Properties

Label 384.6.f.f.191.17
Level $384$
Weight $6$
Character 384.191
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,6,Mod(191,384)]
 
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, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.191");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.f (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: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 5192x^{16} + 8441320x^{12} + 4098006217x^{8} + 8949568544x^{4} + 8386816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{87}\cdot 3^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.17
Root \(0.123732 - 0.123732i\) of defining polynomial
Character \(\chi\) \(=\) 384.191
Dual form 384.6.f.f.191.18

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(14.0881 - 6.67280i) q^{3} -28.8990 q^{5} +44.6454i q^{7} +(153.948 - 188.014i) q^{9} +O(q^{10})\) \(q+(14.0881 - 6.67280i) q^{3} -28.8990 q^{5} +44.6454i q^{7} +(153.948 - 188.014i) q^{9} -152.334i q^{11} +578.038i q^{13} +(-407.132 + 192.837i) q^{15} +82.5350i q^{17} +1232.16 q^{19} +(297.910 + 628.967i) q^{21} +2598.70 q^{23} -2289.85 q^{25} +(914.247 - 3676.01i) q^{27} -53.2122 q^{29} +4031.95i q^{31} +(-1016.50 - 2146.10i) q^{33} -1290.21i q^{35} -3929.98i q^{37} +(3857.13 + 8143.44i) q^{39} -4852.97i q^{41} +14593.3 q^{43} +(-4448.93 + 5433.41i) q^{45} +10997.2 q^{47} +14813.8 q^{49} +(550.740 + 1162.76i) q^{51} +35748.6 q^{53} +4402.31i q^{55} +(17358.8 - 8221.95i) q^{57} -20128.9i q^{59} +28232.8i q^{61} +(8393.94 + 6873.05i) q^{63} -16704.7i q^{65} +1409.94 q^{67} +(36610.7 - 17340.6i) q^{69} -44453.1 q^{71} +17537.7 q^{73} +(-32259.5 + 15279.7i) q^{75} +6801.02 q^{77} -73476.1i q^{79} +(-11649.3 - 57888.5i) q^{81} +37013.3i q^{83} -2385.18i q^{85} +(-749.658 + 355.074i) q^{87} -106980. i q^{89} -25806.7 q^{91} +(26904.4 + 56802.4i) q^{93} -35608.2 q^{95} +32885.5 q^{97} +(-28640.9 - 23451.5i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 44 q^{9} + 3568 q^{15} - 6112 q^{23} + 15228 q^{25} + 7592 q^{33} + 2800 q^{39} - 26112 q^{47} - 81044 q^{49} - 89296 q^{57} + 14816 q^{63} + 72224 q^{71} - 61256 q^{73} + 89588 q^{81} + 145648 q^{87} - 385504 q^{95} + 92808 q^{97}+O(q^{100}) \) 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
\(3\) 14.0881 6.67280i 0.903750 0.428060i
\(4\) 0 0
\(5\) −28.8990 −0.516961 −0.258481 0.966016i \(-0.583222\pi\)
−0.258481 + 0.966016i \(0.583222\pi\)
\(6\) 0 0
\(7\) 44.6454i 0.344375i 0.985064 + 0.172187i \(0.0550835\pi\)
−0.985064 + 0.172187i \(0.944917\pi\)
\(8\) 0 0
\(9\) 153.948 188.014i 0.633529 0.773719i
\(10\) 0 0
\(11\) 152.334i 0.379591i −0.981824 0.189796i \(-0.939217\pi\)
0.981824 0.189796i \(-0.0607825\pi\)
\(12\) 0 0
\(13\) 578.038i 0.948632i 0.880355 + 0.474316i \(0.157305\pi\)
−0.880355 + 0.474316i \(0.842695\pi\)
\(14\) 0 0
\(15\) −407.132 + 192.837i −0.467204 + 0.221291i
\(16\) 0 0
\(17\) 82.5350i 0.0692654i 0.999400 + 0.0346327i \(0.0110261\pi\)
−0.999400 + 0.0346327i \(0.988974\pi\)
\(18\) 0 0
\(19\) 1232.16 0.783039 0.391519 0.920170i \(-0.371950\pi\)
0.391519 + 0.920170i \(0.371950\pi\)
\(20\) 0 0
\(21\) 297.910 + 628.967i 0.147413 + 0.311229i
\(22\) 0 0
\(23\) 2598.70 1.02432 0.512162 0.858889i \(-0.328845\pi\)
0.512162 + 0.858889i \(0.328845\pi\)
\(24\) 0 0
\(25\) −2289.85 −0.732751
\(26\) 0 0
\(27\) 914.247 3676.01i 0.241354 0.970437i
\(28\) 0 0
\(29\) −53.2122 −0.0117494 −0.00587471 0.999983i \(-0.501870\pi\)
−0.00587471 + 0.999983i \(0.501870\pi\)
\(30\) 0 0
\(31\) 4031.95i 0.753548i 0.926305 + 0.376774i \(0.122967\pi\)
−0.926305 + 0.376774i \(0.877033\pi\)
\(32\) 0 0
\(33\) −1016.50 2146.10i −0.162488 0.343056i
\(34\) 0 0
\(35\) 1290.21i 0.178028i
\(36\) 0 0
\(37\) 3929.98i 0.471940i −0.971760 0.235970i \(-0.924173\pi\)
0.971760 0.235970i \(-0.0758266\pi\)
\(38\) 0 0
\(39\) 3857.13 + 8143.44i 0.406072 + 0.857327i
\(40\) 0 0
\(41\) 4852.97i 0.450867i −0.974259 0.225433i \(-0.927620\pi\)
0.974259 0.225433i \(-0.0723798\pi\)
\(42\) 0 0
\(43\) 14593.3 1.20360 0.601800 0.798646i \(-0.294450\pi\)
0.601800 + 0.798646i \(0.294450\pi\)
\(44\) 0 0
\(45\) −4448.93 + 5433.41i −0.327510 + 0.399983i
\(46\) 0 0
\(47\) 10997.2 0.726169 0.363084 0.931756i \(-0.381724\pi\)
0.363084 + 0.931756i \(0.381724\pi\)
\(48\) 0 0
\(49\) 14813.8 0.881406
\(50\) 0 0
\(51\) 550.740 + 1162.76i 0.0296497 + 0.0625986i
\(52\) 0 0
\(53\) 35748.6 1.74811 0.874057 0.485824i \(-0.161480\pi\)
0.874057 + 0.485824i \(0.161480\pi\)
\(54\) 0 0
\(55\) 4402.31i 0.196234i
\(56\) 0 0
\(57\) 17358.8 8221.95i 0.707671 0.335188i
\(58\) 0 0
\(59\) 20128.9i 0.752818i −0.926454 0.376409i \(-0.877159\pi\)
0.926454 0.376409i \(-0.122841\pi\)
\(60\) 0 0
\(61\) 28232.8i 0.971471i 0.874106 + 0.485736i \(0.161448\pi\)
−0.874106 + 0.485736i \(0.838552\pi\)
\(62\) 0 0
\(63\) 8393.94 + 6873.05i 0.266449 + 0.218171i
\(64\) 0 0
\(65\) 16704.7i 0.490406i
