Properties

Label 825.4.c.b.199.1
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,4,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.b.199.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.00000i q^{2} +3.00000i q^{3} -17.0000 q^{4} +15.0000 q^{6} -3.00000i q^{7} +45.0000i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-5.00000i q^{2} +3.00000i q^{3} -17.0000 q^{4} +15.0000 q^{6} -3.00000i q^{7} +45.0000i q^{8} -9.00000 q^{9} -11.0000 q^{11} -51.0000i q^{12} +32.0000i q^{13} -15.0000 q^{14} +89.0000 q^{16} -33.0000i q^{17} +45.0000i q^{18} -47.0000 q^{19} +9.00000 q^{21} +55.0000i q^{22} +113.000i q^{23} -135.000 q^{24} +160.000 q^{26} -27.0000i q^{27} +51.0000i q^{28} +54.0000 q^{29} +178.000 q^{31} -85.0000i q^{32} -33.0000i q^{33} -165.000 q^{34} +153.000 q^{36} -19.0000i q^{37} +235.000i q^{38} -96.0000 q^{39} +139.000 q^{41} -45.0000i q^{42} -308.000i q^{43} +187.000 q^{44} +565.000 q^{46} -195.000i q^{47} +267.000i q^{48} +334.000 q^{49} +99.0000 q^{51} -544.000i q^{52} +152.000i q^{53} -135.000 q^{54} +135.000 q^{56} -141.000i q^{57} -270.000i q^{58} +625.000 q^{59} +320.000 q^{61} -890.000i q^{62} +27.0000i q^{63} +287.000 q^{64} -165.000 q^{66} -200.000i q^{67} +561.000i q^{68} -339.000 q^{69} -947.000 q^{71} -405.000i q^{72} -448.000i q^{73} -95.0000 q^{74} +799.000 q^{76} +33.0000i q^{77} +480.000i q^{78} +721.000 q^{79} +81.0000 q^{81} -695.000i q^{82} +142.000i q^{83} -153.000 q^{84} -1540.00 q^{86} +162.000i q^{87} -495.000i q^{88} -404.000 q^{89} +96.0000 q^{91} -1921.00i q^{92} +534.000i q^{93} -975.000 q^{94} +255.000 q^{96} -79.0000i q^{97} -1670.00i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 34 q^{4} + 30 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 34 q^{4} + 30 q^{6} - 18 q^{9} - 22 q^{11} - 30 q^{14} + 178 q^{16} - 94 q^{19} + 18 q^{21} - 270 q^{24} + 320 q^{26} + 108 q^{29} + 356 q^{31} - 330 q^{34} + 306 q^{36} - 192 q^{39} + 278 q^{41} + 374 q^{44} + 1130 q^{46} + 668 q^{49} + 198 q^{51} - 270 q^{54} + 270 q^{56} + 1250 q^{59} + 640 q^{61} + 574 q^{64} - 330 q^{66} - 678 q^{69} - 1894 q^{71} - 190 q^{74} + 1598 q^{76} + 1442 q^{79} + 162 q^{81} - 306 q^{84} - 3080 q^{86} - 808 q^{89} + 192 q^{91} - 1950 q^{94} + 510 q^{96} + 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5.00000i − 1.76777i −0.467707 0.883883i \(-0.654920\pi\)
0.467707 0.883883i \(-0.345080\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −17.0000 −2.12500
\(5\) 0 0
\(6\) 15.0000 1.02062
\(7\) − 3.00000i − 0.161985i −0.996715 0.0809924i \(-0.974191\pi\)
0.996715 0.0809924i \(-0.0258089\pi\)
\(8\) 45.0000i 1.98874i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) − 51.0000i − 1.22687i
\(13\) 32.0000i 0.682708i 0.939935 + 0.341354i \(0.110885\pi\)
−0.939935 + 0.341354i \(0.889115\pi\)
\(14\) −15.0000 −0.286351
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) − 33.0000i − 0.470804i −0.971898 0.235402i \(-0.924359\pi\)
0.971898 0.235402i \(-0.0756407\pi\)
\(18\) 45.0000i 0.589256i
\(19\) −47.0000 −0.567502 −0.283751 0.958898i \(-0.591579\pi\)
−0.283751 + 0.958898i \(0.591579\pi\)
\(20\) 0 0
\(21\) 9.00000 0.0935220
\(22\) 55.0000i 0.533002i
\(23\) 113.000i 1.02444i 0.858854 + 0.512220i \(0.171177\pi\)
−0.858854 + 0.512220i \(0.828823\pi\)
\(24\) −135.000 −1.14820
\(25\) 0 0
\(26\) 160.000 1.20687
\(27\) − 27.0000i − 0.192450i
\(28\) 51.0000i 0.344218i
\(29\) 54.0000 0.345778 0.172889 0.984941i \(-0.444690\pi\)
0.172889 + 0.984941i \(0.444690\pi\)
\(30\) 0 0
\(31\) 178.000 1.03128 0.515641 0.856805i \(-0.327554\pi\)
0.515641 + 0.856805i \(0.327554\pi\)
\(32\) − 85.0000i − 0.469563i
\(33\) − 33.0000i − 0.174078i
\(34\) −165.000 −0.832273
\(35\) 0 0
\(36\) 153.000 0.708333
\(37\) − 19.0000i − 0.0844211i −0.999109 0.0422106i \(-0.986560\pi\)
0.999109 0.0422106i \(-0.0134400\pi\)
\(38\) 235.000i 1.00321i
\(39\) −96.0000 −0.394162
\(40\) 0 0
\(41\) 139.000 0.529467 0.264734 0.964322i \(-0.414716\pi\)
0.264734 + 0.964322i \(0.414716\pi\)
\(42\) − 45.0000i − 0.165325i
\(43\) − 308.000i − 1.09232i −0.837682 0.546158i \(-0.816090\pi\)
0.837682 0.546158i \(-0.183910\pi\)
\(44\) 187.000 0.640712
\(45\) 0 0
\(46\) 565.000 1.81097
\(47\) − 195.000i − 0.605185i −0.953120 0.302592i \(-0.902148\pi\)
0.953120 0.302592i \(-0.0978520\pi\)
\(48\) 267.000i 0.802878i
\(49\) 334.000 0.973761
\(50\) 0 0
\(51\) 99.0000 0.271819
\(52\) − 544.000i − 1.45075i
\(53\) 152.000i 0.393940i 0.980410 + 0.196970i \(0.0631101\pi\)
−0.980410 + 0.196970i \(0.936890\pi\)
\(54\) −135.000 −0.340207
\(55\) 0 0
\(56\) 135.000 0.322145
\(57\) − 141.000i − 0.327647i
\(58\) − 270.000i − 0.611254i
\(59\) 625.000 1.37912 0.689560 0.724229i \(-0.257804\pi\)
0.689560 + 0.724229i \(0.257804\pi\)
\(60\) 0 0
\(61\) 320.000 0.671669 0.335834 0.941921i \(-0.390982\pi\)
0.335834 + 0.941921i \(0.390982\pi\)
\(62\) − 890.000i − 1.82307i
\(63\) 27.0000i 0.0539949i
\(64\) 287.000 0.560547
\(65\) 0 0
\(66\) −165.000 −0.307729
