Properties

Label 1344.4.p.b.223.15
Level $1344$
Weight $4$
Character 1344.223
Analytic conductor $79.299$
Analytic rank $0$
Dimension $16$
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] = [1344,4,Mod(223,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.223");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.p (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} + 58 x^{13} + 178264 x^{12} - 331354 x^{11} + 307862 x^{10} - 610 x^{9} + 8375926786 x^{8} - 15937543350 x^{7} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 223.15
Root \(4.51719 - 4.51719i\) of defining polynomial
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.b.223.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +16.7103 q^{5} +(-16.4816 - 8.44725i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +16.7103 q^{5} +(-16.4816 - 8.44725i) q^{7} -9.00000 q^{9} +8.98444 q^{11} +6.28960 q^{13} +50.1310i q^{15} +106.695i q^{17} -25.1841i q^{19} +(25.3417 - 49.4449i) q^{21} +19.5261i q^{23} +154.235 q^{25} -27.0000i q^{27} -132.780i q^{29} -120.761 q^{31} +26.9533i q^{33} +(-275.413 - 141.156i) q^{35} +72.1865i q^{37} +18.8688i q^{39} +410.552i q^{41} -64.1310 q^{43} -150.393 q^{45} -281.172 q^{47} +(200.288 + 278.449i) q^{49} -320.085 q^{51} +401.935i q^{53} +150.133 q^{55} +75.5523 q^{57} +584.836i q^{59} +736.702 q^{61} +(148.335 + 76.0252i) q^{63} +105.101 q^{65} +643.821 q^{67} -58.5782 q^{69} -186.578i q^{71} +64.5717i q^{73} +462.705i q^{75} +(-148.078 - 75.8938i) q^{77} -50.5244i q^{79} +81.0000 q^{81} +17.9983i q^{83} +1782.91i q^{85} +398.341 q^{87} +378.330i q^{89} +(-103.663 - 53.1298i) q^{91} -362.282i q^{93} -420.834i q^{95} -593.957i q^{97} -80.8599 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 144 q^{9} + 56 q^{13} + 36 q^{21} + 80 q^{25} + 392 q^{49} + 336 q^{57} + 184 q^{61} - 1536 q^{65} + 864 q^{69} - 240 q^{77} + 1296 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 16.7103 1.49462 0.747309 0.664477i \(-0.231346\pi\)
0.747309 + 0.664477i \(0.231346\pi\)
\(6\) 0 0
\(7\) −16.4816 8.44725i −0.889924 0.456109i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 8.98444 0.246265 0.123132 0.992390i \(-0.460706\pi\)
0.123132 + 0.992390i \(0.460706\pi\)
\(12\) 0 0
\(13\) 6.28960 0.134186 0.0670931 0.997747i \(-0.478628\pi\)
0.0670931 + 0.997747i \(0.478628\pi\)
\(14\) 0 0
\(15\) 50.1310i 0.862918i
\(16\) 0 0
\(17\) 106.695i 1.52220i 0.648635 + 0.761099i \(0.275340\pi\)
−0.648635 + 0.761099i \(0.724660\pi\)
\(18\) 0 0
\(19\) 25.1841i 0.304086i −0.988374 0.152043i \(-0.951415\pi\)
0.988374 0.152043i \(-0.0485852\pi\)
\(20\) 0 0
\(21\) 25.3417 49.4449i 0.263334 0.513798i
\(22\) 0 0
\(23\) 19.5261i 0.177020i 0.996075 + 0.0885102i \(0.0282106\pi\)
−0.996075 + 0.0885102i \(0.971789\pi\)
\(24\) 0 0
\(25\) 154.235 1.23388
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 132.780i 0.850231i −0.905139 0.425116i \(-0.860233\pi\)
0.905139 0.425116i \(-0.139767\pi\)
\(30\) 0 0
\(31\) −120.761 −0.699653 −0.349826 0.936815i \(-0.613759\pi\)
−0.349826 + 0.936815i \(0.613759\pi\)
\(32\) 0 0
\(33\) 26.9533i 0.142181i
\(34\) 0 0
\(35\) −275.413 141.156i −1.33010 0.681708i
\(36\) 0 0
\(37\) 72.1865i 0.320740i 0.987057 + 0.160370i \(0.0512688\pi\)
−0.987057 + 0.160370i \(0.948731\pi\)
\(38\) 0 0
\(39\) 18.8688i 0.0774724i
\(40\) 0 0
\(41\) 410.552i 1.56384i 0.623379 + 0.781920i \(0.285759\pi\)
−0.623379 + 0.781920i \(0.714241\pi\)
\(42\) 0 0
\(43\) −64.1310 −0.227439 −0.113720 0.993513i \(-0.536277\pi\)
−0.113720 + 0.993513i \(0.536277\pi\)
\(44\) 0 0
\(45\) −150.393 −0.498206
\(46\) 0 0
\(47\) −281.172 −0.872619 −0.436310 0.899797i \(-0.643715\pi\)
−0.436310 + 0.899797i \(0.643715\pi\)
\(48\) 0 0
\(49\) 200.288 + 278.449i 0.583930 + 0.811804i
\(50\) 0 0
\(51\) −320.085 −0.878842
\(52\) 0 0
\(53\) 401.935i 1.04170i 0.853648 + 0.520850i \(0.174385\pi\)
−0.853648 + 0.520850i \(0.825615\pi\)
\(54\) 0 0
\(55\) 150.133 0.368071
\(56\) 0 0
\(57\) 75.5523 0.175564
\(58\) 0 0
\(59\) 584.836i 1.29049i 0.763974 + 0.645247i \(0.223245\pi\)
−0.763974 + 0.645247i \(0.776755\pi\)
\(60\) 0 0
\(61\) 736.702 1.54631 0.773156 0.634216i \(-0.218677\pi\)
0.773156 + 0.634216i \(0.218677\pi\)
\(62\) 0 0
\(63\) 148.335 + 76.0252i 0.296641 + 0.152036i
\(64\) 0 0
\(65\) 105.101 0.200557
\(66\) 0 0
\(67\) 643.821 1.17396 0.586980 0.809602i \(-0.300317\pi\)
0.586980 + 0.809602i \(0.300317\pi\)
\(68\) 0 0
\(69\) −58.5782 −0.102203
\(70\) 0 0
