Properties

Label 384.4.c.d.383.6
Level $384$
Weight $4$
Character 384.383
Analytic conductor $22.657$
Analytic rank $0$
Dimension $12$
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] = [384,4,Mod(383,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.383");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.c (of order \(2\), degree \(1\), 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: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.6
Root \(2.36157i\) of defining polynomial
Character \(\chi\) \(=\) 384.383
Dual form 384.4.c.d.383.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.556921 + 5.16622i) q^{3} +10.6077i q^{5} +7.90379i q^{7} +(-26.3797 + 5.75435i) q^{9} +O(q^{10})\) \(q+(0.556921 + 5.16622i) q^{3} +10.6077i q^{5} +7.90379i q^{7} +(-26.3797 + 5.75435i) q^{9} +11.7652 q^{11} +30.5992 q^{13} +(-54.8017 + 5.90765i) q^{15} +118.124i q^{17} +66.4680i q^{19} +(-40.8327 + 4.40179i) q^{21} -166.335 q^{23} +12.4768 q^{25} +(-44.4197 - 133.079i) q^{27} -111.999i q^{29} -224.770i q^{31} +(6.55231 + 60.7818i) q^{33} -83.8410 q^{35} -70.6099 q^{37} +(17.0414 + 158.082i) q^{39} +247.184i q^{41} -98.7603i q^{43} +(-61.0404 - 279.828i) q^{45} -189.577 q^{47} +280.530 q^{49} +(-610.257 + 65.7860i) q^{51} -529.370i q^{53} +124.802i q^{55} +(-343.388 + 37.0174i) q^{57} -811.889 q^{59} +833.410 q^{61} +(-45.4812 - 208.499i) q^{63} +324.587i q^{65} -2.83446i q^{67} +(-92.6357 - 859.325i) q^{69} -796.627 q^{71} -875.489 q^{73} +(6.94861 + 64.4581i) q^{75} +92.9900i q^{77} +656.826i q^{79} +(662.775 - 303.596i) q^{81} +237.058 q^{83} -1253.03 q^{85} +(578.610 - 62.3744i) q^{87} +868.956i q^{89} +241.850i q^{91} +(1161.21 - 125.179i) q^{93} -705.072 q^{95} +755.048 q^{97} +(-310.363 + 67.7014i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 10 q^{3} + 36 q^{11} + 84 q^{15} + 136 q^{21} - 120 q^{23} - 300 q^{25} - 266 q^{27} - 116 q^{33} + 432 q^{35} + 528 q^{37} + 620 q^{39} + 440 q^{45} - 1248 q^{47} - 948 q^{49} - 1072 q^{51} - 172 q^{57} + 2508 q^{59} + 624 q^{61} + 2744 q^{63} - 24 q^{69} - 2040 q^{71} - 216 q^{73} - 3894 q^{75} - 1076 q^{81} + 4572 q^{83} + 480 q^{85} + 4156 q^{87} - 112 q^{93} - 5448 q^{95} - 48 q^{97} - 6044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 0.556921 + 5.16622i 0.107180 + 0.994240i
\(4\) 0 0
\(5\) 10.6077i 0.948781i 0.880315 + 0.474390i \(0.157332\pi\)
−0.880315 + 0.474390i \(0.842668\pi\)
\(6\) 0 0
\(7\) 7.90379i 0.426764i 0.976969 + 0.213382i \(0.0684480\pi\)
−0.976969 + 0.213382i \(0.931552\pi\)
\(8\) 0 0
\(9\) −26.3797 + 5.75435i −0.977025 + 0.213124i
\(10\) 0 0
\(11\) 11.7652 0.322487 0.161243 0.986915i \(-0.448450\pi\)
0.161243 + 0.986915i \(0.448450\pi\)
\(12\) 0 0
\(13\) 30.5992 0.652823 0.326412 0.945228i \(-0.394160\pi\)
0.326412 + 0.945228i \(0.394160\pi\)
\(14\) 0 0
\(15\) −54.8017 + 5.90765i −0.943316 + 0.101690i
\(16\) 0 0
\(17\) 118.124i 1.68526i 0.538495 + 0.842629i \(0.318993\pi\)
−0.538495 + 0.842629i \(0.681007\pi\)
\(18\) 0 0
\(19\) 66.4680i 0.802568i 0.915954 + 0.401284i \(0.131436\pi\)
−0.915954 + 0.401284i \(0.868564\pi\)
\(20\) 0 0
\(21\) −40.8327 + 4.40179i −0.424306 + 0.0457404i
\(22\) 0 0
\(23\) −166.335 −1.50797 −0.753985 0.656891i \(-0.771871\pi\)
−0.753985 + 0.656891i \(0.771871\pi\)
\(24\) 0 0
\(25\) 12.4768 0.0998147
\(26\) 0 0
\(27\) −44.4197 133.079i −0.316614 0.948555i
\(28\) 0 0
\(29\) 111.999i 0.717159i −0.933499 0.358580i \(-0.883261\pi\)
0.933499 0.358580i \(-0.116739\pi\)
\(30\) 0 0
\(31\) 224.770i 1.30226i −0.758968 0.651128i \(-0.774296\pi\)
0.758968 0.651128i \(-0.225704\pi\)
\(32\) 0 0
\(33\) 6.55231 + 60.7818i 0.0345640 + 0.320629i
\(34\) 0 0
\(35\) −83.8410 −0.404906
\(36\) 0 0
\(37\) −70.6099 −0.313735 −0.156868 0.987620i \(-0.550140\pi\)
−0.156868 + 0.987620i \(0.550140\pi\)
\(38\) 0 0
\(39\) 17.0414 + 158.082i 0.0699693 + 0.649063i
\(40\) 0 0
\(41\) 247.184i 0.941551i 0.882253 + 0.470775i \(0.156026\pi\)
−0.882253 + 0.470775i \(0.843974\pi\)
\(42\) 0 0
\(43\) 98.7603i 0.350251i −0.984546 0.175126i \(-0.943967\pi\)
0.984546 0.175126i \(-0.0560332\pi\)
\(44\) 0 0
\(45\) −61.0404 279.828i −0.202208 0.926983i
\(46\) 0 0
\(47\) −189.577 −0.588355 −0.294178 0.955751i \(-0.595046\pi\)
−0.294178 + 0.955751i \(0.595046\pi\)
\(48\) 0 0
\(49\) 280.530 0.817872
\(50\) 0 0
\(51\) −610.257 + 65.7860i −1.67555 + 0.180625i
\(52\) 0 0
\(53\) 529.370i 1.37197i −0.727614 0.685987i \(-0.759371\pi\)
0.727614 0.685987i \(-0.240629\pi\)
\(54\) 0 0
\(55\) 124.802i 0.305969i
\(56\) 0 0
\(57\) −343.388 + 37.0174i −0.797945 + 0.0860189i
\(58\) 0 0
\(59\) −811.889 −1.79151 −0.895753 0.444551i \(-0.853363\pi\)
−0.895753 + 0.444551i \(0.853363\pi\)
\(60\) 0 0
\(61\) 833.410 1.74930 0.874650 0.484756i \(-0.161091\pi\)
0.874650 + 0.484756i \(0.161091\pi\)
\(62\) 0 0
\(63\) −45.4812 208.499i −0.0909538 0.416960i
