \(0.586191\pi\)
\(180\) 0 0
\(181\) −90807.3 −0.206027 −0.103014 0.994680i \(-0.532849\pi\)
−0.103014 + 0.994680i \(0.532849\pi\)
\(182\) 70226.2 0.157152
\(183\) 359442. 0.793416
\(184\) −14147.5 −0.0308060
\(185\) 0 0
\(186\) 178196. 0.377673
\(187\) 24241.5 0.0506939
\(188\) −18728.6 −0.0386465
\(189\) −28705.1 −0.0584528
\(190\) 0 0
\(191\) −496794. −0.985356 −0.492678 0.870212i \(-0.663982\pi\)
−0.492678 + 0.870212i \(0.663982\pi\)
\(192\) 317738. 0.622036
\(193\) −271362. −0.524391 −0.262196 0.965015i \(-0.584447\pi\)
−0.262196 + 0.965015i \(0.584447\pi\)
\(194\) 61529.7 0.117376
\(195\) 0 0
\(196\) −507498. −0.943614
\(197\) 190972. 0.350594 0.175297 0.984516i \(-0.443911\pi\)
0.175297 + 0.984516i \(0.443911\pi\)
\(198\) 79178.7 0.143531
\(199\) 602637. 1.07876 0.539378 0.842064i \(-0.318660\pi\)
0.539378 + 0.842064i \(0.318660\pi\)
\(200\) 0 0
\(201\) −346258. −0.604519
\(202\) 1.01761e6 1.75470
\(203\) 216830. 0.369300
\(204\) −59978.6 −0.100907
\(205\) 0 0
\(206\) 1.33476e6 2.19146
\(207\) −112192. −0.181986
\(208\) −216778. −0.347422
\(209\) 42391.8 0.0671299
\(210\) 0 0
\(211\) 918176. 1.41978 0.709888 0.704315i \(-0.248746\pi\)
0.709888 + 0.704315i \(0.248746\pi\)
\(212\) 1.23753e6 1.89111
\(213\) 123528. 0.186559
\(214\) 106252. 0.158600
\(215\) 0 0
\(216\) −7446.11 −0.0108591
\(217\) −96505.0 −0.139123
\(218\) 306753. 0.437167
\(219\) −357627. −0.503872
\(220\) 0 0
\(221\) 44228.7 0.0609149
\(222\) −295227. −0.402044
\(223\) −959605. −1.29220 −0.646101 0.763252i \(-0.723601\pi\)
−0.646101 + 0.763252i \(0.723601\pi\)
\(224\) −325231. −0.433083
\(225\) 0 0
\(226\) 688217. 0.896301
\(227\) 289954. 0.373477 0.186739 0.982410i \(-0.440208\pi\)
0.186739 + 0.982410i \(0.440208\pi\)
\(228\) −104886. −0.133623
\(229\) 111161. 0.140076 0.0700379 0.997544i \(-0.477688\pi\)
0.0700379 + 0.997544i \(0.477688\pi\)
\(230\) 0 0
\(231\) −42880.5 −0.0528725
\(232\) 56245.6 0.0686071
\(233\) 453216. 0.546909 0.273455 0.961885i \(-0.411834\pi\)
0.273455 + 0.961885i \(0.411834\pi\)
\(234\) 144462. 0.172470
\(235\) 0 0
\(236\) 71772.5 0.0838838
\(237\) −321056. −0.371287
\(238\) 63730.1 0.0729294
\(239\) 1.26387e6 1.43123 0.715614 0.698496i \(-0.246147\pi\)
0.715614 + 0.698496i \(0.246147\pi\)
\(240\) 0 0
\(241\) 134350. 0.149003 0.0745013 0.997221i \(-0.476264\pi\)
0.0745013 + 0.997221i \(0.476264\pi\)
\(242\) 118279. 0.129829
\(243\) −59049.0 −0.0641500
\(244\) −1.32851e6 −1.42853
\(245\) 0 0
\(246\) −38337.6 −0.0403912
\(247\) 77343.8 0.0806646
\(248\) −25033.4 −0.0258458
\(249\) −719997. −0.735923
\(250\) 0 0
\(251\) −1.43370e6 −1.43639 −0.718197 0.695840i \(-0.755032\pi\)
−0.718197 + 0.695840i \(0.755032\pi\)
\(252\) 106095. 0.105243
\(253\) −167596. −0.164612
\(254\) 1.14100e6 1.10969
\(255\) 0 0
\(256\) 960873. 0.916360
\(257\) 953935. 0.900920 0.450460 0.892797i \(-0.351260\pi\)
0.450460 + 0.892797i \(0.351260\pi\)
\(258\) −878226. −0.821404
\(259\) 159885. 0.148101
\(260\) 0 0
\(261\) 446038. 0.405294
\(262\) 1.70030e6 1.53029
\(263\) −322280. −0.287306 −0.143653 0.989628i \(-0.545885\pi\)
−0.143653 + 0.989628i \(0.545885\pi\)
\(264\) −11123.2 −0.00982245
\(265\) 0 0
\(266\) 111446. 0.0965744
\(267\) 340047. 0.291918
\(268\) 1.27978e6 1.08843
\(269\) 808737. 0.681438 0.340719 0.940165i \(-0.389329\pi\)
0.340719 + 0.940165i \(0.389329\pi\)
\(270\) 0 0
\(271\) −909303. −0.752117 −0.376058 0.926596i \(-0.622721\pi\)
−0.376058 + 0.926596i \(0.622721\pi\)
\(272\) −196726. −0.161227
\(273\) −78235.5 −0.0635327
\(274\) −158059. −0.127187
\(275\) 0 0
\(276\) 414667. 0.327662
\(277\) −236933. −0.185535 −0.0927675 0.995688i \(-0.529571\pi\)
−0.0927675 + 0.995688i \(0.529571\pi\)
\(278\) 1.07040e6 0.830681
\(279\) −198519. −0.152684
\(280\) 0 0
