Properties

Label 224.6.p.a.31.11
Level $224$
Weight $6$
Character 224.31
Analytic conductor $35.926$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,6,Mod(31,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.31");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 224.p (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(35.9259756381\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.11
Character \(\chi\) \(=\) 224.31
Dual form 224.6.p.a.159.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-8.16376 + 14.1401i) q^{3} +(19.2452 - 11.1112i) q^{5} +(103.004 - 78.7217i) q^{7} +(-11.7941 - 20.4280i) q^{9} +(-230.097 - 132.846i) q^{11} -31.5899i q^{13} +362.837i q^{15} +(110.876 + 64.0143i) q^{17} +(-1375.26 - 2382.02i) q^{19} +(272.226 + 2099.15i) q^{21} +(-2222.74 + 1283.30i) q^{23} +(-1315.58 + 2278.66i) q^{25} -3582.45 q^{27} +1603.73 q^{29} +(-349.938 + 606.110i) q^{31} +(3756.91 - 2169.05i) q^{33} +(1107.64 - 2659.51i) q^{35} +(-2421.84 - 4194.76i) q^{37} +(446.683 + 257.893i) q^{39} -5910.34i q^{41} -15217.5i q^{43} +(-453.959 - 262.093i) q^{45} +(-49.5838 - 85.8817i) q^{47} +(4412.79 - 16217.4i) q^{49} +(-1810.33 + 1045.20i) q^{51} +(17857.4 - 30929.9i) q^{53} -5904.34 q^{55} +44909.2 q^{57} +(3443.80 - 5964.84i) q^{59} +(-3111.85 + 1796.62i) q^{61} +(-2822.97 - 1175.72i) q^{63} +(-351.002 - 607.953i) q^{65} +(-42290.4 - 24416.4i) q^{67} -41906.2i q^{69} -31698.2i q^{71} +(27063.6 + 15625.2i) q^{73} +(-21480.2 - 37204.8i) q^{75} +(-34158.9 + 4429.84i) q^{77} +(-20637.0 + 11914.8i) q^{79} +(32112.3 - 55620.1i) q^{81} +18288.6 q^{83} +2845.11 q^{85} +(-13092.5 + 22676.8i) q^{87} +(60686.8 - 35037.5i) q^{89} +(-2486.81 - 3253.90i) q^{91} +(-5713.62 - 9896.28i) q^{93} +(-52934.2 - 30561.6i) q^{95} +122256. i q^{97} +6267.22i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q - 3240 q^{9} + 6856 q^{21} + 27728 q^{25} - 15920 q^{29} - 28296 q^{33} - 2152 q^{37} + 35400 q^{45} + 29280 q^{49} - 17512 q^{53} - 108368 q^{57} + 37704 q^{61} + 18216 q^{65} + 157704 q^{73} + 28224 q^{77}+ \cdots + 39400 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −8.16376 + 14.1401i −0.523706 + 0.907085i 0.475913 + 0.879492i \(0.342118\pi\)
−0.999619 + 0.0275929i \(0.991216\pi\)
\(4\) 0 0
\(5\) 19.2452 11.1112i 0.344268 0.198763i −0.317890 0.948128i \(-0.602974\pi\)
0.662158 + 0.749364i \(0.269641\pi\)
\(6\) 0 0
\(7\) 103.004 78.7217i 0.794530 0.607224i
\(8\) 0 0
\(9\) −11.7941 20.4280i −0.0485354 0.0840658i
\(10\) 0 0
\(11\) −230.097 132.846i −0.573362 0.331031i 0.185129 0.982714i \(-0.440730\pi\)
−0.758491 + 0.651684i \(0.774063\pi\)
\(12\) 0 0
\(13\) 31.5899i 0.0518430i −0.999664 0.0259215i \(-0.991748\pi\)
0.999664 0.0259215i \(-0.00825199\pi\)
\(14\) 0 0
\(15\) 362.837i 0.416374i
\(16\) 0 0
\(17\) 110.876 + 64.0143i 0.0930498 + 0.0537223i 0.545803 0.837914i \(-0.316225\pi\)
−0.452753 + 0.891636i \(0.649558\pi\)
\(18\) 0 0
\(19\) −1375.26 2382.02i −0.873979 1.51378i −0.857847 0.513905i \(-0.828198\pi\)
−0.0161318 0.999870i \(-0.505135\pi\)
\(20\) 0 0
\(21\) 272.226 + 2099.15i 0.134704 + 1.03871i
\(22\) 0 0
\(23\) −2222.74 + 1283.30i −0.876130 + 0.505834i −0.869381 0.494143i \(-0.835482\pi\)
−0.00674965 + 0.999977i \(0.502148\pi\)
\(24\) 0 0
\(25\) −1315.58 + 2278.66i −0.420986 + 0.729170i
\(26\) 0 0
\(27\) −3582.45 −0.945738
\(28\) 0 0
\(29\) 1603.73 0.354108 0.177054 0.984201i \(-0.443343\pi\)
0.177054 + 0.984201i \(0.443343\pi\)
\(30\) 0 0
\(31\) −349.938 + 606.110i −0.0654014 + 0.113278i −0.896872 0.442290i \(-0.854166\pi\)
0.831471 + 0.555569i \(0.187499\pi\)
\(32\) 0 0
\(33\) 3756.91 2169.05i 0.600546 0.346725i
\(34\) 0 0
\(35\) 1107.64 2659.51i 0.152838 0.366971i
\(36\) 0 0
\(37\) −2421.84 4194.76i −0.290832 0.503735i 0.683175 0.730255i \(-0.260599\pi\)
−0.974007 + 0.226519i \(0.927265\pi\)
\(38\) 0 0
\(39\) 446.683 + 257.893i 0.0470260 + 0.0271505i
\(40\) 0 0
\(41\) 5910.34i 0.549101i −0.961573 0.274551i \(-0.911471\pi\)
0.961573 0.274551i \(-0.0885291\pi\)
\(42\) 0 0
\(43\) 15217.5i 1.25508i −0.778585 0.627540i \(-0.784062\pi\)
0.778585 0.627540i \(-0.215938\pi\)
\(44\) 0 0
\(45\) −453.959 262.093i −0.0334184 0.0192941i
\(46\) 0 0
\(47\) −49.5838 85.8817i −0.00327413 0.00567095i 0.864384 0.502833i \(-0.167709\pi\)
−0.867658 + 0.497162i \(0.834376\pi\)
\(48\) 0 0
\(49\) 4412.79 16217.4i 0.262557 0.964916i
\(50\) 0 0
\(51\) −1810.33 + 1045.20i −0.0974614 + 0.0562694i
\(52\) 0 0
\(53\) 17857.4 30929.9i 0.873228 1.51248i 0.0145896 0.999894i \(-0.495356\pi\)
0.858638 0.512582i \(-0.171311\pi\)
\(54\) 0 0
\(55\) −5904.34 −0.263187
\(56\) 0 0
\(57\) 44909.2 1.83083
\(58\) 0 0
\(59\) 3443.80 5964.84i 0.128798 0.223084i −0.794413 0.607378i \(-0.792222\pi\)
0.923211 + 0.384293i \(0.125555\pi\)
\(60\) 0 0
\(61\) −3111.85 + 1796.62i −0.107076 + 0.0618205i −0.552582 0.833459i \(-0.686357\pi\)
0.445505 + 0.895279i \(0.353024\pi\)
\(62\) 0 0
\(63\) −2822.97 1175.72i −0.0896097 0.0373209i
\(64\) 0 0
\(65\) −351.002 607.953i −0.0103045 0.0178479i
\(66\) 0 0
\(67\) −42290.4 24416.4i −1.15095 0.664499i −0.201829 0.979421i \(-0.564689\pi\)
−0.949118 + 0.314921i \(0.898022\pi\)
\(68\) 0 0
\(69\) 41906.2i 1.05963i
\(70\) 0 0
\(71\) 31698.2i 0.746257i −0.927780 0.373128i \(-0.878285\pi\)
0.927780 0.373128i \(-0.121715\pi\)
\(72\) 0 0
\(73\) 27063.6 + 15625.2i 0.594399 + 0.343176i 0.766835 0.641844i \(-0.221831\pi\)
