Properties

Label 384.7.l.b.31.16
Level $384$
Weight $7$
Character 384.31
Analytic conductor $88.341$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,7,Mod(31,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.31");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.l (of order \(4\), degree \(2\), not 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: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 31.16
Character \(\chi\) \(=\) 384.31
Dual form 384.7.l.b.223.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(11.0227 + 11.0227i) q^{3} +(95.7535 + 95.7535i) q^{5} +338.697 q^{7} +243.000i q^{9} +O(q^{10})\) \(q+(11.0227 + 11.0227i) q^{3} +(95.7535 + 95.7535i) q^{5} +338.697 q^{7} +243.000i q^{9} +(1594.02 - 1594.02i) q^{11} +(-602.697 + 602.697i) q^{13} +2110.92i q^{15} -1410.09 q^{17} +(6642.31 + 6642.31i) q^{19} +(3733.36 + 3733.36i) q^{21} +14966.5 q^{23} +2712.47i q^{25} +(-2678.52 + 2678.52i) q^{27} +(25687.3 - 25687.3i) q^{29} -52243.3i q^{31} +35140.9 q^{33} +(32431.4 + 32431.4i) q^{35} +(11201.3 + 11201.3i) q^{37} -13286.7 q^{39} -78047.4i q^{41} +(-62949.8 + 62949.8i) q^{43} +(-23268.1 + 23268.1i) q^{45} -59196.3i q^{47} -2933.21 q^{49} +(-15543.0 - 15543.0i) q^{51} +(45616.2 + 45616.2i) q^{53} +305266. q^{55} +146432. i q^{57} +(-40115.2 + 40115.2i) q^{59} +(68015.9 - 68015.9i) q^{61} +82303.4i q^{63} -115421. q^{65} +(-75821.1 - 75821.1i) q^{67} +(164972. + 164972. i) q^{69} -520740. q^{71} +349935. i q^{73} +(-29898.7 + 29898.7i) q^{75} +(539891. - 539891. i) q^{77} -584164. i q^{79} -59049.0 q^{81} +(75532.5 + 75532.5i) q^{83} +(-135021. - 135021. i) q^{85} +566286. q^{87} +881047. i q^{89} +(-204132. + 204132. i) q^{91} +(575863. - 575863. i) q^{93} +1.27205e6i q^{95} +463594. q^{97} +(387347. + 387347. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 2720 q^{11} + 3936 q^{19} + 26240 q^{23} - 66400 q^{29} - 162336 q^{35} + 7200 q^{37} - 340704 q^{43} + 806736 q^{49} - 80352 q^{51} - 443680 q^{53} + 232704 q^{55} + 886144 q^{59} + 326496 q^{61} - 372832 q^{65} + 962112 q^{67} - 541728 q^{69} + 534016 q^{71} + 1073088 q^{75} + 932960 q^{77} - 2834352 q^{81} + 2497760 q^{83} + 372000 q^{85} - 775008 q^{91} + 660960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) 11.0227 + 11.0227i 0.408248 + 0.408248i
\(4\) 0 0
\(5\) 95.7535 + 95.7535i 0.766028 + 0.766028i 0.977405 0.211377i \(-0.0677947\pi\)
−0.211377 + 0.977405i \(0.567795\pi\)
\(6\) 0 0
\(7\) 338.697 0.987455 0.493728 0.869617i \(-0.335634\pi\)
0.493728 + 0.869617i \(0.335634\pi\)
\(8\) 0 0
\(9\) 243.000i 0.333333i
\(10\) 0 0
\(11\) 1594.02 1594.02i 1.19761 1.19761i 0.222733 0.974879i \(-0.428502\pi\)
0.974879 0.222733i \(-0.0714979\pi\)
\(12\) 0 0
\(13\) −602.697 + 602.697i −0.274327 + 0.274327i −0.830839 0.556512i \(-0.812139\pi\)
0.556512 + 0.830839i \(0.312139\pi\)
\(14\) 0 0
\(15\) 2110.92i 0.625459i
\(16\) 0 0
\(17\) −1410.09 −0.287011 −0.143506 0.989650i \(-0.545838\pi\)
−0.143506 + 0.989650i \(0.545838\pi\)
\(18\) 0 0
\(19\) 6642.31 + 6642.31i 0.968408 + 0.968408i 0.999516 0.0311079i \(-0.00990356\pi\)
−0.0311079 + 0.999516i \(0.509904\pi\)
\(20\) 0 0
\(21\) 3733.36 + 3733.36i 0.403127 + 0.403127i
\(22\) 0 0
\(23\) 14966.5 1.23009 0.615046 0.788491i \(-0.289137\pi\)
0.615046 + 0.788491i \(0.289137\pi\)
\(24\) 0 0
\(25\) 2712.47i 0.173598i
\(26\) 0 0
\(27\) −2678.52 + 2678.52i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) 25687.3 25687.3i 1.05323 1.05323i 0.0547301 0.998501i \(-0.482570\pi\)
0.998501 0.0547301i \(-0.0174298\pi\)
\(30\) 0 0
\(31\) 52243.3i 1.75366i −0.480799 0.876831i \(-0.659653\pi\)
0.480799 0.876831i \(-0.340347\pi\)
\(32\) 0 0
\(33\) 35140.9 0.977847
\(34\) 0 0
\(35\) 32431.4 + 32431.4i 0.756418 + 0.756418i
\(36\) 0 0
\(37\) 11201.3 + 11201.3i 0.221138 + 0.221138i 0.808977 0.587840i \(-0.200021\pi\)
−0.587840 + 0.808977i \(0.700021\pi\)
\(38\) 0 0
\(39\) −13286.7 −0.223987
\(40\) 0 0
\(41\) 78047.4i 1.13242i −0.824261 0.566209i \(-0.808409\pi\)
0.824261 0.566209i \(-0.191591\pi\)
\(42\) 0 0
\(43\) −62949.8 + 62949.8i −0.791751 + 0.791751i −0.981779 0.190027i \(-0.939142\pi\)
0.190027 + 0.981779i \(0.439142\pi\)
\(44\) 0 0
\(45\) −23268.1 + 23268.1i −0.255343 + 0.255343i
\(46\) 0 0
\(47\) 59196.3i 0.570166i −0.958503 0.285083i \(-0.907979\pi\)
0.958503 0.285083i \(-0.0920211\pi\)
\(48\) 0 0
\(49\) −2933.21 −0.0249319
\(50\) 0 0
\(51\) −15543.0 15543.0i −0.117172 0.117172i
\(52\) 0 0
\(53\) 45616.2 + 45616.2i 0.306402 + 0.306402i 0.843512 0.537110i \(-0.180484\pi\)
−0.537110 + 0.843512i \(0.680484\pi\)
\(54\) 0 0
\(55\) 305266. 1.83481
\(56\) 0 0
\(57\) 146432.i 0.790702i
\(58\) 0 0
\(59\) −40115.2 + 40115.2i −0.195323 + 0.195323i −0.797992 0.602669i \(-0.794104\pi\)
0.602669 + 0.797992i \(0.294104\pi\)
\(60\) 0 0
\(61\) 68015.9 68015.9i 0.299654 0.299654i −0.541224 0.840878i \(-0.682039\pi\)
0.840878 + 0.541224i \(0.182039\pi\)
\(62\) 0 0
\(63\) 82303.4i 0.329152i
\(64\) 0 0
\(65\) −115421. −0.420285
\(66\) 0 0
\(67\) −75821.1 75821.1i −0.252096 0.252096i 0.569734 0.821829i \(-0.307046\pi\)
−0.821829 + 0.569734i \(0.807046\pi\)
\(68\) 0 0
\(69\) 164972. + 164972.i 0.502183 + 0.502183i
