Properties

Label 384.4.f.f.191.4
Level $384$
Weight $4$
Character 384.191
Analytic conductor $22.657$
Analytic rank $0$
Dimension $4$
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] = [384,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.4
Root \(2.54951 - 2.54951i\) of defining polynomial
Character \(\chi\) \(=\) 384.191
Dual form 384.4.f.f.191.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.09902 + 1.00000i) q^{3} +10.1980 q^{5} -10.1980i q^{7} +(25.0000 + 10.1980i) q^{9} +O(q^{10})\) \(q+(5.09902 + 1.00000i) q^{3} +10.1980 q^{5} -10.1980i q^{7} +(25.0000 + 10.1980i) q^{9} +46.0000i q^{11} +44.0000i q^{13} +(52.0000 + 10.1980i) q^{15} +20.3961i q^{17} -10.1980 q^{19} +(10.1980 - 52.0000i) q^{21} +88.0000 q^{23} -21.0000 q^{25} +(117.277 + 77.0000i) q^{27} +254.951 q^{29} -214.159i q^{31} +(-46.0000 + 234.555i) q^{33} -104.000i q^{35} -332.000i q^{37} +(-44.0000 + 224.357i) q^{39} +489.506i q^{41} -234.555 q^{43} +(254.951 + 104.000i) q^{45} -384.000 q^{47} +239.000 q^{49} +(-20.3961 + 104.000i) q^{51} +458.912 q^{53} +469.110i q^{55} +(-52.0000 - 10.1980i) q^{57} -630.000i q^{59} +236.000i q^{61} +(104.000 - 254.951i) q^{63} +448.714i q^{65} +50.9902 q^{67} +(448.714 + 88.0000i) q^{69} -680.000 q^{71} +422.000 q^{73} +(-107.079 - 21.0000i) q^{75} +469.110 q^{77} -744.457i q^{79} +(521.000 + 509.902i) q^{81} -186.000i q^{83} +208.000i q^{85} +(1300.00 + 254.951i) q^{87} +958.616i q^{89} +448.714 q^{91} +(214.159 - 1092.00i) q^{93} -104.000 q^{95} -1062.00 q^{97} +(-469.110 + 1150.00i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 100 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 100 q^{9} + 208 q^{15} + 352 q^{23} - 84 q^{25} - 184 q^{33} - 176 q^{39} - 1536 q^{47} + 956 q^{49} - 208 q^{57} + 416 q^{63} - 2720 q^{71} + 1688 q^{73} + 2084 q^{81} + 5200 q^{87} - 416 q^{95} - 4248 q^{97}+O(q^{100}) \) 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
\(3\) 5.09902 + 1.00000i 0.981307 + 0.192450i
\(4\) 0 0
\(5\) 10.1980 0.912140 0.456070 0.889944i \(-0.349257\pi\)
0.456070 + 0.889944i \(0.349257\pi\)
\(6\) 0 0
\(7\) 10.1980i 0.550642i −0.961352 0.275321i \(-0.911216\pi\)
0.961352 0.275321i \(-0.0887842\pi\)
\(8\) 0 0
\(9\) 25.0000 + 10.1980i 0.925926 + 0.377705i
\(10\) 0 0
\(11\) 46.0000i 1.26087i 0.776244 + 0.630433i \(0.217123\pi\)
−0.776244 + 0.630433i \(0.782877\pi\)
\(12\) 0 0
\(13\) 44.0000i 0.938723i 0.883006 + 0.469362i \(0.155516\pi\)
−0.883006 + 0.469362i \(0.844484\pi\)
\(14\) 0 0
\(15\) 52.0000 + 10.1980i 0.895089 + 0.175541i
\(16\) 0 0
\(17\) 20.3961i 0.290987i 0.989359 + 0.145493i \(0.0464770\pi\)
−0.989359 + 0.145493i \(0.953523\pi\)
\(18\) 0 0
\(19\) −10.1980 −0.123136 −0.0615682 0.998103i \(-0.519610\pi\)
−0.0615682 + 0.998103i \(0.519610\pi\)
\(20\) 0 0
\(21\) 10.1980 52.0000i 0.105971 0.540349i
\(22\) 0 0
\(23\) 88.0000 0.797794 0.398897 0.916996i \(-0.369393\pi\)
0.398897 + 0.916996i \(0.369393\pi\)
\(24\) 0 0
\(25\) −21.0000 −0.168000
\(26\) 0 0
\(27\) 117.277 + 77.0000i 0.835928 + 0.548839i
\(28\) 0 0
\(29\) 254.951 1.63252 0.816262 0.577682i \(-0.196042\pi\)
0.816262 + 0.577682i \(0.196042\pi\)
\(30\) 0 0
\(31\) 214.159i 1.24078i −0.784295 0.620388i \(-0.786975\pi\)
0.784295 0.620388i \(-0.213025\pi\)
\(32\) 0 0
\(33\) −46.0000 + 234.555i −0.242654 + 1.23730i
\(34\) 0 0
\(35\) 104.000i 0.502263i
\(36\) 0 0
\(37\) 332.000i 1.47515i −0.675266 0.737574i \(-0.735971\pi\)
0.675266 0.737574i \(-0.264029\pi\)
\(38\) 0 0
\(39\) −44.0000 + 224.357i −0.180657 + 0.921176i
\(40\) 0 0
\(41\) 489.506i 1.86458i 0.361706 + 0.932292i \(0.382195\pi\)
−0.361706 + 0.932292i \(0.617805\pi\)
\(42\) 0 0
\(43\) −234.555 −0.831844 −0.415922 0.909400i \(-0.636541\pi\)
−0.415922 + 0.909400i \(0.636541\pi\)
\(44\) 0 0
\(45\) 254.951 + 104.000i 0.844574 + 0.344520i
\(46\) 0 0
\(47\) −384.000 −1.19175 −0.595874 0.803078i \(-0.703194\pi\)
−0.595874 + 0.803078i \(0.703194\pi\)
\(48\) 0 0
\(49\) 239.000 0.696793
\(50\) 0 0
\(51\) −20.3961 + 104.000i −0.0560004 + 0.285547i
\(52\) 0 0
\(53\) 458.912 1.18937 0.594683 0.803960i \(-0.297278\pi\)
0.594683 + 0.803960i \(0.297278\pi\)
\(54\) 0 0
\(55\) 469.110i 1.15009i
\(56\) 0 0
\(57\) −52.0000 10.1980i −0.120835 0.0236976i
\(58\) 0 0
\(59\) 630.000i 1.39015i −0.718936 0.695076i \(-0.755371\pi\)
0.718936 0.695076i \(-0.244629\pi\)
\(60\) 0 0
\(61\) 236.000i 0.495356i 0.968842 + 0.247678i \(0.0796675\pi\)
−0.968842 + 0.247678i \(0.920333\pi\)
\(62\) 0 0
\(63\) 104.000 254.951i 0.207980 0.509854i
\(64\) 0 0
\(65\) 448.714i 0.856247i
\(66\) 0 0
\(67\) 50.9902 0.0929768 0.0464884 0.998919i \(-0.485197\pi\)
0.0464884 + 0.998919i \(0.485197\pi\)
\(68\) 0 0
\(69\) 448.714 + 88.0000i 0.782881 + 0.153536i
\(70\) 0 0
\(71\) −680.000 −1.13664 −0.568318 0.822809i \(-0.692406\pi\)