\(71\) 186.578i 0.311870i −0.987767 0.155935i \(-0.950161\pi\)
0.987767 0.155935i \(-0.0498390\pi\)
\(72\) 0 0
\(73\) 64.5717i 0.103528i 0.998659 + 0.0517640i \(0.0164844\pi\)
−0.998659 + 0.0517640i \(0.983516\pi\)
\(74\) 0 0
\(75\) 462.705i 0.712381i
\(76\) 0 0
\(77\) −148.078 75.8938i −0.219157 0.112323i
\(78\) 0 0
\(79\) 50.5244i 0.0719549i −0.999353 0.0359775i \(-0.988546\pi\)
0.999353 0.0359775i \(-0.0114545\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 17.9983i 0.0238020i 0.999929 + 0.0119010i \(0.00378830\pi\)
−0.999929 + 0.0119010i \(0.996212\pi\)
\(84\) 0 0
\(85\) 1782.91i 2.27510i
\(86\) 0 0
\(87\) 398.341 0.490881
\(88\) 0 0
\(89\) 378.330i 0.450594i 0.974290 + 0.225297i \(0.0723353\pi\)
−0.974290 + 0.225297i \(0.927665\pi\)
\(90\) 0 0
\(91\) −103.663 53.1298i −0.119415 0.0612035i
\(92\) 0 0
\(93\) 362.282i 0.403945i
\(94\) 0 0
\(95\) 420.834i 0.454492i
\(96\) 0 0
\(97\) 593.957i 0.621724i −0.950455 0.310862i \(-0.899382\pi\)
0.950455 0.310862i \(-0.100618\pi\)
\(98\) 0 0
\(99\) −80.8599 −0.0820882
\(100\) 0 0
\(101\) −107.294 −0.105704 −0.0528522 0.998602i \(-0.516831\pi\)
−0.0528522 + 0.998602i \(0.516831\pi\)
\(102\) 0 0
\(103\) 1748.70 1.67286 0.836430 0.548073i \(-0.184639\pi\)
0.836430 + 0.548073i \(0.184639\pi\)
\(104\) 0 0
\(105\) 423.469 826.240i 0.393584 0.767931i
\(106\) 0 0
\(107\) −1818.36 −1.64288 −0.821439 0.570297i \(-0.806828\pi\)
−0.821439 + 0.570297i \(0.806828\pi\)
\(108\) 0 0
\(109\) 1966.38i 1.72794i 0.503544 + 0.863969i \(0.332029\pi\)
−0.503544 + 0.863969i \(0.667971\pi\)
\(110\) 0 0
\(111\) −216.559 −0.185179
\(112\) 0 0
\(113\) −1754.77 −1.46084 −0.730419 0.682999i \(-0.760675\pi\)
−0.730419 + 0.682999i \(0.760675\pi\)
\(114\) 0 0
\(115\) 326.287i 0.264578i
\(116\) 0 0
\(117\) −56.6064 −0.0447287
\(118\) 0 0
\(119\) 901.281 1758.51i 0.694288 1.35464i
\(120\) 0 0
\(121\) −1250.28 −0.939354
\(122\) 0 0
\(123\) −1231.66 −0.902883
\(124\) 0 0
\(125\) 488.527 0.349562
\(126\) 0 0
\(127\) 1974.40i 1.37952i 0.724037 + 0.689761i \(0.242284\pi\)
−0.724037 + 0.689761i \(0.757716\pi\)
\(128\) 0 0
\(129\) 192.393i 0.131312i
\(130\) 0 0
\(131\) 1566.88i 1.04503i 0.852631 + 0.522513i \(0.175006\pi\)
−0.852631 + 0.522513i \(0.824994\pi\)
\(132\) 0 0
\(133\) −212.736 + 415.075i −0.138696 + 0.270613i
\(134\) 0 0
\(135\) 451.179i 0.287639i
\(136\) 0 0
\(137\) 1903.82 1.18726 0.593630 0.804738i \(-0.297694\pi\)
0.593630 + 0.804738i \(0.297694\pi\)
\(138\) 0 0
\(139\) 935.422i 0.570802i −0.958408 0.285401i \(-0.907873\pi\)
0.958408 0.285401i \(-0.0921267\pi\)
\(140\) 0 0
\(141\) 843.515i 0.503807i
\(142\) 0 0
\(143\) 56.5085 0.0330453
\(144\) 0 0
\(145\) 2218.80i 1.27077i
\(146\) 0 0
\(147\) −835.346 + 600.864i −0.468695 + 0.337132i
\(148\) 0 0
\(149\) 474.828i 0.261070i 0.991444 + 0.130535i \(0.0416695\pi\)
−0.991444 + 0.130535i \(0.958330\pi\)
\(150\) 0 0
\(151\) 1097.08i 0.591254i −0.955303 0.295627i \(-0.904471\pi\)
0.955303 0.295627i \(-0.0955286\pi\)
\(152\) 0 0
\(153\) 960.256i 0.507400i
\(154\) 0 0
\(155\) −2017.95 −1.04571
\(156\) 0 0
\(157\) 2782.88 1.41464 0.707320 0.706894i \(-0.249904\pi\)
0.707320 + 0.706894i \(0.249904\pi\)
\(158\) 0 0
\(159\) −1205.81 −0.601425
\(160\) 0 0
\(161\) 164.942 321.822i 0.0807405 0.157535i
\(162\) 0 0
\(163\) −58.9320 −0.0283185 −0.0141592 0.999900i \(-0.504507\pi\)
−0.0141592 + 0.999900i \(0.504507\pi\)
\(164\) 0 0
\(165\) 450.399i 0.212506i
\(166\) 0 0
\(167\) 1532.53 0.710123 0.355062 0.934843i \(-0.384460\pi\)
0.355062 + 0.934843i \(0.384460\pi\)
\(168\) 0 0
\(169\) −2157.44 −0.981994
\(170\) 0 0
\(171\) 226.657i 0.101362i
\(172\) 0 0
\(173\) 1729.85 0.760219 0.380109 0.924942i \(-0.375886\pi\)
0.380109 + 0.924942i \(0.375886\pi\)
\(174\) 0 0
\(175\) −2542.04 1302.86i −1.09806 0.562784i
\(176\) 0 0
\(177\) −1754.51 −0.745067
\(178\) 0 0
\(179\) −3137.12 −1.30994 −0.654971 0.755654i \(-0.727319\pi\)
−0.654971 + 0.755654i \(0.727319\pi\)
\(180\) 0 0
\(181\) 3422.20 1.40536 0.702681 0.711506i \(-0.251986\pi\)
0.702681 + 0.711506i \(0.251986\pi\)
\(182\) 0 0
\(183\) 2210.10i 0.892763i
\(184\) 0 0
\(185\) 1206.26i 0.479384i
\(186\) 0 0
\(187\) 958.596i 0.374864i
\(188\) 0 0
\(189\) −228.076 + 445.004i −0.0877781 + 0.171266i
\(190\) 0 0
\(191\) 2442.02i 0.925123i 0.886587 + 0.462562i \(0.153070\pi\)
−0.886587 + 0.462562i \(0.846930\pi\)
\(192\) 0 0
\(193\) 3281.21 1.22377 0.611883 0.790948i \(-0.290412\pi\)
0.611883 + 0.790948i \(0.290412\pi\)
\(194\) 0 0
\(195\) 315.304i 0.115792i
\(196\) 0 0