\(70\) 0 0
\(71\) −520740. −1.45494 −0.727472 0.686137i \(-0.759305\pi\)
−0.727472 + 0.686137i \(0.759305\pi\)
\(72\) 0 0
\(73\) 349935.i 0.899537i 0.893145 + 0.449768i \(0.148494\pi\)
−0.893145 + 0.449768i \(0.851506\pi\)
\(74\) 0 0
\(75\) −29898.7 + 29898.7i −0.0708710 + 0.0708710i
\(76\) 0 0
\(77\) 539891. 539891.i 1.18259 1.18259i
\(78\) 0 0
\(79\) 584164.i 1.18482i −0.805635 0.592412i \(-0.798176\pi\)
0.805635 0.592412i \(-0.201824\pi\)
\(80\) 0 0
\(81\) −59049.0 −0.111111
\(82\) 0 0
\(83\) 75532.5 + 75532.5i 0.132099 + 0.132099i 0.770065 0.637966i \(-0.220224\pi\)
−0.637966 + 0.770065i \(0.720224\pi\)
\(84\) 0 0
\(85\) −135021. 135021.i −0.219859 0.219859i
\(86\) 0 0
\(87\) 566286. 0.859960
\(88\) 0 0
\(89\) 881047.i 1.24977i 0.780718 + 0.624884i \(0.214854\pi\)
−0.780718 + 0.624884i \(0.785146\pi\)
\(90\) 0 0
\(91\) −204132. + 204132.i −0.270886 + 0.270886i
\(92\) 0 0
\(93\) 575863. 575863.i 0.715929 0.715929i
\(94\) 0 0
\(95\) 1.27205e6i 1.48366i
\(96\) 0 0
\(97\) 463594. 0.507952 0.253976 0.967211i \(-0.418262\pi\)
0.253976 + 0.967211i \(0.418262\pi\)
\(98\) 0 0
\(99\) 387347. + 387347.i 0.399204 + 0.399204i
\(100\) 0 0
\(101\) −457625. 457625.i −0.444166 0.444166i 0.449243 0.893409i \(-0.351694\pi\)
−0.893409 + 0.449243i \(0.851694\pi\)
\(102\) 0 0
\(103\) 1.49249e6 1.36584 0.682921 0.730492i \(-0.260709\pi\)
0.682921 + 0.730492i \(0.260709\pi\)
\(104\) 0 0
\(105\) 714964.i 0.617613i
\(106\) 0 0
\(107\) −1.61763e6 + 1.61763e6i −1.32047 + 1.32047i −0.407072 + 0.913396i \(0.633450\pi\)
−0.913396 + 0.407072i \(0.866550\pi\)
\(108\) 0 0
\(109\) 212604. 212604.i 0.164170 0.164170i −0.620241 0.784411i \(-0.712965\pi\)
0.784411 + 0.620241i \(0.212965\pi\)
\(110\) 0 0
\(111\) 246937.i 0.180558i
\(112\) 0 0
\(113\) 1.23333e6 0.854760 0.427380 0.904072i \(-0.359437\pi\)
0.427380 + 0.904072i \(0.359437\pi\)
\(114\) 0 0
\(115\) 1.43310e6 + 1.43310e6i 0.942285 + 0.942285i
\(116\) 0 0
\(117\) −146455. 146455.i −0.0914425 0.0914425i
\(118\) 0 0
\(119\) −477592. −0.283411
\(120\) 0 0
\(121\) 3.31025e6i 1.86855i
\(122\) 0 0
\(123\) 860294. 860294.i 0.462308 0.462308i
\(124\) 0 0
\(125\) 1.23642e6 1.23642e6i 0.633047 0.633047i
\(126\) 0 0
\(127\) 2.86669e6i 1.39949i 0.714394 + 0.699744i \(0.246703\pi\)
−0.714394 + 0.699744i \(0.753297\pi\)
\(128\) 0 0
\(129\) −1.38775e6 −0.646462
\(130\) 0 0
\(131\) 1.18710e6 + 1.18710e6i 0.528046 + 0.528046i 0.919989 0.391943i \(-0.128197\pi\)
−0.391943 + 0.919989i \(0.628197\pi\)
\(132\) 0 0
\(133\) 2.24973e6 + 2.24973e6i 0.956260 + 0.956260i
\(134\) 0 0
\(135\) −512955. −0.208486
\(136\) 0 0
\(137\) 4.54167e6i 1.76626i −0.469132 0.883128i \(-0.655433\pi\)
0.469132 0.883128i \(-0.344567\pi\)
\(138\) 0 0
\(139\) −3.47603e6 + 3.47603e6i −1.29431 + 1.29431i −0.362218 + 0.932093i \(0.617980\pi\)
−0.932093 + 0.362218i \(0.882020\pi\)
\(140\) 0 0
\(141\) 652504. 652504.i 0.232769 0.232769i
\(142\) 0 0
\(143\) 1.92143e6i 0.657076i
\(144\) 0 0
\(145\) 4.91929e6 1.61361
\(146\) 0 0
\(147\) −32331.9 32331.9i −0.0101784 0.0101784i
\(148\) 0 0
\(149\) −1.14413e6 1.14413e6i −0.345873 0.345873i 0.512697 0.858570i \(-0.328646\pi\)
−0.858570 + 0.512697i \(0.828646\pi\)
\(150\) 0 0
\(151\) −2.06680e6 −0.600300 −0.300150 0.953892i \(-0.597037\pi\)
−0.300150 + 0.953892i \(0.597037\pi\)
\(152\) 0 0
\(153\) 342651.i 0.0956704i
\(154\) 0 0
\(155\) 5.00248e6 5.00248e6i 1.34335 1.34335i
\(156\) 0 0
\(157\) 994443. 994443.i 0.256969 0.256969i −0.566851 0.823820i \(-0.691839\pi\)
0.823820 + 0.566851i \(0.191839\pi\)
\(158\) 0 0
\(159\) 1.00563e6i 0.250176i
\(160\) 0 0
\(161\) 5.06912e6 1.21466
\(162\) 0 0
\(163\) 477011. + 477011.i 0.110145 + 0.110145i 0.760032 0.649886i \(-0.225183\pi\)
−0.649886 + 0.760032i \(0.725183\pi\)
\(164\) 0 0
\(165\) 3.36486e6 + 3.36486e6i 0.749058 + 0.749058i
\(166\) 0 0
\(167\) 566969. 0.121733 0.0608667 0.998146i \(-0.480614\pi\)
0.0608667 + 0.998146i \(0.480614\pi\)
\(168\) 0 0
\(169\) 4.10032e6i 0.849489i
\(170\) 0 0
\(171\) −1.61408e6 + 1.61408e6i −0.322803 + 0.322803i
\(172\) 0 0
\(173\) −6.06937e6 + 6.06937e6i −1.17221 + 1.17221i −0.190528 + 0.981682i \(0.561020\pi\)
−0.981682 + 0.190528i \(0.938980\pi\)
\(174\) 0 0
\(175\) 918705.i 0.171420i
\(176\) 0 0
\(177\) −884356. −0.159480
\(178\) 0 0
\(179\) 6.12901e6 + 6.12901e6i 1.06864 + 1.06864i 0.997464 + 0.0711761i \(0.0226752\pi\)
0.0711761 + 0.997464i \(0.477325\pi\)
\(180\) 0 0
\(181\) 3.73717e6 + 3.73717e6i 0.630242 + 0.630242i 0.948129 0.317886i \(-0.102973\pi\)
−0.317886 + 0.948129i \(0.602973\pi\)
\(182\) 0 0
\(183\) 1.49944e6 0.244667
\(184\) 0 0
\(185\) 2.14512e6i 0.338795i
\(186\) 0 0
\(187\) −2.24771e6 + 2.24771e6i −0.343728 + 0.343728i
\(188\) 0 0
\(189\) −907206. + 907206.i −0.134376 + 0.134376i
\(190\) 0 0
\(191\) 7.48444e6i 1.07414i 0.843539 + 0.537068i \(0.180468\pi\)
−0.843539 + 0.537068i \(0.819532\pi\)
\(192\) 0 0
\(193\) 1.84618e6 0.256804 0.128402 0.991722i \(-0.459015\pi\)
0.128402 + 0.991722i \(0.459015\pi\)
\(194\) 0 0
\(195\) −1.27225e6 1.27225e6i −0.171581 0.171581i
\(196\) 0 0
\(197\) 5.01650e6 + 5.01650e6i 0.656149 + 0.656149i 0.954467 0.298318i \(-0.0964255\pi\)