\(64\) 0 0
\(65\) 324.587i 0.619386i
\(66\) 0 0
\(67\) 2.83446i 0.00516843i −0.999997 0.00258421i \(-0.999177\pi\)
0.999997 0.00258421i \(-0.000822582\pi\)
\(68\) 0 0
\(69\) −92.6357 859.325i −0.161624 1.49928i
\(70\) 0 0
\(71\) −796.627 −1.33158 −0.665790 0.746139i \(-0.731905\pi\)
−0.665790 + 0.746139i \(0.731905\pi\)
\(72\) 0 0
\(73\) −875.489 −1.40367 −0.701837 0.712338i \(-0.747637\pi\)
−0.701837 + 0.712338i \(0.747637\pi\)
\(74\) 0 0
\(75\) 6.94861 + 64.4581i 0.0106981 + 0.0992397i
\(76\) 0 0
\(77\) 92.9900i 0.137626i
\(78\) 0 0
\(79\) 656.826i 0.935427i 0.883880 + 0.467713i \(0.154922\pi\)
−0.883880 + 0.467713i \(0.845078\pi\)
\(80\) 0 0
\(81\) 662.775 303.596i 0.909156 0.416455i
\(82\) 0 0
\(83\) 237.058 0.313500 0.156750 0.987638i \(-0.449898\pi\)
0.156750 + 0.987638i \(0.449898\pi\)
\(84\) 0 0
\(85\) −1253.03 −1.59894
\(86\) 0 0
\(87\) 578.610 62.3744i 0.713028 0.0768648i
\(88\) 0 0
\(89\) 868.956i 1.03493i 0.855703 + 0.517467i \(0.173125\pi\)
−0.855703 + 0.517467i \(0.826875\pi\)
\(90\) 0 0
\(91\) 241.850i 0.278602i
\(92\) 0 0
\(93\) 1161.21 125.179i 1.29476 0.139575i
\(94\) 0 0
\(95\) −705.072 −0.761461
\(96\) 0 0
\(97\) 755.048 0.790345 0.395173 0.918607i \(-0.370685\pi\)
0.395173 + 0.918607i \(0.370685\pi\)
\(98\) 0 0
\(99\) −310.363 + 67.7014i −0.315078 + 0.0687297i
\(100\) 0 0
\(101\) 1159.48i 1.14230i 0.820846 + 0.571149i \(0.193502\pi\)
−0.820846 + 0.571149i \(0.806498\pi\)
\(102\) 0 0
\(103\) 1426.76i 1.36488i −0.730942 0.682440i \(-0.760919\pi\)
0.730942 0.682440i \(-0.239081\pi\)
\(104\) 0 0
\(105\) −46.6928 433.141i −0.0433976 0.402574i
\(106\) 0 0
\(107\) 1503.34 1.35826 0.679128 0.734020i \(-0.262358\pi\)
0.679128 + 0.734020i \(0.262358\pi\)
\(108\) 0 0
\(109\) −446.011 −0.391928 −0.195964 0.980611i \(-0.562784\pi\)
−0.195964 + 0.980611i \(0.562784\pi\)
\(110\) 0 0
\(111\) −39.3242 364.787i −0.0336260 0.311928i
\(112\) 0 0
\(113\) 1413.62i 1.17684i 0.808557 + 0.588418i \(0.200249\pi\)
−0.808557 + 0.588418i \(0.799751\pi\)
\(114\) 0 0
\(115\) 1764.44i 1.43073i
\(116\) 0 0
\(117\) −807.198 + 176.079i −0.637825 + 0.139132i
\(118\) 0 0
\(119\) −933.630 −0.719208
\(120\) 0 0
\(121\) −1192.58 −0.896002
\(122\) 0 0
\(123\) −1277.00 + 137.662i −0.936127 + 0.100915i
\(124\) 0 0
\(125\) 1458.31i 1.04348i
\(126\) 0 0
\(127\) 1315.32i 0.919020i 0.888173 + 0.459510i \(0.151975\pi\)
−0.888173 + 0.459510i \(0.848025\pi\)
\(128\) 0 0
\(129\) 510.218 55.0017i 0.348234 0.0375398i
\(130\) 0 0
\(131\) 2183.27 1.45613 0.728067 0.685506i \(-0.240419\pi\)
0.728067 + 0.685506i \(0.240419\pi\)
\(132\) 0 0
\(133\) −525.349 −0.342508
\(134\) 0 0
\(135\) 1411.66 471.190i 0.899971 0.300397i
\(136\) 0 0
\(137\) 751.608i 0.468717i −0.972150 0.234359i \(-0.924701\pi\)
0.972150 0.234359i \(-0.0752990\pi\)
\(138\) 0 0
\(139\) 1745.14i 1.06490i 0.846463 + 0.532448i \(0.178728\pi\)
−0.846463 + 0.532448i \(0.821272\pi\)
\(140\) 0 0
\(141\) −105.580 979.399i −0.0630596 0.584966i
\(142\) 0 0
\(143\) 360.008 0.210527
\(144\) 0 0
\(145\) 1188.05 0.680427
\(146\) 0 0
\(147\) 156.233 + 1449.28i 0.0876591 + 0.813161i
\(148\) 0 0
\(149\) 1859.32i 1.02229i 0.859494 + 0.511145i \(0.170779\pi\)
−0.859494 + 0.511145i \(0.829221\pi\)
\(150\) 0 0
\(151\) 967.660i 0.521504i 0.965406 + 0.260752i \(0.0839704\pi\)
−0.965406 + 0.260752i \(0.916030\pi\)
\(152\) 0 0
\(153\) −679.730 3116.08i −0.359169 1.64654i
\(154\) 0 0
\(155\) 2384.30 1.23556
\(156\) 0 0
\(157\) 3639.08 1.84987 0.924936 0.380123i \(-0.124118\pi\)
0.924936 + 0.380123i \(0.124118\pi\)
\(158\) 0 0
\(159\) 2734.84 294.817i 1.36407 0.147047i
\(160\) 0 0
\(161\) 1314.68i 0.643548i
\(162\) 0 0
\(163\) 337.359i 0.162110i −0.996710 0.0810551i \(-0.974171\pi\)
0.996710 0.0810551i \(-0.0258290\pi\)
\(164\) 0 0
\(165\) −644.755 + 69.5049i −0.304207 + 0.0327936i
\(166\) 0 0
\(167\) 2686.84 1.24499 0.622497 0.782622i \(-0.286118\pi\)
0.622497 + 0.782622i \(0.286118\pi\)
\(168\) 0 0
\(169\) −1260.69 −0.573822
\(170\) 0 0
\(171\) −382.480 1753.40i −0.171047 0.784129i
\(172\) 0 0
\(173\) 1970.03i 0.865773i 0.901449 + 0.432886i \(0.142505\pi\)
−0.901449 + 0.432886i \(0.857495\pi\)
\(174\) 0 0
\(175\) 98.6143i 0.0425974i
\(176\) 0 0
\(177\) −452.158 4194.40i −0.192013 1.78119i
\(178\) 0 0
\(179\) 836.542 0.349308 0.174654 0.984630i \(-0.444119\pi\)
0.174654 + 0.984630i \(0.444119\pi\)
\(180\) 0 0
\(181\) −1716.39 −0.704853 −0.352426 0.935840i \(-0.614643\pi\)
−0.352426 + 0.935840i \(0.614643\pi\)
\(182\) 0 0
\(183\) 464.144 + 4305.58i 0.187489 + 1.73922i
\(184\) 0 0
\(185\) 749.009i 0.297666i
\(186\) 0 0
\(187\) 1389.76i 0.543473i
\(188\) 0 0
\(189\) 1051.82 351.084i 0.404809 0.135119i
\(190\) 0 0
\(191\) −358.884 −0.135958 −0.0679788 0.997687i \(-0.521655\pi\)
−0.0679788 + 0.997687i \(0.521655\pi\)
\(192\) 0 0
\(193\) 2670.56 0.996015 0.498007 0.867173i \(-0.334065\pi\)