\(281\) −450779. −0.340563 −0.170282 0.985395i \(-0.554468\pi\)
−0.170282 + 0.985395i \(0.554468\pi\)
\(282\) 40936.1 0.0306538
\(283\) −683413. −0.507245 −0.253622 0.967303i \(-0.581622\pi\)
−0.253622 + 0.967303i \(0.581622\pi\)
\(284\) −456563. −0.335896
\(285\) 0 0
\(286\) 215801. 0.156005
\(287\) 20762.3 0.0148789
\(288\) −669028. −0.475295
\(289\) −1.37972e6 −0.971731
\(290\) 0 0
\(291\) −68547.1 −0.0474522
\(292\) 1.32180e6 0.907213
\(293\) 1.81068e6 1.23218 0.616088 0.787678i \(-0.288717\pi\)
0.616088 + 0.787678i \(0.288717\pi\)
\(294\) 1.10927e6 0.748459
\(295\) 0 0
\(296\) 41474.1 0.0275136
\(297\) −88209.0 −0.0580259
\(298\) −1.23859e6 −0.807953
\(299\) −305779. −0.197801
\(300\) 0 0
\(301\) 475617. 0.302581
\(302\) 2.00472e6 1.26484
\(303\) −1.13367e6 −0.709380
\(304\) −344019. −0.213500
\(305\) 0 0
\(306\) 131099. 0.0800376
\(307\) 3.00860e6 1.82187 0.910937 0.412546i \(-0.135361\pi\)
0.910937 + 0.412546i \(0.135361\pi\)
\(308\) 158488. 0.0951962
\(309\) −1.48698e6 −0.885952
\(310\) 0 0
\(311\) −2.04083e6 −1.19648 −0.598241 0.801316i \(-0.704134\pi\)
−0.598241 + 0.801316i \(0.704134\pi\)
\(312\) −20294.3 −0.0118029
\(313\) 1.73940e6 1.00355 0.501774 0.864999i \(-0.332681\pi\)
0.501774 + 0.864999i \(0.332681\pi\)
\(314\) 1.45975e6 0.835513
\(315\) 0 0
\(316\) 1.18663e6 0.668497
\(317\) −106704. −0.0596392 −0.0298196 0.999555i \(-0.509493\pi\)
−0.0298196 + 0.999555i \(0.509493\pi\)
\(318\) −2.70495e6 −1.50000
\(319\) 666304. 0.366603
\(320\) 0 0
\(321\) −118370. −0.0641181
\(322\) −440604. −0.236815
\(323\) 70189.3 0.0374338
\(324\) 218247. 0.115501
\(325\) 0 0
\(326\) 1.42220e6 0.741170
\(327\) −341738. −0.176736
\(328\) 5385.74 0.00276415
\(329\) −22169.6 −0.0112919
\(330\) 0 0
\(331\) 1.54759e6 0.776402 0.388201 0.921575i \(-0.373097\pi\)
0.388201 + 0.921575i \(0.373097\pi\)
\(332\) 2.66114e6 1.32502
\(333\) 328898. 0.162536
\(334\) −2.37365e6 −1.16426
\(335\) 0 0
\(336\) 347985. 0.168156
\(337\) −1.22550e6 −0.587814 −0.293907 0.955834i \(-0.594956\pi\)
−0.293907 + 0.955834i \(0.594956\pi\)
\(338\) −2.60581e6 −1.24066
\(339\) −766708. −0.362352
\(340\) 0 0
\(341\) −296553. −0.138107
\(342\) 229255. 0.105987
\(343\) −1.26253e6 −0.579440
\(344\) 123375. 0.0562123
\(345\) 0 0
\(346\) 167914. 0.0754042
\(347\) −3.82684e6 −1.70615 −0.853074 0.521789i \(-0.825265\pi\)
−0.853074 + 0.521789i \(0.825265\pi\)
\(348\) −1.64857e6 −0.729727
\(349\) 456968. 0.200827 0.100413 0.994946i \(-0.467983\pi\)
0.100413 + 0.994946i \(0.467983\pi\)
\(350\) 0 0
\(351\) −160937. −0.0697251
\(352\) −999413. −0.429920
\(353\) −4.32830e6 −1.84876 −0.924380 0.381474i \(-0.875417\pi\)
−0.924380 + 0.381474i \(0.875417\pi\)
\(354\) −156877. −0.0665352
\(355\) 0 0
\(356\) −1.25683e6 −0.525594
\(357\) −70998.5 −0.0294835
\(358\) −1.85264e6 −0.763982
\(359\) −845380. −0.346191 −0.173095 0.984905i \(-0.555377\pi\)
−0.173095 + 0.984905i \(0.555377\pi\)
\(360\) 0 0
\(361\) −2.35336e6 −0.950429
\(362\) −733599. −0.294230
\(363\) −131769. −0.0524864
\(364\) 289161. 0.114390
\(365\) 0 0
\(366\) 2.90380e6 1.13309
\(367\) −2.96016e6 −1.14723 −0.573615 0.819125i \(-0.694460\pi\)
−0.573615 + 0.819125i \(0.694460\pi\)
\(368\) 1.36008e6 0.523534
\(369\) 42709.9 0.0163291
\(370\) 0 0
\(371\) 1.46491e6 0.552555
\(372\) 733735. 0.274904
\(373\) −1.02299e6 −0.380713 −0.190357 0.981715i \(-0.560964\pi\)
−0.190357 + 0.981715i \(0.560964\pi\)
\(374\) 195839. 0.0723968
\(375\) 0 0
\(376\) −5750.80 −0.00209777
\(377\) 1.21567e6 0.440517
\(378\) −231898. −0.0834772
\(379\) −4.19402e6 −1.49980 −0.749899 0.661552i \(-0.769898\pi\)
−0.749899 + 0.661552i \(0.769898\pi\)
\(380\) 0 0
\(381\) −1.27113e6 −0.448620
\(382\) −4.01342e6 −1.40720
\(383\) 1.64287e6 0.572276 0.286138 0.958188i \(-0.407628\pi\)
