Properties

Label 2304.3.h.k.2177.7
Level $2304$
Weight $3$
Character 2304.2177
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(2177,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.2177");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (of order \(2\), degree \(1\), 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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2177.7
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2177
Dual form 2304.3.h.k.2177.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.87832 q^{5} +11.7980 q^{7} +O(q^{10})\) \(q+4.87832 q^{5} +11.7980 q^{7} -9.75663 q^{11} -22.6969i q^{13} +22.1988i q^{17} +17.7980i q^{19} +14.1421i q^{23} -1.20204 q^{25} +20.0061 q^{29} +39.7980 q^{31} +57.5542 q^{35} +2.40408i q^{37} +64.3395i q^{41} +3.19184i q^{43} -41.8549i q^{47} +90.1918 q^{49} +55.5043 q^{53} -47.5959 q^{55} +111.423 q^{59} -10.8082i q^{61} -110.723i q^{65} +18.2020i q^{67} +34.7983i q^{71} -87.5959 q^{73} -115.108 q^{77} +151.394 q^{79} +61.8112 q^{83} +108.293i q^{85} -72.9532i q^{89} -267.778i q^{91} +86.8241i q^{95} -87.3735 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} - 88 q^{25} + 240 q^{31} + 408 q^{49} - 224 q^{55} - 544 q^{73} + 976 q^{79} + 320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 4.87832 0.975663 0.487832 0.872938i \(-0.337788\pi\)
0.487832 + 0.872938i \(0.337788\pi\)
\(6\) 0 0
\(7\) 11.7980 1.68542 0.842711 0.538366i \(-0.180958\pi\)
0.842711 + 0.538366i \(0.180958\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −9.75663 −0.886966 −0.443483 0.896283i \(-0.646257\pi\)
−0.443483 + 0.896283i \(0.646257\pi\)
\(12\) 0 0
\(13\) − 22.6969i − 1.74592i −0.487793 0.872959i \(-0.662198\pi\)
0.487793 0.872959i \(-0.337802\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 22.1988i 1.30581i 0.757438 + 0.652907i \(0.226451\pi\)
−0.757438 + 0.652907i \(0.773549\pi\)
\(18\) 0 0
\(19\) 17.7980i 0.936735i 0.883534 + 0.468367i \(0.155158\pi\)
−0.883534 + 0.468367i \(0.844842\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 14.1421i 0.614875i 0.951568 + 0.307438i \(0.0994716\pi\)
−0.951568 + 0.307438i \(0.900528\pi\)
\(24\) 0 0
\(25\) −1.20204 −0.0480816
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 20.0061 0.689865 0.344932 0.938628i \(-0.387902\pi\)
0.344932 + 0.938628i \(0.387902\pi\)
\(30\) 0 0
\(31\) 39.7980 1.28381 0.641903 0.766786i \(-0.278145\pi\)
0.641903 + 0.766786i \(0.278145\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 57.5542 1.64440
\(36\) 0 0
\(37\) 2.40408i 0.0649752i 0.999472 + 0.0324876i \(0.0103429\pi\)
−0.999472 + 0.0324876i \(0.989657\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 64.3395i 1.56926i 0.619967 + 0.784628i \(0.287146\pi\)
−0.619967 + 0.784628i \(0.712854\pi\)
\(42\) 0 0
\(43\) 3.19184i 0.0742287i 0.999311 + 0.0371144i \(0.0118166\pi\)
−0.999311 + 0.0371144i \(0.988183\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 41.8549i − 0.890531i −0.895399 0.445265i \(-0.853109\pi\)
0.895399 0.445265i \(-0.146891\pi\)
\(48\) 0 0
\(49\) 90.1918 1.84065
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 55.5043 1.04725 0.523625 0.851949i \(-0.324579\pi\)
0.523625 + 0.851949i \(0.324579\pi\)
\(54\) 0 0
\(55\) −47.5959 −0.865380
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 111.423 1.88852 0.944260 0.329200i \(-0.106779\pi\)
0.944260 + 0.329200i \(0.106779\pi\)
\(60\) 0 0
\(61\) − 10.8082i − 0.177183i −0.996068 0.0885915i \(-0.971763\pi\)
0.996068 0.0885915i \(-0.0282366\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 110.723i − 1.70343i
\(66\) 0 0
\(67\) 18.2020i 0.271672i 0.990731 + 0.135836i \(0.0433721\pi\)
−0.990731 + 0.135836i \(0.956628\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 34.7983i 0.490117i 0.969508 + 0.245059i \(0.0788072\pi\)
−0.969508 + 0.245059i \(0.921193\pi\)
\(72\) 0 0
\(73\) −87.5959 −1.19994 −0.599972 0.800021i \(-0.704822\pi\)
−0.599972 + 0.800021i \(0.704822\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −115.108 −1.49491
\(78\) 0 0
\(79\) 151.394 1.91638 0.958189 0.286136i \(-0.0923708\pi\)