−0.568318 + 0.822809i \(0.692406\pi\)
\(72\) 0 0
\(73\) 422.000 0.676594 0.338297 0.941039i \(-0.390149\pi\)
0.338297 + 0.941039i \(0.390149\pi\)
\(74\) 0 0
\(75\) −107.079 21.0000i −0.164860 0.0323316i
\(76\) 0 0
\(77\) 469.110 0.694286
\(78\) 0 0
\(79\) 744.457i 1.06023i −0.847927 0.530114i \(-0.822149\pi\)
0.847927 0.530114i \(-0.177851\pi\)
\(80\) 0 0
\(81\) 521.000 + 509.902i 0.714678 + 0.699454i
\(82\) 0 0
\(83\) 186.000i 0.245978i −0.992408 0.122989i \(-0.960752\pi\)
0.992408 0.122989i \(-0.0392479\pi\)
\(84\) 0 0
\(85\) 208.000i 0.265421i
\(86\) 0 0
\(87\) 1300.00 + 254.951i 1.60201 + 0.314179i
\(88\) 0 0
\(89\) 958.616i 1.14172i 0.821048 + 0.570860i \(0.193390\pi\)
−0.821048 + 0.570860i \(0.806610\pi\)
\(90\) 0 0
\(91\) 448.714 0.516901
\(92\) 0 0
\(93\) 214.159 1092.00i 0.238787 1.21758i
\(94\) 0 0
\(95\) −104.000 −0.112318
\(96\) 0 0
\(97\) −1062.00 −1.11165 −0.555824 0.831300i \(-0.687597\pi\)
−0.555824 + 0.831300i \(0.687597\pi\)
\(98\) 0 0
\(99\) −469.110 + 1150.00i −0.476235 + 1.16747i
\(100\) 0 0
\(101\) 10.1980 0.0100470 0.00502348 0.999987i \(-0.498401\pi\)
0.00502348 + 0.999987i \(0.498401\pi\)
\(102\) 0 0
\(103\) 1988.62i 1.90237i 0.308618 + 0.951186i \(0.400133\pi\)
−0.308618 + 0.951186i \(0.599867\pi\)
\(104\) 0 0
\(105\) 104.000 530.298i 0.0966606 0.492874i
\(106\) 0 0
\(107\) 234.000i 0.211417i 0.994397 + 0.105709i \(0.0337111\pi\)
−0.994397 + 0.105709i \(0.966289\pi\)
\(108\) 0 0
\(109\) 724.000i 0.636208i −0.948056 0.318104i \(-0.896954\pi\)
0.948056 0.318104i \(-0.103046\pi\)
\(110\) 0 0
\(111\) 332.000 1692.87i 0.283892 1.44757i
\(112\) 0 0
\(113\) 1019.80i 0.848983i −0.905432 0.424492i \(-0.860453\pi\)
0.905432 0.424492i \(-0.139547\pi\)
\(114\) 0 0
\(115\) 897.427 0.727700
\(116\) 0 0
\(117\) −448.714 + 1100.00i −0.354561 + 0.869188i
\(118\) 0 0
\(119\) 208.000 0.160230
\(120\) 0 0
\(121\) −785.000 −0.589782
\(122\) 0 0
\(123\) −489.506 + 2496.00i −0.358839 + 1.82973i
\(124\) 0 0
\(125\) −1488.91 −1.06538
\(126\) 0 0
\(127\) 2253.77i 1.57472i −0.616493 0.787360i \(-0.711447\pi\)
0.616493 0.787360i \(-0.288553\pi\)
\(128\) 0 0
\(129\) −1196.00 234.555i −0.816294 0.160088i
\(130\) 0 0
\(131\) 1742.00i 1.16183i −0.813966 0.580913i \(-0.802696\pi\)
0.813966 0.580913i \(-0.197304\pi\)
\(132\) 0 0
\(133\) 104.000i 0.0678041i
\(134\) 0 0
\(135\) 1196.00 + 785.249i 0.762484 + 0.500618i
\(136\) 0 0
\(137\) 2406.74i 1.50089i −0.660935 0.750443i \(-0.729840\pi\)
0.660935 0.750443i \(-0.270160\pi\)
\(138\) 0 0
\(139\) 764.853 0.466719 0.233360 0.972390i \(-0.425028\pi\)
0.233360 + 0.972390i \(0.425028\pi\)
\(140\) 0 0
\(141\) −1958.02 384.000i −1.16947 0.229352i
\(142\) 0 0
\(143\) −2024.00 −1.18360
\(144\) 0 0
\(145\) 2600.00 1.48909
\(146\) 0 0
\(147\) 1218.67 + 239.000i 0.683768 + 0.134098i
\(148\) 0 0
\(149\) −479.308 −0.263533 −0.131767 0.991281i \(-0.542065\pi\)
−0.131767 + 0.991281i \(0.542065\pi\)
\(150\) 0 0
\(151\) 1499.11i 0.807920i 0.914777 + 0.403960i \(0.132367\pi\)
−0.914777 + 0.403960i \(0.867633\pi\)
\(152\) 0 0
\(153\) −208.000 + 509.902i −0.109907 + 0.269432i
\(154\) 0 0
\(155\) 2184.00i 1.13176i
\(156\) 0 0
\(157\) 1948.00i 0.990238i 0.868825 + 0.495119i \(0.164875\pi\)
−0.868825 + 0.495119i \(0.835125\pi\)
\(158\) 0 0
\(159\) 2340.00 + 458.912i 1.16713 + 0.228894i
\(160\) 0 0
\(161\) 897.427i 0.439299i
\(162\) 0 0
\(163\) −3395.95 −1.63185 −0.815924 0.578160i \(-0.803771\pi\)
−0.815924 + 0.578160i \(0.803771\pi\)
\(164\) 0 0
\(165\) −469.110 + 2392.00i −0.221334 + 1.12859i
\(166\) 0 0
\(167\) −2008.00 −0.930441 −0.465221 0.885195i \(-0.654025\pi\)
−0.465221 + 0.885195i \(0.654025\pi\)
\(168\) 0 0
\(169\) 261.000 0.118798
\(170\) 0 0
\(171\) −254.951 104.000i −0.114015 0.0465092i
\(172\) 0 0
\(173\) −1213.57 −0.533328 −0.266664 0.963790i \(-0.585921\pi\)
−0.266664 + 0.963790i \(0.585921\pi\)
\(174\) 0 0
\(175\) 214.159i 0.0925079i
\(176\) 0 0
\(177\) 630.000 3212.38i 0.267535 1.36417i
\(178\) 0 0
\(179\) 2382.00i 0.994632i −0.867570 0.497316i \(-0.834319\pi\)
0.867570 0.497316i \(-0.165681\pi\)
\(180\) 0 0
\(181\) 2652.00i 1.08907i −0.838738 0.544535i \(-0.816706\pi\)
0.838738 0.544535i \(-0.183294\pi\)
\(182\) 0 0
\(183\) −236.000 + 1203.37i −0.0953313 + 0.486096i
\(184\) 0 0
\(185\) 3385.75i 1.34554i
\(186\) 0 0
\(187\) −938.220 −0.366895
\(188\) 0 0
\(189\) 785.249 1196.00i 0.302214 0.460297i
\(190\) 0 0
\(191\) −4784.00 −1.81235 −0.906173 0.422907i \(-0.861010\pi\)
−0.906173 + 0.422907i \(0.861010\pi\)
\(192\) 0 0
\(193\) −3074.00 −1.14648 −0.573242 0.819386i \(-0.694314\pi\)
−0.573242 + 0.819386i \(0.694314\pi\)
\(194\) 0 0
\(195\) −448.714 + 2288.00i −0.164785 + 0.840241i
\(196\) 0 0
\(197\) −4844.07 −1.75191 −0.875953 0.482396i \(-0.839767\pi\)
−0.875953 + 0.482396i \(0.839767\pi\)
\(198\) 0 0