\(67\) − 200.000i − 0.364685i −0.983235 0.182342i \(-0.941632\pi\)
0.983235 0.182342i \(-0.0583679\pi\)
\(68\) 561.000i 1.00046i
\(69\) −339.000 −0.591461
\(70\) 0 0
\(71\) −947.000 −1.58293 −0.791466 0.611213i \(-0.790682\pi\)
−0.791466 + 0.611213i \(0.790682\pi\)
\(72\) − 405.000i − 0.662913i
\(73\) − 448.000i − 0.718280i −0.933284 0.359140i \(-0.883070\pi\)
0.933284 0.359140i \(-0.116930\pi\)
\(74\) −95.0000 −0.149237
\(75\) 0 0
\(76\) 799.000 1.20594
\(77\) 33.0000i 0.0488402i
\(78\) 480.000i 0.696786i
\(79\) 721.000 1.02682 0.513410 0.858143i \(-0.328382\pi\)
0.513410 + 0.858143i \(0.328382\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 695.000i − 0.935975i
\(83\) 142.000i 0.187789i 0.995582 + 0.0938947i \(0.0299317\pi\)
−0.995582 + 0.0938947i \(0.970068\pi\)
\(84\) −153.000 −0.198734
\(85\) 0 0
\(86\) −1540.00 −1.93096
\(87\) 162.000i 0.199635i
\(88\) − 495.000i − 0.599627i
\(89\) −404.000 −0.481168 −0.240584 0.970628i \(-0.577339\pi\)
−0.240584 + 0.970628i \(0.577339\pi\)
\(90\) 0 0
\(91\) 96.0000 0.110588
\(92\) − 1921.00i − 2.17694i
\(93\) 534.000i 0.595411i
\(94\) −975.000 −1.06983
\(95\) 0 0
\(96\) 255.000 0.271102
\(97\) − 79.0000i − 0.0826931i −0.999145 0.0413466i \(-0.986835\pi\)
0.999145 0.0413466i \(-0.0131648\pi\)
\(98\) − 1670.00i − 1.72138i
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) −545.000 −0.536926 −0.268463 0.963290i \(-0.586516\pi\)
−0.268463 + 0.963290i \(0.586516\pi\)
\(102\) − 495.000i − 0.480513i
\(103\) − 1306.00i − 1.24936i −0.780881 0.624680i \(-0.785230\pi\)
0.780881 0.624680i \(-0.214770\pi\)
\(104\) −1440.00 −1.35773
\(105\) 0 0
\(106\) 760.000 0.696394
\(107\) − 1938.00i − 1.75097i −0.483247 0.875484i \(-0.660543\pi\)
0.483247 0.875484i \(-0.339457\pi\)
\(108\) 459.000i 0.408956i
\(109\) 576.000 0.506154 0.253077 0.967446i \(-0.418557\pi\)
0.253077 + 0.967446i \(0.418557\pi\)
\(110\) 0 0
\(111\) 57.0000 0.0487405
\(112\) − 267.000i − 0.225260i
\(113\) − 1104.00i − 0.919076i −0.888158 0.459538i \(-0.848015\pi\)
0.888158 0.459538i \(-0.151985\pi\)
\(114\) −705.000 −0.579204
\(115\) 0 0
\(116\) −918.000 −0.734777
\(117\) − 288.000i − 0.227569i
\(118\) − 3125.00i − 2.43796i
\(119\) −99.0000 −0.0762632
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) − 1600.00i − 1.18735i
\(123\) 417.000i 0.305688i
\(124\) −3026.00 −2.19147
\(125\) 0 0
\(126\) 135.000 0.0954504
\(127\) 1739.00i 1.21505i 0.794301 + 0.607525i \(0.207837\pi\)
−0.794301 + 0.607525i \(0.792163\pi\)
\(128\) − 2115.00i − 1.46048i
\(129\) 924.000 0.630649
\(130\) 0 0
\(131\) 1818.00 1.21251 0.606257 0.795269i \(-0.292670\pi\)
0.606257 + 0.795269i \(0.292670\pi\)
\(132\) 561.000i 0.369915i
\(133\) 141.000i 0.0919267i
\(134\) −1000.00 −0.644678
\(135\) 0 0
\(136\) 1485.00 0.936307
\(137\) − 870.000i − 0.542548i −0.962502 0.271274i \(-0.912555\pi\)
0.962502 0.271274i \(-0.0874450\pi\)
\(138\) 1695.00i 1.04557i
\(139\) 636.000 0.388092 0.194046 0.980992i \(-0.437839\pi\)
0.194046 + 0.980992i \(0.437839\pi\)
\(140\) 0 0
\(141\) 585.000 0.349403
\(142\) 4735.00i 2.79826i
\(143\) − 352.000i − 0.205844i
\(144\) −801.000 −0.463542
\(145\) 0 0
\(146\) −2240.00 −1.26975
\(147\) 1002.00i 0.562201i
\(148\) 323.000i 0.179395i
\(149\) 239.000 0.131407 0.0657035 0.997839i \(-0.479071\pi\)
0.0657035 + 0.997839i \(0.479071\pi\)
\(150\) 0 0
\(151\) 1208.00 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) − 2115.00i − 1.12861i
\(153\) 297.000i 0.156935i
\(154\) 165.000 0.0863382
\(155\) 0 0
\(156\) 1632.00 0.837593
\(157\) 1874.00i 0.952621i 0.879277 + 0.476310i \(0.158026\pi\)
−0.879277 + 0.476310i \(0.841974\pi\)
\(158\) − 3605.00i − 1.81518i
\(159\) −456.000 −0.227441
\(160\) 0 0
\(161\) 339.000 0.165944
\(162\) − 405.000i − 0.196419i
\(163\) − 1904.00i − 0.914925i −0.889229 0.457463i \(-0.848758\pi\)
0.889229 0.457463i \(-0.151242\pi\)
\(164\) −2363.00 −1.12512
\(165\) 0 0
\(166\) 710.000 0.331968
\(167\) 1180.00i 0.546773i 0.961904 + 0.273387i \(0.0881438\pi\)
−0.961904 + 0.273387i \(0.911856\pi\)
\(168\) 405.000i 0.185991i
\(169\) 1173.00 0.533910
\(170\) 0 0
\(171\) 423.000 0.189167
\(172\) 5236.00i 2.32117i
\(173\) − 3177.00i − 1.39620i −0.716000 0.698101i \(-0.754029\pi\)
0.716000 0.698101i \(-0.245971\pi\)
\(174\) 810.000 0.352908
\(175\) 0 0
\(176\) −979.000 −0.419289
\(177\) 1875.00i 0.796235i
\(178\) 2020.00i 0.850592i
\(179\) −1787.00 −0.746182 −0.373091 0.927795i \(-0.621702\pi\)
−0.373091 + 0.927795i \(0.621702\pi\)
\(180\) 0 0
\(181\) −835.000 −0.342901 −0.171450 0.985193i \(-0.554845\pi\)
−0.171450 + 0.985193i \(0.554845\pi\)
\(182\) − 480.000i − 0.195494i
\(183\) 960.000i 0.387788i
\(184\) −5085.00 −2.03734
\(185\) 0 0
\(186\) 2670.00 1.05255
\(187\) 363.000i 0.141953i
\(188\) 3315.00i 1.28602i
\(189\) −81.0000 −0.0311740
\(190\) 0 0
\(191\) 3613.00 1.36873 0.684365 0.729139i \(-0.260079\pi\)
0.684365 + 0.729139i \(0.260079\pi\)
\(192\) 861.000i 0.323632i
\(193\) 4204.00i 1.56793i 0.620805 + 0.783965i \(0.286806\pi\)