0.958189 + 0.286136i \(0.0923708\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 61.8112 0.744714 0.372357 0.928090i \(-0.378550\pi\)
0.372357 + 0.928090i \(0.378550\pi\)
\(84\) 0 0
\(85\) 108.293i 1.27403i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 72.9532i − 0.819699i −0.912153 0.409850i \(-0.865581\pi\)
0.912153 0.409850i \(-0.134419\pi\)
\(90\) 0 0
\(91\) − 267.778i − 2.94261i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 86.8241i 0.913937i
\(96\) 0 0
\(97\) −87.3735 −0.900757 −0.450379 0.892838i \(-0.648711\pi\)
−0.450379 + 0.892838i \(0.648711\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.63573 0.0161953 0.00809766 0.999967i \(-0.497422\pi\)
0.00809766 + 0.999967i \(0.497422\pi\)
\(102\) 0 0
\(103\) 38.9898 0.378542 0.189271 0.981925i \(-0.439388\pi\)
0.189271 + 0.981925i \(0.439388\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −205.303 −1.91872 −0.959362 0.282179i \(-0.908943\pi\)
−0.959362 + 0.282179i \(0.908943\pi\)
\(108\) 0 0
\(109\) − 33.3031i − 0.305533i −0.988262 0.152766i \(-0.951182\pi\)
0.988262 0.152766i \(-0.0488182\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 93.8951i − 0.830930i −0.909609 0.415465i \(-0.863619\pi\)
0.909609 0.415465i \(-0.136381\pi\)
\(114\) 0 0
\(115\) 68.9898i 0.599911i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 261.901i 2.20085i
\(120\) 0 0
\(121\) −25.8082 −0.213291
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −127.822 −1.02257
\(126\) 0 0
\(127\) 45.8184 0.360775 0.180387 0.983596i \(-0.442265\pi\)
0.180387 + 0.983596i \(0.442265\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 194.304 1.48324 0.741619 0.670821i \(-0.234058\pi\)
0.741619 + 0.670821i \(0.234058\pi\)
\(132\) 0 0
\(133\) 209.980i 1.57879i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 90.3668i 0.659612i 0.944049 + 0.329806i \(0.106983\pi\)
−0.944049 + 0.329806i \(0.893017\pi\)
\(138\) 0 0
\(139\) − 8.22245i − 0.0591543i −0.999563 0.0295772i \(-0.990584\pi\)
0.999563 0.0295772i \(-0.00941608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 221.446i 1.54857i
\(144\) 0 0
\(145\) 97.5959 0.673075
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −79.2457 −0.531851 −0.265925 0.963994i \(-0.585677\pi\)
−0.265925 + 0.963994i \(0.585677\pi\)
\(150\) 0 0
\(151\) −14.9898 −0.0992702 −0.0496351 0.998767i \(-0.515806\pi\)
−0.0496351 + 0.998767i \(0.515806\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 194.147 1.25256
\(156\) 0 0
\(157\) − 267.576i − 1.70430i −0.523295 0.852151i \(-0.675298\pi\)
0.523295 0.852151i \(-0.324702\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 166.848i 1.03633i
\(162\) 0 0
\(163\) − 261.171i − 1.60228i −0.598478 0.801139i \(-0.704228\pi\)
0.598478 0.801139i \(-0.295772\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 300.070i 1.79683i 0.439150 + 0.898414i \(0.355280\pi\)
−0.439150 + 0.898414i \(0.644720\pi\)
\(168\) 0 0
\(169\) −346.151 −2.04823
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 31.2909 0.180872 0.0904362 0.995902i \(-0.471174\pi\)
0.0904362 + 0.995902i \(0.471174\pi\)
\(174\) 0 0
\(175\) −14.1816 −0.0810379
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 66.4825 0.371410 0.185705 0.982606i \(-0.440543\pi\)
0.185705 + 0.982606i \(0.440543\pi\)
\(180\) 0 0
\(181\) − 235.464i − 1.30091i −0.759546 0.650454i \(-0.774579\pi\)
0.759546 0.650454i \(-0.225421\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 11.7279i 0.0633939i
\(186\) 0 0
\(187\) − 216.586i − 1.15821i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 140.535i − 0.735787i −0.929868 0.367893i \(-0.880079\pi\)
0.929868 0.367893i \(-0.119921\pi\)
\(192\) 0 0
\(193\) 293.151 1.51892 0.759459 0.650556i \(-0.225464\pi\)
0.759459 + 0.650556i \(0.225464\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 60.4324 0.306763 0.153382 0.988167i \(-0.450984\pi\)
0.153382 + 0.988167i \(0.450984\pi\)
\(198\) 0 0
\(199\) −143.757 −0.722398 −0.361199 0.932489i \(-0.617632\pi\)
−0.361199 + 0.932489i \(0.617632\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 236.031 1.16271
\(204\) 0 0
\(205\) 313.868i 1.53107i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 173.648i − 0.830852i
