Properties

Label 320.3.r.a.271.9
Level $320$
Weight $3$
Character 320.271
Analytic conductor $8.719$
Analytic rank $0$
Dimension $32$
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] = [320,3,Mod(111,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.111");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.r (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 271.9
Character \(\chi\) \(=\) 320.271
Dual form 320.3.r.a.111.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.374900 + 0.374900i) q^{3} +(-1.58114 - 1.58114i) q^{5} -2.42442 q^{7} -8.71890i q^{9} +O(q^{10})\) \(q+(0.374900 + 0.374900i) q^{3} +(-1.58114 - 1.58114i) q^{5} -2.42442 q^{7} -8.71890i q^{9} +(-13.8058 + 13.8058i) q^{11} +(8.90470 - 8.90470i) q^{13} -1.18554i q^{15} -27.7570 q^{17} +(-7.49556 - 7.49556i) q^{19} +(-0.908914 - 0.908914i) q^{21} -13.5071 q^{23} +5.00000i q^{25} +(6.64281 - 6.64281i) q^{27} +(-10.5369 + 10.5369i) q^{29} -46.4696i q^{31} -10.3516 q^{33} +(3.83334 + 3.83334i) q^{35} +(-4.68607 - 4.68607i) q^{37} +6.67674 q^{39} -38.6588i q^{41} +(-45.6877 + 45.6877i) q^{43} +(-13.7858 + 13.7858i) q^{45} -35.3503i q^{47} -43.1222 q^{49} +(-10.4061 - 10.4061i) q^{51} +(61.0885 + 61.0885i) q^{53} +43.6578 q^{55} -5.62017i q^{57} +(9.21882 - 9.21882i) q^{59} +(20.3084 - 20.3084i) q^{61} +21.1383i q^{63} -28.1591 q^{65} +(-3.47945 - 3.47945i) q^{67} +(-5.06382 - 5.06382i) q^{69} +8.72384 q^{71} +23.1038i q^{73} +(-1.87450 + 1.87450i) q^{75} +(33.4711 - 33.4711i) q^{77} -73.9752i q^{79} -73.4893 q^{81} +(50.7301 + 50.7301i) q^{83} +(43.8877 + 43.8877i) q^{85} -7.90060 q^{87} +51.4759i q^{89} +(-21.5887 + 21.5887i) q^{91} +(17.4215 - 17.4215i) q^{93} +23.7030i q^{95} +82.8950 q^{97} +(120.372 + 120.372i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 32 q^{11} + 32 q^{19} + 128 q^{23} + 96 q^{27} + 32 q^{29} - 96 q^{37} - 384 q^{39} - 96 q^{43} + 224 q^{49} + 256 q^{51} - 160 q^{53} + 352 q^{59} - 32 q^{61} - 160 q^{67} + 96 q^{69} - 256 q^{71} + 224 q^{77} - 288 q^{81} + 480 q^{83} + 160 q^{85} + 384 q^{91} + 96 q^{93} - 608 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 0.374900 + 0.374900i 0.124967 + 0.124967i 0.766824 0.641857i \(-0.221836\pi\)
−0.641857 + 0.766824i \(0.721836\pi\)
\(4\) 0 0
\(5\) −1.58114 1.58114i −0.316228 0.316228i
\(6\) 0 0
\(7\) −2.42442 −0.346345 −0.173173 0.984891i \(-0.555402\pi\)
−0.173173 + 0.984891i \(0.555402\pi\)
\(8\) 0 0
\(9\) 8.71890i 0.968767i
\(10\) 0 0
\(11\) −13.8058 + 13.8058i −1.25507 + 1.25507i −0.301658 + 0.953416i \(0.597540\pi\)
−0.953416 + 0.301658i \(0.902460\pi\)
\(12\) 0 0
\(13\) 8.90470 8.90470i 0.684977 0.684977i −0.276141 0.961117i \(-0.589056\pi\)
0.961117 + 0.276141i \(0.0890556\pi\)
\(14\) 0 0
\(15\) 1.18554i 0.0790358i
\(16\) 0 0
\(17\) −27.7570 −1.63276 −0.816382 0.577512i \(-0.804024\pi\)
−0.816382 + 0.577512i \(0.804024\pi\)
\(18\) 0 0
\(19\) −7.49556 7.49556i −0.394503 0.394503i 0.481786 0.876289i \(-0.339988\pi\)
−0.876289 + 0.481786i \(0.839988\pi\)
\(20\) 0 0
\(21\) −0.908914 0.908914i −0.0432816 0.0432816i
\(22\) 0 0
\(23\) −13.5071 −0.587266 −0.293633 0.955918i \(-0.594864\pi\)
−0.293633 + 0.955918i \(0.594864\pi\)
\(24\) 0 0
\(25\) 5.00000i 0.200000i
\(26\) 0 0
\(27\) 6.64281 6.64281i 0.246030 0.246030i
\(28\) 0 0
\(29\) −10.5369 + 10.5369i −0.363343 + 0.363343i −0.865042 0.501699i \(-0.832708\pi\)
0.501699 + 0.865042i \(0.332708\pi\)
\(30\) 0 0
\(31\) 46.4696i 1.49902i −0.661993 0.749510i \(-0.730289\pi\)
0.661993 0.749510i \(-0.269711\pi\)
\(32\) 0 0
\(33\) −10.3516 −0.313685
\(34\) 0 0
\(35\) 3.83334 + 3.83334i 0.109524 + 0.109524i
\(36\) 0 0
\(37\) −4.68607 4.68607i −0.126650 0.126650i 0.640940 0.767591i \(-0.278545\pi\)
−0.767591 + 0.640940i \(0.778545\pi\)
\(38\) 0 0
\(39\) 6.67674 0.171198
\(40\) 0 0
\(41\) 38.6588i 0.942897i −0.881894 0.471449i \(-0.843731\pi\)
0.881894 0.471449i \(-0.156269\pi\)
\(42\) 0 0
\(43\) −45.6877 + 45.6877i −1.06251 + 1.06251i −0.0645935 + 0.997912i \(0.520575\pi\)
−0.997912 + 0.0645935i \(0.979425\pi\)
\(44\) 0 0
\(45\) −13.7858 + 13.7858i −0.306351 + 0.306351i
\(46\) 0 0
\(47\) 35.3503i 0.752134i −0.926592 0.376067i \(-0.877276\pi\)
0.926592 0.376067i \(-0.122724\pi\)
\(48\) 0 0
\(49\) −43.1222 −0.880045
\(50\) 0 0
\(51\) −10.4061 10.4061i −0.204041 0.204041i
\(52\) 0 0
\(53\) 61.0885 + 61.0885i 1.15261 + 1.15261i 0.986027 + 0.166586i \(0.0532745\pi\)
0.166586 + 0.986027i \(0.446726\pi\)
\(54\) 0 0
\(55\) 43.6578 0.793779
\(56\) 0 0
\(57\) 5.62017i 0.0985994i
\(58\) 0 0
\(59\) 9.21882 9.21882i 0.156251 0.156251i −0.624652 0.780903i \(-0.714759\pi\)
0.780903 + 0.624652i \(0.214759\pi\)
\(60\) 0 0
\(61\) 20.3084 20.3084i 0.332925 0.332925i −0.520771 0.853696i \(-0.674356\pi\)
0.853696 + 0.520771i \(0.174356\pi\)
\(62\) 0 0
\(63\) 21.1383i 0.335528i
\(64\) 0 0
\(65\) −28.1591 −0.433217
\(66\) 0 0
\(67\) −3.47945 3.47945i −0.0519320 0.0519320i 0.680664 0.732596i \(-0.261691\pi\)
−0.732596 + 0.680664i \(0.761691\pi\)
\(68\) 0 0
\(69\) −5.06382 5.06382i −0.0733887 0.0733887i
