Properties

Label 2304.3.h.c.2177.1
Level $2304$
Weight $3$
Character 2304.2177
Analytic conductor $62.779$
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] = [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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2177.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2177
Dual form 2304.3.h.c.2177.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.24264 q^{5} -4.00000 q^{7} +O(q^{10})\) \(q-4.24264 q^{5} -4.00000 q^{7} -16.9706 q^{11} -8.00000i q^{13} +12.7279i q^{17} -16.0000i q^{19} +16.9706i q^{23} -7.00000 q^{25} -4.24264 q^{29} -44.0000 q^{31} +16.9706 q^{35} -34.0000i q^{37} +46.6690i q^{41} +40.0000i q^{43} -84.8528i q^{47} -33.0000 q^{49} +38.1838 q^{53} +72.0000 q^{55} -33.9411 q^{59} -50.0000i q^{61} +33.9411i q^{65} +8.00000i q^{67} +50.9117i q^{71} +16.0000 q^{73} +67.8823 q^{77} +76.0000 q^{79} +118.794 q^{83} -54.0000i q^{85} +12.7279i q^{89} +32.0000i q^{91} +67.8823i q^{95} +176.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{7} - 28 q^{25} - 176 q^{31} - 132 q^{49} + 288 q^{55} + 64 q^{73} + 304 q^{79} + 704 q^{97}+O(q^{100}) \) 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
\(5\) −4.24264 −0.848528 −0.424264 0.905539i \(-0.639467\pi\)
−0.424264 + 0.905539i \(0.639467\pi\)
\(6\) 0 0
\(7\) −4.00000 −0.571429 −0.285714 0.958315i \(-0.592231\pi\)
−0.285714 + 0.958315i \(0.592231\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −16.9706 −1.54278 −0.771389 0.636364i \(-0.780438\pi\)
−0.771389 + 0.636364i \(0.780438\pi\)
\(12\) 0 0
\(13\) − 8.00000i − 0.615385i −0.951486 0.307692i \(-0.900443\pi\)
0.951486 0.307692i \(-0.0995567\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 12.7279i 0.748701i 0.927287 + 0.374351i \(0.122134\pi\)
−0.927287 + 0.374351i \(0.877866\pi\)
\(18\) 0 0
\(19\) − 16.0000i − 0.842105i −0.907036 0.421053i \(-0.861661\pi\)
0.907036 0.421053i \(-0.138339\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 16.9706i 0.737851i 0.929459 + 0.368925i \(0.120274\pi\)
−0.929459 + 0.368925i \(0.879726\pi\)
\(24\) 0 0
\(25\) −7.00000 −0.280000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.24264 −0.146298 −0.0731490 0.997321i \(-0.523305\pi\)
−0.0731490 + 0.997321i \(0.523305\pi\)
\(30\) 0 0
\(31\) −44.0000 −1.41935 −0.709677 0.704527i \(-0.751159\pi\)
−0.709677 + 0.704527i \(0.751159\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 16.9706 0.484873
\(36\) 0 0
\(37\) − 34.0000i − 0.918919i −0.888199 0.459459i \(-0.848043\pi\)
0.888199 0.459459i \(-0.151957\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 46.6690i 1.13827i 0.822244 + 0.569135i \(0.192722\pi\)
−0.822244 + 0.569135i \(0.807278\pi\)
\(42\) 0 0
\(43\) 40.0000i 0.930233i 0.885250 + 0.465116i \(0.153987\pi\)
−0.885250 + 0.465116i \(0.846013\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 84.8528i − 1.80538i −0.430293 0.902690i \(-0.641590\pi\)
0.430293 0.902690i \(-0.358410\pi\)
\(48\) 0 0
\(49\) −33.0000 −0.673469
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 38.1838 0.720448 0.360224 0.932866i \(-0.382700\pi\)
0.360224 + 0.932866i \(0.382700\pi\)
\(54\) 0 0
\(55\) 72.0000 1.30909
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −33.9411 −0.575273 −0.287637 0.957740i \(-0.592870\pi\)
−0.287637 + 0.957740i \(0.592870\pi\)
\(60\) 0 0
\(61\) − 50.0000i − 0.819672i −0.912159 0.409836i \(-0.865586\pi\)
0.912159 0.409836i \(-0.134414\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 33.9411i 0.522171i
\(66\) 0 0
\(67\) 8.00000i 0.119403i 0.998216 + 0.0597015i \(0.0190149\pi\)
−0.998216 + 0.0597015i \(0.980985\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 50.9117i 0.717066i 0.933517 + 0.358533i \(0.116723\pi\)
−0.933517 + 0.358533i \(0.883277\pi\)
\(72\) 0 0
\(73\) 16.0000 0.219178 0.109589 0.993977i \(-0.465047\pi\)
0.109589 + 0.993977i \(0.465047\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 67.8823 0.881588
\(78\) 0 0
\(79\) 76.0000 0.962025 0.481013 0.876714i \(-0.340269\pi\)
0.481013 + 0.876714i \(0.340269\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 118.794 1.43125 0.715626 0.698484i \(-0.246141\pi\)
0.715626 + 0.698484i \(0.246141\pi\)
\(84\) 0 0
\(85\) − 54.0000i − 0.635294i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.7279i 0.143010i 0.997440 + 0.0715052i \(0.0227802\pi\)
−0.997440 + 0.0715052i \(0.977220\pi\)
\(90\) 0 0