−0.172436 + 0.985021i \(0.555164\pi\)
\(74\) 0 0
\(75\) −21480.2 37204.8i −0.440946 0.763741i
\(76\) 0 0
\(77\) −34158.9 + 4429.84i −0.656563 + 0.0851455i
\(78\) 0 0
\(79\) −20637.0 + 11914.8i −0.372031 + 0.214792i −0.674345 0.738416i \(-0.735574\pi\)
0.302314 + 0.953208i \(0.402241\pi\)
\(80\) 0 0
\(81\) 32112.3 55620.1i 0.543824 0.941931i
\(82\) 0 0
\(83\) 18288.6 0.291398 0.145699 0.989329i \(-0.453457\pi\)
0.145699 + 0.989329i \(0.453457\pi\)
\(84\) 0 0
\(85\) 2845.11 0.0427121
\(86\) 0 0
\(87\) −13092.5 + 22676.8i −0.185449 + 0.321206i
\(88\) 0 0
\(89\) 60686.8 35037.5i 0.812118 0.468876i −0.0355730 0.999367i \(-0.511326\pi\)
0.847691 + 0.530491i \(0.177992\pi\)
\(90\) 0 0
\(91\) −2486.81 3253.90i −0.0314803 0.0411908i
\(92\) 0 0
\(93\) −5713.62 9896.28i −0.0685021 0.118649i
\(94\) 0 0
\(95\) −52934.2 30561.6i −0.601766 0.347430i
\(96\) 0 0
\(97\) 122256.i 1.31929i 0.751579 + 0.659644i \(0.229293\pi\)
−0.751579 + 0.659644i \(0.770707\pi\)
\(98\) 0 0
\(99\) 6267.22i 0.0642668i
\(100\) 0 0
\(101\) −91505.7 52830.9i −0.892575 0.515329i −0.0177913 0.999842i \(-0.505663\pi\)
−0.874784 + 0.484513i \(0.838997\pi\)
\(102\) 0 0
\(103\) −90044.5 155962.i −0.836304 1.44852i −0.892964 0.450127i \(-0.851379\pi\)
0.0566605 0.998394i \(-0.481955\pi\)
\(104\) 0 0
\(105\) 28563.1 + 37373.8i 0.252832 + 0.330822i
\(106\) 0 0
\(107\) −130537. + 75365.5i −1.10223 + 0.636375i −0.936807 0.349847i \(-0.886234\pi\)
−0.165427 + 0.986222i \(0.552900\pi\)
\(108\) 0 0
\(109\) −39532.1 + 68471.7i −0.318701 + 0.552007i −0.980217 0.197924i \(-0.936580\pi\)
0.661516 + 0.749931i \(0.269913\pi\)
\(110\) 0 0
\(111\) 79085.5 0.609241
\(112\) 0 0
\(113\) 100179. 0.738040 0.369020 0.929421i \(-0.379693\pi\)
0.369020 + 0.929421i \(0.379693\pi\)
\(114\) 0 0
\(115\) −28518.0 + 49394.6i −0.201082 + 0.348285i
\(116\) 0 0
\(117\) −645.318 + 372.575i −0.00435822 + 0.00251622i
\(118\) 0 0
\(119\) 16460.0 2134.60i 0.106552 0.0138181i
\(120\) 0 0
\(121\) −45229.2 78339.2i −0.280837 0.486425i
\(122\) 0 0
\(123\) 83572.5 + 48250.6i 0.498082 + 0.287568i
\(124\) 0 0
\(125\) 127916.i 0.732233i
\(126\) 0 0
\(127\) 133657.i 0.735329i −0.929959 0.367664i \(-0.880158\pi\)
0.929959 0.367664i \(-0.119842\pi\)
\(128\) 0 0
\(129\) 215176. + 124232.i 1.13846 + 0.657292i
\(130\) 0 0
\(131\) 64841.1 + 112308.i 0.330120 + 0.571785i 0.982535 0.186077i \(-0.0595773\pi\)
−0.652415 + 0.757862i \(0.726244\pi\)
\(132\) 0 0
\(133\) −329174. 137096.i −1.61360 0.672039i
\(134\) 0 0
\(135\) −68944.9 + 39805.4i −0.325588 + 0.187978i
\(136\) 0 0
\(137\) 23507.7 40716.5i 0.107006 0.185340i −0.807550 0.589799i \(-0.799207\pi\)
0.914556 + 0.404459i \(0.132540\pi\)
\(138\) 0 0
\(139\) 323808. 1.42151 0.710756 0.703439i \(-0.248353\pi\)
0.710756 + 0.703439i \(0.248353\pi\)
\(140\) 0 0
\(141\) 1619.16 0.00685871
\(142\) 0 0
\(143\) −4196.61 + 7268.73i −0.0171616 + 0.0297248i
\(144\) 0 0
\(145\) 30864.0 17819.4i 0.121908 0.0703837i
\(146\) 0 0
\(147\) 193289. + 194792.i 0.737759 + 0.743494i
\(148\) 0 0
\(149\) −88736.6 153696.i −0.327444 0.567150i 0.654560 0.756010i \(-0.272854\pi\)
−0.982004 + 0.188860i \(0.939521\pi\)
\(150\) 0 0
\(151\) 132002. + 76211.4i 0.471128 + 0.272006i 0.716712 0.697370i \(-0.245646\pi\)
−0.245584 + 0.969375i \(0.578980\pi\)
\(152\) 0 0
\(153\) 3019.97i 0.0104297i
\(154\) 0 0
\(155\) 15552.9i 0.0519976i
\(156\) 0 0
\(157\) −269397. 155536.i −0.872254 0.503596i −0.00415732 0.999991i \(-0.501323\pi\)
−0.868097 + 0.496395i \(0.834657\pi\)
\(158\) 0 0
\(159\) 291567. + 505008.i 0.914629 + 1.58418i
\(160\) 0 0
\(161\) −127928. + 307163.i −0.388957 + 0.933908i
\(162\) 0 0
\(163\) 152716. 88170.5i 0.450210 0.259929i −0.257709 0.966223i \(-0.582968\pi\)
0.707919 + 0.706294i \(0.249634\pi\)
\(164\) 0 0
\(165\) 48201.6 83487.6i 0.137832 0.238733i
\(166\) 0 0
\(167\) 469993. 1.30407 0.652035 0.758189i \(-0.273916\pi\)
0.652035 + 0.758189i \(0.273916\pi\)
\(168\) 0 0
\(169\) 370295. 0.997312
\(170\) 0 0
\(171\) −32439.9 + 56187.6i −0.0848378 + 0.146943i
\(172\) 0 0
\(173\) −674698. + 389537.i −1.71393 + 0.989540i −0.784839 + 0.619700i \(0.787254\pi\)
−0.929095 + 0.369841i \(0.879412\pi\)
\(174\) 0 0
\(175\) 43868.9 + 338276.i 0.108283 + 0.834981i
\(176\) 0 0
\(177\) 56228.8 + 97391.1i 0.134904 + 0.233661i
\(178\) 0 0
\(179\) −535593. 309225.i −1.24940 0.721343i −0.278412 0.960462i \(-0.589808\pi\)
−0.970990 + 0.239119i \(0.923141\pi\)
\(180\) 0 0
\(181\) 178785.i 0.405635i 0.979217 + 0.202818i \(0.0650098\pi\)
−0.979217 + 0.202818i \(0.934990\pi\)
\(182\) 0 0
\(183\) 58668.9i 0.129503i
\(184\) 0 0
\(185\) −93217.6 53819.2i −0.200248 0.115613i
\(186\) 0 0
\(187\) −17008.1 29459.0i −0.0355675 0.0616047i
\(188\) 0 0
\(189\) −369008. + 282017.i −0.751418 + 0.574275i
\(190\) 0 0
\(191\) 9396.44 5425.04i 0.0186372 0.0107602i −0.490652 0.871355i \(-0.663242\pi\)
0.509290 + 0.860595i \(0.329908\pi\)
\(192\) 0 0
\(193\) −473563. + 820235.i −0.915133 + 1.58506i −0.108427 + 0.994104i \(0.534581\pi\)
−0.806706 + 0.590953i \(0.798752\pi\)
\(194\) 0 0
\(195\) 11462.0 0.0215861
\(196\) 0 0
\(197\) −81454.8 −0.149538 −0.0747689 0.997201i \(-0.523822\pi\)
−0.0747689 + 0.997201i \(0.523822\pi\)
\(198\) 0 0
\(199\) −491764. + 851761.i −0.880287 + 1.52470i −0.0292649 + 0.999572i \(0.509317\pi\)
−0.851022 + 0.525130i \(0.824017\pi\)
\(200\) 0 0