−0.620805 + 0.783965i \(0.713194\pi\)
\(194\) −395.000 −0.146182
\(195\) 0 0
\(196\) −5678.00 −2.06924
\(197\) 4517.00i 1.63362i 0.576908 + 0.816809i \(0.304259\pi\)
−0.576908 + 0.816809i \(0.695741\pi\)
\(198\) − 495.000i − 0.177667i
\(199\) −4164.00 −1.48331 −0.741654 0.670783i \(-0.765958\pi\)
−0.741654 + 0.670783i \(0.765958\pi\)
\(200\) 0 0
\(201\) 600.000 0.210551
\(202\) 2725.00i 0.949160i
\(203\) − 162.000i − 0.0560107i
\(204\) −1683.00 −0.577616
\(205\) 0 0
\(206\) −6530.00 −2.20858
\(207\) − 1017.00i − 0.341480i
\(208\) 2848.00i 0.949391i
\(209\) 517.000 0.171108
\(210\) 0 0
\(211\) 4660.00 1.52042 0.760208 0.649680i \(-0.225097\pi\)
0.760208 + 0.649680i \(0.225097\pi\)
\(212\) − 2584.00i − 0.837122i
\(213\) − 2841.00i − 0.913907i
\(214\) −9690.00 −3.09530
\(215\) 0 0
\(216\) 1215.00 0.382733
\(217\) − 534.000i − 0.167052i
\(218\) − 2880.00i − 0.894762i
\(219\) 1344.00 0.414699
\(220\) 0 0
\(221\) 1056.00 0.321422
\(222\) − 285.000i − 0.0861619i
\(223\) 3560.00i 1.06904i 0.845157 + 0.534518i \(0.179507\pi\)
−0.845157 + 0.534518i \(0.820493\pi\)
\(224\) −255.000 −0.0760621
\(225\) 0 0
\(226\) −5520.00 −1.62471
\(227\) 4678.00i 1.36780i 0.729577 + 0.683898i \(0.239717\pi\)
−0.729577 + 0.683898i \(0.760283\pi\)
\(228\) 2397.00i 0.696251i
\(229\) 4447.00 1.28326 0.641629 0.767015i \(-0.278259\pi\)
0.641629 + 0.767015i \(0.278259\pi\)
\(230\) 0 0
\(231\) −99.0000 −0.0281979
\(232\) 2430.00i 0.687661i
\(233\) 411.000i 0.115560i 0.998329 + 0.0577801i \(0.0184022\pi\)
−0.998329 + 0.0577801i \(0.981598\pi\)
\(234\) −1440.00 −0.402290
\(235\) 0 0
\(236\) −10625.0 −2.93063
\(237\) 2163.00i 0.592835i
\(238\) 495.000i 0.134815i
\(239\) 6380.00 1.72673 0.863364 0.504582i \(-0.168353\pi\)
0.863364 + 0.504582i \(0.168353\pi\)
\(240\) 0 0
\(241\) 7282.00 1.94637 0.973184 0.230027i \(-0.0738813\pi\)
0.973184 + 0.230027i \(0.0738813\pi\)
\(242\) − 605.000i − 0.160706i
\(243\) 243.000i 0.0641500i
\(244\) −5440.00 −1.42730
\(245\) 0 0
\(246\) 2085.00 0.540385
\(247\) − 1504.00i − 0.387438i
\(248\) 8010.00i 2.05095i
\(249\) −426.000 −0.108420
\(250\) 0 0
\(251\) −4728.00 −1.18896 −0.594480 0.804111i \(-0.702642\pi\)
−0.594480 + 0.804111i \(0.702642\pi\)
\(252\) − 459.000i − 0.114739i
\(253\) − 1243.00i − 0.308880i
\(254\) 8695.00 2.14792
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) − 5418.00i − 1.31504i −0.753437 0.657521i \(-0.771605\pi\)
0.753437 0.657521i \(-0.228395\pi\)
\(258\) − 4620.00i − 1.11484i
\(259\) −57.0000 −0.0136749
\(260\) 0 0
\(261\) −486.000 −0.115259
\(262\) − 9090.00i − 2.14344i
\(263\) − 3354.00i − 0.786375i −0.919458 0.393187i \(-0.871372\pi\)
0.919458 0.393187i \(-0.128628\pi\)
\(264\) 1485.00 0.346195
\(265\) 0 0
\(266\) 705.000 0.162505
\(267\) − 1212.00i − 0.277802i
\(268\) 3400.00i 0.774955i
\(269\) −1062.00 −0.240711 −0.120356 0.992731i \(-0.538403\pi\)
−0.120356 + 0.992731i \(0.538403\pi\)
\(270\) 0 0
\(271\) −4821.00 −1.08065 −0.540323 0.841458i \(-0.681698\pi\)
−0.540323 + 0.841458i \(0.681698\pi\)
\(272\) − 2937.00i − 0.654712i
\(273\) 288.000i 0.0638482i
\(274\) −4350.00 −0.959099
\(275\) 0 0
\(276\) 5763.00 1.25685
\(277\) − 4.00000i 0 0.000867642i −1.00000 0.000433821i \(-0.999862\pi\)
1.00000 0.000433821i \(-0.000138089\pi\)
\(278\) − 3180.00i − 0.686057i
\(279\) −1602.00 −0.343761
\(280\) 0 0
\(281\) 4647.00 0.986537 0.493268 0.869877i \(-0.335802\pi\)
0.493268 + 0.869877i \(0.335802\pi\)
\(282\) − 2925.00i − 0.617664i
\(283\) − 4283.00i − 0.899639i −0.893119 0.449820i \(-0.851488\pi\)
0.893119 0.449820i \(-0.148512\pi\)
\(284\) 16099.0 3.36373
\(285\) 0 0
\(286\) −1760.00 −0.363885
\(287\) − 417.000i − 0.0857656i
\(288\) 765.000i 0.156521i
\(289\) 3824.00 0.778343
\(290\) 0 0
\(291\) 237.000 0.0477429
\(292\) 7616.00i 1.52634i
\(293\) − 6811.00i − 1.35803i −0.734124 0.679015i \(-0.762407\pi\)
0.734124 0.679015i \(-0.237593\pi\)
\(294\) 5010.00 0.993841
\(295\) 0 0
\(296\) 855.000 0.167891
\(297\) 297.000i 0.0580259i
\(298\) − 1195.00i − 0.232297i
\(299\) −3616.00 −0.699394
\(300\) 0 0
\(301\) −924.000 −0.176938
\(302\) − 6040.00i − 1.15087i
\(303\) − 1635.00i − 0.309994i
\(304\) −4183.00 −0.789183
\(305\) 0 0
\(306\) 1485.00 0.277424
\(307\) 460.000i 0.0855166i 0.999085 + 0.0427583i \(0.0136145\pi\)
−0.999085 + 0.0427583i \(0.986385\pi\)
\(308\) − 561.000i − 0.103786i
\(309\) 3918.00 0.721318
\(310\) 0 0
\(311\) 8328.00 1.51845 0.759224 0.650829i \(-0.225579\pi\)
0.759224 + 0.650829i \(0.225579\pi\)
\(312\) − 4320.00i − 0.783884i
\(313\) − 5929.00i − 1.07069i −0.844633 0.535346i \(-0.820181\pi\)
0.844633 0.535346i \(-0.179819\pi\)
\(314\) 9370.00 1.68401
\(315\) 0 0
\(316\) −12257.0 −2.18199
\(317\) − 5040.00i − 0.892980i −0.894789 0.446490i \(-0.852674\pi\)
0.894789 0.446490i \(-0.147326\pi\)
\(318\) 2280.00i 0.402063i
\(319\) −594.000 −0.104256
\(320\) 0 0
\(321\) 5814.00 1.01092
\(322\) − 1695.00i − 0.293350i
\(323\) 1551.00i 0.267183i
