Properties

Label 2175.2.c.p.349.16
Level $2175$
Weight $2$
Character 2175.349
Analytic conductor $17.367$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(349,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
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: \( x^{16} + 28x^{14} + 308x^{12} + 1671x^{10} + 4568x^{8} + 5616x^{6} + 2105x^{4} + 256x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.16
Root \(2.72810i\) of defining polynomial
Character \(\chi\) \(=\) 2175.349
Dual form 2175.2.c.p.349.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.72810i q^{2} +1.00000i q^{3} -5.44251 q^{4} -2.72810 q^{6} +4.61695i q^{7} -9.39150i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+2.72810i q^{2} +1.00000i q^{3} -5.44251 q^{4} -2.72810 q^{6} +4.61695i q^{7} -9.39150i q^{8} -1.00000 q^{9} -1.87758 q^{11} -5.44251i q^{12} -5.48400i q^{13} -12.5955 q^{14} +14.7359 q^{16} -5.74056i q^{17} -2.72810i q^{18} +5.40455 q^{19} -4.61695 q^{21} -5.12221i q^{22} +0.137016i q^{23} +9.39150 q^{24} +14.9609 q^{26} -1.00000i q^{27} -25.1278i q^{28} -1.00000 q^{29} -8.18167 q^{31} +21.4180i q^{32} -1.87758i q^{33} +15.6608 q^{34} +5.44251 q^{36} -4.45785i q^{37} +14.7441i q^{38} +5.48400 q^{39} -0.215243 q^{41} -12.5955i q^{42} -7.17115i q^{43} +10.2187 q^{44} -0.373793 q^{46} -9.91293i q^{47} +14.7359i q^{48} -14.3162 q^{49} +5.74056 q^{51} +29.8467i q^{52} +1.14754i q^{53} +2.72810 q^{54} +43.3601 q^{56} +5.40455i q^{57} -2.72810i q^{58} +0.244913 q^{59} +8.75284 q^{61} -22.3204i q^{62} -4.61695i q^{63} -28.9585 q^{64} +5.12221 q^{66} -6.53295i q^{67} +31.2431i q^{68} -0.137016 q^{69} -0.277412 q^{71} +9.39150i q^{72} +11.4933i q^{73} +12.1614 q^{74} -29.4143 q^{76} -8.66866i q^{77} +14.9609i q^{78} -1.66223 q^{79} +1.00000 q^{81} -0.587203i q^{82} -9.58514i q^{83} +25.1278 q^{84} +19.5636 q^{86} -1.00000i q^{87} +17.6333i q^{88} -16.3332 q^{89} +25.3193 q^{91} -0.745712i q^{92} -8.18167i q^{93} +27.0434 q^{94} -21.4180 q^{96} +8.04128i q^{97} -39.0559i q^{98} +1.87758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9} + 12 q^{11} - 18 q^{14} + 64 q^{16} - 4 q^{21} - 6 q^{24} + 36 q^{26} - 16 q^{29} + 16 q^{31} + 26 q^{34} + 24 q^{36} + 12 q^{39} + 4 q^{41} + 30 q^{44} + 48 q^{46} - 76 q^{49} + 24 q^{51} - 4 q^{54} + 116 q^{56} - 36 q^{59} + 24 q^{61} - 42 q^{64} - 6 q^{66} - 28 q^{69} + 48 q^{71} + 44 q^{74} - 20 q^{79} + 16 q^{81} + 28 q^{84} + 16 q^{86} - 68 q^{89} + 52 q^{91} + 86 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.72810i 1.92906i 0.263981 + 0.964528i \(0.414964\pi\)
−0.263981 + 0.964528i \(0.585036\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −5.44251 −2.72126
\(5\) 0 0
\(6\) −2.72810 −1.11374
\(7\) 4.61695i 1.74504i 0.488577 + 0.872521i \(0.337516\pi\)
−0.488577 + 0.872521i \(0.662484\pi\)
\(8\) − 9.39150i − 3.32040i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.87758 −0.566110 −0.283055 0.959104i \(-0.591348\pi\)
−0.283055 + 0.959104i \(0.591348\pi\)
\(12\) − 5.44251i − 1.57112i
\(13\) − 5.48400i − 1.52099i −0.649345 0.760494i \(-0.724957\pi\)
0.649345 0.760494i \(-0.275043\pi\)
\(14\) −12.5955 −3.36628
\(15\) 0 0
\(16\) 14.7359 3.68398
\(17\) − 5.74056i − 1.39229i −0.717901 0.696145i \(-0.754897\pi\)
0.717901 0.696145i \(-0.245103\pi\)
\(18\) − 2.72810i − 0.643019i
\(19\) 5.40455 1.23989 0.619944 0.784646i \(-0.287155\pi\)
0.619944 + 0.784646i \(0.287155\pi\)
\(20\) 0 0
\(21\) −4.61695 −1.00750
\(22\) − 5.12221i − 1.09206i
\(23\) 0.137016i 0.0285698i 0.999898 + 0.0142849i \(0.00454719\pi\)
−0.999898 + 0.0142849i \(0.995453\pi\)
\(24\) 9.39150 1.91703
\(25\) 0 0
\(26\) 14.9609 2.93407
\(27\) − 1.00000i − 0.192450i
\(28\) − 25.1278i − 4.74870i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −8.18167 −1.46947 −0.734736 0.678353i \(-0.762694\pi\)
−0.734736 + 0.678353i \(0.762694\pi\)
\(32\) 21.4180i 3.78620i
\(33\) − 1.87758i − 0.326844i
\(34\) 15.6608 2.68581
\(35\) 0 0
\(36\) 5.44251 0.907085
\(37\) − 4.45785i − 0.732866i −0.930444 0.366433i \(-0.880579\pi\)
0.930444 0.366433i \(-0.119421\pi\)
\(38\) 14.7441i 2.39181i
\(39\) 5.48400 0.878142
\(40\) 0 0
\(41\) −0.215243 −0.0336153 −0.0168076 0.999859i \(-0.505350\pi\)
−0.0168076 + 0.999859i \(0.505350\pi\)
\(42\) − 12.5955i − 1.94352i
\(43\) − 7.17115i − 1.09359i −0.837266 0.546795i \(-0.815848\pi\)
0.837266 0.546795i \(-0.184152\pi\)
\(44\) 10.2187 1.54053
\(45\) 0 0
\(46\) −0.373793 −0.0551128
\(47\) − 9.91293i − 1.44595i −0.690874 0.722975i \(-0.742774\pi\)
0.690874 0.722975i \(-0.257226\pi\)
\(48\) 14.7359i 2.12695i
\(49\) −14.3162 −2.04517
\(50\) 0 0
\(51\) 5.74056 0.803839
\(52\) 29.8467i 4.13899i
\(53\) 1.14754i 0.157627i 0.996889 + 0.0788134i \(0.0251131\pi\)
−0.996889 + 0.0788134i \(0.974887\pi\)
\(54\) 2.72810 0.371247
\(55\) 0 0
\(56\) 43.3601 5.79423
\(57\) 5.40455i 0.715850i
\(58\) − 2.72810i − 0.358217i
\(59\) 0.244913 0.0318849 0.0159425 0.999873i \(-0.494925\pi\)
0.0159425 + 0.999873i \(0.494925\pi\)
\(60\) 0 0