0.498007 + 0.867173i \(0.334065\pi\)
\(194\) 0 0
\(195\) −1676.89 + 180.770i −0.615819 + 0.0663855i
\(196\) 0 0
\(197\) 597.410i 0.216059i 0.994148 + 0.108030i \(0.0344542\pi\)
−0.994148 + 0.108030i \(0.965546\pi\)
\(198\) 0 0
\(199\) 3336.44i 1.18851i −0.804276 0.594256i \(-0.797447\pi\)
0.804276 0.594256i \(-0.202553\pi\)
\(200\) 0 0
\(201\) 14.6435 1.57857i 0.00513865 0.000553949i
\(202\) 0 0
\(203\) 885.213 0.306058
\(204\) 0 0
\(205\) −2622.05 −0.893326
\(206\) 0 0
\(207\) 4387.87 957.153i 1.47333 0.321385i
\(208\) 0 0
\(209\) 782.012i 0.258818i
\(210\) 0 0
\(211\) 1234.24i 0.402695i 0.979520 + 0.201348i \(0.0645321\pi\)
−0.979520 + 0.201348i \(0.935468\pi\)
\(212\) 0 0
\(213\) −443.658 4115.55i −0.142718 1.32391i
\(214\) 0 0
\(215\) 1047.62 0.332312
\(216\) 0 0
\(217\) 1776.54 0.555757
\(218\) 0 0
\(219\) −487.578 4522.97i −0.150445 1.39559i
\(220\) 0 0
\(221\) 3614.52i 1.10018i
\(222\) 0 0
\(223\) 1647.48i 0.494723i −0.968923 0.247362i \(-0.920436\pi\)
0.968923 0.247362i \(-0.0795635\pi\)
\(224\) 0 0
\(225\) −329.135 + 71.7962i −0.0975215 + 0.0212729i
\(226\) 0 0
\(227\) 340.265 0.0994897 0.0497449 0.998762i \(-0.484159\pi\)
0.0497449 + 0.998762i \(0.484159\pi\)
\(228\) 0 0
\(229\) 4031.94 1.16349 0.581743 0.813373i \(-0.302371\pi\)
0.581743 + 0.813373i \(0.302371\pi\)
\(230\) 0 0
\(231\) −480.407 + 51.7881i −0.136833 + 0.0147507i
\(232\) 0 0
\(233\) 1464.29i 0.411710i 0.978582 + 0.205855i \(0.0659976\pi\)
−0.978582 + 0.205855i \(0.934002\pi\)
\(234\) 0 0
\(235\) 2010.98i 0.558220i
\(236\) 0 0
\(237\) −3393.31 + 365.800i −0.930038 + 0.100259i
\(238\) 0 0
\(239\) −3357.75 −0.908765 −0.454382 0.890807i \(-0.650140\pi\)
−0.454382 + 0.890807i \(0.650140\pi\)
\(240\) 0 0
\(241\) −2736.93 −0.731541 −0.365771 0.930705i \(-0.619195\pi\)
−0.365771 + 0.930705i \(0.619195\pi\)
\(242\) 0 0
\(243\) 1937.56 + 3254.96i 0.511499 + 0.859284i
\(244\) 0 0
\(245\) 2975.78i 0.775981i
\(246\) 0 0
\(247\) 2033.87i 0.523935i
\(248\) 0 0
\(249\) 132.023 + 1224.70i 0.0336008 + 0.311694i
\(250\) 0 0
\(251\) 5959.74 1.49871 0.749354 0.662170i \(-0.230364\pi\)
0.749354 + 0.662170i \(0.230364\pi\)
\(252\) 0 0
\(253\) −1956.98 −0.486301
\(254\) 0 0
\(255\) −697.837 6473.42i −0.171374 1.58973i
\(256\) 0 0
\(257\) 1384.26i 0.335982i −0.985788 0.167991i \(-0.946272\pi\)
0.985788 0.167991i \(-0.0537280\pi\)
\(258\) 0 0
\(259\) 558.086i 0.133891i
\(260\) 0 0
\(261\) 644.480 + 2954.49i 0.152844 + 0.700683i
\(262\) 0 0
\(263\) −6426.06 −1.50664 −0.753322 0.657651i \(-0.771550\pi\)
−0.753322 + 0.657651i \(0.771550\pi\)
\(264\) 0 0
\(265\) 5615.40 1.30170
\(266\) 0 0
\(267\) −4489.22 + 483.940i −1.02897 + 0.110924i
\(268\) 0 0
\(269\) 1594.21i 0.361341i −0.983544 0.180671i \(-0.942173\pi\)
0.983544 0.180671i \(-0.0578268\pi\)
\(270\) 0 0
\(271\) 7502.42i 1.68170i 0.541271 + 0.840848i \(0.317943\pi\)
−0.541271 + 0.840848i \(0.682057\pi\)
\(272\) 0 0
\(273\) −1249.45 + 134.691i −0.276997 + 0.0298604i
\(274\) 0 0
\(275\) 146.793 0.0321889
\(276\) 0 0
\(277\) 1099.05 0.238395 0.119198 0.992871i \(-0.461968\pi\)
0.119198 + 0.992871i \(0.461968\pi\)
\(278\) 0 0
\(279\) 1293.41 + 5929.37i 0.277542 + 1.27234i
\(280\) 0 0
\(281\) 6848.46i 1.45390i 0.686692 + 0.726949i \(0.259062\pi\)
−0.686692 + 0.726949i \(0.740938\pi\)
\(282\) 0 0
\(283\) 4354.29i 0.914614i 0.889309 + 0.457307i \(0.151186\pi\)
−0.889309 + 0.457307i \(0.848814\pi\)
\(284\) 0 0
\(285\) −392.669 3642.56i −0.0816131 0.757075i
\(286\) 0 0
\(287\) −1953.69 −0.401820
\(288\) 0 0
\(289\) −9040.37 −1.84009
\(290\) 0 0
\(291\) 420.502 + 3900.74i 0.0847088 + 0.785793i
\(292\) 0 0
\(293\) 2845.10i 0.567278i −0.958931 0.283639i \(-0.908458\pi\)
0.958931 0.283639i \(-0.0915417\pi\)
\(294\) 0 0
\(295\) 8612.27i 1.69975i
\(296\) 0 0
\(297\) −522.608 1565.70i −0.102104 0.305896i
\(298\) 0 0
\(299\) −5089.74 −0.984439
\(300\) 0 0
\(301\) 780.581 0.149475
\(302\) 0 0
\(303\) −5990.11 + 645.737i −1.13572 + 0.122431i
\(304\) 0 0
\(305\) 8840.56i 1.65970i
\(306\) 0 0
\(307\) 7798.95i 1.44987i −0.688819 0.724934i \(-0.741870\pi\)
0.688819 0.724934i \(-0.258130\pi\)
\(308\) 0 0
\(309\) 7370.95 794.592i 1.35702 0.146287i
\(310\) 0 0
\(311\) −4799.43 −0.875084 −0.437542 0.899198i \(-0.644151\pi\)
−0.437542 + 0.899198i \(0.644151\pi\)
\(312\) 0 0
\(313\) −2400.65 −0.433524 −0.216762 0.976224i \(-0.569550\pi\)
−0.216762 + 0.976224i \(0.569550\pi\)
\(314\) 0 0
\(315\) 2211.70 482.451i 0.395603 0.0862953i
\(316\) 0 0
\(317\) 5599.88i 0.992179i −0.868271 0.496090i \(-0.834769\pi\)
0.868271 0.496090i \(-0.165231\pi\)
\(318\) 0 0
\(319\) 1317.69i 0.231274i
\(320\) 0 0
\(321\) 837.242 + 7766.59i 0.145577 + 1.35043i
\(322\) 0 0
\(323\) −7851.49 −1.35253
\(324\) 0 0
\(325\) 381.782 0.0651614
\(326\) 0 0
\(327\) −248.393 2304.19i −0.0420067 0.389670i
\(328\) 0 0
\(329\) 1498.38i 0.251089i
\(330\) 0 0