\(324\) −1377.00 −0.236111
\(325\) 0 0
\(326\) −9520.00 −1.61737
\(327\) 1728.00i 0.292228i
\(328\) 6255.00i 1.05297i
\(329\) −585.000 −0.0980307
\(330\) 0 0
\(331\) 10396.0 1.72633 0.863166 0.504920i \(-0.168478\pi\)
0.863166 + 0.504920i \(0.168478\pi\)
\(332\) − 2414.00i − 0.399053i
\(333\) 171.000i 0.0281404i
\(334\) 5900.00 0.966568
\(335\) 0 0
\(336\) 801.000 0.130054
\(337\) 7236.00i 1.16964i 0.811162 + 0.584822i \(0.198836\pi\)
−0.811162 + 0.584822i \(0.801164\pi\)
\(338\) − 5865.00i − 0.943828i
\(339\) 3312.00 0.530629
\(340\) 0 0
\(341\) −1958.00 −0.310943
\(342\) − 2115.00i − 0.334404i
\(343\) − 2031.00i − 0.319719i
\(344\) 13860.0 2.17233
\(345\) 0 0
\(346\) −15885.0 −2.46816
\(347\) − 1468.00i − 0.227108i −0.993532 0.113554i \(-0.963777\pi\)
0.993532 0.113554i \(-0.0362234\pi\)
\(348\) − 2754.00i − 0.424224i
\(349\) −5690.00 −0.872718 −0.436359 0.899773i \(-0.643732\pi\)
−0.436359 + 0.899773i \(0.643732\pi\)
\(350\) 0 0
\(351\) 864.000 0.131387
\(352\) 935.000i 0.141579i
\(353\) 5376.00i 0.810582i 0.914188 + 0.405291i \(0.132830\pi\)
−0.914188 + 0.405291i \(0.867170\pi\)
\(354\) 9375.00 1.40756
\(355\) 0 0
\(356\) 6868.00 1.02248
\(357\) − 297.000i − 0.0440306i
\(358\) 8935.00i 1.31908i
\(359\) −3734.00 −0.548950 −0.274475 0.961594i \(-0.588504\pi\)
−0.274475 + 0.961594i \(0.588504\pi\)
\(360\) 0 0
\(361\) −4650.00 −0.677941
\(362\) 4175.00i 0.606169i
\(363\) 363.000i 0.0524864i
\(364\) −1632.00 −0.235000
\(365\) 0 0
\(366\) 4800.00 0.685519
\(367\) 10274.0i 1.46130i 0.682750 + 0.730652i \(0.260784\pi\)
−0.682750 + 0.730652i \(0.739216\pi\)
\(368\) 10057.0i 1.42461i
\(369\) −1251.00 −0.176489
\(370\) 0 0
\(371\) 456.000 0.0638122
\(372\) − 9078.00i − 1.26525i
\(373\) 13662.0i 1.89649i 0.317537 + 0.948246i \(0.397144\pi\)
−0.317537 + 0.948246i \(0.602856\pi\)
\(374\) 1815.00 0.250940
\(375\) 0 0
\(376\) 8775.00 1.20355
\(377\) 1728.00i 0.236065i
\(378\) 405.000i 0.0551083i
\(379\) 7906.00 1.07151 0.535757 0.844372i \(-0.320026\pi\)
0.535757 + 0.844372i \(0.320026\pi\)
\(380\) 0 0
\(381\) −5217.00 −0.701509
\(382\) − 18065.0i − 2.41960i
\(383\) − 3168.00i − 0.422656i −0.977415 0.211328i \(-0.932221\pi\)
0.977415 0.211328i \(-0.0677788\pi\)
\(384\) 6345.00 0.843208
\(385\) 0 0
\(386\) 21020.0 2.77174
\(387\) 2772.00i 0.364105i
\(388\) 1343.00i 0.175723i
\(389\) −10770.0 −1.40375 −0.701877 0.712298i \(-0.747655\pi\)
−0.701877 + 0.712298i \(0.747655\pi\)
\(390\) 0 0
\(391\) 3729.00 0.482311
\(392\) 15030.0i 1.93656i
\(393\) 5454.00i 0.700046i
\(394\) 22585.0 2.88786
\(395\) 0 0
\(396\) −1683.00 −0.213571
\(397\) − 5670.00i − 0.716799i −0.933568 0.358399i \(-0.883323\pi\)
0.933568 0.358399i \(-0.116677\pi\)
\(398\) 20820.0i 2.62214i
\(399\) −423.000 −0.0530739
\(400\) 0 0
\(401\) 832.000 0.103611 0.0518056 0.998657i \(-0.483502\pi\)
0.0518056 + 0.998657i \(0.483502\pi\)
\(402\) − 3000.00i − 0.372205i
\(403\) 5696.00i 0.704064i
\(404\) 9265.00 1.14097
\(405\) 0 0
\(406\) −810.000 −0.0990139
\(407\) 209.000i 0.0254539i
\(408\) 4455.00i 0.540577i
\(409\) 5712.00 0.690563 0.345281 0.938499i \(-0.387783\pi\)
0.345281 + 0.938499i \(0.387783\pi\)
\(410\) 0 0
\(411\) 2610.00 0.313240
\(412\) 22202.0i 2.65489i
\(413\) − 1875.00i − 0.223396i
\(414\) −5085.00 −0.603657
\(415\) 0 0
\(416\) 2720.00 0.320574
\(417\) 1908.00i 0.224065i
\(418\) − 2585.00i − 0.302480i
\(419\) 4559.00 0.531555 0.265778 0.964034i \(-0.414371\pi\)
0.265778 + 0.964034i \(0.414371\pi\)
\(420\) 0 0
\(421\) 6855.00 0.793568 0.396784 0.917912i \(-0.370126\pi\)
0.396784 + 0.917912i \(0.370126\pi\)
\(422\) − 23300.0i − 2.68774i
\(423\) 1755.00i 0.201728i
\(424\) −6840.00 −0.783443
\(425\) 0 0
\(426\) −14205.0 −1.61557
\(427\) − 960.000i − 0.108800i
\(428\) 32946.0i 3.72081i
\(429\) 1056.00 0.118844
\(430\) 0 0
\(431\) 10770.0 1.20365 0.601824 0.798628i \(-0.294441\pi\)
0.601824 + 0.798628i \(0.294441\pi\)
\(432\) − 2403.00i − 0.267626i
\(433\) 8498.00i 0.943159i 0.881824 + 0.471579i \(0.156316\pi\)
−0.881824 + 0.471579i \(0.843684\pi\)
\(434\) −2670.00 −0.295309
\(435\) 0 0
\(436\) −9792.00 −1.07558
\(437\) − 5311.00i − 0.581372i
\(438\) − 6720.00i − 0.733091i
\(439\) −9835.00 −1.06925 −0.534623 0.845091i \(-0.679546\pi\)
−0.534623 + 0.845091i \(0.679546\pi\)
\(440\) 0 0
\(441\) −3006.00 −0.324587
\(442\) − 5280.00i − 0.568199i
\(443\) − 10745.0i − 1.15239i −0.817311 0.576197i \(-0.804536\pi\)
0.817311 0.576197i \(-0.195464\pi\)
\(444\) −969.000 −0.103574
\(445\) 0 0
\(446\) 17800.0 1.88981
\(447\) 717.000i 0.0758679i
\(448\) − 861.000i − 0.0908001i
\(449\) −8356.00 −0.878272 −0.439136 0.898421i \(-0.644715\pi\)
−0.439136 + 0.898421i \(0.644715\pi\)
\(450\) 0 0
\(451\) −1529.00 −0.159640
\(452\) 18768.0i 1.95304i
\(453\) 3624.00i 0.375873i
\(454\) 23390.0 2.41795
\(455\) 0 0
\(456\) 6345.00 0.651605
\(457\) 7058.00i 0.722449i 0.932479 + 0.361225i \(0.117641\pi\)
−0.932479 + 0.361225i \(0.882359\pi\)
\(458\) − 22235.0i − 2.26850i
\(459\) −891.000 −0.0906064