\(210\) 0 0
\(211\) − 255.778i − 1.21222i −0.795382 0.606108i \(-0.792730\pi\)
0.795382 0.606108i \(-0.207270\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 15.5708i 0.0724222i
\(216\) 0 0
\(217\) 469.535 2.16375
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 503.845 2.27984
\(222\) 0 0
\(223\) 248.969 1.11645 0.558227 0.829688i \(-0.311482\pi\)
0.558227 + 0.829688i \(0.311482\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 277.600 1.22291 0.611454 0.791280i \(-0.290585\pi\)
0.611454 + 0.791280i \(0.290585\pi\)
\(228\) 0 0
\(229\) − 193.666i − 0.845704i −0.906199 0.422852i \(-0.861029\pi\)
0.906199 0.422852i \(-0.138971\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 150.435i 0.645643i 0.946460 + 0.322821i \(0.104631\pi\)
−0.946460 + 0.322821i \(0.895369\pi\)
\(234\) 0 0
\(235\) − 204.182i − 0.868858i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 140.593i 0.588255i 0.955766 + 0.294128i \(0.0950291\pi\)
−0.955766 + 0.294128i \(0.904971\pi\)
\(240\) 0 0
\(241\) −228.182 −0.946812 −0.473406 0.880844i \(-0.656976\pi\)
−0.473406 + 0.880844i \(0.656976\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 439.984 1.79585
\(246\) 0 0
\(247\) 403.959 1.63546
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 214.546 0.854766 0.427383 0.904071i \(-0.359436\pi\)
0.427383 + 0.904071i \(0.359436\pi\)
\(252\) 0 0
\(253\) − 137.980i − 0.545374i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 382.652i 1.48892i 0.667669 + 0.744458i \(0.267292\pi\)
−0.667669 + 0.744458i \(0.732708\pi\)
\(258\) 0 0
\(259\) 28.3633i 0.109511i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 164.563i − 0.625713i −0.949800 0.312856i \(-0.898714\pi\)
0.949800 0.312856i \(-0.101286\pi\)
\(264\) 0 0
\(265\) 270.767 1.02176
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −239.480 −0.890262 −0.445131 0.895465i \(-0.646843\pi\)
−0.445131 + 0.895465i \(0.646843\pi\)
\(270\) 0 0
\(271\) 153.778 0.567445 0.283722 0.958906i \(-0.408431\pi\)
0.283722 + 0.958906i \(0.408431\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 11.7279 0.0426468
\(276\) 0 0
\(277\) 135.242i 0.488238i 0.969745 + 0.244119i \(0.0784987\pi\)
−0.969745 + 0.244119i \(0.921501\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 63.6396i 0.226475i 0.993568 + 0.113238i \(0.0361222\pi\)
−0.993568 + 0.113238i \(0.963878\pi\)
\(282\) 0 0
\(283\) 333.576i 1.17871i 0.807873 + 0.589356i \(0.200618\pi\)
−0.807873 + 0.589356i \(0.799382\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 759.075i 2.64486i
\(288\) 0 0
\(289\) −203.788 −0.705148
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −278.822 −0.951610 −0.475805 0.879551i \(-0.657843\pi\)
−0.475805 + 0.879551i \(0.657843\pi\)
\(294\) 0 0
\(295\) 543.555 1.84256
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 320.983 1.07352
\(300\) 0 0
\(301\) 37.6571i 0.125107i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 52.7256i − 0.172871i
\(306\) 0 0
\(307\) 99.7367i 0.324875i 0.986719 + 0.162438i \(0.0519356\pi\)
−0.986719 + 0.162438i \(0.948064\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 313.927i − 1.00941i −0.863292 0.504705i \(-0.831601\pi\)
0.863292 0.504705i \(-0.168399\pi\)
\(312\) 0 0
\(313\) 52.4245 0.167490 0.0837452 0.996487i \(-0.473312\pi\)
0.0837452 + 0.996487i \(0.473312\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −68.8888 −0.217315 −0.108657 0.994079i \(-0.534655\pi\)
−0.108657 + 0.994079i \(0.534655\pi\)
\(318\) 0 0
\(319\) −195.192 −0.611887
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −395.094 −1.22320
\(324\) 0 0
\(325\) 27.2827i 0.0839466i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 493.803i − 1.50092i
\(330\) 0 0
\(331\) 114.202i 0.345021i 0.985008 + 0.172511i \(0.0551879\pi\)
−0.985008 + 0.172511i \(0.944812\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 88.7953i 0.265061i
\(336\) 0 0
\(337\) −203.192 −0.602943 −0.301472 0.953475i \(-0.597478\pi\)
−0.301472 + 0.953475i \(0.597478\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −388.294 −1.13869
\(342\) 0 0
\(343\) 485.980 1.41685