\(197\) 2405.89i 0.870114i 0.900403 + 0.435057i \(0.143272\pi\)
−0.900403 + 0.435057i \(0.856728\pi\)
\(198\) 0 0
\(199\) −949.634 −0.338280 −0.169140 0.985592i \(-0.554099\pi\)
−0.169140 + 0.985592i \(0.554099\pi\)
\(200\) 0 0
\(201\) 1931.46i 0.677786i
\(202\) 0 0
\(203\) −1121.63 + 2188.44i −0.387798 + 0.756641i
\(204\) 0 0
\(205\) 6860.46i 2.33734i
\(206\) 0 0
\(207\) 175.735i 0.0590068i
\(208\) 0 0
\(209\) 226.265i 0.0748855i
\(210\) 0 0
\(211\) −3535.42 −1.15350 −0.576750 0.816921i \(-0.695679\pi\)
−0.576750 + 0.816921i \(0.695679\pi\)
\(212\) 0 0
\(213\) 559.735 0.180058
\(214\) 0 0
\(215\) −1071.65 −0.339935
\(216\) 0 0
\(217\) 1990.33 + 1020.09i 0.622638 + 0.319118i
\(218\) 0 0
\(219\) −193.715 −0.0597719
\(220\) 0 0
\(221\) 671.069i 0.204258i
\(222\) 0 0
\(223\) −6109.42 −1.83460 −0.917302 0.398192i \(-0.869638\pi\)
−0.917302 + 0.398192i \(0.869638\pi\)
\(224\) 0 0
\(225\) −1388.12 −0.411293
\(226\) 0 0
\(227\) 560.043i 0.163750i −0.996643 0.0818752i \(-0.973909\pi\)
0.996643 0.0818752i \(-0.0260909\pi\)
\(228\) 0 0
\(229\) −4491.68 −1.29615 −0.648075 0.761576i \(-0.724426\pi\)
−0.648075 + 0.761576i \(0.724426\pi\)
\(230\) 0 0
\(231\) 227.681 444.234i 0.0648499 0.126530i
\(232\) 0 0
\(233\) 2031.76 0.571266 0.285633 0.958339i \(-0.407796\pi\)
0.285633 + 0.958339i \(0.407796\pi\)
\(234\) 0 0
\(235\) −4698.47 −1.30423
\(236\) 0 0
\(237\) 151.573 0.0415432
\(238\) 0 0
\(239\) 6026.95i 1.63117i 0.578634 + 0.815587i \(0.303586\pi\)
−0.578634 + 0.815587i \(0.696414\pi\)
\(240\) 0 0
\(241\) 2390.47i 0.638938i −0.947597 0.319469i \(-0.896496\pi\)
0.947597 0.319469i \(-0.103504\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 3346.88 + 4652.97i 0.872752 + 1.21334i
\(246\) 0 0
\(247\) 158.398i 0.0408041i
\(248\) 0 0
\(249\) −53.9949 −0.0137421
\(250\) 0 0
\(251\) 1488.61i 0.374344i 0.982327 + 0.187172i \(0.0599322\pi\)
−0.982327 + 0.187172i \(0.940068\pi\)
\(252\) 0 0
\(253\) 175.431i 0.0435938i
\(254\) 0 0
\(255\) −5348.73 −1.31353
\(256\) 0 0
\(257\) 3311.30i 0.803708i 0.915704 + 0.401854i \(0.131634\pi\)
−0.915704 + 0.401854i \(0.868366\pi\)
\(258\) 0 0
\(259\) 609.777 1189.75i 0.146292 0.285434i
\(260\) 0 0
\(261\) 1195.02i 0.283410i
\(262\) 0 0
\(263\) 1881.16i 0.441055i −0.975381 0.220527i \(-0.929222\pi\)
0.975381 0.220527i \(-0.0707778\pi\)
\(264\) 0 0
\(265\) 6716.47i 1.55694i
\(266\) 0 0
\(267\) −1134.99 −0.260151
\(268\) 0 0
\(269\) −3965.12 −0.898726 −0.449363 0.893349i \(-0.648349\pi\)
−0.449363 + 0.893349i \(0.648349\pi\)
\(270\) 0 0
\(271\) −885.508 −0.198490 −0.0992450 0.995063i \(-0.531643\pi\)
−0.0992450 + 0.995063i \(0.531643\pi\)
\(272\) 0 0
\(273\) 159.389 310.988i 0.0353358 0.0689446i
\(274\) 0 0
\(275\) 1385.72 0.303861
\(276\) 0 0
\(277\) 3749.11i 0.813221i −0.913602 0.406610i \(-0.866711\pi\)
0.913602 0.406610i \(-0.133289\pi\)
\(278\) 0 0
\(279\) 1086.85 0.233218
\(280\) 0 0
\(281\) −3695.89 −0.784620 −0.392310 0.919833i \(-0.628324\pi\)
−0.392310 + 0.919833i \(0.628324\pi\)
\(282\) 0 0
\(283\) 8615.09i 1.80959i 0.425848 + 0.904795i \(0.359976\pi\)
−0.425848 + 0.904795i \(0.640024\pi\)
\(284\) 0 0
\(285\) 1262.50 0.262401
\(286\) 0 0
\(287\) 3468.03 6766.56i 0.713281 1.39170i
\(288\) 0 0
\(289\) −6470.86 −1.31709
\(290\) 0 0
\(291\) 1781.87 0.358953
\(292\) 0 0
\(293\) 4271.95 0.851775 0.425888 0.904776i \(-0.359962\pi\)
0.425888 + 0.904776i \(0.359962\pi\)
\(294\) 0 0
\(295\) 9772.80i 1.92879i
\(296\) 0 0
\(297\) 242.580i 0.0473936i
\(298\) 0 0
\(299\) 122.811i 0.0237537i
\(300\) 0 0
\(301\) 1056.98 + 541.731i 0.202404 + 0.103737i
\(302\) 0 0
\(303\) 321.881i 0.0610284i
\(304\) 0 0
\(305\) 12310.5 2.31114
\(306\) 0 0
\(307\) 5029.94i 0.935095i 0.883968 + 0.467547i \(0.154862\pi\)
−0.883968 + 0.467547i \(0.845138\pi\)
\(308\) 0 0
\(309\) 5246.10i 0.965826i
\(310\) 0 0
\(311\) 4564.45 0.832238 0.416119 0.909310i \(-0.363390\pi\)
0.416119 + 0.909310i \(0.363390\pi\)
\(312\) 0 0
\(313\) 7217.45i 1.30337i −0.758490 0.651684i \(-0.774063\pi\)
0.758490 0.651684i \(-0.225937\pi\)
\(314\) 0 0
\(315\) 2478.72 + 1270.41i 0.443365 + 0.227236i
\(316\) 0 0
\(317\) 5136.65i 0.910104i 0.890465 + 0.455052i \(0.150379\pi\)
−0.890465 + 0.455052i \(0.849621\pi\)
\(318\) 0 0
\(319\) 1192.96i 0.209382i
\(320\) 0 0
\(321\) 5455.09i 0.948516i
\(322\) 0 0
\(323\) 2687.02 0.462879
\(324\) 0 0
\(325\) 970.076 0.165570
\(326\) 0 0
\(327\) −5899.15 −0.997626
\(328\) 0 0
\(329\) 4634.17 + 2375.13i 0.776565 + 0.398009i
\(330\) 0 0