\(66\) 0 0
\(67\) 1409.94 0.0383720 0.0191860 0.999816i \(-0.493893\pi\)
0.0191860 + 0.999816i \(0.493893\pi\)
\(68\) 0 0
\(69\) 36610.7 17340.6i 0.925733 0.438472i
\(70\) 0 0
\(71\) −44453.1 −1.04654 −0.523270 0.852167i \(-0.675288\pi\)
−0.523270 + 0.852167i \(0.675288\pi\)
\(72\) 0 0
\(73\) 17537.7 0.385181 0.192590 0.981279i \(-0.438311\pi\)
0.192590 + 0.981279i \(0.438311\pi\)
\(74\) 0 0
\(75\) −32259.5 + 15279.7i −0.662224 + 0.313661i
\(76\) 0 0
\(77\) 6801.02 0.130722
\(78\) 0 0
\(79\) 73476.1i 1.32458i −0.749247 0.662291i \(-0.769584\pi\)
0.749247 0.662291i \(-0.230416\pi\)
\(80\) 0 0
\(81\) −11649.3 57888.5i −0.197282 0.980347i
\(82\) 0 0
\(83\) 37013.3i 0.589743i 0.955537 + 0.294871i \(0.0952768\pi\)
−0.955537 + 0.294871i \(0.904723\pi\)
\(84\) 0 0
\(85\) 2385.18i 0.0358075i
\(86\) 0 0
\(87\) −749.658 + 355.074i −0.0106185 + 0.00502946i
\(88\) 0 0
\(89\) 106980.i 1.43162i −0.698293 0.715812i \(-0.746057\pi\)
0.698293 0.715812i \(-0.253943\pi\)
\(90\) 0 0
\(91\) −25806.7 −0.326685
\(92\) 0 0
\(93\) 26904.4 + 56802.4i 0.322564 + 0.681019i
\(94\) 0 0
\(95\) −35608.2 −0.404801
\(96\) 0 0
\(97\) 32885.5 0.354875 0.177438 0.984132i \(-0.443219\pi\)
0.177438 + 0.984132i \(0.443219\pi\)
\(98\) 0 0
\(99\) −28640.9 23451.5i −0.293697 0.240482i
\(100\) 0 0
\(101\) 177795. 1.73426 0.867131 0.498079i \(-0.165961\pi\)
0.867131 + 0.498079i \(0.165961\pi\)
\(102\) 0 0
\(103\) 10133.0i 0.0941118i 0.998892 + 0.0470559i \(0.0149839\pi\)
−0.998892 + 0.0470559i \(0.985016\pi\)
\(104\) 0 0
\(105\) −8609.29 18176.5i −0.0762069 0.160893i
\(106\) 0 0
\(107\) 168254.i 1.42072i 0.703841 + 0.710358i \(0.251467\pi\)
−0.703841 + 0.710358i \(0.748533\pi\)
\(108\) 0 0
\(109\) 171330.i 1.38123i 0.723220 + 0.690617i \(0.242661\pi\)
−0.723220 + 0.690617i \(0.757339\pi\)
\(110\) 0 0
\(111\) −26224.0 55365.9i −0.202018 0.426515i
\(112\) 0 0
\(113\) 172128.i 1.26811i −0.773289 0.634054i \(-0.781390\pi\)
0.773289 0.634054i \(-0.218610\pi\)
\(114\) 0 0
\(115\) −75100.0 −0.529536
\(116\) 0 0
\(117\) 108679. + 88987.5i 0.733975 + 0.600986i
\(118\) 0 0
\(119\) −3684.81 −0.0238532
\(120\) 0 0
\(121\) 137845. 0.855911
\(122\) 0 0
\(123\) −32382.9 68369.0i −0.192998 0.407471i
\(124\) 0 0
\(125\) 156484. 0.895765
\(126\) 0 0
\(127\) 133394.i 0.733882i −0.930244 0.366941i \(-0.880405\pi\)
0.930244 0.366941i \(-0.119595\pi\)
\(128\) 0 0
\(129\) 205592. 97378.2i 1.08775 0.515214i
\(130\) 0 0
\(131\) 53705.5i 0.273426i −0.990611 0.136713i \(-0.956346\pi\)
0.990611 0.136713i \(-0.0436539\pi\)
\(132\) 0 0
\(133\) 55010.3i 0.269659i
\(134\) 0 0
\(135\) −26420.9 + 106233.i −0.124771 + 0.501679i
\(136\) 0 0
\(137\) 352598.i 1.60501i −0.596643 0.802507i \(-0.703499\pi\)
0.596643 0.802507i \(-0.296501\pi\)
\(138\) 0 0
\(139\) 60518.3 0.265674 0.132837 0.991138i \(-0.457591\pi\)
0.132837 + 0.991138i \(0.457591\pi\)
\(140\) 0 0
\(141\) 154929. 73382.1i 0.656275 0.310844i
\(142\) 0 0
\(143\) 88055.0 0.360092
\(144\) 0 0
\(145\) 1537.78 0.00607400
\(146\) 0 0
\(147\) 208698. 98849.4i 0.796571 0.377295i
\(148\) 0 0
\(149\) 66567.9 0.245640 0.122820 0.992429i \(-0.460806\pi\)
0.122820 + 0.992429i \(0.460806\pi\)
\(150\) 0 0
\(151\) 496298.i 1.77133i 0.464323 + 0.885666i \(0.346298\pi\)
−0.464323 + 0.885666i \(0.653702\pi\)
\(152\) 0 0
\(153\) 15517.7 + 12706.1i 0.0535919 + 0.0438816i
\(154\) 0 0
\(155\) 116519.i 0.389555i
\(156\) 0 0
\(157\) 231714.i 0.750245i −0.926975 0.375123i \(-0.877601\pi\)
0.926975 0.375123i \(-0.122399\pi\)
\(158\) 0 0
\(159\) 503629. 238543.i 1.57986 0.748298i
\(160\) 0 0
\(161\) 116020.i 0.352751i
\(162\) 0 0
\(163\) 189838. 0.559646 0.279823 0.960052i \(-0.409724\pi\)
0.279823 + 0.960052i \(0.409724\pi\)
\(164\) 0 0
\(165\) 29375.7 + 62020.1i 0.0839999 + 0.177346i
\(166\) 0 0
\(167\) −475478. −1.31929 −0.659644 0.751578i \(-0.729293\pi\)
−0.659644 + 0.751578i \(0.729293\pi\)
\(168\) 0 0
\(169\) 37165.2 0.100097
\(170\) 0 0
\(171\) 189688. 231663.i 0.496078 0.605852i
\(172\) 0 0
\(173\) 445145. 1.13080 0.565400 0.824817i \(-0.308722\pi\)
0.565400 + 0.824817i \(0.308722\pi\)
\(174\) 0 0
\(175\) 102231.i 0.252341i
\(176\) 0 0
\(177\) −134316. 283577.i −0.322251 0.680360i
\(178\) 0 0
\(179\) 183022.i 0.426945i 0.976949 + 0.213472i \(0.0684773\pi\)
−0.976949 + 0.213472i \(0.931523\pi\)
\(180\) 0 0
\(181\) 297588.i 0.675180i 0.941293 + 0.337590i \(0.109612\pi\)
−0.941293 + 0.337590i \(0.890388\pi\)
\(182\) 0 0
\(183\) 188392. + 397746.i 0.415848 + 0.877967i
\(184\) 0 0
\(185\) 113573.i 0.243975i
\(186\) 0 0
\(187\) 12572.9 0.0262925
\(188\) 0 0
\(189\) 164117. + 40816.9i 0.334194 + 0.0831162i
\(190\) 0 0
\(191\) 190595. 0.378031 0.189016 0.981974i \(-0.439470\pi\)
0.189016 + 0.981974i \(0.439470\pi\)
\(192\) 0 0
\(193\) 320590. 0.619521 0.309761 0.950815i \(-0.399751\pi\)
0.309761 + 0.950815i \(0.399751\pi\)
\(194\) 0 0
\(195\) −111467. 235337.i −0.209923 0.443205i