\(70\) 0 0
\(71\) 8.72384 0.122871 0.0614355 0.998111i \(-0.480432\pi\)
0.0614355 + 0.998111i \(0.480432\pi\)
\(72\) 0 0
\(73\) 23.1038i 0.316490i 0.987400 + 0.158245i \(0.0505836\pi\)
−0.987400 + 0.158245i \(0.949416\pi\)
\(74\) 0 0
\(75\) −1.87450 + 1.87450i −0.0249933 + 0.0249933i
\(76\) 0 0
\(77\) 33.4711 33.4711i 0.434689 0.434689i
\(78\) 0 0
\(79\) 73.9752i 0.936394i −0.883624 0.468197i \(-0.844904\pi\)
0.883624 0.468197i \(-0.155096\pi\)
\(80\) 0 0
\(81\) −73.4893 −0.907276
\(82\) 0 0
\(83\) 50.7301 + 50.7301i 0.611206 + 0.611206i 0.943260 0.332055i \(-0.107742\pi\)
−0.332055 + 0.943260i \(0.607742\pi\)
\(84\) 0 0
\(85\) 43.8877 + 43.8877i 0.516325 + 0.516325i
\(86\) 0 0
\(87\) −7.90060 −0.0908115
\(88\) 0 0
\(89\) 51.4759i 0.578381i 0.957272 + 0.289191i \(0.0933861\pi\)
−0.957272 + 0.289191i \(0.906614\pi\)
\(90\) 0 0
\(91\) −21.5887 + 21.5887i −0.237238 + 0.237238i
\(92\) 0 0
\(93\) 17.4215 17.4215i 0.187328 0.187328i
\(94\) 0 0
\(95\) 23.7030i 0.249506i
\(96\) 0 0
\(97\) 82.8950 0.854588 0.427294 0.904113i \(-0.359467\pi\)
0.427294 + 0.904113i \(0.359467\pi\)
\(98\) 0 0
\(99\) 120.372 + 120.372i 1.21587 + 1.21587i
\(100\) 0 0
\(101\) 90.9079 + 90.9079i 0.900078 + 0.900078i 0.995442 0.0953644i \(-0.0304016\pi\)
−0.0953644 + 0.995442i \(0.530402\pi\)
\(102\) 0 0
\(103\) 32.2822 0.313419 0.156710 0.987645i \(-0.449911\pi\)
0.156710 + 0.987645i \(0.449911\pi\)
\(104\) 0 0
\(105\) 2.87424i 0.0273737i
\(106\) 0 0
\(107\) −39.5799 + 39.5799i −0.369906 + 0.369906i −0.867443 0.497537i \(-0.834238\pi\)
0.497537 + 0.867443i \(0.334238\pi\)
\(108\) 0 0
\(109\) −1.28042 + 1.28042i −0.0117469 + 0.0117469i −0.712956 0.701209i \(-0.752644\pi\)
0.701209 + 0.712956i \(0.252644\pi\)
\(110\) 0 0
\(111\) 3.51361i 0.0316542i
\(112\) 0 0
\(113\) 181.626 1.60731 0.803654 0.595096i \(-0.202886\pi\)
0.803654 + 0.595096i \(0.202886\pi\)
\(114\) 0 0
\(115\) 21.3566 + 21.3566i 0.185710 + 0.185710i
\(116\) 0 0
\(117\) −77.6392 77.6392i −0.663582 0.663582i
\(118\) 0 0
\(119\) 67.2945 0.565500
\(120\) 0 0
\(121\) 260.201i 2.15042i
\(122\) 0 0
\(123\) 14.4932 14.4932i 0.117831 0.117831i
\(124\) 0 0
\(125\) 7.90569 7.90569i 0.0632456 0.0632456i
\(126\) 0 0
\(127\) 44.6769i 0.351787i 0.984409 + 0.175893i \(0.0562814\pi\)
−0.984409 + 0.175893i \(0.943719\pi\)
\(128\) 0 0
\(129\) −34.2566 −0.265555
\(130\) 0 0
\(131\) −151.968 151.968i −1.16006 1.16006i −0.984461 0.175602i \(-0.943813\pi\)
−0.175602 0.984461i \(-0.556187\pi\)
\(132\) 0 0
\(133\) 18.1724 + 18.1724i 0.136634 + 0.136634i
\(134\) 0 0
\(135\) −21.0064 −0.155603
\(136\) 0 0
\(137\) 106.085i 0.774342i 0.922008 + 0.387171i \(0.126548\pi\)
−0.922008 + 0.387171i \(0.873452\pi\)
\(138\) 0 0
\(139\) 111.976 111.976i 0.805586 0.805586i −0.178376 0.983962i \(-0.557084\pi\)
0.983962 + 0.178376i \(0.0570844\pi\)
\(140\) 0 0
\(141\) 13.2528 13.2528i 0.0939917 0.0939917i
\(142\) 0 0
\(143\) 245.873i 1.71939i
\(144\) 0 0
\(145\) 33.3208 0.229798
\(146\) 0 0
\(147\) −16.1665 16.1665i −0.109976 0.109976i
\(148\) 0 0
\(149\) −67.4923 67.4923i −0.452969 0.452969i 0.443370 0.896339i \(-0.353783\pi\)
−0.896339 + 0.443370i \(0.853783\pi\)
\(150\) 0 0
\(151\) 154.364 1.02228 0.511140 0.859498i \(-0.329223\pi\)
0.511140 + 0.859498i \(0.329223\pi\)
\(152\) 0 0
\(153\) 242.010i 1.58177i
\(154\) 0 0
\(155\) −73.4749 + 73.4749i −0.474032 + 0.474032i
\(156\) 0 0
\(157\) 85.0691 85.0691i 0.541842 0.541842i −0.382227 0.924069i \(-0.624843\pi\)
0.924069 + 0.382227i \(0.124843\pi\)
\(158\) 0 0
\(159\) 45.8041i 0.288076i
\(160\) 0 0
\(161\) 32.7469 0.203397
\(162\) 0 0
\(163\) 114.693 + 114.693i 0.703636 + 0.703636i 0.965189 0.261553i \(-0.0842346\pi\)
−0.261553 + 0.965189i \(0.584235\pi\)
\(164\) 0 0
\(165\) 16.3673 + 16.3673i 0.0991958 + 0.0991958i
\(166\) 0 0
\(167\) −279.035 −1.67087 −0.835435 0.549590i \(-0.814784\pi\)
−0.835435 + 0.549590i \(0.814784\pi\)
\(168\) 0 0
\(169\) 10.4128i 0.0616142i
\(170\) 0 0
\(171\) −65.3530 + 65.3530i −0.382182 + 0.382182i
\(172\) 0 0
\(173\) −217.656 + 217.656i −1.25813 + 1.25813i −0.306145 + 0.951985i \(0.599039\pi\)
−0.951985 + 0.306145i \(0.900961\pi\)
\(174\) 0 0
\(175\) 12.1221i 0.0692691i
\(176\) 0 0
\(177\) 6.91227 0.0390523
\(178\) 0 0
\(179\) −203.712 203.712i −1.13805 1.13805i −0.988799 0.149256i \(-0.952312\pi\)
−0.149256 0.988799i \(-0.547688\pi\)
\(180\) 0 0
\(181\) −175.707 175.707i −0.970755 0.970755i 0.0288295 0.999584i \(-0.490822\pi\)
−0.999584 + 0.0288295i \(0.990822\pi\)
\(182\) 0 0
\(183\) 15.2272 0.0832090
\(184\) 0 0
\(185\) 14.8186i 0.0801008i
\(186\) 0 0
\(187\) 383.208 383.208i 2.04924 2.04924i
\(188\) 0 0
\(189\) −16.1050 + 16.1050i −0.0852114 + 0.0852114i
\(190\) 0 0
\(191\) 350.356i 1.83433i −0.398512 0.917163i \(-0.630473\pi\)
0.398512 0.917163i \(-0.369527\pi\)
\(192\) 0 0
\(193\) −134.033 −0.694470 −0.347235 0.937778i \(-0.612879\pi\)
−0.347235 + 0.937778i \(0.612879\pi\)
\(194\) 0 0
\(195\) −10.5568 10.5568i −0.0541377 0.0541377i
\(196\) 0 0
\(197\) −149.471 149.471i −0.758736 0.758736i 0.217356 0.976092i \(-0.430257\pi\)