−0.298318 + 0.954467i \(0.596425\pi\)
\(198\) 0 0
\(199\) 3.90212e6 0.495155 0.247578 0.968868i \(-0.420365\pi\)
0.247578 + 0.968868i \(0.420365\pi\)
\(200\) 0 0
\(201\) 1.67151e6i 0.205835i
\(202\) 0 0
\(203\) 8.70020e6 8.70020e6i 1.04002 1.04002i
\(204\) 0 0
\(205\) 7.47331e6 7.47331e6i 0.867464 0.867464i
\(206\) 0 0
\(207\) 3.63687e6i 0.410031i
\(208\) 0 0
\(209\) 2.11760e7 2.31956
\(210\) 0 0
\(211\) −9.10818e6 9.10818e6i −0.969581 0.969581i 0.0299699 0.999551i \(-0.490459\pi\)
−0.999551 + 0.0299699i \(0.990459\pi\)
\(212\) 0 0
\(213\) −5.73997e6 5.73997e6i −0.593978 0.593978i
\(214\) 0 0
\(215\) −1.20553e7 −1.21301
\(216\) 0 0
\(217\) 1.76947e7i 1.73166i
\(218\) 0 0
\(219\) −3.85723e6 + 3.85723e6i −0.367234 + 0.367234i
\(220\) 0 0
\(221\) 849855. 849855.i 0.0787350 0.0787350i
\(222\) 0 0
\(223\) 3.47876e6i 0.313697i 0.987623 + 0.156848i \(0.0501334\pi\)
−0.987623 + 0.156848i \(0.949867\pi\)
\(224\) 0 0
\(225\) −659129. −0.0578660
\(226\) 0 0
\(227\) 8.39488e6 + 8.39488e6i 0.717690 + 0.717690i 0.968132 0.250441i \(-0.0805758\pi\)
−0.250441 + 0.968132i \(0.580576\pi\)
\(228\) 0 0
\(229\) −9.40100e6 9.40100e6i −0.782830 0.782830i 0.197477 0.980308i \(-0.436725\pi\)
−0.980308 + 0.197477i \(0.936725\pi\)
\(230\) 0 0
\(231\) 1.19021e7 0.965580
\(232\) 0 0
\(233\) 1.24039e7i 0.980599i 0.871554 + 0.490299i \(0.163113\pi\)
−0.871554 + 0.490299i \(0.836887\pi\)
\(234\) 0 0
\(235\) 5.66826e6 5.66826e6i 0.436763 0.436763i
\(236\) 0 0
\(237\) 6.43907e6 6.43907e6i 0.483702 0.483702i
\(238\) 0 0
\(239\) 3.38975e6i 0.248299i 0.992264 + 0.124149i \(0.0396202\pi\)
−0.992264 + 0.124149i \(0.960380\pi\)
\(240\) 0 0
\(241\) −7.75493e6 −0.554022 −0.277011 0.960867i \(-0.589344\pi\)
−0.277011 + 0.960867i \(0.589344\pi\)
\(242\) 0 0
\(243\) −650880. 650880.i −0.0453609 0.0453609i
\(244\) 0 0
\(245\) −280865. 280865.i −0.0190985 0.0190985i
\(246\) 0 0
\(247\) −8.00660e6 −0.531322
\(248\) 0 0
\(249\) 1.66515e6i 0.107858i
\(250\) 0 0
\(251\) −6.71306e6 + 6.71306e6i −0.424521 + 0.424521i −0.886757 0.462236i \(-0.847047\pi\)
0.462236 + 0.886757i \(0.347047\pi\)
\(252\) 0 0
\(253\) 2.38570e7 2.38570e7i 1.47317 1.47317i
\(254\) 0 0
\(255\) 2.97659e6i 0.179514i
\(256\) 0 0
\(257\) −1.78180e7 −1.04969 −0.524844 0.851198i \(-0.675876\pi\)
−0.524844 + 0.851198i \(0.675876\pi\)
\(258\) 0 0
\(259\) 3.79384e6 + 3.79384e6i 0.218364 + 0.218364i
\(260\) 0 0
\(261\) 6.24200e6 + 6.24200e6i 0.351077 + 0.351077i
\(262\) 0 0
\(263\) 1.27225e7 0.699365 0.349683 0.936868i \(-0.386289\pi\)
0.349683 + 0.936868i \(0.386289\pi\)
\(264\) 0 0
\(265\) 8.73583e6i 0.469425i
\(266\) 0 0
\(267\) −9.71152e6 + 9.71152e6i −0.510215 + 0.510215i
\(268\) 0 0
\(269\) −1.77250e7 + 1.77250e7i −0.910604 + 0.910604i −0.996320 0.0857153i \(-0.972682\pi\)
0.0857153 + 0.996320i \(0.472682\pi\)
\(270\) 0 0
\(271\) 4.03947e6i 0.202963i 0.994837 + 0.101481i \(0.0323582\pi\)
−0.994837 + 0.101481i \(0.967642\pi\)
\(272\) 0 0
\(273\) −4.50017e6 −0.221178
\(274\) 0 0
\(275\) 4.32373e6 + 4.32373e6i 0.207903 + 0.207903i
\(276\) 0 0
\(277\) −2.71660e6 2.71660e6i −0.127816 0.127816i 0.640305 0.768121i \(-0.278808\pi\)
−0.768121 + 0.640305i \(0.778808\pi\)
\(278\) 0 0
\(279\) 1.26951e7 0.584554
\(280\) 0 0
\(281\) 3.00632e6i 0.135493i 0.997703 + 0.0677464i \(0.0215809\pi\)
−0.997703 + 0.0677464i \(0.978419\pi\)
\(282\) 0 0
\(283\) 7.04093e6 7.04093e6i 0.310650 0.310650i −0.534512 0.845161i \(-0.679504\pi\)
0.845161 + 0.534512i \(0.179504\pi\)
\(284\) 0 0
\(285\) −1.40214e7 + 1.40214e7i −0.605700 + 0.605700i
\(286\) 0 0
\(287\) 2.64344e7i 1.11821i
\(288\) 0 0
\(289\) −2.21492e7 −0.917625
\(290\) 0 0
\(291\) 5.11006e6 + 5.11006e6i 0.207370 + 0.207370i
\(292\) 0 0
\(293\) 1.09177e6 + 1.09177e6i 0.0434037 + 0.0434037i 0.728476 0.685072i \(-0.240229\pi\)
−0.685072 + 0.728476i \(0.740229\pi\)
\(294\) 0 0
\(295\) −7.68234e6 −0.299245
\(296\) 0 0
\(297\) 8.53923e6i 0.325949i
\(298\) 0 0
\(299\) −9.02029e6 + 9.02029e6i −0.337448 + 0.337448i
\(300\) 0 0
\(301\) −2.13209e7 + 2.13209e7i −0.781819 + 0.781819i
\(302\) 0 0
\(303\) 1.00885e7i 0.362660i
\(304\) 0 0
\(305\) 1.30255e7 0.459087
\(306\) 0 0
\(307\) −3.35575e7 3.35575e7i −1.15978 1.15978i −0.984524 0.175252i \(-0.943926\pi\)
−0.175252 0.984524i \(-0.556074\pi\)
\(308\) 0 0
\(309\) 1.64513e7 + 1.64513e7i 0.557603 + 0.557603i
\(310\) 0 0
\(311\) −7.25043e6 −0.241036 −0.120518 0.992711i \(-0.538456\pi\)
−0.120518 + 0.992711i \(0.538456\pi\)
\(312\) 0 0
\(313\) 5.86126e6i 0.191143i −0.995423 0.0955714i \(-0.969532\pi\)
0.995423 0.0955714i \(-0.0304678\pi\)
\(314\) 0 0
\(315\) −7.88084e6 + 7.88084e6i −0.252139 + 0.252139i
\(316\) 0 0
\(317\) −1.42197e7 + 1.42197e7i −0.446388 + 0.446388i −0.894152 0.447764i \(-0.852221\pi\)
0.447764 + 0.894152i \(0.352221\pi\)
\(318\) 0 0
\(319\) 8.18921e7i 2.52273i
\(320\) 0 0
\(321\) −3.56613e7 −1.07816
\(322\) 0 0
\(323\) −9.36623e6 9.36623e6i −0.277944 0.277944i
\(324\) 0 0
\(325\) −1.63480e6 1.63480e6i −0.0476226 0.0476226i
\(326\) 0 0
\(327\) 4.68695e6 0.134044
\(328\) 0 0
\(329\) 2.00496e7i 0.563013i
\(330\) 0 0
\(331\) −4.55754e7 + 4.55754e7i −1.25674 + 1.25674i −0.304106 + 0.952638i \(0.598358\pi\)