\(331\) −6895.65 −1.14507 −0.572537 0.819879i \(-0.694041\pi\)
−0.572537 + 0.819879i \(0.694041\pi\)
\(332\) 0 0
\(333\) 649.678i 0.106913i
\(334\) 0 0
\(335\) 10758.5 1.75462
\(336\) 0 0
\(337\) −9227.06 −1.49148 −0.745742 0.666235i \(-0.767905\pi\)
−0.745742 + 0.666235i \(0.767905\pi\)
\(338\) 0 0
\(339\) 5264.30i 0.843415i
\(340\) 0 0
\(341\) −1084.97 −0.172300
\(342\) 0 0
\(343\) −948.944 6281.17i −0.149382 0.988779i
\(344\) 0 0
\(345\) −978.862 −0.152754
\(346\) 0 0
\(347\) −1912.98 −0.295948 −0.147974 0.988991i \(-0.547275\pi\)
−0.147974 + 0.988991i \(0.547275\pi\)
\(348\) 0 0
\(349\) −115.346 −0.0176915 −0.00884577 0.999961i \(-0.502816\pi\)
−0.00884577 + 0.999961i \(0.502816\pi\)
\(350\) 0 0
\(351\) 169.819i 0.0258241i
\(352\) 0 0
\(353\) 5218.75i 0.786872i 0.919352 + 0.393436i \(0.128714\pi\)
−0.919352 + 0.393436i \(0.871286\pi\)
\(354\) 0 0
\(355\) 3117.78i 0.466126i
\(356\) 0 0
\(357\) 5275.53 + 2703.84i 0.782103 + 0.400847i
\(358\) 0 0
\(359\) 9771.78i 1.43659i −0.695740 0.718294i \(-0.744923\pi\)
0.695740 0.718294i \(-0.255077\pi\)
\(360\) 0 0
\(361\) 6224.76 0.907532
\(362\) 0 0
\(363\) 3750.84i 0.542336i
\(364\) 0 0
\(365\) 1079.01i 0.154735i
\(366\) 0 0
\(367\) 8207.66 1.16740 0.583701 0.811969i \(-0.301604\pi\)
0.583701 + 0.811969i \(0.301604\pi\)
\(368\) 0 0
\(369\) 3694.97i 0.521280i
\(370\) 0 0
\(371\) 3395.25 6624.55i 0.475128 0.927033i
\(372\) 0 0
\(373\) 13506.3i 1.87488i −0.348152 0.937438i \(-0.613191\pi\)
0.348152 0.937438i \(-0.386809\pi\)
\(374\) 0 0
\(375\) 1465.58i 0.201820i
\(376\) 0 0
\(377\) 835.135i 0.114089i
\(378\) 0 0
\(379\) −2631.57 −0.356662 −0.178331 0.983971i \(-0.557070\pi\)
−0.178331 + 0.983971i \(0.557070\pi\)
\(380\) 0 0
\(381\) −5923.19 −0.796467
\(382\) 0 0
\(383\) 13787.0 1.83938 0.919692 0.392641i \(-0.128439\pi\)
0.919692 + 0.392641i \(0.128439\pi\)
\(384\) 0 0
\(385\) −2474.43 1268.21i −0.327555 0.167880i
\(386\) 0 0
\(387\) 577.179 0.0758131
\(388\) 0 0
\(389\) 4491.05i 0.585360i 0.956210 + 0.292680i \(0.0945471\pi\)
−0.956210 + 0.292680i \(0.905453\pi\)
\(390\) 0 0
\(391\) −2083.34 −0.269460
\(392\) 0 0
\(393\) −4700.63 −0.603347
\(394\) 0 0
\(395\) 844.279i 0.107545i
\(396\) 0 0
\(397\) −7456.08 −0.942595 −0.471297 0.881974i \(-0.656214\pi\)
−0.471297 + 0.881974i \(0.656214\pi\)
\(398\) 0 0
\(399\) −1245.22 638.209i −0.156239 0.0800762i
\(400\) 0 0
\(401\) −8374.63 −1.04292 −0.521458 0.853277i \(-0.674612\pi\)
−0.521458 + 0.853277i \(0.674612\pi\)
\(402\) 0 0
\(403\) −759.535 −0.0938837
\(404\) 0 0
\(405\) 1353.54 0.166069
\(406\) 0 0
\(407\) 648.555i 0.0789869i
\(408\) 0 0
\(409\) 6640.80i 0.802852i −0.915891 0.401426i \(-0.868515\pi\)
0.915891 0.401426i \(-0.131485\pi\)
\(410\) 0 0
\(411\) 5711.47i 0.685464i
\(412\) 0 0
\(413\) 4940.25 9639.04i 0.588605 1.14844i
\(414\) 0 0
\(415\) 300.757i 0.0355749i
\(416\) 0 0
\(417\) 2806.27 0.329553
\(418\) 0 0
\(419\) 6063.13i 0.706929i −0.935448 0.353465i \(-0.885004\pi\)
0.935448 0.353465i \(-0.114996\pi\)
\(420\) 0 0
\(421\) 12873.6i 1.49031i −0.666890 0.745157i \(-0.732375\pi\)
0.666890 0.745157i \(-0.267625\pi\)
\(422\) 0 0
\(423\) 2530.55 0.290873
\(424\) 0 0
\(425\) 16456.1i 1.87821i
\(426\) 0 0
\(427\) −12142.0 6223.10i −1.37610 0.705286i
\(428\) 0 0
\(429\) 169.525i 0.0190787i
\(430\) 0 0
\(431\) 8064.52i 0.901286i −0.892704 0.450643i \(-0.851195\pi\)
0.892704 0.450643i \(-0.148805\pi\)
\(432\) 0 0
\(433\) 7184.20i 0.797345i −0.917093 0.398673i \(-0.869471\pi\)
0.917093 0.398673i \(-0.130529\pi\)
\(434\) 0 0
\(435\) 6656.41 0.733680
\(436\) 0 0
\(437\) 491.747 0.0538294
\(438\) 0 0
\(439\) 1863.44 0.202590 0.101295 0.994856i \(-0.467701\pi\)
0.101295 + 0.994856i \(0.467701\pi\)
\(440\) 0 0
\(441\) −1802.59 2506.04i −0.194643 0.270601i
\(442\) 0 0
\(443\) 576.702 0.0618509 0.0309254 0.999522i \(-0.490155\pi\)
0.0309254 + 0.999522i \(0.490155\pi\)
\(444\) 0 0
\(445\) 6322.02i 0.673466i
\(446\) 0 0
\(447\) −1424.48 −0.150729
\(448\) 0 0
\(449\) 8358.04 0.878487 0.439243 0.898368i \(-0.355247\pi\)
0.439243 + 0.898368i \(0.355247\pi\)
\(450\) 0 0
\(451\) 3688.58i 0.385118i
\(452\) 0 0
\(453\) 3291.25 0.341361
\(454\) 0 0
\(455\) −1732.24 887.816i −0.178480 0.0914757i
\(456\) 0 0
\(457\) 3756.51 0.384512 0.192256 0.981345i \(-0.438420\pi\)
0.192256 + 0.981345i \(0.438420\pi\)
\(458\) 0 0
\(459\) 2880.77 0.292947
\(460\) 0 0
\(461\) −866.025 −0.0874942 −0.0437471 0.999043i \(-0.513930\pi\)
−0.0437471 + 0.999043i \(0.513930\pi\)
\(462\) 0 0