−0.976092 + 0.217356i \(0.930257\pi\)
\(198\) 0 0
\(199\) −307.365 −1.54455 −0.772274 0.635290i \(-0.780881\pi\)
−0.772274 + 0.635290i \(0.780881\pi\)
\(200\) 0 0
\(201\) 2.60889i 0.0129795i
\(202\) 0 0
\(203\) 25.5460 25.5460i 0.125842 0.125842i
\(204\) 0 0
\(205\) −61.1249 + 61.1249i −0.298170 + 0.298170i
\(206\) 0 0
\(207\) 117.767i 0.568924i
\(208\) 0 0
\(209\) 206.965 0.990262
\(210\) 0 0
\(211\) −76.7992 76.7992i −0.363977 0.363977i 0.501298 0.865275i \(-0.332856\pi\)
−0.865275 + 0.501298i \(0.832856\pi\)
\(212\) 0 0
\(213\) 3.27056 + 3.27056i 0.0153548 + 0.0153548i
\(214\) 0 0
\(215\) 144.477 0.671987
\(216\) 0 0
\(217\) 112.662i 0.519179i
\(218\) 0 0
\(219\) −8.66161 + 8.66161i −0.0395507 + 0.0395507i
\(220\) 0 0
\(221\) −247.168 + 247.168i −1.11841 + 1.11841i
\(222\) 0 0
\(223\) 108.117i 0.484830i −0.970173 0.242415i \(-0.922060\pi\)
0.970173 0.242415i \(-0.0779396\pi\)
\(224\) 0 0
\(225\) 43.5945 0.193753
\(226\) 0 0
\(227\) −6.15669 6.15669i −0.0271220 0.0271220i 0.693416 0.720538i \(-0.256105\pi\)
−0.720538 + 0.693416i \(0.756105\pi\)
\(228\) 0 0
\(229\) −110.025 110.025i −0.480459 0.480459i 0.424819 0.905278i \(-0.360338\pi\)
−0.905278 + 0.424819i \(0.860338\pi\)
\(230\) 0 0
\(231\) 25.0966 0.108643
\(232\) 0 0
\(233\) 109.457i 0.469772i −0.972023 0.234886i \(-0.924528\pi\)
0.972023 0.234886i \(-0.0754717\pi\)
\(234\) 0 0
\(235\) −55.8938 + 55.8938i −0.237846 + 0.237846i
\(236\) 0 0
\(237\) 27.7333 27.7333i 0.117018 0.117018i
\(238\) 0 0
\(239\) 418.232i 1.74992i 0.484192 + 0.874962i \(0.339114\pi\)
−0.484192 + 0.874962i \(0.660886\pi\)
\(240\) 0 0
\(241\) −240.522 −0.998015 −0.499007 0.866598i \(-0.666302\pi\)
−0.499007 + 0.866598i \(0.666302\pi\)
\(242\) 0 0
\(243\) −87.3364 87.3364i −0.359409 0.359409i
\(244\) 0 0
\(245\) 68.1822 + 68.1822i 0.278295 + 0.278295i
\(246\) 0 0
\(247\) −133.491 −0.540451
\(248\) 0 0
\(249\) 38.0374i 0.152761i
\(250\) 0 0
\(251\) −55.0912 + 55.0912i −0.219487 + 0.219487i −0.808282 0.588795i \(-0.799602\pi\)
0.588795 + 0.808282i \(0.299602\pi\)
\(252\) 0 0
\(253\) 186.477 186.477i 0.737063 0.737063i
\(254\) 0 0
\(255\) 32.9070i 0.129047i
\(256\) 0 0
\(257\) 0.417209 0.00162338 0.000811690 1.00000i \(-0.499742\pi\)
0.000811690 1.00000i \(0.499742\pi\)
\(258\) 0 0
\(259\) 11.3610 + 11.3610i 0.0438648 + 0.0438648i
\(260\) 0 0
\(261\) 91.8706 + 91.8706i 0.351995 + 0.351995i
\(262\) 0 0
\(263\) −53.6800 −0.204106 −0.102053 0.994779i \(-0.532541\pi\)
−0.102053 + 0.994779i \(0.532541\pi\)
\(264\) 0 0
\(265\) 193.179i 0.728977i
\(266\) 0 0
\(267\) −19.2983 + 19.2983i −0.0722783 + 0.0722783i
\(268\) 0 0
\(269\) 229.835 229.835i 0.854404 0.854404i −0.136268 0.990672i \(-0.543511\pi\)
0.990672 + 0.136268i \(0.0435110\pi\)
\(270\) 0 0
\(271\) 180.605i 0.666439i 0.942849 + 0.333220i \(0.108135\pi\)
−0.942849 + 0.333220i \(0.891865\pi\)
\(272\) 0 0
\(273\) −16.1872 −0.0592938
\(274\) 0 0
\(275\) −69.0291 69.0291i −0.251015 0.251015i
\(276\) 0 0
\(277\) −255.348 255.348i −0.921834 0.921834i 0.0753246 0.997159i \(-0.476001\pi\)
−0.997159 + 0.0753246i \(0.976001\pi\)
\(278\) 0 0
\(279\) −405.164 −1.45220
\(280\) 0 0
\(281\) 397.276i 1.41379i −0.707317 0.706897i \(-0.750095\pi\)
0.707317 0.706897i \(-0.249905\pi\)
\(282\) 0 0
\(283\) 130.023 130.023i 0.459444 0.459444i −0.439029 0.898473i \(-0.644678\pi\)
0.898473 + 0.439029i \(0.144678\pi\)
\(284\) 0 0
\(285\) −8.88627 + 8.88627i −0.0311799 + 0.0311799i
\(286\) 0 0
\(287\) 93.7250i 0.326568i
\(288\) 0 0
\(289\) 481.451 1.66592
\(290\) 0 0
\(291\) 31.0773 + 31.0773i 0.106795 + 0.106795i
\(292\) 0 0
\(293\) −170.949 170.949i −0.583443 0.583443i 0.352405 0.935848i \(-0.385364\pi\)
−0.935848 + 0.352405i \(0.885364\pi\)
\(294\) 0 0
\(295\) −29.1525 −0.0988219
\(296\) 0 0
\(297\) 183.419i 0.617572i
\(298\) 0 0
\(299\) −120.277 + 120.277i −0.402264 + 0.402264i
\(300\) 0 0
\(301\) 110.766 110.766i 0.367994 0.367994i
\(302\) 0 0
\(303\) 68.1627i 0.224959i
\(304\) 0 0
\(305\) −64.2208 −0.210560
\(306\) 0 0
\(307\) 108.448 + 108.448i 0.353251 + 0.353251i 0.861318 0.508066i \(-0.169640\pi\)
−0.508066 + 0.861318i \(0.669640\pi\)
\(308\) 0 0
\(309\) 12.1026 + 12.1026i 0.0391669 + 0.0391669i
\(310\) 0 0
\(311\) 436.855 1.40468 0.702340 0.711842i \(-0.252139\pi\)
0.702340 + 0.711842i \(0.252139\pi\)
\(312\) 0 0
\(313\) 328.972i 1.05103i 0.850785 + 0.525514i \(0.176127\pi\)
−0.850785 + 0.525514i \(0.823873\pi\)
\(314\) 0 0
\(315\) 33.4225 33.4225i 0.106103 0.106103i
\(316\) 0 0
\(317\) −306.697 + 306.697i −0.967498 + 0.967498i −0.999488 0.0319904i \(-0.989815\pi\)
0.0319904 + 0.999488i \(0.489815\pi\)
\(318\) 0 0
\(319\) 290.942i 0.912045i
\(320\) 0 0
\(321\) −29.6770 −0.0924517
\(322\) 0 0
\(323\) 208.054 + 208.054i 0.644131 + 0.644131i
\(324\) 0 0
\(325\) 44.5235 + 44.5235i 0.136995 + 0.136995i
\(326\) 0 0
\(327\) −0.960056 −0.00293595
\(328\) 0 0
\(329\) 85.7039i 0.260498i
\(330\) 0 0
\(331\) −245.686 + 245.686i −0.742253 + 0.742253i −0.973011 0.230758i \(-0.925879\pi\)