−0.952638 + 0.304106i \(0.901642\pi\)
\(332\) 0 0
\(333\) −2.72191e6 + 2.72191e6i −0.0737125 + 0.0737125i
\(334\) 0 0
\(335\) 1.45203e7i 0.386225i
\(336\) 0 0
\(337\) −5.70761e7 −1.49130 −0.745650 0.666338i \(-0.767861\pi\)
−0.745650 + 0.666338i \(0.767861\pi\)
\(338\) 0 0
\(339\) 1.35946e7 + 1.35946e7i 0.348954 + 0.348954i
\(340\) 0 0
\(341\) −8.32771e7 8.32771e7i −2.10021 2.10021i
\(342\) 0 0
\(343\) −4.08409e7 −1.01207
\(344\) 0 0
\(345\) 3.15932e7i 0.769373i
\(346\) 0 0
\(347\) 1.33582e7 1.33582e7i 0.319713 0.319713i −0.528944 0.848657i \(-0.677412\pi\)
0.848657 + 0.528944i \(0.177412\pi\)
\(348\) 0 0
\(349\) 5.81390e7 5.81390e7i 1.36770 1.36770i 0.503993 0.863708i \(-0.331864\pi\)
0.863708 0.503993i \(-0.168136\pi\)
\(350\) 0 0
\(351\) 3.22867e6i 0.0746625i
\(352\) 0 0
\(353\) −3.25741e7 −0.740539 −0.370269 0.928924i \(-0.620735\pi\)
−0.370269 + 0.928924i \(0.620735\pi\)
\(354\) 0 0
\(355\) −4.98627e7 4.98627e7i −1.11453 1.11453i
\(356\) 0 0
\(357\) −5.26436e6 5.26436e6i −0.115702 0.115702i
\(358\) 0 0
\(359\) −1.13980e7 −0.246346 −0.123173 0.992385i \(-0.539307\pi\)
−0.123173 + 0.992385i \(0.539307\pi\)
\(360\) 0 0
\(361\) 4.11947e7i 0.875628i
\(362\) 0 0
\(363\) 3.64880e7 3.64880e7i 0.762833 0.762833i
\(364\) 0 0
\(365\) −3.35075e7 + 3.35075e7i −0.689071 + 0.689071i
\(366\) 0 0
\(367\) 6.66527e6i 0.134840i −0.997725 0.0674201i \(-0.978523\pi\)
0.997725 0.0674201i \(-0.0214768\pi\)
\(368\) 0 0
\(369\) 1.89655e7 0.377473
\(370\) 0 0
\(371\) 1.54501e7 + 1.54501e7i 0.302559 + 0.302559i
\(372\) 0 0
\(373\) 3.13045e7 + 3.13045e7i 0.603226 + 0.603226i 0.941167 0.337941i \(-0.109731\pi\)
−0.337941 + 0.941167i \(0.609731\pi\)
\(374\) 0 0
\(375\) 2.72574e7 0.516881
\(376\) 0 0
\(377\) 3.09633e7i 0.577860i
\(378\) 0 0
\(379\) 3.59578e7 3.59578e7i 0.660503 0.660503i −0.294995 0.955499i \(-0.595318\pi\)
0.955499 + 0.294995i \(0.0953181\pi\)
\(380\) 0 0
\(381\) −3.15986e7 + 3.15986e7i −0.571338 + 0.571338i
\(382\) 0 0
\(383\) 4.29186e6i 0.0763923i 0.999270 + 0.0381962i \(0.0121612\pi\)
−0.999270 + 0.0381962i \(0.987839\pi\)
\(384\) 0 0
\(385\) 1.03393e8 1.81179
\(386\) 0 0
\(387\) −1.52968e7 1.52968e7i −0.263917 0.263917i
\(388\) 0 0
\(389\) 2.81391e7 + 2.81391e7i 0.478037 + 0.478037i 0.904503 0.426467i \(-0.140242\pi\)
−0.426467 + 0.904503i \(0.640242\pi\)
\(390\) 0 0
\(391\) −2.11041e7 −0.353050
\(392\) 0 0
\(393\) 2.61700e7i 0.431148i
\(394\) 0 0
\(395\) 5.59358e7 5.59358e7i 0.907608 0.907608i
\(396\) 0 0
\(397\) 102931. 102931.i 0.00164504 0.00164504i −0.706284 0.707929i \(-0.749630\pi\)
0.707929 + 0.706284i \(0.249630\pi\)
\(398\) 0 0
\(399\) 4.95963e7i 0.780783i
\(400\) 0 0
\(401\) −9.77419e7 −1.51582 −0.757910 0.652359i \(-0.773779\pi\)
−0.757910 + 0.652359i \(0.773779\pi\)
\(402\) 0 0
\(403\) 3.14869e7 + 3.14869e7i 0.481077 + 0.481077i
\(404\) 0 0
\(405\) −5.65415e6 5.65415e6i −0.0851142 0.0851142i
\(406\) 0 0
\(407\) 3.57102e7 0.529674
\(408\) 0 0
\(409\) 1.16185e8i 1.69817i −0.528255 0.849086i \(-0.677153\pi\)
0.528255 0.849086i \(-0.322847\pi\)
\(410\) 0 0
\(411\) 5.00615e7 5.00615e7i 0.721071 0.721071i
\(412\) 0 0
\(413\) −1.35869e7 + 1.35869e7i −0.192873 + 0.192873i
\(414\) 0 0
\(415\) 1.44650e7i 0.202383i
\(416\) 0 0
\(417\) −7.66304e7 −1.05680
\(418\) 0 0
\(419\) −1.00557e8 1.00557e8i −1.36701 1.36701i −0.864669 0.502341i \(-0.832472\pi\)
−0.502341 0.864669i \(-0.667528\pi\)
\(420\) 0 0
\(421\) −6.79990e7 6.79990e7i −0.911289 0.911289i 0.0850844 0.996374i \(-0.472884\pi\)
−0.996374 + 0.0850844i \(0.972884\pi\)
\(422\) 0 0
\(423\) 1.43847e7 0.190055
\(424\) 0 0
\(425\) 3.82481e6i 0.0498245i
\(426\) 0 0
\(427\) 2.30368e7 2.30368e7i 0.295895 0.295895i
\(428\) 0 0
\(429\) −2.11793e7 + 2.11793e7i −0.268250 + 0.268250i
\(430\) 0 0
\(431\) 8.48038e7i 1.05921i 0.848243 + 0.529607i \(0.177660\pi\)
−0.848243 + 0.529607i \(0.822340\pi\)
\(432\) 0 0
\(433\) 1.35872e8 1.67366 0.836830 0.547463i \(-0.184406\pi\)
0.836830 + 0.547463i \(0.184406\pi\)
\(434\) 0 0
\(435\) 5.42239e7 + 5.42239e7i 0.658753 + 0.658753i
\(436\) 0 0
\(437\) 9.94124e7 + 9.94124e7i 1.19123 + 1.19123i
\(438\) 0 0
\(439\) −2.21120e7 −0.261357 −0.130679 0.991425i \(-0.541716\pi\)
−0.130679 + 0.991425i \(0.541716\pi\)
\(440\) 0 0
\(441\) 712770.i 0.00831063i
\(442\) 0 0
\(443\) 1.39108e7 1.39108e7i 0.160008 0.160008i −0.622562 0.782570i \(-0.713908\pi\)
0.782570 + 0.622562i \(0.213908\pi\)
\(444\) 0 0
\(445\) −8.43634e7 + 8.43634e7i −0.957357 + 0.957357i
\(446\) 0 0
\(447\) 2.52228e7i 0.282404i
\(448\) 0 0
\(449\) 3.72059e7 0.411029 0.205515 0.978654i \(-0.434113\pi\)
0.205515 + 0.978654i \(0.434113\pi\)
\(450\) 0 0
\(451\) −1.24409e8 1.24409e8i −1.35620 1.35620i
\(452\) 0 0
\(453\) −2.27818e7 2.27818e7i −0.245071 0.245071i
\(454\) 0 0
\(455\) −3.90927e7 −0.415013
\(456\) 0 0
\(457\) 1.71918e8i 1.80124i −0.434603 0.900622i \(-0.643111\pi\)
0.434603 0.900622i \(-0.356889\pi\)
\(458\) 0 0
\(459\) 3.77694e6 3.77694e6i 0.0390573 0.0390573i
\(460\) 0 0
\(461\) 5.67322e7 5.67322e7i 0.579064 0.579064i −0.355581 0.934645i \(-0.615717\pi\)
0.934645 + 0.355581i \(0.115717\pi\)
\(462\) 0 0
\(463\) 1.56654e8i 1.57833i −0.614180 0.789166i \(-0.710513\pi\)