\(331\) 5568.28i 0.924654i 0.886709 + 0.462327i \(0.152985\pi\)
−0.886709 + 0.462327i \(0.847015\pi\)
\(332\) 0 0
\(333\) 1862.67 406.315i 0.306527 0.0668646i
\(334\) 0 0
\(335\) 30.0671 0.00490370
\(336\) 0 0
\(337\) −1366.22 −0.220839 −0.110420 0.993885i \(-0.535220\pi\)
−0.110420 + 0.993885i \(0.535220\pi\)
\(338\) 0 0
\(339\) −7303.09 + 787.276i −1.17006 + 0.126133i
\(340\) 0 0
\(341\) 2644.48i 0.419961i
\(342\) 0 0
\(343\) 4928.25i 0.775803i
\(344\) 0 0
\(345\) 9115.46 982.651i 1.42249 0.153345i
\(346\) 0 0
\(347\) −9688.05 −1.49879 −0.749397 0.662121i \(-0.769657\pi\)
−0.749397 + 0.662121i \(0.769657\pi\)
\(348\) 0 0
\(349\) −11427.5 −1.75272 −0.876361 0.481654i \(-0.840036\pi\)
−0.876361 + 0.481654i \(0.840036\pi\)
\(350\) 0 0
\(351\) −1359.21 4072.10i −0.206693 0.619239i
\(352\) 0 0
\(353\) 630.357i 0.0950440i −0.998870 0.0475220i \(-0.984868\pi\)
0.998870 0.0475220i \(-0.0151324\pi\)
\(354\) 0 0
\(355\) 8450.37i 1.26338i
\(356\) 0 0
\(357\) −519.958 4823.34i −0.0770843 0.715065i
\(358\) 0 0
\(359\) 7429.40 1.09222 0.546112 0.837712i \(-0.316107\pi\)
0.546112 + 0.837712i \(0.316107\pi\)
\(360\) 0 0
\(361\) 2441.01 0.355884
\(362\) 0 0
\(363\) −664.172 6161.13i −0.0960331 0.890841i
\(364\) 0 0
\(365\) 9286.92i 1.33178i
\(366\) 0 0
\(367\) 11698.9i 1.66397i 0.554797 + 0.831986i \(0.312796\pi\)
−0.554797 + 0.831986i \(0.687204\pi\)
\(368\) 0 0
\(369\) −1422.38 6520.62i −0.200667 0.919919i
\(370\) 0 0
\(371\) 4184.03 0.585510
\(372\) 0 0
\(373\) 5197.32 0.721466 0.360733 0.932669i \(-0.382527\pi\)
0.360733 + 0.932669i \(0.382527\pi\)
\(374\) 0 0
\(375\) −7533.96 + 812.165i −1.03747 + 0.111840i
\(376\) 0 0
\(377\) 3427.07i 0.468178i
\(378\) 0 0
\(379\) 2510.74i 0.340284i 0.985420 + 0.170142i \(0.0544227\pi\)
−0.985420 + 0.170142i \(0.945577\pi\)
\(380\) 0 0
\(381\) −6795.22 + 732.528i −0.913726 + 0.0985001i
\(382\) 0 0
\(383\) 4920.99 0.656530 0.328265 0.944586i \(-0.393536\pi\)
0.328265 + 0.944586i \(0.393536\pi\)
\(384\) 0 0
\(385\) −986.409 −0.130577
\(386\) 0 0
\(387\) 568.302 + 2605.27i 0.0746471 + 0.342204i
\(388\) 0 0
\(389\) 3477.90i 0.453308i −0.973975 0.226654i \(-0.927221\pi\)
0.973975 0.226654i \(-0.0727786\pi\)
\(390\) 0 0
\(391\) 19648.3i 2.54132i
\(392\) 0 0
\(393\) 1215.91 + 11279.3i 0.156068 + 1.44775i
\(394\) 0 0
\(395\) −6967.41 −0.887515
\(396\) 0 0
\(397\) 2710.55 0.342667 0.171333 0.985213i \(-0.445192\pi\)
0.171333 + 0.985213i \(0.445192\pi\)
\(398\) 0 0
\(399\) −292.578 2714.07i −0.0367098 0.340535i
\(400\) 0 0
\(401\) 4661.17i 0.580468i −0.956956 0.290234i \(-0.906267\pi\)
0.956956 0.290234i \(-0.0937331\pi\)
\(402\) 0 0
\(403\) 6877.80i 0.850144i
\(404\) 0 0
\(405\) 3220.45 + 7030.51i 0.395125 + 0.862590i
\(406\) 0 0
\(407\) −830.743 −0.101175
\(408\) 0 0
\(409\) 12349.9 1.49307 0.746533 0.665348i \(-0.231717\pi\)
0.746533 + 0.665348i \(0.231717\pi\)
\(410\) 0 0
\(411\) 3882.97 418.586i 0.466017 0.0502369i
\(412\) 0 0
\(413\) 6417.00i 0.764551i
\(414\) 0 0
\(415\) 2514.64i 0.297443i
\(416\) 0 0
\(417\) −9015.76 + 971.903i −1.05876 + 0.114135i
\(418\) 0 0
\(419\) −528.053 −0.0615682 −0.0307841 0.999526i \(-0.509800\pi\)
−0.0307841 + 0.999526i \(0.509800\pi\)
\(420\) 0 0
\(421\) 8788.97 1.01745 0.508727 0.860928i \(-0.330116\pi\)
0.508727 + 0.860928i \(0.330116\pi\)
\(422\) 0 0
\(423\) 5000.99 1090.90i 0.574838 0.125393i
\(424\) 0 0
\(425\) 1473.82i 0.168213i
\(426\) 0 0
\(427\) 6587.10i 0.746539i
\(428\) 0 0
\(429\) 200.496 + 1859.88i 0.0225642 + 0.209314i
\(430\) 0 0
\(431\) −3660.00 −0.409040 −0.204520 0.978862i \(-0.565563\pi\)
−0.204520 + 0.978862i \(0.565563\pi\)
\(432\) 0 0
\(433\) −4322.55 −0.479742 −0.239871 0.970805i \(-0.577105\pi\)
−0.239871 + 0.970805i \(0.577105\pi\)
\(434\) 0 0
\(435\) 661.648 + 6137.71i 0.0729278 + 0.676508i
\(436\) 0 0
\(437\) 11056.0i 1.21025i
\(438\) 0 0
\(439\) 12788.3i 1.39032i 0.718854 + 0.695161i \(0.244667\pi\)
−0.718854 + 0.695161i \(0.755333\pi\)
\(440\) 0 0
\(441\) −7400.29 + 1614.27i −0.799082 + 0.174308i
\(442\) 0 0
\(443\) −5382.41 −0.577259 −0.288630 0.957441i \(-0.593200\pi\)
−0.288630 + 0.957441i \(0.593200\pi\)
\(444\) 0 0
\(445\) −9217.62 −0.981926
\(446\) 0 0
\(447\) −9605.65 + 1035.49i −1.01640 + 0.109569i
\(448\) 0 0
\(449\) 9465.38i 0.994875i −0.867500 0.497437i \(-0.834274\pi\)
0.867500 0.497437i \(-0.165726\pi\)
\(450\) 0 0
\(451\) 2908.17i 0.303638i
\(452\) 0 0
\(453\) −4999.15 + 538.910i −0.518500 + 0.0558945i
\(454\) 0 0
\(455\) −2565.47 −0.264332
\(456\) 0 0
\(457\) −716.322 −0.0733219 −0.0366610 0.999328i \(-0.511672\pi\)
−0.0366610 + 0.999328i \(0.511672\pi\)
\(458\) 0 0
\(459\) 15719.8 5247.05i 1.59856 0.533575i
\(460\) 0 0
\(461\) 19191.7i 1.93893i 0.245226 + 0.969466i \(0.421138\pi\)
−0.245226 + 0.969466i \(0.578862\pi\)
\(462\) 0 0
\(463\) 222.541i 0.0223377i −0.999938 0.0111689i \(-0.996445\pi\)