\(199\) 1927.43i 0.686592i −0.939227 0.343296i \(-0.888457\pi\)
0.939227 0.343296i \(-0.111543\pi\)
\(200\) 0 0
\(201\) 260.000 + 50.9902i 0.0912387 + 0.0178934i
\(202\) 0 0
\(203\) 2600.00i 0.898937i
\(204\) 0 0
\(205\) 4992.00i 1.70076i
\(206\) 0 0
\(207\) 2200.00 + 897.427i 0.738698 + 0.301331i
\(208\) 0 0
\(209\) 469.110i 0.155258i
\(210\) 0 0
\(211\) 3436.74 1.12130 0.560651 0.828052i \(-0.310551\pi\)
0.560651 + 0.828052i \(0.310551\pi\)
\(212\) 0 0
\(213\) −3467.33 680.000i −1.11539 0.218746i
\(214\) 0 0
\(215\) −2392.00 −0.758758
\(216\) 0 0
\(217\) −2184.00 −0.683224
\(218\) 0 0
\(219\) 2151.79 + 422.000i 0.663946 + 0.130211i
\(220\) 0 0
\(221\) −897.427 −0.273156
\(222\) 0 0
\(223\) 1233.96i 0.370548i −0.982687 0.185274i \(-0.940683\pi\)
0.982687 0.185274i \(-0.0593173\pi\)
\(224\) 0 0
\(225\) −525.000 214.159i −0.155556 0.0634545i
\(226\) 0 0
\(227\) 1270.00i 0.371334i 0.982613 + 0.185667i \(0.0594446\pi\)
−0.982613 + 0.185667i \(0.940555\pi\)
\(228\) 0 0
\(229\) 4556.00i 1.31471i −0.753580 0.657356i \(-0.771675\pi\)
0.753580 0.657356i \(-0.228325\pi\)
\(230\) 0 0
\(231\) 2392.00 + 469.110i 0.681308 + 0.133615i
\(232\) 0 0
\(233\) 877.031i 0.246593i 0.992370 + 0.123297i \(0.0393467\pi\)
−0.992370 + 0.123297i \(0.960653\pi\)
\(234\) 0 0
\(235\) −3916.05 −1.08704
\(236\) 0 0
\(237\) 744.457 3796.00i 0.204041 1.04041i
\(238\) 0 0
\(239\) 1920.00 0.519642 0.259821 0.965657i \(-0.416336\pi\)
0.259821 + 0.965657i \(0.416336\pi\)
\(240\) 0 0
\(241\) −1618.00 −0.432467 −0.216233 0.976342i \(-0.569377\pi\)
−0.216233 + 0.976342i \(0.569377\pi\)
\(242\) 0 0
\(243\) 2146.69 + 3121.00i 0.566708 + 0.823919i
\(244\) 0 0
\(245\) 2437.33 0.635573
\(246\) 0 0
\(247\) 448.714i 0.115591i
\(248\) 0 0
\(249\) 186.000 948.418i 0.0473384 0.241380i
\(250\) 0 0
\(251\) 1586.00i 0.398834i −0.979915 0.199417i \(-0.936095\pi\)
0.979915 0.199417i \(-0.0639049\pi\)
\(252\) 0 0
\(253\) 4048.00i 1.00591i
\(254\) 0 0
\(255\) −208.000 + 1060.60i −0.0510803 + 0.260459i
\(256\) 0 0
\(257\) 3956.84i 0.960392i 0.877161 + 0.480196i \(0.159435\pi\)
−0.877161 + 0.480196i \(0.840565\pi\)
\(258\) 0 0
\(259\) −3385.75 −0.812279
\(260\) 0 0
\(261\) 6373.77 + 2600.00i 1.51160 + 0.616613i
\(262\) 0 0
\(263\) 4344.00 1.01849 0.509244 0.860622i \(-0.329925\pi\)
0.509244 + 0.860622i \(0.329925\pi\)
\(264\) 0 0
\(265\) 4680.00 1.08487
\(266\) 0 0
\(267\) −958.616 + 4888.00i −0.219724 + 1.12038i
\(268\) 0 0
\(269\) 2743.27 0.621785 0.310893 0.950445i \(-0.399372\pi\)
0.310893 + 0.950445i \(0.399372\pi\)
\(270\) 0 0
\(271\) 3253.17i 0.729211i 0.931162 + 0.364606i \(0.118796\pi\)
−0.931162 + 0.364606i \(0.881204\pi\)
\(272\) 0 0
\(273\) 2288.00 + 448.714i 0.507238 + 0.0994776i
\(274\) 0 0
\(275\) 966.000i 0.211825i
\(276\) 0 0
\(277\) 4956.00i 1.07501i −0.843261 0.537504i \(-0.819367\pi\)
0.843261 0.537504i \(-0.180633\pi\)
\(278\) 0 0
\(279\) 2184.00 5353.97i 0.468648 1.14887i
\(280\) 0 0
\(281\) 1978.42i 0.420009i 0.977700 + 0.210005i \(0.0673479\pi\)
−0.977700 + 0.210005i \(0.932652\pi\)
\(282\) 0 0
\(283\) 7985.06 1.67725 0.838627 0.544706i \(-0.183359\pi\)
0.838627 + 0.544706i \(0.183359\pi\)
\(284\) 0 0
\(285\) −530.298 104.000i −0.110218 0.0216155i
\(286\) 0 0
\(287\) 4992.00 1.02672
\(288\) 0 0
\(289\) 4497.00 0.915327
\(290\) 0 0
\(291\) −5415.16 1062.00i −1.09087 0.213937i
\(292\) 0 0
\(293\) 1845.85 0.368039 0.184019 0.982923i \(-0.441089\pi\)
0.184019 + 0.982923i \(0.441089\pi\)
\(294\) 0 0
\(295\) 6424.76i 1.26801i
\(296\) 0 0
\(297\) −3542.00 + 5394.76i −0.692012 + 1.05399i
\(298\) 0 0
\(299\) 3872.00i 0.748908i
\(300\) 0 0
\(301\) 2392.00i 0.458048i
\(302\) 0 0
\(303\) 52.0000 + 10.1980i 0.00985915 + 0.00193354i
\(304\) 0 0
\(305\) 2406.74i 0.451834i
\(306\) 0 0
\(307\) 8291.01 1.54134 0.770672 0.637232i \(-0.219921\pi\)
0.770672 + 0.637232i \(0.219921\pi\)
\(308\) 0 0
\(309\) −1988.62 + 10140.0i −0.366112 + 1.86681i
\(310\) 0 0
\(311\) −10104.0 −1.84227 −0.921134 0.389246i \(-0.872736\pi\)
−0.921134 + 0.389246i \(0.872736\pi\)
\(312\) 0 0
\(313\) 1198.00 0.216342 0.108171 0.994132i \(-0.465501\pi\)
0.108171 + 0.994132i \(0.465501\pi\)
\(314\) 0 0
\(315\) 1060.60 2600.00i 0.189707 0.465058i
\(316\) 0 0
\(317\) 5027.63 0.890789 0.445394 0.895335i \(-0.353063\pi\)
0.445394 + 0.895335i \(0.353063\pi\)
\(318\) 0 0
\(319\) 11727.7i 2.05839i
\(320\) 0 0
\(321\) −234.000 + 1193.17i −0.0406872 + 0.207465i
\(322\) 0 0
\(323\) 208.000i 0.0358311i
\(324\) 0 0
\(325\) 924.000i 0.157706i
\(326\) 0 0
\(327\) 724.000 3691.69i 0.122438 0.624315i
\(328\) 0 0
\(329\) 3916.05i 0.656227i
\(330\) 0 0
\(331\) −6129.02 −1.01777 −0.508884 0.860835i \(-0.669942\pi\)
−0.508884 + 0.860835i \(0.669942\pi\)
\(332\) 0 0
\(333\) 3385.75 8300.00i 0.557171 1.36588i
\(334\) 0 0
\(335\) 520.000 0.0848079
\(336\) 0 0