0.614180 0.789166i \(-0.289487\pi\)
\(464\) 0 0
\(465\) 1.10282e8 1.09684
\(466\) 0 0
\(467\) 1.11254e8 + 1.11254e8i 1.09236 + 1.09236i 0.995276 + 0.0970811i \(0.0309506\pi\)
0.0970811 + 0.995276i \(0.469049\pi\)
\(468\) 0 0
\(469\) −2.56804e7 2.56804e7i −0.248933 0.248933i
\(470\) 0 0
\(471\) 2.19229e7 0.209814
\(472\) 0 0
\(473\) 2.00687e8i 1.89642i
\(474\) 0 0
\(475\) −1.80170e7 + 1.80170e7i −0.168114 + 0.168114i
\(476\) 0 0
\(477\) −1.10847e7 + 1.10847e7i −0.102134 + 0.102134i
\(478\) 0 0
\(479\) 4.71347e7i 0.428878i −0.976737 0.214439i \(-0.931208\pi\)
0.976737 0.214439i \(-0.0687924\pi\)
\(480\) 0 0
\(481\) −1.35020e7 −0.121328
\(482\) 0 0
\(483\) 5.58754e7 + 5.58754e7i 0.495883 + 0.495883i
\(484\) 0 0
\(485\) 4.43907e7 + 4.43907e7i 0.389105 + 0.389105i
\(486\) 0 0
\(487\) −1.00829e8 −0.872972 −0.436486 0.899711i \(-0.643777\pi\)
−0.436486 + 0.899711i \(0.643777\pi\)
\(488\) 0 0
\(489\) 1.05159e7i 0.0899333i
\(490\) 0 0
\(491\) −9.12387e7 + 9.12387e7i −0.770788 + 0.770788i −0.978244 0.207457i \(-0.933481\pi\)
0.207457 + 0.978244i \(0.433481\pi\)
\(492\) 0 0
\(493\) −3.62212e7 + 3.62212e7i −0.302289 + 0.302289i
\(494\) 0 0
\(495\) 7.41798e7i 0.611603i
\(496\) 0 0
\(497\) −1.76373e8 −1.43669
\(498\) 0 0
\(499\) −7.61214e7 7.61214e7i −0.612640 0.612640i 0.330994 0.943633i \(-0.392616\pi\)
−0.943633 + 0.330994i \(0.892616\pi\)
\(500\) 0 0
\(501\) 6.24953e6 + 6.24953e6i 0.0496975 + 0.0496975i
\(502\) 0 0
\(503\) 1.11499e8 0.876127 0.438063 0.898944i \(-0.355665\pi\)
0.438063 + 0.898944i \(0.355665\pi\)
\(504\) 0 0
\(505\) 8.76383e7i 0.680487i
\(506\) 0 0
\(507\) −4.51966e7 + 4.51966e7i −0.346802 + 0.346802i
\(508\) 0 0
\(509\) −7.40728e7 + 7.40728e7i −0.561701 + 0.561701i −0.929790 0.368089i \(-0.880012\pi\)
0.368089 + 0.929790i \(0.380012\pi\)
\(510\) 0 0
\(511\) 1.18522e8i 0.888253i
\(512\) 0 0
\(513\) −3.55831e7 −0.263567
\(514\) 0 0
\(515\) 1.42911e8 + 1.42911e8i 1.04627 + 1.04627i
\(516\) 0 0
\(517\) −9.43603e7 9.43603e7i −0.682838 0.682838i
\(518\) 0 0
\(519\) −1.33802e8 −0.957106
\(520\) 0 0
\(521\) 8.30371e7i 0.587163i −0.955934 0.293582i \(-0.905153\pi\)
0.955934 0.293582i \(-0.0948472\pi\)
\(522\) 0 0
\(523\) −8.42929e7 + 8.42929e7i −0.589231 + 0.589231i −0.937423 0.348192i \(-0.886796\pi\)
0.348192 + 0.937423i \(0.386796\pi\)
\(524\) 0 0
\(525\) −1.01266e7 + 1.01266e7i −0.0699820 + 0.0699820i
\(526\) 0 0
\(527\) 7.36676e7i 0.503320i
\(528\) 0 0
\(529\) 7.59612e7 0.513127
\(530\) 0 0
\(531\) −9.74799e6 9.74799e6i −0.0651076 0.0651076i
\(532\) 0 0
\(533\) 4.70390e7 + 4.70390e7i 0.310653 + 0.310653i
\(534\) 0 0
\(535\) −3.09788e8 −2.02303
\(536\) 0 0
\(537\) 1.35117e8i 0.872541i
\(538\) 0 0
\(539\) −4.67561e6 + 4.67561e6i −0.0298587 + 0.0298587i
\(540\) 0 0
\(541\) 2.21061e8 2.21061e8i 1.39611 1.39611i 0.585287 0.810826i \(-0.300982\pi\)
0.810826 0.585287i \(-0.199018\pi\)
\(542\) 0 0
\(543\) 8.23875e7i 0.514591i
\(544\) 0 0
\(545\) 4.07152e7 0.251517
\(546\) 0 0
\(547\) 1.41321e8 + 1.41321e8i 0.863464 + 0.863464i 0.991739 0.128275i \(-0.0409440\pi\)
−0.128275 + 0.991739i \(0.540944\pi\)
\(548\) 0 0
\(549\) 1.65279e7 + 1.65279e7i 0.0998848 + 0.0998848i
\(550\) 0 0
\(551\) 3.41246e8 2.03992
\(552\) 0 0
\(553\) 1.97855e8i 1.16996i
\(554\) 0 0
\(555\) −2.36451e7 + 2.36451e7i −0.138313 + 0.138313i
\(556\) 0 0
\(557\) 4.48748e7 4.48748e7i 0.259679 0.259679i −0.565244 0.824924i \(-0.691218\pi\)
0.824924 + 0.565244i \(0.191218\pi\)
\(558\) 0 0
\(559\) 7.58793e7i 0.434398i
\(560\) 0 0
\(561\) −4.95517e7 −0.280653
\(562\) 0 0
\(563\) −6.88093e7 6.88093e7i −0.385586 0.385586i 0.487523 0.873110i \(-0.337900\pi\)
−0.873110 + 0.487523i \(0.837900\pi\)
\(564\) 0 0
\(565\) 1.18096e8 + 1.18096e8i 0.654770 + 0.654770i
\(566\) 0 0
\(567\) −1.99997e7 −0.109717
\(568\) 0 0
\(569\) 2.18716e7i 0.118726i 0.998236 + 0.0593628i \(0.0189069\pi\)
−0.998236 + 0.0593628i \(0.981093\pi\)
\(570\) 0 0
\(571\) −6.28910e7 + 6.28910e7i −0.337816 + 0.337816i −0.855545 0.517729i \(-0.826778\pi\)
0.517729 + 0.855545i \(0.326778\pi\)
\(572\) 0 0
\(573\) −8.24987e7 + 8.24987e7i −0.438514 + 0.438514i
\(574\) 0 0
\(575\) 4.05962e7i 0.213541i
\(576\) 0 0
\(577\) 1.05532e8 0.549359 0.274680 0.961536i \(-0.411428\pi\)
0.274680 + 0.961536i \(0.411428\pi\)
\(578\) 0 0
\(579\) 2.03499e7 + 2.03499e7i 0.104840 + 0.104840i
\(580\) 0 0
\(581\) 2.55827e7 + 2.55827e7i 0.130442 + 0.130442i
\(582\) 0 0
\(583\) 1.45427e8 0.733902
\(584\) 0 0
\(585\) 2.80472e7i 0.140095i
\(586\) 0 0
\(587\) −2.86519e6 + 2.86519e6i −0.0141657 + 0.0141657i −0.714154 0.699988i \(-0.753188\pi\)
0.699988 + 0.714154i \(0.253188\pi\)
\(588\) 0 0
\(589\) 3.47016e8 3.47016e8i 1.69826 1.69826i
\(590\) 0 0
\(591\) 1.10591e8i 0.535743i
\(592\) 0 0
\(593\) 7.90158e7 0.378922 0.189461 0.981888i \(-0.439326\pi\)
0.189461 + 0.981888i \(0.439326\pi\)
\(594\) 0 0
\(595\) −4.57311e7 4.57311e7i −0.217101 0.217101i
\(596\) 0 0
\(597\) 4.30119e7 + 4.30119e7i 0.202146 + 0.202146i
\(598\) 0 0
\(599\) 3.85910e8 1.79558 0.897792 0.440419i \(-0.145170\pi\)
0.897792 + 0.440419i \(0.145170\pi\)
\(600\) 0 0
\(601\) 2.00376e8i 0.923044i 0.887129 + 0.461522i \(0.152697\pi\)