0.999938 0.0111689i \(-0.00355523\pi\)
\(464\) 0 0
\(465\) 1327.86 + 12317.8i 0.132426 + 1.22844i
\(466\) 0 0
\(467\) −5784.61 −0.573190 −0.286595 0.958052i \(-0.592523\pi\)
−0.286595 + 0.958052i \(0.592523\pi\)
\(468\) 0 0
\(469\) 22.4030 0.00220570
\(470\) 0 0
\(471\) 2026.68 + 18800.3i 0.198268 + 1.83922i
\(472\) 0 0
\(473\) 1161.94i 0.112951i
\(474\) 0 0
\(475\) 829.310i 0.0801081i
\(476\) 0 0
\(477\) 3046.18 + 13964.6i 0.292401 + 1.34045i
\(478\) 0 0
\(479\) 9218.02 0.879294 0.439647 0.898171i \(-0.355103\pi\)
0.439647 + 0.898171i \(0.355103\pi\)
\(480\) 0 0
\(481\) −2160.61 −0.204814
\(482\) 0 0
\(483\) 6791.93 732.173i 0.639841 0.0689752i
\(484\) 0 0
\(485\) 8009.32i 0.749864i
\(486\) 0 0
\(487\) 17851.9i 1.66109i −0.556955 0.830543i \(-0.688030\pi\)
0.556955 0.830543i \(-0.311970\pi\)
\(488\) 0 0
\(489\) 1742.87 187.882i 0.161176 0.0173749i
\(490\) 0 0
\(491\) 12979.1 1.19295 0.596473 0.802633i \(-0.296568\pi\)
0.596473 + 0.802633i \(0.296568\pi\)
\(492\) 0 0
\(493\) 13229.8 1.20860
\(494\) 0 0
\(495\) −718.155 3292.24i −0.0652095 0.298940i
\(496\) 0 0
\(497\) 6296.37i 0.568271i
\(498\) 0 0
\(499\) 17513.6i 1.57117i 0.618752 + 0.785586i \(0.287639\pi\)
−0.618752 + 0.785586i \(0.712361\pi\)
\(500\) 0 0
\(501\) 1496.36 + 13880.8i 0.133438 + 1.23782i
\(502\) 0 0
\(503\) 7405.59 0.656459 0.328230 0.944598i \(-0.393548\pi\)
0.328230 + 0.944598i \(0.393548\pi\)
\(504\) 0 0
\(505\) −12299.4 −1.08379
\(506\) 0 0
\(507\) −702.103 6512.98i −0.0615019 0.570516i
\(508\) 0 0
\(509\) 2162.88i 0.188346i 0.995556 + 0.0941728i \(0.0300206\pi\)
−0.995556 + 0.0941728i \(0.969979\pi\)
\(510\) 0 0
\(511\) 6919.68i 0.599038i
\(512\) 0 0
\(513\) 8845.46 2952.48i 0.761280 0.254104i
\(514\) 0 0
\(515\) 15134.6 1.29497
\(516\) 0 0
\(517\) −2230.42 −0.189737
\(518\) 0 0
\(519\) −10177.6 + 1097.15i −0.860786 + 0.0927931i
\(520\) 0 0
\(521\) 13662.4i 1.14887i −0.818550 0.574435i \(-0.805222\pi\)
0.818550 0.574435i \(-0.194778\pi\)
\(522\) 0 0
\(523\) 3574.54i 0.298860i 0.988772 + 0.149430i \(0.0477438\pi\)
−0.988772 + 0.149430i \(0.952256\pi\)
\(524\) 0 0
\(525\) −509.463 + 54.9204i −0.0423520 + 0.00456556i
\(526\) 0 0
\(527\) 26550.9 2.19464
\(528\) 0 0
\(529\) 15500.5 1.27398
\(530\) 0 0
\(531\) 21417.4 4671.89i 1.75035 0.381813i
\(532\) 0 0
\(533\) 7563.63i 0.614666i
\(534\) 0 0
\(535\) 15947.0i 1.28869i
\(536\) 0 0
\(537\) 465.888 + 4321.76i 0.0374386 + 0.347295i
\(538\) 0 0
\(539\) 3300.51 0.263753
\(540\) 0 0
\(541\) −5965.93 −0.474114 −0.237057 0.971496i \(-0.576183\pi\)
−0.237057 + 0.971496i \(0.576183\pi\)
\(542\) 0 0
\(543\) −955.895 8867.26i −0.0755458 0.700793i
\(544\) 0 0
\(545\) 4731.15i 0.371854i
\(546\) 0 0
\(547\) 6960.57i 0.544081i −0.962286 0.272040i \(-0.912302\pi\)
0.962286 0.272040i \(-0.0876985\pi\)
\(548\) 0 0
\(549\) −21985.1 + 4795.74i −1.70911 + 0.372818i
\(550\) 0 0
\(551\) 7444.32 0.575569
\(552\) 0 0
\(553\) −5191.41 −0.399207
\(554\) 0 0
\(555\) 3869.54 417.139i 0.295951 0.0319037i
\(556\) 0 0
\(557\) 6929.11i 0.527102i −0.964645 0.263551i \(-0.915106\pi\)
0.964645 0.263551i \(-0.0848937\pi\)
\(558\) 0 0
\(559\) 3021.99i 0.228652i
\(560\) 0 0
\(561\) −7179.82 + 773.988i −0.540343 + 0.0582492i
\(562\) 0 0
\(563\) −20207.5 −1.51269 −0.756344 0.654175i \(-0.773016\pi\)
−0.756344 + 0.654175i \(0.773016\pi\)
\(564\) 0 0
\(565\) −14995.3 −1.11656
\(566\) 0 0
\(567\) 2399.56 + 5238.43i 0.177728 + 0.387995i
\(568\) 0 0
\(569\) 14954.6i 1.10181i −0.834567 0.550906i \(-0.814282\pi\)
0.834567 0.550906i \(-0.185718\pi\)
\(570\) 0 0
\(571\) 11065.5i 0.810996i −0.914096 0.405498i \(-0.867098\pi\)
0.914096 0.405498i \(-0.132902\pi\)
\(572\) 0 0
\(573\) −199.870 1854.07i −0.0145719 0.135174i
\(574\) 0 0
\(575\) −2075.34 −0.150518
\(576\) 0 0
\(577\) 20741.1 1.49647 0.748235 0.663433i \(-0.230901\pi\)
0.748235 + 0.663433i \(0.230901\pi\)
\(578\) 0 0
\(579\) 1487.29 + 13796.7i 0.106752 + 0.990277i
\(580\) 0 0
\(581\) 1873.66i 0.133791i
\(582\) 0 0
\(583\) 6228.17i 0.442443i
\(584\) 0 0
\(585\) −1867.79 8562.51i −0.132006 0.605156i
\(586\) 0 0
\(587\) 8044.96 0.565675 0.282837 0.959168i \(-0.408724\pi\)
0.282837 + 0.959168i \(0.408724\pi\)
\(588\) 0 0
\(589\) 14940.0 1.04515
\(590\) 0 0
\(591\) −3086.35 + 332.710i −0.214815 + 0.0231571i
\(592\) 0 0
\(593\) 11007.3i 0.762255i −0.924522 0.381128i \(-0.875536\pi\)
0.924522 0.381128i \(-0.124464\pi\)
\(594\) 0 0
\(595\) 9903.66i 0.682371i
\(596\) 0 0
\(597\) 17236.8 1858.13i 1.18167 0.127384i
\(598\) 0 0
\(599\) 15693.5 1.07048 0.535242 0.844699i \(-0.320220\pi\)
0.535242 + 0.844699i \(0.320220\pi\)
\(600\) 0 0
\(601\) 6984.08 0.474021 0.237010 0.971507i \(-0.423832\pi\)
0.237010 + 0.971507i \(0.423832\pi\)
\(602\) 0 0
\(603\) 16.3105 + 74.7722i 0.00110152 + 0.00504968i
\(604\) 0 0
\(605\) 12650.5i 0.850110i
\(606\) 0 0
\(607\) 2251.13i 0.150528i −0.997164 0.0752641i \(-0.976020\pi\)