\(61\) 8.75284 1.12069 0.560343 0.828260i \(-0.310669\pi\)
0.560343 + 0.828260i \(0.310669\pi\)
\(62\) − 22.3204i − 2.83469i
\(63\) − 4.61695i − 0.581680i
\(64\) −28.9585 −3.61981
\(65\) 0 0
\(66\) 5.12221 0.630500
\(67\) − 6.53295i − 0.798127i −0.916923 0.399064i \(-0.869335\pi\)
0.916923 0.399064i \(-0.130665\pi\)
\(68\) 31.2431i 3.78878i
\(69\) −0.137016 −0.0164948
\(70\) 0 0
\(71\) −0.277412 −0.0329227 −0.0164614 0.999865i \(-0.505240\pi\)
−0.0164614 + 0.999865i \(0.505240\pi\)
\(72\) 9.39150i 1.10680i
\(73\) 11.4933i 1.34519i 0.740011 + 0.672594i \(0.234820\pi\)
−0.740011 + 0.672594i \(0.765180\pi\)
\(74\) 12.1614 1.41374
\(75\) 0 0
\(76\) −29.4143 −3.37405
\(77\) − 8.66866i − 0.987886i
\(78\) 14.9609i 1.69399i
\(79\) −1.66223 −0.187015 −0.0935076 0.995619i \(-0.529808\pi\)
−0.0935076 + 0.995619i \(0.529808\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 0.587203i − 0.0648457i
\(83\) − 9.58514i − 1.05211i −0.850452 0.526053i \(-0.823671\pi\)
0.850452 0.526053i \(-0.176329\pi\)
\(84\) 25.1278 2.74167
\(85\) 0 0
\(86\) 19.5636 2.10960
\(87\) − 1.00000i − 0.107211i
\(88\) 17.6333i 1.87971i
\(89\) −16.3332 −1.73132 −0.865659 0.500634i \(-0.833100\pi\)
−0.865659 + 0.500634i \(0.833100\pi\)
\(90\) 0 0
\(91\) 25.3193 2.65419
\(92\) − 0.745712i − 0.0777458i
\(93\) − 8.18167i − 0.848400i
\(94\) 27.0434 2.78932
\(95\) 0 0
\(96\) −21.4180 −2.18596
\(97\) 8.04128i 0.816468i 0.912877 + 0.408234i \(0.133855\pi\)
−0.912877 + 0.408234i \(0.866145\pi\)
\(98\) − 39.0559i − 3.94525i
\(99\) 1.87758 0.188703
\(100\) 0 0
\(101\) −12.3239 −1.22627 −0.613135 0.789978i \(-0.710092\pi\)
−0.613135 + 0.789978i \(0.710092\pi\)
\(102\) 15.6608i 1.55065i
\(103\) 0.660912i 0.0651216i 0.999470 + 0.0325608i \(0.0103663\pi\)
−0.999470 + 0.0325608i \(0.989634\pi\)
\(104\) −51.5030 −5.05028
\(105\) 0 0
\(106\) −3.13060 −0.304071
\(107\) − 14.5868i − 1.41016i −0.709128 0.705079i \(-0.750911\pi\)
0.709128 0.705079i \(-0.249089\pi\)
\(108\) 5.44251i 0.523706i
\(109\) −15.5050 −1.48511 −0.742557 0.669783i \(-0.766387\pi\)
−0.742557 + 0.669783i \(0.766387\pi\)
\(110\) 0 0
\(111\) 4.45785 0.423120
\(112\) 68.0349i 6.42869i
\(113\) 7.80660i 0.734383i 0.930145 + 0.367191i \(0.119681\pi\)
−0.930145 + 0.367191i \(0.880319\pi\)
\(114\) −14.7441 −1.38091
\(115\) 0 0
\(116\) 5.44251 0.505325
\(117\) 5.48400i 0.506996i
\(118\) 0.668145i 0.0615078i
\(119\) 26.5038 2.42960
\(120\) 0 0
\(121\) −7.47471 −0.679519
\(122\) 23.8786i 2.16187i
\(123\) − 0.215243i − 0.0194078i
\(124\) 44.5289 3.99881
\(125\) 0 0
\(126\) 12.5955 1.12209
\(127\) 19.7106i 1.74903i 0.484999 + 0.874515i \(0.338820\pi\)
−0.484999 + 0.874515i \(0.661180\pi\)
\(128\) − 36.1656i − 3.19662i
\(129\) 7.17115 0.631385
\(130\) 0 0
\(131\) −1.05738 −0.0923834 −0.0461917 0.998933i \(-0.514709\pi\)
−0.0461917 + 0.998933i \(0.514709\pi\)
\(132\) 10.2187i 0.889426i
\(133\) 24.9525i 2.16366i
\(134\) 17.8225 1.53963
\(135\) 0 0
\(136\) −53.9125 −4.62296
\(137\) 3.21467i 0.274648i 0.990526 + 0.137324i \(0.0438501\pi\)
−0.990526 + 0.137324i \(0.956150\pi\)
\(138\) − 0.373793i − 0.0318194i
\(139\) −8.48410 −0.719612 −0.359806 0.933027i \(-0.617157\pi\)
−0.359806 + 0.933027i \(0.617157\pi\)
\(140\) 0 0
\(141\) 9.91293 0.834820
\(142\) − 0.756807i − 0.0635098i
\(143\) 10.2966i 0.861046i
\(144\) −14.7359 −1.22799
\(145\) 0 0
\(146\) −31.3548 −2.59494
\(147\) − 14.3162i − 1.18078i
\(148\) 24.2619i 1.99432i
\(149\) 5.93205 0.485972 0.242986 0.970030i \(-0.421873\pi\)
0.242986 + 0.970030i \(0.421873\pi\)
\(150\) 0 0
\(151\) 8.06281 0.656142 0.328071 0.944653i \(-0.393601\pi\)
0.328071 + 0.944653i \(0.393601\pi\)
\(152\) − 50.7568i − 4.11692i
\(153\) 5.74056i 0.464097i
\(154\) 23.6490 1.90569
\(155\) 0 0
\(156\) −29.8467 −2.38965
\(157\) − 9.87127i − 0.787813i −0.919151 0.393906i \(-0.871123\pi\)
0.919151 0.393906i \(-0.128877\pi\)
\(158\) − 4.53472i − 0.360763i
\(159\) −1.14754 −0.0910058
\(160\) 0 0
\(161\) −0.632596 −0.0498555
\(162\) 2.72810i 0.214340i
\(163\) − 1.25711i − 0.0984643i −0.998787 0.0492321i \(-0.984323\pi\)
0.998787 0.0492321i \(-0.0156774\pi\)
\(164\) 1.17146 0.0914757
\(165\) 0 0
\(166\) 26.1492 2.02957
\(167\) − 15.1462i − 1.17205i −0.810294 0.586024i \(-0.800692\pi\)
0.810294 0.586024i \(-0.199308\pi\)
\(168\) 43.3601i 3.34530i
\(169\) −17.0742 −1.31340
\(170\) 0 0
\(171\) −5.40455 −0.413296
\(172\) 39.0291i 2.97594i
\(173\) − 2.34929i − 0.178613i −0.996004 0.0893065i \(-0.971535\pi\)
0.996004 0.0893065i \(-0.0284651\pi\)
\(174\) 2.72810 0.206816
\(175\) 0 0
\(176\) −27.6678 −2.08554
\(177\) 0.244913i 0.0184088i
\(178\) − 44.5586i − 3.33981i
\(179\) 17.7814 1.32904 0.664520 0.747270i \(-0.268636\pi\)
0.664520 + 0.747270i \(0.268636\pi\)
\(180\) 0 0
\(181\) −2.69871 −0.200594 −0.100297 0.994958i \(-0.531979\pi\)
−0.100297 + 0.994958i \(0.531979\pi\)
\(182\) 69.0735i 5.12007i
\(183\) 8.75284i 0.647029i
\(184\) 1.28679 0.0948632
\(185\) 0 0
\(186\) 22.3204 1.63661
\(187\) 10.7783i 0.788190i