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −79.5524 −0.229258 −0.114629 0.993408i \(-0.536568\pi\)
−0.114629 + 0.993408i \(0.536568\pi\)
\(348\) 0 0
\(349\) 479.898i 1.37507i 0.726153 + 0.687533i \(0.241306\pi\)
−0.726153 + 0.687533i \(0.758694\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 132.922i − 0.376549i −0.982116 0.188274i \(-0.939711\pi\)
0.982116 0.188274i \(-0.0602894\pi\)
\(354\) 0 0
\(355\) 169.757i 0.478189i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 660.308i 1.83930i 0.392742 + 0.919649i \(0.371527\pi\)
−0.392742 + 0.919649i \(0.628473\pi\)
\(360\) 0 0
\(361\) 44.2327 0.122528
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −427.320 −1.17074
\(366\) 0 0
\(367\) −281.373 −0.766685 −0.383343 0.923606i \(-0.625227\pi\)
−0.383343 + 0.923606i \(0.625227\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 654.837 1.76506
\(372\) 0 0
\(373\) − 617.959i − 1.65673i −0.560191 0.828364i \(-0.689272\pi\)
0.560191 0.828364i \(-0.310728\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 454.077i − 1.20445i
\(378\) 0 0
\(379\) 695.292i 1.83454i 0.398262 + 0.917272i \(0.369613\pi\)
−0.398262 + 0.917272i \(0.630387\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 315.269i − 0.823156i −0.911375 0.411578i \(-0.864978\pi\)
0.911375 0.411578i \(-0.135022\pi\)
\(384\) 0 0
\(385\) −561.535 −1.45853
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −539.364 −1.38654 −0.693270 0.720677i \(-0.743831\pi\)
−0.693270 + 0.720677i \(0.743831\pi\)
\(390\) 0 0
\(391\) −313.939 −0.802912
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 738.547 1.86974
\(396\) 0 0
\(397\) 490.404i 1.23527i 0.786463 + 0.617637i \(0.211910\pi\)
−0.786463 + 0.617637i \(0.788090\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 460.891i 1.14935i 0.818380 + 0.574677i \(0.194872\pi\)
−0.818380 + 0.574677i \(0.805128\pi\)
\(402\) 0 0
\(403\) − 903.292i − 2.24142i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 23.4557i − 0.0576308i
\(408\) 0 0
\(409\) −535.292 −1.30878 −0.654391 0.756156i \(-0.727075\pi\)
−0.654391 + 0.756156i \(0.727075\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1314.56 3.18296
\(414\) 0 0
\(415\) 301.535 0.726590
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −405.993 −0.968958 −0.484479 0.874803i \(-0.660991\pi\)
−0.484479 + 0.874803i \(0.660991\pi\)
\(420\) 0 0
\(421\) − 70.6969i − 0.167926i −0.996469 0.0839631i \(-0.973242\pi\)
0.996469 0.0839631i \(-0.0267578\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 26.6839i − 0.0627856i
\(426\) 0 0
\(427\) − 127.514i − 0.298628i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 334.239i − 0.775497i −0.921765 0.387749i \(-0.873253\pi\)
0.921765 0.387749i \(-0.126747\pi\)
\(432\) 0 0
\(433\) 88.3429 0.204025 0.102013 0.994783i \(-0.467472\pi\)
0.102013 + 0.994783i \(0.467472\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −251.701 −0.575975
\(438\) 0 0
\(439\) −231.353 −0.527000 −0.263500 0.964659i \(-0.584877\pi\)
−0.263500 + 0.964659i \(0.584877\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −715.091 −1.61420 −0.807101 0.590414i \(-0.798965\pi\)
−0.807101 + 0.590414i \(0.798965\pi\)
\(444\) 0 0
\(445\) − 355.889i − 0.799750i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 576.455i − 1.28386i −0.766761 0.641932i \(-0.778133\pi\)
0.766761 0.641932i \(-0.221867\pi\)
\(450\) 0 0
\(451\) − 627.737i − 1.39188i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 1306.30i − 2.87100i
\(456\) 0 0
\(457\) 168.990 0.369781 0.184890 0.982759i \(-0.440807\pi\)
0.184890 + 0.982759i \(0.440807\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −772.653 −1.67604 −0.838019 0.545641i \(-0.816286\pi\)
−0.838019 + 0.545641i \(0.816286\pi\)
\(462\) 0 0
\(463\) −461.818 −0.997448 −0.498724 0.866761i \(-0.666198\pi\)
−0.498724 + 0.866761i \(0.666198\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −516.016 −1.10496 −0.552480 0.833526i \(-0.686318\pi\)
−0.552480 + 0.833526i \(0.686318\pi\)
\(468\) 0 0
\(469\) 214.747i 0.457883i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 31.1416i − 0.0658384i
\(474\) 0 0