0.286138 + 0.958188i \(0.407628\pi\)
\(384\) 188120. 0.0651039
\(385\) 0 0
\(386\) −2.19223e6 −0.748891
\(387\) 978387. 0.332073
\(388\) 253353. 0.0854371
\(389\) 769820. 0.257938 0.128969 0.991649i \(-0.458833\pi\)
0.128969 + 0.991649i \(0.458833\pi\)
\(390\) 0 0
\(391\) −277493. −0.0917933
\(392\) −155832. −0.0512203
\(393\) −1.89422e6 −0.618657
\(394\) 1.54279e6 0.500688
\(395\) 0 0
\(396\) 326024. 0.104475
\(397\) −4.86631e6 −1.54961 −0.774806 0.632199i \(-0.782153\pi\)
−0.774806 + 0.632199i \(0.782153\pi\)
\(398\) 4.86848e6 1.54059
\(399\) −124157. −0.0390426
\(400\) 0 0
\(401\) 717064. 0.222688 0.111344 0.993782i \(-0.464484\pi\)
0.111344 + 0.993782i \(0.464484\pi\)
\(402\) −2.79730e6 −0.863323
\(403\) −541062. −0.165953
\(404\) 4.19007e6 1.27723
\(405\) 0 0
\(406\) 1.75169e6 0.527402
\(407\) 491316. 0.147019
\(408\) −18417.0 −0.00547733
\(409\) −3.80091e6 −1.12352 −0.561758 0.827302i \(-0.689875\pi\)
−0.561758 + 0.827302i \(0.689875\pi\)
\(410\) 0 0
\(411\) 176086. 0.0514185
\(412\) 5.49595e6 1.59514
\(413\) 84959.4 0.0245096
\(414\) −906361. −0.259896
\(415\) 0 0
\(416\) −1.82343e6 −0.516601
\(417\) −1.19248e6 −0.335823
\(418\) 342468. 0.0958691
\(419\) −1.55888e6 −0.433787 −0.216894 0.976195i \(-0.569592\pi\)
−0.216894 + 0.976195i \(0.569592\pi\)
\(420\) 0 0
\(421\) 4.13561e6 1.13719 0.568596 0.822617i \(-0.307487\pi\)
0.568596 + 0.822617i \(0.307487\pi\)
\(422\) 7.41761e6 2.02760
\(423\) −45604.9 −0.0123925
\(424\) 379997. 0.102651
\(425\) 0 0
\(426\) 997935. 0.266427
\(427\) −1.57260e6 −0.417397
\(428\) 437502. 0.115444
\(429\) −240413. −0.0630687
\(430\) 0 0
\(431\) −5.25900e6 −1.36367 −0.681836 0.731505i \(-0.738818\pi\)
−0.681836 + 0.731505i \(0.738818\pi\)
\(432\) 715836. 0.184546
\(433\) 694549. 0.178026 0.0890130 0.996030i \(-0.471629\pi\)
0.0890130 + 0.996030i \(0.471629\pi\)
\(434\) −779629. −0.198684
\(435\) 0 0
\(436\) 1.26308e6 0.318210
\(437\) −485260. −0.121554
\(438\) −2.88914e6 −0.719586
\(439\) −7.80969e6 −1.93407 −0.967036 0.254640i \(-0.918043\pi\)
−0.967036 + 0.254640i \(0.918043\pi\)
\(440\) 0 0
\(441\) −1.23578e6 −0.302583
\(442\) 357307. 0.0869934
\(443\) 5.43876e6 1.31671 0.658356 0.752707i \(-0.271252\pi\)
0.658356 + 0.752707i \(0.271252\pi\)
\(444\) −1.21562e6 −0.292644
\(445\) 0 0
\(446\) −7.75229e6 −1.84541
\(447\) 1.37985e6 0.326635
\(448\) −1.39014e6 −0.327238
\(449\) 5.46277e6 1.27878 0.639392 0.768881i \(-0.279186\pi\)
0.639392 + 0.768881i \(0.279186\pi\)
\(450\) 0 0
\(451\) 63801.3 0.0147703
\(452\) 2.83378e6 0.652409
\(453\) −2.23336e6 −0.511343
\(454\) 2.34243e6 0.533368
\(455\) 0 0
\(456\) −32206.3 −0.00725318
\(457\) −3.07144e6 −0.687941 −0.343970 0.938980i \(-0.611772\pi\)
−0.343970 + 0.938980i \(0.611772\pi\)
\(458\) 898028. 0.200044
\(459\) −146050. −0.0323572
\(460\) 0 0
\(461\) 3.85641e6 0.845143 0.422572 0.906329i \(-0.361127\pi\)
0.422572 + 0.906329i \(0.361127\pi\)
\(462\) −346416. −0.0755080
\(463\) −3.20594e6 −0.695029 −0.347514 0.937675i \(-0.612974\pi\)
−0.347514 + 0.937675i \(0.612974\pi\)
\(464\) −5.40721e6 −1.16594
\(465\) 0 0
\(466\) 3.66136e6 0.781049
\(467\) 2.43952e6 0.517622 0.258811 0.965928i \(-0.416669\pi\)
0.258811 + 0.965928i \(0.416669\pi\)
\(468\) 594830. 0.125539
\(469\) 1.51492e6 0.318023
\(470\) 0 0
\(471\) −1.62623e6 −0.337777
\(472\) 22038.5 0.00455330
\(473\) 1.46154e6 0.300371
\(474\) −2.59369e6 −0.530240
\(475\) 0 0
\(476\) 262413. 0.0530846
\(477\) 3.01344e6 0.606411
\(478\) 1.02104e7 2.04396
\(479\) 4.69186e6 0.934342 0.467171 0.884167i \(-0.345273\pi\)
0.467171 + 0.884167i \(0.345273\pi\)
\(480\) 0 0
\(481\) 896406. 0.176662
\(482\) 1.08536e6 0.212793
\(483\) 490854. 0.0957381
\(484\) 487023. 0.0945010
\(485\) 0 0