\(188\) 53.9512i 3.93480i
\(189\) 4.61695 0.335833
\(190\) 0 0
\(191\) 10.7997 0.781442 0.390721 0.920509i \(-0.372226\pi\)
0.390721 + 0.920509i \(0.372226\pi\)
\(192\) − 28.9585i − 2.08990i
\(193\) 4.18447i 0.301205i 0.988594 + 0.150602i \(0.0481213\pi\)
−0.988594 + 0.150602i \(0.951879\pi\)
\(194\) −21.9374 −1.57501
\(195\) 0 0
\(196\) 77.9160 5.56543
\(197\) 5.76856i 0.410993i 0.978658 + 0.205496i \(0.0658809\pi\)
−0.978658 + 0.205496i \(0.934119\pi\)
\(198\) 5.12221i 0.364019i
\(199\) 5.45094 0.386407 0.193203 0.981159i \(-0.438112\pi\)
0.193203 + 0.981159i \(0.438112\pi\)
\(200\) 0 0
\(201\) 6.53295 0.460799
\(202\) − 33.6207i − 2.36554i
\(203\) − 4.61695i − 0.324046i
\(204\) −31.2431 −2.18745
\(205\) 0 0
\(206\) −1.80303 −0.125623
\(207\) − 0.137016i − 0.00952328i
\(208\) − 80.8117i − 5.60328i
\(209\) −10.1474 −0.701914
\(210\) 0 0
\(211\) 0.863332 0.0594342 0.0297171 0.999558i \(-0.490539\pi\)
0.0297171 + 0.999558i \(0.490539\pi\)
\(212\) − 6.24550i − 0.428943i
\(213\) − 0.277412i − 0.0190080i
\(214\) 39.7942 2.72027
\(215\) 0 0
\(216\) −9.39150 −0.639011
\(217\) − 37.7743i − 2.56429i
\(218\) − 42.2993i − 2.86487i
\(219\) −11.4933 −0.776645
\(220\) 0 0
\(221\) −31.4812 −2.11766
\(222\) 12.1614i 0.816223i
\(223\) 0.615197i 0.0411967i 0.999788 + 0.0205983i \(0.00655712\pi\)
−0.999788 + 0.0205983i \(0.993443\pi\)
\(224\) −98.8856 −6.60707
\(225\) 0 0
\(226\) −21.2972 −1.41667
\(227\) − 23.7230i − 1.57455i −0.616602 0.787275i \(-0.711491\pi\)
0.616602 0.787275i \(-0.288509\pi\)
\(228\) − 29.4143i − 1.94801i
\(229\) 20.9808 1.38645 0.693226 0.720721i \(-0.256189\pi\)
0.693226 + 0.720721i \(0.256189\pi\)
\(230\) 0 0
\(231\) 8.66866 0.570356
\(232\) 9.39150i 0.616582i
\(233\) 7.95482i 0.521138i 0.965455 + 0.260569i \(0.0839101\pi\)
−0.965455 + 0.260569i \(0.916090\pi\)
\(234\) −14.9609 −0.978023
\(235\) 0 0
\(236\) −1.33294 −0.0867670
\(237\) − 1.66223i − 0.107973i
\(238\) 72.3051i 4.68684i
\(239\) −17.4888 −1.13125 −0.565627 0.824661i \(-0.691366\pi\)
−0.565627 + 0.824661i \(0.691366\pi\)
\(240\) 0 0
\(241\) −14.7743 −0.951699 −0.475849 0.879527i \(-0.657859\pi\)
−0.475849 + 0.879527i \(0.657859\pi\)
\(242\) − 20.3917i − 1.31083i
\(243\) 1.00000i 0.0641500i
\(244\) −47.6375 −3.04968
\(245\) 0 0
\(246\) 0.587203 0.0374387
\(247\) − 29.6385i − 1.88585i
\(248\) 76.8382i 4.87923i
\(249\) 9.58514 0.607433
\(250\) 0 0
\(251\) −3.30996 −0.208923 −0.104461 0.994529i \(-0.533312\pi\)
−0.104461 + 0.994529i \(0.533312\pi\)
\(252\) 25.1278i 1.58290i
\(253\) − 0.257258i − 0.0161737i
\(254\) −53.7723 −3.37398
\(255\) 0 0
\(256\) 40.7463 2.54664
\(257\) − 30.8668i − 1.92542i −0.270539 0.962709i \(-0.587202\pi\)
0.270539 0.962709i \(-0.412798\pi\)
\(258\) 19.5636i 1.21798i
\(259\) 20.5816 1.27888
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) − 2.88463i − 0.178213i
\(263\) − 19.4833i − 1.20139i −0.799477 0.600697i \(-0.794890\pi\)
0.799477 0.600697i \(-0.205110\pi\)
\(264\) −17.6333 −1.08525
\(265\) 0 0
\(266\) −68.0728 −4.17381
\(267\) − 16.3332i − 0.999577i
\(268\) 35.5557i 2.17191i
\(269\) −20.8385 −1.27055 −0.635273 0.772287i \(-0.719113\pi\)
−0.635273 + 0.772287i \(0.719113\pi\)
\(270\) 0 0
\(271\) 24.5849 1.49343 0.746713 0.665146i \(-0.231631\pi\)
0.746713 + 0.665146i \(0.231631\pi\)
\(272\) − 84.5924i − 5.12916i
\(273\) 25.3193i 1.53239i
\(274\) −8.76994 −0.529811
\(275\) 0 0
\(276\) 0.745712 0.0448866
\(277\) 1.89904i 0.114102i 0.998371 + 0.0570510i \(0.0181698\pi\)
−0.998371 + 0.0570510i \(0.981830\pi\)
\(278\) − 23.1455i − 1.38817i
\(279\) 8.18167 0.489824
\(280\) 0 0
\(281\) 32.4916 1.93828 0.969142 0.246503i \(-0.0792814\pi\)
0.969142 + 0.246503i \(0.0792814\pi\)
\(282\) 27.0434i 1.61041i
\(283\) 11.9151i 0.708279i 0.935193 + 0.354140i \(0.115226\pi\)
−0.935193 + 0.354140i \(0.884774\pi\)
\(284\) 1.50982 0.0895912
\(285\) 0 0
\(286\) −28.0902 −1.66101
\(287\) − 0.993764i − 0.0586600i
\(288\) − 21.4180i − 1.26207i
\(289\) −15.9540 −0.938472
\(290\) 0 0
\(291\) −8.04128 −0.471388
\(292\) − 62.5524i − 3.66060i
\(293\) − 20.7359i − 1.21140i −0.795692 0.605702i \(-0.792892\pi\)
0.795692 0.605702i \(-0.207108\pi\)
\(294\) 39.0559 2.27779
\(295\) 0 0
\(296\) −41.8659 −2.43341
\(297\) 1.87758i 0.108948i
\(298\) 16.1832i 0.937467i
\(299\) 0.751396 0.0434544
\(300\) 0 0
\(301\) 33.1088 1.90836
\(302\) 21.9961i 1.26574i
\(303\) − 12.3239i − 0.707987i
\(304\) 79.6409 4.56772
\(305\) 0 0
\(306\) −15.6608 −0.895268
\(307\) − 16.9755i − 0.968842i −0.874835 0.484421i \(-0.839030\pi\)
0.874835 0.484421i \(-0.160970\pi\)
\(308\) 47.1793i 2.68829i
\(309\) −0.660912 −0.0375980
\(310\) 0 0
\(311\) −20.9019 −1.18524 −0.592619 0.805483i \(-0.701906\pi\)
−0.592619 + 0.805483i \(0.701906\pi\)
\(312\) − 51.5030i − 2.91578i
\(313\) − 12.7754i − 0.722110i −0.932545 0.361055i \(-0.882417\pi\)
0.932545 0.361055i \(-0.117583\pi\)
\(314\) 26.9298 1.51973
\(315\) 0 0
\(316\) 9.04669 0.508916
\(317\) 15.0552i 0.845582i 0.906227 + 0.422791i \(0.138950\pi\)