0.997164 0.0752641i \(-0.0239800\pi\)
\(608\) 0 0
\(609\) 492.994 + 4573.21i 0.0328032 + 0.304295i
\(610\) 0 0
\(611\) −5800.93 −0.384092
\(612\) 0 0
\(613\) −4638.50 −0.305624 −0.152812 0.988255i \(-0.548833\pi\)
−0.152812 + 0.988255i \(0.548833\pi\)
\(614\) 0 0
\(615\) −1460.27 13546.1i −0.0957462 0.888180i
\(616\) 0 0
\(617\) 16218.9i 1.05826i −0.848540 0.529132i \(-0.822518\pi\)
0.848540 0.529132i \(-0.177482\pi\)
\(618\) 0 0
\(619\) 11838.8i 0.768727i −0.923182 0.384363i \(-0.874421\pi\)
0.923182 0.384363i \(-0.125579\pi\)
\(620\) 0 0
\(621\) 7388.56 + 22135.7i 0.477444 + 1.43039i
\(622\) 0 0
\(623\) −6868.05 −0.441673
\(624\) 0 0
\(625\) −13909.7 −0.890222
\(626\) 0 0
\(627\) −4040.04 + 435.519i −0.257327 + 0.0277399i
\(628\) 0 0
\(629\) 8340.76i 0.528724i
\(630\) 0 0
\(631\) 14214.6i 0.896791i 0.893835 + 0.448395i \(0.148004\pi\)
−0.893835 + 0.448395i \(0.851996\pi\)
\(632\) 0 0
\(633\) −6376.36 + 687.375i −0.400375 + 0.0431607i
\(634\) 0 0
\(635\) −13952.5 −0.871949
\(636\) 0 0
\(637\) 8584.01 0.533926
\(638\) 0 0
\(639\) 21014.8 4584.07i 1.30099 0.283792i
\(640\) 0 0
\(641\) 13075.6i 0.805704i 0.915265 + 0.402852i \(0.131981\pi\)
−0.915265 + 0.402852i \(0.868019\pi\)
\(642\) 0 0
\(643\) 5272.61i 0.323377i 0.986842 + 0.161689i \(0.0516940\pi\)
−0.986842 + 0.161689i \(0.948306\pi\)
\(644\) 0 0
\(645\) 583.441 + 5412.23i 0.0356170 + 0.330398i
\(646\) 0 0
\(647\) 32739.3 1.98936 0.994679 0.103027i \(-0.0328527\pi\)
0.994679 + 0.103027i \(0.0328527\pi\)
\(648\) 0 0
\(649\) −9552.07 −0.577737
\(650\) 0 0
\(651\) 989.391 + 9177.99i 0.0595657 + 0.552555i
\(652\) 0 0
\(653\) 14895.3i 0.892649i −0.894871 0.446324i \(-0.852733\pi\)
0.894871 0.446324i \(-0.147267\pi\)
\(654\) 0 0
\(655\) 23159.5i 1.38155i
\(656\) 0 0
\(657\) 23095.1 5037.87i 1.37142 0.299157i
\(658\) 0 0
\(659\) 13070.1 0.772595 0.386297 0.922374i \(-0.373754\pi\)
0.386297 + 0.922374i \(0.373754\pi\)
\(660\) 0 0
\(661\) −24322.7 −1.43123 −0.715616 0.698494i \(-0.753854\pi\)
−0.715616 + 0.698494i \(0.753854\pi\)
\(662\) 0 0
\(663\) −18673.4 + 2013.00i −1.09384 + 0.117916i
\(664\) 0 0
\(665\) 5572.74i 0.324965i
\(666\) 0 0
\(667\) 18629.3i 1.08146i
\(668\) 0 0
\(669\) 8511.23 917.515i 0.491873 0.0530242i
\(670\) 0 0
\(671\) 9805.27 0.564126
\(672\) 0 0
\(673\) −1854.52 −0.106221 −0.0531104 0.998589i \(-0.516914\pi\)
−0.0531104 + 0.998589i \(0.516914\pi\)
\(674\) 0 0
\(675\) −554.217 1660.40i −0.0316027 0.0946797i
\(676\) 0 0
\(677\) 6481.24i 0.367938i −0.982932 0.183969i \(-0.941105\pi\)
0.982932 0.183969i \(-0.0588946\pi\)
\(678\) 0 0
\(679\) 5967.74i 0.337291i
\(680\) 0 0
\(681\) 189.501 + 1757.88i 0.0106633 + 0.0989166i
\(682\) 0 0
\(683\) −7618.89 −0.426836 −0.213418 0.976961i \(-0.568460\pi\)
−0.213418 + 0.976961i \(0.568460\pi\)
\(684\) 0 0
\(685\) 7972.83 0.444710
\(686\) 0 0
\(687\) 2245.47 + 20829.9i 0.124702 + 1.15678i
\(688\) 0 0
\(689\) 16198.3i 0.895657i
\(690\) 0 0
\(691\) 8461.55i 0.465836i 0.972496 + 0.232918i \(0.0748273\pi\)
−0.972496 + 0.232918i \(0.925173\pi\)
\(692\) 0 0
\(693\) −535.097 2453.05i −0.0293314 0.134464i
\(694\) 0 0
\(695\) −18511.9 −1.01035
\(696\) 0 0
\(697\) −29198.4 −1.58676
\(698\) 0 0
\(699\) −7564.82 + 815.492i −0.409339 + 0.0441269i
\(700\) 0 0
\(701\) 19113.8i 1.02984i 0.857238 + 0.514921i \(0.172179\pi\)
−0.857238 + 0.514921i \(0.827821\pi\)
\(702\) 0 0
\(703\) 4693.30i 0.251794i
\(704\) 0 0
\(705\) 10389.2 1119.96i 0.555005 0.0598298i
\(706\) 0 0
\(707\) −9164.25 −0.487493
\(708\) 0 0
\(709\) 7854.59 0.416058 0.208029 0.978123i \(-0.433295\pi\)
0.208029 + 0.978123i \(0.433295\pi\)
\(710\) 0 0
\(711\) −3779.61 17326.9i −0.199362 0.913935i
\(712\) 0 0
\(713\) 37387.3i 1.96377i
\(714\) 0 0
\(715\) 3818.85i 0.199744i
\(716\) 0 0
\(717\) −1870.00 17346.9i −0.0974010 0.903530i
\(718\) 0 0
\(719\) −27011.1 −1.40103 −0.700517 0.713635i \(-0.747047\pi\)
−0.700517 + 0.713635i \(0.747047\pi\)
\(720\) 0 0
\(721\) 11276.8 0.582482
\(722\) 0 0
\(723\) −1524.26 14139.6i −0.0784062 0.727327i
\(724\) 0 0
\(725\) 1397.39i 0.0715830i
\(726\) 0 0
\(727\) 22178.9i 1.13146i −0.824591 0.565729i \(-0.808595\pi\)
0.824591 0.565729i \(-0.191405\pi\)
\(728\) 0 0
\(729\) −15736.8 + 11822.6i −0.799512 + 0.600651i
\(730\) 0 0
\(731\) 11666.0 0.590264
\(732\) 0 0
\(733\) 9103.17 0.458708 0.229354 0.973343i \(-0.426339\pi\)
0.229354 + 0.973343i \(0.426339\pi\)
\(734\) 0 0
\(735\) −15373.5 + 1657.27i −0.771512 + 0.0831693i
\(736\) 0 0
\(737\) 33.3481i 0.00166675i
\(738\) 0 0
\(739\) 19226.3i 0.957036i −0.878078 0.478518i \(-0.841174\pi\)
0.878078 0.478518i \(-0.158826\pi\)
\(740\) 0 0
\(741\) −10507.4 + 1132.70i −0.520917 + 0.0561551i
\(742\) 0 0
\(743\) −22501.9 −1.11106 −0.555528 0.831498i \(-0.687484\pi\)
−0.555528 + 0.831498i \(0.687484\pi\)
\(744\) 0 0
\(745\) −19723.1 −0.969930
\(746\) 0 0