0.230758 + 0.973011i \(0.425879\pi\)
\(332\) 0 0
\(333\) −40.8574 + 40.8574i −0.122695 + 0.122695i
\(334\) 0 0
\(335\) 11.0030i 0.0328447i
\(336\) 0 0
\(337\) 205.265 0.609095 0.304548 0.952497i \(-0.401495\pi\)
0.304548 + 0.952497i \(0.401495\pi\)
\(338\) 0 0
\(339\) 68.0915 + 68.0915i 0.200860 + 0.200860i
\(340\) 0 0
\(341\) 641.551 + 641.551i 1.88138 + 1.88138i
\(342\) 0 0
\(343\) 223.343 0.651145
\(344\) 0 0
\(345\) 16.0132i 0.0464151i
\(346\) 0 0
\(347\) 82.7836 82.7836i 0.238570 0.238570i −0.577688 0.816258i \(-0.696045\pi\)
0.816258 + 0.577688i \(0.196045\pi\)
\(348\) 0 0
\(349\) 89.8282 89.8282i 0.257387 0.257387i −0.566603 0.823991i \(-0.691743\pi\)
0.823991 + 0.566603i \(0.191743\pi\)
\(350\) 0 0
\(351\) 118.304i 0.337050i
\(352\) 0 0
\(353\) −104.126 −0.294974 −0.147487 0.989064i \(-0.547119\pi\)
−0.147487 + 0.989064i \(0.547119\pi\)
\(354\) 0 0
\(355\) −13.7936 13.7936i −0.0388552 0.0388552i
\(356\) 0 0
\(357\) 25.2287 + 25.2287i 0.0706687 + 0.0706687i
\(358\) 0 0
\(359\) 372.933 1.03881 0.519405 0.854528i \(-0.326154\pi\)
0.519405 + 0.854528i \(0.326154\pi\)
\(360\) 0 0
\(361\) 248.633i 0.688735i
\(362\) 0 0
\(363\) 97.5494 97.5494i 0.268731 0.268731i
\(364\) 0 0
\(365\) 36.5303 36.5303i 0.100083 0.100083i
\(366\) 0 0
\(367\) 283.076i 0.771325i −0.922640 0.385663i \(-0.873973\pi\)
0.922640 0.385663i \(-0.126027\pi\)
\(368\) 0 0
\(369\) −337.062 −0.913447
\(370\) 0 0
\(371\) −148.104 148.104i −0.399202 0.399202i
\(372\) 0 0
\(373\) 48.6420 + 48.6420i 0.130407 + 0.130407i 0.769298 0.638890i \(-0.220606\pi\)
−0.638890 + 0.769298i \(0.720606\pi\)
\(374\) 0 0
\(375\) 5.92769 0.0158072
\(376\) 0 0
\(377\) 187.657i 0.497763i
\(378\) 0 0
\(379\) 120.242 120.242i 0.317262 0.317262i −0.530453 0.847715i \(-0.677978\pi\)
0.847715 + 0.530453i \(0.177978\pi\)
\(380\) 0 0
\(381\) −16.7494 + 16.7494i −0.0439616 + 0.0439616i
\(382\) 0 0
\(383\) 46.9823i 0.122669i −0.998117 0.0613346i \(-0.980464\pi\)
0.998117 0.0613346i \(-0.0195357\pi\)
\(384\) 0 0
\(385\) −105.845 −0.274922
\(386\) 0 0
\(387\) 398.347 + 398.347i 1.02932 + 1.02932i
\(388\) 0 0
\(389\) 103.685 + 103.685i 0.266542 + 0.266542i 0.827705 0.561163i \(-0.189646\pi\)
−0.561163 + 0.827705i \(0.689646\pi\)
\(390\) 0 0
\(391\) 374.917 0.958867
\(392\) 0 0
\(393\) 113.946i 0.289938i
\(394\) 0 0
\(395\) −116.965 + 116.965i −0.296114 + 0.296114i
\(396\) 0 0
\(397\) 290.747 290.747i 0.732361 0.732361i −0.238726 0.971087i \(-0.576730\pi\)
0.971087 + 0.238726i \(0.0767299\pi\)
\(398\) 0 0
\(399\) 13.6256i 0.0341495i
\(400\) 0 0
\(401\) −5.07753 −0.0126622 −0.00633108 0.999980i \(-0.502015\pi\)
−0.00633108 + 0.999980i \(0.502015\pi\)
\(402\) 0 0
\(403\) −413.798 413.798i −1.02679 1.02679i
\(404\) 0 0
\(405\) 116.197 + 116.197i 0.286906 + 0.286906i
\(406\) 0 0
\(407\) 129.390 0.317912
\(408\) 0 0
\(409\) 254.647i 0.622610i 0.950310 + 0.311305i \(0.100766\pi\)
−0.950310 + 0.311305i \(0.899234\pi\)
\(410\) 0 0
\(411\) −39.7712 + 39.7712i −0.0967669 + 0.0967669i
\(412\) 0 0
\(413\) −22.3503 + 22.3503i −0.0541169 + 0.0541169i
\(414\) 0 0
\(415\) 160.423i 0.386560i
\(416\) 0 0
\(417\) 83.9599 0.201343
\(418\) 0 0
\(419\) 257.710 + 257.710i 0.615059 + 0.615059i 0.944260 0.329201i \(-0.106779\pi\)
−0.329201 + 0.944260i \(0.606779\pi\)
\(420\) 0 0
\(421\) 280.397 + 280.397i 0.666026 + 0.666026i 0.956794 0.290768i \(-0.0939108\pi\)
−0.290768 + 0.956794i \(0.593911\pi\)
\(422\) 0 0
\(423\) −308.216 −0.728643
\(424\) 0 0
\(425\) 138.785i 0.326553i
\(426\) 0 0
\(427\) −49.2361 + 49.2361i −0.115307 + 0.115307i
\(428\) 0 0
\(429\) −92.1778 + 92.1778i −0.214867 + 0.214867i
\(430\) 0 0
\(431\) 203.154i 0.471355i 0.971831 + 0.235677i \(0.0757308\pi\)
−0.971831 + 0.235677i \(0.924269\pi\)
\(432\) 0 0
\(433\) 19.3904 0.0447816 0.0223908 0.999749i \(-0.492872\pi\)
0.0223908 + 0.999749i \(0.492872\pi\)
\(434\) 0 0
\(435\) 12.4919 + 12.4919i 0.0287171 + 0.0287171i
\(436\) 0 0
\(437\) 101.243 + 101.243i 0.231678 + 0.231678i
\(438\) 0 0
\(439\) 286.595 0.652835 0.326418 0.945226i \(-0.394158\pi\)
0.326418 + 0.945226i \(0.394158\pi\)
\(440\) 0 0
\(441\) 375.978i 0.852558i
\(442\) 0 0
\(443\) −234.141 + 234.141i −0.528535 + 0.528535i −0.920136 0.391600i \(-0.871922\pi\)
0.391600 + 0.920136i \(0.371922\pi\)
\(444\) 0 0
\(445\) 81.3906 81.3906i 0.182900 0.182900i
\(446\) 0 0
\(447\) 50.6057i 0.113212i
\(448\) 0 0
\(449\) −586.663 −1.30660 −0.653300 0.757099i \(-0.726616\pi\)
−0.653300 + 0.757099i \(0.726616\pi\)
\(450\) 0 0
\(451\) 533.716 + 533.716i 1.18341 + 1.18341i
\(452\) 0 0
\(453\) 57.8711 + 57.8711i 0.127751 + 0.127751i
\(454\) 0 0
\(455\) 68.2695 0.150043
\(456\) 0 0
\(457\) 284.950i 0.623524i 0.950160 + 0.311762i \(0.100919\pi\)
−0.950160 + 0.311762i \(0.899081\pi\)
\(458\) 0 0
\(459\) −184.385 + 184.385i −0.401709 + 0.401709i
\(460\) 0 0
\(461\) −443.411 + 443.411i −0.961845 + 0.961845i −0.999298 0.0374533i \(-0.988075\pi\)
0.0374533 + 0.999298i \(0.488075\pi\)
\(462\) 0 0
\(463\) 725.563i 1.56709i −0.621335 0.783545i \(-0.713409\pi\)
0.621335 0.783545i \(-0.286591\pi\)