\(196\) 0 0
\(197\) −508933. −0.934319 −0.467160 0.884173i \(-0.654723\pi\)
−0.467160 + 0.884173i \(0.654723\pi\)
\(198\) 0 0
\(199\) 738755.i 1.32242i 0.750203 + 0.661208i \(0.229956\pi\)
−0.750203 + 0.661208i \(0.770044\pi\)
\(200\) 0 0
\(201\) 19863.4 9408.26i 0.0346787 0.0164255i
\(202\) 0 0
\(203\) 2375.68i 0.00404620i
\(204\) 0 0
\(205\) 140246.i 0.233081i
\(206\) 0 0
\(207\) 400064. 488592.i 0.648939 0.792539i
\(208\) 0 0
\(209\) 187700.i 0.297235i
\(210\) 0 0
\(211\) −358830. −0.554858 −0.277429 0.960746i \(-0.589482\pi\)
−0.277429 + 0.960746i \(0.589482\pi\)
\(212\) 0 0
\(213\) −626258. + 296626.i −0.945811 + 0.447982i
\(214\) 0 0
\(215\) −421732. −0.622215
\(216\) 0 0
\(217\) −180008. −0.259503
\(218\) 0 0
\(219\) 247072. 117025.i 0.348107 0.164880i
\(220\) 0 0
\(221\) −47708.4 −0.0657074
\(222\) 0 0
\(223\) 1.33973e6i 1.80408i −0.431658 0.902038i \(-0.642071\pi\)
0.431658 0.902038i \(-0.357929\pi\)
\(224\) 0 0
\(225\) −352516. + 430523.i −0.464219 + 0.566943i
\(226\) 0 0
\(227\) 1.11010e6i 1.42987i 0.699191 + 0.714935i \(0.253544\pi\)
−0.699191 + 0.714935i \(0.746456\pi\)
\(228\) 0 0
\(229\) 28453.2i 0.0358544i −0.999839 0.0179272i \(-0.994293\pi\)
0.999839 0.0179272i \(-0.00570671\pi\)
\(230\) 0 0
\(231\) 95813.3 45381.8i 0.118140 0.0559567i
\(232\) 0 0
\(233\) 274080.i 0.330741i 0.986232 + 0.165370i \(0.0528819\pi\)
−0.986232 + 0.165370i \(0.947118\pi\)
\(234\) 0 0
\(235\) −317808. −0.375401
\(236\) 0 0
\(237\) −490291. 1.03514e6i −0.567000 1.19709i
\(238\) 0 0
\(239\) −736307. −0.833805 −0.416902 0.908951i \(-0.636884\pi\)
−0.416902 + 0.908951i \(0.636884\pi\)
\(240\) 0 0
\(241\) −1.01712e6 −1.12805 −0.564027 0.825757i \(-0.690748\pi\)
−0.564027 + 0.825757i \(0.690748\pi\)
\(242\) 0 0
\(243\) −550394. 737804.i −0.597941 0.801540i
\(244\) 0 0
\(245\) −428104. −0.455653
\(246\) 0 0
\(247\) 712235.i 0.742816i
\(248\) 0 0
\(249\) 246982. + 521446.i 0.252445 + 0.532980i
\(250\) 0 0
\(251\) 1.60216e6i 1.60517i 0.596540 + 0.802584i \(0.296542\pi\)
−0.596540 + 0.802584i \(0.703458\pi\)
\(252\) 0 0
\(253\) 395872.i 0.388824i
\(254\) 0 0
\(255\) −15915.8 33602.6i −0.0153278 0.0323611i
\(256\) 0 0
\(257\) 1.78274e6i 1.68366i 0.539739 + 0.841832i \(0.318523\pi\)
−0.539739 + 0.841832i \(0.681477\pi\)
\(258\) 0 0
\(259\) 175456. 0.162524
\(260\) 0 0
\(261\) −8191.89 + 10004.6i −0.00744360 + 0.00909075i
\(262\) 0 0
\(263\) −1.84815e6 −1.64759 −0.823793 0.566891i \(-0.808146\pi\)
−0.823793 + 0.566891i \(0.808146\pi\)
\(264\) 0 0
\(265\) −1.03310e6 −0.903707
\(266\) 0 0
\(267\) −713858. 1.50715e6i −0.612821 1.29383i
\(268\) 0 0
\(269\) −1.48588e6 −1.25200 −0.625999 0.779824i \(-0.715308\pi\)
−0.625999 + 0.779824i \(0.715308\pi\)
\(270\) 0 0
\(271\) 2.00499e6i 1.65840i −0.558950 0.829201i \(-0.688796\pi\)
0.558950 0.829201i \(-0.311204\pi\)
\(272\) 0 0
\(273\) −363567. + 172203.i −0.295242 + 0.139841i
\(274\) 0 0
\(275\) 348822.i 0.278146i
\(276\) 0 0
\(277\) 2.23674e6i 1.75152i −0.482745 0.875761i \(-0.660360\pi\)
0.482745 0.875761i \(-0.339640\pi\)
\(278\) 0 0
\(279\) 758062. + 620709.i 0.583034 + 0.477394i
\(280\) 0 0
\(281\) 1.98583e6i 1.50029i 0.661272 + 0.750146i \(0.270017\pi\)
−0.661272 + 0.750146i \(0.729983\pi\)
\(282\) 0 0
\(283\) −562336. −0.417379 −0.208689 0.977982i \(-0.566920\pi\)
−0.208689 + 0.977982i \(0.566920\pi\)
\(284\) 0 0
\(285\) −501651. + 237606.i −0.365839 + 0.173279i
\(286\) 0 0
\(287\) 216663. 0.155267
\(288\) 0 0
\(289\) 1.41304e6 0.995202
\(290\) 0 0
\(291\) 463294. 219439.i 0.320719 0.151908i
\(292\) 0 0
\(293\) −868343. −0.590911 −0.295456 0.955356i \(-0.595471\pi\)
−0.295456 + 0.955356i \(0.595471\pi\)
\(294\) 0 0
\(295\) 581706.i 0.389178i
\(296\) 0 0
\(297\) −559983. 139271.i −0.368369 0.0916158i
\(298\) 0 0
\(299\) 1.50215e6i 0.971707i
\(300\) 0 0
\(301\) 651524.i 0.414490i
\(302\) 0 0
\(303\) 2.50478e6 1.18639e6i 1.56734 0.742369i
\(304\) 0 0
\(305\) 815901.i 0.502213i
\(306\) 0 0
\(307\) 655310. 0.396827 0.198413 0.980118i \(-0.436421\pi\)
0.198413 + 0.980118i \(0.436421\pi\)
\(308\) 0 0
\(309\) 67615.3 + 142754.i 0.0402855 + 0.0850535i
\(310\) 0 0
\(311\) −2.43617e6 −1.42826 −0.714130 0.700013i \(-0.753178\pi\)
−0.714130 + 0.700013i \(0.753178\pi\)
\(312\) 0 0
\(313\) 538383. 0.310621 0.155310 0.987866i \(-0.450362\pi\)
0.155310 + 0.987866i \(0.450362\pi\)
\(314\) 0 0
\(315\) −242577. 198624.i −0.137744 0.112786i
\(316\) 0 0
\(317\) −2.44244e6 −1.36514 −0.682568 0.730822i \(-0.739137\pi\)
−0.682568 + 0.730822i \(0.739137\pi\)
\(318\) 0 0
\(319\) 8106.05i 0.00445998i
\(320\) 0 0
\(321\) 1.12273e6 + 2.37038e6i 0.608152 + 1.28397i
\(322\) 0 0
\(323\) 101696.i 0.0542374i
\(324\) 0 0
\(325\) 1.32362e6i 0.695111i
\(326\) 0 0
\(327\) 1.14325e6 + 2.41371e6i 0.591252 + 1.24829i
\(328\) 0 0
\(329\) 490974.i 0.250074i
\(330\) 0 0
\(331\) 1.40659e6 0.705662 0.352831 0.935687i \(-0.385219\pi\)