\(201\) 690498. 398659.i 1.20551 0.696004i
\(202\) 0 0
\(203\) 165191. 126248.i 0.281350 0.215023i
\(204\) 0 0
\(205\) −65671.0 113745.i −0.109141 0.189038i
\(206\) 0 0
\(207\) 52430.4 + 30270.7i 0.0850467 + 0.0491017i
\(208\) 0 0
\(209\) 730793.i 1.15725i
\(210\) 0 0
\(211\) 427608.i 0.661210i −0.943769 0.330605i \(-0.892747\pi\)
0.943769 0.330605i \(-0.107253\pi\)
\(212\) 0 0
\(213\) 448214. + 258776.i 0.676918 + 0.390819i
\(214\) 0 0
\(215\) −169084. 292863.i −0.249464 0.432084i
\(216\) 0 0
\(217\) 11668.9 + 89979.7i 0.0168221 + 0.129716i
\(218\) 0 0
\(219\) −441881. + 255120.i −0.622580 + 0.359447i
\(220\) 0 0
\(221\) 2022.21 3502.56i 0.00278513 0.00482398i
\(222\) 0 0
\(223\) 60238.8 0.0811174 0.0405587 0.999177i \(-0.487086\pi\)
0.0405587 + 0.999177i \(0.487086\pi\)
\(224\) 0 0
\(225\) 62064.5 0.0817310
\(226\) 0 0
\(227\) 27615.1 47830.8i 0.0355699 0.0616088i −0.847692 0.530488i \(-0.822009\pi\)
0.883262 + 0.468879i \(0.155342\pi\)
\(228\) 0 0
\(229\) −3063.24 + 1768.56i −0.00386004 + 0.00222859i −0.501929 0.864909i \(-0.667376\pi\)
0.498069 + 0.867138i \(0.334043\pi\)
\(230\) 0 0
\(231\) 216227. 519172.i 0.266612 0.640150i
\(232\) 0 0
\(233\) −599259. 1.03795e6i −0.723143 1.25252i −0.959734 0.280912i \(-0.909363\pi\)
0.236590 0.971610i \(-0.423970\pi\)
\(234\) 0 0
\(235\) −1908.50 1101.87i −0.00225435 0.00130155i
\(236\) 0 0
\(237\) 389078.i 0.449952i
\(238\) 0 0
\(239\) 531464.i 0.601837i −0.953650 0.300918i \(-0.902707\pi\)
0.953650 0.300918i \(-0.0972932\pi\)
\(240\) 0 0
\(241\) −487249. 281313.i −0.540391 0.311995i 0.204846 0.978794i \(-0.434331\pi\)
−0.745237 + 0.666799i \(0.767664\pi\)
\(242\) 0 0
\(243\) 89046.0 + 154232.i 0.0967384 + 0.167556i
\(244\) 0 0
\(245\) −95269.3 361137.i −0.101400 0.384377i
\(246\) 0 0
\(247\) −75247.8 + 43444.3i −0.0784786 + 0.0453096i
\(248\) 0 0
\(249\) −149304. + 258602.i −0.152607 + 0.264323i
\(250\) 0 0
\(251\) 1.06837e6 1.07037 0.535187 0.844734i \(-0.320241\pi\)
0.535187 + 0.844734i \(0.320241\pi\)
\(252\) 0 0
\(253\) 681926. 0.669786
\(254\) 0 0
\(255\) −23226.8 + 40229.9i −0.0223686 + 0.0387435i
\(256\) 0 0
\(257\) 1.16285e6 671372.i 1.09822 0.634060i 0.162470 0.986714i \(-0.448054\pi\)
0.935754 + 0.352654i \(0.114721\pi\)
\(258\) 0 0
\(259\) −579679. 241427.i −0.536955 0.223633i
\(260\) 0 0
\(261\) −18914.6 32761.0i −0.0171868 0.0297684i
\(262\) 0 0
\(263\) −1.39724e6 806697.i −1.24561 0.719153i −0.275378 0.961336i \(-0.588803\pi\)
−0.970231 + 0.242183i \(0.922136\pi\)
\(264\) 0 0
\(265\) 793667.i 0.694263i
\(266\) 0 0
\(267\) 1.14415e6i 0.982213i
\(268\) 0 0
\(269\) 1.89088e6 + 1.09170e6i 1.59325 + 0.919864i 0.992744 + 0.120245i \(0.0383679\pi\)
0.600507 + 0.799619i \(0.294965\pi\)
\(270\) 0 0
\(271\) 203437. + 352363.i 0.168270 + 0.291452i 0.937812 0.347144i \(-0.112849\pi\)
−0.769542 + 0.638597i \(0.779515\pi\)
\(272\) 0 0
\(273\) 66312.0 8599.58i 0.0538500 0.00698346i
\(274\) 0 0
\(275\) 605422. 349541.i 0.482755 0.278719i
\(276\) 0 0
\(277\) 254717. 441182.i 0.199461 0.345477i −0.748893 0.662691i \(-0.769414\pi\)
0.948354 + 0.317215i \(0.102748\pi\)
\(278\) 0 0
\(279\) 16508.8 0.0126971
\(280\) 0 0
\(281\) −1.11878e6 −0.845240 −0.422620 0.906307i \(-0.638889\pi\)
−0.422620 + 0.906307i \(0.638889\pi\)
\(282\) 0 0
\(283\) −134959. + 233755.i −0.100169 + 0.173498i −0.911754 0.410736i \(-0.865272\pi\)
0.811585 + 0.584234i \(0.198605\pi\)
\(284\) 0 0
\(285\) 864285. 498995.i 0.630296 0.363902i
\(286\) 0 0
\(287\) −465272. 608790.i −0.333428 0.436278i
\(288\) 0 0
\(289\) −701733. 1.21544e6i −0.494228 0.856028i
\(290\) 0 0
\(291\) −1.72870e6 998066.i −1.19671 0.690918i
\(292\) 0 0
\(293\) 1.92194e6i 1.30789i −0.756542 0.653945i \(-0.773113\pi\)
0.756542 0.653945i \(-0.226887\pi\)
\(294\) 0 0
\(295\) 153059.i 0.102401i
\(296\) 0 0
\(297\) 824311. + 475916.i 0.542250 + 0.313068i
\(298\) 0 0
\(299\) 40539.3 + 70216.1i 0.0262239 + 0.0454212i
\(300\) 0 0
\(301\) −1.19794e6 1.56747e6i −0.762115 0.997199i
\(302\) 0 0
\(303\) 1.49406e6 862597.i 0.934894 0.539761i
\(304\) 0 0
\(305\) −39925.3 + 69152.7i −0.0245753 + 0.0425657i
\(306\) 0 0
\(307\) −940708. −0.569651 −0.284825 0.958579i \(-0.591936\pi\)
−0.284825 + 0.958579i \(0.591936\pi\)
\(308\) 0 0
\(309\) 2.94041e6 1.75191
\(310\) 0 0
\(311\) 365640. 633306.i 0.214364 0.371290i −0.738712 0.674022i \(-0.764565\pi\)
0.953076 + 0.302732i \(0.0978987\pi\)
\(312\) 0 0
\(313\) 271139. 156542.i 0.156434 0.0903171i −0.419740 0.907645i \(-0.637879\pi\)
0.576173 + 0.817327i \(0.304545\pi\)
\(314\) 0 0
\(315\) −67392.2 + 8739.66i −0.0382678 + 0.00496270i
\(316\) 0 0
\(317\) 1.65827e6 + 2.87221e6i 0.926846 + 1.60534i 0.788565 + 0.614951i \(0.210824\pi\)
0.138281 + 0.990393i \(0.455842\pi\)
\(318\) 0 0
\(319\) −369013. 213050.i −0.203032 0.117221i
\(320\) 0 0
\(321\) 2.46107e6i 1.33309i
\(322\) 0 0
\(323\) 352145.i 0.187809i
\(324\) 0 0
\(325\) 71982.5 + 41559.1i 0.0378023 + 0.0218252i
\(326\) 0 0
\(327\) −645462. 1.11797e6i −0.333812 0.578179i
\(328\) 0 0
\(329\) −11868.1 4942.87i −0.00604493 0.00251761i
\(330\) 0 0
\(331\) 908606. 524584.i 0.455833 0.263175i −0.254458 0.967084i \(-0.581897\pi\)
0.710290 + 0.703909i \(0.248564\pi\)
\(332\) 0 0
\(333\) −57127.0 + 98946.8i −0.0282313 + 0.0488980i
\(334\) 0 0
\(335\) −1.08518e6 −0.528312
\(336\) 0 0
\(337\) −2.90507e6 −1.39342 −0.696710 0.717353i \(-0.745353\pi\)