\(464\) 0 0
\(465\) −55.0915 −0.118476
\(466\) 0 0
\(467\) −105.560 105.560i −0.226039 0.226039i 0.584997 0.811036i \(-0.301096\pi\)
−0.811036 + 0.584997i \(0.801096\pi\)
\(468\) 0 0
\(469\) 8.43563 + 8.43563i 0.0179864 + 0.0179864i
\(470\) 0 0
\(471\) 63.7848 0.135424
\(472\) 0 0
\(473\) 1261.51i 2.66705i
\(474\) 0 0
\(475\) 37.4778 37.4778i 0.0789006 0.0789006i
\(476\) 0 0
\(477\) 532.625 532.625i 1.11661 1.11661i
\(478\) 0 0
\(479\) 606.909i 1.26703i −0.773729 0.633516i \(-0.781611\pi\)
0.773729 0.633516i \(-0.218389\pi\)
\(480\) 0 0
\(481\) −83.4560 −0.173505
\(482\) 0 0
\(483\) 12.2768 + 12.2768i 0.0254178 + 0.0254178i
\(484\) 0 0
\(485\) −131.068 131.068i −0.270244 0.270244i
\(486\) 0 0
\(487\) −961.389 −1.97411 −0.987053 0.160396i \(-0.948723\pi\)
−0.987053 + 0.160396i \(0.948723\pi\)
\(488\) 0 0
\(489\) 85.9966i 0.175862i
\(490\) 0 0
\(491\) −564.547 + 564.547i −1.14979 + 1.14979i −0.163197 + 0.986594i \(0.552181\pi\)
−0.986594 + 0.163197i \(0.947819\pi\)
\(492\) 0 0
\(493\) 292.474 292.474i 0.593254 0.593254i
\(494\) 0 0
\(495\) 380.648i 0.768986i
\(496\) 0 0
\(497\) −21.1502 −0.0425558
\(498\) 0 0
\(499\) −5.48141 5.48141i −0.0109848 0.0109848i 0.701593 0.712578i \(-0.252472\pi\)
−0.712578 + 0.701593i \(0.752472\pi\)
\(500\) 0 0
\(501\) −104.610 104.610i −0.208803 0.208803i
\(502\) 0 0
\(503\) 558.604 1.11054 0.555272 0.831669i \(-0.312614\pi\)
0.555272 + 0.831669i \(0.312614\pi\)
\(504\) 0 0
\(505\) 287.476i 0.569259i
\(506\) 0 0
\(507\) −3.90376 + 3.90376i −0.00769972 + 0.00769972i
\(508\) 0 0
\(509\) −170.507 + 170.507i −0.334985 + 0.334985i −0.854476 0.519491i \(-0.826122\pi\)
0.519491 + 0.854476i \(0.326122\pi\)
\(510\) 0 0
\(511\) 56.0132i 0.109615i
\(512\) 0 0
\(513\) −99.5832 −0.194119
\(514\) 0 0
\(515\) −51.0426 51.0426i −0.0991118 0.0991118i
\(516\) 0 0
\(517\) 488.040 + 488.040i 0.943985 + 0.943985i
\(518\) 0 0
\(519\) −163.199 −0.314448
\(520\) 0 0
\(521\) 248.276i 0.476537i 0.971199 + 0.238269i \(0.0765798\pi\)
−0.971199 + 0.238269i \(0.923420\pi\)
\(522\) 0 0
\(523\) −53.5442 + 53.5442i −0.102379 + 0.102379i −0.756441 0.654062i \(-0.773064\pi\)
0.654062 + 0.756441i \(0.273064\pi\)
\(524\) 0 0
\(525\) 4.54457 4.54457i 0.00865632 0.00865632i
\(526\) 0 0
\(527\) 1289.86i 2.44755i
\(528\) 0 0
\(529\) −346.558 −0.655118
\(530\) 0 0
\(531\) −80.3779 80.3779i −0.151371 0.151371i
\(532\) 0 0
\(533\) −344.245 344.245i −0.645862 0.645862i
\(534\) 0 0
\(535\) 125.163 0.233949
\(536\) 0 0
\(537\) 152.743i 0.284438i
\(538\) 0 0
\(539\) 595.337 595.337i 1.10452 1.10452i
\(540\) 0 0
\(541\) 356.241 356.241i 0.658486 0.658486i −0.296536 0.955022i \(-0.595831\pi\)
0.955022 + 0.296536i \(0.0958314\pi\)
\(542\) 0 0
\(543\) 131.745i 0.242624i
\(544\) 0 0
\(545\) 4.04903 0.00742942
\(546\) 0 0
\(547\) −205.888 205.888i −0.376394 0.376394i 0.493405 0.869799i \(-0.335752\pi\)
−0.869799 + 0.493405i \(0.835752\pi\)
\(548\) 0 0
\(549\) −177.067 177.067i −0.322526 0.322526i
\(550\) 0 0
\(551\) 157.961 0.286680
\(552\) 0 0
\(553\) 179.347i 0.324316i
\(554\) 0 0
\(555\) −5.55551 + 5.55551i −0.0100099 + 0.0100099i
\(556\) 0 0
\(557\) −378.558 + 378.558i −0.679638 + 0.679638i −0.959918 0.280280i \(-0.909573\pi\)
0.280280 + 0.959918i \(0.409573\pi\)
\(558\) 0 0
\(559\) 813.670i 1.45558i
\(560\) 0 0
\(561\) 287.329 0.512173
\(562\) 0 0
\(563\) −600.937 600.937i −1.06738 1.06738i −0.997559 0.0698249i \(-0.977756\pi\)
−0.0698249 0.997559i \(-0.522244\pi\)
\(564\) 0 0
\(565\) −287.176 287.176i −0.508276 0.508276i
\(566\) 0 0
\(567\) 178.169 0.314231
\(568\) 0 0
\(569\) 394.322i 0.693009i −0.938048 0.346504i \(-0.887369\pi\)
0.938048 0.346504i \(-0.112631\pi\)
\(570\) 0 0
\(571\) −86.9337 + 86.9337i −0.152248 + 0.152248i −0.779121 0.626873i \(-0.784334\pi\)
0.626873 + 0.779121i \(0.284334\pi\)
\(572\) 0 0
\(573\) 131.349 131.349i 0.229230 0.229230i
\(574\) 0 0
\(575\) 67.5356i 0.117453i
\(576\) 0 0
\(577\) −398.959 −0.691436 −0.345718 0.938338i \(-0.612365\pi\)
−0.345718 + 0.938338i \(0.612365\pi\)
\(578\) 0 0
\(579\) −50.2488 50.2488i −0.0867855 0.0867855i
\(580\) 0 0
\(581\) −122.991 122.991i −0.211688 0.211688i
\(582\) 0 0
\(583\) −1686.75 −2.89323
\(584\) 0 0
\(585\) 245.517i 0.419686i
\(586\) 0 0
\(587\) −332.530 + 332.530i −0.566490 + 0.566490i −0.931143 0.364653i \(-0.881188\pi\)
0.364653 + 0.931143i \(0.381188\pi\)
\(588\) 0 0
\(589\) −348.316 + 348.316i −0.591368 + 0.591368i
\(590\) 0 0
\(591\) 112.073i 0.189633i
\(592\) 0 0
\(593\) −154.212 −0.260054 −0.130027 0.991510i \(-0.541506\pi\)
−0.130027 + 0.991510i \(0.541506\pi\)
\(594\) 0 0
\(595\) −106.402 106.402i −0.178827 0.178827i
\(596\) 0 0
\(597\) −115.231 115.231i −0.193017 0.193017i
\(598\) 0 0
\(599\) 382.295 0.638223 0.319111 0.947717i \(-0.396616\pi\)
0.319111 + 0.947717i \(0.396616\pi\)
\(600\) 0 0
\(601\) 84.1090i 0.139948i −0.997549 0.0699742i \(-0.977708\pi\)
0.997549 0.0699742i \(-0.0222917\pi\)
\(602\) 0 0
\(603\) −30.3369 + 30.3369i −0.0503100 + 0.0503100i
\(604\) 0 0
\(605\) −411.414 + 411.414i −0.680024 + 0.680024i