−0.906227 + 0.422791i \(0.861050\pi\)
\(318\) − 3.13060i − 0.175555i
\(319\) 1.87758 0.105124
\(320\) 0 0
\(321\) 14.5868 0.814155
\(322\) − 1.72578i − 0.0961741i
\(323\) − 31.0251i − 1.72628i
\(324\) −5.44251 −0.302362
\(325\) 0 0
\(326\) 3.42951 0.189943
\(327\) − 15.5050i − 0.857431i
\(328\) 2.02145i 0.111616i
\(329\) 45.7675 2.52324
\(330\) 0 0
\(331\) −17.7325 −0.974668 −0.487334 0.873216i \(-0.662031\pi\)
−0.487334 + 0.873216i \(0.662031\pi\)
\(332\) 52.1672i 2.86305i
\(333\) 4.45785i 0.244289i
\(334\) 41.3203 2.26095
\(335\) 0 0
\(336\) −68.0349 −3.71161
\(337\) − 17.5905i − 0.958217i −0.877756 0.479108i \(-0.840960\pi\)
0.877756 0.479108i \(-0.159040\pi\)
\(338\) − 46.5801i − 2.53363i
\(339\) −7.80660 −0.423996
\(340\) 0 0
\(341\) 15.3617 0.831883
\(342\) − 14.7441i − 0.797271i
\(343\) − 33.7784i − 1.82386i
\(344\) −67.3479 −3.63115
\(345\) 0 0
\(346\) 6.40909 0.344555
\(347\) − 21.0789i − 1.13157i −0.824552 0.565787i \(-0.808573\pi\)
0.824552 0.565787i \(-0.191427\pi\)
\(348\) 5.44251i 0.291749i
\(349\) 0.723154 0.0387096 0.0193548 0.999813i \(-0.493839\pi\)
0.0193548 + 0.999813i \(0.493839\pi\)
\(350\) 0 0
\(351\) −5.48400 −0.292714
\(352\) − 40.2139i − 2.14341i
\(353\) − 5.93891i − 0.316096i −0.987431 0.158048i \(-0.949480\pi\)
0.987431 0.158048i \(-0.0505201\pi\)
\(354\) −0.668145 −0.0355115
\(355\) 0 0
\(356\) 88.8938 4.71136
\(357\) 26.5038i 1.40273i
\(358\) 48.5092i 2.56379i
\(359\) 2.30973 0.121903 0.0609514 0.998141i \(-0.480587\pi\)
0.0609514 + 0.998141i \(0.480587\pi\)
\(360\) 0 0
\(361\) 10.2091 0.537323
\(362\) − 7.36234i − 0.386956i
\(363\) − 7.47471i − 0.392321i
\(364\) −137.801 −7.22272
\(365\) 0 0
\(366\) −23.8786 −1.24815
\(367\) 22.9235i 1.19660i 0.801274 + 0.598298i \(0.204156\pi\)
−0.801274 + 0.598298i \(0.795844\pi\)
\(368\) 2.01906i 0.105251i
\(369\) 0.215243 0.0112051
\(370\) 0 0
\(371\) −5.29813 −0.275065
\(372\) 44.5289i 2.30871i
\(373\) 2.83182i 0.146626i 0.997309 + 0.0733129i \(0.0233572\pi\)
−0.997309 + 0.0733129i \(0.976643\pi\)
\(374\) −29.4043 −1.52046
\(375\) 0 0
\(376\) −93.0973 −4.80113
\(377\) 5.48400i 0.282440i
\(378\) 12.5955i 0.647841i
\(379\) 29.2539 1.50267 0.751335 0.659921i \(-0.229410\pi\)
0.751335 + 0.659921i \(0.229410\pi\)
\(380\) 0 0
\(381\) −19.7106 −1.00980
\(382\) 29.4627i 1.50744i
\(383\) 2.99063i 0.152814i 0.997077 + 0.0764070i \(0.0243448\pi\)
−0.997077 + 0.0764070i \(0.975655\pi\)
\(384\) 36.1656 1.84557
\(385\) 0 0
\(386\) −11.4156 −0.581040
\(387\) 7.17115i 0.364530i
\(388\) − 43.7648i − 2.22182i
\(389\) 8.57339 0.434688 0.217344 0.976095i \(-0.430261\pi\)
0.217344 + 0.976095i \(0.430261\pi\)
\(390\) 0 0
\(391\) 0.786549 0.0397775
\(392\) 134.451i 6.79078i
\(393\) − 1.05738i − 0.0533376i
\(394\) −15.7372 −0.792828
\(395\) 0 0
\(396\) −10.2187 −0.513510
\(397\) 9.29605i 0.466556i 0.972410 + 0.233278i \(0.0749451\pi\)
−0.972410 + 0.233278i \(0.925055\pi\)
\(398\) 14.8707i 0.745400i
\(399\) −24.9525 −1.24919
\(400\) 0 0
\(401\) 19.5392 0.975739 0.487870 0.872916i \(-0.337774\pi\)
0.487870 + 0.872916i \(0.337774\pi\)
\(402\) 17.8225i 0.888907i
\(403\) 44.8683i 2.23505i
\(404\) 67.0728 3.33700
\(405\) 0 0
\(406\) 12.5955 0.625103
\(407\) 8.36995i 0.414883i
\(408\) − 53.9125i − 2.66907i
\(409\) −37.9582 −1.87691 −0.938456 0.345398i \(-0.887744\pi\)
−0.938456 + 0.345398i \(0.887744\pi\)
\(410\) 0 0
\(411\) −3.21467 −0.158568
\(412\) − 3.59702i − 0.177213i
\(413\) 1.13075i 0.0556405i
\(414\) 0.373793 0.0183709
\(415\) 0 0
\(416\) 117.456 5.75876
\(417\) − 8.48410i − 0.415468i
\(418\) − 27.6832i − 1.35403i
\(419\) 10.0952 0.493184 0.246592 0.969119i \(-0.420689\pi\)
0.246592 + 0.969119i \(0.420689\pi\)
\(420\) 0 0
\(421\) 13.8521 0.675111 0.337556 0.941306i \(-0.390400\pi\)
0.337556 + 0.941306i \(0.390400\pi\)
\(422\) 2.35525i 0.114652i
\(423\) 9.91293i 0.481983i
\(424\) 10.7771 0.523384
\(425\) 0 0
\(426\) 0.756807 0.0366674
\(427\) 40.4114i 1.95565i
\(428\) 79.3888i 3.83740i
\(429\) −10.2966 −0.497125
\(430\) 0 0
\(431\) 18.2149 0.877380 0.438690 0.898638i \(-0.355443\pi\)
0.438690 + 0.898638i \(0.355443\pi\)
\(432\) − 14.7359i − 0.708982i
\(433\) − 6.73476i − 0.323652i −0.986819 0.161826i \(-0.948262\pi\)
0.986819 0.161826i \(-0.0517383\pi\)
\(434\) 103.052 4.94666
\(435\) 0 0
\(436\) 84.3864 4.04137
\(437\) 0.740510i 0.0354234i
\(438\) − 31.3548i − 1.49819i
\(439\) −30.9590 −1.47759 −0.738795 0.673930i \(-0.764605\pi\)
−0.738795 + 0.673930i \(0.764605\pi\)
\(440\) 0 0
\(441\) 14.3162 0.681723
\(442\) − 85.8838i − 4.08508i
\(443\) − 21.4432i − 1.01880i −0.860531 0.509398i \(-0.829868\pi\)
0.860531 0.509398i \(-0.170132\pi\)
\(444\) −24.2619 −1.15142
\(445\) 0 0
\(446\) −1.67832 −0.0794707
\(447\) 5.93205i 0.280576i
\(448\) − 133.700i − 6.31672i
\(449\) −25.6895 −1.21236 −0.606180 0.795327i \(-0.707299\pi\)
−0.606180 + 0.795327i \(0.707299\pi\)
\(450\) 0 0
\(451\) 0.404134 0.0190299
\(452\) − 42.4875i − 1.99844i
\(453\) 8.06281i 0.378824i
\(454\) 64.7186 3.03739
\(455\) 0 0