\(463\) 3348.04i 0.336062i −0.985782 0.168031i \(-0.946259\pi\)
0.985782 0.168031i \(-0.0537408\pi\)
\(464\) 0 0
\(465\) 6053.85i 0.603743i
\(466\) 0 0
\(467\) 12122.4i 1.20119i 0.799553 + 0.600596i \(0.205070\pi\)
−0.799553 + 0.600596i \(0.794930\pi\)
\(468\) 0 0
\(469\) −10611.2 5438.52i −1.04474 0.535453i
\(470\) 0 0
\(471\) 8348.65i 0.816742i
\(472\) 0 0
\(473\) −576.181 −0.0560102
\(474\) 0 0
\(475\) 3884.27i 0.375205i
\(476\) 0 0
\(477\) 3617.42i 0.347233i
\(478\) 0 0
\(479\) 8827.69 0.842061 0.421031 0.907046i \(-0.361668\pi\)
0.421031 + 0.907046i \(0.361668\pi\)
\(480\) 0 0
\(481\) 454.024i 0.0430389i
\(482\) 0 0
\(483\) 965.465 + 494.825i 0.0909527 + 0.0466156i
\(484\) 0 0
\(485\) 9925.22i 0.929240i
\(486\) 0 0
\(487\) 12562.9i 1.16895i 0.811412 + 0.584475i \(0.198699\pi\)
−0.811412 + 0.584475i \(0.801301\pi\)
\(488\) 0 0
\(489\) 176.796i 0.0163497i
\(490\) 0 0
\(491\) −3335.07 −0.306536 −0.153268 0.988185i \(-0.548980\pi\)
−0.153268 + 0.988185i \(0.548980\pi\)
\(492\) 0 0
\(493\) 14167.0 1.29422
\(494\) 0 0
\(495\) −1351.20 −0.122690
\(496\) 0 0
\(497\) −1576.07 + 3075.11i −0.142247 + 0.277541i
\(498\) 0 0
\(499\) 17947.1 1.61007 0.805033 0.593230i \(-0.202148\pi\)
0.805033 + 0.593230i \(0.202148\pi\)
\(500\) 0 0
\(501\) 4597.58i 0.409990i
\(502\) 0 0
\(503\) −3789.15 −0.335885 −0.167942 0.985797i \(-0.553712\pi\)
−0.167942 + 0.985797i \(0.553712\pi\)
\(504\) 0 0
\(505\) −1792.92 −0.157987
\(506\) 0 0
\(507\) 6472.32i 0.566955i
\(508\) 0 0
\(509\) −10041.2 −0.874399 −0.437199 0.899365i \(-0.644030\pi\)
−0.437199 + 0.899365i \(0.644030\pi\)
\(510\) 0 0
\(511\) 545.453 1064.25i 0.0472200 0.0921321i
\(512\) 0 0
\(513\) −679.971 −0.0585213
\(514\) 0 0
\(515\) 29221.4 2.50029
\(516\) 0 0
\(517\) −2526.17 −0.214895
\(518\) 0 0
\(519\) 5189.54i 0.438912i
\(520\) 0 0
\(521\) 16041.1i 1.34890i 0.738322 + 0.674448i \(0.235618\pi\)
−0.738322 + 0.674448i \(0.764382\pi\)
\(522\) 0 0
\(523\) 3627.15i 0.303259i 0.988437 + 0.151629i \(0.0484520\pi\)
−0.988437 + 0.151629i \(0.951548\pi\)
\(524\) 0 0
\(525\) 3908.59 7626.13i 0.324923 0.633965i
\(526\) 0 0
\(527\) 12884.6i 1.06501i
\(528\) 0 0
\(529\) 11785.7 0.968664
\(530\) 0 0
\(531\) 5263.52i 0.430165i
\(532\) 0 0
\(533\) 2582.21i 0.209846i
\(534\) 0 0
\(535\) −30385.5 −2.45547
\(536\) 0 0
\(537\) 9411.37i 0.756295i
\(538\) 0 0
\(539\) 1799.47 + 2501.71i 0.143801 + 0.199919i
\(540\) 0 0
\(541\) 181.348i 0.0144118i −0.999974 0.00720588i \(-0.997706\pi\)
0.999974 0.00720588i \(-0.00229372\pi\)
\(542\) 0 0
\(543\) 10266.6i 0.811386i
\(544\) 0 0
\(545\) 32858.9i 2.58261i
\(546\) 0 0
\(547\) 18922.2 1.47908 0.739539 0.673114i \(-0.235044\pi\)
0.739539 + 0.673114i \(0.235044\pi\)
\(548\) 0 0
\(549\) −6630.31 −0.515437
\(550\) 0 0
\(551\) −3343.96 −0.258543
\(552\) 0 0
\(553\) −426.792 + 832.724i −0.0328193 + 0.0640344i
\(554\) 0 0
\(555\) −3618.78 −0.276772
\(556\) 0 0
\(557\) 11700.6i 0.890075i −0.895512 0.445037i \(-0.853190\pi\)
0.895512 0.445037i \(-0.146810\pi\)
\(558\) 0 0
\(559\) −403.358 −0.0305192
\(560\) 0 0
\(561\) −2875.79 −0.216428
\(562\) 0 0
\(563\) 22989.7i 1.72096i −0.509485 0.860479i \(-0.670164\pi\)
0.509485 0.860479i \(-0.329836\pi\)
\(564\) 0 0
\(565\) −29322.7 −2.18339
\(566\) 0 0
\(567\) −1335.01 684.227i −0.0988805 0.0506787i
\(568\) 0 0
\(569\) −837.958 −0.0617382 −0.0308691 0.999523i \(-0.509827\pi\)
−0.0308691 + 0.999523i \(0.509827\pi\)
\(570\) 0 0
\(571\) 16018.3 1.17399 0.586993 0.809592i \(-0.300311\pi\)
0.586993 + 0.809592i \(0.300311\pi\)
\(572\) 0 0
\(573\) −7326.07 −0.534120
\(574\) 0 0
\(575\) 3011.61i 0.218422i
\(576\) 0 0
\(577\) 1914.68i 0.138144i −0.997612 0.0690722i \(-0.977996\pi\)
0.997612 0.0690722i \(-0.0220039\pi\)
\(578\) 0 0
\(579\) 9843.64i 0.706542i
\(580\) 0 0
\(581\) 152.036 296.641i 0.0108563 0.0211820i
\(582\) 0 0
\(583\) 3611.16i 0.256534i
\(584\) 0 0
\(585\) −945.911 −0.0668523
\(586\) 0 0
\(587\) 17100.9i 1.20243i −0.799086 0.601217i \(-0.794683\pi\)
0.799086 0.601217i \(-0.205317\pi\)
\(588\) 0 0
\(589\) 3041.25i 0.212754i
\(590\) 0 0
\(591\) −7217.66 −0.502360
\(592\) 0 0
\(593\) 5989.45i 0.414768i 0.978260 + 0.207384i \(0.0664949\pi\)
−0.978260 + 0.207384i \(0.933505\pi\)
\(594\) 0 0
\(595\) 15060.7 29385.3i 1.03769 2.02467i
\(596\) 0 0
\(597\) 2848.90i 0.195306i
\(598\) 0 0
\(599\) 6751.78i 0.460551i 0.973125 + 0.230276i \(0.0739628\pi\)
−0.973125 + 0.230276i \(0.926037\pi\)
\(600\) 0 0
\(601\) 24461.5i 1.66024i −0.557582 0.830122i \(-0.688271\pi\)