0.352831 + 0.935687i \(0.385219\pi\)
\(332\) 0 0
\(333\) −738891. 605011.i −0.365149 0.298987i
\(334\) 0 0
\(335\) −40745.9 −0.0198368
\(336\) 0 0
\(337\) −3.23749e6 −1.55287 −0.776433 0.630200i \(-0.782973\pi\)
−0.776433 + 0.630200i \(0.782973\pi\)
\(338\) 0 0
\(339\) −1.14858e6 2.42496e6i −0.542827 1.14605i
\(340\) 0 0
\(341\) 614204. 0.286040
\(342\) 0 0
\(343\) 1.41172e6i 0.647909i
\(344\) 0 0
\(345\) −1.05801e6 + 501127.i −0.478568 + 0.226673i
\(346\) 0 0
\(347\) 734757.i 0.327582i −0.986495 0.163791i \(-0.947628\pi\)
0.986495 0.163791i \(-0.0523722\pi\)
\(348\) 0 0
\(349\) 1.99425e6i 0.876426i 0.898871 + 0.438213i \(0.144388\pi\)
−0.898871 + 0.438213i \(0.855612\pi\)
\(350\) 0 0
\(351\) 2.12487e6 + 528470.i 0.920588 + 0.228956i
\(352\) 0 0
\(353\) 1.67494e6i 0.715423i 0.933832 + 0.357712i \(0.116443\pi\)
−0.933832 + 0.357712i \(0.883557\pi\)
\(354\) 0 0
\(355\) 1.28465e6 0.541021
\(356\) 0 0
\(357\) −51911.8 + 24588.0i −0.0215574 + 0.0102106i
\(358\) 0 0
\(359\) 131434. 0.0538233 0.0269116 0.999638i \(-0.491433\pi\)
0.0269116 + 0.999638i \(0.491433\pi\)
\(360\) 0 0
\(361\) −957880. −0.386851
\(362\) 0 0
\(363\) 1.94197e6 919813.i 0.773529 0.366381i
\(364\) 0 0
\(365\) −506821. −0.199123
\(366\) 0 0
\(367\) 1.39215e6i 0.539535i 0.962925 + 0.269768i \(0.0869469\pi\)
−0.962925 + 0.269768i \(0.913053\pi\)
\(368\) 0 0
\(369\) −912425. 747103.i −0.348844 0.285637i
\(370\) 0 0
\(371\) 1.59601e6i 0.602006i
\(372\) 0 0
\(373\) 1.31485e6i 0.489333i −0.969607 0.244667i \(-0.921321\pi\)
0.969607 0.244667i \(-0.0786785\pi\)
\(374\) 0 0
\(375\) 2.20455e6 1.04418e6i 0.809548 0.383441i
\(376\) 0 0
\(377\) 30758.7i 0.0111459i
\(378\) 0 0
\(379\) 1.11465e6 0.398602 0.199301 0.979938i \(-0.436133\pi\)
0.199301 + 0.979938i \(0.436133\pi\)
\(380\) 0 0
\(381\) −890110. 1.87926e6i −0.314146 0.663246i
\(382\) 0 0
\(383\) −2.84613e6 −0.991419 −0.495710 0.868488i \(-0.665092\pi\)
−0.495710 + 0.868488i \(0.665092\pi\)
\(384\) 0 0
\(385\) −196543. −0.0675780
\(386\) 0 0
\(387\) 2.24660e6 2.74374e6i 0.762516 0.931249i
\(388\) 0 0
\(389\) 226921. 0.0760328 0.0380164 0.999277i \(-0.487896\pi\)
0.0380164 + 0.999277i \(0.487896\pi\)
\(390\) 0 0
\(391\) 214484.i 0.0709502i
\(392\) 0 0
\(393\) −358366. 756607.i −0.117043 0.247109i
\(394\) 0 0
\(395\) 2.12339e6i 0.684757i
\(396\) 0 0
\(397\) 2.18565e6i 0.695992i 0.937496 + 0.347996i \(0.113138\pi\)
−0.937496 + 0.347996i \(0.886862\pi\)
\(398\) 0 0
\(399\) 367072. + 774988.i 0.115430 + 0.243704i
\(400\) 0 0
\(401\) 3.78893e6i 1.17667i −0.808616 0.588336i \(-0.799783\pi\)
0.808616 0.588336i \(-0.200217\pi\)
\(402\) 0 0
\(403\) −2.33062e6 −0.714840
\(404\) 0 0
\(405\) 336653. + 1.67292e6i 0.101987 + 0.506801i
\(406\) 0 0
\(407\) −598671. −0.179144
\(408\) 0 0
\(409\) 2.94246e6 0.869765 0.434882 0.900487i \(-0.356790\pi\)
0.434882 + 0.900487i \(0.356790\pi\)
\(410\) 0 0
\(411\) −2.35282e6 4.96743e6i −0.687042 1.45053i
\(412\) 0 0
\(413\) 898663. 0.259252
\(414\) 0 0
\(415\) 1.06965e6i 0.304874i
\(416\) 0 0
\(417\) 852586. 403826.i 0.240103 0.113725i
\(418\) 0 0
\(419\) 5.05386e6i 1.40633i 0.711026 + 0.703166i \(0.248231\pi\)
−0.711026 + 0.703166i \(0.751769\pi\)
\(420\) 0 0
\(421\) 1.45462e6i 0.399984i −0.979798 0.199992i \(-0.935908\pi\)
0.979798 0.199992i \(-0.0640917\pi\)
\(422\) 0 0
\(423\) 1.69299e6 2.06762e6i 0.460049 0.561850i
\(424\) 0 0
\(425\) 188993.i 0.0507543i
\(426\) 0 0
\(427\) −1.26047e6 −0.334550
\(428\) 0 0
\(429\) 1.24053e6 587573.i 0.325434 0.154141i
\(430\) 0 0
\(431\) −3.82711e6 −0.992380 −0.496190 0.868214i \(-0.665268\pi\)
−0.496190 + 0.868214i \(0.665268\pi\)
\(432\) 0 0
\(433\) 4.03269e6 1.03365 0.516827 0.856090i \(-0.327113\pi\)
0.516827 + 0.856090i \(0.327113\pi\)
\(434\) 0 0
\(435\) 21664.4 10261.3i 0.00548938 0.00260004i
\(436\) 0 0
\(437\) 3.20202e6 0.802085
\(438\) 0 0
\(439\) 2.40698e6i 0.596090i −0.954552 0.298045i \(-0.903665\pi\)
0.954552 0.298045i \(-0.0963345\pi\)
\(440\) 0 0
\(441\) 2.28055e6 2.78520e6i 0.558396 0.681960i
\(442\) 0 0
\(443\) 1.08258e6i 0.262090i 0.991376 + 0.131045i \(0.0418332\pi\)
−0.991376 + 0.131045i \(0.958167\pi\)
\(444\) 0 0
\(445\) 3.09163e6i 0.740094i
\(446\) 0 0
\(447\) 937813. 444194.i 0.221997 0.105149i
\(448\) 0 0
\(449\) 521460.i 0.122069i −0.998136 0.0610344i \(-0.980560\pi\)
0.998136 0.0610344i \(-0.0194399\pi\)
\(450\) 0 0
\(451\) −739274. −0.171145
\(452\) 0 0
\(453\) 3.31169e6 + 6.99188e6i 0.758237 + 1.60084i
\(454\) 0 0
\(455\) 745789. 0.168884
\(456\) 0 0
\(457\) 1.05419e6 0.236117 0.118058 0.993007i \(-0.462333\pi\)
0.118058 + 0.993007i \(0.462333\pi\)
\(458\) 0 0
\(459\) 303400. + 75457.4i 0.0672177 + 0.0167175i
\(460\) 0 0
\(461\) 7.05120e6 1.54529 0.772647 0.634836i \(-0.218933\pi\)
0.772647 + 0.634836i \(0.218933\pi\)
\(462\) 0 0
\(463\) 4.69949e6i 1.01882i 0.860523 + 0.509411i \(0.170137\pi\)
−0.860523 + 0.509411i \(0.829863\pi\)