−0.696710 + 0.717353i \(0.745353\pi\)
\(338\) 0 0
\(339\) −817837. + 1.41653e6i −0.386516 + 0.669465i
\(340\) 0 0
\(341\) 161039. 92976.0i 0.0749973 0.0432997i
\(342\) 0 0
\(343\) −822120. 2.01784e6i −0.377311 0.926086i
\(344\) 0 0
\(345\) −465628. 806492.i −0.210616 0.364798i
\(346\) 0 0
\(347\) −391122. 225815.i −0.174377 0.100677i 0.410271 0.911964i \(-0.365434\pi\)
−0.584648 + 0.811287i \(0.698767\pi\)
\(348\) 0 0
\(349\) 1.27258e6i 0.559269i −0.960107 0.279634i \(-0.909787\pi\)
0.960107 0.279634i \(-0.0902132\pi\)
\(350\) 0 0
\(351\) 113169.i 0.0490299i
\(352\) 0 0
\(353\) 1.73409e6 + 1.00118e6i 0.740686 + 0.427636i 0.822319 0.569027i \(-0.192680\pi\)
−0.0816324 + 0.996663i \(0.526013\pi\)
\(354\) 0 0
\(355\) −352205. 610037.i −0.148328 0.256912i
\(356\) 0 0
\(357\) −104193. + 250172.i −0.0432679 + 0.103889i
\(358\) 0 0
\(359\) −1.09761e6 + 633703.i −0.449480 + 0.259507i −0.707610 0.706603i \(-0.750227\pi\)
0.258131 + 0.966110i \(0.416893\pi\)
\(360\) 0 0
\(361\) −2.54463e6 + 4.40743e6i −1.02768 + 1.77999i
\(362\) 0 0
\(363\) 1.47696e6 0.588305
\(364\) 0 0
\(365\) 694457. 0.272843
\(366\) 0 0
\(367\) −2.18287e6 + 3.78084e6i −0.845985 + 1.46529i 0.0387779 + 0.999248i \(0.487654\pi\)
−0.884763 + 0.466041i \(0.845680\pi\)
\(368\) 0 0
\(369\) −120736. + 69707.1i −0.0461607 + 0.0266509i
\(370\) 0 0
\(371\) −595464. 4.59167e6i −0.224606 1.73195i
\(372\) 0 0
\(373\) 1.74926e6 + 3.02980e6i 0.651001 + 1.12757i 0.982880 + 0.184245i \(0.0589840\pi\)
−0.331879 + 0.943322i \(0.607683\pi\)
\(374\) 0 0
\(375\) −1.80874e6 1.04427e6i −0.664198 0.383475i
\(376\) 0 0
\(377\) 50661.7i 0.0183580i
\(378\) 0 0
\(379\) 4.42455e6i 1.58224i 0.611663 + 0.791118i \(0.290501\pi\)
−0.611663 + 0.791118i \(0.709499\pi\)
\(380\) 0 0
\(381\) 1.88991e6 + 1.09114e6i 0.667006 + 0.385096i
\(382\) 0 0
\(383\) 2.38842e6 + 4.13686e6i 0.831980 + 1.44103i 0.896466 + 0.443113i \(0.146126\pi\)
−0.0644852 + 0.997919i \(0.520541\pi\)
\(384\) 0 0
\(385\) −608172. + 464799.i −0.209110 + 0.159814i
\(386\) 0 0
\(387\) −310862. + 179476.i −0.105509 + 0.0609158i
\(388\) 0 0
\(389\) −1.27234e6 + 2.20375e6i −0.426312 + 0.738395i −0.996542 0.0830906i \(-0.973521\pi\)
0.570230 + 0.821485i \(0.306854\pi\)
\(390\) 0 0
\(391\) −328598. −0.108698
\(392\) 0 0
\(393\) −2.11739e6 −0.691544
\(394\) 0 0
\(395\) −264775. + 458604.i −0.0853856 + 0.147892i
\(396\) 0 0
\(397\) −988211. + 570544.i −0.314683 + 0.181682i −0.649020 0.760771i \(-0.724821\pi\)
0.334337 + 0.942454i \(0.391488\pi\)
\(398\) 0 0
\(399\) 4.62584e6 3.53533e6i 1.45465 1.11173i
\(400\) 0 0
\(401\) −2.23541e6 3.87184e6i −0.694217 1.20242i −0.970444 0.241327i \(-0.922417\pi\)
0.276227 0.961093i \(-0.410916\pi\)
\(402\) 0 0
\(403\) 19147.0 + 11054.5i 0.00587269 + 0.00339060i
\(404\) 0 0
\(405\) 1.42722e6i 0.432369i
\(406\) 0 0
\(407\) 1.28693e6i 0.385097i
\(408\) 0 0
\(409\) −497871. 287446.i −0.147167 0.0849666i 0.424609 0.905377i \(-0.360412\pi\)
−0.571775 + 0.820410i \(0.693745\pi\)
\(410\) 0 0
\(411\) 383823. + 664801.i 0.112080 + 0.194127i
\(412\) 0 0
\(413\) −114836. 885507.i −0.0331285 0.255456i
\(414\) 0 0
\(415\) 351968. 203209.i 0.100319 0.0579192i
\(416\) 0 0
\(417\) −2.64349e6 + 4.57866e6i −0.744454 + 1.28943i
\(418\) 0 0
\(419\) 2.06791e6 0.575434 0.287717 0.957715i \(-0.407104\pi\)
0.287717 + 0.957715i \(0.407104\pi\)
\(420\) 0 0
\(421\) −4.36685e6 −1.20078 −0.600390 0.799708i \(-0.704988\pi\)
−0.600390 + 0.799708i \(0.704988\pi\)
\(422\) 0 0
\(423\) −1169.59 + 2025.80i −0.000317822 + 0.000550484i
\(424\) 0 0
\(425\) −291733. + 168432.i −0.0783454 + 0.0452327i
\(426\) 0 0
\(427\) −179100. + 430030.i −0.0475364 + 0.114138i
\(428\) 0 0
\(429\) −68520.2 118680.i −0.0179753 0.0311341i
\(430\) 0 0
\(431\) 5.19547e6 + 2.99961e6i 1.34720 + 0.777806i 0.987852 0.155398i \(-0.0496659\pi\)
0.359348 + 0.933204i \(0.382999\pi\)
\(432\) 0 0
\(433\) 5.69706e6i 1.46026i −0.683307 0.730131i \(-0.739459\pi\)
0.683307 0.730131i \(-0.260541\pi\)
\(434\) 0 0
\(435\) 581892.i 0.147441i
\(436\) 0 0
\(437\) 6.11368e6 + 3.52974e6i 1.53144 + 0.884176i
\(438\) 0 0
\(439\) 2.73386e6 + 4.73519e6i 0.677042 + 1.17267i 0.975867 + 0.218364i \(0.0700718\pi\)
−0.298825 + 0.954308i \(0.596595\pi\)
\(440\) 0 0
\(441\) −383333. + 101125.i −0.0938598 + 0.0247606i
\(442\) 0 0
\(443\) 5.51411e6 3.18357e6i 1.33495 0.770735i 0.348898 0.937161i \(-0.386556\pi\)
0.986054 + 0.166426i \(0.0532225\pi\)
\(444\) 0 0
\(445\) 778618. 1.34861e6i 0.186391 0.322838i
\(446\) 0 0
\(447\) 2.89770e6 0.685938
\(448\) 0 0
\(449\) 4.50171e6 1.05381 0.526904 0.849925i \(-0.323353\pi\)
0.526904 + 0.849925i \(0.323353\pi\)
\(450\) 0 0
\(451\) −785167. + 1.35995e6i −0.181769 + 0.314834i
\(452\) 0 0
\(453\) −2.15527e6 + 1.24434e6i −0.493464 + 0.284902i
\(454\) 0 0
\(455\) −84013.8 34990.4i −0.0190249 0.00792355i
\(456\) 0 0
\(457\) 718680. + 1.24479e6i 0.160970 + 0.278808i 0.935217 0.354075i \(-0.115204\pi\)
−0.774247 + 0.632884i \(0.781871\pi\)
\(458\) 0 0
\(459\) −397208. 229328.i −0.0880008 0.0508073i
\(460\) 0 0
\(461\) 6.30635e6i 1.38206i −0.722828 0.691028i \(-0.757158\pi\)
0.722828 0.691028i \(-0.242842\pi\)
\(462\) 0 0
\(463\) 9.12301e6i 1.97782i 0.148531 + 0.988908i \(0.452545\pi\)
−0.148531 + 0.988908i \(0.547455\pi\)
\(464\) 0 0
\(465\) −219919. 126970.i −0.0471662 0.0272314i
\(466\) 0 0
\(467\) −664462. 1.15088e6i −0.140987 0.244196i 0.786882 0.617104i \(-0.211694\pi\)