0.557582 0.830122i \(-0.311729\pi\)
\(602\) 0 0
\(603\) −5794.39 −0.391320
\(604\) 0 0
\(605\) −20892.6 −1.40397
\(606\) 0 0
\(607\) −22238.9 −1.48707 −0.743533 0.668699i \(-0.766852\pi\)
−0.743533 + 0.668699i \(0.766852\pi\)
\(608\) 0 0
\(609\) −6565.31 3364.89i −0.436847 0.223895i
\(610\) 0 0
\(611\) −1768.46 −0.117093
\(612\) 0 0
\(613\) 10848.5i 0.714792i −0.933953 0.357396i \(-0.883665\pi\)
0.933953 0.357396i \(-0.116335\pi\)
\(614\) 0 0
\(615\) −20581.4 −1.34946
\(616\) 0 0
\(617\) 19769.2 1.28991 0.644957 0.764219i \(-0.276875\pi\)
0.644957 + 0.764219i \(0.276875\pi\)
\(618\) 0 0
\(619\) 26002.5i 1.68842i 0.536016 + 0.844208i \(0.319929\pi\)
−0.536016 + 0.844208i \(0.680071\pi\)
\(620\) 0 0
\(621\) 527.204 0.0340676
\(622\) 0 0
\(623\) 3195.85 6235.49i 0.205520 0.400995i
\(624\) 0 0
\(625\) −11115.9 −0.711419
\(626\) 0 0
\(627\) 678.795 0.0432352
\(628\) 0 0
\(629\) −7701.95 −0.488230
\(630\) 0 0
\(631\) 420.513i 0.0265299i −0.999912 0.0132650i \(-0.995778\pi\)
0.999912 0.0132650i \(-0.00422249\pi\)
\(632\) 0 0
\(633\) 10606.3i 0.665974i
\(634\) 0 0
\(635\) 32992.8i 2.06186i
\(636\) 0 0
\(637\) 1259.73 + 1751.33i 0.0783553 + 0.108933i
\(638\) 0 0
\(639\) 1679.20i 0.103957i
\(640\) 0 0
\(641\) 14251.4 0.878154 0.439077 0.898449i \(-0.355306\pi\)
0.439077 + 0.898449i \(0.355306\pi\)
\(642\) 0 0
\(643\) 22847.4i 1.40126i −0.713523 0.700632i \(-0.752902\pi\)
0.713523 0.700632i \(-0.247098\pi\)
\(644\) 0 0
\(645\) 3214.95i 0.196261i
\(646\) 0 0
\(647\) 17057.9 1.03650 0.518248 0.855230i \(-0.326584\pi\)
0.518248 + 0.855230i \(0.326584\pi\)
\(648\) 0 0
\(649\) 5254.42i 0.317803i
\(650\) 0 0
\(651\) −3060.28 + 5970.99i −0.184243 + 0.359480i
\(652\) 0 0
\(653\) 23995.8i 1.43802i −0.694999 0.719010i \(-0.744595\pi\)
0.694999 0.719010i \(-0.255405\pi\)
\(654\) 0 0
\(655\) 26183.0i 1.56192i
\(656\) 0 0
\(657\) 581.145i 0.0345093i
\(658\) 0 0
\(659\) 29296.8 1.73177 0.865887 0.500239i \(-0.166754\pi\)
0.865887 + 0.500239i \(0.166754\pi\)
\(660\) 0 0
\(661\) 20769.2 1.22213 0.611066 0.791580i \(-0.290741\pi\)
0.611066 + 0.791580i \(0.290741\pi\)
\(662\) 0 0
\(663\) −2013.21 −0.117928
\(664\) 0 0
\(665\) −3554.89 + 6936.04i −0.207298 + 0.404463i
\(666\) 0 0
\(667\) 2592.68 0.150508
\(668\) 0 0
\(669\) 18328.2i 1.05921i
\(670\) 0 0
\(671\) 6618.85 0.380802
\(672\) 0 0
\(673\) 15326.2 0.877835 0.438918 0.898527i \(-0.355362\pi\)
0.438918 + 0.898527i \(0.355362\pi\)
\(674\) 0 0
\(675\) 4164.35i 0.237460i
\(676\) 0 0
\(677\) 3440.17 0.195298 0.0976488 0.995221i \(-0.468868\pi\)
0.0976488 + 0.995221i \(0.468868\pi\)
\(678\) 0 0
\(679\) −5017.31 + 9789.38i −0.283574 + 0.553287i
\(680\) 0 0
\(681\) 1680.13 0.0945413
\(682\) 0 0
\(683\) −31000.6 −1.73676 −0.868378 0.495902i \(-0.834837\pi\)
−0.868378 + 0.495902i \(0.834837\pi\)
\(684\) 0 0
\(685\) 31813.5 1.77450
\(686\) 0 0
\(687\) 13475.0i 0.748333i
\(688\) 0 0
\(689\) 2528.01i 0.139782i
\(690\) 0 0
\(691\) 19379.3i 1.06689i −0.845834 0.533447i \(-0.820897\pi\)
0.845834 0.533447i \(-0.179103\pi\)
\(692\) 0 0
\(693\) 1332.70 + 683.044i 0.0730522 + 0.0374411i
\(694\) 0 0
\(695\) 15631.2i 0.853130i
\(696\) 0 0
\(697\) −43803.9 −2.38047
\(698\) 0 0
\(699\) 6095.27i 0.329820i
\(700\) 0 0
\(701\) 35585.2i 1.91731i −0.284572 0.958655i \(-0.591852\pi\)
0.284572 0.958655i \(-0.408148\pi\)
\(702\) 0 0
\(703\) 1817.95 0.0975325
\(704\) 0 0
\(705\) 14095.4i 0.752999i
\(706\) 0 0
\(707\) 1768.38 + 906.338i 0.0940688 + 0.0482126i
\(708\) 0 0
\(709\) 12432.6i 0.658558i 0.944233 + 0.329279i \(0.106806\pi\)
−0.944233 + 0.329279i \(0.893194\pi\)
\(710\) 0 0
\(711\) 454.719i 0.0239850i
\(712\) 0 0
\(713\) 2357.98i 0.123853i
\(714\) 0 0
\(715\) 944.275 0.0493900
\(716\) 0 0
\(717\) −18080.8 −0.941759
\(718\) 0 0
\(719\) 25374.8 1.31616 0.658082 0.752946i \(-0.271368\pi\)
0.658082 + 0.752946i \(0.271368\pi\)
\(720\) 0 0
\(721\) −28821.4 14771.7i −1.48872 0.763006i
\(722\) 0 0
\(723\) 7171.42 0.368891
\(724\) 0 0
\(725\) 20479.4i 1.04908i
\(726\) 0 0
\(727\) 28559.3 1.45696 0.728478 0.685069i \(-0.240228\pi\)
0.728478 + 0.685069i \(0.240228\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 6842.47i 0.346208i
\(732\) 0 0
\(733\) −31475.5 −1.58605 −0.793025 0.609189i \(-0.791495\pi\)
−0.793025 + 0.609189i \(0.791495\pi\)
\(734\) 0 0
\(735\) −13958.9 + 10040.6i −0.700520 + 0.503883i
\(736\) 0 0
\(737\) 5784.37 0.289105
\(738\) 0 0
\(739\) 17164.7 0.854416 0.427208 0.904153i \(-0.359497\pi\)