\(91\) 32.0000i 0.351648i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 67.8823i 0.714550i
\(96\) 0 0
\(97\) 176.000 1.81443 0.907216 0.420664i \(-0.138203\pi\)
0.907216 + 0.420664i \(0.138203\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 29.6985 0.294044 0.147022 0.989133i \(-0.453031\pi\)
0.147022 + 0.989133i \(0.453031\pi\)
\(102\) 0 0
\(103\) −28.0000 −0.271845 −0.135922 0.990719i \(-0.543400\pi\)
−0.135922 + 0.990719i \(0.543400\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) − 56.0000i − 0.513761i −0.966443 0.256881i \(-0.917305\pi\)
0.966443 0.256881i \(-0.0826948\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 156.978i 1.38918i 0.719404 + 0.694592i \(0.244415\pi\)
−0.719404 + 0.694592i \(0.755585\pi\)
\(114\) 0 0
\(115\) − 72.0000i − 0.626087i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 50.9117i − 0.427829i
\(120\) 0 0
\(121\) 167.000 1.38017
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 135.765 1.08612
\(126\) 0 0
\(127\) −92.0000 −0.724409 −0.362205 0.932099i \(-0.617976\pi\)
−0.362205 + 0.932099i \(0.617976\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −169.706 −1.29546 −0.647731 0.761869i \(-0.724282\pi\)
−0.647731 + 0.761869i \(0.724282\pi\)
\(132\) 0 0
\(133\) 64.0000i 0.481203i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 156.978i 1.14582i 0.819617 + 0.572911i \(0.194186\pi\)
−0.819617 + 0.572911i \(0.805814\pi\)
\(138\) 0 0
\(139\) − 152.000i − 1.09353i −0.837288 0.546763i \(-0.815860\pi\)
0.837288 0.546763i \(-0.184140\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 135.765i 0.949402i
\(144\) 0 0
\(145\) 18.0000 0.124138
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 275.772 1.85082 0.925408 0.378972i \(-0.123722\pi\)
0.925408 + 0.378972i \(0.123722\pi\)
\(150\) 0 0
\(151\) −148.000 −0.980132 −0.490066 0.871685i \(-0.663027\pi\)
−0.490066 + 0.871685i \(0.663027\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 186.676 1.20436
\(156\) 0 0
\(157\) 82.0000i 0.522293i 0.965299 + 0.261146i \(0.0841006\pi\)
−0.965299 + 0.261146i \(0.915899\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 67.8823i − 0.421629i
\(162\) 0 0
\(163\) 56.0000i 0.343558i 0.985135 + 0.171779i \(0.0549515\pi\)
−0.985135 + 0.171779i \(0.945048\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 33.9411i − 0.203240i −0.994823 0.101620i \(-0.967597\pi\)
0.994823 0.101620i \(-0.0324026\pi\)
\(168\) 0 0
\(169\) 105.000 0.621302
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 173.948 1.00548 0.502741 0.864437i \(-0.332325\pi\)
0.502741 + 0.864437i \(0.332325\pi\)
\(174\) 0 0
\(175\) 28.0000 0.160000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −203.647 −1.13769 −0.568846 0.822444i \(-0.692610\pi\)
−0.568846 + 0.822444i \(0.692610\pi\)
\(180\) 0 0
\(181\) − 232.000i − 1.28177i −0.767638 0.640884i \(-0.778568\pi\)
0.767638 0.640884i \(-0.221432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 144.250i 0.779729i
\(186\) 0 0
\(187\) − 216.000i − 1.15508i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 33.9411i − 0.177702i −0.996045 0.0888511i \(-0.971680\pi\)
0.996045 0.0888511i \(-0.0283195\pi\)
\(192\) 0 0
\(193\) 206.000 1.06736 0.533679 0.845687i \(-0.320809\pi\)
0.533679 + 0.845687i \(0.320809\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 165.463 0.839914 0.419957 0.907544i \(-0.362045\pi\)
0.419957 + 0.907544i \(0.362045\pi\)
\(198\) 0 0
\(199\) 20.0000 0.100503 0.0502513 0.998737i \(-0.483998\pi\)
0.0502513 + 0.998737i \(0.483998\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 16.9706 0.0835988
\(204\) 0 0
\(205\) − 198.000i − 0.965854i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 271.529i 1.29918i
\(210\) 0 0
\(211\) 296.000i 1.40284i 0.712746 + 0.701422i \(0.247451\pi\)
−0.712746 + 0.701422i \(0.752549\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 169.706i − 0.789328i
\(216\) 0 0
\(217\) 176.000 0.811060
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 101.823 0.460739
\(222\) 0 0
\(223\) 436.000 1.95516 0.977578 0.210571i \(-0.0675325\pi\)
0.977578 + 0.210571i \(0.0675325\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.9706 0.0747602 0.0373801 0.999301i \(-0.488099\pi\)
0.0373801 + 0.999301i \(0.488099\pi\)
\(228\) 0 0
\(229\) 8.00000i 0.0349345i 0.999847 + 0.0174672i \(0.00556028\pi\)
−0.999847 + 0.0174672i \(0.994440\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 12.7279i − 0.0546263i −0.999627 0.0273131i \(-0.991305\pi\)