\(747\) −6253.52 + 1364.12i −0.306298 + 0.0668145i
\(748\) 0 0
\(749\) 11882.1i 0.579655i
\(750\) 0 0
\(751\) 2080.69i 0.101099i −0.998722 0.0505495i \(-0.983903\pi\)
0.998722 0.0505495i \(-0.0160973\pi\)
\(752\) 0 0
\(753\) 3319.10 + 30789.3i 0.160631 + 1.49007i
\(754\) 0 0
\(755\) −10264.6 −0.494793
\(756\) 0 0
\(757\) −26222.3 −1.25900 −0.629501 0.777000i \(-0.716741\pi\)
−0.629501 + 0.777000i \(0.716741\pi\)
\(758\) 0 0
\(759\) −1089.88 10110.2i −0.0521215 0.483499i
\(760\) 0 0
\(761\) 11443.7i 0.545118i 0.962139 + 0.272559i \(0.0878700\pi\)
−0.962139 + 0.272559i \(0.912130\pi\)
\(762\) 0 0
\(763\) 3525.18i 0.167261i
\(764\) 0 0
\(765\) 33054.5 7210.36i 1.56220 0.340773i
\(766\) 0 0
\(767\) −24843.2 −1.16954
\(768\) 0 0
\(769\) 18991.7 0.890582 0.445291 0.895386i \(-0.353100\pi\)
0.445291 + 0.895386i \(0.353100\pi\)
\(770\) 0 0
\(771\) 7151.37 770.921i 0.334047 0.0360104i
\(772\) 0 0
\(773\) 36956.6i 1.71958i −0.510645 0.859791i \(-0.670593\pi\)
0.510645 0.859791i \(-0.329407\pi\)
\(774\) 0 0
\(775\) 2804.42i 0.129984i
\(776\) 0 0
\(777\) 2883.20 310.810i 0.133120 0.0143504i
\(778\) 0 0
\(779\) −16429.8 −0.755659
\(780\) 0 0
\(781\) −9372.51 −0.429417
\(782\) 0 0
\(783\) −14904.6 + 4974.94i −0.680265 + 0.227062i
\(784\) 0 0
\(785\) 38602.2i 1.75512i
\(786\) 0 0
\(787\) 19962.2i 0.904162i −0.891977 0.452081i \(-0.850682\pi\)
0.891977 0.452081i \(-0.149318\pi\)
\(788\) 0 0
\(789\) −3578.81 33198.4i −0.161481 1.49797i
\(790\) 0 0
\(791\) −11173.0 −0.502232
\(792\) 0 0
\(793\) 25501.7 1.14198
\(794\) 0 0
\(795\) 3127.33 + 29010.4i 0.139516 + 1.29420i
\(796\) 0 0
\(797\) 6131.27i 0.272498i 0.990675 + 0.136249i \(0.0435047\pi\)
−0.990675 + 0.136249i \(0.956495\pi\)
\(798\) 0 0
\(799\) 22393.7i 0.991530i
\(800\) 0 0
\(801\) −5000.28 22922.8i −0.220570 1.01116i
\(802\) 0 0
\(803\) −10300.3 −0.452666
\(804\) 0 0
\(805\) 13945.7 0.610586
\(806\) 0 0
\(807\) 8236.05 887.850i 0.359260 0.0387284i
\(808\) 0 0
\(809\) 18915.6i 0.822050i 0.911624 + 0.411025i \(0.134829\pi\)
−0.911624 + 0.411025i \(0.865171\pi\)
\(810\) 0 0
\(811\) 8699.98i 0.376692i −0.982103 0.188346i \(-0.939687\pi\)
0.982103 0.188346i \(-0.0603127\pi\)
\(812\) 0 0
\(813\) −38759.2 + 4178.26i −1.67201 + 0.180243i
\(814\) 0 0
\(815\) 3578.60 0.153807
\(816\) 0 0
\(817\) 6564.40 0.281101
\(818\) 0 0
\(819\) −1391.69 6379.92i −0.0593768 0.272201i
\(820\) 0 0
\(821\) 21629.4i 0.919453i 0.888061 + 0.459726i \(0.152052\pi\)
−0.888061 + 0.459726i \(0.847948\pi\)
\(822\) 0 0
\(823\) 17966.4i 0.760960i 0.924789 + 0.380480i \(0.124241\pi\)
−0.924789 + 0.380480i \(0.875759\pi\)
\(824\) 0 0
\(825\) 81.7521 + 758.365i 0.00344999 + 0.0320035i
\(826\) 0 0
\(827\) 19001.1 0.798951 0.399475 0.916744i \(-0.369192\pi\)
0.399475 + 0.916744i \(0.369192\pi\)
\(828\) 0 0
\(829\) 25624.2 1.07354 0.536770 0.843728i \(-0.319644\pi\)
0.536770 + 0.843728i \(0.319644\pi\)
\(830\) 0 0
\(831\) 612.083 + 5677.93i 0.0255511 + 0.237022i
\(832\) 0 0
\(833\) 33137.4i 1.37832i
\(834\) 0 0
\(835\) 28501.2i 1.18123i
\(836\) 0 0
\(837\) −29912.1 + 9984.23i −1.23526 + 0.412312i
\(838\) 0 0
\(839\) −39037.6 −1.60635 −0.803176 0.595742i \(-0.796858\pi\)
−0.803176 + 0.595742i \(0.796858\pi\)
\(840\) 0 0
\(841\) 11845.3 0.485683
\(842\) 0 0
\(843\) −35380.7 + 3814.05i −1.44552 + 0.155828i
\(844\) 0 0
\(845\) 13373.0i 0.544431i
\(846\) 0 0
\(847\) 9425.89i 0.382382i
\(848\) 0 0
\(849\) −22495.2 + 2425.00i −0.909346 + 0.0980279i
\(850\) 0 0
\(851\) 11744.9 0.473104
\(852\) 0 0
\(853\) −10697.0 −0.429377 −0.214688 0.976683i \(-0.568874\pi\)
−0.214688 + 0.976683i \(0.568874\pi\)
\(854\) 0 0
\(855\) 18599.6 4057.23i 0.743967 0.162286i
\(856\) 0 0
\(857\) 3546.91i 0.141377i 0.997498 + 0.0706886i \(0.0225197\pi\)
−0.997498 + 0.0706886i \(0.977480\pi\)
\(858\) 0 0
\(859\) 37273.1i 1.48049i −0.672336 0.740246i \(-0.734709\pi\)
0.672336 0.740246i \(-0.265291\pi\)
\(860\) 0 0
\(861\) −1088.05 10093.2i −0.0430669 0.399506i
\(862\) 0 0
\(863\) −12710.9 −0.501372 −0.250686 0.968068i \(-0.580656\pi\)
−0.250686 + 0.968068i \(0.580656\pi\)
\(864\) 0 0
\(865\) −20897.5 −0.821429
\(866\) 0 0
\(867\) −5034.77 46704.5i −0.197220 1.82949i
\(868\) 0 0
\(869\) 7727.72i 0.301663i
\(870\) 0 0
\(871\) 86.7324i 0.00337407i
\(872\) 0 0
\(873\) −19917.9 + 4344.81i −0.772187 + 0.168442i
\(874\) 0 0
\(875\) −11526.2 −0.445322
\(876\) 0 0
\(877\) −13273.5 −0.511078 −0.255539 0.966799i \(-0.582253\pi\)
−0.255539 + 0.966799i \(0.582253\pi\)
\(878\) 0 0
\(879\) 14698.4 1584.49i 0.564010 0.0608005i
\(880\) 0 0
\(881\) 24760.3i 0.946875i −0.880828 0.473437i \(-0.843013\pi\)
0.880828 0.473437i \(-0.156987\pi\)
\(882\) 0 0
\(883\) 46347.6i 1.76639i 0.469009 + 0.883193i \(0.344611\pi\)
−0.469009 + 0.883193i \(0.655389\pi\)
\(884\) 0 0
\(885\) 44492.9 4796.35i 1.68996 0.182178i
\(886\) 0 0
\(887\) 12454.7 0.471461 0.235731 0.971818i \(-0.424252\pi\)