\(486\) −477035. −0.0916136
\(487\) −5.73305e6 −1.09538 −0.547688 0.836683i \(-0.684492\pi\)
−0.547688 + 0.836683i \(0.684492\pi\)
\(488\) −407932. −0.0775423
\(489\) −1.58441e6 −0.299636
\(490\) 0 0
\(491\) −693872. −0.129890 −0.0649449 0.997889i \(-0.520687\pi\)
−0.0649449 + 0.997889i \(0.520687\pi\)
\(492\) −157858. −0.0294004
\(493\) 1.10322e6 0.204430
\(494\) 624832. 0.115198
\(495\) 0 0
\(496\) 2.40660e6 0.439238
\(497\) −540448. −0.0981438
\(498\) −5.81659e6 −1.05098
\(499\) 1.23153e6 0.221407 0.110704 0.993853i \(-0.464690\pi\)
0.110704 + 0.993853i \(0.464690\pi\)
\(500\) 0 0
\(501\) 2.64436e6 0.470681
\(502\) −1.15823e7 −2.05133
\(503\) 7.61551e6 1.34208 0.671041 0.741420i \(-0.265847\pi\)
0.671041 + 0.741420i \(0.265847\pi\)
\(504\) 32577.6 0.00571272
\(505\) 0 0
\(506\) −1.35395e6 −0.235085
\(507\) 2.90300e6 0.501566
\(508\) 4.69816e6 0.807734
\(509\) −1.55496e6 −0.266026 −0.133013 0.991114i \(-0.542465\pi\)
−0.133013 + 0.991114i \(0.542465\pi\)
\(510\) 0 0
\(511\) 1.56466e6 0.265074
\(512\) 8.43141e6 1.42143
\(513\) −255402. −0.0428480
\(514\) 7.70649e6 1.28662
\(515\) 0 0
\(516\) −3.61616e6 −0.597892
\(517\) −68125.8 −0.0112095
\(518\) 1.29165e6 0.211505
\(519\) −187064. −0.0304840
\(520\) 0 0
\(521\) −3.26766e6 −0.527403 −0.263701 0.964604i \(-0.584943\pi\)
−0.263701 + 0.964604i \(0.584943\pi\)
\(522\) 3.60338e6 0.578807
\(523\) 8.35303e6 1.33533 0.667667 0.744460i \(-0.267293\pi\)
0.667667 + 0.744460i \(0.267293\pi\)
\(524\) 7.00112e6 1.11388
\(525\) 0 0
\(526\) −2.60358e6 −0.410305
\(527\) −491012. −0.0770133
\(528\) 1.06934e6 0.166928
\(529\) −4.51787e6 −0.701931
\(530\) 0 0
\(531\) 174769. 0.0268985
\(532\) 458888. 0.0702956
\(533\) 116405. 0.0177482
\(534\) 2.74711e6 0.416892
\(535\) 0 0
\(536\) 392970. 0.0590810
\(537\) 2.06393e6 0.308858
\(538\) 6.53349e6 0.973172
\(539\) −1.84604e6 −0.273696
\(540\) 0 0
\(541\) 4.72814e6 0.694540 0.347270 0.937765i \(-0.387109\pi\)
0.347270 + 0.937765i \(0.387109\pi\)
\(542\) −7.34593e6 −1.07411
\(543\) 817266. 0.118950
\(544\) −1.65476e6 −0.239738
\(545\) 0 0
\(546\) −632036. −0.0907320
\(547\) 3.84639e6 0.549648 0.274824 0.961495i \(-0.411380\pi\)
0.274824 + 0.961495i \(0.411380\pi\)
\(548\) −650819. −0.0925782
\(549\) −3.23498e6 −0.458079
\(550\) 0 0
\(551\) 1.92923e6 0.270710
\(552\) 127328. 0.0177859
\(553\) 1.40466e6 0.195325
\(554\) −1.91409e6 −0.264965
\(555\) 0 0
\(556\) 4.40745e6 0.604644
\(557\) −1.34095e7 −1.83137 −0.915684 0.401900i \(-0.868350\pi\)
−0.915684 + 0.401900i \(0.868350\pi\)
\(558\) −1.60376e6 −0.218050
\(559\) 2.66658e6 0.360932
\(560\) 0 0
\(561\) −218174. −0.0292682
\(562\) −3.64168e6 −0.486363
\(563\) −1.51163e6 −0.200990 −0.100495 0.994938i \(-0.532043\pi\)
−0.100495 + 0.994938i \(0.532043\pi\)
\(564\) 168557. 0.0223126
\(565\) 0 0
\(566\) −5.52105e6 −0.724403
\(567\) 258346. 0.0337477
\(568\) −140192. −0.0182328
\(569\) 1.39221e7 1.80271 0.901353 0.433085i \(-0.142575\pi\)
0.901353 + 0.433085i \(0.142575\pi\)
\(570\) 0 0
\(571\) −3.44073e6 −0.441632 −0.220816 0.975316i \(-0.570872\pi\)
−0.220816 + 0.975316i \(0.570872\pi\)
\(572\) 888574. 0.113554
\(573\) 4.47115e6 0.568896
\(574\) 167731. 0.0212488
\(575\) 0 0
\(576\) −2.85964e6 −0.359133
\(577\) −3.96778e6 −0.496144 −0.248072 0.968742i \(-0.579797\pi\)
−0.248072 + 0.968742i \(0.579797\pi\)
\(578\) −1.11463e7 −1.38774
\(579\) 2.44226e6 0.302758
\(580\) 0 0
\(581\) 3.15007e6 0.387151
\(582\) −553767. −0.0677672
\(583\) 4.50156e6 0.548519
\(584\) 405872. 0.0492445
\(585\) 0 0
\(586\) 1.46278e7 1.75969
\(587\) −3.64595e6 −0.436733 −0.218366 0.975867i \(-0.570073\pi\)
−0.218366 + 0.975867i \(0.570073\pi\)
\(588\) 4.56748e6 0.544796
\(589\) −858645. −0.101982
\(590\) 0 0
\(591\) −1.71875e6 −0.202416