\(460\) 0 0
\(461\) 646.000 0.0652651 0.0326326 0.999467i \(-0.489611\pi\)
0.0326326 + 0.999467i \(0.489611\pi\)
\(462\) 495.000i 0.0498474i
\(463\) − 8982.00i − 0.901574i −0.892631 0.450787i \(-0.851143\pi\)
0.892631 0.450787i \(-0.148857\pi\)
\(464\) 4806.00 0.480847
\(465\) 0 0
\(466\) 2055.00 0.204283
\(467\) 13476.0i 1.33532i 0.744466 + 0.667661i \(0.232704\pi\)
−0.744466 + 0.667661i \(0.767296\pi\)
\(468\) 4896.00i 0.483585i
\(469\) −600.000 −0.0590734
\(470\) 0 0
\(471\) −5622.00 −0.549996
\(472\) 28125.0i 2.74271i
\(473\) 3388.00i 0.329345i
\(474\) 10815.0 1.04799
\(475\) 0 0
\(476\) 1683.00 0.162059
\(477\) − 1368.00i − 0.131313i
\(478\) − 31900.0i − 3.05245i
\(479\) −12996.0 −1.23967 −0.619835 0.784732i \(-0.712801\pi\)
−0.619835 + 0.784732i \(0.712801\pi\)
\(480\) 0 0
\(481\) 608.000 0.0576350
\(482\) − 36410.0i − 3.44073i
\(483\) 1017.00i 0.0958077i
\(484\) −2057.00 −0.193182
\(485\) 0 0
\(486\) 1215.00 0.113402
\(487\) 6026.00i 0.560707i 0.959897 + 0.280353i \(0.0904516\pi\)
−0.959897 + 0.280353i \(0.909548\pi\)
\(488\) 14400.0i 1.33577i
\(489\) 5712.00 0.528232
\(490\) 0 0
\(491\) 11698.0 1.07520 0.537600 0.843200i \(-0.319331\pi\)
0.537600 + 0.843200i \(0.319331\pi\)
\(492\) − 7089.00i − 0.649587i
\(493\) − 1782.00i − 0.162794i
\(494\) −7520.00 −0.684900
\(495\) 0 0
\(496\) 15842.0 1.43413
\(497\) 2841.00i 0.256411i
\(498\) 2130.00i 0.191662i
\(499\) 17052.0 1.52976 0.764882 0.644170i \(-0.222797\pi\)
0.764882 + 0.644170i \(0.222797\pi\)
\(500\) 0 0
\(501\) −3540.00 −0.315680
\(502\) 23640.0i 2.10180i
\(503\) − 932.000i − 0.0826160i −0.999146 0.0413080i \(-0.986848\pi\)
0.999146 0.0413080i \(-0.0131525\pi\)
\(504\) −1215.00 −0.107382
\(505\) 0 0
\(506\) −6215.00 −0.546029
\(507\) 3519.00i 0.308253i
\(508\) − 29563.0i − 2.58198i
\(509\) −4384.00 −0.381763 −0.190882 0.981613i \(-0.561135\pi\)
−0.190882 + 0.981613i \(0.561135\pi\)
\(510\) 0 0
\(511\) −1344.00 −0.116350
\(512\) 24475.0i 2.11260i
\(513\) 1269.00i 0.109216i
\(514\) −27090.0 −2.32469
\(515\) 0 0
\(516\) −15708.0 −1.34013
\(517\) 2145.00i 0.182470i
\(518\) 285.000i 0.0241741i
\(519\) 9531.00 0.806097
\(520\) 0 0
\(521\) −2322.00 −0.195257 −0.0976283 0.995223i \(-0.531126\pi\)
−0.0976283 + 0.995223i \(0.531126\pi\)
\(522\) 2430.00i 0.203751i
\(523\) − 9749.00i − 0.815094i −0.913184 0.407547i \(-0.866384\pi\)
0.913184 0.407547i \(-0.133616\pi\)
\(524\) −30906.0 −2.57659
\(525\) 0 0
\(526\) −16770.0 −1.39013
\(527\) − 5874.00i − 0.485532i
\(528\) − 2937.00i − 0.242077i
\(529\) −602.000 −0.0494781
\(530\) 0 0
\(531\) −5625.00 −0.459707
\(532\) − 2397.00i − 0.195344i
\(533\) 4448.00i 0.361471i
\(534\) −6060.00 −0.491090
\(535\) 0 0
\(536\) 9000.00 0.725263
\(537\) − 5361.00i − 0.430809i
\(538\) 5310.00i 0.425521i
\(539\) −3674.00 −0.293600
\(540\) 0 0
\(541\) 4208.00 0.334410 0.167205 0.985922i \(-0.446526\pi\)
0.167205 + 0.985922i \(0.446526\pi\)
\(542\) 24105.0i 1.91033i
\(543\) − 2505.00i − 0.197974i
\(544\) −2805.00 −0.221072
\(545\) 0 0
\(546\) 1440.00 0.112869
\(547\) − 10179.0i − 0.795654i −0.917461 0.397827i \(-0.869764\pi\)
0.917461 0.397827i \(-0.130236\pi\)
\(548\) 14790.0i 1.15292i
\(549\) −2880.00 −0.223890
\(550\) 0 0
\(551\) −2538.00 −0.196229
\(552\) − 15255.0i − 1.17626i
\(553\) − 2163.00i − 0.166329i
\(554\) −20.0000 −0.00153379
\(555\) 0 0
\(556\) −10812.0 −0.824696
\(557\) 2314.00i 0.176028i 0.996119 + 0.0880138i \(0.0280519\pi\)
−0.996119 + 0.0880138i \(0.971948\pi\)
\(558\) 8010.00i 0.607689i
\(559\) 9856.00 0.745732
\(560\) 0 0
\(561\) −1089.00 −0.0819565
\(562\) − 23235.0i − 1.74397i
\(563\) 24330.0i 1.82129i 0.413188 + 0.910646i \(0.364415\pi\)
−0.413188 + 0.910646i \(0.635585\pi\)
\(564\) −9945.00 −0.742482
\(565\) 0 0
\(566\) −21415.0 −1.59035
\(567\) − 243.000i − 0.0179983i
\(568\) − 42615.0i − 3.14804i
\(569\) −3445.00 −0.253817 −0.126909 0.991914i \(-0.540505\pi\)
−0.126909 + 0.991914i \(0.540505\pi\)
\(570\) 0 0
\(571\) −13056.0 −0.956877 −0.478438 0.878121i \(-0.658797\pi\)
−0.478438 + 0.878121i \(0.658797\pi\)
\(572\) 5984.00i 0.437419i
\(573\) 10839.0i 0.790237i
\(574\) −2085.00 −0.151614
\(575\) 0 0
\(576\) −2583.00 −0.186849
\(577\) 17347.0i 1.25159i 0.779989 + 0.625793i \(0.215225\pi\)
−0.779989 + 0.625793i \(0.784775\pi\)
\(578\) − 19120.0i − 1.37593i
\(579\) −12612.0 −0.905245
\(580\) 0 0
\(581\) 426.000 0.0304190
\(582\) − 1185.00i − 0.0843983i
\(583\) − 1672.00i − 0.118777i
\(584\) 20160.0 1.42847
\(585\) 0 0
\(586\) −34055.0 −2.40068
\(587\) − 8379.00i − 0.589162i −0.955626 0.294581i \(-0.904820\pi\)
0.955626 0.294581i \(-0.0951801\pi\)
\(588\) − 17034.0i − 1.19468i
\(589\) −8366.00 −0.585255
\(590\) 0 0
\(591\) −13551.0 −0.943170
\(592\) − 1691.00i − 0.117398i
\(593\) 1958.00i 0.135591i 0.997699 + 0.0677955i \(0.0215965\pi\)
−0.997699 + 0.0677955i \(0.978403\pi\)
\(594\) 1485.00 0.102576
\(595\) 0 0
\(596\) −4063.00 −0.279240
\(597\) − 12492.0i − 0.856388i
\(598\) 18080.0i 1.23636i