\(337\) 390.000 0.0630405 0.0315203 0.999503i \(-0.489965\pi\)
0.0315203 + 0.999503i \(0.489965\pi\)
\(338\) 0 0
\(339\) 1019.80 5200.00i 0.163387 0.833113i
\(340\) 0 0
\(341\) 9851.31 1.56445
\(342\) 0 0
\(343\) 5935.26i 0.934326i
\(344\) 0 0
\(345\) 4576.00 + 897.427i 0.714097 + 0.140046i
\(346\) 0 0
\(347\) 4366.00i 0.675444i 0.941246 + 0.337722i \(0.109656\pi\)
−0.941246 + 0.337722i \(0.890344\pi\)
\(348\) 0 0
\(349\) 5492.00i 0.842350i −0.906979 0.421175i \(-0.861618\pi\)
0.906979 0.421175i \(-0.138382\pi\)
\(350\) 0 0
\(351\) −3388.00 + 5160.21i −0.515208 + 0.784705i
\(352\) 0 0
\(353\) 81.5843i 0.0123011i −0.999981 0.00615056i \(-0.998042\pi\)
0.999981 0.00615056i \(-0.00195780\pi\)
\(354\) 0 0
\(355\) −6934.67 −1.03677
\(356\) 0 0
\(357\) 1060.60 + 208.000i 0.157234 + 0.0308362i
\(358\) 0 0
\(359\) −2216.00 −0.325783 −0.162891 0.986644i \(-0.552082\pi\)
−0.162891 + 0.986644i \(0.552082\pi\)
\(360\) 0 0
\(361\) −6755.00 −0.984837
\(362\) 0 0
\(363\) −4002.73 785.000i −0.578757 0.113504i
\(364\) 0 0
\(365\) 4303.57 0.617149
\(366\) 0 0
\(367\) 7128.43i 1.01390i 0.861976 + 0.506950i \(0.169227\pi\)
−0.861976 + 0.506950i \(0.830773\pi\)
\(368\) 0 0
\(369\) −4992.00 + 12237.6i −0.704263 + 1.72647i
\(370\) 0 0
\(371\) 4680.00i 0.654915i
\(372\) 0 0
\(373\) 2028.00i 0.281517i −0.990044 0.140759i \(-0.955046\pi\)
0.990044 0.140759i \(-0.0449541\pi\)
\(374\) 0 0
\(375\) −7592.00 1488.91i −1.04546 0.205032i
\(376\) 0 0
\(377\) 11217.8i 1.53249i
\(378\) 0 0
\(379\) 1315.55 0.178298 0.0891492 0.996018i \(-0.471585\pi\)
0.0891492 + 0.996018i \(0.471585\pi\)
\(380\) 0 0
\(381\) 2253.77 11492.0i 0.303055 1.54528i
\(382\) 0 0
\(383\) −2128.00 −0.283905 −0.141953 0.989873i \(-0.545338\pi\)
−0.141953 + 0.989873i \(0.545338\pi\)
\(384\) 0 0
\(385\) 4784.00 0.633286
\(386\) 0 0
\(387\) −5863.87 2392.00i −0.770226 0.314192i
\(388\) 0 0
\(389\) 1927.43 0.251220 0.125610 0.992080i \(-0.459911\pi\)
0.125610 + 0.992080i \(0.459911\pi\)
\(390\) 0 0
\(391\) 1794.85i 0.232148i
\(392\) 0 0
\(393\) 1742.00 8882.49i 0.223594 1.14011i
\(394\) 0 0
\(395\) 7592.00i 0.967076i
\(396\) 0 0
\(397\) 6220.00i 0.786330i 0.919468 + 0.393165i \(0.128620\pi\)
−0.919468 + 0.393165i \(0.871380\pi\)
\(398\) 0 0
\(399\) −104.000 + 530.298i −0.0130489 + 0.0665366i
\(400\) 0 0
\(401\) 1937.63i 0.241298i 0.992695 + 0.120649i \(0.0384976\pi\)
−0.992695 + 0.120649i \(0.961502\pi\)
\(402\) 0 0
\(403\) 9422.99 1.16475
\(404\) 0 0
\(405\) 5313.18 + 5200.00i 0.651886 + 0.638000i
\(406\) 0 0
\(407\) 15272.0 1.85996
\(408\) 0 0
\(409\) −1270.00 −0.153539 −0.0767695 0.997049i \(-0.524461\pi\)
−0.0767695 + 0.997049i \(0.524461\pi\)
\(410\) 0 0
\(411\) 2406.74 12272.0i 0.288846 1.47283i
\(412\) 0 0
\(413\) −6424.76 −0.765477
\(414\) 0 0
\(415\) 1896.84i 0.224366i
\(416\) 0 0
\(417\) 3900.00 + 764.853i 0.457995 + 0.0898202i
\(418\) 0 0
\(419\) 9126.00i 1.06404i 0.846731 + 0.532022i \(0.178568\pi\)
−0.846731 + 0.532022i \(0.821432\pi\)
\(420\) 0 0
\(421\) 3436.00i 0.397768i −0.980023 0.198884i \(-0.936268\pi\)
0.980023 0.198884i \(-0.0637317\pi\)
\(422\) 0 0
\(423\) −9600.00 3916.05i −1.10347 0.450129i
\(424\) 0 0
\(425\) 428.318i 0.0488858i
\(426\) 0 0
\(427\) 2406.74 0.272764
\(428\) 0 0
\(429\) −10320.4 2024.00i −1.16148 0.227785i
\(430\) 0 0
\(431\) 1920.00 0.214578 0.107289 0.994228i \(-0.465783\pi\)
0.107289 + 0.994228i \(0.465783\pi\)
\(432\) 0 0
\(433\) −15766.0 −1.74981 −0.874903 0.484299i \(-0.839075\pi\)
−0.874903 + 0.484299i \(0.839075\pi\)
\(434\) 0 0
\(435\) 13257.5 + 2600.00i 1.46126 + 0.286576i
\(436\) 0 0
\(437\) −897.427 −0.0982375
\(438\) 0 0
\(439\) 7311.99i 0.794949i −0.917613 0.397474i \(-0.869887\pi\)
0.917613 0.397474i \(-0.130113\pi\)
\(440\) 0 0
\(441\) 5975.00 + 2437.33i 0.645179 + 0.263182i
\(442\) 0 0
\(443\) 8190.00i 0.878372i 0.898396 + 0.439186i \(0.144733\pi\)
−0.898396 + 0.439186i \(0.855267\pi\)
\(444\) 0 0
\(445\) 9776.00i 1.04141i
\(446\) 0 0
\(447\) −2444.00 479.308i −0.258607 0.0507170i
\(448\) 0 0
\(449\) 13767.4i 1.44704i −0.690303 0.723521i \(-0.742523\pi\)
0.690303 0.723521i \(-0.257477\pi\)
\(450\) 0 0
\(451\) −22517.3 −2.35099
\(452\) 0 0
\(453\) −1499.11 + 7644.00i −0.155484 + 0.792818i
\(454\) 0 0
\(455\) 4576.00 0.471486
\(456\) 0 0
\(457\) 10682.0 1.09340 0.546699 0.837329i \(-0.315884\pi\)
0.546699 + 0.837329i \(0.315884\pi\)
\(458\) 0 0
\(459\) −1570.50 + 2392.00i −0.159705 + 0.243244i
\(460\) 0 0
\(461\) −3090.01 −0.312182 −0.156091 0.987743i \(-0.549889\pi\)
−0.156091 + 0.987743i \(0.549889\pi\)
\(462\) 0 0
\(463\) 2661.69i 0.267169i −0.991037 0.133584i \(-0.957351\pi\)
0.991037 0.133584i \(-0.0426487\pi\)
\(464\) 0 0
\(465\) 2184.00 11136.3i 0.217808 1.11061i
\(466\) 0 0
\(467\) 12246.0i 1.21344i 0.794915 + 0.606721i \(0.207515\pi\)