\(475\) − 21.3939i − 0.0450397i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 175.706i − 0.366818i −0.983037 0.183409i \(-0.941287\pi\)
0.983037 0.183409i \(-0.0587133\pi\)
\(480\) 0 0
\(481\) 54.5653 0.113441
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −426.235 −0.878836
\(486\) 0 0
\(487\) 694.100 1.42526 0.712628 0.701542i \(-0.247505\pi\)
0.712628 + 0.701542i \(0.247505\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 512.173 1.04312 0.521561 0.853214i \(-0.325350\pi\)
0.521561 + 0.853214i \(0.325350\pi\)
\(492\) 0 0
\(493\) 444.111i 0.900834i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 410.549i 0.826054i
\(498\) 0 0
\(499\) 132.849i 0.266230i 0.991101 + 0.133115i \(0.0424980\pi\)
−0.991101 + 0.133115i \(0.957502\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 243.986i 0.485063i 0.970144 + 0.242531i \(0.0779777\pi\)
−0.970144 + 0.242531i \(0.922022\pi\)
\(504\) 0 0
\(505\) 7.97959 0.0158012
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 396.530 0.779038 0.389519 0.921018i \(-0.372641\pi\)
0.389519 + 0.921018i \(0.372641\pi\)
\(510\) 0 0
\(511\) −1033.45 −2.02241
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 190.205 0.369329
\(516\) 0 0
\(517\) 408.363i 0.789871i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 76.4527i 0.146742i 0.997305 + 0.0733711i \(0.0233757\pi\)
−0.997305 + 0.0733711i \(0.976624\pi\)
\(522\) 0 0
\(523\) − 298.524i − 0.570793i −0.958410 0.285396i \(-0.907875\pi\)
0.958410 0.285396i \(-0.0921252\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 883.468i 1.67641i
\(528\) 0 0
\(529\) 329.000 0.621928
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1460.31 2.73979
\(534\) 0 0
\(535\) −1001.53 −1.87203
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −879.968 −1.63259
\(540\) 0 0
\(541\) − 566.070i − 1.04634i −0.852228 0.523170i \(-0.824749\pi\)
0.852228 0.523170i \(-0.175251\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 162.463i − 0.298097i
\(546\) 0 0
\(547\) 500.727i 0.915405i 0.889105 + 0.457702i \(0.151328\pi\)
−0.889105 + 0.457702i \(0.848672\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 356.067i 0.646220i
\(552\) 0 0
\(553\) 1786.14 3.22991
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −224.568 −0.403174 −0.201587 0.979471i \(-0.564610\pi\)
−0.201587 + 0.979471i \(0.564610\pi\)
\(558\) 0 0
\(559\) 72.4449 0.129597
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −708.092 −1.25771 −0.628856 0.777521i \(-0.716477\pi\)
−0.628856 + 0.777521i \(0.716477\pi\)
\(564\) 0 0
\(565\) − 458.050i − 0.810708i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 627.453i 1.10273i 0.834264 + 0.551365i \(0.185893\pi\)
−0.834264 + 0.551365i \(0.814107\pi\)
\(570\) 0 0
\(571\) 25.2531i 0.0442260i 0.999755 + 0.0221130i \(0.00703937\pi\)
−0.999755 + 0.0221130i \(0.992961\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 16.9994i − 0.0295642i
\(576\) 0 0
\(577\) 158.000 0.273830 0.136915 0.990583i \(-0.456281\pi\)
0.136915 + 0.990583i \(0.456281\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 729.246 1.25516
\(582\) 0 0
\(583\) −541.535 −0.928876
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1060.63 −1.80687 −0.903434 0.428728i \(-0.858962\pi\)
−0.903434 + 0.428728i \(0.858962\pi\)
\(588\) 0 0
\(589\) 708.322i 1.20258i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1051.50i 1.77319i 0.462545 + 0.886596i \(0.346936\pi\)
−0.462545 + 0.886596i \(0.653064\pi\)
\(594\) 0 0
\(595\) 1277.63i 2.14729i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 52.0835i − 0.0869507i −0.999054 0.0434754i \(-0.986157\pi\)
0.999054 0.0434754i \(-0.0138430\pi\)
\(600\) 0 0
\(601\) −66.0000 −0.109817 −0.0549085 0.998491i \(-0.517487\pi\)
−0.0549085 + 0.998491i \(0.517487\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −125.900 −0.208100
\(606\) 0 0
\(607\) −210.627 −0.346996 −0.173498 0.984834i \(-0.555507\pi\)
−0.173498 + 0.984834i \(0.555507\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −949.979 −1.55479
\(612\) 0 0
\(613\) − 320.706i − 0.523175i −0.965180 0.261587i \(-0.915754\pi\)
0.965180 0.261587i \(-0.0842460\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 233.316i − 0.378146i −0.981963 0.189073i \(-0.939452\pi\)