\(592\) −3.98714e6 −0.467582
\(593\) 486759. 0.0568431 0.0284215 0.999596i \(-0.490952\pi\)
0.0284215 + 0.999596i \(0.490952\pi\)
\(594\) −712608. −0.0828676
\(595\) 0 0
\(596\) −5.09997e6 −0.588101
\(597\) −5.42373e6 −0.622820
\(598\) −2.47027e6 −0.282483
\(599\) 1.37413e7 1.56481 0.782404 0.622771i \(-0.213993\pi\)
0.782404 + 0.622771i \(0.213993\pi\)
\(600\) 0 0
\(601\) 7.84470e6 0.885911 0.442955 0.896544i \(-0.353930\pi\)
0.442955 + 0.896544i \(0.353930\pi\)
\(602\) 3.84234e6 0.432120
\(603\) 3.11633e6 0.349019
\(604\) 8.25457e6 0.920666
\(605\) 0 0
\(606\) −9.15848e6 −1.01308
\(607\) 3.41681e6 0.376399 0.188200 0.982131i \(-0.439735\pi\)
0.188200 + 0.982131i \(0.439735\pi\)
\(608\) −2.89371e6 −0.317465
\(609\) −1.95147e6 −0.213215
\(610\) 0 0
\(611\) −124296. −0.0134695
\(612\) 539807. 0.0582586
\(613\) 1.32402e7 1.42313 0.711564 0.702621i \(-0.247987\pi\)
0.711564 + 0.702621i \(0.247987\pi\)
\(614\) 2.43054e7 2.60184
\(615\) 0 0
\(616\) 48665.3 0.00516735
\(617\) −1.72143e7 −1.82044 −0.910220 0.414126i \(-0.864087\pi\)
−0.910220 + 0.414126i \(0.864087\pi\)
\(618\) −1.20128e7 −1.26524
\(619\) 1.60142e7 1.67988 0.839940 0.542679i \(-0.182590\pi\)
0.839940 + 0.542679i \(0.182590\pi\)
\(620\) 0 0
\(621\) 1.00973e6 0.105069
\(622\) −1.64871e7 −1.70871
\(623\) −1.48774e6 −0.153571
\(624\) 1.95100e6 0.200584
\(625\) 0 0
\(626\) 1.40520e7 1.43318
\(627\) −381526. −0.0387574
\(628\) 6.01061e6 0.608162
\(629\) 813486. 0.0819830
\(630\) 0 0
\(631\) −7.75049e6 −0.774918 −0.387459 0.921887i \(-0.626647\pi\)
−0.387459 + 0.921887i \(0.626647\pi\)
\(632\) 364368. 0.0362867
\(633\) −8.26358e6 −0.819708
\(634\) −862021. −0.0851716
\(635\) 0 0
\(636\) −1.11378e7 −1.09183
\(637\) −3.36810e6 −0.328879
\(638\) 5.38282e6 0.523550
\(639\) −1.11175e6 −0.107710
\(640\) 0 0
\(641\) 8.77648e6 0.843675 0.421838 0.906671i \(-0.361385\pi\)
0.421838 + 0.906671i \(0.361385\pi\)
\(642\) −956272. −0.0915680
\(643\) 1.51011e7 1.44040 0.720198 0.693768i \(-0.244051\pi\)
0.720198 + 0.693768i \(0.244051\pi\)
\(644\) −1.81422e6 −0.172375
\(645\) 0 0
\(646\) 567034. 0.0534598
\(647\) −1.57910e7 −1.48303 −0.741516 0.670936i \(-0.765893\pi\)
−0.741516 + 0.670936i \(0.765893\pi\)
\(648\) 67015.0 0.00626952
\(649\) 261075. 0.0243306
\(650\) 0 0
\(651\) 868545. 0.0803230
\(652\) 5.85603e6 0.539491
\(653\) −1.33260e7 −1.22297 −0.611484 0.791256i \(-0.709427\pi\)
−0.611484 + 0.791256i \(0.709427\pi\)
\(654\) −2.76078e6 −0.252399
\(655\) 0 0
\(656\) −517762. −0.0469754
\(657\) 3.21864e6 0.290910
\(658\) −179100. −0.0161262
\(659\) −882213. −0.0791334 −0.0395667 0.999217i \(-0.512598\pi\)
−0.0395667 + 0.999217i \(0.512598\pi\)
\(660\) 0 0
\(661\) −5.59700e6 −0.498255 −0.249127 0.968471i \(-0.580144\pi\)
−0.249127 + 0.968471i \(0.580144\pi\)
\(662\) 1.25024e7 1.10879
\(663\) −398058. −0.0351692
\(664\) 817128. 0.0719234
\(665\) 0 0
\(666\) 2.65704e6 0.232120
\(667\) −7.62720e6 −0.663820
\(668\) −9.77367e6 −0.847454
\(669\) 8.63644e6 0.746053
\(670\) 0 0
\(671\) −4.83250e6 −0.414348
\(672\) 2.92708e6 0.250041
\(673\) −1.29320e6 −0.110060 −0.0550300 0.998485i \(-0.517525\pi\)
−0.0550300 + 0.998485i \(0.517525\pi\)
\(674\) −9.90040e6 −0.839466
\(675\) 0 0
\(676\) −1.07296e7 −0.903062
\(677\) −1.48798e7 −1.24775 −0.623873 0.781526i \(-0.714442\pi\)
−0.623873 + 0.781526i \(0.714442\pi\)
\(678\) −6.19395e6 −0.517480
\(679\) 299902. 0.0249634
\(680\) 0 0
\(681\) −2.60958e6 −0.215627
\(682\) −2.39575e6 −0.197233
\(683\) −8.62925e6 −0.707817 −0.353909 0.935280i \(-0.615148\pi\)
−0.353909 + 0.935280i \(0.615148\pi\)
\(684\) 943974. 0.0771471
\(685\) 0 0
\(686\) −1.01996e7 −0.827506
\(687\) −1.00045e6 −0.0808729
\(688\) −1.18607e7 −0.955302
\(689\) 8.21310e6 0.659112
\(690\) 0 0