−0.794915 + 0.606721i \(0.792485\pi\)
\(468\) 0 0
\(469\) 520.000i 0.0511969i
\(470\) 0 0
\(471\) −1948.00 + 9932.89i −0.190571 + 0.971727i
\(472\) 0 0
\(473\) 10789.5i 1.04884i
\(474\) 0 0
\(475\) 214.159 0.0206869
\(476\) 0 0
\(477\) 11472.8 + 4680.00i 1.10126 + 0.449230i
\(478\) 0 0
\(479\) 15008.0 1.43159 0.715796 0.698309i \(-0.246064\pi\)
0.715796 + 0.698309i \(0.246064\pi\)
\(480\) 0 0
\(481\) 14608.0 1.38476
\(482\) 0 0
\(483\) 897.427 4576.00i 0.0845432 0.431087i
\(484\) 0 0
\(485\) −10830.3 −1.01398
\(486\) 0 0
\(487\) 5721.10i 0.532336i −0.963927 0.266168i \(-0.914242\pi\)
0.963927 0.266168i \(-0.0857576\pi\)
\(488\) 0 0
\(489\) −17316.0 3395.95i −1.60134 0.314049i
\(490\) 0 0
\(491\) 5018.00i 0.461220i 0.973046 + 0.230610i \(0.0740722\pi\)
−0.973046 + 0.230610i \(0.925928\pi\)
\(492\) 0 0
\(493\) 5200.00i 0.475043i
\(494\) 0 0
\(495\) −4784.00 + 11727.7i −0.434394 + 1.06489i
\(496\) 0 0
\(497\) 6934.67i 0.625880i
\(498\) 0 0
\(499\) 16082.3 1.44277 0.721386 0.692533i \(-0.243506\pi\)
0.721386 + 0.692533i \(0.243506\pi\)
\(500\) 0 0
\(501\) −10238.8 2008.00i −0.913048 0.179064i
\(502\) 0 0
\(503\) 10632.0 0.942460 0.471230 0.882010i \(-0.343810\pi\)
0.471230 + 0.882010i \(0.343810\pi\)
\(504\) 0 0
\(505\) 104.000 0.00916424
\(506\) 0 0
\(507\) 1330.84 + 261.000i 0.116578 + 0.0228628i
\(508\) 0 0
\(509\) −3579.51 −0.311707 −0.155854 0.987780i \(-0.549813\pi\)
−0.155854 + 0.987780i \(0.549813\pi\)
\(510\) 0 0
\(511\) 4303.57i 0.372561i
\(512\) 0 0
\(513\) −1196.00 785.249i −0.102933 0.0675820i
\(514\) 0 0
\(515\) 20280.0i 1.73523i
\(516\) 0 0
\(517\) 17664.0i 1.50263i
\(518\) 0 0
\(519\) −6188.00 1213.57i −0.523358 0.102639i
\(520\) 0 0
\(521\) 1590.89i 0.133778i 0.997760 + 0.0668890i \(0.0213073\pi\)
−0.997760 + 0.0668890i \(0.978693\pi\)
\(522\) 0 0
\(523\) −4109.81 −0.343613 −0.171806 0.985131i \(-0.554960\pi\)
−0.171806 + 0.985131i \(0.554960\pi\)
\(524\) 0 0
\(525\) −214.159 + 1092.00i −0.0178032 + 0.0907786i
\(526\) 0 0
\(527\) 4368.00 0.361049
\(528\) 0 0
\(529\) −4423.00 −0.363524
\(530\) 0 0
\(531\) 6424.76 15750.0i 0.525068 1.28718i
\(532\) 0 0
\(533\) −21538.3 −1.75033
\(534\) 0 0
\(535\) 2386.34i 0.192842i
\(536\) 0 0
\(537\) 2382.00 12145.9i 0.191417 0.976039i
\(538\) 0 0
\(539\) 10994.0i 0.878562i
\(540\) 0 0
\(541\) 9484.00i 0.753695i 0.926275 + 0.376848i \(0.122992\pi\)
−0.926275 + 0.376848i \(0.877008\pi\)
\(542\) 0 0
\(543\) 2652.00 13522.6i 0.209592 1.06871i
\(544\) 0 0
\(545\) 7383.38i 0.580311i
\(546\) 0 0
\(547\) −5476.35 −0.428065 −0.214033 0.976827i \(-0.568660\pi\)
−0.214033 + 0.976827i \(0.568660\pi\)
\(548\) 0 0
\(549\) −2406.74 + 5900.00i −0.187098 + 0.458663i
\(550\) 0 0
\(551\) −2600.00 −0.201023
\(552\) 0 0
\(553\) −7592.00 −0.583806
\(554\) 0 0
\(555\) 3385.75 17264.0i 0.258950 1.32039i
\(556\) 0 0
\(557\) −1091.19 −0.0830076 −0.0415038 0.999138i \(-0.513215\pi\)
−0.0415038 + 0.999138i \(0.513215\pi\)
\(558\) 0 0
\(559\) 10320.4i 0.780871i
\(560\) 0 0
\(561\) −4784.00 938.220i −0.360037 0.0706090i
\(562\) 0 0
\(563\) 17898.0i 1.33981i −0.742449 0.669903i \(-0.766336\pi\)
0.742449 0.669903i \(-0.233664\pi\)
\(564\) 0 0
\(565\) 10400.0i 0.774392i
\(566\) 0 0
\(567\) 5200.00 5313.18i 0.385149 0.393532i
\(568\) 0 0
\(569\) 21089.5i 1.55381i −0.629616 0.776907i \(-0.716788\pi\)
0.629616 0.776907i \(-0.283212\pi\)
\(570\) 0 0
\(571\) 3334.76 0.244405 0.122203 0.992505i \(-0.461004\pi\)
0.122203 + 0.992505i \(0.461004\pi\)
\(572\) 0 0
\(573\) −24393.7 4784.00i −1.77847 0.348786i
\(574\) 0 0
\(575\) −1848.00 −0.134029
\(576\) 0 0
\(577\) −5066.00 −0.365512 −0.182756 0.983158i \(-0.558502\pi\)
−0.182756 + 0.983158i \(0.558502\pi\)
\(578\) 0 0
\(579\) −15674.4 3074.00i −1.12505 0.220641i
\(580\) 0 0
\(581\) −1896.84 −0.135446
\(582\) 0 0
\(583\) 21109.9i 1.49963i
\(584\) 0 0
\(585\) −4576.00 + 11217.8i −0.323409 + 0.792822i
\(586\) 0 0
\(587\) 20042.0i 1.40924i 0.709587 + 0.704618i \(0.248882\pi\)
−0.709587 + 0.704618i \(0.751118\pi\)
\(588\) 0 0
\(589\) 2184.00i 0.152785i
\(590\) 0 0
\(591\) −24700.0 4844.07i −1.71916 0.337155i
\(592\) 0 0
\(593\) 26351.7i 1.82485i −0.409244 0.912425i \(-0.634208\pi\)
0.409244 0.912425i \(-0.365792\pi\)
\(594\) 0 0
\(595\) 2121.19 0.146152
\(596\) 0 0
\(597\) 1927.43 9828.00i 0.132135 0.673758i
\(598\) 0 0
\(599\) 4344.00 0.296312 0.148156 0.988964i \(-0.452666\pi\)
0.148156 + 0.988964i \(0.452666\pi\)
\(600\) 0 0
\(601\) −18250.0 −1.23866 −0.619329 0.785132i \(-0.712595\pi\)
−0.619329 + 0.785132i \(0.712595\pi\)
\(602\) 0 0
\(603\) 1274.75 + 520.000i 0.0860896 + 0.0351178i
\(604\) 0 0
\(605\) −8005.46 −0.537964
\(606\) 0 0
\(607\) 5272.39i 0.352553i −0.984341 0.176276i \(-0.943595\pi\)
0.984341 0.176276i \(-0.0564053\pi\)
\(608\) 0 0
\(609\) 2600.00 13257.5i 0.173001 0.882133i
\(610\) 0 0