\(456\) 50.7568 2.37691
\(457\) 30.3278i 1.41868i 0.704868 + 0.709338i \(0.251006\pi\)
−0.704868 + 0.709338i \(0.748994\pi\)
\(458\) 57.2377i 2.67454i
\(459\) −5.74056 −0.267946
\(460\) 0 0
\(461\) −31.0026 −1.44393 −0.721967 0.691928i \(-0.756762\pi\)
−0.721967 + 0.691928i \(0.756762\pi\)
\(462\) 23.6490i 1.10025i
\(463\) 27.6990i 1.28728i 0.765328 + 0.643640i \(0.222577\pi\)
−0.765328 + 0.643640i \(0.777423\pi\)
\(464\) −14.7359 −0.684097
\(465\) 0 0
\(466\) −21.7015 −1.00530
\(467\) − 4.03243i − 0.186598i −0.995638 0.0932992i \(-0.970259\pi\)
0.995638 0.0932992i \(-0.0297413\pi\)
\(468\) − 29.8467i − 1.37966i
\(469\) 30.1623 1.39277
\(470\) 0 0
\(471\) 9.87127 0.454844
\(472\) − 2.30010i − 0.105871i
\(473\) 13.4644i 0.619093i
\(474\) 4.53472 0.208286
\(475\) 0 0
\(476\) −144.248 −6.61157
\(477\) − 1.14754i − 0.0525422i
\(478\) − 47.7110i − 2.18225i
\(479\) −1.32186 −0.0603973 −0.0301986 0.999544i \(-0.509614\pi\)
−0.0301986 + 0.999544i \(0.509614\pi\)
\(480\) 0 0
\(481\) −24.4468 −1.11468
\(482\) − 40.3058i − 1.83588i
\(483\) − 0.632596i − 0.0287841i
\(484\) 40.6812 1.84915
\(485\) 0 0
\(486\) −2.72810 −0.123749
\(487\) − 9.89390i − 0.448335i −0.974551 0.224168i \(-0.928034\pi\)
0.974551 0.224168i \(-0.0719663\pi\)
\(488\) − 82.2024i − 3.72113i
\(489\) 1.25711 0.0568484
\(490\) 0 0
\(491\) 17.1305 0.773089 0.386545 0.922271i \(-0.373669\pi\)
0.386545 + 0.922271i \(0.373669\pi\)
\(492\) 1.17146i 0.0528135i
\(493\) 5.74056i 0.258542i
\(494\) 80.8568 3.63792
\(495\) 0 0
\(496\) −120.564 −5.41350
\(497\) − 1.28080i − 0.0574516i
\(498\) 26.1492i 1.17177i
\(499\) 7.95477 0.356104 0.178052 0.984021i \(-0.443020\pi\)
0.178052 + 0.984021i \(0.443020\pi\)
\(500\) 0 0
\(501\) 15.1462 0.676683
\(502\) − 9.02989i − 0.403024i
\(503\) − 13.3471i − 0.595119i −0.954703 0.297559i \(-0.903827\pi\)
0.954703 0.297559i \(-0.0961727\pi\)
\(504\) −43.3601 −1.93141
\(505\) 0 0
\(506\) 0.701825 0.0311999
\(507\) − 17.0742i − 0.758293i
\(508\) − 107.275i − 4.75956i
\(509\) 5.55324 0.246143 0.123071 0.992398i \(-0.460726\pi\)
0.123071 + 0.992398i \(0.460726\pi\)
\(510\) 0 0
\(511\) −53.0639 −2.34741
\(512\) 38.8286i 1.71600i
\(513\) − 5.40455i − 0.238617i
\(514\) 84.2076 3.71424
\(515\) 0 0
\(516\) −39.0291 −1.71816
\(517\) 18.6123i 0.818567i
\(518\) 56.1487i 2.46703i
\(519\) 2.34929 0.103122
\(520\) 0 0
\(521\) 41.5095 1.81857 0.909283 0.416179i \(-0.136631\pi\)
0.909283 + 0.416179i \(0.136631\pi\)
\(522\) 2.72810i 0.119406i
\(523\) 11.6261i 0.508375i 0.967155 + 0.254188i \(0.0818081\pi\)
−0.967155 + 0.254188i \(0.918192\pi\)
\(524\) 5.75479 0.251399
\(525\) 0 0
\(526\) 53.1525 2.31756
\(527\) 46.9674i 2.04593i
\(528\) − 27.6678i − 1.20409i
\(529\) 22.9812 0.999184
\(530\) 0 0
\(531\) −0.244913 −0.0106283
\(532\) − 135.804i − 5.88786i
\(533\) 1.18039i 0.0511284i
\(534\) 44.5586 1.92824
\(535\) 0 0
\(536\) −61.3543 −2.65010
\(537\) 17.7814i 0.767322i
\(538\) − 56.8495i − 2.45096i
\(539\) 26.8797 1.15779
\(540\) 0 0
\(541\) −5.29830 −0.227792 −0.113896 0.993493i \(-0.536333\pi\)
−0.113896 + 0.993493i \(0.536333\pi\)
\(542\) 67.0700i 2.88090i
\(543\) − 2.69871i − 0.115813i
\(544\) 122.951 5.27149
\(545\) 0 0
\(546\) −69.0735 −2.95607
\(547\) − 6.78093i − 0.289932i −0.989437 0.144966i \(-0.953693\pi\)
0.989437 0.144966i \(-0.0463072\pi\)
\(548\) − 17.4959i − 0.747387i
\(549\) −8.75284 −0.373562
\(550\) 0 0
\(551\) −5.40455 −0.230241
\(552\) 1.28679i 0.0547693i
\(553\) − 7.67441i − 0.326349i
\(554\) −5.18076 −0.220109
\(555\) 0 0
\(556\) 46.1748 1.95825
\(557\) 0.186307i 0.00789410i 0.999992 + 0.00394705i \(0.00125639\pi\)
−0.999992 + 0.00394705i \(0.998744\pi\)
\(558\) 22.3204i 0.944898i
\(559\) −39.3266 −1.66334
\(560\) 0 0
\(561\) −10.7783 −0.455062
\(562\) 88.6401i 3.73906i
\(563\) 27.1382i 1.14374i 0.820345 + 0.571869i \(0.193781\pi\)
−0.820345 + 0.571869i \(0.806219\pi\)
\(564\) −53.9512 −2.27176
\(565\) 0 0
\(566\) −32.5056 −1.36631
\(567\) 4.61695i 0.193893i
\(568\) 2.60532i 0.109317i
\(569\) −22.8795 −0.959157 −0.479578 0.877499i \(-0.659210\pi\)
−0.479578 + 0.877499i \(0.659210\pi\)
\(570\) 0 0
\(571\) 38.3621 1.60541 0.802703 0.596379i \(-0.203394\pi\)
0.802703 + 0.596379i \(0.203394\pi\)
\(572\) − 56.0395i − 2.34313i
\(573\) 10.7997i 0.451165i
\(574\) 2.71108 0.113158
\(575\) 0 0
\(576\) 28.9585 1.20660
\(577\) − 12.0497i − 0.501637i −0.968034 0.250818i \(-0.919300\pi\)
0.968034 0.250818i \(-0.0806997\pi\)
\(578\) − 43.5241i − 1.81036i
\(579\) −4.18447 −0.173901
\(580\) 0 0
\(581\) 44.2541 1.83597
\(582\) − 21.9374i − 0.909334i
\(583\) − 2.15459i − 0.0892341i
\(584\) 107.939 4.46656
\(585\) 0 0
\(586\) 56.5695 2.33687
\(587\) − 23.6704i − 0.976982i −0.872569 0.488491i \(-0.837547\pi\)
0.872569 0.488491i \(-0.162453\pi\)
\(588\) 77.9160i 3.21320i
\(589\) −44.2183 −1.82198
\(590\) 0 0
\(591\) −5.76856 −0.237287
\(592\) − 65.6905i − 2.69986i
\(593\) − 25.9414i − 1.06529i −0.846340 0.532643i \(-0.821199\pi\)
0.846340 0.532643i \(-0.178801\pi\)