−0.887129 + 0.461522i \(0.847303\pi\)
\(602\) 0 0
\(603\) 1.84245e7 1.84245e7i 0.0840320 0.0840320i
\(604\) 0 0
\(605\) 3.16968e8 3.16968e8i 1.43136 1.43136i
\(606\) 0 0
\(607\) 1.61943e7i 0.0724094i 0.999344 + 0.0362047i \(0.0115268\pi\)
−0.999344 + 0.0362047i \(0.988473\pi\)
\(608\) 0 0
\(609\) 1.91799e8 0.849172
\(610\) 0 0
\(611\) 3.56775e7 + 3.56775e7i 0.156412 + 0.156412i
\(612\) 0 0
\(613\) −1.23680e8 1.23680e8i −0.536932 0.536932i 0.385695 0.922626i \(-0.373962\pi\)
−0.922626 + 0.385695i \(0.873962\pi\)
\(614\) 0 0
\(615\) 1.64752e8 0.708282
\(616\) 0 0
\(617\) 9.47161e7i 0.403244i 0.979463 + 0.201622i \(0.0646213\pi\)
−0.979463 + 0.201622i \(0.935379\pi\)
\(618\) 0 0
\(619\) −9.92581e7 + 9.92581e7i −0.418499 + 0.418499i −0.884686 0.466187i \(-0.845627\pi\)
0.466187 + 0.884686i \(0.345627\pi\)
\(620\) 0 0
\(621\) −4.00881e7 + 4.00881e7i −0.167394 + 0.167394i
\(622\) 0 0
\(623\) 2.98408e8i 1.23409i
\(624\) 0 0
\(625\) 2.79165e8 1.14346
\(626\) 0 0
\(627\) 2.33417e8 + 2.33417e8i 0.946955 + 0.946955i
\(628\) 0 0
\(629\) −1.57948e7 1.57948e7i −0.0634690 0.0634690i
\(630\) 0 0
\(631\) −1.48727e8 −0.591975 −0.295987 0.955192i \(-0.595649\pi\)
−0.295987 + 0.955192i \(0.595649\pi\)
\(632\) 0 0
\(633\) 2.00793e8i 0.791659i
\(634\) 0 0
\(635\) −2.74495e8 + 2.74495e8i −1.07205 + 1.07205i
\(636\) 0 0
\(637\) 1.76784e6 1.76784e6i 0.00683950 0.00683950i
\(638\) 0 0
\(639\) 1.26540e8i 0.484981i
\(640\) 0 0
\(641\) 1.07227e8 0.407127 0.203564 0.979062i \(-0.434748\pi\)
0.203564 + 0.979062i \(0.434748\pi\)
\(642\) 0 0
\(643\) 1.39954e8 + 1.39954e8i 0.526444 + 0.526444i 0.919510 0.393067i \(-0.128586\pi\)
−0.393067 + 0.919510i \(0.628586\pi\)
\(644\) 0 0
\(645\) −1.32882e8 1.32882e8i −0.495208 0.495208i
\(646\) 0 0
\(647\) 1.74204e8 0.643198 0.321599 0.946876i \(-0.395780\pi\)
0.321599 + 0.946876i \(0.395780\pi\)
\(648\) 0 0
\(649\) 1.27889e8i 0.467842i
\(650\) 0 0
\(651\) 1.95043e8 1.95043e8i 0.706948 0.706948i
\(652\) 0 0
\(653\) −2.12946e8 + 2.12946e8i −0.764769 + 0.764769i −0.977180 0.212411i \(-0.931868\pi\)
0.212411 + 0.977180i \(0.431868\pi\)
\(654\) 0 0
\(655\) 2.27337e8i 0.808997i
\(656\) 0 0
\(657\) −8.50342e7 −0.299846
\(658\) 0 0
\(659\) 1.00416e8 + 1.00416e8i 0.350870 + 0.350870i 0.860433 0.509563i \(-0.170193\pi\)
−0.509563 + 0.860433i \(0.670193\pi\)
\(660\) 0 0
\(661\) 1.71149e8 + 1.71149e8i 0.592611 + 0.592611i 0.938336 0.345725i \(-0.112367\pi\)
−0.345725 + 0.938336i \(0.612367\pi\)
\(662\) 0 0
\(663\) 1.87354e7 0.0642869
\(664\) 0 0
\(665\) 4.30839e8i 1.46504i
\(666\) 0 0
\(667\) 3.84449e8 3.84449e8i 1.29557 1.29557i
\(668\) 0 0
\(669\) −3.83454e7 + 3.83454e7i −0.128066 + 0.128066i
\(670\) 0 0
\(671\) 2.16838e8i 0.717740i
\(672\) 0 0
\(673\) −4.72365e8 −1.54965 −0.774823 0.632178i \(-0.782161\pi\)
−0.774823 + 0.632178i \(0.782161\pi\)
\(674\) 0 0
\(675\) −7.26539e6 7.26539e6i −0.0236237 0.0236237i
\(676\) 0 0
\(677\) 1.35510e8 + 1.35510e8i 0.436721 + 0.436721i 0.890907 0.454186i \(-0.150070\pi\)
−0.454186 + 0.890907i \(0.650070\pi\)
\(678\) 0 0
\(679\) 1.57018e8 0.501580
\(680\) 0 0
\(681\) 1.85069e8i 0.585992i
\(682\) 0 0
\(683\) −1.86498e8 + 1.86498e8i −0.585344 + 0.585344i −0.936367 0.351023i \(-0.885834\pi\)
0.351023 + 0.936367i \(0.385834\pi\)
\(684\) 0 0
\(685\) 4.34881e8 4.34881e8i 1.35300 1.35300i
\(686\) 0 0
\(687\) 2.07249e8i 0.639178i
\(688\) 0 0
\(689\) −5.49856e7 −0.168109
\(690\) 0 0
\(691\) −2.83368e8 2.83368e8i −0.858849 0.858849i 0.132354 0.991203i \(-0.457746\pi\)
−0.991203 + 0.132354i \(0.957746\pi\)
\(692\) 0 0
\(693\) 1.31194e8 + 1.31194e8i 0.394196 + 0.394196i
\(694\) 0 0
\(695\) −6.65684e8 −1.98296
\(696\) 0 0
\(697\) 1.10054e8i 0.325017i
\(698\) 0 0
\(699\) −1.36725e8 + 1.36725e8i −0.400328 + 0.400328i
\(700\) 0 0
\(701\) 2.30227e8 2.30227e8i 0.668347 0.668347i −0.288986 0.957333i \(-0.593318\pi\)
0.957333 + 0.288986i \(0.0933182\pi\)
\(702\) 0 0
\(703\) 1.48805e8i 0.428303i
\(704\) 0 0
\(705\) 1.24959e8 0.356615
\(706\) 0 0
\(707\) −1.54996e8 1.54996e8i −0.438594 0.438594i
\(708\) 0 0
\(709\) −4.24526e7 4.24526e7i −0.119115 0.119115i 0.645037 0.764152i \(-0.276842\pi\)
−0.764152 + 0.645037i \(0.776842\pi\)
\(710\) 0 0
\(711\) 1.41952e8 0.394941
\(712\) 0 0
\(713\) 7.81902e8i 2.15717i
\(714\) 0 0
\(715\) −1.83983e8 + 1.83983e8i −0.503339 + 0.503339i
\(716\) 0 0
\(717\) −3.73642e7 + 3.73642e7i −0.101367 + 0.101367i
\(718\) 0 0
\(719\) 2.52511e8i 0.679351i −0.940543 0.339676i \(-0.889683\pi\)
0.940543 0.339676i \(-0.110317\pi\)
\(720\) 0 0
\(721\) 5.05503e8 1.34871
\(722\) 0 0
\(723\) −8.54803e7 8.54803e7i −0.226178 0.226178i
\(724\) 0 0
\(725\) 6.96758e7 + 6.96758e7i 0.182839 + 0.182839i
\(726\) 0 0
\(727\) −5.00217e8 −1.30183 −0.650916 0.759150i \(-0.725615\pi\)
−0.650916 + 0.759150i \(0.725615\pi\)
\(728\) 0 0
\(729\) 1.43489e7i 0.0370370i
\(730\) 0 0
\(731\) 8.87646e7 8.87646e7i 0.227241 0.227241i
\(732\) 0 0
\(733\) −3.76387e8 + 3.76387e8i −0.955703 + 0.955703i −0.999060 0.0433569i \(-0.986195\pi\)
0.0433569 + 0.999060i \(0.486195\pi\)
\(734\) 0 0
\(735\) 6.19179e6i 0.0155939i
\(736\) 0 0
\(737\) −2.41721e8 −0.603826
\(738\) 0 0