0.999627 0.0273131i \(-0.00869512\pi\)
\(234\) 0 0
\(235\) 360.000i 1.53191i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 135.765i 0.568052i 0.958817 + 0.284026i \(0.0916703\pi\)
−0.958817 + 0.284026i \(0.908330\pi\)
\(240\) 0 0
\(241\) 32.0000 0.132780 0.0663900 0.997794i \(-0.478852\pi\)
0.0663900 + 0.997794i \(0.478852\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 140.007 0.571458
\(246\) 0 0
\(247\) −128.000 −0.518219
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −50.9117 −0.202835 −0.101418 0.994844i \(-0.532338\pi\)
−0.101418 + 0.994844i \(0.532338\pi\)
\(252\) 0 0
\(253\) − 288.000i − 1.13834i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 182.434i 0.709858i 0.934893 + 0.354929i \(0.115495\pi\)
−0.934893 + 0.354929i \(0.884505\pi\)
\(258\) 0 0
\(259\) 136.000i 0.525097i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 373.352i − 1.41959i −0.704408 0.709795i \(-0.748787\pi\)
0.704408 0.709795i \(-0.251213\pi\)
\(264\) 0 0
\(265\) −162.000 −0.611321
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 343.654 1.27752 0.638762 0.769404i \(-0.279447\pi\)
0.638762 + 0.769404i \(0.279447\pi\)
\(270\) 0 0
\(271\) −380.000 −1.40221 −0.701107 0.713056i \(-0.747310\pi\)
−0.701107 + 0.713056i \(0.747310\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 118.794 0.431978
\(276\) 0 0
\(277\) − 328.000i − 1.18412i −0.805896 0.592058i \(-0.798316\pi\)
0.805896 0.592058i \(-0.201684\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 284.257i 1.01159i 0.862654 + 0.505795i \(0.168801\pi\)
−0.862654 + 0.505795i \(0.831199\pi\)
\(282\) 0 0
\(283\) 208.000i 0.734982i 0.930027 + 0.367491i \(0.119783\pi\)
−0.930027 + 0.367491i \(0.880217\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 186.676i − 0.650440i
\(288\) 0 0
\(289\) 127.000 0.439446
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −436.992 −1.49144 −0.745720 0.666259i \(-0.767894\pi\)
−0.745720 + 0.666259i \(0.767894\pi\)
\(294\) 0 0
\(295\) 144.000 0.488136
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 135.765 0.454062
\(300\) 0 0
\(301\) − 160.000i − 0.531561i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 212.132i 0.695515i
\(306\) 0 0
\(307\) − 520.000i − 1.69381i −0.531743 0.846906i \(-0.678463\pi\)
0.531743 0.846906i \(-0.321537\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 373.352i 1.20049i 0.799816 + 0.600245i \(0.204930\pi\)
−0.799816 + 0.600245i \(0.795070\pi\)
\(312\) 0 0
\(313\) 94.0000 0.300319 0.150160 0.988662i \(-0.452021\pi\)
0.150160 + 0.988662i \(0.452021\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −335.169 −1.05731 −0.528657 0.848835i \(-0.677304\pi\)
−0.528657 + 0.848835i \(0.677304\pi\)
\(318\) 0 0
\(319\) 72.0000 0.225705
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 203.647 0.630485
\(324\) 0 0
\(325\) 56.0000i 0.172308i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 339.411i 1.03165i
\(330\) 0 0
\(331\) − 536.000i − 1.61934i −0.586889 0.809668i \(-0.699647\pi\)
0.586889 0.809668i \(-0.300353\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 33.9411i − 0.101317i
\(336\) 0 0
\(337\) −208.000 −0.617211 −0.308605 0.951190i \(-0.599862\pi\)
−0.308605 + 0.951190i \(0.599862\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 746.705 2.18975
\(342\) 0 0
\(343\) 328.000 0.956268
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −288.500 −0.831411 −0.415705 0.909499i \(-0.636465\pi\)
−0.415705 + 0.909499i \(0.636465\pi\)
\(348\) 0 0
\(349\) 238.000i 0.681948i 0.940073 + 0.340974i \(0.110757\pi\)
−0.940073 + 0.340974i \(0.889243\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 224.860i 0.636997i 0.947923 + 0.318499i \(0.103179\pi\)
−0.947923 + 0.318499i \(0.896821\pi\)
\(354\) 0 0
\(355\) − 216.000i − 0.608451i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 560.029i 1.55997i 0.625799 + 0.779984i \(0.284773\pi\)
−0.625799 + 0.779984i \(0.715227\pi\)
\(360\) 0 0
\(361\) 105.000 0.290859
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −67.8823 −0.185979
\(366\) 0 0
\(367\) −284.000 −0.773842 −0.386921 0.922113i \(-0.626461\pi\)
−0.386921 + 0.922113i \(0.626461\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −152.735 −0.411685
\(372\) 0 0
\(373\) − 190.000i − 0.509383i −0.967022 0.254692i \(-0.918026\pi\)
0.967022 0.254692i \(-0.0819740\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 33.9411i 0.0900295i