\(594\) −5.12221 −0.210167
\(595\) 0 0
\(596\) −32.2852 −1.32245
\(597\) 5.45094i 0.223092i
\(598\) 2.04988i 0.0838259i
\(599\) 17.2886 0.706393 0.353196 0.935549i \(-0.385095\pi\)
0.353196 + 0.935549i \(0.385095\pi\)
\(600\) 0 0
\(601\) −30.8997 −1.26042 −0.630212 0.776423i \(-0.717032\pi\)
−0.630212 + 0.776423i \(0.717032\pi\)
\(602\) 90.3240i 3.68133i
\(603\) 6.53295i 0.266042i
\(604\) −43.8820 −1.78553
\(605\) 0 0
\(606\) 33.6207 1.36575
\(607\) 4.07152i 0.165258i 0.996580 + 0.0826290i \(0.0263317\pi\)
−0.996580 + 0.0826290i \(0.973668\pi\)
\(608\) 115.754i 4.69446i
\(609\) 4.61695 0.187088
\(610\) 0 0
\(611\) −54.3625 −2.19927
\(612\) − 31.2431i − 1.26293i
\(613\) − 32.3595i − 1.30699i −0.756931 0.653494i \(-0.773302\pi\)
0.756931 0.653494i \(-0.226698\pi\)
\(614\) 46.3108 1.86895
\(615\) 0 0
\(616\) −81.4118 −3.28017
\(617\) − 33.4957i − 1.34849i −0.738509 0.674243i \(-0.764470\pi\)
0.738509 0.674243i \(-0.235530\pi\)
\(618\) − 1.80303i − 0.0725286i
\(619\) 16.6857 0.670656 0.335328 0.942101i \(-0.391153\pi\)
0.335328 + 0.942101i \(0.391153\pi\)
\(620\) 0 0
\(621\) 0.137016 0.00549827
\(622\) − 57.0224i − 2.28639i
\(623\) − 75.4096i − 3.02122i
\(624\) 80.8117 3.23506
\(625\) 0 0
\(626\) 34.8526 1.39299
\(627\) − 10.1474i − 0.405250i
\(628\) 53.7245i 2.14384i
\(629\) −25.5905 −1.02036
\(630\) 0 0
\(631\) 9.91590 0.394746 0.197373 0.980328i \(-0.436759\pi\)
0.197373 + 0.980328i \(0.436759\pi\)
\(632\) 15.6108i 0.620965i
\(633\) 0.863332i 0.0343143i
\(634\) −41.0719 −1.63118
\(635\) 0 0
\(636\) 6.24550 0.247650
\(637\) 78.5099i 3.11068i
\(638\) 5.12221i 0.202790i
\(639\) 0.277412 0.0109742
\(640\) 0 0
\(641\) −22.9862 −0.907902 −0.453951 0.891027i \(-0.649986\pi\)
−0.453951 + 0.891027i \(0.649986\pi\)
\(642\) 39.7942i 1.57055i
\(643\) 12.9117i 0.509190i 0.967048 + 0.254595i \(0.0819421\pi\)
−0.967048 + 0.254595i \(0.918058\pi\)
\(644\) 3.44291 0.135670
\(645\) 0 0
\(646\) 84.6395 3.33010
\(647\) 16.6096i 0.652989i 0.945199 + 0.326494i \(0.105867\pi\)
−0.945199 + 0.326494i \(0.894133\pi\)
\(648\) − 9.39150i − 0.368933i
\(649\) −0.459842 −0.0180504
\(650\) 0 0
\(651\) 37.7743 1.48049
\(652\) 6.84182i 0.267947i
\(653\) 21.5638i 0.843858i 0.906629 + 0.421929i \(0.138647\pi\)
−0.906629 + 0.421929i \(0.861353\pi\)
\(654\) 42.2993 1.65403
\(655\) 0 0
\(656\) −3.17180 −0.123838
\(657\) − 11.4933i − 0.448396i
\(658\) 124.858i 4.86747i
\(659\) −31.7863 −1.23822 −0.619110 0.785305i \(-0.712506\pi\)
−0.619110 + 0.785305i \(0.712506\pi\)
\(660\) 0 0
\(661\) 30.0046 1.16704 0.583521 0.812098i \(-0.301674\pi\)
0.583521 + 0.812098i \(0.301674\pi\)
\(662\) − 48.3760i − 1.88019i
\(663\) − 31.4812i − 1.22263i
\(664\) −90.0189 −3.49341
\(665\) 0 0
\(666\) −12.1614 −0.471246
\(667\) − 0.137016i − 0.00530529i
\(668\) 82.4334i 3.18944i
\(669\) −0.615197 −0.0237849
\(670\) 0 0
\(671\) −16.4341 −0.634432
\(672\) − 98.8856i − 3.81460i
\(673\) 35.0842i 1.35240i 0.736719 + 0.676199i \(0.236374\pi\)
−0.736719 + 0.676199i \(0.763626\pi\)
\(674\) 47.9886 1.84845
\(675\) 0 0
\(676\) 92.9267 3.57410
\(677\) 42.1433i 1.61970i 0.586638 + 0.809849i \(0.300451\pi\)
−0.586638 + 0.809849i \(0.699549\pi\)
\(678\) − 21.2972i − 0.817912i
\(679\) −37.1261 −1.42477
\(680\) 0 0
\(681\) 23.7230 0.909067
\(682\) 41.9082i 1.60475i
\(683\) 31.8895i 1.22022i 0.792318 + 0.610109i \(0.208874\pi\)
−0.792318 + 0.610109i \(0.791126\pi\)
\(684\) 29.4143 1.12468
\(685\) 0 0
\(686\) 92.1508 3.51833
\(687\) 20.9808i 0.800468i
\(688\) − 105.673i − 4.02876i
\(689\) 6.29311 0.239748
\(690\) 0 0
\(691\) 11.2963 0.429732 0.214866 0.976644i \(-0.431069\pi\)
0.214866 + 0.976644i \(0.431069\pi\)
\(692\) 12.7860i 0.486052i
\(693\) 8.66866i 0.329295i
\(694\) 57.5052 2.18287
\(695\) 0 0
\(696\) −9.39150 −0.355984
\(697\) 1.23561i 0.0468022i
\(698\) 1.97283i 0.0746729i
\(699\) −7.95482 −0.300879
\(700\) 0 0
\(701\) −1.39354 −0.0526332 −0.0263166 0.999654i \(-0.508378\pi\)
−0.0263166 + 0.999654i \(0.508378\pi\)
\(702\) − 14.9609i − 0.564662i
\(703\) − 24.0927i − 0.908672i
\(704\) 54.3717 2.04921
\(705\) 0 0
\(706\) 16.2019 0.609767
\(707\) − 56.8986i − 2.13989i
\(708\) − 1.33294i − 0.0500949i
\(709\) −5.24967 −0.197156 −0.0985778 0.995129i \(-0.531429\pi\)
−0.0985778 + 0.995129i \(0.531429\pi\)
\(710\) 0 0
\(711\) 1.66223 0.0623384
\(712\) 153.394i 5.74867i
\(713\) − 1.12102i − 0.0419826i
\(714\) −72.3051 −2.70595
\(715\) 0 0
\(716\) −96.7752 −3.61666
\(717\) − 17.4888i − 0.653130i
\(718\) 6.30116i 0.235157i
\(719\) −46.4801 −1.73341 −0.866707 0.498817i \(-0.833768\pi\)
−0.866707 + 0.498817i \(0.833768\pi\)
\(720\) 0 0
\(721\) −3.05140 −0.113640
\(722\) 27.8515i 1.03653i
\(723\) − 14.7743i − 0.549464i
\(724\) 14.6878 0.545866
\(725\) 0 0
\(726\) 20.3917 0.756808
\(727\) 43.3101i 1.60628i 0.595790 + 0.803141i \(0.296839\pi\)
−0.595790 + 0.803141i \(0.703161\pi\)
\(728\) − 237.786i − 8.81295i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −41.1664 −1.52259
\(732\) − 47.6375i − 1.76073i