\(691\) −1.79734e7 −1.43198 −0.715988 0.698113i \(-0.754023\pi\)
−0.715988 + 0.698113i \(0.754023\pi\)
\(692\) 691396. 0.0548860
\(693\) 385925. 0.0305260
\(694\) −3.09157e7 −2.43658
\(695\) 0 0
\(696\) −506211. −0.0396103
\(697\) 105638. 0.00823639
\(698\) 3.69167e6 0.286804
\(699\) −4.07894e6 −0.315758
\(700\) 0 0
\(701\) −2.13485e7 −1.64086 −0.820432 0.571744i \(-0.806267\pi\)
−0.820432 + 0.571744i \(0.806267\pi\)
\(702\) −1.30015e6 −0.0995754
\(703\) 1.42256e6 0.108563
\(704\) −4.27181e6 −0.324848
\(705\) 0 0
\(706\) −3.49667e7 −2.64024
\(707\) 4.95992e6 0.373187
\(708\) −645953. −0.0484303
\(709\) 6.90604e6 0.515957 0.257979 0.966151i \(-0.416944\pi\)
0.257979 + 0.966151i \(0.416944\pi\)
\(710\) 0 0
\(711\) 2.88950e6 0.214363
\(712\) −385921. −0.0285298
\(713\) 3.39466e6 0.250076
\(714\) −573571. −0.0421058
\(715\) 0 0
\(716\) −7.62837e6 −0.556095
\(717\) −1.13749e7 −0.826320
\(718\) −6.82951e6 −0.494400
\(719\) −1.71863e7 −1.23983 −0.619913 0.784670i \(-0.712832\pi\)
−0.619913 + 0.784670i \(0.712832\pi\)
\(720\) 0 0
\(721\) 6.50573e6 0.466077
\(722\) −1.90119e7 −1.35732
\(723\) −1.20915e6 −0.0860266
\(724\) −3.02065e6 −0.214167
\(725\) 0 0
\(726\) −1.06451e6 −0.0749566
\(727\) 2.43516e7 1.70880 0.854401 0.519615i \(-0.173925\pi\)
0.854401 + 0.519615i \(0.173925\pi\)
\(728\) 88789.8 0.00620919
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.41992e6 0.167497
\(732\) 1.19566e7 0.824765
\(733\) 1.93102e7 1.32747 0.663737 0.747966i \(-0.268969\pi\)
0.663737 + 0.747966i \(0.268969\pi\)
\(734\) −2.39141e7 −1.63838
\(735\) 0 0
\(736\) 1.14403e7 0.778472
\(737\) 4.65525e6 0.315700
\(738\) 345038. 0.0233199
\(739\) −9.23336e6 −0.621940 −0.310970 0.950420i \(-0.600654\pi\)
−0.310970 + 0.950420i \(0.600654\pi\)
\(740\) 0 0
\(741\) −696094. −0.0465717
\(742\) 1.18345e7 0.789112
\(743\) −8.02302e6 −0.533170 −0.266585 0.963811i \(-0.585895\pi\)
−0.266585 + 0.963811i \(0.585895\pi\)
\(744\) 225300. 0.0149221
\(745\) 0 0
\(746\) −8.26434e6 −0.543702
\(747\) 6.47998e6 0.424885
\(748\) 806379. 0.0526969
\(749\) 517885. 0.0337309
\(750\) 0 0
\(751\) 1.19930e6 0.0775938 0.0387969 0.999247i \(-0.487647\pi\)
0.0387969 + 0.999247i \(0.487647\pi\)
\(752\) 552856. 0.0356507
\(753\) 1.29033e7 0.829302
\(754\) 9.82096e6 0.629109
\(755\) 0 0
\(756\) −954858. −0.0607623
\(757\) 1.40172e7 0.889040 0.444520 0.895769i \(-0.353374\pi\)
0.444520 + 0.895769i \(0.353374\pi\)
\(758\) −3.38820e7 −2.14188
\(759\) 1.50836e6 0.0950389
\(760\) 0 0
\(761\) −1.98659e7 −1.24350 −0.621750 0.783215i \(-0.713578\pi\)
−0.621750 + 0.783215i \(0.713578\pi\)
\(762\) −1.02690e7 −0.640681
\(763\) 1.49514e6 0.0929761
\(764\) −1.65255e7 −1.02429
\(765\) 0 0
\(766\) 1.32721e7 0.817275
\(767\) 476331. 0.0292361
\(768\) −8.64785e6 −0.529060
\(769\) −2.09738e7 −1.27897 −0.639486 0.768803i \(-0.720853\pi\)
−0.639486 + 0.768803i \(0.720853\pi\)
\(770\) 0 0
\(771\) −8.58542e6 −0.520146
\(772\) −9.02668e6 −0.545111
\(773\) −7.69041e6 −0.462914 −0.231457 0.972845i \(-0.574349\pi\)
−0.231457 + 0.972845i \(0.574349\pi\)
\(774\) 7.90403e6 0.474238
\(775\) 0 0
\(776\) 77794.4 0.00463761
\(777\) −1.43896e6 −0.0855062
\(778\) 6.21909e6 0.368365
\(779\) 184731. 0.0109068
\(780\) 0 0
\(781\) −1.66076e6 −0.0974270
\(782\) −2.24177e6 −0.131091
\(783\) −4.01434e6 −0.233997
\(784\) 1.49810e7 0.870466
\(785\) 0 0
\(786\) −1.53027e7 −0.883513
\(787\) 1.39975e6 0.0805589 0.0402794 0.999188i \(-0.487175\pi\)
0.0402794 + 0.999188i \(0.487175\pi\)
\(788\) 6.35256e6 0.364446
\(789\) 2.90052e6 0.165876
\(790\) 0 0
\(791\) 3.35443e6 0.190624
\(792\) 100109. 0.00567100
\(793\) −8.81689e6 −0.497889
\(794\) −3.93131e7 −2.21302
\(795\) 0 0
\(796\) 2.00463e7 1.12138
\(797\) −1.22917e7 −0.685432 −0.342716 0.939439i \(-0.611347\pi\)