\(378\) 0 0
\(379\) 160.000i 0.422164i 0.977468 + 0.211082i \(0.0676986\pi\)
−0.977468 + 0.211082i \(0.932301\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 271.529i − 0.708953i −0.935065 0.354477i \(-0.884659\pi\)
0.935065 0.354477i \(-0.115341\pi\)
\(384\) 0 0
\(385\) −288.000 −0.748052
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −403.051 −1.03612 −0.518060 0.855344i \(-0.673346\pi\)
−0.518060 + 0.855344i \(0.673346\pi\)
\(390\) 0 0
\(391\) −216.000 −0.552430
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −322.441 −0.816306
\(396\) 0 0
\(397\) − 146.000i − 0.367758i −0.982949 0.183879i \(-0.941135\pi\)
0.982949 0.183879i \(-0.0588655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 326.683i − 0.814672i −0.913278 0.407336i \(-0.866458\pi\)
0.913278 0.407336i \(-0.133542\pi\)
\(402\) 0 0
\(403\) 352.000i 0.873449i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 576.999i 1.41769i
\(408\) 0 0
\(409\) −368.000 −0.899756 −0.449878 0.893090i \(-0.648532\pi\)
−0.449878 + 0.893090i \(0.648532\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 135.765 0.328728
\(414\) 0 0
\(415\) −504.000 −1.21446
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 390.323 0.931558 0.465779 0.884901i \(-0.345774\pi\)
0.465779 + 0.884901i \(0.345774\pi\)
\(420\) 0 0
\(421\) − 40.0000i − 0.0950119i −0.998871 0.0475059i \(-0.984873\pi\)
0.998871 0.0475059i \(-0.0151273\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 89.0955i − 0.209636i
\(426\) 0 0
\(427\) 200.000i 0.468384i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 152.735i − 0.354374i −0.984177 0.177187i \(-0.943300\pi\)
0.984177 0.177187i \(-0.0566997\pi\)
\(432\) 0 0
\(433\) 542.000 1.25173 0.625866 0.779931i \(-0.284746\pi\)
0.625866 + 0.779931i \(0.284746\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 271.529 0.621348
\(438\) 0 0
\(439\) −4.00000 −0.00911162 −0.00455581 0.999990i \(-0.501450\pi\)
−0.00455581 + 0.999990i \(0.501450\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −322.441 −0.727857 −0.363929 0.931427i \(-0.618565\pi\)
−0.363929 + 0.931427i \(0.618565\pi\)
\(444\) 0 0
\(445\) − 54.0000i − 0.121348i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 216.375i − 0.481904i −0.970537 0.240952i \(-0.922540\pi\)
0.970537 0.240952i \(-0.0774596\pi\)
\(450\) 0 0
\(451\) − 792.000i − 1.75610i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 135.765i − 0.298384i
\(456\) 0 0
\(457\) 400.000 0.875274 0.437637 0.899152i \(-0.355816\pi\)
0.437637 + 0.899152i \(0.355816\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 301.227 0.653422 0.326711 0.945124i \(-0.394060\pi\)
0.326711 + 0.945124i \(0.394060\pi\)
\(462\) 0 0
\(463\) 604.000 1.30454 0.652268 0.757989i \(-0.273818\pi\)
0.652268 + 0.757989i \(0.273818\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 356.382 0.763130 0.381565 0.924342i \(-0.375385\pi\)
0.381565 + 0.924342i \(0.375385\pi\)
\(468\) 0 0
\(469\) − 32.0000i − 0.0682303i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 678.823i − 1.43514i
\(474\) 0 0
\(475\) 112.000i 0.235789i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 526.087i 1.09830i 0.835723 + 0.549152i \(0.185049\pi\)
−0.835723 + 0.549152i \(0.814951\pi\)
\(480\) 0 0
\(481\) −272.000 −0.565489
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −746.705 −1.53960
\(486\) 0 0
\(487\) 596.000 1.22382 0.611910 0.790928i \(-0.290402\pi\)
0.611910 + 0.790928i \(0.290402\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 271.529 0.553012 0.276506 0.961012i \(-0.410823\pi\)
0.276506 + 0.961012i \(0.410823\pi\)
\(492\) 0 0
\(493\) − 54.0000i − 0.109533i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 203.647i − 0.409752i
\(498\) 0 0
\(499\) 224.000i 0.448898i 0.974486 + 0.224449i \(0.0720582\pi\)
−0.974486 + 0.224449i \(0.927942\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 865.499i − 1.72067i −0.509726 0.860337i \(-0.670253\pi\)
0.509726 0.860337i \(-0.329747\pi\)
\(504\) 0 0
\(505\) −126.000 −0.249505
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −479.418 −0.941883 −0.470941 0.882164i \(-0.656086\pi\)
−0.470941 + 0.882164i \(0.656086\pi\)
\(510\) 0 0
\(511\) −64.0000 −0.125245
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 118.794 0.230668
\(516\) 0 0
\(517\) 1440.00i 2.78530i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 521.845i − 1.00162i −0.865557 0.500811i \(-0.833035\pi\)