\(611\) 16896.0i 1.11872i
\(612\) 0 0
\(613\) 11748.0i 0.774058i 0.922068 + 0.387029i \(0.126499\pi\)
−0.922068 + 0.387029i \(0.873501\pi\)
\(614\) 0 0
\(615\) −4992.00 + 25454.3i −0.327312 + 1.66897i
\(616\) 0 0
\(617\) 3895.65i 0.254186i −0.991891 0.127093i \(-0.959435\pi\)
0.991891 0.127093i \(-0.0405647\pi\)
\(618\) 0 0
\(619\) −7189.62 −0.466842 −0.233421 0.972376i \(-0.574992\pi\)
−0.233421 + 0.972376i \(0.574992\pi\)
\(620\) 0 0
\(621\) 10320.4 + 6776.00i 0.666899 + 0.437861i
\(622\) 0 0
\(623\) 9776.00 0.628679
\(624\) 0 0
\(625\) −12559.0 −0.803776
\(626\) 0 0
\(627\) 469.110 2392.00i 0.0298795 0.152356i
\(628\) 0 0
\(629\) 6771.50 0.429248
\(630\) 0 0
\(631\) 26831.0i 1.69275i 0.532585 + 0.846376i \(0.321221\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(632\) 0 0
\(633\) 17524.0 + 3436.74i 1.10034 + 0.215795i
\(634\) 0 0
\(635\) 22984.0i 1.43637i
\(636\) 0 0
\(637\) 10516.0i 0.654096i
\(638\) 0 0
\(639\) −17000.0 6934.67i −1.05244 0.429313i
\(640\) 0 0
\(641\) 23639.1i 1.45661i 0.685254 + 0.728305i \(0.259691\pi\)
−0.685254 + 0.728305i \(0.740309\pi\)
\(642\) 0 0
\(643\) −16245.5 −0.996359 −0.498180 0.867074i \(-0.665998\pi\)
−0.498180 + 0.867074i \(0.665998\pi\)
\(644\) 0 0
\(645\) −12196.9 2392.00i −0.744575 0.146023i
\(646\) 0 0
\(647\) −7384.00 −0.448679 −0.224339 0.974511i \(-0.572022\pi\)
−0.224339 + 0.974511i \(0.572022\pi\)
\(648\) 0 0
\(649\) 28980.0 1.75280
\(650\) 0 0
\(651\) −11136.3 2184.00i −0.670452 0.131486i
\(652\) 0 0
\(653\) −15082.9 −0.903889 −0.451944 0.892046i \(-0.649269\pi\)
−0.451944 + 0.892046i \(0.649269\pi\)
\(654\) 0 0
\(655\) 17765.0i 1.05975i
\(656\) 0 0
\(657\) 10550.0 + 4303.57i 0.626476 + 0.255553i
\(658\) 0 0
\(659\) 20434.0i 1.20788i 0.797028 + 0.603942i \(0.206404\pi\)
−0.797028 + 0.603942i \(0.793596\pi\)
\(660\) 0 0
\(661\) 12148.0i 0.714830i 0.933946 + 0.357415i \(0.116342\pi\)
−0.933946 + 0.357415i \(0.883658\pi\)
\(662\) 0 0
\(663\) −4576.00 897.427i −0.268050 0.0525689i
\(664\) 0 0
\(665\) 1060.60i 0.0618468i
\(666\) 0 0
\(667\) 22435.7 1.30242
\(668\) 0 0
\(669\) 1233.96 6292.00i 0.0713120 0.363621i
\(670\) 0 0
\(671\) −10856.0 −0.624577
\(672\) 0 0
\(673\) 12730.0 0.729131 0.364566 0.931178i \(-0.381217\pi\)
0.364566 + 0.931178i \(0.381217\pi\)
\(674\) 0 0
\(675\) −2462.83 1617.00i −0.140436 0.0922050i
\(676\) 0 0
\(677\) 19672.0 1.11678 0.558388 0.829580i \(-0.311420\pi\)
0.558388 + 0.829580i \(0.311420\pi\)
\(678\) 0 0
\(679\) 10830.3i 0.612120i
\(680\) 0 0
\(681\) −1270.00 + 6475.75i −0.0714633 + 0.364393i
\(682\) 0 0
\(683\) 6558.00i 0.367401i 0.982982 + 0.183701i \(0.0588077\pi\)
−0.982982 + 0.183701i \(0.941192\pi\)
\(684\) 0 0
\(685\) 24544.0i 1.36902i
\(686\) 0 0
\(687\) 4556.00 23231.1i 0.253016 1.29014i
\(688\) 0 0
\(689\) 20192.1i 1.11649i
\(690\) 0 0
\(691\) 30461.5 1.67701 0.838503 0.544896i \(-0.183431\pi\)
0.838503 + 0.544896i \(0.183431\pi\)
\(692\) 0 0
\(693\) 11727.7 + 4784.00i 0.642857 + 0.262235i
\(694\) 0 0
\(695\) 7800.00 0.425713
\(696\) 0 0
\(697\) −9984.00 −0.542570
\(698\) 0 0
\(699\) −877.031 + 4472.00i −0.0474569 + 0.241984i
\(700\) 0 0
\(701\) 31461.0 1.69510 0.847549 0.530717i \(-0.178077\pi\)
0.847549 + 0.530717i \(0.178077\pi\)
\(702\) 0 0
\(703\) 3385.75i 0.181644i
\(704\) 0 0
\(705\) −19968.0 3916.05i −1.06672 0.209201i
\(706\) 0 0
\(707\) 104.000i 0.00553228i
\(708\) 0 0
\(709\) 25396.0i 1.34523i 0.739993 + 0.672614i \(0.234829\pi\)
−0.739993 + 0.672614i \(0.765171\pi\)
\(710\) 0 0
\(711\) 7592.00 18611.4i 0.400453 0.981692i
\(712\) 0 0
\(713\) 18846.0i 0.989884i
\(714\) 0 0
\(715\) −20640.8 −1.07961
\(716\) 0 0
\(717\) 9790.12 + 1920.00i 0.509928 + 0.100005i
\(718\) 0 0
\(719\) −7888.00 −0.409142 −0.204571 0.978852i \(-0.565580\pi\)
−0.204571 + 0.978852i \(0.565580\pi\)
\(720\) 0 0
\(721\) 20280.0 1.04753
\(722\) 0 0
\(723\) −8250.21 1618.00i −0.424383 0.0832283i
\(724\) 0 0
\(725\) −5353.97 −0.274264
\(726\) 0 0
\(727\) 6475.75i 0.330361i 0.986263 + 0.165181i \(0.0528207\pi\)
−0.986263 + 0.165181i \(0.947179\pi\)
\(728\) 0 0
\(729\) 7825.00 + 18060.7i 0.397551 + 0.917580i
\(730\) 0 0
\(731\) 4784.00i 0.242056i
\(732\) 0 0
\(733\) 14772.0i 0.744361i −0.928160 0.372180i \(-0.878610\pi\)
0.928160 0.372180i \(-0.121390\pi\)
\(734\) 0 0
\(735\) 12428.0 + 2437.33i 0.623692 + 0.122316i
\(736\) 0 0
\(737\) 2345.55i 0.117231i
\(738\) 0 0
\(739\) 22364.3 1.11324 0.556620 0.830767i \(-0.312098\pi\)
0.556620 + 0.830767i \(0.312098\pi\)
\(740\) 0 0
\(741\) 448.714 2288.00i 0.0222455 0.113430i
\(742\) 0 0
\(743\) 19080.0 0.942096 0.471048 0.882108i \(-0.343876\pi\)
0.471048 + 0.882108i \(0.343876\pi\)
\(744\) 0 0
\(745\) −4888.00 −0.240379
\(746\) 0 0
\(747\) 1896.84 4650.00i 0.0929071 0.227757i
\(748\) 0 0
\(749\) 2386.34 0.116415
\(750\) 0 0