\(464\) 0 0
\(465\) −777510. 1.64153e6i −0.166753 0.352061i
\(466\) 0 0
\(467\) 1.38535e6i 0.293946i 0.989140 + 0.146973i \(0.0469531\pi\)
−0.989140 + 0.146973i \(0.953047\pi\)
\(468\) 0 0
\(469\) 62947.4i 0.0132143i
\(470\) 0 0
\(471\) −1.54618e6 3.26440e6i −0.321150 0.678034i
\(472\) 0 0
\(473\) 2.22306e6i 0.456876i
\(474\) 0 0
\(475\) −2.82146e6 −0.573772
\(476\) 0 0
\(477\) 5.50341e6 6.72123e6i 1.10748 1.35255i
\(478\) 0 0
\(479\) −3.72473e6 −0.741748 −0.370874 0.928683i \(-0.620942\pi\)
−0.370874 + 0.928683i \(0.620942\pi\)
\(480\) 0 0
\(481\) 2.27168e6 0.447697
\(482\) 0 0
\(483\) 774179. + 1.63450e6i 0.150999 + 0.318799i
\(484\) 0 0
\(485\) −950360. −0.183457
\(486\) 0 0
\(487\) 4.89476e6i 0.935209i −0.883938 0.467605i \(-0.845117\pi\)
0.883938 0.467605i \(-0.154883\pi\)
\(488\) 0 0
\(489\) 2.67445e6 1.26675e6i 0.505781 0.239562i
\(490\) 0 0
\(491\) 6.81907e6i 1.27650i 0.769829 + 0.638250i \(0.220342\pi\)
−0.769829 + 0.638250i \(0.779658\pi\)
\(492\) 0 0
\(493\) 4391.87i 0.000813828i
\(494\) 0 0
\(495\) 827695. + 677725.i 0.151830 + 0.124320i
\(496\) 0 0
\(497\) 1.98462e6i 0.360402i
\(498\) 0 0
\(499\) −8.21107e6 −1.47621 −0.738105 0.674686i \(-0.764279\pi\)
−0.738105 + 0.674686i \(0.764279\pi\)
\(500\) 0 0
\(501\) −6.69857e6 + 3.17277e6i −1.19231 + 0.564734i
\(502\) 0 0
\(503\) −5.77094e6 −1.01701 −0.508507 0.861058i \(-0.669802\pi\)
−0.508507 + 0.861058i \(0.669802\pi\)
\(504\) 0 0
\(505\) −5.13809e6 −0.896547
\(506\) 0 0
\(507\) 523586. 247996.i 0.0904624 0.0428474i
\(508\) 0 0
\(509\) 7.52490e6 1.28738 0.643689 0.765287i \(-0.277403\pi\)
0.643689 + 0.765287i \(0.277403\pi\)
\(510\) 0 0
\(511\) 782975.i 0.132646i
\(512\) 0 0
\(513\) 1.12650e6 4.52943e6i 0.188989 0.759890i
\(514\) 0 0
\(515\) 292833.i 0.0486522i
\(516\) 0 0
\(517\) 1.67525e6i 0.275647i
\(518\) 0 0
\(519\) 6.27123e6 2.97036e6i 1.02196 0.484050i
\(520\) 0 0
\(521\) 9.63123e6i 1.55449i −0.629199 0.777244i \(-0.716617\pi\)
0.629199 0.777244i \(-0.283383\pi\)
\(522\) 0 0
\(523\) 3.69103e6 0.590057 0.295028 0.955488i \(-0.404671\pi\)
0.295028 + 0.955488i \(0.404671\pi\)
\(524\) 0 0
\(525\) −682167. 1.44024e6i −0.108017 0.228053i
\(526\) 0 0
\(527\) −332777. −0.0521948
\(528\) 0 0
\(529\) 316921. 0.0492394
\(530\) 0 0
\(531\) −3.78451e6 3.09880e6i −0.582470 0.476932i
\(532\) 0 0
\(533\) 2.80520e6 0.427707
\(534\) 0 0
\(535\) 4.86239e6i 0.734455i
\(536\) 0 0
\(537\) 1.22127e6 + 2.57843e6i 0.182758 + 0.385851i
\(538\) 0 0
\(539\) 2.25665e6i 0.334574i
\(540\) 0 0
\(541\) 7.19936e6i 1.05755i 0.848762 + 0.528775i \(0.177348\pi\)
−0.848762 + 0.528775i \(0.822652\pi\)
\(542\) 0 0
\(543\) 1.98575e6 + 4.19245e6i 0.289018 + 0.610194i
\(544\) 0 0
\(545\) 4.95127e6i 0.714045i
\(546\) 0 0
\(547\) 1.16775e7 1.66871 0.834353 0.551230i \(-0.185841\pi\)
0.834353 + 0.551230i \(0.185841\pi\)
\(548\) 0 0
\(549\) 5.30816e6 + 4.34638e6i 0.751646 + 0.615455i
\(550\) 0 0
\(551\) −65566.0 −0.00920025
\(552\) 0 0
\(553\) 3.28037e6 0.456152
\(554\) 0 0
\(555\) 757847. + 1.60002e6i 0.104436 + 0.220492i
\(556\) 0 0
\(557\) 2.59886e6 0.354932 0.177466 0.984127i \(-0.443210\pi\)
0.177466 + 0.984127i \(0.443210\pi\)
\(558\) 0 0
\(559\) 8.43548e6i 1.14177i
\(560\) 0 0
\(561\) 177128. 83896.5i 0.0237619 0.0112548i
\(562\) 0 0
\(563\) 6.44592e6i 0.857065i −0.903526 0.428533i \(-0.859031\pi\)
0.903526 0.428533i \(-0.140969\pi\)
\(564\) 0 0
\(565\) 4.97434e6i 0.655563i
\(566\) 0 0
\(567\) 2.58445e6 520087.i 0.337607 0.0679389i
\(568\) 0 0
\(569\) 3.43660e6i 0.444988i −0.974934 0.222494i \(-0.928580\pi\)
0.974934 0.222494i \(-0.0714198\pi\)
\(570\) 0 0
\(571\) −6.87411e6 −0.882320 −0.441160 0.897429i \(-0.645433\pi\)
−0.441160 + 0.897429i \(0.645433\pi\)
\(572\) 0 0
\(573\) 2.68511e6 1.27180e6i 0.341646 0.161820i
\(574\) 0 0
\(575\) −5.95063e6 −0.750574
\(576\) 0 0
\(577\) 929033. 0.116169 0.0580847 0.998312i \(-0.481501\pi\)
0.0580847 + 0.998312i \(0.481501\pi\)
\(578\) 0 0
\(579\) 4.51649e6 2.13923e6i 0.559892 0.265192i
\(580\) 0 0
\(581\) −1.65247e6 −0.203092
\(582\) 0 0
\(583\) 5.44574e6i 0.663568i
\(584\) 0 0
\(585\) −3.14072e6 2.57165e6i −0.379437 0.310687i
\(586\) 0 0
\(587\) 6.26726e6i 0.750728i −0.926878 0.375364i \(-0.877518\pi\)
0.926878 0.375364i \(-0.122482\pi\)
\(588\) 0 0
\(589\) 4.96801e6i 0.590057i
\(590\) 0 0
\(591\) −7.16989e6 + 3.39601e6i −0.844391 + 0.399945i
\(592\) 0 0
\(593\) 3.55168e6i 0.414760i 0.978260 + 0.207380i \(0.0664936\pi\)
−0.978260 + 0.207380i \(0.933506\pi\)
\(594\) 0 0
\(595\) 106487. 0.0123312
\(596\) 0 0
\(597\) 4.92956e6 + 1.04076e7i 0.566073 + 1.19513i
\(598\) 0 0
\(599\) 1.62084e7 1.84575 0.922876 0.385098i \(-0.125832\pi\)
0.922876 + 0.385098i \(0.125832\pi\)
\(600\) 0 0
\(601\) 6.56876e6 0.741818 0.370909 0.928669i \(-0.379046\pi\)
0.370909 + 0.928669i \(0.379046\pi\)
\(602\) 0 0
\(603\) 217057. 265088.i 0.0243098 0.0296891i
\(604\) 0 0