0.865557 0.500811i \(-0.166965\pi\)
\(522\) 0 0
\(523\) 736.000i 1.40727i 0.710564 + 0.703633i \(0.248440\pi\)
−0.710564 + 0.703633i \(0.751560\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 560.029i − 1.06267i
\(528\) 0 0
\(529\) 241.000 0.455577
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 373.352 0.700474
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 560.029 1.03901
\(540\) 0 0
\(541\) 808.000i 1.49353i 0.665088 + 0.746765i \(0.268394\pi\)
−0.665088 + 0.746765i \(0.731606\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 237.588i 0.435941i
\(546\) 0 0
\(547\) 536.000i 0.979890i 0.871753 + 0.489945i \(0.162983\pi\)
−0.871753 + 0.489945i \(0.837017\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 67.8823i 0.123198i
\(552\) 0 0
\(553\) −304.000 −0.549729
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 165.463 0.297061 0.148531 0.988908i \(-0.452546\pi\)
0.148531 + 0.988908i \(0.452546\pi\)
\(558\) 0 0
\(559\) 320.000 0.572451
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −322.441 −0.572719 −0.286359 0.958122i \(-0.592445\pi\)
−0.286359 + 0.958122i \(0.592445\pi\)
\(564\) 0 0
\(565\) − 666.000i − 1.17876i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 156.978i 0.275883i 0.990440 + 0.137942i \(0.0440487\pi\)
−0.990440 + 0.137942i \(0.955951\pi\)
\(570\) 0 0
\(571\) − 368.000i − 0.644483i −0.946657 0.322242i \(-0.895564\pi\)
0.946657 0.322242i \(-0.104436\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 118.794i − 0.206598i
\(576\) 0 0
\(577\) −142.000 −0.246101 −0.123050 0.992400i \(-0.539268\pi\)
−0.123050 + 0.992400i \(0.539268\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −475.176 −0.817858
\(582\) 0 0
\(583\) −648.000 −1.11149
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −373.352 −0.636035 −0.318017 0.948085i \(-0.603017\pi\)
−0.318017 + 0.948085i \(0.603017\pi\)
\(588\) 0 0
\(589\) 704.000i 1.19525i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1107.33i − 1.86733i −0.358142 0.933667i \(-0.616590\pi\)
0.358142 0.933667i \(-0.383410\pi\)
\(594\) 0 0
\(595\) 216.000i 0.363025i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 797.616i − 1.33158i −0.746139 0.665790i \(-0.768095\pi\)
0.746139 0.665790i \(-0.231905\pi\)
\(600\) 0 0
\(601\) −158.000 −0.262895 −0.131448 0.991323i \(-0.541963\pi\)
−0.131448 + 0.991323i \(0.541963\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −708.521 −1.17111
\(606\) 0 0
\(607\) −332.000 −0.546952 −0.273476 0.961879i \(-0.588173\pi\)
−0.273476 + 0.961879i \(0.588173\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −678.823 −1.11100
\(612\) 0 0
\(613\) 578.000i 0.942904i 0.881892 + 0.471452i \(0.156270\pi\)
−0.881892 + 0.471452i \(0.843730\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 55.1543i − 0.0893911i −0.999001 0.0446956i \(-0.985768\pi\)
0.999001 0.0446956i \(-0.0142318\pi\)
\(618\) 0 0
\(619\) − 896.000i − 1.44750i −0.690064 0.723748i \(-0.742418\pi\)
0.690064 0.723748i \(-0.257582\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 50.9117i − 0.0817202i
\(624\) 0 0
\(625\) −401.000 −0.641600
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 432.749 0.687996
\(630\) 0 0
\(631\) 20.0000 0.0316957 0.0158479 0.999874i \(-0.494955\pi\)
0.0158479 + 0.999874i \(0.494955\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 390.323 0.614682
\(636\) 0 0
\(637\) 264.000i 0.414443i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 258.801i − 0.403746i −0.979412 0.201873i \(-0.935297\pi\)
0.979412 0.201873i \(-0.0647028\pi\)
\(642\) 0 0
\(643\) 728.000i 1.13219i 0.824339 + 0.566096i \(0.191547\pi\)
−0.824339 + 0.566096i \(0.808453\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 458.205i 0.708200i 0.935208 + 0.354100i \(0.115213\pi\)
−0.935208 + 0.354100i \(0.884787\pi\)
\(648\) 0 0
\(649\) 576.000 0.887519
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −301.227 −0.461298 −0.230649 0.973037i \(-0.574085\pi\)
−0.230649 + 0.973037i \(0.574085\pi\)
\(654\) 0 0
\(655\) 720.000 1.09924
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1052.17 −1.59662 −0.798312 0.602244i \(-0.794273\pi\)
−0.798312 + 0.602244i \(0.794273\pi\)
\(660\) 0 0
\(661\) 62.0000i 0.0937973i 0.998900 + 0.0468986i \(0.0149338\pi\)
−0.998900 + 0.0468986i \(0.985066\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 271.529i − 0.408314i