\(599\) −23583.0 −1.60864 −0.804320 0.594196i \(-0.797470\pi\)
−0.804320 + 0.594196i \(0.797470\pi\)
\(600\) 0 0
\(601\) −15328.0 −1.04034 −0.520168 0.854064i \(-0.674131\pi\)
−0.520168 + 0.854064i \(0.674131\pi\)
\(602\) 4620.00i 0.312786i
\(603\) 1800.00i 0.121562i
\(604\) −20536.0 −1.38344
\(605\) 0 0
\(606\) −8175.00 −0.547998
\(607\) − 160.000i − 0.0106988i −0.999986 0.00534942i \(-0.998297\pi\)
0.999986 0.00534942i \(-0.00170278\pi\)
\(608\) 3995.00i 0.266478i
\(609\) 486.000 0.0323378
\(610\) 0 0
\(611\) 6240.00 0.413164
\(612\) − 5049.00i − 0.333486i
\(613\) − 5948.00i − 0.391904i −0.980613 0.195952i \(-0.937220\pi\)
0.980613 0.195952i \(-0.0627798\pi\)
\(614\) 2300.00 0.151173
\(615\) 0 0
\(616\) −1485.00 −0.0971304
\(617\) − 334.000i − 0.0217931i −0.999941 0.0108965i \(-0.996531\pi\)
0.999941 0.0108965i \(-0.00346855\pi\)
\(618\) − 19590.0i − 1.27512i
\(619\) 7202.00 0.467646 0.233823 0.972279i \(-0.424876\pi\)
0.233823 + 0.972279i \(0.424876\pi\)
\(620\) 0 0
\(621\) 3051.00 0.197154
\(622\) − 41640.0i − 2.68426i
\(623\) 1212.00i 0.0779418i
\(624\) −8544.00 −0.548131
\(625\) 0 0
\(626\) −29645.0 −1.89274
\(627\) 1551.00i 0.0987894i
\(628\) − 31858.0i − 2.02432i
\(629\) −627.000 −0.0397458
\(630\) 0 0
\(631\) 10306.0 0.650199 0.325099 0.945680i \(-0.394602\pi\)
0.325099 + 0.945680i \(0.394602\pi\)
\(632\) 32445.0i 2.04208i
\(633\) 13980.0i 0.877812i
\(634\) −25200.0 −1.57858
\(635\) 0 0
\(636\) 7752.00 0.483313
\(637\) 10688.0i 0.664794i
\(638\) 2970.00i 0.184300i
\(639\) 8523.00 0.527644
\(640\) 0 0
\(641\) −1228.00 −0.0756678 −0.0378339 0.999284i \(-0.512046\pi\)
−0.0378339 + 0.999284i \(0.512046\pi\)
\(642\) − 29070.0i − 1.78707i
\(643\) − 18454.0i − 1.13181i −0.824470 0.565906i \(-0.808527\pi\)
0.824470 0.565906i \(-0.191473\pi\)
\(644\) −5763.00 −0.352630
\(645\) 0 0
\(646\) 7755.00 0.472316
\(647\) − 17647.0i − 1.07230i −0.844124 0.536148i \(-0.819879\pi\)
0.844124 0.536148i \(-0.180121\pi\)
\(648\) 3645.00i 0.220971i
\(649\) −6875.00 −0.415820
\(650\) 0 0
\(651\) 1602.00 0.0964475
\(652\) 32368.0i 1.94422i
\(653\) 25918.0i 1.55322i 0.629984 + 0.776608i \(0.283061\pi\)
−0.629984 + 0.776608i \(0.716939\pi\)
\(654\) 8640.00 0.516591
\(655\) 0 0
\(656\) 12371.0 0.736290
\(657\) 4032.00i 0.239427i
\(658\) 2925.00i 0.173295i
\(659\) −12864.0 −0.760410 −0.380205 0.924902i \(-0.624147\pi\)
−0.380205 + 0.924902i \(0.624147\pi\)
\(660\) 0 0
\(661\) −11419.0 −0.671933 −0.335966 0.941874i \(-0.609063\pi\)
−0.335966 + 0.941874i \(0.609063\pi\)
\(662\) − 51980.0i − 3.05175i
\(663\) 3168.00i 0.185573i
\(664\) −6390.00 −0.373464
\(665\) 0 0
\(666\) 855.000 0.0497456
\(667\) 6102.00i 0.354228i
\(668\) − 20060.0i − 1.16189i
\(669\) −10680.0 −0.617209
\(670\) 0 0
\(671\) −3520.00 −0.202516
\(672\) − 765.000i − 0.0439145i
\(673\) 15784.0i 0.904054i 0.892004 + 0.452027i \(0.149299\pi\)
−0.892004 + 0.452027i \(0.850701\pi\)
\(674\) 36180.0 2.06766
\(675\) 0 0
\(676\) −19941.0 −1.13456
\(677\) 26050.0i 1.47885i 0.673238 + 0.739426i \(0.264903\pi\)
−0.673238 + 0.739426i \(0.735097\pi\)
\(678\) − 16560.0i − 0.938028i
\(679\) −237.000 −0.0133950
\(680\) 0 0
\(681\) −14034.0 −0.789698
\(682\) 9790.00i 0.549675i
\(683\) − 15095.0i − 0.845672i −0.906206 0.422836i \(-0.861035\pi\)
0.906206 0.422836i \(-0.138965\pi\)
\(684\) −7191.00 −0.401981
\(685\) 0 0
\(686\) −10155.0 −0.565189
\(687\) 13341.0i 0.740889i
\(688\) − 27412.0i − 1.51900i
\(689\) −4864.00 −0.268946
\(690\) 0 0
\(691\) 15896.0 0.875126 0.437563 0.899188i \(-0.355842\pi\)
0.437563 + 0.899188i \(0.355842\pi\)
\(692\) 54009.0i 2.96693i
\(693\) − 297.000i − 0.0162801i
\(694\) −7340.00 −0.401473
\(695\) 0 0
\(696\) −7290.00 −0.397021
\(697\) − 4587.00i − 0.249275i
\(698\) 28450.0i 1.54276i
\(699\) −1233.00 −0.0667187
\(700\) 0 0
\(701\) 10529.0 0.567296 0.283648 0.958928i \(-0.408455\pi\)
0.283648 + 0.958928i \(0.408455\pi\)
\(702\) − 4320.00i − 0.232262i
\(703\) 893.000i 0.0479092i
\(704\) −3157.00 −0.169011
\(705\) 0 0
\(706\) 26880.0 1.43292
\(707\) 1635.00i 0.0869738i
\(708\) − 31875.0i − 1.69200i
\(709\) 16087.0 0.852130 0.426065 0.904693i \(-0.359900\pi\)
0.426065 + 0.904693i \(0.359900\pi\)
\(710\) 0 0
\(711\) −6489.00 −0.342274
\(712\) − 18180.0i − 0.956916i
\(713\) 20114.0i 1.05649i
\(714\) −1485.00 −0.0778358
\(715\) 0 0
\(716\) 30379.0 1.58564
\(717\) 19140.0i 0.996927i
\(718\) 18670.0i 0.970415i
\(719\) −24336.0 −1.26228 −0.631140 0.775669i \(-0.717413\pi\)
−0.631140 + 0.775669i \(0.717413\pi\)
\(720\) 0 0
\(721\) −3918.00 −0.202377
\(722\) 23250.0i 1.19844i
\(723\) 21846.0i 1.12374i
\(724\) 14195.0 0.728664
\(725\) 0 0
\(726\) 1815.00 0.0927837
\(727\) − 13960.0i − 0.712170i −0.934454 0.356085i \(-0.884111\pi\)
0.934454 0.356085i \(-0.115889\pi\)
\(728\) 4320.00i 0.219931i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −10164.0 −0.514267
\(732\) − 16320.0i − 0.824050i
\(733\) 9252.00i 0.466208i 0.972452 + 0.233104i \(0.0748882\pi\)