−0.342716 + 0.939439i \(0.611347\pi\)
\(798\) −1.00302e6 −0.0557573
\(799\) −112798. −0.00625078
\(800\) 0 0
\(801\) −3.06042e6 −0.168539
\(802\) 5.79290e6 0.318024
\(803\) 4.80810e6 0.263138
\(804\) −1.15181e7 −0.628404
\(805\) 0 0
\(806\) −4.37104e6 −0.236999
\(807\) −7.27863e6 −0.393429
\(808\) 1.28660e6 0.0693292
\(809\) −1.19607e7 −0.642515 −0.321258 0.946992i \(-0.604106\pi\)
−0.321258 + 0.946992i \(0.604106\pi\)
\(810\) 0 0
\(811\) −1.15112e7 −0.614565 −0.307282 0.951618i \(-0.599420\pi\)
−0.307282 + 0.951618i \(0.599420\pi\)
\(812\) 7.21270e6 0.383891
\(813\) 8.18373e6 0.434235
\(814\) 3.96916e6 0.209961
\(815\) 0 0
\(816\) 1.77053e6 0.0930846
\(817\) 4.23177e6 0.221803
\(818\) −3.07062e7 −1.60451
\(819\) 704120. 0.0366806
\(820\) 0 0
\(821\) 2.77684e7 1.43778 0.718890 0.695124i \(-0.244651\pi\)
0.718890 + 0.695124i \(0.244651\pi\)
\(822\) 1.42253e6 0.0734315
\(823\) 2.63447e7 1.35580 0.677898 0.735156i \(-0.262891\pi\)
0.677898 + 0.735156i \(0.262891\pi\)
\(824\) 1.68758e6 0.0865860
\(825\) 0 0
\(826\) 686356. 0.0350025
\(827\) 3.32551e7 1.69081 0.845406 0.534125i \(-0.179359\pi\)
0.845406 + 0.534125i \(0.179359\pi\)
\(828\) −3.73200e6 −0.189176
\(829\) 2.84581e7 1.43820 0.719101 0.694905i \(-0.244554\pi\)
0.719101 + 0.694905i \(0.244554\pi\)
\(830\) 0 0
\(831\) 2.13239e6 0.107119
\(832\) −7.79391e6 −0.390344
\(833\) −3.05654e6 −0.152622
\(834\) −9.63360e6 −0.479594
\(835\) 0 0
\(836\) 1.41013e6 0.0697822
\(837\) 1.78667e6 0.0881519
\(838\) −1.25936e7 −0.619498
\(839\) −9.47931e6 −0.464913 −0.232457 0.972607i \(-0.574676\pi\)
−0.232457 + 0.972607i \(0.574676\pi\)
\(840\) 0 0
\(841\) 9.81196e6 0.478372
\(842\) 3.34101e7 1.62404
\(843\) 4.05701e6 0.196624
\(844\) 3.05425e7 1.47587
\(845\) 0 0
\(846\) −368425. −0.0176980
\(847\) 576505. 0.0276118
\(848\) −3.65312e7 −1.74451
\(849\) 6.15072e6 0.292858
\(850\) 0 0
\(851\) −5.62411e6 −0.266213
\(852\) 4.10907e6 0.193930
\(853\) −2.13145e7 −1.00300 −0.501501 0.865157i \(-0.667219\pi\)
−0.501501 + 0.865157i \(0.667219\pi\)
\(854\) −1.27045e7 −0.596090
\(855\) 0 0
\(856\) 134339. 0.00626640
\(857\) −2.21044e7 −1.02808 −0.514039 0.857767i \(-0.671851\pi\)
−0.514039 + 0.857767i \(0.671851\pi\)
\(858\) −1.94221e6 −0.0900693
\(859\) −7.07193e6 −0.327006 −0.163503 0.986543i \(-0.552279\pi\)
−0.163503 + 0.986543i \(0.552279\pi\)
\(860\) 0 0
\(861\) −186861. −0.00859035
\(862\) −4.24855e7 −1.94748
\(863\) 5.45080e6 0.249134 0.124567 0.992211i \(-0.460246\pi\)
0.124567 + 0.992211i \(0.460246\pi\)
\(864\) 6.02125e6 0.274412
\(865\) 0 0
\(866\) 5.61101e6 0.254241
\(867\) 1.24175e7 0.561029
\(868\) −3.21017e6 −0.144620
\(869\) 4.31642e6 0.193898
\(870\) 0 0
\(871\) 8.49351e6 0.379351
\(872\) 387840. 0.0172727
\(873\) 616924. 0.0273966
\(874\) −3.92023e6 −0.173593
\(875\) 0 0
\(876\) −1.18962e7 −0.523780
\(877\) 2.02823e7 0.890467 0.445234 0.895414i \(-0.353121\pi\)
0.445234 + 0.895414i \(0.353121\pi\)
\(878\) −6.30917e7 −2.76208
\(879\) −1.62961e7 −0.711397
\(880\) 0 0
\(881\) −6.58178e6 −0.285696 −0.142848 0.989745i \(-0.545626\pi\)
−0.142848 + 0.989745i \(0.545626\pi\)
\(882\) −9.98340e6 −0.432123
\(883\) 1.63860e7 0.707247 0.353623 0.935388i \(-0.384949\pi\)
0.353623 + 0.935388i \(0.384949\pi\)
\(884\) 1.47124e6 0.0633217
\(885\) 0 0
\(886\) 4.39378e7 1.88042
\(887\) 2.92929e6 0.125013 0.0625063 0.998045i \(-0.480091\pi\)
0.0625063 + 0.998045i \(0.480091\pi\)
\(888\) −373267. −0.0158850
\(889\) 5.56136e6 0.236008
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −3.19206e7 −1.34326
\(893\) −197252. −0.00827739
\(894\) 1.11473e7 0.466472
\(895\) 0 0
\(896\) −823047. −0.0342495
\(897\) 2.75201e6 0.114201
\(898\) 4.41317e7 1.82625
\(899\) −1.34960e7 −0.556936
\(900\) 0 0