\(739\) 1.52550e8 + 1.52550e8i 0.377989 + 0.377989i 0.870376 0.492387i \(-0.163876\pi\)
−0.492387 + 0.870376i \(0.663876\pi\)
\(740\) 0 0
\(741\) −8.82544e7 8.82544e7i −0.216911 0.216911i
\(742\) 0 0
\(743\) −5.90150e8 −1.43878 −0.719392 0.694604i \(-0.755580\pi\)
−0.719392 + 0.694604i \(0.755580\pi\)
\(744\) 0 0
\(745\) 2.19109e8i 0.529897i
\(746\) 0 0
\(747\) −1.83544e7 + 1.83544e7i −0.0440330 + 0.0440330i
\(748\) 0 0
\(749\) −5.47887e8 + 5.47887e8i −1.30390 + 1.30390i
\(750\) 0 0
\(751\) 9.72140e7i 0.229514i −0.993394 0.114757i \(-0.963391\pi\)
0.993394 0.114757i \(-0.0366089\pi\)
\(752\) 0 0
\(753\) −1.47992e8 −0.346620
\(754\) 0 0
\(755\) −1.97904e8 1.97904e8i −0.459847 0.459847i
\(756\) 0 0
\(757\) 9.05372e7 + 9.05372e7i 0.208708 + 0.208708i 0.803718 0.595010i \(-0.202852\pi\)
−0.595010 + 0.803718i \(0.702852\pi\)
\(758\) 0 0
\(759\) 5.25937e8 1.20284
\(760\) 0 0
\(761\) 1.81374e8i 0.411549i −0.978599 0.205774i \(-0.934029\pi\)
0.978599 0.205774i \(-0.0659713\pi\)
\(762\) 0 0
\(763\) 7.20085e7 7.20085e7i 0.162110 0.162110i
\(764\) 0 0
\(765\) 3.28100e7 3.28100e7i 0.0732862 0.0732862i
\(766\) 0 0
\(767\) 4.83546e7i 0.107165i
\(768\) 0 0
\(769\) 3.12737e8 0.687701 0.343851 0.939024i \(-0.388269\pi\)
0.343851 + 0.939024i \(0.388269\pi\)
\(770\) 0 0
\(771\) −1.96403e8 1.96403e8i −0.428533 0.428533i
\(772\) 0 0
\(773\) −5.33319e8 5.33319e8i −1.15464 1.15464i −0.985610 0.169035i \(-0.945935\pi\)
−0.169035 0.985610i \(-0.554065\pi\)
\(774\) 0 0
\(775\) 1.41708e8 0.304432
\(776\) 0 0
\(777\) 8.36368e7i 0.178293i
\(778\) 0 0
\(779\) 5.18415e8 5.18415e8i 1.09664 1.09664i
\(780\) 0 0
\(781\) −8.30072e8 + 8.30072e8i −1.74246 + 1.74246i
\(782\) 0 0
\(783\) 1.37608e8i 0.286653i
\(784\) 0 0
\(785\) 1.90443e8 0.393691
\(786\) 0 0
\(787\) 2.52168e7 + 2.52168e7i 0.0517328 + 0.0517328i 0.732500 0.680767i \(-0.238353\pi\)
−0.680767 + 0.732500i \(0.738353\pi\)
\(788\) 0 0
\(789\) 1.40236e8 + 1.40236e8i 0.285515 + 0.285515i
\(790\) 0 0
\(791\) 4.17726e8 0.844037
\(792\) 0 0
\(793\) 8.19859e7i 0.164407i
\(794\) 0 0
\(795\) −9.62925e7 + 9.62925e7i −0.191642 + 0.191642i
\(796\) 0 0
\(797\) 2.19556e7 2.19556e7i 0.0433682 0.0433682i −0.685090 0.728458i \(-0.740237\pi\)
0.728458 + 0.685090i \(0.240237\pi\)
\(798\) 0 0
\(799\) 8.34719e7i 0.163644i
\(800\) 0 0
\(801\) −2.14094e8 −0.416589
\(802\) 0 0
\(803\) 5.57805e8 + 5.57805e8i 1.07730 + 1.07730i
\(804\) 0 0
\(805\) 4.85386e8 + 4.85386e8i 0.930464 + 0.930464i
\(806\) 0 0
\(807\) −3.90755e8 −0.743505
\(808\) 0 0
\(809\) 4.91857e7i 0.0928951i −0.998921 0.0464476i \(-0.985210\pi\)
0.998921 0.0464476i \(-0.0147900\pi\)
\(810\) 0 0
\(811\) 1.05349e8 1.05349e8i 0.197501 0.197501i −0.601427 0.798928i \(-0.705401\pi\)
0.798928 + 0.601427i \(0.205401\pi\)
\(812\) 0 0
\(813\) −4.45259e7 + 4.45259e7i −0.0828592 + 0.0828592i
\(814\) 0 0
\(815\) 9.13510e7i 0.168749i
\(816\) 0 0
\(817\) −8.36264e8 −1.53348
\(818\) 0 0
\(819\) −4.96040e7 4.96040e7i −0.0902953 0.0902953i
\(820\) 0 0
\(821\) 1.42572e8 + 1.42572e8i 0.257635 + 0.257635i 0.824091 0.566457i \(-0.191686\pi\)
−0.566457 + 0.824091i \(0.691686\pi\)
\(822\) 0 0
\(823\) 8.79812e8 1.57830 0.789152 0.614199i \(-0.210521\pi\)
0.789152 + 0.614199i \(0.210521\pi\)
\(824\) 0 0
\(825\) 9.53184e7i 0.169752i
\(826\) 0 0
\(827\) 2.06154e8 2.06154e8i 0.364481 0.364481i −0.500979 0.865460i \(-0.667027\pi\)
0.865460 + 0.500979i \(0.167027\pi\)
\(828\) 0 0
\(829\) 2.98660e7 2.98660e7i 0.0524219 0.0524219i −0.680410 0.732832i \(-0.738198\pi\)
0.732832 + 0.680410i \(0.238198\pi\)
\(830\) 0 0
\(831\) 5.98884e7i 0.104361i
\(832\) 0 0
\(833\) 4.13608e6 0.00715573
\(834\) 0 0
\(835\) 5.42893e7 + 5.42893e7i 0.0932512 + 0.0932512i
\(836\) 0 0
\(837\) 1.39935e8 + 1.39935e8i 0.238643 + 0.238643i
\(838\) 0 0
\(839\) −7.67224e8 −1.29908 −0.649541 0.760327i \(-0.725039\pi\)
−0.649541 + 0.760327i \(0.725039\pi\)
\(840\) 0 0
\(841\) 7.24847e8i 1.21859i
\(842\) 0 0
\(843\) −3.31378e7 + 3.31378e7i −0.0553147 + 0.0553147i
\(844\) 0 0
\(845\) −3.92620e8 + 3.92620e8i −0.650732 + 0.650732i
\(846\) 0 0
\(847\) 1.12117e9i 1.84511i
\(848\) 0 0
\(849\) 1.55220e8 0.253644
\(850\) 0 0
\(851\) 1.67644e8 + 1.67644e8i 0.272020 + 0.272020i
\(852\) 0 0
\(853\) −8.60861e8 8.60861e8i −1.38703 1.38703i −0.831489 0.555542i \(-0.812511\pi\)
−0.555542 0.831489i \(-0.687489\pi\)
\(854\) 0 0
\(855\) −3.09108e8 −0.494552
\(856\) 0 0
\(857\) 5.10264e8i 0.810685i −0.914165 0.405343i \(-0.867152\pi\)
0.914165 0.405343i \(-0.132848\pi\)
\(858\) 0 0
\(859\) 2.06652e8 2.06652e8i 0.326032 0.326032i −0.525043 0.851076i \(-0.675951\pi\)
0.851076 + 0.525043i \(0.175951\pi\)
\(860\) 0 0
\(861\) 2.91379e8 2.91379e8i 0.456509 0.456509i
\(862\) 0 0
\(863\) 1.09999e9i 1.71142i 0.517456 + 0.855710i \(0.326879\pi\)
−0.517456 + 0.855710i \(0.673121\pi\)
\(864\) 0 0
\(865\) −1.16233e9 −1.79589
\(866\) 0 0
\(867\) −2.44144e8 2.44144e8i −0.374619 0.374619i
\(868\) 0 0
\(869\) −9.31171e8 9.31171e8i −1.41896 1.41896i
\(870\) 0 0
\(871\) 9.13943e7 0.138314
\(872\) 0 0
\(873\) 1.12653e8i 0.169317i
\(874\) 0 0
\(875\) 4.18772e8 4.18772e8i 0.625106 0.625106i
\(876\) 0 0
\(877\) −9.87963e7 + 9.87963e7i −0.146468 + 0.146468i −0.776538 0.630070i \(-0.783026\pi\)