−0.927868 + 0.372908i \(0.878361\pi\)
\(468\) 0 0
\(469\) −6.27820e6 + 814179.i −1.31796 + 0.170918i
\(470\) 0 0
\(471\) 4.39858e6 2.53952e6i 0.913609 0.527472i
\(472\) 0 0
\(473\) −2.02159e6 + 3.50149e6i −0.415470 + 0.719615i
\(474\) 0 0
\(475\) 7.23707e6 1.47173
\(476\) 0 0
\(477\) −842447. −0.169530
\(478\) 0 0
\(479\) −2.00910e6 + 3.47987e6i −0.400095 + 0.692985i −0.993737 0.111744i \(-0.964356\pi\)
0.593642 + 0.804729i \(0.297690\pi\)
\(480\) 0 0
\(481\) −132512. + 76505.8i −0.0261151 + 0.0150776i
\(482\) 0 0
\(483\) −3.29892e6 4.31652e6i −0.643435 0.841910i
\(484\) 0 0
\(485\) 1.35841e6 + 2.35283e6i 0.262226 + 0.454188i
\(486\) 0 0
\(487\) −7.43986e6 4.29540e6i −1.42148 0.820695i −0.425059 0.905166i \(-0.639747\pi\)
−0.996426 + 0.0844709i \(0.973080\pi\)
\(488\) 0 0
\(489\) 2.87921e6i 0.544505i
\(490\) 0 0
\(491\) 1.57232e6i 0.294332i 0.989112 + 0.147166i \(0.0470151\pi\)
−0.989112 + 0.147166i \(0.952985\pi\)
\(492\) 0 0
\(493\) 177815. + 102662.i 0.0329497 + 0.0190235i
\(494\) 0 0
\(495\) 69636.4 + 120614.i 0.0127739 + 0.0221250i
\(496\) 0 0
\(497\) −2.49533e6 3.26505e6i −0.453145 0.592924i
\(498\) 0 0
\(499\) −1.25743e6 + 725978.i −0.226065 + 0.130518i −0.608755 0.793358i \(-0.708331\pi\)
0.382691 + 0.923877i \(0.374998\pi\)
\(500\) 0 0
\(501\) −3.83692e6 + 6.64573e6i −0.682948 + 1.18290i
\(502\) 0 0
\(503\) −3.59897e6 −0.634247 −0.317123 0.948384i \(-0.602717\pi\)
−0.317123 + 0.948384i \(0.602717\pi\)
\(504\) 0 0
\(505\) −2.34806e6 −0.409714
\(506\) 0 0
\(507\) −3.02300e6 + 5.23599e6i −0.522298 + 0.904647i
\(508\) 0 0
\(509\) −1.76938e6 + 1.02155e6i −0.302711 + 0.174770i −0.643660 0.765312i \(-0.722585\pi\)
0.340949 + 0.940082i \(0.389252\pi\)
\(510\) 0 0
\(511\) 4.01770e6 521030.i 0.680653 0.0882695i
\(512\) 0 0
\(513\) 4.92680e6 + 8.53347e6i 0.826555 + 1.43164i
\(514\) 0 0
\(515\) −3.46584e6 2.00101e6i −0.575825 0.332453i
\(516\) 0 0
\(517\) 26348.1i 0.00433534i
\(518\) 0 0
\(519\) 1.27204e7i 2.07291i
\(520\) 0 0
\(521\) 6.36726e6 + 3.67614e6i 1.02768 + 0.593332i 0.916320 0.400448i \(-0.131145\pi\)
0.111362 + 0.993780i \(0.464479\pi\)
\(522\) 0 0
\(523\) 4.56417e6 + 7.90537e6i 0.729638 + 1.26377i 0.957036 + 0.289968i \(0.0936445\pi\)
−0.227399 + 0.973802i \(0.573022\pi\)
\(524\) 0 0
\(525\) −5.14138e6 2.14130e6i −0.814107 0.339062i
\(526\) 0 0
\(527\) −77599.5 + 44802.1i −0.0121712 + 0.00702703i
\(528\) 0 0
\(529\) 75537.1 130834.i 0.0117360 0.0203274i
\(530\) 0 0
\(531\) −162466. −0.0250050
\(532\) 0 0
\(533\) −186707. −0.0284670
\(534\) 0 0
\(535\) −1.67480e6 + 2.90085e6i −0.252976 + 0.438167i
\(536\) 0 0
\(537\) 8.74491e6 5.04887e6i 1.30864 0.755542i
\(538\) 0 0
\(539\) −3.16979e6 + 3.14534e6i −0.469957 + 0.466332i
\(540\) 0 0
\(541\) −5.66666e6 9.81493e6i −0.832403 1.44176i −0.896128 0.443797i \(-0.853631\pi\)
0.0637246 0.997968i \(-0.479702\pi\)
\(542\) 0 0
\(543\) −2.52804e6 1.45956e6i −0.367946 0.212433i
\(544\) 0 0
\(545\) 1.75700e6i 0.253385i
\(546\) 0 0
\(547\) 9.79610e6i 1.39986i 0.714211 + 0.699930i \(0.246786\pi\)
−0.714211 + 0.699930i \(0.753214\pi\)
\(548\) 0 0
\(549\) 73402.9 + 42379.2i 0.0103940 + 0.00600097i
\(550\) 0 0
\(551\) −2.20555e6 3.82012e6i −0.309483 0.536040i
\(552\) 0 0
\(553\) −1.18775e6 + 2.85186e6i −0.165163 + 0.396565i
\(554\) 0 0
\(555\) 1.52201e6 878735.i 0.209742 0.121095i
\(556\) 0 0
\(557\) −3.23228e6 + 5.59847e6i −0.441439 + 0.764595i −0.997797 0.0663481i \(-0.978865\pi\)
0.556357 + 0.830943i \(0.312199\pi\)
\(558\) 0 0
\(559\) −480718. −0.0650670
\(560\) 0 0
\(561\) 555402. 0.0745076
\(562\) 0 0
\(563\) 627220. 1.08638e6i 0.0833967 0.144447i −0.821310 0.570482i \(-0.806756\pi\)
0.904707 + 0.426035i \(0.140090\pi\)
\(564\) 0 0
\(565\) 1.92796e6 1.11311e6i 0.254084 0.146695i
\(566\) 0 0
\(567\) −1.07080e6 8.25704e6i −0.139879 1.07862i
\(568\) 0 0
\(569\) −2.77361e6 4.80403e6i −0.359140 0.622050i 0.628677 0.777667i \(-0.283597\pi\)
−0.987817 + 0.155617i \(0.950263\pi\)
\(570\) 0 0
\(571\) −5.57019e6 3.21595e6i −0.714956 0.412780i 0.0979373 0.995193i \(-0.468776\pi\)
−0.812893 + 0.582412i \(0.802109\pi\)
\(572\) 0 0
\(573\) 177155.i 0.0225407i
\(574\) 0 0
\(575\) 6.75314e6i 0.851797i
\(576\) 0 0
\(577\) 9.92482e6 + 5.73010e6i 1.24103 + 0.716510i 0.969304 0.245864i \(-0.0790715\pi\)
0.271728 + 0.962374i \(0.412405\pi\)
\(578\) 0 0
\(579\) −7.73211e6 1.33924e7i −0.958521 1.66021i
\(580\) 0 0
\(581\) 1.88381e6 1.43971e6i 0.231524 0.176944i
\(582\) 0 0
\(583\) −8.21784e6 + 4.74457e6i −1.00135 + 0.578130i
\(584\) 0 0
\(585\) −8279.51 + 14340.5i −0.00100026 + 0.00173251i
\(586\) 0 0
\(587\) 1.54952e7 1.85610 0.928052 0.372451i \(-0.121483\pi\)
0.928052 + 0.372451i \(0.121483\pi\)
\(588\) 0 0
\(589\) 1.92502e6 0.228638
\(590\) 0 0
\(591\) 664978. 1.15178e6i 0.0783138 0.135644i
\(592\) 0 0
\(593\) 7.54037e6 4.35344e6i 0.880554 0.508388i 0.00971325 0.999953i \(-0.496908\pi\)
0.870841 + 0.491564i \(0.163575\pi\)
\(594\) 0 0
\(595\) 293058. 223971.i 0.0339361 0.0259358i
\(596\) 0 0
\(597\) −8.02929e6 1.39071e7i −0.922023 1.59699i
\(598\) 0 0
\(599\) −9.92434e6 5.72982e6i −1.13015 0.652490i −0.186175 0.982517i \(-0.559609\pi\)
−0.943972 + 0.330026i \(0.892942\pi\)
\(600\) 0 0
\(601\) 9.83228e6i 1.11037i −0.831727 0.555185i \(-0.812647\pi\)
0.831727 0.555185i \(-0.187353\pi\)
\(602\) 0 0
\(603\) 1.15188e6i 0.129007i