\(666\) 0 0
\(667\) − 72.0000i − 0.107946i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 848.528i 1.26457i
\(672\) 0 0
\(673\) −670.000 −0.995542 −0.497771 0.867308i \(-0.665848\pi\)
−0.497771 + 0.867308i \(0.665848\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1294.01 −1.91138 −0.955691 0.294372i \(-0.904889\pi\)
−0.955691 + 0.294372i \(0.904889\pi\)
\(678\) 0 0
\(679\) −704.000 −1.03682
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 560.029 0.819954 0.409977 0.912096i \(-0.365537\pi\)
0.409977 + 0.912096i \(0.365537\pi\)
\(684\) 0 0
\(685\) − 666.000i − 0.972263i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 305.470i − 0.443353i
\(690\) 0 0
\(691\) − 40.0000i − 0.0578871i −0.999581 0.0289436i \(-0.990786\pi\)
0.999581 0.0289436i \(-0.00921431\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 644.881i 0.927887i
\(696\) 0 0
\(697\) −594.000 −0.852224
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −954.594 −1.36176 −0.680880 0.732395i \(-0.738403\pi\)
−0.680880 + 0.732395i \(0.738403\pi\)
\(702\) 0 0
\(703\) −544.000 −0.773826
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −118.794 −0.168025
\(708\) 0 0
\(709\) 968.000i 1.36530i 0.730744 + 0.682652i \(0.239173\pi\)
−0.730744 + 0.682652i \(0.760827\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 746.705i − 1.04727i
\(714\) 0 0
\(715\) − 576.000i − 0.805594i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1170.97i 1.62861i 0.580439 + 0.814304i \(0.302881\pi\)
−0.580439 + 0.814304i \(0.697119\pi\)
\(720\) 0 0
\(721\) 112.000 0.155340
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 29.6985 0.0409634
\(726\) 0 0
\(727\) −508.000 −0.698762 −0.349381 0.936981i \(-0.613608\pi\)
−0.349381 + 0.936981i \(0.613608\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −509.117 −0.696466
\(732\) 0 0
\(733\) 1144.00i 1.56071i 0.625337 + 0.780355i \(0.284961\pi\)
−0.625337 + 0.780355i \(0.715039\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 135.765i − 0.184212i
\(738\) 0 0
\(739\) − 304.000i − 0.411367i −0.978619 0.205683i \(-0.934058\pi\)
0.978619 0.205683i \(-0.0659417\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 848.528i − 1.14203i −0.820940 0.571015i \(-0.806550\pi\)
0.820940 0.571015i \(-0.193450\pi\)
\(744\) 0 0
\(745\) −1170.00 −1.57047
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −188.000 −0.250333 −0.125166 0.992136i \(-0.539946\pi\)
−0.125166 + 0.992136i \(0.539946\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 627.911 0.831670
\(756\) 0 0
\(757\) − 1240.00i − 1.63804i −0.573761 0.819022i \(-0.694516\pi\)
0.573761 0.819022i \(-0.305484\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 156.978i − 0.206278i −0.994667 0.103139i \(-0.967111\pi\)
0.994667 0.103139i \(-0.0328887\pi\)
\(762\) 0 0
\(763\) 224.000i 0.293578i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 271.529i 0.354014i
\(768\) 0 0
\(769\) −910.000 −1.18336 −0.591678 0.806175i \(-0.701534\pi\)
−0.591678 + 0.806175i \(0.701534\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1387.34 −1.79475 −0.897376 0.441266i \(-0.854529\pi\)
−0.897376 + 0.441266i \(0.854529\pi\)
\(774\) 0 0
\(775\) 308.000 0.397419
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 746.705 0.958543
\(780\) 0 0
\(781\) − 864.000i − 1.10627i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 347.897i − 0.443180i
\(786\) 0 0
\(787\) − 1360.00i − 1.72808i −0.503422 0.864041i \(-0.667926\pi\)
0.503422 0.864041i \(-0.332074\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 627.911i − 0.793819i
\(792\) 0 0
\(793\) −400.000 −0.504414
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 106.066 0.133082 0.0665408 0.997784i \(-0.478804\pi\)
0.0665408 + 0.997784i \(0.478804\pi\)
\(798\) 0 0
\(799\) 1080.00 1.35169
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −271.529 −0.338143
\(804\) 0 0
\(805\) 288.000i 0.357764i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 1107.33i − 1.36876i −0.729124 0.684381i \(-0.760072\pi\)
0.729124 0.684381i \(-0.239928\pi\)
\(810\) 0 0
\(811\) 160.000i 0.197287i 0.995123 + 0.0986436i \(0.0314504\pi\)
−0.995123 + 0.0986436i \(0.968550\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 237.588i − 0.291519i
\(816\) 0 0
\(817\) 640.000 0.783354
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 436.992 0.532268 0.266134 0.963936i \(-0.414254\pi\)