−0.972452 + 0.233104i \(0.925112\pi\)
\(734\) 51370.0 2.58324
\(735\) 0 0
\(736\) 9605.00 0.481039
\(737\) 2200.00i 0.109957i
\(738\) 6255.00i 0.311992i
\(739\) −28453.0 −1.41632 −0.708160 0.706052i \(-0.750474\pi\)
−0.708160 + 0.706052i \(0.750474\pi\)
\(740\) 0 0
\(741\) 4512.00 0.223688
\(742\) − 2280.00i − 0.112805i
\(743\) 512.000i 0.0252806i 0.999920 + 0.0126403i \(0.00402363\pi\)
−0.999920 + 0.0126403i \(0.995976\pi\)
\(744\) −24030.0 −1.18412
\(745\) 0 0
\(746\) 68310.0 3.35256
\(747\) − 1278.00i − 0.0625965i
\(748\) − 6171.00i − 0.301650i
\(749\) −5814.00 −0.283630
\(750\) 0 0
\(751\) 772.000 0.0375109 0.0187554 0.999824i \(-0.494030\pi\)
0.0187554 + 0.999824i \(0.494030\pi\)
\(752\) − 17355.0i − 0.841585i
\(753\) − 14184.0i − 0.686446i
\(754\) 8640.00 0.417308
\(755\) 0 0
\(756\) 1377.00 0.0662447
\(757\) − 8058.00i − 0.386886i −0.981111 0.193443i \(-0.938034\pi\)
0.981111 0.193443i \(-0.0619655\pi\)
\(758\) − 39530.0i − 1.89419i
\(759\) 3729.00 0.178332
\(760\) 0 0
\(761\) 18650.0 0.888386 0.444193 0.895931i \(-0.353490\pi\)
0.444193 + 0.895931i \(0.353490\pi\)
\(762\) 26085.0i 1.24010i
\(763\) − 1728.00i − 0.0819893i
\(764\) −61421.0 −2.90855
\(765\) 0 0
\(766\) −15840.0 −0.747157
\(767\) 20000.0i 0.941536i
\(768\) − 24837.0i − 1.16696i
\(769\) 7144.00 0.335005 0.167503 0.985872i \(-0.446430\pi\)
0.167503 + 0.985872i \(0.446430\pi\)
\(770\) 0 0
\(771\) 16254.0 0.759239
\(772\) − 71468.0i − 3.33185i
\(773\) − 1904.00i − 0.0885927i −0.999018 0.0442963i \(-0.985895\pi\)
0.999018 0.0442963i \(-0.0141046\pi\)
\(774\) 13860.0 0.643653
\(775\) 0 0
\(776\) 3555.00 0.164455
\(777\) − 171.000i − 0.00789523i
\(778\) 53850.0i 2.48151i
\(779\) −6533.00 −0.300474
\(780\) 0 0
\(781\) 10417.0 0.477272
\(782\) − 18645.0i − 0.852614i
\(783\) − 1458.00i − 0.0665449i
\(784\) 29726.0 1.35414
\(785\) 0 0
\(786\) 27270.0 1.23752
\(787\) − 7555.00i − 0.342194i −0.985254 0.171097i \(-0.945269\pi\)
0.985254 0.171097i \(-0.0547311\pi\)
\(788\) − 76789.0i − 3.47144i
\(789\) 10062.0 0.454014
\(790\) 0 0
\(791\) −3312.00 −0.148876
\(792\) 4455.00i 0.199876i
\(793\) 10240.0i 0.458554i
\(794\) −28350.0 −1.26713
\(795\) 0 0
\(796\) 70788.0 3.15203
\(797\) − 24950.0i − 1.10888i −0.832225 0.554438i \(-0.812933\pi\)
0.832225 0.554438i \(-0.187067\pi\)
\(798\) 2115.00i 0.0938223i
\(799\) −6435.00 −0.284924
\(800\) 0 0
\(801\) 3636.00 0.160389
\(802\) − 4160.00i − 0.183160i
\(803\) 4928.00i 0.216570i
\(804\) −10200.0 −0.447421
\(805\) 0 0
\(806\) 28480.0 1.24462
\(807\) − 3186.00i − 0.138975i
\(808\) − 24525.0i − 1.06781i
\(809\) −19893.0 −0.864525 −0.432262 0.901748i \(-0.642285\pi\)
−0.432262 + 0.901748i \(0.642285\pi\)
\(810\) 0 0
\(811\) 34503.0 1.49391 0.746957 0.664872i \(-0.231514\pi\)
0.746957 + 0.664872i \(0.231514\pi\)
\(812\) 2754.00i 0.119023i
\(813\) − 14463.0i − 0.623911i
\(814\) 1045.00 0.0449966
\(815\) 0 0
\(816\) 8811.00 0.377998
\(817\) 14476.0i 0.619891i
\(818\) − 28560.0i − 1.22075i
\(819\) −864.000 −0.0368628
\(820\) 0 0
\(821\) 16890.0 0.717984 0.358992 0.933341i \(-0.383120\pi\)
0.358992 + 0.933341i \(0.383120\pi\)
\(822\) − 13050.0i − 0.553736i
\(823\) 34692.0i 1.46936i 0.678411 + 0.734682i \(0.262669\pi\)
−0.678411 + 0.734682i \(0.737331\pi\)
\(824\) 58770.0 2.48465
\(825\) 0 0
\(826\) −9375.00 −0.394913
\(827\) − 41424.0i − 1.74178i −0.491476 0.870891i \(-0.663543\pi\)
0.491476 0.870891i \(-0.336457\pi\)
\(828\) 17289.0i 0.725645i
\(829\) 18494.0 0.774817 0.387408 0.921908i \(-0.373370\pi\)
0.387408 + 0.921908i \(0.373370\pi\)
\(830\) 0 0
\(831\) 12.0000 0.000500933 0
\(832\) 9184.00i 0.382690i
\(833\) − 11022.0i − 0.458451i
\(834\) 9540.00 0.396095
\(835\) 0 0
\(836\) −8789.00 −0.363605
\(837\) − 4806.00i − 0.198470i
\(838\) − 22795.0i − 0.939666i
\(839\) −6680.00 −0.274874 −0.137437 0.990511i \(-0.543886\pi\)
−0.137437 + 0.990511i \(0.543886\pi\)
\(840\) 0 0
\(841\) −21473.0 −0.880438
\(842\) − 34275.0i − 1.40284i
\(843\) 13941.0i 0.569577i
\(844\) −79220.0 −3.23088
\(845\) 0 0
\(846\) 8775.00 0.356608
\(847\) − 363.000i − 0.0147259i
\(848\) 13528.0i 0.547822i
\(849\) 12849.0 0.519407
\(850\) 0 0
\(851\) 2147.00 0.0864844
\(852\) 48297.0i 1.94205i
\(853\) 43358.0i 1.74039i 0.492711 + 0.870193i \(0.336006\pi\)
−0.492711 + 0.870193i \(0.663994\pi\)
\(854\) −4800.00 −0.192333
\(855\) 0 0
\(856\) 87210.0 3.48222
\(857\) − 15585.0i − 0.621206i −0.950540 0.310603i \(-0.899469\pi\)
0.950540 0.310603i \(-0.100531\pi\)
\(858\) − 5280.00i − 0.210089i
\(859\) 17036.0 0.676672 0.338336 0.941025i \(-0.390136\pi\)
0.338336 + 0.941025i \(0.390136\pi\)
\(860\) 0 0
\(861\) 1251.00 0.0495168
\(862\) − 53850.0i − 2.12777i
\(863\) 28064.0i 1.10696i 0.832861 + 0.553482i \(0.186701\pi\)
−0.832861 + 0.553482i \(0.813299\pi\)
\(864\) −2295.00 −0.0903675
\(865\) 0 0
\(866\) 42490.0 1.66729
\(867\) 11472.0i 0.449377i
\(868\) 9078.00i 0.354985i
\(869\) −7931.00 −0.309598
\(870\) 0 0
\(871\) 6400.00 0.248973
\(872\) 25920.0i 1.00661i
\(873\) 711.000i 0.0275644i