0.981963 0.189073i \(-0.0605484\pi\)
\(618\) 0 0
\(619\) − 206.020i − 0.332828i −0.986056 0.166414i \(-0.946781\pi\)
0.986056 0.166414i \(-0.0532188\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 860.699i − 1.38154i
\(624\) 0 0
\(625\) −593.504 −0.949607
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −53.3678 −0.0848455
\(630\) 0 0
\(631\) 42.5857 0.0674892 0.0337446 0.999430i \(-0.489257\pi\)
0.0337446 + 0.999430i \(0.489257\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 223.516 0.351994
\(636\) 0 0
\(637\) − 2047.08i − 3.21362i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 160.021i − 0.249643i −0.992179 0.124821i \(-0.960164\pi\)
0.992179 0.124821i \(-0.0398358\pi\)
\(642\) 0 0
\(643\) 997.980i 1.55207i 0.630691 + 0.776034i \(0.282771\pi\)
−0.630691 + 0.776034i \(0.717229\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 552.628i − 0.854140i −0.904219 0.427070i \(-0.859546\pi\)
0.904219 0.427070i \(-0.140454\pi\)
\(648\) 0 0
\(649\) −1087.11 −1.67505
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 374.272 0.573158 0.286579 0.958057i \(-0.407482\pi\)
0.286579 + 0.958057i \(0.407482\pi\)
\(654\) 0 0
\(655\) 947.878 1.44714
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 251.759 0.382032 0.191016 0.981587i \(-0.438822\pi\)
0.191016 + 0.981587i \(0.438822\pi\)
\(660\) 0 0
\(661\) − 339.535i − 0.513668i −0.966456 0.256834i \(-0.917321\pi\)
0.966456 0.256834i \(-0.0826794\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1024.35i 1.54037i
\(666\) 0 0
\(667\) 282.929i 0.424181i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 105.451i 0.157155i
\(672\) 0 0
\(673\) 1049.03 1.55873 0.779367 0.626567i \(-0.215541\pi\)
0.779367 + 0.626567i \(0.215541\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −310.390 −0.458479 −0.229240 0.973370i \(-0.573624\pi\)
−0.229240 + 0.973370i \(0.573624\pi\)
\(678\) 0 0
\(679\) −1030.83 −1.51816
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −746.233 −1.09258 −0.546291 0.837596i \(-0.683961\pi\)
−0.546291 + 0.837596i \(0.683961\pi\)
\(684\) 0 0
\(685\) 440.838i 0.643559i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 1259.78i − 1.82841i
\(690\) 0 0
\(691\) − 776.241i − 1.12336i −0.827355 0.561679i \(-0.810155\pi\)
0.827355 0.561679i \(-0.189845\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 40.1117i − 0.0577147i
\(696\) 0 0
\(697\) −1428.26 −2.04916
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 801.223 1.14297 0.571486 0.820612i \(-0.306367\pi\)
0.571486 + 0.820612i \(0.306367\pi\)
\(702\) 0 0
\(703\) −42.7878 −0.0608645
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.2982 0.0272959
\(708\) 0 0
\(709\) 1198.25i 1.69006i 0.534719 + 0.845030i \(0.320417\pi\)
−0.534719 + 0.845030i \(0.679583\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 562.828i 0.789380i
\(714\) 0 0
\(715\) 1080.28i 1.51088i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 1206.22i − 1.67764i −0.544409 0.838820i \(-0.683246\pi\)
0.544409 0.838820i \(-0.316754\pi\)
\(720\) 0 0
\(721\) 460.000 0.638003
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −24.0481 −0.0331698
\(726\) 0 0
\(727\) −503.394 −0.692426 −0.346213 0.938156i \(-0.612533\pi\)
−0.346213 + 0.938156i \(0.612533\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −70.8550 −0.0969289
\(732\) 0 0
\(733\) 380.736i 0.519421i 0.965687 + 0.259711i \(0.0836272\pi\)
−0.965687 + 0.259711i \(0.916373\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 177.591i − 0.240964i
\(738\) 0 0
\(739\) − 717.775i − 0.971279i −0.874159 0.485640i \(-0.838587\pi\)
0.874159 0.485640i \(-0.161413\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 239.675i − 0.322577i −0.986907 0.161288i \(-0.948435\pi\)
0.986907 0.161288i \(-0.0515649\pi\)
\(744\) 0 0
\(745\) −386.586 −0.518907
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2422.16 −3.23386
\(750\) 0 0
\(751\) −481.010 −0.640493 −0.320246 0.947334i \(-0.603766\pi\)
−0.320246 + 0.947334i \(0.603766\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −73.1249 −0.0968542
\(756\) 0 0