\(901\) 7.45337e6 0.305873
\(902\) 515427. 0.0210936
\(903\) −4.28056e6 −0.174695
\(904\) 870140. 0.0354134
\(905\) 0 0
\(906\) −1.80425e7 −0.730256
\(907\) 7.88236e6 0.318154 0.159077 0.987266i \(-0.449148\pi\)
0.159077 + 0.987266i \(0.449148\pi\)
\(908\) 9.64512e6 0.388234
\(909\) 1.02030e7 0.409561
\(910\) 0 0
\(911\) −2.16713e7 −0.865146 −0.432573 0.901599i \(-0.642394\pi\)
−0.432573 + 0.901599i \(0.642394\pi\)
\(912\) 3.09617e6 0.123264
\(913\) 9.67996e6 0.384323
\(914\) −2.48130e7 −0.982458
\(915\) 0 0
\(916\) 3.69769e6 0.145610
\(917\) 8.28745e6 0.325460
\(918\) −1.17989e6 −0.0462097
\(919\) 4.32037e7 1.68746 0.843728 0.536771i \(-0.180356\pi\)
0.843728 + 0.536771i \(0.180356\pi\)
\(920\) 0 0
\(921\) −2.70774e7 −1.05186
\(922\) 3.11545e7 1.20696
\(923\) −3.03006e6 −0.117070
\(924\) −1.42639e6 −0.0549616
\(925\) 0 0
\(926\) −2.58996e7 −0.992580
\(927\) 1.33829e7 0.511504
\(928\) −4.54827e7 −1.73371
\(929\) 3.70288e7 1.40767 0.703834 0.710364i \(-0.251470\pi\)
0.703834 + 0.710364i \(0.251470\pi\)
\(930\) 0 0
\(931\) −5.34505e6 −0.202105
\(932\) 1.50759e7 0.568518
\(933\) 1.83675e7 0.690790
\(934\) 1.97080e7 0.739224
\(935\) 0 0
\(936\) 182648. 0.00681438
\(937\) −1.22598e7 −0.456180 −0.228090 0.973640i \(-0.573248\pi\)
−0.228090 + 0.973640i \(0.573248\pi\)
\(938\) 1.22385e7 0.454173
\(939\) −1.56546e7 −0.579399
\(940\) 0 0
\(941\) −1.25646e7 −0.462565 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(942\) −1.31377e7 −0.482384
\(943\) −730335. −0.0267450
\(944\) −2.11868e6 −0.0773812
\(945\) 0 0
\(946\) 1.18073e7 0.428964
\(947\) −3.16122e7 −1.14546 −0.572730 0.819744i \(-0.694116\pi\)
−0.572730 + 0.819744i \(0.694116\pi\)
\(948\) −1.06797e7 −0.385957
\(949\) 8.77237e6 0.316192
\(950\) 0 0
\(951\) 960334. 0.0344327
\(952\) 80576.6 0.00288148
\(953\) 2.84509e7 1.01476 0.507380 0.861722i \(-0.330614\pi\)
0.507380 + 0.861722i \(0.330614\pi\)
\(954\) 2.43445e7 0.866024
\(955\) 0 0
\(956\) 4.20419e7 1.48778
\(957\) −5.99673e6 −0.211658
\(958\) 3.79038e7 1.33435
\(959\) −770395. −0.0270500
\(960\) 0 0
\(961\) −2.26225e7 −0.790190
\(962\) 7.24174e6 0.252293
\(963\) 1.06533e6 0.0370186
\(964\) 4.46905e6 0.154890
\(965\) 0 0
\(966\) 3.96543e6 0.136725
\(967\) 4.73439e7 1.62816 0.814081 0.580751i \(-0.197241\pi\)
0.814081 + 0.580751i \(0.197241\pi\)
\(968\) 149545. 0.00512961
\(969\) −631704. −0.0216124
\(970\) 0 0
\(971\) −4.33248e7 −1.47465 −0.737324 0.675539i \(-0.763911\pi\)
−0.737324 + 0.675539i \(0.763911\pi\)
\(972\) −1.96423e6 −0.0666846
\(973\) 5.21723e6 0.176668
\(974\) −4.63152e7 −1.56432
\(975\) 0 0
\(976\) 3.92168e7 1.31780
\(977\) 4.64678e7 1.55745 0.778727 0.627363i \(-0.215866\pi\)
0.778727 + 0.627363i \(0.215866\pi\)
\(978\) −1.27998e7 −0.427915
\(979\) −4.57174e6 −0.152449
\(980\) 0 0
\(981\) 3.07564e6 0.102038
\(982\) −5.60553e6 −0.185498
\(983\) 1.02443e7 0.338141 0.169071 0.985604i \(-0.445923\pi\)
0.169071 + 0.985604i \(0.445923\pi\)
\(984\) −48471.7 −0.00159588
\(985\) 0 0
\(986\) 8.91250e6 0.291949
\(987\) 199527. 0.00651940
\(988\) 2.57279e6 0.0838517
\(989\) −1.67303e7 −0.543892
\(990\) 0 0
\(991\) −6.52638e6 −0.211100 −0.105550 0.994414i \(-0.533660\pi\)
−0.105550 + 0.994414i \(0.533660\pi\)
\(992\) 2.02431e7 0.653127
\(993\) −1.39283e7 −0.448256
\(994\) −4.36608e6 −0.140161
\(995\) 0 0
\(996\) −2.39502e7 −0.765000
\(997\) −1.08535e7 −0.345804 −0.172902 0.984939i \(-0.555314\pi\)
−0.172902 + 0.984939i \(0.555314\pi\)
\(998\) 9.94904e6 0.316195
\(999\) −2.96008e6 −0.0938403
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 825.6.a.i.1.3 3
5.4 even 2 165.6.a.b.1.1 3
15.14 odd 2 495.6.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.b.1.1 3 5.4 even 2
495.6.a.d.1.3 3 15.14 odd 2
825.6.a.i.1.3 3 1.1 even 1 trivial