\(874\) −26555.0 −1.02773
\(875\) 0 0
\(876\) −22848.0 −0.881236
\(877\) 22654.0i 0.872259i 0.899884 + 0.436130i \(0.143651\pi\)
−0.899884 + 0.436130i \(0.856349\pi\)
\(878\) 49175.0i 1.89018i
\(879\) 20433.0 0.784059
\(880\) 0 0
\(881\) −22380.0 −0.855847 −0.427924 0.903815i \(-0.640755\pi\)
−0.427924 + 0.903815i \(0.640755\pi\)
\(882\) 15030.0i 0.573794i
\(883\) 35174.0i 1.34054i 0.742116 + 0.670271i \(0.233822\pi\)
−0.742116 + 0.670271i \(0.766178\pi\)
\(884\) −17952.0 −0.683022
\(885\) 0 0
\(886\) −53725.0 −2.03716
\(887\) − 30868.0i − 1.16848i −0.811579 0.584242i \(-0.801392\pi\)
0.811579 0.584242i \(-0.198608\pi\)
\(888\) 2565.00i 0.0969322i
\(889\) 5217.00 0.196820
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) − 60520.0i − 2.27170i
\(893\) 9165.00i 0.343443i
\(894\) 3585.00 0.134117
\(895\) 0 0
\(896\) −6345.00 −0.236575
\(897\) − 10848.0i − 0.403795i
\(898\) 41780.0i 1.55258i
\(899\) 9612.00 0.356594
\(900\) 0 0
\(901\) 5016.00 0.185469
\(902\) 7645.00i 0.282207i
\(903\) − 2772.00i − 0.102155i
\(904\) 49680.0 1.82780
\(905\) 0 0
\(906\) 18120.0 0.664455
\(907\) − 10070.0i − 0.368654i −0.982865 0.184327i \(-0.940990\pi\)
0.982865 0.184327i \(-0.0590105\pi\)
\(908\) − 79526.0i − 2.90657i
\(909\) 4905.00 0.178975
\(910\) 0 0
\(911\) 1885.00 0.0685542 0.0342771 0.999412i \(-0.489087\pi\)
0.0342771 + 0.999412i \(0.489087\pi\)
\(912\) − 12549.0i − 0.455635i
\(913\) − 1562.00i − 0.0566207i
\(914\) 35290.0 1.27712
\(915\) 0 0
\(916\) −75599.0 −2.72692
\(917\) − 5454.00i − 0.196409i
\(918\) 4455.00i 0.160171i
\(919\) −23703.0 −0.850805 −0.425403 0.905004i \(-0.639867\pi\)
−0.425403 + 0.905004i \(0.639867\pi\)
\(920\) 0 0
\(921\) −1380.00 −0.0493730
\(922\) − 3230.00i − 0.115374i
\(923\) − 30304.0i − 1.08068i
\(924\) 1683.00 0.0599206
\(925\) 0 0
\(926\) −44910.0 −1.59377
\(927\) 11754.0i 0.416453i
\(928\) − 4590.00i − 0.162364i
\(929\) −53804.0 −1.90016 −0.950082 0.312001i \(-0.899001\pi\)
−0.950082 + 0.312001i \(0.899001\pi\)
\(930\) 0 0
\(931\) −15698.0 −0.552611
\(932\) − 6987.00i − 0.245565i
\(933\) 24984.0i 0.876677i
\(934\) 67380.0 2.36054
\(935\) 0 0
\(936\) 12960.0 0.452576
\(937\) − 1326.00i − 0.0462311i −0.999733 0.0231155i \(-0.992641\pi\)
0.999733 0.0231155i \(-0.00735856\pi\)
\(938\) 3000.00i 0.104428i
\(939\) 17787.0 0.618165
\(940\) 0 0
\(941\) −27109.0 −0.939137 −0.469569 0.882896i \(-0.655591\pi\)
−0.469569 + 0.882896i \(0.655591\pi\)
\(942\) 28110.0i 0.972265i
\(943\) 15707.0i 0.542408i
\(944\) 55625.0 1.91784
\(945\) 0 0
\(946\) 16940.0 0.582206
\(947\) − 31143.0i − 1.06865i −0.845279 0.534325i \(-0.820566\pi\)
0.845279 0.534325i \(-0.179434\pi\)
\(948\) − 36771.0i − 1.25977i
\(949\) 14336.0 0.490375
\(950\) 0 0
\(951\) 15120.0 0.515562
\(952\) − 4455.00i − 0.151667i
\(953\) − 879.000i − 0.0298779i −0.999888 0.0149389i \(-0.995245\pi\)
0.999888 0.0149389i \(-0.00475539\pi\)
\(954\) −6840.00 −0.232131
\(955\) 0 0
\(956\) −108460. −3.66930
\(957\) − 1782.00i − 0.0601921i
\(958\) 64980.0i 2.19145i
\(959\) −2610.00 −0.0878846
\(960\) 0 0
\(961\) 1893.00 0.0635427
\(962\) − 3040.00i − 0.101885i
\(963\) 17442.0i 0.583656i
\(964\) −123794. −4.13603
\(965\) 0 0
\(966\) 5085.00 0.169366
\(967\) 14824.0i 0.492976i 0.969146 + 0.246488i \(0.0792766\pi\)
−0.969146 + 0.246488i \(0.920723\pi\)
\(968\) 5445.00i 0.180794i
\(969\) −4653.00 −0.154258
\(970\) 0 0
\(971\) 34089.0 1.12664 0.563320 0.826239i \(-0.309524\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(972\) − 4131.00i − 0.136319i
\(973\) − 1908.00i − 0.0628650i
\(974\) 30130.0 0.991199
\(975\) 0 0
\(976\) 28480.0 0.934040
\(977\) − 33446.0i − 1.09522i −0.836733 0.547611i \(-0.815537\pi\)
0.836733 0.547611i \(-0.184463\pi\)
\(978\) − 28560.0i − 0.933792i
\(979\) 4444.00 0.145077
\(980\) 0 0
\(981\) −5184.00 −0.168718
\(982\) − 58490.0i − 1.90070i
\(983\) − 52025.0i − 1.68804i −0.536315 0.844018i \(-0.680184\pi\)
0.536315 0.844018i \(-0.319816\pi\)
\(984\) −18765.0 −0.607933
\(985\) 0 0
\(986\) −8910.00 −0.287781
\(987\) − 1755.00i − 0.0565980i
\(988\) 25568.0i 0.823306i
\(989\) 34804.0 1.11901
\(990\) 0 0
\(991\) −41260.0 −1.32257 −0.661285 0.750135i \(-0.729989\pi\)
−0.661285 + 0.750135i \(0.729989\pi\)
\(992\) − 15130.0i − 0.484252i
\(993\) 31188.0i 0.996698i
\(994\) 14205.0 0.453275
\(995\) 0 0
\(996\) 7242.00 0.230393
\(997\) 190.000i 0.00603547i 0.999995 + 0.00301773i \(0.000960576\pi\)
−0.999995 + 0.00301773i \(0.999039\pi\)
\(998\) − 85260.0i − 2.70427i
\(999\) −513.000 −0.0162468
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 825.4.c.b.199.1 2
5.2 odd 4 825.4.a.j.1.1 yes 1
5.3 odd 4 825.4.a.a.1.1 1
5.4 even 2 inner 825.4.c.b.199.2 2
15.2 even 4 2475.4.a.a.1.1 1
15.8 even 4 2475.4.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.a.1.1 1 5.3 odd 4
825.4.a.j.1.1 yes 1 5.2 odd 4
825.4.c.b.199.1 2 1.1 even 1 trivial
825.4.c.b.199.2 2 5.4 even 2 inner
2475.4.a.a.1.1 1 15.2 even 4
2475.4.a.k.1.1 1 15.8 even 4