\(606\) 0 0
\(607\) 848.304i 1.39754i −0.715348 0.698768i \(-0.753732\pi\)
0.715348 0.698768i \(-0.246268\pi\)
\(608\) 0 0
\(609\) 19.1544 0.0314521
\(610\) 0 0
\(611\) −314.784 314.784i −0.515194 0.515194i
\(612\) 0 0
\(613\) −216.458 216.458i −0.353112 0.353112i 0.508154 0.861266i \(-0.330328\pi\)
−0.861266 + 0.508154i \(0.830328\pi\)
\(614\) 0 0
\(615\) −45.8314 −0.0745226
\(616\) 0 0
\(617\) 654.303i 1.06046i −0.847854 0.530229i \(-0.822106\pi\)
0.847854 0.530229i \(-0.177894\pi\)
\(618\) 0 0
\(619\) −61.5563 + 61.5563i −0.0994448 + 0.0994448i −0.755079 0.655634i \(-0.772401\pi\)
0.655634 + 0.755079i \(0.272401\pi\)
\(620\) 0 0
\(621\) −89.7253 + 89.7253i −0.144485 + 0.144485i
\(622\) 0 0
\(623\) 124.799i 0.200320i
\(624\) 0 0
\(625\) −25.0000 −0.0400000
\(626\) 0 0
\(627\) 77.5910 + 77.5910i 0.123750 + 0.123750i
\(628\) 0 0
\(629\) 130.071 + 130.071i 0.206790 + 0.206790i
\(630\) 0 0
\(631\) −532.078 −0.843230 −0.421615 0.906775i \(-0.638537\pi\)
−0.421615 + 0.906775i \(0.638537\pi\)
\(632\) 0 0
\(633\) 57.5840i 0.0909700i
\(634\) 0 0
\(635\) 70.6404 70.6404i 0.111245 0.111245i
\(636\) 0 0
\(637\) −383.990 + 383.990i −0.602810 + 0.602810i
\(638\) 0 0
\(639\) 76.0623i 0.119033i
\(640\) 0 0
\(641\) 85.5372 0.133443 0.0667217 0.997772i \(-0.478746\pi\)
0.0667217 + 0.997772i \(0.478746\pi\)
\(642\) 0 0
\(643\) 405.524 + 405.524i 0.630675 + 0.630675i 0.948237 0.317562i \(-0.102864\pi\)
−0.317562 + 0.948237i \(0.602864\pi\)
\(644\) 0 0
\(645\) 54.1645 + 54.1645i 0.0839760 + 0.0839760i
\(646\) 0 0
\(647\) 778.109 1.20264 0.601321 0.799008i \(-0.294641\pi\)
0.601321 + 0.799008i \(0.294641\pi\)
\(648\) 0 0
\(649\) 254.547i 0.392214i
\(650\) 0 0
\(651\) −42.2369 + 42.2369i −0.0648800 + 0.0648800i
\(652\) 0 0
\(653\) 561.135 561.135i 0.859319 0.859319i −0.131939 0.991258i \(-0.542120\pi\)
0.991258 + 0.131939i \(0.0421204\pi\)
\(654\) 0 0
\(655\) 480.566i 0.733689i
\(656\) 0 0
\(657\) 201.440 0.306605
\(658\) 0 0
\(659\) 212.956 + 212.956i 0.323150 + 0.323150i 0.849974 0.526824i \(-0.176617\pi\)
−0.526824 + 0.849974i \(0.676617\pi\)
\(660\) 0 0
\(661\) 250.037 + 250.037i 0.378270 + 0.378270i 0.870478 0.492208i \(-0.163810\pi\)
−0.492208 + 0.870478i \(0.663810\pi\)
\(662\) 0 0
\(663\) −185.326 −0.279527
\(664\) 0 0
\(665\) 57.4661i 0.0864151i
\(666\) 0 0
\(667\) 142.324 142.324i 0.213379 0.213379i
\(668\) 0 0
\(669\) 40.5331 40.5331i 0.0605876 0.0605876i
\(670\) 0 0
\(671\) 560.749i 0.835691i
\(672\) 0 0
\(673\) 363.444 0.540036 0.270018 0.962855i \(-0.412970\pi\)
0.270018 + 0.962855i \(0.412970\pi\)
\(674\) 0 0
\(675\) 33.2141 + 33.2141i 0.0492060 + 0.0492060i
\(676\) 0 0
\(677\) −628.852 628.852i −0.928880 0.928880i 0.0687532 0.997634i \(-0.478098\pi\)
−0.997634 + 0.0687532i \(0.978098\pi\)
\(678\) 0 0
\(679\) −200.972 −0.295982
\(680\) 0 0
\(681\) 4.61628i 0.00677869i
\(682\) 0 0
\(683\) 321.108 321.108i 0.470144 0.470144i −0.431817 0.901961i \(-0.642128\pi\)
0.901961 + 0.431817i \(0.142128\pi\)
\(684\) 0 0
\(685\) 167.735 167.735i 0.244868 0.244868i
\(686\) 0 0
\(687\) 82.4969i 0.120083i
\(688\) 0 0
\(689\) 1087.95 1.57903
\(690\) 0 0
\(691\) 451.940 + 451.940i 0.654038 + 0.654038i 0.953963 0.299925i \(-0.0969615\pi\)
−0.299925 + 0.953963i \(0.596962\pi\)
\(692\) 0 0
\(693\) −291.831 291.831i −0.421112 0.421112i
\(694\) 0 0
\(695\) −354.101 −0.509497
\(696\) 0 0
\(697\) 1073.05i 1.53953i
\(698\) 0 0
\(699\) 41.0354 41.0354i 0.0587059 0.0587059i
\(700\) 0 0
\(701\) 252.343 252.343i 0.359976 0.359976i −0.503828 0.863804i \(-0.668075\pi\)
0.863804 + 0.503828i \(0.168075\pi\)
\(702\) 0 0
\(703\) 70.2494i 0.0999280i
\(704\) 0 0
\(705\) −41.9091 −0.0594456
\(706\) 0 0
\(707\) −220.399 220.399i −0.311738 0.311738i
\(708\) 0 0
\(709\) −401.245 401.245i −0.565931 0.565931i 0.365055 0.930986i \(-0.381050\pi\)
−0.930986 + 0.365055i \(0.881050\pi\)
\(710\) 0 0
\(711\) −644.982 −0.907148
\(712\) 0 0
\(713\) 627.671i 0.880324i
\(714\) 0 0
\(715\) 388.760 388.760i 0.543720 0.543720i
\(716\) 0 0
\(717\) −156.795 + 156.795i −0.218682 + 0.218682i
\(718\) 0 0
\(719\) 518.286i 0.720843i −0.932790 0.360422i \(-0.882633\pi\)
0.932790 0.360422i \(-0.117367\pi\)
\(720\) 0 0
\(721\) −78.2654 −0.108551
\(722\) 0 0
\(723\) −90.1715 90.1715i −0.124719 0.124719i
\(724\) 0 0
\(725\) −52.6847 52.6847i −0.0726686 0.0726686i
\(726\) 0 0
\(727\) 1097.86 1.51012 0.755059 0.655657i \(-0.227608\pi\)
0.755059 + 0.655657i \(0.227608\pi\)
\(728\) 0 0
\(729\) 595.919i 0.817447i
\(730\) 0 0
\(731\) 1268.15 1268.15i 1.73482 1.73482i
\(732\) 0 0
\(733\) −749.818 + 749.818i −1.02294 + 1.02294i −0.0232137 + 0.999731i \(0.507390\pi\)
−0.999731 + 0.0232137i \(0.992610\pi\)
\(734\) 0 0
\(735\) 51.1230i 0.0695551i
\(736\) 0 0
\(737\) 96.0732 0.130357
\(738\) 0 0
\(739\) −687.419 687.419i −0.930201 0.930201i 0.0675168 0.997718i \(-0.478492\pi\)
−0.997718 + 0.0675168i \(0.978492\pi\)
\(740\) 0 0
\(741\) −50.0459 50.0459i −0.0675383 0.0675383i
\(742\) 0 0
\(743\) 825.630 1.11121 0.555605 0.831446i \(-0.312487\pi\)