\(605\) −3.98359e6 −0.442473
\(606\) 0 0
\(607\) 1.54789e7i 1.70518i 0.522584 + 0.852588i \(0.324968\pi\)
−0.522584 + 0.852588i \(0.675032\pi\)
\(608\) 0 0
\(609\) −15852.4 33468.7i −0.00173202 0.00365676i
\(610\) 0 0
\(611\) 6.35680e6i 0.688867i
\(612\) 0 0
\(613\) 1.23906e7i 1.33181i −0.746038 0.665903i \(-0.768046\pi\)
0.746038 0.665903i \(-0.231954\pi\)
\(614\) 0 0
\(615\) 935834. + 1.97580e6i 0.0997725 + 0.210647i
\(616\) 0 0
\(617\) 5.76077e6i 0.609211i 0.952479 + 0.304606i \(0.0985247\pi\)
−0.952479 + 0.304606i \(0.901475\pi\)
\(618\) 0 0
\(619\) 1.06686e7 1.11913 0.559564 0.828787i \(-0.310969\pi\)
0.559564 + 0.828787i \(0.310969\pi\)
\(620\) 0 0
\(621\) 2.37586e6 9.55287e6i 0.247225 0.994042i
\(622\) 0 0
\(623\) 4.77618e6 0.493015
\(624\) 0 0
\(625\) 2.63354e6 0.269675
\(626\) 0 0
\(627\) −1.25249e6 2.64434e6i −0.127234 0.268626i
\(628\) 0 0
\(629\) 324361. 0.0326891
\(630\) 0 0
\(631\) 1.14991e7i 1.14972i −0.818253 0.574858i \(-0.805057\pi\)
0.818253 0.574858i \(-0.194943\pi\)
\(632\) 0 0
\(633\) −5.05522e6 + 2.39440e6i −0.501453 + 0.237513i
\(634\) 0 0
\(635\) 3.85495e6i 0.379389i
\(636\) 0 0
\(637\) 8.56293e6i 0.836130i
\(638\) 0 0
\(639\) −6.84344e6 + 8.35778e6i −0.663013 + 0.809728i
\(640\) 0 0
\(641\) 2.17687e6i 0.209261i −0.994511 0.104630i \(-0.966634\pi\)
0.994511 0.104630i \(-0.0333660\pi\)
\(642\) 0 0
\(643\) 1.45127e7 1.38427 0.692133 0.721770i \(-0.256671\pi\)
0.692133 + 0.721770i \(0.256671\pi\)
\(644\) 0 0
\(645\) −5.94139e6 + 2.81413e6i −0.562327 + 0.266346i
\(646\) 0 0
\(647\) 1.07629e7 1.01081 0.505405 0.862882i \(-0.331343\pi\)
0.505405 + 0.862882i \(0.331343\pi\)
\(648\) 0 0
\(649\) −3.06632e6 −0.285763
\(650\) 0 0
\(651\) −2.53596e6 + 1.20116e6i −0.234526 + 0.111083i
\(652\) 0 0
\(653\) −8.99624e6 −0.825616 −0.412808 0.910818i \(-0.635452\pi\)
−0.412808 + 0.910818i \(0.635452\pi\)
\(654\) 0 0
\(655\) 1.55204e6i 0.141351i
\(656\) 0 0
\(657\) 2.69988e6 3.29732e6i 0.244023 0.298021i
\(658\) 0 0
\(659\) 1.19620e7i 1.07298i −0.843908 0.536488i \(-0.819751\pi\)
0.843908 0.536488i \(-0.180249\pi\)
\(660\) 0 0
\(661\) 2.07936e7i 1.85109i −0.378640 0.925544i \(-0.623608\pi\)
0.378640 0.925544i \(-0.376392\pi\)
\(662\) 0 0
\(663\) −672119. + 318348.i −0.0593830 + 0.0281267i
\(664\) 0 0
\(665\) 1.58974e6i 0.139403i
\(666\) 0 0
\(667\) −138283. −0.0120352
\(668\) 0 0
\(669\) −8.93974e6 1.88742e7i −0.772253 1.63043i
\(670\) 0 0
\(671\) 4.30083e6 0.368762
\(672\) 0 0
\(673\) −1.54212e7 −1.31244 −0.656221 0.754569i \(-0.727846\pi\)
−0.656221 + 0.754569i \(0.727846\pi\)
\(674\) 0 0
\(675\) −2.09349e6 + 8.41750e6i −0.176852 + 0.711089i
\(676\) 0 0
\(677\) 1.02029e6 0.0855565 0.0427783 0.999085i \(-0.486379\pi\)
0.0427783 + 0.999085i \(0.486379\pi\)
\(678\) 0 0
\(679\) 1.46819e6i 0.122210i
\(680\) 0 0
\(681\) 7.40746e6 + 1.56391e7i 0.612070 + 1.29225i
\(682\) 0 0
\(683\) 1.00685e7i 0.825873i −0.910760 0.412936i \(-0.864503\pi\)
0.910760 0.412936i \(-0.135497\pi\)
\(684\) 0 0
\(685\) 1.01897e7i 0.829730i
\(686\) 0 0
\(687\) −189862. 400851.i −0.0153478 0.0324034i
\(688\) 0 0
\(689\) 2.06641e7i 1.65832i
\(690\) 0 0
\(691\) −2.49758e7 −1.98987 −0.994934 0.100531i \(-0.967946\pi\)
−0.994934 + 0.100531i \(0.967946\pi\)
\(692\) 0 0
\(693\) 1.04700e6 1.27869e6i 0.0828160 0.101142i
\(694\) 0 0
\(695\) −1.74892e6 −0.137343
\(696\) 0 0
\(697\) 400540. 0.0312294
\(698\) 0 0
\(699\) 1.82888e6 + 3.86126e6i 0.141577 + 0.298907i
\(700\) 0 0
\(701\) −1.23032e7 −0.945631 −0.472816 0.881161i \(-0.656762\pi\)
−0.472816 + 0.881161i \(0.656762\pi\)
\(702\) 0 0
\(703\) 4.84237e6i 0.369547i
\(704\) 0 0
\(705\) −4.47731e6 + 2.12067e6i −0.339269 + 0.160694i
\(706\) 0 0
\(707\) 7.93770e6i 0.597236i
\(708\) 0 0
\(709\) 1.44444e7i 1.07915i −0.841936 0.539577i \(-0.818584\pi\)
0.841936 0.539577i \(-0.181416\pi\)
\(710\) 0 0
\(711\) −1.38145e7 1.13115e7i −1.02485 0.839161i
\(712\) 0 0
\(713\) 1.04778e7i 0.771877i
\(714\) 0 0
\(715\) −2.54470e6 −0.186154
\(716\) 0 0
\(717\) −1.03732e7 + 4.91323e6i −0.753551 + 0.356919i
\(718\) 0 0
\(719\) −4.61863e6 −0.333189 −0.166595 0.986025i \(-0.553277\pi\)
−0.166595 + 0.986025i \(0.553277\pi\)
\(720\) 0 0
\(721\) −452391. −0.0324097
\(722\) 0 0
\(723\) −1.43293e7 + 6.78704e6i −1.01948 + 0.482875i
\(724\) 0 0
\(725\) 121848. 0.00860940
\(726\) 0 0
\(727\) 681858.i 0.0478473i −0.999714 0.0239237i \(-0.992384\pi\)
0.999714 0.0239237i \(-0.00761587\pi\)
\(728\) 0 0
\(729\) −1.26772e7 6.72157e6i −0.883497 0.468438i
\(730\) 0 0
\(731\) 1.20446e6i 0.0833679i
\(732\) 0 0
\(733\) 1.19274e7i 0.819945i 0.912098 + 0.409972i \(0.134462\pi\)
−0.912098 + 0.409972i \(0.865538\pi\)
\(734\) 0 0
\(735\) −6.03116e6 + 2.85665e6i −0.411796 + 0.195047i
\(736\) 0 0
\(737\) 214783.i 0.0145657i
\(738\) 0 0
\(739\) 1.82646e7 1.23027 0.615133 0.788424i \(-0.289103\pi\)
0.615133 + 0.788424i \(0.289103\pi\)
\(740\) 0 0