\(751\) 20324.7i 0.987561i −0.869586 0.493781i \(-0.835615\pi\)
0.869586 0.493781i \(-0.164385\pi\)
\(752\) 0 0
\(753\) 1586.00 8087.04i 0.0767557 0.391379i
\(754\) 0 0
\(755\) 15288.0i 0.736937i
\(756\) 0 0
\(757\) 4188.00i 0.201077i −0.994933 0.100539i \(-0.967943\pi\)
0.994933 0.100539i \(-0.0320566\pi\)
\(758\) 0 0
\(759\) −4048.00 + 20640.8i −0.193588 + 0.987108i
\(760\) 0 0
\(761\) 30757.3i 1.46511i 0.680706 + 0.732556i \(0.261673\pi\)
−0.680706 + 0.732556i \(0.738327\pi\)
\(762\) 0 0
\(763\) −7383.38 −0.350323
\(764\) 0 0
\(765\) −2121.19 + 5200.00i −0.100251 + 0.245760i
\(766\) 0 0
\(767\) 27720.0 1.30497
\(768\) 0 0
\(769\) −10370.0 −0.486283 −0.243142 0.969991i \(-0.578178\pi\)
−0.243142 + 0.969991i \(0.578178\pi\)
\(770\) 0 0
\(771\) −3956.84 + 20176.0i −0.184828 + 0.942440i
\(772\) 0 0
\(773\) 10779.3 0.501559 0.250780 0.968044i \(-0.419313\pi\)
0.250780 + 0.968044i \(0.419313\pi\)
\(774\) 0 0
\(775\) 4497.34i 0.208450i
\(776\) 0 0
\(777\) −17264.0 3385.75i −0.797095 0.156323i
\(778\) 0 0
\(779\) 4992.00i 0.229598i
\(780\) 0 0
\(781\) 31280.0i 1.43315i
\(782\) 0 0
\(783\) 29900.0 + 19631.2i 1.36467 + 0.895993i
\(784\) 0 0
\(785\) 19865.8i 0.903236i
\(786\) 0 0
\(787\) −23384.1 −1.05915 −0.529576 0.848262i \(-0.677649\pi\)
−0.529576 + 0.848262i \(0.677649\pi\)
\(788\) 0 0
\(789\) 22150.1 + 4344.00i 0.999450 + 0.196008i
\(790\) 0 0
\(791\) −10400.0 −0.467486
\(792\) 0 0
\(793\) −10384.0 −0.465002
\(794\) 0 0
\(795\) 23863.4 + 4680.00i 1.06459 + 0.208783i
\(796\) 0 0
\(797\) 3232.78 0.143677 0.0718387 0.997416i \(-0.477113\pi\)
0.0718387 + 0.997416i \(0.477113\pi\)
\(798\) 0 0
\(799\) 7832.09i 0.346783i
\(800\) 0 0
\(801\) −9776.00 + 23965.4i −0.431233 + 1.05715i
\(802\) 0 0
\(803\) 19412.0i 0.853094i
\(804\) 0 0
\(805\) 9152.00i 0.400703i
\(806\) 0 0
\(807\) 13988.0 + 2743.27i 0.610162 + 0.119663i
\(808\) 0 0
\(809\) 33857.5i 1.47140i 0.677305 + 0.735702i \(0.263148\pi\)
−0.677305 + 0.735702i \(0.736852\pi\)
\(810\) 0 0
\(811\) 10493.8 0.454361 0.227180 0.973853i \(-0.427049\pi\)
0.227180 + 0.973853i \(0.427049\pi\)
\(812\) 0 0
\(813\) −3253.17 + 16588.0i −0.140337 + 0.715580i
\(814\) 0 0
\(815\) −34632.0 −1.48847
\(816\) 0 0
\(817\) 2392.00 0.102430
\(818\) 0 0
\(819\) 11217.8 + 4576.00i 0.478612 + 0.195236i
\(820\) 0 0
\(821\) 15103.3 0.642032 0.321016 0.947074i \(-0.395976\pi\)
0.321016 + 0.947074i \(0.395976\pi\)
\(822\) 0 0
\(823\) 17958.7i 0.760635i −0.924856 0.380317i \(-0.875815\pi\)
0.924856 0.380317i \(-0.124185\pi\)
\(824\) 0 0
\(825\) 966.000 4925.65i 0.0407658 0.207866i
\(826\) 0 0
\(827\) 6598.00i 0.277430i −0.990332 0.138715i \(-0.955703\pi\)
0.990332 0.138715i \(-0.0442973\pi\)
\(828\) 0 0
\(829\) 41028.0i 1.71889i −0.511227 0.859446i \(-0.670809\pi\)
0.511227 0.859446i \(-0.329191\pi\)
\(830\) 0 0
\(831\) 4956.00 25270.7i 0.206885 1.05491i
\(832\) 0 0
\(833\) 4874.66i 0.202758i
\(834\) 0 0
\(835\) −20477.7 −0.848693
\(836\) 0 0
\(837\) 16490.2 25116.0i 0.680987 1.03720i
\(838\) 0 0
\(839\) 17816.0 0.733107 0.366553 0.930397i \(-0.380538\pi\)
0.366553 + 0.930397i \(0.380538\pi\)
\(840\) 0 0
\(841\) 40611.0 1.66514
\(842\) 0 0
\(843\) −1978.42 + 10088.0i −0.0808308 + 0.412158i
\(844\) 0 0
\(845\) 2661.69 0.108361
\(846\) 0 0
\(847\) 8005.46i 0.324759i
\(848\) 0 0
\(849\) 40716.0 + 7985.06i 1.64590 + 0.322788i
\(850\) 0 0
\(851\) 29216.0i 1.17686i
\(852\) 0 0
\(853\) 28308.0i 1.13628i 0.822932 + 0.568140i \(0.192337\pi\)
−0.822932 + 0.568140i \(0.807663\pi\)
\(854\) 0 0
\(855\) −2600.00 1060.60i −0.103998 0.0424229i
\(856\) 0 0
\(857\) 33735.1i 1.34466i 0.740254 + 0.672328i \(0.234705\pi\)
−0.740254 + 0.672328i \(0.765295\pi\)
\(858\) 0 0
\(859\) 336.535 0.0133672 0.00668361 0.999978i \(-0.497873\pi\)
0.00668361 + 0.999978i \(0.497873\pi\)
\(860\) 0 0
\(861\) 25454.3 + 4992.00i 1.00753 + 0.197592i
\(862\) 0 0
\(863\) −48208.0 −1.90153 −0.950764 0.309915i \(-0.899700\pi\)
−0.950764 + 0.309915i \(0.899700\pi\)
\(864\) 0 0
\(865\) −12376.0 −0.486470
\(866\) 0 0
\(867\) 22930.3 + 4497.00i 0.898216 + 0.176155i
\(868\) 0 0
\(869\) 34245.0 1.33680
\(870\) 0 0
\(871\) 2243.57i 0.0872795i
\(872\) 0 0
\(873\) −26550.0 10830.3i −1.02930 0.419875i
\(874\) 0 0
\(875\) 15184.0i 0.586643i
\(876\) 0 0
\(877\) 42844.0i 1.64965i 0.565391 + 0.824823i \(0.308725\pi\)
−0.565391 + 0.824823i \(0.691275\pi\)
\(878\) 0 0
\(879\) 9412.00 + 1845.85i 0.361159 + 0.0708291i
\(880\) 0 0
\(881\) 6975.46i 0.266753i −0.991065 0.133376i \(-0.957418\pi\)
0.991065 0.133376i \(-0.0425819\pi\)
\(882\) 0 0
\(883\) 4803.28 0.183061 0.0915306 0.995802i \(-0.470824\pi\)
0.0915306 + 0.995802i \(0.470824\pi\)
\(884\) 0 0
\(885\) 6424.76 32760.0i 0.244029 1.24431i
\(886\) 0 0
\(887\) −37336.0 −1.41333 −0.706663 0.707550i \(-0.749800\pi\)
−0.706663 + 0.707550i \(0.749800\pi\)