\(733\) − 18.6793i − 0.689936i −0.938614 0.344968i \(-0.887890\pi\)
0.938614 0.344968i \(-0.112110\pi\)
\(734\) −62.5375 −2.30830
\(735\) 0 0
\(736\) −2.93461 −0.108171
\(737\) 12.2661i 0.451828i
\(738\) 0.587203i 0.0216152i
\(739\) −41.3256 −1.52019 −0.760093 0.649815i \(-0.774847\pi\)
−0.760093 + 0.649815i \(0.774847\pi\)
\(740\) 0 0
\(741\) 29.6385 1.08880
\(742\) − 14.4538i − 0.530616i
\(743\) − 32.3390i − 1.18640i −0.805054 0.593202i \(-0.797864\pi\)
0.805054 0.593202i \(-0.202136\pi\)
\(744\) −76.8382 −2.81703
\(745\) 0 0
\(746\) −7.72547 −0.282849
\(747\) 9.58514i 0.350702i
\(748\) − 58.6612i − 2.14487i
\(749\) 67.3464 2.46078
\(750\) 0 0
\(751\) −9.34454 −0.340987 −0.170494 0.985359i \(-0.554536\pi\)
−0.170494 + 0.985359i \(0.554536\pi\)
\(752\) − 146.076i − 5.32685i
\(753\) − 3.30996i − 0.120622i
\(754\) −14.9609 −0.544843
\(755\) 0 0
\(756\) −25.1278 −0.913888
\(757\) 5.01932i 0.182430i 0.995831 + 0.0912151i \(0.0290751\pi\)
−0.995831 + 0.0912151i \(0.970925\pi\)
\(758\) 79.8074i 2.89873i
\(759\) 0.257258 0.00933788
\(760\) 0 0
\(761\) −34.9103 −1.26550 −0.632748 0.774358i \(-0.718073\pi\)
−0.632748 + 0.774358i \(0.718073\pi\)
\(762\) − 53.7723i − 1.94797i
\(763\) − 71.5859i − 2.59159i
\(764\) −58.7777 −2.12650
\(765\) 0 0
\(766\) −8.15872 −0.294787
\(767\) − 1.34310i − 0.0484965i
\(768\) 40.7463i 1.47031i
\(769\) −37.4889 −1.35189 −0.675943 0.736954i \(-0.736263\pi\)
−0.675943 + 0.736954i \(0.736263\pi\)
\(770\) 0 0
\(771\) 30.8668 1.11164
\(772\) − 22.7740i − 0.819655i
\(773\) − 33.7332i − 1.21330i −0.794970 0.606649i \(-0.792513\pi\)
0.794970 0.606649i \(-0.207487\pi\)
\(774\) −19.5636 −0.703199
\(775\) 0 0
\(776\) 75.5197 2.71100
\(777\) 20.5816i 0.738362i
\(778\) 23.3890i 0.838538i
\(779\) −1.16329 −0.0416792
\(780\) 0 0
\(781\) 0.520862 0.0186379
\(782\) 2.14578i 0.0767330i
\(783\) 1.00000i 0.0357371i
\(784\) −210.962 −7.53436
\(785\) 0 0
\(786\) 2.88463 0.102891
\(787\) − 11.5508i − 0.411740i −0.978579 0.205870i \(-0.933998\pi\)
0.978579 0.205870i \(-0.0660025\pi\)
\(788\) − 31.3955i − 1.11842i
\(789\) 19.4833 0.693626
\(790\) 0 0
\(791\) −36.0426 −1.28153
\(792\) − 17.6333i − 0.626571i
\(793\) − 48.0006i − 1.70455i
\(794\) −25.3605 −0.900012
\(795\) 0 0
\(796\) −29.6668 −1.05151
\(797\) − 9.11795i − 0.322974i −0.986875 0.161487i \(-0.948371\pi\)
0.986875 0.161487i \(-0.0516291\pi\)
\(798\) − 68.0728i − 2.40975i
\(799\) −56.9058 −2.01318
\(800\) 0 0
\(801\) 16.3332 0.577106
\(802\) 53.3047i 1.88226i
\(803\) − 21.5795i − 0.761525i
\(804\) −35.5557 −1.25395
\(805\) 0 0
\(806\) −122.405 −4.31153
\(807\) − 20.8385i − 0.733551i
\(808\) 115.740i 4.07171i
\(809\) 11.9809 0.421227 0.210613 0.977569i \(-0.432454\pi\)
0.210613 + 0.977569i \(0.432454\pi\)
\(810\) 0 0
\(811\) −19.2172 −0.674807 −0.337403 0.941360i \(-0.609549\pi\)
−0.337403 + 0.941360i \(0.609549\pi\)
\(812\) 25.1278i 0.881812i
\(813\) 24.5849i 0.862230i
\(814\) −22.8340 −0.800332
\(815\) 0 0
\(816\) 84.5924 2.96132
\(817\) − 38.7568i − 1.35593i
\(818\) − 103.554i − 3.62067i
\(819\) −25.3193 −0.884728
\(820\) 0 0
\(821\) −10.5611 −0.368585 −0.184292 0.982871i \(-0.558999\pi\)
−0.184292 + 0.982871i \(0.558999\pi\)
\(822\) − 8.76994i − 0.305887i
\(823\) − 29.1655i − 1.01665i −0.861166 0.508323i \(-0.830266\pi\)
0.861166 0.508323i \(-0.169734\pi\)
\(824\) 6.20696 0.216230
\(825\) 0 0
\(826\) −3.08479 −0.107334
\(827\) − 40.5140i − 1.40881i −0.709798 0.704405i \(-0.751214\pi\)
0.709798 0.704405i \(-0.248786\pi\)
\(828\) 0.745712i 0.0259153i
\(829\) −51.3925 −1.78494 −0.892468 0.451111i \(-0.851028\pi\)
−0.892468 + 0.451111i \(0.851028\pi\)
\(830\) 0 0
\(831\) −1.89904 −0.0658769
\(832\) 158.808i 5.50569i
\(833\) 82.1829i 2.84747i
\(834\) 23.1455 0.801461
\(835\) 0 0
\(836\) 55.2276 1.91009
\(837\) 8.18167i 0.282800i
\(838\) 27.5407i 0.951379i
\(839\) 30.2352 1.04383 0.521917 0.852996i \(-0.325217\pi\)
0.521917 + 0.852996i \(0.325217\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 37.7899i 1.30233i
\(843\) 32.4916i 1.11907i
\(844\) −4.69869 −0.161736
\(845\) 0 0
\(846\) −27.0434 −0.929773
\(847\) − 34.5103i − 1.18579i
\(848\) 16.9100i 0.580693i
\(849\) −11.9151 −0.408925
\(850\) 0 0
\(851\) 0.610797 0.0209379
\(852\) 1.50982i 0.0517255i
\(853\) 0.163336i 0.00559252i 0.999996 + 0.00279626i \(0.000890079\pi\)
−0.999996 + 0.00279626i \(0.999110\pi\)
\(854\) −110.246 −3.77255
\(855\) 0 0
\(856\) −136.992 −4.68229
\(857\) 0.869730i 0.0297094i 0.999890 + 0.0148547i \(0.00472857\pi\)
−0.999890 + 0.0148547i \(0.995271\pi\)
\(858\) − 28.0902i − 0.958983i
\(859\) 50.5362 1.72427 0.862136 0.506677i \(-0.169126\pi\)
0.862136 + 0.506677i \(0.169126\pi\)
\(860\) 0 0
\(861\) 0.993764 0.0338674
\(862\) 49.6920i 1.69251i
\(863\) 43.1139i 1.46761i 0.679358 + 0.733807i \(0.262258\pi\)
−0.679358 + 0.733807i \(0.737742\pi\)
\(864\) 21.4180 0.728654
\(865\) 0 0
\(866\) 18.3731 0.624342
\(867\) − 15.9540i − 0.541827i
\(868\) 205.587i 6.97809i
\(869\) 3.12096 0.105871
\(870\) 0 0