\(741\) 4.75260e6 + 1.00340e7i 0.317970 + 0.671320i
\(742\) 0 0
\(743\) −9.74762e6 −0.647778 −0.323889 0.946095i \(-0.604990\pi\)
−0.323889 + 0.946095i \(0.604990\pi\)
\(744\) 0 0
\(745\) −1.92375e6 −0.126986
\(746\) 0 0
\(747\) 6.95900e6 + 5.69811e6i 0.456295 + 0.373619i
\(748\) 0 0
\(749\) −7.51178e6 −0.489259
\(750\) 0 0
\(751\) 1.40856e7i 0.911330i −0.890151 0.455665i \(-0.849402\pi\)
0.890151 0.455665i \(-0.150598\pi\)
\(752\) 0 0
\(753\) 1.06909e7 + 2.25713e7i 0.687108 + 1.45067i
\(754\) 0 0
\(755\) 1.43425e7i 0.915710i
\(756\) 0 0
\(757\) 1.59982e7i 1.01469i −0.861744 0.507344i \(-0.830627\pi\)
0.861744 0.507344i \(-0.169373\pi\)
\(758\) 0 0
\(759\) −2.64157e6 5.57707e6i −0.166440 0.351400i
\(760\) 0 0
\(761\) 6.23238e6i 0.390115i 0.980792 + 0.195057i \(0.0624893\pi\)
−0.980792 + 0.195057i \(0.937511\pi\)
\(762\) 0 0
\(763\) −7.64910e6 −0.475663
\(764\) 0 0
\(765\) −448447. 367193.i −0.0277050 0.0226851i
\(766\) 0 0
\(767\) 1.16353e7 0.714148
\(768\) 0 0
\(769\) −6.40292e6 −0.390447 −0.195224 0.980759i \(-0.562543\pi\)
−0.195224 + 0.980759i \(0.562543\pi\)
\(770\) 0 0
\(771\) 1.18959e7 + 2.51154e7i 0.720710 + 1.52161i
\(772\) 0 0
\(773\) −1.25165e7 −0.753416 −0.376708 0.926332i \(-0.622944\pi\)
−0.376708 + 0.926332i \(0.622944\pi\)
\(774\) 0 0
\(775\) 9.23255e6i 0.552163i
\(776\) 0 0
\(777\) 2.47183e6 1.17078e6i 0.146881 0.0695701i
\(778\) 0 0
\(779\) 5.97964e6i 0.353046i
\(780\) 0 0
\(781\) 6.77173e6i 0.397257i
\(782\) 0 0
\(783\) −48649.1 + 195609.i −0.00283577 + 0.0114021i
\(784\) 0 0
\(785\) 6.69631e6i 0.387848i
\(786\) 0 0
\(787\) −2.29675e7 −1.32183 −0.660917 0.750459i \(-0.729832\pi\)
−0.660917 + 0.750459i \(0.729832\pi\)
\(788\) 0 0
\(789\) −2.60369e7 + 1.23323e7i −1.48901 + 0.705266i
\(790\) 0 0
\(791\) 7.68474e6 0.436705
\(792\) 0 0
\(793\) −1.63196e7 −0.921569
\(794\) 0 0
\(795\) −1.45544e7 + 6.89367e6i −0.816726 + 0.386841i
\(796\) 0 0
\(797\) −1.69135e7 −0.943164 −0.471582 0.881822i \(-0.656317\pi\)
−0.471582 + 0.881822i \(0.656317\pi\)
\(798\) 0 0
\(799\) 907654.i 0.0502983i
\(800\) 0 0
\(801\) −2.01138e7 1.64694e7i −1.10767 0.906975i
\(802\) 0 0
\(803\) 2.67159e6i 0.146211i
\(804\) 0 0
\(805\) 3.35287e6i 0.182359i
\(806\) 0 0
\(807\) −2.09332e7 + 9.91498e6i −1.13149 + 0.535930i
\(808\) 0 0
\(809\) 2.09637e7i 1.12615i 0.826406 + 0.563075i \(0.190382\pi\)
−0.826406 + 0.563075i \(0.809618\pi\)
\(810\) 0 0
\(811\) −5.01493e6 −0.267740 −0.133870 0.990999i \(-0.542740\pi\)
−0.133870 + 0.990999i \(0.542740\pi\)
\(812\) 0 0
\(813\) −1.33789e7 2.82465e7i −0.709896 1.49878i
\(814\) 0 0
\(815\) −5.48613e6 −0.289316
\(816\) 0 0
\(817\) 1.79813e7 0.942466
\(818\) 0 0
\(819\) −3.97288e6 + 4.85202e6i −0.206965 + 0.252762i
\(820\) 0 0
\(821\) −3.05082e6 −0.157964 −0.0789821 0.996876i \(-0.525167\pi\)
−0.0789821 + 0.996876i \(0.525167\pi\)
\(822\) 0 0
\(823\) 2.19672e7i 1.13051i 0.824916 + 0.565255i \(0.191222\pi\)
−0.824916 + 0.565255i \(0.808778\pi\)
\(824\) 0 0
\(825\) 2.32762e6 + 4.91423e6i 0.119063 + 0.251374i
\(826\) 0 0
\(827\) 2.60374e7i 1.32384i −0.749576 0.661918i \(-0.769743\pi\)
0.749576 0.661918i \(-0.230257\pi\)
\(828\) 0 0
\(829\) 1.21363e7i 0.613336i −0.951817 0.306668i \(-0.900786\pi\)
0.951817 0.306668i \(-0.0992141\pi\)
\(830\) 0 0
\(831\) −1.49253e7 3.15113e7i −0.749756 1.58294i
\(832\) 0 0
\(833\) 1.22266e6i 0.0610509i
\(834\) 0 0
\(835\) 1.37409e7 0.682021
\(836\) 0 0
\(837\) 1.48215e7 + 3.68620e6i 0.731271 + 0.181872i
\(838\) 0 0
\(839\) 1.94937e7 0.956069 0.478034 0.878341i \(-0.341349\pi\)
0.478034 + 0.878341i \(0.341349\pi\)
\(840\) 0 0
\(841\) −2.05083e7 −0.999862
\(842\) 0 0
\(843\) 1.32510e7 + 2.79765e7i 0.642215 + 1.35589i
\(844\) 0 0
\(845\) −1.07404e6 −0.0517461
\(846\) 0 0
\(847\) 6.15415e6i 0.294754i
\(848\) 0 0
\(849\) −7.92224e6 + 3.75236e6i −0.377206 + 0.178663i
\(850\) 0 0
\(851\) 1.02129e7i 0.483419i
\(852\) 0 0
\(853\) 1.87188e7i 0.880858i −0.897788 0.440429i \(-0.854826\pi\)
0.897788 0.440429i \(-0.145174\pi\)
\(854\) 0 0
\(855\) −5.48180e6 + 6.69483e6i −0.256453 + 0.313202i
\(856\) 0 0
\(857\) 5.40764e6i 0.251510i 0.992061 + 0.125755i \(0.0401354\pi\)
−0.992061 + 0.125755i \(0.959865\pi\)
\(858\) 0 0
\(859\) −1.14398e7 −0.528977 −0.264489 0.964389i \(-0.585203\pi\)
−0.264489 + 0.964389i \(0.585203\pi\)
\(860\) 0 0
\(861\) 3.05236e6 1.44575e6i 0.140323 0.0664637i
\(862\) 0 0
\(863\) 3.07572e7 1.40579 0.702895 0.711294i \(-0.251890\pi\)
0.702895 + 0.711294i \(0.251890\pi\)
\(864\) 0 0
\(865\) −1.28642e7 −0.584580
\(866\) 0 0
\(867\) 1.99071e7 9.42896e6i 0.899414 0.426006i
\(868\) 0 0
\(869\) −1.11929e7 −0.502799
\(870\) 0 0
\(871\) 815000.i 0.0364009i
\(872\) 0 0
\(873\) 5.06265e6 6.18293e6i 0.224824 0.274574i
\(874\) 0 0
\(875\) 6.98628e6i 0.308479i
\(876\) 0 0
\(877\) 1.93601e7i 0.849981i 0.905198 + 0.424991i \(0.139723\pi\)
−0.905198 + 0.424991i \(0.860277\pi\)
\(878\) 0 0
\(879\) −1.22333e7 + 5.79428e6i −0.534036 + 0.252946i