\(604\) 0 0
\(605\) −1.74089e6 1.00510e6i −0.193367 0.111640i
\(606\) 0 0
\(607\) 6.70187e6 + 1.16080e7i 0.738286 + 1.27875i 0.953267 + 0.302130i \(0.0976977\pi\)
−0.214981 + 0.976618i \(0.568969\pi\)
\(608\) 0 0
\(609\) 436576. + 3.36647e6i 0.0476998 + 0.367817i
\(610\) 0 0
\(611\) −2712.99 + 1566.35i −0.000293999 + 0.000169740i
\(612\) 0 0
\(613\) −838947. + 1.45310e6i −0.0901744 + 0.156187i −0.907584 0.419870i \(-0.862076\pi\)
0.817410 + 0.576056i \(0.195409\pi\)
\(614\) 0 0
\(615\) 2.14449e6 0.228631
\(616\) 0 0
\(617\) 1.48745e7 1.57300 0.786499 0.617592i \(-0.211891\pi\)
0.786499 + 0.617592i \(0.211891\pi\)
\(618\) 0 0
\(619\) −3.56661e6 + 6.17755e6i −0.374136 + 0.648022i −0.990197 0.139676i \(-0.955394\pi\)
0.616061 + 0.787698i \(0.288727\pi\)
\(620\) 0 0
\(621\) 7.96285e6 4.59735e6i 0.828590 0.478387i
\(622\) 0 0
\(623\) 3.49279e6 8.38638e6i 0.360539 0.865674i
\(624\) 0 0
\(625\) −2.68990e6 4.65904e6i −0.275445 0.477085i
\(626\) 0 0
\(627\) −1.03335e7 5.96603e6i −1.04973 0.606061i
\(628\) 0 0
\(629\) 620131.i 0.0624967i
\(630\) 0 0
\(631\) 9.49875e6i 0.949715i −0.880063 0.474857i \(-0.842500\pi\)
0.880063 0.474857i \(-0.157500\pi\)
\(632\) 0 0
\(633\) 6.04639e6 + 3.49089e6i 0.599773 + 0.346279i
\(634\) 0 0
\(635\) −1.48509e6 2.57225e6i −0.146156 0.253150i
\(636\) 0 0
\(637\) −512305. 139400.i −0.0500241 0.0136117i
\(638\) 0 0
\(639\) −647530. + 373852.i −0.0627347 + 0.0362199i
\(640\) 0 0
\(641\) 7.67359e6 1.32910e7i 0.737655 1.27766i −0.215894 0.976417i \(-0.569266\pi\)
0.953549 0.301239i \(-0.0974002\pi\)
\(642\) 0 0
\(643\) −988668. −0.0943025 −0.0471512 0.998888i \(-0.515014\pi\)
−0.0471512 + 0.998888i \(0.515014\pi\)
\(644\) 0 0
\(645\) 5.52146e6 0.522582
\(646\) 0 0
\(647\) −5.51747e6 + 9.55655e6i −0.518179 + 0.897512i 0.481598 + 0.876392i \(0.340056\pi\)
−0.999777 + 0.0211198i \(0.993277\pi\)
\(648\) 0 0
\(649\) −1.58482e6 + 914994.i −0.147695 + 0.0852720i
\(650\) 0 0
\(651\) −1.36758e6 569574.i −0.126474 0.0526742i
\(652\) 0 0
\(653\) 2.55762e6 + 4.42993e6i 0.234722 + 0.406550i 0.959192 0.282756i \(-0.0912488\pi\)
−0.724470 + 0.689306i \(0.757915\pi\)
\(654\) 0 0
\(655\) 2.49576e6 + 1.44093e6i 0.227300 + 0.131232i
\(656\) 0 0
\(657\) 737139.i 0.0666248i
\(658\) 0 0
\(659\) 1.13694e7i 1.01982i −0.860227 0.509911i \(-0.829678\pi\)
0.860227 0.509911i \(-0.170322\pi\)
\(660\) 0 0
\(661\) −1.41836e7 8.18893e6i −1.26265 0.728993i −0.289066 0.957309i \(-0.593345\pi\)
−0.973587 + 0.228316i \(0.926678\pi\)
\(662\) 0 0
\(663\) 33017.6 + 57188.2i 0.00291717 + 0.00505269i
\(664\) 0 0
\(665\) −7.85832e6 + 1.01909e6i −0.689089 + 0.0893635i
\(666\) 0 0
\(667\) −3.56467e6 + 2.05806e6i −0.310245 + 0.179120i
\(668\) 0 0
\(669\) −491775. + 851779.i −0.0424816 + 0.0735803i
\(670\) 0 0
\(671\) 954701. 0.0818580
\(672\) 0 0
\(673\) 9.53588e6 0.811564 0.405782 0.913970i \(-0.366999\pi\)
0.405782 + 0.913970i \(0.366999\pi\)
\(674\) 0 0
\(675\) 4.71301e6 8.16317e6i 0.398143 0.689604i
\(676\) 0 0
\(677\) 1.06224e7 6.13284e6i 0.890740 0.514269i 0.0165554 0.999863i \(-0.494730\pi\)
0.874184 + 0.485594i \(0.161397\pi\)
\(678\) 0 0
\(679\) 9.62416e6 + 1.25929e7i 0.801103 + 1.04821i
\(680\) 0 0
\(681\) 450887. + 780959.i 0.0372563 + 0.0645298i
\(682\) 0 0
\(683\) −2.91586e6 1.68347e6i −0.239175 0.138088i 0.375623 0.926773i \(-0.377429\pi\)
−0.614797 + 0.788685i \(0.710762\pi\)
\(684\) 0 0
\(685\) 1.04480e6i 0.0850756i
\(686\) 0 0
\(687\) 57752.4i 0.00466851i
\(688\) 0 0
\(689\) −977071. 564112.i −0.0784112 0.0452707i
\(690\) 0 0
\(691\) −2.65246e6 4.59420e6i −0.211327 0.366028i 0.740803 0.671722i \(-0.234445\pi\)
−0.952130 + 0.305694i \(0.901112\pi\)
\(692\) 0 0
\(693\) 493366. + 645551.i 0.0390244 + 0.0510620i
\(694\) 0 0
\(695\) 6.23174e6 3.59789e6i 0.489381 0.282544i
\(696\) 0 0
\(697\) 378346. 655315.i 0.0294990 0.0510938i
\(698\) 0 0
\(699\) 1.95688e7 1.51486
\(700\) 0 0
\(701\) 3.03260e6 0.233088 0.116544 0.993186i \(-0.462818\pi\)
0.116544 + 0.993186i \(0.462818\pi\)
\(702\) 0 0
\(703\) −6.66133e6 + 1.15378e7i −0.508362 + 0.880508i
\(704\) 0 0
\(705\) 31161.1 17990.8i 0.00236124 0.00136326i
\(706\) 0 0
\(707\) −1.35844e7 + 1.76168e6i −1.02210 + 0.132549i
\(708\) 0 0
\(709\) −7.72319e6 1.33770e7i −0.577007 0.999406i −0.995820 0.0913332i \(-0.970887\pi\)
0.418813 0.908072i \(-0.362446\pi\)
\(710\) 0 0
\(711\) 486790. + 281049.i 0.0361134 + 0.0208501i
\(712\) 0 0
\(713\) 1.79630e6i 0.132329i
\(714\) 0 0
\(715\) 186517.i 0.0136444i
\(716\) 0 0
\(717\) 7.51492e6 + 4.33874e6i 0.545917 + 0.315185i
\(718\) 0 0
\(719\) 8.99286e6 + 1.55761e7i 0.648748 + 1.12366i 0.983422 + 0.181330i \(0.0580402\pi\)
−0.334675 + 0.942334i \(0.608626\pi\)
\(720\) 0 0
\(721\) −2.15525e7 8.97627e6i −1.54405 0.643070i
\(722\) 0 0
\(723\) 7.95557e6 4.59315e6i 0.566012 0.326787i
\(724\) 0 0
\(725\) −2.10984e6 + 3.65435e6i −0.149075 + 0.258205i
\(726\) 0 0
\(727\) −1.55969e7 −1.09447 −0.547234 0.836980i \(-0.684319\pi\)
−0.547234 + 0.836980i \(0.684319\pi\)
\(728\) 0 0
\(729\) 1.26988e7 0.884998
\(730\) 0 0
\(731\) 974136. 1.68725e6i 0.0674258 0.116785i
\(732\) 0 0
\(733\) −1.11442e7 + 6.43409e6i −0.766104 + 0.442310i −0.831483 0.555550i \(-0.812508\pi\)
0.0653790 + 0.997861i \(0.479174\pi\)
\(734\) 0 0
\(735\) 5.88426e6 + 1.60113e6i 0.401766 + 0.109322i
\(736\) 0 0
\(737\) 6.48726e6 + 1.12363e7i 0.439939 + 0.761997i
\(738\) 0 0