0.266134 + 0.963936i \(0.414254\pi\)
\(822\) 0 0
\(823\) 332.000 0.403402 0.201701 0.979447i \(-0.435353\pi\)
0.201701 + 0.979447i \(0.435353\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −101.823 −0.123124 −0.0615619 0.998103i \(-0.519608\pi\)
−0.0615619 + 0.998103i \(0.519608\pi\)
\(828\) 0 0
\(829\) − 632.000i − 0.762364i −0.924500 0.381182i \(-0.875517\pi\)
0.924500 0.381182i \(-0.124483\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 420.021i − 0.504227i
\(834\) 0 0
\(835\) 144.000i 0.172455i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 729.734i − 0.869767i −0.900487 0.434883i \(-0.856790\pi\)
0.900487 0.434883i \(-0.143210\pi\)
\(840\) 0 0
\(841\) −823.000 −0.978597
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −445.477 −0.527192
\(846\) 0 0
\(847\) −668.000 −0.788666
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 576.999 0.678025
\(852\) 0 0
\(853\) 446.000i 0.522860i 0.965222 + 0.261430i \(0.0841941\pi\)
−0.965222 + 0.261430i \(0.915806\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 428.507i − 0.500008i −0.968245 0.250004i \(-0.919568\pi\)
0.968245 0.250004i \(-0.0804319\pi\)
\(858\) 0 0
\(859\) − 728.000i − 0.847497i −0.905780 0.423749i \(-0.860714\pi\)
0.905780 0.423749i \(-0.139286\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 916.410i − 1.06189i −0.847407 0.530945i \(-0.821837\pi\)
0.847407 0.530945i \(-0.178163\pi\)
\(864\) 0 0
\(865\) −738.000 −0.853179
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1289.76 −1.48419
\(870\) 0 0
\(871\) 64.0000 0.0734788
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −543.058 −0.620638
\(876\) 0 0
\(877\) 910.000i 1.03763i 0.854887 + 0.518814i \(0.173626\pi\)
−0.854887 + 0.518814i \(0.826374\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 929.138i 1.05464i 0.849667 + 0.527320i \(0.176803\pi\)
−0.849667 + 0.527320i \(0.823197\pi\)
\(882\) 0 0
\(883\) 1064.00i 1.20498i 0.798125 + 0.602492i \(0.205825\pi\)
−0.798125 + 0.602492i \(0.794175\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1391.59i 1.56887i 0.620212 + 0.784434i \(0.287047\pi\)
−0.620212 + 0.784434i \(0.712953\pi\)
\(888\) 0 0
\(889\) 368.000 0.413948
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1357.65 −1.52032
\(894\) 0 0
\(895\) 864.000 0.965363
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 186.676 0.207649
\(900\) 0 0
\(901\) 486.000i 0.539401i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 984.293i 1.08762i
\(906\) 0 0
\(907\) 1768.00i 1.94928i 0.223771 + 0.974642i \(0.428163\pi\)
−0.223771 + 0.974642i \(0.571837\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 237.588i − 0.260799i −0.991462 0.130399i \(-0.958374\pi\)
0.991462 0.130399i \(-0.0416260\pi\)
\(912\) 0 0
\(913\) −2016.00 −2.20811
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 678.823 0.740264
\(918\) 0 0
\(919\) 380.000 0.413493 0.206746 0.978395i \(-0.433712\pi\)
0.206746 + 0.978395i \(0.433712\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 407.294 0.441271
\(924\) 0 0
\(925\) 238.000i 0.257297i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 666.095i − 0.717002i −0.933529 0.358501i \(-0.883288\pi\)
0.933529 0.358501i \(-0.116712\pi\)
\(930\) 0 0
\(931\) 528.000i 0.567132i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 916.410i 0.980118i
\(936\) 0 0
\(937\) 178.000 0.189968 0.0949840 0.995479i \(-0.469720\pi\)
0.0949840 + 0.995479i \(0.469720\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 436.992 0.464391 0.232196 0.972669i \(-0.425409\pi\)
0.232196 + 0.972669i \(0.425409\pi\)
\(942\) 0 0
\(943\) −792.000 −0.839873
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1798.88 1.89956 0.949778 0.312924i \(-0.101309\pi\)
0.949778 + 0.312924i \(0.101309\pi\)
\(948\) 0 0
\(949\) − 128.000i − 0.134879i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1310.98i 1.37563i 0.725886 + 0.687815i \(0.241430\pi\)
−0.725886 + 0.687815i \(0.758570\pi\)
\(954\) 0 0
\(955\) 144.000i 0.150785i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 627.911i − 0.654756i
\(960\) 0 0
\(961\) 975.000 1.01457
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −873.984 −0.905683
\(966\) 0 0
\(967\) 1700.00 1.75801 0.879007 0.476808i \(-0.158206\pi\)
0.879007 + 0.476808i \(0.158206\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −458.205 −0.471890 −0.235945 0.971766i \(-0.575819\pi\)
−0.235945 + 0.971766i \(0.575819\pi\)