0.555605 + 0.831446i \(0.312487\pi\)
\(744\) 0 0
\(745\) 213.429i 0.286482i
\(746\) 0 0
\(747\) 442.310 442.310i 0.592116 0.592116i
\(748\) 0 0
\(749\) 95.9582 95.9582i 0.128115 0.128115i
\(750\) 0 0
\(751\) 1237.27i 1.64749i 0.566958 + 0.823747i \(0.308120\pi\)
−0.566958 + 0.823747i \(0.691880\pi\)
\(752\) 0 0
\(753\) −41.3073 −0.0548570
\(754\) 0 0
\(755\) −244.071 244.071i −0.323273 0.323273i
\(756\) 0 0
\(757\) 606.680 + 606.680i 0.801427 + 0.801427i 0.983319 0.181892i \(-0.0582220\pi\)
−0.181892 + 0.983319i \(0.558222\pi\)
\(758\) 0 0
\(759\) 139.820 0.184216
\(760\) 0 0
\(761\) 1024.72i 1.34655i 0.739393 + 0.673274i \(0.235113\pi\)
−0.739393 + 0.673274i \(0.764887\pi\)
\(762\) 0 0
\(763\) 3.10426 3.10426i 0.00406850 0.00406850i
\(764\) 0 0
\(765\) 382.652 382.652i 0.500199 0.500199i
\(766\) 0 0
\(767\) 164.182i 0.214057i
\(768\) 0 0
\(769\) 161.299 0.209752 0.104876 0.994485i \(-0.466555\pi\)
0.104876 + 0.994485i \(0.466555\pi\)
\(770\) 0 0
\(771\) 0.156411 + 0.156411i 0.000202868 + 0.000202868i
\(772\) 0 0
\(773\) 679.758 + 679.758i 0.879377 + 0.879377i 0.993470 0.114093i \(-0.0363963\pi\)
−0.114093 + 0.993470i \(0.536396\pi\)
\(774\) 0 0
\(775\) 232.348 0.299804
\(776\) 0 0
\(777\) 8.51846i 0.0109633i
\(778\) 0 0
\(779\) −289.769 + 289.769i −0.371976 + 0.371976i
\(780\) 0 0
\(781\) −120.440 + 120.440i −0.154212 + 0.154212i
\(782\) 0 0
\(783\) 139.990i 0.178787i
\(784\) 0 0
\(785\) −269.012 −0.342691
\(786\) 0 0
\(787\) 441.767 + 441.767i 0.561331 + 0.561331i 0.929685 0.368355i \(-0.120079\pi\)
−0.368355 + 0.929685i \(0.620079\pi\)
\(788\) 0 0
\(789\) −20.1246 20.1246i −0.0255065 0.0255065i
\(790\) 0 0
\(791\) −440.337 −0.556684
\(792\) 0 0
\(793\) 361.681i 0.456091i
\(794\) 0 0
\(795\) 72.4227 72.4227i 0.0910977 0.0910977i
\(796\) 0 0
\(797\) −99.4692 + 99.4692i −0.124804 + 0.124804i −0.766750 0.641946i \(-0.778127\pi\)
0.641946 + 0.766750i \(0.278127\pi\)
\(798\) 0 0
\(799\) 981.219i 1.22806i
\(800\) 0 0
\(801\) 448.814 0.560317
\(802\) 0 0
\(803\) −318.967 318.967i −0.397219 0.397219i
\(804\) 0 0
\(805\) −51.7774 51.7774i −0.0643198 0.0643198i
\(806\) 0 0
\(807\) 172.330 0.213544
\(808\) 0 0
\(809\) 1353.26i 1.67276i −0.548153 0.836378i \(-0.684669\pi\)
0.548153 0.836378i \(-0.315331\pi\)
\(810\) 0 0
\(811\) 78.4824 78.4824i 0.0967724 0.0967724i −0.657063 0.753836i \(-0.728202\pi\)
0.753836 + 0.657063i \(0.228202\pi\)
\(812\) 0 0
\(813\) −67.7088 + 67.7088i −0.0832826 + 0.0832826i
\(814\) 0 0
\(815\) 362.690i 0.445019i
\(816\) 0 0
\(817\) 684.910 0.838323
\(818\) 0 0
\(819\) 188.230 + 188.230i 0.229829 + 0.229829i
\(820\) 0 0
\(821\) −574.440 574.440i −0.699684 0.699684i 0.264659 0.964342i \(-0.414741\pi\)
−0.964342 + 0.264659i \(0.914741\pi\)
\(822\) 0 0
\(823\) 918.417 1.11594 0.557969 0.829862i \(-0.311581\pi\)
0.557969 + 0.829862i \(0.311581\pi\)
\(824\) 0 0
\(825\) 51.7580i 0.0627370i
\(826\) 0 0
\(827\) −675.019 + 675.019i −0.816226 + 0.816226i −0.985559 0.169333i \(-0.945839\pi\)
0.169333 + 0.985559i \(0.445839\pi\)
\(828\) 0 0
\(829\) −312.622 + 312.622i −0.377107 + 0.377107i −0.870057 0.492950i \(-0.835918\pi\)
0.492950 + 0.870057i \(0.335918\pi\)
\(830\) 0 0
\(831\) 191.460i 0.230397i
\(832\) 0 0
\(833\) 1196.94 1.43691
\(834\) 0 0
\(835\) 441.193 + 441.193i 0.528375 + 0.528375i
\(836\) 0 0
\(837\) −308.689 308.689i −0.368804 0.368804i
\(838\) 0 0
\(839\) −612.003 −0.729443 −0.364722 0.931117i \(-0.618836\pi\)
−0.364722 + 0.931117i \(0.618836\pi\)
\(840\) 0 0
\(841\) 618.945i 0.735964i
\(842\) 0 0
\(843\) 148.939 148.939i 0.176677 0.176677i
\(844\) 0 0
\(845\) 16.4641 16.4641i 0.0194841 0.0194841i
\(846\) 0 0
\(847\) 630.837i 0.744789i
\(848\) 0 0
\(849\) 97.4909 0.114830
\(850\) 0 0
\(851\) 63.2953 + 63.2953i 0.0743775 + 0.0743775i
\(852\) 0 0
\(853\) −184.397 184.397i −0.216175 0.216175i 0.590709 0.806884i \(-0.298848\pi\)
−0.806884 + 0.590709i \(0.798848\pi\)
\(854\) 0 0
\(855\) 206.664 0.241713
\(856\) 0 0
\(857\) 602.644i 0.703202i −0.936150 0.351601i \(-0.885637\pi\)
0.936150 0.351601i \(-0.114363\pi\)
\(858\) 0 0
\(859\) −628.796 + 628.796i −0.732009 + 0.732009i −0.971018 0.239008i \(-0.923178\pi\)
0.239008 + 0.971018i \(0.423178\pi\)
\(860\) 0 0
\(861\) −35.1375 + 35.1375i −0.0408101 + 0.0408101i
\(862\) 0 0
\(863\) 1015.61i 1.17684i 0.808555 + 0.588420i \(0.200250\pi\)
−0.808555 + 0.588420i \(0.799750\pi\)
\(864\) 0 0
\(865\) 688.290 0.795711
\(866\) 0 0
\(867\) 180.496 + 180.496i 0.208184 + 0.208184i
\(868\) 0 0
\(869\) 1021.29 + 1021.29i 1.17524 + 1.17524i
\(870\) 0 0
\(871\) −61.9668 −0.0711445
\(872\) 0 0
\(873\) 722.753i 0.827896i
\(874\) 0 0
\(875\) −19.1667 + 19.1667i −0.0219048 + 0.0219048i
\(876\) 0 0
\(877\) 395.253 395.253i 0.450687 0.450687i −0.444895 0.895583i \(-0.646759\pi\)
0.895583 + 0.444895i \(0.146759\pi\)
\(878\) 0 0
\(879\) 128.177i 0.145822i
\(880\) 0 0
\(881\) 1316.34 1.49415 0.747073 0.664742i \(-0.231459\pi\)
0.747073 + 0.664742i \(0.231459\pi\)
\(882\) 0 0
\(883\) 433.169 + 433.169i 0.490565 + 0.490565i 0.908484 0.417919i \(-0.137241\pi\)