0.630070 + 0.776538i \(0.283026\pi\)
\(878\) 0 0
\(879\) 2.40685e7i 0.0354390i
\(880\) 0 0
\(881\) 7.70267e8 1.12645 0.563227 0.826302i \(-0.309560\pi\)
0.563227 + 0.826302i \(0.309560\pi\)
\(882\) 0 0
\(883\) −8.65914e8 8.65914e8i −1.25775 1.25775i −0.952167 0.305578i \(-0.901150\pi\)
−0.305578 0.952167i \(-0.598850\pi\)
\(884\) 0 0
\(885\) −8.46802e7 8.46802e7i −0.122166 0.122166i
\(886\) 0 0
\(887\) −4.32770e8 −0.620136 −0.310068 0.950714i \(-0.600352\pi\)
−0.310068 + 0.950714i \(0.600352\pi\)
\(888\) 0 0
\(889\) 9.70939e8i 1.38193i
\(890\) 0 0
\(891\) −9.41254e7 + 9.41254e7i −0.133068 + 0.133068i
\(892\) 0 0
\(893\) 3.93200e8 3.93200e8i 0.552153 0.552153i
\(894\) 0 0
\(895\) 1.17375e9i 1.63722i
\(896\) 0 0
\(897\) −1.98856e8 −0.275525
\(898\) 0 0
\(899\) −1.34199e9 1.34199e9i −1.84701 1.84701i
\(900\) 0 0
\(901\) −6.43228e7 6.43228e7i −0.0879409 0.0879409i
\(902\) 0 0
\(903\) −4.70028e8 −0.638353
\(904\) 0 0
\(905\) 7.15695e8i 0.965567i
\(906\) 0 0
\(907\) −1.78126e8 + 1.78126e8i −0.238729 + 0.238729i −0.816324 0.577595i \(-0.803991\pi\)
0.577595 + 0.816324i \(0.303991\pi\)
\(908\) 0 0
\(909\) 1.11203e8 1.11203e8i 0.148055 0.148055i
\(910\) 0 0
\(911\) 1.02858e9i 1.36045i −0.733002 0.680226i \(-0.761882\pi\)
0.733002 0.680226i \(-0.238118\pi\)
\(912\) 0 0
\(913\) 2.40801e8 0.316407
\(914\) 0 0
\(915\) 1.43576e8 + 1.43576e8i 0.187422 + 0.187422i
\(916\) 0 0
\(917\) 4.02066e8 + 4.02066e8i 0.521422 + 0.521422i
\(918\) 0 0
\(919\) 6.52230e7 0.0840339 0.0420169 0.999117i \(-0.486622\pi\)
0.0420169 + 0.999117i \(0.486622\pi\)
\(920\) 0 0
\(921\) 7.39788e8i 0.946953i
\(922\) 0 0
\(923\) 3.13849e8 3.13849e8i 0.399131 0.399131i
\(924\) 0 0
\(925\) −3.03831e7 + 3.03831e7i −0.0383890 + 0.0383890i
\(926\) 0 0
\(927\) 3.62676e8i 0.455281i
\(928\) 0 0
\(929\) −7.88148e8 −0.983016 −0.491508 0.870873i \(-0.663554\pi\)
−0.491508 + 0.870873i \(0.663554\pi\)
\(930\) 0 0
\(931\) −1.94833e7 1.94833e7i −0.0241442 0.0241442i
\(932\) 0 0
\(933\) −7.99193e7 7.99193e7i −0.0984027 0.0984027i
\(934\) 0 0
\(935\) −4.30452e8 −0.526611
\(936\) 0 0
\(937\) 5.31183e8i 0.645692i 0.946451 + 0.322846i \(0.104640\pi\)
−0.946451 + 0.322846i \(0.895360\pi\)
\(938\) 0 0
\(939\) 6.46069e7 6.46069e7i 0.0780337 0.0780337i
\(940\) 0 0
\(941\) −1.68586e8 + 1.68586e8i −0.202327 + 0.202327i −0.800996 0.598670i \(-0.795696\pi\)
0.598670 + 0.800996i \(0.295696\pi\)
\(942\) 0 0
\(943\) 1.16810e9i 1.39298i
\(944\) 0 0
\(945\) −1.73736e8 −0.205871
\(946\) 0 0
\(947\) 1.02795e8 + 1.02795e8i 0.121039 + 0.121039i 0.765031 0.643993i \(-0.222723\pi\)
−0.643993 + 0.765031i \(0.722723\pi\)
\(948\) 0 0
\(949\) −2.10905e8 2.10905e8i −0.246768 0.246768i
\(950\) 0 0
\(951\) −3.13479e8 −0.364475
\(952\) 0 0
\(953\) 9.78404e8i 1.13042i 0.824947 + 0.565210i \(0.191205\pi\)
−0.824947 + 0.565210i \(0.808795\pi\)
\(954\) 0 0
\(955\) −7.16661e8 + 7.16661e8i −0.822818 + 0.822818i
\(956\) 0 0
\(957\) 9.02673e8 9.02673e8i 1.02990 1.02990i
\(958\) 0 0
\(959\) 1.53825e9i 1.74410i
\(960\) 0 0
\(961\) −1.84186e9 −2.07533
\(962\) 0 0
\(963\) −3.93084e8 3.93084e8i −0.440156 0.440156i
\(964\) 0 0
\(965\) 1.76778e8 + 1.76778e8i 0.196719 + 0.196719i
\(966\) 0 0
\(967\) 7.24279e8 0.800989 0.400495 0.916299i \(-0.368838\pi\)
0.400495 + 0.916299i \(0.368838\pi\)
\(968\) 0 0
\(969\) 2.06482e8i 0.226940i
\(970\) 0 0
\(971\) −2.79870e6 + 2.79870e6i −0.00305702 + 0.00305702i −0.708634 0.705577i \(-0.750688\pi\)
0.705577 + 0.708634i \(0.250688\pi\)
\(972\) 0 0
\(973\) −1.17732e9 + 1.17732e9i −1.27807 + 1.27807i
\(974\) 0 0
\(975\) 3.60397e7i 0.0388837i
\(976\) 0 0
\(977\) 8.62168e8 0.924503 0.462252 0.886749i \(-0.347042\pi\)
0.462252 + 0.886749i \(0.347042\pi\)
\(978\) 0 0
\(979\) 1.40441e9 + 1.40441e9i 1.49674 + 1.49674i
\(980\) 0 0
\(981\) 5.16629e7 + 5.16629e7i 0.0547232 + 0.0547232i
\(982\) 0 0
\(983\) 6.29606e8 0.662839 0.331419 0.943484i \(-0.392472\pi\)
0.331419 + 0.943484i \(0.392472\pi\)
\(984\) 0 0
\(985\) 9.60696e8i 1.00526i
\(986\) 0 0
\(987\) 2.21001e8 2.21001e8i 0.229849 0.229849i
\(988\) 0 0
\(989\) −9.42140e8 + 9.42140e8i −0.973927 + 0.973927i
\(990\) 0 0
\(991\) 1.51486e9i 1.55651i −0.627947 0.778256i \(-0.716105\pi\)
0.627947 0.778256i \(-0.283895\pi\)
\(992\) 0 0
\(993\) −1.00473e9 −1.02613
\(994\) 0 0
\(995\) 3.73642e8 + 3.73642e8i 0.379303 + 0.379303i
\(996\) 0 0
\(997\) −8.68297e7 8.68297e7i −0.0876159 0.0876159i 0.661940 0.749556i \(-0.269733\pi\)
−0.749556 + 0.661940i \(0.769733\pi\)
\(998\) 0 0
\(999\) −6.00057e7 −0.0601860
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 384.7.l.b.31.16 48
4.3 odd 2 384.7.l.a.31.9 48
8.3 odd 2 192.7.l.a.79.16 48
8.5 even 2 48.7.l.a.43.17 yes 48
16.3 odd 4 inner 384.7.l.b.223.16 48
16.5 even 4 192.7.l.a.175.16 48
16.11 odd 4 48.7.l.a.19.17 48
16.13 even 4 384.7.l.a.223.9 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.7.l.a.19.17 48 16.11 odd 4
48.7.l.a.43.17 yes 48 8.5 even 2
192.7.l.a.79.16 48 8.3 odd 2
192.7.l.a.175.16 48 16.5 even 4
384.7.l.a.31.9 48 4.3 odd 2
384.7.l.a.223.9 48 16.13 even 4
384.7.l.b.31.16 48 1.1 even 1 trivial
384.7.l.b.223.16 48 16.3 odd 4 inner