\(888\) 0 0
\(889\) −22984.0 −0.867108
\(890\) 0 0
\(891\) −23455.5 + 23966.0i −0.881917 + 0.901112i
\(892\) 0 0
\(893\) 3916.05 0.146747
\(894\) 0 0
\(895\) 24291.7i 0.907244i
\(896\) 0 0
\(897\) −3872.00 + 19743.4i −0.144127 + 0.734909i
\(898\) 0 0
\(899\) 54600.0i 2.02560i
\(900\) 0 0
\(901\) 9360.00i 0.346090i
\(902\) 0 0
\(903\) −2392.00 + 12196.9i −0.0881515 + 0.449486i
\(904\) 0 0
\(905\) 27045.2i 0.993384i
\(906\) 0 0
\(907\) 16490.2 0.603692 0.301846 0.953357i \(-0.402397\pi\)
0.301846 + 0.953357i \(0.402397\pi\)
\(908\) 0 0
\(909\) 254.951 + 104.000i 0.00930274 + 0.00379479i
\(910\) 0 0
\(911\) 35360.0 1.28598 0.642991 0.765874i \(-0.277693\pi\)
0.642991 + 0.765874i \(0.277693\pi\)
\(912\) 0 0
\(913\) 8556.00 0.310145
\(914\) 0 0
\(915\) −2406.74 + 12272.0i −0.0869555 + 0.443388i
\(916\) 0 0
\(917\) −17765.0 −0.639751
\(918\) 0 0
\(919\) 4395.35i 0.157769i 0.996884 + 0.0788843i \(0.0251358\pi\)
−0.996884 + 0.0788843i \(0.974864\pi\)
\(920\) 0 0
\(921\) 42276.0 + 8291.01i 1.51253 + 0.296632i
\(922\) 0 0
\(923\) 29920.0i 1.06699i
\(924\) 0 0
\(925\) 6972.00i 0.247825i
\(926\) 0 0
\(927\) −20280.0 + 49715.4i −0.718536 + 1.76146i
\(928\) 0 0
\(929\) 1815.25i 0.0641081i −0.999486 0.0320541i \(-0.989795\pi\)
0.999486 0.0320541i \(-0.0102049\pi\)
\(930\) 0 0
\(931\) −2437.33 −0.0858005
\(932\) 0 0
\(933\) −51520.5 10104.0i −1.80783 0.354545i
\(934\) 0 0
\(935\) −9568.00 −0.334660
\(936\) 0 0
\(937\) 34034.0 1.18660 0.593299 0.804982i \(-0.297825\pi\)
0.593299 + 0.804982i \(0.297825\pi\)
\(938\) 0 0
\(939\) 6108.63 + 1198.00i 0.212298 + 0.0416350i
\(940\) 0 0
\(941\) 36967.9 1.28068 0.640339 0.768092i \(-0.278794\pi\)
0.640339 + 0.768092i \(0.278794\pi\)
\(942\) 0 0
\(943\) 43076.5i 1.48756i
\(944\) 0 0
\(945\) 8008.00 12196.9i 0.275662 0.419856i
\(946\) 0 0
\(947\) 24862.0i 0.853122i −0.904459 0.426561i \(-0.859725\pi\)
0.904459 0.426561i \(-0.140275\pi\)
\(948\) 0 0
\(949\) 18568.0i 0.635135i
\(950\) 0 0
\(951\) 25636.0 + 5027.63i 0.874137 + 0.171432i
\(952\) 0 0
\(953\) 22802.8i 0.775085i −0.921852 0.387542i \(-0.873324\pi\)
0.921852 0.387542i \(-0.126676\pi\)
\(954\) 0 0
\(955\) −48787.4 −1.65311
\(956\) 0 0
\(957\) −11727.7 + 59800.0i −0.396138 + 2.01992i
\(958\) 0 0
\(959\) −24544.0 −0.826452
\(960\) 0 0
\(961\) −16073.0 −0.539525
\(962\) 0 0
\(963\) −2386.34 + 5850.00i −0.0798533 + 0.195757i
\(964\) 0 0
\(965\) −31348.8 −1.04575
\(966\) 0 0
\(967\) 40068.1i 1.33247i −0.745740 0.666237i \(-0.767904\pi\)
0.745740 0.666237i \(-0.232096\pi\)
\(968\) 0 0
\(969\) 208.000 1060.60i 0.00689569 0.0351613i
\(970\) 0 0
\(971\) 43522.0i 1.43840i −0.694803 0.719201i \(-0.744508\pi\)
0.694803 0.719201i \(-0.255492\pi\)
\(972\) 0 0
\(973\) 7800.00i 0.256995i
\(974\) 0 0
\(975\) 924.000 4711.49i 0.0303504 0.154758i
\(976\) 0 0
\(977\) 17683.4i 0.579060i −0.957169 0.289530i \(-0.906501\pi\)
0.957169 0.289530i \(-0.0934991\pi\)
\(978\) 0 0
\(979\) −44096.3 −1.43956
\(980\) 0 0
\(981\) 7383.38 18100.0i 0.240299 0.589081i
\(982\) 0 0
\(983\) −42360.0 −1.37444 −0.687220 0.726450i \(-0.741169\pi\)
−0.687220 + 0.726450i \(0.741169\pi\)
\(984\) 0 0
\(985\) −49400.0 −1.59798
\(986\) 0 0
\(987\) −3916.05 + 19968.0i −0.126291 + 0.643960i
\(988\) 0 0
\(989\) −20640.8 −0.663640
\(990\) 0 0
\(991\) 25199.4i 0.807754i 0.914813 + 0.403877i \(0.132338\pi\)
−0.914813 + 0.403877i \(0.867662\pi\)
\(992\) 0 0
\(993\) −31252.0 6129.02i −0.998743 0.195870i
\(994\) 0 0
\(995\) 19656.0i 0.626268i
\(996\) 0 0
\(997\) 5356.00i 0.170137i −0.996375 0.0850683i \(-0.972889\pi\)
0.996375 0.0850683i \(-0.0271109\pi\)
\(998\) 0 0
\(999\) 25564.0 38936.1i 0.809619 1.23312i
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.f.f.191.4 yes 4
3.2 odd 2 384.4.f.e.191.3 yes 4
4.3 odd 2 384.4.f.e.191.1 4
8.3 odd 2 384.4.f.e.191.4 yes 4
8.5 even 2 inner 384.4.f.f.191.1 yes 4
12.11 even 2 inner 384.4.f.f.191.2 yes 4
16.3 odd 4 768.4.c.f.767.1 2
16.5 even 4 768.4.c.g.767.1 2
16.11 odd 4 768.4.c.e.767.2 2
16.13 even 4 768.4.c.d.767.2 2
24.5 odd 2 384.4.f.e.191.2 yes 4
24.11 even 2 inner 384.4.f.f.191.3 yes 4
48.5 odd 4 768.4.c.e.767.1 2
48.11 even 4 768.4.c.g.767.2 2
48.29 odd 4 768.4.c.f.767.2 2
48.35 even 4 768.4.c.d.767.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.f.e.191.1 4 4.3 odd 2
384.4.f.e.191.2 yes 4 24.5 odd 2
384.4.f.e.191.3 yes 4 3.2 odd 2
384.4.f.e.191.4 yes 4 8.3 odd 2
384.4.f.f.191.1 yes 4 8.5 even 2 inner
384.4.f.f.191.2 yes 4 12.11 even 2 inner
384.4.f.f.191.3 yes 4 24.11 even 2 inner
384.4.f.f.191.4 yes 4 1.1 even 1 trivial
768.4.c.d.767.1 2 48.35 even 4
768.4.c.d.767.2 2 16.13 even 4
768.4.c.e.767.1 2 48.5 odd 4
768.4.c.e.767.2 2 16.11 odd 4
768.4.c.f.767.1 2 16.3 odd 4
768.4.c.f.767.2 2 48.29 odd 4
768.4.c.g.767.1 2 16.5 even 4
768.4.c.g.767.2 2 48.11 even 4