0.427208 + 0.904153i \(0.359497\pi\)
\(740\) 0 0
\(741\) 475.193 0.0235582
\(742\) 0 0
\(743\) 29385.8i 1.45095i −0.688247 0.725477i \(-0.741619\pi\)
0.688247 0.725477i \(-0.258381\pi\)
\(744\) 0 0
\(745\) 7934.54i 0.390200i
\(746\) 0 0
\(747\) 161.985i 0.00793401i
\(748\) 0 0
\(749\) 29969.6 + 15360.2i 1.46204 + 0.749330i
\(750\) 0 0
\(751\) 2143.28i 0.104140i −0.998643 0.0520701i \(-0.983418\pi\)
0.998643 0.0520701i \(-0.0165819\pi\)
\(752\) 0 0
\(753\) −4465.83 −0.216128
\(754\) 0 0
\(755\) 18332.6i 0.883698i
\(756\) 0 0
\(757\) 5880.09i 0.282319i 0.989987 + 0.141160i \(0.0450831\pi\)
−0.989987 + 0.141160i \(0.954917\pi\)
\(758\) 0 0
\(759\) −526.292 −0.0251689
\(760\) 0 0
\(761\) 36013.8i 1.71551i −0.514062 0.857753i \(-0.671860\pi\)
0.514062 0.857753i \(-0.328140\pi\)
\(762\) 0 0
\(763\) 16610.5 32409.2i 0.788128 1.53773i
\(764\) 0 0
\(765\) 16046.2i 0.758368i
\(766\) 0 0
\(767\) 3678.38i 0.173166i
\(768\) 0 0
\(769\) 16133.5i 0.756555i −0.925692 0.378277i \(-0.876517\pi\)
0.925692 0.378277i \(-0.123483\pi\)
\(770\) 0 0
\(771\) −9933.89 −0.464021
\(772\) 0 0
\(773\) −33017.3 −1.53629 −0.768143 0.640278i \(-0.778819\pi\)
−0.768143 + 0.640278i \(0.778819\pi\)
\(774\) 0 0
\(775\) −18625.5 −0.863288
\(776\) 0 0
\(777\) 3569.25 + 1829.33i 0.164796 + 0.0844619i
\(778\) 0 0
\(779\) 10339.4 0.475541
\(780\) 0 0
\(781\) 1676.30i 0.0768025i
\(782\) 0 0
\(783\) −3585.07 −0.163627
\(784\) 0 0
\(785\) 46502.9 2.11434
\(786\) 0 0
\(787\) 28920.2i 1.30990i 0.755672 + 0.654951i \(0.227311\pi\)
−0.755672 + 0.654951i \(0.772689\pi\)
\(788\) 0 0
\(789\) 5643.48 0.254643
\(790\) 0 0
\(791\) 28921.4 + 14823.0i 1.30003 + 0.666301i
\(792\) 0 0
\(793\) 4633.56 0.207494
\(794\) 0 0
\(795\) −20149.4 −0.898901
\(796\) 0 0
\(797\) 12634.5 0.561529 0.280764 0.959777i \(-0.409412\pi\)
0.280764 + 0.959777i \(0.409412\pi\)
\(798\) 0 0
\(799\) 29999.7i 1.32830i
\(800\) 0 0
\(801\) 3404.97i 0.150198i
\(802\) 0 0
\(803\) 580.140i 0.0254953i
\(804\) 0 0
\(805\) 2756.23 5377.74i 0.120676 0.235454i
\(806\) 0 0
\(807\) 11895.3i 0.518880i
\(808\) 0 0
\(809\) −41369.9 −1.79788 −0.898941 0.438070i \(-0.855662\pi\)
−0.898941 + 0.438070i \(0.855662\pi\)
\(810\) 0 0
\(811\) 32.6724i 0.00141465i 1.00000 0.000707326i \(0.000225149\pi\)
−1.00000 0.000707326i \(0.999775\pi\)
\(812\) 0 0
\(813\) 2656.52i 0.114598i
\(814\) 0 0
\(815\) −984.773 −0.0423253
\(816\) 0 0
\(817\) 1615.08i 0.0691610i
\(818\) 0 0
\(819\) 932.965 + 478.168i 0.0398052 + 0.0204012i
\(820\) 0 0
\(821\) 33836.1i 1.43835i 0.694828 + 0.719176i \(0.255481\pi\)
−0.694828 + 0.719176i \(0.744519\pi\)
\(822\) 0 0
\(823\) 9891.61i 0.418955i −0.977813 0.209477i \(-0.932824\pi\)
0.977813 0.209477i \(-0.0671763\pi\)
\(824\) 0 0
\(825\) 4157.15i 0.175434i
\(826\) 0 0
\(827\) −32492.4 −1.36623 −0.683114 0.730311i \(-0.739375\pi\)
−0.683114 + 0.730311i \(0.739375\pi\)
\(828\) 0 0
\(829\) 5778.92 0.242111 0.121056 0.992646i \(-0.461372\pi\)
0.121056 + 0.992646i \(0.461372\pi\)
\(830\) 0 0
\(831\) 11247.3 0.469513
\(832\) 0 0
\(833\) −29709.1 + 21369.8i −1.23573 + 0.888857i
\(834\) 0 0
\(835\) 25609.0 1.06136
\(836\) 0 0
\(837\) 3260.54i 0.134648i
\(838\) 0 0
\(839\) 26855.3 1.10506 0.552532 0.833492i \(-0.313662\pi\)
0.552532 + 0.833492i \(0.313662\pi\)
\(840\) 0 0
\(841\) 6758.35 0.277107
\(842\) 0 0
\(843\) 11087.7i 0.453001i
\(844\) 0 0
\(845\) −36051.5 −1.46771
\(846\) 0 0
\(847\) 20606.6 + 10561.4i 0.835954 + 0.428447i
\(848\) 0 0
\(849\) −25845.3 −1.04477
\(850\) 0 0
\(851\) −1409.52 −0.0567776
\(852\) 0 0
\(853\) 32717.6 1.31328 0.656641 0.754203i \(-0.271977\pi\)
0.656641 + 0.754203i \(0.271977\pi\)
\(854\) 0 0
\(855\) 3787.51i 0.151497i
\(856\) 0 0
\(857\) 16998.3i 0.677538i −0.940870 0.338769i \(-0.889989\pi\)
0.940870 0.338769i \(-0.110011\pi\)
\(858\) 0 0
\(859\) 7023.27i 0.278965i 0.990225 + 0.139482i \(0.0445439\pi\)
−0.990225 + 0.139482i \(0.955456\pi\)
\(860\) 0 0
\(861\) 20299.7 + 10404.1i 0.803498 + 0.411813i
\(862\) 0 0
\(863\) 21301.7i 0.840231i 0.907471 + 0.420115i \(0.138010\pi\)
−0.907471 + 0.420115i \(0.861990\pi\)
\(864\) 0 0
\(865\) 28906.3 1.13624
\(866\) 0 0
\(867\) 19412.6i 0.760422i
\(868\) 0 0
\(869\) 453.933i 0.0177199i
\(870\) 0 0
\(871\) 4049.38 0.157529
\(872\) 0 0
\(873\) 5345.62i 0.207241i
\(874\) 0 0
\(875\) −8051.73 4126.71i −0.311083 0.159438i
\(876\) 0 0
\(877\) 6855.04i 0.263943i 0.991253 + 0.131972i \(0.0421308\pi\)
−0.991253 + 0.131972i \(0.957869\pi\)
\(878\) 0 0
\(879\) 12815.9i 0.491773i