\(880\) 0 0
\(881\) 2.36498e7i 1.02657i 0.858219 + 0.513284i \(0.171571\pi\)
−0.858219 + 0.513284i \(0.828429\pi\)
\(882\) 0 0
\(883\) −1.50443e7 −0.649336 −0.324668 0.945828i \(-0.605253\pi\)
−0.324668 + 0.945828i \(0.605253\pi\)
\(884\) 0 0
\(885\) 3.88160e6 + 8.19511e6i 0.166592 + 0.351720i
\(886\) 0 0
\(887\) 1.89243e7 0.807628 0.403814 0.914841i \(-0.367684\pi\)
0.403814 + 0.914841i \(0.367684\pi\)
\(888\) 0 0
\(889\) 5.95542e6 0.252731
\(890\) 0 0
\(891\) −8.81841e6 + 1.77459e6i −0.372131 + 0.0748864i
\(892\) 0 0
\(893\) 1.35503e7 0.568618
\(894\) 0 0
\(895\) 5.28917e6i 0.220714i
\(896\) 0 0
\(897\) 1.00235e7 + 2.11624e7i 0.415949 + 0.878180i
\(898\) 0 0
\(899\) 214549.i 0.00885375i
\(900\) 0 0
\(901\) 2.95051e6i 0.121084i
\(902\) 0 0
\(903\) 4.34748e6 + 9.17871e6i 0.177427 + 0.374595i
\(904\) 0 0
\(905\) 8.60001e6i 0.349042i
\(906\) 0 0
\(907\) 4.26513e7 1.72153 0.860765 0.509003i \(-0.169986\pi\)
0.860765 + 0.509003i \(0.169986\pi\)
\(908\) 0 0
\(909\) 2.73710e7 3.34278e7i 1.09871 1.34183i
\(910\) 0 0
\(911\) 2.50247e7 0.999018 0.499509 0.866309i \(-0.333514\pi\)
0.499509 + 0.866309i \(0.333514\pi\)
\(912\) 0 0
\(913\) 5.63839e6 0.223861
\(914\) 0 0
\(915\) −5.44434e6 1.14945e7i −0.214977 0.453875i
\(916\) 0 0
\(917\) 2.39770e6 0.0941612
\(918\) 0 0
\(919\) 3.02276e7i 1.18063i 0.807172 + 0.590316i \(0.200997\pi\)
−0.807172 + 0.590316i \(0.799003\pi\)
\(920\) 0 0
\(921\) 9.23206e6 4.37275e6i 0.358632 0.169866i
\(922\) 0 0
\(923\) 2.56955e7i 0.992781i
\(924\) 0 0
\(925\) 8.99906e6i 0.345814i
\(926\) 0 0
\(927\) 1.90514e6 + 1.55995e6i 0.0728161 + 0.0596225i
\(928\) 0 0
\(929\) 3.85644e7i 1.46604i −0.680204 0.733022i \(-0.738109\pi\)
0.680204 0.733022i \(-0.261891\pi\)
\(930\) 0 0
\(931\) 1.82530e7 0.690175
\(932\) 0 0
\(933\) −3.43210e7 + 1.62561e7i −1.29079 + 0.611381i
\(934\) 0 0
\(935\) −363345. −0.0135922
\(936\) 0 0
\(937\) 4.10429e7 1.52717 0.763587 0.645705i \(-0.223436\pi\)
0.763587 + 0.645705i \(0.223436\pi\)
\(938\) 0 0
\(939\) 7.58478e6 3.59252e6i 0.280724 0.132964i
\(940\) 0 0
\(941\) 2.86355e7 1.05422 0.527110 0.849797i \(-0.323276\pi\)
0.527110 + 0.849797i \(0.323276\pi\)
\(942\) 0 0
\(943\) 1.26114e7i 0.461833i
\(944\) 0 0
\(945\) −4.74282e6 1.17957e6i −0.172765 0.0429679i
\(946\) 0 0
\(947\) 4.29903e7i 1.55774i −0.627183 0.778872i \(-0.715792\pi\)
0.627183 0.778872i \(-0.284208\pi\)
\(948\) 0 0
\(949\) 1.01374e7i 0.365395i
\(950\) 0 0
\(951\) −3.44093e7 + 1.62979e7i −1.23374 + 0.584360i
\(952\) 0 0
\(953\) 1.53032e7i 0.545822i −0.962039 0.272911i \(-0.912014\pi\)
0.962039 0.272911i \(-0.0879864\pi\)
\(954\) 0 0
\(955\) −5.50800e6 −0.195428
\(956\) 0 0
\(957\) 54090.0 + 114199.i 0.00190914 + 0.00403070i
\(958\) 0 0
\(959\) 1.57419e7 0.552726
\(960\) 0 0
\(961\) 1.23725e7 0.432166
\(962\) 0 0
\(963\) 3.16341e7 + 2.59024e7i 1.09923 + 0.900064i
\(964\) 0 0
\(965\) −9.26472e6 −0.320268
\(966\) 0 0
\(967\) 5.08678e7i 1.74935i 0.484711 + 0.874674i \(0.338925\pi\)
−0.484711 + 0.874674i \(0.661075\pi\)
\(968\) 0 0
\(969\) 678599. + 1.43271e6i 0.0232169 + 0.0490171i
\(970\) 0 0
\(971\) 3.01232e7i 1.02530i 0.858597 + 0.512652i \(0.171337\pi\)
−0.858597 + 0.512652i \(0.828663\pi\)
\(972\) 0 0
\(973\) 2.70186e6i 0.0914916i
\(974\) 0 0
\(975\) −8.83223e6 1.86472e7i −0.297549 0.628207i
\(976\) 0 0
\(977\) 5.06872e7i 1.69888i −0.527689 0.849438i \(-0.676941\pi\)
0.527689 0.849438i \(-0.323059\pi\)
\(978\) 0 0
\(979\) −1.62968e7 −0.543432
\(980\) 0 0
\(981\) 3.22124e7 + 2.63759e7i 1.06869 + 0.875052i
\(982\) 0 0
\(983\) −5.20972e6 −0.171961 −0.0859806 0.996297i \(-0.527402\pi\)
−0.0859806 + 0.996297i \(0.527402\pi\)
\(984\) 0 0
\(985\) 1.47077e7 0.483007
\(986\) 0 0
\(987\) 3.27617e6 + 6.91688e6i 0.107047 + 0.226005i
\(988\) 0 0
\(989\) 3.79237e7 1.23288
\(990\) 0 0
\(991\) 3.30327e7i 1.06846i −0.845338 0.534232i \(-0.820601\pi\)
0.845338 0.534232i \(-0.179399\pi\)
\(992\) 0 0
\(993\) 1.98161e7 9.38588e6i 0.637743 0.302066i
\(994\) 0 0
\(995\) 2.13493e7i 0.683638i
\(996\) 0 0
\(997\) 700904.i 0.0223316i 0.999938 + 0.0111658i \(0.00355426\pi\)
−0.999938 + 0.0111658i \(0.996446\pi\)
\(998\) 0 0
\(999\) −1.44467e7 3.59298e6i −0.457988 0.113904i
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 384.6.f.f.191.17 yes 20
3.2 odd 2 384.6.f.e.191.18 yes 20
4.3 odd 2 384.6.f.e.191.4 yes 20
8.3 odd 2 384.6.f.e.191.17 yes 20
8.5 even 2 inner 384.6.f.f.191.4 yes 20
12.11 even 2 inner 384.6.f.f.191.3 yes 20
24.5 odd 2 384.6.f.e.191.3 20
24.11 even 2 inner 384.6.f.f.191.18 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.f.e.191.3 20 24.5 odd 2
384.6.f.e.191.4 yes 20 4.3 odd 2
384.6.f.e.191.17 yes 20 8.3 odd 2
384.6.f.e.191.18 yes 20 3.2 odd 2
384.6.f.f.191.3 yes 20 12.11 even 2 inner
384.6.f.f.191.4 yes 20 8.5 even 2 inner
384.6.f.f.191.17 yes 20 1.1 even 1 trivial
384.6.f.f.191.18 yes 20 24.11 even 2 inner