\(757\) 338.656i 0.447366i 0.974662 + 0.223683i \(0.0718080\pi\)
−0.974662 + 0.223683i \(0.928192\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 525.945i − 0.691123i −0.938396 0.345561i \(-0.887688\pi\)
0.938396 0.345561i \(-0.112312\pi\)
\(762\) 0 0
\(763\) − 392.908i − 0.514952i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 2528.95i − 3.29720i
\(768\) 0 0
\(769\) 248.506 0.323155 0.161577 0.986860i \(-0.448342\pi\)
0.161577 + 0.986860i \(0.448342\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −402.701 −0.520958 −0.260479 0.965479i \(-0.583881\pi\)
−0.260479 + 0.965479i \(0.583881\pi\)
\(774\) 0 0
\(775\) −47.8388 −0.0617275
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1145.11 −1.46998
\(780\) 0 0
\(781\) − 339.514i − 0.434717i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1305.32i − 1.66283i
\(786\) 0 0
\(787\) 1258.97i 1.59971i 0.600195 + 0.799853i \(0.295090\pi\)
−0.600195 + 0.799853i \(0.704910\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1107.77i − 1.40047i
\(792\) 0 0
\(793\) −245.312 −0.309347
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 878.734 1.10255 0.551276 0.834323i \(-0.314141\pi\)
0.551276 + 0.834323i \(0.314141\pi\)
\(798\) 0 0
\(799\) 929.131 1.16287
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 854.641 1.06431
\(804\) 0 0
\(805\) 813.939i 1.01110i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 1199.39i − 1.48256i −0.671195 0.741281i \(-0.734218\pi\)
0.671195 0.741281i \(-0.265782\pi\)
\(810\) 0 0
\(811\) − 30.2837i − 0.0373412i −0.999826 0.0186706i \(-0.994057\pi\)
0.999826 0.0186706i \(-0.00594338\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1274.08i − 1.56328i
\(816\) 0 0
\(817\) −56.8082 −0.0695326
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −144.441 −0.175933 −0.0879665 0.996123i \(-0.528037\pi\)
−0.0879665 + 0.996123i \(0.528037\pi\)
\(822\) 0 0
\(823\) −795.839 −0.966997 −0.483499 0.875345i \(-0.660634\pi\)
−0.483499 + 0.875345i \(0.660634\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 567.541 0.686265 0.343133 0.939287i \(-0.388512\pi\)
0.343133 + 0.939287i \(0.388512\pi\)
\(828\) 0 0
\(829\) − 1188.54i − 1.43370i −0.697228 0.716849i \(-0.745584\pi\)
0.697228 0.716849i \(-0.254416\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2002.15i 2.40354i
\(834\) 0 0
\(835\) 1463.84i 1.75310i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 598.256i 0.713058i 0.934284 + 0.356529i \(0.116040\pi\)
−0.934284 + 0.356529i \(0.883960\pi\)
\(840\) 0 0
\(841\) −440.757 −0.524087
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1688.63 −1.99838
\(846\) 0 0
\(847\) −304.484 −0.359485
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −33.9989 −0.0399517
\(852\) 0 0
\(853\) − 337.192i − 0.395301i −0.980273 0.197651i \(-0.936669\pi\)
0.980273 0.197651i \(-0.0633311\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 913.497i 1.06592i 0.846139 + 0.532962i \(0.178921\pi\)
−0.846139 + 0.532962i \(0.821079\pi\)
\(858\) 0 0
\(859\) − 748.645i − 0.871531i −0.900060 0.435765i \(-0.856478\pi\)
0.900060 0.435765i \(-0.143522\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 460.632i 0.533757i 0.963730 + 0.266879i \(0.0859923\pi\)
−0.963730 + 0.266879i \(0.914008\pi\)
\(864\) 0 0
\(865\) 152.647 0.176470
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1477.09 −1.69976
\(870\) 0 0
\(871\) 413.131 0.474318
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1508.04 −1.72347
\(876\) 0 0
\(877\) 1066.32i 1.21588i 0.793985 + 0.607938i \(0.208003\pi\)
−0.793985 + 0.607938i \(0.791997\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 586.899i − 0.666173i −0.942896 0.333087i \(-0.891910\pi\)
0.942896 0.333087i \(-0.108090\pi\)
\(882\) 0 0
\(883\) 1242.95i 1.40764i 0.710378 + 0.703820i \(0.248524\pi\)
−0.710378 + 0.703820i \(0.751476\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 37.8259i 0.0426447i 0.999773 + 0.0213224i \(0.00678763\pi\)
−0.999773 + 0.0213224i \(0.993212\pi\)
\(888\) 0 0
\(889\) 540.563 0.608058
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 744.933 0.834191
\(894\) 0 0
\(895\) 324.322 0.362371