\(739\) −1.72415e7 9.95440e6i −1.16135 0.670508i −0.209726 0.977760i \(-0.567257\pi\)
−0.951628 + 0.307252i \(0.900591\pi\)
\(740\) 0 0
\(741\) 1.41868e6i 0.0949157i
\(742\) 0 0
\(743\) 2.89554e7i 1.92423i −0.272640 0.962116i \(-0.587897\pi\)
0.272640 0.962116i \(-0.412103\pi\)
\(744\) 0 0
\(745\) −3.41550e6 1.97194e6i −0.225457 0.130168i
\(746\) 0 0
\(747\) −215698. 373600.i −0.0141431 0.0244966i
\(748\) 0 0
\(749\) −7.51297e6 + 1.80391e7i −0.489336 + 1.17492i
\(750\) 0 0
\(751\) 56095.8 32386.9i 0.00362936 0.00209541i −0.498184 0.867071i \(-0.666000\pi\)
0.501814 + 0.864976i \(0.332666\pi\)
\(752\) 0 0
\(753\) −8.72188e6 + 1.51067e7i −0.560561 + 0.970920i
\(754\) 0 0
\(755\) 3.38720e6 0.216259
\(756\) 0 0
\(757\) −1.28686e7 −0.816190 −0.408095 0.912940i \(-0.633807\pi\)
−0.408095 + 0.912940i \(0.633807\pi\)
\(758\) 0 0
\(759\) −5.56709e6 + 9.64248e6i −0.350771 + 0.607553i
\(760\) 0 0
\(761\) −1.58510e7 + 9.15160e6i −0.992193 + 0.572843i −0.905929 0.423429i \(-0.860826\pi\)
−0.0862641 + 0.996272i \(0.527493\pi\)
\(762\) 0 0
\(763\) 1.31822e6 + 1.01649e7i 0.0819742 + 0.632110i
\(764\) 0 0
\(765\) −33555.5 58119.8i −0.00207305 0.00359063i
\(766\) 0 0
\(767\) −188429. 108789.i −0.0115654 0.00667726i
\(768\) 0 0
\(769\) 1.68118e7i 1.02518i 0.858634 + 0.512589i \(0.171314\pi\)
−0.858634 + 0.512589i \(0.828686\pi\)
\(770\) 0 0
\(771\) 2.19237e7i 1.32824i
\(772\) 0 0
\(773\) 2.16919e7 + 1.25238e7i 1.30571 + 0.753854i 0.981378 0.192088i \(-0.0615259\pi\)
0.324336 + 0.945942i \(0.394859\pi\)
\(774\) 0 0
\(775\) −920744. 1.59478e6i −0.0550662 0.0953774i
\(776\) 0 0
\(777\) 8.14615e6 6.22574e6i 0.484061 0.369946i
\(778\) 0 0
\(779\) −1.40785e7 + 8.12825e6i −0.831216 + 0.479903i
\(780\) 0 0
\(781\) −4.21099e6 + 7.29365e6i −0.247034 + 0.427875i
\(782\) 0 0
\(783\) −5.74528e6 −0.334894
\(784\) 0 0
\(785\) −6.91278e6 −0.400386
\(786\) 0 0
\(787\) 1.99502e6 3.45547e6i 0.114818 0.198870i −0.802889 0.596128i \(-0.796705\pi\)
0.917707 + 0.397258i \(0.130038\pi\)
\(788\) 0 0
\(789\) 2.28135e7 1.31714e7i 1.30467 0.753249i
\(790\) 0 0
\(791\) 1.03189e7 7.88625e6i 0.586395 0.448156i
\(792\) 0 0
\(793\) 56755.2 + 98302.9i 0.00320496 + 0.00555115i
\(794\) 0 0
\(795\) 1.12225e7 + 6.47931e6i 0.629755 + 0.363589i
\(796\) 0 0
\(797\) 2.69992e7i 1.50558i 0.658259 + 0.752792i \(0.271293\pi\)
−0.658259 + 0.752792i \(0.728707\pi\)
\(798\) 0 0
\(799\) 12696.3i 0.000703575i
\(800\) 0 0
\(801\) −1.43149e6 826473.i −0.0788330 0.0455142i
\(802\) 0 0
\(803\) −4.15149e6 7.19060e6i −0.227204 0.393528i
\(804\) 0 0
\(805\) 950949. + 7.33284e6i 0.0517211 + 0.398825i
\(806\) 0 0
\(807\) −3.08735e7 + 1.78248e7i −1.66879 + 0.963476i
\(808\) 0 0
\(809\) 8.61028e6 1.49134e7i 0.462536 0.801136i −0.536550 0.843868i \(-0.680273\pi\)
0.999087 + 0.0427321i \(0.0136062\pi\)
\(810\) 0 0
\(811\) 1.90148e7 1.01517 0.507587 0.861601i \(-0.330538\pi\)
0.507587 + 0.861601i \(0.330538\pi\)
\(812\) 0 0
\(813\) −6.64325e6 −0.352496
\(814\) 0 0
\(815\) 1.95936e6 3.39371e6i 0.103329 0.178970i
\(816\) 0 0
\(817\) −3.62483e7 + 2.09280e7i −1.89991 + 1.09691i
\(818\) 0 0
\(819\) −37140.9 + 89177.3i −0.00193483 + 0.00464563i
\(820\) 0 0
\(821\) −1.54355e6 2.67352e6i −0.0799216 0.138428i 0.823294 0.567615i \(-0.192134\pi\)
−0.903216 + 0.429186i \(0.858800\pi\)
\(822\) 0 0
\(823\) 2.03659e7 + 1.17582e7i 1.04810 + 0.605122i 0.922117 0.386910i \(-0.126458\pi\)
0.125984 + 0.992032i \(0.459791\pi\)
\(824\) 0 0
\(825\) 1.14143e7i 0.583866i
\(826\) 0 0
\(827\) 2.15389e7i 1.09512i 0.836767 + 0.547559i \(0.184443\pi\)
−0.836767 + 0.547559i \(0.815557\pi\)
\(828\) 0 0
\(829\) −1.14533e7 6.61258e6i −0.578823 0.334183i 0.181843 0.983328i \(-0.441794\pi\)
−0.760665 + 0.649144i \(0.775127\pi\)
\(830\) 0 0
\(831\) 4.15889e6 + 7.20342e6i 0.208918 + 0.361856i
\(832\) 0 0
\(833\) 1.52742e6 1.51563e6i 0.0762684 0.0756801i
\(834\) 0 0
\(835\) 9.04510e6 5.22219e6i 0.448949 0.259201i
\(836\) 0 0
\(837\) 1.25364e6 2.17136e6i 0.0618526 0.107132i
\(838\) 0 0
\(839\) 1.67107e7 0.819575 0.409788 0.912181i \(-0.365603\pi\)
0.409788 + 0.912181i \(0.365603\pi\)
\(840\) 0 0
\(841\) −1.79392e7 −0.874607
\(842\) 0 0
\(843\) 9.13348e6 1.58196e7i 0.442657 0.766704i
\(844\) 0 0
\(845\) 7.12639e6 4.11442e6i 0.343343 0.198229i
\(846\) 0 0
\(847\) −1.08258e7 4.50876e6i −0.518503 0.215948i
\(848\) 0 0
\(849\) −2.20354e6 3.81664e6i −0.104918 0.181724i
\(850\) 0 0
\(851\) 1.07662e7 + 6.21590e6i 0.509613 + 0.294225i
\(852\) 0 0
\(853\) 484559.i 0.0228020i 0.999935 + 0.0114010i \(0.00362914\pi\)
−0.999935 + 0.0114010i \(0.996371\pi\)
\(854\) 0 0
\(855\) 1.44179e6i 0.0674506i
\(856\) 0 0
\(857\) −2.82629e7 1.63176e7i −1.31451 0.758935i −0.331674 0.943394i \(-0.607613\pi\)
−0.982840 + 0.184459i \(0.940947\pi\)
\(858\) 0 0
\(859\) −5.10980e6 8.85044e6i −0.236277 0.409243i 0.723366 0.690465i \(-0.242594\pi\)
−0.959643 + 0.281221i \(0.909261\pi\)
\(860\) 0 0
\(861\) 1.24067e7 1.60894e6i 0.570359 0.0739662i
\(862\) 0 0
\(863\) 2.91512e7 1.68305e7i 1.33239 0.769253i 0.346721 0.937968i \(-0.387295\pi\)
0.985665 + 0.168715i \(0.0539617\pi\)
\(864\) 0 0
\(865\) −8.65645e6 + 1.49934e7i −0.393369 + 0.681334i
\(866\) 0 0
\(867\) 2.29151e7 1.03532
\(868\) 0 0
\(869\) 6.33135e6 0.284411
\(870\) 0 0
\(871\) −771312. + 1.33595e6i −0.0344496 + 0.0596685i
\(872\) 0 0
\(873\) 2.49744e6 1.44190e6i 0.110907 0.0640322i
\(874\) 0 0