\(880\) 0 0
\(881\) 12391.1i 0.473855i 0.971527 + 0.236927i \(0.0761404\pi\)
−0.971527 + 0.236927i \(0.923860\pi\)
\(882\) 0 0
\(883\) 7078.73 0.269783 0.134891 0.990860i \(-0.456931\pi\)
0.134891 + 0.990860i \(0.456931\pi\)
\(884\) 0 0
\(885\) −29318.4 −1.11359
\(886\) 0 0
\(887\) 47526.1 1.79906 0.899532 0.436854i \(-0.143907\pi\)
0.899532 + 0.436854i \(0.143907\pi\)
\(888\) 0 0
\(889\) 16678.2 32541.3i 0.629212 1.22767i
\(890\) 0 0
\(891\) 727.739 0.0273627
\(892\) 0 0
\(893\) 7081.06i 0.265351i
\(894\) 0 0
\(895\) −52422.4 −1.95786
\(896\) 0 0
\(897\) −368.433 −0.0137142
\(898\) 0 0
\(899\) 16034.6i 0.594867i
\(900\) 0 0
\(901\) −42884.6 −1.58567
\(902\) 0 0
\(903\) −1625.19 + 3170.95i −0.0598926 + 0.116858i
\(904\) 0 0
\(905\) 57186.1 2.10048
\(906\) 0 0
\(907\) −31328.6 −1.14691 −0.573457 0.819236i \(-0.694398\pi\)
−0.573457 + 0.819236i \(0.694398\pi\)
\(908\) 0 0
\(909\) 965.644 0.0352348
\(910\) 0 0
\(911\) 21649.9i 0.787367i −0.919246 0.393684i \(-0.871201\pi\)
0.919246 0.393684i \(-0.128799\pi\)
\(912\) 0 0
\(913\) 161.705i 0.00586160i
\(914\) 0 0
\(915\) 36931.6i 1.33434i
\(916\) 0 0
\(917\) 13235.8 25824.7i 0.476646 0.929995i
\(918\) 0 0
\(919\) 39980.3i 1.43507i −0.696522 0.717535i \(-0.745270\pi\)
0.696522 0.717535i \(-0.254730\pi\)
\(920\) 0 0
\(921\) −15089.8 −0.539877
\(922\) 0 0
\(923\) 1173.50i 0.0418486i
\(924\) 0 0
\(925\) 11133.7i 0.395755i
\(926\) 0 0
\(927\) −15738.3 −0.557620
\(928\) 0 0
\(929\) 50671.5i 1.78954i 0.446531 + 0.894768i \(0.352659\pi\)
−0.446531 + 0.894768i \(0.647341\pi\)
\(930\) 0 0
\(931\) 7012.48 5044.07i 0.246858 0.177565i
\(932\) 0 0
\(933\) 13693.3i 0.480493i
\(934\) 0 0
\(935\) 16018.5i 0.560277i
\(936\) 0 0
\(937\) 25149.9i 0.876853i 0.898767 + 0.438427i \(0.144464\pi\)
−0.898767 + 0.438427i \(0.855536\pi\)
\(938\) 0 0
\(939\) 21652.4 0.752500
\(940\) 0 0
\(941\) −3742.64 −0.129656 −0.0648282 0.997896i \(-0.520650\pi\)
−0.0648282 + 0.997896i \(0.520650\pi\)
\(942\) 0 0
\(943\) −8016.47 −0.276832
\(944\) 0 0
\(945\) −3811.22 + 7436.16i −0.131195 + 0.255977i
\(946\) 0 0
\(947\) 37380.8 1.28270 0.641348 0.767250i \(-0.278375\pi\)
0.641348 + 0.767250i \(0.278375\pi\)
\(948\) 0 0
\(949\) 406.130i 0.0138920i
\(950\) 0 0
\(951\) −15409.9 −0.525449
\(952\) 0 0
\(953\) 2567.62 0.0872753 0.0436376 0.999047i \(-0.486105\pi\)
0.0436376 + 0.999047i \(0.486105\pi\)
\(954\) 0 0
\(955\) 40807.0i 1.38270i
\(956\) 0 0
\(957\) 3578.87 0.120887
\(958\) 0 0
\(959\) −31378.1 16082.1i −1.05657 0.541519i
\(960\) 0 0
\(961\) −15207.9 −0.510486
\(962\) 0 0
\(963\) 16365.3 0.547626
\(964\) 0 0
\(965\) 54830.2 1.82906
\(966\) 0 0
\(967\) 20902.7i 0.695126i −0.937657 0.347563i \(-0.887009\pi\)
0.937657 0.347563i \(-0.112991\pi\)
\(968\) 0 0
\(969\) 8061.06i 0.267243i
\(970\) 0 0
\(971\) 52144.6i 1.72338i 0.507436 + 0.861689i \(0.330593\pi\)
−0.507436 + 0.861689i \(0.669407\pi\)
\(972\) 0 0
\(973\) −7901.74 + 15417.3i −0.260348 + 0.507970i
\(974\) 0 0
\(975\) 2910.23i 0.0955917i
\(976\) 0 0
\(977\) 25818.9 0.845466 0.422733 0.906254i \(-0.361071\pi\)
0.422733 + 0.906254i \(0.361071\pi\)
\(978\) 0 0
\(979\) 3399.08i 0.110965i
\(980\) 0 0
\(981\) 17697.4i 0.575980i
\(982\) 0 0
\(983\) 27741.0 0.900104 0.450052 0.893002i \(-0.351405\pi\)
0.450052 + 0.893002i \(0.351405\pi\)
\(984\) 0 0
\(985\) 40203.2i 1.30049i
\(986\) 0 0
\(987\) −7125.38 + 13902.5i −0.229791 + 0.448350i
\(988\) 0 0
\(989\) 1252.23i 0.0402614i
\(990\) 0 0
\(991\) 39220.7i 1.25720i 0.777729 + 0.628600i \(0.216372\pi\)
−0.777729 + 0.628600i \(0.783628\pi\)
\(992\) 0 0
\(993\) 20687.0i 0.661108i
\(994\) 0 0
\(995\) −15868.7 −0.505599
\(996\) 0 0
\(997\) 25324.2 0.804440 0.402220 0.915543i \(-0.368239\pi\)
0.402220 + 0.915543i \(0.368239\pi\)
\(998\) 0 0
\(999\) 1949.04 0.0617265
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 1344.4.p.b.223.15 yes 16
4.3 odd 2 inner 1344.4.p.b.223.8 yes 16
7.6 odd 2 1344.4.p.a.223.2 yes 16
8.3 odd 2 1344.4.p.a.223.10 yes 16
8.5 even 2 1344.4.p.a.223.1 16
28.27 even 2 1344.4.p.a.223.9 yes 16
56.13 odd 2 inner 1344.4.p.b.223.16 yes 16
56.27 even 2 inner 1344.4.p.b.223.7 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1344.4.p.a.223.1 16 8.5 even 2
1344.4.p.a.223.2 yes 16 7.6 odd 2
1344.4.p.a.223.9 yes 16 28.27 even 2
1344.4.p.a.223.10 yes 16 8.3 odd 2
1344.4.p.b.223.7 yes 16 56.27 even 2 inner
1344.4.p.b.223.8 yes 16 4.3 odd 2 inner
1344.4.p.b.223.15 yes 16 1.1 even 1 trivial
1344.4.p.b.223.16 yes 16 56.13 odd 2 inner