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 796.201 0.885652
\(900\) 0 0
\(901\) 1232.13i 1.36751i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 1148.67i − 1.26925i
\(906\) 0 0
\(907\) 763.837i 0.842157i 0.907024 + 0.421079i \(0.138348\pi\)
−0.907024 + 0.421079i \(0.861652\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 378.008i 0.414937i 0.978242 + 0.207469i \(0.0665225\pi\)
−0.978242 + 0.207469i \(0.933478\pi\)
\(912\) 0 0
\(913\) −603.069 −0.660536
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2292.39 2.49988
\(918\) 0 0
\(919\) 206.141 0.224310 0.112155 0.993691i \(-0.464225\pi\)
0.112155 + 0.993691i \(0.464225\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 789.815 0.855704
\(924\) 0 0
\(925\) − 2.88981i − 0.00312411i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1793.22i − 1.93027i −0.261752 0.965135i \(-0.584300\pi\)
0.261752 0.965135i \(-0.415700\pi\)
\(930\) 0 0
\(931\) 1605.23i 1.72420i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 1056.57i − 1.13002i
\(936\) 0 0
\(937\) −322.767 −0.344469 −0.172234 0.985056i \(-0.555099\pi\)
−0.172234 + 0.985056i \(0.555099\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −805.965 −0.856499 −0.428249 0.903661i \(-0.640870\pi\)
−0.428249 + 0.903661i \(0.640870\pi\)
\(942\) 0 0
\(943\) −909.898 −0.964897
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1197.31 −1.26432 −0.632158 0.774839i \(-0.717831\pi\)
−0.632158 + 0.774839i \(0.717831\pi\)
\(948\) 0 0
\(949\) 1988.16i 2.09500i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1109.71i − 1.16444i −0.813030 0.582222i \(-0.802184\pi\)
0.813030 0.582222i \(-0.197816\pi\)
\(954\) 0 0
\(955\) − 685.576i − 0.717880i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1066.14i 1.11172i
\(960\) 0 0
\(961\) 622.878 0.648156
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1430.08 1.48195
\(966\) 0 0
\(967\) −1075.72 −1.11243 −0.556213 0.831040i \(-0.687746\pi\)
−0.556213 + 0.831040i \(0.687746\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −958.163 −0.986779 −0.493390 0.869808i \(-0.664242\pi\)
−0.493390 + 0.869808i \(0.664242\pi\)
\(972\) 0 0
\(973\) − 97.0081i − 0.0997000i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 116.792i − 0.119542i −0.998212 0.0597709i \(-0.980963\pi\)
0.998212 0.0597709i \(-0.0190370\pi\)
\(978\) 0 0
\(979\) 711.778i 0.727046i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 209.817i 0.213446i 0.994289 + 0.106723i \(0.0340358\pi\)
−0.994289 + 0.106723i \(0.965964\pi\)
\(984\) 0 0
\(985\) 294.808 0.299298
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −45.1394 −0.0456414
\(990\) 0 0
\(991\) 359.798 0.363066 0.181533 0.983385i \(-0.441894\pi\)
0.181533 + 0.983385i \(0.441894\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −701.293 −0.704817
\(996\) 0 0
\(997\) − 1628.06i − 1.63296i −0.577377 0.816478i \(-0.695924\pi\)
0.577377 0.816478i \(-0.304076\pi\)
\(998\) 0 0
\(999\) 0 0
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 2304.3.h.k.2177.7 8
3.2 odd 2 inner 2304.3.h.k.2177.1 8
4.3 odd 2 2304.3.h.i.2177.7 8
8.3 odd 2 2304.3.h.i.2177.2 8
8.5 even 2 inner 2304.3.h.k.2177.2 8
12.11 even 2 2304.3.h.i.2177.1 8
16.3 odd 4 1152.3.e.h.1025.1 yes 4
16.5 even 4 1152.3.e.b.1025.4 yes 4
16.11 odd 4 1152.3.e.f.1025.4 yes 4
16.13 even 4 1152.3.e.d.1025.1 yes 4
24.5 odd 2 inner 2304.3.h.k.2177.8 8
24.11 even 2 2304.3.h.i.2177.8 8
48.5 odd 4 1152.3.e.b.1025.1 4
48.11 even 4 1152.3.e.f.1025.1 yes 4
48.29 odd 4 1152.3.e.d.1025.4 yes 4
48.35 even 4 1152.3.e.h.1025.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.e.b.1025.1 4 48.5 odd 4
1152.3.e.b.1025.4 yes 4 16.5 even 4
1152.3.e.d.1025.1 yes 4 16.13 even 4
1152.3.e.d.1025.4 yes 4 48.29 odd 4
1152.3.e.f.1025.1 yes 4 48.11 even 4
1152.3.e.f.1025.4 yes 4 16.11 odd 4
1152.3.e.h.1025.1 yes 4 16.3 odd 4
1152.3.e.h.1025.4 yes 4 48.35 even 4
2304.3.h.i.2177.1 8 12.11 even 2
2304.3.h.i.2177.2 8 8.3 odd 2
2304.3.h.i.2177.7 8 4.3 odd 2
2304.3.h.i.2177.8 8 24.11 even 2
2304.3.h.k.2177.1 8 3.2 odd 2 inner
2304.3.h.k.2177.2 8 8.5 even 2 inner
2304.3.h.k.2177.7 8 1.1 even 1 trivial
2304.3.h.k.2177.8 8 24.5 odd 2 inner