−0.417919 + 0.908484i \(0.637241\pi\)
\(884\) 0 0
\(885\) −10.9293 10.9293i −0.0123494 0.0123494i
\(886\) 0 0
\(887\) −1519.98 −1.71362 −0.856810 0.515633i \(-0.827557\pi\)
−0.856810 + 0.515633i \(0.827557\pi\)
\(888\) 0 0
\(889\) 108.315i 0.121840i
\(890\) 0 0
\(891\) 1014.58 1014.58i 1.13870 1.13870i
\(892\) 0 0
\(893\) −264.970 + 264.970i −0.296719 + 0.296719i
\(894\) 0 0
\(895\) 644.193i 0.719769i
\(896\) 0 0
\(897\) −90.1835 −0.100539
\(898\) 0 0
\(899\) 489.648 + 489.648i 0.544659 + 0.544659i
\(900\) 0 0
\(901\) −1695.63 1695.63i −1.88195 1.88195i
\(902\) 0 0
\(903\) 83.0524 0.0919739
\(904\) 0 0
\(905\) 555.633i 0.613959i
\(906\) 0 0
\(907\) −183.726 + 183.726i −0.202564 + 0.202564i −0.801098 0.598534i \(-0.795750\pi\)
0.598534 + 0.801098i \(0.295750\pi\)
\(908\) 0 0
\(909\) 792.617 792.617i 0.871966 0.871966i
\(910\) 0 0
\(911\) 194.108i 0.213071i −0.994309 0.106535i \(-0.966024\pi\)
0.994309 0.106535i \(-0.0339758\pi\)
\(912\) 0 0
\(913\) −1400.74 −1.53422
\(914\) 0 0
\(915\) −24.0764 24.0764i −0.0263130 0.0263130i
\(916\) 0 0
\(917\) 368.435 + 368.435i 0.401783 + 0.401783i
\(918\) 0 0
\(919\) 100.398 0.109247 0.0546237 0.998507i \(-0.482604\pi\)
0.0546237 + 0.998507i \(0.482604\pi\)
\(920\) 0 0
\(921\) 81.3144i 0.0882893i
\(922\) 0 0
\(923\) 77.6831 77.6831i 0.0841637 0.0841637i
\(924\) 0 0
\(925\) 23.4303 23.4303i 0.0253301 0.0253301i
\(926\) 0 0
\(927\) 281.465i 0.303630i
\(928\) 0 0
\(929\) 546.652 0.588430 0.294215 0.955739i \(-0.404942\pi\)
0.294215 + 0.955739i \(0.404942\pi\)
\(930\) 0 0
\(931\) 323.225 + 323.225i 0.347180 + 0.347180i
\(932\) 0 0
\(933\) 163.777 + 163.777i 0.175538 + 0.175538i
\(934\) 0 0
\(935\) −1211.81 −1.29605
\(936\) 0 0
\(937\) 1158.63i 1.23653i −0.785971 0.618264i \(-0.787836\pi\)
0.785971 0.618264i \(-0.212164\pi\)
\(938\) 0 0
\(939\) −123.331 + 123.331i −0.131343 + 0.131343i
\(940\) 0 0
\(941\) 218.412 218.412i 0.232106 0.232106i −0.581465 0.813571i \(-0.697520\pi\)
0.813571 + 0.581465i \(0.197520\pi\)
\(942\) 0 0
\(943\) 522.169i 0.553732i
\(944\) 0 0
\(945\) 50.9283 0.0538924
\(946\) 0 0
\(947\) −1323.25 1323.25i −1.39731 1.39731i −0.807665 0.589642i \(-0.799269\pi\)
−0.589642 0.807665i \(-0.700731\pi\)
\(948\) 0 0
\(949\) 205.732 + 205.732i 0.216788 + 0.216788i
\(950\) 0 0
\(951\) −229.961 −0.241810
\(952\) 0 0
\(953\) 1435.67i 1.50647i 0.657751 + 0.753235i \(0.271508\pi\)
−0.657751 + 0.753235i \(0.728492\pi\)
\(954\) 0 0
\(955\) −553.962 + 553.962i −0.580065 + 0.580065i
\(956\) 0 0
\(957\) 109.074 109.074i 0.113975 0.113975i
\(958\) 0 0
\(959\) 257.194i 0.268190i
\(960\) 0 0
\(961\) −1198.43 −1.24706
\(962\) 0 0
\(963\) 345.093 + 345.093i 0.358352 + 0.358352i
\(964\) 0 0
\(965\) 211.924 + 211.924i 0.219611 + 0.219611i
\(966\) 0 0
\(967\) 1507.98 1.55945 0.779723 0.626125i \(-0.215360\pi\)
0.779723 + 0.626125i \(0.215360\pi\)
\(968\) 0 0
\(969\) 155.999i 0.160990i
\(970\) 0 0
\(971\) 137.835 137.835i 0.141951 0.141951i −0.632560 0.774511i \(-0.717996\pi\)
0.774511 + 0.632560i \(0.217996\pi\)
\(972\) 0 0
\(973\) −271.478 + 271.478i −0.279011 + 0.279011i
\(974\) 0 0
\(975\) 33.3837i 0.0342397i
\(976\) 0 0
\(977\) −1819.85 −1.86269 −0.931346 0.364137i \(-0.881364\pi\)
−0.931346 + 0.364137i \(0.881364\pi\)
\(978\) 0 0
\(979\) −710.667 710.667i −0.725912 0.725912i
\(980\) 0 0
\(981\) 11.1638 + 11.1638i 0.0113800 + 0.0113800i
\(982\) 0 0
\(983\) 644.917 0.656070 0.328035 0.944666i \(-0.393614\pi\)
0.328035 + 0.944666i \(0.393614\pi\)
\(984\) 0 0
\(985\) 472.669i 0.479867i
\(986\) 0 0
\(987\) −32.1304 + 32.1304i −0.0325536 + 0.0325536i
\(988\) 0 0
\(989\) 617.110 617.110i 0.623973 0.623973i
\(990\) 0 0
\(991\) 1660.16i 1.67524i −0.546252 0.837621i \(-0.683946\pi\)
0.546252 0.837621i \(-0.316054\pi\)
\(992\) 0 0
\(993\) −184.215 −0.185514
\(994\) 0 0
\(995\) 485.987 + 485.987i 0.488429 + 0.488429i
\(996\) 0 0
\(997\) 1038.21 + 1038.21i 1.04134 + 1.04134i 0.999108 + 0.0422293i \(0.0134460\pi\)
0.0422293 + 0.999108i \(0.486554\pi\)
\(998\) 0 0
\(999\) −62.2573 −0.0623197
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 320.3.r.a.271.9 32
4.3 odd 2 80.3.r.a.11.1 32
8.3 odd 2 640.3.r.a.31.9 32
8.5 even 2 640.3.r.b.31.8 32
16.3 odd 4 inner 320.3.r.a.111.9 32
16.5 even 4 640.3.r.a.351.9 32
16.11 odd 4 640.3.r.b.351.8 32
16.13 even 4 80.3.r.a.51.1 yes 32
20.3 even 4 400.3.k.g.299.10 32
20.7 even 4 400.3.k.h.299.7 32
20.19 odd 2 400.3.r.f.251.16 32
80.13 odd 4 400.3.k.h.99.7 32
80.29 even 4 400.3.r.f.51.16 32
80.77 odd 4 400.3.k.g.99.10 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.3.r.a.11.1 32 4.3 odd 2
80.3.r.a.51.1 yes 32 16.13 even 4
320.3.r.a.111.9 32 16.3 odd 4 inner
320.3.r.a.271.9 32 1.1 even 1 trivial
400.3.k.g.99.10 32 80.77 odd 4
400.3.k.g.299.10 32 20.3 even 4
400.3.k.h.99.7 32 80.13 odd 4
400.3.k.h.299.7 32 20.7 even 4
400.3.r.f.51.16 32 80.29 even 4
400.3.r.f.251.16 32 20.19 odd 2
640.3.r.a.31.9 32 8.3 odd 2
640.3.r.a.351.9 32 16.5 even 4
640.3.r.b.31.8 32 8.5 even 2
640.3.r.b.351.8 32 16.11 odd 4