\(871\) −35.8267 −1.21394
\(872\) 145.616i 4.93117i
\(873\) − 8.04128i − 0.272156i
\(874\) −2.02018 −0.0683337
\(875\) 0 0
\(876\) 62.5524 2.11345
\(877\) − 3.23901i − 0.109374i −0.998504 0.0546868i \(-0.982584\pi\)
0.998504 0.0546868i \(-0.0174160\pi\)
\(878\) − 84.4590i − 2.85035i
\(879\) 20.7359 0.699404
\(880\) 0 0
\(881\) 9.49218 0.319800 0.159900 0.987133i \(-0.448883\pi\)
0.159900 + 0.987133i \(0.448883\pi\)
\(882\) 39.0559i 1.31508i
\(883\) 40.4268i 1.36047i 0.732994 + 0.680235i \(0.238122\pi\)
−0.732994 + 0.680235i \(0.761878\pi\)
\(884\) 171.337 5.76268
\(885\) 0 0
\(886\) 58.4991 1.96532
\(887\) 17.5249i 0.588430i 0.955739 + 0.294215i \(0.0950582\pi\)
−0.955739 + 0.294215i \(0.904942\pi\)
\(888\) − 41.8659i − 1.40493i
\(889\) −91.0026 −3.05213
\(890\) 0 0
\(891\) −1.87758 −0.0629011
\(892\) − 3.34822i − 0.112107i
\(893\) − 53.5749i − 1.79282i
\(894\) −16.1832 −0.541247
\(895\) 0 0
\(896\) 166.975 5.57823
\(897\) 0.751396i 0.0250884i
\(898\) − 70.0833i − 2.33871i
\(899\) 8.18167 0.272874
\(900\) 0 0
\(901\) 6.58752 0.219462
\(902\) 1.10252i 0.0367098i
\(903\) 33.1088i 1.10179i
\(904\) 73.3157 2.43844
\(905\) 0 0
\(906\) −21.9961 −0.730773
\(907\) 34.8878i 1.15843i 0.815174 + 0.579216i \(0.196641\pi\)
−0.815174 + 0.579216i \(0.803359\pi\)
\(908\) 129.113i 4.28475i
\(909\) 12.3239 0.408757
\(910\) 0 0
\(911\) 43.5212 1.44192 0.720960 0.692976i \(-0.243701\pi\)
0.720960 + 0.692976i \(0.243701\pi\)
\(912\) 79.6409i 2.63717i
\(913\) 17.9968i 0.595608i
\(914\) −82.7373 −2.73671
\(915\) 0 0
\(916\) −114.188 −3.77289
\(917\) − 4.88185i − 0.161213i
\(918\) − 15.6608i − 0.516883i
\(919\) 38.2543 1.26189 0.630946 0.775827i \(-0.282667\pi\)
0.630946 + 0.775827i \(0.282667\pi\)
\(920\) 0 0
\(921\) 16.9755 0.559361
\(922\) − 84.5780i − 2.78543i
\(923\) 1.52133i 0.0500751i
\(924\) −47.1793 −1.55209
\(925\) 0 0
\(926\) −75.5655 −2.48324
\(927\) − 0.660912i − 0.0217072i
\(928\) − 21.4180i − 0.703079i
\(929\) −40.3492 −1.32381 −0.661906 0.749586i \(-0.730252\pi\)
−0.661906 + 0.749586i \(0.730252\pi\)
\(930\) 0 0
\(931\) −77.3725 −2.53578
\(932\) − 43.2942i − 1.41815i
\(933\) − 20.9019i − 0.684298i
\(934\) 11.0009 0.359959
\(935\) 0 0
\(936\) 51.5030 1.68343
\(937\) − 4.95384i − 0.161835i −0.996721 0.0809174i \(-0.974215\pi\)
0.996721 0.0809174i \(-0.0257850\pi\)
\(938\) 82.2857i 2.68672i
\(939\) 12.7754 0.416910
\(940\) 0 0
\(941\) −47.5220 −1.54917 −0.774587 0.632467i \(-0.782042\pi\)
−0.774587 + 0.632467i \(0.782042\pi\)
\(942\) 26.9298i 0.877419i
\(943\) − 0.0294917i 0 0.000960383i
\(944\) 3.60901 0.117463
\(945\) 0 0
\(946\) −36.7321 −1.19426
\(947\) − 37.1547i − 1.20736i −0.797225 0.603682i \(-0.793700\pi\)
0.797225 0.603682i \(-0.206300\pi\)
\(948\) 9.04669i 0.293823i
\(949\) 63.0292 2.04601
\(950\) 0 0
\(951\) −15.0552 −0.488197
\(952\) − 248.911i − 8.06725i
\(953\) − 44.5351i − 1.44263i −0.692606 0.721316i \(-0.743537\pi\)
0.692606 0.721316i \(-0.256463\pi\)
\(954\) 3.13060 0.101357
\(955\) 0 0
\(956\) 95.1828 3.07843
\(957\) 1.87758i 0.0606934i
\(958\) − 3.60616i − 0.116510i
\(959\) −14.8420 −0.479272
\(960\) 0 0
\(961\) 35.9398 1.15935
\(962\) − 66.6933i − 2.15028i
\(963\) 14.5868i 0.470053i
\(964\) 80.4095 2.58982
\(965\) 0 0
\(966\) 1.72578 0.0555262
\(967\) 48.7291i 1.56702i 0.621379 + 0.783511i \(0.286573\pi\)
−0.621379 + 0.783511i \(0.713427\pi\)
\(968\) 70.1988i 2.25627i
\(969\) 31.0251 0.996671
\(970\) 0 0
\(971\) −51.7448 −1.66057 −0.830285 0.557339i \(-0.811822\pi\)
−0.830285 + 0.557339i \(0.811822\pi\)
\(972\) − 5.44251i − 0.174569i
\(973\) − 39.1706i − 1.25575i
\(974\) 26.9915 0.864864
\(975\) 0 0
\(976\) 128.981 4.12858
\(977\) 37.7726i 1.20845i 0.796813 + 0.604227i \(0.206518\pi\)
−0.796813 + 0.604227i \(0.793482\pi\)
\(978\) 3.42951i 0.109664i
\(979\) 30.6669 0.980117
\(980\) 0 0
\(981\) 15.5050 0.495038
\(982\) 46.7337i 1.49133i
\(983\) 21.1318i 0.673999i 0.941505 + 0.336999i \(0.109412\pi\)
−0.941505 + 0.336999i \(0.890588\pi\)
\(984\) −2.02145 −0.0644416
\(985\) 0 0
\(986\) −15.6608 −0.498742
\(987\) 45.7675i 1.45679i
\(988\) 161.308i 5.13189i
\(989\) 0.982563 0.0312437
\(990\) 0 0
\(991\) 11.0241 0.350191 0.175096 0.984551i \(-0.443977\pi\)
0.175096 + 0.984551i \(0.443977\pi\)
\(992\) − 175.235i − 5.56371i
\(993\) − 17.7325i − 0.562725i
\(994\) 3.49414 0.110827
\(995\) 0 0
\(996\) −52.1672 −1.65298
\(997\) 28.2785i 0.895589i 0.894136 + 0.447795i \(0.147790\pi\)
−0.894136 + 0.447795i \(0.852210\pi\)
\(998\) 21.7014i 0.686945i
\(999\) −4.45785 −0.141040
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 2175.2.c.p.349.16 16
5.2 odd 4 2175.2.a.bd.1.1 yes 8
5.3 odd 4 2175.2.a.bc.1.8 8
5.4 even 2 inner 2175.2.c.p.349.1 16
15.2 even 4 6525.2.a.by.1.8 8
15.8 even 4 6525.2.a.bz.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.bc.1.8 8 5.3 odd 4
2175.2.a.bd.1.1 yes 8 5.2 odd 4
2175.2.c.p.349.1 16 5.4 even 2 inner
2175.2.c.p.349.16 16 1.1 even 1 trivial
6525.2.a.by.1.8 8 15.2 even 4
6525.2.a.bz.1.1 8 15.8 even 4