\(972\) 0 0
\(973\) 608.000i 0.624872i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 759.433i − 0.777311i −0.921383 0.388655i \(-0.872940\pi\)
0.921383 0.388655i \(-0.127060\pi\)
\(978\) 0 0
\(979\) − 216.000i − 0.220633i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1052.17i 1.07037i 0.844734 + 0.535186i \(0.179758\pi\)
−0.844734 + 0.535186i \(0.820242\pi\)
\(984\) 0 0
\(985\) −702.000 −0.712690
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −678.823 −0.686373
\(990\) 0 0
\(991\) 772.000 0.779011 0.389506 0.921024i \(-0.372646\pi\)
0.389506 + 0.921024i \(0.372646\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −84.8528 −0.0852792
\(996\) 0 0
\(997\) 194.000i 0.194584i 0.995256 + 0.0972919i \(0.0310180\pi\)
−0.995256 + 0.0972919i \(0.968982\pi\)
\(998\) 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.c.2177.1 4
3.2 odd 2 inner 2304.3.h.c.2177.3 4
4.3 odd 2 2304.3.h.f.2177.1 4
8.3 odd 2 2304.3.h.f.2177.4 4
8.5 even 2 inner 2304.3.h.c.2177.4 4
12.11 even 2 2304.3.h.f.2177.3 4
16.3 odd 4 18.3.b.a.17.1 2
16.5 even 4 576.3.e.f.449.1 2
16.11 odd 4 576.3.e.c.449.1 2
16.13 even 4 144.3.e.b.17.2 2
24.5 odd 2 inner 2304.3.h.c.2177.2 4
24.11 even 2 2304.3.h.f.2177.2 4
48.5 odd 4 576.3.e.f.449.2 2
48.11 even 4 576.3.e.c.449.2 2
48.29 odd 4 144.3.e.b.17.1 2
48.35 even 4 18.3.b.a.17.2 yes 2
80.3 even 4 450.3.b.b.449.2 4
80.13 odd 4 3600.3.c.b.449.2 4
80.19 odd 4 450.3.d.f.251.2 2
80.29 even 4 3600.3.l.d.1601.2 2
80.67 even 4 450.3.b.b.449.3 4
80.77 odd 4 3600.3.c.b.449.4 4
112.3 even 12 882.3.s.d.863.2 4
112.19 even 12 882.3.s.d.557.1 4
112.51 odd 12 882.3.s.b.557.1 4
112.67 odd 12 882.3.s.b.863.2 4
112.83 even 4 882.3.b.a.197.1 2
144.13 even 12 1296.3.q.f.593.2 4
144.29 odd 12 1296.3.q.f.1025.2 4
144.61 even 12 1296.3.q.f.1025.1 4
144.67 odd 12 162.3.d.b.107.2 4
144.77 odd 12 1296.3.q.f.593.1 4
144.83 even 12 162.3.d.b.53.2 4
144.115 odd 12 162.3.d.b.53.1 4
144.131 even 12 162.3.d.b.107.1 4
176.131 even 4 2178.3.c.d.485.2 2
208.51 odd 4 3042.3.c.e.1691.2 2
208.83 even 4 3042.3.d.a.3041.1 4
208.99 even 4 3042.3.d.a.3041.4 4
240.29 odd 4 3600.3.l.d.1601.1 2
240.77 even 4 3600.3.c.b.449.3 4
240.83 odd 4 450.3.b.b.449.4 4
240.173 even 4 3600.3.c.b.449.1 4
240.179 even 4 450.3.d.f.251.1 2
240.227 odd 4 450.3.b.b.449.1 4
336.83 odd 4 882.3.b.a.197.2 2
336.131 odd 12 882.3.s.d.557.2 4
336.179 even 12 882.3.s.b.863.1 4
336.227 odd 12 882.3.s.d.863.1 4
336.275 even 12 882.3.s.b.557.2 4
528.131 odd 4 2178.3.c.d.485.1 2
624.83 odd 4 3042.3.d.a.3041.3 4
624.467 even 4 3042.3.c.e.1691.1 2
624.515 odd 4 3042.3.d.a.3041.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.b.a.17.1 2 16.3 odd 4
18.3.b.a.17.2 yes 2 48.35 even 4
144.3.e.b.17.1 2 48.29 odd 4
144.3.e.b.17.2 2 16.13 even 4
162.3.d.b.53.1 4 144.115 odd 12
162.3.d.b.53.2 4 144.83 even 12
162.3.d.b.107.1 4 144.131 even 12
162.3.d.b.107.2 4 144.67 odd 12
450.3.b.b.449.1 4 240.227 odd 4
450.3.b.b.449.2 4 80.3 even 4
450.3.b.b.449.3 4 80.67 even 4
450.3.b.b.449.4 4 240.83 odd 4
450.3.d.f.251.1 2 240.179 even 4
450.3.d.f.251.2 2 80.19 odd 4
576.3.e.c.449.1 2 16.11 odd 4
576.3.e.c.449.2 2 48.11 even 4
576.3.e.f.449.1 2 16.5 even 4
576.3.e.f.449.2 2 48.5 odd 4
882.3.b.a.197.1 2 112.83 even 4
882.3.b.a.197.2 2 336.83 odd 4
882.3.s.b.557.1 4 112.51 odd 12
882.3.s.b.557.2 4 336.275 even 12
882.3.s.b.863.1 4 336.179 even 12
882.3.s.b.863.2 4 112.67 odd 12
882.3.s.d.557.1 4 112.19 even 12
882.3.s.d.557.2 4 336.131 odd 12
882.3.s.d.863.1 4 336.227 odd 12
882.3.s.d.863.2 4 112.3 even 12
1296.3.q.f.593.1 4 144.77 odd 12
1296.3.q.f.593.2 4 144.13 even 12
1296.3.q.f.1025.1 4 144.61 even 12
1296.3.q.f.1025.2 4 144.29 odd 12
2178.3.c.d.485.1 2 528.131 odd 4
2178.3.c.d.485.2 2 176.131 even 4
2304.3.h.c.2177.1 4 1.1 even 1 trivial
2304.3.h.c.2177.2 4 24.5 odd 2 inner
2304.3.h.c.2177.3 4 3.2 odd 2 inner
2304.3.h.c.2177.4 4 8.5 even 2 inner
2304.3.h.f.2177.1 4 4.3 odd 2
2304.3.h.f.2177.2 4 24.11 even 2
2304.3.h.f.2177.3 4 12.11 even 2
2304.3.h.f.2177.4 4 8.3 odd 2
3042.3.c.e.1691.1 2 624.467 even 4
3042.3.c.e.1691.2 2 208.51 odd 4
3042.3.d.a.3041.1 4 208.83 even 4
3042.3.d.a.3041.2 4 624.515 odd 4
3042.3.d.a.3041.3 4 624.83 odd 4
3042.3.d.a.3041.4 4 208.99 even 4
3600.3.c.b.449.1 4 240.173 even 4
3600.3.c.b.449.2 4 80.13 odd 4
3600.3.c.b.449.3 4 240.77 even 4
3600.3.c.b.449.4 4 80.77 odd 4
3600.3.l.d.1601.1 2 240.29 odd 4
3600.3.l.d.1601.2 2 80.29 even 4