\(875\) 1.00697e7 + 1.31759e7i 0.444630 + 0.581781i
\(876\) 0 0
\(877\) 3.88346e6 + 6.72635e6i 0.170498 + 0.295311i 0.938594 0.345023i \(-0.112129\pi\)
−0.768096 + 0.640335i \(0.778796\pi\)
\(878\) 0 0
\(879\) 2.71764e7 + 1.56903e7i 1.18637 + 0.684950i
\(880\) 0 0
\(881\) 3.44116e7i 1.49370i −0.664990 0.746852i \(-0.731564\pi\)
0.664990 0.746852i \(-0.268436\pi\)
\(882\) 0 0
\(883\) 4.33214e7i 1.86982i 0.354880 + 0.934912i \(0.384522\pi\)
−0.354880 + 0.934912i \(0.615478\pi\)
\(884\) 0 0
\(885\) 2.16427e6 + 1.24954e6i 0.0928865 + 0.0536280i
\(886\) 0 0
\(887\) −40392.9 69962.6i −0.00172384 0.00298577i 0.865162 0.501492i \(-0.167215\pi\)
−0.866886 + 0.498506i \(0.833882\pi\)
\(888\) 0 0
\(889\) −1.05217e7 1.37672e7i −0.446510 0.584241i
\(890\) 0 0
\(891\) −1.47779e7 + 8.53200e6i −0.623616 + 0.360045i
\(892\) 0 0
\(893\) −136381. + 236219.i −0.00572303 + 0.00991258i
\(894\) 0 0
\(895\) −1.37434e7 −0.573506
\(896\) 0 0
\(897\) −1.32381e6 −0.0549345
\(898\) 0 0
\(899\) −561206. + 972037.i −0.0231592 + 0.0401129i
\(900\) 0 0
\(901\) 3.95991e6 2.28625e6i 0.162507 0.0938237i
\(902\) 0 0
\(903\) 3.19438e7 4.14258e6i 1.30367 0.169064i
\(904\) 0 0
\(905\) 1.98652e6 + 3.44076e6i 0.0806254 + 0.139647i
\(906\) 0 0
\(907\) 9.05859e6 + 5.22998e6i 0.365630 + 0.211097i 0.671548 0.740961i \(-0.265630\pi\)
−0.305917 + 0.952058i \(0.598963\pi\)
\(908\) 0 0
\(909\) 2.49237e6i 0.100047i
\(910\) 0 0
\(911\) 1.12503e7i 0.449127i −0.974459 0.224563i \(-0.927904\pi\)
0.974459 0.224563i \(-0.0720956\pi\)
\(912\) 0 0
\(913\) −4.20816e6 2.42958e6i −0.167076 0.0964616i
\(914\) 0 0
\(915\) −651882. 1.12909e6i −0.0257405 0.0445838i
\(916\) 0 0
\(917\) 1.55200e7 + 6.46383e6i 0.609493 + 0.253844i
\(918\) 0 0
\(919\) −4.64723e6 + 2.68308e6i −0.181512 + 0.104796i −0.588003 0.808859i \(-0.700086\pi\)
0.406491 + 0.913655i \(0.366752\pi\)
\(920\) 0 0
\(921\) 7.67972e6 1.33017e7i 0.298329 0.516722i
\(922\) 0 0
\(923\) −1.00134e6 −0.0386882
\(924\) 0 0
\(925\) 1.27445e7 0.489745
\(926\) 0 0
\(927\) −2.12399e6 + 3.67886e6i −0.0811807 + 0.140609i
\(928\) 0 0
\(929\) −2.14406e7 + 1.23787e7i −0.815074 + 0.470583i −0.848715 0.528851i \(-0.822623\pi\)
0.0336410 + 0.999434i \(0.489290\pi\)
\(930\) 0 0
\(931\) −4.46988e7 + 1.17917e7i −1.69014 + 0.445864i
\(932\) 0 0
\(933\) 5.96999e6 + 1.03403e7i 0.224527 + 0.388893i
\(934\) 0 0
\(935\) −654649. 377962.i −0.0244895 0.0141390i
\(936\) 0 0
\(937\) 1.46850e6i 0.0546419i 0.999627 + 0.0273210i \(0.00869762\pi\)
−0.999627 + 0.0273210i \(0.991302\pi\)
\(938\) 0 0
\(939\) 5.11189e6i 0.189198i
\(940\) 0 0
\(941\) 5.16498e6 + 2.98200e6i 0.190149 + 0.109783i 0.592052 0.805899i \(-0.298318\pi\)
−0.401903 + 0.915682i \(0.631651\pi\)
\(942\) 0 0
\(943\) 7.58472e6 + 1.31371e7i 0.277754 + 0.481084i
\(944\) 0 0
\(945\) −3.96808e6 + 9.52758e6i −0.144544 + 0.347059i
\(946\) 0 0
\(947\) 4.13119e7 2.38514e7i 1.49692 0.864250i 0.496931 0.867790i \(-0.334460\pi\)
0.999994 + 0.00353991i \(0.00112679\pi\)
\(948\) 0 0
\(949\) 493597. 854935.i 0.0177913 0.0308154i
\(950\) 0 0
\(951\) −5.41510e7 −1.94158
\(952\) 0 0
\(953\) −4.82889e7 −1.72232 −0.861162 0.508330i \(-0.830263\pi\)
−0.861162 + 0.508330i \(0.830263\pi\)
\(954\) 0 0
\(955\) 120557. 208812.i 0.00427746 0.00740877i
\(956\) 0 0
\(957\) 6.02507e6 3.47858e6i 0.212658 0.122778i
\(958\) 0 0
\(959\) −783878. 6.04455e6i −0.0275234 0.212235i
\(960\) 0 0
\(961\) 1.40697e7 + 2.43694e7i 0.491445 + 0.851208i
\(962\) 0 0
\(963\) 3.07913e6 + 1.77774e6i 0.106995 + 0.0617735i
\(964\) 0 0
\(965\) 2.10474e7i 0.727580i
\(966\) 0 0
\(967\) 5.23912e7i 1.80174i −0.434089 0.900870i \(-0.642930\pi\)
0.434089 0.900870i \(-0.357070\pi\)
\(968\) 0 0
\(969\) 4.97935e6 + 2.87483e6i 0.170358 + 0.0983565i
\(970\) 0 0
\(971\) −1.39434e7 2.41507e7i −0.474592 0.822017i 0.524985 0.851112i \(-0.324071\pi\)
−0.999577 + 0.0290944i \(0.990738\pi\)
\(972\) 0 0
\(973\) 3.33536e7 2.54907e7i 1.12943 0.863176i
\(974\) 0 0
\(975\) −1.17530e6 + 678558.i −0.0395946 + 0.0228599i
\(976\) 0 0
\(977\) 2.87737e7 4.98375e7i 0.964405 1.67040i 0.253201 0.967414i \(-0.418517\pi\)
0.711204 0.702985i \(-0.248150\pi\)
\(978\) 0 0
\(979\) −1.86184e7 −0.620850
\(980\) 0 0
\(981\) 1.86499e6 0.0618732
\(982\) 0 0
\(983\) 5.32884e6 9.22981e6i 0.175893 0.304656i −0.764577 0.644532i \(-0.777052\pi\)
0.940470 + 0.339877i \(0.110385\pi\)
\(984\) 0 0
\(985\) −1.56761e6 + 905061.i −0.0514811 + 0.0297226i
\(986\) 0 0
\(987\) 166781. 127463.i 0.00544946 0.00416478i
\(988\) 0 0
\(989\) 1.95285e7 + 3.38244e7i 0.634862 + 1.09961i
\(990\) 0 0
\(991\) −5.15687e6 2.97732e6i −0.166802 0.0963034i 0.414275 0.910152i \(-0.364035\pi\)
−0.581077 + 0.813848i \(0.697369\pi\)
\(992\) 0 0
\(993\) 1.71303e7i 0.551306i
\(994\) 0 0
\(995\) 2.18564e7i 0.699875i
\(996\) 0 0
\(997\) −3.30824e7 1.91001e7i −1.05404 0.608552i −0.130265 0.991479i \(-0.541583\pi\)
−0.923779 + 0.382927i \(0.874916\pi\)
\(998\) 0 0
\(999\) 8.67614e6 + 1.50275e7i 0.275051 + 0.476402i
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 224.6.p.a.31.11 80
4.3 odd 2 inner 224.6.p.a.31.30 yes 80
7.5 odd 6 inner 224.6.p.a.159.30 yes 80
28.19 even 6 inner 224.6.p.a.159.11 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.6.p.a.31.11 80 1.1 even 1 trivial
224.6.p.a.31.30 yes 80 4.3 odd 2 inner
224.6.p.a.159.11 yes 80 28.19 even 6 inner
224.6.p.a.159.30 yes 80 7.5 odd 6 inner