0.235731 + 0.971818i \(0.424252\pi\)
\(888\) 0 0
\(889\) −10396.0 −0.392205
\(890\) 0 0
\(891\) 7797.71 3571.88i 0.293191 0.134301i
\(892\) 0 0
\(893\) 12600.8i 0.472195i
\(894\) 0 0
\(895\) 8873.78i 0.331416i
\(896\) 0 0
\(897\) −2834.58 26294.7i −0.105512 0.978768i
\(898\) 0 0
\(899\) −25174.0 −0.933925
\(900\) 0 0
\(901\) 62531.5 2.31213
\(902\) 0 0
\(903\) 434.722 + 4032.65i 0.0160206 + 0.148614i
\(904\) 0 0
\(905\) 18207.0i 0.668751i
\(906\) 0 0
\(907\) 31110.3i 1.13892i 0.822020 + 0.569459i \(0.192847\pi\)
−0.822020 + 0.569459i \(0.807153\pi\)
\(908\) 0 0
\(909\) −6672.04 30586.6i −0.243452 1.11605i
\(910\) 0 0
\(911\) 2882.81 0.104843 0.0524214 0.998625i \(-0.483306\pi\)
0.0524214 + 0.998625i \(0.483306\pi\)
\(912\) 0 0
\(913\) 2789.05 0.101100
\(914\) 0 0
\(915\) −45672.3 + 4923.49i −1.65014 + 0.177886i
\(916\) 0 0
\(917\) 17256.1i 0.621426i
\(918\) 0 0
\(919\) 6249.30i 0.224315i 0.993690 + 0.112157i \(0.0357761\pi\)
−0.993690 + 0.112157i \(0.964224\pi\)
\(920\) 0 0
\(921\) 40291.1 4343.40i 1.44152 0.155396i
\(922\) 0 0
\(923\) −24376.2 −0.869287
\(924\) 0 0
\(925\) −880.989 −0.0313154
\(926\) 0 0
\(927\) 8210.07 + 37637.4i 0.290889 + 1.33352i
\(928\) 0 0
\(929\) 32605.2i 1.15150i 0.817627 + 0.575749i \(0.195289\pi\)
−0.817627 + 0.575749i \(0.804711\pi\)
\(930\) 0 0
\(931\) 18646.3i 0.656398i
\(932\) 0 0
\(933\) −2672.91 24794.9i −0.0937910 0.870043i
\(934\) 0 0
\(935\) −14742.2 −0.515637
\(936\) 0 0
\(937\) 15176.1 0.529116 0.264558 0.964370i \(-0.414774\pi\)
0.264558 + 0.964370i \(0.414774\pi\)
\(938\) 0 0
\(939\) −1336.97 12402.3i −0.0464649 0.431027i
\(940\) 0 0
\(941\) 37892.6i 1.31271i −0.754450 0.656357i \(-0.772096\pi\)
0.754450 0.656357i \(-0.227904\pi\)
\(942\) 0 0
\(943\) 41115.4i 1.41983i
\(944\) 0 0
\(945\) 3724.19 + 11157.4i 0.128199 + 0.384075i
\(946\) 0 0
\(947\) 54773.3 1.87951 0.939754 0.341853i \(-0.111054\pi\)
0.939754 + 0.341853i \(0.111054\pi\)
\(948\) 0 0
\(949\) −26789.3 −0.916351
\(950\) 0 0
\(951\) 28930.2 3118.69i 0.986464 0.106341i
\(952\) 0 0
\(953\) 5158.12i 0.175328i 0.996150 + 0.0876642i \(0.0279403\pi\)
−0.996150 + 0.0876642i \(0.972060\pi\)
\(954\) 0 0
\(955\) 3806.93i 0.128994i
\(956\) 0 0
\(957\) 6807.48 733.850i 0.229942 0.0247879i
\(958\) 0 0
\(959\) 5940.55 0.200032
\(960\) 0 0
\(961\) −20730.7 −0.695872
\(962\) 0 0
\(963\) −39657.6 + 8650.76i −1.32705 + 0.289477i
\(964\) 0 0
\(965\) 28328.4i 0.945000i
\(966\) 0 0
\(967\) 46108.3i 1.53334i 0.642039 + 0.766672i \(0.278089\pi\)
−0.642039 + 0.766672i \(0.721911\pi\)
\(968\) 0 0
\(969\) −4372.66 40562.5i −0.144964 1.34474i
\(970\) 0 0
\(971\) 5437.59 0.179712 0.0898561 0.995955i \(-0.471359\pi\)
0.0898561 + 0.995955i \(0.471359\pi\)
\(972\) 0 0
\(973\) −13793.2 −0.454460
\(974\) 0 0
\(975\) 212.622 + 1972.37i 0.00698396 + 0.0647860i
\(976\) 0 0
\(977\) 21599.2i 0.707288i −0.935380 0.353644i \(-0.884942\pi\)
0.935380 0.353644i \(-0.115058\pi\)
\(978\) 0 0
\(979\) 10223.5i 0.333753i
\(980\) 0 0
\(981\) 11765.6 2566.51i 0.382924 0.0835294i
\(982\) 0 0
\(983\) −37178.0 −1.20630 −0.603151 0.797627i \(-0.706088\pi\)
−0.603151 + 0.797627i \(0.706088\pi\)
\(984\) 0 0
\(985\) −6337.14 −0.204993
\(986\) 0 0
\(987\) 7740.96 834.479i 0.249643 0.0269116i
\(988\) 0 0
\(989\) 16427.3i 0.528169i
\(990\) 0 0
\(991\) 8124.54i 0.260428i −0.991486 0.130214i \(-0.958434\pi\)
0.991486 0.130214i \(-0.0415665\pi\)
\(992\) 0 0
\(993\) −28767.0 + 3101.09i −0.919328 + 0.0991040i
\(994\) 0 0
\(995\) 35391.9 1.12764
\(996\) 0 0
\(997\) 32053.1 1.01819 0.509093 0.860711i \(-0.329981\pi\)
0.509093 + 0.860711i \(0.329981\pi\)
\(998\) 0 0
\(999\) 3136.47 + 9396.67i 0.0993328 + 0.297595i
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 384.4.c.d.383.6 yes 12
3.2 odd 2 384.4.c.a.383.8 yes 12
4.3 odd 2 384.4.c.a.383.7 12
8.3 odd 2 384.4.c.c.383.6 yes 12
8.5 even 2 384.4.c.b.383.7 yes 12
12.11 even 2 inner 384.4.c.d.383.5 yes 12
16.3 odd 4 768.4.f.e.383.11 12
16.5 even 4 768.4.f.f.383.11 12
16.11 odd 4 768.4.f.h.383.2 12
16.13 even 4 768.4.f.g.383.2 12
24.5 odd 2 384.4.c.c.383.5 yes 12
24.11 even 2 384.4.c.b.383.8 yes 12
48.5 odd 4 768.4.f.e.383.12 12
48.11 even 4 768.4.f.g.383.1 12
48.29 odd 4 768.4.f.h.383.1 12
48.35 even 4 768.4.f.f.383.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.c.a.383.7 12 4.3 odd 2
384.4.c.a.383.8 yes 12 3.2 odd 2
384.4.c.b.383.7 yes 12 8.5 even 2
384.4.c.b.383.8 yes 12 24.11 even 2
384.4.c.c.383.5 yes 12 24.5 odd 2
384.4.c.c.383.6 yes 12 8.3 odd 2
384.4.c.d.383.5 yes 12 12.11 even 2 inner
384.4.c.d.383.6 yes 12 1.1 even 1 trivial
768.4.f.e.383.11 12 16.3 odd 4
768.4.f.e.383.12 12 48.5 odd 4
768.4.f.f.383.11 12 16.5 even 4
768.4.f.f.383.12 12 48.35 even 4
768.4.f.g.383.1 12 48.11 even 4
768.4.f.g.383.2 12 16.13 even 4
768.4.f.h.383.1 12 48.29 odd 4
768.4.f.h.383.2 12 16.11 odd 4